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2,877,628,090,054 | arxiv | \section{Introduction}
The presence of ubiquitous small-amplitude propagating magnetohydrodynamic (MHD) waves in magnetic waveguides of the solar corona was first observed with the Coronal Multi-channel Polarimeter (CoMP) \citep[see][]{tomczyk07,tomczyk09}. The observations have inspired a number of recent theoretical works in which the properties of propagating waves are studied. Based on MHD wave theory \citep[e.g.,][]{erdelyi2007,tom08,tom08b,VTG} the observations have been interpreted as propagating kink waves, i.e., transverse MHD waves with mixed fast and Alfv\'enic properties whose dominating restoring force is magnetic tension \citep[see, e.g.,][]{edwinroberts,goossens2009}. In particular, \citet{tom08b} showed that field-aligned density enhancements in the corona act as natural waveguides for MHD waves. Apart from the CoMP observations in coronal loops, propagating kink waves have also been observed in chromospheric spicules \citep[e.g.,][]{depontieu07,he1,he2} and in thin threads of solar prominences \citep[e.g.,][]{lin07,lin09}. In the same manner as standing kink MHD waves are damped in time by resonant absorption \citep[see, e.g.,][]{goossens2002,rudermanroberts}, propagating kink MHD waves are damped in space due to naturally occurring plasma inhomogeneity in the direction transverse to the the magnetic field \citep{TGV}. \citet{TGV}, hereafter TGV, obtained the important result that the damping length by resonant absorption is inversely proportional to the wave frequency. This means that high-frequency waves are damped in shorter length scales than low-frequency waves. \citet{VTG} showed that the analytical theory of propagating resonant kink waves developed by TGV is fully consistent with the CoMP observations. For the time-dependent, driven problem \citet{pascoe} studied numerically the spatial damping of resonant kink waves and obtained equivalent results to those analytically predicted by TGV. Importantly, \citet{solerspatial} have shown that the results of TGV still hold when the plasma is partially ionized, so that the theory can be applied to kink waves propagating in the chromosphere and in prominences as well. Recently, \citet{stratified} extended the results of TGV by taking into account for the first time density variation both transversely and along the magnetic field direction.
Field-aligned flows are also ubiquitous in magnetic structures in the solar atmosphere \citep[see the observational reports by, e.g.,][]{brekke,zirker,winebarger01,winebarger02,okamoto,chae2008,ofmanwang}. Typically, the observed flow velocities are smaller than 10\% of the plasma Alfv\'en speed. Faster flows of the order of the Alfv\'en speed are much less frequent and are related to energetic events as, e.g., flares and coronal mass ejections \citep[see, e.g,][]{innes}. The investigation of the effect of flow on the properties of the waves is therefore of evident interest. For example, the influence of flow and implications for MHD seismology of standing kink waves in coronal loops have been recently discussed by \citet{rudermanflow} and \citet{terradasletterflow}. In the case of prominences, \citet{solerthread} investigated standing kink waves in coronal flux tubes partially filled with flowing threads of prominence material. Nevertheless, none of these works took damping into account. Also for standing waves, \citet{terradasflow} studied temporal damping by resonant absorption in the presence of flow and found corrections to the damping time due to the flow with respect to the static case \citep{goossens2002}. However, in the case of spatial damping of propagating kink waves the effect of flow has not been investigated. TGV did not include flow in their study. To our knowledge there is no work in the literature that studies in detail the influence of flow on resonantly damped propagating waves in solar magnetic waveguides. Existing investigations of surface waves in the solar wind as, e.g., \citet{evans} did not perform a rigorous treatment of the process of resonant absorption and considered very simplified expressions for the damping length of the waves. Thus, a detailed investigation of propagating resonant MHD waves in the presence of flows in needed.
Here we investigate the effect of flow on the spatial damping of resonant kink waves in transversely nonuniform solar waveguides. We attack the problem both analytically and numerically. The analytical theory uses the thin tube (TT) and thin boundary (TB) approximations to obtain expressions for the wavelength and the damping length. We determine the influence of flow and compare our expressions with those obtained by TGV in the static case. Later, we use numerical methods to study the propagation and spatial damping of kink waves beyond the TT and TB approximations. The full numerical computations enable us to test the validity of the analytical expressions. Finally, we discuss the implications of our results for MHD seismology.
\section{Model and governing equations}
\label{sec:model}
The equilibrium configuration is a straight cylindrical magnetic flux tube of
radius $R$ embedded in a magnetized plasma environment. For convenience, we use
cylindrical coordinates, namely $r$, $\varphi$, and $z$ for the radial, azimuthal,
and longitudinal coordinates, respectively. The $z$-direction is set along the axis of the cylinder.
We use the $\beta = 0$ approximation, where $\beta$ is the ratio of gas pressure to magnetic pressure. The $\beta = 0$ approximation enables us to arbitrarily choose the density profile. In what follows, subscripts $\rm i$ and $\rm e$ refer to the internal and external plasmas, respectively. For example, we denote by $\rho_{\rm i}$ and
$\rho_{\rm e}$ the internal and external densities, respectively. Both of these quantities are
constants. There is a nonuniform transitional layer in the transverse
direction that continuously connects the internal density to the external density.
The layer has a thickness $l$ and covers the interval $R - l/2 \leq r \leq R
+ l/2$. The equilibrium magnetic field is straight, ${\bf B} = B
\hat{e}_z$, with $B$ constant. We assume a background flow along the magnetic field direction, ${\bf U} = U
\hat{e}_z$. We denote by $U_{\mathrm{i}}$ and $U_{\mathrm{e}}$ the internal and external flow velocities, respectively, which are constants. We take $U_{\mathrm{i}}$ and $U_{\mathrm{e}}$ as positive quantities. As for the density, we allow the flow velocity to change continuously in the radial direction from its internal to its external values within a transitional layer. The transitional layer for the flow velocity extends in the interval $R - l^\star/2 \leq r \leq R
+ l^\star/2$, with $l^\star$ the thickness of the transition.
Linear, ideal MHD waves propagating in our model are governed by the following set of equations,
\begin{eqnarray}
\rho \left( \frac{\partial {\bf v}}{\partial t} + {\bf U} \cdot \nabla {\bf v} + {\bf v} \cdot \nabla {\bf U} \right) &=& \frac{1}{\mu} \left( \nabla
\times {\bf b} \right) \times {\bf B}, \label{eq:b1} \\
\frac{\partial {\bf b}}{\partial t} -\nabla \times \left( {\bf U} \times {\bf b} \right) &=&
\nabla \times \left( {\bf v} \times {\bf B} \right), \label{eq:b2}
\end{eqnarray}
where $\rho$ is the plasma density, ${\bf v}$ is the velocity perturbation, ${\bf b}$ is the magnetic field perturbation, and $\mu$ is the magnetic permittivity. As the equilibrium is uniform in both $\varphi$- and $z$-directions and we consider waves propagating along the tube with a fixed frequency, we write all
perturbations proportional to $\exp \left( i m \varphi + i k_z z - i \omega t \right)$, where
$m$ is the azimuthal wavenumber, $k_z$ is the longitudinal wavenumber, and $\omega$ is the wave angular frequency. Due to the presence of
a transverse inhomogeneous transitional layer, wave modes with $m \neq 0$ are
spatially damped due to resonant absorption. Here we are interested in kink waves, which are described by $m=1$. Kink waves have mixed Alfv\'enic and fast MHD properties. They are the only wave modes that can displace the magnetic cylinder axis and so produce transverse motions of the whole flux tube \citep[see, e.g.,][]{edwinroberts,goossens2009}. As a result of the process of resonant absorption, transverse kink motions of the flux tube are damped and azimuthal motions within the transitional layer are amplified as the wave propagates along the magnetic cylinder.
For real $\omega$, resonant damping causes $k_z$ to be complex, $k_z = k_{z \rm R} + i k_{z \rm I}$, with $k_{z \rm R}$ and $k_{z \rm I}$ the real and imaginary parts of $k_z$, respectively. For fixed and positive $\omega$, the direction of wave propagation is determined by the sign of $k_{z \rm R}$. For $k_{z \rm R} > 0$ the wave propagates towards the positive $z$-direction (forward waves), whereas for $k_{z \rm R} < 0$ the wave propagates towards the negative $z$-direction (backward waves). In the absence of flow both directions of propagation are equivalent. In the presence of flow forward and backward waves have not the same properties and both directions of propagation must be taken into account \citep[see, e.g.,][]{nakaroberts,terra,solerflow,vashe}. Regarding the imaginary part of $k_z$, resonant absorption spatially damps the wave, so we expect $k_{z \rm I} > 0$. However, strong flows may trigger the Kelvin-Helmholtz Instability (KHI) \citep[see, e.g.,][]{chandra,drazin}, causing modes to be amplified in $z$, i.e., $k_{z \rm I} < 0$. From the real and imaginary parts of $k_z$ we compute the wavelength, $\lambda$, and the damping length, $L_{\rm D}$, as
\begin{equation}
\lambda = \frac{2 \pi}{k_{z \rm R} }, \qquad L_{\rm D} = \frac{1}{k_{z \rm I}}.
\end{equation}
\section{Analytical investigation}
\label{sec:analytics}
To study analytically the effect of flow on the resonantly damped kink waves we use the TT and TB approximations. In the TT approximation we restrict ourselves to waves with $\lambda / R \gg 1$. In terms of frequency, the TT approximation is equivalent to the low-frequency approximation, i.e., $\omega \tau_{\mathrm{A}} \ll 1$, with $\tau_{\mathrm{A}} = R / v_{\mathrm{A}}$ the Alfv\'en travel time and $v_{\mathrm{A}} = B/\sqrt{\mu \rho}$ the Alfv\'en velocity. The TB approximation is used here to include the effect of resonant absorption in the inhomogeneous layer, and is valid for $l/R \ll 1$ and $l^\star/R \ll 1$. In the TB approximation, the jump of the perturbations across the inhomogeneous layer is assumed to be the same as their jump across the resonant layer. The expressions for the jump conditions can be found in, e.g., \citet{SGH91,goossens95,tirry} for the static case, and in \citet{goossens92,erdelyi} for the stationary case. Then, the connection formulae at the Alfv\'en resonance are used as jump conditions for the perturbations at the tube boundary \citep[see extensive details about the method in][]{goossens06,goossensIAU,goossensSSR}. The dispersion relation for kink and fluting MHD waves in the TT and TB approximations is \citep[see, e.g.,][]{goossens92}
\begin{eqnarray}
\rho_{\rm i} \left( \Omega_{\rm i}^2 - \omega_{\rm A i}^2 \right) &+& \rho_{\rm e} \left( \Omega_{\rm e}^2 - \omega_{\rm A e}^2 \right) = \nonumber \\
&& i \pi \frac{m/ r_{\rm A}}{\rho \left(r_{\rm A} \right) \left| \Delta_{\rm A} \right|} \rho_{\rm i} \left( \Omega_{\rm i}^2 - \omega_{\rm A i}^2 \right)\rho_{\rm e} \left( \Omega_{\rm e}^2 - \omega_{\rm A e}^2 \right), \label{eq:reldisper}
\end{eqnarray}
where $\Omega = \omega - U k_z$ is the Doppler-shifted frequency, $\omega_{\rm A}^2 = k_z^2 v_{\mathrm{A}}^2$ is the square of the Alfv\'en frequency, $r_{\rm A}$ is the Alfv\'en resonance position, and
\begin{equation}
\left| \Delta_{\rm A} \right| = \left| \frac{\rm d}{{\rm d}r} \left[ \Omega^2 - \omega_{\rm A}^2 \right]_{r_{\rm A}} \right|. \label{eq:delta}
\end{equation}
The term on the right-hand side of Equation~(\ref{eq:reldisper}) contains the effect of resonant absorption. Note that the dependence on $m$ of the dispersion relation is only present in this term. This means that in the TT approximation the effect of the azimuthal wavenumber, $m$, is only felt in the damping of the wave, not in its propagation.
In the case of temporal damping of standing waves, i.e., real $k_z$ and complex $\omega$, the solutions to Equation~(\ref{eq:reldisper}) have been investigated in detail by \citet{goossens92} and \citet{terradasflow}. In the case of spatial damping of propagating waves, i.e., real $\omega$ and complex $k_z$, Equation~(\ref{eq:reldisper}) has been explored by TGV in the absence of flow. Here our purpose is to investigate the effect of flow on the results obtained by TGV in the static case.
\subsection{No resonant damping}
In the absence of resonant damping, i.e., for $l/R = l^\star/R = 0$, Equation~(\ref{eq:reldisper}) becomes
\begin{equation}
\rho_{\rm i} \left( \Omega_{\rm i}^2 - \omega_{\rm A i}^2 \right) + \rho_{\rm e} \left( \Omega_{\rm e}^2 - \omega_{\rm A e}^2 \right) = 0. \label{eq:reldisper0}
\end{equation}
Equation~(\ref{eq:reldisper0}) is independent of $m$. In the static case, $U_{\mathrm{i}} = U_{\mathrm{e}} = 0$ and $\Omega_{\rm i} = \Omega_{\rm e} = \omega$. The solution to Equation~(\ref{eq:reldisper0}) is
\begin{equation}
k_z = \pm \frac{\omega}{v_{\rm k}} \equiv \pm k_0, \label{eq:kznoflow}
\end{equation}
with
\begin{equation}
v_{\rm k} = \left( \frac{\rho_{\rm i} v_{\mathrm{Ai}}^2 + \rho_{\rm e} v_{\mathrm{Ae}}^2}{\rho_{\rm i} + \rho_{\rm e}} \right)^{1/2},
\end{equation}
the kink velocity. In Equation~(\ref{eq:kznoflow}) the $+$ sign stands for the forward wave and the $-$ sign for the backward wave. In the absence of flow both forward and backward waves are equivalent and have the same wavelength, $\lambda = 2\pi / k_0$.
In the presence of flow, we rewrite Equation~(\ref{eq:reldisper0}) as a second-order polynomial in $k_z$, namely
\begin{equation}
\left( v_{\rm k}^2 - v_{\rm KE}^2 \right) k_z^2 + 2 v_{\rm cm} \omega\, k_z - \omega^2 = 0, \label{eq:poli}
\end{equation}
where $v_{\rm cm}$ and $v_{\rm KE}$ are defined as
\begin{equation}
v_{\rm cm} = \frac{\rho_{\rm i} U_{\mathrm{i}} + \rho_{\rm e} U_{\mathrm{e}}}{\rho_{\rm i} + \rho_{\rm e}}, \qquad
v_{\rm KE} =\left( \frac{\rho_{\rm i} U_{\mathrm{i}}^2 + \rho_{\rm e} U_{\mathrm{e}}^2}{\rho_{\rm i} + \rho_{\rm e}} \right)^{1/2}.
\end{equation}
$v_{\rm cm}$ is the center-of-mass velocity. $v_{\rm KE}$ is the velocity associated to the kinetic energy of the flow. Note that, similarly, the kink velocity, $v_{\rm k}$, is the velocity associated to the energy of the magnetic field. The two solutions to Equation~(\ref{eq:poli}) for $v_{\rm KE} \ne v_{\rm k}$ are
\begin{equation}
k_z = - \frac{v_{\rm cm}}{v_{\rm k}^2 - v_{\rm KE}^2} \omega \pm k_0 \frac{v_{\rm k}^2}{v_{\rm k}^2 - v_{\rm KE}^2} \left[ 1 - \frac{\rho_{\rm i} \rho_{\rm e}}{\left( \rho_{\rm i} + \rho_{\rm e} \right)^2} \frac{\left( U_{\mathrm{i}} - U_{\mathrm{e}} \right)^2}{v_{\rm k}^2} \right]^{1/2}, \label{eq:kzsol}
\end{equation}
where the $+$ and $-$ signs in front of the second term stand for forward and backward waves, respectively. If $U_{\mathrm{i}} = U_{\mathrm{e}} = 0$, Equation~(\ref{eq:kzsol}) simply reduces to Equation~(\ref{eq:kznoflow}). We clearly see in Equation~(\ref{eq:kzsol}) that the equivalence between both directions of propagation is broken by the flow. For $v_{\rm KE} < v_{\rm k}$ forward and backward waves propagate in opposite directions. For $v_{\rm KE} > v_{\rm k}$ both waves propagate in the same direction, i.e., they both are forward waves in practice because the flow is strong enough to force the backward wave to reverse its direction of propagation. In the particular case $v_{\rm KE} = v_{\rm k}$, the solution to Equation~(\ref{eq:poli}) is
\begin{equation}
k_z = \frac{\omega}{2 v_{\rm cm}},
\end{equation}
which corresponds to the forward wave, while the backward wave does not propagate in the static reference frame. For $U_{\mathrm{e}} = 0$ the condition $v_{\rm KE} = v_{\rm k}$ is equivalent to
\begin{equation}
U_{\mathrm{i}}^2 = 2 v_{\mathrm{Ai}}^2. \label{eq:inversion}
\end{equation}
When the argument of the square root in Equation~(\ref{eq:kzsol}) is negative, $k_z$ becomes complex. This is the classical KHI \citep[see, e.g.,][]{chandra,drazin}. Then, the two solutions correspond to a spatially damped mode and a spatially amplified mode, respectively. The KHI appears for a critical velocity shear, $\Delta U = U_{\mathrm{i}} - U_{\mathrm{e}}$, defined as
\begin{equation}
\left( \Delta U\right)^2 > \frac{\left( \rho_{\rm i} + \rho_{\rm e} \right)^2}{\rho_{\rm i} \rho_{\rm e}} v_{\rm k}^2 \equiv v_{\rm KH}^2. \label{eq:velKHI}
\end{equation}
Again, in the reference frame where $U_{\mathrm{e}} = 0$ Equation~(\ref{eq:velKHI}) can be rewritten as
\begin{equation}
U_{\mathrm{i}}^2 > 2 \left( 1 + \frac{\rho_{\rm i}}{\rho_{\rm e}} \right) v_{\mathrm{Ai}}^2. \label{eq:velKHI2}
\end{equation}
From Equations~(\ref{eq:inversion}) and (\ref{eq:velKHI2}) we see that the KHI requires a faster flow velocity than the one needed to reverse the propagation of the backward wave.
Equivalently to the case without flow (see Equation~(\ref{eq:kznoflow})), we can rewrite Equation~(\ref{eq:kzsol}) as
\begin{equation}
k_z = \frac{\omega}{{v_{\rm kf}^{\pm}}} \label{eq:kzgen}
\end{equation}
where we have introduced the effective kink velocity modified by the flow, ${v_{\rm kf}^{\pm}}$, defined by
\begin{eqnarray}
{v_{\rm kf}^{\pm}} = \left\{ - \frac{v_{\rm cm}}{v_{\rm k}^2 - v_{\rm KE}^2} \pm \frac{v_{\rm k}}{v_{\rm k}^2 - v_{\rm KE}^2} \left[ 1 - \frac{\rho_{\rm i} \rho_{\rm e}}{\left( \rho_{\rm i} + \rho_{\rm e} \right)^2} \frac{\left( U_{\mathrm{i}} - U_{\mathrm{e}} \right)^2}{v_{\rm k}^2} \right]^{1/2} \right\}^{-1}. \nonumber \\ \label{eq:vkf}
\end{eqnarray}
Note that the effective kink velocity is different for forward ($+$ sign) and backward ($-$ sign) waves. This means that the phase speed of the waves depends on their direction of propagation. Also note that ${v_{\rm kf}^{\pm}}$ becomes complex beyond the critical velocity shear for the KHI. For slow, sub-Alfv\'enic flows we may drop the quadratic terms in $U_{\mathrm{i}}$ and $U_{\mathrm{e}}$ from Equation~(\ref{eq:vkf}) to obtain a first-order approximation for ${v_{\rm kf}^{\pm}}$, namely
\begin{equation}
{v_{\rm kf}^{\pm}} \approx \pm v_{\rm k} + v_{\rm cm} . \label{eq:vkfapp}
\end{equation}
In the absence of flow, Equation~(\ref{eq:vkf}) reduces to ${v_{\rm kf}^{\pm}} = \pm v_{\rm k}$. When the effective kink velocity is equal to the external Alfv\'en velocity, the wave becomes leaky in the external medium. In terms of $k_z$, waves are leaky for wavenumbers larger than
\begin{equation}
k_z = \pm \frac{\omega}{v_{\mathrm{Ae}}} \equiv \pm k_{\rm L}.
\end{equation}
The forward wave becomes leaky for much slower flow velocities than the backward wave. By using Equation~(\ref{eq:vkfapp}) and taking $U_{\mathrm{e}} = 0$, we find that the forward wave becomes leaky for
\begin{equation}
U_{\mathrm{i}} \gtrsim \left( \frac{\rho_{\rm i} + \rho_{\rm e} }{\sqrt{\rho_{\rm i} \rho_{\rm e}}} - \sqrt{\frac{2 \left( \rho_{\rm i} + \rho_{\rm e} \right)}{\rho_{\rm i}}} \right) v_{\mathrm{Ai}}. \label{eq:uileaky}
\end{equation}
Again, we use the approximation of Equation~(\ref{eq:vkfapp}) in Equation~(\ref{eq:kzgen}) to obtain a first-order approximation of the wavelength as
\begin{equation}
\lambda \approx \lambda_0 \left( \pm 1 + \frac{v_{\rm cm}}{v_{\rm k}} \right), \label{eq:lam}
\end{equation}
with $\lambda_0 = 2\pi / k_0$. We consider the particular case $U_{\mathrm{e}} = 0$ and use the dimensionless notation of TGV to rewrite Equation~(\ref{eq:lam}) as
\begin{equation}
\frac{\lambda}{R} \approx 2\pi \sqrt{\frac{2\zeta}{\zeta + 1}} \frac{1}{f} \left( \pm 1 + \sqrt{\frac{\zeta}{2 \left( \zeta + 1 \right)}} \bar{U_{\mathrm{i}}} \right), \label{eq:lam2}
\end{equation}
with
\begin{equation}
\zeta = \frac{\rho_{\rm i}}{\rho_{\rm e}}, \qquad f = \frac{\omega R}{v_{\mathrm{Ai}}}, \qquad \bar{U_{\mathrm{i}}} = \frac{U_{\mathrm{i}}}{v_{\mathrm{Ai}}}, \label{eq:dimension}
\end{equation}
the density contrast, the dimensionless frequency, and the dimensionless flow velocity, respectively.
We plot in Figure~\ref{fig:tt}(a) $|\lambda|/R$ versus $\bar{U_{\mathrm{i}}}$ computed from the full Equation~(\ref{eq:kzsol}) for the particular case $U_{\mathrm{e}} = 0$. As predicted by Equation~(\ref{eq:inversion}), the backward wave reverts its direction of propagation for $\bar{U_{\mathrm{i}}} = \sqrt{2}$. For the set of parameters used in Figure~\ref{fig:tt}, the forward wave becomes leaky for a flow velocity slightly sub-Alfv\'enic (Equation~(\ref{eq:uileaky})). When the threshold velocity of the KHI is reached (Equation~(\ref{eq:velKHI2})), both forward and backward waves merge. We compare the full result with the approximation for slow flows given by Equation~(\ref{eq:lam2}) (see the symbols in Fig.~\ref{fig:tt}(a)), and obtain a good agreement for sub-Alfv\'enic flows, i.e., $\bar{U_{\mathrm{i}}} \lesssim 1$. On the other hand, Figure~\ref{fig:tt}(b) displays $k_{z \rm I} R$ versus $\bar{U_{\mathrm{i}}}$. In the absence of resonant damping, the imaginary part of $k_z$ is zero for flow velocities slower than the critical velocity shear of the KHI. For larger velocities, one damped solution and one overstable solution are present.
\begin{figure}[!t]
\centering
\includegraphics[width=0.85\columnwidth]{f01a.eps}
\includegraphics[width=0.85\columnwidth]{f01b.eps}
\caption{(a) $\left| \lambda \right| / R$ and (b) $k_{z \rm I} R$ versus $\bar{U_{\mathrm{i}}}$ corresponding to the forward (solid line) and backward (dashed line) kink waves in the absence of resonant damping. The vertical dotted lines in both panels correspond to the different critical flow velocities indicated in the text. The horizontal dotted line in panel (a) is the wavelength for $\bar{U_{\mathrm{i}}} = 0$, with $\lambda_0 = 2\pi/k_0$. The symbols correspond to the approximation for slow flows given in Equation~(\ref{eq:lam2}). In panel (b) the horizontal dotted line denotes $k_{z \rm I} = 0$. In this plot, $f = 0.1$, $\zeta = 3$, and $U_{\mathrm{e}} = 0$. \label{fig:tt}}
\end{figure}
\subsection{Effect of resonant damping}
\label{sec:resonant}
Here we incorporate the damping due to resonant absorption. We take into account the full expression of the dispersion relation (Equation~(\ref{eq:reldisper})). We write $k_z = k_{z \rm R} + i k_{z \rm I}$ in Equation~(\ref{eq:reldisper}) and consider weak damping, so we neglect terms of $\mathcal{O} \left( k_{z \rm I}^2 \right)$. In addition, we implicitly assume that the flow velocities are slower than the critical velocity of the KHI. After long but straightforward analytical manipulations, we obtain from Equation~(\ref{eq:reldisper}) the expression for the ratio $k_{z \rm I} / k_{z \rm R}$, namely
\begin{equation}
\left| \frac{k_{z \rm I}}{k_{z \rm R}}\right| = \frac{\pi}{2} \frac{m}{r_{\rm A}} \frac{\rho_{\rm i}^2}{\rho_{\rm i}+\rho_{\rm e}}\frac{1}{\rho \left(r_{\rm A} \right) \left| \Delta_{\rm A} \right|} \frac{\left( \Omega^2_{\rm i} - \omega_{\rm Ai} \right)^2}{\omega \left( \omega - \omega_{\rm cm} \right)}, \label{eq:ratio}
\end{equation}
with $\omega_{\rm cm} = k_{z \rm R} v_{\rm cm}$. In the absence of flows, Equation~(\ref{eq:ratio}) reduces to Equation~(8) of TGV. Due to the different values of $\Omega_{\rm i}$ and $\omega_{\rm cm}$ for forward and backward waves, Equation~(\ref{eq:ratio}) predicts that both waves have different damping ratios. The expression for the ratio $k_{z \rm I} / k_{z \rm R}$ is more complicated in the presence of flows compared to the static case of TGV. Let us find a more simple expression for $k_{z \rm I} / k_{z \rm R}$ in the limit of slow, sub-Alfv\'enic flows. First, we evaluate $\rho \left(r_{\rm A} \right) \left| \Delta \right|_{\rm A}$ to explicitly take into account the radial variation of the density and the flow velocity at the resonance position. From Equation~(\ref{eq:delta}) we get
\begin{equation}
\rho \left(r_{\rm A} \right) \left| \Delta_{\rm A} \right| = \left| \Omega^2 \left(r_{\rm A}\right) \left( \frac{{\rm d} \rho}{{\rm d}r} \right)_{r_{\rm A}} - 2 \rho\left(r_{\rm A}\right) \Omega \left(r_{\rm A}\right)k_{z \rm R} \left( \frac{{\rm d} U}{{\rm d}r} \right)_{r_{\rm A}} \right|, \label{eq:deltafull}
\end{equation}
where we have used the resonant condition $\Omega^2 \left(r_{\rm A}\right)= k_{z \rm R}^2 v_{\mathrm{A}}^2 \left(r_{\rm A}\right)$. The quantities present in Equation~(\ref{eq:deltafull}) have to be evaluated at the resonance position, $r_{\rm A}$. In the TB approximation it is reasonable to assume $r_{\rm A} \approx R$. There are two terms in the right-hand side of Equation~(\ref{eq:deltafull}). The first term is due to the variation of density and the second term is due to the variation of flow velocity. The direction of wave propagation is also important in Equation~(\ref{eq:deltafull}) because of the sign of $k_{z \rm R}$. In the absence of flow, Equation~(\ref{eq:deltafull}) simplifies to $\rho \left(r_{\rm A} \right) \left| \Delta_{\rm A} \right| = \omega^2 \left( \frac{{\rm d}\rho}{{\rm d}r} \right)_{r_{\rm A}}$.
For smooth profiles, the derivatives of the density and the flow velocity profiles at the resonance position can be cast as
\begin{equation}
\left( \frac{{\rm d}\rho}{{\rm d}r} \right)_{r_{\rm A}} \approx \mathcal{F} \frac{\pi^2}{4} \frac{\rho_{\rm i} - \rho_{\rm e}}{l}, \qquad \left( \frac{{\rm d}U}{{\rm d}r} \right)_{r_{\rm A}} \approx \mathcal{F} \frac{\pi^2}{4} \frac{U_{\mathrm{i}} - U_{\mathrm{e}}}{l^\star} \label{eq:deriv}
\end{equation}
with $\mathcal{F}$ a factor that depends on the form of the transverse profile. For example, $\mathcal{F} = 4/\pi^2$ for a linear profile \citep{goossens2002} and $\mathcal{F} = 2/\pi$ for a sinusoidal profile \citep{rudermanroberts}. For simplicity we assume the same profile for both density and flow velocity, but we keep $l \ne l^\star$. As values of density and flow velocity at the resonance position we take
\begin{equation}
\rho \left(r_{\rm A} \right) = \frac{\rho_{\rm i} + \rho_{\rm e}}{2}, \qquad U \left(r_{\rm A} \right) = \frac{U_{\mathrm{i}} + U_{\mathrm{e}}}{2}. \label{eq:valsra}
\end{equation}
We use in Equation~(\ref{eq:deltafull}) the expressions given in Equations~(\ref{eq:deriv}) and (\ref{eq:valsra}) to obtain
\begin{eqnarray}
\rho \left(r_{\rm A} \right) \left| \Delta_{\rm A} \right| &=& \mathcal{F} \frac{\pi^2}{4} \frac{\rho_{\rm i} - \rho_{\rm e}}{l} \Omega^2 \left(r_{\rm A}\right) \nonumber \\
&\times& \left| 1 - \frac{k_{z \rm R}}{\Omega\left(r_{\rm A}\right)} \frac{\rho_{\rm i} + \rho_{\rm e}}{\rho_{\rm i}-\rho_{\rm e}} \left( U_{\mathrm{i}} - U_{\mathrm{e}} \right) \frac{l}{l^\star} \right|. \label{eq:deltafull2}
\end{eqnarray}
For our following analysis, we assume that the second term within the absolute value in Equation~(\ref{eq:deltafull2}) is smaller than one, so that we can drop the absolute value sign. For slow flows, this is equivalent to assume $l^\star \gtrsim l$. In the limit of slow, sub-Alfv\'enic flows, we neglect the quadratic terms in the flow velocities. In addition, we write $k_{z \rm R} \approx \omega / {v_{\rm kf}^{\pm}}$ and use the first-order approximation for ${v_{\rm kf}^{\pm}}$ given in Equation~(\ref{eq:vkfapp}). Equation~(\ref{eq:deltafull2}) can then be rewritten as
\begin{eqnarray}
\rho \left(r_{\rm A} \right) \left| \Delta_{\rm A} \right| &\approx& \mathcal{F} \frac{\pi^2}{4} \frac{\rho_{\rm i} - \rho_{\rm e}}{l} \omega^2 \nonumber \\
&\times& \left\{ 1 \mp \frac{U_{\mathrm{i}} + U_{\mathrm{e}} }{v_{\rm k}} \left[ 1 + \frac{\rho_{\rm i} + \rho_{\rm e}}{\rho_{\rm i} - \rho_{\rm e}} \frac{U_{\mathrm{i}} - U_{\mathrm{e}}}{U_{\mathrm{i}} + U_{\mathrm{e}}} \frac{l}{l^\star} \right] \right\}. \label{eq:detafull3}
\end{eqnarray}
Next, we use Equation~(\ref{eq:detafull3}) in Equation~(\ref{eq:ratio}) and perform a first-order expansion in the flow velocities. Equation~(\ref{eq:ratio}) becomes
\begin{eqnarray}
\left| \frac{k_{z \rm I}}{k_{z \rm R}}\right| &\approx& \frac{1}{2 \pi} \frac{m}{\mathcal{F}} \frac{l}{R} \frac{ \rho_{\rm i}-\rho_{\rm e} }{\rho_{\rm i}+\rho_{\rm e}} \left\{ 1 \pm \frac{v_{\rm cm}}{v_{\rm k}} \left[ 1 - 4 \frac{\rho_{\rm i} + \rho_{\rm e}}{\rho_{\rm i} - \rho_{\rm e}} \frac{ \rho_{\rm i} U_{\mathrm{i}} - \rho_{\rm e} U_{\mathrm{e}}}{\rho_{\rm i} U_{\mathrm{i}} + \rho_{\rm e} U_{\mathrm{e}}} \right. \right. \nonumber \\
&+& \left. \left. \frac{U_{\mathrm{i}} - U_{\mathrm{e}}}{v_{\rm cm}} \left( 1 + \frac{\rho_{\rm i} + \rho_{\rm e}}{\rho_{\rm i} - \rho_{\rm e} } \frac{U_{\mathrm{i}} - U_{\mathrm{e}}}{U_{\mathrm{i}} + U_{\mathrm{e}}} \frac{l}{l^\star} \right) \right] \right\}. \label{eq:ratioapp}
\end{eqnarray}
As in previous expressions, the $+$ and $-$ signs in Equation~(\ref{eq:ratioapp}) stand for forward and backward waves, respectively. In the particular case $U_{\mathrm{e}} = 0$, Equation~(\ref{eq:ratioapp}) simplifies to
\begin{eqnarray}
\left| \frac{k_{z \rm I}}{k_{z \rm R}}\right| &\approx& \frac{1}{2 \pi} \frac{m}{\mathcal{F}} \frac{l}{R} \frac{ \rho_{\rm i}-\rho_{\rm e} }{\rho_{\rm i}+\rho_{\rm e}} \nonumber \\
&\times& \left[ 1 \pm \frac{U_{\mathrm{i}}}{v_{\rm k}} \left( 1 + \frac{\rho_{\rm i}}{\rho_{\rm i} + \rho_{\rm e}} - \frac{4\rho_{\rm i}}{\rho_{\rm i} - \rho_{\rm e}} +\frac{\rho_{\rm i} + \rho_{\rm e}}{\rho_{\rm i} - \rho_{\rm e} } \frac{l}{l^\star} \right) \right]. \label{eq:ratiofin}
\end{eqnarray}
We use again the approximation $k_{z \rm R} \approx \omega / {v_{\rm kf}^{\pm}}$ and compute from Equation~(\ref{eq:ratiofin}) the damping length, $L_{\rm D} = 1/k_{z \rm I}$, as
\begin{equation}
L_{\rm D} \approx 2\pi \frac{\mathcal{F}}{m} \frac{R}{l} \frac{\rho_{\rm i} + \rho_{\rm e}}{\rho_{\rm i} - \rho_{\rm e}} \frac{v_{\rm k}}{\omega} \left[ 1 \pm \frac{U_{\mathrm{i}}}{v_{\rm k}} \left( \frac{3\rho_{\rm i} + \rho_{\rm e}}{\rho_{\rm i}-\rho_{\rm e}} - \frac{\rho_{\rm i} + \rho_{\rm e}}{\rho_{\rm i}-\rho_{\rm e}} \frac{l}{l^\star} \right) \right]. \label{eq:ld}
\end{equation}
Finally, we express Equation~(\ref{eq:ld}) using the dimensionless quantities defined in Equation~(\ref{eq:dimension}), namely
\begin{eqnarray}
\frac{L_{\rm D}}{R} &\approx& 2\pi \xi_{\rm E} \sqrt{\frac{2\zeta}{\zeta + 1}} \frac{1}{f} \left[ 1 \pm \bar{U_{\mathrm{i}}} \sqrt{\frac{\zeta + 1}{2\zeta}} \left( \frac{3 \zeta + 1}{\zeta - 1} - \frac{\zeta + 1}{\zeta - 1} \frac{l}{l^\star} \right) \right], \nonumber \\ \label{eq:ldtt}
\end{eqnarray}
with
\begin{eqnarray}
\xi_{\rm E} = \frac{\mathcal{F}}{m} \frac{R}{l} \frac{\zeta + 1}{\zeta - 1}.
\end{eqnarray}
Equation~(\ref{eq:ldtt}) is the key equation of this investigation and contains basic properties on the spatial damping of propagating kink MHD waves. Several important results can be extracted from Equation~(\ref{eq:ldtt}). Equation~(\ref{eq:ldtt}) predicts that backward and forward propagating waves are damped on length scales that are inversely proportional to the frequency, $f$. This is the same dependence found by TGV in the static case. However, the factor of proportionality depends on the characteristics of the flow and the density contrast, so that the damping length differs from its static analogue. As for the wavelength (see Equation~(\ref{eq:lam2})), the damping length for forward and backward waves is different.
To shed more light on this result, let us consider the case $l^\star \gg l$. Equation~(\ref{eq:ldtt}) reduces to
\begin{equation}
L_{\rm D} \approx 2\pi \xi_{\rm E} \sqrt{\frac{2\zeta}{\zeta + 1}} \frac{1}{f} \left( 1 \pm \bar{U_{\mathrm{i}}} \sqrt{\frac{\zeta + 1}{2\zeta}} \frac{3 \zeta + 1}{\zeta - 1} \right) \label{eq:ld2}
\end{equation}
According to Equation~(\ref{eq:ld2}) the backward propagating wave ($-$ sign) gets damped on a shorter length scale than in the absence of flow, while for the forward propagating wave ($+$ sign) the damping length is longer. For $l \approx l^\star$ the damping length for the forward wave remains longer than that of the backward wave, but we must note that for $l^\star \ll l$ the situation may be the opposite. From Equation~(\ref{eq:ldtt}) we can assess the relation between $l$ and $l^\star$ for which the two terms multiplying the flow velocity cancel each other, namely
\begin{equation}
l^\star = \frac{\zeta + 1}{3\zeta+ 1} l. \label{eq:llstar}
\end{equation}
Thus, the backward wave damping length becomes longer than that of the forward wave for $l^\star$ smaller than the value given in Equation~(\ref{eq:llstar}). Strictly, Equation~(\ref{eq:ldtt}) is not valid in the limit $l^\star \ll l$, because we have to properly take into account the absolute value in Equation~(\ref{eq:deltafull2}). For slow flows and $l^\star \ll l$, Equation~(\ref{eq:deltafull2}) can be approximated as
\begin{equation}
\rho \left(r_{\rm A} \right) \left| \Delta_{\rm A} \right| \approx \mathcal{F} \frac{\pi}{4} \frac{\rho_{\rm i} + \rho_{\rm e}}{l^\star} \omega^2 \frac{U_{\mathrm{i}} - U_{\mathrm{e}}}{v_{\rm k}}. \label{eq:deltalim}
\end{equation}
Importantly, we find that Equation~(\ref{eq:deltalim}) is independent of $l$ and the same expression holds for forward and backward waves. Now, from Equation~(\ref{eq:deltalim}) it is straightforward to obtain that $L_{\rm D} \sim 1/l^\star$ for both forward and backward waves in the limit $l^\star \ll l$. As discussed by \citet{terradasflow} in the case of temporal damping of standing waves, multiple resonances may occur within the inhomogeneous transitional layer in the limit $l^\star \ll l$ \citep[see Fig.~5 of][]{terradasflow}. In such a case, the total damping rate is the sum of the contributions from each resonance. We do not study this peculiar situation which takes place for very small, probably not realistic $l^\star$. Instead, we refer the reader to \citet{terradasflow} for details.
\section{Numerical computations}
\label{sec:numerics}
Here, we verify the validity of the analytical expressions derived in Section~\ref{sec:analytics} in the TT and TB approximations. To do so, we numerically solve the full eigenvalue problem by means of the PDE2D code \citep{sewell}. The numerical scheme is similar to that used by TGV. The code implements a method based on finite elements to numerically integrate Equations~(\ref{eq:b1}) and (\ref{eq:b2}) in the radial direction from the cylinder axis, $r=0$, to the edge of the numerical domain, $r=r_{\rm max}$. The boundary conditions at $r=0$ are set according the symmetry arguments, while we impose all perturbations to vanish at $r=r_{\rm max}$. In order to obtain a good convergence of the solutions, $r_{\rm max}$ is located far enough from the magnetic tube to avoid numerical errors. We take $r_{\rm max} = 100R$. We use a nonuniform grid with a large density of grid points within the inhomogeneous layer in order to correctly describe the small spatial scales of the eigenfunctions due to the Alfv\'en resonance. To avoid the singularity of the ideal MHD equations at the resonance position, we add to the induction equation (Equation~(\ref{eq:b2})) a resistive term, i.e., $\eta \nabla^2 {\vec b}$, with $\eta$ the magnetic diffusivity. The output of the code is the complex eigenfunctions and their corresponding eigenvalues. In the limit of large $R_{\rm m}$, with $R_{\rm m} =v_{\mathrm{A}} R / \eta$ the magnetic Reynolds number, the eigenvalues are independent of $R_{\rm m}$. In such a case, wave damping is due to resonant absorption exclusively. In our computations, we have considered a sufficiently large $R_{\rm m}$ and have checked that the eigenvalues are indeed independent of $R_{\rm m}$. We typically take $R_{\rm m} \approx 10^7$ in the computations.
The PDE2D code solves the eigenvalue problem for the temporal damping, i.e., for complex $\omega$ provided a fixed and real $k_z$. As in TGV, we need to convert the results from temporal damping to spatial damping. We use Equation~(40) of TGV to perform the conversion from temporal damping (complex $\omega$ and real $k_z$) to spatial damping (real $\omega$ and complex $k_z$). From Equation~(40) of TGV it is straightforward to obtain the relation between the imaginary parts of $\omega$ and $k_z$, namely
\begin{equation}
k_{z \rm I} = \omega_{\rm I} \left( \frac{\partial \omega_{\rm R}}{\partial k_{z \rm R}} \right)^{-1}, \label{eq:relationwk}
\end{equation}
where $\omega_{\rm R}$ and $\omega_{\rm I}$ are the real and imaginary parts of the frequency in the temporal damping case. In the spatial damping case $\omega = \omega_{\rm R}$. The factor within the parenthesis is the group velocity. Note that the flow does not explicitly appear in Equation~(\ref{eq:relationwk}), but its effect is contained in the value of group velocity numerically computed.
First of all, we test the numerical code by considering the case without flow. In this case we fully recover the results of TGV. Thus, we are confident that the code works properly. Hereon we incorporate the effect of flow. Regarding the wavelength, our numerical results indicate that the presence of the transitional layer has a small impact on the value of the wavelength. The wavelength in the case without resonant damping (see Fig.~\ref{fig:tt}(a)) is a very good approximation to the wavelength in the resonant case. The approximation for slow flows given in Equation~(\ref{eq:lam2}) also holds in the case with damping. Therefore, our following analysis is focused on the behavior of $L_{\rm D}$ in the presence of flows. The numerical results are used to test the approximations behind Equation~(\ref{eq:ldtt}), i.e., the TT and TB approximations, and the assumption of slow flows.
In Figure~\ref{fig:ld_f} we display $L_{\rm D}/R$ as a function of the dimensionless frequency, $f$, for $\bar{U_{\mathrm{i}}} = 0.1$ (the rest of parameters are indicated in the caption of the Fig.~\ref{fig:ld_f}). As in the static case of TGV, the higher the frequency, the shorter the damping length by resonant absorption. We obtain the analytically predicted result that forward and backward waves have different damping lengths. In this example, the damping length of the forward wave is longer than that of the backward wave. The equivalent damping length in the absence of flow is in between both values (the dotted line in Fig.~\ref{fig:ld_f}). We compare the numerical results with those in the TT approximation and for slow flows (Equation~(\ref{eq:ldtt})). The TT approximation applies when $f \ll 1$. In Figure~\ref{fig:ld_f}(a) we plot the results for $f \leq 0.1$. An excellent agreement is found between numerical and analytical solutions for both forward and backward waves. The agreement between approximate and numerical results is also remarkably good even when the condition $f \ll 1$ of the TT approximation is not strictly fulfilled. This can be seen in Figure~\ref{fig:ld_f}(b), where results are plotted for $f \leq 1$.
\begin{figure}[!t]
\centering
\includegraphics[width=0.85\columnwidth]{f02a.eps}
\includegraphics[width=0.85\columnwidth]{f02b.eps}
\caption{(a) Ratio of the damping length to the radius, $L_{\rm D} / R$, versus the dimensionless frequency, $f$, corresponding to the forward (solid line) and backward (dashed line) kink waves for $\bar{U_{\mathrm{i}}} = 0.1$, $U_{\mathrm{e}} = 0$, $\zeta = 3$, and $l / R = l^\star / R = 0.1$. The symbols correspond to the approximation for slow flows given in Equation~(\ref{eq:ldtt}). The dotted line is the result in the absence of flow (see TGV). (b) Same as panel (a) but for larger $f$. \label{fig:ld_f}}
\end{figure}
Next we determine the influence of $l$ and $l^\star$, and test the TB approximation used to derive the analytical expressions. First we consider the case $l = l^\star$. Figure~\ref{fig:ld_l}(a) shows the dependence of the damping length on $l/R$. As for the frequency, $L_{\rm D}/R$ decreases as $l/R$ increases. The dependence of the damping length on $l/R$ for both forward and backward waves is the same. As before, a very good agreement between Equation~(\ref{eq:ldtt}) and the numerical result is found even when the thickness of the transitional layer departs from the limit $l/R \ll 1$. This means that the TB approximation is sufficiently accurate when the condition $l/R \ll 1$ is slightly relaxed. This result enables us to confidently use Equation~(\ref{eq:ldtt}) beyond the condition of validity of the TB approximation. Alternately, in Figure~\ref{fig:ld_l}(b) we keep $l/R$ constant and vary $l^\star /R$, so that the spatial scales for the variation of density and flow velocity are different. We recover the analytically predicted result that for $l^\star \gg l$ the results are independent of $l^\star/R$. For $l^\star \ll l$ Equation~(\ref{eq:ldtt}) does not correctly describe the damping length of the forward wave. As discussed at the end of Section~\ref{sec:resonant}, $L_{\rm D} \sim 1/l^\star$ in the limit $l^\star \ll l$, and so $L_{\rm D}/R$ increases as $l^\star/R \to 0$.
\begin{figure}[!t]
\centering
\includegraphics[width=0.85\columnwidth]{f03a.eps}
\includegraphics[width=0.85\columnwidth]{f03b.eps}
\caption{(a) Ratio of the damping length to the radius, $L_{\rm D} / R$, versus $l/R$ for the forward (solid line) and backward (dashed line) kink waves in the case $ l^\star / R = l/R$. The dotted line is the result in the absence of flow. (b) Dependence on $l^\star / R$ for $l/R = 0.15$. The vertical dotted line denotes the value of $l^\star / R$ for which both forward and backward waves have the same $L_{\rm D} / R$ (Equation~(\ref{eq:llstar})). In both panels the symbols are the approximation of Equation~(\ref{eq:ldtt}). In these plots, $\bar{U_{\mathrm{i}}} = 0.05$, $U_{\mathrm{e}} = 0$, $f = 0.1$, and $\zeta = 3$.\label{fig:ld_l}}
\end{figure}
Finally, we assess the effect of the flow velocity. This is done in Figure~\ref{fig:ld_ui}. Again, the linear approximation (Equation~(\ref{eq:ldtt})) is quite accurate and agrees well with the full numerical results for $\bar{U_{\mathrm{i}}} \lesssim 0.1$. As expected, the difference between the numerical results and the linear approximation increases as the flow velocity gets faster. However, for $\bar{U_{\mathrm{i}}} = 0.2$ the relative difference between the full solution and the approximation is only around 10\% for the forward wave and 30\% for the backward wave. This means that for realistic flow velocities observed in coronal magnetic loops \citep[e.g.,][]{brekke,winebarger01,winebarger02}, Equation~(\ref{eq:ldtt}) correctly describes the behavior of the damping length.
\begin{figure}[!t]
\centering
\includegraphics[width=0.85\columnwidth]{f04.eps}
\caption{Ratio of the damping length to the radius, $L_{\rm D} / R$, versus the flow velocity normalized to the internal Alfv\'en velocity, $\bar{U_{\mathrm{i}}}$, for the forward (solid line) and backward (dashed line) kink waves. The symbols are the linear approximation given in Equation~(\ref{eq:ldtt}). We have used $l/R = l^\star /R = 0.1$, $U_{\mathrm{e}} = 0$, $f = 0.1$, and $\zeta = 3$.\label{fig:ld_ui}}
\end{figure}
In summary, in this Section we have confirmed that the analytical expressions obtained in the TT and TB approximations and for slow, sub-Alfv\'enic flows are very accurate even when these expressions are used outside their domain of strict validity. This result enables us to use Equation~(\ref{eq:ldtt}), i.e., the key equation of this investigation, when realistic values of frequency, flow velocity, and the rest of relevant parameters obtained from the observations are used.
\section{Discussion}
\label{sec:discussion}
Naturally, kink waves propagating in nonuniform magnetic flux tubes are spatially damped by resonant absorption. In the static case, TGV showed that the damping length is inversely proportional to the frequency. Here we have investigated analytically and numerically the spatial damping of resonant kink waves in a transversely nonuniform magnetic waveguide in the presence of longitudinal background flow. Longitudinal flow breaks the equivalence between forward and backward propagating waves with respect to the flow direction. The wavelength and the damping length due to resonant absorption are both affected by the flow. For sub-Alfv\'enic flows, the backward wavelength is shorter than that of the forward wave, and backward waves are damped in shorter length scales than forward waves. However, as in TGV we have found that the damping length of both forward and backward propagating waves is inversely proportional to the frequency.
MHD seismology based on propagating waves has attracted limited attention and definitely less than its counterpart based on standing kink waves. Standing kink MHD waves are rare phenomena as they need a violent and energetic event such as a solar flare for their excitation \citep[see, e.g.,][]{ash,naka}. In the absence of flow and for coronal loop standing oscillations \citep[see, e.g.,][]{nakaofman,goossens2002,arregui07,arregui08,goossens08}, MHD seismology has been used to obtain information of the plasma physical conditions. Particularly, for a given set of parameters provided by the observations, i.e., period, damping time, and wavelength in the case of standing waves, \citet{arregui07} and \citet{goossens08} showed that the possible values of $v_{\mathrm{Ai}}$, $\zeta$, and $l/R$ which are consistent with the theory form a one-dimensional curve in the three-dimensional parameter space. In principle, any point of this curve can equally explain the observations. \citet{solerfine} and \citet{arreguiballester} showed that more constrained estimations of $v_{\mathrm{Ai}}$ and $l/R$ can be given in the case of prominence thread oscillations as the limit $\zeta \gg 1$ can be adopted. More recently, \citet{bayesian} found that more accurate estimations of the parameters are possible by combining the analytical theory of \citet{goossens08} with statistical Bayesian analysis. In the presence of flows, \citet{terradasletterflow} have recently explained also for standing waves that the flow velocity can be estimated from the wave phase difference along the magnetic loop.
On the contrary, propagating MHD waves are ubiquitous in the solar atmosphere \citep[see, e.g.,][]{tomczyk07,tomczyk09} and provide a huge reservoir of possibilities for seismology. Some examples of MHD seismology based on propagating waves are, e.g., \citet{tom08b} using numerical simulations of guided MHD waves by density enhancements in the solar corona, \citet{lin09} using observations of kink waves in prominence threads, \citet{VTG} using resonantly damped kink waves in coronal loops, and \citet{verthspicule} exploiting the properties of kink waves in chromospheric spicules. However, none of these works included flow in their analysis. Our theoretical results given in Equations~(\ref{eq:lam2}) and (\ref{eq:ldtt}) have direct implications for MHD seismology based on propagating waves in a flowing medium, and could be used to infer information about the plasma properties. Therefore, the potential application of MHD seismology to the case of resonantly damped propagating kink waves in a flowing medium must be explored.
In the presence of flow two waves with different wavelengths and damping lengths but with the same frequency (or period) are simultaneously present. We denote as $\lambda^+$ and ${L_{\rm D}}^+$ the wavelength and damping length of the forward wave, respectively, and as $\lambda^-$ and ${L_{\rm D}}^-$ the equivalent quantities of the backward wave. Observationally, this means that it is possible to measure five quantities, namely $\lambda^+$, $\lambda^-$, ${L_{\rm D}}^+$, ${L_{\rm D}}^-$, and the period, while the rest of parameters, i.e., $v_{\mathrm{Ai}}$, $U_{\mathrm{i}}$, $l/R$, and $\zeta$ are in principle unknown. In this analysis we assume $l/R = l^\star /R$ for simplicity. We also take $\lambda^-$ as a positive quantity and so we perform the absolute value of Equation~(\ref{eq:lam2}) when the $-$ sign is used. If observations can provide us with reliable values for the wavelengths and damping lengths of the two waves, then we can use our theoretical results (Equations~(\ref{eq:lam2}) and (\ref{eq:ldtt})) to obtain seismological estimations for the unknown quantities $v_{\mathrm{Ai}}$, $U_{\mathrm{i}}$, $l/R$, and $\zeta$ as
\begin{eqnarray}
v_{\mathrm{A}} &=& \frac{1}{P} \sqrt{\frac{\zeta + 1}{2 \zeta}} \frac{\lambda^+ + \lambda^-}{2}, \label{eq:seis1} \\
U_{\mathrm{i}} &=& \frac{1}{P} \frac{\zeta + 1}{\zeta} \frac{\lambda^+ - \lambda^-}{2}, \label{eq:seis2}\\
\frac{l}{R} &=& \mathcal{F} \frac{\zeta + 1}{\zeta - 1} \frac{\lambda^+ + \lambda^-}{{L_{\rm D}}^+ + {L_{\rm D}}^-}, \label{eq:seis3} \\
\zeta &=& \frac{1 + 2 \gamma}{1 - 2 \gamma}, \label{eq:seis4}
\end{eqnarray}
with $P = 2\pi / \omega$ the period and
\begin{equation}
\gamma = \frac{\lambda^+ - \lambda^-}{\lambda^+ + \lambda^-} \frac{{L_{\rm D}}^+ + {L_{\rm D}}^-}{{L_{\rm D}}^+ - {L_{\rm D}}^-}.
\end{equation}
In the absence of flow, $\lambda^+ = \lambda^-$ and ${L_{\rm D}}^+ = {L_{\rm D}}^-$. Then, Equations~(\ref{eq:seis1}) and (\ref{eq:seis3}) are equivalent to the expressions studied by \citet{goossens08}, and the density contrast (Equation~(\ref{eq:seis4})) becomes indeterminate. Thanks to the flow, the density contrast can be determined if reliable measures of the wavelengths and damping lengths of both forward and backward waves are available.
The theoretical results of the present paper offer new and exciting opportunities for MHD seismology in plasma structures with equilibrium flows. For the first time we have shown that an estimation of the density contrast, $\zeta = \rho_{\rm i}/\rho_{\rm e}$, is possible. MHD seismology requires theory and observations. The required observations might not be available at present time. However, the seismological tool provided in Equations~(\ref{eq:seis1})--(\ref{eq:seis4}) could be used in the future when the required observations become available.
The investigation performed in this paper may be improved in the future by incorporating additional physics in the MHD wave model. Effects that come to mind are the variation of the plasma parameters along the magnetic field direction as in \citet{stratified} and magnetic expansion and twist of the flux tube. These and other effects might be included in forthcoming investigations on propagating resonant kink waves.
\acknowledgements{
This manuscript was finished during a visit of MG to the Solar Physics Group of UIB. MG is happy to acknowledge the hospitality of the Solar Physics Group and the financial support from UIB through grant 40/2010 under the program ``Estades breus de professors convidats''. We thank I. Arregui for useful comments. RS acknowledges support from a postdoctoral fellowship within the EU Research and Training Network ``SOLAIRE'' (MTRN-CT-2006-035484). MG acknowledges support from K.U. Leuven via GOA/2009-009. JT acknowledges support from the Spanish Ministerio de Educaci\'on y Ciencia through a Ram\'on y Cajal grant and funding provided under projects AYA2006-07637 and FEDER funds.}
|
2,877,628,090,055 | arxiv | \section{Introduction}
\label{intro}
This paper presents safety critical control for a plant with sector-bounded input uncertainties. There are many applications including autonomous driving, medical or industrial robotics, and aerospace vehicles that require prioritizing safety over performance objectives~\cite{knight2002safety}. One of the popular methods to encode safety is by means of Control Barrier Functions (CBF), which can be used as a constraint in a quadratic program to modify control actions to adhere to safety specifications~\cite{ames2016control,ames2019control}. CBFs can be designed using a representative or a surrogate model of the system~\cite{ames2019control} or can be learned online~\cite{taylor2020learning,choi2020reinforcement}. Often, for simplicity, actuator nonlinearities are ignored, which raises robustness concerns.
Our approach is to characterize the input-output behavior of nonlinearities at the plant input using point-wise in time quadratic constraints. We then use the robust control barrier functions (RCBFs) presented in Section~\ref{sec:RCBF} to provide safety guarantees for the entire uncertainty set. This is done by ensuring that a safe action always exists for all nonlinearites at the modeled uncertainty level. This combines a traditional robust control approach with the CBF methods for safety critical control. As a result, we obtain more cautious trajectories when close to the unsafe region.
There are three main contributions of the paper. First, we present a new robust control barrier function based approach to handle sector-bounded uncertainties at the plant input. This allows us to handle nonlinearities and time-varying memoryless uncertainties described by a quadratic constraint. Second, we formulate an optimization problem that minimally alters the control command to guarantee safety in the presence of modeled input uncertainty. This optimization problem can be rewritten in terms of a second-order cone program (SOCP) to be solved online. Finally, the proposed approach is demonstrated using a lateral vehicle control example to study robust safety.
There is a large body of literature on CBFs with a good overview provided in~\cite{ames2019control}. Only the most closely related work is summarized here. Robust control barrier functions are presented for guaranteeing safety in the presence of $\mathcal{L}_\infty$ bounded disturbances in~\cite{xu2015robustness,garg2021robust,breeden2021robust} and stochastic disturbances in~\cite{takano2018application}. The work in~\cite{nguyen2021robust} also considers robust CBFs to account for the changes in the dynamics as a perturbation to the vector field. A key distinction is that the input nonlinearities in our work lead to uncertainties that depend on the control decisions, which is not allowed in the framework of~\cite{xu2015robustness,garg2021robust,breeden2021robust,nguyen2021robust}. In this paper, we provide safety guarantees for static nonlinearities and/or time-varying memoryless uncertainties at the plant input. The most recent work in~\cite{pete2021arxiv} considers a more general class of unmodeled dynamics (e.g. unknown time-delays, actuation lag, etc.) using $\alpha$-IQCs and CBFs. The price for this generality is that the trajectories tend to be even more conservative than those obtained with the method presented here. Other related work on robust CBFs includes~\cite{jankovic2018robust,choi2021robust,dean2020guaranteeing}.
\noindent\textbf{Notation:} Let $\field{R}^{n \times m}$ and $\field{S}^{n}$ denote the sets of $n$-by-$m$ real matrices and $n$-by-$n$ real, symmetric matrices. The Euclidean norm of a vector $\mathbf{v}\in\field{R}^m$ is defined as $\|\mathbf{v}\|_2 := \sqrt{\mathbf{v}^\top \mathbf{v}}$. A continuous function $\eta :\field{R}\rightarrow\field{R}$ is called extended class-$\mathcal{K}_{\infty}$ $(\mathcal{K}_{\infty,e})$ if it is strictly monotonically increasing and satisfies $\eta(0) = 0, \lim_{r\rightarrow -\infty} \eta(r) = -\infty$, and $\lim_{r\rightarrow \infty} \eta(r) = \infty$.
\section{Preliminaries}
\label{sec:prelim}
\subsection{Problem Formulation}
\label{sec:problem}
Consider the design interconnection as shown in~\figref{fig:designic}. The uncertain plant $P$ is described as a series interconnection of known part $G$ and an unknown perturbation $\phi$ at the plant input.
\begin{figure}
\centering
\begin{tikzpicture}[thick,scale=1.1,every node/.style={scale=0.95}]
\draw [dashed,green!50!black, fill=green!5!white] (-4.2,1.1) rectangle (-8,-0.3);
\draw [dashed,red,fill=red!5!white] (-4,1.1) rectangle (-1.3,-0.3);
\draw (-2.4,0) rectangle node{$G$}(-1.48,0.8);
\node at (-6.6,0.2) {$\mathbf{u_0}$};
\node at (-3.8,0.2) {$\mathbf{u}$};
\node at (-1.1,0.2) {$\mathbf{x}$};
\node at (-2.7,0.2) {$\mathbf{v}$};
\draw (-7.8,0.7) rectangle node{$\mathbf{k_0}(\mathbf{x})$}(-6.9,0);
\draw (-6.3,0.7) rectangle node{Safety Filter}(-4.4,0);
\draw (-3.5,0.7) rectangle node{$\phi$}(-2.9,0.1);
\draw [->](-1.5,0.4) -- (-0.92,0.4) -- (-0.9,-0.5) -- (-8.2,-0.5) -- (-8.2,0.4) -- (-7.8,0.4);
\draw [->](-6.9,0.4) -- (-6.3,0.4);
\draw [->](-2.9,0.4) -- (-2.4,0.4);
\draw [->](-4.4,0.4) -- (-3.5,0.4);
\node [red] at (-3.8,0.9) {$P$};
\node [green!50!black] at (-7.6,0.9) {$\mathbf{k(x)}$};
\end{tikzpicture}
\caption{Uncertain State-Feedback Design Interconnection}
\label{fig:designic}
\end{figure}
This perturbation represents nonlinearities and/or time-varying, memoryless uncertainties. We will refer to $\phi$ as a ``nonlinearity" for simplicity. Let $P$ be given with the following input-affine dynamics:
\begin{align}
\label{eq:system}
\begin{split}
\dot{\mathbf{x}}(t) &= \mathbf{f}(\mathbf{x}(t)) + \mathbf{g}(\mathbf{x}(t)) \,\mathbf{v}(t), \hspace{0.2in} \mathbf{x}(0) = \mathbf{x}_0\\
\mathbf{v}(t) &= \phi(\mathbf{u}(t),t)
\end{split}
\end{align}
where $\mathbf{x}(t) \in D\subset \field{R}^{n}$ is the state, $\mathbf{u}(t) \in \mathcal{U} \subset \field{R}^{m}$ is the admissible control input and $\mathbf{v}(t) \in \mathcal{V}(\mathbf{u}(t)) \subset \field{R}^{m}$ is the uncertain input. Moreover, $\mathbf{f}:D\subset \field{R}^n \rightarrow \field{R}^n$ and $\mathbf{g}:D\subset\field{R}^n \rightarrow \field{R}^{n\times m}$ are locally Lipschitz continuous functions of the state $\mathbf{x}$. It is assumed that the dynamics given by~\eqref{eq:system} are defined on open set $D\subset \field{R}^{n}$ and are forward complete, i.e. for every initial condition $\mathbf{x}(0) \in D$, there exists a unique solution $\mathbf{x}(t)$ for all $t\geq 0$. The nonlinearity $\phi$ is assumed to lie in a sector $\left[\alpha,\beta\right]$ with $0 < \alpha \leq 1 \leq\beta$ so that the sector-bound contains the nominal case $\mathbf{v}=\mathbf{u}$. This sector-bound can be written as the following point-wise in time quadratic constraint (Section 6.1 of~\cite{khalil2002nonlinear}):
\begin{align}
\label{eq:sectorqc}
\left[\mathbf{v}(t)-\alpha \mathbf{u}(t)\right]^\top \left[\beta \mathbf{u}(t)-\mathbf{v}(t)\right]\geq 0,\,\, \forall t \geq 0.
\end{align}
If $m=1$ then the single control channel sector-bound can be illustrated as shown in~\figref{fig:sectorbnd}. The uncertainty $\phi$ lies in a sector $\left[\alpha,\beta\right]$ when it can be bounded by two lines with slopes of $\alpha$ and $\beta$, respectively. The shaded gray region represents the allowable uncertainty set. In more general setting when $m\neq 1$, the constraint~\eqref{eq:sectorqc} allows cross-coupling between the input channels.
\begin{figure}
\centering
\begin{tikzpicture}[thick,scale=1,rounded corners = 0.5mm,every node/.style={scale=0.9}]
\draw [->,axis](2.9,-1) -- (7.1,-1);
\draw [->,axis](5,-2.8) -- (5,0.9);
\draw[white,fill=gray!15!white] (6.9,0.1) -- ++(-0.8,0.7) -- ++(-1.1,-1.8);
\draw[white,fill=gray!15!white] (3.9,-2.8) -- ++(1.1,1.8) -- ++(-1.9,-1.1);
\draw [-,black](6.1,0.8) -- (3.9,-2.8);
\draw [-,black](6.9,0.1) -- (3.1,-2.1);
\node at (7,-1.2) {$u$};
\node at (4.8,0.8) {$v$};
\node [black] at (7.1,-0.2) {$\alpha u $};
\node [black] at (6,1) {$\beta u$};
\node [red!90!black] at (7.1,0.6) {$v = \phi(u,t)$};
\draw [red!90!black,very thick] plot[smooth, tension=.7] coordinates { (3.4,-2.4) (3.9,-2.2) (4.1,-1.7) (4.7,-1.3) (5.3,-0.7) (5.7,-0.3) (6.1,0.2) (6.7,0.4)};
\end{tikzpicture}
\caption{Illustration of SISO Nonlinearity $\phi \in \left[\alpha,\beta\right]$}
\label{fig:sectorbnd}
\end{figure}
We assume that a locally Lipschitz continuous function $\mathbf{k_0}:D\subset\field{R}^n\rightarrow\mathcal{U} \subset\field{R}^m$ is given such that the baseline (not necessarily safe) control law is $\mathbf{u_0} = \mathbf{k_0}(\mathbf{x})$. The notion of safety is formalized by defining a safe set $\mathcal{C} \subset D \subset \field{R}^n$ in the state space that the system must remain within~\cite{ames2016control}. In particular, consider the set $\mathcal{C}$ as the zero-superlevel set of a continuously differentiable function $h : D\subset\field{R}^n \rightarrow \field{R}$:
\begin{align}
\label{eq:safeset}
\mathcal{C} &\triangleq \{ \mathbf{x} \in D \subset \field{R}^n : h(\mathbf{x}) \geq 0\}
\end{align}
The boundary and interior of the safe set are denoted as $\partial\mathcal{C}$ and $\mbox{Int}(\mathcal{C})$, respectively. It is assumed that zero is a regular value of $h$ and $\mathcal{C}$ is non-empty with no isolated points. These assumptions mean that $h(\mathbf{x}) = 0$ implies $\frac{\partial h}{\partial \mathbf{x}} (\mathbf{x}) \neq 0$, Int$(\mathcal{C}) \neq \emptyset$, and $\overline{\mbox{Int}(\mathcal{C})} = \mathcal{C}$. Explicit time dependence of variables are omitted when it is clear from the context. It is assumed that the initial condition is in the safe set i.e. $\mathbf{x}_0 \in \mathcal{C}$.
Our primary goal is to design a safety filter in~\figref{fig:designic} that minimally alters the baseline control command $\mathbf{u_0}$ so the state of the closed-loop system remains safe even in the presence of the nonlinearity. The proposed solution in Section~\ref{sec:optcontrol} is an optimization problem that can be solved online to compute safe control action $\mathbf{u}\in\mathcal{U}$.
\subsection{Background}
\label{sec:back}
This section provides a brief summary on set-invariance using control barrier functions~\cite{ames2016control}. First, consider the nominal case i.e. without nonlinearities at the plant input. In this case, $\mathbf{v} = \mathbf{u}$. If we let $\mathbf{u} = \mathbf{k(x)}$ then the closed-loop dynamics are given by:
\begin{align}
\label{eq:clsys}
\dot{\mathbf{x}} = \mathbf{f_{cl}}(\mathbf{x}) = \mathbf{f}(\mathbf{x}) + \mathbf{g}(\mathbf{x}) \,\mathbf{k}(\mathbf{x}), \hspace{0.2in} \mathbf{x}(0) = \mathbf{x}_0
\end{align}
In the context of the autonomous system above, safety is synonymous with the forward invariance of~$\mathcal{C}$~\cite{ames2016control}:
\begin{defn}[Forward Invariance and Safety]
A set $\mathcal{C} \subset D \subset \field{R}^n$ is forward invariant if for every initial condition $\mathbf{x_0} \in \mathcal{C}$, the solution to the closed-loop system~\eqref{eq:clsys} satisfies $\mathbf{x}(t) \in \mathcal{C}$ for all $t \geq 0$. The system~\eqref{eq:clsys} is safe with respect to $\mathcal{C}$ if $\mathcal{C}$ is forward invariant.
\end{defn}
As mentioned in the previous section, the baseline control law $\mathbf{u_0}=\mathbf{k_0(x)}$ is not necessarily safe. In this case, we would like to ensure that a safe action $\mathbf{u}$ exists which can steer the system to remain within the safe set $\mathcal{C}$. This is formalized by defining the notion of control invariance.
\begin{defn}[Control Invariance]
A set $\mathcal{C}$ is control invariant if there exists a controller $\mathbf{k}:D\subset\field{R}^n\rightarrow\mathcal{U}\subset\field{R}^m$ such that $\mathcal{C}$ is forward invariant with respect to the system~\eqref{eq:clsys}.
\end{defn}
Control barrier functions are used to ensure control invariance for the nominal plant. Let $L_\mathbf{f}h:= \frac{\partial h}{\partial \mathbf{x}} \mathbf{f}$ and $L_\mathbf{g}h:=\frac{\partial h}{\partial \mathbf{x}} \mathbf{g}$ denote the Lie derivatives of $h$ with respect to $\mathbf{f}$ and $\mathbf{g}$.
\begin{defn}[Control Barrier Function]
Let $\mathcal{C} \subset D \subset \field{R}^n$ be the zero-superlevel set of a continuously differentiable function $h : D\subset\field{R}^n \rightarrow \field{R}$. Then $h$ is a Control Barrier Function if there exists $\eta \in \mathcal{K}_{\infty,e}$ such that for all $\mathbf{x}\in D$:
\begin{align}
\label{eq:CBF}
\sup_{\mathbf{u}\in \mathcal{U}}
\left[ L_\mathbf{f}h(\mathbf{x}) + L_\mathbf{g}h(\mathbf{x})\, \mathbf{u}\right] \geq -\eta(h(\mathbf{x}))
\end{align}
\end{defn}
The existence of a control barrier function $h$ satisfying constraint~\eqref{eq:CBF} implies that if the state reaches the boundary of~$\mathcal{C}$, then the control input $\mathbf{u}$ can be used to prevent the state from entering the unsafe region. The main result in~\cite{ames2016control,ames2019control} can be used to design a Lipschitz continuous safety-filter that yields safety for the nominal closed-loop. The specific implementation involves solving the following quadratic program online with CBF constraint to minimally alter the baseline action $\mathbf{u_0}$.
\begin{align}
\tag{CBF-QP}
\label{eq:CBFQP}
\mathbf{u}^*(\mathbf{x}) =\, &\arg \min_{\mathbf{u}\in\mathcal{U}} \frac{1}{2} \|\mathbf{u} - \mathbf{u_0}\|^2_2\\
&\mbox{s.t. } L_\mathbf{f}h(\mathbf{x}) + L_\mathbf{g}h(\mathbf{x})\, \mathbf{u} \geq -\eta(h(\mathbf{x}))\nonumber
\end{align}
In general, constructing a CBF $h$ is not straightforward and often requires careful consideration~\cite{ames2019control}.
Next, consider the uncertain case with the sector-bounded nonlinearity $\phi$ at the plant input. This nonlinearity can alter the control command resulting in a safety violation. The uncertain closed-loop dynamics with the nonlinearity can be written as follows:
\begin{align}
\label{eq:uclsys}
\dot{\mathbf{x}} = \mathbf{f_{ucl}}(\mathbf{x}) = \mathbf{f}(\mathbf{x}) + \mathbf{g}(\mathbf{x}) \,\phi(\mathbf{k}(\mathbf{x}),t), \hspace{0.1in} \mathbf{x}(0) = \mathbf{x}_0
\end{align}
The notion of robust control invariance~(Definition 4.4 in~\cite{blanchini2015set}) is useful to guarantee safety in the presence of uncertainties. A specific definition for the sector-bounded nonlinearity is given next.
\begin{defn}[Robust Control Invariance and Robust Safety]
A set $\mathcal{C}$ is robust control invariant if there exists a controller $\mathbf{k}:D\subset\field{R}^n\rightarrow\mathcal{U}\subset\field{R}^m$, such that $\mathcal{C}$ is forward invariant with respect to the uncertain closed-loop~\eqref{eq:uclsys} for all nonlinearities $\phi$ in a sector $[\alpha,\beta]$. The system~\eqref{eq:uclsys} is robustly safe with respect to $\mathcal{C}$ if $\mathcal{C}$ is robust control invariant.
\end{defn}
Robust control invariance can be ensured by showing the existence of a robust control barrier function as presented in the following section.
\section{Main Results}
\label{sec:mainres}
\subsection{Uncertainty Mapping}
\label{sec:uncmapping}
The first step is to use a loop-shifting transformation (Section 6.5 of~\cite{khalil2002nonlinear}) to map the nonlinearity $\phi \in [\alpha,\beta]$ into a normalized, input additive form, as in standard robust control workflow. To make this precise, define $\Delta:\field{R}^m\times\field{R} \rightarrow \field{R}^m$ such that $\mathbf{v}(t)=\phi(\mathbf{u}(t),t)$ is mapped to:
\begin{align}
\mathbf{v}(t) = \frac{1}{2}\, (\alpha+\beta) (\mathbf{u}(t) + \Delta(\mathbf{u}(t),t))
\end{align}
The mapped nonlinearity satisfies $\Delta \in [-\theta, +\theta]$ where $\theta:= (\beta-\alpha)/(\beta+\alpha)$. This re-centers the sector-bound to $0$ and separates the nominal control action $\mathbf{u}(t)$ and the uncertain control command $\Delta(\mathbf{u}(t),t)$. The factor $\frac{1}{2}\, (\alpha+\beta)$ scales the input function $\mathbf{g}(\mathbf{x})$ to be $\mathbf{\tilde{g}}(\mathbf{x}) := \frac{1}{2}\,(\alpha+\beta)\,\mathbf{g}(\mathbf{x})$. This yields the following input-affine system with mapped nonlinearity:
\begin{align}
\label{eq:mappedsys}
\begin{split}
\dot{\mathbf{x}}(t) &= \mathbf{f}(\mathbf{x}(t)) + \mathbf{\tilde{g}}(\mathbf{x}(t)) \,(\mathbf{u}(t) + \mathbf{w}(t)), \hspace{0.1in} \mathbf{x}(0) = \mathbf{x}_0\\
\mathbf{w}(t) &= \Delta(\mathbf{u}(t),t)
\end{split}
\end{align}
Assume the uncertainty level satisfies $0\leq\theta<1$. The symmetric sector constraint on the mapped nonlinearity $\Delta$ corresponds to a norm bound:
\begin{align}
\label{eq:normbnd}
\|\mathbf{w}(t)\|_2 \leq \theta \|\mathbf{u}(t)\|_2, \,\, \forall t \geq 0.
\end{align}
Note that as $\theta\rightarrow 0$, i.e. as $\alpha$ and $\beta$ both tend to $1$, then $\mathbf{w}(t) \rightarrow 0$ and we recover the nominal plant $G$. For simplicity, the remainder of the paper considers the system~\eqref{eq:mappedsys} with mapped nonlinearity $\Delta$ instead of the original system~\eqref{eq:system} with $\phi$.
Let $\mathcal{W}(\mathbf{u}(t))$ denote the set of uncertain inputs $\mathbf{w}(t)$ satisfying the norm bound constraint~\eqref{eq:normbnd}. The set $\mathcal{W}(\mathbf{u}(t))$ depends on the control input $\mathbf{u}$ at time $t$. Thus, the uncertain input $\mathbf{w}(t)$ can not be simply treated as an exogenous disturbance input, as it depends on $\mathbf{u}(t)$ through the nonlinearity $\Delta$.
\subsection{Robust Control Barrier Functions (RCBF)}
\label{sec:RCBF}
Robust control barrier functions defined in this section can be used to synthesize controllers ensuring the safety of the uncertain closed-loop system with respect to a given set $\mathcal{C}$.
\begin{defn}
\label{def:RCBF}
Let $\mathcal{C} \subset D \subset \field{R}^n$
be a safe set given by~\eqref{eq:safeset}. The function $h$ is a Robust Control
Barrier Function for~\eqref{eq:mappedsys} if there exists $\eta \in \mathcal{K}_{\infty,e}$
such that for all $x\in D$:
\begin{align}
\label{eq:RCBF}
\sup_{\mathbf{u}\in \mathcal{U}}\, \inf_{\mathbf{w}\in\mathcal{W}} \left[ L_\mathbf{f}h(\mathbf{x}) + L_\mathbf{\tilde{g}}h(\mathbf{x}) (\mathbf{u} + \mathbf{w}) \right] \geq -\eta(h(\mathbf{x}))
\end{align}
\end{defn}
The nonlinearity $\Delta$ can, in the worst case, yield an uncertain input $\mathbf{w}$ that minimizes the left side of inequality~\eqref{eq:RCBF}. We aim to choose a single control input $\mathbf{u} \in\mathcal{U}$ for all uncertain inputs $\mathbf{w}\in\mathcal{W}$ satisfying the constraint~\eqref{eq:normbnd}. The worst-case uncertain input $\mathbf{w}^*(\mathbf{u})$ after solving the inner optimization problem is given by:
\begin{align}
\label{eq:wcw}
\mathbf{w}^*(\mathbf{u}) = - \theta \|\mathbf{u}\|_2 \frac{L_\mathbf{\tilde{g}}h(\mathbf{x})^\top}{\|L_\mathbf{\tilde{g}}h(\mathbf{x})\|_2}
\end{align}
This follows from the linear cost in~\eqref{eq:RCBF}, but a more formal argument using Lagrange relaxation is given in Appendix~\ref{sec:wcw}. It can be verified that as $\theta\rightarrow0$, we have $\mathbf{w}^*(\mathbf{u})\rightarrow0$. Plugging in for $\mathbf{w^*}(\mathbf{u})$ in condition~\eqref{eq:RCBF} yields:
\begin{align}
\label{eq:RCBF_wc}
\sup_{\mathbf{u}\in \mathcal{U}}\, \left[ L_\mathbf{f}h(\mathbf{x}) + L_\mathbf{\tilde{g}}h(\mathbf{x}) (\mathbf{u} + \mathbf{w^*}(\mathbf{u})) \right] \geq -\eta(h(\mathbf{x}))
\end{align}
Define $p(\mathbf{x}) := L_\mathbf{f}h(\mathbf{x}) + \eta(h(\mathbf{x}))$ and the set of all control actions that render the set $\mathcal{C}$ robustly safe as follows:
\begin{align}
\mathcal{U}_{RCBF}(\mathbf{x}) := \{ \mathbf{u} \in \mathcal{U} : p(\mathbf{x}) + L_\mathbf{\tilde{g}}h(\mathbf{x}) (\mathbf{u} + \mathbf{w^*}(\mathbf{u}))\nonumber \geq 0 \}
\end{align}
The feasibility of RCBF constraint~\eqref{eq:RCBF} ensures that the above set is nonempty. This implies that if the state reaches the boundary of $\mathcal{C}$ then there exists a control input to prevent the state of the uncertain closed-loop from crossing out of the safe set. Thus, the existence of a robust control barrier function implies that the system is robustly safe. This statement is formalized in the next theorem, which can be viewed as a robust version of the main result in~\cite{ames2016control}.
\begin{theorem}
Let $\mathcal{C} \subset D$ be a safe set defined using~\eqref{eq:safeset} as the superlevel set of a continuously differentiable function $h : D\subset\field{R}^n \rightarrow \field{R}$. If $h$ is a robust control barrier function on $D$ and $\frac{\partial h}{\partial \mathbf{x}}(\mathbf{x}) \neq 0$ for all $\mathbf{x} \in \partial \mathcal{C}$, then any Lipschitz continuous controller $\mathbf{u}(\mathbf{x}) \in \mathcal{U}_{RCBF}(\mathbf{x})$ for the system~\eqref{eq:mappedsys} renders the set $\mathcal{C}$ robust control invariant.
\end{theorem}
\begin{proof} If $h$ is a RCBF on open set $D$ then for any $\mathbf{x}\in\partial\mathcal{C}$ and for all $\mathbf{w}\in\mathcal{W}$, there exists a control input $\mathbf{u}\in\mathcal{U}$ such that
$\dot{h}(\mathbf{x,w,u}) \geq \eta(h(\mathbf{x})) = 0$. For a Lipschitz continuous control law $\mathbf{u}(\mathbf{x})\in \mathcal{U}_{RCBF}(\mathbf{x})$, according to generalizations of Nagumo’s theorem (Theorem $4.10$ in~\cite{blanchini2015set}) the closed set $\mathcal{C}$ is robust control invariant.
\end{proof}
\subsection{Optimization-Based Control}
\label{sec:optcontrol}
We can solve the following optimization online to compute a control input $\mathbf{u}$ that minimally alters the baseline input $\mathbf{u}_0$ to ensure safety:
\begin{align}
\label{eq:RCBFOpt}
\mathbf{u}^*(\mathbf{x}) =\, &\arg \min_{\mathbf{u}\in\mathcal{U}} \frac{1}{2} \|\mathbf{u} - \mathbf{u_0}\|^2_2\\
&\mbox{ s.t. } p(\mathbf{x}) + L_\mathbf{\tilde{g}}h(\mathbf{x}) (\mathbf{u} + \mathbf{w}^*(\mathbf{u})) \geq 0\nonumber
\end{align}
Plugging in for $\mathbf{w}^*(\mathbf{u})$ and expanding the cost function yields the following problem with an equivalent optimizer.
\begin{align}
\label{eq:RCBFOpt1}
\mathbf{u}^*(\mathbf{x}) =\, &\arg \min_{\mathbf{u}\in\mathcal{U}} \left[ \frac{1}{2}\mathbf{u}^\top \mathbf{u} - \mathbf{u_0}^\top \mathbf{u} \right]\\
&\mbox{ s.t. } p(\mathbf{x}) + L_\mathbf{\tilde{g}}h(\mathbf{x}) \mathbf{u} - \theta \|\mathbf{u}\|_2 \|L_\mathbf{\tilde{g}}h(\mathbf{x})\|_2 \geq 0\nonumber
\end{align}
Again, the last term drops out as the uncertainty level $\theta \to 0$, yielding an affine constraint in $\mathbf{u}$. In this case, the above optimization problem is simply a (nominal) \ref{eq:CBFQP}. However, for $\theta>0$, the decision variable $\mathbf{u}$ in the constraint of~\eqref{eq:RCBFOpt1} appears as the Euclidean norm $\|\mathbf{u}\|_2$. Reformulate the problem~\eqref{eq:RCBFOpt1} using a slack variable $q$ as follows:
\begin{align}
\label{eq:RCBFOpt2}
\begin{smallmatrix}\mathbf{u}^*(\mathbf{x})\\ q^*(\mathbf{x}) \end{smallmatrix} \right] =\, &\arg \min_{ \mathbf{u}\in\mathcal{U},q} \left[q - \mathbf{u_0}^\top \mathbf{u} \right]\\
&\mbox{ s.t. } p(\mathbf{x}) + L_\mathbf{\tilde{g}}h(\mathbf{x}) \mathbf{u} \geq \theta \|L_\mathbf{\tilde{g}}h(\mathbf{x})\|_2 \|\mathbf{u}\|_2 \nonumber\\
&\hspace{0.2in} 2q \geq \|\mathbf{u}\|_2^2\nonumber
\end{align}
This yields a minimization problem over $\mathbf{u}\in\mathcal{U}$ and $q>0$. The optimal solutions are related by $2q^*={\mathbf{u}^*}^\top \mathbf{u}^*$. The second constraint can be rewritten as a rotated second-order cone (SOC) condition as in Section 10.1 of~\cite{calafiore2014optimization}, which yields the optimization~\eqref{eq:RCBFOpt2} as follows:
\begin{align}\tag{RCBF-SOCP}
\label{eq:SOCP}
\begin{smallmatrix}\mathbf{u}^*(\mathbf{x})\\ q^*(\mathbf{x}) \end{smallmatrix} \right] =\, &\arg \min_{ \mathbf{u}\in\mathcal{U}, q} \left[q - \mathbf{u_0}^\top \mathbf{u} \right]\\
&\mbox{ s.t. } \theta \|L_\mathbf{\tilde{g}}h(\mathbf{x})\|_2 \|\mathbf{u}\|_2\leq p(\mathbf{x}) + L_\mathbf{\tilde{g}}h(\mathbf{x}) \mathbf{u} \nonumber\\
&\hspace{0.2in} \Big\|\begin{smallmatrix} \sqrt{2}\,\mathbf{u}\\ q - 1 \end{smallmatrix} \right]\Big\|_2 \leq q + 1\nonumber
\end{align}
This problem falls under a special class of convex optimization problems known as second-order cone programs (SOCP)~\cite{boyd2004convex}, which can be solved online using existing numerical solvers. At higher uncertainty levels, the~\ref{eq:SOCP} problem (if feasible) yields more cautious (conservative and safe) control actions to prevent the states from entering the unsafe region. For the point-wise feasibility of the~\ref{eq:SOCP}, it is assumed that the set of control inputs $\mathcal{U}$ is not overly restrictive, thus allowing us to have sufficient control authority to maintain safety in the presence of modeled uncertainty. However, an approach similar to~\cite{zeng2021safety} can also be used to relax this assumption.
\subsection{Lipschitz Continuity}
\label{sec:lipcont}
This section discusses a key Lipschitz continuity property of the~\ref{eq:SOCP} problem. In the nominal case ($\theta =0$), if $\mathcal{U}\equiv\field{R}^m$, then the~\ref{eq:CBFQP} has only a single linear constraint in $\mathbf{u}$, and in this special case, there is an explicit solution. This solution can be used to show that the resulting safety filter is a locally Lipschitz continuous function of the state $\mathbf{x}\in D$ (Theorem~$8$ of~\cite{xu2015robustness}). For the robust case ($\theta \neq 0$), the optimization problem~\eqref{eq:RCBFOpt} can compactly be written as:
\begin{align}
\label{eq:RCBFOpt3}
\mathbf{u}^*(\mathbf{x}) =\, &\arg \min_{\mathbf{u}\in\mathcal{U}_{RCBF}(\mathbf{x})} \frac{1}{2} \|\mathbf{u} - \mathbf{u_0}\|^2_2
\end{align}
To the best of our knowledge, there is no explicit solution to this general problem. However, it is a standard projection problem over the parameterized non-empty closed convex set, i.e. the optimizer $\mathbf{u}^*(\mathbf{x})$ is a projection of $\mathbf{u}_0 = \mathbf{k}_0(\mathbf{x})$ onto the set $\mathcal{U}_{RCBF}(\mathbf{x})$. The main results in Section~$6$ of~\cite{bednarczuk2020lipschitz} show that $\mathbf{u^*(x)}$ is a locally Lipschitz continuous function of $\mathbf{x}$, if the set $\mathcal{U}_{RCBF}(\mathbf{x})$ is described by polyhedral constraints parameterized by the state $\mathbf{x}$. We conjecture that $\mathbf{u^*(x)}$ remains locally Lipschitz when $\mathcal{U}_{RCBF}(\mathbf{x})$ is described by the two SOC constraints as in the~\ref{eq:SOCP} problem. Future work will focus on investigating this further.
The remainder of this section presents a Lipschitz continuity result for the special case of scalar control input $u\in\field{R}$. For this case $m=1$ and $L_{\tilde{\mathbf{g}}} h(\mathbf{x})\in\field{R}$. The optimization~\eqref{eq:RCBFOpt} can be written as follows:
\begin{align}
\label{eq:sclarporb}
u^*(\mathbf{x}) =\, &\arg \min_{u} \frac{1}{2} (u - u_0)^2 \\
&\mbox{s.t. } p(\mathbf{x}) + L_{\tilde{\mathbf{g}}} h(\mathbf{x})\, u - \theta |u| |L_{\tilde{\mathbf{g}}}h(\mathbf{x})| \geq 0\nonumber
\end{align}
The next theorem makes a formal statement about the Lipschitz continuity of the function $u^*$. A direct and more concise (independent of~\cite{bednarczuk2020lipschitz}) proof is provided in this case.
\begin{theorem}
Assume $\mathbf{f}:D\subset \field{R}^n \rightarrow \field{R}^n$, $\mathbf{g}:D\subset \field{R}^n \rightarrow \field{R}^{n\times1}$, $\eta\in\mathcal{K}_{\infty,e}$ and $u_0$ are given locally Lipschitz continuous functions. Let $u\in\field{R}$ and $h : D\subset \field{R}^n \rightarrow \field{R}$ be a locally Lipschitz continuous robust control barrier function. Moreover, let $L_{\tilde{\mathbf{g}}}h(\mathbf{x})\neq0$. Then the solution, $u^*(\mathbf{x})$ of~\eqref{eq:sclarporb} is a locally Lipschitz continuous function of $\mathbf{x}\in D$.
\end{theorem}
\begin{proof}
Assume $L_{\tilde{\mathbf{g}}}h(\mathbf{x}) > 0$ and $0\leq\theta<1$. The constraint from~\eqref{eq:sclarporb} can be written as:
\begin{align}
\label{eq:constraint1}
u - \theta |u| \geq \frac{-p(\mathbf{x})}{L_{\tilde{\mathbf{g}}}h(\mathbf{x})}
\end{align}
If $\frac{-p(\mathbf{x})}{L_{\tilde{\mathbf{g}}}h(\mathbf{x})}\geq 0$ then $u\geq0$. Hence the constraint~\eqref{eq:constraint1} is:
\begin{align}
\label{eq:constraint2}
u \geq \frac{-p(\mathbf{x})}{(1-\theta)L_{\tilde{\mathbf{g}}}h(\mathbf{x})}
\end{align}
If $\frac{-p(\mathbf{x})}{L_{\tilde{\mathbf{g}}}h(\mathbf{x})}< 0$ then~\eqref{eq:constraint1} is satisfied for any $u\geq0$. Moreover it is satisfied for negative values of $u$ that satisfy:
\begin{align}
\label{eq:constraint3}
u \geq \frac{-p(\mathbf{x})}{(1+\theta)L_{\tilde{\mathbf{g}}}h(\mathbf{x})}
\end{align}
Thus the constraint in~\eqref{eq:sclarporb} is a parameterized interval of the form $u(\mathbf{x})\in [u_l(\mathbf{x}),+\infty)$ where $u_l(\mathbf{x})$ is given by:
\begin{align}
u_l(\mathbf{x}) = \max\left\{
\frac{-p(\mathbf{x})}{(1-\theta)L_{\tilde{\mathbf{g}}}h(\mathbf{x})},\, \frac{-p(\mathbf{x})}{(1+\theta)L_{\tilde{\mathbf{g}}}h(\mathbf{x})}\right\}
\end{align}
The functions $p$ and $L_{\tilde{\mathbf{g}}}h$ are locally Lipschitz continuous, since $\mathbf{f}$, $\mathbf{g}$, $\eta$ and $h$ are locally Lipschitz continuous by assumption. Note that $L_{\tilde{\mathbf{g}}}h(\mathbf{x})\neq0$, because $L_{\tilde{\mathbf{g}}}h(\mathbf{x})$ is Lipschitz continuous, so there exists an $\epsilon>0$ and a ball around $\mathbf{x}$ such that $|L_{\tilde{\mathbf{g}}}h(\mathbf{x})|\geq\epsilon>0$. Thus, by Proposition~$1.30$ and Corollary~$1.31$ of~\cite{weaver2018lipschitz}, the ratio $\frac{-p(\mathbf{x})}{L_{\tilde{\mathbf{g}}}h(\mathbf{x})}$ is also locally Lipschitz continuous. The boundary function $u_l$ is locally Lipschitz continuous, because the point-wise maximum/minimum of two Lipschitz functions is also Lipschitz by Proposition $1.32$ of~\cite{weaver2018lipschitz}. Finally, the optimal solution of~\eqref{eq:sclarporb} can be written as $u^*(\mathbf{x}) = \max \{u_l(\mathbf{x}),u_0\}$. The baseline controller $u_0$ is assumed to be locally Lipschitz continuous. Thus, again using the Proposition~$1.32$ of~\cite{weaver2018lipschitz}, we have that $u^*$ is a locally Lipschitz continuous function of $\mathbf{x}\in D$. The case $L_{\tilde{\mathbf{g}}}h(\mathbf{x}) < 0$ can be handled similarly.
\end{proof}
\subsection{Extensions}
\label{sec:ext}
\subsubsection{Sector-Bound for Individual Control Channels}
\label{sec:sectbndforindcontrolchannel}
Consider the case where the control input $\mathbf{u}(t) \in \field{R}^m$ is vector valued ($m>1$). The sector bound corresponding to the constraint~\eqref{eq:sectorqc} allows for uncertainty that is coupled across input channels. An alternative model is to treat each input as having its own sector-bounded nonlinearity with no cross-coupling. In this case, the dynamics~\eqref{eq:mappedsys} can be written as:
\begin{align*}
\dot{\mathbf{x}}(t) = \mathbf{f}(\mathbf{x}(t)) + \sum_{i=1}^{m} \tilde{\mathbf{g}}_i(\mathbf{x}(t)) \,(u_i(t) + w_i(t)), \hspace{0.1in} \mathbf{x}(0) = \mathbf{x}_0
\end{align*}
where $w_i(t)= \Delta_i(u_i(t),t)$, and each $\Delta_i$ satisfies the constraint $|w_i(t)| \leq \theta_i |u_i(t)|$, $\forall t \geq 0$. The worst-case uncertain input for an individual channel can be obtained as $w^*_i = -\theta_i |u_i| \mbox{sgn}(L_{\tilde{\mathbf{g}}_i}h(x))$. The linear programming trick (presented in Appendix~\ref{sec:linprogtrick} for a scalar input) can be used to separate positive and negative parts of the individual control input $u_i$. This yields a quadratic program for the safety filter.
\subsubsection{Robust Exponential CBF}
\label{sec:RECBF}
The previous section presented robust control barrier functions with relative degree one, i.e. control input $\mathbf{u}$ shows up after differentiating the function $h$ once, which implies that $L_\mathbf{\tilde{g}}h(\mathbf{x}) \neq 0$ in Definition~\ref{def:RCBF}. However, in general $h$ can have a higher relative degree. Note that the dynamics in Equation~\eqref{eq:mappedsys} has the uncertain input $\mathbf{w}$ matched with the control input $\mathbf{u}$. This matching condition allows us to naturally extend the RCBF framework to higher relative degree robust CBFs (referred to as robust exponential CBFs or RECBF). The theory is similar to that of the (nominal) exponential CBF combined with the robustness argument already presented in this paper before. More details on (nominal) ECBF are provided in~\cite{nguyen2016exponential} with an overview in~\cite{ames2019control}. The example in Section~\ref{sec:example} demonstrates the RECBF design approach for the relative degree of two.
\subsubsection{Unifying with RCLF}
\label{sec:RCLF}
Sometimes stability and safety objectives are in direct conflict~\cite{ames2019control}. If the baseline controller is not designed with robustness as a consideration, uncertainty may lead to unstable behavior before safety violation becomes an issue. A careful design should first consider providing a robust stability guarantee for the baseline controller using a robust control Lyapunov function (RCLF) (see Chapter 3 of~\cite{freeman2008robust}). The stability and safety objectives can also be combined in a single multi-objective optimization problem to design the controller $\mathbf{k(x)}$ in~\figref{fig:designic}. For nominal design ($\theta=0$) this problem is referred to as CLF-CBF QP~\cite{ames2019control}. A similar design can also be considered for the robust ($\theta>0$) counterpart, which yields a RCLF-RCBF SOCP problem. A specific implementation uses a hard constraint with a RCBF to enforce robust safety and a soft constraint with a RCLF (using a slack variable) to approximately enforce robust stability.
\subsubsection{Parametric Uncertainties}
\label{sec:param}
The robust design approach presented in this paper can also be extended to systems with uncertain parameters in the input function $\mathbf{g}(\mathbf{x})$. Let the system dynamics be given by:
\begin{align*}
\mathbf{\dot{x}} = \mathbf{f}_0(\mathbf{x})
+ \left[\mathbf{g}_0(\mathbf{x}) + \sum_{i=1}^{n_p} \mathbf{g}_i(\mathbf{x})\delta_i\right]\mathbf{u}, \hspace{0.1in} \mathbf{x}(0) = \mathbf{x}_0
\end{align*}
where the functions $\mathbf{f}_0$ and $\mathbf{g}_0$ capture the nominal dynamics. The remaining terms capture the effects of uncertain, real parameters $\{ \delta_i \}_{i=1}^{n_p}$. Each parameter variation is assumed to be normalized such that $|\delta_{i}| \leq \theta_i$.
Define $\mathbf{w}_i := \delta_i\mathbf{u}$ and rewrite the constraint on $\delta_i$ as an individual norm-bound constraint $\|\mathbf{w}_i\| \leq \theta_i \|\mathbf{u}\|_2$. Definition~\ref{def:RCBF} can be modified to consider inner optimization over each $\mathbf{w}_i$. The worst-case $\mathbf{w}^*_i$ for the inner optimization is then given by an expression similar to that of~\eqref{eq:wcw} using each $\theta_i$ and $L_{\tilde{\mathbf{g}}_i}h(\mathbf{x})$.
\begin{rem}
If $n_p=1$, $\mathbf{f_1(x)} = 0$ and $\mathbf{g_1(x)} = \mathbf{g_0(x)}$, then $\mathbf{\dot{x}} = \mathbf{f_0(x)} + \mathbf{g_0(x)\,(u + w)}$, with $\mathbf{w} = \delta_1\mathbf{u}$ and $|\delta_1 | \leq \theta_1$. These dynamics are similar to those in~\eqref{eq:mappedsys}. This special case corresponds to a gain variation at the plant input as appears in the classical gain margin calculation.
\end{rem}
\section{Example: Vehicle Lateral Control}
\label{sec:example}
Consider a vehicle being driven on a straight road that must avoid a stationary obstacle. The lateral dynamics of vehicle are linearized at a constant longitudinal speed to obtain the following linear time-invariant (LTI) model~\cite{alleyne1997comparison}.
\begin{align}
\dot{\mathbf{x}}(t) &= \mathbf{A}\mathbf{x}(t) + \mathbf{B}(u(t)+w(t)),\,\,\, \mathbf{x}(0) = \mathbf{x_0}\\
|w(t)| &\leq \theta\,|u(t)| \nonumber
\end{align}
where $\mathbf{x}(t) = \begin{smallmatrix} e(t) & \dot{e}(t) & \psi(t) & \dot{\psi}(t)\end{smallmatrix} \right]^\top\in\field{R}^4$ is the linearized state and the control $u(t)\in \field{R}$ is the front wheel steering angle input. The model from~\cite{alleyne1997comparison} is slightly modified to include the uncertain input $w(t)\in\field{R}$, which represents nonlinearities (e.g. allowable saturation) and/or time-varying uncertainties. Here, $e$ is the lateral distance to the lane center and $\psi$ is the vehicle heading relative to the path. The longitudinal distance and velocity along the road are denoted by $s$ and $\dot{s} = 28\, m/s$ respectively. The longitudinal dynamics of the vehicle are not controlled. The state-space matrices and the vehicle parameters are given in~\cite{alleyne1997comparison} and are provided in Appendix~\ref{sec:vehicledynparam} for completeness. A baseline state-feedback controller was designed using linear quadratic regulator with cost matrices $Q=diag(10,1,\frac{1}{30},1)$ and $R=5$.
This was implemented to track the reference command $\mathbf{r}(t)\in\field{R}^4$ as:
\begin{align*}
u_0 = \mathbf{K}\cdot\mathbf{(r - x)},\,\,\,
\mbox{where } \mathbf{K} =\begin{smallmatrix} 1.41 & 0.41 & 3.30 & 0.24 \end{smallmatrix} \right].
\end{align*}
This differs slightly from the feedback diagram in~\figref{fig:designic} due to
the inclusion of the reference command, i.e. the baseline
controller is of the form $u_0 = k_0(x, r)$.
A stationary obstacle of radius $d$ is assumed at the origin. The safe set $\mathcal{C}$ is defined by Equation~\eqref{eq:safeset} with $h(\mathbf{x}) = e^2 + s^2 -d^2 \geq 0$ where $d$ is chosen based on geometries of the vehicle and the obstacle. The first and second time-derivatives of $h$ along a trajectory of state $\mathbf{x}$ are given by:
\begin{align*}
\dot{h}(\mathbf{x}) = 2 e \dot{e} + 2 s \dot{s}, \,\,\,\,
\ddot{h}(\mathbf{x}, u, w) = 2 e \ddot{e}(u,w) + 2\dot{e}^2 + 2\dot{s}^2
\end{align*}
Since the control input $u$ and uncertain input $w$ both appear in $\ddot{e}$, the system has relative degree two. This requires the robust exponential CBF as discussed in Section~\ref{sec:RECBF} to ensure safety. The set $\mathcal{U}\equiv\field{R}$ is considered for simplicity.
The initial state of the vehicle is $\mathbf{x}_0 = \begin{smallmatrix} 2 & 0 & 0 & 0 \end{smallmatrix} \right]^\top$ with $s(0) = -20$. The safe distance $d$ is chosen as $3$ meters. The reference trajectory is selected to track the center of the lane, i.e. $\mathbf{r}(t) = \mathbf{0}\in\field{R}^4$. The safety filter is designed using the RECBF-SOCP approach. The MATLAB implementation using the function~\texttt{coneprog} is available online at~\cite{github}.
Let the uncertainty level be $\theta = 0.5$ for the simulation study. \figref{fig:VLC_CBF_comparison} shows the simulation results for the uncertain plant with the worst-case uncertainty as in Equation (\ref{eq:wcw}). The shaded red circle represents the unsafe region. The ECBF and RECBF design poles are chosen to be two repeated poles at $-30$. Note that the nominal LQR controller runs into the obstacle due to its lack of safety consideration. The (nominal) ECBF-QP safety filter does not explicitly account for the uncertainty. Thus, its trajectory slightly violates the safety requirement around $s=0$ or $t\approx0.7$ seconds. The RECBF-SOCP trajectory avoids the obstacle successfully by choosing cautious control input to account for the uncertainty.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{Fig1.pdf}
\includegraphics[width=\linewidth]{Fig2.pdf}
\caption{LQR, ECBF, and RECBF simulations with worst-case plant.}
\label{fig:VLC_CBF_comparison}
\end{figure}
Next, the simulations are performed on the nominal plant. In other words the RECBF-SOCP controller is designed assuming $\theta>0$ but the simulations are performed on a nominal plant without the nonlinearity. \figref{fig:VLC_uncertainLevel} shows the RECBF-SOCP simulations with the same initial condition, but with the safety filter designed at different assumed values for the uncertainty bound $\theta$. Only the zoomed region around the obstacle is shown. It is observed that, as the model uncertainty level in design increases from $\theta = 0.2$ to $0.8$ the RECBF-SOCP generates more cautious trajectories around the obstacle. This is due to the fact that the proposed design explicitly consider the uncertainty at the plant input and yields robustly-safe control actions.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{Fig3.pdf}
\caption{RECBF trajectories with varying uncertainty level $\theta$.}
\label{fig:VLC_uncertainLevel}
\end{figure}
\section{Conclusion}
\label{sec:conclusions}
This paper presents a robust control barrier function approach to handle sector-bounded uncertainties at the plant input. The proposed optimization problem can be written in terms of the second-order cone-program to be solved online. The robustness of the designed controller was studied in a lateral vehicle control example.
\section{Acknowledgments}
We thank Prof. Murat Arcak, Kate Schweidel and Adnane Saoud at the University of California, Berkeley for valuable discussions. We also thank Doug Philbrick at the NAWCWD China Lake for helpful comments.
\bibliographystyle{ieeetr}
|
2,877,628,090,056 | arxiv | \section{Introduction}
The symmetric inverse semigroup $\mathcal{IS}_n$, also known as the rook monoid, consists of the partial injective transformations of $\{1,2,\ldots,n\}$ \cite{Ganyushkin,solomon}. Any element of $\mathcal{IS}_n$ can be represented as an $n\times n$ matrix whose entries are $0$ or $1$, with at most one $1$ in every row and every column.
In a previous paper \cite{fikioris-fikioris}, we introduced a submonoid $M_n$ of $\mathcal{IS}_n$ and studied its properties. The monoid $M_n$ consists of the zero matrix together with those matrices of $\mathcal{IS}_n$ whose $1$s lie on a single diagonal and form an uninterrupted block (i.e., no $0$ lies between any two $1$s). Let $d$ be the said diagonal ($d=-n+1,\ldots, n-1$, with $d=0$ being the main diagonal), let $k$ be the row of the northwestern $1$, and let $m$ be the row of the southeastern $1$. The study of \cite{fikioris-fikioris} was facilitated by representing the elements of $M_n$ as triplets $\langle d,k,m\rangle \in\mathbb{Z}^3$ ($d$, $k$, and $m$ are appropriately restricted), and developing a closed-form expression representing the product of two elements.
This short note is an extension that allows $\langle d,k,m\rangle \in\mathbb{R}^3$; the restrictions on the parameters $d$, $k$, $m$, as well as the product formula, remain unaltered. We thus study a new monoid $\overline{M}_n$, of which the
$M_n$ of \cite{fikioris-fikioris} is a submonoid.
To facilitate comparisons with ``the integer case,'' we maintain much of the notation of \cite{fikioris-fikioris}; for example, we retain the symbol $x^T$ for the semigroup inverse\footnote{The underlying reason for the symbol is that inverting $x\in M_n$ amounts to transposing the matrix represented by $x$.} of $x\in \overline{M}_n$. As in \cite{fikioris-fikioris}, $\mathbf{0}$ and $\mathbf{1}$ denote monoid zero and identity, and ideal means two-sided ideal. We use the traditional notations
for Green’s relations, associated equivalence classes, and principal ideals. A $j$th root of $x\in \overline{M}_n$ is a $y\in \overline{M}_n$ such that $x= y^j$ ($j \in \mathbb{N}$).
\section{Definitions; the height function}
\label{section:definitions}
Let $n\in\mathbb{Z}$ with $n\ge 2$. Our definition of $\overline{M}_n$ is
\begin{equation}
\label{eq:m-definition}
\begin{split}
\overline{M}_n=\{\mathbf{0}\}
\cup
\{\langle& d,k,m\rangle:\ d,k,m\in \mathbb{R};\\ &1-\min(0,d)\le k \le m\le n-\max(0,d)\}. \end{split}
\end{equation}
Note that the restrictions in (\ref{eq:m-definition}) further imply
\begin{equation}
-(n-1)\le d\le n-1\quad \mathrm{and}\quad 1\le k\le m\le n.
\end{equation}
As in \cite{fikioris-fikioris}, the formula for the product of two nonzero elements is
\begin{equation}
\label{eq:multiplication-finite}
\langle d,k,m\rangle \langle d',k',m'\rangle=
\begin{cases}
\langle d'',k'',m''\rangle,\quad k'' \le m'',\\\
\mathbf{0},\quad k''> m'',
\end{cases}
\end{equation}
in which the parameters $d''$, $k''$, and $m''$ are
\begin{equation}
\label{eq:doubleprime}
d''=d+d',\quad
k''=\max(k,k'-d),\quad
m''=\min(m,m'-d).
\end{equation}
We can use the definitions (\ref{eq:m-definition}), (\ref{eq:multiplication-finite}), and (\ref{eq:doubleprime}) to show that $\overline{M}_n$ is a noncommutative monoid, whose identity is $\mathbf{1}=\langle 0,1,n\rangle$.
We obtain the submonoid $M_n$ if, in (\ref{eq:m-definition}), we replace the condition $d,k,m\in\mathbb{R}$ by the more restrictive one $d,k,m\in\mathbb{Z}$. In $M_n$, the triplet of integers represents an $n\times n$ matrix. Analogously, we can interpret the triplets of $\overline{M}_n$ as line segments that are contained within a $(n-1)\times (n-1)$ square and are parallel to the diagonal shown in Fig.~\ref{fig:square}. Segment endpoints are permitted to lie on the square boundary, while $\mathbf{0}$ corresponds to the square being empty.
\begin{figure}
\centering
\includegraphics[width=.5\textwidth]{pics/square.pdf}
\caption{Any triplet $\langle d, k, m\rangle$ of $\overline{M}_n\setminus \{\mathbf{0}\}$ corresponds to a line segment similar to the depicted $x$, whose height is $h(x)$. The element $\mathbf{0}$ corresponds to the empty square, with $h(\mathbf{0})=-1$.}
\label{fig:square}
\end{figure}
Since (\ref{eq:m-definition}) allows $m=k$, our line segments can reduce to points within the aforementioned closed square. We use $\overline{P}_n$ to denote the set of points, viz.,
\begin{equation}
\label{eq:points}
\overline{P}_n=\{\langle d,k,m\rangle\in \overline{M}_n\setminus\{\mathbf{0}\}:\ \ m=k\}.
\end{equation}
Let $h(x)$ denote the height of the segment $x\in \overline{M}_n$ (see Fig.~\ref{fig:square}), so that
\begin{equation}
\label{eq:v}
h(x)=
\begin{cases}
-1,\quad x=\mathbf{0},\\
m-k, \quad x=\langle d,k,m \rangle\in \overline{M}_n\setminus\{\mathbf{0}\}.
\end{cases}
\end{equation}
The height arises in a natural manner throughout this paper so we now give some of its properties. By (\ref{eq:m-definition}) and (\ref{eq:v}),
\begin{equation}
\label{eq:rank-range}
0\le h(x)\le n-\abs{d}-1\le n-1,\quad x\ne\mathbf{0},
\end{equation}
while $h$ assumes the particular values $-1$, $0$, and $n-1$ according to
\begin{equation}
\label{eq:rank-special-cases}
h(x)=-1\Leftrightarrow x=\mathbf{0},
\quad h(x)=0\Leftrightarrow x\in \overline{P}_n,
\quad h(x)=n-1\Leftrightarrow x=\mathbf{1}.
\end{equation}
It follows from (\ref{eq:multiplication-finite}), (\ref{eq:doubleprime}), and (\ref{eq:v}) that $h(xy)\le \min\left(h(x),h(y)\right)$. By induction, we then get
\begin{equation}
\label{eq:rank-property}
h(x_1x_2\ldots x_j)\le h(x_i),\mathrm{\ for\ all\ } i\in \{1,2,\ldots,j\},\quad x_i\in \overline{M}_n.
\end{equation}
\begin{remark}
\label{remark:rnk}
Ref.~\cite{fikioris-fikioris} uses
the symbol $\mathrm{rnk}(x)$ for the rank of the partial transformation represented by $x\in M_n$. Thus in the integer case we have
\begin{equation}
\label{eq:rnk}
\mathrm{rnk}(x)=h(x)+1,\quad x\in M_n,
\end{equation}
which
shows why we chose the seemingly arbitrary value $h(\mathbf{0})=-1$ in (\ref{eq:v}).
\end{remark}
The first of the two lemmas that follow gives the principal ideals of $\overline{M}_n$, as further explained in Section~\ref{section:ideals}; the second will help us in Section~\ref{section:brandt}, where we discuss the subsemigroup $\{\mathbf{0}\}\cup\overline{P}_n$.
\begin{lemma}
\label{lemma:yequalszxw-1}
Let $x,y\in\overline{M}_n$. Then $h(y)\le h(x)$ iff there exist $z,w\in\overline{M}_n$ such that
\begin{equation}
\label{eq:yeqzxw}
y=zxw.
\end{equation}
Furthermore, if $0\le h(y)\le h(x)$ with $x=\langle d,k,m\rangle$ and $y=\langle d',k',m'\rangle$, then (\ref{eq:yeqzxw}) is satisfied by the nonzero elements $z=\langle d_z,k_z,m_z\rangle$ and $w=\langle d_w,k_w,m_w\rangle$ where
\begin{equation}
\label{eq:zdefinition}
d_z=k-k',\quad k_z=k',\quad m_z=m';
\end{equation}
\begin{equation}
\label{eq:wdefinition}
d_w=k'+d'-k-d,\quad k_w=k+d,\quad m_w=k+d+m'-k'.
\end{equation}
\end{lemma}
\begin{proof}
If (\ref{eq:yeqzxw}) holds, then $h(y)\le h(x)$ by (\ref{eq:rank-property}). Conversely, suppose that $h(y)\le h(x)$. If $x=\mathbf{0}$ or $y=\mathbf{0}$, (\ref{eq:yeqzxw}) is trivial. We thus take $x,y\in \overline{M}_n\setminus\{\mathbf{0}\}$; call $x=\langle d,k,m\rangle$, $y=\langle d',k',m'\rangle$; and define $z$, $w$ by (\ref{eq:zdefinition}), (\ref{eq:wdefinition}).
By (\ref{eq:v}), the assumption $h(y)\le h(x)$ amounts to
\begin{equation}
\label{eq:vylessthanvx}
m'-k'\le m-k.
\end{equation}
Write the conditions in (\ref{eq:m-definition}) for $d$, $k$, $m$, and again for $d'$, $k'$, $m'$. Upon invoking (\ref{eq:zdefinition})--(\ref{eq:vylessthanvx}), we can easily deduce identical conditions for $d_z$, $k_z$, $m_z$ and for $d_w$, $k_w$, $m_w$. Thus $z$ and $w$ are well-defined elements of $\overline{M}_n\setminus\{\mathbf{0}\}$. Finally, a quick calculation based on the multiplication formula (\ref{eq:multiplication-finite}) verifies (\ref{eq:yeqzxw}).
\end{proof}
\begin{lemma}
\label{lemma:yequalszxw-2}
Let $y\in \{\mathbf{0}\}\cup\overline{P}_n$. Let $x\in\overline{M}_n\setminus\{\mathbf{0}\}$. Then there exist $z,w\in \{\mathbf{0}\}\cup\overline{P}_n$ such that $y=zxw$.
\end{lemma}
\begin{proof}
If $y=0$, the statement is trivial. Otherwise $y\in\overline{P}_n$, so $0=h(y)\le h(x)$ by (\ref{eq:rank-range}) and (\ref{eq:rank-special-cases}). Thus (\ref{eq:yeqzxw}) holds, where $z,w\in \overline{M}_n$ are given by (\ref{eq:zdefinition}) and (\ref{eq:wdefinition}) with $m'=k'$. It follows that $k_z=m_z$ and $k_w=m_w$, so that $z,w\in \overline{P}_n$.
\end{proof}
\section{Basic results}
This section discusses certain properties of $\overline{M}_n$ that readily follow from the definitions of Section~\ref{section:definitions}.
Our first result has no counterpart in the integer case. By means of an affine transformation $\varphi$ (easily visualized by means of Fig.~\ref{fig:square}), we demonstrate that all $\overline{M}_n$ are isomorphic:
\begin{proposition}
\label{prop:isomorphism}
Let $n,q$ be integers $\ge 2$. The map $\varphi: \overline{M}_n\rightarrow \overline{M}_q$ given by
\begin{equation}
\mathbf{0}\mapsto \mathbf{0}, \quad x_n=\langle d_n,k_n,m_n \rangle\mapsto x_q=\langle d_q,k_q,m_q \rangle,
\end{equation}
where
\begin{equation}
d_q=\frac{q-1}{n-1} d_n,\quad k_q-1=\frac{q-1}{n-1} (k_n-1),\quad m_q-1=\frac{q-1}{n-1} (m_n-1),
\end{equation}
is a monoid isomorphism. In particular any $\overline{M}_n$ is isomorphic to $\overline{M}_2$.
\end{proposition}
\begin{proof}
$\varphi$ is bijective by (\ref{eq:m-definition}), while $\varphi(x_ny_n)=\varphi(x_n)\varphi(y_n)$ and $\varphi(\langle 0, 1,n\rangle)=\langle 0, 1,q\rangle$ follow from (\ref{eq:multiplication-finite}).
\end{proof}
\begin{remark}
\label{remark:no-n}
By Proposition~\ref{prop:isomorphism}, a stand-alone study of $\overline{M}_n$ would be facilitated if one took $n=2$, corresponding to segments lying in a $1\times 1$ closed square. However, we retain the parameter $n$ in order to draw upon and compare to results from \cite{fikioris-fikioris}.
\end{remark}
\begin{remark}
\label{remark:h-instead-of-rnk}
Eqn. (\ref{eq:v}) and Proposition~\ref{prop:isomorphism} imply that, for $x_n \in \overline{M}_n\setminus\{\mathbf{0}\}$,
\begin{equation}
\label{eq:v-isomorphism}
h(x_q)=\frac{q-1}{n-1} h(x_n).
\end{equation}
Eqn. (\ref{eq:v-isomorphism}) shows why, for $\overline{M}_n$, we use the height $h(x)$ instead of extending (to the non-integer case) the quantity $\mathrm{rnk}(x)=m-k+1$ mentioned in Remark~\ref{remark:rnk}: In $\overline{M}_n$, the latter quantity would scale in an unnatural manner.
\end{remark}
The two propositions that follow resemble results of \cite{fikioris-fikioris}. The first is a formula for powers which can be verified from (\ref{eq:multiplication-finite}) by induction.
\begin{proposition}
\label{prop:powers}
For $x=\langle d,k,m\rangle\in \overline{M}_n\setminus\{\mathbf{0}\}$ and $j\in\mathbb{N}$ we have
\begin{equation}
\label{eq:powers-1}
x^j=\begin{cases}
\left\langle d^{(j)},k^{(j)},m^{(j)}\right\rangle,\quad \textrm{if}\quad k^{(j)}\le m^{(j)},
\\
\mathbf{0},\quad \textrm{if}\quad k^{(j)}>m^{(j)},
\end{cases}
\end{equation}
where
\begin{equation}
\label{eq:powers-2}
d^{(j)}=jd,\quad k^{(j)}=k-(j-1)\min(0,d),\quad m^{(j)}=m-(j-1)\max(0,d).
\end{equation}
\end{proposition}
The next proposition states that $\overline{M}_n$ consists solely of idempotents and nilpotents and gives the nilpotent indexes $i(x)$. In contrast to the integer case of $M_n$ (and as expected from the aforementioned isomorphism, which leaves $i(x)$ unaltered), $i(x)$ can take on values larger than $n$.
\begin{proposition}
\label{prop:idempotent-nilpotent}
An element $x=\langle d,k,m \rangle\in \overline{M}_n\setminus \{\mathbf{0}\}$ is idempotent if $d=0$ and nilpotent if $d\ne 0$. When $d\ne 0$, the index $i(x)$ of the nilpotent is given by
\begin{equation}
\label{eq:index-of-nilpotent}
i(x)
=2+\left\lfloor \frac{m-k}{\abs d}
\right\rfloor=2+\left\lfloor \frac{h(x)}{\abs d}
\right\rfloor,
\end{equation}
where $\lfloor \beta \rfloor$ denotes the floor of $\beta\in\mathbb{R}$. In particular, $i(x)=2$ when $x\in \overline{P}_n$; and $i(x)\to\infty $ as $d\to 0$ (with $h(x)=m-k$ held fixed and positive).
\end{proposition}
\begin{proof}
Proposition~\ref{prop:powers} gives $\langle 0,k,m\rangle^2=\langle 0,k,m\rangle$, so $x$ is idempotent when $d=0$. When $d>0$, the $m^{(j)}$ in (\ref{eq:powers-2}) decreases linearly with $j$, while $k^{(j)}=k$ remains constant. Thus $k^{(j)}>m^{(j)}$ for large enough $j$, in which case $x^j=\mathbf{0}$ by (\ref{eq:powers-1}). The index $i(x)$ is the smallest such $j$ and is given by (\ref{eq:index-of-nilpotent}). The proof for $d<0$ is similar.
\end{proof}
\begin{remark}
In the special case of integer parameters ($x\in M_n$) we can show that (\ref{eq:index-of-nilpotent}) reduces to formula (28) of \cite{fikioris-fikioris} (which involves the ceiling rather than the floor function). However, (28) of \cite{fikioris-fikioris} \textit{does not hold} for the more general case $x\in\overline{M}_n$.
\end{remark}
We close this section by giving a number of semigroup classes to which $\overline{M}_n$ and $\overline{S}_n=\overline{M}_n\setminus \{\mathbf{1}\}$ belong.
\begin{theorem}
\label{th:periodic}
Both
$\overline{M}_n$ and
\begin{equation}
\label{eq:sn-semigroup-1}
\overline{S}_n=\overline{M}_n\setminus \{\mathbf{1}\}=\overline{M}_n\setminus \left\{\langle 0,1,n\rangle\right\}
\end{equation}
are noncommutative, inverse, periodic, combinatorial semigroups. In both $\overline{M}_n$ and $\overline{S}_n$, the unique inverse of $x$ is
\begin{equation}
\label{eq:transpose}
x^T=
\begin{cases}
\mathbf{0},\quad x=\mathbf{0},\\
\langle -d,k+d,m+d\rangle, \quad x=\langle d,k,m\rangle\ne \mathbf{0}.
\end{cases}
\end{equation}
\end{theorem}
\begin{proof}
We first prove the assertions for $\overline{M}_n$. A semigroup is \textit{periodic} when all of its elements are of finite order, i.e., when the monogenic subsemigroup generated by any semigroup element has finite cardinality \cite{Howie}. As the elements of $\overline{M}_n$ are either idempotent or nilpotent (Proposition~\ref{prop:idempotent-nilpotent}), $\overline{M}_n$ is periodic. For $x\in \overline{M}_n$ and for the $x^T$ defined in (\ref{eq:transpose}), we can use (\ref{eq:multiplication-finite}) to show $xx^Tx=x$ and $x^Txx^T=x^T$. Thus $x^T$ is an inverse of $x$ and $\overline{M}_n$ is an regular semigroup. By Proposition~\ref{prop:idempotent-nilpotent} the idempotents are $\mathbf{0}$ together with the elements $\langle 0,k,m\rangle$; and by (\ref{eq:multiplication-finite}), all these idempotents commute. Accordingly \cite{Lawson}, $\overline{M}_n$ is an inverse semigroup and the inverse $x^T$ is unique (in $\overline{M}_n$). As shown in Theorem~\ref{th:green} below, $\mathcal{H}$ is the equality relation. Therefore \cite{Lawson} $\overline{M}_n$ is a combinatorial semigroup.
If $\mathbf{1}=xy$, then $h(\mathbf{1})\le h(x)$ by (\ref{eq:rank-property}), so that $x=\mathbf{1}$ by (\ref{eq:rank-range}) and (\ref{eq:rank-special-cases}). Similarly, $y=\mathbf{1}$. We have thus shown
\begin{equation}
\label{eq:sn-semigroup-2}
xy=\mathbf{1}\implies x=y=\mathbf{1}, \quad x,y\in\overline{M}_n.
\end{equation}
This implies that $\overline{M}_n\setminus\{\mathbf{1}\}=\overline{S}_n$ is a semigroup and that our theorem (already proved for $\overline{M}_n$) carries over to $\overline{S}_n$.
\end{proof}
Inverse semigroups are associated with a natural partial order \cite{Lawson,Howie} which, for our nonzero elements, can be formulated in terms of triplet parameters:
\begin{corollary}
\label{corollary:natural-partial-order}
Let $\le$ be the natural partial order in $\overline{M}_n\setminus\{\mathbf{0}\}$. Then
\begin{equation}
\label{eq:natural-partial-order}
\langle d,k,m\rangle \le \langle d',k',m'\rangle\iff d=d',\ k\ge k',\ \mathrm{and}\ m\le m'.
\end{equation}
\end{corollary}
\begin{proof}
As $x\le y$ iff $x=xx^Ty$ \cite{Lawson}, the assertion follows easily from (\ref{eq:multiplication-finite}) and (\ref{eq:transpose}).
\end{proof}
Therefore two segments are comparable iff one lies upon and is contained within the other, in which case the shorter segment is $\le$ the longer one.
\section{\texorpdfstring{$j$}{j}th roots}
Theorem~6 of \cite{fikioris-fikioris} discusses $j$th roots for the integer case: In $M_n$, a nonzero element $x=\langle d,k,m\rangle$ has a $j$th root iff $d$ is an integer multiple of $j$; and the $j$th root, when it exists, is unique. The theorem that follows shows that, in $\overline{M}_n$, a unique root $y$ \textit{always} exists.
In other words (and in complete analogy to the case of $\mathbb{R}_{>0}$ and its subset $\mathbb{N}$) any nonzero element $x\in \overline{M}_n$ ($x\in \mathbb{R}_{>0}$) has a unique root $y\in\overline{M}_n$ ($y\in \mathbb{R}_{>0}$); but in the special case $x\in M_n$ ($x\in \mathbb{N}$), the said root $y$ is not necessarily in $M_n$ (in $\mathbb{N}$).
\begin{theorem}
\label{th:roots}
Let $j\in\mathbb{N}$. The element $x=\langle d,k,m\rangle\in \overline{M}_n\setminus\{\mathbf{0}\}$ has a unique $j$th root in $\overline{M}_n$. It is given by $y=\langle d',k',m'\rangle\in \overline{M}_n\setminus\{\mathbf{0}\}$, where
\begin{equation}
\label{eq:root-parameters}
d'=\frac{d}{j},\quad k'=k+(j-1)\min(0,d'),\quad m'=m+(j-1)\max(0,d').
\end{equation}
\end{theorem}
\begin{proof}
Assume $d\ge 0$, so that (\ref{eq:m-definition}) implies
\begin{equation}
\label{eq:root-condition-on-x}
1\le k\le m\le n-d.
\end{equation}
We seek $y\in \overline{M}_n$ such that $x=y^j$. As $y\ne \mathbf{0}$, we set $y=\langle d', k', m'\rangle$. By Proposition~\ref{prop:powers}, $d'=d/j\ge 0$. Invoking (\ref{eq:m-definition}), we thus require
\begin{equation}
\label{eq:root-condition-2}
1\le k'\le m'\le n-d'.
\end{equation}
By Proposition~\ref{prop:powers}, $x=y^j$ is equivalent to the three equations
\begin{equation*}
\label{eq:root-condition-1}
d=jd',\quad k=k',\quad m=m'-(j-1)d'.
\end{equation*}
These are uniquely solvable for $d'$, $k'$, $m'$ and the solution is given in (\ref{eq:root-parameters}). Eqns. (\ref{eq:root-parameters}) and (\ref{eq:root-condition-on-x}) then imply (\ref{eq:root-condition-2}), completing the proof for $d\ge 0$. We can extend to $d<0$ by taking the inverse.
\end{proof}
\begin{remark}
One could also consider the submonoid $A_n$ of $\overline{M}_n$ in which $d,k,m\in\mathbb{Q}$. For $x\in A_n\setminus\{\mathbf{0}\}$, the unique $j$th root $y$ given in (\ref{eq:root-parameters}) also belongs to $A_n\setminus\{\mathbf{0}\}$. Thus in $A_n\setminus\{\mathbf{0}\}$, a unique root $y$ always exists. Consequently, despite the aforementioned analogy of $\overline{M}_n$ to $\mathbb{R}_{>0}$ and $M_n$ to $\mathbb{N}$, the submonoid $A_n$ is not analogous to $\mathbb{Q}_{>0}$.
\end{remark}
\section{Green's relations}
\label{section:green}
The theorem below gives Green's relations on $\overline{M}_n$, which turn out to be very similar to those in $M_n$.
\edit{
\begin{theorem}
\label{th:green}
In the inverse monoid $\overline{M}_n$, Green's relations for any two nonzero elements $x=\langle d,k,m\rangle$ and $y=\langle d',k',m'\rangle$ are as follows.
\begin{equation}
\label{eq:green-R}
x\mathcal{R}y\iff k=k'\ \mathrm{and}\ m=m',
\end{equation}
\begin{equation}
\label{eq:green-L}
x\mathcal{L}y\iff k+d=k'+d'\ \mathrm{and}\ m+d=m'+d',
\end{equation}
\begin{equation}
\label{eq:green-H}
x\mathcal{H}y\iff\ x=y,
\end{equation}
\begin{equation}
\label{eq:green-D-or-J}
x\mathcal{D}y\iff x\mathcal{J}y\iff h(x)=h(y).
\end{equation}
In all cases, $\mathbf{0}$ forms a class of its own,
\begin{equation}
\label{eq:green-zero}
R_\mathbf{0}=L_\mathbf{0}=H_\mathbf{0}=D_\mathbf{0}=J_\mathbf{0}=\{\mathbf{0}\}.
\end{equation}
\end{theorem}
}
\begin{proof}
The proof is identical to the proof of Theorem~12 of \cite{fikioris-fikioris} with two exceptions: (i)~The condition $m-k=m'-k'$ translates to $h(x)=h(y)$ (rather than $\mathrm{rnk}(x)=\mathrm{rnk}(y)$, see Remark~\ref{remark:h-instead-of-rnk}); (ii) The equality $\mathcal{J}=\mathcal{D}$ holds by virtue of Theorem~\ref{th:periodic},
because $\mathcal{J}=\mathcal{D}$ in any semigroup that is periodic \cite{Howie} ($\overline{M}_n$ is not finite as is $M_n$).
\end{proof}
Our theorem has simple graphical interpretations. The equivalence class $R_x$ ($L_x$) consists of all horizontal (vertical) translations of the segment $x$. Furthermore, the result that $J_x$ consists of all segments of height $h(x)$ means that elements whose heights are equal generate the same principal ideal. The next section goes beyond this observation and explicitly describes all ideals, whether principal or not.
\section{Ideals of \texorpdfstring{$\overline{M}_n$}{Mn}}
\label{section:ideals}
As opposed to $M_n$ (see Theorem~13 of \cite{fikioris-fikioris}), $\overline{M}_n$ has (two-sided) ideals that are not principal. In the theorem that follows, these non-principal ideals are denoted by $K_\mu$.
\begin{theorem}
\label{th:principal-ideals}
The principal ideals of $\overline{M}_n$ are precisely the following sets ${I}_\mu$,
\begin{equation}
\label{eq:principal-ideals-definition}
{I}_\mu=\{y\in \overline{M}_n: h(y)\le \mu\},\quad \mu\in \{-1\}\cup[0,n-1].
\end{equation}
In particular,
\begin{equation}
\label{eq:ideals-special-cases}
{I}_{-1}=\{\mathbf{0}\},\quad {I}_{0}=\{\mathbf{0}\}\cup \overline{P}_n,\quad {I}_{n-1}=\overline{M}_n.
\end{equation}
The ${I}_\mu$ defined in (\ref{eq:principal-ideals-definition}) are also given by
\begin{equation}
\label{eq:principal-ideals-given-by}
{I}_\mu=\overline{M}_n x\overline{M}_n,
\end{equation}
in which $x$ is any element of $\overline{M}_n$ with $h(x)= \mu$.
The non-principal ideals of $\overline{M}_n$ are precisely the following sets
${K}_\mu$,
\begin{equation}
\label{eq:ideals-definition}
{K}_\mu=\{y\in \overline{M}_n: h(y)< \mu\},\quad \mu\in (0,n-1].
\end{equation}
It follows that the collections $\{{I}_\mu\}$ and $\{{K}_\mu\}$ are both strictly totally ordered; that is,
${I}_\mu\subset {I}_\xi$ and ${K}_\mu\subset {K}_\xi$ whenever $\mu<\xi$.
\end{theorem}
\begin{proof} Define the sets $I_\mu$ by (\ref{eq:principal-ideals-definition}) and choose an $x\in\overline{M}_n$ such that $h(x)=\mu$. The iff statement of Lemma~\ref{lemma:yequalszxw-1} can then be rephrased as: $y\in I_\mu\iff y\in \overline{M}_n x \overline{M}_n$. We have thus shown (\ref{eq:principal-ideals-given-by}). Therefore all principal ideals are given in (\ref{eq:principal-ideals-definition}).
The special cases in (\ref{eq:ideals-special-cases}) follow from (\ref{eq:rank-special-cases}) and (\ref{eq:principal-ideals-definition}).
We now let ${I}$ be an \textit{arbitrary} ideal. From
\begin{equation}
\label{eq:ideals-union}
{I}=\overline{M}_n{I}\overline{M}_n=\cup_{x\in {I}}\overline{M}_n x\overline{M}_n,
\end{equation}
we see that ${I}$ is a union of principal ideals. By (\ref{eq:principal-ideals-definition}), these are totally ordered sets. If ${I}$ contains an element $x$ such that $h(y)\le h(x)$ for all $y\in{I}$, then the union in (\ref{eq:ideals-union}) equals ${I}_\mu$, where $\mu=h(x)=\mathrm{max}_{y\in I}\{h(y)\}$, so that ${I}$ is itself a principal ideal. If there is no such element $x\in I$---i.e., if the subset $\{h(y): y\in{I}\}$ of $\mathbb{R}$
has no maximum---then the union in (\ref{eq:ideals-union}) is one of the totally ordered sets in (\ref{eq:ideals-definition}), namely ${K}_\mu$, where $\mu=\mathrm{sup}_{y\in I}\{h(y)\}$.
It remains to show, conversely, that all the ${K}_\mu$ defined in (\ref{eq:ideals-definition}) are ideals. Let $y\in\overline{M}_n{K}_\mu$, so that $y=zw$ with $z\in \overline{M}_n$ and $w\in {K}_\mu$. It follows from (\ref{eq:rank-property}) that $h(y)\le h(w)$. Since $h(w)<\mu$, we have $h(y)<\mu$, so that $y\in {K}_\mu$. Hence $\overline{M}_n{K}_\mu
\subseteq {K}_\mu$, so ${K}_\mu$ is a left ideal by definition. Similarly, ${K}_\mu$ is a right ideal. Thus ${K}_\mu$ is a two-sided ideal, completing our proof.
\end{proof}
\section{The Brandt semigroup as a subsemigroup of \texorpdfstring{$\overline{M}_n$}M}
\label{section:brandt}
By (\ref{eq:m-definition}) and (\ref{eq:points}), the set $\overline{B}_n=\{\mathbf{0}\}\cup\overline{P}_n$ is given by
\begin{equation}
\label{eq:brandt-definition}
\overline{B}_n=\{\mathbf{0}\}\cup\overline{P}_n=\{\mathbf{0}\}
\cup
\{\langle d,k,k\rangle: \ 1-\min(0,d)\le k \le n-\max(0,d)\}.
\end{equation}
Example~2 of \cite{fikioris-fikioris} shows that, in the integer case, the subsemigroup $B_n$ of $\overline{B}_n$ is isomorphic to a certain Brandt semigroup of finite cardinality. The theorem that follows is a generalization that can be proved in a number of ways. We give a proof that builds upon previous results in the present paper, as well as concepts and results on inverse semigroups that can be found in \cite{Lawson}.
\begin{theorem}
$\overline{B}_n$ is a Brandt semigroup.
\end{theorem}
\begin{proof}
By (\ref{eq:multiplication-finite}), (\ref{eq:transpose}), and (\ref{eq:brandt-definition}), $\overline{B}_n$ is an inverse subsemigroup of $\overline{M}_n$. Therefore $\overline{B}_n$ inherits its natural partial order $\le$ from $\overline{M}_n$. By (\ref{eq:brandt-definition}) and Corollary~\ref{corollary:natural-partial-order}, $x\le y$ iff $x=y$ ($x,y\in \overline{P}_n$), meaning that in $\overline{P}_n=\overline{B}_n\setminus\{\mathbf{0}\}$, the $\le$ reduces to an equality. Equivalently \cite{Lawson}, all idempotents of $\overline{B}_n\setminus\{\mathbf{0}\}$ are primitive.
Now let $I\subseteq \overline{B}_n$ be an ideal of $\overline{B}_n$. Assume $I\ne \{\mathbf{0}\}$, so that some nonzero $x$ belongs to $I$. Choose any $y$ in $\overline{B}_n$. By Lemma~\ref{lemma:yequalszxw-2}, this $y$
belongs to the principal ideal $\overline{B}_nx\overline{B}_n$, so that $\overline{B}_n\subseteq\overline{B}_nx\overline{B}_n$. As $\overline{B}_nx\overline{B}_n\subseteq I$, we further have $\overline{B}_n\subseteq I$, so $I=\overline{B}_n$. Therefore the only ideals of $\overline{B}_n$ are $\{\mathbf{0}\}$ and $\overline{B}_n$ itself, meaning that $\overline{B}_n$ is $0$-simple.
Inverse, $0$-simple semigroups with at least one primitive idempotent are Brandt semigroups \cite{Lawson}, completing our proof.
\end{proof}
\section{Sierpi\'{n}ski rank of \texorpdfstring{$\overline{M}_n$}M}
\label{section:sierpinski}
Corollary~6 of \cite{fikioris-fikioris} determines a minimal generating set for $M_n$ that, for any $n$, consists of only three elements. Thus the rank of $M_n$ (integer case) is 3. Since $\overline{M}_n$ is uncountable, the situation is very different. In what follows, we prove that $\overline M_n$ has infinite Sierpi\'{n}ski rank \cite{peresse2006,peresse2009generating,east2012}, meaning that there are countable subsets of $\overline M_n$ that cannot be generated by finitely many elements of $\overline M_n$.
\begin{theorem}
The Sierpi\'{n}ski rank of $\overline M_n$ is infinite.
\end{theorem}
\begin{proof}
It suffices to prove that the Sierpi\'{n}ski rank of $\overline{S}_n=\overline{M}_n\setminus\{\mathbf{1}\}$ is infinite, see (\ref{eq:sn-semigroup-1}) or (\ref{eq:sn-semigroup-2}). By (\ref{eq:m-definition}), the countable set $A_n = \{y_i : i \in \mathbb{N}\}$ with elements
\begin{equation*}
y_i=\langle 2^{-i}, 1,n-2^{-i}\rangle,
\end{equation*}
is a well-defined subset of $\overline{S}_n$. By (\ref{eq:v}), the sequence of heights $h(y_i)$ increases, with
\begin{equation}
\label{eq:sierpinski-nminus1}
\sup_{i\in\mathbb{N}}h(y_i)=\lim_{i\to\infty} \left(n-2^{-i}-1\right)=n-1.
\end{equation}
Assume that $A_n$ is generated by a finite set with $r$ elements
$G_n = \{g_1, \ldots, g_r\}$. For every $i \in \mathbb{N}$ this implies that $y_i = g_{i_1}g_{i_2}\ldots g_{i_s}$ for some $i_1, i_2,\ldots, i_s \in \{1,\ldots, r\}$. By \eqref{eq:rank-property} this means that $h(y_i) \le \min\{h(g_{i_1}),h(g_{i_2}),\ldots, h(g_{i_s})\} \le h_{\max}$, where $h_{\max} = \max\{h(g_j) : j \in \{1,\ldots, r\}\}$. Since $\mathbf{1}\notin G_n\subset\overline{S}_n$, (\ref{eq:rank-range}) and (\ref{eq:rank-special-cases}) give $h_{\max}<n-1$, which contradicts (\ref{eq:sierpinski-nminus1}).
\end{proof}
|
2,877,628,090,057 | arxiv | \section{Introduction}
Metamaterials are artificial composite materials which possess
unusual electromagnetic properties not normally found in natural
materials. Electromagnetic properties of nanostructured
metamaterials in the optical range are one of the foci of interest
in modern electromagnetics. Traditionally, metamaterials and
metasurfaces composed of small individual resonant inclusions are
realized as periodical arrays. However, most recently, random or
amorphous metamaterials start to attract attention, see
Refs.~\cite{Helg}--\cite{Chen}. This is due to novel technological
possibilities to manufacture amorphous structures cheaply and on a
large scale, using advanced self-assembly techniques. In addition,
effects of strong spatial dispersion (often undesirable) can be in
some cases suppressed in disordered structures. It is generally
accepted that the electromagnetic properties of both regular and
random arrays of scatterers are quite similar if the distances
between inclusions are electrically small. The main difference in
electromagnetic response comes from scattering on the lattice
inhomogeneities. This apparently results in additional loss in
amorphous metamaterials, and for this reason regular metamaterial
lattices have been the preferred choice if low-loss response is
desired.
However, it appears that in metamaterial structures exhibiting
resonant responses in several modes, the effects due to position
randomness of inclusions are more complicated. In a recent
paper~\cite{Helg} by Helgert et al. reflection and transmission
properties of regular and random (amorphous) planar arrays of
cut-wire particles were studied both numerically and experimentally.
Specially introduced position disorder of individual scatterers
allowed to study the effect of distortion of periodicity on the
electromagnetic response of the array. It was found that position
randomness drastically affects the electromagnetic behavior at the
electric resonance, but makes little impact at the array properties
near the magnetic resonance of the particles. These results were
validated by numerical simulations and confirmed in posterior work
\cite{R2011}.
The authors of paper \cite{Helg} put forward a hypothesis that the
discovered dramatic difference between scattering properties in
electric and magnetic modes is caused by difference in
electromagnetic interactions between particles in different modes.
It was based on an observation that magnetic dipoles as well as
electric quadrupoles do not generate tangential electric fields in
the array plane, and it was assumed that this means that magnetic
scatterers are not interacting with each other, so that the exciting
field acting on a single particle is solely the external
illumination. On the other hand, the electric-dipole scatterers
interact strongly and the exciting field is affected by positional
disorder, which leads to resonance broadening and damping. However,
from the duality principle it is known that in fact magnetic dipole
particles interact via their magnetic fields exactly as strongly as
electric dipoles interact via their electric fields, which means
that the phenomenon discovered in Ref.~\cite{Helg} must have some
other physical reasons.
The goal of this paper is to study the phenomenon of resonance
damping and broadening theoretically and explain the strong
differences in resonance broadening in different resonant modes. To
this end, we analytically study the effect of positional randomness
on electromagnetic behavior of grids of resonant particles which can
exhibit both electric and magnetic resonant responses. We introduce
a simple model, which allows us to analyze the reflective,
transmitting, and absorptive properties of multi-resonant grids,
both in the regular and amorphous states. The theory is confirmed by
numerical simulations using an example of the same metasurface as
that studied in Ref.~\cite{Helg}. The results reveal the mechanisms
of resonance broadening and damping in amorphous structures and
explain the earlier discovered differences in the cases of electric
(symmetric) and magnetic (anti-symmetric) resonances. Understanding
physical phenomena which define the differences between effective
electromagnetic responses of regular and disordered metamaterials is
urgently needed before the emerging amorphous metamaterials can find
applications. Developing analytical models of disordered structures
will allow the design and optimization of future composite materials
with desired performance.
\section{Analytical theory of planar arrays with electrically and magnetically resonant inclusions}
Let us consider an optically dense planar array of optically small
resonant particles excited by normally incident plane waves. We
assume that the distance between the particles in the grid $a$ is
smaller than the wavelength. We are interested in the case when each
particle exhibits both electric and magnetic responses, that is,
both electric and magnetic moments are induced by local electric and
magnetic fields, respectively. We also assume that bi-anisotropic
magnetoelectric coupling is either forbidden due to the particle
symmetry or it is negligible. Many widely-studied infra-red and
optical metamaterial structures like the cut-wire pairs considered
in Ref.~\cite{Helg} belong to this class. In this paper we consider
only electric and magnetic dipole moments of particles, neglecting
quadrupoles and higher-order moments, concentrating on the influence
of array randomness on the reflection and transmission coefficients.
Relative strengths of dipolar and higher-order effects in cut-wire
pairs have been analyzed in Ref.~\cite{Pet}.
Assuming for simplicity that no cross-polarized dipole moments in
the array plane are induced (the particles have the form of discs or
squares, for example) and considering the excitation by normally
incident plane waves, we can write the relations between the induced
electric dipole moment $p$, magnetic dipole moment $m$, and the
incident fields $E_{\rm inc}$ and $H_{\rm inc}$ as scalar relations
\begin{equation} p=\alpha_{ee}(E_{\rm inc} + \beta_{ee} p), \qquad
m=\alpha_{mm}(H_{\rm inc} + \beta_{mm} m)\l{moments}\end{equation} Here
$\alpha_{ee}$ and $\alpha_{mm}$ are the electric and magnetic
polarizabilities of individual inclusions, respectively. Parameters
$\beta_{ee}$ and $\beta_{mm}$ are called \emph{interaction
constants} and they measure contributions of the fields created by
all other particles of the array into the local field $E_{\rm
loc}=E_{\rm inc} + \beta_{ee} p$ exciting each particle (see
e.g.~Ref.~\cite{modeboo}). The interaction constants for electric
and magnetic dipoles are related simply as \begin{equation} \beta_{mm}={1\over
\eta_0^2}\, \beta_{ee} \end{equation} where $\eta_0=\sqrt{\mu_0/\epsilon_0}$ is
the wave impedance of the surrounding space. Fields created by
magnetic dipoles do not contribute to the electric local field
exciting electric dipoles because the tangential component of the
electric field of the magnetic dipole grid equals zero in the array
plane. Likewise, fields scattered by electric dipoles do not excite
magnetically polarizable particles positioned in the same plane.
Most often, both moments are actually induced in the same particles,
but the two modes have resonances at different frequencies.
Next, we calculate the plane-wave electric fields created by the
surface averaged electric current sheet $J_e=-{i\omega p\over a^2}$
and the magnetic current sheet $J_m=-{i\omega m\over a^2}$ (the
harmonic time dependence assumption is of the form $e^{-i\omega
t}$):
\begin{equation} E^{\rm e}_{\rm ref} = -{\eta_0 \over {2}}J_e, \quad H^{\rm
m}_{\rm ref} = -{1 \over {2\eta_0}}J_m \l{EeHm}\end{equation} \begin{equation} E^{\rm m}_{\rm
ref} = -{\eta_0} H^{\rm m}_{\rm ref}, \quad E_{\rm ref} = E^{\rm
e}_{\rm ref}+E^{\rm m}_{\rm ref} \l{EH}\end{equation} Here $E^{\rm e}_{\rm ref}$
and $E^{\rm m}_{\rm ref}$ are reflected electric fields created by
the induced electric and magnetic currents $J_e$ and $J_m$,
respectively, and $H^{\rm m}_{\rm ref}$ is the reflected magnetic
field created by the induced magnetic current $J_m$. Solving
\r{moments} for the induced dipole moments in terms of the incident
fields and using \r{EeHm} and \r{EH} we find the reflection and
transmission coefficients in the simple form
\begin{equation} R = \frac{E_{\rm ref}}{E_{\rm inc}}= R_e + R_m
={i\omega \eta_0 \over 2 a^2}{{1\over {1\over
\alpha_{ee}}-\beta_{ee}}} - {i \omega \over 2 \eta_0 a^2}{{1\over
{1\over \alpha_{mm}}-\beta_{mm}}} \l{R}\end{equation} \begin{equation} T=1+R_e-R_m\l{T}\end{equation} Here
we have used the plane-wave relation between the electric and
magnetic incident fields: $H_{\rm inc}=E_{\rm inc}/\eta_0$. The two
partial reflections coefficients $R_e$ and $R_m$ correspond to the
fields created by the induced electric and magnetic currents,
respectively. Since $\beta_{ee}$ has the dimension of $1/(\epsilon_0
a^3)$ and $\beta_{mm}$ has the dimension of $1/(\mu_0 a^3)$, it is
convenient to multiply and divide the reflection coefficients by
$\epsilon_0 a^3$ or $\mu_0 a^3$. The result is \begin{equation} R_e={i k_0 a
\over 2 }{{1\over {\epsilon_0 a^3 \over \alpha_{ee}}- \beta}}
\l{Ten}\end{equation} \begin{equation} R_m=-{ik_0 a \over 2 }{{1\over {\mu_0 a^3\over
\alpha_{mm}}-\beta}} \l{Tmn}\end{equation} where
$k_0=\omega\sqrt{\epsilon_0\mu_0}$ is the wave number in the
surrounding space. The normalized dimensionless interaction
constants are the same for both electric and magnetic particles, and
we denote them as $\beta$: \begin{equation} \beta=\epsilon_0 a^3 \beta_{ee}=\mu_0
a^3 \beta_{mm}\end{equation}
Let us assume a simple Lorentz-type resonant response model of
individual particles. This type of resonant response is very common
and approximates very well the particle response near their
resonances. Let us write down the {\itshape inverse} values of the
normalized polarizabilities to make it easy to discuss the radiation
loss factor: \begin{equation} {\epsilon_0 a^3 \over \alpha_{ee}} =\left({A_e\over
{\omega_{0e}^2-\omega^2 -i\omega \Gamma_e}}\right)^{-1}-i{k_0^3
a^3\over 6\pi}\l{inv_ee}\end{equation} \begin{equation} {\mu_0 a^3 \over \alpha_{mm}}
=\left({A_m\over {\omega_{0m}^2-\omega^2 -i\omega
\Gamma_m}}\right)^{-1} -i{k_0^3 a^3\over 6\pi}\l{inv_mm}\end{equation} Here
$\Gamma_{e,m}$ model the dissipation losses in the particle (in
respective modes), while the last imaginary term is due to the
scattering (re-radiation of power) loss \cite{modeboo}. In case of
regular or ``totally random'' (on the wavelength scale) grids there
is no scattering loss, when the array period is smaller than the
wavelength. In this case spherical-wave scattering from individual
particles is suppressed by interactions between the particles in the
array. Correspondingly, the imaginary parts of the interaction
constants $\beta_{ee}$ and $\beta_{mm}$ contain terms proportional
to $k_0^3$ which compensate the corresponding terms in the inverse
polarizabilities (see e.g. Ref.~\cite{modeboo}): \begin{equation}
{\beta_{\rm{regular}}= {\rm Re} ( \beta)-i\frac{ k_0^3a^3}{6\pi}+
i\frac{ k_0a}{2}} \l{be_reg}\end{equation} The other imaginary term corresponds
to the plane waves created by the surface-averaged currents. In case
of amorphous (on the wavelength scale) arrays particles scatter
individually, and there is no corresponding term in the interaction
constants: \begin{equation} {\beta_{\rm{amorph}}= {\rm Re}( \beta)+i\frac{
k_0a}{2}} \l{be_amorph}\end{equation} In the quasi-static limit $\rm{Re}( \beta)
\approx 0.36$ (see Ref.~\cite{modeboo}).
Next, we substitute these interaction constants and the Lorentz
particle polarizabilities \r{inv_ee} and \r{inv_mm} in the general
formulas for the reflection coefficients \r{Ten} and \r{Tmn}.
For regular or totally random (on the wavelength scale) arrays we
get \begin{equation} R_{e\, \rm regular}= i{k_0 a\over 2}{A_e\over \tilde
\omega_{0e}^2-\omega^2 -i\omega \Gamma_e-i\frac{
k_0a}{2}A_e}\end{equation} \begin{equation} R_{m\, \rm
regular}=-i{k_0 a\over 2}{A_m\over \tilde \omega_{0m}^2-\omega^2
-i\omega \Gamma_m-i\frac{
k_0a}{2}A_m}\end{equation} Here $\tilde \omega_0$ denotes the resonant
frequency shifted due to interactions between the particles in the
grid. In the quasi-static approximation for the real part of the
interaction constant $\tilde\omega_{0e,m}^2\approx
\omega_{0e,m}^2-0.36 A_{e,m}$. For amorphous grids we get \begin{equation} R_{e\,
\rm amorph}= i{k_0 a\over 2}{A_e\over \tilde \omega_{0e}^2-\omega^2
-i\omega \Gamma_e - {{ik_0^3a^3\over {6\pi}} A_e}-i\frac{
k_0a}{2}A_e}\end{equation} \begin{equation} R_{m\, \rm amorph}= - i{k_0
a\over 2}{A_m\over \tilde \omega_{0m}^2-\omega^2 -i\omega
\Gamma_m - {{ik_0^3a^3\over {6\pi}} A_m}-i\frac{
k_0a}{2}A_m}\end{equation}
Let us consider the case when electric and magnetic resonances occur
at different frequencies. Then in the vicinity of one of the
resonances the non-resonant moment varies weakly with the frequency
and we can find a simple estimation of the resonant curve width (on
the field-strength scale): \begin{equation} 2\Delta \omega_{e,m\, \rm
regular}=\Gamma_{e,m}+\frac{
k_0a}{2}{A_m\over {\tilde \omega_{0e,m}}} \end{equation} for regular grids and \begin{equation} 2\Delta
\omega_{e,m\, \rm amorph}=\Gamma_{e,m}+{k_0^3a^3\over {6\pi}} {A_{e,m}\over
{\tilde \omega_{0e,m}}}+\frac{
k_0a}{2}{A_m\over {\tilde \omega_{0e,m}}}\end{equation} for amorphous grids.
We now see that if the condition \begin{equation} \tilde
\omega_{0e,m}{\Gamma_{e,m}\over A_{e,m}}+\frac{ k_0a}{2}\gg
{k_0^3a^3\over {6\pi}}\l{condition}\end{equation} is satisfied, near the
corresponding resonant frequency $\tilde \omega_{0e,m}$ the effect
of inclusion position randomness is negligible, and the response of
regular and amorphous structures is nearly the same. Physically,
this condition means that absorption (the first member of the
left-hand side) and coherent plane-wave reflection (the second
member on the left) dominate over scattering (the right-hand side
term). The above relation shows that this is the case of high
dissipative losses, low resonance strength, and small electrical
size of the unit cell. Note that for the case of negligible
absorption, this condition simply tells that scattering loss is
negligible in random arrays if the distance between particles is
optically very small ($k_0^2a^2\ll 3\pi$).
From the above results we can conclude that the effect of strong
widening of the resonant curve of the electric-dipole mode and
hardly any effect of array randomness on the magnetic mode
discovered in Ref.~\cite{Helg} can be due to two reasons:
\begin{enumerate}
\item At the frequency of the magnetic resonance the grid is
practically homogeneous on the wavelength scale (``totally
random''). Then the scattering term cancels out just like for
periodical grids, and there is no difference in the resonant
curve widths for regular and amorphous layers.
\item At the magnetic resonance the particles are considerably more
lossy and weaker excited than at the electric resonance, that
is, \r{condition} is satisfied near the magnetic resonance but
not satisfied near the electric-mode resonance.
\end{enumerate}
\section{Example: Arrays of cut-wire pairs}
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{Cut_wire_Model.eps}
\caption{(Color online) Geometry of the unit cell of the cut-wire array.} \label{figmodel}
\end{figure}
As an example we consider the cut-wire pair structure which was
studied in Refs.~\cite{Helg, Shal, Dol}. A unit cell of the infinite
regular array is depicted in Fig.~\ref{figmodel}. The dimensions are
the same as in Ref.~\cite{Helg}. The square lattice has the period
of $a=512$~nm along two transverse directions. The width of the
cut-wire pairs in both lateral directions is $W_c=180$~nm. The
height of each gold pair is $H_c=30$~nm. The gap between the two
elements in each pair equals $g=45$~nm and it is filled with a
material with the relative permittivity equal to
$\epsilon_{r1}=1.72$. The structure is placed on top of a substrate
with the permittivity of $\epsilon_{r2}=1.5$. The permittivity of
gold is taken from Ref.~\cite{John}.
\begin{figure}[h!]
\centering
\subfigure[]{
\includegraphics[width=.45\textwidth]{alphaE.eps}
\label{figalphaE}
}
\subfigure[]{
\includegraphics[width=.45\textwidth]{alphaM.eps}
\label{figalphaM}
}
\caption[Optional caption for list of figures]{\subref{figalphaE}: Electric polarizability of a single cut-wire pair
}
and {\subref{figalphaM}: Magnetic polarizability of the same particle
}
\label{figalpha}
\end{figure}
\begin{figure}[h]
\centering
\subfigure[]{
\includegraphics[width=.45\textwidth]{alphaEfreespace.eps}
\label{figalphaEfree}
}
\subfigure[]{
\includegraphics[width=.45\textwidth]{alphaMfreespace.eps}
\label{figalphaMfree}
}
\caption[Optional caption for list of figures]{\subref{figalphaE}: Electric polarizability of a single cut-wire pair in free space and
\subref{figalphaM}:
Magnetic polarizability of the same particle}
\label{figalphaFree}
\end{figure}
\begin{figure*}
\begin{minipage}[h]{0.49\linewidth}
\center
\epsfig{file=Rcomparison.eps,width=1\linewidth, height=0.7\linewidth}
\end{minipage}
\hfill
\begin{minipage}[h]{0.49\linewidth}
\center
\epsfig{file=Tcomparison.eps,width=1\linewidth, height=0.7\linewidth}
\end{minipage}
\begin{minipage}[h]{1\linewidth}
\begin{tabular}{p{0.49\linewidth}p{0.49\linewidth}}
\centering a) & \centering b)
\end{tabular}
\end{minipage}
\vfill
\begin{minipage}[h]{0.49\linewidth}
\center
\epsfig{file=Ranglecomparison.eps,width=1\linewidth, height=0.7\linewidth}
\end{minipage}
\hfill
\begin{minipage}[h]{0.49\linewidth}
\center
\epsfig{file=Tanglecomparison.eps,width=1\linewidth, height=0.7\linewidth}
\end{minipage}
\begin{minipage}[h]{1\linewidth}
\begin{tabular}{p{0.49\linewidth}p{0.49\linewidth}}
\centering c) & \centering d)
\end{tabular}
\end{minipage}
\caption{(Color online) (a) -- Amplitude of the reflection coefficient; (b) -- Amplitude of the transmission coefficient; (c) -- Phase of the reflection coefficient; and (d) -- Phase of the transmission coefficient for grids with different randomness levels$(r_n)$} \label{figRTcomp}
\end{figure*}
\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{Ecomparison.eps}
\caption{{(Color online)}~Absorption in the array in transition from regular to amorphous states} \label{figEcomp}
\end{figure}
First we calculate the reflection and transmission coefficients for
the regular array using the full-wave numerical simulator Ansoft
HFSS. Substituting the numerical data for reflection and
transmission coefficients in \r{Ten} and \r{Tmn} and using the
quasi-static approximation for the real part of the interaction
constants $\beta\approx 0.36$, we extract the polarizabilities of
individual inclusions. The results are shown in Fig.~\ref{figalpha},
and it is apparent that the array has electric and magnetic
resonances in different frequency regions, as expected. In fact the
array is weakly bi-anisotropic due to the presence of the substrate
(omega-type magnetoelectric coupling~\cite{biama,Mo}), which has
been neglected in the theory and in the parameter extraction. We
have checked that this approximation is valid by repeating the
simulations and parameter extraction for the same array in free
space. The results are presented in Fig.~\ref{figalphaFree} and they
show that this simplifying assumption is reasonable: The substrate
effect is quite small. Behavior of the extracted electric and
magnetic polarizabilities is very close to the canonical
Lorentz-type resonant response.
Scattering losses which appear in transition from regular to
amorphous grids we model by the randomness parameter
$0~\leq~{r_n}~\leq~1$, where unity corresponds to the case where the
scattering loss is completely compensated {\em(regular array)} and
$r_n=0$ means that the scattering loss is not compensated at all
{\em (amorphous array, each inclusion scatters individually)}.
Transition from regular to amorphous state we model modifying the
interaction constant \r{be_reg} as follows:
\begin{equation} \beta={{{\rm Re\mit}\{\beta\}}-{r_n}{i k_0^3a^3\over {6\pi}}+{i k_0a\over
{2}}} \l{betaE}\end{equation}
which corresponds to a continuous transition from \r{be_reg} to
\r{be_amorph} with $r_n$ changing from unity to zero. It should be
noted that for simplicity the randomness factor $r_n$ is assumed to
be the same for both electric and magnetic interaction constants.
Due to differences in resonant frequencies, this means that the same
value of $r_n$ may correspond to somewhat different degrees of
geometrical randomness of particle positions.
Next we investigate how the reflection, transmission, and extinction
change in transition from regular to amorphous states, using the
analytical formulas \r{Ten} and \r{Tmn} with the extracted values of
the polarizabilities and the interaction constant \r{betaE}.
Figs.~\ref{figRTcomp} and \ref{figEcomp} show the randomness
effects. One can see that the developed simple model gives very good
agreement with the experimental and numerical data from
Ref.~\cite{Helg}. Electrical response of the grid is strongly
influenced by randomness, while close to the magnetic resonance
there is almost no dependence on randomness. The reason for this
phenomenon is the difference in the ratio of the absorption and
scattering losses. Fig.~\ref{figEcomp} shows that in the periodical
case the absorption is much stronger at the magnetic resonance than
at the electric one. Higher losses are mainly due to larger
imaginary part of gold permittivity, which is more than two times
higher at the magnetic resonance: $\rm{Im}(\epsilon)_{1022 \,
\rm{nm}} \approx 3.2$, $\rm{Im}(\epsilon)_{797\, \rm{nm}} \approx
1.5$.
In addition, for the case of the grid in free space
(Fig.~\ref{figalphaFree}) we have fitted the numerically extracted
polarizability curves to the Lorentz model \r{inv_ee} and \r{inv_mm}
and extracted parameters $\Gamma_{e,m}$ and $A_{e,m}$. This allowed
us to find the values in inequality \r{condition}. At the resonant
frequency of the electric polarizability we find that the left-hand
side equals $0.4+2.3$ while the right-hand side equals $5.6$.
Scattering effects are clearly dominating and position randomness
changes the array response quite significantly. At the resonant
frequency of the magnetic polarizability the left-hand side reads
$2.3+2$, while the right-hand side equals $3$. In this case the
terms are of the same order and the randomness effect is much
weaker. Note that condition \r{condition} is a simple approximation
which assumes that the two resonances are sharp and well separated.
In this particular example, in the frequency region of the magnetic
resonance the electric dipoles in fact give a significant
contribution to the total absorption and coherent reflection.
\section{Conclusions}
In this paper we have developed a simple model which explains the
electromagnetic effects in transition from regular to random states
of resonant particle arrays. We have derived a general condition
under which randomizing particle positions gives only negligible
effects on the reflection and transmission coefficients and
explained the earlier discovered dramatic differences in resonance
damping for electric and magnetic modes of particles. We have also
shown that the physical phenomena leading to the resonance damping
in amorphous structures are the same for electrically or
magnetically polarizable particles. The widening of the resonances
takes place due to additional scattering losses, which are
compensated in the case of electrically dense periodical grids.
Studying transition to the amorphous state for a particular example
of cut-wire pairs we have found that the reason for the much weaker
resonance widening and damping in the magnetic mode is strong
absorption in that frequency range. When scattering losses are much
smaller than the dissipation losses, they make little impact on the
total extinction. On the contrary, at the higher-frequency electric
resonance scattering losses are stronger than the dissipation ones,
which leads to strong resonance damping and distortion in the random
case. In other situations, different transition effects in different
resonant modes can also be caused by differences in the electrical
size of the unit cell. In the considered example, the array period
is comparable with the wavelength, thus, even for geometrically
random positions of the particles with respect to the cell centers,
the array cannot be made homogeneous on the wavelength scale.
Our findings can have important implications in understanding the
physical differences in electromagnetic responses of regular and
amorphous structures, in design of various metamaterial structures
for such applications as subwavelength imaging, control of thermal
radiation, microwave, terahertz and optical absorbers, and others.
Using the developed model it is possible to predict and engineer the
effects of randomness, relaxing conventional requirements on strong
periodicity and make use of inexpensive self-assembly techniques in
production of metamaterials.
\subsection*{Acknowledgements}
This study has been done as a student research project within the
Aalto University course on analytical modeling in applied
electromagnetics. One of the authors (ST) wants to acknowledge
enlightening discussions with C. Rockstuhl within the frame of the
EU-funded FP7 project NANOGOLD.
|
2,877,628,090,058 | arxiv | \section{Introduction}
Video data is reported to occupy more than 82\% of all consumer Internet traffic~\cite{cisco2020cisco},
and is expected to see the rapid rate of growth in the next few years, especially the high-definition videos and ultra high-definition videos.
Therefore, video compression is a key requirement for the bandwidth-limited Internet.
During the past decades, several video coding standards were developed, such as H.264~\cite{wiegand2003overview}, H.265~\cite{sullivan2012overview}, and H.266~\cite{bross2021developments}.
These methods are based on hand-designed modules such as block partition, inter prediction and transform~\cite{ahmed1974discrete}, etc.
While these traditional video compression methods have made a promising performance,
their performance are limited since the modules are artificially designed and optimized separately.
Recently, learned image compression~\cite{cui2021asymmetric,cheng2020learned,minnen2018joint,guo2020variable} based on variational auto-encoder~\cite{kingma2013auto} has shown great potential,
achieving better performance than traditional image codecs~\cite{bellard2016bpg,wallace1992jpeg,bross2021developments}.
Inspired by the learned image compression, and combined with the idea of traditional video codecs,
many learning-based video compression approaches~\cite{agustsson2020scale,guo2021learning,pourreza2021extending,feng2021versatile,habibian2019video,hu2021fvc,lu2020end,li2021deep} were proposed.
Given the reference frame, variant kinds of motion compensation (alignment) methods were proposed like
scale-space alignment~\cite{agustsson2020scale}, feature-based alignment~\cite{hu2021fvc}, multi-scale feature-based alignment~\cite{sheng2021temporal}.
These methods aim to improve the diversity of motion compensation and result in more compression-friendly predictions.
However, such methods increase the complexity on both encoder and decoder side.
Inspired by AMVP (Advanced Motion Vector Prediction) on traditional video compression methods~\cite{sullivan2012overview},
we expect the encoder side to predict a more accurate motion information.
Further, at the encoder side of AlphaVC, we propose a pixel-to-feature motion prediction method that can obtain high-quality
motion information without increasing the complexity of the decoder.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/figure1-eps-converted-to.pdf}
\caption{
(a): BD-rate against VTM in terms of PSNR (Lower is better). (b): BD-rate against VTM as a function of encoding/decoding time on 1080p videos.
}
\label{fig:fig1}
\end{figure}
Existing learned video compression can be divided into two categories: Low-Delay P mode and Low-Delay B/Random-Access mode.
For the Low-Delay P mode, the methods~\cite{agustsson2020scale,guo2021learning,hu2021fvc,sheng2021temporal} only include the P(predictive)-frames and I(image)-frames.
For the Low-Delay B or Random-Access mode, the methods~\cite{feng2021versatile,pourreza2021extending} insert the B(bidirectional predictive) frames into the GoP to improve compression performance.
AlphaVC focuses on the Low-Delay P mode. In this mode, due to the accumulation error in P-frame~\cite{lu2020content},
most existing methods have to use the inefficient I-frame as the first frame in limited length GoP.
Unlike the existing methods, we overcome this issue by introducing a conditional I-frame (cI-frame) as the first frame in the GoP,
which stabilizes the reconstructed quality and achieves better performance.
In addition, we all know that the entropy coding~\cite{howard1994arithmetic,duda2009asymmetric} can only run serially will increase the runtime.
Moreover, the auto-regressive entropy module~\cite{minnen2018joint}, which significantly increase the decoding time,
is always used on learned image codecs for a higher compression ratio.
We found that most elements of the latents usually have very low information entropy, which means the probability distributions of these elements estimated by entropy module always is highly concentrated.
Inspired by this, we propose an efficient probability-based entropy skipping method (Skip) which can significantly save runtime in entropy coding,
and achieve higher performance without auto-regressive.
With the help of the above technologies, AlphaVC achieves the highest E2E compression performance while being very efficient.
As shown in Fig.~\ref{fig:fig1},
the proposed AlphaVC outperforms VTM-IPP/VTM-LDP by 28.2\%/6.59\% ,
where the VTM is the official software of H.266/VVC,
the IPP denotes the configuration using one reference frame and flat QP,
and the LDP denotes the better configuration using multiple references and dynamic QP.
Note the configuration of AlphaVC is the same as IPP.
To the best of our knowledge, AlphaVC is the only learning-based video codec that can consistently achieve comparable or better performance with VTM-LDP in terms of PSNR on all common test datasets.
Comparing with the state-of-the-art learning-based video codecs~\cite{sheng2021temporal}, AlphaVC reduces the BD-rate by about 25\% while faster encoding and decoding.
Our contributions are summarized as follows:
\begin{enumerate}
\item We introduce a new type of frame named conditional-I frame (cI-frame) and propose a new coding mode for learned video compression.
It can effectively save the bit rate of I-frame and alleviate the problem of accumulated error.
\item The proposed motion prediction method, utilizing the idea of pixel-to-feature and global-to-local,
can significantly improve the accuracy of inter-frame prediction without increasing decoding complexity.
\item An efficient method in entropy estiamtion module and entropy coding have higher performance and faster encoding and decoding time.
\end{enumerate}
\section{Related Work}
\subsection{Image Compression}
In the past decades, the traditional image compression methods like
JPEG~\cite{wallace1992jpeg}, JPEG2000~\cite{christopoulos2000jpeg2000} and BPG~\cite{bellard2016bpg} can efficiently reduce the image size.
Those methods have achieved a high performance by exploiting the hand-crafted techniques, such as DCT~\cite{ahmed1974discrete}.
Recently, thanks to variational autoencoder~(VAE)~\cite{kingma2013auto} and scalar quantization assumption~\cite{balle2016end},
the learning-based image compression methods have achieved great progress.
With the optimization of entropy estimation modules~\cite{balle2018variational,minnen2018joint} and network structure~\cite{cui2021asymmetric,cheng2020learned},
the learning-based image compression methods have achieved better performance than the traditional image compression codecs on common metrics,
such as PSNR and MS-SSIM~\cite{wang2003multiscale}.
\subsection{Video Compression}
Video compression is a more challenging problem compared to image compression.
There is a long history of progress for hand-designed video compression methods,
and several video coding standards have been proposed,
such as H.264(JM)~\cite{wiegand2003overview}, H.265(HM)~\cite{sullivan2012overview} and more recently H.266(VTM)~\cite{bross2021developments}.
With the development of video coding standards,
the traditional video compression methods made significant improvements and provided a strong baseline.
Even they have shown a good performance, these algorithms are limited by the hand-designed strategy and the difficult to optimize jointly.
Recently, learning-based video compression has become a new direction.
Following the traditional video compression framework,
Lu et al. proposed the end-to-end optimized video compression framework DVC~\cite{lu2020end},
in which the neural networks are used to replace all the critical components in traditional video compression codec.
Then, the exploration direction of existing approaches can be classified into three categories.
One category of approaches focuses on the motion compensation (alignment) method to improve the accuracy of inter prediction.
For example, SSF~\cite{agustsson2020scale} designed a scale-space flow to replace the bilinear warping operation.
Hu et al.~\cite{hu2021fvc} propose the FVC framework, which apply transformation in feature space with deformable convolution~\cite{dai2017deformable}.
Later Sheng et al. introduce multi-scale in feature space transformation~\cite{sheng2021temporal}.
Another popular direction is the design of auto-encoder module.
Such as Habibian et al.~\cite{habibian2019video} use a 3D spatio-temporal autoencoder network to directly compress multiple frames.
Li et al.~\cite{li2021deep} use the predicted frame as the input of encoder, decoder, instead of explicitly computing the residual.
The third category extends the learned video compression to more codec functions, like B-frame~\cite{pourreza2021extending,feng2021versatile}, utilizing multiple reference frames~\cite{hu2021fvc}.
\section{Method}
\subsection{Overview}
Let $\mathcal{X}=\{\mathbf{X}_1, \mathbf{X}_2, \dots\}$ denote a video sequence,
video codecs usually break the full sequence into groups of pictures (GoP).
Due to the accumulative error of P-frames, in low delay P mode, which is AlphaVC adopted,
each group needs to start with an I-frame and then follow P-frames.
In AlphaVC, we propose a new codecing mode in GoP, including three types of frames.
As shown in Fig.~\ref{fig:framework}(a), the I-frame is only used for the first frame.
For other groups, we propose to start with conditional-I-frame instead of I-frame.
The Conditional-I-frame (named cI-frame),
which uses the reference frame as condition of entropy to reduce the bit-rate,
stabilises the reconstructed quality like I-frame, and meanwhile has a high compression rate.
The details of each type of our P-frame and cI-frame are summarized as follows:
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/system-framework-eps-converted-to.pdf}
\caption{Overview of our proposed video compression scheme. (a): Two kinds of GoP. (b): The framework of P-frame. (c): The framework of cI-frame.}
\label{fig:framework}
\end{figure}
\subsubsection{P-Frame}
First of all, we define the P-Frame in learned video compression as a class of methods that has the following form on decoder side:
\begin{equation} \label{eq:p-frame}
\begin{array}{rl}
\hat{\bf{X}}_t = D_{p}(H_{\text{align}}(\hat{\bf{X}}_{t-1}, \hat{\bf{m}}_t), \hat{\bf{r}}_t)
\end{array}
\end{equation}
where $D_{p}(\cdot), H_{\text{align}}(\cdot)$ denote the method of reconstruction and alignment,
$\hat{\bf{m}}_t, \hat{\bf{r}}_t$ are the quantized latent representation of motion, residual.
Note that the quantized latent representation is the features to be encoded after the encoder and quantization.
That is, the reference frame $\hat{\bf{X}}_{t-1}$ will participate in and affect the reconstruction of current frame,
which means that the consecutive P-frame will generate cumulative errors.
In this paper, we use the feature-align based P-frame framework,
Fig.~\ref{fig:framework}(b) sketches our P-frame compression framework.
We first transform $\hat{\bf{X}}_{t-1}, \bf{X}_t$ into feature space $\hat{\bf{F}}_{t-1}$, $\bf{F}_t$.
Then motion predictor will generate the predicted motion ${\bf{M}}_t$
and the predicted motion will be compressed by motion compression model.
The predicted feature $\tilde{\bf{F}}_t$ is generated by deformable alignment~\cite{dai2017deformable}
with the reconstructed motion $\hat{\bf{M}}_t$ and reference feature $\hat{\bf{F}}_{t-1}$.
Finally, the residual in feature-based $\bf{R}_t=\bf{F}_t-\tilde{\bf{F}}_t$ will be compressed by residual compression model.
The reconstructed feature $\hat{\bf{F}}_t=\hat{\bf{R}}_t + \tilde{\bf{F}}_t$ is transformed into the current reconstruct frame $\hat{\bf{X}}_t$ with frame generator.
Both the motion compression model and residual compression model are implemented by auto-encoder structure~\cite{balle2018variational},
including an encoder module, decoder module and the proposed entropy estiamtion module.
The newtork structure of auto-encoder part is the same as FVC~\cite{hu2021fvc}. To further reduce redundant information, we introduce the temporal and structure prior
for the entropy estimation module in both motion and residual compression models:
\begin{equation} \label{eq:entropy-prior}
\begin{array}{rl}
& \mathbb{E}_{\hat{\bf{m}}_t \sim p_t} [-\log_2{q_t(\hat{\bf{m}}_t | \hat{\bf{F}}_{t-1}, \hat{\bf{m}}_{t-1})}] \\
& \mathbb{E}_{\hat{\bf{r}}_t \sim p_t} [-\log_2{q_t(\hat{\bf{r}}_t | \tilde{\bf{F}}_{t}, \hat{\bf{r}}_{t-1})}]
\end{array}
\end{equation}
the reference feature $\hat{\bf{F}}_{t-1}$ and previous quantized motion latent representation $\hat{\bf{m}}_{t-1}$ are structure and temporal priors of $\hat{\bf{m}}_t$ respectively, and
the predicted feature $\tilde{\bf{F}}_{t}$ and previous quantized residual latent representation $\hat{\bf{r}}_{t-1}$ are structure and temporal priors of $\hat{\bf{r}}_t$ respectively.
\subsubsection{Conditional-I-Frame (cI-frame)}
We introduce a new type of frame called the cI-frame like~\cite{liu2020conditional}, which can be formulated as:
\begin{equation} \label{eq:cI-frame}
\begin{array}{rl}
& \text{Auto-Encoder}: \hat{\bf{y}}_t = Q(E_{cI}({\bf{X}_t})), \hat{\bf{X}}_t = D_{cI}(\hat{\bf{y}}_t), \\
& \text{Entropy}: R(\hat{\bf{y}}_t|\hat{\bf{X}}_{t-1}) = \mathbb{E}_{\hat{\bf{y}}_t \sim p_t}[-\log_2{q_t(\hat{\bf{y}}_t | H_{\text{align}}(\hat{\bf{X}}_{t-1}, \hat{\bf{m}}_t))}], \\
\end{array}
\end{equation}
where $\hat{\bf{y}}_t$ is the quantized latent representation of $\bf{X}_t$,
$E_{cI}(\cdot), Q(\cdot), D_{cI}(\cdot)$ denote the function of cI encoder module, quantization and reconstruction.
That is, cI-frame reduces the inter redundant information through the entropy conditioned on $\hat{\bf{X}}_{t-1}$.
For cI-frame, the input of the autoencoder does not use the reference frames, thus make the reconstructed quality stable.
Further, we use cI-frame as the first frame in the GoP excluding the first GoP, which not only stabilizes the sequence quality like I-frame,
but also improves the compression ratio, thereby alleviating the problem of accumulated errors.
The framework for cI-frame is shown in Fig.~\ref{fig:framework}(c).
The feature extractor, motion prediction and motion compression part share the same structure with P-frame framework.
$\tilde{\bf{F}}_t$ is only used as the prior, the current feature $\bf{F}_t$ will be the only input of the encoder.
Furthermore, we propose two novel strategies in both P-frame and cI-frame, named pixel-to-feature motion prediction (P2F MP) and probability-based entropy skipping method (Skip),
to improve the accuracy of inter prediction and coding efficiency.
\subsection{Pixel-to-Feature Motion Prediction}
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/pixel-to-feature-eps-converted-to.pdf}
\caption{Illustration of our proposed pixel-to-feature motion prediction module.}
\label{fig:pixel-to-feature}
\end{figure}
Inter-frame prediction is a critical module to improve the efficiency of inter-frame coding, since it determines the accuracy of the predicted frame.
We propose pixel-to-feature motion prediction to fully exploit
the diversity of feature-based alignment and the state-of-the-art optical flow network.
The illustration is shown in Fig.~\ref{fig:pixel-to-feature}.
Given the previous reconstructed frame $\hat{\bf{X}}_{t-1}$ and the current frame ${\bf{X}}_t$,
the optical flow in pixel space $\bf{M}_t^{\text{pixel}}$ will be generated by a state-of-the-art optical flow network~\cite{sun2018pwc,teed2020raft}.
The pixel space motion $\bf{M}_t^\text{pixel}$ is then used to initialize a motion in feature space $\bf{M}_t^{\text{init}}$.
Then, we apply the deformable alignment ${D(\cdot, \cdot)}$ to the reference feature $\hat{\bf{F}}_{t-1}$ by $\bf{M}_t^{\text{init}}$:
\begin{equation} \label{eq:align1}
\begin{array}{rl}
\bar{\bf{F}}_t = {D}(\hat{\bf{F}}_{t-1}, \bf{M}_t^{\text{init}})
\end{array}
\end{equation}
After initial alignment, the motion local refinement network will refine the initial motion locally
according to the initially aligned feature $\bar{\bf{F}}_t$ and the target feature $\bf{F}_t$,
and then generate the final predicted motion $\bf{M}_t$.
\begin{equation} \label{eq:align2}
\begin{array}{rl}
\bf{M}_t = \text{Refine}(\bar{\bf{F}}_t, F_t) + \bf{M}_t^{\text{init}}
\end{array}
\end{equation}
Finally, the predicted motion will be compressed to reconstruct motion $\hat{\bf{M}}_t$ through motion compression model.
Unlike existing methods, AlphaVC neither learn motion directly from features~\cite{hu2021fvc} that are difficult to fit through convolutions
nor compress the generated optical flow directly~\cite{lu2020end}. We follow pixel-to-feature and global-to-local principles,
first generate the feature space motion before coding with optical flow,
then performing further fine-tuning through alignment feedback.
Experiments show that this method greatly improves the accuracy of inter-frame prediction without affecting the decoding complexity and running time.
\subsection{Probability-base Entropy Skipping Method}
For a latent representation variable $\mathbf{v}$ in learned image or video compression, we first quantize it with round-based quantization $\hat{\mathbf{v}}=[\mathbf{v}]$,
and estimate the probability distribution of ${\mathbf{v}}$ by an entropy estimation module with some priors, such as hyper~\cite{balle2018variational}, context~\cite{minnen2018joint}, etc.
Then $\hat{\mathbf{v}}$ is compressed into the bitstream by entropy coding like arithmetic coding~\cite{howard1994arithmetic}, asymmetric numeral system~\cite{duda2009asymmetric}.
In video compression, due to the introduction of the reference frame, the entropy of quantized latent representation variables like
$\hat{\mathbf{m}_t}, \hat{\mathbf{r}_t}$ in P-frame is very small, especially in low bit-rate.
That means the probability distributions of most elements in the latent variable are concentrated.
If it is slightly off-center for such an element, we will encode it to bitstream with a high cost.
In other words, if we skip these elements without encoding/decoding and replace them with the peak of probability distribution,
we can save both bit-rate and runtime of entropy coding with little error expectations.
Inspired by this idea, we propose an efficient probability-based entropy skipping method (Skip).
For a latent representation variable ${\mathbf{v}}$,
we define $\mathcal{Q}$ as the probability density set of ${\mathbf{v}}$ estimated by its entropy module.
The value which has the maximum probability density of the $i$-th element is calculated as:
\begin{equation} \label{eq:skip1}
\theta_{i} = \displaystyle\mathop{\arg \max}_{\theta_{i}} q_i(\theta_{i})
\end{equation}
The probability that the element ${v}_i$ is close to $\theta_i$ can be computed by:
\begin{equation} \label{eq:skip2}
q_{i}^{\text{max}} = \displaystyle\int_{\theta_{i}-0.5}^{\theta_{i}+0.5} q_i(x) \,dx \\
\end{equation}
If the probability $q_{i}^\text{max}$ is high enough, we will not encode/decode the element to/from the bitstream, and replace the value with $\theta_{i}$.
After this operation, the quantized latent representation will become $\hat{\mathbf{v}}^\text{s}$:
\begin{equation} \label{eq:skip3}
\hat{v_{i}}^\text{s} =
\left\{
\begin{array}{rl}
\theta_{i}, & \quad q_{i}^\text{max} >= \tau \\
\left[v_i\right], & \quad q_{i}^\text{max} < \tau
\end{array}
\right.
\end{equation}
where $\tau$ is a threshold to determine whether to skip.
In our paper, we use gaussian distribution as the estimated probability density of all the quantized latent representations.
Hence the Eq.~\ref{eq:skip1} and Eq.~\ref{eq:skip2} can be easily solved as:
\begin{equation} \label{eq:skip4}
\theta_{i} = \mu_i, q_i^\text{max} = \text{erf}(\frac{1}{2\sqrt{2}\sigma_i}).
\end{equation}
It can be seen that $q_i^\text{max}$ is the monotone function of $\sigma_i$,
we use $\sigma_i$ as the condition of Eq.~\ref{eq:skip3} to further reduce the computational complexity:
\begin{equation} \label{eq:skip5}
\hat{v_{i}}^\text{s} =
\left\{
\begin{array}{rl}
\mu_{i}, & \quad \sigma_{i} < {\tau}_{\sigma} \\
\left[{v}_i\right], & \quad \sigma_{i} >= {\tau}_{\sigma}
\end{array}
\right.
\end{equation}
There are two benefits of Skip. First, it can dynamically reduce the number of elements that need to be entropy encoded, significantly reducing the serial CPU runtime.
Second, we can better trade-off errors and bit rates for elements with high determinism, thereby achieving high compression performance.
\subsection{Loss Function}
Our proposed AlphaVC targets to jointly optimize the rate-distortion (R-D) cost.
\begin{equation} \label{eq:loss-func}
L=R + \lambda \cdot D =
(R_0^\text{I}+\lambda \cdot D_0^\text{I}) + \displaystyle \sum_{t=1}^{\text{T}-1}(R_t^\text{p}+\lambda \cdot D_t^\text{p}) + (R_{\text{T}}^{\text{cI}} + \lambda \cdot D_{\text{T}}^{\text{cI}})
\end{equation}
where the training GoP size is $\text{T}$, $\lambda$ controls the trade-off,
$R_0^\text{I}-D_0^\text{I}$, $R_t^{\text{p}}-D_t^\text{p}$ and $R_\text{T}^{\text{cI}}-D_\text{T}^\text{cI}$ represent the rate-distortion of the $0$-th I-frame, the $t$-th P-frame and the $\text{T}$-th cI-frame, respectively.
\section{Experiments}
\subsection{Setup}
\subsubsection{Training.}
We train our model on the Vimeo-90k dataset. This dataset consists of 4278 videos with 89800
independent shots that are different from each other in content.
We randomly crop the frames to patches of size $256\times 256$, and start training from scratch.
We train the models with Adam optimizer for 60 epochs,
where the batchsize was set to 8 and learning rate was initially set to $1e-4$ and reduced to half for 30 epochs.
The skip operation will been enabled during training.
The loss function is the joint rate-distortion loss as shown in Eq.~\ref{eq:loss-func},
where the multiplier $\lambda$ is chosen from (0.07, 0.05, 0.01, 0.005, 0.001, 0.0007) for the MSE optimization.
The the MS-SSIM optimized models are finetuned from MSE-optimized model with $\lambda=$ 0.03, 0.01, 0.007, 0.005, 0.001.
\subsubsection{Testing.}
We evaluate our proposed algorithm on the HEVC datasets~\cite{bossen2012common} (Class B,C,D,E), the UVG datasets~\cite{mercat2020uvg}, and the MCL-JCV datasets~\cite{wang2016mcl}.
The HEVC datasets contain 16 videos with different resolution $416\times 240$, $832\times 480$ and $1920\times 1080$.
The UVG and MCL-JVC datasets contain 7 and 30 1080p videos, respectively.
The GoP size in AlphaVC is set to 20 for all testing datasets.
\subsubsection{Camparision.}
Both IPP and LDP configuration of VTM-10.0 and HM-16.20 are used for comparision.
The IPP only references the previous frame, and each P-frame has the flat QP, which is the same configuration with AlphaVC.
The LDP is the default low-delay P configuration that references multiple previous frames and has dynamic QP for each P-frame.
In addition, state-of-the-art learning-based video compression methods, i.e.,
FVC (CVPR'21)~\cite{hu2021fvc}, DCVC (NIPS'21)~\cite{li2021deep}, B-EPIC (ICCV'21)~\cite{pourreza2021extending}, VLVC (2021)~\cite{feng2021versatile}, TCMVC (2021)~\cite{sheng2021temporal}.
Note that, B-EPIC and VLVC don't belong to IPPP mode, due to the introduction of B-frame.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{figures/rdcurve-psnr-eps-converted-to.pdf}
\caption{PSNR based R-D Curves of traditional codecs and state-of-the-art learning-based codecs on each datasets.
The red solid line is AlphaVC. Traditional codecs are all represented by solid lines, and other learning-based codecs are represented by dotted lines.}
\label{fig:rdcurve}
\end{figure}
\subsection{Experiment results}
\subsubsection{Performance}
Fig.~\ref{fig:rdcurve},~\ref{fig:rdcurve-msssim} shows the experimental results on all testing datasets.
It is obvious that AlphaVC achieves the bset performance of all methods.
In terms of MS-SSIM, AlphaVC significantly outperforms all the other methods over the entire bitrate range and on all the datasets.
In terms of PSNR, AlphaVC significantly outperforms all the learning-based codecs and VTM-IPP, and even outperforms VTM-LDP in most situations.
As mentioned before, VTM-LDP references multiple previous frames and has dynamic QP for each P-frame.
which is not adopted by AlphaVC.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{figures/rdcurve-msssim-eps-converted-to.pdf}
\caption{MS-SSIM based R-D Curves.}
\label{fig:rdcurve-msssim}
\end{figure}
Table~\ref{table:bdrate-psnr} and Table~\ref{table:bdrate-msssim} show the BD-rate savings in PSNR and MS-SSIM that anchored by VTM-IPP.
In terms of PSNR, AlphaVC achieves an average 28.2\% bitrate saving compared to VTM-IPP, outperforming all the reported methods,
including the stronger VTM-LDP (23.5\% bitrate saving).
In the worst case, AlphaVC also achieves a BD-rate saving of 14.9\% showing a good stability.
In terms of MS-SSIM, learning-based codecs generally have better performances than traditional codecs,
among with AlphaVC performing the best, by saving an additional 8\% bitrate over the best SOTA TCMVC.
\setlength{\tabcolsep}{4pt}
\begin{table}[h]
\begin{center}
\caption{BD-rate calculated by PSNR with the anchor of VTM-IPP.
\textcolor{red}{Red} means more bits ($> 3\%$) required. \textcolor{teal}{Green} means fewer bits ($< -3\%$) required.}
\label{table:bdrate-psnr}
\resizebox{\textwidth}{!}{
\begin{tabular}{cccccccccccc}
\hline\noalign{\smallskip}
& VTM-IPP & VTM-LDP & HM-IPP & HM-LDP & SSF & FVC & DCVC & VLVC & TCMVC & B-EPIC & AlphaVC \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
HEVC\_B & 0 & \textcolor{teal}{-17.9\%} & \textcolor{red}{55.2\%} & \textcolor{red}{24.0\%} & - & \textcolor{red}{75.4\%}
& \textcolor{red}{43.7\%} & \textcolor{red}{27.1\%} &\textcolor{teal}{-6.92\%}& \textcolor{red}{42.5\%} & \textcolor{teal}{\bf{-22.5\%}} \\
\noalign{\smallskip}
HEVC\_C & 0 & \textcolor{teal}{\bf{-23.1\%}} & \textcolor{red}{38.6\%} & \textcolor{red}{27.1\%} & - & \textcolor{red}{40.9\%}
& \textcolor{red}{42.8\%} & \textcolor{red}{40.8\%} & \textcolor{red}{10.2\%} & \textcolor{red}{35.6\%} & \textcolor{teal}{-14.9\%} \\
\noalign{\smallskip}
HEVC\_D & 0 & \textcolor{teal}{-17.9\%} & \textcolor{red}{35.7\%} & \textcolor{red}{24.9\%} & - & \textcolor{red}{47.9\%}
& \textcolor{red}{38.6\%} & \textcolor{red}{30.5\%} &\textcolor{teal}{-6.61\%}& \textcolor{red}{117.\%} & \textcolor{teal}{\bf{-29.0\%}} \\
\noalign{\smallskip}
UVG & 0 & \textcolor{teal}{-31.9\%} & \textcolor{red}{18.5\%} & \textcolor{black}{1.99\%} & \textcolor{red}{57.7\%} & \textcolor{red}{28.4\%}
& \textcolor{red}{24.0\%} & 2.15\% &\textcolor{teal}{-17.3\%}& \textcolor{red}{3.78\%} & \textcolor{teal}{\bf{-41.7\%}} \\
\noalign{\smallskip}
MCL-JCV & 0 & \textcolor{teal}{-26.6\%} & \textcolor{red}{26.3\%} & \textcolor{red}{15.2\%} & \textcolor{red}{50.6\%} & \textcolor{red}{29.3\%}
& \textcolor{red}{43.8\%} & - &2.32\%& \textcolor{red}{50.6\%} & \textcolor{teal}{\bf{-32.9\%}} \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
Avg & 0 & \textcolor{teal}{-23.5\%} & \textcolor{red}{35.6\%} & \textcolor{red}{19.7\%} & \textcolor{red}{54.2\%} & \textcolor{red}{44.4\%}
& \textcolor{red}{38.6\%} & \textcolor{red}{25.1\%} &\textcolor{teal}{-3.66\%}& \textcolor{red}{49.9\%} & \textcolor{teal}{\bf{-28.2\%}} \\
\noalign{\smallskip}
\hline
\end{tabular}
}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\setlength{\tabcolsep}{4pt}
\begin{table}[h]
\begin{center}
\caption{BD-rate calculated by MS-SSIM with the anchor of VTM-PVC-IPP.
\textcolor{red}{Red} means more bits ($> 3\%$) required. \textcolor{teal}{Green} means fewer bits ($< -3\%$) required.}
\label{table:bdrate-msssim}
\resizebox{\textwidth}{!}{
\begin{tabular}{cccccccccccc}
\hline\noalign{\smallskip}
& VTM-IPP & VTM-LDP & HM-IPP & HM-LDP & SSF & FVC & DCVC & VLVC & TCMVC & B-EPIC & AlphaVC \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
HEVC\_B & 0 & \textcolor{teal}{-20.5\%} & \textcolor{red}{54.6\%} & \textcolor{red}{17.4\%} & - & \textcolor{teal}{-21.3\%}
& \textcolor{teal}{-16.0\%} & \textcolor{teal}{-42.5\%} &\textcolor{teal}{-53.5\%}& \textcolor{teal}{-7.1\%} & \textcolor{teal}{\bf{-61.6\%}} \\
\noalign{\smallskip}
HEVC\_C & 0 & \textcolor{teal}{-20.7\%} & \textcolor{red}{53.6\%} & \textcolor{red}{12.8\%} & - & \textcolor{teal}{-22.2\%}
& \textcolor{teal}{-12.8\%} & \textcolor{teal}{-41.6\%} & \textcolor{teal}{-47.6\%} & \textcolor{red}{-15.4\%} & \textcolor{teal}{\bf{-58.9\%}} \\
\noalign{\smallskip}
HEVC\_D & 0 & \textcolor{teal}{-27.2\%} & \textcolor{red}{39.3\%} & {-1.5\%} & - & \textcolor{teal}{-34.7\%}
& \textcolor{teal}{-33.0\%} & \textcolor{teal}{-49.6\%} &\textcolor{teal}{-60.7\%}& \textcolor{teal}{-21.5\%} & \textcolor{teal}{\bf{-67.2\%}} \\
\noalign{\smallskip}
UVG & 0 & \textcolor{teal}{-26.7\%} & \textcolor{red}{56.3\%} & \textcolor{red}{20.2\%} & \textcolor{red}{33.9\%} & \textcolor{red}{11.5\%}
& \textcolor{red}{10.9\%} & \textcolor{teal}{-12.9\%} &\textcolor{teal}{-22.0\%}& -1.63\% & \textcolor{teal}{\bf{-32.9\%}} \\
\noalign{\smallskip}
MCL-JCV & 0 & \textcolor{teal}{-26.0\%} & \textcolor{red}{49.6\%} & \textcolor{red}{14.5\%} & \textcolor{teal}{-4.5\%}& \textcolor{teal}{-18.8\%}
&\textcolor{teal}{-17.9\%}& - &\textcolor{teal}{-38.8\%}&\textcolor{teal}{-19.9\%}& \textcolor{teal}{\bf{-40.5\%}} \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
Avg & 0 & \textcolor{teal}{-24.2\%} & \textcolor{red}{49.9\%} & \textcolor{red}{11.5\%} & \textcolor{red}{14.7\%} & \textcolor{teal}{-17.1\%}
& \textcolor{teal}{-13.7\%} & \textcolor{teal}{-36.6\%} &\textcolor{teal}{-44.5\%}& \textcolor{teal}{-13.1\%} & \textcolor{teal}{\bf{-52.2\%}} \\
\noalign{\smallskip}
\hline
\end{tabular}
}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\subsubsection{Complexity}
The MAC(Multiply Accumulate) of the P-frame at the decoding side is about 1.13M/pixel, and the cI-frame is about 0.98M/pixel.
We use arithmetic coding for the complete entropy encoding and decoding process, and 1080p videos to evaluate the runtime.
The runtime of the encoding side includes model inference, data transmission from GPU to CPU and entropy encoding,
and the runtime of the decoding side includes entropy decoding, data transmission and model inference.
The comparison results are shown in Table~\ref{table:runtime},
in which running platform of AlphaVC is Intel(R) Xeon(R) Gold 6278C CPU and NVIDIA V100 GPU.
The encoding and decoding times of AlphaVC on a 1080p frame average about 715ms and 379ms.
The encoding time is about 1000x faster than VTM, and the decoding time is similar to VTM (1.69x).
Even though AlphaVC uses more parameters than TCMVC, it is still faster.
The main reason is the proposed probability-based skip entropy technique,
which significantly reduces the running time on CPU.
In addition, we can find that the cI-frame is slower than P-frame although the cI-frame has less complexity.
This is also because the bit-rate in the cI-frame is higher, and the number of skipping elements in the cI-frame is fewer.
\setlength{\tabcolsep}{4pt}
\begin{table}[h]
\begin{center}
\caption{
Complexity on 1080p video. We compare our AlphaVC including cI-Frame and p-Frame with traditional codecs and TCMVC.
The time ratio is calculated with the anchor of VTM.
}
\label{table:runtime}
\resizebox{0.8\textwidth}{!}{
\begin{tabular}{cccccc}
\hline\noalign{\smallskip}
Method & Params. & Enc-T (s) & Dec-T (s) & Enc-T ratio & Dec-T ratio \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
VTM-10.0-IPP & - & 661.9 & 0.224 & 1.0000 & 1.0000 \\
\noalign{\smallskip}
HM-16.40-IPP & - & 26.47 & 0.140 & 0.0400 & 0.6250 \\
\noalign{\smallskip}
TCMVC & 10.7M & 0.827 & 0.472 & 0.0012 & 2.1071 \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
AlphaVC & 63.7M & 0.715 & 0.379 & 0.0011 & 1.6920 \\
\noalign{\smallskip}
AlphaVC-cI & 29.9M & 0.733 & 0.580 & 0.0011 & 2.5893 \\
\noalign{\smallskip}
AlphaVC-P & 33.8M & 0.685 & 0.365 & 0.0010 & 1.6295 \\
\noalign{\smallskip}
\hline
\end{tabular}
}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\subsection{Ablation Study and Analysis}
\subsubsection{Frame Analysis}
We use three types of frame in AlphaVC:I-frame, cI-frame and P-frame.
To justify this approach and evaluate each type of frame,
we train two additional models AlphaVC-P and AlphaVC-cI.
AlphaVC-P only includes I-frame and P-frame, and the GoP size is the same with AlphaVC in the test phase.
AlphaVC-cI only includes I-frame and cI-frame, and there is no group in AlphaVC-cI, I-frame is only used in the first frame and all subsequent frames are cI-frames.
The R-D performance is shown in Fig.~\ref{fig:curve-frames}(a), AlphaVC-P achieves comparable performance with VTM\_IPP, and AlphaVC-cI only achieves comparable performance with HM\_IPP.
The reason may be that cI-frame utilizes reference frames in a more implicityly way: as the condition of entropy.
The reason is that, although the cI-frame is not good enough, it is stable and has no accumulated error as shown in Fig.~\ref{fig:curve-frames}(b).
By combining these two types of frame, AlphaVC achieves better R-D performance for the following two reasons:
\begin{enumerate}
\item The accumulated error of P-frame in AlphaVC is smaller than the P-frame in AlphaVC-P. (see in Fig.~\ref{fig:curve-frames}(b)).
\item The performance of cI-frame is much better than I-frame (see in Fig.~\ref{fig:curve-frames}, similar distortion with smaller rate).
\end{enumerate}
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{figures/curve-frames-eps-converted-to.pdf}
\caption{
Comparison with each type of frame in AlphaVC.
AlphaVC-P only include P-frame and I-frame, the GoP size is 20 samed as AlphaVC.
AlphaVC-cI only include cI-frame and I-frame, only the first frame uses the I-frame.
(a): R-D performance of AlphaVC, AlphaVC-P and AlphaVC-cI under PSNR on HEVC class B dataset.
(b): Example of performance comparison for each type of frame, the tested sequence is BQTerrace in class B.
The solid line indicates the curve of distortion, the dashed line indicates the curve of rate.
}
\label{fig:curve-frames}
\end{figure}
\subsubsection{Effectiveness of Different Components.}
We demonstrate the effectiveness of our proposed components with AlphaVC-P as the anchor.
We gradually remove the P2F MP, Skip in $\bf{\hat{m}}$ and Skip in $\bf{\hat{r}}$ from AlphaVC-P.
Note that, without P2F MP, the current feature and reference feature will be fed to the motion compression module directly.
The BD-Rate savings against AlphaVC-P are presented in Table~\ref{table:ablation}(b).
Moreover, a more intuitive analysis for the proposed methods is shown in Fig.~\ref{fig:skip-prob}.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/skip-prob-eps-converted-to.pdf}
\caption{
Analysis of methods.
(a): Two adjacent original frames of HEVC classB BasketballDrive.
(b): Left/Right: The compressed motion wo/w our motion prediction module.
(c): Visualization of variance of gaussian distortion $\bf{\sigma}$ and error after replacement.
(d): Example result of the average skip ratio and arithmetic decoding time at 4 different bit rates,
the ratio is calculated by skipped elements / total elements.
The motion and residual latents are shown in the red and yellow curve, respectively.
The solid and dotted curves represent ratio and time, respectively.
The number on curves indicates bit-rate(BPP).
}
\label{fig:skip-prob}
\end{figure}
As shown in Table~\ref{table:ablation}(b), P2F MP brings 10.4\% BD-rate saving.
From Fig.~\ref{fig:skip-prob}(b), we can see that the compressed motion with P2F MP is more accurate and with smaller entropy.
\setlength{\tabcolsep}{4pt}
\begin{table}[htb]
\caption{
Effectiveness of our different components.
The BD-rate values are computed under PSNR on HEVC class B dataset.
}
\label{table:ablation}
\begin{subtable}{0.45\linewidth}
\caption{}
\centering
\resizebox{0.8\textwidth}{!}{
\begin{tabular}{lccc}
\hline\noalign{\smallskip}
I-frame & \checkmark & \checkmark & \checkmark \\
P-frame & \checkmark & \checkmark & \\
cI-frame & \checkmark & & \checkmark \\
\hline\noalign{\smallskip}
BD-Rate & 0\% & 21.4\% & 92.7\% \\
\hline\noalign{\smallskip}
\end{tabular}
}
\end{subtable}
\begin{subtable}{0.55\linewidth}
\caption{}
\centering
\resizebox{0.85\textwidth}{!}{
\begin{tabular}{lcccc}
\hline\noalign{\smallskip}
P2F MP & \checkmark & & & \\
Skip in M. & \checkmark & \checkmark & & \\
Skip in R.& \checkmark & \checkmark & \checkmark & \\
\hline\noalign{\smallskip}
BD-Rate & 0\% & 10.4\% & 18.6\% & 37.5\% \\
\hline\noalign{\smallskip}
\end{tabular}
}
\end{subtable}
\end{table}
\setlength{\tabcolsep}{1.4pt}
To analyze Skip,
we first explore the relationship between the replacement error, and the variance of Gaussian distribution as shown in Fig.~\ref{fig:skip-prob}(c).
Notice that the replacement error is highly correlated with variance, and elements with smaller variance have small errors.
Therefore, skipping the entropy coding of these elements will not cause any loss, and may even improve performance.
Due to the smoothness of motion information,
the Skip ratio of motion latents is as high as 90\% at each quality level as shown in Fig.~\ref{fig:skip-prob}(d),
The Skip ratio of residual latents gradually increases (60\% -- 90\%) with the decrease of quality.
With the number of skipped elements increases, we can clearly see in Fig.~\ref{fig:skip-prob}(d) that the runtime of entropy coding on CPU is greatly reduced.
In addition, as shown in Table~\ref{table:ablation}(b), the probability-based skip entropy method can also improve performance obviously.
\section{Conclusion}
This paper proposed a high-performance and efficient learned video compression approach named AlphaVC.
Specifically, we designed a new coding mode including three types of frame: I-frame, P-frame, and cI-frame,
to reduce the bit rate of I-frame and mitigate the accumulative error.
We then proposed two efficient techniques:
P2F MP for improving the accuracy of inter-frame prediction at the encoder side,
and Skip for reducing entropy and speeding up runtime.
Experimental results show that AlphaVC outperforms H.266/VVC in terms of PSNR by 28\% under the same configuration,
meanwhile AlphaVC has the comparable decoding time compared with VTM.
To the best of our knowledge, AlphaVC is the first learned video compression scheme achieving such a milestone result that
outperforms VTM-IPP over the entire bitrate range and on all common test datasets.
We believe that our proposed AlphaVC provides some novel and useful techniques that
can help researcheres to further develop the next generation video codecs with more powerful compression.
\clearpage
\bibliographystyle{splncs04}
\section{Introduction}
This document serves as an example submission. It illustrates the format
we expect authors to follow when submitting a paper to ECCV.
At the same time, it gives details on various aspects of paper submission,
including preservation of anonymity and how to deal with dual submissions,
so we advise authors to read this document carefully.
\section{Initial Submission}
\subsection{Language}
All manuscripts must be in English.
\subsection{Paper length}
Papers submitted for review should be complete.
The length should match that intended for final publication.
Papers accepted for the conference will be allocated 14 pages (plus additional pages for references) in the proceedings.
Note that the allocated 14 pages do not include the references. The reason for this policy
is that we do not want authors to omit references for sake of space limitations.
Papers with more than 14 pages (excluding references) will be rejected without review.
This includes papers where the margins and
formatting are deemed to have been significantly altered from those
laid down by this style guide. Do not use the TIMES, or any other font than the default. The reason such papers will not be reviewed is that there is no provision for supervised revisions of manuscripts. The reviewing process cannot determine the suitability of the paper for presentation in 14 pages if it is reviewed in 16.
\subsection{Paper ID}
It is imperative that the paper ID is mentioned on each page of the manuscript.
The paper ID is a number automatically assigned to your submission when
registering your paper submission on the submission site.
All lines should be numbered in the initial submission, as in this example document. This makes reviewing more efficient, because reviewers can refer to a line on a page. Line numbering is removed in the camera-ready.
\subsection{Mathematics}
Please number all of your sections and displayed equations. Again,
this makes reviewing more efficient, because reviewers can refer to a
line on a page. Also, it is important for readers to be able to refer
to any particular equation. Just because you didn't refer to it in
the text doesn't mean some future reader might not need to refer to
it. It is cumbersome to have to use circumlocutions like ``the
equation second from the top of page 3 column 1''. (Note that the
line numbering will not be present in the final copy, so is not an
alternative to equation numbers). Some authors might benefit from
reading Mermin's description of how to write mathematics:
\url{www.pamitc.org/documents/mermin.pdf}.
\section{Policies}
To avoid confusion, in case of discrepancies between policies mentioned here and those in the ECCV 2022 webpage, the web page is the one that is updated regularly and its policies shall overrule those appearing here.
\subsection{Review Process}
By submitting a paper to ECCV, the authors agree to the review process and understand that papers are processed by the Toronto system to match each manuscript to the best possible chairs and reviewers.
\subsection{Confidentiality}
The review process of ECCV is confidential. Reviewers are volunteers not part of the ECCV organisation and their efforts are greatly appreciated. The standard practice of keeping all information confidential during the review is part of the standard communication to all reviewers. Misuse of confidential information is a severe professional failure and appropriate measures will be taken when brought to the attention of ECCV organizers. It should be noted, however, that the organisation of ECCV is not and cannot be held responsible for the consequences when reviewers break confidentiality.
Accepted papers will be published by Springer (with appropriate copyrights) electronically up to three weeks prior to the main conference. Please make sure to discuss this issue with your legal advisors as it pertains to public disclosure of the contents of the papers submitted.
\subsection{Dual and Double Submissions}
By submitting a manuscript to ECCV 2022, authors acknowledge that it has not been previously published or accepted for publication in substantially similar form in any peer-reviewed venue including journal, conference, or workshop. Furthermore, no paper substantially similar in content has been or will be submitted to a journal, another conference or workshop during the review period (March 07, 2022 – July 3, 2022). The authors also attest that they did not submit substantially similar submissions to ECCV 2022. Violation of any of these conditions will lead to rejection and the violation will be reported to the other venue or journal, which will typically lead to rejection there as well.
The goals of the dual submission policy are (i) to have exciting new work be published for the first time at ECCV 2022, and (ii) to avoid duplicating the efforts of the reviewers.
Therefore, all papers under review are checked for dual submissions and this is not allowed, independent of the page size of submissions.
For already published papers, our policy is based upon the following particular definition of ``publication''. A publication, for the purposes of the dual submission policy, is defined to be a written work longer than four pages that was submitted for review by peers for either acceptance or rejection, and, after review, was accepted. In particular, this definition of publication does not depend upon whether such an accepted written work appears in a formal proceedings or whether the organizers declare that such work ``counts as a publication''.
An arXiv.org paper does not count as a publication because it was not peer-reviewed for acceptance. The same is true for university technical reports. However, this definition of publication does include peer-reviewed workshop papers, even if they do not appear in a proceedings, if their length is more than 4 pages including citations. Given this definition, any submission to ECCV 2022 should not have substantial overlap with prior publications or other concurrent submissions. As a rule of thumb, the ECCV 2022 submission should contain no more than 20 percent of material from previous publications.
\subsection{Requirements for publication}
Publication of the paper in the ECCV 2022 proceedings of Springer requires that at least one of the authors registers for the conference and present the paper there. It also requires that a camera-ready version that satisfies all formatting requirements is submitted before the camera-ready deadline.
\subsection{Double blind review}
\label{sec:blind}
ECCV reviewing is double blind, in that authors do not know the names of the area chair/reviewers of their papers, and the area chairs/reviewers cannot, beyond reasonable doubt, infer the names of the authors from the submission and the additional material. Avoid providing links to websites that identify the authors. Violation of any of these guidelines may lead to rejection without review. If you need to cite a different paper of yours that is being submitted concurrently to ECCV, the authors should (1) cite these papers, (2) argue in the body of your paper why your ECCV paper is non trivially different from these concurrent submissions, and (3) include anonymized versions of those papers in the supplemental material.
Many authors misunderstand the concept of anonymizing for blind
review. Blind review does not mean that one must remove
citations to one's own work. In fact it is often impossible to
review a paper unless the previous citations are known and
available.
Blind review means that you do not use the words ``my'' or ``our''
when citing previous work. That is all. (But see below for
technical reports).
Saying ``this builds on the work of Lucy Smith [1]'' does not say
that you are Lucy Smith, it says that you are building on her
work. If you are Smith and Jones, do not say ``as we show in
[7]'', say ``as Smith and Jones show in [7]'' and at the end of the
paper, include reference 7 as you would any other cited work.
An example of a bad paper:
\begin{quote}
\begin{center}
An analysis of the frobnicatable foo filter.
\end{center}
In this paper we present a performance analysis of our
previous paper [1], and show it to be inferior to all
previously known methods. Why the previous paper was
accepted without this analysis is beyond me.
[1] Removed for blind review
\end{quote}
An example of an excellent paper:
\begin{quote}
\begin{center}
An analysis of the frobnicatable foo filter.
\end{center}
In this paper we present a performance analysis of the
paper of Smith [1], and show it to be inferior to
all previously known methods. Why the previous paper
was accepted without this analysis is beyond me.
[1] Smith, L. and Jones, C. ``The frobnicatable foo
filter, a fundamental contribution to human knowledge''.
Nature 381(12), 1-213.
\end{quote}
If you are making a submission to another conference at the same
time, which covers similar or overlapping material, you may need
to refer to that submission in order to explain the differences,
just as you would if you had previously published related work. In
such cases, include the anonymized parallel
submission~\cite{Authors14} as additional material and cite it as
\begin{quote}
1. Authors. ``The frobnicatable foo filter'', BMVC 2014 Submission
ID 324, Supplied as additional material {\tt bmvc14.pdf}.
\end{quote}
Finally, you may feel you need to tell the reader that more
details can be found elsewhere, and refer them to a technical
report. For conference submissions, the paper must stand on its
own, and not {\em require} the reviewer to go to a techreport for
further details. Thus, you may say in the body of the paper
``further details may be found in~\cite{Authors14b}''. Then
submit the techreport as additional material. Again, you may not
assume the reviewers will read this material.
Sometimes your paper is about a problem which you tested using a tool which
is widely known to be restricted to a single institution. For example,
let's say it's 1969, you have solved a key problem on the Apollo lander,
and you believe that the ECCV audience would like to hear about your
solution. The work is a development of your celebrated 1968 paper entitled
``Zero-g frobnication: How being the only people in the world with access to
the Apollo lander source code makes us a wow at parties'', by Zeus.
You can handle this paper like any other. Don't write ``We show how to
improve our previous work [Anonymous, 1968]. This time we tested the
algorithm on a lunar lander [name of lander removed for blind review]''.
That would be silly, and would immediately identify the authors. Instead
write the following:
\begin{quotation}
\noindent
We describe a system for zero-g frobnication. This
system is new because it handles the following cases:
A, B. Previous systems [Zeus et al. 1968] didn't
handle case B properly. Ours handles it by including
a foo term in the bar integral.
...
The proposed system was integrated with the Apollo
lunar lander, and went all the way to the moon, don't
you know. It displayed the following behaviours
which show how well we solved cases A and B: ...
\end{quotation}
As you can see, the above text follows standard scientific convention,
reads better than the first version, and does not explicitly name you as
the authors. A reviewer might think it likely that the new paper was
written by Zeus, but cannot make any decision based on that guess.
He or she would have to be sure that no other authors could have been
contracted to solve problem B. \\
For sake of anonymity, it's recommended to omit acknowledgements
in your review copy. They can be added later when you prepare the final copy.
\section{Manuscript Preparation}
This is an edited version of Springer LNCS instructions adapted
for ECCV 2022 first paper submission.
You are strongly encouraged to use \LaTeX2$_\varepsilon$ for the
preparation of your
camera-ready manuscript together with the corresponding Springer
class file \verb+llncs.cls+.
We would like to stress that the class/style files and the template
should not be manipulated and that the guidelines regarding font sizes
and format should be adhered to. This is to ensure that the end product
is as homogeneous as possible.
\subsection{Printing Area}
The printing area is $122 \; \mbox{mm} \times 193 \;
\mbox{mm}$.
The text should be justified to occupy the full line width,
so that the right margin is not ragged, with words hyphenated as
appropriate. Please fill pages so that the length of the text
is no less than 180~mm.
\subsection{Layout, Typeface, Font Sizes, and Numbering}
Use 10-point type for the name(s) of the author(s) and 9-point type for
the address(es) and the abstract. For the main text, please use 10-point
type and single-line spacing.
We recommend using Computer Modern Roman (CM) fonts, which is the default font in this template.
Italic type may be used to emphasize words in running text. Bold
type and underlining should be avoided.
With these sizes, the interline distance should be set so that some 45
lines occur on a full-text page.
\subsubsection{Headings.}
Headings should be capitalized
(i.e., nouns, verbs, and all other words
except articles, prepositions, and conjunctions should be set with an
initial capital) and should,
with the exception of the title, be aligned to the left.
Words joined by a hyphen are subject to a special rule. If the first
word can stand alone, the second word should be capitalized.
The font sizes
are given in Table~\ref{table:headings}.
\setlength{\tabcolsep}{4pt}
\begin{table}
\begin{center}
\caption{Font sizes of headings. Table captions should always be
positioned {\it above} the tables. The final sentence of a table
caption should end without a full stop}
\label{table:headings}
\begin{tabular}{lll}
\hline\noalign{\smallskip}
Heading level & Example & Font size and style\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Title (centered) & {\Large \bf Lecture Notes \dots} & 14 point, bold\\
1st-level heading & {\large \bf 1 Introduction} & 12 point, bold\\
2nd-level heading & {\bf 2.1 Printing Area} & 10 point, bold\\
3rd-level heading & {\bf Headings.} Text follows \dots & 10 point, bold
\\
4th-level heading & {\it Remark.} Text follows \dots & 10 point,
italic\\
\hline
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
Here are some examples of headings: ``Criteria to Disprove Context-Freeness of
Collage Languages'', ``On Correcting the Intrusion of Tracing
Non-deterministic Programs by Software'', ``A User-Friendly and
Extendable Data Distribution System'', ``Multi-flip Networks:
Parallelizing GenSAT'', ``Self-determinations of Man''.
\subsubsection{Lemmas, Propositions, and Theorems.}
The numbers accorded to lemmas, propositions, and theorems etc. should
appear in consecutive order, starting with the number 1, and not, for
example, with the number 11.
\subsection{Figures and Photographs}
\label{sect:figures}
Please produce your figures electronically and integrate
them into your text file. For \LaTeX\ users we recommend using package
\verb+graphicx+ or the style files \verb+psfig+ or \verb+epsf+.
Check that in line drawings, lines are not
interrupted and have constant width. Grids and details within the
figures must be clearly readable and may not be written one on top of
the other. Line drawings should have a resolution of at least 800 dpi
(preferably 1200 dpi).
For digital halftones 300 dpi is usually sufficient.
The lettering in figures should have a height of 2~mm (10-point type).
Figures should be scaled up or down accordingly.
Please do not use any absolute coordinates in figures.
Figures should be numbered and should have a caption which should
always be positioned {\it under} the figures, in contrast to the caption
belonging to a table, which should always appear {\it above} the table.
Please center the captions between the margins and set them in
9-point type
(Fig.~\ref{fig:example} shows an example).
The distance between text and figure should be about 8~mm, the
distance between figure and caption about 5~mm.
\begin{figure}
\centering
\includegraphics[height=6.5cm]{eijkel2}
\caption{One kernel at $x_s$ ({\it dotted kernel}) or two kernels at
$x_i$ and $x_j$ ({\it left and right}) lead to the same summed estimate
at $x_s$. This shows a figure consisting of different types of
lines. Elements of the figure described in the caption should be set in
italics,
in parentheses, as shown in this sample caption. The last
sentence of a figure caption should generally end without a full stop}
\label{fig:example}
\end{figure}
If possible (e.g. if you use \LaTeX) please define figures as floating
objects. \LaTeX\ users, please avoid using the location
parameter ``h'' for ``here''. If you have to insert a pagebreak before a
figure, please ensure that the previous page is completely filled.
\subsection{Formulas}
Displayed equations or formulas are centered and set on a separate
line (with an extra line or halfline space above and below). Displayed
expressions should be numbered for reference. The numbers should be
consecutive within the contribution,
with numbers enclosed in parentheses and set on the right margin.
For example,
\begin{align}
\psi (u) & = \int_{0}^{T} \left[\frac{1}{2}
\left(\Lambda_{0}^{-1} u,u\right) + N^{\ast} (-u)\right] dt \; \\
& = 0 ?
\end{align}
Please punctuate a displayed equation in the same way as ordinary
text but with a small space before the end punctuation.
\subsection{Footnotes}
The superscript numeral used to refer to a footnote appears in the text
either directly after the word to be discussed or, in relation to a
phrase or a sentence, following the punctuation sign (comma,
semicolon, or full stop). Footnotes should appear at the bottom of
the
normal text area, with a line of about 2~cm in \TeX\ and about 5~cm in
Word set
immediately above them.\footnote{The footnote numeral is set flush left
and the text follows with the usual word spacing. Second and subsequent
lines are indented. Footnotes should end with a full stop.}
\subsection{Program Code}
Program listings or program commands in the text are normally set in
typewriter font, e.g., CMTT10 or Courier.
\noindent
{\it Example of a Computer Program}
\begin{verbatim}
program Inflation (Output)
{Assuming annual inflation rates of
years};
const
MaxYears = 10;
var
Year: 0..MaxYears;
Factor1, Factor2, Factor3: Real;
begin
Year := 0;
Factor1 := 1.0; Factor2 := 1.0; Factor3 := 1.0;
WriteLn('Year
repeat
Year := Year + 1;
Factor1 := Factor1 * 1.07;
Factor2 := Factor2 * 1.08;
Factor3 := Factor3 * 1.10;
WriteLn(Year:5,Factor1:7:3,Factor2:7:3,Factor3:7:3)
until Year = MaxYears
end.
\end{verbatim}
\noindent
{\small (Example from Jensen K., Wirth N. (1991) Pascal user manual and
report. Springer, New York)}
\subsection{Citations}
The list of references is headed ``References" and is not assigned a
number
in the decimal system of headings. The list should be set in small print
and placed at the end of your contribution, in front of the appendix,
if one exists.
Please do not insert a pagebreak before the list of references if the
page is not completely filled.
An example is given at the
end of this information sheet. For citations in the text please use
square brackets and consecutive numbers: \cite{Alpher02},
\cite{Alpher03}, \cite{Alpher04} \dots
\section{Submitting a Camera-Ready for an Accepted Paper}
\subsection{Converting Initial Submission to Camera-Ready}
To convert a submission file into a camera-ready for an accepted paper:
\begin{enumerate}
\item First comment out \begin{verbatim}
\usepackage{ruler}
\end{verbatim} and the line that follows it.
\item The anonymous title part should be removed or commented out, and a proper author block should be inserted, for which a skeleton is provided in a commented-out version. These are marked in the source file as \begin{verbatim}
\end{verbatim} and \begin{verbatim}
\end{verbatim}
\item Please write out author names in full in the paper, i.e. full given and family names. If any authors have names that can be parsed into FirstName LastName in multiple ways, please include the correct parsing in a comment to the editors, below the \begin{verbatim}\author{}\end{verbatim} field.
\item Make sure you have inserted the proper Acknowledgments.
\end{enumerate}
\subsection{Preparing the Submission Package}
We need all the source files (LaTeX files, style files, special fonts, figures, bib-files) that are required to compile papers, as well as the camera ready PDF. For each paper, one ZIP-file called XXXX.ZIP (where XXXX is the zero-padded, four-digit paper ID) has to be prepared and submitted via the ECCV 2022 Submission Website, using the password you received with your initial registration on that site. The size of the ZIP-file may not exceed the limit of 60 MByte. The ZIP-file has to contain the following:
\begin{enumerate}
\item All source files, e.g. LaTeX2e files for the text, PS/EPS or PDF/JPG files for all figures.
\item PDF file named ``XXXX.pdf" that has been produced by the submitted source, where XXXX is the four-digit paper ID (zero-padded if necessary). For example, if your paper ID is 24, the filename must be 0024.pdf. This PDF will be used as a reference and has to exactly match the output of the compilation.
\item PDF file named ``XXXX-copyright.PDF": a scanned version of the signed copyright form (see ECCV 2022 Website, Camera Ready Guidelines for the correct form to use).
\item If you wish to provide supplementary material, the file name must be in the form XXXX-supp.pdf or XXXX-supp.zip, where XXXX is the zero-padded, four-digit paper ID as used in the previous step. Upload your supplemental file on the ``File Upload" page as a single PDF or ZIP file of 100 MB in size or less. Only PDF and ZIP files are allowed for supplementary material. You can put anything in this file – movies, code, additional results, accompanying technical reports–anything that may make your paper more useful to readers. If your supplementary material includes video or image data, you are advised to use common codecs and file formats. This will make the material viewable by the largest number of readers (a desirable outcome). ECCV encourages authors to submit videos using an MP4 codec such as DivX contained in an AVI. Also, please submit a README text file with each video specifying the exact codec used and a URL where the codec can be downloaded. Authors should refer to the contents of the supplementary material appropriately in the paper.
\end{enumerate}
Check that the upload of your file (or files) was successful either by matching the file length to that on your computer, or by using the download options that will appear after you have uploaded. Please ensure that you upload the correct camera-ready PDF–renamed to XXXX.pdf as described in the previous step as your camera-ready submission. Every year there is at least one author who accidentally submits the wrong PDF as their camera-ready submission.
Further considerations for preparing the camera-ready package:
\begin{enumerate}
\item Make sure to include any further style files and fonts you may have used.
\item References are to be supplied as BBL files to avoid omission of data while conversion from BIB to BBL.
\item Please do not send any older versions of papers. There should be one set of source files and one XXXX.pdf file per paper. Our typesetters require the author-created pdfs in order to check the proper representation of symbols, figures, etc.
\item Please remove unnecessary files (such as eijkel2.pdf and eijkel2.eps) from the source folder.
\item You may use sub-directories.
\item Make sure to use relative paths for referencing files.
\item Make sure the source you submit compiles.
\end{enumerate}
Springer is the first publisher to implement the ORCID identifier for proceedings, ultimately providing authors with a digital identifier that distinguishes them from every other researcher. ORCID (Open Researcher and Contributor ID) hosts a registry of unique researcher identifiers and a transparent method of linking research activities to these identifiers. This is achieved through embedding ORCID identifiers in key workflows, such as research profile maintenance, manuscript submissions, grant applications and patent applications.
\subsection{Most Frequently Encountered Issues}
Please kindly use the checklist below to deal with some of the most frequently encountered issues in ECCV submissions.
{\bf FILES:}
\begin{itemize}
\item My submission package contains ONE compiled pdf file for the camera-ready version to go on Springerlink.
\item I have ensured that the submission package has all the additional files necessary for compiling the pdf on a standard LaTeX distribution.
\item I have used the correct copyright form (with editor names pre-printed), and a signed pdf is included in the zip file with the correct file name.
\end{itemize}
{\bf CONTENT:}
\begin{itemize}
\item I have removed all \verb| \vspace| and \verb|\hspace| commands from my paper.
\item I have not used \verb|\thanks| or \verb|\footnote| commands and symbols for corresponding authors in the title (which is processed with scripts) and (optionally) used an Acknowledgement section for all the acknowledgments, at the end of the paper.
\item I have not used \verb|\cite| command in the abstract.
\item I have read the Springer author guidelines, and complied with them, including the point on providing full information on editors and publishers for each reference in the paper (Author Guidelines – Section 2.8).
\item I have entered a correct \verb|\titlerunning{}| command and selected a meaningful short name for the paper.
\item I have entered \verb|\index{Lastname,Firstname}| commands for names that are longer than two words.
\item I have used the same name spelling in all my papers accepted to ECCV and ECCV Workshops.
\item I have inserted the ORCID identifiers of the authors in the paper header (see http://bit.ly/2H5xBpN for more information).
\item I have not decreased the font size of any part of the paper (except tables) to fit into 14 pages, I understand Springer editors will remove such commands.
\end{itemize}
{\bf SUBMISSION:}
\begin{itemize}
\item All author names, titles, and contact author information are correctly entered in the submission site.
\item The corresponding author e-mail is given.
\item At least one author has registered by the camera ready deadline.
\end{itemize}
\section{Conclusions}
The paper ends with a conclusion.
\clearpage\mbox{}Page \thepage\ of the manuscript.
\clearpage\mbox{}Page \thepage\ of the manuscript.
This is the last page of the manuscript.
\par\vfill\par
Now we have reached the maximum size of the ECCV 2022 submission (excluding references).
References should start immediately after the main text, but can continue on p.15 if needed.
\clearpage
\bibliographystyle{splncs04}
|
2,877,628,090,059 | arxiv | \section{Channel Model}
\label{sec:Channel Model}
\begin{figure}
\centering
\begin{tikzpicture}[node distance=2cm,auto,>=latex]
\node at (-5,0) (source) {$W$};
\node [int, right of = source,node distance = 1.4 cm](enc){Enc.};
%
\node [sum, right of = enc,node distance = 1.7 cm](enc1){};
%
\node [right of = enc1,node distance = 1cm](p11){+};
\node [joint,right of = enc1,node distance = 1cm](p11){};
\node [right of = p11,node distance = .75 cm](p12){+};
\node [joint,right of = p11,node distance = .75 cm](p12){};
\node [int, right of = p12,node distance = 1.8 cm](dec1){Dec. 2};
\node [ right of = dec1,node distance = 1.5 cm](sink1){$\Wh_2$};
%
\node [above of = p11,node distance = 1 cm](S1){$S_2^N$};
\node [above of = p12,node distance = 1 cm](Z1){$Z_2^N$};
%
\draw[->,line width=1pt] (S1) -- (p11) ;
\draw[->,line width=1pt] (Z1) -- (p12) ;
%
\draw[-,line width=1pt] (p11) -- (p12) ;
\draw[->,line width=1pt] (p12) node[above, xshift =0.7 cm] {$Y_2^N$}-- (dec1) ;
%
%
\node [above of = p11,node distance = 2 cm](p21){+};
\node [joint,above of = p11,node distance = 2 cm](p21){};
%
\node [right of = p21,node distance = .75 cm](p22){+};
\node [joint,right of = p21,node distance = .75 cm](p22){};
%
\node [int, right of = p22,node distance = 1.8 cm](dec2){Dec. 1};
\node [right of = dec2,node distance = 1.5 cm](sink2){$\Wh_1$};
%
\node [above of = p21,node distance = 1 cm](S2){$S_1^N$};
\node [above of = p22,node distance = 1 cm](Z2){$Z_1^N$};
%
\draw[->,line width=1pt] (S2) -- (p21) ;
\draw[->,line width=1pt] (Z2) -- (p22) ;
%
\draw[-,line width=1pt] (p21) -- (p22) ;
\draw[->,line width=1pt] (p22) node[above, xshift =0.7 cm] {$Y_1^N$} -- (dec2) ;
%
%
\node [below of = p11,node distance = 1 cm](p31){};
\node [right of = p31,node distance = 1.5 cm](p32){$\vdots$};
\node [below of = sink1,node distance = 1 cm](sink3){$\vdots$};
\node [below of = dec1,node distance = 1 cm](dec3){$\vdots$};
%
\node [below of = p31,node distance = 1.5 cm](p41){+};
\node [joint,below of = p31,node distance = 1.5 cm](p41){};
%
\node [right of = p41,node distance = .75 cm](p42){+};
\node [joint,right of = p41,node distance = .75 cm](p42){};
%
\node [int, right of = p42,node distance = 1.8 cm](dec4){Dec. M};
\node [right of = dec4,node distance = 1.5 cm](sink4){$\Wh_M$};
%
\draw[-,line width=1pt] (p41) -- (p42) ;
\draw[->,line width=1pt] (p42) node[above, xshift =0.7 cm] {$Y_M^N$} -- (dec4) ;
%
\node [above of = p41,node distance = 1 cm](SM){$S_M^N$};
\node [above of = p42,node distance = 1 cm](ZM){$Z_M^N$};
%
\draw[->,line width=1pt] (SM) -- (p41) ;
\draw[->,line width=1pt] (ZM) -- (p42) ;
%
%
\draw[-,line width=1pt] (enc) node[above, xshift =1.25 cm] {$X^N$}--(enc1) {};
\draw[->,line width=1pt, bend left=90 ] (enc1) |- (p21);
\draw[->,line width=1pt] (enc1) -- (p11);
\draw[->,line width=1pt] (enc1) |- (p41);
%
\draw[->,line width=.5 pt] (source) -- (enc);
\draw[->,line width=.5 pt] (dec1) -- (sink1);
\draw[->,line width=.5 pt] (dec2) -- (sink2);
\draw[->,line width=.5 pt] (dec4) -- (sink4);
%
\draw[->,densely dotted,line width=.5 pt] (S1) -| (enc);
\draw[->,densely dotted,line width=.5 pt] (S2) -| (enc);
\draw[->,densely dotted,line width=.5 pt] (SM) -| (enc);
\end{tikzpicture}
\caption{The ``Carbon Copying on Dirty Paper'' (CCDP).}
\label{fig:CCDP channel}
\vspace{-.5 cm}
\end{figure}
\label{def:M-user multicast channel Gelfand-Pisker with additive noise and additive state}
The $M$-user ``Carbon Copying on Dirty Paper'' (CCDP) channel, also depicted in Fig. \ref{fig:CCDP channel}, is the compound GP channel
in which
the channel outputs are obtained as
\ea{
Y_m^N=X^N+c S_m^N+Z_m^N, \quad \quad m \in [1 \ldots M],
\label{eq:CCDP definition}
}
where $Z_m^N, \ \forall \ m$ is an iid Gaussian sequence with zero mean and unitary variance and $\{S_m^N, \ m \in [1 \ldots M] \}$ is an iid jointly Gaussian sequence with zero mean %
and covariance matrix $\Sigma_S$ with
\ea{
1=\var[S_1] \leq \var[S_2] \ldots \leq \var[S_M],
\label{eq:state variance assumption}
}
where \eqref{eq:state variance assumption} is assumed without loss of generality.
The transmitter has anti-causal knowledge of $\{S_m^N, \ m \in [1 \ldots M] \}$ and is subject to the average power constraint $\sum_{n=1}^N \Ebb \lsb |X_n|^2 \rsb \leq N P$.
In the following we focus on the CCDP in which each state has unitary variance and each two states have the same correlation.
We term this model as ``Carbon Copying on Dirty Paper with Equivalent States'' (CCDP-ES), since all the channel states are statistically equivalent.
The range of feasible values for the correlation $\rho$ is shown by the next lemma.
\begin{lem
\label{lem:Feasible CCDP-ES}
Let the matrix $\Sigma_S$ be equal to
\ea{
\Sigma_S = (1-\rho) \Iv_{M,M} + \rho \ones_{M,M} = \lsb \p{
1 & \rho & \ldots \\
\rho & 1 & \ddots \\
\vdots & \ddots & \ddots \\
} \rsb,
\label{eq:ES sigma matrix}
}
where $\Iv_{M,M}$ is the identity matrix of size $M$ and $\ones_{M,M}$ is the matrix of all ones of size $M \times M$,
then $\Sigma_S$ is positive defined for
\ea{
- 1/(M-1) \leq \rho \leq 1.
\label{eq:positive defined covariance matrix}
}
\end{lem}
\begin{IEEEproof}
See App. \ref{app:Feasible CCDP-ES}.
\end{IEEEproof}
\begin{lem
\label{lem:capacity decreasing scaling fading}
The capacity of the CCDP channel is decreasing in $c$.
\end{lem}
\begin{IEEEproof}
See App. \ref{app:capacity decreasing scaling fading}.
\end{IEEEproof}
This result is rather intuitive since capacity can only increase if we reduce the variance of the state.
\section{Related Results}
\label{sec:Related Results}
\medskip
\noindent
$\bullet$ {\bf Carbon Copy onto Dirty Paper (CCDP) channel.}
The channel model in \eqref{eq:CCDP definition} was originally introduced in \cite{LapidothCarbonCopying},
in which the authors derive a number of inner and outer bounds to capacity
\begin{thm}{\bf Inner and outer bounds for the 2-CCDP channel with independent states \cite[Th. 3, Th. 4]{LapidothCarbonCopying}.}
\label{th:Inner and outer bounds 2-CCDP with independent states }
Consider the CCDP channel in \eqref{eq:CCDP definition} for $M=2$ and $\Sigma_S=\Iv_{2,2}$, then capacity is upper bounded as
\ea{
\Ccal \leq R^{\rm OUT} = \lcb \p{
\f 14 \log\lb\f{1+P}{ c^2/4+1} \rb \\
\quad +\f 1 4 \lb \f{1+P+c^2+2 c \sqrt{P}}{c^2/4 +1}\rb & c^2 < 4 \\
\f 14 \log(1+P)-\f 14 \log(c^2) \\
\quad +\f 14 \log (1+P+c^2+2 c \sqrt{P} ) & c^2 \geq 4
} \rnone
\label{eq:outer lapidoth}
}
and lower bounded as
\ea{
\Ccal \geq R^{\rm IN} = \lcb \p{
\f 12 \log \lb 1 + \f {P}{c^2/2+1}\rb & c^2/2 \leq 1 \\
\f 12 \log \lb \f{P+c^2/2+1}{c^2}\rb \\
\quad + \f 14 \log \lb \f {c^2} 2 \rb & 1 \leq c^2/2 <P+1 \\
\f 14 \log (P+1) & c^2/2 \geq P+1
}
\rnone
\label{eq:inner lapidoth}
}
\end{thm}
A powerful bounding techniques is introduced in \cite{LapidothCarbonCopying} to derive the outer bound in \eqref{eq:outer lapidoth} while the inner bound in \eqref{eq:inner lapidoth} is obtained by having the transmitter pre-code against two linear combinations of the state sequences.
The outer bounding technique for the case of $M=2$ is also extended to the case of a general $M$.
\begin{thm}{\bf Outer bounds for the M-CCDP channel with independent states \cite[Eq. (31)]{LapidothCarbonCopying}.}
Consider the CCDP channel in \eqref{eq:CCDP definition} for and $\Sigma_S=\Iv_{M,M}$, then capacity can be upper bounded as
\ea{
\Ccal
& \leq R^{\rm OUT} = \f 12 \log\lb P+c^2+2c \sqrt{P} \rb - \f{M-1}{2 M} \log c^2 \nonumber \\
& \quad \quad -\f 1 {2 M}\log M - \lsb \f 1 {2M} \log \lb \f {c^2}{M(P+1)} \rb \rsb^+.
\label{eq:outer bound M independent states}
}
\end{thm}
Inner and outer bounds for the case $M=2$ are close for small values of $P$ but otherwise no capacity characterization is possible using the bounds in Th. \ref{th:Inner and outer bounds 2-CCDP with independent states }.
By generalizing the inner bound in \eqref{eq:inner lapidoth} to any $M$, we can again show that inner and outer bound are close
only for small values of $P$.
\medskip
\noindent
$\bullet$ {\bf Compound GP.}
The compound GP is a more general channel model than the CCDP:
in \cite{nair2010achievability} an attainable rate region for this model is obtained as:
\ea{
R^{\rm IN} \leq \max_{P_{X,V,U_1,U_2}} \min \lcb I_1, I_2, \f 12 \lb I_1+I_2 - I(U_1;U_2| V, S_1,S_2) \rb \rcb,
\label{eq:nair compound GP 1}
}
for $I_i = I(Y_i;U_i,V)-I(V,U_i;S_1,S_2), \ i\in\{1,2\}$. The variable $V$ is a common message decoded at both receivers, while $U_1$ and $U_2$ are pre-coded against $S_1$ and $S_2$
respectively as in the GP channel.
\section{The 2-CCDP channel with independent, equal-variance states}
\label{sec: 2WRDP}
We begin by deriving the approximate capacity for 2-CCDP-ES for $\rho=0$: this is allows us to illustrate the main inner and outer bounding techniques while deferring more complex derivations to the latter sections.
In the derivation of the inner bound, we consider the same attainable strategy as in \cite{rini2014impact}, also depicted in Fig. \ref{fig:achievableSchemeSuperposition}:
the channel input is obtained as the superposition of three codewords:
(i) a bottom common codeword, $X_{\rm SAN}^N$ ($\rm SAN$ for \emph{State As Noise}) with power $\al P$, carries the message $W_{\rm SAN}$ with rate $R_{\rm SAN}$ and treats the state sequences $S_1^N$ and $S_2^N$ as additional noise while, and
(ii) two top private codewords, $X_{\rm PAS-1}^N$,$X_{\rm PAS-2}^N$ ($\rm PAS-i$ for \emph{Pre-coded Against State $S_i^N$}),
with power $\alb P$ for $\alb=1-\al$, pre-coded against $S_1^N$ and $S_2^N$ respectively and transmitted for half of the time each.
Since the $\var[S_1]=\var[S_2]$, the codeword $X_{\rm SAN}^N$ can be decoded at both receivers simultaneously.
On the other hand, $X_{\rm PAS-i}^N$ is decoded only at receiver $i$ since it is pre-coded against the state $S_i^N$.
In order for the both decoders to decode the same amount of common information, these codewords carry the same message $W_{\rm PAS}$ at rate $R_{\rm PAS}$.
As a result of these consideration, both receivers are able to correctly decode both $W_{\rm SAN}$ and $W_{\rm PAS}$.
thus attaining the transmission rate
\ea{
R^{\rm IN} = \f 12 \log \lb 1+ \f {\al P}{ c^2 +\alb P + 1}\rb + \f 1 4 \log \lb 1+ \alb P \rb.
\label{eq:inner bound before power optimization}
}
The expression in \eqref{eq:inner bound before power optimization} can be maximized over $\al$, the ratio between the power of the common and the private codewords.
When $P+1\geq c^2$, the optimal value of $\alb$ is $(c^2-1)/P$, which corresponds to fixing the power of the private codewords to the same power
as the state sequence.
When $c^2>P+1$, instead, all the power is allocated to the private codewords and the scheme reduces to pre-coding for receiver~1 half of the time and pre-coding for receiver~2 the remaining portion of the time.
\begin{figure}
\centering
\begin{tikzpicture}
\node at (0,0) (w){};
\node [left of = w, distance=-2 cm ]{\includegraphics[width=0.52 \textwidth]{achievableSchemeSuperposition2-5A}};
\end{tikzpicture}
\vspace{-0.5 cm}
\caption{A graphical representation of the capacity approaching scheme in Th. \ref{th:Approximate capacity for the 2-CCDP with Gaussian independent states}.}
\vspace{-0.3 cm}
\label{fig:achievableSchemeSuperposition}
\end{figure}
With respect to the outer bound, we are able to improve on the result of Th. \ref{th:Inner and outer bounds 2-CCDP with independent states } using the observation in Lem. \ref{lem:capacity decreasing scaling fading}: note that the outer bound expression in \eqref{eq:outer lapidoth} for $c>4$ is not decreasing increasing in $c$, as shown Fig. \ref{fig:outer bound}.
For this reason it is possible to improve the outer bound by considering a channel with a parameter $c'=\min\{\sqrt{P+1},c\}\leq c$:
this channel has a larger capacity than the original channel but provides a tighter outer bound.
By comparing these inner and outer bound expressions, we can bound the capacity to within $1 \ \bpcu$.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\node at (0,0) {\includegraphics[trim=0cm 0cm 0cm 0cm,clip=true,scale=0.45]{outerFig1.eps}};
\vspace{-1 cm}
\node[rotate=90] at (-4.2,0.3) {{$R~ [\bpcu]$ }} ;
\node at (-1,-3.5) {$\sqrt{P+1}$};
\node at (0,-3.7) {$c$};
\draw[dashed,line width=1pt] (-1,-2.55) -- (+3.3,-2.55);
\draw[line width=.5 pt] (-1,-2.55) -- (-1,-3.2);
\node[rotate = 0, text width=3 cm] at (0,0) {\color{blue} \small Original outer bound in Th. \ref{th:Inner and outer bounds 2-CCDP with independent states }.};
\node[rotate = 0, text width=2 cm] at (2,-2) {\small optimized outer bound};
\vspace{-1 cm}
\end{tikzpicture}
\caption{ The outer bound in \eqref{eq:outer lapidoth} for $P=10$ and $c\in[0,10]$. }
\label{fig:outer bound}
\end{center}
\vspace{-.8 cm}
\end{figure}
\begin{thm}{\bf Approximate capacity for the 2-CCDP with independent, equal-variance states. \\}
\label{th:Approximate capacity for the 2-CCDP with Gaussian independent states}
Consider the 2-CCDP-IS channel in Fig. \ref{fig:CCDP channel} for $\Sigma_S=\Iv_{2,2}$, then an outer bound to capacity is
\ea{
& \Ccal \leq R^{\rm OUT} = \nonumber \\
& \lcb \p{
\f 1 2 \log(P+1)
& c^2 \leq 1 \\
%
\f 12 \log(P+c^2+1) \\
\quad - \f 14 \log(c^2+1)+\f1 2
& 1 < c^2 < P+1 \\
\f 1 4 \log(P+1)+1
& c^2 \geq P+1 \\
} \rnone
\label{eq: outer bound CCDP bernoulli}
}
and the exact capacity $\Ccal$ is to within a gap of $1 \ \bpcu$ from the outer bound in \eqref{eq: outer bound CCDP bernoulli}.
\end{thm}
\begin{IEEEproof}
See. App. \ref{app:Approximate capacity for the 2-CCDP with Gaussian independent states}.
\end{IEEEproof}
The result in Th. \ref{th:Approximate capacity for the 2-CCDP with Gaussian independent states} is somewhat expected:
when the states in the 2-CCDP channel are independent, the best strategy is to send a common codeword at a power level larger than the channel state that can be decoded at both users
and a private codeword for each user, pre-coded against the state realization in the corresponding channel output.
In order for the private codeword to communicate the same message at the two receiver, this codeword must be time-shared between the
two receivers.
The major difficulty in proving theorem is therefore in deriving an outer bound which matches this intuitively optimal solution.
Before showing the approximate capacity of the CCDP-ES, we first show how to extend of the result it Thm. \ref{th:Approximate capacity for the 2-CCDP with Gaussian independent states} from the case of $M=2$ to the case of any number of users.
\section{The M-CCDP channel with independent, equal-variance states}
\label{sec:The M-WRDP channel}
The approximate capacity of the M-CCDP channel with independent, equal-variance states is obtained through the appropriate extension of the inner and outer bounds in
Sec. \ref{sec: 2WRDP}.
A generalization of the inner bound in Fig. \ref{fig:achievableSchemeSuperposition} to the case of any number of users is rather straightforward:
we can modify the attainable strategy in Fig. \ref{fig:achievableSchemeSuperposition} as shown in Fig. \ref{fig:achievableSchemeSuperpositionM} and employ one common codeword
$X_{\rm SAN}^N$ at power $\al P$ and $M$ time-shared codewords $X_{\rm PAS-m}^N, \ m \in [1 \ldots M]$ of power $\alb P$, each pre-coded against the state sequence $S_m^N$.
All the codewords $X_{\rm PAS-m}^N$ convey the same message $W_{\rm PAS}$ and receiver $m$ decodes both the codeword $X_{\rm SAN}^N$ and $X_{\rm PAS-m}^N$ so that,
at the end of the transmission, all the decoders can correctly decode both $W_{\rm SAN}$ and $W_{\rm PAS}$.
The rate that we can attain with this strategy is
\ea{
R^{\rm IN} = \f 12 \log \lb 1+ \f {\al P}{ c^2 +\alb P + 1}\rb + \f 1 {2M} \log \lb 1+ \alb P \rb,
\label{eq:attainable rate WRDP with M}
}
which can again be maximized over $\al$.
In this case the optimal value of $\alb$ is
\ea{
\alb^*= \max \lcb 0,\min \lcb 1, \f{c^2+1-M}{P(M-1)} \rcb \rcb,
}
and the above scheme reduces to simple time-sharing and Costa pre-coding when $c^2>(M-1)(P+1)$.
The generalization of the outer bound in Th. \ref{th:Approximate capacity for the 2-CCDP with Gaussian independent states} is rather more involved:
this can be accomplished by establishing a recursive bounding of the mutual information terms obtained from Fano's inequality and
using a very carefully-chosen genie side information for each decoder.
We refer the interested reader to \cite[App. D]{RiniISITstate-16} for the complete proof.
Again, the observation in Lem. \ref{lem:capacity decreasing scaling fading} is employed to tighten the outer bound expression by optimizing over the state gain $c$.
\begin{figure}
\centering
\includegraphics[width=.52 \textwidth]{achievableSchemeSuperpositionMA}
\vspace{-1 cm}
\caption{A graphical representation of the capacity approaching scheme in Th. \ref{th:Approximate capacity for the 2-CCDP with Gaussian independent states}.}
\label{fig:achievableSchemeSuperpositionM}
\vspace{-0.8 cm}
\end{figure}
\begin{thm}{\bf Approximate capacity M-user CCDP with independent, equal-variance states. \\}
\label{th:Approximate capacity M-CCDP with Gaussian independent states}
Consider the M-CCDP-IS channel in Fig. \ref{fig:CCDP channel} for $\Sigma_S=\Iv_{M,M}$
then an outer bound to capacity is
\ea{
& \Ccal \leq R^{\rm OUT} = \nonumber \\
& \lcb \p{
\f 12 \log \lb 1+\f {P}{1+ c^2} \rb+\f 9 4& M-1 \geq c^2 \\
\f 1 {2M} \log(1+P) & \small {M-1 < c^2 \leq (M-1)(P+1)} \\
\quad +\f {M-1}{2 M } \log \lb {c^2} \rb+ \f 3 2 & \\
\f 1 {2M} \log(1+P) +2 & c^2 > (M-1)(P+1)
} \rnone
\label{eq: outer bound CCDP M}
}
and the exact capacity $\Ccal$ is to within a gap of $2.25 \ \bpcu$ from the outer bound in \eqref{eq: outer bound CCDP M}.
\end{thm}
\begin{IEEEproof}
See App. \ref{app:Approximate capacity M-CCDP with Gaussian independent states}.
\end{IEEEproof}
It is interesting to notice that pure time-sharing with no common codeword is approximatively optimal when $c^2 > (M-1)(P+1)$
that is when the state variance is roughly $M$ times stronger than the transmit power.
This occurs, intuitively, because the pre-log of the rate of the codeword $X_{\rm SAN}^N$ is $1/2$ while the pre-log of the codewords $X_{\rm PAS-m}$ is $1/2M$.
\section{The CCDP-ES channel}
\label{sec:General CCDP channel}
In this section we finally derive the approximate capacity of the CCDP-ES channel: the result relies, from a high-level viewpoint, on two observations:
(i) positive correlation among the states implies that there exists a common component which can be pre-coded against in the common codeword $X_{SAN}^N$, and
(ii) negative correlation among the states does not allow any improvement in the attainable rates with respect to the case of independent channel states.
To illustrate these points, note that the output of the 2-CCDP-ES can be equivalently expressed as
\eas{
Y_1^N & = X + c \lb a S_c^N + \sqrt{1-a} \St_1^N \rb + Z_1^N \\
Y_2^N & = X + c \lb \f{\rho} {a} S_c^N + \sqrt{1-\f {\rho^2} {a^2} } \St_2^N \rb + Z_2^N,
}{\label{eq:common noise}}
for some $S_c,\St_1,\St_2 \sim \Ncal(0,1), \ iid$, and any $a \in [-1,+1]$.
The choice $a=\sqrt{|\rho|}$ makes the term $S_c$ have the same scaling in both channel outputs: for the case of positive correlation this term can be simultaneously
pre-coded at both receivers as in the WDP channel.
For of negative correlation, since the common term appears in with opposite sign in the two outputs, no coding advantage is possible.
\begin{thm}{\bf Approximate capacity for the general 2-CCDP-ES. \\}
\label{th:Approximate capacity for the 2-CCDP correlated}
Consider the general 2-CCDP channel with state covariance matrix $\Sigma_S$ as in \eqref{eq:ES sigma matrix} for $\rho$ satisfying \eqref{eq:positive defined covariance matrix},
then capacity can be upper bounded as
\ea{
& \Ccal \leq R^{\rm OUT} = \\
& \lcb \p{
\f 1 2 \log(P+1)
& c^2 \rhob^+ \leq 1 \\
%
\f 12 \log(P+c^2 \rhob^++1) & 1 < c^2 \rhob^+ < P+1 \\
\quad - \f 14 \log(\rhob^+ c^2)+\f1 2
\\
\f 1 4 \log(P+1)+\f 12
& c^2\rhob \geq P+1
} \rnone
\label{eq: outer bound 2-CCDP correlated}
}
for $\rhob^+=1-\max\{\rho,0\}$
and the exact capacity is to within $2.25 \ \bpcu$ from the outer bound in \eqref{eq: outer bound 2-CCDP correlated}.
\end{thm}
\begin{IEEEproof}
See App. \ref{app:Approximate capacity for the 2-CCDP correlated}.
\end{IEEEproof}
The outer bound in \eqref{eq: outer bound 2-CCDP correlated} for $\rho>0$ is obtained by providing the common state $S_c$ as a side information to the receiver: the resulting channel
is then the same model as in Th. \ref{th:Approximate capacity for the 2-CCDP with Gaussian independent states} but with $c'=\rhob c$.
For the case of $\rho<0$, we rely on the fact that outer bound in Th. \ref{th:Approximate capacity for the 2-CCDP with Gaussian independent states}, when adapted to the case of correlated states, is increasing in the parameter $\rho$ and thus the case of $\rho=0$ provides a looser outer bound than the case of $\rho<0$.
The achievability proof for the case $\rho<0$ is the same as the achievability proof in Th. \ref{th:Approximate capacity for the 2-CCDP with Gaussian independent states},
since this scheme is not affected by correlation among the states.
For the case of $\rho>0$ we adapt the scheme in Th. \ref{th:Approximate capacity for the 2-CCDP with Gaussian independent states} by having the common codeword $X_{SAN}^N$
pre-coded against the common state sequence $c \sqrt{\rho} S_c^N$.
The decomposition of the channel outputs in \eqref{eq:common noise} in terms of a common component can be extended to the case of any users,
and the distinction between positive and negative pairwise correlation becomes clearer in this context.
For the case of positive correlation, a common term with variance $\rho$ can be extracted from all channel outputs by representing the channel states as
\ea{
S_m = \sqrt{\rho} S_c + \sqrt{1-\rho} \St_m \quad m \in [1 \ldots M],
\label{eq:M positive correlation}
}
for $S_c,\St_m \sim \Ncal(0,1), iid$.
As for the proof of Th. \ref{th:Approximate capacity for the 2-CCDP correlated}, the transmitter can simultaneously pre-code against the term $\sqrt{\rho} S_c$ at all the users as in
the WDP channel.
The case of negative correlation is more intriguing, since, in this case, the channel states can be represented as
\ea{
S_m= \sum_{j=m+1}^{N} \sqrt{\rho} \Sh_{mj} - \sum_{j=1}^{m-1} \sqrt{\rho}\Sh_{jm} + \sqrt{1-(N-1)\rho } \St_m,
\label{eq:negative rho case}
}
for $\Sh_{mj},\St_m \sim \Ncal(0,1) , [m,j]\in [1 \ldots M]^2, \ m>j$.
The representation in \eqref{eq:negative rho case} provides some intuition on the result in Lem. \ref{lem:Feasible CCDP-ES}: in order for the two states, $S_j$ and $S_k$ with $k>j$, to be
negatively correlated, they must share a term $\Sh_{jk}$ that does not appear in any other $S_m$.
This must be the case, otherwise this term would affect the correlation among $S_j,S_k$ and $S_m$.
Since each $S_m$ must be negatively correlated with other $N-1$ states, it must contain $N-1$ terms $\Sh_{mj}$ or $\Sh_{jm}$, each with variance $|\rho|$.
Given that the variance of $S_m$ is equal to one, we necessarily have that $|\rho|\leq 1/(N-1)$ or $\rho>-1/(N-1)$.
With the considerations in \eqref{eq:M positive correlation} and \eqref{eq:negative rho case} we can finally state the main result of the paper.
\begin{thm}{\bf Approximate capacity for the M-CCDP-ES. \\}
\label{th:Approximate capacity for the M-CCDP with Gaussian independent states}
Consider the M-CCDP channel in Fig. \ref{fig:CCDP channel} for $\Sigma_S$ as in \eqref{eq:ES sigma matrix} for $\rho$ satisfying \eqref{eq:positive defined covariance matrix},
then capacity can be upper bounded as
\ea{
& \Ccal \leq R^{\rm OUT} = \nonumber \\
& \lcb \p{
\f 12 \log \lb 1+\f {P}{1+ \rhob c^2} \rb+\f 9 4 & M-1 \geq c^2 \rhob \\
\f 1 {2M} \log(1+P) & {\small M-1 < c^2 \rhob \leq (M-1)(P+1) }\\
\quad +\f {M-1}{2 M } \log \lb {\rhob c^2} \rb+ \f 3 2 & \\
\f 1 {2M} \log(1+P) +2 & \rhob c^2 > (M-1)(P+1)
} \rnone
\label{eq:outer M CCDP rho}
}
for $\rhob=1-\max\{0,\rho\}$ and the exact capacity is to within $2.25 \ \bpcu$ from the outer bound in \eqref{eq:outer M CCDP rho}.
\end{thm}
\begin{IEEEproof}
{app:Approximate capacity for the M-CCDP with Gaussian independent states}
\end{IEEEproof}
The difficulty in extending the result of Th. \ref{th:Approximate capacity for the M-CCDP with Gaussian independent states} to the case of any correlation matrix $\Sigma_S$
lays in the fact that, in this case, decoders have different decoding capabilities and therefore there are a number of ways in which the same set of public bits can be transmitted to
each receiver.
This can be accomplished by varying the time-sharing ratio for the private codeword for each receiver in the scheme in Fig. \ref{fig:achievableSchemeSuperpositionM}.
This optimization quickly becomes untractable and deriving a matching outer bound is challenging.
\section{Conclusion}
\label{sec:Conclusion}
In this paper we study the capacity of the carbon copying onto dirty paper channel with equivalent states,
a variation of the classic dirty paper channel in which the transmitted message is decoded at $M$ receivers,
each observing a linear combination of the input, Gaussian noise and one of $M$ possible state sequences.
These state sequences are non-causally known at the transmitter and are statistically equivalent, being jointly Gaussian-distributed, with unitary variance and identical pairwise correlation.
Although inner and outer bounds to the capacity of this channel are available in the literature, no characterization of capacity was known.
We derive the capacity of this model to within 2.25 bits-per-channel-use for any channel and any pairwise correlation among the states.
In this model capacity can be approached with a rather simple strategy in which the input is composed of the superposition of two codewords:
a bottom, common codeword decoded at all users and a top, private codeword decoded at each receiver for a portion $1/M$ of the time and pre-coded against the
channel state experienced at the given receiver.
The major contribution of the paper is in the derivation of an outer bound which closely approaches this intuitive inner bound.
Despite of our progress, the capacity of the channel in which the states have any jointly Gaussian distribution remains unknown.
\bibliographystyle{IEEEtran}
|
2,877,628,090,060 | arxiv | \section{Introduction}\noindent
Oscillating spatial propagators have been the subject of several studies and carry important information about the underlying physics.
Patel~\cite{Patel:1983sc,Patel:1983qc} has argued that many-body correlations among the hadrons produced in heavy-ion collisions
may be oscillatory and has shown how those signals can be related to hadronization properties of the quark-gluon plasma (QGP)~\cite{Patel:2011dp}.
The underlying idea is that the QGP can be described as a network of quarks and of flux tubes into which the gluonic degrees of
freedom are concentrated. The flux tubes are assumed to interact mainly via three-point vertices, from here on called junctions,
where three flux tubes join together to form an $SU(3)$ singlet. It has been suggested that this system behaves like a liquid with
spatially oscillating two-body correlation between junctions and this structure might remain as the QGP hadronizes. This would be the case
if the string network breaks up via pair production rather than via coalescence of junctions. If that happens, then
the oscillatory signature should persist also in the two-body correlations of transversely outgoing hadrons.
Another situation with oscillating spatial correlation functions is in a possible crystalline phase in the QCD phase diagram,
which may occur at high density and low temperature. The existence of such a phase is supported by the exact solution of
the $(1+1)$-dimensional Gross-Neveu model at high density~\cite{Thies:2006ti,deForcrand:2006zz}. While the system described above may show
liquid-like correlations, i.e. exponential decay modulated by a cosine, the signature of a crystalline phase would be a purely
trigonometric correlation function.
When we talk about liquid-like behavior above and below, we have in mind a system where the spatial correlation functions are exponentially damped,
but with an oscillating modulation. The typical example of such behavior is the hard spheres model~\cite{Hoover:1968}: below the jamming transition
characteristic of the solid phase, one observes a liquid phase where spheres like to form spherical shells, causing oscillations in the
density-density correlation. However, both liquids and gases are fluids and
they are typically analytically connected through a cross-over, like in the case of water. A rigorous distinction is therefore ambiguous, although
in the presence of a first order transition, it is of course easy to identify the liquid as the denser, less compressible state.
To understand better when to expect such non-monotonic behavior, Ogilvie et al. have in a series of
papers~\cite{Meisinger:2010be,Nishimura:2014kla,Nishimura:2015lit} studied models which break charge conjugation $\mathcal{C}$,
but remain invariant under the combined action of $\mathcal{C}$ and complex conjugation $\mathcal{K}$. QCD at
nonzero chemical potential $\mu$ has this property, but also simpler models like the Polyakov-Nambu-Jona Lasinio (PNJL)
model with nonzero $\mu$, $SU(3)$ (Polyakov loop) spin models with nonzero $\mu$, and even the three-state Potts model
with nonzero $\mu$ have the same property. So before tackling full QCD one can hope to learn the implications of this
symmetry pattern from simpler models, which may even in some cases be mapped to limiting cases of QCD itself. It is well
known that for QCD, the expectation value of the Polyakov loop differs from the expectation value of its Hermitian conjugate,
$\expv{\Tr_FL}\neq\expv{\Tr_FL^\dagger}$ when the chemical potential $\mu$ is nonzero. However, the free energies are real
because of the $\mathcal{C}\mathcal{K}$ symmetry. As a further consequence of the breaking of $\mathcal{C}$, the transfer matrix
$T$ is not Hermitian, which means that the eigenvalues are not all necessarily real. Because of the invariance under
$\mathcal{C}\mathcal{K}$, however, if $\lambda$ is an eigenvalue of $T$, then so is $\lambda^*$, i.e. the eigenvalues are
either real or occur in complex conjugate pairs. This is interesting because it implies, in the case where complex eigenvalues
occur, that the Polyakov loop correlator is non-monotonic.
The dependence of a correlation function on the eigenvalues of the transfer matrix is most easily demonstrated with a one-dimensional
lattice model, but generalizes to any correlator with fixed momenta in the orthogonal directions. Let $\phi_i,\; i=1,\ldots,N,$ be a field living on
a circle with $N$ sites. If $T$ is the transfer matrix connecting neighboring sites, then the correlation function of $\phi$ is given by
\begin{equation}
\label{eq:corr_1d}
\expv{\phi(x)\phi(0)} = \frac{\Tr\left(T^{N-x}\phi{}T^x\phi\right)}{\Tr\left(T^N\right)} =
\frac{\Tr\Big(\Lambda^{N-x}\overbrace{P^{-1}\phi{}P}^{\tilde\phi}\Lambda^x\overbrace{P^{-1}\phi{}P}^{\tilde\phi}\Big)}{\Tr\left(\Lambda^N\right)},
\end{equation}
where $\left(P^{-1}TP\right)_{ij}=\Lambda_{ij} = \lambda_i\delta_{ij}$ is the diagonalized transfer matrix with the eigenvalues $\lambda$ of $T$
sorted in magnitude such that
$\Re{\lambda_0}\geq\Re{\lambda_1}\geq\cdots$ and $\tilde{\phi}_{ij}$ is $\phi$ in the eigenbasis of $T$. For simplicity we assume
a discrete spectrum in the description below. We also consider the
$N\to\infty$ limit. In general three scenarios are possible\footnote{The eigenvalues of the transfer matrix are either real or come in complex conjugate
pairs, since the model is invariant under the simultaneous action of charge and complex conjugation.}. Firstly, all eigenvalues can be real
and the correlator is a conventional, exponentially decaying function,
\begin{equation}
\label{eq:corr_real}
\expv{\phi(x)\phi(0)} = \sum_n\abs{\tilde{\phi}_{0n}}^2\left(\frac{\lambda_n}{\lambda_0}\right)^x =
\abs{\tilde{\phi}_{00}}^2+\abs{\tilde{\phi}_{01}}^2e^{-m_1x} + \mathcal{O}(e^{-m_2x}).
\end{equation}
Here we parametrize $\lambda_n/\lambda_0=e^{-m_n}$, with $m_n\geq0$.
Secondly, if the largest eigenvalue is real and the next two are
a complex-conjugate pair\footnote{Strictly speaking, there can be more real eigenvalues above the complex-conjugate pair,
with a consequently weaker oscillation in the correlator. We do not treat that case separately here.}, then the correlator also decays exponentially but is modulated by a cosine,
and the system behaves as a liquid,
\begin{equation}
\label{eq:corr_liquid}
\expv{\phi(x)\phi(0)} \approx \abs{\tilde{\phi}_{00}}^2 + \abs{\tilde{\phi}_{01}}^2\left(\left(\frac{\lambda_1}{\lambda_0}\right)^x+
\left(\frac{\lambda_1^*}{\lambda_0}\right)^x\right) = \abs{\tilde{\phi}_{00}}^2 + 2\abs{\tilde{\phi}_{01}}^2e^{-m_Rx}\cos{}m_Ix,
\end{equation}
where $\lambda_1/\lambda_0=e^{-m_R+im_I}$.
Finally, if the eigenvalue with the largest real part is part of a conjugate pair $\lambda_1/\lambda_0=e^{im_I}$, then the correlator is a pure
trigonometric function and a crystalline behavior is observed,
\begin{align}
\label{eq:corr_crystal}
\expv{\phi(x)\phi(0)} &\approx \frac{\lambda_0^{N-x}\sum_n\abs{\tilde{\phi}_{0n}}^2\lambda_n^x+(\lambda_0^*)^{N-x}\sum_n\abs{\tilde{\phi}_{1n}}^2\lambda_n^x}
{\lambda_0^N+(\lambda_0^*)^N}\nonumber\\
&\approx \frac{e^{-im_IN/2}\left(\abs{\tilde{\phi}_{00}}^2+\abs{\tilde{\phi}_{01}}^2e^{im_Ix}\right)+
e^{im_IN/2}\left(\abs{\tilde{\phi}_{10}}^2e^{-im_Ix}+\abs{\tilde{\phi}_{11}}^2\right)}{2\cos(m_IN/2)}\\
&=\abs{\tilde{\phi}_{00}}^2+\frac{\cos\left(m_Ix-m_IN/2\right)}{\cos(m_IN/2)}\abs{\tilde{\phi}_{01}}^2,\nonumber
\end{align}
of the form $\abs{\tilde\phi_{00}}^2+A\cos(m_Ix+\theta)$, which reveals long-range
order for arbitrarily large system size $N$. If $m_I$ is small in units of the lattice spacing then one oscillation spans several
lattice spacings and has nothing to do with the underlying structure of the lattice. For a continuous spectrum, the above categorization is
still valid. In case one, the eigenvalues are all distributed on the real line whereas in case two, the eigenvalues branch off into the complex plane
somewhere below the largest eigenvalue. The transition between cases one and two occurs at a so called disorder line. Case three is obtained when
the branching point reaches the largest real eigenvalue.
All these three cases have been found by Ogilvie et al~\cite{Meisinger:2010be}
in $1$-dimensional models, where the complete phase diagram can be obtained using transfer-matrix methods. Such a $1$-dimensional model can
for example serve as a dimensionally reduced effective models of $1+1$-dimensional QCD at finite temperature. Recently~\cite{Nishimura:2015lit}
it has been proposed that also higher dimensional models show these characteristics, based on the fact that the $1$-dimensional solution
can be seen as the first order in a character expansion. It has, however, to our knowledge, not been demonstrated with first-principles
lattice simulations that this is actually the case.
As mentioned above, it has been suggested~\cite{Patel:2011dp} that the conditions in the fireball after a heavy-ion collision might be such that
the baryon-number correlations have an oscillatory character. This conjecture is based on an effective flux-tube model
introduced in~\cite{Patel:1983sc,Patel:1983qc} which can be mapped into an $XY$-model with external magnetic fields which break
charge symmetry, such that it falls in the same category of models discussed above. Another flux-tube model, which can be
mapped into a three-state Potts model, is treated in~\cite{Condella:1999bk}. In general, the Hamiltonian and partition function
for such a flux-tube model are given by
\begin{equation}
\label{eq:ham_fluxtube}
H = \sigma\sum_{x,\nu}\abs{l_{x,\nu}} + m\sum_x\abs{q_x} + v\sum_x\abs{j_x},\quad Z = \sum_{\mathclap{\{l_{x,\nu},q_x,j_x\}}}e^{-\beta(H-\mu \sum_x q_x)},
\end{equation}
where $l_{x,\nu}$ denote flux tubes with string tension $\sigma$ living on the links, $q_x$ denote quarks with mass $m$ and chemical
potential $\mu$ living on the sites and $j_x$ denote junctions with vertex energy $v$ living on the sites. All occupation
numbers are integer valued and, depending on their allowed range and on whether $v$ is zero or nonzero, the model can be mapped to
either an $XY$ model (for $v\neq0$) or a $Z_N$ spin model (for $v=0$). The junctions $j$ call for further explanation. In $SU(N)$,
they are related to the invariant
$\epsilon$-tensor, i.e. $N$ flux lines emanating from $N$ (anti-)quarks join at a junction and form an $SU(N)$ singlet, and thus the
(anti-)quarks together with the flux lines are identified with a (anti-)baryon.
In this report we study the $\mathbb{Z}_3$ spin model with nonzero chemical potential $\mu$ in 1 and 3 dimensions and show that
a complex mass spectrum can occur in both cases. The rest of the report is organized as follows. In section~\ref{sec:model} we
define the model and review the solution in 1 dimension using the transfer matrix as well as the approximate solution in any dimension
using Extended Mean Field Theory (EMFT)~\cite{Akerlund:2014mea}. In section~\ref{sec:results} we present our Monte Carlo results
and then we draw our conclusions in section~\ref{sec:conclusions}.
\section{Model}\noindent
\label{sec:model}
The model we will be studying is the three-states Potts model with nonzero chemical potential or, more accurately, the $\ensuremath{\mathbb{Z}}_3$ spin model with
complex external fields\footnote{It may be worth pointing out that this type of model is often called a 3-state Potts model.
This is not entirely accurate since the $\mathbb{Z}_3$ spin model~\eqref{eq:S_potts} is only equivalent to a 3-state Potts model if $h_I=0$.}.
In $d$ dimensions, it can be seen as the crudest approximation of $(d+1)$-dimensional QCD in the static-dense limit.
The action is given by
\begin{equation}
\label{eq:S_potts}
S = -\beta\sum_{\expv{i,j}}\left(P_iP^\dagger_j + P^\dagger_iP_j\right) - 2\sum_i\left(h_R\Re{}P_i + ih_I\Im{}P_i\right),
\end{equation}
and the $\mathbb{Z}_3$ spins $P\in\{1,e^{i\frac{2\pi{}}{3}},e^{-i\frac{2\pi{}}{3}}\}$ at each site represent the center of the Polyakov loops $\Tr_FL$.
The usual interpretation of the external fields is $h_R = e^{-M/T}\cosh(\mu/T), h_I = e^{-M/T}\sinh(\mu/T)$, where $M$ and $\mu$ are
the mass and chemical potential of the quarks respectively~\cite{Alford:2001ug}, but as mentioned in the introduction, it is also possible to map
$(\beta,h_R,h_I)$ of eq.~\eqref{eq:S_potts} into $(\sigma,m,\mu)$ of eq.~\eqref{eq:ham_fluxtube}, as described in~\cite{Condella:1999bk} (see eqs.~(14-18)).
We will primarily use the first mapping but will evaluate the results also in the light of the second one. Note, however, that the mapping between
eq.~\eqref{eq:ham_fluxtube} and eq.~\eqref{eq:S_potts} is not possible for all parameter values (see Fig.~\ref{fig:pd_3d_z3_emft}).
In the formulation \eqref{eq:S_potts} the action is complex, and the model clearly suffers from a sign problem, but as long as $h_R,h_I\in\mathbb{R}$
and $h_R>\abs{h_I}$, which corresponds to the physical case of $M,\mu\in\mathbb{R}$, there exists a sign-problem-free
representation\footnote{This is essentially going back to the representation in terms of flux-tube variables}
that can be sampled by a worm algorithm. The model can however be interesting in its own right also in the unphysical region $h_I>h_R$,
but it is a shortcoming that it does not have a continuum limit in three dimensions, which could make it harder to clearly separate the
lattice spacing $a$ from the correlation length $\xi$ and in extension, the wavelength $\lambda$ of the oscillations we are looking for.
In one dimension the model can be solved for general external fields using a transfer-matrix method and we can use EMFT to obtain an approximate
solution in any number of dimensions.
\subsection{Transfer matrix}\noindent
\label{sec:transfer_matrix}
In $1d$ the partition function of a chain of $N$ $\ensuremath{\mathbb{Z}}_3$ spins with periodic boundary conditions is given by
\begin{equation}
\label{eq:Z_1d}
Z = \Tr{}T^N, \quad T \propto \begin{pmatrix} e^{2\beta+2h_R} & e^{-\beta+\frac{h_R}{2}}e^{i\frac{\sqrt{3}h_I}{2}} & e^{-\beta+\frac{h_R}{2}}e^{-i\frac{\sqrt{3}h_I}{2}} \\
e^{-\beta+\frac{h_R}{2}}e^{i\frac{\sqrt{3}h_I}{2}} & e^{2\beta-h_R}e^{i\sqrt{3}h_I} & e^{-\beta-h_R}\\
e^{-\beta+\frac{h_R}{2}}e^{-i\frac{\sqrt{3}h_I}{2}} & e^{-\beta-h_R} & e^{2\beta-h_R}e^{-i\sqrt{3}h_I}\end{pmatrix}
\end{equation}
where $T$ is the transfer matrix. It is easy to verify that the characteristic polynomial of $T$ is a cubic polynomial with real coefficients
so there are either three real roots or one real root and a pair of complex conjugate roots, as claimed above. For a given $\beta$, it is now
straightforward to determine the phase diagram which contains the three phases described in the introduction. The phase diagram at fixed
$\beta=0.08$ can be seen in Fig.~\ref{fig:pd_1d_z3_tm}. The color coding and labels are as follows: I (blue) marks the region where all eigenvalues
of $T$ are real. In Ia they are all positive and the connected correlator is a pure sum of exponentials. In Ib two eigenvalues are negative (the product of
all three, i.e. the determinant of $T$, is always positive) and the connected correlator is in general a sum of two oscillating functions with wavelength 2,
due to factors $(-1)^x$.
Depending on how the signs and magnitudes of the eigenvalues are distributed, this may or may not be detectable on a discrete lattice.
II (green) denotes the region where the largest eigenvalue is real and the other two are a complex conjugate pair. The connected correlator is a
cosine-modulated exponential, this is characteristic of a liquid. III (red) marks the region where the complex conjugate pair is larger in magnitude than the
real eigenvalue and the connected correlator at long distance is a pure trigonometric function, this is the long-range order characteristic of a crystal.
The two black lines bound the wedge
where $h_R>\abs{h_I}$ and mark the region where the flux-variables representation is sign-problem free and the worm algorithm can be used.
It is evident that the crystalline phase is out of reach of the worm algorithm but some parts of the liquid phase lie within the physical region
$h_R>\abs{h_I}$, so that the non-monotonic behavior of the connected correlator there can be reproduced by lattice simulations. Initially the transfer-matrix
method is only defined for integer separations but it is straight forward to extend it to any real separation via the matrix power-function.
In the liquid phase, the connected correlator is given by eq.~\eqref{eq:corr_liquid}, which is made periodic ($\exp\to\cosh$) at finite $N$ to obtain
\begin{align}
\label{eq:corr_exact}
\langle{}f(P(x))f(P(0))^\dagger\rangle_c = &\,a_f\left(\cosh\left(m_R\left(x-\frac{N}{2}\right)\right)\cos\left(m_I\left(x-\frac{N}{2}\right)\right)
\cos\left(\phi_f\right)\right.\\
&\phantom{a_f\Bigg(}\left.+\sinh\left(m_R\left(x-\frac{N}{2}\right)\right)\sin\left(m_I\left(x-\frac{N}{2}\right)\right)\sin\left(\phi_f\right)\right),\nonumber
\end{align}
where $f(P)$ is either $P,\Re{}P$ or $\Im{}P$. The parameters $a_f$ and $\phi_f$ can be calculated from the eigenvectors of $T$.
These functions can be directly compared to the correlators obtained by the worm algorithm and will serve as a consistency check for the algorithm
before going on to three dimensions where no exact results are available.
\begin{figure}[htp]
\centering
\includegraphics[width=0.6\linewidth]{pd_1d}
\caption{Phase diagram of the $1d$ $\mathbb{Z}_3$ spin model in the $(h_R,h_I)$-plane for fixed $\beta=0.08$. The crystalline phase III is outside the
region of parameter space where the worm algorithm can be applied but the liquid phase II is susceptible to lattice simulations. For a more
detailed description of the phases see the text. The phase diagram is periodic in $h_I$ with period $\pi/\sqrt{3}$}
\label{fig:pd_1d_z3_tm}
\end{figure}
A comment about this ``phase diagram'' is in order. Actually, the different phases are not separated by phase transitions in the strict
sense; there is no singularity in the free energy anywhere in the $(h_R,h_I)$-plane, since the zeros of the characteristic polynomial of $T$
are smooth functions over the whole plane. Instead, the boundary of the different phases are \emph{disorder lines}, which mark a smooth change
in the characteristic of the correlator, for example from a non-oscillatory exponential decay to an oscillatory exponential decay. In general,
however, it is not necessarily so that the change from non-oscillatory to oscillatory behavior take place at a disorder line, it can also
occur at a first order transition, as is evident from for example the water-vapor transition.
\subsection{EMFT}\noindent
\label{sec:EMFT}
In more than one dimension, and especially in the physically interesting case of three dimensions, the transfer-matrix method is not practical
anymore. It is reasonable to assume that the structure of the phase diagram will remain~\cite{Nishimura:2015lit} but one is totally at
a loss when it comes to the exact location of the disorder lines. In the light of the one-dimensional results, it is unlikely that the
crystalline phase can be probed by lattice simulations, but one may hope to find evidence of a liquid phase. In this case the
three largest eigenvalues of the transfer matrix will be given by (up to a trivial overall multiplicative, real constant)
$\lambda_0 = 1, \lambda_1 = e^{-m_R-im_I}, \lambda_2 = e^{-m_R+im_I}$, where $m_R,m_I>0$ are real numbers chosen to paramterize the eigenvalues.
The decay of the spin-spin correlator will thus be governed by $\expv{P(0)P^\dagger(r)}\sim e^{-m_Rr}\cos(m_Ir)$. It becomes clear that our
prospects for detecting this characteristic behavior of the correlator depend rather sensitively on $m_R$ and $m_I$; we require a
point in phase space where $m_R$ is not too large at the same time as $m_I$ is not too small, so that the first maximum in the correlator
occurs before the signal is too damped. Much time and effort can be saved by quickly,
albeit approximately, solving the model for extended regions of parameter space. Mean field theory is one candidate which falls short
since it does not give access to the mass spectrum. EMFT~\cite{Akerlund:2015fya} on the other hand does exactly that and is thus an apt choice.
It will be useful to consider the real part $\Re{}P$ and the imaginary part $\Im{}P$ of the Potts spin $P$ as independent variables here.
Since the imaginary part of the action \eqref{eq:S_potts} is odd in $\Im{}P$, the expectation value of $i\Im{}P$ will be real and
we have $\expv{P}=\expv{\Re{}P}+\expv{i\Im{}P}\neq\expv{\Re{}P}-\expv{i\Im{}P}=\expv{P^\dagger}$. The $\mathbb{Z}_3$ spin $P$ is then decomposed
into its mean value and fluctuations around the mean,
\begin{align}
\label{eq:P_split}
P &= \expv{\Re{}P} + \delta\Re{}P+\expv{i\Im{}P} + i\delta\Im{}P,\\
P^\dagger &= \expv{\Re{}P} + \delta\Re{}P-\expv{i\Im{}P} +-i\delta\Im{}P.\nonumber
\end{align}
We now formally integrate out all fields except the one at the origin and assume that this amounts to the introduction of effective couplings
for the bilinears $\delta\Re{}P\delta\Re{}P, \delta\Im{}P\delta\Im{}P$ and $\delta\Re{}P\delta\Im{}P$~\cite{Akerlund:2014mea}. The effective
EMFT action can then be written
\begin{align}
\label{eq:S_EMFT}
S_\text{EMFT} = &-\left(\Re{}P\right)^2\Delta_1-\left(\Im{}P\right)^2\Delta_2 - 2i\Re{}P\Im{}P\Delta_3 \nonumber\\
&-2\Re{}P\left(h_R+\expv{\Re{}P}\left(2d\beta-\Delta_1\right)+\expv{i\Im{}P}\Delta_3\right)\\
&-2i\Im{}P\left(h_I+\expv{i\Im{}P}\left(2d\beta-\Delta_2\right)-\expv{\Re{}P}\Delta_3\right).\nonumber
\end{align}
So far we have not assumed anything about the variables $P$, so the effective action above is generally valid for any action of the
form~\eqref{eq:S_potts}. For $P\in\mathbb{Z}_3$ the action can be simplified slightly by using $\left(\Im{}P\right)^2=1-\left(\Re{}P\right)^2$
and $\Re{}P=-\frac{1}{2}$ whenever $\Im{}P\neq0$. We then obtain
\begin{align}
\label{eq:S_EMFT_simpl}
S_\text{EMFT}&=-\left(\Re{}P\right)^2(\Delta_1-\Delta_2)-2\Re{}P\tilde{h}_R-\frac{2}{\sqrt{3}}i\Im{}P\tilde{h}_I,\\
\tilde{h}_R &= h_R+\expv{\Re{}P}\left(2d\beta-\Delta_1\right)+\expv{i\Im{}P}\Delta_3,\\
\frac{\tilde{h}_I}{\sqrt{3}} &= h_I+\expv{i\Im{}P}\left(2d\beta-\Delta_2\right)-\left(\expv{\Re{}P}+\frac{1}{2}\right)\Delta_3.
\end{align}
Defining $\log\gamma=-\frac{3}{4}\left(\Delta_1-\Delta_2\right)-3\tilde{h}_R$, it is straightforward to calculate all expectation values of the model
\begin{align}
\expv{\Re{}P}&=\frac{1-\gamma\cos\tilde{h}_I}{1+2\gamma\cos\tilde{h}_I} &
\expv{i\Im{}P}&=\frac{\sqrt{3}\gamma\sin\tilde{h}_I}{1+2\gamma\cos\tilde{h}_I} & & \\
\expv{\left(\Re{}P\right)^2}&=\frac{1+\frac{1}{2}\gamma\cos\tilde{h}_I}{1+2\gamma\cos\tilde{h}_I} &
\expv{\left(\Im{}P\right)^2}&=-\frac{\frac{3}{2}\gamma\cos\tilde{h}_I}{1+2\gamma\cos\tilde{h}_I} &
\expv{i\Im{}P\Re{}P}&=\frac{-\frac{\sqrt{3}}{2}\gamma\sin\tilde{h}_I}{1+2\gamma\cos\tilde{h}_I}.\nonumber
\end{align}
It is obvious how to self-consistently determine the linear expectation values, whereas the bilinears may need some more explanation.
The details of their determination will reveal how a complex spectrum can arise. As usual in EMFT~\cite{Akerlund:2014mea},
we fix the effective quadratic couplings $\Delta_i$ by matching the bilinear expectation values to an approximation to the point-to-point
correlator of the full model,
\begin{equation}
\label{eq:dyson_emft}
G_{\text{EMFT},c} = \int\ensuremath{\mathrm{d}}^dk\,G_c(k) = \int\ensuremath{\mathrm{d}}^dk\,\left[G_{0,c}^{-1}(k)+\Sigma(k)\right]^{-1} \approx
\int\ensuremath{\mathrm{d}}^dk\,\left[G_{0,c}^{-1}(k)+\Sigma_\text{EMFT}\right]^{-1}.
\end{equation}
This is a matrix equation where $G_{0,c}(k)$ is the connected Green's function of the free theory. It is not immediately clear what the free theory of
a spin model is, but the approximation above is in fact valid for any choice. A good choice will be close to the model we want to study
and at the same time allow for an efficient numerical treatment. We have chosen the free model to have the same action as the original
$\mathbb{Z}_3$ model, eq~\eqref{eq:S_potts}, but with the variables $P$ ranging freely over the complex plane. With this choice the
free connected Green's function is given by $G_{0,c}^{-1} = -2\beta{}\rm{Id}_2\sum_{\nu}\cos{}k_\nu$. The self-energy $\Sigma(k)$ in eq.~\eqref{eq:dyson_emft}
then arises due to the restriction of the field to take values in $\mathbb{Z}_3$. The EMFT self-energy $\Sigma_\text{EMFT}$ is likewise identified
as the difference between the variance of eq.~\eqref{eq:S_EMFT} with $(\Re{}P,\Im{}P)\in\mathbb{R}^2$ and the variance when $P\in\mathbb{Z}_3$
and is given by $G_{\text{EMFT},c}^{-1}+\Delta$, with
\begin{align}
G_{\text{EMFT},c} &= 2\begin{pmatrix}\expv{\left(\Re{}P\right)^2} - \expv{\Re{}P}^2 & -i\left(\expv{i\Im{}P\Re{}P}-\expv{i\Im{}P}\expv{\Re{}P}\right)\\
-i\left(\expv{i\Im{}P\Re{}P}-\expv{i\Im{}P}\expv{\Re{}P}\right) & \expv{\left(\Im{}P\right)^2} + \expv{i\Im{}P}^2 \end{pmatrix},\\
\Delta &= \begin{pmatrix}\Delta_1 & i\Delta_3\\ i\Delta_3 & \Delta_2 \end{pmatrix}.
\end{align}
Hence, the final self-consistency equation becomes
\begin{equation}
\label{eq:dyson_emft_2}
G_{\text{EMFT},c} = \int\ensuremath{\mathrm{d}}^dk\,\left[G_{\text{EMFT},c}^{-1}+\Delta-2\beta{}\rm{Id}_2\sum_\nu\cos{}k_\nu\right]^{-1}.
\end{equation}
It is clear that $G_{\text{EMFT},c}^{-1}+\Delta-2\beta{}\rm{Id}_2 \equiv \beta M$ plays the role of a mass matrix and we should diagonalize it to obtain
the mass spectrum. It will also be vastly more efficient to integrate over the momenta when $M$ is diagonal. The mass matrix can be parametrized as
\begin{equation}
\label{eq:Mass_matrix}
M = \begin{pmatrix} a+b & ic\\ic&a-b\end{pmatrix},
\end{equation}
where $a,b,c\in\mathbb{R}$. The eigenvalues are then given by $m_\pm=a\pm\sqrt{b^2-c^2}$, such that if $\abs{c}>\abs{b}$ the spectrum will consist
of a pair of complex conjugated masses $m_R\pm im_I$ with $m_R = a$ and $m_I = \sqrt{c^2-b^2}$. This implies cosine-modulated exponential fall-off
in the correlators in the $\left(\Re{}P,\Im{}P\right)$ basis, as expected. By solving the model in the $(h_R,h_I)$-plane, a phase diagram
analogous to what was obtained in one dimension with the transfer matrix, Fig.~\ref{fig:pd_1d_z3_tm}, can be constructed by studying the
behavior of the masses. In Fig.~\ref{fig:pd_3d_z3_emft} we show the results for fixed $\beta=0.08$, with the most interesting features being the
disorder lines (in red), where the masses are degenerate, and the blue dashed lines where the real part of the complex masses vanishes. Beyond these lines
the momentum
integral in the self-consistency equation no longer converges, since the integrand is no longer decaying at large distances.
One may guess that with purely imaginary masses, the system would enter a crystalline phase with a purely trigonometric correlator but
there is no way to verify that using EMFT. This phase diagram can then be compared both to the mapping
$(h_R,h_I)\to(e^{-M/T}\cosh(\mu/T),e^{-M/T}\sinh(\mu/T))$ and to the alternative mapping in~\cite{Condella:1999bk}. It is found that the second case
covers a subspace of $M,\mu\in\mathbb{R}$ and there are indeed regions in parameter space where the mapping is valid and where one expects
a complex spectrum. However, in that region $h_R$ is substantially larger than $h_I$ which means in the $M,\mu$ variables that both $M$ and $\mu$
are rather small, which is presumably far away from the region where the model is expected to be a valid approximation of QCD.
\begin{figure}[htp]
\centering
\includegraphics[width=0.6\linewidth]{pd_3d}
\caption{Phase diagram of the $3d$ three-states Potts model at fixed $\beta=0.08$ obtained by EMFT. The thick red lines are disorder lines where
the mass spectrum turns complex and on the dashed blue lines the real part of the mass vanishes. Those lines bound the region of convergence
of EMFT. Inside the wedge bounded by the thin black lines the model~\eqref{eq:S_potts} is sign-problem free and the blue region marks the image
of the map from the standard $\mathbb{Z}_3$ model to the flux-tube model of~\cite{Condella:1999bk}, eq.~\eqref{eq:ham_fluxtube} with $v=0$.}
\label{fig:pd_3d_z3_emft}
\end{figure}
Now that an approximate phase diagram has been obtained, we can select points in the liquid phase which are favorable in terms of $m_R$
and $m_I$ where full Monte Carlo simulations using the worm algorithm will be performed.
\section{Results}\noindent
\label{sec:results}
For our lattice simulations we used the flux-variables representation described in~\cite{Mercado:2012yf}, with quark occupation number
$n_x\in\{-1,0,1\}$ on each site and flux occupation number $l_{x,\nu}\in\{-1,0,1\}$ on each link. Gauss' law requires that the flux at each
site is a multiple of three. Allowed configurations consist of flux-tube networks with or without attached quarks. If there are no quarks
attached the flux-network can be thought of as a glueball. There are also neutral networks with any number of quarks and an equal number
of anti-quarks attached, for example networks connecting one quark with an anti-quark can be thought of as mesons. The third possibility
is to have a surplus of $3n$ (anti-)quarks. This is equivalent to having the junctions of the network to sum up to $n$, we say that the
network has junction charge $n$. These charged networks are associated with baryons.
The worm algorithm generates a Markov chain of allowed configurations by temporarily violating the constraint, something which can be
exploited to obtain improved estimators for spin-spin correlation functions. In addition to the usual $\expv{P(0)P^\dagger(x)}$ we use
a modification introduced in~\cite{Rindlisbacher:2015xku,Rindlisbacher:2016zht}, which allows us access to improved estimators of also
$\expv{\Re{}P(0)\Re{}P(x)}$ and
$\expv{\Im{}P(0)\Im{}P(x)}$. This is crucial because the best signal-to-noise ratio will be found in the correlator of the imaginary
parts of the spins, since it has the smallest constant background. We first reproduced the results obtained by the transfer matrix
method in one dimension in order to verify that the algorithm was properly implemented. A typical correlator in the liquid phase is
shown in Fig.~\ref{fig:1d_corr} and there is perfect agreement with the analytic result for all three propagators. It should be noted that
the real part of the
mass $m_R$ is in general always large when the imaginary part $m_I$ is of order one or larger, this makes it very difficult to resolve
the first local maximum of the correlator.
\begin{figure}[htp]
\centering
\includegraphics[width=0.49\linewidth]{1d_imim}
\includegraphics[width=0.49\linewidth]{1d_rere}
\caption{Two components of the spin-spin correlator in one dimension for $\beta=0.5,e^{-M/T}=0.02$ and $\mu/T=3.689$. There is a clear oscillation
in both correlators and the result agrees perfectly with the exact result obtained using the transfer matrix. The complex mass is
given by $m_\pm \approx 1.306\pm0.663i$.}
\label{fig:1d_corr}
\end{figure}
We also measured the junction-junction correlator on the configurations generated by the worm algorithm. The junction $j_x$ takes the value
$n$ if $3n,\,n\in\mathbb{Z}$ units of flux flow into the site $x$. With the flux
variables described above there are in general four types of junctions, depicted in Fig.~\ref{fig:junctions} but in one dimension only junction A
with one quark and two in-going fluxes (or its reverse) attached to the site is possible. In Fig.~\ref{fig:1d_junction} we show the correlation
between positive $j_+$ and negative $j_-$ junctions for two different parameter values. Here the oscillation is even clearer due to a less noisy
observable, although we do not have an improved estimator for this correlator. The dashed line is obtained by fitting the amplitude and phase in
eq.~\eqref{eq:corr_exact} while keeping the masses fixed at the exact values obtained by the transfer matrix. The mass is the same as for the
spin-spin correlator since the junction is a local object and there is only one (complex) mass in the one-dimensional case. It should be noted that these
parameter values have been selected to give a maximally clear first maximum in the oscillation. For general parameter values in the liquid phase
it is only possible to see the first minimum, while the first maximum is drowned in noise. This will be especially true in three dimensions where
the real part of the mass is larger than in the one-dimensional case.
\begin{figure}[htp]
\centering
\includegraphics[width=0.6\linewidth]{junctions_c}
\caption{The different junctions allowed in the flux-variable representation of the $\mathbb{Z}_3$ model described in the text. The red crosses
represent quarks and the lines represent the directed flux-tubes. The junction is located in the center of each network where the flux sums up to
three. Note that the quarks bounding the network may also be replaced by arbitrary larger networks of charge one. In one dimension only junction
A is possible. The three-dimensional junction D is only present in dimension three or higher.}
\label{fig:junctions}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=0.49\linewidth]{1d_junction_mu_3689}
\includegraphics[width=0.49\linewidth]{1d_junction_mu_4000}
\caption{The junction-junction correlator in one dimension for $\beta=0.5,e^{-M/T}=0.02$ and $\mu/T=3.689$ (\emph{left panel}) and for
$\beta=1.2,e^{-M/T}=0.0042$ and $\mu/T=4$ (\emph{right panel}). The signal of oscillation is even clearer than in the spin-spin correlator
since this observable is less noisy, cf. Fig.~\ref{fig:1d_corr}. The fits are given by eq.~\eqref{eq:corr_exact} with the mass fixed at
the value obtained with the transfer matrix.}
\label{fig:1d_junction}
\end{figure}
We then move to the physically interesting case of three
dimensions, and guided by the phase diagram calculated by EMFT we select a few points assumed to be in the liquid phase and look
for the corresponding signals in the correlators. Also here, however, the damping of the correlator is always strong, as is illustrated
in Fig.~\ref{fig:masses_emft}. In the left panel $m_R$ and $m_I$, obtained by EMFT, are plotted as a function of $\tanh\mu/T$ for fixed $M/T$ and $\beta$
and in the right panel $m_I$ is plotted as a function of $m_R$ to emphasize the approximately linear growth relation. Both masses increase with $\mu/T$.
\begin{figure}[htp]
\centering
\includegraphics[width=0.49\linewidth]{emft_masses}
\includegraphics[width=0.49\linewidth]{emft_masses_cmp}
\caption{\emph{Left}: the masses obtained by EMFT as a function of $\tanh\mu/T=h_I/h_R$ for fixed $\beta=0.08$ and $e^{-M/T}=0.05$.
\emph{Right}: imaginary part versus real part of the complex mass, for the same
parameters as in the left panel. The feature that the real part rises approximatively linearly with the imaginary part is generic,
as is the fairly large value of the real part at $\mu=0$. In the part of the curve to the left of the cusp, both masses are real and
their half-difference is shown as a function of their half-sum.}
\label{fig:masses_emft}
\end{figure}
As a consequence, it is typically only possible to resolve the first minimum of the oscillating correlator. In Fig.~\ref{fig:3d_corr}
we show correlators of $\Im{}P$, as a function of the Euclidean distance $r=\sqrt{x^2+y^2+z^2}$, obtained by our worm simulations for three different
values of the chemical potential $\mu/T\in\{2.0,2.5,3.3\}$
at fixed $\beta=0.08$ and $e^{-M/T}=0.05$. There is a clear staggered component in the correlators, which makes it very hard to fit the data
to a simple ansatz. This short-distance effect, whose sign is $\mu$-dependent (Fig.~\ref{fig:3d_corr}, blue vs red), stems from the size and
shape of the junctions, shown in Fig.~\ref{fig:junctions}: in $d=1$,
only A is possible. For all $\mu$ there is a clear minimum whose position moves toward zero and whose width decreases
as the chemical potential increases. This suggests that the imaginary part of the mass increases with $\mu$, as it does in one dimension and
as EMFT predicts.
Also, in neither of these correlators is it possible to see a maximum. This is not very surprising, but a discernible maximum would
be indisputable evidence of a complex spectrum.
\begin{figure}[htp]
\centering
\includegraphics[width=0.6\linewidth]{3d_imim}
\caption{The correlator of the imaginary part of the spins for three different chemical potentials $\mu$ at fixed $\beta=0.08$ and $e^{-M/T}=0.05$.
The minimum of the correlator moves towards zero and its width decreases as $\mu$ increases, suggesting that the imaginary part of the
mass increases with $\mu$, as expected. The significant staggered contribution to the correlator makes a fit to the data difficult.
The data sets are shifted vertically for clarity and the data points at $r\leq1$ are far above the shown data points, i.e. we have zoomed
in on the minimum of the correlators.}
\label{fig:3d_corr}
\end{figure}
If the mass spectrum is complex and the system behaves like a liquid then the junction-junction correlator should also show the characteristic,
oscillatory behavior seen in the $1d$ model. Since the junction-junction correlator is less noisy than the spin-spin one, one may even hope
that a maximum of the oscillating correlator can be resolved, thus establishing the complex spectrum without doubt.
In Fig.~\ref{fig:3d_jct} we show two junction-junction correlators for $\beta=0.08,e^{-M/T}$ and $\mu/T=3.3$. In the left panel we show the
correlator of the absolute values of the junctions whereas in the right panel the sign of the junctions is also taken into account.
The difference of scales of the two correlators comes from the fact that they are both normalized to one at large distances and that
$\expv{\abs{j}}\sim3\expv{j}$. To emphasize the staggered component of the correlators we plot the correlators on the two different
sub-lattices with different colors. Inspecting first the charge-insensitive correlator (\emph{left panel}) we see that there is indeed
a depletion in the density of junctions of any type within distance $[1,2.5]$ of a junction but it is not possible to tell if this minimum
is followed by a maximum. In the charge-sensitive correlator (\emph{right panel}) there is a clear maximum in the correlator in roughly the
same interval, but only in one of the sub-lattices. This strong staggered dependence is of course a lattice artifact. It should however be
noted that for smaller values of $\mu/T$ (and thus longer wavelength oscillations, cf. Fig~\ref{fig:3d_corr}) there is a clear, broader, minimum
in both correlators and on both sub-lattices, which indicates that the effect is not merely a staggered effect, although the maximum which is
predicted to follow is completely damped away. All in all, the behavior of the different correlators strongly suggests that there is a complex
mass spectrum at the investigated parameter values, and that the prediction in~\cite{Nishimura:2015lit} that the phase structure observed in
one-dimension has an analogue also in three dimensions is correct.
\begin{figure}[htp]
\centering
\includegraphics[width=0.49\linewidth]{3d_jct_abs_mu33}
\includegraphics[width=0.49\linewidth]{3d_jct_mu33}
\caption{The correlator of the absolute value of the junction number (\emph{left panel}) and the correlator of the junction number
(\emph{right panel}) for $\beta=0.08,e^{-M/T}=0.05$ and $\mu/T=3.3$ on a $12^3$ lattice. The depletion in the left correlator and the
enhancement in the right correlator around distance $2$ support the proposition that the system behaves as a liquid. However,
the strong staggered character still leaves some doubt. In the right panel, the data point at $\sqrt{2}$ and $2$ are far below $0.99$
and are omitted such that it can be clearly seen that the points at $\sqrt{3}$ and $\sqrt{5}$ are above $1$.}
\label{fig:3d_jct}
\end{figure}
Finally, we measured some statistics of the flux-tube networks and the junctions. Using the labeling of Fig.~\ref{fig:junctions} we
find that the ratio of C to D junctions is very close to $3/2$ and the ratio of B to A junctions is very close to 4, both in full
agreement with entropic arguments, i.e. their relative abundance is obtained by counting the number of possible different orientations
for each type of junction, assuming all orientations appear with equal probability.
The ratio of pure-flux junctions (C\&D) to flux-quark junction (A\&B) depends on the parameters but
for the parameter values we used the flux-quark junctions typically outnumber the pure-flux junctions by a factor 10, reflecting the energy cost
of the additional flux tube.
In Fig.~\ref{fig:cluster_stats} we show the histograms of the distribution of the flux-network size, the number of junctions in a network
and the network charge for $\beta=0.08,e^{-M/T}=0.05$ and $\mu=2.0$ on a $12^3$ lattice.
\begin{figure}[htp]
\centering
\includegraphics[width=0.6\linewidth]{cluster_stats}
\caption{Histograms of the distribution of flux-network size, the number of junctions in a network
and the network charge for $\beta=0.08,e^{-M/T}=0.05$ and $\mu=2.0$ on a $12^3$ lattice.}
\label{fig:cluster_stats}
\end{figure}
\section{Conclusions}\label{sec:conclusions}\noindent
Using unbiased Monte Carlo simulations, we have shown that the $\mathbb{Z}_3$ spin model in three dimensions with charge-symmetry breaking
external fields has non-monotonic correlators, both in the original spin variables and in the flux variables, for some regions in parameter
space. This strongly suggests that the spectrum in these regions is complex and this claim is also backed up by EMFT calculations of the
three-dimensional model. Of special interest is the oscillatory nature of the
junction-junction correlator, which is the analogue of the baryon-baryon correlator in heavy-dense QCD, for some regions of parameter space.
The possibility that models with complex saddle points may have a complex mass spectrum and thus non-monotonic correlators has previously been established
analytically in the one-dimensional case
and has been argued to also hold true in higher dimensions. We have shown that the worm algorithm is capable of reproducing these results, even though
the original spin model suffers from a strong sign problem. The phase diagram in one dimension contains regions where the system behaves like
a liquid, with exponentially damped oscillations, and like a crystal with a purely oscillatory correlator. In general, it is
expected~\cite{Nishimura:2015lit} that these features carry over also to higher dimensions. For those regions of
parameter space where the model has a sign-problem free representation we have only found evidence of the liquid phase with exponentially
damped but oscillating correlations between spins, as well as junctions. We have found no evidence of a crystalline phase and it is probable that it lies
beyond the reach of the worm algorithm in three dimensions, as it does in one dimension. It should also be noted that even the liquid phase
may lie in an unphysical region of parameter space. At least according to EMFT it lies outside of the region of parameter space which
can be mapped to the more physical flux-tube model of Condella and Detar~\cite{Condella:1999bk}. A complex mass spectrum can only be found in a
parameter region where the mass $M$ of the underlying heavy quark satisfies $M\ll T$, whereas the validity of the effective description of QCD
by a Potts model requires $M\gg T$. This situation may change if the junctions are given a nonzero weight as in~\cite{Patel:2011dp}, but this
possibility has not been investigated here. However, whatever the values of the other parameters, introducing a junction weight will further damp the
signal we want to measure, making the search yet more difficult.
Our findings supports the claim that in general, it is plausible that models without charge-conjugation symmetry, but invariance under the combined
action of charge conjugation and complex conjugation, will have regions with a complex mass spectrum in their phase diagram. However, more work is
needed before precise statements can be made about whether or not this is a phenomenon which occurs under physical conditions.
This first proof of principle should encourage the study of more realistic models, and the search for experimental signals in heavy-ion
collisions as advocated in~\cite{Patel:2011dp}.
\bibliographystyle{JHEP.bst}
|
2,877,628,090,061 | arxiv | \section{Introduction}
The precise measurement of the binding and unbinding kinetics of single biological bonds has been the ambition of an active and rapidly developing field for over two decades now~\cite{Robert2007}. A multitude of experimental methods to probe bonds at the single-molecule level are currently available, and include AFM~\cite{florin1994adhesion}, optical tweezers~\cite{thoumine2000short}, magnetic tweezers~\cite{danilowicz2005dissociation}, \cite{jacob2012quantification}, laminar flow chambers~\cite{kaplanski1993granulocyte}, total internal reflection fluorescence microscopy~\cite{schneckenburger2005total}, \cite{jungmann2010single}, interferometric imaging~\cite{piliarik2014direct}, plasmonic sensing~\cite{beuwer2015stochastic} and acoustic force spectroscopy~\cite{sitters2015acoustic}. Moreover, the focus on further development of the single-molecule toolbox is projected to intensify in the direct future~\cite{VanOijen2011}. In this paper, we direct our attention to a single-molecule property for which currently few tools are available: the association kinetics of a single pair of noncovalently bonding molecules. A first complicating factor in measuring these properties is the generic difficulty to disentangle extrinsic and intrinsic factors: association is a strongly distance-dependent process that, trivially, may only occur when the two binding partners are within touching proximity. A second complicating factor is that molecular association is a stochastic process, so one needs to be able to gather statistical data of repeated events in order to be able to extract association parameters with sufficient precision. This requires a stable molecular system and a readout arrangement suited for long observation times.
\\[2mm]
Here we propose a method based on Tethered Particle Motion (TPM) to address the abovementioned problems. The method allows for separation of the encounter and association processes, it allows for repeated probing of the same system and it allows for long observation times. The method is based on measuring bond kinetics by tracking the motion of a molecularly tethered particle that can form secondary bonds with a substrate, as sketched in Fig~\ref{fig:fig1}. In its rawest form, the data collected in such an experiment consist of a time series $\vec R(t_i)$ of the particle position projected onto the substrate. We will call such a trace a {\em motion pattern}. The basic idea is straightforward: if a secondary bond is present, the motion pattern of the bead will change as it is now confined by two, rather than one, bonds to the substrate~\cite{Visser}. If no secondary bond is present, the motion pattern is that of a regular noninteracting TPM system. Thus, the motion pattern itself reports on the binding state. There is, however, more information in the time-varying motion pattern: the dynamics of the {\em switching} between the different (bound and free) motion patters reports on the kinetics of the secondary bond.
\\[2mm]
In what follows, we present first the basic concepts and definitions required to measure association kinetics by TPM, and then present Molecular Dynamics (MD) simulations of the process. The insight from these simulations permit us to devise a protocol to extract the association rates from raw experimental data. We provide proof-of-principle for our method, validating the protocol using simulated raw data.
\section{Motion of tethered particles with secondary bonds}\label{sec:motion}
Tethered particle motion is a proven tool in biophysics. It has been used to determine the transcription of RNA polymerase~\cite{DorothyA.SchaferJeffGellesMichaelP.Sheetz1991}, the persistence length of DNA~\cite{Brinkers2009} and the looping kinetics of DNA~\cite{milstein2011bead}. In all previous TPM-based research, the focus has been on properties and interactions of the tether, rather than those of the particle. For the most part, the particle has been a large (and therefore easily visualized) marker for the end of the tether.
There is, however, no reason why one should not assign further functionality to the particle itself. In particular, it will prove useful to consider particles that may form additional bonds with the substrate, besides the one effected by the tether. In what follows, we will assume the dsDNA tether not to change, and use it solely as a means to keep the functionalized particle close to the substrate and confine its random thermal motion. As explained, there is useful information in both the instantaneous magnitude and the time-dependence of the raw signal $\vec R(t_i)$. However, while all this information is undoubtedly present, a crucial question is whether $\vec R(t_i)$ may be dissected to isolate the association rate. We show that this is indeed possible.
A clear advantage of our method is that it is easily parallelized: multiple particles can be tethered to the same substrate, the measurement of their trajectories may be done robustly in wide field optics, and the change in motion pattern of even a single particle due to binding may constitute a detection and permit the determination of rates --- multiple detections obviously improving the accuracy of the method.\\[2mm]
Fig.~\ref{fig:fig1}a presents the basic concept of TPM-based single molecule measurements that we propose. The particle is in one of three states: free, encounter, or bound. The {\em free} state is one in which the particle is only bound to the substrate by the tether. The {\em encounter} state is a conformation in which the particle and the substrate are close enough for a secondary bond to form, but this bond is not actually connected. The {\em bound} state, finally, is where the secondary bond is formed. Four distinct rates characterize the transitions back and forth between the three particle states. By observing the motion pattern we may distinguish only between states where the secondary bond is not present (free, encounter) and the one where it is (bound). The entangled, distance-dependent character of this process is obvious: The transition from the free to the bound state is a two-step process, that must necessarily pass through the encounter state. An experiment where no changes in binding pattern are observed may either have a complexation rate $k_{\rm c}$ that is low with respect to the reciprocal timescale of the experiment, a dissociation rate $k_{\rm off}$ that is high with respect to the reciprocal time resolution of the experimental detection method, or an encounter rate $k_{\rm enc}$ that is so low that no opportunity for binding occurs on the timescale of the experiment. The first two relate to an intrinsic property of the single bond, the third relates to an extrinsic effect, dependent on the geometry of the particle-tether construct and its Brownian motion. Clearly it is important to separate the intrinsic and extrinsic effects. The point of this paper is to demonstrate that the contribution of the particle's Brownian motion to the overall motion can be modeled and understood using MD simulations. Applying this principle allows otherwise inaccessible single-bond properties to be extracted from raw experimental data.
\section{TPM and changing motion patterns}
To be able to put real numbers on our axes, and to remain close to experimentally feasible dimensions we will consider for the most part a 50 nm double-stranded DNA (dsDNA) tether that attaches a 1 $\mu$m diameter particle to a substrate~\cite{Visser}. In typical TPM systems the in-plane motion of this particle is tracked over time, so that a two-dimensional projection $\vec R(t_i)$ of the movement of the particles is obtained~\cite{milstein2011bead},~\cite{Fan2012}, \cite{kumar2014enhanced}. When such a particle is repeatedly imaged within several consecutive time intervals $\Delta t=t_{i+1}-t_i$, the combined result is a motion pattern like the one shown in the left panel of Fig.~\ref{fig:fig1}b.
We consider now the case where both the particle and the substrate are coated with complementary binding molecules. This results in the possibility of a tethered particle to form the secondary bond with the substrate. When such a secondary bond is formed, the motion pattern is significantly altered: It is no longer axisymmetric, and much more localized as shown in the right panel of Fig.~\ref{fig:fig1}b.\\[2mm]
The in-plane distance that the particle travels between two frames is indicative of the presence of a secondary bond. We will refer to this traveled distance as the {\em step size}, defined as $R_{step}(t) = |\vec{R}(t_{i+1}) - \vec{R}(t_i)|$. Since a secondary bond results in additional confinement of the motion of our tethered particle, the average step size is reduced by secondary bonds. This is graphically represented in the distribution of step sizes in Fig.~\ref{fig:fig1}. Moreover, the shape of the motion pattern in the bound state is the result of the specific position of the binding molecules. The typical time between formation and dissociation of these secondary bonds is the kinetic data that we aim to relate to the kinetic binding properties of the individual molecules that form the bond.\\[2mm]
A more detailed description on how the step size may be used to distinguish between states can be found in~\cite{Max}. Alternatively, one might consider the Brownian motion amplitude as indicator of the formation of a secondary bond, as was previously used to study the looping state of DNA~\cite{Fan2012}.\\[2mm]
The step size itself is a potential reporter for the presence or absence of a secondary bond. There is, however, additional information in the switches in magnitude of the step size, and this additional information has not been tapped into yet. To extract it, we must understand the temporal behavior of this system. This is why we now turn to molecular dynamics (MD) simulations.
\section{Simulation methods}
We perform Langevin dynamics simulations using LAMMPS molecular dynamics package~\cite{plimpton1995fast} with one spatially extended spherical particle, the TPM particle, that is connected to a flexible tether, which is represented by a string of $N$ point particles in a bead-spring model. Our simulation is thus a coarse-grained MD simulation to describe the motion of a TPM system.
\subsection{MD algorithm}
In the Langevin dynamics method, each particle is subject to conservative, drag and random forces, $\vec F_c$, $\vec F_d$ and $\vec F_r$, respectively, and obeys the following translational equation of motion:
\begin{equation}
M \ddot{\vec r} = \vec F_c + \vec F_d + \vec F_r, \label{eq:sum_forces}
\end{equation}
where $M$ is the particle's mass and $\ddot{\vec r}$ is the particle's acceleration. Excluded volume, bonding and angle-bend interactions are explicitly included in $\vec F_c$, as described in section~\ref{sec:interactions}. The drag force is given by $\vec F_d = - \gamma \dot{\vec r}$, where $\gamma$ is the drag coefficient and $\dot{\vec r}$ is the particle's velocity. The drag coefficient is described in more detail in section~\ref{sec:drag}. The drag force $\vec F_d$ and the random force $\vec F_r$ are both the result of the interaction with the solvent and, by extension, these forces are related. In particular, the fluctuation-dissipation theorem tells us that~\cite{Zhang2003}
\begin{equation}
\avg{\vec F_r(t) \cdot \vec F_r(t')} = 6 k_B T \gamma \delta(t-t').\label{eq:flucdis},
\end{equation}
with $k_B$ Boltzmann's constant and $T$ the temperature.\\[2mm]
For a spatially extended particle the rotational motion has to be taken into account as well. This results in the rotational counterpart of equation~\ref{eq:sum_forces}:
\begin{equation}
I \ddot{\vec \phi} = \vec \tau_c + \vec \tau_d + \vec \tau_r,
\end{equation}
where $I$ denotes the moment of inertia, $\ddot{\vec \phi}$ is the angular acceleration and the conservative, drag and random torque are given by $\vec \tau_c$, $\vec \tau_d$ and $\vec \tau_r$ respectively. Similar to the translational force, the translational drag is given by $\vec \tau_d = - \gamma_{rot} \dot{\vec \phi}$, where $\gamma_{rot}$ is the rotational drag coefficient and $\dot{\vec \phi}$ is the angular velocity. Once more, the fluctuation-dissipation theorem provides us a relation for the effects caused by the solvent, \emph{i.e.,}
\begin{equation}
\avg{\vec \tau_r(t) \cdot \vec \tau_r(t')} = 6 k_B T \gamma_{rot} \delta(t-t').
\end{equation}
\subsection{Interactions}\label{sec:interactions}
There are two main types of interactions included in our simulations: firstly, interactions that prescribe the behavior of the tether and secondly, steric exclusion effects.\\[2mm]
A bead-spring model is used to represent the tether in the simulations. These beads are held together by a harmonic bond potential given by
\begin{equation}
U_{bond} = K_b(r-r_0)^2, \label{eq:bond_potential}
\end{equation}
with bond coefficient $K_b$, $r_0$ the rest distance between two beads and $r$ the distance between two beads. To ensure that bond lengths are essentially fixed during the simulations, we choose a large value for the strength of the bond potential, $K_b = 50 k_B T/r_0^2$~\cite{Naderi2014}, where $k_B T$ is the thermal energy.
\\[2mm]
To include the limited flexibility of a typical dsDNA tether, we include an angle-bending potential
\begin{equation}
U_{angle} = K_a \theta^
\end{equation}
where $K_a$ is the angle-bending coefficient and $\theta$ is the angle between two adjacent springs. In the limit of $r_0/l_p \rightarrow 0$, the angle-bending coefficient can be related to the thermal energy $k_B T$, the persistence length $l_p$ and the rest distance $r_0$ by $K_a= \frac{k_B T l_p}{2 r_0}$~\cite{Underhill2004}. We construct our model so that $r_0/l_p \ll 1$, which justifies the use of this angle-bend potential.\\[2mm]
Moreover, three relevant steric exclusion mechanisms are present in a TPM system: tether-substrate exclusion, particle-substrate exclusion, and tether-particle exclusion. For the tether-substrate exclusion we use the repulsive part of the Lennard-Jones potential
\begin{equation}
U_{LJ}(r) = 4 \epsilon \left[ \left( \frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6\right], \quad \; \; r < r_c\label{eq:LJ}
\end{equation}
where the energy $\epsilon$ and distance $\sigma$ are the parameters that define the potential and $r_c=2^{1/6}\sigma$ so that only the repulsive part is used. $r$ describes the distance beween the interacting elements.\\[2mm]
To ensure a steep potential at the edge of the particle, an extra parameter is required. Therefore, for the interactions that involve the particle we use
\begin{equation}
U_{LJ,exp}(r) = 4 \epsilon \left[ \left( \frac{\sigma}{r-\Delta}\right)^{12} - \left(\frac{\sigma}{r-\Delta}\right)^6\right], \quad \; \; r < r_c + \Delta \label{eq:LJ_expand},
\end{equation}
as the expanded potential with extra parameter $\Delta$. \\[2mm]
For $\epsilon$ we have used a value of 100 $k_B T$ to ensure repulsion and minimize the overlap of particles. For $\sigma$ we have chosen 1 nm, which is small enough to reproduce geometrically sensible results, but large enough to create a steepness of the potential that is computationally acceptable. To model a particle with radius 500 nm, we have chosen $\Delta = 499$ nm.
\subsection{Drag coefficients}\label{sec:drag}
\subsubsection{Drag on particle}\label{sec:drag_bead}
An important feature of a relatively big particle on a small tether is that it is at all times close to the substrate, \emph{i.e.,} the distance between the substrate and the edge of the particle is much smaller than the radius of the particle. Hydrodynamic wall effects are therefore important~\cite{Brenner1961},~\cite{happel1983low}, \emph{i.e.,} the drag on a sphere is no longer given by the isotropic Stokes drag, but the drag coefficient is elevated and no longer isotropic. \\[2mm]
For a spherical particle with radius $R$ and distance $z$ between the particle center and the substrate, the parallel and perpendicular drag coefficients are given by~\cite{Scha2007}
\begin{align}
\gamma_{\parallel} &= \frac{\gamma_0}{1-\frac{9}{16}q + \frac{1}{8}q^3-\frac{45}{256}q^4-\frac{1}{16}q^5},\\
\gamma_{\bot} &= \frac{\gamma_0}{1-\frac{9}{8}q + \frac{1}{2}q-\frac{57}{100}q^4+\frac{1}{5}q^5},
\end{align}
where $q=R/z$, $\eta$ is the dynamic viscosity of the solvent and $\gamma_0 = 6 \pi \eta R$, the Stokes drag in an unbounded liquid.
\subsubsection{Drag on tether}
The tether is modeled by a bead-spring system consisting of $N_{beads}$ beads, where every bead is subject to a Stokes drag force with drag coefficient
\begin{equation}
\gamma_{teth} = 6 \pi \eta R_{hy},
\end{equation}
where $R_{hy}$ is the so-called hydrodynamic radius of the beads: an effective parameter that determines the amount of drag on the tether.\\[2mm]
In general, a polymer in solution experiences different drag in the direction parallel and perpendicular to its axis. In a TPM experiment, the ends of the tether are attached to the bead and the substrate, so that the tether predominantly moves in the direction perpendicular to its axis. Due to the fact that the drag on the particle, more than the drag on the tether, provides the dominant contribution to the observable timescales in the system and due to considerations of computational efficiency, we have approximated the drag on the tether by the drag on a cylinder in the perpendicular direction~\cite{doi1986theory}, distributed evenly across all beads without preferential direction. This leads to
\begin{equation}
\gamma_{teth} = \frac{4 \pi\eta l}{N_{beads} \ln(l/b)},
\end{equation}
where $l$ is the contour length of the tether, $b$ is the width of the cylinder and for dsDNA we use $b=2$ nm~\cite{phillips2012physical}. Since the hydrodynamic radius $R_{hy}$ is typically much smaller than the average distance of the beads to the substrate and those beads that are close to the surface move very little, we neglect the hydrodynamic surface effects on the tether and apply a homogeneous Stokes drag to each bead.
\subsection{Validation of the spatial encounter distribution and diffusion kinetics}
We now compare spatial and kinetic results from our MD simulations to results from reference methods. First, we compare our simulation results to results from previously developed Monte Carlo (MC) simulations, which yield the probability distribution of particle positions in the equilibrium state~\cite{Visser}. The spatial encounter distribution describes the probability as function of the in-plane position that a particle is within the encounter interaction range of the substrate. In Fig.~\ref{fig:benchmark}a the spatial encounter distribution is compared for the MC results and for MD results with a varying number of beads that make up the tether. We observe that for an increasing number of beads the two distributions converge. Based on this comparison, we have used a tether consisting of 10 beads in our MD simulations.
\\[2mm]
Second, we compare results from our MD simulations to the analytical expression for Brownian motion. From conventional Brownian motion theory, it is known that the in-plane mean-squared distance traveled by a free diffusing particle is given by
\begin{equation}
\langle R_{\parallel}^2 \rangle = 4 D t
\end{equation}
as opposed to the factor 6 for 3D. Here $D$ stands for the diffusion coefficient and $t$ for time.\\[2mm]
In the TPM system, the drag on the particle is increased near the surface, as described in section~\ref{sec:drag_bead}. This results in a lower effective diffusion coefficient $D_{\parallel}$. We have performed 3000 MD simulations starting out with a TPM system in upright position and we plot the simulated $\langle R_{\parallel}^2 \rangle$ and the calculated $4 D_{\parallel} t$ in one figure. The result are presented in Fig.~\ref{fig:benchmark}b.\\[2mm]
For small times (see the inset) the MD simulation results indeed correspond to the analytical relation for a free particle. For larger times differences occur, caused by the tether that pulls the particle on average closer to the substrate, which results in a higher drag and a lower diffusivity.
\section{Determining association kinetics}
\subsection{General algorithm}\label{sec:algorithm}
Now that we have introduced the relevant concepts and properties that are involved in the measurement of single-bond kinetics using TPM, we outline an algorithm that allows one to process and interpret measurement data. A schematic outline of the algorithm can be found in Fig.~\ref{fig:scheme}.
We will further expand on either of these steps in this section and we will clarify how these steps enable the extraction of single-bond data from a TPM experiment.
\\[2mm]
The starting point of our algorithm is the data of a TPM experiment. This is generally the 2-D projected position of the particle captured multiple times in a sequence of time intervals.
The initial analysis of this data aims to isolate the captured positions that correspond to the particle being in the bound state. A means of doing this is by reviewing the step size: the in-plane distance the particle travels between frames. By averaging this over multiple frames and using two separate thresholds, as described in~\cite{Max}, individual binding events can be isolated from the experimental data, an example of this can be found in Fig~\ref{fig:analysis}a.
\\[2mm]
Having extracted the frames that correspond to the bound state, these frames are put together to compose the time-independent bound motion pattern. More sophisticated metrics may be employed, but for now we characterize the bound motion pattern using three parameters: its length $L$, its width $W$ and its average distance to the central axis $D$. These parameters are indicated by arrows in Fig.~\ref{fig:analysis}b.
\vspace{2mm}
\noindent
With the bound motion patterns thus characterized, the next step is to determine the location of the binding molecules that is most likely to correspond to that particular binding event. This is where simulation data comes in: using the results from simulations with varying binding spots we can compile the functional relation between binding spot and motion pattern, allowing us to translate the values $(L,W,D)$ into values for $d_s$ and $d_p$, the distances along substrate and particle where binding most likely occurred. Thus, our simulations permit us to extract new information from experimental data.\\[2mm]
Subsequently, simulations are used once more to determine the probability $P_{\rm enc}$ that these two binding spots are within the interaction range. We consider here the case where only one molecule is present on the particle and only one on the substrate, but our method is easily extended to include different coverages. Whether this is required of course depends on the specific experimental settings, and $P_{\rm enc}$ should be determined accordingly. Finally, by acquiring the distribution of times between binding events and factoring out $P_{\rm enc}$ we can isolate the molecular complexation rate $k_{\rm c}$. If the apparent binding rate is given by $\kappa$, then we can find $k_{\rm c}$ using the relation
\begin{equation}
\kappa = P_{\rm enc} \, k_{\rm c}.
\end{equation}
\\[2mm]
In summary, by using simulation results we have created the possibility to extract otherwise inaccessible single-molecule data from TPM experiments.
\subsection{Example application}
As a proof-of-principle and as a means of demonstrating our algorithm we have constructed a mock experiment to generate pseudo-TPM data. This model requires three geometric parameters and four rates as input. The geometric parameters $L$, $W$ and $D$ describe the shape of the bound motion pattern. It uses preset values for the rates $k_{\rm enc}$, $k_{\rm sep}$, $k_{\rm c}$ and $k_{\rm off}$, and generates a motion pattern in time as output. The challenge to our analysis protocol is now to back out the value of $k_{\rm c}$ in the manner described above. The input values are listed in table~\ref{tab:mock}.
\\[2mm]
The mock model steps through time, and at every time point the system is in either the `free', `encounter' or `bound' state. Every step the system may change state and this happens with a probability determined by the relevant transition rates. \\[2mm]
In the simulations of the system with a secondary bond, the binding molecule had a Y-shape and a fully-stretched length of 15 nm. In the simulations without secondary bond we considered this 15 nm as the encounter range $d_{\rm enc}$.\\[2mm]
In the mock model, every time step a $X$- and $Y$-coordinate of the particle are generated. When the system is in the `free'-state, a random point in a circle with a radius of 220 nm is generated, \emph{i.e.,} within the typical radius of `free' motion patterns in our system. When the system is not in the free state, and thus in the `encounter'- or `bound'-state, a random point within a confining ellipse, with given $L$, $W$ and $D$, is generated. Clearly this results in points that are, on average, closer to each other in the bound state than in any of the unbound states. This can be seen in Fig.~\ref{fig:analysis}a, where the step size obtained from our mock model is represented. The data points are generated with a frame rate of 30 Hz, so the step size corresponds to the distance traveled in 0.033 s. An averaging window of 30 frames is used, so the black line corresponds to the distance traveled in 1 s.
\\[2mm]
Using the obtained step size and averaged step size, we now isolate the points of the motion pattern that are classified as belonging to the `bound' state so that we end up with a motion pattern similar to the pattern in Fig.~\ref{fig:analysis}b. As is indicated in this figure, we can extract the $L$, $W$ and $D$ from this figure. This leads to a value of $L=247$ nm, $W=141$ nm and $D=150$ nm, leading to the most probable binding spots $d_{p} = 160$ nm and $d_{s} = 200$ nm.\\[2mm]
Using simulation results of a free tethered particle we know that the probability of a binder on the substrate and on the particle being within interaction range $d_{\rm enc}$ is $P_{\rm enc} = 1.2 \cdot 10^{-4}$. This result is obtained by dividing the average number of frames in a simulation in which the two binders are within interaction range through the total number of frames in a simulation. We measure the time it takes for a free particle to become bound throughout the whole experiment to come up with a Cumulative Distribution Function (CDF). This distribution is fit to an exponential function, and the result is represented in Fig.~\ref{fig:analysis}c. From the fit we obtain the fitting parameter $\kappa=(1.7 \pm 0.3) \cdot 10^{-3}$ s$^{-1}$ (mean $\pm$ standard deviation).\\[2mm]
Factoring out the encounter probability $P_{\rm enc}$, we end up with a value of $k_{\rm c} = (1.4 \pm 0.2) \cdot 10^1$ s$^{-1}$, which is to be compared to the input value $k_{\rm c} = 1.7 \cdot 10^1$ s$^{-1}$ that was not used to generate the data. We hypothesize that the underestimation compared to the actual value may be attributed to a structural issue, also present in experiments, which is, that some events are too short lived to be resolved (shorter than one time step) and thus there is a systematic underestimation of both the bound time and the association rate. An averaging window to detect bonds of 1 s has been used, while the typical dissociation time is 10 s ($1/k_{\rm off}$). Therefore, a detectable difference between the apparent association rate and the actual association rate is expected. In further refinements, this too may be corrected for using the simulations.
\begin{table}[]
\centering
\caption{Input values for the mock model}\label{tab:mock}
\label{my-label}
\begin{tabular}{c|c|c}
parameter & value & units \\
\hline
$k_{\rm enc}$ & 1.0 & s$^{-1}$ \\
$k_{\rm sep}$ & 8.3$\cdot 10^3$ & s$^{-1}$\\
$k_{\rm c} $ & 1.7 $\cdot 10^1$ & s$^{-1}$\\
$k_{\rm off}$ & 0.1 & s$^{-1}$\\
$L$ & 247 & nm\\
$W$ & 141 & nm\\
$D$ & 150 & nm\\
\end{tabular}
\end{table}
\section{Conclusions and Outlook}
We have shown that by understanding the contribution of Brownian motion to the overall motion of a tethered particle, a TPM system allows for the probing of single bonds. We have focused on the basic principles of this novel approach. Several opportunities for future research arise. On the one hand, the understanding of the system and the corresponding simulations could be further developed. On the other hand, experimental validation of this method would be a crucial next cornerstone.\\[2mm]
The parameter that we only have a rough approximation for is the encounter distance $d_{\rm enc}$. The next step in developing a TPM based association measurement would be to increase our understanding of the encounter distance.
Two potential experiments come to mind. Firstly, one could investigate the influence of the encounter distance by using linker molecules with different lengths. The length of linker molecules between binder A and the substrate, and between binder B and the particle, will profoundly influence the encounter distance corresponding to the binder complex. If the apparent association rate $\kappa$ in an experiment with varying linker lengths $\ell$ would display comparable behavior to the encounter probability $P_{\rm enc}$ as function of the encounter distance $d_{\rm enc}$ in the simulations, that would be a strong experimental justification of our approach.
Secondly, by using several different binding molecules, a range of complexation rates $k_{\rm c}$ may be probed. In the upper limit, every encounter will lead to a bond, so for high complexation rates the apparent association rate should converge to $\kappa = P_{\rm enc}$.\\[2mm]
On the simulations side, the next step would be to incorporate the molecular complexation process. The complexation rate $k_{\rm c}$ is the result of distance-dependent physicochemical molecular interactions: charge interactions, van der Waals interactions, steric effects, hydrogen bonds, hydrophobic effects, etc. Simulations with a higher level of detail could provide more insight in the molecular processes involved in TPM experiments.\\[2mm]
In summary, what we show here is that the measurement and analysis of surface-binding data in TPM experiments provides a new window on single-molecule association dynamics, adding a novel modality to establish these important properties in experiments.\\
Given the new measurement concept proposed in this paper and the results from the simulations, we believe that the measurement and kinetic analysis of surface-binding data in TPM experiments holds promise to become a new modality for studies on single-molecule association kinetics.\\[2mm]
\section{Acknowledgments}
We thank Max Scheepers, Emiel Visser and Leo van IJzendoorn for numerous valuable discussions. We thank Emiel Visser for providing us the code for the MC simulations and Stefan Paquay for his continuous helpfulness in developing MD simulations.
|
2,877,628,090,062 | arxiv | \section*{introduction}
\label{sec:introduction}
The global geometry of a given Hilbert scheme is generally very difficult to study. Recently, the theory of Bridgeland stability has provided a new set of tools to study the geometry of these Hilbert schemes. For instance, the study of the Hilbert scheme of points on surfaces has benefited from these new tools (see \cite{ABCH13, BM14, CHW17, LZ16, MM13, Nue16, YY14}). A sensible step forward is now to apply these tools to examine families of curves contained in threefolds. The first instance of this was carried out by the last author in \cite{Sch15}, where he studies the Hilbert scheme of twisted cubics. This paper continues this investigation about curves in $\ensuremath{\mathbb{P}}^3$ and analyzes the global geometry, as well as wall-crossing phenomena, of the Hilbert scheme $\mathop{\mathrm{Hilb}}\nolimits^{4t}(\ensuremath{\mathbb{P}}^3)$, which parametrizes subschemes of $\ensuremath{\mathbb{P}}^3$ of genus $1$ and degree $4$.
A smooth curve of genus $1$ and degree $4$ in $\ensuremath{\mathbb{P}}^3$, which we refer to as an elliptic quartic, is the transversal intersection of two quadric surfaces. By considering the pencil that these quadrics generate, we realize the family of smooth elliptic quartics as an open subset of $\ensuremath{\mathbb{G}}(1,9)$, the Grassmannian of lines in the space $ |\mathcal{O}_{\ensuremath{\mathbb{P}}^3}(2)|$ of quadric surfaces in $\ensuremath{\mathbb{P}}^3$. We show that the Hilbert scheme $\mathop{\mathrm{Hilb}}\nolimits^{4t}(\ensuremath{\mathbb{P}}^3)$ is a moduli space of Bridgeland stable objects, and moreover, one of its components is related through birational transformations to the Grassmannian $\ensuremath{\mathbb{G}}(1,9)$ via wall-crossing.
Let us recall the notion of Bridgeland stability in order to state this result precisely. For classical slope stability with respect to a given polarization $H$ on a smooth projective complex variety $X$, one defines a number
$\mu_H(E) = \tfrac{H^{n-1} \cdot \mathop{\mathrm{ch}}\nolimits_1(E)}{H^n \cdot \mathop{\mathrm{ch}}\nolimits_0(E)}$
called the slope for any coherent sheaf $E \in \mathop{\mathrm{Coh}}\nolimits(X)$. A coherent sheaf is then called slope semistable if all proper non trivial subsheaves have smaller slope. For Bridgeland stability, one replaces the category of coherent sheaves with a different abelian subcategory $\ensuremath{\mathcal A} \subset D^b(X)$ and replaces the slope with a homomorphism $Z: K_0(X) \to \ensuremath{\mathbb{C}}$, mapping $\ensuremath{\mathcal A}$ to the upper half plane or the negative real line, where $K_0(X)$ is the Grothendieck group. The slope is then given by
\[
\mu(E) = -\frac{\Re{Z(E)}}{\Im{Z(E)}}
\]
for any $E \in \ensuremath{\mathcal A}$. In addition, one demands that every object in $D^b(X)$ has a canonical filtration into semistable factors called the Harder-Narasimhan filtration and the so called support property, which ensures that the set of stability conditions $\mathop{\mathrm{Stab}}\nolimits(X)$ can be naturally given the structure of a complex manifold.
We can now state our main result. Let us fix a class $v \in K_0(X)$, then there is a locally finite wall and chamber structure in $\mathop{\mathrm{Stab}}\nolimits(X)$, such that the set of semistable objects of class $v$ is constant within each chamber. Our main result describes the wall and chamber structure of a subspace of $\mathop{\mathrm{Stab}}\nolimits(\ensuremath{\mathbb{P}}^3)$ as well as the corresponding moduli spaces of semistable objects in the case of elliptic quartics in $\ensuremath{\mathbb{P}}^3$.
\newtheorem*{thm:wall_crossings}{Theorem A}
\begin{thm:wall_crossings}
Let $v = (1, 0, -4, 8) = \mathop{\mathrm{ch}}\nolimits(\ensuremath{\mathcal I}_C)$, where $C \subset \ensuremath{\mathbb{P}}^3$ is an elliptic quartic curve. There is a path $\gamma: [0,1] \to \ensuremath{\mathbb{R}}_{>0} \times \ensuremath{\mathbb{R}} \subset \mathop{\mathrm{Stab}}\nolimits(\ensuremath{\mathbb{P}}^3)$ such that the moduli spaces of semistable objects with Chern character $v$ in its image outside of walls are given in the following order.
\begin{enumerate}
\setcounter{enumi}{-1}
\item The empty space $M_0 = \emptyset$.
\item The Grassmannian $M_1 = \ensuremath{\mathbb{G}}(1,9)$ parametrizing pencils of quadrics. The only non-ideal sheaves in the moduli space come from the case, where a $2$-plane is contained in the base locus of the pencil.
\item The second moduli space $M_2$ is the blow up of $\ensuremath{\mathbb{G}}(1,9)$ along a smooth locus isomorphic to $\ensuremath{\mathbb{G}}(1,3) \times (\ensuremath{\mathbb{P}}^3)^{\vee}$ parametrizing the non-ideal sheaves in $M_1$. The exceptional divisor generically parametrizes unions of a line and a plane cubic intersecting themselves in a single point. The only non-ideal sheaves in this moduli space come from the case when the line is contained in the plane.
\item The third moduli space $M_3$ has two irreducible components $M_3^1$ and $M_3^2$. The first component $M_3^1$ is the blow up of $M_2$ along the smooth incidence variety parametrizing length two subschemes in a plane in $\ensuremath{\mathbb{P}}^3$. The second component $M_3^2$ is a $\ensuremath{\mathbb{P}}^{14}$-bundle over $\mathop{\mathrm{Hilb}}\nolimits^2(\ensuremath{\mathbb{P}}^3) \times (\ensuremath{\mathbb{P}}^3)^{\vee}$. It generically parametrizes unions of plane quartics with two points, either outside the curve or embedded. The two components intersect transversally along the exceptional locus of the blow up. The only non-ideal sheaves occur in the case where at least one of the two points is not scheme-theoretically contained in the plane.
\item The fourth moduli space $M_4$ has two irreducible components $M_4^1$ and $M_4^2$. The first component is equal to $M_3^1$. The second component is birational to $M_3^2$. The new locus parametrizes plane quartics with two points, such that exactly one point is scheme-theoretically contained in the plane.
\item The fifth moduli space is the Hilbert scheme $\mathop{\mathrm{Hilb}}\nolimits^{4t}(\ensuremath{\mathbb{P}}^3)$, which has two components: $\mathop{\mathrm{Hilb}}\nolimits_1^{4t}$ and $\mathop{\mathrm{Hilb}}\nolimits_2^{4t}$. The principal component $\mathop{\mathrm{Hilb}}\nolimits_1^{4t}$ contains an open subset of elliptic quartic curves and is equal to $M_3^1$. The second component is of dimension $23$ and is birational to $M_3^2$. Moreover, the two components intersect transversally along a locus of dimension $15$. The component $\mathop{\mathrm{Hilb}}\nolimits_2^{4t}$ differs from $M_4^2$ in the locus of plane cubics together with two points scheme-theoretically contained in the plane.
\end{enumerate}
\end{thm:wall_crossings}
\begin{figure}
\centering
\begin{overpic}[scale=0.75,unit=1 mm]{bwalls}
\put(12, 50) {$\mathop{\mathrm{Hilb}}\nolimits^{4t}(\ensuremath{\mathbb{P}}^3)$}
\put(12, 12) {$M_4$}
\put(23, 12) {$M_3$}
\put(60, 12) {$M_2$}
\put(95, 12) {$\ensuremath{\mathbb{G}}(1,9)$}
\put(130, 12) {$\emptyset$}
\end{overpic}
\caption{Wall and chamber structure in a subspace of $\mathop{\mathrm{Stab}}\nolimits(\ensuremath{\mathbb{P}}^3)$ for $\mathop{\mathrm{Hilb}}\nolimits^{4t}(\ensuremath{\mathbb{P}}^3)$ and their associated models according to Theorem A}
\label{fig:bridgeland_walls_elliptic}
\end{figure}
As a consequence of Theorem A, we obtain that the Hilbert scheme $\mathop{\mathrm{Hilb}}\nolimits^{4t}(\ensuremath{\mathbb{P}}^3)$ has two components. This is a well known fact (see \cite{CN12, Got08}). More interestingly, the previous result describes what is called the principal component, which parametrizes smooth elliptic curves along with their flat limits. We will denote this component by $\mathop{\mathrm{Hilb}}\nolimits_1^{4t}$, and our next result describes its global geometry.
\newtheorem*{thm:va92}{Theorem B}
\begin{thm:va92}[\cite{VA92}]
The closure of the family of smooth elliptic quartics in the Hilbert scheme $\mathop{\mathrm{Hilb}}\nolimits^{4t}(\ensuremath{\mathbb{P}}^3)$, is a double blowup of the Grassmannian $\ensuremath{\mathbb{G}}(1,9)$ along smooth centers.
\end{thm:va92}
A comment is in order about the previous theorem. The description of $\mathop{\mathrm{Hilb}}\nolimits_1^{4t}(\ensuremath{\mathbb{P}}^3)$ above was proved in \cite{VA92} by Vainsencher and Avritzer using classical methods. Our techniques to reprove their result are distinct, as we make use of the bounded derived category of coherent sheaves on $\ensuremath{\mathbb{P}}^3$ and Bridgeland stability.
Since the principal component $\mathop{\mathrm{Hilb}}\nolimits_1^{4t}(\ensuremath{\mathbb{P}}^3)$ is a double blowup, it is natural to ask what are the subschemes of $\ensuremath{\mathbb{P}}^3$ that the exceptional divisors parametrize and whether they span extremal rays in the cone of effective divisors $\mathop{\mathrm{Eff}}\nolimits(\mathop{\mathrm{Hilb}}\nolimits_1^{4t})$. Proposition \ref{prop:E1_E2_description}, and the following result answer these two questions. Consequently, we have a moduli interpretation for the generators of $\mathop{\mathrm{Eff}}\nolimits(\mathop{\mathrm{Hilb}}\nolimits_1^{4t})$, which is the following.
Let $E_1$ be the closure of the locus parametrizing subschemes of $\ensuremath{\mathbb{P}}^3$ that are the union of a plane cubic and an incident line. By $E_2$ we denote the closure of the locus parametrizing plane quartics with two nodes and two embedded points at such nodes. Let $\Delta$ denote the closure of the locus of nodal elliptic curves.
\newtheorem*{thm:cones}{Theorem C}
\begin{thm:cones}
The cone of effective divisors of $\mathop{\mathrm{Hilb}}\nolimits_1^{4t}$ is generated by $\mathop{\mathrm{Eff}}\nolimits(\mathop{\mathrm{Hilb}}\nolimits_1^{4t})=\langle E_1, E_2, \Delta\rangle$.
\end{thm:cones}
\subsection*{Ingredients}
The notion of tilt stability on a smooth projective threefold was introduced in \cite{BMT14}. It is defined in a similar way one defines Bridgeland stability on a surface. Thus, these two notions of stability share computational properties. Tilt stability is intended as a stepping stone to Bridgeland stability. The proof of Theorem A is mostly based on this theory.
In contrast to the surface case, computing which objects destabilize at a given wall is difficult due to the lack of unique stable factors in the Jordan-H\"older filtration of a strictly semistable object. Computing the walls numerically in tilt stability is of similar difficulty as in the surface case and often times possible. On the other hand, while it is generally difficult to determine all walls in Bridgeland stability on a given path, it is not so difficult to determine which objects destabilize at a given wall. In order to resolve this issue we apply a technique from \cite{Sch15} that allows to translate walls from tilt stability into Bridgeland stability.
In order to identify the global structure of the Bridgeland moduli spaces, a careful analysis of its singularities is necessary. We apply deformation theory to these problems, and large parts of it reduce to heavy $\mathop{\mathrm{Ext}}\nolimits$-computation. Even though this can be done by hand, computer calculations with \cite{M2} turn out to be tremendously helpful. The situation is more involved when it comes to the intersection of the two components. We reduce the question to a single ideal in that case and apply the technique of \cite{PS85}. We make use of the Macaulay2 implementation \cite{Ilt12} of this technique.
The proof of Theorem C uses the description of the exceptional divisors provided in Proposition \ref{prop:E1_E2_description}, and exhibits the dual curves to them in order to conclude.
\subsection*{Organization}
In Section 1, we recall basic definitions about stability conditions. In Section 2, we carry out numerical computations in tilt stability needed to understand walls in Bridgeland stability. In Section 3, we describe the equations of some ideals depending on the exact sequences they fit in. We use this description to understand the local geometry of the intersection of the two components of our Hilbert scheme. In Section 4, we translate the computations in tilt stability to Bridgeland stability. Furthermore, we analyze singularities to provide a proof of Theorem A and Theorem B. In Section 5, we prove Theorem C. Appendix A contains our Macaulay2 code.
\subsection*{Acknowledgements}
We would like to thank Francesco Cavazzani, Dawei Chen, Izzet Coskun, Joe Harris, Sean Keel, Emanuele Macr\`i, and Edoardo Sernesi for insightful discussions about this work. We are also grateful to the referee for carefully going through the article. C. Lozano Huerta and B. Schmidt would also like to thank the organizers of the II ELGA school in Cabo Frio, Brazil for organizing a wonderful conference supported by NSF grant DMS-1502154 at which parts of this work was done. P. Gallardo is supported by NSF grant DMS-1344994 of the RTG in Algebra, Algebraic Geometry, and Number Theory, at the University of Georgia. C. Lozano Huerta thanks the Department of Mathematics at Harvard University for providing ideal working conditions. He is funded by CONACYT Grant CB-2015-01 No. 253061. B. Schmidt has been partially supported by NSF grant DMS-1523496 (PI Emanuele Macr\`i) and a Presidential Fellowship of the Ohio State University. He also wants to thank Northeastern University, where part of this work was done, for their hospitality
\subsection*{Notation}
We work over the field of the complex numbers throughout. We also use the following notation.
\begin{center}
\begin{tabular}{ r l }
$\ensuremath{\mathcal I}_{Z/X}$, $\ensuremath{\mathcal I}_Z$ & ideal sheaf of a closed subscheme $Z \subset X$ \\
$D^b(X)$ & bounded derived category of coherent sheaves on $X$ \\
$\mathop{\mathrm{ch}}\nolimits_X(E)$, $\mathop{\mathrm{ch}}\nolimits(E)$ & Chern character of an object $E \in D^b(X)$ \\
$\mathop{\mathrm{ch}}\nolimits_{\leq l,X}(E)$, $\mathop{\mathrm{ch}}\nolimits_{\leq l}(E)$ & $(\mathop{\mathrm{ch}}\nolimits_{0,X}(E), \ldots,
\mathop{\mathrm{ch}}\nolimits_{l,X}(E))$ \\
$\ensuremath{\mathbb{G}}(r,k)$ & the Grassmannian parametrizing subspaces $\ensuremath{\mathbb{P}}^r \subset \ensuremath{\mathbb{P}}^k$ \\
$\mathop{\mathrm{Hilb}}\nolimits_1^{4t}$ & closure of the locus of elliptic quartic curves in $\mathop{\mathrm{Hilb}}\nolimits(\ensuremath{\mathbb{P}}^3)$ \\
$\mathop{\mathrm{Hilb}}\nolimits_2^{4t}$ & closure in $\mathop{\mathrm{Hilb}}\nolimits(\ensuremath{\mathbb{P}}^3)$ of the locus of unions of plane \\
& quartic curves with two points in $\ensuremath{\mathbb{P}}^3$ \\
\end{tabular}
\end{center}
\section{Preliminaries on Stability Conditions}
\label{sec:prelim}
This section recalls the construction of Bridgeland stability conditions on $\ensuremath{\mathbb{P}}^3$ due to \cite{BMT14, MacE14}. We refer the reader to \cite{Bri07} for a detailed introduction to the theory of Bridgeland stability. Let $X$ be a smooth projective threefold. A \textit{Bridgeland stability condition} on $D^b(X)$ is a pair $(Z,\ensuremath{\mathcal A})$, where $\ensuremath{\mathcal A}$ is the heart of a bounded t-structure and $Z: K_0(X) = K_0(\ensuremath{\mathcal A}) \to \ensuremath{\mathbb{C}}$ is an additive homomorphism that maps any non trivial object in $\ensuremath{\mathcal A}$ to the upper half-plane or the negative real line. Additionally, technical properties such as the existence of Harder-Narasimhan filtrations and the support property have to be fulfilled. Bridgeland's main result is that the set of stability condition can be given the structure of a complex manifold. We will denote this \textit{stability manifold} by $\mathop{\mathrm{Stab}}\nolimits(X)$.
Let $H$ be the very ample generator of $\mathop{\mathrm{Pic}}\nolimits(\ensuremath{\mathbb{P}}^3)$. Due to the simplicity of the cohomology of $\ensuremath{\mathbb{P}}^3$, we will abuse notation by writing $\mathop{\mathrm{ch}}\nolimits_i(E) = H^{3-i} \mathop{\mathrm{ch}}\nolimits_i(E)$ for any $E \in D^b(X)$. If $\beta \in \ensuremath{\mathbb{R}}$, we define the \textit{twisted Chern character} by $\mathop{\mathrm{ch}}\nolimits^{\beta} := e^{-\beta H} \cdot \mathop{\mathrm{ch}}\nolimits$. In more detail, we have
\begin{align*}
\mathop{\mathrm{ch}}\nolimits^{\beta}_0 &= \mathop{\mathrm{ch}}\nolimits_0, \, \, \mathop{\mathrm{ch}}\nolimits^{\beta}_1 = \mathop{\mathrm{ch}}\nolimits_1 - \beta \mathop{\mathrm{ch}}\nolimits_0, \, \, \mathop{\mathrm{ch}}\nolimits^{\beta}_2 = \mathop{\mathrm{ch}}\nolimits_2 - \beta \mathop{\mathrm{ch}}\nolimits_1 + \frac{\beta^2}{2} \mathop{\mathrm{ch}}\nolimits_0, \\
\mathop{\mathrm{ch}}\nolimits^{\beta}_3 &= \mathop{\mathrm{ch}}\nolimits_3 - \beta \mathop{\mathrm{ch}}\nolimits_2 + \frac{\beta^2}{2} \mathop{\mathrm{ch}}\nolimits_1 - \frac{\beta^3}{6} \mathop{\mathrm{ch}}\nolimits_0.
\end{align*}
We write a twisted version of the classical \textit{slope function} as
\[
\mu_{\beta}(\mathop{\mathrm{ch}}\nolimits_0, \mathop{\mathrm{ch}}\nolimits_1) := \frac{\mathop{\mathrm{ch}}\nolimits^{\beta}_1}{\mathop{\mathrm{ch}}\nolimits^{\beta}_0} = \frac{\mathop{\mathrm{ch}}\nolimits_1}{\mathop{\mathrm{ch}}\nolimits_0} - \beta,
\]
where division by $0$ is interpreted as $+\infty$. In \cite{BMT14} the notion of \textit{tilt stability} has been introduced as an auxilliary notion in between classical slope stability and Bridgeland stability on threefolds. We will recall this construction and a few of its properties. Tilting is used to obtain a new heart of a bounded t-structure. For more information on the general theory of tilting we refer to \cite{HRS96}. A torsion pair is defined by
\begin{align*}
\ensuremath{\mathcal T}_{\beta} &:= \{E \in \mathop{\mathrm{Coh}}\nolimits(\ensuremath{\mathbb{P}}^3) : \text{any quotient $E \ensuremath{\twoheadrightarrow} G$ satisfies $\mu_{\beta}(G) > 0$} \}, \\
\ensuremath{\mathcal F}_{\beta} &:= \{E \in \mathop{\mathrm{Coh}}\nolimits(\ensuremath{\mathbb{P}}^3) : \text{any subsheaf $F \subset E$ satisfies $\mu_{\beta}(F) \leq 0$} \}.
\end{align*}
A new heart of a bounded t-structure is given by the extension closure $\mathop{\mathrm{Coh}}\nolimits^{\beta}(\ensuremath{\mathbb{P}}^3) := \langle \ensuremath{\mathcal F}_{\beta}[1], \ensuremath{\mathcal T}_{\beta} \rangle$. Equivalently, the objects in $\mathop{\mathrm{Coh}}\nolimits^{\beta}(\ensuremath{\mathbb{P}}^3)$ are complexes $E \in D^b(X)$ satisfying $H^i(E) = 0$ for $i \neq 0,-1$, $H^{-1}(E) \in \ensuremath{\mathcal F}_{\beta}$ and $H^0(E) \in \ensuremath{\mathcal T}_{\beta}$. Let $\alpha > 0$ be a positive real number. The new slope function is
\[
\nu_{\alpha, \beta}(\mathop{\mathrm{ch}}\nolimits_0, \mathop{\mathrm{ch}}\nolimits_1, \mathop{\mathrm{ch}}\nolimits_2) := \frac{\mathop{\mathrm{ch}}\nolimits^{\beta}_2 - \frac{\alpha^2}{2} \mathop{\mathrm{ch}}\nolimits^{\beta}_0}{\mathop{\mathrm{ch}}\nolimits^{\beta}_1} = \frac{\mathop{\mathrm{ch}}\nolimits_2 - \beta \mathop{\mathrm{ch}}\nolimits_1 + \frac{\beta^2}{2} \mathop{\mathrm{ch}}\nolimits_0 - \frac{\alpha^2}{2} \mathop{\mathrm{ch}}\nolimits_0}{\mathop{\mathrm{ch}}\nolimits_1 - \beta \mathop{\mathrm{ch}}\nolimits_0}.
\]
As in classical slope stability an object $E \in \mathop{\mathrm{Coh}}\nolimits^{\beta}(\ensuremath{\mathbb{P}}^3)$ is called \textit{$\nu_{\alpha, \beta}$-(semi)stable} or \textit{tilt (semi)stable} with respect to $(\alpha, \beta)$ if for all short exact sequences $0 \to F \to E \to G \to 0$ in $\mathop{\mathrm{Coh}}\nolimits^{\beta}(\ensuremath{\mathbb{P}}^3)$ the inequality $\nu_{\alpha, \beta}(F) < (\leq) \nu_{\alpha, \beta}(G)$ holds. Note that in regard to \cite{BMT14} this slope has been modified by switching $\alpha$ with $\sqrt{3} \alpha$. We prefer this point of view because it will make the walls semicircular. In concrete computations it becomes relevant to restrict the Chern characters of semistable objects. One of the main tools to perform this restriction is the following inequality for semistable objects.
\begin{thm}[{Bogomolov-Gieseker Inequality for Tilt Stability, \cite[Corollary 7.3.2]{BMT14}}]
\label{thm:bg_inequality}
Any $\nu_{\alpha, \beta}$-semistable object $E \in \mathop{\mathrm{Coh}}\nolimits^{\beta}(\ensuremath{\mathbb{P}}^3)$ satisfies
\begin{align*}
Q^{\mathop{\mathrm{tilt}}}(E) &:= (\mathop{\mathrm{ch}}\nolimits_1^{\beta}(E))^2 - 2\mathop{\mathrm{ch}}\nolimits_0^{\beta}(E)\mathop{\mathrm{ch}}\nolimits_2^{\beta}(E) \\
&= (\mathop{\mathrm{ch}}\nolimits_1(E))^2 - 2\mathop{\mathrm{ch}}\nolimits_0(E)\mathop{\mathrm{ch}}\nolimits_2(E) \geq 0.
\end{align*}
\end{thm}
Let $v = \mathop{\mathrm{ch}}\nolimits_{\leq 2}(E) = (v_0, v_1, v_2)$ for some object $E \in D^b(\ensuremath{\mathbb{P}}^3)$. A \textit{numerical wall} in tilt stability for $v$ is by definition induced by a class $(r,c,d) \in \ensuremath{\mathbb{Z}}^2 \times \tfrac{1}{2} \ensuremath{\mathbb{Z}}$ as the set of solutions $(\alpha, \beta)$ to the equation $\nu_{\alpha, \beta}(v) = \nu_{\alpha, \beta}(r,c,d)$, where we assume that this is a non trivial proper solution set. For example throughout this article, we will always choose $v = \mathop{\mathrm{ch}}\nolimits_{\leq 2}(\ensuremath{\mathcal I}_C)$, where $C \subset \ensuremath{\mathbb{P}}^3$ is an elliptic quartic curve and study moduli spaces involving these objects.
A subset of a numerical wall is an \textit{actual wall} if the set of stable or semistable objects with class $v$ changes at it. Numerical walls in tilt stability satisfy Bertram's Nested Wall Theorem. For surfaces it was proved in \cite{MacA14}. A proof in the threefold case can be found in \cite{Sch15}.
\begin{thm}[Structure Theorem for Walls in Tilt Stability]
All numerical walls in the following statements are for fixed $v = (v_0, v_1, v_2)$.
\begin{enumerate}
\item Numerical walls in tilt stability are of the form
\[x\alpha^2 + x\beta^2 + y\beta + z = 0\]
for $x = v_0c - v_1r$, $y = 2(v_2r - v_0d)$ and $z = 2(v_1d - v_2c)$. In particular, they are either semicircular walls with center on the $\beta$-axis or vertical rays.
\item If two numerical walls given by $\nu_{\alpha, \beta}(r,c,d) = \nu_{\alpha, \beta}(v)$ and $\nu_{\alpha, \beta}(r',c',d') = \nu_{\alpha, \beta}(v)$ intersect for any $\alpha \geq 0$, then $(r,c,d)$, $(r',c',d')$ and $v$ are linearly dependent. In particular, the two walls are completely identical.
\item The curve $\nu_{\alpha, \beta}(v) = 0$ is given by the hyperbola
\[v_0\alpha^2 - v_0\beta^2 + 2v_1\beta - 2v_2 = 0.\]
Moreover, this hyperbola intersects all semicircular walls at their top point.
\item If $v_0 \neq 0$, there is exactly one vertical numerical wall given by $\beta = v_1/v_0$. If $v_0 = 0$ there is no vertical wall.
\item If a numerical wall has a single point at which it is an actual wall, then all of it is an actual wall.
\end{enumerate}
\end{thm}
On smooth projective surfaces tilt stability is enough to get a Bridgeland stability condition (see \cite{Bri08, AB13}). On threefolds Bayer, Macr\`i and Toda proposed another tilt to obtain a suitable category to define a Bridgeland stability condition as follows. Let
\begin{align*}
\ensuremath{\mathcal T}'_{\alpha, \beta} &:= \{E \in \mathop{\mathrm{Coh}}\nolimits^{\beta}(\ensuremath{\mathbb{P}}^3) : \text{any quotient $E \ensuremath{\twoheadrightarrow} G$ satisfies $\nu_{\alpha, \beta}(G) > 0$} \}, \\
\ensuremath{\mathcal F}'_{\alpha, \beta} &:= \{E \in \mathop{\mathrm{Coh}}\nolimits^{\beta}(\ensuremath{\mathbb{P}}^3) : \text{any subobject $F \ensuremath{\hookrightarrow} E$ satisfies $\nu_{\alpha, \beta}(F) \leq 0$} \}
\end{align*}
and set $\ensuremath{\mathcal A}^{\alpha, \beta} := \langle \ensuremath{\mathcal F}'_{\alpha, \beta}[1], \ensuremath{\mathcal T}'_{\alpha, \beta} \rangle $. For any $s>0$ they define
\begin{align*}
Z_{\alpha,\beta,s} &:= -\mathop{\mathrm{ch}}\nolimits^{\beta}_3 + (s+\tfrac{1}{6})\alpha^2 \mathop{\mathrm{ch}}\nolimits^{\beta}_1 + i (\mathop{\mathrm{ch}}\nolimits^{\beta}_2 - \frac{\alpha^2}{2} \mathop{\mathrm{ch}}\nolimits^{\beta}_0), \\
\lambda_{\alpha,\beta,s} &:= -\frac{\Re(Z_{\alpha,\beta,s})}{\Im(Z_{\alpha,\beta,s})}.
\end{align*}
In order to prove that this yields a Bridgeland stability condition, Bayer, Macr\`i, and Toda conjectured a generalized Bogomolov-Gieseker inequality involving third Chern characters for tilt semistable objects with $\nu_{\alpha, \beta} = 0$. In \cite{BMS16} it was shown that the conjecture is equivalent to a more general inequality that drops the hypothesis $\nu_{\alpha, \beta} = 0$. In the case of $\ensuremath{\mathbb{P}}^3$ the inequality was proved in \cite{MacE14}. Recall the definition of $Q^{\mathop{\mathrm{tilt}}}$ from Theorem \ref{thm:bg_inequality}.
\begin{thm}[BMT Inequality]
Any $\nu_{\alpha, \beta}$-stable object $E \in \mathop{\mathrm{Coh}}\nolimits^{\beta}(\ensuremath{\mathbb{P}}^3)$ satisfies
\[ \alpha^2 Q^{\mathop{\mathrm{tilt}}}(E) + 4(\mathop{\mathrm{ch}}\nolimits_2^{\beta}(E))^2 - 6 \mathop{\mathrm{ch}}\nolimits_1^{\beta}(E) \mathop{\mathrm{ch}}\nolimits_3^{\beta}(E) \geq 0. \]
\end{thm}
Similar inequalities were proved for the smooth quadric threefold \cite{Sch14} and all abelian threefolds \cite{BMS16, MP13a, MP13b}. Recently, the inequality has also been generalized to all Fano threefolds of Picard rank $1$ in \cite{Li15}. By using the definition of $\mathop{\mathrm{ch}}\nolimits^{\beta}(E)$, one finds $x(E),y(E) \in \ensuremath{\mathbb{R}}$ such that the BMT Inequality becomes
\[\alpha^2 Q^{\mathop{\mathrm{tilt}}}(E) + \beta^2 Q^{\mathop{\mathrm{tilt}}}(E) + x(E)\beta + y(E) \geq 0.\]
This means the solution set is given by the complement of a semi-disc with center on the $\beta$-axis or a quadrant to one side of a vertical line.
Using the same proof as in the surface case in \cite[Proposition 14.1]{Bri08} leads to the following lemma. It allows to identify the moduli space of slope stable sheaves as a moduli space of tilt stable objects.
\begin{lem}
\label{lem:stable_sheaves_stable_in_tilt}
Let $v = (v_0, v_1, v_2, v_3) \in K_0(\ensuremath{\mathbb{P}}^3)$ such that $\beta < \mu(v)$ and $(v_0, v_1)$ is primitive. Then an object $E$ with $\mathop{\mathrm{ch}}\nolimits(E) = v$ is $\nu_{\alpha, \beta}$-stable for all $\alpha \gg 0$ if and only if $E$ is a slope stable sheaf.
\end{lem}
An important question is how moduli spaces change set theoretically at walls in Bridgeland stability. In case the destabilizing subobject and quotient are both stable this has a satisfactory answer, and a proof can for example be found in \cite[Lemma 3.10]{Sch15}. Note that this does not work in the case of tilt stability due to the lack of unique Jordan-H\"older filtrations.
\begin{lem}
\label{lem:wall_crossing}
Let $\sigma = (\ensuremath{\mathcal A}, Z) \in \mathop{\mathrm{Stab}}\nolimits(\ensuremath{\mathbb{P}}^3)$ such that there are stable object $F,G \in \ensuremath{\mathcal A}$ with $\mu_{\sigma}(F) = \mu_{\sigma}(G)$. Then there is an open neighborhood $U$ around $\sigma$ where non trivial extensions $0 \to F \to E \to G \to 0$ are stable for all $\sigma' \in U$ where $F \ensuremath{\hookrightarrow} E$ does not destabilize $E$.
\end{lem}
Another crucial issue is the construction of reasonably behaved moduli spaces of Bridgeland stable objects. A recent result by Piyaratne and Toda is a major step towards this. It applies in particular to the case of $\ensuremath{\mathbb{P}}^3$, since the conjectural BMT-inequality is known.
\begin{thm}[{\cite{PT15}}]
Let $X$ be a smooth projective threefold such that the conjectural construction of Bridgeland stability from \cite{BMT14} works. Then any moduli space of semistable objects for such a Bridgeland stability condition is a universally closed algebraic stack of finite type over $\ensuremath{\mathbb{C}}$.
\end{thm}
If there are no strictly semistable objects, the moduli space becomes a proper algebraic space of finite type over $\ensuremath{\mathbb{C}}$.
Our strategy to compute concrete wall crossing follows that of \cite{Sch15}. We do numerical computations in tilt stability and then translate them into Bridgeland stability. Let $v = (v_0, v_1, v_2, v_3)$ be the Chern character of an object in $D^b(X)$. For any $\alpha > 0$, $\beta \in \ensuremath{\mathbb{R}}$ and $s > 0$ we denote the set of $\lambda_{\alpha, \beta, s}$-semistable objects with Chern character $v$ by $M_{\alpha, \beta, s}(v)$ and the set of $\nu_{\alpha, \beta}$-semistable objects with Chern character $v$ by $M^{\mathop{\mathrm{tilt}}}_{\alpha, \beta, s}(v)$. Analogously to our notation for twisted Chern characters, we write $v^{\beta} := (v^{\beta}_0, v^{\beta}_1, v^{\beta}_2, v^{\beta}_3) = v \cdot e^{-\beta H}$. We also write
\[
P_v := \{(\alpha, \beta) \in \ensuremath{\mathbb{R}}_{\geq 0} \times \ensuremath{\mathbb{R}} : \nu_{\alpha, \beta}(v) > 0 \}.
\]
We need the following technical statement. Under mild hypotheses, it says that on one side of the hyperbola $\{ \nu_{\alpha, \beta}(v) = 0 \}$ all the chambers and walls of tilt stability occur in Bridgeland stability. Note that $\nu_{\alpha, \beta}(v) = 0$ implies $\lambda_{\alpha, \beta, s}(v) = \infty$. This is a crucial fact in establishing the following relation between walls in tilt stability and walls in Bridgeland stability.
\begin{thm}[{\cite[Theorem 6.1]{Sch15}}]
\label{thm:wall_intersecting_hyperbola}
Let $\alpha_0 > 0$, $\beta_0 \in \ensuremath{\mathbb{R}}$ and $s > 0$ such that $\nu_{\alpha_0, \beta_0}(v) = 0$ and $v^{\beta_0}_1 > 0$.
\begin{enumerate}
\item Assume there is an actual wall in Bridgeland stability for $v$ at $(\alpha_0, \beta_0)$ given by
\[0 \to F \to E \to G \to 0.\]
That means $\lambda_{\alpha_0, \beta_0, s}(F) = \lambda_{\alpha_0, \beta_0, s}(G)$ and $\mathop{\mathrm{ch}}\nolimits(E) = -v$ for semistable $E,F,G \in \ensuremath{\mathcal A}^{\alpha_0, \beta_0}(\ensuremath{\mathbb{P}}^3)$. Further assume there is a neighborhood $U$ of $(\alpha_0, \beta_0)$ such that the same sequence also defines an actual wall in $U \cap P_v$, i.e. $E,F,G$ remain semistable in $U \cap P_v \cap \{ \lambda_{\alpha, \beta, s}(F) = \lambda_{\alpha, \beta, s}(G)\}$. Then $E[-1]$, $F[-1]$, $G[-1] \in \mathop{\mathrm{Coh}}\nolimits^{\beta_0}(\ensuremath{\mathbb{P}}^3)$ are $\nu_{\alpha_0, \beta_0}$-semistable. In particular, there is an actual wall in tilt stability at $(\alpha_0, \beta_0)$.
\item Assume that all $\nu_{\alpha_0, \beta_0}$-semistable objects with class $v$ are stable. Then there is a neighborhood $U$ of $(\alpha_0, \beta_0)$ such that
\[M_{\alpha, \beta, s}(v) = M^{\mathop{\mathrm{tilt}}}_{\alpha, \beta}(v)\]
for all $(\alpha, \beta) \in U \cap P_v$. Moreover, in this case all objects in $M_{\alpha, \beta, s}(v)$ are $\lambda_{\alpha, \beta, s}$-stable.
\item Assume there is an actual wall in tilt stability for $v$ at $(\alpha_0, \beta_0)$ given by
\[0 \to F^n \to E \to G^m \to 0\]
such that $F, G \in \mathop{\mathrm{Coh}}\nolimits^{\beta_0}(\ensuremath{\mathbb{P}}^3)$ are $\nu_{\alpha_0, \beta_0}$-stable objects, $\mathop{\mathrm{ch}}\nolimits(E) = v$ and $\nu_{\alpha_0, \beta_0}(F) = \nu_{\alpha_0, \beta_0}(G)$. Assume further that the set
\[P_v \cap P_{\mathop{\mathrm{ch}}\nolimits(F)} \cap P_{\mathop{\mathrm{ch}}\nolimits(G)} \cap \{ \lambda_{\alpha, \beta, s}(F) = \lambda_{\alpha, \beta, s}(G)\}\]
is non-empty. Then there is a neighborhood $U$ of $(\alpha_0, \beta_0)$ such that $F,G$ are $\lambda_{\alpha, \beta, s}$-stable for all $(\alpha, \beta) \in U \cap P_v \cap \{ \lambda_{\alpha, \beta, s}(F) = \lambda_{\alpha, \beta, s}(G)\}$. In particular, there is an actual wall in Bridgeland stability in $U \cap P_v$ defined by the same sequence.
\end{enumerate}
\end{thm}
This Theorem will be used as follows in the the remainder of the article. Assume that we have determined all exact sequences that give walls in tilt stability for objects with a fixed Chern character $v$. By part (1) of the Theorem, we know that on one side of the hyperbola $\nu_{\alpha, \beta}(v) = 0$ the only walls in Bridgeland stability have to be defined by an exact sequence giving a wall in tilt stability. We will then use part (3) to show that every such sequence does indeed define a wall in Bridgeland stability. At this point we know all exact sequences defining walls on a path close to one side of the hyperbola $\nu_{\alpha, \beta}(v) = 0$. Finally, we have to use part (2) to show that all the moduli spaces of tilt stable objects actually occur in Bridgeland stability on this path.
By doing this, we can translate simple computations in tilt stability into the more complicated framework of Bridgeland stability. Sometimes there are exact sequences giving identical numerical walls in tilt stability, but different numerical walls in Bridgeland stability. Therefore, this translation allows us to observe additional chambers that are hidden in tilt stability.
\section{Tilt Stability for Elliptic Quartics}
Let $C$ be the complete intersection of two quadrics in $\ensuremath{\mathbb{P}}^3$, i.e. an elliptic quartic curve. We will compute all walls in tilt stability for $\beta < 0$ with respect to $v = \mathop{\mathrm{ch}}\nolimits(\ensuremath{\mathcal I}_C)$. There is a locally free resolution $0 \to \ensuremath{\mathcal O}(-4) \to \ensuremath{\mathcal O}(-2)^{\oplus 2} \to \ensuremath{\mathcal I}_C \to 0$. This leads to
\[\mathop{\mathrm{ch}}\nolimits^{\beta}(\ensuremath{\mathcal I}_C) = \left(1, -\beta, \frac{\beta^2}{2} - 4, -\frac{\beta^3}{6} + 4\beta + 8\right).\]
We denote the set of tilt semistable objects with respect to $(\alpha, \beta)$ and class $v$ by $M^{\mathop{\mathrm{tilt}}}_{\alpha, \beta}(v)$.
\begin{thm}
\label{thm:tiltStabilityEllipticQuartics}
There are three walls for $M^{\mathop{\mathrm{tilt}}}_{\alpha, \beta}(1,0,-4,8)$ for $\alpha > 0$ and $\beta < 0$. Moreover, the following table lists pairs of tilt semistable objects whose extensions completely describe all strictly semistable objects at each of the corresponding walls. Let $L$ be a line in $\ensuremath{\mathbb{P}}^3$, $V$ a plane in $\ensuremath{\mathbb{P}}^3$, $Z \subset \ensuremath{\mathbb{P}}^3$ a length two zero dimensional subscheme, $Z' \subset
V$ a length two zero dimensional subscheme and $P \in \ensuremath{\mathbb{P}}^3$, $Q \in V$ be points.
\begin{center}
\begin{tabular}{ r | l }
& \\
$\alpha^2 + (\beta + 3)^2 = 1$ & $\ensuremath{\mathcal O}(-2)^{\oplus
2}$, $\ensuremath{\mathcal O}(-4)[1]$ \\
& \\
\hline
& \\
$\alpha^2 + \left(\beta + \frac{7}{2} \right)^2 = \frac{17}{4}$ &
$\ensuremath{\mathcal I}_L(-1)$, $\ensuremath{\mathcal O}_V(-3)$ \\
& \\
\hline
& \\
\ & $\ensuremath{\mathcal I}_Z(-1)$, $\ensuremath{\mathcal O}_V(-4)$ \\
& \\
$\alpha^2 + \left(\beta + \frac{9}{2}\right)^2 =
\left(\frac{7}{2}\right)^2$ & $\ensuremath{\mathcal I}_P(-1)$, $\ensuremath{\mathcal I}_{Q/V}(-4)$ \\
& \\
\ & $\ensuremath{\mathcal O}(-1)$, $\ensuremath{\mathcal I}_{Z'/V}(-4)$ \\
\end{tabular}
\end{center}
The hyperbola $\nu_{\alpha, \beta}(1,0,-4) = 0$ is given by the equation
\[\beta^2 - \alpha^2 = 8.\]
Moreover, there are no semistable objects for $(\alpha, \beta)$ inside the smallest semicircle.
\end{thm}
It is interesting to note that all relevant objects in this Theorem are sheaves and no actual 2-term complexes. The key difference to the classical picture, as we will see later, is that some sheaves of positive rank with torsion will turn out to be stable and replace ideal sheaves of heavily singular curves in some chambers.
The fact that the smallest wall is given by the equation $\alpha^2 + (\beta + 3)^2 = 1$ was already proved in \cite[Theorem 5.1]{Sch15} in more generality. Moreover, it was shown there that all semistable objects $E$ at the wall are given by extensions of the form $0 \to \ensuremath{\mathcal O}(-2)^{\oplus 2} \to E \to \ensuremath{\mathcal O}(4)[1] \to 0$ and that there are no tilt semistable objects inside this semicircle.
In order to prove the remainder of Theorem \ref{thm:tiltStabilityEllipticQuartics} we need to put numerical restrictions on potentially destabilizing objects. This can be done by the following two lemmas.
\begin{lem}[{\cite[Lemma 5.4]{Sch14}}]
\label{lem:idealSheafRestriction}
Let $E \in \mathop{\mathrm{Coh}}\nolimits^{\beta}(\ensuremath{\mathbb{P}}^3)$ be tilt semistable with respect to some $\beta \in \ensuremath{\mathbb{Z}}$ and $\alpha \in \ensuremath{\mathbb{R}}_{> 0}$.
\begin{enumerate}
\item If $\mathop{\mathrm{ch}}\nolimits^{\beta}(E) = (1,1,d,e)$ then $d - 1/2 \in \ensuremath{\mathbb{Z}}_{\leq 0}$. In the case $d = -1/2$, we get $E \cong \ensuremath{\mathcal I}_L(\beta + 1)$ where $L$ is a line plus $1/6-e$ (possibly embedded) points in $\ensuremath{\mathbb{P}}^3$. If $d = 1/2$, then $E \cong \ensuremath{\mathcal I}_Z(\beta + 1)$ for a zero dimensional subscheme $Z \subset \ensuremath{\mathbb{P}}^3$ of length $1/6 - e$.
\item If $\mathop{\mathrm{ch}}\nolimits^{\beta}(E) = (0,1,d,e)$, then $d - 1/2 \in \ensuremath{\mathbb{Z}}$ and $E \cong I_{Z/V}(\beta + d + 1/2)$ where $Z$ is a dimension zero subscheme of length $1/24 + d^2/2 - e$.
\end{enumerate}
\end{lem}
The next lemma determines the Chern characters of possibly destabilizing objects for $\beta = -2$.
\begin{lem}
\label{lem:chAtMinusTwo}
If an exact sequence $0 \to F \to E \to G \to 0$ in $\mathop{\mathrm{Coh}}\nolimits^{-2}(\ensuremath{\mathbb{P}}^3)$ defines a wall for $\beta = -2$ with $\mathop{\mathrm{ch}}\nolimits_{\leq 2}(E) = (1, 0, -4)$ then
\[\mathop{\mathrm{ch}}\nolimits^{-2}_{\leq 2}(F), \mathop{\mathrm{ch}}\nolimits^{-2}_{\leq 2}(G) \in \left\{\left(1, 1, -\frac{1}{2}\right), \left(0, 1, -\frac{3}{2}\right), \left(1, 1, \frac{1}{2}\right), \left(0,1, -\frac{5}{2}\right) \right\}.\]
\begin{proof}
The four possible Chern characters group into two cases that add up to $\mathop{\mathrm{ch}}\nolimits^{-2}_{\leq 2}(E) = (1, 2, -2)$.
Let $\mathop{\mathrm{ch}}\nolimits^{-2}_{\leq 2}(F) = (r,c,d)$. By definition of $\mathop{\mathrm{Coh}}\nolimits^{-2}(\ensuremath{\mathbb{P}}^3)$, we have $0 \leq c \leq 2$. If $c=0$, then $\nu_{\alpha, -2}(F) = \infty$ and this is in fact no wall for any $\alpha > 0$. If $c=2$, then the same argument for the quotient $G$ shows there is no wall. Therefore, $c=1$ must hold. We can compute
\begin{align*}
\nu_{\alpha, -2}(E) = -1 - \frac{\alpha^2}{4}, \ \nu_{\alpha, -2}(F) = d - \frac{r \alpha^2}{2}.
\end{align*}
The wall is defined by $\nu_{\alpha, -2}(E) = \nu_{\alpha, -2}(F)$. This leads to
\begin{align}
\label{eq:alphaPositive}
\alpha^2 = \frac{4d+4}{2r-1} > 0.
\end{align}
The next step is to rule out the cases $r \geq 2$ and $r \leq -1$. If $r \geq 2$, then $\mathop{\mathrm{ch}}\nolimits_0(G) \leq -1$. By exchanging the roles of $F$ and $G$ in the following argument, it is enough to deal with the situation $r \leq -1$. In that case we use (\ref{eq:alphaPositive}) and the Bogomolov Gieseker inequality to get the contradiction $2rd \leq 1$, $d < -1$ and $r \leq -1$.
Therefore, we know $r=0$ or $r=1$. By again interchanging the roles of $F$ and $G$ if necessary we only have to handle the case $r=1$. Equation (\ref{eq:alphaPositive}) implies $d > - 1$. By Lemma
\ref{lem:idealSheafRestriction} we get $d - 1/2 \in \ensuremath{\mathbb{Z}}_{\leq 0}$. Therefore, we are left with the cases claimed.
\end{proof}
\end{lem}
\begin{proof}[Proof of Theorem \ref{thm:tiltStabilityEllipticQuartics}]
By assumption we are only dealing with walls that intersect the branch of the hyperbola with $\beta < 0$. As explained before, we already know the smallest wall. This semicircle intersects the $\beta$-axis at $\beta = -4$ and $\beta = -2$. Therefore, all other walls intersecting this branch of the hyperbola also have to intersect the ray $\beta = -2$. By Lemma \ref{lem:chAtMinusTwo} there are at most two walls intersecting the line $\beta = -2$. They correspond to the two solutions claimed to exist.
Let $0 \to F \to E \to G \to 0$ define a wall in $\mathop{\mathrm{Coh}}\nolimits^{-2}(\ensuremath{\mathbb{P}}^3)$ with $\mathop{\mathrm{ch}}\nolimits(E) = (1,0,-4,8)$. One can compute $\mathop{\mathrm{ch}}\nolimits^{-2}(E) = (1, 2, -2, \tfrac{4}{3})$. A direct computation shows that the middle wall is given by $\mathop{\mathrm{ch}}\nolimits^{-2}(F) = (1, 1, -1/2, e)$ and $\mathop{\mathrm{ch}}\nolimits^{-2}(G) = (0, 1, -3/2, 4/3 - e)$. By Lemma \ref{lem:idealSheafRestriction} we get $F \cong \ensuremath{\mathcal I}_L(-1)$ where $L$ is a line plus $1/6-e$ (possibly embedded) points in $\ensuremath{\mathbb{P}}^3$. In particular, the inequality $e \leq 1/6$ holds. The same lemma also implies that $G \cong I_{Z/V}(-3)$ where $Z$ is a dimension zero subscheme of length $e-1/6$. Overall this shows $e=1/6$. Therefore, $L$ is a just a line and $E \cong \ensuremath{\mathcal O}_V(-3)$.
The outermost wall is given by $\mathop{\mathrm{ch}}\nolimits^{-2}(F) = (1, 1, 1/2, e)$ and $\mathop{\mathrm{ch}}\nolimits^{-2}(G) = (0, 1, -5/2, 4/3 - e)$. We use again Lemma \ref{lem:idealSheafRestriction} to get $F \cong \ensuremath{\mathcal I}_Z(-1)$ for a zero
dimensional subscheme $Z \subset \ensuremath{\mathbb{P}}^3$ of length $1/6 - e$. Therefore, we have $e-1/6 \in \ensuremath{\mathbb{Z}}_{\geq 0}$. The lemma also shows $G \cong I_{Z/V}(-4)$ where $Z$ is a dimension zero subscheme of length $e + 11/6$. Overall, we get $e \in \{-11/6, -5/6, 1/6\}$. That corresponds exactly to the three cases in the Theorem.
\end{proof}
\section{Curves on the intersection of the two components}
Let $\mathop{\mathrm{Hilb}}\nolimits^{4t}_1 \subset \mathop{\mathrm{Hilb}}\nolimits^{4t}(\ensuremath{\mathbb{P}}^3)$ be the closure of the locus of smooth elliptic quartic curves. By $\mathop{\mathrm{Hilb}}\nolimits^{4t}_2 \subset \mathop{\mathrm{Hilb}}\nolimits^{4t}(\ensuremath{\mathbb{P}}^3)$ we denote the closure of the locus of plane quartics curves plus two disjoint points. A straightforward dimension count shows $\mathop{\mathrm{dim}}\nolimits \mathop{\mathrm{Hilb}}\nolimits^{4t}_1 = 16$ and $\mathop{\mathrm{dim}}\nolimits \mathop{\mathrm{Hilb}}\nolimits^{4t}_2 = 23$. In this section, we will prove some preliminary results about the intersection of the two components. We will do this following the approach of Piene and Schlessinger in \cite{PS85}, which requires a careful analysis of the equations of the curves along this intersection.
\begin{prop}
\label{prop:ideal_sequence_three}
Let $I_C$ be the ideal of a subscheme $C \subset \ensuremath{\mathbb{P}}^3$ of dimension $1$, which fits into an exact sequence of the form $0 \to \ensuremath{\mathcal I}_{Z'}(-1) \to I_C \to \ensuremath{\mathcal O}_V(-4) \to 0$, where $V$ is a plane in $\ensuremath{\mathbb{P}}^3$ and $Z' \subset V$ is a zero dimensional subscheme of length two.
\begin{enumerate}
\item The ideal $I_C$ is projectively equivalent to one of the ideals
\begin{align*}
(x^2, xy, xzw, f_4(x,y,z,w)), \\
(x^2, xy, xz^2, g_4(x,y,z,w)),
\end{align*}
where $f_4 \in (x, y, zw)$, respectively $g_4 \in (x, y, z^2)$ is of degree $4$.
\item The ideal
\[ (x^2, xy, xz^2, y^4) \]
lies in the closure of the orbit of $\ensuremath{\mathcal I}_C$ under the action of $\mathop{\mathrm{PGL}}(4)$ for any $\ensuremath{\mathcal I}_C$ as above.
\end{enumerate}
\begin{proof}
Up to the action of $\mathop{\mathrm{PGL}}(4)$ we can assume that either $I_{Z'} = (x, y, zw)$ or $I_{Z'} = (x, y, z^2)$ and $I_V = (x)$. The exact sequence $0 \to \ensuremath{\mathcal I}_{Z'}(-1) \to I_C \to \ensuremath{\mathcal O}_V(-4) \to 0$ implies that either $l(x, y, z, w) \cdot (x, y, zw) \subset I_C$ or $l(x, y, z, w) \cdot (x, y, z^2) \subset I_C$ for a linear polynomial $l(x, y, z, w) \in \ensuremath{\mathbb{C}}[x,y,z,w]$. Since the quotient is supported on $V$, we must have $l = x$. Therefore, either $(x^2, xy, xzw) \subset I_C$ or $(x^2, xy, xz^2) \subset I_C$. Since the quotient is $\ensuremath{\mathcal O}_V(-4)$, there has to be another degree $4$ generator $f_4(x,y,z,w)$ with $x f_4(x,y,z,w) \in (x^2, xy, xzw)$, respectively $g_4(x,y,z,w)$ with $x g_4(x,y,z,w) \in (x^2, xy, xz^2)$. That proves (1).
By (1), we can assume that either $I_C = (x^2, xy, xzw, f_4(x,y,z,w))$ for $f_4 \in (x, y, zw)$ or $I_C = (x^2, xy, xz^2, g_4(x,y,z,w))$ for $g_4 \in (x, y, z^2)$. We can take the limit $t \to 0$ for the action of the element $g_t \in \mathop{\mathrm{PGL}}(4)$ that fixes $x$, $y$, $z$ and maps $w \mapsto (1-t)z + tw$. Thus, we can assume that $I_C = (x^2, xy, xz^2, g_4(y,z))$ where $g_4 \in \ensuremath{\mathbb{C}}[y,z]$. Pick $\lambda \in \ensuremath{\mathbb{C}} \backslash \{ 0 \}$ such that $g_4(\lambda, 1) \neq 0$. We analyze the action of $g_t \in \mathop{\mathrm{PGL}}(4)$ that fixes $x$, $w$, maps $y \mapsto \lambda y$ and maps $z \mapsto (1-t)y + tz$. We get
\begin{align*}
g_t \cdot (x^2, xy, xz^2, g_4(y,z)) &= (x^2, \lambda xy, (1-t)^2 xy^2 +2(1-t)txyz + t^2 xz^2, g_4(\lambda y, (1-t)y + tz)) \\
&= (x^2, xy, xz^2, g_4(\lambda y, (1-t)y + tz)).
\end{align*}
Since $g_4(\lambda, 1) \neq 0$, we have $g_4(\lambda y, y) \neq 0$ and we can finish the proof of (2) by taking the limit $t \to 0$.
\end{proof}
\end{prop}
Next we want to analyze the singularities of the point on the Hilbert scheme corresponding to $(x^2, xy, xz^2, y^4)$. We will use \cite{M2} and the techniques developed in \cite{PS85}.
\begin{prop}
\label{prop:piene-schlessinger}
If $I_C = (x^2, xy, xz^2, y^4)$, then $I_C$ lies on the intersection of two irreducible components of $\mathop{\mathrm{Hilb}}\nolimits(\ensuremath{\mathbb{P}}^3)$ and is a smooth point on each of them. Moreover, the intersection is locally of dimension $15$ and transversal.
\begin{proof}
Let $p_C \in \mathop{\mathrm{Hilb}}\nolimits(\ensuremath{\mathbb{P}}^3)$ be the point parametrizing $C$. Next, we use the comparison theorem \cite[p. 764]{PS85} which claims the Hilbert scheme $\mathop{\mathrm{Hilb}}\nolimits(\ensuremath{\mathbb{P}}^3)$ and the universal deformation space which parametrizes all homogeneous ideals with Hilbert function equal to that of $I_C$
are isomorphic in an \'etale neighborhood of the point $p_C$ if
\[
\left( \frac{\mathbb{C}[x,y,z,w]}{I_C} \right)_d \cong H^0(C, \mathcal{O}_C(d))
\]
for $d=\mathop{\mathrm{deg}}\nolimits(f_i)$ where $f_i$ are generators of $I_C$. For our particular ideal, this equality can for example directly be checked with help of Macaulay2 or by hand. The comparison theorem allows us to find local equations of the Hilbert scheme near $p_C$ by using the same strategy than the proof of \cite[Lemma 6]{PS85}. In fact, this procedure has been implemented in the Macaulay2 Package ``VersalDeformations" (see \cite{Ilt12}). In particular, the routine localHilbertScheme generates an ideal of the form (see Appendix A)
\[
\left( -t_5t_{24}, -t_6t_{24} ,-t_7t_{24}, -t_8t_{24},t_{15}t_{24} ,t_{16}t_{24},t_{17}t_{24}-2t_{22}t_{24},t_{18}t_{24}-2t_{23}t_{24} \right) \in \mathbb{C}[t_1, \ldots, t_{24}.]
\]
Then, \'etale locally at $p_C$, the Hilbert scheme is the transversal intersection of the hyperplane $(t_{24}=0)$ and a $16$-dimensional linear subspace.
\end{proof}
\end{prop}
It is not hard to see that the two components $(x^2, xy, xz^2, y^4)$ is lying on are $\mathop{\mathrm{Hilb}}\nolimits^{4t}_1$ and $\mathop{\mathrm{Hilb}}\nolimits^{4t}_2$ by giving explicit degenerations. However, it is also a direct consequence of the results in the next section.
\section{Bridgeland stability}
\label{sec:bridgeland}
The goal of this section is to translate the computations in tilt stability to actual wall crossings in Bridgeland stability. We will analyze the singular loci of the occurring moduli spaces and use this to reprove the global description of the main component of the Hilbert scheme as in \cite{VA92}.
As a consequence of Theorem \ref{thm:wall_intersecting_hyperbola} and Theorem \ref{thm:tiltStabilityEllipticQuartics}, we obtain the following corollary. In this application of Theorem \ref{thm:wall_intersecting_hyperbola} all exact sequences giving walls in tilt stability to the left hand side of the unique vertical wall are of the form in (3). Therefore, we do not have more sequences giving walls in tilt stability than in Bridgeland stability to the left hand side of the left branch of the hyperbola.
\begin{cor}
\label{cor:allwalls}
There is a path $\gamma: [0,1] \to \ensuremath{\mathbb{R}}_{>0} \times \ensuremath{\mathbb{R}} \subset \mathop{\mathrm{Stab}}\nolimits(\ensuremath{\mathbb{P}}^3)$ that crosses the following walls for $v = (1, 0, -4, 8)$ in the given order. The walls are defined by the two given objects having the same slope. Moreover, all strictly semistable objects at each of the walls are extensions of those two objects. Let $L$ be a line in $\ensuremath{\mathbb{P}}^3$, $V$ a plane in $\ensuremath{\mathbb{P}}^3$, $Z \subset \ensuremath{\mathbb{P}}^3$ a length two zero dimensional subscheme, $Z' \subset V$ a length two zero dimensional subscheme and $P \in \ensuremath{\mathbb{P}}^3$, $Q \in V$ be points.
\begin{enumerate}
\item $\ensuremath{\mathcal O}(-2)^{\oplus 2}$, $\ensuremath{\mathcal O}(-4)[1]$
\item $\ensuremath{\mathcal I}_L(-1)$, $\ensuremath{\mathcal O}_V(-3)$
\item $\ensuremath{\mathcal I}_Z(-1)$, $\ensuremath{\mathcal O}_V(-4)$
\item $\ensuremath{\mathcal I}_P(-1)$, $\ensuremath{\mathcal I}_{Q/V}(-4)$
\item $\ensuremath{\mathcal O}(-1)$, $\ensuremath{\mathcal I}_{Z'/V}(-4)$
\end{enumerate}
\end{cor}
We denote the moduli space of Bridgeland stable objects with Chern character $(1,0,-4,8)$ in the chambers from inside the smallest wall to outside the largest wall by $M_0, \ldots, M_5$. The goal of this section is to give some description of these spaces. By Theorem \ref{thm:tiltStabilityEllipticQuartics} we have $M_0 = \emptyset$. After the largest wall we must have $M_5 = \mathop{\mathrm{Hilb}}\nolimits^{4t}(\ensuremath{\mathbb{P}}^3)$. More precisely, it is the moduli of ideal sheaves which is the same as the Hilbert scheme due to \cite[p. 1265]{MNOP06}. See Figure \ref{fig:bridgeland_walls_elliptic} for a visualization of the walls.
\begin{prop}
The first moduli space $M_1$ is isomorphic to the Grassmannian $\ensuremath{\mathbb{G}}(1,9)$.
\begin{proof}
All extensions in $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}(-4)[1], \ensuremath{\mathcal O}(-2)^{\oplus 2})$ are cokernels of morphisms $\ensuremath{\mathcal O}(-4) \to \ensuremath{\mathcal O}(-2)^{\oplus 2}$. The stability condition ensures that the two quadrics defining it are not collinear. Therefore, these extensions parametrize pencils of quadrics and the moduli space is the Grassmannian $\ensuremath{\mathbb{G}}(1,9)$.
\end{proof}
\end{prop}
The tangent space of a moduli space of Bridgeland stable objects at any stable complex $E$ is given by $\mathop{\mathrm{Ext}}\nolimits^1(E,E)$ (see \cite{Ina02} and \cite{Lie06} for the deformation theory of moduli spaces of complexes). Obtaining these groups requires a substantial amount of diagram chasing and computations. In order to minimize the distress on the reader and the authors, we will prove the following lemma with heavy usage of \cite{M2}.
\begin{lem}
\label{lem:ext-computations}
Let notation be as in Theorem \ref{cor:allwalls}. The equalities
\begin{align*}
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_L(-1), \ensuremath{\mathcal O}_V(-3)) = \ensuremath{\mathbb{C}} &, \ \mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-3), \ensuremath{\mathcal I}_L(-1)) = \ensuremath{\mathbb{C}}^9, \\
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_L(-1), \ensuremath{\mathcal I}_L(-1)) = \ensuremath{\mathbb{C}}^4 &, \ \mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-3), \ensuremath{\mathcal O}_V(-3)) = \ensuremath{\mathbb{C}}^3, \\
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_Z(-1), \ensuremath{\mathcal O}_V(-4)) = \begin{cases} \ensuremath{\mathbb{C}} &, \ Z \subset V \\ 0 &, \text{ otherwise} \end{cases} &, \ \mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-4), \ensuremath{\mathcal I}_Z(-1)) = \ensuremath{\mathbb{C}}^{15}, \\
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_Z(-1), \ensuremath{\mathcal I}_Z(-1)) = \ensuremath{\mathbb{C}}^6 &, \ \mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-4), \ensuremath{\mathcal O}_V(-4)) = \ensuremath{\mathbb{C}}^3, \\
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_P(-1), \ensuremath{\mathcal I}_{Q/V}(-4)) = \begin{cases} \ensuremath{\mathbb{C}}^3 &, \ P = Q \\ \ensuremath{\mathbb{C}} &, \ P \neq Q \end{cases} &, \ \mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_{Q/V}(-4), \ensuremath{\mathcal I}_P(-1)) = \begin{cases} \ensuremath{\mathbb{C}}^{17} &, \ P = Q \\ \ensuremath{\mathbb{C}}^{15} &, \ P \neq Q, \end{cases} \\
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_P(-1), \ensuremath{\mathcal I}_P(-1)) = \ensuremath{\mathbb{C}}^3 &, \ \mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_{Q/V}(-4), \ensuremath{\mathcal I}_{Q/V}(-4)) = \ensuremath{\mathbb{C}}^5, \\
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}(-1), \ensuremath{\mathcal I}_{Z'/V}(-4)) = \ensuremath{\mathbb{C}}^2 &, \ \mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_{Z'/V}(-4), \ensuremath{\mathcal O}(-1)) = \ensuremath{\mathbb{C}}^{15}, \\
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}(-1), \ensuremath{\mathcal O}(-1)) = 0 &, \ \mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_{Z'/V}(-4), \ensuremath{\mathcal I}_{Z'/V}(-4)) = \ensuremath{\mathbb{C}}^7
\end{align*}
hold.
If $Z \subset V$ is a double point supported at $P$, then
\begin{align*}
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_Z(-1), \ensuremath{\mathcal I}_{P/V}(-4)) = \ensuremath{\mathbb{C}}^3 &, \ \mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-4)), \ensuremath{\mathcal I}_{P/V}(-4)) = \ensuremath{\mathbb{C}}^2, \\
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_Z(-1), \ensuremath{\mathcal I}_P(-1)) = \ensuremath{\mathbb{C}}^3 &, \ \mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-4)), \ensuremath{\mathcal I}_P(-1)) = \ensuremath{\mathbb{C}}^{15}.
\end{align*}
\begin{proof}
Under the action of $\mathop{\mathrm{PGL}}(4)$ there are two orbits of pairs of a line and a plane $(L,V)$. Either we have $L \subset V$ or not. By choosing representatives defined over $\ensuremath{\mathbb{Q}}$, we can use \cite{M2} to compute $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_L(-1), \ensuremath{\mathcal O}_V(-3)) = \ensuremath{\mathbb{C}}$, $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-3), \ensuremath{\mathcal I}_L(-1)) = \ensuremath{\mathbb{C}}^9$, $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-3), \ensuremath{\mathcal O}_V(-3) = \ensuremath{\mathbb{C}}^3$ and $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_L(-1), \ensuremath{\mathcal I}_L(-1) = \ensuremath{\mathbb{C}}^4$. All other equalities follow in the same way. The Macaulay2 code can be found in Appendix A.
\end{proof}
\end{lem}
Since the dimension of tangent spaces is bounded from below by the dimension of the space, the following lemma can sometimes simplify computations.
\begin{lem}
\label{lem:ext_estimate}
Let $0 \to F^n \to E \to G^m \to 0$ be an exact sequence at a wall in Bridgeland stability where $F$ and $G$ are distinct stable objects of the same Bridgeland slope and $E$ is semistable to one side of the wall. Then the following inequality holds,
\[\mathop{\mathrm{ext}}\nolimits^1(E,E) \leq n^2 \mathop{\mathrm{ext}}\nolimits^1(F,F) + m^2 \mathop{\mathrm{ext}}\nolimits^1(G,G) + nm \mathop{\mathrm{ext}}\nolimits^1(F,G) + nm\mathop{\mathrm{ext}}\nolimits^1(G,F) - n^2. \]
\begin{proof}
Stability to one side of the wall implies $\mathop{\mathrm{Hom}}\nolimits(E,F) = 0$. Since $F$ is stable, we also know $\mathop{\mathrm{Hom}}\nolimits(F,F) = \ensuremath{\mathbb{C}}$. By the long exact sequence coming from applying $\mathop{\mathrm{Hom}}\nolimits(\cdot, F)$ to the above exact sequence, we get $\mathop{\mathrm{ext}}\nolimits^1(E, F) \leq m \mathop{\mathrm{ext}}\nolimits^1(G, F) + n \mathop{\mathrm{ext}}\nolimits^1(F, F) - n$. Moreover, we can use $\mathop{\mathrm{Hom}}\nolimits(\cdot, G)$ to get $\mathop{\mathrm{ext}}\nolimits^1(E, G) \leq m \mathop{\mathrm{ext}}\nolimits^1(G, G) + n \mathop{\mathrm{ext}}\nolimits^1(F, G)$. These two inequalities together with applying $\mathop{\mathrm{Hom}}\nolimits(E, \cdot)$ lead to the claim.
\end{proof}
\end{lem}
We also have to handle the issue of potentially new components after crossing a wall. The following result will solve this issue in some cases.
\begin{lem}
\label{lem:new_components}
Let $M$ and $N$ be two moduli spaces of Bridgeland semistable objects separated by a single wall. Assume that $A \subset M$ and $B \subset N$ are the loci destabilized at the wall. If $A$ intersects an irreducible component $H$ of $M$ non trivially and $H$ is not contained in $A$, then $B$ must intersect the closure of $H \backslash A$ inside $N$.
\begin{proof}
This follows from the fact that moduli spaces of Bridgeland semistable objects are universally closed. If $B$ would not intersect the closure of $H \backslash A$ inside $N$, then this would correspond to a component in $N$ that is not universally closed.
\end{proof}
\end{lem}
In order to identify the global structure of some of the moduli spaces as blow ups we need the following classical result by Moishezon. Recall that the analytification of a smooth proper algebraic spaces of finite type over $\ensuremath{\mathbb{C}}$ of dimension $n$ is a complex manifold with $n$ independent meromorphic functions.
\begin{thm}[\cite{Moi67}]
\label{thm:blow_up}
Any birational morphism $f: X \to Y$ between smooth proper algebraic spaces of finite type over $\ensuremath{\mathbb{C}}$ such that the contracted locus $E$ is irreducible and the image $f(E)$ is smooth is the blow up of $Y$ in $f(E)$.
\end{thm}
\begin{prop}
\label{prop:second_moduli}
The second moduli space $M_2$ is the blow up of $\ensuremath{\mathbb{G}}(1,9)$ along the smooth locus $\ensuremath{\mathbb{G}}(1,3) \times (\ensuremath{\mathbb{P}}^3)^{\vee}$ parametrizing pairs $(\ensuremath{\mathcal I}_L(-1), \ensuremath{\mathcal O}_V(-3))$. The center of the blow up parametrizes pencils whose base locus is not of dimension one. A generic point of the exceptional divisor parametrizes the union of a line and a plane cubic which intersect themselves at a point. The only non-ideal sheaves in the moduli space come from the case when the line is contained in the plane.
\begin{proof}
We know that $M_1$ is smooth. The wall separating $M_1$ and $M_2$ has strictly semistable objects given by extensions between $\ensuremath{\mathcal I}_L(-1)$ and $\ensuremath{\mathcal O}_V(-3)$. By Lemma \ref{lem:ext-computations} we have $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_L(-1), \ensuremath{\mathcal O}_V(-3)) = \ensuremath{\mathbb{C}}$, $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-3), \ensuremath{\mathcal I}_L(-1)) = \ensuremath{\mathbb{C}}^9$, $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-3), \ensuremath{\mathcal O}_V(-3)) = \ensuremath{\mathbb{C}}^3$, and $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_L(-1), \ensuremath{\mathcal I}_L(-1)) = \ensuremath{\mathbb{C}}^4$.
This means the locus of semistable objects occurring as extensions in $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_L(-1), \ensuremath{\mathcal O}_V(-3))$ for any $L$ and $V$ is isomorphic to $\ensuremath{\mathbb{G}}(1,3) \times (\ensuremath{\mathbb{P}}^3)^{\vee}$, i.e. is smooth and irreducible. By Lemma \ref{lem:wall_crossing} this is the locus destabilized at the wall in $\ensuremath{\mathbb{G}}(1,9)$. By Lemma \ref{lem:ext_estimate} any extension $E$ in $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-3), \ensuremath{\mathcal I}_L(-1))$ satisfies $\mathop{\mathrm{ext}}\nolimits^1(E,E) \leq 16$. Lemma \ref{lem:new_components} shows that $M_2$ has to be connected, i.e. is smooth and irreducible. The locus of semistable objects that can be written as extensions in $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-3), \ensuremath{\mathcal I}_L(-1))$ for any $L$ and $V$ is irreducible of dimension $15$, i.e. is a divisor in $M_2$. An immediate application of Theorem \ref{thm:blow_up} implies the fact that $M_2$ is the blow up of $\ensuremath{\mathbb{G}}(1,9)$ in the smooth locus $\ensuremath{\mathbb{G}}(1,3) \times (\ensuremath{\mathbb{P}}^3)^{\vee}$.
The description of the exceptional divisor is immediate from the fact that curves $C$ with ideal sheaves fitting into an exact sequence
\[
0 \to \ensuremath{\mathcal I}_L(-1) \to \ensuremath{\mathcal I}_C \to \ensuremath{\mathcal O}_V(-3) \to 0
\]
have to be unions of lines with a plane cubic intersecting in one point. If $L \subset V$, then no such extension can be an ideal sheaf, since the line would intersect the cubic in three points giving the wrong genus.
\end{proof}
\end{prop}
The next moduli space will acquire a second component.
\begin{prop}
\label{prop:third_moduli}
The third moduli space $M_3$ has two irreducible components $M_3^1$ and $M_3^2$. The first component $M_3^1$ is the blow up of $M_2$ in the smooth incidence variety parametrizing length two subschemes in a plane in $\ensuremath{\mathbb{P}}^3$. The second component $M_3^2$ is a $\ensuremath{\mathbb{P}}^{14}$-bundle over $\mathop{\mathrm{Hilb}}\nolimits^2(\ensuremath{\mathbb{P}}^3) \times (\ensuremath{\mathbb{P}}^3)^{\vee}$ parametrizing pairs $(\ensuremath{\mathcal I}_Z(-1)$, $\ensuremath{\mathcal O}_V(-4))$. It generically parametrizes unions of plane quartics with two generic points in $\ensuremath{\mathbb{P}}^3$. The two components intersect transversally along the exceptional locus of the blow up. The only non-ideal sheaves occur in the case where at least one of the two points is not scheme-theoretically contained in the plane.
\begin{proof}
By Lemma \ref{lem:ext-computations} we have
\begin{align*}
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_Z(-1), \ensuremath{\mathcal O}_V(-4)) &= \begin{cases} \ensuremath{\mathbb{C}} &, \ Z \subset V \\ 0 &, \text{ otherwise,} \end{cases} \\
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-4), \ensuremath{\mathcal I}_Z(-1)) &= \ensuremath{\mathbb{C}}^{15}.
\end{align*}
This means the locus destabilized in $M_2$ is of dimension $7$, and the new locus appearing in $M_3$ is of dimension $23$. Since $M_2$ is of dimension $16$, the locus appearing in $M_3$ must be a new component $M_3^2$. The closure of what is left of $M_2$ is denoted by $M_3^1$. If $M_3^2$ is reduced, it is a $\ensuremath{\mathbb{P}}^{14}$-bundle over $\mathop{\mathrm{Hilb}}\nolimits^2(\ensuremath{\mathbb{P}}^3) \times (\ensuremath{\mathbb{P}}^3)^{\vee}$ parametrizing pairs $(\ensuremath{\mathcal I}_Z(-1)$, $\ensuremath{\mathcal O}_V(-4))$. We will more strongly show that it is smooth.
Assume $Z$ is not scheme theoretically contained in $V$. Then Lemma \ref{lem:ext_estimate} implies that any non-trivial extension $E$ in $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-4), \ensuremath{\mathcal I}_Z(-1))$ satisfies $\mathop{\mathrm{ext}}\nolimits^1(E,E) \leq 23$. Therefore, it is a smooth point and can in particular not lie on $M_3^1$. Let $E$ be an extensions of the form $0 \to \ensuremath{\mathcal I}_Z(-1) \to E \to \ensuremath{\mathcal O}_V(-4) \to 0$, where $Z \subset V$. Any point on the intersection must satisfy $\mathop{\mathrm{ext}}\nolimits^1(E,E) \geq 24$. Assume $E$ is not an ideal sheaf. If $E$ fits into an exact sequence $0 \to \ensuremath{\mathcal I}_{Z/V}(-4) \to E \to \ensuremath{\mathcal O}(-1) \to 0$ or $0 \to \ensuremath{\mathcal I}_{Q/V}(-4) \to E \to \ensuremath{\mathcal I}_P(-1) \to 0$ for $P \neq Q$, then a direct application of Lemma \ref{lem:ext_estimate} to these sequences shows $\mathop{\mathrm{ext}}\nolimits^1(E,E) \leq 23$, a contradiction. Therefore, $E$ must fit into an exact sequence $0 \to \ensuremath{\mathcal I}_{P/V}(-4) \to E \to \ensuremath{\mathcal I}_P(-1) \to 0$. Then we have the following commutative diagram with short exact rows and columns.
\centerline{
\xymatrix{
0 \ar@{^{(}->}[r] \ar@{^{(}->}[d] & \ensuremath{\mathcal I}_{P/V}(-4) \ar@{->>}[r] \ar@{^{(}->}[d] & \ensuremath{\mathcal I}_{P/V}(-4) \ar@{^{(}->}[d] \\
\ensuremath{\mathcal I}_Z(-1) \ar@{^{(}->}[r] \ar@{->>}[d] & E \ar@{->>}[r] \ar@{->>}[d] & \ensuremath{\mathcal O}_V(-4) \ar@{->>}[d] \\
\ensuremath{\mathcal I}_Z(-1) \ar@{^{(}->}[r] & \ensuremath{\mathcal I}_P(-1) \ar@{->>}[r] & \ensuremath{\mathcal O}_P \\
}}
Therefore, $Z$ has to be a double point supported at $P$. By Lemma \ref{lem:ext-computations} we have
\begin{align*}
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_Z(-1), \ensuremath{\mathcal I}_{P/V}(-4)) = \ensuremath{\mathbb{C}}^3 &, \ \mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-4)), \ensuremath{\mathcal I}_{P/V}(-4)) = \ensuremath{\mathbb{C}}^2, \\
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_Z(-1), \ensuremath{\mathcal I}_P(-1)) = \ensuremath{\mathbb{C}}^3 &, \ \mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}_V(-4)), \ensuremath{\mathcal I}_P(-1)) = \ensuremath{\mathbb{C}}^{15}.
\end{align*}
Next, we apply $\mathop{\mathrm{Hom}}\nolimits(\cdot, \ensuremath{\mathcal I}_{P/V}(-4))$ to $0 \to \ensuremath{\mathcal I}_Z(-1) \to E \to \ensuremath{\mathcal O}_V(-4) \to 0$ to get $\mathop{\mathrm{ext}}\nolimits^1(E, \ensuremath{\mathcal I}_{P/V}(-4)) \leq 5$. By applying $\mathop{\mathrm{Hom}}\nolimits(\cdot, \ensuremath{\mathcal I}_P(-1))$ to the same sequence we get $\mathop{\mathrm{ext}}\nolimits^1(E, \ensuremath{\mathcal I}_P(-1)) \leq 18$. Finally, we can apply $\mathop{\mathrm{Hom}}\nolimits(E, \cdot)$ to $0 \to \ensuremath{\mathcal I}_{P/V} \to E \to \ensuremath{\mathcal I}_P(-1) \to 0$ to get $\mathop{\mathrm{ext}}\nolimits^1(E,E) \leq 23$.
Therefore, the intersection of $M_3^1$ and $M_3^2$ parametrizes some of the ideals fitting into an exact sequence $0 \to \ensuremath{\mathcal I}_Z(-1) \to I_C \to \ensuremath{\mathcal O}_V(-4) \to 0$, where $Z \subset V$. The intersection must have a closed orbit. By Proposition \ref{prop:ideal_sequence_three}, there is precisely one such closed orbit. If the intersection was disconnected, it would have at least two closed orbits. If it is reducible, then the closed orbit must lie on the intersection of all irreducible components. By Proposition \ref{prop:piene-schlessinger} the intersection along the closed orbit is transversal of dimension $15$, and its points are smooth on both components. That would be impossible if the intersection is not irreducible at the closed orbit. The singular locus on either component is closed and must therefore contain a closed orbit. Thus, the whole intersection must consist of points that are smooth on each of the two components individually. The induced map $M_3^1 \to M_2$ contracts the intersection, which is an irreducible divisor, onto a locus isomorphic to the smooth incidence variety parametrizing length two subschemes in a plane in $\ensuremath{\mathbb{P}}^3$. Theorem \ref{thm:blow_up} implies the description of $M_3^1$.
The description of the curves parametrized by $M_3^2$ is again a consequence of the exact sequence that the ideal sheaves fit into.
\end{proof}
\end{prop}
In order to reprove the description of the main component of the Hilbert scheme from \cite{VA92}, we have to make sure that none of the remaining walls modify the first component.
\begin{prop}
The fourth moduli space $M_4$ has two irreducible components $M_4^1$ and $M_4^2$. The first component is equal to $M_3^1$. The second component is birational to $M_3^2$.
\begin{proof}
Lemma \ref{lem:ext-computations} says
\begin{align*}
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_P(-1), \ensuremath{\mathcal I}_{Q/V}(-4)) &= \begin{cases} \ensuremath{\mathbb{C}}^3 &, \ P = Q \\ \ensuremath{\mathbb{C}} &, \ P \neq Q \end{cases}, \\
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_{Q/V}(-4), \ensuremath{\mathcal I}_P(-1)) &= \begin{cases} \ensuremath{\mathbb{C}}^{17} &, \ P = Q \\ \ensuremath{\mathbb{C}}^{15} &, \ P \neq Q \end{cases}.
\end{align*}
Moreover, the moduli space of pairs $(\ensuremath{\mathcal I}_P(-1), \ensuremath{\mathcal I}_{Q/V}(-4))$ is irreducible of dimension $8$, while the sublocus where $P = Q$ is of dimension $5$. Therefore, the closure of the locus of extensions in $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_{Q/V}(-4), \ensuremath{\mathcal I}_P(-1))$ for $P \neq Q$ is irreducible of dimension $22$. The locus of extensions in $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_{P/V}(-4), \ensuremath{\mathcal I}_P(-1))$ for $P \in V$ is irreducible of dimension $21$. Let $M_4^1$ be the closure of what is left from $M_3^1$ in $M_4$ and $M_4^2$ be the closure of what is left from $M_3^2$.
If $P \neq Q$, then Lemma \ref{lem:ext_estimate} implies smoothness. In particular, we can use Lemma \ref{lem:new_components} to show that all points in $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_{Q/V}(-4), \ensuremath{\mathcal I}_P(-1))$ for $P \neq Q$ are in $M_4^2$ and no other component. Assume we have a general non trivial extension $0 \to \ensuremath{\mathcal I}_P(-1) \to E \to \ensuremath{\mathcal I}_{P/V}(-4) \to 0$. Then $E = I_C$ is an ideal sheaf of a plane quartic curve plus a double point in the plane. We can assume that the double point is not an embedded point due to the fact that $E$ is general. Clearly, $I_C$ is the flat limit of elements in $\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_{Q/V}(-4), \ensuremath{\mathcal I}_P(-1))$ by choosing $P \notin V$ and regarding the limit $P \to Q$. Therefore, $E$ is a part of $M_4^2$.
We showed $M_4 = M_4^1 \cup M_4^1$ and that $M_4^2$ is birational to $M_3^2$. We are left to show $M_4^1 = M_4^2$. If not, there is an object $E$ with a non trivial exact sequence $0 \to \ensuremath{\mathcal I}_P(-1) \to E \to \ensuremath{\mathcal I}_{P/V}(-4) \to 0$ in $M_4^1$. By Lemma \ref{lem:new_components} this implies that there is also an object $E'$ with non trivial exact sequence $0 \to \ensuremath{\mathcal I}_{P/V}(-4) \to E' \to \ensuremath{\mathcal I}_P(-1) \to 0$ lying on $M_3^1$. But we already established that all those extensions are smooth points on $M_3^2$ in the previous proof.
\end{proof}
\end{prop}
We can now prove the following theorem.
\begin{thm}
\label{thm:last_wall}
The Hilbert scheme $\mathop{\mathrm{Hilb}}\nolimits^{4t}(\ensuremath{\mathbb{P}}^3)$ has two components $\mathop{\mathrm{Hilb}}\nolimits^{4t}_1$ and $\mathop{\mathrm{Hilb}}\nolimits^{4t}_2$. The main component $\mathop{\mathrm{Hilb}}\nolimits^{4t}_1$ contains an open subset of elliptic quartic curves and is a smooth double blow up of the Grassmannian $\ensuremath{\mathbb{G}}(1,9)$. The second component is of dimension $23$. Moreover, the two components intersect transversally in a locus of dimension $15$.
\begin{proof}
By Lemma \ref{lem:ext-computations} we have
\begin{align*}
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal O}(-1), \ensuremath{\mathcal I}_{Z'/V}(-4)) = \ensuremath{\mathbb{C}}^2, \\
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_{Z'/V}(-4), \ensuremath{\mathcal O}(-1)) = \ensuremath{\mathbb{C}}^{15}, \\
\mathop{\mathrm{Ext}}\nolimits^1(\ensuremath{\mathcal I}_{Z'/V}(-4), \ensuremath{\mathcal I}_{Z'/V}(-4)) = \ensuremath{\mathbb{C}}^7.
\end{align*}
The moduli space of objects $\ensuremath{\mathcal I}_{Z'/V}$ is irreducible of dimension $5$. Lemma \ref{lem:ext_estimate} implies that all strictly semistable objects at the largest wall are smooth points on either $M_4$ or $M_5 = \mathop{\mathrm{Hilb}}\nolimits^{4t}(\ensuremath{\mathbb{P}}^3)$. Therefore, we can again use Lemma \ref{lem:new_components} to see that $\mathop{\mathrm{Hilb}}\nolimits^{4t}(\ensuremath{\mathbb{P}}^3)$ has exactly two components birational to $M_4^1$ and $M_4^2$. Moreover, this argument shows that the ideals that destabilize at the largest wall cannot lie on the intersection of the two components and we have $M_5^1 = M_4^1$.
\end{proof}
\end{thm}
We denote the exceptional divisor of the first blow up of the main component by $E_1$ and the exceptional divisor of the second blow up by $E_2$. We finish this section by describing which curves actually lie in $E_1$ and $E_2$.
\begin{prop}
\label{prop:E1_E2_description}
The general point in $E_1$ parametrizes subschemes of $\ensuremath{\mathbb{P}}^3$ that are the union of a plane cubic and an incident line. The general point in $E_2$ parametrizes subschemes of $\ensuremath{\mathbb{P}}^3$ that are plane quartics with two nodes and two embedded points at such nodes.
\begin{proof}
By Corollary \ref{cor:allwalls}, any ideal sheaf $I_C$ of a scheme in $E_1$ fits into an exact sequence of the form $0 \to \ensuremath{\mathcal I}_L(-1) \to \ensuremath{\mathcal I}_C \to \ensuremath{\mathcal O}_V(-3) \to 0$, where $L \subset \ensuremath{\mathbb{P}}^3$ is a line and $V \subset \ensuremath{\mathbb{P}}^3$ is a plane. By Proposition \ref{prop:second_moduli} the reverse holds, i.e. all ideal sheaves fitting into this sequence correspond to curves in $E_1$. For the general element in $E_1$ the line $L$ is not contained $V$. Then the above sequence implies that $C \subset L \cup V$. If $C \subset V$, then there would be a morphism $\ensuremath{\mathcal O}(-1) \to \ensuremath{\mathcal I}_C$ destabilizing the curve earlier, a contradiction. Thus, $L$ is an irreducible component of $C$ and another component of degree $3$ lies in $V$.
By Theorem \ref{thm:last_wall}, the last two walls do not modify the main component. Therefore, Corollary \ref{cor:allwalls} implies that any ideal sheaf $I_C$ of a scheme in $E_2$ fits into an exact sequence of the form $0 \to \ensuremath{\mathcal I}_Z(-1) \to \ensuremath{\mathcal I}_C \to \ensuremath{\mathcal O}_V(-4) \to 0$, where $Z \subset \ensuremath{\mathbb{P}}^3$ is a zero dimensional subscheme of length $2$ and $V \subset \ensuremath{\mathbb{P}}^3$ is a plane. This implies that $C$ is plane quartic curve plus two points. The two points have to be embedded, since otherwise the curve cannot be smoothened. Moreover, a classical result by Hironaka \cite[p. 360]{Hir58} implies that the two embedded points must occur at singularities of the plane quartic.
\end{proof}
\end{prop}
\section{Effective divisors of the Principal Component $\mathop{\mathrm{Hilb}}\nolimits_1^{4t}$}
In this section we compute the cone of effective divisors $\mathop{\mathrm{Eff}}\nolimits(\mathop{\mathrm{Hilb}}\nolimits_1^{4t})$, where $\mathop{\mathrm{Hilb}}\nolimits_1^{4t}$ denotes the principal component of the Hilbert scheme $\mathop{\mathrm{Hilb}}\nolimits^{4t}(\ensuremath{\mathbb{P}}^3)$.
By Theorem B, there is an isomorphism $\mathop{\mathrm{Pic}}\nolimits(\mathop{\mathrm{Hilb}}\nolimits_1^{4t})\cong \ensuremath{\mathbb{Z}}^3$, with generators $H$, $E_1$ and $E_2$. Here, $H$ denotes the pullback of the class $\sigma_1\in A^1(\ensuremath{\mathbb{G}}(1,9))$, whereas $E_1$ is the exceptional divisor of the first blow up and $E_2$ is the exceptional divisor of the second blow up. Set-theoretically, $E_1$ is the closure, in $\mathop{\mathrm{Hilb}}\nolimits_1^{4t}$, of the locus parametrizing subschemes of $\ensuremath{\mathbb{P}}^3$ that consist of a smooth plane cubic with an incident line. Moreover, $E_2$ is the closure, in $\mathop{\mathrm{Hilb}}\nolimits_1^{4t}$, of the locus parametrizing plane quartics with two nodes and two embedded points at such nodes.
As a consequence of Theorem B, we also have that $\mathop{\mathrm{Pic}}\nolimits(\mathop{\mathrm{Hilb}}\nolimits_1^{4t}) \otimes \ensuremath{\mathbb{Q}} \cong N^1(\mathop{\mathrm{Hilb}}\nolimits_1^{4t})\otimes \ensuremath{\mathbb{Q}}$, where $N^1(\mathop{\mathrm{Hilb}}\nolimits_1^{4t})\otimes \ensuremath{\mathbb{Q}}$ denotes the N\'eron-Severi group of Cartier divisors with rational coefficients up to numerical equivalence.
In order to describe the cone of effective divisors $\mathop{\mathrm{Eff}}\nolimits(\mathop{\mathrm{Hilb}}\nolimits_1^{4t})$, we need an additional divisor $\Delta$ defined as the closure of the locus of irreducible nodal elliptic quartics.
\begin{thm:cones}
The cone of effective divisors of $\mathop{\mathrm{Hilb}}\nolimits_1^{4t}$ is generated by $\mathop{\mathrm{Eff}}\nolimits(\mathop{\mathrm{Hilb}}\nolimits_1^{4t})=\langle\Delta, E_1,E_2\rangle$.
\end{thm:cones}
The strategy of the proof is to construct a dual basis of curves to $\Delta$, $E_1$, and $E_2$ in $N_1(\mathop{\mathrm{Hilb}}\nolimits_1^{4t})$, the space of 1-cycles up to numerical equivalence. Since the closure of the convex cone of movable curves is dual to the effective cone, we will then observe that these curves are movable; which allows us to conclude the proof. In our context, a curve in a smooth algebraic variety $X$ is called \emph{movable}, if it lies in a family that covers a dense open subset of $X$. We refer the reader to \cite{BDPP13} for a detailed exposition on movable curves.
Before proceeding with the proof, we will define and describe some properties of our movable curves. Let $Q\subset \ensuremath{\mathbb{P}}^3$ be a a fixed smooth quadric. Then, the curve $\gamma_1$ is a general pencil in $|\mathcal{O}_Q(2)|$. This curve is movable because a generic curve parametrized by $\mathop{\mathrm{Hilb}}\nolimits_1^{4t}$ is the transversal intersection of two quadric hypersurfaces $Q_1,Q_2$ where we can assume one of these quadrics is smooth. Moreover, by construction $\gamma_1 \cdot E_1=\gamma_1 \cdot E_2=0$.
On the other hand, the intersection $\gamma_1 \cdot \Delta=12$ holds. This follows from the fact that the parameter space of plane curves of degree $d$ in $\ensuremath{\mathbb{P}}^2$ contains a divisor of degree $3(d-1)^2$ of singular curves (see \cite[Ch 13.D]{GKZ08}). The following geometric argument is self contained.
The base locus of a general pencil in $|\mathcal{O}_Q(2)|$, where $Q$ stands for a smooth quadric, consists of $8$ points. This means that the total space of this pencil $\mathcal{X}$ is the blowup of $Q$ on these $8$ points, and this implies that its topological Euler characteristic $\chi_{\tiny{top}}(\mathcal{X})=12$. Observe that the pencil $\mathcal{X}$ is not a fibration over $\ensuremath{\mathbb{P}}^1$ due to the presence of singular fibers: if $\mathcal{X}$ were a fibration over $\ensuremath{\mathbb{P}}^1$, then the topological Euler characteristic $\chi_{\tiny{top}}(\mathcal{X})$ would be zero. This means that we may count the singular fibers of $\mathcal{X}$ (which are the singular members of the pencil), by computing the topological Euler characteristic $\chi_{\tiny{top}}(\mathcal{X})$. Since we are considering a general pencil, Bertini's Theorem guarantees that the singular fibers of $\mathcal{X}$ are all nodal curves.
We now define two more curves $\gamma_2$ and $\gamma_3$. Let $\Lambda_1$ and $\Lambda_2$ be two $3$-planes in $\ensuremath{\mathbb{P}}^7$. Let $s: \ensuremath{\mathbb{P}}^3 \times \ensuremath{\mathbb{P}}^1 \to \ensuremath{\mathbb{P}}^7$ be the Segre embedding, and for any $t \in \ensuremath{\mathbb{P}}^1$ we write $s_t: \ensuremath{\mathbb{P}}^3 \to \ensuremath{\mathbb{P}}^7$ for the restriction of $s$ to $\ensuremath{\mathbb{P}}^3 \times \{ t \}$.
We have a projection $\pi: \ensuremath{\mathbb{P}}^7 \backslash \Lambda_1 \to \Lambda_2$. To summarize, we have the following diagram of maps with vertical projections
\[
\xymatrix{
\ensuremath{\mathbb{P}}^3 \times \ensuremath{\mathbb{P}}^1 \ar[r]^{s \times \mathop{\mathrm{id}}\nolimits} \ar[d] & \ensuremath{\mathbb{P}}^7 \times \ensuremath{\mathbb{P}}^1 \ar@{-->}[r]^{\pi \times \mathop{\mathrm{id}}\nolimits} \ar[d] & \Lambda_2 \times \ensuremath{\mathbb{P}}^1 \ar[d] \\
\ensuremath{\mathbb{P}}^3 \ar[r]^{s_t} & \ensuremath{\mathbb{P}}^7 \ar@{-->}[r]^{\pi} & \Lambda_2 \cong \ensuremath{\mathbb{P}}^3.
}
\]
Observe that both $s_t$ and $\pi$ are linear maps.
\begin{lem}
\label{lem:isomorphic_projection}
Let $t \in \ensuremath{\mathbb{P}}^1$, and let $\Lambda_2$ be general. If $\Lambda_1 \cap s_t(\ensuremath{\mathbb{P}}^3) = \emptyset$, then $\pi \circ s_t$ is an isomorphism. If $\Lambda_1 \cap s_t(\ensuremath{\mathbb{P}}^3)$ is a point, then the image of $\pi \circ s_t$ is a plane in $\Lambda_2$.
\begin{proof}
The image of $\pi \circ s_t$ is the intersection of $\Lambda_2$ with the linear subspace generated by $\Lambda_1$ and $s_t(\ensuremath{\mathbb{P}}^3)$.
\end{proof}
\end{lem}
The image of the Segre embedding $s(\ensuremath{\mathbb{P}}^3 \times \ensuremath{\mathbb{P}}^1)$ has degree four. Hence, $\Lambda_1$ can be chosen general such that it intersects the Segre embedding in exactly four points. If we also choose $\Lambda_2$ general, then by Lemma \ref{lem:isomorphic_projection}, we have that $\pi \circ s_t: \ensuremath{\mathbb{P}}^3 \to \Lambda_2 \cong \ensuremath{\mathbb{P}}^3$ is an isomorphism except for four values.
\begin{defn}
Let $E$ be a smooth elliptic quartic in $\ensuremath{\mathbb{P}}^3$. Let $\Lambda_2$ be a general $3$-plane in $\ensuremath{\mathbb{P}}^7$.
\begin{enumerate}
\item Let $\Lambda_1$ be another general $3$-plane in $\ensuremath{\mathbb{P}}^7$. Then $\gamma_2$ is the image $(\pi \times \mathop{\mathrm{id}}\nolimits) \circ (s \times \mathop{\mathrm{id}}\nolimits) (E \times \ensuremath{\mathbb{P}}^1)$. It is a flat family of smooth curves isomorphic to $E$ everywhere, except for four special fibers.
\item Consider four general points in $s(E \times \ensuremath{\mathbb{P}}^1)$ and let $\Lambda'_1$ be the unique $3$-plane generated by them. Then $\gamma_3$ is the image $(\pi \times \mathop{\mathrm{id}}\nolimits) \circ (s \times \mathop{\mathrm{id}}\nolimits) (E \times \ensuremath{\mathbb{P}}^1)$. It is a flat family of smooth curves isomorphic to $E$ everywhere except for four special fibers.
\end{enumerate}
\end{defn}
\begin{lem}
\label{lem:singular_fibers_curves}
The four singular fibers for $\gamma_2$ are plane quartic curves with only two nodes and embedded points at them. For $\gamma_3$ these four fibers are smooth plane cubic curves together with an incident line. Both $\gamma_2$ and $\gamma_3$ are movable.
\begin{proof}
Let $t \in \ensuremath{\mathbb{P}}^1$ correspond to one of the four singular fibers of $\gamma_2$. Since $\Lambda_1$ is chosen general it will not intersect $s(E \times \ensuremath{\mathbb{P}}^1)$. Therefore, Lemma \ref{lem:isomorphic_projection} implies that the image $\pi(s_t(E))$ is a plane curve. Since $\pi \circ s_t$ is defined on all of $E$, the set-theoretic support of $\gamma_2$ at $t$ is a plane curve of degree four with $2$ nodes and no other singularities. Hence, we get a plane quartic with two embedded points at the only $2$ nodes.
Let $t \in \ensuremath{\mathbb{P}}^1$ correspond to one of the four singular fibers $\gamma_3$. By definition the intersection of $\Lambda'_1$ with $E \times \ensuremath{\mathbb{P}}^1$ contains four points one of which is of the form $(x,t)$. Choose a plane $\ensuremath{\mathbb{P}}^2 \subset \Lambda'_1$ that does not intersect the Segre embedding $s(\ensuremath{\mathbb{P}}^3 \times \ensuremath{\mathbb{P}}^1)$ and a general $\ensuremath{\mathbb{P}}^4 \subset \ensuremath{\mathbb{P}}^7$. Then the projection of $s_t(\ensuremath{\mathbb{P}}^3)$ away from $\ensuremath{\mathbb{P}}^2$ onto $\ensuremath{\mathbb{P}}^4$ is the intersection of this $\ensuremath{\mathbb{P}}^4$ with the linear span of $s_t(\ensuremath{\mathbb{P}}^3)$ and $\ensuremath{\mathbb{P}}^2$ which is a $\ensuremath{\mathbb{P}}^6$. In particular, it is of dimension $3$, i.e. $E$ is projected isomorphically onto $\ensuremath{\mathbb{P}}^3 \subset \ensuremath{\mathbb{P}}^4$. Let $P \in \ensuremath{\mathbb{P}}^4$ be the image of $(x,t)$ via this projection. Then we project from this point onto a general $\Lambda_2 \subset \ensuremath{\mathbb{P}}^4$. The image is isomorphic to $E$. Hence, we get in $\mathop{\mathrm{Hilb}}\nolimits_1^{4t}$ a smooth plane cubic together with an incident line.
Both curve classes $\gamma_2$ and $\gamma_3$ are movable. Indeed, every smooth curve parametrized in $\mathop{\mathrm{Hilb}}\nolimits_1^{4t}$ is contained in some representative of $\gamma_2$ and $\gamma_3$ by varying the curve $E$ used to define them.
\end{proof}
\end{lem}
\begin{proof}[Proof of Theorem C]
Since $E_1$, $E_2$, and $\Delta$ are effective, we only need to show the containment $\mathop{\mathrm{Eff}}\nolimits(\mathop{\mathrm{Hilb}}\nolimits_1^{4t})\subset \langle E_1, E_2,\Delta\rangle$. Observe that this latter containment is equivalent to $\langle E_1,E_2,\Delta\rangle^{\vee} \subset \mathop{\mathrm{Eff}}\nolimits(\mathop{\mathrm{Hilb}}\nolimits_1^{4t})^{\vee}$ of dual cones. Since $\mathop{\mathrm{Eff}}\nolimits(\mathop{\mathrm{Hilb}}\nolimits_1^{4t})^{\vee}$ is the cone of movable curves, it suffices to exhibit that the dual cone $\langle E_1, E_2, \Delta\rangle^{\vee}$ is generated by movable curves. We already proved that $\gamma_1,\gamma_2,\gamma_3$ are movable curves. We will
This means we are left to show they generate the dual cone $\langle E_1$, $E_2, \Delta\rangle^{\vee}$. It suffices to check that the following intersection numbers hold. Note that for our purposes it is enough to show that the intersections are zero or positive, and therefore, we will skip proving that the intersections are transversal.
\begin{equation*}
\begin{aligned}
\gamma_1 \cdot E_1=0,\quad & \gamma_2 \cdot E_1=0, \quad & \gamma_3 \cdot E_1=4, \\
\gamma_1 \cdot E_2=0, \quad & \gamma_2 \cdot E_2=4, \quad & \gamma_3 \cdot E_2 =0,\\
\gamma_1 \cdot \Delta=12,\quad & \gamma_2 \cdot \Delta=0, \quad & \gamma_3 \cdot \Delta=0. \\
\end{aligned}
\end{equation*}
The intersections with $E_1$ and $E_2$ follow directly from the definitions and Lemma \ref{lem:singular_fibers_curves}. The intersection numbers $\gamma_1 \cdot \Delta = 12$ is also discussed above. We are left to show $\gamma_2 \cdot \Delta=\gamma_3 \cdot \Delta=0$.
Suppose $\gamma_2 \cdot \Delta \neq 0$. Then there is a flat family $\pi: S \to Z$ for a smooth curve $Z$ such that for general $z \in Z$ the fiber $S_z$ is a nodal complete intersection in $\Delta$, and the special fiber $S_0$ is a curve in $\gamma_2 \cap E_2$. Therefore, $S_0$ is a plane quartic curve with exactly two nodes and simple embedded points at both nodes. The normalization $\tilde{S}$ smooths out the nodes in the general fibers by making them into $\ensuremath{\mathbb{P}}^1$. By \cite[Theorem III.7]{BEAU} this means $\tilde{S}$ is birational over $Z$ to $\ensuremath{\mathbb{P}}^1 \times Z$. We can resolve the rational map from $\ensuremath{\mathbb{P}}^1 \times Z$ to $S$ by successively blowing up points. That leads to a family $X \to Z$ factoring through $S \to Z$ such that every fiber is a union of rational curves $\ensuremath{\mathbb{P}}^1$. That means the special fiber $S_0$ is the set theoretic image of such a union of rational curves. Every $\ensuremath{\mathbb{P}}^1$ must map to the normalization of the reduced structure of $S_0$. But the normalization of the reduced structure of $S_0$ is a smooth curve of genus $1$, and $\ensuremath{\mathbb{P}}^1$ has no non trivial maps to an elliptic curve.
Suppose $\gamma_3 \cdot \Delta \neq 0$. Then there is a flat family $\pi: S \to Z$ for a smooth curve $Z$ such that for general $z \in Z$ the fiber $S_z$ is a nodal complete intersection in $\Delta$, and the special fiber $S_0$ is a curve in $\gamma_3 \cap E_1$. This means $S_0$ is the union of a smooth plane cubic with an incident line. With the exact same argument as for $\gamma_2$, we can create a family $X \to Z$ factoring through $S \to Z$ such that every fiber is a union of rational curves $\ensuremath{\mathbb{P}}^1$. As previously, the special fiber $S_0$ is the image of such a union of rational curves. Since there is no non-trivial map from $\ensuremath{\mathbb{P}}^1$ to any elliptic curve, they must all map to the incident line, a contradiction.
\end{proof}
\section*{Appendix A: Macaulay2 Code}
This appendix contains all Macaulay2 code used in Proposition \ref{prop:piene-schlessinger} and Lemma \ref{lem:ext-computations}.
{\tiny
\begin{multicols}{2}
\begin{lstlisting}
-------------------------------------
-- Computation for Proposition 3.2 --
-------------------------------------
needsPackage "VersalDeformations";
S = QQ[x,y,z,w];
F0 = matrix {{x^2,x*y,x*z^2,y^4}};
(F,R,G,C) = localHilbertScheme(F0,Verbose=>2);
T = ring first G;
sum G
-------------------------------
-- Computation for Lemma 4.3 --
-------------------------------
-- A = O_V(-3)
-- B = I_L(-1) L is not contained in V
-- C = I_L(-1) L is contained in V
X = Proj(QQ[x,y,z,w]);
A = (OO_X(0) / sheaf module ideal(x))**OO_X(-3);
B = (sheaf module ideal (y,z))**OO_X(-1);
C = (sheaf module ideal (x,y))**OO_X(-1);
Ext^1(B,A)
Ext^1(C,A)
Ext^1(A,B)
Ext^1(A,C)
Ext^1(A,A)
Ext^1(B,B)
Ext^1(C,C)
-------------------------------
-- A = O_V(-4)
-- B = I_Z(-1) Two separate points
-- outside V
-- C = I_Z(-1) Double point outside V
-- D = I_Z(-1) One point inside,
-- one point outside V
-- E = I_Z(-1) Two separate points
-- inside V
-- F = I_Z(-1) Double point scheme
-- theoretically in V
-- G = I_Z(-1) Double point set but
-- not scheme theoretically in V
X = Proj(QQ[x,y,z,w]);
A = (OO_X(0) / sheaf module ideal(x))**OO_X(-4);
B = (sheaf module ideal (y*(x-y),z,w))**OO_X(-1);
C = (sheaf module ideal (y^2,z,w))**OO_X(-1);
D = (sheaf module ideal (x*y,z,w))**OO_X(-1);
E = (sheaf module ideal (x,y*z,w))**OO_X(-1);
F = (sheaf module ideal (x,y,z^2))**OO_X(-1);
G = (sheaf module ideal (x^2,y,z))**OO_X(-1);
Ext^1(A,B)
Ext^1(A,C)
Ext^1(A,D)
Ext^1(A,E)
Ext^1(A,F)
Ext^1(A,G)
Ext^1(B,A)
Ext^1(C,A)
Ext^1(D,A)
Ext^1(E,A)
Ext^1(F,A)
Ext^1(G,A)
Ext^1(A,A)
Ext^1(B,B)
Ext^1(C,C)
-------------------------------
-- A = I_{Q/V}(-4)
-- B = I_P(-1) P \notin V
-- C = I_P(-1) P \in V, P \neq Q
-- D = I_P(-1) P = Q
X = Proj(QQ[x,y,z,w]);
A = (sheaf module ideal (x,y,z)
/ sheaf module ideal(x))**OO_X(-4);
B = (sheaf module ideal (y,z,w))**OO_X(-1);
C = (sheaf module ideal (x,y,w))**OO_X(-1);
D = (sheaf module ideal (x,y,z))**OO_X(-1);
Ext^1(A,B)
Ext^1(A,C)
Ext^1(A,D)
Ext^1(B,A)
Ext^1(C,A)
Ext^1(D,A)
Ext^1(A,A)
Ext^1(B,B)
Ext^1(D,D)
-------------------------------
-- A = O(-1)
-- B = I_{Z'/V}(-4) Two separate points
-- C = I_{Z'/V}(-4) Double point
X = Proj(QQ[x,y,z,w]);
A = OO_X(-1);
B = (sheaf module ideal (x,y,z^2)
/ sheaf module ideal(x))**OO_X(-4);
C = (sheaf module ideal (x,y,w*z)
/ sheaf module ideal(x))**OO_X(-4);
Ext^1(B,A)
Ext^1(C,A)
Ext^1(A,B)
Ext^1(A,C)
Ext^1(B,B)
Ext^1(C,C)
-------------------------------
-- A = I_Z(-1), Z \subset V double point at P
-- B = O_V(-4)
-- C = I_{P/V}(-4)
-- D = I_P(-1)
X = Proj(QQ[x,y,z,w]);
A = (sheaf module ideal(x,y,z^2))**OO_X(-1);
B = (OO_X(0) / sheaf module ideal(x))**OO_X(-4);
C = (sheaf module ideal(x,y,z)
/ sheaf module ideal(x))**OO_X(-4);
D = (sheaf module ideal(x,y,z))**OO_X(-1);
Ext^1(A,C)
Ext^1(B,C)
Ext^1(A,D)
Ext^1(B,D)
\end{lstlisting}
\end{multicols}
}
|
2,877,628,090,063 | arxiv | \section{Introduction}
For a polynomial $f(x_1, \ldots, x_n)$ with rational coefficients and an integer $a$, we say that $f$ represents $a$ if the diophantine equation
\begin{equation} \label{1steqn}
f(x_1, \ldots, x_n) = a
\end{equation}
is soluble in the integers. The {\em representation problem} asks for a complete determination of the set of integers represented by a given polynomial. This problem is considered to be untractable in general in view of Matiyasevich's negative answer to Hilbert's tenth problem \cite{ma}. Moreover, Jones \cite{j} has shown that whether a general single diophantine equation of degree four or higher is soluble in the positive integers is already undecidable. However, the linear and the quadratic cases have been studied extensively. The linear case is elementary and its solution is a consequence of the Euclidean algorithm. For the quadratic case, the representation problem for homogeneous quadratic polynomials, or quadratic forms in other words, has a long history and it still garners a lot of attention from mathematicians across many areas. For accounts of more recent development of the subject, the readers are referred to the surveys \cite{h, sp} and the references therein. In this paper, we will discuss a couple of questions which are related to the representation problem of quadratic polynomials in general, namely {\em universality} and {\em regularity}, which we will explain below.
A quadratic polynomial $f(\bx) = f(x_1, \ldots, x_n)$ can be written as
$$f(\bx) = Q(\bx) + L(\bx) + c$$
where $Q(\bx)$ is a quadratic form, $L(\bx)$ is a linear form, and $c$ is a constant. Unless stated otherwise {\em we assume that $Q$ is positive definite}. This in particular implies that there exists a unique vector $\bv \in \q^n$ such that $L(\bx) = 2B(\bv, \bx)$, where $B$ is the bilinear form such that $B(\bx, \bx) = Q(\bx)$. As a result,
$$f(\bx) = Q(\bx + \bv) - Q(\bv) + c \geq -Q(\bv) + c,$$
and so $f(\bx)$ attains an absolute minimum on $\z^n$. We denote this minimum by $m_f$ and will simply call it the minimum of $f(\bx)$. We call $f(\bx)$ {\em positive} if $m_f \geq 0$.
In this paper, we call a quadratic polynomial $f(\bx)$ {\em integral} if it is integer-valued, that is, $f(\bx) \in \z$ for all $\bx \in \z^n$. A positive integral quadratic polynomial $f(\bx)$ is called {\em universal} if it represents all nonnegative integers. Positive definite universal integral quadratic forms have been studied for many years by many authors and have become a popular topic in the recent years. It is known that positive definite universal integral quadratic forms must have at least four variables, and there are only finitely many equivalence classes of such universal quadratic forms in four variables. Moreover, a positive definite integral quadratic form is universal if and only if it represents all positive integers up to 290 \cite{bh}. However, Bosma and Kane \cite{bk} show that this kind of finiteness theorem does not exist for positive integral quadratic polynomials in general. More precisely, given any finite subset $T$ of $\mathbb N$ and a positive integer $n \not \in T$, Bosma and Kane construct explicitly a positive integral quadratic polynomial with minimum 0 which represents every integer in $T$ but not $n$.
An integral quadratic polynomial is called {\em almost universal} if it represents all but finitely many positive integers. A classical theorem of Tartakovski \cite{t} implies that a positive definite integral quadratic form in five or more variables is almost universal provided it is universal over $\z_p$ for every prime $p$. An effective procedure for deciding whether a positive definite integral quadratic form in four variables is almost universal is given in \cite{bo}.
Unlike positive definite universal or almost universal quadratic forms, positive universal and almost universal integral quadratic polynomials do exist in three variables. One well-known example of universal quadratic polynomial is the sum of three triangular numbers
$$\frac{x_1(x_1+1)}{2} + \frac{x_2(x_2+1)}{2} + \frac{x_3(x_3 + 1)}{2}.$$
Given positive integers $a_1, \ldots, a_n$, we follow the terminology used in \cite{co} and call the polynomial
$$\Delta(a_1, \ldots, a_n): = a_1\frac{x_1(x_1 + 1)}{2} + \cdots + a_n\frac{x_n(x_n + 1)}{2}$$
a triangular form. There are only seven universal ternary triangular forms and they were found by Liouville in 1863 \cite{li}. Bosma and Kane \cite{bk} have a simple criterion--the Triangular Theorem of Eight--to determine the universality of a triangular form: a triangular form is universal if and only if it represents the integers 1, 2, 4, 5, and 8. In \cite{co}, the present authors give a complete characterization of triples of positive integers $a_1, a_2, a_3$ for which $\Delta(a_1, a_2, a_3)$ are almost universal. Particularly, it is shown there that there are infinitely many almost universal ternary triangular forms. Almost universal integral quadratic polynomials in three variables that are mixed sums of squares and triangular numbers are determined in \cite{ch} and \cite{ks}.
Two quadratic polynomials $f(\bx)$ and $g(\bx)$ are said to be {\em equivalent} if there exists $T \in \text{GL}_n(\z)$ and $\bx_0 \in \z^n$ such that
\begin{eqnarray} \label{equiv}
g(\bx) = f(\bx T + \bx_0).
\end{eqnarray}
One can check readily that this defines an equivalence relation on the set of quadratic polynomials, and equivalent quadratic polynomials represent the same set of integers. In Section \ref{universal}, we will prove the following finiteness result on almost universal integral quadratic polynomials in three variables. It, in particular, implies that given a nonnegative integer $k$, there are only finitely many almost universal ternary triangular forms that represent all integers $\geq k$.
\begin{thm} \label{thmin3}
Let $k$ be a nonnegative integer. There are only finitely many equivalence classes of positive integral quadratic polynomials in three variables that represent all integers $\geq k$.
\end{thm}
An integral polynomial is called {\em regular} if it represents all the integers that are represented by the polynomial itself over $\z_p$ for every prime $p$ including $p = \infty$ (here $\z_\infty = \mathbb R$ by convention). In other words, $f(\bx)$ is regular if
\begin{equation} \label{hasse}
(\ref{1steqn}) \mbox{ is soluble in $\z_p$ for every $p \leq \infty$ } \Longrightarrow (\ref{1steqn}) \mbox{ is soluble in $\z$}.
\end{equation}
Watson \cite{w1, w2} showed that up to equivalence there are only finitely many primitive positive definite regular integral quadratic forms in three variables. A list containing all possible candidates of equivalence classes of these regular quadratic forms is compiled by Jagy, Kaplansky, and Schiemann in \cite{jks}. This list contains 913 candidates and all but twenty two of them are verified to be regular. Recently Oh \cite{o} verifies the regularity of eight of the remaining twenty two forms. As a first step to understand regular quadratic polynomials in three variables, we prove the following in Section \ref{regularpoly}.
\begin{thm} \label{regular3}
There are only finitely many primitive regular triangular forms in three variables.
\end{thm}
A quadratic polynomial $f(\bx)$ is called {\em complete} if it takes the form
$$f(\bx) = Q(\bx) + 2B(\bv, \bx) + Q(\bv) = Q(\bx + \bv).$$
Every quadratic polynomial is complete after adjusting the constant term suitably. In Section \ref{coset}, we will describe a geometric approach of studying the arithmetic of complete quadratic polynomials. In a nut shell, a complete integral quadratic polynomial $f(\bx)$ is just a coset $M + \bv$ of an integral $\z$-lattice $M$ on a quadratic $\q$-space with a quadratic map $Q$, and solving the diophantine equation $f(\bx) = a$ is the same as finding a vector $\be$ in $M$ such that $Q(\be + \bv) = a$. The definition of the class number of a coset will be introduced, and it will be shown in Section \ref{coset} that this class number is always finite and can be viewed as a measure of obstruction of the local-to-global implication in (\ref{hasse}).
In the subsequent sections, especially in Section \ref{coset}, we will complement our discussion with the geometric language of quadratic spaces and lattices. Let $R$ be a PID. If $M$ is a $R$-lattice on some quadratic space over the field of fractions of $R$ and $A$ is a symmetric matrix, we shall write ``$M \cong A$" if $A$ is the Gram matrix for $M$ with respect to some basis of $M$. The discriminant of $M$ is the determinant of one of its Gram matrices. An $n \times n$ diagonal matrix with $a_1, \ldots, a_n$ as its diagonal entries is written as $\langle a_1, \ldots, a_n \rangle$. Any other unexplained notation and terminology in the language of quadratic spaces and lattices used in this paper can be found in \cite{ca}, \cite{ki}, and \cite{om}.
\section{Universal Ternary Quadratic Polynomials} \label{universal}
We start this section with a technical lemma which will be used in the proof of Theorem \ref{thmin3}.
\begin{lem} \label{2variables}
Let $q(\bx)$ be a positive definite binary quadratic form and $b$ be the associated bilinear form. For $i = 1, \ldots, t$, let $f_i(\bx) = q(\bx) + 2b(\bw_i, \bx) + c_i$ be a positive integral quadratic polynomial with quadratic part $q(\bx)$. For any integer $k \geq 0$, there exists a positive integer $N\geq k$, bounded above by a constant depending only on $q(\bx)$, $k$, and $t$, such that $N$ is not represented by $f_i(\bx)$ for every $i = 1, \ldots, t$.
\end{lem}
\begin{proof}
Let $d$ be the discriminant of $q(\bx)$. Choose odd primes $p_1 < \cdots < p_t$ such that $-d$ is a nonresidue mod $p_i$ for all $i$. Then for every $i = 1, \ldots, t$, $q(\bx)$ is anisotropic $\z_{p_i}$-unimodular. In particular, $q(\bx) \in \z_{p_i}$, and hence $2b(\bw_i, \bx)$ as well, are in $\z_{p_i}$ for all $\bx \in \z_{p_i}^2$. This implies that $\bw_i \in \z_{p_i}^2$ and so $q(\bw_i) \in \z_{p_i}$. Let $N$ be the smallest positive integer satisfying $N \geq k$ and
$$N \equiv p_i + c_i - q(\bw_i) \mod p_i^2, \quad \mbox{ for } i = 1, \ldots, t.$$
Then for every $i$, $\ord_{p_i}(N - c_i + q(\bw_i)) = 1$ and so $N - c_i + q(\bw_i)$ is not represented by $q(\bx + \bw_i)$ over $\z_{p_i}$. Thus $N$ is not represented by $f_i(\bx)$.
\end{proof}
A positive ternary quadratic polynomial $f(\bx) = Q(\bx) + 2B(\bv, \bx) + m$ is called {\em Minkowski reduced}, or simply {\em reduced}, if its quadratic part is Minkowski reduced and it attains its minimum at the zero vector. This means that the quadratic part $Q(\bx)$ is of the form $\bx A \bx^t$, where $A$ is a Minkowski reduced symmetric matrix. So, if $\be_1, \be_2, \be_3$ is the standard basis for $\z^3$, then $Q(\be_1) \leq Q(\be_2) \leq Q(\be_3)$. Also, $Q(\bx) + 2B(\bv, \bx) \geq 0$ for all $\bx \in \z^3$, and hence
\begin{equation} \label{inequality}
2\vert B(\bv, \be_i) \vert \leq Q(\be_i) \mbox{ for } i = 1, 2, 3.
\end{equation}
\begin{lem} \label{reduced}
Every positive ternary quadratic polynomial is equivalent to a reduced ternary quadratic polynomial.
\end{lem}
\begin{proof}
Let $f(\bx)$ be a positive ternary quadratic polynomial. It follows from reduction theory that there exists $T \in \text{GL}_n(\z)$ such that the quadratic part of $f(\bx T)$ is Minkowski reduced. If $f(\bx T)$ attains its minimum at $\bx_0$, then the polynomial $g(\bx):= f(\bx T + \bx_0)$, which is equivalent to $f(\bx)$, is reduced.
\end{proof}
\begin{lem} \label{reduction}
Let $Q(\bx)$ be a positive definite reduced ternary quadratic form. Then for any $(x_1, x_2, x_3) \in \z^3$,
$$Q(x_1\be_1 + x_2\be_2 + x_3\be_3) \geq \frac{1}{6}(Q(\be_1)x_1^2 + Q(\be_2)x_2^2 + Q(\be_3)x_3^2).$$
\end{lem}
\begin{proof} Let $C_{ij} = Q(\be_i)Q(\be_j) - B(\be_i, \be_j)^2$, which is positive if $i \neq j$ because $Q(\bx)$ is reduced. For any permutation $i, j, k$ of the integers $1, 2, 3$, we have
$$Q(\be_k)C_{ij} \leq Q(\be_1)Q(\be_2)Q(\be_3) \leq 2 D,$$
where $D$ is the discriminant of $Q$. Now, by completing the squares,
\begin{eqnarray*}
Q(x_1\be_1 + x_2\be_2 + x_3\be_3) & \geq & Q(\be_i)(x_i + \cdots)^2 + \frac{C_{ij}}{Q(\be_j)}(x_j + \cdots )^2 + \frac{D}{C_{ij}} x_k^2\\
& \geq & \frac{Q(\be_k)}{2} x_k^2.
\end{eqnarray*}
Thus
$$3(Q(x_1\be_1 + x_2\be_2 + x_3\be_3)) \geq \frac{1}{2}(Q(\be_1)x_1^2 + Q(\be_2)x_2^2 + Q(\be_3)x_3^2),$$
and the lemma follows immediately.
\end{proof}
We are now ready to prove Theorem \ref{thmin3}.
\begin{proof}[Proof of Theorem \ref{thmin3}]
Let $k$ be a fixed nonnegative integer. By virtue of Lemma \ref{reduced}, it suffices to show that there are only finitely many reduced positive ternary integral quadratic polynomials which represent all positive integers $\geq k$. By adjusting the constant terms of these quadratic polynomials, we may assume that their minimum is 0.
Let $f(\bx) = Q(\bx) + 2B(\bv, \bx)$ be a reduced positive ternary integral quadratic polynomial with minimum $0$. Let $\be_1, \be_2, \be_3$ be the standard basis for $\z^3$. For simplicity, for each $i = 1, 2, 3$, we denote $Q(\be_i)$ by $\mu_i$ and $B(\bv, \be_i)$ by $w_i$. Furthermore, for $i \neq j$, let $a_{ij}$ be $B(\be_i, \be_j)$. We assume throughout below that $f(\bx)$ represents all integers $\geq k$. The proof will be complete if we can show that $\mu_3$ is bounded above by a constant depending only on $k$. From now on, $(x_1, x_2, x_3)$ always denotes a vector in $\z^3$.
By (\ref{inequality}) and Lemma \ref{reduction},
\begin{eqnarray*}
f(x_1, x_2, x_3) & \geq & \sum_{i=1}^3 \left(\frac{1}{6} \mu_i x_i^2 - 2\vert w_i x_i \vert\right)\\
& \geq & \sum_{i = 1}^3 \mu_i \left(\frac{1}{6} x_i^2 - \vert x_i \vert \right),
\end{eqnarray*}
and so if $\vert x_3 \vert \geq 9$, we have
$$f(x_1, x_2, x_3) \geq -\frac{3}{2}\mu_1 - \frac{3}{2}\mu_2 + \frac{9}{2} \mu_3 \geq \frac{3}{2}\mu_3.$$
Suppose that $\vert x_3 \vert \leq 8$. Since $2\vert a_{12} \vert \leq \mu_1$, one obtains $\frac{\mu_1}{2}x_1^2 + 2a_{12}x_1x_2 + \frac{\mu_2}{2}x_2^2 \geq 0$ for all $(x_1, x_2) \in \z^2$. So, if $\vert x_2 \vert \geq 22$, then
\begin{eqnarray*}
f(x_1, x_2, x_3) & \geq & \frac{\mu_1}{2}x_1^2 + 2(a_{13}x_3 + w_1)x_1 + \frac{\mu_2}{2} x_2^2 + 2(a_{23}x_3 + w_2)x_2 + f(0,0,x_3) \\
& \geq & - \frac{81}{2}\mu_1 + 44\mu_2\\
& \geq & \frac{7}{2}\mu_2.
\end{eqnarray*}
Let us assume further that $\vert x_2 \vert \leq 21$. If, in addition, $\vert x_1 \vert \geq 31$, then
\begin{eqnarray*}
f(x_1, x_2, x_3) & = & \mu_1 x_1^2 + 2(a_{12}x_2 + a_{13}x_3 + w_1)x_1 + f(0, x_2, x_3)\\
& \geq & \mu_1(x_1^2 - 30 \vert x_1\vert)\\
& \geq & 31 \mu_1.
\end{eqnarray*}
Therefore, we have
$$f(x_1, x_2, x_3) \geq \gamma(f): = \min \left \{ \frac{3}{2}\mu_3, \frac{7}{2}\mu_2, 31\mu_1 \right \}$$
unless
$$\vert x_1 \vert \leq 30, \quad \vert x_2 \vert \leq 21, \,\, \mbox{ and }\,\, \vert x_3 \vert \leq 8.$$
In particular, this means that there are at most $61\times 43 \times 17$ choices of $(x_1, x_2, x_3)$ for which $f(x_1, x_2, x_3) < \gamma(f)$, and thus there are at most $61\times 43\times 17$ distinct positive integers less than $\gamma(f)$ which may be represented by $f$. So, if $\gamma(f) \geq 61\times 43 \times 17 + 2 + k$, then $f(x_1, x_2, x_3)$ does not represent at least one integer among $k + 1, k + 2, \ldots, k + 61\times 43 \times 17 + 1$. Consequently,
$$\frac{3}{2}\mu_1 \leq \gamma(f) \leq k + 61 \times 43 \times 17 + 1.$$
Let $\eta$ be the smallest positive integer satisfying
$$43 \times 17 \times [2(15 + \sqrt{225 + k + \eta}) + 1] < \eta.$$
Suppose that $\frac{3}{2}\mu_2 > k + \eta$. Let $s$ be a positive integer $\leq k + \eta$. If $f(x_1, x_2, x_3) = s$, then $\vert x_2 \vert \leq 21$ and $\vert x_3 \vert \leq 8$; thus, as shown before,
\begin{eqnarray*}
f(x_1, x_2, x_3) & = & \mu_1 x_1^2 + 2(a_{12}x_2 + a_{13}x_3 + w_1)x_1 + f(0, x_2, x_3)\\
& \geq & \mu_1(x_1^2 - 30 \vert x_1\vert)\\
& \geq & x_1^2 - 30 \vert x_1 \vert.
\end{eqnarray*}
So, if $\vert x_1 \vert > 15 + \sqrt{225 + k + \eta}$, then $f(x_1, x_2, x_3) > k + \eta$. Therefore, the number of vectors $(x_1, x_2, x_3) \in \z^3$ satisfying $k + 1 \leq f(x_1, x_2, x_3) \leq k + \eta$ is not bigger than
$$43 \times 17 \times [2(15 + \sqrt{225 + k + \eta}) + 1],$$
which is strictly less than $\eta$. This is impossible, which means that
$$\mu_2 \leq \frac{2(k + \eta)}{3}.$$
Recall that if $\vert x_3 \vert \geq 9$, then $f(x_1, x_2, x_3) \geq \frac{3}{2}\mu_3$. It follows from Lemma \ref{2variables} that there exists a positive integer $N \geq k$ which is not represented by $f(x_1, x_2, t)$ for any integer $t \in [-8, 8]$, and this $N$ is bounded above by a constant depending only on $k, \mu_1, \mu_2$, and $a_{12}$ (note that $2\vert a_{12}\vert \leq \mu_1$). This means that whenever $f(x_1, x_2, x_3) = N$, we must have $\vert x_3 \vert \geq 9$ and so
$$\mu_3 \leq \frac{2N}{3}.$$
This completes the proof.
\end{proof}
\section{Regular Ternary Triangular Forms} \label{regularpoly}
A triangular form $\Delta(\alpha_1, \ldots, \alpha_n)$ is said to be {\em primitive} if $\gcd(\alpha_1, \ldots, \alpha_n) = 1$. Its discriminant, denoted $d(\Delta)$, is defined to be the product $\alpha_1\cdots \alpha_n$. By completing the squares, it is easy to see that $\Delta(\alpha_1, \ldots, \alpha_n)$ represents an integer $m$ if and only if the equation
\begin{equation} \label{3to2}
\alpha_1(2x_1 + 1)^2 + \cdots + \alpha_n(2x_n + 1)^2 = 8m + (\alpha_1 + \cdots + \alpha_n)
\end{equation}
is soluble in $\z$. Let $M$ be the $\z$-lattice with quadratic map $Q$ and an orthogonal basis $\{\be_1, \ldots, \be_n\}$ such that $M \cong \langle 4\alpha_1, \ldots, 4\alpha_n \rangle$. Then (\ref{3to2}) is soluble in $\z$ if and only if $8m + (\alpha_1 + \cdots + \alpha_n)$ is represented by the coset $M + \bv$, where $\bv = (\be_1 + \cdots + \be_n)/2$, that is, there exists a vector $\bx \in M$ such that $Q(\bx + \bv) = 8m + (\alpha_1 + \cdots + \alpha_n)$.
Let $p$ be an odd prime. If $M_p$ is the $\z_p$-lattice $\z_p\otimes M$, then $M_p + \bv = M_p$. Therefore, (\ref{3to2}) is soluble in $\z_p$ if and only if $M_p$ represents $8m + (\alpha_1 + \cdots + \alpha_n)$. In particular, $\Delta(\alpha_1, \ldots, \alpha_n)$ is universal over $\z_p$ if and only if $M_p$ is universal.
\begin{lem}\label{at2}
A primitive triangular form is universal over $\z_2$.
\end{lem}
\begin{proof}
It suffices to prove that for an odd integer $\alpha$, the polynomial $\alpha x(x + 1)/2$ is universal over $\z_2$. But this is clear by the Local Square Theorem \cite[63:1]{om} or \cite[Lemma 1.6, page 40]{ca}.
\end{proof}
\begin{lem} \label{atodd}
Let $p$ be an odd prime and $\alpha, \beta, \gamma$ be $p$-adic units. Then over $\z_p$,
\begin{enumerate}
\item[(1)] $\Delta(\alpha,\beta)$ represents all integers $m$ for which $8m + \alpha + \beta \not \equiv 0$ mod $p$;
\item[(2)] $\Delta(\alpha, \beta)$ is universal if $\alpha + \beta \equiv 0 \mod p$;
\item[(3)] $\Delta(\alpha, \beta, \gamma)$ is universal.
\end{enumerate}
\end{lem}
\begin{proof}
The binary $\z_p$-lattice $\langle \alpha, \beta \rangle$ represents all $p$-adic units \cite[92:1b]{om}. Therefore, it represents all integers $m$ for which $8m + \alpha + \beta \not \equiv 0 \mod p$. This proves (1).
In (2), the condition on $\alpha$ and $\beta$ implies that the $\z_p$-lattice $\langle \alpha,\beta \rangle$ is isometric to the hyperbolic plane which is universal. For (3), it follows from \cite[92:1b]{om} that any unimodular $\z_p$-lattice of rank at least three is universal.
\end{proof}
Recall that a triangular form is regular if it represents all positive integers that are represented by the triangular form itself over $\z_p$ for all primes $p$. For example, every universal triangular form is regular. The following lemma is a ``descending trick" which transforms a regular ternary triangular form to another one with smaller discriminant.
\begin{lem} \label{watson}
Let $q$ be an odd prime and $a, b, c$ be positive integers which are not divisible by $q$. Suppose that $\Delta(a, q^rb, q^sc)$ is regular, with $1 \leq r \leq s$. Then $\Delta(q^{2-\delta}a, q^{r - \delta}b, q^{s - \delta}c)$ is also regular, where $\delta = \min\{2, r\}$.
\end{lem}
\begin{proof}
It suffices to show that $\Delta(q^2a, q^rb, q^sc)$ is regular. Suppose that $m$ is a positive integer represented by $\Delta(q^2a, q^rb, q^sc)$ over $\z_p$ for all primes $p$. Then the equation
\begin{equation} \label{1}
8m + (q^2a + q^rb + q^sc) = q^2a (2x_1 + 1)^2 + q^rb (2x_2 + 1)^2 + q^sc (2x_3 + 1)^2
\end{equation}
is soluble in $\z_p$ for every prime $p$. Since $q$ is odd, we can say that
\begin{equation}\label{2}
8m + (q^2a + q^rb + q^sc) = a(2x_1 + 1)^2 + q^rb (2x_2 + 1)^2 + q^sc (2x_3 + 1)^2
\end{equation}
is also soluble in $\z_p$ for every prime $p$. Notice that $q^2 \equiv 1$ mod 8, and so $8m + (q^2a + q^rb + q^sc) = 8m' + (a + q^rb + q^sc)$ for some integer $m'$. Thus, the regularity of $\Delta(a, q^rb, q^sc)$ implies that (\ref{2}) is soluble in $\z$. Let $(x_1, x_2, x_3) \in \z^3$ be a solution to (\ref{2}). Then $(2x_1 + 1)$ must be divisible by $q$ because $q \mid m$ by (\ref{1}), and we can write $(2x_1 + 1)$ as $q(2y_1 + 1)$ for some $y_1 \in \z$. So $(y_1, x_2, x_3)$ is an integral solution to (\ref{1}), which means that $m$ is in fact represented by $\Delta(q^2a, q^rb, q^sc)$.
\end{proof}
The following lemma will be used many times in the subsequent discussion. It is a reformulation of \cite[Lemma 3]{kko}.
\begin{lem}\label{kkolemma}
Let $T$ be a finite set of primes and $a$ be an integer not divisible by any prime in $T$. For any integer $d$, the number of integers in the set $\{d, a + d, \ldots, (n-1)a + d \}$ that are not divisible by any prime in $T$ is at least
$$n\frac{\tilde{p}-1}{\tilde{p} + t - 1} - 2^t + 1,$$
where $t = \vert T \vert$ and $\tilde{p}$ is the smallest prime in $T$.
\end{lem}
For the sake of convenience, we say that a ternary triangular form $\Delta(\alpha, \beta, \gamma)$ {\em behaves well} if the unimodular Jordan component of the $\z_p$-lattice $\langle \alpha, \beta, \gamma \rangle$ has rank at least two, or equivalently, $p$ does not divide at least two of $\alpha, \beta$, and $\gamma$. For a ternary triangular form $\Delta$, we can rearrange the variables so that $\Delta = \Delta(\mu_1, \mu_2, \mu_3)$ with $\mu_1 \leq \mu_2 \leq \mu_3$. Collectively, we call these $\mu_i$ the successive minima of $\Delta$.
In what follows, an inequality of the form $A \ll B$ always means that there exists a constant $k > 0$ such that $\vert A \vert \leq k\vert B\vert$. A real-valued function in several variables is said to be bounded if its absolute value is bounded above by a constant independent of the variables.
\begin{prop} \label{well}
There exists an absolute constant $C$ such that if $\Delta$ is a primitive regular ternary triangular form which behaves well at all odd primes, then $d(\Delta) \leq C$.
\end{prop}
\begin{proof}
Let $\mu_1\leq \mu_2\leq \mu_3$ be the successive minima of $\Delta$, and let $M$ be the $\z$-lattice $\langle 4\mu_1, 4\mu_2, 4\mu_3 \rangle$. Let $T$ be the set of odd primes $p$ for which $M_p$ is not split by the hyperbolic plane. Then $T$ is a finite set. Let $t$ be the size of $T$, $\tilde{p}$ be the smallest prime in $T$, and $\omega = (\tilde{p} + t - 1)/(\tilde{p} - 1)$. Note that, since $\tilde{p} \geq 2$, we have $\omega \leq t + 1$. Let $\eta = (\mu_1 + \mu_2 + \mu_3)$ and $\mathfrak T$ be the product of primes in $T$. It follows from Lemmas \ref{at2} and \ref{atodd} and the regularity of $\Delta$ that $\Delta$ represents every positive integer $m$ for which $8m + \eta$ is relatively prime to $\mathfrak T$.
By Lemma \ref{kkolemma}, there exists a positive integer $k_1 < (t+1)2^t$ such that $8k_1 + \eta$ is relatively prime to $\mathfrak T$. Therefore, $k_1$ is represented by $\Delta$ and hence
$$\mu_1 \leq (t+1)2^t \ll t2^t.$$
For any positive integer $n$, the number of integers between 1 and $n$ that are represented by the triangular form $\Delta(\mu_1)$ is at most $2\sqrt{n}$. Therefore, by virtue of Lemma \ref{kkolemma}, if $n \geq 4(t+1)^2 + 3(t+1)2^t$, there must be a positive integer $k_2 \leq n$ such that $8k_2 + \eta$ is relatively prime to $\mathfrak T$ and $k_2$ is not represented by $\Delta(\mu_1)$. This implies that
$$\mu_2 \leq 4(t+1)^2 + 3(t+1)2^t \ll t2^t.$$
Let $\mathfrak A$ be the product of primes in $T$ that do not divide $\mu_1\mu_2$. Following the argument in \cite[page 862]{e}, we find that there must be an odd prime $q$ outside $T$ such that $-\mu_1\mu_2$ is a nonresidue mod $q$ and $q \ll (\mu_1\mu_2)^{\frac{7}{8}} {\mathfrak A}^{\frac{1}{4}}$. Since $\mathfrak A \leq \mathfrak T$, we have
$$q \ll (\mu_1\mu_2)^{\frac{7}{8}} {\mathfrak T}^{\frac{1}{4}} \ll (t2^t)^{\frac{7}{8}}{\mathfrak T}^{\frac{1}{4}}.$$
Fix a positive integer $m \leq q^2$ such that
$$8m + \mu_1 + \mu_2 \equiv q \mod q^2.$$
For any integer $\lambda$, $8(m + \lambda q^2) + \mu_1 + \mu_2$ is not represented by the binary lattice $\langle \mu_1, \mu_2 \rangle$, which means that $m + \lambda q^2$ is not represented by $\Delta(\mu_1, \mu_2)$. However, by Lemma \ref{kkolemma}, there must be a positive integer $k_3 \leq (t+1)2^t$ such that $8q^2k_3 + 8m + \eta$ is relatively prime to $\mathfrak T$. Then $m + q^2k_3$ is an integer represented by $\Delta$ but not by $\Delta(\mu_1,\mu_2)$. As a result,
$$\mu_3 \leq m + q^2k_3 \ll (t 2^t)^{\frac{11}{4}} {\mathfrak T}^{\frac{1}{2}},$$
and hence
$$\mathfrak T \leq d(\Delta) = \mu_1\mu_2\mu_3 \ll (t 2^t)^{\frac{19}{4}} {\mathfrak T}^{\frac{1}{2}}.$$
Since $\mathfrak T$, a product of $t$ distinct primes, grows at least as fast as $t!$, the above inequality shows that $t$, and hence $\mathfrak T$ as well, must be bounded. This means that $d(\Delta)$ is also bounded.
\end{proof}
Starting with a primitive regular ternary triangular form $\Delta$, we may apply Lemma \ref{watson} successively at suitable odd primes and eventually obtain a primitive regular ternary triangular form $\overline{\Delta}$ which behaves well at all odd primes. It is also clear from Lemma \ref{watson} that $d(\overline{\Delta})$ divides $d(\Delta)$. Let $\ell$ be an odd prime divisor of $d(\Delta)$. If $\ell$ divides $d(\overline{\Delta})$, then $\ell$ is bounded by Proposition \ref{well}. So we assume from now on that $\ell$ does not divide $d(\overline{\Delta})$. Our next goal is to bound $\ell$.
When we obtain $\overline{\Delta}$ from $\Delta$, we may first apply Lemma \ref{watson} at all primes $p$ not equal to $\ell$. So, there is no harm to assume from the outset that $\Delta$ behaves well at all primes $p \neq \ell$. Then, by Lemma \ref{watson}, $\Delta$ can be transformed to a primitive regular ternary triangular form $\tilde{\Delta} = \tilde{\Delta}(a, \ell^2b, \ell^2c)$, with $\ell \nmid abc$, which behaves well at all primes $p \neq \ell$. Since further application of Lemma \ref{watson} to $\tilde{\Delta}$ results in the triangular form $\overline{\Delta}$, therefore all the prime divisors of $d(\tilde{\Delta})$, except $\ell$, are bounded.
Let $\tilde{T}$ be the set of odd prime divisors of $d(\tilde{\Delta})$ that are not $\ell$. By Lemmas \ref{at2} and \ref{atodd}, we see that
$\tilde{\Delta}$ represents all positive integers $m$ for which $8m + a + \ell^2b + \ell^2c$ is relatively prime to every prime in $\tilde{T}$ and $(8m + a + \ell^2b + \ell^2c)a$ is a quadratic residue modulo $\ell$.
In order to find integers represented by $\tilde{\Delta}$, we need a result which is a slight generalization of Proposition 3.2 and Corollary 3.3 in \cite{e}. Let $\chi_1, \ldots, \chi_r$ be Dirichlet characters modulo $k_1, \ldots, k_r$, respectively, $u_1, \ldots, u_r$ be values taken from the set $\{\pm 1\}$, and $\Gamma$ be the least common multiple of $k_1, \ldots, k_r$. Given a nonnegative number $s$ and a positive number $H$, let $S_s(H)$ be the set of integers $n$ in the interval $(s, s + H)$ which satisfy the conditions
$$\chi_i(n) = u_i \quad \mbox{ for } i = 1, \ldots, r
\text{ and } \gcd(n, X) = 1,$$
where $X$ is a positive integer relatively prime to $\Gamma$.
\begin{prop} \label{eresult}
Suppose that $\chi_1, \ldots, \chi_r$ are independent. Let $h = \min \{H : S_s(H) > 0 \}$ and $\omega(\Gamma)$ denote the number of distinct prime divisor of $\Gamma$. Then
\begin{equation} \label{e3.2}
S_s(H) = 2^{-r}\frac{\phi(\Gamma X)}{\Gamma X}H + O\left(H^{\frac{1}{2}}\Gamma^{\frac{3}{16} + \epsilon} X^\epsilon\right),
\end{equation}
and if $r \leq \omega(\Gamma) + 1$, we have
\begin{equation} \label{e3.3}
h \ll \Gamma^{\frac{3}{8} + \epsilon}X^\epsilon,
\end{equation}
where $\phi$ is the Euler's phi-function and the implied constants in both \textnormal{(\ref{e3.2})} and \textnormal{(\ref{e3.3})} depend only on $\epsilon$.
\end{prop}
\begin{proof}
We may proceed as in the proofs for Proposition 3.2 and Corollary 3.3 in \cite{e}, but notice that \cite[Lemma 3.1]{e} remains valid if we replace ``$0 < n < H$" by ``$s < n < s + H$" in the summations since Burgess's estimate for character sums \cite[Theorem 2]{b} holds for any interval of length $H$.
\end{proof}
\begin{lem} \label{primel}
The prime $\ell$ is bounded.
\end{lem}
\begin{proof}
Let $\tilde{\mu_1} \leq \tilde{\mu_2}$ be the first two successive minima of $\tilde{\Delta}$. Let $s = a + \ell^2 b + \ell^2 c$ and write $s = 2^\kappa s_0$ with $2 \nmid s_0$. Suppose that $\kappa \geq 3$. We apply Proposition \ref{eresult} to the quadratic residue mod $\ell$ character $\chi_\ell$, taking $\epsilon = 1/8$ and $X$ to be the product of the primes in $\tilde{T}$. So, there is a positive integer $h \ll \ell^{\frac{1}{2}}$ such that $\chi_\ell(h + 2^{\kappa - 3}s_0) = \chi_\ell(2a)$ and $h + 2^{\kappa - 3} s_0$ is not divisible by any prime in $\tilde{T}$. Then $\tilde{\Delta}$ represents $h$ and hence $\tilde{\mu_1} \ll \ell^{\frac{1}{2}}$.
If $\kappa < 3$, then we apply Proposition \ref{eresult} again but this time to $\chi_\ell$ and possibly the mod 4 character $\left(\frac{-1}{*} \right)$ and the mod 8 character $\left(\frac{2}{*} \right)$. We obtain a positive integer $n > s_0$ such that $\chi_\ell(n) = \chi_\ell(2^{\kappa} a)$, $n$ is not divisible by any prime in $\tilde{T}$, $n \equiv s_0$ mod $2^{3 - \kappa}$, and $n - s_0 \ll \ell^{\frac{1}{2}}$. Then we can write $2^\kappa n = 8m + s$, where $m$ is represented by $\tilde{\Delta}$ and $m \ll \ell^{\frac{1}{2}}$. So, $\tilde{\mu_1} \ll \ell^{\frac{1}{2}}$ in this case as well.
Now, for any $H > 0$, the number of integers in the interval $(s, s + H)$ that are represented by the triangular form $\Delta(\tilde{\mu_1})$ is equal to $O(\sqrt{H})$. Thus, by Proposition \ref{eresult} and an argument similar to the one above, we must have $\tilde{\mu_2} \ll \ell^{\frac{1}{2}}$. Then $\ell^2 \leq \tilde{\mu_1}\tilde{\mu_2} \ll \ell$, and hence $\ell$ is bounded.
\end{proof}
We now present the proof of Theorem \ref{regular3} which asserts that there are only finitely many primitive regular ternary triangular forms.
\begin{proof}[Proof of Theorem \ref{regular3}]
Let $\Delta$ be a primitive regular ternary triangular form, and $\mu_1\leq \mu_2\leq \mu_3$ be its successive minima. It suffices to show that these successive minima are bounded. Let $S$ be the set of odd prime divisors of $d(\Delta)$. It follows from Proposition \ref{well} and Lemma \ref{primel} that all the primes in $S$ are bounded. Let $\mathfrak S$ be the product of these primes. It is clear from Lemma \ref{at2} and Lemma \ref{atodd}(3) that $\Delta$ represents $\mathfrak S$ over $\mathbb Z_p$ for all $p \not \in S$. Also, Lemma \ref{atodd}(1) (if $\mu_1 + \mu_2 \not \equiv 0$ mod $p$) or Lemma \ref{atodd}(2) (if $\mu_1 + \mu_2 \equiv 0$ mod $p$) shows that $\Delta$ represents $\mathfrak S$ over $\mathbb Z_p$ for all primes $p \in S$. Consequently, $\Delta$ represents $\mathfrak S$ over $\mathbb Z_p$ for all primes $p$. Since $\Delta$ is regular, it must represent $\mathfrak S$. This shows that $\mu_1$ is bounded.
Let $q_1$ be the smallest odd prime not dividing $3\mu_1\mathfrak S$, and $q_2$ be the smallest odd prime not dividing $q_1\mu_1\mathfrak S$ for which $8q_2 \mathfrak S\mu_1 + \mu_1^2$ is a nonresidue mod $q_1$. Such $q_2$ exists because there are at least two nonresidues mod $q_1$. Note that $q_2\mathfrak S$ is represented by $\Delta$ but not by $\Delta(\mu_1)$. Therefore, $\mu_2$ is also bounded.
Now, let $q_3$ be the smallest odd prime not dividing $\mathfrak S$ for which $-\mu_1\mu_2$ is a nonresidue mod $q_3$, and $q_4$ be the smallest odd prime not dividing $\mathfrak S$ which satisfies
$$-8q_4 \mathfrak S \equiv \mu_1 + \mu_2 + q_3 \mod q_3^2.$$
Then $q_4\mathfrak S$ is represented by $\Delta$ but not by $\Delta(\mu_1, \mu_2)$, which means that $\mu_3$ is bounded. This completes the proof.
\end{proof}
\section{Representations of Cosets} \label{coset}
In the previous sections we have seen some connection between the diophantine aspect of quadratic polynomials and the geometric theory of quadratic spaces and lattices. In this section we will amplify this connection by describing a geometric approach of a special, but yet general enough for most practical purpose, family of quadratic polynomials. Since it will not present any additional difficulty, we shall consider quadratic polynomials over global fields and the Dedekind domains inside. For simplicity, the quadratic map and its associated bilinear form on any quadratic space will be denoted by $Q$ and $B$ respectively.
Now, unless stated otherwise, $F$ is assumed to be a global field of characteristic not 2 and $\ring$ is a Dedekind domain inside $F$ defined by a Dedekind set of places $\Omega$ on $F$ (see, for example, \cite[\S 21]{om}). We call a quadratic polynomial $f(\bx)$ over $F$ in variables $\bx = (x_1, \ldots, x_n)$ {\em complete} if $f(\bx) = (\bx + \bv)A (\bx + \bv)^t$, where $A$ is an $n\times n$ nonsingular symmetric matrix over $F$ and $\bv \in F^n$. It is called {\em integral} if $f(\bx) \in \ring$ for all $\bx \in \ring^n$. Two quadratic polynomials $f(\bx)$ and $g(\bx)$ are said to be {\em equivalent} if there exist $T \in \text{GL}_n(\ring)$ and $\bx_0 \in \ring^n$ such that $g(\bx) = f(\bx T + \bx_0)$.
On the geometric side, an $\ring$-{\em coset} is a set $M + \bv$, where $M$ is an $\ring$-lattice on an $n$-dimensional nondegenerate quadratic space $V$ over $F$ and $\bv$ is a vector in $V$. An $\ring$-coset $M + \bv$ is called {\em integral} if $Q(M + \bv) \subseteq \ring$, and is {\em free} if $M$ is a free $\ring$-lattice. Two $\ring$-cosets $M + \bv$ and $N + \bw$ on two quadratic spaces $V$ and $W$, respectively, are said to be {\em isometric}, written $M + \bv \cong N + \bw$, if there exists an isometry $\sigma: V \longrightarrow W$ such that $\sigma(M + \bv) = N + \bw$. This is the same as requiring $\sigma(M) = N$ and $\sigma(\bv) \in \bw + N$. For each $\p \in \Omega$, $\ring_\p$-cosets and isometries between $\ring_\p$-cosets are defined analogously.
As in the case of quadratic forms and lattices, there is a one-to-one correspondence between the set of equivalence classes of complete quadratic polynomials in $n$ variables over $F$ and the set of isometry classes of free cosets on $n$-dimensional quadratic spaces over $F$. Under this correspondence, integral complete quadratic polynomials corresponds to integral free cosets.
\begin{defn}
Let $M + \bv$ be an $\ring$-coset on a quadratic space $V$. The genus of $M + \bv$ is the set
$$\text{gen}(M + \bv) = \{K + \bw \mbox{ on } V : K_\p + \bw \cong M_\p + \bv \mbox{ for all } \p \in \Omega\}.$$
\end{defn}
\begin{lem}
Let $M + \bv$ be an $\ring$-coset on a quadratic space $V$ and let $S$ be a finite subset of $\Omega$. Suppose that an $\ring_\p$-coset $M(\p) + \bx_\p$ on $V_\p$ is given for each $\p \in
S$. Then there exists an $\ring$-coset $K + \bz$ on $V$ such that
$$K_\p + \bz = \left \{
\begin{array}{ll}
M(\p) + \bx_\p & \mbox{ if $\p \in S$};\\
M_\p + \bv & \mbox{ if $\p \in \Omega \setminus S$}.
\end{array}
\right .$$ \label{crt}
\end{lem}
\begin{proof}
Let $T$ be the set of places $\p \in \Omega \setminus S$ for which $\bv \not \in M_\p$. Then $T$ is a finite set. For each $\p \in T$, let $M(\p) = M_\p$ and $\bx_\p = \bv$. Let $K$ be an $\ring$-lattice on $V$ such that
$$K_\p = \left \{
\begin{array}{ll}
M(\p) & \mbox{ if $\p \in S\cup T$};\\
M_\p & \mbox{ if $\p \in \Omega\setminus (S\cup T)$}.
\end{array}
\right .$$
By the strong approximation property of $V$, there exists $\bz \in V$ such that $\bz \equiv \bx_\p$ mod $M(\p)$ for all $\p
\in S\cup T$, and $\bz \in M_\p$ for all $\p \in \Omega \setminus (S\cup T)$. Then $K + \bz$ is the desired $\ring$-coset.
\end{proof}
Let $O_\ad(V)$ be the adelization of the orthogonal group of $V$. Let $\Sigma$ be an element in $O(V)_\ad$. The $\p$-component of $\Sigma$ is denoted by $\Sigma_\p$. Given an $\ring$-coset $M + \bv$ on $V$, $\Sigma_\p(M_\p + \bv) = \Sigma_\p(M_\p) = M_\p$ for almost all finite places $\p$. By Lemma \ref{crt}, there exists an $\ring$-coset $K + \bz$ on $V$ such that $K_\p + \bz = \Sigma_\p(M_\p + \bv)$ for all $\p \in \Omega$. Therefore, we can define
$\Sigma(M + \bv)$ to be $K + \bz$, and so $O(V)_\ad$ acts transitively on $\text{gen}(M + \bv)$. As a result,
$$\text{gen}(M + \bv) = O_\ad(V) \cdot (M + \bv).$$
Let $O_\ad(M + \bv)$ be the stabilizer of $M + \bv$ in $O_\ad(V)$. Then the (isometry) classes in $\text{gen}(M + \bv)$ can be
identified with
$$O(V)\setminus O_\ad(V) / O_\ad(M + \bv).$$
The group $O_\ad(M + \bv)$ is clearly a subgroup of $O_\ad(M)$. For each $\p \in \Omega$, we have
\begin{eqnarray*}
O(M_\p + \bv) & = & \{ \sigma \in O(V_\p) : \sigma(M_\p) = M_\p \mbox{ and } \sigma(\bv) \equiv \bv \mbox{ mod } M_\p \}\\
& \subseteq & O(M_\p)\cap O(M_\p + \ring_\p \bv).
\end{eqnarray*}
\begin{lem}
For any $\p \in \Omega$, the group index $[O(M_\p) : O(M_\p + \bv)]$ is finite.
\end{lem}
\begin{proof}
There is the natural map
$$O(M_\p)\cap O(M_\p + \ring_\p \bv) \longrightarrow \text{Aut}_{\ring_\p}((M_\p + \ring_\p \bv) / M_\p)$$
whose kernel is precisely $O(M_\p + \bv)$. Since $(M_\p + \ring_\p \bv)/ M_\p$ is a finite group, the index $[O(M_\p)\cap O(M_\p + \ring_\p \bv) : O(M_\p + \bv)]$ is finite. But it is known \cite[30.5]{kn} that the index $[O(M_\p) : O(M_\p)\cap O(M_\p + \ring_\p \bv)]$ is always finite. This proves the lemma.
\end{proof}
Since $M_\p = M_\p + \bv$ for almost all $\p \in \Omega$, the index $[O_\ad(M) : O_\ad(M + \bv)]$ is finite. The set
$O(V)\setminus O_\ad(V) / O_\ad(M)$ is finite (which is the class number of $M$), hence the set $O(V)\setminus O_\ad(V) / O_\ad(M + \bv)$ is also finite. Let $h(M + \bv)$ be the number of elements in this set, which is also the number of classes of in $\text{gen}(M +\bv)$. We call it the {\em class number} of $M + \bv$.
\begin{cor} \label{classnumber}
The class number $h(M + \bv)$ is finite, and $h(M + \bv) \geq h(M)$.
\end{cor}
If we replace the orthogonal groups by the special orthogonal groups in the above discussion, then we have the definitions for the proper genus $\text{gen}^+(M + \bv)$, which can be identified with $O^+(V)\setminus O^+_\ad(V)/O^+_\ad(M + \bv)$, and the proper class number $h^+(M + \bv)$ which is also finite. Unlike the case of lattices, it is not true in general that $\text{gen}(M + \bv)$ coincides with $\text{gen}^+(M + \bv)$. The following example illustrates this phenomenon. It also shows that in general $h(M + \bv)$ and $h(M)$ are not equal.
\begin{exam}
Let $W$ be the hyperbolic plane over $\mathbb Q$, and let $H$ be the $\mathbb Z$-lattice on $W$ spanned by two linear independent isotropic vectors $\be$ and $\mathbf f$ such that $B(\be,\mathbf f) = 1$. Consider the $\z$-coset $H + \bv$, where $\bv = \frac{1}{p}\be$ for some odd prime $p$. Suppose that
$\sigma_p$ is an improper isometry of $H_p + \bv$. Then $\sigma_p$ must send $\be$ to $\epsilon \mathbf f$ and $\mathbf f$ to $\epsilon^{-1}\be$ for some unit $\epsilon$ in $\mathbb Z_p$.
Then
$$\bv = \frac{1}{p}\be \equiv \sigma_p(\bv) \equiv \frac{\epsilon}{p} \mathbf f \mbox{ mod } H_p.$$
This implies that $\frac{1}{p}\be - \frac{\epsilon}{p}\mathbf f$ is in $H_p$, which is absurd. Therefore, $H_p + \bv$ does not have
any improper isometry and hence $\text{gen}(H + \bv)$ is not the same as $\text{gen}^+(H + \bv)$.
Now, suppose in addition that $p > 3$. Let $q$ be an integer such that $q \not \equiv \pm 1$ mod $p$. Let $\bu$ be the vector $\frac{q}{p}\mathbf e$. Then the coset $H + \bu$ is in $\text{gen}^+(H + \bv)$. To see this, observe that $H_\ell + \bu = H_\ell + \bv$ for all primes $\ell \neq p$. At $p$, the isometry that sends $\be$ to $q^{-1}\be$ and $\mathbf f$ to $q\mathbf f$, whose determinant is 1, would
send $H_p + \bu$ to $H_p + \bv$. Suppose that there exists $\sigma \in O(W)$ which sends $H + \bu$ to $H + \bv$. Then $\sigma$ necessarily sends $H$ to $H$ itself; hence the matrix for $\sigma$ relative to the basis $\{\be, \mathbf f\}$ is one of the
following:
$$\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix},\quad
\begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix},\quad
\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix},\quad
\begin{bmatrix} 0 & -1\\ -1 & 0 \end{bmatrix}.$$
But a simple calculation shows that none of the above sends $H + \bu$ to $H + \bv$. Hence $H + \bu$ is not in the same class of $H + \bv$. As a result, both $h^+(H + \bv)$ and $h(H + \bv)$ are greater than 1, while $h(H)$ and $h^+(H)$ are 1.
\end{exam}
Of course, there are $\ring$-cosets, which are not $\ring$-lattices themselves, whose class numbers are 1. Here is an example:
\begin{exam}
Let $M$ be the $\mathbb Z$-lattice whose Gram matrix is $\langle 4, 4, 4 \rangle$ relative to a basis $\{\be, \mathbf f, \mathbf g\}$, and let $\bv$ be $\frac{\be + \mathbf f + \mathbf g}{2}$. The class number of $M$ is 1. The lattice $M + \mathbb Z \bv$ is isometric to
$$\begin{pmatrix}
3 & 1 & -1\\
1 & 3 & 1\\
-1 & 1 & 3
\end{pmatrix}$$
whose class number is also 1. Since $h(M) = 1$, any $\z$-coset in $\text{gen}(M + \bv)$ has an isometric copy of the form $M + \bx$ for some $\bx \in \mathbb Q M$. If $M + \bx \in \text{gen}(M + \bv)$, then the lattice $M + \mathbb Z \bx$ is in $\text{gen}(M + \mathbb Z\bv)$ which has only one class. Therefore, there exists an isometry $\sigma \in O(\mathbb Q M)$ such that $\sigma(\bx) \in M + \mathbb Z\bv$. Thus $\sigma(\bx) = \bz + a\bv$, where $\bz \in M$ and $a \in \mathbb Z$. But $Q(\bx)$ must be odd; therefore $a$ must be odd and hence $\sigma(\bx) \equiv \bv$ mod $M$. This shows that $\sigma(M + \bx) = M + \bv$ and so $h(M + \bv) = 1$.
\end{exam}
\begin{prop}
Let $\bx$ be a vector in $V$. Suppose that for each $\p \in \Omega$, there exists $\sigma_\p \in O(V_\p)$ such that $\bx \in \sigma_\p(M_\p + \bv)$. Then there exists $K + \bz \in \textnormal{gen}(M + \bv)$ such that $\bx \in K + \bz$.
\end{prop}
\begin{proof}
This follows from Lemma \ref{crt} since $\bx \in M_\p = M_\p + \bv$ for almost all $\p$.
\end{proof}
Let $a \in F$. We say that $M + \bv$ represents $a$ if there exists a nonzero vector $\bz \in M + \bv$ such that $Q(\bz) = a$, and that $\gen(M + \bv)$ represents $a$ if $V_\ell$ represents $a$ for all places $\ell \not \in \Omega$ and $M_\p + \bv$ represents $a$ for all places $\p \in \Omega$. The following corollary shows that the class number of a coset can be viewed as a measure of the obstruction of the local-to-global implication in (\ref{hasse}).
\begin{cor}
Let $a \in F^\times$. Suppose that $\textnormal{gen}(M_\p + \bv)$ represents $a$. Then there
exists $K + \bz \in \textnormal{gen}(M + \bv)$ which represents $a$.
\end{cor}
\begin{proof}
The hypothesis says that for each $\p \in \Omega$ there is a vector $\bz_\p \in M_\p + \bv$ such that $Q(\bz_\p) = a$. By virtue of the Hasse Principle, there exists a vector $\bz \in V$ such that $Q(\bz) = a$. At each $\p\in \Omega$, it follows from Witt's extension theorem that there is an isometry $\sigma_\p \in O(V_\p)$ such that $\sigma_\p(\bz_\p) = \bz$. Then for each $\p \in \Omega$,
$$\bz = \sigma_\p(\bz_\p) \in \sigma_\p(M_\p + \bv).$$
By the previous proposition, $\bz$ is contained in some coset $K + \bz \in \text{gen}(M + \bv)$. Equivalently, $a$ is represented by $K + \bz$.
\end{proof}
When $F$ is a number field, the obstruction of the local-to-global principle for representations of cosets may be overcome by applying the results on representations of quadratic lattices with approximation properties.
\begin{thm} \label{rep}
Let $M+\bv$ be an $\ring$-coset on a positive definite quadratic space over a totally real number field $F$. Suppose that $a \in F^\times$ is represented by $\textnormal{gen}(M+\bv)$.
\begin{enumerate}
\item[(1)] If $\dim(M) \geq 5$, then there exists a constant $C = C(M , \bv)$ such that $a$ is represented by $M + \bv$ provided $\mathbb N_{F/\q}(a) > C$.
\item[(2)] Suppose that $\dim(M) = 4$ and $a$ is primitively represented by $M_\p + \ring_\p\bv$ whenever $M_\p$ is anisotropic. Then there exists a constant $C^* = C^*(M, \bv)$ such that $a$ is represented by $M + \bv$ provided $\mathbb N_{F/\q}(a) > C^*$.
\end{enumerate}
\end{thm}
\begin{proof}
(1) Let $S$ be the subset of $\Omega$ containing all $\p$ for which $M_\p + \bv \neq M_\p$ or $M_\p$ is not unimodular. This $S$ is a finite set. For each $\p \in S$, let $\bx_\p \in M_\p$ such that $Q(\bx_\p + \bv) = a$. Choose an integer $s$ large enough so that $\p^s\bv \in M_\p$ for all $\p \in S$. Let $C$ be the constant obtained from applying the number field version of the main theorem in \cite{jk} (see \cite[Remark (ii)]{jk}) to $M + \ring\bv$, $S$, and $s$. If $\mathbb N_{F/\q}(a) > C$, then there exists $\bw \in M + \ring \bv$ such that $Q(\bw) = a$ and $\bw \equiv \bx_\p + \bv \equiv \bv$ mod $\p^s(M_\p + \ring_\p \bv)$ for every $\p \in S$. Since $\p^s(M_\p + \ring_\p \bv) \subseteq M_\p$, it follows that $\bw$ is in $M + \bv$, which means that $M + \bv$ represents $a$.
Part (2) can be proved in the same manner, except that we need to replace the main theorem in \cite{jk} by \cite[Appendix]{ch}.
\end{proof}
When $V$ is indefinite, we need to take into account of the orthogonal complement of a vector representing $a$. Since $a$ is represented by $\gen(M + \bv)$, $a$ must be represented by $\gen(M + \ring \bv)$, and it follows from the Hasse Principle that there exists $\bz \in V$ such that $Q(\bz) = a$. Let $W$ be the orthogonal complement of $\bz$ in $V$. The following theorem is an immediate consequence of \cite[Corollary 2.6]{bc}.
\begin{thm} \label{rep2}
Let $M + \bv$ be an $\ring$-coset on an indefinite quadratic space over a number field $F$. Suppose that $a \in F^\times$ is represented by $\textnormal{gen}(M + \bv)$.
\begin{enumerate}
\item[(1)] If $\dim(M) \geq 4$ or $W$ is isotropic, then $a$ is represented by $M + \bv$.
\item[(2)] Suppose that $\dim(M) = 3$, $W$ is anisotropic, and $M + \ring\bv$ represents $a$. Then $M + \bv$ represents $a$ if either $a$ is a spinor exception of $\textnormal{gen}(M + \ring \bv)$ or there exists $\p \not \in \Omega$ for which $W_\p$ is anisotropic and additionally $V_\p$ is isotropic if $\p$ is a real place.
\end{enumerate}
\end{thm}
\begin{proof}
Let $T$ be the set of places $\p$ for which $\bv \not \in M_\p$. By \cite[Corollary 2.6]{bc}, the hypothesis in either (1) or (2) implies that $M + \ring \bv$ represents $a$ with approximation at $T$. Therefore, there exists $\bw \in M + \ring \bv$ such that $Q(\bw) = a$ and $\bw \equiv \bv$ mod $M_\p$ for all $\p \in T$. Consequently, $M + \bv$ represents $a$.
\end{proof}
We conclude this paper by offering a few comments on the additional hypothesis placed in Theorem \ref{rep2}(2). First, there is an effective procedure \cite{sp1} to decide whether $a$ is a spinor exception of $\gen(M + \ring \bv)$. It depends on the knowledge of the local relative spinor norm groups $\theta(M_\p + \ring_\p\bv, a)$. These groups have been computed in \cite{sp1} when $\p$ is nondyadic or 2-adic, and in \cite{x} when $\p$ is general dyadic. When $a$ is a spinor exception of $\gen(M + \ring \bv)$, it is also possible to determine if $M + \ring\bv$ itself represents $a$; see \cite[Theorem 3.6]{cx} for example.
\bibliographystyle{amsplain}
|
2,877,628,090,064 | arxiv | \section*{Results}\label{results}
We describe the dynamics of the \textit{E. coli} TTM by means of a minimal out-of-equilibrium statistical-physics model. We first derive dynamical equations for the currents of DNA segments, mRNAs and ribosomes, incorporating steric effects and using the virial expansion to compute the local free energy.
This procedure is illustrated for a toy system of a binary mixture of hard spheres in \textit{SI Appendix}, Section \ref{toy_model}, and then for the full TTM in Section \ref{currents_model}.
By observing that \textit{E. coli} cells have an approximately cylindrical shape and symmetry, we reduce the three-dimensional cytoplasm to a single dimension
along the long cell axis (see Fig. \ref{fig1}A and B) and describe the TTM in terms of the one-dimensional concentrations of DNA segments, mRNAs, and ribosomes. Namely, we denote by $c_{\smdna}(x,t)$ the concentration of DNA plectoneme segments at position $x$ along the long cell axis and time $t$, by $\rho_n(x,t)$ that of polysomes composed of an mRNA and $n$ ribosomes, and by $c_{\smf}(x,t)$ that of freely diffusing ribosomes, see Fig. \ref{fig1}A and B.
We then consider the reaction-diffusion equations for these concentrations, where we incorporate the currents and the chemical reactions, i.e., ribosome-mRNA binding and unbinding, mRNA synthesis and degradation:
\begin{align}
\partial_t c_{\smdna}(x,t) =& -\partial_x J_{\smdna} (x,t), \label{RD_eqs_C}\\
\partial_t \rho_n (x,t) =& -\partial_x J_n (x,t) - k_{\textrm{on}} c_{\smf} (x,t) \rho_{n}(x,t) \nonumber \\ &-k_{\textrm{off}} \, n \, \rho_n(x,t) + k_{\textrm{on}} c_{\smf} (x,t) \rho_{n-1}(x,t) \nn \\
&+ k_{\textrm{off}} (n+1)\rho_{n+1}(x,t) + \alpha\, c_{\smdna}(x,t) \delta_{n,0} \nonumber \\ & -\beta \rho_{n} (x,t), \label{RD_eqs_n} \\ \label{reac_diff}
\partial_t c_{\smf} (x,t) =& -\partial_x J_{\smf}(x,t)-k_{\textrm{on}} c_{\smf}(x,t)\sum_{n} \rho_n(x,t) \nonumber \\ &
+ k_{\textrm{off}}\sum_{n} n\rho_n(x,t) + \beta\sum_{n} n\rho_n(x,t).
\end{align}
\noindent
In \crefrange{RD_eqs_C}{reac_diff}, $J_{\smdna}$, $J_n$, and $J_{\smf}$ denote the particle currents (derived in \textit{SI Appendix}, Sections \ref{toy_model} and \ref{currents_model}), $k_{\textrm{on}}$ and $k_{\textrm{off}}$ the rate constants for ribosome binding and unbinding due to completion of translation, respectively, $\alpha$ the rate at which mRNAs are created locally by transcription, and $\beta$ the mRNA degradation rate.
Regarding the steric interactions, as shown in Fig. \ref{fig1}A and B, we consider ribosomes as spheres of radius $R$ and, because mRNAs and polysomes with $n$ ribosomes are globular polymer coils, we also approximate them as spheres of radius $R$ and $R_n$, respectively. Because the \textit{E. coli} DNA has a branched, plectonemic structure with a well-defined persistence length and transverse radius \cite{odijk2000dynamics}, we consider the chromosome as a set of cylindrical segments, where the length of each segment corresponds to the persistence length. For the sake of computational tractability, we treat the DNA segments as disconnected as in Fig. \ref{fig1}B.
When deriving the particle currents, the quantity of interest is the free energy of the particles. For independent spherical particles (ribosomes and polysomes), the entropic term in the free energy is included in the virial expansion. However, for the DNA plectoneme the situation is different,
due to the connectivity between the DNA ``cylinders''. While connectivity should not have a large effect on the virial terms, it is not clear that this is the case for entropy. Nevertheless, we find that this entropic term is small compared to the virial terms of the DNA free energy and therefore we neglect it (see \textit{SI Appendix}, Section \ref{DNA_entropy}).
\subsection*{Model parameters}
We fix the model parameters from experiments as follows. First, we consider the parameters on a molecular scale: The radius and length of DNA cylinders are $\rho = 10 \, \rm nm$ and $L = 200 \, \rm nm$ \cite{mondal2011entropy,odijk2000dynamics}, respectively, where $L/2$ is approximately the persistence length of a DNA plectoneme \cite{cunha2001polymer,odijk2000dynamics}. However, two overlapping DNA plectonemes may be nested into each other, as discussed in \cite{mondal2011entropy}. To model this nesting, while we use the radius $\rho$ to describe overlaps between a DNA cylinder and ribosomes or mRNAs in the virial expansion, we use a smaller, effective radius $\rho'<\rho$ for overlaps between two DNA cylinders \cite{mondal2011entropy}, see \textit{SI Appendix}, Section \ref{currents_model} for details.
We take the ribosome radius to be $R = 10 \, \rm nm$ \cite{mondal2011entropy}, and the radius of a ribosome-free mRNA to be $R_0 = 20 \, \rm nm$ \cite{kaczanowska2007ribosome}. The radius $R_n$ of an mRNA loaded with $n$ ribosomes is estimated as the sum of the volume of a bare mRNA and $n$ times the volume of a ribosome, i.e., $4/3 \pi(R_0^3+n R^3)$ yielding $R_n = (R_0^3+n R^3)^{1/3}$.
We estimated the diffusion constant of the different species as follows: $D_{\smf} = 0.4 \, \mu \rm m^2/s$ for ribosomes, and $D_n = 5 \times 10^{-2} \mu \rm m^2/s$ for bare mRNAs and polysomes \cite{bakshi2012superresolution,sanamrad2014single}. Because DNA segments have a linear dimension similar to that of polysomes, we assume that their diffusion coefficients will also be similar and take $D_{\smdna} = 10^{-2} \, \mu \rm m^2/s$.
The parameters relative to the cellular scale are the total number of ribosomes per cell $N_{\smf}$, the cell half-length $\ell$, both of which will be varied, and the radius of the cellular cross section, which is held constant.
Because a central aim is to compare to the experiments in Ref. \cite{wu2019cell}, we are interested in values for a doubling time of $\sim 2 \, \rm hr$ as in that study.
We thus interpolated experimental data points for different growth rates, to obtain the parameter values for the desired growth rate (see \textit{SI Appendix}, Sections \ref{experiments} and \ref{parameters_est}) and obtained a total number $N_{\smf}\sim 7300$ ribosomes, a cross-sectional radius $R_{\rm cell} \approx 0.4 \, \rm \mu m,$ and a cell half-length $\ell \sim 0.9 \, \mu \rm m$ for a reference cell.
In addition, the total mRNA concentration for the reference cell was fixed at
$\rho_{\rm tot} = \sum_n \rho_n = 2400 \, \mu \textrm{m}^{-3}$ \cite{bartholomaus2016mRNA}.
The total number of DNA cylinders for the reference cell was taken to be $N_{\smdna} \sim 6700$ segments \cite{mondal2011entropy}.
Finally, we set the reaction rates to $k_{\rm on} = 6\times 10^{-4} \mu \rm m/s$, $k_{\rm off} = 2.5\times 10^{-2} \rm /s$ \cite{castellana2016spatial}, the mRNA degradation rate $\beta = 3 \times 10^{-3}/\rm s$ corresponds to an mRNA half life of $\sim 5\, \rm min$ \cite{bernstein2004global}, and the mRNA synthesis rate $\alpha$ is estimated from the global steady-state condition of \cref{RD_eqs_n},
$\alpha N_{\rm DNA} = \beta N_{\rm mRNA}$ \cite{castellana2016spatial}, where $N_{\rm mRNA}$ is the total number of mRNA molecules in the cell, i.e., $\rho_{\rm tot}$ times the cell volume.
In order to understand the compaction and localization of the bacterial nucleoid, we solved the one-dimensional reaction-diffusion \crefrange{RD_eqs_C}{reac_diff}. To compare the predictions of our model with experimental data, in what follows, we consider two scenarios for how the concentrations of the molecular species scale with cell length.
\begin{figure}
\includegraphics[scale=1.44]{non_eq_a.eps}
\figCaption{Out-of-equilibrium mechanisms split the nucleoid into two lobes located at 1/4 and 3/4 of the cell length} {\label{fig3} (\textit{A}) Concentration profiles for a filamentous cell, obtained from the equilibrium profile at $t=0$ by integrating forward in time the reaction-diffusion \crefrange{RD_eqs_C}{reac_diff} in the presence of the out-of-equilibrium processes until $t =20 \, \rm min$ and $t =40 \, \rm min$, for a cell with a half-length $ \ell = 4.05 \, \mu \rm m$ shown in the lowest panel of Fig. \ref{fig1}C. (\textit{B}) Positions of the center of mass of the left (dashed red curve) and right (red curve) halves of the DNA along the long axis of the cell as fraction of the total cell length, as functions of time.}
\end{figure}
\subsection*{Filamentous growth}
In filamentous growth, the total number of DNA segments, mRNAs, and ribosomes is proportional to the cell length. For each cell length, we first determined the equilibrium steady state of the system by minimizing the free energy (\textit{SI Appendix}, \cref{F_tot}), and then numerically integrated the reaction-diffusion \crefrange{RD_eqs_C}{reac_diff} forward in time to reach an out-of-equilibrium steady state---see \textit{SI Appendix}, Section \ref{numerics} for details. The minimization of the free energy takes into account the steric interactions between particles and predicts the existence of a phase-separated nucleoid in the cell. The out-of-equilibrium steady state is obtained by switching on the chemical reactions. The results are shown in Fig. \ref{fig1}C and D for different cell lengths, up to the cell length at which the nucleoid spontaneously splits into two lobes, and for different total mRNA densities.
The relation between the nucleoid length and cell length appears to be roughly linear up until the cell length at which the nucleoid begins to split in two, as seen in Fig. \ref{fig1}C and D. In our model, the nucleoid size (provided the nucleoid is single lobed) is mainly set by the balance of osmotic pressures between the nucleoid and the peripheral cytoplasm. These pressures solely stem from the entropy and steric interactions of the components of the mixture, making the nucleoid size a consequence of equilibrium physics: in fact, in \textit{SI Appendix}, Fig. \ref{equil_const_crowd} we show the steady-state profile of the system in the absence of out-of-equilibrium terms, displaying a linear relation between nucleoid size and cell size analogous to that of the nonequilibrium case. Moreover, as shown in Fig. \ref{fig1}D, the higher the total mRNA density, the smaller the nucleoid, implying that a high mRNA density increases the osmotic pressure on the nucleoid, thus making it shrink.
The dependence of the nucleoid size on the cell length can be quite accurately understood from a simplified model as follows. Consider the cell as a cylindrical container, divided in three parts by two movable walls. The chromosome is confined in the central container, while mRNAs and ribosomes are equally divided in the flanking ones. The walls will reach an equilibrium position at the point where the osmotic pressures between the compartments are balanced. In view of the steric interactions in each compartment, the osmotic pressure exerted by the $i$th compartment, to first order in the virial expansion (see \textit{SI Appendix}, Section \ref{estimates}), can be written as:
\begin{equation}
\label{pressures}
P_i=\frac{k_B T N_i}{V_i}\left(1+\frac{N_iB_i}{2V_i}\right),
\end{equation}
where $N_i$ is the number of particles in compartment $i$, $V_i=\sigma L_i$ the volume of the compartment, $\sigma$ the cross-section of the cell cylinder, $L_i$ the compartment length, and $B_i$ the pairwise virial coefficient associated with the interaction among particles. In the central compartment, $B_i$ corresponds to the virial coefficient between DNA segments, while for the flanking compartments we take $B_i$ as an effective virial coefficient, obtained by assuming that all ribosomes are bound to mRNAs, and equally distributed among them. By equating the pressures of the different compartments, we obtain the gray lines in Fig. \ref{fig1}D---see \textit{SI Appendix}, Section \ref{estimates} for details.
\begin{figure*}
\centering
\includegraphics[width=0.75\textwidth]{var_ratio.eps}
\figCaption{Steady-state concentration profiles, for single-chromosome filamentous growth}{
\label{fig2}(\textit{A}) Concentrations of DNA, $c_{\smdna}(x)$, free ribosomes $c_{\rm F}(x)$, and polysomes $\rho_n(x)$, shown as in Fig. \ref{fig1}C. (\textit{B}) Nucleoid length and standard deviation as a function of cell length from Ref. \cite{wu2019cell} (red) and from the model (black).}
\end{figure*}
In \cref{pressures}, the two terms in $1+N_iB_i/(2V_i)$ encode the two different factors that contribute to the osmotic pressure. The first term stems from the entropic pressure of an ideal gas while the second one, $N_iB_i/V_i$, comes from the inter-particle interactions. In the parameter range of Fig. \ref{fig1}D, the interaction term is typically between $10\%$ and $20\%$ larger than the entropic one (however, the effect of introducing a third-order virial coefficient is small, it is of the order of a tenth of the entropic term). We find that the inclusion of steric terms in both the nucleoid and mRNA/ribosome compartments makes the nucleoid swell compared to what its size would be with only entropic terms (ideal gas contribution). This is due to the nature of the nucleoid, a long relatively stiff polymer with little entropy per segment compared to all ribosomes and mRNAs collectively.
While the linear increase of nucleoid length with cell length is the result of equilibrium osmotic pressure balance, the splitting of the nucleoid is entirely due to out-of-equilibrium processes. In fact, for cells with $\rho_\textrm{tot}=2400/\mu \textrm{m}$ and a half length of $\sim 4\, \mu {\rm m}$ or larger, the equilibrium steady state used as the initial condition for the reaction-diffusion equations yields a nucleoid with a single lobe. By contrast, the nucleoid splits into two identical lobes positioned at $1/4$ and $3/4$ of the long cell axis when the reaction-diffusion \crefrange{RD_eqs_C}{reac_diff} are integrated forward in time, see Fig. \ref{fig3} and \textit{Movie S1}. Such a $1/4$ and $3/4$ positioning of the daughter nucleoids has been ubiquitously observed in experiments \cite{wu2019cell}.
In what follows, we present a simple argument to explain the dependence of the length at which the nucleoid splits with respect to the underlying parameters, e.g., the mRNA synthesis rate. We take the nucleoid to be a region with homogeneous DNA-segment concentration which extends from $x=-L_{\textrm{nucl}}/2$ to $x=L_{\textrm{nucl}}/2$, with interfaces that are perfectly sharp. The mRNAs synthesized within the nucleoid will diffuse until they reach the nucleoid boundaries and, because it is energetically favorable, they will then automatically escape the nucleoid and not return. As a result, the steady-state concentration of mRNAs within the nucleoid can be modeled by the following diffusion equation with a uniform source term due to mRNA synthesis and absorbing boundary conditions that represent mRNA escaping from the nucleoid:
\begin{equation}
D_n\frac{\partial^2 \rho_{\rm tot}(x)}{\partial x^2}=\alpha c_{\smdna}, \qquad \rho_{\rm tot}\left(\pm \frac{L_{\textrm{nucl}}}{2}\right)=0,
\end{equation}
where $\rho_{\rm tot}(x)$ is the total mRNA concentration at position $x$, $D_n$ the mRNA diffusion constant (as defined in \textit{Model parameters}), and $\alpha$ the rate of synthesis of mRNA. The solution to the above equation is $\rho_{\rm tot}(x)=(L_{\textrm{nucl}}^2/2-x^2)\alpha c_{\smdna}/D_n$, whose maximum at $x=0$ is $L_{\textrm{nucl}}^2\alpha c_{\smdna}/(2D_n)$.
We hypothesize that when the mRNA concentration at the center becomes larger than a given threshold, $\rho_{\rm tot}^*$, spinodal decomposition takes place due to steric interactions between mRNAs and DNA, causing the nucleoid to split into two lobes. We thus expect $\rho_{\rm tot}^*$ to roughly correspond to the spinodal line of the phase diagram, but, given the out-of-equilibrium nature of the system due to, e.g. mRNA synthesis, it could differ from the equilibrium spinodal boundary. Whatever value $\rho_{\rm tot}^*$ takes (provided its dependency on $\alpha$ is negligible), this simple model predicts a scaling for the critical length $L_{\textrm{nucl}}^*$ at which the nucleoid starts to divide of the form $L_{\textrm{nucl}}^* \propto (\alpha c_{\smdna})^{-1/2}$, obtained from equating the maximum of the mRNA concentration profile to a fixed value $\rho_{\rm tot}^*$. To test the prediction of this simple model, we numerically obtained the length at which the nucleoid divides for different values of $\alpha$, see the inset in Fig. \ref{fig1}D, and found a good agreement with the proposed scaling.
\subsection*{Single-chromosome filamentous growth}
So far we have analyzed the scaling of nucleoid size with cell size by assuming that the number of DNA segments is proportional to cell length. In this section we analyze another case of biological interest, namely, the case of a cell with a fixed amount of DNA and varying cell size. This scenario was recently analyzed in a dynamic imaging study of the \textit{E. coli} chromosome \cite{wu2019cell}, where the initiation of DNA replication and cell division were halted, yielding a single chromosome in a filamentously growing cell. We model this case by fixing the number of DNA segments, but allowing the cell size to vary. In addition, the mRNA and ribosome number are no longer be proportional to cell length: based on the data in Ref. \cite{hanna20xx}, we assume that the total concentrations of mRNAs and ribosomes decrease linearly with cell length, approaching zero at $30\, \mu {\rm m}$---see \textit{SI Appendix}, Section \ref{scaling_single_chromosome} for details.
Results are shown in Fig. \ref{fig2}: our model again predicts a roughly linear scaling of the nucleoid size with respect to cell length, while the DNA segment concentration decreases with cell size. This indicates that the decrease in DNA-segment concentration with cell size is balanced by the decrease of mRNA and ribosome concentrations, so as to keep nucleoid size a linear function of cell size. This can be seen clearly in Fig. \ref{fig2}A where the concentrations of all components of the TTM decrease as the cell size increases. While the model prediction for nucleoid versus cell length agrees reasonably well with experiments \cite{wu2019cell} for cell lengths smaller than $\sim 10 \, \rm \mu m$, there is a discrepancy for larger cells, see \textit{Discussion}.
\begin{figure}
\includegraphics[scale=1.45]{non_eq_c.eps}
\figCaption{ \label{fig5} Out-of-equilibrium processes center the nucleoid at midcell}{(\textit{A}) Concentration profiles obtained by initially shifting the steady-state profiles towards the right cell pole at $t=0$, and then integrating forward in time the reaction-diffusion \crefrange{RD_eqs_C}{reac_diff} in the presence of out-of-equilibrium processes to $t= 5 \, \rm min$ and $t =20\, \rm min$, for a cell with half-length $\ell = 1.8 \, \mu \rm m$, shown as in Fig. \ref{fig1}C. (\textit{B}) In red, location of the center of mass of the nucleoid along the long cell axis, as a function of time. In gray, the analytical lower bound obtained by neglecting nucleoid drag. Inset: The quantities depicted are the same as in \textit{B}, with the \textit{y}-axis is in logarithmic scale.
}
\end{figure}
\subsection*{Nucleoid centering}
As observed in Ref. \cite{wu2019cell}, a single bacterial nucleoid has a strong tendency to localize at midcell for all cell sizes. Following the recent suggestion that the central positioning of the nucleoid is regulated by an active process \cite{joyeux_microorganism}, we investigated whether the out-of-equilibrium process of mRNA production, diffusion, ribosome binding, and mRNA degradation can account for nucleoid centering.
We consider the case of a nucleoid that, due to a fluctuation, is not initially at the center of the cell, and test whether the out-of-equilibrium effects in our model can push the nucleoid back to the cell center. To model this, we use the steady-state profiles obtained for filamentous growth, and shift the concentration profiles towards the right cell pole.
The resulting configuration has a nucleoid displaced from the center, and equal mRNA and ribosome concentrations on both sides of the nucleoid. This concentration profile is used as the initial condition for \crefrange{RD_eqs_C}{reac_diff}, which we integrate forward in time in the presence of the out-of-equilibrium terms. As shown in Fig. \ref{fig5} and \textit{Movie S2}, the nucleoid is centered at midcell after $\sim 30 \, \rm min$.
The physical origin of this centering is mRNA synthesis in the nucleoid: The nascent mRNAs diffuse in the nucleoid until they reach one of its boundaries and then escape, with an equal flux to the left and right of the nucleoid. If the nucleoid is not centered, the accumulating mRNAs occupy a greater fraction of the available volume on one side of the nucleoid and thus create a higher osmotic pressure on that side. The resulting pressure difference ultimately drives the nucleoid back to the center of the cell.
The rate at which the nucleoid moves toward the cell center depends on both the pressure difference due to mRNA accumulation, and on the effective viscous drag experienced by the nucleoid. We can establish a lower bound for the time it takes the nucleoid to center by assuming that the response of the nucleoid to an osmotic pressure difference is fast (low drag), such that the nucleoid is always located at a position where the osmotic pressure difference vanishes. Then, the centering process is only limited by the speed at which mRNAs accumulate on either side of the nucleoid, which sets the pressure differences. The kinetics obtained in this limit are shown in Fig. \ref{fig5}B, and they are given by an exponential relaxation with timescale $\beta^{-1}$, set by the rate of mRNA degradation (see \textit{SI Appendix}, Section \ref{estimates}). As shown in the Figure, the nucleoid centering obtained from the full model lags behind the lower bound, showing that there is a non-negligible contribution from drag on the nucleoid. Both the lower bound and the result from the full model show an exponential relaxation of the nucleoid position for early times in the centering process.
\subsection*{Nucleoid expansion due to halt of mRNA synthesis}
It has been shown experimentally that when \textit{E. coli} transcription is halted, e.g. by treatment with rifampicin, the nucleoid expands \cite{bakshi2012superresolution,Cabrera2009ActiveTranscription}. A halt of mRNA synthesis depletes polysomes, and thus results in a lower osmotic pressure on the nucleoid. We tested this scenario with our model by using the out-of-equilibrium steady state shown in Fig. \ref{fig1}C as the initial condition for \crefrange{RD_eqs_C}{reac_diff}, switching off mRNA synthesis, and integrating forward in time. As shown in Fig. \ref{fig4}, the nucleoid expands and spreads over most of the intracellular space. The nucleoid does not take over the entire cell because there are pockets of free ribosomes at both cell poles, which prevent the DNA from occupying these spaces.
The nucleoid reaches its expanded steady state in $\sim 30 \, \rm min$, which is in good agreement with experimental data \cite{Cabrera2009ActiveTranscription}. However, the bulk of the expansion happens in the first $10 \, \rm min$---a timescale consistent with the half-life of mRNA ($5 \, \rm min$), whose degradation drives the expansion process.
\section*{Discussion}
In this study, we investigated the physical origins of the intracellular localization of DNA, messenger RNAs (mRNAs), and ribosomes in bacteria. This is a topic of general interest due to its far-reaching consequences, e.g., the spatial organization of transcription and translation \cite{gray2019nucleoid, Surovtsev2018SubcellularOrganization, Weng2019SpatialRNAP}, chromosome positioning and segregation \cite{joshi2011escherichia,wu2019cell}, and a wide range of cellular processes regulated by the nucleoid that excludes many macromolecules from the volume which it occupies \citep{Coquel2013LocalizationProtein,Janissen2018DNACompaction}.
We developed a model for the spatial organization of the bacterial nucleoid based on steric interactions among DNA, mRNAs, and ribosomes. The model predicts the formation of a phase-separated nucleoid, whose size is in agreement with experimental measurements \cite{wu2019cell} for cells smaller than $10\, \mu\textrm{m}$ (Fig. \ref{fig1}). Beyond this cell length, our model is no longer accurate, for reasons that may include the lack of connectivity among modeled DNA segments, uncertainties in the concentration of crowders, and molecular components not considered in the model, such as nucleoid-associated proteins \cite{Dame2020} or topoisomerases that control DNA supercoiling \cite{Stuger2002}. The model also accounts for nucleoid expansion as a result of a halt in mRNA synthesis, demonstrating that the progressive degradation of crowders could be the physical cause of the expansion. Indeed, the timescales on which such expansion happens matches the one observed experimentally \cite{Cabrera2009ActiveTranscription}, and coincides with the timescales of mRNA turnover.
\begin{figure}
\includegraphics[scale=1.45]{non_eq_b.eps}
\figCaption{ \label{fig4} The nucleoid expands in the absence of mRNAs synthesis}{(\textit{A}) Steady-state profile including mRNA synthesis ($t=0$) and profiles obtained by integrating forward in time from
the steady-state profile at $t=0$ in the absence of mRNA synthesis ($t =4 \, \rm mins$ and $t =20 \, \rm mins$), for a cell with half length $\ell = 3.6 \, \mu \rm m$. The concentration profiles are shown as in Fig. \ref{fig1}C.
(\textit{B}) Fraction of the cell volume occupied by the nucleoid, as a function of time (computed as the fraction of length along the axis of the cell with a DNA segment concentration $c_\smdna > 1000\, \, \mu\textrm{m}^{-1})$. The step-like shape of the graph is due to the spatial discretization in our numerical solutions (see \textit{SI Appendix}, Section \ref{numerics}).
}
\end{figure}
Our results underline the importance of out-of-equilibrium effects in the regulation of nucleoid size and position. The nucleoid is known to localize at midcell \cite{wu2019cell}, and we demonstrate that the synthesis of mRNAs and their expulsion from the nucleoid caused by steric effects is sufficient to give rise to this positioning---see Fig. \ref{fig5}. In fact, a perturbation from the central position of the nucleoid induces an osmotic-pressure difference between the two cell poles, which pushes the nucleoid back to midcell. The timescale for this centering depends on both the time it takes to establish an osmotic-pressure difference, which is set by the mRNA turnover time, and the drag experienced by the nucleoid. This drag may be underestimated in our model, because we do not include effects that could slow down nucleoid centering, e.g., the transient attachment of the nucleoid to the membrane by proteins that are simultaneously being transcribed, translated, and inserted in the membrane, also known as transertion \cite{Gorle2017}.
Furthermore, our model shows that out-of-equilibrium effects are responsible for the ubiquitous nucleoid splitting and localization at $1/4$ and $3/4$ positions along the long cell axis. Thus, our analysis shows that the synthesis of mRNAs within the nucleoid, without additional active processes, is a robust mechanism to make the daughter nucleoids localize at $1/4$ and $3/4$ positions, as observed experimentally \cite{wu2019cell}.
Our study implies that steric interactions make the bacterial cytoplasm a poor solvent for the chromosome, as recently indicated by experiments \cite{Xiang2020SolventQuality}. However, steric interactions may not be the only contribution to the poor-solvent quality of the cytoplasm. Other types of intermolecular interactions \cite{Odijk1998OsmoticCompaction} or the effect of nucleoid-associated proteins \cite{Dame2020} could also affect the solvent quality of the cytoplasm and the organization of the nucleoid in the cell. For future studies, both theoretical and experimental, research into these other regulators of the nucleoid size could yield a more complete picture of its organization, and improve the accuracy of the results presented here.
Finally, we observe that Turing patterns \cite{Turing1952patterns} display out-of-equilibrium patterning features that could seem similar to the ones produced by our model, see \textit{SI Appendix} Fig. \ref{fig_lobes}. These out-of-equilibrium patterns have been used to investigate many biological features on a cellular scale, such as the positioning of protein clusters in \textit{E. coli} \cite{murray2017self}. However, unlike Turing patterns, our model predicts phase separation exclusively due to steric interactions and in the absence of out-of-equilibrium effects, see Fig. \ref{fig3}. While the patterns produced by our model could be related to other out-of-equilibrium phase-separation models \cite{Li2020noneq} such as models of growing droplets \cite{Zwicker2017division}, our model provides a conceptually simpler framework to produce these patterns. In fact, unlike a model of physically growing droplets, our analysis involves a conserved order parameter---the total number of DNA segments---and the effect of out-of-equilibrium terms---mRNA production and degradation---is limited to nucleoid reshaping and division. Despite its simplicity, our model produces a number of experimentally observed patterning effects, such as nucleoid centering at midcell, and splitting and positioning of sister lobes during cell division. In addition, the patterning of our model is not limited to nucleoid splitting into two sister lobes, because our model predicts that the nucleoid can split into more than two lobes, whose size is given by a characteristic length and whose positions are tightly controlled, see \textit{SI Appendix}, Fig. \ref{fig_lobes}. Similar patterns for ordered nucleoid positioning have been found in long filamentously growing \textit{E. coli} cells \cite{Wehrens2018}, albeit with a shorter characteristic length.
Given its generality, our analysis is not restricted to the nucleoid of prokaryotic cells \cite{Pappu2020Condensates}. Indeed, division of certain phase-separated condensates has been experimentally related to out-of-equilibrium processes, as is the case for the ParABS partition system, which creates phase-separated condensates of DNA and ParB, around \textit{par}S sites, whose division is controlled by the activity of ParB's ATPase activity on ParA \cite{Guilhas2020ATPDriven}.
The activity-driven nucleoid division described in our model may thus constitute a general strategy employed by cells to control the structure of membraneless compartments.
\begin{acknowledgments}
We thank J. Prost, A. Sclocchi, P. Sens, H. Salman, and J. Wagner for valuable conversations and suggestions. This study was supported in part by Agence nationale de la recherche (ANR), grant ANR-17-CE11-0004, and the National Science Foundation, through the Center for the Physics of Biological Function (PHY-1734030).
\end{acknowledgments}
|
2,877,628,090,065 | arxiv | \section{Introduction}
In recent years, large-scale pre-training has become the new paradigm in the natural language processing (NLP) field. These models have demonstrated surprisingly good generalization abilities and can be applied to different downstream tasks by a simple fine-tuning. Several comprehensive benchmarks are constructed to evaluate such powerful models, including GLUE \cite{DBLP:journals/corr/abs-1804-07461} and SuperGLUE \cite{DBLP:journals/corr/abs-1905-00537} for evaluating monolingual natural language understanding systems, XGLUE \cite{liang2020xglue} and XTREME \cite{hu2020xtreme} for evaluating multilingual natural language understanding and generation systems. Such pre-trained models have also been extended to vision-language scenarios \citep{lu2019vilbert,chen2019uniter,li2019unicodervl,li2020oscar,ni2021m3p,sun2019videobert,sun2019cbt,luo2020univl} to handle multimodal tasks such as image(or video)-text retrieval and image (or video) captioning. However, there is still no comprehensive benchmark dataset for evaluating such multimodal pre-trained models. Besides, most existing vision-language datasets are labeled in English only, which cannot be used to evaluate the qualities of such models on other languages.
Motivated by this, we present \textbf{GEM}, a \textbf{G}eneral \textbf{E}valuation benchmark for \textbf{M}ultimodal tasks. Comparing with GLUE, SuperGLUE, XGLUE and XTREME, GEM is designed for evaluating the generalization capabilities of vision-language models and consists of two subsets: GEM-I, which evaluates text-to-image retrieval and image captioning capabilities, and GEM-V, which evaluates text-to-video retrieval and video captioning capabilities. Besides, it is also a multilingual dataset, where the natural language contexts are collected from a commercial search engine. We describe two vision-language pre-trained models, M$^3$P \cite{ni2021m3p} and m-UniVL, as the baselines for GEM-I and GEM-V, respectively, where M$^3$P is an existing multilingual image-language pre-trained model, m-UniVL is a multilingual extension of UniVL \cite{luo2020univl} for multilingual video-language tasks. The evaluation results of these two models on GEM are reported in the experiment part.
The key contribution of this paper is twofold: (1) we build GEM as the first large-scale multilingual multimodal benchmark, which can be used to evaluate the generalization capabilities of vision-language pre-trained models on a set of diversified multimodal tasks. (2) we provide two multilingual multimodal pre-trained models, M$^3$P and m-UniVL, as the baselines of GEM for image-language and video-language tasks, respectively. We hope GEM can further advance the research in the multimodal community, just as its predecessors did in the NLP community.
\section{Dataset Construction}
\label{sec:dataset-construction}
\begin{figure*} [tp]
\centering
\includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{image_case.png}
\caption{Examples in GEM-I dataset: Q: Query, T: Title.}
\label{GEMI-case}
\end{figure*}
\begin{table}
\scalebox{0.7}{
\centering
\begin{tabular}{lrrrr}
\hline \textbf{Language} & \textbf{Train} & \textbf{Dev} & \textbf{Test} & \textbf{Total} \\ \hline
English (en) & 998,000 & 1,000 & 1,000 & 1,000,000 \\
Spanish (es) & 18,000 & 1,000 & 1,000 & 20,000 \\
French (fr) & 18,000 & 1,000 & 1,000 & 20,000 \\
Italian (it) & 18,000 & 1,000 & 1,000 & 20,000 \\
Portuguese (pt) & 18,000 & 1,000 & 1,000 & 20,000 \\
German (de) & 18,000 & 1,000 & 1,000 & 20,000 \\
Korean (ko) & 8,000 & 1,000 & 1,000 & 10,000 \\
Polish (pl) & 8,000 & 1,000 & 1,000 & 10,000 \\
Catalan (ca) & 2,000 & 1,000 & 1,000 & 4,000 \\
Dutch (nl) & 2,000 & 1,000 & 1,000 & 4,000 \\
Japanese (ja) & 2,000 & 1,000 & 1,000 & 4,000 \\
Indonesian (id) & 2,000 & 1,000 & 1,000 & 4,000 \\
Vietnamese (vi) & 2,000 & 1,000 & 1,000 & 4,000 \\
Czech (cs) & 2,000 & 1,000 & 1,000 & 4,000 \\
Romanian (ro) & 2,000 & 1,000 & 1,000 & 4,000 \\
Turkish (tr) & 0 & 0 & 1,000 & 1,000 \\
Galician (gl) & 0 & 0 & 1,000 & 1,000 \\
Croatian (hr) & 0 & 0 & 1,000 & 1,000 \\
Hungarian (hu) & 0 & 0 & 1,000 & 1,000 \\
Malay (ms) & 0 & 0 & 1,000 & 1,000 \\
\hline
\textbf{Total} & \textbf{1,118,000} & \textbf{15,000} & \textbf{20,000} & \textbf{1,153,000} \\
\hline
\end{tabular}
}
\caption{\label{GEMI-table} Language distribution and data statistics of GEM-I for multilingual image-language tasks.}
\end{table}
\begin{table}
\scalebox{0.7}{
\centering
\begin{tabular}{lrrrr}
\hline \textbf{Language} & \textbf{Train} & \textbf{Dev} & \textbf{Test} & \textbf{Total} \\ \hline
German (de) & 3,316 & 1,000 & 1,000 & 5,316 \\
Portuguese (pt) & 3,258 & 1,000 & 1,000 & 5,258 \\
Dutch (nl) & 2,961 & 1,000 & 1,000 & 4,961 \\
Spanish (pt) & 2,894 & 1,000 & 1,000 & 4,894 \\
Russian (ru) & 2,804 & 1,000 & 1,000 & 4,804 \\
French (fr) & 2,776 & 1,000 & 1,000 & 4,776 \\
Italian (it) & 2,589 & 1,000 & 1,000 & 4,589 \\
Korean (ko) & 2,452 & 1,000 & 1,000 & 4,452 \\
English (en) & 2,426 & 1,000 & 1,000 & 4,426 \\
Japanese (ja) & 2,000 & 1,000 & 1,000 & 4,000 \\
Arabic (ar) & 2,000 & 1,000 & 1,000 & 4,000 \\
Polish (pl) & 2,000 & 1,000 & 1,000 & 4,000 \\
Chinese-Traditional (zh-t) & 2,000 & 1,000 & 1,000 & 4,000 \\
Farsi (fa) & 2,000 & 1,000 & 1,000 & 4,000 \\
Indonesian (id) & 2,000 & 1,000 & 1,000 & 4,000 \\
Turkish (tr) & 2,000 & 1,000 & 1,000 & 4,000 \\
Vietnamese (vi) & 2,000 & 1,000 & 1,000 & 4,000 \\
Hebrew (he) & 1,807 & 1,000 & 1,000 & 3,807 \\
Romanian (ro) & 1,441 & 1,000 & 1,000 & 3,441 \\
Swedish (sv) & 1,419 & 1,000 & 1,000 & 3,419 \\
Filipino (tl) & 1,294 & 1,000 & 1,000 & 3,294 \\
Malay (ms) & 0 & 0 & 1,000 & 2,668 \\
Norwegian (no) & 0 & 0 & 1,000 & 1,098 \\
Catalan (ca) & 0 & 0 & 1,000 & 1,002 \\
Croatian (hr) & 0 & 0 & 907 & 907 \\
Georgian (ka) & 0 & 0 & 863 & 863 \\
Chinese-Simplified (zh-s) & 0 & 0 & 833 & 833 \\
Hungarian (hu) & 0 & 0 & 811 & 811 \\
Albanian (sq) & 0 & 0 & 809 & 809 \\
Serbian-Latin (sr-l) & 0 & 0 & 774 & 774 \\
\hline
\textbf{Total} & \textbf{47,437} & \textbf{21,000} & \textbf{28,997} & \textbf{99,202} \\
\hline
\end{tabular}
}
\caption{\label{GEMV-table} Language distribution and data statistics of GEM-V for multilingual video-language tasks. }
\end{table}
To the best of our knowledge, GEM dataset is the first multilingual vision-language dataset constructed for image-language and video-language tasks as the same time. GEM-I contains 1.2 million \{\textit{Query, Image, Title}\} triplets in 20 different languages for text-to-image retrieval and image captioning tasks. GEM-V contains 99K \{\textit{Query, Video, Title}\} triplets in 30 languages for text-to-video retrieval and video captioning tasks. In both GEM-I and GEM-V, \textit{Title} denotes the title of the web page where each image (or video) is extracted. This signal can be used as the auxiliary information in all GEM tasks, as it is usually highly relevant to the corresponding image (or video).
Next, we will describe how GEM-I and GEM-V are collected from a commercial search engine.
\subsection{GEM-I Construction}
First, we collect several billion images with Creative Commons licenses from the Internet, and discard images that contain pornographic or racy content. We also discard images with human faces, to avoid revealing privacy or introducing bias to our data. Besides, we only keep images which are larger than 300$\times$300 pixels to guarantee high image quality. The pornographic classifier, racy classifier, and human face classifier are trained and evaluated on human-labeled data. The (precision, recall) of them are (0.85, 0.92), (0.79, 0.94), and (0.85, 0.92), respectively.
Then, we collect user queries from a commercial search engine for each image based on user historical clicks. We also collect the title of the Web page that contains the image as the additional context, forming \{\textit{Query, Image, Title}\} triplets. Some text cleanup work is done to only keep high quality queries and contexts, including removing pornographic words and meaningless strings, and discarding very short queries or titles in that they are less likely to depict the image content, etc. We also apply an in-house GBDT model to filter out potentially highly irrelevant \{\textit{Query, Image, Title}\} triplets, which is trained using a small amount of human-labeled data, to predict the similarity between each \{\textit{Query}\} and \{\textit{Image, Title}\} pair.
Finally, we only keep the top 20 languages which have more than 1000 images, and sample 1.2 million \{\textit{Query, Image, Title}\} triplets in total. The average length of query in GEM-I is 5.5 terms, which is shorter than 10.6 in MSCOCO \cite{chen2015microsoft} and 12.3 in Flicker30K \cite{vinyals2015tell}. Also, the average length of title is 10.1 terms. This makes GEM-I a more practical benchmark, since all data in GEM-I come from the real world, where the language configuration truly differs from the queries in existing datasets. For example, the queries in GEM were shorter and more concise, without perfect grammar or syntax structure. This makes GEM queries more "natural". Therefore, our benchmark can evaluate the models on data closer to real-world scenarios, so that the performance of the models will be more convincing in terms of being used in real-world applications. Based on human assessment on sampled query-image pairs, 83\% of the them are well matched pairs in that the query is a plausible caption of its paired image. We randomly split the data into train, dev and test sets within each language. The data statistics and language distribution of GEM-I can be found in Table~\ref{GEMI-table}. Figure~\ref{GEMI-case} gives some examples.
\subsection{GEM-V Construction}
We collect several billion videos from the Internet, and discard videos with pornographic or racy contents. We also discard very long videos to save storage and transfer expenses. For each video, its query and title are collected from a commercial search engine and cleaned-up according to a similar process as we described in GEM-I, where another in-house model is trained to filter out potentially irrelevant \{\textit{Query, Video, Title}\} triplets.
Finally, we only keep the top 30 languages which have more than 700 videos, and sample 99K \{\textit{Query, Video, Title}\} triplets in total. The total video length of GEM-V is 2,049 hours, and the average video length is 1.2 minutes. The average length of query in GEM-V is 5.3 terms, and that of title is 8.5 terms. We also conduct human evaluation on some sampled query-video pairs, and find 70\% of them are plausible matched pairs. We randomly split the data into train, dev and test sets within each language. The data statistics and language distribution of GEM-V can be found in Table~\ref{GEMV-table}. Figure~\ref{GEMV-case} gives some examples.
\begin{figure*} [tp]
\centering
\includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{video_case.png}
\caption{Examples in GEM-V dataset: Q: Query, T: Title.}
\label{GEMV-case}
\end{figure*}
\section{Baseline Models}
\label{sec:baseline-models}
This section will introduce two baseline models for GEM, including M$^3$P, which is a multilingual multimodal pre-trained model for image-language tasks, and m-UniVL, which is a multilingual extension of UniVL \cite{luo2020univl} for video-language tasks.
\subsection{M$^3$P as Baseline of GEM-I}
We select M$^3$P \cite{ni2021m3p} as the baseline model for tasks in GEM-I, as it is the state-of-the-art multilingual image-language pre-trained model for both image-language understanding and generation tasks.
The M$^3$P model uses the model architecture of BERT for understanding tasks and a BERT-based encoder-decoder architecture for generation tasks.
For understanding tasks, multilingual masked language modeling, multimodal masked language modeling, masked region modeling and visual-linguistic matching are used as pre-training tasks to train a Transformer-based encoder.
For generation tasks, multilingual denoising auto-encoding, image captioning and denoising image captioning are used as pre-training tasks to train a Transformer-based encoder-decoder. By training the encoder and the encoder-decoder with a multitask learning framework, universal representations are learned to map objects occurred in different modalities or expressed in different languages to vectors in a common semantic space.
\textbf{Fine-tune Tasks} Based on the pre-trained M$^3$P model, we further finetune it on our GEM-I data. For the text-image retrieval task, we adopt the BCE loss and NCE loss \cite{pmlr-v9-gutmann10a} (with equal weights) to learn the instance-level alignment between texts and images. The negative samples are generated by randomly forming text-image pairs from different training samples in the same batch. For the image captioning task, we directly learn captioning loss on GEM-I data.
\textbf{Side-Information}
Since title is considered as the side-information of the image, we concatenate it together with the image and feed them into the model. During the negative sampling process in the retrieval task, we treat title and image as a whole, i.e., for a certain query, the titles and images from other samples are considered as negatives.
\subsection{m-UniVL as Baseline of GEM-V}
We adopt the same model structure with the unified video and language pre-trained model UniVL \cite{luo2020univl}, which can perform both multi-modal understanding and generation tasks. Specifically, we extend the pre-trained UniVL model from monolingual to multilingual by replacing the BERT-based module with XLM-R \cite{DBLP:journals/corr/abs-1911-02116}, and call the new model m-UniVL. m-UniVL adopts an encoder-decoder architecture, including two single-modal encoders to encode the multilingual text and the visual features respectively, and one cross-modal encoder to learn the interactions between the two modalities, and finally an optional decoder for generation tasks. To better leverage the pre-trained models, we initialize each module with different pre-trained weights: for the multilingual text encoder, we directly initialize it with the pre-trained XLM-R\footnote{https://huggingface.co/xlm-roberta-base} \cite{DBLP:journals/corr/abs-1911-02116}, and for other modules including the visual encoder, the cross encoder and the decoder, we initialize them with the weights of the pre-trained UniVL\footnote{https://github.com/microsoft/UniVL}.
\textbf{Fine-tune Tasks} Based on the pre-trained m-UniVL, we further finetune it using GEM-V data. For the text-video retrieval task, we only employ encoders in m-UniVL in the finetuning stage and use them to predict the matching score between text and video. There are two baseline models for the retrieval task: 1) m-UniVL(loose), the loosely coupled model that only uses the single-modal encoders; 2) m-UniVL(tight), the loosely plus tightly coupled model that includes both the single-modal encoders and the cross-modal encoder. We adopt the NCE loss \cite{pmlr-v9-gutmann10a} to learn to discriminate the positive video-text pairs against negative ones. The negative video-text samples are created by replacing the text or video in a positive sample with randomly-selected text or video from other samples. For video captioning task, we employ all modules including all the encoders and decoder to learn the caption generation task.
\textbf{Side-Information} In regard to the titles, we use them as the side-information of the videos for an efficient text-video retrieval. In details, we first map the embedding of the title to the same dimension with the video embedding, and then concatenate them together. Then we encode them using the visual encoder to generate the enhanced video features for further processing.
\section{Related Work}
\subsection{Natural Language Benchmarks}
GLUE \cite{DBLP:journals/corr/abs-1804-07461} and SuperGLUE \cite{DBLP:journals/corr/abs-1905-00537} are two comprehensive datasets that can be used to train and evaluate natural language understanding systems. GLGE \cite{liu2020glge} is another comprehensive dataset for natural language generation evaluation. XGLUE \cite{liang2020xglue} and XTREME \cite{hu2020xtreme} are two recent benchmark efforts that extend the evaluation scenarios from monolingual to multilingual. Recent pre-trained language models benefit a lot from these datasets, by evaluating their effectiveness under a relatively fair environment.
\subsection{Vision-Language Benchmarks}
A number of vision-language datasets have been widely used in the multimodal research.
MSCOCO \cite{chen2015microsoft} and Flicker30K \cite{vinyals2015tell} are two datasets for image-text retrieval and image-captioning tasks. These two benchmarks have been extended to multilingual tasks \citep{elliott2016multi30k,elliott2017findings,mcoco1,mcoco2} as well. VQA \cite{VQA} and GQA \cite{hudson2018gqa} are two datasets for visual question answering. VCR \cite{DBLP:journals/corr/abs-1811-10830} is another dataset for visual commonsense reasoning. Comparing with all these existing datasets, GEM-I has unique characteristics. First, it is a large-scale multilingual image-text dataset covering 20 different languages. Second, the query-image pairs in GEM-I come from a commercial search engine. Therefore, it has big practical values. Third, for each query-image pair, the title of the Web page that contains the image is also included as the additional context, which makes GEM-I different from all existing datasets.
HowTo100M \cite{DBLP:journals/corr/abs-1906-03327}, YouCook2 \cite{ZhXuCoCVPR18}, and MSR-VTT \cite{xu2016msr-vtt} are three typical benchmarks for video-text retrieval and video captioning tasks. TVQA \cite{DBLP:journals/corr/abs-1809-01696} and ActivityNet-QA \cite{DBLP:journals/corr/abs-1906-02467} are two typical benchmarks for video question answering. Comparing with all these existing datasets, GEM-V is the first video-language benchmark supporting multilingual scenarios. Similar to GEM-I, it also has big practical values, as all data in GEM-V come from a real-world search engine with massive users.
\section{Experiments}
In this section, we evaluate two baseline pre-trained models (described in Section~\ref{sec:baseline-models}) on GEM. Specifically, M$^3$P is evaluated on GEM-I for multilingual image-language tasks and m-UniVL is evaluated on GEM-V for multilingual video-language tasks. For both baseline models, we fine-tune them on downstream tasks directly.
\subsection{Image-Language Evaluation on GEM-I}
\begin{table*}[tp]
\centering
\scalebox{0.64}{
\begin{tabular}{l|cccccccccc|c}
\toprule
Setting &en &es &fr &it & pt &de &ko &pl &ca &nl &-\\
\midrule
\multicolumn{11}{c}{\textit{Zero-Shot}} \\
M$^3$P (Q$\rightarrow$I) &22.08 &8.31 &8.51 &7.16 &7.05 &9.31 &4.68 &4.38 &9.10 &9.09 &-\\
M$^3$P (Q$\rightarrow$I+T) &5.80 &4.16 &3.98 &2.9 &3.71 &3.36 &2.68 &4.07 &2.86 &3.96 &-\\
\multicolumn{11}{c}{\textit{Fine-tune on ALL}} \\
M$^3$P (Q$\rightarrow$I) &43.85 &26.15 &24.83 &22.72 &27.05 &27.18 &15.80 &32.80 &12.83 &20.78 &-\\
M$^3$P (Q$\rightarrow$I+T) &93.75 &93.16 &95.15 &93.26 &93.73 &89.37 &67.46 &82.67 &90.78 &90.37 &-\\
\midrule \midrule
&ja &id &vi &cs &ro &tr &gl & hr & hu & ms &AVG\\
\midrule
\multicolumn{11}{c}{\textit{Zero-Shot}} \\
M$^3$P (Q$\rightarrow$I) &7.68 &12.06 &5.00 &5.32 &5.81 &5.13 &4.72 &5.30 &4.30 &8.08 &7.65\\
M$^3$P (Q$\rightarrow$I+T) &4.5 &4.53 &2.08 &3.72 &3.61 &3.28 &2.55 &2.58 &3.26 &3.41 &3.55\\
\multicolumn{11}{c}{\textit{Fine-tune on ALL}}\\
M$^3$P (Q$\rightarrow$I) &18.88 &22.13 &10.10 &13.33 &16.10 &13.23 &10.38 &13.51 &11.11 &14.33 &19.85 \\
M$^3$P (Q$\rightarrow$I+T) &71.90 &81.98 &54.83 &76.23 &64.35 &71.97 &75.00 &71.28 &63.43 &74.18 &79.74\\
\bottomrule
\end{tabular}
}
\caption{Evaluation results of M$^3$P on GEM-I test set for text-to-image retrieval tasks where Mean-Recall is taken as metric. \textbf{Q$\rightarrow$I} denotes
the setting where only image (I) is used to compute its similarity with query (Q), \textbf{Q$\rightarrow$I+T} denotes
the setting where both image (I) and title (T) are used to compute the similarity with query (Q). The average score is computed over all 20 languages.}
\label{tab:result_of_image_text_retrieval}
\end{table*}
\begin{table*}[tp]
\centering
\scalebox{0.64}{
\begin{tabular}{l|l|cccccccccc|c}
\toprule
Setting & Metric &en &es &fr &it &pt &de &ko &pl &ca &nl &- \\
\midrule
M$^3$P & ROUGE-L &6.97 &13.86 &10.47 &9.13 &8.35 &8.67 & 3.27 & 9.31 &12.84 &3.62 &- \\
M$^3$P & METEOR &3.21 &5.84 &4.91 &4.14 &3.67 &3.81 & 2.21 & 4.01 &5.7 &1.54 &- \\
M$^3$P & CIDEr &17.89 &8.68 &14.92 &11.98 &7.93 &20.18 & 7.33 & 8.44 &7.59 &6.79 &- \\
\midrule \midrule
&ja &id &vi &cs &ro &tr &gl & hr & hu & ms &AVG\\
\midrule
M$^3$P & ROUGE-L &4.10 &0.96 & 0.66 & 4.58 & 3.98 &0.32 &9.84&0.25 &0.57&0.39 &5.61\\
M$^3$P & METEOR &1.26 &0.47 & 0.22 & 1.97 & 1.78 &0.15 &4.43&0.11 &0.26&0.18 &2.49\\
M$^3$P & CIDEr &5.01 &2.03 & 1.14 & 5.72 & 2.96 &0.79 &6.86&0.72 &1.43&0.91 &6.97\\
\bottomrule
\end{tabular}
}
\caption{Evaluation results of M$^3$P on GEM-I test set for image captioning task where ROUGE-L, METEOR and CIDEr are taken as metrics. Only images (without title) are used to test its multilingual multimodal captioning ability. The average score is computed over all 20 languages.}
\label{tab:result_of_image_captioning}
\end{table*}
\subsubsection{Experimental Settings}
We select the open-source version\footnote{https://github.com/microsoft/M3P} of M$^3$P \cite{ni2021m3p} for the image-language evaluation on GEM-I. It uses 101G sentences (in 100 languages) extracted from Wikipedia as the multilingual pre-training corpus, and uses 3.3 million English image-caption pairs in Conceptual Captions \cite{sharma2018conceptual} as the multimodal pre-training corpus.
For text-to-image retrieval, the hyper-parameters of the encoder are set as follows: 768 hidden units, 12 heads, GELU activation, a dropout rate of 0.1, 128 max input length, 12 layers in encoder. For image captioning, the hyper-parameters of the encoder-decoder are set as follows: 768 hidden units, 8 heads, GELU activation, a dropout rate of 0.1, 128 max input length, 12 layers in both encoder and decoder. The transformer parameters between the encoder and decoder are shared, including embedding modules and self-attention modules.
We fine-tune M$^3$P on text-to-image retrieval and image captioning tasks. For retrieval, we use Adam optimizer with $\beta_1=0.9$, $\beta_2=0.98$, an initial learning rate of 5e-5, a weight decay of 1e-4 and a batch size of 64 to fine-tune M$^3$P for 30 epochs. For captioning, a learning rate of 1e-4 and a batch size of 16 are used to fine-tune M$^3$P for 20 epochs. All above calculations are carried on 4 NVIDIA Tesla P100 GPUs.
\subsubsection{Text-to-Image Retrieval Results}
We follow the same evaluation metric, mean-Recall (average score of R@1, R@5, R@10), in M$^3$P to report its the performance on text-to-image retrieval task on GEM-I dataset.
From the results reported in Table \ref{tab:result_of_image_text_retrieval},
we have several observations:
1) When applying M$^3$P to GEM-I without fine-tuning (i.e. the zero-shot setting), the general performance is poor. The major reason is that M$^3$P is pre-trained on a monolingual multimodal corpus and a multilingual monomodal corpus, and both datasets have very different data distributions comparing with GEM-I.
2) By fine-tuning M$^3$P using all labeled data in different languages (i.e. the fine-tune on all setting), better performance can be obtained. This shows the strong transfer ability of M$^3$P, when there is a moderate amount of labeled data for fine-tuning.
3) By furthering considering the title signal in this retrieval task, the general performance can be improved significantly. This indicates the strong correlation between the query and the title. Besides, when taking the title signal into the zero-shot setting, we can observe a performance drop. It is due to that M$^3$P is pretrained with input paradigm Q-I, thus making it not suitable for evaluating Q-I-T paradigm directly.
\subsubsection{Image Captioning Results}
As in Table~\ref{tab:result_of_image_captioning}, we report the performance of image captioning tasks on GEM-I test set with M$^3$P model where ROUGE-L \cite{lin-och-2004-automatic}, METEOR \cite{banerjee2005meteor} and CIDEr \cite{vedantam2015cider} are taken as the evaluation metrics. To study the image captioning ability of M$^3$P, we only use images (without title) to generate queries in GEM-I dataset. In general, the M$^3$P model performs relatively poor on this task, due to that most search queries are short keywords instead of a complete sentence, and they differ from our pre-training data a lot.
From the above results from text-to-image retrieval task and the image captioning task, we can conclude that our proposed GEM-I dataset can demonstrate a model's image understanding and generation ability in 20 different languages.
\subsection{Video-Language Evaluation on GEM-V}
\begin{table*}[tp]
\centering
\scalebox{0.63}{
\begin{tabular}{l|ccccccccccccccc|c}
\toprule
Setting & en & id & pt & vi & ro & ko & ja & fr & ar & de & tl & sv & fa & he & it &- \\
\midrule
\multicolumn{17}{c}{\textit{Fine-tune on ALL}} \\
m-UniVL(loose) (Q$\rightarrow$V) & 23.27 & 17.67 & 23.50 & 12.83 & 19.90 & 12.83 & 12.37 & 24.77 & 9.03 & 22.27 & 11.23 & 16.07 & 9.87 & 9.50 & 23.60 &-\\
m-UniVL(loose) (Q$\rightarrow$V+T) & 71.83 & 57.03 & 62.90 & 38.97 & 53.57 & 42.09 & 38.87 & 62.83 & 36.50 & 56.67 & 43.37 & 40.20 & 39.67 & 35.20 & 56.80 &-\\
m-UniVL(tight) (Q$\rightarrow$V+T) & 83.87 & 69.97 & 61.20 & 52.07 & 59.08 & 63.17 & 66.10 & 74.47 & 46.92 & 70.27 & 59.23 & 51.60 & 58.10 & 56.60 & 58.90 &-\\
\multicolumn{17}{c}{\textit{Fine-tune on EACH}} \\
m-UniVL(loose) (Q$\rightarrow$V+T) & 28.90 & 19.97 & 21.23 & 12.33 & 14.93 & 16.87 & 13.30 & 20.73 & 10.57 & 21.70 & 14.60 & 12.23 & 12.03 & 10.27 & 18.70 &-\\
\midrule \midrule
& tr & ru & nl & pl & zh-t & es & ms & no & ca & hr & ka & zh-s & hu & sq & sr-l &AVG\\
\midrule
\multicolumn{17}{c}{\textit{Fine-tune on ALL}} \\
m-UniVL(loose) (Q$\rightarrow$V) & 18.70 & 18.10 & 20.00 & 22.50 & 16.00 & 25.13 & 8.10 & 16.70 & 9.05 & 13.38 & 7.67 & 9.64 & 11.56 & 6.80 & 11.03 &15.44\\
m-UniVL(loose) (Q$\rightarrow$V+T) & 51.43 & 57.73 & 49.73 & 49.80 & 45.70 & 66.03 & 43.97 & 40.60 & 41.40 & 38.37 & 28.00 & 38.38 & 40.69 & 33.00 & 35.70 &46.57\\
m-UniVL(tight) (Q$\rightarrow$V+T) & 63.13 & 70.57 & 59.07 & 50.67 & 69.10 & 64.97 & 64.60 & 54.40 & 59.37 & 31.50 & 39.05 & 64.31 & 58.73 & 50.97 & 32.86 &58.83 \\
\multicolumn{17}{c}{\textit{Fine-tune on EACH}} \\
m-UniVL(loose) (Q$\rightarrow$V+T) & 21.00 & 17.70 & 21.70 & 19.50 & 17.40 & 19.83 & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} &17.40\\
\bottomrule
\end{tabular}
}
\caption{Evaluation results of m-UniVL on GEM-V for text-to-video retrieval, where Mean-Recall is used as the metric. \textbf{Q$\rightarrow$V} denotes the setting where only video (V) is used to compute its similarity with query (Q). \textbf{Q$\rightarrow$V+T} denotes the setting where both video (V) and title (T) are used to compute their similarity with query (Q). The average score is computed over all 30 languages.
}
\label{tab:result_of_video_retrieval}
\end{table*}
\subsubsection{Experimental Settings}
We select the open-source version\footnote{https://github.com/microsoft/UniVL} of UniVL \cite{luo2020univl} and replace the original text encoder with XLM-R \cite{conneau-etal-2020-unsupervised}, to support the multilingual video-language evaluation on GEM-V. The original UniVL is pre-trained on 1.2 million instructional videos with ASR transcripts in HowTo100M \cite{DBLP:journals/corr/abs-1906-03327}.
\begin{table*}[tbp]
\centering
\scalebox{0.54}{
\begin{tabular}{l|l|ccccccccccccccc|c}
\toprule
Setting & Metric & en & id & pt & vi & ro & ko & ja & fr & ar & de & tl & sv & fa & he & it &- \\
\midrule
\multicolumn{17}{c}{\textit{Fine-tune on ALL}} \\
m-UniVL (V$\rightarrow$Q) & ROUGE-L & 9.43 & 10.95 & 14.63 & 6.64 & 9.96 & 3.41 & 3.10 & 16.87 & 3.80 & 8.18 & 9.72 & 8.55 & 8.25 & 2.28 & 10.65 &-\\
m-UniVL (T$\rightarrow$Q) & ROUGE-L & 46.06 & 35.89 & 36.68 & 27.36 & 35.08 & 21.01 & 17.14 & 41.64 & 22.40 & 30.30 &43.36 & 26.10 & 30.58 & 33.07 & 34.43 &-\\
m-UniVL (V+T$\rightarrow$Q) & ROUGE-L & 46.01 & 36.73 & 37.59 & 26.75 & 36.41 & 20.71 & 17.12 & 41.90 & 23.45 & 30.24 & 44.40 & 26.38 & 30.21 & 32.82 & 34.98 &- \\
m-UniVL (V$\rightarrow$Q) & METEOR & 3.93 & 4.51 & 6.09 & 2.39 & 4.09 & 2.99 & 3.38 & 8.06 & 15.16 & 3.71 & 4.31 & 3.62 & 17.06 & 12.04 & 4.65 &-\\
m-UniVL (T$\rightarrow$Q) & METEOR & 23.99 & 17.01 & 17.40 & 13.13 & 17.10 & 20.77 & 18.33 & 21.25 & 26.47 & 14.40 & 22.67 & 12.23 & 26.25 & 29.33 & 16.60 &-\\
m-UniVL (V+T$\rightarrow$Q) & METEOR & 23.98 & 17.44 & 18.01 & 12.52 & 18.09 & 20.34 & 18.24 & 21.39 & 26.76 & 14.52 & 23.00 & 12.26 & 26.17 & 28.91 & 16.94 &- \\
m-UniVL (V$\rightarrow$Q) & CIDEr & 19.16 & 18.35 & 23.58 & 13.53 & 18.56 & 7.12 & 4.20 & 43.82 & 5.72 & 18.23 & 23.40 & 8.99 & 14.88 & 4.75 & 17.53 &-\\
m-UniVL (T$\rightarrow$Q) & CIDEr & 256.65 & 155.05 & 164.26 & 116.46 & 164.89 & 74.95 & 57.15 & 223.09 & 86.57 & 138.50 & 220.58 & 101.64 & 119.49 & 139.39 &170.02 &-\\
m-UniVL (V+T$\rightarrow$Q) & CIDEr & 255.04 & 157.64 & 167.79 & 108.66 & 174.59 & 74.01 & 58.73 & 223.67 & 90.70 & 138.66& 223.21 & 100.61 & 115.37 & 136.24 &174.44 &- \\
\multicolumn{17}{c}{\textit{Fine-tune on EACH}} \\
m-UniVL (V+T$\rightarrow$Q) & ROUGE-L & 21.30 & 12.25 & 18.36 & 6.45 & 9.09 & 7.59 & 3.77 & 15.20 & 4.92 & 8.94 & 14.92 & 8.11 & 7.31 & 1.76 & 10.05 &-\\
m-UniVL (V+T$\rightarrow$Q) & METEOR & 9.50 & 4.92 & 7.99 & 2.44 & 3.67 & 6.79 & 3.48 & 7.01 & 17.21 & 3.92 & 7.12 & 3.47 & 16.66 & 12.75 & 4.22 &-\\
m-UniVL (V+T$\rightarrow$Q) & CIDEr &65.52 & 24.26 & 44.69 & 15.47 & 21.45 & 23.40 & 9.50 & 43.71 & 8.35 & 21.32 & 41.04 & 9.89 & 13.65 & 3.71 & 18.12 &-\\
\midrule \midrule
& tr & ru & nl & pl & zh-t & es & ms & no & ca & hr & ka & zh-s & hu & sq & sr-l &AVG\\
\midrule
m-UniVL (V$\rightarrow$Q) & ROUGE-L & 11.53 & 14.44& 9.83 & 14.19 & 3.95 & 18.07 & 0.15 & 0.91 & 0.27 & 2.49 & 0.00 & 0.11 & 2.68 & 1.29 & 0.33 &6.89\\
m-UniVL (T$\rightarrow$Q) & ROUGE-L & 32.19 & 35.41 & 27.77 & 29.99 & 15.72 & 41.35 & 34.44 & 21.35 & 38.29 & 24.94 & 1.70 & 4.63 & 30.13 & 27.22 & 28.28 &29.15\\
m-UniVL (V+T$\rightarrow$Q) & ROUGE-L & 34.43 & 35.84 & 28.31 & 29.82 & 15.50 & 42.29 & 36.36 & 21.59 & 38.09 & 26.09 & 2.92 & 4.55 & 30.58 & 28.37 & 29.51 &29.67 \\
m-UniVL (V$\rightarrow$Q) & METEOR & 5.41 & 6.11 & 4.47 & 6.06 & 2.65 & 8.21 & 0.07 & 0.39 & 0.13 & 1.05 & 0.00 & 0.61 & 1.18 & 0.67 & 0.13 & 4.44 \\
m-UniVL (T$\rightarrow$Q) & METEOR &15.53 & 16.79 & 13.21 & 13.91 & 16.03 & 20.50 & 16.88 & 9.77 & 19.21 & 12.00 & 1.74& 12.24 & 13.66 & 13.17 & 13.53 & 16.84\\
m-UniVL (V+T$\rightarrow$Q) & METEOR &16.54 & 17.12 & 13.62 & 13.86 & 15.48 & 20.83 & 17.92 & 9.83 & 18.95 & 12.47 & 2.76 & 11.59 & 13.78 & 13.72 & 13.81 & 17.03\\
m-UniVL (V$\rightarrow$Q) & CIDEr & 28.84 & 24.05 & 17.35 & 17.37 & 8.00 & 32.89 & 0.15 & 1.65 & 0.45 & 2.20 & 0.00 & 0.29 & 2.71 & 1.70 & 0.73 & 12.67\\
m-UniVL (T$\rightarrow$Q) & CIDEr & 149.09 & 161.47 & 123.79 & 117.37 & 47.94 & 203.88 & 156.38 & 84.81 & 192.65 & 102.38 & 3.95 & 13.58 & 129.47 & 109.78 & 119.64 &130.16\\
m-UniVL (V+T$\rightarrow$Q) & CIDEr & 156.35 & 166.33 & 124.20 & 116.29 & 47.13 & 204.85 & 166.13 & 82.70 & 189.35 &106.02 & 8.01 & 12.85 & 128.94 & 114.98 & 118.01 & 131.38\\
\multicolumn{17}{c}{\textit{Fine-tune on EACH}} \\
m-UniVL (V+T$\rightarrow$Q) & ROUGE-L & 11.46 & 17.52 & 9.16 & 13.31 & 3.15 & 24.21 & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} &10.90\\
m-UniVL (V+T$\rightarrow$Q) & METEOR &5.20 & 7.44 & 3.93 & 5.65 & 2.08 & 11.19 & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} &6.98\\
m-UniVL (V+T$\rightarrow$Q) & CIDEr &29.62 & 36.11 & 16.48 & 21.15 & 6.56 & 62.26& \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} & \textit{NA} &25.54\\
\bottomrule
\end{tabular}
}
\caption{Evaluation results of m-UniVL on GEM-V for video captioning, where ROUGE-L, METEOR and CIDEr are taken as metrics. \textbf{V$\rightarrow$Q} and \textbf{T$\rightarrow$Q} denote the video caption is generated based on video (V) and title (T), respectively. \textbf{V+T$\rightarrow$Q} denotes the video caption is generated based on both video (V) and title (T). The average score is computed over all 30 languages.
}
\label{tab:caption_result}
\end{table*}
For text-to-video retrieval task, m-UniVL extracts video features using the off-the-shelf pre-trained S3D \cite{DBLP:journals/corr/abs-1912-06430} model. The FPS of the 3D feature extractor is 16 and the dimension is 1,024. The hyper-parameters of the video encoder are set as follows: 768 hidden units, 12 heads, 6 layers of of Transformer blocks to capture the sequential information on the 3D features. The hyper-parameters of the text encoder are identical to the ones in XLM-R: 768 hidden units, 12 heads, 12 layers of Transformer blocks. The cross encoder on the top of the text and video encoders has 2 layers with 768 hidden units and 12 heads. For video captioning, the decoder is with 3 layers, 768 hidden units and 12 heads.
We finetune m-UniVL on text-to-video retrieval and video captioning tasks. For retrieval, a learning rate of 1e-4 and a batch size of 128 are used to fine-tune m-UniVL for 50 epochs. For captioning, a learning rate of 3e-5 and a batch size of 16 are used to fine-tune m-UniVL for 5 epochs. All above calculations are carried on 4 NVIDIA Tesla V100 GPUs.
\subsubsection{Text-to-Video Retrieval Results}
Following official UniVL on retrieval task, we evaluate the text-to-video retrieval task on our GEM-V using two variants. One is m-UniVL (loose), which encodes the input text query and candidate video clips (and optional title) through the text encoder and video encoder respectively and finally calculates the matching score through dot product. The other is m-UniVL (tight), based on m-UniVL (loose), m-UniVL (tight) further concatenates the encoded features and feeds them to the cross encoder to get unified representation and predict the matching score on the first token `$\langle$s$\rangle$'. The evaluation metric is mean-Recall (arithmetic mean of Recall@$K$ for $K\in \{1,5,10\}$).
Tables \ref{tab:result_of_video_retrieval} lists the retrieval results. The results can be divided into two groups. One is from the fine-tuning on all training set across linguistic type (\textit{Fine-tune on ALL}), and the other is from the fine-tuning on individual training set of each language (\textit{Fine-tune on Each}). The target of such a division is to explore whether one language can benefit from other wide languages. Besides, there are 9 languages without training set. We keep such a zero-shot evaluation to explore the transfer ability of the proposed model.
We can get three conclusions from the results:
1) The m-UniVL (tight) outperforms m-UniVL (loose) at the same retrieval settings. It proves that the cross encoder of UniVL enables the multi-modality features to fully interact with each other to capture better alignment.
2) The title of the video introduces a large performance gain and is a good semantic feature of the video. This metadata is especially useful for zero shot setting with a significant improvement. They demonstrate that improving the retrieval performance on pure videos without titles is still a challenge. Our proposed GEM develops a chance to push such a multimodal challenge.
3) Fine-tune on all can achieve better results than fine-tune on each. The reason is the former can effectively leverage the data from all languages and benefit the task rather than the latter. Besides, for zero shot languages, fine-tune on all is also very effective. It demonstrates that our proposed GEM can also be used on multilingual research besides multimodal research.
\noindent
\subsubsection{Video Captioning Results}
The captioning task aims to generate a caption given a video clip (and optional title) in our setting. Such a generation task is from our practical application. We adopt whole m-UniVL including encoders and decoder to finish the task. The evaluation metric are ROUGE-L \cite{lin-och-2004-automatic}, METEOR \cite{banerjee2005meteor} and CIDEr \cite{vedantam2015cider}, whose values are obtained from an open-source tool\footnote{https://github.com/Maluuba/nlg-eval}.
Table \ref{tab:caption_result} lists the experimental results. Similar conclusions can be drawn as the retrieval task, and there are two more observations:
1) For captioning task, the performance of the generation on pure videos is low. The reason is that the search queries sometimes are the keywords instead of a whole sentence, thus the task of V$\rightarrow$Q is relatively hard.
2) Title is especially important due to the characteristic of this data collection process.
From the above results from the text-to-video retrieval task and the video captioning task, we can conclude that our proposed GEM-V can improve video understanding and generation under the multilingual and multimodal perspective.
\section{Conclusion}
This paper presents GEM as a benchmark for evaluating the generalization capabilities of vision-language models on image-language and video-language tasks. GEM is also the first large-scale multilingual multimodal dataset, where the natural language contexts are collected from a commercial search engine in 20 and 30 languages for image-related and video-related tasks, respectively. We describe two vision-language pre-trained models for GEM and hope these efforts can advance the development of multilingual multimodal research.
\section{Ethical Considerations}
We have reviewed our data release process and it has been approved by our institutional review board. Specifically, (a) In GEM-I, all of the images are with proper Creative Commons Licenses, so that they are safe to be distributed without violating any policies or intellectual rights. Also, we discarded images with human faces to avoid revealing privacy. (b) In GEM-V, all of the videos were originated from Youtube, and we will only provide Youtube URLs to the researchers. We have confirmed with our institutional review board that distributing URLs does not violate any policies or intellectual rights. We didn't do anything specific for human faces in the videos, since we are only distributing video URLs, and modifying the original videos (such as blurring the faces) might violate the copyright of the videos. When releasing GEM to the public, we will indicate the data source, emphasize that the dataset is for research purposes only, and provide an email address for people to contact us to delete any data if any infringement. During data collection, we didn't collect, store, or distribute any private information of the users.
To measure the quality of our dataset, we employed crowd-sourcing judges in the United States and provided labeling guidelines for them. The compensation given to the workers is 15 USD per hour for GEM-I and 25 USD per hour for GEM-V. The level of compensation is determined by: (a) Market price according to similar labeling tasks in the US. (b) The difficulty and labeling speed of this task. This task involves labeling if a query is related to an image or video, so it is considered as a relatively easy task. The labeling speed is about 300 query-image pairs per hour and 180 query-video pairs per hour.
\bibliographystyle{acl_natbib}
|
2,877,628,090,066 | arxiv | \section{Introduction}
The ongoing pandemics of COVID-19, has claimed millions of human lives, caused stagnation of the global economy and excessive load on the healthcare systems throughout the world and changed the normal life. Mathematical models of epidemic spreading are important tools for predicting the effects that the pandemics can have on each segment of the society. They provide support for policy-makers to make adequate decisions in order to partially mitigate the consequences by planning various social distancing measures, preparation of healthcare facilities and appropriate adaptation of the economy.
The spectrum of mathematical models applied for the COVID-19 pandemic ranges from the simplest SIR to rather complex SIDARTHE \cite{roda2020difficult, zhao2020modeling, calafiore2020time, giordano2020modelling, gatto2020spread}, which are used for assessment of different aspects of the epidemics. One of the major features of these models is their Markovian nature, which considers transitions from one state to another to be independent on the past. As an example, when Markovian property is assumed to hold, an individual that has just become infected can proceed to recovered state with the same probability as another one which has been infected for longer period. This Markovian assumption, encapsulated in constant transition probabilities, or rates, makes the models easier to study analytically. The outcomes of these studies with Markovian approach offer some, and in certain instances satisfactory, assessment of the spreading dynamics. However, growing body of evidence, particularly for the COVID-19, suggests existence of incubation period and certain infectivity patterns, with possibility for spreading the pathogen before onset of the symptoms, to which correspond functions that are rather distinct from the exponential distribution which the Markovian models rely on \cite{Qin2020, qin2020estimation}. Although adding one or more compartments for the Exposed, Asymptomatic, Presymptomatic, or Quarantined persons or considering various kinds of delay \cite{liu2020covid, dell2020solvable, rong2020effect}
address such observations to certain extent, they cannot systematically incorporate the observed distributions of the incubation period and the healing process.
There are different approaches of non-Markovian modeling of epidemic processes. In one attempt \cite{boguna1} is proposed Gillespie algorithm as an adequate tool for numerical analysis of non-Markovian spreading models. The effects of the form of distribution of infection and curing (recovery) times on SIS epidemic model occurring on complex networks in continuous time has been analyzed in several studies \cite{starnini,delft_nm1, delft_nm2, delft_nm3, Feng2019, krylova2013effects}. With the introduction of SI*V* model \cite{Nowzari} it was suggested that non-Markovian spreading models have capacity to be extended to cover a wide variety of spreading sub-models and variants. Nontrivial distribution of infectious period in an integro-differential SIR model was considered in \cite{riano2020epidemic}. In a recent study, non-Markovian SIS model on complex networks, with arbitrary function for infectivity and recovery was proposed \cite{tomovski2021epidemic}, in which control theory was successfully applied for determination of epidemic threshold. Our approach adds to these pioneering contributions by providing general framework for incorporation of various
distributions of infectivity and healing in a SIR model. By similar approach as in \cite{tomovski2021epidemic} we show how these functions determine the epidemic threshold. The relevance of the model, besides by numerical simulations, is verified by fitting to the observations of the first wave of the epidemic in Italy, in the spring, 2020.
The paper is organized as follows. After providing initial setting of the model in Section \ref{sec:preliminaries}, we introduce the discrete-time and continuous-time models in Sections \ref{sec:discrete} and \ref{sec:cont}, respectively, where we also derive the epidemic threshold relationships. The reduction to Markovian case of the model is presented in Section \ref{sec:Markov}, while numerical simulations and discussions are given in Section \ref{sec:numerics}. The paper concludes with Section \ref{sec:conclusions}.
\section{Preliminaries} \label{sec:preliminaries}
We consider SIR model that has three compartments: Susceptible - S, Infected - I and Recovered - R, with the usual transition $S \to I \to R$. Let the corresponding variables $S$, $I$ and $R$ denote the fractions of the population that are in the given state, and under assumption without births and deaths, one has the normalization condition $S(t)+I(t)+R(t) = 1$ at each moment $t$. To capture the nontrivial dependence of the healing period and the different contagiousness of the infected individual in different stages of the disease we introduce two functions. The infectivity function $\beta(\tau)$ captures the rate, or probability at which individuals that became infected before time $\tau$ are spreading the disease to the susceptible ones. Thus, by simply taking $\beta(\tau) = 0$ for $\tau < T_0$, one is able to introduce incubation period with length $T_0$. Another important function is the healing function $\gamma(\tau)$ that denotes the probability with which individual can heal at moment $\tau$ after contracting the disease. To account for asymptomatic transmitters and existence of certain time window when presence of pathogen can be confirmed, one can introduce a reporting function $\rho(\tau)$. It is associated to the probability that the presence of the pathogen can be confirmed at moment $\tau$ after contraction with it. The asymptomatic cases are conveniently handled by normalizing the reporting function to value smaller than unity. We pursue by considering discrete- and continuous-time models separately, and provide more details about these functions.
\section{Discrete-time version}
\label{sec:discrete}
In this section we consider evolution in discrete time $t$ and denote the fraction of individuals that have become infected within the continuous-time interval $[t-1, t]$ with $I_d(t)$, where for simplicity the unit interval is taken to be 1. This can be relevant for situations like those when cases are considered on daily basis. In such scenario, we have discrete-time healing function $\gamma(\tau)$ and infectivity one $\beta(\tau)$, on which we put the constraint $\beta(0) = 0$. The probability that the individual will heal within $\tau$ time units is $\Gamma(\tau) = \sum_{\nu=0}^{\tau} \gamma(\tau)$. We further assume finite duration $T$ of the disease, what implies $\Gamma(T) = 1$ and for practical reasons introduce its complement $\overline{\Gamma}(\tau) = 1 - \Gamma(\tau)$, to denote the probability that individual has not healed yet for $\tau$ time units. The function $\gamma(\tau)$ also has a meaning of fraction of individuals that have contracted the disease within the same unit time interval, to become healed later within another unit interval $[\tau-1, \tau]$. Similar reasoning holds for the cumulative functions $\Gamma(\tau)$ and $\overline{\Gamma}(\tau)$. On base on the classical SIR model, the proposed model of evolution of the compartments is given with the system
\begin{eqnarray}
S(t + 1) &=& S(t) \left[1 - \sum_{\tau=0}^{T-1} \beta(\tau) \overline{\Gamma}(\tau) I_d(t-\tau)\right] \nonumber \\
I_d(t + 1) &=& S(t) \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) I_d(t-\tau) \nonumber \\
R(t + 1) &=& R(t) +\sum_{\tau=0}^{T-1} \gamma(\tau) I_d(t + 1 -\tau). \label{eq:discrete_model}
\end{eqnarray}
One can note that the infected individuals that have contracted the pathogen up to $T$ periods before the current moment $t$, and which are not healed yet, can contribute to spreading of the disease, with appropriate intensity captured in the function $\beta(\tau)$. We note that in order to determine the infected fraction at given moment, one should sum those infected in the past, but did not heal up to the given moment
\begin{equation}
I(t) = \sum_{\tau=0}^{T-1} I_d(t-\tau) \overline{\Gamma}(\tau).
\label{eq:discrete_total_inf}
\end{equation}
To make the problem completely defined one has to specify the initial conditions for $I_d(t)$. We assume that they are given for $\tau = T-1, T-2, \dots, 0$. In general this model cannot be solved analytically and should be studied by application of numerical simulations.
To get insight of the conditions when epidemic can emerge, one can determine the stability of the disease free state $S^*=1, I^*=I_d^* = R^*=0 $, that is an equilibrium point of the system. Its local stability is established by linearizing the dynamical equations (\ref{eq:discrete_model}) in its neighborhood. By making the linearization in vicinity of $S^*=1, I^*=R^*=0$, one can observe the dynamical evolution of the perturbations $\delta S = S - S^*, \delta I_d =I_d - I_d^*, \delta R = R- R^*$. Under linearization, the perturbations are related with
\begin{eqnarray}
\delta S(t+1) &=& \delta S(t) -\sum_{\tau=0}^{T-1} \beta(\tau) \overline{\Gamma}(\tau) \delta I_d(t-\tau), \nonumber \\
\delta I_d(t+1) &=& \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) \delta I_d(t-\tau), \nonumber\\
\delta R(t + 1) &=& \delta R(t) + \sum_{\tau=0}^{T-1} \gamma(\tau) \delta I_d(t + 1-\tau). \label{eq:discrete_model_pert}
\end{eqnarray}
Let us focus on the infected fraction and make $Z$-transform on the second equation in (\ref{eq:discrete_model_pert}). To do so, multiply first both sides of that equation by $z^{-t}$ and sum to obtain
\begin{equation}
\sum_{t=0}^{\infty} \delta I_d(t+1) z^{-t} = \sum_{t=0}^{\infty} \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) \delta I_d(t-\tau) z^{-t}. \label{eq:Z_transform_disc}
\end{equation}
By using the $Z$-transform of the fraction of the population that become infected at unit interval $I_d(t)$, given as $\mathcal{I}(z) = \sum_{t=0}^{\infty} I_d(t) z^{-t}$, the left hand side of (\ref{eq:Z_transform_disc}) will become
\begin{equation}
\sum_{t=0}^{\infty} \delta I_d(t+1) z^{-t} = z\sum_{t=0}^{\infty} \delta I_d(t+1) z^{-(t+1)} = z\left[\mathcal{I}(z)- \delta I_d(0)\right]. \label{eq:delta_ID_t_left}
\end{equation}
Accordingly, the right-hand side of (\ref{eq:Z_transform_disc}) can be rearranged as
\begin{equation}
\sum_{t=0}^{\infty} \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) \delta I_d(t-\tau) z^{-t}= \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) z^{-\tau} \sum_{t=0}^{\infty} \delta I_d(t-\tau) z^{-(t-\tau)}.
\end{equation}
By using substitution $\nu = t-\tau$, the last sum for $\tau \leq -1$ can be expressed as
\begin{equation}
\sum_{\nu=-\tau}^{\infty} \delta I_d(\nu) z^{-\nu} =\sum_{\nu=-\tau}^{-1} \delta I_d(\nu) z^{-\nu} + \mathcal{I}(z) =
\mathcal{I}_{0}(\tau, z) + \mathcal{I}(z), \label{eq:delta_ID_t_right}
\end{equation}
where we have introduced a function $\mathcal{I}_0(\tau, z)$ that corresponds to the initial conditions. Now, combining the relationships (\ref{eq:delta_ID_t_left}) -- (\ref{eq:delta_ID_t_right}) one has
\begin{equation}
z\left[\mathcal{I}(z)- \delta I_d(0)\right]= \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) \left[\mathcal{I}_0(\tau, z) + \mathcal{I}(z)\right] z^{-\tau} .
\end{equation}
To shorten the notation, one can introduce the following two complex functions
\begin{eqnarray}
\mathcal{E}(z) &=& \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) z^{-\tau}, \nonumber\\
\mathcal{E}_0(z) &=& \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) \mathcal{I}_0(\tau, z) z^{-\tau}.
\end{eqnarray}
The first one is simply the $Z$-transform $\mathcal{E}(z)$ of what might be called epidemic function $E(\tau) = \beta(\tau)\overline{\Gamma}(\tau)$, that is a combination of the infecting and healing functions because $\sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) z^{-\tau}=\sum_{\tau=0}^{\infty} \beta(\tau)\overline{\Gamma}(\tau) z^{-\tau}$. The second complex function $\mathcal{E}_0(z)$ is related to the initial conditions. Now, one has the following relationship
\begin{equation}
z\left[\mathcal{I}(z)- \delta I_d(0)\right] = \mathcal{I}(z)\mathcal{E}(z) + \mathcal{E}_0(z),
\end{equation}
from where
\begin{equation}
\mathcal{I}(z) = \frac{z\delta I_d(0)+\mathcal{E}_0(z)}{z-\mathcal{E}(z)}.\label{eq:infected_Z_transform}
\end{equation}
From a result in theory of discrete linear time-invariant systems, a sequence (the impulse response of such system) is decaying if the poles of its $Z$-transform are within the unit circle \cite{oppenheim2013signals}. Thus, when the poles of the function $\mathcal{I}(z)$ of the complex function (\ref{eq:infected_Z_transform}), or the roots of the polynomial $z-\mathcal{E}(z)$ lie within the unit circle, the perturbation dies out at infinity. So, the epidemic threshold can be obtained by taking $z=1$ in the denominator in (\ref{eq:infected_Z_transform}), that results in
\begin{equation}
\sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) = 1, \label{eq:threshold_discrete}
\end{equation}
which obviously depends on the functional forms of the infectivity and healing functions.
We should finally note that any initial infection would not shift back the population to the disease-free state $S=1, I=R=0$, but to some endemic $S_e^*, I_e^* = 0, R_e^* = 1-S^*$. However, if the conditions are not favoring epidemic both equilibria will be rather close $S_e^* \approx 1$.
\section{Continuous-time version}\label{sec:cont}
We will pursue similarly to the discrete-time approach, where the fractions of individuals within given compartment and the functions modeling the infectivity, healing and reporting are defined for continuous time $t$ and we use the same notation. Thus, $S(t)$ is the fraction of susceptible individuals at given moment $t$ and $R(t)$ corresponds to the recovered and again assume finite healing period $T$. The fraction of infected individuals is conveniently modeled with the rate of infection, or the fraction of newly infected individuals $I_d(t)$ within the infinitesimal interval $(t-dt, t)$. The total fraction of infected persons is given with the integral
\begin{equation}
I(t) = \int_{0}^{T} I_d(t-\tau) \overline{\Gamma}(\tau) d\tau, \label{eq:total_I}
\end{equation}
which accounts for those that had become infected in the past and have not healed yet. Now, the dynamical evolution of the respective fractions is given with
\begin{eqnarray}
\dot{S} &=& - S(t)\int_{0}^{T} \beta(\tau) \overline{\Gamma}(\tau) I_d(t-\tau)d\tau \nonumber \\
I_d(t) &=& S(t) \int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) I_d(t-\tau) d\tau \nonumber \\
\dot{R} &=& \int_{0}^{T} \gamma(\tau) I_d(t-\tau) d\tau. \label{eq:continuous_model}
\end{eqnarray}
In order to determine whether the initial perturbation will grow to epidemics, one could focus on the second equation in the vicinity of the disease-free state $S^*=1, R^*=I^*=0$. Then, the perturbation of newly infected individuals will evolve as
\begin{equation}
\delta I_d(t) = \int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) \delta I_d(t-\tau)d\tau, \label{eq:pert_infect_rate}
\end{equation}
where it is assumed that in vicinity of the disease-free state $S(t)\approx 1$. Now, make Laplace transform of the perturbation of the rate of infection, $\mathcal{I}(s) = \int_0^{\infty}\delta I_d(t)e^{-st}dt$ and use it in the last equation (\ref{eq:pert_infect_rate}). To do that, we will follow the same approach as in the discrete-time version. Multiply both sides with $e^{-st}$ and integrate. The left hand side will result in the Laplace transform of $\delta I_d(t)$, while the right hand one will be
\begin{eqnarray}
A &= &\int_0^{\infty}\int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) \delta I_d(t-\tau)e^{-st}d\tau dt = \int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) e^{-s\tau} \int_0^{\infty} \delta I_d(t-\tau) e^{-s(t-\tau)} dt \nonumber \\
&=&\int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) e^{-s\tau} \int_{-\tau}^{\infty}\delta I_d(\nu) e^{-s\nu} d\nu
\end{eqnarray}
The last integral can be expressed with
\begin{equation}
\int_{-\tau}^{\infty}I_d(\nu) e^{-s\nu} d\nu = \int_{-\tau}^{0}I_d(\nu) e^{-s\nu} d\nu + \mathcal{I}(s) = \mathcal{I}_0(\tau,s) + \mathcal{I}(s).
\end{equation}
Now, one has
\begin{equation}
A=\int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) e^{-s\tau} \left[\mathcal{I}_0(\tau,s) + \mathcal{I}(s) \right] d\tau.
\end{equation}
Similarly to the discrete-time case we can introduce the Laplace transform of the epidemic function $E(\tau)=\beta(\tau)\overline{\Gamma}(\tau)$ and its initial conditions contribution
\begin{eqnarray}
\mathcal{E}(s) &=& \int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) e^{-s\tau} d\tau, \\ \nonumber
\mathcal{E}_0(s) &=& \int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) \mathcal{I}_0(\tau,s) e^{-s\tau} d\tau.
\end{eqnarray}
Finally, one obtains
\begin{equation}
\mathcal{I}(s) = \mathcal{I}(s) \mathcal{E}(s) + \mathcal{E}_0(s),
\end{equation}
from where the Laplace transform of the perturbation of the infection rate is
\begin{equation}
\mathcal{I}(s) = \frac{\mathcal{E}_0(s)}{1-\mathcal{E}(s)}. \label{eq:Laplace_transf_pert}
\end{equation}
From the results of control theory, a continuous-time linear time-invariant system is stable if the poles of its transfer function, or Laplace transform of its impulse response have negative real part \cite{oppenheim2013signals}. Thus, the perturbations $\delta I_d(t)$ will decay if the poles of its Laplace transform $\mathcal{I}(s)$ (\ref{eq:Laplace_transf_pert}), or eigenvalues of the system (\ref{eq:continuous_model}) lie within negative half-plane $Re\{s\} < 0$. Then, the epidemic threshold can be obtained with $s=0$ which leads to
\begin{equation}
\int_0^T \beta(\tau)\overline{\Gamma}(\tau) d\tau = 1, \label{eq:Threshold_continuous}
\end{equation}
that represents the relationship, which corresponds to the discrete-time case (\ref{eq:threshold_discrete}).
\section{Markovian SIR model}\label{sec:Markov}
In order to obtain the classical Markovian SIR model, from the non-Markovian case (\ref{eq:discrete_model}), one should consider taking $T \rightarrow \infty$, $\beta(\tau)=\beta$, $\gamma(0)=0$, $\gamma(\tau)=\gamma(1-\gamma)^{\tau-1}$ where $\beta$ and $\gamma$ are constants. This further yields $\Gamma(0)=0$, $\Gamma(\tau)=1-(1-\gamma)^{\tau}$ and $\overline{\Gamma}(\tau)=(1-\gamma)^{\tau}=\gamma(\tau+1)/\gamma$. First, one could observe that by using constant infectivity $\beta(\tau)=\beta$ in the first relationship of the model (\ref{eq:discrete_model}) and using (\ref{eq:discrete_total_inf}) one will obtain the classical form for evolution of the susceptible population
\begin{equation}
S(t + 1) = S(t) \left[1 - \beta I(t)\right]. \label{eq:SIR_S_classical}
\end{equation}
Next, by implementing the condition $\gamma(0) = 0$, and the relationship $\overline{\Gamma}(\tau)=\gamma(\tau+1)/\gamma$ one can drop the first term in the sum in the recovered population in (\ref{eq:discrete_model}), and further obtain
\begin{equation}
\sum_{\tau=0}^{T-2} \gamma(\tau+1) I_d(t-\tau) = \gamma\sum_{\tau=0}^{T-1} \overline{\Gamma}(\tau) I_d(t-\tau)- \gamma \overline{\Gamma}(T-1) I_d(t-T+1) = \gamma I(t)- \gamma (1-\gamma)^{T-1}I_d(t-T+1),
\end{equation}
from where, for $T \rightarrow \infty$, the recovered population evolves as
\begin{equation}
R(t+1) = R(t) + \gamma I(t). \label{eq:SIR_R_classical}
\end{equation}
Finally, from the conservation relationship $I(t) + S(t) + R(t) = 1$, one can find that the infected fraction is given as
\begin{equation}
I(t+1) = \beta S(t)I(t)+(1-\gamma) I(t).
\label{eq:SIR_I_classical}
\end{equation}
The relationships (\ref{eq:SIR_S_classical}), (\ref{eq:SIR_R_classical}) and (\ref{eq:SIR_I_classical}) represent the classical SIR model in discrete time.
As an example, in the figure \ref{Mar_vs_nonMar} we make a comparison between numerical solutions of the discrete classical SIR model and the classical SIR - equivalent model obtained from the non-Markovian form.
\begin{figure}[h]
\includegraphics[width=0.6\columnwidth]{SIR_comparison_T=150_b=0k2_g=0k03_tnr}
\centering
\caption{Comparison between the discrete classical SIR model and the classical SIR - equivalent model obtained from the non-Markovian form, for $\beta=0.2$, $\gamma=0.03$. It is used rather large finite duration of the healing $T=150$, as a proxy for $T\to\infty$. }
\label{Mar_vs_nonMar}
\end{figure}
Similarly to the discrete-time version, to verify that the proposed continuous model is generalization of the classical, Markovian SIR model, one should consider two characteristics of the latter: 1. The infection rate is independent on the moment when the disease was contracted $\beta(\tau) = \beta$; and 2. The duration of infectivity is infinite and exponentially distributed which implies that the healing function is $\gamma(\tau) = \lambda e^{-\lambda \tau}$. We note that the respective cumulative distribution is $\Gamma(\tau) = 1 - e^{-\lambda\tau}$, and accordingly $\overline{\Gamma}(\tau) = e^{-\lambda\tau}$. By using the functional form of the healing function, the total infectious population will be
\begin{equation}
I(t) = \int_0^{\infty} \overline{\Gamma}(\tau)I_d(t-\tau)d\tau = \int_0^{\infty} e^{-\lambda \tau}I_d(t-\tau)d\tau.
\label{eq:I_total_classical}
\end{equation}
Similarly, by using $\beta(\tau) = \beta$, for the dynamics of the susceptible fraction one has
\begin{equation}
\dot{S} = -S(t)\beta \int_0^{\infty} e^{-\lambda\tau} I_d(t-\tau) d\tau = -\beta S I,
\label{eq:S_classical}
\end{equation}
that represents the corresponding relationship in the classical SIR model. Furthermore, by applying the functional form for the healing function, the dynamics of the recovered population will be as follows
\begin{eqnarray}
\dot{R} = \int_0^{\infty} \lambda e^{-\lambda \tau} I_d(t-\tau) d\tau = \lambda I,
\label{eq:R_classical}
\end{eqnarray}
that is the respective relationship in the classical SIR model. Finally, by using the conservation principle $S(t)+I(t)+R(t) = 1$, the total infectious fraction will evolve as
\begin{equation}
\dot{I} = -\dot{S} - \dot{R} = \beta SI - \lambda I,
\label{eq:I_classical}
\end{equation}
that is the remaining familiar relationship from the classical case. As a final note, we just mention that using respective forms for the infectivity and recovery functions for the Markovian case in the epidemic threshold relationships (\ref{eq:threshold_discrete}) and (\ref{eq:Threshold_continuous}), one will obtain the familiar threshold $\beta_{\mathrm th} = \gamma$.
\section{Numerical experiments and discussion}\label{sec:numerics}
Our numerical experiments with the proposed model were based on solution of the integro-differential equations for the continuous-time case. We have used the Euler method with step $\Delta t = 0.01$. Although in the model can be used arbitrary functions of infection and recovery, we have chosen to use those that can been found in the literature as appropriate for the COVID-19 pandemic. As suggested in \cite{Qin2020, qin2020estimation} the infectivity function $\beta(\tau)$ is conveniently represented with Weibull probability density function, with parameters $\alpha=2.04$ and $\lambda=0.103$, which is further truncated to 35 days and normalized. The daily recovering probabilities were modeled with log-normal probability density function $L(\tau; \mu; \sigma) = 1/(\tau\sigma \sqrt {2\pi }) \exp(-\left(\ln \tau-\mu \right)^{2}/(2\sigma ^{2}))$, with parameters $\mu =\ln (\mu _{X}^{2}/(\sqrt {\mu _{X}^{2}+\sigma _{X}^{2}})$, $\sigma ^{2}=\ln \left (1+\sigma _{X}^{2}/\mu _{X}^{2}\right)$ chosen to match a mean value of $\mu _{X}=21$ and standard deviation $\sigma _{X}=6$. The distribution is then normalized to 61 days, and time-shifted for 4 days in order to exclude immediate recovery. This results in the healing function $\gamma(\tau)$ with mean recovery time of $25 \pm 6$ days, in the following fashion
\begin{eqnarray}
\gamma(\tau)= \begin{cases}
\frac{L(\tau-4; \mu; \sigma)}{\int_{0}^{61}L(\tau; \mu; \sigma)d\tau},& 4\le \tau \le 65,\\
0,& \text{otherwise}.
\end{cases}
\end{eqnarray}
This construct was based on the results from \cite{sreevalsan2020analysis, faes2020time}, assuming that: 1. Onset of symptoms (on average) occurs after four days (the time shift); 2. It takes another 7-10 days from onset of symptoms to diagnosis confirmation and hospitalization; 3. Another 10-11 days, on average, are needed from hospitalization to recovery. The period of $T=65$ days is considered in order to include even most extreme cases in which hospitalization exceeded 40 days.
Furthermore, we have chosen to scale the infectivity function with a parameter $\beta$ given in terms of the epidemic threshold $\beta_{\mathrm{th}}$. The threshold value was obtained from the condition (\ref{eq:Threshold_continuous})
\begin{equation}
\beta_{\mathrm{th}}\int_0^T \beta(\tau)\overline{\Gamma}(\tau) d\tau = 1. \label{eq:beta_critical}
\end{equation}
To verify the value of the epidemic threshold we have varied the infectivity parameter in vicinity of the critical value obtained from (\ref{eq:beta_critical}) and run the continuous-time model for total time equal to 5000. The final values of the susceptible and recovered fraction are plotted as function of the infectivity parameter in the figure \ref{fig:Threshold}. As one can see, once $\beta$ is larger than its critical value, the epidemic emerges.
\begin{figure}[h]
\includegraphics[width=0.6\columnwidth]{Threshold.png}
\centering
\caption{Fractions of susceptible (red stars) and recovered (blue dots) individuals at the end of the epidemic as a function of the scaling of the infectivity function $\beta$ given in terms of its threshold value $\beta_{\mathrm{th}}$.}
\label{fig:Threshold}
\end{figure}
In order to verify how well the approach can be used to model the COVID-19 pandemic we have chosen to use value of the infectivity parameter $\beta$ that nearly matches the growth patterns of the epidemic in the countries before countermeasures were applied. As was obtained in a detailed study \cite{pellis2020challenges}, the epidemic doubling time in many countries is approximately three days. For that reason, we have opted to use the value $\beta = 4.85\beta_{\mathrm{th}}$ that produces such growth. We have numerically verified that in the initial stage of the epidemic, the newly confirmed daily cases and the total number of infected individuals grow with the same rate, and have the same doubling time. Also, by running the model with $\beta = 4.85\beta_{\mathrm{th}}$ for very long time, we have obtained that at the end less than 1\% of the population will remain susceptible! This result means that, if the doubling time is three days in case of free spreading of the virus, then prevention of the epidemic would need nearly everyone should be either vaccinated or had healed from the virus. This is particular challenge of the model that should be addressed carefully.
We have finally attempted to check how well the model can explain the observations. To do so, we have used the COVID-19 data from Our World in Data, for Italy. Our focus was put on the first wave of the pandemic, since in its beginning no preventive measures were used. We have chosen to study the epidemic in Italy, where the wave was the strongest. The window of data under study starts from February 21, 2020, that is the date from which every day were reported new cases. The countrywide lockdown started on March 10, 2020, that corresponds to day 19 in this study. We have used value of infectivity $\beta \approx 3.7\beta_{\mathrm{th}}$ that provided good fit to the observed data for the period from February 21 until March 9. This value was used until the start of the lockdown, when it was set to certain value smaller than the threshold. The initial condition was set to $I_d(0) = 10^{-7}$, that for Italy would mean about 6 persons infected at the starting day of simulation. We have chosen to apply detection of the infected individuals on based on a function that has identical form as the infectivity one, but which is delayed for certain number of days. This corresponds to situation that only those with symptoms are tested, and their appearance is delayed few days after the onset of infectivity. Also, there is certain delay that corresponds to the whole process from onset of symptoms, to visit to hospital to obtaining positive result. We note that the testing function was normalized to 0.8 that corresponds to assuming existence of 20\% asymptomatic cases \cite{buitrago2020occurrence}. To reach a good fit to the observations we had to take the start of the simulation, that is the day when the initial seed of infection was set, to be approximately 60 days before the day 1, when comparison with the real data starts. Its exact value was obtained by fitting the logarithms of the daily detected cases from the simulation to the respective ones from the data. More precisely, we have looked for a shift $s$, that will result in minimal squared error as follows
\begin{equation}
\epsilon = \argmin_{s} \left\{\frac{1}{19} \sum_{k=1}^{19} \left[\ln(I_d^{\mathrm{data}}(k)) - \ln(I_d(k+s)) \right]\right\}.
\end{equation}
We report in the top panel of figure \ref{fig:Italy} two simulated scenarios compared to the observations. In the first case we took testing function that is delayed after the infectivity one for two days, that actually becomes nonzero at the possible onset of the symptoms \cite{he2020temporal}, while the other case corresponds to delay of five days. The latter scenario provides much better fit to the observations, particularly in the period after the lockdown has started, and even further in the period after the peak, as one can notice in the figure \ref{fig:Italy}. We have tried with all integer values of the delay from two to ten (not shown) and five days correspond to the best fit. We remind that the lockdown corresponds to day 19 in the plot, while the peaks of the daily reported cases are delayed: at day 27 and day 30, for scenario one, and two, respectively. The peak at the latter case, appears at March 21, the day when largest number of new cases were registered. This fit to the peak and beyond of the model simulation with the observation, makes a good basis for the relevance of the proposed framework. In the bottom panel in figure \ref{fig:Italy} we show how modification of the value of infectivity parameter $\beta$ during the lockdown phase influences the daily cases.
\begin{figure}[h]
\includegraphics[width=0.6\columnwidth]{DailyItaly.png}
\includegraphics[width=0.6\columnwidth]{BetaPoVrv.png}
\centering
\caption{Daily confirmed cases in the first epidemic wave in Italy in spring 2020 (in blue squares), compared to numerical simulations of the model. Top panel: Confirmation function is delayed for two days after onset of infectivity (green circles) and five days (red stars) and $\beta = 0.75\beta_{\mathrm{th}}$. Bottom panel: Confirmation function is delayed for five days, while the infectivity parameter is: $\beta = 0.5\beta_{\mathrm{th}}$ (green circles), $\beta = 0.75\beta_{\mathrm{th}}$ (red stars), and $\beta = \beta_{\mathrm{th}}$ (magenta crosses)}
\label{fig:Italy}
\end{figure}
Although providing natural framework for incorporation of observed distributions of the infectiousness of the infected individuals and the typical development of the disease, the proposed model has drawbacks as well. First, before using it, one needs to specify the functions modeling the infectiousness, healing and discovering the infected individuals. Their determination is a serious issue by their own and needs careful study. As more complex one, the tuning of the model would need in general more data than the classical Markovian counterparts. Also, its full specification needs providing initial conditions that represent a high-dimensional vector, or an interval of values. How all these factors shape the outcome of the model, and how much is it robust to perturbations of any kind is unknown. We believe that their understanding could provide the epidemiologists with valuable information for better understanding of the possible outcomes of epidemics with pronounced non-Markovian nature.
\section{Conclusions}\label{sec:conclusions}
The proposed general non-Markovian epidemic spreading model captures the typical patterns of the disease in person infected with SARS-CoV-2: delayed onset of symptoms and infectivity and impossibility of immediate cure of those that will become sick. We have studied both discrete- and continuous-time versions and derived analytically the relationships for determination of the epidemic threshold. The model reduces to the classical SIR model with the corresponding choice of the functions of infection and healing. The theoretical analysis was supported by numerical confirmation of the epidemic threshold values. The good fit of the model to the real data shows its promising potential for application for modeling the spread of other infectious diseases. By introducing other appropriate functions one could possibly generalize this model to versions that include other compartments that correspond to hospitalized, quarantined, or deceased persons.
Although the epidemic threshold as key quantity was determined, we did not calculated the basic reproduction number $R_0$, that represents another important quantity. Furthermore, the relationship between the scaling of the infectivity function $\beta/\beta_{\mathrm{th}}$ from one side and $R_0$ and the doubling time, from another should be explored as well. With this regard, we think that it is even more important to determine the herd immunity level needed to prevent the epidemic. Finally, analysis of epidemic spreading by nontrivial contact patterns, modeled with complex networks, and by incorporating the proposed approach could provide further insight in the evolution of the epidemics. These issues could provide better understanding of the non-Markovian setting in modeling the epidemic spreading.
\section{Acknowledgement}
This research was partially supported by the Faculty of Computer Science and Engineering, at the Ss. Cyril and Methodius University in Skopje, Macedonia. The Authors acknowledge support by the German Science Foundation (DFG, Grant number ME 1535/12-1).
\section{Introduction}
The ongoing pandemics of COVID-19, has claimed millions of human lives, caused stagnation of the global economy and excessive load on the healthcare systems throughout the world and changed the normal life. Mathematical models of epidemic spreading are important tools for predicting the effects that the pandemics can have on each segment of the society. They provide support for policy-makers to make adequate decisions in order to partially mitigate the consequences by planning various social distancing measures, preparation of healthcare facilities and appropriate adaptation of the economy.
The spectrum of mathematical models applied for the COVID-19 pandemic ranges from the simplest SIR to rather complex SIDARTHE \cite{roda2020difficult, zhao2020modeling, calafiore2020time, giordano2020modelling, gatto2020spread}, which are used for assessment of different aspects of the epidemics. One of the major features of these models is their Markovian nature, which considers transitions from one state to another to be independent on the past. As an example, when Markovian property is assumed to hold, an individual that has just become infected can proceed to recovered state with the same probability as another one which has been infected for longer period. This Markovian assumption, encapsulated in constant transition probabilities, or rates, makes the models easier to study analytically. The outcomes of these studies with Markovian approach offer some, and in certain instances satisfactory, assessment of the spreading dynamics. However, growing body of evidence, particularly for the COVID-19, suggests existence of incubation period and certain infectivity patterns, with possibility for spreading the pathogen before onset of the symptoms, to which correspond functions that are rather distinct from the exponential distribution which the Markovian models rely on \cite{Qin2020, qin2020estimation}. Although adding one or more compartments for the Exposed, Asymptomatic, Presymptomatic, or Quarantined persons or considering various kinds of delay \cite{liu2020covid, dell2020solvable, rong2020effect}
address such observations to certain extent, they cannot systematically incorporate the observed distributions of the incubation period and the healing process.
There are different approaches of non-Markovian modeling of epidemic processes. In one attempt \cite{boguna1} is proposed Gillespie algorithm as an adequate tool for numerical analysis of non-Markovian spreading models. The effects of the form of distribution of infection and curing (recovery) times on SIS epidemic model occurring on complex networks in continuous time has been analyzed in several studies \cite{starnini,delft_nm1, delft_nm2, delft_nm3, Feng2019, krylova2013effects}. With the introduction of SI*V* model \cite{Nowzari} it was suggested that non-Markovian spreading models have capacity to be extended to cover a wide variety of spreading sub-models and variants. Nontrivial distribution of infectious period in an integro-differential SIR model was considered in \cite{riano2020epidemic}. In a recent study, non-Markovian SIS model on complex networks, with arbitrary function for infectivity and recovery was proposed \cite{tomovski2021epidemic}, in which control theory was successfully applied for determination of epidemic threshold. Our approach adds to these pioneering contributions by providing general framework for incorporation of various
distributions of infectivity and healing in a SIR model. By similar approach as in \cite{tomovski2021epidemic} we show how these functions determine the epidemic threshold. The relevance of the model, besides by numerical simulations, is verified by fitting to the observations of the first wave of the epidemic in Italy, in the spring, 2020.
The paper is organized as follows. After providing initial setting of the model in Section \ref{sec:preliminaries}, we introduce the discrete-time and continuous-time models in Sections \ref{sec:discrete} and \ref{sec:cont}, respectively, where we also derive the epidemic threshold relationships. The reduction to Markovian case of the model is presented in Section \ref{sec:Markov}, while numerical simulations and discussions are given in Section \ref{sec:numerics}. The paper concludes with Section \ref{sec:conclusions}.
\section{Preliminaries} \label{sec:preliminaries}
We consider SIR model that has three compartments: Susceptible - S, Infected - I and Recovered - R, with the usual transition $S \to I \to R$. Let the corresponding variables $S$, $I$ and $R$ denote the fractions of the population that are in the given state, and under assumption without births and deaths, one has the normalization condition $S(t)+I(t)+R(t) = 1$ at each moment $t$. To capture the nontrivial dependence of the healing period and the different contagiousness of the infected individual in different stages of the disease we introduce two functions. The infectivity function $\beta(\tau)$ captures the rate, or probability at which individuals that became infected before time $\tau$ are spreading the disease to the susceptible ones. Thus, by simply taking $\beta(\tau) = 0$ for $\tau < T_0$, one is able to introduce incubation period with length $T_0$. Another important function is the healing function $\gamma(\tau)$ that denotes the probability with which individual can heal at moment $\tau$ after contracting the disease. To account for asymptomatic transmitters and existence of certain time window when presence of pathogen can be confirmed, one can introduce a reporting function $\rho(\tau)$. It is associated to the probability that the presence of the pathogen can be confirmed at moment $\tau$ after contraction with it. The asymptomatic cases are conveniently handled by normalizing the reporting function to value smaller than unity. We pursue by considering discrete- and continuous-time models separately, and provide more details about these functions.
\section{Discrete-time version}
\label{sec:discrete}
In this section we consider evolution in discrete time $t$ and denote the fraction of individuals that have become infected within the continuous-time interval $[t-1, t]$ with $I_d(t)$, where for simplicity the unit interval is taken to be 1. This can be relevant for situations like those when cases are considered on daily basis. In such scenario, we have discrete-time healing function $\gamma(\tau)$ and infectivity one $\beta(\tau)$, on which we put the constraint $\beta(0) = 0$. The probability that the individual will heal within $\tau$ time units is $\Gamma(\tau) = \sum_{\nu=0}^{\tau} \gamma(\tau)$. We further assume finite duration $T$ of the disease, what implies $\Gamma(T) = 1$ and for practical reasons introduce its complement $\overline{\Gamma}(\tau) = 1 - \Gamma(\tau)$, to denote the probability that individual has not healed yet for $\tau$ time units. The function $\gamma(\tau)$ also has a meaning of fraction of individuals that have contracted the disease within the same unit time interval, to become healed later within another unit interval $[\tau-1, \tau]$. Similar reasoning holds for the cumulative functions $\Gamma(\tau)$ and $\overline{\Gamma}(\tau)$. On base on the classical SIR model, the proposed model of evolution of the compartments is given with the system
\begin{eqnarray}
S(t + 1) &=& S(t) \left[1 - \sum_{\tau=0}^{T-1} \beta(\tau) \overline{\Gamma}(\tau) I_d(t-\tau)\right] \nonumber \\
I_d(t + 1) &=& S(t) \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) I_d(t-\tau) \nonumber \\
R(t + 1) &=& R(t) +\sum_{\tau=0}^{T-1} \gamma(\tau) I_d(t + 1 -\tau). \label{eq:discrete_model}
\end{eqnarray}
One can note that the infected individuals that have contracted the pathogen up to $T$ periods before the current moment $t$, and which are not healed yet, can contribute to spreading of the disease, with appropriate intensity captured in the function $\beta(\tau)$. We note that in order to determine the infected fraction at given moment, one should sum those infected in the past, but did not heal up to the given moment
\begin{equation}
I(t) = \sum_{\tau=0}^{T-1} I_d(t-\tau) \overline{\Gamma}(\tau).
\label{eq:discrete_total_inf}
\end{equation}
To make the problem completely defined one has to specify the initial conditions for $I_d(t)$. We assume that they are given for $\tau = T-1, T-2, \dots, 0$. In general this model cannot be solved analytically and should be studied by application of numerical simulations.
To get insight of the conditions when epidemic can emerge, one can determine the stability of the disease free state $S^*=1, I^*=I_d^* = R^*=0 $, that is an equilibrium point of the system. Its local stability is established by linearizing the dynamical equations (\ref{eq:discrete_model}) in its neighborhood. By making the linearization in vicinity of $S^*=1, I^*=R^*=0$, one can observe the dynamical evolution of the perturbations $\delta S = S - S^*, \delta I_d =I_d - I_d^*, \delta R = R- R^*$. Under linearization, the perturbations are related with
\begin{eqnarray}
\delta S(t+1) &=& \delta S(t) -\sum_{\tau=0}^{T-1} \beta(\tau) \overline{\Gamma}(\tau) \delta I_d(t-\tau), \nonumber \\
\delta I_d(t+1) &=& \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) \delta I_d(t-\tau), \nonumber\\
\delta R(t + 1) &=& \delta R(t) + \sum_{\tau=0}^{T-1} \gamma(\tau) \delta I_d(t + 1-\tau). \label{eq:discrete_model_pert}
\end{eqnarray}
Let us focus on the infected fraction and make $Z$-transform on the second equation in (\ref{eq:discrete_model_pert}). To do so, multiply first both sides of that equation by $z^{-t}$ and sum to obtain
\begin{equation}
\sum_{t=0}^{\infty} \delta I_d(t+1) z^{-t} = \sum_{t=0}^{\infty} \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) \delta I_d(t-\tau) z^{-t}. \label{eq:Z_transform_disc}
\end{equation}
By using the $Z$-transform of the fraction of the population that become infected at unit interval $I_d(t)$, given as $\mathcal{I}(z) = \sum_{t=0}^{\infty} I_d(t) z^{-t}$, the left hand side of (\ref{eq:Z_transform_disc}) will become
\begin{equation}
\sum_{t=0}^{\infty} \delta I_d(t+1) z^{-t} = z\sum_{t=0}^{\infty} \delta I_d(t+1) z^{-(t+1)} = z\left[\mathcal{I}(z)- \delta I_d(0)\right]. \label{eq:delta_ID_t_left}
\end{equation}
Accordingly, the right-hand side of (\ref{eq:Z_transform_disc}) can be rearranged as
\begin{equation}
\sum_{t=0}^{\infty} \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) \delta I_d(t-\tau) z^{-t}= \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) z^{-\tau} \sum_{t=0}^{\infty} \delta I_d(t-\tau) z^{-(t-\tau)}.
\end{equation}
By using substitution $\nu = t-\tau$, the last sum for $\tau \leq -1$ can be expressed as
\begin{equation}
\sum_{\nu=-\tau}^{\infty} \delta I_d(\nu) z^{-\nu} =\sum_{\nu=-\tau}^{-1} \delta I_d(\nu) z^{-\nu} + \mathcal{I}(z) =
\mathcal{I}_{0}(\tau, z) + \mathcal{I}(z), \label{eq:delta_ID_t_right}
\end{equation}
where we have introduced a function $\mathcal{I}_0(\tau, z)$ that corresponds to the initial conditions. Now, combining the relationships (\ref{eq:delta_ID_t_left}) -- (\ref{eq:delta_ID_t_right}) one has
\begin{equation}
z\left[\mathcal{I}(z)- \delta I_d(0)\right]= \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) \left[\mathcal{I}_0(\tau, z) + \mathcal{I}(z)\right] z^{-\tau} .
\end{equation}
To shorten the notation, one can introduce the following two complex functions
\begin{eqnarray}
\mathcal{E}(z) &=& \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) z^{-\tau}, \nonumber\\
\mathcal{E}_0(z) &=& \sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) \mathcal{I}_0(\tau, z) z^{-\tau}.
\end{eqnarray}
The first one is simply the $Z$-transform $\mathcal{E}(z)$ of what might be called epidemic function $E(\tau) = \beta(\tau)\overline{\Gamma}(\tau)$, that is a combination of the infecting and healing functions because $\sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) z^{-\tau}=\sum_{\tau=0}^{\infty} \beta(\tau)\overline{\Gamma}(\tau) z^{-\tau}$. The second complex function $\mathcal{E}_0(z)$ is related to the initial conditions. Now, one has the following relationship
\begin{equation}
z\left[\mathcal{I}(z)- \delta I_d(0)\right] = \mathcal{I}(z)\mathcal{E}(z) + \mathcal{E}_0(z),
\end{equation}
from where
\begin{equation}
\mathcal{I}(z) = \frac{z\delta I_d(0)+\mathcal{E}_0(z)}{z-\mathcal{E}(z)}.\label{eq:infected_Z_transform}
\end{equation}
From a result in theory of discrete linear time-invariant systems, a sequence (the impulse response of such system) is decaying if the poles of its $Z$-transform are within the unit circle \cite{oppenheim2013signals}. Thus, when the poles of the function $\mathcal{I}(z)$ of the complex function (\ref{eq:infected_Z_transform}), or the roots of the polynomial $z-\mathcal{E}(z)$ lie within the unit circle, the perturbation dies out at infinity. So, the epidemic threshold can be obtained by taking $z=1$ in the denominator in (\ref{eq:infected_Z_transform}), that results in
\begin{equation}
\sum_{\tau=0}^{T-1} \beta(\tau)\overline{\Gamma}(\tau) = 1, \label{eq:threshold_discrete}
\end{equation}
which obviously depends on the functional forms of the infectivity and healing functions.
We should finally note that any initial infection would not shift back the population to the disease-free state $S=1, I=R=0$, but to some endemic $S_e^*, I_e^* = 0, R_e^* = 1-S^*$. However, if the conditions are not favoring epidemic both equilibria will be rather close $S_e^* \approx 1$.
\section{Continuous-time version}\label{sec:cont}
We will pursue similarly to the discrete-time approach, where the fractions of individuals within given compartment and the functions modeling the infectivity, healing and reporting are defined for continuous time $t$ and we use the same notation. Thus, $S(t)$ is the fraction of susceptible individuals at given moment $t$ and $R(t)$ corresponds to the recovered and again assume finite healing period $T$. The fraction of infected individuals is conveniently modeled with the rate of infection, or the fraction of newly infected individuals $I_d(t)$ within the infinitesimal interval $(t-dt, t)$. The total fraction of infected persons is given with the integral
\begin{equation}
I(t) = \int_{0}^{T} I_d(t-\tau) \overline{\Gamma}(\tau) d\tau, \label{eq:total_I}
\end{equation}
which accounts for those that had become infected in the past and have not healed yet. Now, the dynamical evolution of the respective fractions is given with
\begin{eqnarray}
\dot{S} &=& - S(t)\int_{0}^{T} \beta(\tau) \overline{\Gamma}(\tau) I_d(t-\tau)d\tau \nonumber \\
I_d(t) &=& S(t) \int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) I_d(t-\tau) d\tau \nonumber \\
\dot{R} &=& \int_{0}^{T} \gamma(\tau) I_d(t-\tau) d\tau. \label{eq:continuous_model}
\end{eqnarray}
In order to determine whether the initial perturbation will grow to epidemics, one could focus on the second equation in the vicinity of the disease-free state $S^*=1, R^*=I^*=0$. Then, the perturbation of newly infected individuals will evolve as
\begin{equation}
\delta I_d(t) = \int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) \delta I_d(t-\tau)d\tau, \label{eq:pert_infect_rate}
\end{equation}
where it is assumed that in vicinity of the disease-free state $S(t)\approx 1$. Now, make Laplace transform of the perturbation of the rate of infection, $\mathcal{I}(s) = \int_0^{\infty}\delta I_d(t)e^{-st}dt$ and use it in the last equation (\ref{eq:pert_infect_rate}). To do that, we will follow the same approach as in the discrete-time version. Multiply both sides with $e^{-st}$ and integrate. The left hand side will result in the Laplace transform of $\delta I_d(t)$, while the right hand one will be
\begin{eqnarray}
A &= &\int_0^{\infty}\int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) \delta I_d(t-\tau)e^{-st}d\tau dt = \int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) e^{-s\tau} \int_0^{\infty} \delta I_d(t-\tau) e^{-s(t-\tau)} dt \nonumber \\
&=&\int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) e^{-s\tau} \int_{-\tau}^{\infty}\delta I_d(\nu) e^{-s\nu} d\nu
\end{eqnarray}
The last integral can be expressed with
\begin{equation}
\int_{-\tau}^{\infty}I_d(\nu) e^{-s\nu} d\nu = \int_{-\tau}^{0}I_d(\nu) e^{-s\nu} d\nu + \mathcal{I}(s) = \mathcal{I}_0(\tau,s) + \mathcal{I}(s).
\end{equation}
Now, one has
\begin{equation}
A=\int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) e^{-s\tau} \left[\mathcal{I}_0(\tau,s) + \mathcal{I}(s) \right] d\tau.
\end{equation}
Similarly to the discrete-time case we can introduce the Laplace transform of the epidemic function $E(\tau)=\beta(\tau)\overline{\Gamma}(\tau)$ and its initial conditions contribution
\begin{eqnarray}
\mathcal{E}(s) &=& \int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) e^{-s\tau} d\tau, \\ \nonumber
\mathcal{E}_0(s) &=& \int_{0}^{T} \beta(\tau)\overline{\Gamma}(\tau) \mathcal{I}_0(\tau,s) e^{-s\tau} d\tau.
\end{eqnarray}
Finally, one obtains
\begin{equation}
\mathcal{I}(s) = \mathcal{I}(s) \mathcal{E}(s) + \mathcal{E}_0(s),
\end{equation}
from where the Laplace transform of the perturbation of the infection rate is
\begin{equation}
\mathcal{I}(s) = \frac{\mathcal{E}_0(s)}{1-\mathcal{E}(s)}. \label{eq:Laplace_transf_pert}
\end{equation}
From the results of control theory, a continuous-time linear time-invariant system is stable if the poles of its transfer function, or Laplace transform of its impulse response have negative real part \cite{oppenheim2013signals}. Thus, the perturbations $\delta I_d(t)$ will decay if the poles of its Laplace transform $\mathcal{I}(s)$ (\ref{eq:Laplace_transf_pert}), or eigenvalues of the system (\ref{eq:continuous_model}) lie within negative half-plane $Re\{s\} < 0$. Then, the epidemic threshold can be obtained with $s=0$ which leads to
\begin{equation}
\int_0^T \beta(\tau)\overline{\Gamma}(\tau) d\tau = 1, \label{eq:Threshold_continuous}
\end{equation}
that represents the relationship, which corresponds to the discrete-time case (\ref{eq:threshold_discrete}).
\section{Markovian SIR model}\label{sec:Markov}
In order to obtain the classical Markovian SIR model, from the non-Markovian case (\ref{eq:discrete_model}), one should consider taking $T \rightarrow \infty$, $\beta(\tau)=\beta$, $\gamma(0)=0$, $\gamma(\tau)=\gamma(1-\gamma)^{\tau-1}$ where $\beta$ and $\gamma$ are constants. This further yields $\Gamma(0)=0$, $\Gamma(\tau)=1-(1-\gamma)^{\tau}$ and $\overline{\Gamma}(\tau)=(1-\gamma)^{\tau}=\gamma(\tau+1)/\gamma$. First, one could observe that by using constant infectivity $\beta(\tau)=\beta$ in the first relationship of the model (\ref{eq:discrete_model}) and using (\ref{eq:discrete_total_inf}) one will obtain the classical form for evolution of the susceptible population
\begin{equation}
S(t + 1) = S(t) \left[1 - \beta I(t)\right]. \label{eq:SIR_S_classical}
\end{equation}
Next, by implementing the condition $\gamma(0) = 0$, and the relationship $\overline{\Gamma}(\tau)=\gamma(\tau+1)/\gamma$ one can drop the first term in the sum in the recovered population in (\ref{eq:discrete_model}), and further obtain
\begin{equation}
\sum_{\tau=0}^{T-2} \gamma(\tau+1) I_d(t-\tau) = \gamma\sum_{\tau=0}^{T-1} \overline{\Gamma}(\tau) I_d(t-\tau)- \gamma \overline{\Gamma}(T-1) I_d(t-T+1) = \gamma I(t)- \gamma (1-\gamma)^{T-1}I_d(t-T+1),
\end{equation}
from where, for $T \rightarrow \infty$, the recovered population evolves as
\begin{equation}
R(t+1) = R(t) + \gamma I(t). \label{eq:SIR_R_classical}
\end{equation}
Finally, from the conservation relationship $I(t) + S(t) + R(t) = 1$, one can find that the infected fraction is given as
\begin{equation}
I(t+1) = \beta S(t)I(t)+(1-\gamma) I(t).
\label{eq:SIR_I_classical}
\end{equation}
The relationships (\ref{eq:SIR_S_classical}), (\ref{eq:SIR_R_classical}) and (\ref{eq:SIR_I_classical}) represent the classical SIR model in discrete time.
As an example, in the figure \ref{Mar_vs_nonMar} we make a comparison between numerical solutions of the discrete classical SIR model and the classical SIR - equivalent model obtained from the non-Markovian form.
\begin{figure}[h]
\includegraphics[width=0.6\columnwidth]{SIR_comparison_T=150_b=0k2_g=0k03_tnr}
\centering
\caption{Comparison between the discrete classical SIR model and the classical SIR - equivalent model obtained from the non-Markovian form, for $\beta=0.2$, $\gamma=0.03$. It is used rather large finite duration of the healing $T=150$, as a proxy for $T\to\infty$. }
\label{Mar_vs_nonMar}
\end{figure}
Similarly to the discrete-time version, to verify that the proposed continuous model is generalization of the classical, Markovian SIR model, one should consider two characteristics of the latter: 1. The infection rate is independent on the moment when the disease was contracted $\beta(\tau) = \beta$; and 2. The duration of infectivity is infinite and exponentially distributed which implies that the healing function is $\gamma(\tau) = \lambda e^{-\lambda \tau}$. We note that the respective cumulative distribution is $\Gamma(\tau) = 1 - e^{-\lambda\tau}$, and accordingly $\overline{\Gamma}(\tau) = e^{-\lambda\tau}$. By using the functional form of the healing function, the total infectious population will be
\begin{equation}
I(t) = \int_0^{\infty} \overline{\Gamma}(\tau)I_d(t-\tau)d\tau = \int_0^{\infty} e^{-\lambda \tau}I_d(t-\tau)d\tau.
\label{eq:I_total_classical}
\end{equation}
Similarly, by using $\beta(\tau) = \beta$, for the dynamics of the susceptible fraction one has
\begin{equation}
\dot{S} = -S(t)\beta \int_0^{\infty} e^{-\lambda\tau} I_d(t-\tau) d\tau = -\beta S I,
\label{eq:S_classical}
\end{equation}
that represents the corresponding relationship in the classical SIR model. Furthermore, by applying the functional form for the healing function, the dynamics of the recovered population will be as follows
\begin{eqnarray}
\dot{R} = \int_0^{\infty} \lambda e^{-\lambda \tau} I_d(t-\tau) d\tau = \lambda I,
\label{eq:R_classical}
\end{eqnarray}
that is the respective relationship in the classical SIR model. Finally, by using the conservation principle $S(t)+I(t)+R(t) = 1$, the total infectious fraction will evolve as
\begin{equation}
\dot{I} = -\dot{S} - \dot{R} = \beta SI - \lambda I,
\label{eq:I_classical}
\end{equation}
that is the remaining familiar relationship from the classical case. As a final note, we just mention that using respective forms for the infectivity and recovery functions for the Markovian case in the epidemic threshold relationships (\ref{eq:threshold_discrete}) and (\ref{eq:Threshold_continuous}), one will obtain the familiar threshold $\beta_{\mathrm th} = \gamma$.
\section{Numerical experiments and discussion}\label{sec:numerics}
Our numerical experiments with the proposed model were based on solution of the integro-differential equations for the continuous-time case. We have used the Euler method with step $\Delta t = 0.01$. Although in the model can be used arbitrary functions of infection and recovery, we have chosen to use those that can been found in the literature as appropriate for the COVID-19 pandemic. As suggested in \cite{Qin2020, qin2020estimation} the infectivity function $\beta(\tau)$ is conveniently represented with Weibull probability density function, with parameters $\alpha=2.04$ and $\lambda=0.103$, which is further truncated to 35 days and normalized. The daily recovering probabilities were modeled with log-normal probability density function $L(\tau; \mu; \sigma) = 1/(\tau\sigma \sqrt {2\pi }) \exp(-\left(\ln \tau-\mu \right)^{2}/(2\sigma ^{2}))$, with parameters $\mu =\ln (\mu _{X}^{2}/(\sqrt {\mu _{X}^{2}+\sigma _{X}^{2}})$, $\sigma ^{2}=\ln \left (1+\sigma _{X}^{2}/\mu _{X}^{2}\right)$ chosen to match a mean value of $\mu _{X}=21$ and standard deviation $\sigma _{X}=6$. The distribution is then normalized to 61 days, and time-shifted for 4 days in order to exclude immediate recovery. This results in the healing function $\gamma(\tau)$ with mean recovery time of $25 \pm 6$ days, in the following fashion
\begin{eqnarray}
\gamma(\tau)= \begin{cases}
\frac{L(\tau-4; \mu; \sigma)}{\int_{0}^{61}L(\tau; \mu; \sigma)d\tau},& 4\le \tau \le 65,\\
0,& \text{otherwise}.
\end{cases}
\end{eqnarray}
This construct was based on the results from \cite{sreevalsan2020analysis, faes2020time}, assuming that: 1. Onset of symptoms (on average) occurs after four days (the time shift); 2. It takes another 7-10 days from onset of symptoms to diagnosis confirmation and hospitalization; 3. Another 10-11 days, on average, are needed from hospitalization to recovery. The period of $T=65$ days is considered in order to include even most extreme cases in which hospitalization exceeded 40 days.
Furthermore, we have chosen to scale the infectivity function with a parameter $\beta$ given in terms of the epidemic threshold $\beta_{\mathrm{th}}$. The threshold value was obtained from the condition (\ref{eq:Threshold_continuous})
\begin{equation}
\beta_{\mathrm{th}}\int_0^T \beta(\tau)\overline{\Gamma}(\tau) d\tau = 1. \label{eq:beta_critical}
\end{equation}
To verify the value of the epidemic threshold we have varied the infectivity parameter in vicinity of the critical value obtained from (\ref{eq:beta_critical}) and run the continuous-time model for total time equal to 5000. The final values of the susceptible and recovered fraction are plotted as function of the infectivity parameter in the figure \ref{fig:Threshold}. As one can see, once $\beta$ is larger than its critical value, the epidemic emerges.
\begin{figure}[h]
\includegraphics[width=0.6\columnwidth]{Threshold.png}
\centering
\caption{Fractions of susceptible (red stars) and recovered (blue dots) individuals at the end of the epidemic as a function of the scaling of the infectivity function $\beta$ given in terms of its threshold value $\beta_{\mathrm{th}}$.}
\label{fig:Threshold}
\end{figure}
In order to verify how well the approach can be used to model the COVID-19 pandemic we have chosen to use value of the infectivity parameter $\beta$ that nearly matches the growth patterns of the epidemic in the countries before countermeasures were applied. As was obtained in a detailed study \cite{pellis2020challenges}, the epidemic doubling time in many countries is approximately three days. For that reason, we have opted to use the value $\beta = 4.85\beta_{\mathrm{th}}$ that produces such growth. We have numerically verified that in the initial stage of the epidemic, the newly confirmed daily cases and the total number of infected individuals grow with the same rate, and have the same doubling time. Also, by running the model with $\beta = 4.85\beta_{\mathrm{th}}$ for very long time, we have obtained that at the end less than 1\% of the population will remain susceptible! This result means that, if the doubling time is three days in case of free spreading of the virus, then prevention of the epidemic would need nearly everyone should be either vaccinated or had healed from the virus. This is particular challenge of the model that should be addressed carefully.
We have finally attempted to check how well the model can explain the observations. To do so, we have used the COVID-19 data from Our World in Data, for Italy. Our focus was put on the first wave of the pandemic, since in its beginning no preventive measures were used. We have chosen to study the epidemic in Italy, where the wave was the strongest. The window of data under study starts from February 21, 2020, that is the date from which every day were reported new cases. The countrywide lockdown started on March 10, 2020, that corresponds to day 19 in this study. We have used value of infectivity $\beta \approx 3.7\beta_{\mathrm{th}}$ that provided good fit to the observed data for the period from February 21 until March 9. This value was used until the start of the lockdown, when it was set to certain value smaller than the threshold. The initial condition was set to $I_d(0) = 10^{-7}$, that for Italy would mean about 6 persons infected at the starting day of simulation. We have chosen to apply detection of the infected individuals on based on a function that has identical form as the infectivity one, but which is delayed for certain number of days. This corresponds to situation that only those with symptoms are tested, and their appearance is delayed few days after the onset of infectivity. Also, there is certain delay that corresponds to the whole process from onset of symptoms, to visit to hospital to obtaining positive result. We note that the testing function was normalized to 0.8 that corresponds to assuming existence of 20\% asymptomatic cases \cite{buitrago2020occurrence}. To reach a good fit to the observations we had to take the start of the simulation, that is the day when the initial seed of infection was set, to be approximately 60 days before the day 1, when comparison with the real data starts. Its exact value was obtained by fitting the logarithms of the daily detected cases from the simulation to the respective ones from the data. More precisely, we have looked for a shift $s$, that will result in minimal squared error as follows
\begin{equation}
\epsilon = \argmin_{s} \left\{\frac{1}{19} \sum_{k=1}^{19} \left[\ln(I_d^{\mathrm{data}}(k)) - \ln(I_d(k+s)) \right]\right\}.
\end{equation}
We report in the top panel of figure \ref{fig:Italy} two simulated scenarios compared to the observations. In the first case we took testing function that is delayed after the infectivity one for two days, that actually becomes nonzero at the possible onset of the symptoms \cite{he2020temporal}, while the other case corresponds to delay of five days. The latter scenario provides much better fit to the observations, particularly in the period after the lockdown has started, and even further in the period after the peak, as one can notice in the figure \ref{fig:Italy}. We have tried with all integer values of the delay from two to ten (not shown) and five days correspond to the best fit. We remind that the lockdown corresponds to day 19 in the plot, while the peaks of the daily reported cases are delayed: at day 27 and day 30, for scenario one, and two, respectively. The peak at the latter case, appears at March 21, the day when largest number of new cases were registered. This fit to the peak and beyond of the model simulation with the observation, makes a good basis for the relevance of the proposed framework. In the bottom panel in figure \ref{fig:Italy} we show how modification of the value of infectivity parameter $\beta$ during the lockdown phase influences the daily cases.
\begin{figure}[h]
\includegraphics[width=0.6\columnwidth]{DailyItaly.png}
\includegraphics[width=0.6\columnwidth]{BetaPoVrv.png}
\centering
\caption{Daily confirmed cases in the first epidemic wave in Italy in spring 2020 (in blue squares), compared to numerical simulations of the model. Top panel: Confirmation function is delayed for two days after onset of infectivity (green circles) and five days (red stars) and $\beta = 0.75\beta_{\mathrm{th}}$. Bottom panel: Confirmation function is delayed for five days, while the infectivity parameter is: $\beta = 0.5\beta_{\mathrm{th}}$ (green circles), $\beta = 0.75\beta_{\mathrm{th}}$ (red stars), and $\beta = \beta_{\mathrm{th}}$ (magenta crosses)}
\label{fig:Italy}
\end{figure}
Although providing natural framework for incorporation of observed distributions of the infectiousness of the infected individuals and the typical development of the disease, the proposed model has drawbacks as well. First, before using it, one needs to specify the functions modeling the infectiousness, healing and discovering the infected individuals. Their determination is a serious issue by their own and needs careful study. As more complex one, the tuning of the model would need in general more data than the classical Markovian counterparts. Also, its full specification needs providing initial conditions that represent a high-dimensional vector, or an interval of values. How all these factors shape the outcome of the model, and how much is it robust to perturbations of any kind is unknown. We believe that their understanding could provide the epidemiologists with valuable information for better understanding of the possible outcomes of epidemics with pronounced non-Markovian nature.
\section{Conclusions}\label{sec:conclusions}
The proposed general non-Markovian epidemic spreading model captures the typical patterns of the disease in person infected with SARS-CoV-2: delayed onset of symptoms and infectivity and impossibility of immediate cure of those that will become sick. We have studied both discrete- and continuous-time versions and derived analytically the relationships for determination of the epidemic threshold. The model reduces to the classical SIR model with the corresponding choice of the functions of infection and healing. The theoretical analysis was supported by numerical confirmation of the epidemic threshold values. The good fit of the model to the real data shows its promising potential for application for modeling the spread of other infectious diseases. By introducing other appropriate functions one could possibly generalize this model to versions that include other compartments that correspond to hospitalized, quarantined, or deceased persons.
Although the epidemic threshold as key quantity was determined, we did not calculated the basic reproduction number $R_0$, that represents another important quantity. Furthermore, the relationship between the scaling of the infectivity function $\beta/\beta_{\mathrm{th}}$ from one side and $R_0$ and the doubling time, from another should be explored as well. With this regard, we think that it is even more important to determine the herd immunity level needed to prevent the epidemic. Finally, analysis of epidemic spreading by nontrivial contact patterns, modeled with complex networks, and by incorporating the proposed approach could provide further insight in the evolution of the epidemics. These issues could provide better understanding of the non-Markovian setting in modeling the epidemic spreading.
\section{Acknowledgement}
This research was partially supported by the Faculty of Computer Science and Engineering, at the Ss. Cyril and Methodius University in Skopje, Macedonia. The Authors acknowledge support by the German Science Foundation (DFG, Grant number ME 1535/12-1).
|
2,877,628,090,067 | arxiv | \section{BRIEF DESCRIPTION OF THE METHOD}
In the nuclear mean-field approach, the energy of the nucleus is computed as the expectation value of a two-body Hamiltonian on a trial wave-function \cite{RingSchuck}. Besides the relativistic approaches\cite{Ring_Rev}, there exists two main families of two-body effective forces to this date, the zero-range Skyrme \cite{skyrme_review} interaction and the finite-range Gogny one \cite{D1}. Both are empirical effective forces and there exists a number of realistic parametrizations. Skyrme forces lead to a local energy density which is the basic building block of the nuclear Energy Density Functional (EDF) theory \cite{hfb_review}. The Gogny interaction, because of its finite range, is non-local and computationally more involved, for this reason the calculations are most conveniently carried out in configuration space, i.e. the solutions to the Hartree-Fock (HF) or Hartree-Fock-Bogoliubov (HFB) problem are expanded on a given basis.
The harmonic oscillator (HO) basis has always played a special role in configuration space calculations, as its eigenfunctions are given analytically and are separable. However, a well-known deficiency of this basis is that it is made exclusively of bound-states since the underlying potential has infinite walls. In practice, since calculations are always performed in a given truncation scheme (for a fixed cut-off of the basis) the localization of all HO basis states imply that the physical wave-functions of the system will always acquire a Gaussian asymptotic, including the weakly-bound and positive-energy states. This is clearly unrealistic, as spherical continuum states should be spherical waves. The consequences of this deficiency become more serious close to the drip line, as pairing correlations can couple discrete bound-states to the continuum. It is thus critical to properly describe the continuous spectrum even at the level of ground-state calculations \cite{Doba-continuum}.
There exists a number of techniques to take into account the continuum in nuclear structure calculations, and it is not the purpose of this article to discuss in detail the merits of every one of them. Let us just mention briefly for completeness: coordinate-space Skyrme HFB theory with either vanishing \cite{Doba-continuum} or outgoing-wave boundary conditions \cite{grasso}, coordinate-space Relativistic Hartree-Bogoliubov with finite-element method \cite{RHB_FiniteElem}, the use of the Gamow basis in the Skyrme HFB theory \cite{GamowHFB}, in the Continuum Shell Model \cite{SMEC} and in the Gamow Shell Model \cite{GamowSM}. At the present time technical difficulties in including the full continuum with the exact resonant and non-resonant spectrum lead to the consequence that the most advanced theories are only applied with simple model interactions that are tailored to capture the main physical properties of the system. Only in the coordinate-space HFB approach realistic Skyrme interactions were employed with density-dependence zero-range forces in the pairing channel (requiring the introduction of a cut-off in the quasi-particle spectrum or a regularization procedure \cite{regularization}). Moreover, as far as mean-field based theories are concerned, no attempt has been done to include with the coupling to the continuum the restoration of broken symmetries or collective motion.
Therefore, in order to combine the flexibility of configuration space calculations with the necessary inclusion of the continuum, it has been proposed in \cite{ZMR.03,nous} to work in a basis made of the eigenstates of the Woods-Saxon potential. The latter are obtained by integrating the Schr\"{o}dinger equation in a box of size $R_{box}$ with a mesh size $h$. In practice $R_{box} = 20$ fm and $h = 0.1$ fm are sufficient to obtain a good convergence of the solutions. Boundary conditions are set on the walls of the box. As usual, several choices are possible. Outgoing wave boundary conditions lead to wave-functions that are not square-integrable, and special techniques must be employed to overcome this difficulty \cite{nonHermitian-1,nonHermitian-2,nonHermitian-3,nonHermitian-4}. Vanishing boundary conditions guarantee that the basis functions are square-integrable and can thus be normalized at the price of eliminating all the continuum states that do not have a node on the walls of the box. It was shown in \cite{BoxVsGamow} that both techniques essentially lead to very similar results as far as bound-states and bulk properties of nuclei are concerned. In the following, we use vanishing box boundary conditions.
In our calculations we use the finite-range Gogny interaction \cite{D1}. The same interaction is used in the particle-hole channel (mean-field) and particle-particle channel (pairing) and both the direct and exchange contributions coming from {\it all} the terms of the interaction are taken into account in the calculation. The finite-range of the force in the pairing channel allows to avoid the divergence problem (in momentum space) and cut-off dependence of zero-range forces. All of our calculations are performed in spherical symmetry.
To obtain quantitative information on the neutron halo in neutron-rich nuclei we make use of the Helm method \cite{Helm1,Helm2,Helm3,HelmDoba}. Firstly, the neutrons (protons) form factor is computed as the Fourier transform of the neutrons (protons) density. In spherical symmetry, this leads to:
\begin{equation}
F(q) = 4\pi\int_{0}^{\infty} j_{0}(qr)\rho_{\tau}(r)r^{2}dr
\label{formfactor}
\end{equation}
where $q$ is the momentum, $j_{0}$ is the spherical Bessel function of order 0 and $\rho_{\tau}(r)$ is the density ($\tau$ standing for neutron or proton). This form factor built out of the realistic one-body density, in our case calculated with the
Gogny force, is then compared to the Helm form-factor obtained from the convolution of the Gaussian profile
\begin{equation}
f_G(r) = \frac{e^{{-r^2/(2\sigma^2)}}}{(2\pi)^{3/2}\sigma^3}
\end{equation}
with a sharp density profile: $\rho(r) = \rho_{0}$ for $r \leq R_{0}$ and $\rho(r) = 0$ elsewhere. Since this model is presented in details in the references quoted, we simply recall the formulas we are going to use. The two parameters $R_0$ and $\sigma$ of the model are determined in the following way.
The rms radius $R_{rms}$ is defined as the squared root of the mean-value of the operator $\hat{r}^{2}$. It is extracted from the nucleonic density:
\begin{equation}
R_{rms} = \sqrt{\langle \hat{r}^{2}\rangle} =
\sqrt{\frac{\displaystyle\int d^{3}\vec{r}\;r^{2}\rho(\vec{r})}{\displaystyle\int d^{3}\vec{r}\;\rho(\vec{r})}}
\end{equation}
For the Helm radius one straightforwardly obtains
\begin{equation}
R_{rms}^{H} = \sqrt{\frac{3}{5}(R_{0}^{2} + 5\sigma^{2})}
\end{equation}
where $R_{0}$ is the diffraction radius:
\begin{equation}
R_{0} = 4.49341/q_{1}
\end{equation}
and $q_{1}$ is the first zero of the realistic form-factor (\ref{formfactor}) obtained in our theoretical approach. The surface thickness $\sigma$ is defined as:
\begin{equation}
\sigma^{2} = \frac{2}{q_{m}^{2}}\ln \frac{3N j_{1}(q_{m}R_{0})}{R_{0}q_{m}F(q_{m})}
\end{equation}
where $N$ is the number of particles, $j_{1}$ the spherical Bessel function of order 1 and $q_{m}$ is the first maximum of the realistic form-factor (\ref{formfactor}).
At this point we should note that the method does not carry out any information on the eventual decorrelation between a core and a few valence particles. It only provides a simple and fast method to assess the spatial extension of the nucleus and an excellent starting point to determine the best halo candidates. However, in few-body nuclear models, the nuclear halo is often interpreted as one single nucleon or a pair of nucleons orbiting around a core, see e.g. \cite{Halo-2body-1,Halo-2body-2} for two-body models and \cite{Halo-3body-1,Halo-3body-2,Halo-3body-3} for 3-body models. In order to reconcile these cluster approaches with a mean-field description of the nucleus, a more detailed analysis of the density should be carried out. Alternative techniques have been proposed to cure this deficiency \cite{HaloDuguet}.
It is usually convenient to multiply the rms and the Helm radius by $\sqrt{5/3}$. The quantities
\begin{equation}
R_{geom} = R_{rms}\sqrt{\frac{5}{3}}; \hspace{0.5cm}R_{Helm} = R_{rms}^{H}\sqrt{\frac{5}{3}}.
\end{equation}
are related to the underlying shape of the nucleus.
A measure of the nuclear halo is then provided by the quantity:
\begin{equation}
\delta R_{halo} = R_{geom} - R_{Helm}
\label{deltaR}
\end{equation}
The neutron skin can be defined in various ways depending on which type of radius is considered. Its general expression is:
\begin{equation}
\Delta R = R^{(n)} - R^{(p)}
\end{equation}
where $R$ can be either the geometrical radius, the Helm radius or the diffraction radius. It was argued in \cite{HelmDoba} that the best approximation to the neutron skin is obtained when taking the Helm radius, as the latter is somewhat rid of spurious contributions coming from the neutron halo.
\section{NUCLEAR SKINS AND HALOS WITH FINITE-RANGE INTERACTIONS}
Systematic calculations near the neutron drip line have been carried out using the spherical HFB code in the Woods-Saxon basis that was presented in \cite{nous}. The basis was constructed from the eigenstates of the WS potential with the universal parametrization of \cite{WSUniversal} applied to the $Z=126$ and $N=184$ nucleus. The Schr\"{o}dinger equation was integrated in a box of $R_{box} = 20$ fm with vanishing boundary conditions. All eigenstates with $\ell \leq 15$ and $n\leq 18$ were retained in the basis. As was shown in \cite{nous}, such a choice guarantees a good convergence of the subsequent HFB calculation.
\subsection{Determination and properties of the neutron drip line}
There exist few parametrizations of the Gogny interaction: in our calculations we considered the parametrizations D1 of \cite{D1} and D1S of \cite{D1S}. For each of them the neutron drip line was calculated based on the requirement that the one-neutron $S_{n} = B(N,Z) - B(N-1,Z)$ separation energy must be negative for bound nuclei. Since $S_{2n} = S_{n} + S_{n-1}$, the criterion $S_{n} < 0$ is stricter than the condition that the two-neutron separation energy is negative. In the HFB theory the one-neutron separation energy $S_n$ is approximated by the neutrons Fermi energy $\lambda_n = dE/dN \approx -S_{n} $. A nearly equivalent condition to define the one neutron drip line is therefore $\lambda_n > 0$. When HFB pairing correlations vanish (case of closed shells), the value of the chemical potential $\lambda$ is meaningless and can not be used to define the drip line any more (Hartree-Fock limit). However, in the HF approach and within the approximation of the validity of Koopmans' theorem \cite{Koopman}, the stability of a nucleus is simply governed by the position of the last occupied level: if it has positive energy, then the nucleus is unbound with respect to particle emission.
We display in Table \ref{table01} the one neutron drip line nuclei obtained with both interactions. In the presence of pairing correlations the criterion $\lambda_n > 0$ has been used. For the neutron shell closures $N = 82$ (D1S: elements $Z = 36$ to $Z=40$, D1: elements $Z = 36$ and $Z=38$), $N = 126$ (D1S: elements $Z = 52$ to $Z=64$, D1: elements $Z=54$ to $Z=62$) and $N = 184$ (D1S: elements $Z = 80$ to $Z=92$, D1: elements $Z=80$ to $Z=90$) the neutron pairing correlations vanishes and we have to rely on Koopmans' theorem. The columns corresponding to the D1S interaction were already presented in \cite{nous} and are recalled for comparison. We would like to comment at this point that, due to some technical problems with our previous codes at the aforementioned shell closures, in \cite{nous} the drip line was predicted with two neutron less for the elements Kr, Te, Xe, Ba, Hg and Pb. We also show in Table \ref{table01} the difference in the number of neutrons between the drip line nuclei with the D1 and D1S interaction: $\Delta N = N_{drip line}(D1) - N_{drip line}(D1S)$. In general the D1 parametrization predicts a drip line with more neutrons, probably due to the fact that it provides more pairing correlations than the D1S one. Let us emphasize that all calculations performed in this work are restricted to spherical symmetry. Several of the nuclei listed in Table \ref{table01} may be deformed in their ground-state \cite{Gogny-MassTable}. Symmetry-unrestricted HFB calculations of neutron-rich nuclei would most likely shift the position of the drip line in several places.
\begin{table}[h]
\begin{center}
\caption{Table of spherical HFB one neutron drip line nuclei obtained with the D1S and D1 interactions. The columns marked $\Delta N$ represent the shift of the drip line (in number of neutrons) when using the D1 interaction compared to the D1S.}
\begin{ruledtabular}
\begin{tabular}{c|cc|cc||c|cc|cc}
Z & N & D1S & D1 & $\Delta N$ & Z& N& D1S & D1 & $\Delta N$ \\
\hline
6 &14 & $^{~20}$C & $^{~20}$C & 0 & 50& 120 & $^{170}$Sn & $^{168}$Sn & -2 \\
8 &18 & $^{~26}$O & $^{~26}$O & 0 & 52& 126 & $^{178}$Te & $^{178}$Te & 0 \\
10 &20 & $^{~30}$Ne & $^{~30}$Ne & 0 & 54& 126 & $^{180}$Xe & $^{180}$Xe & 0 \\
12 &28 & $^{~40}$Mg & $^{~42}$Mg & +2 & 56& 126 & $^{182}$Ba & $^{182}$Ba & 0 \\
14 &32 & $^{~46}$Si & $^{~46}$Si & 0 & 58& 126 & $^{184}$Ce & $^{184}$Ce & 0 \\
16 &34 & $^{~50}$S & $^{~52}$S & +2 & 60& 126 & $^{186}$Nd & $^{186}$Nd & 0 \\
18 &38 & $^{~56}$Ar & $^{~58}$Ar & +2 & 62& 126 & $^{188}$Sm & $^{188}$Sm & 0 \\
20 &44 & $^{~64}$Ca & $^{~62}$Ca & -2 & 64& 126 & $^{190}$Gd & $^{194}$Gd & +4 \\
22 &50 & $^{~72}$Ti & $^{~72}$Ti & 0 & 66& 132 & $^{198}$Dy & $^{204}$Dy & +6 \\
24 &52 & $^{~76}$Cr & $^{~78}$Cr & +2 & 68& 138 & $^{206}$Er & $^{216}$Er & +10\\
26 &56 & $^{~82}$Fe & $^{~82}$Fe & 0 & 70& 150 & $^{220}$Yb & $^{230}$Yb & +10\\
28 &58 & $^{~86}$Ni & $^{~88}$Ni & +2 & 72& 168 & $^{240}$Hf & $^{244}$Hf & +4 \\
30 &62 & $^{~92}$Zn & $^{~98}$Zn & +6 & 74& 178 & $^{252}$W & $^{254}$W & +2 \\
32 &72 & $^{104}$Ge & $^{104}$Ge & 0 & 76& 182 & $^{258}$Os & $^{258}$Os & 0 \\
34 &80 & $^{114}$Se & $^{114}$Se & 0 & 78& 182 & $^{260}$Pt & $^{260}$Pt & 0 \\
36 &82 & $^{118}$Kr & $^{118}$Kr & 0 & 80& 184 & $^{264}$Hg & $^{264}$Hg & 0 \\
38 &82 & $^{120}$Sr & $^{120}$Sr & 0 & 82& 184 & $^{266}$Pb & $^{266}$Pb & 0 \\
40 &82 & $^{122}$Zr & $^{124}$Zr & +2 & 84& 184 & $^{268}$Po & $^{268}$Po & 0 \\
42 &88 & $^{130}$Mo & $^{130}$Mo & 0 & 86& 184 & $^{270}$Rn & $^{270}$Rn & 0 \\
44 &92 & $^{136}$Ru & $^{138}$Ru & +2 & 88& 184 & $^{272}$Ra & $^{272}$Ra & 0 \\
46 &94 & $^{140}$Pd & $^{148}$Pd & +8 & 90& 184 & $^{274}$Th & $^{274}$Th & 0 \\
48 &104 & $^{152}$Cd & $^{158}$Cd & +6 & 92& 184 & $^{276}$U & $^{280}$U & +4 \\
& & & & & 94& 188 & $^{282}$Pu & $^{294}$Pu & +12 \\
\end{tabular}
\label{table01}
\end{ruledtabular}
\vspace*{-0.5cm}
\end{center}
\end{table}
For each interaction, the quantity $\delta R_{halo}$ of Eq. (\ref{deltaR}) was then computed at the drip line, i.e. for each element listed in Table \ref{table01}. The results are shown in Fig.~\ref{fig01}. We find a downward trend of $\delta R_{halo}$ as a function of $Z$ superimposed with oscillations. Both features are well understood. The decreasing behavior has to do with the well-known fact that light nuclei have larger halos. The oscillations are related to the neutron magic numbers: To the five minima (for the D1S, for example, Z$_{\rm min}=10, 22, 40, 64 $ and 92 ) correspond to neutron number 20, 50, 82, 126 and 184, see Table \ref{table01}. For proton numbers two (four or six) units larger than a given Z$_{\rm min}$ a few neutrons occupy a new large j-shell thereby inducing pairing correlations and producing a halo. For Z values much larger than a given Z$_{\rm min}$ the number of neutrons in the shell becomes large and the halo disappears.
\begin{figure}[h]
\includegraphics[height=7.0cm,width=9.0cm]{deltaHalo_HFB.eps}
\caption{Measure of the neutron halo: $\delta R_{halo} = R_{geom} - R_{Helm}$ in fm for spherical Gogny HFB calculations in the WS basis with the D1S (plain squares) and D1 (open circles) interactions.
}
\label{fig01}
\end{figure}
As noticed in \cite{HelmDoba}, the size of the halos is correlated with the corresponding chemical potential of the HFB solutions: Larger halos correspond to nuclei with values of $\lambda_n$ close to zero and smaller ones to large $\lambda_n$ values. As far as the effect of the parametrization of the Gogny force is concerned, we observe that the largest difference takes place in $^{42}$Mg which is not bound with D1S interaction while it is bound for the D1 interaction. Apart from this particular nucleus, both parametrizations of the Gogny force give very similar results, even though the isotopes of the drip line elements are sometimes very different, for example $^{216}$Er with the D1 interaction and $^{206}$Er with the D1S.
It is instructive to compare our results with the work of \cite{HelmDoba}. It was pointed out in this reference that the size of the halo, as measured by the quantity $\delta R_{halo}$, significantly depends on the interaction used. A similar conclusion was reached in \cite{HaloDuguet} using a slightly different analysis procedure. In our case both parametrizations provide rather similar results in spite of the fact that the numerical values of the D1S and D1 parametrizations are quite different. It is also interesting to note that both parametrizations can lead in some cases to significantly different drip lines - in the case of the D1 interaction for example, the drip line near Palladium isotopes ($Z=46$) and Erbium ($Z=68$) and Ytterbium ($Z=70$) is located 8 and 10 neutrons further away, respectively, than in the case of the D1S interaction. Yet, as mentioned, the size of the halo remains very similar.
\begin{figure}[h]
\includegraphics[height=7.0cm,width=9.0cm]{Radii_Ni.eps}
\includegraphics[height=7.0cm,width=9.0cm]{Radii_Ni_halo.eps}
\caption{Upper panel: Neutron $R_{geom}(n)$ and $R_{Helm}(n)$ and proton $R_{geom}(p)$ and $R_{Helm}(p)$ radii for the Ni isotopes calculated with the D1S Gogny interaction. Lower panel: Neutron and proton halo parameter $\delta R_{halo}$ along this isotopic chain.
}
\label{fig02}
\end{figure}
Figure \ref{fig01} shows which elements can be considered as the best halo candidates. For each such candidate, the inspection of the full isotopic sequence from drip line to drip line can provide information on the swelling of the nuclear skin and the transition skin to halo. As a first example, we show in Fig. \ref{fig02} the case of the isotopic chain for Nickel element. For the D1S interaction, this element has one of the largest halo. Furthermore, the same isotopic line was also studied in the framework of the Skyrme-HFB (SLy4 and SKP interactions) and RHB (NLSH and NL3 lagrangians) theories \cite{HelmDoba}, which therefore gives us results from three different sorts of mean-fields. In the upper panel of figure \ref{fig02} both the geometrical and Helm radii are plotted for the neutron and proton along the Ni isotopic chain. In the lower panel, the halo parameter $\delta R_{halo}$ is plotted for the neutrons and protons. In nuclei far away from the drip line, the difference between geometrical and Helm radius is very small, reflecting the neglegible coupling to the continuum near the valley of stability. At $N=50$ we observe the last shell closure and immediately after the onset of pairing correlations, which translates into a rapid increase of the halo parameter. As was pointed out in \cite{HaloDuguet}, a shortcoming of the Helm method is that in some cases, the quantity $\delta R_{halo}$ is non-zero even in the middle of the valley of stability. This appears clearly in Fig. \ref{fig02} for the protons along the entire isotopic chain.
\begin{figure}[h]
\includegraphics[height=7.0cm,width=9.0cm]{Radii_Sn.eps}
\includegraphics[height=7.0cm,width=9.0cm]{Radii_Sn_halo.eps}
\caption{Upper panel: Neutron $R_{geom}(n)$ and $R_{Helm}(n)$ and proton $R_{geom}(p)$ and $R_{Helm}(p)$ radii for the Sn isotopes calculated with the D1S Gogny interaction. Lower panel: Neutron and proton halo parameter $\delta R_{halo}$ along this isotopic chain.
}
\label{fig03}
\end{figure}
Another important remark is that both parametrizations of the Gogny force tend to give "compact" nuclei, with relatively small halos in agreement with those obtained in \cite{HelmDoba} with the Skyrme SkP and relativistic mean-field NLSH and NL3 parametrizations and in contrast to the Skyrme/SLy4 interaction.
The case of Tin isotopes is even more enlightening. For this particular element, the position of the drip line is nearly identical in spherical HFB calculations with Gogny/D1S and Skyrme/SLy4 interactions, which facilitates the comparison. In the upper panel of Fig. \ref{fig03} we plot the neutron and proton geometrical and Helm radius from the proton to the neutron drip line. Note that near the neutron drip line, the halo is only about 0.15 fm while Skryme/SLy4 results reported in \cite{HelmDoba} indicate a size of about 0.8 fm.
It could also be tempting to apply our method in some of the {\it experimental} cases of nuclear halos. However, as hinted in the introduction, we are faced with one major difficulty: most of the halo candidates are very light nuclei with $Z\leq 6$ at the drip line like $^{11}$Li, $^{14}$Be and $^{19}$B. For all the elements with $Z\leq 8$ the experimental drip line is rigorously established, in the sense that isotopes beyond the drip line are proved to be particle-unstable \cite{exp_dripline}. The application of our spherical Gogny-HFB calculations, whether the D1 or D1S interaction is used, gives the correct drip line isotope for Lithium (see next section) but fails to reproduce the experimental data for elements B, C and O. Three main mechanisms, possibly combined, can be the source of this discrepancy: (i) the interactions used are not extrapolable in these light nuclei (ii) additional mean-field symmetries must be broken, e.g. rotational invariance (iii) correlations beyond the HFB level must be included. It is almost certain that the fit of the interactions can be improved, but it is today difficult to assess to which extent this would affect the predictions of nuclear halos in very light nuclei. Similarly, it is not very clear at the moment how halos are formed in deformed nuclei.
\subsection{Discussion on giant halos}
Since our description contains the main ingredients for a proper description of the halo phenomenon, namely, a good pairing force, the incorporation of the continuum and eventually particle number projection (in the VAP approach) indispensable in a weak pairing regime we can confront our model with recent spectacular predictions about the existence of several giant halos in light and medium-mass nuclei. In Neon isotopes, {\it spherical} coordinate-space Relativistic Hartree-Bogoliubov (RHB) calculations predicted that giant halos could develop for a number of neutrons around 30 \cite{PRL_RMF_1}. In our {\it spherical} Gogny-HFB calculations the drip line is positioned at $N = 20$ for both D1 and D1S interactions, which falls a bit short of the last known bound Neon isotope at $N=24$ \cite{MgNature} and references therein. When deformation is included in the calculation, the position of the (current) drip line shifts to $N=24$ \cite{Gogny-MassTable}. The very stretched drip line reported in \cite{PRL_RMF_1} is somewhat surprising since the pairing channel was treated by using the D1S finite-range interaction. Moreover, up to $N=20$, RHB results for the r.m.s. radii are very similar to ours: for example at $N=20$ we find a neutron r.m.s radius of $r_{n} = 3.39$ fm with the D1S interaction, to compare with RHB results of $r_{n} \approx 3.42$ fm.
\begin{figure}[h]
\includegraphics[height=7.0cm,width=9.30cm]{Zr_S2n.eps}
\includegraphics[height=7.0cm,width=9.50cm]{Zr_spLevels.eps}
\caption{(color on-line) Upper panel: Two-neutron separation energy for the Zr isotopic chain with the D1 and D1S parametrization. Lower panel: Neutron single-particle levels in the canonical basis for Zr isotopes (D1S interaction). The bullets represent the position of the Fermi level.}
\label{fig04}
\end{figure}
The application of the spherical coordinate-space RHB, with a zero-range density-dependent force in the particle-particle channel, led to another prediction of giant halos in Zirconium isotopes \cite{PRL_RMF_2}. Similarly as in the case of Neon isotopes, such predictions are rooted in the existence of a very stretched drip line at $N = 100$ corresponding to element $^{140}$Zr. Results showed in Table \ref{table01} and upper panel of Fig.~\ref{fig04} show that the drip line in spherical Gogny-HFB calculations is at $N=82$ for the D1S and $N=84$ for the D1 interaction. Deformed Gogny-HFB calculations with the D1S interaction also suggest a drip line at $N=82$ \cite{Gogny-MassTable}. These results are in agreement e.g. with Skyrme HFB calculations with the SLy4 interaction \cite{stoitsov_dripline} which predict the drip line at $N=84$. Other parametrizations of the Skyrme interaction have slightly more extended drip lines, at N = 92 for SKP and N = 94 for SKM* \cite{stoitsov-masstable}. For the Gogny interaction, isotopes with $N\geq 82$ (D1S) and $N\geq 84$ (D1) are unbound with respect to two neutron emission, see upper panel of Fig. \ref{fig04}. Beyond drip line HFB calculations, although not realistic, can be pedagogical: in Fig. \ref{fig05} we show the evolution of the neutron geometrical and Helm radius beyond the drip line for both the D1 and D1S interactions. As we increase the number of neutrons, delocalized orbitals corresponding to discretized continuum states become occupied and cause a very marked increase of the neutron radius.
\begin{figure}[t]
\includegraphics[height=6.5cm,width=9.0cm]{Zr_Radii.eps}
\caption{ Neutron $R_{geom}(n)$ and $R_{Helm}(n)$ radii for the Zr isotopes calculated with the D1S (plain symbols) and D1 (open symbols) Gogny interaction.
}
\label{fig05}
\end{figure}
If we restrict ourselves to physical solutions at the HFB level we find very small halos: $\delta R_{halo} \approx 0.06$ fm for the D1S interaction in $^{122}$Zr and $\delta R_{halo} \approx 0.11$ fm for the D1 interaction in $^{124}$Zr. One may be tempted to attribute this small value to the collapse of pairing correlations which occurs at $N = 82$. This collapse of pairing correlation can be inferred from the lower panel of Fig. \ref{fig04}, where the gap between the (occupied) h$_{11/2}$ and (empty) 2f$_{7/2}$ orbital is very large. However, particle-number projection before variation applied to this nucleus provides the same drip line Zr isotope and leads essentially to the same value of $\delta R_{halo}$ even though pairing correlations do not vanish any more.
It should be noted that our results agree with previous works from \cite{PRL_RMF_2} (RHB), \cite{stoitsov-masstable} (SLY4, SKM* and SKP) as far as the main features of the shell structure of Zr isotopes are concerned, cf. for example the neutron single-particle levels in the canonical basis, Fig. 1 in \cite{PRL_RMF_2} and Fig. \ref{fig04} in the present work. In particular, the inflection point in the neutron radius at $N = 82$ is reproduced by all models. However, all three realizations of the nuclear mean-field differ as to the exact location of the drip line for Zirconium element, and it is this uncertainty that causes the widely different predictions of halo sizes in this particular element.
\begin{figure}[h]
\includegraphics[height=6.5cm,width=9.0cm]{Skins_Ni.eps}
\includegraphics[height=6.5cm,width=9.0cm]{Skins_Sn.eps}
\caption{Upper panel: Neutron skins for the Ni isotopes calculated with the D1S Gogny interaction and expressed as the difference of geometrical radii (plain squares), Helm radii (open circles) and diffraction radii (plain triangles). Lower Panel: Same figure as the upper panel for Sn element.
}
\label{skins_Sn}
\end{figure}
\subsection{Neutron skins}
One of the main features of neutron-rich nuclei is the development of neutron skins as the asymmetry between the number of neutrons and protons increases. The method that we developed to include the continuum in our calculations allow us to compute neutron skins up to the drip line. As an illustration, we display in Fig. (\ref{skins_Sn}) the neutron skins calculated from the geometrical, Helm and diffraction radius for the two isotopic chains of Nickel and Tin.
As expected all three definitions of the neutron skin give a smooth increase with the neutron number. As noticed in \cite{HelmDoba}, however, the skin calculated from geometrical radii shows a clear inflection point at $N=50$ (Ni isotopes) and $N=82$ (Sn isotopes), which is directly related to the one marking the appearance of the neutron halo, cf. Fig. \ref{fig03}. By contrast, the neutron skin calculated from the Helm radius is a more regular function of the neutron number. Interestingly, although the size of the halo with our Gogny/D1S interaction is markedly smaller than with e.g. Skyrme/SLy4, the values for the neutron skin are much closer: In $^{170}$Sn, $\Delta R_{Helm} \approx 0.57$ fm for D1S and $\Delta R_{Helm} \approx 0.70$ fm for Skyrme/SLy4 (a similar number is also obtained in Skyrme/SKP), cf. \cite{HelmDoba}.
\begin{figure}[h]
\includegraphics[height=7.0cm,width=9.50cm]{NeutronSkins.eps}
\includegraphics[height=7.0cm,width=9.50cm]{NeutronSkins_Tz.eps}
\caption{Upper panel: Neutron skins along the neutron drip line calculated with the Gogny interaction and expressed as the difference of Helm radii with the D1S (plain squares) and D1 (open circles) interactions. Lower panel: Quantity $(N-Z)/A$ at the drip line for the D1S (plain squares) and D1 (open circles) interactions
}
\label{neut_skins}
\end{figure}
It is instructive to compute the neutron skin for all the elements located at the drip line. In the upper panel of Fig.(\ref{neut_skins}) we plot the neutron skin for the nuclei listed in Table \ref{table01}. We find an oscillatory behavior relatively similar to the one found for the quantity $\delta R_{halo}$ plotted in Fig. \ref{fig01}. In both cases, halos and skins, these oscillations can be somewhat related to neutron magic numbers, but the underlying physics is quite different.
Skins are defined as the difference between the neutron and proton radius. Therefore, variations in the shape of the skins measures the relative increase or decrease of neutrons versus protons. At a neutron shell closure, one can add several protons without changing the position of the neutron drip line, i.e., as a function of Z, the proton radius increase and the neutron one remains constant, which produces a decrease in the neutron skin. This effect is very clearly seen in Fig. \ref{neut_skins}: the minima at $Z = 40$, $Z = 62-64$ (D1-D1S) and $Z = 90-92$ (D1-D1S) correspond to the last isotone with (magic) neutron number $N = 82$, $N = 126$ and $N=184$ respectively. Once beyond the neutron magic number, the neutron radius increases very rapidly, and this translates into a quick increase of the neutron skin for the next few elements. This sharp rise is also visible in Fig. \ref{neut_skins} in the range $40 \leq Z \leq 50$, $62-64 \leq Z \leq 72$.
In fact, the oscillations of the neutron skin can be correlated very neatly to the quantity (N-Z)/A. While the neutron excess N-Z increases with the mass number A (an upward trend not observed in Fig. \ref{neut_skins}), the ratio (N-Z)/A fluctuates around some average value of 0.37. In the lower panel of Fig.(\ref{neut_skins}) we plot (N-Z)/A as a function of Z at the drip line for the two interactions D1S and D1 considered in this study. We observe that the maxima and minima of the neutron skin correspond almost exactly to the maxima and minima of (N-Z)/A, especially for heavy nuclei. In light nuclei, this correspondence remains, although it is a little less obvious. We should like to stress that the quantity (N-Z)/A is a direct measure of the ratio between the iso-vector and iso-scalar (integrated) densities. Neutron skins could therefore prove particularly useful to obtain experimental constraints on the corresponding terms of the interaction/functional.
It also follows from this observation that we do not observe for the neutron skins the clear downward trend as function of Z that was observed for the halos, cf. Fig. \ref{fig01}. Neutron skins are rather a mass independent observable, which implies that the skin in a very light nucleus such as, e.g. Si ($Z=14$) is of comparable size as the skin in W ($Z=74$). The amplitude of the oscillations is also much smaller for the skins than for the halos, reflecting the fact that the (N-Z)/A ratio does not vary too much along the neutron drip line. Also, the main maxima for halos and skins do not exactly coincide: For the D1S interaction, for example, the skins peak at $Z=14, 32, 50$ and 74 and the halos at $Z=16, 28, 44$ and 74.
\section{INFLUENCE OF SYMMETRY-RESTORATION ON NUCLEAR HALOS}
In this section, we discuss another mechanism that can affect the position of the drip line, namely the restoration of broken symmetries. We focus on the projection on good particle number before variation and examine several of its conceptual as well as practical consequences. The method we use to include continuum effects into our description of weakly-bound nuclei is indeed particularly suitable to include extensions beyond the mean-field.
\subsection{The RVAP approach}
In \cite{nous} we briefly described how we can simulate the Variation After Projection (VAP) of the HFB solutions by means of the restricted-VAP (RVAP) method. Since along the drip lines some conceptual difficulties may arise we discuss the method a bit more at length. To illustrate how the RVAP method works we shall assume a generic two body Hamiltonian
\begin{eqnarray}
\hat{H}
=
\sum_{lq}t_{lq}c_{l}^{\dagger}c_{q}+\frac{1}{4}\sum_{lql'q'}\bar{v}_{lql'q'}c_{l}^{\dagger}c_{q}^{\dagger}c_{q'}c_{l'}
\label{hamil_c}
\end{eqnarray}
with $\bar{v}_{lql'q'}$ the antisymmetric matrix element
\begin{eqnarray}
\bar{v}_{lql'q'}=v_{lql'q'}-v_{lqq'l'}
\label{potencialant}
\end{eqnarray}
and $(c_{i}^{\dagger},c_{i})$ the single-particle creation and annihilation operators in a given basis. Given the most general Bogoliubov transformation
\begin{equation}
\beta_{k}^{\dagger} = \sum_{l}U_{lk}c_{l}^{\dagger}+V_{lk}c_{l}
\label{betas}
\end{equation}
the HFB method provides the product wave function
\begin{equation}
|\Phi\rangle=\prod_{q}\beta_{q}|-\rangle
\end{equation}
that minimizes the expectation value of the Hamiltonian $\hat{H}$. The matrices $U$ and $V$ that fix the Bogoliubov transformation of Eq. (\ref{betas}) are determined by minimization of the functional
\begin{equation}
E_{HFB}'\left[|\Phi\rangle\right]=\frac{\langle\Phi|\hat{H}-\lambda_{N}\hat{N}-\lambda_{Z}\hat{Z}|\Phi\rangle}{\langle\Phi|\Phi\rangle}
\label{funcE}
\end{equation}
with $\lambda_N$ and $\lambda_Z$ the Lagrange parameters that adjust the average number of neutron and proton.
It can be shown \cite{RingSchuck} that the minimization of Eq.~(\ref{funcE}) amounts to the diagonalization of the matrix
\begin{eqnarray}
\left(
\begin{array}{cc}
h' & \Delta \\
-\Delta^{*} & -h'^{*} \\
\end{array}
\right)\left(
\begin{array}{c}
U_{k}\\
V_{k}\\
\end{array}
\right) & = & E_{k} \left(
\begin{array}{c}
U_{k}\\
V_{k}\\
\end{array}
\right)
\label{hfb_eq_diag}
\end{eqnarray}
with $E_{k}$ the quasi-particle energies and $h'=t+\Gamma-\lambda_{N}-\lambda_{Z}$. The Hartree-Fock field $\Gamma$ and the pairing field $\Delta$ are given by
\begin{eqnarray}
\Gamma_{ll'}&=&\sum_{qq'}\bar{v}_{lql'q'}\rho_{q'q} \label{Gamma}\\
\Delta_{ll'}&=&\frac{1}{2}\sum_{qq'}\bar{v}_{ll'qq'}\kappa_{qq'} \label{Delta}
\end{eqnarray}
with $\rho$ the density matrix and $\kappa$ the pairing tensor defined by
\begin{eqnarray}
\rho_{ll'} & = & \langle\Phi|c_{l'}^{\dagger}c_{l}|\Phi\rangle=\left(V^{*}V^{T}\right)_{ll'} \nonumber\\
\kappa_{ll'} & = & \langle\Phi|c_{l'}c_{l}|\Phi\rangle=\left(V^{*}U^{T}\right)_{ll'}.
\end{eqnarray}
The particle number projected energy is given by
\begin{eqnarray}
E^{N}\left[|\Phi\rangle\right]
=
\frac{\langle\Phi^{N}|\hat{H}|\Phi^{N}\rangle}{\langle\Phi^{N}|\Phi^{N}\rangle}
=
\frac{\langle\Phi |\hat{H}\hat{P}^{N}|\Phi\rangle}{\langle\Phi|\hat{P}^{N}|\Phi\rangle}
\label{E_N}
\end{eqnarray}
with $\hat{P}^{N}$ the particle number projector and
\begin{equation}
|\Phi^{N}\rangle = \hat{P}^{N}|\Phi\rangle.
\label{WF_2}
\end{equation}
To avoid cumbersome formula we do not distinguish in Eq. (\ref{WF_2}) between protons and neutrons. The simplicity of projection techniques lies in the fact that, while $|\Phi^{N}\rangle$ is a correlated many-body wave function, the intrinsic wave function $|\Phi\rangle$ remains a product wave function, i.e. the variational parameters to be determined are the matrices $U$ and $V$ of Eq.~(\ref{betas}).
In the Variation After Projection (VAP) approach the projected energy $E^{N}$, see Eq.~(\ref{E_N}), is minimized directly. In the Projection After Variation (PAV) approach the HFB energy $E_{HFB}'$, see Eq.~(\ref{funcE}), is minimized first and the projection is carried out on the HFB wave-function after convergence. The difference is clear: in the VAP method we minimize the energy of the one nucleus (Z,N) we are interested in while in the PAV, the energy of a superposition of nuclei with numbers of particle Z and N around the actual values. Though the variational parameters are the same, the solution of the VAP equations is numerically much more involved than the PAV one. In a strong pairing regime the PAV solution might be a good approximation but in the general case, and in particular along the drip lines, the VAP one is much better. With finite range forces the solution of the VAP equations is rather involved, see \cite{PNP-Gogny}. Considering the additional difficulties inherent to a proper treatment of the coupling to the continuum, it is clear that a full VAP solution is beyond the actual numerical capabilities.
A way out of this problem is the Restricted VAP. In the VAP method the whole Hilbert space associated to the transformation Eq.~(\ref{betas}) is scanned in the variational procedure. In the RVAP approach, however, only a restricted variational space of highly correlated wave-functions is allowed. In our case, since we are interested in pairing correlations, our restricted space should contain a whole set of paired wave-functions $|\Phi(\delta)\rangle$ which parametrically depend on the real number $\delta$. To generate such wave-functions with different pairing content and that simultaneously are consistent with our Hamiltonian, we proceed in the following way: Instead of iterating Eq.~\ref{hfb_eq_diag} together with Eqs.~(\ref{Gamma},\ref{Delta}) as in the usual HFB case, we now iterate
\begin{eqnarray}
\left(
\begin{array}{cc}
h' & \delta \cdot \Delta \\
-\delta \cdot \Delta^{*} & -h'^{*} \\
\end{array}
\right)\left(
\begin{array}{c}
U_{k}\\
V_{k}\\
\end{array}
\right) & = & E_{k} \left(
\begin{array}{c}
U_{k}\\
V_{k}\\
\end{array}
\right)
\label{hfb_eq_diagm}
\end{eqnarray}
together with Eqs.~(\ref{Gamma},\ref{Delta}) until the convergence is achieved. The matrices $U(\delta)$ and $V(\delta)$ obtained in this way determine the wave-functions $|\Phi(\delta)\rangle$. Performing the same procedure for different $\delta$ values we generate the restricted correlated Hilbert space. We then project these wave-functions onto good-particle number and obtain a family of particle-number projected states $|\Phi^N(\delta)\rangle = \hat{P}^{N}|\Phi(\delta)\rangle$, for $\delta = 1.0, \dots, \delta_{max}$. The range of values for $\delta$ is chosen in such a way that at least several $|\Phi^N(\delta)\rangle$ wave functions correspond to highly-paired states. We can then take the expectation value of the Hamiltonian with this set of wave-functions, i.e, using eq.~(\ref{E_N}). This gives us a curve $E^{N}(\delta)$ where, at each point $\delta$, the particle number is conserved. The variational principle guarantees that such a curve has a minimum, which approaches the VAP result \cite{restricted-VAP} .
To illustrate the procedure with a numerical application in Fig.~\ref{int_proj_ener} we display the un-projected energy
\begin{equation}
E^{HFB}(\delta)= \frac {\langle \Phi(\delta )|\hat{H}|\Phi(\delta )\rangle}{\langle \Phi(\delta )|\Phi(\delta )\rangle}
\end{equation}
and the projected one
\begin{equation}
E^{PNP}(\delta)=\frac{\langle \Phi(\delta )|\hat{H}\hat{P}^N|\Phi(\delta )\rangle}{\langle \Phi(\delta )|\hat{P}^{N}|\Phi(\delta )\rangle}
\end{equation}
for the drip line nucleus $^{62}$Ca with the D1S interaction. When computing the density-dependent contribution to the projected energy $E^{PNP}(\delta)$, the projected density $\rho^{PNP}$ has been used (prescription 1 in \cite{PNP-Gogny}).
Since the HFB self-consistent minimum is obtained, by definition, at $\delta =1$ for $E^{HFB}(\delta)$ we expect a parabolic behavior around this value for increasing or decreasing $\delta$ values. Concerning $E^{PNP}(\delta)$, at $\delta = 1$ projecting the HFB solution onto good particle number lowers the energy and for increasing pairing correlations, i.e., values of $\delta$ larger than 1, we first observe a decrease of the projected energy up to a minimum around $\delta =1.12$ followed by a rapid increase. Obviously the solution of the RVAP approach is $|\Phi(\delta=1.12)\rangle$.
\begin{figure}[h]
\includegraphics[height=7.0cm,width=9.0cm]{Ca62_RVAP_Illustration.eps}
\caption{Intrinsic (plain squares) and projected (open circles) energy as a function of the parameter $\delta$ for the nucleus $^{62}$Ca.}
\label{int_proj_ener}
\end{figure}
In a PNP approach the drip lines are defined in terms of the projected separation energies, i.e. in terms of $S^N_n =B^{N}(N,Z)-B^{N-1}(N-1,Z)$ and $S^N_{2n} =B^{N}(N,Z)-B^{N-2}(N-2,Z)$. In the HFB approach $S_n\approx - \lambda_n$ and the one neutron drip line can be easily calculated. This approximation is not valid any more in a projected theory and $S^N_n$ must be explicitly calculated. Since for the moment we are not able to project on an odd number of particles we cannot calculate the one neutron drip line and will therefore focus on the 2-neutron drip line. In the next section we discuss the meaning of $\lambda$ and other quantities in the particular context of a particle number projected theory.
\subsection{On the RVAP approach and the number of particles of the intrinsic wave function}
Let $|\Phi \rangle$ be a HFB wave-function, i.e. a particle number symmetry violating wave function. We will now show that the particle number projected energy is invariant under transformations that change the particle number of the underlying HFB wave function. We define
\begin{equation}
|\tilde{\Phi} \rangle = e^{\alpha \Delta \hat{N}} | \Phi \rangle
\label{Transfo_N}
\end{equation}
with $\Delta \hat{N} = \hat{N} -N_0$, $N_0= \langle \Phi |\hat{N} |\Phi\rangle$ and $\alpha$ is a real number and we assume that
$\langle \Phi | \Phi\rangle=1$. The wave function $| \Phi \rangle$ can be written as \cite{RingSchuck}
\begin{equation}
|{\Phi} \rangle = \sum_{\beta^{\prime},N^{\prime}} C_{\beta^{\prime},N^{\prime}} |\beta^{\prime}, N^{\prime} \rangle
\end{equation}
where $|\beta^{\prime}, N^{\prime} \rangle$ is an eigenstate of $\hat{N}$ with particle number $N'$ and $\beta'$ stands for all other necessary quantum numbers. The transformed wave function reads:
\begin{eqnarray}
|\tilde{\Phi} \rangle
& = &
e^{\alpha \Delta \hat{N}} \sum_{\beta^{\prime},N^{\prime}} C_{\beta^{\prime},N^{\prime}} |\beta^{\prime}, N^{\prime} \rangle \\
& = &
\sum_{\beta^{\prime},N^{\prime}} C_{\beta^{\prime},N^{\prime}} e^{\alpha {(N^{\prime}-N_0)}} |\beta^{\prime}, N^{\prime}\rangle
\end{eqnarray}
The projected energy is given by
\begin{eqnarray}
E^N
& = &
\frac{\langle \tilde{\Phi}| \hat{H} \hat{P}^{N} | \tilde{\Phi}\rangle}{\langle \tilde{\Phi}| \hat{P}^{N} | \tilde{\Phi}\rangle} \\
& = &
\frac{\sum_{\beta{\prime}{\prime},\beta^\prime} C_{\beta^{\prime \prime},N}^* C_{\beta^\prime,N}\langle \beta^{\prime\prime}N | \hat{H} |\beta^\prime N \rangle}
{\sum_{\beta^{\prime \prime},\beta\prime} C_{\beta^{\prime \prime},N}^* C_{\beta^\prime,N}\langle \beta^{\prime \prime} N |\beta^\prime N \rangle} \\
& =&
\frac{\langle {\Phi}| \hat{H} \hat{P}^{N} | {\Phi}\rangle}{\langle {\Phi}| \hat{P}^{N} | {\Phi}\rangle}
\end{eqnarray}
in an obvious way.
The wave functions $ |\tilde{\Phi} \rangle$ and $ |{\Phi} \rangle$ have different number of particles on the average. This can be easily shown assuming the parameter $\alpha$ small enough, in this case
\begin{equation}
\frac{\langle \tilde{\Phi}| \hat{N} | \tilde{\Phi}\rangle} {\langle \tilde{\Phi}| \tilde{\Phi}\rangle}
=
N_0 + 2\alpha \langle \Phi | (\Delta \hat{N})^2 | {\Phi}\rangle.
\label{end_Transfo_N}
\end{equation}
up to $\alpha^2$ terms. Since $|\Phi \rangle$ is per definition a symmetry violating wave function, $\langle \Phi |(\Delta \hat{N})^2 | {\Phi}\rangle\neq 0$, the wave functions $|\Phi \rangle$ and $|\tilde{\Phi} \rangle$ do have on the average different number of particles.
We have therefore demonstrated that we can change the average number of particles of the intrinsic wave function without changing the value of the projected energy: The Lagrange parameter $\lambda$ is therefore superfluous. As a matter of fact, in a VAP approach one uses a Lagrange parameter only to speed up the convergence of the iterative procedure.
It is important to realize that in a projected theory the only meaningful quantities are the projected ones. For example, the intrinsic density $\rho(\vec{r})$ is not invariant under the transformations of Eq. (\ref{Transfo_N}). This is simply the mathematical transcription of the fact that changing the particle number affects the intrinsic density. Conversely the projected density $\rho^N(\vec{r})$ is consistently invariant under the aforementioned transformation.
In the demonstration above, Eqs.(\ref{Transfo_N})-(\ref{end_Transfo_N}) we have assumed that the coefficients of the Bogoliubov transformation (\ref{betas}) are known. In the case of the full VAP approach, this is automatically the case because the $U$ and $V$ matrices are determined self-consistently by minimizing the projected energy, which is invariant under transformations that change the number of particles. In the RVAP approach, however, to determine the $U(\delta)$ and $V(\delta)$ matrices one solves the standard HFB equations with a constraint on the number of particles. The latter equations are obviously not invariant under the transformations of Eq. (\ref{Transfo_N}). This may generate a dependence of the RVAP solution on the Lagrange parameter $\lambda$ (or equivalently on $\langle \hat{N} \rangle$). Obviously
\begin{equation}
\lambda_n (\delta) = \frac{d\langle \Phi (\delta)| \hat{H} |\Phi (\delta)\rangle}{d\langle \Phi (\delta)| \hat{N} |\Phi (\delta)\rangle}
\approx - S_n ^{HFB} (\delta) \neq - S_n ^{PNP} (\delta),
\end{equation}
which illustrates that $\lambda_n$ can not be used to define the one neutron drip line in a PNP approach. It is interesting to realize that this dependence on $\lambda$ could be eventually used to generate additional correlated wave functions $|\Phi(\delta,\lambda)\rangle$ in the RVAP approach, thereby lowering further the projected energy \cite{FE.05}.
\begin{figure}[h]
\includegraphics[height=7.0cm,width=9.0cm]{258Os_Lagrange_Energy.eps}
\includegraphics[height=7.0cm,width=9.0cm]{258Os_Lagrange_Lambda.eps}
\caption{Upper panel: Projected energy for intrinsic wave functions with different average number of particles $N$ around the actual particle number $N_{0} = 182$ in $^{258}$Os. This corresponds to intrinsic wave functions with different Lagrange multipliers. Lower panel: corresponding Fermi level $\lambda$.
}
\label{fig06}
\end{figure}
As an illustration we show in the upper panel of Fig.~\ref{fig06} the projected energy of $^{258}$Os as a function of $\delta$ for different values of the number of particles (or $\lambda$) of the intrinsic wave function. As expected, we find that the minimum of the projected energy does not always correspond to the constraint $\langle N\rangle = N_0$, and for a given constraint on the average particle number, the position of the minimum depends on $\delta$. In the lower panel the corresponding chemical potentials $\lambda (\delta)$ are plotted.
If one restricts oneself to "one-dimensional" RVAP wave functions of the type $|\Phi(\delta)\rangle$, it may happen, in particular near the drip lines, that in the RVAP minimum, the underlying HFB wave function $|\Phi(\delta)\rangle$ corresponds to a positive value of $\lambda$. As emphasized before, this is with no consequence since this $\lambda$ parameter does not define the drip line. If one insists, however, in having a negative Fermi energy, it is always possible to slightly change the average number of particles of the intrinsic wave function in such a way that $\lambda$ becomes negative, with the eventual cost of a small energy loss. In the illustrative case of $^{258}$Os displayed in Fig.~\ref{fig06} the energy cost is approximately 15 keV to go from the RVAP minimum (at $\delta = 1.09$) built on the HFB solution with average number of particles $\langle \hat{N} \rangle = N_{0}$, to the RVAP minimum (at $\delta = 1.06$) with $\langle \hat{N} \rangle = N_{0} - 1.0$. The Fermi energy of the underlying HFB solution goes from +165 keV to -36 keV. The drip line, defined from the two-neutron separation energy $S_{2n}$, remains unchanged.
\subsection{Nuclear halos and drip lines in a symmetry conserving approach}
In this section we investigate the effect of the particle number projection on the size of the halo along the neutron drip line. As mentioned in the Introduction we should distinguish between halos in very light nuclei and in the heavier ones. We are aware that a mean field based approach may not contain enough correlations to describe the halo mechanism in very light nuclei. Nevertheless we shall first discuss the impact of the RVAP procedure on the archetypical case of halo nucleus $^{11}$Li.
\begin{figure}[h]
\includegraphics[height=7.0cm,width=9.0cm]{Li11_WS_RVAP.eps}
\caption{Total projected energy (open squares) and neutron (plain circles) and mass (open triangles) r.m.s radius in $^{11}$Li as a function of the RVAP parameter $\delta$. At each point $\delta$ the HFB solution is projected on good particle number. The minimum is attained at $\delta=1.36$.
}
\label{fig07}
\end{figure}
In the calculation of the nucleus $^{11}$Li, with 3 protons and 8 neutrons, the odd proton was treated in the equal-filling approximation (the 1p$_{1/2}$ state is the blocked state) and only the projection on neutron particle number was carried out. Since the neutron number corresponds to a shell closure it is obvious that the HFB solution is not a super-fluid one. Figure \ref{fig07} shows the total projected energy and the neutron and mass r.m.s radius in $^{11}$Li as function of the RVAP variational parameter $\delta$. All calculations are done in the WS basis with the D1S interaction. The RVAP minimum always corresponds to a paired solution. In $^{11}$Li, our original spherical HFB calculations with the D1S or D1 interactions do not produce any halo. In fact, pairing correlations do not set in at all in this nucleus in the HFB calculations, even when the size of the box is increased up to 30 fm (thereby increasing the level density of continuum states). This is clearly viewed in Fig. \ref{fig07} since the projected energy remains constant at $E^{N} = -47.48 $ MeV for $1.0 \leq \delta \leq 1.18$. In spite of multiplying the pairing field by the factor $\delta$ during the iterations, pairing correlations are still identically 0 at convergence. Only for $\delta>1.18$ do we observe the onset of significant pairing correlations. The total projected energy therefore decreases, continuum states begin to have a non-zero occupation probability, which contributes to the increase of the r.m.s neutron radius. At the minimum of the RVAP curve, both the neutron and mass radius have increased by about 2 \%. The effect is marked but it is clearly not enough to reproduce the experimental halo in this nucleus \cite{halo_11Li}.
\begin{figure}[h]
\includegraphics[height=7.0cm,width=9.0cm]{Li11_Density_WS_RVAP.eps}
\caption{Neutron HFB density $\rho^{HFB}(r)$ in $^{11}$Li at the HFB minimum (dotted line) and at the RVAP minimum (plain line). The dashed line shows the projected density $\rho^{PNP}(r)$ at the RVAP minimum. Calculations are done for the D1S interaction in the WS basis with $R_{box}=20$ fm.
}
\label{fig08}
\end{figure}
To better grasp the impact of particle-number projection we show in Fig. \ref{fig08} the neutron density in $^{11}$Li in 3 different cases. The dotted line corresponds to the standard HFB calculation. The plain line corresponds to the density of the intrinsic HFB wave function $|\Phi(\delta)\rangle$ at the RVAP minimum $\delta=1.36$. We clearly see the formation of a "bump" which is a visual trademark of the nuclear halo. However, this solution is not physical since it is only used to generate the variational space used in the RVAP procedure. Only the projected solution $|\Phi^{N}(\delta=\delta_{min})\rangle$ at the minimum is physical. The corresponding projected density (dashed line) is slightly less extended than the underlying HFB solution.
\begin{table}[h]
\begin{center}
\caption{Table of two neutron drip line nuclei obtained using the D1S parametrization at the HFB and RVAP level.}
\begin{ruledtabular}
\begin{tabular}{cc|cc}
HFB & RVAP & HFB & RVAP \\
\hline
$^{~20}$C & $^{~20}$C & $^{170}$Sn & $^{172}$Sn \\
$^{~26}$O & $^{~26}$O & $^{178}$Te & $^{178}$Te \\
$^{~30}$Ne & $^{~30}$Ne & $^{180}$Xe & $^{180}$Xe \\
$^{~40}$Mg & $^{~42}$Mg & $^{182}$Ba & $^{182}$Ba \\
$^{~46}$Si & $^{~46}$Si & $^{184}$Ce & $^{184}$Ce \\
$^{~50}$S & $^{~50}$S & $^{186}$Nd & $^{186}$Nd \\
$^{~58}$Ar & $^{~58}$Ar & $^{188}$Sm & $^{188}$Sm \\
$^{~64}$Ca & $^{~62}$Ca & $^{192}$Gd & $^{192}$Gd \\
$^{~72}$Ti & $^{~72}$Ti & $^{200}$Dy & $^{200}$Dy \\
$^{~78}$Cr & $^{~74}$Cr & $^{206}$Er & $^{206}$Er \\
$^{~84}$Fe & $^{~84}$Fe & $^{220}$Yb & $^{220}$Yb \\
$^{~86}$Ni & $^{~86}$Ni & $^{242}$Hf & $^{242}$Hf \\
$^{~94}$Zn & $^{~94}$Zn & $^{254}$W & $^{254}$W \\
$^{104}$Ge & $^{104}$Ge & $^{258}$Os & $^{258}$Os \\
$^{114}$Se & $^{114}$Se & $^{260}$Pt & $^{262}$Pt \\
$^{118}$Kr & $^{118}$Kr & $^{264}$Hg & $^{264}$Hg \\
$^{120}$Sr & $^{120}$Sr & $^{266}$Pb & $^{266}$Pb \\
$^{122}$Zr & $^{122}$Zr & $^{268}$Po & $^{268}$Po \\
$^{130}$Mo & $^{130}$Mo & $^{270}$Rn & $^{270}$Rn \\
$^{136}$Ru & $^{136}$Ru & $^{272}$Ra & $^{272}$Ra \\
$^{140}$Pd & $^{140}$Pd & $^{274}$Th & $^{274}$Th \\
$^{152}$Cd & $^{152}$Cd & $^{278}$U & $^{278}$U \\
& & $^{284}$Pu & $^{282}$Pu \\
\end{tabular}
\label{table02}
\end{ruledtabular}
\end{center}
\end{table}
Figures \ref{fig07} and \ref{fig08} suggest that the impact of particle-number projection may be instrumental in the formation of sizeable halos, since the RVAP mechanism always guarantees a solution with non-zero pairing correlations. Since the chemical potential is irrelevant in a projected theory we should therefore, in principle, compute this quantity using {\it projected} energies and compare it from the results obtained using un-projected quantities. As emphasized earlier, the application of particle number projection in odd nuclei is not possible at the moment, hence the one neutron RVAP drip line is not accessible.
We therefore carried out systematic RVAP calculations of the two neutron separation energies, $S_{2n}$, near the drip line using the D1S interaction. The particular choice of the interaction is secondary in this study, since the focus is on the particular role of particle number projection. The procedure was as follows: for a given drip line element $(Z,N)$ from Table \ref{table01} the isotopes with $N-4$, $N-2$, $N$ and $N+2$ neutrons were considered. For each isotope, the RVAP procedure was carried with $\delta=1.0, 1.05, \dots, 1.50$. The minimum of the RVAP curve was retained as the physical solution for every isotope. The two-neutron separation energy was calculated from the total RVAP-projected energies: $S_{2n} = B^{PNP}(N,Z) - B^{PNP}(N-2,Z)$. The criterion $S_{2n} < 0$ was used to define the position of the new drip line. Table \ref{table02} shows the two-neutron drip line nuclei with and without the particle-number projection. These two drip lines differ by the isotopes of 6 elements: $^{42}$Mg$_{30}$, $^{62}$Ca$_{42}$, $^{74}$Cr$_{50}$, $^{172}$Sn$_{122}$, $^{262}$Pt$_{184}$ and $^{282}$Pu$_{188}$. As we can read in the neutron number of these nuclei the differences arise always close to the shell closures, where the pairing correlations are either very weak or vanishing.
For all the elements located at the RVAP drip line, the Helm radius was computed, for the protons and the neutrons, based on the projected density $\rho^{PNP}(r)$. The quantity $\delta R_{halo}^{PNP}(n) = R_{geom}^{PNP}(n) - R_{Helm}^{PNP}(n)$ obtained from these calculations is reported in Fig. \ref{fig09}, together with the original $\delta R_{halo}(n)$ of the two un-projected drip lines ($S_{2n}$ and $S_{n}$ drip line).
\begin{figure}[h]
\includegraphics[height=7.0cm,width=9.0cm]{deltaHalo_RVAP.eps}
\caption{Measure of the halo: $\delta R_{halo}(n) = R_{geom}(n) - R_{Helm}(n)$ for RVAP-projected (plain squares), $S_{2n}$-unprojected (open circles) and $S_{n}$-unprojected (plain circles) drip lines. All results are based on spherical Gogny HFB calculations in the WS basis with the D1S interaction.
}
\label{fig09}
\end{figure}
The impact of particle-number projection is only significant in those nuclei that are unbound at the HFB level but bound in the RVAP-HFB. As can be seen from Table \ref{table02}, there are many nuclei that are particle-unstable ($-S_{1n} \approx \lambda_{n} > 0$) but two-particle-stable ($S_{2n} < 0$). In such cases, the halo is of course larger, sometimes significantly larger like e.g. Cr or Fe, than the corresponding particle-stable isotope. Moreover the value of the halo calculated at the RVAP minimum closely follows the one calculated at the $S_{2n}$ drip line. The case of Cr is singular, in that the RVAP mechanism changes the two neutron drip line by 4 units, thereby considerably lowering the halo.
However, beyond $Z\approx 30$, the differences between all approaches become relatively negligible. This goes along a very clear and definite trend towards smaller halos as the mass of the nucleus increases. Combining this observation with the fact that our mean-field approach, which includes the continuum and uses the best possible treatment of pairing correlations, fails to produce halos in light nuclei, it is tempting to conclude that halos are a trademark of few-body correlations only. As was recognized early, pairing correlations are a pre-requisite to the formation of halos in a mean-field approach indeed. However, our work seems to further indicate that additional correlations beyond the symmetry conserving mean-field approximation are also mandatory.
Figure \ref{fig08} may suggest that for RVAP solutions the projected density profile is markedly different from the unprojected density. As mentioned already, only the projected density bears a physical meaning by construction. The underlying HFB solution is only used to generate a set of highly pair-correlated projected wave functions. Nevertheless one may compare the behavior of $\delta R_{halo}$ when using either the projected density $\rho^{PNP}(r)$ in the RVAP minimum or the underlying un-projected HFB density $\rho^{HFB}(r)$ in this same minimum to visualize the impact of projection itself. The difference in $\delta R_{halo}$ is practically negligible (less than 0.01 fm) and certainly not on the same scale as differences coming from the interaction.
We have also applied the RVAP formalism to the calculation of neutron skins and we do not find any remarkable difference as compared with the HFB ones. This confirms the observation that neutron skins are, from a theoretical point of view, mostly sensitive to the details of the interaction (iso-scalar vs. iso-vector content), and from an experimental point of view, to the neutron excess but are not directly affected by the vicinity of the continuum.
In conclusion, we applied our method to include continuum effects in spherical self-consistent HFB calculations with finite-range forces of the Gogny type to the case of nuclear halos and skins. Our calculations show that both the D1 and D1S parametrizations of the Gogny force lead to relatively small mean-field halos, that are of comparable size to most of the results obtained in Skyrme-HFB or relativistic Hartree-Bogoliubov theories. In particular, we do not find the giant halos in Neon and Zirconium isotopes that were reported in several publications. As a rule of thumb we observe that the size of the halo tends to decrease as the mass of the nucleus increases and only light nuclei feature decent-sized halos. By contrast, neutron skins are found to be very clearly related to the ratio (N-Z)/A.
We also show that the impact of particle number projection, before variation, is relatively important since it can change the position of the drip line.
However, we find that particle-number projected continuum-coupled HFB theory, employing the most realistic form of the pairing interaction, cannot reproduce the large halos observed experimentally in very light nuclei such as $^{11}$Li. This suggests a series of necessary conditions for a successful description of nuclear halos in the framework of mean-field theory: (i) the continuum must be properly included in the formalism (ii) the shell structure must be realistic enough (iii) pairing correlations must be present (iv) symmetry-breaking mean-field calculations, including all relevant deformation degrees of freedom, are probably mandatory (v) all such broken symmetries (in particular particle number) should then be restored (vi) probably configuration mixing such as e.g. GCM should also be included.
{\bf Acknowledgment - } One of us (N.S.) acknowledges financial support of the spanish Ministerio de Educacion y Ciencia (Ref. SB2004-0024). This work has been supported in part by the Spanish Ministerio de Educaci\'on y Ciencia under contract FPA2007-66069, by the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042) as well as by the U.S. Department of Energy under Contract Nos. DE-FC02-07ER41457 (University of Washington), DEFG02- 96ER40963 (University of Tennessee), and DE-AC05- 00OR22725 with UT-Battelle, LLC (Oak Ridge National Laboratory).
|
2,877,628,090,068 | arxiv | \section{Expected behavior of Bayesian model average on InD/OOD uncertainty quantification}
\label{sec:gp_uncertainty}
As a motivating example, we consider uncertainty quantification on InD and OOD data using a Bayesian model average, and relate our findings back to the implications presented in \cref{sec:implications_epis_alea}. An ideal Bayesian model average should express higher posterior variance on OOD data than InD data, even after controlling for other sources of uncertainty.
To demonstrate this desired behavior in practice, we consider Gaussian processes, a class of models well regarded for its uncertainty quantification capabilities \cite{williams2006gaussian}.
The Gaussian process model $f(\cdot)$ is defined by the following generative process:
\begin{equation}
\begin{split}
p(f(\cdot)) &= \mathcal{GP}, \\
p(y \mid f(x)) &= \mathcal{N}(0, \sigma^2(x))
\end{split}
\label{eqn:gp_process}
\end{equation}
where $\sigma^2(x)$ is a heteroskedastic noise function defined as $\sigma^2(x) = \sin^2(x) + 0.01$.
After conditioning on training data $\mathcal D$, the BMA at a test point $x$ is given by:
\begin{equation}
p(y \mid x, \mathcal D) = \mathcal N( \mu_{f \mid \mathcal D}(x), \mathrm{Var}_{f \mid \mathcal D}(x) + \sigma^2(x) ),
\label{eqn:gp_posterior}
\end{equation}
where $\mu_{\mid \mathcal D}(\cdot)$ and $\mathrm{Var}_{f \mid \mathcal D}(\cdot)$ are the posterior predictive GP mean and variance, respectively, which can both be computed in closed form.
(See \cite{williams2006gaussian} for closed-form expressions for these two functions).
Crucially, the predictive variance in \cref{eqn:gp_posterior} is a uncertainty estimate that decomposes into epistemic and aleatoric components:
the {\bf posterior variance} term ($\mathrm{Var}_{f \mid \mathcal D}(\cdot)$)
and the {\bf likelihood variance} term ($\sigma^2(x)$), respectively.
\begin{figure}[t!]
\centering
\includegraphics[width=\linewidth]{figs/gp_conditional_example.pdf}
\caption{
Example of a model where OOD predictions have higher posterior variance, even after controlling for other sources of uncertainty.
{\bf Left:} The predictive uncertainty expressed by a Gaussian process model on a toy regression dataset.
OOD data (orange) express higher posterior variance than InD data (blue).
{\bf Right:} The expected posterior variance ) conditioned on a prediction's likelihood variance is also significantly larger for OOD data.
}
\label{fig:gp}
\end{figure}
In \cref{fig:gp} (left), we generate a one-dimensional dataset by drawing 25 random data points over $x \in [0, 5]$ using the generative process defined in \cref{eqn:gp_process}.\footnote{
In all experiments, the prior GP model has zero mean and a RBF covariance function with a lengthscale of 1.
}
After fitting a GP model to these data, we compute the predictive posterior over the range $x \in [-5, 5]$.
The points in $[0, 5]$ represent InD data---as they share the same domain as the training data---while the points in $[-5, 0]$ (orange) represent OOD data.
In \cref{fig:gp} (right), we observe that OOD predictions have much higher expected posterior variance, even after conditioning on a prediction's likelihood uncertainty.
Note that this is in stark contrast to the analogous deep ensemble results in \cref{sec:uncertainty}, where there is little to no conditional difference between OOD and InD predictions.
\section{Scratch}
Why do we see that the pattern of ensemble generalization is so well determined by single model generalization?
We’d like to be able to understand this phenomenon at the level of the data distribution next.
\paragraph{Notation.} \gp{Move this paragraph somewhere else.}
Let ${\bm{x}} \in \mathbb{R}^D$ be an input and $y \in [1, C]$ be its target, where $D$ is the number of features and $C$ is the number of classes.
A single model ${\bm{f}} : \mathbb{R}^D \to \Delta^C$ maps an input to the $C$-class probability simplex.
Given a set of models ${\bm{f}}_1, \ldots, {\bm{f}}_M$,
let $\bar {\bm{f}}({\bm{x}}) = 1/M \sum_{i=1}^M {\bm{f}}_i({\bm{x}})$ represent the ensemble of models.
We will also represent ensemble members as a discrete distribution over possible models:
$p({\bm{f}}) = \text{Unif.} [ {\bm{f}}_1, \ldots, {\bm{f}}_M ] $.
One reasonable definition for the total uncertainty of an ensemble is $U({\bm{x}}) = 1 - \Vert \bar{\bm{f}}({\bm{x}}) \Vert_2^2$.
$U({\bm{x}})$ will be small when all ensemble members agree and express high confidence in a single class.
Conversely, $U({\bm{x}})$ will be large when ensemble members disagree, or when ensemble members express low confidence in all classes.
Furthermore, $U({\bm{x}})$ can be decomposed into interpretable terms:
\begin{align}
U({\bm{x}}) &\triangleq 1 - \Vert \bar{\bm{f}}({\bm{x}}) \Vert_2^2
\nonumber \\
&= \underbracket{
\left( \E_{{\bm{f}}} \left[ \Vert {\bm{f}}({\bm{x}}) \Vert_2^2 \right] - \Vert \bar{\bm{f}}({\bm{x}}) \Vert_2^2 \right)
}_{\Var[ \Vert {\bm{f}}({\bm{x}}) \Vert_2 ]}
\label{eqn:sq_epistemic1}
\\
&\phantom{=} + \E_{{\bm{f}}} \left[ \left( 1 - \Vert {\bm{f}}({\bm{x}}) \Vert_2^2 \right) \right]
\label{eqn:sq_aleatoric1}
\end{align}
Note that \cref{eqn:sq_aleatoric} measures disagreement between ensembles,
and thus can be interpreted as a notion of epistemic uncertainty.
Conversely, \cref{eqn:sq_epistemic} captures the average aleatoric uncertainty expressed by ensemble members.
Consider the average Brier score across individual models:
\begin{align}
\E_{\bm{f}} \left[ B_p({\bm{f}}) \right] &= \E_{p({\bm{x}}, y)} \E_{\bm{f}} \left[ \Vert {\bm{f}}({\bm{x}}) - {\bm{1}}_y \Vert_2^2 \right]
\label{eqn:avg_brier} \\
&= \E_{p({\bm{x}}, y)} \left[
\E_{\bm{f}} \left[ \Vert {\bm{f}}({\bm{x}}) \Vert_2^2 \right]
+ 2 \bar {\bm{f}}({\bm{x}})^\top {\bm{1}}_y
+ 1
\right]
\nonumber
\end{align}
and the Brier score of the ensemble:
\begin{align}
B_p(\bar {\bm{f}}) &= \E_{p({\bm{x}}, y)} \left[ \Vert \bar {\bm{f}}({\bm{x}}) - {\bm{1}}_y \Vert_2^2 \right]
\label{eqn:ens_brier} \\
&= \E_{p({\bm{x}}, y)} \left[
\Vert \bar {\bm{f}}({\bm{x}}) \Vert_2^2
+ 2 \bar {\bm{f}}({\bm{x}})^\top {\bm{1}}_y
+ 1
\right]
\nonumber
\end{align}
Note that \cref{eqn:avg_brier} and \cref{eqn:ens_brier} only differ by a single term:
\begin{align*}
B_p(\bar {\bm{f}}) &= \E_{\bm{f}} \left[ B_p({\bm{f}}) \right]
+ \E_{p({\bm{x}})} \left[ \Vert \bar {\bm{f}}({\bm{x}}) \Vert_2^2 \right]
\\ &\phantom{=}
- \E_{p({\bm{x}})} \E_{\bm{f}} \left[ \Vert {\bm{f}}({\bm{x}}) \Vert_2^2 \right].
\end{align*}
Now assume that we see a linear relationship between InD and OOD Brier score for individual models:
\[
B_q(\bar {\bm{f}}) = c_0 B_p(\bar {\bm{f}}) + c_1
\]
where $p({\bm{x}}, y)$ and $q({\bm{x}}, y)$ represent the InD and OOD data distributions, respectively,
and $c_0$ and $c_1$ represent some constants.
Consequentially, the OOD Brier score can be written as a linear combination of the InD terms in \cref{eqn:avg_brier}.
\begin{align*}
B_q(\bar {\bm{f}})
= c'_0 \E_{p({\bm{x}})} \left[
\E_{\bm{f}} \left[ \Vert {\bm{f}}({\bm{x}}) \Vert_2^2 \right]
\right]
\\
+ c'_1 \E_{p({\bm{x}}, y)} \left[
\bar {\bm{f}}({\bm{x}})^\top {\bm{1}}_y
\right]
+ c'_2,
\end{align*}
where $c'_0 = $
\section{Details of models for robustness experiments}
\label{sec:app_training_details}
We followed many of the same experimental procedures as \cite{miller2021accuracy} in order to generate ensembles for our experiments. We denote four main groups of models below:
\subsection{CIFAR10 models trained from scratch}
We trained 10 different classes of models on CIFAR10, noted below. We used implementations from \url{https://github.com/huyvnphan/PyTorch_CIFAR10} in order to train convolutional models adapted for CIFAR10 data sizes, with default hyperparameters, and manually extended existing implementations in this repo to create a WideResNet 18 with width 4.
\begin{itemize}
\item ResNet 18 \cite{he2016deep}
\item WideResNet 18-2, 18-4, 28-10 \cite{zagoruyko2016wide}
\item GoogleNet, Inception v3 \cite{szegedy2015going}
\item VGG with 11 and 19 layers \cite{simonyan2014very}
\item DenseNet 121 and 169 \cite{huang2017densely}
\end{itemize}
We trained five independent instances of each of these architectures with random seeds for 100 epochs each (see code repo defaults for other hyper parameters.)
\subsection{CIFAR10 pretrained ensembles}
We use the models trained by \citet{miller2021accuracy}, and we thank the authors for graciously sharing these results with us.
\subsection{ImageNet models trained from scratch}
We additionally trained two sets of ensembles from scratch on the ImageNet dataset. In particular, we trained 5 model ensembles of AlexNet and ResNet 101 models using implementations available at \url{https://pytorch.org/vision/stable/models.html} for 90 epochs each.
\subsection{Imagenet pretrained models}
We use 5 of the ResNet50 models trained by \cite{ashukha2020pitfalls} and the standard 78 trained models provided by \citet{taori2020measuring}.
\section{Additional generalization trend results}
\label{sec:additional_ltrend}
In this section, we report test statistics for the results we show in \cref{fig:ltrend_metrics}, and we extend the results from \cref{fig:ltrend_metrics} to additional OOD datasets, namely CIFAR10.1 and ImageNet-C \cite{hendrycks2018benchmarking}, illustrating generalization trends for ensembles and individual models for various distortions at different intensity levels. The results in this section show that for high intensity distortions, single models can break away from a well defined linear trend, as reported in \cite{miller2021accuracy}. However, even at the highest distortion levels, the generalization performance for ensembles and individual models heavily overlap, suggesting the lack of effective robustness demonstrated by deep ensembles is not dependent upon the same phenomena that generate strong trends in single models to begin with.
\subsection{Test statistics for generalization performance trends}
\label{sec:ltrend_r2_tables}
In each table we report the regression coefficient (Coefficient), the standard error (Std. error) t-statistic, p-value and $R^2$ to reject the null hypothesis that there is no relation between InD and OOD performance for the different metrics considered (left column). The last column indicates the number of models (markers) for each model class depicted in Fig~\ref{fig:ltrend_metrics}.
Note that we do not apply logit scaling to our axes as in \citep{taori2020measuring}, which was found to increase the fit of linear trend lines. Furthermore, we do not consider non-linear parametrizations of NLL, which could potentially improve the quantification of overlap between single models and ensembles. We consider such parameterizations to be beyond the scope of this work.
\begin{table*}[hbt!]
\centering{
\caption{\textbf{ $R^2$ for InD vs OOD generalization trend fits for different metrics}: CIFAR10 vs CINIC10 in \cref{fig:ltrend_cinic10}.}
\resizebox{.9\textwidth}{!}{%
\begin{tabular}{llrrrrrc}
\toprule
& & Coefficient & Std. error & t-statistic & p-value & R\textasciicircum 2 & Number of models \\
Metric & Type & & & & & & \\
\midrule
\multirow{3}{*}{0-1 Error} & All & 0.038 & 0.002 & 18.981 & 0.0 & 0.853 & 434 \\
& Single Model & 0.029 & 0.006 & 5.038 & 0.0 & 0.883 & 54 \\
& Ensemble & 0.039 & 0.002 & 18.349 & 0.0 & 0.848 & 380 \\
\cline{1-8}
\multirow{3}{*}{NLL} & All & 0.116 & 0.006 & 18.285 & 0.0 & 0.894 & 434 \\
& Single Model & 0.120 & 0.022 & 5.511 & 0.0 & 0.864 & 54 \\
& Ensemble & 0.116 & 0.007 & 17.559 & 0.0 & 0.896 & 380 \\
\cline{1-8}
\multirow{3}{*}{Brier} & All & 0.051 & 0.003 & 17.415 & 0.0 & 0.876 & 434 \\
& Single Model & 0.042 & 0.009 & 4.754 & 0.0 & 0.890 & 54 \\
& Ensemble & 0.052 & 0.003 & 16.754 & 0.0 & 0.873 & 380 \\
\cline{1-8}
\multirow{3}{*}{rESCE} & All & 0.009 & 0.002 & 4.712 & 0.0 & 0.791 & 434 \\
& Single Model & 0.026 & 0.007 & 3.755 & 0.0 & 0.632 & 54 \\
& Ensemble & 0.007 & 0.002 & 3.860 & 0.0 & 0.801 & 380 \\
\bottomrule
\end{tabular}
}
}
\label{tab:r2_cinic10}
\end{table*}
\begin{table*}[!ht]
\centering{
\caption{\textbf{ $R^2$ for InD vs OOD generalization trend fits for different metrics}: ImageNet vs ImageNetV2 in \cref{fig:ltrend_imagenetv2}.}
\resizebox{.9\textwidth}{!}{%
\begin{tabular}{llrrrrrc}
\toprule
& & Coefficient & Std. error & t-statistic & p-value & R\textasciicircum 2 & Number of models \\
Metric & Type & & & & & & \\
\midrule
\multirow{3}{*}{0-1 Error} & All & 0.102 & 0.001 & 89.935 & 0.0 & 0.995 & 367 \\
& Single Model & 0.105 & 0.002 & 43.326 & 0.0 & 0.994 & 93 \\
& Ensemble & 0.101 & 0.001 & 78.643 & 0.0 & 0.995 & 274 \\
\cline{1-8}
\multirow{3}{*}{NLL} & All & 0.432 & 0.008 & 54.749 & 0.0 & 0.989 & 367 \\
& Single Model & 0.443 & 0.018 & 24.091 & 0.0 & 0.984 & 93 \\
& Ensemble & 0.428 & 0.009 & 49.622 & 0.0 & 0.991 & 274 \\
\cline{1-8}
\multirow{3}{*}{Brier} & All & 0.156 & 0.002 & 77.827 & 0.0 & 0.989 & 367 \\
& Single Model & 0.159 & 0.005 & 34.540 & 0.0 & 0.985 & 93 \\
& Ensemble & 0.156 & 0.002 & 69.984 & 0.0 & 0.991 & 274 \\
\cline{1-8}
\multirow{3}{*}{rESCE} & All & 0.060 & 0.003 & 19.723 & 0.0 & 0.111 & 367 \\
& Single Model & 0.067 & 0.006 & 10.871 & 0.0 & 0.090 & 93 \\
& Ensemble & 0.058 & 0.004 & 16.342 & 0.0 & 0.113 & 274 \\
\bottomrule
\end{tabular}%
}
}
\label{tab:r2_imagenetv2}
\end{table*}
\begin{table*}[htb!]
\centering{
\caption{\textbf{ $R^2$ for InD vs OOD generalization trend fits for different metrics}: CIFAR10 vs CIFAR10.1 in \cref{fig:ltrend_cifar10.1}}
\resizebox{.9\textwidth}{!}{%
\begin{tabular}{llrrrrrc}
\toprule
& & Coefficient & Std. error & t-statistic & p-value & R\textasciicircum 2 & Number of models \\
Metric & Type & & & & & & \\
\midrule
\multirow{3}{*}{0-1 Error} & All & 0.038 & 0.002 & 18.981 & 0.0 & 0.853 & 434 \\
& Single Model & 0.029 & 0.006 & 5.038 & 0.0 & 0.883 & 54 \\
& Ensemble & 0.039 & 0.002 & 18.349 & 0.0 & 0.848 & 380 \\
\cline{1-8}
\multirow{3}{*}{NLL} & All & 0.116 & 0.006 & 18.285 & 0.0 & 0.894 & 434 \\
& Single Model & 0.120 & 0.022 & 5.511 & 0.0 & 0.864 & 54 \\
& Ensemble & 0.116 & 0.007 & 17.559 & 0.0 & 0.896 & 380 \\
\cline{1-8}
\multirow{3}{*}{Brier} & All & 0.051 & 0.003 & 17.415 & 0.0 & 0.876 & 434 \\
& Single Model & 0.042 & 0.009 & 4.754 & 0.0 & 0.890 & 54 \\
& Ensemble & 0.052 & 0.003 & 16.754 & 0.0 & 0.873 & 380 \\
\cline{1-8}
\multirow{3}{*}{rESCE} & All & 0.009 & 0.002 & 4.712 & 0.0 & 0.791 & 434 \\
& Single Model & 0.026 & 0.007 & 3.755 & 0.0 & 0.632 & 54 \\
& Ensemble & 0.007 & 0.002 & 3.860 & 0.0 & 0.801 & 380 \\
\bottomrule
\end{tabular}%
}
}
\label{tab:r2_cifar10.1}
\end{table*}
\clearpage
\subsection{Evaluation on other datasets}
\label{sec:ltrend_other_datasets}
In this section we follow the same conventions as in \cref{fig:ltrend_metrics} to analyze the generalization performance for two other OOD datasets for CIFAR10 and ImageNet, namely CIFAR10.1 and ImageNetC \cite{hendrycks2018benchmarking}. For ImageNetC we focus on our distortions from this dataset; namely brightness, contrast, fog and gaussian noise for three different degrees of corruption.
\begin{figure*}[!ht]
\centering
\captionsetup{justification=centering}
\subfloat{\includegraphics[width=1\textwidth]{figs/cifar10.1_metrics_ensemble_all.pdf}
}
\caption{Generalization Trends for CIFAR10 vs CIFAR10.1.\\ Conventions and conclusions as in \cref{fig:ltrend_metrics}.}
\label{fig:ltrend_cifar10.1}
\end{figure*}
\begin{figure*}[!ht]
\centering
\captionsetup{justification=centering}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetc--brightness--1_metrics_ensemble_all.pdf}
}
\hfill
\vspace{-0.35in}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetc--brightness--3_metrics_ensemble_all.pdf}%
}
\hfill
\vspace{-0.2in}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetc--brightness--5_metrics_ensemble_all.pdf}%
}
\caption{Generalization Trends for ImageNet vs ImageNet-C Brightness-1, 3 and 5.\\ Conventions and conclusions as in \cref{fig:ltrend_metrics}.}
\label{fig:imagenetc_brightness1_metrics}
\end{figure*}
\begin{figure*}[!ht]
\centering
\captionsetup{justification=centering}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetc--contrast--1_metrics_ensemble_all.pdf}
}
\hfill
\vspace{-0.35in}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetc--contrast--3_metrics_ensemble_all.pdf}%
}
\hfill
\vspace{-0.2in}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetc--contrast--5_metrics_ensemble_all.pdf}%
}
\caption{Generalization Trends for ImageNet vs ImageNet-C Contrast-1, 3 and 5.\\ Conventions and conclusions as in \cref{fig:ltrend_metrics}.}
\label{fig:imagenetc_contrast1_metrics}
\end{figure*}
\begin{figure*}[!ht]
\centering
\captionsetup{justification=centering}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetc--fog--1_metrics_ensemble_all.pdf}%
}
\hfill
\vspace{-0.35in}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetc--fog--3_metrics_ensemble_all.pdf}%
}
\hfill
\vspace{-0.2in}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetc--fog--5_metrics_ensemble_all.pdf}%
}
\caption{Generalization Trends for ImageNet vs ImageNet-C Fog-1,3, and 5.\\ Conventions and conclusions as in \cref{fig:ltrend_metrics}.}
\label{fig:imagenetc_fog1_metrics}
\end{figure*}
\begin{figure*}[!ht]
\centering
\captionsetup{justification=centering}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetc--gaussian_noise--1_metrics_ensemble_all.pdf}%
}
\hfill
\vspace{-0.35in}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetc--gaussian_noise--3_metrics_ensemble_all.pdf}%
}
\hfill
\vspace{-0.2in}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetc--gaussian_noise--5_metrics_ensemble_all.pdf}%
}
\caption{Generalization Trends for ImageNet vs ImageNet-C Gaussian Noise-1,3, and 5.\\ Conventions and conclusions as in \cref{fig:ltrend_metrics}.}
\label{fig:imagenetc_gaussiann1_metrics}
\end{figure*}
\clearpage
\subsection{Calibration metrics}
\label{sec:app_ece_all}
The calibration error is a frequentist idea to measure the quality of uncertainty given by a model. The calibration error is given by,
\begin{align*}
\text{Calibration Error} = | Confidence - Accuracy |
\label{eq:calibration_error}
\end{align*}
To compare the calibration error across multiple models, practitioners have resorted to the expected calibration error (ECE) \cite{naeini2015obtaining}, which approximates the calibration error by binning the predictive probabilities and taking a weighted average of the calibration errors across bins. ECE provides a scalar summary statistic of the quality of uncertainty \cite{guo2017calibration}. We employ both the ECE and, a smooth approximation, the root of the Expected Squared Calibration Error (rESCE) to compare the quality of uncertainty gained by ensembling over individual models. The rESCE is defined as,
\begin{align*}
\text{rESCE} = \sqrt{\sum \frac{\mid B_m\mid}{n}[acc(B_m)-conf(B_m)]^2}
\end{align*}
\begin{figure*}[h!]
\centering
\captionsetup{justification=centering}
\subfloat{\includegraphics[width=0.9\textwidth]{figs/ece_metrics_ensemble_all.pdf}
}
\vspace{-0.1in}
\caption{Generalization trends for the Expected Calibration Error (ECE) for different datasets.\\ Conventions and conclusions similar to the rESCE (right column) in \cref{fig:ltrend_metrics}.}
\label{fig:ece_all}
\end{figure*}
\cref{fig:ltrend_metrics} provides the rESCE for in distribution vs out of distribution for CIFAR10 vs CIFAR10.1 and Imagenet vs ImagenetV2. (See \cref{sec:ltrend_other_datasets} for additional datasets). In \cref{fig:ece_all}, we evaluate the generalization performance in terms of the ECE, following the conventions in \cref{fig:ltrend_metrics}.
From \cref{fig:ece_all} we see that there is no clear trend for InD versus OOD generalization across different datasets.
Furthermore, we find that---for CIFAR10/CIFAR10.1---ensembles are able to achieve some amount of effective robustness with respect to the ECE metric.
However, for the other two dataset pairs, we find that the ECE performance of ensembles heavily overlaps with the ECE performance of single models for most models.
\if 0
\subsection{Comparison Between Homogeneous and Heterogeneous Deep Ensembles}
\label{sec:app_heter_ensemble}
In this section we split the data from the ensemble model class in \cref{fig:ltrend_metrics} into two sub classes: an ensemble class which now contains only the homogeneous ensembles, and the heterogeneous class; to explore if the ensemble model classes provide different generalization trends. We find that this is not the case for several in distribution and out of distribution pairs illustrated in \cref{fig:het_ensembles_metrics}.
\begin{figure*}[h!]
\centering
\captionsetup{justification=centering}
\subfloat{\includegraphics[width=1\textwidth]{figs/cifar10.1_metrics_ensemble_heter.pdf}
}
\hfill
\vspace{-0.35in}
\subfloat{\includegraphics[width=1\textwidth]{figs/cinic10_metrics_ensemble_heter.pdf}%
}
\hfill
\vspace{-0.2in}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetv2mf_metrics_ensemble_heter.pdf}%
}
\caption{Homogeneous and heterogeneous deep ensembles in \cref{fig:ltrend_metrics} follow similar generalization trends. Conventions and conclusions as in \cref{fig:ltrend_metrics}.}
\label{fig:het_ensembles_metrics}
\end{figure*}
\begin{figure*}[!ht]
\centering
\captionsetup{justification=centering}
\subfloat{\includegraphics[width=1\textwidth]{figs/ece_metrics_ensemble_heter.pdf}
}
\caption{Expected Calibration Error (ECE) for homogeneous and heterogeneous ensembles follow similar trends as rESCE in \cref{fig:ltrend_metrics}.}
\label{fig:ece_heter}
\end{figure*}
\fi
\subsection{Comparison between homogeneous, heterogeneous and implicit deep ensembles}
\label{sec:app_heter_ensemble}
In this section we split the data from the ensemble model class in \cref{fig:ltrend_metrics} into two sub classes: an ensemble class which now contains only the homogeneous ensembles, and the heterogeneous class; to explore if the ensemble model classes provide different generalization trends. We find that this is not the case for several in distribution and out of distribution pairs illustrated in \cref{fig:het_ensembles_metrics}.
\ekb{We were additionally interested in evaluating whether implicit deep ensembles, models which aim to bridge the gap between individual networks and deep ensembles such as MC Dropout \cite{gal2016dropout}, Batch Ensemble \cite{wen2020batchensemble}, and MIMO \cite{havasi2021training}, also follow the same observed trends. We include the performance of implicit ensembles, including MC Dropout and MIMO, in \cref{fig:het_ensembles_metrics}. The implicit deep ensemble models for ImageNet were constructed from a Resnet50 architecture, which was selected given its ubiquitous deployment, and availability in the open source Uncertainty Baselines library \cite{nado2021uncertainty}. The number of implicit models considered for ImageNet is 12 (2 MC dropout models, and 10 for MIMO models). The implicit deep ensemble models for CIFAR10 were constructed from a WideResnet-28 architecture. In total, we considered 6 implicit ensemble models (3 MC dropout models and 3 MIMO models). Overall the results illustrated in \cref{fig:het_ensembles_metrics} show that implicit deep ensembles also fall on the line, along with individual models, heterogeneous, and homogeneous ensembles, and do not constitute an effectively robust model class.}
\begin{figure*}[h!]
\centering
\captionsetup{justification=centering}
\subfloat{\includegraphics[width=1\textwidth]{figs/cifar10.1_metrics_ensemble_implicit.pdf}
}
\hfill
\vspace{-0.35in}
\subfloat{\includegraphics[width=1\textwidth]{figs/cinic10_metrics_ensemble_heter.pdf}%
}
\hfill
\vspace{-0.2in}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetv2mf_metrics_ensemble_implicit.pdf}%
}
\caption{Homogeneous and heterogeneous deep ensembles in \cref{fig:ltrend_metrics} follow similar generalization trends. Conventions and conclusions as in \cref{fig:ltrend_metrics}, including Implicit Deep ensemble models.}
\label{fig:het_ensembles_metrics}
\end{figure*}
\begin{figure*}[!ht]
\centering
\captionsetup{justification=centering}
\subfloat{\includegraphics[width=1\textwidth]{figs/ece_metrics_ensemble_heter.pdf}
}
\caption{Expected Calibration Error (ECE) for homogeneous and heterogeneous ensembles follow similar trends as rESCE in \cref{fig:ltrend_metrics}.}
\label{fig:ece_heter}
\end{figure*}
\clearpage
\clearpage
\section{Code, data and compute}
\label{sec:app_code_data}
\subsection{Code and data}
We provide general directions to reproduce the main results of our paper in the linked directions here: \url{https://github.com/cellistigs/interp_ensembles#readme}.
These directions reference two repositories, corresponding to two separate branches of our codebase. The ``compare\_performance" branch can be found here: \url{https://github.com/cellistigs/interp_ensembles/tree/compare_performance}. Likewise, the ``imagenet\_pl" branch can be found here: \url{https://github.com/cellistigs/interp_ensembles/tree/imagenet_pl}.
This code repository, together with the instructions provided above, specify all training and visualization details relevant to our study.
Finally, we share relevant data as a Zenodo repository: \url{https://zenodo.org/record/6582653#.Yo7R0y-B3fZ}. This data provides the logit outputs from individual models (and some ensembles) on the in and out of distribution data that we consider. These data are referenced in the code above.
\subsection{Compute}
We ran all CIFAR10 model training on Amazon Web Services (AWS), using the ``p3.2xlarge" instance type with a Tesla V100 GPU. We ran half of ImageNet model training on an internal cluster with GeForce RTX 2080 Ti GPUs, and the other half on AWS with the ``p3.8xlarge" instance type, again with Tesla V100 GPUs. Visualization and statistical testing was run on M1 MacBook Airs, and additionally on AWS ``p3.2xlarge" and ``p3.8xlarge" instances when additional capacity was required.
We show results for 50 models trained on CIFAR10, and 15 models trained on ImageNet. We estimate that on average, our CIFAR10 models required 3 hours of compute to train, and our ImageNet models required 48 hours to train. Finally, we estimate an additional 8 hours of compute required to run statistical tests and visualize results, resulting in a total of approximately 878 hours of total compute.
\section{Societal impact}
\label{sec:societal_impact}
Deep ensembles are popular in many real world applications, and a potential negative impact of our work is to expose flaws in applications reliant upon deep ensembles, especially in adversarial settings like fraud detection (although this may lead to improved systems further on as well).
\section{Negative Log Likelihood decomposition into epistemic and aleatoric uncertainty}
We can follow the same rationale for the negative log likelihood as we did for the Brier score in order to relate the ensemble and average single model negative log likelihoods. For a given data point ${x,y}$ as above, we have $p^*_i(x)$ as the likelihood of the true class, given by model $i$, $i\in {1\dots M}$. We can then write:
\begin{align*}
-\log\left(\frac{1}{M}\sum_i p_i^*\right) -\frac{1}{M}\sum -\log (p_i^*) &= - \frac{1}{M} \sum \left[ -\log p_i^* +\log\left(\frac{1}{M}\sum p_i^*\right)\right] \\
&= -\frac{1}{M}\sum \log \left( \frac{\frac{1}{M}}{\frac{p^*_i}{\sum p^*_i}} \right) \\
&= -\sum \frac{1}{M}\log\left(\frac{\frac{1}{M}}{\frac{p^*_i}{\sum p^*_i}}\right) \\
&= -D_{KL}(P\|Q)
\end{align*}
In this expression, $P$ is the uniform distribution over $M$ classes and $Q$ is the distribution of normalized likelihoods from all ensemble members, $\frac{p^*_i}{\sum p^*_i}$.
This final term measures ensemble diversity as the Kullback-Liebler divergence between the maximum entropy distribution on the set of $M$ elements, and the distribution of likelihoods output by ensemble members.
\section{Relating Linear Trends to Ensemble Diversity \label{sec:lindiv}}
Assume that we have constants $c_0, c_1$ such that we have $\E_{p_{ood}({\bm{x}}, y)}[B_{ood}(f_i)] = c_0*\E_{p_{ind}({\bm{x}}, y)}[B_{ind}(f_i)]+c_1$. Then, we can write the ensemble Brier Score as:
\begin{align}
&\E_{p_{ood}({\bm{x}}, y)}[B_{ood}(\bar {\bm{f}})] = c_0*\E_{p({\bm{x}}_{ind}, y)}[B_{ind}(\bar {\bm{f}})]+c_1 \\
\implies &E_{p_{ood}({\bm{x}}, y)}[\E_{\bm{f}} \left[ B_{ood}({\bm{f}}) \right]
- \Var_{ood}\|{\bm{f}}(x)\|_2] = \\
&c_0* [ \E_{p_{ind}({\bm{x}}, y)}[B_{ind}({\bm{f}})
- \Var_{ind}\|{\bm{f}}(x)\|_2] ] +c_1 \\
\implies &\frac{\E_{p_{ood}({\bm{x}}, y)}[\Var_{ood}\|{\bm{f}}(x)\|_2]]}{\E_{p_{ind}({\bm{x}}, y)}[\Var_{ind}\|{\bm{f}}(x)\|_2]} = c_0 \tag{Average model is also collinear} \\
\implies &\frac{\E_{{\bm{f}}(x_{ind})}[\E[\Var_{ood}\|{\bm{f}}(x)\|_2\mid \E_{{\bm{f}}}[1-\|{\bm{f}}(x)\|] ]]}{\E_{{\bm{f}}(x_{ood})}[\E [\Var_{ind}\|{\bm{f}}(x)\|_2\mid \E_{{\bm{f}}}[1-\|{\bm{f}}(x)\|]]]} = c_0 \tag{Law of unconscious statistician, tower property of expectation}
\end{align}
\section{Decomposition for proper scores.}
\label{sec:decomposition_prop_scores}
To better understand the contribution of ensemble diversity to these linear trends, we can apply bias-variance decompositions to these proper scoring rules just as we applied them to the total uncertainty in section \ref{sec:metricepiun}.
In particular, the ensemble Brier Score yields the following decomposition:
\begin{align*}
\E_{p({\bm{x}}, y)}[ B_p(\bar {\bm{f}}) ]&= \E_{p({\bm{x}}, y)}[\E_{\bm{f}} \left[ B_p({\bm{f}}) \right]
- \Var\|{\bm{f}}(x)\|_2]
\end{align*}
the ensemble NLL likewise yields the decomposition:
\begin{align*}
\E_{p({\bm{x}}, y)}[&-\log\left(\frac{1}{M}\sum_i p_i^*\right)] \\ &= \E_{p({\bm{x}}, y)}[\frac{1}{M}\sum -\log (p_i^*)
-D_{KL}(P\|Q)]
\end{align*}
Here $P$ is the uniform distribution over a set of $M$ elements and $Q$ is the normalized distribution of likelihoods, $\frac{p_i^8}{\sum p_i^*}$. See appendix B,C,D for details.
Given the collinearity of ensembles and single models, we can analyze the role of ensemble diversity alone for these metrics. Assume that we have constants $c_0, c_1$ such that we have $\E_{p_{ood}({\bm{x}}, y)}[B_{ood}(f_i)] = c_0*\E_{p_{ind}({\bm{x}}, y)}[B_{ind}(f_i)]+c_1$. Then, for the ensemble Brier Score:
\begin{align}
&\E_{p_{ood}({\bm{x}}, y)}[B_{ood}(\bar {\bm{f}})] = c_0*\E_{p({\bm{x}}_{ind}, y)}[B_{ind}(\bar {\bm{f}})]+c_1 \\
\implies &\frac{\E_{{\bm{f}}(x_{ind})}[\E[\Var_{ood}\|{\bm{f}}(x)\|_2\mid \E_{{\bm{f}}}[1-\|{\bm{f}}(x)\|] ]]}{\E_{{\bm{f}}(x_{ood})}[\E [\Var_{ind}\|{\bm{f}}(x)\|_2\mid \E_{{\bm{f}}}[1-\|{\bm{f}}(x)\|]]]} = c_0
\end{align}
See Appendix D for derivation. This equation tells us that the relationship between the in and out of distribution variance of these ensembles is tightly determined given that ensembles are collinear with single models.
The quantity $\E[\Var\|{\bm{f}}(x)\|_2\mid \E_{{\bm{f}}}[1-\|{\bm{f}}(x)\|]]$ is exactly what we plot in Figure \ref{fig:f0}. We can see from this derivation that a sufficient condition to simultaneously 1) improve the quality of ensemble diversity as a measure of epistemic uncertainty and 2) to increase the robustness of ensembles relative to single models would be to increase $\E[\Var_{ood}\|{\bm{f}}(x)\|_2\mid \E_{{\bm{f}}}[1-\|{\bm{f}}(x)\|]]$ relative to $\E[\Var_{ind}\|{\bm{f}}(x)\|_2\mid \E_{{\bm{f}}}[1-\|{\bm{f}}(x)\|]]$.
Similar derivations can be made for the Negative Log Likelihood \ta{derive/plot these}.
\section{Decompositions for uncertainty metrics\label{sec:uncertdecomp}}
\subsection{Jensen-Shannon divergence and entropic uncertainty}
If we use Jensen-Shannon divergence (Eq.~\ref{eqn:variance}) as a metric for ensemble diversity \citep{lakshminarayanan2016simple,fort2019deep}, we show ensemble uncertainty can be decomposed as:
\begin{align}
\overbracket{
\entropy \left[ y \! \mid \! \bar {\bm{f}}({\bm{x}}) \right]
}^{\text{ens. uncert.}}
= \overbracket{
\jsd_{p({\bm{f}})} \left[ y \! \mid \! {\bm{f}}({\bm{x}}) \right]
}^{\text{ens. diversity}}
+ \overbracket{
\E_{p({\bm{f}})} \left[ \entropy \left[ y \! \mid \! {\bm{f}}({\bm{x}}) \right] \right]
}^{\text{avg. single model uncert.}}.
\label{eqn:jsd_uncertainty}
\end{align}
where $\entropy [ y \! \mid \! \cdot ]$ represents the entropy of a categorical distribution parameterized by $(\cdot)$.
Furthermore, $ \jsd_{p({\bm{f}})} \left[ y \! \mid \! {\bm{f}}({\bm{x}}) \right] = \frac{1}{M}\sum_{m=1}^{M}KL[y \! \mid \! {\bm{f}}({\bm{x}}) \| y \! \mid \! \bar {\bm{f}}({\bm{x}}) ]$, the average KL divergence between individual model predictions and the ensemble prediction.
We write:
\begin{align*}
\entropy \left[ y \! \mid \! \bar {\bm{f}}({\bm{x}}) \right] &= - \frac{1}{C}\sum_{i}p(y_i\mid \bar{{\bm{f}}})\log(p(y_i\mid \bar{{\bm{f}}})) \\
& = -\frac{1}{C}\sum_{i}\frac{1}{M}\sum_j p(y_i\mid {\bm{f}}_j)\log(p(y_i\mid \bar{{\bm{f}}})) \\
& = -\frac{1}{M}\sum_{j}\frac{1}{C}\sum_i p(y_i\mid {\bm{f}}_j)\log(p(y_i\mid \bar{{\bm{f}}})) \\
& = -\frac{1}{M}\sum_{j}\frac{1}{C}\sum_i p(y_i\mid {\bm{f}}_j)\left[\log(p(y_i\mid \bar{{\bm{f}}})) - \log(p(y_i\mid {\bm{f}}_j)) + \log(p(y_i\mid {\bm{f}}_j)) \right] \\
& = -\frac{1}{M}\sum_{j}\frac{1}{C}\sum_i p(y_i\mid {\bm{f}}_j)\left[\log[\frac{p(y_i\mid \bar{{\bm{f}}})}{p(y_i\mid {\bm{f}}_j)}] + \log(p(y_i\mid {\bm{f}}_j)) \right] \\
& = -\frac{1}{M}\sum_{j}\frac{1}{C}\sum_i p(y_i\mid {\bm{f}}_j)\left[\log(\frac{p(y_i\mid \bar{{\bm{f}}})}{p(y_i\mid {\bm{f}}_j})\right] + -\frac{1}{M}\sum_{j}\frac{1}{C}\sum_i p(y_i\mid {\bm{f}}_j) \log p(y_i\mid {\bm{f}}_j)) \\
& = \jsd_{p({\bm{f}})} \left[ y \! \mid \! {\bm{f}}({\bm{x}}) \right] + \E_{p({\bm{f}})} \left[ \entropy \left[ y \! \mid \! {\bm{f}}({\bm{x}}) \right] \right]
\end{align*}
\subsection{Variance and quadratic uncertainty}
As in the main text, we provide a decomposition for a quadratic notion of uncertainty as:
\begin{align}
U \left( \bar {\bm{f}}({\bm{x}}) \right) = \Var_{p({\bm{f}})} \left[ {\bm{f}}({\bm{x}}) \right] + \E_{p({\bm{f}})} \left[ U\left( {\bm{f}}({\bm{x}}) \right) \right]
\end{align}
where $U( {\bm{f}}({\bm{x}}) )$ is a quadratic notion of uncertainty:
\[
U \left( {\bm{f}}({\bm{x}}) \right) \triangleq 1 - \sum_{i=1}^C \bigl[ p(y = i \mid {\bm{f}}({\bm{x}})) \bigr]^2.
\]
And variance is defined as:
\[
\Var_{p({\bm{f}})} \left[ {\bm{f}}({\bm{x}}) \right] = {\textstyle \sum_{i=1}^C } \Var_{p({\bm{f}})} \left[ f^{(i)} ({\bm{x}}) \right]
\]
Then, the ensemble uncertainty can be decomposed as follows:
\begin{align*}
U \left( \bar{{\bm{f}}}({\bm{x}}) \right) &= 1 - \sum_{i=1}^C \bigl[ p(y = i \mid \bar{{\bm{f}}}({\bm{x}})) \bigr]^2 \\
& = 1 - \E_{p({\bm{f}})}[\sum_{i=1}^C \bigl[ p(y = i \mid {\bm{f}}({\bm{x}})) \bigr]]^2 + \E_{p({\bm{f}})}[\sum_{i=1}^C \bigl[ p(y = i \mid {\bm{f}}({\bm{x}})) \bigr]]^2 - \sum_{i=1}^C \bigl[ p(y = i \mid \bar{{\bm{f}}}({\bm{x}})) \bigr]^2 \\
& = \E_{p({\bm{f}})}[1 - \sum_{i=1}^C \bigl[ p(y = i \mid {\bm{f}}({\bm{x}})) \bigr]]^2 + \sum_{i=1}^C\E_{p({\bm{f}})}[\bigl[ p(y = i \mid {\bm{f}}({\bm{x}})) \bigr]]^2- \bigl[ \E_{p({\bm{f}})} [p(y = i \mid {\bm{f}}({\bm{x}}))] \bigr]^2 \\
& = \E_{p({\bm{f}})}[ U \left( {\bm{f}}({\bm{x}}) \right)] + \Var_{p({\bm{f}})} \left[ {\bm{f}}({\bm{x}}) \right]
\end{align*}
\section{Brier score Jensen gap}
\label{sec:app_brier_score_uncertainty_decomposition}
We consider the Brier Score of a single model:
\begin{align}
\E_{\bm{f}} \left[ B_p({\bm{f}}_i) \right] &= \E_{p({\bm{x}}, y)} \E_{\bm{f}} \left[ \Vert {\bm{f}}_i({\bm{x}}) - {\bm{1}}_y \Vert_2^2 \right]
\label{eqn:avg_brier} \\
&= \E_{p({\bm{x}}, y)} \left[
\E_{\bm{f}} \left[ \Vert {\bm{f}}_i({\bm{x}}) \Vert_2^2 \right]
+ 2 \bar {\bm{f}}_({\bm{x}})^\top {\bm{1}}_y
+ 1
\right]
\nonumber
\end{align}
and the Brier score of the ensemble:
\begin{align}
B_p(\bar {\bm{f}}) &= \E_{p({\bm{x}}, y)} \left[ \Vert \bar {\bm{f}}({\bm{x}}) - {\bm{1}}_y \Vert_2^2 \right]
\label{eqn:ens_brier} \\
&= \E_{p({\bm{x}}, y)} \left[
\Vert \bar {\bm{f}}({\bm{x}}) \Vert_2^2
+ 2 \bar {\bm{f}}({\bm{x}})^\top {\bm{1}}_y
+ 1
\right]
\nonumber
\end{align}
Note that \cref{eqn:avg_brier} and \cref{eqn:ens_brier} only differ by a single term:
\begin{align*}
B_p(\bar {\bm{f}}) &= \E_{\bm{f}} \left[ B_p({\bm{f}}) \right]
+ \E_{p({\bm{x}})} \left[ \Vert \bar {\bm{f}}({\bm{x}}) \Vert_2^2 \right]
\\ &\phantom{=}
- \E_{p({\bm{x}})} \E_{\bm{f}} \left[ \Vert {\bm{f}}({\bm{x}}) \Vert_2^2 \right] \\
& = \E_{\bm{f}} \left[ B_p({\bm{f}}) \right] - \E_{p({\bm{x}})}[\Vert \E_{{\bm{f}}}{\bm{f}}({\bm{x}}) \Vert_2^2- \Vert \bar {\bm{f}}({\bm{x}}) \Vert_2^2 ] \\
& = \E_{\bm{f}} \left[ B_p({\bm{f}}) \right] - \E_{p({\bm{x}})}[\E_{{\bm{f}}}[\Vert {\bm{f}}({\bm{x}}) \Vert_2^2]- \Vert \E_{{\bm{f}}}[{\bm{f}}({\bm{x}})] \Vert_2^2 ] \\
& = \E_{\bm{f}} \left[ B_p({\bm{f}}) \right] - \E_{p({\bm{x}})}[\Var_{p({\bm{f}})} \left[ {\bm{f}}({\bm{x}}) \right]]
\end{align*}
Where $\Var_{p({\bm{f}})} \left[ {\bm{f}}({\bm{x}}) \right]$ is:
$$\Var_{p({\bm{f}})} \left[ {\bm{f}}({\bm{x}}) \right]= {\textstyle \sum_{j=1}^C } \Var_{p({\bm{f}})} \left[ f^{(j)} ({\bm{x}}) \right]$$.
We note that this relation holds at the level of individual data points as well.
\section{Quantifying conditional diversity\label{sec:appconddiv}}
In this section, we provide additional experimental details for the results in \cref{fig:f0}, and extend to other datasets and measures of ensemble diversity. We also introduce quantifications and signficance tests to validate the stability of our conclusions across many combinations of OOD dataset and model.
\subsection{Marginal distribution of average single model uncertainty \label{appx:avg_plots}}
We end \cref{sec:diversity_exp} with the surprising conclusion that any changes to ensemble UQ between InD and OOD data must come from changes in the distribution of average single model uncertainty, $p(\mathbb{E}(U(f(x))$. Here we confirm empirically that this distribution does shift towards higher uncertainty on OOD data, for the same models that we present in \cref{sec:diversity_exp}. This shift drives any changes in ensemble diversity that we observe in practice.
\begin{figure*}[!htb]
\centering
\subfloat{\includegraphics[width=0.5\textwidth]{figs/expected_uncert_cifar.png}}
\subfloat{\includegraphics[width=0.5\textwidth]{figs/expected_uncert_imagenet.png}}
\caption{Distributions of average single model uncertainty for the WideResNet 28-10 ensembles trained on CIFAR10 (left) and the AlexNet ensembles (right), as in \cref{fig:f0}. InD and OOD test datasets are CIFAR10 and CINIC10 for the left panel, and ImageNet and ImageNet V2 for the right. }
\label{fig:sigtest_Var}
\end{figure*}
\subsection{Generating conditional distributions and conditional expectations}
In order to depict conditional variance distributions, we fit kernel density estimates to the joint distribution of ensemble diversity and average single model uncertainty for all evaluation datasets. We generated KDEs with the bandwidth suggested by Scott's Rule, and approximate conditional distributions by dividing each column of our KDE grid by the average value.
To validate comparisons between conditional distributions more precisely, we estimate the conditional expectation $\E[\text{Diversity}\mid\text{Avg}]$ by fitting a Kernel Ridge Regression model to these data, giving the best fit curve to predict values of ensemble diversity from a given value of average single model uncertainty. We used a Gaussian kernel, with bandwidth identical to what was used to generate KDE plots.
Strictly to ease visualization, we generated conditional expectation estimates for CINIC10 with a randomly subsampled set of 10000 points when fitting Kernel Ridge Regression. We account for any potential bias this may introduce in our statistical quantifications below.
\subsection{Visualizations for other datasets and metrics}
\begin{figure*}[!htb]
\centering
\captionsetup{justification=centering}
\includegraphics[width=1\textwidth]{figs/cifar10_1_vgg_11_Var.png}]
\includegraphics[width=1\textwidth]{figs/cifar10_1_vgg_11_JS.png}]
\includegraphics[width=1\textwidth]{figs/cinic10_wrn28_JS.pdf}]
\includegraphics[width=1\textwidth]{figs/imagenet_alexnet_JS.pdf}]
\vspace{-0.15in}
\caption{The top panels illustrates the InD vs OOD Variance for Cifar 10 vs Cifar10.1 with an ensemble of 4 VGG-11 networks. The bottom 3 panels illustrate the JS divergence on InD (Blue) and OOD (orange) data for CIFAR10 vs CIFAR10.1 (VGG-11), CIFAR10 vs CINIC10 (WideResNet-28-10) and ImageNet vs ImageNetV2 (AlexNet).\\ Conventions and conclusions as in Figure~\ref{fig:f0}.}
\label{fig:f0_appendix}
\end{figure*}
Figure \ref{fig:f0_appendix} first shows the variance analysis that we conducted extended to CIFAR10/CIFAR10.1, estimated with an ensemble of 5 VGG 11 networks. In the rows below, we show all analogous conclusions for Jensen Shannon Divergence as a measure of ensemble diversity, instead of variance for the same models (ensembles of $M=4$ VGG-11, WideResNet28-10, and AlexNet models for CIFAR10.1, CINIC10 and ImageNet V2 respectively).
Across all datasets, we observe that the same trends hold as reported in \cref{fig:f0}. Namely, ensemble diversity is higher on OOD data than InD data, but that the corresponding conditional distributions are not distinguishable.
\subsection{Large scale quantification and statistical tests}
In order to scale these analyses further, we devised a test statistic to directly compare the conditional expected diversity measures of InD and OOD data. Given conditional expectations for InD and OOD data, consider the following statistic:
\begin{align*}
d(InD,OOD) = \int d\text{Avg} \frac{\E_{OOD}[\text{Diversity}\mid\text{Avg}] -\E_{InD}[\text{Diversity}\mid\text{Avg}]}{\E_{InD}[\text{Diversity}\mid\text{Avg}]}
\end{align*}
Intuitively, this statistic measures the percentage change in area under the conditional expectation curve when we consider an OOD conditional expectation instead of a corresponding InD conditional expectation.
We approximated this percentage increase in expected conditional diversity as sum of pointwise differences between InD and OOD, divided by the sum of the InD curve, and report results for all model and dataset pairs that we tested in \cref{fig:percentincrease_Var_cifar},\cref{fig:percentincrease_JS_cifar}. Altogether, we see that in most cases, the percentage increases in area under the OOD curve are very small (for reference, the main text examples demonstrate changes on the order of $\sim 1\%$.) %
Although there are few sporadic cases where certain datasets demonstrate sizeable increases in our statistic on OOD data (consider variance for DenseNet 169 on CIFAR10-C Gaussian Noise, Severity Level 5), we note that these trends are inconsistent across individual models and datasets, limiting practical use of differences in OOD estimation. Furthermore, we note that our results on natural corruptions (leftmost two columns) are far more consistent than our results on synthetic corruptions (all others). In line with previous work \cite{taori2020measuring}, we prioritize results on natural corruptions in reporting our results.
\begin{figure*}[!htb]
\centering
\includegraphics[width=1\textwidth]{figs/percentage_matrix_Var.png}
\caption{Percent Increase (OOD over InD) for Variance Decomposition}
\label{fig:percentincrease_Var_cifar}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[width=1\textwidth]{figs/percentage_matrix_JS.png}
\caption{Percent Increase (OOD over InD) Jensen Shannon Decomposition}
\label{fig:percentincrease_JS_cifar}
\end{figure*}
Next, we performed Monte Carlo permutation tests to quantify the significance of the statistics that we observed:
\begin{itemize}
\item For each model and dataset upon which we computed a statistic, we first aggregated all datapoints from in and out of distribution model evaluations, and randomly permuted the order of these samples, generating a surrogate sample.
\item We then refit Kernel Ridge Regression to the surrogate sample, and calculated the $d$ statistic that resulted.
\item We calculated if the computed $d$ statistic was greater than or less than what we observed on our original sample.
\item We repeated this process for a total of 100 surrogate samples.
\end{itemize}
From this process, we can treat the proportion of surrogate samples that exceeded the value of our true test statistic as a p value for the null hypothesis that the d statistic we calculated measures a significant difference between our two original samples (and in particular, that the conditional expectation of ensemble diversity on OOD data is significantly greater than that of ensemble diversity in InD data.)
In order to compute kernel ridge regression efficiently, we used GPytorch \cite{gardner2018gpytorch} with kernel partitioning to refit models many times on a GPU. This process allowed us to compute statistics on the entire CINIC10 evaluation set, alleviating all possibilities for error in visualization due to subsampling.
In \cref{fig:sigtest_Var} and \cref{fig:sigtest_JS}, we report the estimated p values from this process. Our main goal is to communicate that in many cases, we found that the differences between conditional expectations for in and out of distribution data were almost certainly not significant, regardless of their absolute magnitude.
Finally, we show percentage increases for Imagenet on analogous $M=5$ ensembles of AlexNet, ResNet 50, and ResNet 101 models \cref{fig:sigtest_Var}, \cref{fig:sigtest_JS}- on ImageNet V2, we once again fail to see any considerable increase on the conditional distributions of OOD data relative to InD data, regardless of metric.
\begin{figure*}[!htb]
\centering
\subfloat{\includegraphics[width=1\textwidth]{figs/pval_matrix__01_25_Var.png}}
\caption{P values of (OOD over InD) difference for Variance Decomposition}
\label{fig:sigtest_Var}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[width=1\textwidth]{figs/pval_matrix__01_25_JS.png}
\caption{P values of (OOD over InD) difference for Jensen Shannon Decomposition}
\label{fig:sigtest_JS}
\end{figure*}
\clearpage
We can replicate the finding that differences between in and out of distribution test sets are quite small in the ImageNet dataset as well:
\begin{figure*}[!htb]
\centering
\subfloat{\includegraphics[width=0.3\textwidth]{figs/percentage_matrix_imagenet_Var.png}}
\subfloat{\includegraphics[width=0.3\textwidth]{figs/percentage_matrix_imagenet_JS.png}}
\caption{Percent Increase (OOD over InD) Variance Decomposition (left) and JS Divergence (right)}
\label{fig:percentincrease_imagenet}
\end{figure*}
\section{Quantifying improvement similarity between single models and ensembles\label{sec:appmmd}}
In order to validate the statistical significance of the correlations that we observe between the improvements made by ensembles and single models, we considered as baselines the distribution of improvements we would expect from comparing \textit{within} each model type- i.e. improvement correlations comparing the improvement of two performance matched single models, or two performance matched ensembles.
Although Pearson's $R$ provides a good visual aid to interpret the trends visually, we wanted to be more agnostic when validating the trends that we see.
We directly compared the improvement correlations that we see between ensemble/larger model pairs with improvement correlations that resulted when we substituted one member of these pairs with another, similarly performing model from the opposite model type (i.e., replace the ensemble with a control single model, or vice versa). We compared the resulting pair of improvement correlations with a kernel two sample test \cite{gretton2012kernel}. We calculated the unbiased test statistic $MMD_u^{2}$ from this paper for each combination of improvement correlations, and determined an appropriate threshold for these statistics based upon which we would reject the null hypothesis (Corollary 11 in \cite{gretton2012kernel}).
For each entry in the tables shown here, we consider the performance of four different kinds of predictions: an ensemble, an average single model with similar performance, a ``control" set of average single models or ensembles (again with similar performance) and finally a base model, against which we are comparing the performance of all other models. Each entry compares one of the improvement correlations shown in \cref{fig:perfcomp} (given by the row) against the ``control" improvement correlation listed in the column. In all comparisons, across InD and OOD data, CINIC10 and CIFAR10.1, and using NLL or Brier Score as metrics, we failed to reject the null hypothesis that the distributions we compared were significantly different. Each table shows the statistic value that we computed from any given pair of improvement correlations, along with the threshold statistic value we would have to exceed to reject the null hypothesis in parentheses. We list details of each comparison with each table.
Finally, at the end of this section we list the accuracies of the models that we compare, ensuring that the overall performance of these models does not differ too significantly, regardless of the metric under consideration.
{\renewcommand{\arraystretch}{1.5}
\begin{table*}[h!]
\centering{
\caption{\textbf{ $MMD^2_u$ to compare performance gains: CINIC10 NLL.} Base network: VGG-11. Average Model: WideResNet-18-4. Control Single Model: WideResNet-18-2. Control Ensemble: GoogleNet}
\resizebox{.9\textwidth}{!}{%
\begin{tabular}{l c c c c}
\hline
&
& & \multicolumn{1}{l}{$\Delta$ Single/$\Delta$ Ctrl. Single} & \multicolumn{1}{l}{$\Delta$ Ensemble/$\Delta$ Ctrl. Ensemble} \\ \hline
\multirow{2}{*}{$\Delta$ Ensemble/$\Delta$ Single}
& CIFAR10 (InD) & & $2.2\mathrm{e}{-3} (0.069)$ & $2.1\mathrm{e}{-2} (0.069)$ \\
& CINIC10 (OOD) & & $3.3\mathrm{e}{-3} (0.031)$ & $2.0\mathrm{e}{-3} (0.031)$ \\
\hline
\end{tabular}%
}
}
\label{tab:MMD1}
\end{table*}
}
{\renewcommand{\arraystretch}{1.5}
\begin{table*}[h!]
\centering{
\caption{\textbf{ $MMD^2_u$ to compare performance gains: CINIC10 Brier Score}. Base network: VGG-11. Average Model: WideResNet-18-4. Control Single Model: WideResNet-18-2. Control Ensemble: GoogleNet }
\resizebox{.9\textwidth}{!}{%
\begin{tabular}{l c c c c}
\hline
&
& & \multicolumn{1}{l}{$\Delta$ Single/$\Delta$ Ctrl. Single} & \multicolumn{1}{l}{$\Delta$ Ensemble/$\Delta$ Ctrl. Ensemble} \\ \hline
\multirow{2}{*}{$\Delta$ Ensemble/$\Delta$ Single}
& CIFAR10 (InD) & & $2.7\mathrm{e}{-4} (0.069)$ & $1.3\mathrm{e}{-2} (0.069)$ \\
& CINIC10 (OOD) & & $4.2\mathrm{e}{-3} (0.031)$ & $4.8\mathrm{e}{-3} (0.031)$ \\
\hline
\end{tabular}%
}
}
\label{tab:MMD2}
\end{table*}
}
{\renewcommand{\arraystretch}{1.5}
\begin{table*}[h!]
\centering{
\caption{\textbf{ $MMD^2_u$ to compare performance gains: CIFAR10.1 NLL}. Base network: ResNet 18. Average Model: WideResNet 18-4. Control Single Model: WideResNet 18. Control Ensemble: VGG 11 }
\resizebox{.9\textwidth}{!}{%
\begin{tabular}{l c c c c}
\hline
&
& & \multicolumn{1}{l}{$\Delta$ Single/$\Delta$ Ctrl. Single} & \multicolumn{1}{l}{$\Delta$ Ensemble/$\Delta$ Ctrl. Ensemble} \\ \hline
\multirow{2}{*}{$\Delta$ Ensemble/$\Delta$ Single}
& CIFAR10 (InD) & & $2.4\mathrm{e}{-2} (0.069)$ & $5.1\mathrm{e}{-3} (0.069)$ \\
& CIFAR10.1 (OOD) & & $8.1\mathrm{e}{-3} (0.15)$ & $2.0\mathrm{e}{-3} (0.15)$ \\
\hline
\end{tabular}%
}
}
\label{tab:MMD3}
\end{table*}
}
{\renewcommand{\arraystretch}{1.5}
\begin{table*}[h!]
\centering{
\caption{\textbf{ $MMD^2_u$ to compare performance gains: CIFAR10.1 Brier Score}. Base network: ResNet 18. Average Model: WideResNet 18-4. Control Single Model: WideResNet 18. Control Ensemble: VGG 11}
\resizebox{.9\textwidth}{!}{%
\begin{tabular}{l c c c c}
\hline
&
& & \multicolumn{1}{l}{$\Delta$ Single/$\Delta$ Ctrl. Single} & \multicolumn{1}{l}{$\Delta$ Ensemble/$\Delta$ Ctrl. Ensemble} \\ \hline
\multirow{2}{*}{$\Delta$ Ensemble/$\Delta$ Single}
& CIFAR10 (InD) & & $2.1\mathrm{e}{-2} (0.069)$ & $2.5\mathrm{e}{-3} (0.069)$ \\
& CIFAR10.1 (OOD) & & $8.1\mathrm{e}{-3} (0.15)$ & $8.7\mathrm{e}{-4} (0.15)$ \\
\hline
\end{tabular}%
}
}
\label{tab:MMD4}
\end{table*}
}
{\renewcommand{\arraystretch}{1.5}
\begin{table*}[h!]
\centering{
\caption{\textbf{Corresponding accuracies for models compared on CIFAR10/CINIC10. } Architectures of models left to right: VGG-11, WideResNet-18-4, WideResNet-18-2, GoogleNet}
\resizebox{1.0\textwidth}{!}{%
\begin{tabular}{l c c c c c c}
\hline
Dataset & Ensemble & Single Model & Single Model Control & Ensemble Control \\ \hline
CIFAR10 (InD) & $93.44\%$ & $94.32\pm0.1495\%$ & $93.93\pm0.0804\%$ & $93.68\%$ & \\
CINIC10 (OOD) & $67.88\%$ & $68.84\pm0.3936\%$ & $68.59\pm0.2762\%$ & $69.10\%$ & \\
\hline
\end{tabular}%
}
}
\label{tab:MMD5}
\end{table*}
}
{\renewcommand{\arraystretch}{1.5}
\begin{table*}[h!]
\centering{
\caption{\textbf{Corresponding accuracies for models compared on CIFAR10/CIFAR10.1. } Architectures of models left to right: ResNet18, WideResNet18-4, WideResNet18, VGG-11}
\resizebox{1.0\textwidth}{!}{%
\begin{tabular}{l c c c c c c }
\hline
Dataset & Ensemble & Single Model & Single Model Control & Ensemble Control \\ \hline
CIFAR10 (InD) & $94.26\%$ & $94.28\pm0.1430\%$ & $92.77\pm0.1839\%$ & $93.72\%$ & \\
CIFAR10.1 (OOD) & $86.10\%$ & $86.26\pm0.1727\%$ & $84.61\pm0.4242\%$ & $84.12\%$ & \\
\hline
\end{tabular}%
}
}
\label{tab:MMD6}
\end{table*}
}
\section{Discussion}
\label{sec:discussion}
In this work, we rigorously test common intuitions about the benefits of deep ensembles to UQ and robustness, and find these explanations wanting.
Below, we lay out limitations of our study, summarize our conclusions, and indicate important lines of future work.
\textbf{Ensembling in the overparametrized regime.} We emphasize that our analysis only focuses on ensembles of neural networks, and does not necessarily apply to ensembling techniques in general (e.g. random forests or gradient boosted decision trees).
Indeed, we predict that many of our results are direct consequences of the fact that we are ensembling high-capacity ``interpolating" models, which seem to generalize well despite being massively overparametrized \citep{belkin2019reconciling,adlam2020understanding,nakkiran2021deep,hastie2022surprises}. In future work, we will examine the effect of overparametrization directly by replicating these experiments with ensembles of weak learners.
\textbf{Neural network uncertainty quantification.}
In examining the conditional distributions in \cref{fig:f0}, we see that OOD uncertainty quantification is not directly impacted by ensemble diversity.
These findings show that the role of ensemble diversity in deep ensemble UQ is far more limited than previously hypothesized \citep[e.g.][]{lakshminarayanan2016simple,fort2019deep,gustafsson2020evaluating}.
\textbf{Effective robustness.} Our results in Figure~\ref{fig:ltrend_metrics} show that ensemble diversity does not yield improvements to robustness that cannot be explained by InD performance. This finding is in line with other results demonstrating that effective robustness is very difficult to achieve \cite{andreassen2021evolution}.
\textbf{When should we use deep ensembles?}
Despite our results, we maintain that
ensembling can be viewed as a reliable ``black box'' method of improving neural network performance,
both InD and OOD.
It is simple (though potentially expensive) to improve upon a model through ensembling, and training a single model that matches the performance of an ensemble is not always straightforward \citep{kondratyuk2020ensembling,lobacheva2020power,wasay2020more}.
However we caution that deep ensembles are not a panacea for the issues faced by single models.
In particular, it is dangerous to assume that deep ensembles mitigate the robustness issues of single models in contexts where we can expect dataset shift, or that ensemble diversity provides a reliable baseline for model uncertainty in the absence of ground truth.
Thus, for many practitioners, the choice of using a deep ensemble versus a performance matched single model may ultimately be dictated by practical considerations, such as performance given a pre-determined parameter/FLOP budget for model training and evaluation \citep{kondratyuk2020ensembling,lobacheva2020power,wasay2020more}.
Beyond these practical concerns, we have yet to find evidence for any reason to prefer the use of deep ensembles over an appropriately chosen single model.
\section{Hypothesis: ensemble diversity is responsible for improved UQ}
\label{sec:uncertainty}
The ability of deep ensembles to produce higher estimates of uncertainty on OOD data has been attributed to ensemble diversity \citep{lakshminarayanan2016simple,fort2019deep,wilson2020bayesian}.
In particular, ensemble diversity is hypothesized to increase on OOD data, where
one would expect that OOD predictions from individual ensemble members are less constrained by their shared training data
\citep{lakshminarayanan2016simple}.
This hypothesis is attractive because it suggests that deep ensembles offer an additional mechanism for uncertainty quantification beyond what is afforded by any single model.
In this section, we test this hypothesis by quantifying the contribution of ensemble diversity to a deep ensemble's total predictive uncertainty on both InD and OOD data.
\subsection{Metrics for ensemble diversity\label{sec:metricepiun}}
\ta{Common metrics for ensemble diversity provide interpretable decompositions of uncertainty: \emph{ensemble uncertainty $=$ ensemble diversity $+$ average single model uncertainty}.}
For example,
if we use variance (Eq.~\ref{eqn:variance}) as a metric for ensemble diversity \citep{kendall2017uncertainties,gustafsson2020evaluating}, then we show ensemble uncertainty can be decomposed as:
\begin{align}
\overbracket{
\: U \left( \bar {\bm{f}}({\bm{x}}) \right) \:
}^{\text{ens. uncert.}}
= \overbracket{
\: \Var_{p({\bm{f}})} \left[ {\bm{f}}({\bm{x}}) \right] \:
}^{\text{ens. diversity}}
+ \overbracket{
\E_{p({\bm{f}})} \left[ U\left( {\bm{f}}({\bm{x}}) \right) \right] \:
}^{\text{avg. single model uncert.}}.
\label{eqn:variance_uncertainty}
\end{align}
where ${\bm{f}}({\bm{x}}) \in \Delta^C$ is a probabilistic prediction, and $U( {\bm{f}}({\bm{x}}) )$ is a quadratic notion of uncertainty:
\[
U \left( {\bm{f}}({\bm{x}}) \right) \triangleq 1 - \textstyle{\sum_{i=1}^C }\bigl[ p(y = i \mid {\bm{f}}({\bm{x}})) \bigr]^2.
\]
See derivation in \cref{sec:uncertdecomp}. Intuitively, $U$ will be small when most probability is placed on a single class,
and will be large when probability is distributed amongst classes. See \cref{sec:uncertdecomp} for analogous results with Jensen Shannon divergence as the diversity metric (Eq.~\ref{eqn:variance}).
Based on our hypothesis,
ensemble diversity ($\Var$ in Eq.~\ref{eqn:variance_uncertainty}) should increase on OOD data \textit{independently} of average single model uncertainty ($\E[ U ]$).
In other words, given \textit{any} level of $\E[ U ]$, we would expect more ensemble diversity for OOD data than InD data.
\subsection{Experiment: InD vs OOD ensemble diversity\label{sec:diversity_exp}}
We test 10 different ensembles of size $M=5$ trained on CIFAR10, and three ensembles trained on ImageNet.
We evaluate these ensembles on their respective InD (CIFAR10, Imagenet) and OOD (CIFAR10.1, CINIC10, CIFAR10C, ImageNet V2, ImageNetC) test sets.
\ekb{In \cref{fig:f0}, we analyze the variance of two of these deep ensembles, evaluated on CIFAR10 vs CINIC10 (top row) and ImageNet vs ImageNetV2 (bottom row), see \cref{sec:appconddiv} for a complete set of results. The left panel of \cref{fig:f0} } \ta{shows the distribution $p(\mathrm{Var})$ for InD and OOD data.} \ekb{Ensembles tend to express higher variance on OOD data than InD data; a finding
consistent with previous work
\citep{lakshminarayanan2016simple,fort2019deep}.}
However, we emphasize this result is not sufficient to directly attribute UQ improvements to ensemble diversity.
\textbf{Controlling for single model uncertainty. }
A different picture emerges when we control for single model uncertainty.
\cref{fig:f0} (middle) shows histograms of $p(\Var\mid \mathbb{E}[U])$ i.e. the ensemble variance \textit{conditioned on} average single model uncertainty as given by \cref{eqn:variance_uncertainty}.
Surprisingly, we see that the OOD and InD conditional distributions are very similar.
We \ta{further study} this similarity in \cref{fig:f0} (right), which plots expected ensemble variance conditioned on average single model uncertainty: \ta{$\E[\,{ \Var \mid \mathbb{E}[U]}\,]$}.
Far from what our hypothesis would suggest (i.e. higher OOD diversity across all levels of average single model uncertainty)
we observe that the conditional \ta{expectation} of ensemble diversity on InD vs OOD data is nearly identical.
In \cref{sec:appconddiv} (\cref{fig:percentincrease_Var_cifar}-\cref{fig:percentincrease_imagenet}), we \ta{offer statistical validation of these observations, and further} demonstrate that this phenomenon holds across various architectures, InD, and OOD datasets. \ta{In all cases}, the difference between the InD and OOD expected variance is only a few percentage points, and/or not statistically significant.
\textbf{Understanding the relationship between ensemble diversity and average single model uncertainty.} By controlling for average single model uncertainty, we see that ensemble diversity does not differ significantly for InD versus OOD data.
In turn,
these results imply that the
InD/OOD difference
we see in \cref{fig:f0} (left) must be due entirely to a change in the distribution of \textit{average single model uncertainty}, \ta{$p(\mathbb{E}[U])$}.
From these results, we can conclude that surprisingly, the UQ benefits of ensemble diversity are dictated by
the corresponding average single model uncertainty.
In \cref{appx:avg_plots} we plot the differences in $p(\mathbb{E}[U])$ that drive the changes in ensemble diversity observed in \cref{fig:f0} (left).
\begin{figure*}[!t]
\centering
\subfloat{\includegraphics[width=0.5\linewidth]{figs/cifar10_bs_square.pdf}}
\subfloat{\includegraphics[width=0.5\linewidth]{figs/cinic_bs_square.pdf}}
\caption{\textbf{Ensemble diversity is \ta{meaningfully} correlated with the expected improvements from increasing model capacity}.
(Left) Panels illustrate the per-datapoint gains in Brier score over a single ResNet 18 model by either forming a deep ensemble of ResNet 18 models (x-axis), or by increasing single model capacity, here with a WideResNet 18-4 (y-axis).
The ResNet-18 ensemble and WideResNet 18-4 achieve nearly identical performance and strongly correlated improvements on both CIFAR10 and CIFAR10.1.
Colors indicate the Brier score achieved by the single ResNet 18 model on each datapoint.
(Right) We repeat the experiment
for CIFAR10/CINIC10, showing the gains in Brier score over a VGG11 model, using either an ensemble of VGG11, or a WideResNet 18-4 model.
Improvements are indistinguishable from relevant controls, and corresponding model accuracies are well matched, as shown in \cref{sec:appmmd}.
}
\label{fig:perfcomp}
\end{figure*}
\subsection{What does ensemble diversity actually measure?}
\label{sec:correlated_improvemenets_pp}
Our analysis above shows that ensemble diversity is not directly responsible for the improved OOD uncertainty estimates offered by ensembles.
To begin to understand why this might be the case,
it is useful to consider the link between ensemble diversity and performance.
It has long been established that diversity amongst ensemble members is a necessary and sufficient condition for the superior performance of ensembles \citep[e.g.][]{dietterich2000ensemble}.
To demonstrate this, consider any strictly convex loss function, such as negative log likelihood (NLL) or the multiclass Brier score (B) \citep{brier1950verification}:
\begin{align}
\text{NLL}({\bm{f}}({\bm{x}}), y) \triangleq - \log \left( f^{(y)} ({\bm{x}})\right), \quad
\mathrm{B}({\bm{f}}({\bm{x}}), y) \triangleq \Vert {\bm{f}}({\bm{x}}) - {\bm{1}}_y \Vert_2^2.
\label{eqn:brier}
\end{align}
(Here, ${\bm{1}}_y$ represents a one-hot encoding of $y$.)
Recall that the ensemble prediction $\bar {\bm{f}}({\bm{x}})$ is the average of all model predictions (i.e. $\E_{p({\bm{f}})}[{\bm{f}}({\bm{x}})]$).
By Jensen's inequality:
\begin{equation}
\begin{split}
\text{NLL}(\bar {\bm{f}}({\bm{x}}), y) \leq \E_{p({\bm{f}})} \left[ \text{NLL}( {\bm{f}}({\bm{x}}), y ) \right], \quad
\mathrm{B}(\bar {\bm{f}}({\bm{x}}), y) \leq \E_{p({\bm{f}})} \left[ \mathrm{B}( {\bm{f}}({\bm{x}}), y ) \right]
\end{split}
\label{eqn:jensen}
\end{equation}
In other words, the performance of the ensemble (as measured by NLL or Brier score) must be better than the average performance of ensemble members.
Because both NLL and Brier score are strictly convex, the Jensen gap in \cref{eqn:jensen} will grow as $p({\bm{f}})$ becomes less constant, or more ``diverse.''
In particular, the Jensen gap for Brier score is exactly equal to the ensemble variance (Eq.~\ref{eqn:variance}):
\begin{equation}
\mathrm B(\bar{{\bm{f}}}({\bm{x}}),y) - \E_{p({\bm{f}})}[ \mathrm B({\bm{f}}({\bm{x}}),y)] = \Var_{p({\bm{f}})}[{\bm{f}}({\bm{x}})].
\label{eqn:brier_variance}
\end{equation}
(Similar results are well known in the regression context---\citep[e.g.][]{krogh1995cross,masegosa2020learning}---see \cref{sec:app_brier_score_uncertainty_decomposition} for a short derivation).
In other words, $\Var_{p({\bm{f}})}[{\bm{f}}({\bm{x}})]$ measures the expected predictive improvement we obtain through ensembling.
We can use these results to investigate our UQ findings.
Hypothetically, if $\Var_{p({\bm{f}})}[{\bm{f}}({\bm{x}})]$ were also responsible for improving UQ,
this would imply that the performance gains from ensembling are somehow fundamentally different than the performance gains from increasing a single model's capacity, as the latter can hurt uncertainty estimates \citep{guo2017calibration}.
However, in the next section we demonstrate that these two methods of increasing performance are in fact correlated.
\subsection{Ensembling versus increasing model capacity}
In \cref{fig:perfcomp}, we compare the expected per-datapoint performance improvement gained through ensembling (x-axis) to the performance improvement gained through increasing model capacity (y-axis).
Specifically, we compare an ensemble of 4 CIFAR10 models (ResNet18) with a single large model (WideResNet-18-4).
The ensemble and the large single model achieve comparable Brier Score: $0.084 \pm 0.002$ on the InD test dataset and $0.210\pm0.002$ on the CIFAR10.1 OOD dataset.
In \cref{fig:perfcomp} (left), we plot the Brier score of the ensemble versus the large model on a per-datapoint level, \ta{depicting the \textit{improvement correlation} across the dataset}.
Surprisingly, we find that increasing model capacity and ensembling yield very similar performance improvements \emph{on most datapoints}.
The ensemble improvements and large model improvements have a Pearson's correlation of 0.81 on the InD test set.
Importantly, we see that this correlation is preserved even on OOD data (Pearson's correlation: 0.76).
We replicate this result for a different ensemble/larger model pair (VGG-11 ensemble versus WideResNet-18-4) that again have nearly identical InD and OOD performance: $0.093\pm0.004$ CINIC10 InD Brier Score; $0.48\pm0.02$ CINIC10 OOD Brier Score (\cref{fig:perfcomp}, right).
\ta{We compare each improvement correlation in \cref{fig:perfcomp} to relevant controls, and ensure comparable accuracies (\cref{sec:appmmd}). In all cases we find that improvements are as similar as we might expect if comparing two performance matched ensembles, or two single models.}
This result is unexpected, because the ensemble and the large model represent two distinct architectures (ResNet versus WideResNet) and two different modes of training (independent training of separate models versus training one large model).
Recalling the relationship between ensemble diversity and relative performance gains,
these results suggest that \emph{ensemble diversity \ta{estimates} the improvement we should expect by increasing model capacity.}
We conclude that, with regards to UQ and performance improvements, ensemble diversity offers no significant benefit over what can be obtained with single models.
\subsection{Implications for uncertainty estimation}
\label{sec:implications_epis_alea}
\textbf{Epistemic vs. aleatoric uncertainty.} Uncertainty is often categorized as coming from one of two components \cite[e.g.][]{hullermeier2021aleatoric}. The \emph{epistemic} component is said to capture uncertainty due to a limited number of observations, or uncertainty that the model accurately and uniquely captures the ground truth labeling process.
Apparently, it can be reduced by collecting more data.
In contrast, the \emph{aleatoric} component is described as capturing the inherent ambiguity in the data (e.g. a blurry image) and is considered to be irreducible noise.
In decision making applications such as active learning \cite{settles2009active,gal2017deep} or model-based reinforcement learning \citep{kurutach2018model,yu2020mopo}, this uncertainty decomposition is employed to identify informative datapoints for our model to sample next \citep{depeweg2018decomposition}.
Previous work has interpreted ensemble diversity as in \cref{eqn:variance} as epistemic uncertainty \citep{malinin2018predictive,gustafsson2020evaluating,yu2020mopo}, with average single model uncertainty in \cref{eqn:jsd_uncertainty,eqn:variance_uncertainty} identified as aleatoric uncertainty correspondingly \cite{smith2018understanding}.
Our results in \cref{fig:f0} demonstrate that there is a limitation to this interpretation,
as we would expect more ensemble variance (the proxy for epistemic uncertainty) for OOD data than for InD data, independent of single model uncertainty (the proxy for aleatoric uncertainty).
We therefore suggest caution when using ensembles to differentiate sources of uncertainty in downstream applications.
\textbf{Bayesian perspective.} Bayesian model averaging, or BMA
integrates predictions against a posterior distribution over models.
Given training data $\mathcal{D}$, BMA forms the prediction
$
p(y\mid {\bm{x}}, \mathcal{D}) = \int {\bm{f}}({\bm{x}}) \: p({\bm{f}} \mid \mathcal{D}) \: \mathrm d \! {\bm{f}}.
$
The advantage of BMA is the ability to consider all possible predictions given a prior and conditioned on training data, thereby mitigating the risk in estimating the ``true'' model from limited data.
A recent line of work argues that modern deep ensembles (unlike classic ensembles---see \citet{minka2000bayesian}) can be viewed as approximate BMA \citep{hoffmann2021deep,wilson2020bayesian}, although we also note that concurrent work emphasizes differences between deep ensembles and Bayesian inference in the infinite width limit \citep{NEURIPS2020_0b1ec366}.
Our results in \cref{fig:f0} identify a limitation of ensembles as approximate Bayesian inference.
The posterior predictive distribution should express higher variance for OOD data than InD data, which is not the case for the deep ensemble predictive distribution.
In \cref{sec:gp_uncertainty}, we demonstrate that exact Bayesian inference does yield higher OOD posterior variance, even after conditioning on observational noise.
We emphasize that our results neither agree nor disagree with the BMA interpretation of ensembling.
Rather they suggest that ensemble members should not be interpreted as true posterior samples,
and that (as with many approximate Bayesian methods) the ensemble approximation to BMA is biased.
\section{Related work}
\label{sec:relatedwork}
Ensembling is an established technique to improve generalization \citep[e.g.][]{schapire1990strength,perrone1992networks,Domingos1997WhyDB,opitz1999popular}, where the predictions of multiple models are aggregated to reach a consensus.
It is well established that diversity amongst ensemble members is necessary to improve performance \citep{dietterich2000ensemble}.
This diversity can be achieved through many means.
Randomization approaches introduce diversity by training each model on a random subset of data \citep{breiman1996bagging} or a random subset of features \citep{breiman2001random}.
Alternatively, boosting approaches \citep{freund1995boosting,friedman2001greedy} achieve diversity by manipulating the weighting of training data.
Other methods include using a diverse set of model classes \citep[e.g.][]{caruana2004ensemble} or joint training objectives \citep[e.g.][]{munro1997competition}.
\textbf{Ensembles of neural networks.} %
Historically, neural network ensembles have relied on a variety of mechanisms to introduce diversity \citep[e.g.][]{hansen1990neural,perrone1992networks,moghimi2016boosted,zaidi2020neural}.
Recently, diversity is often obtained by training multiple copies of the same neural network architecture with different intializations and minibatch orderings,
as the inherent randomness of SGD has been shown to introduce a sufficient amount of diversity in these (non-convex) models \cite{lee2015m,goodfellow2016deep,fort2019deep}.
Importantly, this approach can exploit parallel computation \cite{lakshminarayanan2016simple},
because none of the ensemble members depend on one another.
\textbf{Deep ensembles for predictive uncertainty.}
It has been suggested that ensembles of neural networks not only improve accuracy but also estimates of predictive uncertainty \cite{lakshminarayanan2016simple}.
Some research aims to connect ensembles and Bayesian neural networks,
suggesting that these improved uncertainty estimates are the result of performing approximate Bayesian model averaging \cite{gal2016dropout,wilson2020bayesian}.
Although prior work has described shortcomings in the uncertainty estimates derived from deep ensembles \cite[e.g.][]{liu2020simple,ciosek2019conservative,he2020bayesian,osband2021epistemic},
they remain a gold standard in high risk and safety critical settings %
\cite[e.g.][]{ovadia2019can,gustafsson2020evaluating,tran2022plex}.
\textbf{Deep ensembles and robustness.}
Robustness is the ability to maintain good accuracy and calibration under conditions of distributional shift.
Deep ensembles outperform other approaches in maintaining both accuracy and calibration on OOD data \cite{ovadia2019can,gustafsson2020evaluating}, although their limitations have also been demonstrated \cite{kumar2021calibrated,rahaman2020uncertainty}.
This robustness is attributed to the diversity between ensemble members \citep{fort2019deep}.
\textbf{Other related work.}
Recent work investigates whether it is possible to achieve the benefits of an ensemble with reduced computation during training and/or test time \cite{huang2017snapshot,maddox2019simple,wen2020batchensemble,havasi2021training}.
Additionally many works have proposed numerous diversity metrics for ensembles similar to those we examine here \citep[e.g.][]{krogh1995cross,masegosa2020learning,andres2022diversity}.
\section{Introduction}
\label{sec:intro}
In many real-world settings, practitioners deploy ensembles of neural networks that combine the outputs of several individual models \citep[e.g.][]{szegedy2015going,kurutach2018model,yu2020mopo}.
Though training and evaluating multiple models is computationally expensive,
a wide body of research demonstrates that ensembles achieve better performance (as measured by accuracy, negative log likelihood, or a variety of other metrics) than their constituent single models, provided that these models make diverse errors \citep{dietterich2000ensemble}.
This benefit is well-established in the literature:
theoretically proven for ensembles formed via boosting or bagging \citep{schapire1990strength,breiman1996bagging},
and demonstrated for \textit{deep ensembles} that solely rely on the randomness of SGD coupled with non-convex loss surfaces \citep{lee2015m,fort2019deep}.
Of course, ensembling is not the only way to increase performance;
one could also increase the depth or width of a single neural network.
In many settings, a single large model performs similarly to an ensemble of (smaller) models with a similar number of parameters \citep{lobacheva2020power,kondratyuk2020ensembling,wasay2020more}.
This observation poses a natural question:
are there reasons to choose a deep ensemble over a single (larger) neural network with comparable performance?
Recent research suggests that deep ensembles may be preferable to single models in
safety-critical applications and settings where data shifts significantly away from the training distribution.
First, \citet{lakshminarayanan2016simple} demonstrate that deep ensembles provide \emph{well-calibrated estimates of uncertainty} on classification and regression tasks.
Compared with other uncertainty quantification (UQ) methods, ensembles offer better (i.e. less overconfident) uncertainty estimates on out-of-distribution ({OOD}) or shifted data \citep{ovadia2019can}.
Second, recent work indicates that---beyond calibration---ensemble performance (as measured by accuracy, NLL, or other metrics) also tends to be \emph{robust against dataset shift},
again often outperforming other methods in these regimes \citep{gustafsson2020evaluating}.
Intuitions in recent papers \citep[e.g.][]{lee2015m,fort2019deep} attribute these UQ/robustness benefits to the fact that ensembles produce multiple diverse predictions, rather than a single point prediction.
If diversity does in fact explain UQ/robustness improvements, this would suggest that deep ensembles indeed offer benefits that cannot be obtained by (standard) single neural networks.
In this paper, we rigorously test hypotheses that formalize this intuition.
Surprisingly, after controlling for factors related to the performance of an ensemble's component models,
we find no evidence that having a diverse set of predictions is responsible for these purported benefits.
Put differently, we find that these UQ/robustness benefits are not unique to deep ensembles, \emph{as they can be replicated through the use of (larger) single models}.
We confirm these results for a wide variety of model architectures, as well as for \emph{heterogeneous deep ensembles} that combine multiple different neural network architectures and \emph{implicit deep ensembles} like MC Dropout \citep{gal2016dropout}, BatchEnsemble \citep{wen2020batchensemble}, and MIMO \citep{havasi2021training} (\cref{sec:app_heter_ensemble}).
\textbf{Hypothesis: ensemble diversity is responsible for improved UQ.}
Two components contribute to ensemble uncertainty estimates:
the uncertainties expressed by individual ensemble members,
and diversity among ensemble member predictions.
Recent work suggests that the diversity component is primarily responsible for better calibrated OOD uncertainty estimates, as ensemble members should agree less (i.e. offer more diverse predictions) as data shift away from the training distribution \citep{lakshminarayanan2016simple,fort2019deep,gustafsson2020evaluating}.
In contrast, we find that---after conditioning on the uncertainty of individual ensemble members---%
\emph{the level of ensemble disagreement does not statistically differ between InD and OOD data} (\cref{fig:f0}),
and thus ensemble diversity is not directly responsible for larger OOD uncertainty estimates.
\gp{Furthermore, ensemble diversity---on a per-datapoint basis---is correlated with the expected improvement we obtain by increasing model capacity (\cref{fig:perfcomp}), implying that ensemble diversity does not capture a quantity inaccessible to a single (larger) model.}
\textbf{\ta{Hypothesis: ensemble diversity is responsible for improved robustness.}}
\ta{Independent work} demonstrates a deterministic relationship between a (single) neural network's 0-1 accuracy on InD and OOD datasets \cite{taori2020measuring,miller2021accuracy},
whereby \ta{the OOD performance of a model can be predicted from its InD performance}.
It is therefore natural to ask whether having multiple diverse predictions contributes to additional OOD robustness (as suggested by \citep{fort2019deep,gustafsson2020evaluating}), beyond what is expected given performance improvements on InD data.
Our results demonstrate that deep ensembles are not ``effectively robust'' relative to single models---i.e. their OOD performance (as measured by accuracy, NLL, Brier score, and calibration error) follows the same deterministic relationship to InD performance as single models (\cref{fig:ltrend_metrics}).
Therefore, ensemble diversity does not yield additional robustness over what standard single networks achieve.
\textbf{Implications.}
Overall, this paper does not disagree with prior claims about the benefits of deep ensembles relative to an ensemble's component models.
Indeed, in our experiments we confirm that ensembling is a convenient mechanism to improve predictive performance, UQ, and robustness relative to this baseline.
At the same time, our results also indicate that---after controlling for individual model uncertainty and InD performance---
ensembles do not obtain UQ/robustness benefits beyond what can already be obtained from the properties of an appropriately chosen single model.
\section*{Checklist}
\begin{enumerate}
\item For all authors...
\begin{enumerate}
\item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
\answerYes{We relate our claims to two hypotheses in the introduction \cref{sec:intro}, and test each hypothesis in the corresponding results sections \cref{sec:uncertainty,sec:ltrend}.}
\item Did you describe the limitations of your work?
\answerYes{We specify in the discussion \cref{sec:discussion} that our results are limited to neural network ensembles, as opposed to more general ensembles.}
\item Did you discuss any potential negative societal impacts of your work?
\answerYes{We do so in \cref{sec:societal_impact}}
\item Have you read the ethics review guidelines and ensured that your paper conforms to them?
\answerYes{}
\end{enumerate}
\item If you are including theoretical results...
\begin{enumerate}
\item Did you state the full set of assumptions of all theoretical results?
\answerYes{}
\item Did you include complete proofs of all theoretical results?
\answerYes{}
\end{enumerate}
\item If you ran experiments...
\begin{enumerate}
\item Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
\answerYes{We provide a link to a repository in the supplemental material section \cref{sec:app_code_data}. This repository contains instructions to reproduce main figures and to download relevant data. }
\item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
\answerYes{We provide instructions in the code \cref{sec:app_code_data} which specify internally the data splits and hyperparameters we used. We further specify in \cref{sec:app_training_details} that we chose default hyperparameters as specified in a separate code repo.}
\item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
\answerYes{In accordance with checklist guidelines, we report the fact that we ran statistical significance tests for our main results here- in particular, \cref{sec:appconddiv} describes tests for \cref{fig:f0} and related results, \cref{sec:appmmd} describes tests for \cref{fig:perfcomp} and related results, and \cref{sec:additional_ltrend} describes tests for \cref{fig:ltrend_metrics} and related results.}
\item Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
\answerYes{We do so in \cref{sec:app_code_data}}
\end{enumerate}
\item If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
\begin{enumerate}
\item If your work uses existing assets, did you cite the creators?
\answerYes{In \cref{sec:setup}, we reference the origin of the models \cite{miller2021accuracy} and \cite{taori2020measuring}, and provide further details in \cref{sec:app_code_data}.}
\item Did you mention the license of the assets?
\answerNo{We provide links to publicly released assets with relevant licenses, but do not have a license for models that we were provided by the authors of \cite{miller2021accuracy}.}
\item Did you include any new assets either in the supplemental material or as a URL?
\answerNA{We do not provide new assets.}
\item Did you discuss whether and how consent was obtained from people whose data you're using/curating?
\answerYes{In our acknowledgements we thank the authors of \cite{miller2021accuracy} for agreeing to share their data with us- all other data is released under a pubic license.}
\item Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?
\answerNA{We do not believe this to apply to our data, which consists of deep network models trained using popular deep learning frameworks on benchmark datasets.}
\end{enumerate}
\item If you used crowdsourcing or conducted research with human subjects...
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\answerNA{}
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\answerNA{}
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\answerNA{}
\end{enumerate}
\end{enumerate}
\section{Hypothesis: ensemble diversity is responsible for improved robustness} %
\label{sec:ltrend}
Beyond uncertainty quantification, ensembles have been shown to often achieve better predictive performance than single networks (as measured by 0-1 accuracy, NLL, or Brier score) on OOD or shifted datasets \citep{lakshminarayanan2016simple,ovadia2019can,gustafsson2020evaluating}.
In this section, we test the hypothesis that ensemble diversity improves robustness over what single neural networks can offer.
\begin{wrapfigure}{R}{0.40\textwidth}
\centering
\vspace{-25pt}
\includegraphics[width=0.9\linewidth]{figs/FigEffRob_draft6.pdf}%
\caption{\textbf{Cartoon of effectively robust deep ensembles (what we want).}
}
\label{fig:ltrend_schematic}
\vspace{-10pt}
\end{wrapfigure}
\subsection{Effective robustness}
We use the concept of ``effective robustness'' as introduced by \citet{taori2020measuring}.
These authors note that there is often a deterministic relationship between a neural network's accuracy on InD data and its accuracy on an OOD dataset (green line in \cref{fig:ltrend_schematic}).
In other words, any improvements in OOD performance can be entirely explained by improvements in OOD performance.
A model is considered to be \emph{effectively robust} only if it achieves better OOD accuracy than what is predicted by its InD accuracy.
In general, there are very few neural networks or training procedures that exhibit effective robustness against any OOD dataset \citep{taori2020measuring,miller2021accuracy}.
To measure the role that ensemble diversity plays in robustness, we quantify to what extent deep ensemble OOD performance can be explained by InD performance (as measured by the deterministic relationship derived from single models).
If the performance of deep ensembles follows the same deterministic relationship, then deep ensembles are not effectively robust (i.e. multiple diverse predictors offer no additional robustness over what a single neural network provides).
\subsection{Experiment: measuring effective robustness of deep ensembles across metrics}
\textbf{Ensembles are not effectively robust with respect to 0-1 accuracy.}
Following \citet{miller2021accuracy}, we measure the InD and OOD error for all the models described in \cref{sec:setup}.
The top left of \cref{fig:ltrend_metrics} compares the error of models on CIFAR10 (InD) versus CINIC10 (OOD),
and the bottom left plot compares the error of models on ImageNet (InD) versus ImageNetV2 (OOD).
From these plots, we observe several trends.
In agreement with \citet{taori2020measuring} and \citet{miller2021accuracy}, we observe that single models (green dots) follow a colinear relationship for InD versus OOD accuracy.
Additionally, we
find that \emph{ensembles (orange dots) do not deviate from this colinear InD/OOD relationship.}
In \cref{sec:ltrend_r2_tables}, we evaluate the quality of these linear trends.
In particular, we fit separate linear trend lines for individual models and deep ensembles.
All trend lines achieve correlations of $R>0.84$, and their coefficients only differ by $1\%$ at most.
This suggests that, after controlling for InD accuracy, the OOD accuracy of ensembles is nearly identical to that expected of single models.
(See \cref{sec:additional_ltrend} for CIFAR10.1/CIFAR10C/ImageNetC results.)
\textbf{Ensembles are not effectively robust with respect to NLL or Brier score.}
Although deep ensembles are not effectively robust in terms of predictive accuracy, many of their robustness benefits have been reported in terms of probabilistic metrics, such as NLL or Brier score \citep{ovadia2019can}.
We therefore extend our investigation of deep ensemble effective robustness to these metrics.
\cref{fig:ltrend_metrics} (middle left) plots the InD NLL and OOD NLL of various ensembles and single models.
To the best of our knowledge, this is the first time that the effective robustness experiments of \citet{taori2020measuring} and \citet{miller2021accuracy} have been extended to metrics other than 0-1 accuracy.
We observe that the relationship between InD NLL and OOD NLL is not as linear as the accuracy trend.
Nevertheless, we observe no discernible difference between the performance of single networks and ensembles (see \cref{sec:ltrend_r2_tables} for a quantitative analysis).
We observe a similiar phenomenon when we plot InD versus OOD Brier score (\cref{fig:ltrend_metrics}, middle right)---%
ensembles and single models obtain similar OOD Brier score, after controlling for InD Brier score.
Our key conclusion is that deep ensembles fail to demonstrate effective robustness when evaluated on probabilistic performance metrics, just as they do with 0-1 accuracy.
(See \cref{sec:additional_ltrend} for CIFAR10.1/CIFAR10C/ImageNetC results.)
\begin{figure*}[!tb]
\centering
\subfloat{\includegraphics[width=1\textwidth]{figs/cinic10_metrics_ensemble_all.pdf}\label{fig:ltrend_cinic10}}
\hfill
\vspace{-0.2in}
\subfloat{\includegraphics[width=1\textwidth]{figs/imagenetv2mf_metrics_ensemble_all.pdf}\label{fig:ltrend_imagenetv2}}
\caption{\textbf{Deep ensembles are not ``effectively robust" across a variety of performance metrics}.
Panels illustrate InD vs OOD performance metrics, from left to right: 0-1 Error, NLL, Brier Score, and rESCE. The model types considered are single models and ensembles.
Linear trend lines are shown in solid lines, and black dotted lines indicate perfect robustness. We find that, conditioned on InD performance, ensembles offer no better OOD performance than single models.
See \cref{sec:additional_ltrend} for additional corruptions.}
\label{fig:ltrend_metrics}
\end{figure*}
\textbf{Ensembles do not offer effectively robust calibration.}
We also compare InD and OOD calibration for various single models and ensembles.
We consider various metrics for measuring and comparing calibration used throughout the literature.
Expected Calibration Error (ECE) \citep{naeini2015obtaining} is a standard metric for measuring calibration of neural networks.
As we show in \cref{sec:app_ece_all}, there is little correlation between a single model’s InD ECE and OOD ECE, which precludes any discussion of “effective robustness” using this metric.
Conversely, \cref{fig:ltrend_metrics} depicts a strong correlation between a model’s InD/OOD square root of the Expected \emph{Squared} Calibration Error (rESCE) \citep{degroot1983comparison,murphy1977reliability}, which appears in a common decomposition of the Brier score \citep{brocker2009reliability}.
We therefore expect that any InD/OOD trend for the rESCE should be qualitatively similar to the InD/OOD trends observed for Brier score.
In \cref{fig:ltrend_metrics} (top right), we observe a linear trend relating the CIFAR10 (InD) and CINIC10 (OOD) rESCE of single models.
The rESCE of the ImageNet models, \cref{fig:ltrend_metrics} (bottom right), follows a bimodal trend, where---depending on the model architecture---InD rESCE is correlated with either low or high OOD calibration.
Nevertheless, for both datasets we find that ensembles do not achieve better OOD calibration that single models with similar InD calibration.
(See \cref{sec:additional_ltrend} for CIFAR10.1/CIFAR10C/ImageNetC results.)
\subsection{Heterogeneous and implicit ensembles}
\label{sec:heterogeneous}
From the previous results, it is clear that---by many metrics---ensembling multiple copies of the same model architecture confers no additional robustness over single models.
A natural question is whether we could achieve more robustness by ensembling different model architectures together.
To test this hypothesis, we repeat the same robustness experiments with \emph{heterogeneous ensembles}: ensembles that combine multiple architectures, and \emph{implicit ensembles}: single models that approximate deep ensembles, usually through parameter sampling \cite{gal2016dropout}. To construct heterogeneous ensembles,
we divide the 137 CIFAR10 models and 78 ImageNet models from \cref{sec:setup} based on their InD accuracy.
Ensembles are then formed by randomly selecting 4 models from each bin.
This procedure ensures that all ensemble members will have similar accuracy, even though the ensemble members may represent different architectures and training regimens.
Despite their additional diversity, these heterogeneous ensembles do not provide effective robustness, as shown in
\cref{sec:app_heter_ensemble}. Finally, we investigate if these results also follow for three implicit ensembling mechanisms: Monte Carlo Dropout \cite{gal2016dropout}, multiple-input-multiple-output (MIMO) \cite{havasi2021training}, and Batch Ensembles \cite{wen2020batchensemble}. We find that implicit ensembles are also not effectively robust, as depicted in \cref{sec:app_heter_ensemble}.
\subsection{Implications.}
As discussed in \cref{sec:correlated_improvemenets_pp}, ensemble diversity is responsible for improved NLL and Brier score relative to constituent models.
In this sense, ensemble diversity is responsible for improved OOD performance.
However, these OOD improvements exactly follow the deterministic trends predicted by (standard) single models, and thus ensembling multiple diverse predictors does not yield any ``effective robustness'' over what could be achieved by a better performing single model.
Unlike prior research \citep[e.g.][]{ovadia2019can,gustafsson2020evaluating}, these results suggest that ensembles are a tool of convenience for obtaining better OOD performance, but not qualitatively different from single models in this respect.
\section{Setup}
\label{sec:setup}
Consider multiclass classification: inputs ${\bm{x}} \in \mathbb{R}^D$ with targets $y \in [1,\dots,C]$, where $D$ is the number of features and $C$ is the number of classes.
We assume that we have access to $M$ distinct neural networks ${\bm{f}}_1, \ldots, {\bm{f}}_M$,
where each model ${\bm{f}}_i : \mathbb{R}^D \to \Delta^C$ maps an input to the $C$-class probability simplex.
We will primarily focus on the common case of {\bf homogeneous ensembles}, where ${\bm{f}}_1, \ldots, {\bm{f}}_M$ represent the same neural network architecture and training procedure, relying on the inherent randomness of initialization and SGD to produce diverse models (see Sec.~\ref{sec:relatedwork} for a broad discussion).
However, in \cref{sec:heterogeneous} we will also consider {\bf heterogeneous ensembles} where ${\bm{f}}_1, \ldots, {\bm{f}}_M$ represent different architectures or training procedures, \ekb{and {\bf implicit ensembles}, where ${\bm{f}}_1, \ldots, {\bm{f}}_M$ are approximated by changes to a single model \cite{gal2016dropout,havasi2021training,wen2020batchensemble}}.
Throughout the paper, we will also represent these member networks as a discrete distribution of models:
$p({\bm{f}}) = \text{Unif.} [ {\bm{f}}_1, \ldots, {\bm{f}}_M ] $.
The ensemble prediction $\bar {\bm{f}}({\bm{x}})$ is given by the arithmetic mean of the ensemble member \emph{probabilities}:\footnote{
While it is also possible to average the logits (log probabilities) of each model,
we note that probability averaging is far more common in the literature
\citep[e.g.][]{lakshminarayanan2016simple}.
}
\begin{equation}
\textstyle{
\bar {\bm{f}}({\bm{x}}) = \E_{p({\bm{f}})}[{\bm{f}}({\bm{x}})] = \frac{1}{M} \sum_{i=1}^M {\bm{f}}_i({\bm{x}})
}
\label{eqn:ensemble}
\end{equation}
\textbf{Metrics for ensemble diversity.}
Two metrics of ensemble diversity are
\begin{enumerate*}[label=(\arabic*)]
\item variance \citep[e.g.][]{kendall2017uncertainties}, and
\item Jensen-Shannon divergence \citep[e.g.][]{lakshminarayanan2016simple,fort2019deep}.
\end{enumerate*}
Mathematically, they are (respectively) defined as:
\begin{align}
\Var_{p({\bm{f}})} \left[ {\bm{f}}({\bm{x}}) \right] &= {\textstyle \sum_{i=1}^C } \Var_{p({\bm{f}})} \left[ f^{(i)} ({\bm{x}}) \right], \quad
\label{eqn:variance}
\jsd_{p({\bm{f}})} \left[ y \! \mid \! {\bm{f}}({\bm{x}}) \right] = \entropy \bigl[ y \! \mid \! \bar {\bm{f}}({\bm{x}}) \bigr] -
\E_{p({\bm{f}})} \bigl[ \entropy \left[ y \! \mid \! {\bm{f}}({\bm{x}}) \right] \bigr]
\end{align}
\ta{where $f^{(i)}$ refers to the probability assigned by a model to the $i$-th output class, and H is the entropy. Both metrics} are always positive and minimized when the predictions from ensemble members are the same, i.e. not diverse.
\textbf{Models and training datasets.}
We reuse and train a variety of neural networks on two benchmark image classification datasets:
{\bf CIFAR10} \citep{krizhevsky2009learning} and {\bf ImageNet} \citep{deng2009imagenet}.
In particular, we include the 137 CIFAR10 models trained by \citet{miller2021accuracy}, corresponding to 32 different architectures each trained for 2-5 seeds; as well as the ``standard" 78 ImageNet models curated by \citet{taori2020measuring}, each corresponding to a different architecture trained for 1 seed. To form homogeneous ensembles, we additionally train 10 network architectures on CIFAR10
and three on ImageNet.
We train 5 independent instances of each model architecture, where each instance differs only in terms of initialization and minibatch ordering.
We form homogeneous deep ensembles by combining 4 out of the 5 random seeds.
From this process, we can consider 5 single model replicas and 5 ensemble replicas for each model architecture.
Unless otherwise stated, ensembles are formed following \cref{eqn:ensemble}.
\textbf{OOD datasets.}
A majority of our analysis compares deep ensembles on InD versus OOD test data.
To that end, we consider three different catagories of OOD datasets as suggested by \citep{miller2021accuracy}:
\emph{Shifted reproduction datasets.}
This category includes the {\bf CIFAR10.1} and {\bf ImageNetV2} datasets \citep{recht2019imagenet},
both of which were collected and labeled following the same curation processes of the original CIFAR10 and ImageNet datasets, respectively.
Neural networks (trained on the original datasets) tend to achieve worse performance on these new test sets.
\emph{Alternative benchmark datasets.}
The {\bf CINIC10} dataset \citep{darlow2018cinic} shares the same classes as CIFAR10 but uses images drawn and downsampled from the ImageNet dataset.
Because ImageNet and CIFAR10 images were collected using different curation procedures, models trained on CIFAR10 tend to achieve worse performance on CINIC10.
\emph{Synthetically corrupted datasets.}
The {\bf CIFAR10C} and {\bf ImageNetC} datasets \cite{hendrycks2018benchmarking},
apply synthetic perturbations to CIFAR10 and ImageNet images (e.g. Gaussian blur, fog effects, etc.).
Due to their synthetic nature, these datasets offer shifts of various intensity (e.g. mild blur versus heavy blur).
We relegate most of our analysis of these datasets to the Appendix.
|
2,877,628,090,069 | arxiv | \section{Introduction}
In \cite{Karamehmedovic2015} Karamehmedovi\'c defines a class of analytic symbols. These symbols are a subspace of the analytic-type symbols in the sense of Tr\`eves \cite{TI}.
The aim was to obtain holomorphic mapping properties for the associated operators, and apply them to Calder\'on projectors in a (local) Helmholtz-type Dirichlet problem, where the boundary is a piece of a hyperplane.\\
In this way, Karamehmedovi\'c constructs the Dirichlet-to-Neumann (DN) map, and obtains a result on how well it preserves domains of holomorphic extendibility.
That is, how far Neumann data extends given this information about Dirichlet data, and in fact vice-versa, by the same system of equations for the Calder\'on projectors.
It was done by showing that the symbols of the Calder\'on projectors are of that class. The class of the "analytic symbols" was first introduced by Boutet de Monvel in \cite{BoutetDeMonvel1972}, and \cite{Karamehmedovic2015} essentially reuses these, but introduces constraints \cite[pp. 3-4, Definition 2.1]{Karamehmedovic2015}.
The domains obtained in \cite[pp. 10, Theorem 2.9]{Karamehmedovic2015} are larger than those we get here,
and \cite{Karamehmedovic2015} has the advantage of being adapted to poly-rectangular shapes.\\
The aim of this paper is to remove the strong constraints on the symbols in \cite{Karamehmedovic2015}, and reduce them to analytic symbols in the sense of Tr\`eves \cite[pp. 262, Definition 2.2]{TI}. In the process, we will also obtain a general domain-of-extension mapping theorem.
This features in Winterrose \cite{Winterrose2021}. Let $n\in \mathbb{N}$ be the dimension throughout.
\section{Notation}
Denote by $S^d(\mathbb{R}^n \times \mathbb{R}^n)$ order $d\in \mathbb{R}$ $(1,0)$ H\"ormander symbols. These are the $p\in C^\infty(\mathbb{R}^n \times \mathbb{R}^n)$ satisfying for any $\alpha,\beta \in \mathbb{N}^n_0$ the estimates
\begin{align*}
\sup_{(x,\xi)\in \mathbb{R}^n \times \mathbb{R}^n} \langle \xi \rangle^{|\alpha|-d} | \partial_x^\beta \partial_\xi^\alpha p(x,\xi) | < \infty,
\end{align*}
where we use the notation $\langle \xi \rangle = (1+ |\xi|^2)^\frac{1}{2}$ for $\xi \in \mathbb{R}^n$, and put $S^{-\infty}= \cap_{d\in \mathbb{R}}S^d$.
Associated to $p$ is $\textnormal{Op}(p)$,
defined by its action on $u\in C^\infty_0(\mathbb{R}^n)$ via
\begin{align*}
\textnormal{Op}(p)u(x) = \frac{1}{(2\pi)^n}\int_{\mathbb{R}^n} e^{i x \cdot \xi} p(x,\xi) \mathcal{F}u(\xi) \, d\xi
\quad
\textnormal{for all}
\quad
x\in \mathbb{R}^n,
\end{align*}
and on a compactly supported distribution $u\in \mathcal{E}'(\mathbb{R}^n)$ we use the bracket notation.
We use the notation $d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi = (2\pi)^{-n} d\xi$ for the scaled standard Lebesgue measure $d\xi$.
Finally, $B(x,r)$ denotes the open ball in $\mathbb{R}^n$ with center at $x\in \mathbb{R}^n$ and radius $r>0$.
See Grubb \cite{GG} for details on the local theory of pseudodifferential operators.
\newpage
\section{Contour Deformation}
\begin{theorem}
Fix $R>0$, $\epsilon>0$, and $p\in S^d(\mathbb{R}^n \times \mathbb{R}^n)$ a symbol with $d\in \mathbb{R}$. Assume $p|_{B(0,r_0)\times \mathbb{R}^n}$ extends holomorphically into $(B(0,r_0)+iB(0,\delta_0))\times W_\epsilon$, where
\begin{align*}
W_{\epsilon} = \{ \zeta \in \mathbb{C}^n \, | \, |\textnormal{Im}\, \zeta| < \epsilon |\textnormal{Re}\, \zeta | \} \cap \{ \zeta \in \mathbb{C}^n \, | \, |\textnormal{Re}\, \zeta | > R \},
\end{align*}
and satisfies
\begin{align*}
\sup_{(x,\zeta) \in K\times W_\epsilon }\langle \textnormal{Re}\, \zeta \rangle^{-d} |p(x,\zeta)| < \infty
\quad
\textnormal{for any}
\quad
K\subset\subset
B(0,r_0)+iB(0,\delta_0).
\end{align*}
Let $u\in C^\infty_0(\mathbb{R}^n)$. Suppose $u|_{B(0,r)}$ extends holomorphically into $B(0,r)+iB(0,\delta)$. Choose $r>r'>0$ and $\delta \geq \delta' >0$ so that
\begin{align*}
\frac{\delta'}{r-r'} < \epsilon.
\end{align*}
Then $\textnormal{Op}(p)u|_{B(0,\min\{r',r_0\})} $ likewise extends to $B(0,\min\{r',r_0\})+iB(0,\min\{\delta',\delta_0\})$. In particular, it is real-analytic on its restricted domain.
\end{theorem}
This result is similar to Theorem 2.9 in \cite{Karamehmedovic2015}, but without extra constraints on $u$ and $p$.\\
A deformation of $\mathbb{R}^n \times \mathbb{R}^n$ into $\mathbb{C}^n \times \mathbb{C}^n$ allows us to continue $\textnormal{Op}(p)u$ explicitly.
The argument is a detailed run-through of deformations by Boutet de Monvel in \cite{BoutetDeMonvel1972}.
The main idea is to split the oscillatory integral, and apply Stokes' theorem.\\
\begin{proof}
Take $\chi_2\in C^\infty_0(\mathbb{R}^n)$ to be $1$ on $ \overline{B(0,2R)}$
but $\chi_2(\xi)\in [0,1)$ for $\xi \not\in \overline{B(0,2R)}$.
Let $\chi_1\in C^\infty_0(B(0,r))$ be a cutoff with $\chi_1(y)=1$ when $y\in B(0,r'')$, else in $[0,1)$, where $r>r''>r'$ are chosen so that
\begin{align*}
\frac{\delta'}{r-r'}<\frac{\delta'}{r''-r'} < \epsilon.
\end{align*}
Now let $\sigma : [0,1] \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}^n \times \mathbb{C}^n$ be defined by
\begin{align*}
(t,y,\xi) \mapsto
\Big(y-it\delta' \chi_1(y)(1-\chi_2(\xi)) \frac{\xi}{|\xi|}, \xi - i t \frac{\delta' (1-\chi_1(y))}{r''-r'} (1-\chi_2(\xi))|\xi| \frac{y}{|y|} \Big),
\end{align*}
and let $w$ and $\zeta$ denote the first and second $\mathbb{C}^n$ components of this $\sigma$, respectively.
This type of $\sigma$ is used by Boutet de Monvel in \cite[pp. 243-245]{BoutetDeMonvel1972} with sparse details.
Let us put
\begin{align*}
\mathcal{C}(t) = \sigma(\{t\} \times \mathbb{R}^n \times \mathbb{R}^n )
\quad
\textnormal{for all}
\quad
t\in [0,1].
\end{align*}
Under the $\sigma$ deformation, if $\chi_2(\xi) = 0$ and $|\textnormal{Re}\,(x)|<r'$, we get
\begin{align*}
\textnormal{Re}\, (i(x-w)\cdot \zeta)
&=
-\xi \cdot \Big(\textnormal{Im}\,(x) + t\delta'\chi_1(y)\frac{\xi}{|\xi|}\Big)
+
t \frac{\delta' (1-\chi_1(y))}{r''-r'} |\xi| \frac{y}{|y|}
\cdot (\textnormal{Re}\,(x) - y) \\
&\leq -|\xi|
\Big(\frac{\xi}{|\xi|} \cdot\textnormal{Im}\,(x) + t\delta'\chi_1(y) + t\frac{\delta' (1-\chi_1(y))}{r''-r'} \Big(|y| -|\textnormal{Re}\,(x)| \Big) \Big) \\
&\leq -|\xi|
\Big(-|\textnormal{Im}\,(x)| + t\delta' \chi_1(y) + t\frac{\delta' (1-\chi_1(y))}{r''-r'} \Big(|y| -|\textnormal{Re}\,(x)| \Big) \Big) \\
&\leq -|\xi|
\Big( t\delta' -|\textnormal{Im}\,(x)| \Big).
\end{align*}
It will ensure that deformations by $\sigma(t,\cdot,\cdot)$ give convergent integrals for $|\textnormal{Im}\,(x)|<t\delta$.
Take $x\in B(0,r')+itB(0,\delta')$,
and fix $\rho > 2R$ and $1\geq t_2>t_1\geq 0$. Put
\begin{align*}
Q(\rho) = (t_1,t_2) \times \mathbb{R}^n \times (B(0,\rho) \setminus \overline{B(0,2R)}),
\end{align*}
and note then that $\sigma$ is injective on $Q(\rho) $, and
\begin{align*}
\sigma(\overline{Q(\rho)}) \subset \mathbb{C}^n \times W_\epsilon
\quad
\textnormal{for all}
\quad
\rho >2R.
\end{align*}
In the following, we will put $dw = dw_1 \wedge \cdots \wedge dw_n$ and $d\hspace*{-0.08em}\bar{}\hspace*{0.1em} \zeta = (2\pi)^{-n} d\zeta_1 \wedge \cdots \wedge d\zeta_n$.
Define for $(w,\zeta)\in \mathbb{C}^n \times W_\epsilon$ the $2n$-form
\begin{align*}
\mu_x = G_x(w,\zeta) \, dw \wedge d\hspace*{-0.08em}\bar{}\hspace*{0.1em} \zeta = e^{i\zeta\cdot(x-w)} p (x,\zeta) u(w) \, dw \wedge d\hspace*{-0.08em}\bar{}\hspace*{0.1em} \zeta,
\end{align*}
where $\sigma^*\mu_x$ is smooth and compactly supported in $\overline{Q(\rho)}$, and
\begin{align*}
d\mu_x = \sum_{j=1}^n \partial_{\overline{w}_j} \big[ e^{i\zeta\cdot(x-w)} p (x,\zeta) u(w) \big] \, d\overline{w}_j \wedge \, dw \wedge d\hspace*{-0.08em}\bar{}\hspace*{0.1em} \zeta.
\end{align*}
Then $\sigma^* d \mu_x|_{Q(\rho)}=0$ by holomorphy in $y\in B(0,r)$, and reality in $w$ if $y\not\in B(0,r)$.\\
Next, we show $\sigma$ is an injective immersion, and calculate its pullbacks at fixed $t$. In order to shorten expressions, we write
\begin{align*}
s(y,\xi) &= \delta'\chi_1(y)(1-\chi_2(\xi)), \\
\eta(y,\xi) &= \frac{\delta' (1-\chi_1(y))}{r''-r'} (1-\chi_2(\xi)).
\end{align*}
Then we can calculate
\begin{align*}
dw_j &=
dy_j-it \frac{\xi_j}{|\xi|} \sum_{i=1}^n \partial_{y_i} s(y,\xi) dy_i
-it \sum_{i=1}^n \partial_{\xi_i} \Big( s(y,\xi) \frac{\xi_j}{|\xi|} \Big) d\xi_i -i s(y,\xi) dt, \\
d\zeta_j &=
d\xi_j-it \frac{y_j}{|y|} \sum_{i=1}^n \partial_{\xi_i} \Big( \eta(y,\xi) |\xi| \Big) d\xi_i
-it |\xi|\sum_{i=1}^n \partial_{y_i} \Big( \eta(y,\xi) \frac{y_j}{|y|} \Big) dy_i - i\eta(y,\xi) dt.
\end{align*}
It follows then that the real Jacobian of $\sigma$ has rank $2n$, so $\sigma$ is an injective immersion. But with $t$ kept fixed, $\det d_{(y,\xi)}\sigma(t,y,\xi)$ equals the determinant of
\begin{align*}
\renewcommand\arraystretch{2.5}
\begin{bmatrix}
\Big[ \delta_{ij} - it \frac{\xi_j}{|\xi|} \partial_{y_i} s(y,\xi) \Big]_{i,j=1}^n & \Big[ -it \partial_{\xi_i} ( s(y,\xi) \frac{\xi_j}{|\xi|} ) \Big]_{i,j=1}^n \\
\Big[ -it |\xi| \partial_{y_i} ( \eta(y,\xi) \frac{y_j}{|y|} ) \Big]_{i,j=1}^n & \Big[\delta_{ij} -it \frac{y_j}{|y|} \partial_{\xi_i} ( \eta(y,\xi) |\xi| ) \Big]_{i,j=1}^n
\end{bmatrix},
\end{align*}
which is always bounded in $(y,\xi)$, unlike the "product-like" choice of contour in \cite{Karamehmedovic2015}.
This term appears when pulling back
\begin{align*}
\sigma(t,\cdot,\cdot)^*(dw \wedge d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\zeta) = \det d_{(y,\xi)}\sigma(t,y,\xi) \, dy \wedge d\hspace*{-0.08em}\bar{}\hspace*{0.1em} \xi,
\end{align*}
and will feature next in the deformation of oscillatory integrals.
\newpage
Using the above, we can now, without convergence issues, apply Stoke's theorem.
Thus, by Stokes' theorem for manifolds with corners applied to $\overline{Q(\rho)}$, we get
\begin{align*}
0 = \int_{Q(\rho)} \sigma^*d\mu_x = \int_{Q(\rho)} d(\sigma^*\mu_x) = \int_{\partial Q(\rho)} \sigma^*\mu_x.
\end{align*}
Also, by the above estimate, there is some $C>0$ such that
\begin{align*}
|(G_x\circ \sigma)(t,y,\xi) \det \, d_{(y,\xi)} \sigma(t,y,\xi)|
\leq C
e^{- |\xi|(t\delta' - |\textnormal{Im}\,(x)|)}
\langle \xi \rangle^{d}
1_{\textnormal{supp}(u)}(y),
\end{align*}
which ensures existence of $\int_{\mathcal{C}(t)} \mu_x$ when $t>0$.
If $t=0$, it is meaningful if $p\in S^{-\infty}$, but $x$ must then have zero imaginary part. The aim is to show equivalence with $t=1$.
Let $\sigma_\rho : [t_1,t_2] \times \mathbb{R}^n \times \mathbb{S}^{n-1} \to\mathbb{C}^n \times \mathbb{C}^n$ be defined by
\begin{align*}
(t,y,\omega)
\mapsto
\Big(y-it\delta' \chi_1(y) \omega, \rho \Big[ \omega - i t \frac{\delta' (1-\chi_1(y))}{r''-r'} \frac{y}{|y|} \Big] \Big).
\end{align*}
Similarly, if $x\in B(0,r')$, we get $C'>0$ such that
\begin{align*}
|(G_x\circ \sigma_\rho)(t,y,\omega) \det ( d \sigma_\rho )(t,y,\omega)|
\leq
C' e^{- \rho t\delta'}
\langle \rho \rangle^{d+n}
1_{\textnormal{supp}(u)}(y),
\end{align*}
and as $\sigma(t,y,\xi)=(y,\xi)$ for $\xi\in \overline{B(0,2R)}$, $\sigma^*\mu_x$ vanishes on $(t_1,t_2) \times \mathbb{R}^n \times \partial B(0,2R)$. Combining integrals of opposite orientation, by dominated convergence, we get
\begin{align*}
\int_{\mathcal{C}(t_2)} \mu_x - \int_{\mathcal{C}(t_1)} \mu_x
&= \lim_{\rho \to \infty}\int_{t_1}^{t_2} \int_{y\in \mathbb{R}^n}
\int_{\xi\in\partial B(0,\rho)} (\sigma^* \mu_x)(t,y,\xi) \\
&= \lim_{\rho \to \infty}\int_{t_1}^{t_2} \int_{\mathbb{R}^n}
\int_{ \mathbb{S}^{n-1}}
[(G_x\circ \sigma_\rho) \det ( d \sigma_\rho ) ] (t,y,\omega) \,
\textnormal{vol}_{\mathbb{S}^{n-1}}(\omega)
\, dy \, dt,
\end{align*}
where the integrand is compactly supported in $y$, bounded as above for every $\rho>R$.
It follows that the limit is zero, and we obtain that
\begin{align*}
\int_{\mathcal{C}(t_2)} \mu_x = \int_{\mathcal{C}(t_1)} \mu_x
\quad
\textnormal{if}
\quad
x\in B(0,r').
\end{align*}
Pick $t_0 \in (0,1)$ so that $\mathcal{C}(t_0)\subset \mathbb{C}^n \times W_{\frac{1}{2}}$.
By dominated convergence, we get
\begin{align*}
\textnormal{Op}(p)u(x)
&=
\lim_{\lambda \to 0}
\int_{\mathbb{R}^n} \int_{\mathbb{R}^n}
e^{i\xi\cdot(x-y)} [e^{-\lambda^2 \xi\cdot \xi}p (x,\xi)] u(y) \, dy \,d\hspace*{-0.08em}\bar{}\hspace*{0.1em}\xi \\
&=
\lim_{\lambda \to 0}
\int_{\mathcal{C}({t_0})}
e^{i\zeta\cdot(x-w)} [e^{-\lambda^2 \zeta\cdot \zeta} p(x,\zeta)] u(w) \, dw \wedge d\hspace*{-0.08em}\bar{}\hspace*{0.1em} \zeta \\
&=
\int_{\mathcal{C}({t_0})}
e^{i\zeta\cdot(x-w)} p(x,\zeta) u(w) \, dw \wedge d\hspace*{-0.08em}\bar{}\hspace*{0.1em} \zeta \\
&=
\int_{\mathcal{C}(1)}
e^{i\zeta\cdot(x-w)} p (x,\zeta) u(w) \, dw \wedge d\hspace*{-0.08em}\bar{}\hspace*{0.1em} \zeta,
\end{align*}
which makes sense, because if $\lambda\in \mathbb{R}$, we have
\begin{align*}
|e^{-\lambda^2 \zeta \cdot \zeta}| \leq
e^{-\lambda^2(|\textnormal{Re}\, \zeta |^2-|\textnormal{Im}\, \zeta |^2)}
\leq
e^{-\frac{1}{2}\lambda^2|\textnormal{Re}\, \zeta |^2}
\quad
\textnormal{if}
\quad
|\textnormal{Im}\, \zeta |< \frac{1}{2}|\textnormal{Re}\, \zeta |.
\end{align*}
But now the last integral extends holomorphically in $x$ to the right open set.
\end{proof}
\newpage
Note that for $y\not \in B(0,r)$ the function $u$ in $\mu_x$ may fail to extend holomorphically. But this is not an issue, as deformation then only happens in the $\zeta$-variable.
\begin{corollary}
The same holds if $u\in \mathcal{E}'(\mathbb{R}^n)$ instead of merely $u\in C^\infty_0(\mathbb{R}^n)$.
\end{corollary}
\begin{proof}
First pick a $\chi \in C^\infty_0(B(0,r))$
such that $\chi(y)=1$ for every $y\in \textnormal{supp}(\chi_1)$.
Define $\sigma_y : [0,1] \times \mathbb{R}^n \to \mathbb{C}^n$ by
\begin{align*}
(t,\xi) \mapsto \zeta
=
\xi - i t \frac{\delta' (1-\chi_1(y))}{r''-r'} (1-\chi_2(\xi))|\xi| \frac{y}{|y|},
\end{align*}
and put
\begin{align*}
\mathcal{C}_y (1) = \sigma_y(\{1\} \times \mathbb{R}^n).
\end{align*}
As before, if $\chi_2(\xi)=0$ and $|\textnormal{Re}\,(x)| < r'$, we get
\begin{align*}
\textnormal{Re}\, (i(x-y)\cdot \zeta)
&=
-\xi \cdot \textnormal{Im}\,(x)
+
t \frac{\delta' (1-\chi_1(y))}{r''-r'} |\xi| \frac{y}{|y|}
\cdot (\textnormal{Re}\,(x) - y) \\
&\leq -|\xi|
\Big(\frac{\xi}{|\xi|} \cdot\textnormal{Im}\,(x) + t\frac{\delta' (1-\chi_1(y))}{r''-r'} \Big(|y| -|\textnormal{Re}\,(x)| \Big) \Big) \\
&\leq -|\xi|
\Big( t \delta' (1-\chi_1(y)) - |\textnormal{Im}\,(x)| \Big).
\end{align*}
Taking $\varphi \in C^\infty_0(B(0,r'))$, we have
\begin{align*}
\langle \textnormal{Op}(p)u , \varphi \rangle
&=
\langle \textnormal{Op}(p)(\chi u) , \varphi \rangle
+
\langle \textnormal{Op}(p)((1-\chi)u) , \varphi \rangle \\
&=
\langle \textnormal{Op}(p)(\chi u) , \varphi \rangle
+
\int_{\mathbb{R}^n} \Big\langle u(y), K(x,y) \Big\rangle \, \varphi(x) \, dx,
\end{align*}
where $K: B(0,r')\times \mathbb{R}^n \to \mathbb{C}$ is smooth, and is rapidly decaying in $y$ uniformly in $x$.
Note that, by the above theorem, $\textnormal{Op}(p)(\chi u)$ extends holomorphically to
\begin{align*}
B(0,\min\{r',r_0\}) + iB(0,\min\{\delta',\delta_0\})
\quad
\textnormal{when}
\quad
\frac{\delta'}{r-r'}<\frac{\delta'}{r''-r'}<\epsilon,
\end{align*}
and so it remains to show that the residual term extends holomorphically to this set.
Pick $t_0 \in (0,1)$ so that $\mathcal{C}_y (t_0)\subset W_{\frac{1}{2}}$. We deform from $0$ to $t=t_0$ with $\mathcal{C}_y (t_0)\subset W_{\frac{1}{2}}$, where $p$ is multiplied by a Gaussian symbol, and finally from $t=t_0$ to $t=1$ directly.
This is facilitated by an argument via Stokes' theorem very similar to the above one.
Then, by dominated convergence, $K$ has the form
\begin{align*}
K(x,y)
&=
\lim_{\lambda \to 0}\int_{\mathbb{R}^n} e^{i(x-y)\cdot \xi} [e^{-\lambda^2 \xi \cdot \xi}(1-\chi)(y)p(x,\xi) ] \, d\hspace*{-0.08em}\bar{}\hspace*{0.1em} \xi \\
&=
\lim_{\lambda \to 0}
\int_{\mathcal{C}_y({t_0})}
e^{i(x-y)\cdot \zeta} [e^{-\lambda^2 \zeta \cdot \zeta}(1-\chi)(y)p(x,\zeta) ] \, d\hspace*{-0.08em}\bar{}\hspace*{0.1em} \zeta \\
&=
\int_{\mathcal{C}_y(1)}
e^{i(x-y)\cdot \zeta} (1-\chi)(y)p(x,\zeta) \, d\hspace*{-0.08em}\bar{}\hspace*{0.1em} \zeta,
\end{align*}
and $K$ vanishes unless $y\not \in \textnormal{supp}(\chi_1)$, in which case
\begin{align*}
\textnormal{Re}\, (i(x-y)\cdot \zeta)
\leq -|\xi| ( t \delta' - |\textnormal{Im}\,(x)| ).
\end{align*}
This means that the last deformed integral is absolutely convergent if $|\textnormal{Im}\, (x)|<\delta'$, and thus
extends holomorphically in $x$ for each $y\in \mathbb{R}^n$ to the same open set.
\end{proof}
\newpage
\begin{theorem}
Let $U\subset \mathbb{R}^n$ be open, and $U_\mathbb{C}$ be a tube-domain about $U$ in $\mathbb{C}^n$. (This means $z \in U_\mathbb{C}$ implies $\textnormal{Re}\, z \in U$ and $\textnormal{Re}\,(z)+iy \in U_\mathbb{C}$ for all $|y|\leq |\textnormal{Im}\,(z)|$.)
Assume $p|_{U \times \mathbb{R}^n}$ extends holomorphically into
$U_\mathbb{C} \times W_\epsilon$, and satisfies
\begin{align*}
\sup_{(x,\zeta) \in K\times W_\epsilon }\langle \textnormal{Re}\, \zeta \rangle^{-d} |p(x,\zeta)| < \infty
\quad
\textnormal{for any}
\quad
K\subset\subset
U_\mathbb{C}.
\end{align*}
Let $u\in \mathcal{E}'(\mathbb{R}^n)$ be real-analytic on $U$, with $u|_U$ extending holomorphically into $U_\mathbb{C}$.
Then $\textnormal{Op}(p)u|_U$ extends holomorphically into
\begin{align*}
\{ z\in U_{\mathbb{C}} \, | \, |\textnormal{Im}\, z | < \epsilon \, \textnormal{dist}(\textnormal{Re}\, z, \partial U) \}.
\end{align*}
\end{theorem}
\begin{proof}
The above corollary is valid over any $x\in U$ by translating to the origin.
This gives a holomorphic extension of $\textnormal{Op}(p)u|_{B(x,r')}$ into $B(x,r')+iB(0,\delta')$ with
\begin{align*}
\delta' < \epsilon (\textnormal{dist}(x, \partial U) - r'),
\end{align*}
and by making $r'$ small, we can make $\delta'$ arbitrarily close to $\epsilon \, \textnormal{dist}(x, \partial U)$.
\end{proof}
\section{Remarks}
This removes the restrictive topology put on the symbols in \cite{Karamehmedovic2015}. It reduces the situation to symbols defined by Boutet de Monvel \cite{BoutetDeMonvel1972} and Tr\`eves \cite{TI}.
However, the approach to the original question raised in \cite{Karamehmedovic2015} has since changed a lot, and in \cite{Karamehmedovic2021}, we approach it via precise local convergence radius estimates instead. \\
The reason is that it is hard to get parametrix symbols in the right analytic class.
It works in \cite{Karamehmedovic2015} because the geometry is simple - the boundary is a piece of a hyperplane.
To overcome this, the analytic symbols are replaced with pseudo-analytic amplitudes, which have weaker conditions imposed on them - analyticity is replaced by an estimate.
But these are equivalent to analytic amplitudes up to an exponentially small error, and gives a way to build pseudo-analytic parametrices from formal asymptotic sums.
This can be exploited to obtain controlled convergence radius estimates.
\bibliographystyle{siamplain}
|
2,877,628,090,070 | arxiv | \section{Dual-Band WiFi Fingerprint Map}
\section{Recovery of Abnormal Signal}
\label{sec:signal_supplement}
Refinement of the fingerprint database is needed to mitigate abnormal signal caused due to system noise in the temporary database. In the building, the fluctuation of the wireless signal sometimes gets high due to the multi-path effect. Also occasional disturbances can increase the inaccurate signal strength in the database. The recovered dual-band signals may also suffer errors. Therefore, {\tt{AuF}}\ needs to identify these abnormal signals and improve them with more reliable ones.
First, abnormal detection based on hypothesis testing is conducted to identify abnormal signals. Then {\tt{AuF}}{} will try to recover them from the previously constructed database.
As for fingerprinting for the first time, the robot just recollects fingerprints thereof to calibrate these signals.
\subsection{Detection of Abnormal Signal}
{\tt{AuF}}'s identification of abnormal signals is grounded on the assumption that signal propagation is smooth, which is justified by Eq.~\ref{eq:shadow_log}. So, the basic idea is to identify samples which obviously deviate from the smooth signal model. Specifically, {\tt{AuF}}'s detection of abnormal signal includes two steps: (i) fitting the spatial signal model with a Gaussian process and estimating the norm value for each measured signal, and (ii) performing the largest residual test to identify abnormal signals.
\vspace{+3pt}
\noindent $\bullet$ {\bf Estimation with Gaussian Process.~}
The Gaussian process regression is used to fit the spatial distribution of the signal strength for two reasons. First, a Gaussian process assumes that measurements are drawn from random variables conforming Gaussian distributions, as analytically/empirically corroborated in Eq.~\ref{eq:shadow_log} and Fig.~\ref{fig:fit}. Second, the Gaussian process regression predicts the distribution of RSSI at a certain point, including the mean and varianc, which facilitates the next step of {\tt{AuF}}, i.e., the largest residue test.
Let the fingerprint training set be a collection of fingerprints $\mathcal{D} = \{ ({x_1},{y_1}),({x_2},{y_2}), \ldots ,({x_n},{y_n})\}$,
where $x$ denotes 2-D coordinates and $y$ denotes the RSSI of an AP. It is assumed that the measured signal strength $y$ consists of a {\em true} signal strength $f(x)$ and an independent Gaussian noise $\omega \in N(0,{\sigma ^2})$, i.e., ${y_i} = f({x_i}) + {\omega _i}$. The collection of $f({x})$ is to be drawn from a Gaussian process, thus it conforms a multivariate Gaussian distribution with mean function $m(\cdot)$ and covariance function $k(\cdot, \cdot)$:
{
\footnotesize
{
\begin{equation}
\begin{bmatrix}
f(x_1)\\
\vdots \\
f(x_n)
\end{bmatrix}
\sim
{N}
\left (\begin{bmatrix}
m(x_1)\\
\vdots \\
m(x_n)
\end{bmatrix}
,
\begin{bmatrix}
k(x_1, x_1) & \cdots & k(x_1, x_n) \\
\vdots & \ddots & \vdots \\
k(x_n, x_1) & \cdots & k(x_n, x_n)
\end{bmatrix}
\right ) .
\label{eq:GP_model}
\end{equation}
}
}
In general, mean function $m(\cdot)$ is set to $0$, and kernel functions are used to represent the covariance $k(\cdot,\cdot)$. Here the squared exponential kernel is adopted:
\begin{equation}
k(x_i, x_j) = \sigma _f \exp\left ( -\frac{1}{2l^2}\left \| x_i-x_j \right \|^2 \right ),
\label{eq:se_kernel}
\end{equation}
where $\sigma _f$ is signal variance and $l$ is a length scale. Both parameters determine the smoothness of the function $f(x)$ estimated by the Gaussian process.
Then we represent the distribution of $y=f(x)+w$. Since it has a zero-mean noise, its mean function is still $0$.
The noise terms can be incorporated into the covariance function:
\begin{equation}
\mathrm{cov} ({y_i},{y_j}) = k(x_i, x_j) + ({\sigma ^2}){\delta _{ij}},
\end{equation}
where ${\delta _{ij}} = 1 $ if $i = j$ and zero otherwise.
Denote the testing set as $\mathcal{T} = \{ ({x^*_1},{y^*_1}),({x^*_2},{y^*_2}), \ldots ,({x^*_m},{y^*_m})\}$, which is drawn from the same unknown distribution as $\mathcal{D}$. For notational convenience, we aggregate $n$ input vectors $x_i$ of $\mathcal{D}$ into $n \times 2$ matrix $X$, $n$ output values $y_i$ of $\mathcal{D}$ into $n \times 1$ vector $Y$, $m$ input vectors $x^*_i$ of $\mathcal{T}$ into $m \times 2$ matrix $X^*$, $m$ output values $y^*_i$ of $\mathcal{T}$ into $m \times 1$ vector $Y^*$. The training points and testing points must have a joint multivariate Gaussian distribution:
{
\small
\begin{equation}
\begin{bmatrix}
Y\\
Y^*
\end{bmatrix}
\sim
N
\left( 0,
\begin{bmatrix}
K(X,X)+\sigma^2 I & K(X,X^*) \\
K(X^*,X) & K(X^*,X^*) + \sigma^2 I
\end{bmatrix}
\right),
\end{equation}
where
\begin{equation}
\begin{aligned}
K(X,X)[i,j]=k(x_i,x_j)&, K(X,X^*)[i,j]=k(x_i,x^*_j),\\ K(X^*,X)[i,j]=k(x^*_i,x_j)&, K(X^*,X^*)[i,j]=k(x^*_i,x^*_j).
\end{aligned}
\end{equation}
}
With the rules for conditional density, we get the predicted value at $X^*$ conditioned on training data $X, R$:
$Y^*|X^*,X,Y \sim \mathcal{N}({\mu ^*}, {\Sigma ^*})$, where
\begin{equation}
\begin{aligned}
{\mu ^*} = &K({X^*},X){(K(X,X) + \sigma^2 I)^{ - 1}}Y\\
\Sigma ^* = &K({X^*},{X^*})+\sigma ^2 I \\
&- K({X^*},X){(K(X,X) + \sigma^2 I)^{ - 1}}K(X,{X^*})
\end{aligned}
\label{eq:GP_prediction}
\end{equation}
As can be seen from Eq.\ref{eq:GP_prediction}, the predicted mean is a linear combination of observed signal strengths $Y$, and the weights depends on covariance $K({X^*},X)$, while the squared exponential kernel determines that nearby function values are highly correlated. On the other words, it is believed that RSSIs are locally smooth, thus the data will be regraded as outliers if it disagrees with our prior knowledge.
To examine the collected fingerprints, we predict signal strengths of the training data, i.e., $X*=X$. Thus, a measured value $y$ and its expectation and variance $\mu^*_x, \Sigma^*_x$ are obtained for each location $x$.
{\tt{AuF}}\ trains the parameters using scikit-learn~\cite{williams2006gaussian}.
\vspace{+3pt}
\noindent $\bullet$ {\bf Largest Normalized Residual Test.~}
The outlier identification is through the analysis of residues. Normalizing residues is necessary for us to find which one most deviate the estimation:
\begin{equation}
r^N = \frac{y-\mu^*_x}{\sqrt{\Sigma^*_x}} \sim \mathcal{N}(0,1).
\end{equation}
Then existence of outliers can be verified by the following hypothesis test:
\begin{itemize}
\item if any $|r^N|>t$ in collected fingerprints, there is a suspicion of bad data.
\item if all $|r^N| \leq t$, the hypothesis that there is no bad data is supported.
\end{itemize}
{\tt{AuF}}\ set $t$ to 1.96.
From \cite{grubbs1950sample}, it is shown that for a measurement set the measurement with the largest normalized residual contains a gross error. As a result, one abnormal signal can be identified by testing $r^N_{max} > t$. To identify all abnormal signal, the largest normal residual test performs within a loop:
\begin{enumerate}
\item Estimate expectations and variances from the training set.
\item If the largest residue exceeds the threshold, withdraw it from the training set, then go to step one. If not, finish the test.
\end{enumerate}
\subsection{Signal Recovery}
The next step is to recover these abnormal signals from the previously constructed fingerprint database.
Although the pattern of WiFi change after a long time is difficult to analyze~\cite{7174948}, the short time change of the WiFi signal is (relatively) predictable. We collect samples along a path in three days with the same collection method in Fig.~\ref{fig:24and5GHz}. The alternation of the signal after three days can be approximated as a shift, as shown in Fig.~\ref{fig:shift}. So we shift the corresponding signals in the past database to recover the abnormal signals.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.98\linewidth]{figure/signal_shift.eps}
\caption{Difference of the signal strength in three days.}
\label{fig:shift}
\end{figure}
\section{Evaluation}
\label{sec:evaluation}
We present our evaluation of {\tt{AuF}}\ in this section. First we explain the experiment settings in Section~\ref{sec:sec:methodlogy}. Then the time and energy efficiency of {\tt{AuF}}{} is compared to the baseline in Section.~\ref{sec:sec:time_energy}. In Section.~\ref{sec:sec:signal_on_fingerprintmap} the impacts of the signal recovery methods on the spatial domain and on the time domain on the fingerprint database are visualized. Finally, the localization accuracy of {\tt{AuF}}{} is evaluated in Section.~\ref{sec:sec:accuracy}.
\begin{figure}[htbp]
\centering
\begin{minipage}[b]{0.4\linewidth}
\begin{subfigure}[t]{1\linewidth}
\centering
\includegraphics[width=1\linewidth]{figure/robot.png}
\caption{}
\end{subfigure}
\end{minipage}
\begin{minipage}[b]{0.58\linewidth}
\begin{subfigure}[t]{1\linewidth}
\centering
\includegraphics[width=1\linewidth]{figure/3F.eps}
\caption{}
\end{subfigure}
\\
\begin{subfigure}[t]{1\linewidth}
\centering
\includegraphics[width=1\linewidth]{figure/6F.eps}
\caption{}
\end{subfigure}
\end{minipage}
\caption{(a) SLAM-enabled robot. (b) (c) are separately 3rd and 6th floors of our Department building.}
\label{fig:robot}
\end{figure}
\subsection{Methodology}
\label{sec:sec:methodlogy}
\vspace{+3pt}
\noindent $\bullet$ {\bf Experiment Settings.~}
We have deployed and evaluated {\tt{AuF}}{} on two sites (the 3rd floor and the 6th floor) of our Department building. These two areas and AP deployment there are shown in Figs.~\ref{fig:robot}(b)(c).
We use a Pioneer-3DX robot in Fig.~\ref{fig:robot} equipped with a HOKUYO UTM-30LX laser module as the agent. A LENOVO ideapad Y700 laptop with Ubuntu 16.04 operating system serves as the upper computer, and ROS (a robot operating system) is adopted to control the robot. A MI Note Pro smartphone is attached to the robot and used as the WiFi scanner during the site survey with 3 scan interval.
The robot surveys the floor at a speed of $0.5$m/s.
Before and after the site survey, we measure the voltage of the robot, the laser and the laptop. Then we use the discharge curves to calculate consumed power.
We update the fingerprint database every three days in twelve days.
\vspace{+3pt}
\noindent $\bullet$ {\bf Baseline.~}
For comparison, we also implement the survey-with-sojourn method: The robot surveys the building according to the same route as {\tt{AuF}}{}'s survey, but sojourns at each grid point generated in Sec.~\ref{sec:floor_recognition} for 10s.
\vspace{+3pt}
\noindent $\bullet$ {\bf Gaussian Process Fingerprint Map.~}
For visualization of the fingerprint database and localization, Gaussian process fingerprint maps are separately generated using the fingerprint database of {\tt{AuF}}{} and the database of the baseline.
Specifically, a Gaussian process model is trained by fingerprints after completing the site survey, which outputs the mean and variance of the signals' RSSI at every grid point generated in Sec.~\ref{sec:floor_recognition}. {\tt{AuF}}\ uses these RSSI statistics as the fingerprint thereof.
\vspace{+3pt}
\noindent $\bullet$ {\bf Localization Method.~}
We implemented the following simple but classic localization methods on top of the fingerprint map constructed above, and examined the resultant localization accuracy when a person holding a smartphone walks around on the two sites.
\vspace{+3pt}
{\em \underline{(1)~~Bayes.}~}
The first localization method exploits the fingerprint map constructed by {\tt{AuF}}\ with Bayes method~\cite{roos2002probabilistic}.
Denote the reference locations as $L = \{ {l_1},{l_2}, \ldots ,{l_m}\}$, and the observation vector as $o_{1\times b}$.
The reference location with the maximum probability $p(l|o)$ is used as the predicted location $\hat l$:
\begin{equation}
\begin{aligned}
&p(l|o) = {\frac{{p(o|l)p(l)}}{{p(o)}}}\propto p(o|l),
\\
&\hat l = \mathop {\arg \max }\limits_{{l_j}\in{L}} p(l_j|o)=\mathop {\arg \max }\limits_{{l_j}\in{L}}\prod\limits_{i = 1}^b {G({o_i}|{l_j})} .
\end{aligned}
\end{equation}
where $G$ is the probability of $v_i$ in the corresponding Gaussian distribution.
We then further improve the thus-obtained location results with particle filter.\footnote{Please see \cite{evennou2006advanced} for details of particle filter.}
\vspace{+3pt}
{\em \underline{(2)~~KNN.}~}
We also implemented a KNN-based localization method on top of {\tt{AuF}}, i.e., locating the online collected WiFi signals to the $K$ reference locations with the closest WiFi fingerprints. We used a K of $2$ unless specified otherwise. Again, particle filter is then used to further improve the localization accuracy.
\begin{table}[htbp]
\caption{Time and energy cost to survey 3F.}
\centering
\begin{tabular}{ccccc}
\hline
Method & Time & Robot & Laser & Laptop \\ \hline
{\tt{AuF}}{} & 43 min & 30 Wh & 7 Wh & 23 Wh \\
Baseline & 121 min & 69 Wh & 17 Wh & 67 Wh \\ \hline
\end{tabular}
\label{table:time_3F}
\end{table}
\begin{table}[htbp]
\caption{Time and energy cost to survey 6F.}
\centering
\begin{tabular}{ccccc}
\hline
Method & Time & Robot & Laser & Laptop \\ \hline
{\tt{AuF}}{} & 51 min & 33 Wh & 8 Wh & 41 Wh \\
Baseline & 177 min & 90 Wh & 25 Wh & 113 Wh \\ \hline
\end{tabular}
\label{table:time_6F}
\end{table}
\subsection{Time Overhead of Site Surveys}
\label{sec:sec:time_energy}
Tables.~\ref{table:time_3F} and \ref{table:time_6F} summarize the time and energy overhead to survey the sites by {\tt{AuF}}{} and the baseline.
Clearly, travel-without-sojourn requires far less time than travel-with-sojourn: {\tt{AuF}}{} saves 64\% time on the 3rd floor when compared to the traditional survey method, while a 71\% reduction is achieved on the 6th floor.
We also compare the energy overhead. On 3F, the energy consumed by the robot motors is reduced by 57\% with {\tt{AuF}}{} compared to the baseline. The de/acceleration is quite frequent when survey the building with sojourn, due to the dense survey locations required for accurately modeling the spatial distribution of signal strength. In our experiments, the robot performs de/acceleration once for every 0.8m on average.
{\tt{AuF}}{} also saves 59\% energy of the laser on 3F, which operates at constant power.
We can see that the laptop is a major part of energy consumption, since it performs intense computation to localize and navigate the robot. On 3F, {\tt{AuF}}{} consumes 66\% energy less than the baseline.
In total, {\tt{AuF}}{} reduces the energy consumption by 61\%.
\subsection{Signal Recovery on Fingerprint Map}
\label{sec:sec:signal_on_fingerprintmap}
\begin{figure}[htbp]
\centering
\begin{subfigure}[t]{0.98\linewidth}
\centering
\includegraphics[width = 1 \linewidth]{figure/fg_without_recover_2G.eps}
\caption{}
\end{subfigure}
\\
\begin{subfigure}[t]{0.98\linewidth}
\centering
\includegraphics[width = 1 \linewidth]{figure/fg_with_recover_2G.eps}
\caption{}
\end{subfigure}
\caption{The fingerprint map constructed w/o and w/ lost signal recovery.}
\label{fig:GP_raw_recovered_model}
\end{figure}
Next we examine the impact of lost signal recovery on the fingerprint map.
Fig.~\ref{fig:GP_raw_recovered_model} shows the fingerprint maps constructed by {\tt{AuF}}{} with and without signal recovery using the dual-band signals, in which {\em Measured Signal} denotes fingerprints with the measured signal, {\em Lost Signal} denotes fingerprints with the lost signal, which is set to -100 dBm, and {\em Recovered Signal} denotes fingerprints with the recovered signal.
{\em No Sojourn Map} in Fig.~\ref{fig:GP_raw_recovered_model} represents the mean of the fingerprint map constructed with only data collected during survey without sojourn, and {\em Signal Recovery Map} represents the mean of the fingerprint map constructed with lost signal recovery, while {\em Baseline Fingerprint map} represents the mean of the fingerprint map constructed with data of the baseline.
Signal loss occurs even when strong signals could be received, causing significant uncertainty in the thus-constructed fingerprint map.
Such uncertainty can be effectively mitigated via {\tt{AuF}}'s signal recovery, as observed in Fig.~\ref{fig:GP_raw_recovered_model}.
\begin{figure}[htbp]
\centering
\begin{subfigure}[t]{0.98\linewidth}
\centering
\includegraphics[width = 1 \linewidth]{figure/fg_without_resurvey.eps}
\caption{}
\end{subfigure}
\\
\begin{subfigure}[t]{0.98\linewidth}
\centering
\includegraphics[width = 1 \linewidth]{figure/fg_with_resurvey.eps}
\caption{}
\end{subfigure}
\caption{Localization w/o and w/ abnormal signal recovery.}
\label{fig:resurvey_fg}
\end{figure}
We next examine the impact of abnormal signal recovery on the fingerprint map.
Fig.~\ref{fig:resurvey_fg} shows fingerprint maps constructed with and without abnormal signal recovery, in which {\em Regular Signal} denotes fingerprints with the regular signal, {\em Abnormal Signal} denotes fingerprints with the identified abnormal signal, and {\em Recovered Signal} denotes fingerprints with signals recovered by previous data.
{\em No Sojourn map} and {\em Baseline Fingerprint map} in Fig.~\ref{fig:resurvey_fg} have the same meaning as in Fig.~\ref{fig:GP_raw_recovered_model}, while {\em Signal Recovery map} represents the mean of fingerprint map constructed with abnormal signal recovery.
As can be seen from Fig.~\ref{fig:resurvey_fg}, non-smooth points are identified as abnormal signals. These signals make the local shapes of the fingerprint map deviate from the ground truth. But abnormal signal recovery calibrates these flaws and make the fingerprint map of {\tt{AuF}}{} almost coincide with the baseline map.
\subsection{Localization Accuracy}
\label{sec:sec:accuracy}
Next we examine {\tt{AuF}}{}'s impact on localization accuracy. The examination is three-fold: (i) examine the impact of signal recovery; (ii) examine the impact of signal completion; (iii) examine the impact of {\tt{AuF}}{} and compare the localization accuracy of {\tt{AuF}}{} with that of the baseline.
\noindent $\bullet$ {\bf Impact of Lost Signal Recovery.~}
\begin{figure}[htbp]
\centering
\begin{subfigure}[t]{0.48\linewidth}
\centering
\includegraphics[width = 1\linewidth]{figure/recover_loc_3F.eps}
\caption{}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\linewidth}
\centering
\includegraphics[width = 1\linewidth]{figure/recover_loc_3F_corridor.eps}
\caption{}
\end{subfigure}
\caption{localization w/o and w/ Lost signal recovery.}
\label{fig:recovery_localization}
\end{figure}
Fig.~\ref{fig:recovery_localization} summarizes the localization results of fingerprint maps constructed with and without signal recovery, where {\em No Sojourn KNN} indicates using data collected during survey without sojourn and KNN method for localization, and the other legends' meaning is interpreted in the same way. In Fig.~\ref{fig:recovery_localization}(a) signal recovery improves mean error of KNN method from 5.4m to 2.3m. And localization in the narrow corridor is more difficult, but signal recovery still works. Seen from Fig.~\ref{fig:recovery_localization}, using Bayes method, mean error decrease from 4.5m to 3.5m.
But the max error does not degrades much with the help of lost signal recovery, e.g. max errors with/without signal recovery are close In Fig.~\ref{fig:recovery_localization}. This can be explained by the error caused by recovery from an abnormal signal, which is solved by abnormal signal recovery.
\noindent $\bullet$ {\bf Impact of abnormal signal recovery.~}
Fig.~\ref{fig:supplement_localization} summarize the localization results of fingerprint maps constructed with and without signal completion. Obviously both the mean error and the max error are depressed by signal completion. Take bayes method in Fig.~\ref{fig:supplement_localization}(a) as an example, the mean error decreases from 3.4m to 2.3m, while the max error decreases from 9.3m to 5.4m.
\begin{figure}[t]
\centering
\begin{subfigure}[t]{0.48\linewidth}
\centering
\includegraphics[width = 1\linewidth]{figure/resurvey_loc_3F.eps}
\caption{}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\linewidth}
\centering
\includegraphics[width = 1\linewidth]{figure/resurvey_loc_3F_corridor.eps}
\caption{}
\end{subfigure}
\caption{Abnormal signal recovery's impact on localization.}
\label{fig:supplement_localization}
\end{figure}
\noindent $\bullet$ {\bf Comparison with Baseline.~}
\begin{figure}[t]
\centering
\begin{subfigure}[t]{0.48\linewidth}
\centering
\includegraphics[width = 1\linewidth]{figure/compare_baseline_3F.eps}
\caption{}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\linewidth}
\centering
\includegraphics[width = 1\linewidth]{figure/compare_baseline_3F_corridor.eps}
\caption{}
\end{subfigure}
\caption{Accuracy comparison between {\tt{AuF}}{} and the baseline.}
\label{fig:compare_baseline_localization}
\end{figure}
The performance comparison in Fig.~\ref{fig:compare_baseline_localization} clearly shows that only improving the survey speed to reduce offline work is not desirable with the severely diminishing localization performance. In Fig.~\ref{fig:compare_baseline_localization}(a), using bayes method the mean error and the max error of raw increase 1.7m and 5.5m compared to the baseline.
On the other hand, the dataset provided by {\tt{AuF}}{} has no weakness in performance compared to the baseline. A slight improvement can be seen in Fig.~\ref{fig:compare_baseline_localization}(a), the point below which there are 80\% errors shifts from 2.1m to 1.9m, when adopts the fingerprint of {\tt{AuF}}{} rather than the baseline. In Fig.~\ref{fig:compare_baseline_localization}(d), {\tt{AuF}}{}'s has the little worse mean error but the little better max error. Using bayes method the baseline's mean error and max error is 2.2m and 7.5m, while {\tt{AuF}}{}'s mean error and max error is 2.4m and 4.9m.
\section{Fingerprint Database Construction}
\label{sec:survey_process}
In this section we describe {\tt{AuF}}'s construction of the fingerprint database. First we verify the feasibility of continues movement of the robot during collecting fingerprints. Then we explain the dual-band (i.e., 2.4GHz and 5GHz) signal recovery and the dependent signal model thereof.
\subsection{Survey without Sojourn}
\label{sec:fast_survey}
Unlike traditional systems~\cite{ocana2005indoor,youssef2005horus,832252} requesting the surveyor/robot stopping at locations and scanning WiFi multiple times, the robot in {\tt{AuF}}\ travels through generated grids without sojourn to shorten the survey process.
{\tt{AuF}}'s survey-without-sojourn has two challenges.
Intuitively, the robot's continuous movement may affect the collected WiFi signals due to Doppler shift.
To closely examine this, we use the robot to survey the area in Fig.~\ref{fig:path_planning}(a) to collects the WiFi signal from a given AP.
We collect three datasets, data collected with a 9s sojourn at each grid, with a 3s sojourn at each grid, and without sojourn.
Fig.~\ref{fig:speed}(a)(b)(c) visualize the heatmaps of the three dataset, and Fig.~\ref{fig:speed}(d) uses a scatter plot to represent relationships of signals collected with or without sojourn, showing no clear dependency of signal strength with the movement.
This is likely due to the relatively slow travel speed of the robot, e.g., 0.5m/s in the above measurements.
Another challenge is even more severe: with one fingerprint per location, how to ensure that this fingerprint can reliably represent the local signal strength. {\tt{AuF}}\ identifies, and then improves, the locations whose signal representation is unreliable. The following secs.~\ref{sec:signal_recovery} and \ref{sec:signal_supplement} separately explains separately two kind of unreliable signals and the corresponding remedies.
\subsection{Fingerprint Map Construction.~}
With the above collected/recovered/unrecoverable signals, {\tt{AuF}}\ constructs the fingerprint map based on the Gaussian process uncovered in Eq.~(\ref{eq:shadow_log}).
Specifically, the Gaussian model takes as input the measured/recovered signals and the location at which they are collected, and outputs the mean and variance of the signals' RSSI at a given reference location, which is used as the reference location's fingerprint.
Fig.~\ref{fig:Gaussian_model} illustrates such a fingerprint map construction process of {\tt{AuF}}.
\section{Initialization}
\label{sec:floor_recognition}
{\tt{AuF}}{} needs to do some preparations before its site survey, which needs to be conducted only once. Also, this process is automatic thanks to the robotics technique, thus making {\tt{AuF}}{} easy to deploy.
The first task of {\tt{AuF}}'s initialization is to discover the floor map of the floor-of-interest, which is the prerequisite for the robot's navigation indoors. Next, {\tt{AuF}}'s divides the floor map into regions with regular shapes. Then in each region, {\tt{AuF}}\ plans the surveying paths for its robot.
\vspace{+3pt}
\noindent $\bullet$ {\bf Floor Map Discovery.~}
{\tt{AuF}}\ discovers the floor map using a SLAM-enabled robot, as shown in Fig.~\ref{fig:robot}. The robot scans its surrounding environment with laser while surveying the building, and constructs the building's floor map based on the scanning results. The spanning-tree algorithm~\cite{gabriely2001spanning} is adopted to achieve an automatic SLAM process.
A gray-scale grid map (e.g., as shown in Fig.~\ref{fig:path_planning}(a)) is obtained after the site survey, in which the pixels with gray-scales smaller than a pre-defined threshold represents the obstacles of the building (e.g., walls).
{\tt{AuF}}\ then increases the obstacle area by the size of the robot, thus obtaining the floor area where the robot can travel freely, as shown in Fig.~\ref{fig:path_planning}(b).
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{0.24\linewidth}
\includegraphics[width=1\linewidth]{figure/mymapG.png}
\caption{The raw floor map identified using laser. The two marks are explained in Sec.\ref{sec:signal_recovery}}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.24\linewidth}
\includegraphics[width=1\linewidth]{figure/costmap.png}
\caption{The reduced floor map in which the robot can travel freely.}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.24\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/fsi.eps}
\caption{The darker the pixel,
the highest its value.}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.24\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/region_group.eps}
\caption{Adjacent pixels with same value are aggregate into a region.}
\end{subfigure}
\\
\begin{subfigure}[t]{0.24\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/remove_ripple.eps}
\caption{Small ripples in Fig.~\ref{fig:path_planning}(d) are removed.}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.24\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/merge_similar.eps}
\caption{Regions with similar value in Fig.~\ref{fig:path_planning}(e) are merged, and final regions are founded.}
\end{subfigure}
\begin{subfigure}[t]{0.24\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/path_plan.eps}
\caption{Each region's direction is determined by its shape.}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.24\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/reference_location.eps}
\caption{The robot needs to visit red dots according to regions' principal axis.}
\end{subfigure}
\caption{The whole process of initialization. }
\label{fig:path_planning}
\end{figure*}
\vspace{+3pt}
\noindent $\bullet$ {\bf Map Segmentation.~}
Instead of directly planning paths for the entire floor, {\tt{AuF}}{} first divides the above-recognized floor into subareas before planning paths for the robot, because of two reasons. First, directly planning path covering a irregular space is a challenging, while identifying the optimal survey of a regular space is much more feasible --- the map segmentation facilitates planning paths to efficiently cover a irregular space \footnote{By ``efficient", we mean the paths have the least turning/overlapping}.
The second reason necessities {\tt{AuF}}'s map segmentation is the computational cost of constructing the fingerprint database, which grows in cubic with the number of collected samples. In fact, site surveys are conducted separately in each region to limit the size of samples during once site survey. The computation complexity of {\tt{AuF}}{} and its solution will be discussed in detail in Sec.~\ref{sec:discussion}.
{\tt{AuF}}\ grounds its map segmentation using MAORIS~\cite{mielle2018method}. Below we briefly describe its major steps:
First, a free space image, whose pixel value represents the size of the region it belongs to, is created based on the distance image of the floor map\footnote{Pixels in the distance image represent Euclidean distance to the nearest obstacle}. The free space image is initiated with an empty image. A circular mask is created for each pixel of the distance image and centered on the pixel, whose radius are determined by the value of the pixel. Then for pixels in the circular mask, if the value of the equivalent pixel in the free space image is less than the circle radius, the value of the pixel in the free space image is changed to the radius. The result is showed in Fig.~\ref{fig:path_planning}(c).
Then, in the free space image, adjacent pixels with same value are grouped into regions, which can be seen in Fig.~\ref{fig:path_planning}(d).
Clearly, Fig.~\ref{fig:path_planning}(d) is over-segmented, and some regions are too small to represent a semantic area. MAORIS removes these ripples with a simple rule: a region is incorporated into its neighbor if they overlap for more than 40\%, as visualized in Fig.~\ref{fig:path_planning}(e).
After removing ripples, some regions are still pieces of the same place like the left corridor in Fig.~\ref{fig:path_planning}(e), resulted from slightly different distances to the obstacle. These regions with similar value are merged, and Fig.~\ref{fig:path_planning}(f) plots the final segmented map.
\vspace{+3pt}
\noindent $\bullet$ {\bf Path Planning.~}
After segmenting the floor map, {\tt{AuF}}\ needs to determine the principal axis direction for each region in the map, moving along which requires the minimum number of robot's turning.
Let us consider Fig.~\ref{fig:path_planning}(g) as an example. Denoting the coordinates of the pixels in one region ${\rm{C}}$ as ${P}$, we get a set of coordinates representing that area:
\begin{equation}
{\rm{C}} = \left\{ {P_1}, {P_2}, \cdots, {P_l} \right\}.
\end{equation}
We calculate the variance of all pixels in $C$ by:
\begin{equation}
Var(C)
=\begin{bmatrix}
cov(x,x) &cov(x,y)\\
cov(y,x) & cov(y,y)
\end{bmatrix},
\end{equation}
where $x, y$ denotes the pixels' coordinates.
Because $Var(C)$ is a real symmetric matrix, its two eigenvectors (denoted as $\rm A_1$ and $\rm A_2$, as shown in Fig.~\ref{fig:path_planning}) are orthogonal, representing the two directions of the spatial distribution of the region's elements. So, the $\rm A_1$ with a higher eigenvalue is the principal axis of region $\rm{C}$.
Next, {\tt{AuF}}\ discretizes each region into grids (we use $0.8m\times0.8m$ grids) according to the corresponding principal axis direction, and combines the grid centers as the traveling paths of the robot.
To acheive this, {\tt{AuF}}\ identifies the minimum rectangles covering each of the region. Denoting $F$ as the world coordinate system, ${{}^F{P}}$ as the pixel point in $F$.
{\tt{AuF}}\ uses the two directions $\rm A_1$ and $\rm A_2$ as the region's coordinate system (i.e., {$F'$} in Fig.~\ref{fig:path_planning}).
The rotation transformation from ${{}^{F'}{P}}$ to ${{}^F{P}}$ is thus:
\begin{equation}
{^{F'}{P}}=\left[ {\begin{array}{*{20}{c}}
{\rm {A_1}}^\mathrm{T}\\
{\rm {A_2}}^\mathrm{T}
\end{array}} \right]\cdot {{}^F{P}}.
\end{equation}
{\tt{AuF}}\ then finds region-C's max/minimum coordinates according to {$F'$}, thus identifying a minimum rectangle covering the area of region-C.
At last, {\tt{AuF}}\ divides each of the above-identified rectangles into grids according to the coordinate system {$F'$}, and ask the robot to pass the grids' center in boustrophedon like Fig.~\ref{fig:path_planning}(h).
\section{Introduction}
\label{sec:introduction}
WiFi fingerprint-based localization systems --- using the signal strengths of WiFi APs to fingerprint the location from which the signal is collected --- have become the mainstream solutions for indoor localization.
These WiFi fingerprint-based localization methods consist of, in general, two phases: an {\em offline} fingerprinting phase to construct the building's WiFi fingerprint map via site survey, and an {\em online} localization phase to position the mobile devices/users by checking the received WiFi signals with the fingerprint map~\cite{832252}.
Significant research has been devoted to the online localization phase, achieving decimeter-level localization accuracy using advanced algorithms to match the online collected WiFi signals with the fingerprint map~\cite{Rajagopal:2018:EIS:3207947.3208003,tsui2009unsupervised,sun2014wifi,xie2016improved}.
However, a critical bottleneck of fingerprint-based indoor localization, i.e., the intensive overhead in constructing/maintaining the fingerprint map, remains unsolved: (i) an agent (e.g., a people carrying a WiFi scanner) needs to survey the building to collect data and construct the fingerprint map, and (ii) after construction, a fingerprint map needs to be updated frequently to mitigate the dynamics of WiFi signals~\cite{7174948}.
A variety of designs are proposed to use SLAM-enabled robots~\footnote{SLAM (Simultaneous Localization And Mapping)-enabled robot is able to construct/update a map of an unknown environment (e.g., a building) while simultaneously keeping track of the robot's location therein.} to facilitate the construction/maintenance of WiFi fingerprint map~\cite{lingemann2005high,biswas2012depth,varveropoulos2005robot}.
Most of these solutions use their robots in a ``travel-with-sojourn'' way: the robots visit, and stop at, each reference locations of the building to collect sufficient WiFi scans thereof, thus being able to fingerprint the reference locations reliably.
The frequent stop of the robot, however, prolongs the time to finish the site survey (and hence fingerprint map construction) and moreover, increases the power consumption of the robot --- which are usually powered by batteries --- due to frequent de/acceleration, limiting the range the robots can cover/survey and impeding their deployments in large buildings.
To mitigate this deficiency, we design and implement an autonomous WiFi fingerprinting system, called {\tt{AuF}}, in which the robot constructs the WiFi fingerprint database by surveying the indoor environment without sojourn, thus expanding the range the robot can cover and shortening the time needed for fingerprinting.
{\tt{AuF}}'s travel-without-sojourn, however, reduces the WiFi scans collected at specific reference locations, and thus degrades the reliability of collected WiFi measurements in the form of both lost and abnormal signals~\cite{chow2018efficient, bose2007practical}.
{\tt{AuF}}\ mitigates this degraded signal quality by using two novel signal recovery methods.
\begin{itemize}
\item {\bf Lost Signal Recovery.~} {\tt{AuF}}\ recovers the lost signal using the strong correlation between 2.4GHz and 5GHz signals: (i) most commodity WiFi APs support both 2.4GHz/5GHz networking; (ii) for a given AP, the strength of the 2.4GHz/5GHz signal at a given location are strongly correlated; (iii) our empirical results show the two signal seldom lose at the same time. {\tt{AuF}}\ exploits this correlation between 2.4/5GHz signal to recover the lost signal during its site surveying, if anyone of them (but not both) is lost.
%
\item {\bf Abnormal Signal Recovery.~} {\tt{AuF}}\ detects abnormal WiFi samples based on a spatial model of signal strength. {\tt{AuF}}\ then uses the correlation between the current WiFi samples and the previously constructed database to recover them.
The key of this solution is that {\tt{AuF}}{}'s fingerprinting allows a short interval between fingerprint updates, making former information remain effective for the current site survey.
\end{itemize}
Also note that {\tt{AuF}}\ recognizes the indoor environment and plans its site survey without requiring human operation, and thus being a fully autonomous system to construct/maintain the WiFi fingerprint database.
The fingerprint database constructed by {\tt{AuF}}\ can then be used to build/update fingerprint maps by existing WiFi-based indoor localization systems.
We have evaluated {\tt{AuF}}\ on two floors of our Department building, as shown in Fig.~\ref{fig:robot}(b)(c). The results show {\tt{AuF}}\ to construct the fingerprint map with 64\%/71\% less time and 61\%/64\% less power on the two floors without degrading localization accuracy, when compared to the traditional site survey method~\cite{mirowski2012depth,nguyen2016low}.
\section{Discussion: Computational Cost}
\label{sec:discussion}
A potential problem for {\tt{AuF}}{} is its computational cost from the signal recovery, especially the Gaussian process regression training process. The iterative training method's complexity with $N$ samples is $\mathcal{O}(N^2)$. Suppose that the number of abnormal signals is proportional to the number of samples, the computation complexity of {\tt{AuF}}{} is $\mathcal{O}(N^3)$, because {\tt{AuF}}{} repeats Gaussian process regression until abnormal signals are all identified.
We solve it by conducting the survey process for each region, thus small-scale training data is used for every site survey. Assuming the number of fingerprints in each region is the same, the computational cost just grows linearly with the survey area rather than in cubic with the survey area. In our experiments, the time needed for signal recovery computation is 14s on the 3th floor and 23s on the 6th floor, which is trivial compared to the site survey time.
\section{Conclusion}
\label{sec:conclusion}
To mitigate the overhead of fingerprint map construction for WiFi-based indoor location systems, the robot are adopted to perform fingerprinting. However, the high time and energy cost make the large deployment of the proposed autonomous fingerprinting systems difficult. {\tt{AuF}}{} is designed as an energy efficient autonomous fingerprinting system. It conducts the site survey without requiring the robot stop at every reference location, thus saving time and power consumed by de/acceleration. To guarantee the quality of {\tt{AuF}}{}'s fingerprint relatively small database, two kinds of signal recovery methods are proposed to solve the unreliable signals during the site survey.
We have deployed and evaluated {\tt{AuF}}{} on two sites of our Department building. The results validate {\tt{AuF}}'s ability in quick finishing the fingerprint map construction and achieving unabated localization accuracy.
\bibliographystyle{IEEEtran}
\section{Methods}
\input{signal_loss.tex}
\input{bad_data.tex}
\section{Overview}
\label{sec:overview}
Fig.~\ref{fig:overview} depicts an overview of {\tt{AuF}}. First, {\tt{AuF}}\ conducts an automatic initialization process. It acquires a building's floor map for the robot's localization and navigation. Then the floor map is segmented to smaller regions with regular shapes, which is essential for an efficient path planning. After map segmentation, the traveling paths of the robot is planned for the site survey.
Survey-without-sojourn is the beginning of {\tt{AuF}}'s fingerprint database construction, during which the robot surveys the floor without sojourn to build a temporary fingerprint database in a short time. Then, lost signal recovery is performed by exploiting the correlation between 2.4/5GHz signals. To further improve the reliability of this signal recovery, {\tt{AuF}}\ also refines the temporary database by recovering abnormal signals.
These abnormal signals are identified by {\tt{AuF}}'s abnormal detection module. Then, we determine whether these signals can be recovered from the previous fingerprints. For fingerprint database construction, i.e. no past information, the robot surveys with sojourn at locations of abnormal signals, and acquire multiple samples for every location to complete the fingerprints for the temporary fingerprint database.
Otherwise, for fingerprint database maintenance, we recover abnormal signals by exploiting the pattern of signals’s short-term correlation. After this database refinement, the temporary fingerprint database is concluded as the constructed fingerprint database.
\section{Dual-Band WiFi Fingerprint Map}
\section{Recovery of Abnormal Signal}
\label{sec:signal_supplement}
Refinement of the fingerprint database is needed to mitigate abnormal signal caused due to system noise in the temporary database. In the building, the fluctuation of the wireless signal sometimes gets high due to the multi-path effect. Also occasional disturbances can increase the inaccurate signal strength in the database. The recovered dual-band signals may also suffer errors. Therefore, {\tt{AuF}}\ needs to identify these abnormal signals and improve them with more reliable ones.
First, abnormal detection based on hypothesis testing is conducted to identify abnormal signals. Then {\tt{AuF}}{} will try to recover them from the previously constructed database.
As for fingerprinting for the first time, the robot just recollects fingerprints thereof to calibrate these signals.
\subsection{Detection of Abnormal Signal}
{\tt{AuF}}'s identification of abnormal signals is grounded on the assumption that signal propagation is smooth, which is justified by Eq.~\ref{eq:shadow_log}. So, the basic idea is to identify samples which obviously deviate from the smooth signal model. Specifically, {\tt{AuF}}'s detection of abnormal signal includes two steps: (i) fitting the spatial signal model with a Gaussian process and estimating the norm value for each measured signal, and (ii) performing the largest residual test to identify abnormal signals.
\vspace{+3pt}
\noindent $\bullet$ {\bf Estimation with Gaussian Process.~}
The Gaussian process regression is used to fit the spatial distribution of the signal strength for two reasons. First, a Gaussian process assumes that measurements are drawn from random variables conforming Gaussian distributions, as analytically/empirically corroborated in Eq.~\ref{eq:shadow_log} and Fig.~\ref{fig:fit}. Second, the Gaussian process regression predicts the distribution of RSSI at a certain point, including the mean and varianc, which facilitates the next step of {\tt{AuF}}, i.e., the largest residue test.
Let the fingerprint training set be a collection of fingerprints $\mathcal{D} = \{ ({x_1},{y_1}),({x_2},{y_2}), \ldots ,({x_n},{y_n})\}$,
where $x$ denotes 2-D coordinates and $y$ denotes the RSSI of an AP. It is assumed that the measured signal strength $y$ consists of a {\em true} signal strength $f(x)$ and an independent Gaussian noise $\omega \in N(0,{\sigma ^2})$, i.e., ${y_i} = f({x_i}) + {\omega _i}$. The collection of $f({x})$ is to be drawn from a Gaussian process, thus it conforms a multivariate Gaussian distribution with mean function $m(\cdot)$ and covariance function $k(\cdot, \cdot)$:
{
\footnotesize
{
\begin{equation}
\begin{bmatrix}
f(x_1)\\
\vdots \\
f(x_n)
\end{bmatrix}
\sim
{N}
\left (\begin{bmatrix}
m(x_1)\\
\vdots \\
m(x_n)
\end{bmatrix}
,
\begin{bmatrix}
k(x_1, x_1) & \cdots & k(x_1, x_n) \\
\vdots & \ddots & \vdots \\
k(x_n, x_1) & \cdots & k(x_n, x_n)
\end{bmatrix}
\right ) .
\label{eq:GP_model}
\end{equation}
}
}
In general, mean function $m(\cdot)$ is set to $0$, and kernel functions are used to represent the covariance $k(\cdot,\cdot)$. Here the squared exponential kernel is adopted:
\begin{equation}
k(x_i, x_j) = \sigma _f \exp\left ( -\frac{1}{2l^2}\left \| x_i-x_j \right \|^2 \right ),
\label{eq:se_kernel}
\end{equation}
where $\sigma _f$ is signal variance and $l$ is a length scale. Both parameters determine the smoothness of the function $f(x)$ estimated by the Gaussian process.
Then we represent the distribution of $y=f(x)+w$. Since it has a zero-mean noise, its mean function is still $0$.
The noise terms can be incorporated into the covariance function:
\begin{equation}
\mathrm{cov} ({y_i},{y_j}) = k(x_i, x_j) + ({\sigma ^2}){\delta _{ij}},
\end{equation}
where ${\delta _{ij}} = 1 $ if $i = j$ and zero otherwise.
Denote the testing set as $\mathcal{T} = \{ ({x^*_1},{y^*_1}),({x^*_2},{y^*_2}), \ldots ,({x^*_m},{y^*_m})\}$, which is drawn from the same unknown distribution as $\mathcal{D}$. For notational convenience, we aggregate $n$ input vectors $x_i$ of $\mathcal{D}$ into $n \times 2$ matrix $X$, $n$ output values $y_i$ of $\mathcal{D}$ into $n \times 1$ vector $Y$, $m$ input vectors $x^*_i$ of $\mathcal{T}$ into $m \times 2$ matrix $X^*$, $m$ output values $y^*_i$ of $\mathcal{T}$ into $m \times 1$ vector $Y^*$. The training points and testing points must have a joint multivariate Gaussian distribution:
{
\small
\begin{equation}
\begin{bmatrix}
Y\\
Y^*
\end{bmatrix}
\sim
N
\left( 0,
\begin{bmatrix}
K(X,X)+\sigma^2 I & K(X,X^*) \\
K(X^*,X) & K(X^*,X^*) + \sigma^2 I
\end{bmatrix}
\right),
\end{equation}
where
\begin{equation}
\begin{aligned}
K(X,X)[i,j]=k(x_i,x_j)&, K(X,X^*)[i,j]=k(x_i,x^*_j),\\ K(X^*,X)[i,j]=k(x^*_i,x_j)&, K(X^*,X^*)[i,j]=k(x^*_i,x^*_j).
\end{aligned}
\end{equation}
}
With the rules for conditional density, we get the predicted value at $X^*$ conditioned on training data $X, R$:
$Y^*|X^*,X,Y \sim \mathcal{N}({\mu ^*}, {\Sigma ^*})$, where
\begin{equation}
\begin{aligned}
{\mu ^*} = &K({X^*},X){(K(X,X) + \sigma^2 I)^{ - 1}}Y\\
\Sigma ^* = &K({X^*},{X^*})+\sigma ^2 I \\
&- K({X^*},X){(K(X,X) + \sigma^2 I)^{ - 1}}K(X,{X^*})
\end{aligned}
\label{eq:GP_prediction}
\end{equation}
As can be seen from Eq.\ref{eq:GP_prediction}, the predicted mean is a linear combination of observed signal strengths $Y$, and the weights depends on covariance $K({X^*},X)$, while the squared exponential kernel determines that nearby function values are highly correlated. On the other words, it is believed that RSSIs are locally smooth, thus the data will be regraded as outliers if it disagrees with our prior knowledge.
To examine the collected fingerprints, we predict signal strengths of the training data, i.e., $X*=X$. Thus, a measured value $y$ and its expectation and variance $\mu^*_x, \Sigma^*_x$ are obtained for each location $x$.
{\tt{AuF}}\ trains the parameters using scikit-learn~\cite{williams2006gaussian}.
\vspace{+3pt}
\noindent $\bullet$ {\bf Largest Normalized Residual Test.~}
The outlier identification is through the analysis of residues. Normalizing residues is necessary for us to find which one most deviate the estimation:
\begin{equation}
r^N = \frac{y-\mu^*_x}{\sqrt{\Sigma^*_x}} \sim \mathcal{N}(0,1).
\end{equation}
Then existence of outliers can be verified by the following hypothesis test:
\begin{itemize}
\item if any $|r^N|>t$ in collected fingerprints, there is a suspicion of bad data.
\item if all $|r^N| \leq t$, the hypothesis that there is no bad data is supported.
\end{itemize}
{\tt{AuF}}\ set $t$ to 1.96.
From \cite{grubbs1950sample}, it is shown that for a measurement set the measurement with the largest normalized residual contains a gross error. As a result, one abnormal signal can be identified by testing $r^N_{max} > t$. To identify all abnormal signal, the largest normal residual test performs within a loop:
\begin{enumerate}
\item Estimate expectations and variances from the training set.
\item If the largest residue exceeds the threshold, withdraw it from the training set, then go to step one. If not, finish the test.
\end{enumerate}
\subsection{Signal Recovery}
The next step is to recover these abnormal signals from the previously constructed fingerprint database.
Although the pattern of WiFi change after a long time is difficult to analyze~\cite{7174948}, the short time change of the WiFi signal is (relatively) predictable. We collect samples along a path in three days with the same collection method in Fig.~\ref{fig:24and5GHz}. The alternation of the signal after three days can be approximated as a shift, as shown in Fig.~\ref{fig:shift}. So we shift the corresponding signals in the past database to recover the abnormal signals.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.98\linewidth]{figure/signal_shift.eps}
\caption{Difference of the signal strength in three days.}
\label{fig:shift}
\end{figure}
\section{Evaluation}
\label{sec:evaluation}
We present our evaluation of {\tt{AuF}}\ in this section. First we explain the experiment settings in Section~\ref{sec:sec:methodlogy}. Then the time and energy efficiency of {\tt{AuF}}{} is compared to the baseline in Section.~\ref{sec:sec:time_energy}. In Section.~\ref{sec:sec:signal_on_fingerprintmap} the impacts of the signal recovery methods on the spatial domain and on the time domain on the fingerprint database are visualized. Finally, the localization accuracy of {\tt{AuF}}{} is evaluated in Section.~\ref{sec:sec:accuracy}.
\begin{figure}[htbp]
\centering
\begin{minipage}[b]{0.4\linewidth}
\begin{subfigure}[t]{1\linewidth}
\centering
\includegraphics[width=1\linewidth]{figure/robot.png}
\caption{}
\end{subfigure}
\end{minipage}
\begin{minipage}[b]{0.58\linewidth}
\begin{subfigure}[t]{1\linewidth}
\centering
\includegraphics[width=1\linewidth]{figure/3F.eps}
\caption{}
\end{subfigure}
\\
\begin{subfigure}[t]{1\linewidth}
\centering
\includegraphics[width=1\linewidth]{figure/6F.eps}
\caption{}
\end{subfigure}
\end{minipage}
\caption{(a) SLAM-enabled robot. (b) (c) are separately 3rd and 6th floors of our Department building.}
\label{fig:robot}
\end{figure}
\subsection{Methodology}
\label{sec:sec:methodlogy}
\vspace{+3pt}
\noindent $\bullet$ {\bf Experiment Settings.~}
We have deployed and evaluated {\tt{AuF}}{} on two sites (the 3rd floor and the 6th floor) of our Department building. These two areas and AP deployment there are shown in Figs.~\ref{fig:robot}(b)(c).
We use a Pioneer-3DX robot in Fig.~\ref{fig:robot} equipped with a HOKUYO UTM-30LX laser module as the agent. A LENOVO ideapad Y700 laptop with Ubuntu 16.04 operating system serves as the upper computer, and ROS (a robot operating system) is adopted to control the robot. A MI Note Pro smartphone is attached to the robot and used as the WiFi scanner during the site survey with 3 scan interval.
The robot surveys the floor at a speed of $0.5$m/s.
Before and after the site survey, we measure the voltage of the robot, the laser and the laptop. Then we use the discharge curves to calculate consumed power.
We update the fingerprint database every three days in twelve days.
\vspace{+3pt}
\noindent $\bullet$ {\bf Baseline.~}
For comparison, we also implement the survey-with-sojourn method: The robot surveys the building according to the same route as {\tt{AuF}}{}'s survey, but sojourns at each grid point generated in Sec.~\ref{sec:floor_recognition} for 10s.
\vspace{+3pt}
\noindent $\bullet$ {\bf Gaussian Process Fingerprint Map.~}
For visualization of the fingerprint database and localization, Gaussian process fingerprint maps are separately generated using the fingerprint database of {\tt{AuF}}{} and the database of the baseline.
Specifically, a Gaussian process model is trained by fingerprints after completing the site survey, which outputs the mean and variance of the signals' RSSI at every grid point generated in Sec.~\ref{sec:floor_recognition}. {\tt{AuF}}\ uses these RSSI statistics as the fingerprint thereof.
\vspace{+3pt}
\noindent $\bullet$ {\bf Localization Method.~}
We implemented the following simple but classic localization methods on top of the fingerprint map constructed above, and examined the resultant localization accuracy when a person holding a smartphone walks around on the two sites.
\vspace{+3pt}
{\em \underline{(1)~~Bayes.}~}
The first localization method exploits the fingerprint map constructed by {\tt{AuF}}\ with Bayes method~\cite{roos2002probabilistic}.
Denote the reference locations as $L = \{ {l_1},{l_2}, \ldots ,{l_m}\}$, and the observation vector as $o_{1\times b}$.
The reference location with the maximum probability $p(l|o)$ is used as the predicted location $\hat l$:
\begin{equation}
\begin{aligned}
&p(l|o) = {\frac{{p(o|l)p(l)}}{{p(o)}}}\propto p(o|l),
\\
&\hat l = \mathop {\arg \max }\limits_{{l_j}\in{L}} p(l_j|o)=\mathop {\arg \max }\limits_{{l_j}\in{L}}\prod\limits_{i = 1}^b {G({o_i}|{l_j})} .
\end{aligned}
\end{equation}
where $G$ is the probability of $v_i$ in the corresponding Gaussian distribution.
We then further improve the thus-obtained location results with particle filter.\footnote{Please see \cite{evennou2006advanced} for details of particle filter.}
\vspace{+3pt}
{\em \underline{(2)~~KNN.}~}
We also implemented a KNN-based localization method on top of {\tt{AuF}}, i.e., locating the online collected WiFi signals to the $K$ reference locations with the closest WiFi fingerprints. We used a K of $2$ unless specified otherwise. Again, particle filter is then used to further improve the localization accuracy.
\begin{table}[htbp]
\caption{Time and energy cost to survey 3F.}
\centering
\begin{tabular}{ccccc}
\hline
Method & Time & Robot & Laser & Laptop \\ \hline
{\tt{AuF}}{} & 43 min & 30 Wh & 7 Wh & 23 Wh \\
Baseline & 121 min & 69 Wh & 17 Wh & 67 Wh \\ \hline
\end{tabular}
\label{table:time_3F}
\end{table}
\begin{table}[htbp]
\caption{Time and energy cost to survey 6F.}
\centering
\begin{tabular}{ccccc}
\hline
Method & Time & Robot & Laser & Laptop \\ \hline
{\tt{AuF}}{} & 51 min & 33 Wh & 8 Wh & 41 Wh \\
Baseline & 177 min & 90 Wh & 25 Wh & 113 Wh \\ \hline
\end{tabular}
\label{table:time_6F}
\end{table}
\subsection{Time Overhead of Site Surveys}
\label{sec:sec:time_energy}
Tables.~\ref{table:time_3F} and \ref{table:time_6F} summarize the time and energy overhead to survey the sites by {\tt{AuF}}{} and the baseline.
Clearly, travel-without-sojourn requires far less time than travel-with-sojourn: {\tt{AuF}}{} saves 64\% time on the 3rd floor when compared to the traditional survey method, while a 71\% reduction is achieved on the 6th floor.
We also compare the energy overhead. On 3F, the energy consumed by the robot motors is reduced by 57\% with {\tt{AuF}}{} compared to the baseline. The de/acceleration is quite frequent when survey the building with sojourn, due to the dense survey locations required for accurately modeling the spatial distribution of signal strength. In our experiments, the robot performs de/acceleration once for every 0.8m on average.
{\tt{AuF}}{} also saves 59\% energy of the laser on 3F, which operates at constant power.
We can see that the laptop is a major part of energy consumption, since it performs intense computation to localize and navigate the robot. On 3F, {\tt{AuF}}{} consumes 66\% energy less than the baseline.
In total, {\tt{AuF}}{} reduces the energy consumption by 61\%.
\subsection{Signal Recovery on Fingerprint Map}
\label{sec:sec:signal_on_fingerprintmap}
\begin{figure}[htbp]
\centering
\begin{subfigure}[t]{0.98\linewidth}
\centering
\includegraphics[width = 1 \linewidth]{figure/fg_without_recover_2G.eps}
\caption{}
\end{subfigure}
\\
\begin{subfigure}[t]{0.98\linewidth}
\centering
\includegraphics[width = 1 \linewidth]{figure/fg_with_recover_2G.eps}
\caption{}
\end{subfigure}
\caption{The fingerprint map constructed w/o and w/ lost signal recovery.}
\label{fig:GP_raw_recovered_model}
\end{figure}
Next we examine the impact of lost signal recovery on the fingerprint map.
Fig.~\ref{fig:GP_raw_recovered_model} shows the fingerprint maps constructed by {\tt{AuF}}{} with and without signal recovery using the dual-band signals, in which {\em Measured Signal} denotes fingerprints with the measured signal, {\em Lost Signal} denotes fingerprints with the lost signal, which is set to -100 dBm, and {\em Recovered Signal} denotes fingerprints with the recovered signal.
{\em No Sojourn Map} in Fig.~\ref{fig:GP_raw_recovered_model} represents the mean of the fingerprint map constructed with only data collected during survey without sojourn, and {\em Signal Recovery Map} represents the mean of the fingerprint map constructed with lost signal recovery, while {\em Baseline Fingerprint map} represents the mean of the fingerprint map constructed with data of the baseline.
Signal loss occurs even when strong signals could be received, causing significant uncertainty in the thus-constructed fingerprint map.
Such uncertainty can be effectively mitigated via {\tt{AuF}}'s signal recovery, as observed in Fig.~\ref{fig:GP_raw_recovered_model}.
\begin{figure}[htbp]
\centering
\begin{subfigure}[t]{0.98\linewidth}
\centering
\includegraphics[width = 1 \linewidth]{figure/fg_without_resurvey.eps}
\caption{}
\end{subfigure}
\\
\begin{subfigure}[t]{0.98\linewidth}
\centering
\includegraphics[width = 1 \linewidth]{figure/fg_with_resurvey.eps}
\caption{}
\end{subfigure}
\caption{Localization w/o and w/ abnormal signal recovery.}
\label{fig:resurvey_fg}
\end{figure}
We next examine the impact of abnormal signal recovery on the fingerprint map.
Fig.~\ref{fig:resurvey_fg} shows fingerprint maps constructed with and without abnormal signal recovery, in which {\em Regular Signal} denotes fingerprints with the regular signal, {\em Abnormal Signal} denotes fingerprints with the identified abnormal signal, and {\em Recovered Signal} denotes fingerprints with signals recovered by previous data.
{\em No Sojourn map} and {\em Baseline Fingerprint map} in Fig.~\ref{fig:resurvey_fg} have the same meaning as in Fig.~\ref{fig:GP_raw_recovered_model}, while {\em Signal Recovery map} represents the mean of fingerprint map constructed with abnormal signal recovery.
As can be seen from Fig.~\ref{fig:resurvey_fg}, non-smooth points are identified as abnormal signals. These signals make the local shapes of the fingerprint map deviate from the ground truth. But abnormal signal recovery calibrates these flaws and make the fingerprint map of {\tt{AuF}}{} almost coincide with the baseline map.
\subsection{Localization Accuracy}
\label{sec:sec:accuracy}
Next we examine {\tt{AuF}}{}'s impact on localization accuracy. The examination is three-fold: (i) examine the impact of signal recovery; (ii) examine the impact of signal completion; (iii) examine the impact of {\tt{AuF}}{} and compare the localization accuracy of {\tt{AuF}}{} with that of the baseline.
\noindent $\bullet$ {\bf Impact of Lost Signal Recovery.~}
\begin{figure}[htbp]
\centering
\begin{subfigure}[t]{0.48\linewidth}
\centering
\includegraphics[width = 1\linewidth]{figure/recover_loc_3F.eps}
\caption{}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\linewidth}
\centering
\includegraphics[width = 1\linewidth]{figure/recover_loc_3F_corridor.eps}
\caption{}
\end{subfigure}
\caption{localization w/o and w/ Lost signal recovery.}
\label{fig:recovery_localization}
\end{figure}
Fig.~\ref{fig:recovery_localization} summarizes the localization results of fingerprint maps constructed with and without signal recovery, where {\em No Sojourn KNN} indicates using data collected during survey without sojourn and KNN method for localization, and the other legends' meaning is interpreted in the same way. In Fig.~\ref{fig:recovery_localization}(a) signal recovery improves mean error of KNN method from 5.4m to 2.3m. And localization in the narrow corridor is more difficult, but signal recovery still works. Seen from Fig.~\ref{fig:recovery_localization}, using Bayes method, mean error decrease from 4.5m to 3.5m.
But the max error does not degrades much with the help of lost signal recovery, e.g. max errors with/without signal recovery are close In Fig.~\ref{fig:recovery_localization}. This can be explained by the error caused by recovery from an abnormal signal, which is solved by abnormal signal recovery.
\noindent $\bullet$ {\bf Impact of abnormal signal recovery.~}
Fig.~\ref{fig:supplement_localization} summarize the localization results of fingerprint maps constructed with and without signal completion. Obviously both the mean error and the max error are depressed by signal completion. Take bayes method in Fig.~\ref{fig:supplement_localization}(a) as an example, the mean error decreases from 3.4m to 2.3m, while the max error decreases from 9.3m to 5.4m.
\begin{figure}[t]
\centering
\begin{subfigure}[t]{0.48\linewidth}
\centering
\includegraphics[width = 1\linewidth]{figure/resurvey_loc_3F.eps}
\caption{}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\linewidth}
\centering
\includegraphics[width = 1\linewidth]{figure/resurvey_loc_3F_corridor.eps}
\caption{}
\end{subfigure}
\caption{Abnormal signal recovery's impact on localization.}
\label{fig:supplement_localization}
\end{figure}
\noindent $\bullet$ {\bf Comparison with Baseline.~}
\begin{figure}[t]
\centering
\begin{subfigure}[t]{0.48\linewidth}
\centering
\includegraphics[width = 1\linewidth]{figure/compare_baseline_3F.eps}
\caption{}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\linewidth}
\centering
\includegraphics[width = 1\linewidth]{figure/compare_baseline_3F_corridor.eps}
\caption{}
\end{subfigure}
\caption{Accuracy comparison between {\tt{AuF}}{} and the baseline.}
\label{fig:compare_baseline_localization}
\end{figure}
The performance comparison in Fig.~\ref{fig:compare_baseline_localization} clearly shows that only improving the survey speed to reduce offline work is not desirable with the severely diminishing localization performance. In Fig.~\ref{fig:compare_baseline_localization}(a), using bayes method the mean error and the max error of raw increase 1.7m and 5.5m compared to the baseline.
On the other hand, the dataset provided by {\tt{AuF}}{} has no weakness in performance compared to the baseline. A slight improvement can be seen in Fig.~\ref{fig:compare_baseline_localization}(a), the point below which there are 80\% errors shifts from 2.1m to 1.9m, when adopts the fingerprint of {\tt{AuF}}{} rather than the baseline. In Fig.~\ref{fig:compare_baseline_localization}(d), {\tt{AuF}}{}'s has the little worse mean error but the little better max error. Using bayes method the baseline's mean error and max error is 2.2m and 7.5m, while {\tt{AuF}}{}'s mean error and max error is 2.4m and 4.9m.
\section{Fingerprint Database Construction}
\label{sec:survey_process}
In this section we describe {\tt{AuF}}'s construction of the fingerprint database. First we verify the feasibility of continues movement of the robot during collecting fingerprints. Then we explain the dual-band (i.e., 2.4GHz and 5GHz) signal recovery and the dependent signal model thereof.
\subsection{Survey without Sojourn}
\label{sec:fast_survey}
Unlike traditional systems~\cite{ocana2005indoor,youssef2005horus,832252} requesting the surveyor/robot stopping at locations and scanning WiFi multiple times, the robot in {\tt{AuF}}\ travels through generated grids without sojourn to shorten the survey process.
{\tt{AuF}}'s survey-without-sojourn has two challenges.
Intuitively, the robot's continuous movement may affect the collected WiFi signals due to Doppler shift.
To closely examine this, we use the robot to survey the area in Fig.~\ref{fig:path_planning}(a) to collects the WiFi signal from a given AP.
We collect three datasets, data collected with a 9s sojourn at each grid, with a 3s sojourn at each grid, and without sojourn.
Fig.~\ref{fig:speed}(a)(b)(c) visualize the heatmaps of the three dataset, and Fig.~\ref{fig:speed}(d) uses a scatter plot to represent relationships of signals collected with or without sojourn, showing no clear dependency of signal strength with the movement.
This is likely due to the relatively slow travel speed of the robot, e.g., 0.5m/s in the above measurements.
Another challenge is even more severe: with one fingerprint per location, how to ensure that this fingerprint can reliably represent the local signal strength. {\tt{AuF}}\ identifies, and then improves, the locations whose signal representation is unreliable. The following secs.~\ref{sec:signal_recovery} and \ref{sec:signal_supplement} separately explains separately two kind of unreliable signals and the corresponding remedies.
\subsection{Fingerprint Map Construction.~}
With the above collected/recovered/unrecoverable signals, {\tt{AuF}}\ constructs the fingerprint map based on the Gaussian process uncovered in Eq.~(\ref{eq:shadow_log}).
Specifically, the Gaussian model takes as input the measured/recovered signals and the location at which they are collected, and outputs the mean and variance of the signals' RSSI at a given reference location, which is used as the reference location's fingerprint.
Fig.~\ref{fig:Gaussian_model} illustrates such a fingerprint map construction process of {\tt{AuF}}.
\section{Initialization}
\label{sec:floor_recognition}
{\tt{AuF}}{} needs to do some preparations before its site survey, which needs to be conducted only once. Also, this process is automatic thanks to the robotics technique, thus making {\tt{AuF}}{} easy to deploy.
The first task of {\tt{AuF}}'s initialization is to discover the floor map of the floor-of-interest, which is the prerequisite for the robot's navigation indoors. Next, {\tt{AuF}}'s divides the floor map into regions with regular shapes. Then in each region, {\tt{AuF}}\ plans the surveying paths for its robot.
\vspace{+3pt}
\noindent $\bullet$ {\bf Floor Map Discovery.~}
{\tt{AuF}}\ discovers the floor map using a SLAM-enabled robot, as shown in Fig.~\ref{fig:robot}. The robot scans its surrounding environment with laser while surveying the building, and constructs the building's floor map based on the scanning results. The spanning-tree algorithm~\cite{gabriely2001spanning} is adopted to achieve an automatic SLAM process.
A gray-scale grid map (e.g., as shown in Fig.~\ref{fig:path_planning}(a)) is obtained after the site survey, in which the pixels with gray-scales smaller than a pre-defined threshold represents the obstacles of the building (e.g., walls).
{\tt{AuF}}\ then increases the obstacle area by the size of the robot, thus obtaining the floor area where the robot can travel freely, as shown in Fig.~\ref{fig:path_planning}(b).
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{0.24\linewidth}
\includegraphics[width=1\linewidth]{figure/mymapG.png}
\caption{The raw floor map identified using laser. The two marks are explained in Sec.\ref{sec:signal_recovery}}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.24\linewidth}
\includegraphics[width=1\linewidth]{figure/costmap.png}
\caption{The reduced floor map in which the robot can travel freely.}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.24\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/fsi.eps}
\caption{The darker the pixel,
the highest its value.}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.24\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/region_group.eps}
\caption{Adjacent pixels with same value are aggregate into a region.}
\end{subfigure}
\\
\begin{subfigure}[t]{0.24\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/remove_ripple.eps}
\caption{Small ripples in Fig.~\ref{fig:path_planning}(d) are removed.}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.24\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/merge_similar.eps}
\caption{Regions with similar value in Fig.~\ref{fig:path_planning}(e) are merged, and final regions are founded.}
\end{subfigure}
\begin{subfigure}[t]{0.24\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/path_plan.eps}
\caption{Each region's direction is determined by its shape.}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.24\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/reference_location.eps}
\caption{The robot needs to visit red dots according to regions' principal axis.}
\end{subfigure}
\caption{The whole process of initialization. }
\label{fig:path_planning}
\end{figure*}
\vspace{+3pt}
\noindent $\bullet$ {\bf Map Segmentation.~}
Instead of directly planning paths for the entire floor, {\tt{AuF}}{} first divides the above-recognized floor into subareas before planning paths for the robot, because of two reasons. First, directly planning path covering a irregular space is a challenging, while identifying the optimal survey of a regular space is much more feasible --- the map segmentation facilitates planning paths to efficiently cover a irregular space \footnote{By ``efficient", we mean the paths have the least turning/overlapping}.
The second reason necessities {\tt{AuF}}'s map segmentation is the computational cost of constructing the fingerprint database, which grows in cubic with the number of collected samples. In fact, site surveys are conducted separately in each region to limit the size of samples during once site survey. The computation complexity of {\tt{AuF}}{} and its solution will be discussed in detail in Sec.~\ref{sec:discussion}.
{\tt{AuF}}\ grounds its map segmentation using MAORIS~\cite{mielle2018method}. Below we briefly describe its major steps:
First, a free space image, whose pixel value represents the size of the region it belongs to, is created based on the distance image of the floor map\footnote{Pixels in the distance image represent Euclidean distance to the nearest obstacle}. The free space image is initiated with an empty image. A circular mask is created for each pixel of the distance image and centered on the pixel, whose radius are determined by the value of the pixel. Then for pixels in the circular mask, if the value of the equivalent pixel in the free space image is less than the circle radius, the value of the pixel in the free space image is changed to the radius. The result is showed in Fig.~\ref{fig:path_planning}(c).
Then, in the free space image, adjacent pixels with same value are grouped into regions, which can be seen in Fig.~\ref{fig:path_planning}(d).
Clearly, Fig.~\ref{fig:path_planning}(d) is over-segmented, and some regions are too small to represent a semantic area. MAORIS removes these ripples with a simple rule: a region is incorporated into its neighbor if they overlap for more than 40\%, as visualized in Fig.~\ref{fig:path_planning}(e).
After removing ripples, some regions are still pieces of the same place like the left corridor in Fig.~\ref{fig:path_planning}(e), resulted from slightly different distances to the obstacle. These regions with similar value are merged, and Fig.~\ref{fig:path_planning}(f) plots the final segmented map.
\vspace{+3pt}
\noindent $\bullet$ {\bf Path Planning.~}
After segmenting the floor map, {\tt{AuF}}\ needs to determine the principal axis direction for each region in the map, moving along which requires the minimum number of robot's turning.
Let us consider Fig.~\ref{fig:path_planning}(g) as an example. Denoting the coordinates of the pixels in one region ${\rm{C}}$ as ${P}$, we get a set of coordinates representing that area:
\begin{equation}
{\rm{C}} = \left\{ {P_1}, {P_2}, \cdots, {P_l} \right\}.
\end{equation}
We calculate the variance of all pixels in $C$ by:
\begin{equation}
Var(C)
=\begin{bmatrix}
cov(x,x) &cov(x,y)\\
cov(y,x) & cov(y,y)
\end{bmatrix},
\end{equation}
where $x, y$ denotes the pixels' coordinates.
Because $Var(C)$ is a real symmetric matrix, its two eigenvectors (denoted as $\rm A_1$ and $\rm A_2$, as shown in Fig.~\ref{fig:path_planning}) are orthogonal, representing the two directions of the spatial distribution of the region's elements. So, the $\rm A_1$ with a higher eigenvalue is the principal axis of region $\rm{C}$.
Next, {\tt{AuF}}\ discretizes each region into grids (we use $0.8m\times0.8m$ grids) according to the corresponding principal axis direction, and combines the grid centers as the traveling paths of the robot.
To acheive this, {\tt{AuF}}\ identifies the minimum rectangles covering each of the region. Denoting $F$ as the world coordinate system, ${{}^F{P}}$ as the pixel point in $F$.
{\tt{AuF}}\ uses the two directions $\rm A_1$ and $\rm A_2$ as the region's coordinate system (i.e., {$F'$} in Fig.~\ref{fig:path_planning}).
The rotation transformation from ${{}^{F'}{P}}$ to ${{}^F{P}}$ is thus:
\begin{equation}
{^{F'}{P}}=\left[ {\begin{array}{*{20}{c}}
{\rm {A_1}}^\mathrm{T}\\
{\rm {A_2}}^\mathrm{T}
\end{array}} \right]\cdot {{}^F{P}}.
\end{equation}
{\tt{AuF}}\ then finds region-C's max/minimum coordinates according to {$F'$}, thus identifying a minimum rectangle covering the area of region-C.
At last, {\tt{AuF}}\ divides each of the above-identified rectangles into grids according to the coordinate system {$F'$}, and ask the robot to pass the grids' center in boustrophedon like Fig.~\ref{fig:path_planning}(h).
\section{Introduction}
\label{sec:introduction}
WiFi fingerprint-based localization systems --- using the signal strengths of WiFi APs to fingerprint the location from which the signal is collected --- have become the mainstream solutions for indoor localization.
These WiFi fingerprint-based localization methods consist of, in general, two phases: an {\em offline} fingerprinting phase to construct the building's WiFi fingerprint map via site survey, and an {\em online} localization phase to position the mobile devices/users by checking the received WiFi signals with the fingerprint map~\cite{832252}.
Significant research has been devoted to the online localization phase, achieving decimeter-level localization accuracy using advanced algorithms to match the online collected WiFi signals with the fingerprint map~\cite{Rajagopal:2018:EIS:3207947.3208003,tsui2009unsupervised,sun2014wifi,xie2016improved}.
However, a critical bottleneck of fingerprint-based indoor localization, i.e., the intensive overhead in constructing/maintaining the fingerprint map, remains unsolved: (i) an agent (e.g., a people carrying a WiFi scanner) needs to survey the building to collect data and construct the fingerprint map, and (ii) after construction, a fingerprint map needs to be updated frequently to mitigate the dynamics of WiFi signals~\cite{7174948}.
A variety of designs are proposed to use SLAM-enabled robots~\footnote{SLAM (Simultaneous Localization And Mapping)-enabled robot is able to construct/update a map of an unknown environment (e.g., a building) while simultaneously keeping track of the robot's location therein.} to facilitate the construction/maintenance of WiFi fingerprint map~\cite{lingemann2005high,biswas2012depth,varveropoulos2005robot}.
Most of these solutions use their robots in a ``travel-with-sojourn'' way: the robots visit, and stop at, each reference locations of the building to collect sufficient WiFi scans thereof, thus being able to fingerprint the reference locations reliably.
The frequent stop of the robot, however, prolongs the time to finish the site survey (and hence fingerprint map construction) and moreover, increases the power consumption of the robot --- which are usually powered by batteries --- due to frequent de/acceleration, limiting the range the robots can cover/survey and impeding their deployments in large buildings.
To mitigate this deficiency, we design and implement an autonomous WiFi fingerprinting system, called {\tt{AuF}}, in which the robot constructs the WiFi fingerprint database by surveying the indoor environment without sojourn, thus expanding the range the robot can cover and shortening the time needed for fingerprinting.
{\tt{AuF}}'s travel-without-sojourn, however, reduces the WiFi scans collected at specific reference locations, and thus degrades the reliability of collected WiFi measurements in the form of both lost and abnormal signals~\cite{chow2018efficient, bose2007practical}.
{\tt{AuF}}\ mitigates this degraded signal quality by using two novel signal recovery methods.
\begin{itemize}
\item {\bf Lost Signal Recovery.~} {\tt{AuF}}\ recovers the lost signal using the strong correlation between 2.4GHz and 5GHz signals: (i) most commodity WiFi APs support both 2.4GHz/5GHz networking; (ii) for a given AP, the strength of the 2.4GHz/5GHz signal at a given location are strongly correlated; (iii) our empirical results show the two signal seldom lose at the same time. {\tt{AuF}}\ exploits this correlation between 2.4/5GHz signal to recover the lost signal during its site surveying, if anyone of them (but not both) is lost.
%
\item {\bf Abnormal Signal Recovery.~} {\tt{AuF}}\ detects abnormal WiFi samples based on a spatial model of signal strength. {\tt{AuF}}\ then uses the correlation between the current WiFi samples and the previously constructed database to recover them.
The key of this solution is that {\tt{AuF}}{}'s fingerprinting allows a short interval between fingerprint updates, making former information remain effective for the current site survey.
\end{itemize}
Also note that {\tt{AuF}}\ recognizes the indoor environment and plans its site survey without requiring human operation, and thus being a fully autonomous system to construct/maintain the WiFi fingerprint database.
The fingerprint database constructed by {\tt{AuF}}\ can then be used to build/update fingerprint maps by existing WiFi-based indoor localization systems.
We have evaluated {\tt{AuF}}\ on two floors of our Department building, as shown in Fig.~\ref{fig:robot}(b)(c). The results show {\tt{AuF}}\ to construct the fingerprint map with 64\%/71\% less time and 61\%/64\% less power on the two floors without degrading localization accuracy, when compared to the traditional site survey method~\cite{mirowski2012depth,nguyen2016low}.
\section{Discussion: Computational Cost}
\label{sec:discussion}
A potential problem for {\tt{AuF}}{} is its computational cost from the signal recovery, especially the Gaussian process regression training process. The iterative training method's complexity with $N$ samples is $\mathcal{O}(N^2)$. Suppose that the number of abnormal signals is proportional to the number of samples, the computation complexity of {\tt{AuF}}{} is $\mathcal{O}(N^3)$, because {\tt{AuF}}{} repeats Gaussian process regression until abnormal signals are all identified.
We solve it by conducting the survey process for each region, thus small-scale training data is used for every site survey. Assuming the number of fingerprints in each region is the same, the computational cost just grows linearly with the survey area rather than in cubic with the survey area. In our experiments, the time needed for signal recovery computation is 14s on the 3th floor and 23s on the 6th floor, which is trivial compared to the site survey time.
\section{Conclusion}
\label{sec:conclusion}
To mitigate the overhead of fingerprint map construction for WiFi-based indoor location systems, the robot are adopted to perform fingerprinting. However, the high time and energy cost make the large deployment of the proposed autonomous fingerprinting systems difficult. {\tt{AuF}}{} is designed as an energy efficient autonomous fingerprinting system. It conducts the site survey without requiring the robot stop at every reference location, thus saving time and power consumed by de/acceleration. To guarantee the quality of {\tt{AuF}}{}'s fingerprint relatively small database, two kinds of signal recovery methods are proposed to solve the unreliable signals during the site survey.
We have deployed and evaluated {\tt{AuF}}{} on two sites of our Department building. The results validate {\tt{AuF}}'s ability in quick finishing the fingerprint map construction and achieving unabated localization accuracy.
\bibliographystyle{IEEEtran}
\section{Methods}
\input{signal_loss.tex}
\input{bad_data.tex}
\section{Overview}
\label{sec:overview}
Fig.~\ref{fig:overview} depicts an overview of {\tt{AuF}}. First, {\tt{AuF}}\ conducts an automatic initialization process. It acquires a building's floor map for the robot's localization and navigation. Then the floor map is segmented to smaller regions with regular shapes, which is essential for an efficient path planning. After map segmentation, the traveling paths of the robot is planned for the site survey.
Survey-without-sojourn is the beginning of {\tt{AuF}}'s fingerprint database construction, during which the robot surveys the floor without sojourn to build a temporary fingerprint database in a short time. Then, lost signal recovery is performed by exploiting the correlation between 2.4/5GHz signals. To further improve the reliability of this signal recovery, {\tt{AuF}}\ also refines the temporary database by recovering abnormal signals.
These abnormal signals are identified by {\tt{AuF}}'s abnormal detection module. Then, we determine whether these signals can be recovered from the previous fingerprints. For fingerprint database construction, i.e. no past information, the robot surveys with sojourn at locations of abnormal signals, and acquire multiple samples for every location to complete the fingerprints for the temporary fingerprint database.
Otherwise, for fingerprint database maintenance, we recover abnormal signals by exploiting the pattern of signals’s short-term correlation. After this database refinement, the temporary fingerprint database is concluded as the constructed fingerprint database.
\subsection{Floor Recognition}
\label{sec:floor_recognition}
As an end-to-end system, the first task of {\tt{AuF}}\ is to discover the floor map of the floor-of-interest and then identify the reference locations thereof.
\vspace{+3pt}
\noindent $\bullet$ {\bf Floor Map Discovery.~}
{\tt{AuF}}\ discovers the floor map using a SLAM-enabled robot, as shown in Fig.~\ref{fig:robot}. The robot scans its surrounding environment with laser while surveying the building, and constructs the building's floor map based on the scanning results (see Fig.~\ref{fig:robot}). Through the spanning-tree algorithm~\cite{gabriely2001spanning}, this SLAM process is automatic.
A gray-scale grid map (e.g., Fig.~\ref{fig:floormapgray}(a)) is obtained after the site survey, in which the pixels with gray-scales smaller than a pre-defined threshold represents the obstacles in the building (e.g., walls).
{\tt{AuF}}\ then increases the obstacle area by the size of the robot, thus obtaining the floor area where the robot can travel freely, as shown in Fig.~\ref{fig:floormapgray}(b). We will explain the two locations marked in Fig.~\ref{fig:floormapgray}(b) in Sec.~\ref{subsec:fingerprintmapconstruction}.
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/fsi.eps}
\caption{}
\label{fig:segmented_map}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/region_group.eps}
\caption{}
\label{fig:referencelocations}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/remove_ripple.eps}
\caption{}
\label{fig:path_segment}
\end{subfigure}
\\
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/merge_similar.eps}
\caption{}
\label{fig:path_segment}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/path_plan.eps}
\caption{}
\label{fig:path_segment}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/reference_location.eps}
\caption{}
\label{fig:path_segment}
\end{subfigure}
\caption{(a) Map segmentation. (b) {\tt{AuF}} discretizes each region according to their respective coordinate systems. The red star represents the reference locations that the robot needs to visit. (c) {\tt{AuF}}\ greedily plans the trajectory of the robot with straight lines, facilitating quicker site survey.}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{stableduration.eps}
\caption{WiFi signal shows limited short-term dynamics.}
\label{fig:stableduration}
\end{figure}
At the same time, {\tt{AuF}}\ has another task in this process: determine the location of each AP. The extremely precise location of the AP is not necessary, so {\tt{AuF}}\ takes the location with the highest signal strength as the location of the corresponding AP.
\vspace{+3pt}
\noindent $\bullet$ {\bf Map Segmentation.~}
Dividing the floor plan into subareas with regular shapes is important for a efficient survey, since coverage of a whole complex indoor building is hard. And although the RSSI of different areas in the building cannot be depicted by a single signal propagation model, the signals in the same regular area can be considered to conform the same model. This point is used in resurvey. Therefore, we need the map segmentation before motion planning for the robot.
the method of MAORIS~\ref{mielle2018method} is adopted in {\tt{AuF}}\. In this method, the floor plan is processed with four major steps: calculate free space image, group adjacent pixels of same value in regions, remove ripples, merge regions with similar values. And Fig.~ shows the corresponding image results of each step.
Then {\tt{AuF}}\ needs to determine the principal axis direction for each part in the map, because moving along this orientation can minimize the number of turning, which brings extra survey efforts.
Denoting the coordinates of the pixels in one region ${\rm{C}}$ as ${P}$, we get a set of coordinates representing that area:
\begin{equation}
{\rm{C}} = \left\{ {P_1}, {P_2}, \cdots, {P_l} \right\}.
\end{equation}
We calculate the variance of all pixels in $C$:
\begin{equation}
Var(C)
=\begin{bmatrix}
cov(x,x) &cov(x,y)\\
cov(y,x) & cov(y,y)
\end{bmatrix},
\end{equation}
where $x, y$ denotes the pixels' coordinates.
Because $Var(C)$ is a real symmetric matrix, its two eigenvectors (denoted as $\rm A_1$ and $\rm A_2$, as shown in Fig.~\ref{fig:referencelocations}) are orthogonal, representing the two directions of the spatial distribution of the region's elements. So $\rm A_1$ with the higher eigenvalue is the principal axis of region $\rm{C}$.
\vspace{+3pt}
\noindent $\bullet$ {\bf Reference Location Identification.~}
{\tt{AuF}}\ then discretizes the floor map into grids, and uses the grids (or the grids' centers more specifically) as the reference locations in the fingerprint map. Also {\tt{AuF}} needs to survey all these grids to collect fingerprints.
Corresponding to the map segmentation, {\tt{AuF}}\ discretizes each region according to the corresponding principal axis direction.
To do this, {\tt{AuF}}\ identifies the minimum rectangles covering each of the region. Denoting $F$ as the world coordinate system, ${{}^F{P}}$ as the pixel point in $F$.
{\tt{AuF}}\ uses the two directions $\rm A_1$ and $\rm A_2$ as the region's coordinate system (i.e., {$F'$} in Fig.~\ref{fig:referencelocations}).
The rotation transformation from ${{}^{F'}{P}}$ to ${{}^F{P}}$ is thus:
\begin{equation}
{^{F'}{P}}=\left[ {\begin{array}{*{20}{c}}
{\rm {A_1}}^\mathrm{T}\\
{\rm {A_2}}^\mathrm{T}
\end{array}} \right]\cdot {{}^F{P}}.
\end{equation}
{\tt{AuF}}\ then finds region-C's max/minimum coordinates according to {$F'$}, thus identifying a minimum rectangle covering the area of region-C.
At last, {\tt{AuF}}\ divides each of the above-identified rectangles into grids according to coordinate system {$F'$}, and use the grids' center as the reference locations.
\subsection{Overview of R-Map}
\begin{figure*}[t]
\centering
\begin{minipage}{2\columnwidth}
\begin{subfigure}[t]{0.49\linewidth}
\centering
\includegraphics[width=1\linewidth]{2RSSI_speed.eps}
\label{2GHz_speed}
\caption{2.4 GHz RSSI variation at different speeds}
\end{subfigure}
\hfill
\begin{subfigure}[t] {0.49\linewidth}
\centering
\includegraphics[width=1\linewidth]{RSSI_speed.eps}
\label{5GHz_speed}
\caption{5 GHz RSSI variation at different speeds}
\end{subfigure}
\caption{No clear dependency between robot's travel speed and the received WiFi signals is observed.}
\label{fig:speed}
\end{minipage}
\end{figure*}
\section{Problem Statement}
\label{sec:problem_statement}
It is worth noting that during the survey, Traditional methods~\cite{ocana2005indoor,youssef2005horus,832252} stop at the reference locations to obtain multiple fingerprints and get mean values of RSSIs at each location, since measured RSSIs are thought to have high noise.
{\tt{AuF}}\ abandons this method to minimize the heavy work during offline phase, which is the core target for this paper. There is no doubt that sojourning in each reference location will incur additional time, which includes the sojourn time and the time that the robot accelerates and decelerates. In fact, the main time spent in traditional methods stems from this, which makes fingerprint-based methods impractical. Of course, traveling without sojourn has some potential flaws, which will be solved one by one.
Intuitively, the robot's continuous movement may affect the collected WiFi signals due to Doppler shift.
Also, traveling without sojourn collects limited samples of WiFi signals --- deficient signal samples may degrade the reliability of the thus-constructed fingerprint map, especially in view of the temporal dynamics of WiFi signals.
First, to examine the impact of robot's travel speed on the received WiFi signals, we evenly selected 54 locations along a 53m straight path. The robot travels along this path with speed varying from 0.5--1.5m/s\footnote{Note the robot's travel speed will be limited in indoor environment for safety.}, and collects the WiFi signal from a given AP.
Fig.~\ref{fig:speed} plots the thus-collected WiFi signals, showing no clear dependency of signal strength with the travel speed.
Second, WiFi signal, albeit well-known for its temporal dynamics When first adopted in localization~\cite{ocana2005indoor,youssef2005horus,832252}, now shows a short-term stability because of the advancement of chips and power grids.
To corroborate this, we collected the WiFi signal from a given AP at a fixed location for about 9 hours.
Fig.~\ref{fig:stableduration} plots the CDF of the durations for the signal's RSSI to keep stable, showing RSSI changes about every $65$s on average. In conclusion, in usual cases, once scan suffices to represent the "true" RSSI at the location.
Of course, in Fig.~\ref{fig:stableduration} durations of 20\% WiFi signal are less than $5$s, and in the actual fingerprint acquisition process, the wifi signal may not be so ideal due the possible moving objects and the multipath effect. Therefore, the key to achieve continuous survey in {\tt{AuF}}\ is to solve the problem that how to ensure the accuracy of the fingerprint with with some erroneous fingerprints existing.
\begin{figure}[!htb]
\centering
\includegraphics[width=1\linewidth]{stableduration.eps}
\caption{WiFi signal shows limited short-term dynamics.}
\label{fig:stableduration}
\end{figure}
\section{Related Work}
\label{sec:related_work}
Fingerprint-based localization has been extensively explored, and achieves fine-grained accuracy. The localization algorithms can be divided into two types: deterministic and probabilistic algorithms. Deterministic algorithms represent the signal strength as a scalar at a location. For example, RADAR~\cite{832252} takes nearest neighbor method to search the user's location from the database. Probabilistic algorithms establish distributions of signal strengths in database. Horus~\cite{youssef2005horus} is a representative instance, which uses a Bayesian network model. Furthermore, sensor fusion~\cite{priyantha2005mobile} is studied to achieve further improved localization accuracy. {\tt{AuF}}{}, as a fingerprint collection system, can be deployed to support all these localization methods.
Also researches are carried out for the overhead of fingerprint map construction.
Crowdsourcing the signals from the users has also attracted much attention.
The early work, OIL~\cite{Park:2010:GOI:1814433.1814461} and Mole\cite{doi:10.1080/17489725.2012.692617} are designed to get fingerprints from users, but users are required to explicitly label his or her location for the collected signals.
To solve human labelling problem, Unloc~\cite{wang2012no} and WiFi-SLAM~\cite{ferris2007wifi} combine dead-reckoning and WiFi signal patterns to localize walking users, and frees users from labelling their ground truth.
But the high dependence on the inertial sensor and the assumed walking pattern reduces accuracy of the fingerprint map.
Admittedly, in scenarios where only rough locations are needed, radio model-based approach and crowdsourcing approach are both convenient.
In contrast, {\tt{AuF}}{} can quickly construct fingerprint database while maintaining the localization performance.
Using robots as professional surveyors has clear advantages. The robots free the human labor, and carry multiple devices to survey the floor while precise ground truth can be provided with a laser~\cite{lingemann2005high}, a depth camera~\cite{biswas2012depth} or just some sonars~\cite{varveropoulos2005robot}.
The authors of \cite{mirowski2012depth,nguyen2016low} describe a process of WiFi mapping using an autonomous robot. However, they just simply make the robot survey with sojourn, thus the site survey is still time-consuming. More importantly, The robot's power consumption is non-negligible, rendering the deployment of the robot surveyor on the large buildings unacceptable. {\tt{AuF}}{} uses a more efficient survey method, facilitating its deployment in large space.
\subsection{Recovery of Lost Signal}
\label{sec:signal_recovery}
\begin{figure}[htbp]
\centering
\begin{minipage}{1\columnwidth}
\begin{subfigure}{.49\columnwidth}
\includegraphics[width=1\linewidth]{figure/loss_24and5_location1.eps}
\caption{}
\end{subfigure}
\hfill
\begin{subfigure}{.49\columnwidth}
\includegraphics[width=1\linewidth]{figure/ave_24_5_RSSI.eps}
\caption{}
\end{subfigure}
\caption{Loss ratio and the signal strength of WiFi signals received at a fixed location.}
\label{fig:loss_24and5}
\end{minipage}
\end{figure}
\vspace{+3pt}
\noindent $\bullet$ {\bf Signal Loss.~}
The first type of unreliable signals is random signal loss. Lost signals are common during the survey process, which are usually indicators of long distances to the AP and poor signal strengths. We set the RSSIs of missed signals a low value (i.e. -100dBm).
Random signal loss occurs even when APs are just nearby, caused due to a variety of reasons such as obstruction of APs, scanning duration.
Unlike traditional survey method, {\tt{AuF}}{} does not rely on multiple collected samples at a location to mitigate the random signal loss. Instead, {\tt{AuF}}\ recovers the lost signal by exploiting the correlation between 2.4GHz signals and 5GHz signals from the same physical AP.
To corroborate feasibility of this signal recovery, we collect the dual-band WiFi signals from the 10 APs on the 6th floor at two fixed locations, marked in Fig.~\ref{fig:path_planning}, for about 2 hours.
Fig.~\ref{fig:loss_24and5} summarizes the ratios of signal losses, and Fig.~\ref{fig:loss_24and5} plots the signals' average RSSI.
Comparison of Figs.~\ref{fig:loss_24and5} and \ref{fig:loss_24and5} shows: (i) loss is observed at both 2.4GHz and 5GHz signals; (ii) the loss of a single frequency signal from a given AP is not necessarily caused by too weak a signal (e.g., with AP4 and AP5); (iii) the loss of both 2.4GHz and 5GHz signals from a given AP, however, does indicate a weak signal strength (e.g., with AP7--AP10).
\vspace{+3pt}
\noindent $\bullet$ {\bf Signal Correlation.~}
{\tt{AuF}}\ recovers the lost signal using the correlation between 2.4GHz and 5GHz signals.
To corroborate such a signal correlation, we evenly select 66 locations along a 65m straight path, and stop the robot 10s at each location to collect the 2.4GHz and 5GHz signals of an AP located at the 67m location. The average RSSI at each point is plotted in Fig.~\ref{fig:24and5GHz}.
The two traces of WiFi signals have a correlation coefficient of $0.92$, implying the feasibility to recover the lost 2.4GHz signal based on the 5GHz signal, and vice versa. Furthermore, we find that the spatial distribution of the difference between the two signals is regular, as shown in Fig.~\ref{fig:24and5GHz}.
\begin{figure}[htbp]
\centering
\begin{subfigure}[t]{1\linewidth}
\centering
\includegraphics[width=1\linewidth]{figure/plot_dual.eps}
\end{subfigure}
\caption{The two signals received at the same location show clear correlation.}
\label{fig:24and5GHz}
\end{figure}
The correlation between 2.4GHz and 5GHz signals can be explained analytically.
According to the log-normal shadowing model~\cite{rappaport1996wireless}, the RSSI of wireless signals can be expressed as
\begin{eqnarray}
P(d) &=& 10\cdot {\log _{10}}(\frac{{{W}{G_{AP}}{G_{MT}}\lambda ^2}}{{16{\pi ^2}d_0^2L}})
- 10\cdot \beta \cdot {\log _{10}}(\frac{d}{{{d_0}}}) \nonumber \\
&&+ X(0,{\delta ^2})
\label{eq:shadow_log}
\end{eqnarray}
where $P(d)$ is the RSSI measured at a distance $d$ from a given AP, with transmission power $W$.
$G_{AP}$ and $G_{MT}$ are the antenna gains on the AP and the mobile terminal, respectively. $L$ is the system's loss factor, $\lambda$ is the carrier's wavelength, $\beta$ is the path loss exponent, and $X(0,{\delta ^2})$ is a zero-mean Gaussian distributed random variable, capturing the shadowing effect~\cite{rappaport1996wireless}.
Denote ${P_{2.4}}(d)$ and ${P_{5}}(d)$ as the signal strength of the 2.4GHz and 5GHz signal measured at a distance $d$ of the given AP, respectively. The difference between the two signals' RSSI can be calculated based on Eq.~\ref{eq:shadow_log} as
{
\begin{equation}
\begin{aligned}
&{P_{2.4}}(d) - {P_5}(d)= 10\cdot {\log _{10}}(\frac{{\lambda _{2.4}^2}}{{\lambda _5^2}})
\\
&- 10({\beta _{2.4}} - {\beta _5}){\log _{10}}(\frac{d}{{{d_0}}})
+ X(0,\delta _{2.4}^2) - X(0,\delta _5^2),
\end{aligned}
\label{eq:difference}
\end{equation}
}
which can be further simplified to
\begin{equation}
{P_{2.4}}(d) - {P_5}(d)= f(d)+X(\mu ,{\delta ^2}),
\label{eq:simplifieddifference}
\end{equation}
where
\begin{equation}
f(d) = -10\cdot ({\beta _{2.4}} - {\beta _5})\cdot {\log _{10}}(\frac{d}{{{d_0}}}).
\end{equation}
Eq.~(\ref{eq:simplifieddifference}) implies that the RSSI difference of 2.4GHz and 5GHz signals at the same location can be approximated as a Gaussian variable, thus explaining their correlation.
Fig.~\ref{fig:fit} plots the results when fitting the difference between the 2.4GHz and 5GHz signals in Fig.~\ref{fig:loss_24and5} as Gaussian, showing high fitting goodness and thus verifying the above reasoning. Note the signals received from AP2 and AP3 are used here because of their relatively low loss ratios (and thus sufficient samples).
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figure/Gaussian_fit.eps}
\caption{Fitting the difference between 2.4GHz and 5GHz signals as Gaussian.}
\label{fig:fit}
\end{figure}
\vspace{+3pt}
\noindent $\bullet$ {\bf Signal Recovery.~}
{\tt{AuF}}, inspired by the correlated signals, recovers the lost 2.4GHz signal based on the 5GHz signal collected at the same scan, and vise versa.
We use the recovery of lost 2.4GHz signals with 5GHz signals to walk through {\tt{AuF}}'s signal recovery.
Eq.~(\ref{eq:simplifieddifference}) indicates the difference between the 2.4GHz and 5GHz signals consists of $f(d)$ and a Gaussian noise, where $f(d)$ is a function of the signal's propagation distance $d$.
Inspired by this, {\tt{AuF}}\ recovers the lost 2.4GHz signal by training a SVR (Support Vector Regression) model for each AP, with the location's 2-D coordinates as input and the signal difference thereat as output.
Fig.~\ref{fig:recovery_model} summarizes such a signal recovery process of {\tt{AuF}}.
Clearly, {\tt{AuF}}'s signal recovery requires at least a valid signal (i.e., either 2.4GHz or 5GHz) is received.
In case of both signals are lost, {\tt{AuF}}\ will use a weak signal (e.g., assuming a -100dBm RSSI) to fingerprint that location, inspired by the empirical observation uncovered in Figs.~\ref{fig:loss_24and5}(a)(b).
\begin{figure}[htbp]
\centering
\includegraphics[width=0.98\linewidth]{figure/dual_signal_recovery.jpg}
\caption{Flow chart of {\tt{AuF}}'s signal recovery, with recovering the lost 2.4GHz signals using received 5GHz signals as an example. }
\label{fig:recovery_model}
\end{figure}
\subsection{Floor Recognition}
\label{sec:floor_recognition}
As an end-to-end system, the first task of {\tt{AuF}}\ is to discover the floor map of the floor-of-interest and then identify the reference locations thereof.
\vspace{+3pt}
\noindent $\bullet$ {\bf Floor Map Discovery.~}
{\tt{AuF}}\ discovers the floor map using a SLAM-enabled robot, as shown in Fig.~\ref{fig:robot}. The robot scans its surrounding environment with laser while surveying the building, and constructs the building's floor map based on the scanning results (see Fig.~\ref{fig:robot}). Through the spanning-tree algorithm~\cite{gabriely2001spanning}, this SLAM process is automatic.
A gray-scale grid map (e.g., Fig.~\ref{fig:floormapgray}(a)) is obtained after the site survey, in which the pixels with gray-scales smaller than a pre-defined threshold represents the obstacles in the building (e.g., walls).
{\tt{AuF}}\ then increases the obstacle area by the size of the robot, thus obtaining the floor area where the robot can travel freely, as shown in Fig.~\ref{fig:floormapgray}(b). We will explain the two locations marked in Fig.~\ref{fig:floormapgray}(b) in Sec.~\ref{subsec:fingerprintmapconstruction}.
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/fsi.eps}
\caption{}
\label{fig:segmented_map}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/region_group.eps}
\caption{}
\label{fig:referencelocations}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/remove_ripple.eps}
\caption{}
\label{fig:path_segment}
\end{subfigure}
\\
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/merge_similar.eps}
\caption{}
\label{fig:path_segment}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/path_plan.eps}
\caption{}
\label{fig:path_segment}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.32\linewidth}
\centering
\includegraphics[width=1\linewidth]{./figure/reference_location.eps}
\caption{}
\label{fig:path_segment}
\end{subfigure}
\caption{(a) Map segmentation. (b) {\tt{AuF}} discretizes each region according to their respective coordinate systems. The red star represents the reference locations that the robot needs to visit. (c) {\tt{AuF}}\ greedily plans the trajectory of the robot with straight lines, facilitating quicker site survey.}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{stableduration.eps}
\caption{WiFi signal shows limited short-term dynamics.}
\label{fig:stableduration}
\end{figure}
At the same time, {\tt{AuF}}\ has another task in this process: determine the location of each AP. The extremely precise location of the AP is not necessary, so {\tt{AuF}}\ takes the location with the highest signal strength as the location of the corresponding AP.
\vspace{+3pt}
\noindent $\bullet$ {\bf Map Segmentation.~}
Dividing the floor plan into subareas with regular shapes is important for a efficient survey, since coverage of a whole complex indoor building is hard. And although the RSSI of different areas in the building cannot be depicted by a single signal propagation model, the signals in the same regular area can be considered to conform the same model. This point is used in resurvey. Therefore, we need the map segmentation before motion planning for the robot.
the method of MAORIS~\ref{mielle2018method} is adopted in {\tt{AuF}}\. In this method, the floor plan is processed with four major steps: calculate free space image, group adjacent pixels of same value in regions, remove ripples, merge regions with similar values. And Fig.~ shows the corresponding image results of each step.
Then {\tt{AuF}}\ needs to determine the principal axis direction for each part in the map, because moving along this orientation can minimize the number of turning, which brings extra survey efforts.
Denoting the coordinates of the pixels in one region ${\rm{C}}$ as ${P}$, we get a set of coordinates representing that area:
\begin{equation}
{\rm{C}} = \left\{ {P_1}, {P_2}, \cdots, {P_l} \right\}.
\end{equation}
We calculate the variance of all pixels in $C$:
\begin{equation}
Var(C)
=\begin{bmatrix}
cov(x,x) &cov(x,y)\\
cov(y,x) & cov(y,y)
\end{bmatrix},
\end{equation}
where $x, y$ denotes the pixels' coordinates.
Because $Var(C)$ is a real symmetric matrix, its two eigenvectors (denoted as $\rm A_1$ and $\rm A_2$, as shown in Fig.~\ref{fig:referencelocations}) are orthogonal, representing the two directions of the spatial distribution of the region's elements. So $\rm A_1$ with the higher eigenvalue is the principal axis of region $\rm{C}$.
\vspace{+3pt}
\noindent $\bullet$ {\bf Reference Location Identification.~}
{\tt{AuF}}\ then discretizes the floor map into grids, and uses the grids (or the grids' centers more specifically) as the reference locations in the fingerprint map. Also {\tt{AuF}} needs to survey all these grids to collect fingerprints.
Corresponding to the map segmentation, {\tt{AuF}}\ discretizes each region according to the corresponding principal axis direction.
To do this, {\tt{AuF}}\ identifies the minimum rectangles covering each of the region. Denoting $F$ as the world coordinate system, ${{}^F{P}}$ as the pixel point in $F$.
{\tt{AuF}}\ uses the two directions $\rm A_1$ and $\rm A_2$ as the region's coordinate system (i.e., {$F'$} in Fig.~\ref{fig:referencelocations}).
The rotation transformation from ${{}^{F'}{P}}$ to ${{}^F{P}}$ is thus:
\begin{equation}
{^{F'}{P}}=\left[ {\begin{array}{*{20}{c}}
{\rm {A_1}}^\mathrm{T}\\
{\rm {A_2}}^\mathrm{T}
\end{array}} \right]\cdot {{}^F{P}}.
\end{equation}
{\tt{AuF}}\ then finds region-C's max/minimum coordinates according to {$F'$}, thus identifying a minimum rectangle covering the area of region-C.
At last, {\tt{AuF}}\ divides each of the above-identified rectangles into grids according to coordinate system {$F'$}, and use the grids' center as the reference locations.
\subsection{Overview of R-Map}
\begin{figure*}[t]
\centering
\begin{minipage}{2\columnwidth}
\begin{subfigure}[t]{0.49\linewidth}
\centering
\includegraphics[width=1\linewidth]{2RSSI_speed.eps}
\label{2GHz_speed}
\caption{2.4 GHz RSSI variation at different speeds}
\end{subfigure}
\hfill
\begin{subfigure}[t] {0.49\linewidth}
\centering
\includegraphics[width=1\linewidth]{RSSI_speed.eps}
\label{5GHz_speed}
\caption{5 GHz RSSI variation at different speeds}
\end{subfigure}
\caption{No clear dependency between robot's travel speed and the received WiFi signals is observed.}
\label{fig:speed}
\end{minipage}
\end{figure*}
\section{Problem Statement}
\label{sec:problem_statement}
It is worth noting that during the survey, Traditional methods~\cite{ocana2005indoor,youssef2005horus,832252} stop at the reference locations to obtain multiple fingerprints and get mean values of RSSIs at each location, since measured RSSIs are thought to have high noise.
{\tt{AuF}}\ abandons this method to minimize the heavy work during offline phase, which is the core target for this paper. There is no doubt that sojourning in each reference location will incur additional time, which includes the sojourn time and the time that the robot accelerates and decelerates. In fact, the main time spent in traditional methods stems from this, which makes fingerprint-based methods impractical. Of course, traveling without sojourn has some potential flaws, which will be solved one by one.
Intuitively, the robot's continuous movement may affect the collected WiFi signals due to Doppler shift.
Also, traveling without sojourn collects limited samples of WiFi signals --- deficient signal samples may degrade the reliability of the thus-constructed fingerprint map, especially in view of the temporal dynamics of WiFi signals.
First, to examine the impact of robot's travel speed on the received WiFi signals, we evenly selected 54 locations along a 53m straight path. The robot travels along this path with speed varying from 0.5--1.5m/s\footnote{Note the robot's travel speed will be limited in indoor environment for safety.}, and collects the WiFi signal from a given AP.
Fig.~\ref{fig:speed} plots the thus-collected WiFi signals, showing no clear dependency of signal strength with the travel speed.
Second, WiFi signal, albeit well-known for its temporal dynamics When first adopted in localization~\cite{ocana2005indoor,youssef2005horus,832252}, now shows a short-term stability because of the advancement of chips and power grids.
To corroborate this, we collected the WiFi signal from a given AP at a fixed location for about 9 hours.
Fig.~\ref{fig:stableduration} plots the CDF of the durations for the signal's RSSI to keep stable, showing RSSI changes about every $65$s on average. In conclusion, in usual cases, once scan suffices to represent the "true" RSSI at the location.
Of course, in Fig.~\ref{fig:stableduration} durations of 20\% WiFi signal are less than $5$s, and in the actual fingerprint acquisition process, the wifi signal may not be so ideal due the possible moving objects and the multipath effect. Therefore, the key to achieve continuous survey in {\tt{AuF}}\ is to solve the problem that how to ensure the accuracy of the fingerprint with with some erroneous fingerprints existing.
\begin{figure}[!htb]
\centering
\includegraphics[width=1\linewidth]{stableduration.eps}
\caption{WiFi signal shows limited short-term dynamics.}
\label{fig:stableduration}
\end{figure}
\section{Related Work}
\label{sec:related_work}
Fingerprint-based localization has been extensively explored, and achieves fine-grained accuracy. The localization algorithms can be divided into two types: deterministic and probabilistic algorithms. Deterministic algorithms represent the signal strength as a scalar at a location. For example, RADAR~\cite{832252} takes nearest neighbor method to search the user's location from the database. Probabilistic algorithms establish distributions of signal strengths in database. Horus~\cite{youssef2005horus} is a representative instance, which uses a Bayesian network model. Furthermore, sensor fusion~\cite{priyantha2005mobile} is studied to achieve further improved localization accuracy. {\tt{AuF}}{}, as a fingerprint collection system, can be deployed to support all these localization methods.
Also researches are carried out for the overhead of fingerprint map construction.
Crowdsourcing the signals from the users has also attracted much attention.
The early work, OIL~\cite{Park:2010:GOI:1814433.1814461} and Mole\cite{doi:10.1080/17489725.2012.692617} are designed to get fingerprints from users, but users are required to explicitly label his or her location for the collected signals.
To solve human labelling problem, Unloc~\cite{wang2012no} and WiFi-SLAM~\cite{ferris2007wifi} combine dead-reckoning and WiFi signal patterns to localize walking users, and frees users from labelling their ground truth.
But the high dependence on the inertial sensor and the assumed walking pattern reduces accuracy of the fingerprint map.
Admittedly, in scenarios where only rough locations are needed, radio model-based approach and crowdsourcing approach are both convenient.
In contrast, {\tt{AuF}}{} can quickly construct fingerprint database while maintaining the localization performance.
Using robots as professional surveyors has clear advantages. The robots free the human labor, and carry multiple devices to survey the floor while precise ground truth can be provided with a laser~\cite{lingemann2005high}, a depth camera~\cite{biswas2012depth} or just some sonars~\cite{varveropoulos2005robot}.
The authors of \cite{mirowski2012depth,nguyen2016low} describe a process of WiFi mapping using an autonomous robot. However, they just simply make the robot survey with sojourn, thus the site survey is still time-consuming. More importantly, The robot's power consumption is non-negligible, rendering the deployment of the robot surveyor on the large buildings unacceptable. {\tt{AuF}}{} uses a more efficient survey method, facilitating its deployment in large space.
\subsection{Recovery of Lost Signal}
\label{sec:signal_recovery}
\begin{figure}[htbp]
\centering
\begin{minipage}{1\columnwidth}
\begin{subfigure}{.49\columnwidth}
\includegraphics[width=1\linewidth]{figure/loss_24and5_location1.eps}
\caption{}
\end{subfigure}
\hfill
\begin{subfigure}{.49\columnwidth}
\includegraphics[width=1\linewidth]{figure/ave_24_5_RSSI.eps}
\caption{}
\end{subfigure}
\caption{Loss ratio and the signal strength of WiFi signals received at a fixed location.}
\label{fig:loss_24and5}
\end{minipage}
\end{figure}
\vspace{+3pt}
\noindent $\bullet$ {\bf Signal Loss.~}
The first type of unreliable signals is random signal loss. Lost signals are common during the survey process, which are usually indicators of long distances to the AP and poor signal strengths. We set the RSSIs of missed signals a low value (i.e. -100dBm).
Random signal loss occurs even when APs are just nearby, caused due to a variety of reasons such as obstruction of APs, scanning duration.
Unlike traditional survey method, {\tt{AuF}}{} does not rely on multiple collected samples at a location to mitigate the random signal loss. Instead, {\tt{AuF}}\ recovers the lost signal by exploiting the correlation between 2.4GHz signals and 5GHz signals from the same physical AP.
To corroborate feasibility of this signal recovery, we collect the dual-band WiFi signals from the 10 APs on the 6th floor at two fixed locations, marked in Fig.~\ref{fig:path_planning}, for about 2 hours.
Fig.~\ref{fig:loss_24and5} summarizes the ratios of signal losses, and Fig.~\ref{fig:loss_24and5} plots the signals' average RSSI.
Comparison of Figs.~\ref{fig:loss_24and5} and \ref{fig:loss_24and5} shows: (i) loss is observed at both 2.4GHz and 5GHz signals; (ii) the loss of a single frequency signal from a given AP is not necessarily caused by too weak a signal (e.g., with AP4 and AP5); (iii) the loss of both 2.4GHz and 5GHz signals from a given AP, however, does indicate a weak signal strength (e.g., with AP7--AP10).
\vspace{+3pt}
\noindent $\bullet$ {\bf Signal Correlation.~}
{\tt{AuF}}\ recovers the lost signal using the correlation between 2.4GHz and 5GHz signals.
To corroborate such a signal correlation, we evenly select 66 locations along a 65m straight path, and stop the robot 10s at each location to collect the 2.4GHz and 5GHz signals of an AP located at the 67m location. The average RSSI at each point is plotted in Fig.~\ref{fig:24and5GHz}.
The two traces of WiFi signals have a correlation coefficient of $0.92$, implying the feasibility to recover the lost 2.4GHz signal based on the 5GHz signal, and vice versa. Furthermore, we find that the spatial distribution of the difference between the two signals is regular, as shown in Fig.~\ref{fig:24and5GHz}.
\begin{figure}[htbp]
\centering
\begin{subfigure}[t]{1\linewidth}
\centering
\includegraphics[width=1\linewidth]{figure/plot_dual.eps}
\end{subfigure}
\caption{The two signals received at the same location show clear correlation.}
\label{fig:24and5GHz}
\end{figure}
The correlation between 2.4GHz and 5GHz signals can be explained analytically.
According to the log-normal shadowing model~\cite{rappaport1996wireless}, the RSSI of wireless signals can be expressed as
\begin{eqnarray}
P(d) &=& 10\cdot {\log _{10}}(\frac{{{W}{G_{AP}}{G_{MT}}\lambda ^2}}{{16{\pi ^2}d_0^2L}})
- 10\cdot \beta \cdot {\log _{10}}(\frac{d}{{{d_0}}}) \nonumber \\
&&+ X(0,{\delta ^2})
\label{eq:shadow_log}
\end{eqnarray}
where $P(d)$ is the RSSI measured at a distance $d$ from a given AP, with transmission power $W$.
$G_{AP}$ and $G_{MT}$ are the antenna gains on the AP and the mobile terminal, respectively. $L$ is the system's loss factor, $\lambda$ is the carrier's wavelength, $\beta$ is the path loss exponent, and $X(0,{\delta ^2})$ is a zero-mean Gaussian distributed random variable, capturing the shadowing effect~\cite{rappaport1996wireless}.
Denote ${P_{2.4}}(d)$ and ${P_{5}}(d)$ as the signal strength of the 2.4GHz and 5GHz signal measured at a distance $d$ of the given AP, respectively. The difference between the two signals' RSSI can be calculated based on Eq.~\ref{eq:shadow_log} as
{
\begin{equation}
\begin{aligned}
&{P_{2.4}}(d) - {P_5}(d)= 10\cdot {\log _{10}}(\frac{{\lambda _{2.4}^2}}{{\lambda _5^2}})
\\
&- 10({\beta _{2.4}} - {\beta _5}){\log _{10}}(\frac{d}{{{d_0}}})
+ X(0,\delta _{2.4}^2) - X(0,\delta _5^2),
\end{aligned}
\label{eq:difference}
\end{equation}
}
which can be further simplified to
\begin{equation}
{P_{2.4}}(d) - {P_5}(d)= f(d)+X(\mu ,{\delta ^2}),
\label{eq:simplifieddifference}
\end{equation}
where
\begin{equation}
f(d) = -10\cdot ({\beta _{2.4}} - {\beta _5})\cdot {\log _{10}}(\frac{d}{{{d_0}}}).
\end{equation}
Eq.~(\ref{eq:simplifieddifference}) implies that the RSSI difference of 2.4GHz and 5GHz signals at the same location can be approximated as a Gaussian variable, thus explaining their correlation.
Fig.~\ref{fig:fit} plots the results when fitting the difference between the 2.4GHz and 5GHz signals in Fig.~\ref{fig:loss_24and5} as Gaussian, showing high fitting goodness and thus verifying the above reasoning. Note the signals received from AP2 and AP3 are used here because of their relatively low loss ratios (and thus sufficient samples).
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figure/Gaussian_fit.eps}
\caption{Fitting the difference between 2.4GHz and 5GHz signals as Gaussian.}
\label{fig:fit}
\end{figure}
\vspace{+3pt}
\noindent $\bullet$ {\bf Signal Recovery.~}
{\tt{AuF}}, inspired by the correlated signals, recovers the lost 2.4GHz signal based on the 5GHz signal collected at the same scan, and vise versa.
We use the recovery of lost 2.4GHz signals with 5GHz signals to walk through {\tt{AuF}}'s signal recovery.
Eq.~(\ref{eq:simplifieddifference}) indicates the difference between the 2.4GHz and 5GHz signals consists of $f(d)$ and a Gaussian noise, where $f(d)$ is a function of the signal's propagation distance $d$.
Inspired by this, {\tt{AuF}}\ recovers the lost 2.4GHz signal by training a SVR (Support Vector Regression) model for each AP, with the location's 2-D coordinates as input and the signal difference thereat as output.
Fig.~\ref{fig:recovery_model} summarizes such a signal recovery process of {\tt{AuF}}.
Clearly, {\tt{AuF}}'s signal recovery requires at least a valid signal (i.e., either 2.4GHz or 5GHz) is received.
In case of both signals are lost, {\tt{AuF}}\ will use a weak signal (e.g., assuming a -100dBm RSSI) to fingerprint that location, inspired by the empirical observation uncovered in Figs.~\ref{fig:loss_24and5}(a)(b).
\begin{figure}[htbp]
\centering
\includegraphics[width=0.98\linewidth]{figure/dual_signal_recovery.jpg}
\caption{Flow chart of {\tt{AuF}}'s signal recovery, with recovering the lost 2.4GHz signals using received 5GHz signals as an example. }
\label{fig:recovery_model}
\end{figure}
|
2,877,628,090,071 | arxiv | \section{Introduction}
\begin{figure}
\centering
\subfigure[]{
\includegraphics[width=.3\textwidth]{flower.png}
\label{fig:intro:flower}
}
\subfigure[]{
\includegraphics[width=.3\textwidth]{bunny.png}
\label{fig:intro:bunny}
}
\subfigure[]{
\includegraphics[width=.3\textwidth]{calci.png}
\label{fig:intro:plant}
}
\label{fig:intro}
\caption{Example models created with the software described in this paper. (a) A Waratah (\emph{Telopea}) flower created with the extrusion script described in Section \ref{ss:example:extrusion}. (b) A stalk-covered rabbit generated by applying the extrusion script to the Stanford Bunny model. (c) A \emph{Calcispongiae} design modelled with timed L-systems, generalised cylinders, and Lua scripting.}
\end{figure}
Modelling of complex, organic forms presents an enormous challenge in an architectural and design context. One successful approach has been the use of procedural modelling, where the designer supplies a procedure that the computer executes to build and animate a geometric model \cite{Ebert2003}. Typically the designer specifies form and behaviour at a high-level and the computer ``fills in the details'', generating the necessary geometric, behavioural and surface complexity automatically. Popular procedural techniques that generate complex organic structures include: L-systems \cite{Prusinkiewicz1996}, CDM \cite{McCormack2005}, Cellular models \cite{Fleischer1995b}, Physical Models \cite{Combaz2006}, Vertex-Vertex Systems \cite{Smith2006}, and the Simplicial Developmental System \cite{Porter2010}, to name just a few.
\subsection{Script-based Modelling and Development}
The recent proliferation of code-based creative systems suggests that programming is now increasingly embraced by architects and designers as a creative medium \cite{Reas2007,openFrameworks,Smith2008}. These systems, ideal for procedural modelling, have enabled many new and creatively interesting results, e.g.\ \cite{Reas2010}. They are successful because they combine the flexibility of programming with a specialised palette of custom functions and support libraries. They hide the underlying complexity of library functionality from the user, presenting a simple, but powerful interface that makes design and experimentation easy.
Professional animation packages support scripting languages too, using either a custom language (e.g. Maya's \emph{MEL}, Cinema 4D's \emph{COFFEE}) or a general purpose language, such as Python or Java. General purpose languages have the advantage of a broad user community, extensive use and testing, good documentation, examples, and potentially, user familiarity. Irrespective of their origins however, when employed in 3D design systems these languages must balance programming functionality with the interactive modelling and animation components of the software, where scripting is just one feature of many. So while they typically support the creation and manipulation of 3D form, a procedural approach is not necessarily their main focus. For example, the scripting of meshes with changing topologies is not readily supported in a number of systems. This makes the flow of ideas between script and dynamic form less effective than it could be.
This paper describes \emph{Fugu}, a script-driven 3D modelling system developed by the authors. Fugu permits flexible, procedural modelling of dynamic geometric meshes. It supports the generation, manipulation and animation of 3D form using scripts written in the programming language \emph{Lua} \cite{Ierusalimschy2006}, with the specific goal of easily creating the complex organic features found in natural forms and other complex geometries. The system is simple: a modelling program is a single Lua module, and the user interface provides the means to program and execute the code, then visualise and interact with the resulting model cohesively.
We will illustrate Fugu's approach to modelling using examples that highlight different aspects of its functionality and features. Fugu is designed to support multiple levels of model representation and control using the modularity provided by Lua. The goal is to provide a tool that allows easy definition and manipulation of a multi-representation model (for example, mesh and armature representations) within a single script.
After describing the operational and interface basics of Fugu in the next section, we will illustrate different aspects of its functionality in Sections \ref{ss:example:extrusion} and \ref{ss:example:lsystem}, before discussing the effectiveness of our approach and looking at how the system could be further developed in Section \ref{s:conclusions}.
\section{Fugu}
The basis of modelling in Fugu is a Lua script that generates and manipulates 3D triangular meshes via discrete simulation. Lua is a powerful, efficient, lightweight scripting language, frequently used in computationally demanding real-time applications such as games and multimedia environments \cite{Smith2008}. We chose Lua because of its simple syntax, extensible semantics and the ease with which it can be embedded in other software.
Lua scripts interface to a dynamic 3D runtime engine, written in C++, which handles the generation and display of geometry, along with the Fugu user interface (Section \ref{ss:fugu:interface}). Fugu scripts are single file Lua modules containing module-scoped variables and two special callback functions, \func{setup()} and \func{update()} (Section \ref{ss:fugu:script}). An array of functions and libraries (Section \ref{ss:fugu:functionality}) are exposed to the scripting system to manipulate and control this runtime engine.
\subsection{Interface}\label{ss:fugu:interface}
Fugu's interactive, code-oriented interface consists primarily of a code pane (Figure \ref{fig:screenshot}, left) and a 3D view (Figure \ref{fig:screenshot}, right). The design will be familiar to anyone who has used other popular creative-coding systems such as \emph{Processing}. Users can edit a script, press PLAY and see its effect immediately. They can also interact with the generated model while the simulation is running.
The code pane supports standard code editing functionality including syntax highlighting, line numbering, and multi-file editing with tabs. The 3D view uses a trackball-style manipulation of the scene viewport, and viewing modes range from wireframe to ambient occlusion shaded and textured modes. Additionally the user can select smoothly subdivided versions of the geometry (using Butterfly subdivision \cite{Dyn1990}). A console window at the bottom of the screen presents script syntax and run-time errors.
\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{screenshot.png}
\caption{A screenshot of Fugu showing the code pane with syntax colouring (left), 3D interactive view of the model the script generates (right) and the console window for syntax and runtime error reporting (bottom).}\label{fig:screenshot}
\end{figure}
\subsection{Script Format}\label{ss:fugu:script}
A Fugu script is a single file of Lua code written in the form of a Lua module. When a script is loaded, Fugu looks for two special functions in the module: \func{setup()} and \func{update()}.
The setup function is run once at the start of a simulation to specify initial conditions and create any initial meshes. For example, the setup function of Listing \ref{code:intro} creates a new spherical mesh, stores it in a module-scoped variable \emph{m}, then adds the mesh to the Fugu \emph{universe} (see Section \ref{ss:fugu:functionality}).
The update function is called repeatedly and at regular intervals from the time the user presses the PLAY button. Updating continues until the PAUSE button is pressed. The update function receives a parameter specifying the time elapsed since the last update, \emph{dt}. In Listing \ref{code:intro}, the update function iterates over all the vertices in the mesh \emph{m} and modifies their positions every time step.
\begin{lstlisting}[float,caption={An example Fugu script. The setup function creates a spherical mesh. During each update, all the vertices of this mesh are perturbed by a sine function that varies over time and local vertex position.},label=code:intro]
module(...,package.seeall)
local m
function setup()
m = sphere()
fgu:add(meshnode(m))
end
function update(dt)
for _,v in ipairs(vertexlist(m)) do
v.p = v.p + vec3(0,0.01*sin(v.p.x+fgu.t),0)
end
end
\end{lstlisting} \label{code:intro}
\subsection{Geometric Modelling Functionality}\label{ss:fugu:functionality}
In addition to the standard Lua library, Fugu provides access to a range of functions and datatypes that support 3D mesh modelling. Fugu's primary object is the 3D triangular mesh, composed of vertices, edges, and triangles. Fugu simplifies working with mesh geometry by facilitating mesh creation and manipulation through an extensive set of functions and datatypes. A summary, classified according to type, is given below.
The \func{universe} is a simple scene graph for organising multiple mesh objects and their display. The userdata universe object \func{fgu}, contains all the objects of the scene and provides two member functions \func{add(n:node)} and \func{make\_child\_of(parent:node, child:node)} which allow a script to add a node to the universe and to anchor a node's position to another node respectively. There are two types of nodes: \emph{abstract nodes} which can be used as invisible anchors (e.g., as a pivot for an object), and \emph{mesh nodes}, which transform meshes in the scene. Mesh objects must be wrapped in a \func{meshnode} datatype before they are added to the universe (Listing \ref{code:intro}).
Affine \textbf{transformations} (stored as homogeneous matrices) transform nodes in the scene graph, or can be applied directly to a mesh to permanently transform its geometry. Fugu provides a shorthand for the standard geometric transformation matrices: \func{T(t:vec3)} translates a point by the given vector; \func{S(s:vec3)} scales by the vector \code{s}; \func{R(rad:double, axis:vec3)} rotates a point a number of radians around the specified axis; and, \func{Rv(a:vec3,b:vec3)} rotates vector \code{a} to align with vector \code{b}.
A number of \textbf{mesh creation} functions are provided. Standard primitives are created using \func{cube()}, \func{sphere()}, \func{icosahedron()}, etc. The \func{iso(r:int, f:function)} function generates an isosurface mesh by sampling the function \code{f} using the marching cubes algorithm within a 2x2x2 cube with resolution \code{r}. For instance, a sphere could be generated with the following statement:
\begin{lstlisting}
iso(16, function(x,y,z)
return distance(vec3(x,y,z),vec3(0,0,0)) - 1
end)
\end{lstlisting}
This example also illustrates the use of Lua's anonymous functions. Meshes can be loaded from files (in the popular .obj format), useful for applying functions to pre-built geometry (such as the Stanford Bunny in Figure \ref{fig:intro} (b)).
The creation of \textbf{generalised cylinders}---geometric surfaces defined by connecting a series of cross-section curves distributed along a \emph{carrier curve} \cite{Agin1972}---is also supported. A triangular mesh approximating the surface is constructed from the generalised cylinder, using a novel \emph{curve-morphing} technique \cite{Wetter2011}.
Cylinders are defined using a Logo-inspired turtle interface, in which a co-ordinate reference frame, the \func{turtle}, is created and transformed through space, tracing out cross-sections and carrier curves \cite{McCormack2004}. The turtle's member functions change its position and orientation, these include \func{move(d:double)}, \func{roll(a:double)}, \func{pitch(a:double)} and \func{yaw(a:double)}. Cross-section and carrier curves are modelled as piecewise cubic B\'{e}zier curves, the control points of which are added by the turtle as it moves through space, using \func{add\_point()}. After defining the cylinder, the member function \func{mesh()} creates and returns a triangular mesh version of the cylinder, on which additional mesh-based operations can then be performed. The example in Figure \ref{fig:complex} illustrates a complex organic form created in Fugu with generalised cylinders.
\textbf{Mesh Implementation}: Rather than design yet another mesh representation and manipulation library of our own, we used \emph{VCGLib} to provide the mesh functionality required in Fugu. VCGLib is a comprehensive C++ template library that provides a flexible framework for creating and manipulating triangular meshes (\url{http://vcg.sourceforge.net}). Using VCGLib permitted rapid development, allowing us to focus on designing new geometric operators and manipulators, leaving VCGLib to take care of mesh representation and geometric integrity. Additionally, VCGLib provides many useful operations, such as \emph{Loop subdivision}, and has been extensively used and tested in the popular program \emph{MeshLab} (\url{http://meshlab.sourceforge.net/}). Two alternative libraries, \emph{CGAL} (\url{http://www.cgal.org}) and \emph{OpenMesh} (\url{http://www.openmesh.org}), provide similar functionality and were considered for our application. CGAL is oriented towards meshes with a fixed topology and is more focused on guaranteeing correctness of algorithms with its precise kernels, rather than being an efficient real-time format. OpenMesh is a polygonal mesh library based around a half-edge data structure and associated operations. In retrospect OpenMesh may have been a better choice due to its simpler API and more liberal license than VCGLib (LGPL vs GPL).
The lowest-level of mesh manipulation occurs on \emph{vertices} and \emph{faces}. Following the design inherited from our mesh representation library, \emph{VCGLib} (see sidebar \textbf{Mesh Implementation}), edges are manipulated implicitly by modifying vertices and faces. The \func{vertexlist(m:mesh)} function provides access to a mesh's vertices as a Lua array. A \func{vertex} is a user-data object, and has a number of \emph{attributes} including a position, \code{p}, colour, \code{c}, and normal, \code{n}. Listing \ref{code:intro} illustrates one way a vertex position can be modified in a script. Should a script retain a reference to a vertex for some time, there is the possibility the reference may become invalid if the vertex is deleted by another part of the script. To protect against such issues, vertices (and faces) are modelled as \emph{proxies} (see sidebar \textbf{Proxies}). Faces have a similar set of functions, for example \func{facelist(m:mesh)} returns a list of faces in the mesh, \func{face.n} returns the normal of a face, and \func{face:v(i)} returns the i'th vertex of the face.
\begin{figure}[htbp]
\centering
\includegraphics[width=.5\textwidth]{pos_diagram.png}
\caption{Using a \func{pos} to navigate a mesh. This figure shows two faces of a mesh and four pos'es, drawn as triangles connecting the vertex, edge and face referenced by each pos. From the pos \code{<v,0,f>} (indicated in the figure), we can move to the vertex above it with \func{flip\_v}, the opposing edge within the face with \func{flip\_e}, or to the adjacent face with \func{flip\_f}. If the mesh is fully connected we can reach any position on the mesh using a sequence of these operations.}\label{fig:pos}
\end{figure}
When modifying mesh geometry or vertex positions, normals may need to be recalculated to ensure correct shading and display. This is performed automatically between update calls or by calling \func{mesh:sync()} explicitly.
Fugu provides a simple mechanism for navigating around a mesh: the \func{pos} object, which models a cell-tuple \cite{Brisson1989}. A pos is a \code{<vertex,edge,face>} tuple, where vertex and face are the data structures introduced above, and edge is either 0 or 1 (referencing one of two possible edges). A pos uniquely identifies a position on a mesh and provides a set of member functions (\func{flip\_v()}, \func{flip\_e()}, and \func{flip\_f()}) for navigating it over the mesh (see Figure \ref{fig:pos}). Many of the functions in Fugu return a pos or a list of pos'es. Given a pos, \code{p}, the vertex or face it points to can be retrieved as \code{p.v} and \code{p.f}.
\textbf{Proxies}: Fugu is designed to allow multiple scripts to run simultaneously so that, for example, a physics script could simulate the effects of soft-body dynamics on a developing form, while a sprouting script could be sprouting florets from that form's surface. For simplicity we assume that scripts are not running truly concurrently, but rather that they have \code{update()} functions that are called sequentially. However, what happens when a script stores a reference to a vertex, and another script deletes that vertex? To safeguard against accessing an invalid object, vertices and faces are stored by proxy. This adds a safe layer of indirection, and offers a \func{valid()} function to check if the element targeted for access still exists. VCGLib also shuffles vertices and faces around in memory as elements are deleted, for space efficiency -- another reason that necessitates this safety mechanism. The vertex and face proxies implemented in Fugu also ensure they are updated to point to valid memory locations.
A number of \textbf{mesh operations} for assisting with modelling are provided. \emph{Accessor} functions return sets of elements: \func{loopv(v:vertex)}, for example, returns an ordered list of vertices that loop around \code{v}, and \func{nearbyv(v:vertex, n:int)} returns a list of vertices that are n edges or less away from v. Other functions, such as \func{mesh:smooth\_subdivide(levels:int)} operate on an entire mesh. VCGLib provides a large number of these functions, several of which we expose as Lua functions.
Local operations, such as \func{inset(v:vertex,s:double) }(demonstrated in Section \ref{ss:example:extrusion}), insets the faces surrounding a vertex and scales them by a given amount, \code{s}. \func{extrude(v:vertex, d:vec3, m:double)} extrudes the faces surrounding a vertex \code{v} in direction \code{d}, by magnitude \code{m}. Given a list of vertices, \code{vl}, and a plane defined with position \code{p} and normal \code{n}, \func{flattenvl(m:mesh, vl:list, p:vec3, n:vec3)} flattens all the vertices in the list to lie on that plane.
\textbf{Mathematics} functions include trigonometric functions, linear algebra, Perlin noise and a variety of interpolative functions. Datatypes for 3D vectors, homogeneous matrices, and quaternions are available as the userdata types \func{vec3}, \func{mat4}, and \func{quat} respectively. Lua's operator overloading features allow standard mathematical operators (addition, multiplication, etc.) to work on these new types (see Listing \ref{code:intro}).
A set of \textbf{geometric functions} are available to assist with performing geometric queries, collision tests, etc. For example, the function \func{distribute\_\ points\_sphere(n:int)} returns \code{n} equally distributed points on a sphere and \func{perp(v:vec3)} returns a normalised vector perpendicular to \code{v}.
A number of \textbf{utility} functions for manipulating Lua structures are provided, most of which are adopted from the \emph{Underscore.lua} library (\url{http://mirven.github.com/underscore.lua/}). These functions provide a simplified syntax for iterating over Lua tables and performing functional programming. For example, the \func{each(t:table, f:function)} utility function applies \code{f} to each member of \code{t}.
Higher-level functions are implemented directly in Lua, while lower-level, CPU intensive functions are implemented in C++. The support library, \emph{Luabind}, was invaluable in allowing easy binding between C++ datatypes and variables and Lua. Our goal is to eventually provide enough functionality so that most mesh operations can be coded purely on the Lua-side, allowing user-created libraries to be easily shared amongst the user community.
Having now described the basics of Fugu operation, along with its principal functions and datatypes, we will illustrate how these features can be used to model 3D form using a series of examples.
\section{Example Applications}\label{s:example}
In this section we describe two applications that highlight the different modelling features available within Fugu. The first example demonstrates a mesh operation for creating continuous animated extrusions. This compound extrusion operation (implemented as a Lua function that combines several lower-level operations) is used to generate a variety of tubular organic structures over existing meshes. The example also illustrates the general structure of a Fugu script and key mesh-level operations. The second example describes a timed L-system simulation, based on \emph{Calcispongiae} development. Unlike previous L-system modelling tools, the geometry generated by the L-system's development can be subject to further modification using Fugu's mesh operations.
\subsection{Extrusion}\label{ss:example:extrusion}
Extrusion is a fundamental operation in 3D mesh modelling that involves the translation of a group of triangles, typically along their normal, to generate a new form over the region that is swept out. A continuous extrusion can be used to animate the outward growth of a limb, spike, or thorn. This section illustrates how continuous extrusions are modelled and applied over arbitrary meshes in our system. By changing extrusion parameters a variety of different effects can be achieved.
The \func{extrude(v:vertex,dir:vec3,m:double)} function extrudes a set of faces adjacent to a specified vertex, along a supplied direction by amount \code{m}. To effectively model continuous growth, discontinuities in the surface geometry must be minimised as it undergoes extrusion. One method of achieving this is to use very small extrusion steps. However, in the model described below, we use the function, \func{inset}, which performs a zero-distance extrusion, followed by scaling of the end faces.
A continuous extrusion operator can be modelled with a \emph{move phase}, during which the vertices of the extrusion are translated continuously in a specified direction; and an \emph{inset phase}, where the faces at the end of the extrusion are inset, forming a new cap to translate. It is useful to consider growth as a single entity, so to model this in Lua we create a Lua object with an internal state and an \func{update} function that is responsible for performing the extrusion. This \func{update} function returns \code{false} when the extrusion is complete, and \code{true} otherwise. The internal logic is modelled using a state machine with three states: \code{move}, \code{inset}, and \code{done}. The script for this object is shown in Listing \ref{code:new_spike}.
\begin{lstlisting}[float,caption={This function creates a Lua object which, by repeatedly calling its update member function, will create an extrusion at the supplied vertex.},label=code:new_spike]
function new_spike(the_mesh,the_vertex)
-- the possible states
local states = {
move = 1,
inset = 2,
done = 3
}
-- constants for the operation
local SPEED = 4
local SEG_LENGTH = .1
local NUM_SEGS = 5
local SHRINK = .8
-- create the object and its initial instance variables
local obj = {
m=the_mesh,
v=the_vertex,
n=the_vertex.n,
seg = 1,
distance = 0,
cap = nil,
state=states.move
}
-- define the actions
local actions = {}
actions[states.move] = function(self,dt) ... end
actions[states.inset] = function(self,dt) ... end
-- the update function executes the action based on its state
obj.update = function(self,dt)
if self.state==states.done then
return false
else
actions[self.state](self,dt)
return true
end
end
-- return the new object
return obj
end
\end{lstlisting}
The inset state executes the \func{inset} function (Listing \ref{code:actions}) and then returns the machine to the move state. This function doesn't change \code{self.v} (the vertex currently being extruded), which always refers to the vertex at the end of the extrusion. Additionally, the \func{inset} function returns the \emph{cap}: a fan of pos'es (cell-tuples) iterating over the faces located at the active end of the extrusion. This is used in the \code{move} state to access the vertices surrounding \code{self.v}.
While in the move state, the extrusion shifts \code{self.v} by a small amount in the normal direction, then moves the adjacent vertices extracted from \code{self.cap} using the function \func{capov(cap:list)}, which, for a given pos-fan, returns the outer vertices as a Lua array. The vertices are first moved in the extrusion direction, and then scaled from the center so that they shrink. The final step calls \func{flattenvl(m:mesh,vl:list,p:vec3,n:vec3)}, which flattens the end cap vertices in the list \code{vl}, so they sit on the plane defined by position \code{p} with normal \code{n}. If the vertices have been shifted more than \code{SEG\_LENGTH}, then the state is either changed to \code{inset} to generate a new segment, or to \code{done} if the required number of segments have been generated. Figure \ref{fig:extrusionexample} illustrates these stages in the extrusion process.
\begin{lstlisting}[float,caption={The two actions corresponding to the inset and move phases.},label=code:actions]
actions[states.inset] = function(self,dt)
self.cap = inset(self.m,self.v,.99)
self.state = states.move
self.distance = 0
end
actions[states.move] = function(self,dt)
local dist = SPEED*dt
self.v.p = self.v.p + self.n*dist
if (self.cap) then
local outer = capov(self.cap)
local center = vec3(0,0,0)
for _,ov in ipairs(outer) do
ov.p = ov.p + self.n*dist
center = center + ov.p
end
center = center/#outer
for _,ov in ipairs(outer) do
ov.p = center + (ov.p-center)*SHRINK
end
flattenvl(self.m,outer,self.v.p,self.n)
end
self.distance = self.distance + dist
if (self.distance > SEG_LENGTH) then
self.seg = self.seg + 1
if (self.seg <= NUM_SEGS) then
self.state = states.inset
else
self.state = states.done
end
end
end
\end{lstlisting}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{extrusion.png}
\caption{A sequence generated by applying the \code{new\_spike} operation in Listing \ref{code:new_spike} to vertex \code{v}. The sequence alternates between moving the vertex (and its neighbours) and insetting a new set of faces on the end of the protrusion. The variable \code{outer} in Listing \ref{code:actions} refers to the vertices at the end of the cap (as shown).
}\label{fig:extrusionexample}
\end{figure}
Useful modifications to the continuous extrusion operation include rotating the end cap each segment, making the end cap more circular (to reduce the effect of starting conditions), and creating heterogeneous forms by modifying the extrusion object parameters based on vertex height (see Figure \ref{fig:intro} (a) and Figure \ref{fig:extrusions} (a,b)). By extruding outwards and then \emph{inwards}, we can create more interesting shapes, such as suckers (Figure \ref{fig:extrusions} (c)). This extrusion operation is general enough to be performed on any smooth manifold mesh. Figure \ref{fig:intro} (b) illustrates the operation on the Stanford Bunny, for example.
\begin{figure}
\centering
\subfigure[]{
\includegraphics[width=.3\textwidth]{spiky_wireframe.png}
\label{fig:extrusions:a}
}
\subfigure[]{
\includegraphics[width=.3\textwidth]{spiky_green.png}
\label{fig:extrusions:b}
}
\subfigure[]{
\includegraphics[width=.3\textwidth]{suckers.png}
\label{fig:extrusions:c}
}
\label{fig:extrusions}
\caption{(a, b) Forms created by growing tapered extrusions out of a sphere. The spikes curve to orient towards a point above the sphere. (c) The suckers on this object are generated by extruding outwards and then inwards. A collision routine ensures that the suckers only grow until they touch a neighbouring sucker.}
\end{figure}
\subsection{Generalised Cylinders}\label{ss:example:lsystem}
In addition to mesh modification functions, Fugu contains a number of mesh generation methods, the most sophisticated of which is the \emph{generalised cylinder}. In this section we demonstrate how we can easily model timed, parametric L-systems in Lua and use them to drive Fugu's generalised cylinder routines, creating animated growth of organic forms. We then show how this geometry can be manipulated further, by combining the L-system model with the extrusion model introduced in the previous example.
\subsubsection{Modelling L-systems with Fugu}
L-systems, introduced by Lindenmayer \cite{Lindenmayer1968}, are string rewriting grammars commonly used to model the development of cellular structures, herbaceous plants and trees \cite{Prusinkiewicz1996}. \emph{Parametric L-systems} represent models as symbolic strings with associated scalar parameters, which develop from an initial symbol string (an \emph{axiom}) according to a set of rewrite or \emph{production rules}. To obtain geometry from an L-system, the produced string must be interpreted by a geometry building mechanism.
Lua tables are a convenient and efficient data structures for storing an L-system string and specification, but from a user perspective they are too verbose. Therefore, we allow a user to define an L-system using Lua strings (and arrays of strings), which are parsed using Lua's string matching library (see Listing \ref{code:lsys}).
\begin{lstlisting}[float,caption={Creating a parametric L-system in Lua.},label=code:lsys]
axiom = 'B(2) A(4,pi+1)'
rules = {
'A(x,y) : y <= 3 -> B(x) A(x*2,x+y)',
'B(x) : x < 1 -> C(noise(x))',
'B(x) : x >= 1 -> B(x-1)'
}
lsys = new_lsystem(axiom,rules)
\end{lstlisting}
The function \func{new\_lsystem} creates an L-system object containing the current string (a table of symbols and a table of associated parameters) and the production rules. For example, the first L-system string contained in the object, \code{lsys}, in Listing \ref{code:lsys} would be represented as a table with contents:
\begin{lstlisting}
symbols = {'B','A'},
parameters = {{'2'},{'4','pi+1'}}
\end{lstlisting}
Production rules are also stored using Lua tables. The third rule in Listing \ref{code:lsys} would thus be a table containing the following:
\begin{lstlisting}
pred = 'B'
parameters = {'x'}
condition = 'x>=1'
succ = {'B'}
succ_par = {{'x-1'}}
\end{lstlisting}
The L-system object has a member function, \func{derive()}, which produces a derivation string by iterating through the current string and, for each symbol, checking if both the rule predecessor symbol matches and the production conditions are met. If this is the case, the symbol is replaced according to the matched rule.
Parameters and conditions are stored as strings so they can be evaluated dynamically by the Lua interpreter as production rules are matched and applied. This has the benefit of allowing any valid Lua expression (including Lua or Fugu functions) to be used in a parameter expression or condition; for example, the second rule in Listing \ref{code:lsys} illustrates the use of Fugu's \func{noise} function.
At this stage the L-system is purely symbolic, it is then interpreted to generate mesh geometry using the \emph{turtle interpretation} provided by Fugu's generalised cylinder functionality (Section \ref{ss:fugu:functionality}).
\subsubsection{Timed Development}
Parametric L-systems provide a discrete-time model of development, making continuous temporal development difficult. To overcome this limitation, \emph{timed parametric L-systems} were introduced \cite{Prusinkiewicz1996}. Timed L-system symbols are assigned an \emph{age} -- a continuous variable representing the time the symbol has been active in the derivation string. This variable also determines when a production rule should be applied to its associated symbol. A symbol's age can also be used to drive other animation parameters, such as a primitive's scale or length \cite[Chapter 6]{Prusinkiewicz1996}.
Our Lua-based implementation of parametric L-systems can be easily extended to include timed symbols. The L-system object stores an additional table with the age of each developing symbol in the produced string. Additionally, production rules may include a \emph{terminal} age, and a \emph{minimum age} of the predecessor in order for the rule to be applied.
Timed, parametric L-systems can be used to generate animated meshes. A Fugu script first defines an L-system object in its \func{setup} function. The L-system's member function \func{derive(dt:double)} is then called within the script's \func{update} function. The \func{derive} function updates each active symbol's age by \code{dt}, and then applies the appropriate production rules as necessary. Once the state of the L-system has been updated, it is re-interpreted to generate a new mesh.
Using this method, we designed a timed, parametric L-system in Fugu to produce a form inspired by \emph{Calcispongiae} \cite{Haeckel1904}. The result is an animated sequence of geometric models with smooth and fluid continuous development. Figure \ref{fig:calci} shows a sequence of still images from this development. The appendix contains details of the L-system used.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{calci_sequence.png}
\caption{A growth sequence of a timed, parametric L-system built in Fugu.}\label{fig:calci}
\end{figure}
The L-system generates a triangular mesh, which can then have other mesh operations applied to it. Trivially we can just return the plain mesh and operate on it, but more interesting behaviour can be modelled if information is shared between the L-system model and Lua. For the model shown in Figure \ref{fig:complex}, we used texture coordinates assigned to the generalised cylinders to control the placement and properties of extrusions grown on the mesh.
This idea could be extended to allow general tagging or identification of mesh parts. For example, the bulbs in Figure \ref{fig:calci} are created by a substring in an L-system, we could delimit that substring with special symbols that cause the bulb part of the mesh to be tagged (either with a per-vertex attribute or by returning a list of faces associated with tags). The bulb could be tagged with any information, including a symbols age or parameter, with which a mesh operation could act. For instance, the bulb's age could affect a hypothetical \emph{wrinkle} operation, causing the older buds to wrinkle more than the younger ones.
As these examples show, the procedural flexibility of scripts, combined with powerful mesh manipulation and development functions makes the specification of complex models relatively easy.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{complex.png}
\caption{A flower-like form built with a combination of L-systems, generalised cylinders, and mesh extrusions. This hybrid model combines the models used to generate Figure \ref{fig:calci} and Figure \ref{fig:intro} (a).}\label{fig:complex}
\end{figure}
\section{Conclusions}\label{s:conclusions}
The benefits of procedural design are well known, but the accessibility of advanced procedural modelling tools to designers and architects has, until recently, been limited. With new code-based creative tools such as Fugu, this goal becomes increasingly viable. However, there are many additional features that could extend the capabilities of our system to increase its versatility.
The most useful enhancement would be multiple, complementary model representations. For example: a geometric mesh, an armature for skeletal animation and construction, and a physical simulation system. Ideally, these different representations can be flexibly connected together. This could, for example, allow a user to procedurally generate physical structures that could be built using automated building techniques, such as multi-axis robots. The armatures could in turn define geometric primitives used in a physical simulation, preventing the developing structure from intersecting and allowing it to move in response to physical changes.
Morphogen simulation (based on biological patterning) is often used to generate a variety of self-organising natural structures and patterns, and has been widely used in modelling development \cite{Porter2010}. Allowing user-defined attributes for the vertices and faces would facilitate this feature in Fugu, and permit greater flexibility in modelling development. For instance, a user could specify that their model requires a new, per-vertex variable \emph{morphogen}. This could also extend to per-face attributes, for instance to define material properties for rendering. Simulation of morphogen diffusion could be implemented as additional Lua functions, allowing users to easily modify developmental behaviour or implement unusual effects within scripts.
The functionality in Fugu has been oriented towards modelling of specific target forms, such as abstract organic shapes. A number of additional operations and features would expand Fugu's modelling possibilities, while still operating within the script-oriented paradigm. Low-level mesh operations, such as adding and removing vertices and faces, welding vertices, and flipping edges, are necessary for more complex operations to be specified at the Lua level. Here a formal algebra could be employed, such as \emph{Vertex-Vertex} systems \cite{Smith2006}. Higher-level mesh operations such as bevelling, re-triangulation, and edge ring selection could easily be added based on functionality already present in VCGLib.
There are also some technical challenges that need to be overcome to support the real-time creation of more complex models. The smooth-subdivision visualisation used in the current system is performed in a single-thread on the CPU, and thus impacts significantly on overall performance when active (viewing more than 2 levels of subdivision is not currently practical for interactive performance on standard desktop computers). GPU-based smooth subdivision using tessellation shaders found in recent hardware would dramatically improve visualisation performance (see e.g.\ \url{www.opengl.org/registry/specs/ARB/tessellation_shader.txt}).
Lua co-routines offer a parallelism mechanism that could be used to simplify the specification of models \cite{Moura04}. For instance, an operation that performs a sequence of actions, such as \emph{grow stalk}, \emph{grow flower}, \emph{detach petals}, each requiring a second of simulation time to complete, could be implemented like this:
\begin{lstlisting}
obj.update = update(self,dt)
for _,f in ipairs{grow_stalk,grow_flower,detach_petals} do
local time = 0
while (time < 1) do
time = time + dt
f(self,dt)
coroutine.yield(true)
end
end
return false
end
\end{lstlisting}
Where \func{grow\_stalk}, etc., are member functions of the object. This is more concise and arguably a lot more readable than manually maintaining a state and state machine as is done in Listing \ref{code:new_spike}.
The core of Fugu has been designed to be independent of any visual representation, making it easy to incorporate it into a real-time game engine, and could take advantage of its visualization features (e.g.\ shadows, post-processing, shaders). Further development would be required to ensure real-time frame rates however.
The novelty of our system derives from its combination of a rich set of geometric mesh operations with a fast and flexible scripting language. As similar systems have demonstrated in more general ``creative coding'' environments, the use of relatively simple function libraries combined with an initialisation and per-frame step programming model makes it easy for artists and designers to rapidly create animated sequences. Fugu brings this ease of use and experimental flexibility to 3D animated mesh modelling. We hope that as its development continues, Fugu will be able to tackle even more complex modelling tasks in the design of procedurally generated form for design and architecture applications.
You can download a copy of Fugu, including source code, from \url{www.csse.monash.edu.au/cema/fugu}.
\section{Acknowledgements}
This research was supported by an Australian Research Council Discovery Projects grant DP1094064.
\section{Appendix}
The timed, parametric L-system used to generate the \emph{Calcispongiae} shown in Figure \ref{fig:calci} is shown below. A symbol $X$ with parameters $p_1 \ldots p_n$ and age $\alpha$ is denoted by $(X(p_1,\ldots,p_n),\alpha)$. Non-timed symbols are also allowed, in which case the outer parentheses and age are omitted. The turtle commands for each symbol are specified in the table.
The L-system consists of three main components, modelled with symbols $A$, $B$ and $C$. Symbol $A$ generates the first section of the stalk before any branching has occurred, symbol $B$ is responsible for placing the branches in a helical pattern based on the golden angle, and symbol $C$ is responsible for generating each branch and its terminal bulbs.
\noindent \begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}ll}
\hline
\multicolumn{2}{l}{$\mbox{\textbf{axiom}}: \wedge(-.5\pi) G\#(1) (Gsc(3),0) Gs G\#(0) (f(6),0) (A(2,1,10,0.02),0) Ge $} \\
\multicolumn{2}{l}{\textbf{production rules}} \\
$(A(l,w,n,b),1) : n>0 $&$\rightarrow (S(l,b,w),0) (A(l,w,{n-1},b),0)$ \\
$(A(l,w,n,b),1) : n=0 $&$\rightarrow (B(2,0.7,25,0.09),0)$ \\
$(B(l,w,n,b),1) : n>0 $&$\rightarrow [\backslash(2.39996n) (Gsc(w),0) Gs (f(l),0) (C(l,w,10,b),0) Ge]$\\
&$\ldots (S(1,1,1),0) (B(l,w,n-1,0.9b),0)$ \\
$(C(l,w,n,b),1) : n>0 $&$\rightarrow (S(l,b,w),0) (C(l,w,n-1,b),0)$ \\
$(C(l,w,n,b),1) : n=0 $&$\rightarrow (S(1.5l,0,1.1l),0) (S(1.5l,0,l),0)$\\
&$\ldots (S(l,0,0.3l),0) (S(.5l,0,0.6l),0)$ \\
$(S(l,b,w),30)$&$\rightarrow S(l,b,w)$ \\
$(Gsc(x),10) $&$\rightarrow Gsc(x)$ \\
$(f(x),15) $&$\rightarrow f(x)$ \\
\multicolumn{2}{l}{\textbf{symbol interpretations}} \\
\multicolumn{2}{l}{$S(l,b,w)$: Add a segment to the generalised cylinder of length $l$}\\
\multicolumn{2}{l}{pitched by angle $b$, and ending with scale $w$}\\
\multicolumn{2}{l}{$G\#(n)$: Set the cross section}\\
\multicolumn{2}{l}{$Gsc(x)$: Set the scale for the next cylinder control point}\\
\multicolumn{2}{l}{$Gs$: Begin a generalised cylinder}\\
\multicolumn{2}{l}{$Ge$: End a generalised cylinder}\\
\multicolumn{2}{l}{$\backslash(x)$: Roll by $x$ radians}\\
\multicolumn{2}{l}{$f(x)$: Move forward $x$ units}\\
\multicolumn{2}{l}{$\wedge(x):$ pitch by $x$ radians}\\
\hline
\end{tabular*}
\bibliographystyle{plainnat}
|
2,877,628,090,072 | arxiv |
\section{Conclusions and Future Directions}
\label{sec:conclusions}
Clustering over Riemannian manifolds plays an important role in the automatic analysis of images and videos~\cite{Turagaetal2011,shirazietal2012}.
As discussed before, in general, the existing methods suffer from either poor performance or high computational complexity.
In this paper, we propose a novel framework with random projection to tackle the clustering problems over Riemannian manifolds.
Based on the framework, we present three random projection methods for manifold points:~KGRP, KORP and KPCA-RP.
Through experiments on several computer vision applications, we demonstrate that our proposed framework achieves significant speed increases while maintaining clustering performance in comparison to the other conventional methods such Kernel K-means.
Furthermore, we analyse the parameters that impact the performance and run time of our proposed methods.
In the proposed framework, we carry out random projection for manifold points with the aid of RKHS.
In other words, we first project manifold points into RKHS.
One promising future direction is to study the intrinsic random projection, which directly maps manifold points into the random projection space.
To this end, one needs to define the notions of projection and hyperplane generation process in the manifold space.
\subsubsection{}
\vspace{8pt}
{
\textbf{Kun Zhao} received her MSc from University of Electronic Science and Technology of China in 2013.
Currently, she is a PhD student in The University of Queensland (UQ).
Her research interests are in the areas of computer vision, machine learning and pattern recognition.
\textbf{Azadeh Alavi} currently is a research fellow at University of Maryland. She received her PhD from UQ in 2014. She obtained her Bachelor of Applied Mathematics degree in 2002 and worked in industries for about 2 years before commencing her Master of IT-advanced program (Research Stream) at Griffith University. Currently, she is a research fellow at University of Maryland.
Her interests are in the areas of machine learning, pattern recognition and image processing.
\textbf{Arnold Wiliem} is a research fellow at UQ. He received his PhD in 2010 from Queensland University of Technology. He is a member of the IEEE and served as reviewer in various computer vision venues. His current research interests are in the areas of statistical methods in non-linear manifolds, machine learning and pattern recognition.
\textbf{Brian Lovell} received his PhD in 1991 from UQ. Professor Lovell is Director of the Advanced Surveillance Group at UQ. He was President of the International Association for Pattern Recognition (IAPR) [2008-2010], and is Fellow of the IAPR, Senior Member of the IEEE.
His interests include Biometrics, Nonlinear Manifold Learning, and Pattern Recognition.}
\section{Introduction}
\label{sec1}
Clustering analysis is an automated process that groups unlabelled data into subsets (here called clusters) that may express the underlying structure of the data.
It is one of the most critical tools for understanding visual data~\cite{jain2010,dhillon2004kernel}.
For instance, significant amounts of visual data such as videos and pictures are uploaded every second~\cite{Doe2012Online}.
Indeed, this is the case for YouTube where 100 hours of video are uploaded every minute~\cite{Doe2014youtube}.
Although these videos have titles and some additional meta-information, it is often desirable to automatically group the videos in terms of
specific criteria such as visual similarity or detected objects.
In recent years, modelling visual data in analytical manifolds such as Riemannian manifolds has enjoyed success in
various computer vision application domains such as face recognition~\cite{Turagaetal2011},
action recognition~\cite{harandi2013kernel} and pedestrian detection~\cite{tuzel2008pedestrian}.
This is because visual features and models often possess special structures which Euclidean space fails to capture.
Riemannian manifolds which form curved spaces are a more appropriate approach to model problems in various computer vision tasks.
Unfortunately, despite the fact that clustering methods have been studied since the $1950s$~\cite{jain2010,filippone2008survey},
applying such methods directly on data represented on Riemannian manifolds is not trivial.
Riemannian manifolds generally do not conform to Euclidean space~\cite{Turagaetal2011,pennecetal2006}. To address this,
one could use manifold tangent spaces which are locally homeomorphic to Euclidean space~\cite{pennecetal2006}.
However, this brings another challenge to applying existing clustering algorithms as some general algebraic operations
are not well defined~\cite{pennec2006intrinsic}.
For instance, K-means requires the computation of the mean within a cluster which cannot be computed directly.
To this end, Pennec~et al.~\cite{pennec2006intrinsic} reformulated the computation of mean as a solution to an optimisation problem.
Using this formulation, the mean point is considered as the point over the manifold minimising the geodesic distance (i.e.~ the true distance on the manifold between two points) from the mean point to all other points.
The algorithm to solve this problem is called Karcher mean~\cite{pennec2006intrinsic}.
Thanks to the Karcher mean, Turaga~et al.~\cite{Turagaetal2011} extended the K-means algorithm into the Riemannian manifold, which is regarded as intrinsic K-means and has been applied to activity-based video clustering.
Intrinsic K-means has further demonstrated better performance than Euclidean-based methods (for example,
Protein Clustering~\cite{suryanto2012protein}).
Generally, methods that completely honour the manifold topology lead to higher accuracy.
We shall categorise these methods as intrinsic methods.
Unfortunately, the computational cost of intrinsic methods is extremely high since these need to map all of the data to tangent spaces repeatedly.
Extrinsic methods, on the other hand, seek solutions that may not completely consider the manifold topology~\cite{faraki2014log,faraki2014fisher,yuanetal2010,jayasumanaetal2013,jayasumana2013framework,harandi2014expanding}.
The most simplistic way, here called Log Euclidean methods, is to embed all of the points into a designated tangent space at the identity point~\cite{arsignyetal2006}.
Log Euclidean methods can be considered as flattening the manifold space. It has been used in various computer vision applications, such as human action recognition~\cite{faraki2014fisher} and cell classification~\cite{yuanetal2010}.
This addresses the computational cost issues suffered by the intrinsic methods, as the tangent space is homeomorphic to the Euclidean space and well-known Euclidean clustering approaches such as K-means can be directly applied.
Unfortunately, as the flattening step distorts the pair-wise distances in regions far from the origin of the tangent space, accuracy is severely compromised.
So much of the value of the manifold approach is lost.
Other approaches that fall in the extrinsic method category are kernel-based approaches~\cite{jayasumanaetal2013,jayasumana2013framework,harandi2014expanding}, such as Kernel K-means.
In essence, the data in manifold space are first embedded into the Reproducing Kernel Hilbert Space (RKHS)~\cite{shaweCristianini2004}.
As the embedding function is defined implicitly, generally kernel-based approaches make use of the inner products in the RKHS in their formulation.
These inner products are then arranged in a Gram matrix.
It is often observed that the right choice of kernel could significantly improve the performance~\cite{jayasumanaetal2013}.
Furthermore, in general, kernel inner products with specified metrics have much less computational complexity than geodesic distances~\cite{alavietal2014,shirazietal2012}.
With these properties, kernel-based approaches could be suitable to address issues suffered in both the intrinsic approach and the Log Euclidean approach.
Unfortunately, the kernel-based approaches cannot scale easily,
as the Gram matrix computation is $O(n^2)$ where $n$ is the number of data points.
Also, it is often quite challenging to kernelise the existing algorithms that do not have known kernelised versions~\cite{caseiro2013rolling}.
Furthermore, Nikhil~et al.~ demonstrate that clustering data in the RKHS may lead to unexpected results since the clusters obtained in the RKHS may not exhibit the structure of the original data\cite{pal2014and}.
\textbf{Contributions}
We summarise the advantages and shortcomings of the existing approaches in Table~\ref{tab:related work}.
Our goal is to develop an efficient clustering algorithm for Riemannian manifolds, which significantly reduces the computational complexity, but still maintains acceptable performance.
The inspirations are drawn from the random projection for Euclidean spaces which has enjoyed success in various domains~\cite{bingham2001random,kushilevitz2000efficient,goel2005face}
due to its simplicity and theoretical guarantees~\cite{achlioptas2003database}.
We list our contributions as follows:
\begin{enumerate}
\item We propose a random projection framework for manifold features.
In general, the term projection is not well defined in Riemannian manifolds.
Therefore, we address this via
the RKHS constructed from a small subset of data.
Once projected, we choose to apply the
K-means algorithm.
\item From our framework, it becomes clear that
random hyperplane generation is essential. Thus, we describe three generation algorithms which are followed in our framework:
(1) Kernelised Gaussian Random Projection (KGRP);
(2) Kernelised Orthonormal Random Projection (KORP)
and (3) Kernel Principal Component Analysis
Random Projection (KPCA-RP).
\end{enumerate}
We note that our method is different from manifold learning approaches for clustering analysis described in~\cite{elhamifar2011sparse}.
Manifold learning is the collection of
non-linear dimensionality reduction (NLDR) techniques that seek for a low dimensional
representation of a set of high-dimensional points lying on a non-linear manifold~\cite{lin2006riemannian}.
They assume the structure of the underlying manifold was unknown.
Contrary to this, in our paper, we
are interested in Riemannian manifolds whose underlying geometry is known.
\begin{table}
\centering
\caption
{Summary of the existing works compared to our proposal.}
\label{tab:related work}
\vspace{0.5ex}
{
\begin{tabular}{cccp{2cm}}
\toprule
~{Approach}~&
{ Exploits Manifold Structure}&
{Accuracy}~&
{Computational} \\
{}&&&{Complexity}\\
\toprule
{Intrinsic Methods\cite{Turagaetal2011,suryanto2012protein}}
& {Yes}
&{ High}
&{ High}
\\
{Log-Euclidean Methods~\cite{faraki2014log,faraki2014fisher,yuanetal2010}}
&{Minimal}
&{Low}
&{Low}
\\
{Kernel Methods~\cite{jayasumanaetal2013,jayasumana2013framework,harandi2014expanding}}
&{Approximately}
&{High}
&{Moderate}\\
{Our proposal}
&{Approximately}
&{High}
&{Low}\\
\bottomrule
\end{tabular}
}
\end{table}
We continue the paper as follows.
Section~\ref{sec:riemannian_geometry} provides a brief mathematical background of Riemannian manifolds.
Section~\ref{sec:proposed_framework} details the proposed random projection framework for manifold points and develops three different random projection methods for clustering points on manifold spaces.
The proposed methods are then contrasted with the state-of-the-art methods in Section~\ref{sec:experimental_results}.
The conclusions and future directions are summarised in Section~\ref{sec:conclusions}.
\section{Proposed Framework}
\label{sec:proposed_framework}
As mentioned in Section~\ref{sec1}, the goal of our work is to significantly reduce clustering computational complexity for manifold features while maintaining reasonable clustering performance.
We address this by adopting a random projection approach to Riemannian manifolds.
In this section, we first discuss the overview of random projection in Euclidean space.
We then extend the notion into the Riemannian manifold space.
\subsection{Random Projection in Euclidean Space}
In Euclidean space, the random projection embeds original data into a much
lower dimensional space whilst preserving the geometric structure~\cite{santosh2004}.
This can significantly reduce the computational complexity of learning algorithms, such as classification or clustering.
For instance, as a result, this is used to achieve real time performance in object tracking~\cite{salaheldin2013robust}.
A point $\Vec{x} \in \mathbb{R}^d$ in Euclidean space can be projected into a random k-dimensional subspace $(k<<d)$ via a set of randomly generated hyperplanes $\left\lbrace \Vec{r}_1\right\rbrace_{i=1}^k$ where $\Vec{r}_i \in \mathbb{R}^d$.
This can be formulated as:
\begin{equation}
\label{eqn:RP}
f(x)=\Vec{x}^\top \Mat{R}\textrm{ ,}
\end{equation}
\noindent
where $\Mat{R}$ is the random matrix that arranges the random hyperplanes as column vectors.
Note that in order to minimise distortions produced by the projection, the matrix $\Mat{R}$ should possess a particular property.
We introduce this property in Definition~\ref{jl_projection}.
When the random projection matrix $\Mat{R}$ possesses such a property, then the Johnson-Lindenstrauss Lemma (JL-Lemma)~\cite{johnson1984extensions} applies.
\begin{lemma}
\label{JL}[Johnson-Lindenstrauss Lemma~\cite{johnson1984extensions}]
For any $\epsilon$ such that \mbox{$ \epsilon > 0$}, and any set of points $\mathcal{X} $ with $|\mathcal{X}| = n$ upon projection to a uniform random k-dimension subspace where $k \geq \operatorname{O}(\epsilon^{-2} \operatorname {log}\ n)$, the following property holds for every pair $\Vec{u}, \Vec{v} \in \mathcal{X}$,
$(1-\epsilon) || \Vec{u} - \Vec{v}||^2 \leq ||\operatorname{f}(\Vec{u}) - \operatorname{f}(\Vec{v})||^2 \leq (1+\epsilon) ||\Vec{u} - \Vec{v}||^2$, where $\operatorname{f}(\Vec{u}), \operatorname{f}(\Vec{v})$ are the projections of $\Vec{u},\Vec{v}$.
\end{lemma}
\noindent
\textit{Remarks}
The JL-Lemma principally states that a set of high dimensional points can be embedded using a set of uniform random hyperplanes into lower dimensional space wherein the pairwise distance between two points is well preserved (with high probability).
The original proof of JL-Lemma uses quite challenging geometric approximation machinery~\cite{johnson1984extensions}.
Frankl and Meahara~\cite{frankl1988johnson} simplified that proof by considering a projection into $k$
random orthonormal vectors.
Recently there have been several properties of the random matrix where JL-Lemma still applies.
We shall call the type of projection wherein the random matrix has properties that allow the JL-Lemma to be applied as a JL-Type projection.
\begin{definition}[JL-Type projection]
\label{jl_projection}
Let $\Mat{R} = [\Vec{r}_1 \cdots \Vec{r}_k], \Vec{r}_i \in \mathbb{R}^d$ be a random matrix whose columns are the random hyperplanes. The projection $\operatorname{f}(\Vec{u}) = \Mat{R}^{\top} \Vec{u}, \Vec{u} \in \mathbb{R}^d, \operatorname{f}(\Vec{u}) \in \mathbb{R}^k$ is called JL-Type projection when the matrix $\Mat{R}$ possesses at least one of the following properties:
\begin{enumerate}
\item\label{orthonormal} The columns of $\Mat{R}$ are orthogonal unit-length vectors~\cite{frankl1988johnson};
\item\label{Gaussian} Each element in $\Mat{R}$ is selected independently from a standard Gaussian distribution $N(0,1)$ or uniform distribution $U(-1,1)$~\cite{arriaga1999algorithmic};
\item\label{sparse} $\Mat{R}$ is a sparse matrix whose elements belong to $\left\lbrace -1,0,+1\right\rbrace $ with probability $\left\lbrace 1/6,2/3,1/6 \right\rbrace $~\cite{li2006very}.
\end{enumerate}
\end{definition}
\noindent
We note that Property \ref{orthonormal} in Definition~\ref{jl_projection} considers columns of the random matrix $\Mat{R}$ as the basis of a random space, thus they are required to be pairwise orthogonal~\cite{frankl1988johnson}.
To this end, one needs to apply an orthogonalisation technique such as the Gram-Schmidt method~\cite{watkins2004} on $\Mat{R}$.
Arriaga~et al.~\cite{arriaga1999algorithmic} proved that it suffices to use random non-orthonormal matrices
with independent elements chosen from some distributions which are listed in Property \ref{Gaussian} of Definition~\ref{jl_projection}.
Recently, Li~et al.~\cite{li2006very} proposed a
sparse random projection matrix presented in Property~\ref{sparse} of Definition~\ref{jl_projection}.
The sparse random projection achieves a further threefold speed-up as only $1/3$ of the matrix have non-zero elements.
We note that the random projection is not data driven.
It means that it does not need a set of labelled training data, making it suitable for unsupervised learning scenarios such as clustering~\cite{boutsidis2010random,sakai2009fast}.
\subsection{Random Projection in Riemannian Manifolds via RKHS}
As mentioned in Section~\ref{sec1}, applying the random projection on points residing in the Riemannian manifold space is not trivial, due to the notion of projection itself being generally not well defined.
We approach this problem by reformulating the problem in the RKHS.
Recall that, the random matrix containing column vector of hyperplanes $\Vec{r}_i$ should be generated from a particular process.
Thus, the projection of each individual dimension into the projected space is carried out as follows:
\begin{equation}
f_i(\Vec{x}) = \Vec{x}^\top \Vec{r}_i\textrm{ ,}
\end{equation}
\noindent
where $f_i(\cdot)$ is the $i$-th dimension of the projected vector $\Vec{x}$.
In the RKHS, the above formulation can be rewritten as:
\begin{equation}
f_i(\Vec{x}) = \phi(\Vec{x})^\top \Vec{r}_i\textrm{ ,}
\label{eq:rp_in_kernel}
\end{equation}
\noindent
where $\phi(\cdot)$ is the function that embeds the input space into the RKHS.
Note that, in this case, the hyperplane $\Vec{r}_i$ is now defined in the RKHS, $\Vec{r}_i \in \mathcal{H}$.
The projection in the RKHS can be considered as the inner product which is defined as the kernel similarity function.
Eqn.~\ref{eq:rp_in_kernel} provides insight that the JL-Type projection could be achieved
as long as one could generate the hyperplanes that follow one of the above properties in Definition~\ref{jl_projection} in the RKHS.
In similar fashion, when the data point $\Vec{x}$ is replaced by a point $\Mat{X}$ in manifold $\Mat{X} \in \Mat{\mathcal{M}}$, then one could use Eqn.~\ref{eq:rp_in_kernel} as the framework to achieve JL-Type projection in the manifold space.
As such, we propose a framework for clustering manifold points, which is briefly illustrated in Figure~\ref{fig:STEP1}.
This hyperplane generation is the central idea in our work.
First, we generate the hyperplanes over the RKHS.
The points over the manifold space are then projected into the projected space by using the specified kernel similarity function, such as the Gaussian kernel or projection kernel.
Once the manifold points have been embedded into the projected space, we apply the general K-means algorithm to perform clustering.
\begin{figure*}
\centerline{\includegraphics[width=\textwidth,keepaspectratio]{STEP2.pdf}}
\caption{
The illustration of our proposed framework.
We first generate the hyperplanes in RKHS.
Each point in the manifold space is then mapped into the projected space via the kernel inner product. Finally we apply K-means in the projected space.
}
\label{fig:STEP1}
\end{figure*}
In this paper, we explore three hyperplane generation methods for manifold points: (1) KGRP; (2) KORP and (3) KPCA-RP.
The diagram of our proposed generation methods is illustrated in Figure~\ref{fig:process}. Briefly speaking, the hyperplanes are generated using a randomly selected subset from the entire dataset.
The projection made by the hyperplanes will follow one of the properties in Definition~\ref{jl_projection}. We will elaborate on the generation process and theoretical analysis in the following section.
\begin{figure*}
\centerline{\includegraphics[width=0.90\textwidth,keepaspectratio]{process1.pdf}}
\caption{The diagram of our proposed generation methods: KORP, KGRP and KPCA-RP.}
\label{fig:process}
\end{figure*}
\input{subsec_GRP}
\input{subsec_ORP}
\input{subsec_KPCARP}
\section{Experimental Results}
\label{sec:experimental_results}
We evaluate our proposal using \rev{six} benchmark datasets: (1)~Ballet dataset~\cite{wang2009human};
(2)~UCSD traffic dataset~\cite{chan2005probabilistic};
(3)~UCF101 Human actions dataset~\cite{soomro2012ucf101};
\rev{(4)}~Brodatz texture dataset~\cite{randenHusoy1999};~\rev{(5)}~KTH-TIPS2b material dataset~\cite{caputoetal2005}
and \rev{(6)}~HEp-2 Cell ICIP2013 dataset~\cite{foggia2013benchmarking}.
In our evaluation, we consider each video of the first three datasets (i.e.~ Ballet, UCSD and UCF101) as an image set which can be effectively modelled as a point on Grassmannian manifolds.
In addition, we use SPD manifold to model images of the latter three datasets (i.e.~ Brodatz, KTH-TIPS2b and HEp-2 Cell ICIP2013).
To demonstrate the efficacy of our framework, we report the clustering performance and the run time.
\subsection{Datasets and Feature Extraction}
\noindent
\textbf{Ballet action dataset (Ballet)~\cite{wang2009human} -}
The Ballet dataset presents sequences of videos of ballet actions.
More precisely, it comprises 44 sequences with 8 different actions: R-L presenting, L-R presenting,
Presenting, Jump \& swing, Jump, Turn, Step, and Stand still~(see Figure~\ref{fig:Ballet} for examples).
These ballet actions were performed by two men and one woman, resulting in significant intra-class variations such as speed, clothing and movements.
In this evaluation, each video is considered as an image set.
We then represent each image set as a point in the Grassmannian manifold.
To that end, all the videos are down sampled to $16 \times 16$ pixels.
A Grassmannian point is extracted for every $6$ consecutive frames.
Technically, we first vectorise each frame into a column vector and arrange them into a $256 \times 6$ tall matrix (i.e.~ $256 = 16 \times 16$).
The matrix can be considered as a subspace and the orthonormal bases spanning the subspace can be determined by applying the Singular Value Decomposition (SVD).
The set of orthonormal bases is considered as a Grassmannian point~\cite{shirazietal2012}.
We use the projection kernel~(see Eqn.~\ref{eqn:G_kernel}) in this evaluation.
\noindent
\noindent
\textbf{UCSD traffic dataset (UCSD)~\cite{chan2005probabilistic} - }
The UCSD traffic dataset consists of 254 video sequences
collected from the highway traffic over two days in Seattle~(see Figure~\ref{fig:UCSD} for examples).
It contains a variety of traffic patterns and weather conditions
~(i.e.~ overcast, raining, sunny).
In total, there
are 44 sequences of heavy traffic (slow, stop and go
speeds), 45 sequences of medium traffic (reduced speed), and 165 sequences of
light traffic (normal speed).
To extract a Grassmannian point, we first randomly select half the number of frames from each video.
Each frame in each sequence is downsized to $140\times161$ pixels and further normalised by subtracting the mean frame and dividing the variance.
Then, we apply the two dimensional Discrete Cosine Transform (DCT) on the frame and use the DCT coefficients as the feature vector for each frame.
SVD is applied on the feature vectors of the frames to obtain the set of orthonormal bases.
We also choose the projection kernel~(see Eqn.~\ref{eqn:G_kernel}) for this dataset.
\noindent
\textbf{UCF101 Human Actions dataset (UCF101)~\cite{soomro2012ucf101} - }
This dataset consists of $13,320$ videos that belong to 101 categories.
For example, Applying \rev{E}ye \rev{M}akeup, Blow Dry Hair and Mixing Batter (refer to Figure~\ref{fig:ucf}).
For each video, we first extract the normalised pixel intensities as features for all the frames.
Then the SVD is applied on these features of each video to obtain the Grassmannian manifold point.
Thus, in this dataset, there are $13,320$ manifold points in total. Projection kernel (see Eqn.~\ref{eqn:G_kernel}) is used.
\noindent
\textbf{Brodatz texture dataset (Brodatz)~\cite{randenHusoy1999} - }
For the Brodatz dataset~(refer to Figure~\ref{fig:BRODATZ_examples} for examples)
we follow the protocol presented in~\cite{sivalingametal2010}.
The protocol
includes 3 subsets with different numbers of classes: 5-class-texture (‘5c’, ‘5m’, ‘5v’, ‘5v2’, ‘5v3’), 10-class-texture
(‘10’, ‘10v’) and 16-class-texture (‘16c’, ‘16v’).
Each image is down-sampled to $256\times 256$ pixels and divided into 64 $32\times 32$ pixel size regions.
A feature vector $F(x, y)$ for each pixel is calculated using
the grayscale intensities and absolute values of the first- and second-order
derivatives of spatial feature vectors.
It can be illustrated as:
$
F(x, y)
\mbox{~=~}
\left[
I\left(x,y\right),
\left| \frac{\partial I} {\partial x} \right|, \left|\frac{\partial I} {\partial y} \right|,
\left| \frac{\partial^2 I}{\partial x^2}\right|, \left|\frac{\partial^2 I}{\partial y^2}\right|
\right] \nonumber
$.
Each region is represented by a covariance matrix~(SPD matrix)~formed
from these feature vectors.
The Gaussian Kernel with Log-Euclidean distance~(see Eqn.~\ref{eqn:log_kernel}) is used for this dataset.
\noindent
\textbf{KTH-TIPS2b material dataset (KTH-TIPS2b)~\cite{caputoetal2005} - } This dataset contains 11 material categories captured under 4 different illuminations, in 3 poses and at 9 scales~(refer to Figure~\ref{fig:KTH-TIPS2b}).
Thus, there are $3\times4\times9=108$ images for each sample in one category, with $4$ samples per material.
We extract a 20-dimensional feature vector for each pixel in the image:
\begin{equation}
\label{eq:cov_features}
[ \Mat{I}(x,y), \Mat{Y}(x,y), \Mat{C}_b(x,y), \Mat{C}_r{(x,y)}, F^1_{(x,y)}(\Mat{Y}) \cdots F^{16}_{(x,y)}(\Mat{Y})],
\end{equation}
\noindent
where $\Mat{I}(x,y)$ is the image grey level value at location $(x,y)$; $\Mat{Y}$, $\Mat{C}_b$ and $\Mat{C}_r$ are the perceptually uniform CIELab colour space; The filter banks $F^i$ consist of different of offset Gaussians applied on the luminance channel $\Mat{Y}$~\cite{tosatoetal2013}.
The covariance matrix is computed once the feature vectors are extracted from every pixel location.
This becomes the image representation over a SPD manifold.
For the manifold kernel in this dataset, we use Gaussian kernel with the Stein Divergence~(see Eqn.~\ref{eqn:SD_kernel}) as this has been shown to be effective in various classification problem domains~\cite{alavietal2014, alavi2014multi}.
\noindent
\textbf{HEp-2 Cell ICIP2013 dataset~\cite{foggia2013benchmarking} - }
This dataset contains $13,596$ cell images that include six cell patterns namely Centromere, Golgi, Homogeneous, Nucleolar, Nuclear Membrane, and Speckled (refer to Figure~\ref{fig:cell_examples}).
The cell boundary of every cell image is described by a mask image of the same size.
For each cell image, we first extract the following feature vector of each pixel that belongs to the cell content:
$
F(x, y)
\mbox{~=~}
\left[
\left| \frac{\partial I} {\partial x} \right|, \left|\frac{\partial I} {\partial y} \right|,
I\left(x,y\right),
\left| \frac{\partial^2 I}{\partial x^2}\right|,
\left|\frac{\partial^2 I}{\partial y^2}\right|
,
\operatorname{arctan}( \left| \frac{\partial I} {\partial x} \right| /\left|\frac{\partial I} {\partial y} \right|)\right]
\nonumber
$.
Then, the covariance matrix~(SPD matrix) is formed
from these feature vectors extracted from each image.
We also use Gaussian kernel with the Stein Divergence~(see Eqn.~\ref{eqn:SD_kernel}) for the evaluation on this dataset.
\begin{figure}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=3.5cm,height=2cm]{Ballet_1.pdf}
\caption{
}
\label{fig:Ballet}
\end{subfigure}
~~
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=3.5cm,height=2cm]{UCSD.pdf}
\caption{}
\label{fig:UCSD}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=3.5cm,height=2cm]{ucf.pdf}
\caption{
}
\label{fig:ucf}
\end{subfigure}
\caption{Examples from~(a)~Ballet action dataset~\cite{wang2009human}~(b)~UCSD traffic dataset~\cite{chan2005probabilistic} and (c)~UCF101 dataset~\cite{soomro2012ucf101}}
\label{fig:dataset1}
\end{figure}
\begin{figure}[!b]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=3.5cm,height=1.5cm]{BRO_1.pdf}
\caption{}
\label{fig:BRODATZ_examples}
\end{subfigure}
~~~~
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=3.5cm,height=1.5cm]{KTH_1.pdf}
\caption{}
\label{fig:KTH-TIPS2b}
\end{subfigure}
~~~~\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=3.5cm,height=1.5cm]{cell_1.pdf}
\caption{}
\label{fig:cell_examples}
\end{subfigure}
\caption{Examples from~(a)~BRODATZ texture dataset~\cite{randenHusoy1999},~(b)~KTH-TIPS2b material dataset~\cite{caputoetal2005} and (c)~HEp-2 Cell ICIP2013 dataset~\cite{foggia2013benchmarking}.}
\label{fig:dataset}
\end{figure}
\subsection{Experimental Settings}
\label{sec:ex_setting}
As illustrated in Figure~\ref{fig:STEP1}, we first randomly project the points and then apply K-means.
As such, for each dataset, we first run each proposed projection method 10 times to generate 10 different random projection representations.
Then, for each representation, we run the K-means algorithm 10 times, resulting in each method being repeated 100 times for each evaluation.
The average of clustering performance and run time were reported.
As the source of variation for the other approaches is predominantly on the initial cluster seeds of K-means, we only repeat the experiment 10 times to obtain the average clustering performance and run time.
All of the approaches are tuned to give the best performance.
We find the optimum size of set $\mathcal{S}$ as follows: (1)~Ballet: 100; (2)~UCSD: 90; (3)~UCF101: 101;~(4)~Brodatz: 100; (5)~KTH-TIPS2b: 48 and~(6)~HEp-2 Cell ICIP2013: 60.
In addition, for KGRP, we set the number of dimensionality, $b$, to 300.
To measure the clustering quality, there are two main types of metrics: internal metrics based on the \rev{distances} between data points in the space, and external metrics based on the labels of the data~\cite{manningetal2008}.
The clustering task in our proposed framework is performed in a transformed space which may have different scale to other spaces used by \rev{comparable} methods such as LogE \rev{(see below for further discussion on LogE)}. This may make the internal metrics such as Dunn Index unsuitable in our case. Thus, we choose four external metrics to measure the clustering quality: Rand Index (RI), Cluster Purity (CP), F-Measure and Normalized Mutual Information~(NMI). Interested readers are referred to~\cite{manningetal2008} for further explanation of each metric.
In addition,
we also measure the run time (in seconds) of each approach on every evaluation.
The run time is measured from the kernel matrix
computation until the completion of the clustering process.
Finally, we report the average run time of the approaches.
Our proposal is contrasted to six approaches:
(1)~Intrinsic K-means~\cite{Turagaetal2011}; (2)~Log-Euclidean K-means~\cite{faraki2014fisher}; (3)~Kernel K-means~\cite{dhillon2004kernel,jayasumanaetal2013};
(4)~KPCA K-means~\cite{scholkopfetal1998,jayasumanaetal2013}; (5)~Sigma set K-means~\cite{hongetal2009} \rev{and}
(6)~Grassmanian clustering~\cite{shirazietal2012}.
The following is the brief description of each approach.
\noindent
{\bf Intrinsic K-means (Intrinsic):} To cluster a set of manifold points, Intrinsic K-means works directly on the manifold space using the appropriate geodesic distance~\cite{Turagaetal2011}. We note that as the intrinsic approach is generally very slow, we stop the Intrinsic K-means after 100 iterations.
\noindent
{\bf Log-Euclidean K-means~(LogE):} We first project all of the manifold points into the tangent space at the identity~\cite{arsignyetal2006}.
Once projected, each point will be vectorised into a column vector.
As for SPD manifolds, we follow the work in~\cite{pennecetal2006} that uses only the upper triangular elements.
This trick will reduce the final representation dimensionality, markedly reducing the run time on the subsequent process.
Unfortunately the trick cannot be used on Grassmannian manifolds since the representation for a point on the Grassmannian manifold is not a symmetric matrix.
In this case, all the elements are used in the final representation.
This could adversely affect the overall run time when the manifold dimensionality is high.
In the final step, K-means algorithm is applied.
Log-Euclidean k-means has been used for clustering large amount of manifold data~\cite{faraki2014fisher}.
\noindent
{\bf Kernel K-means:} This approach embeds manifold points into RKHS.
Then Kernel K-means is applied to perform clustering~\cite{dhillon2004kernel,jayasumanaetal2013}.
\noindent
{\bf KPCA K-means (KPCA)}:
All manifold points are first embedded into RKHS.
Then, KPCA
is used for projecting the points in RKHS into the space spanned by the principal components~\cite{scholkopfetal1998,jayasumanaetal2013}.
Finally, the K-means is applied.
\noindent
{\bf Sigma set K-means~(SIS):} Hong~et al.~\cite{hongetal2009} proposed a novel descriptor for SPD manifolds which simplifies the computations of distance and mean.
Using their proposed descriptors, we apply K-means with novel efficient computations of mean and distance.
\noindent
{\bf Grassmanian clustering~(G-clustering)}
Shirazi~et al.~\cite{shirazietal2012} proposed a clustering method for Grassmanian manifolds which use the eigenvectors of the normalised projection kernel matrix as the new features of Grassmanian points.
\subsection{Comparative Analysis on Clustering Quality}
\begin{table}
\centering
\caption
{ The clustering quality with variance (in \%) measured by Rand Index (RI), Cluster Purity (CP), F-Measure and Normalized Mutual Information~(NMI) on Ballet dataset. The best performance is in bold.
We refer to Section~\ref{sec:ex_setting} for further explanation of each approach.}
\label{tab:ballet}
\vspace{0.5ex}
\scalebox{1.0}{
\begin{tabular}{c|cccc}
\toprule
{\bf Methods/Measurements}
&{\bf RI}
&{\bf CP}
&{\bf F-Measure}
&{\bf NMI}\\
\midrule
{\bf Intrinsic~\cite{Turagaetal2011}}~&$73.68{\pm0.00} $&$34.92{\pm0.00}$&${33.81\pm0.00}$&${21.73\pm0.00}$\\
{\bf LogE~\cite{faraki2014fisher}}
&$78.23{\pm 0.15}$&$20.85 {\pm2.66}$&${14.81\pm0.37 }$&${3.91\pm0.81}$\\
{\bf G-clustering~\cite{shirazietal2012}}&$76.41{\pm0.07}$&$18.63{\pm0.58}$&${16.39\pm0.25}$&${3.51\pm0.47}$\\
{\bf Kernel K-means~\cite{dhillon2004kernel,jayasumanaetal2013} }~&$\bf{79.89{\pm0.80}}$&$ 40.86{\pm3.06}$&${32.92\pm3.21}$&${32.00\pm2.73}$\\
{\bf KPCA~\cite{scholkopfetal1998,jayasumanaetal2013} }~&$78.62{\pm 2.14}$&$42.30{\pm 3.33}$&${36.27\pm2.68}$&${{34.80\pm3.48}}$\\
\bottomrule
{\bf KGRP}&$78.02{\pm 1.79}$&$41.89{\pm2.43}$&${37.98\pm2.79}$&${34.05\pm2.41}$\\
{\bf KORP}&$78.28 {\pm1.68}$&$\bf{42.54{\pm2.37}}$&$\bf{{38.68\pm2.81}}$&$\bf{{35.30\pm2.80}}$\\
{\bf KPCA-RP}&$77.81{\pm1.94}$&$41.90{\pm2.31}$&${{38.23\pm3.11}}$&${34.64\pm2.75}$\\
\bottomrule
\end{tabular}
}
\end{table}
\begin{table}
\centering
\caption
{
The clustering quality with variance (in \%) measured by Rand Index (RI), Cluster Purity (CP), F-Measure and Normalized Mutual Information~(NMI) on UCSD dataset. The best performance is in bold.
We refer to Section~\ref{sec:ex_setting} for further explanation of each approach.
}
\label{tab:UCSD}
\vspace{0.5ex}
\scalebox{1.0}{
\begin{tabular}{c|cccc}
\toprule
{\bf Methods/Measurements}
&{\bf RI}
&{\bf CP}
&{\bf F-Measure}
&{\bf NMI}\\
\midrule
{\bf Intrinsic~\cite{Turagaetal2011}}~&$73.26{\pm0.00} $&$74.70{\pm0.00}$&$\bf{{75.15\pm0.00}}$&${36.18\pm0.00}$\\
{\bf LogE~\cite{faraki2014fisher}}
&$55.24{\pm 3.25}$&$67.23 {\pm2.66}$&${40.39\pm2.81 }$&${19.11\pm3.59}$\\
{\bf G-clustering~\cite{shirazietal2012}}&$50.68{\pm0.11}$&$64.82{\pm0.00}$&${34.31\pm0.12}$&${0.92\pm0.29}$\\
{\bf Kernel K-means~\cite{dhillon2004kernel,jayasumanaetal2013} }~&$69.98{\pm7.06}$&$ 77.96{\pm4.77}$&${57.34\pm10.22}$&${45.50\pm9.71}$\\
{\bf KPCA~\cite{scholkopfetal1998,jayasumanaetal2013} }~&$\bf{77.90{\pm 5.97}}$&$80.08{\pm 2.96}$&$\bf{{69.29\pm7.56}}$&$\bf{{51.31\pm6.09}}$\\
\bottomrule
{\bf KGRP}&$75.61{\pm 3.48}$&$79.64{\pm2.07}$&${66.97\pm5.17}$&${48.29\pm3.80}$\\
{\bf KORP}&$77.25 {\pm1.25}$&$\bf{80.18{\pm0.74}}$&${{68.99\pm1.62}}$&${{50.58\pm1.67}}$\\
{\bf KPCA-RP}&$76.46{\pm2.79}$&$79.64{\pm1.68}$&${68.60\pm3.50}$&${49.74\pm3.02}$\\
\bottomrule
\end{tabular}
}
\end{table}
\begin{table}
\centering
\caption
{
The clustering quality with variance (in \%) measured by Rand Index (RI), Cluster Purity (CP), F-Measure and Normalized Mutual Information~(NMI) on UCF101 dataset. The best performance is in bold.
We refer to Section~\ref{sec:ex_setting} for further explanation of each approach.
}
\label{tab:UCF101}
\vspace{0.5ex}
\scalebox{1.0}{
\begin{tabular}{c|ccc|ccc}
\toprule
{\bf Methods/Measurements}
&{\bf RI}
&{\bf CP}
&{\bf F-Measure}
&{\bf NMI}\\
\midrule
{\bf Intrinsic~\cite{Turagaetal2011}}~&${97.53\pm0.00} $&${12.94\pm0.00}$&${7.43\pm0.00}$&${27.65\pm0.00}$\\
{\bf LogE~\cite{faraki2014fisher}}
&${97.89\pm 0.02}$&$ {8.21\pm0.15}$&${3.62\pm0.06 }$&${18.68\pm0.07}$\\
{\bf Kernel K-means~\cite{dhillon2004kernel,jayasumanaetal2013} }~&${97.71\pm0.06}$&$ {15.97\pm0.48}$&${8.80\pm0.35}$&${32.35\pm0.31}$\\
{\bf KPCA~\cite{scholkopfetal1998,jayasumanaetal2013} }~&${97.69\pm0.02 }$&$\bf{{17.66\pm0.33 }}$&$\bf{{9.47\pm0.19}}$&$\bf{{33.66\pm0.18}}$\\
\bottomrule
{\bf KGRP}&$\bf{{97.90\pm0.03 }}$&${15.38\pm0.28}$&${7.40\pm0.15}$&${30.96\pm0.21}$\\
{\bf KORP}&$\bf{97.90\pm0.02}$&${15.69\pm0.28}$&${7.62\pm0.17}$&${31.47\pm0.17}$\\
{\bf KPCA-RP}&${97.89\pm0.03}$&${15.66\pm0.33}$&${7.59\pm0.17}$&${31.38\pm0.23}$\\
\bottomrule
\end{tabular}
}
\end{table}
\begin{table}
\centering
\caption
{
The clustering quality with variance (in \%) measured by Rand Index (RI), Cluster Purity (CP), F-Measure and Normalized Mutual Information~(NMI) on BRODATZ dataset. The best performance is in bold.
We refer to Section~\ref{sec:ex_setting} for further explanation of each approach.
}
\label{tab:BRODATZ}
\vspace{0.5ex}
\scalebox{1.0}{
\begin{tabular}{c|cccc}
\toprule
{\bf Methods/Measurements}
&{\bf RI}
&{\bf CP}
&{\bf F-Measure}
&{\bf NMI}\\
\midrule
{\bf Intrinsic~\cite{Turagaetal2011}}~&$92.29{\pm0.00} $&$79.05{\pm0.00}$&${74.20\pm0.00}$&${75.94\pm0.00}$\\
{\bf SIS~\cite{hongetal2009} }&$91.42{\pm0.00}$&$76.99{\pm0.00}$&${69.68\pm0.00}$&${72.84\pm0.00}$\\
{\bf LogE~\cite{faraki2014fisher}}
&$92.04{\pm0.78}$& $78.34{\pm2.34}$&${74.10 \pm2.10}$&${76.13\pm1.45}$\\
{\bf Kernel K-means~\cite{dhillon2004kernel,jayasumanaetal2013} }~&$93.15{\pm0.95}$&$81.40{\pm2.75}$&${75.62\pm2.13}$&${78.19\pm1.83}$\\
{\bf KPCA~\cite{scholkopfetal1998,jayasumanaetal2013} }~&$\bf{93.89{\pm0.22}}$&${82.60{\pagebreak\pm1.14}}$&$\bf{{76.64\pm0.66}}$&$\bf{{79.44\pm0.57}}$\\
\bottomrule
{\bf KGRP}&$93.47{\pm0.78}$&$82.22{\pm2.34}$&${75.84\pm1.82}$&${78.49\pm1.49}$\\
{\bf KORP}&$93.66 {\pm0.77}$&$82.58{\pm2.32}$&${76.30\pm1.81}$&${79.11\pm1.50}$\\
{\bf KPCA-RP}&$93.77{\pm0.84}$&$\bf{82.81{\pm2.49}}$&${76.39\pm1.93}$&${79.16\pm1.56}$\\
\bottomrule
\end{tabular}
}
\end{table}
\begin{table}
\centering
\caption
{
The clustering quality with variance (in \%) measured by Rand Index (RI), Cluster Purity (CP), F-Measure and Normalized Mutual Information~(NMI) on KTH-TIPS2b dataset. The best performance is in bold.
We refer to Section~\ref{sec:ex_setting} for further explanation of each approach.
}
\label{tab:KTH-TIPS2b}
\vspace{0.5ex}
\scalebox{1.0}{
\begin{tabular}{c|cccc}
\toprule
{\bf Methods/Measurements}
&{\bf RI}
&{\bf CP}
&{\bf F-Measure}
&{\bf NMI}\\
\midrule
{\bf Intrinsic~\cite{Turagaetal2011}}~&$86.99{\pm0.00} $&$49.45{\pm0.00}$&${36.19\pm0.00}$&${44.20\pm0.00}$\\
{\bf SIS~\cite{hongetal2009} }&$80.81{\pm0.00}$&$41.62{\pm0.00}$&$\bf{{44.45\pm0.00}}$&${40.47\pm0.00}$\\
{\bf LogE~\cite{faraki2014fisher}}
&$85.94{\pm0.60 }$&$45.19{\pm1.32}$&${33.48\pm1.01 }$&${40.69\pm0.82}$\\
{\bf Kernel K-means~\cite{dhillon2004kernel,jayasumanaetal2013} }~&$88.35{\pm0.35}$&$ 52.59{\pm1.37}$&${41.11\pm1.01}$&$\bf{{51.08\pm0.82}}$\\
{\bf KPCA~\cite{scholkopfetal1998,jayasumanaetal2013} }~&$\bf{88.48}{\pm 0.40}$&$\bf{53.38{\pm 1.53}}$&${{41.22\pm1.35}}$&${50.97\pm0.90}$\\
\bottomrule
{\bf KGRP}&$88.41{\pm 0.42}$&$53.15{\pm1.34}$&${40.48\pm0.99}$&${49.87\pm1.01}$\\
{\bf KORP}&${88.36}{\pm0.39}$&$53.04{\pm1.10}$&${40.61\pm0.93}$&${50.06 \pm0.92}$\\
{\bf KPCA-RP}&$88.35{\pm0.44}$&$53.45{\pm1.35}$&${40.21\pm0.91}$&${49.97\pm1.09}$\\
\bottomrule
\end{tabular}
}
\end{table}
\begin{table}[!t]
\centering
\caption
{
The clustering quality with variance (in \%) measured by Rand Index (RI), Cluster Purity (CP), F-Measure and Normalized Mutual Information~(NMI) on HEp-2 Cell ICIP2013 dataset. The best performance is in bold.
We refer to Section~\ref{sec:ex_setting} for further explanation of each approach.
}
\label{tab:cell}
\vspace{0.5ex}
\scalebox{1.0}{
\begin{tabular}{c|cccc}
\toprule
{\bf Methods/Measurements}
&{\bf RI}
&{\bf CP}
&{\bf F-Measure}
&{\bf NMI}\\
\midrule
{\bf Intrinsic~\cite{Turagaetal2011}}~&${73.96\pm0.00} $&${44.02\pm0.00}$&${35.69\pm0.00}$&${22.65\pm0.00}$\\
{\bf SIS~\cite{hongetal2009} }&${74.50\pm0.00}$&${39.50\pm0.00}$&${27.32\pm0.00}$&${18.01\pm0.00}$\\
{\bf LogE~\cite{faraki2014fisher}}
&${74.80\pm0.95}$&${46.00 \pm2.37}$&${34.75\pm0.86 }$&${23.64\pm1.29}$\\
{\bf Kernel K-means~\cite{dhillon2004kernel,jayasumanaetal2013}}&${73.96\pm2.14}$&${46.45\pm3.30}$&$\bf{{37.29\pm3.29}}$&${24.29\pm1.95}$\\
{\bf KPCA~\cite{scholkopfetal1998,jayasumanaetal2013} }~&$\bf{{75.74\pm 2.87}}$&${48.48\pm1.94 }$&${34.23\pm2.20}$&${25.29\pm0.00}$\\
\bottomrule
{\bf KGRP}&${75.72\pm0.31 }$&$\bf{{49.05\pm1.10}}$&${34.83\pm0.84}$&$\bf{{25.74\pm0.82}}$\\
{\bf KORP}&${75.63 \pm0.62}$&${48.70\pm2.34}$&${34.73\pm1.67}$&${25.49\pm1.72}$\\
{\bf KPCA-RP}&${75.72\pm0.41}$&${48.70\pm2.56}$&${34.48\pm1.74}$&${25.46\pm1.93}$\\
\bottomrule
\end{tabular}
}
\end{table}
Tables~\ref{tab:ballet}, \ref{tab:UCSD},~\ref{tab:UCF101},~\ref{tab:BRODATZ},~\ref{tab:KTH-TIPS2b} and~\ref{tab:cell} report the average clustering quality of each individual approach applied on each dataset.
In general, our proposed methods perform reasonably well and show a close match to KPCA K-means and Kernel K-means.
Also, the performance of the proposed approaches is similar to each other.
These factors suggest that the proposed projection approaches possess the JL-Type projection properties.
Furthermore, we find that the proposed approaches in some cases have markedly
better performance than the Kernel K-means.
One of the possible reasons could be that the random projection
reduces the eccentricity of original Gaussian-distributed clusters and make clusters in projected spaces more spherical~\cite{Dasgupta00}.
Intrinsic K-means gives us reasonable results as it directly works on manifold space.
Compared to the intrinsic approach, LogE has a worse performance in most of datasets.
An exception is on the Ballet dataset where the intrinsic approach has a worse Rand Index than the LogE.
We conjecture that this is caused by the failure of the intrinsic algorithm to converge in 100 iterations.
Nevertheless, the other performance metrics such as CP, F-Measure and NMI for the intrinsic approach in this dataset still show reasonable performance.
The worse performance for LogE is due to significant distortion of the pairwise distance produced
when the points are projected into a tangent space.
The G-clustering has a better Rand Index than the intrinsic approach in the Ballet dataset,
which is a similar conclusion drawn in the original work proposing the approach~\cite{shirazietal2012}.
Note that the measurements for clustering performance are different from that in~\cite{shirazietal2012}.
In most cases, the G-clustering is not robust as the performance of G-clustering measured by CP, F-measure and NMI is usually low.
In addition, we do not report the G-clustering results for the UCF101 dataset, as the K-means does not converge within a specified amount of time.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{p_Kpca_rp_ballet_1.pdf}
\caption
{ Clustering quality of the proposed KPCA-RP when the kernel parameter $\beta$ was varied on the Ballet dataset.
The clustering quality is measured by: Rand Index (RI), Cluster Purity (CP), F-Measure and Normalized Mutual Information~(NMI). }
\label{fig:p_setting_ballet}
\end{figure}
We found the performance of our proposed methods does not change significantly, when the parameters are varied.
Figures~\ref{fig:p_setting_ballet} and~\ref{fig:p_setting_cell} show two examples of the clustering results of KPCA-RP and KORP with different parameters on the Ballet and HEp-2 Cell ICIP2013 dataset, respectively. We note that the results on the other datasets also exhibit similar trends. This suggests that the issue raised in~\cite{pal2014and}\rev{, where different parameters may adversely alter the kernel space,} may not have significant effect \rev{on} our work. We conjecture that this might be due to the selected manifold kernels crafted to capture the manifold intrinsic structure.
However, if in the case where the parameter choice of the manifold kernel significantly contributes to the clustering results, one could use a randomly selected small subset of data to perform the parameter search.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{P_KORP_Cell.pdf}
\caption{ Clustering quality of the proposed KORP when the parameter $\beta$ is varied on the HEp-2 Cell ICIP2013 dataset.
The clustering quality is measured by: Rand Index (RI), Cluster Purity (CP), F-Measure and Normalized Mutual Information~(NMI).}
\label{fig:p_setting_cell}
\end{figure}
The evaluation has clearly shown that our proposal has similar performance to the kernel methods such as KPCA K-means and Kernel K-means.
Indeed, these results alone do not give us much advantage over the other methods.
However, we now present the main advantage of our proposal which is a direct consequence of applying random projection.
\subsection{Run Time Comparative Analysis}
\begin{table}
\centering
\caption
{
The run time (in seconds) of the approaches on each dataset. Lower run time is better. As in each iteration of K-means, the run time is extremely similar, we report the average run time of each approach without variance. The datasets presented in the first three columns~(i.e.~ Ballet, UCSD and UCF101) are modelled in Grassmannian manifolds, whilst the other three (i.e.~ Brodatz, KTH-TIPS2b and HEp-2 Cell ICIP2013 (shorten as Cell)) are modelled in SPD manifolds. The last three rows are the proposed approaches. SIS and G-clustering are only applicable for SPD manifolds and Grassmannian manifolds, respectively. We refer to Section~\ref{sec:ex_setting} for further explanation of each approach.
}
\label{tab:time}
\vspace{0.5ex}
\scalebox{0.9}{
\begin{tabular}{c|ccc|ccc}
\toprule
{\bf Methods/Dataset}
&{\bf Ballet}
&{\bf UCSD}
&{\bf {UCF101}}
&{\bf Brodatz}
&{\bf KTH-TIPS2b}
&{\bf {Cell} } \\
\toprule
{\bf Intrinsic~\cite{Turagaetal2011}}~&$3966.49$&$1990.02$&${1.64\times 10^5}$&$24.63$&$938.95$&${564.49}$\\
{\bf SIS~\cite{hongetal2009} }&$N/A$&$N/A$&${N/A}$&$4.77$&$60.43$&${185.81}$\\
{\bf LogE~\cite{faraki2014fisher} }&$3.35$&$1.55$&${9088.11}$&$\textbf{0.15}$&$\textbf{4.85}$&$\bf{{2.32}}$\\
{\bf G-clustering~\cite{shirazietal2012}}&$2.81$&$0.74$&${N/A}$&$N/A$&$N/A$&${N/A}$\\
{\bf Kernel K-means~\cite{dhillon2004kernel,jayasumanaetal2013} }~&$1.06$&$0.70$&${2019.55}$&$22.57$&$675.75$&${2172.87}$\\%&$7.61$
{\bf KPCA~\cite{scholkopfetal1998,jayasumanaetal2013} }~&$1.47$&$0.73$&${6.11\times 10^4}$&$22.42$&$699.34$&${2881.10}$\\
\bottomrule
{\bf KGRP}&$\textbf{0.51}$&$0.53$&${238.64}$&$7.08$&$14.61$&${21.95}$\\
{\bf KORP}&$0.58$&$\textbf{0.49}$&$\bf{{101.87}}$&$7.03$&$11.75$&${17.73}$\\
{\bf KPCA-RP}&$0.60$ &$\textbf{0.49}$&${102.79}$&$7.75$&$12.28$&${17.73}$\\
\bottomrule
\end{tabular}
}
\end{table}
Table~\ref{tab:time} presents the average run time of the individual approach on each dataset.
One of the striking observations from this table is that our proposed approaches have very fast run times.
In some cases (i.e.~ Ballet, UCSD and \rev{UCF101} datasets) they outperform the LogE which is expected to be the fastest method.
The bottleneck suffered by LogE in these datasets is from the high dimensionality of the feature vectors significantly slowing
the K-means algorithm.
Note that, although the run time of LogE on Brodatz, KTH-TIPS2b and HEp-2 Cell ICIP2013 dataset is quicker than our proposed methods,
the clustering quality shown in Tables~\ref{tab:BRODATZ},~\ref{tab:KTH-TIPS2b} and~\ref{tab:cell} is much worse than that of ours.
The proposed approaches are considerably faster than the kernel approaches such as KPCA K-means and Kernel K-means.
This is because the proposed approaches only compute the kernel matrix on a small subset of data points.
The benefit will become more pronounced for large datasets such as KTH-TIPS2b, UCF101 and HEp-2 Cell \rev{ICIP2013} datasets where our proposed approach achieves $57.5$ (i.e.~ $\frac{675.75}{11.75} \approx 57.5$), $19.8$ (i.e.~ $\frac{2019.55}{101.87} \approx 19.8$) and $ 122.5$ times (i.e.~ $\frac{2172.87}{17.73} \approx 112.5$) speed up, respectively.
Thus, the proposed approaches will contribute significantly to the clustering of large amount of images or video data for practical applications.
The speed up gained by the proposed approaches is attributed to the effect of
applying random projection into a reduced projection space.
The proposed approaches also have additional advantages over the kernel approaches as they do not need to compute the kernel matrix on the entire dataset.
In addition, we analyse the computational complexity of each method in Table~\ref{tab:cc}.
In general, each method has two main steps: (1) Data pre-processing and (2) K-means steps. Data pre-processing may include kernel computation and/or projection. Whilst, K-means step comprises cluster membership and cluster mean computations.
In Intrinsic K-means, the pre-processing step is not required. To calculate mean of each cluster, one need to use the intrinsic mean, denoted Karcher mean~\cite{pennec2006intrinsic} that requires multiple iterations to converge. The intrinsic distance is also used for membership computation.
For LogE, each manifold point needs to be projected onto the Log-Euclidean space. This projection is done once. Then, K-means is applied in the Log-Euclidean space.
The computational complexity of KPCA and Kernel K-means follows quadratic and cubic growth, respectively. However, our proposed methods have linear growth, as the number of data points, $n$, is much bigger than the size of subset, $p$. This further corroborates the results presented in Table~\ref{tab:time}.
\begin{table}
\centering
\caption
{
Computational complexity of the approaches on each dataset.
The dimensionality of SPD and Grassmannian points is $d\times d$ and $q\times d$, respectively. For convenience, $\mathcal{G}$ is used to represent Grassmannian manifold in this table. Note that: $n$ is the number of points; $m$ is the number of clusters; $\ell$ is the number of iterations of K-means; $\ell_{kar}$ is the number of iterations of Karcher mean; $b$ is the dimensionality of the random projection space generated by KGRP \rev{and} $p$ is the dimensionality of the random projection space generated by KORP and KPCA-RP ($p=|\mathcal{S}|$).}
\label{tab:cc}
\vspace{0.5ex}
\scalebox{0.7}{
\begin{tabular}{|c|ccccc|}
\toprule {\bf }&{\bf {Compute}}&{\bf{Compute}}&\bf{{Compute}}&{\bf {Compute}}&{\bf {Overall}}\\
{\bf}&{\bf {Kernel }}&{\bf{Projection}}&\bf{{Mean }}&{\bf {Membership}}&{\bf {Complexity}}\\
\toprule
{\bf {Intrinsic(SPD)~\cite{Turagaetal2011}}}~&${N/A}$&${N/A}$&${O(\ell \ell_{kar}nd^3)}$&${O(\ell nmd^3)}$&${O(\ell \ell_{kar}nd^3+\ell nmd^3)}$\\
{\bf {Intrinsic($\mathcal{G}$)~\cite{Turagaetal2011}}}~&${N/A}$&${N/A}$&${O(\ell \ell_{kar}n(qd^2+d^3))}$&${O(\ell nm(qd^2+d^3))}$&${O((\ell \ell_{kar}n+\ell nm)(qd^2+d^3))}$\\
{\bf{SIS~\cite{hongetal2009} }}&${N/A}$&${O(nd^3)}$&${O(\ell nd^2)}$&${O(\ell nmd^3)}$&${O(\ell nmd^3)}$\\
{\bf {LogE(SPD)~\cite{faraki2014fisher}} }&${N/A}$&${O(nd^3)}$&${O(\ell nd^2)}$&${O(\ell nmd^2)}$&${O(nd^3+\ell nmd^2)}$\\
{\bf {LogE($\mathcal{G}$)~\cite{faraki2014fisher}} }&${N/A}$&${O(nqd^2)}$&${O(\ell nqd)}$&${O(\ell nmqd)}$&${O(nqd^2+\ell nmqd)}$\\
{\bf {G-clustering~\cite{shirazietal2012}}}&${O(n^2)}$&${O(n^3)}$&${O(\ell n^2)}$&${O(\ell n^2m)}$&${O(n^3+\ell n^2m)}$\\
{\bf{ Kernel K-means~\cite{dhillon2004kernel,jayasumanaetal2013}} }~&${O(n^2)}$&${N/A}$&${N/A}$&${O(\ell n^2m)}$&${O(\ell n^2m)}$\\
{\bf {KPCA~\cite{scholkopfetal1998,jayasumanaetal2013}} }&${O(n^2)}$&${O(n^3)}$&${O(\ell n^2)}$&${O(\ell n^2m)}$&${O(n^3+\ell n^2m)}$\\
\bottomrule
{\bf{ KGRP}}&${O(np)}$&${O(p^3+np^2)}$&${O(\ell npb)}$&${O(\ell nmb)}$&${O(np+p^3+np^2+\ell nmb)}$\\
{\bf{ KORP}}&${O(np)}$&${O(p^3+np^2)}$&${O(\ell np)}$&${O(\ell nmp)}$&${O(np+p^3+np^2+\ell nmp)}$\\
{\bf {KPCA-RP}}&${O(np)}$&${O(p^3+np^2)}$&${O(\ell np)}$&${O(\ell nmp)}$&${O(np+p^3+np^2+\ell nmp)}$\\
\bottomrule
\end{tabular}
}
\end{table}
\subsection{Further Analysis}
In this section, we analyse the parameters contributing to the performance and run time of the proposed methods.
\rev{Due to space limitations, we only show the performance measured by RI and CP. Note that the performance measured by F-Measure and NMI also follows the same trends.} An obvious parameter is the projected space dimensionality, $k$.
When $k$ is small, each data point will be represented in a much smaller feature vector, resulting in faster K-means clustering processes.
Another parameter is $|\mathcal{S}|$, the size of set $\mathcal{S}$ which determines the run time of the kernel matrix computation.
As $|\mathcal{S}|$ gets larger, it takes longer to compute the kernel matrix.
Smaller $|\mathcal{S}|$ gives more advantage to the proposed methods over the kernel approaches such as Kernel K-means and KPCA that require kernel computation on the entire data points.
We note that $k$ and $|\mathcal{S}|$ have an interesting relationship.
More precisely, for KORP and KPCA-RP, $|\mathcal{S}|$ determines the projected space dimensionality, $k$.
Therefore, it is desirable to make $|\mathcal{S}|$ as small as possible whilst still preserving as much of the pairwise distance.
\begin{figure}[!t]
\centering
\includegraphics[width=0.8\columnwidth]{KTH_RI_2}
\caption
{
The Rand Index (in \%) of the proposed approaches when the size of set $\mathcal{S}$ is progressively increased on the KTH-TIPS2b dataset. KGRP: Kernelised Gaussian Random Projection; KORP: Kernelised Orthonormal Random Projection; KPCA-RP: Kernel PCA based Random Projection.
}
\label{fig:KTH_RI}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=0.8\columnwidth]{KTH_CP_2}
\caption
{
The Cluster Purity (in \%) of the proposed approaches when the size of set $\mathcal{S}$ is progressively increased on the KTH-TIPS2b dataset. KGRP: Kernelised Gaussian Random Projection; KORP: Kernelised Orthonormal Random Projection; KPCA-RP: Kernel PCA based Random Projection.
}
\label{fig:KTH_CP}
\end{figure}
In contrast to KORP and KPCA-RP, KGRP separates the projected space dimensionality to $|\mathcal{S}|$.
Nevertheless, we found that $|\mathcal{S}|$ still plays an important role in the overall system performance.
To verify this, we vary $|\mathcal{S}|$ on the KTH-TIPS2b.
As we can see from Figures~\ref{fig:KTH_RI} and~\ref{fig:KTH_CP}, the performance of the proposed approaches
increases as $|\mathcal{S}|$ is progressively increased.
The performance increase stops when $|\mathcal{S}|$ reaches a particular value.
In this analysis we also found that the performance of KORP and KPCA-RP is
markedly better than KGRP when $|\mathcal{S}|$ is considerably small.
A possible reason is that the CLT requires the set $\mathcal{S}$ to have a minimum number of elements~(normally 30) in order
to make the theorem applicable.
The above observation suggests the following facts about $|\mathcal{S}|$: (1)~$|\mathcal{S}|$ determines the run time for all the
proposed approaches (i.e.~ on the kernel computation); (2)~$|\mathcal{S}|$ also contributes to the K-means run time for KORP and KPCA-RP;
(3)~the lower bound of $|\mathcal{S}|$ in the KGRP is related to the
lower bound of the CLT and (4) the lower bound of $|\mathcal{S}|$ for KORP and KPCA-RP is related to the lower bound of $k$.
The JL-Lemma relates $k$ to the total number of data points, $n$ (refer to Lemma~\ref{JL}).
This relationship seems unfavourable for KORP and KPCA-RP as this would mean $|\mathcal{S}|$ increases as $n$ increases.
Fortunately, Lemma~\ref{blum} and Theorem~\ref{theorem:kpcarp} suggest that $k$ is related to the margin between classes.
This means that we now need only consider the separating margin to select $|\mathcal{S}|$.
To further corroborate this empirically, we apply the proposed approaches by varying the dataset size of the KTH-TIPS2b.
We assume that the margin is relatively unchanged though the dataset size is varied.
More precisely, we first fix $|\mathcal{S}|$ for each proposed approach.
Then we randomly select the data points from the KTH-TIPS2b to create a smaller version of the dataset.
The proposed approaches are applied on these smaller subsets of the dataset.
Note that although $|\mathcal{S}|$ is fixed, we still select the elements of $\mathcal{S}$ from the given subset.
The results shown in Figures~\ref{fig:KTH_RI_2} and~\ref{fig:KTH_CP_2} suggest that the proposed approaches still have on par performance with both Kernel K-means and KPCA K-means,
suggesting that $|\mathcal{S}|$ relates to the margin separation between classes.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{KTH_RI_exp2_1}
\caption
{
The Rand Index (in \%) of the proposed approaches, Kernel K-Means and KPCA applied on subsets of KTH-TIPS2b with various sizes.
We fix $|\mathcal{S}|$ for all subsets.
}
\label{fig:KTH_RI_2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=0.8\columnwidth]{KTH_CP_exp2_1}
\caption
{
The Cluster Purity (in \%) of the proposed approaches, Kernel K-Means and KPCA applied on subsets of KTH-TIPS2b with various sizes.
We fix $|\mathcal{S}|$ for all subsets.
}
\label{fig:KTH_CP_2}
\end{figure}
\section{The Geometry of Riemannian Manifolds}
\label{sec:riemannian_geometry}
A differentiable manifold $\Mat{\mathcal{M}}$ is a topological space that is locally similar to Euclidean space~\cite{tu2008introduction}.
One can use the tangent space to model the neighbourhood structure on a differentiable manifold.
The tangent space at a point $\Mat{X}$ on the manifold, $\Mat{T_{\Mat{X}}\mathcal{M}}$, is a vector space that contains all possible directions tangentially passing through $\Mat{X}$~\cite{tu2008introduction}.
A Riemannian manifold is a differentiable manifold, endowed with a Riemannian metric.
The Riemannian metric is the family of inner products on all of the tangent spaces~\cite{jost2008riemannian}.
This metric enables us to define geometric concepts such as lengths, angles and distances.
The geodesic distance between two points $\Mat{X},\Mat{Y}$ is defined as the length of the shortest curve between
$\Mat{X}$ and $\Mat{Y}$~\cite{jost2008riemannian}.
In this section, we briefly introduce two well known Riemannian manifolds used in the computer vision community, namely Symmetric Positive Definite~(SPD) manifold and Grassmannian manifold.
\subsection{SPD Manifolds}
To compute a compact representation of an image, one method is to calculate the covariance matrix of a set of $d$-dimensional vector features extracted from the image~\cite{tuzel2006region}.
Covariance matrices naturally arise in the form of SPD matrices, which can be considered as points on SPD manifolds~\cite{tuzel2008pedestrian}.
The geodesic distance between points on SPD manifolds then
can be calculated through an affine invariant Riemannian metric:
\begin{equation}
\label{Eqn:dist_SPD}
\operatorname{dist}(\Mat{X},\Mat{Y})=||\operatorname{log}(\Mat{X}^{- \frac{1}{2}}\Mat{Y} \Mat{X}^{- \frac{1}{2}})||^2_F \textrm{ ,}
\end{equation}
\noindent
where $\Mat{X},\Mat{Y} \in \Mat{\mathcal{M}}$ are two points over the SPD manifold.
For further discussions on SPD manifolds, the readers are referred to~\cite{pennecetal2006}.
To further improve clustering performance,
SPD manifolds could be projected into RKHS by Mercer kernels.
In this paper, we use one of the popular kernels for SPD manifolds, namely the Gaussian kernel, which is defined by:
\begin{equation}
\label{SPD_kernel}
\operatorname{K}(\Mat{X},\Mat{Y})=\operatorname{exp}(-\beta \cdot\operatorname{dist}(\Mat{X},\Mat{Y}))\textrm{ ,}
\end{equation}
where $\operatorname{dist}(\Mat{X},\Mat{Y})$ is the geodesic distance between point $\Mat{X}$ and $\Mat{Y}$ from Eqn.\ref{Eqn:dist_SPD}.
Since the geodesic distance is computationally demanding, several methods for computing the approximate distance have been developed~\cite{arsignyetal2006,sra2011,wang2004affine}.
In this paper, we use two popular approximate distance functions: Log Euclidean Distance~(LED)~\cite{arsignyetal2006} and Stein Divergence~(SD)~\cite{sra2011}.
The Gaussian kernel with LED and SD then can be respectively formulated by:
\begin{equation}
\label{eqn:log_kernel}
\operatorname{K}_{LED}(\Mat{X},\Mat{Y})=\operatorname{exp}(-\beta \cdot ||\log(\Mat{X})-\log(\Mat{Y})||^2_F)
\end{equation}
and
\begin{equation}
\label{eqn:SD_kernel}
\operatorname{K}_{SD}(\Mat{X},\Mat{Y})= \operatorname{exp}(-\beta \cdot \log \left( \det \left( \frac{\Mat{X}+\Mat{Y}}{2}\right) \right)
- \frac{1}{2} \log \left( \det \left( \Mat{X}\Mat{Y} \right) \right))\textrm{ .}
\end{equation}
Note that, in order to become a Mercer kernel, the Gaussian kernel with SD requires $\beta$ to be of the form: $\beta\in\left\lbrace \frac{1}{2},\frac{2}{2},...,\frac{d-1}{2} \right\rbrace $.
\subsection{Grassmannian Manifolds}
The Grassmannian Manifold $\Mat{\mathcal{G}_{q,d}}$, is the set of all $d$-dimensional subspaces of $\mathbb{R}^q$.
A point on the Grassmann manifold, $\Mat{X}\in \Mat{\mathcal{G}_{q,d}}$, can be denoted by an orthonormal matrix in $\mathbb{R}^{q\times d}$.
The geodesic distance between points $\Mat{X}$ and $\Mat{Y}$ on a Grassmannian manifold is defined as:
\begin{equation}
\operatorname{dist}(\Mat{X},\Mat{Y})=\sqrt{\theta_1^2+...+\theta_d^2}\textrm{ ,}
\end{equation}
\noindent
where $\theta_i$ is the principal angle between $\Mat{X}$ and $\Mat{Y}$.
The angle $\theta_i$ can be calculated by $\theta_i=cos^{-1}(\xi_i)$ wherein $\xi_i$ are singular values of $\Mat{X}^\top\Mat{Y}$.
We refer readers to~\cite{absiletal2004} for further treatment on Grassmannian manifolds.
A popular kernel used over Grassmannian manifolds is known as the Projection kernel~\cite{hammLee2008, vemulapalliPillai2013}, which can be formulated as:
\begin{equation}
\label{eqn:G_kernel}
\operatorname{K}(\Mat{X},\Mat{Y})=\beta \cdot||\Mat{X}^{\top}\Mat{Y}||^2_F\textrm{ .}
\end{equation}
\subsubsection{Kernelised Gaussian Random Projection (KGRP)}
\label{sec:GRP}
In the KGRP method, the hyperplanes are generated from the standard Gaussian distribution $\mathcal{N}(\Vec{0},\Vec{I})$.
Each hyperplane $\Vec{r}_i \in \mathcal{H}$ is assumed to be spanned by a group of data points randomly selected.
To this end, first a subset $\mathcal{S}$
containing $p$ points $\{ \phi(\Mat{X}_1), \dots, \phi(\Mat{X}_p) \}$ is randomly chosen from the entire dataset, $\phi(\Mat{X}_i)$ is the representation of manifold points $\Mat{X}_i$ in the RKHS.
Each data point $\operatorname{\phi}(\Mat{X}_i)$ from the subset is considered as a vector generated from a particular distribution $D$ with unknown mean $\Vec{\mu}$ and unknown covariance $\Mat{\Sigma}$.
Thanks to the Central Limit
Theorem (CLT)~\cite{rice2006mathematical}, one can still produce standard Gaussian distribution data points from these data.
More precisely, the CLT states that when the number of data points grows larger, the difference between the population mean and the sample mean approximates the normal
distribution $\mathcal{N}(\Vec{0}$,$\Mat{\Sigma}$).
As such, we first randomly select t, $t < p$, data points from $\mathcal{S}$ and let these points be the set $\mathcal{S}_1 \subset \mathcal{S}$.
Let $\Vec{z}_t = \frac{1}{t}\sum_{i\in \mathcal{S}_{1}} \operatorname{\phi}(\Mat{X}_i)$ be the sample mean over $\mathcal{S}_{1}$.
By applying the CLT and the Whitening transform~\cite{duda2012pattern}, the vector $\Vec{r}_i = \Mat{\Sigma} ^{-\frac{1}{2}} \sqrt t (\Vec{z}_t - \Vec{\mu})$ can be considered as the point generated from a standard Gaussian distribution; thus $\Vec{r}_i$ could be used as a random projection hyperplane.
Therefore, we denote our embedding function that projects data points in the RKHS to the random projection space by:
\begin{equation}
f({\phi}(\Vec{X}_i))=\operatorname{\phi}(\Vec{X}_i)^{T} \Mat{\Sigma} ^{-\frac{1}{2}} \sqrt t (\Vec{z}_t - \Vec{\mu})\textrm{ .}
\label{eqn:mapping_function}
\end{equation}
\noindent
The mean is implicitly estimated as $\Vec{\mu} = \frac{1}{p}\sum_{i=1}^{p} \operatorname{\phi}(\Vec{X}_i)$, and the covariance matrix $\Mat{\Sigma}$ is also formed over the $p$ data points.
In order to compute Eqn.~\ref{eqn:mapping_function}, one could use a similar approach to that of Kernel Principal Component Analysis~(KPCA)~\cite{scholkopfetal1998}.
Specifically, let the Eigen-decomposition of the covariance matrix $\Mat{\Sigma}$ and the kernel matrix over $p$ data points $\Mat{K}_\mathcal{S}$, be $\Mat{V}\Mat{\Lambda}\Mat{V}^{\top}$ and $\Mat{U} \Mat{\Theta} \Mat{U}^{\top}$ respectively.
Based on the fact that the non-zero eigenvalues of $\Mat{V}$ are equal to the non-zero eigenvalues of $\Mat{\Theta}$,
Kulis-Grauman~\cite{kulisGrauman2012} proved that Eqn.~\ref{eqn:mapping_function} is the same as:
\noindent
\begin{equation}
\sum\nolimits_{i=1}^{p} \Vec{w}(i) ( {\operatorname{\phi}(\Mat{X}_i)^{T}} {\operatorname{\phi}(\Mat{X})})
\label{eqn:KLSH_embedding_3}\textrm{ ,}
\end{equation}%
\noindent
where
\begin{equation}
\Vec{w}(i) = \frac{1}{t}\sum\limits_{j = 1}^p {\sum\limits_{l \in {\mathcal{S}_1}}^{} {{\Mat{K}_{ij}}^{ - \frac{3}{2}}} } {\Mat{K}_{jl}}\textrm{ .}
\label{eqn:KLSH_embedding_4}
\end{equation}%
\noindent
Note that $\mathcal{S}_1$ is the set of $t$ points which are randomly selected from $\mathcal{S}$.
Further, defining $\Vec{e}$ as a vector of all ones, and $\Vec{e}_{\mathcal{S}_{1}}$ as a zero vector with ones in the entries corresponding to the indices of $\mathcal{S}_1$,
the expression in Eqn.~\ref{eqn:KLSH_embedding_4} can be further simplified to:
\begin{equation}
\Vec{w} = \sqrt {\frac{{p - 1}}{t}} {\Mat{K}_{\mathcal{S}}^{ - \frac{1}{2}}}{\Vec{e}_{{\mathcal{S}_1}}}\textrm{ .}
\label{eqn:KLSH_embedding_5}
\end{equation}%
We note that the above formulation was first described for developing the kernelise locality sensitive hashing method in Euclidean scenarios~\cite{kulisGrauman2012}.
We then adapted the method in our previous work~\cite{alavietal2014} to perform random projection on SPD manifolds for classification purposes.
Here we apply the method for clustering on Riemannian manifold problems.
The pseudo code for KGRP is summarised in Algorithm~\ref{alg:pseudocode_KGRP}.
\begin{algorithm}[!tb]
\caption{ Kernelised Gaussian Random Projection (KGRP)}
\label{alg:pseudocode_KGRP}
\begin{algorithmic}[1]
\REQUIRE the entire dataset: a set of manifold-valued data points $\left\lbrace \Mat{X}_{i} \right\rbrace^{n}_{i=1}$, $\Mat{X}_i \in \Mat{\mathcal{M}}$; the size of $\mathcal{S}$~: p; the desired projected space dimensionality~: $b$
\ENSURE $\{\Vec{x}_i\}_{i=1}^{n}$, $\Vec{x}_i \in \mathbb{R}^p$ the data points in the projected space
\STATE Randomly select $p$ points $\left\lbrace \Mat{X}_{i} \right\rbrace^{p}_{i=1}$ from the entire dataset
\STATE Compute the Kernel Gram matrix $\Mat{K}_{\mathcal{S}}$ over points $\left\lbrace \Mat{X}_{i} \right\rbrace^{p}_{i=1}$,
$\Mat{K}_\mathcal{S}=\operatorname{\phi}(\Mat{X}_{{i}})^{\top}\operatorname{\phi}(\Mat{X}_{{j}})$, $\forall \Mat{X}_{{i}}, \forall \Mat{X}_{{j}}\in \left\lbrace \Mat{X}_{i} \right\rbrace^{p}_{i=1}$,
let $\mathcal{S}=\left\lbrace \phi(\Mat{X}_{i})\right\rbrace^{p}_{i=1}$ denote the representations for these $p$ points in the RKHS
\STATE Compute the projection matrix
$\Mat{W}= \left\lbrace \Vec{w}_1,...,\Vec{w}_b \right\rbrace $, $\forall \Vec{w}_i\in \mathbb{R}^p$
\FOR{$i = 1 \to b$}
\STATE $\mathcal{S}_1 \gets$ Randomly select $t$ data points from $\mathcal{S}$
\STATE $\Vec{e_\mathcal{S}}=\left[\Delta_1,...,\Delta_p \right]$
if $\phi(\Mat{X}_i)\in \mathcal{S}_1$, $\Delta_i=1$; otherwise $\Delta_i=0$
\STATE$\Vec{w}_i = \sqrt {\frac{{p - 1}}{t}} {\Mat{K}_\mathcal{S}^{ - \frac{1}{2}}}{\Vec{e}_{{\mathcal{S}}}} $
\ENDFOR
\STATE Project each point $\Mat{X}_i$ into the random projection space: $\Vec{x}_i = \tilde{\Mat{K}} \Mat{W}$, where $\tilde{\Mat{K}}$ is the Gram matrix between $\Mat{X}_i$ and the points $\left\lbrace \Mat{X}_{i} \right\rbrace^{p}_{i=1}$
\end{algorithmic}
\end{algorithm}
We note that the total computational complexity of the KGRP algorithm is $O(np+p^3+np^2+\ell nmp)$.
Specifically, there are four factors contributing to the computational complexity:
\begin{enumerate}
\item
Computing the kernel Gram matrix $\Mat{K}_{n,p}$ between $n$ points and $p$ selected points which requires $O(np)$ operations~$(p<<n)$;
\item Generating the random hyperplanes, necessitates calculation of the kernel matrix $\Mat{K}^{-1/2}_\mathcal{S}$ for the $p$ points in $\mathcal{S}$ which requires $O(p^3)$ operations;
\item Projecting all of the data points into the random projection space which requires $O(np^2)$ operations;
\item Applying K-means to get the clustering results which requires $O(\ell nmp)$ operations~($\ell$ is the number of iterations of K-means, $m$ is the number of clusters and $b$ is the dimension of the projected space).
\end{enumerate}
\subsection{KPCA-based Random Projection (KPCA-RP)}
\label{SEC_KPCAA-RP}
Inspired by the previous method, one can derive orthonormal projections using the Kernel PCA (KPCA).
More precisely, after generating random projection hyperplanes by randomly selecting the subset $\mathcal{S}$, one can obtain the principal components of the data points in $\mathcal{S}$ by applying the KPCA.
The principal components of $\mathcal{S}$ are then considered as the set of orthogonal random projection hyperplanes.
Finally, following Eqn.~\ref{eq:rp_in_kernel}, the entire data points can be projected into the random projection space using the hyperplanes.
Let us suppose $\Mat{C}$ is the covariance matrix of the points in $\mathcal{S}$ which have been centred:
\begin{equation}
\Mat{C}=\frac{1}{p}\sum_{i=1}^{p}{\phi}(\Mat{X}_{{i}}){\phi}(\Mat{X}_{{i}})^{\top}.
\end{equation}
To apply KPCA, one needs to solve the generalised eigen-decomposition problem:
\begin{equation}
\label{eq:PCA_V}
\tau \Vec{V}=\Mat{C}\Vec{V}\textrm{ .}
\end{equation}
Following the same argument as KPCA~\cite{scholkopfetal1998}, the eigenvectors of the covariance matrix $C$ lie in the span of
${\phi}(\Mat{X}_{{1}}),{\phi}(\Mat{X}_{2}),..,{\phi}(\Mat{X}_{p})$:
\begin{equation}
\Vec{V}_k=\sum_{i=1}^{p}\alpha^k_{i}{\phi}(\Mat{X}_{{i}})\textrm{ ,}
\label{eq:KPCA_linearcombination}
\end{equation}
\noindent
where the set $\{ \alpha^k_i \}_{i=1}^{p}$ can be determined by solving the following equation:
\begin{equation}
p \tau \Vec{\alpha}=\Mat{K}_\mathcal{S} \Vec{\alpha}\textrm{ ,}
\label{eq:KPCA}
\end{equation}
\noindent
where $\Mat{\alpha} = [\Vec{\alpha}^1 \cdots \Vec{\alpha}^k]$ is a matrix wherein each column represents the vector $\Vec{\alpha}^k = [\alpha^k_1 \cdots \alpha^k_p]^{\top}$ whose elements are the linear combination coefficients presented in Eqn.~\ref{eq:KPCA_linearcombination} and $\Mat{K}_\mathcal{S}$ is the kernel matrix of the set $\mathcal{S}$.
Note that the above equation suggests that the vector $\Vec{\alpha}^k$ is one of the eigenvectors of $\Mat{K}_\mathcal{S}$.
Let $\{\Vec{V}_k\}_{k=1}^{p}$ be the set of principal components extracted from Eqn.~\ref{eq:PCA_V}.
To project a point into the principal component $\Vec{V}_k$, we perform:
\begin{equation}
\label{kpca_proj}
\phi(\Mat{X})^{\top} \cdot \Vec{V}_{k}=\sum_{i=1}^{p}\alpha_{i}^{k}\phi(\Mat{X})^{\top} {\phi}(\Mat{X}_{{i}})\textrm{ .}
\end{equation}
In the following, we present a theorem that guarantees that projections into the principal components of the subset $\mathcal{S}$ achieves JL-Type projection.
\begin{theorem}
If a set of points can be separated by a margin $\lambda$ in the RKHS, then with probability $\geq 1-\delta$,
if $\mathcal{S}=\left\lbrace \phi(\Mat{X}_1),...,\phi(\Mat{X}_p)\right\rbrace$, $\Mat{X}_i \in \Mat{\mathcal{M}}$, $\phi(\Mat{X}_i) \in \mathcal{H}$ are drawn from the same distribution for
$p=\frac{8}{\varepsilon}\left[\frac{1}{\lambda^2}\operatorname{ln}\frac{1}{\delta} \right] $,
the mapping $\operatorname{F_{2}}(x)=\operatorname{F_{1}}(x)[\Vec{\alpha}^1 \cdots \Vec{\alpha}^p]$,
where $\Vec{\alpha}^k$ is the $k$-th eigenvector of $\Mat{K}_\mathcal{S}$, achieves JL-Type projection with error at most $\varepsilon$.
\label{theorem:kpcarp}
\end{theorem}
\textit{Proof.}
As presented in Corollary~\ref{cor}, $\operatorname{F_{1}}(x)$ is the function that maps a point into a random projection space wherein the set of hyperplanes $\mathcal{S}$ is randomly selected from a set of given points.
It is known that principal components of $\mathcal{S}$ represent the orthonormal bases spanning the subspace spanned by $\mathcal{S}$.
Henceforth, computing the principal components of $\mathcal{S}$ can be considered as orthogonalisation of the hyperplanes.
\noindent
\textit{Remarks.}
The above theorem states that applying KPCA on $\mathcal{S}$ means orthogonalising the hyperplanes in $\mathcal{S}$.
Therefore, the difference between KPCA-RP and KORP is related to how the hyperplanes are orthogonalised.
We present the KPCA-RP pseudo code in Algorithm~\ref{kpca-rp}.
\begin{algorithm}
\caption{KPCA-based Random Projection (KPCA-RP)}
\label{kpca-rp}
\begin{algorithmic}[1]
\REQUIRE the entire dataset: a set of manifold-valued data points $\left\lbrace \Mat{X}_{i} \right\rbrace^{n}_{i=1}$, $\Mat{X}_i \in \Mat{\mathcal{M}}$; the desired projected space dimensionality~: $p$
\ENSURE $\{\Vec{x}_i\}_{i=1}^{n}$ the data points in the projected space
\STATE Randomly select $p$ points $\left\lbrace \Mat{X}_{i} \right\rbrace^{p}_{i=1}$ from the entire dataset
\STATE Compute the kernel Gram matrix $\Mat{K}_{\mathcal{S}}$ over points $\left\lbrace \Mat{X}_{i} \right\rbrace^{p}_{i=1}$
$\Mat{K}_\mathcal{S}=\operatorname{\phi}(\Mat{X}_{{i}})^{\top}\operatorname{\phi}(\Mat{X}_{{j}})$, $\forall \Mat{X}_{{i}}, \forall \Mat{X}_{{j}}\in \left\lbrace \Mat{X}_{i} \right\rbrace^{p}_{i=1}$
\STATE Apply KPCA to kernel matrix $\Mat{K}_\mathcal{S}$ to obtain the eigenvectors $\alpha$.
\STATE Project each point $\Mat{X}_i$ into the random projection space: $\Vec{x}_i = \tilde{\Mat{K}} \alpha$, where $\tilde{\Mat{K}}$ is the Gram matrix between $\Mat{X}_i$ and the $\left\lbrace \Mat{X}_{i} \right\rbrace^{p}_{i=1}$
\end{algorithmic}
\end{algorithm}
In terms of calculating the computational complexity of the KPCA-RP algorithm, one needs to consider four factors:
\begin{enumerate}
\item Computing the kernel Gram matrix between the entire dataset and the subset $\mathcal{S}$, which requires $O(np)$ operations;
\item Applying KPCA on the kernel Gram matrix of subset $\mathcal{S}$, which requires $O(p^3)$ operations;
\item Projecting all of the data points into the orthonormal space, which requires $O(np^2)$ operations;
\item Applying K-means to get the clustering results, which requires $O(\ell nmp)$ operations~($\ell$ is the number of iterations of K-means, $m$ is the number of clusters).
\end{enumerate}
Hence, the total computational complexity is $O(np+p^3+np^2+\ell nmp)$.
\subsubsection{Kernelised Orthonormal Random Projection (KORP)}
In the second method, we generate orthonormal random hyperplanes (i.e.~ the first property).
We first present the following Lemma that relates the JL-Lemma to the margin of the linear hyperplane in supervised learning settings~\cite{blum2006}.
\begin{lemma} \label{blum}
Consider any distribution over labelled examples in Euclidean space such that there exists a linear separator $\Vec{w}^{\top}\cdot \Vec{x}=0$ with margin $\lambda$. If we draw $d \geq \frac{8}{\varepsilon}\left[\frac{1}{\lambda^2}\operatorname{ln}\frac{1}{\delta} \right]$ examples $\Vec{z}_1,\cdots,\Vec{z}_d$ iid from this distribution, with probability $\geq 1-\delta$, there exists a vector $\Vec{w}'$ in span $(\Vec{z}_1,\cdots,\Vec{z}_d)$ that has error at most $\varepsilon$ at margin $\frac{\lambda}{2}$~\cite{blum2006}.
\end{lemma}
\noindent
\textit{Proof.}
We refer the readers to~\cite{blum2006} for the proof of this Lemma.
\noindent
\textit{Remarks.}
Lemma~\ref{blum} essentially states that, with a high probability, the margin is still well preserved (with error at most $\varepsilon$) when the hyperplane $\Vec{w}'$ is selected from the space spanned by a subset of the data points.
Note that, as suggested in~\cite{ShiICML2012}, when the margin is well preserved, then the angle and distance between points are also well preserved.
This Lemma can also be applied for cases where the data points are in the RKHS.
This is because the RKHS is essentially an infinite-dimensional Euclidean space~\cite{blum2006}.
Given a set of points which are linearly separable with margin $\lambda$ under a particular kernel function, we draw $d$ random examples $\Vec{x}_1,\cdots,\Vec{x}_d$ from the same distribution.
Then, according to Lemma~\ref{blum}, with probability $\geq 1-\delta$, there exists a separator in RKHS $\Vec{w}' \in \mathcal{H}$ and $\Vec{w}'=\alpha_1 \phi(\Vec{x}_1)+ \cdots +\alpha_d \phi(\Vec{x}_d)$ with error rate at most $\varepsilon$.
Note that as \mbox{$\Vec{w}'^{\top} \cdot \phi(\Vec{x})$} $=\alpha_1 \operatorname{K}(\Vec{x},\Vec{x}_1)+...+\alpha_d\operatorname{K}(\Vec{x},\Vec{x}_d)$, we then can simply consider the vector of $[\operatorname{K}(\Vec{x},\Vec{x}_1) \cdots \operatorname{K}(\Vec{x},\Vec{x}_d)]$ as the feature representation of $\Vec{x}$ in the space spanned by $\{ \phi(\Vec{x}_i) \}_{i=1}^{d}$.
In other words, the $\operatorname{K}(\Vec{x},\Vec{x}_i)$ is considered as the i-th feature of $\Vec{x}$.
We can further formalise this observation with the following Corollary~\cite{blum2006}.
\begin{corollary}\label{cor}
If distribution $P$ has margin $\lambda$ in the RKHS, then with probability $\geq 1-\delta$, if $\Vec{x}_1,\cdots,\Vec{x}_d$ are drawn from the same distribution, for $d=\frac{8}{\varepsilon}\left[\frac{1}{\lambda^2}\operatorname{ln}\frac{1}{\delta} \right] $, the mapping $\operatorname{F_{1}}(\Vec{x})=[\operatorname{K}(\Vec{x},\Vec{x}_1) \cdots \operatorname{K}(\Vec{x},\Vec{x}_d)]$ produces a distribution $\operatorname{F_{1}}(P)$ on labelled examples in $\mathbb{R}^{d}$ that is linearly separable with error at most $\varepsilon$~\cite{blum2006}.
\end{corollary}
\noindent
\noindent
\textit{Remarks.} The above Corollary suggests the following points: (1)~one could generate random projection hyperplanes by randomly selecting a subset of data points in RKHS and then projecting a point into this space by using $\operatorname{F_{1}}(\Vec{x})$; (2)~this projection is a JL-Type projection.
In light of these facts, for our case, we randomly select $p$ points, here denote $\mathcal{S}=\{\phi(\Mat{X}_{1}),\cdots,\phi(\Mat{X}_{p})\}$ as the implicit representations of the $p$ points in RKHS.
However, as it is possible that some hyperplanes are not linearly independent, then the hyperplanes could be highly correlated.
To that end, one needs to orthogonalise the hyperplane set $\mathcal{S}$~\cite{blum2006}.
In this work, we apply QR decomposition~\cite{watkins2004} to construct a set of orthonormal basis from the original basis spanning the same subspace.
Let us arrange the original basis $\{ \operatorname{\phi}(\Mat{X}_i) \}_{i=1}^{p}$ into a matrix $\Mat{A}$.
Then the matrix $\Mat{A}$ can be decomposed into $\Mat{Q}$ and $\tilde{\Mat{R}}$ as follows:
\begin{equation}
\Mat{A} = [\operatorname{\phi}(\Mat{X}_{1}),\cdots,\operatorname{\phi}(\Mat{X}_{p})]=\Mat{Q}\tilde{\Mat{R}}\textrm{ ,}
\end{equation}
\noindent
where $\Mat{Q}$ is the orthonormal basis and $\tilde{\Mat{R}}$ is the upper triangular matrix.
Assuming that we have the orthonormal basis $\Mat{Q}$, then we can observe the following when a data point $\phi(\Mat{X})$ is projected into the orthonormal basis $\Mat{Q}$:
\begin{equation}
\begin{split}
\operatorname{\phi}(\Mat{X})^{\top}\Mat{Q}
&=\operatorname{\phi}(\Mat{X)}^{\top}\Mat{Q}\tilde{\Mat{R}}\tilde{\Mat{R}}^{-1} \\
&=\operatorname{\phi}(\Mat{X})^{\top} [\operatorname{\phi}(\Mat{X}_{{1}}),...,\operatorname{\phi}(\Mat{X}_{{p}})]\tilde{\Mat{R}}^{-1}\\
&=[\operatorname{\phi}(\Mat{X})^{\top}\operatorname{\phi}(\Mat{X}_{{1}}),...,\operatorname{\phi}(\Mat{X})^{\top}\operatorname{\phi}(\Mat{X}_{{p}})]\tilde{\Mat{R}}^{-1}\\
&=[\operatorname{K}(\Mat{X},\Mat{X}_{{1}}),...,\operatorname{K}(\Mat{X},\Mat{X}_{{p}})]\tilde{\Mat{R}}^{-1}\textrm{ .}
\end{split}
\label{eq:QR_proj}
\end{equation}
\noindent
In other words, one only needs to determine the upper triangular $\tilde{\Mat{R}}$ in order to do the projection.
We note that as the original basis $\{ \operatorname{\phi}(\Mat{X}_i) \}_{i=1}^{p}$ are in the RKHS then it is not trivial to apply the QR decomposition to matrix $\Mat{A}$.
To that end, we first multiply the matrix $\Mat{A}$ by its transpose.
By doing this, we will get the kernel matrix $\Mat{K}_\mathcal{S}$, where $\Mat{K}_\mathcal{S}(i,j) = \operatorname{\phi}(\Mat{X}_i)^{\top}\operatorname{\phi}(\Mat{X}_j)$, $\forall \operatorname{\phi}(\Mat{X}_i)$ and $\forall \operatorname{\phi}(\Mat{X}_j) \in \mathcal{S}$.
Thus:
\begin{equation}
\begin{split}
\Mat{K}_\mathcal{S} &= \Mat{A}^{\top}\Mat{A} \\
&= (\Mat{Q}\tilde{\Mat{R}})^{\top}\Mat{Q}\tilde{\Mat{R}} \\
&= \tilde{\Mat{R}}^{\top}\Mat{Q}^{\top}\Mat{Q}\tilde{\Mat{R}} \\
&= \tilde{\Mat{R}}^{\top}\tilde{\Mat{R}}\textrm{ .}
\end{split}
\end{equation}
\noindent
We can employ the Cholesky Factorisation~\cite{watkins2004} on the kernel matrix $\Mat{K}_\mathcal{S}$, in order to compute the upper triangular $\tilde{\Mat{R}}$.
Algorithm~\ref{kop} outlines the algorithm for the proposed Kernelised Orthonormal Random Projection (KORP).
\begin{algorithm}
\caption{Kernelised Orthonormal Random Projection~(KORP)}
\label{kop}
\begin{algorithmic}[1]
\REQUIRE the entire dataset: a set of manifold-valued data points $\left\lbrace \Mat{X}_{i} \right\rbrace^{n}_{i=1}$, $\Mat{X}_i \in \Mat{\mathcal{M}}$; the desired projected space dimensionality~: $p$
\ENSURE $\{\Vec{x}_i\}_{i=1}^{n}$, $\Vec{x}_i \in \mathbb{R}^p$ the data points in the projected space
\STATE Randomly select $p$ points $\left\lbrace \Mat{X}_{i} \right\rbrace^{p}_{i=1}$ from the entire dataset
\STATE Compute the kernel Gram matrix $\Mat{K}_{\mathcal{S}}$ over points $\left\lbrace \Mat{X}_{i} \right\rbrace^{p}_{i=1}$
$\Mat{K}_\mathcal{S}=\operatorname{\phi}(\Mat{X}_{{i}})^{\top}\operatorname{\phi}(\Mat{X}_{{j}})$, $\forall \Mat{X}_{{i}}, \forall \Mat{X}_{{j}}\in \left\lbrace \Mat{X}_{i} \right\rbrace^{p}_{i=1}$
\STATE Apply Cholesky Factorisation to the kernel matrix $\Mat{K}_{\mathcal{S}} = \tilde{\Mat{R}}\tilde{\Mat{R}}^{\top}$
\STATE Project each point $\Mat{X}_i$ into the random projection space: $\Vec{x}_i = \tilde{\Mat{K}} \tilde{\Mat{R}}^{-1}$, where $\tilde{\Mat{K}}$ is the Gram matrix between $\Mat{X}_i$ and the points $\left\lbrace \Mat{X}_{i} \right\rbrace^{p}_{i=1}$,
\end{algorithmic}
\end{algorithm}
The computational complexity of KORP depends on the following steps:
\begin{enumerate}
\item Computing the kernel Gram matrix between the entire dataset and the subset $\mathcal{S}$ which requires $O(np)$ operations;
\item Applying Cholesky Factorisation on the kernel Gram matrix of the $p$ points in $\mathcal{S}$ which requires $O(p^3)$ operations;
\item Applying the matrix inverse of the right triangular matrix $\tilde{\Mat{R}}$ which demands $O(p^3)$ operations;
\item Projecting all of the data points into the orthonormal space with $O(np^2)$ operations;
\item Applying K-means to get the clustering results which demands $O(\ell nmp)$ operations~($\ell$ is the number of iterations of K-means, $m$ is the number of clusters).
\end{enumerate}
Hence, the total computational complexity is $O(np+p^3+np^2+\ell nmp)$. |
2,877,628,090,073 | arxiv | \section{Introduction}
Weak gravitational lensing (WL) is a powerful tool in constraining cosmology \citep{Refregier:2003ct,Mandelbaum:2017jpr,Blake:2020mzy,Jullo:2019lgq,Zhang:2007nk}, however, the so-called ``$S_8$ tension'' between WL and cosmic microwave background (CMB) anisotropy observations prevent us from using their synergy \citep{Planck:2018lbu,Planck:2018vyg,DES:2021wwk,Heymans:2020gsg,KiDS:2020suj,HSC:2018mrq,Hamana:2019etx}. Whether this tension is due to new physics beyond the standard $\Lambda$CDM model will require us to carefully address all kinds of possible systematics \citep{Wright:2020ppw,Yao:2020jpj,Yao:2017dnt,Kannawadi:2018moi,mead2021hmcode,DES:2021vln}. The cross-correlations between different tracers are important tools in investigating such problems, as they are immune to many systematics and can bring extra cosmological information.
Cross-correlations of galaxy surveys with CMB lensing provide a powerful way to probe the large-scale structure (LSS) of the Universe (\citealt{robertson2021strong}; \citealt{zhang2021transitioning}; \citealt{omori2022joint}). CMB lensing can be reconstructed by observing temperature and polarization anisotropies, it allows us to reconstruct a map of the integrated over the line of sight overdensity of intervening matter \citep{Hu:2001kj,Okamoto:2003zw}. Galaxy image surveys provide a trace of LSS, WL can be detected by capturing the images of a large sample of galaxies, usually called source galaxies, and performing shape measurements that can then be analyzed statistically \citep{camacho2021cosmic,Mandelbaum:2017jpr,Bartelmann:1999yn}.
The correlations between the CMB lensing and cosmic shear\footnote{WL generated by LSS.} have been extensively measured, however with low significance, due to the limit of the current observations \citep{Robertson:2020xom,DES:2015wix,Harnois-Deraps:2016huu,DES:2018nnc,Hand:2013xua,Marques:2020dsb,Liu:2015xfa,Harnois-Deraps:2017kfd,POLARBEAR:2019phb,Singh:2016xey}.
Several CMB experiments have reconstructed the CMB lensing signals, such as Planck \citep{Planck:2018lbu}, ACT \citep{Mallaby-Kay:2021tuk,Darwish:2020fwf} and SPTpol \citep{DES:2017fyz,Wu:2019hek}. The next stage CMB observations like CMB-S4 \citep{abazajian2016cmb} and Simons Observertory \citep{2019JCAP...02..056A} will also do this observations with high significance.
AliCPT-1 \citep{li2018tibet} is the first Chinese CMB experiment aiming for high precision measurement of Cosmic Microwave Background B-mode polarization. The telescope, currently under deployment in Tibet, will observe in two frequency bands centered at 90 and 150 GHz. As predicted in Refs. \citep{liu2022forecasts,disandai}, AliCPT-1 has the potential to reconstruct the CMB lensing with a relatively high signal-to-noise ratio.
Considering two different stages of AliCPT-1, namely ``4 modules*yr'' and ``48 modules*yr'', we are able to measure the lensing signal at $15\sigma$ and $31\sigma$ significance, respectively.
The high resolution imaging in optical and infrared bands are one of the major scientific tasks of many galaxy surveys, such as Rubin Observatory \citep{abell2009lsst}, Euclid \citep{laureijs2011euclid} and Roman Space Telescope \citep{spergel2015wide}.
The Chinese Survey Space Telescope (CSST) \citep{zhanhu,Gong:2019yxt,2018MNRAS.480.2178C} project also belongs to the stage-IV galaxy surveys. It is a 2 meter space telescope in the same orbit as the China Manned Space Station. It will cover $17,500$ deg$^2$ sky area in about 10 years with field of view (FOV) 1.1 deg$^2$. It will
simultaneously perform both photometric imaging and slitless grating spectroscopic surveys with high spatial resolution $0.15^{''}$ ($80\%$ energy concentration region) and wide wavelength coverage. There are seven photometric imaging bands and three spectroscopic bands covering $255\sim1000$ nm. We can use this type of data to carry out WL studies.
In this paper, we will present the cross-correlation studies of the cosmic shear from CSST and CMB lensing from AliCPT-1. For this purpose, we build
a simulation pipeline based on the pure Gaussian signal and noise ingredients. For cosmic shear, we consider three types of noises or biases, namely photo-$z$ error, galaxy shape noise as well as intrinsic alignment. For CMB lensing, we consider the noise from the disconnected primary CMB.
We explore the impact of systematics, mainly focusing on the photo-$z$ and intrinsic alignment, as well as the impact from different observational noise.
Then, we perform the standard Pseudo-$C_\ell$ spectrum estimation and compute the corresponding signal-to-noise ratio (SNR). Finally, we investigate the cosmological implication of this cross-correlated signals.
This paper is organized as follows. In Section \ref{sec2} we will introduce the basics of CMB lensing and WL. In Section \ref{sec3}, we will present the method of simulating the CMB lensing $\kappa$ maps and the cosmic shear $\gamma$ maps, including both the cosmological signal and the systematic effects. In Section \ref{sec4} we will present the cross-correlation pseudo-$C_\ell$ measurements method and describe the covariance matrix, computing SNR. In Section \ref{sec5} we will present the cosmological parameter constraint results based on the simulated maps.
\section{Theoretical Modelling}\label{sec2}
\subsection{CMB lensing}
CMB lensing signal reconstructed from CMB temperature and polarization maps traces the integrated matter field along the line-of-sight direction from the current observer to the last scatter surface with redshift $z_*\simeq1100$
\eqsali{
\kappa_{\mathrm{CMB}}(\bvec{\theta})=\displaystyle\int^{z_{*}}_{0}\mathrm{d} z q_\mathrm{CMB}(z)\delta(\chi(z)\bvec{\theta},z)\;,
}
where lensing kernel $q_\mathrm{CMB}(z)$ reads
\eqsali{
q_\mathrm{CMB}(z)=\dfrac{3\Omega_{m0}}{2c}\dfrac{H^{2}_{0}}{H(z)}(1+z)\chi(z)\;,\label{qcmb}
}
$\chi$ is the comoving distance, $H(z)$ is the Hubble parameter.
As we know, the lensing efficiency reaches the maximum when the lenses are placed halfway between the source and the observer. For the CMB lensing, the background light is emitted from the last scattering surface. Hence, the lensing efficiency gradually arises from the low redshift and arrives at a plateau at $z\approx1$. This plateau keeps constant until the very high redshift. Unlike
cosmic shear, CMB lensing signals compress almost all the cosmological matter distribution information into a single sphere. Whereas, for the former, we can do tomography analysis according to the source galaxy distribution.
For CMB lensing reconstruction, we follow the Planck 2018 lensing paper \citep{Planck:2018lbu,Carron:2017mqf} formalism, which is close to the original Hu-Okamoto formalism \citep{Hu:2001kj,Okamoto:2003zw}.
Because lensing induces correlations between different multipoles, the basic idea of the quadratic estimator is to utilize these induced correlations in an (almost) optimal and unbiased way. The quadratic estimator spectrum contains the sought-after signal, but also unavoidably Gaussian reconstruction noise sourced by the CMB and instrumental noise (N0 bias), as well as the non-primary couplings of the connected 4-point function \citep{Kesden:2003cc} (N1 bias). Compared to these two biases, the former is completely dominant in the AliCPT-1 experimental setup \citep{liu2022forecasts}. Hence, in this work, we only consider the N0 term in the noise budget of CMB lensing.
\begin{figure}
\centering
\includegraphics[width=0.6\columnwidth]{noz.pdf}
\caption{The mock galaxy redshift distribution in the photometric imaging survey of the CSST. The black line denotes the total redshift distribution $n(z)$, which is obtained from the COSMOS 2015 catalog \citep{2018MNRAS.480.2178C}. The blue, orange, green, and red curves are the true redshift distribution $n_i(z)$ in each of the four photo-$z$ bins. The four photo-$z$ bins are divided by the gray vertical dashed curves.}
\label{noz}
\end{figure}
\subsection{Cosmic shear}
The weak lensing effect of source galaxies dues to the intervening large-scale structure is called cosmic shear. The corresponding lensing potential $\phi(\boldsymbol{\theta})$ is defined as
\eqsali{
\phi(\boldsymbol{\theta})=-\dfrac{2}{c^2}\displaystyle\int^{\chi_{*}}_{0}\mathrm{d} \chi\dfrac{\chi_{*}-\chi}{\chi_{*}\chi}\Psi(\boldsymbol{\theta},\chi)\label{lensing_potential}
}
where $\Psi(\boldsymbol{\theta},\chi)$ is the 3D Weyl gravitational potential. It induces a distortion of the shape and provides a mapping from lens plane position $\bvec\theta$ to the source plane position $\bvec\beta$. The local properties of a gravitational lens are characterized by the Jacobian matrix $\mathcal{A}$ of the mapping given by \citep{BARTELMANN2001291}
\eqsali{ \mathcal{A}=\derivp{\bvec\beta}{\bvec\theta}
=\delta_{ij}+\dfrac{\partial^2\phi}{\partial\theta_i\partial\theta_j}
=
\left[
\begin{array}{cc}
1-\kappa-\gamma_1 & -\gamma_2 \\
-\gamma_2 & 1-\kappa+\gamma_1
\end{array}
\right]\label{Jacobian}
}
with
\eqsali{
\kappa=-\dfrac{1}{2}\nabla^2\phi ,\quad \gamma_1=\dfrac{1}{2}\left(\derivpp{\phi}{\theta_1}-\derivpp{\phi}{\theta_2}\right) ,\quad \gamma_2=\dfrac{\partial^2\phi}{\partial\theta_1\partial\theta_2}\;,\label{kg1g2}
}
where $\kappa$ is the lensing convergence and $\gamma=\gamma_1+i\gamma_2$ is the complex lensing shear field.
Due to the intrinsic ellipticity of the source galaxies, the observed galaxy shape is actually the mixture of the cosmic shear and the intrinsic ellipticity \citep{bonnet1995statistical}.
Assuming
$\kappa \ll 1$ and $\gamma_1,\gamma_2 \ll 1$, up to the first order, the lensed ellipticity reads
\eqsali{\varepsilon = \varepsilon_s + \gamma\;,
}
where $\varepsilon_s$ is intrinsic ellipticity of the source galaxy.
Using the Poisson equation one can write the convergence in terms of the density perturbation $\delta(\bvec\theta,z)$
\eqsali{
\kappa_i(\bvec\theta)=\displaystyle\int^{z_*}_0\dfrac{c\mathrm{d} z}{H(z)}q_i(z)\delta(\bvec\theta,z)\;,
}
where lensing efficiency $q_i(z)$ in cosmic shear measurement is defined
\eqsali{
q_i(z)=\dfrac{3\Omega_{m0}}{2}\dfrac{H^2_0}{c^2}(1+z)\chi(z)\displaystyle\int^{z_*}_{z}n_i(z')\dfrac{\chi(z')-\chi(z)}{\chi(z')}\mathrm{d} z'\;.\label{qwl}
}
The source galaxies distribution in the $i$th redshift bin is described as a normalized distribution $n_i(z)$.
In harmonic space, we have the following relation between the convergence field, shear field and the lensing potential field
\eqsali{
\kappa(\bvec\ell) &=-\dfrac{|\bvec\ell|^2}{2}\phi(\bvec\ell)\;,\\
\gamma(\bvec\ell)&=\left(\dfrac{\ell^{2}_{1}-\ell^{2}_{2}+2i\ell_{1}\ell_{2}}{|\boldsymbol{\ell}|^{2}}\right)\kappa(\bvec\ell)=\kappa(\bvec\ell)e^{2i\beta}\;,
}
where $\beta$ is the polar angle of $\boldsymbol{\ell}$ \citep{schneider2002b}.
For galaxy surveys, we can not directly measure the convergence field, because we can not determine the intrinsic luminosity of the source galaxies.
However, we measure the galaxy ellipticity, which carries the information of the cosmic shear field $\gamma^G$.
Unfortunately, the observed ellipticity is contaminated by the galaxy's intrinsic shape as well as the intrinsic alignment induced by the local environment.
As a result, the measured shear can be written as $\varepsilon=\varepsilon_s+\gamma^G+\gamma^I$, where $\gamma^I$ is the intrinsic alignment.
Furthermore, by assuming a white noise statistical property, the intrinsic shape noise power spectrum can be expressed as
\eqsali{
N^{\varepsilon\varepsilon}_i(\ell)=\dfrac{4\pi f_{\rm sky}}{N_i}\sigma^2_\varepsilon\;,
\label{nggl}
}
where the ellipticity dispersion $\sigma_\varepsilon\approx0.3$ \citep{miao2022cosmological}, $f_{\rm sky}$ is the fraction of overlapped sky coverage between two experiments,
and $N_i$ is total number of galaxies in the $i$th redshift bin.
Besides the dominant intrinsic galaxy ellipticity, there exists second contamination to the cosmic shear, namely intrinsic alignment.
This signal can correlate with the gravitational tidal field of large-scale structure (GI term) or in local environments with the intrinsic alignment of other galaxies (II term) \citep{Troxel:2014dba}.
Hence, the observed angular power spectrum of shear measurement is composed of four components \citep{Bridle_2007}
\eqsali{
\hat C^{\kappa\kappa}_{ij}(\ell)=C^{\kappa\kappa}_{ij}(\ell)+C^{II}_{ij}(\ell)+C^{GI}_{ij}(\ell)+N^{\varepsilon\varepsilon}_i(\ell)\delta_{ij}\;,\label{eq:Cii}
}
where $C^{\kappa\kappa}_{ij}(\ell)$ is the convergence power spectrum, $C^{II}_{ij}(\ell)$ and $C^{GI}_{ij}(\ell)$ are the Intrinsic-Intrinsic (II) and Gravitational-Intrinsic (GI) power and cross spectra, respectively.
Due to the relevant angular scales are much smaller than 1 radian (multipoles $\ell > 100$), the theoretical angular spectra can be computed using the Limber approximation \citep{limber1953analysis}:
\eqsali{
C^{\kappa\kappa}_{ij}(\ell)=\displaystyle\int_0^{z_*}\dfrac{c\mathrm{d} z}{H(z)}\dfrac{q_i(z)q_j(z)}{\chi^2}P_{\delta}(k=\dfrac{\ell+1/2}{\chi},z)\;,
\label{eq:conv1}
}
The Intrinsic-Intrinsic and Gravitational-Intrinsic power and cross spectra are given
\eqsali{
C^{II}_{ij}(\ell)=\displaystyle\int_0^{z_*}\dfrac{c\mathrm{d} z}{H(z)}\dfrac{n_i(z)n_j(z)f_i(z)f_j(z)}{\chi^2}P_{\delta}(k=\dfrac{\ell+1/2}{\chi},z)\;,
}
and
\eqsali{
C^{GI}_{ij}(\ell)=\displaystyle\int_0^{z_*}\dfrac{c\mathrm{d} z}{H(z)}\dfrac{n_i(z)f_i(z)q_j(z)}{\chi^2}P_{\delta}(k=\dfrac{\ell+1/2}{\chi},z)+
\displaystyle\int_0^{z_*}\dfrac{c\mathrm{d} z}{H(z)}\dfrac{n_j(z)f_j(z)q_i(z)}{\chi^2}P_{\delta}(k=\dfrac{\ell+1/2}{\chi},z)\;,
\label{eq:conv2}
}
where $f_i(z)$ is written as
\eqsali{
f_i(z)=-A_{\rm IA}C_1 \rho_{\rm crit} \dfrac{\Omega_m}{D(z)}\left(\dfrac{1+z}{1+z_0}\right)^{\eta_{\rm IA}}\left(\dfrac{L_i}{L_0}\right)^{\beta_{\rm IA}}\;, \label{eq NLA IA}
}
and $C_1=5\times 10^{-14} h^{-2}M_{\odot}^{-1}{\rm Mpc}^{3}$~\citep{Bridle_2007}, $\rho_{\rm crit}$ is the present critical density, $D(z)$ is the linear growth factor normalized to unity at $z=0$, and $z_0=0.6$ and $L_0$ are pivot redshift and luminosity, respectively. Since the change of average luminosity can be ignored, we don't consider luminosity dependence and the fiducial values of $A_{\rm IA}$, $\eta_{\rm IA}$ and $\beta_{\rm IA}$ are set to be 1, 0, 0, respectively \citep{joudaki2016cfhtlens}.
In Fig. \ref{NeGI}, we show the auto- and cross-spectra of the cosmic shear and intrinsic alignment in 4 photo-z bins, which are defined in the following sections. First of all, the intrinsic alignment model we adopted is anti-correlated with cosmic shear. Second, the amplitude of II correlation is much smaller than GG. Taking these two effects together into account, we can conclude that the $\hat C^{\kappa\kappa}_{\ell}$ in the Eq. (\ref{eq:Cii}) gets smaller once we consider the intrinsic alignment.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{NeGI1x4.pdf}
\caption{Auto- and cross- spectra of cosmic shear (G), intrinsic alignment (I), and shape noise in 4 photo-z bins.
The definition of the photo-z bins is described in Fig. \ref{noz}.
Red, blue, and yellow curves denote the GG, II, and GI spectra. Black dashed curves denote shape noise. Since the cosmic shear and intrinsic alignment is anti-correlated,
to avoid negative values in the log-log plot, we show the absolute value of the GI cross-spectrum.}
\label{NeGI}
\end{figure}
\subsection{Weak lensing-CMB lensing cross-correlation}
Since convergence field and shear field are both determined by the gravitational potential $\phi$, they are not independent. Furthermore, we can find a linear combinations of $\gamma_1$ and $\gamma_2$ to convert shear field into E and B mode ($\gamma_E$, $\gamma_\mathrm{B}$)
\eqsali{
\gamma_E = \gamma_1(\bvec\ell)\dfrac{\ell^2_1-\ell^2_2}{\ell^2_1+\ell^2_2}+\gamma_2(\bvec\ell)\dfrac{2\ell_1\ell_2}{\ell^2_1+\ell^2_2}=\kappa(\bvec\ell);\quad \gamma_\mathrm{B}=0\;.\label{spin-2}
}
We are interest in the cross spectrum of weak lensing and CMB lensing field $C^{\kappa_{\rm CMB}\gamma_E}_i(\ell)$
\eqsali{
\langle\kappa_{\rm CMB}(\bvec\ell)\gamma^*_\mathrm{E,i}(\bvec\ell')\rangle=(2\pi)^2\delta_D(\bvec\ell-\bvec\ell')C^{\kappa_{\rm CMB}\gamma_E}_i(\ell)\;,\label{comcl}
}
where $\delta_D$ is the 2-dimensional Dirac function and the sub-index $i$ denotes for the $i$th redshift bin. Under the Limber approximation, the theoretical cross spectrum reads
\eqsali{
C^{\kappa_{\rm CMB}\gamma_E}_i(\ell)=\displaystyle\int_0^{z_*}\dfrac{c\mathrm{d} z}{H(z)}\dfrac{q_\mathrm{CMB}(z)q_i(z)}{\chi^2}P_{\delta}(k=\dfrac{\ell+1/2}{\chi},z)\;,
}
where $P_\delta(k,z)$ is the nonlinear matter power spectrum, which is calculated with the Boltzmann code \verb'pyccl' \citep{Chisari_2019} and with the \verb'HALOFIT' method for describing the nonlinear part \citep{smith2003stable,takahashi2012revising}.
In Fig. \ref{kernels}, we show the lensing efficiency of CMB lensing
and cosmic shear\footnote{calculated using the CSST source mock
redshift distribution as described in Fig.\ref{noz}} as a function of redshift, as well as the combined weight given by
\eqsali{
q^{\rm combined}(z)=\dfrac{H(z)}{c}D^2(z)q_{\rm CMB}(z)q_{\gamma}(z)\;,
\label{eq:combine}
}
where $D(z)$ is the growth function normalized to 1 at $z=0$, accounts for the growth of matter perturbations with redshifts, and $q_{\gamma}(z)$ is lensing efficiency of cosmic shear with source distribution in entire range of redshift $n(z)$.
\begin{figure}
\centering
\includegraphics[width=0.6\columnwidth]{kernels.png}
\caption{Lensing efficiencies that enter the cosmic shear-CMB lensing correlations as defined in Eq. (\ref{qcmb}) and Eq. (\ref{qwl}). The vertical gray line marks the effective redshift $z\approx0.73$ for the cosmic shear-CMB lensing cross-correlation signal, measured from the nominal CSST number density. For cosmic shear we used the CSST source sample redshift distribution shown in Fig. \ref{noz}. The combined lensing efficiency is defined in Eq. (\ref{eq:combine}).
}
\label{kernels}
\end{figure}
\section{Simulation}\label{sec3}
In this section, we describe our simulations. We select a 30$\times$30 deg$^2$ sky area for the cross-correlation computation.
Due to the unmatched lensing kernel between WL and CMB lensing, it is unrealistic to do a large sky coverage ray tracing through N-body simulation until the redshift range where CMB lensing efficiency is enough high. To evade this obstacle, we generate the correlated cosmic shear and CMB lensing signals from the Gaussian realizations based on the inputted auto- and cross-spectra.
We generate both CMB lensing $\kappa$ map and cosmic shear with \verb'HEALPix'\citep{2005ApJ...622..759G,Zonca2019}. The resolution is chosen as $N_{\rm side}=512$, corresponding to a pixel size about $7'$.
The beam size of both the CMB lensing convergence and WL shear maps are the same, which equals $\ell_*=800$, $B_\ell=\exp[-\ell^2/\ell_*^2]$.
To avoid the leakage from the sharp edges due to the Fourier transformation, we apodize the maps with a cosine function and apodization scale $2^\circ$.
\subsection{CMB lensing map}
For CMB lensing, the convergence signal spectrum is generated via the linear Boltzmann code \verb'pyccl' and the noise spectrum is obtained from AliCPT mocks used both temperature and polarization \citep{liu2022forecasts,disandai}.
For AliCPT, the observed patch covers about 12\% of the sky \citep{li2018tibet}. In this work, we consider two different noise levels, as shown in Fig.\ref{Nphi}. The orange dashed and green dotted curves represent the ``4 module*yr'' and ``48 module*yr'', repsectively. For simplicity, we only consider the leading N0 noise and ignore the sub-leading N1 noise, etc.
We generate the harmonic coefficients of convergence map as the followings
\eqsali{\kappa_{\rm CMB}(\boldsymbol{\ell})=\zeta_{1}(\boldsymbol{\ell})(C^{\kappa\kappa}_{\rm CMB}(\ell))^{1/2}\;,}
and the noise map
\eqsali{n_\kappa(\boldsymbol{\ell})=\zeta'_{1}(\boldsymbol{\ell})(N^{\kappa\kappa}_{\rm CMB}(\ell))^{1/2}\;,}
where $\zeta_{1}(\ell)$ and $\zeta'_{1}(\ell)$ are two independent complex numbers drawn from a Gaussian distribution with zero mean and unit variance. $C^{\kappa\kappa}_{\rm CMB}(\ell)$ is theoretical convergence power spectrum and $N^{\kappa\kappa}_{\rm CMB}(\ell)$ is noise power spectrum.
The noise spectrum is directly read from the AliCPT-1 lensing reconstruction results \citep{liu2022forecasts}.
The map of CMB lensing convergence is shown in the first row of Fig.\ref{k_g_maps}. The simulated CMB lensing convergence map in harmonic space reads $\hat\kappa_{\rm CMB}(\boldsymbol{\ell})=\kappa_{\rm CMB}(\boldsymbol{\ell})+n_\kappa(\boldsymbol{\ell})$.
\begin{figure}
\centering
\includegraphics[width=0.6\columnwidth]{Nphi.pdf}
\caption{The angular power spectrum of CMB lensing and reconstruction noise of AliCPT-1. The blue curve denotes for the theoretical angular power spectrum of the CMB lensing convergence, the dashed orange and green dotted curves are the angular power spectrum of reconstruction noise from ``4 module*yr'' and ``48 module*yr''.}
\label{Nphi}
\end{figure}
\subsection{Galaxy samples and cosmic shear}
In this section, we describe the methodology for generating our cosmic shear maps.
In this paper, we use Eq. (\ref{eq:conv1})-(\ref{eq:conv2}) to generate the simulated convergence and shear maps. And we validate these maps by comparing the reconstructed auto-spectrum with the theoretical ones.
We consider three types of errors or biases, namely photo-$z$ error, galaxy shape error as well as intrinsic alignment. Since we focus on the linear scale, the errors originating from different types of sources are independent. Unlike the photo-$z$ and shape errors, we do not model the intrinsic alignment at the map level. Instead, we treat it as a part of the shear signal and model it in the angular power spectrum in Eq. (\ref{eq:Cii}). In this work, we consider two types of shear power spectrum models, namely only cosmic shear signals without intrinsic alignment (model-I) and with intrinsic alignment (model-II). As demonstrated previously, since the anti-correlation between cosmic shear and intrinsic alignment,
the amplitude in model-II is smaller than in model-I.
For source galaxy samples, we adopt the redshift distribution shown in Fig.\ref{noz}. The total galaxy number density is assumed to be $20~{\rm arcmin}^{-2}$. We divide the source galaxy samples into 25 bins as it is shown.
We use this galaxy redshift distribution $n_i(z)$ to calculate the cosmic shear signal according to Eq. (\ref{qwl}). To satisfy both the auto- and the cross-correlation of CMB lensing convergence and cosmic shear, we generate shear maps using following method \citep{kamionkowski1997statistics}
\eqsali{\kappa^{sig}_i(\boldsymbol{\ell})=\zeta_{1}(\boldsymbol{\ell})\dfrac{C^{ \kappa_{\rm CMB}\gamma^{\rm sig}_E}_i(\ell)}{(C^{\kappa\kappa}_{\rm CMB}(\ell))^{1/2}}+\zeta_{2}(\boldsymbol{\ell})\left[C^{\gamma^{\rm sig}_E}_{ii}(\ell)-\dfrac{(C^{ \kappa_{\rm CMB}\gamma^{\rm sig}_E}_{i}(\ell))^{2}}{C^{\kappa\kappa}_{\rm CMB}(\ell)}\right]^{1/2}\;.\label{eq build map}}
In order to satisfy the model predicted structure growth along the redshift evolution, all of the tomographic cosmic shear fields shall share the same initial random seeds, which are given by $\zeta_2(\boldsymbol{\ell})$.
The shape noise is generated according to its power spectrum given by Eq. (\ref{nggl}). The noise level is shown in Fig. \ref{NeGI} and is jointly determined by the galaxy number density ($n_i$) and the ellipticity dispersion of a single galaxy ($\sigma_\varepsilon$).
Due to the errors in determinating photometric redshift, the real galaxy redshift distribution in the $i$th photo-$z$ bin are conventionally expressed as eg. \citep{ma2006effects}
\eqsali{
n_i(z)=\displaystyle\int^{z^P_{i,{\rm max}}}_{z^P_{i,{\rm min}}}\mathrm{d} z^P n(z) p(z^P|z)\;,
}
where $z^p$ is photo-$z$, $n(z)$ is the total redshift distribution, which is calculated from COSMOS-2015 catalog. $p(z^P|z)$ is the photo-$z$ distribution function given the real redshift $z$
\eqsali{
p(z^P|z)=\dfrac{1}{\sqrt{2\pi }\sigma_z(1+z)}\exp\left[-\dfrac{(z-z^P-\Delta^i_z)^2}{2(\sigma_z(1+z))^2}\right]\;,\label{photo-$z$-eq}
}
where $\Delta_z$ and $\sigma_z$ are the redshift bias and scatter, respectively. In this work, we adopt the typical values in the 4th generation surveys as $\Delta_z=0.005$ and $\sigma_z=0.05$.
The photo-$z$ errors in the maps arise via redistributing the true galaxy redshift according to the above probability distribution.
For the pixelized representation of cosmic shear catalogs, we construct re-weighted tomographic maps as
\eqsali{
\hat{\gamma}_i(p)=\dfrac{\displaystyle\sum_{j}w_{j\rightarrow i}(p)(\gamma^{\rm sig}_j(p)+\varepsilon_j(p))}{\displaystyle\sum_{j}w_{j\rightarrow i}(p)}\;,
}
where $p$ denotes pixel index, $w_j$ are the probability of a shear signal leaking from $j$th redshift bin to $i$th photo-$z$ bin. The weights are drawn by multigaussian distribution $w_{j\rightarrow i}\sim M\{N_{bins},[p_{j\rightarrow i}]\}$
\eqsali{
p_{j\rightarrow i}=\displaystyle\int^{z^P_{i,{\rm max}}}_{z^P_{i,{\rm min}}}\mathrm{d} z^P p(z^P|z_j)=\dfrac{1}{2}\left[{\rm erf}{(x_{j\rightarrow i}^{\rm max})}-{\rm erf}{(x_{j\rightarrow i}^{\rm min})}\right]\;,\label{leakex}
}
with
\eqsali{
x_{j\rightarrow i}^{\rm max/min}=\left[\dfrac{(z_j-z^P_{i,{\rm max/min}}-\Delta^j_z)^2}{2(\sigma_z(1+z_j))^2}\right]\;,
}
where ${\rm erf}(x)$ is error function.
After getting these 25 thorough redshift samples, next, we combine them into 4 redshift bins. We do this step for two reasons. First of all, it will help us enhance the SNR in each snapshot. Second, the reduction of the snapshot numbers will significantly reduce the number of simulations to get a converged covariance matrix. The resulted shear maps are shown in the second row and third row in Fig. \ref{k_g_maps}.
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{k_g_nmaps.pdf}
\caption{CMB lensing convergence map and cosmic shear maps in 4 redshift bins. The map in the first row is the CMB convergence. The second and third rows, from left to right, are the real and imaginary parts of the plural cosmic shear from the first to the fourth redshift bins, respectively.}
\label{k_g_maps}
\end{figure*}
\section{pseudo power spectra measurement}\label{sec4}
We created 300 simulated maps of the CMB convergence field and the cosmic shear field using the same method but with different random seeds. We computed the angular cross spectra between CMB and shear maps using a pseudo-$C_\ell$ estimator based on \verb'NaMaster' algorithm \citep{10.1093/mnras/stz093}. The cross spectra are computed using Eq. (\ref{comcl}). For the same reason of binning the redshifts, to accelerate the covariance computation, here we also bin the cross-spectrum in $\ell$. We use 19 linearly multipole bins of width $\Delta\ell=40$ in the range $20\le\ell\le800$. The estimated band powers are $\hat{C}^{\kappa_{\rm CMB}\gamma_{E}}_i(L)$, with $L$ representing the multipole bin and $i$ for the redshift bin.
In order to simplify the index notations, we further merge the multipole index $L$ and redshift index $i$ into a single index $\nu=[L^{(1)}, L^{(2)}, L^{(3)}, L^{(4)}]$, where we sequentially list the banded multipoles in each redshift bins. In order to highlight the higher multipoles visiually, we use the multipole weighted spectrum
$D_\nu=L C_\nu B_L^{-1}$, where $C_\nu=[\hat{C}^{\kappa_{\rm CMB}\gamma_{E}}_1(L),\hat{C}^{\kappa_{\rm CMB}\gamma_{E}}_2(L),\hat{C}^{\kappa_{\rm CMB}\gamma_{E}}_3(L),\hat{C}^{\kappa_{\rm CMB}\gamma_{E}}_4(L)]$ and $B_L$ is the beam function in the band power. In the spectrum estimation, we de-convolved the beam.
The final covariance matrix reads
\eqsali{
\mathbb{C}_{\nu\nu'}=\dfrac{1}{N_{\rm sim}-1}\displaystyle\sum^{N_{\rm sim}}_{\alpha=1}[\hat{D}^{XY,\alpha}_\nu-\bar{D}^{XY}_\nu][\hat{D}^{XY,\alpha}_{\nu'}-\bar{D}^{XY}_{\nu'}]\;,
}
with $X,Y=\{\kappa_{\rm CMB},\gamma_E\}$ and $\alpha$ denotes the number of the simulation.
$\hat{D}^{XY,\alpha}_\nu$ is the estimated cross spectrum from the $\alpha$th simulation and $\bar{D}^{XY}_\nu$ is the mean over 300 simulations
\eqsali{
\bar{D}^{XY}_\nu=\displaystyle\sum^{N_{\rm sim}}_{\alpha=1}\dfrac{\bar{D}^{XY,\alpha}_\nu}{N_{\rm sim}}\;.
}
To calculate the inverse covariance, we adopt the unbiased
estimator used in \citealt{2007A&A...464..399H}, which is given by
\eqsali{\hat{\mathbb{C}}^{-1}_{\nu\nu'}=\dfrac{N_{\rm sim}-N_{\rm bin}-2}{N_{\rm sim}-1}\mathbb{C}_{\nu\nu'}^{-1}\;,}
where $N_{\rm sim}=300$ and $N_{\rm bin}=4\times19$ is the number of data points and $\mathbb{C}_{\nu\nu'}^{-1}$ is
the normal inverse of $\mathbb{C}_{\nu\nu'}$. In order to include the error propagation from the error in the covariance matrix into the fitting parameters \citep{percival2014clustering} we rescale the covariance matrix,
\eqsali{
\Tilde{\mathbb{C}}_{\nu\nu'}^{-1}=\dfrac{1+B(N_{\rm bin}-N_{\rm p})}{1+A+B(N_{\rm p}+1)}\hat{\mathbb{C}}^{-1}_{\nu\nu'}
}
here $N_{\rm p}$ is the number of the fitting parameters, and
\eqsali{
A=\dfrac{2}{(N_{\rm sim}-N_{\rm bin}-1)(N_{\rm sim}-N_{\rm bin}-4)}
}
\eqsali{
B=\dfrac{N_{\rm sim}-N_{\rm bin}-2}{(N_{\rm sim}-N_{\rm bin}-1)(N_{\rm sim}-N_{\rm bin}-4)}
}
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{total-Cls.pdf}
\caption{The estimation of the shear-CMB cross-spectrum in four photo-$z$ bins. The yellow and blue boxes are the binned error bars in multipoles, with two different CMB lensing reconstruction N0 noises. The first row is the pseudo-$C_\ell$ estimation without any biases. The second row is the one with redshift bias $\Delta_z=0.005$. In the third row are the results with intrinsic alignment amplitude $A_{\rm IA}=1$. The solid curves are the theoretical predictions. The spectrum covariances are estimated from 300 mocks with N0-4 and N0-48 noise setup.}
\label{pseudo-Cls}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=0.4\columnwidth]{N04_coe.pdf}
\includegraphics[width=0.4\columnwidth]{N048_coe.pdf}
\caption{Normalized covariance matrix of multipole and redshift bins of cross spectrum. The left and right panels are for the N0-4 and N0-48 CMB lensing noise cases. }
\label{N048-coe}
\end{figure}
Pseudo-$C_\ell$s with different noise types are shown in Fig. \ref{pseudo-Cls}. The yellow and blue boxes denote the error bars from two different AliCPT-1 experimental setups, namely ``4 modules*yr'' and ``48 modules*yr''. Their noise spectra correspond to the orange dashed and green dotted curves in Fig. \ref{Nphi}.
The first row in Fig. \ref{pseudo-Cls} is calculated assuming without any biases. The green solid curve is the theoretical cross-spectrum. In the second row, we added the photo-$z$ bias $\Delta_z=0.005$. By eye, we can not distinguish the differences between the one with (red solid curve) and without (green solid curve) this bias. In the third row, we added the intrinsic alignment effect in the simulations. The green solid curve is still the theoretical shear-CMB cross-correlation; while the red one assumes the intrinsic alignment amplitude $A_{\rm IA}=1$. One can see that the intrinsic alignment can significantly suppress the signal in the low redshift.
\begin{table}
\centering
\begin{tabular}{lccccc}
\hline
S/N\quad & Bin1 & Bin2 & Bin3 & Bin4 & Total \\
\hline
N04 &$ 5.34$&$ 8.23$&$ 11.44$&$ 14.82$&$ 16.94$\\
N04 with bias-$z$ &$ 5.37$&$ 8.27$&$ 11.47$&$ 14.84$&$ 16.96$\\
N04 with IA &$ 4.62$&$ 7.54$&$ 11.04$&$ 14.58$&$ 16.91$\\
N048 &$ 6.94$&$ 13.55$&$ 18.90$&$ 24.00$&$ 25.71$\\
N048 with bias-$z$&$ 7.01$&$ 13.62$&$ 18.95$&$ 24.02$&$ 25.74$\\
N048 with IA &$ 5.50$&$ 12.32$&$ 18.20$&$ 23.56$&$ 25.57$\\
\hline
\end{tabular}
\caption{Signal-to-noise ratio of pseudo-$C_\ell$s measured from 4 redshift bins and the combined one. Different CMB lensing noise, galaxy photo-$z$ bias as well as the intrinsic alignment are considered.}
\label{snr}
\end{table}
\begin{table}
\centering
\begin{tabular}{lccccc}
\hline
$\chi^2/N_{\rm bin}$\quad & Bin1 & Bin2 & Bin3 & Bin4 & Total \\
\hline
N04 &$ 0.81$&$ 0.64$&$ 0.62$&$ 1.18$&$ 1.02$\\
N04 with bias-$z$ &$ 0.81$&$ 0.64$&$ 0.62$&$ 1.19$&$ 1.02$\\
N04 with IA/G &$ 0.78$&$ 0.63$&$ 0.58$&$ 1.14$&$ 1.01$\\
N04 with IA/G+I &$ 0.79$&$ 0.61$&$ 0.61$&$ 1.17$&$ 1.00$\\
N048 &$ 0.87$&$ 0.69$&$ 0.72$&$ 1.32$&$ 1.06$\\
N048 with bias-$z$&$ 0.88$&$ 0.70$&$ 0.72$&$ 1.33$&$ 1.06$\\
N048 with IA/G &$ 0.88$&$ 0.69$&$ 0.68$&$ 1.26$&$ 1.10$\\
N048 with IA/G+I &$ 0.83$&$ 0.65$&$ 0.71$&$ 1.30$&$ 1.05$\\
\hline
\end{tabular}
\caption{Reduced $\chi^2$ measured from 4 redshift bins and the combined one. The name "with IA/G+I" denotes the case that both in the simulation and spectrum estimation procedures, the intrinsic alignment is consistently concluded. The one "with IA/G" denotes the case that only in the simulation step the intrinsic alignment is included but not in the step of spectrum estimation. Here $N_{\rm bin}=4\times19$ is our data number. }
\label{rechi2}
\end{table}
We report the signal to noise ratio (SNR), which is computed as
\eqsali{
{\rm SNR}=\sqrt{\displaystyle\sum_{\nu\nu'}\hat{D}^{XY}_\nu\hat{\mathbb{C}}^{-1}_{\nu\nu'}\hat{D}^{XY}_{\nu'}}\;,
}
where $\hat{D}^{XY}_\nu$ is generated by a new simulation that is independent of the other 300 ones. The normalized covariance matrices of cross-correlation are shown in Fig. \ref{N048-coe}. One can see that the correlations of the same multipoles between different redshift bins are obvious. This reflects that the lensing has a broader kernel in the redshift dimension. We summarise the SNRs are in Tab.\ref{snr} and the reduced $\chi^2$ in Tab. \ref{rechi2}. The total SNR$\simeq17$ and $26$ in the ``4 modules*yr'' and ``48 modules*yr'' cases, respectively. The reduced $\chi^2\simeq1$ suggests no significant deviation between the model and the data and validates that our spectrum estimation binning choices are appropriate.
Finally, we summarize our simulation and spectrum estimation pipeline in the cartoon picture, Fig. \ref{ppl}.
\begin{figure}
\centering
\includegraphics[width=450pt]{ske.png}
\caption{Sketch picture of photo-$z$ error, pseudo-$C_\ell$ measuerent and covariance matrix computing method.}
\label{ppl}
\end{figure}
\section{Cosmological constraints from cosmic shear-CMB lensing cross-correlation}\label{sec5}
The final step is to study the cosmological implications of the shear-CMB cross-correlation signal. We estimate the cosmological parameter constraint ability by using the Markov Chain Monte Carlo (MCMC) method.
We use the \verb'Emcee' code \citep{Foreman_Mackey_2013}, a public implementation of the affine invariant MCMC ensemble sampler \citep{goodman2010ensemble}. We assume a Gaussian likelihood functions for the cross spectrum
\eqsali{
-2\log\mathcal{L}(\hat{D}^{XY}_\nu|\boldsymbol{\theta})=\chi^2
=\displaystyle\sum_{\nu\nu'}\left(\hat{D}^{XY}_\nu-D^{XY}_\nu(\boldsymbol{\theta})\right)^T\Tilde{\mathbb{C}}_{\nu\nu'}^{-1}\left(\hat{D}^{XY}_{\nu'}-D^{XY}_{\nu'}(\boldsymbol{\theta})\right)\;,
}
where $\boldsymbol{\theta}$ represents
the set of parameters, including the cosmological as well as the nuisance parameters. $\mathbb{C}$ is the covariance matrix. $\hat{D}^{XY}_\nu(\boldsymbol{\theta})$ and $D^{XY}_\nu(\boldsymbol{\theta})$ are the measured and theoretical spectra. The priors are summarized in Tab. \ref{prior}. The posterior on the model parameters is then given by
\eqsali{
\mathcal{P}(\boldsymbol{\theta}|\hat{D}^{XY}_\nu) =\mathcal{L}(\hat{D}^{XY}_\nu|\boldsymbol{\theta})\mathcal{P}(\boldsymbol{\theta})\;,
}
where $\mathcal{P}(\boldsymbol{\theta})$ are the priors.
\begin{table}
\centering
\begin{tabular}{ccc}
\hline
Parameters & Fiducial value &Prior\\
\hline
$\Omega_m$ &$0.314$ &$(0.05,0.7)$\\
$h$ &$0.67$ &fixed\\
$\Omega_b$ &$0.049$ &fixed\\
$\sigma_8$ &$0.811$ &$(0.3,1.3)$\\
$n_s$ &$0.96$ &fixed\\
$A_{\rm IA}$ &$1.0$ &$(-5,5)$\\
\hline
$\Delta^1_z$ &$0.005$ &fixed\\
$\Delta^2_z$ &$0.005$ &fixed\\
$\Delta^3_z$ &$0.005$ &fixed\\
$\Delta^4_z$ &$0.005$ &fixed\\
$\sigma_z$ &$0.05$ &fixed\\
\hline
\end{tabular}
\caption{The cosmological and nuisace parameters used in our simulation. The fiducial values are adopted from Planck-2018 \citep{aghanim2020planck} and COSMOS 2015 \citep{2018MNRAS.480.2178C}.}
\label{prior}
\end{table}
In this work, we are interested in the $\Omega_m$ and $\sigma_8$ constraints. Moreover, we convert the $\sigma_8$ constraint into $S_8$.
\footnote{We do not present here the constraints by adding the redshift bias since we find it has negligible effect.} For $S_8$, we use the following definition
\eqsali{
S_8=\sigma_8\left(\dfrac{\Omega_m}{0.3}\right)^\alpha\;.
}
\begin{table*}
\centering
\begin{tabular}{ccccc}
\hline
\quad & I& II& III& IV\\
\hline
\vspace{5pt}
Data&N04+photo-$z$ & N04+photo-$z$+IA& N048+photo-$z$ & N048+photo-$z$+IA\\
Model & shear & shear+IA\\
\hline
\end{tabular}
\caption{Summary of the data vectors and model templates used in the cosmological constraint.}
\label{tab:datavector}
\end{table*}
For the analysis, we considered 4 types of data and 2 types of the theoretical model. The data contains different kinds of noises and biases. The model difference lies in whether including the intrinsic alignment or not. We summarize our data vectors and model templates in Tab. \ref{tab:datavector}.
We run the MCMC chains by combining the above data and model templates.
We find that, with the AliCPT-1 ``4 modules*yr'' setup, the typical 1$\sigma$ errors on $\sigma_8$ is about $0.034$; with the AliCPT-1 ``48 modules*yr'' setup, the typical 1$\sigma$ errors on $\sigma_8$ is about $0.023$. Furthermore, we find that if we include the IA in the data but not in the fitting template, this will induce $\sim0.6~\sigma$ bias in $\sigma_8$ estimation. However, the intrinsic alignment has almost negligible effects on the $S_8$ estimation. We summarize our main results of the parameter estimation in Tab. \ref{conlim} and Fig. \ref{contour_N04}.
\begin{table*}
\centering
\begin{tabular}{ccccccc}
\hline
Parameter &Data(I)+Model(I) & Data(II)+Model(I) &Data(II)+Model(II) &Data(III)+Model(I) & Data(IV)+Model(I) &Data(IV)+Model(II) \\
\hline
\vspace{5pt}
{\boldmath$\Omega_m $} & $0.321^{+0.025}_{-0.031} $& $0.301^{+0.021}_{-0.026} $& $0.327^{+0.029}_{-0.036} $& $0.310^{+0.017}_{-0.019} $& $0.286^{+0.013}_{-0.016} $& $0.312^{+0.019}_{-0.023} $\\
{\boldmath$\sigma_8 $} & $0.780\pm 0.029 $& $0.791\pm 0.027 $& $0.770\pm 0.034 $& $0.805\pm 0.020 $& $0.822^{+0.020}_{-0.018} $& $0.801\pm 0.023 $\\
{\boldmath$S_8 $} & $0.801^{+0.029}_{-0.025} $& $0.792\pm 0.028 $& $0.797\pm 0.028 $& $0.816\pm 0.015 $& $0.804\pm 0.016 $& $0.813\pm 0.016 $\\
{\boldmath$A_{\rm IA} $} & / & / & $1.20\pm 0.57 $& / & / & $1.19\pm 0.40 $\\
{\boldmath$\alpha$} & $0.42 $& $0.44 $& $0.44 $& $0.43 $& $0.44 $& $0.43$ \\
\hline
\end{tabular}
\caption{The parameter estimation results of $\sigma_8$, $\Omega_m$, $S_8$, $A_{\rm IA}$. $1\sigma (68.3\%)$ C.L. are shown. The results of $S_8$ are derived from $\sigma_8$ and $\Omega_m$ posteriors. The power law index $\alpha$ is calculated via the PCA method \citep{abdi2010principal}.}
\label{conlim}
\end{table*}
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{contours_unbias.pdf}
\caption{The constraint results of $\sigma_8$, $\Omega_m$, $S_8$ and $A_{\rm IA}$. $1\sigma (68.3\%)$ C.L. are shown. }
\label{contour_N04}
\end{figure}
As for the power index in $S_8$, we compute it from the posterior directly. In detail, the power law index $\alpha$ was obtained from the correlation matrix of $\sigma_8$ and $\Omega_m$ in logarithmic space with Perform principal component analysis (PCA) algorithm (e.g. \citealt{abdi2010principal}), which gives the eigenvectors and eigenvalues for the normalized variables by using \verb'getdist' code \citep{2019arXiv191013970L}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{GI-G_48.pdf}
\caption{The constraint results of $\sigma_8$ vs. $\Omega_m$ with N048 noise in different redshift bins. $1\sigma (68.3\%)$ C.L. are shown.
The left panels show the results of without properly considering the intrinsic alignment in the template fitting; while the right panels show the correct one. The input parameter $\sigma_8$ and $\Omega_m$ is marked by gray dashed lines.}
\label{GI-G}
\end{figure}
In order to investigate the reasons for intrinsic alignment bias on $\sigma_8$, we analyze the parameter constraints from each of the individual photo-$z$ bins. The corresponding results are shown in Fig. \ref{GI-G}. The left panels show the results without properly considering the intrinsic alignment in the template fitting; while the right panels show the correct one. The dark red and dark green contours in both the left and right panels are the same. We show them for comparison. One can see that in the left panel the first and second photo-$z$ bins deviate from the combined constraint significantly in the direction of decreasing $\sigma_8$ and $\Omega_m$, simultaneously. Once we corrected the intrinsic alignment bias properly, the contours of each individual photo-$z$ bin overlapped along the constant-$S_8$ direction (anti-diagonal direction in $\sigma_8-\Omega_m$ plane).
We compare the constraining power whether including the correct IA model which introduces more nuisance parameters or not by calculating a Figure of Merit (FoM) defined as:
\eqsali{
{\rm FoM}=\dfrac{1}{\sqrt{{\rm det}[\mathbb{C}(\sigma_8,\Omega_m)]}}
}
where $\mathbb{C}$ refers to the (parameter) covariance matrix between
$\sigma_8$ and $\Omega_m$, the results are shown in Tab. \ref{tab:FoM}.
\begin{table*}
\centering
\begin{tabular}{ccccccc}
\hline
\quad& Data(I)+Model(I)&Data(II)+Model(I)&Data(II)+Model(II)& Data(III)+Model(I)&Data(IV)+Model(I)&Data(IV)+Model(II)\\
\hline
\vspace{2pt}
FoM&1524.30&1761.29&1210.81&3969.57&4492.68&3284.04\\
\hline
\end{tabular}
\caption{The figure-of-merit of each data vector and model template.}
\label{tab:FoM}
\end{table*}
\section{Conclusion}
In this work, we explore the cosmological constraints for the future CSST $\times$ AliCPT. We construct simulated maps for cosmic shear and CMB lensing
based on the experimental nominal setup. In order to evade the complicated ray tracing technique, in this paper, we developed a simulation pipeline based on the pure Gaussian signal and noise ingredients.
We forecast the S/N and the constraints on the cosmological parameters for the cross-correlation, considering statistical error from the two observations. We study the impact of the most important lensing systematics, photo-$z$ and intrinsic alignment, on the predicted cosmological parameters.
More specifically, we simulate the maps (Fig.\,\ref{k_g_maps}) according to CSST and AliCPT-1 nominal parameter setup. We consider standard shape noise for CSST cosmic shear (Fig.\,\ref{NeGI}), and the N0 noise from the disconnected primary CMB for AliCPT (Fig.\,\ref{Nphi}). As the map-building method (Eq.\,\eqref{eq build map} and \cite{kamionkowski1997statistics}) is based on Gaussian random fields, the covariance of their cross-correlation will contain the contribution for the above noises and the cosmic variance. We note that for AliCPT CMB lensing, the noise varies for the ``4 modules*yr'' and ``48 modules*yr'' stages. We perform the standard Pseudo-$C_\ell$ spectrum estimation, and find the shear-CMB cross-correlation can reach the SNR$\simeq17$ for the ``4 modules*yr'' case, and the SNR$\simeq26$ for the ``48 modules*yr'' case, as shown in Fig.\,\ref{pseudo-Cls}. We investigate the cosmological implication of these cross-correlated signals in Fig.\,\ref{contour_N04}. We find that for the ``4 modules*yr'' case, the typical 1$\sigma$ errors on $\sigma_8$ is about $0.034$; for the ``48 modules*yr'' case, the typical 1$\sigma$ errors on $\sigma_8$ is about $0.023$, which is promising in investigating the current S8 tension.
As an extension, we also explore the impact of photo-z bias and intrinsic alignment, which are two of the main sources of systematics in weak lensing. In the generated mock data, we shift the mean redshift to represent the photo-z bias and input an intrinsic alignment signal following the NLA model (Eq.\,\eqref{eq NLA IA} and Fig.\,\ref{NeGI}). We show the contamination of these two in the observed power spectra in Fig.\,\ref{pseudo-Cls}. We find that for the required photo-z precision for CSST with $\Delta_z=0.005$, the bias in the power spectrum is negligible, while the IA contamination with $A_{\rm IA}=1$ is more significant. We find that if we do not consider the intrinsic alignment in the spectrum modeling, this will introduce about $0.6\sigma$ shift in $\sigma_8$ but an almost negligible effect on $S_8$ (Fig.\,\ref{contour_N04}). By including the correct IA model while introducing more nuisance parameters, the figure-of-merit in the $\sigma_8-\Omega_m$ space will be reduced from $\simeq4493$ to $\simeq3284$ (Fig.\,\ref{GI-G}), representing the loss in the cosmological constraining power to the IA parameter.
Interestingly, the map-making method of this paper provides not only an alternative check to the conventional Fisher matrix method, but it can also quickly generate correlated maps. This technique is essential in discussing systematic contaminations when combined with future simulations, as we can directly use maps from simulations rather than assume a model for the power spectrum, especially when sometimes the simulation and the model deviate at some level \citep{Jagvaral2022,Schneider2019}. We note that it is also important to include the impact from non-Gaussian covariance and other sources of systematics, but they are beyond the scope of this work and we leave them for future studies.
\section*{Data Availability}
The inclusion of a Data Availability Statement is a requirement for articles published in MNRAS. Data Availability Statements provide a standardized format for readers to understand the availability of data underlying the research results described in the article. The statement may refer to original data generated in the course of the study or to third-party data analyzed in the article. The statement should describe and provide means of access, where possible, by linking to the data or providing the required accession numbers for the relevant databases or DOIs.
\section*{Acknowledgements}
BH and ZYW are supported by the China Manned Space Project with No.CMS-CSST-2021-B01 and the National Natural Science Foundation of China Grants No. 11973016. JY acknowledges the support of the China Postdoctoral Science Foundation (2021T140451). XKL is supported by NSFC of China under Grant No. 11933002 and No. U1931210, and No. 12173033. ZHF and DZL acknowledge the support from NSFC under 11933002, U1931210, and from China Manned Space Project with No.CMS-CSST-2021-A01. DZL is also supported by the NSFC grant 12103043.
\clearpage
\bibliographystyle{mnras}
|
2,877,628,090,074 | arxiv | \section{Introduction and main results}
\label{S:Intr}
Let $(\mathcal N_t)_{t\geq 0}$ be an inhomogeneous Poisson process with rate $\lambda_t$ such that $\Lambda(t) = \int_0^t \lambda_s \, {\mathrm d} s < \infty$ for all $t>0$. The epochs of $\mathcal N$, in increasing order, are denoted by $T_i$, $i=1,2,\ldots$, so that the gaps are given by $R_i=T_i-T_{i-1}$ with~$T_0=0$. The objects of study of the present paper are the longest gap, $L_t$, before time $t$ and its right-end position, $\sigma_t$:
\begin{align}
\label{eq:Lt}
L_t
&=
\max_{i\geq 1}\{R_i:T_i\leq t\}, \\
\label{eq:sigmat}
\sigma_t
&=
\min_{i\geq 1}\{T_i : R_i = L_t\}.
\end{align}
Note that the definition does not include the gap straddling time $t$, but this is in fact
unimportant for our
asymptotic results, see Remark~\ref{rem:count-the-gap}.
In the homogeneous case, the discrete time analogue of the longest gap
is the longest run, $L_n$, of ones
before time~$n$ in a Bernoulli$(p)$
sequence. The study of the longest run has a long history going back to, among others,
\cite{erdos, vonMises}; a recent survey is in \cite{balakrishnan2011runs}.
A main result is that $L_n$ is of order $\log_{1/p} n$. In the homogeneous Poisson case, $\lambda_t \equiv \lambda$, there is a neat analogue of this:
\begin{equation}\label{18.11a}
\lambda L_t-\log (\lambda t)\,\convdistr\,G\ \text{as }t\to\infty\,,
\end{equation}
where $G$ is Gumbel with cumulative distribution function (cdf) ${\mathbb P}(G\le x)=$ $\exp(-\e^{-x})$. The proof is equally neat:
with $M^\pm_t=$ $\max_{i\le \lambda t(1\pm\epsilon)}R_i$ for $\epsilon > 0$ one has $M^-_t\le L_t\le M^+_t$ for large $t$ with high probability.
Further, by standard extreme value theory, the random variables $\lambda M^\pm_t-\log \{\lambda t(1\pm\epsilon)\}$
have Gumbel limits as $t \to \infty$, so one can just let first $t$ tend to infinity and next
$\epsilon$ tend to $0$. We provide some further comments and references in Remark~\ref{rem:homogeneous} below.
As mentioned above, our interest is in time inhomogeneity. This may occur in at least two ways.
Firstly, one may consider fluctuations around a long-term average which is conveniently modelled in a hidden Markov setting,
see \cite{antzoulakos1999waiting,Olebook,fu1994distribution}. Secondly, the rates $\lambda_t$ may exhibit a
systematic deterministic trend. The only reference here seems to be~\cite{AIRN} (though cf.\ also~\cite{Karlin}),
continuing a study of \cite{A5} related to problems from computer reliability. The results in~\cite{AIRN}
are of large deviations type, giving asymptotic estimates of ${\mathbb P}(L_t<\ell)$ in the rare-event
setting where $t\to\infty$ with $\ell$ fixed. Our concern here is the typical behaviour, that is, analogues
of~\eqref{18.11a}.
As in~\cite{AIRN}, the quantitative form of $\lambda_t$ is crucial both for the form of the results and
the difficulty of the analysis. First, we concentrate on what is maybe the simplest form, a power function $\lambda_t = \lambda_1 t^{\alpha-1}$, and then provide extensions to regularly varying functions. The power function is a rather natural choice with which to start the analysis, and already this case presents
substantial challenges. The case $\alpha=1$ is settled by \eqref{18.11a} and the behaviour when
$\alpha>1$ or $\alpha\le 0$ is easily resolved, see Remark~\ref{Rem:18.11a} below.
Thus what is left for analysis is the case $0<\alpha<1$, and here our result
is the following:
\begin{theorem}
\label{thm:main}
Let $({\mathcal N}_t)_{t \ge 0}$ be an inhomogeneous Poisson process with rate $\lambda_t = \lambda_1 t^{\a-1}$ with $\lambda_1 > 0$ and $\a\in(0,1)$. For $L_t$ and $\sigma_t$ as in \eqref{eq:Lt} and \eqref{eq:sigmat}, we have
\[
\left(\lambda_t L_t-b_t, \, \frac{t-\sigma_t}{t}\log t\right)
\convdistr
\left(G,E_{\a(1-\a)}\right)
\qquad \text{ as }t\to\infty,
\]
where $b_t=\a\log t-\log\log t-\log(\a(1-\a)/\lambda_1)$ and $G,E_{\a(1-\a)}$ are independent random variables: $G$ is Gumbel and $E_{\a(1-\a)}$ is exponential with rate $\a(1-\a)$.
\end{theorem}
In fact, we prove a much more general result establishing weak convergence of a sequence of point processes, from which Theorem~\ref{thm:main} easily follows.
Here and as usual, convergence in distribution of point processes is with respect to the vague topology in the space of Radon measures on~$(-\infty,\infty]^2$.
\begin{theorem}\label{thm:point_proc}
Under the assumptions of Theorem~\ref{thm:main} consider the point process~$\xi_t$ on $(-\infty,\infty]^2$ consisting of the points \[\left(\lambda_t R_i-b_t,\frac{t-T_i}{t}\log t\right)\qquad i=1,2,\ldots\]
Then $\xi_t\convdistr \xi$ as $t\rightarrow\infty$, where $\xi$ is a Poisson point process with intensity measure
\[\mu({\mathrm d} x,{\mathrm d} z)=\e^{-x}{\mathrm d} x\times \a(1-\a)\e^{-\a(1-\a) z}{\mathrm d} z\,.\]
\end{theorem}
Importantly, in Theorem~\ref{thm:main} we consider the compactified Euclidean plane $(-\infty,\infty]^2$ so that the set $[x,\infty]\times [-z,\infty]$ is compact. The points of $\xi_t$ in this set are affine transformations of couples $(R_i,T_i)$ such that $R_i\geq (x+b_t)/\lambda_t$ and $T_i\leq t(1+z/\log t)$.
Hence our result concerns all large enough gaps of~$\mathcal N$ up to the time $t+\Oh(t/\log t)$. Furthermore, since vague convergence of point measures implies convergence of the respective points in any compact set~\cite[Prop.\ 3.13]{resnick}, we conclude that the map
\[\sum_i\delta_{(x_i,z_i)}\mapsto (x,z), \qquad x=\max\{x_i:z_i\geq 0\},\,z=\max\{z_i:x_i=x\}\] is continuous apart from possible discontinuities at point measures with $x_i=x_j$ or $z_i=0$ for some $i\neq j$. Since $\xi$ is not of such form a.s., the continuous mapping theorem
gives that $(\lambda_t L_t-b_t,(1-\sigma_t/t)\log t)\convdistr (X,Z)$, where $(X,Z)$ has the distribution arising from the application of the above map to~$\xi$. A standard calculation reveals that for $z>0$ we have
\begin{align}
\label{eq:XZ}
{\mathbb P}(X\in{\mathrm d} x,Z\in{\mathrm d} z)
&={\mathbb P}(\xi({\mathrm d} x\times{\mathrm d} z)=1, \, \xi((x,\infty)\times(0,\infty))=0)\\
\nonumber
&= \mu({\mathrm d} x,{\mathrm d} z)\exp\{-\mu((x,\infty)\times(0,\infty))\} \\
\nonumber
&= \mu({\mathrm d} x,{\mathrm d} z)\exp(-\e^{-x}),
\end{align}
proving Theorem~\ref{thm:main}; see also the light-gray region in Figure~\ref{fig:points}.
\begin{remark}\label{rem:count-the-gap}\rm
In order to give a feeling for some further results we consider the first gap exceeding~$L_t$ and its time of occurrence:
$(L_t^+,\sigma^+_t)=(R_{i^+_t},T_{i^+_t})$, where $i^+_t=\min\{i\geq 1:R_i>L_t\}$ is the corresponding index.
From Theorem~\ref{thm:point_proc} and the continuous mapping theorem applied to the appropriate map, we find that
\[
\left(\lambda_t L_t-b_t,\, \lambda_t L^+_t-b_t, \, \frac{t-\sigma_t}{t}\log t,\, \frac{\sigma^+_t-t}{t}\log t\right)
\convdistr
\left(X,X^+,Z,Z^+\right),
\]
where the conditional distribution of $X^+,Z^+$ is easily identified to be
\begin{align}
\label{eq:XZ:plus}
{\mathbb P}(X^+\in {\mathrm d} x^+&,Z^+\in {\mathrm d} z^+ \mid X=x,Z=z)\\
\nonumber
&={\mathbb P}(\xi({\mathrm d} x^+\times(-{\mathrm d} z^+))=1,\xi((x,\infty)\times(-z^+,0))=0)\\
\nonumber
&=
\mu({\mathrm d} x^+,-{\mathrm d} z^+)\exp(-\e^{-x}(\e^{\a(1-\a)z^+}-1))
\end{align}
for $x^+>x$ and $z^+>0$; see the dark-grey region in Figure~\ref{fig:points}.
\begin{figure}
\includegraphics[width=0.4\textwidth]{points.pdf}
\caption{The points $(x, z)$ and $(x^+, -z^+)$ and the associated empty regions $(x, \infty) \times (0, \infty)$ in~\eqref{eq:XZ} (light-gray) and $(x, \infty) \times (-z^+, 0)$ in~\eqref{eq:XZ:plus} (dark-gray), respectively.}
\label{fig:points}
\end{figure}
In particular, we find after some computation that
$Z^+\eqdistr E_{\a(1-\a)}\eqdistr Z$.
One may proceed even further and obtain convergence of extremal processes (on the Skorokhod space of two-sided paths) identifying the record gaps and their times, see~\cite[Prop.\ 4.20]{resnick} for the classical setting.
Finally, note that $L_t^+=\oh_p(t/\log t)$ and so ${\mathbb P}(\sigma_t^+-L_t^+>t)\to 1$, showing that the corresponding gap does not straddle time~$t$ in the limit.
\end{remark}
When trying to adapt the above proof of \eqref{18.11a}, with scale constant $\lambda_1 = \a$ say, one quite easily gets $\mathcal N_t\approx t^\a$, which gives a rough estimate of $L_t$ in terms of $\max_{i < t^\a} R_i$. The difficulty is that these interarrival times $R_i$ are no longer
independent nor exponentially distributed. Nevertheless, the $R_i$ are not too far from exponential random variables with rates $\lambda_{T_i} \approx \a \, i^{(\a-1)/\alpha}$, because $T_i\approx i^{1/\alpha}$ for large~$i$. Hence our first step is to consider extreme value theory for sequences
of i.i.d.\ random variables equipped with weights. Some references in that direction are
\cite{Kostya99,Gouet15,DeHaan87,Smith88} and, of particular relevance for us,
\cite[Thm.\ 4.1]{WN10}, from which the following result can be extracted:
\begin{proposition}\label{prop:WN}
\label{prop:max}
Let $X_1, X_2, \ldots$ be independent unit exponential random variables and let $\gamma \in (0, \infty)$.
Then with $M_n=\max_{i=1,\ldots,n}\{i^{\gamma}X_i\}$ we have
\[
\frac{M_n}{n^{\gamma}} - \beta_n
\convdistr G, \qquad \text{ as }n \to \infty,
\]
where $\beta_n = \log (n/\gamma) - \log \log n$ and $G$ is a Gumbel random variable.
\end{proposition}
Our analysis supplements this result by identifying the location of the maximum and providing the analogue of Theorem~\ref{thm:point_proc}. This location is trivially uniform for i.i.d.\ sequences or homogeneous Poisson processes, but has an interesting limiting distribution in the nonhomogeneous case. We also give an extension to
weights in Proposition~\ref{prop:WN} and rates in Theorem~\ref{thm:main}
which are regularly varying rather than of simple
power form. Such an extension is of course expected, but the proof is surprisingly complicated,
and in fact, we need some regularity conditions on the slowly varying function.
\begin{remark}\label{rem:homogeneous}\rm
Despite its simplicity, \eqref{18.11a} does not seem to have been formulated in the longest run/gap literature. Note that its analogue fails in the Bernoulli setting, because the extreme value behaviour
of geometric random variables is more complicated than the one of exponential random variables, cf.~\cite[pp.\,24--25]{LLR}.
However, as pointed out by an associate editor and a referee, there are a number of related results in the stochastic geometry literature. Most of these are more general and go deeper, but \eqref{18.11a} can be deduced after some reformulation.
For example, consider the probability of full coverage of the interval $[0,1]$ in the Boolean model~\cite{hall} on $\mathbb R$ with deterministic segments of length $r^{(t)}=(x+\log(\lambda t))/(\lambda t)$ arriving at the rate $\lambda^{(t)}=\lambda t$. By rescaling time we find that
\begin{align*}
{\mathbb P}(\lambda L_t-\log(\lambda t)\leq x)
&={\mathbb P}(L_t/t\leq r^{(t)}) \\
&={\mathbb P}^{(t)}(\text{full coverage of }[0,1])+\oh(1)\quad\text{as }t\to \infty,
\end{align*}
which converges to $\exp(-{\mathrm e}^{-x})$ according to~\cite[Thm.\ 2.5]{hall}. For related results in the nonuniform setting see~\cite{hall_nonuniform,husler} and~\cite{molchanov} for more recent work.
Furthermore, \eqref{18.11a} also follows from~\cite[(2c)]{calka} specifying the limit behaviour of the maximal circumscribed radius of a Poisson--Voronoi tessellation.
\end{remark}
\begin{remark}\label{Rem:18.11a}\rm
When $\alpha>1$,~\cite{AIRN} gives that the increasing process $L_t$ has a proper limiting distribution, of $L_\infty$, say.
That is, from~(7) in \cite{AIRN} it follows that ${\mathbb P}(L_\infty\geq \ell)\to 0$ as $\ell\to \infty$.
The case $\alpha<0$
is trivial since then $\int_1^\infty\lambda_t\,{\mathrm d} t<\infty$, so that the number of epochs in $[1,\infty)$ is finite with probability~1.
The boundary case $\alpha=0$ is also easy: if $\lambda_t = \lambda_1 / t$ for some scale constant $\lambda_1 > 0$, then
\begin{equation}\label{18.11b}
\left(\frac{L_t}{t},\frac{\sigma_t}{t}\right)\stackrel{\mathcal{D}}{=}(L_1,\sigma_1).
\end{equation}
Indeed, fix $t > 0$ and define the time-changed process $\mathcal{N}'$ by $\mathcal N'_x=\mathcal N_{tx}$ for $x \ge 0$. Its intensity measure, $\Lambda'$, satisfies $\Lambda'(x,y)=\Lambda(tx,ty)=\lambda_1 \log ((ty)/(tx)) = \Lambda(x,y)$ for any $0<x<y$. It follows that ${\mathcal N}'$ has the same distribution as ${\mathcal N}$, and it is then clear that $(L_{t}/t,\sigma_t/t)$ has the same distribution as $(L_1,\sigma_1)$.
Finally, observe that $t-\sigma_t$ is of order~$t$ when $\alpha=0$ or $1$, the two boundary cases in Theorem~\ref{thm:main}.
In contrast, Theorem~\ref{thm:main} gives the smaller order $t/\log t$ when $\a\in(0,1)$.
Therefore, it is intuitive that the limiting random variable $E_{\a(1-\a)}$ must increase to
$\infty$ as $\a$ approaches~0 or~1. This is indeed the case.
\end{remark}
\section{Weighted exponentials}
\label{sec:exponential}
As in Proposition~\ref{prop:WN}, we consider a sequence $X_1, X_2, \ldots$ of independent, unit exponential random variables. We fix $\gamma > 0$ and let
\begin{equation}
\label{eq:MnX}
M_n = \max_{i=1,\ldots,n} \{ i^{\gamma} X_i \}
\qquad \text{and} \qquad
\tau_n = \min \{ i = 1, \ldots, n : i^{\gamma} X_i = M_n \},
\end{equation}
denote the partial maximum of the weighted sequence $(i^\gamma X_i)_{i \ge 1}$ and the location of that maximum, respectively. In the i.i.d.\ case, $\gamma = 0$, the random variable $\tau_n $ is uniformly distributed on $\{1,\ldots,n\}$.
Since the weights $i^\gamma$ increase to infinity, one would expect that $\tau_n/n\to 1$ as $n \to \infty$. The following proposition makes this precise.
\begin{proposition}
\label{prop:loc}
For $M_n$ and $\tau_n$ as in \eqref{eq:MnX}, we have
\[
\left(
\frac{M_n}{n^\gamma} - \beta_n,
\frac{n-\tau_n}{n}\log n
\right)
\convdistr
\left(G, E_\gamma\right)
\qquad \text{ as }n\to \infty,
\]
where $\beta_n = \log(n/\gamma) - \log \log(n)$ and where $G,E_\gamma$ are independent random variables: $G$ is Gumbel and $E_\gamma$ is exponential with rate~$\gamma$.
\end{proposition}
We start by proving a lemma which is basic for the proof of Proposition~\ref{prop:loc} and the associated point process result given in Proposition~\ref{prop:point_discrete}.
\begin{lemma}
\label{lem:Mz1z2}
For every $z\in\mathbb R$, we have
\[
\frac{M_{\lfloor n(1 -z / \log n) \rfloor}}{n^\gamma} - \beta_n
\convdistr G -\gamma z,
\qquad \text{ as }n \to \infty,
\]
where $G$ is a Gumbel random variable.
\end{lemma}
\begin{proof}
Letting $M_n(z)=M_{\lfloor n(1 -z / \log n) \rfloor}$ we find from Proposition~\ref{prop:max} that
\begin{align*}
G_n =
\frac{M_n(z)}{\lfloor n - n z / \log n \rfloor^\gamma}
- \beta_{\lfloor n - n z / \log n \rfloor}
\convdistr G, \qquad \text{ as }n \to \infty.
\end{align*}
Further,
\[
\frac{M_n(z)}{n^\gamma} - \beta_n \\
=
\left(
G_n + \beta_{\lfloor n - n z / \log n \rfloor}
\right) \frac{\lfloor n - n z / \log n \rfloor^\gamma}{n^\gamma}
- \beta_n.
\]
An elementary calculation yields
\begin{equation}\label{eq:toextend}
\beta_{\lfloor n - n z / \log n \rfloor} \frac{\lfloor n - n z / \log n \rfloor^\gamma}{n^\gamma} - \beta_n
\to
- \gamma z,
\qquad \text{ as }n \to \infty.
\end{equation}
The result follows by Slutsky's lemma and the fact that $\lfloor n - n z / \log n \rfloor^\gamma \sim n^\gamma$, where $a_n \sim b_n$ means that $a_n / b_n \to 1$ as $n \to \infty$.
\end{proof}
The following result establishing convergence of the underlying point processes is close in spirit to, e.g.,~\cite[Thm.\ 1]{weissman75}, and it serves as the basis for Theorem~\ref{thm:point_proc}.
\begin{proposition}\label{prop:point_discrete}
The point process $\hat \xi_n$ on $(-\infty,\infty]^2$ consisting of the points \[\left(\frac{i^\gamma X_i}{n^\gamma}-\beta_n,\frac{n-i}{n}\log n\right)\qquad i=1,2,\ldots\]
converges in distribution as $n\rightarrow\infty$ to the Poisson point process $\hat \xi$ with mean measure
\[\hat \mu({\mathrm d} x,{\mathrm d} z)=\e^{-x}{\mathrm d} x\times\gamma\e^{-\gamma z}{\mathrm d} z.\]
\end{proposition}
\begin{proof}
Let $Y_{n,i}=i^\gamma X_i/n^\gamma -\beta_n$.
According to the result of Grigelionis, see e.g.~\cite[Thm.\ 16.18]{kallenberg}, applied to a null array of single points, it is only required to show that
\begin{align*}
\sup_{i\geq 1}\,{\mathbb P}\{(Y_{n,i},(1-i/n)\log n)\in B\}&\rightarrow 0,\\
\sum_{i\geq 1}\,{\mathbb P}\{(Y_{n,i},(1-i/n)\log n)\in B\}&\rightarrow \hat\mu(B),
\end{align*}
for any finite union~$B$ of rectangles in~$(\infty,\infty]^2$.
In our setting it is sufficient to check the above limits for~$B=[x,\infty]\times [z,\infty]$. The first limit result follows from the monotonicity of $(i/n)^\gamma$ and
\begin{equation*}{\mathbb P}(Y_{n,\lfloor n(1-z/\log n)\rfloor}\geq x)=\exp\{-(x+\beta_n)(1+o(1))\}\rightarrow 0.\end{equation*}
Using this and Lemma~\ref{lem:Mz1z2} we also find that
\begin{align*}
&\sum_{\substack{i \geq 1\\ (1 - i/n)\log n\geq z}}{\mathbb P}(Y_{n,i}\geq x)=
-(1+\oh(1))\log\prod_{\substack{i \geq 1\\ (1 - i/n)\log n\geq z}}{\mathbb P}(Y_{n,i}< x)\\
&=-(1+\oh(1))\log {\mathbb P}(M_{\lfloor n-nz/\log n\rfloor}/n^\gamma-\beta_n< x)\rightarrow -\log {\mathbb P}(G-\gamma z< x)\\
&=\e^{-x-\gamma z}=\hat\mu([x,\infty]\times [z,\infty]),
\end{align*}
as required.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:loc}]
It follows by the continuous mapping theorem applied to Proposition~\ref{prop:point_discrete} in the same way as Theorem~\ref{thm:main} follows from Theorem~\ref{thm:point_proc}.
Alternatively, one may proceed directly by identifying the limit distribution:
\[\max\{i^\gamma X_i: {i\in\mathbb N}, n(1-z/\log n)< i\leq n\}/n^\gamma-\beta_n\convdistr G+\log(1-\e^{-\gamma z}),\]
and then expressing the distribution of interest using $M_{\lfloor n-nz/\log n\rfloor}/n^\gamma-\beta_n$ and the above quantity.
\end{proof}
\section{Gaps of an inhomogeneous Poisson process}
\label{sec:gaps}
Let $0 < T_1 < T_2 < \ldots$ be the points of a Poisson process $\mathcal{N} = (\mathcal{N}_t)_{t \ge 0}$ with rate $\lambda_t = \alpha \, t^{\alpha - 1}$ for some $0 < \alpha < 1$ and with cumulative rate function $\Lambda(t) = \int_0^t \lambda_s \, {\mathrm d} s = t^\alpha$. Note that we assume that $\lambda_1 = \alpha$; the case $\lambda_t = \lambda_1 t^{\a - 1}$ for general $\lambda_1 > 0$ follows by the time change argument, but see also Section~\ref{sec:reg_var}. Recall that $R_i = T_i - T_{i-1}$ for integer $i \ge 1$, where $T_0 = 0$.
Define $T'_i = T_i^\a$ for integer $i \ge 0$, so that $0 < T'_1 < T'_2 < \ldots$ are the points of a unit-rate homogeneous Poisson process $({\mathcal N}'_t)_{t \ge 0}$. Let $X_i = T'_i - T'_{i-1}$ be its interarrival times, for integer $i \ge 1$. The random variables $X_1, X_2, \ldots$ are independent unit exponentials. Put
\[
\gamma = (1 - \a)/\a \in (0, \infty).
\]
The following result provides the basic approximation.
\begin{lemma}
\label{lem:RbyX}
We have as $i\to\infty$ that
\begin{equation}
\label{eq:X2R}
\left|\frac{T_i}{i^{1/\a}}-1\right|\vee \left|\frac{\a R_i}{i^\gamma X_i}-1\right|=\oh(1/\log i) \quad\text{a.s.}
\end{equation}
\end{lemma}
\begin{proof}
Since $(T_i')_i$ is the partial sum process of a sequence of independent unit exponentials, the law of the iterated logarithm states that
\begin{equation*}
\limsup_{i \to \infty} \frac{T_i'/i - 1}{\sqrt{i^{-1} \log\log i}} = \sqrt{2}
\qquad
\text{a.s.},
\end{equation*}
which further implies
\begin{equation}\label{eq:itlog}
|T_i'/i-1|\log i\rightarrow 0\quad \text{a.s.}
\end{equation}
But then
\[
T_i/i^{1/\a}-1=(T_i'/i)^{1/\a}-1=(T_i'/i-1)(1+\oh(1))/\a \quad \text{a.s.}
\]
and so $|T_i/i^{1/\a}-1|\log i\rightarrow 0$ a.s.\ as required.
Concerning the second part, we write using the mean-value theorem
\begin{align}
\label{eq:XT2R}
R_i
= T_i - T_{i-1}
= (T_i')^{1/\a} - (T_{i-1}')^{1/\a}
= (T_i')^{1/\a} - (T_i' - X_i)^{1/\a}= \a^{-1} \theta_i^{\gamma} X_i
\end{align}
with $T_{i-1}' < \theta_i < T_i'$.
Hence it is left to show that $|(\theta_i/i)^\gamma-1|\log i\rightarrow 0$ a.s., which again follows from~\eqref{eq:itlog}.
\end{proof}
In the following we relate the points of the point process $\xi_t$ in Theorem~\ref{thm:point_proc} to the corresponding points of the process $\hat\xi_{\lceil t^\a\rceil}$ in Proposition~\ref{prop:point_discrete} with rescaled second component.
\begin{lemma}\label{lem:points_close}
Let $B=[x_1,x_2]\times[z_1,z_2]$ and put
\begin{align*}
u_i(t)
&= (\lambda_t R_i-b_t,(1-T_i/t)\log t),\\
v_i(t)
&= (i^\gamma X_i/n^\gamma-\beta_n,(1-i/n)\log(n)/\a^2)
\end{align*}
with $n=n(t)=\lceil t^\a\rceil$.
Then
\[
\sup_i\{\|u_i(t)-v_i(t)\|_1:v_i(t)\in B\text{ or }u_i(t)\in B\}
\rightarrow 0\quad \text{a.s.}
\]
as $t\rightarrow \infty$ with the convention that~$\sup\varnothing = 0$.
\end{lemma}
\begin{proof}
Letting $I_v(t)=\{i\geq 1:v_i(t)\in B\}$ we see that $i/n\to 1$ and hence also $i/t^\a\to 1$ uniformly in $i\in I_v(t)$ as $t\to\infty$.
Now according to Lemma~\ref{lem:RbyX}, for all $i\in I_v(t)$, we have
\begin{equation}\label{eq:approx_main}\a R_i=i^\gamma X_i(1+\eta'_i),\qquad T_i=i^{1/\a}(1+\eta''_i)\end{equation}
where $|\eta'_i|\vee|\eta''_i|=\oh(1/\log t)$ as $t\rightarrow \infty$ a.s.
So we have a.s.
\begin{align}
\label{lim1}\lambda_t R_i-b_t&\leq i^\gamma X_i/t^{\a\gamma}(1+\oh(1/\log t))-b_t\\
&=(i^\gamma X_i/n^\gamma-\beta_n)(1+\oh(1/\log t))+\oh(1),\nonumber
\end{align}
where in the last line we used the facts: $\lceil t^\a\rceil^\gamma/t^{\a\gamma}=1+\oh(1/\log t)$ and $b_t=\beta_{t^\a}+\oh(1)=\beta_n+\oh(1)$. This and the analogous lower bound imply that
\[\sup_{i\in I_v(t)}|(\lambda_t R_i-b_t)-(i^\gamma X_i/n^\gamma-\beta_n)|\to 0\]
as $t\to\infty$ a.s., because $|i^\gamma X_i/n^\gamma-\beta_n|$ is bounded for the indices of interest.
Upon recalling that $i/t^\a-1\to 0$ uniformly in $i\in I_v(t)$, for all such $i$ we find that
\begin{align}
\label{lim2}
\a^2(1-T_i/t)\log t&\leq\a^2(1-i^{1/\a}(1+\oh(1/\log t))/t)\log t\\
&=\a(1-(i/t^\a)^{1/\a})\log (t^\a)+\oh(1) \nonumber\\
&=(1-i/t^\a)\log (t^\a)(1+\oh(1))+\oh(1)\nonumber\\
&=(1-i/n)\log(n)(1+\oh(1))+\oh(1).\nonumber
\end{align}
This and the analogous lower bound yield
\[\sup_{i\in I_v(t)}|\a^2(1-T_i/t)\log t-(1-i/n)\log n|\to 0\]
as $t\to\infty$ a.s., because now $|1 - i/n| \log n$ is bounded for the indices of interest.
Next, consider the set of indices $I_u(t)=\{i\geq 1:u_i(t)\in B\}$. In this case we use the fact that $T_i/t\to 1$ uniformly in $i\in I_u(t)$.
Furthermore, with probability~1 as $t\rightarrow \infty$ the corresponding indices~$i$ converge to $\infty$ too, and since $T_i\sim i^{1/\a}$ we must have that $i/t^\a\to 1$ uniformly in $i\in I_u(t)$. Thus~\eqref{lim1} holds true and hence also
\begin{equation}\label{lim11}i^\gamma X_i/n^\gamma-\beta_n\geq (\lambda_t R_i-b_t)(1+\oh(1/\log t))+\oh(1).\end{equation}
The corresponding upper bound, as well as the bounds on $(1-i/n)\log(n)/\a^2$ stemming from~\eqref{lim2}, complete the proof, because $|\lambda_t R_i-b_t|$ and $|(1-T_i/t)\log t|$ are bounded for all $i\in I_u(t)$.
\end{proof}
\begin{remark}
\label{rem:rescaling}\rm
The point process $\sum_i\delta_{v_i(n)}$ with $v_i(n)$ defined in Lemma~\ref{lem:points_close}
is a rescaled version of $\hat\xi_n$ in Proposition~\ref{prop:point_discrete}, and the proof of the latter easily yields that $\sum_i\delta_{v_i(n)}$ converges in distribution to a Poisson point process with intensity measure for the set~$[x,\infty]\times[z,\infty]$ given by
\[
\hat\mu([x,\infty]\times[\a^2 z,\infty])
=\e^{-x-\a(1-\a)z}
=\mu([x,\infty]\times[z,\infty]).
\]
That is, the corresponding limit is~$\xi$.
\end{remark}
The following lemma shows that compact sets of the form $[x,\infty]\times[z,\infty]$ can be truncated to finite rectangles.
\begin{lemma}\label{lem:bounding}
For any $\epsilon>0$ and $z,x<\infty$ there exist $z'>z$ and $x'>x$ such that
\[
\limsup_{t\rightarrow \infty}\,
{\mathbb P}(\xi_t(([x,\infty]\times[z,\infty]) \setminus ([x,x']\times[z,z']))>0)<\epsilon.
\]
\end{lemma}
\begin{proof}
Put $n=\lceil t^\a\rceil$ and observe using~\eqref{eq:XT2R} that
\begin{align*}
\max_{i\leq n/2} \a R_i\leq \max_{i \leq n/2} (T_i')^\gamma X_i \leq (T'_{\lceil n/2\rceil})^\gamma\max_{i \leq n/2}X_i=(n/2)^\gamma\log n(1+\oh_p(1)),
\end{align*}
where in the last equality we used the law of large numbers applied to $T_i'$ and the fact that $\max_{i=1,\ldots,k} X_i - \log(k)$ is asymptotically Gumbel. But then
\[
\lambda_t \max_{i\leq n/2}R_i-b_t
\leq 2^{-\gamma}\a\log(t)(1+\oh_p(1))-b_t\rightarrow -\infty
\]
in probability.
Thus it is sufficient to restrict our attention to the indices $i>n/2$, in which case we have~\eqref{eq:approx_main} for all such $i$ a.s.
Observe that for $i>n/2$ the bound~\eqref{lim11} is still true.
Letting $I(t,z)$ be the set of indices $i$ such that $(1-T_i/t)\log t\in[z,z+1]$ or $(1-i/n)\log(n)/\a^2\in[z,z+1]$, we see from the proof of Lemma~\ref{lem:points_close} that $i/t^\a-1\to 0$ uniformly in $i\in I(t,z)$, and also that
\[
\sup_{i\in I(t,z)}
\lvert (1-T_i/t)\log t-(1-i/n)\log(n)/\a^2\rvert
\to 0
\qquad\text{as }t\to\infty\qquad\text{ a.s.}
\]
Hence for any fixed $\delta>0$ with arbitrarily high probability the following is true for large enough $t$: if for some $i>n/2$ it is true that \[\lambda_t R_i-b_t\geq x\text{ and } (1-T_i/t)\log t\geq z\] then
\[
i^\gamma X_i/n^\gamma-\beta_n\geq x-\delta
\text{ and }
(1-i/n)\log(n)/\a^2\geq z-\delta,
\]
because for $i\notin I(t,z)$ the monotonicity of $T_i$ implies $(1-i/n)\log(n)/\a^2>z+1$.
Thus it is left to apply Proposition~\ref{prop:point_discrete} and to note that $\hat\mu(B_1),\hat\mu(B_2)\rightarrow 0$ with
\[B_1=[x-\delta,\infty]\times[\a^2(z'-\delta),\infty],\quad B_2=[x'-\delta,\infty]\times[\a^2(z-\delta),\infty]\] as $x',z'\rightarrow \infty$, which implies that ${\mathbb P}(\hat\xi(B_i)>0)\rightarrow 0$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:point_proc}]
According to~\cite[Thm.\ 1]{kallenberg_SPA} it is sufficient to show that
\begin{align}
\label{thm:point_proc:a}
\lim_{t\rightarrow\infty}{\mathbb P}(\xi_t(B)=0)&= {\mathbb P}(\xi(B)=0), \\ \label{thm:point_proc:b}
\limsup_{t\rightarrow\infty}{\mathbb P}(\xi_t(K)>1) &\leq {\mathbb P}(\xi(K)>1),
\end{align}
where $K$ is a compact rectangle in~$(-\infty,\infty]^2$ and $B$ is a finite union of such rectangles.
According to Lemma~\ref{lem:bounding} we may choose $x_1<x_2,z_1<z_2$ such that
\begin{align*}
0
\leq {\mathbb P}(\xi_t(B')=0)-{\mathbb P}(\xi_t(B)=0)
\leq {\mathbb P} \bigl( \xi_t(B \setminus ([x_1,x_2]\times[z_1,z_2]))>0 \bigr)
< \epsilon
\end{align*}
for $B'=B\cap ([x_1,x_2]\times [z_1,z_2])$ and all $t$ large enough. Furthermore, we may additionally ensure that $0\leq {\mathbb P}(\xi(B')=0)-{\mathbb P}(\xi(B)=0)\leq \epsilon$.
A similar observation holds true with respect to ${\mathbb P}(\xi_t(B)>1)-{\mathbb P}(\xi_t(B')>1)$ and the corresponding difference for the process $\xi$. Hence it is sufficient to prove~\eqref{thm:point_proc:a} and~\eqref{thm:point_proc:b} for any finite rectangle $K$ and a finite union $B$ of such rectangles.
Fix $\delta>0$ and define the $\delta$-enlarged set $B_{\delta+}=\{v:d(v,B)<\delta\}$ and $\delta$-narrowed set $B_{\delta-}=\{v:d(v,{B}^c)>\delta\}$, where $d$ is the Euclidean distance. According to Lemma~\ref{lem:points_close} with the respective rectangle chosen to cover $B$, we have
\begin{align*}
{\mathbb P}(\#\{i: v_i(t)\notin B_{\delta+}\}=0)-\epsilon
&\leq {\mathbb P}(\xi_t(B)=0) \\
&\leq {\mathbb P}(\#\{i: v_i(t)\notin B_{\delta-}\}=0)+\epsilon
\end{align*}
for all $t$ large. Noting that $\mu(\partial B)=0$ we obtain \eqref{thm:point_proc:a} from Remark~\ref{rem:rescaling} based on Proposition~\ref{prop:point_discrete}.
In a similar way we also find that ${\mathbb P}(\xi_t(K)>1)\to{\mathbb P}(\xi(K)>1)$. The proof is complete.
\end{proof}
\section{Extensions to regular variation}
\label{sec:reg_var}
Let $\operatorname{RV}_\rho$ denote the set of measurable functions $f : \mathbb R_+ \to \mathbb R_+$ which are regularly varying at $\infty$ with index~$\rho\in\mathbb R$, i.e., satisfying $f(ut)/f(t)\to u^\rho$ as $t\to \infty$ for all $u>0$. Any such $f$ can be represented as $f(t)=t^\rho\ell(t)$ with $\ell\in \operatorname{RV}_0$ a slowly varying function. Regularly varying functions are thus a generalization of the power functions considered above. Let us also recall the basic theorem concerning regularly varying functions $f\in \operatorname{RV}_\rho$, the Uniform Convergence Theorem~\cite[Thm.\ 1.5.2]{bingham1989regular}:
\begin{equation}\label{UCT}\tag{UCT}
f(ut)/f(t)\to u^\rho,\qquad\text{uniformly in }u
\end{equation}
on intervals $[a,b]$ with $0<a\leq b<\infty$ for $\rho\leq 0$,
and on intervals $(0,b]$ for $\rho>0$ if $f$ is locally bounded.
Assume that the rate function $t \mapsto \lambda_t$ is in $\operatorname{RV}_{\alpha-1}$ for some $\alpha \in (0, 1)$, so that $\Lambda \in \operatorname{RV}_\alpha$. Let $V(t) = \Lambda^{-1}(t)$ be the inverse function of $\Lambda$ and let $v(t) = {\mathrm d} V(t) / {\mathrm d} t$ be its derivative. Then $V \in \operatorname{RV}_{1/\alpha}$ and $v = 1 / (\lambda \circ V) \in \operatorname{RV}_\gamma$ with $\gamma = (1-\alpha)/\alpha$.
The point process $\mathcal{N}'_t = \mathcal{N}_{V(t)}$ is a unit-rate homogenous Poisson process with epochs $T_i' = \Lambda(T_i)$.
We generalize our main results imposing just one condition on the slowly varying function $\ell$ associated with $v$, i.e., $v(t) = t^\gamma \ell(t)$; see Condition~\ref{cond:loc} below.
The basis of our analysis will be the approximation $R_i = V(T_i') - V(T_{i-1}') \approx v(i) X_i$ inspired by the mean-value theorem applied to $V$ and the strong law of large numbers applied to the partial sum sequence $T_i'$. Therefore, we first study the behaviour of the maximum of the weighted exponentials $v(i) X_i$.
\subsection{Weighted exponentials}
Let $X_1, X_2, \ldots$ be a sequence of i.i.d.\ unit exponentials and let $v \in \operatorname{RV}_\gamma$ with $\gamma > 0$. Write $v(t) = t^\gamma \ell(t)$ with $\ell \in \operatorname{RV}_0$. We consider the maximum of the weighted exponentials $v(i) X_i$ and the location of that maximum:
\begin{equation*}
M_n^* = \max_{i=1,\ldots,n} \{ v(i) X_i \}
\qquad\text{and}\qquad
\tau_n^* = \min \{ i = 1, \ldots, n : v(i) X_i = M_n^* \}.
\end{equation*}
\begin{lemma}
\label{lem:tauto1}
We have \[\frac{M^*_n}{v(n)} \, \Big/ \, \frac{M_n}{n^\gamma}=1 + \oh_p(1),\qquad \frac{\tau_n^*}{n} = 1 + \oh_p(1)\] as $n \to \infty$.
\end{lemma}
\begin{proof}
Let $0 < h < 1$. First, we prove that ${\mathbb P}( \tau_n^* / n > h ) \to 1$ as $n \to \infty$, or equivalently, that
\begin{equation}
\label{eq:tauto1}
\lim_{n \to \infty} {\mathbb P}( M_{\floor{nh}}^* < M_n^* ) = 1.
\end{equation}
On the one hand, we have
\[
\frac{M_{\floor{nh}}^*}{v(n)}
=
\max_{i=1,\ldots,{\floor{nh}}} \frac{v(i)}{v(n)} X_i
\le
\max_{i=1,\ldots,{\floor{nh}}} \frac{v(i)}{v(n)}
\max_{i=1,\ldots,{\floor{nh}}} X_i.
\]
But $v\in\operatorname{RV}_\gamma$ can be assumed to be locally bounded [otherwise redefine $v$ by $v(t) = v(\floor{t})$], and so by \eqref{UCT} it follows that
\[
\lim_{n \to \infty} \max_{i=1,\ldots,{\floor{nh}}} \frac{v(i)}{v(n)}
=
\max_{u \in [0, h]} u^\gamma
=
h^\gamma.
\]
Since $X_1, X_2, \ldots$ are i.i.d.\ unit exponentials, we have $\max_{i = 1, \ldots, \floor{nh}} X_i = \log \floor{nh} + \Oh_p(1) = \log(n) \{ 1 + \oh_p(1) \}$ and thus
\[
\frac{M_{\floor{nh}}^*}{v(n)}
\le
h^\gamma \log(n) \{1 + \oh_p(1)\}
\qquad\text{ as }n \to \infty.
\]
On the other hand, let $g \in (h, 1)$. By a similar argument as in the previous paragraph, we find
\begin{align*}
\frac{M_n^*}{v(n)}
\ge
\max_{i=\floor{ng},\ldots,n} \frac{v(i)}{v(n)} X_i
&\ge
\min_{i=\floor{ng},\ldots,n} \frac{v(i)}{v(n)}
\max_{i=\floor{ng},\ldots,n} X_i \\
&=
g^\gamma \log(n) \{1 + \oh_p(1)\}
\qquad\text{ as }n \to \infty.
\end{align*}
Since $h^\gamma < g^\gamma$, we obtain \eqref{eq:tauto1}, as required.
Concerning the first statement, observe from above and Proposition~\ref{prop:loc} that with arbitrarily high probability
\[M^*_n=\max_{i=\floor{ng},\ldots,n} \ell(i)i^\gamma X_i,\qquad M_n=\max_{i=\floor{ng},\ldots,n} i^\gamma X_i\]
for large enough~$n$. Hence it is sufficient to show that
\begin{equation}\label{eq:ell_lim}\max_{i=\floor{ng},\ldots,n} \ell(i)/\ell(n)-1\rightarrow 0\qquad\text{ as }n\rightarrow\infty\end{equation}
and the same for $\min$, which again follows from~\eqref{UCT}.
\end{proof}
In order to generalize Proposition~\ref{prop:max} we need a stronger statement than the readily available~\eqref{eq:ell_lim}, and so we assume the following additional condition on the slowly varying function~$\ell$.
\begin{condition}
\label{cond:loc}
Whenever $0 < \epsilon(t) \to 0$ as $t \to \infty$, we have
\[
\log(t) \left( \frac{\ell([1+\epsilon(t)]t)}{\ell(t)} - 1 \right)
\to 0.
\]
\end{condition}
In Section~\ref{sec:condition} we provide a simple sufficient criterion under which Condition~\ref{cond:loc} holds.
It is important to realize that Condition~\ref{cond:loc} is equivalent to a seemingly stronger condition stated in the following lemma.
\begin{lemma}\label{lem:cond_equiv}
Condition~\ref{cond:loc} is equivalent to
\begin{equation}\label{eq:cond_equiv}
\log(t) \sup_{-\epsilon(t)\leq x\leq \epsilon(t)}\left| \frac{\ell((1+x)t)}{\ell(t)} - 1 \right|
\to 0
\end{equation}
for any $0<\epsilon(t)\rightarrow 0$.
\end{lemma}
\begin{proof}
Given in Appendix~\ref{appendix}.
\end{proof}
Recall $\beta_n = \log(n/\gamma) - \log\log(n)$.
\begin{lemma}
\label{lem:Mnstarz1z2}
Assuming Condition~\ref{cond:loc} we have $M_n^*/v(n)-\beta_n\convdistr G$.
\end{lemma}
\begin{proof}
Since $(\tau_n^* \wedge \tau_n) / n = 1 + \oh_p(1)$ as $n \to \infty$ by Lemma~\ref{lem:tauto1} and Proposition~\ref{prop:loc}, we can find $\epsilon_n > 0$ such that $\epsilon_n \to 0$ and ${\mathbb P}[ \tau_n^* \wedge \tau_n > n(1 - \epsilon_n)] \to 1$ as $n \to \infty$.
Hence with arbitrarily high probability we have
\begin{align*}
\left\lvert
\frac{M_n^*}{v(n)} - \frac{M_n}{n^\gamma}
\right\rvert
&=
\left\lvert
\max_{n(1-\epsilon_n)<i\leq n}
\left\{ \frac{i^\gamma \ell(i)}{n^\gamma \ell(n)} X_i \right\}
-
\max_{n(1-\epsilon_n)<i\leq n}
\left\{ \frac{i^\gamma}{n^\gamma} X_i \right\}
\right\rvert
\\
&\le
\max_{n(1-\epsilon_n)<i\leq n}
\left\{
\left\lvert
\frac{\ell(i)}{\ell(n)} - 1
\right\rvert
\frac{i^\gamma}{n^\gamma} X_i
\right\}
\le
\frac{M_n}{n^\gamma}
\sup_{-\epsilon_n<x\leq 0}
\left\lvert
\frac{\ell(n(1+x))}{\ell(n)} - 1
\right\rvert
.
\end{align*}
Lemma~\ref{lem:cond_equiv} and Lemma~\ref{lem:Mz1z2} show that $M_n^*/v(n)=M_n/n^\gamma+\oh_p(1)$ completing the proof.
\end{proof}
\begin{proposition}
Let $v \in \operatorname{RV}_\gamma$ for some $\gamma > 0$ and put $\ell(t) = t^{-\gamma} v(t)$. If $\ell$ satisfies Condition~\ref{cond:loc} then Proposition~\ref{prop:loc} and Proposition~\ref{prop:point_discrete} hold with $M^*_n, \tau^*_n, v(i),v(n)$ in place of $M_n,\tau_n,i^\gamma,n^\gamma$.
\end{proposition}
\begin{proof}One may follow the same steps as in the original proofs. In addition, for the analogue of~\eqref{eq:toextend} we use Lemma~\ref{lem:RV}, whereas the extension of Proposition~\ref{prop:point_discrete} requires showing that
\[\sup_{1\leq i\leq n(1-z/\log n)}\exp(-(x+\beta_n)v(n)/v(i))\to 0,\]
which follows from \eqref{UCT} applied to the function $v(\lfloor t\rfloor)$.
\end{proof}
\subsection{Gaps of a Poisson process}
Let $( \mathcal{N}_t )_{t \ge 0}$ be an inhomogenous Poisson process as in the beginning of this section.
As a consequence of Lemma~\ref{lem:cond_equiv}, we have that\begin{multline}
\label{eq:vloc}
0 < \delta(t) = \oh \left( 1 / \log t \right)
\text{ as $t \to \infty$ implies } \\
\lim_{t \to \infty} \log(t)
\sup_{-\delta(t) \le x \le \delta(t)}
\left\lvert \frac{v((1+x)t)}{v(t)} - 1 \right\rvert
= 0,
\end{multline}
because $(1\pm\delta(t))^\gamma-1=\oh(1/\log t)$.
Let us now provide a generalization of Lemma~\ref{lem:RbyX}.
\begin{lemma}
\label{lem:approxRbyX}
If $\ell$ satisfies Condition~\ref{cond:loc}, then
\begin{equation*}
\left|\frac{T_i}{V(i)}-1\right|\vee \left|\frac{R_i}{v(i)X_i}-1\right|=\oh(1/\log i) \quad\text{a.s.}
\end{equation*}
as $i\rightarrow\infty$.
\end{lemma}
\begin{proof}
From the monotonicity of $V$ and from~\eqref{eq:itlog}, we find that a.s.
\[T_i/V(i)-1=V(T'_i)/V(i)-1\leq V(i(1+\oh(1/\log i)))/V(i)-1,\]
which is $\oh(1/\log i)$ by Lemma~\ref{lem:RV}. A similar bound from below completes the proof of the first part.
For the second part we write
\begin{equation}\label{eq:R_RV}
R_i
= T_i - T_{i-1}
= V(T_i') - V(T_{i-1}')
= \int_{T_{i-1}'}^{T_i'} v(t) \, {\mathrm d} t,
\end{equation}
so that
\[
R_i - v(i) X_i
=
\int_{T_{i-1}'}^{T_i'} \{ v(t) - v(i) \} \, {\mathrm d} t.
\]
But then
\[
\left\lvert \frac{R_i}{v(i) X_i} - 1 \right\rvert
\le
\frac{1}{X_i}
\int_{T_{i-1}'}^{T_i'}
\left\lvert
\frac{v(t)}{v(i)} - 1
\right\rvert \,
{\mathrm d} t
\le
\sup_{T_{i-1}'/i-1 \le x \le T'_i/i-1}
\left\lvert
\frac{v(i(1+x))}{v(i)} - 1
\right\rvert.
\]
From~\eqref{eq:itlog} and \eqref{eq:vloc} we find that the last term is $\oh(1/\log i)$ a.s.\ as required.
\end{proof}
\begin{theorem}\label{thm:RV}
If the rate function $\lambda\in \operatorname{RV}_{\a-1}$, where $\a\in(0,1)$, is such that $\ell$ satisfies Condition~\ref{cond:loc}, then Theorem~\ref{thm:main} and Theorem~\ref{thm:point_proc} hold with such $\lambda_t$ and
\[b_t=\log\Lambda(t)-\log\log t-\log(1-\a).\]
\end{theorem}
\begin{proof}
In this more general setting we use $n=\lceil\Lambda(t)\rceil$ and so $\log n\sim\a\log t$.
Concerning the generalization of Lemma~\ref{lem:points_close} we only need to show that~\eqref{lim1} and~\eqref{lim2} hold when adapted according to Lemma~\ref{lem:approxRbyX}.
That is,
\begin{align*}
&\lambda_t v(i)X_i(1+\oh(1/\log t))-b_t=(v(i)X_i/v(n)-\beta_n)(1+\oh(1/\log t))+\oh(1),\\
&\a^2\log(t)(1-V(i)/t)=\log(n)(1-i/n)(1+\oh(1))+\oh(1).
\end{align*}
This hinges on the following: (i) $\lambda_t v(n)=1+\oh(1/\log t)$, (ii) $b_t=\beta_n+\oh(1)$, and (iii) $\a(1-V(i)/t)=(1-i/n)(1+\oh(1))+\oh(1/\log t)$ uniformly in $i\in I_v(t)$. Identity (i) holds, because
by \eqref{eq:vloc}
\begin{equation}\label{eq:lambdaRV}
\lambda_t \, v(\ceil{\Lambda(t)})
=
\frac{v(\ceil{\Lambda(t)})}{v(\Lambda(t))}
=
1 + \oh(1/\log t),
\qquad \text{ as }t \to \infty,
\end{equation}
where, indeed, $\abs{ \ceil{\Lambda(t)} / \Lambda(t) - 1 } < 1/\Lambda(t) = \oh(1 / \log t)$. Identity (ii) is rather obvious, whereas concerning (iii) we have
\[\a(1-V(i)/t)=\a(1-V(i)/V(\Lambda(t)))=\a(1-V(i)/V(n))(1+\oh(1))+\oh(1/\log t),\]
but $1-V(n(1+i/n-1))/V(n)=(1-i/n)(1+\oh(1))/\a$
by a slight extension of Lemma~\ref{lem:RV} upon noting that $i/n-1=\oh(1)$ uniformly in $i$ concerned.
It is left to show that Lemma~\ref{lem:bounding} still holds, and the only non-trivial step is to show that
\begin{equation}\label{eq:infty}
\lambda_t \max_{i\leq n/2} R_i-b_t\rightarrow-\infty
\end{equation}
in probability and hence a.s., which we obtain in the following.
Observe from~\eqref{eq:R_RV} that
\begin{align*}
\max_{i\leq n/2}R_i\leq \max_{i\leq n/2}X_i\sup_{T_{i-1}'\leq t\leq T_i'}v(t)\leq \max_{i\leq n/2}X_i\sup_{t\leq T_{\lceil n/2\rceil}'}v(t),
\end{align*}
where $\max_{i\leq n/2}X_i=\log n(1+\Oh_p(1))$ and concerning the latter term we have
\[\sup_{t\leq T_{\lceil n/2\rceil}'}v(t)/v(n)=\sup_{t\leq n/2(1+\oh(1))}\frac{v(t)}{v(n)}\rightarrow 2^{-\gamma}\quad \text{a.s.}\]
by \eqref{UCT}, provided that $v$ is locally bounded. This shows that $\max_{i\leq n/2}R_i/v(n)\leq 2^{-\gamma}\log n(1+\Oh_p(1))$ and hence~\eqref{eq:infty} holds in view of~\eqref{eq:lambdaRV}.
In general, however, we only have that $v$ is bounded on $[a,b]$ for some $a$ and all~$b$. With arbitrarily high probability we may choose an index $j$ such that $T_j'>a$, and then the above steps can be repeated for $\max_{j<i\leq n/2}R_i$, whereas obviously $T_j/v(n)\rightarrow 0$ a.s.
\end{proof}
\subsection{Comments on the assumed condition}
\label{sec:condition}
Let us note that virtually all standard examples of slowly varying functions, e.g.\ $\log^u t,u\in\mathbb R$ and $\log\log t$, satisfy Condition~\ref{cond:loc}. This can be easily checked using the following result.
\begin{lemma}
\label{lem:condition}
Condition~\eqref{cond:loc} holds true if $\ell\in \operatorname{RV}_0$ is eventually differentiable and
\begin{equation}
\label{eq:cond_sufficient}
\frac{t \, \ell'(t)}{\ell(t)}
=
\Oh(1/\log t),
\qquad \text{as } t \to \infty.
\end{equation}
\end{lemma}
\begin{proof}
Using the mean value theorem we have
\[ |\ell([1+\epsilon(t)]t)-\ell(t)|\leq t\epsilon(t)\sup_{t \le s \le [1+\epsilon(t)]t}
|\ell'(s)|.\]
Moreover,
\[
\sup_{t \le s \le [1+\epsilon(t)]t} |\ell'(s)|
\leq
\sup_{t \le s \le [1+\epsilon(t)]t}
\left|\frac{s \, \ell'(s)\log s}{\ell(s)}\right|
\sup_{t \le s \le [1+\epsilon(t)]t}
\left|\frac{\ell(s)}{s\log s}\right|,
\]
where the first term on the right hand side is $\Oh(1)$ according to~\eqref{eq:cond_sufficient}.
Hence Condition~\eqref{cond:loc} holds if
\[\sup_{t \le s \le [1+\epsilon(t)]t}\frac{t\log(t)\ell(s)}{s\log(s)\ell(t)}\]
is bounded for large $t$, but this term tends to~1 by \eqref{UCT} applied to the regularly varying function $(t\log(t))^{-1}\ell(t)$.
\end{proof}
Concerning Theorem~\ref{thm:RV} it is more useful to express the sufficient condition of Lemma~\ref{lem:condition} using the slowly varying function associated with the rate function $\lambda_t$ instead of that associated with~$v(t)$, which is the content of the next result.
\begin{proposition}\label{prop:cond}
Let $\lambda_t=t^{\a-1}\ell_\lambda(t)$ with $\a\in(0,1)$ and $\ell_\lambda\in\operatorname{RV}_0$.
If $\ell_\lambda$ is eventually continuously differentiable and if
\[
\frac{t \, \ell_\lambda'(t)}{\ell_\lambda(t)}=\Oh(1/\log t),
\qquad \text{as } t \to \infty,
\]
then Condition~\ref{cond:loc} is satisfied and the result of Theorem~\ref{thm:RV} holds true.
\end{proposition}
\begin{proof}
First, we show that~\eqref{eq:cond_sufficient} is equivalent to
\begin{equation}
\label{eq:cond2}
\frac{\Lambda(t) \, \lambda'_t}{\lambda_t^2}
=
-\gamma+\Oh(1/\log t).
\end{equation}
Since $v(t)=t^\gamma \ell(t)$ and $v(t)=1/\lambda_{V(t)}$ we find that
\[
\frac{t \, \ell'(t)}{\ell(t)}
=
-\gamma-t \, \lambda'_{V(t)}/\lambda^2_{V(t)}.
\]
Plugging in $t=\Lambda(t)$ and noting that $\log \Lambda(t)\sim \a\log t$ we confirm the equivalence.
Thus it is sufficient to establish that
\begin{align*}
\frac{\lambda'_t \, t}{\lambda_t}
&=
\a-1+\Oh(1/\log t),
&
\frac{\Lambda(t)}{\lambda_t t}
&=
1/\a+\Oh(1/\log t).
\end{align*}
The left statement is a result of a simple calculation, and so we concentrate on the right statement.
Using integration by parts we find
\[
\Lambda(t)
=
\int_c^t x^{\a-1}\ell_\lambda(x) \, {\mathrm d} x\
=\
\frac{1}{\a} t^\a \, \ell_\lambda(t)
+ \Oh(1)-\frac{1}{\a} \int_c^t x^{\a} \, \ell'_\lambda(x) \, {\mathrm d} x
\]
for all $t>c$ and some level~$c$ (to be fixed high enough). Hence it is left to show that
\begin{equation}
\label{eq:withKaramata}
\frac{\int_c^t x^{\a} \, \ell'_\lambda(x) \, {\mathrm d} x}{t^\a \, \ell_\lambda(t)}
\log t
= \Oh(1).
\end{equation}
From our assumption we see that $|\ell_\lambda'(x)|\leq C\ell_\lambda(x)/(x\log x)$ for large enough~$x$. Finally, by Karamata's theorem~\cite[Prop.\ 1.5.8]{bingham1989regular} we have
\[
\frac%
{\int_c^t x^{\a} \, C \, \ell_\lambda(x)/(x\log x) \, {\mathrm d} x}%
{t^\a\,\ell_\lambda(t)/\log t}
\to C\a,
\]
because $\ell_\lambda(t)/\log t\in RV_0$, and so~\eqref{eq:withKaramata} follows.
\end{proof}
As mentioned above, virtually all standard examples of slowly varying functions satisfy the assumption of Proposition~\ref{prop:cond}. In particular, so do $\ell_\lambda(t)=a\log^u t$ and $\ell_\lambda(t)=a\log\log t$ for $a>0,u\in\mathbb R$. Hence $\lambda_t=a t^{\a-1}\log^u t$ and $\lambda_t=a t^{\a-1}\log\log t$ are examples of rate functions to which the asymptotic results in this work apply. For a simple example that does not satisfy the assumption of Proposition~\ref{prop:cond}, consider $\ell_\lambda(t)=\e^{(\log\log t)^2}=(\log t)^{\log\log t}$. Indeed, this is a slowly varying function for which $\log(t) \, t \, \ell_\lambda'(t)/\ell_\lambda(t)=2\log\log t$ is unbounded.
|
2,877,628,090,075 | arxiv | \section{Introduction}
The discovery of a SM-like Higgs boson with a mass of about $126~{\rm
GeV}$~\cite{Aad:2012tfa,Chatrchyan:2012ufa} represents a fundamental
step towards a better understanding of the origin of Electroweak
Symmetry Breaking (EWSB). Measuring its couplings with higher
precision will be one of the priorities in the 14~TeV run of the LHC,
and is one of the main motivations for building a future lepton
collider. The phenomenological description of EWSB within the SM
framework provides a benchmark against which any deviations in the
Higgs boson couplings should be compared, as such deviations could
contain the key to a more fundamental understanding of this
phenomenon.
A currently open question is whether this particle is elementary
(\textit{i.e.}~pointlike), down to distance scales much shorter than
the EW scale, or if, on the contrary, it is a composite bound state of
more fundamental degrees of freedom, whose physics should be revealed
at energies not far above the weak scale. In either case the
discovery of this scalar particle is truly remarkable. If it turns
out to be elementary it would be the first and only known example of
this kind in nature. Its existence at energies low compared to
\textit{e.g.}~the Planck scale could indicate that the universe as we
know it results from a rather perplexing fine-tuning, or perhaps more
plausibly that there is a symmetry at work as exemplified by
supersymmetric scenarios. If it turns out that the Higgs boson is a
composite state arising from some underlying strong dynamics, we would
be in a situation that also presents new characteristics compared to
other known composite scalars. For instance, unlike the pions of QCD,
the dynamics of the Higgs boson must lead to EWSB by generating a
non-vanishing vacuum expectation value (vev) for the composite scalar.
The fact that the LHC has not observed any major deviation from the SM
in its 7-8 TeV run indicates that any new physics should be roughly
above 1~TeV (although one can think of specific examples that are less
constrained, and also examples that are significantly more
constrained). In the context of Higgs compositeness, this means that
there must exist a scalar resonance much lighter than the other strong
resonances. It is then natural to interpret the Higgs as a
pseudo-Nambu Goldstone boson (pNGB) arising from the spontaneous
breaking of an approximate global symmetry of the new strong
sector~\cite{GK}. This idea has received considerable attention
lately~\cite{Contino:2003ve}. A question of special importance
centers on the type of deviations in the Higgs properties that would
be expected in such scenarios. This has been studied to some extent
within specific realizations of a Higgs as a pNGB, and also in the
context of an effective low-energy parametrization such as the
SILH~\cite{Giudice:2007fh} and similar
approaches~\cite{Gillioz:2012se,Corbett:2012dm,Carmi:2012in,Jenkins:2013zja}.
We will focus here on the minimal case~\footnote{The terminology
``Minimal Composite Higgs Model (MCHM)" was actually introduced in a
slightly different context in~\cite{Dobrescu:1999gv}. Our study is
limited to more recent models based on the pNGB idea which have also
been named MCHM~\cite{Agashe:2004rs}. Since we consider a variety of
fermionic realizations, here the ``minimality" refers specifically to
the (common) bosonic sector.} based on the $SO(5) \to SO(4)$ symmetry
breaking pattern~\cite{Agashe:2004rs}, which leads to exactly four
Nambu-Goldstone bosons and contains a custodial symmetry that ensures
that the corrections to certain electroweak observables are
sufficiently suppressed. Although the embedding of the SM gauge
sector is fixed by the above assumption, there is still a considerable
arbitrariness in how the SM fermionic sector is embedded into the
framework. This depends, in particular, on which $SO(5)$
representations for the fermionic resonances are generated by the
strong dynamics and would therefore be sensitive to further details of
the specific UV realization of the idea. Our aim is to study in
detail the implications for the properties of the Higgs boson. In
particular, we will show that if one were to measure a robust
deviation from the SM in the rates $h \to \gamma\gamma$, $h \to ZZ$
and $h \to Z\gamma$ and to a lesser extent in $h \to \tau\tau$, one
could gather indirect information regarding the quantum numbers of the
fermionic resonances. One also expects a generic reduction of the
Higgs production cross section (in particular through gluon fusion),
as well as a suppression of all Yukawa couplings w.r.t.~the SM.
There have been a number of studies on the phenomenology of a pNGB
Higgs as well as partial compositeness. Since the pioneer work of
Ref.~\cite{Falkowski:2007hz} studying Higgs production by gluon
fusion, many works have considered the deviations of the Higgs
couplings in this setup, exploring the dependence on the degree of
compositeness of the fermions, the scale of compositeness and their
relation with the spectrum of resonances, among other important
variables~\cite{Low:2009di,Montull:2013mla,Gillioz:2013pba}. However
most of them have considered generic regions of the parameter space,
that could be unphysical, in the sense that either there is no EWSB,
or the decay constant of the Higgs and its vev are not separated
enough to guarantee compatibility with EW precision measurements, or
the spectrum of the lightest level of states does not reproduce the SM
one, to cite a few examples. To ensure that these conditions are
satisfied and therefore make a realistic study of the Higgs
phenomenology, in general requires a full study of the Higgs potential
that can only be performed in a well defined model, with the risk of
loosing some generality. One of the purposes of this work is to make
a step in that direction. We consider a family of well defined
models, with the same pattern of symmetry breaking for the pNGB Higgs
but allowing different representations for the fields of the theory.
This still represents considerable freedom and for this reason we make
some restrictive assumptions that ensure calculability of the Higgs
potential within the framework of a two site model. We will also
assume that at high energies the symmetry behind the pNGB is linearly
realized for the massive resonances, and for that reason we will
include massive resonances in complete SO(5) representations. It is
possible to relax some of these assumptions, for example by
considering models with more sites, or even to allow for logarithmic
divergences of the potential.\footnote{L.D. thanks Gilad Perez for
discussions on this topic.} Nevertheless, we hope that our setup can
still capture generic features of minimal pNGB
models.\footnote{Recently, another class of pNGB models based on
four-fermion interactions has been discussed in~\cite{Cheng:2013qwa}.
Although they rely on a different breaking pattern, in principle they
could be extended to ${\rm SO}(5) / {\rm SO}(4)$, following the
analysis of~\cite{Barnard:2013zea}.} We will show that it can give
information on the size of the corrections that one can expect on the
Higgs phenomenology as well as on the wealth and direction of
corrections that follow by allowing for different representations of
the fields.
This paper is organized as follows. In Sec.~\ref{pNGB} we review the
basic aspects of the effective two-site description of the composite
Higgs scenario. In Sec.~\ref{sec:models} we present the details of
the specific models we study in this work, which differ in the
realization of the fermionic sector. In Sec.~\ref{sec:corrections} we
describe the low-energy consequences of the pNGB nature of the Higgs
and the presence of the composite resonances, while in
Sec.~\ref{sec_div_VH} we discuss the properties of the Higgs
potential. Sec.~\ref{sec:pheno} contains our numerical results, while
Sec.~\ref{sec:tuning} contains some remarks on the tuning of the
phenomenologically viable models. We summarize and conclude in
Sec.~\ref{sec:conclusions}. We also include four appendices:
App.~\ref{app:generators} summarizes several useful group theoretical
results, App.~\ref{app:masses} contains the mass matrices of the gauge
sector of the models, App.~\ref{sec:correlators} contains all the
correlators for the low-energy limit of the various models, and
finally App.~\ref{app:loops} summarizes how we compute the 1-loop
processes $h \to \gamma\gamma$, $h \to ZZ$ and $h \to Z\gamma$.
\section{A minimal pNGB Higgs}
\label{pNGB}
We are interested in the minimal model that can deliver the Higgs as a
pNGB resonance arising from the spontaneous breaking of a global
symmetry in a strongly coupled sector (SCFT). We will assume that the
SCFT has an exact global symmetry that is spontaneously broken to a
subgroup by effects of the strong dynamics, with the Higgs being the
associated Nambu-Goldstone boson (NGB). The interactions of the
fields in the SCFT with the SM fields explicitly break the global
symmetry, leading to a Higgs potential at loop level. In this case
the degeneracy of the vacuum is uplifted and the Higgs becomes a
pNGB, leading to a natural separation between the scale
of the resonances and the Higgs mass. Usually the gauge contributions
to the 1-loop Coleman-Weinberg potential are aligned with the EW gauge
group. However the fermion contributions, that are expected to be
large because of the large top mass, can induce a missalignement of
the vacuum triggering EW symmetry breaking dynamically.
Ref.~\cite{Agashe:2004rs} has shown that the minimal group containing
the SM EW gauge symmetry and an unbroken custodial symmetry that can
lead to a pNGB Higgs is SO(5). This group is spontaneously broken to
${\rm SO(4)} \simeq {\rm SU(2)}_L \times {\rm SU(2)}_R$, with the
Higgs being the NGB in the coset SO(5)/SO(4) that transforms as a
${\bf 4}$ of SO(4). Besides the Higgs, the SCFT is assumed to lead to
vector resonances in the adjoint representation of the global group
(these are created by the Noether currents of this symmetry). In
addition, one assumes the existence of fermion resonances, some of
which can mix with the SM degrees of freedom. We will consider that
all the massive composite resonances are in complete irreducible
representations of SO(5), realizing the symmetry in a linear way. All
the composite states are taken to interact with typical couplings
$g_{\rho}\gg g_{SM}$. The SM gauge and fermion fields can be
considered as external sources probing the SCFT, {\it i.e.}:
elementary fields. The SM particles do not interact with the Higgs at
leading order, but these interactions are mediated by the resonances
of the SCFT that mix with the elementary fields.
The gauge fields of the SM weakly gauge a subgroup of the SCFT global
symmetry. The conserved currents of the SCFT associated to this
subgroup couple linearly with the SM gauge fields, explicitly breaking
the global symmetry. The masses of the EW vector bosons arises from
mixing between the vector resonances created by the SCFT currents and
the SM gauge fields, as well as from the Higgs interactions.
We are also interested in partial compositeness of the SM fermions,
that can be realized if the elementary fermions couple linearly with
operators of the SCFT: ${\cal L}\supset \lambda \bar\psi {\cal
O}_\psi$. The low energy scaling of the coupling $\lambda$ is
controlled by the dimension of the corresponding SCFT operator $D =
\text{dim}[{\cal O}_\psi]$~\cite{Contino:2004vy,Agashe:2004rs}. For
$D>5/2$ the coupling is irrelevant leading to small mixing between the
elementary fermions and the fermionic resonances created by the SCFT
operator. For $D<5/2$ the coupling is relevant leading to large
mixing between the elementary fermion and the resonances, and thus to
a large Yukawa coupling. The former case leads to light states that
are mainly elementary, whereas the latter one can lead to large
fermion masses, as for the top quark, which is associated with a large
degree of compositeness.
The proper normalization of hypercharge for fermions requires the
introduction of an extra ${\rm U(1)}_X$ symmetry in the composite
sector, with the identification $Y=T^{3}_{R}+X$, where $T^{3}_{R}$ is
the diagonal generator of SU(2)$_R$. The SU(2)$_R$ charge of the
composite operators ${\cal O}_\psi$ is not fixed, allowing for
different representations ${\bf r}_{\cal O}$ under SO(5). However,
the stringent constraints on the corrections to the $Zb\bar b$
couplings arising from LEP and SLC require a non-trivial protection of
the $Zb_L\bar{b}_L$ coupling. Ref.~\cite{Agashe:2006at} has shown
that there is a subgroup of the custodial symmetry ${\rm O}(3)$ that
can ensure that the corrections to this coupling are indeed
sufficiently suppressed. This symmetry requires that the
representation ${\bf r}_{{\cal O}_q}$, where ${\cal O}_q$ is coupled
to the doublet of the third generation $q_L$, decompose under SO(4)
as: ${\bf r}_{{\cal O}_q}\simeq{\bf 4}\oplus \dots$. The smallest
representations satisfying this condition are: ${\bf r}={\bf 5},{\bf
10},{\bf 14}$. On the other hand, invariance of the SCFT under ${\rm
SO(5)} \times {\rm U(1)}_X$ restricts the representations of the
operators ${\cal O}_u$ and ${\cal O}_d$, coupled with $t_R$ and $b_R$
respectively. In this work we will consider several representations
${\bf r}_{\cal O}$ subject to the above restrictions, and we will
study their impact in the Higgs phenomenology at the LHC.
The scenario described in the previous paragraphs can be realized by
considering a theory in a slice of a warped five dimensional
space-time, with the metric being AdS$_5$ near the UV. The elementary
fields and resonances can be identified with degrees of freedom on the
UV boundary and Kaluza-Klein states, respectively. However it is
possible to capture most of the essential ingredients by considering a
theory with the first level of resonances only, as in the
elementary/composite description of Ref.~\cite{Contino:2006nn}. At
low energies one considers an effective description with elementary
fields, one level of resonances and linear mixing between them. This
description has more freedom than the full 5D theory, allowing for new
terms~\cite{DeCurtis:2011yx} as well as a lack of correlation between
some parameters, such as the masses of the different resonances. It
also has a cut-off of order a few TeV. However it is able to
parametrize a family of realistic theories with a pNGB Higgs and it is
still predictive enough to explore, at the LHC, the consequences of
the symmetries protecting the Higgs potential. In the next subsections
we will summarize a realization of this effective theory.
\subsection{Effective description: 2-site model}
\label{sec:2site}
We consider the effective description of the Higgs as a pNGB arising
from a strongly coupled sector, as introduced in
Ref.~\cite{DeCurtis:2011yx} (see also~\cite{Panico:2011pw}). The
simplest model has two sites: one called site-0 that describes
\textit{elementary} fields, and another called site-1 describing the
first level of resonances arising from the strongly coupled sector
(the \textit{composite} sector). Site-0 contains a set of gauge and
fermion fields with the same symmetry group and fermionic
representations as the SM. We will call $G_0$ the gauge symmetry of
this site: $G_0 = {\rm SU(2)}_L\times {\rm U(1)}_Y$.\footnote{There is
also a color SU(3)$_C$ on each site, but we omit mentioning these
factors in the following.} Note that there are no elementary
\textit{scalar} fields. On site-1 we consider a \textit{gauge}
symmetry $G_1 = {\rm SO(5)} \times {\rm U(1)}_X$, which allows to
describe effectively the lowest lying spin-1 resonances of the strong
dynamics. Site-1 also contains several multiplets of fermion fields
in various representations of $G_1$, which will be described in detail
later. The two sites are connected by a $\sigma$-model field
$\Omega$,\footnote{Strictly speaking, there are two link fields,
$\Omega$ and $\Omega_X$, for the ${\rm SO(5)}$ and ${\rm U(1)}_X$
factors. These will be described in detail below.} transforming as
$\Omega \to g_0 \Omega g_1^\dagger$, with $g_{0,1} \in G_{0,1}$. In
Fig.~\ref{fig-moose} we show the Moose diagram corresponding to this
theory. We use lower case letters for fields on site-0 and upper case
letters for fields on site-1.
It turns out to be very convenient to extend $G_0$ to a spurious $G'_0
= {\rm SO(5)} \times {\rm U(1)}_x$. This is achieved by introducing
non-dynamical gauge and fermion fields on site-0 that, together with
the dynamical fields that fill representation of $G_0 \subset G'_0$,
complete full representations of $G'_0$. When one considers
\textit{all} the fields on site-0 as non-dynamical, they act as
sources for an exact global $G'_0$ symmetry, which is to be thought as
a global symmetry of the strongly coupled sector. We assume that the
strong dynamics giving rise to the composite resonances spontaneously
breaks the ${\rm SO(5)}$ global factor down to SO(4), thus delivering
a set of NGB's in the coset SO(5)/SO(4). These will be identified as
the composite Higgs, and are described by a field $\Phi_1$ as shown in
Fig.~\ref{fig-moose}. The presence of the dynamical fields on site-0
explicitly breaks $G'_0$ (e.g.~by their kinetic terms, which are not
present for the spurious fields on site-0), and therefore generates a
potential for the Higgs, which becomes a pNGB. This potential is often
calculable and is one of the attractive theoretical features of these
scenarios. The observation of a Higgs boson at the LHC and
the measurement of its mass and couplings then imposes non-trivial
constraints on the parameters of the model.
As we will see in detail below, the presence of $\Omega$ allows to
realize partial compositeness of the fermions through bilinear terms
involving a fermion $\psi$ at site-0 and a fermion $\Psi$ at site-1.
It also leads to non-zero masses for the \textit{axial} combination of
the gauge fields in sites-0 and 1, and contains the would-be NGB's
that are eaten in this process.
\begin{figure}[t]
\centering
\includegraphics[width=0.6\textwidth]{Figures/Moose.pdf}
\caption{Moose diagram of the two site theory describing the model.
$G_0$ is the SM gauge symmetry and $G_1 = {\rm SO(5)} \times {\rm
U(1)}_X$. The (spontaneous) breaking of $G_1$ down to ${\cal H}_1 =
{\rm SO(4)} \times {\rm U(1)}_X$, is parametrized by a field $\Phi_1$,
that transforms under $G_1$ as a $5$ of SO(5) with $Q_X = 0$, and
whose vev is $\langle \Phi_1 \rangle = \{ 0,0,0,0,1 \}^T$. The link
field transforms like $\Omega \to g_0 \Omega g_1^\dagger$, with
$g_{0,1} \in G_{0,1}$.}
\label{fig-moose}
\end{figure}
\subsection{Bosonic Sector}
\label{BosonicSector}
Let us consider first the Lagrangian describing the fields that
parametrize the ${\rm SO(5)} \to {\rm SO(4)}$ breaking.\footnote{Since
the NGB fields are neutral under $U(1)_X$, we omit this factor for
simplicity in this discussion, but it should be understood.} We denote
the unbroken generators of SO(5) [{\it i.e.}~of ${\cal H}_1 \equiv
SO(4) \simeq {\rm SU(2)}_L \times {\rm SU(2)}_R$] by $T^a$, while the
broken ones are denoted by $T^{\hat{a}}$. For reference, we give
their explicit expressions in a convenient basis in
Appendix~\ref{app:generators}. The NGB's are parametrized by
\begin{equation}
U(\Pi)=e^{i\Pi/f_1}\ , \qquad \Pi=\Pi^{\hat a}T^{\hat a} \ ,
\end{equation}
where $f_1$ is the corresponding decay constant. The $G_1 = {\rm
SO(5)}$ symmetry is non-linearly realized, that is, under a $g_1 \in
G_1$ we have $U \to g_1 \, U \, h_1(g_1;\Pi)^\dagger$, where
$h_1(g_1;\Pi) \in {\cal H}_1$ is an element of the unbroken group,
that depends on the SO(5) transformation $g_1$ and the NGB fields
$\Pi$. The leading order Lagrangian of these NGB's is
\begin{equation}
{\mathcal L}_{\rm NGB}=\frac{f_1^2}{2} \, {\cal D}^{\hat a}_\mu{\cal D}^{\mu\hat a}~,
\label{LNGB}
\end{equation}
with ${\cal D}^{\hat a}_\mu$ implicitly defined by $U^\dagger D_\mu
U=i{\cal E}^{a}_\mu T^a+{\cal D}^{\hat a}_\mu T^{\hat a}$. The
covariant derivative contains the composite spin-1 resonances,
$A_\mu$, and leads to the interactions between these and the NGB's.
We defer the description of the interactions between the NGB's and the
fermions $\Psi$ on site-1 to the next section.
One can obtain a simpler and more explicit description of the above
sector by defining $\Phi_1=U\phi_1$, with $\phi_1^B = \delta^{B \, 5}$
($B = 1, \ldots 5$). Under a $g_1 \in G_1$ one simply has $\Phi_1 \to
g_1 \Phi_1$, and it can be checked that the above Lagrangian can be
written as
\begin{equation}
{\mathcal L}_{\rm NGB} = \frac{f_1^2}{2}|D_\mu\Phi_1|^2~.
\label{LNGBSimple}
\end{equation}
In this form, the breaking of SO(5) down to SO(4) is simply
parametrized by $\langle \Phi_1 \rangle = \{ 0,0,0,0,1 \}^T$.
As for the Lagrangian describing the $\sigma$-model connecting the two
sites, at leading order one has:
\begin{equation}
{\mathcal L}_{\Omega} = \frac{f_\Omega^2}{4}{\rm tr}|D_\mu\Omega|^2 + \frac{f_{\Omega_X}^2}{4} |D_\mu\Omega_X|^2,
\label{LOmega}
\end{equation}
with
\begin{eqnarray}
\Omega = e^{\sqrt{2} \, i \Pi_{\Omega} / f_\Omega}~,
\hspace{2cm}
\Omega_X = e^{\sqrt{2} \, i \Pi_{\Omega_X} / f_{\Omega_X}}~,
\end{eqnarray}
where $\Pi_{\Omega} = \Pi_{\Omega}^{b} \, T^b_{0-1}$ and $T^b_{0-1}$
denote the generators of ${\rm SO(5)}_0 \times {\rm SO(5)}_1 / {\rm
SO(5)}_{0+1}$, with ${\rm SO(5)}_{0+1}$ denoting the diagonal (vector)
subgroup of ${\rm SO(5)}_0 \times {\rm SO(5)}_1$. We have also
included an additional link field $\Omega_X$ (with its decay constant
$f_{\Omega_X}$ and charges $Q_x = Q_X = 1$) for the ${\rm U(1)}_x
\times {\rm U(1)}_X$ factors. The covariant derivatives above are
given by
\begin{eqnarray}
D_\mu\Omega = \partial_\mu \Omega - i \tilde{a}_\mu\Omega + i\Omega \tilde{A}_\mu \ ,
\hspace{1cm}
D_\mu\Omega_X = \partial_\mu \Omega_X - i \tilde{x}_\mu\Omega_X + i\Omega_X \tilde{X}_\mu \ ,
\end{eqnarray}
where $\{\tilde{a}_\mu, \tilde{x}_\mu\}$ and $\{\tilde{A}_\mu,
\tilde{X}_\mu \}$ are the gauge fields of site-0 and site-1,
respectively (the tildes denote non-canonical normalization).
Besides the terms above the bosonic Lagrangian includes the kinetic
terms for the gauge fields of $G_0$ and $G_1$:
\begin{equation}
{\cal L}_{\rm gauge} = - \frac{1}{4 g^2_0} \tilde{w}^{j}_{L \, \mu \nu} \tilde{w}^{j \, \mu \nu}_L - \frac{1}{4 g^{\prime 2}_0} \tilde{b}_{\mu \nu} \tilde{b}^{\mu \nu} - \frac{1}{4 g^2_\rho} \tilde{A}^{B}_{\mu \nu} \tilde{A}^{B \, \mu \nu} - \frac{1}{4 g^{2}_X} \tilde{X}_{\mu \nu} \tilde{X}^{\mu \nu}~,
\label{Lgauge}
\end{equation}
where $j = 1,2,3$, $B = 1, \ldots, 10$, and $\tilde{w}^{j}_{L \, \mu
\nu}$, $\tilde{b}_{\mu \nu}$ and $\{\tilde{A}^{B}_{\mu \nu},
\tilde{X}_{\mu\nu} \}$ are the field strengths of ${\rm SU(2)}_L$,
${\rm U(1)}_Y$ and ${\rm SO(5)} \times {\rm U(1)}_X$, respectively.
The embedding of ${\rm U(1)}_Y \subset {\rm SU(2)}_R \times {\rm
U(1)}_x$ on site-0 is obtained by the identifications $\tilde{w}^3_{R
\, \mu} = \tilde{x}_\mu = \tilde{b}_\mu$
so that $b_\mu$ couples to $Y = T^3_R + Q_X$ with coupling $g'_0 = g_0
g_x / \sqrt{g^2_0 + g^2_x}$.~\footnote{Here the fields are normalized
according to ${\cal L}_{\rm gauge} \supset -1 / (4g_0^2) \,
\tilde{w}^{3}_{R \, \mu \nu} \tilde{w}^{3 \, \mu \nu}_R - 1 / (4g_x^2)
\, \tilde{x}_{\mu \nu} \tilde{x}^{\mu \nu}$, while $\tilde{b}_\mu$ is
normalized as in Eq.~(\ref{Lgauge}).} The relation between the
couplings $g_0$ and $g^\prime_0$ and their SM counterparts will be
specified below, and similarly for the relation between the elementary
gauge fields $\tilde{w}^\mu_L$ and $\tilde{b}^\mu$, and the SM gauge
fields $W^\mu_L$ and $B^\mu$. We assume that the couplings
characterizing the interactions of the composite spin-1 fields,
$g_\rho$ and $g_X$, are large but still perturbative.
The physical field content of the theory becomes evident in unitary
gauge, where the would-be NGB's eaten by the composite $A_\mu$'s are
set to zero. This is achieved by a gauge transformation $g_1=\Omega$
(and using $\Omega_X$ for the $U(1)_X$ factor). The physical NGB's
are then fully parametrized by
\begin{equation}
\Phi \equiv \Omega \, \Phi_1 = \frac{1}{h}\sin \frac{h}{f_h} \, (h_1,h_2,h_3,h_4,h \cot\frac{h}{f_h})^T \ ,
\end{equation}
with
\begin{equation}
\frac{1}{f^2_h}=\frac{1}{f_\Omega^2}+\frac{1}{f_1^2} \ , \qquad h^2=\sum_a h^{\hat a}h^{\hat a}~.
\label{eq:fh}
\end{equation}
The vacuum is characterized by the variable $\epsilon=\sin(v/f_h)$,
with $v=\langle h\rangle$ and $\langle\Phi\rangle^T = (0,0,0,
\epsilon,\sqrt{1-\epsilon^2})$.
The link field Lagrangian in unitary gauge reads
\begin{equation}
{\cal L}_{\Omega} = \frac{1}{4} \, f^2_\Omega \left( \tilde{a}^B_\mu - \tilde{A}^B_\mu \right)^2 + \frac{1}{4} \, f^2_{\Omega_X} \left( \tilde{x}_\mu - \tilde{X}_\mu \right)^2~,
\end{equation}
where we allowed for all possible external source fields on site-0.
Turning on only those that are dynamical as in Eq.~(\ref{Lgauge}), we
have
\begin{eqnarray}
{\cal L}_{\Omega} &=& \frac{1}{2} \, m^2_\rho \sum^3_{i = 1} \left( t_\theta w^{i}_{L \, \mu} - A^i_{L \, \mu} \right)^2 + \frac{1}{2} \, m^2_\rho \left( t_\theta w^{3}_{R \, \mu} - A^3_{R \, \mu} \right)^2 + \frac{1}{2} \, m^2_X \left[ (g_x / g_X) \, x_\mu - X_\mu \right]^2
\nonumber \\
& & \mbox{} + \frac{1}{2} \, m^2_\rho \sum^2_{k = 1} A^k_{R \, \mu} A^{k \, \mu}_{R} + \frac{1}{2} \, m^2_\rho \sum^4_{a = 1} A^{\hat{a}}_{\mu} A^{\hat{a} \, \mu}~,
\end{eqnarray}
where we denoted by $A^i_{L \, \mu}$ and $A^i_{R \, \mu}$ the
composite spin-1 fields associated with the ${\rm SU(2)}_L$ and ${\rm
SU(2)}_R$ factors in ${\rm SO(5)}$, respectively, defined
\begin{eqnarray}
t_\theta ~=~ \frac{g_0}{g_\rho}~,
\end{eqnarray}
and
\begin{eqnarray}
m^2_\rho ~=~ \frac{1}{2} \, g^2_\rho f^2_\Omega~,
\hspace{1.5cm}
m^2_X ~=~ \frac{1}{2} \, g^2_X f^2_{\Omega_X}~,
\label{mrhomX}
\end{eqnarray}
and rescaled the fields according to $\tilde{w}_{L, R} = g_0 w_{L,
R}$, $\tilde{A}_{L, R} = g_\rho A_{L, R}$, $\tilde{x} = g_x x$ and
$\tilde{X} = g_X X$ for canonical normalization. Recall that
$\tilde{w}^{3}_{R \, \mu}$ and $\tilde{x}_\mu$ are written in terms of
$\tilde{b}_\mu$ as given after Eq.~(\ref{Lgauge}). By going to the
mass eigenbasis, we can then identify (in the limit that $\langle h
\rangle = 0$), the following massless fields:
\begin{equation}
W^{i}_{L \, \mu} = c_\theta w^{i}_{L \, \mu} + s_\theta A^{i}_{L \, \mu}~,
\hspace{1cm}
\textrm{for}~~i = 1,2,3~,
\label{Wboson}
\end{equation}
and
\begin{equation}
B_{\mu} = \frac{1}{\sqrt{1 + t^2_{\theta'_\rho} + t^2_{\theta'_X}}} \left[ b_{\mu} + t_{\theta'_\rho} A^{3}_{R \, \mu} + t_{\theta'_X} X_{\mu} \right]~,
\label{Bboson}
\end{equation}
where $t_{\theta'_\rho} = g'_0/g_\rho$ and $t_{\theta'_X} =g'_0 / g_X$.
These are then identified with the SM gauge fields, and acquire masses
when $\langle h \rangle = v$. Indeed, one finds that
\begin{equation}
\label{eq-vSM}
m_Z \approx \frac{1}{2} \sqrt{g^2 + g^{\prime
2}} \, \epsilon \, f_h~,
\hspace{1cm}
\textrm{hence}
\hspace{1cm}
v_{SM} = 246~{\rm GeV} \simeq \epsilon \, f_h~.
\end{equation}
One can also identify the SM gauge couplings:
\begin{equation}
g = c_\theta g_0 = \left( \frac{1}{g^{2}_0} + \frac{1}{g^2_\rho} \right)^{-1/2}~,
\hspace{1cm}
g' = \frac{g'_0}{\sqrt{1 + t^2_{\theta'_\rho} + t^2_{\theta'_X}}}
= \left( \frac{1}{g^{\prime 2}_0} + \frac{1}{g^2_\rho} + \frac{1}{g^2_X} \right)^{-1/2}~.
\label{SMggp}
\end{equation}
We note here, for later use, that in the case that $g_X = g'_0 g_\rho
/ \sqrt{g^2_0 - g^{\prime 2}_0}$ one has that $t_\theta = g_0 / g_\rho
= g_x / g_X$, \textit{i.e.} the ratios of elementary to composite
couplings in the two sites coincide for the SO(5) and ${\rm U(1)}_X$
factors. In this case the usual Weinberg angle coincides with the
naive elementary Weinberg angle: $t_W = g' / g = g'_0 / g_0$.
The combinations orthogonal to Eqs.~(\ref{Wboson}) and (\ref{Bboson})
are massive even in the absence of the Higgs vev. For the ${\rm
SU(2)}_L \times {\rm SO(5)}$ factor one finds states
$\tilde{\rho}^i_{L \, \mu} = c_\theta A^{i}_{L \, \mu} - s_\theta
w^{i}_{L \, \mu}$ (i = 1,2,3) with mass $m_{\tilde{\rho}}^2 = (1 +
t^2_\theta) m^2_\rho$; the other fields in ${\rm SO(5)}$, that do not
mix with elementary fields, correspond to two (charged) fields in
${\rm SU(2)}_R \subset {\rm SO(5)}$ with mass $m_\rho$, and four
fields associated with the broken ${\rm SO(5)}/{\rm SO(4)}$
generators, with squared masses
\begin{equation}
m^2_a = \frac{1}{2} \, g^2_\rho (f^2_\Omega + f_1^2)~,
\label{ma}
\end{equation}
the latter term arising from ${\cal L}_{\rm NGB}$ in
Eq.~(\ref{LNGBSimple}). There are also two massive neutral resonances
arising from the ``hypercharge" gauge sector. Assuming that $m_X =
m_\rho$, the expressions for the latter simplify considerably and one
finds that the state $\propto t_{\theta'_X} A^{3}_{R \, \mu} -
t_{\theta'_\rho} X_\mu$ has mass $m_\rho$ while the state $\propto
t_{\theta'_\rho} A^{3}_{R \, \mu} + t_{\theta'_X} X_\mu -
(t^2_{\theta'_\rho} + t^2_{\theta'_X}) \, b_{\mu}$ has mass squared
$[1 + t^2_{\theta'_\rho} + t^2_{\theta'_X} ] \, m^2_\rho$. All of the
above states receive small corrections when $\langle h \rangle$ is
turned on. For completeness, we give the full mass matrices in
App.~\ref{app:masses}.
\subsection{Fermionic Sector}
On site-0 we consider a set of massless chiral fields $\psi$ with the
same quantum numbers as the fermions of the SM. As explained earlier,
often these will be extended to full $G'_0$ multiplets by the
introduction of additional fermionic sources. On site-1 we include a
set of massive Dirac fermions $\Psi^{(r)}$ arising from the strong
dynamics, transforming in different representations $r$ of $G_1$. The
fermions on site-0 and site-1 can be connected by the $\sigma$-model
fields $\Omega$ and $\Omega_X$. Similarly, fermions in different
representations on site-1 can be connected by the NGB fields in $U$.
The generic form of the fermion Lagrangian at quadratic order in the
fermion fields that we consider in this work is
\begin{equation}
{\cal L}_f=i\bar\psi {\not \!\!D}_0 \psi+\bar\Psi^{(r)}(i{\not \!\!D}_1-m_r)\Psi^{(r)}+m^{(rs)}\bar\Psi^{(r)}_LUP^{(rs)}U^\dagger\Psi^{\prime (s)}_R+\Delta^{(r)}\bar\psi^{(r)} \Omega \, [\Omega_X]^{q_r} \Psi^{(r)}+{\rm h.c.}~
\label{LfermionGeneric}
\end{equation}
Here $D_{\mu \, 0}$ and $D_{\mu \, 1}$ are the covariant derivatives
on sites-0 and 1 ({\it i.e.}~carrying the corresponding elementary or
composite gauge fields) and $P^{(rs)}$ is a projector in the space of
representations of ${\cal H}_1$. Note that besides the ``diagonal"
fermion masses, $m_r$, the NGB's can allow additional ``non-diagonal"
mass terms coupling different fermion representations. From the point
of view of the fermion field content, these bear some similarity with
the Yukawa terms of the SM. By some simple algebraic manipulations,
this term can be written in terms of the field $\Phi$ plus mixing
terms between composite fermions in the same representation of SO(5).
In the next section we will show them explicitly for each fermion
embedding. The last term in Eq.~(\ref{LfermionGeneric}) leads to
mixing between the elementary and composite fields, and realizes the
idea of partial compositeness in the fermion sector. This term is
only written for pairs of elementary and composite fermions with the
same quantum numbers under $G'_0$ and $G_1$ [here $q_r$ denotes the
common charge of $\psi^{(r)}$ under $U(1)_x$ and $\Psi^{(r)}$ under
$U(1)_X$]. Note that this last term violates the $G'_0 \times G_1$
symmetry explicitly only after the non-dynamical source fields in
$\psi^{(r)}$ are set to zero. The precise form of the above
Lagrangian depends on the representations of the fermionic resonances
which would be determined by the strongly coupled UV completion. In
the absence of such an explicit theory, we will study several
possibilities based on the lowest dimensional representations of
SO(5). We will provide the detailed forms of the Lagrangians in
Sec.~\ref{sec:models}.
A comment regarding the structure of the third term that contains the
interactions between fermions and the NGB's parametrized by $U$ is in
order. As will be discussed in Sec.~\ref{sec_div_VH} and
explicitly shown in Sec.~\ref{sec:models}, we will not consider
the most general mass terms. Rather, in order to obtain a finite
Higgs potential $V_H$ we have imposed some constraints. By $\Psi_L$
we mean the Left-handed component of the fields $\Psi$ on site-1 that
mix with the fields $\psi_L$ on site-0, whereas $\Psi'_R$ is the
Right-handed component of the fields $\Psi$ on site-1 that mix with
$\psi_R$ on site-0. Therefore $m^{(rs)}$ will only connect
$\Psi^{(r)}_L$ and $\Psi^{\prime (s)}_R$, but there are neither terms
of type $\bar\Psi^{(r)}_RUP^{(rs)} U^\dagger\Psi^{\prime (s)}_L$ nor
of type $\bar\Psi^{(r)}_L U P^{(rs)}U^\dagger\Psi^{(s)}_R$.
Also, to avoid large corrections to $Zb_L\bar b_L$ we will embed $Q$,
the composite multiplet mixing with $q_L$, in a multiplet such that:
$T_L=T_R$ and $T^{3}_L = T^{3}_R = -1/2$ for $Q_d$, with $T_{L,R}$ the
${\rm SU(2)}_{L,R}$ generators and $Q_d$ the component mixing with
$b_L$. This means that $Q$ contains a $({\bf 2},{\bf 2})$ of ${\rm
SU(2)}_L \times {\rm SU(2)}_R$. The smallest irreducible
representations of SO(5) satisfying this condition are the fundamental
${\bf 5}$, the adjoint (antisymmetric) ${\bf 10}$ and the (symmetric)
${\bf 14}$. The U(1)$_X$ charge is fixed by demanding that the
correct hypercharge be reproduced, where $Y=T^{3}_R+X$, leading to
$X=2/3$. For the composite multiplet $U$ ($D$) mixing with $u_R$
($d_R$) we will consider several possibilities, but we will choose
those that allow to write a Yukawa term $\bar Q \Phi^n U$ ($\bar Q
\Phi^n D$) that is a singlet of $G_1$ and contain a ${\bf 1}_{2/3}$
(${\bf 1}_{-1/3}$) of ${\rm SU(2)}_L \times {\rm U(1)}_Y$. We will
consider the following models: MCHM$_5$ (all the fermions in ${\bf
5}$), MCHM$_{10}$ (all the fermions in ${\bf 10}$), and models
involving more than one representation: MCHM$_{10-5-10}$,
MCHM$_{5-5-10}$, MCHM$_{5-10-10}$, MCHM$_{14-14-10}$ and
MCHM$_{14-1-10}$, with notation MCHM$_{Q-U-D}$ (see also
Refs.~\cite{Carena:2006bn,Carena:2007ua,Csaki:2008zd,Pomarol:2012qf,Pappadopulo:2013vca}).
Since the BR of the Higgs decaying to $\tau^+\tau^-$ is not
negligible, we will also consider the leptonic sector. For each
generation we include two multiplets of composite fermions: $L$ and
$E$, mixing with the elementary leptons $\ell_L$ and $e_R$
respectively. These composite leptons are singlets of ${\rm SU(3)}_C$
and, for each model, we choose their SO(5) embedding copying that of
$Q$ and $D$, again with $X$ chosen to obtain $Y=T^{3}_{R}+X$.
\subsection{The Low-Energy Effective Theory}
\label{sec:EFT}
In order to make contact with measurements at current energies, it is
useful to integrate out the heavy resonances in the previous model.
We will present in this section the result of integrating out the
spin-1 resonances, which is common to the various models we consider
and illustrates the general procedure. In Sec.~\ref{sec:models} we
present the result of integrating out the heavy fermionic sector in
the different models of interest.
In order to simplify the computations it is useful to start with all
elementary fields as non-dynamical and filling complete $G'_0 = {\rm
SO(5)} \times {\rm U(1)}_x$ representations, as discussed in
Subsection~\ref{sec:2site} above. Since in this limit the full theory
has an exact global ${\rm SO(5)} \times {\rm U(1)}_X$ symmetry,
corresponding to the diagonal group of $G'_0 \times G_1$ (due to the
vev of the link fields), the effective theory for these external
sources must take a fully ${\rm SO(5)} \times {\rm U(1)}_X$ form.
Listing all the invariant terms that are quadratic in the external
gauge fields, we must obtain (in momentum space):
\begin{equation}
{\cal L}_{\rm eff}^{\rm sources} = \frac{1}{2} \, \Pi^{(0)}_A \, {\rm tr}(\tilde{a}_\mu \tilde{a}^\mu) + \frac{1}{2} \, \Pi^{(2)}_A \, \Phi^T \tilde{a}_\mu \tilde{a}^\mu \Phi + \frac{1}{2} \, \Pi^{(0)}_X \, \tilde{x}_\mu \tilde{x}^\mu~,
\label{LeffGeneral}
\end{equation}
for some functions $\Pi^{(0)}_A(p^2)$, $\Pi^{(2)}_A(p^2)$ and
$\Pi^{(0)}_X(p^2)$. In the limit that $\langle h \rangle = 0$, {\it
i.e.}~$\Phi = \{ 0,0,0,0,1 \}^T$, this becomes
\begin{equation}
\left. {\cal L}_{\rm eff}^{\rm sources} \right|_{h = 0} = \frac{1}{2} \, \Pi^{(0)}_A \, \tilde{a}^j_\mu \tilde{a}^{j \, \mu} + \frac{1}{2} \left( \Pi^{(0)}_A + \frac{1}{2} \, \Pi^{(2)}_A \right) \tilde{a}^{\hat{b}}_\mu \tilde{a}^{\hat{b} \, \mu} + \frac{1}{2} \, \Pi^{(0)}_X \, \tilde{x}_\mu \tilde{x}^\mu~,
\label{Leffheq0}
\end{equation}
where $j = 1, \ldots 6$ and $b = 1,2,3,4$ label the two SO(4)
representations in the adjoint of SO(5): ${\bf 10} = {\bf 6} + {\bf
4}$. We can then integrate out the heavy spin-1 resonances from
${\cal L} = {\cal L}_{\rm gauge} + {\cal L}_{\Omega} + {\cal L}_{\rm
NGB}$ [Eqs.~(\ref{LNGBSimple})--(\ref{Lgauge})] in the limit $\langle h
\rangle = 0$ and in unitary gauge, and identify $\Pi^{(0)}_A$,
$\Pi^{(2)}_A$ and $\Pi^{(0)}_X$. The equations of motion for the
heavy fields simply read
\begin{equation}
\tilde{A}^j_\mu = - \frac{m^2_\rho}{p^2 - m^2_\rho} \, \tilde{a}^j_\mu~,
\hspace{1cm}
\tilde{A}^{\hat{b}}_\mu = - \frac{m^2_\rho}{p^2 - m^2_a} \, \tilde{a}^{\hat{b}}_\mu~,
\hspace{1cm}
\tilde{X}_\mu = - \frac{m^2_X}{p^2 - m^2_X} \, \tilde{x}_\mu~,
\end{equation}
where $m_\rho$ and $m_X$ were defined in Eq.~(\ref{mrhomX}), and
$m_a$ was defined in Eq.~(\ref{ma}). Replacing back
in the original Lagrangian, we find
\begin{equation}
\Pi^{(0)}_A = \hat{\Pi}_6~,
\hspace{1cm}
\Pi^{(2)}_A = 2( \hat{\Pi}_{4} - \hat{\Pi}_6)~,
\hspace{1cm}
\Pi^{(0)}_X = \hat{\Pi}_X~,
\end{equation}
where
\begin{equation}
\hat{\Pi}_6 = \frac{p^2 m^2_\rho}{g^2_\rho (p^2 - m^2_\rho)}~,
\hspace{1cm}
\hat{\Pi}_{4} = \frac{m^2_\rho (p^2 + m^2_\rho - m^2_a)}{g^2_\rho (p^2 - m^2_a)}~,
\hspace{1cm}
\hat{\Pi}_X = \frac{p^2 m^2_X}{g^2_X (p^2 - m^2_X)}~.
\end{equation}
Going back to Eq.~(\ref{LeffGeneral}) evaluated for an arbitrary Higgs
configuration, and keeping only the sources corresponding to the SM
gauge fields, as described after Eq.~(\ref{Lgauge}), one finds in an
obvious notation:
\begin{equation}
{\cal L}_{\rm eff} = \frac{1}{2} \, \sum^3_{i=1} \Pi_{\tilde{w}^i_L} \tilde{w}^i_{L \, \mu} \tilde{w}^{i \, \mu}_L
+ \Pi_{\tilde{w}^3_L \, \tilde{b}} \, \tilde{w}^3_{L \, \mu} b^{\mu}
+ \frac{1}{2} \, \Pi_{\tilde{b}} \, \tilde{b}_{\mu} \tilde{b}^{\mu}~,
\end{equation}
where
\begin{eqnarray}
\Pi_{\tilde{w}^i_L} = \Pi^{(0)}_A + \frac{1}{4} \, \Pi^{(2)}_A \sin^2(h/f_h)~,
\hspace{1cm}
\Pi_{\tilde{w}^3_L \, \tilde{b}} = - \frac{1}{4} \, \Pi^{(2)}_A \sin^2(h/f_h)~,
\nonumber \\ [0.4 em]
\Pi_{\tilde{b}} = \Pi^{(0)}_X + \Pi^{(0)}_A + \frac{1}{4} \, \Pi^{(2)}_A \sin^2(h/f_h)~.
\hspace{2cm}
\end{eqnarray}
These correlators, which are valid to all orders in momentum as well
as on the Higgs vev will be useful when evaluating the Higgs potential
in Sec.~\ref{sec_div_VH}.
\section{Models based on the ${\bf 1}$, ${\bf 5}$, ${\bf 10}$ and ${\bf 14}$ Reps.~of SO(5)}
\label{sec:models}
In this section, we present a summary of the models we consider in
this work, which differ in the SO(5) representations of the fermionic
resonances arising from the strongly interacting sector. We start
with a few general comments, and then describe each model in turn.
The reader may want to read only the first part of this section and
skip to Sec.~\ref{sec:corrections}, coming back to
Subsections~\ref{sec:MCHM5}-\ref{sec:MCHM14110} only if further
details are desired.
In unitary gauge the fermion Lagrangian can be written as:
\begin{eqnarray}
\label{Lfermions}
{\cal L}_f&=&\sum_{\psi=q_L,u_R,d_R} Z_\psi \bar\psi i{\not \!\!D} \psi + \bar q_L \Delta_{q} Q_R + \bar u_R \Delta_u U_L + \bar d_R \Delta_d D_L + {\rm h.c.} \\
&+& \sum_{\Psi=Q,U,D} \bar\Psi (i{\not \!\!D}-m_\Psi) \Psi
+ m_{y_u} \bar Q_L U_R + m_{y_d} \bar Q_L D_R + {\cal L}_y(Q_L,U_R,D_R, \Phi) + {\rm h.c.}
\nonumber
\end{eqnarray}
Depending on the fermion embedding, the terms $m_{y_u} \bar Q_L U_R +
m_{y_d} \bar Q_L D_R$ can contain a gauge singlet or not. They are
present only in the former case. The explicit form of the Yukawa
terms also depends on the fermion embedding, and will be specified for
each model below.\footnote{These Yukawa interactions are not yet the
SM Yukawa interactions, but will give rise to them. Therefore, we
will refer to them as ``proto-Yukawa" interactions.} For the MCHM$_5$
it is necessary to include two different composite fermions $Q^u$ and
$Q^d$ that mix with the elementary doublet $q_L$. In this case, we
replace $\bar q_L \Delta_{q} Q_R \to \bar q_L\Delta_{q^u} Q^u_R + \bar
q_L\Delta_{q^d} Q^d_R$ and $m_{y_u} \bar Q_L U_R + m_{y_d} \bar Q_L
D_R \to m_{y_u} \bar Q^u_L U_R + m_{y_d} \bar Q^d_L D_R$ above.
However, for the other models a single $Q$ is sufficient, as written
in Eq.~(\ref{Lfermions}).
Integrating out the composite resonances we obtain an effective theory
involving the elementary degrees of freedom only, in complete analogy
to the procedure presented in Sec.~\ref{sec:EFT} for the spin-1
case. The fermions are in complete irreducible representations $r_5$
of SO(5). However, due to the spontaneous breaking ${\rm SO(5)} \to
{\rm SO(4)}$ in the composite sector, each fermion is in general split
into several irreducible representations $r_4$ of SO(4):
$\psi^{r_5}=\sum_{r_4}\alpha_{r_5,r_4}\psi^{r_4}$, with
$\alpha_{r_5,r_4}$ the coefficients associated to the decomposition.
Thus, before EWSB, and taking $\langle\Phi\rangle=\Phi_0$ ({\it
i.e.}~$h=0$), one can write the effective Lagrangian as:
\begin{eqnarray}
\label{Leff-fermions-SO4}
\left. {\cal L}_{\rm eff} \right|_{h = 0} = \sum_{\psi=q_L,u_R,d_R}\sum_{r_4}\bar \psi^{(r_4)} {\not \!p\ } (Z_\psi+\hat\Pi_\psi^{(r_4)}) \psi^{(r_4)}
+ \sum_{\psi=u,d}\sum_{r_4}\bar q_L^{(r_4)} \hat M_{\psi^{(r_4)}} \psi_R^{(r_4)} + {\rm h.c.}
\end{eqnarray}
The explicit form of the correlators $\hat\Pi_\psi^{r_4}$ and $\hat
M_\psi^{r_4}$ are given in the appendix for the different models.
It is then simple to compare to the correlators of an effective
Lagrangian, ${\cal L}_{\rm eff}$, written in fully SO(5) invariant
form with the help of an arbitrary $\Phi$ (one should list all
possible ${\rm SO(5)} \times {\rm U(1)}_X$ invariant operators that
are quadratic in the external fermionic sources, which depends on the
specific model in question). If one then retains the SM degrees of
freedom only, the effective Lagrangian for the elementary fermions
takes the form
\begin{eqnarray}
\label{Leff-fermions}
{\cal L}_{\rm eff}&=& \bar u_L {\not \!p\ } (Z_q+\Pi_{u_L}) u_L + \bar d_L {\not \!p\ } (Z_q+\Pi_{d_L}) d_L + \bar u_R {\not \!p\ } (Z_u+\Pi_{u_R}) u_R + \bar d_R {\not \!p\ } (Z_d+\Pi_{d_R}) d_R \nonumber \\ [0.4em]
& & \mbox{} + \bar u_L M_u u_R + \bar d_L M_d d_R + {\rm h.c.}
\end{eqnarray}
The correlators $\Pi_\psi$ and $M_\psi$ can be expressed in terms of
the correlators of the SO(4) symmetric theory $\hat\Pi_\psi^{r_4}$ and
$\hat M_\psi^{r_4}$, and have an explicit (and generally simple)
dependence on $s_h = \sin h / f_h$ and $c_h = \cos h / f_h$. We show
below the full expressions for each specific model.
The spectrum of fermions that mix with the SM ones (as well as the
masses of the SM degrees of freedom) is given by the zeroes of the
quadratic operator
\begin{equation}\label{eq-spectrum}
{\rm Zero}\left\{p^2[Z_q+\Pi_{\psi_L}(p^2)][Z_\psi+\Pi_{\psi_R}(p^2)]-|M_\psi(p^2)|^2\right\} \ , \qquad \psi=u,d \ .
\end{equation}
The SM states, being lighter than the compositeness scale, can be
obtained by expanding Eq.~(\ref{eq-spectrum}) to ${\cal O}(p^2)$,
leading to
\begin{equation}\label{eq-spectrum-0}
m_\psi^{(0)}\simeq |M_\psi(0)|\left\{[Z_q+\Pi_{\psi_L}(0)][Z_\psi+\Pi_{\psi_R}(0)] - \left. 2 |M_\psi(0)| \frac{d |M_\psi(p^2)|}{d p^2} \right|_{p^2 = 0} \right\}^{-1/2} \ , \qquad \psi=u,d \ ,
\end{equation}
We have used the superindex
$(0)$ for the lightest states, since in the absence of mixings they
are massless.
Similarly, the Yukawa coupling of these states to (a single) Higgs
boson can be obtained by differentiating with respect to $v$:
\begin{eqnarray}
\label{eq-y}
y_\psi^{(0)} &\simeq& \frac{dm_\psi^{(0)}}{dv} \ , \qquad \psi=u,d \ .
\end{eqnarray}
This coupling depends on the model, but since the vev dependence of
the correlators is simple (it is encoded in $s_h$ and $c_h$ in the
formulas given in the following subsections), we can derive simple
expression in terms of the correlators, that will be given for each
model below.
A very important combination for the phenomenology is the function
$y^{(0)}_\psi/m^{(0)}_\psi$. To leading order in $\epsilon$ it can be
approximated by:
\begin{eqnarray}
\label{eq-yom}
\frac{y_\psi^{(0)}}{m_\psi^{(0)}} &\simeq& \frac{F_\psi(\epsilon)}{\epsilon \, f_h} \left[1+{\cal O}(\epsilon^2) \right] ~, \qquad \psi=u,d \ .
\end{eqnarray}
where the $F_\psi(\epsilon)$ depends only on $\epsilon$ (as well as on
the fermion representation) and will be given in
Sec.~\ref{sec:corrections}.~\footnote{The are exceptions to this
statement, with additional dependence on the Yukawa couplings on the
r.h.s.~of Eq.~(\ref{eq-yom}). We consider one such detailed example
in this work and mention a few others. However, in certain limits the
above discussion often applies.} The ${\cal O}(\epsilon^2)$ correction
(which also depends on other microscopic parameters) determines the
deviation compared with the simple and compact leading approximation.
The above relation is intimately connected to certain sum rules that
have been already observed in the
literature~\cite{Falkowski:2007hz,Low:2010mr,Azatov:2011qy}. We will
comment further on this in Sec.~\ref{sec:pheno}.
As will be shown below, different models lead to different sizes for
the ${\cal O}(\epsilon^2)$ term. Multiplying Eq.~(\ref{eq-yom}) by
$v_{SM}$, and using Eq.~(\ref{eq-vSM}), we can obtain the ratio
between the Yukawa couplings in the MCHM and in the SM:
\begin{eqnarray}
\label{eq-yoy}
\frac{y_\psi^{(0)}}{y_\psi^{SM}} &\simeq& F_\psi(\epsilon) \left[1+{\cal O}(\epsilon^2) \right] \ , \qquad \psi=u,d \ ,
\end{eqnarray}
showing that deviations from $F_\psi(\epsilon)$ are suppressed by
${\cal O}(\epsilon^2)$. This correction depends also on the fermionic
mixings in the following way: ${\cal O}(\epsilon^2 s_{\psi_L}^2,
\epsilon^2 s_{\psi_R}^2)$, requiring in general the mixing of both
chiralities to be small to ensure extra suppression factors. However,
for some models the structure inherited from the fermion embedding is
such that the correction involves just one chirality to leading order:
${\cal O}(\epsilon^2 s_{\psi_L}^2)$ or ${\cal O}(\epsilon^2
s_{\psi_R}^2)$. In those cases an extra suppression can be achieved
with small mixing for one chirality only. Note also that the above
corrections do not take the form claimed in \cite{Falkowski:2007hz},
i.e.~${\cal O}(\epsilon^2 m_\psi^2) \sim {\cal O}(\epsilon^2 \,
s_{\psi_L}^2 s_{\psi_R}^2)$, where $m_\psi$ denotes the mass of the SM
field (this has also been observed in Ref.~\cite{Montull:2013mla}).
Thus, the bottom quark, in particular, can give corrections that are
larger than expected, as will be illustrated in Sec.~\ref{sec:pheno}.
\subsection{MCHM$_5$}
\label{sec:MCHM5}
In this model we consider 4 composite fermions for each generation:
$Q^u,U\sim {\bf 5}_{2/3}$ and $Q^d,D\sim {\bf 5}_{-1/3}$, where the
subindex denotes the $U(1)_X$ charge. In unitary gauge the Yukawa
terms of the fermion Lagrangian~(\ref{Lfermions}) read:
\begin{eqnarray}
{\cal L}_y &=& y_u (\bar Q^u_L \Phi)(\Phi^\dagger U_R) + y_d (\bar Q^d_L \Phi)(\Phi^\dagger D_R) \ .
\label{LY5}
\end{eqnarray}
In this case $q_L$ mixes with two composite fermions: $Q^u$ and $Q^d$.
The bottom mass can result from small $\Delta_{q^d}$ and/or small
$\Delta_{d}$.
The correlators of the effective Lagrangian~(\ref{Leff-fermions}) are:
\begin{align}
\Pi_{u_L}&= \Pi_{q^u}^0 + \Pi_{q^d}^0 + \Pi_{q^u}^1 \frac{s_h^2}{2} \ ,\qquad &
\Pi_{d_L}&= \Pi_{q^u}^0 + \Pi_{q^d}^0 + \Pi_{q^d}^1 \frac{s_h^2}{2} \ ,\nonumber\\
\Pi_{u_R}&=\Pi_u^0+\Pi_u^1 c_h^2 \ ,\qquad &
\Pi_{d_R}&=\Pi_d^0+\Pi_d^1 c_h^2 \ ,\nonumber\\
M_u&= m_u^1 \, \frac{s_h c_h}{\sqrt{2}} \ ,\qquad &
M_d&= m_d^1 \, \frac{s_h c_h}{\sqrt{2}} \ .
\end{align}
where the $\Pi^i_\psi$ are defined by
\begin{eqnarray}
{\cal L}^{\rm sources}_{\rm eff} &=&
\bar{q}^u_L {\not \!p\ } \Pi^{0}_{q^u} q^u_L + \bar{q}^d_L {\not \!p\ } \Pi^{0}_{q^d} q^d_L + \bar{u}_R {\not \!p\ } \Pi^{0}_{u} u_R + \bar{d}_R {\not \!p\ } \Pi^{0}_{d} d_R
+ (\bar{q}^u_L \Phi) {\not \!p\ } \Pi^{1}_{q^u} (\Phi^\dagger q^u_L)
\nonumber \\ [0.4em]
& & \mbox{}
+ (\bar{q}^d_L \Phi) {\not \!p\ } \Pi^{1}_{q^d} (\Phi^\dagger q^d_L) +
(\bar{u}_R \Phi) {\not \!p\ } \Pi^{1}_{u} (\Phi^\dagger u_R) + (\bar{d}_R \Phi) {\not \!p\ } \Pi^{1}_{d} (\Phi^\dagger d_R)
\label{LeffSO5}
\\ [0.4em]
& & \mbox{}
+ m_u^0 \, \bar q^u_L u_R + m_d^0 \, \bar q^d_L d_R
+ m_u^1 (\bar q^u_L \Phi)(\Phi^\dagger u_R) + m_d^1 (\bar q^d_L \Phi)(\Phi^\dagger d_R) + {\rm h.c.}
\nonumber
\end{eqnarray}
The superindex ``sources" serves as a reminder that here the $q^u_L$,
$q^d_L$, $u_R$ and $d_R$ fill complete SO(5) multiplets and that all
components are to be treated as external sources. One must still add
``bare" kinetic terms for the dynamical fields on site-0,
\textit{i.e.}~those with SM quantum numbers, as in
Eq.~(\ref{Leff-fermions}). Since a ${\bf 5}$ of SO(5) decomposes
under SO(4) as ${\bf 5}\sim {\bf 1}+{\bf 4}$, one finds
\begin{align}
\Pi_{q^u}^0 &= \hat\Pi_{q^{u(4)}}\ ,\qquad &
\Pi_{q^d}^0 &= \hat\Pi_{q^{d(4)}}\ ,\qquad &
\Pi_d^0 &= \hat\Pi_{d^{(4)}}\ ,\nonumber\\
\Pi_{q^u}^1 &= \hat\Pi_{q^{u(1)}}-\hat\Pi_{q^{u(4)}}
\ ,\qquad &
\Pi_{q^d}^1 &= \hat\Pi_{q^{d(1)}}-\hat\Pi_{q^{d(4)}}
\ ,\qquad &
\Pi_d^1 &= \hat\Pi_{d^{(1)}}-\hat\Pi_{d^{(4)}}\ ,\nonumber\\
\Pi_u^0 &= \hat\Pi_{u^{(4)}} \ ,\qquad &
m_u^0 &= \hat M_{u^{(4)}} \ , \qquad &
m_d^0 &= \hat M_{d^{(4)}} \ ,\\
\Pi_u^1 &= \hat\Pi_{u^{(1)}}-\hat\Pi_{u^{(4)}}\ , \qquad &
m_u^1 &= \hat M_{u^{(1)}}-\hat M_{u^{(4)}} \ ,\qquad &
m_d^1 &= \hat M_{d^{(1)}}-\hat M_{d^{(4)}}\ .
\nonumber
\end{align}
where the hatted correlators are given in Appendix~\ref{app:MCHM5}.
Using these correlators we can compute the prediction for
$y_\psi^{(0)}/m_\psi^{(0)}$:
\begin{eqnarray}
\label{eq-yomt5}
\frac{y_t^{(0)}}{m_t^{(0)}}-\frac{F_t}{s_h f_h} &\simeq& \frac{s_h}{f_h}\frac{ 2 |m_u^1(0)| |m_u^{1}(0)|^{\prime} - [Z_u+\Pi_u^0(0)+\Pi_u^1(0)]\Pi_{q^u}^1+2[Z_q+\Pi_q^0(0)]\Pi_u^1}{2[Z_u+\Pi_u^0(0)+\Pi_u^1(0)][Z_q+\Pi_q^0(0)]} ~, \\
\label{eq-yomb5}
\frac{y_b^{(0)}}{m_b^{(0)}}-\frac{F_b}{s_h f_h} &\simeq& \frac{s_h}{f_h}\frac{ 2 |m_d^1(0)| |m_d^{1}(0)|^{\prime} - [Z_d+\Pi_d^0(0)+\Pi_d^1(0)]\Pi_{q^d}^1+2[Z_q+\Pi_q^0(0)]\Pi_d^1}{2[Z_d+\Pi_d^0(0)+\Pi_d^1(0)][Z_q+\Pi_q^0(0)]} \ ,
\end{eqnarray}
where $|m_\psi^i(0)|^{\prime} \equiv \left. | d \, m_\psi^i(p^2) | /
dp^2 \right|_{p^2 = 0}$. Taking into account that
$\Pi_\psi^j\sim\Delta_\psi^2$ and $M_\psi\sim\Delta_q\Delta_\psi$,
Eq.~(\ref{eq-yomb5}) shows that the ${\cal O}(s^2_h)$ correction to
$y_b$ in this model is small. By expressing $\Delta_\psi$ in terms of
the elementary-composite mixing angles, one sees that the correction
is suppressed by $s_{q^d}^2$ or $s_d^2$. By choosing both of them
small, we expect $y_b^{(0)} / m_b^{(0)}$ to be well approximated by
$F_b / s_h f_h$ in this model. On the other hand,
Eq.~(\ref{eq-yomt5}) shows that the corrections to $y_t$ do not have
any extra suppression factor in general, since the top mass requires
both, $s_{q^u}$ and $s_u \sim {\cal O}(1)$. This property has
important consequences for the phenomenology: one can expect
corrections to loop-induced processes that depend on $y_t$ [gluon
fusion, $h \to \gamma\gamma$ to be discussed in Sec.~\ref{sec:pheno}]
of ${\cal O}(s^2_h)$. The size of these corrections is similar for
all the models. Since all of them require $s_{q}$ and $s_u\lesssim
1$, there can be differences of ${\cal O}(1)$ between them arising
from the different embeddings and regions of the parameter space
selected.
\subsection{MCHM$_{10}$}
From now on, we consider 3 composite fermions for each generation. In
this model: $Q,U,D\sim {\bf 10}_{2/3}$. In unitary gauge the Yukawa
terms of the fermion Lagrangian~(\ref{Lfermions}) read:
\begin{eqnarray}
{\cal L}_y &=& y_u \Phi^\dagger \bar Q_L U_R \Phi + y_d \Phi^\dagger \bar Q_L D_R \Phi \ .
\label{LY10}
\end{eqnarray}
In this case $q_L$ mixes with a single composite fermion $Q$ and,
therefore, the bottom mass requires small $\Delta_{d}$. In this model
the interactions $\Phi^\dagger \bar U_L D_R \Phi$ and $\Phi^\dagger
\bar U_R D_L \Phi$ are also compatible with the symmetries. However
they lead to a logarithmically divergent Higgs potential, and we do
not include them. Note also that we do not include terms of the form
$\epsilon_{ABCDE} \, \Phi^A \bar{Q}_L^{BC} U_R^{DE}$, etc., which
would break a LR symmetry, and have been studied in \cite{Azatov:2013ura}.
The correlators of the effective Lagrangian~(\ref{Leff-fermions}) are:
\begin{align}
\Pi_{u_L}&=\Pi_q^0+\Pi_{q}^1 \left(\frac{c_h^2}{2}+\frac{s_h^2}{4}\right) \ ,\qquad &
\Pi_{d_L}&=\Pi_q^0+\Pi_{q}^1 \frac{c_h^2}{2} \ ,\nonumber\\
\Pi_{u_R}&=\Pi_u^0+\Pi_u^1 \frac{s_h^2}{4} \ ,\qquad &
\Pi_{d_R}&=\Pi_d^0+\Pi_d^1 \frac{s_h^2}{4} \ ,\\
M_u&= - \, m_u^1 \, \frac{s_h c_h}{4} \ ,\qquad &
M_d&= -m_d^1 \, \frac{s_h c_h}{2\sqrt{2}} \ .
\nonumber
\end{align}
where the $\Pi^i_\psi$ are now defined by [see also comments
following Eq.~(\ref{LeffSO5})]
\begin{eqnarray}
{\cal L}^{\rm sources}_{\rm eff} &=&
{\rm Tr} \left[ \bar{q}_L {\not \!p\ } \Pi^{0}_{q} q_L + \bar{u}_R {\not \!p\ } \Pi^{0}_{u} u_R + \bar{d}_R {\not \!p\ } \Pi^{0}_{d} d_R \right]
\nonumber \\ [0.4em]
& & \mbox{}
+ \Phi^\dagger \bar{q}_L {\not \!p\ } \Pi^{1}_{q} \, q_L \Phi
+ \Phi^\dagger \bar{u}_R {\not \!p\ } \Pi^{1}_{u} \, u_R \Phi + \Phi^\dagger \bar{d}_R {\not \!p\ } \Pi^{1}_{d} \, d_R \Phi
\label{LeffSO10}
\\ [0.4em]
& & \mbox{} +
{\rm Tr} \left[m_u^0 \, \bar q_L u_R + m_d^0 \, \bar q_L d_R \right]
+ m_u^1 \, \Phi^\dagger \bar q_L u_R \Phi + m_d^1 \, \Phi^\dagger \bar q_L d_R\Phi + {\rm h.c.}
\nonumber
\end{eqnarray}
Since a ${\bf 10}$ of SO(5)
decomposes under SO(4) as ${\bf 10}\sim {\bf 4}+{\bf 6}$, we find
\begin{align}
\Pi_q^0&= \hat\Pi_{q^{(6)}}\ ,\qquad &
\Pi_u^0&= \hat\Pi_{u^{(6)}}\ ,\qquad &
\Pi_d^0&= \hat\Pi_{d^{(6)}}\ ,\nonumber\\
\Pi_{q}^1&= 2(\hat\Pi_{q^{(4)}}-\hat\Pi_{q^{(6)}})\ ,\qquad &
\Pi_u^1&= 2(\hat\Pi_{u^{(4)}}-\hat\Pi_{u^{(6)}})\ ,\qquad &
\Pi_d^1&= 2(\hat\Pi_{d^{(4)}}-\hat\Pi_{d^{(6)}})\ ,\nonumber\\
m_u^0&= \hat M_{u^{(6)}} \ ,\qquad &
m_d^0&= \hat M_{d^{(6)}}\ , \nonumber\\
m_u^1&= 2(\hat M_{u^{(4)}}-\hat M_{u^{(6)}}) \ ,\qquad &
m_d^1&= 2(\hat M_{d^{(4)}}-\hat M_{d^{(6)}})\ .
\end{align}
where the hatted correlators are given in Appendix~\ref{app:MCHM10}.
The prediction for $y_\psi/m_\psi$ is:
\begin{eqnarray}
\label{eq-yomt10}
\frac{y_t^{(0)}}{m_t^{(0)}}-\frac{F_t}{s_h f_h} &\simeq& \frac{s_h}{f_h}\frac{|m_u^1(0)| |m_u^{1}(0)|^{\prime}+[2Z_u+2\Pi_u^0(0)-\Pi_u^1(0)]\Pi_q^1-2[Z_q+\Pi_q^0(0)]\Pi_u^1}{4[Z_u+\Pi_u^0(0)][2Z_q+2\Pi_q^0(0)+\Pi_q^1(0)]} ~,\\
\label{eq-yomb10}
\frac{y_b^{(0)}}{m_b^{(0)}}-\frac{F_b}{s_h f_h} &\simeq& \frac{s_h}{f_h}\frac{2 |m_d^1(0)| |m_d^{1}(0)|^{\prime}+4[Z_d+\Pi_d^0(0)]\Pi_q^1-[2Z_q+2\Pi_q^0(0)+\Pi_q^1(0)]\Pi_d^1}{4[Z_d+\Pi_d^0(0)][2Z_q+2\Pi_q^0(0)+\Pi_q^1(0)]} \ .
\end{eqnarray}
Eq.~(\ref{eq-yomb10}) shows that the ${\cal O}(s_h^2)$ corrections to
$y_b$ in this model can be sizable. This is because there is a term
suppressed by $s_q^2$ only, but $s_q\sim 1$ to reproduce the top mass.
Thus, we find a suppression by $s_h^2$ only.
\subsection{MCHM$_{10-5-10}$}
In this model: $Q,D\sim{\bf 10}_{2/3}$ and $U\sim{\bf 5}_{2/3}$. In
unitary gauge the Yukawa terms of the fermion
Lagrangian~(\ref{Lfermions}) read:
\begin{eqnarray}
{\cal L}_y &=& y_u \Phi^\dagger\bar Q_L U_R + y_d \Phi^\dagger\bar Q_L D_R \Phi \ .
\label{LY10510}
\end{eqnarray}
The correlators of the effective Lagrangian~(\ref{Leff-fermions}) are:
\begin{align}
\Pi_{u_L}&=\Pi_q^0+\Pi_{q}^1 \left(\frac{c_h^2}{2}+\frac{s_h^2}{4}\right) ,\qquad &
\Pi_{d_L}&=\Pi_q^0+\Pi_{q}^1 \frac{c_h^2}{2} \ ,\nonumber\\
\Pi_{u_R}&=\Pi_u^0+\Pi_u^1 c_h^2 \ ,\qquad &
\Pi_{d_R}&=\Pi_d^0+\Pi_d^1 \frac{s_h^2}{4} \ ,\\
M_u&= -m_u^1 \, \frac{s_h}{2} \ ,\qquad &
M_d&= -m_d^1 \, \frac{s_h c_h}{2\sqrt{2}} \ .
\nonumber
\end{align}
where the $\Pi^i_\psi$ are defined in analogy to Eqs.~(\ref{LeffSO5})
and (\ref{LeffSO10}), with the $\Phi$-dependent terms following the
structure displayed in Eq.~(\ref{LY10510}) for the Yukawa terms in
this model [see also comments following Eq.~(\ref{LeffSO5})].
Expanding the Higgs potential in powers of $s_h$ and $\Delta_\psi$,
the contribution of $M_u$ to the quartic coupling is of order ${\cal
O}(\Delta^8_\psi)$ and the only contributions of order ${\cal
O}(\Delta^4_\psi)$ are from $\Pi_L$ and $\Pi_R$. Therefore, in this
model we expect a small self-coupling and a very light Higgs. This
fact is reflected in the tuning of the model which, after requiring
the proper Higgs mass, is one order of magnitude larger than in the
other models. A sizable quartic coupling demands very large mixings
for the top quark, inducing departures from the analytical
approximations for the Yukawa couplings. This also affects the bottom
since the $b_L$ mixing is equal to the $t_L$ mixing in this model.
Using the previous decompositions of ${\bf 5}$ and ${\bf 10}$ of SO(5)
under SO(4) one finds:
\begin{align}
\Pi_q^0&= \hat\Pi_{q^{(6)}}\ ,\qquad &
\Pi_u^0&= \hat\Pi_{u^{(4)}}\ ,\qquad &
\Pi_d^0&= \hat\Pi_{d^{(6)}}\ ,\nonumber \\
\Pi_{q}^1&= 2(\hat\Pi_{q^{(4)}}-\hat\Pi_{q^{(6)}})\ ,\qquad &
\Pi_u^1&= \hat\Pi_{u^{(1)}}-\hat\Pi_{u^{(4)}}\ ,\qquad &
\Pi_d^1&= 2(\hat\Pi_{d^{(4)}}-\hat\Pi_{d^{(6)}})\ ,\nonumber \\
m_u^0&= 0 \ ,\qquad &
m_d^0&= \hat M_{d^{(6)}}\ , \nonumber \\
m_u^1&= \sqrt{2}\hat M_{u^{(4)}} \ ,\qquad &
m_d^1&= 2(\hat M_{d^{(4)}}-\hat M_{d^{(6)}})\ .
\end{align}
where the hatted correlators are given in
Appendix~\ref{app:MCHM10510}.
The prediction for $y_\psi/m_\psi$ is:
\begin{eqnarray}
\label{eq-yomt10-5-10}
\frac{y_t^{(0)}}{m_t^{(0)}}-\frac{F_t}{s_h f_h} &\simeq& \frac{s_h}{f_h}\frac{2 |m_u^1(0)| |m_u^{1}(0)|^{\prime}+[Z_u+\Pi_u^0(0)+3\Pi_u^1(0)]\Pi_q^1+4[Z_q+\Pi_q^0(0)]\Pi_u^1}{2[Z_u+\Pi_u^0(0)+\Pi_u^1(0)][2Z_q+2\Pi_q^0(0)+\Pi_q^1(0)]} ~,\\
\label{eq-yomb10-5-10}
\frac{y_b^{(0)}}{m_b^{(0)}}-\frac{F_b}{s_h f_h} &\simeq& \frac{s_h}{f_h}\frac{2 |m_d^1(0)| |m_d^{1}(0)|^{\prime}+4[Z_d+\Pi_d^0(0)]\Pi_q^1-[2Z_q+2\Pi_q^0(0)+\Pi_q^1(0)]\Pi_d^1}{4[Z_d+\Pi_d^0(0)][2Z_q+2\Pi_q^0(0)+\Pi_q^1(0)]} \ .
\end{eqnarray}
$y_b/m_b$ in this model is exactly as in the MCHM$_{10}$ when
expressed in terms of the correlators, although the correlators
themselves are different in both models. This can be understood
because the bottom mass arises from the coupling between $q$ and $d$,
that share the same embedding in both models.
\subsection{MCHM$_{5-5-10}$}
In this model: $Q,U\sim{\bf 5}_{2/3}$ and $D\sim{\bf 10}_{2/3}$. In
unitary gauge the Yukawa terms of the fermion
Lagrangian~(\ref{Lfermions}) read:
\begin{eqnarray}
{\cal L}_y &=& y_u (\bar Q_L \Phi) (\Phi^\dagger U_R) + y_d \bar Q_L D_R \Phi \ .
\label{LY5510}
\end{eqnarray}
The correlators of the effective Lagrangian~(\ref{Leff-fermions}) are:
\begin{align}
\Pi_{u_L}&=\Pi_q^0+\Pi_{q}^1 \frac{s_h^2}{2} ,\qquad &
\Pi_{d_L}&=\Pi_q^0 \ ,\nonumber\\
\Pi_{u_R}&=\Pi_u^0+\Pi_u^1 c_h^2 \ ,\qquad &
\Pi_{d_R}&=\Pi_d^0+\Pi_d^1 \frac{s_h^2}{4} \ ,\\
M_u&= m_u^1 \, \frac{s_h c_h}{\sqrt{2}} \ ,\qquad &
M_d&= m_d^1 \, \frac{s_h}{2} \ .
\nonumber
\end{align}
where the $\Pi^i_\psi$ are defined in analogy to Eqs.~(\ref{LeffSO5})
and (\ref{LeffSO10}), with the $\Phi$-dependent terms following the
structure displayed in Eq.~(\ref{LY5510}) for the Yukawa terms in this
model [see also comments following Eq.~(\ref{LeffSO5})]. Using the
previous decompositions of ${\bf 5}$ and ${\bf 10}$ of SO(5) under
SO(4):
\begin{align}
\Pi_q^0&= \hat\Pi_{q^{(4)}}\ ,\qquad &
\Pi_u^0&= \hat\Pi_{u^{(4)}}\ ,\qquad &
\Pi_d^0&= \hat\Pi_{d^{(6)}}\ ,\nonumber \\
\Pi_{q}^1&= \hat\Pi_{q^{(1)}}-\hat\Pi_{q^{(4)}}\ ,\qquad &
\Pi_u^1&= \hat\Pi_{u^{(1)}}-\hat\Pi_{u^{(4)}}\ ,\qquad &
\Pi_d^1&= 2(\hat\Pi_{d^{(4)}}-\hat\Pi_{d^{(6)}})\ ,\nonumber \\
m_u^0&= \hat M_{u^{(4)}} \ ,\qquad &
m_d^0&=0\ , \nonumber \\
m_u^1&= \hat M_{u^{(1)}}-\hat M_{u^{(4)}} \ ,\qquad &
m_d^1&= \sqrt{2}\hat M_{d^{(4)}}\ .
\end{align}
where the hatted correlators are given in Appendix~\ref{app:MCHM5510}.
The prediction for $y_\psi/m_\psi$ is:
\begin{eqnarray}
\label{eq-yomt5-5-10}
\frac{y_t^{(0)}}{m_t^{(0)}}-\frac{F_t}{s_h f_h} &\simeq& \frac{s_h}{f_h}\frac{2 |m_u^1(0)| |m_u^{1}(0)|^{\prime}+2[Z_q+\Pi_q^0(0)]\Pi_u^1-[Z_u+\Pi_u^0(0)+\Pi_u^1(0)]\Pi_q^1}{2[Z_u+\Pi_u^0(0)+\Pi_u^1(0)][Z_q+\Pi_q^0(0)]} \ ;\\
\label{eq-yomb5-5-10}
\frac{y_b^{(0)}}{m_b^{(0)}}-\frac{F_b}{s_h f_h} &\simeq& \frac{s_h}{f_h}\frac{2 |m_d^1(0)| |m_d^{1}(0)|^{\prime}-[Z_q+\Pi_q^0(0)]\Pi_d^1}{4[Z_d+\Pi_d^0(0)][Z_q+\Pi_q^0(0)]} \ .
\end{eqnarray}
For the top quark we obtain a result similar to the MCHM$_5$.
Eq.~(\ref{eq-yomb5-5-10}) shows that the ${\cal O}(s_h^2)$ corrections
to $y_b^{(0)}/m_b^{(0)}$ in this model is also suppressed by
$s_d^2\ll1$.
\subsection{MCHM$_{5-10-10}$}
In this model: $Q\sim{\bf 5}_{2/3}$ and $U,D\sim{\bf 10}_{2/3}$. In
unitary gauge the Yukawa terms of the fermion
Lagrangian~(\ref{Lfermions}) read:
\begin{eqnarray}
{\cal L}_y &=& y_u \bar Q_L U_R \Phi + y_d \bar Q_L D_R \Phi \ .
\label{LY51010}
\end{eqnarray}
In this model the interactions $\Phi^\dagger \bar U_L D_R \Phi$ and
$\Phi^\dagger \bar U_R D_L \Phi$ are also compatible with the
symmetries. However they lead to a logarithmically divergent Higgs
potential, therefore we will not include them.
The correlators of the effective Lagrangian~(\ref{Leff-fermions}) are:
\begin{align}
\Pi_{u_L}&=\Pi_q^0+\Pi_{q}^1 \frac{s_h^2}{2} ,\qquad &
\Pi_{d_L}&=\Pi_q^0 \ ,\nonumber\\
\Pi_{u_R}&=\Pi_u^0+\Pi_u^1 \frac{s_h^2}{4} \ ,\qquad &
\Pi_{d_R}&=\Pi_d^0+\Pi_d^1 \frac{s_h^2}{4} \ ,\\
M_u&= -m_u^1 \, \frac{s_h}{2\sqrt{2}} \ ,\qquad &
M_d&= m_d^1 \, \frac{s_h}{2} \ .
\nonumber
\end{align}
where the $\Pi^i_\psi$ are defined in analogy to Eqs.~(\ref{LeffSO5})
and (\ref{LeffSO10}), with the $\Phi$-dependent terms following the
structure displayed in Eq.~(\ref{LY51010}) for the Yukawa terms in
this model [see also comments following Eq.~(\ref{LeffSO5})]. Since
the Higgs dependence on $M_u$ is the same as in the MCHM$_{10-5-10}$,
the behavior of the Higgs potential and the top Yukawa are similar.
Using the previous decompositions of ${\bf 5}$ and ${\bf 10}$ of SO(5)
under SO(4):
\begin{align}
\Pi_q^0&= \hat\Pi_{q^{(4)}}\ ,\qquad &
\Pi_u^0&= \hat\Pi_{u^{(6)}}\ ,\qquad &
\Pi_d^0&= \hat\Pi_{d^{(6)}}\ ,\nonumber \\
\Pi_{q}^1&= \hat\Pi_{q^{(1)}}-\hat\Pi_{q^{(4)}}\ ,\qquad &
\Pi_u^1&= 2(\hat\Pi_{u^{(4)}}-\hat\Pi_{u^{(6)}})\ ,\qquad &
\Pi_d^1&= 2(\hat\Pi_{d^{(4)}}-\hat\Pi_{d^{(6)}})\ ,\nonumber \\
m_u^0&= 0 \ ,\qquad &
m_d^0&= 0\ , \nonumber \\
m_u^1&= \sqrt{2}\hat M_{u^{(4)}} \ ,\qquad &
m_d^1&= \sqrt{2}\hat M_{d^{(4)}}\ .
\end{align}
where the hatted correlators are given in Appendix~\ref{app:MCHM51010}.
The prediction for $y_\psi/m_\psi$ is:
\begin{eqnarray}
\label{eq-yomt5-10-10}
\frac{y_t^{(0)}}{m_t^{(0)}}-\frac{F_t}{s_h f_h} &\simeq& \frac{s_h}{f_h}\frac{|m_u^1(0)| |m_u^{1}(0)|^{\prime} - [Z_q+\Pi_q^0(0)]\Pi_u^1-2[Z_u+\Pi_u^0(0)]\Pi_u^1}{4[Z_u+\Pi_u^0(0)][Z_q+\Pi_q^0(0)]} ~,\\
\label{eq-yomb5-10-10}
\frac{y_b^{(0)}}{m_b^{(0)}}-\frac{F_b}{s_h f_h} &\simeq& \frac{s_h}{f_h}\frac{2 |m_d^1(0)| |m_d^{1}(0)|^{\prime}-[Z_q+\Pi_q^0(0)]\Pi_d^1}{4[Z_d+\Pi_d^0(0)][Z_q+\Pi_q^0(0)]} \ .
\end{eqnarray}
$y_b/m_b$ in this model is exactly as in the MCHM$_{5-5-10}$ when
expressed in terms of the correlators, although the correlators
themselves are different in both models. This can be understood,
again, because the bottom mass arises from the coupling between $q$
and $d$, which share the same embedding in both models.
Eq.~(\ref{eq-yomb5-10-10}) shows that the ${\cal O}(s_h^2)$
corrections to $y_b^{(0)}/m_b^{(0)}$ in this model is also suppressed
by $s_d^2\ll1$.
\subsection{MCHM$_{14-14-10}$}
\label{sec14-14-10}
In this model: $Q,U\sim{\bf 14}_{2/3}$ and $D\sim{\bf 10}_{2/3}$. In
unitary gauge the Yukawa term of the fermion
Lagrangian~(\ref{Lfermions}) includes:
\begin{eqnarray}
\label{14y1}
{\cal L}_y &\supset& y_u \Phi^\dagger \bar Q_L U_R \Phi + y_d \Phi^\dagger \bar Q_L D_R \Phi \ .
\label{LY141410}
\end{eqnarray}
The following term is also allowed by the symmetries
\begin{equation}
\label{14y2}
{\cal L}_y\supset \tilde y_u (\Phi^\dagger \bar Q_L\Phi)\ (\Phi^\dagger U_R \Phi) \ ,
\end{equation}
having potentially important consequences for the phenomenology, as
will be discussed in the next section.
The correlators of the effective Lagrangian~(\ref{Leff-fermions}) are:
\begin{align}
\Pi_{u_L}&=\Pi_q^0+\Pi_{q}^1 \left(\frac{c_h^2}{2}+\frac{s_h^2}{4}\right) + \Pi_{q}^2 s_h^2 c_h^2\ ,\qquad &
\Pi_{d_L}&=\Pi_q^0+\Pi_{q}^1 \frac{c_h^2}{2} \ ,\nonumber\\
\Pi_{u_R}&=\Pi_u^0+\Pi_u^1 \left(\frac{4}{5}c_h^2+\frac{s_h^2}{20}\right) + \Pi_u^2 \frac{(4c_h^2-s_h^2)^2}{20} \ ,\qquad &
\Pi_{d_R}&=\Pi_d^0+\Pi_d^1 \frac{s_h^2}{4} \ ,\\
M_u&= i\ m_u^1 \, \frac{3}{4\sqrt{5}}s_h c_h + i\ m_u^2 \, \frac{1}{2\sqrt{5}}s_h c_h (4c_h^2-s_h^2) \ ,\qquad &
M_d&= i m_d^1 \, \frac{s_h c_h}{2\sqrt{2}} \ .
\nonumber
\end{align}
where the $\Pi^i_\psi$ are defined in analogy to Eqs.~(\ref{LeffSO5})
and (\ref{LeffSO10}), with the $\Phi$-dependent terms following the
structure displayed in Eqs.~(\ref{14y1}) and (\ref{14y2}) for the
Yukawa terms in this model [see also comments following
Eq.~(\ref{LeffSO5})]. Since a ${\bf 14}$ of SO(5) decomposes under
SO(4) as ${\bf 14}\sim {\bf 1}+{\bf 4}+{\bf 9}$, we find
\begin{align}
\Pi_q^0&= \hat\Pi_{q^{(9)}}\ ,\qquad &
\Pi_u^0&= \hat\Pi_{u^{(9)}}\ ,\qquad &
\Pi_d^0&= \hat\Pi_{d^{(6)}}\ ,\nonumber \\
\Pi_{q}^1&= 2(\hat\Pi_{q^{(4)}}-\hat\Pi_{q^{(9)}})\ ,\qquad &
\Pi_u^1&= 2(\hat\Pi_{u^{(4)}}-\hat\Pi_{u^{(9)}})\ ,\qquad &
\Pi_d^1&= 2(\hat\Pi_{d^{(4)}}-\hat\Pi_{d^{(6)}})\ ,\nonumber \\
\Pi_q^2&= \frac{1}{4}(5\hat\Pi_{q^{(1)}}-8\hat\Pi_{q^{(4)}}+3\hat\Pi_{q^{(9)}})\ ,\qquad &
\Pi_u^2&= \frac{1}{4}(5\hat\Pi_{u^{(1)}}-8\hat\Pi_{u^{(4)}}+3\hat\Pi_{u^{(9)}})\ , \\
m_u^0&= \hat M_{u^{(1)}} \ ,\qquad &
m_d^0&= 0\ , \nonumber \\
m_u^1&= 2(\hat M_{u^{(4)}}-\hat M_{u^{(9)}}) \ ,\qquad &
m_d^1&= 2i\hat M_{d^{(4)}}\ ,\nonumber \\
m_u^2&= \frac{1}{4}(5\hat M_{u^{(1)}}-8\hat M_{u^{(4)}}+3\hat M_{u^{(9)}}) \ .
\nonumber
\end{align}
where the hatted correlators are given in
Appendix~\ref{app:MCHM141410}.
The prediction for $y_\psi/m_\psi$ is:
\begin{eqnarray}
\label{eq-yomt14}
\frac{y_t^{(0)}}{m_t^{(0)}}-\frac{F_t}{s_h f_h} &\simeq& \frac{s_h}{f_h}\left\{
-2\frac{-3 |m_u^1(0)| [\Pi_q^1(0)-4\Pi_q^2(0)]+16 |m_u^2(0)| [5Z_q+5\Pi_q^0(0)+2\Pi_q^1(0)+2\Pi_q^2(0)]}{[3 |m_u^1(0)| + 8 |m_u^2(0)| ][2Z_q+2\Pi_q^0(0)+\Pi_q^1(0)]}\right.\nonumber\\
& & \hspace{-3cm}
\left. +
\frac{-[3 |m_u^1(0)| + 8|m_u^2(0)| ][3 |m_u^{1}(0)|^{\prime} + 8 |m_u^{2}(0)|^{\prime}] + 5[2Z_q+2\Pi_q^0(0)+\Pi_q^1(0)][3\Pi_u^1(0)+8\Pi_u^2(0)]}{[5Z_u+5\Pi_u^0(0)+4\Pi_u^1(0)+4\Pi_u^2(0)][2Z_q+2\Pi_q^0(0)+\Pi_q^1(0)]}
\right\} ~,\\
\label{eq-yomb14}
\frac{y_b^{(0)}}{m_b^{(0)}}-\frac{F_b}{s_h f_h} &\simeq& \frac{s_h}{f_h}\frac{2 |m_d^1(0)| |m_d^{1}(0)|^{\prime}+4[Z_d+\Pi_d^0(0)]\Pi_q^1-[2Z_q+2\Pi_q^0(0)+\Pi_q^1(0)]\Pi_d^1}{4[Z_d+\Pi_d^0(0)][2Z_q+2\Pi_q^0(0)+\Pi_q^1(0)]} \ .
\end{eqnarray}
$y_b/m_b$ in this model is exactly as in the MCHM$_{10}$ when
expressed in terms of the correlators, although the correlators
themselves are different in both models.
\subsection{MCHM$_{14-1-10}$}
\label{sec:MCHM14110}
In this model: $Q\sim{\bf 14}_{2/3}$, $U\sim{\bf 1}_{2/3}$ and
$D\sim{\bf 10}_{2/3}$: In unitary gauge the Yukawa term of the fermion
Lagrangian~(\ref{Lfermions}) reads:
\begin{eqnarray}
{\cal L}_y &=& y_u (\Phi^\dagger\bar Q_L \Phi) U_R + y_d \Phi^\dagger\bar Q_L D_R \Phi \ .
\label{LY14110}
\end{eqnarray}
The correlators of the effective Lagrangian~(\ref{Leff-fermions}) are:
\begin{align}
\Pi_{u_L}&=\Pi_q^0+\Pi_{q}^1 \left(\frac{c_h^2}{2}+\frac{s_h^2}{4}\right)+\Pi_q^2 c_h^2 s_h^2 ,\qquad &
\Pi_{d_L}&=\Pi_q^0+\Pi_{q}^1 \frac{c_h^2}{2} \ ,\nonumber\\
\Pi_{u_R}&=\Pi_u^0 \ ,\qquad &
\Pi_{d_R}&=\Pi_d^0+\Pi_d^1 \frac{s_h^2}{4} \ ,\\
M_u&= -m_u^1 \, \frac{s_h}{2} \ ,\qquad &
M_d&= -m_d^1 \, \frac{s_h c_h}{2\sqrt{2}} \ .
\nonumber
\end{align}
where the $\Pi^i_\psi$ are defined in analogy to Eqs.~(\ref{LeffSO5})
and (\ref{LeffSO10}), with the $\Phi$-dependent terms following the
structure displayed in Eq.~(\ref{LY14110}) for the Yukawa terms in
this model [see also comments following Eq.~(\ref{LeffSO5})]. Using
the previous decompositions of ${\bf 14}$ and ${\bf 10}$ of SO(5)
under SO(4):
\begin{align}
\Pi_q^0&= \hat\Pi_{q^{(9)}}\ ,\qquad &
\Pi_u^0&= \hat\Pi_{u^{(1)}}\ ,\qquad &
\Pi_d^0&= \hat\Pi_{d^{(6)}}\ ,\nonumber \\
\Pi_{q}^1&= 2(\hat\Pi_{q^{(4)}}-\hat\Pi_{q^{(9)}})\ ,\qquad &
& \qquad & \Pi_d^1&= 2(\hat\Pi_{d^{(4)}}-\hat\Pi_{d^{(6)}})\ ,\nonumber \\
\Pi_{q}^2&= \frac{1}{4}(5\hat\Pi_{q^{(1)}}-8\hat\Pi_{q^{(4)}}+3\hat\Pi_{q^{(9)}})\ , \\
m_u^0&= 0 \ ,\qquad &
m_d^0&= 0 \ , \nonumber \\
m_u^1&= \frac{\sqrt{5}}{2}\hat M_{u^{(1)}} \ ,\qquad &
m_d^1&= 2i\hat M_{d^{(4)}}\ .
\nonumber
\end{align}
where the hatted correlators are given in Appendix~\ref{app:MCHM14110}.
The prediction for $y_\psi/m_\psi$ is:
\begin{eqnarray}
\label{eq-yomt4}
\frac{y_t^{(0)}}{m_t^{(0)}}-\frac{F_t}{s_h f_h} &\simeq& \frac{s_h}{f_h} \frac{-8 |m_u^1(0)| |m_u^{1}(0)|^{\prime}+[Z_u+\Pi_u^0(0)][\Pi_q^1(0)-4\Pi_q^2(0)]}{2[Z_u+\Pi_u^0(0)][2Z_q+2\Pi_q^0(0)+\Pi_q^1(0)]} \ ;\\
\label{eq-yomb4}
\frac{y_b^{(0)}}{m_b^{(0)}}-\frac{F_b}{s_h f_h} &\simeq& \frac{s_h}{f_h}\frac{2 |m_d^1(0)| |m_d^{1}(0)|^{\prime}+4[Z_d+\Pi_d^0(0)]\Pi_q^1-[2Z_q+2\Pi_q^0(0)+\Pi_q^1(0)]\Pi_d^1}{4[Z_d+\Pi_d^0(0)][2Z_q+2\Pi_q^0(0)+\Pi_q^1(0)]} \ .
\end{eqnarray}
The prediction for $y_b/m_b$ in this model is exactly as in the
MCHM$_{10}$ when expressed in terms of the correlators, although the
correlators themselves are different in both models.
\subsection{Other Models Based on the Lowest-dimensional Reps.~of SO(5)}
\label{sec:othermodels}
Although we will not provide all the details, we list here the other
possible models one can consider when using the $\bf 1$, $\bf 5$, $\bf
10$ and $\bf 14$ representations of $SO(5)$ in all possible
combinations for the quark sector (assuming the same assignments for
all the families). Besides the cases given above, one can have an
MCHM$_{5-1-10}$, MCHM$_{14-10-10}$, MCHM$_{10-14-10}$,
MCHM$_{14-5-10}$ and MCHM$_{5-14-10}$. This would exhaust all the
models that allow to write Yukawa couplings (in particular for the top
quark, which is hard to imagine arising from other than tree-level
effects). For instance, the MCHM$_{10-1-X}$ does not allow to write
the operator $y_u (\Phi^\dagger \bar{Q}_L \Phi) U_R + {\rm h.c.}$
since it vanishes due to the antisymmetry of the $\bf 10$. Some of
these models (the MCHM$_{14-5-10}$ and MCHM$_{5-14-10}$), like the
MCHM$_{14-14-10}$ described in detail in Sec.~\ref{sec14-14-10}, allow
for two Yukawa structures in the up sector, which can a priori lead to
qualitative differences with the remaining models that allow only a
single Yukawa structure. We will study in detail only the
MCHM$_{14-14-10}$ to illustrate the possible features in such cases,
and will restrict our comments for the models mentioned in this
subsection to only a few general remarks in the following sections
(but enough to get a feel for their phenomenology).
\section{Corrections to Low-Energy Observables in the MCHM}
\label{sec:corrections}
To analyze the low-energy consequences of the model one can either
diagonalize the gauge and fermion mass matrices, explicitly including
the heavy states and their mixing with the elementary fields. The SM
fields are then identified as the lowest lying states in the presence
of a given $\langle h \rangle$. The latter is actually determined
dynamically as discussed in Sec.~\ref{sec_div_VH}, but the procedure
works for any fixed vev. Finding the Higgs mass, however, requires
the minimization of the potential, and incorporating this information
will be deferred to later sections.
Alternatively, one can obtain an effective theory for the fields on
site-0, as done in Sec.~\ref{sec:EFT} for the gauge fields and in
Sec.~\ref{sec:models} for the fermion sector. The zeroes of the
correlators thus obtained determine the spectrum of the model. The
correlators also encode in their Higgs vev dependence information
regarding the couplings of the physical fields and the Higgs boson, as
discussed in the previous section.
Although the numerical analysis to be presented in
Sec.~\ref{sec:pheno} has been obtained by the previous methods (and we
have checked that they agree), it is useful to have a simple analytic
approximation that captures the main phenomenological features of the
Higgs sector in composite Higgs models. To do so, one starts from the
following relation that holds in the simplest situations, which
includes most of the models we study:
\begin{eqnarray}
\sum_n \frac{y^{(n)}_\psi}{m^{(n)}_\psi} &=& \frac{1}{2} \, \frac{d}{dh} \log {\rm det} (M^\dagger_\psi M_\psi) ~=~ \frac{1}{s_h f_h} \, F_\psi(s_h)~,
\label{BasicRelation}
\end{eqnarray}
where $m^{(n)}_\psi$ and $y^{(n)}_\psi$ are the mass and the Yukawa
coupling of the n-th fermionic resonance to the Higgs, respectively,
and $M_\psi$ is the $h$-dependent mass matrix. The fact that the
above trace depends only on $s_h = \sin(h/f_h)$, but not on other
parameters of the model~\footnote{However, one should remember that
$\langle h \rangle$ itself is determined by the effective potential,
which is calculable and depends on various microscopic parameters.
Therefore, the most precise statement is that the r.h.s.~of
Eq.~(\ref{BasicRelation}) depends on the microscopic parameters only
through $h/f_h$.} is not a general statement, but a consequence of the
particular models considered in this work. In the simplest situation
there is just one Yukawa term that leads only to one non-trivial SO(4)
invariant for each sector, resulting in a determinant that factorizes
as $\det (M^\dagger_\psi M_\psi) = \hat{F}_\psi(s_h)\
h_\psi(y,\Delta,m)$. Therefore, its logarithmic derivative depends
only on $s_h$ and $f_h$. $F_\psi(s_h)$ is a model-dependent function
that depends on the representation of the fermions under
$G_1$~\cite{Falkowski:2007hz,Low:2010mr,Azatov:2011qy}.
In the general situation, for arbitrary representations of the
composite fermions, there is more than one non-trivial SO(4) invariant
arising from the Yukawa interactions in each sector. The determinant
does not factorize in this case and its derivative generically depends
on other microscopic parameters as well, such as the composite Yukawa
couplings. This is the case for the most general MCHM$_{14-14-10}$
discussed in Sec.~\ref{sec14-14-10}. This could be important for the
phenomenology, since in the general case one could in principle obtain
enhancement or suppression of the gluon fusion process in different
regions of the parameter space, while there is no such freedom for the
minimal cases with just one invariant.
Under the assumption that Eq.~(\ref{BasicRelation}) holds, the
additional useful observation is that, to leading order in $\epsilon =
\sin(v/f_h)$, the sum is saturated by the zero-mode term, leading to
\begin{eqnarray}
\frac{y^{(0)}_\psi}{m^{(0)}_\psi} &\approx& \frac{1}{\epsilon f_h} \left[ F_\psi(\epsilon) + {\cal O}(\epsilon^2 s_{\psi_L}^2) + {\cal O}(\epsilon^2 s_{\psi_R}^2) \right]~,
\label{BasicApproximation}
\end{eqnarray}
where $s_{\psi_L}$ and $s_{\psi_R}$ are the LH and RH
elementary-composite mixing angles, respectively. This was explicitly
shown in Sec.~\ref{sec:models} for each model, and in
Sec.~\ref{sec:pheno} we will further show numerically that the above
approximation works reasonably well even in the top sector (we will
also discuss the cases where important deviations arise).
Except for the case considered in Sec.~\ref{sec14-14-10} and two
embeddings described in Sec.~\ref{sec:othermodels}, we find only two
different functions for the models considered in this work:
\begin{equation}
F_1=\frac{1-2\epsilon^2}{\sqrt{1-\epsilon^2}}~, \qquad\qquad
F_2=\sqrt{1-\epsilon^2}~.
\label{F1F2}
\end{equation}
The MCHM$_{14-14-10}$ presented in Sec.~\ref{sec14-14-10} is somewhat
different in that two different Yukawa structures are allowed [see
Eqs.~(\ref{14y1}) and (\ref{14y2})]. As a result, the trace involves
a function with a non-trivial dependence on these Yukawa couplings,
not just on $\epsilon$:
\begin{equation}
\frac{1}{\epsilon f_h} \, F_3 \equiv {\rm tr}(Y_uM_u^{-1}) = \frac{1}{\epsilon f_h} \, \frac{\left(6 \epsilon^2-3\right) y_u-2 \left(20 \epsilon^4-23 \epsilon^2+4\right) \tilde y_u}{\sqrt{1-\epsilon^2} \left(2 \left(5 \epsilon^2-4\right) \tilde y_u-3 y_u\right)} \ ,
\label{F3}
\end{equation}
which can change the size and sign of $F_3$. Being $F_3$ a
homogeneous function of the Yukawa couplings, it depends only on the
ratio $r_y=\tilde y_u/y_u$. For $r_y=0$ one recovers the $F_1$
function of the other models: $F_3|_{r_y=0}=F_1$. In the opposite
limit we define a new function
\begin{eqnarray}
\tilde{F}_3 \equiv \lim\limits_{r_y\to\infty}F_3 =
\frac{4-23\epsilon^2+20\epsilon^4}{\sqrt{1-\epsilon^2} \, (4-5
\epsilon^2)}~.
\label{F3tilde}
\end{eqnarray}
For $r_y\to\infty$ one can obtain in principle a large suppression,
since $\tilde{F}_3$ changes sign for $\epsilon\simeq 0.46$. $F_3$
interpolates between $F_1$ and $\tilde{F}_3$ as $r_y$ varies, thus one
can expect a suppression larger than $F_1$ in the general case (see
right panel of Fig.~\ref{YukApprox}). However there is a small region
of the parameter space where there could be an enhancement and a
violent change of sign of $F_3$, as a consequence of an accidental
cancellation in $\det M_u$ that leads to a singularity of $F_3$
(this has also been observed in Ref.~\cite{Montull:2013mla}). This
is connected to the existence of a very light resonance in this
region. For $\epsilon\in(0,0.5)$ the singularity is present if
$r_y\in(-6/11,-3/8)$, thus for points of the parameter space near the
singularity the value of $F_3$ can be very large, changing sign across
the singularity. Although a large correction in any direction is
possible in this model it requires tuning of the Yukawa couplings.
This large correction, being associated with a zero of $\det M_u$,
signals the presence of a very light mode in the spectrum, that can be
in conflict with bounds on top partners. Moreover, by performing a
random scan we have checked that the points able to reproduce the
spectrum and EW constraints are usually far from the singularity.
Thus, we typically obtain a suppression as opposed to an enhancement
from this more complicated function.
Another important consequence is that the presence of two different
flavor structures leads to missalignement of Higgs coupling in LR
operators~\cite{Mrazek:2011iu}. For anarchic models, these new
sources of flavor violation mediated by Higgs exchange are too large
compared with bounds from flavor physics, requiring extra protection.
For this reason we will perform one scan imposing $\tilde y_u=0$, and
a second one allowing $\tilde y_u\neq0$. It turns out that the latter
ends up preferring regions with $y_u \ll \tilde y_u$, so that it is
effectively described by $\tilde{F}_3(\epsilon)$ given in
Eq.~(\ref{F3tilde}) above.
The other models mentioned in Sec.~\ref{sec:othermodels} can be
described by the same $F_i(\epsilon)$ above, except for the
MCHM$_{14-5-10}$ and MCHM$_{5-14-10}$ which lead to the following new
functions that, like the one for the MCHM$_{14-14-10}$, also depend on
the microscopic Yukawa couplings [$F_4$ and $F_5$ are defined in
analogy to Eq.~(\ref{F3})]:
\begin{eqnarray}
F_4 &=& \frac{\sqrt{1-\epsilon ^2} \left( y_u + 2 \tilde{y}_u - 6 \tilde{y}_u \epsilon^2 \right)}{y_u + 2 \tilde{y}_u\left(1 - \epsilon^2 \right)}~,
\hspace{1cm}
F_5 ~=~ \frac{\sqrt{1-\epsilon ^2} \left(y_u - \tilde{y}_u \left(4-15 \epsilon^2\right)\right)}{y_u - \tilde{y}_u \left(4 - 5 \epsilon ^2\right)}~.
\label{F4F5}
\end{eqnarray}
In the limiting cases where only one of the two Yukawa couplings is
turned on, the above become functions of $\epsilon$ only. In such
limits, they lie between the curves for $F_1$ and $\tilde{F}_3$ in the
right panel of Fig.~\ref{YukApprox} in Sec.~\ref{sec:couplings} (they
are not shown in the figure).
\begin{table}[tb]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\rule{0mm}{5mm}
$r$/ MCHM & 10-5-10 & 5-5-10 &
\begin{tabular}{c} 5-10-10, \\ 5-1-10\end{tabular} &
\begin{tabular}{c} 5,\ 10, \\ 14-1-10 \\ 14-10-10 \\ 10-14-10 \end{tabular} &
14-14-10 &
14-5-10 &
5-14-10 \\ [0.3em]
\hline
\rule{0mm}{5mm}
$r_t$ & $F_2$ & $F_1$ & $F_2$ & $F_1$ & $ F_3$ & $ F_4$ & $ F_5$ \\ [0.3em]
\hline
\rule{0mm}{5mm}
$r_b$ & $F_1$ & $F_2$ & $F_2$ & $F_1$ & $ F_1$ & $ F_1$ & $ F_1$ \\ [0.3em]
\hline
\rule{0mm}{5mm}
$r_V$ & $F_2$ & $F_2$ & $F_2$ & $F_2$ & $F_2$ & $ F_2$ & $ F_2$ \\ [0.3em]
\hline
\rule{0mm}{5mm}
$r_g$ & $F_2$ & $F_1$ & $F_2$ & $F_1$ & $ F_3$ & $ F_4$ & $ F_5$ \\ [0.3em]
\hline
\end{tabular}
\end{center}
\caption{Ratio of Higgs SM and MCHM couplings, $r=c^{MCHM}/c^{SM}$,
approximated by the functions $F_i$. $g$ stands for the loop induced
gluon coupling (we have only considered the top sector effect for
$r_g$ in this table, but in the numerical results we have included the
bottom sector as well), $\psi=t,b$ are the Yukawa couplings and
$V=W,Z$ is the coupling to the massive EW gauge bosons. For
completeness, we include also the result for additional models that
were not described in full detail in the main text.}
\label{table-F}
\end{table}
The $F_i$ functions defined in Eqs.~(\ref{F1F2})-(\ref{F4F5}) encode
the deviations from the SM couplings, $r=c_{\rm MCHM}/c_{\rm SM}$, as
shown in Table~\ref{table-F},~\footnote{Some of these functions have
been shown previously in Refs.~\cite{Pomarol:2012qf}
and~\cite{Montull:2013mla}.} and determine the $c_i$ coefficients of
the following set of operators in the low-energy theory:
\begin{align}
&{\mathcal O}_{g} = h \, G^a_{\mu\nu}G^{a \, \mu\nu}~, \qquad
{\mathcal O}_{\gamma} = h \, A_{\mu\nu}A^{\mu\nu}~, \qquad
{\cal O}_{Z\gamma} = h \, A_{\mu\nu}Z^{\mu\nu}~,
\label{Off} \\
&{\cal O}_{w} = h \, W^+_\mu W^{-\mu}~, \qquad
{\cal O}_{z} = h \, Z_\mu Z^{\mu}~, \\
&{\mathcal O}_f = \bar q_L H f_R+{\rm h.c.}
\label{Of}
\end{align}
These are the leading order operators involved in Higgs production and
decay at the LHC. Since the operators ${\mathcal O}_{g}$ and
${\mathcal O}_{\gamma}$ break the shift symmetry of the pNGB Higgs and
must, therefore, involve the explicit symmetry breaking parameters
such as the SM gauge and Yukawa couplings, they are generated at loop
level. Our computation gives the contributions to the Wilson
coefficients of these operators in the MCHM after EWSB to all order in
the Higgs vev, leading to coefficients $c_{\cal O}(v/f)$. Expanding
these coefficients in powers of $v/f$ one can do the matching to the
Wilson coefficients of dimension-six operators which, in the basis of
Refs.~\cite{Elias-Miro:2013gya,Elias-Miro:2013mua,Pomarol:2013zra},
are
\begin{align}
&{\cal O}_H=\frac{1}{2}\left(\partial_\mu|H|^2\right)^2 ~, & {\cal O}_{y_f}=|H|^2\bar q_LHf_R ~, \nonumber \\
&{\mathcal O}_{GG}=|H|^2 G_{\mu\nu}G^{\mu\nu} ~, & {\mathcal O}_{BB}=|H|^2 B_{\mu\nu}B^{\mu\nu} ~, \nonumber \\
&{\cal O}_W=\frac{i}{2}\left(H^\dagger \sigma^a\overleftrightarrow D_\mu H\right)D^\nu W^a_{\mu\nu} ~,
&{\cal O}_B=\frac{i}{2}\left(H^\dagger \overleftrightarrow D_\mu H\right)\partial^\nu B_{\mu\nu} ~, \nonumber \\
&{\cal O}_{HW}=i\left(D^\mu H\right)^\dagger \sigma^a\left(D^\nu H\right) W^a_{\mu\nu} ~,
&{\cal O}_{HB}=i\left(D^\mu H\right)^\dagger \left(D^\nu H\right) B_{\mu\nu} ~.
\label{OpsDim6}
\end{align}
By redefining the Higgs field one can show that ${\cal O}_H$
renormalizes the Higgs couplings to all the other SM fields. ${\cal
O}_{GG}, {\cal O}_{BB}$ and ${\cal O}_-=({\cal O}_W-{\cal O}_B)-({\cal
O}_{HW}-{\cal O}_{HB})$ enter in the interactions $hgg, h\gamma\gamma$
and $hZ\gamma$, respectively, and ${\cal O}_{y_f}$ enters in $hf\bar
f$~\cite{Giudice:2007fh}. The Wilson coefficients $c_H, c_W$ and
$c_B$ are universal for all the MCHM with SO(5)/SO(4) breaking and
have been computed in the SILH description~\cite{Giudice:2007fh}:
\begin{equation}
c_H=1 \ ; \qquad\qquad
c_W= c_B=\frac{27\pi^2}{256}\simeq 1.0 \ .
\end{equation}
$c_y$ has been computed in~\cite{Giudice:2007fh} for the top sector in
the MCHM$_{5}$. In general it can be obtained from the functions
$F_\psi$ that codify the deviation of the Yukawa coupling, leading to:
\begin{align}
&c_{y_t}=1 \ , \qquad {\rm for\ the\ MCHM_{5,\ 10,\ 14-14-10,\ 14-1-10,\ 5-5-10}} \ , \nonumber \\
&c_{y_t}=0 \ , \qquad {\rm for\ the\ MCHM_{10-5-10,\ 5-10-10}} \ , \nonumber \\
&c_{y_b}=1 \ , \qquad {\rm for\ the\ MCHM_{5,\ 10,\ 14-14-10,\ 14-1-10,\ 10-5-10}} \ , \nonumber \\
&c_{y_b}=0 \ , \qquad {\rm for\ the\ MCHM_{5-5-10,\ 5-10-10}} \ .
\end{align}
The coefficients $c_{g,\gamma}$ and $c_{HW,HB}$ are generated at loop
level. Starting with ${\cal O}_g$, this operator is generated by
fermion loops. For each fermion species there is a contribution (see
App.~\ref{app:loops})
\begin{equation}
c_{g}\propto \sum_n \frac{y_n}{m_n}A_{1/2}(\tau_n) \ , \qquad \tau_n=\frac{m_h^2}{4m_n^2} \ .
\end{equation}
For heavy fermions, $\left.A_{1/2}(\tau)\right|_{\tau \to 0} \to 4/3$. Thus,
considering heavy resonances we obtain:
\begin{equation}
\label{eq_cg}
c_{g} \propto \frac{4}{3}\left[{\rm tr}(Y_\psi M_\psi^{-1})-\frac{y^{(0)}_\psi}{m^{(0)}_\psi}\right] + \frac{y^{(0)}_\psi}{m^{(0)}_\psi}A_{1/2}(\tau_0)~,
\end{equation}
with the index 0 referring to the would-be 0-mode, associated with the
SM mass eigenstate. The last term is similar to the SM one, up to
corrections in the Yukawa coupling. These corrections are important
only if the mixing is large. Since $A_{1/2}(\tau)\to_{\tau\to \infty}
0$, this term is small for light fermions, $m_\psi \ll m_h$. As was
shown in Sec.~\ref{sec:models}, the first term is also small if the
mixing of both, the Left and Right chiralities, is small. For the top
quark one can take the limit $A_{1/2}(\tau_t)\to 4/3$, and
Eq.~(\ref{eq_cg}) is dominated by $4/3\ {\rm tr}(Y_t M_t^{-1})$, which is
the sum considered in Eq.~(\ref{BasicRelation}). Thus, one can also
obtain an approximate expression for the gluon fusion process in terms
of the functions above, as shown in Table~\ref{table-F}. For the
coupling of the Higgs to two photons, there is an additional
contribution due to the heavy spin-1 resonances. However, a similar
sum rule applies which allows to obtain an approximate analytical
expression. These will be studied in more detail in
Sec.~\ref{sec:pheno}, after taking into account the constraints from
the recently measured Higgs mass~\cite{measured-h-mass}, as well as
the masses of the $Z$ gauge boson and the top and bottom quarks, which
have the most important impact on the Higgs potential and the Higgs
phenomenology.
\section{Higgs potential}
\label{sec_div_VH}
Discrete models of pNGB Higgs can lead to a finite Higgs potential
under some suitable assumptions. The degree of divergence of the
Higgs potential depends on the particular mechanism of collective
breaking, being thus model dependent. There are at least two concepts
involved: distance between the sites where the symmetries protecting
the pNGB potential are broken, and number of symmetries broken on each
site.
The Higgs potential can be computed by the holographic method
\begin{eqnarray}
\label{ec_VHholo}
V(h) &=& \int \! \frac{d^4p}{(2\pi)^4} \left[ \frac{6}{2}\sum^2_{i = 1} \log \Pi_{w^i_L} + \frac{3}{2}\log \left[\Pi_{w^3_L} \Pi_{b} - (\Pi_{w^3_L \, b})^2 \right]
\right.
\nonumber \\ [0.4em]
& & \hspace{1cm}
\left. \rule{8mm}{0mm} - 2N_c\sum_\psi \log[p^2\Pi_{\psi_L} \Pi_{\psi_R} - |M_{\psi}|^2] \right] \ ,
\end{eqnarray}
where the correlators are obtained from Secs.~\ref{sec:EFT} and
\ref{sec:models}, taking care to add the ``bare" kinetic terms, as in
Eqs.~(\ref{Lgauge}) and (\ref{Leff-fermions}), which were not included
as part of the definition of the correlators in those sections:
\begin{eqnarray}
\Pi_{w^i_L} = \frac{p^2}{g^2_0} + \Pi_{\tilde{w}^i_L}~,
\hspace{1cm}
\Pi_{w^3_L \, b} = \Pi_{\tilde{w}^3_L \, \tilde{b}}~,
\hspace{1cm}
\Pi_{b} = \frac{p^2}{g^{\prime 2}_0} + \Pi_{\tilde{b}}~,
\end{eqnarray}
and similarly for the fermionic correlators. Equivalently, one can
use the standard expression for the Coleman-Weinberg potential in
terms of determinants involving the Higgs-dependent mass matrices of
the gauge and fermion fields. We have checked that the same results
can be reproduced with either approach. Note that
Eq.~(\ref{ec_VHholo}) contains the photon, although it does not
contribute to the Higgs potential, and one can regularize the
divergent constant terms by subtracting $V(0)$.
\subsection{Finiteness of the 1-loop Higgs potential}
\label{sec:finiteness}
In this subsection we illustrate in a toy example how the
inclusion/exclusion of certain operators in the Lagrangian affects the
divergence structure of the Higgs potential. Our example is based on
the fundamental representation of SO(5), but the conclusion holds for
other representations as well. In order to understand the structure
of divergences of the $h$-dependent terms, let us consider the 2-site
model with the following set of fields:
\noindent {\bf site 0}: An elementary fermion doublet $q_L$ and a
singlet $t_R$ of a global symmetry $G_0 = {\rm SU}(2)_L$.\footnote{For
simplicity we ignore U(1)$_Y$ in this discussion.}
\noindent {\bf site 1}: Four chiral composite fermions $Q_L, Q_R, T_L,
T_R$, each transforming in the fundamental representation of a
different global SO(5), called: $G_{Q_L}$, $G_{Q_R}$, $G_{T_L}$,
$G_{T_R}$. In this site there is also a scalar $\Phi_1$ transforming
in the fundamental of another SO(5), called: $G_1$. The vev of
$\Phi_1$ spontaneously breaks $G_1$ to ${\cal H}_1= {\rm SO(4)}$.
Notice that before introducing fermion masses, each chiral fermion of
the composite sector transforms independently, leading to a large
global symmetry (in fact, the symmetry is much larger, but we need
only focus on this subgroup). The Higgs, being a NGB, is in the coset
$G_1/{\cal H}_1$. The following operators break different symmetries:
\begin{itemize}
\item $m_{Q} \, \bar QQ$: $G_{Q_L} \times G_{Q_R} \to G_{Q_{L+R}} = {\rm SO(5)}$~,
\item $m_T \, \bar TT$: $G_{T_L} \times G_{T_R} \to G_{T_{L+R}} = {\rm SO(5)}$~,
\item $\Delta_q \, \bar q_LQ_R + {\rm h.c.}$: $G_0 \times G_{Q_R} \to G_{Q_R+0} = {\rm SU(2)}$~,
\item $\Delta_t \, \bar t_RT_L + {\rm h.c.}$: $G_0 \times G_{T_L} \to G_{T_L+0} = {\rm SU(2)}$~,
\item $y_T \, \bar Q_L\Phi_1\Phi_1^\dagger T_R + {\rm h.c.}$: $G_{Q_L} \times G_{T_R} \times G_1 \to G_{Q_L+T_R+1} = {\rm SO(5)}$~,
\item $y'_T \, \bar Q_R\Phi_1\Phi_1^\dagger T_L + {\rm h.c.}$: $G_{Q_R} \times G_{T_L} \times G_1 \to G_{Q_R+T_L+1}= {\rm SO(5)}$~.
\end{itemize}
There is some abuse of notation in the previous paragraph, since
$G_{Q_R,T_L}$ and $G_0$ have different dimensions, so that when writing
$G_{T_L+0}$ we really mean the diagonal subgroup $G_0'=$SU(2). In
addition to the above, the symmetries allow operators of the form $\bar
Q_L\Phi_1\Phi_1^\dagger Q_R + {\rm h.c.}$ or $\bar
T_L\Phi_1\Phi_1^\dagger T_R + {\rm h.c.}$, which would also lead to
divergences in the Higgs potential of the 2-site model. With three or
more sites, these would lead to a finite 1-loop
result~\cite{Panico:2011pw,DeCurtis:2011yx}. For illustration, we
limit the following discussion to the operators listed above.
A Higgs potential requires insertions of $y_T$ and/or $y'_T$. Let us
consider the following cases:
\noindent {\bf (a)} $y'_T=0$: The $y_T$ term only preserves the
diagonal subgroup $G_{Q_L+T_R+1}$. The Higgs is in the coset
$G_{Q_L+T_R+1}/{\cal H}_1$, and thus a Higgs potential requires
explicit breaking of $G_{Q_L+T_R+1}$. This necessitates interactions
with the elementary sector, which arise from the $\Delta_q$ and/or
$\Delta_t$ terms. However, due to their chirality structure,
insertions of $\Delta_{q,t}$ still do not break $G_{Q_L+T_R+1}$:
$G_{Q_L+T_R+1}\times$$G_0$ is broken only after additional $m_{Q,T}$
insertions. Thus,
\begin{equation}
V_H\sim (\Delta_{q,t} m_{Q,T} y_T)^2\ .
\end{equation}
\noindent {\bf (b)} $y_T=0$: The $y'_T$ term only preserves the
diagonal subgroup $G_{Q_R+T_L+1}$ and the Higgs is in the coset
$G_{Q_R+T_L+1}/{\cal H}_1$. In this case, insertions of $\Delta_q$
and/or $\Delta_t$ break $G_{Q_R+T_L+1}\times$$G_0$ without the need of
$m_{Q,T}$ insertions:
\begin{equation}
V_H\sim (\Delta_{q,t} y'_T)^2\ .
\end{equation}
The previous arguments show how the dimension of the operators
leading to $V_H$ depends on the presence of $y'_T$, leading to
logarithmic divergences at 1-loop for $y'_T\neq0$. The presence of the
operators $m_{Y_t}\ \bar Q_L T_R$ and $m'_{Y_t}\ \bar Q_R T_L$
modifies the potential but not its degree of divergence.
One can also understand this result from Feynman diagram
considerations. For instance, the contribution to the quartic term in
$\Phi$, at leading order in insertions of $m_\psi$ and $\Delta_\psi$
is given by:
\hspace{1.2cm}
\includegraphics[width=0.85\textwidth]{Figures/Feynman-DIvergence.pdf}
\noindent and similar diagrams changing $q\leftrightarrow t$ and
$Q_{L,R}\leftrightarrow T_{R,L}$. These diagrams allow to understand
the superficial degree of divergence of $V_H$ depending on which
operators are present in the theory.
\section{Higgs Phenomenology}
\label{sec:pheno}
We turn now to the Higgs phenomenology of the composite Higgs models
previously described. We present in this section the results of a
detailed numerical analysis obtained by scanning over a sizeable
region of the parameter space of each model. The minimization of the
Higgs potential will be fully taken into account. Note, however, that
we assume that for the light fermion generations both the LH and RH
chiralities have a small degree of compositeness, as opposed to
allowing one of them to have a large mixing angle with the composite
sector, and the other a very suppressed one that accounts for the
small SM fermion
mass~\cite{Redi:2011zi,DaRold:2012sz,Redi:2013eaa,Delaunay:2013pwa}.
This assumption is more natural given the EW precision tests, which
indicate that the light quarks and leptons are mostly elementary,
although one could imagine exploring the second option. As a result,
the Higgs potential is affected mainly by the top and bottom sectors,
as well as by the gauge sector of the models. Nevertheless, when
discussing the Higgs decays we will take into account some of the
light fermions, most prominently the $\tau$ lepton, as discussed
below.
\subsection{Numerical Scan}\label{sec-num-scan}
The effective description of a composite Higgs described in the
previous sections depends on a number of parameters. The gauge sector
is described at the Lagrangian level by the two decay constants
$\{f_\Omega, f_{\Omega_X}\}$ and gauge couplings $\{g_\rho, g_X\}$
associated with the SO(5) and ${\rm U(1)}_X$ (composite) factors,
while in the elementary sector one has the two gauge couplings $g_0$
and $g'_0$ [see Eqs.~(\ref{LOmega}) and (\ref{Lgauge})]. The latter
are related to the SM gauge couplings as given in Eq.~(\ref{SMggp}),
while it is convenient to parametrize the composite gauge couplings in
terms of the elementary/composite mixing angles of the gauge sector:
$t_\theta = g_0 / g_\rho$ and $t_{\theta'_X} = g'_0 / g_X$. However,
for simplicity, in our scan we will fix $g_X$ by imposing the relation
discussed after Eq.~(\ref{SMggp}), so that there is effectively a
single gauge mixing angle $t_\theta$. The two decay constants can in
turn be exchanged for the two mass scales $m_\rho$ and $m_X$ defined
in Eq.~(\ref{mrhomX}), but it is more convenient to scan over a subset
of the physical masses after taking into account the
elementary/composite mixing effects (before including EWSB
effects). Thus, we choose to scan over $m_{\tilde{\rho}} = \sqrt{1 +
t^2_\theta} \, m_\rho = m_\rho / c_\theta$ [see discussion of the last
paragraph of Sec.~\ref{BosonicSector}], and we also choose the
variable $m_{\tilde X} = m_X / c_\theta$. However, since we focus on
a region of parameter space with $t_\theta \ll 1$, quantitatively
there is not a large difference between $m_{\tilde \rho}$ and $m_\rho$
or $m_{\tilde X}$ and $m_X$.
The fermion sector depends on a set of ``diagonal" masses $m_\Psi$,
one for each composite fermion, and on the ``off-diagonal" masses
$m_{y_u}$ and $m_{y_d}$ of Eq.~(\ref{Lfermions}). The composite
sector also involves a number of ``Yukawa-like" mass parameters that
we have called $y_u$ and $y_d$ [see Eqs.~(\ref{LY5}), (\ref{LY10}),
(\ref{LY10510}), (\ref{LY5510}), (\ref{LY141410}), (\ref{14y2}) and
(\ref{LY14110}) which define these for each model]. In spite of the
notation, the $y_\psi$ have dimensions of mass, although they
represent interactions with the Higgs field $\Phi$. Finally, there
are the mixing parameters, $\Delta_q$, $\Delta_u$ and $\Delta_d$,
which also have mass dimension 1. In practice, the scan will be
restricted to the third generation, so that one should reinterpret the
indices as $u \to t$ and $d \to b$. We find convenient to exchange
the mixing parameters $\Delta_\psi$ for ``mixing angles" defined by
$t_{\psi} \equiv \tan \theta_\psi = \Delta_\psi / m_\Psi$, where
$\Psi$ is the composite fermion associated with the elementary fermion
$\psi$ [for the MCHM$_5$ we introduce two mixing angles $t_{q^u}$ and
$t_{q^d}$ corresponding to $\Delta_{q^u}$ and $\Delta_{q^d}$; see
comments after Eq.~(\ref{Lfermions})]. Analogously to the gauge
sector above, we also prefer to scan over diagonal fermion masses that
have been rescaled according to $m_{\tilde \Psi} = m_\Psi / c_\psi$,
where $c_\psi = \cos \theta_\psi$ involves the corresponding mixing
angle defined above. This choice leads to light custodians when the
mixings are large, since their masses are given by $m_{\rm cust} \sim
{\cal O}(m_{\tilde \Psi} c_\psi$)~\cite{Pomarol:2008bh,DaRold:2010as}.
Thus, the parameters for the fermionic sector consist of $\{ m_{\tilde
\Psi}, t_\psi , m_{y_\psi}, y_\psi \}$, where the indices run over the
field content in each model, as described in Sec.~\ref{sec:models} [we
fix $Z_\psi = 1$ in Eq.~(\ref{Lfermions})].
Since one expects that the masses of the various resonances will be of
the same order, for simplicity we have fixed a common mass scale, by
restricting our scan to $m_{\tilde \rho} = m_{\tilde Q} = m_{\tilde U}
= m_{\tilde D}$ (for the MCHM$_5$ we impose the condition on
$m_{\tilde Q^u}$ and $m_{\tilde Q^d}$). This is not necessary, but we
do not expect that the results will depend on this simplifying
assumption.\footnote{Note that the physical masses are obtained after
taking into account all the mixing effects, as well as EWSB, and will
therefore present a nontrivial spread. It is also worth noting that
by scanning over $m_{\tilde \rho}$, $m_{\tilde X}$ and $m_{\tilde
\Psi}$, \textit{i.e.}~by factoring out the elementary/composite mixing
angles, we are proceeding in analogy to the extra-dimensional
realizations, where the compactification scale and therefore the
overall Kaluza-Klein (KK) scale is treated as an input parameter. The
elementary/composite mixing angles of the 4D realization are related
to the 5D localization parameters and boundary conditions for the
various fields. When obtaining the exact spectrum one can get modes
much lighter than the overall KK scale, typically for large mixing
angles in the third generation fermionic sector.} Thus, the final set
of parameters used in the scan is
\begin{eqnarray}
\{ f_h, m_{\tilde \rho}, t_\theta, t_q, t_t, t_b, m_{y_T}, m_{y_B}, y_{T}, y_{B} \}~,
\end{eqnarray}
where we used the notation $y_T$ and $y_B$ instead of $y_t$ and $y_b$
to avoid confusion with the SM top and bottom Yukawa couplings, and we
also included in the list the Higgs decay constant $f_h$ defined by
Eqs.~(\ref{LNGBSimple}) and (\ref{eq:fh}). We also chose to fix
$m_{\tilde X} = s_{\theta_W} / \sqrt{c_{2\theta_W}} \, m_{\tilde \rho}
\approx 0.65 \, m_{\tilde \rho}$, which amounts to fixing
$f_{\Omega_X} = f_\Omega$ in Eq.~(\ref{LOmega}), given the choice of
$g_X$ described above. We choose $1/5 \leq t_\theta \leq 1/3$, so
that $g_\rho$ is large but perturbative, and scan over the fermionic
mixing angles according to $s_\psi \in [0.4, 1]$, with a uniform
distribution (but we adjust $s_b$ to reproduce the bottom quark mass
with little effect on the EWSB properties of the parameter point).
For the mass parameters, ($m_{\tilde \rho}, m_{y_T}, m_{y_B}, y_{T}
\textrm{ and } y_{B}$), we scan in units of $f_h$ as follows:
\begin{itemize}
\item $m_{\tilde{\rho}} / f_h \in [2.5, 5]$, which is consistent with
the underlying relation $m_{\rho} \sim g_\rho f_h$ with $g_\rho$ in
the range of interest,
\item $| y_\psi / f_h | < 2\pi$, which encodes the idea of having a
perturbative proto-Yukawa coupling,
\item and $| m_{y_T} / f_h |, | m_{y_B} / f_h | \lesssim 2\pi$~,
\end{itemize}
while $f_h$ is scanned over a wide range, but we choose only points
with $\epsilon < 0.5$, which corresponds to $f_h \gtrsim 500~{\rm
GeV}$. The final set of points has $f_h$ as large as $\sim 2.5~{\rm
TeV}$ (except for the MCHM$_{5-10-10}$, which has some points with
$f_h$ as large as $\sim 6~{\rm TeV}$). We also required in the final
set of points that $m_{\tilde \rho} > 2~{\rm TeV}$. This final set of
numbers already assumes that we have normalized to $m_Z$ (see below).
Having chosen a given point in the parameter space described above, we
minimize the 1-loop Higgs potential to select those points that do
break the EW symmetry. For each such point, we can rescale all
parameters with dimension of mass so as to reproduce $m_Z$, thereby
normalizing to the EW scale. We further select those points where the
Higgs mass matches the measured value of $\sim 125~{\rm GeV}$, and
also select those points where the top and bottom quarks match the
experimental observations. In practice, our final points have $m_h
\in [120-130]~{\rm GeV}$, $m_t \in [140-170]~{\rm GeV}$ and $m_b
\approx 2.7~{\rm GeV}$.\footnote{We note that the relevant masses from
the point of view of the scan should be the running masses at the
scale where the heavy resonances are integrated out. These would then
be run down to the weak scale with the SM RGE's to make contact with
the experimental measurements. Since each parameter point has a
different scale for the heavy resonances, we have simply defined
generous windows to capture the spirit of the matching procedure.
Although a more precise analysis is possible, we do not expect that
the conclusions will change.} We can then compute the couplings of the
Higgs to the vector bosons and fermions (both the SM ones as well as
the new resonances), which are then used as input to compute the Higgs
production cross sections and branching fractions. This is done
numerically without any approximations, as is done for the 1-loop
induced couplings ($hgg$, $h\gamma\gamma$ and $hZ\gamma$) which are
computed using the exact spectrum and couplings to the Higgs.
However, we also compare to the analytical approximation described in
Sec.~\ref{sec:corrections}, which in general gives a qualitative
understanding of the numerical scan.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Figures/gmt_vs_mh_Detailed_Models.pdf}
\hspace{3mm}
\includegraphics[width=0.48\textwidth]{Figures/gmt_vs_mh_New_Models.pdf}
\caption{A random subset of the points that present electroweak
symmetry breaking, but without requirements on the Higgs, top or
bottom masses (however, we have normalized to $m_Z$). The vertical
and horizontal bands indicate the windows we have defined for $m_h$
and $m_t$. In the left panel we show the models we have presented in
detail in Sec.~\ref{sec:models}. In the right panel we show the
models mentioned in Sec.~\ref{sec:othermodels}, showing again the
MCHM$_5$ for comparison.}
\label{fig:mtvsmh}
\end{figure}
In Fig.~\ref{fig:mtvsmh} we display a random subset of the scanned
points that display EWSB, in the plane of $m_t$ versus $m_h$ (after
normalization to $m_Z$). We have not imposed here any requirements on
$m_h$, $m_t$ nor $m_b$, only that the desired symmetry breaking
pattern be obtained and that the $b_R$ mixing angle be suppressed (as
is necessary to obtain a light bottom quark in models with just one
operator coupled to $q_L$). In the left panel we present the (color
coded) models described in detail in Sec.~\ref{sec:models}, showing
that some of the models reproduce more naturally the Higgs and top
masses than others. In particular, the models involving the 14
representation have a tendency to produce a too large
$m_h$~\cite{Pomarol:2012qf}, although one can find a few points in the
desired range at the price of tuning (the bands correspond to the
windows we have defined in the previous paragraph).
In the right panel, we show the same information for the models
mentioned without details in Sec.~\ref{sec:othermodels}, together with
the MCHM$_5$ for comparison purposes. We see that these models also
typically do not fall in the phenomenologically desired window: for
the MCHM$_{5-1-10}$ the quartic coupling is usually too small, since
the only source of breaking is the mixing with $q_L$, that leads to a
factor $s_h$ in $\Pi_{u_L}$ and $s_h^2$ in $M_u$, in agreement with
the results found in~\cite{Pomarol:2012qf}. The MCHM$_{10-14-10}$
leads to a heavy Higgs. The MCHM$_{14-5-10}$ and MCHM$_{5-14-10}$
allow for two independent proto-Yukawa interactions: ${\cal
L}_y\supset y_u \bar\Psi_5\Psi_{14}\Phi+\tilde y_u
(\bar\Psi_5\Phi)(\Phi^\dagger\Psi_{14}\Phi)$, similar to the
MCHM$_{14-14-10}$. Both of them generically lead to a heavy Higgs,
while EWSB prefers $\tilde y_u\neq 0$ for the MCHM$_{14-5-10}$ and
$y_u\neq 0$ as well as $\tilde y_u\neq 0$ for the MCHM$_{5-14-10}$.
For the remaining three models we did not find points with the proper
$m_h$ and $m_t$ by performing a random scan. Finally, the
MCHM$_{14-10-10}$ generically does not lead to EWSB.
In all these models there is a correlation between $m_h$ and
$m_t$~\cite{Contino:2006qr}, that can usually be approximated by:
$m_h^2\sim a \frac{N_c}{\pi^2}\frac{m_t^2}{f_h^2}m_\psi^2$, with
$m_\psi$ the scale of the lightest fermionic resonance cutting off the
1-loop potential and $a$ a factor that is model dependent. Usually
$a\sim{\cal O}(1)$, however in some cases it can be suppressed
$a\sim{\cal O}(\epsilon^2)$ or enhanced $a\sim{\cal
O}(\epsilon^{-2})$, as shown in~\cite{Pomarol:2012qf}. The analytical
approximations of~\cite{Pomarol:2012qf} are in qualitative agreement
with the full numerical results of Fig.~\ref{fig:mtvsmh}.
From here on we focus on the models described in detail in
Sec.~\ref{sec:models}, which seem to be phenomenologically preferred
due to the previous observations. As mentioned earlier, we analyze
the MCHM$_{14-14-10}$ in detail, even though it tends to produce too
heavy a Higgs, as it may serve also to illustrate the situation in
those models we do not elaborate any further. All the numerical
results of the following sections correspond to points that lie at the
intersection of two bands of Fig.~\ref{fig:mtvsmh}.
\subsection{Corrections to the Gauge and Yukawa Couplings}
\label{sec:couplings}
We start by comparing the simple analytical approximation described
in Sec.~\ref{sec:corrections} for the deviations in the Higgs
couplings to the SM gauge bosons and fermions w.r.t.~the SM
expectation [see also the discussion after Eq.~(\ref{eq-y})]. As
discussed there, this approximation is expected to work well when the
elementary/composite mixing angles are small, which typically happens
for the light fermions in our scenario. However, we find that even
for the top quark, the approximation $y_t \approx [F_t(\epsilon) /
(\epsilon f_h)] \, m_t$ is reasonably good, even when the mixing
angles are sizeable, provided there are no ``ultra-light" fermionic
resonances. This is illustrated in Fig.~\ref{YukApprox}, where we
show the bottom and top Yukawa couplings as a function of $\epsilon$
in several models (normalized to the corresponding SM Yukawa coupling,
$y^{\rm SM}_\psi \equiv m_\psi / v_{\rm SM}$ with $v_{\rm SM} \approx
246~{\rm GeV}$). The points correspond to a random scan over the
parameter space described in the previous subsection, while the solid
curves correspond to the approximation described in
Sec.~\ref{sec:corrections} (see Table~\ref{table-F}).
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Figures/yb_all_Models_WithScans.pdf}
\hspace{3mm}
\includegraphics[width=0.48\textwidth]{Figures/yt_all_Models_WithScans.pdf}
\caption{Bottom (left panel) and top (right panel) Yukawa couplings in
several models, normalized to the SM (defined as $y_\psi = m_\psi /
v_{\rm SM}$ with $v_{\rm SM} = 246~{\rm GeV}$). The points correspond
to a random scan in parameter space, while the solid curves correspond
to the analytic approximation discussed in the main text.}
\label{YukApprox}
\end{figure}
We see in the left panel of Fig.~\ref{YukApprox} that the
approximation described in Eq.~(\ref{BasicApproximation}) works very
well for the bottom sector all the way up to relatively large values
of $\epsilon$. A notable exception occurs for the ${\rm
MCHM}_{10-5-10}$ (green stars), where the analytic expectation, $F_1 =
\cos(2 v /f_h) / \cos(v/f_h)$, systematically overestimates the
suppression in $y_b$ compared to the SM. The sizeable deviation
observed can be understood by considering the next to leading order
term in the expansion of $y_b/y_b^{SM}$ in powers of $\epsilon$, as
shown in Sec.~\ref{sec:models}. We obtain that, after the selection
of points explained above, the coefficient of the ${\cal
O}(\epsilon^2)$ term for the MCHM$_{10-5-10}$ is of ${\cal O}(0.5)$.
In contrast, the corresponding coefficient for the MCHM$_{10}$,
MCHM$_{14-14-10}$ and MCHM$_{14-14-10}^{\rm simple}$ is of ${\cal
O}(0.1)$,\footnote{MCHM$^{\rm simple}_{14-14-10}$ refers to the model
described in Sec.~\ref{sec14-14-10} with $\tilde{y}_T = 0$ in
Eq.~(\ref{sec14-14-10}) [making $u \to T$]. We refer to the general
model with $y_T$, $y_B$ and $\tilde{y}_T$ turned on as
MCHM$_{14-14-10}$.} for the MCHM$_{5}$ and MCHM$_{14-1-10}$ it is of
${\cal O}(10^{-2})$, and for the MCHM$_{5-5-10}$ and MCHM$_{5-10-10}$
it is ${\cal O}(10^{-4})$, in all the cases increasing with $s_q$ as
expected. Since $h \to b\bar b$ is the dominant decay mode,
deviations of $y_b$ can have a deep impact in the Higgs phenomenology.
It is also interesting to note that the bulk of the points in the ${\rm
MCHM}_{10-5-10}$ display
relatively light ($Q = -1/3$) fermionic resonances, together with
relatively large mixing angles. We illustrate this in the left panel
of Fig.~\ref{sqvsmF}, where we show the largest of the mixing angles
$(s_q, s_t)$ versus the lightest vectorlike resonance mass in the
bottom sector. Indeed, most of the green stars (MCHM$_{10-5-10}$)
exhibit resonances below $1~{\rm TeV}$ and $s_q > 0.9$. Note that the
MCHM$_{10}$ (yellow $+$'s), the MCHM$_{14-14-10}$ (brown $*$'s), and
to a somewhat lesser extent the MCHM$^{\rm simple}_{14-14-10}$
(magenta $*$'s), also contain a subset of points with light states
together with sizeable elementary-composite mixing angles, which is
reflected in the somewhat larger dispersion in Fig.~\ref{YukApprox},
compared to the other models. However, note that the MCHM$_{14-1-10}$
(dark magenta \^{}'s) has light $Q = -1/3$ resonances together with
large mixing angles, and nevertheless follows the naive approximation
from Eq.~(\ref{BasicApproximation}) for the bottom Yukawa coupling
rather well.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Figures/sqst_vs_mb_all_Models.pdf}
\hspace{3mm}
\includegraphics[width=0.48\textwidth]{Figures/sqst_vs_mt_all_Models.pdf}
\caption{The largest of the mixing angles between $s_q$ and $s_t$
versus the lightest $Q = -1/3$ resonance (left panel) and $Q = 2/3$
resonance (right panel) in several models. For the MCHM$_5$ we plot
the largest between $s_{q_u}$, $s_{q_d}$ and $s_t$.}
\label{sqvsmF}
\end{figure}
The right panels of Figs.~\ref{YukApprox} and \ref{sqvsmF} display the
same information for the top sector (using the lightest $Q = 2/3$
fermionic resonance as the relevant variable). Here, the dispersion
of the points around the continuous curves is larger, but the general
behavior is still well described by the simple analytic formulas given
above, again with the exception of the MCHM$_{10-5-10}$ (green stars),
which all fall below the ``expected curve" given by $F_2 =
\cos(v/f_h)$. Thus, the analytic approximation underestimates the
suppression in the top Yukawa coupling compared to the SM in this
model. We also note here that the analytic approximation,
$F_1(\epsilon)$, slightly underestimates the exact result for the
MCHM$_{10}$, MCHM$_{14-14-10}$ and the MCHM$_{14-1-10}$ (with the
effect being more pronounced for the latter two). Finally, we point
out that after imposing the physical conditions described in the
previous section, the points in the MCHM$_{14-14-10}$ typically have
$y_T \ll \tilde{y}_T$. This means that the deviations from the SM in
the top sector are reasonably well described by the function
$\tilde{F}_3(\epsilon)$ [see discussion around Eq.~(\ref{F3tilde})],
as can be seen in the right panel of Fig.~\ref{YukApprox}.
Besides the above resonances, one can also find light exotic
resonances with charge $Q=8/3,5/3$ and $-4/3$, depending on the
fermion representations involved. These resonances are also
custodians, thus their masses are also suppressed if they belong to
SO(5) multiplets with large mixing with the elementary fermions. They
can have a rich and exciting phenomenology at colliders, although we
will not consider this issue in this work.
The Yukawa couplings of the light fermions should be very well
described by the analytical approximations, at least when both LH and
RH mixing angles are small, as we are assuming. In particular, all of
them can be expected to deviate from the SM expectation by the same
order as the couplings of the third generation, reflecting the
``universal" character of the leading order deviations found in
composite Higgs scenarios (those parametrized by the $F_i$ functions
of Table~\ref{table-F}).
\subsection{Higgs Production and Decay}
Based on the above observations, we can write simple analytical
expressions for the Higgs branching fractions and production rates
that allow us to understand the qualitative (and often quantitative)
behavior. However, for the numerical computations in the scan we will
not perform any such approximations, as already mentioned.
For the tree-level Higgs decays, we have
\begin{eqnarray}
\Gamma(h \to b\bar{b},\tau\tau) &\approx& \Gamma_{\rm SM}(h \to b\bar{b}, \tau\tau) \times r^2_{b}(\epsilon)~, \\ [0.4em]
\Gamma(h \to c\bar{c}) &\approx& \Gamma_{\rm SM}(h \to c\bar{c}) \times r^2_{c}(\epsilon)~, \\ [0.4em]
\Gamma(h \to WW, ZZ) &\approx& \Gamma_{\rm SM}(h \to WW, ZZ) \times r^2_{V}(\epsilon)~,
\end{eqnarray}
where $\Gamma_{\rm SM}(h \to i)$ is the SM Higgs partial decay width
in the $i$-th channel. We have assumed here that the leptons (in
particular the $\tau$) are in the same $SO(5)$ representations as the
bottom quark. Similarly, all up-type quarks (in particular, charm and
top) will be assumed to belong to the same $SO(5)$ representation,
hence $r_{c}(\epsilon) = r_{t}(\epsilon)$, which can be read from
Table.~\ref{table-F} for the different models.\footnote{If different
generations are assigned to different $SO(5)$ representations it is
straightforward to generalize our expressions by simply computing the
corresponding $F_\psi(\epsilon)$ from Eq.~(\ref{BasicRelation}),
although it may happen that this function has additional dependence on
other microscopic parameters.}
For the loop-level Higgs decays, we write
\begin{eqnarray}
\frac{\Gamma(h \to gg)}{\Gamma_{\rm SM}(h \to gg)} &\approx& \frac{| r_t(\epsilon) \, A_{1/2}(m^2_h / 4m^2_t) + r_b(\epsilon) \, A_{1/2}(m^2_h / 4m^2_b) |^2}{| A_{1/2}(m^2_h / 4m^2_t) + A_{1/2}(m^2_h / 4m^2_b) |^2}~,
\label{Gamma2g} \\ [0.4em]
\frac{\Gamma(h \to \gamma\gamma)}{\Gamma_{\rm SM}(h \to \gamma\gamma)} &\approx& \frac{| r_V(\epsilon) \, A_{1}(\frac{m^2_h}{4m^2_W}) + N_c Q_t^2 \, r_t(\epsilon) \, A_{1/2}(\frac{m^2_h}{4m^2_t}) + N_c Q_b^2 \, r_b(\epsilon) \, A_{1/2}(\frac{m^2_h}{4m^2_b}) |^2}{| A_{1}(m^2_h / 4m^2_W) + N_c Q_t^2 A_{1/2}(m^2_h / 4m^2_t) + N_c Q_b^2 A_{1/2}(m^2_h / 4m^2_b) |^2}~,
\label{Gamma2gamma}
\end{eqnarray}
where $A_{1/2}(\tau)$ and $A_1(\tau)$ are the well-known loop
functions (see App.~\ref{app:loops}), $N_c = 3$ is the number of
colors and $Q_t = 2/3$, $Q_b = -1/3$ are the top and bottom quark
electric charges, respectively. Note that here we have formally
included only the effects of the zero-modes, since in the limit where
Eq.~(\ref{BasicApproximation}) holds, the contribution of the
associated towers of heavy resonances becomes negligible. However, to
the extent that $A_{1/2}(\frac{m^2_h}{4m^2_t}) \approx 4/3$ (its
asymptotic value for $4m^2_t \gg m_h^2$), and given the sum rule
Eq.~(\ref{BasicRelation}), the above set of approximations effectively
include the effects of the full top tower. For the bottom quark
contribution, the situation is different since $|A_{1/2}(m^2_h /
4m^2_b)| \approx 1/16 \ll 1$ for $m_h \approx 125~{\rm GeV}$ and $m_b
\approx 2.7~{\rm GeV}$. In addition, in some cases (as in the
MCHM$_{10-5-10}$), the contribution of the heavy towers can be as
large as 10\% of the sum in Eq.~(\ref{BasicRelation}). As a result,
the contribution of the heavy $Q =-1/3$ states to the above
loop-induced processes can be of the same order as the actual
contribution of the bottom quark, since although $y_b / m_b$ still
dominates the sum in Eq.~(\ref{BasicRelation}), it has to be
multiplied by the small $A_{1/2}(m^2_h / 4m^2_b)$ for the physical
processes. Given that the contribution of the bottom-like resonances
is not included in Eqs.~(\ref{Gamma2g}) and (\ref{Gamma2gamma}), our
approximation could carry an uncertainty of the same order as the
bottom contribution, which can be as large as 10\%. However, for most
models, the approximation is significantly better.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Figures/BRs_norm_all_Models.pdf}
\hspace{3mm}
\includegraphics[width=0.48\textwidth]{Figures/ggh_all_Models.pdf}
\caption{Left panel: Branching fractions (normalized to the SM) into
fermions and gauge bosons for several models following from the
approximation in Eq.~(\ref{BasicApproximation}). Here $VV = WW, ZZ,
\gamma\gamma, gg$. The color coding of the lines matches the color
coding of the closest legend. Right panel: gluon fusion production
cross section (normalized to the SM) in those models. The vector
boson fusion (VBF) cross section coincides with the curve marked as
``MCHM$_{5-10-10}$, MCHM$_{10-5-10}$".}
\label{BRsandProd}
\end{figure}
In the left panel of Fig.~\ref{BRsandProd}, we show the Higgs
branching fractions into fermion and gauge boson pairs in the
MCHM$_{5}$, MCHM$_{10}$, MCHM$^{\rm Simple}_{14-14-10}$,
MCHM$_{14-1-10}$ (solid lines), MCHM$_{14-14-10}$ (dash-dotted lines),
MCHM$_{5-10-10}$ (short dashed lines), MCHM$_{10-5-10}$ (dotted
lines), and MCHM$_{5-5-10}$ (long dashed lines). We see that in some
cases the BR's are enhanced with respect to the SM while in others
they are suppressed. One should notice that all partial decay widths
always present a suppression, in particular for the $b\bar{b}$ decay
channel. As a result the total decay width is suppressed, and the
BR's in some channels can end up being enhanced due to the smaller
denominator. In contrast, the Higgs production cross sections are
always suppressed with respect to the SM, as shown in the right panel
of Fig.~\ref{BRsandProd} for the gluon fusion Higgs production cross
section, normalized to the SM. We also note that the VBF production
cross section coincides with the upper curve in this plot.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Figures/Exp_ZZ_all_Models_WithScans.pdf}
\hspace{3mm}
\includegraphics[width=0.48\textwidth]{Figures/Exp_gaga_all_Models_WithScans.pdf}
\caption{Left panel: Rates in the $h \to ZZ$ decay channel separated
according to production mode: gluon fusion ($+ t\bar{t}h$) versus VBF
($+h W/Z$). The larger black dots indicate the positions of $\epsilon
= 0.1, 0.3, 0.5$. Right panel: Same for the $h \to \gamma\gamma$
channel. The solid curves correspond to the analytical approximation
discussed in the main text, while the points correspond to a random
scan that reproduces $m_h \sim 125~{\rm GeV}$, $m_t \sim 160~{\rm
GeV}$ and $m_h \sim 4~{\rm GeV}$. The shaded region corresponds to
the current 95\% CL curve by ATLAS. The CMS 95\% CL region would cover
the full area of the figure. The \textit{production} signal strengths
are defined as $\mu_i = \sigma^{\rm Model}(i) / \sigma^{\rm SM}(i)$.
The production cross sections used correspond to the 8~TeV run of the
LHC.}
\label{Exp}
\end{figure}
Consequently, the total cross sections in given channels can be
enhanced or suppressed with respect to the SM, depending on how these
opposing effects play out. We illustrate this in Fig.~\ref{Exp} for
the $ZZ$ (left panel) and $\gamma\gamma$ (right panel) decay modes,
separating the gluon fusion ($+ t\bar{t} h$) production from VBF ($+
hW/Z$), as done by the ATLAS and CMS collaborations~\cite{gg-VBF}.
The continuous lines correspond to the expectation based on the above
analytical approximation. We have superimposed the exact predictions
for the scan in the models we consider. We see that the approximation
tracks well the actual analytical predictions for all models (up to
some dispersion due to the effect of the bottom sector explained
above), except for the MCHM$_{10-5-10}$ on which we comment further
below. One can understand the behavior of these curves from
Fig.~\ref{BRsandProd}. For instance, for the MCHM$_{5-10-10}$, since
\textit{all} channels (gauge, down-type and up-type) are suppressed by
exactly the same $r(\epsilon)$, the BR's remain exactly as in the SM,
while the production in all modes is suppressed identically. Thus,
the curve points at a $45^{\circ}$ angle towards the left-down, as
$\epsilon = \sin(v/f)$ increases and the deviations from the SM
increase. The MCHM$_{5-5-10}$ shows a very mild enhancement in the
$ZZ$ and $\gamma\gamma$ BR's (see left panel of
Fig.~\ref{BRsandProd}), which is not enough to compensate the
suppression in production. Since the latter is more significant in
gluon fusion than in VBF, the curve in Fig.~\ref{Exp} points to the
left-down but closer to the horizontal than for the MCHM$_{5-10-10}$.
For the MCHM$_{5}$, MCHM$_{10}$, MCHM$_{14-14-10}$, MCHM$^{\rm
Simple}_{14-14-10}$ and MCHM$_{14-1-10}$, the left panel of
Fig.~\ref{BRsandProd} shows a stronger enhancement in both ${\rm BR}(h
\to ZZ)$ and ${\rm BR}(h \to \gamma\gamma)$, which is sufficient to
compensate the suppression in the VBF production but not enough to
compensate the significant suppression in gluon fusion (see right
panel of Fig.~\ref{BRsandProd}). As a result, the analytical
prediction curves to the left-up. Note, however, that the scanned
points for the MCHM$^{\rm Simple}_{14-14-10}$ show a more pronounced
tendency to compensate the suppression in gluon fusion by the
enhancement in the branching fractions than the naive analytical
expectation. This can be traced to the systematic (albeit small)
deviations exhibited in Fig.~\ref{YukApprox} for the top and bottom
Yukawa couplings. Finally, we see that the analytical prediction for
the MCHM$_{10-5-10}$ does \textit{not} reproduce the qualitative
behavior of the scan. While a line at $45^{\circ}$ to the right-up is
expected (from Fig.~\ref{BRsandProd} one can see that the enhancement
in BR's dominates over the suppression in production in all the
modes), most of the points actually present a suppression with respect
to the SM. This can be traced back to our previous comments in regards
to this model: the analytical approximation systematically
overestimates the suppression in the $b\bar{b}$ channel [hence
overestimates the enhancement in ${\rm BR}(h \to ZZ)$ and ${\rm BR}(h
\to \gamma\gamma)$], while it systematically underestimates the
suppression in the top Yukawa coupling, which translates into an
overestimate of the gluon fusion Higgs production rate. These ${\cal
O}(10\%)$ errors are sufficient within this model to change the
qualitative behavior. The VBF production is still well described by
the analytic approximation, as is for all the other models, since the
gauge resonances are always heavy.
It is interesting that the different fermionic representations lead to
a different behavior in the plane of Fig.~\ref{Exp}, so that a precise
measurement of these rates could be used to distinguish between
different scenarios (although there could still remain a degeneracy
between the MCHM$_5$, MCHM$_{10}$, MCHM$^{\rm Simple}_{14-14-10}$ and
MCHM$_{14-1-10}$, which in fact could be confused with the more
general MCHM$_{14-14-10}$). We also show the current 95\%
C.L.~ellipse from the ATLAS analysis~\cite{gg-VBF}, and indicate the
position along the solid line in each model that corresponds to
$\epsilon = 0.1, 0.3, 0.5$. We see that the experimental
uncertainties still allow for relatively large values of $\epsilon$.
The 95\% C.L.~ellipse from the CMS analysis would fill the region
shown, so we do not indicate it.
The ATLAS and CMS collaborations have measured other properties of the
125~GeV resonance. For instance, by taking channel by channel ratios
of the $ggH + ttH$ and $qqH + VH$ production modes, and performing a
fit to the data, they can set a bound on $\mu_{qqH + VH} / \mu_{ggH +
ttH}$. This analysis only assumes that the same boson $H$ is
responsible for all observed Higgs-like signals and that the
separation of gluon-fusion like events and VBF-like events, based on
the event kinematics, is valid. For instance, the ATLAS collaboration
sets a bound of $\mu_{qqH + VH} / \mu_{ggH + ttH} =
1.2^{+0.7}_{-0.5}$~\cite{Atlas:couplings}.
The models in our scan have $1 \lesssim \mu_{qqH + VH} / \mu_{ggH +
ttH} \lesssim 1.5$, so that they are not yet probed by these analyses.
However, if a ratio below one was established it would disfavor the
pNGB Higgs scenarios based on the lowest dimensional representation of
SO(5). This is a manifestation of the generally important suppression
in the gluon fusion process w.r.t.~the SM. ATLAS also sets bounds on
the Higgs production by gluon fusion alone, in terms of the rescaling
factor $\kappa_g$. However, the analysis assumes that all the BR's
are as in the SM and therefore does not apply to the present case.
From the LHC data one can also derive bounds on ratios of branching
ratios, e.g. on $\rho_{\gamma\gamma/ZZ} = [{\rm BR}(\gamma\gamma) /
{\rm BR}(\gamma\gamma)_{\rm SM}] / [{\rm BR}(ZZ) / {\rm BR}(ZZ)_{\rm
SM}]$, etc. ATLAS finds $\rho_{\gamma\gamma/ZZ} =
1.1^{+0.4}_{-0.3}$~\cite{Atlas:couplings}. Our scans have $1 \lesssim
\rho_{\gamma\gamma/ZZ} \lesssim 1.1$, so that they are not yet probed
in such measurements. Similarly, due to the custodial symmetry, we
have $\rho_{WW/ZZ} \approx 1$, and it would be very challenging to
differentiate it from the SM at the LHC; a significant deviation from
the custodial limit would disfavor both the SM and the pNGB scenarios
we have studied.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Figures/Exp_2tau_all_Models_WithScans.pdf}
\hspace{4mm}
\includegraphics[width=0.48\textwidth]{Figures/Inclusive_2tau_vs_eps_all_Models.pdf}
\caption{Left panel: Similar to Fig.~\ref{Exp}, but for the $h \to
\tau\tau$ channel. Right panel: we show the total rate
(i.e.~inclusive production) in the $\tau\tau$ channel, normalized to
the SM, versus $\epsilon$. The horizontal bands correspond to the
95\%~C.L. limit set by the ATLAS~\cite{Atlas:taus} and
CMS~\cite{CMS:taus} collaborations.}
\label{fig:LHC-tautau}
\end{figure}
Apart from the indirect sensitivity to the top quark via the loop
processes above, the fermionic channels, in particular $h \to
\tau\tau$ are starting to be measured with interesting
precision~\cite{Atlas:taus,CMS:taus} for the present work, although
the uncertainties are still sufficiently large to be consistent with
the great majority of our parameter point sample. In the left panel
of Fig.~\ref{fig:LHC-tautau} we show the expectations for this
channel, discriminating between the Higgs production by gluon fusion
($+ttH$) and VBF ($+VH$), together with the 95\%~C.L. region from
ATLAS. In the right panel we show the signal strength for the
inclusive $h \to \tau\tau$ production as a function of $\epsilon$.
The horizontal bands correspond to the 95\%~C.L. regions from
ATLAS~\cite{Atlas:taus} and CMS~\cite{CMS:taus}. We note that under
our assumptions, the $\tau\tau$ channel is always suppressed
w.r.t.~the SM. However, one should remember that one may be able to
consider different representations for the $\tau$ sector, without
affecting the properties of the Higgs potential. Hence, establishing
an enhancement in the $\tau\tau$ channel over the SM would be in
conflict with our assumptions, but we cannot claim that it would rule
out the general framework.
In contrast, in models with a minimal content of composite fermion
multiplets, one expects a robust suppression w.r.t. the SM in the $h
\to b \bar{b}$ decay mode, so that this would be an interesting
channel to probe the scenario. We find a suppression of $10-20\%$ for
$\epsilon = 0.3$ and $20-40\%$ for $\epsilon = 0.5$, with smaller
dispersion between different models than in the $\tau\tau$ channel.
This is because at the LHC one must consider $pp \to h + X \to b
\bar{b} + X$ in order to be able to discriminate against the large QCD
background, so that only $VBF$ + $VH$ + $ttH$ contribute, but not
$ggH$ which is most sensitive to the new fermionic resonances that
distinguish between different models. Unfortunately, at the LHC the
precision may not be sufficient to provide a clear test, but its high
luminosity phase or a linear collider could set useful bounds.
\subsection{$h \to Z\gamma$}
We turn now to the last decay channel we consider: $h \to Z\gamma$,
which has not yet been observed, but could be seen in the near future.
The decay of a pNGB Higgs to $Z\gamma$ has received considerable
attention recently. Ref.~\cite{Azatov:2013ura} has shown that there
can be large corrections to this decay, while being simultaneously
compatible with precision EW measurements, thus providing a very
interesting test. In order to obtain a large effect in this decay the
composite sector itself must break the $P_{LR}$ symmetry, otherwise
the only source of $P_{LR}$ breaking is the interaction between the
elementary and composite fields, and the effect is
suppressed~\cite{Elias-Miro:2013mua}. We have not considered breaking
of $P_{LR}$ by the composite sector in our work, so that we expect
small corrections in the $h \to Z\gamma$ channel. We have computed
the corrections to this rate in the models presented in the previous
sections. Below we discuss the main features of this decay and show
our results.
In the SM the interaction $hZ\gamma$ is a radiative effect, generated
at 1-loop by virtual $W$'s and fermions. Similar to
$h\gamma\gamma$, the bosonic and fermionic contributions have opposite
sign. The first one dominates over the second one by a factor $\sim
10$, and the fermionic loop is dominated by the top contribution. In
the MCHM one can distinguish the corrections from the new particles in
the loop from those arising from the modified couplings between the
Higgs and the SM gauge and fermion fields, as was the case for the
$h\gamma\gamma$ process. However, unlike in the $h \to \gamma\gamma$
diagrams, there can be two different particle species running in the
loop, since only one of the external particles is a gauge field of an
unbroken symmetry. Therefore, in theories with extra $W$'s, besides
the loop with a single heavy field there are 1-loop effects
involving two different virtual states. We will refer to these
contributions as ``diagonal" and ``non-diagonal", respectively.
Similarly, in theories with new fermions there are 1-loop effects
involving a single new fermion as well as effects involving
propagators of two different fermion species. We will clarify below
which diagrams give the leading contributions.
As in the SM, in the models we are considering there are no tree-level
contributions to the $h \to Z\gamma$ process, so we focus on the
1-loop effects, starting with those due to bosonic fields. Each
diagonal contribution is suppressed by a factor $(m_W/m_{W_n})^2 \sim
{\cal O}(10^{-3})$. Although there are several charged vectors, whose
contributions add up, we find that the total effect is less
than 1\% of that of the $W$ gauge boson in the SM. Next we consider
the corrections from a loop with a SM-$W$ and a heavy charged vector.
The product of the non-diagonal couplings $ZWW_n$ and $hWW_n$ are
suppressed by a factor $\lesssim{\cal O}(10^{-2})$ compared with the
SM coupling, thus they can be neglected as well. For the non-diagonal
contributions involving heavy fields the product of the couplings
$ZW_mW_n$ and $hW_mW_n$ can be of the same order as in SM. However, as
in the diagonal contribution, in this case there is also an extra
factor $(m_W/m_{W_n})^2\sim{\cal O}(10^{-3})$. Therefore, the leading
correction to $hZ\gamma$ mediated by loops of vector bosons is
captured by the correction to the couplings $hW^+W^-$ and $ZW^+W^-$.
The correction to the first one can be approximated by $F_2(\epsilon)
= \sqrt{1-\epsilon^2}$, whereas the correction to $ZW^+W^-$ is very
small. Thus, one can expect the bosonic 1-loop
correction to $c_{hZ\gamma}$ [see comments after Eq.~(\ref{OpsDim6})]
to be modulated by $F_2(\epsilon)$, leading to a suppression in the
amplitude compared with the SM.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Figures/ghZgamma_vs_eps_Top_all_Models.pdf}
\hspace{3mm}
\includegraphics[width=0.48\textwidth]{Figures/ghZgamma_vs_eps_all_Models.pdf}
\caption{Left panel: the amplitude for $h \to Z\gamma$ due to the top
sector in all the models, normalized to the top-mediated amplitude in
the SM. Right panel: the full amplitude (absolute value), arising from
vector bosons and fermions in the models of Sec.~\ref{sec:models},
normalized to the $hZ\gamma$ amplitude in the SM. The continuous line
corresponds to the SM-W loop, with its modified coupling to the Higgs
as encoded in $F_2(\epsilon)$.}
\label{fig:Wt-hZgamma}
\end{figure}
The correction from the fermionic sector is dominated by the top quark
and its partners. The resonances associated to the light SM fermions
decouple and do not contribute. This can be understood from the fact
that $hZ\gamma$ requires breaking of $P_{LR}$, and in the present
models that breaking arises only from the mixing between the two
sectors of the theory. Since we are assuming that the light fermions
have small mixing for both chiralities, the explicit $P_{LR}$ breaking
is suppressed by these small mixings. The effect from the top
partners can have different signs for different representations. In
the left panel of Fig.~\ref{fig:Wt-hZgamma}. we show the corrections
to the amplitude coming from the top sector of all the models,
normalized to the top contribution in the SM. We have included all the
diagonal and non-diagonal contributions. The corrections to the SM
top result can be of order 50\%, or even larger for $\epsilon\sim 0.5$
and for most of the models there is a suppression. However one should
remember that the bosonic contribution is one order of magnitude
larger that the fermionic one.
In the right panel of Fig.~\ref{fig:Wt-hZgamma}, we show the total
amplitude in the MCHM models normalized to the SM, where we have used
the full diagonalization of the mass matrices and couplings to take
into account all the fermionic contributions, the diagonal spin-1
contributions, and the (small) modification of the $ZW^+W^-$ coupling.
However, we do not include the non-diagonal gauge contributions (which
have been argued to be negligible above). See App.~\ref{app:loops}
for further details of this computation.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Figures/Exp_Zga_all_Models_WithScans.pdf}
\hspace{4mm}
\includegraphics[width=0.48\textwidth]{Figures/Exp_Zgamma_vs_2gamma_all_Models_WithScans.pdf}
\caption{Left panel: Similar to Fig.~\ref{Exp}, but for the $h \to
Z\gamma$ channel. Right panel: we show the total signal strength
$\mu(i) = \sigma(pp \to h \to i) / SM$ (i.e.~inclusive production) in
the $Z\gamma$ channel versus the $\gamma\gamma$ channel, showing a
high degree of correlation. The larger deviations from $(1,1)$
correspond to larger values of $\epsilon$.}
\label{fig:LHC-hZgamma}
\end{figure}
Since for most of the models and regions of parameter space the
leading order effect is captured by the lightest states running in the
loop, either bosons or fermions, the corrections to $c_{hZ\gamma}$ can
be approximated by the corrections to the Higgs couplings with $W$ and
$t$. The left panel of Fig.~\ref{fig:LHC-hZgamma} shows that this
approximation works rather well for the models we are considering.
The deviations arise mainly from the diagonal and non-diagonal
contributions of the top partners. In the right panel we exhibit the
correlation between the rates into $Z\gamma$ and $\gamma\gamma$. We
see that this correlation is slightly different between the
MCHM$_{10-5-10}$ and MCHM$_{5-10-10}$ on the one hand, and the other
models on the other, which could allow for a distinction if sufficient
precision is achieved, depending on the size of the deviations from
the SM. We note that for the MCHM$_{5-10-10}$ there is a very good
agreement between the analytical approximation and the full numerical
result in $\gamma\gamma$. However, the top sector gives contributions
of order 5-10\% to $Z\gamma$, as can be seen in the right panel of
Fig.~\ref{fig:Wt-hZgamma}. Those corrections lead to the small
disagreement between the analytical approximation and the full
numerical result for the MCHM$_{5-10-10}$ seen in the right panel of
Fig.~\ref{fig:LHC-hZgamma} (a similar effect but in the opposite
direction is present for the MCHM$_{14-14-10}^{\rm simple}$).
\section{Tuning in the MCHM}
\label{sec:tuning}
In this section we comment on the degree of fine-tuning associated
with the phenomenologically viable points found above. Consistency
with the EW precisions tests (EWPT) in these models, mainly the
$S$-parameter and the $Zb_L\bar b_L$ coupling, require
$\epsilon\lesssim 0.3$~\cite{Agashe:2004rs}.\footnote{Although we do
not perform a detailed analysis of the EWPT on all the models we
consider, we recall that the presence of light fermionic resonances
can play an important role in opening up the viable region of
parameter space, as studied in~\cite{Anastasiou:2009rv}.} However the
Higgs potential generically leads to no EWSB, $\epsilon=0$, or to
maximal EWSB, $\epsilon=1$.\footnote{For $v=f_h$, besides the problems
with EWPT, many models lead to massless SM fermions, as can be seen
from the cancellation of the $LR$ correlator $M_\psi$. This is a
consequence of the restoration of an accidental chiral
symmetry~\cite{Contino:2006qr}.} A careful analysis of the structure
of the Higgs potential shows that the MCHM requires some tuning in the
parameter space of the theory to produce $\epsilon\lesssim 0.5$, and
the amount of tuning depends on the fermion
embedding~\cite{Agashe:2004rs,Panico:2012uw}. Besides these
conditions, the Higgs potential must also lead to a light Higgs.
Since the top contribution to the 1-loop Higgs potential is cut off by
the fermionic resonances mixing with the top, a light Higgs prefers
light top partners. Ref.~\cite{Contino:2006qr} has shown the
correlation between $m_h$ and the mass of the lightest resonance for
MCHM$_5$ and MCHM$_{10}$. Ref.~\cite{Panico:2012uw} has also
discussed the impact of light fermions in the tuning of the MCHM,
arriving to similar results. In our setup, similar to models in a
slice of AdS$_5$, large compositeness of the SM fermions automatically
lead to light custodians that can alleviate the tuning (see the
discussion in~\ref{sec-num-scan}). Below we show our results for the
tuning of the models presented in the previous sections.
Following Refs.~\cite{Barbieri:1987fn, Anderson:1994dz, Panico:2012uw}
we use the sensitivity parameter
\begin{equation}
\label{eq-tuning}
\Delta={\rm max}_i \left|\frac{\partial \log m_Z|_{\rm phys}}{\partial
\log x_i}\right|
\end{equation}
as a measure of fine-tuning. Here $x_i$ are the parameters of the
effective theory and $m_Z$, as given in Eq.~(\ref{eq-vSM}), depends
explicitly on $f_h$ and $\epsilon$, with $\epsilon$ a function of all
the parameters of the theory. By $m_Z|_{\rm phys}$ we mean that we
have selected a region of parameters of the theory that leads to the
observed Higgs and SM masses. We have followed the procedure of
Ref.~\cite{Panico:2012uw} which has shown that Eq.~(\ref{eq-tuning})
can be rewritten in terms of the Higgs potential, allowing for a
simple calculation of $\Delta$. As explained at the beginning of
Sec.~\ref{sec:pheno}, we have considered the dependence of the
potential on the following parameters: the mass scale of the composite
resonances $m_\rho$, the decay constant of the pNGB $f_h$, the
composite proto-Yukawa couplings, the masses mixing composite fermions
$m_y$, the fermion mixing angles $s_\psi$ and the ratio of gauge
couplings $\tan\theta$.
We have computed the tuning of the models presented in the previous
sections, evaluating $\Delta$ in those points of the parameter space
that were selected after the random scan, as explained at the
beginning of Sec.~\ref{sec:pheno}. We find that the gauge
contribution is subdominant, and the tuning is usually dominated by
the top mixings $s_{q}$, $s_{t}$, the Yukawa $y_T$ and the mixing mass
$m_{y_T}$ when present. Below we comment on the size of the tuning
for the different models and discuss some details about its parameter
dependence.
We find that the MCHM$_5$ and MCHM$_{5-5-10}$ have generically
$\Delta\sim 5-40$, with the sensitivity parameter dominated by
$m_{y_T}$ and sometimes by $s_t$. The second model shows some regions
of parameter space with $\Delta\sim 100$ as well as some points where
$s_q$ dominates the tuning. Notice that the MCHM$_{5-5-10}$ has less
freedom, since there is no $m_{y_B}$ and the $b_L$ mixing is
controlled by the same parameter that controls the $t_L$ mixing,
namely $s_q$, whereas for the MCHM$_5$ there are two mixing
parameters, $s_{q^d}$ and $s_{q^u}$. The MCHM$_{10}$ has $\Delta\sim
5-80$, although there are points with $\Delta\sim 300$. The larger
tuning of the MCHM$_{10}$ could be related with the Clebsch-Gordan
coefficient $\sqrt{2}$ suppressing $m_t$ in the latter model, that
requires larger mixing and Yukawa coupling. In this model $\Delta$ is
usually dominated by $s_q$ and sometimes by $s_t$, $y_T$ or $m_{y_T}$.
As explained in the previous sections, MCHM$_{5-10-10}$ and
MCHM$_{10-5-10}$ require a large degree of compositeness of at least
one of the chiralities of the top, leading to the largest tuning of
the models that we have studied with fermions in representations ${\bf
5}$ and ${\bf 10}$. We find $\Delta\sim 100-1000$, usually dominated
by $s_q$ and sometimes by $s_t$. MCHM$_{14-14-10}^{\rm simple}$ and
MCHM$_{14-1-10}$ have $\Delta\sim 80-300$, dominated by $s_t$ for the
first model and by $s_q$ for the second one. The main reason for the
larger tuning of these models compared with MCHM$_5$ and MCHM$_{10}$
is that they generically predict a larger $m_h$~\cite{Panico:2012uw}.
Thus, requiring $m_h\simeq 125$ GeV selects special regions of the
parameter space with non-natural cancellations in the Higgs potential.
On the other hand, for the MCHM$_{14-14-10}$ that has an extra
proto-Yukawa coupling in the top sector, we find $\Delta\sim 10-150$,
with the tuning dominated by $s_t$ and sometimes by $m_{y_t}$ or
$f_h$.
We find that, after applying our selection criteria over the random
scan, the models with larger tuning also show many points in a region
of the parameter space with large composite scale, $f_h\gtrsim 4$ TeV.
In fact, for these models there are some points where the tuning is
dominated by $f_h$.
\section{Conclusions}
\label{sec:conclusions}
We have used a simple two-site realization of the composite Higgs
scenario~\cite{DeCurtis:2011yx} to systematically investigate the
consequences of several fermion representations of the spontaneously
broken symmetry leading to the Higgs as a pNGB. We have restricted
ourselves to the $SO(5) \to SO(4)$ symmetry breaking pattern, which is
denoted here as the ``Minimal Composite Higgs Model", but we have
explored several combination of the lowest-dimensional representations
of SO(5) in the composite fermion sector. In particular, we have
fully taken into account the dynamically generated Higgs potential,
which receives crucial contributions from the states associated with
the third family, especially the top quark. We can therefore
consistently incorporate the measured mass of the resonance discovered
at the LHC in 2012, interpreted as a SM-like Higgs boson, and
investigate the restrictions imposed by the experimental information.
We have also taken into account the effects of the bottom quark
sector, which, although subdominant in determining the dynamics of
EWSB, can have a non-negligible effect on the resulting Higgs
phenomenology. We have assumed that the light families are mostly
elementary, and therefore have a negligible effect on the Higgs
potential. However, the couplings of a composite Higgs to all
fermions can receive sizeable corrections leading to important
deviations from the SM expectations. This can be important in the
near future, as decays such as those into a $\tau$ pair are being
measured with better precision~\cite{Atlas:taus,CMS:taus}.
By including the ``first level" of heavy (spin-1 and spin-1/2)
resonances, we can also compute in detail the effects on loop-induced
processes, such as the Higgs production through gluon fusion and the
Higgs decays into $\gamma\gamma$ and $Z\gamma$. Such processes
consist of two conceptually different, but related parts. First,
the couplings of the Higgs to the SM fermions are modified w.r.t.~the
SM, and therefore when they run in the loop the corresponding
contribution is different from the SM one. Second, the heavy
resonances give an additional non-SM contribution to the loop diagrams.
At zeroth-order and in the simplest models, the sum of the two effects
for the dominant contributions (from the top-related states, as well
as from the W-related ones in the case of $\gamma\gamma$ or $Z\gamma$)
results in a ``universal modification" that depends on the microscopic
parameters only through $\epsilon = \sin v/f_h$. However, we find
that the corrections to this leading order result, in particular those
of the bottom sector, can have a qualitative impact on the Higgs
properties.
Importantly, we find a generic suppression of the gluon fusion process
in all the models we investigated. This is also the case for the
MCHM$_{14-14-10}$, which presents a richer structure of invariants and
leads in general to a sum rule that has dependence on microscopic
parameters beyond $\epsilon$. Although a priori there exists the
potential for finding regions of parameter space with an enhanced
gluon fusion Higgs production cross section~\cite{Azatov:2011qy}, we
find that all the phenomenologically viable points exhibit a rather
significant suppression instead.
Due to the generic suppression of the various decay widths,
in particular $\Gamma(h \to \bar b b)$ which dominates the total Higgs
decay width, one can often find branching fractions that are larger
than those in the SM. The experimental rates then result from
competing effects between production and decay, and can present
enhancements or suppressions in given channels, depending on the model
under consideration. This offers an interesting handle --were a
robust deviation from the SM to be established-- to get indirect
information about the composite fermion representations, which would
constrain the nature of the underlying strongly interacting theory.
Another interesting decay channel is $h\to Z \gamma$. We have shown
that the deviations are small and dominated by the corrections from
loops of SM weak bosons, as expected if the $P_{LR}$ symmetry is not
broken by the composite sector~\cite{Elias-Miro:2013mua}. Moreover,
the contributions from the heavy resonances are small and the
deviations can be approximated at leading order by the corrections to
the $hW^+W^-$ coupling, that are given by a simple function of
$\epsilon$.
We have also investigated the degree of fine-tuning, which is in
general considerable but seems in most cases to compare favorably
against the simplest SUSY scenarios (although this statement should
not be taken as a rigorous one, given the lack of a proper UV
completion for the composite Higgs scenarios). Interestingly, we find
examples where the sensitivity of the weak scale to the underlying
model parameters is below 10\%. However, models such as the
MCHM$_{5-10-10}$ and the MCHM$_{10-5-10}$ present a sensitivity at the
few per mille level. We also note that the models based on the 14
representation, which have been claimed to present little
tuning~\cite{Panico:2012uw} actually are tuned at the per cent or
worse level (although we have not considered a purely composite
$t_R$). These considerations may be suggestive of which case is more
likely to be realized in nature, although of course experimentally the
approach should be open-minded.
As the LHC and the experimental collaborations prepare for the (close
to) 14 TeV and higher luminosity run, the Higgs sector offers a unique
window into physics beyond the SM. The possibility that the Higgs
boson is a pNGB of some underlying strong dynamics remains as an
attractive framework for understanding the breaking of the EW
symmetry, and the opportunity of learning something about the detailed
properties of such a theory from Higgs measurements can be a realistic
one, as illustrated in this work. Eventually one should be able to
produce the strong resonances, studying their properties directly, and
start cross-checking against the previous low-energy information.
\subsection*{Acknowledgements}
We thank Carlos Wagner for enlightening discussions in the beginning
of this work and \'Alex Pomarol for useful suggestions. We also thank
Abdelhak Djouadi, Joseph Lykken, Giuliano Panico, Gilad Perez and
Francesco Riva for discussions. E.P. and L.D. wish to thank the
Fermilab Theory Group for hospitality during various stages of this
work. M.C.~would like to thank the Aspen Center for Physics, where
part of this work was completed. M.C.~would like to thank ICTP-SAIFR
and Centro At\'omico Bariloche for hospitality. M.C.~and E.P.~would
like to thank the MITP at the Johannes Gutenberg-University Mainz, where part of this work was
completed. Fermilab is operated by Fermi Research Alliance, LLC under
contract no. DE-AC02-07CH11359 with the United States Department of
Energy. L.D. is partly supported by FONCYT-Argentina under the
contract PICT-2010-1737 and CONICET-Argentina under the contract PIP
114220100100319. This work was supported by the S\~ao Paulo Research
Foundation (FAPESP) under grant \#~2011/11973.
|
2,877,628,090,076 | arxiv | \section{Experimental}
Mesas 100$\times $400~\textmu m$^2$ in size with two silver strip electrodes were milled from a Bi-2212 crystal by photolithography and argon milling methods. Figure 1(a) shows an optical microscopy image of the Bi-2212 mesa array. We refer to the two mesas as A1 and A2. Profile measurements using atomic force microscopy (Keyence Corp., Model VN-8000) demonstrate that the mesas vary marginally in size. Widths of A1 and A2 were measured as 94~\textmu m and 91~\textmu m, respectively, with thicknesses of 1.4~\textmu m corresponding to 910 IJJs.
The current-voltage characteristics (IVCs) for A1 and A2 show the large hysteresis typical of underdamped Josephson junctions~\cite{SM}. Simultaneous emission occurs when A1 and A2 are biased in parallel. Hereinafter, we refer to the parallel connection as A1$\parallel $A2. The maximum intensity is obtained at 16.8~mA, which is higher than the sum of the bias currents for the maximum emission powers of A1 and A2 individually. Also, the maximum intensity for A1$\parallel $A2 was almost half of that for A2. This can be explained by considering the local temperature increase~\cite{Wang2009,Wang2010,Gross2012,Benseman2013c,Minami2014,Tsujimoto2014,Benseman2015}. Benseman {\it et al.} demonstrated that the self-heating effect limits the power output and in fact may prevent synchronization among multiple mesas~\cite{Benseman2013b}. A variety of studies on the cavity resonance effect have demonstrated that spontaneous synchronization among stacked IJJs is accompanied by the formation of standing EM waves inside the Bi-2212 mesa~\cite{Ozyuzer2007,Tsujimoto2010a,Kashiwagi2011,Tsujimoto2016,Kashiwagi2018,Zhang2019}. For a thin rectangular mesa of width $w$ and length $\ell $, the cavity frequency for a transverse magnetic $(mp)$ mode is given by $f_{mp}^{r}=(c_{0} / 2 n)\sqrt{(m/w)^{2}+(p/\ell )^{2}}$, where the two indices $m$ and $p$ correspond to the numbers of electric field nodes in the width and length directions, respectively, and $n = 4.2$ is the experimentally obtained refractive index~\cite{Tsujimoto2016,Kadowaki2010}. Here, the radiation frequencies measured using Fourier transform interferometry ranged continuously from 0.56 to 0.66~THz depending on the bias point~\cite{SM}. These values are in good agreement with the calculated $f_{1p}^{r}$ with $0<p<6$. If we assume $m \geq 2$, the calculated $f_{mp}^{r}$ values fail to coincide with the observed values for any $p$. Hence, only one EM node is present along the mesa width, and a non-zero $p$ value is expected to produce an elliptical polarization.
Here, we present the measurements of the Stokes parameters, which allow for the quantitative analysis of polarized photons emitted from individual mesas and from synchronized arrays~\cite{SM}. Figure 1(b) shows a schematic view of the synchronized array. Measurement was performed by allowing the polarized radiation to propagate sequentially through two polarizing elements, a quarter-wave plate (QWP), and a linear wire grid polarizer (WGP). The QWP consists of a stack of parallel metal-plate waveguides~\cite{Nagai2015}. We used terahertz time-domain spectroscopy~\cite{Madeo2010,Maussang2016} to verify that the QWP works correctly in the emission frequency range around 0.6~THz~\cite{SM}. Figure 1(c) shows the measurement configuration. The QWP can be rotated through an angle $\theta $ and is followed by a fixed WGP whose transmission axis is fixed in the width direction ($\theta =0$~deg).
Figures 2(a) and 2(b) show polar plots of the bolometer output for A1 and A1$\parallel $A2, respectively, as a function of $\theta $ at the bias conditions that result in the maximum output powers. The error bars in the radial direction correspond to fluctuations in the bolometer output signal, mostly owing to background noise. The four independent Stokes parameters, $S_0$, $S_1$, $S_2$, and $S_3$, which are summarized in Table SI~\cite{SM}, are obtained from this data. The solid lines shown in Figs.~2(a) and 2(b) represent the calculated results using the four Stokes parameters. The experimental data are slightly asymmetric with respect to both the major and minor axes. This is due to the imperfect alignment of the parallel metal-plate waveguides in the QWP. Nevertheless, the calculation results fit the experimental data within the error bars.
The E-field vector at the detection plane is given by $\bm{B}(t)=E_{0x}e^{i(\omega t + \delta _x)}\bm{i} + E_{0y}e^{i(\omega t + \delta _y)}\bm{j}$, where the $x$- and $y$-axes are parallel to the mesa width and length, respectively, $t$ represents time, $E_{0x}$ and $E_{0y}$ are the respective amplitudes, and $\delta _x$ and $\delta _y$ are the respective phase constants. Figures 2(c) and 2(d) are the corresponding polarization ellipses for A1 (red), A2 (blue), and A1$\parallel $A2 (green). The fourth Stokes parameter $S_3$ determines the helicity of the photons: a positive (negative) $S_3$ indicates left (right)-handed helicity with respect to the direction of propagation. Note that the $E$-field rotates forward (counter-)clockwise for left (right)-handed helicity from the viewpoint of the detector. The arrows on the ellipses indicate the direction of the $E$-field rotation. For both mesas, the major axis of the polarization ellipse is oriented along the $x$ axis, {\it i.e.}, $-\pi /4 < \psi < \pi /4$, which is consistent with excitation of $(1p)$ cavity modes.
In the pioneering study of EM-wave emission from an IJJ mesa, it is demonstrated that the emission from rectangular mesas is linearly polarized along the mesa width~\cite{Ozyuzer2007}. The present results suggest, however, that the emitted waves are elliptically polarized with a finite axial ratio at an arbitrary orientation angle. We stress that these polarization parameters contain information essential for understanding the electromagnetism inside the emitting mesa. For example, the orientation angle $\psi $ is derived from the phase difference $\delta _{xy} = \delta _{x}-\delta _{y}$, namely, $\delta _{yx}= \pm \pi /2$ gives $\psi = 0$ and $\delta _{yx}=0$ gives $\psi = 30$~deg for $E_{0x}^2 / E_{0y}^2 \sim 3$. We also found that the actual polarization parameters are, in fact, dependent on the bias condition and $T_b$. Nevertheless, the observed $E_{0x}$ was greater than $E_{0y}$ in all cases, directly suggesting the predominance of the $(1p)$ cavity mode for elongated rectangular mesas.
When two mesas are biased to emit simultaneously, the far-field waves should be described in terms of the superposition of the $E$-fields generated from each mesa. Thus, the total $E$-field depends on the phase difference between the macroscopic Josephson oscillations. The observed pattern for A1$\parallel $A2 shown in Fig.~2(b) exhibits four-fold symmetry with respect to $\theta $, suggesting that the two mesas generate photons synchronously. It is likely that $\psi $ and the helicity for A1$\parallel $A2 are dominated by the photon from A2, which emits more intensively than A1 (Fig.~2(c)). Most importantly, we found that the axial ratio increases significantly from 2 for the individual emissions to 24 for the simultaneous emission. We propose that this pronounced effect on the axial ratio is an indication of the phase synchronization between the two mesas as a result of EM coupling.
We observed the same behavior from other mesas shown in Fig.~1(a) and another supplemental sample, where each mesa showed slightly different polarization depending on the geometrical configuration~\cite{SM}. The amplitude ratio $E_{0x} ⁄ E_{0y}$ explains the predominance of cavity resonances in the width direction over those in the length direction. Since we injected the DC bias current into A1 and A2 using a left strip electrode as shown in Fig.~1(a), the resonance in the width direction may be degraded by a non-uniform current distribution in the mesa~\cite{Kakeya2012,Tsujimoto2014}. This explanation is supported by the observation $E_{0x} / E_{0y} =1.4$ for A1 and, in contrast, 2.0 for supplemental mesas with a symmetrical electrode configuration~\cite{SM}. This technique allows for dynamic control of polarization by adjusting the current distribution in the mesa.
In Fig.~3(a), we plot the polarization ellipses obtained by calculating the locus of $\bm{E}(t)$. According to antenna theory for a transverse magnetic cavity, $E_{0x}$ (or $E_{0y}$) is proportional to the magnitude of the magnetic currents parallel to the $y$-axis (or $x$-axis). Hence, in order to calculate $\bm{E}(t)$, we assume that the anisotropy is equal to the inverse mesa aspect, {\it i.e.}, $E_{0x} / E_{0y}=\ell /w$. This geometrical effect coincides with the numerical simulation for a locally heated square IJJ mesa~\cite{Asai2017}. By comparing Fig.~3(a) with Fig.~2(c), we can estimate $\delta _{yx}$ to be $3 \pi / 4$ (135~deg) for A1 and $- \pi / 4$ ($-45$~deg) for A2.
Let us describe our results in terms of quantum mechanics. The quantum-superposition state of the photon emitted from the parallel-biased mesa array of A1$\parallel $A2 is described as
\begin{equation}
\ket{\omega _{\textrm{A1}\parallel \textrm{A2}}, \bm{S}_{\textrm{A1}\parallel \textrm{A2}}}=
\alpha \ket{\omega _{\textrm{A1}}, \bm{S}_{\textrm{A1}}} +
\beta \ket{\omega _{\textrm{A2}}, \bm{S}_{\textrm{A2}}}, \notag
\end{equation}
where $\bm{S}_i$ ($i=\textrm{A1, A2, or A1$\parallel $A2}$) represents the Stokes vector as a quantum number. Two complex numbers $\alpha $ and $\beta $ represent the probability amplitudes ({\it i.e.}, $|\alpha |^2 + |\beta |^2 =1$) and phases of the unperturbed states $\ket{\omega _{\textrm{A1}}, \bm{S}_{\textrm{A1}}}$ and $\ket{\omega _{\textrm{A2}}, \bm{S}_{\textrm{A2}}}$, respectively. According to the dispesrsion relation of the transverse JPW~\cite{Kadowaki1997}, angular frequencies $\omega _{\textrm{A1}}$ and $\omega _{\textrm{A2}}$ are determined by the wavenumbers $\bm{k}_{\textrm{A1}}$ and $\bm{k}_{\textrm{A2}}$ of the Josephson plasmons (quantized JPW) independently of the polarization. Our concern is finding the 4$\times $4 perturbation matrix $V_m$ that satisfies
$
\begin{pmatrix}
\ket{\omega _{\textrm{A1}}',\bm{k}_{\textrm{A1}'}} \\
\ket{\omega _{\textrm{A2}}',\bm{k}_{\textrm{A2}'}}
\end{pmatrix}
=V_{m}
\begin{pmatrix}
\ket{\omega _{\textrm{A1}},\bm{k}_{\textrm{A1}}} \\
\ket{\omega _{\textrm{A2}},\bm{k}_{\textrm{A2}}}
\end{pmatrix}.
$
Here, inter-mesa coupling $V_m$ causes perturbation and may include non-diagonal elements, which involve a general question in non-linear physics regarding the symmetry of the matrix.
Figure 3(b) shows polarization $\bm{S}_{\textrm{A1$\parallel $A2}}$ calculated by taking the superposition into consideration. We used the actual intensity ratio obtained by measurement. The orientation angle in the range of $0 < \psi < \pi / 2$ is in good agreement with Fig.~2(d). It is noteworthy that the modulus $|\beta / \alpha |$ represents the degree of interaction between the two mesas and $|\beta / \alpha |=0.9$ coincides with our results. Meanwhile, the argument of $\beta / \alpha $ corresponds to the phase difference between A1 and A2.
Figure 4 shows the variation of axial ratio as a function of $\arg (\beta / \alpha )$. As indicated by arrows, two singular states exhibiting perfect polarization are emitted when $|\beta / \alpha |<0.9$. For example, two singular states with very large axial ratio can be observed at $\arg (\beta / \alpha )=45$~deg and at 135~deg when $|\beta / \alpha |=0.7$. We suggest that such perfect polarization is attributed to coherent excitation of the Josephson plasmon. See Supplemental Material for $|\beta / \alpha |$-dependence of $\bm{S}_{\textrm{A1$\parallel $A2}}$~\cite{SM}.
The origin of inter-mesa coupling $V_m$ arises from the propagation of JPWs through the Bi-2212 base crystal. The inset of Fig.~4 shows a schematic cross-sectional view of the mesa array. The dashed line in Fig.~4 represents the phase delay $2 \pi D/\lambda '$ due to the finite propagation time, where $D=58$~\textmu m is the interspace between the two mesas and $\lambda ' = \lambda /n$ is the effective wavelength. We assume that JPWs can propagate from one mesa to another mesa through the base crystal and diffract at the mesa edge. This situation strongly supports the view that the base crystal can mediate the EM interaction~\cite{Benseman2013b,Lin2013a}, the mechanism of which has been unclear in previous works. Furthermore, we found that the total intensity of simultaneous emission $S_0$ reaches a maximal value as $2 \pi D/\lambda '$ coincides with a multiple of $\arg (\beta / \alpha )$~\cite{SM}.
In conclusion, we demonstrated the synchronization of macroscopic Josephson oscillations in two simultaneously biased Bi-2212 IJJ mesas coupled via a base crystal by measuring the complete Stokes parameters. We used an achromatic QWP and a linear WGP to analyze the orthogonal components of the emitted $E$-fields. We proved that the coherent radiation is elliptically polarized with the major axis oriented in the width direction. Most importantly, we observed a significant increase in the axial ratio for simultaneous emission, suggesting that Josephson plasma in the Bi-2212 base crystal can mediate an interaction between two individual mesas. This finding represents a possible means of manipulating the synchronization of IJJ arrays, and is the most promising way to increase the integrated radiation power.
\section*{Acknowledgment}
This work was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant No.~26286006, No.~15KK0204, and No.~19H02540), JSPS-Centre national de la recherche scientifique (CNRS) Bilateral Program (Grant No.~120192908), and the Program to Disseminate the Tenure Tracking System at the University of Tsukuba. The Bi-2221 single crystal was provided by Y.~Nakagawa at Kyoto University. The authors thank H.~Asai, S.~Kawabata, M.~Machida, and T.~Koyama for their valuable discussions.
\section{Verification of quarter-wave plate device}
In order to measure the Stokes parameters, we used a lab constructed quarter-wave plate (QWP) consisting of a stack of parallel metal-plate waveguides. The thickness of the spacer was set to 2.0~mm so that the bandwidth of the QWP coincides with the frequency range of our superconducting terahertz source.
To characterize the QWP, we performed terahertz time-domain spectroscopy (THz-TDS) using an optical system based on a comb-type photoconductive antenna and ZnTe crystal electro-optic detection with an 800-nm femtosecond pump laser~\cite{Madeo2010,Maussang2016}. In this setup, the QWP and a pair of linear polarizers were placed in the optical path according to~\cite{Nagai2015}. The measurement results shown in Fig.~S1(a) prove that the transmitted $E$-field component parallel to the plate was delayed by $\pi /2$ in the frequency range of 0.3–-0.8~THz. Figure S1(b) shows the trajectory of the $E$-field vector composed of orthogonal time-domain signals. These results demonstrate that the QWP device functions as a $\pi /2$ retarder for parallel transmission.
\section{Formulation of the Stokes parameter analysis}
A useful visual representation of the polarized wave is given by
\begin{equation}
\frac{E_{x}^2(z,t)}{E^2_{0x}}+\frac{E_{y}^2(z,t)}{E^2_{0y}}
- \frac{2E_{x}(z,t)E_{y}(z,t)}{E_{0x}E_{0y}}\cos \delta _{yx}=
\sin ^2 \delta _{yx}
\end{equation}
where $\delta _{yx} = \delta _{y} - \delta _{x}$ is the lateral phase difference. Equation (1) describes the elliptical polarization when $E_{x} \neq E_{y}$ and $\delta _{yx} \neq 0$. Figure S2 shows the polarization ellipse that defines the optical parameters. We define the $x$- and $y$-axes as the directions parallel to the Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta }$ (Bi-2212) mesa width and length, respectively. The EM wave propagates along the $z$-direction.
By taking the time average, Eq.~(1) can be transformed to the intensity domain~\cite{Collett1968},
\begin{equation}
S^2_{0} = S^2_{1} + S^2_{2} + S^2_{3},
\end{equation}
where the set of four independent polarization parameters,
\begin{equation}
\bm{S}=
\begin{pmatrix}
S_0 \\
S_1 \\
S_2 \\
S_3
\end{pmatrix}
=
\begin{pmatrix}
E^2_{0x} + E^2_{0y} \\
E^2_{0x} - E^2_{0y} \\
2E_{0x}E_{0y} \cos \delta \\
2E_{0x}E_{0y} \sin \delta
\end{pmatrix},
\end{equation}
are called the Stokes parameters~\cite{Stokes1851}. These parameters are widely used to describe polarization primarily because the polarization ellipse is not directly accessible by measurement~\cite{Wolf1954}. The parameter $S_0$ represents the total intensity of the optical field, $S_1$ represents the predominance of linearly horizontally polarized light over linearly vertically polarized light, $S_2$ represents the predominance of linear $+45$~deg polarized light over linear $-45$~deg polarized light, and the fourth parameter $S_3$ represents the predominance of right circular polarization over left circular polarization. The advantage of the Stokes parameters is that they can be measured because they are represented in terms of the intensities of specific polarizations.
The polarization ellipse shown in Fig.~S2 can be described in terms of two angles: the orientation angle $\psi $ and ellipticity angle $\chi $. These angles can be determined from the Stokes parameters~\cite{Born1999} as
\begin{equation}
\psi = \frac{1}{2} \tan ^{-1} \left( \frac{S_2}{S_1} \right)
\quad
(0 < \psi < \pi)
\end{equation}
\begin{equation}
\chi = \frac{1}{2} \sin ^{-1} \left( \frac{S_3}{S_0} \right)
\quad
(-\frac{\pi }{4} < \chi < \frac{\pi }{4}).
\end{equation}
The major axis $\xi $ is directed along an axis rotated through $\psi $, and the axial ratio equals $1/ \tan \chi $.
The intensity measured using the bolometer as a function of the QWP angle $\theta $ is given by~\cite{Collett1992}
\begin{equation}
I(\theta )=\frac{1}{2} (S_{0} + S_{1} \cos ^2 2\theta
+ S_{2} \cos 2\theta \sin 2\theta + S_{3} \sin 2\theta ).
\end{equation}
The squared and product terms can be rewritten using the trigonometric half-angle formula to yield
\begin{equation}
I(\theta )=\frac{1}{2} (A + B \sin 2\theta + C \cos 4\theta + D \sin 4\theta ),
\end{equation}
where
\begin{align}
A=S_{0}
& \quad &
B=S_{3}
& \quad &
C=\frac{S_{1}}{2}
& \quad &
D=\frac{S_{2}}{2}.
\end{align}
Equation (7) is a truncated Fourier series consisting of a DC term, a second harmonic term, and two fourth harmonics terms. Thus, the four Fourier coefficients in Eq.~(8) can be determined by the following equations:
\begin{equation}
\begin{split}
&A=\frac{2}{N_m} \sum_{n=1}^N I_{n} \\
&B=\frac{4}{N_m} \sum_{n=1}^N I_{n} \sin 2\theta _{n} \\
&C=\frac{4}{N_m} \sum_{n=1}^N I_{n} \cos 4\theta _{n} \\
&D=\frac{4}{N_m} \sum_{n=1}^N I_{n} \sin 4\theta _{n}
\end{split}
\end{equation}
where $I_{n}$ is the intensity measured at angle $\theta _{n}$ and $N_m$ is the number of measurements. From Eq.~(8), the Stokes parameters are found to be
\begin{equation}
\begin{split}
&S_{0}=A-C
\qquad
S_{1}=2C \\
&S_{2}=2D
\qquad
S_{3}=B.
\end{split}
\end{equation}
We measured $N_{m}=10$ data every 18~deg in the range of $0 \leq \theta <180$ to calculate the Stokes parameters using Eqs.~(8), (9), and (10).
\section{Emission characteristics}
Figure S3(a) shows the current-voltage characteristics (IVCs) of the Bi-2212 mesas. The red and blue curves represent the data obtained from mesas A1 and A2, respectively, and the green curve is the IVC from the two mesas connected in parallel and biased simultaneously using a single voltage source ({\it i.e.}, A1$\parallel $A2). The bath temperature was fixed at $T_{b}= 15.0$~K. Each IVC curve shows the large hysteresis typical of underdamped IJJs. We used a silicon-composite bolometer (Infrared Laboratories, Inc.) and lock-in amplifier with a chopper to detect terahertz radiation. Figure S3(b) shows the bolometer output as a function of applied current. EM radiation was observed in the resistive state, with maximum outputs at 5.4~mA for A1 and 6.0~mA for A2 as indicated by the arrows in Figs.~S3(a) and S3(b).
As shown in Fig.~S3(b), the maximum output for A1 was approximately 13\% of that for A2. This difference arises from in-plane non-uniformity of the superconducting order parameters in the Bi-2212 base crystal. The inset of Fig.~S3(b) shows the temperature dependence of the $c$-axis resistance. The onset $T_c$ for A1 and A2 were 77.7~K and 81.4~K, respectively.
Global emission occurs when A1 and A2 are connected in parallel and biased on the outermost IVC branch (see the green line in Fig.~S3(b)). The maximum intensity is obtained at 16.8~mA, which is higher than the sum of the bias currents (11.4~mA) for the maximum emission powers of A1 and A2 individually. Furthermore, the emission voltage was considerably lower than those of A1 and A2. As indicated by the arrows in Fig.~S3(a), the emission voltage was 0.53~V for A1$\parallel $A2, whereas that for A1 and A2 was 0.74~V. This can be partly explained by considering the local temperature increase, which is more pronounced for mesa arrays~\cite{Benseman2013b} and allows for a reduction in the emission voltage. The effective local temperature near the biased mesas increases considerably owing to significant Joule heating, especially in the high-bias regime~\cite{Wang2009,Wang2010,Gross2012,Benseman2013c,Minami2014,Tsujimoto2014,Benseman2015}.
A lab-constructed Fourier transform interferometer based on split mirrors was used to measure the radiation frequencies. For the configuration employed the frequency resolution was 10~GHz. Figure S4(a) shows the I-V characteristic curve for A2, while Fig.~S4(b) shows the emission spectra obtained at 6.5~mA and 9.9~mA. According to the AC Josephson effect~\cite{Josephson1962}, application of a DC voltage $V$ leads to an AC current and electromagnetic emission at the Josephson frequency in the form of $f_{J}=2eV ⁄ h N$, where $e$ is the elementary charge, $h$ is the Planck constant, and $N$ is the number of junctions contributing to the emission. By substituting the radiation frequencies and voltages into the above equation, we can estimate that at most $N \sim 600$--700 junctions contribute to emission. The small steps in the outermost I-V branches indicate that emission takes place in the inner regions of multiple I-V branches~\cite{Tsujimoto2012a}, where the number of resistive junctions is reduced from the total number.
Figure S5 shows the $T_b$ dependence of the I-V characteristics with color-coded bolometer outputs. The results demonstrate stable emission on the outermost branches in the low-bias regime at temperatures below $T_{b}=55.0$~K.
\section{Supplemental data from a mesa array of differing design}
We present supplemental data obtained from a mesa array of differing design. The size of the mesas were almost identical at 77$\times $350$\times $1.2~\textmu m$^3$ with an interspace of 78~\textmu m. Here, the width was measured at the mesa bottom, and the width at the top was 72~\textmu m.
Figure S6 displays polar plots of the maximum bolometer output for mesa (a) B1 and (b) B2 at $T_{b}= 42.0$~K. Figure S6(c) shows the same plot for global emission from mesa array (B1$\parallel $B2). Data is plotted in the same way as was done in Fig.~2 of the main text.
The obtained Stokes parameters are summarized in Table SI. Figure S6(d) shows the polarization ellipses. In the case of B1$\parallel $B2, we observed a significant increase in axial ratio from $\sim $8 to 20. This behavior is consistent with A1$\parallel $A2, and the other polarization parameters were similar to those for the main sample. Note that we did not examine the possibility of synchronization with more than two mesas.
\section{Calculation of polarization for various coupling strengths}
Figure S7(a–f) shows various polarization characteristics $\bm{S}_{\textrm{A1$\parallel $A2}}$ plotted for different $|\beta / \alpha |$ values. The modulus $|\beta / \alpha |$ is the degree of interaction between the two mesas, $|\beta / \alpha |=0.9$ coinciding with the experimental results. Meanwhile, $\arg (\beta / \alpha )$ corresponds to the phase difference between the two mesas.
As indicated by arrows in Fig.~4 in the main text, two singular states exhibiting perfect linear polarization are emitted when $|\beta / \alpha |<0.9$. For example, two singular states with very large axial ratios can be clearly observed in Fig.~S7(b) at $\arg (\beta / \alpha )=45$~deg and at 135~deg.
\newpage
\section{Calculation of the total intensity as a function of inter-mesa space}
We simulated the total intensity of the global radiation emitted from the mesa arrays. The total intensity was derived from the first Stokes parameter $S_0$ and was measured using a bolometer in our experiment.
The orthogonal components for each mesa are given as follows:
\begin{equation}
\begin{split}
&E_{x1}(t)=E_{0x} \cos (\omega t) \\
&E_{y1}(t)=E_{0y} \cos (\omega t + \delta _{yx1}) \\
&E_{x2}(t)=E_{0x} \cos (\omega t + \delta _{21}) \\
&E_{y2}(t)=E_{0y} \cos (\omega t + \delta _{21} + \delta _{yx2}),
\end{split}
\end{equation}
where 1 and 2 denote mesa A1 and A2, respectively. The relative phase $\delta _{21}$ represents the phase difference between the $E_x$ of each mesa. The linear combination of these fields allows for calculation of the total intensity $S_0$. Using the sum formula for the cosine function, $\cos \phi + \cos (\phi + \gamma ) =
\frac{\sin \gamma }{\sin (\delta _{21}/2)}
\cos (\phi + \frac{\gamma }{2})$, we can obtain the total intensity as
\begin{equation}
S_{0}=
\left\{
E_{0x}
\frac{\sin \delta _{21}}{\sin (\delta _{21}/2)}
\right\}^2
+ \left\{
E_{0y}
\frac{\sin (\delta _{21}+\delta _{yx2}-\delta _{yx1})}{\sin (\frac{\delta _{21}+\delta _{yx2}-\delta _{yx1}}{2})}
\right\}^2 .
\end{equation}
In Fig.~S8, we plot $S_0$ for various $\delta _{yx2}-\delta _{yx1}$ as a function of $D$. To calculate $S_0$, we use $E_{0x} / E_{0y} = \ell / w$ and $D/\lambda '=\delta _{21}/2\pi $. The results clearly indicate that the total intensity has local maxima when $D$ coincides with an integer multiple of $\lambda '$ independently of $\delta _{yx}$. When synchronized, the total intensity depends strongly on $D$, whereas its peak value is only marginally affected by $\delta _{yx}$. This allows for the possibility of increasing the total intensity and controlling the consequent polarization by adjusting $D$.
|
2,877,628,090,077 | arxiv | \section{Introduction}
Fact-checking is an essential part of content moderation pipelines, since it provides ground truth for veracity judgements of a given claim.
However, manual fact-checking is slow and expensive, as it requires human expertise.
This demand has been already identified by a recent survey study \cite{nakov2021automated}, showing that fact-checkers from 24 organizations in 50 countries expressed the need for reliable methods to detect previously fact-checked claims. Recent work in natural language processing (NLP) has focused mainly on the development of automatic systems to identify misinforming claims both in monolingual \cite{shaar-etal-2020-known} and multilingual settings~\cite{kazemi-etal-2021-claim}. However, this research is mostly limited to short claims and does not directly assist the dissemination of existing fact-checks which are usually article-length documents. Moreover, existing work has addressed mainly claims in English with few exceptions such as work by~\citet{kazemi-etal-2021-claim} and the CheckThat! Lab \cite{CheckThat:ECIR2021}, an evaluation lab that included a claim retrieval challenge in English and Arabic in their 2021 edition.
To contribute
to this research direction,
our paper addresses the problem of matching and finding applicable fact-checks to social media posts.
We approach this problem using two strategies (i) fact-check ``matching'', i.e., determining whether a social media post (tweet) and a fact-check pair match or not, and (ii) fact-check ``retrieval'', i.e., given a social media post (tweet), rank and return the most relevant fact-checks discussing the claims made in it. We address the ``matching'' task by building a binary classifier on top of XLM-RoBERTa (XLM-R), a large transformer-based multilingual language model~\cite{conneau-etal-2020-unsupervised}. For the ``retrieval'' task, we build an embedding similarity search system using sentence embeddings from LaBSE \cite{feng2020language}, SBERT \cite{reimers-gurevych-2019-sentence} models and pairwise cosine similarity.
We also analyze these tasks for languages other than English, i.e., Spanish and Portuguese. Further, we investigate a cross-lingual scenario where we seek to identify applicable English fact-checks to Hindi tweets.
\begin{table*}[t]
\caption{An example tweet and a matching fact-check (both in English) from our dataset. The fact-checking article is redacted and can be found at \href{https://www.boomlive.in/kerala-floods-fake-news-about-dam-burst-no-power-create-panic/}{this URL}.}
\begin{tabular}{| p{\columnwidth - 12pt} |}
\hline
\textbf{Tweet \#1:} Heartbreaking to see a barrage of fake WhatsApp forwards, kooky safety instructions, hysteria that \textbf{dams are breaking}, transformers are submerged and \textbf{electricity is being cut off}, hindering rescue efforts in Kerala. \\
\hline
\textbf{Tweet \#2:} Heard a news that a \textbf{shutter of Cheerakuzhy dam, Thrissur broke.} Can someone pls confirm this news. Yet to find any reference in MSM.
If true, pls inform authorities immediately. \#KeralaFloods2018 \\
\hline
\textbf{Fact-Check Report:} The Kerala government already on the back foot trying to battle a massive crisis due to relentless rain and flooding over the past week now have one more big worry - fake news led by incorrect reporting and rumours. The floods have already resulted in the deaths of 324 people in the past 17 days with thousands of people stranded across the state on rooftops and relief camps. ... [redacted]
Worst of all is an audio clip in which a person is heard saying that the Mullaperiyar \textbf{dam has developed cracks} and in the next three hours, the downstream districts of Idukki, Ernakulam, Thrissur and Allapuzha will be washed away. He urges people to take it seriously as the government is hiding the information about the leak and that he got to know of it from a friend who works in Modi's office. (We have not uploaded the audio clip in the story to prevent further panic). ... [redacted]
Yet another audio message was going around claiming that the \textbf{shutters of Cheerakuzhi dam} built across Gayatri river (also know as Bharathpuzha) in Thrissur \textbf{are damaged.} However, regional media Manorama News and Deshabhimani clarified that it is not true and this exaggerated message is meant to create a scare. ... [redacted]
Another message which has created quite a scare is that the \textbf{whole state will have no electricity} tomorrow as the Kerala State Electricity Board (KSEB) will shut down its operations. [redacted] However, KSEB and Kerala Police were quick to respond and call the message fake. KSEB clarified through a Facebook post that KSEB employees are engaged in relentless efforts to restore electricity in the areas facing power cuts. To avoid danger during floods, power supply and production in certain parts have been temporarily discontinued. However, as the water recedes the power supply will be restored. The electricity board also appealed to people to not spread these rumours. ... [redacted] \\
\hline
\end{tabular}
\label{data_example}
\end{table*}
\section{Related Work}
While fully automatic fake news detection and fact-checking systems \cite{perez-rosas-etal-2018-automatic, thorne-vlachos-2018-automated} remain an active research topic within the NLP community, there have been new research fronts in the fight against misinformation, including claim matching \cite{shaar-etal-2020-known, kazemi-etal-2021-claim}, check-worthiness detection \cite{hassan2017toward, konstantinovskiy2021toward}, explanations \cite{kazemi-etal-2021-extractive, atanasova-etal-2020-generating, kotonya-toni-2020-explainable}, and detecting out of context misinformation \cite{da-etal-2021-edited, aneja2021cosmos}.
On the context of claim matching, \citet{shaar-etal-2020-known} introduced a retrieval-based version of the task where, for a given input claim, the goal is to rank similar check-worthy claims based on their relevance to the input claim. For this task, they focus on political related claims in English and a presented a rank model that relied on BERT~\cite{devlin-etal-2019-bert} and BM25 based architectures. More recently, \citet{kazemi-etal-2021-claim} focused on matching claims that can be served with one fact-check in five low and high-resource languages. Similarly \citet{vo-lee-2020-facts} conducted claim matching in a multimodal setting where they find previously debunked texts and images. In addition, The CheckThat! Lab \citeyear{CheckThat:ECIR2021} evaluation presented claim matching as a shared task for English and Arabic.
Although works such as \citet{shaar-etal-2020-known} have matched English tweets and fact-checks, most of prior work has mainly focused on matching claims with other similar claims that are usually short in length. In this paper, we seek to match claims with applicable fact-check reports that are significantly longer and potentially express the claim in different ways. Additionally, we approach the claim matching problem in multilingual and cross-lingual settings and experiment with recent neural models in multilingual NLP.
Among them, we use XLM-RoBERTa \cite{conneau-etal-2020-unsupervised}, a powerful multilingual transformer-based language model that have achieved competitive performance on cross-lingual and multilingual benchmarks. The model is trained on more than 2TBs CommonCrawl data and supports one hundred languages. We also rely on recent language agnostic embedding models such as LaBSE (Language-agnostic BERT Sentence Embedding) \cite{feng2020language}, a sentence embedding model that can produce embeddings in 109 languages. This model was built using a combined pretraining method of masked and translation language modeling trained on 17 billion monolingual sentences from CommonCrawl and 6 billion translated pairs of sentences. Sentence-BERT (SBERT) \cite{reimers-gurevych-2019-sentence} use twin and triplet networks on top of language models for producing sentence embeddings. In their follow up work to SBERT \cite{reimers-gurevych-2020-making}, they also propose an approach to convert monolingual embeddings into multilingual ones. We also use Elasticsearch's implementation of the BM25 retrieval system \cite{robertson2009probabilistic}, which provides fast and scalable text search.
\section{Data}
Our data is derived from 150,000 fact-checks obtained from several sources, including (i) fact-checking organizations certified by the International Fact-Checking Network (IFCN) and (ii) fact-checking aggregators such as Google Fact-check Explorer,\footnote{\url{https://toolbox.google.com/factcheck/explorer}} GESIS~\cite{tchechmedjiev2019claimskg}, and Data Commons.\footnote{\url{https://datacommons.org/factcheck/faq}} The collected fact-checks cover several languages, including English, Spanish, Portuguese, and Hindi. Each fact-check includes a claim and usually a justification article for the claim verdict, and metadata such as publication date, claim veracity and references to the original content that needed the fact-check.
\begin{table*}
\caption{Per language statistics of our (tweet, fact-check) dataset.}
\label{table:dataset}
\begin{tabular}{ c c c }
\toprule
Tweet Language & Article Language & \# of Pairs \\
\midrule
English & English & 4,850 \\
Hindi & English & 664 \\
Spanish & Spanish & 617 \\
Portuguese & Portuguese & 402 \\
\bottomrule
\end{tabular}
\end{table*}
Similar to \citet{shahi2020amused} and \citet{shahi2021exploratory}, we use social media links included in the fact-checks and their original news sources (whenever available) to build a dataset consisting of (tweet, fact-check) pairs. Given that the fact-checks include several languages, we obtain monolingual pairs in English, Spanish, and Portuguese and also cross-lingual pairs consisting of Hindi tweets and English fact-checks. In cases where the tweet contains a link (usually to a news article), we also append the preview text from the link to the tweet text, to capture more of the tweet's context.
Since we match tweets and fact-checks automatically through references in the text we conducted an additional verification step to make sure that the identified pairs are indeed related.
We thus annotate a random sample of 100 English (tweet, fact-check) pairs to verify whether each fact-check is applicable to its matched tweet. The annotation was conducted independently by two annotators, reaching an 87\% agreement between annotator responses. We find that 89\% of the tweets in our sample matched their corresponding fact-checks and in most cases the pairs include at least one fact-check worthy claim. This finding suggests that while there is some degree of noise in the pairing process, most of the pairs are correct matches. Table \ref{table:dataset} shows a summary of the final set of (fact-check,tweet) pairs per language. Sample (fact-check,tweet) pairs are shown in Table \ref{data_example}. Note that multiple tweets can be matched to the same fact-check.
\section{Models \& Baselines}
\subsection{Matching (tweet, fact-check) Pairs}
We address the task of matching (tweet, fact-check) pairs as a binary classification problem using ``match'' or ``not match'' as possible labels.
Our dataset consists of only positive labels since we only collected matching (tweet, fact-check) pairs, and training a binary classifier also requires negative examples, so we explored several strategies to obtain negative samples. Initially, we selected negative examples by randomly pairing non-matching tweets and fact-checks. We then built a binary classifier using an XLM-R model fine-tuned on the resulting dataset. However, preliminary evaluations showed that the resulting classifier was not able to generalize well.
We believe this is due to the classifier's lack of exposure to challenging negative samples, since most of random pairings are easily distinguished from matching (tweet, fact-check) pairs.
In order to get more challenging negative samples, we opted for finding non-matching (tweet, fact-check) pairs based on their pairwise similarity. We start by calculating the pair-wise cosine similarity across all possible (tweet, fact-check) pairs in the dataset, within the same multi/cross-lingual setting. Then, we use LaBSE embeddings \cite{feng2020language} of tweets and fact-check articles and rank non-matching pairs by decreasing cosine similarities. Next, we pick the top negative samples from this set, i.e., pairs with similarities lower than 0.7, to reduce the number of false negatives.
We train our XLM-R classifier with the resulting data and find an 15\% absolute improvement of classification accuracy as compared to training on randomly selected pairs.%
Since our dataset contains multiple languages, we conduct an additional set of experiments where we train separate classifiers for each language pair e.g., English, Spanish, Portuguese, Hindi-English, as well as a classifier that uses pairs in all languages. Results are presented in Table~\ref{table:classifier}
\begin{table*}[t]
\caption{Results from matching (tweet, fact-check) pairs as a binary classification problem. F1+ and F1- refer to the F1 score for the ``match'' and ``not match'' classes.}
\centering
\begin{tabular}{ l c c c c c c }
\toprule
& \multicolumn{3}{c}{\textbf{Trained Separately}} & \multicolumn{3}{c}{\textbf{Trained Altogether}} \\
\textbf{Lang. Pairs} & Acc. & F1+ & F1- & Acc. & F1+ & F1- \\
\midrule
En-En & 88.46\% & 88.72\% & 88.17\% & 88.61\% & 88.66\% & 88.54\% \\
Hi-En & 80.27\% & 80.53\% & 79.71\% & 80.50\% & 81.60\% & 78.90\% \\
Es-Es & 85.82\% & 86.07\% & 85.09\% & 88.57\% & 88.93\% & 88.06\% \\
Pt-Pt & 84.08\% & 83.67\% & 83.59\% & 87.44\% & 87.65\% & 87.25\% \\
\bottomrule
\end{tabular}
\label{table:classifier}
\end{table*}
\subsection{Finding Applicable Fact-Checks for Tweets}
A different perspective on the problem of matching fact-checks with tweets is to retrieve and rank fact-checks based on their relevance to an input tweet. As opposed to binary classification, this approach provides a ranked list of options to choose from and requires human intervention to select the most appropriate fact-check. This strategy makes the search process more scalable since finding applicable fact-checks does not require the quadratic number of computationally expensive comparisons that make the binary classification approach computationally intractable for retrieval.
During our experiments we use BM25 as our baseline retrieval method. We use the implementation provided in Elasticsearch~\cite{robertson2009probabilistic} .
BM25 is inherently language agnostic since it relies on token matching. However, this makes it unable to handle cross-lingual text, which is the case of our set of (Hindi tweets, English fact-checks). To address this issue, we translate the Hindi tweets into English using Google translate before using BM25. Our preliminary experiments show that the use of translated tweets leads to a stronger baseline as compared to just applying BM25 to the original Hindi tweets. The translation is only to accommodate for the lack of cross-lingual operability of BM25 and is necessary for keeping consistent with our comparison methodology.
Additionally, we experiment with multilingual sentence embeddings, namely LaBSE and (multilingual) MPNet-SBERT.
Since these embedding models only support inputs up to 512 tokens and fact-check articles are usually longer, we embed article paragraphs instead of whole articles. Thus, we compare an input tweet with paragraphs from the fact-check reports and not with full-length articles. Note that unlike the embedding-based models, BM25 is able to handle text in arbitrary length, so in order to carry out a fair comparison of the baseline and embeddings, we additionally provide a BM25 baseline using article paragraphs only.
The results for these experiments are presented in tables \ref{table:english_results}, \ref{table:other_language_results} and \ref{table:crosslingual_results}. We discuss them in detail in the next sections of the paper.
\subsection{Experimental Setup}
We use HuggingFace's \textit{transformers} \cite{wolf-etal-2020-transformers} and the SBERT library to implement our models. We run our code on a GPU-enabled server. For the English retrieval experiments, we use LaBSE and the \textit{paraphrase-mpnet-base-v2} model which we call ``MPNet-SBERT'', an SBERT embedding model trained on top of MPNet \cite{song2020mpnet}. We use the multilingual version of the same model (\textit{paraphrase-multilingual-mpnet-base-v2}) for Spanish, Portuguese and Hindi-English pairs.
To evaluate classification tasks, we use accuracy and F1 score as our main metrics. The classification experiments are conducted using 5-fold cross validation. For our retrieval experiments, we use ``mean reciprocal rank'' (MRR) and ``mean average precision'' (MAP@K) for different values of K.
\section{Experiments in English}
Results in Table~\ref{table:classifier} show a promising performance from our XLM-R models in matching tweet-fact check pairs in English, with accuracies of up to 89\%. As observed, there is a slight performance increase when training the model with all languages as compared to using English only. While the increase in performance when training altogether is more significant for other languages, it is worth noting that the English performance remains robust to noise as using training data from other languages can introduce noise for a model applied on English only. Although there is a slight performance decrease in the ``match'' class, the performance gain for the ``not match" class when using the training altogether model is large enough to improve the overall accuracy, which suggests potential benefits from using data in other languages.
Table \ref{table:english_results} presents results for the retrieval-based evaluation. The full-length BM25 baseline achieves 65\% MAP@1 and 72\% MRR scores as the best performing model.
The gap between MAP numbers mostly decreases as K increases which is an expected behavior for mean average precision. At first glance, it seems that feeding paragraphs from the article to the embedding models could account for the performance loss, since the full-length BM25 uses the whole document at once, therefore providing the upper bound performance for this task
While the paragraph BM25 system has a decrease in performance relative to full-length BM25, not all of the performance gap between embedding models and BM25 can be explained by the inability of embedding models to process longer documents. Among the embedding based models, MPNet-SBERT is the best performing model achieving 54\% MAP@1 and 62\% MRR. The second best performing model, LaBSE, is behind MPNet-SBERT by a noticeable margin of more than 8 MAP@1 and MRR points. A potential explanation for this performance decrease is that LaBSE is a multilingual model and performance decrease with respect to single-language models is often observed when a model supports multiple languages (100+ in LaBSE's case) at once.
Even though we see promising classification results, our experiments show that state-of-the-art NLP algorithms are still unable to compete against the BM25 baseline in finding applicable fact-checks.
\begin{table*}[ht]
\caption{Results from retrieval experiments in English.}
\centering
\begin{tabular}{ l c c c c c c }
\toprule
& \multicolumn{5}{c}{\textbf{MAP@K}} & \textbf{MRR} \\
\textbf{Model} & K=1 & K=5 & K=10 & K=20 & K=50 & \\
\midrule
Full-Length BM25 & 64.85\% & 70.82\% & 71.18\% & 71.30\% & 71.39\% & 71.51\% \\
Paragraph BM25 & 62.03\% & 66.68\% & 67.27\% & 67.46\% & 67.57\% & 67.61\% \\
LaBSE & 44.98\% & 51.59\% & 52.24\% & 52.64\% & 52.81\% & 53.00\% \\
MPNet-SBERT & 53.56\% & 60.58\% & 61.20\% & 61.48\% & 61.68\% & 61.84\% \\
\bottomrule
\end{tabular}
\label{table:english_results}
\end{table*}
\section{Experiments in Other Languages}\label{section:nonenglish}
Since our dataset also covers Spanish and Portuguese, we conduct an additional set of experiments to assess the performance of our models in languages other than English. During these experiments, we test the same models used with English, with the exception of English MPNet-SBERT that was replaced with the multilingual version.
The results in Table \ref{table:classifier} indicate that training a single XLM-R model on data from all languages performs more accurately on average (86.28\%) in comparison with training separate models per language (84.66\%) for matching. Particularly, we see a performance increase for Spanish and Portuguese, with accuracies of up to 88.57\% and 87.44\% respectively.
Training a single XLM-R model on all languages leads to a performance improvement up to 3.36\% for Spanish and Portuguese as compared to the single-language models, implying the transfer of task expertise across languages for XLM-R. A potential explanation of the fact that a single model has the leverage of larger data. We believe this is particularly effective when the languages are similar and can learn from each other's data. Also note that classifying (tweet, fact-check) pairs in multiple languages with a single XLM-R model is preferred since it saves computational resources and is easier to use.
We observe mostly similar trends to the English experiements for fact-check retrieval as shown in Table \ref{table:other_language_results}, with two exceptions: (i) multilingual MPNet-SBERT slightly outperforming the paragraph BM25 model by 1.19 MAP@1 and 2.55 MRR points in Spanish and (ii) LaBSE outperforming multilingual MPNet-SBERT by 5 MAP@1 and 2 MRR@1 points for Portuguese. Even though LaBSE performed worse than MPNet-SBERT in Spanish (2\% MAP@1 and MRR), together with the Portuguese results, LaBSE would still be the preferred embedding in non-English languages such as Spanish and Portuguese.
Note that during these experiments, the embedding models mostly underperformed in comparison to both BM25 baselines, with the full-length BM25 outperforming the best embedding model by 14 MAP@1 and 11 MRR points in Spanish in comparison with MPNet-SBERT and 10 MAP@1 and MRR points in Portuguese in comparison with LaBSE.
\begin{table*}[ht]
\caption{Results from retrieval experiments in Spanish and Portuguese. ML in ``ML MPNet-SBERT'' is short for multilingual.}
\centering
\begin{tabular}{ l c c c c c c }
\toprule
\textbf{Model} & \multicolumn{5}{c}{\textbf{MAP@K}} & \textbf{MRR} \\
\midrule
\textbf{Spanish} & K=1 & K=5 & K=10 & K=20 & K=50 & \\
\midrule
Full-Length BM25 & 73.41\% & 78.56\% & 78.78\% & 78.84\% & 78.90\% & 78.54\% \\
Paragraph BM25 & 58.33\% & 63.65\% & 64.38\% & 64.78\% & 64.88\% & 64.56\% \\
LaBSE & 57.14\% & 62.83\% & 63.68\% & 64.01\% & 64.24\% & 64.68\% \\
ML MPNet-SBERT & 59.52\% & 66.28\% & 66.58\% & 66.74\% & 66.90\% & 67.23\% \\
\midrule
\multicolumn{7}{l}{\textbf{Portuguese}} \\
\midrule
Full-Length BM25 & 69.62\% & 74.09\% & 74.73\% & 74.99\% & 75.04\% & 75.04\% \\
Paragraph BM25 & 69.62\% & 72.51\% & 72.97\% & 73.23\% & 73.32\% & 73.32\% \\
LaBSE & 59.49\% & 62.95\% & 63.25\% & 63.53\% & 63.89\% & 63.89\% \\
ML MPNet-SBERT & 54.43\% & 60.06\% & 61.07\% & 61.29\% & 61.55\% & 61.55\% \\
\bottomrule
\end{tabular}
\label{table:other_language_results}
\end{table*}
\section{Cross-Language Experiments}
The retrieval results are presented in Table \ref{table:crosslingual_results}. Unlike the monolingual experiments, models from the previous section outperform BM25 by noticeable margins in the retrieval setting and perform competitively with other language pairs in classification too. The only difference is that for the cross-lingual Hindi-English pairs in retrieval, the tweets are first translated into English. Also, the single XLM-R model trained on data from all language pairs classifies (Hindi tweet, English fact-check) pairs comparably with the monolingual models with 80.57\% accuracy according to Table \ref{table:classifier}. Although there is a 5.8\% accuracy decrease compared to the best mean accuracy (altogether), the Hindi-English XLM-R performance is still considered competitive for the more difficult task of cross-lingual matching.
Furthermore, we observe high cross-lingual performance from LaBSE, better than its performance on English and close with Portuguese and Spanish. LaBSE outperforms the best BM25 system (full-length BM25) by about 7.5 MAP@1 and 7 MRR points. However, there is a large performance gap between the embedding models (12\% MAP@1, 9.5\% MRR) as multilingual MPNet-SBERT has the worst performance of all systems but still performs not too far worse than the BM25 baselines. The improvement of LaBSE over ML MPNet-SBERT can be attributed to the fact that LaBSE was trained specifically for cross-lingual representation learning. We believe that BM25's underperformance can be attributed to translation errors. However, this is one of the few ways (if not the only way) that BM25 can support cross-lingual queries, ultimately making this a downside of using BM25. Overall, the use of XLM-R and LaBSE for cross-lingual matching and retrieval of fact-checks is a promising direction.
\begin{table*}[ht]
\caption{Results from cross-lingual retrieval experiments with tweets in Hindi and fact-check articles in English. For BM25 systems, the tweet is first translated into English before being fed to BM25.}
\centering
\begin{tabular}{ l c c c c c c }
\toprule
& \multicolumn{5}{c}{\textbf{MAP@K}} & \textbf{MRR} \\
\textbf{Model} & K=1 & K=5 & K=10 & K=20 & K=50 & \\
\midrule
Full-Length BM25 & 47.95\% & 52.59\% & 52.99\% & 53.11\% & 53.15\% & 52.80\% \\
Paragraph BM25 & 45.89\% & 50.31\% & 50.78\% & 50.98\% & 51.02\% & 50.48\% \\
LaBSE & 55.48\% & 59.12\% & 59.63\% & 59.92\% & 60.14\% & 59.79\% \\
ML MPNet-SBERT & 43.15\% & 48.72\% & 49.17\% & 49.68\% & 49.87\% & 50.22\% \\
\bottomrule
\end{tabular}
\label{table:crosslingual_results}
\end{table*}
\section{Discussion and Future Work}
Our experiments show promising performance from XLM-R in the matching classification of tweets and fact-check pairs, with the single XLM-R model trained on all data performing on average 86.28\% accurately. While the binary XLM-R classifier performs reasonably well on full-length articles, we found that it does not perform as well in classifying (tweet, fact-check paragraph) pairs when we used it to refine retrieval results. Reranking classifiers have shown promising results in prior work \cite{nogueira2019passage}, however they were not particularly applicable in our case since we do not have paragraph-level labels for the fact-check articles to train a classifier that can rerank paragraphs.
Both BM25 baselines outperform or perform competitively with state-of-the-art multilingual sentence embedding models in monolingual retrieval settings. However, there is a key difference between BM25 and the embedding models: BM25 can handle articles of arbitrary length, whereas both LaBSE and MPNet-SBERT can handle only up to 512 tokens of input. This is a source of performance loss for LaBSE and SBERT in our task. In future work, we plan to explore long document transformers such as Longformer \cite{beltagy2020longformer} to address this problem. We believe that a long document multilingual embedding model can provide improvements not only to our research problem, but to many other areas such as news NLP and legal document processing in multilingual settings.
It is important to note that the input length limit does not explain all of the performance gap. We found that the paragraph BM25 system still outperforms the embedding-based systems by at least 10 MAP@1 and 9.5 MRR points in Portuguese experiments and performs similarly to LaBSE and MPNet SBERT in Spanish. While specialized embeddings like question answering embedding models exist for English through SBERT, they are neither necessarily applicable in searching through fact-checks nor easy to come by in non-English, multilingual and cross-lingual capabilities. Building specialized embedding models for searching through applicable fact-checks is a promising next step in improving the embedding-based retrieval systems.
LaBSE provides impressive results on cross-lingual (Hindi tweet, English fact-check) pairs, outperforming BM25 baselines and MPNet-SBERT as a single multilingual embedding model with support for more than one hundred languages. This is an important problem to solve, since misinformation travels across borders and being able to search through fact-checks across different languages can save a great deal of manual fact-checking efforts. Therefore, cross-lingual search of applicable fact-checks for social media posts has a great potential for extending the reach of manual fact-checking. Furthermore, since LaBSE performs better or comparable with multilingual MPNet-SBERT overall, this makes it the better choice for embedding models when searching for relevant fact-checks on non-English social media.
\section{Conclusion}
In this paper, we approached a new version of the ``claim matching'' problem in which we match applicable fact-checks with social media posts to increase the reach of manual fact-checking. We addressed this problem using classification and retrieval based strategies in multiple languages (English, Hindi, Portuguese and Spanish). We provided new benchmarks for this task, which we will release publicly upon the paper's acceptance.
Our results showed promising performance as
we are able to classify matching (tweet, fact-check) pairs with accuracies of up to 89\% in four language pairs.
From our retrieval experiments we found that monolingual pairs of (tweet, fact-check) are better retrieved by BM25, which meaningfully outperforms state-of-the-art multilingual embedding models in the retrieval task. Despite this, we observe promising performance in cross-lingual settings with LaBSE achieving more than 7.5 MAP@1 points improvement over the best BM25 baseline.
We identified the monolingual retrieval of applicable fact-checks as a challenging area for state-of-the-art NLP and highlighted the need for specialized and long document multilingual embeddings as an important direction for future work.
Our newly curated multi/cross-lingual dataset of (tweet, fact-check) pairs in English, Spanish, Portuguese and Hindi is publicly available at \href{http://lit.eecs.umich.edu}{http://lit.eecs.umich.edu}.
|
2,877,628,090,078 | arxiv | \section{Introduction}
Recall that (\cite{LinTAF1}, \cite{LinTAF2}) a unital simple separable C*-algebra $A$ is said to be tracially AF, or TAF, if for any finite set $\mathcal F\subseteq A$, any $\varepsilon>0$, and any $a\in A^+\setminus\{0\}$, there is a non-zero finite-dimensional C*-subalgebra $F\subseteq A$ such that with $p=1_F$,
\begin{enumerate}
\item $\norm{fp-pf} < \varepsilon$, $f\in\mathcal F$,
\item $pfp \in_\varepsilon F$, $f\in\mathcal F$, and
\item $1-p$ is Murray-von Neumann equivalent to a projection in $\overline{aAa}$.
\end{enumerate}
TAF algebras are relatively well behaved. They always have real rank zero, stable rank one, strict comparison of positive elements, and they are tracially $\mathcal Z$-absorbing (\cite{HO}). The classification of simple separable nuclear TAF algebras which satisfy the Universal Coefficient Theorem (UCT) is one of the milestones in Elliott's classification program for separable nuclear C*-algebras (\cite{LnDuke}), and this class of classifiable TAF algebras coincides with the class of simple AH algebras with real rank zero and with no dimension growth (\cite{EG-RR0AH}). By Corollary 3.1 of \cite{TW1}, it in particular implies that a simple separable nuclear TAF algebra $A$ with the UCT is $\mathcal Z$-absorbing, i.e., $A\cong A\otimes\mathcal Z$, where $\mathcal Z$ is the Jiang-Su algebra.
But even without the UCT assumption, Matui and Sato showed that any simple separable nuclear TAF algebra is $\mathcal Z$-absorbing (\cite{Matui-Sato-CP}). In this note, we show that the nuclearity assumption is necessary for the $\mathcal Z$-absorption: there are non-nuclear (but exact---see Appendix) TAF algebras $A$ such that $A \ncong A\otimes\mathcal Z$. Since any tracially AF algebra is tracially $\mathcal Z$-absorbing (see Definition 2.1 of \cite{HO}), this also gives examples of (non-nuclear) tracially $\mathcal Z$-absorbing C*-algebras which are not $\mathcal Z$-absorbing, in contrast to the nuclear case (see Theorem 4.1 of \cite{HO}). (Among many other things, tracial $\mathcal Z$-absorption is also studied in \cite{Fu}).
The main tool that we use is a version of Property $\Gamma$ for C*-algebras. Recall (\cite{GJS-Z}) that a C*-algebra has Property $\Gamma$ if there is a central sequence of unitaries which vanish under all traces. It is a C*-algebra analog of Property $\Gamma$ of a von Neumann factor of type II$_1$. The reduced group C*-algebra of $\mathbf F_2$ (the free group on two generators) does not have the Property $\Gamma$. In \cite{GJS-Z}, Gong, Jiang and Su showed that all $\mathcal Z$-absorbing C*-algebras have the Property $\Gamma$, and therefore the reduced group C*-algebra of $\mathbf F_2$ is not $\mathcal Z$-absorbing (see Section 2 of \cite{GJS-Z}).
In this note, a modified version of the Property $\Gamma$ is considered (see Definition \ref{prop-gamma}): instead of arbitrary traces, one considers a fixed state. It is shown that for any unital $\mathcal Z$-absorbing C*-algebra and any given state, there exists a central sequence consisting of unitaries which are arbitrarily small under the given state (Corollary \ref{Gamma}). On the other hand, there are TAF algebras in the class constructed by D\u{a}d\u{a}rlat in \cite{MDD-SUBAF} which do not have this property (Proposition \ref{non-Gamma} below), and hence they cannot be $\mathcal Z$-absorbing.
The authors hope that this work could lead to deeper investigations of possible connections between $\mathcal Z$-absorption of (non-nuclear) C*-algebras and central sequences in (type III) von Neumann algebras.
\subsubsection*{Acknowledgements} The research of the first named author is partially supported by a Simons Collaboration Grant (Grant \#317222) and by an NSF grant (DMS-1800882). Part of the results in this paper were obtained during the visits of the second named author to the University of Wyoming in December 2017 and in June 2018, and these visits were also partly supported by the Simons Collaboration Grant \#317222. The second named author would like to thank Chris Phillips for many helpful comments. Both authors are indebted to Caleb Eckhardt for improving the original construction to achieve exactness, see Appendix A. Both authors are also indebted to George Elliott for his careful reading of the paper.
\section{The main result and the proof}
Let $G$ be a countable discrete group. Let $\mathbb C [G]$, $\mathrm{C}^\ast_{\mathrm{red}}(G)$, and $\mathrm{C}^\ast(G)$ denote the group algebra, the reduced group C*-algebra, and the full group C*-algebra of $G$ respectively.
The trace map $ \mathbb C[ G] \ni a \mapsto a(e) \in \mathbb C$ can be extended to a tracial state of $\mathrm{C}^\ast_{\mathrm{red}} (G)$, and it is denoted by $\tau$ throughout this paper.
For $g \in G$, we use $u_g$ to denote the associated standard unitary in
$\mathrm{C}^\ast_{\mathrm{red}} (G)$. We will frequently write $g$ for $u_g$ when there is no confusion.
\subsection{D\u{a}d\u{a}rlat's construction}\label{construction}
The C*-algebras we shall consider in this paper were actually constructed by D\u{a}d\u{a}rlat in \cite{MDD-SUBAF}. We briefly describe the construction for the reader's convenience.
A C*-algebra is called residually finite-dimensional, or RFD, if it has a separating family of finite-dimensional representations. Let $D$ be a separable unital RFD C*-algebra. Denote by $\pi_1, \pi_2,$... a sequence of finite-dimensional representations of $D$ which separates points, and denote by $n_1, n_2, ...$ the dimension of $\pi_1, \pi_2, ...$, respectively. Denote by $A$ the direct limit of $\mathrm{M}_{k_i}(D)$, where $k_1=1$ and $k_i=(n_1+1) \cdots (n_{i-1}+1)$ for $i = 2, 3, ...$, with the connecting map from $\mathrm{M}_{k_i}(D)$ to $\mathrm{M}_{k_{i+1}}(D)$ defined by
$$a\mapsto \mathrm{diag}(a, \pi_i (a)),\quad a \in \mathrm{M}_{k_i}(D).$$
Then $A$ is a simple unital separable TAF algebra. (See, for instance, Proposition 3.7.8 and Theorem 3.7.9 of \cite{Lin_book} or Example 4.16 of \cite{EN-Tapprox}.)
As a TAF algebra, $A$ has many regularity properties: real rank zero, stable rank one, strict order on projections is determined by traces, and any state on the ordered $\textrm{K}_0$-group arises from a trace (\cite{EN-Tapprox}). If $A$ is nuclear, then
$A$ is $\mathcal{Z}$-absorbing, by Theorem 5.4 of \cite{Matui-Sato-CP}. However, this is no longer true without the nuclearity assumption, as is shown by the following result, the main result of the paper.
\begin{thm}\label{main-thm}
There exists a simple separable unital (non-nuclear but exact) tracially AF algebra $A$ which does not absorb the Jiang-Su algebra $\mathcal Z$ tensorially, i.e. $A \ncong A\otimes\mathcal Z$.
More precisely, let $G$ be a discrete group which is not inner amenable (see \cite{Effros} or Definition \ref{def-inn-am} below), and let $D$ be a separable unital RFD C*-algebra such that $\mathrm{C}^\ast_{\mathrm{red}}(G)$ is a quotient of $D$. Denote by $A$ the TAF algebra constructed from $D$ as described above. Then $A$ is not $\mathcal Z$-absorbing, i.e., $A \ncong A\otimes\mathcal Z$. Moreover, with a suitable choice of $D$ and $\mathrm{C}^\ast_{\mathrm{red}}(G)$, the C*-algebra $A$ is exact.
\end{thm}
Let $G$ be a countable discrete group which is not inner amenable. Then, there always exists a (separable unital) RFD C*-algebra $D$ which has $\mathrm{C}^\ast_{\mathrm{red}}(G)$ as a quotient (see Theorem 1.6 of \cite{GM90} or just choose $D$ to be the universal group C*-algebra of $\mathbf F_\infty$, the free group on countably many generators). Thus the pair $(D, \mathrm{C}^\ast_{\mathrm{red}}(G))$ always exists for any discrete non-inner-amenable group $G$. Moreover, if $G$ is exact, $D$ can be chosen to be exact as well (see Proposition \ref{A1} of Appendix, by Caleb Eckhardt).
The following are two concrete constructions of the pair $(D, \mathrm{C}^\ast_{\mathrm{red}}(G))$.
\begin{example}\label{example-F2}
Let $G$ be a countable discrete non-inner-amenable group such that $D:=\mathrm{C}^\ast(G)$ is RFD. Then the pair $(D, \mathrm{C}^\ast_{\mathrm{red}}(G))$ satisfies Theorem \ref{main-thm}. One particular example of such a group is $G=\mathbf F_d$, the free group on $d$ generators, where $d=2, 3, ..., \infty$. The group $\mathbf F_d$ is not inner amenable (see \cite{Effros}), and its full group C*-algebra $D$ is RFD by Theorem 7 of \cite{choi-F_2}.
The C*-algebra $A$ constructed from $G$ and $D$ as in \ref{construction} is not exact. In fact, by Theorem 1.1 of \cite{NP-Diagonal}, the group $G$ is maximally almost periodic. Since $G$ is assumed not to be inner amenable, $\mathrm{C^*}(G)$ is not exact by the main theorem of \cite{Harpe-exact}.
\end{example}
\begin{example}
Let $G$ be a countable discrete group which is not inner amenable. Assume that $\mathrm{C}^\ast_{\mathrm{red}}(G)$ is embedded into $\prod_{i=1}^\infty{\mathrm{M}_{n_i}(\mathbb C)}/\bigoplus_{i=1}^\infty \mathrm{M}_{n_i}(\mathbb C)$ for some matrix algebra $\mathrm{M}_{n_i}(\mathbb C)$, $i=1, 2, ...$ (the MF property). Then, the C*-algebra $D:=\pi^{-1}(\mathrm{C}^\ast_{\mathrm{red}}(G))\subseteq \prod{\mathrm{M}_{n_i}(\mathbb C)}$ is RFD, where $\pi$ is the quotient map $\prod_{i=1}^\infty{\mathrm{M}_{n_i}(\mathbb C)} \to \prod_{i=1}^\infty{\mathrm{M}_{n_i}(\mathbb C)}/\bigoplus_{i=1}^\infty \mathrm{M}_{n_i}(\mathbb C)$. The pair $(D, \mathrm{C}^\ast_{\mathrm{red}}(G))$ satisfies Theorem \ref{main-thm}.
A particular example is $G= \mathbf F_d$. It follows from Corollary 8.4 of \cite{Haag-Thor} that $\mathrm{C}^\ast_{\mathrm{red}}(\mathbf F_d)$ is MF. Since $\mathrm{C}^\ast_{\mathrm{red}}(\mathbf F_d)$ can be embedded into the nuclear C*-algebra $\mathcal O_2$ (see \cite{Choi79}), $\mathrm{C}^\ast_{\mathrm{red}}(\mathbf F_d)$ is exact, and hence $D$ (as an extension of $\mathrm{C}^\ast_{\mathrm{red}}(\mathbf F_d)$ by $\bigoplus_{i=1}^\infty \mathrm{M}_{n_i}(\mathbb C)$) and $A$ (constructed above as an inductive limit of matrix algebras over $D$) are exact.
\end{example}
A more interesting example is given by Caleb Eckhardt (Proposition \ref{A2} of Appendix), where an exact RFD algebra $D$ is constructed between $\mathrm{C}^\ast(\mathbf F_d)$ and $\mathrm{C}^\ast_{\mathrm{red}}(\mathbf F_d)$. Eckhardt also pointed out a general way to produce exact examples (Proposition \ref{A1} of Appendix).
\subsection{Central unitaries in \texorpdfstring{$A$}{A}}
We first introduce the following version of Property $\Gamma$ which is similar to Definition 2.1 of \cite{GJS-Z}:
\begin{defn}\label{prop-gamma}
Let $A$ be a unital C*-algebra and let $S$ be a collection of states on $A$. We shall say that $A$ has \emph{Property $\Gamma$}
with respect to $S$ if there is a central sequence of unitaries $(u_i)$ of $A$ such that
$\abs{\rho(u_i)} \to 0$ as $i\to\infty$ for any $\rho \in S$. If $S$ consists of a single state $\rho$, we shall say that $A$ has Property $\Gamma$ with respect to $\rho$.
\end{defn}
For the C*-algebra $A$ constructed in Theorem \ref{main-thm}, we shall show that there is a state $\rho$ of $A$ such that $A$ does not have Property $\Gamma$ with respect to $\rho$. Let us start with a simple observation.
\begin{lem}\label{pre-est-2}
Let $D$ be a unital C*-algebra and let $n$ be a positive integer.
Let $$u=\left(\begin{array}{cc} a & b \\ c & d \end{array}\right) \in \mathrm{M}_{1+n}(D)$$ be a matrix over $D$ with $a\in D$, $d\in \mathrm{M}_n(D)$, $b\in \mathrm{M}_{1, n}(D)$, and $c\in \mathrm{M}_{n, 1}(D)$. Then $$\norm{a}, \norm{b}, \norm{c}, \norm{d} \leq \|u\|.$$
\end{lem}
\begin{proof}
Let $p=\mathrm{diag}(1, 0)$ and $q=\mathrm{diag}(0, 1_n)$. Identify $a$ with
$\left(\begin{array}{cc} a & 0 \\ 0 & 0 \end{array}\right)$ and similarly for $b, c, d$. (This is justified since the identification does not change the norm.) Then
$$a=pup,\quad b=puq,\quad c=qup,\quad\mathrm{and}\quad d=quq.$$
The lemma follows.
\end{proof}
\begin{ntn}\label{notation}
Let $G$ be any discrete group and let $m, n$ be natural numbers. We shall use $\norm{\cdot}_{\mathrm{red}}$ to denote the operator norm on $\mathrm{M}_{m, n}(\mathrm{C}_{\mathrm{red}}^\ast(G))$ induced by the C*-norm of
$\mathrm{C}_{\mathrm{red}}^\ast(G)$.
Recall that $\tau$ is the canonical trace of $\mathrm{C}_{\mathrm{red}}^\ast(G)$. For $b = (b_1, b_2, ..., b_n) \in \mathrm{M}_{1,n}(\mathrm{C}_{\mathrm{red}}^\ast(G))$, define
\[
\norm{b}_2 = (\tau(b_1b_1^* + b_2 b_2^* + \cdots + b_n b_n^*))^{\frac{1}{2}} = \tau(bb^*)^{\frac{1}{2}},
\]
and define $\tilde{b}$ to be the function
$$\tilde{b}: G \ni \gamma \mapsto \|b(\gamma)\|_2 = \left(\sum_{i=1}^n \abs{b_{i}(\gamma)}^2 \right)^{\frac{1}{2}}\in\mathbb C.$$
\end{ntn}
It is straightforward to check that $\|\tilde{b}\|_2 = \|b\|_2$ for any $b \in \mathrm{M}_{1,n}(\mathrm{C}_{\mathrm{red}}^\ast(G))$. In particular, the function $\tilde{b}$ defined above is in $l^2(G)$.
\begin{lem}\label{norms}
Let $G$ be a discrete group. Let $n \in \mathbb{N}$ and let $a, b$ be
elements in $\mathrm{M}_{1, n}(\mathrm{C}_{\mathrm{red}}^\ast (G))$. With the notation as \ref{notation}, one has
\begin{enumerate}
\item \label{p_1} $\widetilde{bu} = \tilde{b}$ for any unitary $u \in \mathrm{M}_n(\mathbb C)$,
\item \label{p_2} $\|\tilde{b}\|_2 = \|b\|_2 \leq \|b\|_{\mathrm{red}}$ for any $b \in \mathrm{M}_{1, n}(\mathrm{C}_{\mathrm{red}}^\ast (G))$,
\item \label{p_3} $\|bu\|_2 = \|b\|_2$ for any unitary $u \in \mathrm{M}_n(\mathbb C)$,
\item \label{p_4} $\|g\tilde{a} - \tilde{b}\|_2 \leq \|ga - b\|_2$ for any $g \in G$, and
\item \label{p_5} $\|b - bd\|_2^2 \leq 2\|b\|_2(\|b\|_2 - \|bd\|_2)$ for any matrix $d \in \mathrm{M}_n(\mathbb C)$ which is diagonal, positive, and contractive.
\end{enumerate}
\end{lem}
\begin{proof}
(\ref{p_1}) and (\ref{p_2}) follow from straightforward computation. (\ref{p_3}) is a direct consequence of (\ref{p_1}) and (\ref{p_2}). For (\ref{p_4}), note that for each $\gamma \in G$, $a(\gamma)$ is a vector in $\mathbb C^n$ and $\|a(\gamma)\|_2$ is the vector norm. Using triangle inequality at the third step and (1) at the last step, we have
\begin{align*}
\|g\tilde{a} - \tilde{b}\|_2^2
& = \sum_{\gamma \in G} \abs{(g\tilde{a})(\gamma) - \tilde{b}(\gamma)}^2 \\
& = \sum_{\gamma \in G} \left(\norm{a(g^{-1}\gamma)}_2 - \norm{b(\gamma)}_2 \right)^2 \\
& \leq \sum_{\gamma \in G} \norm{a(g^{-1}\gamma)- b(\gamma)}_2^2 \\
& = \sum_{\gamma \in G} \norm{(ga-b)(\gamma)}_2^2 \\
& = \|\widetilde{ga-b}\|_2^2 = \|ga-b\|_2^2.
\end{align*}
For (\ref{p_5}), set $b = (b_1, b_2, ..., b_n)$ and $d = \mathrm{diag}\{\lambda_1, ..., \lambda_n\}$, where $b_i \in \mathrm{C}_{\mathrm{red}}^\ast (G)$ and $\lambda_i \in [0, 1]$ for $i = 1, 2,..., n$. Note that $(1-\lambda_i)^2 \leq (1-\lambda_i)(1+ \lambda_i) = (1 - \lambda_i^2)$, and hence
\begin{align*}
\|b - bd\|_2^2
& = \sum_{\gamma \in G}\sum_{i = 1}^n (1 - \lambda_i)^2|b_i(\gamma)|^2 \\
& \leq \sum_{\gamma \in G}\sum_{i = 1}^n (1 - \lambda_i^2)|b_i(\gamma)|^2 = (\|b\|_2^2 - \|bd\|_2^2) \\
& = (\|b\|_2 + \|bd\|_2)(\|b\|_2 - \|bd\|_2) \leq 2\|b\|_2(\|b\|_2 - \|bd\|_2),
\end{align*}
as desired.
\end{proof}
\begin{lem}\label{inv-mean} Let $G$ be a discrete group, and let $n \in \mathbb{N}$. Let $b$ be an element of $\mathrm{M}_{1, n}(\mathrm{C}_{\mathrm{red}}^\ast (G))$ with $\norm{b}_{\mathrm{red}} \leq 1$, and let $g \in G$. Assume that there are $\varepsilon > 0$ and a matrix $\pi(g) \in \mathrm{M}_n(\mathbb C)$ with norm at most $1$ such that
\begin{equation}\label{comm-0}
\norm{gb - b\pi(g)}_2 < \varepsilon.
\end{equation}
Then, with notation as \ref{notation}, one has
$\|g\tilde{b} - \tilde{b}\|_2 < \varepsilon + \sqrt{2\varepsilon}$.
\end{lem}
\begin{proof}
Applying the polar decomposition to the matrix $\pi(g)$, we have unitaries $u, w\in \mathrm{M}_n(\mathbb C)$ and a diagonal matrix $d = \mathrm{diag}\{\lambda_1, ..., \lambda_n\}$, where $\lambda_i \in [0, 1]$, $i = 1, 2,..., n$, such that $$\pi(g) = u(wdw^*).$$
It follows from Lemma \ref{norms}(\ref{p_3}) and \eqref{comm-0} that
\begin{equation} \label{esti-0}
\norm{g(bw) - buwd}_2 = \norm{(gb)w - buwd}_2 = \norm{gb - buwdw^*}_2 < \varepsilon.
\end{equation}
Since $u, w$ are unitary matrices, by Lemma \ref{norms}(\ref{p_3}) again, we have
\begin{equation}\label{mea-pres}
\norm{buw}_2 = \norm{b}_2= \norm{bw}_2= \norm{g(bw)}_2 \approx_\varepsilon \norm{buwd}_2.
\end{equation}
Since $\|b\|_{\mathrm{red}} \leq 1$, It follows from (\ref{p_2}) and (\ref{p_3}) of Lemma \ref{norms} that
\begin{equation} \label{esti-1}
\|buw\|_2 = \|b\|_2 \leq \|b\|_{\mathrm{red}} \leq 1.
\end{equation}
Using Lemma \ref{norms}(\ref{p_1}) in the first step, triangle inequality in the second step,
Lemma \ref{norms}(\ref{p_4}) in the third step, (\ref{esti-0}) and Lemma \ref{norms}(\ref{p_5}) in the fourth step and (\ref{mea-pres}) and (\ref{esti-1}) in the final step, we get
\begin{align*}
\|g\tilde{b} - \tilde{b}\|_2
& = \|g\widetilde{bw} - \widetilde{buw}\|_2 \\
& \leq \|g\widetilde{bw} - \widetilde{buwd}\|_2 + \|\widetilde{buwd} - \widetilde{buw}\|_2 \\
& \leq \|gbw - buwd\|_2 + \|(buw)d - buw\|_2 \\
& \leq \varepsilon + 2\|buw\|_2^{\frac{1}{2}}(\|(buw)\|_2 - \|buw\|_2)^{\frac{1}{2}} \\
& \leq \varepsilon + 2\sqrt{\varepsilon},
\end{align*}
as desired.
\end{proof}
\begin{lem}\label{small-vec}
Let $ G$ be a countable discrete group which is not amenable. For any $\varepsilon>0$, there exist $\delta>0$ and a finite set $K \subseteq G$ such that if $\xi \in l^2(G)$ satisfies
$$\norm{g\xi - \xi}_2 < \delta,\quad g\in K,$$
then $\norm{\xi}_2 < \varepsilon.$
\end{lem}
\begin{proof}
Let $(K_n)$ be an increasing sequence of finite subsets of $G$ whose union is $G$. If the statement were not true, there will be $\varepsilon_0>0$ such that for any $n \in \mathrm{N}$, there is $\xi_n \in l^2(G)$ with $$\norm{g\xi_n - \xi_n}_2 < \frac{1}{n},\quad g\in K_n,$$ but $$\norm{\xi_n}_2 \geq \varepsilon_0.$$ Then the sequence $\{ \norm{\xi_n}_2^{-1}\xi_n \colon n=1, 2, ... \}$ forms an almost invariant vector for the left regular representation of $G$, which implies that $G$ is amenable, a contradiction.
\end{proof}
The following result is a consequence of Lemma \ref{inv-mean} and Lemma \ref{small-vec}.
\begin{cor}\label{small-offd}
Let $G$ be a countable discrete non-amenable group. For any $\varepsilon>0$, there exist $\delta>0$ and a finite set $K \subseteq G$ with the following property:
For any $n \in \mathbb{N}$ and any $b \in \mathrm{M}_{1, n}(\mathrm{C}^\ast_{\mathrm{red}}(G))$, if $\|b\|_{\mathrm{red}} \leq 1$, and if for each $g\in K$ there is a matrix $\pi(g) \in \mathrm{M}_n(\mathbb C)$ with norm at most $1$ such that
\begin{equation}
\norm{gb - b\pi(g)}_2 < \delta,
\end{equation}
then $\norm{b}_2 < \varepsilon.$
\end{cor}
\begin{proof}
Let $\delta_0>0$ and $K \subseteq G$ denote the constant and finite set provided by Lemma \ref{small-vec} with respect to $\varepsilon$. Pick $\delta> 0$ such that $ \delta + \sqrt{2\delta} \leq \delta_0 $.
With $\tilde{b}$ as \ref{notation}, it follows from Lemma \ref{inv-mean} that, if
$$\norm{gb - b\pi(g)}_{2} < \delta,\quad g\in K,$$ then
$$\norm{g\tilde{b} - \tilde{b}}_2 < \delta + \sqrt{2\delta} \leq \delta_0,\quad g\in K. $$
By the choice of $\delta_0$ and $K$, it follows that $$\norm{b}_2 = \|\tilde{b}\|_2 < \varepsilon,$$ as desired.
\end{proof}
Recall that a \emph{mean} on a countable discrete group $G$ is a positive linear functional $m$ on $l^{\infty}(G)$ with $m(1) = 1$.
Let $e$ be the neutral element of $G$. It is easy to check that the map $d_e \colon l^{\infty}(G) \rightarrow \mathbb{C}$ defined by $d_e(f) = f(e)$, $f \in l^{\infty}(G)$,
is always a mean, which is called the \emph{trivial mean}.
If $\xi$ is a function on $G$ and $g$ is an element of $G$, define $g \xi g^{-1}$ to be the function
\[
(g \xi g^{-1})(x) = \xi(g^{-1}xg),\quad x \in G.
\]
\begin{defn}\label{def-inn-am}
(See \cite{Effros}.) A countable discrete group $G$ is said to be \emph{inner amenable} if there is a nontrivial inner invariant mean $m$, in the sense that
\[
m(g \xi g^{-1}) = m( \xi), \quad \xi \in l^{\infty}(G) \text{\,\,and\,\,} g \in G.
\]
\end{defn}
The following lemma is surely well known. A proof is included for the reader's convenience.
\begin{lem}\label{small-d}
Let $G$ be a countable discrete group which is not inner amenable.
Let $1_e$ denote the identity of $\mathbb C [G]$. For any $\varepsilon>0$, there are $\delta>0$ and a finite set $K \subseteq G$ such that if $\xi \in \mathrm{C}^\ast_{\mathrm{red}}(G)$ satisfies
$$\norm{g \xi g^{-1} - \xi}_2 < \delta,\quad g\in K,$$
then $\norm{\xi - \tau(\xi)1_e}_2 < \varepsilon.$
\end{lem}
\begin{proof}
Assume that the statement were false.
Choose an increasing sequence of finite subsets $(K_n)$ whose union is $G$.
Then there is some $\varepsilon_0 > 0$ such that for any $n \in \mathbb{N}$, there is $\xi_n \in \mathrm{C}^\ast_{\mathrm{red}}(G)$ satisfying
\[
\norm{g \xi_n g^{-1} - \xi_n}_2 < \frac{1}{n},\quad g\in K_n,
\]
but $\|\xi_n - \tau(\xi_n)1_e\|_2 \geq \varepsilon_0$.
Let $\eta_n = \|\xi_n - \tau(\xi_n)1_e\|_2^{-1}(\xi_n - \tau(\xi_n)1_e)$.
Then $\|\eta_n\|_2 = 1$, $\eta_n(e) = 0$, and for any $g\in G$,
\[
\norm{g \eta_n g^{-1} - \eta_n}_2 = \frac{\norm{g \xi_n g^{-1} - \xi_n}_2}{\|\xi_n - \tau(\xi_n)1_e\|_2} \leq \frac{1}{\varepsilon_0 n}
\rightarrow 0 \quad \text{as\,\,} n \rightarrow \infty.
\]
By the main theorem of \cite{Effros}, this implies that $G$ is inner amenable (the statement of the main theorem of \cite{Effros} assumes that $G$ is i.c.c., but the proof of (2)$\Rightarrow$(3)$\Rightarrow$(4) of the main theorem of \cite{Effros} does not use this assumption), which contradicts the assumption.
\end{proof}
\begin{defn}\label{defn-seminorm}
Consider the C*-algebra $\mathrm{M}_{1+n}(\mathrm{C}^\ast_{\mathrm{red}}(G))$, where $G$ is a discrete group and $n$ is a natural number, and consider the state $$\rho: \mathrm{M}_{1+n}(\mathrm{C}^\ast_{\mathrm{red}}(G)) \ni \left(\begin{array}{cc} a & b \\ c & d \end{array} \right) \mapsto \tau(a) \in\mathbb C,$$ where $\tau$ is the canonical trace of $\mathrm{C}^\ast_{\mathrm{red}}(G)$. Define the seminorm $\norm{\cdot}_\rho^\#$ of $\mathrm{M}_{1+n}(\mathrm{C}^\ast_{\mathrm{red}}(G))$ by $$\norm{x}_\rho^\# = (\frac{\rho(xx^*) + \rho(x^*x)}{2})^{\frac{1}{2}} = (\frac{\tau(aa^*) + \tau(bb^*) + \tau(a^*a) + \tau(c^*c)}{2})^{\frac{1}{2}},$$ if $x = \left(\begin{array}{cc} a & b \\ c & d \end{array} \right) \in \mathrm{M}_{1+n}(\mathrm{C}^\ast_{\mathrm{red}}(G)).$
Note that $\norm{\cdot}_\rho^\# \leq \norm{\cdot}_{\mathrm{red}}$.
\end{defn}
\begin{cor}\label{diag}
Let $G$ be a countable discrete group which is not inner amenable.
For any $\varepsilon>0$, there exist $\delta>0$ and a finite set $K \subseteq G$ such that
for any element $$u=\left(\begin{array}{cc} a & b \\ c & d \end{array} \right) \in \mathrm{M}_{1+n}(\mathrm{C}^\ast_{\mathrm{red}}(G))$$ with $a\in \mathrm{C}^\ast_{\mathrm{red}}(G)$ which satisfies $\norm{u}_{\mathrm{red}}\leq 1$, if for each $g\in K$, there is a matrix $\pi(g) \in \mathrm{M}_{n}(\mathbb C)$ with norm at most $1$ such that
\begin{equation}\label{am-comm}
\norm{ \left[\left(\begin{array}{cc} a & b \\ c & d \end{array} \right), \left(\begin{array}{cc} g & 0 \\ 0 & \pi(g) \end{array} \right)\right] }_{\rho}^\#< \delta, \quad g\in K,
\end{equation}
then
$$\norm{b}_2, \norm{c}_2 < \varepsilon$$ and
$$\norm{a-\tau(a)1_e}_2<\varepsilon.$$
\end{cor}
\begin{proof}
Applying Corollary \ref{small-offd} and and Lemma \ref{small-d} to $\varepsilon$, one obtains ($\delta_1$, $K_1$) and ($\delta_2$, $K_2$) respectively. Set $\delta=\frac{1}{\sqrt{2}}\min\{\delta_1, \delta_2\}$ and $K = K_1 \cup K_2$.
Let $\left(\begin{array}{cc} a & b \\ c & d \end{array} \right) \in \mathrm{M}_{1+n}(\mathrm{C}^\ast_{\mathrm{red}}(G))$ satisfy the assumption for this choice of $\delta$ and $K$. It follows from \eqref{am-comm} that
\begin{equation}
\norm{ \left(\begin{array}{cc} ag - ga & b\pi(g) - gb \\ cg - \pi(g)c & d\pi(g) - \pi(g) d \end{array} \right) }_{\rho}^\#< \delta, \quad g\in K,
\end{equation}
and hence, by the definition of $\norm{\cdot}_\rho^\#$, one has that for any $g\in K$,
\begin{eqnarray}
\norm{ga-ag}_2 & < & \delta_2,\label{comm0} \\
\norm{gb - b\pi(g)}_2 & < & \delta_1,\quad\mathrm{and} \label{comm1}\\
\norm{cg- \pi(g) c}_2 & < & \delta_1. \label{comm2}
\end{eqnarray}
Applying Lemma \ref{small-d} to \eqref{comm0}, one obtains
$$\norm{a - \tau(a)1_e}_2<\varepsilon.$$
For the estimates on $b$ and $c$, since $\norm{u}_{\mathrm{red}}\leq 1$, by Lemma \ref{pre-est-2}, one has $\|b\|_{\mathrm{red}} \leq 1$ and $\|c\|_{\mathrm{red}} \leq 1$.
With the choice of $\delta$ and $K$, by \eqref{comm1} and \eqref{comm2}, it follows from Corollary \ref{small-offd} that $$\norm{b}_2<\varepsilon\quad\mathrm{and}\quad\norm{c}_2<\varepsilon,$$
as desired.
\end{proof}
Recall that in D\u{a}d\u{a}rlat's construction (\ref{construction}), the C*-algebra $A$ is the direct limit of $\mathrm{M}_{k_i}(D)$ with the connecting maps
$$a\mapsto \mathrm{diag}(a, \pi_i (a)),$$
and there is a surjective homomorphism $\theta: D \to \mathrm{C}^\ast_{\mathrm{red}}(G)$. Consider the state of $A$ defined by $$\rho_\theta((a_{jk})) = \tau(\theta(a_{11})),\quad (a_{jk}) \in \mathrm{M}_{k_i}(D)$$
where $\tau$ is the canonical trace of $\mathrm{C}^\ast_{\mathrm{red}}(G)$. Similar to Definition \ref{defn-seminorm}, consider the seminorm
$$\norm{a}_{\rho_\theta}^\# := (\frac{\rho_\theta(aa^*) + \rho_\theta(a^*a)}{2})^\frac{1}{2},\quad a\in A.$$
Note that the successive connecting maps $D \rightarrow \mathrm{M}_{k_i}(D)$ always have the form
\begin{equation} \label{embedding}
a \rightarrow \mathrm{diag} (a, \pi(a)),
\end{equation}
where $\pi: D \to \mathrm{M}_{k_i-1}(\mathbb C 1_D) \subseteq \mathrm{M}_{k_i-1}(D)$ is a finite-dimensional representation of $D$. This induces an embedding of $D$ into $A$. We shall identify $D$ as a sub-C*-algebra of $A$ via this embedding.
\begin{prop}\label{non-Gamma}
Let $G$ be a countable discrete group which is not inner amenable, and let $D$ be a separable unital RFD algebra such that $\mathrm{C}^\ast_{\mathrm{red}}(G)$ is a quotient of $D$. Let $A$ be the TAF algebra constructed from $D$ and let $\rho_\theta$ be the state described as above.
For any $g \in G$, pick an element $\check{g}$ of $D$ with norm 1 which lifts $u_g$, and regard $\check{g}$ as an element of $A$ via the embedding induced by the maps \eqref{embedding}. Then, for any $\varepsilon>0$, there exist $\delta>0$ and a finite set $K \subseteq G$ such that if $u\in A$ is a unitary satisfying $$\norm{u\check{g}-\check{g}u}_{\rho_\theta}^\# < \delta,\quad g\in K,$$ then $$|\rho_\theta(u)| > 1-\varepsilon.$$
In particular, since $\norm{\cdot}_{\rho_\theta}^\# \leq \norm{\cdot}$, the TAF algebra $A$ does not have Property $\Gamma$ with respect to $\rho_\theta$.
\end{prop}
\begin{proof}
Let $\varepsilon>0$ be arbitrary. Choose $\varepsilon_0 > 0$ small enough so that
\[
\sqrt{1-\varepsilon_0 -2 \varepsilon_0^2} - \varepsilon_0
> 1 - \varepsilon.
\] Let $\delta_0 > 0$ and $K \subseteq G$ be the constant and finite subset provided by Corollary \ref{diag} with $\varepsilon_0$ in the place of $\varepsilon$. Set
\[
\delta = \min \{ \frac{\delta_0}{3}, \frac{\varepsilon_0}{2}\}.
\]
Let $u$ be a unitary in $A$ satisfying $$\|u\check{g}-\check{g}u\|_{\rho_\theta}^\# < \delta,\quad g \in K.$$ By the construction of $A$, there is $n \in \mathbb{N}$ and
$$v=\left(\begin{array}{cc} a & b \\ c & d \end{array}\right) \in \mathrm{M}_{1+n}(D)\subseteq A$$ with $a\in D$, $\norm{v}=1$, and
$\|v - u\| < \delta$.
Note that $\check{g}$ is identified with $\left(\begin{array}{cc} \check{g} & 0 \\ 0 & \pi({\check{g})} \end{array}\right) \in \mathrm{M}_{1+n}(D) \subseteq A,
$
where $\pi(\check{g})$ is a scalar matrix of norm at most 1; so $\theta(\pi(\check{g})) = \pi(\check{g})$.
Since
\begin{eqnarray*}
\norm{ \left[\left(\begin{array}{cc} \theta(a) & \theta(b) \\ \theta(c) & \theta(d) \end{array} \right), \left(\begin{array}{cc} g & 0 \\ 0 & \pi(\check{g}) \end{array} \right)\right] }_{\rho}^\#
& = &\|\theta(\check{g}v -v\check{g})\|_{\rho}^\# = \|\check{g}v-v\check{g}\|_{\rho_\theta}^\# \\
&\leq &\|\check{g}u-u\check{g}\|_{\rho_\theta}^\# + 2\|u - v\| < 3\delta \leq \delta_0,
\end{eqnarray*}
for $g \in K$,
by the choice of $\delta_0$ and $K$, it follows from Corollary \ref{diag} that
\begin{equation}\label{eq-sm-1}
\norm{\theta(b)}_2, \norm{\theta(c)}_2 < \varepsilon_0
\end{equation} and
\begin{equation}\label{eq-sm-2}
\norm{\theta(a)-\tau(\theta(a))1_e}_2<\varepsilon_0.
\end{equation}
Since $\|u -v\| < \delta \leq {\varepsilon_0}/{2}$ and $u$ is a unitary, one has
$$
\norm{(aa^*+bb^*)-1_D}
\leq \norm{vv^* - 1_{\mathrm{M}_{1+n}(D)}}
\leq \norm{uu^* - 1_A} + 2 \|v-u\|
< \varepsilon_0.
$$
Hence by \eqref{eq-sm-1}, \begin{equation}\label{eq-sm-3}\tau(\theta(aa^*))\geq 1 - \varepsilon_0-\tau(\theta(bb^*)) > 1-\varepsilon_0-\varepsilon_0^2.
\end{equation}
On the other hand, write $a=\lambda1_D + a_0$ with $\lambda = \tau(\theta(a))$ and $a_0=a-\tau(\theta(a))1_D$. Then
$$
aa^* = (\lambda 1_D + a_0) (\bar{\lambda} 1_D + a^*_0)
= \abs{\lambda}^2 1_D + \lambda a^*_0 + \bar{\lambda} a_0 + a_0a_0^*.
$$
Applying the quotient map $\theta$ and the trace $\tau$ on both sides, by \eqref{eq-sm-2}, we have
$$\tau(\theta(aa^*)) = \abs{\lambda}^2 + \tau(\theta(a_0a_0^*)) = \abs{\lambda}^2 + \norm{\theta(a)-\tau(\theta(a))1_e}_2^2 < \abs{\lambda}^2 + \varepsilon_0^2.$$
Together with \eqref{eq-sm-3}, we have $$\abs{\lambda}^2 > 1-\varepsilon_0-2\varepsilon_0^2,$$
and therefore
$$
|\rho_\theta(u)| \geq |\rho_\theta(v)| - \varepsilon_0
= |\tau(\theta(a))| - \varepsilon_0 > \sqrt{1-\varepsilon_0 -2 \varepsilon_0^2} - \varepsilon_0 > 1 - \varepsilon,
$$
as desired.
\end{proof}
As a simple corollary, all unitary central sequences in the von Neumann algebra generated by $A$ under the GNS representation associated to $\rho_\theta$ are trivial:
\begin{cor}
Consider the GNS representation of $A$ associated to the state $\rho_\theta$, and consider the von Neumann algebra $A''_{\rho_\theta}$. Then, if there is a sequence of unitaries $(u_n) \subseteq A''_{\rho_\theta}$ such that for any $x\in A_{\rho_\theta}''$,
$[u_n, x] \to 0$ in the strong* topology as $n\to\infty$, one has
$$(u_n - \rho_\theta(u_n)1_A) \to 0,\quad n\to\infty,$$ in the strong* operator topology.
\end{cor}
\begin{proof}
Since the unitary group of $A$ is strongly* dense in the unitary group of $A_{\rho_\theta}''$ (Kaplansky Density Theorem), there is a sequence of unitaries $(w_n)\subseteq A$ such that $(u_n-w_n) \to 0$ strongly* as $n\to\infty$. Let $x\in A_{\rho_\theta}''$. Since $[u_n, x] \to 0$ in the strong* topology as $n\to\infty$, one has
\begin{equation*}
\norm{(u_nx - xu_n)(\xi_{1_A})}_{\rho_\theta} + \norm{(u_nx - xu_n)^*(\xi_{1_A})}_{\rho_\theta}\to 0\quad \text{as\,\,} n\to\infty,
\end{equation*}
and hence
\begin{equation*}
\norm{(w_nx - xw_n)(\xi_{1_A})}_{\rho_\theta} + \norm{(w_nx - xw_n)^*(\xi_{1_A})}_{\rho_\theta}\to 0 \quad \text{as\,\,} n\to\infty.
\end{equation*}
That is,
$$\rho_\theta((w_nx - xw_n)^*(w_nx - xw_n)) + \rho_\theta((w_nx - xw_n)(w_nx - xw_n)^*) \to 0 \quad \text{as\,\,} n\to\infty,$$ and
$$\norm{w_nx - xw_n}_{\rho_\theta}^\# \to 0 \quad \text{as\,\,} n\to\infty.$$
Since $(w_n)\subseteq A$, by Proposition \ref{non-Gamma}, $$\abs{\rho_\theta(w_n)} \to 1 \quad \text{as\,\,} n\to\infty,$$
and this implies
\begin{eqnarray*}
&& \rho_\theta((w_n-\rho_\theta(w_n)1_A)^*(w_n-\rho_\theta(w_n)1_A)) \\
& = & \rho_\theta(w_n^*w_n - \rho_\theta(w_n)w_n^* -\overline{\rho_\theta(w_n)}w_n +\abs{\rho_\theta(w_n)}^21_A) \\
& = & \rho_\theta(1_A) - \abs{\rho_\theta(w_n)}^2 \to 0 \quad \text{as\,\,} n\to\infty.
\end{eqnarray*}
The same calculation also shows
$$\rho_\theta((w_n-\rho_\theta(w_n)1_A)(w_n-\rho_\theta(w_n)1_A)^*) \to 0 \quad \text{as\,\,} n\to\infty.$$ Therefore
$$\norm{w_n-\rho_\theta(w_n)1_A}_{\rho_\theta}^\# \to 0 \quad \text{as\,\,} n\to\infty,$$ and since $(w_n-u_n) \to 0$ strongly* as $n\to\infty$,
$$\norm{u_n-\rho_\theta(u_n)1_A}_{\rho_\theta}^\# \to 0 \quad \text{as\,\,} n\to\infty.$$
Let $x_1, x_2\in A$ be arbitrary. Set $M = \max \{\norm{x_1}, \norm{x_2}\}$. Then, since $[u_n-\rho_\theta(u_n)1_A, x] \to 0$ strongly* as $n\to\infty$, for any $x\in A$, one has
\begin{eqnarray*}
& &\limsup_{n\to\infty} \left( \norm{(u_n-\rho_\theta(u_n)1_A)x_1}_{\rho_\theta} + \norm{(u_n-\rho_\theta(u_n)1_A)^*x_2}_{\rho_\theta} \right)\\
& \leq & \limsup_{n\to\infty} \left(\norm{x_1} \norm{(u_n-\rho_\theta(u_n)1_A)}_{\rho_\theta} + \norm{x_2} \norm{(u_n-\rho_\theta(u_n)1_A)^*}_{\rho_\theta} \right) \\
& \leq & \limsup_{n\to\infty} M \left(\norm{(u_n-\rho_\theta(u_n)1_A)}_{\rho_\theta} + \norm{(u_n-\rho_\theta(u_n)1_A)^*}_{\rho_\theta} \right) \\
& \leq & \limsup_{n\to\infty} 2 \sqrt{2} M \left(\norm{(u_n-\rho_\theta(u_n)1_A)}_{\rho_\theta}^{\#} \right) \\
& = & 0.
\end{eqnarray*}
Thus, $$(u_n - \rho_\theta(u_n) 1_A) \to 0 \quad \text{as\,\,} n\to\infty$$ in the strong* operator topology.
\end{proof}
\subsection{$\mathcal Z$-absorbing C*-algebras have Property $\Gamma$}
Let us show that if a C*-algebra is $\mathcal Z$-absorbing, then it has Property $\Gamma$ (in the sense of Definition \ref{prop-gamma}) with respect to any given state (Corollary \ref{Gamma}).
\begin{prop}\label{Z-unitary}
Let $p, q \in \mathbb{N}$ be prime numbers and let
$Z_{p, q}$ be the dimension drop algebra. Let $\rho$ be a state on $Z_{p, q}$. Then, for any $\varepsilon>0$, there is a unitary $u\in Z_{p, q}$ such that $|\rho(u)| <\varepsilon$.
\end{prop}
\begin{proof}
Recall (see, for instance, \cite{Ell-Cre}) that for a pair of natural numbers $p, q$ which are relatively prime, the dimension drop algebra $Z_{p, q}$ is defined as $$Z_{p, q}:=\{f\in\mathrm{C}([0, 1], \mathrm{M}_{pq}(\mathbb C)): f(0)\in \mathrm{M}_p(\mathbb C)\otimes 1_q\ \mathrm{and}\ f(1)\in 1_p\otimes\mathrm{M}_q(\mathbb C)\}.$$ We assert that the enveloping Borel *-algebra of $Z_{p, q}$ is isomorphic to
$$\mathcal B_{p, q}=\{f\in\mathrm{L}^\infty([0, 1], \mathrm{M}_{pq}(\mathbb C)): f(0)\in \mathrm{M}_p(\mathbb C)\otimes 1_q\ \mathrm{and}\ f(1)\in 1_p\otimes\mathrm{M}_q(\mathbb C)\}.$$ Indeed, denote by $\mathcal B$ the enveloping Borel *-algebra of $Z_{p, q}$. Since $\mathcal B_{p, q}$ is a monotonic sequential closure of $Z_{p, q}$, by Theorem 4.5.9 of \cite{Ped-book}, there is a surjective homomorphism from $\mathcal B$ to $\mathcal B_{p, q}$. Suppose there is an element $a\in\mathcal B$ which is sent to $0$ under this map. Then $a$ must be $0$ under all the irreducible representations of $Z_{p, q}$, and hence $a$ must be $0$ by Corollary 4.5.13 of \cite{Ped-book}. Therefore, the surjection from $\mathcal B$ to $\mathcal B_{p, q}$ is an isomorphism.
Let $\rho$ be a state of $Z_{p, q}$. Then $\rho$ can be extended to a normal state of $\mathcal B_{p, q}$, which will still be denoted by $\rho$.
Identify the center of $\mathcal B_{p, q}$ with
$L^{\infty}([0,1])$.
The restriction of $\rho$ to the center of $\mathcal B_{p, q}$ is then induced by a probability Borel measure $\mu$ on $[0, 1]$; that is,
$$\rho(f) = \int_{[0, 1]} f d \mu,\quad f\in L^\infty([0, 1]) = \mathrm{Z}(\mathcal B_{p, q}).$$
Let $\mathrm{tr}$ denote the tracial state of $\mathrm{M}_{pq}(\mathbb C)$. Define a (normal) trace of $\mathcal B_{p, q}$ by
$$\phi(f)= \int_{[0, 1]} \mathrm{tr}(f(t)) d\mu(t).$$ We assert that $\rho\ll\phi$. Indeed, if $f \in \mathcal B_{p, q}$ is a positive element such that $\phi(f) = 0$; then, with $E=\{x: f(x) \neq 0\}$, one has that $\mu(E) = 0.$ Set $$\hat{f} = \norm{f}\chi_{E}\in \mathrm{Z}(\mathcal B_{p, q}).$$ It is clear that $f \leq \hat{f}$ and $\rho(\hat{f}) = 0$; hence, $\rho(f) = 0$.
By the Radon-Nikodym Theorem (see, for instance, Theorem 5.3.11 of \cite{Ped-book}), there is a positive (not necessarily bounded) operator $h$ on $H_\phi$ which is affiliated to $\pi_\phi(\mathcal B_{p, q})$ such that
\begin{equation}\label{RN-thm}
\rho(a) = \left<h\pi_\phi(a)\overline{1_{\mathcal B_{p, q}}}, \overline{1_{\mathcal B_{p, q}}}\right>_\phi = \phi(h \pi_\phi(a)),\quad a\in \mathcal B_{p, q},
\end{equation}
where $(H_\phi, \pi_\phi)$ is the GNS representation of $\mathcal B_{p, q}$ induced by $\phi$.
For each $t \in \mathbb{R}$, define a real function $f_t$ by $f_t(x) = \min\{x, t\}$, and set $h_t= f_t(h)$. Note that $h_t\in \pi_\phi(\mathcal B_{p, q})$.
Since $\rho(1)=1$, $$1=\phi(h)=\lim_{t\to\infty}\phi(h_t).$$
Thus, for the given $\varepsilon$, there is a sufficiently large $t$ that for any element $a\in \mathcal B_{p, q}$ with $\|a\| \leq 1$,
\begin{equation}\label{cut-der}
\abs{\phi(h\pi_\phi(a)) - \phi(h_t\pi_\phi(a))}^2 = \abs{\phi((h-h_t)^\frac{1}{2}\pi_\phi(a)(h-h_t)^{\frac{1}{2}})}^2\leq \abs{\phi(h-h_t)}^2 \leq \left(\frac{\varepsilon}{3}\right)^2.
\end{equation}
Regarding $h_t$ as an element of $\mathcal B_{p, q}$ (by picking an element of the pre-image), there is $\bar{h}\in (\mathcal B_{p, q})^+$ satisfying
\begin{equation}\label{simple-function} \norm{h_t-\bar{h}}_\infty < \frac{\varepsilon}{3}, \end{equation}
and
$\bar{h}$ is a simple function; that is, there are disjoint Borel sets $E_1, E_2, ..., E_n\subseteq [0, 1]$ with $\bigsqcup_{i=1}^n E_i=[0, 1]$ and positive matrices $h_1, h_2, ..., h_n\in\mathrm{M}_{pq}(\mathbb C)$ such that $$\bar{h}(t) = h_i,\quad\textrm{if $t\in E_i$}.$$
Write $h_i=u^*_id_iu_i$, $i=1, 2, ..., n$, where $u_i$ are unitary matrics and $d_i$ are diagonal matrices. Note that if $E_i \ni 0$, then $h_i, d_i \in \mathrm{M}_p(\mathbb C)\otimes 1_q$; and if $E_i \ni 1$, then $h_i, d_i \in 1_p\otimes \mathrm{M}_q(\mathbb C)$.
We may require that the unitaries $u_i$ also satisfy the same property: $u_i \in \mathrm{M}_p(\mathbb C)\otimes 1_q$ if $E_i \ni 0$, and $u_i \in 1_p\otimes \mathrm{M}_q(\mathbb C)$ if $E_i \ni 1$. Define a unitary $u'\in \mathcal B_{p, q}$ by
$$u'(t) = u_i^*w_iu_i,\quad \textrm{if $t\in E_i$},$$
where $w_i \in \mathrm{M}_{pq}(\mathbb C)$ is a unitary with all diagonal elements being zero, $w_i \in \mathrm{M}_p(\mathbb C)\otimes 1_q$ if $E_i \ni 0$ and $w_i \in 1_p\otimes \mathrm{M}_q(\mathbb C)$ if $E_i \ni 1$. Then, by \eqref{RN-thm}, \eqref{cut-der}, and \eqref{simple-function}, one has
\begin{eqnarray}
\rho(u') & = & \phi(h\pi_\phi(u'))\approx_{\frac{\varepsilon}{3}} \phi(h_t u') \approx_{\frac{\varepsilon}{3}} \phi(\bar{h} u') \label{almost-0} \\
& = & \sum_{i=1}^n \mathrm{tr}(h_i u_i^*w_iu_i) \mu(E_i) \nonumber \\
& = & \sum_{i=1}^n \mathrm{tr}(u_i^*d_iu_i u_i^*w_iu_i) \mu(E_i) \nonumber\\
& = & \sum_{i=1}^n \mathrm{tr}(d_iw_i) \mu(E_i) = 0. \nonumber
\end{eqnarray}
Consider the GNS representation $(\pi_\rho, H_\rho)$ of $Z_{p,q}$. By Corollary 4.5.10 of \cite{Ped-book}, the homomorphism $\pi_\rho$ extends to a normal surjective homomorphism $\pi_\rho'': \mathcal B_{p, q} \to \pi_\rho(Z_{p, q})''$. By the Kaplansky Density Theorem, there is a unitary $v\in \pi_\rho(Z_{p, q})$ such that
\begin{equation}\label{purt-unitary}
\abs{\left<v \overline{1_{Z_{p, q}}}, \overline{1_{Z_{p, q}}}\right>_\rho - \left<\pi_\rho''(u') \overline{1_{Z_{p, q}}}, \overline{1_{Z_{p, q}}}\right>_\rho} < \frac{\varepsilon}{3}.
\end{equation}
Since $Z_{p. q}$ has stable rank one, it follows from Proposition 4.3 of \cite{Ror-Ann} that there is a unitary $u\in Z_{p, q}$ such that $\pi_\rho(u) = v$. By \eqref{purt-unitary}, we have $\abs{\rho(u) - \rho(u')} < \frac{\varepsilon}{3}$,
and hence by \eqref{almost-0}, $|\rho(u)| < \varepsilon$, as desired.
\end{proof}
\begin{cor}\label{Gamma}
Let $A$ be a unital C*-algebra such that $A\cong A\otimes\mathcal Z$, and let $\rho$ be a state of $A$. Then, for any finite set $\mathcal F \subseteq A$ and any $\varepsilon>0$, there is a unitary $u\in A$ such that
$$\norm{ua -au} < \varepsilon,\quad a\in \mathcal F,\quad\textrm{and}\quad |\rho(u)| < \varepsilon.$$
\end{cor}
\begin{proof}
Since $A$ is $\mathcal Z$-absorbing, for any given $\varepsilon>0$ and any finite set $\mathcal F\subseteq A$, there is a unital embedding $ \iota \colon Z_{p, q} \rightarrow A$ such that
$$\norm{a\iota(c) - \iota(c)a} < \varepsilon,\quad a\in\mathcal F
\text{\,\, and \,\,} c\in Z_{p, q},\ \norm{c}=1.$$
Consider the composition $\rho \circ \iota$, which is a state on $Z_{p, q}$. By Proposition \ref{Z-unitary}, there is a unitary $u\in Z_{p, q}$ satisfying $\abs{(\rho\circ \iota)(u)} < \varepsilon$. Then $\iota(u)$ is a unitary with the desired properties.
\end{proof}
\begin{proof}[Proof of Theorem \ref{main-thm}]
Assume that the TAF algebra $A$ is $\mathcal Z$-absorbing. By Corollary \ref{Gamma}, there is a central sequence consisting of unitaries $(u_n)$ in $A$ with $\rho(u_n) \to 0$. But this contradicts Proposition \ref{non-Gamma} which asserts that $|\rho(u_n)| \to 1$.
\end{proof}
|
2,877,628,090,079 | arxiv | \section{ Introduction}
In the standard scheme \cite{EWL, EW} of a quantized version of a
non-cooperative game \cite{GoogleScholar}, the players share an entangled
state, their strategic moves are local unitary transformations on the state,
and the quantum measurement \cite{Peres} generates the players' payoffs. The
resulting players' payoffs in the quantum game can be understood as the
expected values of the entries in the payoff matrix of the (classical) game
\cite{Binmore, Rasmusen, Osborne} arising from a set of quantum
probabilities \cite{Peres}. The key concerns in determining the players'
payoffs relations in the quantum game are a) What are the players' moves in
the quantum game? b) Which set of quantum \ probabilities is obtained by
quantum measurement? and c) How the players' strategic moves are related to
the set of quantum probabilities?
This brings us to question whether the unitary transformations are really
necessary in the setup of a quantum game. A proposed scheme \cit
{IqbalWeigert, Iqbal, Iqbal1, Chappell} for playing a quantum game in which
players' strategic moves are not unitary transformations uses the setting of
an Einstein-Podolsky-Rosen (EPR) experiment \cite{Peres, Bell1, Bell2,Bell3,
Aspect, CHSH}. Two players are located in spacelike-separated regions and
share a singlet state. In a run of the experiment, each player decides one
out of the two available directions and a quantum measurement is performed.
This leads to obtaining a (normalized) set of quantum probabilities along
with a listing of the directional choices the players make in each run of
the experiment. As the players' directional choices determine the quantum
probability distribution, the setting can be used to develop a quantum
version of a two-player game. A multipartite EPR experiment would then be
required for a multiplayer quantum game.
In this paper, we consider a classical three-player symmetric game, along
with a reported expression for a quantum probability distribution, which is
relevant to the three-partite EPR experiment. We then define players'
directional choices in the experiment as their strategic moves and express
players' payoff relations in the quantum game in terms of the three
directional choices and the entries of the payoff matrix.
This paper thus provides a game-theoretic perspective on the peculiarity of
quantum probabilities. The first perspective along game-theoretical lines on
quantum probabilities that are associated to the GHZ state \cite{Peres} was
provided by Vaidman in Ref. \cite{Vaidman}. Vaidman proposed a set of rules
defining a game that can only be won by a team of three players when they
share a GHZ state. The present paper extends Vaidman's perspective by
considering Nash equilibria in the set of symmetric games played by a team
of three players in a non-cooperative game setting. Vaidman presented his
game without invoking Hilbert space as is the case in the present paper.
\section{Three-player games with mixed-strategies}
Consider a three-player (noncooperative) game in which the players Alice
(A) $, Bob $(B)$, and Chris $(C)$ make their strategic moves simultaneously.
The players are assumed located at distance and are unable to communicate to
one another. They, however, can communicate to a referee who organizes the
game and ensures that the rules of the game are obeyed. Each player has to
decide between two choices, called the \textit{pure strategies}, and in
repeated version of the game they can also play the \textit{mixed strategies
. Their payoff relations are made public by the referee at the start of the
game. The payoff relations depend on the game matrix, the players' pure
strategies, and the probability distribution on pure strategies.
To be specific, we assume that the player $A$'s pure strategies are $S_{1},$
$S_{2}$; the player $B$'s pure strategies are $S_{1}^{\prime },$
S_{2}^{\prime }$; and the player $C$'s pure strategies are $S_{1}^{\prime
\prime },$ $S_{2}^{\prime \prime }$. Also, the game is defined by the
following pure-strategy payoff relations \cite{IqbalToor
\begin{equation}
\begin{array}{l}
\Pi _{A,B,C}(S_{1},S_{1}^{\prime },S_{1}^{\prime \prime })=\alpha _{1},\beta
_{1},\gamma _{1}; \\
\Pi _{A,B,C}(S_{2},S_{1}^{\prime },S_{1}^{\prime \prime })=\alpha _{2},\beta
_{2},\gamma _{2}; \\
\Pi _{A,B,C}(S_{1},S_{2}^{\prime },S_{1}^{\prime \prime })=\alpha _{3},\beta
_{3},\gamma _{3}; \\
\Pi _{A,B,C}(S_{1},S_{1}^{\prime },S_{2}^{\prime \prime })=\alpha _{4},\beta
_{4},\gamma _{4}
\end{array
\begin{array}{l}
\Pi _{A,B,C}(S_{1},S_{2}^{\prime },S_{2}^{\prime \prime })=\alpha _{5},\beta
_{5},\gamma _{5}; \\
\Pi _{A,B,C}(S_{2},S_{1}^{\prime },S_{2}^{\prime \prime })=\alpha _{6},\beta
_{6},\gamma _{6}; \\
\Pi _{A,B,C}(S_{2},S_{2}^{\prime },S_{1}^{\prime \prime })=\alpha _{7},\beta
_{7},\gamma _{7}; \\
\Pi _{A,B,C}(S_{2},S_{2}^{\prime },S_{2}^{\prime \prime })=\alpha _{8},\beta
_{8},\gamma _{8}
\end{array}
\label{payoffs_constants_definitions}
\end{equation}
For example, $\Pi _{A,B,C}(S_{1},S_{2}^{\prime },S_{1}^{\prime \prime
})=\alpha _{3},$ $\beta _{3},$ $\gamma _{3}$ states that the players $A$, $B
, and $C$ obtain the payoffs $\alpha _{3},$ $\beta _{3},$ and $\gamma _{3}$,
respectively, when they play the pure strategies $S_{1},$ $S_{2}^{\prime },$
and $S_{1}^{\prime \prime }$, respectively.
In a repeated version of this game, a player can choose between his/her two
pure strategies with some probability, which defines his/her mixed-strategy.
We specify a mixed-strategy by $x,$ $y,$ $z\in \lbrack 0,1]$ for players $A
, $B$, and $C$, respectively. These are the probabilities with which the
players $A$, $B$, and $C$ play the pure strategies $S_{1},$ $S_{1}^{\prime
}, $ and $S_{1}^{\prime \prime }$, respectively. They, then, play the pure
strategies $S_{2},$ $S_{2}^{\prime },$ and $S_{2}^{\prime \prime }$ with
probabilities $(1-x),$ $(1-y),$ and $(1-z)$, respectively, and the
mixed-strategy payoff relations, therefore, rea
\begin{equation}
\begin{array}{l}
\Pi _{A,B,C}(x,y,z)=xyz\Pi _{A,B,C}(S_{1},S_{1}^{\prime },S_{1}^{\prime
\prime })+x(1-y)z\Pi _{A,B,C}(S_{1},S_{2}^{\prime },S_{1}^{\prime \prime })+
\\
xy(1-z)\Pi _{A,B,C}(S_{1},S_{1}^{\prime },S_{2}^{\prime \prime
})+x(1-y)(1-z)\Pi _{A,B,C}(S_{1},S_{2}^{\prime },S_{2}^{\prime \prime })+ \\
(1-x)yz\Pi _{A,B,C}(S_{2},S_{1}^{\prime },S_{1}^{\prime \prime
})+(1-x)(1-y)z\Pi _{A,B,C}(S_{2},S_{2}^{\prime },S_{1}^{\prime \prime })+ \\
(1-x)y(1-z)\Pi _{A,B,C}(S_{2},S_{1}^{\prime },S_{2}^{\prime \prime
})+(1-x)(1-y)(1-z)\Pi _{A,B,C}(S_{2},S_{2}^{\prime },S_{2}^{\prime \prime })
\end{array}
\label{3coin_mixed_strategy_payoffs}
\end{equation
that can also be written a
\begin{equation}
\begin{array}{l}
\Pi _{A,B,C}(x,y,z)=\dsum\limits_{i,j,k=1,2}\Pr_{c}(S_{i},S_{j}^{\prime
},S_{k}^{\prime \prime })\Pi _{A,B,C}(S_{i},S_{j}^{\prime },S_{k}^{\prime
\prime })
\end{array
\end{equation
where $\Pr_{c}(S_{i},S_{j}^{\prime },S_{k}^{\prime \prime })$ are the
classical factorizable probabilities and for instance,
\Pr_{c}(S_{1},S_{1}^{\prime },S_{2}^{\prime \prime })=xy(1-z)$ and
\Pr_{c}(S_{2},S_{2}^{\prime },S_{1}^{\prime \prime })=(1-x)(1-y)z.$
\subsection{Symmetric three-player games}
Symmetric three-player games are defined by the condition that a player's
payoff is decided by his/her strategic move and not by his/her identity.
Mathematically, this is expressed by the condition
\begin{equation}
\Pi _{A}(x,y,z)=\Pi _{A}(x,z,y)=\Pi _{B}(y,x,z)=\Pi _{B}(z,x,y)=\Pi
_{C}(y,z,x)=\Pi _{C}(z,y,x), \label{constraints_for_symmetric_game}
\end{equation
i.e. the player $A$'s payoff when s/he plays $x$ remains the same either
when player $B$ plays $y$ whereas player $C$ plays $y$ or when player $B$
plays $x$ whereas player $C$ play $x$. The payoff relations (\re
{3coin_mixed_strategy_payoffs}) satisfy the conditions (\re
{constraints_for_symmetric_game}) when \cite{IqbalToor
\begin{equation}
\begin{array}{c}
\begin{array}{cccc}
\beta _{1}=\alpha _{1}, & \beta _{2}=\alpha _{3}, & \beta _{3}=\alpha _{2},
& \beta _{4}=\alpha _{3}, \\
\beta _{5}=\alpha _{6}, & \beta _{6}=\alpha _{5}, & \beta _{7}=\alpha _{6},
& \beta _{8}=\alpha _{8}, \\
\gamma _{1}=\alpha _{1}, & \gamma _{2}=\alpha _{3}, & \gamma _{3}=\alpha
_{3}, & \gamma _{4}=\alpha _{2}, \\
\gamma _{5}=\alpha _{6}, & \gamma _{6}=\alpha _{6}, & \gamma _{7}=\alpha
_{5}, & \gamma _{8}=\alpha _{8}
\end{array}
\\
\begin{array}{cc}
\alpha _{6}=\alpha _{7}, & \alpha _{3}=\alpha _{4}
\end{array
\end{array
\end{equation
A symmetric three-player game can, therefore, be defined by only six
constants $\alpha _{1},$ $\alpha _{2},$ $\alpha _{3},$ $\alpha _{5},$
\alpha _{6},$ and $\alpha _{8}$. In the rest of this paper we will define
these six constants to be $\alpha ,$ $\beta ,$ $\delta ,$ $\epsilon ,$
\theta ,$ and $\omega ,$ where $\alpha _{1}=\alpha ,$ $\alpha _{2}=\beta ,$
\alpha _{3}=\delta ,$ $\alpha _{5}=\epsilon ,$ $\alpha _{6}=\theta ,$ and
\alpha _{8}=\omega $. The pure-strategy payoff relations (\re
{payoffs_constants_definitions}) in this symmetric game are then
re-expressed a
\begin{equation}
\begin{array}{l}
\Pi _{A,B,C}(S_{1},S_{1}^{\prime },S_{1}^{\prime \prime })=\alpha ,\alpha
,\alpha ; \\
\Pi _{A,B,C}(S_{2},S_{1}^{\prime },S_{1}^{\prime \prime })=\beta ,\delta
,\delta ; \\
\Pi _{A,B,C}(S_{1},S_{2}^{\prime },S_{1}^{\prime \prime })=\delta ,\beta
,\delta ; \\
\Pi _{A,B,C}(S_{1},S_{1}^{\prime },S_{2}^{\prime \prime })=\delta ,\delta
,\beta
\end{array
\begin{array}{l}
\Pi _{A,B,C}(S_{1},S_{2}^{\prime },S_{2}^{\prime \prime })=\epsilon ,\theta
,\theta ; \\
\Pi _{A,B,C}(S_{2},S_{1}^{\prime },S_{2}^{\prime \prime })=\theta ,\epsilon
,\theta ; \\
\Pi _{A,B,C}(S_{2},S_{2}^{\prime },S_{1}^{\prime \prime })=\theta ,\theta
,\epsilon ; \\
\Pi _{A,B,C}(S_{2},S_{2}^{\prime },S_{2}^{\prime \prime })=\omega ,\omega
,\omega
\end{array}
\label{symmetric_3player_game_definition}
\end{equation
The mixed-strategy payoff relations in Eq. (\re
{3coin_mixed_strategy_payoffs}) then take the form
\begin{equation}
\begin{array}{l}
\Pi _{A,B,C}(x,y,z)=xyz(\alpha ,\alpha ,\alpha )+x(1-y)z(\delta ,\beta
,\delta )+xy(1-z)(\delta ,\delta ,\beta )+ \\
x(1-y)(1-z)(\epsilon ,\theta ,\theta )+(1-x)yz(\beta ,\delta ,\delta
)+(1-x)(1-y)z(\theta ,\theta ,\epsilon )+ \\
(1-x)y(1-z)(\theta ,\epsilon ,\theta )+(1-x)(1-y)(1-z)(\omega ,\omega
,\omega )
\end{array}
\label{Payoffs_factorizable}
\end{equation}
\section{Quantum probability distribution for a GHZ state}
Consider the GHZ state
\begin{equation}
\left\vert \psi \right\rangle =(\left\vert 0\right\rangle _{1}\left\vert
0\right\rangle _{2}\left\vert 0\right\rangle _{3}+\left\vert 1\right\rangle
_{1}\left\vert 1\right\rangle _{2}\left\vert 1\right\rangle _{3})/\sqrt{2},
\label{GHZ state}
\end{equation
that is shared among three the three players, where $\left\vert
i\right\rangle _{j}$ is the $i$-th state of the $j$-th qubit and the setting
of the generalized EPR experiments. Each player measures the dichotomic
observable $\vec{n}.\vec{\sigma}$ where $\vec{n}=\vec{a},\vec{b},\vec{c}$
and $\vec{\sigma}$ is a vector the components of which are standard Pauli
matrices. The family of observables $\vec{n}.\vec{\sigma}$ covers all
possible dichotomic observables for a qubit system \cite{Peres}.
Kaszlikowski and \.{Z}ukowski \cite{Kaszlikowski} show that the probability
of obtaining the result $m=\pm 1$ for the player $A$, when s/he plays the
strategy $\vec{a}$, the result $l=\pm 1$ for the player $B$, when s/he plays
the strategy $\vec{b}$ and the result $k=\pm 1$ for the player $C$, when
s/he plays the strategy $\vec{c}$ is given by
\begin{equation}
\Pr_{QM}(m,l,k;\vec{a},\vec{b},\vec{c})=\frac{1}{8}\left[
1+mla_{3}b_{3}+mka_{3}c_{3}+lkb_{3}c_{3}+ml
\sum_{r,p,s=1}^{3}M_{rps}a_{r}b_{p}c_{s}\right] ,
\label{KaszlikowskiEquation}
\end{equation
where $a_{r}$, $b_{p}$, $c_{s}$ are components of vectors $\vec{a},\vec{b}
\vec{c}$ and where nonzero elements of the tensor $M_{rps}$ are $M_{111}=1,$
$M_{122}=-1,$ $M_{212}=-1,$ $M_{221}=-1$. In view of this, the only terms in
the product $a_{r}b_{p}c_{s}$ that contribute towards the probability
\Pr_{QM}(m,l,k;\vec{a},\vec{b},\vec{c})$ are $a_{1}b_{1}c_{1},$
a_{1}b_{2}c_{2},$ $a_{2}b_{1}c_{2},$ and $a_{2}b_{2}c_{1}$. Eq. (\re
{KaszlikowskiEquation}) can therefore be written as
\begin{equation}
\Pr_{QM}(m,l,k;\vec{a},\vec{b},\vec{c})=\frac{1}{8}\left[
1+mla_{3}b_{3}+mka_{3}c_{3}+lkb_{3}c_{3}+mlk(a_{1}b_{1}c_{1}-a_{1}b_{2}c_{2}-a_{2}b_{1}c_{2}-a_{2}b_{2}c_{1}
\right] . \label{QProbabilities}
\end{equation
Note that Eq. (\ref{KaszlikowskiEquation}) gives a quantum probability
distribution without reference to the undelying Hilbert space, unitary
transformations, or quantum measurement.
We consider playing a three-player quantum game in which the players Alice,
Bob, and Chris (henceforth, labelled as player $A$, player $B$, and player
C $) moves consist of choosing the directions $\vec{a},\vec{b},$ and $\vec{c}
$, respectively. The players's payoff relations are then expressed in terms
of the quantum probability distribution given in Eq. (\re
{KaszlikowskiEquation}).
\subsection{Players sharing a GHZ state and when choosing a direction is a
player's move}
Let $\vec{a}=\vec{a}(a_{1},a_{2},a_{3}),$ $\vec{b}=\vec{b
(b_{1},b_{2},b_{3}),$ $\vec{c}=\vec{c}(c_{1},c_{2},c_{3})$ be the players'
directional choices that we consider as their strategies. Denoting the
quantum probabilities by $\Pr_{{\small Q}}$, the set of quantum
probabilities can be obtained from Eq. (\ref{QProbabilities}) as follow
\begin{eqnarray}
\Pr_{{\small Q}}(S_{1},S_{1}^{\prime },S_{1}^{\prime \prime }) &=&\Pr_
{\small Q}}[(\vec{a},m=+1),(\vec{b},l=+1),(\vec{c},k=+1)] \notag \\
&=&\frac{1}{8}\left[ 1+a_{3}b_{3}+a_{3}c_{3}+b_{3}c_{3}+\Delta \right] ;
\notag \\
\Pr_{{\small Q}}(S_{1},S_{2}^{\prime },S_{1}^{\prime \prime }) &=&\Pr_
{\small Q}}[(\vec{a},m=+1),(\vec{b},l=-1),(\vec{c},k=+1)] \notag \\
&=&\frac{1}{8}\left[ 1-a_{3}b_{3}+a_{3}c_{3}-b_{3}c_{3}-\Delta \right] ;
\notag \\
\Pr_{{\small Q}}(S_{1},S_{1}^{\prime },S_{2}^{\prime \prime }) &=&\Pr_
{\small Q}}[(\vec{a},m=+1),(\vec{b},l=+1),(\vec{c},k=-1)] \notag \\
&=&\frac{1}{8}\left[ 1+a_{3}b_{3}-a_{3}c_{3}-b_{3}c_{3}-\Delta \right] ;
\notag \\
\Pr_{{\small Q}}(S_{1},S_{2}^{\prime },S_{2}^{\prime \prime }) &=&\Pr_
{\small Q}}[(\vec{a},m=+1),(\vec{b},l=-1),(\vec{c},k=-1)] \notag \\
&=&\frac{1}{8}\left[ 1-a_{3}b_{3}-a_{3}c_{3}+b_{3}c_{3}+\Delta \right] ;
\notag \\
\Pr_{{\small Q}}(S_{2},S_{1}^{\prime },S_{1}^{\prime \prime }) &=&\Pr_
{\small Q}}[(\vec{a},m=-1),(\vec{b},l=+1),(\vec{c},k=+1)] \notag \\
&=&\frac{1}{8}\left[ 1-a_{3}b_{3}-a_{3}c_{3}+b_{3}c_{3}-\Delta \right] ;
\notag \\
\Pr_{{\small Q}}(S_{2},S_{2}^{\prime },S_{1}^{\prime \prime }) &=&\Pr_
{\small Q}}[(\vec{a},m=-1),(\vec{b},l=-1),(\vec{c},k=+1)] \notag \\
&=&\frac{1}{8}\left[ 1+a_{3}b_{3}-a_{3}c_{3}-b_{3}c_{3}+\Delta \right] ;
\notag \\
\Pr_{{\small Q}}(S_{2},S_{1}^{\prime },S_{2}^{\prime \prime }) &=&\Pr_
{\small Q}}[(\vec{a},m=-1),(\vec{b},l=+1),(\vec{c},k=-1)] \notag \\
&=&\frac{1}{8}\left[ 1-a_{3}b_{3}+a_{3}c_{3}-b_{3}c_{3}+\Delta \right] ;
\notag \\
\Pr_{{\small Q}}(S_{2},S_{2}^{\prime },S_{2}^{\prime \prime }) &=&\Pr_
{\small Q}}[(\vec{a},m=-1),(\vec{b},l=-1),(\vec{c},k=-1)] \notag \\
&=&\frac{1}{8}\left[ 1+a_{3}b_{3}+a_{3}c_{3}+b_{3}c_{3}-\Delta \right] ;
\label{QProbabs}
\end{eqnarray
where $\Delta
=a_{1}b_{1}c_{1}-a_{1}b_{2}c_{2}-a_{2}b_{1}c_{2}-a_{2}b_{2}c_{1}.$ We define
players $A$'s, $B$'s, $C$'s payoff relations in the quantum game as follow
\begin{equation}
\begin{array}{l}
\Pi _{A,B,C}(\vec{a},\vec{b},\vec{c})=\sum_{i,j,k=1}^{2
\Pr_{Q}(S_{i},S_{j}^{\prime },S_{k}^{\prime \prime })\Pi
_{A,B,C}(S_{i},S_{j}^{\prime },S_{k}^{\prime \prime })
\end{array}
\label{Payoffs}
\end{equation
i.e. these are obtained as the expectation of payoff entries (\re
{symmetric_3player_game_definition}) on the set of quantum probabilities
\ref{QProbabs}). For the symmetric game defined in Eq. (\re
{symmetric_3player_game_definition}), the payoffs to the players $A,$ $B,$
and $C,$ given in (\ref{Payoffs}), can then be expanded as follows:
\begin{equation}
\begin{array}{l}
\Pi _{A,B,C}(\vec{a},\vec{b},\vec{c})= \\
\frac{1}{8}\{\left[ 1+a_{3}b_{3}+a_{3}c_{3}+b_{3}c_{3}+\Delta \right]
(\alpha ,\alpha ,\alpha )+\left[ 1-a_{3}b_{3}+a_{3}c_{3}-b_{3}c_{3}-\Delta
\right] (\delta ,\beta ,\delta )+ \\
\left[ 1+a_{3}b_{3}-a_{3}c_{3}-b_{3}c_{3}-\Delta \right] (\delta ,\delta
,\beta )+\left[ 1-a_{3}b_{3}-a_{3}c_{3}+b_{3}c_{3}+\Delta \right] (\epsilon
,\theta ,\theta )+ \\
\left[ 1-a_{3}b_{3}-a_{3}c_{3}+b_{3}c_{3}-\Delta \right] (\beta ,\delta
,\delta )+\left[ 1+a_{3}b_{3}-a_{3}c_{3}-b_{3}c_{3}+\Delta \right] (\theta
,\theta ,\epsilon )+ \\
\left[ 1-a_{3}b_{3}+a_{3}c_{3}-b_{3}c_{3}+\Delta \right] (\theta ,\epsilon
,\theta )+\left[ 1+a_{3}b_{3}+a_{3}c_{3}+b_{3}c_{3}-\Delta \right] (\omega
,\omega ,\omega )\}
\end{array}
\label{Payoffs_ABC}
\end{equation
Let $a_{3}=b_{3}=c_{3}=0$ i.e. when the players' unit vectors are confined
to the X-Y plane, the payoff relations (\ref{Payoffs_ABC}) can be written as
\begin{equation}
\begin{array}{l}
\Pi _{A,B,C}(\vec{a},\vec{b},\vec{c})= \\
\frac{1}{8}\{(1+\Delta )(\alpha ,\alpha ,\alpha )+(1-\Delta )(\delta ,\beta
,\delta )+(1-\Delta )(\delta ,\delta ,\beta )+(1+\Delta )(\epsilon ,\theta
,\theta )+ \\
(1-\Delta )(\beta ,\delta ,\delta )+(1+\Delta )(\theta ,\theta ,\epsilon
)+(1+\Delta )(\theta ,\epsilon ,\theta )+(1-\Delta )(\omega ,\omega ,\omega
)\}
\end{array}
\label{ConfinedDirections}
\end{equation
It is apparent from above that the resulting payoff relations (\re
{ConfinedDirections}) in the quantum game cannot be put into a form that is
same as for the classical mixed-strategy game i.e. Eq. (\re
{Payoffs_factorizable}). This raises the question whether there exist
constraints that can be placed on the players' directional choices, i.e. the
unit vectors $\vec{a},$ $\vec{b},$ and $\vec{c},$ such that the payoff
relations (\ref{Payoffs_ABC}) in the quantum game are reduced to the
players' payoffs in the classical game allowing mixed strategies (\re
{Payoffs_factorizable}). In order to find an answer to this we set
\begin{equation}
\Pi _{A,B,C}(\vec{a},\vec{b},\vec{c})=\Pi _{A,B,C}(x,y,z),
\end{equation
and equate the right sides of Eqs. (\ref{Payoffs_ABC}, \re
{Payoffs_factorizable}) i.e.
\begin{eqnarray}
\frac{1}{8}\left[ 1+a_{3}b_{3}+a_{3}c_{3}+b_{3}c_{3}+\Delta \right] &=&xyz,
\label{Eq_1} \\
\frac{1}{8}\left[ 1-a_{3}b_{3}+a_{3}c_{3}-b_{3}c_{3}-\Delta \right]
&=&x(1-y)z, \label{Eq_2} \\
\frac{1}{8}\left[ 1+a_{3}b_{3}-a_{3}c_{3}-b_{3}c_{3}-\Delta \right]
&=&xy(1-z), \label{Eq_3} \\
\frac{1}{8}\left[ 1-a_{3}b_{3}-a_{3}c_{3}+b_{3}c_{3}+\Delta \right]
&=&x(1-y)(1-z), \label{Eq_4} \\
\frac{1}{8}\left[ 1-a_{3}b_{3}-a_{3}c_{3}+b_{3}c_{3}-\Delta \right]
&=&(1-x)yz, \label{Eq_5} \\
\frac{1}{8}\left[ 1+a_{3}b_{3}-a_{3}c_{3}-b_{3}c_{3}+\Delta \right]
&=&(1-x)(1-y)z, \label{Eq_6} \\
\frac{1}{8}\left[ 1-a_{3}b_{3}+a_{3}c_{3}-b_{3}c_{3}+\Delta \right]
&=&(1-x)y(1-z), \label{Eq_7} \\
\frac{1}{8}\left[ 1+a_{3}b_{3}+a_{3}c_{3}+b_{3}c_{3}-\Delta \right]
&=&(1-x)(1-y(1-z). \label{Eq_8}
\end{eqnarray
Now, by adding Eqs. (\ref{Eq_1}) and (\ref{Eq_2}) we obtain
\begin{equation}
\frac{1}{4}(1+a_{3}c_{3})=xz, \label{Eq_1.1}
\end{equation
adding Eqs. (\ref{Eq_1}) and (\ref{Eq_3}) gives
\begin{equation}
\frac{1}{4}(1+a_{3}b_{3})=xy, \label{Eq_9}
\end{equation
adding Eqs. (\ref{Eq_1}) and (\ref{Eq_5}) gives
\begin{equation}
\frac{1}{4}(1+b_{3}c_{3})=yz, \label{Eq_10}
\end{equation
adding Eqs. (\ref{Eq_3}) and (\ref{Eq_4}) gives
\begin{equation}
\frac{1}{4}(1-a_{3}c_{3})=x(1-z). \label{Eq_11}
\end{equation}
Now, we add Eqs. (\ref{Eq_1.1}) and (\ref{Eq_11}) to obtain $x=\frac{1}{2}.$
Adding Eqs. (\ref{Eq_7}) and (\ref{Eq_8}) gives
\begin{equation}
\frac{1}{4}(1+a_{3}c_{3})=(1-x)(1-z), \label{Eq_12}
\end{equation
and substitution from Eq. (\ref{Eq_1.1}) and $x=\frac{1}{2}$ gives $z=\frac{
}{2}.$ Similarly, adding Eqs. (\ref{Eq_2}) and (\ref{Eq_4}) gives
\begin{equation}
\frac{1}{4}(1-a_{3}b_{3})=x(1-y), \label{Eq_13}
\end{equation
and adding Eqs. (\ref{Eq_6}) and (\ref{Eq_8}) gives
\begin{equation}
\frac{1}{4}(1+a_{3}b_{3})=(1-x)(1-y). \label{Eq_14}
\end{equation
By adding Eqs. (\ref{Eq_13}) and (\ref{Eq_14}) we obtain $y=\frac{1}{2}$ and
thus $(x,y,z)=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ is obtained as the
solution of the Eqs. (\ref{Eq_1}) to (\ref{Eq_8}).
Therefore, the mixed-strategy payoff relations (\ref{Payoffs_factorizable})
can be recovered from the payoffs relations (\ref{Payoffs_ABC}) for the
quantum game only for the special case when $(x,y,z)=(\frac{1}{2},\frac{1}{2
,\frac{1}{2})$. This is because the quantum probability distribution for the
GHZ state, from which the payoff relations (\ref{Payoffs_ABC}) are
constructed, are inherently non-factorizable. In the research area of
quantum games, recovering the mixed strategy classical payoff relations from
the payoff relations for a quantum game is quite often considered an
essential requirement. When the underlying quantum probabilities in a
quantum game are obtained from the GHZ state, this requirement is not
satisfied except for a very special case, i.e. $(x,y,z)=(\frac{1}{2},\frac{
}{2},\frac{1}{2})$.
Considering the payoff relations (\ref{Payoffs_ABC}) in the quantum game, a
Nash equilibrium (NE) is a directional triple $(\vec{a}^{\ast },\vec{b
^{\ast },\vec{c}^{\ast })$ that satisfies the following constraints:
\begin{eqnarray}
\Pi _{A}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{A}(\vec{a}
\vec{b}^{\ast },\vec{c}^{\ast }) &\geq &0, \notag \\
\Pi _{B}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{B}(\vec{a
^{\ast },\vec{b},\vec{c}^{\ast }) &\geq &0, \notag \\
\Pi _{C}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{C}(\vec{a
^{\ast },\vec{b}^{\ast },\vec{c}) &\geq &0,
\end{eqnarray
for all $\vec{a},$ $\vec{b},$ and $\vec{c}$. For the symmetric game, these
Nash inequalities take the form
\begin{eqnarray}
\Pi _{A}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{A}(\vec{a}
\vec{b}^{\ast },\vec{c}^{\ast }) &=&\frac{1}{8}\left[ (a_{3}^{\ast
}-a_{3})\Delta _{1}\gamma _{1}+\gamma _{2}(a_{1}^{\ast }-a_{1})\Delta
_{2}-\gamma _{2}(a_{2}^{\ast }-a_{2})\Delta _{3}\right] \geq 0, \notag \\
\Pi _{A}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{A}(\vec{a
^{\ast },\vec{b},\vec{c}^{\ast }) &=&\frac{1}{8}\left[ (b_{3}^{\ast
}-b_{3})\Delta _{1}^{\prime }\gamma _{1}-\gamma _{2}(b_{2}^{\ast
}-b_{2})\Delta _{2}^{\prime }+\gamma _{2}(b_{1}^{\ast }-b_{1})\Delta
_{3}^{\prime }\right] \geq 0, \notag \\
\Pi _{A}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{A}(\vec{a
^{\ast },\vec{b}^{\ast },\vec{c}) &=&\frac{1}{8}\left[ (c_{3}^{\ast
}-c_{3})\Delta _{1}^{\prime \prime }\gamma _{1}-\gamma _{2}(c_{2}^{\ast
}-c_{2})\Delta _{2}^{\prime \prime }+\gamma _{2}(c_{1}^{\ast }-c_{1})\Delta
_{3}^{\prime \prime }\right] \geq 0, \notag \\
&& \label{NE_ABC}
\end{eqnarray
where
\begin{equation}
\gamma _{1}=\alpha -\beta -\epsilon +\omega \text{ and }\gamma _{2}=\alpha
-2\delta -\beta +\epsilon +2\theta -\omega ,
\end{equation
and
\begin{eqnarray}
\Delta _{1} &=&b_{3}+c_{3},\text{ }\Delta _{2}=b_{1}c_{1}-b_{2}c_{2},\text{
\Delta _{3}=b_{1}c_{2}+b_{2}c_{1}, \notag \\
\Delta _{1}^{\prime } &=&a_{3}+c_{3},\text{ }\Delta _{2}^{\prime
}=a_{1}c_{2}+a_{2}c_{1},\text{ }\Delta _{3}^{\prime }=a_{1}c_{1}-a_{2}c_{2},
\notag \\
\Delta _{1}^{\prime \prime } &=&a_{3}+b_{3},\text{ }\Delta _{2}^{\prime
\prime }=a_{1}b_{2}+a_{2}b_{1},\text{ }\Delta _{3}^{\prime \prime
}=a_{1}b_{1}-a_{2}b_{2}. \label{Deltas}
\end{eqnarray}
\subsection{Three-player Prisoners' Dilemma}
Prisoner's Dilemma (PD) is a noncooperative game \cit
{Binmore,Rasmusen,Osborne} that is widely known in the areas of economics,
social, and political sciences. In recent years, quantum physics has been
added to this list. It was investigated early in the history of quantum
games and provided significant motivation for further work in this area.
Two-player PD is about two suspects, considered here as the players in a
game, who have been arrested on the allegations of having committed a crime
but there not not enough available evidence to convict them. The
investigators come up with an ingenious plan to make the suspects confess
their crime.
They are taken to separate cells and are not allowed to communicate. They
are contacted individually and, along with being dictated a set of rules,
are asked to choose between two choices (strategies): \emph{to Confess} $
\mathfrak{D})$ and \emph{Not to Confess} $(\mathfrak{C})$, where $\mathfrak{
}$ and $\mathfrak{D}$ stand for Cooperation and Defection. These are the
well-known wordings for the available choices for them and refer to the
choice they make to the fellow prisoner, and not to the authorities.
The rules state that if neither prisoner confesses, i.e. $(\mathfrak{C}
\mathfrak{C})$, both are given freedom; when one prisoner confesses $
\mathfrak{D})$ and the other does not $(\mathfrak{C})$, i.e. $(\mathfrak{C}
\mathfrak{D})$ or $(\mathfrak{D},\mathfrak{C})$, the prisoner who confesses
(\mathfrak{D})$ gets freedom as well as a financial reward, while the
prisoner who did not confess ends up in prison for a longer term. If both
prisoners confess, i.e. $(\mathfrak{D},\mathfrak{D})$, both are given a
reduced term.
In the two-player case, involving the players $A$ and $B$ the strategy pair
(\mathfrak{D},\mathfrak{D})$ comes out as the unique NE (and the rational
outcome) of the game, leading to the situation of both ending up in jail
with reduced term. The game offers a dilemma as the rational outcome $
\mathfrak{D},\mathfrak{D})$ differs from the outcome $(\mathfrak{C}
\mathfrak{C})$, which is an available choice, and for which both prisoners
obtain freedom.
With the above notation, the three-player PD can be defined by making the
following association
\begin{equation}
S_{1}\sim \mathfrak{C},\text{ }S_{2}\sim \mathfrak{D},\text{ }S_{1}^{\prime
}\sim \mathfrak{C},\text{ }S_{2}^{\prime }\sim \mathfrak{D},\text{
S_{1}^{\prime \prime }\sim \mathfrak{C},\text{ }S_{2}^{\prime \prime }\sim
\mathfrak{D},
\end{equation
and afterwards imposing the following conditions \cite{3playerPD}:
a) The strategy $S_{2}$ is a dominant choice \cite{Rasmusen} for each
player. For Alice this require
\begin{equation}
\begin{array}{l}
\Pi _{A}(S_{2},S_{1}^{\prime },S_{1}^{\prime \prime })>\Pi
_{A}(S_{1},S_{1}^{\prime },S_{1}^{\prime \prime }), \\
\Pi _{A}(S_{2},S_{2}^{\prime },S_{2}^{\prime \prime })>\Pi
_{A}(S_{1},S_{2}^{\prime },S_{2}^{\prime \prime }), \\
\Pi _{A}(S_{2},S_{1}^{\prime },S_{2}^{\prime \prime })>\Pi
_{A}(S_{1},S_{1}^{\prime },S_{2}^{\prime \prime })
\end{array
\end{equation
and similar inequalities hold for players Bob and Chris.
b) A player is better off if more of his/her opponents choose to cooperate.
For Alice this require
\begin{equation}
\begin{array}{l}
\Pi _{A}(S_{2},S_{1}^{\prime },S_{1}^{\prime \prime })>\Pi
_{A}(S_{2},S_{1}^{\prime },S_{2}^{\prime \prime })>\Pi
_{A}(S_{2},S_{2}^{\prime },S_{2}^{\prime \prime }), \\
\Pi _{A}(S_{1},S_{1}^{\prime },S_{1}^{\prime \prime })>\Pi
_{A}(S_{1},S_{1}^{\prime },S_{2}^{\prime \prime })>\Pi
_{A}(S_{1},S_{2}^{\prime },S_{2}^{\prime \prime })
\end{array
\end{equation
and similar relations hold for Bob and Chris.
c) If one player's choice is fixed, the other two players are left in the
situation of a two-player PD. For Alice this require
\begin{equation}
\begin{array}{l}
\Pi _{A}(S_{1},S_{1}^{\prime },S_{2}^{\prime \prime })>\Pi
_{A}(S_{2},S_{2}^{\prime },S_{2}^{\prime \prime }), \\
\Pi _{A}(S_{1},S_{1}^{\prime },S_{1}^{\prime \prime })>\Pi
_{A}(S_{2},S_{1}^{\prime },S_{2}^{\prime \prime }), \\
\Pi _{A}(S_{1},S_{1}^{\prime },S_{2}^{\prime \prime })>(1/2)\left\{ \Pi
_{A}(S_{1},S_{2}^{\prime },S_{2}^{\prime \prime })+\Pi
_{A}(S_{2},S_{1}^{\prime },S_{2}^{\prime \prime })\right\} , \\
\Pi _{A}(S_{1},S_{1}^{\prime },S_{1}^{\prime \prime })>(1/2)\left\{ \Pi
_{A}(S_{1},S_{1}^{\prime },S_{2}^{\prime \prime })+\Pi
_{A}(S_{2},S_{1}^{\prime },S_{1}^{\prime \prime })\right\}
\end{array
\end{equation
and similar relations hold for Bob and Chris.
Translating the above conditions while using the notation introduced in (\re
{symmetric_3player_game_definition}) require
\begin{equation}
\begin{array}{l}
\text{a) }\beta >\alpha ,\ \ \omega >\epsilon ,\ \ \theta >\delta , \\
\text{b) }\beta >\theta >\omega ,\ \ \alpha >\delta >\epsilon , \\
\text{c) }\delta >\omega ,\ \ \alpha >\theta ,\ \ \delta >(1/2)(\epsilon
+\theta ),\ \ \alpha >(1/2)(\delta +\beta )
\end{array
\end{equation
which defines the generalized three-player PD. For example \cite{3playerPD},
by letting
\begin{equation}
\alpha =7,\text{ }\beta =9,\text{ }\delta =3,\text{ }\epsilon =0,\ \omega =1
\text{ }\theta =5, \label{PD_values}
\end{equation
all of these conditions hold.
\section{Three-player quantum Prisoners' Dilemma with GHZ state}
The values in (\ref{PD_values}) give $\gamma _{1}=-1$ and $\gamma _{2}=1$.
With the deltas given in (\ref{Deltas}), the Nash inequalities (\ref{NE_ABC
) take the form
\begin{gather}
\Pi _{A}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{A}(\vec{a}
\vec{b}^{\ast },\vec{c}^{\ast })= \notag \label{NE_ABC_1} \\
\frac{1}{8}\left[ -(a_{3}^{\ast }-a_{3})(b_{3}+c_{3})+(a_{1}^{\ast
}-a_{1})(b_{1}c_{1}-b_{2}c_{2})-(a_{2}^{\ast }-a_{2})(b_{1}c_{2}+b_{2}c_{1}
\right] \geq 0, \notag \\
\Pi _{A}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{A}(\vec{a
^{\ast },\vec{b},\vec{c}^{\ast })= \notag \\
\frac{1}{8}\left[ -(b_{3}^{\ast }-b_{3})(a_{3}+c_{3})-(b_{2}^{\ast
}-b_{2})(a_{1}c_{2}+a_{2}c_{1})+(b_{1}^{\ast }-b_{1})(a_{1}c_{1}-a_{2}c_{2}
\right] \geq 0, \notag \\
\Pi _{A}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{A}(\vec{a
^{\ast },\vec{b}^{\ast },\vec{c})= \notag \\
\frac{1}{8}\left[ -(c_{3}^{\ast }-c_{3})(a_{3}+b_{3})-(c_{2}^{\ast
}-c_{2})(a_{1}b_{2}+a_{2}b_{1})+(c_{1}^{\ast }-c_{1})(a_{1}b_{1}-a_{2}b_{2}
\right] \geq 0. \notag \\
\end{gather
These inequalities show that for the PD game defined in (\ref{PD_values}),
no directional triplet can exist as a NE when the three players have the
choice to direct their respective unit vector along any direction i.e. there
are no restrictions placed on the players' directional choices.
The inequalities (\ref{NE_ABC}) suggest the following cases:
\subsection{Case (a)}
Consider $a_{3}=b_{3}=c_{3}=0$. Nash inequalities (\ref{NE_ABC}) then take
the form
\begin{eqnarray}
\Pi _{A}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{A}(\vec{a}
\vec{b}^{\ast },\vec{c}^{\ast }) &=&\frac{1}{8}\gamma _{2}\left[
(a_{1}^{\ast }-a_{1})\Delta _{2}-(a_{2}^{\ast }-a_{2})\Delta _{3}\right]
\geq 0, \notag \\
\Pi _{A}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{A}(\vec{a
^{\ast },\vec{b},\vec{c}^{\ast }) &=&\frac{1}{8}\gamma _{2}\left[
-(b_{2}^{\ast }-b_{2})\Delta _{2}^{\prime }+(b_{1}^{\ast }-b_{1})\Delta
_{3}^{\prime }\right] \geq 0, \notag \\
\Pi _{A}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{A}(\vec{a
^{\ast },\vec{b}^{\ast },\vec{c}) &=&\frac{1}{8}\gamma _{2}\left[
-(c_{2}^{\ast }-c_{2})\Delta _{2}^{\prime \prime }+(c_{1}^{\ast
}-c_{1})\Delta _{3}^{\prime \prime }\right] \geq 0, \notag \\
&&
\end{eqnarray
that can also be expressed as
\begin{eqnarray}
\Pi _{A}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{A}(\vec{a}
\vec{b}^{\ast },\vec{c}^{\ast }) &=&\frac{1}{8}\gamma _{2}\left[ a_{1}^{\ast
}\Delta _{2}-a_{2}^{\ast }\Delta _{3}+\varsigma \right] \geq 0, \notag \\
\Pi _{A}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{A}(\vec{a
^{\ast },\vec{b},\vec{c}^{\ast }) &=&\frac{1}{8}\gamma _{2}\left[
-b_{2}^{\ast }\Delta _{2}^{\prime }+b_{1}^{\ast }\Delta _{3}^{\prime
}+\varsigma \right] \geq 0, \notag \\
\Pi _{A}(\vec{a}^{\ast },\vec{b}^{\ast },\vec{c}^{\ast })-\Pi _{A}(\vec{a
^{\ast },\vec{b}^{\ast },\vec{c}) &=&\frac{1}{8}\gamma _{2}\left[
-c_{2}^{\ast }\Delta _{2}^{\prime \prime }+c_{1}^{\ast }\Delta _{3}^{\prime
\prime }+\varsigma \right] \geq 0,
\end{eqnarray
where
\begin{equation}
\varsigma =a_{1}b_{2}c_{2}+a_{2}b_{1}c_{2}-a_{1}b_{1}c_{1}+a_{2}b_{2}c_{1}.
\end{equation
Consider the case when $\gamma _{2}>0,$ then for given $a_{1}^{\ast },$
b_{1}^{\ast },$ and $c_{1}^{\ast }$, the restrictions on the directions that
the unit vectors $\vec{a},$ $\vec{b},$ and $\vec{c}$ can take can be
determined. For instance, for $a_{2}^{\ast },=b_{1}^{\ast },=c_{1}^{\ast }=1
, i.e. then $a_{2}^{\ast }=b_{2}^{\ast }=c_{2}^{\ast }=0,$these constraints
become
\begin{equation}
\Delta _{2}+\varsigma \geq 0,\text{ }\Delta _{3}^{\prime }+\varsigma \geq 0
\text{ }\Delta _{3}^{\prime \prime }+\varsigma \geq 0.
\end{equation}
\subsection{Case (b)}
Consider $\gamma _{2}=0$ and $a_{3}=b_{3}=c_{3}=0.$ With these constraints,
the allowed directions are confined to the X-Y plane and any directional
triplet then exists as a NE. In this case, from Eqs. (\ref{Deltas}) we then
have $\Delta _{1}=\Delta _{1}^{\prime }=\Delta _{1}^{\prime \prime }=0.$ As
\vec{a},$ $\vec{b},$ and $\vec{c}$ are unit vectors, we also have $a_{2}=\pm
\sqrt{1-a_{1}^{2}},$ $b_{2}=\pm \sqrt{1-b_{1}^{2}},$ and $c_{2}=\pm \sqrt
1-c_{1}^{2}}$.
\section{Discussion}
We present an analysis of the three-partite EPR experiments that use a GHZ
state and is considered as a three-player non-cooperative quantum game. The
players' strategic choices are the three directions $\vec{a},$ $\vec{b},$
and $\vec{c}$ along which the dichotomic observables $\vec{n}.\vec{\sigma}$\
are measured, where $\vec{n}=\vec{a},\vec{b},\vec{c}$ and $\vec{\sigma}$ is
a vector whose components are the standard Pauli matrices. Using
Kaszlikowski and \.{Z}ukowski's results \cite{Kaszlikowski} for the quantum
probabilities involved in such experiments, we develop a three-player
quantum \ game, with the underlying three-partite EPR experiment. This
extends an approach to quantum games by Vaidman \cite{Vaidman} that does not
involve Hilbert space, and/or quantum measurement, and shows how
three-player quantum games with EPR experiments can be developed. Players'
strategies are their directions in terms of which their payoffs are
expressed using Eq. (\ref{QProbabilities}). Nash inequalities are used to
obtain Nash equilibria as direction triples and the players' payoffs are
then compared to their payoffs for the Nash equilibria in the classical game.
For a three-player Prisoners' Dilemma game, defined in (\ref{PD_values}), we
conclude that no directional triplet can exist as a NE when no restrictions
are placed on the players' directional choices. A directional triplet,
however, can exist as a NE under constraints placed on the directions
allowed to the players. This is in accordance with Eisert et al.'s\emph{\
result in Ref. \cite{EWL} showing that a pair of unitary transformations $
\hat{Q},\hat{Q})$, where $\hat{Q}\sim \hat{U}(0,\pi /2),$ exists as a NE in
PD when the players' allowed actions are restricted to certain subsets of
the set SU(2) consisting of all unitary transformations.
As is known \cite{Benjamin1, FlitneyHollenberg} that the particular subset
of unitary transformations that Eisert et al. used in order to obtain the NE
of $(\hat{Q},\hat{Q})$ in two-player quantum Prisoners' Dilemma is not even
closed under composition.\emph{\ }In particular, in Eisert et al.'s protocol
for $2\times 2$ quantum games \cite{EWL}, the new Nash equilibria, and the
classical-quantum transitions that occur, are the outcomes of the particular
strategy space chosen that is a two-parameter subset of single qubit unitary
operators. By choosing a different, but equally plausible, two-parameter
strategy a different Nash equilibria with different classical-quantum
transitions can arise.
Using an EPR setting, and a shared GHZ state, for a three-player quantum
Prisoners' Dilemma game, we present an approach that is driven along purely
probabilistic lines with only an implicit reference to the mathematical
formalism of quantum theory, and showing the constraints on the players'
directional choices under which a particular triplet can exist as a NE in
the game.
|
2,877,628,090,080 | arxiv | \section{Introduction}
One problem with quantum gravity is that we don't know what the theory
should compute. In particle physics, the most precise observable is
the S-matrix. But this quantity seems ill-suited to cosmology, where
the observer is not outside the system, initial states cannot be set
up, and experiments cannot be arbitrarily repeated to gain
statistically significant results.
This ignorance is not especially unusual or embarrassing. It is
rarely clear at the outset what a theory should compute. For example,
the insight that gravity is a theory of a symmetric,
diffeomorphism-invariant tensor field in itself already constituted a
significant part of the development of general relativity. But once a
theory is in its final form, the observables should be apparent.
If string theory is the correct quantum theory of gravity, then
whatever it computes presumably are the observables. But string
theory---perhaps because it is {\em not\/} in its final form---has so
far sidestepped the problem of cosmological observables. It defines
quantum gravity for certain classes of geometries characterized by
asymptotic conditions, such as asymptotically flat or Anti-de~Sitter
spacetimes. In these geometries an S-matrix happens to make sense,
and string theory computes its matrix elements. (In the case of AdS,
it computes boundary correlators, which are a close analogue of the
S-matrix.)
However, we have yet to learn how to apply string theory to cosmology
or to an observer inside a black hole, with the same level of rigor as
in Anti-de Sitter space. Hence, it would be premature to conclude
that the S-matrix will remain the only well-defined object. It is too
early to know what, if anything, string theory has to say about
cosmological observables.
Fortunately, classical and quantum properties of cosmological
solutions impose significant constraints on possible observables, and
may even hint at some of the principles on which a theory computing
them must be based. De~Sitter space is a case in point.
Semi-classical analysis has provided overwhelming evidence that {\em
no\/} exact observables exist in eternal de~Sitter space---at least,
none that correspond to experiments that can be performed by an
observer inside the universe. This is related to the presence of a
cosmological event horizon in de~Sitter space, which limits the
accessible information and emits pernicious thermal radiation.
In this paper we use similar semi-classical reasoning to characterize
constraints on exact observables in other cosmological solutions.
Does the universe contain regions where fluctuations, including those
of the gravitational field, become arbitrarily weak? Accurate
measurements take a long time\footnote{For this reason, we shall use
the terms ``asymptotic observable'' and ``exact observable''
interchangeably.}, and they require devices with a large number of
states. Does the universe last long enough, i.e., does it contain
geodesics of infinite proper time in the future? Does the causally
accessible region have enough quantum states? According to entropy
bounds~\cite{Tho93,Sus95,CEB1}, this translates into a minimum
size for the region. By asking whether such requirements are met, one
can investigate whether exact quantum mechanical observables exist in
a given cosmology, without knowledge of the full theory.
By an observable we mean a quantity or limit of quantities that can
actually be measured by an observer inside the universe, without
violating laws of physics such as causality or entropy bounds. For
example, it may turn out that an S-matrix for de~Sitter space can be
formally computed as a useful ``meta-observable''~\cite{Wit01}, from
which predictions for true, operationally defined observables can be
extracted by further processing. The restrictions derived below apply
only to the latter, operationally meaningful quantities.
Our conclusions for different classes of universes vary in their
details, but they do strike two common chords. First: Aside from
de~Sitter space and the obvious case of crunching universes, our
necessary conditions for exact observables are satisfied in all the
other cases considered. Surprisingly, this includes universes with a
cosmological event horizon. Second: Observables that invoke a global
out-state (such as the S-matrix) do not seem to describe any
experiment in cosmology. We find that the information content of a
generic out-state is causally inaccessible even in a universe with a
null infinity and no horizon.
We present a number of intermediate results that are of interest in
their own right: an analysis of the thermodynamics and the fluctuation
spectrum of quintessence universes; an argument demonstrating that an
open universe resides inside the Farhi-Guth solution; and entropic
reasoning suggesting that the global state of a non-compact universe
is not accessible to experiment, independently of event horizons.
This paper does not tackle the actual definition of any asymptotic
cosmological observables (see Ref.~\cite{BanFis01a,FreSus04} for
recent approaches). Even that challenge, in turn, will only be an
intermediate goal. In our view, asymptotic observables are at best a
crutch. The description of a real experiment involving gravity
requires well-defined (but necessarily imprecise?) local observables.
This is a famously difficult problem in the presence of gravity. It
is further complicated, but perhaps also helpfully constrained, by the
counter-intuitive holographic restriction on bulk degrees of
freedom~\cite{Bek81,Tho93,Sus95,FisSus98,CEB1,CEB2,RMP,Bou03}. This
task will have to be confronted eventually.
\paragraph*{Relation to other work}
For a review of the difficulties with physics in de~Sitter space, see,
e.g., Ref.~\cite{Bou02b}. A broad discussion of the problem of
observables in cosmologies with a non-positive cosmological constant
was given by Banks and Fischler~\cite{BanFis01a}, who noted that in a
non-compact universe, an S-matrix description must restrict to states
with a finite number of extra particles, and that those states are
very special. While this restriction is necessary, it is not
sufficient: as shown below, the unobserved region can have infinite
entropy even if no particles are added, because of the internal states
of the matter already present.
Our analysis of the thermodynamics of Q-space builds on
Refs.~\cite{HelKal01,FisKas01}, who derived its global structure and
pointed out that its event horizon obstructs the definition of an
S-matrix. We do not question this conclusion; indeed, we find that
the difficulties with an S-matrix are quite general in cosmology. We
do argue, however, that other asymptotic observables may exist in
Q-space. This possibility was first raised by Witten~\cite{Wit01},
who noted that observers will not be thermalized in Q-space. The
existence of accessible high entropy states was not demonstrated
there.
The problem of defining cosmological observables is closely related to
the challenge of describing physics from the point of view of an
observer falling into a black hole. In both cases, some type of local
observables will eventually be required, but in both cases, one can
hope to make progress by asking how some of the information in the
gravity-dominated region may be encoded in asymptotic
data~\cite{KraOog03,FidHub04}.
The recent discovery that the universe is accelerating has turned the
cosmological constant problem into the (worse) problem of small
positive vacuum energy. Its possible resolution by a discretuum of
meta-stable vacua in string theory~\cite{BouPol00}, populated by
cosmological dynamics, makes it all the more urgent to understand
string theory observables in cosmology. Explicit constructions of
de~Sitter vacua have been proposed (e.g., Ref.~\cite{KKLT}), and
sophisticated counting arguments (e.g.,
Refs.~\cite{AshDou03,GirKac04}) broadly confirm the original estimates
of the vast number of such vacua. The present discussion does not
address specifically the development of a theoretical
framework~\cite{FreSus04,Ban04,BouFre05} describing this
``landscape''~\cite{Sus03}. But the question of observables is a part
of this challenge, so our results may have some implications in this
context.
\paragraph*{Outline}
The paper is structured as follows. In the first sections we mainly
consider spatially flat FRW universes with fixed equation of state
$w=p/\rho$. They are especially simple and suffice for deriving our
main results. Moreover, their late time behavior is a good
approximation to other classes of cosmologies, including some we
discuss at the end of the paper.
Sec.~\ref{sec-frw}, aside from a review of the flat FRW solutions and
their causal structure, contains our main observation about
decelerating universes ($w>-1/3$): All observers, at all times, lack
information about infinitely large regions of the universe, even
though there is no event horizon. If such regions contain any
non-redundant information, then the global out-state
computed by an S-matrix cannot be measured.
Next, we turn to eternally accelerating universes, de~Sitter space
($w=-1$) and ``Q-space'' ($-1<w<-1/3$) \cite{RatPee88}\footnote{For a
subset of this range, quintessence has been proposed as a model of
dark energy~\cite{WanCal99}. Here we study these solutions simply
as instructive examples to understand conditions for asymptotic
observables.}, which have a cosmological event
horizon~\cite{HelKal01,FisKas01}. We show in Sec.~\ref{sec-temp} that
Q-space exhibits thermodynamic properties similar to those of the
de~Sitter horizon. The horizon radius in Q-space grows linearly with
time, and consequently the temperature slowly decreases. We find that
this behavior is consistent with the first law of thermodynamics: the
temperature and entropy respond appropriately to the flux of
quintessence stress-energy across the horizon.
Sec.~\ref{sec-fluc} contains our main results for accelerating
universes. They support the existence of asymptotic observables in
Q-space. We study specific aspects of the thermal spectrum emitted by
the horizon. The time-dependence of the temperature leads to
significant differences between de~Sitter space and Q-space. In the
semi-classical theory, an infinite number of Hawking quanta are
produced (and re-absorbed) by the horizon. In de~Sitter space, the
total energy thus emitted diverges, whereas in Q-space the energy per
quantum decreases rapidly enough to render the total energy finite.
Hence, observers in Q-space will not be thermalized.
We ask whether observers will be destroyed by rare massive
fluctuations, such as black holes. We consider objects of fixed
energy and compute the rate at which they are emitted, according to
standard statistical mechanics. If the energy is much larger than the
temperature, the rate will be miniscule. However, in de~Sitter space
the rate is constant, so all fluctuations that are not completely
forbidden will occur. This guarantees that any observer who survives
the thermal radiation long enough will eventually be swallowed by a
large black hole emitted by the horizon. In Q-space, the rate of such
violent processes decreases exponentially with time. The integrated
probability is therefore finite and can be exceedingly small. It
follows that experiments in Q-space can last for an arbitrarily long
time.
But the classical supply of matter in Q-space is bounded, seemingly
ruling out exact measurements. Yet, we show in
Sec.~\ref{sec-qentropy} that arbitrarily complex matter configurations
are quantum mechanically {\em produced\/} by the Q-space horizon: The
rate for a fluctuation of a given fixed entropy---no matter how
large---is constant and non-vanishing at late times. This contrasts
pleasantly with de~Sitter space, where the entropy is strictly bounded
by the inverse of the (fixed) cosmological constant.
In Sec.~\ref{sec-discussion} we draw conclusions on the nature of
observables in the universes we have studied. In particular, we argue
that no direct analogue of an S-matrix can be defined in any flat FRW
universe unless the set of allowed states is severely restricted.
In Sec.~\ref{sec-other} we extend the discussion to open and closed
FRW solutions. We also study composite universes that feature an
asymptotically flat region on the far side of a black hole. We show
that the Farhi-Guth solution can be regarded as an example of this
setup in which the black hole resides inside an open universe produced
by the decay of meta-stable de~Sitter space. Because an open universe
has infinite entropy, one would not expect generic microstates to be
represented on the far side of the black hole, or on the asymptotic
boundary.
\section{Spatially flat universes}
\label{sec-frw}
In this section we review various classical properties of flat FRW
universes---in particular, the results of
Ref.~\cite{HelKal01,FisKas01} on the causal structure of accelerating
cosmologies. We ask how much matter and information is causally
accessible to an observer in the classical evolution. We find that
this amount is finite in accelerating universes and unbounded in
decelerating universes. However, even in the latter case, no more
than an infinitely small fraction of the matter is ever observable.
\subsection{Metric and causal structure}
The metric of a spatially flat FRW universe is given by
\begin{equation}
ds^2 = -dt^2+a(t)^2 (dr^2+r^2 d\Omega^2) ~.
\label{eq-flatfrw}
\end{equation}
A quick way to obtain its causal structure is to transform to
conformal time, defined by $d\eta=dt/a(t)$. This shows that the
metric is conformal to Minkowsi space: $ds^2 = a(\eta)^2
d\tilde{s}^2$, where
\begin{equation}
d\tilde{s}^2 = -d\eta^2+dr^2+r^2 d\Omega^2 ~.
\end{equation}
Hence, the conformal diagram is a subset of the Minkowski space
Penrose diagram (Fig.~\ref{fig-penfrw}), selected by the range of
$\eta$. The existence of horizons is determined as follows.
\begin{figure}
\includegraphics[width=8.5cm]{penfrw}
\caption{\label{fig-penfrw} Conformal diagrams of Minkowski space (left) and a
decelerating flat FRW universe (right). In the FRW case, any
infinitesimal neighborhood of spatial infinity (circle) contains an
infinite amount of matter and potentially an infinite amount of
information, whereas the observer's causal past is a finite region.}
\end{figure}
If and only if $\eta$ is bounded from above ($\eta\to\eta_{\rm
max}<\infty$ as $t\to\infty$), then an observer at $r=0$ is
surrounded by a future event horizon.\footnote{By homogeneity, all
comoving observers are equivalent, so we consider an observer at
$r=0$. Any non-comoving observer whose spatial position remains
finite at late times has the same horizon as a comoving observer
located at the same asymptotic spatial position.} The horizon is
located at $r=\eta_{\rm max}-\eta$. Light-rays originating beyond
this hypersurface never reach the observer. Similarly, if $\eta$ is
bounded from below, then there exists a past horizon.\footnote{This
assumes that the FRW solution in question is past inextendible.
Hence this analysis does not apply to the flat slicing of de Sitter
space.} Events beyond this horizon cannot be influenced by the
observer.
The dynamical evolution of the scale factor and the matter density is
determined by the equations
\begin{eqnarray}
\frac{\dot{a}^2}{a^2} & = & \frac{8\pi\rho}{3}\ ,\\
\frac{\ddot{a}}{a} & = & -\frac{4\pi}{3}(\rho+3p)\ . \label{eq-rho}
\end{eqnarray}
We will assume that the energy density, $\rho$, and pressure, $p$,
obey the equation of state
\begin{equation}
p=w\rho
\end{equation}
with constant $w$. It will be more convenient to work with the
parameter
\begin{equation}
\epsilon = \frac{3}{2}(w+1)\ .
\end{equation}
Thus one obtains a family of solutions
parameterized by $\epsilon$,
\begin{eqnarray}
a(t) & = & t^{1/\epsilon} ~, \label{eq-a} \\
\rho(t) & = & \frac{3}{8\pi \epsilon ^2 t^2} ~.
\end{eqnarray}
except for $\epsilon=0$, which corresponds to a cosmological constant
$\Lambda$. In that case a solution is given by
$a(t)=\exp[(\Lambda/3)^{1/2}t]$, $\rho=\Lambda/8\pi$.
We assume the dominant energy condition, which restricts $\epsilon$ to
the range $0\leq \epsilon\leq 3$. From Eq.~(\ref{eq-rho}) we can see
directly that for $\epsilon>1$, the expansion of the universe
decelerates: $\ddot{a}<0$. This includes the familiar cases of matter
domination ($\epsilon=3/2$) and radiation domination ($\epsilon=2$).
For $\epsilon<1$, on the other hand, the scale factor grows
increasingly rapidly: $\ddot{a}>0$. The degenerate case $\epsilon=1$
will not be considered here.
As discussed more generally above, we transform to conformal time,
\begin{equation}
\eta=\frac{\epsilon}{\epsilon-1}\ t^\frac{\epsilon-1}{\epsilon} ~,
\end{equation}
to reveal the causal structure. For decelerating universes
($\epsilon>1$), this expression shows that conformal time is bounded
below but unbounded above. There is no future event horizon. The
conformal diagram is given by the upper half ($\eta>0$) of the Penrose
diagram of Minkowski space (Fig.~\ref{fig-penfrw}).
For accelerating universes with $0<\epsilon<1$, the situation is
reversed. Conformal time ranges from $-\infty$ to $0$, and so is
bounded above but not below. Hence, the conformal diagram is the
lower half of the Minkowski wedge (Fig.~\ref{fig-penfrwb}). There is
a future event horizon at $r+\eta=0$, whose area, $A_{\rm E}$, grows
quadratically with time. The proper horizon area-radius, $R_{\rm
E}=(A_{\rm E}/4\pi)^{1/2}$, is given by
\begin{equation}
R_{\rm E} = -\frac{\epsilon}{\epsilon-1}\ t~.
\label{eq-re}
\end{equation}
\begin{figure}
\includegraphics[width=8.5cm]{penfrwb}
\caption{\label{fig-penfrwb}
Conformal diagrams of flat Q-space~\cite{HelKal01,FisKas01} (left)
and de~Sitter space (right). Past and future cosmological event
horizons are shown. The area of the de~Sitter horizon is constant,
whereas the area of the Q-space horizon grows without bound at late
times~\cite{HelKal01}. The Q-space initial singularity is not
really null, since the curvature already becomes Planckian on a
nearby spacelike slice (see Fig.~3).}
\end{figure}
In the case of de Sitter space, $\epsilon=0$, the metric (\ref{eq-flatfrw})
is geodesically incomplete and extendible. The maximal extension has
closed spatial slices and is given by
\begin{equation}
ds^2 = \frac{3/\Lambda}{\sin^2\eta} (-d\eta^2 + d\chi^2 + \sin^2\chi
d\Omega^2)~.
\label{eq-global}
\end{equation}
Hence, the conformal diagram is a square (Fig.~\ref{fig-penfrwb}), and
de Sitter space has both past and future event horizons of constant
radius $\sqrt{3/\Lambda}$.
\subsection{Classical observable matter content}
\label{sec-class}
The maximum spacetime region probed by an experiment is called the causal
diamond~\cite{Bou00b}. It is generally defined as the causal past of
the future endpoint of the observer's worldline, intersected with the
causal future of the past endpoint. (Note that the latter is crucial:
events lying in the observer's past but outside the bottom cone cannot
be probed directly and may not send any signals in the observer's
direction. If a signal is sent, then what information can be gleaned
about the event is precisely what passes through the bottom cone.)
How much matter enters an observer's causal diamond? We restrict for
now to the classical evolution of the cosmological fluid, and postpone
the inclusion of the thermal properties of the horizon until
Sec.~\ref{sec-fluc}.
\paragraph*{de~Sitter space}
In eternal de Sitter space ($\epsilon=0$), the causal diamond is the
region limited by both the past and future event horizons. The
maximum amount of matter that can enter is the largest black hole
allowed in asymptotically de Sitter space, the Nariai black hole. Its
entropy is one third that of the empty de Sitter horizon. But to
arrange for matter to enter, one must either include thermal effects,
or set up appropriate initial conditions in the infinite past.
It is more interesting to consider a universe such as ours, which
contains an era of matter- or radiation-domination before the
cosmological constant takes over. The Penrose diagram for this type
of solution is shown in Fig.~\ref{fig-penfrwc}. In that case, the
bottom cone of the causal diamond is the future light-cone of a point
at the big bang (usually called the particle horizon). Its structure
depends on the details of the matter content. But as long as the
universe is asymptotically de~Sitter in the future, the amount of
information inside the causal patch is bounded by the entropy at late
times, which is that of empty de Sitter space.
\begin{figure}
\includegraphics[width=8.5cm]{penfrwc}
\caption{\label{fig-penfrwc}
This conformal diagram can be interpreted in three ways. It
represents pure Q-space, with a spacelike singularity reflecting a
Planck scale cutoff of the classical metric (see Fig.~2). It also
corresponds to a big bang universe initially dominated by matter or
radiation, which asymptotes to Q-space or de~Sitter space at late
times.---The causal diamond of the observer at $r=0$ is shown. The
bottom cone (B) has finite maximal area, indicating that only a
finite amount of entropy enters the observable region by classical
evolution. In asymptotically Q-space, however, the top cone (T)
allows arbitrarily large entropy. Indeed, an unbounded number of
states can be accessed by quantum fluctuations of the horizon
(Sec.~4.4).}
\end{figure}
To summarize, an observer in asymptotically de~Sitter space can access
at most an entropy of order the inverse cosmological
constant~\cite{Ban00}. This conclusion is independent of whether
thermal effects are included, and may extend to a larger class of
universes with positive cosmological
constant~\cite{Bou00a}.\footnote{If the requirement of a future
asymptotic region dominated by the vacuum energy is dropped,
examples with greater entropy are known in more than four spacetime
dimensions~\cite{BouDew02}.}
\paragraph*{Q-space}
In an accelerating universe with $\epsilon>0$, the largest possible
causal diamond is the intersection of the past of the point $t=t_{\rm
late}$, $r=0$ with the future light-cone of the point $t=1$, $r=0$,
in the limit $t_{\rm late}\to\infty$ (Fig.~\ref{fig-penfrwc}). (We
follow Ref.~\cite{KalLin99} in excising the high curvature region
prior to the Planck time. This replaces the null singularity with a
more standard, spacelike big bang.) The lower cone is again the
particle horizon. The upper cone, in the limit taken, is the future
event horizon (and so is a cone only conformally).
The amount of information entering the causal diamond from the past,
$S_{\rm in}$, is bounded by the maximal area of the lower
cone~\cite{FisSus98,CEB1}. One thus finds that
\begin{equation}
S_{\rm in} \leq \pi \left(\frac{\epsilon}{1-\epsilon}\
2^{\frac{\epsilon}{1-\epsilon}}\right)^2~.
\end{equation}
Unless $\epsilon$ is very close to $1$, this is at most of order
unity, indicating that virtually no information enters the observer's
causal diamond. Note that this result applies strictly to an
accelerating $\epsilon>0$ fluid with no other matter present.
The conclusion changes somewhat if other types of matter dominate at
early times. If quintessence were the source of the vacuum energy in
our universe, for example, our particle horizon would intersect our
future event horizon about now (Fig.~\ref{fig-penfrwc}). Its maximal
area would be quite large: about $10^{123}$ in Planck units. Still,
like in de Sitter space, and unlike the decelerating universes, only a
finite amount of matter and information ever enter the causal diamond
by conventional evolution~\cite{GudBjo01,Loe01,KalKle04}. To show
that Q-space exhibits unbounded complexity, one needs to include
thermal fluctuations (Sec.~\ref{sec-fluc}).
\paragraph*{Decelerating FRW}
In a decelerating universe, the bottom cone extends all the way to
future infinity and has infinite maximal area. There is no bound on
the entropy that can enter. Indeed any comoving particle {\em will\/}
enter it sooner or later. Thus, in decelerating universes any
observer has access to arbitrarily large amounts of matter and
entropy.
However, there is an important order-of-limits issue. Let us ask how
much of the universe is seen by an observer at some finite time $t$.
One finds that the sphere at the edge of the causal diamond has area
\begin{equation}
A_{\rm edge}=\pi\left(\frac{\epsilon t}{\epsilon-1}\right)^2~.
\label{eq-finite}
\end{equation}
This area is a bound on the amount of entropy that has entered the
region observed by the time $t$. Note that this does not diverge at
finite $t$. But finite $t$ is all an observer can ever attain.
Hence, the number of accessible degrees of freedom is, at all times,
an infinitely small fraction of the total number of degrees of freedom
in the universe. This is shown in Fig.~\ref{fig-penfrw}; only the
past light-cone is shown (rather than the stronger restriction to the
causal diamond), since this already suffices to illustrate the
problem. In Sec.~\ref{sec-discussion} we will argue that this
limitation is an important criterion distinguishing the observations
made in a decelerating FRW universe from the S-matrix of
asymptotically flat space.
\section{Temperature and entropy of accelerating universes}
\label{sec-temp}
In this section we obtain the basic thermodynamic properties of
Q-space: entropy, energy, and temperature. We demonstrate that they
satisfy the first law of thermodynamics. We begin by reviewing the
thermodynamics of de~Sitter space.
\subsection{Thermodynamics in de~Sitter space}
\label{sec-dstemp}
De Sitter space has an event horizon of radius $R_0=\sqrt{3/\Lambda}$.
Its area is $A=4\pi R_0^2=12\pi/\Lambda$. It is also a Killing
horizon with surface gravity $\kappa=R_0^{-1}$, with respect to the
the usual timelike Killing vector field normalized at the origin.
Consider an object of mass $M$ in an otherwise empty asymptotically de
Sitter universe. In the presence of this object, the cosmological
horizon will be smaller than that of empty de Sitter space. One way
to estimate its size is to model the object as a small black hole.
For this case an exact solution is known: the Schwarzschild-de Sitter
black holes, with metric
\begin{equation}
ds^2= -V(r) dt^2+\frac{dr^2}{V(r)}+r^2 d\Omega_2^2\ ,
\end{equation}
where
\begin{equation}
V(r)=1-\frac{2M}{r}-\frac{\Lambda}{3} r^2\ .
\end{equation}
For $0<M<1/(3\sqrt{\Lambda})$, $V(r)$ has two positive roots. (The
maximal case $M=1/(3\sqrt{\Lambda})$ is known as the Nariai solution;
larger black holes do not exist in de Sitter space.) The smaller root
is the black hole horizon; it obeys $R_{\rm B}\approx 2M$ for small
$M$. The larger root is the cosmological event horizon. For small
$M$, it obeys
\begin{equation}
R_{\rm C}^2\approx R_0^2-2R_0 M\ ,
\label{eq-rc}
\end{equation}
and it decreases monotonically over the whole range of $M$.
Now suppose that a black hole, or any other object of small mass $M$,
falls across the cosmological horizon, restoring the observer's patch
to empty de~Sitter space. (This can be achieved simply by the
observer moving away from the object.) By Eq.~(\ref{eq-rc}), this
process increases the cosmological horizon area by $\Delta A=-8\pi
R_0\, M$. Thus, the cosmological horizon satisfies the usual first
law of horizon dynamics~\cite{BarCar73}:
\begin{equation}
-dE = \frac{\kappa\, dA}{8\pi}\ ,
\label{eq-bhd}
\end{equation}
where we have defined $dE$ to be the change in the mass of the matter
present on the observer's side of the horizon.\footnote{In black hole
mechanics~\cite{BarCar73}, $dE$ thus corresponds to the change in
mass of matter remaining {\em outside\/} the black hole, which is
minus the change of black hole mass, and hence is negative when
matter is added to the black hole. Hence Eq.~(\ref{eq-bhd}) takes
the same form for black holes and for de Sitter space.} As in the
case black holes, this classical relation betrays the semiclassical
thermodynamic properties exhibited by the de~Sitter horizon. Analysis
of quantum field theory in a de Sitter
background~\cite{GibHaw77a,BirDav} shows that a freely falling
detector will measure a temperature proportional to the surface
gravity
\begin{equation}
T=\frac{\kappa}{2\pi}~.
\end{equation}
Moreover, in order to avoid a decrease of observable entropy in the
above process, it is natural to propose that the horizon area
represents a true contribution to the total entropy, as originally
suggested for black holes~\cite{Bek72}:
\begin{equation}
S=\frac{A}{4}~.
\end{equation}
Consistency requires that these quantities satisfy the first law of
thermodynamics, which is ensured by Eq.~(\ref{eq-bhd}).
\subsection{Thermodynamics in Q-space}
\label{sec-qtemp}
Slow-roll inflation can be thought of as a de~Sitter-like era with
slowly decreasing effective cosmological constant. It is well-known
that the apparent cosmological horizon during inflation has
thermodynamic properties akin to those of de Sitter
space~\cite{GibHaw77a}. Indeed, this temperature is the origin of
density fluctuations and so is responsible for all structure in the
universe.
One would expect similar considerations to apply to an eternally
accelerating universe with $w$ sufficiently close to $-1$. In this
case, the universe is also locally similar to de~Sitter space, with
slowly decreasing vacuum energy. In fact, since a $w>-1$ fluid can be
modeled by a scalar field with custom-designed
potential~\cite{RatPee88}, it can be thought of as a special case of
slow-roll inflation. Thus, horizon thermodynamics should apply in
Q-space.
We will now verify this expectation. Our arguments will be rigorous
only for $w$ very close to a cosmological constant:
\begin{equation}
0<w+1 \ll 1~,
\end{equation}
though we expect our results to be qualitatively correct at least in
the range $-1<w<-2/3$. The parameter $\epsilon = {3\over 2}(w+1)$
will be small and positive for the accelerating universes studied
here. However, all classical formulas below are exact in $\epsilon$.
The radius of the event horizon was given in Eq.~(\ref{eq-re}). We
will also be interested in the apparent horizon.\footnote{On each
constant time slice, the apparent horizon of an observer at $r=0$ is
the sphere whose orthogonal ingoing future-directed light-rays have
vanishing expansion.} In any FRW universe, its proper radius is
directly related to the energy density:
\begin{equation}
R_{\rm A} = \left(\frac{3}{8\pi\rho}\right)^{1/2}~.
\label{eq-rarho}
\end{equation}
For a flat universe, the apparent horizon radius is thus equal to the
Hubble scale, $t_{\rm H}=a/\dot{a}$, and is given by
\begin{equation}
R_{\rm A}=t_{\rm H} = \epsilon t\ .
\label{eq-rat}
\end{equation}
The two horizons satisfy the following key properties. First, they
are approximately equal in the regime we study. More precisely, the
apparent horizon is smaller than the event horizon by a fixed ratio
close to unity:
\begin{equation}
\frac{R_{\rm A}}{R_{\rm E}}= 1-\epsilon\ ,
\end{equation}
Second, neither horizon changes significantly over one Hubble time:
\begin{equation}
\frac{t_{\rm H} \dot{R}_{\rm X}}{R_{\rm X}}
= \epsilon \ll 1;~~~\mbox{X=A,E}~.
\end{equation}
Hence, a thermodynamic description of the horizon will be
approximately valid, and it will not matter much whether we use the
apparent or the event horizon for this purpose.
We will work with the apparent horizon, since this approach is more
general. (For example, in slow-roll inflation, there may be no event
horizon, but one would still like to describe the approximate thermal
state during inflation.) Hence, an observer at $r=0$ will perceive a
thermal heat bath with slowly time-dependent temperature
\begin{equation}
T= \frac{1}{2\pi R_{\rm A}}
\label{eq-temp}
\end{equation}
and will ascribe to the apparent horizon a Bekenstein-Hawking entropy
\begin{equation}
S= \pi R_{\rm A}^2~.
\label{eq-entropy}
\end{equation}
As a consistency check, let us verify that the first law of
thermodynamics is satisfied. We follow Ref.~\cite{BouHaw96}, where a
similar check was performed for slow-roll inflation. Consider an
infinitesimal time interval $dt$. The amount of energy crossing the
horizon during this time is obtained by integrating the flux of the
stress tensor across the surface, contracted with the (approximate)
generators of the horizon, the future directed ingoing null vector
field $k^a$:
\begin{equation}
-dE = 4\pi R_{\rm A}^2 \, T_{ab} k^a k^b \, dt
= 4\pi R_{\rm A}^2 \, \rho (1+w) \, dt
= \epsilon \, dt~.
\end{equation}
In the last equality we have used Eq.~(\ref{eq-rarho}). By
Eqs.~(\ref{eq-entropy}) and (\ref{eq-rat}), the horizon entropy
increases by
\begin{equation}
dS = (2\pi R_{\rm A})\, \dot{R}_{\rm A} dt
= (2\pi R_{\rm A})\, \epsilon\, dt~.
\end{equation}
The term in parentheses is the inverse temperature,
Eq.~(\ref{eq-temp}). Thus we confirm the first law,
\begin{equation}
-dE=T\, dS~.
\end{equation}
\section{Thermal fluctuations in accelerating universes}
\label{sec-fluc}
Both in Q-space and in de Sitter space, the thermal horizon produces
fluctuations---but as we shall see in this section, their implications
are quite different in the two cases. Fluctuations in de~Sitter space
are fatal to experiments. We show, however, that in Q-space
fluctuations are benign: entropic enough to produce complex systems,
but not energetic enough to destroy an observer measuring them.
\subsection{Typical quanta}
\label{sec-typ}
We begin by asking: What is the total number of quanta emitted by the
horizon? For de Sitter as for Q-space, the expected rate is one
quantum (typically with wavelength of order $R_{\rm A}$), per Hubble
time $R_{\rm A}$. In de~Sitter space, $R_{\rm A}=R_0$ is a constant,
and an infinite number of quanta are emitted in total. (No observer
will last long enough to notice more than a finite number, however, as
we shall see shortly.) In Q-space, we see from Eq.~(\ref{eq-rat})
that $R_{\rm A}$ grows linearly with time. However, the integrated
number of quanta still diverges (though only logarithmically, not
linearly as in the de Sitter case):
\begin{equation}
\int \frac{dt}{R_{\rm A}} \sim \log t \to \infty~.
\end{equation}
What is the total energy radiated? In de Sitter space, the typical
energy of each quantum is fixed, so the radiated power integrates to
infinite energy, suggesting that it will erode any physical structure.
Any observer in de Sitter space will be thermalized by the steady
stream of radiation from the horizon.
In Q-space, the rate of emission of quanta and the energy per quantum
each go like $R_{\rm A}^{-1}$. Hence, the radiated power drops off
like the inverse square of time, and it integrates to a finite total
radiated energy. Quantitatively the total energy radiated after the
time $t=t_0$ is
\begin{equation}
E \approx \int_{t_0}^\infty \frac{dt}{R_{\rm A}^2}
= \frac{1}{\epsilon} \int_{R_{\rm A}(t_0)}^\infty
\frac{dR_{\rm A}}{R_{\rm A}^2}
= \frac{1}{\epsilon R_{\rm A}(t_0)}~.
\end{equation}
For example, taking $t_0$ to be the time at which dark energy began to
dominate the evolution of our universe, the total energy (to be)
radiated by the cosmological horizon would be comparable to that of a
single quantum with wavelength of order the present Hubble scale.
Thus, the Q-space horizon falls far short of thermalizing the matter
it contains, in stark contrast with the de Sitter horizon.
\subsection{Large energy fluctuations in de~Sitter space}
\label{sec-dsenergy}
What is the probability for a state of specified energy $E$ to be
radiated by the horizon? Aside from a slow death by thermalization,
observers in de Sitter space also face the threat of collisions with
objects of greater energy than the typical Hawking quanta. Though
exponentially suppressed, such objects will eventually appear as rare
fluctuations in the thermal spectrum. A particularly destructive
example is that of a nearly maximal Schwarzschild-de Sitter black
hole, which will swallow the observer.
For small energy, $E\ll R_0$, the problem is approximately equivalent
to that of a hot cavity at temperature $T=(2\pi R_0)^{-1}$. The
horizon provides the heat bath. For larger energies, gravitational
backreaction can change the volume of the cavity and the temperature
of the horizon by factors of order unity. In particular, there is a
largest possible energy, corresponding to a black hole that just fits
inside the cosmological horizon. We will take these finiteness
effects into account but we begin by considering small energies.
The probability to find in the cavity a particular state $|i\rangle$,
of energy $E_i$, is given by
\begin{equation}
P(|i\rangle) = \frac{1}{Q}\, e^{-E_i/T}
= \frac{1}{Q}\, \exp(-2\pi E_i R_0)~.
\label{eq-pi}
\end{equation}
For a cavity with radius of order the inverse temperature (and a
reasonable number of species), we can neglect factors of the partition
function,
\begin{equation}
Q=\sum_{|i\rangle} \exp(-E_i/T)~,
\end{equation}
since it is dominated by a few states of energy $T$ and so is of order
unity. Note that the probability $P(|i\rangle)$ is really a rate per
time interval of order the interaction time of the heat bath, $R_0$.
The probability to find an arbitrary state with energy $E$ is larger
than (\ref{eq-pi}) by a factor of the number of such states,
$N(E)=e^{S(E)}$:
\begin{equation}
\frac{P_E}{R_0} = \exp[S(E)-2\pi E R_0]~.
\label{eq-pe}
\end{equation}
A de Sitter space variant of the Bekenstein bound~\cite{Bek74,Bek81},
the D-bound~\cite{Bou00b}, guarantees that the exponent will be
non-positive. For high energies compared to the thermal energy
$R_0^{-1}$, the second term in the exponent is large. Thus the
rate of the corresponding fluctuations will be exponentially
suppressed, unless the entropy enhancement factor $e^{S(E)}$ nearly
cancels the suppression term, leaving an exponent of order unity. We
now estimate $S(E)$ to argue that this is not the case.
In a quantum field theory coupled to gravity (a description which
should be locally valid at late times), the objects of highest entropy
for a given energy $E$ are either a black hole, or a radiation gas
with temperature $\tau$ and radius $\chi$ such that
$E\approx\chi^3\tau^4$. (We assume that the number of species with
mass less than $\tau$ is not significant, i.e., less than $10^4$).
The entropy of the black hole is of order $E^2$. The entropy of the
thermal radiation is $\chi^3\tau^3\approx (E\chi)^{3/4}$. This is
maximized by choosing the radius occupied by the gas as large as
possible, $\chi=R_0$. Thus the maximal entropy of thermal
radiation is
\begin{equation}
S_{\rm therm} \approx (ER_0)^{3/4}~.
\end{equation}
Whether this is larger than the black hole entropy $E^2$ depends on
the size of the horizon.
For $R_0^{-1}\lesssim E\lesssim R_0^{3/5}$, thermal
radiation wins. Hence, in this regime we obtain the following upper
bound for the logarithm of the production rate:
\begin{equation}
S(E)-2\pi E R_0 \leq -2\pi (1-\delta) E R_0
\label{eq-exp}
\end{equation}
for some small number $\delta \approx (E R_0)^{-1/4}\ll 1$. At
the level of accuracy required below, the $S(E)$ term can clearly be
dropped altogether ($\delta\approx 0$).
For $R_0^{3/5}\lesssim E\lesssim R_0$, a black hole dominates the
ensemble. Near the lower end of this range, the black hole will have
radius $R_{\rm B}\approx 2E$. For larger energy, however, the
backreaction on the cosmological horizon is significant, and the
definition of energy itself becomes ambiguous. We will simply use the
black hole radius, $R_{\rm B}$, as an energy-like parameter and
abandon the estimate (\ref{eq-pe}) in favor of a direct computation of
the rate of black hole nucleation~\cite{Cha97,BouHaw98}:
\begin{equation}
\frac{P_{\rm B}}{R_0} = \exp[S_{\rm SdS}(R_{\rm B})-S_{\rm dS}]~.
\label{eq-pr}
\end{equation}
$S_{\rm SdS}(R_{\rm B})$ is the total entropy of a Schwarzschild-de
Sitter geometry with a black hole of radius $R_{\rm B}$. It is given
by a quarter of the sum of the black hole and the cosmological horizon
area. $S_{\rm dS}=\pi R_0^2$ is the entropy of the empty de Sitter
solution with the same cosmological constant. Einstein's equation
implies for any static spherically symmetric vacuum
solution~\cite{BouHaw98}:
\begin{equation}
R_{\rm B}^2+R_{\rm C}^2+R_{\rm B} R_{\rm C}= R_0^2~.
\label{eq-radii}
\end{equation}
Here, $R_{\rm C}$ is the radius of the cosmological horizon. Hence
the creation rate (\ref{eq-pr}) is simply
\begin{equation}
\frac{P_{\rm B}}{R_0} = \exp[-\pi R_{\rm B} R_{\rm C}]~.
\label{eq-pr2}
\end{equation}
The exponent agrees well with Eq.~(\ref{eq-exp}) in a large region of
overlap: For $R_{\rm B}\ll R_0$, one can take $R_{\rm B}\approx 2E$.
Moreover, the contribution $S(E)$ from the black hole entropy is
subleading.
Already the smallest black holes, with $R_{\rm B}\approx 1$ and
$P_{\rm B}\sim\exp(-\pi R_0)$, are exponentially suppressed and thus
very unlikely to arise in the thermal spectrum. At fixed cosmological
constant, one finds from Eq.~(\ref{eq-radii}) that $R_{\rm C}(R_{\rm
B})$ is a monotonically decreasing function: the cosmological
horizon gets smaller for larger black holes. But $R_{\rm B} R_{\rm
C}(R_{\rm B})$ grows monotonically, so larger black holes are more
and more unlikely. The biggest black hole allowed by
Eq.~(\ref{eq-radii}) has $R_{\rm B}=R_0/\sqrt{3}$ and is suppressed by
$\exp(-\pi R_0^2/3)$.
However, no matter how small the rate of such fluctuations, in de
Sitter space it is independent of time. Hence, even the most unlikely
fluctuation will eventually occur, on a timescale of order $R_0/P$.
\subsection{Large energy fluctuations in Q-space}
\label{sec-qenergy}
In a $w>-1$ accelerating universe, Eqs.~(\ref{eq-pi}), (\ref{eq-pe}),
and (\ref{eq-pr}) still describe the probability for the corresponding
fluctuations, if we substitute $R_{\rm A}$ for $R_0$. But as we shall
see now, violent events of a specified energy $E$ are not likely to
ever occur at late times, no matter how long one waits.
First consider a fluctuation of less than the Planck energy, $E\leq
1$. Its rate is given by Eq.~(\ref{eq-pe}). Let $t_0$ be a
sufficiently late time so that the temperature of the horizon has
become small compared to the energy of the fluctuation: $R_{\rm
A}(t_0)=\epsilon t_0\gg E^{-1}$. What is the total probability
${\cal P}(E)$ for the fluctuation of energy $E$ to occur after the
time $t_0$?
\begin{eqnarray}
{\cal P}(E) & = & \int_{t_0}^{\infty}
dt\,\frac{P_E(t)}{R_{\rm A}(t)}\\
& \leq & \frac{1}{R_0}\int_{t_0}^\infty
dt\, \exp[S(E)-2\pi ER_{\rm A}(t)].\\
& \leq & \frac{1}{\epsilon R_0}\int_{R_{\rm A}(t_0)}^\infty
dR_{\rm A}\,
\exp[-(2\pi-\delta) ER_{\rm A}]\\
& = & \frac{1}{(2\pi-\delta)\epsilon ER_{\rm A}(t_0)}
\exp[-(2\pi-\delta) ER_{\rm A}(t_0)].
\end{eqnarray}
(Here $\delta \approx [E R_{\rm A}(t_0)]^{-1/4}$.) Since $ER_{\rm
A}(t_0)\gg 1$, the total probability is exponentially small.
Fluctuations greater than the Planck energy cannot be considered until
the horizon has grown large enough to contain a black hole of energy
$E$. During the period $E/\epsilon\lesssim t\lesssim
E^{5/3}/\epsilon$, a fluctuation of energy $E$ is most likely to occur
in the form of a black hole. But this power-law time interval is
insufficient to overcome the exponential suppression in
Eq.~(\ref{eq-pr}), so the fluctuation is extremely unlikely to occur
during this period. Thereafter, the thermal ensemble begins to
dominate, and the fluctuation rate is given by Eq.~(\ref{eq-pe}).
Then the analysis of the previous paragraph applies, with
$t_0=E^{5/3}/\epsilon$. Since $ER_{\rm A}(t_0)$ is again large, the
integrated probability remains negligible for all times.
We conclude that large energy fluctuations inevitably occur in de
Sitter space (if only after an exponentially large time), guaranteeing
the destruction of any observer. In Q-space, however, the temperature
falls monotonically. After it drops below a given energy $E$,
fluctuations of that energy become virtually impossible.
\subsection{Large entropy fluctuations in de~Sitter space}
\label{sec-dsentropy}
What is the probability for a state of specified entropy $S$ to be
radiated by the de Sitter horizon? We have seen in the previous
subsection that the probability of fluctuations is mainly determined
by their energy; the entropy factor in Eq.~(\ref{eq-pe}) turned out to
be negligible. Hence the question is, what is the lightest object
with entropy $S$?
For $S\lesssim R_0^{6/5}$ the lightest object is a thermal state with
temperature $\tau\approx S^{1/3}/R_0$ and energy $E\approx
S^{4/3}/R_0$.\footnote{At least it is the lightest object that we are
sure exists. If a lighter object has the same entropy,
Eqs.~(\ref{eq-pst}), (\ref{eq-pps}) and (\ref{eq-ppps}) still
provide lower bounds on its rate of production, leaving our
conclusions intact.} It is radiated with a probability derived from
Eq.~(\ref{eq-pe}):
\begin{equation}
\frac{P_S}{R_0} = \exp(S-\alpha S^{4/3})~.
\label{eq-pst}
\end{equation}
(Here $\alpha$ is a numerical coefficient involving Stefan's constant
and the effective number of species with mass below $\tau$; for small
species number, it will be on the order of $10$.) For $S\gtrsim
R_0^{6/5}$ the lightest object is a black hole with radius
$R_{\rm B}=(S/\pi)^{1/2}$. Thus Eq.~(\ref{eq-pr2}) implies
\begin{equation}
\frac{P_S}{R_0} = \exp[-(\pi S)^{1/2} R_{\rm C}]~.
\label{eq-psb}
\end{equation}
From Eq.~(\ref{eq-radii}) it follows that the suppression becomes
stronger if $S$ is increased at fixed cosmological constant. Since
these rates are constant, the situation is similar to the case of
large-energy fluctuations: All events that can occur in de~Sitter
space, will occur.
However, there is an absolute entropy bound in de~Sitter
space~\cite{Ban00,Bou00a}. There are no states with entropy greater
than that of the horizon of empty de~Sitter, $\pi R_0^2$. This bound
refers to the combined entropy of the cosmological horizon and of the
matter it encloses. If we ask about the entropy only of systems
contained {\em within\/} the horizon, the limit is more stringent:
there can be no objects with entropy greater than that of the Nariai
black hole ($\pi R_0^2/3$). Fluctuations with greater entropy cannot
occur; their probability is exactly zero. This fundamentally limits
the complexity and accuracy of experiments in de Sitter space.
\subsection{Large entropy fluctuations in Q-space}
\label{sec-qentropy}
In Q-space the horizon grows linearly. A fluctuation of entropy $S$
first becomes possible (in the form of a maximal black hole) when the
horizon reaches a radius of order $S^{1/2}$. Thus, the rate begins at
$e^{-S}$, the suppression of a Nariai black hole. Thereafter the
required black hole radius remains constant. But the corresponding
radius of the cosmological horizon, $R_{\rm C}$, increases as the
effective cosmological constant decreases, according to
Eq.~(\ref{eq-psb}). Hence the fluctuation becomes more and more
unlikely. Eventually the horizon radius satisfies
\begin{equation}
R_{\rm A}\gtrsim S^{5/6}~.
\end{equation}
In this regime, the lightest object of entropy $S$ is an ordinary
thermal state. For all remaining time, the rate of such a fluctuation
is given by Eq.~(\ref{eq-pst}).
What matters about this asymptotic rate is not its (miniscule) value,
but that it is both constant and non-zero, however large one chooses
$S$. It depends on $R_{\rm A}$ only in that $R_{\rm A}$ is the time
interval for which $P_S$ represents the probability of one
fluctuation. Hence the integrated probability ${\cal P}$ for a
fluctuation of entropy $S$ diverges logarithmically:
\begin{eqnarray}
{\cal P}(S) & = & \int_{t_0}^{\infty} dt\,\frac{P_S}{R_{\rm A}}
\label{eq-pps}\\
& \leq & \epsilon^{-1} \exp(S-\alpha S^{4/3})
\int_{S^{5/6}}^\infty \frac{dR_{\rm A}}{R_{\rm A}} \to\infty.
\end{eqnarray}
This is a remarkable result: objects of any complexity, no matter how
large, will eventually be emitted by the horizon. The key observation
is that at fixed entropy, there are many ``scaling states'' whose
energy is inversely proportional to their linear size. As the horizon
grows, these states become energetically cheaper at the same rate as
the temperature drops, leaving their probability invariant. This
restricts consideration to massless fields at late times.\footnote{In
our estimates, this restriction is implemented by using the number
of {\em massless\/} species in the thermodynamic formulas for the
energy and entropy of a thermal cavity. Note that other types of
universes will also have only massless particles at late times, if
massive particles are unstable or processed by black holes.}
Indeed, the stronger statement holds that each scaling microstate
individually is produced with certainty:
\begin{equation}
{\cal P}(|i\rangle) = \epsilon^{-1}\exp(-\alpha S^{4/3})
\int \frac{dR_{\rm A}}{R_{\rm A}} \to\infty~.
\label{eq-ppps}
\end{equation}
This includes highly structured, irregular configurations.
\section{Asymptotic observables}
\label{sec-discussion}
In this section we will compare the various cosmological solutions
studied above, with an eye on the complexity and precision of
measurements that can be achieved, and on the possibility of defining
exact asymptotic observables or an S-matrix.
By an asymptotic observable, we mean any quantity that can be measured
with arbitrary precision at sufficiently late times. We expect that
asymptotic observables exist only in spacetimes where experiments of
arbitrarily long duration can be made and an arbitrarily large amount
of entropy can be accessed. An S-matrix is a special case of an
asymptotic observable, consisting of matrix elements between the
complete initial and final asymptotic states of a closed isolated
system.
The limitations on observation discussed here are imposed by
fundamental aspects of the cosmological solution, such as its causal
and thermal properties, and its information content. This allows us
to proceed without any assumptions about the nature of experiments or
observers.\footnote{Additional restrictions may arise, for example,
from a limited supply of free energy or inability to harvest this
energy for experiments (see, e.g.,
Refs.~\cite{Dys79,Wit01,KraSta00,BusAda03}). To the extent that
they are insurmountable, they may further constrain the asymptotic
observables.}
\subsection{De~Sitter}
Asymptotically de Sitter spacetimes ($w=-1$) are particularly hostile
to observers. There is no S-matrix, since the observer's causal
diamond misses almost all of the asymptotic regions, $\eta\to 0$ and
$\eta\to\pi$ in the global metric (\ref{eq-global}), on which the
global in and out-states might be defined. An S-matrix between such
states would, at best, be a ``meta-observable''~\cite{Wit01}: It would
relate a state no one can set up (because of the past event horizon
$\eta=\chi$) to a state no one can measure (because of the future
event horizon $\eta=\pi-\chi$). This conclusion does not improve if
the past asymptotic region is replaced by a big bang; this only trades
the past event horizon for a particle horizon, and does not affect the
future event horizon.
Nor are there any other asymptotic observables in asymptotically
de~Sitter space. The total accessible entropy is bounded
(Sec.~\ref{sec-class}), and the duration of any experiment
fundamentally limited by thermal erosion (Sec.~\ref{sec-typ}) and
by collisions with black holes (Sec.~\ref{sec-dsenergy}).
\subsection{Q-space}
Like de~Sitter space, quintessence dominated universes ($-1<w<-1/3$)
have a cosmological event horizon~\cite{HelKal01,FisKas01} (see
Sec.~\ref{sec-frw}). Hence, the global state in the asymptotic future
cannot be measured, and there is no S-matrix.
However, some other asymptotic observables may well exist. We have
seen in the previous section that Q-space is significantly more
welcoming to physicists than de~Sitter space. Thermal
fluctuations\footnote{We should emphasize again that the thermal
properties of Q-space discussed in Secs.~\ref{sec-temp} and
\ref{sec-fluc} were rigorously derived only in the limit of small
$\epsilon$ ($w\to -1$). But we expect no qualitative transitions at
least in the range $-1<w<-2/3$.} are present but are too weak to
terminate experiments by erosion (Sec.~\ref{sec-typ}) or by black hole
production (Sec.~\ref{sec-qenergy}). Because the cosmological horizon
becomes arbitrarily large~\cite{HelKal01}, there is no absolute
entropy bound. What we showed in Sec.~\ref{sec-qentropy} is that an
unbounded number of different states are actually produced at late
times. Thus, observers can experience arbitrarily complex events
(and, one might imagine, store large amounts of information for long
times).\footnote{Whether these fluctuations, which involve only
massless fields, can give rise to an apparatus capable of precise
measurements is another question, and we do not claim to have proven
that this will happen.}
\subsection{Decelerating FRW}
Decelerating universes ($-1/3<w<1$) clearly satisfy important
conditions for the existence of asymptotic observables, as noted by
several authors~\cite{BanFis01a,HelKal01,FisKas01,Wit01,FreSus04}. As
the particle horizon grows, the amount of entropy allowed in the
causal diamond increases without bound, as does its actual matter
content (Sec.~\ref{sec-class}). This may include massive particles,
if they are stable and if they are not converted to radiation by black
holes.
But is there an S-matrix? At first sight, the
situation looks promising. There is no future event horizon
(Sec.~\ref{sec-frw}). Every timelike geodesic eventually enters the
causal diamond of the observer. But this does not mean that the
global state of the universe is observable.
At any finite time, only a finite portion of the universe is in the
observer's causal diamond, by Eq.~(\ref{eq-finite}). Beyond lies a
non-compact region, which has {\em at all times\/} infinite volume (as
measured on the homogeneous spacelike slices). This unobserved region
contains an infinite amount of matter and, potentially, an infinite
amount of information. Thus, the decelerating universe never reveals
more than an infinitely small fraction of itself to the observer (see
Fig.~\ref{fig-penfrw}).
Whether all or only part of a system is measured, makes an enormous
difference. Page~\cite{Pag93} has shown that in order to obtain at
least one bit of information about a system in a typical pure state,
one must perform a measurement on more than half of the degrees of
freedom constituting the system. Thus, even a measurement of one half
of a finite system by the other half will reveal practically no
information whatsoever about the global state. The situation in a
flat FRW universe is far more problematic yet: the number of degrees
of freedom available for measurement are finite, and the total system
is infinite.
Let us compare this to a real S-matrix experiment, such as the
scattering of particles in an accelerator. Here, the entire system
hits the detector by some finite time.\footnote{The fact that the
complete system is observable does not rely on any limiting
procedure. The usual limit of late times and large detectors is
taken only in order to refine the separation of particles and make
sure that they have stopped interacting. This is a separate
requirement related to our preference for expressing the out-state
in a convenient Hilbert space basis (the Fock space of the
noninteracting theory).} The key difference is that in
asymptotically flat space, there exists a region near spatial infinity
(i.e., outside a sufficiently large sphere) that is devoid of matter
and energy. Entropy bounds, such as the Bekenstein bound and the
generalized covariant bound~\cite{Bek81,FMW,Bou03}, imply that this
region contains no information. In an FRW solution, on the other
hand, the density at fixed time is asymptotically constant and
non-zero. In this case, entropy bounds permit an arbitrarily large
information content. Therefore, the asymptotic structure of an FRW
universe does not guarantee that an S-matrix exists.
The situation could be improved by restricting to a set of states such
that the (infinite) exterior of some finite region contains either no
information or only redundant information. (See, e.g.,
Ref.~\cite{FreSus04} for an approach to constructing an appropriate
reference state.) We emphasize that this is a strong additional
constraint. The appropriate states would form a set of measure zero
in the Hilbert space of states of the FRW universe. It is possible,
but not obvious, that suitable states are selected as initial
conditions by theory.
\subsection{Discussion: Cosmology vs.\ the S-matrix}
We have found that the entropy of observable matter is unbounded in
any flat FRW universe dominated by a $w>-1$ fluid in the asymptotic
future, accelerating or not. In particular, we conclude that a future
event horizon does not in itself impose a significant restriction.
Its absence is neither necessary for the existence of asymptotic
observables nor sufficient for the existence of an S-matrix. Indeed,
we argued that an S-matrix is not a natural observable in any of the
cosmologies considered.
We based our argument on the combination of two observations: the
late-time global state of the universe is never fully contained in any
observer's causal past; and unlike asymptotically flat space, the
unobserved portion of a cosmological universe can contain
information---in the case of decelerating universes, it contains an
infinite number of degrees of freedom. Page's theorem~\cite{Pag93}
then implies that no information can be gleaned about a generic global
pure quantum state, if by information we mean finding a density matrix
of sub-maximal entropy.
This is just a particularly bad version of a more general problem that
arises whenever one part of a closed system measures another part.
This includes any measurement of the global state of the universe,
independently of causal restrictions. Obviously, the apparatus must
have at least as many degrees of freedom as the system whose quantum
state it attempts to establish (in practice it usually has orders of
magnitude more). This means that at most half of the degrees of
freedom can be observed, just below the Page cutoff for obtaining the
first bit of information about the complete system.
Aside from the problem of measuring the out-state, an S-matrix
description of cosmology is emptied of operational meaning by our
inability to control the initial state and to repeat experiments that
extend over cosmological time and distance scales.
It is conceivable that these problems could be
circumvented.\footnote{I would like to thank L.~Susskind for
discussions of this issue.} Suppose for example that theory
restricts to states with a high degree of spatial symmetry and no
entanglement between distant degrees of freedom. In an infinite
universe without event horizons, it might then be possible to perform
independent but equivalent measurements on an arbitrary number of
disentangled identical subsystems. At all times, however, the
construction of a global state would still require infinite
extrapolation. In any case, this procedure will not resemble an
S-matrix experiment.
If not an S-matrix, what asymptotic observables should one expect to
find? In any large spacetime, local high-energy scattering
experiments will admit an S-matrix description to a good
approximation. But we are interested in measurements probing the
state and the evolution of the universe. We are fortunate to inhabit
a fairly symmetric (part of the) universe, and we can learn aspects of
its early quantum fluctuations by measuring correlations in the cosmic
microwave background. They are limited by cosmic variance, but as our
horizon grows we can obtain more and more data points. In
decelerating universes such correlations may become exact quantum
observables in the asymptotic future. In accelerating universes, one
would measure correlations in fluctuations of the approximately
thermal radiation from the horizon.
Whether or not asymptotic observables exist, they do not correspond
precisely to the experiments we perform {\em today}. Even in an
asymptotically empty spacetime (suppose, for instance, that our
universe is a mere resonance in a giant scattering event in
asymptotically flat space), one would require approximate local
observables to describe measurements by parts of the system on one
another at finite time. They would seem just as likely to be
computable from an initial state directly, than to be derived from
S-matrix. (Indeed, the former option has a chance of applying in
general spacetimes, whereas the latter requires assumptions about the
asymptotic structure.) We do not even know whether such bulk
observables are quantum mechanical.
\section{Other universes}
\label{sec-other}
The discussion above was restricted to flat FRW universes, which may
admit some asymptotic observables though not, in any operational
sense, an S-matrix. In this section, we extend the discussion to
other universes. We focus in particular on the Farhi-Guth solution,
which connects a cosmological region, through the interior of a black
hole, to an asymptotically flat or AdS spacetime, raising anew the
question of S-matrix observables for cosmology.
\subsection{Crunching universes}
Many other cosmological solutions do not admit
asymptotic observables at all.
This includes all universes with a big crunch, such as closed FRW
solutions with decelerating matter content. Open and flat FRW
solutions can also crunch, even if they are initially expanding, if
they contain negative vacuum energy~\cite{KalLin99}. In this case,
spatial infinity exists. However, the largest causally accessible
region has finite entropy~\cite{CEB1,CEB2}, so there will be no exact
observables.
There are also spacetimes that do not crunch globally but in which all
observers must end up inside a black hole. In a dust-dominated flat
universe, for example, gravitational collapse can occur at arbitrarily
large scales. Our own universe was probably produced by a period of
inflation. Then the assumption of an infinite homogeneous flat FRW
universe is not appropriate at very large scales. Assuming a generic
chaotic inflation potential, the density fluctuations produced by
inflation grow logarithmically with the scale. Exponentially many
years from now, the fluctuations entering the horizon will be of order
unity. On some scales this will lead to large voids, on other scales
to overdensities. Sooner or later, any given observer will find
themselves in a large region bound to collapse into a black hole.
\subsection{Asymptotically expanding, non-flat FRW universes}
Open universes have a similar causal structure to flat universes. The
main difference is that in the accelerating (decelerating) cases, the
future conformal boundary is isomorphic only to a portion of the
infinity of de Sitter (Minkowski) space. Hence, the spatial infinity
of an open universe is a conformal two-sphere rather than a point.
The discussion of observables in flat FRW universes, in
Sec.~\ref{sec-discussion}, applies to open universes as well. In the
accelerating case ($w<-1/3$), this is because the spatial curvature
vanishes at late times anyway. In the decelerating case, asymptotic
observables may exist, since the particle horizon diverges; but the
global state of the universe is again unobservable. Closed Q-space
also asymptotes locally to flat Q-space, as the curvature inflates
away.
\subsection{Coleman-De Luccia}
One might hope to gain control over a poorly behaved universe, such as
de Sitter space, by grafting onto it another solution which possesses
asymptotic observables or perhaps even an S-matrix. An example of
such a hybrid is the Coleman-De Luccia (CDL) solution~\cite{ColDel80},
which describes a bubble of true vacuum expanding inside a false
vacuum. Of particular interest is the case where the true vacuum
energy vanishes (so the false one is positive). The expanding portion
of this solution describes an open expanding FRW universe joined to a
meta-stable de Sitter space (Fig.~\ref{fig-penfrwd}). Unlike pure
de~Sitter space, which appears to allow no exact observables, the CDL
solution should therefore admit the same asymptotic observables that
can be defined in an open FRW universe.\footnote{One can also consider
the fully extended CDL solution, which contains a collapsing FRW
universe in the past. Freivogel and Susskind~\cite{FreSus04}
recently proposed that asymptotic observables defined in the two FRW
regions encode information not only about the metastable de Sitter
region, but even about other stable and metastable
vacua~\cite{BouPol00,KKLT} in causally disconnected regions. This
raises a number of subtle questions~\cite{BouFre05} which are
outside the scope of this paper.}
\begin{figure}
\includegraphics[width=8.5cm]{penfrwd}
\caption{\label{fig-penfrwd}
A scalar field potential involving a false vacuum (left) gives rise
to the Coleman-De Luccia solution, which describes a de Sitter
region joined to an open FRW universe, in our example with vanishing
vacuum energy. A conformal diagram of the expanding portion of this
spacetime is shown on the right. Some examples of the orbits of the
symmetry group $O(3,1)$ are shown: the domain wall, the light-cone
starting at $P$, and the hyperbolic time slice that includes
$Q$.---The region enclosed in the dotted circle is asymptotically
identical with a corresponding region of the Farhi-Guth solution
(Fig.~5).}
\end{figure}
It is important that the $\Lambda=0$ region of the CDL solution is an
FRW cosmology~\cite{ColDel80} and not, as one might have expected,
empty Minkowski space. That is, it contains infinite hyperbolic
slices with constant positive energy density. For later use, we
briefly review where this energy comes from.
The true and false vacuum can be modeled by a scalar field potential
with a local minimum at $\phi_+$ and a global minimum at $\phi_-$
(Fig.~\ref{fig-penfrwd}). The domain wall of the CDL solution is a
spherical shell in which the field $\phi$ crosses the barrier between
the vacua. Consider the closed, time-symmetric slice on which the
domain wall radius takes its minimum value. If the energy of the
false vacuum is small compared to the height of the barrier, then the
thickness of the domain wall will be small compared to its radius.
This limit is known as the thin-wall approximation.
Inside the wall, the field value is approximately $\phi_-$, differing
from the exact vacuum value only by an amount exponentially small in
the distance from the wall. The wall is only of finite size, so at
its center (at the event $P$ in Fig.~\ref{fig-penfrwd}) the field
value $\phi_0$ still differs from $\phi_-$ by an exponentially small
amount. (This can be avoided only by infinite fine-tuning of the
potential.) Hence, the energy density at $P$ is not exactly zero. By
continuity, the energy density will also be nonzero at some point $Q$
infinitesimally later than $P$. The $O(3,1)$ symmetry of the CDL
solution guarantees that $Q$ is equivalent to all other events on the
spatial hyperbolic slice generated as its orbit. Hence, the entire
infinite slice has constant positive energy density. This is what
distinguishes a cosmology from asymptotically flat space.
By the same token, if the true vacuum has negative cosmological
constant, the bubble interior will not be Anti-de Sitter space. It
will again be an open FRW solution~\cite{ColDel80,Ban02}. Like any FRW
solution with negative cosmological constant and an admixture of
$w>-1/3$ matter~\cite{KalLin99}, it will only expand for a finite
amount of time and then collapse in a big crunch.
\subsection{Farhi-Guth}
The Coleman-DeLuccia solution is a limiting case of a larger class of
domain wall solutions with spherical symmetry, found by Blau,
Guendelman, and Guth~\cite{BlaGue87} (see also
Refs.~\cite{AurDen85,BerKuz83}). This class includes another
composite cosmology, the Farhi-Guth solution\footnote{We refer to it
by the authors of Ref.~\cite{FarGut87}, who investigated features of
this solution, to distinguish it from the larger class of which it
is a special case.}, which does contain a true asymptotically flat
region. This suggests that it may allow the description of cosmology
using an S-matrix. A similar solution can be constructed with
Anti-de~Sitter asymptotics; it has been suggested that aspects of the
cosmological regions could thus be described via the AdS/CFT
correspondence~\cite{AlbLow99}.\footnote{I have enjoyed discussions
with B.~Freivogel, V.~Hubeny, M.~Rangamani, and S.~Shenker, who are
independently investigating questions that overlap with some of the
topics in this subsection~\cite{FreHub05}.}
This possibility has met with some scepticism (see, e.g.,
Ref.~\cite{Ban02}). Here we demonstrate a feature of the Farhi-Guth
solution which has not, to our knowledge, been previously noted in the
literature: the fact that it contains an open, asymptotically FRW
universe with a black hole. We will argue that this exacerbates the
difficulties with using the Farhi-Guth solution for an S-matrix
description of cosmology.
\paragraph*{Global structure}
The Farhi-Guth solution describes an expanding bubble of de Sitter
space topologically ``inside'' an asymptotically flat universe. The
only way~\cite{FarGut87} this can be achieved is to place the de
Sitter bubble on the far side of a black hole/white hole region, as
shown in Fig.~\ref{fig-penfrwe}. To allow for quick orientation, let
us call this the ``cosmological side'', separated by an Einstein-Rosen
bridge from the ``asymptotic side''. Points on opposite sides are
necessarily spacelike separated, so one cannot travel between them.
Note that the cosmological side is very similar to the CDL solution:
it contains a meta-stable de~Sitter region separated by a domain wall
from a region of vanishing cosmological constant.
\begin{figure}
\includegraphics[width=8.5cm]{penfrwe}
\caption{\label{fig-penfrwe}
Conformal diagram of the fully extended Farhi-Guth solution. In the
text it is shown that the region marked by the dotted circle agrees
asymptotically with the marked region in Fig.~4; this implies that
it contains an open universe. The above diagram corresponds to the
case of vanishing cosmological constant in the true minimum; for
negative cosmological constant, the same argument shows that the
open universe crunches.}
\end{figure}
At late times, the Farhi-Guth bubble grows large, and its wall will be
far from the black hole. Thus, it can be expected to behave
asymptotically like the CDL domain wall. We will now verify this.
The dynamics of the Farhi-Guth wall is governed by the Israel junction
conditions, which yield the equation~\cite{FarGut90}
\begin{equation}
(\frac{dr}{d\tau}) ^2 + V(r,q) =-1\ .
\end{equation}
Here, $r$ is the radius of the bubble, $\tau$ is the proper time on
the bubble trajectory, and $q$ is the radius of the black hole. The
potential $V(r,q)$ is given by
\begin{equation}
V(r,q) = -\frac{q}{r} -
\frac{[q-(\chi^2+\kappa^2) r^3]^2}{4\kappa^2 r^4}~,
\end{equation}
where $\chi$ is the Hubble scale of the meta-stable de~Sitter region,
and $\kappa/4\pi$ is the surface tension of the domain wall. These
latter quantities are determined by the shape of the scalar field
potential.
The CDL solution corresponds to setting $q=0$, so $V(r)= -
\frac{(\chi^2+\kappa^2)^2 r^2}{4\kappa^2}$. This admits a growing and
a decaying exponential solution. The particular, time-symmetric
initial conditions of CDL select the linear combination
\begin{equation}
r = r_0 \cosh \frac{\tau}{r_0}~,
\end{equation}
where $r_0 = 2\kappa/(\chi^2+\kappa^2)$~.
The Farhi-Guth solution is more complicated but we are interested only
in the large radius limit. For $r\to\infty$ one finds that $V(r,q)\to
- \frac{(\chi^2+\kappa^2)^2 r^2}{4\kappa^2}$, which coincides with the
$q\to 0$ limit. Hence, growing bubbles are all attracted to the CDL
solution at late times, independently of the black hole mass:
\begin{equation}
r \to C \exp \frac{\tau}{r_0}~,
\end{equation}
Differences in the prefactor can be absorbed into a shift of the time
variable $\tau$.
This universality has an important consequence: the formation of an
expanding FRW universe outside the bubble will also be universal. In
the CDL case, we exploited the full $O(3,1)$ symmetry to show that the
domain wall dumps constant energy density into the hyperbolic slices.
The details of this mechanism, and the character of the matter it
produces, will vary depending on the potential and couplings. The
Farhi-Guth solution has only spherical symmetry. But at large radius
the initial conditions for the hyperbolic slices are provided mainly
by the domain wall. They are identical to those in the CDL solution,
so the same mechanism will operate, creating asymptotically hyperbolic
slices with asymptotically constant, positive energy density.
Hence, the region of vanishing vacuum energy on the cosmological side
of the Farhi-Guth solution is not empty flat space but is again a
cosmology. It is asymptotic to an open FRW universe. The only
difference to the CDL solution is that the asymptotic FRW universe
contains a black hole, though which it connects to an asymptotically
flat region.
\paragraph*{Discussion}
The Farhi-Guth solution has it all: an asymptotically flat or AdS
region, meta-stable de~Sitter space, a black hole, an open FRW
cosmology, and in the case of AdS asymptotics, even a big crunch.
Could it be that all these interesting regions are described by an
S-matrix, or CFT correlators, defined on the asymptotic
side?\footnote{See Refs.~\cite{Ban02,Ban04} for other discussions of
this issue.}
The region behind the horizon is causally inaccessible to an observer
at infinity. To argue that information about the cosmological side
can be retrieved on the asymptotic side, one would have to appeal
either to black hole complementarity~\cite{AlbLow99} or to subtle
effects of analyticity~\cite{KraOog03,FidHub04}. But complementarity
is a stronger conjecture here than in the case of classical black hole
formation, since the matter on the cosmological side was never present
on the asymptotic side.
Let us restate this in the language of holographic
screens~\cite{CEB2}. For a black hole formed in asymptotically flat
space by the collapse of matter, the past null infinity is a screen
whose light-sheets reach inside the black hole, covering all of the
spacetime. They are appropriate for a description of the infalling
observer; all information that went into the black hole can
unambiguously be stored there. If this trivial fact was not true,
then the assumption that the S-matrix is unitary would not require
that the same information be present at future infinity, and we would
not be led to complementarity. In the Farhi-Guth solution, the
light-sheets off of the asymptotic boundary enter the black hole/white
hole region, but they cannot reach into the cosmological regions on
the far side of the black hole.
A signal that complementarity may not work here is the fact that the
black hole area can be made arbitrarily small, while the de~Sitter
region on the cosmological side can have large entropy. This
contrasts with the usual case, where the black hole area grows large
in response to matter crossing the horizon, and becomes small only as
it returns matter to the asymptotic region in the form of Hawking
radiation.
The infinite open FRW universe, which we have argued is always present
on the cosmological side, exacerbates the entropy mismatch, leaving
little hope that it could be resolved by some unknown macroscopic
constraint on the solutions. The cosmological side would have to be
in one of a small number of very special microstates, if it were to
be described by boundary data defined on the asymptotic side. Another
logical possibility would be that the boundary theory makes more
degrees of freedom available for the cosmological side of the black
hole than it does for the black hole itself; this would lead to a
breakdown of the UV/IR correspondence of AdS/CFT~\cite{SusWit98}.
These entropic considerations are complemented by an aesthetic
objection: If nature had wanted us to use a boundary theory to
describe cosmology, it would have given the universe a nicer boundary.
After all, the Farhi-Guth solutions are rather artificial constructs.
Their description reads like a cocktail recipe: A de Sitter region
separated by a domain wall from an open universe containing an eternal
black hole, on the far side of whose Einstein-Rosen bridge resides the
desired asymptotically flat region. (The idea of the Einstein-Rosen
bridge is logically independent of the de~Sitter region; for example,
one could connect any FRW universe to an asymptotic region this way.)
Unlike the CDL solution, which arises naturally from the decay of
false vacuum, spacetimes containing an eternal black hole are not of
obvious physical relevance. They cannot be created classically from
regular initial conditions~\cite{FarGut87}. It has been argued that
the Farhi-Guth geometry might arise
semiclassically~\cite{FarGut90,FisMor90a,FisMor90b} from a small
bubble of false vacuum, but no regular instanton exists for this
process. Moreover, it is not clear how an observer would distinguish
this type of transition from other exponentially rare events resulting
in the spontaneous formation of a black hole.
In summary, it is questionable whether Farhi-Guth solutions exist in a
full quantum theory of gravity; and even if they do, holographic
considerations suggest that the asymptotic and the cosmological side
will be completely decoupled. That said, we are unable to rule out
that some aspects of cosmological evolution are encoded in boundary
data via the Farhi-Guth solution.
\paragraph*{Acknowledgements}
I would like to thank T.~Banks, B.~Freivogel, V.~Hubeny, N.~Kaloper,
A.~Linde, A.~Mints, J.~Polchinski, M.~Rangamani, S.~Shenker,
L.~Susskind, and E.~Witten for discussions. This work was supported
by the Berkeley Center for Theoretical Physics, by a CAREER grant of
the National Science Foundation, and by DOE grant DE-AC03-76SF00098.
\bibliographystyle{board}
|
2,877,628,090,081 | arxiv | \section{Introduction}
\label{sec_introduction}
\subsection{The RAND Health Insurance Experiment}
In the 1970's, the challenge of financing and delivering high-quality and affordable health care to all Americans was at the center of national policy debate.
At the time, two central questions were ``How much more medical care would people use if it is provided free of charge?'' and ``What are the consequences of using more medical care on their health?''
To address these and other related questions, an interdisciplinary team of researchers led by Joseph P. Newhouse at RAND designed and conducted the Health Insurance Experiment (HIE), a large-scale, multi-year, randomized public policy experiment developed and completed between 1971 and 1982.
To this day, the HIE is one of the largest and most comprehensive social science experiments ever conducted in the U.S.
Even now, four decades after its completion, evidence from the HIE is still fundamental to the national discussion on health care cost sharing and health care reform.
In the HIE, a representative sample of 2,750 families comprising more than 7,700 individuals were chosen from six urban and rural sites across the United States.
At the beginning of the study, participants completed a baseline survey providing numerous demographic, medical, and socioeconomic measurements.
Families were then assigned to health insurance plans that varied substantially in their coinsurance rates and out-of-pocket expenditure maxima, for a total of 13 possible treatment groups. The goal of the study was to estimate the marginal averages of utilization and health outcomes in each of the six sites under each plan.
To make evidence on health utilization and outcomes as strong as possible, the study had to be randomized. However, achieving balance for numerous continuous and categorical covariates through randomization is challenging in contexts with so many treatment groups and implementation sites.
\subsection{Toward balanced, efficient, and robust experimental designs}
Randomized experiments are considered to be the gold standard for causal inference, as randomization provides an unequivocal basis for both inference and control.
In randomized experiments, the act of randomization ensures \textit{balance} on both observed and unobserved covariates on average.
However, a given realization of the random assignment mechanism may produce substantial imbalances on one or more covariates.
This imbalance problem can be exacerbated in settings like the HIE,
where treatments are multi-valued and many baseline covariates exist, leading to loss in efficiency of the effect estimates.
A variety of methods have been proposed in the literature to address this problem, such as blocking (\citealt{fisher1925statistical}, \citealt{fisher1935design}, \citealt{cochran1957experimental}), optimal pair-matching (\citealt{greevy2004optimal}), greedy pair-switching (\citealt{krieger2019nearly}),
and designs using mixed-integer programming (\citealt{bertsimas2015power}).
In particular, rerandomization (\citealt{morgan2012rerandomization}) has gained popularity over the last few years and has become commonplace in experiments.
However, rerandomization may not protect against and be robust to chance imbalances in functions of the covariates that are not explicitly addressed by the rerandomization criterion \citep{banerjee2017decision}, especially in experiments with multi-valued ($>$2) treatments.
Defining the rerandomization criterion requires selection of a tuning parameter governing the acceptable degree of imbalance, which may require iteration in practice.
Moreover, rerandomization rules out imbalanced assignments ex post, which may complicate inference (\citealt{athey2017econometrics}).
To overcome these and other related challenges, we consider the Finite Selection Model (FSM) for experimental design. The original version of the FSM was proposed and developed by Carl Morris in the design of the HIE (\citealt{morris1979finite}, \citealt{newhouse1993free}, \citealt{morris1993the}).
The idea behind the FSM is that each treatment group takes turns in a fair and random order to select units from a pool of available units such that, at each stage, each treatment group selects the unit that maximally improves the combined quality of its current group of units.
The criterion for measuring quality is flexible, and in this paper, we develop a new criterion based on the concept of D-optimality, which does not require tuning parameters.
To illustrate, Figure \ref{fig:simu_boxplot} exhibits the performance of complete randomization, rerandomization, and the FSM in a version of the HIE data with four treatment groups and 20 covariates.
For rerandomization we compute the maximum Mahalanobis distance (using the 20 covariates) across all possible pairs of treatment groups and accept 0.1\% of the assignments with the smallest covariate distance (see Section \ref{sec_hiedata} for details).
The figure displays the distribution of absolute standardized mean differences (ASMD; \citealt{rosenbaum1985constructing})\footnote{The absolute standardized mean difference for a single covariate $X$ between treatment groups $g$ and $g'$ is $\text{ASMD}(X) = {|\bar{X}_g - \bar{X}_{g'} |}/{\sqrt{(s^2_g + s^2_{g'})/{2}}}$, where $\bar{X}_g$ and $s^2_g$ are the mean and variance of $X$ in treatment group $g$, respectively.
Please see \cite{rosenbaum1985constructing}) for details.} in covariates and their second order transformations across multiple realizations of the randomization mechanisms for the three designs.
Lower values of ASMD indicate better balance on the covariates (or transformations thereof).
We observe that rerandomization substantially outperforms complete randomization in terms of imbalances on the main covariates, but not in terms of their squares and interactions.
In contrast, the FSM markedly outperforms both methods without requiring tuning parameters.
This analysis reveals that, while rerandomization performs well by common standards (the majority of the ASMD is smaller than 0.1), there is room for improvement.
As we explain in Section \ref{sec_thehie}, in experiments like the HIE the space of possible assignments is vast and the FSM can improve the assignment of units into treatment groups to achieve better covariate balance.
Better balance can improve the validity and credibility of a study, and also translates into increased efficiency and robustness.
\begin{figure}[!ht]
\centering
\includegraphics[scale =0.45]{graphics/hie_boxplot_wide.pdf}
\caption{Distributions of ASMD for complete randomization, rerandomization, and the FSM, for 20 baseline covariates in the HIE data. Without tuning parameters, the FSM handles multiple ($>$2) treatment groups and substantially improves covariate balance and efficiency.}
\label{fig:simu_boxplot}
\end{figure}
\subsection{Contribution and outline}
In this paper, we revisit, formalize, and extend the FSM for experimental design.
We show that the FSM can be used for balanced, efficient, and robust random treatment assignment, outperforming common assignment methods on these three dimensions.
In particular, we describe the FSM under the potential outcomes framework (\citealt{neyman1923application}, \citealt{rubin1974estimating}).
We use the sequentially controlled Markovian random sampling (SCOMARS, \citealt{morris1983sequentially}) algorithm to determine the selection order of treatments for two-group experiments and develop its extensions to multi-group experiments.
We propose a new selection criterion for treatments based on the idea of D-optimality and discuss its theoretical properties.
In particular, we show that the FSM using the D-optimal selection function is affine invariant and achieves near-exact balance on a class of covariate transformations. The FSM using the D-optimal selection function is also shown to retrieve several classical designs such as randomized block and matched-pair designs.
We analyze the FSM's performance both theoretically and empirically and compare it to common assignment methods.
We discuss model-based approaches to inference to the FSM and develop randomization-based alternatives.
In addition, we discuss potential extensions of the FSM to more complex experimental design settings, such as stratified experiments and experiments with sequential arrival of units in batches.
In an accompanying paper \citep{chattopadhyay2021randomized},
we describe how these methods can be implemented in the new \texttt{FSM} package for \texttt{R}, which is publicly available on CRAN.
The paper proceeds as follows. In Section \ref{sec_hiedesign} we describe the design of the RAND Health Insurance Experiment, focusing on the assignment of each family to a single one of 13 health insurance plans.
In Section \ref{sec_foundations} we present the setup, notation, and main components of the FSM.
In Section \ref{sec_theD}, we propose a selection criterion based on D-optimality and analyze its theoretical properties.
In Section \ref{sec_inference} we discuss inference under the FSM.
In Section \ref{sec_simulation}, we evaluate the performance of the FSM and compare it to standard methods such as complete randomization and rerandomization.
In Section \ref{sec_thehie}, we perform a similar comparison using the HIE data.
Finally, in Section \ref{sec_practical} we consider extensions of the FSM to other settings such as multi-group, stratified, and sequential experiments.
In Section \ref{sec_summary} we conclude with a summary and remarks.
\section{Design of the Health Insurance Experiment}
\label{sec_hiedesign}
In the HIE, families were assigned to different health insurance plans using the original version of the FSM \citep{morris1979finite}. Initially, assignments were made in each of the six HIE sites to 12 or 13 fee-for-service plans with varying combinations of coinsurance (cost sharing) rates and income-related deductibles. Coinsurance plans consisted of $0\%$ (free care), $25\%$, $50\%$, or $95\%$ coinsurance rates, plus a plan with mixed coinsurance rates, and an individual deductible plan. Within the cost sharing plans, families were further assigned to different out-of-pocket maxima where the out-of-pocket expenditures were capped at 5\%, 10\%, or 15\% of family income, with an annual maximum of \$1,000 \citep{brook2006health}. To ensure that the treatment groups across the insurance plans were balanced relative to the overall population, the FSM considered a discard group of study non-participants as an additional treatment group in its assignment process.
The HIE spanned six U.S. sites tracked over several years, listed here in chronological order of study initiation: Dayton, OH; Seattle, WA; Fitchburg, MA; Franklin County, MA; Charleston, SC; and Georgetown County, SC. The FSM was used, independently in each of the sites, to make random assignments to improve balance on up to 22 family-level baseline covariates across treatment groups. In each of the first two sites, the FSM was used multiple times for separate independent subsets of families to maintain baseline data schedules.
In addition to estimating overall marginal effects of health insurance plan design on healthcare utilization and outcomes, the HIE team also sought to understand how particular design choices would affect experimental results. Specifically, in each HIE site, the team conducted four additional randomized sub-experiments to estimate the impact of alternative choices addressing the following questions: (i) which families would undergo shorter enrollment durations (three years or five); (ii) which would receive participation incentives (yes or no); (iii) which would receive pre-experimental physician visits (yes or no); (iv) and which would have higher interviewing frequency (weekly or biweekly) \citep{newhouse1993free}. For each of these four sub-experiments, after the insurance treatments were determined, families were randomized to the sub-treatment groups using the FSM.
\section{Foundations and overview of the FSM}
\label{sec_foundations}
\subsection{Setup and notation}
\label{sec_setup}
Consider a sample of $N$ units indexed by $i = 1, ..., N$.
Each of these units is to be assigned into one of $G$ treatment groups labelled by $g$, with $g = 1, ..., G$.
Write $n_g$ for the pre-specified size of group $g$.
Denote $Z_i \in \{1, 2, ..., G\}$ as the assigned treatment group label of unit $i$ and $\bm{Z} = (Z_1,...,Z_N)^\top$ as the vector of treatment group labels.
Following the potential outcomes framework for causal inference \citep{neyman1923application, rubin1974estimating}, each unit $i$ has a potential outcome under each treatment $g$, $Y_i(g)$, but only one of these outcomes is observed: $Y^{\text{obs}}_i = \sum_{g = 1}^{G} \mathbbm{1}(Z_i = g) Y_i(g)$. Denote $\bm{Y}(g) = (Y_1(g),...,Y_N(g))^\top$ as the vector of potential outcomes under treatment $g$.
Each unit has a vector of $K$ observed covariates, $\boldsymbol{X}_{i}$.
We write $\underline{\bm{X}}_{\text{full}}$ for the $N \times k$ matrix of observed covariates and $\bar{\bm{X}}_{\text{full}}$ and $\underline{\bm{S}}_{\text{full}}$ for the mean vector and covariance matrix of these covariates in the full sample, respectively.
For reference, in Table \ref{tab_notation} of the Online Supplementary Materials we provide a list of the notation used in this paper.
Based on this notation, $Y_i(g') - Y_i(g'')$ is the causal effect of treatment $g'$ relative to treatment $g''$ for unit $i$.
We are interested in estimating the sample average treatment effect $\text{SATE}_{g',g''} = \frac{1}{N}\sum_{i=1}^{N} \{Y_i(g') - Y_i(g'') \}$ and the population average treatment effect $\text{PATE}_{g',g''} = \mathbb{E} [Y_i(g') - Y_i(g'')]$.
For this, we will randomly assign the units into treatment groups using the FSM.
\subsection{Components of the FSM}
\label{sec_components}
In the FSM, the $G$ treatment groups take turns to select units in a random but controlled order while optimizing a certain criterion.
This is accomplished by means of a \emph{selection order matrix} (SOM), which determines the order in which the treatment groups select the units, and a \emph{selection function}, which provides the optimality criterion.
A good SOM guarantees that the selection of units is fair, so that no single treatment group selects all the units of a given type, and random, so that both observed and unobserved covariates are balanced in expectation and there is a basis for inference.
A good selection function will produce efficient and robust inferences under a wide class of possible outcome functions.
To illustrate, Table \ref{tab1}(a) presents an example data set with 12 observations and one covariate, age. We consider assigning these 12 units into two groups of equal sizes using the FSM. Table \ref{tab1}(b) shows an example of an SOM in this setting. The SOM determines the order in which each treatment selects a unit at each stage. In the example, treatment group 2 selects first in stage 1, treatment group 1 selects in stage 2, and so on. Treatment groups select units based on the selection function.
In general, it is crucial that the order of selection is random, but that no group chooses in a disproportionate manner.
For two treatment groups of arbitrary sizes, this can be accomplished by means of the Sequentially Controlled Markovian Random Sampling (SCOMARS) algorithm \citep{morris1983sequentially}.
In the FSM, SCOMARS specifies the probability of a treatment group selecting at stage $r$ ($r \in \{1,2,...,N\}$), conditional on the number of selections made by that group up to stage $r-1$. See the Online Supplementary Materials for a formal description of the algorithm. SCOMARS satisfies the sequentially controlled condition (\citealt{morris1983sequentially}), which requires the deviation of the observed number of selections made by a treatment group up to stage $r$ from its expectation to be strictly less than one. Intuitively, this condition ensures that throughout the selection process, no treatment group departs too much from its expected fair share of choices. Moreover, SCOMARS is Markovian because for each group, the probability of selection at stage $r$ depends solely on the number of selections made up to stage $r-1$.\footnote{In fact, SCOMARS is the \textit{unique} randomized algorithm for generating an SOM that is both Markovian and sequentially controlled.}
For two groups of equal sizes (as in the example in Table \ref{tab1}), generating an SOM under SCOMARS boils down to successively generating $N/2$ independent random permutations of the treatment labels $(1, 2)$.
In Section \ref{sec_multi} we describe this and other extensions of SCOMARS to multi-group experiments. Unless otherwise specified, in the rest of the paper, we will use SCOMARS to generate the SOM for experiments with two treatment groups.
The selection function gives a value to each of the units available for selection at each stage.
This value depends on the characteristics of each available unit in addition to those already assigned to the choosing treatment group.
In principle, any criterion can be used in the selection function.
For example, if the selection function is constant, units are randomly assigned.
Alternatively, the selection function can compute the contribution of each unit to a measure of accuracy of the estimators. In this spirit, we propose the \textit{D-optimal} selection function, which, at each stage, minimizes the generalized variance of the estimated regression coefficients in a linear potential outcome model.
To build intuition, we discuss the special case of $k=1$ covariate. With the D-optimal selection function, the choosing group, in its first choice, selects the unit whose covariate value is farthest from the full-sample mean of the covariate; and in the subsequent choices, selects the unit whose covariate value is farthest from its current mean of the covariate. In the example in Table \ref{tab1}, treatment $2$ selects unit $1$ with age $24$, the farthest age from the full-sample mean $43$. In the next stage, treatment $1$ selects unit $12$ with age $60$, the farthest age from $43$. Next, treatment $1$ selects unit $2$ with age $30$, the farthest age from its current mean age $60$. The process continues until all the 12 units are selected.
\begin{singlespacing}
\begin{table}[H]
\caption{(a) Example data set; (b) selection order matrix and an assignment using the FSM.}
\begin{subtable}{.45\linewidth}
\centering
\caption{\footnotesize Data set}
\scalebox{0.65}{
\setlength\extrarowheight{-2pt}
\begin{tabular}{cc}
\toprule
Index & Age \\
\hline
1 & 24 \\
2 & 30 \\
3 & 34 \\
4 & 36 \\
5 & 40 \\
6 & 41 \\
7 & 45 \\
8 & 46 \\
9 & 50 \\
10 & 54 \\
11 & 56 \\
12 & 60 \\
\hline
Mean & 43 \\
\bottomrule
\end{tabular}
}
\end{subtable}
\begin{subtable}{.45\linewidth}
\centering
\caption{\footnotesize Selection order matrix and assignment}
\scalebox{0.65}{
\setlength\extrarowheight{-2pt}
\begin{tabular}{ccccc}
\toprule
\multicolumn{2}{c}{Selection order matrix} & \multicolumn{3}{c}{Unit selected}\\
\cmidrule(r){1-2} \cmidrule(r){3-5}
Stage & Treatment & Index & Age\\
\toprule
1 & 2 & 1 & 24 \\
2 & 1 & 12 & 60 \\
3 & 1 & 2 & 30 \\
4 & 2 & 11 & 56 \\
5 & 1 & 3 & 34 \\
6 & 2 & 10 & 54 \\
7 & 1 & 9 & 50 \\
8 & 2 & 4 & 36 \\
9 & 1 & 5 & 40 \\
10 & 2 & 8 & 46 \\
11 & 2 & 6 & 41 \\
12 & 1 & 7 & 45 \\
\bottomrule
\end{tabular}
}
\end{subtable}
\label{tab1}
\end{table}
\end{singlespacing}
\section{The D-optimal selection function}
\label{sec_theD}
\subsection{Definition and behavior}
\label{sec_definition}
In this section, we formally define the D-optimal selection function and provide an equivalent, closed-form characterization that explains how this selection function governs the selection of units at each stage.
Without loss of generality, we assume that treatment 1 gets to select at the $r$th stage, $r \in \{1,2,...,N\}$.
Let $\tilde{n}_{r-1}$ be the number of units already belonging to treatment group 1 after the $(r-1)$th stage.
We denote $\mathcal{R}_{r-1}$ as the remaining set of unselected units after the $(r-1)$th stage. Let $\bar{\bm{X}}_{r-1}$ and $\underline{\bm{S}}_{r-1}$ be the mean vector and covariance matrix of the covariates in treatment group 1, respectively, after the $(r-1)$th stage. Also, let $\underline{\tilde{\bm{X}}}_{r-1}$ be the $\tilde{n}_{r-1} \times (k+1)$ design matrix in treatment 1 after the $(r-1)$th stage.\footnote{The design matrix includes a column of all 1's (corresponding to the intercept) and $k$ columns of covariates.} Finally, let $\underline{\tilde{\bm{X}}}_{\text{full}}$ be the design matrix in the full sample. We assume that $\underline{\tilde{\bm{X}}}_{\text{full}}$ has full column rank.
To define the selection function, we implicitly consider a linear potential outcome model of $Y_i(1)$ on $\bm{X}_i$, i.e., $Y_i(1) = \bm{\beta}^\top (1, \bm{X}_i^\top)^\top + \eta_i$, where $\eta_i$ is an error term satisfying $\mathbb{E}[\eta_i|\bm{X}_i] = 0$.\footnote{More generally, one can consider a linear model of $Y_i(1)$ on a vector of basis functions $\bm{B}(\bm{X}_i)$ of the covariates.} For unit $i \in \mathcal{R}_{r-1}$, let $\underline{\tilde{\bm{X}}}_{r,i}$ be the resulting design matrix in treatment group 1 if unit $i$ is selected. We first consider the case where $\tilde{n}_{r-1} \geq 1$ (i.e., treatment 1 has made at least one selection) and $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is invertible. The D-optimal selection function selects unit $i' \in \mathcal{R}_{r-1}$, where $i' \in \arg\!\max\limits_{i \in \mathcal{R}_{r-1}}\det(\underline{\tilde{\bm{X}}}^\top_{r,i}\underline{\tilde{\bm{X}}}_{r,i})$.\footnote{Ties in the values of $\det(\underline{\tilde{\bm{X}}}^\top_{r,i}\underline{\tilde{\bm{X}}}_{r,i})$ are resolved randomly.} In other words, at the $r$th stage, the D-optimal selection function chooses the unit among $\mathcal{R}_{r-1}$ that optimally decreases the generalized variance of the estimated regression coefficients of the fitted linear model in treatment 1. In Lemma \ref{lemma:dopt} in the Online Supplementary Materials, we show that
maximizing $\det(\underline{\tilde{\bm{X}}}^\top_{r,i}\underline{\tilde{\bm{X}}}_{r,i})$ is equivalent to maximizing $(1, \bm{X}^\top_i) (\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1})^{-1} \begin{psmallmatrix}
1\\
\bm{X}_i
\end{psmallmatrix}$, where $\bm{X}_i$ is the covariate vector of a the $i$th unit in $\mathcal{R}_{r-1}$.
In stages where $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is not invertible, we augment $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ by a scalar multiple of $\underline{\tilde{\bm{X}}}^\top_{\text{full}} \underline{\tilde{\bm{X}}}_{\text{full}}$ (akin to ridge augmentation) and consider the objective function
$(1, \bm{X}^\top_i) \Big(\frac{1}{\tilde{n}_{r-1}}\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1} + \frac{\epsilon}{N} \underline{\tilde{\bm{X}}}^\top_{\text{full}} \underline{\tilde{\bm{X}}}_{\text{full}} \Big)^{-1} \begin{psmallmatrix}
1\\
\bm{X}_i
\end{psmallmatrix}$, where $\epsilon >0$ is a fixed constant. Finally, when $\tilde{n}_{r-1} = 0$ (i.e., treatment 1 has not made any selections yet), the objective function takes the form $(1, \bm{X}^\top_i) \Big( \underline{\tilde{\bm{X}}}^\top_{\text{full}} \underline{\tilde{\bm{X}}}_{\text{full}} \Big)^{-1} \begin{psmallmatrix}
1\\
\bm{X}_i
\end{psmallmatrix}$.
In Theorem \ref{thm:mahal}, we provide an equivalent characterization of the D-optimal selection function that provides more insight into the selection made by the choosing treatment group at each stage.
\begin{theorem}\normalfont
Let treatment 1 be the choosing group at the $r$th stage. The D-optimal selection function chooses unit $i'$ with covariate vector $\bm{X}_{i'} \in \mathbb{R}^k$, where
\begin{equation*}
i' \in \arg\!\max\limits_{i \in \mathcal{R}_{r-1}} (\bm{X}_i - \bar{\bm{X}}^*_{r-1})^\top (\underline{\bm{S}}^*_{r-1})^{-1} (\bm{X}_i - \bar{\bm{X}}^*_{r-1}),
\end{equation*} \begin{singlespacing}
where {\small
\begin{equation*}
\bar{\bm{X}}^*_{r-1} =
\begin{cases}
\bar{\bm{X}}_{\text{full}} & \text{if $\tilde{n}_{r-1} = 0$}\\
\frac{\bar{\bm{X}}_{r-1}+\epsilon\bar{\bm{X}}_{\text{full}}}{1+\epsilon} & \text{if $\tilde{n}_{r-1} \geq 1$ and $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is not invertible}\\
\bar{\bm{X}}_{r-1} & \text{if $\tilde{n}_{r-1} \geq 1$ and $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is invertible}
\end{cases}
\end{equation*}
}
and {\small
\begin{equation*}
\underline{\bm{S}}^*_{r-1} =
\begin{cases}
\underline{\bm{S}}_{\text{full}} & \text{if $\tilde{n}_{r-1} = 0$}\\
(\frac{1}{\tilde{n}_{r-1}}\underline{\bm{X}}_{r-1}^\top \underline{\bm{X}}_{r-1} + \frac{\epsilon}{N} \underline{\bm{X}}_{\text{full}}^\top \underline{\bm{X}}_{\text{full}}) - (1+\epsilon)\bar{\bm{X}}^*_{r-1}\bar{\bm{X}}^{*\top}_{r-1} & \text{if $\tilde{n}_{r-1} \geq 1$ and $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is not invertible}\\
\underline{\bm{S}}_{r-1} & \text{if $\tilde{n}_{r-1} \geq 1$ and $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is invertible.}
\end{cases}
\end{equation*}
}
\end{singlespacing}
\label{thm:mahal}
\end{theorem}
Theorem \ref{thm:mahal} shows that at every stage, the D-optimal selection function selects that unit among the remaining pool whose covariate vector maximizes a type of Mahalanobis distance. In its first choice, treatment 1 aims to maximize the Mahalanobis distance from the covariate distribution in the full sample (in particular, from $\bar{\bm{X}}_{\text{full}}$), thereby choosing the most outlying unit available in the full sample. For the subsequent stages where $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is not invertible, treatment 1 aims to maximize the Mahalanobis distance from a mixture covariate distribution between treatment group 1 and the full sample, where $\epsilon$ determines the mixing rate. Finally, the latter selections by treatment 1 aim at maximizing the Mahalanobis distance from the covariate distribution in treatment group 1. Therefore, with every selection, treatment 1 maximizes the overall separation of the covariates from its current mean, which helps increase the efficiency of the estimated regression coefficients.
\subsection{Properties}
\label{sec_properties}
By definition, the D-optimal selection function improves the estimation accuracy of the fitted linear model in each treatment group by sequentially minimizing the generalized variance of the estimated regression coefficients. With the D-optimal selection function, we can also establish several additional desirable properties of the FSM. In particular, Theorem \ref{thm:dopt_properties} leverages the connection between D-optimality and Mahalanobis distance (as in Theorem \ref{thm:mahal}) and presents two key properties of the FSM with the D-optimal selection function.
\begin{theorem}\normalfont
\begin{enumerate}[label=(\alph*)]
\item The FSM with the D-optimal selection function is invariant under affine transformations of the covariate vector.
\item For continuous, symmetrically distributed covariates and two groups of equal size, the FSM with the D-optimal selection function \emph{almost always} produces exact mean-balance on all even transformations of the centered covariate vector.
\end{enumerate}
\label{thm:dopt_properties}
\end{theorem}
It follows from Theorem \ref{thm:dopt_properties}(a) that, for any SOM, the choices made by each treatment group remain unchanged even if the covariate vectors are transformed via an affine transformation (e.g., changing the units of measurement of the covariates).
Therefore, the FSM with the D-optimal selection function self-standardizes the covariates. Thus, without loss of generality, we can assume that the covariates have mean zero in the full sample. In addition, if the covariate vector is symmetrically distributed in the sample, then by Theorem \ref{thm:dopt_properties}(b), the FSM exactly balances even transformations such as the second, fourth order moments, and the pairwise products of the covariates. An implication of Theorem \ref{thm:dopt_properties}(b) is that, for covariates drawn from symmetric continuous distributions (such as the Normal, t, and Laplace distributions), the FSM tends to balance all these transformations because of the approximate symmetry of the covariates in the sample. The choice of the D-optimal selection function is thus robust in the sense that it allows the FSM to balance a family of transformations of the covariate vector by design, without explicitly including them in the assumed linear model nor requiring the specification of tuning parameters.
The FSM with D-optimal selection function is also attractive because it can encompass several classical designs, such as randomized blocked and matched-pair designs. Theorem \ref{thm:retrieve} formalizes this result.
In the traditional randomized block design (RBD), the units are grouped into blocks of size $G$ according to a categorical \textit{blocking variable} and each treatment is randomly applied to exactly one unit within each block (see, e.g., \citealt{cox2000theory}, Section 3.4). Here we consider a more general version of an RBD where the blocks are of size $c \times G$ (where $c$ is a fixed positive integer) and each treatment is applied to $c$ units within each block. This is a special case of a stratified randomized experiment with strata of equal size and equal allocation among treatments per stratum. In a matched-pair design with $G=2$ treatments, similar units are grouped into pairs and each treatment is randomly applied to one unit within each pair. This is also a special case of a stratified randomized experiment with equal allocation per strata, where the size of each stratum equals two.
\begin{theorem}\normalfont
\begin{enumerate}[label=(\alph*)]
\item Consider a setting where $N = cBG$ units belonging to $B$ blocks of equal size are to be randomly assigned into $G$ treatment groups of equal size, where $c$ is a fixed positive integer.
Then, if the linear model in the FSM consists of an intercept and indicators of any $B-1$ levels of the blocking variable, the FSM with the D-optimal selection function produces the same assignment as an RBD.
\item Consider a setting where $N/2$ identical pairs of units in terms of baseline covariates $\bm{X}_i$ are to be assigned into $G = 2$ treatment groups of equal size.
Assume $\bm{X}_i$ is drawn from a continuous distribution.
Then, if the linear model in the FSM consists of the intercept and the covariates $\bm{X}_i$, then the FSM \textit{almost surely} produces the same assignment mechanism as a matched-pair design.
\end{enumerate}
\label{thm:retrieve}
\end{theorem}
In the first setting, Theorem \ref{thm:retrieve}(a) states that, by including the levels of a blocking variable as regressors, the FSM with the D-optimal selection function automatically blocks on that variable. Thus, the FSM retrieves an RBD without explicitly performing separate randomizations within each block.
In the second setting, Theorem \ref{thm:retrieve}(b) states that, by including the covariates as regressors, the FSM with the D-optimal selection function produces the same assignment as a matched-pair experiment, without explicitly performing separate randomizations in each pair. This phenomenon is particularly useful when the sample consists of near-identical twins but that are difficult to identify a priori due to multiple covariates.
\subsection{Connection to A-optimality}
\label{sec_connection}
The original FSM used a criterion based on A-optimality as the selection function (see \citealt{morris1979finite}). In this section, we compare the A-and D-optimal selection functions. The A-optimal selection function requires prespecifying a \textit{policy matrix} $\underline{\bm{P}}_{p \times (k+1)}$ and a corresponding vector of \textit{policy weights} $\bm{w}_{p\times 1}$. Here, $\underline{\bm{P}}$ transforms the original vector of regression coefficients to a vector of $p$ linear combinations that are of policy interest, and $\bm{w}$ assigns weights to each combination according to their importance.
If treatment 1 gets to choose at the $r$th stage, then the A-optimality criterion selects the unit that minimizes the resulting $\text{trace}\Big(\underline{\bm{T}} (\underline{\tilde{\bm{X}}}^\top_{r,i} \underline{\tilde{\bm{X}}}_{r,i})^{-1}\Big)$, where $\underline{\bm{T}} = \underline{\bm{P}}^\top \text{diag}(\bm{w}) \underline{\bm{P}}$.\footnote{For ease of exposition, we only discuss the case where $(\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1})$ is invertible.} Proposition \ref{prop:aopt} shows an equivalent characterization of the A-optimal selection function.
\begin{proposition} \normalfont
Let treatment 1 be the choosing group at the $r$th stage. Assume that $\tilde{n}_{r-1} \geq 1$ and $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is invertible. The A-optimal selection function chooses unit $i'$ with covariate vector $\bm{X}_{i'} \in \mathbb{R}^k$, where
\begin{equation}
i' \in \arg\!\max\limits_{i \in \mathcal{R}_{r-1}}\frac{(1, \bm{X}^\top_i) (\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1})^{-1} \underline{\bm{T}} (\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1})^{-1} \begin{psmallmatrix}
1\\
\bm{X}_i
\end{psmallmatrix}}{1+(1, \bm{X}^\top_i) (\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1})^{-1} \begin{psmallmatrix}
1\\
\bm{X}_i
\end{psmallmatrix}}.
\end{equation}
\label{prop:aopt}
\end{proposition}
The A-optimality criterion provides a family of selection functions depending on $\underline{\bm{P}}$ and $\bm{w}$. In general, the A-optimality criterion is not invariant with respect to affine transformations of the covariate vector. For instance, setting $\underline{\bm{P}} = \bm{I}$ and $\bm{w} = (1,1,...,1)^\top$ produces a selection function that is not affine invariant. On the other hand, setting $\underline{\bm{P}} = \underline{\tilde{\bm{X}}}_{\text{full}}$ and $\bm{w} = (1,1,...,1)^\top$ yields an affine invariant selection function. In fact, for the latter choice, the A-optimal selection function is closely related to the D-optimal selection function. To see this, consider a case where in the selection process, the design matrices in each treatment group scale similarly relative to the design matrix in the full sample. In particular, for treatment 1 (the choosing group at stage $r$), $(\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1})^{-1} = c_r (\underline{\tilde{\bm{X}}}^\top_{\text{full}} \underline{\tilde{\bm{X}}}_{\text{full}})^{-1}$ for some constant $c_r>0$. In this case, the A-optimal selection function chooses unit $i'$ such that
\begin{align}
& i' \in \arg\!\max\limits_{i \in \mathcal{R}_{r-1}}(1, \bm{X}^\top_i) (\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1})^{-1} \begin{psmallmatrix}
1\\
\bm{X}_i
\end{psmallmatrix} \\
& \iff i' \in \arg\!\max\limits_{i \in \mathcal{R}_{r-1}} (\bm{X}_i - \bar{\bm{X}}_{r-1})^\top (\underline{\bm{S}}_{r-1})^{-1} (\bm{X}_i - \bar{\bm{X}}_{r-1}),
\end{align}
which is same as the D-optimal selection function. Hence, in this case, the FSM under the D-optimal and A-optimal selection functions make similar choices of units.
\section{Inference under the FSM}
\label{sec_inference}
Using the FSM we can make both model- and randomization-based inferences. Both modes of inference are feasible for any selection function and any randomized SOM.
In model-based inference, the sample is typically assumed to be drawn randomly from some superpopulation and inference for the PATE is done by modeling the observed outcome distribution conditional on the treatment indicators and the covariates.
For instance, let the potential outcome model under treatment $g$ be $Y_i(g) = \bm{\beta}^\top_{g}\bm{B}(\bm{X}_i) + \epsilon_{ig}$, where $\bm{B}(\bm{X}_i) = \{ B_1(\bm{X}_i),...,B_b(\bm{X}_i) \}^\top$ is a vector of $b$ basis functions of the covariates, and $\epsilon_{ig}$, $i \in \{1,2,...,N\}$ are mutually independent errors, independent of the covariates.
Under this model, $\text{PATE}_{g',g''}$ can be unbiasedly estimated by $\widehat{\text{PATE}}_{g',g''} = \hat{\bm{\beta}}^\top_{g'}\overline{\bm{B}(\bm{X})} - \hat{\bm{\beta}}^\top_{g''}\overline{\bm{B}(\bm{X})}$, where $\overline{\bm{B}(\bm{X})} = \frac{1}{N}\sum_{i=1}^{N}\bm{B}(\bm{X}_i)$ and $\hat{\bm{\beta}}_g$ is the OLS estimator of $\bm{\beta}_g$ obtained by fitting a linear regression of $Y^{\text{obs}}_i$ on $\bm{B}(\bm{X}_i)$ in treatment group $g = g', g''$.
We call this the regression imputation estimator of $\text{PATE}_{g',g''}$.
The standard error of this estimator and the corresponding confidence interval for $\text{PATE}_{g',g''}$ can be obtained using standard OLS theory.
We note that, in model-based inference, the standard errors and confidence intervals do not take into account the randomness stemming from the assignment mechanism.
Moreover, often the regression models proposed at the design stage are considered misspecified and are later modified at the analysis stage by, e.g., incorporating covariates (or transformations thereof) that are deemed important predictors for the outcome.
Due to the balancing properties of the FSM, the regression imputation estimators tend to exhibit sufficient precision even when the model posited by the FSM is misspecified (see sections \ref{sec_definition}, \ref{sec_simulation}, and \ref{sec_thehie}).
In randomization-based inference, the potential outcomes and the covariates are typically considered as fixed and the assignment mechanism is the only source of randomness (see Chapter 2 of \citealt{rosenbaum2002observational} and chapters 5--7 of \citealt{imbens2015causal} for overviews).
Inference for causal effects can be done via exact randomization tests for sharp null hypotheses on unit-level causal effects (\citealt{fisher1935design}), or via estimation under Neyman's repeated sampling approach (\citealt{neyman1923application}).
Under the FSM, randomization tests for sharp null hypotheses can be performed by approximating the distribution of the test statistic through repeated realizations of the FSM.
To illustrate, consider testing the sharp null hypothesis of zero unit-level causal effects, i.e., $H_0: Y_i(2) - Y_i(1) = 0$ for all $i$, at level $\alpha$ using the FSM. While any choice of test statistic preserves the validity of the test, a common choice is the absolute difference-in-means statistic $|\frac{1}{n_2}\sum_{i:Z_i = 2}Y^{\text{obs}}_i - \frac{1}{n_1}\sum_{i:Z_i = 1}Y^{\text{obs}}_i| = |\frac{1}{n_2}\sum_{i:Z_i = 2}Y_i(2) - \frac{1}{n_1}\sum_{i:Z_i = 1}Y_i(1)| = : T(\bm{Z},\bm{Y}(1),\bm{Y}(2))$.
Large values of $T(\bm{Z},\bm{Y}(1),\bm{Y}(2))$ are considered evidence against $H_0$.
Under $H_0$, $Y_i(2) = Y_i(1) = Y^{\text{obs}}_i$ and the vectors of potential outcomes $\bm{Y}(1)$ and $\bm{Y}(2)$ are known and fixed.
The $p$-value of the test is given by $p = P_{H_0}(T(\bm{Z},\bm{Y}(1),\bm{Y}(2))\geq t_{\text{obs}})$, where $t_{\text{obs}}$ is the value of the test statistic for the observed realization of $\bm{Z}$ under the FSM.
We can compute this $p$-value by Monte Carlo approximation, i.e., we generate independent vectors of assignments $\bm{Z}^{(m)} = (Z^{(m)}_1,...,Z^{(m)}_N)^\top$, $m \in \{1,2,...,M\}$ using the FSM and approximate the $p$-value as $\hat{p} = \frac{1}{M}\sum_{m = 1}^{M}\mathbbm{1}\big(T(\bm{Z}^{(m)},\bm{Y}(1),\bm{Y}(2))\geq t_{\text{obs}}\big)$. We reject $H_0$ at level $\alpha$ if $\hat{p}\leq \alpha$.
Similar tests can be applied for more general sharp hypotheses of treatment effects (e.g., dilated and tobit effects; \citealt{rosenbaum2002observational, rosenbaum2010design2}).
We can invert these tests to obtain a confidence interval for the hypothesized effect (\citealt{rosenbaum2002observational}, Section 2.6.1). Moreover, we can get a point estimate of the effect by solving a Hodges-Lehmann estimating equation corresponding to these tests (\citealt{rosenbaum2002observational}, Section 2.7.2).
Finally, under Neyman's approach, we can estimate the sample average treatment effect $\text{SATE}_{g',g''}$ by the difference-in-means statistic.
In particular, for groups of equal size, this difference-in-means statistic is unbiased for $\text{SATE}_{g',g''}$ under the FSM (see Proposition \ref{fsm_prop:unbiased} for a proof).
\section{A simulation study}
\label{sec_simulation}
\subsection{Setup}
We now compare the performance of the FSM to complete randomization and rerandomization in a simulation study.
Here, $N=120$, $G=2$, $n_1 = n_2 = 60$, and $k=6$.
The covariates are generated following the design of \cite{hainmueller2012balancing}:
\begin{equation}
\begin{psmallmatrix}
X_{1}\\
X_{2}\\
X_3
\end{psmallmatrix} \sim
\mathcal{N}_3\left[\begin{psmallmatrix}
0\\
0\\
0
\end{psmallmatrix},\begin{psmallmatrix}
2 & 1 & -1\\
1 & 1 & -0.5\\
-1 & -0.5 & 1
\end{psmallmatrix}\right],\hspace{0.1cm}
X_4 \sim \text{Unif}[-3,3], \hspace{0.1cm} X_5 \sim \chi^2_1, \hspace{0.1cm} X_6 \sim \text{Bernoulli}(0.5). \label{dgp}
\end{equation}
\noindent In this design, $X_4$, $X_5$, and $X_6$ are mutually independent and separately independent of $(X_1,X_2,X_3)^\top$.
We draw a sample of 120 units once from the data generating mechanism in (\ref{dgp}).
Conditional on this sample, we compare four different assignment methods, namely a completely randomized design (CRD), rerandomization with 0.01 acceptance rate (RR 0.01), rerandomization with 0.001 acceptance rate (RR 0.001), and the FSM.
Both RR 0.01 and RR 0.001 use as rerandomization criteria the Mahalanobis distance between the two treatment groups on the original covariates.
The FSM uses a linear potential outcome model on the original covariates and the D-optimal selection function.
For each design we draw 800 independent assignments. The assignments under the FSM are generated using the open source R package \texttt{FSM} available on CRAN.
The total runtime of the FSM for the 800 simulated experiments was about one and a half minutes on a Windows 64-bit computer with an Intel(R) Core i7 processor.
See \cite{chattopadhyay2021randomized} for detailed step-by-step instructions and vignettes on the use of FSM package.
\subsection{Balance}
We evaluate balance on the main and transformed covariates.
Figures \ref{fig:simu_asmd}(a) and \ref{fig:simu_asmd}(b) show density plots of the Absolute Standardized Mean Differences (ASMD; \citealt{rosenbaum1985constructing}, \citealt{stuart2010matching}) of the six main covariates and their second-order transformations (including squares and pairwise products), respectively.
A smaller ASMD for a covariate indicates better mean-balance on that covariate between the two treatment groups.
Figure \ref{fig:simu_asmd}(a) indicates that both rerandomization methods improve balance on the means of the original covariates over CRD.
As expected, the ASMD distribution under RR 0.001 is more concentrated than that of RR 0.01, with 32\% smaller mean ASMD than RR 0.01.
Both the FSM and RR 0.001 have similar distributions of the ASMD with FSM having moderately (9\%) smaller mean ASMD.
See Table \ref{tab_app:simu_asmd_org} in the Online Supplementary Materials for a comparison of the average ASMD of each covariate.
\begin{figure}[H]
\begin{subfigure}{.52\textwidth}
\includegraphics[scale =0.42]{graphics/simu_asmd_org_emp.pdf}
\caption{\footnotesize Main covariates}
\end{subfigure}%
\begin{subfigure}{.52\textwidth}
\includegraphics[scale =0.42]{graphics/simu_asmd_sqint_emp.pdf}
\caption{\footnotesize Squares and pairwise products}
\end{subfigure}
\caption{Distributions of absolute standardized mean differences (ASMD) of the main covariates and all their second-order transformations. In the top right corners the legends present the average ASMD across simulations for the four methods.
On average, the FSM achieves better covariate balance.
In terms of the main covariates, the FSM marginally outperforms RR 0.001.
In terms of the second-order transformations, the FSM substantially outperforms RR 0.001.}
\label{fig:simu_asmd}
\end{figure}
Figure \ref{fig:simu_asmd}(b) shows that the imbalances of covariate transformations are substantially smaller with the FSM than with CRD, RR 0.01, and RR 0.001.
In fact, the FSM achieves a 70\% reduction in the mean ASMD with respect to RR 0.001.
Thus, although the FSM and RR 0.001 exhibit comparable balance in terms of the main covariates, the FSM balances these transformations of the covariates much better than RR 0.001.
This highlights the improved robustness of the FSM against model misspecification, as discussed previously in the context of Theorem \ref{thm:dopt_properties}(b).\footnote{For the FSM, the implicit potential outcome model is the same as the model used to specify the D-optimal selection function. Although rerandomization does not explicitly model the potential outcomes, an implicit model can be conceptualized from the covariates (or transformations thereof) used to construct the Mahalanobis distance.}
Moreover, reducing the tuning parameter of rerandomization from 0.01 to 0.001 yields only 2\% improvement in the mean ASMD.\footnote{In fact, for some covariate transformations, reducing this tuning parameter exacerbates imbalance (see Table \ref{tab_app:simu_asmd_sqint} in the Online Supplementary Materials).}
In Figure \ref{fig:simu_asmd}(b), both RR 0.01 and RR 0.001 often produce ASMD larger than 0.1, and in some cases, larger than 0.5, indicative of substantial imbalances on these covariate transformations.
Under rerandomization, balance on the squares and pairwise products of the covariates can be improved by explicitly incorporating these transformations in the Mahalanobis distance.
For instance, with $k$ continuous covariates, a Mahalanobis distance needs to include $\frac{k(k+3)}{2}$ variables to control the imbalances on the means of all the covariates and their squares and pairwise products.
However, with large $k$, calculating the Mahalanobis distance becomes computationally expensive and, in the extreme case (when $\frac{k(k+3)}{2}>N$), infeasible.
The FSM, by contrast, only requires $k$ main covariates for computing the D-optimality criterion (see Theorem \ref{thm:mahal}) to produce adequate balance on these transformations.
For each method, we also compare balance in the overall correlation structure of the covariates.
Let $\underline{\bm{R}}_g$ denote the sample correlation matrix in group $g$, $g \in \{1,2\}$.
As a measure of imbalance, we consider the Frobenius norm of $\underline{\bm{R}}_1 - \underline{\bm{R}}_2$, denoted by $||\underline{\bm{R}}_1 - \underline{\bm{R}}_2||_F$.\footnote{The Frobenius norm of a matrix is the square root of the sum of squares of all its elements.} Smaller values of $||\underline{\bm{R}}_1 - \underline{\bm{R}}_2||_F$ are indicative of better balance on the correlation matrix of the covariates between the two groups.
Figure \ref{fig:simu_frob_cor} shows the boxplots of the distributions of $||\underline{\bm{R}}_1 - \underline{\bm{R}}_2||_F$.
The FSM outperforms the other three designs with at least 75\% smaller average $||\underline{\bm{R}}_1 - \underline{\bm{R}}_2||_F$.
In particular, among the 800 randomizations, the highest value of $||\underline{\bm{R}}_1 - \underline{\bm{R}}_2||_F$ under FSM is smaller than the corresponding lowest value under the other three designs, indicating that in terms of the correlation structure (and hence the interactions) of the covariates, the least balanced realization of the 800 FSMs exhibits better balance than the best balanced realization of the 800 complete randomizations and rerandomizations.
\begin{figure}[H]
\centering
\includegraphics[scale =0.42]{graphics/simu_frob_cor.pdf}
\caption{Distributions of discrepancies between the correlation matrices of the covariates in the treatment and the control group (as measured by the Frobenius norm, $||\underline{\bm{R}}_1 - \underline{\bm{R}}_2||_F$) across 800 randomizations.
The FSM produces substantially lower discrepancies than the other three methods, indicating markedly improved balance on the correlations of the covariates.}
\label{fig:simu_frob_cor}
\end{figure}
Finally, we evaluate balance on the joint distribution of the covariates.
To this end, we use two recently proposed non-parametric graph-based tests for equality of multivariate distributions (\citealt{agarwal2019distribution}).
In Table \ref{tab_app:simu_mcm} of the Online Supplementary Materials, we show the average p-values of the two tests for each design.
Since each method ensures covariate balance in expectation, the average p-values for both tests are all substantially greater than the typical 0.05 level.
Nevertheless, the average p-value for the FSM is the highest among the four designs, indicating improved covariate balance in aggregate on the joint distributions.
\subsection{Efficiency}
We now compare the efficiency of the methods under both model- and randomization-based approaches to inference.
Under the model-based approach, we consider a potential outcome model where $\mathbb{E}[Y_i(g)|\bm{X}_i]$ is linear in $\bm{X}_i$ (Model A1) and another model where $\mathbb{E}[Y_i(g)|\bm{X}_i]$ is linear in $\bm{X}_i$ and all its second-order transformations (Model A2).
For each potential outcome model, we fit the corresponding observed outcome model by OLS and estimate $\text{PATE}_{2,1}$ using the regression imputation method described in Section \ref{sec_inference}.
Tables \ref{tab:simu_var_model}(a) and \ref{tab:simu_var_model}(b) show the average and maximum model-based standard error (SE) of the regression imputation estimator relative to the FSM across 800 randomizations under the two models.
\begin{singlespacing}
\begin{table}[H]
\caption{Average and maximum model-based standard errors relative to the FSM across randomizations. Under Model A1 (linear model on the main covariates), the FSM and RR exhibit similar performance, improving over CRD. Under Model A2 (linear model on the main covariates and their second-order transformations), the FSM is considerably more efficient than both CRD and RR.}
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model A1}
\scalebox{0.75}{
\begin{tabular}{p{2.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR 0.01 & RR 0.001 & FSM\\
\toprule
Average SE & 1.03 & 1.00 & 1.00 & 1.00 \\
Maximum SE & 1.13 & 1.00 & 1.00 & 1.00 \\
\bottomrule
\end{tabular}
}
\end{subtable}%
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model A2}
\scalebox{0.75}{
\begin{tabular}{p{2.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR 0.01 & RR 0.001 & FSM\\
\toprule
Average SE & 1.39 & 1.27 & 1.26 & 1.00 \\
Maximum SE & 3.61 & 1.97 & 1.80 & 1.00 \\
\bottomrule
\end{tabular}
}
\end{subtable}
\label{tab:simu_var_model}
\end{table}
\end{singlespacing}
Under Model A1, since both rerandomization and the FSM are able to adequately balance the means of the original covariates,
they lead to lower SE (hence, higher efficiency) than CRD.
Across randomizations, the worst case SE under RR 0.01, RR 0.001, and the FSM are 13\% smaller than under CRD.
Under Model A1, the FSM has similar model-based SE as the two rerandomization methods.
However, under Model A2, the FSM uniformly outperforms the other three designs, with a 26\% reduction in average SE and an 80\% reduction in maximum SE than RR 0.001.
This improvement in efficiency can be attributed to the balance achieved by the FSM on the main covariates and their squares and pairwise products.
In sum, when the model assumed at the design stage is correct and is used at the analysis stage, the FSM is as efficient as the two rerandomizations for estimating the treatment effect.
However, when the model assumed at the design stage is misspecified and later corrected by augmenting transformations of the covariates (e.g., squares and pairwise products), the FSM is considerably more efficient and robust than the other designs.
Under the randomization-based approach, we compare the standard errors of the difference-in-means statistic under each design. Following \cite{hainmueller2012balancing}, the potential outcomes are generated using the models: $Y(1) = X_1 + X_2 + X_3 - X_4 +X_5 + X_6 + \eta$, $Y(2) = Y(1)$ (Model B1) and $Y(1) = (X_1 + X_2 + X_5)^2 + \eta$, $Y(2) = Y(1)$ (Model B2), where $\eta \sim \mathcal{N}(0,1)$.
Both generative models satisfy the sharp-null hypothesis of zero treatment effect for every unit and hence, $\text{SATE}_{2,1} = 0$. Conditional on these potential outcomes, $\text{SATE}_{2,1}$ is estimated under each design using the standard difference-in-means estimator. The corresponding randomization-based SE of this estimator is obtained by generating 800 randomizations of the design and computing the standard deviation of the difference-in-means estimator across these 800 randomizations.
Table \ref{tab:simu_var_rand} shows the randomization-based SE of the difference-in-means statistic for $\text{SATE}_{2,1}$ under each model.
\begin{singlespacing}
\begin{table}[H]
\caption{Average randomization-based standard errors relative to the FSM. The standard error for the FSM is 0.2 under Model B1 (linear model on the main covariates) and 0.43 under Model B2 (linear model on the main covariates and their second-order transformations). Especially under Model B2, the FSM is considerably more efficient than both CRD and RR.}
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model B1}
\scalebox{0.75}{
\begin{tabular}{p{1.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR 0.01 & RR 0.001 & FSM\\
\toprule
SE & 2.72 & 1.26 & 1.08 & 1 \\
\bottomrule
\end{tabular}
}
\end{subtable}%
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model B2}
\scalebox{0.75}{
\begin{tabular}{p{1.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR 0.01 & RR 0.001 & FSM\\
\toprule
SE & 5.69 & 4.56 & 4.47 & 1 \\
\bottomrule
\end{tabular}
}
\end{subtable}
\label{tab:simu_var_rand}
\end{table}
\end{singlespacing}
Under Model B1, the potential outcomes depend linearly on the covariates and therefore balancing the means of the covariates improves efficiency.
This is reflected in Table \ref{tab:simu_var_rand} as the FSM has the smallest SE, closely followed by RR 0.001.
Under Model B2, the potential outcomes depend linearly on the squares and pairwise products of the covariates.
By better balancing these transformations, the FSM yields a considerably smaller SE than the other designs.
In particular, under Model B2, the SE under the FSM is 67\% smaller than the SE under RR 0.001.
Therefore, in a similar way as in the model-based approach, in randomization-based approach the FSM exhibits comparable efficiency to rerandomization under correct-specification of the outcome model, and considerable robustness under model misspecification.
\section{The Health Insurance Experiment}
\label{sec_thehie}
\subsection{Data}
\label{sec_hiedata}
We evaluate and compare the performance of the FSM with standard designs using the baseline data of the HIE. To this end, we consider a version of the HIE data presented in \cite{aron2013rand}. This version includes data on six cost sharing plans described in Section \ref{sec_hiedesign}. To make the group sizes more homogeneous, we consider combining the groups with $25\%$, $50\%$, and mixed coinsurance plans. Thus, in our analysis, we have $G = 4$ treatment groups corresponding to $g = 1$, ``free care'' ($n_1 = 564$); $g = 2$, ``$25\%, 50\%$, or mixed coinsurance'' ($n_2 = 456$); $g = 3$, ``$95\%$ coinsurance'' ($n_3 = 372$); and $g = 4$, ``individual deductible'' ($n_4 = 495$). In total, there are $N = n_1 + ... + n_4 = 1,887$ families. We consider assigning all $N$ families to the four treatment groups and hence do not consider a discard group of non-participants.
Moreover, in this version, the units (i.e., families) across five of the six sites are pooled and we consider randomly assigning all the families in this pooled set to the four treatment groups at once. Due to loss of data, the Dayton site is excluded from this analysis.
We consider $k = 20$ family-level baseline covariates, where $X_1, ..., X_5$ are scaled covariates, $X_6, ..., X_{14}$ are binary covariates, and $X_{15}, ..., X_{20}$ are binary covariates indicating missing data (see Table \ref{tab:hie1} for a description of each baseline covariate). Using this data, we compare complete randomization, rerandomization, and the FSM in terms of balance and efficiency. For the FSM, we generate the SOM by first using SCOMARS on the combined groups $\{1,2\}$ and $\{3,4\}$, and then using SCOMARS again to split each combined group into its component groups.
For rerandomization, we use two balance criteria based on Wilks' lambda statistic (\citealt{lock2011rerandomization}, Section 5.2) and the maximum pairwise Mahalanobis distance between any two treatment groups (\citealt{morgan2012rerandomization}).
For each design, we draw 800 independent assignments.
The runtime of each of these assignments with the FSM was less than one minute on a Windows 64-bit laptop computer with an Intel(R) Core i7 processor.
\subsection{Balance}
Figure \ref{fig:test1} displays the ASMD distributions across randomizations for the main covariates and second-order transformations of the scaled covariates in the HIE data.
In both cases we see that the FSM outperforms complete randomization and rerandomization.
While rerandomization balances the main covariates better than complete randomization, this advantage is less marked than in the previous simulation study and disappears for the transformations of the covariates.
In fact, the average imbalances for these transformations are very similar between complete randomization (0.055) and rerandomization (0.052), and with both methods it is common to see imbalances greater than 0.1 ASMD.
With the FSM, however, the average imbalance is less than half (0.02) of those under CRD and RR, and extreme imbalances are non-existent after the assignments.
\begin{figure}[H]
\begin{subfigure}{.52\textwidth}
\includegraphics[scale =0.5]{graphics/hie_asmd_org_emp.pdf}
\caption{\footnotesize Main covariates}
\label{fig:sub1}
\end{subfigure}%
\begin{subfigure}{.52\textwidth}
\includegraphics[scale =0.5]{graphics/hie_asmd_sqint_emp.pdf}
\caption{\footnotesize Squares and pairwise products}
\label{fig:sub2}
\end{subfigure}
\caption{
Distributions of absolute standardized mean differences (ASMD) of the main covariates and their second-order transformations in the HIE data. The legends present the average ASMD across simulations for the four methods.
On average, the FSM substantially outperforms CRD and RR in terms of both the main covariates and their second-order transformations.}
\label{fig:test1}
\end{figure}
A related question is how well the methods balance all second-order features of the joint distribution of the covariates.
Figure \ref{figfrob} provides an answer to this question in the boxplots of the discrepancies between correlation matrices ($||\underline{\bm{R}}_{g} - \underline{\bm{R}}_{g'} ||_F$) across randomizations.
As in the aforementioned second-order transformations, we see a similar performance between complete randomization and rerandomization, which is considerably improved by the FSM with a median about three times smaller.
Arguably, one could improve the performance of rerandomization; for example, by restricting imbalances on these transformations via the rerandomization criterion; however, unlike the FSM, this may incur increased computational cost and may require additional tuning parameters. Moreover, these transformations can also be included in the FSM model, which would then also improve balance on higher order transformations of them.
\begin{figure}[H]
\begin{subfigure}{.52\textwidth}
\includegraphics[scale =0.5]{graphics/hie_frob_cor12_emp.pdf}
\caption{\footnotesize $||\underline{\bm{R}}_1 - \underline{\bm{R}}_2||_F$}
\label{fig:simu_asmd_org}
\end{subfigure}%
\begin{subfigure}{.52\textwidth}
\includegraphics[scale =0.5 ]{graphics/hie_frob_cor34_emp.pdf}
\caption{\footnotesize $||\underline{\bm{R}}_3 - \underline{\bm{R}}_4||_F$}
\label{fig:simu_asmd_sqint}
\end{subfigure}
\caption{Distributions of discrepancies of the correlation matrices of the covariates in the treatment groups of the HIE data across randomizations.
The discrepancies are measured by $||\underline{\bm{R}}_{g} - \underline{\bm{R}}_{g'}||_F$, where $\underline{\bm{R}}_g$ is the sample correlation matrix of the covariates in treatment group $g$ and $||\cdot||_F$ is the Frobenius norm.
The FSM systematically produces lower discrepancies than the other methods, exhibiting substantially improved balance on the correlations of the covariates.
}
\label{figfrob}
\end{figure}
\subsection{Efficiency}
As in the simulation study, we evaluate efficiency under model- and randomization-based approaches to inference.
The main differences between the model- and randomization-based standard errors is that in the model-based approach, the variance calculation does not explicitly take into account the variability arising through the randomization distribution, whereas in the randomization-based approach it does. For illustration, here we consider estimating the average treatment effect of treatment 3 relative to treatment 2, i.e., $\text{SATE}_{3,2}$ and $\text{PATE}_{3,2}$.
Under the model-based approach, we consider two potential outcome models, one that is linear on the main covariates (Model A3), and another that is linear on the main covariates and the second-order transformations of the scaled covariates (Model A4).
The results are summarized in Table 4.
While the performance of the three methods is similar under Model A3, under Model A4 there are substantial differences with the FSM outperforming both complete randomization and rerandomization.
In fact, under Model A4, there is a 13-15\% reduction in the average standard error, and a 50-69\% reduction in the maximum standard error, with the FSM.
\begin{singlespacing}
\begin{table}[H]
\caption{Average and maximum model-based standard errors relative to the FSM across randomizations. Under Model A3 (linear model on the covariates), the FSM is slightly more efficient than RR and CRD. Under Model A4 (linear model on the covariates and their second-order transformations), the FSM is considerably more efficient than CRD and RR.}
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model A3}
\scalebox{0.72}{
\begin{tabular}{p{2.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR Wilks & RR Mahalanobis & FSM\\
\toprule
Average SE & 1.02 & 1.01 & 1.01 & 1.00 \\
Maximum SE & 1.04 & 1.02 & 1.02 & 1.00 \\
\bottomrule
\end{tabular}
}
\end{subtable}%
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model A4}
\scalebox{0.72}{
\begin{tabular}{p{2.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR Wilks & RR Mahalanobis & FSM\\
\toprule
Average SE & 1.15 & 1.13 & 1.13 & 1.00 \\
Maximum SE & 1.69 & 1.50 & 1.55 & 1.00 \\
\bottomrule
\end{tabular}
}
\end{subtable}
\label{tab:hie_var_model}
\end{table}
\end{singlespacing}
Under the randomization-based approach, we consider the generative models $Y(3) = 10 + 2X_1 + 3X_2 + 0.5X_3 + 0.3X_4 + \eta$ (Model B3) and $Y(3) = 10 + 2X_1 + 2X_2X_3 - X_4X_5 + \eta$ (Model B4) where $Y(3) = Y(2)$ and $\eta \sim \mathcal{N}(0,1.5^2)$. Similar to the simulation study, both generative models satisfy the sharp-null hypothesis of zero treatment effect for every unit and hence, $\text{SATE}_{3,2} = 0$. Under each design, $\text{SATE}_{3,2}$ is estimated using the standard difference-in-means estimator and the corresponding randomization-based SE is obtained by generating 800 randomizations and computing the standard deviation of the estimator across these 800 randomizations.
The results are summarized in Table \ref{tab:hie_var_rand}.
In terms of efficiency, we see again a clear advantage of the FSM.
Under Model B3, the average standard errors of rerandomization are 73\% and 83\% larger than the one of the FSM.
Under Model B4, this difference is accentuated and the average standard errors of rerandomization are 236\% and 242\% larger.
\begin{singlespacing}
\begin{table}[H]
\caption{Randomization-based standard errors relative to the FSM. The standard error for the FSM is 0.12 under Model B3 (linear model on the covariates) and 0.67 under Model B4 (linear model on the covariates and their second-order transformations). Under both models, the FSM is considerably more efficient than both CRD and RR.}
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model B3}
\scalebox{0.75}{
\begin{tabular}{p{1.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR 0.01 & RR 0.001 & FSM\\
\toprule
SE & 2.36 & 1.73 & 1.83 & 1 \\
\bottomrule
\end{tabular}
}
\end{subtable}%
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model B4}
\scalebox{0.75}{
\begin{tabular}{p{1.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR 0.01 & RR 0.001 & FSM\\
\toprule
SE & 3.95 & 3.42 & 3.36 & 1 \\
\bottomrule
\end{tabular}
}
\end{subtable}
\label{tab:hie_var_rand}
\end{table}
\end{singlespacing}
\subsection{Explanation}
This analysis illustrates some important differences between the FSM and RR.
First is the criterion for assignment.
While RR uses the Mahalanobis distance, the FSM uses the D-optimality criterion, which, coupled with a suitable SOM, leads to robust assignments under a more general class of potential outcome models.
Second is the use of this criterion.
While RR essentially ``constrains'' the allowable treatment assignments, the FSM ``optimizes'' them toward the criterion.
In essence, RR solves a feasibility problem whereas the FSM solves a maximization problem. Furthermore, the feasibility problem solved by RR depends on the balance threshold, which may be difficult to choose in practice.
A very high value of this threshold bears the risk of accepting an assignment with poor covariate balance, whereas a very low value can be computationally onerous.
Third is the step-wise assignment of units into treatment groups.
While RR assigns all units in one step and then discards imbalanced assignments, the FSM assigns units one at a time in a random but optimal fashion governed by the selection order and the selection criterion.
This difference is crucial because in experiments like the HIE with several treatment groups and many covariates, the space of possible treatment assignments is vast. As shown in our analysis, optimally selecting among these assignments in a step-wise manner can make a substantial difference in terms of balance, efficiency, computational time, and, ultimately, in the use of ever-scarce resources available for experimentation.\footnote{Figures \ref{fig:simu_boxplot} and \ref{fig:test1} show that, although RR does well under common balance standards (the mean differences are systematically lower than the typical threshold of 0.1 ASMD), there is room to select better (more balanced) random treatment assignments, which is achieved by the FSM.}
\section{Practical considerations and extensions}
\label{sec_practical}
\subsection{Multi-group experiments}
\label{sec_multi}
As discussed, the FSM can readily handle experiments with multiple treatment groups.
In so doing, the key methodological consideration is the choice of the SOM.
As in two-group experiments, with $G > 2$ groups, it helps to generate an SOM that is randomized and sequentially controlled, so that at every stage of the random selection process, the number of selections made by each treatment group up to that stage is close to its fair share.
Formally, we say that an SOM is sequentially controlled if $|S_{ig} - F_{ig}|<1$ for all $i \in \{1,2,...,N\}$ and $g \in \{1,2,...,G\}$, where $S_{ig}$ is the number of selections made by group $g$ up to stage $i$ and $F_{ig} := \mathbb{E}[S_{ig}]$ is the corresponding expected number of selections.
While the construction of an SOM that is sequentially controlled for multi-group experiments with arbitrary group sizes is an open problem, such constructions are possible for several practically relevant configurations of the group sizes.
For example, in multi-group experiments with groups of equal size, we can generate an SOM that is sequentially controlled by successively generating random permutations of the group labels $(1, 2, ..., G)$.
See Definition \ref{def_randchunk} and Proposition \ref{prop_randchunk} for a formal statement of this algorithm and a proof.
In multi-group experiments with groups having one of two distinct sizes, we can generate a sequentially controlled SOM by combining the groups of equal size, followed by using SCOMARS on the combined group labels and randomly permuting the component group labels within each combined group (see Theorem \ref{thm:multiscomars1} for details). For example, with $G = 5$ groups of sizes $n_1= n_2 = 10$, $n_3 = n_4 = n_5 = 21$, this algorithm first generates an SOM at the level of the combined groups $\{1,2\}$ (of size 20) and $\{3,4,5\}$ (of size 63), and then splits each combined group into its component groups.
Experiments with multiple groups of two sizes arise, for example, in clinical settings where an exploratory treatment is evaluated with greater precision in a larger group size and conventional treatments are applied to smaller groups of equal size.
Finally, in multi-group experiments with groups of more than two distinct sizes such that when combined by groups of equal size they have the same total size, we can again obtain a sequentially controlled SOM by randomly permuting across and within the combined group labels.
See Theorem \ref{thm:multiscomars2} for details. For example, with $G = 6$ groups of sizes $n_1 = n_2 = 30$, $n_3 = n_4 = n_5 = 20$, $n_6 = 60$, this algorithm first generates an SOM at the level of the combined groups $\{1,2\}$, $\{3,4,5\}$, $\{6\}$ (each of size 60), and then splits each combined group into its component groups.
In practice, for more general group size configurations, one strategy to generate an SOM is to first identify one of the previous three configurations that is structurally similar to the configuration at hand, and then use the corresponding SOM-generating algorithm.
The resulting algorithm may not always be sequentially controlled, but is still likely to produce a well-controlled randomized selection order.
\subsection{Stratified experiments}
\label{sec_stratified}
In stratified experiments, units are grouped into two or more predefined strata, and within each stratum units are randomly assigned into treatment groups.
By Theorem \ref{thm:retrieve}, when the strata are of equal size, the FSM with stratification variables as covariates automatically retrieves a stratified randomized experiment, without explicitly randomizing units within each stratum.
However, in more general stratified experiments, the FSM needs to be extended to explicitly account for the possibly varying stratum sizes and shares of the treatment groups within each stratum.
Here we discuss a family of such extensions of the FSM, which we term as \textit{stratified FSM}.
We consider stratified experiments where the treatment group sizes within each stratum are set by the investigator beforehand. To accommodate the FSM to such experiments, we again need to carefully construct an SOM.
In the stratified FSM, we append the SOM with an additional column of stratum labels, indicating which stratum the treatment group selects from at each stage of the selection process. This column of stratum labels is specified in such a way that by construction, the resulting SOM is consistent with the pre-fixed treatment group sizes within each stratum. To this end, we discuss two potential approaches below.
Conceptually, the most straightforward approach is to generate a separate SOM for each stratum. This is equivalent to setting the column of stratum labels as $(\underbrace{1,...,1}_{m_1},\underbrace{2,...,2}_{m_2},...,\underbrace{S,....,S}_{m_S})^\top$, where $S$ is the number of strata and $m_s$ is the size of $s$th stratum, $s \in \{1,2,...,S\}$.
This approach is easy to implement and can be useful if, e.g., data on each stratum is available at different stages of the experiment, akin to a sequential experiment (see Section \ref{sec_sequential}). However, in this approach, the treatment groups only get to explore the covariate space of a single stratum for a number of successive stages of selection and hence may not make the most efficient choices.
We address this issue with an alternative approach.
For ease of exposition, we consider two strata: 1 and 2. Let $n_{1g}$ and $n_{2g}$ be the (fixed) sizes of treatment group $g \in \{1,2,...,G\}$ in strata 1 and 2, respectively, where $n_{1g} + n_{2g} = n_g$.
In this approach, we first generate a usual SOM with group sizes $n_1,...,n_G$. For $g \in \{1,2,...,G\}$, we then select the order of the strata that treatment $g$ chooses from by running a SCOMARS algorithm with group sizes $n_{1g}$ and $n_{2g}$.
By allowing the treatment groups to select units from different strata in a balanced manner, this approach mimics the unstratified FSM where the covariate space of the entire sample is explored for choosing units.
Also, by design, this approach satisfies the size requirement of each treatment group within each stratum.
\subsection{Sequential experiments}
\label{sec_sequential}
Sequential experiments are experiments where units arrive sequentially in batches, possibly of varying sizes. In this section, we discuss extensions of the FSM to sequential experiments, which we term as \textit{batched FSM}. For simplicity of exposition, we consider assigning units into equal-sized groups. Let $b_j$ denote the size of the $j$th batch. Using random permutations of the group labels, we can generate a sufficiently large SOM and use the first $b_1$ rows as an SOM for the first batch, the next $b_2$ rows as an SOM for the second batch, and so on. Given a new batch and its corresponding SOM, the simplest approach is to treat the batch as a fresh new sample and assign units using the usual FSM. However, this approach completely ignores the covariate information of the units already assigned. Therefore, in general, this approach fails to correct for any existing covariate imbalances among the treatment groups.
To alleviate this, we consider an alternative approach that explicitly takes into account the covariate information of both the current and previous batches. Given a new batch and its corresponding SOM, we run the FSM as if the new batch is a continuation of the previous batch.
In other words, we use the design matrix based on all the units already assigned to the choosing treatment group to evaluate the D-optimal selection function for each unit in the new batch, and select the unit that maximizes the selection function. By carrying over the existing design matrix to the new batch, this approach tends to correct for any existing covariate imbalances.
An important practical consideration for batched FSM is the size of the batches. For a fixed total number of units enrolled in the experiment, increasing the batch size (and hence, reducing the number of batches) tends to increase the overall balance and efficiency properties of the FSM. On one extreme, a batched FSM with a single batch containing all the units enrolled in the experiment reduces to the usual FSM, which tends to have the highest efficiency among all possible batched FSMs. On the other extreme, a batched FSM with multiple batches of size one each is equivalent to CRD, which ignores the covariate information. How to optimally determine the batch sizes for the FSM is an important open question for practice.
\section{Summary and remarks}
\label{sec_summary}
We revisited, formalized, and extended the FSM for experimental design. We proposed a new selection function based on D-optimality that requires no tuning parameters. We showed that, equipped with this selection function, the FSM has a number of appealing properties. First, the FSM is affine invariant and hence, it self-standardizes covariates with possibly different units of measurements. Second, for approximately symmetric data, the FSM yields near-exact balance on a large class of covariate transformations, including transformations that are not part of the assumed linear model under the FSM. Third, the FSM produces randomized block designs without explicitly randomizing in each block. Fourth, the FSM also produces matched-pair designs without explicitly constructing the matched pairs beforehand and randomizing within each pair. We described how both model-based and randomization-based inference on treatment effects can be conducted using the FSM. For a range of practically relevant configurations of group sizes in multi-group experiments, we proposed new algorithms to generate a fair and random selection order of treatments under the FSM. We also discussed potential extensions of the FSM to stratified and sequential experiments.
In a simulation study and a case study on the RAND Health Insurance Experiment, we showed that the FSM is a robust approach to randomization, exhibiting better performance than complete randomization and rerandomization in terms of balance and efficiency.
While there are settings where complete randomization may perform better than the FSM in terms of efficiency, such settings are less common and involve jagged, i.e., highly non-smooth, potential outcome models.
In practice, where investigators believe there is reasonable smoothness in the models, the FSM is expected to perform well.
Overall, through our extensive explorations with real and simulated experimental data, the FSM has consistently stood out as a robust design that can handle multiple treatment groups and a fairly large number of categorical and continuous covariates without requiring tuning parameters and without the need to coarsen covariates.
We recommend giving strong considerations to the FSM in experimental design for its conceptual simplicity, practicality, and robustness.
\vspace{1cm}
\onehalfspacing
\bibliographystyle{asa}
\section{Introduction}
\label{sec_introduction}
\subsection{The RAND Health Insurance Experiment}
In the 1970's, the challenge of financing and delivering high-quality and affordable health care to all Americans was at the center of national policy debate.
At the time, two central questions were ``How much more medical care would people use if it is provided free of charge?'' and ``What are the consequences of using more medical care on their health?''
To address these and other related questions, an interdisciplinary team of researchers led by Joseph P. Newhouse at RAND designed and conducted the Health Insurance Experiment (HIE), a large-scale, multi-year, randomized public policy experiment developed and completed between 1971 and 1982.
To this day, the HIE is one of the largest and most comprehensive social science experiments ever conducted in the U.S.
Even now, four decades after its completion, evidence from the HIE is still fundamental to the national discussion on health care cost sharing and health care reform.
In the HIE, a representative sample of 2,750 families comprising more than 7,700 individuals were chosen from six urban and rural sites across the United States.
At the beginning of the study, participants completed a baseline survey providing numerous demographic, medical, and socioeconomic measurements.
Families were then assigned to health insurance plans that varied substantially in their coinsurance rates and out-of-pocket expenditure maxima, for a total of 13 possible treatment groups. The goal of the study was to estimate the marginal averages of utilization and health outcomes in each of the six sites under each plan.
To make evidence on health utilization and outcomes as strong as possible, the study had to be randomized. However, achieving balance for numerous continuous and categorical covariates through randomization is challenging in contexts with so many treatment groups and implementation sites.
\subsection{Toward balanced, efficient, and robust experimental designs}
Randomized experiments are considered to be the gold standard for causal inference, as randomization provides an unequivocal basis for both inference and control.
In randomized experiments, the act of randomization ensures \textit{balance} on both observed and unobserved covariates on average.
However, a given realization of the random assignment mechanism may produce substantial imbalances on one or more covariates.
This imbalance problem can be exacerbated in settings like the HIE,
where treatments are multi-valued and many baseline covariates exist, leading to loss in efficiency of the effect estimates.
A variety of methods have been proposed in the literature to address this problem, such as blocking (\citealt{fisher1925statistical}, \citealt{fisher1935design}, \citealt{cochran1957experimental}), optimal pair-matching (\citealt{greevy2004optimal}), greedy pair-switching (\citealt{krieger2019nearly}),
and designs using mixed-integer programming (\citealt{bertsimas2015power}).
In particular, rerandomization (\citealt{morgan2012rerandomization}) has gained popularity over the last few years and has become commonplace in experiments.
However, rerandomization may not protect against and be robust to chance imbalances in functions of the covariates that are not explicitly addressed by the rerandomization criterion \citep{banerjee2017decision}, especially in experiments with multi-valued ($>$2) treatments.
Defining the rerandomization criterion requires selection of a tuning parameter governing the acceptable degree of imbalance, which may require iteration in practice.
Moreover, rerandomization rules out imbalanced assignments ex post, which may complicate inference (\citealt{athey2017econometrics}).
To overcome these and other related challenges, we consider the Finite Selection Model (FSM) for experimental design. The original version of the FSM was proposed and developed by Carl Morris in the design of the HIE (\citealt{morris1979finite}, \citealt{newhouse1993free}, \citealt{morris1993the}).
The idea behind the FSM is that each treatment group takes turns in a fair and random order to select units from a pool of available units such that, at each stage, each treatment group selects the unit that maximally improves the combined quality of its current group of units.
The criterion for measuring quality is flexible, and in this paper, we develop a new criterion based on the concept of D-optimality, which does not require tuning parameters.
To illustrate, Figure \ref{fig:simu_boxplot} exhibits the performance of complete randomization, rerandomization, and the FSM in a version of the HIE data with four treatment groups and 20 covariates.
For rerandomization we compute the maximum Mahalanobis distance (using the 20 covariates) across all possible pairs of treatment groups and accept 0.1\% of the assignments with the smallest covariate distance (see Section \ref{sec_hiedata} for details).
The figure displays the distribution of absolute standardized mean differences (ASMD; \citealt{rosenbaum1985constructing})\footnote{The absolute standardized mean difference for a single covariate $X$ between treatment groups $g$ and $g'$ is $\text{ASMD}(X) = {|\bar{X}_g - \bar{X}_{g'} |}/{\sqrt{(s^2_g + s^2_{g'})/{2}}}$, where $\bar{X}_g$ and $s^2_g$ are the mean and variance of $X$ in treatment group $g$, respectively.
Please see \cite{rosenbaum1985constructing}) for details.} in covariates and their second order transformations across multiple realizations of the randomization mechanisms for the three designs.
Lower values of ASMD indicate better balance on the covariates (or transformations thereof).
We observe that rerandomization substantially outperforms complete randomization in terms of imbalances on the main covariates, but not in terms of their squares and interactions.
In contrast, the FSM markedly outperforms both methods without requiring tuning parameters.
This analysis reveals that, while rerandomization performs well by common standards (the majority of the ASMD is smaller than 0.1), there is room for improvement.
As we explain in Section \ref{sec_thehie}, in experiments like the HIE the space of possible assignments is vast and the FSM can improve the assignment of units into treatment groups to achieve better covariate balance.
Better balance can improve the validity and credibility of a study, and also translates into increased efficiency and robustness.
\begin{figure}[!ht]
\centering
\includegraphics[scale =0.45]{graphics/hie_boxplot_wide.pdf}
\caption{Distributions of ASMD for complete randomization, rerandomization, and the FSM, for 20 baseline covariates in the HIE data. Without tuning parameters, the FSM handles multiple ($>$2) treatment groups and substantially improves covariate balance and efficiency.}
\label{fig:simu_boxplot}
\end{figure}
\subsection{Contribution and outline}
In this paper, we revisit, formalize, and extend the FSM for experimental design.
We show that the FSM can be used for balanced, efficient, and robust random treatment assignment, outperforming common assignment methods on these three dimensions.
In particular, we describe the FSM under the potential outcomes framework (\citealt{neyman1923application}, \citealt{rubin1974estimating}).
We use the sequentially controlled Markovian random sampling (SCOMARS, \citealt{morris1983sequentially}) algorithm to determine the selection order of treatments for two-group experiments and develop its extensions to multi-group experiments.
We propose a new selection criterion for treatments based on the idea of D-optimality and discuss its theoretical properties.
In particular, we show that the FSM using the D-optimal selection function is affine invariant and achieves near-exact balance on a class of covariate transformations. The FSM using the D-optimal selection function is also shown to retrieve several classical designs such as randomized block and matched-pair designs.
We analyze the FSM's performance both theoretically and empirically and compare it to common assignment methods.
We discuss model-based approaches to inference to the FSM and develop randomization-based alternatives.
In addition, we discuss potential extensions of the FSM to more complex experimental design settings, such as stratified experiments and experiments with sequential arrival of units in batches.
In an accompanying paper \citep{chattopadhyay2021randomized},
we describe how these methods can be implemented in the new \texttt{FSM} package for \texttt{R}, which is publicly available on CRAN.
The paper proceeds as follows. In Section \ref{sec_hiedesign} we describe the design of the RAND Health Insurance Experiment, focusing on the assignment of each family to a single one of 13 health insurance plans.
In Section \ref{sec_foundations} we present the setup, notation, and main components of the FSM.
In Section \ref{sec_theD}, we propose a selection criterion based on D-optimality and analyze its theoretical properties.
In Section \ref{sec_inference} we discuss inference under the FSM.
In Section \ref{sec_simulation}, we evaluate the performance of the FSM and compare it to standard methods such as complete randomization and rerandomization.
In Section \ref{sec_thehie}, we perform a similar comparison using the HIE data.
Finally, in Section \ref{sec_practical} we consider extensions of the FSM to other settings such as multi-group, stratified, and sequential experiments.
In Section \ref{sec_summary} we conclude with a summary and remarks.
\section{Design of the Health Insurance Experiment}
\label{sec_hiedesign}
In the HIE, families were assigned to different health insurance plans using the original version of the FSM \citep{morris1979finite}. Initially, assignments were made in each of the six HIE sites to 12 or 13 fee-for-service plans with varying combinations of coinsurance (cost sharing) rates and income-related deductibles. Coinsurance plans consisted of $0\%$ (free care), $25\%$, $50\%$, or $95\%$ coinsurance rates, plus a plan with mixed coinsurance rates, and an individual deductible plan. Within the cost sharing plans, families were further assigned to different out-of-pocket maxima where the out-of-pocket expenditures were capped at 5\%, 10\%, or 15\% of family income, with an annual maximum of \$1,000 \citep{brook2006health}. To ensure that the treatment groups across the insurance plans were balanced relative to the overall population, the FSM considered a discard group of study non-participants as an additional treatment group in its assignment process.
The HIE spanned six U.S. sites tracked over several years, listed here in chronological order of study initiation: Dayton, OH; Seattle, WA; Fitchburg, MA; Franklin County, MA; Charleston, SC; and Georgetown County, SC. The FSM was used, independently in each of the sites, to make random assignments to improve balance on up to 22 family-level baseline covariates across treatment groups. In each of the first two sites, the FSM was used multiple times for separate independent subsets of families to maintain baseline data schedules.
In addition to estimating overall marginal effects of health insurance plan design on healthcare utilization and outcomes, the HIE team also sought to understand how particular design choices would affect experimental results. Specifically, in each HIE site, the team conducted four additional randomized sub-experiments to estimate the impact of alternative choices addressing the following questions: (i) which families would undergo shorter enrollment durations (three years or five); (ii) which would receive participation incentives (yes or no); (iii) which would receive pre-experimental physician visits (yes or no); (iv) and which would have higher interviewing frequency (weekly or biweekly) \citep{newhouse1993free}. For each of these four sub-experiments, after the insurance treatments were determined, families were randomized to the sub-treatment groups using the FSM.
\section{Foundations and overview of the FSM}
\label{sec_foundations}
\subsection{Setup and notation}
\label{sec_setup}
Consider a sample of $N$ units indexed by $i = 1, ..., N$.
Each of these units is to be assigned into one of $G$ treatment groups labelled by $g$, with $g = 1, ..., G$.
Write $n_g$ for the pre-specified size of group $g$.
Denote $Z_i \in \{1, 2, ..., G\}$ as the assigned treatment group label of unit $i$ and $\bm{Z} = (Z_1,...,Z_N)^\top$ as the vector of treatment group labels.
Following the potential outcomes framework for causal inference \citep{neyman1923application, rubin1974estimating}, each unit $i$ has a potential outcome under each treatment $g$, $Y_i(g)$, but only one of these outcomes is observed: $Y^{\text{obs}}_i = \sum_{g = 1}^{G} \mathbbm{1}(Z_i = g) Y_i(g)$. Denote $\bm{Y}(g) = (Y_1(g),...,Y_N(g))^\top$ as the vector of potential outcomes under treatment $g$.
Each unit has a vector of $K$ observed covariates, $\boldsymbol{X}_{i}$.
We write $\underline{\bm{X}}_{\text{full}}$ for the $N \times k$ matrix of observed covariates and $\bar{\bm{X}}_{\text{full}}$ and $\underline{\bm{S}}_{\text{full}}$ for the mean vector and covariance matrix of these covariates in the full sample, respectively.
For reference, in Table \ref{tab_notation} of the Online Supplementary Materials we provide a list of the notation used in this paper.
Based on this notation, $Y_i(g') - Y_i(g'')$ is the causal effect of treatment $g'$ relative to treatment $g''$ for unit $i$.
We are interested in estimating the sample average treatment effect $\text{SATE}_{g',g''} = \frac{1}{N}\sum_{i=1}^{N} \{Y_i(g') - Y_i(g'') \}$ and the population average treatment effect $\text{PATE}_{g',g''} = \mathbb{E} [Y_i(g') - Y_i(g'')]$.
For this, we will randomly assign the units into treatment groups using the FSM.
\subsection{Components of the FSM}
\label{sec_components}
In the FSM, the $G$ treatment groups take turns to select units in a random but controlled order while optimizing a certain criterion.
This is accomplished by means of a \emph{selection order matrix} (SOM), which determines the order in which the treatment groups select the units, and a \emph{selection function}, which provides the optimality criterion.
A good SOM guarantees that the selection of units is fair, so that no single treatment group selects all the units of a given type, and random, so that both observed and unobserved covariates are balanced in expectation and there is a basis for inference.
A good selection function will produce efficient and robust inferences under a wide class of possible outcome functions.
To illustrate, Table \ref{tab1}(a) presents an example data set with 12 observations and one covariate, age. We consider assigning these 12 units into two groups of equal sizes using the FSM. Table \ref{tab1}(b) shows an example of an SOM in this setting. The SOM determines the order in which each treatment selects a unit at each stage. In the example, treatment group 2 selects first in stage 1, treatment group 1 selects in stage 2, and so on. Treatment groups select units based on the selection function.
In general, it is crucial that the order of selection is random, but that no group chooses in a disproportionate manner.
For two treatment groups of arbitrary sizes, this can be accomplished by means of the Sequentially Controlled Markovian Random Sampling (SCOMARS) algorithm \citep{morris1983sequentially}.
In the FSM, SCOMARS specifies the probability of a treatment group selecting at stage $r$ ($r \in \{1,2,...,N\}$), conditional on the number of selections made by that group up to stage $r-1$. See the Online Supplementary Materials for a formal description of the algorithm. SCOMARS satisfies the sequentially controlled condition (\citealt{morris1983sequentially}), which requires the deviation of the observed number of selections made by a treatment group up to stage $r$ from its expectation to be strictly less than one. Intuitively, this condition ensures that throughout the selection process, no treatment group departs too much from its expected fair share of choices. Moreover, SCOMARS is Markovian because for each group, the probability of selection at stage $r$ depends solely on the number of selections made up to stage $r-1$.\footnote{In fact, SCOMARS is the \textit{unique} randomized algorithm for generating an SOM that is both Markovian and sequentially controlled.}
For two groups of equal sizes (as in the example in Table \ref{tab1}), generating an SOM under SCOMARS boils down to successively generating $N/2$ independent random permutations of the treatment labels $(1, 2)$.
In Section \ref{sec_multi} we describe this and other extensions of SCOMARS to multi-group experiments. Unless otherwise specified, in the rest of the paper, we will use SCOMARS to generate the SOM for experiments with two treatment groups.
The selection function gives a value to each of the units available for selection at each stage.
This value depends on the characteristics of each available unit in addition to those already assigned to the choosing treatment group.
In principle, any criterion can be used in the selection function.
For example, if the selection function is constant, units are randomly assigned.
Alternatively, the selection function can compute the contribution of each unit to a measure of accuracy of the estimators. In this spirit, we propose the \textit{D-optimal} selection function, which, at each stage, minimizes the generalized variance of the estimated regression coefficients in a linear potential outcome model.
To build intuition, we discuss the special case of $k=1$ covariate. With the D-optimal selection function, the choosing group, in its first choice, selects the unit whose covariate value is farthest from the full-sample mean of the covariate; and in the subsequent choices, selects the unit whose covariate value is farthest from its current mean of the covariate. In the example in Table \ref{tab1}, treatment $2$ selects unit $1$ with age $24$, the farthest age from the full-sample mean $43$. In the next stage, treatment $1$ selects unit $12$ with age $60$, the farthest age from $43$. Next, treatment $1$ selects unit $2$ with age $30$, the farthest age from its current mean age $60$. The process continues until all the 12 units are selected.
\begin{singlespacing}
\begin{table}[H]
\caption{(a) Example data set; (b) selection order matrix and an assignment using the FSM.}
\begin{subtable}{.45\linewidth}
\centering
\caption{\footnotesize Data set}
\scalebox{0.65}{
\setlength\extrarowheight{-2pt}
\begin{tabular}{cc}
\toprule
Index & Age \\
\hline
1 & 24 \\
2 & 30 \\
3 & 34 \\
4 & 36 \\
5 & 40 \\
6 & 41 \\
7 & 45 \\
8 & 46 \\
9 & 50 \\
10 & 54 \\
11 & 56 \\
12 & 60 \\
\hline
Mean & 43 \\
\bottomrule
\end{tabular}
}
\end{subtable}
\begin{subtable}{.45\linewidth}
\centering
\caption{\footnotesize Selection order matrix and assignment}
\scalebox{0.65}{
\setlength\extrarowheight{-2pt}
\begin{tabular}{ccccc}
\toprule
\multicolumn{2}{c}{Selection order matrix} & \multicolumn{3}{c}{Unit selected}\\
\cmidrule(r){1-2} \cmidrule(r){3-5}
Stage & Treatment & Index & Age\\
\toprule
1 & 2 & 1 & 24 \\
2 & 1 & 12 & 60 \\
3 & 1 & 2 & 30 \\
4 & 2 & 11 & 56 \\
5 & 1 & 3 & 34 \\
6 & 2 & 10 & 54 \\
7 & 1 & 9 & 50 \\
8 & 2 & 4 & 36 \\
9 & 1 & 5 & 40 \\
10 & 2 & 8 & 46 \\
11 & 2 & 6 & 41 \\
12 & 1 & 7 & 45 \\
\bottomrule
\end{tabular}
}
\end{subtable}
\label{tab1}
\end{table}
\end{singlespacing}
\section{The D-optimal selection function}
\label{sec_theD}
\subsection{Definition and behavior}
\label{sec_definition}
In this section, we formally define the D-optimal selection function and provide an equivalent, closed-form characterization that explains how this selection function governs the selection of units at each stage.
Without loss of generality, we assume that treatment 1 gets to select at the $r$th stage, $r \in \{1,2,...,N\}$.
Let $\tilde{n}_{r-1}$ be the number of units already belonging to treatment group 1 after the $(r-1)$th stage.
We denote $\mathcal{R}_{r-1}$ as the remaining set of unselected units after the $(r-1)$th stage. Let $\bar{\bm{X}}_{r-1}$ and $\underline{\bm{S}}_{r-1}$ be the mean vector and covariance matrix of the covariates in treatment group 1, respectively, after the $(r-1)$th stage. Also, let $\underline{\tilde{\bm{X}}}_{r-1}$ be the $\tilde{n}_{r-1} \times (k+1)$ design matrix in treatment 1 after the $(r-1)$th stage.\footnote{The design matrix includes a column of all 1's (corresponding to the intercept) and $k$ columns of covariates.} Finally, let $\underline{\tilde{\bm{X}}}_{\text{full}}$ be the design matrix in the full sample. We assume that $\underline{\tilde{\bm{X}}}_{\text{full}}$ has full column rank.
To define the selection function, we implicitly consider a linear potential outcome model of $Y_i(1)$ on $\bm{X}_i$, i.e., $Y_i(1) = \bm{\beta}^\top (1, \bm{X}_i^\top)^\top + \eta_i$, where $\eta_i$ is an error term satisfying $\mathbb{E}[\eta_i|\bm{X}_i] = 0$.\footnote{More generally, one can consider a linear model of $Y_i(1)$ on a vector of basis functions $\bm{B}(\bm{X}_i)$ of the covariates.} For unit $i \in \mathcal{R}_{r-1}$, let $\underline{\tilde{\bm{X}}}_{r,i}$ be the resulting design matrix in treatment group 1 if unit $i$ is selected. We first consider the case where $\tilde{n}_{r-1} \geq 1$ (i.e., treatment 1 has made at least one selection) and $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is invertible. The D-optimal selection function selects unit $i' \in \mathcal{R}_{r-1}$, where $i' \in \arg\!\max\limits_{i \in \mathcal{R}_{r-1}}\det(\underline{\tilde{\bm{X}}}^\top_{r,i}\underline{\tilde{\bm{X}}}_{r,i})$.\footnote{Ties in the values of $\det(\underline{\tilde{\bm{X}}}^\top_{r,i}\underline{\tilde{\bm{X}}}_{r,i})$ are resolved randomly.} In other words, at the $r$th stage, the D-optimal selection function chooses the unit among $\mathcal{R}_{r-1}$ that optimally decreases the generalized variance of the estimated regression coefficients of the fitted linear model in treatment 1. In Lemma \ref{lemma:dopt} in the Online Supplementary Materials, we show that
maximizing $\det(\underline{\tilde{\bm{X}}}^\top_{r,i}\underline{\tilde{\bm{X}}}_{r,i})$ is equivalent to maximizing $(1, \bm{X}^\top_i) (\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1})^{-1} \begin{psmallmatrix}
1\\
\bm{X}_i
\end{psmallmatrix}$, where $\bm{X}_i$ is the covariate vector of a the $i$th unit in $\mathcal{R}_{r-1}$.
In stages where $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is not invertible, we augment $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ by a scalar multiple of $\underline{\tilde{\bm{X}}}^\top_{\text{full}} \underline{\tilde{\bm{X}}}_{\text{full}}$ (akin to ridge augmentation) and consider the objective function
$(1, \bm{X}^\top_i) \Big(\frac{1}{\tilde{n}_{r-1}}\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1} + \frac{\epsilon}{N} \underline{\tilde{\bm{X}}}^\top_{\text{full}} \underline{\tilde{\bm{X}}}_{\text{full}} \Big)^{-1} \begin{psmallmatrix}
1\\
\bm{X}_i
\end{psmallmatrix}$, where $\epsilon >0$ is a fixed constant. Finally, when $\tilde{n}_{r-1} = 0$ (i.e., treatment 1 has not made any selections yet), the objective function takes the form $(1, \bm{X}^\top_i) \Big( \underline{\tilde{\bm{X}}}^\top_{\text{full}} \underline{\tilde{\bm{X}}}_{\text{full}} \Big)^{-1} \begin{psmallmatrix}
1\\
\bm{X}_i
\end{psmallmatrix}$.
In Theorem \ref{thm:mahal}, we provide an equivalent characterization of the D-optimal selection function that provides more insight into the selection made by the choosing treatment group at each stage.
\begin{theorem}\normalfont
Let treatment 1 be the choosing group at the $r$th stage. The D-optimal selection function chooses unit $i'$ with covariate vector $\bm{X}_{i'} \in \mathbb{R}^k$, where
\begin{equation*}
i' \in \arg\!\max\limits_{i \in \mathcal{R}_{r-1}} (\bm{X}_i - \bar{\bm{X}}^*_{r-1})^\top (\underline{\bm{S}}^*_{r-1})^{-1} (\bm{X}_i - \bar{\bm{X}}^*_{r-1}),
\end{equation*} \begin{singlespacing}
where {\small
\begin{equation*}
\bar{\bm{X}}^*_{r-1} =
\begin{cases}
\bar{\bm{X}}_{\text{full}} & \text{if $\tilde{n}_{r-1} = 0$}\\
\frac{\bar{\bm{X}}_{r-1}+\epsilon\bar{\bm{X}}_{\text{full}}}{1+\epsilon} & \text{if $\tilde{n}_{r-1} \geq 1$ and $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is not invertible}\\
\bar{\bm{X}}_{r-1} & \text{if $\tilde{n}_{r-1} \geq 1$ and $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is invertible}
\end{cases}
\end{equation*}
}
and {\small
\begin{equation*}
\underline{\bm{S}}^*_{r-1} =
\begin{cases}
\underline{\bm{S}}_{\text{full}} & \text{if $\tilde{n}_{r-1} = 0$}\\
(\frac{1}{\tilde{n}_{r-1}}\underline{\bm{X}}_{r-1}^\top \underline{\bm{X}}_{r-1} + \frac{\epsilon}{N} \underline{\bm{X}}_{\text{full}}^\top \underline{\bm{X}}_{\text{full}}) - (1+\epsilon)\bar{\bm{X}}^*_{r-1}\bar{\bm{X}}^{*\top}_{r-1} & \text{if $\tilde{n}_{r-1} \geq 1$ and $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is not invertible}\\
\underline{\bm{S}}_{r-1} & \text{if $\tilde{n}_{r-1} \geq 1$ and $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is invertible.}
\end{cases}
\end{equation*}
}
\end{singlespacing}
\label{thm:mahal}
\end{theorem}
Theorem \ref{thm:mahal} shows that at every stage, the D-optimal selection function selects that unit among the remaining pool whose covariate vector maximizes a type of Mahalanobis distance. In its first choice, treatment 1 aims to maximize the Mahalanobis distance from the covariate distribution in the full sample (in particular, from $\bar{\bm{X}}_{\text{full}}$), thereby choosing the most outlying unit available in the full sample. For the subsequent stages where $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is not invertible, treatment 1 aims to maximize the Mahalanobis distance from a mixture covariate distribution between treatment group 1 and the full sample, where $\epsilon$ determines the mixing rate. Finally, the latter selections by treatment 1 aim at maximizing the Mahalanobis distance from the covariate distribution in treatment group 1. Therefore, with every selection, treatment 1 maximizes the overall separation of the covariates from its current mean, which helps increase the efficiency of the estimated regression coefficients.
\subsection{Properties}
\label{sec_properties}
By definition, the D-optimal selection function improves the estimation accuracy of the fitted linear model in each treatment group by sequentially minimizing the generalized variance of the estimated regression coefficients. With the D-optimal selection function, we can also establish several additional desirable properties of the FSM. In particular, Theorem \ref{thm:dopt_properties} leverages the connection between D-optimality and Mahalanobis distance (as in Theorem \ref{thm:mahal}) and presents two key properties of the FSM with the D-optimal selection function.
\begin{theorem}\normalfont
\begin{enumerate}[label=(\alph*)]
\item The FSM with the D-optimal selection function is invariant under affine transformations of the covariate vector.
\item For continuous, symmetrically distributed covariates and two groups of equal size, the FSM with the D-optimal selection function \emph{almost always} produces exact mean-balance on all even transformations of the centered covariate vector.
\end{enumerate}
\label{thm:dopt_properties}
\end{theorem}
It follows from Theorem \ref{thm:dopt_properties}(a) that, for any SOM, the choices made by each treatment group remain unchanged even if the covariate vectors are transformed via an affine transformation (e.g., changing the units of measurement of the covariates).
Therefore, the FSM with the D-optimal selection function self-standardizes the covariates. Thus, without loss of generality, we can assume that the covariates have mean zero in the full sample. In addition, if the covariate vector is symmetrically distributed in the sample, then by Theorem \ref{thm:dopt_properties}(b), the FSM exactly balances even transformations such as the second, fourth order moments, and the pairwise products of the covariates. An implication of Theorem \ref{thm:dopt_properties}(b) is that, for covariates drawn from symmetric continuous distributions (such as the Normal, t, and Laplace distributions), the FSM tends to balance all these transformations because of the approximate symmetry of the covariates in the sample. The choice of the D-optimal selection function is thus robust in the sense that it allows the FSM to balance a family of transformations of the covariate vector by design, without explicitly including them in the assumed linear model nor requiring the specification of tuning parameters.
The FSM with D-optimal selection function is also attractive because it can encompass several classical designs, such as randomized blocked and matched-pair designs. Theorem \ref{thm:retrieve} formalizes this result.
In the traditional randomized block design (RBD), the units are grouped into blocks of size $G$ according to a categorical \textit{blocking variable} and each treatment is randomly applied to exactly one unit within each block (see, e.g., \citealt{cox2000theory}, Section 3.4). Here we consider a more general version of an RBD where the blocks are of size $c \times G$ (where $c$ is a fixed positive integer) and each treatment is applied to $c$ units within each block. This is a special case of a stratified randomized experiment with strata of equal size and equal allocation among treatments per stratum. In a matched-pair design with $G=2$ treatments, similar units are grouped into pairs and each treatment is randomly applied to one unit within each pair. This is also a special case of a stratified randomized experiment with equal allocation per strata, where the size of each stratum equals two.
\begin{theorem}\normalfont
\begin{enumerate}[label=(\alph*)]
\item Consider a setting where $N = cBG$ units belonging to $B$ blocks of equal size are to be randomly assigned into $G$ treatment groups of equal size, where $c$ is a fixed positive integer.
Then, if the linear model in the FSM consists of an intercept and indicators of any $B-1$ levels of the blocking variable, the FSM with the D-optimal selection function produces the same assignment as an RBD.
\item Consider a setting where $N/2$ identical pairs of units in terms of baseline covariates $\bm{X}_i$ are to be assigned into $G = 2$ treatment groups of equal size.
Assume $\bm{X}_i$ is drawn from a continuous distribution.
Then, if the linear model in the FSM consists of the intercept and the covariates $\bm{X}_i$, then the FSM \textit{almost surely} produces the same assignment mechanism as a matched-pair design.
\end{enumerate}
\label{thm:retrieve}
\end{theorem}
In the first setting, Theorem \ref{thm:retrieve}(a) states that, by including the levels of a blocking variable as regressors, the FSM with the D-optimal selection function automatically blocks on that variable. Thus, the FSM retrieves an RBD without explicitly performing separate randomizations within each block.
In the second setting, Theorem \ref{thm:retrieve}(b) states that, by including the covariates as regressors, the FSM with the D-optimal selection function produces the same assignment as a matched-pair experiment, without explicitly performing separate randomizations in each pair. This phenomenon is particularly useful when the sample consists of near-identical twins but that are difficult to identify a priori due to multiple covariates.
\subsection{Connection to A-optimality}
\label{sec_connection}
The original FSM used a criterion based on A-optimality as the selection function (see \citealt{morris1979finite}). In this section, we compare the A-and D-optimal selection functions. The A-optimal selection function requires prespecifying a \textit{policy matrix} $\underline{\bm{P}}_{p \times (k+1)}$ and a corresponding vector of \textit{policy weights} $\bm{w}_{p\times 1}$. Here, $\underline{\bm{P}}$ transforms the original vector of regression coefficients to a vector of $p$ linear combinations that are of policy interest, and $\bm{w}$ assigns weights to each combination according to their importance.
If treatment 1 gets to choose at the $r$th stage, then the A-optimality criterion selects the unit that minimizes the resulting $\text{trace}\Big(\underline{\bm{T}} (\underline{\tilde{\bm{X}}}^\top_{r,i} \underline{\tilde{\bm{X}}}_{r,i})^{-1}\Big)$, where $\underline{\bm{T}} = \underline{\bm{P}}^\top \text{diag}(\bm{w}) \underline{\bm{P}}$.\footnote{For ease of exposition, we only discuss the case where $(\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1})$ is invertible.} Proposition \ref{prop:aopt} shows an equivalent characterization of the A-optimal selection function.
\begin{proposition} \normalfont
Let treatment 1 be the choosing group at the $r$th stage. Assume that $\tilde{n}_{r-1} \geq 1$ and $\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1}$ is invertible. The A-optimal selection function chooses unit $i'$ with covariate vector $\bm{X}_{i'} \in \mathbb{R}^k$, where
\begin{equation}
i' \in \arg\!\max\limits_{i \in \mathcal{R}_{r-1}}\frac{(1, \bm{X}^\top_i) (\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1})^{-1} \underline{\bm{T}} (\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1})^{-1} \begin{psmallmatrix}
1\\
\bm{X}_i
\end{psmallmatrix}}{1+(1, \bm{X}^\top_i) (\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1})^{-1} \begin{psmallmatrix}
1\\
\bm{X}_i
\end{psmallmatrix}}.
\end{equation}
\label{prop:aopt}
\end{proposition}
The A-optimality criterion provides a family of selection functions depending on $\underline{\bm{P}}$ and $\bm{w}$. In general, the A-optimality criterion is not invariant with respect to affine transformations of the covariate vector. For instance, setting $\underline{\bm{P}} = \bm{I}$ and $\bm{w} = (1,1,...,1)^\top$ produces a selection function that is not affine invariant. On the other hand, setting $\underline{\bm{P}} = \underline{\tilde{\bm{X}}}_{\text{full}}$ and $\bm{w} = (1,1,...,1)^\top$ yields an affine invariant selection function. In fact, for the latter choice, the A-optimal selection function is closely related to the D-optimal selection function. To see this, consider a case where in the selection process, the design matrices in each treatment group scale similarly relative to the design matrix in the full sample. In particular, for treatment 1 (the choosing group at stage $r$), $(\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1})^{-1} = c_r (\underline{\tilde{\bm{X}}}^\top_{\text{full}} \underline{\tilde{\bm{X}}}_{\text{full}})^{-1}$ for some constant $c_r>0$. In this case, the A-optimal selection function chooses unit $i'$ such that
\begin{align}
& i' \in \arg\!\max\limits_{i \in \mathcal{R}_{r-1}}(1, \bm{X}^\top_i) (\underline{\tilde{\bm{X}}}^\top_{r-1} \underline{\tilde{\bm{X}}}_{r-1})^{-1} \begin{psmallmatrix}
1\\
\bm{X}_i
\end{psmallmatrix} \\
& \iff i' \in \arg\!\max\limits_{i \in \mathcal{R}_{r-1}} (\bm{X}_i - \bar{\bm{X}}_{r-1})^\top (\underline{\bm{S}}_{r-1})^{-1} (\bm{X}_i - \bar{\bm{X}}_{r-1}),
\end{align}
which is same as the D-optimal selection function. Hence, in this case, the FSM under the D-optimal and A-optimal selection functions make similar choices of units.
\section{Inference under the FSM}
\label{sec_inference}
Using the FSM we can make both model- and randomization-based inferences. Both modes of inference are feasible for any selection function and any randomized SOM.
In model-based inference, the sample is typically assumed to be drawn randomly from some superpopulation and inference for the PATE is done by modeling the observed outcome distribution conditional on the treatment indicators and the covariates.
For instance, let the potential outcome model under treatment $g$ be $Y_i(g) = \bm{\beta}^\top_{g}\bm{B}(\bm{X}_i) + \epsilon_{ig}$, where $\bm{B}(\bm{X}_i) = \{ B_1(\bm{X}_i),...,B_b(\bm{X}_i) \}^\top$ is a vector of $b$ basis functions of the covariates, and $\epsilon_{ig}$, $i \in \{1,2,...,N\}$ are mutually independent errors, independent of the covariates.
Under this model, $\text{PATE}_{g',g''}$ can be unbiasedly estimated by $\widehat{\text{PATE}}_{g',g''} = \hat{\bm{\beta}}^\top_{g'}\overline{\bm{B}(\bm{X})} - \hat{\bm{\beta}}^\top_{g''}\overline{\bm{B}(\bm{X})}$, where $\overline{\bm{B}(\bm{X})} = \frac{1}{N}\sum_{i=1}^{N}\bm{B}(\bm{X}_i)$ and $\hat{\bm{\beta}}_g$ is the OLS estimator of $\bm{\beta}_g$ obtained by fitting a linear regression of $Y^{\text{obs}}_i$ on $\bm{B}(\bm{X}_i)$ in treatment group $g = g', g''$.
We call this the regression imputation estimator of $\text{PATE}_{g',g''}$.
The standard error of this estimator and the corresponding confidence interval for $\text{PATE}_{g',g''}$ can be obtained using standard OLS theory.
We note that, in model-based inference, the standard errors and confidence intervals do not take into account the randomness stemming from the assignment mechanism.
Moreover, often the regression models proposed at the design stage are considered misspecified and are later modified at the analysis stage by, e.g., incorporating covariates (or transformations thereof) that are deemed important predictors for the outcome.
Due to the balancing properties of the FSM, the regression imputation estimators tend to exhibit sufficient precision even when the model posited by the FSM is misspecified (see sections \ref{sec_definition}, \ref{sec_simulation}, and \ref{sec_thehie}).
In randomization-based inference, the potential outcomes and the covariates are typically considered as fixed and the assignment mechanism is the only source of randomness (see Chapter 2 of \citealt{rosenbaum2002observational} and chapters 5--7 of \citealt{imbens2015causal} for overviews).
Inference for causal effects can be done via exact randomization tests for sharp null hypotheses on unit-level causal effects (\citealt{fisher1935design}), or via estimation under Neyman's repeated sampling approach (\citealt{neyman1923application}).
Under the FSM, randomization tests for sharp null hypotheses can be performed by approximating the distribution of the test statistic through repeated realizations of the FSM.
To illustrate, consider testing the sharp null hypothesis of zero unit-level causal effects, i.e., $H_0: Y_i(2) - Y_i(1) = 0$ for all $i$, at level $\alpha$ using the FSM. While any choice of test statistic preserves the validity of the test, a common choice is the absolute difference-in-means statistic $|\frac{1}{n_2}\sum_{i:Z_i = 2}Y^{\text{obs}}_i - \frac{1}{n_1}\sum_{i:Z_i = 1}Y^{\text{obs}}_i| = |\frac{1}{n_2}\sum_{i:Z_i = 2}Y_i(2) - \frac{1}{n_1}\sum_{i:Z_i = 1}Y_i(1)| = : T(\bm{Z},\bm{Y}(1),\bm{Y}(2))$.
Large values of $T(\bm{Z},\bm{Y}(1),\bm{Y}(2))$ are considered evidence against $H_0$.
Under $H_0$, $Y_i(2) = Y_i(1) = Y^{\text{obs}}_i$ and the vectors of potential outcomes $\bm{Y}(1)$ and $\bm{Y}(2)$ are known and fixed.
The $p$-value of the test is given by $p = P_{H_0}(T(\bm{Z},\bm{Y}(1),\bm{Y}(2))\geq t_{\text{obs}})$, where $t_{\text{obs}}$ is the value of the test statistic for the observed realization of $\bm{Z}$ under the FSM.
We can compute this $p$-value by Monte Carlo approximation, i.e., we generate independent vectors of assignments $\bm{Z}^{(m)} = (Z^{(m)}_1,...,Z^{(m)}_N)^\top$, $m \in \{1,2,...,M\}$ using the FSM and approximate the $p$-value as $\hat{p} = \frac{1}{M}\sum_{m = 1}^{M}\mathbbm{1}\big(T(\bm{Z}^{(m)},\bm{Y}(1),\bm{Y}(2))\geq t_{\text{obs}}\big)$. We reject $H_0$ at level $\alpha$ if $\hat{p}\leq \alpha$.
Similar tests can be applied for more general sharp hypotheses of treatment effects (e.g., dilated and tobit effects; \citealt{rosenbaum2002observational, rosenbaum2010design2}).
We can invert these tests to obtain a confidence interval for the hypothesized effect (\citealt{rosenbaum2002observational}, Section 2.6.1). Moreover, we can get a point estimate of the effect by solving a Hodges-Lehmann estimating equation corresponding to these tests (\citealt{rosenbaum2002observational}, Section 2.7.2).
Finally, under Neyman's approach, we can estimate the sample average treatment effect $\text{SATE}_{g',g''}$ by the difference-in-means statistic.
In particular, for groups of equal size, this difference-in-means statistic is unbiased for $\text{SATE}_{g',g''}$ under the FSM (see Proposition \ref{fsm_prop:unbiased} for a proof).
\section{A simulation study}
\label{sec_simulation}
\subsection{Setup}
We now compare the performance of the FSM to complete randomization and rerandomization in a simulation study.
Here, $N=120$, $G=2$, $n_1 = n_2 = 60$, and $k=6$.
The covariates are generated following the design of \cite{hainmueller2012balancing}:
\begin{equation}
\begin{psmallmatrix}
X_{1}\\
X_{2}\\
X_3
\end{psmallmatrix} \sim
\mathcal{N}_3\left[\begin{psmallmatrix}
0\\
0\\
0
\end{psmallmatrix},\begin{psmallmatrix}
2 & 1 & -1\\
1 & 1 & -0.5\\
-1 & -0.5 & 1
\end{psmallmatrix}\right],\hspace{0.1cm}
X_4 \sim \text{Unif}[-3,3], \hspace{0.1cm} X_5 \sim \chi^2_1, \hspace{0.1cm} X_6 \sim \text{Bernoulli}(0.5). \label{dgp}
\end{equation}
\noindent In this design, $X_4$, $X_5$, and $X_6$ are mutually independent and separately independent of $(X_1,X_2,X_3)^\top$.
We draw a sample of 120 units once from the data generating mechanism in (\ref{dgp}).
Conditional on this sample, we compare four different assignment methods, namely a completely randomized design (CRD), rerandomization with 0.01 acceptance rate (RR 0.01), rerandomization with 0.001 acceptance rate (RR 0.001), and the FSM.
Both RR 0.01 and RR 0.001 use as rerandomization criteria the Mahalanobis distance between the two treatment groups on the original covariates.
The FSM uses a linear potential outcome model on the original covariates and the D-optimal selection function.
For each design we draw 800 independent assignments. The assignments under the FSM are generated using the open source R package \texttt{FSM} available on CRAN.
The total runtime of the FSM for the 800 simulated experiments was about one and a half minutes on a Windows 64-bit computer with an Intel(R) Core i7 processor.
See \cite{chattopadhyay2021randomized} for detailed step-by-step instructions and vignettes on the use of FSM package.
\subsection{Balance}
We evaluate balance on the main and transformed covariates.
Figures \ref{fig:simu_asmd}(a) and \ref{fig:simu_asmd}(b) show density plots of the Absolute Standardized Mean Differences (ASMD; \citealt{rosenbaum1985constructing}, \citealt{stuart2010matching}) of the six main covariates and their second-order transformations (including squares and pairwise products), respectively.
A smaller ASMD for a covariate indicates better mean-balance on that covariate between the two treatment groups.
Figure \ref{fig:simu_asmd}(a) indicates that both rerandomization methods improve balance on the means of the original covariates over CRD.
As expected, the ASMD distribution under RR 0.001 is more concentrated than that of RR 0.01, with 32\% smaller mean ASMD than RR 0.01.
Both the FSM and RR 0.001 have similar distributions of the ASMD with FSM having moderately (9\%) smaller mean ASMD.
See Table \ref{tab_app:simu_asmd_org} in the Online Supplementary Materials for a comparison of the average ASMD of each covariate.
\begin{figure}[H]
\begin{subfigure}{.52\textwidth}
\includegraphics[scale =0.42]{graphics/simu_asmd_org_emp.pdf}
\caption{\footnotesize Main covariates}
\end{subfigure}%
\begin{subfigure}{.52\textwidth}
\includegraphics[scale =0.42]{graphics/simu_asmd_sqint_emp.pdf}
\caption{\footnotesize Squares and pairwise products}
\end{subfigure}
\caption{Distributions of absolute standardized mean differences (ASMD) of the main covariates and all their second-order transformations. In the top right corners the legends present the average ASMD across simulations for the four methods.
On average, the FSM achieves better covariate balance.
In terms of the main covariates, the FSM marginally outperforms RR 0.001.
In terms of the second-order transformations, the FSM substantially outperforms RR 0.001.}
\label{fig:simu_asmd}
\end{figure}
Figure \ref{fig:simu_asmd}(b) shows that the imbalances of covariate transformations are substantially smaller with the FSM than with CRD, RR 0.01, and RR 0.001.
In fact, the FSM achieves a 70\% reduction in the mean ASMD with respect to RR 0.001.
Thus, although the FSM and RR 0.001 exhibit comparable balance in terms of the main covariates, the FSM balances these transformations of the covariates much better than RR 0.001.
This highlights the improved robustness of the FSM against model misspecification, as discussed previously in the context of Theorem \ref{thm:dopt_properties}(b).\footnote{For the FSM, the implicit potential outcome model is the same as the model used to specify the D-optimal selection function. Although rerandomization does not explicitly model the potential outcomes, an implicit model can be conceptualized from the covariates (or transformations thereof) used to construct the Mahalanobis distance.}
Moreover, reducing the tuning parameter of rerandomization from 0.01 to 0.001 yields only 2\% improvement in the mean ASMD.\footnote{In fact, for some covariate transformations, reducing this tuning parameter exacerbates imbalance (see Table \ref{tab_app:simu_asmd_sqint} in the Online Supplementary Materials).}
In Figure \ref{fig:simu_asmd}(b), both RR 0.01 and RR 0.001 often produce ASMD larger than 0.1, and in some cases, larger than 0.5, indicative of substantial imbalances on these covariate transformations.
Under rerandomization, balance on the squares and pairwise products of the covariates can be improved by explicitly incorporating these transformations in the Mahalanobis distance.
For instance, with $k$ continuous covariates, a Mahalanobis distance needs to include $\frac{k(k+3)}{2}$ variables to control the imbalances on the means of all the covariates and their squares and pairwise products.
However, with large $k$, calculating the Mahalanobis distance becomes computationally expensive and, in the extreme case (when $\frac{k(k+3)}{2}>N$), infeasible.
The FSM, by contrast, only requires $k$ main covariates for computing the D-optimality criterion (see Theorem \ref{thm:mahal}) to produce adequate balance on these transformations.
For each method, we also compare balance in the overall correlation structure of the covariates.
Let $\underline{\bm{R}}_g$ denote the sample correlation matrix in group $g$, $g \in \{1,2\}$.
As a measure of imbalance, we consider the Frobenius norm of $\underline{\bm{R}}_1 - \underline{\bm{R}}_2$, denoted by $||\underline{\bm{R}}_1 - \underline{\bm{R}}_2||_F$.\footnote{The Frobenius norm of a matrix is the square root of the sum of squares of all its elements.} Smaller values of $||\underline{\bm{R}}_1 - \underline{\bm{R}}_2||_F$ are indicative of better balance on the correlation matrix of the covariates between the two groups.
Figure \ref{fig:simu_frob_cor} shows the boxplots of the distributions of $||\underline{\bm{R}}_1 - \underline{\bm{R}}_2||_F$.
The FSM outperforms the other three designs with at least 75\% smaller average $||\underline{\bm{R}}_1 - \underline{\bm{R}}_2||_F$.
In particular, among the 800 randomizations, the highest value of $||\underline{\bm{R}}_1 - \underline{\bm{R}}_2||_F$ under FSM is smaller than the corresponding lowest value under the other three designs, indicating that in terms of the correlation structure (and hence the interactions) of the covariates, the least balanced realization of the 800 FSMs exhibits better balance than the best balanced realization of the 800 complete randomizations and rerandomizations.
\begin{figure}[H]
\centering
\includegraphics[scale =0.42]{graphics/simu_frob_cor.pdf}
\caption{Distributions of discrepancies between the correlation matrices of the covariates in the treatment and the control group (as measured by the Frobenius norm, $||\underline{\bm{R}}_1 - \underline{\bm{R}}_2||_F$) across 800 randomizations.
The FSM produces substantially lower discrepancies than the other three methods, indicating markedly improved balance on the correlations of the covariates.}
\label{fig:simu_frob_cor}
\end{figure}
Finally, we evaluate balance on the joint distribution of the covariates.
To this end, we use two recently proposed non-parametric graph-based tests for equality of multivariate distributions (\citealt{agarwal2019distribution}).
In Table \ref{tab_app:simu_mcm} of the Online Supplementary Materials, we show the average p-values of the two tests for each design.
Since each method ensures covariate balance in expectation, the average p-values for both tests are all substantially greater than the typical 0.05 level.
Nevertheless, the average p-value for the FSM is the highest among the four designs, indicating improved covariate balance in aggregate on the joint distributions.
\subsection{Efficiency}
We now compare the efficiency of the methods under both model- and randomization-based approaches to inference.
Under the model-based approach, we consider a potential outcome model where $\mathbb{E}[Y_i(g)|\bm{X}_i]$ is linear in $\bm{X}_i$ (Model A1) and another model where $\mathbb{E}[Y_i(g)|\bm{X}_i]$ is linear in $\bm{X}_i$ and all its second-order transformations (Model A2).
For each potential outcome model, we fit the corresponding observed outcome model by OLS and estimate $\text{PATE}_{2,1}$ using the regression imputation method described in Section \ref{sec_inference}.
Tables \ref{tab:simu_var_model}(a) and \ref{tab:simu_var_model}(b) show the average and maximum model-based standard error (SE) of the regression imputation estimator relative to the FSM across 800 randomizations under the two models.
\begin{singlespacing}
\begin{table}[H]
\caption{Average and maximum model-based standard errors relative to the FSM across randomizations. Under Model A1 (linear model on the main covariates), the FSM and RR exhibit similar performance, improving over CRD. Under Model A2 (linear model on the main covariates and their second-order transformations), the FSM is considerably more efficient than both CRD and RR.}
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model A1}
\scalebox{0.75}{
\begin{tabular}{p{2.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR 0.01 & RR 0.001 & FSM\\
\toprule
Average SE & 1.03 & 1.00 & 1.00 & 1.00 \\
Maximum SE & 1.13 & 1.00 & 1.00 & 1.00 \\
\bottomrule
\end{tabular}
}
\end{subtable}%
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model A2}
\scalebox{0.75}{
\begin{tabular}{p{2.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR 0.01 & RR 0.001 & FSM\\
\toprule
Average SE & 1.39 & 1.27 & 1.26 & 1.00 \\
Maximum SE & 3.61 & 1.97 & 1.80 & 1.00 \\
\bottomrule
\end{tabular}
}
\end{subtable}
\label{tab:simu_var_model}
\end{table}
\end{singlespacing}
Under Model A1, since both rerandomization and the FSM are able to adequately balance the means of the original covariates,
they lead to lower SE (hence, higher efficiency) than CRD.
Across randomizations, the worst case SE under RR 0.01, RR 0.001, and the FSM are 13\% smaller than under CRD.
Under Model A1, the FSM has similar model-based SE as the two rerandomization methods.
However, under Model A2, the FSM uniformly outperforms the other three designs, with a 26\% reduction in average SE and an 80\% reduction in maximum SE than RR 0.001.
This improvement in efficiency can be attributed to the balance achieved by the FSM on the main covariates and their squares and pairwise products.
In sum, when the model assumed at the design stage is correct and is used at the analysis stage, the FSM is as efficient as the two rerandomizations for estimating the treatment effect.
However, when the model assumed at the design stage is misspecified and later corrected by augmenting transformations of the covariates (e.g., squares and pairwise products), the FSM is considerably more efficient and robust than the other designs.
Under the randomization-based approach, we compare the standard errors of the difference-in-means statistic under each design. Following \cite{hainmueller2012balancing}, the potential outcomes are generated using the models: $Y(1) = X_1 + X_2 + X_3 - X_4 +X_5 + X_6 + \eta$, $Y(2) = Y(1)$ (Model B1) and $Y(1) = (X_1 + X_2 + X_5)^2 + \eta$, $Y(2) = Y(1)$ (Model B2), where $\eta \sim \mathcal{N}(0,1)$.
Both generative models satisfy the sharp-null hypothesis of zero treatment effect for every unit and hence, $\text{SATE}_{2,1} = 0$. Conditional on these potential outcomes, $\text{SATE}_{2,1}$ is estimated under each design using the standard difference-in-means estimator. The corresponding randomization-based SE of this estimator is obtained by generating 800 randomizations of the design and computing the standard deviation of the difference-in-means estimator across these 800 randomizations.
Table \ref{tab:simu_var_rand} shows the randomization-based SE of the difference-in-means statistic for $\text{SATE}_{2,1}$ under each model.
\begin{singlespacing}
\begin{table}[H]
\caption{Average randomization-based standard errors relative to the FSM. The standard error for the FSM is 0.2 under Model B1 (linear model on the main covariates) and 0.43 under Model B2 (linear model on the main covariates and their second-order transformations). Especially under Model B2, the FSM is considerably more efficient than both CRD and RR.}
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model B1}
\scalebox{0.75}{
\begin{tabular}{p{1.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR 0.01 & RR 0.001 & FSM\\
\toprule
SE & 2.72 & 1.26 & 1.08 & 1 \\
\bottomrule
\end{tabular}
}
\end{subtable}%
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model B2}
\scalebox{0.75}{
\begin{tabular}{p{1.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR 0.01 & RR 0.001 & FSM\\
\toprule
SE & 5.69 & 4.56 & 4.47 & 1 \\
\bottomrule
\end{tabular}
}
\end{subtable}
\label{tab:simu_var_rand}
\end{table}
\end{singlespacing}
Under Model B1, the potential outcomes depend linearly on the covariates and therefore balancing the means of the covariates improves efficiency.
This is reflected in Table \ref{tab:simu_var_rand} as the FSM has the smallest SE, closely followed by RR 0.001.
Under Model B2, the potential outcomes depend linearly on the squares and pairwise products of the covariates.
By better balancing these transformations, the FSM yields a considerably smaller SE than the other designs.
In particular, under Model B2, the SE under the FSM is 67\% smaller than the SE under RR 0.001.
Therefore, in a similar way as in the model-based approach, in randomization-based approach the FSM exhibits comparable efficiency to rerandomization under correct-specification of the outcome model, and considerable robustness under model misspecification.
\section{The Health Insurance Experiment}
\label{sec_thehie}
\subsection{Data}
\label{sec_hiedata}
We evaluate and compare the performance of the FSM with standard designs using the baseline data of the HIE. To this end, we consider a version of the HIE data presented in \cite{aron2013rand}. This version includes data on six cost sharing plans described in Section \ref{sec_hiedesign}. To make the group sizes more homogeneous, we consider combining the groups with $25\%$, $50\%$, and mixed coinsurance plans. Thus, in our analysis, we have $G = 4$ treatment groups corresponding to $g = 1$, ``free care'' ($n_1 = 564$); $g = 2$, ``$25\%, 50\%$, or mixed coinsurance'' ($n_2 = 456$); $g = 3$, ``$95\%$ coinsurance'' ($n_3 = 372$); and $g = 4$, ``individual deductible'' ($n_4 = 495$). In total, there are $N = n_1 + ... + n_4 = 1,887$ families. We consider assigning all $N$ families to the four treatment groups and hence do not consider a discard group of non-participants.
Moreover, in this version, the units (i.e., families) across five of the six sites are pooled and we consider randomly assigning all the families in this pooled set to the four treatment groups at once. Due to loss of data, the Dayton site is excluded from this analysis.
We consider $k = 20$ family-level baseline covariates, where $X_1, ..., X_5$ are scaled covariates, $X_6, ..., X_{14}$ are binary covariates, and $X_{15}, ..., X_{20}$ are binary covariates indicating missing data (see Table \ref{tab:hie1} for a description of each baseline covariate). Using this data, we compare complete randomization, rerandomization, and the FSM in terms of balance and efficiency. For the FSM, we generate the SOM by first using SCOMARS on the combined groups $\{1,2\}$ and $\{3,4\}$, and then using SCOMARS again to split each combined group into its component groups.
For rerandomization, we use two balance criteria based on Wilks' lambda statistic (\citealt{lock2011rerandomization}, Section 5.2) and the maximum pairwise Mahalanobis distance between any two treatment groups (\citealt{morgan2012rerandomization}).
For each design, we draw 800 independent assignments.
The runtime of each of these assignments with the FSM was less than one minute on a Windows 64-bit laptop computer with an Intel(R) Core i7 processor.
\subsection{Balance}
Figure \ref{fig:test1} displays the ASMD distributions across randomizations for the main covariates and second-order transformations of the scaled covariates in the HIE data.
In both cases we see that the FSM outperforms complete randomization and rerandomization.
While rerandomization balances the main covariates better than complete randomization, this advantage is less marked than in the previous simulation study and disappears for the transformations of the covariates.
In fact, the average imbalances for these transformations are very similar between complete randomization (0.055) and rerandomization (0.052), and with both methods it is common to see imbalances greater than 0.1 ASMD.
With the FSM, however, the average imbalance is less than half (0.02) of those under CRD and RR, and extreme imbalances are non-existent after the assignments.
\begin{figure}[H]
\begin{subfigure}{.52\textwidth}
\includegraphics[scale =0.5]{graphics/hie_asmd_org_emp.pdf}
\caption{\footnotesize Main covariates}
\label{fig:sub1}
\end{subfigure}%
\begin{subfigure}{.52\textwidth}
\includegraphics[scale =0.5]{graphics/hie_asmd_sqint_emp.pdf}
\caption{\footnotesize Squares and pairwise products}
\label{fig:sub2}
\end{subfigure}
\caption{
Distributions of absolute standardized mean differences (ASMD) of the main covariates and their second-order transformations in the HIE data. The legends present the average ASMD across simulations for the four methods.
On average, the FSM substantially outperforms CRD and RR in terms of both the main covariates and their second-order transformations.}
\label{fig:test1}
\end{figure}
A related question is how well the methods balance all second-order features of the joint distribution of the covariates.
Figure \ref{figfrob} provides an answer to this question in the boxplots of the discrepancies between correlation matrices ($||\underline{\bm{R}}_{g} - \underline{\bm{R}}_{g'} ||_F$) across randomizations.
As in the aforementioned second-order transformations, we see a similar performance between complete randomization and rerandomization, which is considerably improved by the FSM with a median about three times smaller.
Arguably, one could improve the performance of rerandomization; for example, by restricting imbalances on these transformations via the rerandomization criterion; however, unlike the FSM, this may incur increased computational cost and may require additional tuning parameters. Moreover, these transformations can also be included in the FSM model, which would then also improve balance on higher order transformations of them.
\begin{figure}[H]
\begin{subfigure}{.52\textwidth}
\includegraphics[scale =0.5]{graphics/hie_frob_cor12_emp.pdf}
\caption{\footnotesize $||\underline{\bm{R}}_1 - \underline{\bm{R}}_2||_F$}
\label{fig:simu_asmd_org}
\end{subfigure}%
\begin{subfigure}{.52\textwidth}
\includegraphics[scale =0.5 ]{graphics/hie_frob_cor34_emp.pdf}
\caption{\footnotesize $||\underline{\bm{R}}_3 - \underline{\bm{R}}_4||_F$}
\label{fig:simu_asmd_sqint}
\end{subfigure}
\caption{Distributions of discrepancies of the correlation matrices of the covariates in the treatment groups of the HIE data across randomizations.
The discrepancies are measured by $||\underline{\bm{R}}_{g} - \underline{\bm{R}}_{g'}||_F$, where $\underline{\bm{R}}_g$ is the sample correlation matrix of the covariates in treatment group $g$ and $||\cdot||_F$ is the Frobenius norm.
The FSM systematically produces lower discrepancies than the other methods, exhibiting substantially improved balance on the correlations of the covariates.
}
\label{figfrob}
\end{figure}
\subsection{Efficiency}
As in the simulation study, we evaluate efficiency under model- and randomization-based approaches to inference.
The main differences between the model- and randomization-based standard errors is that in the model-based approach, the variance calculation does not explicitly take into account the variability arising through the randomization distribution, whereas in the randomization-based approach it does. For illustration, here we consider estimating the average treatment effect of treatment 3 relative to treatment 2, i.e., $\text{SATE}_{3,2}$ and $\text{PATE}_{3,2}$.
Under the model-based approach, we consider two potential outcome models, one that is linear on the main covariates (Model A3), and another that is linear on the main covariates and the second-order transformations of the scaled covariates (Model A4).
The results are summarized in Table 4.
While the performance of the three methods is similar under Model A3, under Model A4 there are substantial differences with the FSM outperforming both complete randomization and rerandomization.
In fact, under Model A4, there is a 13-15\% reduction in the average standard error, and a 50-69\% reduction in the maximum standard error, with the FSM.
\begin{singlespacing}
\begin{table}[H]
\caption{Average and maximum model-based standard errors relative to the FSM across randomizations. Under Model A3 (linear model on the covariates), the FSM is slightly more efficient than RR and CRD. Under Model A4 (linear model on the covariates and their second-order transformations), the FSM is considerably more efficient than CRD and RR.}
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model A3}
\scalebox{0.72}{
\begin{tabular}{p{2.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR Wilks & RR Mahalanobis & FSM\\
\toprule
Average SE & 1.02 & 1.01 & 1.01 & 1.00 \\
Maximum SE & 1.04 & 1.02 & 1.02 & 1.00 \\
\bottomrule
\end{tabular}
}
\end{subtable}%
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model A4}
\scalebox{0.72}{
\begin{tabular}{p{2.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR Wilks & RR Mahalanobis & FSM\\
\toprule
Average SE & 1.15 & 1.13 & 1.13 & 1.00 \\
Maximum SE & 1.69 & 1.50 & 1.55 & 1.00 \\
\bottomrule
\end{tabular}
}
\end{subtable}
\label{tab:hie_var_model}
\end{table}
\end{singlespacing}
Under the randomization-based approach, we consider the generative models $Y(3) = 10 + 2X_1 + 3X_2 + 0.5X_3 + 0.3X_4 + \eta$ (Model B3) and $Y(3) = 10 + 2X_1 + 2X_2X_3 - X_4X_5 + \eta$ (Model B4) where $Y(3) = Y(2)$ and $\eta \sim \mathcal{N}(0,1.5^2)$. Similar to the simulation study, both generative models satisfy the sharp-null hypothesis of zero treatment effect for every unit and hence, $\text{SATE}_{3,2} = 0$. Under each design, $\text{SATE}_{3,2}$ is estimated using the standard difference-in-means estimator and the corresponding randomization-based SE is obtained by generating 800 randomizations and computing the standard deviation of the estimator across these 800 randomizations.
The results are summarized in Table \ref{tab:hie_var_rand}.
In terms of efficiency, we see again a clear advantage of the FSM.
Under Model B3, the average standard errors of rerandomization are 73\% and 83\% larger than the one of the FSM.
Under Model B4, this difference is accentuated and the average standard errors of rerandomization are 236\% and 242\% larger.
\begin{singlespacing}
\begin{table}[H]
\caption{Randomization-based standard errors relative to the FSM. The standard error for the FSM is 0.12 under Model B3 (linear model on the covariates) and 0.67 under Model B4 (linear model on the covariates and their second-order transformations). Under both models, the FSM is considerably more efficient than both CRD and RR.}
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model B3}
\scalebox{0.75}{
\begin{tabular}{p{1.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR 0.01 & RR 0.001 & FSM\\
\toprule
SE & 2.36 & 1.73 & 1.83 & 1 \\
\bottomrule
\end{tabular}
}
\end{subtable}%
\begin{subtable}{.5\linewidth}
\centering
\caption{\footnotesize Model B4}
\scalebox{0.75}{
\begin{tabular}{p{1.5cm}cccc}
\toprule
\multirow{2}{5cm}{} & \multicolumn{4}{c}{Designs}\\
\cline{2-5}
& CRD & RR 0.01 & RR 0.001 & FSM\\
\toprule
SE & 3.95 & 3.42 & 3.36 & 1 \\
\bottomrule
\end{tabular}
}
\end{subtable}
\label{tab:hie_var_rand}
\end{table}
\end{singlespacing}
\subsection{Explanation}
This analysis illustrates some important differences between the FSM and RR.
First is the criterion for assignment.
While RR uses the Mahalanobis distance, the FSM uses the D-optimality criterion, which, coupled with a suitable SOM, leads to robust assignments under a more general class of potential outcome models.
Second is the use of this criterion.
While RR essentially ``constrains'' the allowable treatment assignments, the FSM ``optimizes'' them toward the criterion.
In essence, RR solves a feasibility problem whereas the FSM solves a maximization problem. Furthermore, the feasibility problem solved by RR depends on the balance threshold, which may be difficult to choose in practice.
A very high value of this threshold bears the risk of accepting an assignment with poor covariate balance, whereas a very low value can be computationally onerous.
Third is the step-wise assignment of units into treatment groups.
While RR assigns all units in one step and then discards imbalanced assignments, the FSM assigns units one at a time in a random but optimal fashion governed by the selection order and the selection criterion.
This difference is crucial because in experiments like the HIE with several treatment groups and many covariates, the space of possible treatment assignments is vast. As shown in our analysis, optimally selecting among these assignments in a step-wise manner can make a substantial difference in terms of balance, efficiency, computational time, and, ultimately, in the use of ever-scarce resources available for experimentation.\footnote{Figures \ref{fig:simu_boxplot} and \ref{fig:test1} show that, although RR does well under common balance standards (the mean differences are systematically lower than the typical threshold of 0.1 ASMD), there is room to select better (more balanced) random treatment assignments, which is achieved by the FSM.}
\section{Practical considerations and extensions}
\label{sec_practical}
\subsection{Multi-group experiments}
\label{sec_multi}
As discussed, the FSM can readily handle experiments with multiple treatment groups.
In so doing, the key methodological consideration is the choice of the SOM.
As in two-group experiments, with $G > 2$ groups, it helps to generate an SOM that is randomized and sequentially controlled, so that at every stage of the random selection process, the number of selections made by each treatment group up to that stage is close to its fair share.
Formally, we say that an SOM is sequentially controlled if $|S_{ig} - F_{ig}|<1$ for all $i \in \{1,2,...,N\}$ and $g \in \{1,2,...,G\}$, where $S_{ig}$ is the number of selections made by group $g$ up to stage $i$ and $F_{ig} := \mathbb{E}[S_{ig}]$ is the corresponding expected number of selections.
While the construction of an SOM that is sequentially controlled for multi-group experiments with arbitrary group sizes is an open problem, such constructions are possible for several practically relevant configurations of the group sizes.
For example, in multi-group experiments with groups of equal size, we can generate an SOM that is sequentially controlled by successively generating random permutations of the group labels $(1, 2, ..., G)$.
See Definition \ref{def_randchunk} and Proposition \ref{prop_randchunk} for a formal statement of this algorithm and a proof.
In multi-group experiments with groups having one of two distinct sizes, we can generate a sequentially controlled SOM by combining the groups of equal size, followed by using SCOMARS on the combined group labels and randomly permuting the component group labels within each combined group (see Theorem \ref{thm:multiscomars1} for details). For example, with $G = 5$ groups of sizes $n_1= n_2 = 10$, $n_3 = n_4 = n_5 = 21$, this algorithm first generates an SOM at the level of the combined groups $\{1,2\}$ (of size 20) and $\{3,4,5\}$ (of size 63), and then splits each combined group into its component groups.
Experiments with multiple groups of two sizes arise, for example, in clinical settings where an exploratory treatment is evaluated with greater precision in a larger group size and conventional treatments are applied to smaller groups of equal size.
Finally, in multi-group experiments with groups of more than two distinct sizes such that when combined by groups of equal size they have the same total size, we can again obtain a sequentially controlled SOM by randomly permuting across and within the combined group labels.
See Theorem \ref{thm:multiscomars2} for details. For example, with $G = 6$ groups of sizes $n_1 = n_2 = 30$, $n_3 = n_4 = n_5 = 20$, $n_6 = 60$, this algorithm first generates an SOM at the level of the combined groups $\{1,2\}$, $\{3,4,5\}$, $\{6\}$ (each of size 60), and then splits each combined group into its component groups.
In practice, for more general group size configurations, one strategy to generate an SOM is to first identify one of the previous three configurations that is structurally similar to the configuration at hand, and then use the corresponding SOM-generating algorithm.
The resulting algorithm may not always be sequentially controlled, but is still likely to produce a well-controlled randomized selection order.
\subsection{Stratified experiments}
\label{sec_stratified}
In stratified experiments, units are grouped into two or more predefined strata, and within each stratum units are randomly assigned into treatment groups.
By Theorem \ref{thm:retrieve}, when the strata are of equal size, the FSM with stratification variables as covariates automatically retrieves a stratified randomized experiment, without explicitly randomizing units within each stratum.
However, in more general stratified experiments, the FSM needs to be extended to explicitly account for the possibly varying stratum sizes and shares of the treatment groups within each stratum.
Here we discuss a family of such extensions of the FSM, which we term as \textit{stratified FSM}.
We consider stratified experiments where the treatment group sizes within each stratum are set by the investigator beforehand. To accommodate the FSM to such experiments, we again need to carefully construct an SOM.
In the stratified FSM, we append the SOM with an additional column of stratum labels, indicating which stratum the treatment group selects from at each stage of the selection process. This column of stratum labels is specified in such a way that by construction, the resulting SOM is consistent with the pre-fixed treatment group sizes within each stratum. To this end, we discuss two potential approaches below.
Conceptually, the most straightforward approach is to generate a separate SOM for each stratum. This is equivalent to setting the column of stratum labels as $(\underbrace{1,...,1}_{m_1},\underbrace{2,...,2}_{m_2},...,\underbrace{S,....,S}_{m_S})^\top$, where $S$ is the number of strata and $m_s$ is the size of $s$th stratum, $s \in \{1,2,...,S\}$.
This approach is easy to implement and can be useful if, e.g., data on each stratum is available at different stages of the experiment, akin to a sequential experiment (see Section \ref{sec_sequential}). However, in this approach, the treatment groups only get to explore the covariate space of a single stratum for a number of successive stages of selection and hence may not make the most efficient choices.
We address this issue with an alternative approach.
For ease of exposition, we consider two strata: 1 and 2. Let $n_{1g}$ and $n_{2g}$ be the (fixed) sizes of treatment group $g \in \{1,2,...,G\}$ in strata 1 and 2, respectively, where $n_{1g} + n_{2g} = n_g$.
In this approach, we first generate a usual SOM with group sizes $n_1,...,n_G$. For $g \in \{1,2,...,G\}$, we then select the order of the strata that treatment $g$ chooses from by running a SCOMARS algorithm with group sizes $n_{1g}$ and $n_{2g}$.
By allowing the treatment groups to select units from different strata in a balanced manner, this approach mimics the unstratified FSM where the covariate space of the entire sample is explored for choosing units.
Also, by design, this approach satisfies the size requirement of each treatment group within each stratum.
\subsection{Sequential experiments}
\label{sec_sequential}
Sequential experiments are experiments where units arrive sequentially in batches, possibly of varying sizes. In this section, we discuss extensions of the FSM to sequential experiments, which we term as \textit{batched FSM}. For simplicity of exposition, we consider assigning units into equal-sized groups. Let $b_j$ denote the size of the $j$th batch. Using random permutations of the group labels, we can generate a sufficiently large SOM and use the first $b_1$ rows as an SOM for the first batch, the next $b_2$ rows as an SOM for the second batch, and so on. Given a new batch and its corresponding SOM, the simplest approach is to treat the batch as a fresh new sample and assign units using the usual FSM. However, this approach completely ignores the covariate information of the units already assigned. Therefore, in general, this approach fails to correct for any existing covariate imbalances among the treatment groups.
To alleviate this, we consider an alternative approach that explicitly takes into account the covariate information of both the current and previous batches. Given a new batch and its corresponding SOM, we run the FSM as if the new batch is a continuation of the previous batch.
In other words, we use the design matrix based on all the units already assigned to the choosing treatment group to evaluate the D-optimal selection function for each unit in the new batch, and select the unit that maximizes the selection function. By carrying over the existing design matrix to the new batch, this approach tends to correct for any existing covariate imbalances.
An important practical consideration for batched FSM is the size of the batches. For a fixed total number of units enrolled in the experiment, increasing the batch size (and hence, reducing the number of batches) tends to increase the overall balance and efficiency properties of the FSM. On one extreme, a batched FSM with a single batch containing all the units enrolled in the experiment reduces to the usual FSM, which tends to have the highest efficiency among all possible batched FSMs. On the other extreme, a batched FSM with multiple batches of size one each is equivalent to CRD, which ignores the covariate information. How to optimally determine the batch sizes for the FSM is an important open question for practice.
\section{Summary and remarks}
\label{sec_summary}
We revisited, formalized, and extended the FSM for experimental design. We proposed a new selection function based on D-optimality that requires no tuning parameters. We showed that, equipped with this selection function, the FSM has a number of appealing properties. First, the FSM is affine invariant and hence, it self-standardizes covariates with possibly different units of measurements. Second, for approximately symmetric data, the FSM yields near-exact balance on a large class of covariate transformations, including transformations that are not part of the assumed linear model under the FSM. Third, the FSM produces randomized block designs without explicitly randomizing in each block. Fourth, the FSM also produces matched-pair designs without explicitly constructing the matched pairs beforehand and randomizing within each pair. We described how both model-based and randomization-based inference on treatment effects can be conducted using the FSM. For a range of practically relevant configurations of group sizes in multi-group experiments, we proposed new algorithms to generate a fair and random selection order of treatments under the FSM. We also discussed potential extensions of the FSM to stratified and sequential experiments.
In a simulation study and a case study on the RAND Health Insurance Experiment, we showed that the FSM is a robust approach to randomization, exhibiting better performance than complete randomization and rerandomization in terms of balance and efficiency.
While there are settings where complete randomization may perform better than the FSM in terms of efficiency, such settings are less common and involve jagged, i.e., highly non-smooth, potential outcome models.
In practice, where investigators believe there is reasonable smoothness in the models, the FSM is expected to perform well.
Overall, through our extensive explorations with real and simulated experimental data, the FSM has consistently stood out as a robust design that can handle multiple treatment groups and a fairly large number of categorical and continuous covariates without requiring tuning parameters and without the need to coarsen covariates.
We recommend giving strong considerations to the FSM in experimental design for its conceptual simplicity, practicality, and robustness.
\vspace{1cm}
\onehalfspacing
\bibliographystyle{asa}
|
2,877,628,090,082 | arxiv | \section{Introduction}
In population genetics, extant (present-time) data are usually used for inference of past population genetic processes. The coalescent \citep{King82} is a stochastic process that describes the genealogy of a sample from a single locus back to the last common ancestor. It allows for convenient simulations of genealogical trees, conditional on the current sample size and either the current (effective) population size or past (effective) population sizes. Along the branches of this tree, mutations may introduce new allelic states. The coalescent has become a pillar of theoretical and empirical population genetics (see \citep[\eg][]{Hein05,Wake09} and \eg\ the software {\it ms} \citep{Huds90}).
While the extant sample is assumed given, many intermediate configurations are possible until the last common ancestor of the sample is reached. Mutations increase this complexity further. Coalescent simulations have been used for inferring parameters of relatively complex population genetic models assuming high-throughput population genetic data. The combinatorial complexity of the coalescent with mutation means that sufficient statistics are often not available. Summary statistics and Approximate Bayesian Computation (ABC) \citep[\eg][]{Beau02,Esto10,Frai17} are then employed for inference of population genetic parameters. Summary statistics are generally not sufficient and ABC is so computationally demanding that often only subsets of the parameter space can be investigated. Inference therefore becomes approximate.
The infinite sites model \citep{Kimu71} is another classic population genetic model; one of its assumptions is complete linkage disequilibrium, \ie\ no recombination \citep[\eg][]{Watt75}. Using a forward algorithm (a dynamic programming method) with the underlying logic of an urn model, \citet{Wu10} showed that inference under the infinite sites model is efficient in analysis of data sets from panmictic or subdivided populations in equilibrium. Forward in time, the sample sizes leading to the extant sample size are random variables. Therefore, the forward approach cannot be extended to account for deviations from equilibrium, \eg\ via changing population sizes. \citet{Fais15} introduced a corresponding backward algorithm; since it proceeds backward in time conditional on the extant sample, modelling changing population sizes is possible.
Note that data from autosomes of some model species are further from complete linkage disequilibrium, which is assumed by the infinite sites model, than from linkage equilibrium: In flies of the genus {\it Drosophila,} deviations from linkage equilibrium are barely noticeable in genomic regions of moderate to high recombination rates \citep{Pars10} and weak enough to be negligible even within short introns \citep{Clem12a}. In great apes, recombination rates are considerably more variable across the genome: while on average neither linkage equilibrium nor complete disequilibrium can be assumed, either assumption may hold approximately in certain genomic regions (see Supplementary Material, Figure S3 in \citep{Myers05}).
In the protein coding genes of eukaryotes, the expected heterozygosity (which is roughly equal to the scaled mutation rate) is approximately $\leq 10^{-2}$ or smaller \citep{Lynch06}. This makes it unlikely to find more than one allele segregating in a small to moderately sized sample. Thus bi-allelic mutation models suffice to capture the true dynamics of a population. In genomic regions where the recombination rate is much greater than the mutation rate, polymorphic sites can additionally be considered independent: The distance between polymorphic loci in small to moderately sized samples is great enough for recombination to break their association. In other words, neighboring polymorphic sites have different genealogies due to recombination. Data from such populations can be represented in a site frequency spectrum (SFS) without loss of information (and data from multiple populations in a joint site frequency spectrum (jSFS)). The SFS records the frequency of the focal allele in a sample. A bi-allelic mutation model that can be represented in this way can also be re-parametrized to a parent-inde\-pen\-dent mutation model; this means that both mutation and coalescent events are uninformative regarding the immediately ancestral allelic state.
In this article, we assume data that can be represented as site frequency spectra without loss of information. We further assume that data are generated by a haploid bi-allelic Moran model of population size $N$ or by the corresponding diffusion model. (Note that the Wright-Fisher model has the same diffusion limit as the Moran model.) In Sec.~(\ref{section:recap_Moran}), we first review the discrete time and discrete space decoupled mutation-drift Moran model and the derivation of its corresponding diffusion equation. Then we describe two known forward-backward algorithms previously applied by \citet{Berg17}. Such algorithms enable inference of probabilities of ancestral population allele frequencies at any time in the past in addition to the probabilities of extant sample configurations.
The first of these algorithms is for the discrete case: Essentially, the algorithm of \citet{RabinerJuang86} can be directly applied \citep{Berg17}. Our representation of this algorithm is based on matrix multiplications and the population size is assumed constant over time. Changes in mutation-drift parameters can however be immediately incorporated and changes in the effective population size can be indirectly modeled through rescaling of time.
The second of these algorithm is for the continuous case: An analytical form of the transition density function is found by decomposition of the diffusion generator into eigenvectors of orthogonal polynomials (more precisely, modified Jacobi polynomials) and their corresponding eigenvectors \citep{Song12,Berg17}. The prior distributions of both the ancestral population allele proportions and the probabilities of extant sample allele configurations can then be extended across time. This orthogonal polynomial approach is computationally efficient compared to the matrix multiplications of the discrete case. Again, changes in mutation-drift parameters can be accommodated, as well as demographic changes through concurrent rescaling of time. These modifications necessitate a change in the base of the orthogonal polynomials through linear transformation. Such transformations can cause numerical inaccuracies.
In Sec.~(\ref{section:particle}), we introduce our so-called particle model: Using coalescent arguments, one can trace the probability of sample configurations backward through the sample history similarly to the algorithm assuming complete linkage and the infinite sites model \citep{Fais15}. By conditioning on the extant sample, changing demography can be accounted for. Augmenting with temporal dynamics, the joint probability of the number of focal alleles in the sample and the past sample sizes can be determined for every point in time: Essentially, the history of the extant sample can be inferred. We show that the marginal distribution of the extant sample configuration derived using this backward particle approach is equivalent to that determined using the backward orthogonal polynomial approach. The sample genealogy can therefore be considered embedded in the spectral decomposition of the backwards diffusion generator similar to how it can be considered embedded within the discrete Moran model \citep[][chapt.~2.8]{Ethe11}.
Assuming equilibrium and reversing time in the coalescent arguments, we obtain a forward particle approach that can also be used to determine the probabilities of sample configurations. It essentially corresponds to an urn model \citep{Step00} and is again similar to the algorithm assuming complete linkage equilibrium and the infinite sites model \citep{Wu10}. Recall that running the particle model forward in time means that past sample sizes become random variables and modelling non-equilibrium is not feasible. For a full forward-backward algorithm using the particle model, the backward particle approach must therefore be combined with the forward orthogonal polynomial approach.
In Sec.~(\ref{section:particle_boundary}), we discuss different forward-backward algorithms for the boundary mutation-drift Moran model \citep{Vogl12}, which is a simplification of the general mutation Moran model for small scaled mutation rates. Inference, especially in non-equilibrium scenarios, becomes particularly efficient using orthogonal polynomials because the change of base required in the general mutation model can be avoided \citep{Vogl16}. The corresponding boundary mutation particle model, which we introduce in this article, allows for simple derivation of non-equilibrium transition probabilities.
\section{Population allele proportions in the Moran and diffusion models}
\label{section:recap_Moran}
We assume a haploid population of size $N$ evolving according to a discrete space, bi-allelic, reversible, decoupled mutation-drift Moran model \citep[section 2.8]{Mora58a, Mora58b, Mora62,Ethe11}. Evolution proceeds stepwise, either by a birth-death event during which a randomly chosen individual is replaced the randomly chosen offspring of another or by an individual mutating. In Sec.~(\ref{section:discrete_Moran}) and Sec.~(\ref{section:diffusion_limit}) we establish our notation for the time discrete and diffusion versions of this Moran model respectively. The diffusion generator can be decomposed into eigenvectors (spectral decomposition) of modified Jacobi polynomials with corresponding eigenvalues. In other words, the transition density function has an explicit spectral representation; we review this in Sec.~(\ref{section:GeneralMutationDiffusion}). To infer population genetic parameters under these models, the marginal distributions of population allele frequencies have to be determined conditional on an extant sample. In the discrete case, a classic forward-backward algorithm on the transition rate matrix can be employed (Sec.~\ref{section:discrete_Moran_for_back}). In the diffusion limit, the Jacobi polynomials allow for efficient representation of the result (Sec.~\ref{section:diffusion_Moran_for_back}).
\subsection{The discrete decoupled Moran model}
\label{section:discrete_Moran}
We consider a discrete time and discrete space, reversible, and decoupled mutation-drift Moran model with haploid population size $N$ \citep[][section 2.8]{Ethe11} to describe the evolution of the proportions of a focal allele $1$ and a non-focal allele $0$. We assume that birth-death events initially occur at rate $1$, while mutations arise at rates $\mu_1$ towards and $\mu_0$ away from the focal allele (so that the total mutation rate is $\mu=\mu_0+\mu_1$). We re-parameterize by setting the mutation bias towards the focal allele to $\alpha=\mu_1/(\mu_0+\mu_1)$ with $0<\alpha<1$, and equivalently define $\beta=1-\alpha$. Fig.~(\ref{fig:Moran_general.eps}) provides a visualisation for $N=6$ individuals, of which $y=4$ are of the focal type at the present time $s=0$.
\newpage
\begin{figure}[!ht]
\centering
\includegraphics[width = 11.5cm]{Plot_Moran_general.eps}
\caption{Schematic plot of a Moran model: The allelic type of the extant individuals is given at the top of the plot; backward in time the arrows indicate birth-death events and the 'x' mark mutation events. Note that the position and numbering of individuals on the x-axis is arbitrary.}
\label{fig:Moran_general.eps}
\end{figure}
\newpage
Let $\x{s}$ ($0 \leq \x{s} \leq 1$) denote the relative frequency of allele $0$ at a non-focal locus at time $s$. The numbering of individuals is arbitrary. The transition rate matrix $\mathbf{T}_{i,j}$ is tridiagonal, aperiodic, and right stochastic:
\begin{equation}\label{eq:transition_decoupled_Moran}
\mathbf{T}_{i,j}=\Pr(\x{s+1}= \tfrac{j}{N} \given \x{s}= \tfrac{i}{N}) = \begin{cases}
\x{s}(1-\x{s})+\beta\mu \x{s} &\text{for j=i-1}\\
1-2 \x{s}(1-\x{s})-\beta\mu \x{s} - \alpha\mu \x{s} &\text{for j=i }\\
\x{s}(1-\x{s})+\alpha\mu \x{s} &\text{for j=i+1}\\
0 \qquad &\text{otherwise}\,.
\end{cases}
\end{equation}
Following convention, we re-scale the Moran events to bring the birth-death rate to $N^2$, and equivalently time becomes $t=s/N$ (note that this differs from the otherwise identical treatment in \citep[][formula~2.10]{Ethe11}, where time is scaled by $\binom{N}{2}$ to match the rate of the coalescent in the Wright-Fisher model). We also set the overall scaled mutation rate to $\theta=N(\mu_0+\mu_1)$. Then, the law of total probability allows us to write the forward N-particle generator of the Moran process as \citep[][formula~75]{Berg17}:
\begin{equation}\label{eq:forw_partly_cont_mutation}
\begin{split}
{\cal L}_N \Pr(N \x{t}) &=
\alpha \theta \big((N-i+1)\Pr(N\x{t}=i-1)- (N-i)\Pr(N\x{t}=i)\big)\\
\qquad+ \beta \theta &\big((i+1)\Pr(N\x{t}=i+1)- i\Pr(N\x{t}=i)\big)\\
\qquad+ &\big((i-1)(N-i+1)\Pr(N\x{t}=i-1)\\
&+(i+1)(N-i-1)\Pr(N\x{t}=i+1) - 2i(N-i)\Pr(N\x{t}=i) \big)\,.
\end{split}
\end{equation}
Note that that the terms have been collected so that the first two summands account for mutation events and the remaining for genetic drift.
Suppose a sample of size $K$, with $0 \leq K \leq N$, is drawn from the population at the current time $t=0$. The likelihood of observing $\y$ of the focal alleles in the sample follows a hypergeometric distribution:
\begin{equation}\label{eq:hypergeometric}
\Pr(y \given K,N, Nx =i,t=0)= \frac{\binom{i}{y}\binom{N-i}{K-y}}{\binom{N}{K}}\,.
\end{equation}
Calculation of sample allele configurations at previous times requires the backward N-particle generator (note the generator ${\cal L}_N^{'}$ operating on the sample allele frequency in Eq.~(\ref{eq:hypergeometric}) below is defined equivalently to the forward generator ${\cal L}_N$ in Eq.~(\ref{eq:forw_partly_cont_mutation}) acting on the transition density from Eq.~(\ref{eq:transition_decoupled_Moran})):
\begin{equation}\label{eq:backw_partly_cont_mutation}
\begin{split}
&{\cal L}_N^{'}\Pr(\y \given K, Nx=i,t) = \\
&\qquad \alpha \theta (N-i) \bigg(\Pr(\y\given K, Nx=i+1, t)-\Pr(y\given K, Nx=i, t)\bigg)\\
&\qquad+ \beta \theta i \bigg(\Pr(\y\given K, Nx=i-1,t)-\Pr(\y\given K, Nx=i, t)\bigg)\\
&\qquad+ i(N-i) \bigg(\Pr(\y\given K, Nx=i+1,t)+(\Pr(\y\given K, Nx=i-1, t)\\
&\qquad\quad-2\Pr(\y\given K, Nx=i, t)\bigg)\,.
\end{split}
\end{equation}
\subsection{The diffusion limit}
\label{section:diffusion_limit}
Passing to the diffusion limit $N \to \infty$ is comparatively straightforward within the decoupled Moran model \emph{vs} the classic Moran model (see \citep[][chapt.~4]{Ewen04} for the derivation assuming the classic Wright-Fisher model, rescaling suffices to obtain equivalent results for the Moran model):
Set $\delta x=1/N$ and denote the transition rate density of continuous allele proportions as $\phi(x\given t)$. Then, the continuous forward N-particle generator is (compare: Eq.~(\ref{eq:forw_partly_cont_mutation}) \citep[][formula~76]{Berg17}:
\begin{equation}\label{eq:forw_partly_cont_mutation_2}
\begin{split}
{\cal L}_N\phi(x\given t) &=
\alpha\theta \bigg(\frac{(1-x+\delta x)\phi(x-\delta x\given t) - (1-x)\phi(x\given t)}{\delta x}\bigg)\\
&\qquad+\beta\theta \bigg(\frac{(x+\delta x)\phi(x+\delta x\given t) - x\phi(x\given t)}{\delta x}\bigg)\\
&\qquad+\bigg(\frac{(x-\delta x)(1-x+\delta x)\phi(x-\delta x\given t)}{\delta x^2}\\
&\qquad+ \frac{(x+\delta x)(1-x-\delta x)\phi(x+\delta x\given t)}{\delta x^2}-\frac{2x(1-x)\phi(x\given t)}{\delta x^2}\bigg).\\
\end{split}
\end{equation}
Taking $\lim_{N\to \infty} {\cal L}_N\phi(x\given t)$ and recognizing that each term defines a first or second derivative with respect to $x$ immediately recovers the infinitesimal operator \citep[][formula 2.11]{Ethe11}
\begin{equation}\label{eq:forw_operator}
{\cal L}\phi(x\given t) = -\frac{\partial}{\partial x}\theta(\alpha-x)\phi(x\given t) +\frac{\partial^2}{\partial x^2}x(1-x)\phi(x\given t)\,
\end{equation}
of the Kolmogorov forward (Fokker-Planck) diffusion equation for a general biallelic mutation-drift model:
\begin{equation}\label{eq:forw_mutdrift}
\frac{\partial}{\partial t}\phi(x\given t)={\cal L} \phi(x\given t).
\end{equation}
Again, let us take a population sample of size $K$, $0 \leq K < \infty$, at the current time $t=0$. The likelihood of a sample of size $K$ with $\y$ alleles of the focal type is binomial:
\begin{equation}\label{eq:binomial_likelihood}
\Pr(y\given K,x=\tfrac{i}{N},t=0)=\binom{K}{y}x^{y} (1-x)^{K-y}\,.
\end{equation}
The trajectory of the sample allele proportions backward in time is described by the Kolmogorov backward equation:
\begin{equation}\label{eq:backw_mutdrift}
-\frac{\partial}{\partial t} \Pr(\y\given K, x,t)={\cal L}^{'}\Pr(\y\given K, x,t),
\end{equation}
with the backwards operator derived from Eq.~(\ref{eq:backw_partly_cont_mutation}) \citep[][formula~80]{Berg17}: \begin{equation}\label{eq:backw_operator}
{\cal L}^{'}\Pr(\y\given K, x, t)=
\theta(\alpha-x)\frac{\partial}{\partial x}\Pr(\y\given K, x, t)+x(1-x)\frac{\partial^2}{\partial x^2}\Pr(\y\given K, x, t)\,.
\end{equation}
The discrete probability distribution $\Pr(y\given K,x,t)$ in Eq.~(\ref{eq:backw_mutdrift}) is interpreted as the probability of obtaining the extant sample configuration $(y,K)$ conditional on an allele proportion $x$ at past times $t$ \citep{Berg17}. Note that the negative sign on the left side of the backward diffusion equation Eq.~(\ref{eq:backw_mutdrift}) (opposite to \citep{Ewen04}), ensures compatibility of the direction of time between the forward and backward Kolmogorov equations \citep{Zhao13a}.
\subsubsection{Modified Jacobi polynomials}
\label{section:GeneralMutationDiffusion}
Obtaining explicit analytical representations of the transition density at different time steps is a non-trivial problem. We will use the method of \citet{Song12} as adapted by \citet{Berg17} and introduce the (modified) Jacobi polynomials (compare also formula~22.3.2 in \citep{Abra70}):
\begin{equation}\label{eq:Jacobi_modified}
R_n^{(\alpha,\theta)}(x)=\sum_{l=0}^n(-1)^l\frac{\Gamma(n-1+l+\theta)\Gamma(n+\alpha\theta)}{\Gamma(n-1+\theta)\Gamma(l+\alpha\theta)l!(n-l)!}x^l\,,
\end{equation}
where $n$, with $0\leq n\leq\infty$, is the order of the polynomial. Note that any regular polynomial of order $n$ can be represented as a weighted sum of the above modified Jacobi polynomials.
The modified Jacobi polynomials for any orders $m,n$, fulfill the following orthogonality relationship with respect to the weight function $w(x,\alpha,\theta)=x^{\alpha\theta-1}(1-x)^{\beta\theta-1}$:
\begin{equation}\label{eq:ortho_Jacobi}
\int_0^1 R_n^{(\alpha,\theta)}(x) R_m^{(\alpha,\theta)}(x)\, w(x,\alpha,\theta)\,dx=\delta_{n,m} \Delta_n^{(\alpha,\theta)}\,,
\end{equation}
where $\delta_{n,m}$ is Kronecker's delta, and
\begin{equation}
\Delta_n^{(\alpha,\theta)}=\frac{\Gamma(n+\alpha\theta)\Gamma(n+\beta\theta)}{(2n+\theta-1)\Gamma(n+\theta-1)\Gamma(n+1)}\,
\end{equation}
is the proportionality constant.
The forward and backward Kolmogorov operators (Eqs.~(\ref{eq:forw_mutdrift}) and (\ref{eq:backw_mutdrift}), respectively) can be conveniently decomposed with the modified Jacobi polynomials as eigenfunctions (for details see \citep{Song12,Berg17}). The forward operator becomes:
\begin{equation}
-\lambda_n w(x,\alpha,\theta) R_n^{(\alpha,\theta)}(x)={\cal L} w(x,\alpha,\theta) R_n^{(\alpha,\theta)}(x)\,,
\end{equation}
and the backward operator:
\begin{equation}
\lambda_n R_n^{(\alpha,\theta)}(x)={\cal L}^{'}R_n^{(\alpha,\theta)}(x)\,,
\end{equation}
with corresponding eigenvalues
\begin{equation}\label{eq:eigenvalues}
\lambda_n=n(n+\theta-1)\,.
\end{equation}
\subsection{Forward-backward algorithm}
Forward-backward algorithms are dynamic programming techniques that enable the efficient calculation of model states at any time from a sequence of observations. In our case, we have information from a population sample at the current time and aim to infer the distribution of past population allele frequencies. As shown in \citet{Berg17}, the forward-backward algorithm classically used for hidden Markov models \citep{RabinerJuang86, Vogl10} can be readily applied to the discrete decoupled Moran model of Sec.~(\ref{section:discrete_Moran}): the population allele proportions are considered `hidden' states and the sample allele configurations `emitted'. In Sec.~(\ref{section:diffusion_Moran_for_back}), we again follow \citet{Berg17} in establishing that a forward-backward algorithm can be constructed for the diffusion model of Sec.~(\ref{section:diffusion_limit}) if the diffusion model is represented using the modified Jacobi polynomials.
\subsubsection{Discrete Moran model}
\label{section:discrete_Moran_for_back}
We here reproduce the outline of the forward-backward algorithm for the discrete decoupled Moran model \citep{Berg17}.
\paragraph{Forward in time}
We start at time $s=S$ and assume that the allele proportions of the ancestral population are distributed according to an arbitrary distribution $\bs{\rho}(x)$. Recall that the beta distribution describes the allele proportions of a bi-allelic, general mutation Moran model in equilibrium \citep{Wrig31}. Multiplying with the binomial sampling likelihood and integrating over allele proportions results a beta-binomial compound distribution as marginal likelihood. For the biallelic Moran model, the following beta-binomial distribution with arbitrary mutation-drift parameters is therefore the standard prior $\bs{\rho}(x)$:
\begin{equation}\label{eq:beta-binomial_prior}
\begin{split}
\Pr(N x = i \given N,\alpha,\theta)\\
&=\binom{N}{i}\,\frac{\Gamma(\theta)}{\Gamma(\alpha\theta)\Gamma(\beta\theta)}\frac{\Gamma(i+\alpha\theta)\Gamma(N-i+\beta\theta)}{\Gamma(N+\theta)}\,.
\end{split}
\end{equation}
Any starting distribution can be represented as a row vector of probabilities $\fv{S}= \bs{\rho}(x)$, where each entry corresponds to the probability of allele proportions being $0$, $1$,...,$N$. The probabilities of allele proportions at any time between $s=S$ and $s=0$ given our prior distribution, $\fv{s}=\Pr(N x_{s} \given \bs{\rho})$, can be determined via:
\begin{equation}
\fv{s+1} = \fv{s}\mathbf{T}_{i,j} \quad (S \le s < 0),
\end{equation}
where ${T}_{i,j}$ is the transition matrix defined in Eq.~(\ref{eq:transition_decoupled_Moran}).
At $s=0$, the entries of row vector $\bv{0_i}$ are given by the hypergeometric sampling scheme from Eq.~(\ref{eq:hypergeometric}) for each possible extant focal allele proportion between $0$ and $N$. The marginal likelihood of the observed sample allele frequency is then:
\begin{equation}\label{eq:marg_lh}
\begin{split}
\Pr(\y \given K,x,\bs{\rho})&=\fv{0}\bv{0}'\\
&= \fv{S} \mathbf{T}_{i,j}^{|S|} \bv{0}'\,.
\end{split}
\end{equation}
\paragraph{Backward in time}
The same marginal likelihood may be obtained by recursing backwards from our sampling step at $s=0$ with initial probabilities $\bv{0_i}$ to the ancestral population state at $s=S$. Define entries of the row vector $\bv{s,i}=\Pr(y \given K ,N, Nx_s=i)$---they can be interpreted as the probability of the data given the population allele proportion at time $s$. We can recurse back in time by:
\begin{equation}
\begin{split}
\bv{s}' = \mathbf{T}_{i,j} \bv{s+1}' \quad (0 \geq s > S)\,,
\end{split}
\end{equation}
At $s=S$, we again obtain the marginal likelihood Eq.~(\ref{eq:marg_lh}).
\paragraph{Joint and conditional probabilities} At any time $s$, the joint probability of the population allele proportion $\x{s} = \tfrac{i}{N}$, and the sample allele frequency $y$ conditional on the starting distribution $\bs{\rho}$ is:
\begin{equation}\label{eq:joint_xy_discr}
\Pr(\x{s}=\tfrac{i}{N},y \given \bs{\rho}) = (\fv{s})_i (\bv{s})_i\,.
\end{equation}
Furthermore, the probability of the population allele proportions $\x{s} = i/N$ conditional on both the sample allele frequency and the starting distribution is:
\begin{equation}\label{eq:cond_x|y_discr}
\Pr(\x{s}=\tfrac{i}{N} \given y ,\bs{\rho}) = \frac{(\fv{s})_i (\bv{s})_i}{\fv{s}\bv{s}'}\,.
\end{equation}
\paragraph{Summary} The forward-backward algorithm with the Moran model conforms to the canonical situation \citep{RabinerJuang86}: At each time point, the population allele proportions comprise $N$ hidden states and the transition matrix $\mathbf{T}_{i,j}$ is of dimension $N\times N$. Conditional on an observed sample at the current time $s=0$ and a prior distribution on the ancestral allele configuration, the distribution of past and current population allele proportions can be determined. In population genetics, population demographic events are usually modeled to occur at a specific time in the past. Note that driving changes in mutation parameters can be incorporated into this approach by assuming different parameters for the prior distribution at $s=S$ than for the transition rate matrix and therefore the times $S < s\leq 0$. Further changes in population demography may be modeled by time-dependent transition matrices.
\subsubsection{Diffusion Model}
\label{section:diffusion_Moran_for_back}
We now adapt the forward-backward algorithm to the diffusion model described in Sec.~(\ref{section:diffusion_limit}) using the modified Jacobi polynomials from Sec.~(\ref{section:GeneralMutationDiffusion}).
\paragraph{Forward in time}
Suppose that at time $t=S$, with $S\leq 0$, the distribution of ancestral allele proportions is given by the arbitrary distribution $\bs{\rho}(x)=\phi(x\given t=S)$. This distribution is often assumed to be the beta equilibrium distribution for the bi-allelic Moran model \citep{Wrig31}:
\begin{equation}\label{eq:bb}
\bs{\rho}(x)=\frac{\Gamma(\theta)}{\Gamma(\alpha\theta)\Gamma(\beta\theta)}\,x^{\alpha\theta-1}(1-x)^{\beta\theta-1}\,.
\end{equation}
(Note that the weight function of the modified Jacobi polynomials is proportional to this beta distribution.) The allele proportions further forward in time, $\phi(x\given t,\bs{\rho})$, are determined by the forward diffusion equation Eq.~(\ref{eq:forw_mutdrift}). The solution to the forward equation can be represented using modified Jacobi polynomials; the ancestral allele proportion distribution is first expanded to:
\begin{equation}
\bs{\rho}(x)=\sum_{n=0}^\infty \rho_n^{(\alpha,\theta)} R_n^{(\alpha,\theta)}(x),
\end{equation}
where the $\rho_n^{(\alpha,\theta)}$ are a possibly infinite number of Jacobi coefficients that depend on $\bs{\rho}(x)$. More explicitly:
\begin{equation}\label{eq:rho_coeffs}
\rho_n^{(\alpha,\theta)}= \frac{1}{\Delta_n^{(\alpha,\theta)}} \int_{0}^1 w(x,\alpha,\theta) R_n^{(\alpha,\theta)}(x)\bs{\rho}(x)\,dx\,.
\end{equation}
We then incorporate temporal dynamics and obtain the full solution:
\begin{equation}\label{eq:forward_given_rho}
\phi(x\given t,\bs{\rho})=w(x, \alpha, \theta) \sum_{n=0}^\infty \rho_n^{(\alpha,\theta)} R_n^{(\alpha,\theta)}(x) e^{\lambda_n (S-t)}\,.
\end{equation}
\paragraph{Backward in time}
Backward in time, we again start with our sample: The binomial likelihood of the sampled allele proportions at time $t=0$ in Eq.~(\ref{eq:binomial_likelihood}) is expressed as a regular polynomial up to order $K$ with coefficients $$a_{j=y+i}(K,y)=(-1)^i\binom{K}{y}\binom{K-y}{i}$$ for $0\leq i\leq K-y$, and zero otherwise.
Let $\mathbf{a}(K,y)$ be the vector of coefficients $a_{j}(K,y)$ and $\mathbf{R}^{(\alpha,\theta)}$ be the matrix of coefficients $R_n^{(\alpha,\theta)}(x)$. Note that this matrix is lower triangular. Then the binomial distribution can be uniquely expanded into Jacobi polynomials via the following linear algebraic equation:
\begin{equation}\label{eq:matr}
\mathbf{d}^{(\alpha,\theta)}(K,y)=\mathbf{a}(K,y)\mathbf{R}^{(\alpha,\theta)}
\end{equation}
Note that the triangular structure of $\mathbf{R}^{(\alpha,\theta)}$ obviates matrix inversion. Now the binomial sampling distribution from Eq.~\ref{eq:binomial_likelihood} can be rewritten:
\begin{equation}\label{eq:binom2jacobi}
\Pr(y\given K,x,\alpha,\theta,t=0)=\binom{K}{y}x^y(1-x)^{K-y}=\sum_{n=0}^K d_n^{(\alpha,\theta)}(K,y) R_n^{(\alpha,\theta)}(x)\,.
\end{equation}
Further back, at times $t$ ($S \le t \le 0$), the distribution of sample proportions is given by:
\begin{equation}\label{eq:binom_cond_t}
\Pr(y\given K,x,\alpha,\theta,t)=\sum_{n=0}^K d_n^{(\alpha,\theta)}(K,y) R_n^{(\alpha,\theta)}(x)e^{\lambda_n t}\,.
\end{equation}
Using the orthogonality of the Jacobi polynomials (Eq.~\ref{eq:ortho_Jacobi}), the continuous marginal likelihood becomes:
\begin{equation}\label{eq:marg_like_general}
\begin{split}
\Pr(\y\given K,\alpha,\theta,S,\bs{\rho})&= \int_0^1\phi(x\given t,\bs{\rho}) \Pr(y\given K,x, \alpha,\theta,t=0)\,dx\\
&=\sum_{n=0}^K \rho_n^{(\alpha,\theta)} d_n^{(\alpha,\theta)}(K,y) \Delta_n^{(\alpha,\theta)} e^{\lambda_n S}\,.
\end{split}
\end{equation}
Note that the expansion $\rho_n^{(\alpha,\theta)}$ may be infinite; however, calculation of the marginal likelihood only requires expansion to the order of the sample size $K$.
As briefly noted in the summary of Sec.~(\ref{section:discrete_Moran_for_back}), it is often convenient to be able to account for population demographics. We will consider this possibility for the diffusion approach: Assume the mutation parameters change from $\alpha$ to $\alpha^{*}$ and from $\theta$ to $\theta^{*}$ at time $t=S$. The coefficients of the new polynomial expansions can be obtained by linear transformation. More explicitly, consider a simple model where:
\begin{itemize}
\item{i)} at the present time $t=0$, the sample allele configuration $(y,K)$ is given;
\item{ii)} between the times $S\leq t\leq 0$, the population genetic parameters $\alpha$ and $\theta$ remain constant; and
\item{iii)} at time $t=S$ in the past the population allele proportion is beta distributed according to:
\begin{equation} \label{eq:bb_time_change}
\bs{\rho}(x)=\frac{\Gamma(\theta^{*})}{\Gamma(\alpha^{*}\theta^{*})\Gamma(\beta^{*}\theta^{*})}\,x^{\alpha^{*}\theta^{*}-1}(1-x)^{\beta^{*}\theta^{*}-1}
\end{equation}
In fact, the allele proportion is beta distributed as above between $-\infty\leq t\leq S$ and can be expressed as the series $\rho_n^{(\alpha^{*},\theta^{*})}R_n^{(\alpha^{*},\theta^{*})}(x)$.
\end{itemize}
The continuous marginal distribution then becomes:
\begin{equation}\label{eq:marg_like_general_tc}
\begin{split}
\Pr(\y\given K,\alpha,\theta,S,\bs{\rho})&= \int_0^1\phi(x\given t,\bs{\rho}) \Pr(y\given K,x,\alpha,\theta,t=0)\,dx\\
&=\sum_{n=0}^K \rho_n^{(\alpha^{*},\theta^{*})} d_n^{(\alpha,\theta)}(K,y) \Delta_n^{(\alpha,\theta)} e^{\lambda_n S}\,.
\end{split}
\end{equation}
\paragraph{Joint and conditional probabilities}
At any time $t$, the joint probability of the population allele proportion $\x{t} = \tfrac{i}{N}$, and the number of focal alleles in the sample $y$ conditional on the starting distribution $\bs{\rho}$ can be determined:
$$\Pr(\x{t} = \tfrac{i}{N}, y \given \bs{\rho})=\phi(x\given t,\bs{\rho}) \Pr(y\given K,x,t=0).$$
The probability of the population allele frequencies $\x{t} = \tfrac{i}{N}$ conditional on the both the sample allele proportions and the starting distribution is:
$$
\Pr(\x{t} = \tfrac{i}{N} \given y ,\bs{\rho})= \frac{\Pr(\x{t} = \tfrac{i}{N}, y \given \bs{\rho})}{\Pr(\y\given K,\alpha,\theta,S,\bs{\rho})}\,.
$$
We will not detail these equations here.
\paragraph{Summary} The forward-backward algorithm with the continuous diffusion mo\-del represented using modified orthogonal Jacobi polynomials deviates from the canonical situation: A transition kernel for population allele proportions is employed, which is expanded into an infinite-dimensional system of eigenfunctions and corresponding eigenvalues. For representing sample allele frequencies, however, only an expansion of the order of the sample size is needed. Indeed most problems only require a polynomial expansion up to the order of the sample size and the temporal system required is diagonal and thus extremely simple. Furthermore, a change in the mutation parameters can also be incorporated.
\section{Particle models and orthogonal polynomials}
\label{section:particle}
In this section, we introduce a novel forward-backward algorithm that harnesses together three components:
\begin{itemize}
\item{i)} An approach based on orthogonal polynomials to describe the evolution of the
population allele frequencies forward in time (Sec.~\ref{section:conditional_upsilon_kappa}),
\item{ii)} a so-called particle model that yields the conditional probabilities of the proportion of the focal allele at any point in the history of the sample by running traditionally backwards-looking coalescent arguments not only backward but also forward in time (Sec.~\ref{section:forward-particle}), (Sec.~\ref{section:backward-particle}); this is augmented by
\item{iii)} backward-in-time temporal dynamics accounting for the effect of changing mutation parameters on the sample sizes (Sec.~\ref{section:backward-particle_temp}) to yield probabilities of all past sample configurations.
In total, we will arrive at the following joint probability:
\begin{equation}
\Pr(\upsilon,\kappa\given K,y,\alpha,\theta,t)\,.
\end{equation}
This is the probability of seeing $\upsilon$ focal alleles in a sample of size $\kappa$ at any time $t<0$ in the past, conditional on an extant sample of size $K$ with $y$ focal alleles (and the underlying mutation-drift parameters $\theta$ and $\alpha$) at time $t=0$.
\end{itemize}
We now begin by motivating the particle model.
\paragraph{Sample genealogy backward in time: coalescent}
In the (decoupled) Moran model of population size $N$, time is conventionally scaled so that the genealogies of all individuals in the population conform to the Kingman coalescent. Furthermore, the genealogy of any sample of size $K<N$ is embedded within the (decoupled) Moran model: The sample probabilities and transition rates of the coalescent remain unaffected by a change in sample size. In other words, the classic (decoupled) Moran model is sample consistent, and is dual to the Kingman coalescent in the sense that the expected population allele frequencies are identical between the two models \citep[][chapt.~2.8]{Ethe11}.
In Fig.~(\ref{fig:Moran_coal.eps}), the history of a sample of size $K=6$ with $y=4$ focal alleles is depicted. Both mutation and coalescent events are uninformative regarding the immediately preceding allelic state. Starting at the present time $t=0$ and looking back, the size of the sample $\kappa$ with $K\geq \kappa \geq 0$ is reduced in discrete stages by these events \citep[compare][for a similar argument in the context of the infinite sites model]{Fais15}. With the usual scaling of time, the rate of a coalescent is $\kappa(\kappa-1)$ and that of a mutation $\kappa\theta$, hence the total rate of reduction events is $\kappa(\kappa-1+\theta)$.
At any time $t < 0$ the number of alleles $\kappa$ remaining from the original sample of size $K$ at $t=0$ is a random variable, as is the number of focal alleles remaining from the $y$ focal alleles at $t=0$, these we denote $\upsilon$. The probability of a reduction in the number of focal alleles from $\upsilon$ to $\upsilon-1$ in each backward time step is proportional to $\upsilon(\upsilon-1+\alpha\theta)$ (the contribution of the coalescent event is $\upsilon(\upsilon-1)$, that of the mutation event $\alpha\theta \upsilon$). The analogous reduction probabilities for the non-focal alleles are $(\kappa-\upsilon)(\kappa-\upsilon-1+\beta\theta)$.
Note that these reduction probabilities are for ordered events. However, we will treat allele 'labels' as interchangeable in each time step, and therefore use unordered reduction probabilities. We obtain these by dividing each of the previous ordered reduction probabilities by the number of potential alleles selected for a reduction event.
Let us now reverse the direction of time: Consider the indicator variable $z_{\kappa+1}$, which is one if the $\kappa+1$st allele is of the focal type and zero otherwise.
In each forward step, the probability of going from $\kappa$ to $\kappa+1$ unordered focal alleles is:
\begin{equation}\label{eq:ratio_general}
\begin{split}
{\Pr}(z_{\kappa+1}\given \kappa,\upsilon,\alpha,\theta)&=\frac{(\upsilon+\alpha\theta)^{1-z_\kappa}(\kappa-\upsilon+\beta\theta)^{z_\kappa}}{\kappa+\theta}\\
&=\frac{\Gamma(\kappa+\theta)}{\Gamma(\kappa+1+\theta)}\frac{\Gamma(\upsilon+z_\kappa+\alpha\theta)}{\Gamma(\upsilon+\alpha\theta)}
\frac{\Gamma(\kappa+1-\upsilon-z_\kappa+\beta\theta)}{\Gamma(\kappa-\upsilon+\beta\theta)}\,.
\end{split}
\end{equation}
\newpage
\begin{figure}[!ht]
\centering
\includegraphics[width = 9cm]{Plot_Moran_coal.eps}
\caption{Schematic plot of a Moran model and embedded sample genealogy: The allelic type of the extant individuals is given at the top of the plot; backwards in time the arrows indicate birth/death events and the 'x' are mutation events. Note that the position and numbering of individuals on the lower x-axis is arbitrary. The bold lines and arrows correspond to the coalescent and mutation history of the sample respectively.}
\label{fig:Moran_coal.eps}
\end{figure}
\newpage
\paragraph{Sample genealogy forward in time: urn models}
Let us briefly view the probability of sample configurations running forward in time as an urn model similar to the Polya- or Hoppe-urns or the urn-models in \citep{Step00}. Hereto, we introduce a new time index $t_{\kappa-1}$ that runs from $t_0=0$, a time in the past at which only one individual from the sample is present in the population, to the present time $t_K$. The urn model is then given by the following algorithm:
\begin{itemize}
\item {Initiation.} Start with a sample of size $\kappa=1$ at time $t_0=0$. The probability of the focal allele is set to $\alpha$.
\item{Recursion.} Add a period of rate $\kappa(\kappa-1+\theta)$ to $t_{\kappa-1}$ obtain $t_{\kappa}$. Increase the sample size from $\kappa$ to $\kappa+1$ by a focal allele with probability
\begin{equation}
p_{\upsilon+1}=\frac{\upsilon + \alpha\theta}{\kappa+\theta}\,,
\end{equation}
or increase the sample size by a non-focal allele with probability $1-p_{\upsilon+1}$. Together this corresponds to the probability of the indicator variable in Eq.~(\ref{eq:ratio_general}).
\item{Stop.} When the sample size $\kappa=K$ is reached, add a period of rate $K(K-1+\theta)$ to $t_{K-1}$ obtain $t_K$.
\end{itemize}
In order to obtain our usual time index with an extant time of zero, $t_K$ must be subtracted from all $t_{\kappa-1}$. Thus running the coalescent process forwards in time as in Eq.~(\ref{eq:ratio_general}) yields an urn model.
\subsection{Particle model: algorithms}
We will now demonstrate how running coalescent arguments for the genealogy of an unordered sample from a decoupled Moran model both forward and backward in time can be used to determine sample allele frequencies for every past sample size (Sec.~\ref{section:forward-particle}), (Sec.~\ref{section:backward-particle}). Note that the past sample sizes are initially assumed given because they are determined by the constant rate of coalescence. Clearly, these sample sizes depend on mutation-drift parameters that are not necessarily constant across time. Note, however, that the number of focal alleles is conditionally independent of time given the (past) sample sizes $\kappa$ and the mutation-drift parameters. Hence we first treat the time independent dynamics, and then augment this with differential backward equations (Sec.~\ref{section:backward-particle_temp}) to account for changing mutation-drift parameters.
\subsection{Particle model: algorithm for forward probabilities}
\label{section:forward-particle}
For every $0\leq \kappa\leq K$, let $f(.,\kappa)$ denote a forward row vector of length $\kappa+1$. Each entry $f(\upsilon,\kappa)$ can be interpreted as a probability $\Pr(\upsilon\given \kappa,\alpha,\theta)$, where $0\leq \upsilon\leq \kappa$.
\begin{itemize}
\item{Initiation.} Start with a sample of size $\kappa=0$. Trivially, the frequency of the focal allele is $\upsilon=0$, so set $f(\upsilon=0,\kappa=0)=1$.
\item{Recursion.} Move from sample size $\kappa$ to $\kappa+1$ by calculating every entry of $f(.,\kappa+1)$:
\begin{equation}
\begin{split}
f(\upsilon,\kappa+1)&= \frac{\kappa-\upsilon-1+\beta\theta}{\kappa+\theta}\,f(\upsilon,\kappa-1)+\frac{\upsilon-1+\alpha\theta}{\kappa+\theta}\,f(\upsilon-1,\kappa-1)\,.
\end{split}
\end{equation}
\item{Stop.} Stop when the sample size $\kappa=K$ is reached.
\end{itemize}
With time independent mutation-drift parameters, $f(\upsilon,\kappa)$ is a beta-binomial compound distribution:
\begin{equation}\label{eq:beta-binomial}
\begin{split}
f(\upsilon,\kappa)&=\Pr(\upsilon\given\kappa,\alpha,\theta)\\
&=\binom{\kappa}{\upsilon}\,\frac{\Gamma(\theta)}{\Gamma(\alpha\theta)\Gamma(\beta\theta)}\frac{\Gamma(\upsilon+\alpha\theta)\Gamma(\kappa-\upsilon+\beta\theta)}{\Gamma(\kappa+\theta)}\,.
\end{split}
\end{equation}
\subsection{Particle model: algorithm for backward probabilities}
\label{section:backward-particle}
For every $0\leq \kappa\leq K$, introduce the backward row vector $b(.,\kappa)$ of length $\kappa+1$. Every entry $b(\upsilon,\kappa)$ corresponds to ${\Pr}(y\given K,\kappa,\upsilon,\alpha,\theta)$ for $0\leq \upsilon\leq \kappa$, which is the probability of the observed number of focal alleles given past allele configurations.
\begin{itemize}
\item {Initiation.} Start with a sample of size $K$ with $y$ focal alleles. Set $\upsilon=y$ and $\kappa=K$, and therefore $b(\upsilon=y,\kappa=K)=1$.
\item{Recursion.} Move from $\kappa+1$ to $\kappa$ by calculating the entries of $b(.,\kappa)$:
\begin{equation}
\begin{split}
b(\upsilon,\kappa)&= \frac{\kappa-\upsilon+\beta\theta}{\kappa+\theta}\,b(\upsilon,\kappa+1)+\frac{\upsilon+\alpha\theta}{\kappa+\theta}\,b(\upsilon+1,\kappa+1)\,.
\end{split}
\end{equation}
\item{Stop.} End the recursion, when the sample size $\kappa=0$ is reached. Note that $b(\upsilon=0,\kappa=0)={\Pr}(y\given K,\upsilon=0,\kappa=0,\alpha,\theta)$ corresponds to the likelihood ${\Pr}(y\given K,\alpha,\theta)$.
\end{itemize}
With time independent mutation-drift parameters, $b(\upsilon,\kappa)$ is again a beta-binomial compound distribution:
\begin{equation}\label{eq:beta_binom_upsilon_kappa}
\begin{split}
b(\upsilon,\kappa)&=\Pr(y\given K,\kappa,\upsilon,\alpha,\theta)\\
&=\binom{K-\kappa}{y-\upsilon}\,\frac{\Gamma(\kappa+\theta)}{\Gamma(\upsilon+\alpha\theta)\Gamma(\kappa-\upsilon+\beta\theta)}\frac{\Gamma(y+\alpha\theta)\Gamma(K-y+\beta\theta)}{\Gamma(K+\theta)}\,.
\end{split}
\end{equation}
\subsection{Particle model: time independent conditional probabilities of focal alleles}
We can now combine the forward and backward algorithms and calculate the likelihood of seeing $y$ focal alleles in a sample of size $K$ given time independent mutation-drift parameters (note the use of Vandermonde's identity to obtain the final result):
\begin{equation}
\begin{split}
\Pr(y\given K,\alpha,\theta)&=\sum_{\upsilon=0}^\y f(\upsilon,\kappa)b(\upsilon,\kappa)\\
&=\sum_{\upsilon=0}^\y
\binom{\kappa}{\upsilon}\,\frac{\Gamma(\theta)}{\Gamma(\alpha\theta)\Gamma(\beta\theta)}\frac{\Gamma(\upsilon+\alpha\theta)\Gamma(\kappa-\upsilon+\beta\theta)}{\Gamma(\kappa+\theta)}\\
&\quad\times\binom{K-\kappa}{y-\upsilon}\,\frac{\Gamma(\kappa+\theta)}{\Gamma(\upsilon+\alpha\theta)\Gamma(\kappa-\upsilon+\beta\theta)}\frac{\Gamma(y+\alpha\theta)\Gamma(K-y+\beta\theta)}{\Gamma(K+\theta)}\\
&=\sum_{\upsilon=0}^\y\binom{\kappa}{\upsilon}\binom{K-\kappa}{y-\upsilon} \frac{\Gamma(\theta)}{\Gamma(\alpha\theta)\Gamma(\beta\theta)}\frac{\Gamma(y+\alpha\theta)\Gamma(K-y+\beta\theta)}{\Gamma(K+\theta)}\\
&=\binom{K}{y} \frac{\Gamma(\theta)}{\Gamma(\alpha\theta)\Gamma(\beta\theta)}\frac{\Gamma(y+\alpha\theta)\Gamma(K-y+\beta\theta)}{\Gamma(K+\theta)}\,.
\end{split}
\end{equation}
Similarly, the probability of seeing $\upsilon$ focal alleles in a given past sample of size $\kappa$ assuming a current sample of size $K$ containing $y$ focal alleles and time independent mutation-drift parameters is:
\begin{equation}\label{eq:conditional_upsilon_given kappa}
\Pr(\upsilon\given K,y,\kappa,\alpha,\theta)=
\frac{f(\upsilon,\kappa)b(\upsilon,\kappa)}{\sum_{\upsilon=0}^\kappa f(\upsilon,\kappa)b(\upsilon,\kappa)}\,.
\end{equation}
We provide a visual example of the likelihood of the focal locus traced backward in time in Fig.~(\ref{fig:num_example_equilibrium}).
\newpage
\begin{figure}[!ht]
\centering
\includegraphics[width = 11.5cm]{plot_coalescent_jacobi.eps}
\caption{Consider a sample of size $K = 10$ with $y=5$ focal alleles, \ie\ a starting configuration of $(5,10)$. Assume a mutation bias of $\alpha = 0.3$ towards the focal allele and an overall scaled mutation rate of $\theta = 0.1$. Here, we show the conditional probabilities of all possible particle configurations between $(5,10)$ and $(0,0)$ as per Eq.~(\ref{eq:conditional_upsilon_given kappa}). This is an application of the forward- backward particle model in Secs.~(\ref{section:forward-particle}) and (\ref{section:backward-particle}).
}
\label{fig:num_example_equilibrium}
\end{figure}
\newpage
\subsection{Particle model: temporal dynamics}
\label{section:backward-particle_temp}
So far, we have always assumed the past sample sizes $\kappa$ as given. However, they can easily be modelled as a pure death process dependent on mutation and drift: Both events simply reduce the sample size by one.
The following system of differential equations describes the evolution of the sample size $\kappa$ backward in time:
\begin{equation}\label{eq:temp_system_back}
\begin{split}
-\frac{d}{dt}\Pr(\kappa=K\given K,\theta,t)&=\lambda_K\,\Pr(\kappa=K\given K,\theta,t)\,, \quad\text{and}\\
-\frac{d}{dt}\Pr(\kappa\given K,\theta,t)&=-\lambda_{\kappa+1}\,\Pr(\kappa+1\given K,\theta,t)\\
&\qquad+\lambda_{\kappa}\,\Pr(\kappa\given K,\theta,t)\,, \quad\text{for $K>\kappa\geq 0$.}
\end{split}
\end{equation}
with corresponding eigenvalues $\lambda_\kappa=\kappa(\kappa-1+\theta)$ (as in Eq.~(\ref{eq:eigenvalues})).
The starting condition is: $\Pr(\kappa=K\given K,\theta, t=0)=1$, and $\Pr(K>\kappa\geq 0\given K,\theta,t=0)=0$. Then the solution of the above system of equations is \citep[][chapter 6, Eq.~2.2]{TaylorKarlin98}
\begin{equation}\label{eq:temp_system_back_solution}
\begin{split}
\Pr(\kappa=K\given K,\theta,t)&=e^{\lambda_K t}\,,\\
\Pr(\kappa\given K,\theta,t)&=\sum_{i=\kappa}^{K}c_{i,\kappa}\,e^{\lambda_i t}\, \text{, for $K-1\geq \kappa\geq 0$}\\
&\text{with $c_{i,\kappa}=\frac{\prod_{j=\kappa+1}^K \lambda_j}{\prod_{j=\kappa,j\neq i}^K(\lambda_j-\lambda_i)}$.}
\end{split}
\end{equation}
\subsection{Particle model: total backward dynamics}
Augmenting the time independent backward variables from Eq.~(\ref{eq:beta_binom_upsilon_kappa}) with the temporal dynamics from the previous subsection, \ie\ Eq.~(\ref{eq:temp_system_back_solution}), we obtain the joint probability of the number of focal alleles in the extant sample and the past sample sizes:
\begin{equation}\label{eq:beta_binom_upsilon_kappa_t}
\Pr(y,\kappa\given K,\upsilon,\alpha,\theta,t)=
\Pr(y\given K,\kappa,\upsilon,\alpha,\theta)\Pr(\kappa\given K,\theta,t)\,.
\end{equation}
\subsection{Particle model: joint and marginal probabilities}
\label{joint_y_upsilon_kappa}
The aim of the forward-backward algorithm is to obtain the probability of ancestral sample configurations $(\upsilon,\kappa)$ at arbitrary times $t$, conditional on the sample configuration $(y,K)$ at time $t=0$, \ie{} $\Pr(\upsilon,\kappa\given K,y,\dots)$. Multiplying the probability in Eq.~(\ref{eq:beta_binom_upsilon_kappa_t}), $\Pr(y,\kappa\given K,\upsilon,\dots)$, with the probability of the frequency of the focal allele $\upsilon$ given a past sample size $\kappa$ and population proportion $x$ gives us the joint probability $\Pr(y,\upsilon,\kappa\given K,x,\dots)$. Recall that the probabilities in Eq.~(\ref{eq:beta_binom_upsilon_kappa_t}) are for unordered samples, since we treat the allele 'labels' as interchangeable in each time step within the sample genealogy. Obtaining the number of focal alleles $\upsilon$ in an ancestral sample from a given population allele proportion $x$, however, requires ordered sampling (instead of binomial \ie{} unordered sampling):
\begin{equation}\label{eq:ordered_bin}
\Pr(\upsilon\given \kappa,x)^o=x^\upsilon(1-x)^{\kappa-\upsilon}\,,
\end{equation}
to obtain:
\begin{equation}\label{eq:joint_dist}
\Pr(y,\upsilon,\kappa\given K,x,\alpha,\theta,t)=
\Pr(y\given K,\kappa,\upsilon,\alpha,\theta)\Pr(\upsilon\given \kappa,x)^o\Pr(\kappa\given K,\theta,t)\,.
\end{equation}
The marginal distribution of the number of focal alleles $y$ in the extant sample of size $K$ is then determined by:
\begin{equation}\label{eq:marg_particle}
\Pr(y\given K,x,\alpha,\theta,t)= \sum_{\kappa=0}^{K}\sum_{\upsilon=\max(\kappa+y-K,0)}^{\min(\kappa,y)} \Pr(y,\upsilon,\kappa\given K,x,\alpha,\theta,t)\,,
\end{equation}
for every $t<0$. This is a sum over all possible $(K+1)(y+1)$ allele configurations. For $t\to -\infty$, the sum becomes dominated by the probability for the case $\kappa=0$ since the likelihood in Eq.~(\ref{eq:ordered_bin}) is trivially one for $\kappa=0$ irrespective of $x$. Hence, this can be used to obtain the likelihood:
\begin{equation}
\Pr(y\given K,x,\theta,\alpha,t\to\infty)=
\Pr(y\given K,\theta,\alpha)\,.
\end{equation}
\subsubsection{Relationship between particle probabilities and Jacobi polynomials}
\label{section:particle_jacobi}
Recall that we previously obtained an expression for Eq.~(\ref{eq:marg_particle}) in terms of modified Jacobi polynomials (Eq.~\ref{eq:binom_cond_t}), which we reproduce here:
\begin{equation}\label{eq:orthopolynomial}
\Pr(y\given K,x,\alpha,\theta,t)=\sum_{n=0}^K e^{\lambda_n t} d_n(K,y)^{(\alpha,\theta)} R_n^{(\alpha,\theta)}(x)\,.
\end{equation}
Clearly, these two representations must be equivalent: The order of expansion of the modified Jacobi polynomials is generally equivalent to the sample size, so we can set $\kappa$ to $n$ in Eq.~(\ref{eq:marg_particle}). Now, for each past sample size $\kappa$ (or now $n$), the terms with the same temporal component $e^{\lambda_n t}$ and the same power of $x$ from Eq.~(\ref{eq:marg_particle})(left below) and Eq.~(\ref{eq:orthopolynomial})(right below) can be equated:
\begin{equation}
\begin{split}
e^{\lambda_n t}x^n c_{n,n}\sum_{\upsilon=\max(n+y-K,0)}^{\min(\kappa,y)}(-1)^{n-\upsilon}b(\upsilon,n)&= e^{\lambda_n t}x^n d_n(K,y)^{(\alpha,\theta)} r_{n,n} \\
c_{n,n}\sum_{\upsilon=\max(n+y-K,0)}^{\min(\kappa,y)}(-1)^{n-\upsilon}b(\upsilon,n)&= d_n(K,y)^{(\alpha,\theta)} r_{n,n} \,
\end{split}
\end{equation}
where the $R_n^{(\alpha,\theta)}(x)$ are written as the term $x^n$ multiplied by the corresponding coefficient $r_{n,l}$, and $c_{n,n}$ are the $n$th coefficients solving the system of temporal differential equations in Eq.~(\ref{eq:temp_system_back}). For $y=0$ or $y=K$, the above formula simplifies as there only a single term remains in the summation.
\subsection{Past allele configurations}
\label{section:conditional_upsilon_kappa}
In order to determine the distribution of allele configurations at any given point in time conditional on a current sample, we must combine the total backwards dynamics of a sample given in Eq.~(\ref{eq:beta_binom_upsilon_kappa_t}) with repeated sampling from the general population forward in time. Note that the latter cannot be modelled using a particle model: The sample sizes in the particle model - both current and past - are assumed fixed until augmented backward in time by temporal dynamics accounting for mutation-drift parameters; importantly, these temporal components are conditional on the extant sample size. When sampling from the population forward in time and agnostic to either past or future sample sizes and configurations, however, the sample size $\kappa$ becomes a random variable.
The population allele proportion can be modelled forward in time by expanding the transition density $\phi(x\given t,\bs{\rho})$ into orthogonal polynomials as in Eq.~(\ref{eq:forward_given_rho}); $\bs{\rho}$ here represented the beta equilibrium distribution of the bi-allelic Moran model.
Further, the likelihood of observing $\upsilon$ alleles of the focal type in a sample of size $\kappa$ conditional on the population allele frequency $x$ can also be expanded into the orthogonal polynomials:
\begin{equation}
\Pr(\upsilon\given \kappa,x)=\binom{\kappa}{\upsilon} x^{\upsilon}(1-x)^{\kappa-\upsilon}=\sum_{n=0}^\kappa {d}_n(\kappa,\upsilon) R_n^{(\alpha,\theta)}(x)\,.
\end{equation}
Then the joint probability of the focal alleles $y$ in the extant sample and the past particle configurations becomes:
\begin{equation}\label{eq:joint_y_i}
\begin{split}
&\Pr(y,\upsilon,\kappa\given K,\alpha,\theta,t,\bs{\rho})\\
&\qquad=\Pr(y,\kappa\given K,\upsilon,\alpha,\theta,t) \int_0^1 \Pr(\upsilon\given \kappa,x)\,\phi(x\given t,\bs{\rho})\,dx\\
&\qquad=\Pr(\upsilon\given K,y,\kappa,\alpha,\theta)\,\Pr(\kappa\given K,\theta,t)\\
&\qquad\qquad \times\sum_{n=0}^\kappa \,e^{\lambda_j(S-t)}\int_0^1 {d}_n(\kappa,\upsilon) R_n^{(\alpha,\theta)}(x) \rho_j^{(\alpha,\theta)} R_n^{(\alpha,\theta)}(x)
x^{\alpha\theta-1}(1-x)^{\beta\theta-1}\,dx\\
&\qquad= \Pr(\upsilon\given K,y,\kappa,\alpha,\theta)\,\Pr(\kappa\given K,\theta,t)\,\sum_{n=0}^\kappa e^{\lambda_j(S-t)} {d}_n(\kappa,\upsilon)\rho_n^{(\alpha,\theta)} \Delta_n^{(\alpha,\theta)}\,.
\end{split}
\end{equation}
Note that the first term can be calculated using our forward-backward particle algorithm in Eq.~(\ref{eq:conditional_upsilon_given kappa}), the second is the solution of the temporal system in Eq.~(\ref{eq:temp_system_back_solution}).
Finally, the probability of ancestral allele configurations at any time, conditional on an extant sample, becomes:
\begin{equation}\label{eq:joint_y_i_2}
\begin{split}
\Pr(\upsilon,\kappa \given K,y, \alpha,\theta,t,\bs{\rho})&=\frac{\Pr(y,\upsilon,\kappa\given K,\alpha,\theta,t,\bs{\rho})}{\Pr(y\given K,\alpha,\theta,t,\bs{\rho})}\,,
\end{split}
\end{equation}
where the denominator can be determined using either
\begin{itemize}
\item{(i)} the particle approach via
$$\Pr(y\given K,\alpha,\theta,t,\bs{\rho})=\Pr(y\given K,x,\alpha,\theta,t)\int_0^1\phi(x\given t,\bs{\rho})\,dx\\$$
with the first term from Eq.~(\ref{eq:marg_particle}), or
\item{(ii)} the equivalent polynomial approach in Eq.~(\ref{eq:marg_like_general}).
\end{itemize}
Changes in mutation parameters over time can also be incorporated in this forward-backward algorithm: Once again consider the scenario in which the mutation parameters $\alpha$ and $\theta$ remain constant between the times $S\leq t\leq 0$, and at time $t=S$ in the past they change to $\alpha^*$ and $\theta^*$. Recall that the ancestral allele configuration $\bs{\rho}$ is then modelled as beta-binomial with parameters $\alpha^*$ and $\theta^*$ instead of $\alpha$ and $\theta$ (Eq.~(\ref{eq:bb_time_change})), and this can be substituted into the equations of this subsection accordingly (directly in Eq.~(\ref{eq:joint_y_i}) and (i), and see Eq.~(\ref{eq:marg_like_general_tc}) for (ii)). An example is shown in Fig.~(\ref{fig:num_example_non-equilibrium}).
\paragraph{Summary} The forward-backward algorithm using a combination of the particle model and orthogonal polynomials deviates considerably from the canonical situation: Within the forward-backward recursions tracing the sample allele configurations described by the particle model, each step leads to a change in the sample size $\kappa$ and hence the dimension of the transition matrix, which is no longer square. Furthermore, the backwards temporal system accounting for the effect of changing mutation-drift parameters on the sample size is not diagonal and thus moderately complex. Calculating the marginal likelihood of the number of focal alleles in the extant sample involves a summation over $(K+1)(y+1)$ allele configurations rather than an eigensystem decomposition as with the orthogonal polynomials---this may sometimes be convenient though it is generally less efficient. However, the two methods yield equivalent results, which is of considerable theoretical interest. To determine the likelihood of any past population allele frequencies, the particle model is combined with forward population dynamics represented as orthogonal polynomials.
\newpage
\begin{figure}[!ht]
\centering
\includegraphics[width = 9cm,angle=-90]{Fig_2.eps}
\caption{Let us again consider a sample of size $K=10$ with $y=5$ focal individuals drawn at the present time $t=0$. We now consider a non-equilibrium situation where the population size changes at $t=S=-0.5$ in the past; this change is driven by a single change in the scaled mutation rate: The ancestral rate $\theta^{*}=0.3$ changes to the current rate $\theta^{*}=0.1$, $\alpha=0.3$ remains constant throughout. We show the probabilities of the various possible allele configurations $\upsilon$ and $\kappa$ from Eq.~(\ref{eq:joint_y_i});
the size of the dots is proportional to the probability of the configuration.}
\label{fig:num_example_non-equilibrium}
\end{figure}
\newpage
\section{Transition probabilities of the boundary mutation-drift Moran model}
\label{section:particle_boundary}
Recall that in synonymous sites of protein coding genes of eukaryotes, the expected heterozygosity, which is roughly equal to the scaled mutation rate, is $\leq 10^{-2}$ \citep{Lynch06}. Note that the expected heterozygosity corresponds to the expected polymorphism in a sample of size two and is thus proportional to the number of polymorphic sites in the sample; increasing the sample size leads to a roughly logarithmic increase in the proportion of polymorphic sites. Population genetic models derived in the limit of small scaled mutation rates, $\theta\to 0$, often approximate general mutation dynamics with sufficient accuracy and have clear numeric advantages \citep{Vogl20}. A first order Taylor expansion in $\theta$ of the beta-binomial equilibrium distribution of a sample from the decoupled bi-allelic Moran model yields the equilibrium distribution of the proportion of alleles $\mathbf{X}$ at any locus in the so-called boundary mutation-drift Moran model \citep{Vogl12}:
\begin{equation}\label{eq:eq_boundary}
{\Pr}(\mathbf{X}=\frac{i}{N}\given N,\alpha,\theta)=\begin{cases}
\beta\big(1-\alpha\theta H_{N-1}\big) &\text{for $i=0$,}\\
\alpha\beta\theta\,\frac{N}{i(N-i)} &\text{for $1\leq i\leq (N-1)$,}\\
\alpha\big(1-\beta\theta H_{N-1}\big) &\text{for $i=N$}\,,
\end{cases}
\end{equation}
where $H_{N-1}=\sum_{i=1}^{N-1}\frac{i}{N}$ is the harmonic number.
In essence, the boundary mutation-drift Moran model assumes that only a single mutation segregates in a population at any given time: Mutations arise exclusively from the monomorphic boundaries rather than from near the boundaries as in the general mutation model with low mutation rates; the polymorphic interior is governed by drift (and selection in non-neutral scenarios \citep{VoglMikula21}). Simulations show that the boundary mutation-drift Moran model is a good approximation for the general mutation Moran model if the expected equilibrium heterozygosity $2\alpha\beta\theta \leq 10^{-2}$ \citep{Vogl12}. Importantly, the boundary mutation-drift Moran model simplifies inference considerably \citep{Vogl14}.
Several approaches have been taken to derive transition probabilities compatible with this equilibrium distribution \citep{Vogl16,BurdenGriffiths19,VoglMikula21}. \citet{BurdenGriffiths19} start their derivations from a Wright-Fisher diffusion with $\theta\to 0$, and obtain the probability of the admissible sample configurations under the constrained mutation rate at any time $t$ by considering all coalescent sub-trees and their scaled branch lengths. The resulting formulae are complex and not easily employed in an inferential framework.
The transition rate matrix of the boundary-mutation Moran model given by \citet{VoglMikula21} is comparatively tractable. To balance the constraint that mutations are only allowed from the boundaries, mutation rates are re-scaled to obtain an equilibrium distribution corresponding to the Taylor series expansion of the general mutation Moran model. Importantly, this transition matrix and in particular its eigenvalues are not consistent with varying $N$, precisely because it is scaled so that mutations segregate at an identical average rate in equilibrium regardless of the sample size. Therefore the embedded genealogies of alleles depend on the sample size. For small scaled mutation rates the deviation of the eigenvalues from those of the general mutation Moran model (which are consistent for varying $N$) is negligible, as shown in Fig.~(\ref{fig:eigenvalue}). For larger values such as $\theta=0.025$, which is close to the limit of feasibility for the first order $\theta$ approximation, the first non-zero eigenvalue $\lambda_1$, which determines the asymptotic speed of approach to equilibrium, is increased by approximately $2.5\%$ to $5\%$. This could become important in a phylogenetic context, as split times between species may considerably exceed the (effective) population size $N$, which in turn constrains the effect of drift.
The stationary distribution of the phylogenetic rendition of the boundary mutation-drift Moran model has traditionally been scaled to maintain the proportion of monomorphic sites across different sample sizes \citep{Schrempf16} (but note that one version can be reparametrized to give the other), and the corresponding transition matrix \citep{Borges19} is therefore similarly inconsistent. (Note, however, that with a given population size $N$ this approach provides a statistically consistent framework for inference \citep{Borges20}.) Divergence estimators based on branch lengths \citep{Schrempf16, Borges19} may be affected by this; note that this also depends on whether the deviation actually exceeds the margin of numerical accuracy of the implementation, which may not be the case. Determining substitution rates by multiplying the mutation rates and fixation/hitting probabilities derived from the continuous approximations avoids the problem entirely \citep{VoglMikula21}.
\newpage
\begin{figure}[!ht]
\centering
\includegraphics[width = 10cm]{eigenvalues_N.eps}
\caption{Percentage bias of the eigenvalues of the boundary mutation-drift Moran model vs. the eigenvalues of the general mutation Moran model in dependence on the haploid population size $N$: Each plot shows a different eigenvalue - A) $\lambda_1$, B) $\lambda_2$, C) $\lambda_3$, D) $\lambda_4$ - of the transition matrix with three different overall scaled mutation rates: $\theta=c(0.1,0.05,0.01)$, whereby the largest overall scaled mutation rate corresponds to the slope showing the highest percentage bias, etc. In all plots, the reference line at 100\% corresponds to the value of the eigenvalue in the limit $\theta\to 0$.}
\label{fig:eigenvalue}
\end{figure}
\newpage
An alternative approach to representing the transition rates of the boundary mutation-drift Moran model has been proposed by \citet{Vogl16}: In the limit of the small scaled mutation rate $\theta\to0$, the interior of the forward (Fokker-Planck) diffusion equation in Eq.~(\ref{eq:forw_mutdrift}) becomes a pure drift diffusion model that can be decomposed into eigenvectors given by (modified) Gegenbauer polynomials and corresponding eigenvalues $\lambda_n=n(n-1)$ for $n\geq 2$, where $0 \leq n \leq \infty$ is the order of the polynomials. The (modified) Gegenbauer polynomials can be obtained from the (modified) Jacobi polynomials by a Taylor series expansion up to zeroth order \citep{Vogl16}. To incorporate boundary dynamics similar to those in the boundary mutation-drift Moran model, boundary terms are introduced to derive a system of inhomogeneous differential equations yielding the first two eigenvalues $\lambda_0=0$, and $\lambda_1=\theta$. Hence, the first non-zero eigenvalue is identical to that of the transition density of the general mutation model represented by orthogonal modified Jacobi polynomials (see Sec.~\ref{section:GeneralMutationDiffusion}). This representation of the transition density of the boundary mutation-drift Moran model can be used for inference with our forward-backward algorithm that combines the particle model and the orthogonal polynomial approach: For this to work, we must establish a continuous prior distribution $\bs{\rho(x)}$ for the boundary mutation-drift Moran model (Sec.~\ref{sec:functional}). We will then review the forward-backward algorithm using orthogonal polynomials for the boundary mutation-drift Moran model (Sec.~\ref{section:for_back_gegenbauer}), and introduce the corresponding particle model (Sec.~\ref{section:particle_gegenbauer}).
\subsection{A functional as an improper prior}
\label{sec:functional}
The beta equilibrium distribution Eq.~(\ref{eq:bb}) \citep{Wrig31} is generally assumed as the prior distribution $\phi(x\given t=S)=\bs{\rho}(x)$ for the bi-allelic Moran model in the continuous forward algorithms (Sec.~\ref{section:diffusion_Moran_for_back}). For the boundary mutation-drift Moran model, we define the functional
\begin{equation}\label{eq:Vogl_Bergman_functional}
eq(x\given \alpha,\theta)=\lim_{N\to\infty}\begin{cases}
\beta\big(1-\alpha\theta \int_{1/N}^{1-1/N} \frac1x\,dx\big) &\text{for $x=0$,}\\
\alpha\beta\theta\,\frac{1}{x(1-x)} &\text{for $1/N \leq x \leq 1-1/N$,}\\
\alpha\big(1-\beta\theta \int_{1/N}^{1-1/N} \frac1x\,dx\big) &\text{for $x=1$,}
\end{cases}
\end{equation}
to replace this beta distribution the prior. For the functional to be a valid prior, taking a sample of size $K$ with $y$ focal alleles with replacement from it should result in a marginal distribution for $y$ that is equal to the stationary distribution in Eq.~(\ref{eq:eq_boundary}). We will prove this here:
\begin{itemize}
\item{(i)} For polymorphic samples $1\leq y\leq (K-1)$, the function
\begin{equation}
\begin{cases}
\alpha\beta\theta\,x^{y-1}(1-x)^{K-y-1} &\text{within $[1/N,1-1/N]$}\\
0 &\text{otherwise}
\end{cases}
\end{equation}
is bounded from above by $\alpha\beta\theta\,x^{y-1}(1-x)^{K-y-1}$ within the interval $[0,1]$ (but converges monotonically towards it with increasing $N$). By the monotone convergence theorem, the order of taking the limit and integration can be reversed and we obtain:
\begin{equation}
\begin{split}
\Pr(1\leq y\leq K-1\given K,\alpha,\theta
&=\alpha\beta\theta\binom{K}{y}\lim_{N\to\infty} \int_{1/N}^{1-1/N} x^{y-1}(1-x)^{K-y-1}\,dx\\
&=\alpha\beta\theta \frac{K}{y(K-y)}\,.
\end{split}
\end{equation}
\item{(ii)} For monomorphic samples, analogous arguments apply. We show the case $y=K$ (equivalent calculations hold for $y=0$):
\begin{equation}
\begin{split}
\Pr(y=K\given K,\alpha,\theta)&= \lim_{N\to\infty} \int_{1/N}^{1-1/N} x^K\alpha\beta\theta \frac{1}{x(1-x)}\\
&\qquad+ \alpha\bigg(1-\beta\theta \int_{1/N}^{1-1/N} \frac1x\,dx\bigg) \,dx\\
&= \lim_{N\to\infty} \int_{1/N}^{1-1/N} \alpha\beta\theta \frac{x^{K-1}}{1-x}+ \alpha\bigg(1-\beta\theta \int_{1/N}^{1-1/N} \frac1{1-x}\,dx\bigg) \,dx\\
&= \alpha+\lim_{N\to\infty}\alpha\beta\theta \int_{1/N}^{1-1/N} (x^{K-1}-1)(1+x+x^2+\dots) \,dx\\
&=\alpha+\lim_{N\to\infty}\alpha\beta\theta \int_{1/N}^{1-1/N} 1+x+x^2+\dots+x^{K-2} \,dx\\
&=\alpha-\alpha\beta\theta H_{K-1}\,.
\end{split}
\end{equation}
\end{itemize}
The functional Eq.~(\ref{eq:Vogl_Bergman_functional}) can thus be used as an improper prior distribution for the boundary mutation-drift Moran model. As long as $K<e^{\frac1{\max(\alpha,\beta)\theta}}$ approximately holds, a proper marginal posterior distribution for the proportion of focal alleles in the sample will result.
\subsection{Transition probabilities with small scaled mutation rates}
\subsubsection{Forward-backward with augmented Gegenbauer polynomials}
\label{section:for_back_gegenbauer}
Recall that the forward and backward operators of the Kolmogorov diffusion equation for the general mutation bi-allelic Moran model can be decomposed into eigenvectors represented as (modified) Jacobi polynomials and corresponding eigenvalues (Sec.~\ref{section:GeneralMutationDiffusion}, Sec.~\ref{section:diffusion_Moran_for_back}). A similar decomposition can be achieved for the boundary mutation-drift Moran model; we will reproduce an outline of the derivation here, details can be found in \citet[][Appendix]{Vogl16}:
\paragraph{Pure drift model}
In a first step, the eigenvectors of a pure drift model are obtained by a zeroth order Taylor series expansion around $\theta$ in Sec.~(\ref{section:GeneralMutationDiffusion}), Sec.~(\ref{section:diffusion_Moran_for_back}).
The backward eigenvectors of the pure drift model are \citep{Berg17}:
\begin{equation}\label{eq:backw_Us}
\begin{cases}
B_0^{(\alpha,\theta)}(x)&=B_0^{(\alpha,0)}(x)=1\\
B_1^{(\alpha,\theta)}(x)&=B_1^{(\alpha,0)}(x)=x-\alpha\\
B_{n\geq2}^{(\alpha,0)}(x)&=w(x)U_n(x)\,,
\end{cases}
\end{equation}
where for $n\geq 2$:
\begin{equation}\label{eq:Gegenbauer_modified}
\begin{split}
U_n(x)&=\sum_{l=0}^{n-2}(-1)^l\binom{n+l}{l}\binom{n-1}{l+1}x^l\,.
\end{split}
\end{equation}
are the modified Gegenbauer polynomials (compare \citep{Song12} Eqs.~(12),(13)) with weight function $w(x)=x(1-x)$. For any $m,n$, the modified Gegenbauer polynomials fulfill the following orthogonality relationship with respect to the weight function:
\begin{equation}\label{eq:ortho_Gegen}
\int_0^1 U_n(x) U_n(x) w(x)\,dx=\delta_{n,m} \Delta_n\,,
\end{equation}
where $\delta_{n,m}$ is Kronecker's delta and $\Delta_n=\frac{(n-1)}{(2n-1)n}$ for $n\geq 2$ are proportionality constants. The forward eigenvectors of the pure drift model are:
\begin{equation}\label{eq:forw_Us}
\begin{cases}
F_0^{(\alpha,0)}(x)&=\beta\delta(x)+\alpha\delta(x-1)\\
F_1^{(\alpha,0)}(x)&=-\delta(x)+\delta(x-1)\\
F_{n\geq2}^{(\alpha,0)}(x)&=-\frac{(-1)^n}n\delta(x)+U_n(x)-\frac{1}n\delta(x-1)\,.
\end{cases}
\end{equation}
The corresponding eigenvalues of the system are $\lambda_0=0$, $\lambda_1=\theta=0$, and $\lambda_n=n(n-1)$ for $n\geq 2$.
\paragraph{Boundary mutation-drift model}
To introduce mutations arising exclusively at the monomorphic boundaries, $\lambda_1$ is set to $\theta$ with $0<\theta<<1$. The discrete boundary mutation-drift model is scaled so that mutations enter the polymorphic region from the respective boundaries at average rates of $\frac{\alpha\theta}{N}$ and $\frac{\beta\theta}{N}$ per Moran drift event in equilibrium; this is required to maintain a constant equilibrium mutation rate across generations. In an orthogonal polynomial representation of the transition density of the continuous boundary mutation-drift model in which the (effective) population size and thus the generation lengths are not necessarily constant, mutations enter the polymorphic region from the respective boundaries at rates $\alpha\theta b_0(x,t)$ and $\beta\theta b_1(t)$: Here, $b_0(t)$ and $b_1(t)$ represent the probability masses both already fixed at the boundary at time $t$ or expected to fix there imminently by drift. In the polymorphic region, drift operates at an exponential rate that depends on the sample size $\kappa$, which is the order of the polynomial expansion and the sample size $K$. Overall, the forward and backward eigenvectors of the pure drift model of order $K$ are multiplied with the solution $\tau_n(t)$, with $0\leq n\leq K$, of the following system of first order linear equations describing the temporal dynamics induced by the boundary mutation (\citep{Vogl16}, but note the different weighting there):
\begin{equation}\label{eq:TauExpansion_origional}
\begin{split}
&\frac{d}{dt} \tau_0(t) = 0\\
&\frac{d}{dt} \tau_1(t) = -\theta\tau_1(t)\\
&\frac{d}{dt} \tau_n(t) = -\lambda_n\tau_n(t) + A_n +B_n e^{\theta t} \text{ for $n\geq 2$}\\
\end{split}
\end{equation}
where
\begin{equation*}\label{eq:TauExpansion_A}
A_n = -\alpha\beta\theta(2n-1)n((-1)^n+1)
\end{equation*}
and
\begin{equation*}\label{eq:TauExpansion_B}
B_n = -\theta(2n-1)n(b_0(t)-\beta)((-1)^n\alpha - \beta)\,.
\end{equation*}
\paragraph{Diagonalized boundary mutation-drift model}
The system (Eq.~\ref{eq:TauExpansion_origional}) corresponds to a triangular matrix with eigenvalues $\lambda_0=0$, $\lambda_1=\theta$, $\lambda_n=n(n-1)$ on the main diagonal. Hence the eigensystem can be easily diagonalized \citep{Berg17}:
The backward eigenfunctions of the boundary mutation-drift model become:
\begin{equation}\label{eq:backw_bound_diag}
\begin{cases}
B_0^{(\alpha,\theta)}(x)&=B_0^{(\alpha,0)}(x)=1\\
B_1^{(\alpha,\theta)}(x)&=B_1^{(\alpha,0)}(x)=x-\alpha\\
B_{n \geq 2}^{(\alpha,\theta)}(x)&=B_n^{(\alpha,0)}(x)-\vartheta \frac{E_n\Delta_n}{\lambda_n} B_0^{(\alpha,0)}(x)-\theta \frac{B_n\Delta_n}{\lambda_n} B_1^{(\alpha,0)}(x)\,.
\end{cases}
\end{equation}
with
\begin{equation}
\begin{split}
\vartheta&=\alpha\beta\theta\,,\\
E_{n}&=-(n-1)\frac{((-1)^n+1)}{\Delta_n}\,,\\
O_{n}&=-(n-1)\frac{(-1)^n\alpha-\beta}{\Delta_n}\,,
\end{split}
\end{equation}
Equivalently, the forward eigensystem becomes \citep{Berg17}:
\begin{equation}\label{eq:forw_bound_diag}
\begin{cases}
F_0^{(\alpha,\theta)}(x)&=eq(x\given \alpha,\theta)\\
F_1^{(\alpha,\theta)}(x)&=neq(x\given \alpha,\theta)\\
F_{n \geq 2}^{(\alpha,\theta)}(x)&=F_n^{(\alpha,0)}(x)\,,
\end{cases}
\end{equation}
with the functional $eq(x\given \alpha,\theta)$ from Eq.~(\ref{eq:Vogl_Bergman_functional}), and the similar functional:
\begin{equation}\label{eq:Vogl_Bergman_functional_1}
neq(x\given \alpha,\theta)=\lim_{N\to\infty}\begin{cases}
-1+\alpha\theta \int_{1/N}^{1-1/N} \frac1x\,dx &\text{for $x=0$,}\\
\theta\,\bigg(-\frac{\alpha}{x} + \frac{\beta}{1-x}\bigg)&\text{for $1/N \leq x \leq 1-1/N$,}\\
1-\beta\theta \int_{1/N}^{1-1/N} \frac1x\,dx &\text{for $x=1$.}
\end{cases}
\end{equation}
The weights are augmented by $\Delta_0=\Delta_1=1$.
Note that the first diagonalized eigenfunction $F_0^{(\alpha,\theta)}(x)$ corresponds to the equilibrium distribution. With this and the second eigenfunction $F_1^{(\alpha,\theta)}(x)$ a quasi-equilibrium state of the allele proportions in the polymorphic interior can be modelled:
\begin{equation}
F_0^{(\alpha,\theta)}(x)+\alpha e^{-\theta t} F_1^{(\alpha,\theta)}(x)\,.
\end{equation}
Observe that the rate of approach to equilibrium depends on the mutation rate $\theta$, which is considerably slower than the rate of drift.
\paragraph{Application of the forward-backward algorithm}
If the ancestral allele configuration \bs{\rho}(x) is taken to be distributed according to the functional in Sec.~(\ref{sec:functional}) with
\begin{equation}
\bs{\rho}(x)=\sum_{n=0}^\infty \rho_n^{(\alpha,\theta)} F_{n}^{(\alpha,\theta)}(x),
\end{equation}
and $\rho_n^{(\alpha,\theta)}$ obtained according to:
\begin{equation}\label{eq:forward_given_rho_coefs_2}
\rho_n^{(\alpha,\theta)}= \frac{1}{\Delta_n^{(\alpha,\theta)}} \int_{x=0}^1 w(x) F_n^{(\alpha,\theta)}(x)\bs{\rho}(x)dx\,.
\end{equation}
The forward transition density of the population can be expanded to either:
\begin{equation}\label{eq:forward_given_rho_geg}
\phi(x\given t,\bs{\rho})=w(x) \sum_{n=0}^\infty \rho_n^{(\alpha,\theta)} F_n^{(\alpha,0)}(x) \tau_n(t)\,,
\end{equation}
or
\begin{equation}\label{eq:forward_given_rho_geg_diag}
\phi(x\given t,\bs{\rho})=w(x) \sum_{n=0}^\infty \rho_n^{(\alpha,\theta)} F_n^{(\alpha,\theta)}(x) (\tau_n(t))^-\,,
\end{equation}
where $(\tau_n(t))^-$ is the solution to
\begin{equation}\label{eq:TauExpansion_2}
\begin{split}
&\frac{d}{dt} \tau_n(t) = -\lambda_n\tau_n(t)
\end{split}
\end{equation}
The marginal likelihood of the number of focal alleles $y$ in an extant sample of size $K$ can similarly be determined via:
\begin{equation}\label{eq:marg_like_general_geg}
\begin{split}
\Pr(\y \given K,\alpha,\theta,t=0,\bs{\rho})=\sum_{n=0}^K \rho_n^{(\alpha,\theta)} d_n(K,y) \Delta_n \tau_n(t=0)\,.
\end{split}
\end{equation}
or
\begin{equation}\label{eq:marg_like_general_geg_diag}
\begin{split}
\Pr(\y \given K,\alpha,\theta,t=0,\bs{\rho})&=\sum_{n=0}^K \rho_n^{(\alpha,\theta)} d_n(K,y) \Delta_n (\tau_n(t=0))^-\,,
\end{split}
\end{equation}
with
\begin{equation}
d_n(K,y)=\frac{1}{\Delta_n}\int_{x=0}^1 \binom{K}{y}x^y(1-x)^{K-y}\,F_n^{(\alpha,0)}(x)\,dx\,.
\end{equation}
\paragraph{Summary} A forward-backward algorithm approach using orthogonal polynomials can easily be applied to the boundary mutation-drift Moran model. Note that using the un-diagonalized version of the eigensystem is particularly efficient when mutation parameters change over time, both with respect to time and numerical accuracy, since this change does not affect the eigenvectors but only the temporal system. Although the boundary mutation-drift Moran model is an approximation to the general mutation bi-allelic Moran model, it may provide more accurate results in an inference framework than direct use of the general mutation model through this numerical advantage. The increased numerical accuracy may allow the use of higher sample sizes, which is necessary for many data analyses.
\subsubsection{Particle transition probabilities}
\label{section:particle_gegenbauer}
As with the general mutation model, Eq.~(\ref{eq:marg_like_general_geg}) and Eq.~(\ref{eq:marg_like_general_geg_diag}) can be determined with an appropriate particle model. This can again be constructed using the ratios of the probabilities of the appropriate configurations of ancestral focal alleles $\upsilon$ in a sample size of $\kappa$ (see Eq.~\ref{eq:ratio_general}).
Recall that the indicator variable $z_{\kappa+1}$ is one if the $\kappa+1$st allele is of the focal type and zero otherwise.
Note that the transition probabilities from $z_{\kappa}$ to $z_{\kappa+1}$ must hold in equilibrium. Recall that for the general mutation model, we have:
\begin{equation}
\begin{split}
{\Pr}(z_{\kappa+1}\given \upsilon,\kappa,\alpha,\theta)&=\frac{{\Pr}(z_{\kappa+1}+\upsilon\given \kappa+1,\alpha,\theta)}{{\Pr}(\upsilon\given \kappa,\alpha,\theta)}\\
&=\frac{(\upsilon+\alpha\theta)^{1-z_\kappa}(\kappa-\upsilon+\beta\theta)^{z_\kappa}}{\kappa+\theta}\,.
\end{split}
\end{equation}
Note that the terms of ${\Pr}(z_{\kappa+1}+\upsilon\given \kappa+1,\alpha,\theta)$ and ${\Pr}(\upsilon\given \kappa,\alpha,\theta)$ that only hold in equilibrium cancel out. Therefore, the transition probabilities for the general mutation model as given in Eq.~(\ref{eq:ratio_general}) hold in non-equilibrium situations.
For the boundary mutation-drift model, transition probabilities need to be consistent with the equilibrium probability Eq.~(\ref{eq:Vogl_Bergman_functional}). For $1\leq \upsilon\leq (\kappa-1)$, no mutations are possible and the transition probabilities are independent of the mutation parameters $\alpha$ and $\theta$:
\begin{equation}\label{eq:trans_poly}
\begin{split}
{\Pr}(z_{\kappa+1}\given \upsilon,\alpha,\theta)&=\frac{{\Pr}(z_{\kappa+1}+\upsilon\given \kappa+1,\alpha,\theta)}{{\Pr}(\upsilon\given \kappa,\alpha,\theta)}\\
&=\frac{(\kappa-\upsilon)^{z_\kappa} \upsilon^{1-z_\kappa}} {\kappa}\,.
\end{split}
\end{equation}
For $\upsilon=0$, the transition probability consistent with that in Eq.~(\ref{eq:trans_poly}) is:
\begin{equation}
\begin{split}
{\Pr}(z_{\kappa+1}\given \upsilon=0,\alpha,\theta)&=\frac{{\Pr}(z_{\kappa+1}\given \kappa+1,\alpha,\theta)}{{\Pr}(\upsilon=0\given \kappa,\alpha,\theta)}\\
&=\frac{(\frac{\alpha\theta}\kappa)^{z_\kappa}(1-\alpha\theta H_{\kappa})^{1-z_\kappa}} {1-\alpha\theta H_{\kappa-1}}\,,
\end{split}
\end{equation}
and the analogous transition rate holds for $\upsilon=\kappa$.
For the transition from $\kappa=1$ to $\kappa=0$ (or the reverse forward in time), the transition probabilities of the boundary mutation-drift models are identical to those of the general mutation model: Specifically, the probabilities are $\alpha$ and $\beta$ to zeroth order in $\theta$.
\paragraph{Temporal dynamics}
The backward temporal dynamics of the particle model are clearly equivalent to those in Sec.~(\ref{section:backward-particle_temp}) however with eigenvalues $\lambda_0=0$, $\lambda_1=\theta$, and
$\lambda_n=n(n-1)$ for $n\geq2$.
\paragraph{Summary}
A particle model can be used to trace sample allele configurations in the boundary mutation-drift Moran model forward and backward in time. The transition probabilities defined through it are sample consistent, and the first non-zero eigenvalue (defining the speed of approach to equilibrium) is identical to that of the general mutation Moran model and independent of the sample size. In phylogenetic settings, where split times (in generations) between species or populations are large compared to effective population sizes, this eigenvalue dominates.
\section{Conclusions}
In population genetics, small to moderately sized extant samples are generally used to infer parameters of past population processes. The evolutionary trajectory of the population can, under certain assumptions, be described by the bi-allelic decoupled Moran model in the diffusion limit.
To determine the marginal likelihood of sample allele proportions in the extant sample, the backwards N-particle diffusion of the extant sampling distribution can be decomposed into eigenvectors of orthogonal polynomials and corresponding eigenvectors. The backward orthogonal polynomial approach is computationally efficient and accommodates temporally changing effective population sizes (and therefore mutation rates) or mutation biases. This is even more true when scaled overall mutation rates are small and a boundary mutation-drift Moran model is assumed to describe the evolutionary dynamics: if the system is not diagonalized, time inhomogeneous dynamics allow for effient numerics.
The probabilities of ancestral sample allele configurations can be inferred with a so-called particle model augmented with backwards temporal dynamics: It directly traces the size and allele proportions of a sample both forward (urn model) and backward (coalescent) in time. Empirically, this approach may be useful for inferring the timing of mutation-drift events at a specific locus. However, its main value lies in its equivalence with the backwards orthogonal polynomial method: The backward orthogonal polynomial expansion of an extant sample to $n$th order has the genealogy of the sample embedded within it. It therefore contains all the information traditionally modelled by the coalescent, but is more tractable and numerically superior.
In order to infer the probability of ancestral population allele proportions at any time in the past, the particle model needs to be complemented with a forward in time orthogonal polynomial approach. The result is a full forward-backward algorithm equivalent to the forward-backward algorithm using only orthogonal polynomials.
\section*{Acknowledgments}
We thank the present and past members of the doctorate college population genetics, especially Juraj Bergman, for stimulating discussions. CV and SP's research was supported by the Austrian Science Fund (FWF): DK W1225-B20; LCM's by the School of Biology at the University of St.Andrews and also partially funded through Vienna Science and Technology Fund (WWTF) [MA016-061].
|
2,877,628,090,083 | arxiv | \section{Introduction}
Over the last years, optical microcavities have attracted researcher's attention in both fundamental and applied physics due to their high quality factor (Q) and small mode volumes \cite{Vahala.2003}. Besides applications in sensing \cite{Foreman.2015}, nonlinear optics \cite{Ilchenko.2004}, light-matter interaction \cite{Xiao.2012} and lasing \cite{Harayama.2011}, they have been used as a research platform to study exciting physical phenomena such as exceptional points \cite{Wiersig.2014,Chen.2017} and optical chirality \cite{Liu.2018}. In recent years, the application range of optical microresonators has been further expanded by introducing systems of deformed or perturbed resonators. This leads to new effects which can be used for applications such as microlasers with directional emission \cite{Wiersig.2006, Harayama.2015} or enhanced coupling \cite{Jiang.2017}. Phase-space analysis is a powerful tool to both describe and understand the optical modes in such deformed resonators, where the chaotic dynamics plays a major role. Instead of studying the real-space mode distribution, one looks at the field intensities as a function of both position and angle of incidence for a reduced subsystem, usually the resonator boundary. This method provides a thorough understanding of the underlying dynamics and is well suited to establish ray-wave correspondence \cite{Hentschel.2002}. A common phase-space representation are Husimi functions, which were introduced for open dielectric systems by Hentschel \textit{et. al.} \cite{Hentschel.2003} and have been used extensively to study the far-field patterns, wave dynamics and ray-wave correspondence. Recently, these phase-space approaches have been extended to include systems with non-homogenous refractive index \cite{Kim.2018} and have been used to study free space coupling into asymmetric cavities \cite{Shu.2013} and the dynamical evolution of light in a deformed cavity by calculating the respective functions at different times and following the evolution in both real space and phase space \cite{Chen.2019,Kwon.2013}. In this paper, we apply phase-space analysis based on generalized Husimi functions to study systems of coupled resonators, providing an intuitive understanding of the involved coupling processes. Coupled resonator systems have shown promising features for lasing systems \cite{Kreismann.2019} and non-hermitian physics \cite{Bosch.2019} and are as such a subject of interest. Firstly we present how the Husimi functions as derived in \cite{Hentschel.2003} can be used to study coupling phenomena in resonator systems and the difficulties involved. We then proceed to analyze the coupling in an illustrative waveguide-resonator system using the described method.
\section{Husimi functions for eigenmodes in dielectric cavities}
The Husimi function was originally defined as a quasi-probability distribution in phase space \cite{Husimi.1940}, given by the overlap of the wavefunction with a Gaussian-type wavepacket (minimal-uncertainty). Applied to optical systems, it allows the representation of the field intensities as a function of position as well as momentum. Mathematically it is a windowed transformation, with the Gaussian window leading to the smallest possible uncertainty linked to such transformations. For 2D optical cavities, the Husimi function is usually calculated on the Poincaré surface of section (SOS) at the system boundary, leading to a phase-space representation of a reduced system which can be easily visualised in two dimensions \cite{Crespi.1993}. The values of the Husimi function correspond to the field intensities for a given boundary position $s$ and angle of incidence $\chi$ with respect to the boundary normal, where $\chi < (>) 0$ indicates (counter-) clockwise propagation direction.
Hentschel \textit{et. al.} \cite{Hentschel.2003} presented four different Husimi functions for open dielectric systems with piecewise constant refractive indices, corresponding to the intensities of the incident (inc) and emerging (em) waves inside ($j=1$) and outside ($j=0$) of the interface (see Fig.~\ref{fig:Skizze System}). The four Husimi functions along the cavity boundary $\Gamma$ with Birkhoff coordinates $(s, \sin \chi)$ are defined as:
\begin{equation}
\label{eq:Husimi}
H_{j}^{inc(em)}(s, \sin\chi_j) = \frac{k_j}{2\pi} \left| {(-1)^j F_j h_j(s,\sin\chi_j) + (-) \frac{i}{k_0 F_j} h'_j(s,\sin\chi_j)} \right|^2,
\end{equation}
with the weighting factors $F_j = \sqrt{n_j \cos{\chi_j}}$, the refractive indices $n_0=1$, $n_1$, the (vacuum) wave number $k_0$, and the overlap functions $h_j$, $h'_j$ given by
\begin{equation}
\label{eq:h}
h_j(s, \sin\chi_j) = \oint_\Gamma{ ds' \psi_j(s') \xi(s'; s, \sin\chi_j)},
\end{equation}
\begin{equation}
\label{eq:h'}
h'_j(s, \sin\chi_j) = \oint_\Gamma{ ds' \psi'_j(s') \xi(s'; s, \sin\chi_j)}.
\end{equation}
The wave functions $\psi$ (and its normal derivative $\psi'$) are taken on the respective side $j$ of the dielectric interface. The minimum-uncertainty wave packet $\xi$ is given by
\begin{equation}
\label{eq:chi}
\xi(s'; s, \sin\chi_j) = (\sigma \pi)^{-\frac{1}{4}} \sum_{l \mathsmaller{\in} \mathbb{Z}} \exp \left[ \frac{-(s' - s + 2\pi l)^2}{2\sigma} - ik_j \sin(\chi_j+2 \pi l) \right],
\end{equation}
which is a periodic function in $s'$ centered around $(s', \sin\chi_j)$. The parameter $\sigma = \sqrt{2}/k_1$ determines the extension along the $s'$ direction and thereby the uncertainty in $\sin\chi_j$. The value $k_j$ is the wavenumber in each region, the angles of incidence are related by Snell's law $n_1 \sin\chi_1 = n_0 \sin\chi_0 $. This phase-space representation gives insight into the wave dynamics and allows the identification of regions in phase-space with high intensity. This approach has been used extensively to study asymmetric cavities \cite{Song.2010, Wiersig.2008, Shinohara.2010}.
The four Husimi functions for an eigenmode of the shortegg cavity \cite{Schermer.2015} can be seen in Fig.~\ref{fig:Skizze_Mode_Husimi}. The whispering gallery mode (WGM) is confined by total internal reflection at the dielectric boundary. This can be seen in the Husimi functions inside the cavity $H^\mathrm{em}_1$ and $H^\mathrm{inc}_1$, which show high intensities for angles close to the boundary tangent only. The influence of the deformation can be seen in the deviation from perfectly straight lines, which is in turn the cause for the directional emission exhibited by this cavity. The differences between $H^\mathrm{inc}_1$ and $H^\mathrm{em}_1$ can be seen clearly for the points in the leaky region (between the dashed lines $\sin \chi_{cr} = \pm 1/n $), for which the light leaves the cavity according to Snell's law. Outside these lines, total reflection occurs and the incoming and emitted components are similar. The high intensity emission points along the boundary can be identified in $H^\mathrm{em}_0$ and $H^\mathrm{inc}_1$. The directional emission can be seen clearly as well in phase-space: the highest outgoing intensities outside of the cavity in $H^\mathrm{em}_0$ are localized along a line joining the positions $(s = 0.35, \sin\chi_0 = 1)$ and $(s = \pi - 0.35, \sin\chi_0 = -1)$. This closely resembles the signature of a plane wave emerging from $s \approx \pi/2$, in accordance with the shortegg's farfield emission characteristics.\\
\begin{figure}
\subfigure[]{\raisebox{8mm}{
\includegraphics[width=0.17\textwidth]{Fig1a.pdf}}
\label{fig:Skizze System}
}
\subfigure[]{\raisebox{8mm}{
\includegraphics[width=0.2\textwidth]{Fig1b.pdf}}
\label{fig:EF_Mode_1907}
}
\subfigure[]{
\includegraphics[width=0.6\textwidth]{Fig1c.pdf}
\label{fig:Husimis_EF_Shortegg_Mode_1907}
}
\caption{(a) Incident and emerging rays at a dielectric boundary $\Gamma$ corresponding to the four Husimi functions and introduction of phase-space coordinates arc length $s$ (with $0 \leq s \leq 2 \pi)$ and angle $\chi$ of incidence. (b) Mode distribution of the shortegg cavity with $n = 1.44$ for a TE whispering-gallery eigenmode with the mode number $kR = 79.935 - 0.003i$. (c) The four Husimi functions calculated for the mode distribution shown in Fig.~\ref{fig:EF_Mode_1907}. Note that the scale depicts a relative intensity.}
\label{fig:Skizze_Mode_Husimi}
\end{figure}%
\section{Husimi functions for waveguide-resonator coupled systems}
When analyzing coupling processes, it is necessary to look at whether the incoming fields excite the resonant mode at the right position as well as with the right angle of incidence. The phase-space analysis of coupling problems via Husimi functions allows to do this naturally by identifying the signatures of the $H^\mathrm{inc}_0$ function for the resonator of interest in a coupled system. $H^\mathrm{inc}_0$ gives insight into how incoming light from neighbouring regions behaves at the resonator boundary and can thus provide an explanation for the coupled system response. The excitation of supported modes is expected to be directly correlated to the overlap between the incoming exciting fields and the resonant eigenmode. A problem with this approach, however, arises due to the relative intensities of the incoming components and the resonant fields. Fig.~\ref{fig:fullCoupled} shows the Husimi functions for a coupled waveguide-shortegg system at an off-resonant as well as for a resonant frequency. The fields in the resonator were excited by an incoming field distribution from the left (red arrow) calculated from a port boundary analysis. The incoming field components $H^\mathrm{inc}_0$ from the waveguide are expected to be located around ($s=\pi/2, \, \sin \chi_0 = -1$, see red arrow), but they can barely be distinguished from the intensities arising due to the intra-cavity field in the resonant case. In the off-resonant case, the waveguide component can be seen in $H^\mathrm{inc}_0$ due to the lower intra-cavity intensity.
For all practical purposes, representing the fully coupled waveguide-resonator system in phase space will cause difficulties related to the dominance of the resonant mode in all the Husimi functions.
One could try to change the Gaussian pulse in order to attain a higher precision in $\sin\chi_j$, but this is only possible through an undesired trade-off with increasing the uncertainty in $s$.\\
\begin{figure}
\subfigure[]{\raisebox{5mm}{
\includegraphics[width=0.26\textwidth]{Fig2a.pdf}}
\label{fig:Full_Coupled_Mode_off}
}
\subfigure[]{
\includegraphics[width=0.7\textwidth]{Fig2b.pdf}
\label{fig:Full_Coupled_Husimi_off}
}\\
\subfigure[]{\raisebox{5mm}{
\includegraphics[width=0.26\textwidth]{Fig2c.pdf}}
\label{fig:Full_Coupled_Mode_res}
}
\subfigure[]{
\includegraphics[width=0.7\textwidth]{Fig2d.pdf}
\label{fig:Full_Coupled_Husimi_res}
}
\caption{ (a) Mode distribution of the waveguide-shortegg system at the off-resonant frequency $\mathrm{Re}(kR) = 80.015$. (b) Husimi functions corresponding to the mode showed in (a). (c) Mode distribution of the waveguide-shortegg system at the resonant frequency $\mathrm{Re}(kR) = 80.211$. (d) Husimi functions corresponding to the mode showed in (c). The width of the waveguide is set to $w_{WG} = 0.565 \lambda / n_{in}$ and the distance between the resonator and the waveguide is $\Delta = \lambda /2$. The waveguide has the same refractive index as the cavity $n_{in}=n=1.44$.}%
\label{fig:fullCoupled}
\end{figure}%
To overcome this issue, one can compute the field components originating from the coupled system (waveguide, neighbouring resonator) separately.
In particular, one can compute the waveguide's field distribution at the resonator boundary without including the resonator in the calculation, hereby preventing the resonant intensity from overshadowing the analysis of the coupling. To this end, the Husimi projection is taken at the boundary where the resonator to be coupled into would be situated. This result is subsequently used to calculate the overlap with the resonant mode's Husimi function (using the Husimi function $H_1^\mathrm{em}$ in both cases).
Although Husimi functions were originally motivated and derived for dielectric boundaries, it is not restricted to this, and the overlap of the wave field with a minimum-uncertainty wave packet can be calculated along an arbitrary boundary $\Gamma$.
This allows us to proceed in calculating the incoming and outgoing field components originating at the waveguide mode along the boundary where the resonator would be while setting $n_{0} = n_{1} = 1$. The overlap $S_{\mathrm{Exc,Res}}$ between the Husimi functions of the exciting field $H^\mathrm{em}_{1,\mathrm{exc}}$ and the Husimi Functions of a chosen resonant mode $H^\mathrm{em}_{1,\mathrm{res}}$ can be calculated via
\begin{equation}
\label{eq:OverlapStrength}
S_{\mathrm{Exc,Res}} = \frac{\int \int ds \, \, d\sin \chi_j H^{em}_{1,\mathrm{exc}}(s,\sin \chi_j) H^{em}_{1,\mathrm{res}}(s,\sin \chi_j) }{\int \int ds \, \, d\sin \chi_j}
\end{equation}
and is correlated to the coupling strength between the incoming field and the resonant mode. Note that both $H^{em}_{1,\mathrm{exc}}$ and $H^\mathrm{em}_{1,\mathrm{res}}$ have to be evaluated along the same boundary $\Gamma$.
The results are not expected to match full-wave calculations perfectly due to two reasons: first, the uncertainty arising from the Husimi projection and second, the fact that interactions between the cavity and the exciting system can not be accounted for. In the weak coupling regime, realized if the distance between the resonator and the waveguide is large enough, the eigenmodes of the coupled resonator remain essentially unchanged. Consequently, the results should correlate well with full numerical wave simulation results of the combined system. More importantly, this analysis gives an intuitive understanding of the involved coupling processes in phase space. If one is interested in a time-dependent analysis, the phase-space description of the incoming fields can be used as time-dependent input for the computation of the dynamical fields inside the resonator.\\
\begin{figure}
\subfigure[]{\raisebox{5mm}{
\includegraphics[width=0.28\textwidth]{Fig3a.pdf}}
\label{fig:H_ModeDistribution}
}
\subfigure[]{
\includegraphics[width=0.7\textwidth]{Fig3b.pdf}
\label{fig:Husimis_EF_Shortegg}
}
\caption{(a) Mode distribution in the shortegg cavity with $n = 1.44$ for a TE eigenmode with the mode number $kR = 80.004 - 0.029i$. (b) The four Husimi functions calculated for the mode distribution showed in \ref{fig:H_ModeDistribution}, with the dashed line marking the critcal angles $\sin \chi_i = 1/n$. The Husimi functions inside show four areas with high intensity along $s$, which can be linked to the four reflection areas on which the mode distribution along the boundary has the highest values.
}
\label{fig:EF_Husimi}
\end{figure}
In order to illustrate this method, we studied the transmission through the waveguide in dependence on the orientation angle of the shortegg resonator and compared the results
from conventional numerical methods for the full system with the phase-space analysis described above. The electric field distributions used for the eigenmodes, the phase-space analysis as well as the transmission curves were calculated by solving the 2D Helmholtz equation in the frequency domain, with outgoing wave conditions imposed at infinity \cite{Jackson.1998}.
We focus here on TE modes, where $E_\mathrm{z}$ and its normal derivative are continuous across dielectric boundaries. The fields were solved for numerically in COMSOL v5.4 \cite{Comsol.2018}, using perfectly matched layers (PMLs) at the edges of the truncated system.
The analyzed system consisted of a waveguide coupled to a shortegg cavity, whose geometric shape is given, in polar coordinates $(r, \phi)$, as
$r(\phi) = R_0(1 + 0.16 \cos(\phi) - 0.022 \cos(2\phi) - 0.05 \cos(3\phi)).$
For both the waveguide and the shortegg cavity the refractive index was set to $n=1.44$. The distance between the resonator and waveguide was held constant at $\Delta = 0.5 \lambda$ for all angles by shifting the center of the resonator.
As known from in-house experimental data \cite{Behrens.2020}, the transmission spectrum (i.e., intensity transmitted through the waveguide as a function of the orientation angle) is highly dependent on the orientation of the waveguide relative to the resonator. Although this is to be expected due to the asymmetry of the resonator, an accurate, quantitative, and predictive explanation is accessible by analyzing the system in phase space.
A typical mode distribution as well as the four Husimi functions can be seen in Fig. \ref{fig:H_ModeDistribution}. Note that the resonant frequency for the coupled system shown in \ref{fig:fullCoupled} and the eigenfrequency vary slightly due to the coupling.
The values $H^\mathrm{em}_{1,\mathrm{res}}$ are computed from eigenvalue calculations. To calculate the fields used for $H^\mathrm{em}_{1,\mathrm{exc}}$ originating at the waveguide, the mode distribution of the waveguide-only system (see Fig. \ref{fig:dashed_mode_wg}) is computed. The waveguide mode corresponds to the fundamental mode for the used frequency and waveguide width and was derived from a numerical boundary mode analysis.
\begin{figure}
\begin{center}
\subfigure[]{\raisebox{5mm}{
\includegraphics[width=0.3\textwidth]{Fig4a.pdf}
\label{fig:dashed_mode_wg}
}}
\subfigure[]{
\includegraphics[width=0.48\textwidth]{Fig4b.pdf}
\label{fig:Husimis_Dashed_Snell}
}
\\
\subfigure[]{{
\includegraphics[width=0.4\textwidth]{Fig4c.pdf}
}
\label{fig:Intensity_Comparision}
}
\subfigure[]{{
\includegraphics[width=0.4\textwidth]{Fig4d.pdf}
}
\label{fig:Transmission_Comparision}
}
\caption{(a) Waveguide-only mode used to compute the exciting fields on the boundary marked by the dashed lines. (b, top) Husimi function $H^\mathrm{inc}_{0,\mathrm{exc}}$ computed from the field distribution in (a) on the dashed boundary. (b, bottom) $H^\mathrm{em}_{1,\mathrm{exc}}$ inside the resonator boundary computed by applying Snells and Fresnel laws to $H^\mathrm{inc}_{0,\mathrm{exc}}$ on (b,top). (c) Comparison of the normalized intracavity energy and the overlap $S_{\mathrm{Exc,Res}}$ for the waveguide-shortegg coupled system. (d) Comparison of the transmission and the reversed normalized overlap for the same system. The black curves (left axis) are obtained from the full-wave simulation, the red curves (right axis) from the phase space analysis.}
\end{center}
\end{figure}
Figure \ref{fig:Husimis_Dashed_Snell} shows its Husimi function $H^\mathrm{inc}_0$ at the boundary marked by the dashed line (top), from which the refracted field can be computed using Snell's and Fresnel's laws (bottom) yielding $H^\mathrm{em}_{1,\mathrm{exc}}$ used in Eq. \eqref{eq:OverlapStrength}. The overlap value $S_{\mathrm{Exc,Res}}$ can be computed for each orientation angle by shifting the center of $H^\mathrm{em}_{1,\mathrm{exc}}$ to the respective angle. This method requires considerably less computational effort than computing the transmission for each orientation angle individually via a full-wave simulation, since here the eigenfrequency and the excitation field have to be computed only once from FEM simulations instead of for each angle separately. The dependence on the orientation angle is accounted for in the overlap computation. Figure.~\ref{fig:Intensity_Comparision} shows the intracavity energy computed with FEM as well as the overlap $S_{\mathrm{Exc,Res}}$ calculated from the phase-space analysis as function of the respective angle, which is $0^o$ for the resonator position depicted in Fig.~\ref{fig:dashed_mode_wg}. In experiments it is common to measure the transmission, since the intracavity energy is not easily accessible and both values are connected due to energy conservation. Note that a big overlap value corresponds to a strong excitation of the mode and therefore a low transmission through the waveguide, thus the overlap values are normalized and substracted from unity (reversed overlap) to yield a comparative measure for the waveguide transmission: $S_{rev}(\theta) = 1 - S_{\mathrm{Exc,Res}(\theta)}/\max(S_{\mathrm{Exc,Res}})$, as displayed in Figure.~\ref{fig:Transmission_Comparision}.
In Fig.~\ref{fig:Transmission_Comparision} one can observe the missing symmetry about the $0^o$-position of the reversed overlap value and the FEM-computed transmission. This is linked to observing only in-coupling components into the cavity and not correcting by how the cavity fields couple back into the waveguide. A close look at $H^\mathrm{em}_{1}$ in Fig.~\ref{fig:Husimis_EF_Shortegg}, shows that the Husimi function is not perfectly symmetric around $(s=\pi/2, \, \sin \chi_1 = -0.8)$, leading to the asymmetry displayed here. As expected, this effects are not relevant when looking at the intra-cavity energy. Nonetheless, the results agree semi quantitatively with the values from full-wave simulations. The differences can be attributed to the uncertainty involved in the calculation of the Husimi function as well as not having corrected for the non-constant curvature of the shortegg cavity. This could be accounted for by computing $H^\mathrm{em}_{1,\mathrm{exc}}$ for each angle, but it comes with a considerable computational effort and thus negates one of the advantages of the used method, since the used method the computation involves only the evaluation of an overlap integral and it provides a reasonable approach to full numerical simulations.
\section{Summary}
In this work we introduce a phase-space approach based on Husimi functions to analyze systems of coupled dielectric resonators. The method involves the computation of the incoming field components at the boundary where the coupling occurs for the individual non-coupled subsystems. From this we compute the respective Husimi functions and use their overlap to characterize the coupling efficiency. The method was verified on a system consisting of an asymmetric resonator excited by coupling to a waveguide. The proposed method provides a good approximation of full FEM calculations with considerably lower computational effort. In addition, this analysis gives intuitive insights into the coupling mechanisms taking place at the cavity boundaries. This method can be applied to more complex coupled optical systems consisting of multiple optical components, enabling a deeper understanding of the coupling processes and an effective approximation of the expected behaviour for the weak coupling regime.
|
2,877,628,090,084 | arxiv | \section{}
\maketitle
Frustrated magnetic systems attract a lot of interest due to the interplay
of frustration and quantum effects which bring about the emergence
of many exotic ground states and fractionalized excitations~\cite{Normand,Balents}. The ($S$\,=\,1/2)
Heisenberg model on the kagome lattice has a particularly high level of frustration
leading to a nonmagnetic ground state and many competing singlet states at low energy ~\cite{Lecheminant}.
The nature of the ground state is currently highly debated~\cite{Yan}. Proposals include various quantum spin liquids
with unbroken lattice symmetry and valence bond solids (VBS)
where the lattice symmetry is
broken. Unfortunately, only a handful of real systems do not
magnetically order at low temperatures, and not all of them are well suited to
study the properties of their ground states. Even fewer can be
grown as single crystals and without the intrinsic disorder which complicates the
interpretation of experimental results.\\
\indent Recently, a singlet ground state has been found in the distorted kagome system
Rb$_2$Cu$_3$SnF$_{12}$ ~\cite{Morita}, where the $S$\,=\,1/2 spin is
carried by the copper Cu$^{2+}$ ion enclosed in an F$_6$
octahedron. The distortion of the kagome lattice can be seen as 6
elongated hexagons surrounding a regular one, and the spins are
connected through the Cu$^{2+}-$F$^--$Cu$^{2+}$ bonds with 4 different
bonding angles, creating 4 different exchange couplings. Magnetization
measurements at high magnetic fields showed a crossover in the behavior
of the system appearing between 10 and 20~T for the magnetic field
perpendicular to the kagome planes (\textbf{H}~$\parallel$~$c$~axis)
and a gradual filling of the triplet band. It was
proposed~\cite{Yang09} that the strongest coupling ($J_1$~=~216\,K~\cite{Matan})
creates valence bonds between spin pairs and forms a ``pinwheel"
pattern around the regular kagome hexagon, with twelve sites in the unit-cell (inset to
Fig.~\ref{spectrum}). The remaining three couplings ($J_2 = 0.95J_1$,
$J_3 = 0.85J_1$, $J_4 = 0.55J_1$) are somewhat weaker,
and what remains of the pure kagome physics is an open question. It is important to note that, for the pure kagome model, a VBS state with a 12-site unit cell was originally described as resonating between the two ``pinwheel'' patterns~\cite{Syromyatnikov}. It was argued that the distortions observed in Rb$_2$Cu$_3$SnF$_{12}$ stabilize the VBS with a single ``pinwheel'' pattern~\cite{Yang09}, and such a chirality breaking has been recently predicted to occur spontaneously in the pure kagome model~\cite{Capponi}. A neutron scattering study~\cite{Matan} has found evidence of a 12-site VBS ground state, separated from the first triplet state with strongly renormalized energy gap $\Delta(H$\,=\,0) = 27\,K $\approx J_1/8$. Measurements in magnetic field up to 6\,T applied along the $c$ axis (perpendicular to the kagome planes) show a reduction of the singlet-triplet gap and lifting of the triplet band degeneracy, accounted for by a \textit{longitudinal} Dzyaloshinskii-Moriya (DM) interaction (\textbf{D} $\parallel$~$c$)~\cite{Matan} which does not break the rotational U(1) symmetry.\\
\indent The availability of Rb$_2$Cu$_3$SnF$_{12}$ in the form of large single
crystals has already stimulated fruitful experimental and theoretical
work~\cite{Morita,Yang09,Matan,Hwang,Khatami}. However, the
microscopic properties of the 12-site ``pinwheel" VBS ground state, as
well as its behavior at high magnetic field, are still unknown. In
this Letter we report an NMR study of the on-site copper $^{63,65}$Cu
nuclei in magnetic fields up to 30\,T, applied parallel to the
$c$ axis (experimental details can be found in~\cite{CF}). We find evidence of an unconventional magnetic lattice with
strong staggered \textit{transverse} magnetic moments. We determine the field
dependence of the singlet-triplet gap, and show that there is no phase
transition connected with the closing of the gap, but rather a
level anticrossing that keeps the gap open. Together
with the field dependence of local spin polarizations, this points to a mixing
\begin{figure*}[t!]
\includegraphics[width=\textwidth]{Fig1.eps}
\caption{\label{spectrum}(color online) $^{63,65}$Cu spectrum of Rb$_2$Cu$_3$SnF$_{12}$ measured at 2.6\,K and 11.568\,T for the magnetic field parallel to the $c$-axis of the crystal. Colors and their hues are used to distinguish positively (red) and negatively (blue) polarized sites, as well as the central lines (dark color) and satellites (light color). Additional lines of $^{87}$Rb and $^{27}$Al (used for magnetic field calibration) are marked in grey. The inset shows the exchange couplings structure of a pinwheel cluster.
}
\end{figure*}
of the singlet and triplet states, typically associated with
DM interaction \textit{perpendicular} to the field and/or a staggered off-diagonal $g$-tensor component $g_s$, that both break the U(1) rotational symmetry.
We argue that the very existence of the anticrossing imposes an additional symmetry condition on the lowest triplet state, which defines the sign of the longitudinal DM interaction. By using exact diagonalization of small clusters, we show that the amplitude of the anticrossing gap is dominantly due to $g_s$ terms.\\
\indent When a single crystal sample is placed in a magnetic field, the NMR spectrum of the copper $^{63,65}$Cu (spin 3/2) nuclei will show 3 lines per isotope for each non-equivalent site: one central line surrounded by two satellites. This sextuplet of lines is multiplied by the number of inequivalent sites, e.g., the spectrum shown in Fig.~\ref{spectrum} presents a very complicated structure spanning over 100 MHz, consisting of 72 NMR lines originating from 12 sites. To analyse such spectra, we follow closely the procedure and nomenclature used in Ref.~\cite{Aimo}, where the technical details are in the second column of page 2. The six NMR frequencies of each sextuplet of lines belonging to one copper site are determined by the quadrupolar coupling tensor, described by its principal component $\nu_Q$ and the asymmetry parameter $\eta$, by the Zeeman coupling to the local magnetic field $H_{\rm{eff}}$, and by the ($\vartheta$,\,$\varphi$) angles defining relative direction of that field with respect to the principal axes of the quadrupolar tensor. Here, these 5 parameters are further constrained by the known nuclear quadrupolar frequency $\nu_{NQR} = \nu_Q \sqrt{1+ \eta^2/3}$, as four different $\nu_{NQR}$ frequencies, all in the narrow range 49.8-55.3~MHz, have been reported in a previous study~\cite{Tashiro}. The main difficulty in the analysis of the spectra lies in the assignments of different sextuplets to lines in the NMR spectrum. This requires numerous trials and errors, where the correct assignment was recognized by successful fit to several complete spectra taken at different field values, leading to reasonable values of the fit parameters ($H_{\rm{eff}}$,~$\vartheta$,~$\varphi$,~$\nu_Q$,~$\eta$)$_i$ , where $i$ denotes different Cu sites. In particular, for nearly all the sites we found $\vartheta \approx 24$\,$\pm$\,2$^\circ$, (while only two sites have somewhat smaller values $13$\,$\pm$\,2$^\circ$) which corresponds precisely to what is expected from the crystal structure~\cite{Morita}. That is, for all CuF$_6$ octahedra, the (average) planes passing through 4 tetragonally placed F$^-$ ions that define the $d_{x^2-y^2}$ orbital of copper are tilted by $22^\circ$-$23^\circ$ with respect to the kagome plane (see inset to Fig.~\ref{Gap}).\\
\indent Among the complete set of parameters, only the $H_{\rm{eff}}$ values are expected to be temperature ($T$) and field dependent. Once determined, the other parameters can be taken as constants, so that in the study of $T$- and $H$-dependence of spectra, from each NMR line position we can calculate the corresponding $H_{\rm{eff}}$ value. For these studies we have thus measured only the central lines of the spectra, meaning that each Cu site was represented by two lines from two $^{65,63}$Cu isotopes which should lead to the same $H_{\rm{eff}}$ value. As shown in Figs.~\ref{Spread}(a) and~\ref{SpreadH}(a) this is indeed the case. In these figures we have plotted the negative of the local field $-H_{\rm{loc}} = H - H_{\rm{eff}}$, where $H$ is the applied field, which is directly proportional to the local spin polarization, $-H_{\rm{loc}} = -\textbf{A} \times 2 \left\langle \textbf{S}\right\rangle - \textbf{K}_{\rm{orb}} \textbf{H}$, up to relatively small constant correction due to the orbital contribution, estimated as $K_{\rm{orb}} \approx 1.5$\,\%. Typical values of the hyperfine coupling tensor \textbf{A} for the Cu nuclei in the CuF$_6$ environment are known~\cite{Kubo}, $A_{\parallel} \approx -18$\,T, $A_{\parallel} /A_{\bot} \approx$~10, where ``$\parallel$" denotes the principal axis of the \textbf{A} tensor, expected to be nearly parallel to the principal axis of the quadrupolar coupling tensor, i.e. tilted by $\approx$$\vartheta$ from \textbf{H} $\parallel c$. In this case, both the longitudinal coupling constant $A_{zz} = A_{\parallel}$cos$^2\vartheta + A_{\bot} $sin$^2\vartheta$, as well as the transverse one $A_{z \bot} = \frac{1}{2}(A_{\parallel} - A_{\bot})$\,sin\,$2\vartheta$\,cos\,$\phi$ are known, and the measured $-H_{\rm{loc}}$ directly reflects the corresponding local spin polarization ($S_z$, $S_{\bot}$, $\phi$), where $\phi$ denotes the azimuthal angle relative to the one defined by the $\vartheta$-tilted $A_{\parallel}$ axis.
It is clear that from one number, $H_{\rm{loc}}$, one cannot {\it{a priori}} deduce three spin components. The necessary information to get $S_z$ and $S_{\bot}$ spin components is thus obtained from the analysis of the $-H_{\rm{loc}}(T, H)$ dependence (Figs.~\ref{Spread} and~\ref{SpreadH}).\\
\indent Our NMR spectrum shown in Fig.~\ref{spectrum} has a large number of lines, corresponding to many different local fields originating from crystallographically and/or magnetically different sites. In contrast to that, the room temperature crystal structure predicts only {\it{two}} inequivalent copper sites in the unit cell. In order to understand the origin of the observed distribution of local fields, we have tracked their temperature dependence, shown in Fig.~\ref{Spread}(a). There, one can identify two families of sites (marked with A (green) and B (orange)) occupied in the ratio $P(A)$\,:\,$P(B)$ = 8\,:\,4 = 2\,:\,1, which can be associated with 12 sites per unit cell of the crystal structure below 215\,K. Close to this temperature a small structural distortion has been observed by X-ray scattering~\cite{Matan}, leading to the enlargement of the lattice cell to $2a$$\times$$2a$, but the superlattice peaks were too weak to fully resolve the new structure at lower $T$. While this structural transition was not detected in the dc susceptibility, such subtle deformations can cause the splitting of the NMR lines. Furthermore, in Fig.~\ref{Spread}(a) it is easy to observe the local ``pairs" of each family that develop opposite polarizations at low temperature, defining thus the local staggered ($H_{A,B}^s$) and uniform ($H_{A,B}^u$) fields,
\begin{figure}[t!]
\includegraphics[width=\columnwidth]{Fig2.eps}
\caption{(color online) (a) Temperature dependence of the local fields at 13.0 T. Two groups of sites are marked by the green (A) and orange (B) symbols. Empty (filled) symbols show the values obtained for $^{63}$Cu ($^{65}$Cu) isotope.
The dashed line marks estimated orbital contribution ($K_{\rm{orb}}~\approx~1.5$\,\%). (b) The average local {\it{staggered}} field at sites A (green, left scale) and B (orange, right scale). (c) The {\it{uniform}} hyperfine field averaged over the A, B and all sites (green, orange and black circles, respectively).}
\label{Spread}
\end{figure}\begin{eqnarray}\label{Eq}
H_{A,B}^s &= (\left\langle H_{i+} \right\rangle_{A,B} - \left\langle H_{i-} \right\rangle_{A,B})/2 ,\nonumber \\
H_{A,B}^u &= (\left\langle H_{i+} \right\rangle_{A,B} + \left\langle H_{i-}\right\rangle_{A,B})/2 ,
\end{eqnarray}
where the average is taken over the upper ($H_{i+}$) or lower ($H_{i-}$) local fields of either A or B family of lines. In Fig.~\ref{Spread}(b) we can see that both A and B sites develop very strong staggered moments below 30\,K ($\approx \Delta(0)$). Additionally (see Fig.~\ref{Spread}(c)), the two families of lines also develop distinct positive ($H_A^u$) and negative ($H_B^u$) uniform magnetizations. When $T$ is raised, the distribution of the local fields is reduced but retains a finite value, even at 120\,K ($\approx 4\Delta(0)$)~\cite{comment1}. This site-inequivalence is attributed to the previously mentioned structural distortion of the lattice~\cite{Matan}. On the other hand, the development of the staggered ($H_{\rm{loc}}^s$) and uniform ($H_{\rm{loc}}^u$) fields at temperatures below the zero field gap in Fig.~\ref{Spread}(b)
\begin{figure}[t!]
\includegraphics[width=\columnwidth]{Fig3.eps}
\caption{(color online) (a) Magnetic field dependence of the local fields at 3.9~K, with the same symbol and color code as in Fig.~\ref{Spread}. (b) The average local staggered field. The thick line is a guide for the eye showing a square root dependence, while the dotted line shows an expected qualitative behavior for lower magnetic fields.}
\label{SpreadH}%
\end{figure}
is clearly related to the formation of the local spin order.\\
\indent As already mentioned, both the transverse and longitudinal magnetization contribute to $H_{\rm{loc}}$, and, due to the uncertainty in the site environments and the complexity of the spectra, we could not find a way to formally separate the two contributions. However, considering their relative size at low temperature, $\left| H^s \right| \gg \left| H^u \right|$, it is natural to associate the staggered (uniform) fields $H^{s(u)}$ defined by Eqs.~(\ref{Eq}) to the transverse (longitudinal) local spin polarization $S_{\bot} (S_z)$, respectively. This is further supported by the field dependence of these values and the physics behind them, as discussed later. Within this attribution we find very big low-$T$ values for the local transverse staggered spin polarization, $S_{\bot}$cos\,$\phi \approx$ 14\,\% and 6\,\% for the A and B sites respectively, while the local uniform fields correspond to much smaller spin polarization of $S_z \approx 1.2$\,\% and $-2$\,\%. While in this way only a projection $S_{\bot}$cos\,$\phi$ is determined, in all cases the spins polarizations lie nearly completely in the kagome plane, $S_{\bot} \gg \left| S_z \right|$.\\
\indent We observe that the total average $\left\langle S_z \right\rangle$ component is almost fully cancelled out, $P(A) \left\langle S_z\right\rangle_A + P(B) \left\langle S_z\right\rangle_B \approx 0$, but the true value cannot be precisely defined, because of the uncertainty in the estimate of the orbital shift $K_{\rm{orb}}$
\begin{figure}[t!]%
\includegraphics[width=\columnwidth]{Fig4.eps}%
\caption{(color online) $H$- dependence of the singlet-triplet energy gap measured by $T^{-1} _1$ (red-yellow circles) and by neutron scattering (black-grey squares, from Ref.~\cite{Matan}). Insets: (top) Arrhenius plot of the measured $T_1^{-1}(T)$ dependence (symbols) and the fit (lines) used to extract the gap values shown in the main figure. Only selected, representative data sets are shown. (bottom-right) Two neighboring CuF$_6$ octahedra.}%
\label{Gap}%
\end{figure}
defining the zero. As regards the transverse staggered components, the smaller value of $H_B^s$ for sites B can be explained either as a weaker polarized moments having equivalent orientations, $\phi_A \approx \phi_B$, or moments of similar size but different $\phi$ values, e.g., $\phi_A \approx 0$ and $\phi_B \approx 65^{\circ}$. We cannot differentiate between these two cases, but the existence of a large staggered magnetization within the kagome planes is not affected by this uncertainty.\\
\indent We have also followed the magnetic field dependence of the local polarizations up to 30\,T (Fig.~\ref{SpreadH}), measured at 3.9\,K where the temperature dependence of the spin polarizations saturates. The field dependence of $H_{A,B}^s$ (Fig.~\ref{SpreadH}(b)) is clearly different from the high field magnetization~\cite{Morita} and follows approximately a square-root dependence, reaching at 30\,T a polarization of $S_{\bot}$cos\,$\phi \approx 30$\,\% for the A sites. Similar temperature and magnetic field dependence of local moments was found~\cite{Kodama} in SrCu$_2$(BO$_3$)$_2$, another 2D frustrated dimer system. There, a moderate in-plane DM interaction ($D/J = 0.034$) and $g_s$ (= 0.023) terms were enough to mix the singlet and triplet states and thus create considerable transverse staggered moments, while the longitudinal moments remained much smaller. In Rb$_2$Cu$_3$SnF$_{12}$ the DM interaction is estimated to be much stronger ($D/J = 0.18-0.2$) \cite{Matan, Hwang}, but only the longitudinal $c$-axis component of the \textbf{D} vector, which does \textit{not} induce level-mixing, has been considered so far.\\
\indent While the application of the critical field in spin-dimer systems is expected to close the singlet-triplet gap by the level-crossing, leading to a new, polarized phase, mixing of the singlet and triplet states will keep the gap open (``anticrossing"). In order to follow the magnetic field dependence of this gap, we have performed measurements of the nuclear spin-lattice relaxation rate $T_{1}^{-1}$, which probes the dynamics of the low-energy excitations. Importantly, no qualitative changes in the spectra nor any $T_1^{-1}$ anomaly were observed down to 1.5\,K, confirming the absence of a phase transition. For a fixed field, the temperature dependence of $T_1^{-1}$ below 6\,K (top inset of Fig.~\ref{Gap}) shows an activated behavior, $\propto e^{-\Delta(H)/T}$, typical of a 2-magnon process, allowing us to determine the value of the gap, $\Delta(H)$~\cite{comment2}. As the field is raised from 8\,T to 13\,T, the gap value is seen to decrease (Fig.~\ref{Gap}), and the NMR data smoothly continue those determined from the neutron-scattering up to 6\,T~\cite{Matan}. Close to 13\,T the gap value passes through a broad minimum after which it again increases at higher fields. The residual value of the gap $\Delta(13~\rm{T}) \approx \Delta(0)/2$ appears to be very large, which is a clear evidence for important U(1)-symmetry breaking terms that induce level anticrossing.\\
\indent In order to describe this anticrossing, we have included the
transverse anisotropies in the Hamiltonian, which was defined previously
by the Heisenberg couplings $J_1 - J_4$ and the $z$ component ($d_z$)
of the \textbf{D} vector~\cite{Matan, Hwang}, and performed exact diagonalizations of this Hamiltonian on clusters of $N$=~12 and 24 sites \cite{CF}. We consider the in-plane DM terms ($d_p = D_{\perp i}/J_i$) perpendicular
to the bonds and respecting the crystal symmetry, as well as the
off-diagonal $g$-tensor terms as generated by the strong tilting of
the CuF$_6$ octahedra, $\vartheta \approx 23^{\circ}$ (inset of
Fig.~\ref{Gap}), leading to $g_s = 0.135$ (see \cite{CF} for
definitions). Remarkably, depending on the helicity of these terms as compared to that defined by the screw axis of the $d_z$ vector, the system presents either level crossing or anticrossing. That is, depending on the sign of $d_z$, the lowest triplet state will be either of $E_g^{+}$ or $E_g^{-}$ symmetry, and only one of these two (in the present case $E_g^{-}$ corresponding to $d_z>0$ \cite{CF}) allows for level anticrossing. The calculated gap presents \textit{small} finite-size effects near the anticrossing, in contrast to the zero-field gap as described in~\mbox{\cite{Matan,Hwang}}. This enabled us to make a theoretical estimate of the residual gap size and position, and conclude that the experimentally observed gap is well accounted for by the $g_s$ terms alone, with calculated values $\Delta_{res}^{th}=13.0$~K and $H_c^{th} = 14.0$~T \cite{CF}, while only minor contribution can come from $d_p$ (estimated to $\left|d_p\right| < 0.012$).\\
\indent In summary, we have described the microscopic properties of Rb$_2$Cu$_3$SnF$_{12}$ by on-site $^{63,65}$Cu NMR.
In the field perpendicular to the kagome planes and at low temperature, the NMR spectra evidence a strong staggered transverse spin polarization, growing approximately as a square-root of magnetic field.
The field dependence of the singlet-triplet gap measured via the $T_1^{-1}$ data presents an anticrossing of the energy levels with a large residual gap value $\Delta(13~\rm{T}) \approx \Delta(0)/2$, well accounted for by the staggered $g$-tensor terms defined by the crystal structure.
We have argued that the observed anticrossing and the absence of phase transition is only compatible with a given sign of $d_z$. The other sign would lead to a phase transition because the spontaneously induced transverse moments would break the rotation symmetry of the crystal. Further theoretical work is needed to fully exploit the available information on spin polarizations.\\
\indent We acknowledge fruitful discussions with Y. Fukumoto, H. Mayaffre and S. Maegawa. We thank W. G. Clark for reviewing the manuscript. Part of this work has been supported by the French ANR project NEMSICOM, by the EuroMagNET network under the EU contract No. 228043, by the ARRS project No. J1-2118, and by the EU FP7 project SOLeNeMaR No. 229390.
|
2,877,628,090,085 | arxiv | \section{Introduction}
\label{sec:intro}
Light-matter systems in which the coherent coupling frequencies exceed the
dissipative loss rates are promising elements for solid state quantum
information circuits~\cite{Wallraff2004, Kubo2010, Putz2014}. Spin ensembles
may couple strongly to electromagnetic modes of a microwave resonator
resulting in hybridized states referred to as magnon
polaritons~\cite{Mills1974, Lehmeyer1985, Cao2015, Zare2015} with the benefit
of long coherence ~\cite{Bar-Gill2013} and short
manipulation~\cite{Childress2006} times. Here a \textquotedblleft
magnon\textquotedblright\ refers to the collective excitation or spin wave of
the polarized spin system. Ferro/ferrimagnets can combine a high spontaneous
spin density with low damping leading to large cooperativities and narrow
linewidths~\cite{Tabuchi2014, Zhang2014}. The strong, and even ultra-strong
coupling regime in which the coupling strength $g$ is comparable to the mode
frequencies~\cite{Niemczyk2010} can therefore be accessed with relative ease.
Furthermore, due to the possibility of coupling magnon modes to photons at
optical frequencies~\cite{Shen1966, Demokritov2001}, magnetic systems are
candidates for coherent conversion of solid state qubits into
\textquotedblleft flying ones\textquotedblright~\cite{Zhang2015, Osada2016}.
On the other hand, controlled creation and read-out of spin-entangled states
in quantum information processing with solid state systems remains a major
challenge. Coherent coupling of spins can be mediated by a variety of physical
mechanisms, such as the magnetic dipolar, exchange, or spin-orbit interaction.
The coupling of spins/pseudospins does not have to be direct, but can be
realized via an intermediary. This can be localized electrons in a filled
shell ion that generate superexchange or the itinerant carriers of metals in
the RKKY interaction~\cite{Ruderman1954, Kasuya1956, Yosida1957}. The
non-local exchange coupling can have either sign; it causes the staggered
magnetization in magnetic multilayers that display the giant
magnetoresistance~\cite{Parkin1990, Bruno1991, Parkin1991}. Quantum systems
can also be coupled radiatively over large distances, i.e. when the
interaction is mediated by virtual photons in a low-loss resonator or
cavity~\cite{Haroche2006, Blais2004}.
Here we address the hybridization of two magnets by cavity photons. Yttrium
iron garnet (YIG), a ferrimagnetic insulator that serves in magnetically
tunable filters and resonators at microwave frequencies, can provide high
coupling strengths and low damping. YIG's spin density is 2$\cdot$10$^{22}$
cm$^{-3}$~\cite{Gilleo1958}, while its Gilbert constant of the magnetization
dynamics typically ranges from 10$^{-3}$ to 10$^{-5}$~\cite{Kajiwara2010,
Heinrich2011, Kurebayashi2011}. Strong coupling between magnons and cavity
photons are manifest in a series of anticrossings in YIG films in coplanar
resonators~\cite{Huebl2013, Stenning2013, Bhoi2014} and YIG spheres in 3D
microwave cavities~\cite{Tabuchi2014, Zhang2014, Goryachev2014}.
Soykal \textit{et al.} \cite{Soykal2010} reported a quantum theory of
photon-magnon coupling in YIG spheres, but this regime has not yet been
reached in experiments. Cao \textit{et al.} modelled the classical
magnon-photon coupling for a thin YIG film in a planar cavity and found strong
coupling even for spin waves beyond the Kittel mode in microwave transmission
and inverse spin Hall effect~\cite{Cao2015}, which was confirmed by
experiments \cite{Bai2015,MaierHaig2016}. Our study of the coherent coupling
between a YIG sphere and microwave cavity modes~\cite{Zare2015} revealed that
YIG\ spheres are efficient antennas for microwaves such that
(ultra)strong-coupling regimes can be achieved in stand-alone magnetic
spheres, as exploited recently~\cite{Bourhill2016}. The long-range strong
coupling of magnons in spatially separated YIG spheres as mediated by a
microwave cavity has been reported~\cite{Zhang2015, Lambert2016}. Electrical
readout of two distant YIG
$\vert$%
Pt bilayers coupled by a microwave cavity mode has been demonstrated
recently~\cite{Bai2017}.
Here we extend the classical model~\cite{Zare2015} to investigate the
long-range coupling of magnons in two spatially separated YIG spheres mediated
by a microwave cavity, producing a delocalized magnon-polariton hybridized
state. The conventional magnetostatic approximation~\cite{Walker1958,
Fletcher1959}, in which the spins interact by the magnetic dipolar field,
disregarding exchange as well as propagation effects, is valid in the Rayleigh
regime $\lambda\gg a$, where $a$ is the radius of the sphere and $\lambda$ the
wavelength of the incident radiation, but breaks down when $\lambda<a$, which
is the regime encountered in sub-mm YIG\ spheres and nanostructured thin
films. We therefore study here the properties of the hybridized
magnon-polaritons, including retardation effects of microwaves, but disregard
the exchange interaction, which is valid for ferromagnets as long as the
exchange length $l_{\mathrm{ex}}=\sqrt{2A/(\mu_{0}M_{s}^{2})}\ll a$, with $A$
and $M_{s}$ being the exchange constant and saturation magnetization,
respectively. Our results help to picture photon-mediated coupling between two
or more magnetic samples in terms of the concept of a chemical bond.
This manuscript is organized as follows. In Sec.~\ref{sec:model}, we introduce
the details of our model and derive the scattered intensity and efficiency
factors for a strongly coupled system of two magnetic spheres in a spherical
microwave cavity. In Sec.~\ref{sec:results}, we present and discuss our
results that demonstrate the effects both due to the dielectric as well as
magnetic effects on the scattering properties and compare our results with
experiments. In Sec.~\ref{sec:concl}, we conclude and summarize our findings.
\section{Model and formalism}
\label{sec:model}
Mie expressed a general scattering problem in terms of a rapidly converging
expansion into spherical multipole partial waves~\cite{Mie1908, Stratton2007}.
Here we model the indirect coupling of the collective excitations of two
magnetic spheres mediated by photons in a spherical cavity by a Mie-like
expansion of the coupled Landau-Lifshitz-Gilbert and Maxwell equations. We
consider a plane electromagnetic wave with arbitrary polarization and wave
vector shining on a cavity loaded by two magnetic spheres with gyromagnetic
permeability tensors $\overleftrightarrow{\mu}_{1}$ and
$\overleftrightarrow{\mu}_{2}$. A thin spherical shell of a material with high
dielectric constant $\epsilon_{c}/\epsilon_{0}\gg1$, radius $R$, and thickness
$\delta$, models a generic resonant cavity. We mimic realistic situations by
adjusting the parameters $R$ and $\delta$ (see Fig. \ref{fig1}) to tune the
frequencies and broadenings of the cavity modes.
The dynamics of the magnetization vector $\mathbf{M}$ is described by the LLG
equation,
\begin{equation}
\partial_{t}\mathbf{M}=-\gamma\mathbf{M}\times\mathbf{H}_{\mathrm{eff}}%
+\frac{\alpha}{M_{s}}\mathbf{M}\times\partial_{t}\mathbf{M}%
\end{equation}
with $\alpha$ and $M_{s}$ being the damping parameter and saturated
magnetization, respectively. Effective field $\mathbf{H}_{\mathrm{eff}%
}=\mathbf{H}_{\mathrm{ext}}+\mathbf{h}$ comprises the external and (collinear)
easy axis anisotropy fields $\mathbf{H}_{\mathrm{ext}}$ as well as a
distributed ac field $\mathbf{h}(\mathbf{r},t)$. We linearize the LLG equation
by considering the magnetization and driving field vectors \begin{figure}[ptb]
\includegraphics[width=8.5cm]{fig1}\caption{(Color online) A plane
electromagnetic wave illuminates a large spherical cavity from an arbitray
direction. The latter is modeled by a dielectric spherical shell of radius
$R$, thickness $\delta$, and permittivity $\epsilon_{c}$. Two magnetic spheres
of radius $a_{1}$ and $a_{2}$ are located at antinodes of the ac magnetic
field of the (2,2) and (2,-2) confinement modes of the cavity, i.e., at
$\mathbf{d}_{1}$ and $\mathbf{d}_{2}$ on the $\mathbf{x}$ axis. A constant
magnetic field $H_{0}$ saturates the equilibrium magnetizations. The scattered
waves are measured by a detector in the far field as a function of the
scattering angles, here $\left( \theta,\varphi\right) =\left( \pi
/2,\pi\right) $.}%
\label{fig1}%
\end{figure}%
\begin{align}
\mathbf{M}(\mathbf{r},t) & =\mathbf{M}_{0}+\mathbf{m}(\mathbf{r},t)\\
\mathbf{H}(\mathbf{r},t) & =\mathbf{H}_{0}+\mathbf{h}(\mathbf{r},t).
\end{align}
To leading order in the small modulations $\mathbf{m}$ and $\mathbf{h}$:
\begin{equation}
\partial_{t}\mathbf{m}=-\gamma(\mathbf{M}_{0}\times\mathbf{H}_{\mathrm{eff}%
}^{(1)}+\mathbf{m}\times\mathbf{H}_{\mathrm{eff}}^{(0)})+\frac{\alpha}{M_{s}%
}\mathbf{M}_{0}\times\partial_{t}\mathbf{m,} \label{eq5}%
\end{equation}
where $\mathbf{H}_{\mathrm{eff}}^{(0)}=\mathbf{H}_{\mathrm{ext}}$ and
$\mathbf{H}_{\mathrm{eff}}^{(1)}=\mathbf{h}$. In the frequency domain, for
$\mathbf{H}_{\mathrm{ext}}$ and $\mathbf{M}_{0}\Vert\hat{z}$,
\begin{equation}
i\omega\mathbf{m}=\mathbf{z}\times\left( \omega_{\mathrm{M}}\mathbf{h}%
-\omega_{\mathrm{H}}\mathbf{m}+i\omega\alpha\mathbf{m}\right)
\end{equation}
with $\omega_{\mathrm{M}}=\gamma M_{s}$ and $\omega_{\mathrm{H}}=\gamma H_{0}%
$. We express Eq. (\ref{eq5}) as $\mathbf{m}=\overleftrightarrow{\chi}%
\cdot\mathbf{h}$ in terms of the magnetic permeability tensor
\begin{align}
\overleftrightarrow{\mu} & =\mu_{0}(\overleftrightarrow{\mathrm{I}%
}+\overleftrightarrow{\chi})\label{eq7}\\
& =\mu_{0}%
\begin{pmatrix}
1+\chi & -i\kappa & 0\\
i\kappa & 1+\chi & 0\\
0 & 0 & 1
\end{pmatrix}
,
\end{align}
where
\begin{align}
\chi & =\frac{(\omega_{\mathrm{H}}-i\alpha\omega)\omega_{\mathrm{M}}}%
{(\omega_{\mathrm{H}}-i\alpha\omega)^{2}-\omega^{2}},\\
\kappa & =\frac{\omega\omega_{\mathrm{M}}}{(\omega_{\mathrm{H}}-i\alpha
\omega)^{2}-\omega^{2}}.
\end{align}
The Maxwell equations inside a homogeneous sphere at frequency $\omega$ read
\begin{align}
\nabla\times\mathbf{E} & =i\omega\mathbf{b};\quad\nabla\times\mathbf{h}%
=-i\omega\epsilon_{\mathrm{sp}}\mathbf{E}\\
\nabla\cdot\mathbf{E} & =0;\quad\nabla\cdot\mathbf{b}=0. \label{eq10}%
\end{align}
The magnetic induction $\mathbf{b}$ and the magnetic field $\mathbf{h}$ inside
this medium are related by
\begin{equation}
\mathbf{b}=\overleftrightarrow{\mu}\cdot\mathbf{h},\quad\mathbf{D}%
=\epsilon_{\mathrm{sp}}\mathbf{E}.
\end{equation}
and $\mathbf{b}$ satisfies the wave equation
\begin{equation}
\nabla\times\nabla\times\left( \mu_{0}\overleftrightarrow{\mu}^{-1}%
\cdot\mathbf{b}\right) -k_{\mathrm{sp}}^{2}\mathbf{b}=0, \label{wave_eq}%
\end{equation}
where $k_{\mathrm{sp}}^{2}=\omega^{2}\epsilon_{\mathrm{sp}}\mu_{0}$ and
$\epsilon_{\mathrm{sp}}$ is the scalar permittivity of the medium. Keeping Eq.
(\ref{eq10}) in mind, we expand $\mathbf{h}$ in terms of vector spherical
waves as
\begin{equation}
\mathbf{h}=\sum_{nm}\bar{\eta}_{nm}\left[ d_{mn}\mathbf{V}_{nm}%
^{(1)}(k,\mathbf{r})+c_{mn}\mathbf{N}_{nm}^{(1)}(k,\mathbf{r})\right]
\label{eq16}%
\end{equation}
where $k$ is as yet undetermined, $n$ runs from $1$ to $\infty,$ and
$m=-n,\cdots,n.$ The prefactors read $\bar{\eta}_{nm}=\eta_{nm}k_{0}%
/(\omega\mu_{0})$ with
\begin{equation}
\eta_{nm}=i^{n}E_{0}\left[ \frac{2n+1}{n(n+1)}\frac{(n-m)!}{(n+m)!}\right]
^{1/2}%
\end{equation}
where $E_{0}$ is the amplitude of the electric field of the incident wave. The
vector spherical wave functions are defined as
\begin{align}
\mathbf{V}_{nm}^{(j)}(k,\mathbf{r}) & =z_{n}^{(j)}(kr)\mathbf{X}%
_{nm}(\mathbf{r}),\nonumber\\
k\mathbf{N}_{nm}^{(j)}(k,\mathbf{r}) & =\nabla\times\mathbf{V}_{nm}%
^{(j)}(k,\mathbf{r}).
\end{align}
where $z_{n}^{(j)}$ are spherical Bessel functions of the $j$-th kind, e.g.,
$z_{n}^{(3)}=h_{n}^{(1)}$ is the spherical Bessel functions of the third kind
(Hankel function). $\mathbf{X}_{nm}=\mathbf{L}Y_{nm}(\hat{r})/\sqrt{n(n+1)}$,
where $Y_{nm}(\hat{r})$ are spherical (surface) harmonics and $\mathbf{L}%
=-i\mathbf{r}\times\nabla_{r}$ is the angular momentum and $\nabla_{r}$ the
gradient operator. By invoking the vector spherical wave function expansion
for $\mathbf{b}$ and $\overleftrightarrow{\mu}^{-1}\cdot\mathbf{b}$ in the
wave equation Eq. (\ref{wave_eq}) leads to the dispersion relation for
$k(\omega)$. We focus in the following on the lowest frequency resonances for
a given angular momentum without radial nodes in the sphere. For simplicity of
notation we therefore omit the \textquotedblleft main quantum
number\textquotedblright\ when labelling the cavity modes.
The electric field distribution is obtained by $\mathbf{E}=(i/\omega
c)\nabla\times\mathbf{h}$. We expand the incident fields $\mathbf{E}%
_{\mathrm{inc}}$, $\mathbf{h}_{\mathrm{inc}}$ and scattered fields
$\mathbf{E}_{s}$, $\mathbf{h}_{s}$ outside the sphere analogously. The
scattered field reads then
\begin{equation}
\mathbf{h}_{s}=\sum_{nm}\bar{\eta}_{nm}\left[ b_{mn}\mathbf{N}_{nm}%
^{(3)}+a_{mn}\mathbf{V}_{nm}^{(3)}\right]
\end{equation}
with $k_{0}^{2}=\omega^{2}\epsilon_{0}\mu_{0}$. The expansion coefficients
$a_{nm}$ and $b_{nm}$ are determined by the boundary conditions. We consider
the situation that the magnetic sphere is illuminated by a plane wave with
arbitrary polarization and direction of incidence as indicated in Fig.
(\ref{fig1}). This incident fields can be expanded as,
\begin{equation}
\mathbf{h}_{\mathrm{inc}}=-\sum_{nm}\bar{\eta}_{nm}\left[ q_{mn}%
\mathbf{N}_{nm}^{(1)}+p_{mn}\mathbf{V}_{nm}^{(1)}\right]
\end{equation}
with coefficients
\begin{align}
p_{mn} & =\frac{\eta_{nm}}{i^{n}E_{0}}\left[ p_{\theta}\tau_{mn}(\cos
\theta_{k})-ip_{\phi}\pi_{mn}(\cos\theta_{k})\right] e^{-im\phi_{k}%
}\label{eq54}\\
q_{mn} & =\frac{\eta_{nm}}{i^{n}E_{0}}\left[ p_{\theta}\pi_{mn}(\cos
\theta_{k})-ip_{\phi}\tau_{mn}(\cos\theta_{k})\right] e^{-im\phi_{k}}
\label{eq55}%
\end{align}
where $\hat{\mathbf{p}}=(p_{\theta}\boldsymbol{\hat{\theta}}_{k}+p_{\phi
}\boldsymbol{\hat{\phi}}_{k})$ is the normalized complex polarization vector,
with unit vectors $\boldsymbol{\hat{\theta}}_{k}$ and $\boldsymbol{\hat{\phi}%
}_{k},$ $\left\vert \hat{\mathbf{p}}\right\vert =1$ and $\theta_{k}(\phi_{k})$
is the polar (azimuthal) angle of incidence. Two auxiliary functions are
defined by
\begin{equation}
\pi_{mn}(\cos\theta)=\frac{m}{\sin\theta}P_{n}^{m}(\cos\theta),~\tau_{mn}%
(\cos\theta)=\frac{d}{d\theta}P_{n}^{m}(\cos\theta)
\end{equation}
All fields of the scattering problem are now expanded in terms of vector
spherical wave functions. The boundary conditions
\begin{align}
\left[ \mathbf{E}_{\mathrm{inc}}+\mathbf{E}_{s}\right] \times\mathbf{e_{r}}
& =\mathbf{E}_{i}\times\mathbf{e_{r},}\\
\left[ \mathbf{h}_{\mathrm{inc}}+\mathbf{h}_{s}\right] \times\mathbf{e_{r}}
& =\mathbf{h}_{i}\times\mathbf{e_{r}}%
\end{align}
can be rewritten in terms of the transmission matrix $\mathcal{T}$ that
relates the scattered to the incoming fields
\begin{equation}%
\begin{pmatrix}
a_{nm}\\
b_{nm}%
\end{pmatrix}
=\mathcal{T}%
\begin{pmatrix}
p_{nm}\\
q_{nm}%
\end{pmatrix}
. \label{sscatcoeff}%
\end{equation}
We are interested in more than one scattering objects in the cavity. In order
to describe the collective excitations of non-overlapping magnetic spheres, we
expand the total incident field striking the surface of the $i$-th sphere, the
initial incident waves, and the scattered field of the other spheres with
index $j\neq i$, in the coordinate systems centered at sphere $i$ as
\begin{equation}
\mathbf{E}_{\mathrm{inc}}^{i}=\mathbf{E}_{\mathrm{inc}}+\sum_{j\neq
i}\mathbf{E}_{s}^{j};\quad\mathbf{h}_{\mathrm{inc}}^{i}=\mathbf{h}%
_{\mathrm{inc}}+\sum_{j\neq i}\mathbf{h}_{s}^{j}%
\end{equation}
The transformation of waves scattered by one sphere into incident waves for
the other one is formulated by the addition theorem of vector spherical
harmonics~\cite{Xu1996}, i.e., the expansion of the basis set in a translated
reference system. By transforming the wave scattered by one sphere to a
coordinate system centered at the other and imposing appropriate boundary
conditions, we arrive at the scattering coefficients
\begin{equation}%
\begin{pmatrix}
a_{nm}^{i}\\
b_{nm}^{i}%
\end{pmatrix}
=\mathcal{T}^{i}\left[
\begin{pmatrix}
p_{nm}^{i}\\
q_{nm}^{i}%
\end{pmatrix}
+\sum_{j\neq i}\mathcal{R}^{ji}%
\begin{pmatrix}
a_{nm}^{j}\\
b_{nm}^{j}%
\end{pmatrix}
\right] ,
\end{equation}
where the superscript indicates the coordinate system centered at sphere $i$
and $\mathcal{R}^{ji}$ is the translation matrix from sphere $j$ to
$i$~\cite{Xu1996}. The second term on the right-hand side represents the
multiple scattering between the objects. The scattering coefficients in the
coordinate system of the cavity can be obtained by the unitary transformation
$\mathcal{R}^{i0}$ defined by the addition theorem
\begin{equation}%
\begin{pmatrix}
a_{nm}^{0}\\
b_{nm}^{0}%
\end{pmatrix}
=\mathcal{R}^{i0}%
\begin{pmatrix}
a_{nm}^{i}\\
b_{nm}^{i}%
\end{pmatrix}
\end{equation}
These expressions are sufficient to compute the scattering matrix for the
entire system.
In order to make contact with experiments, we consider the \textit{far-field}
limit, in which the intensity of the two polarization components $I_{\theta}$
and $I_{\phi}$ are
\begin{equation}
I_{\theta}\sim\frac{E_{0}^{2}}{k_{0}^{2}r^{2}}|S_{1}(\theta,\phi)|^{2},\qquad
I_{\phi}\sim\frac{E_{0}^{2}}{k_{0}^{2}r^{2}}|S_{2}(\theta,\phi)|^{2}%
\end{equation}
where $\theta(\phi)$ is the polar (azimuthal) angle of the observer at
distance $r$ and scattering intensity functions are
\begin{align}
S_{1}(\theta,\phi) & =\sum_{nm}\left[ a_{mn}\tilde{\tau}_{mn}\left(
\cos\theta\right) +b_{mn}\tilde{\pi}_{mn}\left( \cos\theta\right) \right]
e^{im\phi},\qquad\\
S_{2}(\theta,\phi) & =\sum_{nm}\left[ a_{mn}\tilde{\pi}_{mn}\left(
\cos\theta\right) +b_{mn}\tilde{\tau}_{mn}\left( \cos\theta\right) \right]
e^{im\phi},\qquad
\end{align}
We define a dimensionless \textit{scattering efficiency factor }%
$Q_{\mathrm{sca}}$ as the total (i.e. angular integrated) scattering cross
section of the light intensity divided by the geometrical area $\pi R^{2}$
as,
\begin{equation}
Q_{\mathrm{sca}}=\dfrac{4}{k_{0}^{2}R^{2}}\sum_{nm}\left( |a_{nm}%
|^{2}+|b_{nm}|^{2}\right)
\end{equation}
The efficiency factor $Q_{\mathrm{ext}}$ defined analogously for the total
extinction cross section
\begin{equation}
Q_{\mathrm{ext}}=\dfrac{4}{k_{0}^{2}R^{2}}\sum_{nm}\operatorname{Re}\left(
p_{nm}^{\ast}a_{nm}+q_{nm}^{\ast}b_{nm}\right)
\end{equation}
measures the total energy loss of the incident beam by absorption and
scattering.
\begin{equation}
Q_{\mathrm{abs}}=Q_{\mathrm{ext}}-Q_{\mathrm{sca}} \label{Qsca}%
\end{equation}
reflects the loss of intensity due to Gilbert damping in the sample.
\section{Results}
\label{sec:results}
The observables defined above can be computed numerically as a function of
material and cavity parameters. We focus here on a spherical cavity with fixed
radius ($R=4$\thinspace mm) loaded with two dielectric spheres at a fixed
distance $d_{0}=2.5$\thinspace mm, but with adjustable diameter, as in Fig.
\ref{fig1}. We focus on the strong coupling regime in which the polaritonic
mode splitting is comparable or larger than the dissipation, i.e. we have
spectrally sharp cavity modes and not too large Gilbert damping. Without using
the macrospin approximation, we focus our discussion to the nearly uniform
(Kittel) mode that displays the strongest coupling to the
microwaves~\cite{Cao2015}.
Forward scattered intensities, i.e., $\theta=\pi/2,\phi=\pi$, and scattering
efficiency factors are convenient and observable measures of the
microwave-matter coupling. In order to compare our results with recent
experiments, we adopt parameters for YIG with gyromagnetic ratio $\gamma
/(2\pi)=28$ GHz/T, saturation magnetization $\mu_{0}M_{s}=175$ mT~
\cite{Manuilov2009}, Gilbert damping constant $\alpha=3\times10^{-4}%
$~\cite{Kajiwara2010, Heinrich2011, Kurebayashi2011}, and relative
permittivity $\epsilon_{\mathrm{sp}}/\epsilon_{0}=15$~\cite{Sadhana2009}. The
incident microwave radiation comes from the positive $\mathbf{x}$ direction
($\theta_{k}=\pi/2$ and $\phi_{k}=0$) and is linearly polarized such that its
electric/magnetic components are in the $-\mathbf{z}/\mathbf{y}$ directions
(static magnetic field and magnetization $\mathbf{H}_{0}\parallel\mathbf{z}$).
We also investigate the dependence of the observables on the scattering angle
with respect to the outgoing radiation. \begin{figure}[ptb]
\includegraphics[width=8.5cm]{fig2}\caption{(Color online) (a): The scattering
efficiency factor Eq. (\ref{Qsca}) for two non-magnetic dielectric spheres of
radius $a_{1}=a_{2}=1$ mm, cavity radius $R=4$ mm, and asymmetry $\delta n=1$
plotted as a function of frequency $\omega/2\pi$ and average refractive index
$n_{\mathrm{sp}}$. (b) and (c): The scattering intensity $|S_{1}|^{2}$ as
function of scattering angle $\theta$ and frequency $\omega/2\pi$ plotted for
the same spheres ($n_{\mathrm{sp}}=7,$ $\delta n=1$) without and with cavity,
respectively, while (d) and (e) are the corresponding scattering efficiencies.
The anticrossing in (e) reveals the interaction with the cavity field by the
coupling strength $2g_{\mathrm{eff}}^{\mathrm{di}}$, i.e., the frequency
splitting of the modes at $\delta n=0$. The dashed lines are guides for the
eye.}%
\label{fig2}%
\end{figure}
We start by studying the effects of asymmetry on the photon-mediated coupling
of two \emph{non-magnetic} spheres with refractive indices $n_{1}%
=n_{\mathrm{sp}}+\delta n$ and $n_{2}=n_{\mathrm{sp}}-\delta n$. In Fig.
\ref{fig2}(a) the scattering efficiency factor Eq. (\ref{Qsca}) is plotted as
a function of frequency $\omega/\left( 2\pi\right) $ and average refractive
index $n_{\mathrm{sp}}=\sqrt{\epsilon_{\mathrm{sp}}/\epsilon_{0}}$ of the
spheres with $a=1$ mm in a spherical cavity with radius $R=4$ mm and broken
symmetry with $\delta n=1$. The spheres are placed at the local maxima of the
electric field distribution of the cavity, i.e., $\mathbf{d}_{1}%
=d_{0}\mathbf{x}$ and $\mathbf{d}_{2}=-d_{0}\mathbf{x}$, respectively, where
$d_{0}=2.5$\thinspace mm. This ensures a significant coupling strength and
nearly uniform distribution of the cavity field over the spheres.
When $\delta n\neq0$ the individual resonances of the two spheres are
distinguishable in Fig. \ref{fig2}(a). Not only the lowest but also higher
plasmonic modes ($\sim n_{\mathrm{sp}}^{2})$ anticross strongly with the
(constant) cavity resonances. The angular dependence of the scattering without
and with cavity is plotted in panels (b) and (c) of Fig. \ref{fig2},
respectively. The eigenmodes of the two coupled-dielectric spheres have a
predominant $s$-wave character when the wavelength $\lambda\gtrapprox
a\sqrt{\epsilon_{\mathrm{sp}}/\epsilon_{0}}$, i.e. no scattering-angle
dependence in the regime in which no resonant states are formed.
The radiative coupling between two dielectric spheres by the cavity eigenmodes
is revealed by tuning the resonances with the asymmetry parameter $\delta
n\in\lbrack-0.5,0.5]$ for $n_{\mathrm{sp}}=7$. Fig. \ref{fig2}(d) and (e) are
plots of the scattering efficiency factor $Q_{\mathrm{sca}}$ as a function of
frequency $\omega/2\pi$ and asymmetry $\delta n$ in the absence and presence
of the external cavity, respectively. The photon-mediated coupling corresponds
to the splitting at the nominal crossing point ($\delta n=0$) and found to be
$g_{\mathrm{eff}}^{\mathrm{di}}/2\pi\sim0.6$ GHz, which is much larger that
the broadening and therefore \textquotedblleft strong\textquotedblright.
Removing the cavity suppresses the splitting, as seen in Fig. \ref{fig2}(d),
proving that the direct dipolar coupling between the spheres and the multiple
scattering of the microwaves between spheres in the absence of a cavity are
weak. In analogy with plasmonic molecules in metallic
nanostructures~\cite{Prodan}, which are bound by the optical near-fields, we
refer to this hybridized state as a \emph{plasmon-polariton molecule}.
\begin{figure}[ptb]
\includegraphics[width=8.5cm]{fig3}\caption{(Color online) Scattering
efficiency factor $Q_{\mathrm{sca}}$ as function of magnetic field
$H_{0}/M_{s}$ and frequency $\omega/2\pi$ for two YIG spheres of radius
$a_{1}=a_{2}=0.5$ mm and relative permittivity $\epsilon_{\mathrm{sp}%
}/\epsilon_{0}=15$ in a spherical cavity of radius $R=4$ mm on the two
antinodes of the cavity mode at $\omega_{2}/2\pi\sim7.05$ GHz is shown in
bottom panel. The field at each sphere is detuned by $\left\vert \delta
H\right\vert /M_{s}\sim0.2$ with opposite sign. $g_{\mathrm{eff}%
}^{\mathrm{mag}}$ is the magnon-cavity coupling strength. The radial component
of microwave magnetic field $h_{r}$ for the cavity mode frequency $\omega_{2}$
in the equator plane is shown in top panel, the black circles indicate two
spheres.}%
\label{fig3}%
\end{figure}
\begin{figure}[ptb]
\includegraphics[width=8.5cm]{fig4}\caption{(Color online) (a) and (b):
Scattering efficiency factor $Q_{\mathrm{sca}}$ as function of $\omega/2\pi$
and $\delta H/M_{s}$ for the same two spheres as Fig. (\ref{fig3}) without and
with cavity, respectively, but the detuning is much smaller than in Fig.
(\ref{fig3}). $H_{0}/M_{s}=1$ is fixed such that the magnetostatic modes of
each sphere are detuned from $\omega_{1}$. The anticrossing in (b) illustrates
the coupling of the two YIG spheres; the non-local magnon-magnon coupling
strength $g_{\mathrm{eff}}^{\mathrm{ind.mag}}$ is the frequency splitting of
the modes at $\delta H=0$. The azimuthal component of microwave magnetic field
$h_{\phi}$ for the cavity mode frequency $\omega_{1}$ in the equator plane is
shown in top panel, the black circles indicate two spheres.}%
\label{fig4}%
\end{figure}
The magnetism of the spheres affects the microwave scattering properties
strongly, but the plasmonic effects causing hybridization of the resonances of
cavity and sphere remain to be very relevant. Our results help to interpret
recent experimental results on cavity-mediated coupling of two YIG
spheres~\cite{Lambert2016} by taking into acount the finite size of the
spheres and cavity-field distribution. Fig. \ref{fig3} shows the scattering
efficiency factor as a function of frequency $\omega/2\pi$ and uniform
magnetic field $H_{0}/M_{s}$ for our spherical cavity containing now two YIG
spheres with radii $a_{1}=a_{2}=0.5$ mm. The frequency of the microwaves with
wave vector along the $x$-direction is tuned to the 5-fold degenerate cavity
modes with $n=2$ ($d$-wave); $\omega_{2}/2\pi\sim7.05$ GHz, of which only the
$\omega_{2,\pm2}$ states are excited by symmetry. An asymmetry is now induced
by a detuning magnetic field with opposite sign on different spheres $\delta
H=\pm0.2\,M_{s}$. The two spheres occupy antinodes of the $p$ and $d$ cavity
resonances shown in the top panels of Figs. \ref{fig3} and \ref{fig4} with
parameters chosen to be close to the experiment~\cite{Lambert2016}. Two
distinct anticrossings are the signature of mixed \textit{magnon-polariton}
modes with a magnon-photon coupling of $g_{\mathrm{eff}}^{\mathrm{mag}}%
/2\pi\sim150$ MHz between the Kittel modes of both spheres and the cavity
mode. Small satellites indicate the coupling to a higher (\textquotedblleft
Walker\textquotedblright) mode in both spheres. Fig. \ref{fig3} also shows a
cavity mode that is not affected by the magnets~\cite{Zhang2015}. This mode is
a linear combination of the active cavity modes $\omega_{2,\pm2}$ that does
not couple to the sphere. Although the spherical symmetry of the empty cavity
has been broken by the load, the axial symmetry remains intact and is
responsible for this effect. \begin{figure}[ptb]
\includegraphics[width=8.5cm]{fig5}\caption{(Color online) Scattering
efficiency as function of frequency $\omega/2\pi$ and normalized bias field
$\delta H/M_{s}$ for two YIG spheres ($\epsilon_{\mathrm{sp}}/\epsilon_{0}%
=15$) with radius $a_{1}=0.5$ mm and $a_{2}=1$ mm (a) without cavity and (b)
in a spherical cavity of radius $R=4$ mm. The white arrow in (b) illustrates
the \textquotedblleft indirect gap\textquotedblright\ induced by the radiative
coupling.}%
\label{fig5}%
\end{figure}
Next we fix $H_{0}/M_{s}=1$ and study the effect of small detunings $\delta
H/M_{s}$ in the dispersive regime$.$ In Figs. \ref{fig4}(a) and (b) the Kittel
mode lies above the $p$-wave cavity eigenmode $\omega_{1}/2\pi\sim6\,$GHz.
Note that the scattering efficiencies in the dispersive regime are much
smaller than those in Fig. \ref{fig3}. Panel (a) shows results for two YIG
spheres of radii $a=0.5$ mm without cavity, while for panel (b) the spherical
cavity has been added. The anticrossing in Fig. \ref{fig4}(b) illustrates that
the magnons of the two magnets interact over long distances through the
virtual exchange of cavity microwave photons. The coupling strength is given
by the frequency splitting of the modes at $\delta H=0$, giving a value of
$g_{\mathrm{eff}}^{\mathrm{ind.mag}}/2\pi\sim43$ MHz. This coupling requires
an external resonator, cf. Fig. \ref{fig4}(a), and can therefore not be
explained by the direct magnetic dipolar interactions or multiple scattering
between the spheres, as observed \cite{Lambert2016}.
We observe that the upper mode has a relatively large oscillator strength
(\textquotedblleft bright mode \textquotedblright), while the lower mode
intensity is suppressed at $\delta H=0$ (\textquotedblleft dark
mode\textquotedblright). The order and symmetry of these modes depends on the
sign of the magnon-cavity mode detuning as well as the phase relation between
the amplitude of the cavity mode on the spheres. In principle, many modes
contribute, but the ones closest in frequency dominate. The higher frequency
mode in \ref{fig4}(a) is the \textquotedblleft acoustic\textquotedblright%
\ (symmetric) mode that strongly interacts with the low frequency mode
$\omega_{1}$, which has the largest oscillator strength for forward
scattering. The lower \textquotedblleft optical\textquotedblright%
\ (antisymmetric) mode for $\delta H=0$ interacts with (and is pushed to lower
frequencies) by mode $\omega_{2}$. The scattering power of the $\omega_{2}$
mode (without load) is much weaker than that of $\omega_{1},$ which renders
the lower collective magnetic mode to be \textquotedblleft
dark\textquotedblright. We note that the \textquotedblleft
darkness\textquotedblright\ is not absolute, since the remaining intensity
does not vanish for $\delta H=0$ and depends on the details of the system and
scattering configuration.
Lambert et al. \cite{Lambert2016} find that a cavity mode $\omega_{2}/2\pi
\sim7.15$ GHz couples with the Kittel mode of a YIG sphere with $a=0.5$%
\thinspace mm by $g_{2}/2\pi\left( \equiv g_{\mathrm{eff}}^{\mathrm{mag}%
}/2\pi\right) \approx150$ MHz, in excellent agreement with our calculations.
By a dispersive measurement technique they also observe a splitting which they
interpret in terms of in-phase and out-of-phase precessions of the individual
magnetization dynamics. The observed splitting of these two modes agrees well
with the calculated ones, i.e. $2J/2\pi=87~$MHz as compared to our
$2g_{\mathrm{eff}}^{\mathrm{ind.mag}}/2\pi\sim86~$MHz. The order of
\textquotedblleft bright\textquotedblright\ and \textquotedblleft
dark\textquotedblright\ modes is opposite to what we find in Fig. \ref{fig4}.
This discrepancy is caused by the relative low frequency $\omega_{1}/2\pi
\sim3.55\,$GHz in the experiments, which is not reproduced by our spherical
cavity in which $\omega_{1}/2\pi\sim6$ GHz.
For two identical spheres the scattering properties $\mathcal{A}$, such as
$Q_{\mathrm{sca}}$, are parity (mirror) symmetric in parameter space, i.e.,
$\mathcal{A}(\omega,\delta H)=\mathcal{A}(\omega,-\delta H)$. The mode
coupling at $\delta H=0$ therefore must generate a direct gap and parabolic
dependence on small $\delta H,$ as indicated in Fig. \ref{fig4}(b). Different
radii break the symmetry and $\mathcal{A}(\omega,\delta H)\neq\mathcal{A}%
(\omega,-\delta H)$. Fig. \ref{fig5}(b) illustrates the strong magnon-magnon
coupling of two different YIG spheres with $a_{1}=1$ mm and $a_{2}=0.5$ mm
$\left( \epsilon_{\mathrm{sp}}/\epsilon_{0}=15\right) $ in a cavity of
radius $R=4$ mm. The asymmetry generates now an \textquotedblleft
indirect\textquotedblright\ gap.
\begin{figure}[ptb]
\includegraphics[width=8.5cm]{fig6}\caption{(Color online) Energy level
diagram describing the magnon hybridization in analogy with chemical bonds
resulting from the interaction between two spheres via microwave cavity modes.
(a) Magnonic homodimer consists of two similar magnetic spheres subjected to
local magnetic fields $H_{1(2)}=H_{0}\pm\delta H$, and (b) Magnonic
heterodimer consists of two dissimilar magnetic spheres. In (a) magnon
hybridization only occurs between magnonic states of the same angular momentum
denoted by $n_{i}$, while the reduced symmetry in a heterodimer introduces
coupling between all modes. In a homodimer, the bonding level is dark since it
has no dipole moment, while the antibonding level is bright. In a heterodimer
all modes are visible. The arrows in circles indicate the relative magnonic
phase (not spin or equilibrium magnetization).}%
\label{fig6}%
\end{figure}
The radiative coupling transforms the individual magnon (Kittel) modes of the
two-particle system into linear combinations, analogous to the molecular
orbital theory of diatomic molecules, according to which the interaction of
two atoms splits the levels into \textit{bonding }(symmetric) and
\textit{antibonding }(antisymmetric) orbitals. The magnetic spheres can be
interpreted as \textit{magnonic atoms} that are bound into \textit{magnonic
molecules}. Particle arrays will form \textit{magnonic crystals}, although
this term is also used for magnetic structures with periodic variations of
their magnetic properties~\cite{Krawczyk, Chumak} or distributions of
dipolar-coupled constituent materials~\cite{Vasseur}. The magnonic dimer has
bonding and antibonding combinations, where the hybridization depends on the
difference in their energies $\omega_{i}(H_{i})$ and on their interaction. A
homodimer A$_{2}$ corresponds to Fig. \ref{fig6}(a), while the mismatched
spheres in Fig. \ref{fig6}(b) form a heterodimer AB.
In a homodimer with inversion symmetry in which the splitting between internal
modes is large, bonding is dominated by magnons with the same angular momentum
$n$. We may use chemical intuition, however, to maximize the coupling by
varying both the local field and the sphere radius. This may reduces the
splitting between the internal $n=1$ and $n=2$ modes (cf. Fig. \ref{fig7}) and
facilitate an increased bonding via sp-hybrid states.
Bonding and antibonding modes belong to different irreducible representations.
In a heterodimer the lack of a mirror plane reduces the spatial symmetry and
introduces couplings between all modes. Furthermore, energies of the different
shells shift with respect to each other. Fig. \ref{fig6} illustrates that the
lowest-energy (dipolar) magnon of the smaller particle can couple efficiently
to both the dipolar and higher multipolar magnons of the larger particle. The
heterodimer thereby displays a significantly more complex magnon mixing
behavior than the homodimer.
The bonding configuration corresponds to two dipole moments moving out of
phase (optical mode, negative parity of dipole moments, or antisymmetric
magnetic fields), while the antibonding configuration corresponds to the
positive parity of the dipoles (acoustic mode, symmetric fields). In contrast
to the positive parity (symmetric) magnons, the net magnetic moment of the
negative parity (antisymmetric field) magnon vanishes for identical spheres,
and does not interact with the $p$-wave cavity mode in the present
configuration. The former are then \textit{bright}, and the latter the
\textit{dark }states, as shown in Fig. (\ref{fig4}). In the heterodimer, all
magnons mix and contribute to the bonding and antibonding modes. As a
consequence, all modes become bright, see Fig. (\ref{fig5}).
We can parameterize the observations by elementary molecular orbital theory.
The energy gap, $E_{\mathrm{gap}}$, between the bonding and antibonding energy
levels for a diatomic molecule is given by the secular equation
\begin{equation}
E_{\mathrm{gap}}^{2}=(2g)^{2}+\left( E_{A}-E_{B}\right) ^{2}%
\end{equation}
where $g$ is the coupling parameter between the two sites, while $E_{A}$ and
$E_{B}$ refer to their energies. There are two contributions to the energy
gap, the covalent(homopolar) bonding contribution $E_{h}=2g$, and the ionic
contribution, $E_{i}=E_{A}-E_{B}$, due to the difference in \textquotedblleft
electronegativity\textquotedblright\ between the two atoms. For any bond, we
can then define the bond covalency, $\alpha_{c}=E_{h}/E_{\mathrm{gap}}$, and
polarity, $\alpha_{p}=E_{i}/E_{\mathrm{gap}}$, which parametrizes the
continuous transition from covalent to ionic bonding.
\begin{figure}[ptb]
\includegraphics[width=8.5cm]{fig7}\caption{Same as Figs. \ref{fig3} and
\ref{fig4} but for relatively large YIG spheres of radius $a_{1}=a_{2}=1.25$
mm. The cavity modes are strongly mixed with those confined in the two YIG
spheres. In (a) the modes are shifted relative to each other by $\delta
H/M_{s}\sim0.7$ and the uniform field in panels (b) and (c) is fixed at
$H_{0}/M_{s}\sim6$.}%
\label{fig7}%
\end{figure}
In a homodimer at $\delta H=0$ we have a direct gap due to covalent bonding
$E_{\mathrm{gap}}=2g$, see Fig. \ref{fig4}(b) and bonding and anti-bonding
wave functions are equally shared between the two atoms. However, in a
heterodimer, due to the detuning of the atomic levels $\omega_{1}(H_{0}%
)\neq\omega_{2}(H_{0})$, the gap has an \textquotedblleft
ionic\textquotedblright\ contribution, leading to an indirect gap as a
function of $\delta H$ in Fig. \ref{fig5}(b). In a polar molecule, the
amplitude of the bonding state shifts towards the more
\textit{magnon-negative} site referred to as the \textit{magnonic anion}, with
the anti-bonding state shifting towards the less magnon-negative site,
referred to as the \textit{magnonic cation}, a partially polarized molecule.
The covalent bonding strength can be independently modulated by the average
frequency spacing with the dominant cavity mode.
The scattering efficiency factor $Q_{\mathrm{sca}}$ is plotted as a function
of frequency $\omega/2\pi$, uniform field $H_{0}/M_{s}$, and differential
field $\delta H/M_{s}$ for two YIG spheres of radius $a_{1}=a_{2}=1.25$ mm and
relative permittivity $\epsilon_{\mathrm{sp}}/\epsilon_{0}=15,$ placed in a
spherical cavity of radius $R=4$ mm in Fig. \ref{fig7}(a) and (c), and without
cavity in \ref{fig7}(b). Without cavity the system can be interpreted as two
independent antennas operating in the ultrastrong coupling regime, since due
to their relatively large size individual spheres act as efficient microwave
antennas. Many anticrossings in Fig. \ref{fig7}(a) emphasize that the cavity
modes are strongly and even ultrastrongly mixed with the modes in each
individual spheres when detuned by a differential field $\delta H/M_{s}%
\sim0.7$. The large differences between Figs. \ref{fig7}(b) and (c) provide
more evidence for the strong cavity-mode induced coupling between the spheres.
In Fig. (\ref{fig7})(b), beside the main crossing modes in absence of the
cavity, we observe tails from other crossings modes at higher frequencies,
which are standing electromagnetic resonance modes confined by the magnetic
spheres. Strong coupling with cavity mode not only turns the main crossing
modes into anticrossing but also causes the complex anticrossing pattern shown
in Fig. (\ref{fig7})(c) by hybridizing all higher modes.
\section{Conclusion}
\label{sec:concl}
In conclusion, we studied the plasmonics and optomagnonics of two dielectric
and two magnetic spheres in microwave cavities by Mie scattering theory, i.e.
a systematic expansion of the coupled Maxwell and LLG equations for magnetic
systems. We employ the linear and magnetostatic approximations, but otherwise
the treatment is numerically exact. The magnetization dynamics of spatially
separated spheres in cavities can be efficiently coupled over large distances.
The main reason is not the magnetic but the electric-field coupling, since two
dielectric spheres with zero magnetization in a cavity display very similar
dynamic behavior. Both strong and ultrastrong coupling can be realized not
only for individual spheres but also for their mutual interaction. Two
(properly placed) identical spheres form an inversion symmetric system, which
is apparent by an anticrossing that generates a \textquotedblleft
direct\textquotedblright\ gap when plotted as a function of a symmetry
breaking parameter, such as a staggered magnetic field or a size difference.
Spheres with different sizes, however, break the symmetry at constant magnetic
field and lead to an \textquotedblleft indirect gap\textquotedblright\ as a
function of field detuning. Magnon-polaritons within individual magnetic
spheres may also hybridize in cavities, forming a complex mixed state of light
and spin. Our study suggests a new direction for \textquotedblleft spin
cavitronics\textquotedblright, viz. a route towards coherent control of the
dynamics of various systems and materials (magnets, pieozoelectrics,
superconductors, charge density waves, etc.) in microwave cavities via the
non-magnetic (plasmonic) interactions.
\acknowledgments B. Z. R. thanks S. M. Reza Taheri, A. Eskandari-asl, M. F.
Miri and Y. M. Blanter for fruitful discussions. This work was partially
supported by Iran Science Elites Federation (B. Z. R). Our research was
supported by the Dutch NWO and JSPS Grants-in-Aid for Scientific Research
(Grant Nos. 25247056, 25220910, 26103006).
|
2,877,628,090,086 | arxiv | \section{Introduction and Preliminaries}\label{s1}
Let $\left(\mathcal{H}, \left\langle \cdot\mid \cdot\right\rangle \right)$ be a non-trivial complex Hilbert space, and let $\mathcal{B}(\mathcal{H})$
denote the $C^*$-algebra of all bounded linear operators on $\mathcal{H}$ with identity $I_{\mathcal{H}}$ (or $I$ if no confusion arises). If $\mathcal{H}=\mathbb{C}^d$, we identify $\mathcal{B}(\mathbb{C}^d)$ with the matrix algebra $\mathbb{M}_d(\mathbb{C})$ of $d\times d$ complex matrices. Let $\mathcal{B}(\mathcal{H})^+$ be the cone of positive (semi-definite) operators, i.e., $\mathcal{B}(\mathcal{H})^+=\left\{A\in \mathcal{B}(\mathcal{H})\,:\,\langle Ax\mid x\rangle\geq 0,\;\forall\;x\in \mathcal{H}\;\right\}$. Every $A\in \mathcal{B}(\mathcal{H})^+$ defines the following positive semi-definite sesquilinear form:
$$\langle\cdot\mid\cdot\rangle_{A}:\mathcal{H}\times \mathcal{H}\longrightarrow\mathbb{C},\;(x,y)\longmapsto\langle x\mid y\rangle_{A} =\langle Ax\mid y\rangle.$$
Clearly, the induced semi-norm is given by $\|x\|_A=\langle x\mid x\rangle_A^{1/2}$, for every $x\in \mathcal{H}$. This makes $\mathcal{H}$ into a semi-Hilbertian space. One can verify that $\|\cdot\|_A$ is a norm on $\mathcal{H}$ if and only if $A$ is injective, and that $(\mathcal{H},\|\cdot\|_A)$ is complete if and only if the range of $A$ is a closed subspace of $\mathcal{H}$.
Throughout this article, we shall assume that an operator $A\in\mathcal{B}(\mathcal{H})$ is a nonzero positive (semidefinite) operator. Moreover, by an operator we mean a bounded linear operator. In addition, the range and the null space of an operator $T$ are denoted by ${\mathcal R}(T)$ and ${\mathcal N}(T)$, respectively. Also, $T^*$ will be denoted to be the adjoint of $T$.
An operator $S\in\mathcal{B}(\mathcal{H})$ is called an $A$-adjoint of $T$ if for every $x,y\in \mathcal{H}$, the identity $\langle Tx\mid y\rangle_A=\langle x\mid Sy\rangle_A$ holds. The existence of an $A$-adjoint operator is not guaranteed. Observe that $T$ admits an $A$-adjoint operator if and only if the equation $AX = T^*A$ has solution. This kind of equations can be studied by using the next theorem due to Douglas (for its proof see \cite{doug} or \cite{mos2019}).
\begin{theorem}\label{doug}(\cite[Theorem 1]{doug})
If $T, S \in \mathcal{B}(\mathcal{H})$, then the following statements are equivalent:
\begin{enumerate}
\item[{\rm (i)}] $\mathcal{R}(S) \subseteq \mathcal{R}(T)$;
\item[{\rm (ii)}] $TD=S$ for some $D\in \mathcal{B}(\mathcal{H})$;
\item[{\rm (iii)}]$SS^* \leq \lambda^2 TT^*$ for some $\lambda\geq 0$ (or equivalently $\|S^*x\| \leq \lambda\|T^*x\|$ for all $x\in \mathcal{H}$).
\end{enumerate}
If one of these conditions holds, then there exists an unique operator $Q\in\mathcal{B}(\mathcal{H})$ such that $TX=S$ and $\mathcal{R}(Q) \subseteq \overline{\mathcal{R}(T^{*})}$. Furthermore, $\mathcal{N}(Q)=\mathcal{N}(S)$ and
$$\|Q\|^2=\inf\left\{\mu\,;\;SS^*\leq \mu TT^*\right\}.$$
Such $Q$ is called the reduced solution or Douglas solution of $TX=S$.
\end{theorem}
Therefore, if we denote by $\mathcal{B}_{A}(\mathcal{H})$ the subalgebra of $\mathcal{B}(\mathcal{H})$ of all operators which admit an $A$-adjoint
operator, then by Theorem \ref{doug} we see that
$$\mathcal{B}_{A}(\mathcal{H})=\left\{T\in \mathcal{B}(\mathcal{H})\,;\;\mathcal{R}(T^{*}A)\subset \mathcal{R}(A)\right\}.$$
Let $T\in\mathcal{B}_{A}(\mathcal{H})$. The Douglas solution of the equation $AX = T^*A$ is a distinguished $A$-adjoint operator of $T$, which is denoted by $T^{\sharp_A}$. Note that, $T^{\sharp_A} = A^{\dag}T^*A$ in which $A^{\dag}$ is denoted to be the Moore-Penrose inverse of $A$ (see \cite{acg2}). It is important to mention that if $T\in\mathcal{B}_{A}(\mathcal{H})$, then $T^{\sharp_A}\in\mathcal{B}_{A}(\mathcal{H})$, $\|T^{\sharp_A}\|_A=\|T\|_A$ and $(T^{\sharp_A})^{\sharp_A} = P_{\overline{\mathcal{R}(A)}}TP_{\overline{\mathcal{R}(A)}}$. Here, $P_{\overline{\mathcal{R}(A)}}$ denotes the orthogonal projection onto $\overline{\mathcal{R}(A)}$. Furthermore, if $T, S\in\mathcal{B}_{A}(\mathcal{H})$, then $(TS)^{\sharp_A} = S^{\sharp_A}T^{\sharp_A}$. In addition, an operator $U\in \mathcal{B}_A(\mathcal{H})$ is said to be $A$-unitary if $\|U^{\sharp_A}x\|_A= \|Ux\|_A=\|x\|_A$ for all $x\in \mathcal{H}$. For more details, the reader is invited to consult \cite{acg1,acg2,bakfeki01,bakfeki04} and their references.
Furthermore, again by applying Douglas theorem we obtain
\begin{equation}\label{abbbbbbbb}
\mathcal{B}_{A^{1/2}}(\mathcal{H})=\left\{T \in \mathcal{B}(\mathcal{H})\,;\;\exists \,\lambda > 0\,;\;\|Tx\|_{A} \leq \lambda \|x\|_{A},\;\forall\,x\in \mathcal{H} \right\}.
\end{equation}
Operators in $\mathcal{B}_{A^{1/2}}(\mathcal{H})$ are called $A$-bounded. It should be mention here that $\mathcal{B}_{A}(\mathcal{H})$ and $\mathcal{B}_{A^{1/2}}(\mathcal{H}))$ are two subalgebras of $\mathcal{B}(\mathcal{H})$ which are neither closed nor dense in $\mathcal{B}(\mathcal{H})$. Moreover, we have $\mathcal{B}_{A}(\mathcal{H}) \subseteq \mathcal{B}_{A^{1/2}}(\mathcal{H}))$ (see \cite[Proposition 1.2.]{acg3}).
The semi-inner product $\langle\cdot\mid\cdot\rangle_{A}$ induces the following seminorm on $\mathcal{B}_{A^{1/2}}(\mathcal{H})$:
\begin{equation}\label{semiiineq}
\|T\|_A:=\sup_{\substack{x\in \overline{\mathcal{R}(A)},\\ x\not=0}}\frac{\|Tx\|_A}{\|x\|_A}=\sup\left\{\|Tx\|_{A}\,;\;x\in \mathcal{H},\,\|x\|_{A}= 1\right\}<\infty.
\end{equation}
If $A=I$, we get the classical definition of the operator norm of an operator $T$ which will be denoted by $\|T\|$. It was shown in \cite{fg} that for every $T\in\mathcal{B}_{A^{1/2}}(\mathcal{H})$ we have
\begin{equation}\label{fg}
\|T\|_A=\sup\left\{|\langle Tx\mid y\rangle_A|\,;\;x,y\in \mathcal{H},\,\|x\|_{A}=\|y\|_{A}= 1\right\}.
\end{equation}
In addition, for every $T\in\mathcal{B}_{A}(\mathcal{H})$ we have
\begin{equation}\label{diez}
\|T\|_A^2={\|T^{\sharp_A}T\|}_A = {\|TT^{\sharp_A}\|}_A.
\end{equation}
The $A$-the numerical radius and the $A$-spectral radius of an $A$-bounded operator $T\in\mathcal{B}_{A^{1/2}}(\mathcal{H})$ are defined by
\begin{align*}
\omega_A(T) = \sup\Big\{\big|{\langle Tx\mid x\rangle}_A\big|: \,\,x\in \mathcal{H}, \,{\|x\|}_A = 1\Big\}\;\text{ and }
\end{align*}
\begin{equation}\label{newrad}
r_A(T):=\displaystyle\inf_{n\in \mathbb{N}^*}\|T^n\|_A^{\frac{1}{n}}=\displaystyle\lim_{n\to\infty}\|T^n\|_A^{\frac{1}{n}},
\end{equation}
respectively. Notice that the second equality in \eqref{newrad} is proved in \cite{feki01}. If $A=I$, the spectral and numerical radius of $T$ will be simply denoted by $r(T)$ and $\omega(T)$ respectively.
It is well known that $\omega_A(\cdot)$ defines a seminorm on $\mathcal{B}_{A^{1/2}}(\mathcal{H})$, which is equivalent to the $A$-operator seminorm $ \left\| \cdot \right\|_A $, more precisely,
\begin{equation*}
\tfrac{1}{2} \left\|T\right\|_A \leq \omega_A(T)\leq \left\|T\right\|_A,
\end{equation*}
for every $T\in\mathcal{B}_{A^{1/2}}(\mathcal{H})$. Moreover, it was shown in \cite{feki01} that for $T\in\mathcal{B}_{A^{1/2}}(\mathcal{H})$, it holds
\begin{equation*}
\omega_A(T)\leq \frac{1}{2}\left(\|T\|_A+\|T^2\|_A^{1/2}\right).
\end{equation*}
So, clearly, if $T\in\mathcal{B}_{A^{1/2}}(\mathcal{H})$ and satisfies $AT^2=0$, then
\begin{equation}\label{at2}
\omega_A(T)= \frac{1}{2}\|T\|_A.
\end{equation}
It should be emphasized here that for every $T\in\mathcal{B}_{A^{1/2}}(\mathcal{H})$ we have
\begin{equation}\label{dom}
r_A(T) \leq \omega_A(T).
\end{equation}
Also $r_A(\cdot)$ satisfies the commutativity property, which asserts that
\begin{equation}\label{commut}
r_A(TS)=r_A(ST),
\end{equation}
for every $T,S\in \mathcal{B}_{A^{1/2}}(\mathcal{H})$.
An operator $T\in\mathcal{B}(\mathcal{H})$ is said to be $A$-selfadjoint if $AT$ is selfadjoint, that is, $AT = T^*A$. Moreover, it was shown in \cite{feki01} that if $T$ is $A$-self-adjoint, then
\begin{equation}\label{aself1}
\|T\|_{A}=\omega_A(T)=r_A(T).
\end{equation}
For the sequel, for any arbitrary operator $T\in {\mathcal B}_A({\mathcal H})$, we write
$$\Re_A(T):=\frac{T+T^{\sharp_A}}{2}\;\;\text{ and }\;\;\Im_A(T):=\frac{T-T^{\sharp_A}}{2i}.$$
It has recently been shown in \cite[Theorem 2.5]{zamani1} that if $T\in\mathcal{B}_{A}(\mathcal{H})$, then
\begin{align}\label{zamnum}
\omega_A(T) = \displaystyle{\sup_{\theta \in \mathbb{R}}}{\left\|\Re_A(e^{i\theta}T)\right\|}_A.
\end{align}
Recently, many results covering some classes of operators on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle\big)$
are extended to $\big(\mathcal{H}, {\langle \cdot\mid \cdot\rangle}_A\big)$ (see, e.g., \cite{fekisidha2019,feki01,zamani2,bakfeki01,bakfeki04,zamani1,majsecesuci}).
In in this work, we consider the following diagonal operator matrix whose each diagonal entry is $A$:
\begin{equation*}
\mathbb{A}=\left( {\begin{array}{*{20}{c}}
{A} & {} & {0} & {} \\
{} & {A} & & {} \\
{} & & \ddots & {} \\
{} & {0} & {} & {A} \\
\end{array}} \right),
\end{equation*}
acting on the Hilbert space $\mathbb{H}=\oplus_{i=1}^d\mathcal{H}$ equipped with the following inner-product:
$$\langle x, y\rangle=\sum_{k=1}^d\langle x_k\mid y_k\rangle,$$
for all $x=(x_1,\cdots,x_d)\in \mathbb{H}$ and $y=(y_1,\cdots,y_d)\in \mathbb{H}$. The semi-inner product induced by the positive operator $\mathbb{A}$ is given by
$$\langle x, y\rangle_{\mathbb{A}}= \langle \mathbb{A}x, y\rangle=\sum_{k=1}^d\langle Ax_k\mid y_k\rangle=\sum_{k=1}^d\langle x_k\mid y_k\rangle_A,$$
for all $x=(x_1,\cdots,x_d)\in \mathbb{H}$ and $y=(y_1,\cdots,y_d)\in \mathbb{H}$. The purpose of this paper is to establish several inequalities for $\omega_\mathbb{A}(\mathbb{T})$, where $\mathbb{T}=(T_{ij})$ is a $d\times d$ operator matrix with $T_{ij}$ are $A$-bounded operators. The inspiration for our investigation comes from \cite{OK2,S.D.M,BP,bpnayek}.
\section{Results}
In this section, we present our results. To prove our first result, we need the following lemmas.
\begin{lemma}\label{mjom}(\cite{feki03})
Let $\mathbb{T}= (T_{ij})_{d \times d}$ be a $d \times d$ operator matrix be such that $T_{ij}\in \mathcal{B}_{A^{1/2}}(\mathcal{H})$ for all $i,j$. Then, $$r_{\mathbb{A}}(\mathbb{T})\leq r(\|T_{ij} \|_A).$$
\end{lemma}
\begin{lemma}\label{ir2020}(\cite{bhunfekipaul})
Let $\mathbb{T}= (T_{ij})_{d \times d}$ be such that $T_{ij}\in \mathcal{B}_{A}(\mathcal{H})$ for all $i,j$. Then, $\mathbb{T}\in\mathcal{B}_{\mathbb{A}}(\mathbb{H})$ and
$$\mathbb{T}^{\sharp_\mathbb{A}}=(T_{ji}^{\sharp_\mathbb{A}})_{d \times d}.$$
\end{lemma}
\begin{lemma}\label{weak}(\cite{bhunfekipaul})
Let $T\in \mathcal{B}_{A^{1/2}}(\mathcal{H})$. Then,
\begin{equation*}
\omega_A(U^{\sharp_A}TU)=\omega_A(T),
\end{equation*}
for any $A$-unitary operator $U\in\mathcal{B}_A(\mathcal{H})$.
\end{lemma}
Now, we are in a position to prove the following theorem.
\begin{theorem}\label{thf1}
Let $\mathbb{T}=(T_{ij})$ be a ${d\times d}$ operator matrix where $T_{ij}\in \mathcal{B}_{A}(\mathcal{H})$. Then,
\begin{equation*}
\omega_{\mathbb{A}}(\mathbb{T})\leq \frac{1}{2}\sum_{i=1}^d\left(\|T_{ii}\|_A+\sqrt{\left\|T_{ii}T_{ii}^{\sharp_A}+\sum^d_{j=1,j\neq i}T_{ij}T_{ij}^{\sharp_A}\right\|_A}\right).
\end{equation*}
\end{theorem}
\begin{proof}
We first prove that
\begin{equation}\label{first01}
\omega_{\mathbb{A}}(\mathbb{S})\leq \frac{1}{2}\left(\|T_{11}\|_A+\sqrt{\left\|\sum^d_{j=1}T_{1j}T_{1j}^{\sharp_A}\right\|_A}\right),
\end{equation}
where $\mathbb{S}=\begin{pmatrix}
T_{11} & T_{12} &\cdots& T_{1d}\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}$. Let $\theta\in \mathbb{R}$. It is not difficult to verify that $\Re_A(e^{i\theta}\mathbb{S})$ is an $\mathbb{A}$-self-adjoint operator. So, by \eqref{aself1} we have
\begin{align}\label{rr1}
r_{\mathbb{A}}\left(\Re_\mathbb{A}(e^{i\theta}\mathbb{S})\right)=\|\Re_\mathbb{A}(e^{i\theta}\mathbb{S})\|_{\mathbb{A}}.
\end{align}
On the other hand, by using Lemma \ref{ir2020} we see that
\begin{align*}
r_{\mathbb{A}}\left[\Re_A(e^{i\theta}\mathbb{S})\right]
& =\tfrac{1}{2}r_{\mathbb{A}}(e^{i\theta}\mathbb{S}+e^{-i\theta}\mathbb{S}^{\sharp_{\mathbb{A}}})\\
&=\frac{1}{2}r_{\mathbb{A}}\left[e^{i\theta}\begin{pmatrix}
T_{11} & T_{12} &\cdots& T_{1d}\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}+e^{-i\theta}\begin{pmatrix}
T_{11}^{\sharp_{A}} & 0 &\cdots& 0\\
T_{12}^{\sharp_{A}} &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
T_{1d}^{\sharp_{A}} & 0 &\cdots& 0\\
\end{pmatrix}\right]\\
&=\frac{1}{2}r_{\mathbb{A}}\left[\begin{pmatrix}
e^{i\theta}T_{11}+e^{-i\theta}T_{11}^{\sharp_{A}} & e^{i\theta}T_{12} &\cdots& e^{i\theta}T_{1d}\\
e^{-i\theta}T_{12}^{\sharp_{A}} &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
e^{-i\theta}T_{1d}^{\sharp_{A}} & 0 &\cdots& 0\\
\end{pmatrix}\right]\\
&=\frac{1}{2}r_{\mathbb{A}}\left[
\begin{pmatrix}
T_{11}^{\sharp_{A}} & e^{i\theta}I &\cdots& 0\\
T_{12}^{\sharp_{A}} &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
T_{1d}^{\sharp_{A}} & 0 &\cdots& 0\\
\end{pmatrix}
\begin{pmatrix}
e^{-i\theta}I &0 &\cdots& 0\\
T_{11} & T_{12} &\cdots& T_{1d}\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}
\right].
\end{align*}
So, by using \eqref{rr1} together with \eqref{commut} we get
\begin{align*}
\|\Re_A(e^{i\theta}\mathbb{S})\|_{\mathbb{A}}
&=\frac{1}{2}r_{\mathbb{A}}\left[
\begin{pmatrix}
e^{-i\theta}I &0 &\cdots& 0\\
T_{11} & T_{12} &\cdots& T_{1d}\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}
\begin{pmatrix}
T_{11}^{\sharp_{A}} & e^{i\theta}I &\cdots& 0\\
T_{12}^{\sharp_{A}} &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
T_{1d}^{\sharp_{A}} & 0 &\cdots& 0\\
\end{pmatrix}
\right]\\
&=\tfrac{1}{2}r_{\mathbb{A}}\left[
\begin{pmatrix}
e^{-i\theta}T_{11}^{\sharp_{A}} &I &0&\cdots& 0\\
\sum_{k=1}^d T_{1k}T_{1k}^{\sharp_{A}}& e^{i\theta}T_{11} &0&\cdots& 0\\
0 & 0 &0&\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &0&\cdots& 0\\
\end{pmatrix}
\right]\\
&\leq\frac{1}{2}r\left[
\begin{pmatrix}
\|T_{11}\|_A &1 &0&\cdots& 0\\
\left\|\sum_{k=1}^d T_{1k}T_{1k}^{\sharp_{A}}\right\|_A& \|T_{11}\|_A &0&\cdots& 0\\
0 & 0 &0&\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &0&\cdots& 0\\
\end{pmatrix}
\right]\; (\text{by Lemma }\ref{mjom}).
\end{align*}
Hence, we infer that
$$\|\Re_A(e^{i\theta}\mathbb{S})\|_{\mathbb{A}}\leq\frac{1}{2}\left(\|T_{11}\|_A+\sqrt{\left\|\sum^d_{j=1}T_{1j}T_{1j}^{\sharp_A}\right\|_A}\right).$$
So, by taking the supremum over all $\theta\in \mathbb{R}$ in the above inequality and then using \eqref{zamnum} we get \eqref{first01} as desired. Now, for $k\in\{2,\cdots,d\}$, we let
\[
\mathbb{U}_k = \left(\begin{array}{c|c}
\mathbb{J}_{k\times k} & 0_{k\times (d-k)} \\
\hline
0_{(d-k)\times k} & \mathbb{I}_{(d-k)\times (d-k)}
\end{array}\right),
\]
where $\mathbb{J}_{k\times k}$ and $\mathbb{I}_{(d-k)\times (n-k)}$ are $k\times k$ and $(d-k)\times (d-k)$ operator matrices respectively and are defined by
$$
\mathbb{J}_{k\times k}=\begin{pmatrix}
0 &\cdots &0 &I\\
\vdots & \iddots &I &0\\
0 &I\smash{\makebox[0pt][l]{\;\raisebox{0.8em}{$\iddots$}}} & \iddots &\vdots\\
I &0 &\cdots &0
\end{pmatrix}\;\text{ and }\;\mathbb{I}_{(d-k)\times (d-k)}=\begin{pmatrix}
I &0 &\cdots &0\\
0 & I \smash{\makebox[0pt][l]{\;\raisebox{-0.8em}{$\ddots$}}} &\ddots &\vdots\\
\vdots &\ddots &I &0\\
0 &\cdots &0 &I
\end{pmatrix}
$$
In view of Lemma \ref{ir2020}, we have $\mathbb{U}_k\in \mathcal{B}_{\mathbb{A}}(\mathcal{H}\oplus \mathcal{H})$ for all $k$. Moreover, a short calculation shows that $\mathbb{U}_k^{\sharp_{\mathbb{A}}}=\mathbb{P}\mathbb{U}_k$ where $\mathbb{P}=\begin{pmatrix}
P_{\overline{\mathcal{R}(A)}} &0 &\ldots &0\\
0 & P_{\overline{\mathcal{R}(A)}} &\ddots &\vdots\\
\vdots &\ddots &P_{\overline{\mathcal{R}(A)}} &0\\
0 &\ldots &0 &P_{\overline{\mathcal{R}(A)}}
\end{pmatrix}$. So, it is not difficult to verify that $\mathbb{U}_k$ is $\mathbb{A}$-unitary operator for all $k$. Moreover, one can check that
\begin{align*}
\omega_{\mathbb{A}}(\mathbb{T})
& \leq\omega_{\mathbb{A}}\left[\begin{pmatrix}
T_{11} & T_{12} &\cdots& T_{1d}\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\right]+\omega_{\mathbb{A}}\left[\begin{pmatrix}
0 &0 &\cdots& 0\\
T_{21} & T_{22} &\cdots& T_{2d}\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\right] \\
&\;\;\;+\ldots\ldots+\omega_{\mathbb{A}}\left[\begin{pmatrix}
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
T_{d1} & T_{d2} &\cdots& T_{dd}\\
\end{pmatrix}\right] \\
&=\omega_{\mathbb{A}}\left[\begin{pmatrix}
T_{11} & T_{12} &\cdots& T_{1d}\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\right]+\omega_{\mathbb{A}}\left[\mathbb{U}_2^{\sharp_{\mathbb{A}}}\begin{pmatrix}
T_{22} & T_{21} &\cdots& T_{2d}\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\mathbb{U}\right] \\
&\;\;\;+\ldots\ldots+\omega_{\mathbb{A}}\left[\mathbb{U}_d^{\sharp_{\mathbb{A}}}\begin{pmatrix}
T_{dd} & T_{dd-1} &\cdots& T_{d1}\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\mathbb{U}\right] \\
\end{align*}
So, by using Lemma \ref{weak} together with \eqref{first01}, we obtain
\begin{align*}
\omega_{\mathbb{A}}(\mathbb{T})
& \leq\omega_{\mathbb{A}}\left[\begin{pmatrix}
T_{11} & T_{12} &\cdots& T_{1d}\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\right]+\omega_{\mathbb{A}}\left[\begin{pmatrix}
T_{22} & T_{21} &\cdots& T_{2d}\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\right] \\
&\;\;\;+\ldots\ldots+\omega_{\mathbb{A}}\left[\begin{pmatrix}
T_{dd} & T_{dd-1} &\cdots& T_{d1}\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\right] \\
&\leq\frac{1}{2}\left(\|T_{11}\|_A+\sqrt{\left\|\sum^d_{j=1}T_{1j}T_{1j}^{\sharp_A}\right\|_A}\right)+\frac{1}{2}\left(\|T_{22}\|_A+\sqrt{\left\|\sum^d_{j=1,j\neq 2}T_{2j}T_{2j}^{\sharp_A}\right\|_A}\right)\\
&\;\;\;+\ldots\ldots+\frac{1}{2}\left(\|T_{dd}\|_A+\sqrt{\left\|\sum^{d-1}_{j=1}T_{dj}T_{dj}^{\sharp_A}\right\|_A}\right).
\end{align*}
This finishes the proof of the theorem.
\end{proof}
To establish our next result, we shall require the following lemma.
\begin{lemma}\label{maxma}
Let $\mathbb{T}=\begin{pmatrix}
T_1 & 0 & 0\\
0 & \ddots & 0 \\
0 & 0 & T_d
\end{pmatrix}$ and $\mathbb{S}=\left( {\begin{array}{*{20}{c}}
{0} & {} & {} & {{T_1}} \\
& {} & {{T_2}} & {} \\
{} & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu
\raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & {} & {} \\
{{T_d}} & {} & & {0} \\
\end{array}} \right)$be such that $T_i\in \mathcal{B}_{A^{1/2}}(\mathcal{H})$ for all $i\in\{1,\cdots,d\}$. Then, the following assertions hold
\begin{itemize}
\item [(a)] $\|\mathbb{T}\|_\mathbb{A}=\max_{i\in\{1,\cdots,d\}}\|T_i\|_A$.
\item [(b)] $\omega_{\mathbb{A}}(\mathbb{T})=\max_{i\in\{1,\cdots,d\}}\omega_{A}(T_i)$.
\item [(c)] $\|\mathbb{S}\|_\mathbb{A}=\max_{i\in\{1,\cdots,d\}}\|T_i\|_A$.
\end{itemize}
\end{lemma}
\begin{proof}
\noindent (a)\;Let $x=(x_1,\cdots,x_d),y=(y_1,\cdots,y_d)\in \mathbb{H}$. By using the Cauchy-Schwarz inequality and the arithmetic-geometric mean inequality, we obtain
\begin{align}\label{k0}
|\langle \mathbb{T}x,y \rangle_\mathbb{A}|\nonumber
&\leq \sum_{k=1}^d|\langle T_kx_k\mid y_k \rangle_A|\nonumber \\
&\leq \sum_{k=1}^d\|T_k\|_A\|x_k\|_A\|y_k\|_A \nonumber\\
&\leq\left(\max_{i\in\{1,\cdots,d\}}\|T_i\|_A\right)\times\tfrac{1}{2}\sum_{k=1}^d\left(\|x_k\|_A^2+\|y_k\|_A^2\right)\nonumber\\
&=\frac{\|x\|_\mathbb{A}^2+\|y\|_\mathbb{A}^2}{2}\left(\max_{i\in\{1,\cdots,d\}}\|T_i\|_A\right).
\end{align}
By taking the supremum over all $x\in \mathbb{H}$ with $\|x\|_\mathbb{A} =1$ in the inequality \eqref{k0} and then using \eqref{fg}, we get
$$\|\mathbb{T}\|_\mathbb{A}\leq\max_{i\in\{1,\cdots,d\}}\|T_i\|_A.$$
Let $u=(x,0,\cdots,0)\in \mathbb{H}$ and $v=(y,0,\cdots,0)\in \mathbb{H}$ be such that $\|x\|_A=\|y\|_A=1.$ Then $\|u\|_\mathbb{A}=\|v\|_\mathbb{A}=\|x\|_A=\|y\|_A=1.$ Therefore, in view of \eqref{fg}, we have
\begin{align*}
\|\mathbb{T}\|_\mathbb{A}
&\geq |\langle \mathbb{T}u,v\rangle_\mathbb{A}|= |\langle T_1x\mid y\rangle_A|.
\end{align*}
This implies that $\|\mathbb{T}\|_\mathbb{A}\geq \|T_1\|_A$. Similarly, we can show that $\|\mathbb{T}\|_\mathbb{A}\geq \|T_k\|_A$ for all $k\in\{2,3,\ldots,d\}$. This proves the desired equality.
\par \vskip 0.1 cm \noindent (b)\;Follows by using similar arguments as in $(a)$.
\par \vskip 0.1 cm \noindent (c)\;$x=(x_1,\cdots,x_d)\in \mathbb{H}$. By using \eqref{semiiineq}, it can be observed that
\begin{align*}
\left\|\mathbb{S}x\right\|_{\mathbb{A}}^2
&=\|T_1x_d\|_A^2+\|T_2x_{d-1}\|_A^2+\cdots+\|T_dx_1\|_A^2\\
&\leq \left(\max_{i\in\{1,\cdots,d\}}\|T_i\|_A^2\right) \sum_{k=1}^d\|x_k\|_A^2.
\end{align*}
This yields that
$$\left\|\mathbb{S}\right\|_{\mathbb{A}}\leq \max_{i\in\{1,\cdots,d\}}\|T_i\|_A.$$
Let $x_1\in \mathcal{H}$ be such that $\|x\|_A =1$ and $u=(0,0,\cdots,x_1)$. Clearly, $\|u\|_\mathbb{A} =1.$ So, we obtain
$$\left\|\mathbb{S}\right\|_{\mathbb{A}}\geq\left\|\mathbb{S}u\right\|_{\mathbb{A}}=\|T_1x_1\|_A.$$
Thus, by taking the supremum over all $x_1\in \mathcal{H}$ with $\|x_1\|_A=1$, we obtain $\left\|\mathbb{S}\right\|_{\mathbb{A}}\geq\|T_1\|_A.$ Similarly, it is not difficult to prove that $\|\mathbb{S}\|_\mathbb{A}\geq \|T_i\|_A$ for all $i\in\{2,3,\ldots,d\}$. This proves the desired equality.
\end{proof}
Our next result is stated as follows.
\begin{theorem}
Let $\mathbb{T}=(T_{ij})$ be a ${d\times d}$ operator matrix where $T_{ij}\in \mathcal{B}_{A}(\mathcal{H})$. Then,
\begin{equation}\label{r2}
\omega_{\mathbb{A}}(\mathbb{T})\leq \frac{1}{2}\sum_{i=1}^d\omega_A(T_{ii})+\frac{1}{4}\left(d+\sum^d_{i,j=1}\|T_{ij}\|_A^2\right).
\end{equation}
\end{theorem}
\begin{proof}
We first prove that
\begin{equation}\label{first02}
\omega_{\mathbb{A}}(\mathbb{S})\leq \frac{\omega_A(T_{11})}{2}+\frac{1}{4}+\frac{1}{4}\left\|\sum^d_{j=1}T_{1j}T_{1j}^{\sharp_A}\right\|_A,
\end{equation}
where $\mathbb{S}=\begin{pmatrix}
T_{11} & T_{12} &\cdots& T_{1d}\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}$. Let $\theta\in \mathbb{R}$. By proceeding as in the proof of Theorem \ref{thf1} we get
\begin{align*}
\|\Re_A(e^{i\theta}\mathbb{S})\|_{\mathbb{A}}
&=\frac{1}{2}r_{\mathbb{A}}\left[\begin{pmatrix}
e^{i\theta}T_{11}+e^{-i\theta}T_{11}^{\sharp_{A}} & e^{i\theta}T_{12} &\cdots& e^{i\theta}T_{1d}\\
e^{-i\theta}T_{12}^{\sharp_{A}} &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
e^{-i\theta}T_{1d}^{\sharp_{A}} & 0 &\cdots& 0\\
\end{pmatrix}\right]\\
&=\frac{1}{2}r_{\mathbb{A}}\left[
\begin{pmatrix}
T_{11}^{\sharp_{A}} & 0 &\cdots& e^{i\theta}I\\
T_{12}^{\sharp_{A}} &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
T_{1d}^{\sharp_{A}} & 0 &\cdots& 0\\
\end{pmatrix}
\begin{pmatrix}
e^{-i\theta}I &0 &\cdots& 0\\
0 & 0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
T_{11} & T_{12} &\cdots& T_{1d}\\
\end{pmatrix}
\right].
\end{align*}
So, by using \eqref{commut} and \eqref{dom} we get
\begin{align*}
\|\Re_A(e^{i\theta}\mathbb{S})\|_{\mathbb{A}}
&=\frac{1}{2}r_{\mathbb{A}}\left[
\begin{pmatrix}
e^{-i\theta}I &0 &\cdots& 0\\
0 & 0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
T_{11} & T_{12} &\cdots& T_{1d}\\
\end{pmatrix}
\begin{pmatrix}
T_{11}^{\sharp_{A}} & 0 &\cdots& e^{i\theta}I\\
T_{12}^{\sharp_{A}} &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
T_{1d}^{\sharp_{A}} & 0 &\cdots& 0\\
\end{pmatrix}
\right]\\
&=\tfrac{1}{2}r_{\mathbb{A}}\left[
\begin{pmatrix}
e^{-i\theta}T_{11}^{\sharp_{A}} & 0&\cdots&0& I\\
0 & 0 &\cdots&0& 0\\
\vdots & \vdots & \vdots & \vdots& \vdots\\
\sum_{k=1}^d T_{1k}T_{1k}^{\sharp_{A}}& 0 &\cdots&0& e^{i\theta}T_{11}\\
\end{pmatrix}
\right]\\
&\leq\tfrac{1}{2}\omega_{\mathbb{A}}\left[
\begin{pmatrix}
e^{-i\theta}T_{11}^{\sharp_{A}} & 0&\cdots&0& I\\
0 & 0 &\cdots&0& 0\\
\vdots & \vdots & \vdots & \vdots& \vdots\\
\sum_{k=1}^d T_{1k}T_{1k}^{\sharp_{A}}& 0 &\cdots&0& e^{i\theta}T_{11}\\
\end{pmatrix}
\right]\\
&\leq\tfrac{1}{2}\omega_{\mathbb{A}}\left[
\begin{pmatrix}
e^{-i\theta}T_{11}^{\sharp_{A}} & 0&\cdots&0& 0\\
0 & 0 &\cdots&0& 0\\
\vdots & \vdots & \vdots & \vdots& \vdots\\
0& 0 &\cdots&0& e^{i\theta}T_{11}\\
\end{pmatrix}
\right]+\tfrac{1}{2}\omega_{\mathbb{A}}\left[
\begin{pmatrix}
0&\cdots&0& I\\
0 &\cdots&0& 0\\
\vdots & \vdots & \vdots& \vdots\\
0 &\cdots&0& 0\\
\end{pmatrix}
\right]\\
&+ \tfrac{1}{2}\omega_{\mathbb{A}}\left[
\begin{pmatrix}
0 & 0&\cdots&0\\
0 & 0 &\cdots&0\\
\vdots & \vdots & \vdots & \vdots\\
\sum_{k=1}^d T_{1k}T_{1k}^{\sharp_{A}}& 0 &\cdots&0\\
\end{pmatrix}
\right].
\end{align*}
So, by using Lemma \ref{maxma} together with \eqref{at2} we obtain
\begin{equation}
\|\Re_A(e^{i\theta}\mathbb{S})\|_{\mathbb{A}}\leq \frac{\omega_A(T_{11})}{2}+\frac{1}{4}+\frac{1}{4}\left\|\sum^d_{j=1}T_{1j}T_{1j}^{\sharp_A}\right\|_A,
\end{equation}
So, by taking the supremum over all $\theta\in \mathbb{R}$ in the above inequality and then using \eqref{zamnum} we get \eqref{first02}. Finally, by using an argument similar to that used in proof of Theorem \ref{thf1} we reach the desired equality \eqref{r2}.
\end{proof}
In order to prove our next result in this section, we need the following lemmas.
\begin{lemma}(\cite{feki03})\label{lemmajdid}
Let $\mathbb{T}= (T_{ij})_{d \times d}$ be such that $T_{ij}\in \mathcal{B}_{A^{1/2}}(\mathcal{H})$ for all $i,j$. Then, $\mathbb{T}\in\mathcal{B}_{\mathbb{A}^{1/2}}(\mathbb{H})$. Moreover, we have
\begin{equation}\label{tag0}
\|\mathbb{T}\|_{\mathbb{A}}\leq \|\widehat{\mathbb{T}}^{\mathbb{A}}\|,
\end{equation}
where $\widehat{\mathbb{T}}^{\mathbb{A}}=(\|T_{ij} \|_A)_{d\times d}\in \mathbb{M}_d(\mathbb{C})$.
\end{lemma}
\begin{lemma}\label{jdidddd}
Let $T,S\in \mathcal{B}_{A}(\mathcal{H})$ and $\mathbb{B}=\begin{pmatrix}
A &0\\
0 &A
\end{pmatrix}$. Then,
\begin{equation*}
\omega_{\mathbb{B}}\left[\begin{pmatrix}
0&T\\
S &0
\end{pmatrix}\right]=\frac{1}{2}\sup_{\theta\in \mathbb{R}}\left\|e^{i\theta}T+e^{-i\theta}S^{\sharp_A}\right\|_A.
\end{equation*}
\end{lemma}
\begin{proof}
Notice first that, in view of Lemma \ref{maxma} (c) we have
\begin{equation}\label{particular}
\left\|\begin{pmatrix}
0&X\\
X^{\sharp_A} &0
\end{pmatrix}\right\|_{\mathbb{B}}=\|X\|_A,
\end{equation}
for every $X\in \mathcal{B}_{A}(\mathcal{H})$. Now, by using Lemma \ref{ir2020} and \eqref{zamnum}, it follows that
\begin{align*}
\omega_{\mathbb{B}}\left[\begin{pmatrix}
0&T\\
S &0
\end{pmatrix}\right]
& =\omega_{\mathbb{B}}\left[\begin{pmatrix}
0&T\\
S &0
\end{pmatrix}^{\sharp_\mathbb{B}}\right]\\
&=\frac{1}{2}\sup_{\theta\in \mathbb{R}}\left\|e^{i\theta}\begin{pmatrix}
0&S^{\sharp_A}\\
T^{\sharp_A} &0
\end{pmatrix}+e^{-i\theta}\begin{pmatrix}
0& (T^{\sharp_A})^{\sharp_A}\\
(S^{\sharp_A})^{\sharp_A} &0
\end{pmatrix}\right\|_\mathbb{B}\\
&=\frac{1}{2}\sup_{\theta\in \mathbb{R}}\left\|\begin{pmatrix}
0&e^{i\theta}S^{\sharp_A}+e^{-i\theta}(T^{\sharp_A})^{\sharp_A}\\
e^{i\theta}T^{\sharp_A}+e^{-i\theta}(S^{\sharp_A})^{\sharp_A} &0
\end{pmatrix}\right\|_\mathbb{B}\\
&=\frac{1}{2}\sup_{\theta\in \mathbb{R}}\left\|\begin{pmatrix}
0&e^{i\theta}S^{\sharp_A}+e^{-i\theta}(T^{\sharp_A})^{\sharp_A}\\
[e^{i\theta}S^{\sharp_A}+e^{-i\theta}(T^{\sharp_A})^{\sharp_A}]^{\sharp_A} &0
\end{pmatrix}\right\|_\mathbb{B}\\
&=\frac{1}{2}\sup_{\theta\in \mathbb{R}}\left\|e^{i\theta}S^{\sharp_A}+e^{-i\theta}(T^{\sharp_A})^{\sharp_A}\right\|_A,\;(\text{by }\eqref{particular})\\
&=\frac{1}{2}\sup_{\theta\in \mathbb{R}}\left\|e^{i\theta}T^{\sharp_A}+e^{-i\theta}S\right\|_A.
\end{align*}
So, by replacing $\theta$ by $-\theta$ in the above equality, we obtain
\begin{equation}\label{durr2}
\omega_{\mathbb{B}}\left[\begin{pmatrix}
0&T\\
S &0
\end{pmatrix}\right]=\frac{1}{2}\sup_{\theta\in \mathbb{R}}\left\|e^{-i\theta}T^{\sharp_A}+e^{i\theta}S\right\|_A.
\end{equation}
Let $\mathbb{U}=\begin{pmatrix}
0&I\\
I&0
\end{pmatrix}.$ In view of Lemma \ref{ir2020}, we have $\mathbb{U}\in \mathcal{B}_{\mathbb{B}}(\mathcal{H}\oplus \mathcal{H})$ and $\mathbb{U}^{\sharp_{\mathbb{B}}}=\begin{pmatrix}
0&P_{\overline{\mathcal{R}(A)}} \\
P_{\overline{\mathcal{R}(A)}}&0
\end{pmatrix}.$ So, we verify that $\|\mathbb{U}x\|_\mathbb{B}=\|\mathbb{U}^{\sharp_\mathbb{B}}x\|_\mathbb{B}=\|x\|_\mathbb{B}$ for all $x=(x_1,x_2)\in \mathcal{H}\oplus \mathcal{H}$. Hence, $\mathbb{U}$ is $\mathbb{B}$-unitary operator. Thus, by Lemma \ref{weak} we have
\begin{align}\label{offfd}
\omega_{\mathbb{B}}\left[\begin{pmatrix}
0&T\\
S &0
\end{pmatrix}\right]
& =\omega_\mathbb{B}\left[\mathbb{U}^{\sharp_A}\begin{pmatrix}
0&T\\
S &0
\end{pmatrix}\mathbb{U}\right]\nonumber\\
& =\omega_\mathbb{B}\left[\begin{pmatrix}
P_{\overline{\mathcal{R}(A)}}&0\\
0 &P_{\overline{\mathcal{R}(A)}}
\end{pmatrix}\begin{pmatrix}
0&S\\
T &0
\end{pmatrix}\right]\nonumber\\
&=\omega_{\mathbb{B}}\left[\begin{pmatrix}
0&S\\
T &0
\end{pmatrix}\right]
\end{align}
Hence, by combining \eqref{durr2} together with \eqref{offfd} we prove the desired equality.
\end{proof}
Now, we are in a position to establish the following result which generalizes \cite[Theorem 4.17.]{BP}.
\begin{theorem}\label{th-2}
Let $\mathbb{T}=(T_{ij})$ be an $d\times d$ operator matrix with $T_{ij}\in \mathcal{B}_A(\mathcal{H})$. Then,
\[\omega_\mathbb{A}(\mathbb{T})\leq \omega(S),\]
where $S=[s_{ij}]\in \mathbb{M}_d(\mathbb{C})$ is given by
\begin{equation}\label{sij}
s_{ij}=
\begin{cases}
\omega(T_{ij})&,\text{ if }\;i=j,\\
\omega_\mathbb{B}\left[\begin{pmatrix}
0 &T_{ij}\\
T_{ji}&0
\end{pmatrix}\right]\text{ with } \mathbb{B}=\begin{pmatrix}
A&0\\
0&A
\end{pmatrix}&,\text{ if }\;i\neq j.
\end{cases}.
\end{equation}
\end{theorem}
\begin{proof}
Let $\theta\in \mathbb{R}$ and $\mathbb{B}=\begin{pmatrix}
A&0\\
0&A
\end{pmatrix}$. By using \eqref{ir2020}, it can be seen that
\begin{align*}
\Re_\mathbb{A}(e^{i\theta}\mathbb{T})
& = \frac{1}{2}\left(e^{i\theta}\mathbb{T}+e^{-i\theta}\mathbb{T}^{\sharp_\mathbb{A}}\right)\\
&=\begin{pmatrix}
\Re_A(e^{i\theta}T_{11}) &\tfrac{1}{2}(e^{i\theta}T_{12}+e^{-i\theta}T_{21}^{\sharp_A}) &\cdots&\tfrac{1}{2}(e^{i\theta}T_{1d}+e^{-i\theta}T_{d1}^{\sharp_A}) \\
\tfrac{1}{2}(e^{i\theta}T_{21}+e^{-i\theta}T_{12}^{\sharp_A}) & \Re_A(e^{i\theta}T_{22}) &\cdots& \tfrac{1}{2}(e^{i\theta}T_{2d}+e^{-i\theta}T_{d2}^{\sharp_A}) \\
\vdots & \vdots & \vdots & \vdots\\
\tfrac{1}{2}(e^{i\theta}T_{d1}+e^{-i\theta}T_{1d}^{\sharp_A}) & \tfrac{1}{2}(e^{i\theta}T_{d2}+e^{-i\theta}T_{2d}^{\sharp_A}) &\cdots& \Re_A(e^{i\theta}T_{dd})\\
\end{pmatrix}.
\end{align*}
So, by applying Lemma \eqref{lemmajdid} together with the norm monotonicity of matrices with nonnegative entries and then using Lemma \ref{jdidddd} and \eqref{zamnum} we get
\begin{align*}
&\left\|\Re_\mathbb{A}(e^{i\theta}\mathbb{T})\right\|_\mathbb{A}\\
&\leq\left\|\begin{pmatrix}
\omega_A(T_{11}) &\omega_\mathbb{B}\left[\begin{pmatrix}
0 &T_{12}\\
T_{21}&0
\end{pmatrix}\right] &\cdots&\omega_\mathbb{B}\left[\begin{pmatrix}
0 &T_{1d}\\
T_{d1}&0
\end{pmatrix}\right] \\
\omega_\mathbb{B}\left[\begin{pmatrix}
0 &T_{21}\\
T_{12}&0
\end{pmatrix}\right] & \omega_A(T_{22}) &\cdots& \omega_\mathbb{B}\left[\begin{pmatrix}
0 &T_{2d}\\
T_{d2}&0
\end{pmatrix}\right] \\
\vdots & \vdots & \vdots & \vdots\\
\omega_\mathbb{B}\left[\begin{pmatrix}
0 &T_{d1}\\
T_{1d}&0
\end{pmatrix}\right] & \omega_\mathbb{B}\left[\begin{pmatrix}
0 &T_{d2}\\
T_{2d}&0
\end{pmatrix}\right] &\cdots& \omega_A(T_{dd})\\
\end{pmatrix}\right\|.
\end{align*}
So, by taking the supremum over all $\theta\in \mathbb{R}$ in the above inequality we get $\omega_\mathbb{A}(\mathbb{T})\leq \|S\|$ where $S$ is defined in \eqref{sij}. Finally, by \eqref{offfd}, we have $\omega_\mathbb{B}\left[\begin{pmatrix}
0 &T_{ij}\\
T_{ji}&0
\end{pmatrix}\right]=\omega_\mathbb{B}\left[\begin{pmatrix}
0 &T_{ji}\\
T_{ij}&0
\end{pmatrix}\right]$ for all $i,j$. Thus, $S$ is a real symmetric matrix and so $\omega(S)=\|S\|$. Therefore, we get the desired result.
\end{proof}
Next we state from \cite[p. 44]{HJ} the following useful lemma.
\begin{lemma}\label{l-10}
Let $T=(t_{ij})\in \mathbb{M}_d(\mathbb{C})$ such that $t_{ij}\geq 0$ for all $i,j=1,2,\ldots,d.$ Then
$$\omega(T)=\frac{r\left (t_{ij}+t_{ji}\right)}{2}.$$
\end{lemma}
\begin{remark}
Bhunia et al. proved recently in \cite[Theorem 4.12.]{BP} that for a $d \times d$ operator matrix $\mathbb{T}=(T_{ij})$ with $T_{ij}\in \mathcal{B}_A(\mathcal{H})$. Then,
\begin{equation}\label{pint2020}
\omega_\mathbb{A}(\mathbb{T})\leq \omega([t_{ij}])\;\;\text{ where }\;\; t_{ij}= \begin{cases}
\omega_{A}(T_{ij}), & i=j \\
\|T_{ij}\|_A, & i\neq j.
\end{cases}
\end{equation}
Clearly, by Lemma \ref{jdidddd} one observes that
\begin{equation*}
\omega_{\mathbb{B}}\left[\begin{pmatrix}
0&T\\
S &0
\end{pmatrix}\right]\leq\frac{\|T\|_A+\|S\|_A}{2},
\end{equation*}
for every $T,S\in \mathcal{B}_{A}(\mathcal{H})$. So, by taking into consideration Lemma \ref{l-10}, it is not difficult to verify that the inequality proved in Theorem \ref{th-2} refines the inequality \eqref{pint2020}.
\end{remark}
Our next result is stated as follows.
\begin{theorem}
Let $\mathbb{T}=(T_{ij})$ be a ${d\times d}$ operator matrix where $T_{ij}\in \mathcal{B}_{A}(\mathcal{H})$. Then,
\begin{equation*}
\omega_{\mathbb{A}}(\mathbb{T})\leq \frac{1}{2}\sum_{i=1}^d\left(\omega_A(T_{ii})+\sqrt{\omega_A^2(T_{ii})+\sum^d_{j=1,j\neq i}\|T_{ij}\|_A^2}\right).
\end{equation*}
\end{theorem}
\begin{proof}
We first prove that
\begin{equation}\label{first004}
\omega_{\mathbb{A}}\left[\begin{pmatrix}
T_{11} & T_{12} &\cdots& T_{1d}\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\right]\leq \frac{1}{2}\left(\omega_A(T_{11})+\sqrt{\omega_A^2(T_{11}) +\sum^d_{j=2}\|T_{1j}\|_A^2 }\right).
\end{equation}
By applying Theorem \ref{th-2} we obtain
\begin{align*}
&\omega_{\mathbb{A}}\left[\begin{pmatrix}
T_{11} & T_{12} &\cdots& T_{1d}\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\right]\\
&\leq\omega\left[\begin{pmatrix}
\omega_A(T_{11}) &\omega_\mathbb{B}\left[\begin{pmatrix}
0 &T_{12}\\
0&0
\end{pmatrix}\right] &\cdots&\omega_\mathbb{B}\left[\begin{pmatrix}
0 &T_{1d}\\
0&0
\end{pmatrix}\right] \\
\omega_\mathbb{B}\left[\begin{pmatrix}
0 &0\\
T_{12}&0
\end{pmatrix}\right] & 0 &\cdots& 0 \\
\vdots & \vdots & \vdots & \vdots\\
\omega_\mathbb{B}\left[\begin{pmatrix}
0 &0\\
T_{1d}&0
\end{pmatrix}\right] & 0 &\cdots& 0\\
\end{pmatrix}\right].
\end{align*}
Moreover, since $\mathbb{B}\begin{pmatrix}
0 &0\\
T_{1j}&0
\end{pmatrix}^2=\begin{pmatrix}
0 &0\\
0&0
\end{pmatrix}$ for every $j\in\{1,\cdots,d\}$, then by applying \eqref{at2} together with Lemma \ref{maxma} (c) we have
$$\omega_\mathbb{B}\left[\begin{pmatrix}
0 &0\\
T_{1j}&0
\end{pmatrix}\right]=\frac{1}{2}\left\|\begin{pmatrix}
0 &0\\
T_{1j}&0
\end{pmatrix}\right\|_\mathbb{B}=\frac{1}{2}\|T_{1j}\|_A.$$
So, we obtain
\begin{align*}
&\omega_{\mathbb{A}}\left[\begin{pmatrix}
T_{11} & T_{12} &\cdots& T_{1d}\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\right]\\
&\leq\omega\left[\begin{pmatrix}
\omega_A(T_{11}) &\frac{\|T_{12}\|_A}{2} &\cdots&\frac{\|T_{1d}\|_A}{2} \\
\frac{\|T_{12}\|_A}{2} & 0 &\cdots& 0 \\
\vdots & \vdots & \vdots & \vdots\\
\frac{\|T_{1d}\|_A}{2} & 0 &\cdots& 0\\
\end{pmatrix}\right]\\
&=\frac{1}{2}r\left[\begin{pmatrix}
2\omega_A(T_{11}) &\|T_{12}\|_A &\cdots&\|T_{1d}\|_A \\
\|T_{12}\|_A & 0 &\cdots& 0 \\
\vdots & \vdots & \vdots & \vdots\\
\|T_{1d}\|_A & 0 &\cdots& 0\\
\end{pmatrix}\right]\;(\text{ by Lemma }\ref{l-10})\\
&=\frac{1}{2}\left(\omega_A(T_{11})+\sqrt{\omega_A^2(T_{11}) +\sum^d_{j=2}\|T_{1j}\|_A^2 }\right).
\end{align*}
This proves \eqref{first004}. Now, by proceeding as in proof of Theorem \ref{thf1} we get the required result.
\end{proof}
The following lemma is useful in proving our next result.
\begin{lemma}\label{lemma:2}
Let $T \in \mathcal{B}_A(\mathcal{H})$. Then
\begin{equation*}
\omega_A(T) \leq\sqrt{ \|\Re_A(T)\|_A^2+\|\Im_A(T)\|_A^2}.
\end{equation*}
\end{lemma}
\begin{proof}
Let $x\in \mathcal{H}$. Since $\Re_A(T)$ and $\Im_A(T)$ are $A$-selfadjoint operators, then by taking into consideration \eqref{aself1} we see that
\begin{align*}
\big|{\langle Tx\mid x \rangle}_A\big|^2
&= \big|{\langle \Re_A(T)x\mid x \rangle}_A+i{\langle \Im_A(T)x\mid x \rangle}_A\big|^2\\
& = \big|{\langle\Re_A(T)x\mid x\rangle}_A\big|^2 + \big|{\langle \Im_A(T)x\mid x\rangle}_A\big|^2\\
&\leq \|\Re_A(T)\|_A^2+\|\Im_A(T)\|_A^2.
\end{align*}
So, by taking the supremum over all $x\in \mathcal{H}$ with $\|x\|_A=1$ we get required result.
\end{proof}
\begin{theorem} \label{theorem:upper bound oprt 2}
Let $\mathbb{T}=(T_{ij})$ be a ${d\times d}$ operator matrix where $T_{ij}\in \mathcal{B}_{A}(\mathcal{H})$. Then,
\begin{equation*}
\omega_A(\mathbb{T})\leq \frac{1}{2}\sum^d_{i=1}\sqrt{\lambda^2_i+\mu^2_i},
\end{equation*}
where
\begin{align*}
\lambda_i &=\|\Re_A(T_{ii})\|_A+\sqrt{\|\Re_A(T_{ii})\|_A^2+\sum_{j=1,j\neq i}^d\|T_{ij}\|_A^2} \;\;\text{ and }\\
\mu_i &=\|\Im_A(T_{ii})\|_A+\sqrt{\|\Im_A(T_{ii})\|_A^2+\sum_{j=1,j\neq i}^d\|T_{ij}\|_A^2}.
\end{align*}
\end{theorem}
\begin{proof}
Let $\mathbb{S}=\left(\begin{array}{cccc}
T_{11}&T_{12}&\ldots &T_{1d} \\
0&0&\ldots &0\\
\vdots & \vdots & &\vdots \\
0&0&\ldots&0
\end{array}\right).$ It is not difficult to verify that
\begin{align*}
\left\|\Re_\mathbb{A}(\mathbb{T})\right\|_\mathbb{A}
&=r_\mathbb{A}\left[\Re_\mathbb{A}(\mathbb{T})\right] \\
&=r_\mathbb{A}\left[\left(\begin{array}{cccc}
\Re_A(T_{11})&\frac{T_{12}}{2}&\ldots&\frac{T_{1d}}{2} \\
\frac{T_{12}^{\sharp_A}}{2}&0&\dots &0\\
\vdots& \vdots& &\vdots \\
\frac{T_{1d}^{\sharp_A}}{2}&0&\ldots&0
\end{array}\right)\right] \\
&\leq r\left[ \left(\begin{array}{cccc}
\|\Re_A(T_{11})\|_A&\frac{\|T_{12}\|_A}{2}&\ldots&\frac{\|T_{1d}\|_A}{2} \\
\frac{\|T_{12}^{\sharp_A}\|_A}{2}&0&\ldots&0\\
\vdots& \vdots& &\vdots \\
\frac{\|T_{1d}^{\sharp_A}\|_A}{2}&0&\ldots&0
\end{array}\right) \right \|\;(\text{ by Lemma }\ref{mjom})\\
&= \frac{1}{2}\left(\|\Re_A(T_{11})\|_A+\sqrt{\|\Re_A(T_{11})\|_A^2+\sum_{j=2}^d\|T_{1j}\|_A^2}\right).
\end{align*}
Now, it can be seen that
$$\Im_\mathbb{A}(\mathbb{T})=\left(\begin{array}{cccc}
\Im_A(T_{11})&\frac{T_{12}}{2i}&\ldots&\frac{T_{1d}}{2i} \\
-\frac{T^{\sharp_A}_{12}}{2i}&0&\ldots&0\\
\vdots& \vdots& &\vdots \\
-\frac{T^{\sharp_A}_{1d}}{2i}&0&\ldots&0
\end{array}\right).$$
Similarly, we prove that
\begin{align*}
\|\Im_\mathbb{A}(\mathbb{T})\|_\mathbb{A}\leq \frac{1}{2}\left(\|\Im_A(T_{11})\|_A+\sqrt{\|\Im_A(T_{11})\|_A^2+\sum_{j=2}^d\|T_{1j}\|_A^2}\right).
\end{align*}
Hence, by Lemma \ref{lemma:2}, we get
\begin{equation*}
\omega_{\mathbb{A}}\left(\mathbb{S}\right)\leq \frac{1}{2} \sqrt{\lambda^2+\mu^2},
\end{equation*}
where
\begin{align*}
\lambda&=\|\Re_A(T_{11})\|_A+\sqrt{\|\Re_A(T_{11})\|_A^2+\sum_{j=2}^d\|T_{1j}\|_A^2},\\
\mu&=\|\Im_A(T_{11})\|_A+\sqrt{\|\Im_A(T_{11})\|_A^2+\sum_{j=2}^d\|T_{1j}\|_A^2}.
\end{align*}
Finally, by using an argument similar to that used in proof of Theorem \ref{thf1} we reach the desired result.
\end{proof}
Our next result reads as follows.
\begin{theorem}
Let $\mathbb{T}=(T_{ij})$ be a ${d\times d}$ operator matrix where $T_{ij}\in \mathcal{B}_{A}(\mathcal{H})$. Then,
\begin{equation*}
\omega_{\mathbb{A}}(\mathbb{T})\leq \max_{i\in\{1,\cdots,d\}}\omega_A(T_{ii})+\frac{1}{2}\sum^d_{i=1}\sqrt{\left\|\sum^d_{j=1,j\neq i}T_{ij}T_{ij}^{\sharp_A}\right\|_A}.
\end{equation*}
\end{theorem}
\begin{proof}
By using the triangle inequality and Lemma \ref{maxma} (b) we get
\begin{align*}
\omega_{\mathbb{A}}(\mathbb{T})
& \leq\max_{i\in\{1,\cdots,d\}}\omega_A(T_{ii})+\omega_{\mathbb{A}}\left[\begin{pmatrix}
0 &T_{12} &\cdots& T_{1d}\\
0 & 0 &\cdots& 0\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\right]
\\
&+\omega_{\mathbb{A}}\left[\begin{pmatrix}
0 &0 &\cdots& 0& 0\\
T_{21} & 0 &T_{23}&\cdots& T_{2d}\\
0 &0 &\cdots& 0& 0\\
\vdots & \vdots & \vdots & \vdots& \vdots\\
0 & 0 &\cdots& 0& 0\\
\end{pmatrix}\right]+\ldots+\omega_{\mathbb{A}}\left[\begin{pmatrix}
0 &0 &\cdots& 0& 0\\
\vdots & \vdots & \vdots & \vdots& \vdots\\
0 & 0 &\cdots& 0& 0\\
T_{d1} & T_{d2} &\cdots& T_{dd-1}& 0\\
\end{pmatrix}\right].
\end{align*}
On the other hand it can be seen that
\begin{align*}
&\mathbb{A}\begin{pmatrix}
0 &T_{12} &\cdots& T_{1d}\\
0 & 0 &\cdots& 0\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}^2
=\mathbb{A}\begin{pmatrix}
0 &0 &\cdots& 0& 0\\
T_{21} & 0 &T_{23}&\cdots& 0\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots& \vdots\\
0 & 0 &\cdots& 0& 0\\
\end{pmatrix}^2 \\
&=\ldots=\mathbb{A}\begin{pmatrix}
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
T_{d1} & T_{d2} &\cdots& T_{dd}\\
\end{pmatrix}^2=\begin{pmatrix}
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
0 & 0 &\cdots& 0\\
\end{pmatrix}.
\end{align*}
So, by \eqref{at2} we infer that
$$\mathbb{A}\begin{pmatrix}
0 &T_{12} &\cdots& T_{1d}\\
0 & 0 &\cdots& 0\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}=\frac{1}{2}\left\|\begin{pmatrix}
0 &T_{12} &\cdots& T_{1d}\\
0 & 0 &\cdots& 0\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\right\|_\mathbb{A}.$$
Moreover, by using \eqref{diez} and Lemma \ref{maxma}, it can be checked that
\begin{align*}
\left\|\begin{pmatrix}
0 &T_{12} &\cdots& T_{1d}\\
0 & 0 &\cdots& 0\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\right\|_\mathbb{A}^2
&=\left\|\begin{pmatrix}
0 &T_{12} &\cdots& T_{1d}\\
0 & 0 &\cdots& 0\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\begin{pmatrix}
0 &T_{12} &\cdots& T_{1d}\\
0 & 0 &\cdots& 0\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}^{\sharp_{\mathbb{A}}}\right\|_\mathbb{A}\\
& =\left\|\begin{pmatrix}
\sum^d_{k=2}T_{1k}T_{1k}^{\sharp_A} &0 &\cdots& 0\\
0 & 0 &\cdots& 0\\
0 &0 &\cdots& 0\\
\vdots & \vdots & \vdots & \vdots\\
0 & 0 &\cdots& 0\\
\end{pmatrix}\right\|_\mathbb{A}=\left\|\sum^d_{k=2}T_{1k}T_{1k}^{\sharp_A}\right\|_A.
\end{align*}
Hence, by using similar arguments we get
\begin{align*}
\omega_{\mathbb{A}}(\mathbb{T})
& \leq\max_{i\in\{1,\cdots,d\}}\omega_A(T_{ii})+
\frac{1}{2}\sqrt{\left\|\sum^d_{k=2}T_{1k}T_{1k}^{\sharp_A}\right\|_A}
\\
&+\frac{1}{2}\sqrt{\left\|\sum^d_{k=1,k\neq2}T_{2k}T_{2k}^{\sharp_A}\right\|_A}+\ldots+
\frac{1}{2}\sqrt{\left\|\sum^d_{k=1,k\neq d}T_{dk}T_{dk}^{\sharp_A}\right\|_A}.
\end{align*}
This achieves the proof of the theorem.
\end{proof}
|
2,877,628,090,087 | arxiv | \section{Introduction}
Iron based superconductors have been discovered in several families of compounds containing square nets of iron with formal valence near 2+ in tetrahedral coordination. These include the 1111 family based on LaFeAsO \cite{Kamihara2008}, the 122 family based on BaFe$_2$As$_2$ \cite{BaFe2As2-SC}, the 111 family based on LiFeAs \cite{LiFeAs}, the 11 family based on Fe$_{1+\delta}$Se \cite{FeSe-SC}, and the 32522 family based on Sr$_3$Sc$_2$O$_5$Fe$_2$As$_2$ \cite{32522-SC}. In the 1111 and 122 systems substitution of cobalt for iron has been demonstrated as an effective way to induce superconductivity \cite{Sefat-Co1111, Sefat-Co122}. While the main interest in these materials is the superconducting state, non-superconducting members of these families have also shown interesting behavior, and their study may help further our understanding of the superconducting materials. Upon cooling, the pure iron compounds LaFeAsO and BaFe$_2$As$_2$ undergo a spin density wave transition with an accompanying crystallographic distortion \cite{BaFe2As2, McGuire-LaFeAsO, Nature-LaFeAsO}, a ground state that seems to compete with superconductivity. In the pure cobalt compound LaCoAsO, small moments on Co order ferromagnetically below about 60 K with saturation moments of 0.3$-$0.5 $\mu$B per Co \cite{Yanagi-LaCoAsO, Sefat-Co1111, Ohta-LaCoAsO}. Experimental and theoretical studies suggest that BaCo$_2$As$_2$ is a highly renormalized paramagnetic metal, near a ferromagnetic instability \cite {Sefat-BaCo2As2}. It has been proposed that spin fluctuations play an important role in the magnetic behavior of LaCoAsO \cite{Yanagi-LaCoAsO, Ohta-LaCoAsO}, as well as the magnetic and superconducting properties of the iron-based superconductors \cite{Ning}.
In the superconducting 1111 materials, the transition temperature T$_c$ is increased significantly as La is replaced by other lanthanides, from 28 K for LaFeAsO$_{1-x}$F$_x$ to 55 K for SmFeAsO$_{1-x}$F$_x$ \cite{Sefat-F1111, SmFeAsOF}. It has also been shown that the identity of the lanthanide $Ln$ strongly effects the low temperature transport properties of the undoped parent compounds $Ln$FeAsO, and interaction between the $Ln$ and Fe magnetism has been observed below the $Ln$ antiferromagnetic ordering temperatures \cite{McGuire-LnFeAsO}. To further probe the interesting magnetic properties that are found in these layered itinerant magnetic materials, we have begun investigating the effects of replacing La in LaCoAsO with magnetic lanthanides.
Here we report our results for NdCoAsO. It is isostructural to LaFeAsO and LaCoAsO, adopting the tetragonal ZrCuSiAs structure type with space group \textit{P4/nmm }\cite{Quebe}. Our magnetization measurements revealed upon cooling a series of magnetic transitions: a ferromagnetic transition similar to that reported for LaCoAsO, and two antiferromagnetic transitions at lower temperatures. During the course of this work, a report of magnetization measurements on NdCoAsO has been published \cite{Ohta-LCoAsO}. Those results are in good agreement with the results presented here; however, the authors discussed only two of the three magnetic phase transitions. In addition to magnetization analysis, we have used powder neutron diffraction to determine the antiferromagnetic structure of the Nd and Co moments in this material. We also report detailed crystal structure information as well as the effects of the phase transitions on the transport properties and heat capacity.
\section{Experimental Details}
Polycrystalline samples of NdCoAsO were prepared by solid state reactions, using procedures similar to those reported for this and other related compounds \cite{Sefat-Co1111, McGuire-LaFeAsO, McGuire-LnFeAsO, Ohta-LCoAsO, Yanagi-LaCoAsO}. CoAs was first prepared from Co powder and As pieces heated very slowly to 700$^\circ$C and then 1065$^\circ$C. Nd$_2$O$_3$, Nd, and CoAs powders were then thoroughly mixed in an agate mortar and pestle inside a helium filled glovebox. The mixtures were pressed into pellets and sealed in silica tubes which were evacuated and backfilled with about 0.2 atm argon gas. The samples were reacted at 1200$^\circ$C for 12 hours, reground and re-pelletized, and heated for a second time at 1200$^\circ$C for 12 hours. Starting materials were of 99.9\% purity or better. From powder X-ray and neutron diffraction analysis we estimate the purity of our NdCoAsO samples to be $\gtrsim$ 95\%. Observed impurity phases were CoAs (an antiferromagnetic with T$_N$ = 54 K \cite{Lewis-CoAs}) and Nd$_2$SiO$_5$. No contributions from these impurity phases to the measured physical properties were observed.
Heat capacity and transport measurements were performed using a Quantum Design Physical Property Measurement System. SQUID magnetometery measurements were carried out using a Quantum Design Magnetic Property Measurement System. Neutron powder diffraction data were collected from a 9 gram sample of NdCoAsO at temperatures from 300 mK to 300 K on the High-Resolution Powder Diffractometer (HB2A) at the High Flux Isotope Reactor at Oak Ridge National Laboratory, using 12'-31'-6' collimation and with a wavelength of 1.538 {\AA} Ge(115). More details about the HB2A instrument and data collection strategies can be found in Ref.~\onlinecite{HB2A}. The crystallographic structure of NdCoAsO was refined at 300 K, 120 K and 30 K (above the antiferromagnetic ordering temperatures) using the JANA2006 program package \cite{JANA, JANA2}. Effects of the ferromagnetic ordering expected at 30 K could not be reliably observed in the current data, due to the small magnitudes of the moments and their coincidence with the much stronger nuclear Bragg reflections. JANA2006 was also used for the determination of the magnetic structure at lower temperatures (20 K and below). Refinement of the magnetic structure was also performed using the program Fullprof \cite{Fullprof}. These two software packages use different approaches to describe the magnetic structure, and the two programs gave equivalent results. The values of magnetic moments and magnetic refinement results presented in this report were obtained using Fullprof.
\section{Results and Discussion}
\subsection{Magnetization}
\begin{figure}
\includegraphics[width=3.0in]{magnetization.eps}
\caption{\label{fig:mag}
(color online) Magnetic properties of NdCoAsO. (a) Temperature dependence of the magnetization of NdCoAsO at four different applied magnetic fields. (b) Magnetic field dependence of the magnetization at three different temperatures, showing a component which saturates at 30 K, near the center of the ferromagnetic-like temperature regime, and linear behavior in both the paramagnetic state at higher temperature (150 K) and the antiferromagnetic state at lower temperature (2 K). (c) Low field behavior at 30 K which indicates a saturation moment near 0.18 $\mu_B$ per formula unit (FU).
}
\end{figure}
Figure \ref{fig:mag} shows the measured magnetic properties of NdCoAsO. These results are consistent with the report of Ohta and Yoshimura \cite{Ohta-LCoAsO}. Three phase transitions are observed in the temperature dependence of the magnetization (Figure \ref{fig:mag}a). Upon cooling in low applied fields, an abrupt increase is observed near T$_C$ = 69 K, suggestive of ferromagnetic ordering. This is followed by an antiferromagnetic transition at T$_{N1}$ = 14 K, at which point the magnetization sharply decreases (except in the strongest applied fields). A third anomaly, a downturn in the magnetization upon cooling, is observed near T$_{N2}$ = 3.5 K (Figure \ref{fig:mag}a, inset). At high fields (50 kOe) much of this behavior is overwhelmed by Nd paramagnetism; however, a broad feature can be discerned below 100 K, and a sharp cusp is observed at 3 K. The ferromagnetic-like region is extended to cover a broader temperature range as the applied field is increased. A Curie-Weiss fit to data collected at 50 kOe over the temperature range of 200$-$300 K (not shown) gives an effective moment of 3.5 $\mu_B$, close to the expected value of 3.62 $\mu_B$ for Nd$^{3+}$, and in good agreement with the literature report for this material \cite{Ohta-LCoAsO}. If the magnetism in the Co layer is itinerant in nature, a Co contribution to this effective moment may not be expected. However, Curie-Weiss behavior has been observed in LaCoAsO \cite{Ohta-LaCoAsO, Sefat-Co1111} with an effective moment near 1.3 $\mu_B$ per Co. Thus, the effective moment observed here for NdCoAsO may include contributions from both magnetic ions. The fitted Weiss temperature for NdCoAsO is 34 K. Interestingly, this value is positive, which would indicate predominantly ferromagnetic interaction at high temperatures.
The field dependence of the magnetization at 2, 30, and 150 K is shown in Figure \ref{fig:mag}b. Linear behavior is observed at 2 and 150 K. At 30 K, near the middle of the ferromagnetic region, a rapid saturation is observed up to 0.5 kOe, followed by linear region up to 60 kOe. No magnetic hysteresis is observed. A linear fit to the data above 1 kOe reveals a saturated moment of 0.18 $\mu_B$ per formula unit (Figure \ref{fig:mag}c). A similar saturation moment was observed by Ohta and Yoshimura \cite{Ohta-LCoAsO}. Since similar behavior has been observed in LaCoAsO, where Co is the only magnetic atom, it is likely that the saturated component in NdCoAsO arises from Co, and the linear behavior at higher fields originates primarily from the paramagnetic response of Nd. The sharp decrease in magnetization at T$_{N1}$ likely indicates a ferromagnetic-antiferromagnetic transition, as noted previously \cite{Ohta-LCoAsO}. The downturn in magnetization near 3 K suggests antiferromagnetic order of Nd moments. This occurs near 2 K in the closely related compound NdFeAsO \cite{NdFeAsO-Ndmag}. The evolution of the magnetic structure of this material upon cooling through these magnetic phase transitions, determined from neutron diffraction measurements, is discussed below.
\subsection{Neutron diffraction}
\begin{table}
\caption{Crystallographic parameters and agreement factors (R$_{obs}$ and goodness of fit) from Rietveld refinements in space group \textit{P4/nmm} with Nd atoms at (1/4, 1/4, z-Nd), Co atoms at (1/4, 3/4, 1/2), As atoms at (1/4, 1/4, z-As), and O atoms at (1/4, 3/4, 0). U$_{iso}$ are isotropic displacement parameters.}
\begin{tabular*}{3.0in}%
{@{\extracolsep{\fill}}lccc}
\hline
T (K) & 300 & 120 & 30 \\
&&&\\
a ({\AA}) & 3.98423(8) & 3.98299(8) & 3.98237(7) \\
c ({\AA}) & 8.3333(3) & 8.3193(2) & 8.3123(2) \\
&&&\\
z-Nd & 0.1422(2) & 0.1421(4) & 0.1418(2) \\
z-As & 0.6501(3) & 0.6507(3) & 0.6511(3) \\
&&&\\
U$_{iso}$-Nd ({\AA}$^2$) & 0.0123(5) & 0.0041(5) & 0.0019(5) \\
U$_{iso}$-Co ({\AA}$^2$) & 0.0187(11) & 0.0058(12) & 0.0036(11) \\
U$_{iso}$-As ({\AA}$^2$) & 0.0198(6) & 0.0080(6) & 0.0061(5) \\
U$_{iso}$-O ({\AA}$^2$) & 0.017(6) & 0.0088(6) & 0.0079(6) \\
&&&\\
R$_{obs} (\%)$ & 1.97 & 1.72 & 1.68 \\
GoF & 1.73 & 1.47 & 1.98 \\
\hline
\end{tabular*}
\label{table:structure}
\end{table}
\begin{figure}
\includegraphics[width=3.5in]{Rietveld.eps}
\caption{\label{fig:Rietveld}
(color online) Rietveld refinements of powder neutron diffraction data at (a) T = 120 K and (b) T = 1.4 K showing measured data (circles) and fitted and difference curves. Tick marks locate Bragg reflections. In (b) upper ticks represent nuclear reflections and lower ticks indicate magnetic reflections. The inset in (b) shows the low angle region of the 1.4 K data, with peaks labeled by their indices.}
\end{figure}
Results of crystal structure refinements at three temperatures above the antiferromagnetic transitions are reported in Table \ref{table:structure}. The diffraction pattern at T = 120 K is displayed in Figure \ref{fig:Rietveld}a. The tetragonal space group \textit{P4/nmm} was used at all temperatures. We note that the nuclear Bragg reflections at lower temperatures were also well described by this space group (Figure \ref{fig:Rietveld}b). The variation of Bragg peak widths with temperature gave no indication of a structural distortion within the resolution of the current data. As Table \ref{table:structure} indicates, no unusual behavior in the lattice parameters, atomic positions, or isotropic displacement parameter are observed upon cooling from room temperature into the ferromagnetic state.
\begin{figure}
\includegraphics[width=3.0in]{NPD-mag.eps}
\caption{\label{fig:NPD-mag}
(color online) (a) Low angle neutron powder diffraction data from NdCoAsO at 15 K, 3 K, and 1.4 K. The (0 0 2) nuclear Bragg peak appears at 21.3 degrees. Magnetic reflections occur at 15.9 and 22.8 degrees. Data collected at different temperatures are offset vertically for clarity. (b) The temperature dependence of the refined antiferromagnetically ordered moments on Co and Nd for T $\leq$ 10 K.
}
\end{figure}
\begin{figure}
\includegraphics[width=2.5in]{structure.eps}
\caption{\label{fig:structure}
(color online) Crystal and magnetic structure of NdCoAsO in the antiferromagnetic states. Arrows indicate magnetic moments on Nd and Co which lie along the a-axis.
}
\end{figure}
Low angle neutron diffraction patterns collected at 15 K, 3 K, and 1.4 K are shown in Figure \ref{fig:NPD-mag}a. In the angular range displayed, only the nuclear (002) peak is observed at 15 K. At lower temperatures the (0 0 3/2) and (1 0 1/2) magnetic reflections are observed. Representational analysis~\cite{Bertaut1, Bertaut2, Bertaut3, Bertaut4} has been used to determine the symmetry-allowed magnetic structures, given the crystal structure and the propagation vector of the magnetic ordering. The calculations were carried out using the program SARA{\textit{h}}-Representational Analysis.\cite{Sarah} They involve the determination of the space group symmetry elements, $g$, that leave the propagation vector ${\mathbf k}$ invariant, forming the little group $G_{\mathbf k}$.~\cite{Kovalev} The decomposition of the magnetic representation in terms of the non-zero irreducible representations (IRs) of $G_{\textbf{k}}$ for each crystallographic site examined (Co and Nd), and their associated basis vectors, are given in Table \ref{basis-vectors}.
\begin{table}[h]
\caption{Basis vectors for the space group P 4/n m m:2 with ${\bf k}_{19}=( 0,~0,~.5)$. The Co atoms of the nonprimitive basis are defined according to 1: $( .75,~ .25,~ .5)$, 2: $( .25,~ .75,~ .5)$, while the Nd atoms are
1: $( .25,~ .25,~ .141)$, 2: $( .75,~ .75,~ .858)$.}
\label{basis-vectors}
\begin{tabular}{|cc|ccc|cc|ccc|}
\multicolumn{5}{c}{Co sites}& \multicolumn{5}{c}{Nd sites}\\
IR & Atom & $m_{\|a}$ & $m_{\|b}$ & $m_{\|c}$ & IR & Atom & $m_{\|a}$ & $m_{\|b}$ & $m_{\|c}$ \\
\hline
$\Gamma_{2}$ & 1 & 0 & 0 & 1 & $\Gamma_{2}$ & 1 & 0 & 0 & 1 \\
& 2 & 0 & 0 & 1 & & 2 & 0 & 0 & 1 \\
$\Gamma_{7}$ & 1 & 0 & 0 & 1 & $\Gamma_{3}$ & 1 & 0 & 0 & 1 \\
& 2 & 0 & 0 & -1 & & 2 & 0 & 0 & -1 \\
$\Gamma_{9}$ & 1 & 1 & 0 & 0 & $\Gamma_{9}$ & 1 & 1 & 0 & 0 \\
& 2 & -1 & 0 & 0 & & 2 & -1 & 0 & 0 \\
& 1 & 0 & 1 & 0 & & 1 & 0 & -1 & 0 \\
& 2 & 0 & -1 & 0 & & 2 & 0 & 1 & 0 \\
$\Gamma_{10}$ & 1 & 0 & 1 & 0 & $\Gamma_{10}$ & 1 & 0 & 1 & 0 \\
& 2 & 0 & 1 & 0 & & 2 & 0 & 1 & 0 \\
& 1 & 1 & 0 & 0 & & 1 & 1 & 0 & 0 \\
& 2 & 1 & 0 & 0 & & 2 & 1 & 0 & 0 \\
\end{tabular}
\end{table}
Since the neutrons sense only the projections of the magnetic moments in the plane perpendicular to the scattering vector, the presence of a strong (0 0 3/2) reflection is a clear indication that the moments exhibit components lying in the basal plane. Thus, the basis vectors involved in the magnetic structure appeared to be limited to those associated with the IRs $\Gamma_{9}$ and $\Gamma_{10}$. Several magnetic structure models were tested. It was found that the simplest model which would fit the data adequately is one consisting of Co magnetic moments parallel to the tetragonal $a$-axis, which are compensated by Nd moments pointing in opposite direction. Such a model corresponds to the IR $\Gamma_{10}$, that allows ferromagnetic alignment within each magnetic sublattices (Nd or Co), as well as an antiferromagnetic coupling between them. Such a moment configuration corresponds to the magnetic space group $Pcc'n$. A view of the magnetic structure is displayed in Figure \ref{fig:structure}. The Co magnetic moments alternate their directions from layer to layer, so that the magnetic unit cell is twice as large as the chemical cell. It is important to note that the (0 0 1/2) magnetic reflection is much weaker than (0 0 3/2) at all temperatures investigated (see Figure \ref{fig:Rietveld}b). This strongly supports the presence of magnetism on both magnetic ions. Analysis of the 300 mK data yielded the ordered moments: $m_{Co}\approx0.32(5)~\mu _{B}$ and $m _{Nd}\approx1.39(5)~\mu _{B}$.
The temperature dependences of the magnitude of the ordered moments on Co and Nd at T $\leq$ 10 K determined from neutron powder diffraction refinements are shown in Figure \ref{fig:NPD-mag}b. Non-zero values of these moments are not detected at 15 K and above. The magnitude of the ordered moment on Co is temperature independent below T$_{N1}$. The magnitude of the ordered moment on Nd increases slowly with decreasing temperature for $T_{N2} < T < T_{N1}$. Below $T_{N2}$, the Nd ordered moment abruptly increases, and is nearly saturated at the lowest temperature investigated here (300 mK). These results suggest that $T_{N1}$ involves an antiferromagnetic ordering of moments on cobalt, likely a reorientation of the moments which are ordered ferromagnetically at higher temperatures, which induces a partial ordering of Nd magnetic moments. Antiferromagnetic ordering of full Nd moments then sets in below $T_{N2}$. This behavior is quite similar to that observed in Nd$_2$CuO$_4$, in which moments on Nd are induced below one of the Cu spin reorientation temperatures (30 K) and then order at low temperatures ($\sim$ 1 K) \cite{Nd2CuO4}.
\subsection{Heat capacity}
\begin{figure}
\includegraphics[width=3.0in]{cP.eps}
\caption{\label{fig:cP}
(color online) (a) Specific heat c$_P$ of NdCoAsO in units of R per mole of atoms. The insets show the behavior of c$_P$ near the phase transitions. The anomaly near 69 K is emphasized by plotting c$_{P}$/T vs. T.
}
\end{figure}
Results of heat capacity measurements are presented in Figure \ref{fig:cP}. The temperature dependence of the specific heat $c_P$ per mole of atoms is shown in the main panel for temperatures below 150 K. At room temperature, $c_P$ reaches a value of 2.9 R. Heat capacity anomalies associated with magnetic phase transitions are emphasized in the insets of Figure \ref{fig:cP}. The feature at 69 K is subtle, and is most easily observed in a plot of $c_P/T$ vs. T. Another small peak is observed at 13 K, and a large anomaly occurs near 3.5 K.
The peaks in specific heat shown in Figure \ref{fig:cP} were integrated to obtain entropy changes associated with each transition. In the absence of an appropriate non-magnetic analogue for determination of the background lattice/electronic contribution to the heat capacity, analysis of the anomalies was performed using polynomial fits above and below the phase transitions to estimate the non-magnetic portion of the heat capacity. For the peaks at 3.5 K and 14 K, the background was estimated from a single fit to the data from 12.8 to 18 K, excluding the range around the 14 K peak, using the equation $a_1T+a_2T^3$, with fitting coefficients $a_1$ and $a_2$. For the 69 K peak, the background was estimated using a third order polynomial from 60 to 80 K, excluding the temperature range near the peak. The background was subtracted from the data to obtain $\Delta c_P$, and $\Delta c_P/T$ was integrated to obtain the entropy changes $\Delta$S. The integrations result in $\Delta$S = $\sim 1\times10^{-4}\ R$ and $\sim 3\times10^{-4}\ R$ for the transitions at T$_C$ and T$_{N1}$, respectively. The relatively small entropy change associated with the transitions at T$_{C}$ and T$_{N1}$ are consistent with the small sizes of the moments which order at these temperatures.
Since the peak at T$_{N2}$ extends beyond the lowest temperature measured in this work, we can only roughly estimate the entropy associated with this transition. To do this, the low T behavior of the heat capacity anomaly $\Delta c_P$ was interpolated from 0 to 1.87 K. Two interpolation schemes were used: $\Delta c_P = b_1 T$ and $\Delta c_P = b_2 T^3$. The integrated entropy change is of course sensitive to the interpolation scheme, giving $\Delta$S = 0.21 $R$ for linear interpolation and 0.14 $R$ for cubic interpolation.
\subsection{Transport properties}
\begin{figure}
\includegraphics[width=3.0in]{transport.eps}
\caption{\label{fig:transport}
(color online) Transport properties of NdCoAsO. (a) Electrical resistivity $\rho$. The inset shows the Seebeck coefficient and thermal conductivity. (b) Resistivity behavior near the antiferromagnetic phase transitions, including the effect of applied magnetic field. (c) Temperature derivative of the resistivity near the ferromagnetic phase transition and the effects of applied magnetic field.
}
\end{figure}
The temperature dependence from 2 to 150 K of the electrical resistivity $\rho$, Seebeck coefficient $S$, and total thermal conductivity $\kappa$ of a polycrystalline sample of NdCoAsO is shown in Figure \ref{fig:transport}a. The electrical resistivity displays metallic behavior up to room temperature, and reaches a value of 0.36 m$\Omega$ cm at 300 K. The residual resistivity ratio $R(300 K)/R(1.8 K)$ is 29, relatively high for a polycrystalline material. Indications of the low temperature magnetic phase transitions observed in $\rho$ are shown in Figure \ref{fig:transport}b. Upon cooling through T$_{N1}$, a sharp increase in $\rho$ occurs. Since this phase transition involves a reorientation of moments on Co atoms in the conducting CoAs layer, the observed change in $\rho$ is likely due to changes in the magnetic scattering rate of the charge carriers. Below about 4 K a gradual decrease in $\rho$ is seen, and is attributed to the magnetic ordering of larger moments on Nd which occurs near 3 K. A similar decrease in resistivity is also seen in \textit{Ln}FeAsO below the magnetic ordering temperature of the rare earth ions \textit{Ln} \cite{McGuire-LnFeAsO}. The effects of the ferromagnetic transition on $\rho$ are less dramatic, and are best observed in the temperature derivative d$\rho$/dT shown in Figure \ref{fig:transport}c. A local maximum in d$\rho$/dT occurs near T$_C$.
The dependence of the resistivity anomalies on applied magnetic field is also shown in Figure \ref{fig:transport}b and c. An applied field extends the ferromagnetic region to a larger temperature range by pushing T$_{C}$ up and T$_{N1}$ down. This is consistent with the magnetization measurements presented above.
The thermal conductivity (Figure \ref{fig:transport}a, inset) shows behavior typical of crystalline materials. No effects of the phase transitions on $\kappa$ are observed. The Seebeck coefficient (Figure \ref{fig:transport}a, inset) is negative for temperatures above 23 K, indicating predominantly n-type conduction. No effects of the magnetic transitions at T$_{C}$ and T$_{N2}$ can be resolved in the measured behavior of $S$. The maximum that occurs in $S$ near 13 K may be related to the magnetic transition at T$_{N1}$, or may simply be a result of the contribution of multiple bands, including at least one hole band, to the conduction in NdCoAsO.
\section{Conclusions}
We have shown that NdCoAsO undergoes three magnetic phase transitions at low temperatures, and determined the magnetic structure in the antiferromagnetically ordered states. A bulk ferromagnetic phase transition is observed at $T_{C}$ = 69 K by magnetization measurements and heat capacity ($c_P$) measurements, which show a small anomaly at this temperature. Two antiferromagnetic phase transitions are observed, one at $T_{N1}$ = 14 K, with a small $c_P$ anomaly, and another at $T_{N2}$ = 3.5 K, with a larger peak in $c_P$. The observed behavior is attributed to ferromagnetic ordering of small moments on Co at $T_{C}$, a reorientation of these moments into an antiferromagnetically ordered state at $T_{N1}$ with similarly small ordered moments induced on Nd, and a subsequent antiferromagnetic ordering of ``full'' Nd moments at $T_{N2}$. In the antiferromagnetic states, all ordered moments lie in the ab-plane. The Co atoms in each CoAs layer are ferromagnetically ordered, and these layers are ordered antiferromagnetically along the c-direction. The two Nd sites in each NdO layer are arranged antiferromagnetically, and alternate in direction between layers. These arrangements result in a doubling of the chemical unit cell along c, and a propagation vector of (0 0 1/2). No indication of a structural phase transition is observed down to 1.4 K. Seebeck coefficient measurements suggest electron dominated conduction near room temperature, but show small positive values at low temperatures, likely due to multiband effects. Analysis of electrical resistivity reveals anomalies at all three magnetic phase transitions, and show that the ferromagnetic temperature regime is increased in applied magnetic fields, as $T_{C}$ is pushed to high temperature and $T_{N1}$ is lowered, indicating an increased stabilization of the ferromagnetic state.
\\
\textit{Note}: While this manuscript was under review, the authors became aware of another report on NdCoAsO (arXiv:1001.2713v1) which is in general agreement with the results presented here.
\\
We gratefully acknowledge helpful discussions with K. V. Vemuru and V. Petricek. Research supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award \# (synthesis and physical property measurements). The work at Oak Ridge National Laboratory's High Flux Isotope Reactor (neutron diffraction) was sponsored by the U.S. Department of Energy, Office of Basic Energy Sciences, Scientific User Facilities Division under Award \# . Part of this research performed by Eugene P. Wigner Fellows at ORNL.
|
2,877,628,090,088 | arxiv | \section{Detail of Context Entropy Model}
\vspace{10pt}
The loss function Eqn.~(\ref{eqn:loss}) contains three terms. The second term is the entropy loss of the quantized map $\hat{\mathbf{Z}}$, which is defined as $L_{{\rm{ent}},g}=-\sum \log p(\hat{\mathbf{Z}}^{(g)}| \mathbf{\theta}_g)$ and optimized by the context entropy model's parameters $\mathbf{\theta}_g$. $\mathbf{\theta}_g$ represents the $g$-th 3D CNN based context model (please refer to \cite{conditional} for detail) and its illustration is showing in Fig.~\ref{fig:3dconv}. We also evalute the performance of the 3D context entropy model with different $k$. From Tab.~\ref{tab:dk}, we can find that stacking more residual layers (larger $k$) can reduce BPP.
\begin{figure}[h]
\centering
\begin{minipage}[c]{0.4\textwidth}
\scalebox{1}[1] {\includegraphics[width=0.99\textwidth]{./3dconv.pdf}}
\caption{Architecture of the context model copied from Mentzer's paper. ``3D k3 n64'' refers to a 3D masked convolution with filter size 3 and 64 output channels. The last layer outputs $Q_g$ values for each voxel in $\hat{\mathbf{Z}}^{(g)}$.}
\label{fig:3dconv}
\end{minipage}
\hspace{15pt}
\begin{minipage}[c]{0.52\textwidth}
\centering
\renewcommand\arraystretch{1.15}
\begin{tabular}{p{1.4cm}<{\centering}p{1.4cm}<{\centering}p{2.5cm}<{\centering}p{2.5cm}<{\centering}}
\toprule
$\mathbf{q}^{\top}$ & CI Type & MS-SSIM / BPP ($k=1$) & MS-SSIM / BPP ($k=3$)\\
\hline \hline
$[5]$ & None & 0.9651 / 0.2664 & 0.9651 / 0.2595\\
\rowcolor{mygray}
$[3, 5, 7]$ & SE-based & 0.9646 / 0.2608 & 0.9647 / 0.2587 \\
$[3, 5, 7]$& RE-based & 0.9652 / 0.2586 & \textbf{0.9653} / 0.2571 \\
\rowcolor{mygray}
$[3, 5, 7]$ & Predefine & \textbf{0.9653} / \textbf{0.2576} & 0.9652 / \textbf{0.2524}\\
\bottomrule
\end{tabular}
\captionof{table}[foo]{Performance of the 3D context entropy model with different $k$. Evaluation by MS-SSIM and BPP.}
\label{tab:dk}
\end{minipage}
\end{figure}
\section{Comparison on Other Datasets}
\vspace{10pt}
Furthermore, to assess performance on high-quality full-resolution images, we test our proposed methods on the datasets BSDS100 and Urban100, commonly used in the super-resolution task. The experiment results are shown in Fig.~\ref{fig:visual_more}. Our method outperforms BPG and JPEG2000, as well as the neural network-based approach of \cite{conditional} for all tested BPPs, i.e., from 0.1 BPP to 0.6 BPP.
\begin{figure*}[h]
\centering
\begin{multicols}{2}
\centering \includegraphics[width=0.45\textwidth]{./b100.pdf} \hspace{15pt}\\
\includegraphics[width=0.45\textwidth]{./u100.pdf}\\
\end{multicols}
\caption{Performance of our method on the BSDS100 dataset (left) and the Urban100 dataset (right), where we outperform Mentzer's, BPG and JPEG2000 for all tested BPPs, i.e., from 0.1 BPP to 0.6 BPP in MS-SSIM. \textit{Best viewed on-screen.}}
\label{fig:visual_more}
\end{figure*}
\clearpage
\section{PixelShuffle and Inverse PixelShuffle}
\vspace{10pt}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.7\linewidth]{./ps.pdf}
\end{center}
\caption{Illustration of PixelShuffle and Inverse PixelShuffle.}
\label{fig:ps}
\end{figure}
\vspace{-5pt}
\begin{equation}\label{eqn:ps}
\mathcal{PS}(\mathbf{X})_{c, h, w} = \mathbf{X}_{c+C \cdot {\rm{mod}}(h,d)+C \cdot {\rm{mod}}(w,d), \lfloor h/d \rfloor, \lfloor w/d \rfloor}.
\end{equation}
\vspace{3pt}
\begin{equation}\label{eqn:ips}
\mathcal{IPS}(\mathbf{X})_{c(di+j), h, w} = \mathbf{X}_{c, dh + i, dw + j}, \ 1\leq i,j \leq d.
\end{equation}
\vspace{3pt}
PixelShuffle~\cite{pixelshuffle}, i.e., Eqn.~(\ref{eqn:ps}), and Inverse PixelShuffle, i.e., Eqn.~(\ref{eqn:ips}), are very simple operations for down-sampling and up-sampling. We think these two operations are very suitable for deep image compression because they can vary the spatial resolution of the feature map and reconstruct the original input without information loss. (This may share some similar insight with residual learning in ResNet.) Moreover, comparing with down-sampling and up-sampling convolutions, there are more efficient both for computation and memory. In our experiments, we also found that inverse PixelShuffle can improve the stability of training while sometimes down-sampling convolution (stride $>$ 1) fails to converge. Last, PixelShuffle is widely used in the super-resolution task~\cite{rcan,srcliquenet}, which also indicates this simple operation is effective for resolution transform.
\vspace{10pt}
\section{Exploration of different $G$ and $\mathbf{r}$}
\vspace{10pt}
Here we conduct a comparison experiment. We design three variants by varying $G$, $\mathbf{r}$ and $\mathbf{q}$. All variants are trained for 200 epochs under the same setting and evaluated on the Kodak dataset. Tab.~\ref{table:vary_g} shows the detailed results. From it, the trend of BPPs almost follows the values of $\mathbf{r}^\top\log_2\mathbf{q}$, which is consistent with our analysis in Sec.~\ref{controller}. As $G$ becomes larger, BPP can reduce further with negligible loss of MS-SSIM. However, as $G$ becomes larger, the number of channels in each segment will decrease, which may influence the performance of the context
entropy model due to the relevant or dependent information
among the channels is reduced. Additionally, we found that increasing $G$ will make the training of the whole deep compression system unstable.
\begin{table*}[h]
\renewcommand\arraystretch{1.2}
\centering
\setlength{\tabcolsep}{0.3mm}{
\begin{tabular}{p{1.3cm}<{\centering}p{3cm}<{\centering}p{2.3cm}<{\centering}p{2.3cm}<{\centering}p{2.3cm}<{\centering}p{2.3cm}<{\centering}p{2.3cm}<{\centering}p{2.3cm}<{\centering}}
\toprule
$G$ & $\mathbf{r}^\top$ & $\mathbf{q}^\top$ & $\mathbf{r}^\top\log_2\mathbf{q}$ &PSNR & MS-SSI$\rm{M}_{dB}$ & BPP \\
\hline \hline
\rowcolor{mygray}
2 & [1/2, 1/2] & [4, 6] & 2.293 & 27.581 & 14.282 & 0.2570\\
3 & [1/3, 1/3, 1/3] & [3, 5, 7] & 2.238 & 27.422 & 14.272 &0.2518\\
\rowcolor{mygray}
4 & [1/4, 1/4, 1/4, 1/4] & [2, 4, 6, 8] & 2.146 & 27.383 & 14.152 &0.2431\\
\bottomrule
\end{tabular}}
\caption{Investigation of $G$, $\mathbf{r}$ and $\mathbf{q}$. The channels number of quantized feature map $C$ equals to $32$ for all experiments.}
\label{table:vary_g}
\end{table*}
\clearpage
\section{More Visualization Results}
In this section, we provide a few more visualization reconstructed results at a wider BPP range. All figures include the ground truth images (left) and the reconstructed images for BPG (middle) and our proposed method (right).
\begin{figure*}[h!]
\centering
\begin{multicols}{3}
\centering \includegraphics[width=0.32\textwidth]{./2.png}
\centering { MM-SSIM / MM-SSIM (dB) / BPP}\hspace{15pt} \\
\includegraphics[width=0.32\textwidth]{./2bpg0280.png}\\
\centering {BPG: 0.951 / 13.09 / 0.280} \hspace{15pt} \\
\includegraphics[width=0.32\textwidth]{./2my0262.png}\\
\centering {\textbf{Ours: 0.964 / 14.44 / 0.262}}
\end{multicols}
\begin{multicols}{3}
\centering \includegraphics[width=0.32\textwidth]{./4.png} \centering { MM-SSIM / MM-SSIM (dB) / BPP}\hspace{15pt} \\
\includegraphics[width=0.32\textwidth]{./4bpg0399.png}\\
\centering {BPG: 0.970 / 15.23 / 0.399} \hspace{15pt} \\
\includegraphics[width=0.32\textwidth]{./4my0387.png}\\
\centering { \textbf{Ours: 0.975 / 16.02 / 0.387}}
\end{multicols}
\begin{multicols}{3}
\centering \includegraphics[width=0.33\textwidth]{./13.png} \centering { MM-SSIM / MM-SSIM (dB) / BPP}\hspace{15pt} \\
\includegraphics[width=0.33\textwidth]{./13bpg0583.png}\\
\centering {BPG: 0.931 / 11.61 / 0.583} \hspace{15pt}\\
\includegraphics[width=0.33\textwidth]{./13my0534.png} \\
\centering {\textbf{Ours: 0.965 / 14.55 / 0.534}}
\end{multicols}
\caption{Some sample test results from the Kodak dataset (\textit{kodim2}, \textit{kodim4} and \textit{kodim13}). At similar bit rates, our combined method provides the highest visual quality. BPG shows more “classical” compression artifacts. \textit{Best viewed on-screen.}}
\label{fig:vis}
\end{figure*}
\section{Introduction}\label{intro}
Since the development of the internet, increasingly more high-definition digital media data has overwhelmed our daily life. Image compression refers to the task of representing images using as little storage as possible and is an essential topic in computer vision and image processing.
The typical image compression codecs, e.g., JPEG~\cite{jpeg} and JPEG 2000~\cite{jpeg2000}, generally use some transformations such as discrete cosine transform~(DCT) and discrete wavelet transform~(DWT), which are mathematically well-defined. These compression methods do not fully utilize the nature of data and may introduce visible artifacts such as ringing and blocking. In the last several years, deep learning has revolutionized versatile computer vision tasks~\cite{imagenet,srcnn,resnet}. Image compression based on deep learning, or deep image compression for brevity, has become a popular area of research, which can possibly explore the use of the nature of images beyond conventional compression methods.
A deep image compression system is similar to the conventional one, as it usually includes four components, i.e., encoder, quantizer, entropy model, and decoder, to form the final codec. To train a deep image compression system, a rate-distortion trade-off loss function: ${R} + \lambda {D}$ was proposed in \cite{end2end}, where $\lambda$ is the balanced hyper-parameter. The loss function includes two competing terms, i.e., ${R}$ measures the bitrate of the final compressed code, and ${D}$ measures the distortion between the original and reconstructed images. Recently, to improve the performance of the deep image compression system further, researchers proposed many novel and effective derivatives for the above four components.
\paragraph{En/Decoder.} The most popular architecture for en/decoder is based on convolutional neural network~(CNN). E.g., \cite{end2end} proposed a three convolutional layers en/decoder and generalized divisive normalization~(GDN) for activation. \cite{weighted} proposed a nine convolutional layers en/decoder with the residual block~(\cite{resnet}). Google Inc presented three variants (\cite{icrnn,icrnn2,icrnn3}) of a recurrent neural network~(RNN)-based en/decoder to compress progressive images and their residuals. \cite{icgan} proposed a generative adversarial network~(GAN)-based en/decoder for extremely low bitrate image compression, which achieved better user study results.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.92\textwidth]{./toy4x8.pdf} \\
\vspace{-5pt}
\begin{multicols}{2}
\centering
\includegraphics[width=0.45\textwidth]{./channel_psnr.pdf} \\
\includegraphics[width=0.45\textwidth]{./channel_ssim.pdf} \\
\end{multicols}
\vspace{-15pt}
\caption{Illustration of channel influences. Top left: The original example image (\textit{kodim15}) from the Kodak dataset. Top right: The visual results of the quantized feature map (channel by channel, 32 channels in total). Bottom left: PSNR loss of each channel. Bottom right: MS-SSIM loss of each channel in decibels: $-10\log_{10}(1-$MS-SSIM$)$. \textit{Best viewed on screen.}}
\vspace{-15pt}
\label{fig:channel}
\end{figure*}
\vspace{-2pt}
\paragraph{Quantizer.} In conventional codecs, the quantization operation is usually implemented by the round function. However, the gradient of the round function is almost always zero, which is highly unsuitable for deep compression. Thus, many differentiable quantization approaches were proposed by researchers. In \cite{icrnn}, Bernoulli distribution noise was added to implement the map function from continuous values to the fixed set $\{-1,+1\}$. The importance map was proposed in \cite{weighted} to address the spatial inconsistency coding length. Based on the K-means algorithm, the soft quantizer~\cite{conditional} was proposed for the multi-bits quantization case. \cite{end2end} proposed uniformed additive noise for infinite range quantization, whose quantization level is undetermined.
\vspace{-2pt}
\paragraph{Entropy model.} To further compress the quantized code by through entropy coding methods, e.g., Huffman coding and arithmetic coding, the entropy model was proposed to regularize the redundancy among the quantization code. Almost all entropy arithmetic coding models are motivated by the context-based adaptive binary arithmetic coding~(CABAC) framework. Specifically, in \cite{icrnn2}, they used PixelRNN~\cite{pixelrnn} and long short term memory~(LSTM) architectures for entropy estimation. In \cite{conditional}, they utilized a 3D convolutional model to generate the conditional probability of the quantized code. In \cite{vae_entropy,minnen2018joint,lee2018context}, they proposed a variational autoencoder~(VAE) with a scale hyperprior to learn the context distribution, which achieves consequently achieving better compression results.
\vspace{-5pt}
\section{Channel Influences}\label{channel_influence}
\vspace{2pt}
All previous deep image compression systems view all channels as a unified whole and ignore the channel-level influences. However, useful information is unevenly distributed across channels. Channel redundancy and uneven distribution have been widely studied in the field of network compression~\cite{thinet,slim,he2017channel}. In this study, we utilize a toy example model to illustrate its feasibility in deep image compression. We use a simple encoder and quantizer to extract features and quantize them. The final quantized feature map has $32$ channels. We allocate one bit for quantization, i.e., its quantization level is two. Evaluating on the Kodak dataset, this toy model yields an average MS-SSIM~\cite{msssim} of 0.922 at an average rate of 0.112 bits per pixel~(BPP). In the top part of Fig.~\ref{fig:channel}, we present the visual results of the quantized feature map (channel by channel, 32 channels) by using \textit{kodim15} from Kodak. The visual results indicate that Channel-8, 23, and 26 have similar content and profile (similar to low-frequency information) with the original image. By contrast, some visualizations, e.g., Channel-9, 10, and 28 appear disorganized and could not be recognized (similar to high-frequency information). We also make quantitative comparisons. We conduct 32 experiments. In each experiment, we cut one relative channel (set its values to 0) of the quantized feature map to observe the influence of each channel on the final reconstruction results. The bottom of Fig.~\ref{fig:channel} depicts the PSNR loss of each channel on the left and the MS-SSIM loss of each channel on the right. Consistent with the analysis of visual results, Channel-8, 23, and 26 are significant for reconstruction, whereas Channel-9, 10, and 28 are negligible. Moreover, this phenomenon appears on all images of the dataset. Thus, the problem is as follows: Can we design a variable deep image compression system to ensure the allocation of more bits for important channels and the reduction of bitrate for negligible channels? In this paper, we propose a novel network to solve this issue.
\vspace{2pt}
The overall contributions of this study are three-fold:
\vspace{-2pt}
\begin{itemize}
\item We analyze the channel influences in deep image compression. We propose a novel variable channel controller to effectively utilize channel diversity. To the best of our knowledge, we are the first to perform image compression in a channel-level manner.
\vspace{-3pt}
\item We propose a novel quantizer based on Gaussian mixture model~(GMM). This novel quantizer has powerful representation and is a more generalized pattern for the existing finite quantizers.
\vspace{-3pt}
\item Extensive quantitative and qualitative experiments show that our method achieves superior performance over the state-of-the-art methods without a hyperprior VAE.
\end{itemize}
\vspace{-10pt}
\section{Approach}
\vspace{2pt}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=1\linewidth]{./frame1.pdf}
\end{center}
\vspace{-8pt}
\caption{Framework of the channel-level variable quantization network. The entire encoder and decoder bodies both contain four stages. $C=8$, $G=3$, and $\mathbf{r}=[25\%, 50\%, 25\%]^\top$ are selected for the illustration of variable quantization controller. \textit{Best viewed on screen.}}
\label{fig:frame}
\vspace{-12pt}
\end{figure*}
\vspace{-2pt}
The framework of the proposed system is shown in Fig.~\ref{fig:frame}. In this section, we first introduce the channel attention residual network for encoding and decoding. Then, we present a novel quantizer based on GMM. Finally, we illustrate the details of the variable quantization level controller, which makes the entire system able to dynamically alter the quantization levels for each channel.
\vspace{-2pt}
\subsection{Channel Attention Residual En/Decoder}\label{encoder}
Our channel attention residual encoder comprises three parts: head, body, and tail. The head module contains one convolutional layer, which transforms the original image into feature map $\mathbf{X}^{(0)}$ with $C_0$ channels. The body of the encoder is shown in left part of Fig.~\ref{fig:frame}. The entire body includes four stages. In each stage, the output feature map is only half the resolution $(h, w)$ of the input feature map. We denote the input feature map at Stage-$t$ as $\mathbf{X}^{(t)}\in \mathbb{R}^{C_t \times H \times W}$. Motivated by the super-resolution task's method~\cite{pixelshuffle}, we use inverse PixelShuffle to implement the down-sampling operation. It can be expressed as:
\begin{equation}\label{ips}
\mathcal{IPS}(\mathbf{X}^{(t)})_{c(di+j), h, w} = \mathbf{X}^{(t)}_{c, dh + i, dw + j}, \ 1\leq i,j \leq d,
\end{equation}
where $d$ is the down-sampling factor. It is a periodic shuffling operator that rearranges the elements of a $C_t \times H \times W$ tensor to a tensor of shape $d^2C_t \times \frac{1}{d}H \times \frac{1}{d}W$. Notably, this operator preserves all information of the input because the number of elements does not vary. We also found that inverse PixelShuffle can improve the stability of training and reduce the memory costs relative to the down-sampling convolution.
Previous CNN-based image compression methods treat channel-wise features equally, which is not flexible for real cases. To make the network focus on more informative features, we follow \cite{senet,rcan,srcliquenet} and exploit the inter-dependencies among feature channels. We send the feature map to the residual group module, shown in left part of Fig.~\ref{fig:frame}. The residual group consists of $B$ residual channel attention blocks, which are used to extract the inter-dependencies among feature channels and distill the feature map. The residual group does not change the number of channels.
Finally, we add a convolutional layer to alter the number of channels from $C_t$ to $C_{t+1}$ for the next stage. Thus the output of Stage-$t$ is $\mathbf{X}^{(t+1)}\in \mathbb{R}^{C_{t+1} \times \frac{1}{d}H \times \frac{1}{d}W}$, which is also the input of the next stage.
After four stages of processing in the body, a convolutional layer, appended as the tail part, generates the compressed (latent) representation $\mathbf{Z}$ with $C$ channels, where $C$ can be varied manually for different BPPs. Similarly, the architecture of the decoder is simply the inverse version of the encoder. As shown in left part of Fig.~\ref{fig:frame}, we replace inverse PixelShuffle with PixelShuffle for the up-sampling operation.
\subsection{GMM Quantizer}\label{gmm}
For the quantizer, we propose a novel quantization method based on GMM. Concretely, we model the prior distribution $p(\mathbf{Z})$ as a mixture of Gaussian distributions:
\vspace{-2pt}
\begin{equation}
p(\mathbf{Z}) = \prod_{i}\sum_{q=1}^{Q}\pi_q\mathcal{N}(z_i|\mu_q, \sigma_q^2),
\end{equation}
where $\pi_q$, $\mu_q$, and $\sigma_q$ are the learnable mixture parameters and $Q$ is the quantization level.
We obtain the forward quantization result by setting it to the mean that takes the largest responsibility:
\vspace{-1pt}
\begin{equation}
\hat{z}_i \leftarrow \mathop{\arg\max}_{\mu_j}\frac{\pi_j\mathcal{N}(z_i|\mu_j, \sigma_j^2)}{\sum_q^Q\pi_q\mathcal{N}(z_i|\mu_q, \sigma_q^2)}.
\label{eqn:forward}
\end{equation}
\vspace{-1pt}
Obviously, Eqn. (\ref{eqn:forward}) is non-differentiable. We relax $\hat{z}_i$ to $\tilde{z}_i$ to compute its gradients during the backward pass by:
\vspace{-1pt}
\begin{equation}
\tilde{z}_i = \sum_{j=1}^Q\frac{\pi_j\mathcal{N}(z_i|\mu_j, \sigma_j^2)}{\sum_q^Q\pi_q\mathcal{N}(z_i|\mu_q, \sigma_q^2)}\mu_j.
\end{equation}
\vspace{-1pt}
Unlike the conventional GMM, which optimizes $\pi_q$, $\mu_q$, and $\sigma_q$ by using the expectation maximization~(EM) algorithm, we learn the mixture parameters by minimizing the negative likelihood loss function through the network back-propagation. We denote the prior distribution loss function of GMM quantizer as:
\vspace{-5pt}
\begin{equation}
L_{\rm{GMM}} \!=\! - {\rm{log}}\left(p(\mathbf{Z})\right) \!= \! - \sum_{i}{\rm{log}}\sum_{q=1}^{Q}\pi_q\mathcal{N}(z_i|\mu_q, \sigma_q^2).
\end{equation}
\vspace{-5pt}
\iffalse
\begin{figure*}[t]
\begin{center}
\includegraphics[width=1\linewidth]{./frame1.pdf}
\end{center}
\vspace{-10pt}
\caption{Illustration of the variable quantization controller with $C=8$, $G=3$, and $\mathbf{r}=[25\%, 50\%, 25\%]^\top$.}
\vspace{-10pt}
\label{fig:controller}
\end{figure*}
\fi
Here, we would like to make a comparison between the GMM quantizer and the soft quantizer~\cite{soft}. The soft quantizer can be viewed as a differentiable version of the K-means algorithm. If the mixture parameters satisfy: $\pi_1=\pi_2=\dots=\pi_Q=1/Q$ and $\sigma_1=\sigma_2=\dots=\sigma_Q=\sqrt{2}/2$, the GMM quantizer will degenerate to the soft quantizer, which implies that the GMM quantizer has a more powerful representation and is a more generalized model.
\vspace{-1pt}
\subsection{Variable Quantization Controller}\label{controller}
As mentioned in Sec.~\ref{channel_influence}, each channel of the quantized feature map may have a different impact on the final reconstruction results. To allocate appropriate bitrates for different channels, we propose the variable quantization controller model.
The illustration of the variable quantization controller is shown in the right part of Fig.~\ref{fig:frame}. In the variable quantization controller, there are two key components: channel importance module and splitting-merging module. In the following, we will introduce the mechanism of these two modules in detail.
\vspace{-1pt}
\subsubsection{Channel Importance Module}\label{cim}
\vspace{2pt}
The input of the channel importance module is $\mathbf{Z}$, which is the output of the encoder mentioned in Sec.~\ref{encoder}. Let us denote the channel number of $\mathbf{Z}$ as $C$ ($C = 8$ in Fig.~\ref{fig:frame}). We expect the channel importance module to generate a channel importance vector $\mathbf{w} \in \mathbb{R}^C_+$. Each element $\mathbf{w}_c$ represents the reconstruction importance of Channel-$c$.
Here, we design three types of channel importance module:
\paragraph{Sequeze and excitation block-based.} We utilize average pooling and two convolutional layers to operate $\mathbf{Z}$ (refer~\cite{senet}) and get an $M \times C$ matrix, where $M$ is the mini-batch size. We generate a learnable channel importance vector $\mathbf{w}$ by using the mean operation on the matrix by reducing the first dimension ($M$).
\paragraph{Reconstruction error-based.} We perform three steps to implement it: First, we construct a validation dataset by randomly selecting $N$ images from the training dataset. Second, we prune the $c$-th channel of the $n$-th image's feature map $\mathbf{Z}_{n,c}$: $\mathbf{Z}_n(c,:,:)=0$. Last, we represent $\mathbf{w}_c $ by calculating the average MS-SSIM reconstruction error of each channel over the validation dataset:
\vspace{-2pt}
\begin{equation}
\mathbf{w}_c = \frac{1}{N}\sum_{n=1}^{N}d_{\rm MS\textendash SSIM} \left(\mathbf{I}_n,{\rm{Dec}} \left({\rm{Qua}}\left(\mathbf{Z}_{n,c}\right)\right)\right),
\end{equation}
\vspace{-2pt}
\noindent where $\mathbf{I}_n$ is the $n$-th image of the validation dataset, ${\rm{Dec}}$ and ${\rm{Qua}}$ are represent the decoder and quantizer, respectively.
\vspace{2pt}
\paragraph{Predefined.} We directly predefine the channel importance vector $\mathbf{w}$ as $\mathbf{w}_c = c$, which is fixed during the training and evaluation process.
\subsubsection{Splitting-Merging Module}
\vspace{2pt}
At the beginning of the splitting-merging module, we sort the feature map $\mathbf{Z}$ in ascending order according to the channel importance vector $\mathbf{w}$. Because the new feature map is well organized, we split it to $G$ groups ($G = 3$ in Fig.~\ref{fig:frame}). The $G$ portions of the feature map are quantized and encoded using different quantization levels in different groups.
After the splitting operation, the $C$ channels are divided into $G$ groups. We denote the ratio vector of $G$ groups as $\mathbf{r}$, which satisfies: $\sum_{g}^G\mathbf{r}_g=1$, and $\forall g, \mathbf{r}_g>0$. Here, we use the right part of Fig.~\ref{fig:frame} to explain its mechanism. Suppose that the parameters $C=8$, $G=3$, and $\mathbf{r}=[25\%, 50\%, 25\%]^\top$, Channel-1 and 2 will be assigned Group-1 for quantization and encoding, Channel-3, 4, 5, and 6 will be assigned Group-2 and Channel-7 and 8 will be assigned Group-3. On the other hand, because the channel importances of Channel-1 and 2 are smaller than the others, we use smaller quantization level $\mathbf{q}_1$ for quantizing and encoding. Similarly, we apply a larger quantization level $\mathbf{q}_3$ to quantize and encode Channel-7 and 8. At the last step, we merge $G$ groups and reorder the channel dimension to construct the final compressed result.
\begin{figure*}[t!]
\centering
\begin{multicols}{2}
\centering \includegraphics[width=0.54\textwidth]{./visual_channel32.pdf} \\\hspace{10pt}
\includegraphics[width=0.44\textwidth]{./compare4.pdf}\\
\end{multicols}
\vspace{-18pt}
\caption{Left: Visualization results (\textit{kodim15}) of the predefined model's quantized feature map, which contains three quantization levels: 3, 5, and 7. Right: Comparisons of the rate-distortion curves on Kodak. MS-SSIM values are converted into decibels. \textit{Best viewed on screen.}}
\label{fig:visual_compare}
\vspace{-15pt}
\end{figure*}
\vspace{-1pt}
\subsubsection{Analysis}
\vspace{2pt}
Here, we conduct an analysis, examining under what condition the variable quantization controller can theoretically guarantee a better compression rate than that of the original one-group model. We suppose that the feature map $\mathbf{Z}$ is a $C \times H \times W$ tensor and the channel number of Group-$g$ is $C_g=C\mathbf{r}_g$. Obviously, it satisfies $\sum_{g}C_g=C$. $\hat{\mathbf{Z}}$ has the same dimensions as $\mathbf{Z}$ because $\hat{\mathbf{Z}}$ is simply the quantized version of $\mathbf{Z}$. Because the number of dimensions ${\dim(\hat{\mathbf{Z}})}$ and the quantization level $Q$ are finite, the entropy is bounded by $H(\hat{\mathbf{Z}}) \leq \dim(\hat{\mathbf{Z}}){\rm{log_2}}(Q)= CHW{\rm{log_2}}(Q)$ (refer, e.g., \cite{cover2012elements}). Contrastingly, for $G$ groups, suppose that the quantization level vector is $\mathbf{q}=[\mathbf{q}_1, \mathbf{q}_2, ..., \mathbf{q}_G]^\top$, then, the entropy upper-bound of $\{\hat{\mathbf{Z}}_g\}$ is:
\vspace{-5pt}
\begin{equation}
H(\{\hat{\mathbf{Z}}_g\})=\sum_{g=1}^{G}H(\hat{\mathbf{Z}}_g)\leq HWC\sum_{g=1}^{G}\mathbf{r}_g{\rm{log_2}}(\mathbf{q}_g).
\end{equation}
\vspace{-5pt}
Thus, if the $G$ groups satisfy $\mathbf{r}^\top{\rm{log_2}}(\mathbf{q}) < {\rm{log_2}}(Q)$, the variable quantization controller will provide a lower entropy upper-bound than the conventional one-group model. On the other hand, although $\hat{\mathbf{Z}}$ has the same total number of elements as $\{\hat{\mathbf{Z}}_g\}$, $\hat{\mathbf{Z}}$ has only $Q$ values to pick up, whereas $\{\hat{\mathbf{Z}}_g\}$ has $\sum_{g}\mathbf{q}_g$ values, indicating that $\{\hat{\mathbf{Z}}_g\}$ may have better diversity.
Overall, in the variable quantization controller, we choose the GMM quantizer (in Sec.~\ref{gmm}) and the 3D CNN-based context model (refer \cite{conditional}) for quantization, and entropy estimating, respectively. All quantized feature maps $\{\hat{\mathbf{Z}}_k\}$ will concatenate together and be sent to the decoder. The final loss function of the entire system becomes:
\vspace{-10pt}
\begin{equation}\label{eqn:loss}
L =\alpha L_{\rm{dis}} + \frac{1}{G}\sum_{g=1}^{G}L_{{\rm{ent}},g} + \beta \frac{1}{G}\sum_{g=1}^{G}L_{{\rm{GMM}},g}.
\end{equation}
\vspace{5pt}
\section{Experiments}
\vspace{2pt}
\subsection{Implementation and Training Details}
\vspace{2pt}
\paragraph{Datasets.} We merge three common datasets, namely DIK2K~\cite{div2k}, Flickr2K~\cite{flickr}, and CLIC2018, to form our training dataset, which contains approximately 4,000 images in total. Following many deep image compression methods, we evaluate our models on the Kodak dataset with the metrics MS-SSIM for lossy image compression.
\renewcommand\arraystretch{1.1}
\begin{table}[t]
\centering
\vspace{5pt}
\begin{tabular}{p{1.4cm}<{\centering}p{1.4cm}<{\centering}p{1.45cm}<{\centering}p{2.55cm}<{\centering}}
\hline
$\mathbf{q}$ & CI Type & MS-SSIM & BPP \\
\hline \hline
$[5]$ & None & 0.9651 & 0.2664 \\
\rowcolor{mygray}
$[3, 5, 7]^\top$ & SE-based & 0.9646 & 0.2608 ($\downarrow$ 2.11\%) \\
$[3, 5, 7]^\top$& RE-based & 0.9652 & 0.2586 ($\downarrow$ 2.93\%) \\
\rowcolor{mygray}
$[3, 5, 7]^\top$ & Predefine & \textbf{0.9653} & \textbf{0.2576 ($\downarrow$ 3.31\%)} \\
\hline
\end{tabular}
\vspace{-5pt}
\caption{Investigation of channel importance module. We run it three times and show the average results. CI Type denotes the type of channel importance module mentioned in Sec.\ref{cim}. }
\vspace{-10pt}
\label{table:variable}
\end{table}
\vspace{-1pt}
\paragraph{Parameter setting.} In our experiments, we use the Adam optimizer~\cite{adam} with a mini-batch size $M$ of 32 to train our five models on $256\times256$ image patches. We vary the quantized feature map $\hat{\mathbf{Z}}$'s channel number $C$ from 16 to 80 to obtain different BPPs. The total number of training epochs equals to $400$. The initialized learning rates are set to $1 \times 10^{-4}, 1 \times 10^{-4}, 5 \times 10^{-5}$ and $1 \times 10^{-4}$ for the encoder, quantizer, entropy model, and decoder, respectively. We reduce them twice (at Epoch-200 and Epoch-300) by a factor of five during training. In the channel attention residual en/decoder, we set the number of residual channel attention blocks $B=6$ for all stages. The channel numbers for each stage in the encoder are 32, 64, 128, and 192, respectively, whereas those for each stage in the decoder are 192, 128, 64, and 32, respectively. In the variable quantization controller, we set the number of groups $G=3$. The ratio vector $\mathbf{r}=[25\%, 50\%, 25\%]^\top$. For loss function Eqn.~(\ref{eqn:loss}), we choose negative MS-SSIM for the distortion loss $L_{\rm{dis}}$ and $\alpha=128$; we select cross entropy for the entropy estimation loss $L_{\rm{ent}}$ and $\beta=0.001$.
\vspace{-2pt}
\subsection{Ablation Study}
\vspace{1pt}
\begin{table}[t]
\renewcommand\arraystretch{1.1}
\centering
\setlength{\tabcolsep}{0.3mm}{
\vspace{5pt}
\begin{tabular}{p{1.8cm}<{\centering}p{1.8cm}<{\centering}p{1.8cm}<{\centering}p{2.5cm}<{\centering}}
\hline
$\mathbf{q}$ & PSNR & MS-SSIM & BPP \\
\hline \hline
$[5]$ & 27.926 & 0.9651 & 0.2664 \\
\rowcolor{mygray}
$[4, 5, 6]^\top$ & 28.012 & 0.9652 & 0.2639($\downarrow$ 0.94\%) \\
$[3, 5, 7]^\top$ & \textbf{28.024} &\textbf{0.9653}& 0.2576($\downarrow$ 3.31\%) \\
\rowcolor{mygray}
$[2, 5, 8]^\top$ & 27.982& 0.9644 & \textbf{0.2471($\downarrow$ 7.24\%)} \\
\hline
\end{tabular}}
\vspace{-4pt}
\caption{Investigation of the combination in $\mathbf{q}$. We run it three times and show the average results.}
\vspace{-13pt}
\label{table:combination}
\end{table}
\begin{figure*}
\centering
\subfigure{
\begin{minipage}[b]{0.185\linewidth}
\centering {\small \bf Org.}\vspace{1pt}
\includegraphics[width=1\linewidth]{./1_org.png}\vspace{2pt}
\includegraphics[width=1\linewidth]{./1_part_org.png}\vspace{-1pt}
\centering {\small MS-SSIM / BPP}\vspace{4pt}
\includegraphics[width=1\linewidth]{./21_org.png}\vspace{2pt}
\includegraphics[width=1\linewidth]{./21_part_org.png}\vspace{-1pt}
\centering {\small MS-SSIM / BPP}
\end{minipage}}
\subfigure{
\begin{minipage}[b]{0.185\linewidth}
\centering {\small \bf WebP}\vspace{1pt}
\includegraphics[width=1\linewidth]{./1_webp.png}\vspace{2pt}
\includegraphics[width=1\linewidth]{./1_part_webp.png}\vspace{-1pt}
\centering {\small 0.903 / 0.250}\vspace{4pt}
\includegraphics[width=1\linewidth]{./21_webp.png}\vspace{2pt}
\includegraphics[width=1\linewidth]{./21_part_webp.png}\vspace{-1pt}
\centering {\small 0.918 / 0.160}
\end{minipage}}
\subfigure{
\begin{minipage}[b]{0.185\linewidth}
\centering {\small \bf BPG}\vspace{1pt}
\includegraphics[width=1\linewidth]{./1_bpg.png}\vspace{2pt}
\includegraphics[width=1\linewidth]{./1_part_bpg.png}\vspace{-1pt}
\centering {\small 0.927 / 0.246}\vspace{4pt}
\includegraphics[width=1\linewidth]{./21_bpg.png}\vspace{2pt}
\includegraphics[width=1\linewidth]{./21_part_bpg.png}\vspace{-1pt}
\centering {\small 0.931 / 0.137}
\end{minipage}}
\subfigure{
\begin{minipage}[b]{0.185\linewidth}
\centering {\small \bf Mentzer et al.}\vspace{1pt}
\includegraphics[width=1\linewidth]{./1_men.png}\vspace{2pt}
\includegraphics[width=1\linewidth]{./1_part_men.png}\vspace{-1pt}
\centering {\small 0.940 / 0.239}\vspace{4pt}
\includegraphics[width=1\linewidth]{./21_men.png}\vspace{2pt}
\includegraphics[width=1\linewidth]{./21_part_men.png}\vspace{-1pt}
\centering {\small 0.933 / 0.124}
\end{minipage}}
\subfigure{
\begin{minipage}[b]{0.185\linewidth}
\centering {\small \bf Ours}\vspace{1pt}
\includegraphics[width=1\linewidth]{./1_our.png}\vspace{2pt}
\includegraphics[width=1\linewidth]{./1_part_our.png}\vspace{-1pt}
\centering {\small 0.952 / 0.242}\vspace{4pt}
\includegraphics[width=1\linewidth]{./21_our.png}\vspace{2pt}
\includegraphics[width=1\linewidth]{./21_part_our.png}\vspace{-1pt}
\centering {\small 0.943 / 0.125}
\end{minipage}}
\vspace{-10pt}
\caption{Visual comparisons on example images (top: \textit{kodim1}, bottom: \textit{kodim21}) from the Kodak dataset. From left to right: the original images, WebP, BPG, Mentzer's, and ours. Our model achieves the best visual quality, demonstrating the superiority of our model in preserving both sharp edges and detailed textures. \textit{Best viewed on screen.}}
\vspace{-16pt}
\label{fig:examples}
\end{figure*}
\subsubsection{Investigation of Channel Importance Module}
To demonstrate the effectiveness of the variable quantization mechanism and the channel importance module, we design several comparative experiments to evaluate the reconstruction performance. The baseline model generated a quantized feature map with channel number $C=32$. The quantization level vector $\mathbf{q}=[5]$ indicates that there are no splitting and merging operations. Thus, this model just contains one group. By contrast, with the same setting $\mathbf{q}=[3, 5, 7]^\top$ and $\mathbf{r}$, we use three different types of the channel importance module mentioned in Sec.~\ref{cim}, i.e., Sequeze and excitation block~(SE)-based, reconstruction error~(RE)-based, and predefined. We train these four variants for 400 epochs under the same training setting. We run all experiments three times and record the best MS-SSIM on Kodak. The details of the average results are listed in Tab.~\ref{table:variable}. We observe that the channel importance module and the splitting-merging module make the system more effective (smaller BPP) and powerful (better MS-SSIM). Additionally, the predefined channel importance module distinctly outperforms SE and RE-based modules, even SE and RE-based modules are learnable and data-dependent. This may be consistent with the network pruning research~\cite{rethinkpruning}: training predefined target models from scratch can have better performance than pruning algorithms under some conditions. We also visualize the quantized feature map of the predefined model in Fig.~\ref{fig:visual_compare}. Comparing it with Fig.~\ref{fig:channel} (top right), we can see that the channels containing much more profile and context information of the original image are allocated more bits in the new system.
\vspace{-1pt}
\subsubsection{Investigation of the Combination in $\mathbf{q}$}
As mentioned in Sec.~\ref{controller}, if the $G$ groups satisfy $\mathbf{r}^\top{\rm{log_2}}(\mathbf{q}) < {\rm{log_2}}(Q)$, the variable quantization controller will provide a lower theoretical entropy upper-bound. Here, we explore what combination may have better performance. The baseline model only has one quantization level, i.e., $\mathbf{q} = [5]$. We extend it to three types of combinations: $\mathbf{q} = [4, 5, 6]^\top$, $\mathbf{q} = [3, 5, 7]^\top$, and $\mathbf{q} = [2, 5, 8]^\top$. The ratio vectors of the three types of models are the same and equal to $[25\%, 50\%, 25\%]^\top$. Quantitatively, $\log_2(2)+\log_2(8) < \log_2(3)+\log_2(7)<\log_2(4)+\log_2(6) < 2\log_2(5)$, and the experimental results are consistent with the theoretical analysis. Additionally, we find that the odd quantized level may have better performances. Because the odd quantized level more likely contains a quantized value close to 0. This may meet the similar results in research related to network quantization~\cite{zhu2016trained}. If the quantization levels in $\mathbf{q}$ are too different, e.g., $[2, 5, 8]^\top$, the performance will degrade.
\vspace{-2pt}
\subsection{Comparisons}
\vspace{1pt}
In this subsection, we compare the proposed method against three conventional compression techniques, JPEG2000, WebP, and BPG (4:4:4), as well as recent deep learning-based compression work by \cite{icrnn3}, \cite{icml2017}, \cite{weighted}, and \cite{conditional}. We use the best performing configuration we can find of JPEG 2000, WebP, and BPG. Trading off between the distortion and the compression rate, $\mathbf{q}$ is set to $[3,5,7]^\top$ in the following experiments.
\vspace{-1pt}
\subsubsection{Quantitative Evaluation}
\vspace{2pt}
Following \cite{icml2017,conditional}, and because MS-SSIM is more consistent with human visual perception than PSNR, we use MS-SSIM as the performance metric. Fig.~\ref{fig:visual_compare} depicts the rate-distortion curves of these eight methods. Our method outperforms conventional compression techniques JPEG2000, WebP and BPG, as well as the deep learning-based approaches of \cite{icrnn2}, \cite{weighted}, \cite{conditional}, and \cite{icml2017}. This superiority of the proposed method holds for almost all tested BPPs, i.e., from 0.1 BPP to 0.6 BPP. It should be noted that both \cite{weighted} and \cite{conditional} are trained on the Large Scale Visual Recognition Challenge 2012 (ILSVRC2012)~\cite{imagenet2012}, which contains more than one million images. \cite{icml2017} trained their models on the Yahoo Flickr Creative Commons 100 Million dataset~\cite{thomee2016yfcc100m}, which includes approximately 100 million images. While our models are trained using only 4,000 images.
\vspace{-1pt}
\subsubsection{Visual Quality Evaluation}
Owing to the lack of reconstruction results for many deep image compression algorithms and the space limitations of the paper, we present only two reconstruction results of images and compare them with WebP, BPG, and \cite{conditional}. In the first row of Fig.~\ref{fig:examples}, our method accurately reconstructs more clear and textural details of objects, e.g., door and the stripes on the wall. Other results have blocking artifacts more or less. For the second reconstruction results, our method can obtain better visual quality on images of objects such as clouds and water waves. Notably, our method is the only one that succeeds in reconstructing the spire of a lighthouse. Furthermore, the MS-SSIM measurements are also better than other methods in similar BBP ranges.
\vspace{-6pt}
\section{Conclusion}
\vspace{1pt}
In this paper, we propose, to the best of our knowledge, the first channel-level method for deep image compression. Moreover, based on the channel importance module and the splitting-merging module, the entire system can variably allocate different bitrates to different channels, which can further improve the compression rates and performances. Additionally, we formulate the quantizer into a GMM manner, which is a universal pattern for the existing finite range quantizers. Ablation studies validate the effectiveness of the proposed modules. Extensive quantitative and qualitative experiments clearly demonstrate that our method achieves superior performance and generates better visual reconstructed results than the state-of-the-art methods without a hyperprior VAE.
\vspace{-5pt}
\section*{Acknowledgments}
This work was partially supported by JST CREST JPMJ-CR1686, Japan.
\clearpage
\small
\bibliographystyle{named}
|
2,877,628,090,089 | arxiv | \section{Introduction}
\label{sec:intro}
In the heavy ion collision (HIC), a non-vanishing chiral charge $N_5$ may be induced through the Adler-Bell-Jackiw anomaly \cite{Adler:1969gk,Bell:1969ts,Smilga:1991xa} due to topologically non-trivial gluon configurations \cite{McLerran:1990de,Shuryak:2002qz}
\begin{align}
N_5=N_R-N_L=-\frac{g^2 N_F}{32 \pi^2}\int d^4x \epsilon _{\mu \nu \lambda \sigma} F_a^{\mu \nu} F_a^{\lambda \sigma}.
\end{align}
The $N_{R,L}$ denotes the net number of quarks (minus antiquarks) with right- or left-handed chirality, so $N_5$ is the net number of right handed quark over left handed ones. Non-vanishing $N_5$ in a strong magnetic field could result in the chiral magnetic effect (CME) \cite{Fukushima:2008xe,Kharzeev:2007jp,Copinger:2018ftr}, i.e., an electric current can be induced along the direction of the magnetic field. Consequently, there will be a charge separation within the produced fireball. Peripheral HICs provide a good testing ground for the CME as an extremely strong magnetic field $e B$ between several $m_\pi^2$ to $15\ m_\pi^2$ \cite{Skokov:2009qp, Voronyuk:2011jd, Bzdak:2011yy, Deng:2012pc,Cheng:2019qsn,Xu:2020sui} is generated by the colliding ions, in particular the spectator protons. Confirmation of the CME would reflect the local parity and charge-parity violation in quantum chromodynamics (QCD), hence is of great interest. Experimental searches have thus been actively ongoing \cite{Adamczyk:2013hsi,Abelev:2012pa,Adamczyk:2014mzf,Skokov:2016yrj}.
To facilitate the study involving the chiral imbalance, a chiral chemical potential $\mu_5$ is introduced as conjugate to $N_5$. The associated term $\mu_5\bar{\psi}\gamma_4\gamma_5\psi$ is then added to the Lagrangian density \cite{Fukushima:2008xe}. Technically the $N_5$ is not conserved in QCD, so the $\mu_5$ serves to mimic the chiral imbalance. It gains support from arguments that the chiral charge density $n_5$ equilibrates shortly after the collision and stay unchanged in a thermodynamical equilibrium in a longer period \cite{Ruggieri:2016asg,Ruggieri:2016lrn}. Consequently, the phase diagram of the QCD matter gets extended to a new dimension $\mu_5$, in addition to the temperature $T$ and quark number chemical potential $\mu$. However, there has been notable contradictions among different calculations, in particular concerning the chiral transition at finite temperature. The debate is over whether the pseudo-critical temperature $T_c$ (defined as the maxima of susceptibilities at finite $T$ with $\mu=0$) increases with increasing $\mu_5$, or the opposite. The DSEs \cite{Wang:2015tia,Xu:2015vna} and lattice QCD \cite{Braguta:2015zta,Braguta:2015owi} have been giving consistent predictions that $T_c$ increases with $\mu_5$, while for NJL model the results differ by regularization schemes \cite{Ruggieri:2011xc,Yu:2015hym,Farias:2016let,Cui:2016zqp,Khunjua:2018jmn,Yang:2019lyn}. This problem is also connected with the determination of the CEP. Early model studies \cite{Chernodub:2011fr, Ruggieri:2011xc} suggest that the chiral crossover would turn into a first order phase transition at large $\mu_5$, which could be an indirect signal for the existence of CEP. However, lattice simulation finds no signal of phase transition as $\mu_5$ increases \cite{Yamamoto:2011gk, Braguta:2015zta,Braguta:2015owi}, and the DSEs found the crossover behavior persists \cite{Wang:2015tia,Xu:2015vna}. In those work, the separable model \cite{Burden:1996nh} and the Maris-Tandy (MT) model \cite{Maris:1997tm,Maris:1999nt} were employed for gluon propagator. However, the former model is oversimplified for purpose of computation, and the later's infrared momentum behavior contradicts today's gauge sector study \cite{Aguilar:2010gm, Boucaud:2010gr,Oliveira:2010xc,Bowman:2004jm}. So in this work we will check the consistency within DSEs by supplementing a calculation based on a more realistic gluon model, the so called Qin-Chang (QC) model. It has the correct infrared momentum behavior and also had been used extensively in hadron studies and finite temperature QCD \cite{Qin:2011dd}. However, it has never been employed in the study of finite $\mu_5$ before. Model details will be given in later sections.
The fully dressed quark propagator encodes abundant information of the QCD matter's thermodynamical properties. Among them the finite chiral charge density $n_5$ is a novel feature of the chirally imbalanced matter. It is indispensable for the CME. Relating $n_5$ and $\mu_5$ is useful for expressing the induced electric current density as a function of the chirality density \cite{Fukushima:2010fe}. We therefore focus $n_5$ in various conditions, e.g., $T$, $\mu$, $\mu_5$ and also the system size. Note that the chemical potential $\mu$ is directly computable in the DSEs, so it doesn't pose challenge to DSE as for lattice QCD. Combined effort with DSE and lattice QCD had been carried out to locate the CEP \cite{Fischer:2010fx,Fischer:2011mz}. Meanwhile, the finite size effect could be relevant for the CME experiments since the CME is typically investigated with peripheral collisions. Note that the finite size effect on QCD phase diagram on $T-\mu$ plane had been studied with various methods \cite{Bhattacharyya:2014uxa,Tripolt:2013zfa,Braun:2011iz, Shi:2018swj,Shi:2018tsq,Abreu:2019czp,Ya-Peng:2018gkz}, but its influence on the chirally imbalanced medium was seldom studied. For a first investigation, we will consider the system in a cubic box of edge length $L$.
This paper is organized as follows. In section \ref{sec:gap} we introduce the quark's DSE at finite $T$, $\mu$ and $\mu_5$ and its solution. In section \ref{sec:chiral} we study the chiral phase diagram in the presence of $\mu_5$. With two popular gluon propagator models, the catalysis effect of DCSB by $\mu_5$ is examined and the shift of CEP with $\mu_5$ is also shown. The $n_5$ is studied in section \ref{sec:n5}, including the finite volume effects. Finally we summarize in section \ref{sec:summary}.
\section{Quark DSE at finite chiral chemical potential}\label{sec:gap}
The Dyson-Schwinger equation of quark, namely the gap equation, at finite $T$, $\mu$ and $\mu_5$ reads \cite{Wang:2015tia}
\begin{eqnarray}
\label{eq:gapeqinf}
\hspace{-5mm} [ G(\vec{p},\tilde{\omega}_n)]^{-1}&=&[G^0(\vec{p},\tilde{\omega}_n)]^{-1}+T\sum_{l=-\infty}^\infty\int\frac{d^3q}{(2\pi)^3}
\nonumber\\
&&\hspace*{10mm}\times \left[g^2D_{\mu \nu}(\vec{p}\!-\!\vec{q},\tilde{\omega}_n-\tilde{\omega}_l)\frac{\lambda^a}{2}\gamma_{\mu}G(\vec{q},\tilde{\omega}_l)\Gamma_{\nu}^a \right].
\end{eqnarray}
Here $G(\vec{p},\tilde{\omega}_n)$ is the fully dressed quark propagator and $G^0(\vec{p},\tilde{\omega}_n)$ is the free quark propagator. The $D_{\mu \nu}$ is the fully dressed gluon propagator (with color index contracted) and $\Gamma_{\nu}^a$ is the fully dressed quark-gluon vertex. The $\tilde{\omega}_n$ denotes $\omega_n+i \mu$, with $\omega_n$ quark's Matsubara frequency $\{\omega_n$=$(2n+1)\pi T$, $n=0,\pm 1,\pm 2...\}$. The gluon's Matsubara frequency is $\{\Omega_n$=$2 n\pi T$, $n=0,\pm 1,\pm 2...\}$. We've set all the renormalization constants to $1.0$ since we will use gluon models that are heavily suppressed in the ultraviolet region.
The fully dressed quark propagator $G(\vec{p},\tilde{\omega}_n)$ can be generally decomposed as the summation of eight Dirac structures associated with coefficients of scalar functions \cite{Wang:2015tia}
\begin{subequations}\label{eq:Gdecomp}
\begin{align}
\label{eq:Ginv}
G^{-1}(\vec{p},\tilde{\omega}_n)=\sum_{i=1}^{8}T_i(\vec{p},\tilde{\omega}_n) F_i(|\vec{p}|,\tilde{\omega}_n),
\end{align}
or equivalently
\begin{align}
\label{eq:G}
G(\vec{p},\tilde{\omega}_n)=\sum_{i=1}^{8}T_i(\vec{p},\tilde{\omega}_n) \sigma_i(|\vec{p}|,\tilde{\omega}_n).
\end{align}
\end{subequations}
The Dirac bases are $T_i \in \{i\sh{\vec{p}}, I_4, i\gamma_4 \tilde{\omega}_n, \sh{\vec{p}} \gamma_4, i \gamma_5 \sh{\vec{p}}, \gamma_5, \gamma_5 \gamma_4, i \gamma_5 \sh{\vec{p}} \gamma_4 \}$, with the last four Dirac bases brought about by the presence of chiral chemical potential $\mu_5$. The $F_i(|\vec{p}|,\tilde{\omega}_n)$'s and $\sigma_i(|\vec{p}|,\tilde{\omega}_n)$'s are dressed scalar functions. For the free quark propagator $G^0(\vec{p},\tilde{\omega}_n)$, one has $F_1=F_3=1$, $F_2=m$ and $F_7=\mu_5$, with $m$ the current quark mass.
In \cite{Shi:2014zpa} we've studied the QCD chiral phase diagram on the $T-\mu$ plane. As a generalization to the case of nonzero $\mu_5$, we adopt the same setup as in \cite{Shi:2014zpa}, i.e., taking the Rainbow truncation for the quark-gluon vertex
\begin{equation}
\label{eq:rainbow}
\Gamma^a_\nu(p,q)=\frac{\lambda^a}{2}\gamma_\nu,
\end{equation}
and the fully dressed gluon propagator takes the model of
\begin{equation}
\label{eq:MT}
g^2D_{\mu\nu}(k)=\frac{4\pi^2}{\omega^6}D\textrm{e}^{-(k^2+\alpha \mu^2)/\omega^2} k^2\left(\delta_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}\right).
\end{equation}
This model is based on the popular MT model that successfully describes many hadron properties \cite{Maris:2003vk}. The additional term $e^{-\alpha \mu^2/\omega^2}$ mimics the screening effect of finite chemical potential \cite{Chen:2011my,Jiang:2013xwa}. We remind that the preferred parameters $\omega=0.45$ GeV, $D \omega=(0.8 \ \textrm{GeV})^3$ and $\alpha=0.6$ were used in \cite{Shi:2014zpa}, with the current quark mass set to $m=5$ MeV. Note that $\omega$ here is a parameter instead of the Matsubara frequency $\tilde{\omega}_n$. The functions $F_i(\vec{p}, \tilde{\omega}_n)$ can then be solved: In the gap equation (\ref{eq:gapeqinf}), expand $[ G(\vec{p},\tilde{\omega}_n)]^{-1}$ with Eq.~(\ref{eq:Ginv}) and $G(\vec{q},\tilde{\omega}_l)$ with Eq.~(\ref{eq:G}), multiply both sides of Eq.~(\ref{eq:gapeqinf}) with the eight Dirac basis $T_i(\vec{p},\tilde{\omega}_n)$ and then take trace. One finally obtains eight coupled equations
\begin{subequations}\label{eq:sigtoF}
\begin{align}
F_1&=1+\frac{4}{-3\vec{p}^2}\sum \hspace{-0.5cm}\int
\left[\sigma_1 \vec{p}\cdot\vec{q} \left(\tilde{\omega }_{ln }^2+3 \vec{k}^2\right)-2 \sigma_3 \vec{k} \cdot \vec{p}\tilde{\omega }_l \tilde{\omega }_{ln }\right]\times D_{\textrm{MT}}(\vec{k},\tilde{\omega}_{ln}) \\
F_2&=m+\frac{4}{3}\sum \hspace{-0.5cm}\int
\left[3 \sigma_2 \left(\tilde{\omega }_{ln }^2+\vec{k}^2\right)\right] \times D_{\textrm{MT}}(\vec{k},\tilde{\omega}_{ln})\\
F_3&=1+\frac{4}{-3\tilde{\omega }_{n}^2}\sum \hspace{-0.5cm}\int
\left[\sigma_3 \tilde{\omega }_l \tilde{\omega }_n \left(3 \tilde{\omega }_{ln }^2+\vec{k}^2\right)-2 \sigma_1 \vec{k} \cdot \vec{q} \tilde{\omega }_n \tilde{\omega }_{ln }\right]\times D_{\textrm{MT}}(\vec{k},\tilde{\omega}_{ln}) \\
\label{eq:sig4toF4}
F_4&=\frac{4}{-3\vec{p}^2}\sum \hspace{-0.5cm}\int
\left[-\sigma_4 \vec{p}\cdot\vec{q} \left(\tilde{\omega }_{ln }^2+\vec{k}^2\right)\right]\times D_{\textrm{MT}}(\vec{k},\tilde{\omega}_{ln}) \\
F_5&=\frac{4}{3\vec{p}^2}\sum \hspace{-0.5cm}\int
\left[\sigma_5 \vec{p}\cdot\vec{q} \left(\tilde{\omega }_{ln }^2+3 \vec{k}^2\right)+2 i \sigma_7 \vec{k} \cdot \vec{p}\tilde{\omega }_{ln }\right]\times D_{\textrm{MT}}(\vec{k},\tilde{\omega}_{ln}) \\
\label{eq:sig6toF6}
F_6&=\frac{4}{3}\sum \hspace{-0.5cm}\int
\left[-3 \sigma_6 \left(\tilde{\omega }_{ln }^2+\vec{k}^2\right)\right]\times D_{\textrm{MT}}(\vec{k},\tilde{\omega}_{ln}) \\
F_7&=\mu_5+\frac{4}{-3}\sum \hspace{-0.5cm}\int
\left[\sigma_7 \left(-3 \tilde{\omega }_{ln }^2-\vec{k}^2\right)+2 i \sigma_5 \vec{k} \cdot \vec{q} \tilde{\omega }_{ln }\right]\times D_{\textrm{MT}}(\vec{k},\tilde{\omega}_{ln}) \\
F_8&=\frac{4}{3\vec{p}^2}\sum \hspace{-0.5cm}\int
\left[-\sigma_8 \vec{p}\cdot\vec{q} \left(\tilde{\omega }_{ln }^2+\vec{k}^2\right)\right] \times D_{\textrm{MT}}(\vec{k},\tilde{\omega}_{ln})
\end{align}
with
\begin{align}
D_{\textrm{MT}}(\vec{k},\tilde{\omega}_{ln})=\frac{4\pi^2 D}{\omega^6}\textrm{e}^{-(\vec{k}^2+\tilde{\omega}_{ln}^2+\alpha \mu^2)/\omega^2}.\label{eq:D}
\end{align}
\end{subequations}
The abbreviations we used are $\tiny{\sum} \hspace{-0.35cm}\int=T\sum_{n=-\infty}^\infty \int \frac{d^3\vec{p}}{(2\pi)^3}$, $F_i=F_i(|\vec{p}|,\tilde{\omega}_n)$, $\sigma_i=\sigma_i(|\vec{q}|,\tilde{\omega}_l)$, $\vec{k}=\vec{p}-\vec{q}$, $\tilde{\omega}_{ln}=\tilde{\omega}_{l}-\tilde{\omega}_{n}$. The $\sigma_i(\vec{q}_{ l},\tilde{\omega}_l)$ can be obtained with $F_i(\vec{q}_{ l},\tilde{\omega}_l)$ via
\begin{subequations}\label{eq:Ftosig}
\begin{align}
\sigma_1&=-\frac{1}{2 | \vec{q}| } \left(\frac{F_1 | \vec{q}| -F_7}{t_1}+\frac{F_1 | \vec{q}| +F_7}{t_2}\right)\\
\sigma_2&=\frac{1}{2} \left( \frac{F_2-F_8 | \vec{q}| }{t_1}+\frac{F_8 | \vec{q}| +F_2}{t_2} \right)\\
\sigma_3&=-\frac{1}{2 \tilde{\omega }_l} \left( \frac{F_3 \tilde{\omega }_l-i F_5 | \vec{q}| }{t_1}+\frac{F_3 \tilde{\omega }_l+i F_5 | \vec{q}| }{t_2}\right)\\
\label{eq:F46tosig4}
\sigma_4&=-\frac{1}{2 | \vec{q}| } \left( \frac{F_4 | \vec{q}| -i F_6}{t_1}+\frac{F_4 | \vec{q}| +i F_6}{t_2}\right)\\
\sigma_5&=-\frac{1}{2 | \vec{q}| } \left(\frac{F_5 | \vec{q}| -i F_3 \tilde{\omega }_l}{t_1}+\frac{F_5 | \vec{q}| +i F_3 \tilde{\omega }_l}{t_2} \right)\\
\label{eq:F46tosig6}
\sigma_6&=-\frac{1}{2} \left(\frac{F_6+i F_4 | \vec{q}| }{t_1} +\frac{F_6-i F_4 | \vec{q}| }{t_2}\right)\\
\sigma_7&=-\frac{1}{2}\left( \frac{F_7-F_1 | \vec{q}| }{t_1}+\frac{F_1 | \vec{q}| +F_7}{t_2}\right)\\
\sigma_8&=\frac{1}{2 | \vec{q}| }\left(\frac{F_8 | \vec{q}| -F_2}{t_1}+\frac{F_8 | \vec{q}| +F_2}{t_2} \right)\\
t_1&=2 i F_3 F_5 \tilde{\omega }_l | \vec{q}|
+F_3^2 \tilde{\omega }_l^2-2 F_8 F_2 | \vec{q}| -2 i F_4 F_6 | \vec{q}|
+F_2^2-F_6^2+F_7^2 \nonumber\\
&\hspace{20mm}+| \vec{q}|
\left(\left(F_1^2+F_4^2-F_5^2+F_8^2\right) | \vec{q}| -2 F_1 F_7\right)\\
t_2&=-2 i F_3 F_5 \tilde{\omega }_l | \vec{q}|
+F_3^2 \tilde{\omega }_l^2+2 F_8 F_2 | \vec{q}| +2 i F_4 F_6 | \vec{q}|+F_2^2-F_6^2+F_7^2\nonumber \\
&\hspace{20mm}+|\vec{q}|
\left(\left(F_1^2+F_4^2-F_5^2+F_8^2\right) | \vec{q}| +2 F_1 F_7\right)
\end{align}
\end{subequations}
Eqs.~(\ref{eq:sigtoF},\ref{eq:Ftosig}) can be fully solved numerically by iteration. Meanwhile, there are two observations that could further simplify the computation. First, the $F_4$ and $F_6$ can simultaneously be zero, as can be read from Eqs.~(\ref{eq:sig4toF4},\ref{eq:sig6toF6},\ref{eq:F46tosig4},\ref{eq:F46tosig6}). In principle, there is a chance a non-vanishing solution exists, just as $F_2(\vec{p},\tilde{\omega}_n)$ associated with the quark mass function can be nonzero even in the chiral limit $m=0$ due to DCSB. But as we tested they are vanishingly small even if they are kept in the computation. Therefore the terms $F_4$ and $F_6$ can be set to zero. Another useful relation is that the scalar functions satisfy
\begin{align}
F_i(|\vec{p}|,\tilde{\omega}_n)=F^*_i(|\vec{p}|,\tilde{\omega}_{-n-1}),
\end{align}
so the number of scalar functions with different $n$'s to compute are halved.
\section{Chiral phase diagram at finite chiral chemical potential}\label{sec:chiral}
As is well known, the DCSB and confinement are two key properties of QCD at low energy scale. The restoration of chiral symmetry and de-confinement are expected for QCD matter at finite temperature. The influence of $\mu_5$ on the chiral condensate has long been of interest \cite{Ruggieri:2011xc,Fukushima:2010fe}.
The quark condensate $\langle \bar{\psi}\psi \rangle$ is the order parameter of chiral phase transition in chiral limit, and an indicator in the presence of a small current quark mass. It can be calculated as the trace of full quark propagator
\begin{subequations}
\begin{align}
\langle \bar{\psi}\psi \rangle=T\int\frac{d^3\vec{p}}{(2\pi)^3} \sum\limits_{n=-\infty}^{\infty} \textrm{Tr}[G(\vec{p},\tilde{\omega}_n)].
\end{align}
However, this quantity suffers from ultraviolet divergence in the presence of nonzero quark mass, so people define regularized condensates as \cite{Shi:2016koj}
\begin{align}
\langle \bar{\psi}\psi \rangle_r=\langle \bar{\psi} \psi \rangle(T,\mu)-\langle \bar{\psi}_0 \psi_0 \rangle(T,\mu)
\end{align}
or that proposed in lattice calculation\cite{Bali:2011qj}
\begin{align}\label{eq:latcond}
\langle \bar{\psi}\psi \rangle_R=\langle \bar{\psi} \psi \rangle(T,\mu)-\langle \bar{\psi}_0 \psi_0 \rangle(T=0,\mu=0)
\end{align}
\end{subequations}
The $\psi_0$ indicates the free quark field. Meanwhile, a rigorous definition exists as $\langle \bar{\psi}\psi \rangle$ being the first derivative of partition function versus current quark mass $m$, e.g.,
\begin{align}
\label{eq:cjtcond}
\langle\bar{\psi}\psi\rangle=\frac{\partial \textrm{ln} \mathcal{Z}}{\partial m}.
\end{align}
Since the Rainbow truncation renders the partition function calculable in the framework of CJT effective action \cite{Cornwall:1974vz,Roberts:2000aa}, i.e.,
\begin{eqnarray}
\label{eq:d}
\mathcal{P}(T,\mu,\mu_5)&=&\frac{T}{V} \textrm{ln} \mathcal{Z}=\frac{T}{V}\textrm{Tr}\biggr[Ln(G^{-1}G_0)-\frac{1}{2}(1-G_0^{-1}G)\biggr],
\end{eqnarray}
we employ the definition of Eq.~(\ref{eq:cjtcond}) to calculate the quark condensate. This was first done in \cite{Shi:2014zpa} with $\mu_5$ not introduced. We note that the CJT effective action $\mathcal{P}(T,\mu,\mu_5)$ alone is ultra-violate divergent, but the difference of two actions with two masses, i.e., $\mathcal{P}(T,\mu,\mu_5;m+\delta m)-\mathcal{P}(T,\mu,\mu_5;m)$ is finite.
\begin{figure}[h!]
\centering
\includegraphics[width=.49\textwidth]{cond1.pdf}
\hfill
\includegraphics[width=.49\textwidth]{cond0.pdf}
\caption{\label{fig:condsMT} The quark condensate from MT model at finite $T$ with $\mu=0$ GeV and different $\mu_5$'s.}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=.49\textwidth]{condQC2.pdf}
\hfill
\includegraphics[width=.49\textwidth]{condQC.pdf}
\caption{\label{fig:condsQC}The quark condensate from QC model at finite $T$ with $\mu=0$ GeV and different $\mu_5$'s.}
\end{figure}
The obtained $\langle\bar{\psi}\psi\rangle$ is shown in Fig.~\ref{fig:condsMT}. The left panel shows the quark condensate on the $T$ axis with different $\mu_5$'s beyond the chiral limit. An important feature is that $-\langle \bar{\psi} \psi \rangle$ rises with increasing $\mu_5$,as also found by lattice simulation \cite{Braguta:2015zta}. NJL model studies gave similar results \cite{Fukushima:2010fe,Yu:2015hym}. Meanwhile, we find the $-\langle \bar{\psi} \psi \rangle$ always exhibits crossover behavior regardless of increasing $\mu_5$, and the pseudo-transition temperature $T_c$ increases with $\mu_5$. Note that for a first order chiral phase transition the $T_c$ is unique, but for crossover it is defined as the peak location in the susceptibilities, hence definition-dependent \cite{Du:2015psa,Xu:2019ccc}. To remove the ambiguity brought by definition of $T_c$, we supplement with the chiral limit $m=0$ case on the right panel. In this case, the chiral crossover becomes a second order phase transition. We again observe that when $\mu_5$ goes larger, the second order phase transition temperature $T_c$ increases. This is known as the catalysis effect of DCBS by $\mu_5$, as found by lattice result as well \cite{Braguta:2015zta,Braguta:2015owi}. For other model studies as linear sigma model or NJL model, the results differ by regularization schemes \cite{Chernodub:2011fr, Yu:2015hym,Ruggieri:2011xc,Yu:2015hym,Farias:2016let,Cui:2016zqp,Khunjua:2018jmn}.
We further employ an alternative gluon propagator model, i.e., the QC model. It had never been employed in the case of a finite $\mu_5$ before. The model reads \cite{Qin:2011dd}
\begin{equation}
\label{eq:QC}
g^2D_{\mu\nu}(k)=\frac{8\pi^2}{\omega^4}D\textrm{e}^{-k^2/\omega^2} \left(\delta_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}\right).
\end{equation}
Constrained by hadron properties as pion mass and decay constant, the parameters are chosen to be $D=1, \omega=0.6$, same as our early work \cite{Shi:2016koj}. This model is also widely used in hadron study within the DSE approach. Taking away the tensor part $\delta_{\mu \nu}-\frac{k_\mu k_\nu}{k^2}$, it is finite (non-vanishing) at $k^2=0$ as compared to the MT model Eq.~(\ref{eq:MT}), so it better resembles realistic gluon propagator that were produced by gauge sector study from DSE \cite{Aguilar:2010gm, Boucaud:2010gr} and lattice QCD \cite{Oliveira:2010xc,Bowman:2004jm}. The quark condensate from QC model is shown in the Fig.~\ref{fig:condsQC}. We find the QC model gives same behavior as MT model, e.g., i) the crossover behavior remains with increasing $\mu_5$ and ii) the $T_c$ rises with increasing $\mu_5$. The DSE approach therefore gives consistent result concerning the chiral crossover in the presence of chiral chemical potential.
We also show the chiral quark condensate at finite temperature $T$ with different $\mu$'s and $\mu_5$'s in figure~\ref{fig:condsmu}. In the left panel we set $\mu=0.1$ GeV. In this case, the chiral transition is a crossover when $\mu_5=0$ GeV (red solid line), and becomes a first order phase transition when $\mu_5=0.2$ GeV (green dashed line). It remains a first order phase transition when $\mu_5$ further increases to $0.4$ GeV (blue dotted line). Therefore, the increasing $\mu_5$ changes the chiral transition at low $\mu$ from a crossover to first order phase transition. At higher $\mu$ like $\mu=0.2$ GeV in the right panel, the first order phase transition remains in the presence of finite $\mu_5$. One can observe from Fig.~\ref{fig:condsmu} that the $T_c$ generally increases with $\mu_5$ at finite $\mu$, which is similar to the $\mu=0$ case above. Note that Fig.~\ref{fig:condsmu} is plotted using the MT model. The QC model gives qualitatively same result, hence omitted.
\begin{figure}[h!]
\centering
\includegraphics[width=.49\textwidth]{cond2.pdf}
\hfill
\includegraphics[width=.49\textwidth]{cond3.pdf}
\caption{\label{fig:condsmu}The quark condensate from MT model at finite $T$ with different $\mu$'s and $\mu_5$'s.}
\end{figure}
These findings indicate that the location of the critical end point in the $T-\mu$ plane shifts with $\mu_5$. We plot the CEPs for different $\mu_5$'s in Fig.~\ref{fig:CEP}. Results from both models are displayed. The rightmost green dot at $(\mu_T, T_E)=(0.135, 0.127)$ GeV denotes the location of CEP at $\mu_5=0$ GeV using MT model, and rightmost blue dot at $(\mu_T, T_E)=(0.16, 0.116)$ GeV denotes the one with QC model. The red dotted lines represent the corresponding crossover lines. When $\mu_5$ increases, the CEPs shift from lower-right to upper-left until the $\mu_E$'s for both models stay almost constant at $\mu_E \approx 0.05$ GeV. The CEPs therefore won't hit the temperature axis no matter how large $\mu_5$ is. Note that early model studies \cite{Fukushima:2010fe,Chernodub:2011fr,Ruggieri:2011xc} suggest that the CEP and first order phase transition would show up on the $T$ axis as $\mu_5$ increases, and confirmation from lattice QCD simulation could provide a strong evidence for the existence of CEP. However, such signal hasn't been found in existing lattice QCD \cite{Yamamoto:2011gk, Braguta:2015zta,Braguta:2015owi}. Therefore in accordance with lattice QCD at $\mu=0$, the DSE supplements with a possible trajectory of CEP at finite $\mu$, if it exists, in the presence of $\mu_5$.
\begin{figure}[h!]
\centering
\includegraphics[width=.6\textwidth]{CEP.pdf}
\caption{\label{fig:CEP} The shift of CEPs in the presence of $\mu_5$. The green points (empty circle) shifting from right to left indicate the location of CEP using MT model Eq.~(\ref{eq:MT}), when $\mu_5=0, 100, 200, 300, 400, 600, 800$ MeV respectively. The blue points (empty triangle) are obtained using QC model Eq.~(\ref{eq:QC}). The red dotted lines ended with CEPs are the chiral crossover lines at zero $\mu_5$. }
\end{figure}
\section{The chiral charge density}\label{sec:n5}
The non-vanishing finite chiral charge density $n_5$ is a novel feature of hot and dense QCD matter in the presence of $\mu_5$. It is indispensable for the CME effect. Relating $n_5$ and $\mu_5$ is useful for to expressing the induced electric current density as a function of the chirality density \cite{Fukushima:2010fe}. The $n_5=\langle \bar{\psi}\gamma_4\gamma_5 \psi \rangle$ can be calculated with the fully dressed quark propagator
\begin{align}\label{eq:n5}
n_5=-N_c N_f T\int \frac{d^3 \vec{p}}{(2 \pi)^3}\sum_n \textrm{Tr}[\gamma_4 \gamma_5 G(\vec{p},\tilde{\omega}_n)].
\end{align}
However, directly computing Eq.~(\ref{eq:n5}) is very difficult: in computation one always need to truncate the Matsubara frequency $n$ to some maximum value, but Eq.~(\ref{eq:n5}) converges very slowly with $n$. Same situation happens to the computation of quark number density $n=\langle \bar{\psi}\gamma_4 \psi \rangle$ as well. In \cite{Xu:2019ccc}, a numerical technique is proposed to deal with this problem. The basic idea is to pick out the UV behavior of the integration, which converges very slowly but analytically calculable, and compute the remaining integration and summation that converges quickly. Here we take that idea and calculate $n_5$ with
\begin{align}
\label{eq:n5tech}
n_5&=\langle \bar{\psi}\gamma_4\gamma_5 \psi \rangle(T,\mu,\mu_5;m)-\langle \bar{\psi}_0\gamma_4\gamma_5 \psi_0 \rangle(T,\mu,\mu_5;m=0)+\langle \bar{\psi}_0\gamma_4\gamma_5 \psi_0 \rangle(T,\mu,\mu_5;m=0) \nonumber\\
&=N_c N_f\left(-\int \frac{d^3 \vec{p}}{(2 \pi)^3}\sum_n \textrm{Tr}\left[\gamma_4 \gamma_5 (G(\vec{p},\tilde{\omega}_n)-G_0(\vec{p},\tilde{\omega}_n;m=0))\right]\right. \nonumber \\
&\hspace{70mm}\left.+\frac{\mu_5(\pi^2 T^2+3 \mu^2+\mu_5^2)}{3\pi^2}\right).
\end{align}
Namely, we subtract the $n_5$ of free quark propagator and add it back. The summation identity
\begin{align}
\sum\limits_{n=-\infty}^{\infty}\frac{1}{i(2n+1)\pi T-\Delta}=\frac{1}{2T}\frac{1-e^{\Delta/T}}{1+e^{\Delta/T}}
\end{align}
is useful in the derivation. In this way, the asymptotic behavior of full quark propagator $G(\vec{p},\tilde{\omega}_n)$ at large $n$ is separated out. The summation over Matsubara frequency $n$ and integration over $\vec{p}$ then converge quickly enough to validate a practical computation with precision.
We then plot the chiral charge density in Fig.~\ref{fig:n5}. At $\mu_5=0$ GeV, the $n_5$ vanishes, hence not plotted. When $\mu_5=0.2$ GeV, we show the result of $\mu=0$ GeV (blue dotted) and $\mu=0.2$ GeV (green solid) respectively. One can see that $n_5$ generally increases with temperature $T$, i.e., it slightly decreases with $T$ in Nambu phase (phase with DCSB) but increases steadily in the Wigner phase (phase with chiral symmetry partially restored). The Nambu phase is irrelevant for CME, since chiral condensate couple the left-handed and right-handed quarks, and with a large chiral condensate the $n_5$ decays quickly \cite{Fukushima:2008xe}. In such case an equilibrium can not be reached. Note that de-confinement is also a necessary condition for CME, but it's beyond the scope of this work. We see in Fig.~\ref{fig:n5} that at $\mu=0$ the transition is a crossover and the $n_5$ exhibits a smooth continuous curve. At $\mu=0.2$ GeV, a first order phase transition takes place and it becomes discontinuous. Meanwhile the $n_5$ increases with $\mu_5$, see, eg., curves from $\mu_5=0.2$ GeV to $\mu_5=0.4$ GeV. We also observe an increase of $n_5$ with respect to quark chemical potential $\mu$ for Wigner phase in most area. Therefore, we conclude that in the Wigner phase where chiral symmetry gets partially restored, increasing $T$, $\mu_5$ and $\mu$ all result in an increase in the chiral charge density.
\begin{figure}[h!]
\centering
\includegraphics[width=.6\textwidth]{n5.pdf}
\caption{\label{fig:n5} The chiral densities $n_5$ at finite $T$ with different $\mu$'s and $\mu_5$'s, obtained with MT model.}
\end{figure}
Another factor that can potentially influences the $n_5$ is the size of the fireball created in HIC, namely the system volume. Analysis shows that the volume of homogeneity before freeze-out for Au-Au and Pb-Pb collisions ranges between approximately $50\sim 250$ fm$^3$ \cite{Graef:2012sh}, and could go as low as (2 fm)$^3$ \cite{Palhares:2009tf}. Since the CME is typically investigated with peripheral collisions, the finite size effect could be relevant. In \cite{Shi:2018swj,Shi:2018tsq} the finite volume effect on chiral phase diagram in the absence of $\mu_5$ had been studied. The finding was that decreasing the volume would weaken the DCSB, and consequently change the chiral phase diagram in the $\mu-T$ plane. Here we take the formalism and investigate the finite size effect on $n_5$. As a first estimate, we will use a rather simplified approach as in \cite{Bhattacharyya:2014uxa}, which is explained below.
At finite volume, the quark and gluon fields are constrained by certain spatial boundary condition. If we consider a system in a cubic box of size $L$, a popular and practical boundary condition is the anti-periodic boundary condition $\psi(\vec{\textbf x}=\vec{\textbf 0},\tau)=-\psi({\vec{\textbf x}=\vec{\textbf L},\tau})$ for quark fields, and $A^a_\mu(\vec{\textbf x}=\vec{\textbf 0},\tau)=A_\mu^a({\vec{\textbf x}=\vec{\textbf L},\tau})$ for gluon fields.
In this case, the quark and gluon fields take the same boundary condition in their spatial and temporal directions. As pointed out by authors in \cite{Klein:2017shl}, such choice allows a permutation symmetry of the spatial and temporal directions in the effective Lagrangian, rendering temperature- and volume-independent coupling constants. One can then directly employ models that were determined at zero temperature and volume. This boundary condition therefore has been widely employed in model studies as Refs.~\cite{Fischer:2011mz,Tripolt:2013zfa,Abreu:2019czp,Shi:2018swj,Shi:2018tsq}. Mathematically, it leads to the discretization of momentum $\vec{p}$ into modes $\vec{p}_{\textbf n}=\sum_{n_i=0, \pm 1, \pm 2,...}(2 n_i+1)\pi /L\hat{e_i}$ ($\hat{e_i}$ is the Cartesian unit vectors in momentum space) for quarks, and $\vec{p}_{\textbf n}=\sum_{n_i=0, \pm 1, \pm 2,...}2n_i\pi /L\hat{e_i}$ for gluons. So the momentum integration in Eq.~(\ref{eq:gapeqinf}) becomes
\begin{align}\label{eq:rep1}
\int \frac{dq^3}{(2 \pi)^3}(\dotsb) \rightarrow \frac{1}{L^3} \sum_{n_i=0, \pm 1, \pm 2,...} (\dotsb).
\end{align}
However, the breaking of O(3) symmetry leads to scalar functions with more variables, e.g., the $F_i(|\vec{p}|,\tilde{\omega}_n)$ in Eq.~(\ref{eq:Ginv}) now becomes $F_i(n_1,n_2,n_3,\tilde{\omega}_n)$. This calls for a lot more computing power. So for a first qualitative analyze, we use an approximation method employed by \cite{Bhattacharyya:2014uxa,Li:2017zny}, i.e.,
\begin{align}\label{eq:rep2}
\int \frac{dq^3}{(2 \pi)^3}(\dotsb) \rightarrow \int_{|\vec{q}|=\pi/L}^{|\vec{q}|=\infty} \frac{dq^3}{(2 \pi)^3}(\dotsb).
\end{align}
The right hand side (RHS) of Eq.~(\ref{eq:rep2}) intends to approximate the RHS of of Eq.~(\ref{eq:rep1}) by introducing an infrared momentum cutoff. It obviously gets more accurate at larger $L$ but less accurate with smaller $L$. However, for the purpose of a qualitative study, we find it suffices to give a correct picture in the case of chiral phase diagram. For instance, calculation with \cite{Li:2017zny} and without \cite{Shi:2018tsq} approximation give qualitatively same results concerning the chiral phase transition.
Take the replacement Eq.~(\ref{eq:rep2}) in Eq.~(\ref{eq:gapeqinf}) and Eq.~(\ref{eq:n5}), we obtain the $n_5$ at finite volume as shown in Fig.~\ref{fig:n5V}. We remind this result is obtained with MT model at $m=5$ MeV, in correspondence with left plot in Fig.~\ref{fig:condsMT}. There are two set of curves. One is by setting $T=220$ MeV which is just above the pseudo-critical temperature $T_c \approx 150$ MeV (when $\mu=0$ and $\mu_5=0$). We see that as system size decreases, the $n_5$ increases, and the smaller the volume the quicker the increase. At $L=2$ fm, the $n_5$ increases by about 30\% for all $\mu_5$. At high temperature $T=0.4$ GeV, the finite volume effect remains but gets weakened, as can be seen from the dotted purple curve against dot-dashed gray curve. Our result therefore suggests an increase in $n_5$ when the system size decreases.
\begin{figure}[h!]
\centering
\includegraphics[width=.6\textwidth]{n5_fin.pdf}
\caption{\label{fig:n5V} The chiral density $n_5$ at finite $\mu_5$ with different system sizes, obtained by MT model.}
\end{figure}
\section{Summary}\label{sec:summary}
In this paper, we study chirally imbalanced hot and dense strongly interacting matter by means of the Dyson-Schwinger equations. By solving the quark's DSE, the fully dressed quark propagator is obtained with its complete Dirac structures, rendering thermodynamical properties calculable.
The chiral phase diagram is studied in the presence of $\mu_5$. The chiral quark condensate $\langle \bar{\psi} \psi \rangle$ is unambiguously obtained with the CJT effective action, in concert with the Rainbow truncation scheme. Catalysis effect of DCSB by $\mu_5$ is observed. For instance, We find the $-\langle \bar{\psi} \psi \rangle$ increases with $\mu_5$, along with the increase of pseudo-critical temperature $T_c$ of the crossover at $\mu=0$. To avoid ambiguities from definition in $T_c$, we check with the chiral limit when the crossover becomes a second order phase transition. To examine the model dependence within the DSE, we supplemented a new calculation with a more realistic QC gluon propagator model, as compared to the MT model. Consistency is found within the DSE approach, as well as in comparison with lattice QCD. Since finite $\mu$ is directly calculable in the DSE, we further study the influence of $\mu_5$ on the CEP location $(\mu_E, T_E)$ in the $T-\mu$ plane. It is found that the CEP shifts toward larger $T_E$ but constant $\mu_E$ as $\mu_5$ increases.
A technique is then developed to overcome the computational difficulty within $n_5$. We then show our results on $n_5$ with various conditions. We find the $n_5$ generally increases with the increase of $T$, $\mu$ and $\mu_5$. Since the CME is typically investigated with peripheral collisions, finite size effect is then considered based on a specific spatial boundary condition, i.e., anti-periodic for quark fields and periodic for gluon fields. We found an increase in $n_5$ with the decrease of system size.
We remark that to mimic a more realistic condition for the HIC, the magnetic field $B$ should also be taken into account, since it influences the chiral phase diagram as well \cite{Bali:2012zg}. The finite size effect study could also be improved. For instance, based on the present boundary condition and within DSEs, a full computation using Eq.~(\ref{eq:rep1}) rather than the infrared momentum cut off scheme Eq.~(\ref{eq:rep2}) is worth studying. Meanwhile, alternative boundary conditions implementing more realistic physical constraints, such as considering a sphere or a rotating cylinder instead of cubic box, are also worth investigation \cite{Chernodub:2016kxh,Zhang:2019gva,Zhang:2020jux}. These studies could deepen our understanding of chirally imbalanced hot and dense QCD matter produced in HICs.
\acknowledgments
This work is supported in part by the National Natural Science Foundation of China (under Grants No. 11905104, No. 11475085, No. 11535005, No. 11690030, No.11873030 and No. 11905107), the National Major state Basic Research and Development of China (Grant No. 2016YFE0129300), the starting grant of Nanjing University of Aeronautics and Astronautics (under Grant No. 1006-YAH20009), the innovation Program of Jiangsu Province, Jiangsu Province Natural Science Foundation, under grant No. BK20190721,
Nanjing University of Posts and Telecommunications Science Foundation, under grant No. NY129032 and
Natural Science Foundation of the Jiangsu Higher Education Institutions of China 19KJB140016.
\bibliographystyle{JHEP}
|
2,877,628,090,090 | arxiv | \section{Introduction}
\hskip 2em
Identifying the underlying structure of a data matrix and extracting meaningful information
is a crucial problem in data analysis, and most efforts have been focused on manipulating, understanding and interpreting large-scale data matrices.
In many cases, matrix factorization
methods are employed for constructing parsimonious and informative representations to facilitate computation and interpretation.
A principal approach is the CUR decomposition \cite{drineas2008SIAMrelative,sorensen2016SIAMdeim,mahoney2009PNAScur,wang2013JMLRimproving},
which is a low-rank approximation of a matrix $A\in\mathbb{R}^{m\times n}$ of the form
\begin{equation}\label{CUR decomposition}
A \approx CUR,
\end{equation}
where matrices $C\in\mathbb{R}^{m\times k}$ and $R\in\mathbb{R}^{k\times n}$ are subsets of the columns and rows, respectively, of the original matrix $A$.
The $k \times k$ matrix $U$ is constructed to ensure that CUR is a good approximation to $A$.
The CUR factorization is an important tool for handling large-scale data sets,
offering two advantages over the rank-$k$ singular value decomposition (SVD)
$A\approx VSW^{\mathrm{T}}$
: when $A$ is sparse, so too are $C$ and $R$, unlike the matrices $V$ and $W$ of singular vectors;
and the columns and rows that comprise $C$
and $R$ are representative of the data (e.g., sparse, nonnegative, integer valued, etc.).
\hskip 2em
There is extensive work on CUR-type decompositions in both numerical linear algebra and theoretical computer science; see \cite{boutsidis2014optimal,wang2013JMLRimproving,hamm2021SIAMperturbations,cai2021SIAMrobust}.
Recently, in \cite{gidisu2022SIAMgeneralized}, Gidisu and Hochstenbach developed a generalized CUR decomposition (GCUR) for matrix pair $A$ and $B$ with the same number of columns:
$A$ is $m \times n$, $B$ is $d \times n$ and both are of full column rank, which can be viewed as a CUR decomposition of $A$ relative to $B$.
The proposed factorization can be used in situations where a
low-rank matrix is perturbed with noise, where the covariance of the noise is not a multiple
of the identity matrix.
Besides, it may also be appropriate for applications where one is interested in
extracting the most discriminative information from a data set of interest relative to another data set.
Furthermore, in recent times, real-world data sets often comprise different representations or views, which provide information complementary to each other.
The multi-view dimension reduction \cite{xu2013ARXIVsurvey}, and integration of information from multiple views in multi-view learning is a rapidly growing direction in machine learning which involves learning with multiple views to improve the generalization performance.
Motivated by this, in \cite{gidisu2022ARXIVrsvd}, Gidisu and Hochstenbach developed a new coordinated CUR factorization of a matrix
triplet $(A, B, G)$ of compatible dimensions, based on the restricted singular value decomposition (RSVD) \cite{zha1991SIAMrestricted}.
This factorization was called an RSVD based CUR (RSVD-CUR) factorization.
An RSVD-CUR factorization as a tool for multi-view dimension reduction can cope with a two-view case.
In the same context, one can use an RSVD-CUR as a supervised feature selection technique in multilabel classification problems.
It can also be applied for applications where the goal is to select a subset of rows and columns of one data set relative to two other data sets.
There are several index selection strategies proposed in the literature for finding the subsets of the columns and rows while constructing the GCUR and RSVD-CUR decomposition.
Two sampling techniques employed in \cite{gidisu2022ARXIVrsvd,gidisu2022SIAMgeneralized} are named DEIM \cite{barrault2004CRMempirical,chaturantabut2010SIAMnonlinear} and L-DEIM \cite{gidisu2022Arxivhybrid}, which are greedy deterministic procedures and simple to implement.
Specifically, as the inputs, the DEIM and L-DEIM require the generalized SVD (GSVD) of the matrix pair $(A,B)$ and the RSVD of the matrix triplet $(A, B, G)$ for sampling when constructing the GCUR and RSVD-CUR decomposition, respectively.
The overall computational complexity of the algorithms discussed in \cite{gidisu2022ARXIVrsvd,gidisu2022SIAMgeneralized} are dominated by the construction of the GSVD and the RSVD.
However, in practice, this cost can be prohibitively expensive, making it unsuitable for large-scale applications.
\hskip 2em
It is known that randomized algorithms \cite{halko2011SIAMfinding,mahoney2011FTMLrandomized} facilitate the matrix decomposition procedure not only by reducing the computational complexity of deterministic algorithms but also by reducing the communication among different levels of memories, which is the main bottleneck in modern computing environments and architectures for large-scale data matrices.
Based on the framework in \cite{halko2011SIAMfinding}, many computationally efficient methods for implementing large-scale matrix factorizations have been proposed, analyzed, and implemented, such as \cite{wei2016SIAMtikhonov,wei2021CAMCrandomized,saibaba2021NLAArandomized,saibaba2016NLAArandomized}.
Meanwhile, these well-established randomized algorithms have been widely used for many
practical applications, such as the least squares problems \cite{xie2019NLAArandomized,boutsidis2009LAArandom,zhang2020JCAMrandomized} and Tikhonov regularization \cite{xiang2013IPregularization,rachkovskij2012CSArandomized}.
Motivated by this success, in this work we introduce the randomized
schemes for efficiently computing the GCUR and the RSVD-CUR decomposition.
To be specific, there are two main computational stages involved in our randomized algorithms.
In the first stage, we use random projections to identify a subspace that captures most of the action of the input matrix.
Then we project the input matrix onto this subspace and get a reduced matrix which is then manipulated deterministically to obtain the desired low-rank approximation of the GSVD and RSVD.
The second stage can be completed with well-established deterministic methods DEIM and L-DEIM operating on the approximation obtained in the first stage to sample the columns and rows of the original matrices.
Compared with non-random approaches, our algorithms allow for a comparable accuracy with much lower cost and will be more computationally efficient on large-scale data.
Details of the algorithm, theoretical analysis and numerical results are provided to show the effectiveness of our approaches.
\hskip 2em
The rest of this paper is organized as follows.
In Section 2, we first give a brief overview of the GSVD and the RSVD,
then we introduce some basic notation and describe several sampling techniques including the DEIM and L-DEIM.
Next, in Section 3, we present our randomized algorithms for computing the GCUR factorization using the DEIM and L-DEIM procedure, where the probabilistic error bound is also presented in detail.
In Section 4, we first briefly review the literature on existing algorithms for the computation of the RSVD, and develop an efficient method for computing this decomposition.
Then we develop randomized algorithms for computing the RSVD-CUR decomposition based on the sampling procedure L-DEIM, along with detailed probabilistic error analysis.
In Section 5, we test the performance of the proposed algorithms on several synthetic matrices and real-world datasets.
Finally, in Section 6, we end this paper with concluding remarks.
\section{Preliminaries}
\hskip 2em
Throughout this paper, we use the MATLAB notation to index vectors and matrices, so that, e.g., $X(\mathbf{q},:)$ denotes the $k$ rows of $X$ whose indices are
specified by the entries of the vector $\mathbf{q}\in\mathbb{N}^k_+$, while $X(:,\mathbf{p})$ denotes the $k$ columns of $X$ indexed by $\mathbf{p}$.
We denote the 2-norm by $\| \cdot \|$.
$A^{\dagger}$ denotes the Moore-Penrose pseudoinverse \cite{wei2018WSnumerical} of $A$.
\subsection{GSVD and RSVD}
\hskip 2em
We now give a brief introduction to the GSVD and RSVD which are the key building blocks of the proposed algorithms.
The original existence of GSVD was first introduced by Van Loan in \cite{van1976SIAMgeneralizing}.
Paige and Saunders \cite{paige1981SIAMtowards} later presented a more general formulation without any restrictions on the dimensions except for both matrices to have the same number of columns, and other formulations and contributions to the GSVD can be found in \cite{huang2022JSC,stewart1982NMcomputing,van1985NMcomputing,sun1983SIAMperturbation,bai1993SIAMcomputing,zha1996NMcomputing}.
In line with \cite{gidisu2022SIAMgeneralized}, in this paper, we adopt the formulation proposed by Van Loan in \cite{van1985NMcomputing}.
Let
$A\in\mathbb{R}^{m \times n}$ and $B\in\mathbb{R}^{d \times n}$
with both $ m\ge n$ and $d \ge n$, then there exist orthogonal matrices
$U\in\mathbb{R}^{m \times m}$, $V\in\mathbb{R}^{d\times d}$
and a nonsingular $Y\in\mathbb{R}^{n \times n}$ such that
\begin{equation}\label{GSVD of B}
B = V \Sigma Y^{\mathrm{T}},\
\Sigma=\operatorname{diag}(\beta_1,\ldots,\beta_n),\ \beta_i\in\left[0,1\right],
\end{equation}
\begin{equation}\label{GSVD of A}
A = U \Gamma Y^{\mathrm{T}},\
\Gamma=\operatorname{diag}(\gamma_1,\ldots,\gamma_n),\ \gamma_i\in\left[0,1\right],
\end{equation}
where $\gamma_i^2+\beta_i^2=1$ and the ratios $\gamma_i/\beta_i$ are in a non-increasing order for $i=1,\ldots,n.$
Further, nonnegative number pairs $\{\gamma_i,\beta_i\}_{i=1}^n$ are actually the generalized singular values of the matrix pair $(A,B)$ as defined in \cite{sun1983SIAMperturbation}, and
the sensitivity of the generalized singular values of a matrix pair to
perturbations in the matrix elements was analyzed in \cite{sun1982MNSperturbation,li1993SIAMbounds,sun1983SIAMperturbation}.
\hskip 2em
The RSVD \cite{de1991SIAMrestricted,zha1991SIAMrestricted} is the factorization of a given
matrix, relative to two other given matrices, which can be interpreted as the ordinary singular value decomposition
with different inner products in the row and column spaces.
Given a matrix triplet
$A\in\mathbb{R}^{m\times n}$, $B\in\mathbb{R}^{m\times l}$ and $G\in\mathbb{R}^{d\times n}$, with $\ell \ge d \ge m \ge n$ and we assume that $B$ and $G$ are of full rank.
Following the formulation of the RSVD proposed by Zha \cite{zha1991SIAMrestricted}, there exist
orthogonal matrices $U\in\mathbb{R}^{l\times l}$, $V\in\mathbb{R}^{d\times d}$
and nonsingular matrices $Z\in\mathbb{R}^{m\times m}$ and $W\in\mathbb{R}^{n\times n}$ such that
\begin{equation}\label{RSVD of (A,B,G)}
A=ZD_AW^{\mathrm{T}},\ B=ZD_BU^{\mathrm{T}},\ G=VD_GW^{\mathrm{T}},
\end{equation}
or alternatively it can be expressed conveniently as
\begin{equation*}
\left[\begin{array}{ll}
A & B \\
G &
\end{array}\right]=\left[\begin{array}{ll}
Z & \\
& V
\end{array}\right]\left[\begin{array}{ll}
D_A & D_B \\
D_G &
\end{array}\right]\left[\begin{array}{ll}
W & \\
& U
\end{array}\right]^{\mathrm{T}},
\end{equation*}
where $D_A\in\mathbb{R}^{m\times n}$, $D_B\in\mathbb{R}^{m\times l}$, and $D_G\in\mathbb{R}^{d\times n}$ are nonnegative diagonal matrices.
\subsection{Subset Selection Procedure}\label{subsection: Subset selection procedure}
\hskip 2em
We now describe several tools for the subset selection that extract appropriate columns or rows from matrices, that are the deterministic leverage score sampling procedure, the DEIM algorithm and the L-DEIM algorithm.
\hskip 2em
Given $A\in \mathbb{R}^{m \times n}$ with $\mathrm{rank}(A)\ge k$.
Let $V_k$ contain its $k$ leading right singular vectors, and we denote the $i$th row of $V_k$ by $\left[V_k\right]_{i,:}$.
Then the rank-$k$ leverage score of the $i$th column of $A$ is defined as
\begin{equation*}
\ell_i=\left\|\left[V_k\right]_{i,:}\right\|^2, \quad i=1, \ldots, n.
\end{equation*}
The deterministic leverage score sampling procedure \cite{jolliffe1972JRSSdiscarding,papailiopoulos2014ACMprovable}
selects columns of $A$ corresponding to the indices of the largest leverage scores for a given $k.$
From a practical perspective, this deterministic algorithm is extremely simple to implement,
but it does not admit provable performance guarantees.
\hskip 2em
The DEIM selection algorithm was first presented in \cite{chaturantabut2010SIAMnonlinear} in the
context of model order reduction for nonlinear dynamical systems and is a discrete
variant of the empirical interpolation method originally proposed in \cite{barrault2004CRMempirical}.
To derive the method, we elaborate upon the interpolatory projectors.
Given a full column rank matrix $V\in \mathbb{R}^{m \times k}$ and a set of distinct indices $\mathbf{p}$, the interpolatory projector for $\mathbf{p}$ onto the range of $V$ $\mathrm{Ran}(V)$ is
\begin{equation*}
\mathbb{P}=V(P^{\mathrm{T}}V)^{-1}P^{\mathrm{T}},
\end{equation*}
where $P=I(:,\mathbf{p})\in \mathbb{R}^{m \times k}$, provided $P^{\mathrm{T}}V$ is invertible.
In general, $\mathbb{P}$ is an oblique projector, and it has an important property:
for any vector $x\in \mathbb{R}^{m}$,
\begin{equation}
(\mathbb{P} x)(\mathbf{p})
=P^{\mathrm{T}} \mathbb{P} x
=P^{\mathrm{T}} V \left(P^{\mathrm{T}} V\right)^{-1} P^{\mathrm{T}} x
=P^{\mathrm{T}} x
=x(\mathbf{p}),
\end{equation}
so the projected vector $Px$ matches $x$ in the $\mathbf{p}$ entries.
The DEIM algorithm processes the columns of $V$ sequentially starting with the first
dominant singular vector.
Each step processes the next singular vector to produce the next index.
The selected indices are used to compute the interpolatory projector $\mathbb{P}$.
The next index is selected by removing the direction of the interpolatory projection in the previous vectors from the subsequent one and finding the index of the entry with the largest magnitude in the residual vector.
See Algorithm \ref{Al-DEIM} for details.
\begin{algorithm}[htb]
\caption{DEIM index selection \cite{chaturantabut2010SIAMnonlinear} }
\label{Al-DEIM}
\hspace*{0.02in} {\bf Input:}
$V \in \mathbb{R}^{m \times k}$ with $k \leq \mathrm{min}(m,n)$.
\\
\hspace*{0.02in} {\bf Output:}
column index $\mathbf{p}\in\mathbb{N}^k_{+}$, with non-repeating entries,
$V \in \mathbb{R}^{m \times k}$ with $k \leq \mathrm{min}(m,n)$.
\begin{algorithmic}[1]
\State $v=V(:, 1)$.
\State $p_{1}=\operatorname{argmax}_{1 \leq i \leq n}\left|v_{i}\right|$.
\For{$j=2, \ldots, k$}
\State $v=V(:, j)$.
\State $c=V(\mathbf{p}, 1: j-1)^{-1} v(\mathbf{p})$.
\State $r=v-V(:, 1: j-1) {c}$.
\State $p_{j}=\operatorname{argmax}_{1 \leq i \leq m}\left|{r}_{i}\right|$.
\State $\mathbf{p}=\left[\begin{array}{ll}\mathbf{p} & p_{j}\end{array}\right]$.
\EndFor
\end{algorithmic}
\end{algorithm}
\hskip 2em
In \cite{sorensen2016SIAMdeim}, the DEIM algorithm was shown to be a viable index selection method for identifying the most representative and influential subset of columns and rows that define a low-dimensional space of the data.
However, a notable limitation of this index selection algorithm is that the number of indices that can be selected is limited to the number of available singular vectors.
\hskip 2em
Combining the strengths of deterministic leverage score sampling and the DEIM procedure, the authors in \cite{gidisu2022Arxivhybrid} proposed a new variant of DEIM, called L-DEIM (Algorithm \ref{Al-LDEIM}).
This method allows for the selection of a number of indices greater than the number of input singular vectors.
As a result, constructing a rank-${k}$ CUR decomposition of a matrix using the L-DEIM only requires $\widehat{k}$ singular vectors where $k>\widehat{k}$.
To select the first $\widehat{k}$ indices, this method performs the original DEIM while keeping the residual singular vector in each index selection step, which is the error between the input singular vector and its approximation from interpolating the previous singular vectors at the selected indices.
Using the idea of the leverage scores, then it computes the 2-norm of the rows of
the residual singular vectors to select the additional $k-\widehat{k}$ indices.
According to the conclusion summarized in \cite{gidisu2022Arxivhybrid}, the L-DEIM is computationally more efficient than the original DEIM, and the accuracy
of both methods may be comparable when the target rank $k$ is at most twice the available $\widehat{k}$ singular vectors, and empirically, we can set $\widehat{k}=k/2$.
Consequently, this novel selection procedure may be viewed as an approach to reusing the same information to further improve the approximation.
\begin{algorithm}[htb]
\caption{L-DEIM index selection \cite{gidisu2022Arxivhybrid} }
\label{Al-LDEIM}
\hspace*{0.02in} {\bf Input:}
$V \in \mathbb{R}^{m \times \widehat{k}}$, target rank $k$ with $\widehat{k} \leq k \leq \min (m, n)$.
\\
\hspace*{0.02in} {\bf Output:}
column indices $\mathbf{p}\in\mathbb{N}^k_{+}$ with non-repeating entries.
\begin{algorithmic}[1]
\For{$j=1,\ldots,\widehat{k}$}
\State $\mathbf{p}(j)=\operatorname{argmax}_{1 \leq i \leq m}\left|(V(:, j))_{i}\right|$.
\State $V(:, j+1)=V(:, j+1)-V(:, 1: j) \cdot(V(\mathbf{p}, 1: j) \backslash V(\mathbf{p}, j+1))$.
\EndFor
\State Compute $\ell_{i}=\left\|V_{i:}\right\|^2 \quad$ for $i=1, \ldots,m$.
\State Sort $\ell$ in non-increasing order.
\State Remove entries in $\ell$ corresponding to the indices in $\mathbf{p}$.
\State $\mathbf{p}^{\prime}=k-\widehat{k}$ indices corresponding to $k-\widehat{k}$ largest entries of $\ell$.
\State $\mathbf{p}=\left[\mathbf{p} ; \mathbf{p}^{\prime}\right]$.
\end{algorithmic}
\end{algorithm}
\section{Randomization for GCUR}
\hskip 2em
In this section, we first give a brief introduction to the GCUR factorization.
Moreover, by combining the random sampling techniques with the DEIM and L-DEIM procedures, we establish two versions of efficient randomized algorithms for computing this factorization, along with the detailed probabilistic error analysis for our approaches.
\subsection{GCUR}
\hskip 2em
In \cite{gidisu2022SIAMgeneralized}, Gidisu and Hochstenbach developed a GCUR decomposition for two matrices $A$ and $B$ with the same number of columns.
The intuition behind this factorization is that we can view it as a CUR decomposition of $A$ relative to $B$, which is appropriate for applications where one is interested in extracting the most discriminative information from a data set of interest relative to another data set.
Given a matrix pair $(A,B)$, where $A$ is $m \times n$ and $B$ is $d \times n$ and both are of full column ranks with $m\ge n$ and $d\ge n$, then the rank-$k$ GCUR decomposition of $(A,B)$ is a matrix approximation of $A$ and $B$ expressed as
\begin{equation}\label{GCUR of A}
A\approx C_A M_A R_A
=A(:,\mathbf{p})\ M_A\ A(\mathbf{s}_A,:),
\end{equation}
\begin{equation}\label{GCUR of B}
B\approx C_B M_B R_B
=B(:,\mathbf{p})\ M_B\ B(\mathbf{s}_B,:).
\end{equation}
Here matrices $C_A$ and $C_B$ indexed by the vector $\mathbf{p}$ are the subset of the columns of $A$ and $B$, capturing the most relevant information of the original matrix.
Selecting the same columns of $A$ and $B$ gives a coupling between the decomposition of $A$ and $B$.
Meanwhile, $R_A$ and $R_B$ are formed by extracting $k$
rows from $A$ and $B$, where the selected row
indices are stored in the vectors $\mathbf{s}_A$ and $\mathbf{s}_B$, respectively.
Given the row/column indices, the middle matrices $M_A$ and $M_B$ can be constructed in different ways to satisfy certain desirable approximation properties.
Following the work in \cite{sorensen2016SIAMdeim,mahoney2009PNAScur,stewart1999NMfour}, the authors in \cite{gidisu2022SIAMgeneralized} choose to construct the middle matrices $M_A$ and $M_B$ as
\begin{equation*}
M_A=C_A^{\dagger} A R_A^{\dagger}
=(C_A^{\mathrm{T}}C_A)^{-1}C^{\mathrm{T}}_A A R_A^{\mathrm{T}}(R_AR_A^{\mathrm{T}})^{-1},
\end{equation*}
\begin{equation*}
M_B=C_B^{\dagger} B R_B^{\dagger}
=(C_B^{\mathrm{T}}C_B)^{-1}C^{\mathrm{T}}_B B R_B^{\mathrm{T}}(R_B R_B^{\mathrm{T}})^{-1},
\end{equation*}
yielding the GCUR factorization that can be viewed as a two step process:
first the columns of $A$ are projected onto the range of $C_A$;
then the result is projected onto the row space of $R_A$:
\begin{equation*}
(1)\quad X=C_AC_A^{\dagger}A,\ (2)\quad C_AM_AR_A=X R_A R_A^{\dagger}.
\end{equation*}
Both steps are optimal with respect to the two-norm error and as shown by Stewart \cite{stewart1999NMfour}, this option minimizes
$\| A - C_AM_AR_A\|$ and $\| B - C_B M_B R_B\|$ for the given sampling indices.
\hskip 2em
In essence, this factorization is a generalization of the CUR decomposition.
To be specific, when $B$ is square and nonsingular, the GCUR decomposition has a close connection
with the CUR of $AB^{-1}$.
Moreover, in the special case where $B = I$, the GCUR decomposition of $A$ coincides
with the CUR decomposition of $A$ in that the factors $C$ and $R$ of $A$ are the same for both methods: (\ref{GCUR of A}) is equivalent to (\ref{CUR decomposition}).
More generally, the GCUR is also applicable to rectangular matrices $B$, and still has a close connection with the CUR decomposition of $AB^{\dagger}$.
A more detailed discussion of the properties can be found in \cite[Proposition 4.2]{gidisu2022SIAMgeneralized}.
To build this decomposition, it is relevant to know the dominant rows and columns of $A$ and $B$ in their rank-$k$ approximations.
Specifically, given a GSVD for matrix pair of the form (\ref{GSVD of A}) and (\ref{GSVD of B}), the DEIM procedure uses $U$, $V$ and $Y$ to select the indices $\mathbf{s}_A$, $\mathbf{s}_B$ and $\mathbf{p}$ respectively.
Algorithm \ref{Al-DEIM-GCUR} is a summary of this procedure, where the backslash operator is a Matlab-type notation for solving linear systems and least-squares problems.
\begin{algorithm}[htb]
\caption{DEIM-type GCUR decomposition \cite{gidisu2022SIAMgeneralized}}
\label{Al-DEIM-GCUR}
\hspace*{0.02in} {\bf Input:}
$A \in \mathbb{R}^{m \times n}$
and
$B \in \mathbb{R}^{m \times n}$ with $m\ge n$ and $d\ge n$, desired rank $k$.
\\
\hspace*{0.02in} {\bf Output:}
A rank-$k$ GCUR decomposition\\
\hspace*{0.02in}
$A\approx A(:,\mathbf{p}) \cdot M_A \cdot A(\mathbf{s}_A,:)$,
$B\approx B(:,\mathbf{p}) \cdot M_B \cdot B(\mathbf{s}_B,:)$.
\begin{algorithmic}[1]
\State $[U,V,Y]=\mathrm{gsvd}(A,B)$.
\State $y=Y(:, 1)$.
\State $p_{1}=\operatorname{argmax}_{1 \leq i \leq n}\left|y_{i}\right|$.
\For{$j=2, \ldots, k$}
\State $y=Y(:, j)$.
\State $c=Y(\mathbf{p}, 1: j-1)^{-1} y(\mathbf{p})$.
\State $r=y-Y(:, 1: j-1) {c}$.
\State $p_{j}=\operatorname{argmax}_{1 \leq i \leq n}\left|{r}_{i}\right|$.
\State
$\mathbf{p}=\left[\begin{array}{ll}\mathbf{p} & p_{j}\end{array}\right]$.
\EndFor
\State
Perform 2-9 on $U$ and $V$ to obtain the corresponding indices $\mathbf{s}_A$ and $\mathbf{s}_B$.
\State
$M_A=A(:, \mathbf{p}) \backslash\left(A / A\left(\mathbf{s}_A,:\right)\right)$,
$M_B=B(:, \mathbf{p}) \backslash\left(B / B\left(\mathbf{s}_B,:\right)\right)$.
\end{algorithmic}
\end{algorithm}
\hskip 2em
In terms of computational complexity, the computation of the GSVD requires $\mathcal{O}((m+n+d)n^2)$, while the DEIM procedure costs
$\mathcal{O}((m+n+d)k^2)$.
Therefore, the overall complexity of Algorithm \ref{Al-DEIM-GCUR} is dominated by the construction of the GSVD.
Nevertheless, this computational cost can be prohibitively expensive when the dimensions are very large, making it difficult for large-scale applications.
To tackle the large-scale problems where a full GSVD may not be affordable, we turn to
the randomized algorithms \cite{wei2021CAMCrandomized,halko2011SIAMfinding}, which are typically computationally efficient and easy to implement.
Moreover, they have favorable numerical properties such as stability, and allow for restructuring computations in ways that make them amenable to implementation in a variety of settings including parallel computations.
Following this success, and building on the random sampling techniques \cite{halko2011SIAMfinding}, we develop randomized algorithms for efficiently computing the GCUR, and a more exhaustive treatment for our randomized approaches-including pseudocode, and the detailed error analysis will be discussed in the following work.
\subsection{Randomization for DEIM Based GCUR}\label{subsection: Randomization for the DEIM based GCUR}
\hskip 2em
As concluded in \cite{halko2011SIAMfinding}, the task of computing a low-rank
approximation to a given matrix $A$ can be split naturally into two computational stages.
The first stage is to construct a low-dimensional subspace that captures the action of the input matrices, which can be executed very efficiently with random sampling methods.
In other words, we require a matrix $Q$ for which
\begin{equation*
Q\ \mathrm{has}\ \mathrm{orthonormal}\ \mathrm{columns}\ \mathrm{and}\
A\approx QQ^{\mathrm{T}}A.
\end{equation*}
The second is to restrict the matrix to the subspace and then compute a standard factorization (QR, SVD, etc.) of the reduced matrix,
and it can be completed with well-established deterministic methods.
Here we wish to compute the approximate GSVD of the input pair $(A,B)$, where $A\in \mathbb{R}^{m\times n}$, $B\in \mathbb{R}^{d\times n}$ with $m \geq n$, such that
\begin{equation}\label{RGSVD}
\left[\begin{array}{l}
B \\
A
\end{array}\right] \approx\left[\begin{array}{c}
B \\
Q Q^{\mathrm{T}} A
\end{array}\right]=\left[\begin{array}{ll}
V & \\
& U
\end{array}\right]\left[\begin{array}{l}
\Sigma \\
\Gamma
\end{array}\right]
Y^{\mathrm{T}}.
\end{equation}
This goal can be achieved after five simple steps \cite{wei2016SIAMtikhonov}:
\hskip 2em 1. Generate an $n \times (k+p)$ Gaussian random matrix $\Omega$;
\hskip 2em 2. Form the $m\times (k+p)$ matrix $K=A\Omega$;
\hskip 2em 3. Compute the $m\times (k+p)$ orthonormal matrix $Q$ via the QR factorization
$K = QR$;
\hskip 2em 4. Compute the GSVD of $(Q^{\mathrm{T}}A,B)$:
$\left[\begin{array}{c} B \\ Q^{\mathrm{T}} A \end{array}\right]
=\left[\begin{array}{ll} V & \\ & W \end{array}\right]
\left[\begin{array}{l} \Sigma \\ \Gamma \end{array}\right] Y^{\mathrm{T}}$;
\hskip 2em 5. Form the $m\times (r+p)$ matrix $U = QW$.
By \cite{halko2011SIAMfinding}, the above operations generates (\ref{RGSVD}) with the error $E=A-QQ^{\mathrm{T}}A$ saitisfying
\begin{equation}\label{Err of RGSVD}
\left\|E\right\|
\leq
\left(1+6 \sqrt{(k+p) p \log p}\right) {\sigma}_{k+1}(A)
+3 \sqrt{k+p} \sqrt{\sum_{j>k} {\sigma}_j^2(A)}
\end{equation}
with probability not less than $1-3p^{-p}$, where $\sigma_{j}(A)$ is the $j$th largest singular value of $A$.
Here $p$ is the oversampling parameter, which usually determines that small number of columns are added to provide flexibility \cite{halko2011SIAMfinding}, and its selection is crucial for the effectiveness of the randomized algorithms.
The main computational cost for the randomized approach is the computation of GSVD for the much smaller matrix pair $(Q^{\mathrm{T}}A,B)$.
\hskip 2em
Combining the randomized GSVD algorithm with the DEIM technique, we present our randomized algorithm for computing the GCUR decomposition in Algorithm \ref{Al-R-DEIM-GCUR}.
\begin{algorithm}[htb]
\caption{DEIM based GCUR randomized algorithm \cite{gidisu2022SIAMgeneralized}}
\label{Al-R-DEIM-GCUR}
\hspace*{0.02in} {\bf Input:}
$A \in \mathbb{R}^{m \times n}$
and
$B \in \mathbb{R}^{m \times n}$ with $m\ge n$ and $d\ge n$, desired rank $k$, and the oversampling
\hspace*{0.02in} parameter $p$.
\\
\hspace*{0.02in} {\bf Output:}
A rank-$k$ GCUR decomposition\\
\hspace*{0.02in}
$\hat{A}=A(:,\mathbf{p}) \cdot M_A \cdot A(\mathbf{s}_A,:)$,
$\hat{B}=B(:,\mathbf{p}) \cdot M_B \cdot B(\mathbf{s}_B,:)$.
\begin{algorithmic}[1]
\State Generate an $n\times (k+p)$ Gaussian random matrix $\Omega$.
\State Form the $m\times (k+p)$ matrix $K=A\Omega$.
\State Compute the $m\times (k+p)$ orthonormal matrix $Q$ via the QR factorization
$K = QR$.
\State Compute the GSVD of $(B,Q^{\mathrm{T}}A)$:
$\left[\begin{array}{c} B \\ Q^{\mathrm{T}} A \end{array}\right]
=\left[\begin{array}{ll} V & \\ & W \end{array}\right]
\left[\begin{array}{l} \Sigma \\ \Gamma \end{array}\right] Y^{\mathrm{T}}$.
\State Form the $m\times (r+p)$ matrix $U = QW$.
\State $y=Y(:, 1)$.
\State $p_{1}=\operatorname{argmax}_{1 \leq i \leq n}\left|y_{i}\right|$.
\For{$j=2, \ldots, k$}
\State $y=Y(:, j)$.
\State $c=Y(\mathbf{p}, 1: j-1)^{-1} y(\mathbf{p})$.
\State $r=y-Y(:, 1: j-1) {c}$.
\State $p_{j}=\operatorname{argmax}_{1 \leq i \leq n}\left|{r}_{i}\right|$.
\State
$\mathbf{p}=\left[\begin{array}{ll}\mathbf{p} & p_{j}\end{array}\right]$.
\EndFor
\State
Perform 2-9 on $U$ and $V$ to obtain the corresponding indices $\mathbf{s}_A$ and $\mathbf{s}_B$.
\State
Compute
$M_A=A(:, \mathbf{p}) \backslash\left(A / A\left(\mathbf{s}_A,:\right)\right)$,
$M_B=B(:, \mathbf{p}) \backslash\left(B / B\left(\mathbf{s}_B,:\right)\right)$.
\end{algorithmic}
\end{algorithm}
In this algorithm, we first exploit the randomization techniques in \cite{wei2016SIAMtikhonov} to accelerate the process of the GSVD to obtain the generalized singular vectors.
Then we use the DEIM index selection procedure, operating on the approximate generalized singular vector matrices to determine the selected columns and rows.
We note that we can parallelize the work in lines 7 to 15 since it consists of three independent runs of DEIM.
Also, as noted in \cite{gidisu2022SIAMgeneralized}, if we are only
interested in approximating the matrix $A$ from the pair $(A, B)$, we can omit the manipulation on $V$.
Meanwhile, one can observe that the dominant cost of the randomized algorithm lies in computing the GSVD of matrix pair $(Q^{\mathrm{T}}A,B)$, and it is much lower than its counterpart in the non-random algorithm.
Consequently, from a practical perspective, Algorithm \ref{Al-R-DEIM-GCUR} is extremely simple to implement and can greatly reduce the computational time.
The following work will give performance guarantees by quantifying the error of the rank-$k$ GCUR decomposition
$\hat{A}=C_A M_A R_A = A(:, \mathbf{p}) \cdot M_A \cdot A\left(\mathbf{s}_A,:\right)$
and
$\hat{B}=C_B M_B R_B = B(:, \mathbf{p}) \cdot M_B \cdot B\left(\mathbf{s}_B,:\right)$.
\hskip 2em
Consistent with (\ref{GSVD of A}) and (\ref{GSVD of B}), let the number pairs $\{(\gamma_i, \beta_i)\}_{i=1}^n$ be the generalized singular values of the matrix pair $(A,B)$, where we we maintain the ratios $\gamma_i / \beta_i$ in a non-increasing order.
As described in Algorithm \ref{Al-R-DEIM-GCUR}, the matrix pair $(A,B)$ owns the approximate GSVD
\begin{equation*}
QQ^{\mathrm{T}}A=U\Gamma Y^{\mathrm{T}} \qquad
\mathrm{and} \qquad
B=V\Sigma Y^{\mathrm{T}},
\end{equation*}
where
$\Gamma=\operatorname{diag}(\tilde{\gamma}_1, \ldots, \tilde{\gamma}_n)$,
$\Sigma=\operatorname{diag}(\tilde{\beta}_1, \ldots, \tilde{\beta}_n)$,
and the ratios $\tilde{\gamma}_i / \tilde{\beta}_i$ are in a non-increasing order,
and the approximation error satisfies (\ref{Err of RGSVD}) with failure probability not exceeding $3p^{-p}$.
Partition the matrices:
\begin{equation}\label{partition of GCUR eq1}
U=\left[ \begin{array}{ll} U_k &\widehat{U} \end{array}\right],~~~
V=\left[ \begin{array}{ll} V_k & \widehat{V} \end{array}\right],~~~
Y=\left[\begin{array}{ll} Y_k & \widehat{Y} \end{array}\right],
\end{equation}
\begin{equation}\label{partition of GCUR eq2}
\Gamma=\operatorname{diag}\left(\Gamma_k, ~\widehat{\Gamma}\right),~~~
\Sigma=\operatorname{diag}\left(\Sigma_k, ~\widehat{\Sigma}\right),
\end{equation}
where matrices $U_k$, $V_k$, and $Y_k$ contain the first $k$ columns of $U$, $V$, and $Y$ respectively.
For our analysis, instead of $Y$, we use its orthonormal QR factor $H$
from the QR decomposition of $Y$:
\begin{equation}\label{partition of GCUR eq3}
\left[\begin{array}{ll}
Y_k & \widehat{Y}
\end{array}\right]=Y=H T=\left[\begin{array}{ll}
H_k & \widehat{H}
\end{array}\right]\left[\begin{array}{cc}
T_k & T_{12} \\
0 & T_{22}
\end{array}\right]=\left[\begin{array}{ll}
H_k T_k & H \widehat{T}
\end{array}\right],
\end{equation}
with $\widehat{T} =\left[\begin{array}{l}
T_{12} \\
T_{22}
\end{array}\right].$
This implies that
QQ^{\mathrm{T}}A
=U_k \Gamma_k Y_k^{\mathrm{T}}+\widehat{U} \widehat{\Gamma} \widehat{Y}^{\mathrm{T}}
=U_k \Gamma_k T_k^{\mathrm{T}} H_k^{\mathrm{T}}+\widehat{U} \widehat{\Gamma} \widehat{T}^{\mathrm{T}} H^{\mathrm{T}}.
$
With the above preparation, the following theorem derives the error bound for $\|A-\hat{A}\|$.
\begin{theorem}\label{Err of R-DEIM-GCUR}
Suppose $A\in\mathbb{R}^{m\times n},~B\in\mathbb{R}^{d\times n}$ and both are of full column rank, and let matrix pair $(\hat{A}, \hat{B})$ be a rank-$k$ GCUR decomposition for matrix pair $(A,B)$ computed by Algorithm \ref{Al-R-DEIM-GCUR}. Let
$\Theta_k = (1+6 \sqrt{(k+p) p \log p}) \sigma_{k+1}(A)+3 \sqrt{k+p} \sqrt{\sum_{j>k} \sigma_j^2(A)}$, and
$\eta_k = \sqrt{\frac{nk}{3}}2^k + \sqrt{\frac{mk}{3}}2^k$.
Then
\begin{equation}\label{Err of R-DEIM-GCUR for A}
\|\hat{A}-A \|
\le
\eta_k
\left[
\Theta_k
+
\left(\left\|A\right\|+\left\|B\right\|\right)
\left(
\frac{\gamma_{k+1}}{\beta_{k+1}} + \frac{\Theta_k}{\beta_{k+1}}
\left\|
\left(\begin{array}{l}
A \\
B
\end{array}\right)^{\dagger}
\right\|
\right)
\right]
\end{equation}
holds with probability not less than $1-3p^{-p}$,
where the number pair $(\gamma_{k+1},\beta_{k+1})$
is defined in (\ref{GSVD of B})and (\ref{GSVD of A}), and both ${\gamma}_i / {\beta}_i$ and $1/\beta_{i}$ are in a non-increasing order.
\end{theorem}
\begin{proof}
By the definition of $M_A$, we have
\begin{equation*}
A-C_A M_A R_A=A-C_AC_A^{\dagger}AR_A^{\dagger}R_A=
(I-C_A C_A^{\dagger}) A+C_A C_A^{\dagger} A(I-R_A^{\dagger} R_A).
\end{equation*}
Since $C_AC_A^{\dagger}$ is an orthogonal projection, it directly follows that
\begin{equation}\label{Proof R-DEIM-GCUR eq1}
\begin{aligned}
\| A-C_AM_AR_A \|
\le&
\| (I-C_A C_A^{\dagger}) A \|+ \| A(I-R_A^{\dagger} R_A)\|.
\end{aligned}
\end{equation}
According to \cite[Lemma 3.2]{sorensen2016SIAMdeim}, the column and row indices $\mathbf{s}_A$ and $\mathbf{p}$ give the full rank matrices $C_A=AS_A$ and $R_A=P^{\mathrm{T}}A$ where
$S_A = I(:, \mathbf{s}_A)$ and $P=I(:,\mathbf{p})$.
Let
$\mathbb{P}=P(H_k^{\mathrm{T}}P)^{-1}H_k^{\mathrm{T}}$ and
$\mathbb{S}=U_k(S_A^{\mathrm{T}}U_k)^{-1}S_A^{\mathrm{T}}.$
Then using the result in \cite[Proposition 4.7]{gidisu2022SIAMgeneralized}, we get
\begin{equation*}
\|(I-C_AC_A^{\dagger})A\|
\le
\|A(I-\mathbb{P})\|,~~~
\|A(I-R_A^{\dagger} R_A)\|
\le
\|(I-\mathbb{S})A\|.
\end{equation*}
Note that $U_k^{\mathrm{T}}U_k=I$ and $H_k^{\mathrm{T}}H_k=I$.
Then according to \cite[Lemma 4.1]{sorensen2016SIAMdeim}, we obtain that
\begin{equation}\label{Proof R-DEIM-GCUR eq2}
\begin{aligned}
\|(I-C_AC_A^{\dagger})A\|
\le&
\|(H_k^{\mathrm{T}}P)^{-1}\| \|A(I-H_k H_k^{\mathrm{T}})\|
\\ \le &
\|(H_k^{\mathrm{T}}P)^{-1}\| \left( \|E\| +\|QQ^{\mathrm{T}}A\left(I-H_kH_k^{\mathrm{T}}\right)\| \right),
\end{aligned}
\end{equation}
where we use $\left\|I-H_kH_k^{\mathrm{T}}\right\|=1.$
Analogous operation gives that
\begin{equation}\label{Proof R-DEIM-GCUR eq3}
\|A(I-R_A^{\dagger}R_A)\|
\le
\|(S_A^{\mathrm{T}}U_k)^{-1}\|
\left( \|E\| + \|(I-U_kU_k^{\mathrm{T}})QQ^{\mathrm{T}}A\| \right).
\end{equation}
Note that
\begin{equation*}
\begin{aligned}
QQ^{\mathrm{T}}A H_k H_k^{\mathrm{T}}
&=
\left[\begin{array}{ll}
U_k & \widehat{U}
\end{array}\right]
\left[\begin{array}{cc}
\Gamma_k & 0 \\
0 & \widehat{\Gamma}
\end{array}\right]
\left[\begin{array}{cc}
T_k^{\mathrm{T}} & 0 \\
T_{12}^{\mathrm{T}} & T_{22}^{\mathrm{T}}
\end{array}\right]
\left[\begin{array}{c}
I_k \\
0
\end{array}\right] H_k^{\mathrm{T}}
\\&=
U_k \Gamma_k T_k^{\mathrm{T}} H_k^{\mathrm{T}}
+\widehat{U} \widehat{\Gamma} T_{12}^{\mathrm{T}} H_k^{\mathrm{T}},
\end{aligned}
\end{equation*}
and hence,
\begin{equation*}
QQ^{\mathrm{T}}A\left(I-H_k H_k^{\mathrm{T}}\right)
=\widehat{U} \widehat{\Gamma} \widehat{T}^{\mathrm{T}} H^{\mathrm{T}}
-\widehat{U} \widehat{\Gamma} T_{12}^{\mathrm{T}} H_k^{\mathrm{T}}
=\widehat{U} \widehat{\Gamma} T_{22}^{\mathrm{T}} \widehat{H}^{\mathrm{T}}.
\end{equation*}
Similarly, it holds that
\begin{equation*}
\left(I-U_k U_k^T\right)QQ^{\mathrm{T}}A
=QQ^{\mathrm{T}}A-U_k \Gamma_k Y_k^{\mathrm{T}}
=\widehat{U} \widehat{\Gamma} \widehat{Y}^{\mathrm{T}}
=\widehat{U} \widehat{\Gamma} \widehat{T}^T H^{\mathrm{T}}.
\end{equation*}
Therefore,
\begin{equation}\label{Proof R-DEIM-GCUR eq4}
\|QQ^{\mathrm{T}}A(I-HH_k^{\mathrm{T}})\|
=
\|\widehat{U} \widehat{\Gamma} T_{22}^{\mathrm{T}} \widehat{H}^{\mathrm{T}}\|
\le
\tilde{\gamma}_{k+1}\|T_{22}\|
\le
\tilde{\gamma}_{k+1}\|\widehat{T}\|,
\end{equation}
\begin{equation}\label{Proof R-DEIM-GCUR eq5}
\|(I-U_kU_k^{\mathrm{T}})QQ^{\mathrm{T}}A\|
\le
\tilde{\gamma}_{k+1}\|\widehat{T}\|.
\end{equation}
To bound $\|\widehat{T}\|$, recall the result in \cite[Theorem 2.3]{hansen1998SIAMrank} that
$\|Y\|\le\|QQ^{\mathrm{T}}A\|+\|B\|$.
Given the partitioning and QR factorization of $Y$, we have
\begin{equation}\label{Proof R-DEIM-GCUR eq6}
\|\widehat{T}\|=\|H \widehat{T}\|=\|\widehat{Y}\| \leq\|Y\| \leq\|QQ^{\mathrm{T}}A\|+\|B\|
\le \|A\|+\|B\|.
\end{equation}
For the DEIM selection scheme, \cite[Lemma 4.4]{sorensen2016SIAMdeim} derives the bound
\begin{equation}\label{Proof R-DEIM-GCUR eq8}
\left\|\left(H_k^{\mathrm{T}} P\right)^{-1}\right\|<\sqrt{\frac{n k}{3}} 2^k,
\quad
\text { and }
\left\|\left(S_A^{\mathrm{T}} U_k\right)^{-1}\right\|<\sqrt{\frac{m k}{3}} 2^k.
\end{equation}
Inserting (\ref{Proof R-DEIM-GCUR eq2})-(\ref{Proof R-DEIM-GCUR eq8}) and into (\ref{Proof R-DEIM-GCUR eq1}),
we obtain
\begin{equation}\label{Proof R-DEIM-GCUR eq9}
\|\hat{A}-A \|
\le
\eta_k \left(
\|E\| +
\tilde{\gamma}_{k+1}
\left(\left\|A\right\|+\left\|B\right\|\right)
\right),
\end{equation}
where $\tilde{\gamma}_{k+1}$ is the $(k+1)$th diagonal entry of $\Gamma$ with a non-increasing order.
\hskip 2em
Recall the perturbation results for the generalized singular values in \cite[Theorem 3]{sun1982MNSperturbation}
\begin{equation*}
\left|\tilde{\gamma}_i\beta_i-\tilde{\beta}_i\gamma_i\right|
\le
\left\|
\left(\begin{array}{l}
E \\
F
\end{array}\right)
\right\|
\cdot
\left\|
\left(\begin{array}{l}
A \\
B
\end{array}\right)^{\dagger}
\right\|, \quad 1 \leqslant i \leqslant n,
\end{equation*}
where matrices $E$ and $F$ are the perturbations to $A$ and $B$, respectively.
Clearly, we have $F=0$ for our randomized algorithm.
As a result, we have
\begin{equation}\label{Err of tilde alpha_k+1}
\tilde{\gamma}_{k+1}
\le
\frac{1}{\beta_{k+1}}
\left(
\gamma_{k+1}
+
\|E\|
\cdot
\left\|
\left(\begin{array}{l}
A \\
B
\end{array}\right)^{\dagger}
\right\|
\right).
\end{equation}
We finish the proof by combining (\ref{Proof R-DEIM-GCUR eq9}), (\ref{Err of tilde alpha_k+1}) and the probabilistic error bound (\ref{Err of RGSVD}).
\end{proof}
Because the ratios ${\gamma}_i / {\beta}_i$ and $1/\beta_{i}$ are maintained in a non-increasing order, the right-hand side of (\ref{Err of R-DEIM-GCUR for A}) decreases as the target rank $k$ increases.
Note that the randomized GSVD algorithm provides an exact decomposition of $B$, the error bound for $\|B-\hat{B }\|$ in \cite{gidisu2022SIAMgeneralized} still holds that
\begin{equation*}
\begin{aligned}
\left\|B-C_B M_B R_B\right\|
& \leq
\|\left(H_k^{\mathrm{T}} P\right)^{-1}\|
\cdot
\left\|T_{22}\right\|
+\|\left(S_B^{\mathrm{T}} V_k\right)^{-1}\|
\cdot
\|\widehat{T}\| \\
& \leq
\left(\|\left(H_k^{\mathrm{T}} P\right)^{-1}\|
+\|\left(S_B^{\mathrm{T}} V_k\right)^{-1}\|\right)
\cdot\|\widehat{T}\| \\
&\leq
\left(\sqrt{\frac{n k}{3}} 2^k + \sqrt{\frac{d k}{3}} 2^k \right)
\left(\|A\|+\|B\|\right).
\end{aligned}
\end{equation*}
Compared with the error bound of $\|A-C_A M_A R_A\|$ under the non-random scheme in \cite{gidisu2022SIAMgeneralized} that
\begin{equation*}
\|A-C_A M_A R_A\| \le {\gamma}_{k+1}(\|A\|+\|B\|) \cdot \eta_k,
\end{equation*}
(\ref{Err of R-DEIM-GCUR for A}) involves a truncation term
$\Theta_k$ due to the randomization of the GSVD, and consequently, our randomized approach works well for matrices whose singular
values exhibit some decay.
\subsection{Randomization for L-DEIM Based GCUR}
\hskip 2em
To further improve the efficiency of our randomized algorithm, we now turn our gaze to combining the random sampling methods with the L-DEIM algorithm, which we will see can yield acceptable error bounds with high probability at a lower computational cost. We present this scheme in Algorithm \ref{Al-R-LDEIM-GCUR} and give a similar probabilistic error estimate in Theorem \ref{Err of R-LDEIM-GCUR}.
\begin{algorithm}[htb]
\caption{L-DEIM based GCUR randomized algorithm}
\label{Al-R-LDEIM-GCUR}
\hspace*{0.02in} {\bf Input:}
$A \in \mathbb{R}^{m \times n}$
and
$B \in \mathbb{R}^{m \times n}$ with $m\ge n$ and $d\ge n$, desired rank $k$, the oversampling
\hspace*{0.02in} parameter $p$ and the specified parameter $\widehat{k}$.
\\
\hspace*{0.02in} {\bf Output:}
A rank-$k$ GCUR decomposition\\
\hspace*{0.02in}
$\hat{A}=A(:,\mathbf{p}) \cdot M_A \cdot A(\mathbf{s}_A,:)$,
$\hat{B}=B(:,\mathbf{p}) \cdot M_B \cdot B(\mathbf{s}_B,:)$.
\begin{algorithmic}[1]
\State Generate an $n\times (\widehat{k}+p)$ Gaussian random matrix $\Omega$.
\State Form the $m\times (\widehat{k}+p)$ matrix $K=A\Omega$.
\State Compute the $m\times (\widehat{k}+p)$ orthonormal matrix $Q$ via the QR factorization
$K = QR$.
\State Compute the GSVD of $(Q^{\mathrm{T}}A,B)$:
$\left[\begin{array}{c} B \\ Q^{\mathrm{T}} A\end{array}\right]
=\left[\begin{array}{ll} V & \\ & W \end{array}\right]
\left[\begin{array}{l}\Sigma \\ \Gamma \end{array}\right] Y^{\mathrm{T}}$.
\State Form the $m\times (\widehat{k}+p)$ matrix $U = QW$.
%
\For{$j=1,\ldots,\widehat{k}$}
\State $\mathbf{p}(j)=\operatorname{argmax}_{1 \leq i \leq n}\left|(Y(:, j))_{i}\right|$.
\State $Y(:, j+1)=Y(:, j+1)-Y(:, 1: j) \cdot(Y(\mathbf{s}, 1: j) \backslash Y(\mathbf{p}, j+1))$.
\EndFor
\State Compute $\ell_{i}=\left\|[Y]_{i:}\right\|^2 \quad$ for $i=1, \ldots, n$.
\State Sort $\ell$ in non-increasing order.
\State Remove entries in $\ell$ corresponding to the indices in $\mathbf{p}$.
\State $\mathbf{p}^{\prime}=k-\widehat{k}$ indices corresponding to $k-\widehat{k}$ largest entries of $\ell$.
\State $\mathbf{p}=\left[\mathbf{p} ; \mathbf{p}^{\prime}\right]$.
%
\State
Perform 6-14 on $U$ and $V$ to obtain the corresponding indices $\mathbf{s}_A$ and $\mathbf{s}_B$.
\State
Compute
$M_A=A(:, \mathbf{p}) \backslash\left(A / A\left(\mathbf{s}_A,:\right)\right)$,
$M_B=B(:, \mathbf{p}) \backslash\left(B / B\left(\mathbf{s}_B,:\right)\right)$.
\end{algorithmic}
\end{algorithm}
\begin{theorem}\label{Err of R-LDEIM-GCUR}
Let the matrix pair $(\hat{A},\hat{B})$ be a rank-$\widehat{k}$ GCUR approximation for pair $(A,B)$ computed by Algorithm \ref{Al-R-LDEIM-GCUR}.
Suppose that
$\Theta_{\widehat{k}} = \left(1+6 \sqrt{({\widehat{k}}+p) p \log p}\right) \sigma_{{\widehat{k}}+1}(A)+3 \sqrt{{\widehat{k}}+p} \sqrt{\sum_{j>{\widehat{k}}} \sigma_j^2(A)}$, and
$\eta_{\widehat{k}} = \sqrt{\frac{n{\widehat{k}}}{3}}2^{\widehat{k}} + \sqrt{\frac{m{\widehat{k}}}{3}}2^{\widehat{k}}$,
and then the following error bound
\begin{equation*}
\|\hat{A}-A \|
\le
\eta_{\widehat{k}}\left[
\Theta_{\widehat{k}}
+ \left(\left\|A\right\|+\left\|B\right\|\right)
\left(
\frac{\gamma_{\widehat{k}+1}}{\beta_{{\widehat{k}}+1}} + \frac{\Theta_{\widehat{k}}}{\beta_{{\widehat{k}}+1}}
\left\|
\left(\begin{array}{l}
A \\
B
\end{array}\right)^{\dagger}
\right\|
\right)
\right],
\end{equation*}
fails with probability not exceeding than $3p^{-p}$ and ${\gamma_i}/{\beta_i}$ and ${1}/{\beta_i}$ are in a non-increasing order.
\end{theorem}
\section{Randomization for RSVD-CUR}
\hskip 2em
Real-world data sets often comprise different representations or views, which provide information complementary to each other.
The canonical correlation analysis (CCA) \cite{johnson2002PHUSEapplied} is one of the most common and useful techniques for multi-data processing.
Motivated by the CCA, Gidisu and Hochstenbach \cite{gidisu2022ARXIVrsvd} generalized the DEIM-type CUR to
a new coordinated CUR factorization of a matrix triplet $(A, B, G)$ of compatible dimensions based on the RSVD, which was called the RSVD-CUR decomposition.
Analogous to CCA, an RSVD-CUR factorization as a tool for multi-view dimension reduction can cope with a two-view case.
Furthermore, in the same context, one can use an RSVD-CUR as a supervised feature
selection technique in multilabel classification problems and it can also applied to cases where the the goal is to select a subset of rows and or columns of one data set relative to two other data sets.
In this section, we introduce new randomized algorithms for computing the RSVD-CUR decomposition where we apply the L-DEIM scheme and the random sampling techniques.
Detailed error analysis which provides insight into the accuracy of the algorithms and the choice of the algorithmic parameters is given.
\subsection{RSVD-CUR}
\hskip 2em
We now give a brief overview of the RSVD-CUR of a matrix triplet $(A, B, G)$ with
$A\in\mathbb{R}^{m\times n}$
$B\in\mathbb{R}^{m\times \ell}$, and
$G\in\mathbb{R}^{d\times n}$ ($\ell \ge d \ge m \ge n$) where $B$ and $G$ are of full rank.
Then a rank-$k$ RSVD-GCUR approximation of $(A,B,G)$ is defined as
\begin{equation}\label{RSVD-CUR of (A,B,G)}
\begin{aligned}
A & \approx C_A M_A R_A=AP\ M_A\ S^{\mathrm{T}} A, \\
B & \approx C_B M_B R_B=BP_B\ M_B\ S^{\mathrm{T}} B, \\
G & \approx C_G M_G R_G=GP\ M_G\ S_G^{\mathrm{T}} G.
\end{aligned}
\end{equation}
Here
$S\in \mathbb{R}^{m \times k}$, $S_G \in \mathbb{R}^{d \times k},$
$P\in \mathbb{R}^{n \times k}$, and $P_B\in \mathbb{R}^{\ell \times k}$
are index selection matrices with some columns of the identity that select rows and columns of the respective matrices.
It is key that the same rows of $A$ and $B$ are picked and the same columns of $A$ and $G$ are selected; this gives a coupling among the decompositions.
As a result, the RSVD-GCUR may be viewed as a CUR-type
decomposition of a matrix relative to two other matrices of compatible dimensions.
\hskip 2em
The matrices $C_A\in \mathbb{R}^{m \times k}$, $C_B\in \mathbb{R}^{m \times k}$, $C_G\in \mathbb{R}^{d \times k}$
and $R_A\in \mathbb{R}^{k \times n}$, $R_B\in \mathbb{R}^{k \times \ell}$, $R_G\in \mathbb{R}^{k \times n}$ are subsets of the columns and rows, respectively, of the given matrices.
Let the vectors $\mathbf{s}$, $\mathbf{s}_G$, $\mathbf{p}$ and $\mathbf{p}_B$ contain the indices of the selected rows and columns,
such that
$S=I(:,\mathbf{s})$, $S_G=I(:,\mathbf{s}_G)$,
$P=I(:,\mathbf{p})$, and $P_B=I(:,\mathbf{p}_B)$.
In \cite{gidisu2022ARXIVrsvd}, the choice of $\mathbf{s}$, $\mathbf{s}_G$, $\mathbf{p}$ and $\mathbf{p}_B$ is guided by the knowledge
of the orthogonal and nonsingular matrices from the rank-$k$ RSVD,
where the DEIM and L-DEIM algorithms are employed as the index selection strategies for finding the ``best'' row and column indices.
Specifically, suppose that the RSVD of $(A,B,G)$ are available, as shown in (\ref{RSVD of (A,B,G)}).
To construct a DEIM-type RSVD-CUR decomposition of a matrix pair $(A,B,G)$, given the target rank $k$, the DEIM operates on the first $k$ columns on matrices
$W$, $Z$, $U$ and $V$ to obtain the corresponding indices
$\mathbf{p}$, $\mathbf{s}$, $\mathbf{p}_B$ and $\mathbf{s}_G$.
Moreover, by utilizing the L-DEIM, one can use at least the first $k/2$ vectors of $W$, $Z$, $U$, and $V$ to obtain the indices, with the approximation quality as good as that of the DEIM-type RSVD-CUR, which is demonstrated numerically in \cite{gidisu2022ARXIVrsvd}.
\hskip 2em
It is clear that both the DEIM and the L-DEIM type RSVD-CUR decompositions require the inputs of the RSVD.
Nevertheless, computing this factorization can be a significant computational bottleneck in the large-scale applications.
How to reduce this computational cost and still ensure the accuracy of the approximation is our main concern.
Next, we introduce the randomized schemes for computing the RSVD-CUR decomposition, together with a detailed error analysis.
\subsection{Randomization for Restricted SVD}
\hskip 2em
The computation of the RSVD is still an active field of research; see some recent works\cite{chu2000SIAMcomputation,zhang2021NCneural,zwaan2020Arxivtowards}.
The RSVD can be considered as a double GSVD \cite{de1991SIAMrestricted}. We first compute the GSVD of $(A,G)$,
\begin{equation*}
A=U_1 \Gamma_1 Y_1^{\mathrm{T}},
\quad
G=V_1
\left[\begin{array}{l}
\Sigma_1 \\
0_{d-n,n}
\end{array}\right]
Y_1^{\mathrm{T}},
\end{equation*}
and then we compute the GSVD of $(B^\mathrm T U_1,\Sigma_1^{-1} \Gamma_1^\mathrm T)$, so that
\begin{equation*}
B^{\mathrm{T}} U_1 = U_2 \Gamma_2 Y_2^\mathrm{T},
\quad
\Sigma_1^{-1} \Gamma_1^\mathrm T = V_2 \Sigma_2 Y_2^\mathrm T.
\end{equation*}
The above two steps can be summarized in (\ref{matrix expresssion of RSVD}).
\begin{equation}\label{matrix expresssion of RSVD}
\begin{aligned}
{\left[\begin{array}{ll}
A & B \\
G &
\end{array}\right] }
&=
\left[\begin{array}{ll}
U_1 & \\
& V_1
\end{array}\right]
\left[\begin{array}{c:c}
\Gamma_1 & U_1^{\mathrm{T}} B \\
\hdashline \Sigma_1 & \\
0_{d-n, n} &
\end{array}\right]
\left[\begin{array}{ll}
Y_1^{\mathrm{T}} & \\
& I
\end{array}\right]\\
&=\left[\begin{array}{ll}
U_1 & \\
& V_1
\end{array}\right]
\left[\begin{array}{c:c}
\Gamma_1 \Sigma^{-1}_1 & U_1^\mathrm{T} B \\
\hdashline I & \\
0_{d-n, n} &
\end{array}\right]
\left[\begin{array}{ll}
\Sigma_1 Y_1^{\mathrm{T}} & \\
& I
\end{array}\right]\\
&=\left[\begin{array}{ll}
U_1 Y_2 & \\
& V_1
\end{array}\right]\left[\begin{array}{c:c}
\Sigma_2^{\mathrm{T}} & \Gamma_2^{\mathrm{T}} \\
\hdashline V_2 & \\
0_{d-n, n} &
\end{array}\right]\left[\begin{array}{ll}
V_2^{\mathrm{T}} \Sigma_1 Y_1^{\mathrm{T}} & \\
& U_2^{\mathrm{T}}
\end{array}\right]\\
&=\left[\begin{array}{ll}
U_1 Y_2 & \\
& V_1 \widehat{V}_2
\end{array}\right]
\left[\begin{array}{c:c}
\Sigma_2^{\mathrm{T}} \Gamma_G & \Gamma_2^{\mathrm{T}} \\
\hdashline \Gamma_G & \\
0_{d-n, n} &
\end{array}\right]
\left[\begin{array}{cc}
Y_1 \Sigma_1 V_2 \Gamma_G^{-1} & \\
& U_2
\end{array}\right]^{\mathrm{T}},
\end{aligned}
\end{equation}
where $\widehat{V}_2=\mathrm{diag}(V_2,I_{d-n})$.
Moreover, $\Gamma_G = \operatorname{diag}\left(\gamma_1, \ldots, \gamma_n\right) \in \mathbb{R}^{n \times n}$ is a scaling matrix that one can freely select (see, e.g., \cite{zwaan2020Arxivtowards}),
and critically, we keep the diagonal entries of $\Gamma_1$ and $\Gamma_2$ in
non-decreasing order while those of $\Sigma_1$ and $\Sigma_2$ are non-increasing.
In accordance with (\ref{RSVD of (A,B,G)}), one can define
\begin{equation}\label{component of R-RSVD}
Z \triangleq U_1Y_2,\
W \triangleq Y_1\Sigma_1V_2\Gamma_G^{-1},\
V \triangleq V_1 \widehat{V}_2,\
U \triangleq U_2,\
\end{equation}
\begin{equation*}
\left\{
\begin{aligned}
D_A &\triangleq \Sigma_2^{\mathrm{T}} \Gamma_G
=\left[\begin{array}{ccc}
\alpha_1 & & \\
& \ddots & \\
& & \alpha_n \\
\hdashline & 0_{m-n,n} &
\end{array}\right] \in \mathbb{R}^{ m\times n}, \\
D_B &\triangleq \Gamma_2^{\mathrm{T}}
=\left[\begin{array}{lll:l:l}
\beta_1 & & & & \\
& \ddots& & 0_{n, m-n} & 0_{n, l-m} \\
& & \beta_n & & \\
\hdashline
& 0_{m-n, n} & & I_{m-n} & 0_{m-n, l-m}
\end{array}\right] \in \mathbb{R}^{ m\times l}, \\
D_G & \triangleq \left[\begin{array}{c}
\Gamma_G \\
0_{d-n, n}
\end{array}\right]
=\left[\begin{array}{ccc}
\gamma_1 & & \\
& \ddots & \\
& & \gamma_n \\
\hdashline & 0_{d-n,n} &
\end{array}\right] \in \mathbb{R}^{d\times n}.
\end{aligned}
\right.
\end{equation*}
Denote $\Sigma_2 = \operatorname{diag}\left(\sigma_1, \ldots, \sigma_n\right) \in \mathbb{R}^{n \times m}$.
Here we choose
$\gamma_i=\frac{\sigma_i}{\sqrt{\sigma_i^2+1}}$ for $i=1,\ldots, n,$ which are ordered non-increasingly
( since $f(x)=x\left(x^2+1\right)^{-1 / 2}$ is a strictly increasing function)
and it implies that
$\alpha_i=\frac{\sigma_i^2}{\sqrt{\sigma_i^2+1}}$.
Given that $\beta_i^2+\sigma_i^2=1$ from the second GSVD, we have that
$\alpha_i^2+\beta_i^2+\gamma_i^2=1$ for $i=1, \ldots, n$.
Note that $B$ and $G$ are of full rank,
$1>\alpha_i\ge\alpha_{i+1}>0$, $1>\gamma_i\ge\gamma_{i+1}>0$ and $0<\beta_i\le\beta_{i+1}<1$.
\hskip 2em
We now proceed to propose a fast randomized algorithm for computing the RSVD.
The main idea of our approach is to accelerate this computational process by exploiting the randomized GSVD algorithm and its analysis relies heavily on the results introduced in Subsection \ref{subsection: Randomization for the DEIM based GCUR}.
Firstly, an orthonormal matrix $H_1\in\mathbb{R}^{d\times (k+p_1)}$ is generated to satisfy $\left\|G-H_1 H_1^{\mathrm{T}} G\right\| \leq c {\sigma}_{k+1}$
with high probability, where $\sigma_{k+1}$ is the
$(k+1)$th largest singular value of $G$ and $c$ is a constant depending on $k$ and $p_1$.
Here $p_1$ is the oversampling parameter, which is used to provide flexibility \cite{halko2011SIAMfinding}.
According to (\ref{matrix expresssion of RSVD}), $\Sigma_1$ is required to be square, hence, here we fix that $p_1=n-k$.
By performing the GSVD of
$[(H_1^{\mathrm{T}} G)^{\mathrm{T}}, A^{\mathrm{T}}]^{\mathrm{T}}$,
we get the approximate GSVD of $[G^{\mathrm{T}}, A^{\mathrm{T}}]^{\mathrm{T}}$,
\begin{equation}\label{R-RSVD-GSVD Stage1}
\left[\begin{array}{l}
A \\
G
\end{array}\right] \approx\left[\begin{array}{c}
A \\
H_1 H_1^{\mathrm{T}} G
\end{array}\right]=\left[\begin{array}{ll}
U_1 & \\
& V_1
\end{array}\right]\left[\begin{array}{l}
\Gamma_1 \\
\Sigma_1
\end{array}\right] Y_1^{\mathrm{T}}.
\end{equation}
When $m\gg n$, the computational advantage of (\ref{R-RSVD-GSVD Stage1}) becomes much more obvious.
Furthermore, we can formulate the approximate GSVD for the pair
$(B^{\mathrm{T}}U_1,\Sigma_1^{-1}\Gamma_1^\mathrm T)$
by performing the GSVD of the small-scale matrix
$[(H_2^{\mathrm{T}}B^{\mathrm{T}} U_1)^{\mathrm{T}}, (\Sigma_1^{-1} \Gamma_1^\mathrm T)^{\mathrm{T}}]^{\mathrm{T}}$,
where $H_2$ is a ${(k+p_2)\times n}$ orthonormal matrix, and $p_2$ is also an oversampling parameter.
Then we obtain
\begin{equation}\label{R-RSVD-GSVD Stage2}
\left[\begin{array}{c}
B^{\mathrm{T}}U_1 \\
\Sigma_1^{-1} \Gamma_1^\mathrm T
\end{array}\right] \approx\left[\begin{array}{c}
H_2H_2^{\mathrm{T}} (B^{\mathrm{T}}U_1) \\
\Sigma_1^{-1} \Gamma_1^\mathrm T
\end{array}\right]=\left[\begin{array}{ll}
U_2 & \\
& V_2
\end{array}\right]\left[\begin{array}{c}
\Gamma_2 \\
\Sigma_2
\end{array}\right] Y_2^{\mathrm{T}} .
\end{equation}
Finally, we can formulate the corresponding approximate RSVD of $(A,B,G)$,
\begin{equation}\label{R-RSVD}
A=Z D_A W^{\mathrm{T}},
\qquad
B\approx \tilde{B} = Z D_B U^{\mathrm{T}},
\qquad
G\approx \tilde{G} = V D_G W^{\mathrm{T}}.
\end{equation}
To be more clear in presentation, the above process can be expressed as follows:
\begin{equation*}
\begin{aligned}
{\left[\begin{array}{ll}
A & B \\
G &
\end{array}\right] }
&\approx
{\left[\begin{array}{ll}
A & B \\
H_1H_1^{\mathrm{T}}G &
\end{array}\right] }
=\left[\begin{array}{ll}
U_1 & \\
& V_1
\end{array}\right]\left[\begin{array}{ll}
\Gamma_1 & U_1^{\mathrm{T}} B \\
\Sigma_1 &
\end{array}\right]\left[\begin{array}{cc}
Y_1^{\mathrm{T}} & \\
& I
\end{array}\right]
\\
&=\left[\begin{array}{ll}
U_1 & \\
& V_1
\end{array}\right]\left[\begin{array}{cc}
\Gamma_1 \Sigma_1^{-1} & U_1^{\mathrm{T}} B \\
I &
\end{array}\right]\left[\begin{array}{cc}
\Sigma_1 Y_1^{\mathrm{T}} & \\
& I
\end{array}\right]
\\
&\approx
\left[\begin{array}{ll}
U_1 & \\
& V_1
\end{array}\right]\left[\begin{array}{cc}
\Gamma_1 \Sigma_1^{-1} & \left(U_1^{\mathrm{T}} B\right)H_2H_2^\mathrm T\ \\
I &
\end{array}\right]\left[\begin{array}{cc}
\Sigma_1 Y_1^{\mathrm{T}} & \\
& I
\end{array}\right]
\\
&=\left[\begin{array}{ll}
U_1 Y_2 & \\
& V_1
\end{array}\right]\left[\begin{array}{cc}
\Sigma_2^{\mathrm{T}} & \Gamma_2^{\mathrm{T}} \\
V_2 &
\end{array}\right]\left[\begin{array}{ll}
V_2^{\mathrm{T}} \Sigma_1 Y_1^{\mathrm{T}} & \\
& U_2^{\mathrm{T}}
\end{array}\right]\\
&=\left[\begin{array}{ll}
U_1 Y_2 & \\
& V_1 V_2
\end{array}\right]\left[\begin{array}{cc}
\Sigma_2^{\mathrm{T}} \Gamma_G & \Gamma_2^{\mathrm{T}} \\
\Gamma_G &
\end{array}\right]\left[\begin{array}{cc}
Y_1 \Sigma_1 V_2 \Gamma_G^{-1} & \\
& U_2
\end{array}\right]^{\mathrm{T}}\\
&\triangleq \left[\begin{array}{ll}
Z & \\
& V
\end{array}\right]\left[\begin{array}{ll}
D_A & D_B \\
D_G &
\end{array}\right]\left[\begin{array}{ll}
W & \\
& U
\end{array}\right]^{\mathrm{T}} .
\end{aligned}
\end{equation*}
We summarize the details in Algorithm \ref{Al-R-RSVD}.
Notice that (\ref{R-RSVD}) indicates that our randomized approach provides an exact factorization for $A$, which is a direct consequence of (\ref{R-RSVD-GSVD Stage1}),
while it does not hold for matrices $B$ and $G$.
We present a detailed analysis of the approximation error in the following theorem.
\begin{theorem}
Suppose that
$B \in \mathbb{R}^{m \times l}$ and $G \in \mathbb{R}^{d \times n}$ with $l\ge d\ge m \ge n$
and $p$ is an oversampling parameter.
Let $\tilde{B}$ and $\tilde{G}$ be the approximation of $B$ and $G$ computed by Algorithm \ref{Al-R-RSVD},
then
\begin{equation}\label{Err of R-RSVD-B}
\|B-\tilde{B}\|
\le
\left(1+6 \sqrt{(k+p) p \log p}\right) \sigma_{k+1}(B)+ 3 \sqrt{k+p} \sqrt{\sum_{j>k} \sigma_j^2(B)},
\end{equation}
\begin{equation}\label{Err of R-RSVD-G}
\|G-\tilde{G}\|
\le
\left(1+6 \sqrt{n (n-k) \log (n-k)}\right) \sigma_{k+1}(G) +3 \sqrt{n\sum_{j>k} \sigma_j^2(G)}
\end{equation}
hold with probability not less than $1-3p^{-p}$ and $1-(n-k)^{-(n-k)}$ respectively
\end{theorem}
\begin{proof}
Let $E_G$ and $E_B$ be the error matrices such that
\begin{equation}\label{Proof of R-RSVD eq1}
G=V_1\Sigma_1Y_1^{\mathrm{T}}+E_G,
\qquad
B^{\mathrm{T}}U_1=U_2\Gamma_2Y_2^{\mathrm{T}}+E_B,
\qquad
\Sigma_1^{-1} \Gamma_1^\mathrm T = V_2\Sigma_2Y_2^{\mathrm{T}}.
\end{equation}
Inserting (\ref{component of R-RSVD}) and (\ref{Proof of R-RSVD eq1}) into $\tilde{B}$ and $\tilde{G}$, we have
\begin{equation*}
B-\tilde{B}
=
B-ZD_BU^{\mathrm{T}}
=
B-(U_1Y_2)\Gamma_2^{\mathrm{T}}U_2^{\mathrm{T}}
=
B-U_1(U_1^{\mathrm{T}}B-E_B^{\mathrm{T}})
=
U_1E_B^{\mathrm{T}},
\end{equation*}
\begin{equation*}
G-\tilde{G}
=
G-VD_GW^{\mathrm{T}}
=
G-(V_1V_2)\Gamma_G(Y_1\Sigma_1V_2\Gamma_G^{-1})^{\mathrm{T}}
=
G-V_1\Sigma_1Y_1^{\mathrm{T}}
=
E_G.
\end{equation*}
During randomization for the GSVD of $(A, G)$, we set the oversampling parameter $p'=n-k$.
By the probabilistic error bound (\ref{Err of RGSVD}), we have
\begin{equation*}
\|G-\tilde{G}\| \le \|E_G\|
\le
\left(1+6 \sqrt{n (n-k) \log (n-k)}\right) \sigma_{k+1}(G)+3 \sqrt{n\sum_{j>k} \sigma_j^2(G)},
\end{equation*}
which holds with probability not less than $1-3(n-k)^{-(n-k)}$,
and similarly
\begin{equation*}
\begin{aligned}
\|B-\tilde{B}\| \le \|E_B\|
&\le
\left(1+6 \sqrt{(k+p) p \log p}\right) \sigma_{k+1}(U_1^{\mathrm{T}}B)+3 \sqrt{k+p} \sqrt{\sum_{j>k} \sigma_j^2(U_1^{\mathrm{T}}B)}
\\&\le
\left(1+6 \sqrt{(k+p) p \log p}\right) \sigma_{k+1}(B)+ 3 \sqrt{k+p} \sqrt{\sum_{j>k} \sigma_j^2(B)},
\end{aligned}
\end{equation*}
with probability not less than $1-p^{-p}$,
where we apply the result in \cite[Lemma 3.3.1]{horn1991PCtopics} that
$\sigma_j(U_1^{\mathrm{T}}B)\le \sigma_j(B)$
when $U_1$ is orthonormal.
\end{proof}
\begin{algorithm}[htb]
\caption{Randomized RSVD algorithm}
\label{Al-R-RSVD}
\hspace*{0.02in} {\bf Input:}
$A \in \mathbb{R}^{m \times n}$,
$B \in \mathbb{R}^{m \times \ell}$,
and $G \in \mathbb{R}^{d \times n}$, with
with $\ell \ge d \ge m\ge n$, desired rank $k$, and the oversampling parameter $p$.
\\
\hspace*{0.02in} {\bf Output:}
an RSVD of matrix triplet $(A,B,G)$,\\
\hspace*{0.02in}
$A=ZD_AW^{\mathrm{T}}$, $B\approx ZD_BU^{\mathrm{T}}$, $G\approx VD_GW^{\mathrm{T}}$.
\begin{algorithmic}[1]
\State Generate an $n\times n$ Gaussian random matrix $\Omega_1$.
\State Form the $d\times n$ matrix $G\Omega_1$.
\State Compute the $d\times n$ orthonormal matrix $H_1$ via the QR factorization
$G\Omega_1 = H_1R$.
\State Compute the GSVD of $(A,H_1^{\mathrm{T}}G)$:
$\left[\begin{array}{c} A \\ H_1^{\mathrm{T}}G\end{array}\right]
=\left[\begin{array}{ll}U_1 & \\ & \tilde{V}_1\end{array}\right]
\left[\begin{array}{l}\Gamma_1 \\ \Sigma_1 \end{array}\right] Y_1^{\mathrm{T}}$.
\State Form the $d\times n$ orthonormal matrix $V_1=H_1\tilde{V}_1$.
\State Form the $m\times k_2$ matrix $G\Omega_2$.
\State Compute the $(k_2+p)\times n$ orthonormal matrix $H_2$ via the QR factorization
$(B^{\mathrm{T}}U_1)\Omega_2 = H_2R$.
\State Compute the GSVD of
$\left[H_2^{\mathrm{T}}\left(B^{\mathrm{T}}U_1\right),\Sigma_1^{-1}\Gamma_1^{\mathrm{T}}\right]$:
$\left[\begin{array}{c}
H_2^{\mathrm{T}}\left(B^{\mathrm{T}}U_1\right) \\ \Sigma_1^{-1}\Gamma_1^{\mathrm{T}}
\end{array}\right]
=\left[\begin{array}{ll}\tilde{U}_2 & \\ & V_2 \end{array}\right]
\left[\begin{array}{l}\Gamma_2 \\ \Sigma_2 \end{array}\right] Y_2^{\mathrm{T}}$,
where $\Sigma_2=\operatorname{diag}\left(\sigma_1, \ldots, \sigma_n\right)$.
\State Form the $(k_2+p)\times k_2$ orthonormal matrix $U_2=H_2\tilde{U}_2$.
\State Form the diagonal matrix
$\Gamma_G=\mathrm{diag}(\gamma_1,\ldots, \gamma_n)$, $\gamma_i= \frac{\sigma_i}{\sqrt{\sigma_i^2+1}}$.
\State Form the orthonormal matrices
$U=U_2\in\mathbb{R}^{(k_2+p)\times k_2}$,
$V=V_1V_2\in\mathbb{R}^{d\times k_1}$,
diagonal matrices $D_A=\Sigma_2^{\mathrm{T}}\Gamma_G$,
$D_B=\Gamma_2^{\mathrm{T}}$, $D_G=\Gamma_G$,
and the nonsingular matrices $Z=U_1Y_2\in\mathbb{R}^{m\times m}$,
$W=Y_1\Sigma_1V_2\Gamma_G^{-1}\in\mathbb{R}^{n\times n}$.
\end{algorithmic}
\end{algorithm}
\subsection{Randomization for L-DEIM Based RSVD-CUR}
\hskip 2em
Now we are ready to establish an efficient procedure for computing an approximate RSVD-CUR decomposition, along with a theoretical analysis of its error bound.
Given a matrix triplet $(A,B,G)$, with
$A\in\mathbb{R}^{m\times n}$ ,
$B\in\mathbb{R}^{m\times l}$, and
$G\in\mathbb{R}^{d\times n}$ ($\ell \ge d \ge m \ge n$) where $B$ and $G$ are of full rank.
Our approach provides a rank-$k$ RSVD-CUR decomposition of the form (\ref{RSVD of (A,B,G)}), and the choice of indices $\mathbf{s}$, $\mathbf{s}_G$, $\mathbf{p}$, and $\mathbf{p}_B$ is guided by the knowledge of the orthonormal matrices and nonsingular matrices from the approximation of the rank-$\widehat{k}$ RSVD, where $\widehat{k}\le k$.
The details are summarized in Algorithm \ref{Al-R-LDEIM-RSVD-CUR}.
\hskip 2em
The innovation of our approach has two aspects.
First, we leverage the randomized algorithms (Algorithm \ref{Al-R-RSVD}) to accomplish the truncation procedure of the RSVD, where the random sampling technique can be used to identify a subspace that captures most of the action of a matrix.
As a result, a large-scale problem is projected randomly to a smaller subspace that contains the main information, and then we apply the deterministic algorithm to the
associated small-scale problem.
Consequently, an approximate rank-$\widehat{k}$ RSVD of the form (\ref{R-RSVD}) is obtained.
Second,
to further strengthen the efficiency of our algorithm scheme, we adopt the L-DEIM method for sampling instead of the DEIM.
As described in Subsection \ref{subsection: Subset selection procedure}, compared to the DEIM scheme, the L-DEIM procedure is
computationally more efficient and requires less than $k$ input vectors to select the indices.
\hskip 2em
We now provide a rough error analysis that shows that the accuracy of the proposed
algorithm is closely associated with the error of the approximation RSVD.
The analysis follows the results in \cite{sorensen2016SIAMdeim,gidisu2022SIAMgeneralized,gidisu2022ARXIVrsvd} with some necessary modifications.
We begin by partitioning the matrices in (\ref{R-RSVD})
\begin{equation*}
\begin{aligned}
U &=\left[\begin{array}{ll}
U_{\widehat{k}} & \widehat{U}
\end{array}\right], \quad V=\left[\begin{array}{ll}
V_{\widehat{k}} & \widehat{V}
\end{array}\right], \quad W=\left[\begin{array}{ll}
W_{\widehat{k}} & \widehat{W}
\end{array}\right], \quad Z=\left[\begin{array}{ll}
Z_{\widehat{k}} & \widehat{Z}
\end{array}\right], \\
D_A &=\operatorname{diag}\left(D_{A_{\widehat{k}}}, \widehat{D}_A\right), \quad
D_B=\operatorname{diag}\left(D_{B_{\widehat{k}}}, \widehat{D}_B\right), \quad
D_G=\operatorname{diag}\left(D_{G_{\widehat{k}}}, \widehat{D}_G\right),
\end{aligned}
\end{equation*}
where
$\widehat{D}_A \in \mathbb{R}^{(m-\widehat{k}) \times(n-\widehat{k})}$,
$\widehat{D}_B \in \mathbb{R}^{(m-\widehat{k}) \times(l-\widehat{k})}$, and
$\widehat{D}_G \in \mathbb{R}^{(d-\widehat{k}) \times(n-\widehat{k})}$.
As with the DEIM-type GCUR method in \cite{gidisu2022SIAMgeneralized}, the lack of orthogonality of the basis vectors in $W$ and $Z$ from the RSVD
necessitates some additional work.
Mimicking the techniques in \cite{gidisu2022ARXIVrsvd},
here we take a QR factorization of $W$ and $Z$ to obtain an orthonormal basis to facilitate the analysis,
\begin{equation*}
\begin{aligned}
{\left[\begin{array}{ll}
Z_{\widehat{k}} & \widehat{Z}
\end{array}\right]=Z=Q_Z T_Z=\left[\begin{array}{ll}
Q_{Z_{\widehat{k}}} & \widehat{Q}_Z
\end{array}\right]\left[\begin{array}{cc}
T_{Z_{\widehat{k}}} & T_{Z_{12}} \\
0 & T_{Z_{22}}
\end{array}\right]=\left[\begin{array}{ll}
Q_{Z_{\widehat{k}}} T_{Z_{\widehat{k}}} & Q_Z \widehat{T}_Z
\end{array}\right],} \\
{\left[\begin{array}{ll}
W_{\widehat{k}} & \widehat{W}
\end{array}\right]=W=Q_W T_W=\left[\begin{array}{ll}
Q_{W_{\widehat{k}}} & \widehat{Q}_W
\end{array}\right]\left[\begin{array}{cc}
T_{W_{\widehat{k}}} & T_{W_{12}} \\
0 & T_{W_{22}}
\end{array}\right]=\left[\begin{array}{ll}
Q_{W_{\widehat{k}}} T_{W_{\widehat{k}}} & Q_W \widehat{T}_W
\end{array}\right],}
\end{aligned}
\end{equation*}
where we have denoted
\begin{equation*}
\widehat{T}_Z:=\left[\begin{array}{c}
T_{Z_{12}} \\
T_{Z_{22}}
\end{array}\right],
\quad
\widehat{T}_W:=\left[\begin{array}{l}
T_{W_{12}} \\
T_{W_{22}}
\end{array}\right].
\end{equation*}
It is straightforward to check that
\begin{equation*}
\begin{aligned}
B &
=Z_{\widehat{k}} D_{B_k} U_{\widehat{k}}^{\mathrm{T}} + \widehat{Z} \widehat{D}_B \widehat{U}^{\mathrm{T}} + E_B
=Q_{Z_{\widehat{k}}} T_{Z_{\widehat{k}}} D_{B_{\widehat{k}}} U_{\widehat{k}}^{\mathrm{T}}+Q_Z \widehat{T}_Z \widehat{D}_B \widehat{U}^{\mathrm{T}} + E_B,
\\
G &
=V_{\widehat{k}} D_{G_{\widehat{k}}} W_{\widehat{k}}^{\mathrm{T}}+\widehat{V} \widehat{D}_G \widehat{W}^{\mathrm{T}} + E_G
=V_{\widehat{k}} D_{G_{\widehat{k}}} T_{W_{\widehat{k}}}^{\mathrm{T}} Q_{W_{\widehat{k}}}^{\mathrm{T}}+V_{\widehat{k}} \widehat{D}_G \widehat{T}_W^{\mathrm{T}} Q_W^{\mathrm{T}} + E_G,
\end{aligned}
\end{equation*}
where $E_B$ and $E_G$ satisfy the probabilistic error bounds (\ref{Err of R-RSVD-B}) and (\ref{Err of R-RSVD-G}).
Since Algorithm \ref{Al-R-RSVD} provides an exact decomposition of $A,$ the error bound for $A$ in \cite[Proposition 2]{gidisu2022ARXIVrsvd}
\begin{equation}\label{Err of R-LDEIM-RSVD-CUR-A}
\|A-C_A M_A R_A\|
\le
\alpha_{k+1} \cdot
\left(\sqrt{\frac{n {\widehat{k}}}{3}} 2^{\widehat{k}}+\sqrt{\frac{m {\widehat{k}}}{3}} 2^{\widehat{k}}\right) \cdot
\left\|\widehat{T}_W\right\|\left\|\widehat{T}_Z\right\|,
\end{equation}
still holds. Here $\alpha_{k+1}$ is the $(k+1)$th diagonal entry of $D_A$, which is ordered non-increasingly.
The following theorem roughly quantifies the error bounds for
$\|B-C_B M_B R_B\|$ and $\|G-C_G M_G R_G\|.$
\begin{theorem}\label{Th err of R-LDEIM-RSVD-CUR}
Suppose that a rank-$k$ RSVD-CUR decomposition for $(A,B,G)$ of the form (\ref{RSVD-CUR of (A,B,G)}) is produced by Algorithm \ref{Al-R-LDEIM-RSVD-CUR}, where
$S=I(:,\mathbf{s})$, $S_G=I(:,\mathbf{s}_G)$, $P=I(:,\mathbf{p})$ and $P_B=I(:,\mathbf{p}_B)$ are the index selection matrices,
and $p$ is the oversampling parameter. Let $\eta_G = \sqrt{\frac{n {\widehat{k}}}{3}} 2^{\widehat{k}} + \sqrt{\frac{d {\widehat{k}}}{3}} 2^{\widehat{k}}$,
and $\eta_B = \sqrt{\frac{l {\widehat{k}}}{3}} 2^{\widehat{k}} + \sqrt{\frac{m {\widehat{k}}}{3}} 2^{\widehat{k}}$. Then
\begin{equation*}
\|G-C_G M_G R_G\|
\le
\eta_G \cdot
\left(
\|E_G\|
+
\left\|\widehat{T}_W\right\|
\right),
~~~
\|B-C_B M_B R_B\|
\le
\eta_B \cdot
\left(
\|E_B \|
+
\left\|\widehat{T}_Z\right\|
\right),
\end{equation*}
where
\begin{equation*}
\|E_G\| \le \left(1+6 \sqrt{n(n-{\widehat{k}}) \log (n-\widehat{k})}\right) \sigma_{\widehat{k}+1}(G) +3 \sqrt{n \sum_{j>\widehat{k}} \sigma_j^2(G)},
\end{equation*}
\begin{equation*}
\|E_B \| \leq \left(1+6 \sqrt{(\widehat{k}+p) p \log p}\right) \sigma_{\widehat{k}+1}(B)+ 3 \sqrt{(\widehat{k}+p )\sum_{j>\widehat{k}} \sigma_j^2(B)}
\end{equation*}
which hold with probability not less than $1-(n-\widehat{k})^{-(n-\widehat{k})}$ and $1-3 p^{-p},$ respectively.
\end{theorem}
\begin{proof}
It suffices to prove the bound for
$\|G-C_G M_G R_G\|.$
Given the orthogonal projectors $C_G C_G^{\dagger}$ and $R_G R_G^{\dagger}$
and compute $M_G=C_G^{\dagger}GR_G^{\dagger}$, using the result in \cite{mahoney2009PNAScur},
we have
\begin{equation}
G-C_G M_G R_G
=G-C_G C_G^{+} G R_G^{\dagger} R_G
=(I-C_G C_G^{\dagger})G + C_G C_G^{\dagger} G (I-R_G^{\dagger} R_G).
\end{equation}
Then $$\left\|G-C_G M_G R_G\right\| \le
\|(I-C_G C_G^{\dagger}) G\|
+\|G (I-R_G^{\dagger} R_G)\|.$$
Given index selection matrix $P$ from the L-DEIM scheme on matrix $W_{\widehat{k}}$,
and suppose that $Q_{W_{\widehat{k}}}$
is an orthonormal basis for $\mathrm{Ran}(W_{\widehat{k}})$.
We form
$\mathbb{P}=P(Q_{W_{\widehat{k}}}^{\mathrm{T}} P)^{\dagger} Q_{W_{\widehat{k}}}^{\mathrm{T}}$:
an oblique projector with
$P(W_{\widehat{k}}^{\mathrm{T}} P)^{\dagger} W_{\widehat{k}}^{\mathrm{T}}
=P(Q_{W_{\widehat{k}}}^{\mathrm{T}} P)^{\dagger} Q_{W_{\widehat{k}}}^{\mathrm{T}}$
(\cite[Equation 3.6]{chaturantabut2010SIAMnonlinear})
and
we also have
$Q_{W_{\widehat{k}}}^{\mathrm{T}} \mathbb{P}
=Q_{W_{\widehat{k}}}^{\mathrm{T}} P(Q_{W_{\widehat{k}}}^{\mathrm{T}} P)^{\dagger} Q_{W_{\widehat{k}}}^{\mathrm{T}}=Q_{W_{\widehat{k}}}^{\mathrm{T}}$,
which implies $Q_{W_{\widehat{k}}}^{\mathrm{T}}(I-\mathbb{P})=0$.
From \cite[Lemmas 2 and 3]{hendryx2021MLextended}, we obtain that
\begin{equation*}
\|(I-C_G C_G^{\dagger}) G\| \leq \|G(I-{\mathbb{P}})\| =\|G(I-Q_{W_{\widehat{k}}} Q_{W_{\widehat{k}}}^{\mathrm{T}})(I-\mathbb{P})\|
\leq \|G(I-Q_{W_{\widehat{k}}} Q_{W_{\widehat{k}}}^{\mathrm{T}})\|\|I-{\mathbb{P}}\|,
\end{equation*}
\begin{equation*}
\|G(I-R_G^{\dagger} R_G)\| \leq \|(I-{\mathbb{S}}) G\| =\|(I-{\mathbb{S}})(I-V_{\widehat{k}} V_{\widehat{k}}^{\mathrm{T}}) G\|
\leq \|(I-{\mathbb{S}})\| \|(I-V_{\widehat{k}} V_{\widehat{k}}^{\mathrm{T}}) G \|.
\end{equation*}
Since $\widehat{k}<r$, ${\mathbb{P}} \neq 0, {\mathbb{P}} \neq I$ and $\mathbb{S}\neq 0, \mathbb{S}\neq I.$
By \cite[Lemma 4.1]{szyld2006NMmany}, we have
\begin{equation*}
\|I-\mathbb{P}\|
=\|\mathbb{P}\|
=\|(Q_{W_{\widehat{k}}}^{\mathrm{T}} P)^{\dagger}\|,~~~
\|I-\mathbb{S}\|=\|\mathbb{S}\|=\|(S^{\mathrm{T}} V_{\widehat{k}})^{\dagger}\|.
\end{equation*}
Using the partitioning of $G$, we have
\begin{equation*}
\begin{aligned}
G Q_{W_{\widehat{k}}} Q_{W_{\widehat{k}}}^{\mathrm{T}} &
=\left[\begin{array}{ll}
V_{\widehat{k}} & \widehat{V}
\end{array}\right]
\left[\begin{array}{cc}
D_{G_{\widehat{k}}} & 0 \\
0 & \widehat{D}_G
\end{array}\right]
\left[\begin{array}{cc}
T_{W_{\widehat{k}}}^{\mathrm{T}} & 0 \\
T_{W_{12}}^{\mathrm{T}} & T_{W_{22}}^{\mathrm{T}}
\end{array}\right]
\left[\begin{array}{c}
I_{\widehat{k}} \\
0
\end{array}\right]
Q_{W_{\widehat{k}}}^{\mathrm{T}}
+ E_G Q_{W_{\widehat{k}}} Q_{W_{\widehat{k}}}^{\mathrm{T}}
\\
&=
V_{\widehat{k}} D_{G_{\widehat{k}}} T_{W_{\widehat{k}}}^{\mathrm{T}} Q_{W_{\widehat{k}}}^{\mathrm{T}}
+\widehat{V} \widehat{D}_G T_{W_{12}}^{\mathrm{T}} Q_{W_{\widehat{k}}}^{\mathrm{T}}
+ E_G Q_{W_{\widehat{k}}} Q_{W_{\widehat{k}}}^{\mathrm{T}},
\end{aligned}
\end{equation*}
and hence
\begin{equation*}
\begin{aligned}
G(I-Q_{W_{\widehat{k}}} Q_{W_k}^{\mathrm{T}})
&=\widehat{V} \widehat{D}_G \widehat{T}^{\mathrm{T}} Q^{\mathrm{T}}
-\widehat{V} \widehat{D}_G T_{W_{12}}^{\mathrm{T}} Q_{W_{\widehat{k}}}^{\mathrm{T}}
-E_G Q_{W_{\widehat{k}}} Q_{W_{\widehat{k}}}^{\mathrm{T}}
=\widehat{V} \widehat{D}_G T_{W_{22}}^{\mathrm{T}} \widehat{Q}_W^{\mathrm{T}}
- E_G Q_{W_{\widehat{k}}} Q_{W_{\widehat{k}}}^{\mathrm{T}}.
\end{aligned}
\end{equation*}
This implies
\begin{equation*}
\|G(I-Q_{W_{\widehat{k}}} Q_{W_{\widehat{k}}}^{\mathrm{T}})\|
\leq
\gamma_{\widehat{k}+1} \left\|T_{W_{22}}\right\| + \|E_G\|
\leq
\left\|T_{W_{22}}\right\| + \|E_G\|
\end{equation*}
and then
\begin{equation*}
\|(I-C_G C_G^{\dagger}) G\
\leq
\|G(I-Q_{W_{\widehat{k}}} Q_{W_{\widehat{k}}}^{\mathrm{T}})\|\|I-{\mathbb{P}}\|
\leq
\|(Q_{W_{\widehat{k}}}^{\mathrm{T}} P)^{\dagger}\|
\cdot
( \left\|T_{W_{22}}\right\| + \|E_G\| ),
\end{equation*}
Similarly, we have
\begin{equation*}
\|G(I-R_G^{\dagger} R_G)
\le
\|(S^{\mathrm{T}} V_{\widehat{k}})^{\dagger}\| \cdot ( \|\widehat{T}_W\|+\|E_G\|)
\le
\|(S^{\mathrm{T}} V_{\widehat{k}})^{\dagger}\| \cdot ( \|\widehat{T}_W\|+\|E_G\|).
\end{equation*}
Then it follows that
\begin{equation*}\label{Proof of R-LDEIM-RSVD-CUR eq1}
\begin{aligned}
\|G-C_G M_G R_G\|
\le &
\left( \|(Q_{W_{\widehat{k}}}^{\mathrm{T}} P)^{\dagger}\|+ \|(S^{\mathrm{T}} V_{\widehat{k}})^{\dagger}\|\right)
\cdot
(\|\widehat{T}_W\|+\|E_G\|).
\end{aligned}
\end{equation*}
Using the upper bounds \cite{gidisu2022ARXIVrsvd}
\begin{equation*}
\begin{aligned}
&\|(Q^{\mathrm{T}}_{W_{\widehat{k}}}P)^{\dagger}\| < \sqrt{\frac{n\widehat{k}}{3}}2^{\widehat{k}},
\quad
&\|(S_G^{\mathrm{T}} V_{\widehat{k}})^{\dagger}\| < \sqrt{\frac{d{\widehat{k}}}{3}}2^{\widehat{k}}.
\end{aligned}
\end{equation*}
and applying the probabilistic error bound (\ref{Err of RGSVD}),
we obtain the desired result.
\end{proof}
\hskip 2em
Comparing the results of the error bounds in Theorem \ref{Th err of R-LDEIM-RSVD-CUR} to \cite[Theorem 4.3]{gidisu2022ARXIVrsvd} that
\begin{equation*}
\|G-C_G M_G R_G\|
\le
\gamma_{k+1}
\cdot
\eta_G
\cdot
\left\|\widehat{T}_W\right\|,
\|B-C_B M_B R_B\|
\le
\eta_B
\cdot
\left\|\widehat{T}_Z\right\|,
\end{equation*}
our results involve the item
$(1+6 \sqrt{n(n-\widehat{k}) \log (n-\widehat{k})}) \sigma_{\widehat{k}+1}(G)+3 \sqrt{n \sum_{j>\widehat{k}} \sigma_j^2(G)}$
in the error bound of $\| G-C_G M_G R_G\|$
and the item
$(1+6 \sqrt{(\widehat{k}+p) p \log p}) \sigma_{\widehat{k}+1}(B)+3 \sqrt{\widehat{k}+p} \sqrt{\sum_{j>\widehat{k}} \sigma_j^2(B)}$
in the error bound of $\|B-C_B M_B R_B\|$, respectively.
Therefore, our randomized algorithm works well for the matrices whose singular
values exhibit some decay.
\begin{algorithm}[htb]
\caption{L-DEIM based RSVD-CUR randomized algorithm}
\label{Al-R-LDEIM-RSVD-CUR}
\hspace*{0.02in} {\bf Input:}
$A \in \mathbb{R}^{m \times n}$,
$B \in \mathbb{R}^{m \times l}$,
and
$G \in \mathbb{R}^{d \times n}$
with $l=d\ge m\ge n$, desired rank $k$, the \hspace*{0.02in} oversampling
parameter $p$ and the specified parameter $\widehat{k}$.
\\
\hspace*{0.02in} {\bf Output:}
A rank-$k$ RSVD-CUR decomposition\\
\hspace*{0.02in}
$A\approx A(:,\mathbf{p}) \cdot M_A \cdot A(\mathbf{s},:)$,
$B\approx B(:,\mathbf{p}_B) \cdot M_B \cdot B(\mathbf{s},:)$,
$G\approx G(:,\mathbf{p}) \cdot M_G \cdot G(\mathbf{s}_G,:)$.
\begin{algorithmic}[1]
\State Generate an $n\times n$ Gaussian random matrix $\Omega_1$.
\State Form the $d\times n$ matrix $G\Omega_1$.
\State Compute the $d\times n$ orthonormal matrix $H_1$ via the QR factorization
$G\Omega_1 = H_1R_1$.
\State Compute the GSVD of $(H_1^{\mathrm{T}}G,A)$:
$\left[\begin{array}{c} H_1^{\mathrm{T}} G \\ A\end{array}\right]
=\left[\begin{array}{ll}\tilde{V}_1 & \\ & U_1\end{array}\right]
\left[\begin{array}{l}\Sigma_1 \\ \Gamma_1 \end{array}\right] Y_1^{\mathrm{T}}$.
\State Form the $n\times n$ orthogonal matrix $V_1 = H_1\tilde{V}_1$.
\State Generate an $m\times (\widehat{k}+p)$ Gaussian random matrix $\Omega_2$.
\State Form the $l\times (\widehat{k}+p)$ matrix $(B^{\mathrm{T}}U_1)\Omega_2$.
%
\State Compute the $l\times (\widehat{k}+p)$ orthonormal matrix $H_2$ via the QR factorization
$(B^{\mathrm{T}}U_1)\Omega_2= H_2R_2$.
\State Compute the GSVD of
$(H_2^{\mathrm{T}}(B^{\mathrm{T}}U_1),\Sigma_1^{-1}\Gamma_1^{\mathrm{T}})$:
$\left[\begin{array}{c}
H_2^{\mathrm{T}} (B^{\mathrm{T}}U_1) \\ \Sigma_1^{-1}\Gamma_1^{\mathrm{T}}
\end{array}\right]
=\left[\begin{array}{ll}\tilde{U}_2 & \\ & V_2\end{array}\right]
\left[\begin{array}{l}\Gamma_2 \\ \Sigma_2 \end{array}\right] Y_2^{\mathrm{T}}$,
where $\Sigma_2=\operatorname{diag}\left(\sigma_1, \ldots, \sigma_n\right)$.
\State Form the $l\times (\widehat{k}+p)$ orthonormal matrix $U_2 = H_2\tilde{U}_2$.
\State Form the $n\times n$ diagonal matrix
$\Gamma_G=\mathrm{diag}(\gamma_1,\ldots, \gamma_n)$, $\gamma_i=\frac{\sigma_i}{\sqrt{\sigma_i^2+1}}$.
\State Form the orthonormal matrices
$V=V_1V_2$, $U=U_2$ and nonsingular matrices $Z=U_1Y_2$ and $W=Y_1\Sigma_1V_2\Gamma_G^{-1}$.
\For{$j=1,\ldots,\widehat{k}$}
\State $\mathbf{p}_B(j)=\operatorname{argmax}_{1 \leq i \leq l}\left|(U(:, j))_{i}\right|$.
\State $U(:, j+1)=U(:, j+1)-U(:, 1: j) \cdot(U(\mathbf{p}_B, 1: j) \backslash U(\mathbf{p}_B, j+1))$.
\EndFor
\State Compute $\ell_{i}=\left\|[U]_{i:}\right\|^2 \quad$ for $i=1, \ldots, l$.
\State Sort $\ell$ in non-increasing order.
\State Remove entries in $\ell$ corresponding to the indices in $\mathbf{p}_B$.
\State $\mathbf{p}_B^{\prime}=k-\widehat{k}$ indices corresponding to $k-\widehat{k}$ largest entries of $\ell$.
\State $\mathbf{p}_B=\left[\mathbf{p}_B ; \mathbf{p}_B^{\prime}\right]$.
%
\State
Perform 13-21 on $W$, $Z$ and $V$ to obtain the corresponding indices $\mathbf{p}$, $\mathbf{s}$ and $\mathbf{s}_G$.
\State
Compute
$M_A=A(:, \mathbf{p}) \backslash\left(A / A\left(\mathbf{s}_A,:\right)\right)$,
$M_B=B(:, \mathbf{p}) \backslash\left(B / B\left(\mathbf{s}_B,:\right)\right)$.
\end{algorithmic}
\end{algorithm}
\section{Numerical Examples}
\hskip 2em
In this section, we check the accuracy and the computational cost
of our algorithms on several synthetic and real-world datasets.
In Example 5.1, we consider the case where the data matrix $A$ is corrupted by a random additive noise $E$ and the covariance of this noise (the expectation of $E^{\mathrm{T}}E$) is not a multiple of the identity matrix.
\cite[Experiment 5.1]{gidisu2022SIAMgeneralized} demonstrates that
using the SVD-based methods without prewhitening the perturbed data yields less accurate approximation results of the original matrix, while the GCUR technique gives a more accurate low-rank approximation.
In Example 5.1 we show that utilizing the randomized methods yields accurate approximation results compared to the GCUR and causes a dramatic enhancement in the computing speed, which is especially noticeable for large-scale matrices.
For Examples 5.2 and 5.3, we consider testing the performance of the approaches on a set with two data sets collected under different conditions, e.g., treatment and control experiment, where the former has distinct variation caused by the treatment: signal-free and signal recordings with the signal-free data set containing only noise.
We are interested in exploring and identifying patterns and discriminative features that are specific to one data set.
Finally, in Example 5.4, we evaluate the performance of the proposed randomized RSVD-CUR algorithm
for reconstructing a data matrix perturbed with nonwhite noise.
All computations are carried out in MATLAB R2020a on a
computer with an AMD Ryzen 5 processor and 16 GB RAM.
To facilitate the comparison between different algorithms, we define the following acronyms.
\hskip 2em 1. DEIM-GCUR$-$
implements the GCUR algorithm with column subset selection implemented using the DEIM algorithm (Algorithm \ref{Al-DEIM}) labeled ``DEIM-GCUR'' (Algorithm \ref{Al-DEIM-GCUR}).
\hskip 2em 2. R-GCUR $-$
applies the randomized GCUR algorithm with column subset selection implemented using either the DEIM algorithm labeled ``R-DEIM-GCUR'',
summarized in Algorithm \ref{Al-R-DEIM-GCUR},
or the L-DEIM algorithm (Algorithm \ref{Al-LDEIM}) labeled ``R-LDEIM-GCUR'' as summarized in Algorithm \ref{Al-R-LDEIM-GCUR}.
\hskip 2em 3. RSVD-CUR $-$
implements the RSVD-CUR decomposition algorithm by using the DEIM labeled ``DEIM-RSVD-CUR'',
as summarized in \cite[Algorithm 3]{gidisu2022ARXIVrsvd},
or the L-DEIM algorithm, labeled ``LDEIM-RSVD-CUR'', as summarized in \cite[Algorithm 4]{gidisu2022ARXIVrsvd}.
\hskip 2em 4. R-LDEIM-RSVD-CUR $-$
implements the randomized RSVD-CUR algorithm based on the L-DEIM procedure labeled ``R-LDEIM-RSVD-CUR'' (Algorithm \ref{Al-R-LDEIM-RSVD-CUR}) to produce the RSVD-CUR decomposition.
$\mathbf{Example}$ $\mathbf{5.1}$
This experiment is an adaptation of experiments in \cite[Section 3.4.4]{hansen1998SIAMrank}, \cite[Experiment 5.1]{gidisu2022SIAMgeneralized} and \cite[Example 6.1]{sorensen2016SIAMdeim}.
We build a matrix $A\in\mathbb{R}^{m\times n}$ of the form
\begin{equation*}
A=\sum_{j=1}^{10} \frac{2}{j} \mathbf{x}_j \mathbf{y}_j^{\mathrm{T}}+\sum_{j=11}^{50} \frac{1}{j} \mathbf{x}_j \mathbf{y}_j^{\mathrm{T}},
\end{equation*}
where
$\mathbf{x}_j \in \mathbb{R}^{m}$ and $\mathbf{y}_j \in \mathbb{R}^{n}$
are sparse vectors with random nonnegative entries
(in MATLAB, $\mathbf{x}_j=\mathtt{sprand}(m,1,0.025)$ and $\mathbf{y}_j=\mathtt{sprand}(n,1,0.025)$).
Following \cite{gidisu2022SIAMgeneralized}, we then perturb this matrix with a noise matrix $E \in \mathbb{R}^ {m\times n}$ whose entries are correlated.
Given $A_E = A + E$, we evaluate and compare the GCUR, R-GCUR algorithms and the CUR decomposition on $A_E$ in terms of recovering the original matrix $A$.
We present the numerical results for four noise levels; the noise
$E = \varepsilon \frac{\|F\|}{\|A\|}F$,
where $\varepsilon$ is the parameter for the noise level and $F$ is a randomly generated correlated noise.
Just as in \cite{gidisu2022SIAMgeneralized}, we construct a correlated Gaussian noise $E$ whose entries have zero mean and a Toeplitz covariance structure, i.e.,
in MATLAB
desired-cov($F$)=$\mathtt{toeplitz} (0.99^0, \ldots, 0.99^{n-1})$,
$B=\mathtt{chol}$({{desired}-{cov}}$(F)$),
and $F=\mathtt{randn}(m, n) \cdot B$
and $\varepsilon \in\{0.05,0.1,0.15,0.2\}$.
The performance is assessed based on the 2-norm of the relative matrix approximation error, i.e.,
\begin{equation*}
\mathrm{Err}={\|A-\widetilde{A}\|}/{\|A\|},
\end{equation*}
where $\widetilde{A}$ is the approximated low-rank matrix.
\hskip 2em
We first compare the accuracy of the
GCUR algorithms with their randomized counterparts R-GCUR and the standard DEIM-CUR decomposition for reconstructing the low-rank matrix $A$ for different noise levels.
As inputs, we fix $m=10000$, $n=300$ and using the target rank $k$ varies from $1$ to $50$, and the parameter contained in the L-DEIM procedure is $\widehat{k}=k/2$.
The relative errors are plotted in Figure \ref{Fig-Exp1}.
\begin{figure}[H]
\centering
\subfigure[$\varepsilon=0.2$]
{\includegraphics[width=0.46\textwidth,height=0.3\textwidth]{Exp1_varp=0_2.png} }
\subfigure[$\varepsilon=0.15$]
{\includegraphics[width=0.46\textwidth,height=0.3\textwidth]{Exp1_varp=0_15.png} }
\subfigure[$\varepsilon=0.1$]
{\includegraphics[width=0.46\textwidth,height=0.3\textwidth]{Exp1_varp=0_1.png} }
\subfigure[$\varepsilon=0.05$]
{\includegraphics[width=0.46\textwidth,height=0.3\textwidth]{Exp1_varp=0_05.png} }
\caption{Accuracy of the R-GCUR approximations compared with the standard DEIM-CUR approximation and the GCUR decomposition in recovering a sparse,
nonnegative matrix A perturbed with correlated Gaussian noise using exact Cholesky factor of the noise covariance.
The relative errors as a function of rank $k$ for $\varepsilon = 0.2, 0.15, 0.1, 0.05$, respectively.}
\label{Fig-Exp1}
\end{figure}
We observe that the GCUR and R-GCUR techniques achieve a comparable relative error.
Consistent with the results in \cite{gidisu2022SIAMgeneralized}, the R-GCUR algorithm performs significantly well under high noise.
Besides, we observe that, as $k$ approaches $\mathrm{rank}(A)$, however, the relative errors of both the GCUR and the R-GCUR do not decrease any more.
\cite{gidisu2022SIAMgeneralized} attributes this phenomenon to the fact that the relative error is saturated by the noise, considering we pick the columns and rows of the noisy data.
\hskip 2em
The analysis of the proposed algorithms implies that our randomized algorithms are less expensive compared to their deterministic counterparts.
To illustrate this, we record the running time in seconds (denoted as CPU) and the approximation quality Err of
the GCUR and R-GCUR for reconstructing matrix $A$ for different noise levels $\varepsilon=0.2, 0.1, 0.05$ as the dimension and the target rank $k$ increase.
According to the conclusions summarized in \cite{gidisu2022Arxivhybrid}, the L-DEIM procedure may be comparable to the original DEIM method when the target rank $k$ is at most twice the available $\widehat{k}$ singular vectors.
Therefore, here we set the parameter $\widehat{k}$ contained in the L-DEIM to be $\widehat{k}=k/2$, and the oversampling parameter $p=5$.
We record the results in Tables \ref{Table1-Exp1}-\ref{Table3-Exp1}.
It is clear from the running time that the algorithms R-DEIM-GCUR and R-LDEIM-GCUR
have a huge advantage in computing speed over the non-random GCUR method, and the R-LDEIM-GCUR achieves the smallest running time among the three sets of experiments.
\begin{table}[H
\caption{Comparison of GCUR and randomized algorithms (R-DEIM-GCUR and R-LDEIM-GCUR) in CPU and relative error as the dimension and the target rank $k$ increase, with noise level $\varepsilon=0.2$. }
\label{Table1-Exp1}
\setlength{\tabcolsep}{0.7mm}
\centering
\begin{tabular}{c c c c c c }
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{$(m,n,k)$ }
& $(10000,200,20)$ & $(50000,200,20)$ &$(100000,500,30)$ & $(200000,1000,40)$\\
\hline
\multirow{2}{*}{ GCUR }
& Err
& $0.15725$ & $0.14842$ & $0.18058$ & $0.17292$ \\
\cline { 2 - 6 }
& CPU
& $0.10197$ & $0.54697$ & $5.4971$ & $40.001$ \\
\hline \multirow{2}{*}{ R-DEIM-GCUR }
& Err
& $0.14524$ & $0.16584$ & $0.18260$ & $0.17772$ \\
\cline { 2 - 6 }
& CPU
& $0.027867$ & $0.11137$ & $0.49037$ & $2.4217$ \\
\hline \multirow{2}{*}{ R-LDEIM-GCUR }
& Err
& $0.16173$ & $0.14640$ & $0.16955$ & $0.16758$ \\
\cline { 2 - 6 }
& CPU
& $0.018809$ & $0.056930$ & $0.28019$ & $1.5563$ \\
\hline
\end{tabular}
\end{tabular}
\end{table}
\begin{table}[H
\caption{Comparison of GCUR and randomized algorithms (R-DEIM-GCUR and R-LDEIM-GCUR) in CPU and relative error as the dimension and the target rank $k$ increase, with noise level $\varepsilon=0.1$.}
\label{Table2-Exp1}
\setlength{\tabcolsep}{0.5mm}
\centering
\begin{tabular}{c c c c c c}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{$(m,n,k)$ }
& $(10000,1000,30)$ & $(100000,500,30)$ &$(200000,1000,40)$ & $(200000,1000,50)$ \\
\hline
\multirow{2}{*}{ GCUR } & Err
& $0.16493$ & $0.18058$ & $0.17292$ & $0.18699$\\
\cline { 2 - 6 } & CPU
& $2.1302$ & $4.5977$ & $33.783$ & $51.365$ \\
\hline \multirow{2}{*}{ R-DEIM-GCUR } & Err
& $0.16524$ & $0.18260$ & $0.17772$ & $0.18614$\\
\cline { 2 - 6 } & CPU
& $0.56099$ & $0.47876$ & $2.0406$ & $3.7259$ \\
\hline \multirow{2}{*}{ R-LDEIM-GCUR } & Err
& $0.16906$ & $0.16955$ & $0.16758$ & $0.172631$\\
\cline { 2 - 6 } & CPU
& $0.50487$ & $0.27769$ & $1.2856$ & $1.5229$\\
\hline
\end{tabular}
\end{tabular}
\end{table}
\begin{table}[H
\caption{Comparison of GCUR and randomized algorithms (R-DEIM-GCUR and R-LDEIM-GCUR) in CPU and relative error as the dimension and the target rank $k$ increase, with noise level $\varepsilon=0.05$.}
\label{Table3-Exp1}
\setlength{\tabcolsep}{0.5mm}
\centering
\begin{tabular}{c c c c c c}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{$(m,n,k)$ }
& $(10000,500,20)$ & $(100000,500,30)$ &$(150000,1000,40)$ & $(200000,1000,50)$ \\
\hline
\multirow{2}{*}{ GCUR } & Err
& $0.13828$ & $0.18058$ & $0.18230$ & $0.18699$ \\
\cline { 2 - 6 } & CPU
& $0.56859$ & $4.5496$ & $25.523$ & $32.360$ \\
\hline \multirow{2}{*}{ R-DEIM-GCUR } & Err
& $0.13089$ & $0.18260$ & $0.17513$ & $0.18614$ \\
\cline { 2 - 6 } & CPU
& $0.10336$ & $0.48947$ & $1.8595$ & $2.6152$ \\
\hline \multirow{2}{*}{ R-LDEIM-GCUR } & Err
& $0.13807$ & $0.16955$ & $0.17975$ & $0.17263$ \\
\cline { 2 - 6 } & CPU
& $0.079551$ & $0.28887$ & $1.0581$ & $1.4994$ \\
\hline
\end{tabular}
\end{tabular}
\end{table}
$\mathbf{Example}$ $\mathbf{5.2}$
We now test our randomized algorithms on synthetic data sets, as created in \cite[Example 5.3]{gidisu2022SIAMgeneralized} and \cite{abid2018Natureexploring},
which give an intuition for settings where the CUR and GCUR resolve the problem
of subgroups.
Consider a data set of interest (target data) $A$, containing $4m$ data points in
a $3d$-dimensional feature space.
This data set has four subgroups (blue, yellow, orange, and purple), each of $m$ data points.
The first $d$ columns for all $4m$ data points are randomly
sampled from a normal distribution with a mean of $0$ and a variance of $100$.
The next $d$ columns of two of the subgroups (blue and orange) are randomly sampled from a normal distribution with a mean of $0$ and a unit variance,
while the other two subgroups (yellow and purple) are randomly sampled from a normal distribution with a mean of $6$ and a unit variance.
The last $d$ columns of subgroups blue and yellow are sampled from a normal distribution
with a mean of $0$ and a unit variance, and those of purple and orange are sampled from a
normal distribution with a mean of $3$ and a unit variance.
\hskip 2em
Now we are interested in reducing the dimension of $A$ and this can be implemented by the SVD (principal component analysis).
However, if we project the data onto the two leading right singular vectors, we are unable to identify the subgroups because the variation along the first $d$ columns is significantly larger than in any other direction.
\hskip 2em
Following the operations in \cite{gidisu2022SIAMgeneralized}, we construct another data set $B$ (a background data set),
whose first 10 columns are sampled from a normal distribution with a mean of $0$ and a variance of $100$.
The next $10$ columns are sampled from a normal distribution with a mean of $0$ and a variance of $9$,
and the last $d$ columns are sampled from a normal distribution with a mean of $0$ and a unit variance.
The background data set should have the structure we would like to suppress in the target data, which usually corresponds to the direction with high variance but not of interest for the data analysis \cite{abid2018Natureexploring}.
With this new data, one way to extract discriminative features for clustering the subgroups in $A$ is to maximize the variance of $A$ while minimizing that of $B$, which leads to a trace ratio maximization problem \cite{chen2018IEEEnonlinear}
\begin{equation*}
\widehat{U}=
\underset{U \in \mathbb{R}^{n \times k}, U^T U=I_k}{\operatorname{argmax}}
\operatorname{Tr}\left[\left(U^T B^T B U\right)^{-1}\left(U^T A^T A U\right)\right],
\end{equation*}
where $n=3d$.
By doing this, the first dimensions are less likely to be selected because they also have a high variance in data set $B$.
Instead, the middle and
last dimensions of $A$ are likely to be selected, as they have the dimensions with the lowest variance in $B$, thereby allowing us to separate all four subgroups.
The solution $\widehat{U}$ to the above problem is given by the $k$ right eigenvectors of $(B^TB)^{-1}A^TA$ corresponding
to the $k$ largest eigenvalues (cf. \cite[pp. 448--449]{fukunaga2013introduction});
this corresponds to the (``largest") right generalized singular vectors of $(A, B)$.
We perform two sets of experiments where we set $m=2500$, $d=200$ and $m=25000$ and $d=300$, respectively.
\begin{figure}[htbp]
\centering
\subfigure
{\includegraphics[width=0.46\textwidth,height=0.3\textwidth]{png_EX3_1.png} }
\subfigure
{\includegraphics[width=0.46\textwidth,height=0.3\textwidth]{png_EX3_2.png} }
\subfigure
{\includegraphics[width=0.46\textwidth,height=0.3\textwidth]{png_EX3_3.png} }
\subfigure
{\includegraphics[width=0.46\textwidth,height=0.3\textwidth]{png_EX3_4.png} }
\caption{
In the top figure, we visualize the data using the first two columns selected by CUR (left) and GCUR (right), respectively.
In the bottom figure, we visualize the data using the first two columns selected by the R-DEIM-GCUR (left) and R-LDEIM-GCUR (right).
The lower-dimensional representation of the data using the GCUR and R-GCUR methods clearly separates the four clusters, while the DEIM based CUR method fails to do so.
In this experiment, we set $m=2500$, $d=200$ and $k=30$.
The input oversampling parameters for the R-DEIM-GCUR and R-LDEIM-GCUR are set to be $70$ and $100$, respectively, and $\widehat{k}=k/2$.
}
\label{Fig1-Exp2}
\end{figure}
Figure \ref{Fig1-Exp2} is a visualization
of the data using the first two important columns selected using the algorithms DEIM-CUR, GCUR, R-DEIM-GCUR and R-LDEM-GCUR for two of input matrix dimensions, respectively.
It can be seen that the GCUR and R-GCUR methods produce a much clearer subgroup
separation than the CUR. To a large extent, the GCUR and R-GCUR are able to differentiate the subgroups, while the CUR fails to do so.
In terms of the running time, for the case that $m=2500$ and $d=200$, the nonrandom GCUR costs $1.4250$ seconds while the R-DEIM-GCUR and R-LDEIM-GCUR spend $0.83059$ seconds and $0.82679$ seconds.
Meanwhile, for case that $m=25000$ and $d=300$, the nonrandom GCUR costs $1.4250$ seconds while the R-DEIM-GCUR and R-LDEIM-GCUR spend $0.83059$ seconds and $0.82679$ seconds.
\hskip 2em
Emulating the manipulations in \cite{gidisu2022SIAMgeneralized},
we investigate this further by comparing the performance of subset selection via DEIM-CUR on $A$, and GCUR and the R-GCUR on $(A, B)$ in identifying the subgroup or class representatives of $A$;
we select a subset of the columns of $A$ and compare the classification results of each method.
We center the data sets by subtracting the mean of each column from all the entries in that column.
Given the class labels of the subgroups, we perform a 10-fold cross validation, i.e.,
split the data points into $10$ groups and for each unique group take the group as test data and the rest as training \cite[p. 181]{james2013introduction} and apply two classifiers on the reduced data set: ECOC
(Error-Correcting Output Codes) \cite{dietterich1994JAIRsolving} and classification tree \cite{banfield2006IEEEcomparison} using the functions $\mathtt{fitcecoc}$ and
$\mathtt{fitctree}$ with default parameters as implemented in MATLAB.
It is evident from Table \ref{Table1-Exp2} that the R-LDEIM-GCUR achieves the lowest classification error rate, using the ECOC and tree classifier,
while the standard DEIM-CUR method achieves the worst classification error rate.
\begin{table}[htbp
\caption{k-Fold loss is the average classification loss overall 10-fold using CUR, GCUR, and R-GCUR as dimension reduction.
The second and third columns give dimension information $m_1=2500$, $d_1=200$, $m_2=3000$, $d_2=200$,
and the information on the number of columns $k=30$, selected from the data set using GCUR and R-GCUR for the ECOC classifier, likewise for the fifth and sixth columns for the tree classifier.}
\label{Table1-Exp2}
\setlength{\tabcolsep}{0.5mm}
\centering
\begin{tabular}{llllll}
\hline Method & \multicolumn{2}{c}{$k$-Fold Loss } & Method & \multicolumn{2}{c}{$k$-Fold Loss } \\
& \multicolumn{1}{c}{($m_1,d_1$)} & ($m_2,d_2$) & & \multicolumn{1}{c}{($m_1,d_1$)} & ($m_2,d_2$) \\
\hline
CUR+ECOC & $0.7512$ & $0.7521$ & CUR+Tree & $0.7488$ & $0.7465$ \\
GCUR+ECOC & $0.0669$ & $0.0666$ & GCUR+Tree & $0.0986$ & $0.09758$ \\
R-DEIM-GCUR+ECOC & $0.06930$ & $0.06700$ & R-DEIM-GCUR+Tree & $0.1000$ & $0.09558$ \\
R-LDEIM-GCUR+ECOC & $0.06680$ & $0.06358$ & R-LDEIM-GCUR+Tree & $0.0980$ & $0.09691$ \\
\hline
\end{tabular}
\end{table}
$\mathbf{Example}$ $\mathbf{5.3}$
Now we investigate the performance of the R-GCUR compared to the GCUR and the CUR on higher-dimensional public data sets.
Our experiment is adapted from \cite[Experiment 5.4]{gidisu2022SIAMgeneralized}.
The data sets consist of single-cell RNA expression levels of bone marrow mononuclear cells (BMMCs) from an acute myeloid leukemia
(AML) patient and two healthy individuals.
We have data on the BMMCs before stem-cell transplant and the BMMCs after stem-cell transplant.
We preprocess the data sets as described by the authors in \cite{boileau2020BIOexploring} keeping the $1000$ most variable genes measured across all 16856 cells (patient-035: 4501 cells and two healthy individuals;
one of 1985 cells and the other of 2472 cells).
The data from the two healthy patients are combined to create a background data matrix of dimension $4457 \times 1000$, and we use the patient-035 data set as the target data
matrix of dimension $4501\times 1000$.
Both data matrices are sparse: The patient-035 data matrix has $1,628,174$ nonzeros, i.e., about 36\% of all entries are nonzero, and the background data
matrix has $1,496,229$ nonzeros, i.e., about 34\% of all entries are nonzero.
We are interested in exploring the differences in the AML patient's BMMC cells pre- and posttransplant.
We perform CUR, GCUR, and R-GCUR on the target data (AML patient-035) to see if we can capture the biologically meaningful information relating to the treatment status.
For the GCUR and R-GCUR procedures, the background data are taken into account.
As evident in Figure \ref{Fig1-Exp3}, the GCUR and R-GCUR produce almost linearly separable clusters which correspond to pre- and posttreatment cells, while both the R-DEIM-GCUR and R-LDEIM-GCUR beat the GCUR in terms of the running time due to a much lower computational cost, and specifically, the running time of the nonrandom GCUR algorithm is roughly twice that of our randomized algorithms.
Moreover, these methods evidently capture the biologically meaningful information relating to the treatment and are more effective at separating the pre- and posttransplant cell samples.
For the CUR scheme, we observe that it does not give a discernible cluster of the pre- and post transplant cells, and fail to separate the pre- and posttransplant cells.
\begin{figure}[htbp]
\centering
\subfigure
{\includegraphics[width=0.46\textwidth,height=0.3\textwidth]{png_EX4_1.png} }
\subfigure
{\includegraphics[width=0.46\textwidth,height=0.3\textwidth]{png_EX4_2.png} }
\subfigure
{\includegraphics[width=0.46\textwidth,height=0.3\textwidth]{png_EX4_3.png} }
\subfigure
{\includegraphics[width=0.46\textwidth,height=0.3\textwidth]{png_EX4_4.png} }
\caption{
Acute myeloid leukemia patient-035 scRNA-seq data.
In the top figure, we visualize the data using
the first three genes selected by DEIM-CUR (top-left) and GCUR (top-right), and the CUR does not effectively give a discernible cluster of the pre- and post transplant cells.
In the bottom figure, we visualize the data using the first three genes selected by R-DEIM-GCUR (bottom-left) and R-LDEIM-GCUR (bottom-right), which both produce almost linearly separable clusters which correspond to pre- and posttreatment cells.
}
\label{Fig1-Exp3}
\end{figure}
$\mathbf{Example}$ $\mathbf{5.4}$
For our last experiment, we demonstrate the performance of randomized algorithm for producing the RSVD-CUR decomposition.
This test is an adaptation of \cite[Experiment 1]{gidisu2022ARXIVrsvd}, which considers a matrix perturbation problem of the form
$A_E = A + B F G$,
where $A\in\mathbb{R}^{m \times n}$,
matrices $B\in\mathbb{R}^{m \times l}$, $G\in\mathbb{R}^{d \times n}$
are noises distributed normally with mean $0$ and unit variance,
and our goal is to reconstruct a low-rank matrix $A$ from $A_E$.
We evaluate and compare a rank-$k$ RSVD-CUR decomposition of $A_E$, obtained by the nonrandom RSVD-CUR algorithm and its counterpart randomized algorithm, in terms of reconstructing matrix $A$ and the running time.
The approximation quality of the decomposition is assessed by the relative
matrix approximation error, i.e.,
$\|A-\widetilde{A}\| /\|A\|$,
where $\widetilde{A}$ is the reconstructed low-rank matrix.
As an adaptation of the experiment in
\cite[Example 1]{sorensen2016SIAMdeim} and \cite[Experiment 1]{gidisu2022ARXIVrsvd},
we generate a rank-$100$ sparse nonnegative matrix
$A\in \mathbb{R}^{m\times n}$ of the form
\begin{equation*}
A=\sum_{j=1}^{10} \frac{2}{j} \mathbf{x}_j \mathbf{y}_j^{\mathrm{T}}+\sum_{j=11}^{100} \frac{1}{j} \mathbf{x}_j \mathbf{y}_j^{\mathrm{T}}
\end{equation*}
where $\mathbf{x}_j\in\mathbb{R}^{m}$ and $\mathbf{x}_j\in\mathbb{R}^{n}$
are random sparse vectors with nonnegative entries.
We then perturb $A$ with a nonwhite noise matrix $BFG$ \cite{hansen1998SIAMrank}.
The resulting perturbed matrix we use is of the form
\begin{equation*}
A_E=A+\varepsilon \frac{\|A\|}{\|B F G\|} B F G,
\end{equation*}
where $\varepsilon$ is the noise level.
Given each noise level $\varepsilon\in\{0.1, 0.15, 0.2\}$, we generate the RSVD-CUR decomposition computed by the RSVD-CUR algorithms and the randomized algorithm for varying dimensions and the target rank $k$ values.
Here we set the parameter $\widehat{k}$, contained in the L-DEIM to be $\widehat{k} = k/2$ and $\widehat{k}=k$, respectively.
The corresponding results are displayed in Tables \ref{Table1-Exp4}, \ref{Table2-Exp4} and \ref{Table3-Exp4}, where we can see that the randomized algorithms give comparable relative errors at substantially less cost.
It indicates that using the random sampling techniques and L-DEIM
method leads to a dramatic speed-up over classical approaches.
\begin{table}[H
\caption{Comparison of RSVD-CUR and randomized algorithms in CPU and relative error as the dimension $l$, $d$, $m$, $n$ ( we set $m=n$) and the target rank $k$ increase, with noise level $\varepsilon=0.1$. }
\label{Table1-Exp4}
\setlength{\tabcolsep}{0.1mm}
\centering
\begin{tabular}{|ccc|c|c|c|}
\hline
\multicolumn{3}{|c|}{$(l,d,m,k)$}
& $(1000,500,100,10)$ & $(5000,1000,100,20)$ & $(7000,2000,200,30)$ \\ \hline
\multicolumn{2}{|c|}{\multirow{2}{*}{DEIM-RSVD-CUR}}
& Err & $0.095573$ & $0.085198$ & $0.084425$ \\ \cline{3-6}
\multicolumn{2}{|c|}{}
& CPU & $0.099726$ & $8.1768$ & $15.251$ \\ \hline
\multicolumn{2}{|c|}{\multirow{2}{*}{LDEIM-RSVD-CUR}}
& Err & $0.11652$ & $0.094442$ & $0.083729$ \\ \cline{3-6}
\multicolumn{2}{|c|}{}
& CPU & $0.098987$ & $8.5575$ & $15.886$ \\ \hline
\multicolumn{3}{|c|}{oversampling parameter}
& $80$ & $500$ & $500$ \\ \hline
\multicolumn{1}{|c|}{\multirow{4}{*}{R-LDEIM-RSVD-CUR}}
& \multicolumn{1}{c|}{\multirow{2}{*}{$k=\widehat{k}$}}
& Err & $0.095573$ & $0.085198$ & $0.084425$ \\ \cline{3-6}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{}
& CPU & $0.028330$ & $0.057914$ & $0.39295$ \\ \cline{2-6}
\multicolumn{1}{|c|}{}
& \multicolumn{1}{c|}{\multirow{2}{*}{$\widehat{k}=k/2$}}
& Err & $0.095573$ & $0.085198$ & $0.084425$ \\ \cline{3-6}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{}
& CPU & $0.024517$ & $0.056423$ & $0.50397$ \\ \hline
\end{tabular}
\end{table}
\begin{table}[H
\caption{Comparison of RSVD-CUR and randomized algorithms in CPU and relative error as the dimension $l$, $d$, $m$, $n$ ( we set $m=n$) and the target rank $k$ increase, with noise level $\varepsilon=0.15$. }
\label{Table2-Exp4}
\setlength{\tabcolsep}{0.1mm}
\centering
\begin{tabular}{|ccc|c|c|c|}
\hline
\multicolumn{3}{|c|}{$(l,d,m,k)$}
& $(5000,1000,200,20)$ & $(10000,2000,500,30)$ & $(20000,2000,500,40)$ \\ \hline
\multicolumn{2}{|c|}{\multirow{2}{*}{DEIM-RSVD-CUR}}
& Err & $0.13123$ & $0.15709$ & $0.14705$ \\ \cline{3-6}
\multicolumn{2}{|c|}{}
& CPU & $7.5313$ & $62.594$ & $328.99$ \\ \hline
\multicolumn{2}{|c|}{\multirow{2}{*}{LDEIM-RSVD-CUR}}
& Err & $0.13103$ & $0.16492$ & $0.15604$ \\ \cline{3-6}
\multicolumn{2}{|c|}{}
& CPU & $7.3077$ & $61.396$ & $330.20$ \\ \hline
\multicolumn{3}{|c|}{oversampling parameter}
& $500$ & $500$ & $500$ \\ \hline
\multicolumn{1}{|c|}{\multirow{4}{*}{R-LDEIM-RSVD-CUR}}
& \multicolumn{1}{c|}{\multirow{2}{*}{$k=\widehat{k}$}}
& Err & $0.13123$ & $0.15709$ & $0.14705$ \\ \cline{3-6}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{}
& CPU & $0.33164$ & $3.0591$ & $2.7742$ \\ \cline{2-6}
\multicolumn{1}{|c|}{}
& \multicolumn{1}{c|}{\multirow{2}{*}{$\widehat{k}=k/2$}}
& Err & $0.13123$ & $0.15709$ & $0.14705$ \\ \cline{3-6}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{}
& CPU & $0.34972$ & $2.5485$ & $2.9289$ \\ \hline
\end{tabular}
\end{table}
\begin{table}[H
\caption{Comparison of RSVD-CUR and randomized algorithms in CPU and relative error as the dimension $l$, $d$, $m$, $n$ ( we set $m=n$) and the target rank $k$ increase, with noise level $\varepsilon=0.2$. }
\label{Table3-Exp4}
\setlength{\tabcolsep}{0.1mm}
\centering
\begin{tabular}{|ccc|c|c|c|}
\hline
\multicolumn{3}{|c|}{$(l,d,m,k)$}
& $(5000,1000,100,10)$ & $(10000,1000,500,30)$ & $(7000,2000,200,30)$ \\ \hline
\multicolumn{2}{|c|}{\multirow{2}{*}{DEIM-RSVD-CUR}}
& Err & $0.15325$ & $0.18345$ & $0.18429$ \\ \cline{3-6}
\multicolumn{2}{|c|}{}
& CPU & $7.9139$ & $56.001$ & $384.17$ \\ \hline
\multicolumn{2}{|c|}{\multirow{2}{*}{LDEIM-RSVD-CUR}}
& Err & $0.14943$ & $0.21517$ & $0.19300$ \\ \cline{3-6}
\multicolumn{2}{|c|}{}
& CPU & $7.9627$ & $51.571$ & $357.91$ \\ \hline
\multicolumn{3}{|c|}{oversampling parameter}
& $100$ & $500$ & $500$ \\ \hline
\multicolumn{1}{|c|}{\multirow{4}{*}{R-LDEIM-RSVD-CUR}}
& \multicolumn{1}{c|}{\multirow{2}{*}{$k=\widehat{k}$}}
& Err & $0.15325$ & $0.18345$ & $0.18429$ \\ \cline{3-6}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{}
& CPU & $0.079395$ & $2.1681$ & $3.9116$ \\ \cline{2-6}
\multicolumn{1}{|c|}{}
& \multicolumn{1}{c|}{\multirow{2}{*}{$\widehat{k}=k/2$}}
& Err & $0.15325$ & $0.18345$ & $0.18429$ \\ \cline{3-6}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{}
& CPU & $0.06830$ & $2.0079$ & $3.8923$ \\ \hline
\end{tabular}
\end{table}
\section{Conclusion}
\hskip 2em
In this paper, by combining the random sampling techniques with the L-DEIM method, we
develop new efficient randomized algorithms for computing the GCUR decomposition for matrix pairs and the RSVD-CUR decomposition for matrix triplets with a given target rank.
We also provided the detailed probabilistic analysis for the proposed randomized algorithms.
Theoretical analyses and numerical examples illustrate that exploiting the randomized techniques results in a significant improvement in terms of the CPU time while keeping a high degree of accuracy.
Finally, it is natural to consider applying the L-DEIM for developing randomized algorithms that adaptively find a low rank representation satisfying a given tolerance, which is beneficial when the target rank is not known in advance, and it will be discussed in our future work.
\section*{Funding}
Z. Cao is supported by the National Natural Science Foundation of China under Grant 11801534.
Y. Wei is supported by the National Natural Science Foundation of China under Grant 12271108 and the Innovation Program of Shanghai Municipal Education Committee.
P. Xie is supported by the National Natural Science Foundation of China under Grants 12271108, 11801534 and the Fundamental Research Funds for the Central Universities under Grant 202264006.
\section*{Declarations}
The authors have not disclosed any competing interests.
\section*{Data Availability Statements}
All datasets are publicly available.
\bibliographystyle{siam}
|
2,877,628,090,091 | arxiv | \section{Introduction}
\label{sec:intro}
Modern observations unambiguously show that the chromosphere of the Sun is
very dynamic and inhomogeneous
\citep[see the reviews by, e.g.,][]{2001ASPC..223..131S, 2006ASPC..354..259J,
2007ASPC..368...27R, 2009SSRv..144..317W}.
The interpretation of chromospheric observations, which often involves the
construction of numerical models, thus must take into account spatial and temporal
variations.
Unfortunately, many simplifying equilibrium assumptions that can be made for the
low photosphere, are not applicable for the layers above.
A realistic chromosphere model must therefore account for deviations from the
equilibrium state in the context of an atmosphere that changes on short timescales
and small spatial scales.
A successful example is the explanation of the formation of ``calcium grains'' by
\citet{1997ApJ...481..500C}.
Other examples of simulations with a time-dependent non-equilibrium treatment
include the ionisation of hydrogen
(\citeauthor{2002ApJ...572..626C} \citeyear{2002ApJ...572..626C},
hereafter referred to as Paper~I;
\citeauthor{2006A&A...460..301L}
\citeyear{2006A&A...460..301L};
\citeauthor{2007A&A...473..625L}
\citeyear{2007A&A...473..625L})
and the concentration of carbon monoxide \citep{asensio03, 2005A&A...438.1043W}.
When an individual process, be it a chemical reaction or atomic transition, changes
on finite timescales longer than the dynamic timescales of the atmosphere, then it
becomes necessary to solve a system of rate equations instead of the much simpler
statistical equilibrium equations.
In paper~I, it was shown that the resulting variation of the hydrogen ionisation
degree is significantly reduced due to slow recombination rates.
The ionisation degree then depends not only on the local gas
temperature, density, and the
radiation field but also on the history of these properties.
Already \citet{1991A&A...250..212R} considered the non-equilibrium ionisation
of magnesium (\element{Mg}) and calcium (\element{Ca}) in
their ab-initio solar chromosphere models.
Their numerical implementation included a non-local thermodynamic equilibrium
(NLTE) approximation with very simplified two-level atoms for
\ion{Mg}{II}-{III} and \ion{Ca}{II}-{III}.
In this paper, we investigate the \ion{Ca}{II}-{III} ionisation balance in
detail.
The study is based on RADYN simulations similar to the hydrogen case in Paper~I.
The numerical simulations are described in Sect.~\ref{sec:sim}.
The results of this study are presented in Sect.~\ref{sec:res},
followed by a discussion and conclusions in Sect.~\ref{sec:disc}.
\section{Numerical simulations}
\label{sec:sim}
We use one-dimensional radiation hydrodynamic simulations calculated with
RADYN, which are in most aspects very similar to the earlier simulation runs by
\citet[][ paper~I]{1992ApJ...397L..59C, 1994chdy.conf...47C, 1995ApJ...440L..29C}.
Here, we summarise only the most important properties.
More details can be found in paper~I and references therein.
The simulation code RADYN solves the equations of mass, momentum, energy, and
charge conservation together with the non-LTE radiative transfer and population
rate equations, on an adaptive mesh in one spatial dimension.
It takes into account non-equilibrium ionisation, excitation, and radiative
energy exchange from the atomic species H, He, and Ca with back-coupling on the
hydrodynamics.
Also, the effect of motion on the emitted radiation from these species is considered.
Hydrogen and singly ionised calcium are modelled with six-level atoms and helium
with a nine-level atom.
Furthermore, doubly ionised helium is included.
The transitions between all considered atomic levels are treated in detail.
Each line is described with 31-101 frequency points, whereas 4-23
frequency points are used for each continuum.
Other elements than H, He, and Ca are taken into account in the form of background
continua in LTE, which are derived with the Uppsala atmospheres program
\citep{gustafsson73}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=5cm]{16186fig1.eps}
\caption{Calcium model atom with the five lowest energy levels of \ion{Ca}{II},
a continuum level (\ion{Ca}{III}), and all included b-b and b-f transitions.
Energies are not to scale.}
\label{fig:modelatom}
\end{center}
\end{figure}
The boundary conditions are the same as for the simulations in paper~I.
The lower and upper boundaries are both transmitting.
The lower boundary is located at a fixed geometrical depth, which corresponds
to 480\,km below $\tau_{500} = 1$ in the initial atmosphere.
We again use the same piston as in paper~I to excite waves at the bottom, which
then propagate through the atmosphere.
The piston velocity is based on Doppler-shift observations in a \ion{Fe}{I} line
at $\lambda = 396.68$\,nm in the wing of the Ca\,H line \citep{1993ApJ...414..345L}.
The upper boundary condition is located at a height of 10\,Mm.
It represents a corona on top of the simulated layers
with a temperature set to $10^6$\,K and corresponding incident radiation
\citep{1991JATP...53.1005T}.
The dynamic simulation has an overall duration of 3600\,s.
The numerical simulation is characterised by shock waves, which are excited as
acoustic disturbances of small amplitude through the piston at the lower boundary.
They propagate upwards and steepen into shocks with a saw-tooth profile above the
photosphere.
The simulated chromosphere is therefore subject to strong fluctuations in all
hydrodynamic quantities.
It takes a few 100\,s until the first shock waves transform the initially
hydrostatic atmosphere into the characteristic dynamic atmosphere.
The first 600\,s of the simulation are therefore excluded from the analysis.
\label{sec:runb}
A large number of additional short simulation runs are carried out, which are
used for a timescale analysis (see Sect.~\ref{sec:timescales}).
Each run starts from a snapshot of the detailed time-dependent (TD) simulation in
the time window 600\,s to 2350\,s.
The initial atmosphere of the individual runs contains the statistical
equilibrium (SE) solution for the thermodynamic state in that snapshot.
In the next time step, the atmosphere is perturbed by increasing the gas
temperature by 1\,\% and the time-evolution is followed for 50\,s.
In addition, we calculate the statistical equilibrium state of the perturbed
atmosphere.
For each snapshot in the dynamic simulation we thus have the following
data:
(i)~the statistical equilibrium solution of that snapshot,
(ii)~the time-evolution after a perturbation has been applied, and
(iii)~the final equilibrium state corresponding to the perturbed atmosphere.
\begin{figure}[t]
\begin{center}
\includegraphics{16186fig2.eps}
\caption{Ionisation fractions of calcium ($\chi_{\element{Ca}}$, thick solid, see
Eq.~(\ref{eq:ionca})) and of hydrogen (dashed) as a
function of column mass in the initial radiative equilibrium atmosphere.
The height scale $z_0$ of the initial atmosphere is given as reference.
%
The \ion{Ca}{II}-{III} ionisation fraction rises from a minimum of $2\,10^{-4}$
at \mbox{$\lg m_\mathrm{c} = 0.0$} ($z \approx 180$\,km) in the low photosphere
to $\sim 5$\,\% at the base of the transition region.
%
The grey lines represent $\chi_{\element{Ca}}$ in simulation snapshots
at an interval of 10\,s during the whole simulation sequence of 3600\,s.
%
The thin short-dashed lines are the 5\,\% and 95\,\% percentiles.
}
\label{fig:ionfrac}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics{16186fig3.eps}
\caption{Logarithmic \ion{Ca}{II}-{III} ionisation fraction as function of time at
different Lagrangian
locations.
%
The labels refer to the corresponding geometric height in the initial atmosphere.
The lines represent the results from the non-equilibrium simulation (solid) and
for statistical equilibrium (dotted) calculated for the same values of the
hydrodynamic variables.
%
The horizontal lines are the corresponding time-averages.
%
}
\label{fig:ionvstime}
\end{center}
\end{figure}
\subsection{The calcium model atom}
\label{sec:modelatom}
The calcium model atom contains 5 bound states of singly ionised calcium
(\ion{Ca}{II}) and a continuum level ($i = 6$), which represents the next
ionisation stage (\ion{Ca}{III}).
The lowest energy level with $i = 1$ is the ground state of
\ion{Ca}{II} (4s\,$^2$S).
We also include the two important energy level pairs of
4p\,$^2$P ($i = 2, 3$)
and
4d\,$^2$D ($i = 4, 5$).
All five allowed radiative bound-bound (b-b) transitions are considered:
the H and K resonance lines, and the infrared triplet (IRT).
Also all five radiative bound-free (b-f) transitions with the corresponding
photoionisation continua from the five lowest levels to level $i = 6$
are included.
The model atom is illustrated in Fig.~\ref{fig:modelatom}.
\section{Results}
\label{sec:res}
\subsection{Ionisation fraction}
In the solar chromosphere, calcium is mostly present in singly and doubly ionised
form, i.e. as \ion{Ca}{II} and \ion{Ca}{III}, respectively.
We therefore neglect neutral calcium and define
the \ion{Ca}{II}-\ion{Ca}{III} ionisation fraction as the ratio of the population
density of doubly ionised calcium $n_\mathrm{Ca\,III}$ and the population density
of all included Ca ions
($n_\mathrm{Ca,\,total} = n_\mathrm{Ca\,II} + n_\mathrm{Ca\,III}$).
For our six level atom it reduces to
\begin{equation}
\label{eq:ionca}
\chi_{\element{Ca}} =
\frac{n_\mathrm{Ca\,III}}{n_\mathrm{Ca\,II} + n_\mathrm{Ca\,III}}
=
\frac{n_6}{\sum_{i=1}^6 n_i}\enspace,
\end{equation}
where $n_i$ is the population density of level~$i$ and $n_\mathrm{Ca\,III} = n_6$.
In the following, we refer to $\chi_{\element{Ca}}$ as \ion{Ca}{II}-{III} ionisation fraction.
The variation of $\chi_{\element{Ca}}$ is shown in Fig.~\ref{fig:ionfrac} as a function of
column mass density in the initial radiative equilibrium atmosphere (thick line).
It has a minimum of $2\,10^{-4}$ in the middle photosphere at
\mbox{$\lg m_\mathrm{c} = 0.0$}, which corresponds to $z_0 \approx 180$\,km
on the height scale of the initial model.
Like the hydrogen ionisation fraction (dashed line) it rises with height but
less steeply.
At the base of the transition region, $\chi_{\element{Ca}} \approx 5$\,\% is reached.
The grey lines in Fig.~\ref{fig:ionfrac} illustrate the temporal evolution of
the \ion{Ca}{II}-{III} ionisation fraction during the simulation.
The thin dotted lines are the 5\,\% and 95\,\% percentiles, which enclose the typical
data range of $\chi_{\element{Ca}}$.
While the temporal variation is small below the middle
photosphere, the fluctuations cover usually two orders of magnitude in the layers
above.
This is caused by the upward propagating shock waves.
In Fig.~\ref{fig:ionvstime}, the \ion{Ca}{II}-{III} ionisation fraction $\chi_{\element{Ca}}$ is
shown as function of time for three different Lagrangian locations, i.e. at fixed
column mass densities.
The corresponding geometrical heights $z$ vary in time.
For orientation, the labels in each panel specify the geometrical height $z_0$
in the initial atmosphere.
Please note that the fluctuations of the ionisation degree appear much larger
at a fixed geometrical height compared to a fixed column mass density because the
atmospheric stratification and with it the height of the transition region
is influcenced by the propagating shocks.
The ionisation degree in the time-dependent non-equilibrium simulation
(TD, solid line in Fig.~\ref{fig:ionvstime}) is compared to the statistical
equilibrium values (SE, dotted line in Fig.~\ref{fig:ionvstime}).
Both are calculated for the same atmospheric states.
The differences between both cases are hardly discernible in the lower panel, which
refers to the top of the photosphere.
It shows that the ionisation fraction does not deviate considerably from the
statistical equilibrium values in the lower atmosphere.
The deviation is somewhat larger but still moderate in the upper chromosphere
at heights around $z \sim 1400$\,km (upper panel in Fig.~\ref{fig:ionvstime}).
At times of minimal ionisation, the time-dependent and statistical equilibrium
results are very similar.
At phases of higher ionisation in connection with passing shock fronts,
the time-dependent simulation gives slightly smaller peak values, which lag
behind the SE solution by 5 to 10\,s.
The time-average of the SE case is $\sim 27$\,\% larger than the TD result
(see horizontal lines).
In the middle chromosphere at heights around $z \sim 1000$\,km
(see middle panel of Fig.~\ref{fig:ionvstime}), the SE
ionisation fraction (dotted line) varies over two orders of magnitude on time
scales of a few tens of seconds.
In contrast, the detailed time-dependent simulation produces a \ion{Ca}{II}-{III}
ionisation fraction that cannot follow the rapid changes (solid line).
After a strong increase and a shock-induced peak, which is similar in both cases,
the ionisation fraction decays so slowly in the TD simulation that already the next
shock front has arrived before the equilibrium value can be attained.
As we shall see in Sect~\ref{sec:rates}, this behaviour is caused by a relatively
slow net recombination process.
There is also a general time difference of the order of 5 to 10\,s between the
occurrence of the ionisation peaks in the time-dependent
simulation compared to the SE approach.
At heights around $z \sim 1000$\,km, the TD ionisation fraction nevertheless varies
strongly between values of $\sim 10^{-3}$ and $\sim 10^{-1}$.
Although the temporal evolution of $\chi_{\element{Ca}}$ is very different for the TD and SE case
at those heights, the time-averages are relatively similar.
The SE average is only $\sim 14$\,\% smaller than the TD result
(see horizontal lines).
\subsection{Timescales}
\label{sec:timescales}
In the following, we derive the relaxation timescale of the \ion{Ca}{II}-{III}
ionisation fraction (i)~numerically from the temporal evolution of an initially
perturbed atmosphere (Sect.~\ref{sec:numtimescale}) and (ii)~from the eigenvalue
analysis of the rate matrices (Sect.~\ref{sec:eigenvaluetimescale}).
\begin{figure}[t]
\begin{center}
\includegraphics{16186fig4.eps}
\caption{Relaxation timescale (solid lines) and gas temperature (dot-dashed)
as function of column mass and time in the dynamic simulation.
The thick lines represent the relaxation timescale and the temperature
averaged in time over all time steps between $t = 600$\,s and $t = 2350$\,s.
Two individual time steps are drawn as thin lines:
$t = 1600$\,s (grey) and $t = 1620$\,s (black).
%
The dotted vertical lines mark the temperature peak and the minimum timescale
for this time step.
%
For reference, the height scale ($z_0$) of the initial state is given on top.
}
\label{fig:timescale_zt}
\end{center}
\end{figure}
\subsubsection{Numerical relaxation timescale}
\label{sec:numtimescale}
The simulation runs are used to determine a numerical timescale
as function of time by analyzing each time step in the time range from 600\,s to
2350\,s individually (see Sect.~\ref{sec:runb}).
For each time step, the timescale is derived from the exponential relaxation of the
\ion{Ca}{II}-{III} ionisation fraction towards the equilibrium state of the
perturbed atmosphere.
The time-averaged relaxation timescale is then calculated by
averaging over the timescales of the individual time steps.
This average timescale is shown in Fig.~\ref{fig:timescale_zt}
as function of column mass (thick solid line).
It is very small in the photosphere ($< 1$\,s) and increases
strongly with height until a maximum with values of the order of 1\,min is reached
at heights around $z = 900$\,km.
The individual time steps show exactly the same behaviour in the photosphere,
whereas the maximum in the chromosphere can reach values of up to 150\,s
in some cases.
In the middle chromosphere above, the timescale is decreasing towards a minimum for all
time steps.
The average timescale drops to $\sim 10$\,s at $z_0 = 1200$\,km.
The individual time steps also show a minimum but the exact position varies owing to the
passage of shock waves.
Values down to only a few seconds are found.
The timescale starts to rise again at even larger heights but gets very small
at the high-temperature transition region.
A number of successive individual time steps are represented by thin lines in
Fig.~\ref{fig:timescale_zt}.
They demonstrate the response of the relaxation time scale (solid lines) to the passage
of a shock wave (gas temperature as dot-dashed lines).
At first glance, it seems as if the relaxation timescale and gas temperature are neatly
anti-correlated, i.e., the shortest time scales are found at the positions with the
highest gas temperatures.
This would be expected from high temperatures resulting in large ionisation rates
and thus small timescales.
A closer look, however, reveals that the minimum timescale is found at a height
slightly below the temperature peak.
This can be seen at $t = 1620$\,s
(thin black lines in Fig.~\ref{fig:timescale_zt}).
The height difference is marked by dotted vertical lines.
At the beginning of the shock wave passage in the photosphere, when it is still a
disturbance with small temperature amplitude, the timescale at the position of the
disturbance is mostly set by the thermodynamic state of the background atmosphere
and thus the aftermath of previous shock waves.
It is only at heights of $z > 900$\,km, where the shock has steepened
enough to have a significant effect on the timescale.
There, a local timescale minimum develops at the height of the
temperature peak.
While the shock front continues to propagate upwards in the atmosphere, the peak
temperature grows, resulting in shorter and shorter timescales.
The timescale minimum starts to appear lower than the peak temperature.
This height difference grows during the passage through the chromosphere
until values of the order of $\Delta\,z \sim 100$\,km are reached.
From the time at which the locations of the temperature peak and the timescale minimum
coincide, the relaxation timescale drops from a few 10\,s to just a few seconds.
This behaviour is commonly found for all shock events.
When looking at the data as function of time at a fixed geometric height,
the height offset translates into a small time difference between the occurrence
of the temperature peak and the minimum timescale.
This time difference decreases in most cases with height from a few 10\,s around
$z \sim 1000$\,km to only a few seconds around $z \sim 1300$\,km.
There are examples where the time difference does not show such a height
dependence but rather varies around values of just a few seconds.
The time and height offsets are caused by the influence of the hydrogen
ionisation on the recombination rates of calcium via the electron density.
See the analysis of the transition rates in Sect.~\ref{sec:rates}
for more details.
\subsubsection{Timescales derived from eigenvalue analysis}
\label{sec:eigenvaluetimescale}
\begin{figure}[t]
\begin{center}
\includegraphics{16186fig5.eps}
\caption{Numerical relaxation timescale as function of column mass determined
numerically (solid line) compared to the timescale determined from eigenvalues
of the rate matrix in the initial statistical equilibrium states (dot-dashed).
%
The thick lines represent the averages over all time steps between
$t = 600$\,s and $t = 2350$\,s, whereas the thin lines are for a
particular time step at $t = 1620$\,s.
Minimum and maximum values for both dynamic and eigenvalue timescales during
the considered time window are shown as grey dotted lines.
%
For reference, the height scale at $t = 1620$\,s is given on top.
}
\label{fig:timescale_eigenv}
\end{center}
\end{figure}
As in paper~I, we use the eigenvalues of the transition rate matrix to
determine the involved timescales and to
indicate the processes governing the timescales.
The rates in the dynamic simulation are affected by the previous
history of the atmospheric state.
The corresponding timescales are therefore not directly comparable to
timescales derived from the numerical experiments in
Sect.~\ref{sec:numtimescale}.
Therefore, we use the statistical equilibrium state corresponding to
the hydrodynamic variables of a given snapshot
(see Sect.~\ref{sec:runb}).
The Lagrangian time-derivative of the population densities is
given by the rate equations:
\begin{equation}\label{eq:rate}
{D\vec{n}\over Dt}=\vec{P} \, \vec{n}
\end{equation}
with $\vec{n}$ being the vector of the level populations and $\vec{P}$ the
rate matrix.
The matrix element $P_{ij}$ of the rate matrix is the transition rate per atom
from level $i$ to level $j$ with \mbox{$P_{ii} = -\sum_j P_{ij}$}.
The matrix elements depend on the population densities in general
through the non-linear radiation terms. If these non-linearities are
not dominant we may write the solution to
Eq.~(\ref{eq:rate}) in
terms of eigenvectors and eigenvalues of the rate matrix
\citep[Paper~I;][]{2005JQSRT..92..479J}.
For the Ca model atom (Sect.~\ref{sec:modelatom}) with 5 bound states and
a continuum level (\mbox{$i = 6$}, \ion{Ca}{III}), the solution for the
relaxation process can then be written as
\begin{equation}
\vec{n} = \sum_{i=1}^{6}\, c_i \, \vec{v}_i \, e^{\lambda_i t}\quad.
\end{equation}
Here,
$\vec{v}_i$ is the $i$th eigenvector of the rate matrix \vec{P},
$\lambda_i$ is the corresponding eigenvalue, and
$c_i$ is a coefficient, which depends on the initial conditions.
The equilibrium state corresponds to the eigenvector with a zero eigenvalue.
All other eigenvalues are negative and represent the evolution of the level
populations toward their equilibrium values.
The longest timescale, on which the system relaxes towards the equilibrium state,
is then given by the inverse of the smallest (in absolute value) nonzero eigenvalue.
In Fig.~\ref{fig:timescale_eigenv}, the timescales from the eigenvalue calculation
are compared to the numerically determined relaxation timescales described in
Sect.~\ref{sec:numtimescale}.
The timescales averaged over all snapshots between $t = 600$\,s and
$t = 2350$\,s match closely throughout the whole atmosphere.
The eigenvalue and numerical timescales also match very well for the individual
snapshots.
This shows that in the case of calcium ionization, the timescales do
not depend strongly on the non-linear parts of the rate matrix and we
have a convenient way of calculating the ionization/recombination
timescales from an eigenvalue analysis of the rate matrix.
\begin{figure*}[t]
\sidecaption
\includegraphics[width=12cm]{16186fig6.eps}
\caption{\ion{Ca}{II}-{III} ionisation fraction (left column, solid line, left axis), gas
temperature (left column, dot-dashed line, right axis), and normalised rates (right
column) as a function of column mass for four time steps (rows from top to bottom)
in the dynamic simulation.
The rates for the radiative transitions from the bound levels~\mbox{1 - 5} to the
continuum ($i = 6$, abbreviated ``c'') are divided by the total Ca number
densities.
Positive rates indicate ionisation, negative rates recombination.
The sum of all radiative b-f transitions is represented by the thick solid line.
The gas temperature is repeated in the right column as reference (dot-dashed line).
The height axis for the individual time step is shown at the top of each panel.
Please note the different data ranges for the individual time steps.
}
\label{fig:ratesvsheight}
\end{figure*}
\subsubsection{Timescales of individual processes}
\label{sec:timescale_process}
We now investigate which processes dominate the relaxation timescale by modifying the
entries of the rate matrix prior to determining the eigenvalue timescale.
The resulting timescales have qualitatively the same run with height but differ in
their absolute value.
Leaving out the collisional rates does not change the
overall timescale in the middle chromosphere because the collisional transitions
occur on timescales that are up to 5 orders of magnitude larger compared to the
overall timescale.
Also setting the radiative \mbox{b-b} transition rates to zero has no noticeable
effect on the timescale.
Setting the rates $R_{1\,6}$ and $R_{6\,1}$ for the radiative \mbox{b-f}
transition between the ground level (\mbox{$i = 1$}) and
the continuum (\mbox{$i = 6$}, Ca~III) to zero changes the timescale only slightly.
We now set all matrix elements to zero except for the entries of individual \mbox{b-f}
transitions.
A rate matrix with non-zero entries only for $R_{1\,6}$ and $R_{6\,1}$
results in timescales of up to a few thousand seconds in the middle chromosphere.
It implies that this transition, which connects the most populated energy level
(\mbox{$i = 1$}) with the ionisation continuum, does not govern the overall timescale.
It is the relaxation between level \mbox{$i = 3$} and the continuum, which produces
the shortest timescales, followed by the transition between level \mbox{$i = 2$}
and the continuum.
Both transitions have timescales of the order of a few 10\,s.
The ratio of the timescale with just a single transition and the timescale for the
full rate matrix is about 2 for the transition 3-6 and $\sim 3$ for 2-6.
The ratios for the transitions with lower level 4,5,1 are $\sim 20$, $~10$, and
$\sim 70$, respectively.
We conclude that the transitions between the metastable levels (2 and
3) and the continuum are the most important for setting the timescale
for ionization/recombination. Including only these transitions result
in a timescale that is only $\sim 18$\,\% longer than the timescale
derived from the full rate matrix.
\subsection{Transition rate analysis}
\label{sec:rates}
In contrast to the hydrogen case discussed in paper~I,
the shortest timescales do not coincide exactly with the temperature peaks
in the shock fronts (see Sect.~\ref{sec:timescales}).
In this section, we show that this behaviour is caused by the dependence of the
recombination rates on the density of free electrons, which is influenced by
the ionisation/recombination timescale of hydrogen.
For the Ca model atom considered here, the recombination process involves a
collision between a free electron and a
\ion{Ca}{III} ion \citep[see, e.g.,][ p.130f]{mihalas78}.
Calcium is only a minor electron donor in the solar atmosphere, whereas
already small changes in the hydrogen ionisation fraction can lead to significant
fluctuations of the electron density.
As shown in paper~I, the hydrogen ionisation/recombination timescale
is on average on the order of one to several hours in the chromosphere
but can be strongly reduced in hot shock fronts.
The timescales can then get as short as 10\,s to 20\,s.
The resulting delayed release of electrons has a direct effect on the
recombination rates of Ca so that the related timescales reach their
minimum only shortly after the occurrence of the temperature peak.
\begin{figure*}[t]
\sidecaption
\includegraphics[width=12cm]{16186fig7.eps}
\caption{Temporal evolution at a fixed geometric height of $z = 1200$\,km.
\textit{Top:}~total Ca number density $n_\mathrm{Ca,total}$ and relative level
populations $n_i / n_\mathrm{total}$.
\textit{Middle:}~net radiative transition rates $P_{\mathrm{r}\,ij}$ for the
b-f transitions between the bound levels and the continuum together
with the overall sum $\sum_{i=0}^{5} P_{\mathrm{r}\,i6}$.
\textit{Bottom:}~the involved radiative rates $R_{ij}$ and $R_{ji}$ in comparison to
the density of free electrons $n_\mathrm{el}$.
The \ion{Ca}{II}-{III} ionisation fraction $\chi_{\element{Ca}}$ (grey dotted line), the gas
temperature (grey dashed line), and the inversed relaxation timescale (grey solid
line) are shown in all panels as reference.
The rates for the transitions from the bound levels 1 - 5 to the continuum ($i = 6$)
are divided by the total Ca number densities.
Positive rates indicate ionisation, negative rates recombination.
}
\label{fig:rates_time}
\end{figure*}
In the following, the role of the individual ionisation and recombination processes
is illustrated in detail for a typical shock that propagates through the model
chromosphere at $t \sim 2200$\,s in the dynamic simulation.
Most other shocks in the simulation show the same behaviour.
In Fig.~\ref{fig:ratesvsheight}, the gas temperature, \ion{Ca}{II}-{III} ionisation
fraction, and transition rates are shown as function of column mass density for the
selected shock event.
The four different time steps (rows from top to bottom) represent different stages of
the upward propagating shock wave, which is most clearly seen as excess in gas
temperature (dot-dashed line in the left column).
The corresponding increase in ionisation fraction (solid line in the left column) is
comparatively subtle with a peak (if discernible) slightly below the gas temperature
peak.
The normalised net rate $P_{\mathrm{r}, i j}$ for a radiative transition between
level~$i$ and level~$j$ is given by
\begin{equation}
P_{\mathrm{r}, i j} = \frac{ n_i R_{ij} - n_j R_{ji} }{n_\mathrm{Ca, total}}
\label{eq:netrate}
\end{equation}
with the level population densities $n_i$ and $n_j$ and the radiative transition
rates $R_{ij}$ (upwards, $i \rightarrow j$) and $R_{ji}$ (downwards, $j \rightarrow i$).
The bound-free transition rates $P_{\mathrm{r}, i 6}$ are shown in the right
column of Fig.~\ref{fig:ratesvsheight} together with the gas temperature as
reference.
The net rates $P_{\mathrm{r}, i j}$ are thus given per Ca~ion with positive values
indicating net photoionisation and negative values indicating net recombination.
Collisional b-f transitions have much smaller rates and can be neglected for this
analysis.
The sum of the radiative rates (thick solid) is close to zero in the lower parts
of the model atmosphere ($\lg m_\mathrm{c} > -1.0$).
There, the rates are in or close to equilibrium with the rates of the individual
transitions all being small.
The sum of the rates is also close to zero in the upper parts of the model atmosphere
below the transition region ($\lg m_\mathrm{c} \approx -5.6$) that have not yet been
disturbed by the shock wave (e.g., $\lg m_\mathrm{c} < -3.5$ in the upper row).
In the shock front, the sum of the rates is increased, leading to net ionisation
(positive values in the figure).
The largest contribution is due to photoionisation from the meta-stable
level~\mbox{$i = 3$}, followed by photoionisation from level~\mbox{$i = 2$}.
As the lower levels of these transitions belong to the same term and are
energetically close together (see Fig.~\ref{fig:modelatom}), the transition
rates show a very similar behaviour.
The photoionisation from the ground level ($i = 1$) is seen at all heights
below the transition region with a local maximum in the shock front, where it
has a small but positive contribution to the net ionisation.
The normalised rate reaches much larger relative values above the shock front
close to the transition region, compensating the recombination in most of the
other b-f transitions.
These large values are a result of the previous shock wave and long relaxation
timescales for this transition (see Sect.~\ref{sec:timescale_process}).
The transitions from level \mbox{$i = 4$} and \mbox{$i = 5$} have much smaller
rates with recombination prevailing throughout the model atmosphere.
They both behave very similarly because the lower levels are energetically close
together.
As the shock propagates through the chromosphere, the normalised photoionisation
rates increase significantly.
While the sum of the rates (thick solid line) is of the order of $10^{-4}$\,s$^{-1}$
at the shock front at $z \approx 900$\,km (top row of Fig.~\ref{fig:ratesvsheight}),
values of $\approx 2.5\,10^{-3}\,\mathrm{s}^{-1}$ are found at $z = 1150$\,km only
50\,s later (bottom row).
The second row from the bottom ($t = 2250$\,s) shows the instant shortly before
the shock wave reaches the transition region (seen as sharp jump in gas temperature
around $\lg m_\mathrm{c} = -5.6$).
Once the shock reaches the transition region (20\,s later, bottom row), the rates
decrease rapidly.
Except for a very small and smooth photoionisation from the ground level ($i = 1$),
all other transitions and with it the sum of the rates show recombination in almost
the entire model atmosphere from just behind the shock front down to the low
photosphere.
It demonstrates that the timescales on which the ionisation equilibrium is restored
are relatively short -- except for a comparatively sluggish photoionisation from
the ground level.
In Fig.~\ref{fig:rates_time}, the time evolution of the shock event is analysed
at a fixed geometric height of \mbox{$z = 1200$\,km}.
The vertical dotted lines divide the time span into a pre-shock phase,
a phase~I (when the shock occurs) and a phase~II (during the passage of the shock
wake).
As the processes are complex and the timescales are small, the \ion{Ca}{II}-{III}
ionisation fraction, the gas temperature, and the inverse relaxation timescale
are shown in all panels for reference purposes (all scaled to their full data
range).
The gas temperature (grey dashed line) shows a clear shock signature with a
rapid increase, a peak of $T_\mathrm{gas} = 8543$\,K at $t = 2248$\,s and a slower
decay in the shock wake afterwards.
The \ion{Ca}{II}-{III} ionisation fraction (grey dotted line) seems to react
on the shock with some delay and also much smoother.
It peaks only at $t = 2258$\,s with a value of 3.2\,\%, i.e. 10\,s after the
occurrence of the temperature peak.
The relaxation timescale $\tau$ reaches a minimum of
$\tau_\mathrm{min} = 7.6$\,s even later at $t = 2262$\,s.
The inverse timescale $\tau^{-1}$ is shown as solid grey line in the figure.
While the change in phase~I occurs rapidly (with a slope of $\tau^{-1}$
being between those of the gas temperature and the ionisation fraction),
the post-shock evolution in phase~II is much slower.
Then (e.g. at $t = 2300$\,s) the timescale $\tau$ is still reduced while
the gas temperature and the ionisation
fraction are already relatively close to their pre-shock state again.
The uppermost panel of Fig.~\ref{fig:rates_time} displays the total Ca~number density
$n_\mathrm{Ca,total}$,
i.e. the sum of levels 1 to 6 (thick solid line, right axis).
As the element abundance of Ca is constant in the simulation, it behaves exactly
like the gas density.
It decreases before the arrival of the shock front and rises rapidly again in
the shock front and more gradually in the shock wake afterwards.
The level population densities $n_i$, i.e. the number densities of Ca~atoms
with their valence electron in a specific energy level~$i$, are divided by the
total Ca number density $n_\mathrm{Ca, total}$ in the uppermost panel.
It removes the fluctuations due to the change in gas density and reveals the
relative distribution of the Ca atoms over the 6 included levels.
At mid-chromospheric heights, usually 95\,\% or more of all \ion{Ca}{II} ions are
in the ground level ($i = 1$) before and after the shock passage (i.e., in the
cool phases).
During the passage of a shock wave, the ground level is depopulated but usually
maintains at least 70\,\% of $n_\mathrm{Ca,total}$ and
thus remains the most populated level at all times.
The population densities $n_i$ of all other levels with $i > 1$ increase
during the shock passage as
\ion{Ca}{II} atoms are excited from the ground level into a higher level.
The increase of the population densities of levels 2 and 3 correlates with the
change in gas temperature, while levels 4 and 5 react with a delay of 5\,s, shortly
followed by the continuum level ($i = 6$).
The density of the latter, $n_6$, increases at most to a few percent of $n_\mathrm{Ca,total}$.
Most Ca atoms therefore remain singly ionised, even during the shock passages.
The \ion{Ca}{III} ions are actually by a factor of more than 3 less
abundant than the \ion{Ca}{II} ions with an electron in level 3 alone.
Ionisation is thus only a ``minor process'' compared to the bound-bound transitions
within the Ca atom.
The level populations of levels 4 and 5 on the other hand are much smaller than
the \ion{Ca}{III} number density.
Even if all ions in these levels would ionise, the resulting rise of the
\ion{Ca}{III} population would be negligible.
Levels 4 and 5 are nevertheless important as upper levels of radiative b-b
transitions.
\begin{figure}[t]
\begin{center}
\includegraphics{16186fig8.eps}
\caption{
Emergent intensity in the \ion{Ca}{II}\,K line:
temporal evolution of the line core intensity (top row)
and time-averaged spectrum (bottom row) at the solar limb ($\mu = 0.05$, left column)
and at disk-centre ($\mu = 1.0$, right column).
Each panel contains the results from the non-equilibrium simulation
($I_\mathrm{K, TD}$, solid) and from the statistical equilibrium
solution ($I_\mathrm{K, SE}$, red thick dashed) together with the
corresponding difference $(I_\mathrm{K, TD} - I_\mathrm{K, SE})/I_\mathrm{K, SE}$)
between both (solid grey line at the top of each panel).
The grey areas in the bottom panels represent the full data ranges of
all intensity profiles during the simulation sequence between
$t = 600$\,s and $t = 2350$\,s.
}
\label{fig:intensity}
\end{center}
\end{figure}
The net rates $P_{\mathrm{r}, i j}$ for the radiative bound-free transitions are
shown as function of time in the middle panel of Fig.~\ref{fig:rates_time}.
They are again given per Ca~ion with positive values
indicating net photoionisation and negative values indicating net recombination
(see Eq.~(\ref{eq:netrate})).
The sum of the net radiative rates $\sum_{i=1}^5 P_{\mathrm{r}, i 6}$
(thick solid line) gives the change of the \ion{Ca}{III}
number density and thus the \ion{Ca}{II}-{III} ionisation fraction.
Contributions from collisional b-f transitions are again small here.
The sum exhibits a net \mbox{(photo-)}ionisation in phase~I, which peaks shortly
(here 1\,s) before the gas temperature, and rapidly turns into net recombination
for phase~II.
While the shock wake is passing through the analysed height point, the
recombination becomes weaker and the rates approach zero again.
The b-f transitions from levels 3 and 2 contribute most (in that order) and
show the same temporal behaviour as the sum.
The net radiative bound-free rate for the ground level, $P_{\mathrm{r},1 6}$,
exhibits net photoionisation throughout the whole shock passage.
The rate increases with time and reaches a first hardly discernible maximum
together with or shortly before the gas temperature.
It roughly stays at the same level and smoothly declines in phase~II.
At that time, it is the only transition with a positive net contribution.
The bound-free transitions for levels 4 and 5 behave very much oppositely,
producing a net recombination the whole time.
The absolute values, however, stay small and even below the maximum recombination
rates of $P_{\mathrm{r},2 6}$ and $P_{\mathrm{r},3 6}$ that are reached during
phase~II.
The peculiar behaviour of the net rates $P_{\mathrm{r},2 6}$ and
$P_{\mathrm{r},3 6}$ is caused by a temporal delay between the photoionisation
($n_j R_{ji}$) and the recombination part ($n_i R_{ij}$, cf. Eq.~(\ref{eq:netrate})).
The photoionisation terms increase first and produce the strong net photoionisation
in phase~I, while the recombination part reaches a maximum only about 20\,s later.
At that time both parts are equal, resulting in a zero net rate.
Afterwards the recombination part, which lags behind, stays larger than the
ionisation part, causing the net recombination in phase~II.
The differences in the rates $R_{ij}$ are balanced by the number densities
of the lower levels so that the recombination part is only slightly smaller than
the photoionisation part.
In contrast, the recombination into level $i = 1$ is much smaller than the
corresponding photoionisation, which produces the smooth behaviour in time.
Finally, the radiative bound-free transition rates $R_{i\,6}$ for $i = 1, 2, 3$
are shown in Fig.~\ref{fig:rates_time}.
The photoionisation rates $R_{2\,6}$ and $R_{3\,6}$, which are almost identical
due to the similarity of their lower levels, exhibit no sharp response to the
passage of the shock wave.
In contrast, the recombination rates $R_{6\,3}$ and $R_{6\,2}$ increase strongly
during the shock phase~I.
The rate $R_{6\,3}$ is larger than $R_{6\,2}$ but both only differ by a
constant factor.
The strong increase of these rates correlates very well with the inverse
relaxation time scale (grey solid line) and the electron density
(solid black line).
It demonstrates that the relaxation timescale is set by the recombination into the
bound levels 3 and 2, which is directly depending on the density of free electrons.
With hydrogen being the main electron donor in the chromosphere,
the temporal behaviour of the recombination rates $R_{6\,3}$ and $R_{6\,2}$ is
therefore similar to the \ion{H}{II} number density.
This leads to the conclusion that the \ion{Ca}{II}-{III} ionisation in the quiet
solar chromosphere is governed by the finite timescales on which hydrogen reacts
on strong temperature changes.
\section{Discussion and conclusions}
\label{sec:disc}
The ionisation/recombination timescales, which are here derived from 1-D RADYN
simulations, are too long
for the assumption of an instantaneous ionisation equilibrium to be
valid. On the other hand, the timescales are not long enough to
warrant an assumption of a constant ionisation fraction.
We find noticeable deviations from the ionisation equilibrium in the middle model
chromosphere but in general
the \ion{Ca}{II}-{III} ionisation fraction remains small.
The error due to the often made simplifying assumption of statistical equilibrium
is therefore negligible for most applications.
The effect is barely visible in the synthesized intensity for the
diagnostically important spectral lines of \ion{Ca}{II}, i.e.,
the H and K lines and the infrared triplet.
This finding is illustrated for the \ion{Ca}{II}\,K line
in Fig.~\ref{fig:intensity}.
The differences in the emergent intensity between the statistical
equilibrium approach ($I_\mathrm{K, SE}$) and the time-dependent non-equilibrium simulation
($I_\mathrm{K, TD}$) are very small and hardly discernible in the temporal
evolution of the intensity at a fixed wavelength position in the line
core (top row) and even smaller (order of 0.1\,\%) in the
average spectrum (bottom row).
The relative difference $(I_\mathrm{K, TD} - I_\mathrm{K, SE})/I_\mathrm{K, SE}$
reveals peaks with values of up to a few percent for wavelengths
close to the line core.
This is true both at disk-centre (right column) and close to the limb (left column).
The differences are even smaller away from the line core and the emission peaks.
Noticeable deviations occur only in connection with shock fronts and thus sharp
intensity increases.
There, the effect is caused by a small temporal
offset in the evolution of the level populations in the TD case with respect to the
SE case, like
it is seen for the ionisation fraction in Fig.~\ref{fig:ionvstime}.
It is safe to conclude that the time-dependent non-equilibrium treatment
of the \ion{Ca}{II}-{III} ionisation appears to be of minor importance for the
lower atmosphere in quiet Sun regions.
One restriction of the simulations is the use of only one spatial dimension, which
could cause the shock wave profiles to be too extreme with potentially too high
peak temperatures.
In 3-D, the shock fronts are expected to be weaker owing to the larger number of
degrees of freedom.
The consequences of the deviations from the \ion{Ca}{II}-{III} ionisation
equilibrium would be even less important in that case.
Furthermore, we neglect the effect of incident radiation from the corona on
the photoionisation of Ca.
Most important in that respect would be the Lyman alpha line at
$\lambda = 121.5$\,nm.
However, the Lyman $\alpha$ photons have too little energy to ionise
\ion{Ca}{II} from the ground state (11.88\,eV or $\lambda = 104.4$\,nm).
The photoionisation edges for levels 2 and 3, on the other hand, both lie
close to the Lyman alpha line.
The incident radiation certainly matters most at large heights close to the
transition region.
There, however, photoionisation from the ground level dominates, which is
obviously not influenced by Lyman alpha photons.
We conclude that the RADYN simulations employed here are sufficiently
realistic for an evaluation of the importance of non-equilibrium
effects for the ionisation of calcium in the context of a strongly varying
chromosphere in quiet Sun regions.
\begin{acknowledgements}
This work was supported by a Marie Curie Intra-European Fellowship of the
European Commission (6th Framework Programme, FP6-2005-Mobility-5, Proposal
No.~042049) and a grant from the Research Council of Norway
(No.~191814/V30).
\end{acknowledgements}
\bibliographystyle{aa}
|
2,877,628,090,092 | arxiv | \section{Introduction}
It is widely believed that the cubic nonlinear Schr\"{o}dinger equation (NLS)
\begin{equation*}
i\partial _{t}\phi =L\phi +\left\vert \phi \right\vert ^{2}\phi \text{ in }
\mathbb{R}^{n+1},
\end{equation*}
where $L$ is the Laplacian $-\triangle $ or the Hermite operator $-\triangle
+\omega ^{2}\left\vert x\right\vert ^{2},$ describes the physical phenomenon
of Bose-Einstein condensation (BEC). This belief is one of the main
motivations for studying the cubic NLS. BEC is the phenomenon that particles
of integer spin (bosons) occupy a
macroscopic quantum state. This unusual state of matter was first predicted
theoretically by Einstein for non-interacting particles. The first
experimental observation of BEC in an interacting atomic gas did not occur
until 1995 using laser cooling techniques \cite{Anderson, Davis}. E. A.
Cornell, W. Ketterle, and C. E. Wieman were awarded the 2001 Nobel Prize in
physics for observing BEC. Many similar successful experiments \cite{Cornish, Ketterle, Stamper} were
performed later.
Let $t\in \mathbb{R}$ be the time variable and $\mathbf{r}_{N}=\left(
r_{1},r_{2},...,r_{N}\right) \in \mathbb{R}^{nN}$ be the position vector of $
N$ particles in $\mathbb{R}^{n}$. Then BEC naively means that the $N$-body
wave function $\psi _{N}(t,\mathbf{r}_{N})$ satisfies
\begin{equation*}
\psi _{N}(t,\mathbf{r}_{N})\sim \dprod\limits_{j=1}^{N}\varphi (t,r_{j})
\end{equation*}
up to a phase factor solely depending on $t$, for some one particle state $
\varphi .$ In other words, every particle is in the same quantum state.
Equivalently, there is the Penrose-Onsager formulation \cite{Penrose} of
BEC: if we define $\gamma _{N}^{(k)}$ to be the $k$-particle marginal
densities associated with $\psi _{N}$ by
\begin{equation}
\gamma _{N\,}^{(k)}(t,\mathbf{r}_{k};\mathbf{r}_{k}^{\prime })=\int \psi
_{N}(t,\mathbf{r}_{k},\mathbf{r}_{N-k})\overline{\psi _{N}}(t,\mathbf{r}
_{k}^{\prime },\mathbf{r}_{N-k})d\mathbf{r}_{N-k},\quad \mathbf{r}_{k},
\mathbf{r}_{k}^{\prime }\in \mathbb{R}^{nk} \label{E:marginal}
\end{equation}
then, equivalently, BEC means
\begin{equation}
\gamma _{N}^{(k)}(t,\mathbf{r}_{k};\mathbf{r}_{k}^{\prime })\sim
\dprod\limits_{j=1}^{k}\varphi (t,r_{j})\bar{\varphi}(t,r_{j}^{\prime }).
\label{formula:BEC state}
\end{equation}
Gross \cite{Gr1,Gr2} and Pitaevskii \cite{Pitaevskii} proposed that the
many-body effect should be model by a strong on-site interaction and hence
the one-particle state $\varphi $ should be modeled by the a cubic NLS. In a
series of works \cite{Lieb1, LiebAndSeiringer, E-E-S-Y1,
E-S-Y1,E-S-Y2,E-S-Y4, E-S-Y5, E-S-Y3,TChenAndNPSpace-Time, Chen3DDerivation}
, it has been proven rigorously that, under suitable assumptions on the
interaction potential, relation \eqref{formula:BEC state} holds in 3D and the
one-particle state $\varphi $ satisfies the 3D cubic NLS.
It is then natural to believe that the 2D cubic NLS describes the 2D BEC as
well. However, there is no BEC in 2D unless the temperature is absolute zero
(see p. 69 of \cite{Lieb2} and the references within). In other words, 2D
BEC is physically impossible due to the third law of thermodynamics. In a
physically realistic setting, 2D NLS can only arise from a 3D BEC with
strong confining in one direction (which we take to be the $z$-direction).
Such an effective 3D$\ $to 2D phenomenon has been experimentally observed
\cite{Kettle3Dto2DExperiment, FrenchExperiment, Philips, NatureExperiment,
Another2DExperiment}. (See \cite{ReviewFor2DExperiment} for a review.) It is
then natural to consider the derivation of the 2D NLS from a 3D $N$-body
quantum dynamic. Combining \cite{Abdallah1, Abdallah2, Chen3DDerivation}
suggests a route of getting the 2D NLS from 3D. First, a special case of
Theorem 2 in \cite{Chen3DDerivation} establishes the 3D cubic NLS
\begin{equation}
i\partial _{t}\varphi =-\triangle _{x}\varphi +\left( -\partial
_{z}^{2}+\omega ^{2}z^{2}\right) \varphi +\left\vert \varphi \right\vert
^{2}\varphi ,\text{ }\left( x,z\right) \in \mathbb{R}^{2+1}
\label{eqn:3D Cubic NLS}
\end{equation}%
from the 3D $N$-body quantum dynamic as a $N\rightarrow \infty $ limit. Then
the result in \cite{Abdallah1, Abdallah2} shows that the 2D cubic NLS arises
from equation \eqref{eqn:3D Cubic NLS} as a $\omega \rightarrow \infty $
limit. This path corresponds to the iterated limit ($\lim_{\omega
\rightarrow \infty }\lim_{N\rightarrow \infty }$) of the $N$-body dynamic,
thus the 2D cubic NLS coming from such a path approximates the 3D $N$-body
dynamic when $\omega $ is large and $N$ is infinity. In experiments, it is
fully possible to have $N$ and $\omega $ comparable to each other. In fact, $%
N$ is about $10^{4}$ and $\omega $ is about $10^{3}$ in \cite%
{Kettle3Dto2DExperiment, FrenchExperiment, NatureExperiment,
Another2DExperiment}. In this paper, we derive rigorously the 2D cubic NLS
as the double limit ($\lim_{N,\omega \rightarrow \infty }$) of a 3D quantum $%
N$-body dynamic directly, without passing through any 3D cubic NLS.
It is elementary mathematical analysis that $\lim_{\omega \rightarrow \infty
}\lim_{N\rightarrow \infty }$ and $\lim_{N,\omega \rightarrow \infty }$ are
topologically different and one does not imply each other. Let us adopt the
notation
\begin{equation*}
r_{i}=(x_{i},z_{i})\in \mathbb{R}^{2+1}
\end{equation*}%
and investigate the procedure of laboratory experiments of BEC according to
\cite{Kettle3Dto2DExperiment, FrenchExperiment, Philips, NatureExperiment,
Another2DExperiment}.
\noindent \textbf{Step A}. Confine a large number of bosons inside a trap with strong
confining in the $z$-direction. Cool it down so that the many-body system
reaches its ground state. It is expected that this ground state is a BEC
state / factorized state. To formulate the problem mathematically, we use
the quadratic potential $\left\vert \cdot \right\vert ^{2}$ to represent the
trap and
\begin{equation*}
V_{a}\left( r\right) =\frac{1}{a^{3\beta }}V\left( \frac{r}{a^{\beta }}
\right) \text{, }\beta >0
\end{equation*}
to represent the interaction potential. We use the quadratic potential to
represent the trap because this simplified yet reasonably general model is
expected to capture the salient features of the actual trap: on the one hand
the quadratic potential varies slowly, on the other hand it tends to $\infty
$ as $\left\vert x\right\vert \rightarrow \infty $. In the physics
literature, Lieb, Seiringer and Yngvason remarked in \cite{Lieb1} that the
confining potential is typically $\sim \left\vert x\right\vert ^{2}$ in the
available experiments. The review \cite{ReviewFor2DExperiment} on \cite{Kettle3Dto2DExperiment, FrenchExperiment, Philips, NatureExperiment, Another2DExperiment} also mentioned that the trap is harmonic. We use $
V_{a}\left( r\right) $ to represent the interaction potential to match the
Gross-Pitaevskii description \cite{Gr1,Gr2,Pitaevskii} that the many-body
effect should be modeled by an on-site self interaction because $V_{a}$ is
an approximation of the identity as $a\rightarrow 0$. This step then corresponds
to the following mathematical problem:
\begin{problem}
\label{Problem:Schnee-Yngvason}Show that, for large $N$ and large $\omega \gg \omega _{0}$, the ground state of the $N$-body Hamiltonian
\begin{equation}
\sum_{j=1}^{N}\left( -\triangle _{r_{j}}+\omega _{0}^{2}\left\vert
x_{j}\right\vert ^{2}+\omega ^{2}z_{j}^{2}\right) +
\sum_{1\leqslant i<j\leqslant N}\frac{1}{a^{3\beta -1}}V\left( \frac{
r_{i}-r_{j}}{a^{\beta }}\right) \label{E:general-Hamiltonian}
\end{equation}
is a factorized state under proper assumptions on $a$ and $V$.
\end{problem}
\noindent \textbf{Step B}. Switch the trap in order to enable measurement or direct
observation. It is assumed that such a shift of the confining potential is
instant and does not destroy the BEC obtained from Step A. To be more
precise about the word ``switch'': in \cite{Philips, Another2DExperiment}, the
trap in the $x$-spatial directions are tuned very loose to generate a 2D
Bose gas. For mathematical convenience, we can assume $\omega _{0}$ becomes $
0$. The system is then time dependent. Therefore, the factorized structure
obtained in Step A must be preserved in time for the observation of BEC.
Mathematically, this step stands for the following problem.
\begin{problem}
\label{Problem:ours}Take the BEC state obtained in Step A as initial datum,
show that, for large $N$ and $\omega ,$ the solution to the $N-$body Schr\"{o}dinger equation
\begin{equation}
i\partial _{t}\psi _{N,\omega }=\sum_{j=1}^{N}\left( -\frac{1}{2}\triangle
_{r_{j}}+\frac{\omega ^{2}}{2}z_{j}^{2}\right) \psi _{N,\omega }+
\sum_{1\leqslant i<j\leqslant N}\frac{1}{a^{3\beta-1 }}V\left( \frac{
r_{i}-r_{j}}{a^{\beta }}\right) \psi _{N,\omega }
\label{equation:N-Body Schrodinger with anisotropic trap}
\end{equation}
is a BEC state / factorized state under the same assumptions of the
interaction potential $V$ in Problem \ref{Problem:Schnee-Yngvason}.
\end{problem}
We first remark that neither of the problems listed above admits a
factorized state solution. It is also unrealistic to solve the equations in
Problems \ref{Problem:Schnee-Yngvason} and \ref{Problem:ours} for large $N$.
Moreover, both problems are linear so that it is not clear how the 2D cubic
NLS arises from either problem. Therefore, in order to justify the statement
that the 2D cubic NLS depicts the 3D to 2D BEC, we have to show
mathematically that, in an appropriate sense, for some 3D one particle state
$\varphi $ fully described by the 2D cubic NLS
\begin{equation*}
\gamma _{N,\omega }^{(k)}(t,\mathbf{r}_{k};\mathbf{r}_{k}^{\prime })\sim
\dprod\limits_{j=1}^{k}\varphi (t,r_{j})\bar{\varphi}(t,r_{j}^{\prime })
\text{ as }N,\omega \rightarrow \infty
\end{equation*}
where $\gamma _{N,\omega }^{(k)}$ are the $k$-marginal densities associated
with $\psi _{N,\omega }$.
For Problem \ref{Problem:Schnee-Yngvason} (Step A), a satisfying answer has
been found by Schnee and Yngvason. Let $\func{scat}(W)$ denote the 3D
scattering length of the potential $W$. By \cite[Lemma A.1]{E-S-Y2}, for $
0<\beta \leq 1$ and $a\ll 1$, we have
\begin{equation*}
\func{scat}\left( a\cdot \frac{1}{a^{3\beta }}V\left( \frac{r}{a^{\beta }}
\right) \right) \sim \left\{ \begin{aligned} &a \int_{\mathbb{R}^3} V &&
\text{if } 0\leq \beta <1 \\ &a \func{scat}(V) && \text{if }\beta =1 \end{aligned}
\right.
\end{equation*}
Consider $\phi _{\omega _{0},Ng}$, the minimizer to the 2D NLS energy
functional
\begin{equation}
E_{\omega _{0},Ng}=\int_{\mathbb{R}^{2}}\left( |\nabla \phi (x)|^{2}+\omega
_{0}^{2}\left\vert x\right\vert ^{2}|\phi (x)|^{2}+4\pi Ng|\phi
(x)|^{4}\right) \,dx \label{E:GP-Hamiltonian}
\end{equation}
subject to the constraint $\Vert \phi \Vert _{L^{2}(\mathbb{R}^{2})}=1$. The
existence of this nonlinear ground state stems from the presence of the
confining potential $\omega _{0}^{2}\left\vert x\right\vert ^{2}$; otherwise
the nonlinear term is defocusing (as it is called in the NLS literature).
Given parameters $\omega _{0},\omega ,N,a$, Schnee-Yngvason \cite
{SchneeYngvason} define $g=g(\omega _{0},\omega ,N,a)$ and $\bar{\rho}=\bar{
\rho}(\omega _{0},\omega ,N,a)$ by the two simultaneous equations (see
(1.15) and (1.18) in \cite{SchneeYngvason})
\begin{equation*}
g\overset{\mathrm{def}}{=}\left\vert -\log (\frac{\bar{\rho}}{\omega })+
\frac{1}{\sqrt{\omega }a\int_{\mathbb{R}}h_{1}^{4}}\right\vert
^{-1}\,,\qquad \bar{\rho}=N\int |\phi _{\omega _{0},Ng}|^{4}.
\end{equation*}
They argue that this definition for $g$ makes the 2D NLS Hamiltonian
\eqref{E:GP-Hamiltonian} relevant to the analysis of the limiting behavior
of the ground state of \eqref{E:general-Hamiltonian} describing a dilute
interacting Bose gas in a 3D trap that is strongly confining in the $z$-direction. (See also \cite{JunYin} for the case with rotation)
The Gross-Pitaevskii limit means $Ng\sim 1$. We have liberty to fix the
value of $\omega _{0}$ by scaling, so we take $\omega _{0}=1$. Then the
minimizer $\phi _{\omega _{0},Ng}$ is fixed and hence $\bar{\rho}\sim N$.
In this paper, we consider Problem \ref{Problem:ours} (Step B) and offer a
rigorous derivation of the 2D cubic NLS from the 3D quantum many-body
dynamic. For the scaling of the interaction potential, we consider the case
(called Region I in \cite{SchneeYngvason}) in which the term $(\sqrt{\omega }
a)^{-1}$ dominates in the definition of $g$. Then
\begin{equation*}
1\sim Ng\sim Na\sqrt{\omega }\iff a\sim \frac{1}{N\sqrt{\omega }}
\end{equation*}
This then implies that
\begin{equation*}
\frac{1}{\sqrt{\omega }a}\sim N\gg \log \frac{N}{\omega }\sim \log \frac{
\bar{\rho}}{\omega }
\end{equation*}
so that our assumption that the term $(\sqrt{\omega }a)^{-1}$ dominates in
the definition of $g$ is self-consistent.
We will take for mathematical convenience $a=(N\sqrt{\omega })^{-1}$ for
Problem \ref{Problem:ours} (Step B). The Hamiltonian
\eqref{E:general-Hamiltonian} then becomes
\begin{equation}
H_{N,\omega }=\sum_{j=1}^{N}\left( -\triangle _{r_{j}}+\omega
^{2}z_{j}^{2}\right) +\frac{1}{N\sqrt{\omega }}\sum_{1\leqslant i<j\leqslant
N}\left( N\sqrt{\omega }\right) ^{3\beta }V\left( \left( N\sqrt{\omega }
\right) ^{\beta }\left( r_{i}-r_{j}\right) \right)
\label{Hamiltonian:H_N,W,nonscaled}
\end{equation}
Let $h(z)=\pi ^{-1}e^{-z^{2}/2}$ so that $h$ is the normalized ground state
eigenfunction of $-\partial _{z}^{2}+z^{2}$, i.e. it solves $(-1-\partial
_{z}^{2}+z^{2})h=0$. Then the normalized ground state eigenfunction $
h_{\omega }(z)$ of $-\partial _{z}^{2}+\omega ^{2}z^{2}$ is given by $
h_{\omega }(z)=\omega ^{1/4}h(\omega ^{1/2}z)$, i.e. it solves $(-\omega
-\partial _{z}^{2}+\omega ^{2}z^{2})h_{\omega }=0$. In particular, $h_{1}=h$.
We consider initial data that is asymptotically (as $N\rightarrow \infty
,\omega \rightarrow \infty $) factorized in the $x$-direction and in the
ground state in the $z$-direction; in particular we could take
\begin{equation*}
\psi _{N,\omega }(0,\mathbf{r}_{N})=\prod_{j=1}^{N}\phi _{0}(x_{j})h_{\omega
}(z_{j})\,,\qquad \Vert \phi _{0}\Vert _{L^{2}(\mathbb{R}^{2})}=1.
\end{equation*}
Let
\begin{equation}
\psi _{N,\omega }(t,\cdot )=e^{itH_{N,\omega }}\psi _{N,\omega }(0,\cdot )
\label{E:evolution}
\end{equation}
denote the evolution of this initial data according to the Hamiltonian
\eqref{Hamiltonian:H_N,W,nonscaled}. We prove that in a certain sense, as $
N\rightarrow \infty ,\omega \rightarrow \infty $,
\begin{equation}
\psi _{N,\omega }(t,\mathbf{r}_{N})\sim \prod_{j=1}^{N}\phi
(t,x_{j})h_{\omega }(z_{j}) \label{E:informal-conv}
\end{equation}
where $\phi (t)$ solves a 2D cubic NLS with initial data $\phi _{0}(x)$. To make
this statement more precise, we introduce the rescaled solution
\begin{equation}
\tilde{\psi}_{N,\omega }(t,\mathbf{r}_{N})\overset{\mathrm{def}}{=}\frac{1}{
\omega ^{N/4}}\psi _{N,\omega }(t,\mathbf{x}_{N},\frac{\mathbf{z}_{N}}{\sqrt{
\omega }}) \label{E:rescaled}
\end{equation}
and the rescaled Hamiltonian
\begin{equation}
\tilde{H}_{N,\omega }=\sum_{j=1}^{N}(-\Delta _{x_{j}}+\omega (-\partial
_{z_{j}}^{2}+z_{j}^{2}))+\frac{1}{N}\sum_{1\leq i<j\leq N}V_{N,\omega
}(r_{i}-r_{j}) \label{E:rescaled-Hamiltonian}
\end{equation}
where
\begin{equation}
V_{N,\omega }(r)=N^{3\beta }\left( \sqrt{\omega }\right) ^{3\beta -1}V\left(
\left( N\sqrt{\omega }\right) ^{\beta }x,\frac{\left( N\sqrt{\omega }\right)
^{\beta }}{\sqrt{\omega }}z\right) , \label{E:V}
\end{equation}
Then
\begin{equation*}
(\tilde{H}_{N,\omega }\tilde{\psi}_{N,\omega })(t,\mathbf{x}_{N},\mathbf{z}
_{N})=\frac{1}{\omega ^{N/4}}(H_{N,\omega }\psi _{N,\omega })(t,\mathbf{x}
_{N},\frac{\mathbf{z}_{N}}{\sqrt{\omega }})
\end{equation*}
and hence when $\psi _{N,\omega }(t)$ is given by \eqref{E:evolution} and $
\tilde{\psi}_{N,\omega }$ is defined by \eqref{E:rescaled}, we have
\begin{equation*}
\tilde{\psi}_{N,\omega }(t,\mathbf{r}_{N})=e^{it\tilde{H}_{N,\omega }}\tilde{
\psi}(0,\mathbf{r}_{N})
\end{equation*}
The informal statement of convergence given by \eqref{E:informal-conv}
becomes the informal statement
\begin{equation}
\tilde{\psi}(t,\mathbf{r}_{N})\sim \prod_{j=1}^{N}\phi (t,x_{j})h(z_{j})
\label{E:informal-conv-rescaled}
\end{equation}
where $\phi (t)$ solves 2D NLS with initial data $\phi _{0}(x)$. In fact,
the convergence we prove is stated in terms of the associated density
operators with kernels
\begin{equation}
\tilde{\gamma}_{N,\omega }(t,\mathbf{r}_{N},\mathbf{r}_{N}^{\prime })=\tilde{
\psi}(t,\mathbf{r}_{N})\overline{\tilde{\psi}(t,\mathbf{r}_{N}^{\prime })}
\label{E:densities}
\end{equation}
The version of \eqref{E:informal-conv-rescaled} that we prove is the
convergence
\begin{equation*}
\tilde{\gamma}_{N,\omega }^{(k)}(t,\mathbf{r}_{k},\mathbf{r}_{k}^{\prime})\rightarrow \prod_{j=1}^{k}\phi (x_{j})h(z_{j})\overline{\phi (x_{j}^{\prime})} \overline{h(z_{j}^{\prime })}
\end{equation*}
in trace class, for each $k\geq 0$.
We define
\begin{equation}
\label{E:vofbeta}
v(\beta) = \max\left( \frac{1-\beta}{2\beta}, \; \frac{\frac54\beta-\frac1{12}}{1-\frac52\beta}, \; \frac{\frac12\beta + \frac56}{1-\beta}, \; \frac{\beta+\frac13}{1-2\beta}\right)
\end{equation}
(see Fig. \ref{F:vofbeta})
Our main theorem is the following:
\begin{theorem}[main theorem]
\label{Theorem:3D->2D BEC (Nonsmooth)} Assume the pair interaction $V$ is a
nonnegative Schwartz class function. Let $\{\tilde{\gamma}_{N,\omega
}^{(k)}(t,\mathbf{r}_{k};\mathbf{r}_{k}^{\prime })\,\}$ be the family of
marginal densities associated with the 3D rescaled Hamiltonian evolution $
\tilde{\psi}_{N,\omega }(t)=e^{it\tilde{H}_{N,\omega }}\tilde{\psi}
_{N,\omega }(0)$ for some $\beta \in \left( 0,2/5\right) $, (see
\eqref{E:marginal}, \eqref{E:rescaled-Hamiltonian}, \eqref{E:densities}).
Suppose the initial datum $\tilde{\psi}_{N,\omega }(0)$ satisfies the
following:
\textnormal{(a)} $\tilde \psi _{N,\omega }(0)$ is normalized, that is, $\| \tilde \psi
_{N,\omega }(0)\|_{L^{2}}=1$,
\textnormal{(b)} $\tilde{\psi}_{N,\omega }(0)$ is asymptotically factorized in the sense
that
\begin{equation*}
\lim_{N,\omega \rightarrow \infty }\limfunc{Tr}\left\vert \tilde{\gamma}
_{N,\omega }^{(1)}(0,x_{1},z_{1};x_{1}^{\prime },z_{1}^{\prime })-\phi
_{0}(x_{1})\overline{\phi _{0}}(x_{1}^{\prime })h(z_{1})h(z_{1}^{\prime
})\right\vert =0,
\end{equation*}
for some one particle state $\phi _{0}\in H^{1}\left( \mathbb{R}^{2}\right)
, $
\textnormal{(c)} Away from the $z$-directional ground state energy, $\tilde{\psi}
_{N,\omega }(0)$ has finite energy per particle:
\begin{equation*}
\sup_{\omega ,N}\frac{1}{N}\langle \tilde{\psi}_{N,\omega }(0),(\tilde{H}
_{N,\omega }-N\omega )\tilde{\psi}_{N,\omega }(0)\rangle \leqslant C,
\end{equation*}
Then $\forall k\geqslant 1,t\geqslant 0,$ and $\varepsilon >0$, we have the
convergence in trace norm (propagation of chaos) that
\begin{equation*}
\lim_{\substack{ N,\omega \rightarrow \infty \\ N\geqslant \omega ^{v(\beta
)+\varepsilon }}}\limfunc{Tr}\left\vert \tilde{\gamma}_{N,\omega }^{(k)}(t,
\mathbf{x}_{k},\mathbf{z}_{k};\mathbf{x}_{k}^{\prime },\mathbf{z}
_{k}^{\prime })-\dprod\limits_{j=1}^{k}\phi (t,x_{j})\overline{\phi }
(t,x_{j}^{\prime })h_{1}(z_{j})h_{1}(z_{j}^{\prime })\right\vert =0,
\end{equation*}
where $v(\beta )$ is given by \eqref{E:vofbeta} and $\phi (t,x)$ solves the
2D cubic NLS with coupling constant $b_{0}\left( \int \left\vert
h_{1}(z)\right\vert ^{4}dz\right) $ that is
\begin{equation}
i\partial _{t}\phi =-\triangle _{x}\phi +b_{0}\left( \int \left\vert
h_{1}(z)\right\vert ^{4}dz\right) \left\vert \phi \right\vert ^{2}\phi \quad
\text{ in }\mathbb{R}^{2+1} \label{equation:2D Cubic NLS}
\end{equation}
with initial condition $\phi \left( 0,x\right) =\phi _{0}(x)$ and $
b_{0}=\int V\left( r\right) dr$.
\end{theorem}
Theorem \ref{Theorem:3D->2D BEC (Nonsmooth)} is equivalent to the following
theorem.
\begin{theorem}[main theorem]
\label{Theorem:3D->2D BEC} Assume the pair interaction $V$ is a nonnegative
Schwartz class function. Let $\{\tilde{\gamma}_{N,\omega }^{(k)}(t,\mathbf{r}
_{k};\mathbf{r}_{k}^{\prime })\,\}$ be the family of marginal densities
associated with the 3D rescaled Hamiltonian evolution $\tilde{\psi}
_{N,\omega }(t)=e^{it\tilde{H}_{N,\omega }}\tilde{\psi}_{N,\omega }(0)$ for
some $\beta \in \left( 0,2/5\right) $, (see \eqref{E:marginal},
\eqref{E:rescaled-Hamiltonian}, \eqref{E:densities}). Suppose the initial
datum $\tilde{\psi}_{N,\omega }(0)$ is normalized, asymptotically factorized
and satisfies the energy condition that
\textnormal{($\text{c}'$)} there is a $C>0$ such that
\begin{equation}
\langle \tilde{\psi}_{N,\omega }(0),(\tilde{H}_{N,\omega }-N\omega )^{k}
\tilde{\psi}_{N,\omega }(0)\rangle \leqslant C^{k}N^{k}\text{, }\forall
k\geqslant 1, \label{Condition:EnergyBoundOnInitialData}
\end{equation}
Then $\forall k\geqslant 1,t\geqslant 0,$ and $\varepsilon >0$, we have the
convergence in trace norm (propagation of chaos) that
\begin{equation*}
\lim_{\substack{ N,\omega \rightarrow \infty \\ N\geqslant \omega ^{v(\beta
)+\varepsilon }}}\limfunc{Tr}\left\vert \tilde{\gamma}_{N,\omega }^{(k)}(t,
\mathbf{x}_{k},\mathbf{z}_{k};\mathbf{x}_{k}^{\prime },\mathbf{z}
_{k}^{\prime })-\dprod\limits_{j=1}^{k}\phi (t,x_{j})\overline{\phi }
(t,x_{j}^{\prime })h_{1}(z_{j})h_{1}(z_{j}^{\prime })\right\vert =0,
\end{equation*}
where $v(\beta )$ is given by \eqref{E:vofbeta} and $\phi (t,x)$ solves the
2D cubic NLS \eqref{equation:2D Cubic NLS}.
\end{theorem}
\begin{figure}
\includegraphics[scale=0.7]{vofbeta}
\caption{ \label{F:vofbeta} A graph of the various rational functions of $\beta$ appearing in \eqref{E:vofbeta}. In Theorems \ref{Theorem:3D->2D BEC (Nonsmooth)}, \ref{Theorem:3D->2D BEC}, the limit $(N,\omega) \to \infty$ is taken with $N\geq \omega^{v(\beta)+\epsilon}$. As shown here, there are values of $\beta$ for which $v(\beta) \sim 1$, which allows $N\sim \omega$, as in the experimental paper \cite{Kettle3Dto2DExperiment, FrenchExperiment, NatureExperiment,Another2DExperiment}. We conjecture that Theorems \ref{Theorem:3D->2D BEC (Nonsmooth)}, \ref{Theorem:3D->2D BEC} hold with \eqref{E:vofbeta} replaced by the weaker constraint $v(\beta)=\frac{1-\beta}{2\beta}$ for all $0< \beta < 1$.}
\end{figure}
We remark that assumptions (a), (b), and (c) in Theorem \ref{Theorem:3D->2D BEC (Nonsmooth)} are
reasonable assumptions on the initial datum coming from Step A. In
fact, if we assume further that $\phi _{0}$ minimizes the 2D
Gross-Pitaevskii functional \eqref{E:GP-Hamiltonian}, then (a), (b) and (c)
are the conclusion of \cite[Theorem 1.1, 1.3]{SchneeYngvason}. The limit in
Theorem \ref{Theorem:3D->2D BEC (Nonsmooth)}, which is taken as $N,\omega \rightarrow
\infty $ within the subregion $N\geqslant \omega ^{v(\beta )+\varepsilon }$
is optimal in the sense that if $N\leqslant \omega ^{\frac{1}{2\beta }-\frac{
1}{2}}$, then the limit of $V_{N,\omega }$ defined by \eqref{E:V} is not a
delta function.
The equivalence of Theorems \ref{Theorem:3D->2D BEC (Nonsmooth)} and \ref{Theorem:3D->2D BEC} for asymptotically factorized initial data is well-known. In the main part of this paper, we prove Theorem \ref{Theorem:3D->2D BEC} in full detail. For completeness, we
discuss briefly how to deduce Theorem \ref{Theorem:3D->2D BEC (Nonsmooth)} from Theorem \ref{Theorem:3D->2D BEC} in Appendix \ref{A:equivalence}.
The main tool used to prove Theorem \ref{Theorem:3D->2D BEC} is the analysis
of the BBGKY hierarchy of $\left\{ \tilde{\gamma}_{N,\omega }^{(k)}\right\}
_{k=1}^{N}$ as $N,\omega \rightarrow \infty .$ With our definition, the
sequence of the marginal densities $\left\{ \tilde{\gamma}_{N,\omega
}^{(k)}\right\} _{k=1}^{N}$ associated with $\tilde{\psi}_{N,\omega }$
satisfies the BBGKY hierarchy
\begin{equation}
\label{hierarchy:BBGKY hierarchy for scaled marginal densities}
\begin{aligned}
i\partial _{t}\tilde{\gamma}_{N,\omega }^{(k)} =&\sum_{j=1}^{k}\left[
-\triangle _{x_{j}},\tilde{\gamma}_{N,\omega }^{(k)}\right]
+\sum_{j=1}^{k}\omega \left[ -\partial _{z_{j}}^{2}+z_{j}^{2},\tilde{\gamma}
_{N,\omega }^{(k)}\right] +\frac{1}{N}\sum_{i<j}^{k}\left[ V_{N,\omega
}\left( r_{i}-r_{j}\right) ,\tilde{\gamma}_{N,\omega }^{(k)}\right]\\
&+\frac{N-k}{N}\limfunc{Tr}\nolimits_{r_{k+1}}\sum_{j=1}^{k}\left[
V_{N,\omega }\left( r_{j}-r_{k+1}\right) ,\tilde{\gamma}_{N,\omega }^{(k+1)}
\right]
\end{aligned}
\end{equation}
In the classical setting, deriving mean-field type equations by studying the
limit of the BBGKY hierarchy was proposed by Kac and demonstrated by
Landford's work \cite{Lanford} on the Boltzmann equation. In the quantum
setting, the usage of the BBGKY hierarchy was suggested by Spohn \cite{Spohn}
and has been proven to be successful by Elgart, Erd\"{o}s, Schlein, and Yau
in their fundamental papers \cite{E-E-S-Y1, E-S-Y1,E-S-Y2,E-S-Y4, E-S-Y5,
E-S-Y3} which rigorously derives the 3D cubic NLS from a 3D quantum many-body
dynamic without a trap. The Elgart-Erd\"{o}s-Schlein-Yau program consists of
two principal parts: in one part, they consider the sequence of the marginal
densities $\left\{ \gamma _{N}^{(k)}\right\} $ associated with the
Hamiltonian evolution $e^{itH_{N}}\psi _{N}(0)$ where
\begin{equation*}
H_{N}=\sum_{j=1}^{N}-\triangle _{r_{j}}+\frac{1}{N}\sum_{1\leqslant
i<j\leqslant N}N^{3\beta }V(N^{\beta }\left( r_{i}-r_{j}\right) )
\end{equation*}
and prove that an appropriate limit of as $N\rightarrow \infty $ solves the
3D Gross-Pitaevskii hierarchy
\begin{equation}
i\partial _{t}\gamma ^{(k)}+\sum_{j=1}^{k}\left[ \triangle _{r_{k}},\gamma
^{(k)}\right] =b_{0}\sum_{j=1}^{k}\limfunc{Tr}\nolimits_{r_{k+1}}
[\delta(r_j-r_{k+1}),\gamma ^{(k+1)}] ,\text{ for all }k \geq 1 \,.
\label{equation:Gross-Pitaevskii hiearchy without a trap}
\end{equation}
In another part, they show that hierarchy \eqref{equation:Gross-Pitaevskii
hiearchy without a trap} has a unique solution which is therefore a
completely factorized state. However,
the uniqueness theory for hierarchy \eqref{equation:Gross-Pitaevskii hiearchy
without a trap} is surprisingly delicate due to the fact that it is a
system of infinitely many coupled equations over an unbounded number of
variables. In \cite{KlainermanAndMachedon}, by imposing a space-time bound
on the limit of $\left\{ \gamma _{N}^{(k)}\right\} $, Klainerman and
Machedon gave another proof of the uniqueness in \cite{E-S-Y2} through a
collapsing estimate originating from the ordinary multilinear Strichartz
estimates in their null form paper \cite{KlainermanMachedonNullForm} and a
board game argument inspired by the Feynman graph argument in \cite{E-S-Y2}.
Later, the method in Klainerman and Machedon \cite{KlainermanAndMachedon}
was taken up by Kirkpatrick, Schlein, and Staffilani \cite{Kirpatrick}, who
derived the 2D cubic NLS from the 2D quantum many-body dynamic; by Chen and
Pavlovi\'{c} \cite{TChenAndNpGP1, TChenAndNP}, who considered the 1D and 2D
3-body interaction problem and the general existence theory of hierarchy $
\eqref{equation:Gross-Pitaevskii hiearchy without a trap}$; and by X.C. \cite
{ChenAnisotropic}, who investigated the trapping problem in 2D and 3D. In
\cite{TCNPNT, TCNPNT1}, Chen, Pavlovi\'{c} and Tzirakis worked out the
virial and Morawetz identities for hierarchy \eqref{equation:Gross-Pitaevskii
hiearchy without a trap}. In 2011, for the 3D case without traps, Chen and
Pavlovi\'{c} \cite{TChenAndNPSpace-Time} proved that, for $\beta \in (0,1/4)$
, the limit of $\left\{ \gamma _{N}^{(k)}\right\} $ actually satisfies the
space-time bound assumed by Klainerman and Machedon \cite
{KlainermanAndMachedon} as $N\rightarrow \infty $. This has been a
well-known open problem in the field. In 2012, X.C. \cite{Chen3DDerivation}
extended and simplified their method to study the 3D trapping problem for $
\beta \in (0,2/7].$
The $\beta =0$ case has been studied by many authors as well \cite{E-Y1,LChen,KnowlesAndPickl,MichelangeliSchlein,RodnianskiAndSchlein}.
Away from the usage of the BBGKY hierarchy, there has been work by X.C.,
Grillakis, Machedon and Margetis \cite{GMM1,GMM2,Chen2ndOrder,GM1} using the
second order correction which can deal with $e^{itH_{N}}\psi _{N}$ directly.
To our knowledge, this is the first direct rigorous treatment of the 3D to 2D dynamic problem. We now compare our theorem with the known work which derives $n$D cubic NLS from the $n$D quantum many-body dynamic. It is easy to tell that Theorem \ref{Theorem:3D->2D BEC} deals with a different limit than the known work \cite{AGT, E-E-S-Y1,
E-S-Y1,E-S-Y2,E-S-Y4, E-S-Y5,
E-S-Y3,Kirpatrick,TChenAndNP,ChenAnisotropic,TChenAndNPSpace-Time,
Chen3DDerivation} which derives $n$D NLS from $n$D dynamics. On the one
hand, Theorem \ref{Theorem:3D->2D BEC} deals with a 3D to 2D effect. Such a
phenomenon is described by the limit equation \eqref{equation:2D Cubic NLS}
and the coupling constant $\int \left\vert h_{1}(z)\right\vert ^{4}dz.$ The
limit in Theorem \ref{Theorem:3D->2D BEC} is with the scaling
\begin{equation*}
\lim_{\substack{ N,\omega \rightarrow \infty \\ N\geqslant \omega ^{v(\beta
)+\varepsilon }}}N\sqrt{\omega }\limfunc{scat}\left( \frac{V_{N,\omega }}{N}
\right) =\text{constant,}
\end{equation*}
instead of the scaling
\begin{equation*}
\lim_{N\rightarrow \infty }N\limfunc{scat}(N^{n\beta -1}V(N^{\beta }\cdot ))=
\text{constant,}
\end{equation*}
in the known $n$D to $n$D work.
The main idea of the proof of Theorem \ref{Theorem:3D->2D BEC} is to
investigate the limit of hierarchy \eqref{hierarchy:BBGKY hierarchy for scaled
marginal densities} which at a glance is similar to the $n$D to $n$D work.
However, in contrast with the $n$D to $n$D case, even the formal limit of
hierarchy \eqref{hierarchy:BBGKY hierarchy for scaled marginal densities} is
not known.
Heuristically, according to the uncertainty principle, in 3D, as the $z$-component of the particles' position becomes more and more determined to be $0$, the $z$-component of the momentum and thus the energy must blow up. Hence the energy of the system is dominated by its $z$-directional part which is in fact infinity as $N,\omega \rightarrow \infty $. This renders the energy and thus the analysis of the $x-$component intractable.
Technically, it is not clear whether the term
\begin{equation*}
\omega \left[ -\partial _{z_{j}}^{2}+z_{j}^{2},\tilde{\gamma}_{N,\omega
}^{(k)}\right]
\end{equation*}
tends to a limit as $N,\omega \rightarrow \infty $. Since $\tilde{\gamma}
_{N,\omega }^{(k)}\ $is not a factorized state for $t>0$, one cannot expect
the commutator to be zero. Thus we formally have an $\infty -\infty $ in
hierarchy \eqref{hierarchy:BBGKY hierarchy for scaled marginal densities} as $
N,\omega \rightarrow \infty .$ This is the main difficulty we need to
circumvent in the proof of Theorem \ref{Theorem:3D->2D BEC}.
\subsection{Acknowledgements}
J.H. was supported in part by NSF grant DMS-0901582 and a Sloan Research Fellowship (BR-4919). X.C. would like to express his thanks to M. Grillakis, M. Machedon, D. Margetis, W. Strauss, and N. Tzirakis for discussions related to this work, to T. Chen and N. Pavlovi\'{c} for raising the 2D to 1D question during the X.C.'s seminar talk in Austin, to K. Kirkpatrick for encouraging X.C. to work on this problem during X.C.'s visit to Urbana. We thank Christof Sparber for pointing out references \cite{Abdallah1, Abdallah2}.
\section{Outline of the proof of Theorem \ref{Theorem:3D->2D BEC}}
We begin by setting down some notation that will be used in the remainder of the paper. We will always assume $\omega \geq 1$. Note that, as an operator, we have the positivity:
$$-1 - \partial_{z_j}^2 + z_j^2 \geq 0$$
Define
\begin{equation}
\label{E:tilde-S-def}
\tilde S_j \stackrel{\rm{def}}{=} (1-\Delta_{x_j} + \omega( -1 - \partial_{z_j}^2 + z_j^2))^{1/2}
\end{equation}
We have $\tilde S_j^2 (\phi(x_j) h(z_j)) = (1-\Delta_{x_j})\phi(x_j) \, h(z_j)$ and thus the diverging $\omega$ parameter has no consequence when the operator is applied to a tensor product function $\phi(x_j)h(z_j)$ for which the $z_j$-component rests in the ground state.
Let $P_0$ denote the orthogonal projection onto the ground state of $-\partial_z^2 + z^2$ and $P_1$ denote the orthogonal projection onto all higher energy modes, so $I=P_0+P_1$, where $I:L^2(\mathbb{R}^3)\to L^2(\mathbb{R}^3)$. Let $P_0^j$ and $P_1^j$ be the corresponding operators acting on $L^2(\mathbb{R}^{3N})$ in the $z_j$ component, $1\leq j \leq N$. Then
\begin{equation}
\label{E:cpct1}
I = \prod_{j=1}^k (P_0^j+ P_1^j) \,, \quad \text{where} \quad I: L^2(\mathbb{R}^{3N}) \to L^2(\mathbb{R}^{3N})
\end{equation}
For a $k$-tuple $\mathbf{\alpha} = (\alpha_1, \ldots, \alpha_k)$ with $\alpha_j \in \{0,1\}$, let $P_{\mathbf{\alpha}} = P_{\alpha_1}^1 \cdots P_{\alpha_k}^k$. Adopt the notation
$$|\mathbf{\alpha}| = \alpha_1 + \cdots + \alpha_k$$
This leads to the coercivity (operator lower bounds) given in Lemma \ref{L:coercivity}.
We next introduce an appropriate topology on the density matrices as was previously done in \cite{E-E-S-Y1, E-Y1, E-S-Y1,E-S-Y2,E-S-Y4, E-S-Y5, E-S-Y3,Kirpatrick,TChenAndNP,ChenAnisotropic,Chen3DDerivation}. Denote the spaces of compact operators and trace class operators on $L^{2}\left( \mathbb{R}^{3k}\right) $ as $\mathcal{K}_{k}$ and $\mathcal{L}_{k}^{1}$, respectively. Then $\left( \mathcal{K}_{k}\right) ^{\prime }=\mathcal{L}_{k}^{1}$. By the fact that $\mathcal{K}_{k}$ is separable, we select a dense countable subset $\{ J_{i}^{(k)}\} _{i\geqslant 1}\subset \mathcal{K}_{k}$ in the unit ball of $\mathcal{K}_{k}$ (so $\Vert J_{i}^{(k)}\Vert _{\operatorname{op}}\leqslant 1$ where $\left\Vert \cdot \right\Vert_{\operatorname{op}}$ is the operator norm). For $\gamma ^{(k)},\tilde{\gamma}^{(k)}\in \mathcal{L}_{k}^{1}$, we then define a metric $d_{k}$ on $\mathcal{L}_{k}^{1}$ by
\begin{equation*}
d_{k}(\gamma ^{(k)},\tilde{\gamma}^{(k)})=\sum_{i=1}^{\infty
}2^{-i}\left\vert \limfunc{Tr}J_{i}^{(k)}\left( \gamma ^{(k)}-\tilde{\gamma}
^{(k)}\right) \right\vert .
\end{equation*}
A uniformly bounded sequence $\tilde{\gamma}_{N,\omega }^{(k)}\in
\mathcal{L}_{k}^{1}$ converges to $\tilde{\gamma}^{(k)}\in \mathcal{L}
_{k}^{1}$ with respect to the weak* topology if and only if
\begin{equation*}
\lim_{N,\omega \rightarrow \infty }d_{k}(\tilde{\gamma}_{N,\omega }^{(k)},
\tilde{\gamma}^{(k)})=0.
\end{equation*}
For fixed $T>0$, let $C\left( \left[ 0,T\right] ,\mathcal{L}_{k}^{1}\right) $
be the space of functions of $t\in \left[ 0,T\right] $ with values in $
\mathcal{L}_{k}^{1}$ which are continuous with respect to the metric $d_{k}.$
On $C\left( \left[ 0,T\right] ,\mathcal{L}_{k}^{1}\right) ,$ we define the
metric
\begin{equation*}
\hat{d}_{k}(\gamma ^{(k)}\left( \cdot \right) ,\tilde{\gamma}^{(k)}\left(
\cdot \right) )=\sup_{t\in \left[ 0,T\right] }d_{k}(\gamma ^{(k)}\left(
t\right) ,\tilde{\gamma}^{(k)}\left( t\right) ),
\end{equation*}
and denote by $\tau _{prod}$ the topology on the space $\oplus _{k\geqslant
1}C\left( \left[ 0,T\right] ,\mathcal{L}_{k}^{1}\right) $ given by the
product of topologies generated by the metrics $\hat{d}_{k}$ on $C\left(
\left[ 0,T\right] ,\mathcal{L}_{k}^{1}\right) .$
With the above topology on the space of marginal densities, we now outline the proof of Theorem \ref{Theorem:3D->2D BEC}. We divide the proof into five steps.
\medskip
\noindent \textbf{Step I} (Energy estimate). We transform, through Theorem \ref{Theorem:Energy Estimate},
the energy condition \eqref{Condition:EnergyBoundOnInitialData} into an
``easier to use'' $H^{1}$ type energy bound in which the interaction $V$ is not involved. Since the quantity on the left-hand side of energy condition \eqref{Condition:EnergyBoundOnInitialData} is conserved by the evolution, we
deduce the \emph{a priori} bounds on the scaled marginal densities
$$
\sup_{t}\limfunc{Tr}\dprod\limits_{j=1}^{k}\left( 1-\triangle
_{x_{j}}+\omega \left( -1-\partial _{z_{j}}^{2}+z_{j}^{2}\right) \right)
\tilde{\gamma}_{N,\omega }^{(k)} \leqslant C^{k}
$$
$$
\sup_{t}\limfunc{Tr}\dprod\limits_{j=1}^{k}\left( 1-\triangle
_{r_{j}}\right) \tilde{\gamma}_{N,\omega }^{(k)} \leqslant C^{k}
$$
$$
\sup_t \operatorname{Tr} P_{\mathbf{\alpha}} \tilde \gamma_{N,\omega}^{(k)} P_{\mathbf{\beta}} \leq C^k \omega^{-\frac12 |\mathbf{\alpha}| - \frac12 \mathbf{|\beta}|}
$$
via Corollary \ref{Corollary:Energy Bound for Marginal Densities}. We remark
that, in contrast to the $n$D to $n$D work, the quantity
\begin{equation*}
\limfunc{Tr}\left( 1-\triangle _{r_{1}}\right) \tilde{\gamma}_{N,\omega
}^{(1)}
\end{equation*}
is not the one particle kinetic energy of the system; the one particle
kinetic energy of the system is $\limfunc{Tr}\left(
1-\triangle _{x_{1}}-\omega \partial _{z_{1}}^{2}\right) \tilde{\gamma}
_{N,\omega }^{(1)}$ and grows like $\omega$.
\medskip
\noindent\textbf{Step II} (Compactness of BBGKY). We fix $T>0$ and work in the time-interval $t\in \lbrack
0,T].$ In Theorem \ref{Theorem:Compactness of the scaled marginal density},
we establish the compactness of the sequence $\Gamma _{N,\omega }(t)=\left\{
\tilde{\gamma}_{N,\omega }^{(k)}\right\} _{k=1}^{N}\in \oplus _{k\geqslant
1}C\left( \left[ 0,T\right] ,\mathcal{L}_{k}^{1}\right) $ with respect to
the product topology $\tau _{prod}$ even though there is an $\infty -\infty $ in
hierarchy \eqref{hierarchy:BBGKY hierarchy for scaled marginal densities}.
Moreover, in Corollary \ref{Corollary:LimitMustBeAProduct}, we prove that, to
be compatible with the energy bound obtained in Step I, every limit point $
\Gamma (t)=\left\{ \tilde{\gamma}^{(k)}\right\} _{k=1}^{N}$ must take the
form
\begin{equation*}
\tilde{\gamma}^{(k)}\left( t,\left( \mathbf{x}_{k},\mathbf{z}_{k}\right)
;\left( \mathbf{x}_{k}^{\prime },\mathbf{z}_{k}^{\prime }\right) \right) =
\tilde{\gamma}_{x}^{(k)}(t,\mathbf{x}_{k};\mathbf{x}_{k}^{\prime
})\dprod\limits_{j=1}^{k}h_{1}\left( z_{j}\right) h_{1}\left( z_{j}^{\prime
}\right) ,
\end{equation*}
where $\tilde{\gamma}_{x}^{(k)}=\limfunc{Tr}_{z}\tilde{\gamma}^{(k)}$ is the
$x$-component of $\tilde{\gamma}^{(k)}.$
\medskip
\noindent\textbf{Step III} (Limit points of BBGKY satisfy GP). In Theorem \ref{Theorem:Convergence to the Coupled
Gross-Pitaevskii}, we prove that if $\Gamma (t)=\left\{ \tilde{\gamma}
^{(k)}\right\} _{k=1}^{\infty }$ is a $N\geqslant \omega ^{v(\beta)+\varepsilon }$ limit point of $\Gamma _{N,\omega }(t)=\left\{
\tilde{\gamma}_{N,\omega }^{(k)}\right\} _{k=1}^{N}$ with respect to the
product topology $\tau _{prod}$, then $\left\{ \tilde{\gamma}_{x}^{(k)}=
\limfunc{Tr}_{z}\tilde{\gamma}^{(k)}\right\} _{k=1}^{\infty }$ is a solution
to the coupled Gross-Pitaevskii (GP) hierarchy subject to initial data $\tilde{
\gamma}_{x}^{(k)}\left( 0\right) =\left\vert \phi _{0}\right\rangle
\left\langle \phi _{0}\right\vert ^{\otimes k}$ with coupling constant $
b_{0}=$ $\int V\left( r\right) dr$, which written in differential form, is
\begin{equation*}
i\partial _{t}\tilde{\gamma}_{x}^{(k)}=\sum_{j=1}^{k}\left[ -\triangle
_{x_{j}},\tilde{\gamma}_{x}^{(k)}\right] +b_{0}\sum_{j=1}^{k}\limfunc{Tr}
\nolimits_{x_{k+1}}\limfunc{Tr}\nolimits_{z}\left[ \delta \left(
r_{j}-r_{k+1}\right) ,\tilde{\gamma}^{(k+1)}\right] .
\end{equation*}
Together with Corollary \ref{Corollary:LimitMustBeAProduct}, we then deduce
that $\left\{ \tilde{\gamma}_{x}^{(k)}=\limfunc{Tr}_{z}\tilde{\gamma}
^{(k)}\right\} _{k=1}^{\infty }$ is a solution to the well-known 2D
GP hierarchy subject to initial data $\tilde{\gamma}
_{x}^{(k)}\left( 0\right) =\left\vert \phi _{0}\right\rangle \left\langle
\phi _{0}\right\vert ^{\otimes k}$ with coupling constant $b_{0}\left( \int
\left\vert h_{1}\left( z\right) \right\vert ^{4}dz\right) $, which, written
in differential form, is
\begin{equation}
i\partial _{t}\tilde{\gamma}_{x}^{(k)}=\sum_{j=1}^{k}\left[ -\triangle
_{x_{j}},\tilde{\gamma}_{x}^{(k)}\right] +b_{0}\left( \int \left\vert
h_{1}\left( z\right) \right\vert ^{4}dz\right) \sum_{j=1}^{k}\limfunc{Tr}
\nolimits_{x_{k+1}}\left[ \delta \left( x_{j}-x_{k+1}\right) ,\tilde{\gamma}
_{x}^{(k+1)}\right] . \label{hierarchy:2D GP hierarchy in differential form}
\end{equation}
\medskip
\noindent\textbf{Step IV} (GP has a unique solution). When $\tilde{\gamma}_{x}^{(k)}\left( 0\right) =\left\vert
\phi _{0}\right\rangle \left\langle \phi _{0}\right\vert ^{\otimes k},$ we
know one solution to the 2D Gross-Pitaevskii hierarchy \eqref{hierarchy:2D GP
hierarchy in differential form}, namely $\left\vert \phi \right\rangle \left\langle \phi \right\vert ^{\otimes k}$, where $\phi $ solves equation \eqref{equation:2D Cubic NLS}. Since we have the
\emph{a priori} bound
\begin{equation*}
\sup_{t}\limfunc{Tr}\dprod\limits_{j=1}^{k}\left( 1-\triangle
_{x_{j}}\right) \tilde{\gamma}_{x}^{(k)}\leqslant C^{k},
\end{equation*}
the uniqueness theorem $($Theorem \ref{Theorem:CombiningChenAndKirpatrick}$)$
then gives that $\tilde{\gamma}_{x}^{(k)}=\left\vert \phi \right\rangle \left\langle \phi\right\vert ^{\otimes k}$. Thus the compact sequence $\Gamma _{N,\omega }(t)=\left\{ \tilde{\gamma}_{N,\omega }^{(k)}\right\} _{k=1}^{N}$ has only one $N\geqslant \omega^{v(\beta)+\varepsilon }$ limit point, namely
\begin{equation*}
\tilde{\gamma}^{(k)}=\dprod\limits_{j=1}^{k}\phi (t,x_{j})\overline{\phi }
(t,x_{j}^{\prime })h_{1}\left( z_{j}\right) h_{1}(z_{j}^{\prime }) \,.
\end{equation*}
By the definition of the topology, we know, as trace class operators
\begin{equation*}
\tilde{\gamma}_{N,\omega }^{(k)}\rightarrow \dprod\limits_{j=1}^{k}\phi
(t,x_{j})\overline{\phi }(t,x_{j}^{\prime })h_{1}\left( z_{j}\right)
h_{1}(z_{j}^{\prime })\text{ weak*.}
\end{equation*}
\begin{remark}
This is in fact the very first time that the Klainerman-Machedon theory
applies to a 3D many-body system with $\beta \geqslant 1/3$. The previous
best is $\beta \in \left( 0,2/7\right] $ in \cite{Chen3DDerivation} after
the $\beta \in \left( 0,1/4\right) $ work \cite{TChenAndNPSpace-Time}. Of
course, we are not actually using any 3D Gross-Pitaevskii hierarchies here.
\end{remark}
\noindent\textbf{Step V} (Weak convergence upgraded to strong). We use the argument in the bottom of p. 296 of \cite{E-S-Y3} to conclude that the weak* convergence obtained in Step IV is in fact strong. We include this argument for completeness. We test the sequence obtained in Step IV against the compact observable
\begin{equation*}
J^{(k)}=\dprod\limits_{j=1}^{k}\phi (t,x_{j})\overline{\phi }
(t,x_{j}^{\prime })h_{1}\left( z_{j}\right) h_{1}(z_{j}^{\prime }),
\end{equation*}
and notice the fact that $\left( \tilde{\gamma}_{N,\omega }^{(k)}\right)
^{2}\leqslant \tilde{\gamma}_{N,\omega }^{(k)}$ since the initial data is
normalized, we see that as Hilbert-Schmidt operators
\begin{equation*}
\tilde{\gamma}_{N,\omega }^{(k)}\rightarrow \dprod\limits_{j=1}^{k}\phi
(t,x_{j})\overline{\phi }(t,x_{j}^{\prime })h_{1}\left( z_{j}\right)
h_{1}(z_{j}^{\prime })\text{ strongly.}
\end{equation*}
Since $\limfunc{Tr}\tilde{\gamma}_{N,\omega }^{(k)}=\limfunc{Tr}\tilde{\gamma
}^{(k)},$ we deduce the strong convergence
\begin{equation*}
\lim_{\substack{ N,\omega \rightarrow \infty \\ N\geqslant \omega ^{v(\beta
)+\varepsilon }}}\limfunc{Tr}\left\vert
\tilde \gamma _{N,\omega }^{(k)}(t,\mathbf{x}_{k},\mathbf{z}_{k};\mathbf{x}_{k}^{\prime },\mathbf{z}_{k}^{\prime})-\dprod\limits_{j=1}^{k}\phi (t,x_{j})\overline{\phi }
(t,x_{j}^{\prime })h_{1}\left( z_{j}\right) h_{1}(z_{j}^{\prime
})\right\vert =0,
\end{equation*}
via the Gr\"{u}mm's convergence theorem \cite[Theorem 2.19]{Simon}
\section{Energy estimate\label{Section:EnergyEstimate}}
We find it more convenient to prove the energy estimate for $\psi_{N,\omega}$ and then convert it by scaling to an estimate for $\tilde\psi_{N,\omega}$ (see \eqref{E:rescaled}). Note that, as an operator, we have the positivity:
$$-\omega - \partial_{z_j}^2 + \omega^2 z_j^2 \geq 0$$
Define
$$S_j \stackrel{\rm{def}}{=} (1-\Delta_{x_j} - \omega - \partial_{z_j}^2 + \omega^2 z_j^2)^{1/2} = (1-\omega - \Delta_{r_j} + \omega^2z_j^2)^{1/2}$$
\begin{theorem}
\label{Theorem:Energy Estimate}
Let the Hamiltonian be defined as in \eqref{Hamiltonian:H_N,W,nonscaled} with $\beta \in \left( 0,2/5\right)$. Then for all $\varepsilon >0$, there exists a constant $C>0$, and for all $\omega,k\geqslant 0$, there exists $N_{0}(k,\omega)$ such that
\begin{equation}
\label{E:energy-nonscaled}
\left\langle \psi_{N,\omega} ,\left( N+H_{N,\omega }-N\omega \right) ^{k}\psi_{N,\omega} \right\rangle \geqslant C^{k}N^{k} \left\| \prod_{j=1}^k S_j \psi_{N,\omega} \right\|_{L^2(\mathbb{R}^{3N})}^2
\end{equation}
for all $N\geqslant \omega^{v(\beta)+\epsilon}$, and all $\psi \in L_{s}^{2}\left( \mathbb{R}^{3N}\right)\cap \mathcal{D}(H_{N,\omega}^k)$.
\end{theorem}
\begin{proof}
We adapt the proof of \cite[Prop. 3.1]{E-E-S-Y1} to accommodate the operator $-\omega - \partial_{z_j}^2 + \omega^2z_j^2$ in place of $-\partial_{z_j}^2$.
The case $k=0$ is trivial and the case $k=1$ follows from the positivity of $V$ and symmetry of $\psi$. We proceed by induction. Suppose that the result holds for $k=n$, and we will prove it for $k=n+2$. By the induction hypothesis,
\begin{equation}
\label{E:en1}
\begin{aligned}
\hspace{0.3in}&\hspace{-0.3in} \langle \psi, (N-N\omega+H_{N,\omega})^{n+2}\psi \rangle \\
&\geq C^nN^n \langle \psi, (N-N\omega+H_{N,\omega}) \prod_{j=1}^n S_j^2 (N-N\omega+H_{N,\omega}) \psi \rangle
\end{aligned}
\end{equation}
For convenience, let
$$\tilde V(r) = (N\sqrt \omega)^{3\beta-1} V( (N\sqrt \omega)^\beta r)$$
Expand
$$N-N\omega + H_{N,\omega} = \sum_{\ell=n+1}^N S_\ell^2 + \left(\sum_{\ell=1}^n S_\ell^2 + H_{N,\omega}^I\right)$$
and substitute in both occurrences of the operator $N-N\omega + H_{N,\omega}$ in the right side of \eqref{E:en1} to obtain four terms. We ignore the last (positive) one of these terms to obtain
\begin{equation}
\label{E:en2}
\langle \psi, (N-N\omega+H_{N,\omega})^{n+2}\psi \rangle \geq C^nN^n(\text{I}+\text{II}+\text{III})
\end{equation}
We have
$$\text{I} = \sum_{\ell_1,\ell_2 = n+1}^N \langle \psi, S_{\ell_1}^2S_{\ell_2}^2 \prod_{j=1}^n S_j^2 \psi \rangle$$
In this double sum, there are $(N-n)(N-n-1)$ terms where $\ell_1\neq \ell_2$ that are all the same by symmetry, and there are $(N-n)$ terms where $\ell_1=\ell_2$ that are all the same by symmetry. We have
\begin{equation}
\label{E:en11}
\text{I} = (N-n)(N-n-1)\langle \psi , \prod_{j=1}^{n+2} S_j^2 \psi \rangle +(N-n) \langle \psi, S_1^2 \prod_{j=1}^{n+1} S_j^2 \psi \rangle
\end{equation}
the first of which will ultimately fulfill the induction claim.
In \eqref{E:en2}, we also have
$$\text{II}+\text{III} =
\begin{aligned}[t]
&2\sum_{\ell_1=n+1}^N \sum_{\ell_2=1}^n \langle \psi , S_{\ell_1}^2 \prod_{j=1}^n S_j^2 S_{\ell_2}^2 \psi \rangle + \sum_{\ell=n+1}^N \langle \psi, S_\ell^2 \prod_{j=1}^n S_j^2 H_{N,\omega}^I \psi \rangle \\
&+ \sum_{\ell=n+1}^N \langle \psi, H_{N,\omega}^I \prod_{j=1}^n S_j^2 S_\ell^2\psi \rangle
\end{aligned}
$$
Exploiting symmetry this becomes
\begin{equation}
\label{E:en10}
\text{II}+\text{III} = 2(N-n)n \langle \psi, S_1^2 \prod_{j=1}^{n+1} S_j^2 \psi \rangle +2(N-n) \Re\langle \psi, \prod_{j=1}^{n+1} S_j^2 H_{N,\omega}^I \psi \rangle
\end{equation}
In the first term, we have applied the permutation that swaps $\ell_1$ and $n+1$ and $\ell_2$ and $1$. In the second and third terms, we have applied the permutation $\sigma$ that swaps $\ell$ and $n+1$. Strictly speaking, this permutation maps $H_{N,\omega}^I$ to $H_{N,\omega,\sigma}^I$ where
$$H_{N,\omega,\sigma}^I \stackrel{\rm{def}}{=} \frac{1}{N\omega^{1/2}} \sum_{1\leq i < j \leq N} (N\omega^{1/2})^{3\beta} V((\pm 1)(N\omega^{1/2})^\beta (r_i-r_j ))$$
where $\pm 1$ is chosen according to the affect of the permutation on the pair $(i,j)$. The distinction between $H_{N,\omega}^I$ and $H_{N,\omega,\sigma}^I$ is inconsequential for the remainder of the analysis (and in fact $H_{N,\omega}^I=H_{N,\omega,\sigma}^I$ if $V$ is even), so we have ignored it in \eqref{E:en10}.
The first of the terms in \eqref{E:en10} is positive -- it is the second term that requires attention; in particular, we have to manage commutators.
Assuming $N \geq 2n+2$, we substitute \eqref{E:en11}, \eqref{E:en10} into \eqref{E:en2} to obtain
\begin{equation}
\label{E:en12}
\begin{aligned}
\langle \psi, (N-N\omega +H_{N,\omega})^{n+2}\psi \rangle \geq \tfrac14 C^nN^{n+2} \langle \psi, \prod_{j=1}^{n+2} S_j^2 \psi \rangle + C^nN^{n+1} \langle \psi, S_1^2 \prod_{j=1}^{n+1} S_j^2 \psi \rangle &\\
+ 2C^nN^n (N-n) \Re \langle \psi, \prod_{j=1}^{n+1}S_j^2 H_{N,\omega}^I \psi \rangle =: D+E+F &
\end{aligned}
\end{equation}
The first two terms, $D$ and $E$, in \eqref{E:en12} are positive. The third term $F$ will be decomposed into components, some of which are positive and others that can be bounded in terms of the first two terms appearing in \eqref{E:en12}. In the expression for $H_{N,\omega}^I$, there are
\begin{itemize}
\item $\frac12 (n+1)n$ terms of the form $\tilde V(r_i-r_j)$ for $1\leq i< j \leq n+1$.
\item $(n+1)(N-n-1)$ terms of the form $\tilde V(r_i-r_j)$ for $1\leq i \leq n+1$ and $n+2 \leq j \leq N$.
\item $\frac12 (N-n-1)(N-n-2)$ terms of the form $\tilde V(r_i-r_j)$ for $n+2 \leq i<j \leq N$.
\end{itemize}
For convenience, let
$$V_{ij} \stackrel{\rm{def}}{=} (N\omega^{1/2})^{3\beta -1} V( (N\omega^{1/2})^\beta(r_i-r_j))$$
Using symmetry, we obtain
\begin{align*}
F &=
\begin{aligned}[t]
&2C^nN^n (N-n) (n+1)n \Re \langle \psi, \prod_{j=1}^{n+1} S_j^2 V_{12} \psi \rangle \\
&+ 2C^nN^n (N-n) (n+1)(N-n-1) \Re \langle \psi, \prod_{j=1}^{n+1} S_j^2 V_{1(n+2)} \psi \rangle \\
&+ C^nN^n (N-n)(N-n-1)(N-n-2) \Re \langle \psi, \prod_{j=1}^{n+1} S_j^2 V_{(n+2)(n+3)} \psi \rangle
\end{aligned}\\
&=: F_1+F_2+F_3
\end{align*}
The last term $F_3$ is positive since each $S_j$ for $1\leq j \leq n+1$ commutes with $V_{(n+2)(n+3)}$. We will show $F_1 \geq -\frac12 E$ and $F_2 \geq -\frac12D$ provided $N\geq N_0(n)$, which together with \eqref{E:en12} will complete the induction argument. We have
\begin{align*}
F_1 &= 2C^nN^n (N-n) (n+1)n \Re \langle \psi, \prod_{j=1}^{n+1} S_j^2 V_{12} \psi \rangle \\
&= 2C^nN^n (N-n) (n+1)n \Re \int_{r_3,\ldots, r_N} \underbrace{\langle f, S_1^2S_2^2 V_{12} f \rangle_{r_1,r_2}}_{=: \tilde F_1} \, dr_3 \cdots dr_N
\end{align*}
where $f = \prod_{j=3}^{n+1} S_j \psi$. We can regard $r_3, \ldots, r_N$ as frozen in the following computation, so to prove $|F_1| \leq \frac12 E$, it will suffice to show that
\begin{equation}
\label{E:en13}
| \tilde F_1 | \leq \tfrac14 n^{-2} \|S_1^2S_2 f \|_{L_{r_1}^2L_{r_2}^2}^2
\end{equation}
Toward this end, we have
\begin{align*}
|\tilde F_1 |
&= |\langle S_1^2 f, V_{12} S_2^2 f \rangle + 2 \langle S_1^2 f, \nabla_{r_2} V_{12} \cdot \nabla_{r_2} f \rangle + \langle S_1^2 f, (\Delta_{r_2} V_{12}) \, f \rangle|\\
&\lesssim \|S_1^2 f \|_{L_{r_1}^2 L_{r_2}^6} \|V_{12} \|_{L_{r_1}^\infty L_{r_2}^3} \|S_2^2 f\|_{L_{r_1}^2 L_{r_2}^2} + \|S_1^2 f \|_{L_{r_1}^2 L_{r_2}^6} \|\nabla_{r_2} V_{12} \|_{L_{r_1}^\infty L_{r_2}^{3/2}} \|\nabla_{r_2} f\|_{L_{r_1}^2 L_{r_2}^6} \\
& \qquad + \| S_1^2 f\|_{L_{r_1}^2L_{r_2}^6} \| \Delta_{r_2} V_{12} \|_{L_{r_1}^\infty L_{r_2}^{6/5}} \|f\|_{L_{r_1}^2 L_{r_2}^\infty}
\end{align*}
By evaluation of
$$\|V_{12} \|_{ L_{r_2}^3} \sim (N\omega^{1/2})^{2\beta-1}\,, \quad \|\nabla_{r_2} V_{12} \|_{L_{r_2}^{3/2}} \sim (N\omega^{1/2})^{2\beta-1}\,, \quad
\| \Delta_{r_2} V_{12} \|_{L_{r_2}^{6/5}} \sim (N\omega^{1/2})^{\frac52\beta-1}
$$
the above estimate reduces to
\begin{align*}
|\tilde F_1 | &\lesssim (N\omega^{1/2})^{2\beta-1} \|S_1^2 f \|_{L_{r_1}^2 L_{r_2}^6} \|S_2^2 f\|_{L_{r_1}^2 L_{r_2}^2} + (N\omega^{1/2})^{2\beta-1} \|S_1^2 f \|_{L_{r_1}^2 L_{r_2}^6} \|\nabla_{r_2} f\|_{L_{r_1}^2 L_{r_2}^6} \\
& \qquad + (N\omega^{1/2})^{\frac52\beta-1} \| S_1^2 f\|_{L_{r_1}^2L_{r_2}^6} \|f\|_{L_{r_1}^2 L_{r_2}^\infty}
\end{align*}
Applying Lemma \ref{L:Sobolev-with-loss}, this reduces further to
\begin{align*}
|\tilde F_1 | &\lesssim (N\omega^{1/2})^{2\beta-1} \omega^{1/6} \|S_1^2 S_2 f \|_{L_{r_1}^2 L_{r_2}^2} \|S_2^2 f\|_{L_{r_1}^2 L_{r_2}^2} \\
&\qquad + (N\omega^{1/2})^{2\beta-1} \omega^{1/6} \omega^{2/3} \|S_1^2S_2 f \|_{L_{r_1}^2 L_{r_2}^2} \| S_2^2 f\|_{L_{r_1}^2 L_{r_2}^2} \\
& \qquad + (N\omega^{1/2})^{\frac52\beta-1} \omega^{1/6} \omega^{1/4} \| S_1^2S_2 f\|_{L_{r_1}^2L_{r_2}^2} \|S_2^2f\|_{L_{r_1}^2 L_{r_2}^2}
\end{align*}
Hence we need $\beta < \frac25$ and conditions \eqref{E:en20}, \eqref{E:en18} below to achieve \eqref{E:en13}.
Let us now establish $F_2 \geq -\frac12 D$. We have
\begin{align*}
F_2 &= 2C^nN^n (N-n) (n+1)(N-n-1) \Re \langle \psi, \prod_{j=1}^{n+1} S_j^2 V_{1(n+2)} \psi \rangle \\
&= 2C^nN^n (N-n) (n+1)(N-n-1) \int \underbrace{\langle f, S_1^2V_{1(n+2)} f \rangle_{r_1,r_{n+2}}}_{=:\tilde F_2} \, dr_2 \cdots dr_{n+1}dr_{n+3}\cdots dr_N
\end{align*}
where $f = \prod_{j=2}^{n+1} S_j \psi$. Now
\begin{align*}
\tilde F_2 &= \langle f, (-\omega - \partial_{z_1}^2 + \omega^2 z_1^2) V_{1(n+2)} f\rangle_{r_1,r_{n+2}} \\
&= -\omega \langle f, V_{1(n+2)}f \rangle_{r_1r_{n+2}} + \langle \partial_{z_1}f, (\partial_{z_1}V_{1(n+2)})f \rangle_{r_1r_{n+2}} \\
& \qquad + \langle \partial_{z_1}f, V_{1(n+2)} \partial_{z_1}f\rangle_{r_1r_{n+2}} + \langle f, \omega^2 z_1^2 f\rangle_{r_1r_{n+2}} \\
&=: \tilde F_{2,1} + \tilde F_{2,2} + \tilde F_{2,3} + \tilde F_{2,4}
\end{align*}
Note that $\tilde F_{2,3}$ and $\tilde F_{2,4}$ are positive and can thus be disregarded. To prove $F_2 \geq - \frac12 D$, it suffices to prove
\begin{equation}
\label{E:en14}
|\tilde F_{2,1}|+ |\tilde F_{2,2}| \leq \tfrac1{16} n^{-1} \| S_1 S_{n+2} f \|_{L_{r_1}^2 L_{r_{n+2}}^2}^2
\end{equation}
But
$$|\tilde F_{2,1}| \lesssim \omega \| f\|_{L_{r_1}^2L_{r_{n+2}}^6} \|V_{1(n+2)}\|_{L_{r_1}^\infty L_{r_{n+2}}^{3/2}} \|f\|_{L_{r_1}^2 L_{r_{n+2}}^6}$$
By Lemma \ref{L:Sobolev-with-loss} and $\|V_{1(n+2)} \|_{L_{r_1}^\infty L_{r_{n+2}}^{3/2}} \sim (N\omega^{1/2})^{\beta-1}$, we obtain
\begin{equation}
\label{E:en15}
|\tilde F_{2,1}| \lesssim \omega^{4/3} (N\omega^{1/2})^{\beta-1} \| S_{n+2}f\|_{L_{r_1}^2L_{r_{n+2}}^2}^2
\end{equation}
The upper bound in \eqref{E:en14} will be achieved provided \eqref{E:en19} below holds. Also,
$$
|\tilde F_{2,2}| \lesssim \| \partial_{z_1} f\|_{L_{r_1}^2 L_{r_{n+2}}^6} \|\partial_{z_1} V_{1(n+2)} \|_{L_{r_1}^\infty L_{r_{n+2}}^{3/2}} \|f\|_{L_{r_1}^2 L_{r_{n+2}}^6}
$$
Note that $\|\partial_{z_1} V_{1(n+2)} \|_{L_{r_1}^\infty L_{r_{n+2}}^{3/2}} \sim (N\omega^{1/2})^{2\beta-1}$. By Lemma \ref{L:Sobolev-with-loss},
$$\| \partial_{z_1} f \|_{L_{r_1}^2 L_{r_{n+2}}^6} \lesssim \omega^{1/6} \| S_{n+2} \partial_{z_1} f\|_{L_{r_1}^2L_{r_{n+2}}^2} \lesssim \omega^{2/3} \|S_1 S_{n+2} f\|_{L_{r_1}^2 L_{r_{n+2}}^2}$$
and $\|f \|_{L_{r_1}^2 L_{r_{n+2}}^6} \lesssim \omega^{1/6} \|S_{n+2}f\|_{L_{r_1}^2L_{r_{n+2}}^2}$. From this, it follows that
\begin{equation}
\label{E:en16}
|\tilde F_{2,2}| \lesssim \omega^{5/6} (N\omega^{1/2})^{2\beta-1} \| S_1S_{n+2} f\|_{L_{r_1}^2 L_{r_{n+2}}^2} \|S_{n+2} f\|_{L_{r_1}^2 L_{r_{n+2}}^2}
\end{equation}
The upper bound in \eqref{E:en14} will be achieved provided \eqref{E:en20} holds. By \eqref{E:en15}, \eqref{E:en16}, we obtain \eqref{E:en14}, completing the proof. Let us collect the conditions on $N$ and $\omega$. We have
\begin{align}
\label{E:en18} &(N\omega^{1/2})^{\frac52\beta-1}\omega^{5/12} \ll n^{-2} && \iff N \gg \omega^\frac{\frac54 \beta - \frac1{12}}{1-\frac52\beta} n^\frac{2}{1-\frac52\beta} \\
\label{E:en19} &(N\omega^{1/2})^{\beta-1} \omega^{4/3} \ll n^{-1} && \iff N \gg \omega^\frac{ \frac12\beta+\frac56 }{1-\beta} n^\frac{1}{1-\beta}\\
\label{E:en20} &(N\omega^{1/2})^{2\beta-1} \omega^{5/6} \ll n^{-1} && \iff N \gg \omega^ \frac{\beta+\frac13}{1-2\beta} n^\frac{1}{1-2\beta}
\end{align}
The requirement that \eqref{E:en18}, \eqref{E:en19}, and \eqref{E:en20} hold is imposed in the definition \eqref{E:vofbeta} of $v(\beta)$.
\end{proof}
Now consider the rescaled operator \eqref{E:tilde-S-def} so that
$$(S_j \psi)(t,\mathbf{x}_N, \mathbf{z}_N) = \omega^{N/4} (\tilde S_j \tilde \psi)(t, \mathbf{x}_N, \sqrt \omega \mathbf{z}_N)\,.$$
We will convert the conclusions of Theorem \ref{Theorem:Energy Estimate} into statements about $\tilde \psi$, $\tilde S_j$, and $\tilde \gamma_{N,\omega}^{(k)}$ that we will then apply in the remainder of the paper.
\begin{corollary}
\label{Corollary:Energy Bound for Marginal Densities}
Let $\tilde \psi _{N,\omega}( t) =e^{it \tilde H_{N,\omega }}\tilde \psi _{N,\omega }( 0)$ and $\{ \tilde \gamma _{N,\omega }^{(k)}(t)\} $ be the marginal densities associated with it, then for all $\omega \geq 1$ , $k\geq 0$, $N \geq \omega^{v(\beta)+\epsilon}$, we have the uniform-in-time bound
\begin{equation}
\label{E:e-1}
\operatorname{Tr} \prod_{j=1}^k \tilde S_j^2 \tilde \gamma_{N,\omega}^{(k)} = \left\| \prod_{j=1}^k \tilde S_j \tilde \psi_{N,\omega}(t) \right\|_{L^2(\mathbb{R}^{3N})}^2 \leq C^k
\end{equation}
Consequently,
\begin{equation}
\label{E:e-2}
\operatorname{Tr} \prod_{j=1}^k (1-\Delta_{r_j}) \tilde \gamma_{N,\omega}^{(k)} = \left\| \prod_{j=1}^k (1-\Delta_{r_j})^{1/2} \tilde \psi_{N,\omega}(t) \right\|_{L^2(\mathbb{R}^{3N})}^2 \leq C^k
\end{equation}
and
\begin{equation}
\label{E:e-3}
\| P_{\mathbf{\alpha}} \tilde \psi_{N,\omega} \|_{L^2(\mathbb{R}^{3N})} \leq C^k \omega^{-|\mathbf{\alpha}|/2}\,, \qquad \operatorname{Tr} P_{\mathbf{\alpha}} \tilde \gamma_{N,\omega}^{(k)} P_{\mathbf{\beta}} \leq C^k \omega^{-\frac12 |\mathbf{\alpha}| - \frac12 \mathbf{|\beta}|}
\end{equation}
\end{corollary}
\begin{proof}
Substituting \eqref{E:rescaled} into \eqref{E:energy-nonscaled} of Theorem \ref{Theorem:Energy Estimate} and rescaling, we obtain
\begin{equation}
\label{E:en100}
\langle \tilde \psi_{N,\omega}, (N-\tilde H_{N,\omega} - N\omega)^k \tilde \psi_{N,\omega} \rangle \geq C^kN^k \left\| \prod_{j=1}^k \tilde S_j \tilde \psi_{N,\omega} \right\|_{L^2(\mathbb{R}^{3N})}^2
\end{equation}
Since $N-\tilde H_{N,\omega} - N\omega$ is self-adjoint and $[\tilde H_{N,\omega}, N- \tilde H_{N,\omega} - N\omega] =0$,
$$\partial_t \langle \tilde \psi_{N,\omega}, (N-\tilde H_{N,\omega} - N\omega)^k \tilde \psi_{N,\omega} \rangle =0$$
Hence by \eqref{E:en100},
\begin{align*}
C^k N^k \left\| \prod_{j=1}^k \tilde S_j \tilde \psi_{N,\omega}(t) \right\|_{L^2(\mathbb{R}^{3N})}^2 \leq
&\langle \tilde \psi_{N,\omega}(t), (N- \tilde H_{N,\omega}-N\omega)^k \tilde \psi_{N,\omega}(t) \rangle\\
&= \langle \tilde \psi_{N,\omega}(0), (N-\tilde H_{N,\omega} - N\omega)^k \tilde \psi_{N,\omega}(0) \rangle \leq (C')^k N^k
\end{align*}
where the last estimate follows from the hypothesis \eqref{Condition:EnergyBoundOnInitialData} of Theorem \ref{Theorem:3D->2D BEC}.
The inequality \eqref{E:e-2} follows from \eqref{E:e-1} and \eqref{E:tilde-S-1}. The inequality on the left of \eqref{E:e-3} follows from \eqref{E:tilde-S-3} and \eqref{E:e-1}. By Lemma \ref{L:trace-of-tp-kernel}, $\operatorname{Tr} P_{\mathbf{\alpha}} \tilde \gamma_{N,\omega}^{(k)} P_{\mathbf{\beta}} = \langle P_{\mathbf{\alpha}} \tilde \psi_{N,\omega}, P_{\mathbf{\beta}} \tilde\psi_{N,\omega} \rangle$, so the inequality on the right of \eqref{E:e-3} follows by Cauchy-Schwarz.
\end{proof}
\section{Compactness of the BBGKY sequence \label{Section:Compactness} }
\begin{theorem}
\label{Theorem:Compactness of the scaled marginal density}
The sequence
$$\Gamma _{N,\omega }(t)=\left\{ \tilde{\gamma}_{N,\omega }^{(k)}\right\}_{k=1}^{N}\in \bigoplus _{k\geqslant 1}C\left( \left[ 0,T\right] ,\mathcal{L}_{k}^{1}\right)$$
which satisfies the $\infty-\infty$ BBGKY hierarchy \eqref{hierarchy:BBGKY hierarchy for scaled marginal densities}, is compact with respect to the product topology $\tau_{prod}$. For any limit point $\Gamma (t)=\left\{ \tilde{\gamma}^{(k)}\right\} _{k=1}^{N},$ $\tilde{\gamma}^{(k)}$ is a symmetric nonnegative trace class operator with trace bounded by $1$.
\end{theorem}
We establish Theorem \ref{Theorem:Compactness of the scaled marginal density}
at the end of this section. With Theorem \ref{Theorem:Compactness of the
scaled marginal density}, we can start talking about the limit points of $
\Gamma _{N,\omega }(t)=\{ \tilde{\gamma}_{N,\omega }^{(k)}\}
_{k=1}^{N}.$
\begin{corollary}
\label{Corollary:LimitMustBeAProduct}
Let $\Gamma (t)=\{ \tilde{\gamma}^{(k)}\} _{k=1}^{\infty }$ be a limit point of $\Gamma _{N,\omega}(t)=\{ \tilde{\gamma}_{N,\omega }^{(k)}\} _{k=1}^{N}$ with respect to the product topology $\tau _{prod}$, then $\tilde{\gamma}^{(k)}$ satisfies
\begin{equation}
\label{E:e-7}
\limfunc{Tr}\dprod\limits_{j=1}^{k}\left( 1-\triangle _{r_{j}}\right) \tilde{\gamma}^{(k)}\leqslant C^{k}
\end{equation}
\begin{equation}
\label{E:e-8}
\tilde{\gamma}^{(k)}\left( t,\left( \mathbf{x}_{k},\mathbf{z}_{k}\right);\left( \mathbf{x}_{k}^{\prime },\mathbf{z}_{k}^{\prime }\right) \right) = \tilde{\gamma}_{x}^{(k)}(t,\mathbf{x}_{k};\mathbf{x}_{k}^{\prime})\dprod\limits_{j=1}^{k}h_{1}\left( z_{j}\right) h_{1}\left( z_{j}^{\prime}\right)
\end{equation}
\end{corollary}
\begin{proof}
The estimate \eqref{E:e-7} is a direct consequence of \eqref{E:e-2} in Corollary \ref{Corollary:Energy Bound for Marginal Densities} and Theorem \ref{Theorem:Compactness of the scaled marginal density}. The formula \eqref{E:e-8} is equivalent to the statement that if either $\mathbf{\alpha} \neq 0$ or $\mathbf{\beta} \neq 0$, then $P_\mathbf{\alpha} \tilde \gamma^{(k)} P_{\mathbf{\beta}} =0$. This is equivalent to the statement that for any $J^{(k)}\in \mathcal{K}_k$, $\operatorname{Tr} J^{(k)} P_\mathbf{\alpha} \tilde \gamma^{(k)} P_{\mathbf{\beta}} = 0$. However,
\begin{equation}
\label{E:e-4}
\operatorname{Tr} J^{(k)} P_\mathbf{\alpha} \tilde \gamma^{(k)} P_{\mathbf{\beta}}= \lim_{(N,\omega) \to \infty} \operatorname{Tr} J^{(k)} P_\mathbf{\alpha} \tilde \gamma_{N,\omega}^{(k)} P_{\mathbf{\beta}}
\end{equation}
By Lemma \ref{L:trace-of-tp-kernel},
$$\operatorname{Tr} J^{(k)} P_\mathbf{\alpha} \tilde \gamma_{N,\omega}^{(k)} P_{\mathbf{\beta}} = \langle J^{(k)} P_{\mathbf{\alpha}} \tilde\psi_{N,\omega}, P_{\mathbf{\beta}} \tilde\psi_{N,\omega} \rangle_{\mathbf{r}_k} $$
and by Cauchy-Schwarz and \eqref{E:e-3},
$$| \operatorname{Tr} J^{(k)} P_\mathbf{\alpha} \tilde \gamma_{N,\omega}^{(k)} P_{\mathbf{\beta}} | \leq \| J^{(k)} \|_{\operatorname{op}} \| P_{\mathbf{\alpha}} \tilde \psi_{N,\omega}\|_{L^2(\mathbb{R}^{3N})} \| P_{\mathbf{\beta}} \tilde\psi_{N,\omega}\|_{L^2(\mathbb{R}^{3N})} \leq C^k \omega^{-\frac12 |\mathbf{\alpha}| - \frac12 | \mathbf{\beta}|}$$
Hence the right side of \eqref{E:e-4} is $0$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{Theorem:Compactness of the scaled marginal density}]
By the standard diagonalization argument, it suffices to show the compactness of $\tilde{\gamma}_{N,\omega }^{(k)}$ for fixed $k$ with respect to the metric $\hat{d}_{k}$. By the Arzel\`a-Ascoli theorem, this is equivalent to the equicontinuity of $\tilde{\gamma}_{N,\omega }^{(k)}$, and by \cite[Lemma 6.2]{E-S-Y3}, this is equivalent to the statement that for every observable $J^{(k)}$ from a dense subset of $\mathcal{K}( L^2( \mathbb{R}^{3k}))$ and for every $\varepsilon >0$, there exists $\delta(J^{(k)},\varepsilon )$ such that for all $t_{1},t_{2}\in \left[ 0,T\right]$ with $\left\vert t_{1}-t_{2}\right\vert \leqslant \delta$, we have
\begin{equation}
\label{E:cpct5}
\sup_{N,\omega }\left\vert \operatorname{Tr} J^{(k)}\tilde{\gamma}_{N,\omega }^{(k)}(t_{1}) - \operatorname{Tr} J^{(k)} \tilde{\gamma}_{N,\omega}^{(k)}(t_{2}) \right\vert \leqslant \varepsilon\, .
\end{equation}
We assume that our compact operators $J^{(k)}$ have been cutoff as in Lemma \ref{L:compact-operator-truncation}. Assume $t_1\leq t_2$. Inserting the decomposition \eqref{E:cpct1} on the left and right side of $\gamma_{N,\omega}^{(k)}$, we obtain
$$\tilde \gamma_{N,\omega}^{(k)} = \sum_{\mathbf{\alpha}, \mathbf{\beta}} P_{\mathbf{\alpha}} \tilde \gamma_{N,\omega}^{(k)} P_{\mathbf{\beta}}$$
where the sum is taken over all $k$-tuples $\mathbf{\alpha}$ and $\mathbf{\beta}$ of the type described above.
To establish \eqref{E:cpct5} it suffices to establish, for each $\mathbf{\alpha}$ and $\mathbf{\beta}$
\begin{equation}
\label{E:cpct4}
\sup_{N,\omega }\left\vert \operatorname{Tr} J^{(k)}P_{\mathbf{\alpha}} \tilde{\gamma}_{N,\omega }^{(k)}P_{\mathbf{\beta}}(t_{1}) - \operatorname{Tr} J^{(k)} P_{\mathbf{\alpha}}\tilde{\gamma}_{N,\omega}^{(k)}P_{\mathbf{\beta}}(t_{2}) \right\vert \leqslant \varepsilon\, .
\end{equation}
Below, we establish the estimate
\begin{equation}
\label{E:cpct2}
\begin{aligned}
\hspace{0.3in}&\hspace{-0.3in}
| \operatorname{Tr} J^{(k)} P_{\mathbf{\alpha}} \tilde{\gamma}_{N,\omega }^{(k)} P_{\mathbf{\beta}}(t_2) - \operatorname{Tr} J^{(k)} P_{\mathbf{\alpha}} \tilde{\gamma}_{N,\omega }^{(k)} P_{\mathbf{\beta}}(t_1)| \\
&\lesssim |t_2-t_1|\begin{cases}
1 & \text{if both } \mathbf{\alpha}=0 \text{ and }\mathbf{\beta}=0 \\
\max(1,\omega^{1-\frac12 |\mathbf{\alpha}| - \frac12 |\mathbf{\beta}|}) & \text{otherwise}
\end{cases}
\end{aligned}
\end{equation}
Estimate \eqref{E:cpct2} suffices to prove \eqref{E:cpct4} except when $|\mathbf{\alpha}|=0$ and $|\mathbf{\beta}|=1$ or vice versa, in which case it yields the upper bound $\omega^{1/2} |t_2-t_1|$ with the adverse factor $\omega^{1/2}$. On the other hand, we can also prove the (comparatively simpler) bound
\begin{equation}
\label{E:cpct3}
| \operatorname{Tr} J^{(k)} P_{\mathbf{\alpha}} \tilde{\gamma}_{N,\omega}^{(k)} P_{\mathbf{\beta}}(t_2) - \operatorname{Tr} J^{(k)} P_{\mathbf{\alpha}} \tilde{\gamma}_{N,\omega}^{(k)} P_{\mathbf{\beta}}(t_1)| \lesssim \omega^{-\frac12 |\mathbf{\alpha}| - \frac12 |\mathbf{\beta}|}
\end{equation}
that provides no gain as $t_2\to t_1$, but a better power of $\omega$. By averaging \eqref{E:cpct2} and \eqref{E:cpct3} in the case $|\mathbf{\alpha}|=0$ and $|\mathbf{\beta}|=1$ (or vice versa), we obtain
$$| \operatorname{Tr} J^{(k)} P_{\mathbf{\alpha}} \tilde{\gamma}_{N,\omega}^{(k)} P_{\mathbf{\beta}}(t_2) - \operatorname{Tr} J^{(k)} P_{\mathbf{\alpha}} \tilde{\gamma}_{N,\omega}^{(k)} P_{\mathbf{\beta}}(t_1)| \lesssim |t_2-t_1|^{1/2}$$
which suffices to establish \eqref{E:cpct4}.
Thus, it remains to prove both \eqref{E:cpct2} and \eqref{E:cpct3}, and we begin with \eqref{E:cpct2}. Hierarchy \eqref{hierarchy:BBGKY hierarchy for scaled marginal densities} yields
\begin{equation}
\label{E:cpct6}
i\partial _{t} \, P_{\mathbf{\alpha}} \tilde{\gamma}_{N,\omega }^{(k)} P_{\mathbf{\beta}} =
\begin{aligned}[t]
&\sum_{j=1}^{k}\left[-\triangle _{x_{j}}, \; P_{\mathbf{\alpha}} \tilde{\gamma}_{N,\omega }^{(k)} P_{\mathbf{\beta}} \right]
+\sum_{j=1}^{k}\omega \left[ -\partial _{z_{j}}^{2}+z_{j}^{2}, \; P_{\mathbf{\alpha}}\tilde{\gamma}_{N,\omega }^{(k)} P_{\mathbf{\beta}} \right] \\
&+\frac{1}{N}\sum_{i<j}^{k} P_{\mathbf{\alpha}} \left[ V_{N,\omega}\left( r_{i}-r_{j}\right) , \; \tilde{\gamma}_{N,\omega }^{(k)}\right] P_{\mathbf{\beta}} \\
&+\frac{N-k}{N}\limfunc{Tr}\nolimits_{r_{k+1}}\sum_{j=1}^{k} P_{\mathbf{\alpha}} \left[V_{N,\omega }\left( r_{j}-r_{k+1}\right) , \; \tilde{\gamma}_{N,\omega }^{(k+1)}\right] P_{\mathbf{\beta}}
\end{aligned}
\end{equation}
Let
$$\text{I} = - i\sum_{j=1}^k \operatorname{Tr} J^{(k)}[-\Delta_{x_j}, P_{\mathbf{\alpha}} \tilde \gamma_{N,\omega}^{(k)} P_{\mathbf{\beta}}]$$
\begin{equation}
\label{E:cpct7}
\text{II} = -\omega i \sum_{j=1}^k \operatorname{Tr} J^{(k)} [ -\partial_{z_j}^2 + z_j^2, P_{\mathbf{\alpha}} \tilde \gamma_{N,\omega}^{(k)} P_{\mathbf{\beta}}]
\end{equation}
$$\text{III} = -i N^{-1} \sum_{1\leq i < j \leq k} \operatorname{Tr} J^{(k)} P_{\mathbf{\alpha}} [V_{N,\omega}(r_i-r_j), \tilde \gamma_{N,\omega}^{(k)} ] P_{\mathbf{\beta}}$$
$$\text{IV} = - i\frac{N-k}{N} \sum_{j=1}^k \operatorname{Tr} J^{(k)} P_{\mathbf{\alpha}} [ V_{N,\omega}(r_j-r_{k+1}), \tilde \gamma_{N,\omega}^{(k+1)}] P_{\mathbf{\beta}}$$
Then it follows from \eqref{E:cpct6} that
\begin{equation}
\label{E:cpct50}
\partial_t \operatorname{Tr} J^{(k)} P_{\mathbf{\alpha}} \tilde \gamma_{N,\omega}^{(k)} P_{\mathbf{\beta}} = \text{I} + \text{II} + \text{III} + \text{IV}
\end{equation}
First, consider $\text{I}$. Applying Lemma \ref{L:trace-of-tp-kernel} and then integration by parts, we obtain
\begin{align*}
\text{I} &= i\sum_{j=1}^k \left( \langle J^{(k)} \Delta_{x_j} P_{\mathbf{\alpha}} \psi, P_{\mathbf{\beta}} \psi \rangle_{\mathbf{r}_k} - \langle J^{(k)} P_{\mathbf{\alpha}} \psi, P_{\mathbf{\beta}} \Delta_{x_j}\psi\rangle_{\mathbf{r}_k} \right) \\
&= i\sum_{j=1}^k \left( \langle J^{(k)} \Delta_{x_j} P_{\mathbf{\alpha}} \psi, P_{\mathbf{\beta}} \psi \rangle_{\mathbf{r}_k} - \langle \Delta_{x_j} J^{(k)} P_{\mathbf{\alpha}} \psi , P_{\mathbf{\beta}} \psi \rangle_{\mathbf{r}_k} \right)
\end{align*}
Hence
\begin{equation}
\label{E:cpct51}
|\text{I}| \leq \sum_{j=1}^k ( \|J^{(k)} \Delta_{x_j} \|_{\operatorname{op}} + \|\Delta_{x_j} J^{(k)} \|_{\operatorname{op}}) \|P_{\mathbf{\alpha}} \psi \|_{L^2(\mathbb{R}^{3N})} \|P_{\mathbf{\beta}} \psi \|_{L^2(\mathbb{R}^{3N})} \leq C_{k,J^{(k)}}
\end{equation}
where in the last step we applied the energy estimate.
Now, consider \text{II}. When $\mathbf{\alpha}=0$ and $\mathbf{\beta}=0$, we use that
$$\text{II} = -\omega i \sum_{j=1}^k \operatorname{Tr} J^{(k)} [ 1-\partial_{z_j}^2 + z_j^2, P_{\mathbf{\alpha}} \tilde \gamma_{N,\omega}^{(k)} P_{\mathbf{\beta}}]=0$$
Otherwise, we proceed directly from \eqref{E:cpct7}, applying Lemma \ref{L:trace-of-tp-kernel} and integration by parts to obtain ($H_j = -\partial_{z_j}^2+z_j^2$)
\begin{align*}
\text{II}
&= \omega i \sum_{j=1}^k \langle J^{(k)} H_j P_{\mathbf{\alpha}} \psi, P_{\mathbf{\beta}} \psi \rangle - \langle J^{(k)} P_{\mathbf{\alpha}} \psi, H_j P_{\mathbf{\beta}} \psi \rangle \\
&= \omega i \sum_{j=1}^k \langle J^{(k)} H_j P_{\mathbf{\alpha}} \psi, P_{\mathbf{\beta}} \psi \rangle - \langle H_j J^{(k)} P_{\mathbf{\alpha}} \psi, P_{\mathbf{\beta}} \psi \rangle
\end{align*}
Hence
$$|\text{II}| \lesssim \omega \sum_{j=1}^k (\| J^{(k)} H_j\|_{\operatorname{op}} + \|H_j J^{(k)}\|_{\operatorname{op}}) \| P_{\mathbf{\alpha}} \psi \|_{L^2(\mathbb{R}^{3N})} \| P_{\mathbf{\beta}} \psi \|_{L^2(\mathbb{R}^{3N})}$$
By the energy estimates,
\begin{equation}
\label{E:cpct52}
\text{II}
\begin{cases}
=0 & \text{if } \mathbf{\alpha}=0 \text{ and } \mathbf{\beta}=0 \\
\lesssim C_{k, J^{(k)}} \; \omega^{1- \frac12 |\mathbf{\alpha}| - \frac12 |\mathbf{\beta}|} & \text{otherwise}
\end{cases}
\end{equation}
Now, consider $\text{III}$.
$$\text{III}=-iN^{-1} \sum_{1\leq i<j\leq k} \langle J^{(k)} P_{\mathbf{\alpha}} V_{N,\omega}(r_i-r_j) \psi, P_{\mathbf{\beta}} \psi \rangle - \langle J^{(k)} P_{\mathbf{\alpha}} \psi, P_{\mathbf{\beta}} V_{N,\omega}(r_i-r_j) \psi \rangle$$
$$=-iN^{-1} \sum_{1\leq i<j\leq k} \langle J^{(k)} P_{\mathbf{\alpha}} V_{N,\omega}(r_i-r_j) \psi, P_{\mathbf{\beta}} \psi \rangle - \langle P_{\mathbf{\alpha}} \psi, J^{(k)} P_{\mathbf{\beta}} V_{N,\omega}(r_i-r_j) \psi \rangle$$
Let $L_i = (1-\Delta_{r_i})^{1/2}$ and
$$W_{ij} = L_i^{-1}L_j^{-1}V_{N,\omega}(r_i-r_j) L_i^{-1}L_j^{-1} \,.$$
Then
$$\text{III} = -iN^{-1} \sum_{1\leq i<j\leq k} \langle J^{(k)} P_{\mathbf{\alpha}} L_iL_j W_{ij} L_iL_j \psi, P_{\mathbf{\beta}} \psi \rangle - \langle P_{\mathbf{\alpha}} \psi, J^{(k)} P_{\mathbf{\beta}} L_iL_j W_{ij} L_iL_j \psi \rangle$$
Hence
$$|\text{III}| \lesssim
\begin{aligned}[t]
&N^{-1}\| J^{(k)} L_iL_j \|_{\operatorname{op}} \|W_{ij}\|_{\operatorname{op}} \|L_iL_j \psi\|_{L^2(\mathbb{R}^{3N})} \|P_{\mathbf{\beta}} \psi \|_{L^2(\mathbb{R}^{3N})} \\
&+ N^{-1}\|P_{\mathbf{\alpha}} \psi \|_{L^2(\mathbb{R}^{3N})} \|J^{(k)} L_iL_j \|_{\operatorname{op}} \|W_{ij}\|_{\operatorname{op}} \|L_iL_j \psi \|_{L^2(\mathbb{R}^{3N})}
\end{aligned}
$$
By Lemma \ref{Lemma:ESYSoblevLemma}, $\| W_{ij}\|_{\operatorname{op}} \lesssim \|V_{N,\omega} \|_{L^1} = \|V\|_{L^1}$ (independent of $N$, $\omega$), and hence the energy estimates imply that
\begin{equation}
\label{E:cpct53}
| \text{III} | \lesssim C_{k,J^{(k)}} \; N^{-1}
\end{equation}
Now consider $\text{IV}$.
$$\text{IV} = -i\frac{N-k}{N} \sum_{j=1}^k \left( \langle J^{(k)} P_{\mathbf{\alpha}} V_{N,\omega}(r_j-r_{k+1}) \psi, P_{\mathbf{\beta}} \psi \rangle - \langle J^{(k)} P_{\mathbf{\alpha}} \psi, P_{\mathbf{\beta}} V_{N,\omega}(r_j-r_{k+1}) \psi \rangle \right)$$
Then, since $J^{(k)}L_{k+1} = L_{k+1} J^{(k)}$,
$$\text{IV} =
\begin{aligned}[t]
&-i\frac{N-k}{N} \sum_{j=1}^k \langle J^{(k)} L_jP_{\mathbf{\alpha}} W_{j(k+1)} L_jL_{k+1} \psi, P_{\mathbf{\beta}} L_{k+1}\psi \rangle \\
& -i\frac{N-k}{N} \sum_{j=1}^k \langle L_j J^{(k)} P_{\mathbf{\alpha}} L_{k+1}\psi, P_{\mathbf{\beta}} W_{j(k+1)} L_j L_{k+1} \psi \rangle \end{aligned}
$$
Estimating yields
$$ | \text{IV} | \lesssim \sum_{j=1}^k ( \| J^{(k)} L_j \|_{\operatorname{op}} + \| L_j J^{(k)} \|_{\operatorname{op}}) \|W_{j(k+1)} \|_{\operatorname{op}} \| L_jL_{k+1} \psi \|_{L^2(\mathbb{R}^{3N})} \|L_{k+1} \psi \|_{L^2(\mathbb{R}^{3N})}$$
By \eqref{E:e-2},
\begin{equation}
\label{E:cpct54}
| \text{IV} | \lesssim C_{k,J^{(k)}}
\end{equation}
Integrating \eqref{E:cpct50} from $t_1$ to $t_2$ and applying the bounds obtained in \eqref{E:cpct51}, \eqref{E:cpct52}, \eqref{E:cpct53}, and \eqref{E:cpct54}, we obtain \eqref{E:cpct2}.
Finally, we proceed to prove \eqref{E:cpct3}. We have, by Lemma \ref{Lemma:ESYSoblevLemma},
\begin{align*}
\hspace{0.3in}&\hspace{-0.3in} | \operatorname{Tr} J^{(k)} P_{\mathbf{\alpha}} \tilde{\gamma}_{N,\omega}^{(k)} P_{\mathbf{\beta}}(t_2) - \operatorname{Tr} J^{(k)} P_{\mathbf{\alpha}} \tilde{\gamma}_{N,\omega}^{(k)} P_{\mathbf{\beta}}(t_1)| \\
&\leq 2 \sup_t | \langle J^{(k)}P_{\mathbf{\alpha}} \tilde \psi_{N,\omega}(t), P_{\mathbf{\beta}} \tilde \psi_{N,\omega}(t) \rangle_{\mathbf{r}_k} | \\
&\lesssim \| J^{(k)} \|_{\operatorname{op}} \| P_{\mathbf{\alpha}} \tilde \psi_{N,\omega}(t)\|_{L^2(\mathbb{R}^{3N})} \| P_{\mathbf{\beta}} \tilde \psi_{N,\omega}(t)\|_{L^2(\mathbb{R}^{3N})} \\
&\lesssim \omega^{-\frac12 |\mathbf{\alpha}|- \frac12 |\mathbf{\beta}|}
\end{align*}
where in the last step we applied \eqref{E:e-3}.
\end{proof}
According to Corollary \ref{Corollary:LimitMustBeAProduct}, the study of the
limit point of $\Gamma _{N,\omega }(t)=\left\{ \tilde{\gamma}_{N,\omega
}^{(k)}\right\} _{k=1}^{N}$ is directly related to the sequence $\Gamma
_{x,N,\omega }(t)=\left\{ \tilde{\gamma}_{x,N,\omega }^{(k)}=\limfunc{Tr}_{z}%
\tilde{\gamma}_{N,\omega }^{(k)}\right\} _{k=1}^{N}\in \oplus _{k\geqslant
1}C\left( \left[ 0,T\right] ,\mathcal{L}_{k}^{1}\left( \mathbb{R}%
^{2k}\right) \right) .$ We will do so in Section \ref{Section:Convergence of
The Infinite Hierarchy}. We end this section on compactness by proving that $%
\Gamma _{x,N,\omega }(t)$ is compact with respect to the two dimensional
version of the product topology $\tau _{prod}$ used in Theorem \ref%
{Theorem:Compactness of the scaled marginal density}. This proof is not as
delicate as the proof of Theorem \ref{Theorem:Compactness of the scaled
marginal density} because we do not need to deal with $\infty -\infty $ here.
\begin{theorem}
\label{Theorem:Compactness of the x-marginal density}The sequence
\begin{equation*}
\Gamma _{x,N,\omega }(t)=\left\{ \tilde{\gamma}_{x,N,\omega }^{(k)}=\limfunc{%
Tr}\nolimits_{z}\tilde{\gamma}_{N,\omega }^{(k)}\right\} _{k=1}^{N}\in
\bigoplus_{k\geqslant 1}C\left( \left[ 0,T\right] ,\mathcal{L}_{k}^{1}\left(
\mathbb{R}^{2k}\right) \right) .
\end{equation*}%
is compact with respect to the two dimensional version of the product
topology $\tau _{prod}$ used in Theorem \ref{Theorem:Compactness of the
scaled marginal density}.
\end{theorem}
\begin{proof}
Similar to Theorem \ref{Theorem:Compactness of the scaled marginal density},
we show that for every observable $J_{x}^{(k)}$ from a dense subset of $%
\mathcal{K}\left( L^{2}\left( \mathbb{R}^{2k}\right) \right) $ and for every
$\varepsilon >0,$ $\exists \delta (J_{x}^{(k)},\varepsilon )$ s.t. $\forall
t_{1},t_{2}\in \left[ 0,T\right] $ with $\left\vert t_{1}-t_{2}\right\vert
\leqslant \delta ,$ we have%
\begin{equation*}
\sup_{N,\omega }\left\vert \limfunc{Tr}J_{x}^{(k)}\left( \tilde{\gamma}%
_{x,N,\omega }^{(k)}\left( t_{1}\right) -\tilde{\gamma}_{x,N,\omega
}^{(k)}\left( t_{2}\right) \right) \right\vert \leqslant \varepsilon .
\end{equation*}%
We utilize the observables $J_{x}^{(k)}\in \mathcal{K}\left( L^{2}\left(
\mathbb{R}^{2k}\right) \right) $ which satisfy
\begin{equation*}
\left\Vert \left\langle \nabla _{x_{i}}\right\rangle \left\langle \nabla
_{x_{j}}\right\rangle J_{x}^{(k)}\left\langle \nabla _{x_{i}}\right\rangle
^{-1}\left\langle \nabla _{x_{j}}\right\rangle ^{-1}\right\Vert _{\func{op}%
}+\left\Vert \left\langle \nabla _{x_{i}}\right\rangle ^{-1}\left\langle
\nabla _{x_{j}}\right\rangle ^{-1}J_{x}^{(k)}\left\langle \nabla
_{x_{i}}\right\rangle \left\langle \nabla _{x_{j}}\right\rangle \right\Vert
_{\func{op}}<\infty .
\end{equation*}%
Here we choose similar but different observables from the proof of Theorem %
\ref{Theorem:Compactness of the scaled marginal density} since $\tilde{\gamma%
}_{x,N,\omega }^{(k)}$ acts on $L^{2}\left( \mathbb{R}^{2k}\right) $ instead
of $L^{2}\left( \mathbb{R}^{3k}\right) .$ This seems to make a difference
when we deal with the terms involving $\tilde{\gamma}_{N,\omega }^{(k)}$ or $%
\tilde{\gamma}^{(k)}.$ But $J_{x}^{(k)}$ does nothing on the $z$ variable,
hence
\begin{eqnarray*}
\left\Vert L_{j}J_{x}^{(k)}L_{j}^{-1}\right\Vert _{\func{op}} &\sim
&\left\Vert \left( \left\langle \nabla _{x_{j}}\right\rangle +\partial
_{z_{j}}\right) J_{x}^{(k)}\frac{1}{\left( \left\langle \nabla
_{x_{j}}\right\rangle +\partial _{z_{j}}\right) }\right\Vert _{\func{op}} \\
&\leqslant &\left\Vert \left\langle \nabla _{x_{j}}\right\rangle J_{x}^{(k)}%
\frac{1}{\left( \left\langle \nabla _{x_{j}}\right\rangle +\partial
_{z_{j}}\right) }\right\Vert _{\func{op}}+\left\Vert J_{x}^{(k)}\frac{%
\partial _{z_{j}}}{\left( \left\langle \nabla _{x_{j}}\right\rangle
+\partial _{z_{j}}\right) }\right\Vert _{\func{op}} \\
&\leqslant &\left\Vert \left\langle \nabla _{x_{j}}\right\rangle
J_{x}^{(k)}\left\langle \nabla _{x_{j}}\right\rangle ^{-1}\right\Vert _{%
\func{op}}+\left\Vert J_{x}^{(k)}\right\Vert _{\func{op}},
\end{eqnarray*}%
i.e. $\Vert L_{j}J_{x}^{(k)}L_{j}^{-1}\Vert _{\func{op}},\Vert
L_{j}^{-1}J_{x}^{(k)}L_{j}\Vert _{\func{op}},\Vert
L_{i}L_{j}J_{x}^{(k)}L_{i}^{-1}L_{j}^{-1}\Vert _{\func{op}}$ and $\Vert
L_{i}^{-1}L_{j}^{-1}J_{x}^{(k)}L_{i}L_{j}\Vert _{\func{op}}$ are all finite.
It is true that $J_{x}^{(k)}$ and the related operators listed are only in $%
\mathcal{L}^{\infty }\left( L^{2}\left( \mathbb{R}^{3k}\right) \right) $,
but this is good enough for our purpose here.
Taking $\limfunc{Tr}_{z}$ on both sides of hierarchy \eqref{hierarchy:BBGKY
hierarchy for scaled marginal densities}, we have that $\tilde{\gamma}%
_{x,N,\omega }^{(k)}$ satisfies the coupled BBGKY hierarchy:%
\begin{eqnarray}
i\partial _{t}\tilde{\gamma}_{x,N,\omega }^{(k)} &=&\sum_{j=1}^{k}\left[
-\triangle _{x_{j}},\tilde{\gamma}_{x,N,\omega }^{(k)}\right] +\frac{1}{N}%
\sum_{i<j}^{k}\limfunc{Tr}\nolimits_{z}\left[ V_{N,\omega }\left(
r_{i}-r_{j}\right) ,\tilde{\gamma}_{N,\omega }^{(k)}\right]
\label{hierarchy:coupled BBGKY for the x-component} \\
&&+\frac{N-k}{N}\sum_{j=1}^{k}\limfunc{Tr}\nolimits_{x_{k+1}}\limfunc{Tr}%
\nolimits_{z}\left[ V_{N,\omega }\left( r_{j}-r_{k+1}\right) ,\tilde{\gamma}%
_{N,\omega }^{(k+1)}\right] . \notag
\end{eqnarray}
Assume $t_{1}\leqslant t_{2},$ the above hierarchy yields%
\begin{eqnarray*}
&&\left\vert \limfunc{Tr}J_{x}^{(k)}\left( \tilde{\gamma}_{x,N,\omega
}^{(k)}\left( t_{1}\right) -\tilde{\gamma}_{x,N,\omega }^{(k)}\left(
t_{2}\right) \right) \right\vert \\
&\leqslant &\sum_{j=1}^{k}\int_{t_{1}}^{t_{2}}\left\vert \limfunc{Tr}%
J_{x}^{(k)}\left[ -\triangle _{x_{j}},\tilde{\gamma}_{x,N,\omega }^{(k)}%
\right] \right\vert dt+\frac{1}{N}\sum_{i<j}^{k}\int_{t_{1}}^{t_{2}}\left%
\vert \limfunc{Tr}J_{x}^{(k)}\left[ V_{N,\omega }\left( r_{i}-r_{j}\right) ,%
\tilde{\gamma}_{N,\omega }^{(k)}\right] \right\vert dt \\
&&+\frac{N-k}{N}\sum_{j=1}^{k}\int_{t_{1}}^{t_{2}}\left\vert \limfunc{Tr}%
J_{x}^{(k)}\left[ V_{N,\omega }\left( r_{j}-r_{k+1}\right) ,\tilde{\gamma}%
_{N,\omega }^{(k+1)}\right] \right\vert dt. \\
&=&\sum_{j=1}^{k}\int_{t_{1}}^{t_{2}}\text{I}\left( t\right) dt+\frac{1}{N}%
\sum_{i<j}^{k}\int_{t_{1}}^{t_{2}}\text{II}\left( t\right) dt+\frac{N-k}{N}%
\sum_{j=1}^{k}\int_{t_{1}}^{t_{2}}\text{III}\left( t\right) dt.
\end{eqnarray*}%
For I, we have
\begin{eqnarray*}
&& \hspace{-0.5in} \left\vert \limfunc{Tr}J_{x}^{(k)}\left[ -\triangle _{x_{j}},\tilde{\gamma}%
_{x,N,\omega }^{(k)}\right] \right\vert \\
&=&\left\vert \limfunc{Tr}J_{x}^{(k)}\left[ \left\langle \nabla
_{x_{j}}\right\rangle ^{2},\tilde{\gamma}_{x,N,\omega }^{(k)}\right]
\right\vert \text{ (}1\text{ commutes with everything)} \\
&=&\left\vert \limfunc{Tr}\left\langle \nabla _{x_{j}}\right\rangle
^{-1}J_{x}^{(k)}\left\langle \nabla _{x_{j}}\right\rangle ^{2}\tilde{\gamma}%
_{x,N,\omega }^{(k)}\left\langle \nabla _{x_{j}}\right\rangle -\limfunc{Tr}%
\left\langle \nabla _{x_{j}}\right\rangle J_{x}^{(k)}\left\langle \nabla
_{x_{j}}\right\rangle ^{-1}\left\langle \nabla _{x_{j}}\right\rangle \tilde{%
\gamma}_{x,N,\omega }^{(k)}\left\langle \nabla _{x_{j}}\right\rangle
\right\vert \\
&\leqslant &\left( \left\Vert \left\langle \nabla _{x_{j}}\right\rangle
^{-1}J_{x}^{(k)}\left\langle \nabla _{x_{j}}\right\rangle \right\Vert
_{\operatorname{op}}+\left\Vert \left\langle \nabla _{x_{j}}\right\rangle
J_{x}^{(k)}\left\langle \nabla _{x_{j}}\right\rangle ^{-1}\right\Vert
_{\operatorname{op}}\right) \limfunc{Tr}\left\langle \nabla _{x_{j}}\right\rangle \tilde{%
\gamma}_{x,N,\omega }^{(k)}\left\langle \nabla _{x_{j}}\right\rangle \\
&\leqslant &C_{J}\limfunc{Tr}\left\langle \nabla _{x_{j}}\right\rangle ^{2}%
\tilde{\gamma}_{N,\omega }^{(k)} \\
&\leqslant &C_{J}\text{ (Corollary \ref{Corollary:Energy Bound for Marginal
Densities}).}
\end{eqnarray*}
for II and III, we have%
\begin{eqnarray*}
\text{II} &=&\left\vert \limfunc{Tr}J_{x}^{(k)}\left[ V_{N,\omega }\left(
r_{i}-r_{j}\right) ,\tilde{\gamma}_{N,\omega }^{(k)}\right] \right\vert \\
&=&|\limfunc{Tr}L_{i}^{-1}L_{j}^{-1}J_{x}^{(k)}L_{i}L_{j}W_{ij}L_{i}L_{j}%
\tilde{\gamma}_{N,\omega }^{(k)}L_{i}L_{j}-\limfunc{Tr}%
L_{i}L_{j}J_{x}^{(k)}L_{i}^{-1}L_{j}^{-1}L_{i}L_{j}\tilde{\gamma}_{N,\omega
}^{(k)}L_{i}L_{j}W_{ij}| \\
&\leqslant &\left( \left\Vert
L_{i}^{-1}L_{j}^{-1}J_{x}^{(k)}L_{i}L_{j}\right\Vert _{\operatorname{op}}+\left\Vert
L_{i}L_{j}J_{x}^{(k)}L_{i}^{-1}L_{j}^{-1}\right\Vert _{\operatorname{op}}\right) \left\Vert
W_{ij}\right\Vert _{\operatorname{op}}\limfunc{Tr}L_{i}L_{j}\tilde{\gamma}_{N,\omega
}^{(k)}L_{i}L_{j} \\
&\leqslant &C_{J}\text{,}
\end{eqnarray*}%
and similarly,%
\begin{eqnarray*}
\text{III} &=&\left\vert \limfunc{Tr}J_{x}^{(k)}\left[ V_{N,\omega }\left(
r_{j}-r_{k+1}\right) ,\tilde{\gamma}_{N,\omega }^{(k+1)}\right] \right\vert
\\
&=&|\limfunc{Tr}%
L_{j}^{-1}L_{k+1}^{-1}J_{x}^{(k)}L_{j}L_{k+1}W_{j(k+1)}L_{j}L_{k+1}\tilde{%
\gamma}_{N,\omega }^{(k+1)}L_{j}L_{k+1} \\
&&-\limfunc{Tr}L_{j}L_{k+1}J_{x}^{(k)}L_{j}^{-1}L_{k+1}^{-1}L_{j}L_{k+1}%
\tilde{\gamma}_{N,\omega }^{(k+1)}L_{j}L_{k+1}W_{j(k+1)}| \\
&\leqslant &\left( \left\Vert L_{j}^{-1}J_{x}^{(k)}L_{j}\right\Vert
_{\operatorname{op}}+\left\Vert L_{j}J_{x}^{(k)}L_{j}^{-1}\right\Vert _{\operatorname{op}}\right)
\left\Vert W_{j(k+1)}\right\Vert _{\operatorname{op}}\limfunc{Tr}L_{j}L_{k+1}\tilde{\gamma}%
_{N,\omega }^{(k+1)}L_{j}L_{k+1} \\
&\leqslant &C_{J}.
\end{eqnarray*}%
Up to this point, we have proven uniform in time bounds for I - III,
thus we conclude the compactness of the sequence $\Gamma _{x,N,\omega
}(t)=\left\{ \tilde{\gamma}_{x,N,\omega }^{(k)}\right\} _{k=1}^{N}$.
\end{proof}
\section{Limit points satisfy GP hierarchy\label{Section:Convergence of The Infinite Hierarchy}}
\begin{theorem}
\label{Theorem:Convergence to the Coupled Gross-Pitaevskii}
Let $\Gamma(t)=\left\{ \tilde{\gamma}^{(k)}\right\} _{k=1}^{\infty }$ be a $N\geqslant \omega ^{v(\beta)+\varepsilon }$ limit point of $\Gamma_{N,\omega }(t)=\left\{ \tilde{\gamma}_{N,\omega }^{(k)}\right\} _{k=1}^{N}$ with respect to the product topology $\tau _{prod}$, then $\left\{ \tilde{\gamma}_{x}^{(k)}=\limfunc{Tr}_{z}\tilde{\gamma}^{(k)}\right\}_{k=1}^{\infty }$ is a solution to the coupled Gross-Pitaevskii hierarchy subject to initial data $\tilde{\gamma}_{x}^{(k)}\left( 0\right) =\left\vert\phi _{0}\right\rangle \left\langle \phi _{0}\right\vert ^{\otimes k}$ with coupling constant $b_{0}=$ $\int V\left( r\right) dr$, which, written in integral form, is
\begin{equation}
\label{hierarchy:coupled Gross-Pitaevskii}
\tilde{\gamma}_{x}^{(k)}=U^{(k)}(t)\tilde{\gamma}_{x}^{(k)}\left( 0\right)-ib_{0}\sum_{j=1}^{k}\int_{0}^{t}U^{(k)}(t-s)\limfunc{Tr}\nolimits_{x_{k+1}}\limfunc{Tr}\nolimits_{z}\left[ \delta \left( r_{j}-r_{k+1}\right) ,\tilde{\gamma}^{(k+1)}\left( s\right) \right] ds,
\end{equation}
where
\begin{equation*}
U^{(k)}=\dprod\limits_{j=1}^{k}e^{it\triangle _{x_{j}}}e^{-it\triangle_{x_{j}^{\prime }}}.
\end{equation*}
\end{theorem}
We prove Theorem \ref{Theorem:Convergence to the Coupled Gross-Pitaevskii} below. Combining Corollary \ref{Corollary:LimitMustBeAProduct} and Theorem \ref{Theorem:Convergence to the Coupled Gross-Pitaevskii}, we see that $\tilde{\gamma}_{x}^{(k)}$ in fact solves the 2D Gross-Pitaevskii hierarchy with the desired coupling constant $b_{0}\left( \int \left\vert h_{1}\left( z\right)\right\vert ^{4}dz\right) .$
\begin{corollary}
\label{Theorem:Convergence to the 2D Gross-Pitaevskii}Let $\Gamma(t)=\left\{ \tilde{\gamma}^{(k)}\right\} _{k=1}^{\infty }$ be a $N\geqslant \omega ^{v(\beta)+\varepsilon }$ limit point of $\Gamma_{N,\omega }(t)=\left\{ \tilde{\gamma}_{N,\omega }^{(k)}\right\} _{k=1}^{N}$ with respect to the product topology $\tau _{prod}$, then $\left\{ \tilde{\gamma}_{x}^{(k)}=\limfunc{Tr}_{z}\tilde{\gamma}^{(k)}\right\}_{k=1}^{\infty }$ is a solution to the 2D Gross-Pitaevskii hierarchy subject to initial data $\tilde{\gamma}_{x}^{(k)}\left( 0\right) =\left\vert \phi_{0}\right\rangle \left\langle \phi _{0}\right\vert ^{\otimes k}$ with coupling constant $b_{0}\left( \int \left\vert h_{1}\left( z\right)\right\vert ^{4}dz\right) $, which, written in integral form, is
\begin{equation}
\tilde{\gamma}_{x}^{(k)}=U^{(k)}(t)\tilde{\gamma}_{x}^{(k)}\left( 0\right)
-ib_{0}\left( \int \left\vert h_{1}\left( z\right) \right\vert ^{4}dz\right)
\sum_{j=1}^{k}\int_{0}^{t}U^{(k)}(t-s)\limfunc{Tr}\nolimits_{x_{k+1}}\left[
\delta \left( x_{j}-x_{k+1}\right) ,\tilde{\gamma}_{x}^{(k+1)}\left(
s\right) \right] ds. \label{hierarchy:2D Gross-Pitaevskii}
\end{equation}
\end{corollary}
\begin{proof}
We compute the $k=1$ case explicitly here. Written in kernels, the inhomogeneous term in hierarchy \eqref{hierarchy:coupled Gross-Pitaevskii} is
\begin{align*}
&ib_{0}\int U^{(1)}(t-s)ds\int \delta ( z_{1}-z_{1}^{\prime })dz_{1}dz_{1}^{\prime }\int \delta ( r_{1}-r_{2}) \tilde{\gamma}^{(2)}( r_{1},r_{2},r_{1}^{\prime },r_{2}) dr_{2} \\
&\quad -ib_{0}\int U^{(1)}(t-s)ds\int \delta ( z_{1}-z_{1}^{\prime })dz_{1}dz_{1}^{\prime }\int \delta ( r_{1}^{\prime }-r_{2}) \tilde{\gamma}^{(2)}( r_{1},r_{2},r_{1}^{\prime },r_{2}) dr_{2}
\end{align*}
which, by Corollary \ref{Corollary:LimitMustBeAProduct}, is
\begin{align*}
= &
\begin{aligned}[t]
ib_{0}\int U^{(1)}(t-s)ds\int \delta ( z_{1}-z_{1}^{\prime })\delta ( r_{1}-r_{2}) \tilde{\gamma}_{x}^{(2)}(x_{1},x_{2},x_{1}^{\prime },x_{2}) &\\
\times h_{1}( z_{1}) h_{1}( z_{2}) h_{1}( z_{1}^{\prime }) & h_{1}(z_{2}) dr_{2}dz_{1}dz_{1}^{\prime }
\end{aligned}\\
&\quad
\begin{aligned}
-ib_{0}\int U^{(1)}(t-s)ds\int \delta ( z_{1}-z_{1}^{\prime })\delta ( r_{1}^{\prime }-r_{2}) \tilde{\gamma}_{x}^{(2)}(x_{1},x_{2},x_{1}^{\prime },x_{2}) &\\
\times h_{1}( z_{1})h_{1}( z_{2}) h_{1}( z_{1}^{\prime }) & h_{1}(z_{2}) dr_{2}dz_{1}dz_{1}^{\prime }
\end{aligned}
\end{align*}
Further simplifications lead to
\begin{align*}
= & \, ib_{0}\int U^{(1)}(t-s)ds\int \delta ( x_{1}-x_{2}) \tilde{\gamma}_{x}^{(2)}( x_{1},x_{2},x_{1}^{\prime },x_{2}) \vert h_{1}( z_{1}) \vert ^{4}dx_{2}dz_{1} \\
&\quad -ib_{0}\int U^{(1)}(t-s)ds\int \delta ( x_{1}^{\prime }-x_{2}) \tilde{\gamma}_{x}^{(2)}( x_{1},x_{2},x_{1}^{\prime },x_{2})\vert h_{1}( z_{1}^{\prime }) \vert^{4}dx_{2}dz_{1}^{\prime }.
\end{align*}
In summary, we have
\begin{align*}
\hspace{0.3in}&\hspace{-0.3in} ib_{0}\int U^{(1)}(t-s)\limfunc{Tr}\nolimits_{x_{2}}\limfunc{Tr}
\nolimits_{z}\left[ \delta \left( r_{1}-r_{2}\right) ,\tilde{\gamma}
^{(2)}\left( s\right) \right] ds \\
&=ib_{0}\left( \int \left\vert h_{1}\left( z\right) \right\vert
^{4}dz\right) \int U^{(2)}(t-s)\limfunc{Tr}\nolimits_{x_{2}}\left[ \delta
\left( x_{1}-x_{2}\right) ,\tilde{\gamma}_{x}^{(2)}\left( s\right) \right]
ds.
\end{align*}
\end{proof}
\begin{proof}[Proof of Theorem \ref{Theorem:Convergence to the Coupled Gross-Pitaevskii}]
By Theorems \ref{Theorem:Compactness of the scaled marginal density}, \ref{Theorem:Compactness of the x-marginal density}, passing to subsequences if necessary, we have
\begin{equation}
\label{condition:fast convergence}
\begin{aligned}
\lim_{\substack{ N,\omega \rightarrow \infty \\ N\geqslant \omega ^{v(\beta
)+\varepsilon }}}\sup_{t}\limfunc{Tr}J^{(k)}\left( \tilde{\gamma}_{N,\omega
}^{(k)}\left( t\right) -\tilde{\gamma}^{(k)}\left( t\right) \right) &=&0,
\quad \forall \; J^{(k)}\in \mathcal{K}\left( L^{2}\left( \mathbb{R}
^{3k}\right) \right) , \\
\lim_{\substack{ N,\omega \rightarrow \infty \\ N\geqslant \omega ^{v(\beta
)+\varepsilon }}}\sup_{t}\limfunc{Tr}J_{x}^{(k)}\left( \tilde{\gamma}
_{x,N,\omega }^{(k)}\left( t\right) -\tilde{\gamma}_{x}^{(k)}\left( t\right)
\right) &=&0,\quad \forall \; J_{x}^{(k)}\in \mathcal{K}\left( L^{2}\left(
\mathbb{R}^{2k}\right) \right) .
\end{aligned}
\end{equation}
We establish \eqref{hierarchy:coupled Gross-Pitaevskii} by testing the limit
point against the observables $J_{x}^{(k)}\in \mathcal{K}\left( L^{2}\left(
\mathbb{R}^{2k}\right) \right) $ as in the proof of Theorem \ref{Theorem:Compactness of the x-marginal density}. We will prove that the limit point satisfies
\begin{equation}
\label{equality:testing the limit pt with initial data}
\limfunc{Tr}J_{x}^{(k)}\tilde{\gamma}_{x}^{(k)}\left( 0\right) = \limfunc{Tr} J_{x}^{(k)}\left\vert \phi _{0}\right\rangle \left\langle \phi_{0}\right\vert ^{\otimes k}
\end{equation}
and
\begin{equation}
\label{hierarchy:testing the limit point}
\begin{aligned}
\limfunc{Tr}J_{x}^{(k)}\tilde{\gamma}_{x}^{(k)}\left( t\right) = &\limfunc{Tr}
J_{x}^{(k)}U^{(k)}\left( t\right) \tilde{\gamma}_{x}^{(k)}\left( 0\right) \\
& -ib_{0}\sum_{j=1}^{k}\int_{0}^{t}\limfunc{Tr}J_{x}^{(k)}U^{(k)}(t-s)\left[
\delta \left( r_{j}-r_{k+1}\right) ,\tilde{\gamma}^{(k+1)}\left( s\right)
\right] ds.
\end{aligned}
\end{equation}
To this end, we use the coupled BBGKY hierarchy \eqref{hierarchy:coupled BBGKY for the x-component} satisfied by $\tilde{\gamma}_{x,N,\omega }^{(k)}$, which, written in the form needed here, is
\begin{align*}
\limfunc{Tr}J_{x}^{(k)}\tilde{\gamma}_{x,N,\omega }^{(k)}\left( t\right) = &
\limfunc{Tr}J_{x}^{(k)}U^{(k)}\left( t\right) \tilde{\gamma}_{x,N,\omega
}^{(k)}\left( 0\right) \\
&-\frac{i}{N}\sum_{i<j}^{k}\int_{0}^{t}\limfunc{Tr}
J_{x}^{(k)}U^{(k)}\left( t-s\right) \left[ V_{N,\omega }\left(
r_{i}-r_{j}\right) ,\tilde{\gamma}_{N,\omega }^{(k)}\left( s\right) \right]
ds \\
&-i\left( \frac{N-k}{N}\right) \sum_{j=1}^{k}\int_{0}^{t}\limfunc{Tr}
J_{x}^{(k)}U^{(k)}\left( t-s\right) \left[ V_{N,\omega }\left(
r_{j}-r_{k+1}\right) ,\tilde{\gamma}_{N,\omega }^{(k+1)}\left( s\right)
\right] ds \\
=&A-\frac{i}{N}\sum_{i<j}^{k}B-i\left( 1-\frac{k}{N}\right) \sum_{j=1}^{k}D.
\end{align*}
By \eqref{condition:fast convergence}, we know
\begin{eqnarray*}
\lim_{\substack{ N,\omega \rightarrow \infty \\ N\geqslant \omega ^{v(\beta
)+\varepsilon }}}\limfunc{Tr}J_{x}^{(k)}\tilde{\gamma}_{x,N,\omega
}^{(k)}\left( t\right) &=&\limfunc{Tr}J_{x}^{(k)}\tilde{\gamma}
_{x}^{(k)}\left( t\right) , \\
\lim_{\substack{ N,\omega \rightarrow \infty \\ N\geqslant \omega ^{v(\beta
)+\varepsilon }}}\limfunc{Tr}J_{x}^{(k)}U^{(k)}\left( t\right) \tilde{\gamma}
_{x,N,\omega }^{(k)}\left( 0\right) &=&\limfunc{Tr}J_{x}^{(k)}U^{(k)}\left(
t\right) \tilde{\gamma}_{x}^{(k)}\left( 0\right) .
\end{eqnarray*}
By the argument that appears between Theorem 1 and Corollary 1 in \cite{LiebAndSeiringer}, we know that assumption (b) in Theorem \ref{Theorem:3D->2D BEC (Nonsmooth)},
\begin{equation*}
\tilde{\gamma}_{N,\omega }^{(1)}\left( 0\right) \rightarrow \left\vert \phi
_{0}\otimes h_{1}\right\rangle \left\langle \phi _{0}\otimes h_{1}\right\vert \,,
\quad \text{strongly in trace norm}\,,
\end{equation*}
in fact implies
\begin{equation*}
\tilde{\gamma}_{N,\omega }^{(k)}\left( 0\right) \rightarrow \left\vert \phi
_{0}\otimes h_{1}\right\rangle \left\langle \phi _{0}\otimes
h_{1}\right\vert ^{\otimes k} \,,
\quad \text{strongly in trace norm}\,.
\end{equation*}
Thus we have tested relation \eqref{equality:testing the limit pt with initial
data}, the left-hand side of \eqref{hierarchy:testing the limit point}, and the first
term on the right-hand side of \eqref{hierarchy:testing the limit point} for the limit
point. We are left to prove that
\begin{eqnarray*}
\lim_{\substack{ N,\omega \rightarrow \infty \\ N\geqslant \omega ^{v(\beta
)+\varepsilon }}}\frac{B}{N} &=&0, \\
\lim_{\substack{ N,\omega \rightarrow \infty \\ N\geqslant \omega ^{v(\beta
)+\varepsilon }}}\left( 1-\frac{k}{N}\right) D &=&b_{0}\int_{0}^{t}\limfunc{
Tr}J_{x}^{(k)}U^{(k)}(t-s)\left[ \delta \left( r_{j}-r_{k+1}\right) ,\tilde{
\gamma}^{(k+1)}\left( s\right) \right] ds.
\end{eqnarray*}
First of all, we can use an argument similar to the estimate of $\text{III}$ and $\text{IV}$ in the proof of Theorem \ref{Theorem:Compactness of the scaled marginal density} to show the boundedness of $\left\vert B\right\vert $ and $
\left\vert D\right\vert $ for every finite time $t$. In fact, noticing that $
U^{(k)}$ commutes with Fourier multipliers, we have
\begin{eqnarray*}
\left\vert B\right\vert &\leqslant &\int_{0}^{t}\left\vert \limfunc{Tr}
J_{x}^{(k)}U^{(k)}\left( t-s\right) \left[ V_{N,\omega }\left(
r_{i}-r_{j}\right) ,\tilde{\gamma}_{N,\omega }^{(k)}\left( s\right) \right]
\right\vert ds \\
&=&\int_{0}^{t}ds|\limfunc{Tr}
L_{i}^{-1}L_{j}^{-1}J_{x}^{(k)}L_{i}L_{j}U^{(k)}\left( t-s\right)
W_{ij} L_iL_j\tilde{\gamma}_{N,\omega }^{(k)}\left(
s\right) L_{i}L_{j} \\
&&-\limfunc{Tr}L_{i}L_{j}J_{x}^{(k)}L_{i}^{-1}L_{j}^{-1}U^{(k)}\left(
t-s\right) L_{i}L_{j}\tilde{\gamma}_{N,\omega }^{(k)}\left( s\right)
L_{i}L_{j}W_{ij}| \\
&\leqslant &\int_{0}^{t}ds\left\Vert
L_{i}^{-1}L_{j}^{-1}J_{x}^{(k)}L_{i}L_{j}\right\Vert _{\operatorname{op}}\left\Vert
U^{(k)}\right\Vert _{\operatorname{op}}\left\Vert W_{ij} \right\Vert \limfunc{Tr}
L_{i}^{2}L_{j}^{2}\tilde{\gamma}_{N,\omega }^{(k)}\left( s\right) \\
&&+\int_{0}^{t}ds\left\Vert
L_{i}L_{j}J_{x}^{(k)}L_{i}^{-1}L_{j}^{-1}\right\Vert _{\operatorname{op}}\left\Vert
U^{(k)}\right\Vert _{\operatorname{op}}\left\Vert W_{ij} \right\Vert \limfunc{Tr}
L_{i}^{2}L_{j}^{2}\tilde{\gamma}_{N,\omega }^{(k)}\left( s\right) \\
&\leqslant &C_{J}t.
\end{eqnarray*}
Hence
\begin{equation*}
\lim_{\substack{ N,\omega \rightarrow \infty \\ N\geqslant \omega ^{v(\beta
)+\varepsilon }}}\frac{B}{N}=\lim_{\substack{ N,\omega \rightarrow \infty
\\ N\geqslant \omega ^{v(\beta )+\varepsilon }}}\frac{kD}{N}=0.
\end{equation*}
To prove
\begin{equation}
\lim_{\substack{ N,\omega \rightarrow \infty \\ N\geqslant \omega ^{v(\beta
)+\varepsilon }}}D=\int_{0}^{t}\limfunc{Tr}J_{x}^{(k)}U^{(k)}(t-s)\left[
\delta \left( r_{j}-r_{k+1}\right) ,\tilde{\gamma}^{(k+1)}\left( s\right)
\right] ds, \label{limit:converges to delta function}
\end{equation}
we need Lemma \ref{Lemma:ComparingDeltaFunctions} (stated and proved in Appendix \ref{A:Sobolev}) which compares the $\delta -$function and its approximation. We choose a probability measure $\rho \in L^{1}\left( \mathbb{R}^{3}\right) $ and define $\rho _{\alpha
}\left( r\right) =\alpha ^{-3}\rho \left( \frac{r}{\alpha }\right) .$ In
fact, $\rho $ can be the square of any 3D Hermite function. Write $
J_{s-t}^{(k)}=J_{x}^{(k)}U^{(k)}\left( t-s\right) $, we then have
\begin{align*}
\hspace{0.3in}&\hspace{-0.3in} \left\vert \limfunc{Tr}J_{x}^{(k)}U^{(k)}\left( t-s\right) \left(
V_{N,\omega }\left( r_{j}-r_{k+1}\right) \tilde{\gamma}_{N,\omega
}^{(k+1)}\left( s\right) -b_{0}\delta \left( r_{j}-r_{k+1}\right) \tilde{
\gamma}^{(k+1)}\left( s\right) \right) \right\vert \\
&\leqslant \left\vert \limfunc{Tr}J_{s-t}^{(k)}\left( V_{N,\omega }\left(
r_{j}-r_{k+1}\right) -b_{0}\delta \left( r_{j}-r_{k+1}\right) \right) \tilde{
\gamma}_{N,\omega }^{(k+1)}\left( s\right) \right\vert \\
&\quad +b_{0}\left\vert \limfunc{Tr}J_{s-t}^{(k)}\left( \delta \left(
r_{j}-r_{k+1}\right) -\rho _{\alpha }\left( r_{j}-r_{k+1}\right) \right)
\tilde{\gamma}_{N,\omega }^{(k+1)}\left( s\right) \right\vert \\
&\quad +b_{0}\left\vert \limfunc{Tr}J_{s-t}^{(k)}\rho _{\alpha }\left(
r_{j}-r_{k+1}\right) \left( \tilde{\gamma}_{N,\omega }^{(k+1)}\left(
s\right) -\tilde{\gamma}^{(k+1)}\left( s\right) \right) \right\vert \\
&\quad +b_{0}\left\vert \limfunc{Tr}J_{s-t}^{(k)}\left( \rho _{\alpha }\left(
r_{j}-r_{k+1}\right) -\delta \left( r_{j}-r_{k+1}\right) \right) \tilde{
\gamma}^{(k+1)}\left( s\right) \right\vert \\
&=\text{I}+\text{II}+\text{III}+\text{IV}
\end{align*}
We take care of $\text{I}$ first because it is a term which requires $N>\omega ^{\frac{1}{2\beta }-\frac{1}{2}}$. Write $V_{\omega }(r)=\frac{1}{\sqrt{\omega
}}V(x,\frac{z}{\sqrt{\omega }}),$ we have $V_{N,\omega }=\left( N\sqrt{
\omega }\right) ^{3\beta }V_{\omega }(\left( N\sqrt{\omega }\right) ^{\beta
}r)$, Lemma \ref{Lemma:ComparingDeltaFunctions} then yields
\begin{eqnarray*}
\text{I} &\leqslant &\frac{Cb_{0}}{\left( N\sqrt{\omega }\right) ^{\beta \kappa }}
\left( \int V_{\omega }(r)\left\vert r\right\vert ^{\kappa }dr\right) \\
&&\times \left( \left\Vert L_{j}J_{x}^{(k)}L_{j}^{-1}\right\Vert
_{\operatorname{op}}+\left\Vert L_{j}^{-1}J_{x}^{(k)}L_{j}\right\Vert _{\operatorname{op}}\right) \limfunc{
Tr}L_{j}L_{k+1}\tilde{\gamma}_{N,\omega }^{(k+1)}\left( s\right) L_{j}L_{k+1}
\\
&=&C_{J}\frac{\left( \int V_{\omega }(r)\left\vert r\right\vert ^{\kappa
}dr\right) }{\left( N\sqrt{\omega }\right) ^{\beta \kappa }}.
\end{eqnarray*}
Notice that $\left( \int V_{\omega }(r)\left\vert r\right\vert ^{\kappa
}dr\right) $ grows like $\left( \sqrt{\omega }\right) ^{\kappa }$, so $
I\leqslant C_{J}\left( \frac{\left( \sqrt{\omega }\right) ^{1-\beta }}{
N^{\beta }}\right) ^{\kappa }$ which converges to zero as $N,\omega
\rightarrow \infty $ in the way that $N\geqslant \omega ^{\frac{1}{2\beta }-
\frac{1}{2}+\varepsilon }.$ More precisely,
\begin{equation*}
\lim_{\substack{ N,\omega \rightarrow \infty \\ N\geqslant \omega ^{v(\beta
)+\varepsilon }}}I=0.
\end{equation*}
So we have handled $\text{I}$.
For $\text{II}$ and $\text{IV}$, we have
\begin{eqnarray*}
\text{II} &\leqslant &Cb_{0}\alpha ^{\kappa }\left( \left\Vert
L_{j}J_{x}^{(k)}L_{j}^{-1}\right\Vert _{\operatorname{op}}+\left\Vert
L_{j}^{-1}J_{x}^{(k)}L_{j}\right\Vert _{\operatorname{op}}\right) \limfunc{Tr}L_{j}L_{k+1}
\tilde{\gamma}_{N,\omega }^{(k+1)}\left( s\right) L_{j}L_{k+1}\text{ (Lemma
\ref{Lemma:ComparingDeltaFunctions})} \\
&\leqslant &C_{J}\alpha ^{\kappa }\text{ (Corollary \ref{Corollary:Energy
Bound for Marginal Densities})} \\
\text{IV} &\leqslant &Cb_{0}\alpha ^{\kappa }\left( \left\Vert
L_{j}J_{x}^{(k)}L_{j}^{-1}\right\Vert _{\operatorname{op}}+\left\Vert
L_{j}^{-1}J_{x}^{(k)}L_{j}\right\Vert _{\operatorname{op}}\right) \limfunc{Tr}L_{j}L_{k+1}
\tilde{\gamma}^{(k+1)}\left( s\right) L_{j}L_{k+1}\text{ (Lemma \ref
{Lemma:ComparingDeltaFunctions})} \\
&\leqslant &C_{J}\alpha ^{\kappa }\text{ (Corollary \ref
{Corollary:LimitMustBeAProduct})}
\end{eqnarray*}
which converges to $0$ as $\alpha \rightarrow 0$, uniformly in $N,\omega .$
For $\text{III}$,
\begin{eqnarray*}
\text{III} &\leqslant &b_{0}\left\vert \limfunc{Tr}J_{s-t}^{(k)}\rho _{\alpha
}\left( r_{j}-r_{k+1}\right) \frac{1}{1+\varepsilon L_{k+1}}\left( \tilde{
\gamma}_{N,\omega }^{(k+1)}\left( s\right) -\tilde{\gamma}^{(k+1)}\left(
s\right) \right) \right\vert \\
&&+b_{0}\left\vert \limfunc{Tr}J_{s-t}^{(k)}\rho _{\alpha }\left(
r_{j}-r_{k+1}\right) \frac{\varepsilon L_{k+1}}{1+\varepsilon L_{k+1}}\left(
\tilde{\gamma}_{N,\omega }^{(k+1)}\left( s\right) -\tilde{\gamma}
^{(k+1)}\left( s\right) \right) \right\vert .
\end{eqnarray*}
The first term in the above estimate goes to zero as $N,\omega \rightarrow
\infty $ for every $\varepsilon >0$, since we have assumed condition \eqref
{condition:fast convergence} and $J_{s-t}^{(k)}\rho _{\alpha }\left(
r_{j}-r_{k+1}\right) \left( 1+\varepsilon L_{k+1}\right) ^{-1}$ is a compact
operator. Due to the energy bounds on $\tilde{\gamma}_{N,\omega }^{(k+1)}$
and $\tilde{\gamma}^{(k+1)}$, the second term tends to zero as $\varepsilon
\rightarrow 0$, uniformly in $N$.
Combining the estimates for $\text{I}$-$\text{IV}$, we have justified limit \eqref
{limit:converges to delta function}. Hence, we have obtained Theorem \ref
{Theorem:Convergence to the Coupled Gross-Pitaevskii}.
\end{proof}
\section{Uniqueness of the 2D GP hierarchy\label{Appendix: Uniqueness of 2D GP}}
For completeness, we discuss the uniqueness theory of the 2D Gross-Pitaevskii hierarchy. To be specific, we have the following theorem.
\begin{theorem}[{\cite[Theorem 3]{ChenAnisotropic}}]
\label{Theorem:Uniqueness of 2D GP} Define the
collision operator $B_{j,k+1}$ by
\begin{equation*}
B_{j,k+1}\gamma _{x}^{(k+1)}=\limfunc{Tr}\nolimits_{k+1}\left[ \delta \left(
x_{j}-x_{k+1}\right) ,\gamma _{x}^{(k+1)}\right] .
\end{equation*}
Suppose that $\left\{ \gamma _{x}^{(k)}\right\} _{k=1}^{\infty }$ solves the 2D
constant coefficient Gross-Pitaevskii hierarchy
\begin{equation}
i\partial _{t}\gamma _{x}^{(k)}+\sum_{j=1}^{k}\left[ -\triangle
_{x_{j}},\gamma _{x}^{(k)}\right] =c_{0}\sum_{j=1}^{k}B_{j,k+1}\left( \gamma
_{x}^{(k+1)}\right) , \label{hierarchy:2D GP in appendix, general coupling}
\end{equation}
subject to zero initial data and the space-time bound
\begin{equation}
\int_{0}^{T}\left\Vert \prod_{j=1}^{k}\left( \left\vert \nabla _{x
_{j}}\right\vert ^{\frac{1}{2}}\left\vert \nabla _{x_{j}^{\prime
}}\right\vert ^{\frac{1}{2}}\right) B_{j,k+1}\gamma _{x}^{(k+1)}(t,\mathbf{
\cdot };\mathbf{\cdot })\right\Vert _{L^{2}(\mathbb{R}^{2k}\times \mathbb{R}
^{2k})}dt\leqslant C^{k} \label{Condition:2D Space-Time Bound}
\end{equation}
for some $C>0$ and all $1\leqslant j\leqslant k.$ Then $\forall k,t\in
\lbrack 0,T]$,
\begin{equation*}
\left\Vert \prod_{j=1}^{k}\left( \left\vert \nabla _{x
_{j}}\right\vert ^{\frac{1}{2}}\left\vert \nabla _{x_{j}^{\prime
}}\right\vert ^{\frac{1}{2}}\right) \gamma _{x}^{(k)}(t,\mathbf{\cdot };
\mathbf{\cdot })\right\Vert _{L^{2}(\mathbb{R}^{2k}\times \mathbb{R}
^{2k})}=0.
\end{equation*}
\end{theorem}
\begin{proof}
This is the constant coefficient version of \cite[Theorem 3]{ChenAnisotropic}. W. Beckner obtained the key estimate of this theorem independently in \cite{Beckner}. Some other estimates of this type can be found in \cite{ChenDie, GM}. K. Kirpatrick, G. Staffilani and B. Schlein
are the first to obtain uniqueness theorems for 2D Gross-Pitaevskii hierarchies. One will find their Theorem 7.1 in \cite{Kirpatrick} by replacing $\left\vert \nabla \right\vert ^{\frac{1}{2}}$ by $\left\langle \nabla \right\rangle ^{\frac{1}{2}+\varepsilon }$ in the statement of the above theorem.
\end{proof}
To apply Theorem \ref{Theorem:Uniqueness of 2D GP} to our problem here, it
is necessary to prove that both the known solution to the 2D
Gross-Pitaevskii hierarchy (namely $\left\vert \phi \right\rangle
\left\langle \phi \right\vert ^{\otimes k}$, where $\phi $ solves the 2D
cubic NLS) and the limit obtained from the coupled BBGKY hierarchy \eqref
{hierarchy:coupled BBGKY for the x-component}, satisfy the space-time bound
\eqref{Condition:2D Space-Time Bound}. It is easy to see that $\left\vert \phi
\right\rangle \left\langle \phi \right\vert ^{\otimes k}$ verifies the
space-time bound \eqref{Condition:2D Space-Time Bound} because it is part of
the standard procedure of proving well-posedness of the 2D cubic NLS. We use
the following trace theorem to prove the space-time bound \eqref{Condition:2D
Space-Time Bound} for the limit.
\begin{theorem}
[{\cite[Theorem 5.2]{Kirpatrick}}] For every $\alpha <1,$ there is a $C_{\alpha }>0$ such that
\begin{equation*}
\left\Vert \prod_{j=1}^{k}\left( \left\langle \nabla _{x
_{j}}\right\rangle ^{\alpha }\left\langle \nabla _{x_{j}^{\prime
}}\right\rangle ^{\alpha }\right) B_{j,k+1}\gamma _{x}^{(k+1)}\right\Vert
_{L^{2}(\mathbb{R}^{2k}\times \mathbb{R}^{2k})}\leqslant C_{\alpha }\limfunc{
Tr}\left( \prod_{j=1}^{k+1}\left( 1-\triangle _{x_{j}}\right) \right) \gamma
_{x}^{(k+1)}
\end{equation*}
for all nonnegative $\gamma _{x}^{(k+1)}\in \mathcal{L}^{1}\left(
L^{2}\left( \mathbb{R}^{2k}\right) \right) $.
\end{theorem}
We can combine the above theorems so that it is easy to see how they apply to our problem.
\begin{theorem}
\label{Theorem:CombiningChenAndKirpatrick}There is at most one nonnegative
operator sequence
$$\left\{ \gamma _{x}^{(k)}\right\} _{k=1}^{\infty }\in
\bigoplus _{k\geqslant 1}C\left( \left[ 0,T\right] ,\mathcal{L}_{k}^{1}\left(
\mathbb{R}^{2k}\right) \right)$$
that solves the 2D Gross-Pitaevskii hierarchy \eqref{hierarchy:2D GP in appendix, general coupling} subject to the
energy condition
\begin{equation*}
\limfunc{Tr}\left( \prod_{j=1}^{k}\left( 1-\triangle _{x_{j}}\right) \right)
\gamma _{x}^{(k)}\leqslant C^{k}.
\end{equation*}
\end{theorem}
\section{Conclusion}
In this paper, by proving the limit of a BBGKY hierarchy whose limit is not
even formally known since it contains $\left( \infty -\infty \right) ,$ we
have rigorously derived the 2D cubic nonlinear Schr\"{o}dinger equation from
a 3D quantum many-body dynamic and we have accurately described the 3D to 2D
phenomenon by establishing the exact emergence of the coupling constant $
\left( \int \left\vert h_{1}(z)\right\vert ^{4}dz\right) $. This is the
first direct rigorous treatment of the 3D to 2D dynamic problem in the literature.
|
2,877,628,090,093 | arxiv | \section{Introduction}
Electron cooling that results when a bunch of electrons overlaps a bunch of ions ,
with both bunches moving at the same velocity, may be considered to be an
intrabeam scattering process. The process is similar to
the usual intrabeam scattering, Ref.[1]
where the ions scatter from each other and usually results
in beam growth. An important
difference is that in electron cooling the mass of the ion is different from and
much larger than the mass of the electron. This difference
considerably complicates
the intrabeam scattering theory. It introduces a new term in the emittance growth
rate, which vanishes when the particles are identical and
their masses are equal, and
can give rise to emittance cooling of the heavier particles . The term that gives
rise to beam growth for the usual intrabeam scattering is also present
but is much smaller than the cooling term when one particle is
much heavier than the other.
This paper derives the results found for the emittance cooling rates
due to the scattering of the ions in the ion bunch by the electons
in the electron bunch. The derivations given below makes
considerable use of the results found in two
previous papers, Ref.[2] and Ref.[3]
\section{The $f(x,p)$ distribution and the scattering rate $\delta N$}
The ions are contained within a bunch
and their distibution is given by $f_a(x_a,p_a)$ where $N_a f_a(x_a,p_a)$
is the number
of ions in $d^3x_ad^3p_a$. $N_a$ is the number of ions in the bunch.
\[ \int d^3x_ad^3p_a \; f_a(x_a,p_a)=1 \]
The distribution of the electrons in the electon bunch is
given by $f_b(x_b,p_b)$ and $N_b$ is the number of
electrons in the electron bunch.
Let $\delta N_a$ be the number of ions with momentum, $p_a$ in $d^3p_a$
and space coordinate $x$ in $d^3x$
which are scattered by the electrons with momentum $p_b$ in $d^3p_b$
which are also in $d^3x$, in the time interval $dt$ , into the
solid angle $d\Omega'$ corresponding to the
direction $\hat{p_a'}$. Then $\delta N_a$ is given by, Ref.[2],
\begin{eqnarray}
\delta N_a &=& N_a N_b \sigma_{ab} d\Omega' \frac {d^3p_a}{\gamma_a}
\frac {d^3p_b}{\gamma_b} f_a(x,p_a)f_b(x,p_b) F(p_a,p_b) d^3x dt \nonumber\\
F(p_a,p_b) &=& \frac {[(p_ap_b)^2-m_a^2 m_b^2 ]^{1/2}}{m_a m_b}
\end{eqnarray}
$\sigma_{ab}$ is the scattering cross section for the scattering
of the ions from the electrons.In the expression for $F(p_a,p_b)$, we have put
$c=1$. $F(p_a,p_b)$ has the dimensions of a velocity.
For completeness sake this result is given in the form which is valid in any CS.
For the electron cooling problem for RHIC, one can do all the calclations
in the Rest CS, which is the CS moving along with the two bunches. In the
Rest CS, the central particle in either bunch is at rest and the motion of the
motion of the particles may be treated non-reletavistically.In the
Rest CS , one may put $\gamma_a=\gamma_b=1$ and
\[ F(p_a,p_b)=|\vec{v_a}-\vec{v_b}| \]
\section{Growth rates for $<p_{ia}p_{ja}>$}
Following Bjorken and Mtingwa, Ref.[4], cooling rates will first be
given for $<p_{ia}p_{ja}>$. where the $<>$ indicate an
average over all the particles in the bunch. From these one can compute the
growth rates for the average emittances of the ions, $<\epsilon_{ia}>$.
In a scattering event, where an ion with momentum $p_a$ scatters off an
electron with momentum $p_b$, the momenta will change to $p_a'$ and $p_b'$.
Let $\delta p_{ia}$ represent the change in $p_{ia}$ in the collision,
and similarly for $\delta (p_{ia}p_{ja})$. Then
\begin{eqnarray}
\delta p_{ia} &=& p_{ia}'-p_{ia} \nonumber\\
\delta (p_{ia}p_{ja}) &=& p_{ia}' p_{ja}'-p_{ia}p_{ja}
\end{eqnarray}
Using the scattering rate given by Eq.(1), one can now compute
$ \delta <p_{ia}p_{ja}>$ in the Rest CS,
\begin{eqnarray}
\delta <(p_{ia}p_{ja}) > &=& N_b \int \spc d^3x d^3p_{a}
d^3p_{b} f_a(x,p_a) f_b(x,p_b) |\vec{v_a}-\vec{v_b}| \nonumber\\
& & \spc \sigma_{ab} d\Omega' \spc \delta (p_{ia}p_{ja}) \spc dt \nonumber\\
\delta (p_{ia}p_{ja}) &=& (p_{ia}' p_{ja}'-p_{ia}p_{ja})
\end{eqnarray}
The 11-dimensional integral in Eq.3 can be reduced to a 3-dimenional integral for
gaussian distributions, if one notes that in the Rest CS
$\spc \sigma_{ab}$ depends on
$\vec{v_a}-\vec{v_b}$ and one transforms from the momentum variables
$p_a,p_b$ to two new variables one of which is $\vec{v_a}-\vec{v_b}$.
This can be done by the transformation
\begin{eqnarray}
\pb_{ia} &=& W_i+\frac{\mu}{m_a} \Delta_i \nonumber\\
\pb_{ib} &=& W_i-\frac{\mu}{m_b} \Delta_i \nonumber\\
W_i &=& \frac{p_{ia}+p_{ib}}{\gamma_0 \beta_0 (m_a+m_b)c} \nonumber\\
\Delta_i &=& \pb_{ia}-\pb_{ib}= \frac{v_{ia}-v_{ib}}{\gamma_0 \beta_0 c} \nonumber\\
d^3\pb_a d^3\pb_b &=& d^3W d^3\Delta \nonumber\\
\pb_{ia} &=& \frac{p_{ia}}{\gamma_0 \beta_0 m_ac} \nonumber\\
\pb_{ib} &=& \frac{p_{ib}}{\gamma_0 \beta_0 m_bc} \nonumber\\
\frac{1}{\mu} &=& \frac{1}{m_a}+\frac{1}{m_b} \nonumber\\
d^3 \pb_a d^3 \pb_b &=& d^3 W d^3 \De \nonumber\\
\end{eqnarray}
$\Delta_i$ is proportional to the relative velocity, $\vec{v_a}-\vec{v_b} \spc$
when the velocities are non-relativistic. A similar transformation
is used in Ref.1 and Ref.4 except that for them the particles are
identical and the transformation is simpler.
$\delta (p_{ia}p_{ja})$ can be written as
\begin{eqnarray}
\delta (p_{ia}p_{ja}) &=& p_{ia}q_{ja}+p_{ja}q_{ia}+q_{ia}q_{ja} \nonumber\\
q_{ia}&=& p_{ia}'-p_{ia}
\end{eqnarray}
This result can written as
\begin{eqnarray}
\delta (\pb_{ia}\pb_{ja}) &=& [(W_{i}\qb_{ja}+W_{j}\qb_{ia}) \muma]+
[(\muma)^2 (\De_i \qb_{ja}+\De_j \qb_{ia}+\qb_{ia}\qb_{ja})] \nonumber\\
\qb_{ia} &=& q_{ia} /(\gamma_0 \beta_0 \mu c) \nonumber\\
\end{eqnarray}
Eq.3 can be rewritten in terms of $W,\De$ as
\begin{eqnarray}
<\delta (\pb_{ia}\pb_{ja}) > &=& N_b \int \spc d^3x d^3W
d^3\De f_a(x,p_a) f_b(x,p_b) |\vec{v_a}-\vec{v_b}| \nonumber\\
& & \sigma_{ab} d\Omega' \spc \delta (\pb_{ia}\pb_{ja}) \spc dt
\nonumber\\
\delta (\pb_{ia}\pb_{ja}) &=& [(W_{i}\qb_{ja}+W_{j}\qb_{ia})\muma]+
[(\muma)^2 (\De_i \qb_{ja}+\De_j \qb_{ia}+\qb_{ia}\qb_{ja})] \nonumber\\
\end{eqnarray}
One may note that $\sigma_{ab}$ depends only on $\De$ and not on $W$.
In the expression for $\delta (\pb_{ia}\pb_{ja})$ the second term will be seen
to depend only on $\De$ and gives rise to the usual intrabeam scattering
growth rate, while the first term depends on $W$ and will be seen to vanish for
identical particles and gives rise to the cooling rates
for ion electron scattering.
The transformation from $\pv_a,\pv_b$ to $\Wv,\Dev$ allows us to do
the integral over $d\Omega'$. Eq.7 holds in any CS where the particle
motion is non-relativistic. For each $\pv_a,\pv_b$ one can define a
center of mass CS, called the CMS, in which $\pv_a+\pv_b=0$.
In the CMS
\[\Delta_i = \pb_{ia}-\pb_{ib}
=p_{ia}/(\gamma_0 \beta_0 \mu c) \]
In the CMS, $\Dev$ and $\pv_a$ have the same direction, and $\pv_a$
is scattered by the electrons to $\pv_a \; '$ which is along the direction
given by the polar angles $\theta , \phi$ relative to the direction of
$\pv_a$ or $\Dev$.
In Eq.7, only the $\qb_{ia}$ depend on the scattering angles $\theta ,\phi \spc$.
To do the integral over $d\Omega'$ in the Rest CS one has to evaluate
the integrals
\begin{eqnarray}
d_i &=& \int d\Omega' \sigma_{ab} \qb_{ia} \nonumber\\
c_{ij} &=& \int d\Omega' \sigma_{ab} [(\De_i \qb_{ja}+\De_j \qb_{ia})
+\qb_{ia}\qb_{ja}] \nonumber\\
\end{eqnarray}
$d\Omega' \sigma_{ab}$ is an invariant and
$\Dev,\qv_a$ are both the same in the CMS and the Rest CS as they are both the
difference of 2 vectors that are proportional to a velocity. $d_i$, $c_{ij}$
are tensors in 3-space. If these integrals are evaluated in the CMS
and the result is written in terms of tensors in 3-space then the
result will also hold in the Rest CS.
In the CMS, we introduce a polar coordinate system $\theta,\phi$
where $\theta$ is measured relative to the direction of
$\vec{p_a}$ or $\vec{\Delta}$
and we assume that $\sigma_{ab}(\theta,\phi)$ is a fumction of $\theta$ only.
we can then write
\begin{eqnarray}
\Dev &=& (0,0,1)|\vec{\Delta}| \nonumber\\
\vec{p_a} &=& (0,0,1)|\vec{\Delta}| (\gamma_0 \beta_0 \mu c) \nonumber\\
\vec{p_a \; '} &=& (\sin \theta \cos \phi,\sin \theta \sin \phi,
\cos \theta)|\vec{\Delta}| (\gamma_0 \beta_0 \mu c) \nonumber\\
\vec{q_a} &=& (\sin \theta \cos \phi,\sin \theta \sin \phi,
\cos \theta-1)|\vec{\Delta}|(\gamma_0 \beta_0 \mu c)
\end{eqnarray}
In the CMS,
using Eq.9, one finds
\begin{eqnarray}
d_i &=& -2 \pi \int d\theta sin\theta
(1-cos\theta) \sigma_{ab} (0,0,1) |\vec{\Delta}| \nonumber\\
c_{ij} &=& \pi \int_{0}^{\pi} d\theta \sin^3 \theta \sigma_{ab}
\;|\vec{\Delta}|^2
\left( \begin{array}{ccr} 1&0&0 \\
0&1&0 \\
0&0&-2
\end{array} \right ) \nonumber\\
\end{eqnarray}
In computing $c_{ij}$ one may note that the $\De_i \qb_{ja}+\De_j \qb_{ia}$
term in Eq.8 only contributes to $c_{33}$ while the $\qb_{ia}\qb_{ja}$
term contributes to to all 3 diagonal elements of $c_{ij}$.
These results for $d_i,c_{ij}$ in the CMS can be rewritten in terms of
tensors in 3-space as
\[ \]
\begin{eqnarray}
d_i &=& -2 \pi \int d\theta sin\theta
(1-cos\theta) \sigma_{ab} \De_i \nonumber\\
c_{ij} &=& \pi \int_{0}^{\pi} d\theta \sin^3 \theta \sigma_{ab} \;
(|\vec{\Delta}|^2 \delta_{ij}-3 \De_i \De_j ) \nonumber\\
\end{eqnarray}
In this form the results will also hold in the Rest CS. Eq. 7 can now
be rewritten as
\begin{eqnarray}
<\delta (\pb_{ia}\pb_{ja}) > &=& N_b \int \spc d^3x d^3W
d^3\De f_a(x,p_a) f_b(x,p_b) |\vec{v_a}-\vec{v_b}| \nonumber\\
& & ( [-2\pi \muma (W_i \De_j+W_j \De_i) \int d\theta sin\theta
(1-cos\theta) \sigma_{ab} ]_1 \nonumber\\
& & +[\pi (\muma)^2 (|\vec{\Delta}|^2 \delta_{ij}-3 \De_i \De_j )
\int d\theta sin^3 \theta \sigma_{ab}]_2 ) \spc dt \nonumber\\
\end{eqnarray}
Eq.12 can be used to compute either intrabeam scattering
for identical particles or
electron cooling. If the $a$ and $b$ particles are identical, then the second term
indicated by $[\spc ]_2$ and called the $\De$-term gives the growth rates for
intrabeam scattering. In this case, the first term, indicated by $[\spc ]_1$
and called the W-term, will vanish. This is shown below for gaussian distributions
and also can be shown to hold for any distribution because of the symmetry
of the $a$ and $b$ particles. If the $b$ particle is much
lighter than the $a$ particle,
the W-term gives rise to cooling of the $a$ particles and the $\De$-term
is smaller than the W-term by the factor $m_b/m_a$. This is shown below
for gaussian distributions. Eq. 12 holds for any distibutions,
$f_a(x,p_a), f_b(x,p_b)$. In the next section, we will specialize to gaussian
distributions.
it is often assumed that $\sigma_{ab}$ is given by the Coulonb cross-section in
the CMS CS for the $a$ and $b$ particles. This is given by
\begin{eqnarray}
\sigma_{ab} &=& (\frac {r_{ab}} {\beta_{ab}^2})^2 \frac{1}{(1-cos \theta)^2}
\nonumber\\
r_{ab} &=& \frac{Z_aZ_b e^2}{\mu c^2} \nonumber\\
\beta_{ab} c &=& |\vec{v_a}-\vec{v_b}| \nonumber\\
\end{eqnarray}
The integrals over $\theta$ in Eq.12 can then be written as
\[ \]
\begin{eqnarray}
\int d\theta sin\theta (1-cos\theta) \frac{1}{(1-cos \theta)^2} &=&
ln \lbr 1+\left (\frac{\beta_{ab}^2 b_{maxab}}{r_{ab}} \right )^2\rbr \nonumber\\
\int d\theta sin^3 \theta \frac{1}{(1-cos \theta)^2} &=& 2\lbr
ln \lbr 1+\lpar \frac{\beta_{ab}^2 b_{maxab}}{r_{ab}}\rpar^2 \rbr
-\frac{1}{1+( r_{ab}/(\beta_{ab}^2 b_{maxab}))^2} \rbr \nonumber\\
tan(\theta_{min}) &=& \frac{r_{ab}}{\beta_{ab}^2 b_{maxab}} \nonumber\\
\end{eqnarray}
$b_{maxab}$ is the maximun allowed impact parameter in the CMS. $\theta_{min}$
is the smallest allowed scattering angle in the CMS.
It will be seen below that to compute the cooling rates for the emittances
one will also need the cooling rates for $<x_{ia}p_{ja}>$. When the $a$ and $b$
particles are identical, the $<x_{ia}p_{ja}>$ are zero , but not zero
when the particles are different. Using Eq.7, one finds
\begin{eqnarray}
<\delta (x_{i}\pb_{ja}) > &=& N_b \int \spc d^3x d^3W
d^3\De f_a(x,p_a) f_b(x,p_b) |\vec{v_a}-\vec{v_b}| \nonumber\\
& & \sigma_{ab} d\Omega' \spc \delta (x_{ia}\pb_{ja}) \spc dt
\nonumber\\
\delta (x_{i}\pb_{ja}) &=& x_{i} \delta \pb_{ja}= x_{i} \qb_{ja} \muma
\nonumber\\
\mbox{From Eq.11 one has} & & \nonumber\\
\int d\Omega' \sigma_{ab} \qb_{ja} &=& -2 \pi \int d\theta sin\theta
(1-cos\theta) \sigma_{ab} \De_j \nonumber\\
\mbox{which gives} & & \nonumber\\
<\delta (x_{i}\pb_{ja}) > &=& N_b \int d^3x d^3W
d^3\De \spc f_a(x,p_a) f_b(x,p_b) |\vec{v_a}-\vec{v_b}| \nonumber\\
& & [-2 \pi \int d\theta sin\theta
(1-cos\theta) \sigma_{ab} \spc x_{i} \De_j \muma]
\spc dt
\nonumber\\
\end{eqnarray}
Eq.15 shows that $<\delta (x_{i}\pb_{ja}) >$ gives rise to a cooling term
which vanishes when the particles are identical , or when $\alpha_i=0$ for the
ion partcle for a gaussian distribution.
\section{Cooling rates for $<p_{ia}p_{ja}>$ in the Rest CS for Gaussian
distributions}
In this section, we will find the cooling rates due to the scattering
of the ions by the electrons in the cooling section when the ion and
electron bunches have gaussian distributions. In Eq.12 , we will keep only the
$W$-term as the $\Delta-$term , discussed later, is smaller by the factor
$m_b/m_a$ In this paper, it will be assumed that the dispersion is zero in
the cooling section.
For a gaussian distribution, $f_a(x,p_a)$ is given for the ion bunch for zero dispersion by Ref.[3],
\begin{eqnarray}
f_a(x,p_a) &=& \frac{1}{\Gamma_a} exp[-S_a(x,p_a)] \nonumber\\
\Gamma_a &=& \int d^3xd^3p \; exp[-S_a (x,p_a)] \nonumber\\
\Gamma_a &=& \pi^3 \epb_{xa} \epb_{sa} \epb_{ya} \nonumber\\
\end{eqnarray}
\begin{eqnarray}
S_a &=& S_{xa}+S_{ya}+S_{sa} \nonumber\\
& & \nonumber\\
S_{xa} &=& \frac{1}{\bar{\epsilon_{xa}}} \epsilon_{xa} (x,x'_a) \spc
x_a'=p_{xa}/p_{0a} \nonumber\\
\epsilon_{xa} (x,x'_a) &=& [x^2+(\beta_x x'_a+\alpha_{xa} x)^2]/\beta_{xa}
\nonumber\\
& & \nonumber\\
S_{ya} &=& \frac{1}{\bar{\epsilon_{ya}}} \epsilon_{ya} (y,y'_a) \spc
y_a'=p_{ya}/p_{0a} \nonumber\\
\epsilon_{ya} (y,y'_a) &=& [y^2+(\beta_y y'_a+\alpha_{ya} y)^2]/\beta_{ya}
\nonumber\\
& & \nonumber\\
S_s &=& \frac{1}{\bar{\epsilon_s}} \epsilon_s (s,p_s/p_{0a}) \nonumber\\
\epsilon_s (s,p_s/p_{0a})&=& \frac{s^2}{2\sigma_s^2}+\frac{(p_s/p_{0a})^2}
{2 \sigma_p^2} \nonumber\\
\epsilon_s (s,p_s/p_{0a}) &=& \frac{1}{\beta_s} (s)^2+\beta_s (p_s/p_{0a})^2
\nonumber\\
\epsilon_s (s,p_s/p_{0a}) &=& [(s)^2+(\beta_s (p_s/p_{0a}))^2]/\beta_s \nonumber\\
\beta_s &=& \sigma_s/\sigma_p \nonumber\\
\bar \epsilon_s &=& 2 \sigma_s \sigma_p
\end{eqnarray}
A longitudinal emittance has been introduced
so that the longitudinal motion and the transverse motions can be treated in
a similar manner. $\beta_s$ in the Rest CS is larger than $\beta_s$
in the Laboratory CS by the factor $\gamma_0^2$. $s,p_s$ are the paricle
longitudinal position and momentum in the Rest CS.
In Eq.12 we will now do the integration over $d^3x d^3W$ using the above gaussian
ditributions. Because there is no dispersion in the cooling section the
integral over $dxdW_x$ or $dsdW_s$ or $dydW_y$ can each be treated
in a similar way.
Eq.12 can now be written using the Coulomb cross-section as
\begin{eqnarray}
\delta <(\pb_{ia}\pb_{ja}) > &=& \frac {N_b}{\Gamma_a \Gamma_b}\int \spc d^3x d^3W
d^3\De exp[-(S_a+S_b)] |\vec{v_a}-\vec{v_b}| \nonumber\\
& & \muma \Wb_{ij} \spc
(\frac {r_{ab}} {\beta_{ab}^2})^2 \spc
ln \lbr 1+\lpar \frac{\beta_{ab}^2 b_{maxab}}{r_{ab}}\rpar^2 \rbr
\spc dt \nonumber\\
\Wb_{ij} &=& -2 \pi (W_i \De_j+W_j \De_i)
\end{eqnarray}
We rewrite $S_a+S_b$ as
\begin{eqnarray}
S_a+S_b &=& \Sigma_i(S_{ia}+S_{ib}) \spc i=x,y,s \nonumber\\
S_{ia} &=& \frac {1} {\epb_{ia}}
\lbr
\frac{x_{ia}^2}{\beta_{ia}}+
( \be_{ia}^{1/2} \pb_{ia}+\frac {\al_{ia} x_{ia}} {\be_{ia}^{1/2}} )^2
\rbr
\nonumber\\
S_{ia} &=& \frac {1} {\epb_{ia}}
\lbr
\frac{x_{ia}^2}{\beta_{ia}}+
( \be_{ia}^{1/2} (W_i+\frac{\mu}{m_a} \Delta_i) +\frac {\al_{ia} x_{ia}}
{\be_{ia}^{1/2}} )^2
\rbr
\nonumber\\
& & \nonumber\\
S_{ia}+S_{ib} &=& A_{11i} x_i^2 +A_{22i} W_i^2 +2 A_{12i} x_i W_i
+(A_{10i} x_i +A_{01i} W_i) \De_i +A_{00i} \De_i^2 \nonumber\\
& & \nonumber\\
A_{11i} &=& \lbr \frac{1+\al_{i}^2}{\be_{i} \epb_{i}} \rbr_+ \spc
A_{22i}=\lbr \frac{\be_{i}}{\epb_{i}} \rbr_+ \nonumber\\
A_{12i} &=& \lbr \frac{\al_{i}}{\epb_{i}} \rbr_+ \spc
A_{10i}=\lbr 2 \mum \frac{\al_{i}}{\epb_{i}} \rbr_- \nonumber\\
A_{01i} &=& \lbr 2 \mum \frac{\be_{i}}{\epb_{i}} \rbr_- \spc
A_{00i}=\lbr (\mum)^2 \frac{\be_{i}}{\epb_{i}} \rbr_+ \nonumber\\
\end{eqnarray}
The symbols $[ (\spc ) ]_+ $ and $[ (\spc ) ]_-$ are defined by
\[[ (\spc ) ]_+ =(\spc)_a+(\spc)_b \]
\[[ ( \spc) ]_- =(\spc)_a-(\spc)_b \]
We will now make a transformation to eliminate the $2 A_{12i} x_i W_i$ term
in $S_{ia}+S_{ib}$. We rewrite $S_{ia}+S_{ib}$ as
\begin{eqnarray}
S_{ia}+S_{ib} &=& A_{11i} x_i^2 +A_{22i} W_i^2 +2 A_{12i} x_i W_i
+(A_{10i}x_i +A_{01i} W_i) \De_i +A_{00i} \De_i^2 \nonumber\\
&=& [A_{11} x^2 +A_{22} W^2 +2 A_{12} x W
+(A_{10}x +A_{01} W) \De +A_{00} \De^2]_i \nonumber\\
&=& [x^2(A_{11}-\frac{A_{12}^2}{A_{22}})+
(A_{22}^{1/2}W+\frac{A_{12}}{A_{22}^{1/2}} x)^2 \nonumber\\
& & +(A_{10}x +A_{01} W) \De +A_{00} \De^2 ]_i \nonumber\\
& & \nonumber\\
\eta_i &=& \lbr \frac {\Ab^{1/2}}{A_{22}^{1/2}} x \rbr_i \spc \spc
p_{\eta i} = \lbr A_{22}^{1/2}W+\frac{A_{12}}{A_{22}^{1/2}} x\rbr_i \nonumber\\
\Ab_i &=& [A_{11}A_{22}- A_{12}^2]_i \nonumber\\
x_i &=& [x_{\eta}\eta]_i \spc \spc W_i=[(W_{\eta}\eta+W_{p_{\eta}} p_{\eta} )]_i
\nonumber\\
dx_i dw_i &=& \lbr \frac{1}{\Ab^{1/2}} d\eta dp_{\eta}\rbr_i \nonumber\\
x_{\eta i} &=& \lbr \frac{A_{22}^{1/2}}{\Ab^{1/2}}\rbr_i \spc \spc W_{\eta i}=
\lbr -\frac {A_{12}}{\Ab^{1/2}} \rbr_i
\spc \spc W_{p_{\eta i}}= \lbr \frac{1}{A_{22}^{1/2}}\rbr_i
\nonumber\\
& & \nonumber\\
S_{ia}+S_{ib} &=& [\eta^2+p_{\eta}^2
+(A_{10}x +A_{01} W) \De +A_{00} \De^2 ]_i \nonumber\\
&=& [\eta^2+p_{\eta}^2
+(B_{10}\eta +B_{01}p_{\eta} ) \De +A_{00} \De^2 ]_i \nonumber\\
B_{10i} &=& [A_{10}x_{\eta}+A_{01} W_{\eta}]_i
\spc \spc B_{01i}= [A_{01} W_{p_{\eta}}]_i
\nonumber\\
B_{10i} &=& \lbr A_{10} \frac{A_{22}^{1/2}}{\Ab^{1/2}}
-A_{01} \frac {A_{12}}{\Ab^{1/2}}\rbr_i
\spc \spc B_{01i}= \lbr A_{01} \frac{1}{A_{22}^{1/2}}\rbr_i
\nonumber\\
& & \nonumber\\
\Wb_{ij} &=& -2 \pi [
(W_{\eta}\eta +W_{p_{\eta}} p_{\eta})_i \De_j
+(W_{\eta}\eta +W_{p_{\eta}} p_{\eta})_j\De_i
] \nonumber\\
\end{eqnarray}
In the expression for $S_{ia}+S_{ib}$ given at the end of Eq.19,
the linear terms in $\eta,p_{\eta}$ can be eliminated by the transformation
\begin{eqnarray}
\etb_i &=& \lbr \eta+\frac{B_{10}}{2} \De\rbr _i \spc \spc
\petb_i=\lbr p_{\eta}+\frac{B_{01}}{2} \De\rbr _i \nonumber\\
S_{ia}+S_{ib} &=& [\etb^2+\petb^2
+(A_{00}-B_{10}^2/4-B_{01}^2/4) \De^2 ]_i \nonumber\\
\Wb_{ij} &=& -2 \pi [
[W_{\eta}(\etb-\frac{B_{10}}{2}\De)]_i \De_j +[W_{p_{\eta}}
(\petb-\frac{B_{01}}{2} \De)]_i \De_j \nonumber\\
& & +[W_{\eta}(\etb-\frac{B_{10}}{2}\De)]_j \De_i +[W_{p_{\eta}}
(\petb-\frac{B_{01}}{2} \De)]_j \De_i
] \nonumber\\
\end{eqnarray}
Eq.17 can now be rewritten as
\begin{eqnarray}
<\delta (\pb_{ia}\pb_{ja}) > &=& \frac {N_b}{\Gamma_a \Gamma_b}
\frac{1}{\Ab^{1/2}_p}
\int \spc d^3\etb d^3\petb d^3\De \spc exp[-(S_a+S_b)]
|\vec{v_a}-\vec{v_b}|
\nonumber\\
& & \muma \Wb_{ij} (\frac {r_{ab}} {\beta_{ab}^2})^2 \spc
ln \lbr 1+\lpar \frac{\beta_{ab}^2 b_{maxab}}{r_{ab}}\rpar^2 \rbr
\spc dt
\nonumber\\
\Wb_{ij} &=& -2 \pi [
[W_{\eta}(\etb-\frac{B_{10}}{2}\De)]_i \De_j +[W_{p_{\eta}}
(\petb-\frac{B_{01}}{2} \De)]_i \De_j \nonumber\\
& & +[W_{\eta}(\etb-\frac{B_{10}}{2}\De)]_j \De_i +[W_{p_{\eta}}
(\petb-\frac{B_{01}}{2} \De)]_j \De_i
] \nonumber\\
\Ab^{1/2}_p &=& \Ab_x^{1/2}\Ab_y^{1/2}\Ab_s^{1/2} \nonumber\\
\end{eqnarray}
Using Eq.20 for $S_{ia}+S_{ib}$ and for $\Wb_{ij}$ , one can do the
integral over $d^3\etb d^3\petb$ and get
\[ \]
\[ \]
\[ \]
\[ \]
\begin{eqnarray}
\delta <(\pb_{ia}\pb_{ja}) > &=& \frac {N_b}{\Gamma_a \Gamma_b}
\frac{1}{\Ab^{1/2}_p} \pi^3 r_{ab}^2 c \muma \Wh_{ij} \nonumber\\
& &
\int \spc d^3\De \frac{ exp[-(\la_x \De_x^2+\la_y \De_y^2+\la_s \De_s^2)]}
{\beta_{ab}^3} \De_i\De_j
\nonumber\\
& &
ln \lbr 1+\lpar \frac{\beta_{ab}^2 b_{maxab}}{r_{ab}}\rpar^2 \rbr
\spc dt
\nonumber\\
& & \nonumber\\
\Wh_{ij} &=& 2 \pi \lbr
(W_{\eta}\frac{B_{10}}{2} +W_{p_{\eta}} \frac{B_{01}}{2})_i
+(W_{\eta}\frac{B_{10}}{2} +W_{p_{\eta}} \frac{B_{01}}{2})_j
\rbr \nonumber\\
& & \nonumber\\
\beta_{ab} &=& \gamma_0 \beta_0 (\De_x^2+\De_y^2+\De_s^2)^{1/2} \nonumber\\
\lambda_i &=& \lbr A_{00}-(\frac{B_{10}}{2})^2-(\frac{B_{01}}{2})^2\rbr_i \nonumber\\
\Ab_i &=& [A_{11}A_{22}- A_{12}^2]_i \nonumber\\
\Ab^{1/2}_p &=& \Ab_x^{1/2}\Ab_y^{1/2}\Ab_s^{1/2} \nonumber\\
& & \nonumber\\
x_{\eta i} &=& \lbr \frac{A_{22}^{1/2}}{\Ab^{1/2}}\rbr_i \spc \spc W_{\eta i}=
\lbr -\frac {A_{12}}{\Ab^{1/2}} \rbr_i
\spc \spc W_{p_{\eta i}}= \lbr \frac{1}{A_{22}^{1/2}}\rbr_i
\nonumber\\
B_{10i} &=& [A_{10}x_{\eta}+A_{01} W_{\eta}]_i
\spc \spc B_{01i}= [A_{01} W_{p_{\eta}}]_i
\nonumber\\
B_{10i} &=& \lbr A_{10} \frac{A_{22}^{1/2}}{\Ab^{1/2}}
-A_{01} \frac {A_{12}}{\Ab^{1/2}}\rbr_i
\spc \spc B_{01i}= \lbr A_{01} \frac{1}{A_{22}^{1/2}}\rbr_i
\nonumber\\
& & \nonumber\\
A_{11i} &=& \lbr \frac{1+\al_{i}^2}{\be_{i} \epb_{i}} \rbr_+ \spc
A_{22i}=\lbr \frac{\be_{i}}{\epb_{i}} \rbr_+ \nonumber\\
A_{12i} &=& \lbr \frac{\al_{i}}{\epb_{i}} \rbr_+ \spc
A_{10i}=\lbr 2 \mum \frac{\al_{i}}{\epb_{i}} \rbr_- \nonumber\\
A_{01i} &=& \lbr 2 \mum \frac{\be_{i}}{\epb_{i}} \rbr_- \spc
A_{00i}=\lbr (\mum)^2 \frac{\be_{i}}{\epb_{i}} \rbr_+ \nonumber\\
\end{eqnarray}
Eq.23 is our final result for the cooling rates for $<p_{ia}p_{ja}>$
in the Rest CS, for two overlapping
gaussian bunches , with no dispersion in the cooling section.
For this case one gets zero results when $i \neq j$. The remaining
3-dimensional integral over $d^3\De$ is an integral over the relative
velocities of the ions and electrons.
It will be seen below that to compute the cooling rates for the emittances
one will also need the cooling rates for $<x_{ia}p_{ja}>$. For gaussian
distributioins, using the coulomb cross section and Eq.15, Eq.18 is replaced by
\begin{eqnarray}
\delta <(x_i\pb_{ja}) > &=& \frac {N_b}{\Gamma_a \Gamma_b}\int \spc d^3x d^3W
d^3\De exp[-(S_a+S_b)] |\vec{v_a}-\vec{v_b}| \nonumber\\
& & \muma \xb_{ij} \spc
\lbr \frac {r_{ab}} {\beta_{ab}^2}\rbr^2 \spc
ln \lbr 1+\lpar \frac{\beta_{ab}^2 b_{maxab}}{r_{ab}}\rpar^2 \rbr
\spc dt \nonumber\\
\xb_{ij} &=& -2 \pi x_i \De_j
\end{eqnarray}
After going from the $x,W$ coordinates to $\eta,p_{\eta}$ and integrating over
$\eta,p_{\eta}$ Eq.23 is replaced by
\begin{eqnarray}
\delta <(x_i\pb_{ja}) > &=& \frac {N_b}{\Gamma_a \Gamma_b}
\frac{1}{\Ab^{1/2}_p} \pi^3 r_{ab}^2 c \muma \xh_{ij} \nonumber\\
& &
\int \spc d^3\De \frac{ exp[-(\la_x \De_x^2+\la_y \De_y^2+\la_s \De_s^2)]}
{\beta_{ab}^3} \De_i\De_j
\nonumber\\
& &
ln \lbr 1+\lpar \frac{\beta_{ab}^2 b_{maxab}}{r_{ab}}\rpar^2 \rbr
\spc dt
\nonumber\\
& & \nonumber\\
\xh_{ij} &=& 2 \pi \lbr x_{\eta}\frac{B_{10}}{2}\rbr_i
\nonumber\\
& & \nonumber\\
x_{\eta i} &=& \lbr \frac{A_{22}^{1/2}}{\Ab^{1/2}}\rbr_i \spc \spc W_{\eta i}=
\lbr -\frac {A_{12}}{\Ab^{1/2}} \rbr_i
\spc \spc W_{p_{\eta i}}= \lbr \frac{1}{A_{22}^{1/2}}\rbr_i
\nonumber\\
B_{10i} &=& [A_{10}x_{\eta}+A_{01} W_{\eta}]_i
\spc \spc B_{01i}= [A_{01} W_{p_{\eta}}]_i
\nonumber\\
B_{10i} &=& \lbr A_{10} \frac{A_{22}^{1/2}}{\Ab^{1/2}}
-A_{01} \frac {A_{12}}{\Ab^{1/2}}\rbr_i
\spc \spc B_{01i}= \lbr A_{01} \frac{1}{A_{22}^{1/2}}\rbr_i
\nonumber\\
\end{eqnarray}
\section{Emittance growth rates}
One can compute growth rates for the average emittances, $<\epsilon_{ia}>$ in the
Laboratory Coordinate System, from the growth rates for $<p_{ia}p_{ja}>$ in the
Rest Coordinate System. In the following , $dt$ is the time interval in the
Laboratory System and $d\tilde{t}$ is the time interval in the Rest System.
$dt=\gamma d\tilde{t}$. The final results are, for zero dispersion,
\begin{eqnarray}
\frac{d}{dt} <\epb_{ia}> &=& \frac{\beta_{ia}}{\gamma} \frac{d}{d\tilde{t}}
<\pb_{ia}^2>
+ \frac{2 \alpha_{ia} }{\gamma} \frac{d}{d\tilde{t}} <x_i \pb_{ia}>
\spc i=x,y,s \nonumber\\
\end{eqnarray}
To derive the above results, the simplest case to treat is that of the vertical
emittance. The verical emmitance is given by
\begin{eqnarray}
\epb_{ya}(y,y_a') &=& [y^2+(\beta_{ya} y'_a+\alpha_{ya} y)^2]/\beta_{ya}
\spc y_a'=\pb_{ya} \nonumber\\
\delta \epb_{ya} &=& \beta_{ya} \delta (\pb_{ya}^2)
+\delta (2 \alpha_{ya} y (\pb_{ya}) \nonumber\\
\frac{d}{dt} <\epb_{ya}> &=& \frac{\beta_{ya}}{\gamma} \frac{d}{d\tilde{t}}
<\pb_{ya}^2>
+ \frac{2 \alpha_{ya} }{\gamma} \frac{d}{d\tilde{t}} <y \pb_{ya}>
\nonumber\\
\end{eqnarray}
In Eq.(27), $y_a'=\pb_{ya}$, $\delta \epsilon_{ya}$ is the change in
$\epb_{ya}$ in a scattering event. Similar results will hold for
$\epb_{xa}$ and $\epb_{sa}$ for zero dispersion.
\section*{The $\Delta$ term in electron cooling}
In the previous section it was assumed that in Eq.12 one could drop the
second term or $\De$ term compared to the first term or $W$ term.This is true when
$m_b<<m_a$ and $\pb_a \simeq \pb_b$ in the Rest CS. Using Eq.4, one can write
\begin{eqnarray}
W_i &=& \lbr \pb_a \frac{m_a}{m_a+m_b}+\pb_b \frac{m_b}{m_a+m_b} \rbr_i \nonumber\\
\De_i &=& [\pb_a-\pb_b]_i \nonumber\\
& & \nonumber\\
W_i &\simeq & [ \pb_a]_i \nonumber\\
\De_i &=& [\pb_a-\pb_b]_i \nonumber\\
\end{eqnarray}
Thus $W$ and $\De$ are both of the same order as $\pb_a$ . If the motion is
non-relativistic in the Rest CS, $\qb_a \simeq \De \simeq \pb_a $. From this it
follows that the $\De$ term in Eq.12 is smaller than the $W$ term by the factor
$m_b/m_a$.
It has also been assumed that the motion in the Rest CS is non-relativistic.
In the Laboratory CS, the rms spread in the relative momentum is given by
\begin{eqnarray}
\sigma_{pi} &=& \lbr \frac{\epb_i}{2 \beta_i}\rbr^{1/2} \spc i=x,y,s \nonumber\\
\end{eqnarray}
For gold ions in RHIC at $\gamma=100$
\[\spc \epb_x=\epb_y=5e-8,\beta_x=\beta_y=50 \spc and \spc \sigma_{px}=\sigma_{py}=2.24e-5 \]
\[\spc \epb_s=1.8e-4,\beta_s=300m \spc and \spc \sigma_{ps}=.55e-3 \]
In the Rest CS $\sigma_{px},\sigma_{py}$ are unchanged at 2.24e-5 And
$\sigma_{ps}$ is reduced by the factor $\gamma$ to .55e-5 The spread in
each of the momenta in the Rest CS is of the order of $1e-3 m_ac$
since $\gamma=100$ and the
ion velocities are of the order of 1e-3c. Similar numbers hold for the
electrons in the electron bunch.
\section*{References}
\noindent
1. A. Piwinski Proc. 9th Int. Conf. on High Energy Accelerators (1974) 405
\noindent
2. G. Parzen BNL report C-A/AP/N0.150 (2004)
and at http://arxiv.org/ps\uu cache/physics/pdf/0405/0405019.pdf
\noindent
3. G. Parzen BNL report C-A/AP/N0.169 (2004)
and at http://arxiv.org/ps\uu cache/physics/pdf/0410/0410028.pdf
\noindent
4. J.D. Bjorken and S.K. Mtingwa, Part. Accel.13 (1983) 115
\end {document}
****************************************
We will now further evaluate $C_{ij}$ by first evaluating $C_{ij}$
for some particular values of $p_1$,$p_2$ in the CMS corresponding to
$p_1$,$p_2$ and and then using the tensor properties of $C_{ij}$
to find a result that holds in any other CS. We are particularly interested
in finding a result in the Rest CS, which is the CS which moves
along with the bunch. In the CMS,
\begin{eqnarray*}
\vec{p_2} &=& -\vec{p_1} \nonumber\\
\vec{\Delta} &=& \frac{1}{2} (\vec{p_1}-\vec{p_2})=\vec{p_1} \nonumber\\
\vec{q_1} &=& \vec{p_1'}-\vec{p_1} \nonumber\\
\vec{q_2} &=& -\vec{q_1}
\end{eqnarray*}
Using $\vec{q_1} = \vec{p_1'}-\vec{p_1}$ and $\vec{q_2} = -\vec{q_1}$,
one can show that
\begin{eqnarray}
\frac{1}{2} (\delta (p_{1i}p_{1j})+\delta (p_{2i}p_{2j}) &=&
q_{1i} q_{1j}+\Delta_i q_{1j}+ \Delta_j q_{1i}
\end{eqnarray}
In the CMS, we introduce a polar coordinate system $\theta,\phi$
where $\theta$ is measured relative to the direction of
$\vec{p_1}$ or $\vec{\Delta}$
and we assume that $\sigma(\theta,\phi)$ is a fumction of $\theta$ only.
we can then write
\begin{eqnarray*}
\vec{p_1} &=& (0,0,1)|\vec{\Delta}| \nonumber\\
\vec{p_1'} &=& (\sin \theta \cos \phi,\sin \theta \sin \phi,
\cos \theta)|\vec{\Delta}| \nonumber\\
\vec{q_1} &=& (\sin \theta \cos \phi,\sin \theta \sin \phi,
\cos \theta-1)|\vec{\Delta}|
\end{eqnarray*}
Considering $p_1$,$p_2$ to be 4-vectors, and $\Delta=.5(p_1-p_2)$,
$q_1=p_1'-p_1$, then $\Delta$, $q_1$ are also 4-vectors and
in the CMS, $\Delta_4=0,q_{14}=0$.
Using Eqs(10) and (12), one now finds for $C_{ij}$ in the CMS
\[ C_{ij} = \pi \int_{0}^{\pi} d\theta \sigma \sin^3 \theta \;|\vec{\Delta}|^2
\left( \begin{array}{ccrc} 1&0&0&0 \\
0&1&0&0 \\
0&0&-2&0 \\
0&0&0&0
\end{array} \right ) \]
To find $C_{ij}$ in the Rest CS or in the Lab CS, we will try to
find an expression for $C_{ij}$ in terms of the 4-vectors
$p_{1i},p_{2i}$ which gives the above result for $C_{ij}$ in the
CMS. The expression that does this is given by
\begin{eqnarray}
C_{ij} &=& \pi \int_{0}^{\pi} d\theta \sigma \sin^3 \theta \;
\Delta^2 [\delta_{ij}-3\frac{\Delta_i \Delta_j}{\Delta^2}
+\frac{W_iW_j}{W^2}] \;\; i,j=1,4 \nonumber\\
\Delta_i &=& \frac{1}{2}(p_{1i}-p_{2i}) \nonumber\\
W_i &=& p_{1i}+p_{2i}
\end{eqnarray}
where $\sigma$ is the cross section in the CMS. Let us now verify that
this expression gives the correct result for $C_{ij}$ in the CMS.
In the CMS,
\begin{eqnarray*}
\Delta^2 &=& |\vec{\Delta}|^2 +\Delta_4^2=|\vec{\Delta}|^2 \nonumber\\
W_i &=& 0 \;\; i=1,3 \nonumber\\
\Delta_i &=& 0 \;\; i=1,2 \;\;\Delta_3=|\vec{\Delta}|
\end{eqnarray*}
so that Eq.(13) does give the correct result in the CMS.
An important further simplification results from the fact that
the particle motion is non-relativistic in the CMS and also in the Rest CS
which moves along with the bunch. For RHIC parameters, for $\gamma=100$,
one finds that $p \simeq 1e-3\;\;mc$. One can then drop the
$W_iW_j/W^2$ term. Also $\Delta^2=|\vec{\Delta}|^2$ in the CMS and in the
Rest CS and one can evaluate $F(p_1,p_2)$ using Eq.(4) as
$F(p_1,p_2)=2 c \bar{\beta}$ where $\bar{\beta} c$ is the velocity of either
particle in the CMS . In the Rest CS, one can write
\begin{eqnarray}
C_{ij} &=& \pi \int_{0}^{\pi} d\theta \sigma \sin^3 \theta \;
[|\vec{\Delta}|^2 \delta_{ij}-3\Delta_i \Delta_j
] \;\; i,j=1,3 \nonumber\\
\Delta_i &=& \frac{1}{2}(p_{1i}-p_{2i}) \nonumber\\
<\delta (p_{1i}p_{1j}) > &=& N \int \;\: d^3x d^3p_1
d^3p_2 f(x,p_1)f(x,p_2) 2 \bar{\beta}c \; C_{ij}\;dt
\nonumber\\
\bar{\beta}c &=& |\vec{\Delta}|/m
\end{eqnarray}
\section{Growth rates for $<p_{ia}p_{ja}>$ in the Rest CS for Gaussian
distributions}
\section*{References}
\noindent
1. A. Piwinski Proc. 9th Int. Conf. on High Energy Accelerators (1974) 405
\noindent
2. G. Parzen BNL report C-A/AP/N0.150 (2004)
and at http://arxiv.org/ps\uu cache/physics/pdf/0405/0405019.pdf
\noindent
3. G. Parzen BNL report C-A/AP/N0.169 (2004)
and at http://arxiv.org/ps\uu cache/physics/pdf/0410/0410028.pdf
\noindent
4. J.D. Bjorken and S.K. Mtingwa, Part. Accel.13 (1983) 115
\end {document}
\section{Results}
Ions are indicated by the subscrpt a and the electrons by the subscript b.
The cooling rate for $<p_{ia}^2/p_{0a}^2> , \spc i=x,s,y$ in the Rest CS
is due to the scattering of the
ions from the electrons. This cooling rate has to be added to
the growth rate due to the scattering of the ions from each other
to find the actual growth rate of the ions. $p_{0a}$ is the central
momentum of the ion bunch.
The assumed gaussian distributions of the bunches are described by the
parameters
\[ \epb_{ia},\beta_{ia},\alpha_{ia},\spc
\epb_{ib},\beta_{ib},\alpha_{ib}, \spc i=x,s,y \]
The gaussian distribution is
further defined using these parameters in Ref.[1].
The symbols $[ (\spc ) ]_+ $ and $[ (\spc ) ]_-$ are defined by
\[[ (\spc ) ]_+ =(\spc)_a+(\spc)_b \]
\[[ ( \spc) ]_- =(\spc)_a-(\spc)_b \]
\subsection{ $d<p_{ia}^2>/dt$, $\spc i=x,s,y$ in the Rest CS. }
\begin{eqnarray}
\frac{d} {dt}<p_{ia}^2/p_{0a}^2> &=&
\frac{N_b}{\Gamma_a \Gamma_b} \frac{\pi^3 c}
{(\Ab^{1/2})_p} r_{ab}^2 \spc \Wb_{i} \spc (intdelta)_i \spc
\frac{\mu}{m_a} \nonumber\\
& & \nonumber\\
\Gamma_a &=& \pi^3 \epb_{xa} \epb_{sa} \epb_{ya} \nonumber\\
\Gamma_b &=& \pi^3 \epb_{xb} \epb_{sb} \epb_{yb} \nonumber\\
(\Ab^{1/2})_p &=& (\Ab_{xa} \Ab_{ya} \Ab_{sa})^{1/2} \nonumber\\
\Ab_{i} &=& \left[\frac{1+\alpha_{i}^2}{\beta_{i} \epb_{i}}\right]_+
\left[\frac{\beta_{i}}{\epb_{i}}\right]_+
- \left[ \frac{\alpha_{i}}{\epb_{i}} \right]_+ ^2 \nonumber\\
& & \nonumber\\
r_{ab} &=& \frac{Z e^2}{\mu c^2} \nonumber\\
\frac{1}{\mu} &=& \frac{1}{m_a}+\frac{1}{m_b} \nonumber\\
& & \nonumber\\
\Wb_{i} &=& W_{\eta i}(B10)_i +W_{p\eta i}(B01)_i \nonumber\\
(B01)_i &=& (A01)_i W_{p \eta i} \nonumber\\
(B100_i &=& (A10)_i x_{\eta i}+(A01)_i W_{\eta i} \nonumber\\
(A01)_i &=& [2 \frac{\mu}{m} \frac{\beta_i}{\epb_i}]_ - \nonumber\\
(A10)_i &=& [2\frac{\mu}{m}\frac{\alpha_i}{\epb_i}]_ - \nonumber\\
& & \nonumber\\
W_{\eta i} &=& -\lbr \frac{\alpha_i}{\epb_i}\rbr_+
/\left (\lbr \frac{\beta_i}{\epb_i}\rbr_+ \Ab_{i}\right )^{1/2} \nonumber\\
W_{p \eta i} &=& 1/\lbr \frac{\beta_{i}}{\epb_{i}} \rbr_+ \nonumber\\
x_{\eta i }&=& \left (\lbr\frac{\beta_i}{\epb_i}\rbr_+ / \Ab_{i}\right )^{1/2} \nonumber\\
& & \nonumber\\
(intdelta)_i &=& \int d^3\De \spc \frac
{exp[-(\la_x\De_x^2+\la_s\De_s^2+\la_y\De_y^2)]}
{\beta_{ab}^3 }\spc \De_i^2 \spc (inttheta) \nonumber\\
\beta_{ab} &=& \gamma_0 \beta_0 (\De_x^2+\De_s^2+\De_y^2)^{1/2} \nonumber\\
& & \nonumber\\
\la_i &=& (A00)_i-(B10)_i^2/4-(B01)_i^2)/4 \nonumber\\
(A00)_i &=& \lbr (\frac{\mu}{m})^2\frac{\beta_i}{\epb_i} \rbr_+ \nonumber\\
& & \nonumber\\
(inttheta) &=& 2\pi \spc ln [1+(\beta_{ab}^2 (bmax)_{ab}/r_{ab})^2]
\end{eqnarray}
The symbols $[ (\spc ) ]_+ $ and $[ (\spc ) ]_-$ are defined by
\[[ (\spc ) ]_+ =(\spc)_a+(\spc)_b \]
\[[ ( \spc) ]_- =(\spc)_a-(\spc)_b \]
\subsection{ $d<x_{ia}p_{ia}>/dt$, $\spc i=x,s,y$ in the Rest CS. }
\subsection{$d<\epsilon_{ia}>/dt$ in the Laboratory CS $\spc i=x,s,y$}
In the following , $d \tti$ is the time interval in the
Laboratory System and $dt$ is the time interval in the Rest System.
$d\tti=\gamma dt$. $\beta_s$ is the longitudinal beta function in the
Laboratory System. The dispersion is assumed to be zero in the cooling section.
\begin{eqnarray}
\frac{d}{d\tti} <\epsilon_{xa}> &=& \frac{\beta_x}{\gamma}
\frac{d}{dt} <p_{xa}^2/p_{0a}^2>+
2 \frac{\alpha_x}{\gamma}
\frac{d}{dt} <x_{a} p_{xa}/p_{0a}> \nonumber\\
\frac{d}{d\tti} <\epsilon_{ya}> &=& \frac{\beta_y}{\gamma}
\frac{d}{dt} <p_{ya}^2/p_{0a}^2> \nonumber\\
\frac{d}{d\tti} <\epsilon_{sa}> &=& \beta_s \gamma \frac{d}{dt}
<p_{sa}^2/p_{0a}^2>
\end{eqnarray}
next---Lai,B01,B10, d ep/dt in lab cs.
The integral over $d^3 \Delta$ is an integral over all possible values
of the relative velocity of any two particles in a bunch. $\beta_0,\gamma_0$
are the beta and gamma of the center of the
bunches in the Laboratory Coordinate System.
\end {document}
OUTLINE--
abstract--
introduction--
results--
$ f(XP)$-- omit
gaussian --a,b particles -- omit
$\delta Na$-- omit
$\sigma_{ab}$ -- omit
$d<p_i p_j>/dt$---
$\delta(p_ip_j)$---ion emittance cooling rate
$W_{ij},\Delta_{ij}$--
$\int d\omega W_{ij} $--
\noindent
$d<p_i p_j>/dt,d<x_ip_i>/dt $---
$d<\epsilon_i>/dt$
abstract
the intrabeam scattering growth rates
for a bi-gaussian distribution.
The bi-gaussian distribution is interesting for studying the
possibility of using electron cooling in RHIC. Experiments and computer studies
indicate that in the presence of electron cooling, the beam distribution
changes so that it developes a strong core and a long tail which is not described
well by a gaussian, but may be better described by a bi-gaussian.
Being able to compute the effects of intrabeam scattering for a bi-gaussian
distribution would be useful in computing the effects of electron cooling,
which depend critically on the details of the intrabeam scattering.
The calculation is done using the reformulation of intrabeam
scattering theory given in [1] based on the treatments given
by A. Piwinski [2] and J. Bjorken and S.K.
Mtingwa [3].
The bi-gaussian distribution is defined below as the sum of two gaussians in the
particle coordinates $x,y,s,p_x,p_y,p_s$. The gaussian with the smaller dimensions
produces most of the core of the beam, and the gaussian with the larger
dimensions largely produces the long tail of the beam. The final
result for the growth rates are expressed as the sum of three terms which can be
interperted respectively as the contribution to the growth rates due
to the scattering of the particles in the first gaussian from themselves,
the scattering of the particles in the second gaussian from themselves, and
the scattering of the particles in the first gaussian from the
particles in the second gaussian.
This note finds results for the intrabeam scattering growth rates
for a bi-gaussian distribution.
The bi-gaussian distribution is interesting for studying the
possibility of using electron cooling in RHIC. Experiments and computer studies
indicate that in the presence of electron cooling, the beam distribution
changes so that it developes a strong core and a long tail which is not described
well by a gaussian, but may be better described by a bi-gaussian.
Being able to compute the effects of intrabeam scattering for a bi-gaussian
distribution would be useful in computing the effects of electron cooling,
which depend critically on the details of the intrabeam scattering.
The calculation is done using the reformulation of intrabeam scattering theory
given in [1] based on the treatments given by
A. Piwinski [2] and by J. Bjorken and S.
Mtingwa [3].
The bi-gaussian distribution is defined below as the sum of two gaussians in the
particle coordinates $x,y,s,p_x,p_y,p_s$. The gaussian with the smaller dimensions
produces most of the core of the beam, and the gaussian with the larger
dimensions largely produces the long tail of the beam. The final
result for the growth rates are expressed as the sum of three terms which can be
interperted respectively as the contribution to the growth rates due
to the scattering of the particles in the first gaussian from themselves,
the scattering of the particles in the second gaussian from themselves, and
the scattering of the particles in the first gaussian from the
particles in the second gaussian.
\section{Basic results for intrabeam scattering}
This section lists some general results which can be used to find growth rates
for a beam with any particle distribution $f(x.p)$. Following [3],
growth rates will be computed for $<p_{i}p_{j}>$ , where the $<>$ indicates an
average over all the particles in the bunch. From these one can compute the
growth rates for the emittances, $<\epsilon_i>$. A result that holds in any
coordinate system and for any particle distribution $f(x.p)$ is given in [1]
as
\begin{eqnarray}
\delta <(p_{i}p_{j}) > &=& N \int \;\: d^3x \frac {d^3p_1}{\gamma_1}
\frac {d^3p_2}{\gamma_2} f(x,p_1)f(x,p_2) F(p_1,p_2) C_{ij} dt \nonumber\\
C_{ij} &=& \pi \int_{0}^{\pi} d\theta \sigma(\theta) \sin^3 \theta \;
\Delta^2 [\delta_{ij}-3\frac{\Delta_i \Delta_j}{\Delta^2}
+\frac{W_iW_j}{W^2}] \;\; i,j=1,3 \nonumber\\
\Delta_i &=& \frac{1}{2}(p_{1i}-p_{2i}) \nonumber\\
W_i &=& p_{1i}+p_{2i}
\end{eqnarray}
$Nf(x,p)$ gives the number of particles in $d^3xd^3p$, where N is the
number of particles in a bunch. $\delta <(p_{i}p_{j}) >$ is the change in
$<(p_{i}p_{j}) >$ due to all particle collisions in the time interval dt.
The invariants $F(p_1,p_2),\Delta^2,W^2$ are given by
\begin{eqnarray*}
F(p_1,p_2) &=& c \frac {[(p_1p_2)^2-m_1^2m_2^2c^4]^{1/2}}{m_1m_2 c^2} \nonumber\\
F(p_1,p_2) &=& \gamma_1 \gamma_2 c [(\vec{\beta_1}-\vec{\beta_2})^2 -(\vec{\beta_1} \times \vec{\beta_2})^2]^{1/2} \nonumber\\
\Delta^2 &=& \vec{\Delta}^2-\Delta_0^2, \;\:\;\:\Delta_0=(E_1-E_2)/(2c) \nonumber\\
W^2 &=& \vec{W}^2-W_0^2, \;\:\;\:W_0=(E_1+E_2)/c
\end{eqnarray*}
Eq.(1) is considerably simplified by going to the rest CS , which is the CS
moving along with the bunch and the particle motion is non-relativistic, and
putting $\sigma$ equal to the Coulomb cross section. One gets
\begin{eqnarray}
\frac{1}{p_0^2}<\delta (p_{1i}p_{1j}) > &=& N \int \;\: d^3x d^3p_1
d^3p_2 f(x,p_1)f(x,p_2) 2 \bar{\beta}c \; C_{ij}\;dt
\nonumber\\
\Delta_i &=& \frac{1}{2}(p_{1i}-p_{2i}) \nonumber\\
\bar{\beta}c &=& |\vec{\Delta}|/m \nonumber\\
C_{ij} &=& \frac{2 \pi}{p_0^2} (r_0/2 \bar{\beta}^2)^2
\ln(1+(2 \bar{\beta}^2 b_{max}/r_0)^2) \;\; \nonumber\\
& & [|\vec{\Delta}|^2 \delta_{ij}-3\Delta_i \Delta_j ] \;\; i,j=1,3 \nonumber\\
r_0 &=& Z^2e^2/mc^2 \nonumber\\
\sigma(\theta) &=& [\frac{r_0}{2\bar{\beta}^2 }]^2 \frac{1}{(1-\cos \theta)^2} \nonumber\\
\cot (\theta_{min}/2) &=& 2 \bar{\beta}^2 b_{max}/r_0
\end{eqnarray}
$b_{max}$ is the largest allowed impact parameter in the center of mass CS. It has been asumed that
one can replace $\ln(1+(2 \bar{\beta}^2 b_{max}/r_0)-1$ by $\ln(1+(2 \bar{\beta}^2 b_{max}/r_0)$.
In Eq.(1), the original 11-dimensional integral which arises from
intrabeam scattering theory has been reduced in [1] to a 9-dimensional integral by
integrating over all possible scattering angles. In [1] this reduction was
done for any
particle distribution, $f(x,p)$. In [3], Bjorken and Mtingwa first do the
integration over $x,p_1,p_2$ using a simple gaussian distribution
before doing the integration over the scattering angles and no general result for
doing this reduction for any $f(x,p)$ is given.
In [2] Piwinski computes the growth rates for the emittances $<\epsilon_i>$
instead of for $<p_{i}p_{j}>$. A general result for reducing the integral by
integrating over all possible scattering angles, for any $f(x,p)$, for the
growth rates of $<\epsilon_i>$ is given. However, using this result for a
complicated distribution like the bi-gaussian would be difficult.
\section{Gaussian distribution}
We will first consider the case of a gaussian particle distribution. This will
provide a more simple example of using the results in the reformulation given
in [1]
and of the methods used to evaluate the integrals. Afterwards, the same procedures
will be applied to the case of the bi-gaussian distribution.
Let $Nf(x,p)$ gives the number of particles in $d^3xd^3p$, where N is the
number of particles in a bunch.
For a gaussian distribution, $f(x,p)$ ls given by
\begin{eqnarray}
f(x,p)&=&\frac{1}{\Gamma} exp[-S(x,p)] \nonumber\\
\Gamma &=& \int d^3xd^3p \; exp[-S(x,p)]
\end{eqnarray}
\begin{eqnarray}
S &=& S_x+S_y+S_s \nonumber\\
& & \nonumber\\
S_x &=&\frac{1}{\bar{\epsilon_x}} \epsilon_x (x_\beta,x_{\beta}') \nonumber\\
x_\beta &=& x-D(p-p_0)/p_0 \nonumber\\
x_{\beta}' &=& x'-D'(p-p_0)/p_0 \;\;\; x'=p_x/p_0 \nonumber\\
\epsilon_x (x,x') &=& [x^2+(\beta_x x'+\alpha_x x)^2]/\beta_x \nonumber\\
& & \nonumber\\
S_y &=& \frac{1}{\bar{\epsilon_y}} \epsilon_y (y,y')\;\;\; y'=p_y/p_0 \nonumber\\
\epsilon_y (y,y') &=& [y^2+(\beta_y y'+\alpha_y y)^2]/\beta_y \nonumber\\
& & \nonumber\\
S_s &=& \frac{1}{\bar{\epsilon_s}} \epsilon_s (s-s_c,(p-p_0)/p_0) \nonumber\\
\epsilon_s (s-s_c,(p-p_0)/p_0)&=& \frac{(s-s_c)^2}{2\sigma_s^2}+\frac{((p-p_0)/p_0)^2}{2 \sigma_p^2} \nonumber\\
\epsilon_s (s-s_c,(p-p_0)/p_0)&=& \frac{1}{\beta_s} (s-s_c)^2+\beta_s ((p-p_0)/p_0)^2 \nonumber\\
\epsilon_s (s-s_c,(p-p_0)/p_0) &=& [(s-s_c)^2+(\beta_s ((p-p_0)/p_0))^2]/\beta_s \nonumber\\
\beta_s &=& \sigma_s/\sigma_p \nonumber\\
\bar \epsilon_s &=& 2 \sigma_s \sigma_p
\end{eqnarray}
$D$ is the horizontal dispersion. $D'=dD/ds$.
A longitudinal emittance has been introduced
so that the longitudinal motion and the transverse motions can be treated in
a similar manner. $s_c$ locates the center of the bunch.
$\Gamma$ can now be
computed using Eq.(1).This will provide an example how the integrals are done in
this paper. The integration methods used here are somewhat more complicated
than those used in [3] but they will also work for the more complicated
bi-gaussian distribution.
\begin{eqnarray*}
\Gamma &=& \int d^3xd^3p \; exp[-S_x-S_y-S_s]
\end{eqnarray*}
Writing $\Gamma$ as $\Gamma=\Gamma_y \Gamma_{xs}$ and computing $\Gamma_y$
first because this part is simpler,
\begin{eqnarray}
\Gamma_y &=& \int dydp_y \; exp[-S_y] \nonumber\\
S_y &=& \frac{1}{\bar{\epsilon_y}} \epsilon_y (y,y') \;\;\; y'=p_y/p_0 \nonumber\\
\epsilon_y (y,y') &=& [y^2+2 +(\beta_y y'+\alpha_y y)^2]/\beta_y \nonumber\\
\eta_y&=& y/\sqrt{\beta_y}, \;\;\; p_{\eta y}= (\beta_y y'+\alpha_y y)/\sqrt{\beta_y}
\nonumber\\
dy dp_y &=& p_0 d\eta_y dp_{\eta y} \nonumber\\
\Gamma_y &=& p_0 \int d\eta_y dp_{\eta y} \; exp[-(\eta_y^2+p_{\eta y}^2)/\bar{\epsilon_y}] \nonumber\\
\Gamma_y &=& \pi \bar{\epsilon_y} p_0
\end{eqnarray}
Now for the remaining integral we have
\begin{eqnarray}
\Gamma_{xs} &=& \int dxdp_x dsdp_s\; exp[-S_x-S_s] \nonumber\\
\Gamma_{xs} &=& \int dsdp_s\; exp[-S_s] \int dxdp_x \; exp[-S_x] \nonumber\\
\mbox{Make the transformation} & & \nonumber\\
x_{\beta} &=& x-D(p-p_0)/p_0 \nonumber\\
x'_{\beta }&=& x' -D'(p-p_0)/p_0 \nonumber\\
x' &=& p_x/p_0, \;\;\; x'_{\beta }=p_{\beta x}/p_0 \nonumber\\
dxdp_x&=&p_0 dx_{\beta}dx_{\beta }' \nonumber\\
& & \nonumber\\
\int dxdp_x \; exp[-S_x] &=& p_0 \int dx_{\beta}dx_{\beta}' \; exp[-S_x] \nonumber\\
S_x &=&\frac{1}{\bar{\epsilon_x}} \epsilon_x (x_\beta,x_{\beta}') \nonumber\\
\int dxdp_x \; exp[-S_x] &=& \pi \bar{\epsilon_x} p_0 \mbox{ as in evaluating $\Gamma_y$ } \nonumber\\
& & \nonumber\\
\mbox{ $p\sim p_s$ in the Lab. CS and} & & \nonumber\\
\Gamma_{xs} &=& \pi^2 \bar{\epsilon_s} \bar{\epsilon_x} p_0^2 \nonumber\\
& & \nonumber\\
\Gamma &=& \pi^3 \bar{\epsilon_s} \bar{\epsilon_x} \bar{\epsilon_y} p_0^3
\end{eqnarray}
\section{Growth rates for a Gaussian distribution}
In the following,the growth rates are given in the Rest Coordinate System,
which is the coordinate system moving along with the bunch.
Growth rates are given for $<p_i p_j>$. From these one can compute the
growth rates for $<\epsilon_i>$. Using the general result, Eq.(2), one gets
\begin{eqnarray}
\frac{1}{p_0^2}<\delta (p_{i}p_{j}) > &=& \frac {N}{\Gamma^2} \int \;\: d^3x d^3p_1
d^3p_2 exp[-S(x,p_1)-S(x,p_2)] 2 \bar{\beta}c \; C_{ij}\;dt
\nonumber\\
\vec{\Delta} &=& \frac{1}{2}(\vec{p_{1}}-\vec{p_{2}}) \nonumber\\
\bar{\beta}c &=& |\vec{\Delta}|/m \nonumber\\
C_{ij} &=& \frac{2 \pi}{p_0^2} (r_0/2 \bar{\beta}^2)^2
\ln(1+(2 \bar{\beta}^2 b_{max}/r_0)^2) \;\;
[|\vec{\Delta}|^2 \delta_{ij}-3\Delta_i \Delta_j
] \;\; i,j=1,3 \nonumber\\
r_0 &=& Z^2e^2/mc^2 \nonumber\\
\Gamma &=& \pi^3 \bar{\epsilon_s} \bar{\epsilon_x} \bar{\epsilon_y} p_0^3
\end{eqnarray}
Transform to $W,\Delta$
\begin{eqnarray}
p_1 &=& \frac{W}{2}+\Delta, \;\;\;\;\;\;\;\; p_2=\frac{W}{2}-\Delta \nonumber\\
W &=& p_1+p_2, \;\;\;\;\;\;\;\:\;\;\;\; \Delta=\frac {p_1-p_2}{2} \nonumber\\
d^3p_1 d^3p_2 &=& d^3Wd^3\Delta
\end{eqnarray}
We will first do the integral over $d^3x$ and over $d^3W$.
For the y part of the integral
\begin{eqnarray*}
S_y(y,p_{1y}) &=& \frac{1}{\bar{\epsilon_y}} \epsilon_y (y,y_1'),\;\;\; y_1'=p_{1y}/p_0 \nonumber\\
\epsilon_y (y,y_1') &=& [y^2+(\beta_y y_1'+\alpha_y y)^2]/\beta_y \nonumber\\
S_y(y,p_{1y}) &=& [y^2+(\beta_y (\frac{W_y}{2}+\Delta_y)/p_0 +\alpha_y y)^2]/(\beta_y \bar{\epsilon_y} )
\end{eqnarray*}
\begin{eqnarray}
S_y(y,p_{1y})+S_y(y,p_{2y})&=&(2y^2/\beta_y+2(\beta_y (W_y/p_0)/2+\alpha_y y)^2/\beta_y \nonumber\\
& & +2\beta_y^2 (\Delta_y/p_0)^2/\beta_y)) / \bar{\epsilon_y} \nonumber\\
\mbox{Make the transformation } & & \nonumber\\
\eta_y &=& \sqrt{2}y/\sqrt{\beta_y}, \;\;\;\;\;\; p_{\eta y}=\sqrt{2} (\beta_y(W_y/p_0)/2+\alpha_y y) /\sqrt{\beta_y} \nonumber\\
dydW_y &=& p_0 d\eta_y dp_{\eta y}
\end{eqnarray}
Integrate over $dy,dW_y$
\begin{eqnarray}
\int dy dW_y exp[-S_y(y,p_{1y})-S_y(y,p_{2y})]&=& p_0 \int d\eta_y dp_{\eta y}
\nonumber\\
& & exp[-\frac {\eta_y^2+p_{\eta y}^2+2\beta_y^2 (\Delta_y/p_0)^2/\beta_y } {\bar{\epsilon_y}}] \nonumber\\
&=& p_0 \pi \bar{\epsilon_y} exp[-\frac{2 \beta_y}{\bar{\epsilon_y}}
(\Delta_y/p_0)^2] \nonumber\\
&=& p_0 \pi \bar{\epsilon_y} sxp[-R_y] \nonumber\\
& & \nonumber\\
R_y &=& \frac{2 \beta_y}{\bar{\epsilon_y}}(\Delta_y/p_0)^2
\end{eqnarray}
In doing the remainder of the integral, the integral over $dxdW_xdsdW_s$ we will
do the integral over $dxdW_x$ first and then the integral over $dsdW_s$. Note
that the integral is being done in the Rest CS and in the expression for $S_x$
one has to replace $p-p_0 \sim p_s-p_0$ in the Lab CS by $\gamma p_s$ in the Rest
CS. Remember also that $f(x,p)$ is an invariant (see [1])
One finds for $S_x(x,p_{1x})$
\begin{eqnarray}
S_x(x,p_{1x}) &=& \{ [x-\gamma D\bar{W}_s/2-\gamma D \bar{\Delta}_s]^2+
[ \beta_x (\bar{W}_x/2+\bar{\Delta}_x-\gamma D'\bar{W}_s/2-\gamma D'
\bar{\Delta}_s) + \nonumber\\
& & \alpha_x (x-\gamma D\bar{W}_s/2-\gamma D \bar{\Delta}_s) ]^2 \} /
(\beta_x \bar{\epsilon_x} ) \nonumber\\
& & \nonumber\\
\bar{W}_i &=& W_i/p_0 \;\;\;\;\;\;\bar{\Delta}_i=\Delta/p_0 \nonumber\\
& & \nonumber\\
S_x(y,p_{1x}) &=& \{ [x-\gamma D\bar{W}_s/2-\gamma D \bar{\Delta}_s]^2+
[ \beta_x (\bar{W}_x/2- \gamma D'\bar{W}_s/2) +\nonumber\\
& & \alpha_x (x-\gamma D\bar{W}_s/2)+
(\beta_x\bar{\Delta}_x-\gamma \bar{D} \bar{\Delta}_s) ]^2 \} /
(\beta_x \bar{\epsilon_x} ) \nonumber\\
\bar{D} &=& \beta_x D'+\alpha_x D
\end{eqnarray}
we then find for $S_x(x,p_{1x})+S_x(x,p_{2x})$
\begin{eqnarray}
S_x(x,p_{1x})+S_x(x,p_{2x}) &=& \{ 2[x-\gamma D\bar{W}_s/2]^2+2 \gamma^2 D^2\bar{\Delta}_s^2+ \nonumber\\
& & 2[ \beta_x (\bar{W}_x/2-\gamma D'\bar{W}_s/2) +
\alpha_x (x-\gamma D\bar{W}_s/2) ]^2+ \nonumber\\
& & 2[\beta_x\bar{\Delta}_x-\gamma \bar{D} \bar{\Delta}_s ]^2 \} /
(\beta_x \bar{\epsilon_x} )
\end{eqnarray}
Now make the transformations
\begin{eqnarray}
x^* &=& \sqrt{2}x-\gamma D \Wb_s/\sqrt{2} \;\;\;\;\;\; p_x^*=\Wb_x/\sqrt{2}-\gamma D'\Wb_s/\sqrt{2} \nonumber\\
\eta_x &=& x^*/\sqrt{\beta_x}\;\;\;\;\;p_{\eta_x x}=(\beta_x p_x^*+\alpha_x x^*)/
\sqrt{\beta_x} \nonumber\\
dxdW_x &=& p_0 dx^*dp_x^*=p_0 d\eta_x dp_{\eta_x x}
\end{eqnarray}
Doing the integral over $dxdW_x$ one finds
\begin{eqnarray}
\int dx dW_x exp[-S_x(x,p_{1x})-S_x(x,p_{2x})]&=&
p_0 \int d\eta_x dp_{\eta_x x} \nonumber\\
& & exp[- \{ \eta_x^2+p_{\eta_x}^2+ \nonumber\\
& & 2 [\gamma^2 D^2 \bar{\Delta}_s^2+(\beta_x \bar{\Delta}_x-\gamma \bar{D} \bar{\Delta}_s)^2]
/ \beta_x )\} / \bar{\epsilon_x} ] \nonumber\\
&=& p_0 \pi \bar{\epsilon_x} exp[-R_x] \nonumber\\
& & \nonumber\\
R_x &=& 2 [\gamma^2 D^2\bar{\Delta}_s^2+(\beta_x\bar{\Delta}_x-\gamma \bar{D} \bar{\Delta}_s)^2]
/ (\beta_x \bar{\epsilon_x} ) \nonumber\\
\end{eqnarray}
Now do the integral over $dsdW_s$. One may note that the form of the intgral here
is similar to the integral done over $dydW_y$. The result is then the same with
the proper sustitutions of $s$ for $y$.
\begin{eqnarray}
\int dx dW_s exp[-S_s(s,p_{1s})-S_s(s,p_{2s})]&=& p_0 \pi \bar{\epsilon}_s
exp[-R_s] \nonumber\\
& & \nonumber\\
R_s &=& \frac{2 \gamma^2 \beta_s}{\bar{\epsilon_s}}(\Delta_s/p_0)^2
\end{eqnarray}
Note that the term $\beta_s ((p-p_0)/p_0)^2$ in $S_s$ in the Lab. CS
has to be replaced by $\gamma^2 \beta_s (p_s/p_0)^2$ in the Rest CS.
Using Eq.(7), one gets the result for the growth rates in the Rest CS
for a gauusian distribution.
\begin{eqnarray}
\frac{1}{p_0^2} \frac{d} {dt}<p_i p_j> &=& \frac{N}{\Gamma} \int d^3\Delta \;
exp[-R] C_{ij} \nonumber\\
C_{ij}&=& \frac{2 \pi}{p_0^2} (r_0/2\bar{\beta}^2)^2
(|\Delta|^2 \delta_{ij}-3 \Delta_i \Delta_j ) 2\bar{\beta}c \;\;
ln[1+(2\bar{\beta}^2 b_{max}/r_0)^2] \nonumber\\
\bar{\beta} &=& \beta_0 \gamma_0|\Delta/p_0| \nonumber\\
& & \nonumber\\
r_0 &=& Z^2e^2/Mc^2 \nonumber\\
\Gamma &=& \pi^3 \bar{\epsilon_s} \bar{\epsilon_x} \bar{\epsilon_y} p_0^3 \nonumber\\
& & \nonumber\\
R &=& R_x+R_y+R_s \nonumber\\
R_x &=& \frac{2}{\beta_x \bar{\epsilon_x}} [\gamma^2 D^2 \Delta_s^2 +
(\beta_x \Delta_x-\gamma \tilde{D} \Delta_s)^2 ]/p_0^2 \nonumber\\
\tilde{D} &=& \beta_x D'+\alpha_x D \nonumber\\
R_y &=& \frac{2\beta_y}{ \bar{\epsilon_y}} \Delta_y^2/p_0^2 \nonumber\\
R_s &=& \frac{2\beta_s}{ \bar{\epsilon_s}} \gamma^2 \Delta_s^2/p_0^2
\end{eqnarray}
The integral over $d^3 \Delta$ is an integral over all possible values
of the relative momemtum for any two particles in a bunch. $\beta_0,\gamma_0$
are the beta and gamma corresponding to $p_0$, the central momemtum of the
bunch in the Laboratory Coordinate System. $\gamma=\gamma_0$
The above 3-dimensional integral can be reduced to a 2-dimensional integral
by integrating over $|\Delta|$ and using $d^3\Delta=|\Delta|^2 d|\Delta|
sin\theta d\theta d\phi$. This gives
\begin{eqnarray}
\frac{1}{p_0^2} \frac{d} {dt}<p_i p_j> &=& \frac{N}{\Gamma}
2 \pi p_0^3 \left( \frac{r_0}{2\gamma_0^2 \beta_0^2}\right)^2 2\beta_0 \gamma_0c
\int sin\theta d\theta d\phi \; (\delta_{ij}-3 g_ig_j) \nonumber\\
& & \frac{1}{F} ln\left[\frac{\hat{C}}{F}\right] \nonumber\\
g_3&=&cos\theta=g_s \nonumber\\
g_1&=&sin\theta cos\phi=g_x \nonumber\\
g_2&=&sin\theta sin\phi=g_y \nonumber\\
\hat{C}&=&2 \gamma_0^2 \beta_0^2 b_{max}/r_0 \nonumber\\
& & \nonumber\\
F&=&R/(|\Delta|/p_0)^2 \nonumber\\
F &=& F_x+F_y+F_s \nonumber\\
F_x &=& \frac{2}{\beta_x \bar{\epsilon_x}} [\gamma^2 D^2 g_s^2 +
(\beta_x g_x-\gamma \bar{D} g_s)^2 ] \nonumber\\
F_y &=& \frac{2}{ \bar{\epsilon_y}} \beta_y g_y^2 \nonumber\\
F_s &=& \frac{2}{\bar{\epsilon_s}} \beta_s \gamma^2 g_s^2
\end{eqnarray}
In obtaining the above, one uses $z=|\Delb |^2,dz=2|\Delb | d|\Delb |$ and
\[\int_ {0}^{\infty} dz \;\; exp[-Fz] ln[\hat{C}z]=\frac {1}{F}[ln\left[\frac{\hat{C}}{F}\right]-.5772] \]
For $Z=80,A=200,\gamma=100,b_{max}=1 cm$, $\log_{10} \hat{C}=18.6$
\section{Bi-Gaussian distribution}
The bi-gaussian distribution will be assumed to have the form given
by the following.
$Nf(x,p)$ gives the number of particles in $d^3xd^3p$, where N is the
number of particles in a bunch.
For a bi-gaussian distribution, $f(x,p)$ ls given by
\begin{eqnarray}
f(x,p) &=& \frac{N_a}{N}\frac{1}{\Gamma_a} exp[-S_a(x,p)]+
\frac{N_b}{N}\frac{1}{\Gamma_b} exp[-S_b(x,p)] \nonumber\\
\Gamma_a &=& \pi^3 \bar{\epsilon_{sa}} \bar{\epsilon_{xa}} \bar{\epsilon_{ya}} p_0^3 \nonumber\\
\Gamma_b &=& \pi^3 \bar{\epsilon_{sb}} \bar{\epsilon_{xb}} \bar{\epsilon_{yb}} p_0^3
\end{eqnarray}
In the first gaussian,to find $\Gamma_a,S_a$ then in the expressions
for $\Gamma,S$, given above for the gaussian distribution, replace
$\bar{\epsilon_x},\bar{\epsilon_y},\bar{\epsilon_s}$ by $\bar{\epsilon_{xa}},
\bar{\epsilon_{ya}},\bar{\epsilon_{sa}}$.
In the second gaussian, in the expressions for $\Gamma,S$, replace
$\bar{\epsilon_x},\bar{\epsilon_y},\bar{\epsilon_s}$ by $\bar{\epsilon_{xb}},
\bar{\epsilon_{yb}},\bar{\epsilon_{sb}}$.
In addition. $N_a+N_b=N$. This bi-gaussian has 7 parameters instead of
the three parameters of a gaussian.
\section{Growth rates for a Bi- Gaussian distribution}
In the following,the growth rates are given in the Rest Coordinate System,
which is the coordinate system moving along with the bunch.
Growth rates are given for $<p_i p_j>$. From these one can compute the
growth rates for $<\epsilon_i>$.Starting with Eq.2 and using the $f(x,p)$
from Eq.18, one gets
\begin{eqnarray}
& & \nonumber\\
\frac{1}{p_0^2}<\delta (p_{i}p_{j}) > &=& \int \;\: d^3x d^3p_1 d^3p_2
\left [ \frac{N_a}{N}\frac{1}{\Gamma_a} exp[-S_a(x,p_1)]+
\frac{N_b}{N}\frac{1}{\Gamma_b} exp[-S_b(x,p_1)] \right ] \nonumber\\
& & \left [ \frac{N_a}{N}\frac{1}{\Gamma_a} exp[-S_a(x,p_2)]+
\frac{N_b}{N}\frac{1}{\Gamma_b} exp[-S_b(x,p_2)] \right ] \nonumber\\
& & 2 \bar{\beta}c \; C_{ij}\;dt
\nonumber\\
\vec{\Delta} &=& \frac{1}{2}(\vec{p_{1}}-\vec{p_{2}}) \nonumber\\
\bar{\beta}c &=& |\vec{\Delta}|/m \nonumber\\
C_{ij} &=& \frac{2 \pi}{p_0^2} (r_0/2 \bar{\beta}^2)^2
\ln(1+(2 \bar{\beta}^2 b_{max}/r_0)^2) \;\;
[|\vec{\Delta}|^2 \delta_{ij}-3\Delta_i \Delta_j
] \;\; i,j=1,3 \nonumber\\
r_0 &=& Z^2e^2/mc^2
\end{eqnarray}
The term in the integrand which contains $exp[-S_a(x,p_1)-S_a(x,p_2)]$ is similar
to the integrand for the gaussian distribution except that $\bar{\epsilon}_{i}$
are replaced by $\bar{\epsilon}_{ia}$ and leads to the same result as that
given by Eq.(16) for the gaussian beam except that $R$ has to be replaced by $R_a$
where $R_a$ is obtained from $R$ by replacing $\bar{\epsilon}_{i}$ by
$\bar{\epsilon}_{ia}$. The term containing $exp[-S_b(x,p_1)-S_b(x,p_2)]$ can be
evaluated in the same way leading to the same result as that
given by Eq.(16) for the gaussian beam except that $R$ has to be replaced by $R_b$
where $R_b$ is obtained from $R$ by replacing $\bar{\epsilon}_{i}$ by
$\bar{\epsilon}_{ib}$. The only terms that need further evaluation are the the two
cross product terms. The two cross product terms are equal because of the symmetry
of $p_1$ and $p_2$ in the rest of the integrand. This leads to the remaining
integral to be evaluated
\[ \int \;\: d^3x d^3p_1 d^3p_2
\frac{2N_aN_b}{N^2}\frac{1}{\Gamma_a \Gamma_b}
exp[-S_a(x,p_1)-S_b(x,p_2)] \; 2 \bar{\beta}c \; C_{ij} \]
In evaluating this integral, we will use the same procedure as was used for the
gaussian distribution. We will first transform to $W,\Delta$ from $p_1,p_2$
(see Eq.(8). We will then do the integral over $d^3x$ and over $d^3W$.
For the y part of the integral one finds ,
\begin{eqnarray}
S_{ya}(y,p_{1y}) &=& \{ y^2+[\beta_y (\Wb_y/2+\Delb_y) +\alpha_y y ]^2 \}
/(\beta_y \bar{\epsilon}_{ya} ) \nonumber\\
\Wb_y &=& W_y /p_0 \;\;\;\;\;\; \Delb_y =\Delta_y /p_0
\end{eqnarray}
One then finds that
\begin{eqnarray}
& & \nonumber\\
& & \nonumber\\
S_{ya}(y,p_{1y})+S_{yb}(y,p_{2y})&=&\{ 2y^2/\beta_y +2[\beta_y (\Wb_y/2+\alpha_y y) ]^2/\beta_y \nonumber\\
& & +2\beta_y^2 \Delb_y^2/\beta_y) \} / \bar{\epsilon}_{yc}+ \nonumber\\
& & \{ 4 (\beta_y \Delb_y) (\beta_y \Wb_y/2+\alpha_y y)/\beta_y \} / \bar{\epsilon}_{yd} \nonumber\\
\frac{1}{\bar{\epsilon}_{yc}} &=& \frac{1}{2} (\frac{1}{\bar{\epsilon}_{ya}}+
\frac{1}{\bar{\epsilon}_{yb} } ) \nonumber\\
\frac{1}{\bar{\epsilon}_{yd}} &=& \frac{1}{2} (\frac{1}{\bar{\epsilon}_{ya}}-
\frac{1}{\bar{\epsilon}_{yb}} ) \nonumber\\
& &
\end{eqnarray}
Make the transformation
\begin{eqnarray}
\eta_y &=& \sqrt{2}y/\sqrt{\beta_y}, \;\;\;\;\;\; p_{\eta y}=\sqrt{2} (\beta_y \Wb_y/2+\alpha_y y) /\sqrt{\beta_y} \nonumber\\
dydW_y &=& p_0 d\eta_y dp_{\eta y}
\end{eqnarray}
Integrate over $dy,dW_y$
\begin{eqnarray}
\int dy dW_y ecp[-S_{ya}(y,p_{1y})-S_{yb}(y,p_{2y})] &=& p_0 \int d\eta_y dp_{\eta y}
\nonumber\\
& & exp[-\frac {\eta_y^2+p_{\eta y}^2+2\beta_y^2 (\Delta_y/p_0)^2/\beta_y } {\bar{\epsilon}_{yc}}+ \nonumber\\
& & 4 \beta_y \Delb_y \frac{p_{\eta y}/(\sqrt{2} \sqrt{\beta_y})} { \bar{\epsilon}_{yd} }] \nonumber\\
&=& p_0 \pi \bar{\epsilon_{yc}} exp[-\frac{2 \beta_y}{\bar{\epsilon}_{yc}}
\Delb_y^2 +\frac{2 \beta_y} {\bar{\epsilon}_{yd}^2/\bar{\epsilon}_{yc}} \Delb_y^2] \nonumber\\
&=& p_0 \pi \bar{\epsilon}_{yc} exp[-R_{yc}+R_{yd}] \nonumber\\
& & \nonumber\\
R_{yc} &=& \frac{2 \beta_y} {\bar{\epsilon}_{yc}}\Delb_y^2 \nonumber\\
R_{yd} &=& \frac{2 \beta_y} {\bar{\epsilon}_{yd}^2/\bar{\epsilon}_{yc}} \Delb_y^2
\end{eqnarray}
The exponent $R_{yc}-R_{yd}$ has to be positive. This can be made more obvious by noting that
\[ \frac {1}{\bar{\epsilon}_{yc}^2}-\frac{1}{\bar{\epsilon}_{yd}^2}= \frac{1}{\bar{\epsilon}_{ya}\bar{\epsilon}_{yb}} \]
In doing the remainder of the integral, the integral over $dxdW_xdsdW_s$ we will
do the integral over $dxdW_x$ first and then the integral over $dsdW_s$. Note
that the integral is being done in the Rest CS and in the expression for $S_x$
one has to replace $p-p_0 \sim p_s-p_0$ in the Lab. CS by $\gamma p_s$ in the Rest
CS. Remember also that $f(x,p)$ is an invariant (see [1])
One finds for $S_{xa}(x,p_{1x})$
\begin{eqnarray}
S_{xa}(x,p_{1x}) &=& \{ [x-\gamma D\bar{W}_s/2-\gamma D \bar{\Delta}_s]^2+
[\beta_x (\bar{W}_x/2+\bar{\Delta}_x-\gamma D'\bar{W}_s/2-\gamma D'
\bar{\Delta}_s) + \nonumber\\
& & \alpha_x (x-\gamma D\bar{W}_s/2-\gamma D \bar{\Delta}_s) ]^2 \}/
(\beta_x \bar{\epsilon_xa} ) \nonumber\\
& & \nonumber\\
\bar{W}_i &=& W_i/p_0 \;\;\;\;\;\;\bar{\Delta}_i=\Delta/p_0 \nonumber\\
& & \nonumber\\
S_{xa}(y,p_{1x}) &=& \{ [x-\gamma D\bar{W}_s/2-\gamma D \bar{\Delta}_s]^2+
[ \beta_x (\bar{W}_x/2- \gamma D'\bar{W}_s/2) +\nonumber\\
& & \alpha_x (x-\gamma D\bar{W}_s/2)+
(\beta_x\bar{\Delta}_x-\gamma \bar{D} \bar{\Delta}_s) ]^2 \} /
(\beta_x \bar{\epsilon_{xa}} ) \nonumber\\
\bar{D} &=& \beta_x D'+\alpha_x D
\end{eqnarray}
we then find for $S_{xa}(x,p_{1x})+S_{xb}(x,p_{2x})$
\begin{eqnarray}
S_{xa}(x,p_{1x})+S_{xb}(x,p_{2x}) &=& \{2[x-\gamma D\bar{W}_s/2]^2+2 \gamma^2 D^2\bar{\Delta}_s^2+ \nonumber\\
& & 2[ \beta_x (\bar{W}_x/2-\gamma D'\bar{W}_s/2) +
\alpha_x (x-\gamma D\bar{W}_s/2) ]^2+ \nonumber\\
& & 2[\beta_x\bar{\Delta}_x-\gamma \bar{D} \bar{\Delta}_s]^2 \} /
(\beta_x \bar{\epsilon}_{xc} ) + \nonumber\\
& & \{ -4 \gamma D \bar{\Delta}_s [x-\gamma D\bar{W}_s/2]/\beta_x \nonumber\\
& & +4 (\beta_x\bar{\Delta}_x-\gamma \bar{D} \bar{\Delta}_s) [ \beta_x (\bar{W}_x/2- \gamma D'\bar{W}_s/2) + \nonumber\\
& & \alpha_x (x-\gamma D\bar{W}_s/2) ] /\beta_x \} /\bar{\epsilon}_{xd} \nonumber\\
& &
\end{eqnarray}
Now make the transformations
\begin{eqnarray}
x^* &=& \sqrt{2}x-\gamma D \bar{W}_s/\sqrt{2} \;\;\;\;\;\; p_x^*=\bar{W}_x/\sqrt{2}-\gamma D'\bar{W}_s/\sqrt{2} \nonumber\\
\eta_x &=& x^*/\sqrt{\beta_x}\;\;\;\;\;p_{\eta_x x}=(\beta_x p_x^*+\alpha_x x^*)/
\sqrt{\beta_x} \nonumber\\
dxdW_x &=& p_0 dx^*dp_x^*=p_0 d\eta_x dp_{\eta_x x}
\end{eqnarray}
Doing the integral over $dxdW_x$ one finds
\begin{eqnarray}
\int dx dW_x exp[-S_{xa}(x,p_{1x})-S_{xb}(x,p_{2x})]&=&
p_0 \int d\eta_x dp_{\eta_x x} \nonumber\\
& & exp[-\{ \eta_x^2+p_{\eta_x}^2 \nonumber\\
& & +2[\gamma^2 D^2\bar{\Delta}_s^2+(\beta_x\bar{\Delta}_x-\gamma \bar{D} \bar{\Delta}_s)^2 ]/\beta_x \} /
\bar{\epsilon}_{xc} \nonumber\\
& & \nonumber\\
& & +\{-4 \gamma D \Delb_s \eta_x/\sqrt{2 \beta_x} \nonumber\\
& & +4 (\beta_x\bar{\Delta}_x-\gamma \bar{D} \bar{\Delta}_s) p_{\eta_x x}/\sqrt{2 \beta_x} \} /\bar{\epsilon}_{xd} ] \nonumber\\
& & \nonumber\\
&=& p_0 \pi \bar{\epsilon_{xc}} exp[-R_{xc}+R_{xd}] \nonumber\\
& & \nonumber\\
R_{xc} &=& 2[\gamma^2 D^2\bar{\Delta}_s^2+(\beta_x\bar{\Delta}_x-\gamma \bar{D} \bar{\Delta}_s)^2 ] / (\beta_x \bar{\epsilon}_{xc} ) \nonumber\\
R_{xd} &=&
2 \{ [-\gamma D \Delb_s ]^2 \nonumber\\
& & +[(\beta_x\bar{\Delta}_x-\gamma \bar{D} \bar{\Delta}_s) ]^2 \} \nonumber\\
& & /(\beta_x \bar{\epsilon}_{xd}^2/ \bar{\epsilon}_{xc}) \nonumber\\
& &
\end{eqnarray}
Now do the integral over $dsdW_s$. One may note that the form of the intgral here
is similar to the integral done over $dydW_y$. The result is then the same with
the proper sustitutions of $s$ for $y$.
\begin{eqnarray}
\int dx dW_s exp[-S_{sa}(s,p_{1s})-S_{sb}(s,p_{2s})]&=& p_0 \pi
\bar{\epsilon}_{sc} exp[-R_{sc}+R_{sd}] \nonumber\\
& & \nonumber\\
R_{sc} &=& \frac{2 \beta_s} {\bar{\epsilon}_{sc}}\Delb_s^2 \nonumber\\
R_{sd} &=& \frac{2 \beta_s} {\bar{\epsilon}_{sd}^2/\bar{\epsilon}_{sc}} \Delb_s^2
\end{eqnarray}
Note that the term $\beta_s ((p-p_0)/p_0)^2$ in $S_s$ in the Lab. CS
has to be replaced by $\gamma^2 \beta_s (p_s/p_0)^2$ in the Rest CS.
Putting all the above results, for the bi-gaussian distribution, together one gets the final result
\begin{eqnarray}
\frac{1}{p_0^2} \frac{d} {dt}<p_i p_j> &=& N \int d^3\Delta \;
C_{ij} [ \left(\frac{N_a}{N}\right)^2 \frac{exp(-R_a)}{\Gamma_a}+
\left(\frac{N_b}{N}\right)^2 \frac{exp(-R_b)}{\Gamma_b} \nonumber\\
& & +2\frac{N_a N_b}{N^2} \frac{\Gamma_c}{\Gamma_a \Gamma_b} exp(-T) ] \nonumber\\
& & \nonumber\\
C_{ij}&=& \frac{2 \pi}{p_0^2} (r_0/2\bar{\beta}^2)^2
(|\Delta|^2 \delta_{ij}-3 \Delta_i \Delta_j ) 2\bar{\beta}c \;\;
ln[1+(2\bar{\beta}^2 b_{max}/r_0)^2] \nonumber\\
\bar{\beta} &=& \beta_0 \gamma_0|\Delta/p_0| \nonumber\\
r_0 &=& Z^2e^2/Mc^2 \nonumber\\
& & \nonumber\\
\frac{1}{\bar{\epsilon_{ic}}}&=& \frac{1}{2}\left(\frac{1}{\bar{\epsilon_{ia}}}+
\frac{1}{\bar{\epsilon_{ib}}} \right) \;\;i=x,y,s \nonumber\\
\frac{1}{\bar{\epsilon}_{id}} &=& \frac{1}{2} (\frac{1}{\bar{\epsilon}_{ia}}-
\frac{1}{\bar{\epsilon}_{ib}} ) \nonumber\\
& & \nonumber\\
r_0 &=& Z^2e^2/Mc^2 \nonumber\\
\Gamma_a &=& \pi^3 \bar{\epsilon_{sa}} \bar{\epsilon_{xa}} \bar{\epsilon_{ya}} p_0^3 \nonumber\\
& & \nonumber\\
R_a &=& R_{xa}+R_{ya}+R_{sa} \nonumber\\
R_{xa} &=& \frac{2}{\beta_x \bar{\epsilon_{xa}}} [\gamma^2 D^2 \Delta_s^2 +
(\beta_x \Delta_x-\gamma \tilde{D} \Delta_s)^2 ]/p_0^2 \nonumber\\
\tilde{D} &=& \beta_x D'+\alpha_x D \nonumber\\
R_{ya} &=& \frac{2}{\beta_y \bar{\epsilon_{ya}}} \beta_y^2 \Delta_y^2/p_0^2 \nonumber\\
R_{sa} &=& \frac{2}{\beta_s \bar{\epsilon_{sa}}} \beta_s^2 \gamma^2 \Delta_s^2/p_0^2 \nonumber\\
& & \nonumber\\
T&=& T_x+T_y+T_s \nonumber\\
T_x&=& R_{xc}- R_{xd} \nonumber\\
T_y&=& R_{yc}- R_{yd} \nonumber\\
T_s&=& R_{sc}-R_{sd} \nonumber\\
& & \nonumber\\
& & \nonumber\\
R_{xd} &=&
2 \{ [-\gamma D \Delb_s ]^2 \nonumber\\
& & +[(\beta_x\bar{\Delta}_x-\gamma \bar{D} \bar{\Delta}_s) ]^2 \} \nonumber\\
& & /(\beta_x \bar{\epsilon}_{xd}^2/ \bar{\epsilon}_{xc}) \nonumber\\
R_{yd} &=& \frac{2 \beta_y} {\bar{\epsilon}_{yd}^2/\bar{\epsilon}_{yc}} \Delb_y^2 \nonumber\\
R_{sd} &=& \frac{2 \beta_s} {\bar{\epsilon}_{sd}^2/\bar{\epsilon}_{sc}} \Delb_s^2 \nonumber\\
\Delb_i &=& \Delta_i/p_0
\end{eqnarray}
$R_a,R_b,R_c$ are each the same as $R_a$ given above
except that $\bar{\epsilon_{ia}}$ are replaced by
$\bar{\epsilon_{ia}},\bar{\epsilon_{ib}},\bar{\epsilon_{ic}}$ respectively.
The same remarks apply to $\Gamma_a,\Gamma_b,\Gamma_c$
The above 3-dimensional integral can be reduced to a 2-dimensional integral
by integrating over $|\Delta|$ and using $d^3\Delta=|\Delta|^2 d|\Delta|
sin\theta d\theta d\phi$. This gives
\begin{eqnarray}
\frac{1}{p_0^2} \frac{d} {dt}<p_i p_j> &=& 2 \pi p_0^3 \left( \frac{r_0}
{2\gamma_0^2 \beta_0^2}\right)^2 2\beta_0 \gamma_0c
\int sin\theta d\theta d\phi \; (\delta_{ij}-3 g_ig_j) \nonumber\\
& &N [ \left(\frac{N_a}{N}\right)^2 \frac{1}{\Gamma_a F_a} ln[\frac{\hat{C}}{F_a}]
+\left(\frac{N_b}{N}\right)^2 \frac{1}{\Gamma_b F_b} ln[\frac{\hat{C}}{F_b}] \nonumber\\
& & +2 \frac{N_aN_b}{N^2} \frac{\Gamma_c}{\Gamma_a\Gamma_b}
\frac{1}{G} ln[\frac{\hat{C}}{G} ] ] \nonumber\\
& & \nonumber\\
g_3&=&cos\theta=g_s \nonumber\\
g_1&=&sin\theta cos\phi=g_x \nonumber\\
g_2&=&sin\theta sin\phi=g_y \nonumber\\
\hat{C}&=&2 \gamma_0^2 \beta_0^2 b_{max}/r_0 \nonumber\\
& & \nonumber\\
F_i &=& R_i/(|\Delta|/p_0)^2 \;\;\;\;i=a,b,c \nonumber\\
G&=&T/(|\Delta|/p_0)^2 \nonumber\\
\end{eqnarray}
$F_a,F_b,F_c$ are each the same F that was defined for the Gaussian
distribution except that the $\bar{\epsilon_{i}}$ are replaced by
$\bar{\epsilon_{ia}},\bar{\epsilon_{ib}},\bar{\epsilon_{ic}}$ respectively.
The above results for the growth rates for a bi-gaussian distribution
are expressed as an integral which contains 3 terms, each of which is similar
to the one term in the results for the gaussian distribution.
These three terms may be given a simple interpertation. The first term represents
the contribution to the growth rates due to the scattering of the $N_a$ particles
of the first gaussian from themselves, the seond term the contribution
due to the scattering of the $N_b$ particles of the
second gaussian from themselves,
and the third term the contribution due to the scattering of the $N_a$ particles
of the first gaussian from the $N_b$ partcles of the second gaussian.
\section{Emittance growth rates}
One can compute growth rates for the average emittances, $<\epsilon_i>$ in the
Laboratory Coordinate System, from the growth rates for $<p_ip_j>$ in the
Rest Coordinate System.In the following , $dt$ is the time interval in the
Laboratory System and $d\tilde{t}$ is the time interval in the Rest System.
$dt=\gamma d\tilde{t}$
\begin{eqnarray}
\frac{d}{dt} <\epsilon_x> &=& \frac{\beta_x}{\gamma}
\frac{d}{d\tilde{t}} <p_x^2/p_0^2> + \frac{D^2+\tilde{D}^2}{\beta_x} \gamma
\frac{d}{d\tilde{t}}<p_s^2/p_0^2>
-2 \tilde{D} \frac{d}{d\tilde{t}}<p_x p_s/p_0^2> \nonumber\\
\frac{d}{dt} <\epsilon_y> &=& \frac{\beta_y}{\gamma} \frac{d}{d\tilde{t}}
<p_y^2/p_0^2> \nonumber\\
\frac{d}{dt} <\epsilon_s> &=& \beta_s \gamma \frac{d}{d\tilde{t}}
<p_s^2/p_0^2>
\end{eqnarray}
To derive the above results, the simplest case to treat is that of the vertical
emittance. The verical emmitance is given by
\begin{eqnarray}
\epsilon_y (y,y') &=& [y^2+(\beta_y y'+\alpha_y y)^2]/\beta_y \nonumber\\
\delta \epsilon_y &=& \beta_y \delta (y'^2) \nonumber\\
\frac{d}{dt} <\epsilon_y> &=& \frac{\beta_y}{\gamma} \frac{d}{d\tilde{t}}
<p_y^2/p_0^2> \nonumber\\
\end{eqnarray}
In Eq.(32), $y'=p_y/p_0$, $\delta \epsilon_y$ is the change in $\epsilon_y$ in a
scattering event.
For the longitudinal emittance one finds
\begin{eqnarray}
\epsilon_s &=& [s^2/\gamma^2+(\beta_s \gamma p_s/p_0)^2]/\beta_s \nonumber\\
\delta \epsilon_s &=& \beta_s \delta (\gamma p_s/p_0)^2 \nonumber\\
\frac{d}{dt} <\epsilon_s> &=& \beta_s \gamma \frac{d}{d\tilde{t}}
<p_s^2/p_0^2> \nonumber\\
\end{eqnarray}
In Eq.(33), $s,p_s$ are the coordinates in the rest system and I have used the relationship
$(p-p_0)_{LAB}=(\gamma p_s)_{REST}$
For the horizontal emittance one finds
\begin{eqnarray}
\epsilon_x &=& \{ [x-\gamma D p_s/p_0]^2 +[\beta_x (p_x/p_0-\gamma D'p_s/p_0)+
\alpha_x (x-\gamma D p_s/p_0)]^2 \} /\beta_x \nonumber\\
&=& \{ [x-\gamma D p_s/p_0]^2 +[\beta_x p_x/p_0+ \alpha_x x -
\bar{D}\gamma p_s/p_0]^2 \} /\beta_x \nonumber\\
&=& \{ x^2+(\gamma D p_s/p_0)^2-2x\gamma D p_s/p_0+(\beta_x p_x/p_0+ \alpha_x x)^2+
(\bar{D}\gamma p_s/p_0)^2- \nonumber\\
& & 2(\beta_x p_x/p_0+\alpha_x x)(\bar{D}\gamma p_s/p_0) \} /\beta_x \nonumber\\
\delta \epsilon_x &=& \delta \{\beta_x^2 (p_x/p_0)^2 + \gamma^2 (D^2+\bar{D}^2) (p_s/p_0)^2
-2 \beta_x \bar{D} \gamma (p_x/p_0)( p_s/p_0) \} / \beta_x \nonumber\\
\frac{d}{dt} <\epsilon_x> &=& \frac{\beta_x}{\gamma}
\frac{d}{d\tilde{t}} <p_x^2/p_0^2> + \frac{D^2+\tilde{D}^2}{\beta_x} \gamma
\frac{d}{d\tilde{t}}<p_s^2/p_0^2>
-2 \tilde{D} \frac{d}{d\tilde{t}}<p_x p_s/p_0^2> \nonumber\\
\end{eqnarray}
In the result for $\delta \epsilon_x$, the terms that are linear in $p_x$ or $p_s$ have been dropped
as they do not contribute to $<\delta \epsilon_x>$ . In a scattering event involving two particles
, the $\delta p_x$ of one
particle is equal and opposite to the $\delta p_x$ of the other particle. This is also true
for $p_s$.
\section*{Acknowledgements}
I thank I. Ben-Zvi for his comments and encouragement.
I also thank A. Fedotov and Y. Eidelman for information regarding their results.
\section*{References}
1. G.Parzen, BNL report C-A/AP/N0.150 (2004)
2. A. Piwinski Proc. 9th Int. Conf. on High Energy Accelerators (1974) 405,
M. Martini CERN PS/84-9 (1984), A. Piwinski Cern 87-03 (1987) 402,
A. Piwinski CERN 92-01 (1992) 226
3. J.D. Bjorken and S.K. Mtingwa, Part. Accel.13 (1983) 115, K. Kubo and
K. Oide Phys. Rev. S.T.A.B., 4, (2001) 124401
\end{document}
|
2,877,628,090,094 | arxiv | \section{Introduction}
From the latest Planck data \cite{Planck},
it is believed that dark matter makes up for
around 27 \% of the energy density of the universe
while luminous matter makes up around 4.5 \%,
the rest being in the form of Dark Energy. Furthermore, structure
formation in the Universe is generally believed to be driven
by Dark Matter.
The fact that Dark Matter constitutes the
dominant form of matter in the present time is remarkable.
If it is so now, it is extremely reasonable to think that it
was perhaps also dominant in the early universe.
And perhaps, it was the only matter that existed in the
very early universe. The generation of luminous matter
would come about when a fraction of dark matter converted
into luminous matter. The size of that fraction would
depend on the efficiency of the conversion process.
The temperature (or energy) where this conversion took
place would naturally depend on the dark matter mass(es)
and its conversion efficiency would depend on another mass
scale which governs the strength of the interaction.
It is necessary that dark matter is unified with luminous
matter in the underlying gauge theory. We require that
dark matter interacts with luminous matter strongly enough
to deplete the initial amount of dark antimatter
(and hence dark anti-matter) leaving an excess in dark matter
which leads eventually to an excess in luminous matter.
However, it should at the same time be weakly interacting
enough to escape direct detection at the present sensitivity.
For a recent summary of the status of non-WIMP Dark Matter, one can consult \cite{kusenko}.
\bigskip
\noindent
Our model is based partially on the scenario
presented in \cite{paulpqdm}
where it was proposed that dark matter particles
(fermions) come in {\em two} species: $\chi_l$ and $\chi_q$
which transform under the product of a dark matter gauge group with
the standard model (SM),
namely $SU(4)_{DM} \times SU(3)_c \times SU(2)_L \times U(1)_Y$,
as $(4,1,1,0)_{L,R}$ for $\chi_l$ and $(4,3,1,0)_{L,R}$
for $\chi_q$. The particles
$\chi_l$ and $\chi_q$ will be referred to as
"leptonic" (color singlet) and "baryonic" (color triplet)
dark matter respectively. In the present model
the dark matter carries no SM
quantum numbers so that only the field $\chi_l$ is involved.
A similarly motivated but technically completely different
model has been recently and presciently built in \cite{nath}.
Our model treats the symmetric and asymmetric dark matter differently
from the conventional approaches. Only
the asymmetric part of the dark matter survives to the present
time: all the symmetric part annihilates long ago. Some
14 \% of the asymmetric dark matter transmutes into luminous
matter via our luminogenesis mechanism. In particular, we do
not need the conventional WIMP annihilation cross section value
to obtain the correct relic density which is obtained correctly
by another mechanism. Again, we would like to stress that the usual mechanism
to determine the relic density which relies on the annihilation cross section, even in the presence of an asymmetric part
of DM \cite{asym}, does not apply to our model. This point will be clarified further below.
\section{Inflationary Dark Matter}
\subsection{Dark Matter in SU(6)}
If the dark matter field $\chi_l$ is to be a singlet under the
SM gauge group and if it were to be unified with luminous matter,
its own gauge group $G_{DM}$ (if there were one) should be
embedded in a larger dark unification group
$G_{DUT}$ which contains the SM group $G_{SM}$,
namely $G_{DUT} \rightarrow G_{DM} \times G_{SM}$.
This unified group would be one on which inflation is based such that
the inflaton will decay into dark matter during the reheating process.
\bigskip
\noindent
We use the model proposed to unify dark matter with luminous
matter in \cite{paulpqdm}. The unification of the two
sectors proceeds via the embedding of $SU(2)_L$ into a unifying
group $SU(n+2)$ with the following breaking path
$SU(n+2) \times U(1)_Y \rightarrow
SU(n)_{DM} \times SU(2)_L \times U(1)_{DM} \times U(1)_Y$.
Including QCD, the unifying group would be
$SU(3)_C \times SU(n+2) \times U(1)_Y$
("unifying" solely in the sense of dark and luminous matter
unification and not in the usual sense of gauge unification).
In \cite{paulpqdm}, arguments were given for the selection of
the preferred value
$n=4$ for the dark matter gauge group and our final choice is
\begin{equation}
\label{group}
SU(3)_C \times SU(6) \times U(1)_Y
\end{equation}
\noindent
with $SU(6)$ subsequently breaking according to
\begin{equation}
SU(6) \rightarrow
SU(4)_{DM} \times U(1)_{DM} \times SU(2)_L \, .
\label{su6}
\end{equation}
Here $G_{DUT}$ is $SU(6)$ and $G_{DM}$ is $SU(4)$.
It is convenient to show the various useful representations
of $SU(6) \supset SU(4)_{DM} \times SU(2)_L \times U(1)_{DM}$.
\begin{eqnarray}
\label{rep}
6& =& (1,2)_2 + (4,1)_{-1} \nonumber \\
20 &=& (4,1)_3 + (4^\ast,1)_{-3} + (6,2)_0 \nonumber \\
35 &=& (1,1)_0 + (15,1)_0 +(1,3)_0 + (4,2)_{-3} + (4^\ast ,2)_3 \nonumber \\
\end{eqnarray}
where $U(1)_{DM}$ quantum numbers are indicated by subscripts.
Note that $U(1)_{DM}$ will be spontaneously broken at a
scale $\Lambda_{DM}$ which will be constrained by experimental
direct detection limits. The associated massive gauge boson,
$\gamma_{DM}$, is the oft-discussed "dark photon". We shall
come back to this important point below.
At a scale $\Lambda_4$, $SU(4)_{DM}$ will become confining
and DM hadrons form as has been discussed in \cite{paulpqdm}.
The fact that our model contains strongly self-interacting
dark matter is an interesting feature which might resolve
the well-known $\Lambda$CDM problems \cite{CDMissues}
of dwarf galaxy structures
and of dark matter cusps at the centers of galaxies. This discussion is presented below.
From Eq. (\ref{rep}), the representations that contain
singlets under the SM $SU(2)_L$ gauge group are
$\underline{6}$ and $\underline{20}$. These are the
representations that could contain the desired dark
matter particles, namely $(4,1)$ which appears in
both $\underline{6}$ and $\underline{20}$. To see
where the dark matter belongs, it is important to
classify the fermion representations using Eq. (\ref{rep}).
As discussed above, our unified gauge group is
$SU(3)_C \times SU(6) \times U(1)_Y$. The fermion
representations are required to be anomaly-free.
Representations containing the left-handed SM
quark and lepton doublets are respectively
$(3,6,Y_{6q} /2)_L$ and $(1,6,Y_{6l} /2)_L$
where $Y_{6q,l} /2$ are the $U(1)_Y$ quantum numbers
of the quarks and leptons respectively. In addition,
the $SU(2)_L$ quark and lepton singlets are written
as $(3,1,(Y_{u} /2, Y_{d} /2)_R$ and
$(1,1,Y_{l} /2)_R$ respectively. The $U(1)_Y$ quantum
numbers are, as usual, $Y_{6q} /2= 1/6$
and $Y_{6l} /2=-1/2$ for $SU(6)$ non-singlets
and $Y_{u} /2=2/3$, $Y_{d} /2=-1/3$, and $Y_{l} /2=-1$.
Since $\underline{3}$ and $\underline{6}$
are complex representations, the minimal anomaly-free
representations are given by
\begin{eqnarray}
\label{rep2}
&&(3,6,Y_{6q} /2)_{L,R} + (1,6,Y_{6l} /2)_{L,R}
+ (3,1,(Y_{6u} /2, Y_{6d} /2)_{R,L} \nonumber \\
&&+ (1,1,Y_{l} /2)_{R,L} \, .
\end{eqnarray}
\noindent
As we have mentioned in \cite{paulpqdm}, the right-handed
quark and lepton doublets,
$(3,6,Y_{6q} /2)_{R} + (1,6,Y_{6l} /2)_{R} $,
and left-handed singlets,
$(3,1,(Y_{6u} /2, Y_{6d} /2)_{L} + (1,1,Y_{6l} /2)_{L}$,
are in fact the mirror fermions (distinct from SM fermions)
of the model of electroweak-scale right-handed neutrinos in
\cite{hung}. The details of how right-handed neutrinos, which
are members of doublets along with their mirror charged lepton
partners, can acquire electroweak-scale mass can be found
in \cite{hung}, where arguments were given for assigning
the mirror sector a global symmetry. As discussed in \cite{hung}, this mirror sector can be tested experimentally
at the LHC by looking for lepton-number violating processes through the production of electroweak-scale right-handed
neutrinos. This model fits rather well the electroweak precision parameter constraints as shown in \cite{hung2}.
In light of the
newly-discovered SM-like 126 GeV scalar, an extension of
the model of \cite{hung} to endow {\em separately} the SM sector and the
mirror sector with a global symmetry is needed \cite{ajinkya}:
a global $U(1)_{SM} \times U(1)_{MF}$ ("MF" stands for the mirror sector) is imposed, with two Higgs doublets, one for each sector. As shown in \cite{ajinkya},
a small mixing allowed by the present data on the 126-GeV scalar breaks explicitly this global $U(1)_{SM} \times U(1)_{MF}$ symmetry
with an interesting indirect implication on luminogenesis as we will discuss below.
From Eqs.(\ref{rep},\ref{rep2}), one can see that the
$SU(2)_L$-singlet and $SU(4)$-non-singlet particles
transform under
$SU(3) \times SU(4)_{DM} \times SU(2)_L \times U(1)_Y
\times U(1)_{DM} $ as (for both left and right-handed fermions)
\begin{equation}
\label{rep3}
(1, 4, 1, -\frac{1}{2})_{-1} + (3, 4, 1, \frac{1}{6})_{-1} \, ,
\end{equation}
where the subscripts $U(1)_{DM}$ quantum numbers.
It is clear from (\ref{rep3}) that these particles
which belong to the $6$ of $SU(6)$ {\em cannot} be
candidates for dark matter since they carry $U(1)_Y$
quantum numbers and are therefore electrically charged.
In fact, the color-singlet and colored particles carry
charges $\pm 1/2$ and $\pm 1/6$ respectively.
A suitable representation which is color-singlet and carries
no $U(1)_Y$ quantum number is the following real representation:
\begin{equation}
\label{rep4}
(1, 20, 0) = (1,4,1,0)_3 + (1, 4^\ast,1,0)_{-3} + (1, 6, 2, 0)_0 \, ,
\end{equation}
where the right-hand side represents decompositions under
$SU(3) \times SU(4) \times SU(2)_L \times U(1)_Y \times U(1)_{DM} $.
One notices that $(1,4,1,0)_3 + (1, 4^\ast,1,0)_{-3}$
are {\em inert} under the SM gauge group
$SU(3) \times SU(2)_L \times U(1)_Y$ but not under
$U(1)_{DM} $. These particle are the dark matter in our model:
note that, when one represents fermions in terms of left-handed
Weyl fields, we have $\chi_{L,R}= (1, 4^\ast,1,0)_{-3}$
and $\chi^{c}_{L} = \sigma_2 \chi^{\ast}_{R}=(1,4,1,0)_3$.
How the dark matter candidates are produced in the early
universe and how luminogenesis, the generation of luminous
matter from dark matter, occurs will be discussed in the next
two subsections.
For the sake of clarity, two tables are given below in order to list the different particle contents of the model.
\begin{table}[h]
\label{Tab1}
\begin{tabular}{|l|l|lr|||} \hline
$SU(6)$ & $SU(4) \times SU(2) \times U(1)_{DM}$ \\ \hline
${\bf 6}$ & ${\bf (1,2)_2 + (4,1)_{-1}}$ \\
$ {\bf 20}$ & ${\bf (4,1)_3 + (4^\ast , 1)_{-3} + (6,2)_{0} }$ \\
${\bf 35}$ & ${\bf (1,1)_0 + (15,1)_0 + (1,3)_0 +(4,2)_{-3}}$ \\
& ${\bf + (4^\ast , 2)_3}$ \\ \hline
\end{tabular}
\caption{The ${\bf (1,2)_2}$'s represent luminous matter while ${\bf (4,1)_3 + (4^\ast , 1)_{-3}}$ represent dark matter}
\end{table}
\begin{table}[h]
\label{Tab2}
\begin{tabular}{|l|l|lr|||} \hline
& $SU(3)_c \times SU(6) \times U(1)_{Y}$ \\ \hline
R $\supset$ SM fermions & ${\bf (3,6, 1/6)_L + (1,6, -1/2)_L }$ \\
&${\bf + (3,1,2/3)_R + (3,1,-1/3)_R }$ \\
&${\bf + (1,1,-1)_R}$ \\
\hline
R $\supset$ Mirror fermions & ${\bf (3,6, 1/6)_R + (1,6, -1/2)_R }$ \\
&${\bf + (3,1,2/3)_L + (3,1,-1/3)_L }$ \\
&${\bf + (1,1,-1)_L}$ \\
\hline
R $\supset$ dark matter fermions & ${\bf (1,20,0)}$ \\ \hline
\end{tabular}
\caption{R denotes representation. SM left-handed doublets and right-handed singlets are parts of the first entry, Mirror right-handed doublets and left-handed singlets are parts of the second entry, and dark matter left and right-handed fermions belong to the last entry.}
\end{table}
\subsection{Dark matter genesis}
We assume that the potential for the the adjoint scalar field
$\underline{35}$ of $SU(6)$ is {\em sufficiently flat} so as
to generate sufficient inflation at the scale of $SU(6)$ breaking.
It is beyond the scope of this article to treat this aspect of
inflation and we will restrict ourselves to its group theoretic
aspects. The inflaton field is the $\phi_{inf} =(1,1,1,0)_0$ of
\begin{eqnarray}
\label{adj}
(1,35,0)& =& (1,1,1,0)_0 + (1,15,1,0)_0 +(1,1,3,0)_0 \nonumber \\
&&+ (1,4,2,0)_{-3} + (1,4^\ast ,2,0)_3 \,,
\end{eqnarray}
where the right-hand-side shows the transformation under
$SU(3) \times SU(4) \times SU(2)_L \times U(1)_Y \times U(1)_{DM} $.
The fermions that can couple to the adjoint scalar will come from
$ 20 \times 20 = 1_s + 35_a + 175_s + 189_a$ and
$6 \times \bar{6} = 1 + 35$. Denoting $(1, 20, 0)$ by
$\Psi_{20}$ and $(1,35,0)$ by $\phi_{35}$, one can write the
following coupling
\begin{equation}
\label{yuk20}
g_{20} \, \Psi_{20}^{T} \sigma_2 \Psi_{20} \, \phi_{35} \, .
\end{equation}
From Eq.~(\ref{yuk20}), one can deduce the
coupling of the inflaton to dark matter
\begin{equation}
\label{infdm}
g_{20} \, \chi_{L}^{T} \sigma_2 \chi^{c}_{L} \phi_{inf} \, .
\end{equation}
\noindent
Since $\psi^{c}_L = \sigma_2 \psi^{\ast}_R$, it is clear
that $\bar{6} \sim \psi^{c}_{6,L} $ comes from mirror
fermions and $6 \sim \psi_{6,L}$ contains SM fermions.
As a result, a coupling such as
$\psi^{c}_{6,L} \sigma_2 \psi_{6,L} \, \phi_{35}$ is forbidden
at tree-level by the $U(1)_{SM} \times U(1)_{mirror}$ symmetry.
The inflaton will decay mainly into dark matter while its
decay into luminous matter will be highly suppressed by the
aforementioned symmetry.
Another interesting point that one could point out here is
quantum fluctuations during the inflationary period can create
seeds of structure formation but in our scenario, it is
structures of dark matter that were formed first. This is
actually the current view of structures in the universe.
In our model, structures involving luminous matter came only
later when approximately 14 \% of dark matter is converted
into luminous matter.
The next section discusses this conversion of some of the
dark matter energy density into luminous matter, a process
we call {\em luminogenesis}. In particular, we will present arguments showing that
the symmetric part of DM annihilates "almost completely" into the symmetric part of luminous
matter which, in turn, transforms into radiation. The asymmetric part of DM transforms a small part
of its number density into the asymmetric part of luminous matter through an {\em entirely different} mechanism
and is unrelated to and unconstrained by the annihilation cross section used in the symmetric part.
\subsection{Luminogenesis}
This section deals with the {\em fate} of the asymmetric and symmetric parts of DM. A few words concerning a possible origin of the excess of dark matter over anti-dark matter (the asymmetric part) is in order here.
First, we will assume that there is a global $U(1)_{\chi}$ symmetry for dark matter. The interactions involving the gauge bosons of the coset group $SU(6)/SU(4) \times SU(2) \times U(1)_{DM}$ (similar to X and Y gauge bosons of $SU(5)$) will explicitly violate the $U(1)_{\chi}$ symmetry and their decays involving the interferences between the tree-level and one-loop diagrams will ultimately generate a net DM number assuming the presence of CP violation in the DM sector. This problem will be treated in a separate paper. For the present purpose, we will assume that the asymmetric part is generated by the aforementioned mechanism.
Let us denote the asymmetric number density by $\Delta n_{\chi} = n_{\chi} - n_{\bar{\chi}}$ and the symmetric number density simply by $n_{sym} = n_{\bar{\chi}}$ since the symmetric part is composed of an equal number of DM and anti-DM. It is assumed that $\Delta n_{\chi} \ll n_{sym}$. What we will show below will be that the symmetric part will annihilate through the massive dark photon of $U(1)_{DM}$ into the symmetric part of luminous matter which will eventually transform into radiation, leaving practically very little symmetric DM. As we will also show, this has {\em no bearing} on the relic density of DM and hence that of its luminous off-spring. A fraction ($\sim 14 \%$) of the asymmetric part of DM, $\Delta n_{\chi}$, is transmuted into the asymmetric part of luminous matter through an {\em entirely different} mechanism coming from an exchange of a massive scalar. This is {\em different} from the usual approach whereas the same annihilation process determines both the relic asymmetric and symmetric densities which are therefore intrinsically linked \cite{asym}.
The simplified discussion below will go as follows. There are two different interactions which operate on the total DM density $n_{tot} = n_{sym} + \Delta n_{\chi}$.
The exchange of the massive scalar (to be discussed next) affects both the symmetric and asymmetric parts. Constraints on the scalar mass and the Yukawa couplings are imposed in such a way that 14 \% of the DM (i.e. 14 \% of $n_{sym}$ and 14 \% of $\Delta n_{\chi}$) is converted into luminous matter. The decoupling of the scalar exchange interaction should happen soon after DM becomes non relativistic. At the same time the symmetric part will annihilate into luminous matter via the massive dark photon. This will reduce the symmetric part to slightly below 86 \% by the time of the scalar exchange decoupling. For our simplified discussion, we will ignore this difference. The 86 \% of the symmetric DM will continue to annihilate via the massive dark photon until very little is left as we will show below. What is {\em unaffected} by the $U(1)_{DM}$ interactions is the 86 \% of the asymmetric part and the 14 \% of the asymmetric luminous matter. This is the essence of our luminogenesis. It goes without saying that a more accurate analysis involving a numerical study of the Boltzmann equation as well as the inclusion of chemical equilibrium is needed and this will be carried out in a future work. For this paper, we present a simplified discussion in order to lay out the essence of our model.
\bigskip
\noindent
\bigskip
\noindent
{\bf I}. {\bf The conversion of the symmetric and asymmetric parts via a scalar exchange:}
\bigskip
We present in this section the main mechanism for luminogenesis: The conversion of a small fraction of DM into luminous matter by the exchange of a heavy scalar. We will show the constraints on the Yukawa couplings and the mass of the heavy scalar coming from the requirement that 14 \% of DM is converted into luminous matter. In the next section, we will discuss how most of the 86 \% of the symmetric part of DM annihilates via a massive dark photon into the symmetric luminous matter which eventually transforms into radiation. The 86 \% of asymmetric DM and the 14 \% of asymmetric luminous matter are unaffected by this annihilation process.
\bigskip
Ia) {\bf The interaction Lagrangian and related features:}
Since the dark and luminous sectors
belong to different representations of $SU(6)$, the conversion can be
achieved only through a coupling of the dark and luminous
sectors with a scalar field.
Since $20 \times \bar{6} = 15 + 105$ and
$20 \times 6 = \bar{15} + \bar{105}$ ($20$ is real),
the appropriate scalars transform as $\bar{15}$ and $15$ respectively.
We denote these scalars as $\Phi_{15}^{(L)} (1/2)$ and
$\Phi_{\bar{15}}^{(R)} (-1/2)$ where $\pm 1/2$ denotes the
$U(1)_Y$ quantum number. We have the following Yukawa couplings
\begin{equation}
\label{DMLUM}
g_{6L} \, \Psi_{20}^{T} \sigma_2 \psi_{6,L} \, \Phi_{15}^{(L)} + g_{6R} \, \Psi_{20}^{T} \sigma_2 \psi^{c}_{6,L} \, \Phi_{\bar{15}}^{(R)} \, ,
\end{equation}
with $\Phi_{15}^{(L)}$ and $\Phi_{\bar{15}}^{(R)}$ carrying
appropriate global $U(1)_{SM}$ and $U(1)_{mirror}$ quantum numbers
respectively.
A mass mixing between $\Phi_{15}^{(L)}$ and
$\Phi_{\bar{15}}^{(R)}$ will break the global
$U(1)_{SM} \times U(1)_{MF}$ symmetry and thus
allows for the following conversion process to occur:
$\chi_L + \chi_R \rightarrow l_L + l^{M}_R $
where $l_L $ and $l^{M}_R$ refer to SM and mirror leptons
respectively as in \cite{hung}. The same goes for the anti-DM particles. This process can be represented
by the following effective Lagrangian
\begin{equation}
\label{eff}
\frac{g_{6}^2}{M_{15}^2} \, (\chi^{T}_{L} \sigma_2 l_L)\, (\chi^{c,T}_{L} \sigma_2 l^{M,c}_{L}) + H.c. \, ,
\end{equation}
where $l^{M,c}_{L} = \sigma_{2} l^{M \ast}_{R}$.
The various mixing coefficients are embedded in the prefactor
of Eq.~(\ref{eff}). How effective the conversion of dark matter
into luminous SM {\em and} mirror leptons as represented by
Eq.~(\ref{eff}) will depend on this prefactor, especially
the luminogenesis scale $M_{15}$ which in the next subsection we shall estimate
to be $M_{15} \sim 10^9$ GeV. Notice that mixing between $\Phi_{15}^{(L)}$ and
$\Phi_{\bar{15}}^{(R)}$ has to be sufficiently small so that only a small fraction of DM is converted into luminous matter, namely $\sim$ 14 \%.
As we have mentioned above, it is interesting to note that an extended version of \cite{hung} to include two Higgs doublets (with similar
$U(1)_{SM} \times U(1)_{MF}$ assignments as $\Phi_{15}^{(L)}$ and
$\Phi_{\bar{15}}^{(R)}$) in
order to describe the 126-GeV Higgs-like object also requires the mixing between these two doublets to be small as constrained by the data
\cite{ajinkya}.
At this point we would like to point out an important fact that comes out of the model of \cite{hung}, namely the decay of the
mirror lepton $l^{M}_R$ into a SM lepton $l_L$. This decay proceeds through a Yukawa interaction
\begin{equation}
\label{mirrortol}
g_{sl} \, \bar{l}_{L}\, \phi_S \, l^{M}_{R} + H.c. \, ,
\end{equation}
where $\phi_S$ is the SM-singlet Higgs field with $l_L$ and $l_{R}^{M}$ being SM left-handed and mirror right-handed doublets
respectively. Notice that under $U(1)_{MF}$, $l^{M}_{R} \rightarrow \exp (\imath \alpha_{MF}) l^{M}_{R}$ while under $U(1)_{SM}$, $l_{L} \rightarrow \exp (\imath \alpha_{SM}) l_{L} $. As a result, $\phi_{S} \rightarrow \exp (\imath (\alpha_{SM} - \alpha_{MF}))$. This point is explained in \cite{hung} for $U(1)_{MF}$ and extended to $U(1)_{SM} \times U(1)_{MF}$ in \cite{ajinkya}.
The physical reason for writing down Eq.~(\ref{mirrortol}) is given in \cite{hung} where a model for an electroweak-scale Majorana mass
for the right-handed neutrinos was constructed. The singlet Higgs boson $\phi_S$ is vey light ($\sim 100 keV$ or even $O(~{\rm MeV})$), as discussed in \cite{hung}. (Notice that in \cite{hung}, $g_{sl} \, v_S \sim 10^{5} eV$ and for simplicity it was assumed that $g_{sl} \sim O(1)$ giving $v_{S} \sim 10^{5} eV$. However one can have $g_{sl} \sim 10^{-2}$ which gives $v_{S} \sim 10 ~{\rm MeV}$ and hence a mass of $\phi_S$ in the MeV range. (Or it could even be in the tens of keVs.) Eq.~(\ref{mirrortol}) gives rise to the decay $l_{R}^{M} \rightarrow l_L + \phi_S$. (This includes decays of mirror charged and neutral leptons.)
Depending on the size of the coupling $g_{Sl}$, $l_{R}^{M} $ could be relatively long-lived with distinct signatures at the LHC. However, on a cosmic scale, the mirror leptons $l_{R}^{M}$ promptly decay into SM leptons. The end product of Eq.~(\ref{eff}) will be the conversion of a fraction of the DM particles into SM leptons only with no mirror leptons left. The details of this conversion process are important and will be treated elsewhere.
The conversion of the SM leptonic asymmetry to the baryonic asymmetry can proceed via the well-known sphaleron process \cite{yanagida}.
Although the fate of $\phi_S$ was discussed in \cite{hung}, it is useful to repeat it here. From \cite{hung}, one can obtain the interactions between $\phi_S$, $\nu_R$ (with $M_R \sim O(\Lambda_{EW})$), and $\nu_{L}$, as well as with $e^{M}_R$ and $e_L$ as $g_{Sl} \bar{\nu}_L \phi_{S} \nu_R + H. c.$ and $g_{Sl} \bar{e}_L \phi_{S} e^{M}_R + H. c.$, coming from $g_{Sl} \bar{l}_L \phi_{S} l_R + H. c.$. $\phi_S$ is in thermal equilibrium with luminous matter through the reactions $\phi_S + \phi_S^{\ast} \leftrightarrow \bar{\nu}_L + \nu_L$, $\phi_S + \nu_L \leftrightarrow \phi_S + \nu_L$, $\phi_S + \phi_S^{\ast} \leftrightarrow \bar{e}_L + e_L$, $\phi_S + e_L \leftrightarrow \phi_S + e_L$. To see their effects on BBN, we shall use the analysis of \cite{dreiner} for some particular range of values for $m_{\phi_S} $, namely in the tens of keVs. Without repeating what has been done in \cite{dreiner}, it is illuminating to stress that if there are light particles that couple to matter and if these light particles decouple close to the temperature where neutrinos decouple, they can be counted toward the effective number of neutrinos which may exceed the cosmological constraints of BBN. In a nutshell, if the rate is comparable to the weak interaction rate, the effective number of neutrinos might exceed the current bound. Notice that $C_{x} G_{x}$ of \cite{dreiner} is identified in the model of \cite{hung} and hereon as $\sim G_{\phi_S} = g_{sl}^2/(4\sqrt{2} M_R^2)$ where $M_R \sim O(\Lambda_{EW})$ is a typical mirror fermion mass. The constraint obtained by \cite{dreiner} by requiring the effective number of neutrinos $N_{eff} < 4$ with the neutrino decoupling temperature $T_{\nu} \sim 3\, ~{\rm TeV}$, for a light complex scalar (our case), is then
\begin{equation}
\label{phi}
G_{\phi_S} \alt 4.1 \times 10^{-2} C_V \, G_F \,,
\end{equation}
where $C_V \sim O(1)$ and $G_F = g^2/(4\sqrt{2} M_W^2) = 1.166 \times 10^{-5} ~{\rm GeV}^{-2}$ is the Fermi constant. The constraint (\ref{phi}) can be easily satisfied for $g_{sl}^2 < 10^{-2} g^2$. It is interesting to note that a very small value for $g_{sl}$ can lead to a long-lived mirror lepton and this could have interesting implications at the LHC \cite{hung} as well as facilities searching for rare decays such as $\mu \rightarrow e \gamma$ and $\tau \rightarrow \mu \gamma$ \cite{hung2}. A more comprehensive study of the cosmological implications of $\phi_S$ is beyond the scope of this paper and will be presented elsewhere.
We now proceed to discuss the conversion of part of the asymmetric DM into the asymmetric luminous matter.
\bigskip
Ib) {\bf Dark and Luminous matter densities:}
\bigskip
In what follows we will present a simplified version of luminogenesis which captures the essence of the process. A more complete numerical analysis of the Boltzmann equation and chemical potential equilibrium will be presented elsewhere.
Recall that $n_{tot} = n_{sym} + \Delta n_{\chi}$. We assume that the decoupling occurs fast enough so that the universe is still matter-dominated. The interaction (\ref{eff}) decouples from the DM when the interaction rate $\Gamma \approx (\alpha_{6}^2/ M^{2}_{15}) n_{tot} $ is less than the Hubble rate $H= \sqrt{8 \pi/3} (1/m_{pl}) \sqrt{\rho_m} = \sqrt{8 \pi/3} (1/m_{pl}) \sqrt{n_{tot} m_{\chi}}$. The total density at decoupling is obtained from $\Gamma = H$ giving
\begin{equation}
\label{15}
n_{tot,D} \approx (\frac{8 \pi}{3})(\frac{1}{\alpha_{6}^4})(\frac{M_{15}}{m_{pl}})^2 m_{\chi} M^{2}_{15} \,.
\end{equation}
Let us define
\begin{equation}
\label{ratio1}
r= \frac{n_{tot,D}}{n_{tot,0}} \,.
\end{equation}
where $n_{tot,0}$ is the number density at $T \sim m_{\chi}$ and is given by $n_{tot,0} \sim C m_{\chi}^3$. For the sake of estimate, we will not be very precise about the exact value of $C$. We obtain
\begin{equation}
\label{ratio2}
r \approx \tilde{C} (\frac{1}{\alpha_{6}^4})(\frac{M_{15}}{m_{pl}})^2 (\frac{M_{15}}{m_{\chi}})^2 \,,
\end{equation}
where $\tilde{C} \sim O(10^2)$. Eq.~(\ref{ratio2}) tells us about the relationship between the coupling $\alpha_{6}$ and the mass $M_{15}$ for a given DM mass $m_{\chi}$ with $r$ set to $r= 86 \%$. From Eq.~(\ref{eff}), one can see that the cross section would increase as $M_{15}$ decreases and if we want to keep $r$ fixed, one has to decrease $\alpha_{6}$ as well. For $r= 86 \%$ one can solve for $\alpha_6$ as a function of $M_{15}$ and $m_{\chi}$ and which goes like
\begin{equation}
\label{alpha6}
\alpha_6 \approx 1.16 \tilde{C} \sqrt{\frac{M_{15}}{m_{pl}}} \sqrt{\frac{M_{15}}{m_{\chi}}} \,.
\end{equation}
As an example, let us take $m_{\chi} \sim 1\, ~{\rm TeV}$. One gets $\alpha_6 \sim 10^{-2}, 10^{-3}, 10^{-4}, 10^{-5}$ for $M_{15} \sim 10^{9}, 10^{8}, 10^{7}, 10^{6}$ GeV respectively. Notice that what we call by $\alpha_6$ is actually a quantity which contains various factors such as the square of the Yukawa couplings and, most importantly, the mixing angle between $\Phi_{15}^{(L)}$ and $\Phi_{\bar{15}}^{(R)}$ as discussed above. At this point, it is interesting to note that it is plausible that the mixing between $\Phi_{15}^{(L)}$ and $\Phi_{\bar{15}}^{(R)}$ is similar to that between the two Higgs doublets in an extension \cite{ajinkya} of \cite{hung}. This opens up the possibility that luminogenesis could be indirectly tested at the LHC.
The final accounting goes as follows. At decoupling, there are 86 \% asymmetric DM, 14 \% asymmetric luminous matter and the same goes for the symmetric parts. The 86 \% of symmetric DM will in turn annihilate into symmetric luminous matter and quickly gets depleted as we will show in the next section.
\bigskip
\noindent
{\bf II}. {\bf The symmetric part:}
Since both dark and luminous matter carry nonzero $U(1)_{DM}$ quantum
number, dark matter can annihilate via the $\gamma_{DM}$ massive gauge
boson into particle-antiparticle pairs of the luminous sector. This can be represented by an effective interaction
\begin{equation}
\label{gamma}
\frac{g^2}{M_{\gamma_{DM}}}(\bar{\chi} \gamma_{\mu} \chi)(\bar{f} \gamma^{\mu} f) \,.
\end{equation}
The amount of symmetric DM which remains after the above decoupling is actually lower than 86 \% due to annihilation via $\gamma_{DM}$. However, the number left over after the $U(1)_{DM}$ interaction decouples is so small that we will ignore this difference. The argument goes as follows.
Following a similar reasoning to the above analysis, we look for the number density of the symmetric part of DM at the time of decoupling i.e. its value when the interaction rate is equal to the Hubble rate. There is however a difference with the above analysis concerning the Hubble rate at decoupling. It is reasonable to assume that the temperature at which the $U(1)_{DM}$ interaction goes out of thermal equilibrium to be much lower than $m_{\chi}$. For example, the energy density ratio for $T= m_{\chi}/10$ is roughly $\rho_{\chi}/\rho_{R} < \exp(-10) \approx 4.5 \times 10^{-5}$, implying a radiation-dominated universe. The density of symmetric DM at decoupling is determined by
\begin{equation}
\label{dark}
\frac{\alpha^2}{M_{\gamma_{DM}}^2}\, n_{sym,D} \approx \frac{T_{D}^2}{m_{pl}} \,,
\end{equation}
where $D$ again stands for "decoupling". Again using $n_{tot,0} \sim n_{sym,0} \sim C m_{\chi}^3$, one obtains
\begin{equation}
\label{ratio3}
\frac{n_{sym,D}}{n_{sym,0}} \approx \frac{1}{C\, \alpha^2} \frac{M_{\gamma_{DM}}^2 \, T_{D}^2}{m_{pl}\, m_{\chi}^3} \,.
\end{equation}
If the decoupling temperature is say $m_{\chi}/10$ (just an example) with $m_{\chi} \sim 1 ~{\rm TeV}$ and if $M_{\gamma_{DM}} \sim O(~{\rm TeV})$ (see the section on direct detection), the density at decoupling would be $n_{sym,D} \sim 10^{-16} n_{sym,0} $. Since one expects $\Delta n_{\chi} \sim 10^{-9} n_{sum,0}$ and that 86 \% of the asymmetric part remains, one can see that the number density of symmetric DM at $U(1)_{DM}$ decoupling is negligibly small compared with the asymmetric relic density.
Needless to say, a detailed analysis of luminogenesis is indeed extremely important. This will be treated elsewhere. What has been presented here
could be considered to be the first steps of an extended program of luminogenesis.
\section{Dark matter hadrons and small scale structure problem}
In a subsequent paper \cite{kevin}, we will show that, starting from the $SU(2)_L$ coupling at the
electroweak scale and running it toward the unification scale $M_{DUT}$, one can deduce the
value of the DM gauge coupling of $SU(4)$ at that scale. From hereon, we shall call $SU(4)$ by the name Dark QCD (
DQCD).
Running it backward, one can look for the energy scale
at which $\alpha_4 \sim 1$. This will be, to a good approximation, the scale where DQCD confinement occurs.
As we have discussed briefly in \cite{paulpqdm}, dark baryons are formed by a bound state of {\em four} $\chi$'s resulting in massive {\em bosons}.
For definiteness, we shall call these dark baryons by the name $\chi$ Massive Particle or CHIMP. To a first approximation, the CHIMPs
will have a mass of approximately {\em four} times the DQCD confinement scale $\Lambda_4$.
As in \cite{paulpqdm}, there are three flavors
of dark matter fermion $\chi$ (one for each family). In the absence of explicit mass terms, there is a chiral symmetry $SU(3)_L \times SU(3)_R$ among $\chi$'s. From QCD (restricting oneself to two flavors for simplicity), we learn that $\langle \bar{q} q \rangle \neq 0$ spontaneously breaks the quark chiral symmetry resulting in the appearance of Nambu-Goldstone (NG) bosons which however acquire a small mass due to the explicit breaking of that chiral symmetry coming from the small masses of the up and down quarks and thus becoming what are called pseudo NG bosons. We expect a similar phenomenon to occur for DQCD. $\langle \bar{\chi} \chi \rangle \neq 0$ would yield {\em massless} NG bosons. It will be seen below that one has to break explicitly the DM chiral symmetry by a tiny amount in order to endow these NG bosons with a tiny mass. To be specific, the explicit breaking of the $\chi$-chiral symmetry can be parametrized by a term $m_0 \bar{\chi} \chi$ with $m_0 \ll \Lambda_4$. $m_0$ will be the free parameter of the model which will be determined by the fit to the small scale, Milky Way and possibly cluster scale anomalies as discussed below. For definiteness, we shall denote these pseudo-NG bosons as $\pi_{DM}$. The arguments go as follows.
Below $\Lambda_4$, one can write down an effective theory of CHIMPs interacting with the pseudo-NG bosons, very much in the same vein as nucleon-pion interactions with the difference being that in our case the CHIMPs are {\em bosons} instead of being fermions. In some sense, the dynamics of DQCD would presumably be simpler than that of QCD since CHIMPs carry no spin. One can write down a non-relativistic potential between two CHIMPs exchanging a $\pi_{DM}$ as
\begin{equation}
\label{DMpotential}
V= - \frac{\alpha_{DM}}{r} \exp(-m_{\pi_{DM}} r) \, .
\end{equation}
Such a potential has been investigated phenomenologically by \cite{zurek} although the Lagrangian is for a Yukawa interaction between a fermionic DM and a scalar. Non-relativistically it is the same. Here, we provide an explicit model for the spin-0 field, namely the pseudo-NG bosons- the dark pions- of DQCD.
As an example (although a more detailed investigation is surely needed), one can use Fig. 6 of \cite{zurek} to get a very rough estimate of the parameter range allowed to solve the small scale structure anomalies. First, a word of caution is in order here. Our CHIMP-dark pion interactions are presumably strong judging from what we know about pion-nucleon interactions although QCD can be {\em very different} from DQCD. Results shown in Fig. 6 of \cite{zurek} are for perturbative values of $\alpha_{DM}$ up to $\alpha_{DM} =0.1$. For the sake of argument, let us take $\alpha_{DM} =0.1$ to make our estimate. Taking into account only small scale structure anomalies, one can extrapolate to see that masses of CHIMPs ranging up to 100 TeV or so and $m_{\pi_{DM}}< 1 MeV$. If one would like to accommodate also Milky Way and cluster bounds, the CHIMP mass is seen to be lower, in the range of a few hundreds of GeV to a few TeVs, with
$m_{\pi_{DM}}> 1 MeV$. Notice again that the dark pion mass $m_{\pi_{DM}}$ is related $m_0$ which appears in the explicit breaking term of the $\chi$-chiral symmetry $m_0 \bar{\chi} \chi$ with $m_0$ being a free parameter.
It goes without saying that much remains to be done to tackle these issues in the framework of strong coupling regime as in our model. But it is encouraging that light pseudo scalars appear naturally due to the chiral symmetry of the model and this lightness appear to be what might be needed to solve the small scale structure anomalies and perhaps larger scales as well.
\section{Grand Unification Reconsidered}
Since 1974, a great deal of research has proceeded based on the
idea that the SM gauge group is contained is a
larger grand unified GUT
group $G_{GUT}$. The simplest GUT model is based
\cite{GeorgiGlashow} on $G_{GUT} \equiv SU(5)$ which, in its
minimal form, makes a sharp prediction for the proton decay
lifetime
based\cite{GeorgiQuinnWeinberg} on a GUT scale
$M_{GUT} \gtrsim 10^{14}$ GeV. Experimental searches excluded this
prediction already in 1984 but many alternative GUT
theories are viable which survive this test.
Accurate unification of
the SM couplings at $M_{GUT} \sim 2 \times 10^{16}$ GeV
has frequently been cited
\cite{Amaldi1,Amaldi2} as evidence for supersymmetry, and
GUT theories are an intermediate goal in much of
string theory phenomenology.\\
\noindent
By contrast, in the present luminogenesis model there is no luminous matter
with mass above $M_{15} \sim 10^9$ GeV, so that extrapolation
of the SM gauge couplings to orders
of magnitude above the $T_{15}$ scale, while including only
luminous matter states in the calculation of the
renormalization group flow, is rendered
physically inappropriate. This provides a plausible rationale for the
non-confirmation of the proton lifetime predicted on the basis of such
an extrapolation in e.g. $SU(5)$. In the present model based on the gauge group of Eq.~(\ref{group}), proton decay is absent.
\section{Direct detection}
The direct detection of dark matter in our model
can come about by the exchange of the dark photon,
$\gamma_{DM}$. Dark matter can interact with luminous
matter in the direct detection search through the
exchange in the t-channel of the massive dark photon,
namely through the use of Eq.~(\ref{gamma}). An estimate
of the mass $M_{\gamma_{DM}}$ assuming $g = O(1)$ using the
bound by XENON100 \cite{XENON100} for the cross section for
a dark matter mass of e.g. 1 TeV, namely
$\sigma < 10^{-44} cm^2$ gives $M_{\gamma_{DM}} > O(2 \, TeV)$.
Nevertheless, we can eagerly await results from
the upgraded version of XENON100 to XENON1T
being planned \cite{XENON1T}
for direct detection of dark matter particles.
\section{Discussion}
Our principal underlying assumption is that in the very early
universe the inflaton decays into only dark matter
and that at a later, though still early, cosmological era,
luminogenesis converted some 14 \% of this
dark matter into luminous matter. Our specific model
gives rise naturally to strongly-interacting dark matter
which can overcome some important short-range problems
confronting cold dark matter. Luminogenesis occurs via
an extremely weak interaction characterized
by a mass scale $\sim 10^7-10^9 ~{\rm GeV}$. The possible irrelevance
of grand unified GUT models which include
only luminous matter is clarified
in this broader perspective. Finally, higher sensitivity
direct detection of dark matter will be of crucial importance
in sharpening our understanding of the
luminogenesis stage in the early universe.
\section{Acknowledgments}
PHF was supported in part by US DOE grant DE-FG02-06ER41418.
PQH was supported in part by US DOE grant DE-FG02-97ER41027.
PQH would like to thank Alexander Kusenko for the stimulating atmosphere at PACIFIC 2013 where this work was completed.
|
2,877,628,090,095 | arxiv | \section{Introduction}}
\label{sec:intro}
\input{1_Intro}
\section{Template Matching \& Regression on $\mathbb{R}^2$}
\label{sec:templateMatchingR2}
\input{2_TMR2}
\section{\mbox{Template Matching \& Regression on $SE(2)$}}
\label{sec:templateMatchingSE2}
\input{3_TMSE2}
\section{Applications}
\label{sec:applications}
\input{5_1_Details}
\input{5_2_ONH}
\input{5_3_Fovea}
\input{5_4_Pupil}
\input{5_5_GeneralDiscussion}
\color{black}
\section{Conclusion}
\label{sec:discussionAndConclusion}
\input{6_DiscussionAndConclusion}
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
The authors would like to thank the groups that kindly made available the benchmark datasets and annotations.
The authors gratefully acknowledge Gonzalo Sanguinetti (TU/e) for fruitful discussions and feedback on this manuscript. The research leading to the results of this article has received funding from the European Research Council under the European Community's 7th Framework Programme (FP7/2007–2014)/ERC grant agreement No. 335555. This work is also part of the H\'{e} Programme of Innovation Cooperation, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO).
\clearpage
\appendices
\section{Probabilistic Interpretation of the Smoothing Prior in $SE(2)$
\label{sec:stochasticProcess}
\input{S1_Regularizer}
\section{The Smoothing Regularization Matrix R}
\label{sec:regMatrix}
\input{S2_DerivationOfR}
\section{Normalized Cross Correlation}
\label{sec:normalizedCrossCorrelation}
\input{4_NormalizedCC}
\section{Additional Details on the Detection Problems}
\label{sec:additionalDetails}
In this section we describe additional details about the implementation and results of the three detection problems discussed in the main article.
\begin{figure*}
\begin{center}
\includegraphics[width=\linewidth]{Figs/TrainingSamples}
\end{center}
\caption{A selection of positive and negative image patches $f_i$ used in the training of templates.
}
\label{fig:TrainingSamples}
\end{figure*}
\begin{table*
\centering
\caption{Average processing times. For optic nerve head detection (ONH) the average is taken over 1529 images of the TC, MESSIDOR, DRIVE and STARE database. For fovea detection the average is taken over 1408 images of the TC and MESSIDOR database. For pupil detection the average is taken over 1521 images of the BioID database.}
\begin{tabular}{lll|ll|ll}
\cmidrule[1.5pt]{1-7}
& \multicolumn{2}{c|}{ONH} & \multicolumn{2}{c|}{Fovea} & \multicolumn{2}{c}{\;\;Pupil (left \& right)\;\;}\\
& \multicolumn{1}{l}{$\mathbb{R}^2$} & \multicolumn{1}{l|}{$SE(2)$} & \multicolumn{1}{l}{$\mathbb{R}^2$} & \multicolumn{1}{l|}{$SE(2)$} & \multicolumn{1}{l}{$\mathbb{R}^2$} & \multicolumn{1}{l}{$SE(2)$} \\
\cmidrule{2-7}
& \multicolumn{6}{c}{{Timings (ms)}}\\
\cmidrule{2-7}
\multicolumn{1}{l}{1. Rescaling} & 106\hspace{3em} & 106\hspace{3em} & 111\hspace{3em} & 111\hspace{3em} & 0\hspace{3em} & 0\hspace{3em} \\
\multicolumn{1}{l}{2. $\mathbb{R}^2$-Processing} & 66 & 66 & 64 & 64 & 71 & 71 \\
\multicolumn{1}{l}{3. OS Transform} & 0 & 108 & 0 & 108 & 0 & 121 \\
\multicolumn{1}{l}{4. $SE(2)$-Processing} & 0 & 5 & 0 & 5 & 0 & 6 \\
\multicolumn{1}{l}{5. Template Matching} & 20 & 195 & 19 & 190 & 26 & 116 \vspace{\smallspacing}\\
\multicolumn{1}{l}{Total} & 192 & 479 & 195 & 477 & 97 & 313 \\
\cmidrule{2-7}
& \multicolumn{6}{c}{{Combined Total Timings (ms) - $\mathbb{R}^2$ and $SE(2)$}}\\
\cmidrule{2-7}
\multicolumn{1}{c}{ }& \multicolumn{2}{c|}{497} & \multicolumn{2}{c|}{501} & \multicolumn{2}{c}{420} \\
\cmidrule{2-7}
& \multicolumn{6}{c}{{Combined Total Timings (ms) - Fovea and ONH}}\\
\cmidrule{2-7}
\multicolumn{1}{c}{ }& \multicolumn{4}{c|}{730} & \\
\cmidrule[1.5pt]{1-7}
\end{tabular}
\label{tab:timings}
\end{table*}
\begin{table*
\centering
\caption{Success rates for optic nerve head detection ($\pm$ standard deviation, number of fails in parenthesis) with varying accuracy requirements in 5-fold cross validation. Maximum distance to ground truth location is expressed in optic disk radius $R$.}
\begin{tabular}{l|lllll}
\toprule
& \multicolumn{5}{c}{Maximum distance to ground truth}\\
Database (\# of images) & \;\;\;\;\;\;\;\;\;\;\;\;R/8\;\;\;\;\;\;\;\;\;\;\;\; & \;\;\;\;\;\;\;\;\;\;\;\;R/4\;\;\;\;\;\;\;\;\;\;\;\; & \;\;\;\;\;\;\;\;\;\;\;\;R/2\;\;\;\;\;\;\;\;\;\;\;\; & \;\;\;\;\;\;\;\;\;\;\;\;R\;\;\;\;\;\;\;\;\;\;\;\; & \;\;\;\;\;\;\;\;\;\;\;\;2R\;\;\;\;\;\;\;\;\;\;\;\;\\
\midrule
ES (SLO)\;\;\;\;\;\;\;(208) & 98.05\% {\tiny $\pm$ 2.04\%} (4) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 100.0\% {\tiny $\pm$ 0.00\%} (0)\\
TC\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(208) & 84.19\% {\tiny $\pm$ 4.34\%} (33) & 94.54\% {\tiny $\pm$ 3.51\%} (11) & 99.52\% {\tiny $\pm$ 1.06\%} (1) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 100.0\% {\tiny $\pm$ 0.00\%} (0)\\
MESSIDOR \;\;(1200) & 73.07\% {\tiny $\pm$ 3.69\%} (323) & 94.41\% {\tiny $\pm$ 1.47\%} (67) & 99.50\% {\tiny $\pm$ 0.46\%} (6) & 99.92\% {\tiny $\pm$ 0.19\%} (1) & 100.0\% {\tiny $\pm$ 0.00\%} (0)\\
DRIVE \;\;\;\;\;\;\;\;\;\,(40) & 70.84\% {\tiny $\pm$ 26.0\%} (13) & 91.69\% {\tiny $\pm$ 12.3\%} (4) & 98.18\% {\tiny $\pm$ 4.07\%} (1) & 98.18\% {\tiny $\pm$ 4.07\%} (1) & 100.0\% {\tiny $\pm$ 0.00\%} (0)\\
STARE \;\;\;\;\;\;\;\;\;\,(81) & 48.12\% {\tiny $\pm$ 10.27\%} (42) & 74.94\% {\tiny $\pm$ 6.52\%} (20) & 89.39\% {\tiny $\pm$ 8.16\%} (9) & 98.67\% {\tiny $\pm$ 2.98\%} (1) & 98.67\% {\tiny $\pm$ 2.98\%} (1)\\
&&&&&\\
All Images\;\;\;\;(1737) & 76.11\% {\tiny $\pm$ 2.58\%} (415) & 94.13\% {\tiny $\pm$ 0.79\%} (102) & 99.02\% {\tiny $\pm$ 0.26\%} (17) & 99.83\% {\tiny $\pm$ 0.26\%} (3) & 99.94\% {\tiny $\pm$ 0.13\%} (1)\\
\bottomrule
\end{tabular}
\label{tab:resultsONHAccuracy}
\end{table*}
\subsection{Training Samples}
In all three applications training samples were used to compute the templates. Positive training samples were centered around the object of interest. Negative training samples were centered around random locations in the image, but not within a certain distances to the true positive object location. In the retinal applications this distance was one optic disk radius, in the pupil detection application this was a normalized distance of 0.1 (cf. Eq.(39) of the main article). An selection of the 2D image pathes that were used in the experiments are shown in Fig.~\ref{fig:TrainingSamples}.
\subsection{Processing Pipeline, Settings and Timings}
\label{subsec:implementationdetails}
\subsubsection{Processing Pipeline}
\label{subsec:ProcessingPipeline}
In all three application the same processing pipeline was used. The pipeline can be divided into the following 5 steps:
\begin{enumerate}
\item \emph{Resizing}. Each input image is resized to a certain operating resolution and cropped to remove large regions with value 0 (outside the field of view mask in retinal images, see e.g. Fig.~\ref{fig:resultsOverviewImageONH}). The retinal images are resized such that the pixel size was approximately $40 \mu m/pix$. In the pupil detection application no rescaling or cropping was done.
\item \emph{$\mathbb{R}^2$-Processing}. In all three applications we applied a local intensity and contrast normalization step using an adaptation of \cite{Foracchia2005} which we explain below. The locally normalized image $\hat{f}$ is then mapped through an error function via $\operatorname{erf}(8 \hat{f})$ to dampen outliers.
\item \emph{Orientation score transform}. The processed image is then taken as input for an orientation score transform using Eq.~(23) of the main article. For the oriented wavelets we used cake wavelets \cite{Duits2007a,Bekkers2014} of size $[51 \times 51]$ and with angular resolution $s_\theta = \pi/12$, and with sampling $\theta$ from $0$ to $\pi$.
\item \emph{$SE(2)$-Processing}. For phase-invariant, nonlinear, left-invariant \cite{Duits2010}, and contractive \cite{Bruna2013} processing on SE(2), we work with the modulus of the complex valued orientation scores rather than with the complex-valued scores themselves (taking the modulus of quadrature filter responses is an effective technique for line detection, see e.g. Freeman et al. \cite{Freeman1991}).
\item \emph{Template Matching}. Finally we perform template matching using respectively Eqs.~(3),(4) and (5) of the main article for the $\mathbb{R}^2$ case and Eqs.~(3),(25) and (26) of the main article for the $SE(2)$ case.
\end{enumerate}
Regarding the image resolutions (step 1) we note that the average image size after rescaling was $[300\times 300]$. The average image resolutions in each database were as follows:
\begin{itemize}
\item \emph{ES (SLO)} contained images of average resolution $13.9\mu m /pix$.
\item \emph{TC} contained images of average resolution $9.4 \mu m /pix$.
\item \emph{MESSIDOR} contained images of 3 cameras with average resolutions $13.6\mu m /pix$, $9.1\mu m /pix$ and $8.6\mu m /pix$.
\item \emph{DRIVE} contained images of average resolution $21.9\mu m /pix$.
\item \emph{STARE} contained images of average resolution $17.6\mu m /pix$.
\end{itemize}
Regarding local image normalization (step 2) we note the following. Local image normalization was done using an adaptation of \cite{Foracchia2005}. The method first computes a local average and standard deviation of pixel intensities, and the image is locally normalized to zero mean and unit standard deviation. This is done via Eq.~(\ref{eq:fnormedapprox}). Then a background mask is construct by setting pixels with a larger distance than 1 standard deviation to the average (Mahalanobis distance) to 0, and other pixels to 1. This mask is then used to ignore outliers in a second computation of the local average and standard deviation. The final normalized image is again computed via Eq.~(\ref{eq:fnormedapprox}) but now with the inclusion of the background mask, see Eq.~(\ref{eq:productwithmask}).
\subsubsection{Template Settings}
In the retinal applications we used $\mathbb{R}^2$ templates of size $[N_x \times N_y] = [251 \times 251]$ which were covered by a grid of B-spline basis functions of size $[N_k \times N_l] = [51 \times 51]$, the $SE(2)$ templates were of size $[N_x \times N_y \times N_\theta] = [251 \times 251 \times 12]$ and were covered by a grid of B-spline basis functions of size $[N_k \times N_l \times N_m] = [51 \times 51 \times 12]$.
In the pupil detection application we used $\mathbb{R}^2$ templates of size $[N_x \times N_y] = [101 \times 101]$ which were also covered by a grid of B-spline basis functions of size $[N_k \times N_l] = [51 \times 51]$, the $SE(2)$ templates were of size $[N_x \times N_y \times N_\theta] = [101 \times 101 \times 12]$ and were also covered by a grid of B-spline basis functions of size $[N_k \times N_l \times N_m] = [51 \times 51 \times 12]$.
The regularization parameters ($\lambda$, $\mu$ and $D_{\theta\theta}$) for the different template types were automatically optimized using generalized cross validation.
\subsubsection{Timings}
We computed the average time for detecting one (or two) object(s) in an image and tabulated the results in Tab.~\ref{tab:timings}. Here we sub-divided the timings into the 5 processing steps explained in Subsec.~\ref{subsec:ProcessingPipeline}. The average (full) processing time on the retinal images was in both applications approximately $500ms$. When both the ONH and fovea are detected by the same processing pipeline the processing took $730ms$. For pupil detection the average time to detect $\emph{both}$ the left and right pupil on the \emph{full} images was $420ms$.
The retinal images were on average of size $[1230 \times 1792]$, and $[300\times300]$ after cropping and resizing. The images in the pupil detection application were not resized or cropped and were of size $[286 \times 384]$.
All experiments were performed using Wolfram \emph{Mathematica} 10.4, on a computer with an Intel Core i703612QM CPU and 8GB memory.
\subsection{Detection Results}
In this section we provide the results for the three separate applications. A general discussion of these results can be found in the main article.
\subsubsection{Optic Nerve Head Detection}
A Table of detection performance for each type of template is provided in Tab.~1 of the main article. In Fig.~\ref{fig:resultsOverviewImageONH} we show the 3 failed cases for ONH detection, and a selection of correct ONH localizations in difficult images. In Table \ref{tab:resultsONHAccuracy} we show detection results for varying accuracy criteria. Note that detection results are typically reported for the accuracy requirement of 1 optic disk radius with the target (see also state-of-the-art comparison in Table~2 of the main article).
\begin{figure*}
\begin{center}
\includegraphics[width=\linewidth]{Figs/resultsOverviewImageONH}
\end{center}
\caption{Detection results of our best method for optic nerve head detection in retinal images. Successful detection are indicated with a green frame around the image, failed detections are indicated with a red frame. In the ONH detection application there were only 3 fails in a set of 1737 images.
}
\label{fig:resultsOverviewImageONH}
\end{figure*}
\subsubsection{Fovea Detection}
A Table of detection performance for each type of template is provided in Tab.~\ref{tab:resultsFovea}. In Fig.~\ref{fig:resultsOverviewImageFovea} we show next to a selection of successful detections the only 5 failed cases on images from conventional fundus (CF) cameras (TC, MESSIDOR, DRIVE, STARE), and 3 of the failed detections in images coming from an scanning laser ophthalmoscopy (SLO) camera.
As can also be read from Tab.~\ref{tab:resultsFovea}, we found that fovea detection in SLO images was significantly more difficult than fovea detection in CF images. The reason for this is that on SLO images the clear dark blob-like shape is not always present on these images. Compare for example the positive fovea patches from Fig.~\ref{fig:TrainingSamples} (where one generally sees a dark blob at the center) with the fovea locations in the bottom row of images in Figs.~\ref{fig:resultsOverviewImageONH} and \ref{fig:resultsOverviewImageFovea}.
Additionally, the ES (SLO) and CF databases are also more difficult than the MESSIDOR database for fovea detection, as these two databases contain a mix of both fovea centered and ONH centered images. The MESSIDOR database contains only fovea centered images, in which case the fovea is always located around the center of the image. Therefore, even though MESSIDOR is one of the most used databases, it might not be the most representative database for fovea detection benchmarking.
We show detection performance for a range of accuracy requirements in Table~\ref{tab:resultsFoveaAccuracy} for the different databases used in our experiments, and in Table~\ref{tab:resultsFoveaAccuracyStateOfArt} a comparison to the state of the art. There we see that for the stricter requirement of detection within half an optic disk radius our method still outperforms the state of the art. We also see that with further decreasing the acceptance distance ($R/4$ or lower) none of the methods provided acceptable results.
\begin{table*
\centering
\caption{Average template matching results ($\pm$ standard deviation) for fovea detection in 5-fold cross validation, number of failed detections in parentheses.}
\begin{tabular}{l|lll|l}
\toprule
\multicolumn{1}{l|}{Template} & ES (SLO) & TC & MESSIDOR & All Images\\
ID & 208 & 208 & 1200 & 1616\\
\midrule
\multicolumn{5}{c}{{$\mathbb{R}^2$ templates}}\\
\midrule
$A_{\mathbb{R}^2}$ & 76.36\% {\tiny $\pm$ 6.79\%} (49) & 98.24\% {\tiny $\pm$ 2.74\%} (3) & 98.41\% {\tiny $\pm$ 0.22\%} (19) & \cellcolor{rowcolor}95.60\% {\tiny $\pm$ 0.98\%} (71) \vspace{\smallspacing}\\
$B_{lin:\mathbb{R}^2}$ & 23.50\% {\tiny $\pm$ 3.81\%} (159) & 31.66\% {\tiny $\pm$ 9.03\%} (142) & 51.19\% {\tiny $\pm$ 5.97\%} (587) & 45.07\% {\tiny $\pm$ 3.33\%} (888) \\
$C_{lin:\mathbb{R}^2}$ & 45.65\% {\tiny $\pm$ 8.61\%} (113) & 98.24\% {\tiny $\pm$ 2.74\%} (3) & 98.59\% {\tiny $\pm$ 0.36\%} (17) & 91.77\% {\tiny $\pm$ 1.26\%} (133) \\
$D_{lin:\mathbb{R}^2}$ & 44.21\% {\tiny $\pm$ 4.62\%} (116) & 99.49\% {\tiny $\pm$ 1.14\%} (1) & 98.84\% {\tiny $\pm$ 0.31\%} (14) & 91.90\% {\tiny $\pm$ 0.59\%} (131) \\
$E_{lin:\mathbb{R}^2}$ & 46.10\% {\tiny $\pm$ 8.11\%} (112) & 98.86\% {\tiny $\pm$ 1.57\%} (2) & 98.67\% {\tiny $\pm$ 0.34\%} (16) & 91.95\% {\tiny $\pm$ 1.18\%} \cellcolor{rowcolor}(130) \vspace{\smallspacing}\\
$B_{log:\mathbb{R}^2}$ & 1.43\% {\tiny $\pm$ 1.31\%} (205) & 10.27\% {\tiny $\pm$ 5.09\%} (185) & 20.07\% {\tiny $\pm$ 3.00\%} (959) & 16.53\% {\tiny $\pm$ 2.52\%} (1349) \\
$C_{log:\mathbb{R}^2}$ & 9.59\% {\tiny $\pm$ 3.74\%} (188) & 70.30\% {\tiny $\pm$ 8.57\%} (61) & 77.61\% {\tiny $\pm$ 4.64\%} (267) & 68.06\% {\tiny $\pm$ 3.53\%} (516) \\
$D_{log:\mathbb{R}^2}$ & 11.48\% {\tiny $\pm$ 4.70\%} (184) & 83.47\% {\tiny $\pm$ 7.80\%} (32) & 88.22\% {\tiny $\pm$ 2.81\%} (141) & 77.90\% {\tiny $\pm$ 2.00\%} \cellcolor{rowcolor}(357) \\
$E_{log:\mathbb{R}^2}$ & 2.86\% {\tiny $\pm$ 2.62\%} (202) & 79.68\% {\tiny $\pm$ 7.92\%} (40) & 84.79\% {\tiny $\pm$ 5.16\%} (181) & 73.82\% {\tiny $\pm$ 2.62\%} (423) \\
\midrule
\multicolumn{5}{c}{{$SE(2)$ templates}}\\
\midrule
$A_{SE(2)}$ & 67.81\% {\tiny $\pm$ 4.69\%} (67) & 79.13\% {\tiny $\pm$ 9.11\%} (40) & 98.25\% {\tiny $\pm$ 0.68\%} (21) & \cellcolor{rowcolor}92.08\% {\tiny $\pm$ 0.84\%} (128) \vspace{\smallspacing}\\
$B_{lin:SE(2)}$ & 83.19\% {\tiny $\pm$ 2.76\%} (35) & 71.53\% {\tiny $\pm$ 7.36\%} (58) & 91.31\% {\tiny $\pm$ 0.68\%} (104) & 87.81\% {\tiny $\pm$ 1.25\%} (197) \\
$C_{lin:SE(2)}$ & 83.65\% {\tiny $\pm$ 3.18\%} (34) & 84.13\% {\tiny $\pm$ 6.25\%} (32) & 98.23\% {\tiny $\pm$ 1.04\%} (21) & \cellcolor{rowcolor}94.62\% {\tiny $\pm$ 0.36\%} (87) \\
$D_{lin:SE(2)}$ & 73.57\% {\tiny $\pm$ 4.71\%} (55) & 83.69\% {\tiny $\pm$ 6.83\%} (33) & 97.88\% {\tiny $\pm$ 1.17\%} (25) & 93.01\% {\tiny $\pm$ 1.09\%} (113) \\
$E_{lin:SE(2)}$ & 77.83\% {\tiny $\pm$ 4.29\%} (46) & 84.88\% {\tiny $\pm$ 6.69\%} (30) & 98.22\% {\tiny $\pm$ 1.23\%} (21) & 94.00\% {\tiny $\pm$ 0.93\%} (97) \vspace{\smallspacing}\\
$B_{log:SE(2)}$ & 75.49\% {\tiny $\pm$ 5.73\%} (51) & 60.80\% {\tiny $\pm$ 5.68\%} (80) & 92.79\% {\tiny $\pm$ 1.98\%} (86) & 86.56\% {\tiny $\pm$ 2.20\%} (217) \\
$C_{log:SE(2)}$ & 79.33\% {\tiny $\pm$ 6.57\%} (43) & 70.87\% {\tiny $\pm$ 10.28\%} (59) & 96.90\% {\tiny $\pm$ 0.71\%} (37) & 91.39\% {\tiny $\pm$ 1.36\%} \cellcolor{rowcolor}(139) \\
$D_{log:SE(2)}$ & 62.09\% {\tiny $\pm$ 6.66\%} (79) & 72.57\% {\tiny $\pm$ 8.59\%} (54) & 96.64\% {\tiny $\pm$ 1.05\%} (40) & 89.30\% {\tiny $\pm$ 0.63\%} (173) \\
$E_{log:SE(2)}$ & 68.34\% {\tiny $\pm$ 8.59\%} (66) & 72.20\% {\tiny $\pm$ 8.53\%} (55) & 96.57\% {\tiny $\pm$ 0.96\%} (41) & 89.98\% {\tiny $\pm$ 1.25\%} (162) \\
\midrule
\multicolumn{5}{c}{{Template combinations (sorted on performance)}}\\
\midrule
$C_{lin:\mathbb{R}^2}+C_{log:SE(2)}$ & 97.17\% {\tiny $\pm$ 3.01\%} (6) & 99.17\% {\tiny $\pm$ 1.13\%} (2) & 99.74\% {\tiny $\pm$ 0.38\%} (3) & \cellcolor{rowcolor}99.32\% {\tiny $\pm$ 0.26\%} (11) \\%\vspace{\smallspacing}\\
\hspace{-0.8em}$^*$ $A_{\mathbb{R}^2} \;\;\;\;\; +C_{lin:SE(2)}$ & 98.08\% {\tiny $\pm$ 2.03\%} (4) & 98.07\% {\tiny $\pm$ 1.95\%} (4) & 99.68\% {\tiny $\pm$ 0.33\%} (4) & 99.26\% {\tiny $\pm$ 0.47\%} (12) \\%\vspace{\smallspacing}\\
$E_{lin:\mathbb{R}^2}+C_{log:SE(2)}$ & 96.20\% {\tiny $\pm$ 3.15\%} (8) & 99.17\% {\tiny $\pm$ 1.13\%} (2) & 99.75\% {\tiny $\pm$ 0.23\%} (3) & 99.20\% {\tiny $\pm$ 0.35\%} (13) \\
$E_{lin:\mathbb{R}^2}+C_{lin:SE(2)}$ & 96.65\% {\tiny $\pm$ 2.13\%} (7) & 99.17\% {\tiny $\pm$ 1.13\%} (2) & 99.66\% {\tiny $\pm$ 0.36\%} (4) & 99.19\% {\tiny $\pm$ 0.42\%} (13) \\%\vspace{\smallspacing}\\
$C_{lin:\mathbb{R}^2}+C_{lin:SE(2)}$ & 97.14\% {\tiny $\pm$ 1.97\%} (6) & 98.78\% {\tiny $\pm$ 1.78\%} (3) & 99.58\% {\tiny $\pm$ 0.31\%} (5) & 99.13\% {\tiny $\pm$ 0.40\%} (14) \\
$A_{\mathbb{R}^2} \;\;\;\;\; +E_{lin:SE(2)}$ & 97.59\% {\tiny $\pm$ 1.73\%} (5) & 98.07\% {\tiny $\pm$ 1.95\%} (4) & 99.59\% {\tiny $\pm$ 0.28\%} (5) & 99.13\% {\tiny $\pm$ 0.25\%} (14) \\%\vspace{\smallspacing}\\
$E_{lin:\mathbb{R}^2}+E_{lin:SE(2)}$ & 96.16\% {\tiny $\pm$ 2.76\%} (8) & 99.17\% {\tiny $\pm$ 1.13\%} (2) & 99.58\% {\tiny $\pm$ 0.31\%} (5) & 99.07\% {\tiny $\pm$ 0.38\%} (15) \\%\vspace{\smallspacing}\\
$E_{lin:\mathbb{R}^2}+D_{lin:SE(2)}$ & 95.71\% {\tiny $\pm$ 3.07\%} (9) & 99.17\% {\tiny $\pm$ 1.13\%} (2) & 99.58\% {\tiny $\pm$ 0.31\%} (5) & 99.01\% {\tiny $\pm$ 0.40\%} (16) \\
$C_{lin:\mathbb{R}^2}+E_{lin:SE(2)}$ & 96.16\% {\tiny $\pm$ 2.76\%} (8) & 98.78\% {\tiny $\pm$ 1.78\%} (3) & 99.58\% {\tiny $\pm$ 0.31\%} (5) & 99.01\% {\tiny $\pm$ 0.51\%} (16) \\
$A_{\mathbb{R}^2} \;\;\;\;\; +C_{log:SE(2)}$ & 96.65\% {\tiny $\pm$ 2.13\%} (7) & 98.07\% {\tiny $\pm$ 1.95\%} (4) & 99.58\% {\tiny $\pm$ 0.42\%} (5) & 99.01\% {\tiny $\pm$ 0.26\%} (16) \\
\multicolumn{1}{l|}{{\;\;\;\;\;\;\;\;\;\;\;\;\;...}} & \multicolumn{3}{c|}{{...}} & \multicolumn{1}{c}{{...}}\\
\hspace{-0.8em}$^\dagger$ $A_{\mathbb{R}^2} \;\;\;\;\; +A_{SE(2)}$ & 92.85 \% {\tiny $\pm$ 4.68\%} (15) & 95.84\% {\tiny $\pm$ 2.58\%} (8) & 99.58\% {\tiny $\pm$ 0.30\%} (5) & 98.27\% {\tiny $\pm$ 0.70\%} (28) \\
\multicolumn{1}{l|}{{\;\;\;\;\;\;\;\;\;\;\;\;\;...}} & \multicolumn{3}{c|}{{...}} & \multicolumn{1}{c}{{...}}\\
\bottomrule
\multicolumn{5}{l}{$^*$\emph{Best template combination that does not rely on logistic regression.}}\\
\multicolumn{5}{l}{$^\dagger$\emph{Best template combination that does not rely on template optimization.}}
\end{tabular}
\label{tab:resultsFovea}
\end{table*}
\begin{table*
\centering
\caption{Success rates for fovea detection ($\pm$ standard deviation, number of fails in parenthesis) with varying accuracy requirements in 5-fold cross validation. Maximum distance to ground truth location is expressed in optic disk radius $R$.}
\begin{tabular}{l|lllll}
\toprule
& \multicolumn{5}{c}{Maximum distance to ground truth}\\
Database (\# of images) & \;\;\;\;\;\;\;\;\;\;\;\;R/8\;\;\;\;\;\;\;\;\;\;\;\; & \;\;\;\;\;\;\;\;\;\;\;\;R/4\;\;\;\;\;\;\;\;\;\;\;\; & \;\;\;\;\;\;\;\;\;\;\;\;R/2\;\;\;\;\;\;\;\;\;\;\;\; & \;\;\;\;\;\;\;\;\;\;\;\;R\;\;\;\;\;\;\;\;\;\;\;\; & \;\;\;\;\;\;\;\;\;\;\;\;2R\;\;\;\;\;\;\;\;\;\;\;\;\\
\midrule
ES (SLO)\;\;\;\;\;\;\;(208) & 66.91\% {\tiny $\pm$ 4.64\%} (69) & 92.85\% {\tiny $\pm$ 3.16\%} (15) & 94.74\% {\tiny $\pm$ 1.93\%} (11) & 97.17\% {\tiny $\pm$ 3.01\%} (6) & 97.66\% {\tiny $\pm$ 3.28\%} (5)\\
TC\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(208) & 49.51\% {\tiny $\pm$ 4.07\%} (106) & 80.33\% {\tiny $\pm$ 3.22\%} (40) & 95.41\% {\tiny $\pm$ 1.77\%} (9) & 99.17\% {\tiny $\pm$ 1.13\%} (2) & 99.61\% {\tiny $\pm$ 0.88\%} (1)\\
MESSIDOR \;\;(1200) & 61.81\% {\tiny $\pm$ 2.64\%} (459) & 90.56\% {\tiny $\pm$ 1.31\%} (113) & 98.07\% {\tiny $\pm$ 0.87\%} (23) & 99.74\% {\tiny $\pm$ 0.38\%} (3) & 100.0\% {\tiny $\pm$ 0.00\%} (0)\\
&&&&&\\
All Images\;\;\;\;(1616) & 60.78\% {\tiny $\pm$ 1.84\%} (634) & 89.60\% {\tiny $\pm$ 0.80\%} (168) & 97.34\% {\tiny $\pm$ 0.65\%} (43) & 99.32\% {\tiny $\pm$ 0.26\%} (11) & 99.63\% {\tiny $\pm$ 0.40\%} (6)\\
\bottomrule
\end{tabular}
\label{tab:resultsFoveaAccuracy}
\end{table*}
\begin{table*
\centering
\caption{Success rates for fovea detection (number of fails in parenthesis) with varying accuracy requirements; a comparison to literature using the MESSIDOR database. Maximum distance to ground truth location is expressed in optic disk radius $R$.}
\begin{tabular}{l|lllll}
\toprule
& \multicolumn{5}{c}{Maximum distance to ground truth}\\
Method & \;\;\;\;\;\;\;\;\;\;\;\;R/8\;\;\;\;\;\;\;\;\;\;\;\; & \;\;\;\;\;\;\;\;\;\;\;\;R/4\;\;\;\;\;\;\;\;\;\;\;\; & \;\;\;\;\;\;\;\;\;\;\;\;R/2\;\;\;\;\;\;\;\;\;\;\;\; & \;\;\;\;\;\;\;\;\;\;\;\;R\;\;\;\;\;\;\;\;\;\;\;\; & \;\;\;\;\;\;\;\;\;\;\;\;2R\;\;\;\;\;\;\;\;\;\;\;\;\\
\midrule
Niemeijer {\tiny et al. \cite{Niemeijer2009,GegundezArias2013} }& 75.67\% (292)& 93.50\% (78) & 96.83\% (38) & 97.92\% (25) & - \\
Yu et al. {\tiny et al. \cite{Yu2011}} & - & - & 95.00\% (60) & - & - \\
Gegundez-Arias {\tiny et al. \cite{GegundezArias2013}} & 80.42\% (235) & 93.90\% (73)& 96.08\% (47)& 96.92\% (37)& 97.83\% (26)\\
Giachetti {\tiny et al. \cite{Giachetti2013}} & - & - & - & 99.10\% (11) & - \\
Aquino {\tiny \cite{Aquino2014}} & - & - & - & 98.20\% (21) & - \\
&&&&&\\
Proposed & 61.81\% (459) & 90.56\% (113) & 98.07\% (23) & 99.74\% (3) & 100.0\% (0)\\
\bottomrule
\end{tabular}
\label{tab:resultsFoveaAccuracyStateOfArt}
\end{table*}
\begin{figure*}
\begin{center}
\includegraphics[width=\linewidth]{Figs/resultsOverviewImageFovea}
\end{center}
\caption{Detection results of our best method for fovea detection in retinal images. Successful detection are indicated with a green frame around the image, failed detections are indicated with a red frame. In the fovea detection application there were only 5 fails in a set of 1408 conventional fundus (CF) camera images. Out of the 208 scanning laser ophthalmoscopy (SLO) images there were 6 fails, 3 of them are shown in this figure.
}
\label{fig:resultsOverviewImageFovea}
\end{figure*}
\subsubsection{Pupil Detection}
\label{subsec:pupilResults}
A Table of detection performance for each type of template is provided in Tab.~\ref{tab:resultsPupil}. In Fig.~\ref{fig:resultsOverviewImagePupil} we show a selection of failed and successful detections. By inspection of the failed cases we found that a main source of failed detections was due to rotations of the head. As stated in the previous section \ref{subsec:implementationdetails} we did not employ a rotation invariant detection scheme. Doing so might improve the results. Other failed detections could be attributed to closed eyes, reflection of glasses, distracting background objects and different scales (object distance to camera).
\begin{table}
\centering
\caption{Average template matching results ($\pm$ standard deviation) for pupil detection in 5-fold cross validation, number of failed detections in parentheses. A successful detection has a normalized error $e\le 0.1$.}
\begin{tabular}{l|ll}
\toprule
\multicolumn{1}{l|}{Template} & BioID (Full image)\;\;\; & BioID (Periocular image)\\
ID & 1521 & 1521 \\
\midrule
\multicolumn{3}{c}{{$\mathbb{R}^2$ templates}}\\
\midrule
$A_{\mathbb{R}^2}$ & \cellcolor{rowcolor}41.03\% {\tiny $\pm$ 1.45\%} (897) & \cellcolor{rowcolor}59.70\% {\tiny $\pm$ 1.52\%} (613) \vspace{\smallspacing}\\
$B_{lin:\mathbb{R}^2}$ & 0.00\% \hspace{0.5em}{\tiny $\pm$ 0.00\%} (1521) & 3.62\% \hspace{0.5em}{\tiny $\pm$ 1.09\%} (1466) \\
$C_{lin:\mathbb{R}^2}$ & \cellcolor{rowcolor}12.95\% {\tiny $\pm$ 2.22\%} (1324) & 67.26\% {\tiny $\pm$ 2.55\%} (498) \\
$D_{lin:\mathbb{R}^2}$ & 8.28\% \hspace{0.5em}{\tiny $\pm$ 1.80\%} (1395) & \cellcolor{rowcolor}75.68\% {\tiny $\pm$ 2.33\%} (370) \\
$E_{lin:\mathbb{R}^2}$ & 11.51\% {\tiny $\pm$ 2.25\%} (1346) & 71.47\% {\tiny $\pm$ 2.76\%} (434) \vspace{\smallspacing}\\
$B_{log:\mathbb{R}^2}$ & 0.00\%\hspace{0.5em} {\tiny $\pm$ 0.00\%} (1521) & 0.00\% \hspace{0.5em}{\tiny $\pm$ 0.00\%} (1521) \\
$C_{log:\mathbb{R}^2}$ & \cellcolor{rowcolor}12.89\% {\tiny $\pm$ 2.06\%} (1325) & \cellcolor{rowcolor}39.91\% {\tiny $\pm$ 3.37\%} (914) \\
$D_{log:\mathbb{R}^2}$ & 1.84\% \hspace{0.5em}{\tiny $\pm$ 0.95\%} (1493) & 22.09\% {\tiny $\pm$ 2.37\%} (1185) \\
$E_{log:\mathbb{R}^2}$ & 10.39\% {\tiny $\pm$ 2.26\%} (1363) & 37.21\% {\tiny $\pm$ 4.37\%} (955) \\
\midrule
\multicolumn{3}{c}{{$SE(2)$ templates}}\\
\midrule
$A_{SE(2)}$ & \cellcolor{rowcolor}57.72\% {\tiny $\pm$ 1.68\%} (643) & \cellcolor{rowcolor}75.34\% {\tiny $\pm$ 1.31\%} (375) \vspace{\smallspacing}\\
$B_{lin:SE(2)}$ & 8.74\% \hspace{0.5em}{\tiny $\pm$ 2.00\%} (1388) & 41.81\% {\tiny $\pm$ 5.04\%} (885) \\
$C_{lin:SE(2)}$ & 84.61\% {\tiny $\pm$ 4.19\%} (234) & 86.78\% {\tiny $\pm$ 3.68\%} (201) \\
$D_{lin:SE(2)}$ & \cellcolor{rowcolor}85.53\% {\tiny $\pm$ 3.44\%} (220) & \cellcolor{rowcolor}87.18\% {\tiny $\pm$ 3.71\%} (195) \\
$E_{lin:SE(2)}$ & 85.47\% {\tiny $\pm$ 3.82\%} (221) & 87.11\% {\tiny $\pm$ 3.87\%} (196) \vspace{\smallspacing}\\
$B_{log:SE(2)}$ & 0.00\% \hspace{0.5em}{\tiny $\pm$ 0.00\%} (1521) & 0.13\%\hspace{0.5em} {\tiny $\pm$ 0.29\%} (1519) \\
$C_{log:SE(2)}$ & 86.52\% \cellcolor{rowcolor}{\tiny $\pm$ 0.77\%} (205) & \cellcolor{rowcolor}93.95\% {\tiny $\pm$ 1.33\%} (92) \\
$D_{log:SE(2)}$ & 75.21\% {\tiny $\pm$ 2.18\%} (377) & 89.48\% {\tiny $\pm$ 2.27\%} (160) \\
$E_{log:SE(2)}$ & 83.30\% {\tiny $\pm$ 1.68\%} (254) & 92.77\% {\tiny $\pm$ 1.02\%} (110) \\
\midrule
\multicolumn{3}{c}{{Template combinations (sorted on performance full image)}}\\
\midrule
\hspace{-0.6em}$^*$$C_{lin:\mathbb{R}^2} + E_{lin:SE(2)}$ & \cellcolor{rowcolor}93.49\% {\tiny $\pm$ 1.49\%} (99) & 95.60\% {\tiny $\pm$ 1.46\%} (67) \\
$C_{lin:\mathbb{R}^2} + D_{lin:SE(2)}$ & 93.16\% {\tiny $\pm$ 1.54\%} (104) & 95.00\% {\tiny $\pm$ 1.15\%} (76) \\
$E_{lin:\mathbb{R}^2} + E_{lin:SE(2)}$ & 93.10\% {\tiny $\pm$ 1.04\%} (105) & 95.59\% {\tiny $\pm$ 0.89\%} (67) \\
$E_{lin:\mathbb{R}^2} + D_{lin:SE(2)}$ & 92.97\% {\tiny $\pm$ 1.62\%} (107) & 95.27\% {\tiny $\pm$ 1.31\%} (72) \\
$C_{lin:\mathbb{R}^2} + C_{lin:SE(2)}$ & 92.70\% {\tiny $\pm$ 1.41\%} (111) & 95.33\% {\tiny $\pm$ 0.97\%} (71) \\
$E_{lin:\mathbb{R}^2} + C_{lin:SE(2)}$ & 92.64\% {\tiny $\pm$ 0.94\%} (112) & 95.33\% {\tiny $\pm$ 0.94\%} (71) \\
$D_{lin:\mathbb{R}^2} + D_{lin:SE(2)}$ & 92.51\% {\tiny $\pm$ 0.96\%} (114) & 95.79\% {\tiny $\pm$ 0.82\%} (64) \\
$D_{lin:\mathbb{R}^2} + E_{lin:SE(2)}$ & 92.24\% {\tiny $\pm$ 1.23\%} (118) & 95.86\% {\tiny $\pm$ 0.89\%} (63) \\
$E_{log:\mathbb{R}^2} + D_{lin:SE(2)}$ & 92.11\% {\tiny $\pm$ 2.26\%} (120) & 93.23\% {\tiny $\pm$ 1.93\%} (103) \\
$D_{lin:\mathbb{R}^2} + C_{log:SE(2)}$ & 92.05\% {\tiny $\pm$ 1.52\%} (121) & 95.14\% {\tiny $\pm$ 0.78\%} (74) \\
\multicolumn{1}{l|}{{\;\;\;\;\;\;\;\;\;\;\;\;\;...}} & \multicolumn{2}{c}{{...\;\;\;\;\;\;\;\;\;\;}}\\
\midrule
\multicolumn{3}{c}{{Template combinations (sorted on performance periocular image)}}\\
\midrule
\hspace{-0.6em}$^*$$D_{lin:\mathbb{R}^2} + E_{lin:SE(2)}$ & 92.24\% {\tiny $\pm$ 1.23\%} (118) &\cellcolor{rowcolor}95.86\% {\tiny $\pm$ 0.89\%} (63) \\
$D_{lin:\mathbb{R}^2} + D_{lin:SE(2)}$ & 92.51\% {\tiny $\pm$ 0.96\%} (114) & 95.79\% {\tiny $\pm$ 0.82\%} (64) \\
$D_{lin:\mathbb{R}^2} + C_{lin:SE(2)}$ & 91.52\% {\tiny $\pm$ 1.25\%} (129) & 95.73\% {\tiny $\pm$ 0.77\%} (65) \\
$E_{lin:\mathbb{R}^2} + E_{lin:SE(2)}$ & 93.10\% {\tiny $\pm$ 1.04\%} (105) & 95.59\% {\tiny $\pm$ 0.89\%} (67) \\
$C_{lin:\mathbb{R}^2} + E_{lin:SE(2)}$ & 93.49\% {\tiny $\pm$ 1.49\%} (99) & 95.60\% {\tiny $\pm$ 1.46\%} (67) \\
$E_{lin:\mathbb{R}^2} + C_{lin:SE(2)}$ & 92.64\% {\tiny $\pm$ 0.94\%} (112) & 95.33\% {\tiny $\pm$ 0.94\%} (71) \\
$C_{lin:\mathbb{R}^2} + C_{lin:SE(2)}$ & 92.70\% {\tiny $\pm$ 1.41\%} (111) & 95.33\% {\tiny $\pm$ 0.97\%} (71) \\
$E_{lin:\mathbb{R}^2} + D_{lin:SE(2)}$ & 92.97\% {\tiny $\pm$ 1.62\%} (107) & 95.27\% {\tiny $\pm$ 1.31\%} (72) \\
$D_{lin:\mathbb{R}^2} + E_{log:SE(2)}$ & 91.72\% {\tiny $\pm$ 1.23\%} (126) & 95.27\% {\tiny $\pm$ 0.79\%} (72) \\
$D_{lin:\mathbb{R}^2} + C_{log:SE(2)}$ & 92.05\% {\tiny $\pm$ 1.52\%} (121) & 95.14\% {\tiny $\pm$ 0.78\%} (74) \\
\multicolumn{1}{l|}{{\;\;\;\;\;\;\;\;\;\;\;\;\;...}} & \multicolumn{2}{c}{{...\;\;\;\;\;\;\;\;\;\;}}\\
\hspace{-0.6em}$^\dagger$ $A_{\mathbb{R}^2} \;\;\;\;\,+ A_{SE(2)}$ & 61.34\% {\tiny $\pm$ 1.54\%} (588) & 68.18\% {\tiny $\pm$ 1.25\%} (484) \\
\multicolumn{1}{l|}{{\;\;\;\;\;\;\;\;\;\;\;\;\;...}} & \multicolumn{2}{c}{{...\;\;\;\;\;\;\;\;\;\;}}\\
\bottomrule
\multicolumn{3}{l}{$^*$\emph{Best template combination that does not rely on logistic regression.}}\\
\multicolumn{3}{l}{$^\dagger$\emph{Best template combination that does not rely on template optimization.}}
\end{tabular}
\label{tab:resultsPupil}
\end{table}
\begin{figure*}
\begin{center}
\includegraphics[width=\linewidth]{Figs/resultsOverviewImagePupil}
\end{center}
\caption{Detection results of our best method for pupil detection. Successful detection are indicated with a green frame around the image, failed detections are indicated with a red frame.
}
\label{fig:resultsOverviewImagePupil}
\end{figure*}
\section{Rotation-Scale Invariant Matching}
\subsection{A Basic Extension}
\label{subsec:basicExtension}
The extension to rotation and scale invariant object localization of the 2D cross-correlation based template matching approach, described in Eqs.~(3)-(5) of the main article, is as follows. For the linear potential function (Eq.~(4) of the main article) we can define
\begin{equation}
\label{eq:linearFunctionalInv}
P_{lin,inv}^{\mathbb{R}^2}(\mathbf{x}) := \underset{\begin{array}{c}a\in[a_-,a_+],\\\alpha \in [0,2\pi)\end{array}}{\operatorname{max}}( \mathcal{T}_\mathbf{x} \mathcal{S}_a \mathcal{R}_\alpha \;t , f)_{\mathbb{L}_2(\mathbb{R}^2)},
\end{equation}
and for the logistic regression case (Eq.~(5) of the main article) we define
\begin{equation}
\label{eq:logisticFunctionalInv}
P_{log,inv}^{\mathbb{R}^2}(\mathbf{x}) := \underset{\begin{array}{c}a\in[a_-,a_+],\\\alpha \in [0,2\pi)\end{array}}{\operatorname{max}} \sigma \left( ( \mathcal{T}_\mathbf{x} \mathcal{S}_a \mathcal{R}_\alpha \;t , f)_{\mathbb{L}_2(\mathbb{R}^2)} \right),
\end{equation}
with $\sigma$ the logistic sigmoid function defined in Eq.~(5) of the main article, and with rotation operator $\mathcal{R}_\alpha$ and scaling operator $\mathcal{S}_a$ defined by
\begin{align}
(\mathcal{R}_\alpha t)(\mathbf{x}) &= t(\mathbf{R}_\alpha^{-1} \mathbf{x}),\\
(\mathcal{S}_a t)(\mathbf{x}) &= a^{-1} t(a \mathbf{x}),
\end{align}
with rotation matrix $\mathbf{R}_\alpha$ representing a counter clockwise rotation of angle $\alpha$.
By taking the maximum over scales $a$ (in a suitable range $[a_-,a_+]$) and rotations $\alpha$, the response of the best matching template is obtained at each location $\mathbf{x}$, and invariance is obtained with respect to scaling and rotation of the object of interest.
The rotation/scale invariant extension of the $SE(2)$ cross-correlation template matching case (Eqs.~(25)-(26) of the main article) is equally straightforward; for the linear potential we define
\begin{equation}
\label{eq:linearFunctionalInvSE2}
{P}_{lin,inv}^{SE(2)}(\mathbf{x}) := \underset{\begin{array}{c}a\in[a_-,a_+],\\\alpha \in [0,2\pi)\end{array}}{\operatorname{max}}( \mathcal{T}_{\mathbf{x}} \mathcal{S}_a \mathcal{R}_\alpha \; T , U_{f})_{\mathbb{L}_2(SE(2))},
\end{equation}
and for the logistic potential we define
\begin{equation}
\label{eq:logisticFunctionalInvSE2}
{P}_{log,inv}^{SE(2)}(\mathbf{x}) := \underset{\begin{array}{c}a\in[a_-,a_+],\\\alpha \in [0,2\pi)\end{array}}{\operatorname{max}} \sigma \left( ( \mathcal{T}_{\mathbf{x}} \mathcal{S}_a \mathcal{R}_\alpha \; T , U_{f})_{\mathbb{L}_2(SE(2))} \right),
\end{equation}
with for orientation score objects $T,U_f \in \mathbb{L}_2(SE(2))$ the rotation and scaling operators defined respectively by
\begin{align}
(\mathcal{R}_\alpha T)(\mathbf{x},\theta) &= T(\mathbf{R}_\alpha^{-1} \mathbf{x}, \theta - \alpha),\\
(\mathcal{S}_a T)(\mathbf{x},\theta) &= a^{-1} T(a \mathbf{x},\theta).
\end{align}
It depends on the addressed template matching problem whether or not such invariance is desirable or not. In many applications the object is to be found in a human environment context, in which some objects tend to appear in specific orientations or at typical scales, and in which case rotation/scale invariance might not be desirable. E.g. the sizes of anatomical structures in the retina are relatively constant among different subjects (constant scale) and retinal images are typically taken at a fixed orientation (constant rotation). In the pupil detection problem the subjects typically appear in upright position behind the camera (constant rotation), and within a reasonable distance to the camera (constant scale). In the next Subsec.~\ref{subsubsec:detectionResults} we indeed show that in the applications considered in this manuscript rotation/scale invariance is not necessarily a desired property, and that computation time linearly increases with the number of rotations/scalings tested for (cf. Subsec.~\ref{subsubsec:timings}).
\subsection{Results with Rotation and Scale Invariance}
Here we perform rotation and scale invariant template matching via the extension described in Subsec.~\ref{subsec:basicExtension}. We selected the best template combination for each specific application and compared non-invariant template matching (as described in the main article) to rotation and/or scale invariant template matching (Subsec.~\ref{subsec:basicExtension}). The best template combination for ONH detection was $A_{\mathbb{R}^2}+C_{log:SE(2)}$, for fovea detection this was $C_{lin:\mathbb{R}^2}+C_{log:SE(2)}$, and for pupil detection this was $D_{lin:\mathbb{R}^2}+E_{lin:SE(2)}$.
For the retinal applications we only tested for rotation invariance with
$$
\alpha \in \{-\frac{\pi}{6}, -\frac{\pi}{8}, -\frac{\pi}{12}, -\frac{\pi}{24}, 0, \frac{\pi}{24} , \frac{\pi}{12}, \frac{\pi}{8}, \frac{\pi}{6}\},
$$
and did not included scale invariance since each retinal image was already rescaled to a standardized resolution (see Subsec.~\ref{subsec:ProcessingPipeline}). In pupil detection we tested for a range of scalings with
$$
a \in \{0.7,0.8,0.9,1.0,1.1,1.2,1.3\}
$$
to deal with varying pupil sizes caused by varying distances to the camera; and we tested for a range of rotations with
$$
\alpha \in \{-\frac{\pi}{4}, -\frac{\pi}{8}, -\frac{\pi}{16}, 0, \frac{\pi}{16} , \frac{\pi}{8}, \frac{\pi}{4}\}
$$
to deal with rotations of the head.
\subsubsection{Detection Results}
\label{subsubsec:detectionResults}
The detection results are shown in Table.~\ref{tab:resultsInvariant}. Here we can see that in all three applications the inclusion of a rotation/scale invariant matching scheme results in a slight decrease in performance.
This can be explained by the fact that variations in scale an rotation within the databases are small, and that the trained templates can already deal robustly with these variations (due to the presence of such variations in the training set). By introducing rotation/scale invariance one then only increases the likelihood of false positive detections.
\subsubsection{Computation Time}
\label{subsubsec:timings}
The effect on computation time of rotation/scale invariant matching is shown in Fig.~\ref{fig:RotScalTimings}. Here one sees that computation time linearly increases with the number of template rotations and scalings tested for. This timings-experiment is performed on the pupil detection application, and the shown timings are only of \emph{step 5} of the full detection pipeline (see Subsec.~\ref{subsec:ProcessingPipeline} and Table 1) as this is the only step that is affected by the rotation/scale invariant extension.
\begin{table}
\centering
\caption{Average template matching results ($\pm$ standard deviation, number of fails between parenthesis) for optic nerve head (ONH), fovea, and pupil detection in 5-fold cross validation.
}
\begin{tabular}{ll}
\toprule
Method & Success rate\\
\midrule
\midrule
\multicolumn{2}{l}{\;\;\;\;\;\;\;ONH Detection \;\;\; (1737 images)}\\
\midrule
No invariance & 99.83\% {\tiny $\pm$ 0.26\%} (3)\\
Rotation invariance & 99.60\% {\tiny $\pm$ 0.16\%} (7)\\
\midrule
\multicolumn{2}{l}{\;\;\;\;\;\;\;Fovea Detection \;\; (1616 images)}\\
\midrule
No invariance & 99.32\% {\tiny $\pm$ 0.26\%} (11)\\
Rotation invariance & 97.10\% {\tiny $\pm$ 0.65\%} (47)\\
\midrule
\multicolumn{2}{l}{\;\;\;\;\;\;\;Pupil Detection \;\;\; (1521 images)}\\
\midrule
No invariance & 95.86\% {\tiny $\pm$ 0.89\%} (63)\\
Rotation invariance & 94.48\% {\tiny $\pm$ 1.62\%} (84)\\
Scale invariance & 95.33\% {\tiny $\pm$ 1.46\%} (71)\\
Rotation + scale invariance & 94.28\% {\tiny $\pm$ 2.10\%} (87)\\
\bottomrule
\end{tabular}
\label{tab:resultsInvariant}
\end{table}
\begin{figure}
\centerline{
\includegraphics[width=\hsize]{Figs/RotScalTimings.png}
}
\caption{
Average computation times for the detection of one pupil/eye per image, using $SE(2)$ or $\mathbb{R}^2$ templates, testing for different template orientations or scalings. Two experiments are shown, in blue the number of template orientations $N_\alpha=1$ and the number of scalings $N_a$ is varied, in orange-dashed the number of scalings $N_a=1$ and the number of rotations is $N_\alpha$ is varied.
\label{fig:RotScalTimings}}
\end{figure}
\bibliographystyle{IEEEtran}
\subsection{Paper Outline}
\emph{\textbf{Paper Outline.}}
The remainder of this paper is organized as follows. In Sec.~\ref{sec:templateMatchingR2} we provide the theory for template matching and template construction in the $\mathbb{R}^2$-case. The theory is then extended to the $SE(2)$-case in Sec.~\ref{sec:templateMatchingSE2}.
Additionally, in Sec.~\ref{sec:stochasticProcess} we provide a probabilistic interpretation of the proposed $SE(2)$ prior, and relate it to Brownian motions on $SE(2)$.
In Sec.~\ref{sec:applications} we apply the method to retinal images for ONH (Subsec.~\ref{subsec:ONHDetection}) and fovea detection (Subsec.~\ref{subsec:FoveaDetection}), and to regular camera images for pupil detection (Subsec.~\ref{subsec:PupilDetection}). Finally, we conclude the paper in Sec.~\ref{sec:discussionAndConclusion}.
\subsection{Object Detection via Cross-Correlation}
\label{subsec:objectDetectionR2}
We are considering the problem of finding the location of objects (with specific orientation patterns) in an image. While in principle an image may contain multiple objects of interest, the applications discussed
in this paper only require the detection of one object per image. We search for the most likely location
\begin{equation}
\label{eq:objectDetection}
\mathbf{x}^* = \underset{\mathbf{x} \in \mathbb{R}^2 }{\operatorname{argmax}} \;\;P(\mathbf{x}),
\end{equation}
with $P(\mathbf{x}) \in \mathbb{R}$ denoting the objective functional for finding the object of interest at location $\mathbf{x}$. We define $P$ based on inner products in a \emph{linear regression} and \emph{logistic regression} context, where we respectively define $P$ by
\begin{equation}
\label{eq:linearFunctional}
P(\mathbf{x}) = P_{lin}^{\mathbb{R}^2}(\mathbf{x}) := ( \mathcal{T}_\mathbf{x} \;t , f)_{\mathbb{L}_2(\mathbb{R}^2)},
\end{equation}
or
\begin{equation}
\label{eq:logisticFunctional}
\begin{aligned}
& P(\mathbf{x}) = P_{log}^{\mathbb{R}^2}(\mathbf{x}) := \sigma \left( ( \mathcal{T}_\mathbf{x} \; t , f)_{\mathbb{L}_2(\mathbb{R}^2)} \right),\\
& \text{with} \;\;\;\; \sigma(x) =e^x/(1+e^x),
\end{aligned}
\end{equation}
where $\mathcal{T}_\mathbf{x}$ denotes translation by $\mathbf{x}$ via
$$
( \mathcal{T}_\mathbf{x} t )(\tilde{\mathbf{x}}) = t(\tilde{\mathbf{x}} - \mathbf{x}),
$$
and where the $\mathbb{L}_2(\mathbb{R}^2)$ inner product is given by
\begin{equation}
(t,f)_{\mathbb{L}_2 (\mathbb{R}^2)} := \int_{\mathbb{R}^2} \overline{t(\tilde{\mathbf{x}})} f(\tilde{\mathbf{x}}) {\rm d}\tilde{\mathbf{x}},
\end{equation}
with associated norm $\lVert \cdot \rVert_{\mathbb{L}_2(\mathbb{R}^2)} = \sqrt{ (\cdot , \cdot )_{\mathbb{L}_2(\mathbb{R}^2)} }$.
Note that the inner-product based potentials $P(\mathbf{x})$ can be efficiently evaluated for each $\mathbf{x}$ using convolutions.
For a generalization of cross-correlation based template matching to \emph{normalized} cross correlation, we refer the reader to the supplementary materials. For speed considerations we will however not use normalized cross correlation, but instead use a (fast) preprocessing step to locally normalize the images (cf. Subsec.~\ref{subsubsec:processingPipeline}).
\subsection{Optimizing $t$ Using Linear Regression}
\label{subsec:linearRegresionR2}
Our aim is to construct templates $t$ that are ``aligned'' with image patches that contain the object of interest, and which are orthogonal to non-object patches. Hence, template $t$ is found via the minimization of the following energy
\begin{multline}
\label{eq:energyR2LinearRegression}
E_{lin}(t) = \sum\limits_{i=1}^N \left( ( t , f_i )_{\mathbb{L}_2(\mathbb{R}^2)} - y_i \right)^2 \\
+ \lambda \; \int_{\mathbb{R}^2} \lVert \nabla t(\tilde{\mathbf{x}}) \rVert^2 {\rm d}\tilde{\mathbf{x}} + \mu \; \lVert t \rVert^2_{\mathbb{L}_2(\mathbb{R}^2)},
\end{multline}
with $f_i$ one of the $N$ training patches extracted from an image $f_\mathbf{x}$, and $y_i$ the corresponding label ($y_i=1$ for \emph{objects} and $y_i=0$ for \emph{non-objects}).
In (\ref{eq:energyR2LinearRegression}), the data-term (first term) aims for alignment of template $t$ with object patches, in which case the inner product $( t , f_i )_{\mathbb{L}_2(\mathbb{R}^2)}$ is ideally one, and indeed aims orthogonality to non-object patches (in which case the inner product is zero). The second term enforces spatial smoothness of the template by penalizing its gradient, controlled by $\lambda$. The third (ridge) term improves stability by dampening the $\mathbb{L}_2$-norm of $t$, controlled by $\mu$.
\subsection{Optimizing $t$ Using Logistic Regression}
\label{subsec:logisticRegresionR2}
In object detection we are essentially considering a two-class classification problem: the object is either present or it is not. In this respect, the quadratic loss term in (\ref{eq:energyR2LinearRegression}) might not be the best choice as it penalizes any deviation from the desired response $y_i$, regardless of whether or not the response $( t , f_i )_{\mathbb{L}_2(\mathbb{R}^2)}$ is on the correct side of a decision boundary.
In other words, the aim is not necessarily to construct a template that best maps an image patch $f_i$ to a response $y_i\in\{0,1\}$, but rather the aim is to construct a template that best makes the separation between \emph{object} and \emph{non-object} patches. With this in mind we resort to the logistic regression model, in which case we interpret the non-linear objective functional given in (\ref{eq:logisticFunctional}) as a probability, and define
\begin{equation}
\begin{array}{rl}
p_{1}( f_i \; ; \; t ) &= p( f_i \; ; \; t ),\\
p_{0}( f_i \; ; \; t ) &= 1 - p( f_i \; ; \; t),\\
\multicolumn{2}{c}{\text{with } p( f_i \; ; \; t) = \sigma \left( ( t , f_i)_{\mathbb{L}_2(\mathbb{R}^2)} \right),}\\
\end{array}
\end{equation}
with $p_{1}( f_i ; t )$ and $p_{0}( f_i ; t )$ denoting respectively the probabilities of a patch $f_i$ being an \emph{object} or \emph{non-object} patch. Our aim is now to maximize the likelihood (of each patch $f_i$ having maximum probability $p_{y_i}(f_i ; t)$ for correct label $y_i$):
\begin{equation}
\ell(t) = \prod_{i=1}^N p_{y_i}(f_i;t) = \prod_{i=1}^N p( f_i ; t)^{y_i} (1 - p( f_i ; t))^{1-y_i}.
\end{equation}
We maximize the log-likelood instead,
which is given by
\begin{equation}
\label{eq:loglikelihood}
\begin{array}{l}
\!\! \ell_{log}(t) := \log ( \; \ell(t) \; ) \\
\;\;= \sum\limits_{i=1}^N \log ( \; p( f_i ; t)^{y_i} (1 - p( f_i ; t))^{1-y_i} \; )\\
\;\;= \sum\limits_{i=1}^N y_i ( t , f_i )_{\mathbb{L}_2(\mathbb{R}^2)}
- \log\left(
1 + e^{
( t , f_i )_{ \mathbb{L}_2( \mathbb{R}^2 ) }
}
\right).
\end{array}
\end{equation}%
Maximizing (\ref{eq:loglikelihood}) is known as the problem of logistic regression. Similar to the linear regression case, we impose additional regularization and define the following regularized logistic regression energy, which we aim to \emph{maximize}:
\begin{equation}
\label{eq:energyR2LogisticRegressionLogLikelihood}
E^{\ell}_{log}(t) = \ell_{log}(t) - \lambda \int_{\mathbb{R}^2} \lVert \nabla t(\tilde{\mathbf{x}}) \rVert^2 {\rm d}\tilde{\mathbf{x}}
- \; \mu \; \lVert t \rVert^2_{ \mathbb{L}_2( \mathbb{R}^2 ) }.
\end{equation}
\subsection{Template Optimization in a B-Spline Basis}
\label{subsec:splineBasisR2}
\textbf{\emph{Templates in a B-Spline Basis.}}
In order to solve the optimizations (\ref{eq:energyR2LinearRegression}) and (\ref{eq:energyR2LogisticRegressionLogLikelihood}), the template is described in a basis of direct products of $n$-th order B-splines~$B^n$:
\begin{equation}
\label{eq:tBsplineR2}
t(x,y) = \sum \limits_{k=1}^{N_k} \sum \limits_{l=1}^{N_l} c_{k,l} \;B^n\!\left( \frac{x}{s_k} - k \right)B^n\!\left( \frac{y}{s_l} - l \right),
\end{equation}
with $B^n(x) = \left(1_{\left[-\frac{1}{2},\frac{1}{2}\right]}*^{(n)}1_{\left[-\frac{1}{2},\frac{1}{2}\right]}\right)(x)$ a $n$-th order B-spline obtained by $n$-fold convolution of the indicator function $1_{\left[-\frac{1}{2},\frac{1}{2}\right]}$, and $c_{k,l}$ the coefficients belonging to the shifted B-splines. Here $s_k$ and $s_l$ scale the B-splines and typically depend on the number $N_k$ and $N_l$ of B-splines.
\textbf{\emph{Linear Regression. \label{ch:linearregression}}}
\label{thm:R2Data}
By substitution of (\ref{eq:tBsplineR2}) in (\ref{eq:energyR2LinearRegression}), the energy functional can be expressed in matrix-vector form (see Section 2 of the supplementary materials):
\begin{equation}
\label{eq:energyR2LinearRegressionDiscrete}
E_{lin}^{B}(\mathbf{c}) = \lVert S \mathbf{c} - \mathbf{y} \rVert^2 + \lambda \; \mathbf{c}^\dagger R \mathbf{c} + \mu \; \mathbf{c}^\dagger I \mathbf{c}.
\end{equation}
Regarding our notations we note that for spatial template $t$ given by (\ref{eq:tBsplineR2}) we have $E_{lin}(t)= E^{B}_{lin}(\mathbf{c})$, and label `B' indicates finite expansion in the B-spline basis.
The minimizer of (\ref{eq:energyR2LinearRegressionDiscrete}) is given by
\begin{equation}
\label{eq:minimizerR2LinearRegression}
(S^\dagger S + \lambda R + \mu I)\mathbf{c} = S^\dagger \mathbf{y},
\end{equation}
with $^\dagger$ denoting the conjugate transpose, and $I$ denoting the identity matrix.
Here $S$ is a $[N \times N_k N_l]$ matrix given by
\begin{equation}
\begin{array}{rl}
S &= \{(s_{1,1}^i,...,s_{1,N_l}^i,s_{2,1}^i,...,s_{2,N_l}^i,...,...,s_{N_k,N_l}^i)\}_{i=1}^N,\\
s_{k,l} &= (\; B_{s_ks_l}^n * {f}_i \; )(k,l),
\end{array}
\end{equation}
with $B_{s_ks_l}^n (x,y)= B^n\!\left( \frac{x}{s_k} \right)B^n\!\left( \frac{y}{s_l} \right)$, for all (x,y) on the discrete spatial grid on which the input image ${f}_D:\{1,N_x\}\times\{1,N_y\}\rightarrow\mathbb{R}$ is defined. Here $N_k$ and $N_l$ denote the number of splines in resp. $x$ and $y$ direction, and $s_k=\frac{N_x}{N_k}$ and $s_l=\frac{N_y}{N_l}$ are the corresponding resolution parameters. The $[N_k N_l \times 1]$ column vector $\mathbf{c}$ contains the B-spline coefficients, and the $[N \times 1]$ column vector $\mathbf{y}$ contains the labels, stored in the following form
\begin{equation}
\begin{array}{rl}
\mathbf{c} &= (c_{1,1},...,c_{1,N_l},c_{2,1},...,c_{2,N_{l}},...,...,c_{N_k,N_l})^T\\
\mathbf{y} &= (y_1,y_2,...,y_N)^T.
\end{array}
\end{equation}
The $[N_k N_l \times N_k N_l]$ regularization matrix $R$ is given by
\begin{equation}
\label{eq:regularizationMatrixR2}
R = R_x^{s_k} \otimes R_x^{s_l} + R_y^{s_k} \otimes R_y^{s_l},
\end{equation}
where $\otimes$ denotes the Kronecker product, and with
\begin{equation}
\label{eq:regularizationMatrixR2Elements}
\begin{array}{ll}
R_x^{s_k}(k,k') &= -\frac{1}{s_k}\frac{ \partial^2 B^{2n+1}}{\partial x^2}(k'-k), \\
R_x^{s_l}(l,l') &= s_l B^{2n+1}(l'-l), \\
R_y^{s_k}(k,k') &= s_k B^{2n+1}(k'-k),\\
R_y^{s_l}(l,l') &= -\frac{1}{s_l}\frac{ \partial^2 B^{2n+1}}{\partial y^2}(l'-l),
\end{array}
\end{equation}
with $k,k'={1,2,...,N_k}$ and $l,l'={1,2,...,N_l}$. The coefficients $\mathbf{c}$ can then be computed by solving (\ref{eq:minimizerR2LinearRegression}) directly, or via linear system solvers such as conjugate gradient descent. For a derivation of the regularization matrix $R$ we refer to supplementary materials, Sec.~2.
\textbf{\emph{Logistic Regression. \label{ch:log}}}
The logistic regression log-likelihood functional (\ref{eq:energyR2LogisticRegressionLogLikelihood}) can be expressed in matrix-vector notations as follows:
\begin{multline}
\label{eq:energyR2LogisticRegressionLogLikelihoodDiscrete}
E_{log}^{\ell,B}(\mathbf{c}) = \left[ \mathbf{y}^\dagger S \mathbf{c} - \mathbf{1}_N^\dagger \log(\mathbf{1}_N + \exp( S \mathbf{c} ) ) \right] \\
- \lambda \; \mathbf{c}^\dagger R \mathbf{c} - \mu \; \mathbf{c}^\dagger I \mathbf{c},
\end{multline}
where $\mathbf{1}_N = \{1,1,...,1\}^T \in \mathbb{R}^{N \times 1}$, and where the exponential and logarithm are evaluated element-wise. We follow a standard approach for the optimization of (\ref{eq:energyR2LogisticRegressionLogLikelihoodDiscrete}), see e.g. \cite{HastieBook}, and find the minimizer by settings the derivative to $\mathbf{c}$ to zero
\begin{equation}
\label{eq:minimizerR2LogisticRegression}
\nabla_{\mathbf{c}} E_{log}^{\ell,B}(\mathbf{c}) = S^T (\mathbf{y} - \mathbf{p}) - \lambda \; R \mathbf{c} - \mu \; I \mathbf{c} = \mathbf{0},
\end{equation}
with $\mathbf{p}= (p_1,...,p_N)^T \in \mathbb{R}^{N \times 1}$, with $p_i = \sigma ( (S\mathbf{c})_i)$. To solve (\ref{eq:minimizerR2LogisticRegression}), we use a Newton-Raphson optimization scheme. This requires computation of the Hessian matrix, given by
\begin{equation}
\mathcal{H}(E_{log}^{\ell,B}) = - (S^T W S + \lambda \; R + \mu \; I),
\end{equation}
with diagonal matrix $W = \underset{i \in \{1,...,N\}}{\operatorname{diag}} \left\{ p_i (1 - p_i) \right\}$.
The Newton-Raphson update rule is then given by
\begin{equation}
\label{eq:updateRule}
\begin{aligned}
\mathbf{c}^{new} & = \mathbf{c}^{old} - \mathcal{H}(E_{log}^{\ell,D})^{-1} (\nabla_{\mathbf{c}} E_{log}^{\ell,D}(\mathbf{c})) \\
& = (S^T W S + \lambda \; R + \mu \; I)^{-1} S^T W \mathbf{z},
\end{aligned}
\end{equation}
with $\mathbf{z} = S \mathbf{c}^{old} + W^{-1} (\mathbf{y} - \mathbf{p})$, see e.g. \cite[ch. 4.4]{HastieBook}.
Optimal coefficients found at convergence are denoted with $\mathbf{c}^*$.
Summarizing, we obtain the solution of (\ref{eq:objectDetection}) by substituting the optimized B-spline coefficients $\mathbf{c}^*$ into (\ref{eq:tBsplineR2}), and the resulting $t$ enters (\ref{eq:linearFunctional}) or (\ref{eq:logisticFunctional}). The most likely object location $\mathbf{x}^*$ is then found via (\ref{eq:objectDetection}).
\subsection{Orientation Scores on $SE(2)$}
\label{subsec:orientationScores}
\textbf{\emph{Transformation.}}
An orientation score, constructed from image $f:\mathbb{R}^2 \to \mathbb{R}$, is defined as a function $U_f : \mathbb{R}^2 \rtimes S^1 \rightarrow \mathbb{C}$ and depends on two variables ($\mathbf{x},\theta$), where $\mathbf{x}=(x,y) \in \mathbb{R}^2$ denotes position and $\theta \in [0,2\pi)$ denotes the orientation variable. An orientation score $U_f$ of image $f$ can be constructed by means of correlation with some anisotropic wavelet $\psi$ via
\begin{equation}
\label{eq:ostransform}
U_f(\mathbf{x},\theta) = (\mathcal{W}_\psi f)(\mathbf{x},\theta) = \int_{\mathbb{R}^2}\overline{\psi(\mathbf{R}_\theta^{-1}(\tilde{\mathbf{x}}-\mathbf{x}))}f(\tilde{\mathbf{x}}){\rm d}\tilde{\mathbf{x}},
\end{equation}
where $\psi \in \mathbb{L}_2 (\mathbb{R}^2)$ is the correlation kernel, aligned with the $x$-axis, where $\mathcal{W}_\psi$ denotes the transformation between image $f$ and orientation score $U_f$, $\psi_\theta(\mathbf{x}) = \psi(\mathbf{R}_\theta^{-1}\mathbf{x})$, and $\mathbf{R}_\theta$ is a counter clockwise rotation over angle $\theta$.
In this work we choose cake wavelets \cite{Duits2007a,Bekkers2014} for $\psi$. While in general any kind of anisotropic wavelet could be used to lift the image to $SE(2)$, cake wavelets ensure that no data-evidence is lost during the transformation: By design the set of all rotated wavelets uniformly cover the full Fourier domain of disk-limited functions with zero mean, and have thereby the advantage over other oriented wavelets (s.a. Gabor wavelets for specific scales) that they capture all scales and allow for a stable inverse transformation $\mathcal{W}_\psi^*$ from the score back to the image \cite{Duits2010,Duits2007a}.
\textbf{\emph{Left-Invariant Derivatives.}}
The domain of an orientation score is essentially the classical Euclidean motion group $SE(2)$ of planar translations and rotations, and is equipped with group product $g \cdot g' = (\mathbf{x},\theta)\cdot(\mathbf{x}',\theta') = (\mathbf{R}_\theta \mathbf{x}' + \mathbf{x}, \theta + \theta')$. Here, we can recognize a curved geometry (cf. Fig.~\ref{fig:osFrame}), and it is therefore useful to work in rotating frame of reference. As such, we use the left invariant derivative frame \cite{Duits2010,ZhangDuits2014}:
\begin{equation}
\label{eq:leftInvariantDerivatives}
\left\{ \partial_{\xi} := \cos \theta \, \partial_{x} +\sin \theta \, \partial_{y}, \partial_{\eta} := -\sin \theta \, \partial_{x} + \cos \theta \, \partial_{y},
\partial_{\theta} \right\}.
\end{equation}
Using this derivative frame we will construct in Subsec.~\ref{subsec:linearRegressionSE2} a regularization term in which we can control the amount of (anisotropic) smoothness along line structures.
\subsection{Object Detection via Cross-Correlation}
\label{subsec:objectDetectionSE2}
As in Section \ref{sec:templateMatchingR2}, we search for the most likely {object} location $\mathbf{x}^*$ via (\ref{eq:objectDetection}), but now we define functional $P$ respectively for the linear and logistic regression case in $SE(2)$ by\footnote{Since both the inner product and the construction of orientation scores $U_f$ from images $f$ are linear, template matching might as well be performed directly on the 2D images (likewise (\ref{eq:linearFunctional}) and (\ref{eq:logisticFunctional})). Hence, here we take the modulus of the score as a non-linear intermediate step \cite{Bekkers2015EMMCVPR}.}:
\begin{align}
\label{eq:linearFunctionalSE2}
P(\mathbf{x}) = {P}_{lin}^{SE(2)}(\mathbf{x}) := & ( \mathcal{T}_{\mathbf{x}} \; T , \left|U_{f}\right|)_{\mathbb{L}_2(SE(2))},\;\;\;\;\text{or}\\
\label{eq:logisticFunctionalSE2}
P(\mathbf{x}) = {P}_{log}^{SE(2)}(\mathbf{x}) := & \sigma \left( ( \mathcal{T}_{\mathbf{x}} \; T , \left|U_{f}\right|)_{\mathbb{L}_2(SE(2))} \right),
\end{align}
with $(\mathcal{T}_{\mathbf{x}} T)(\tilde{\mathbf{x}},\tilde{\theta}) = T(\tilde{\mathbf{x}} - \mathbf{x}, \tilde{\theta})$. The $\mathbb{L}_2(SE(2))$-inner product is defined by
\begin{equation}
(T,\left|U_f\right|)_{\mathbb{L}_2 (SE(2))} :=
\int_{\mathbb{R}^2}\int_{0}^{2\pi} \overline{T(\tilde{\mathbf{x}},\tilde{\theta})} \left|U_f\right|(\tilde{\mathbf{x}},\tilde{\theta}) {\rm d}\tilde{\mathbf{x}}{\rm d}\tilde{\theta},
\end{equation}
with norm $\lVert \cdot \rVert_{\mathbb{L}_2(SE(2))} = \sqrt{ (\cdot , \cdot )_{\mathbb{L}_2(SE(2))} }$.
\subsection{Optimizing $T$ Using Linear Regression}
\label{subsec:linearRegressionSE2}
Following the same reasoning as in Section \ref{subsec:linearRegresionR2} we search for the template that minimizes
\begin{multline}
\label{eq:energySE2LinearRegression}
\mathcal{E}_{lin}(T) = \sum\limits_{i=1}^N \left( ( T , \left|U_{f_i}\right| )_{\mathbb{L}_2(SE(2))} - y_i \right)^2 \\
+ \lambda \int_{\mathbb{R}^2} \int_0^{2\pi} \lVert \nabla T(\tilde{\mathbf{x}},\tilde{\theta}) \rVert^2_{D}{\rm d}\tilde{\mathbf{x}}{\rm d}\tilde{\theta}
+ \mu \lVert T \rVert^2_{\mathbb{L}_2(SE(2))},
\end{multline}
with smoothing term:
\begin{equation}
\lVert \nabla T (g) \rVert_{D}^2 = D_{\xi\xi} \left| \frac{\partial T}{\partial \xi}(g)\right|^2 + D_{\eta\eta} \left| \frac{\partial T}{\partial \eta}(g)\right|^2 + D_{\theta\theta} \left| \frac{\partial T}{\partial \theta}(g)\right|^2.
\end{equation}
Here, $\nabla T = (\frac{\partial T}{\partial \xi},\frac{\partial T}{\partial \eta},\frac{\partial T}{\partial \theta})^T$ denotes the left-invariant gradient. Note that $\partial_\xi$ gives the spatial derivative in the direction aligned with the orientation score kernel used at layer $\theta$, recall Fig.~\ref{fig:osFrame}. The parameters $D_{\xi\xi}$, $D_{\eta\eta}$ and $D_{\theta\theta} \geq 0$ are then used to balance the regularization in the three directions. Similar to this problem, first order Tikhonov-regularization on $SE(2)$ is related, via temporal Laplace transforms, to left--invariant diffusions on the group $SE(2)$ (Sec.~\ref{sec:stochasticProcess}), in which case $D_{\xi\xi}$, $D_{\eta\eta}$ and $D_{\theta\theta}$ denote the diffusion constants in $\xi$, $\eta$ and $\theta$ direction. Here we set $D_{\xi\xi}=1$, $D_{\eta\eta}=0$, and thereby we get Laplace transforms of hypo-elliptic
diffusion processes \cite{Citti2006,Duits2010}.
Parameter $D_{\theta\theta}$ can be used to tune between isotropic (large $D_{\theta\theta}$) and anisotropic (low $D_{\theta\theta}$) diffusion (see e.g. \cite[Fig.~3]{Bekkers2015EMMCVPR}). Note that anisotropic diffusion, via a low $D_{\theta\theta}$, is preferred as we want to maintain line structures in orientation scores.
\subsection{Optimizing $T$ Using Logistic Regression}
\label{subsec:logisticRegressionSE2}
Similarly to what is done in Subsec.~\ref{subsec:logisticRegresionR2} we can change the quadratic loss of (\ref{eq:energySE2LinearRegression}) to a logistic loss, yielding the following energy functional
\begin{multline}
\label{eq:energySE2LogisticRegression}
\mathcal{E}_{log}(T) = \mathscr{L}_{log}(T)
-\lambda \int_{\mathbb{R}^2} \int_0^{2\pi} \lVert \nabla T(\tilde{\mathbf{x}},\tilde{\theta}) \rVert^2_{D} {\rm d}\tilde{\mathbf{x}}{\rm d}\tilde{\theta}\\
- \mu \lVert T \rVert^2_{\mathbb{L}_2(SE(2))},
\end{multline}
with log-likelihood (akin to (\ref{eq:loglikelihood}) for the $\mathbb{R}^2$ case)
\begin{equation}
\begin{array}{rl}
\mathscr{L}_{log}(T) = &\sum\limits_{i=1}^N y_i ( T , \left|U_{f_i}\right| )_{\mathbb{L}_2(SE(2))}\\
&\;\;\;\;\;\;\;- \log\left(
1 + e^{
( T , \left|U_{f_i}\right| )_{ \mathbb{L}_2( SE(2) ) }
} \right).
\end{array}
\end{equation}
The optimization of (\ref{eq:energySE2LinearRegression}) and (\ref{eq:energySE2LogisticRegression}) follows quite closely the procedure as described in Sec.~\ref{sec:templateMatchingR2} for the 2D case. In fact, when $T$ is expanded in a B-spline basis, the exact same matrix-vector formulation can be used.
\subsection{Template Optimization in a B-Spline Basis}
\label{subsec:splineBasisSE2}
\textbf{\emph{Templates in a B-Spline Basis.}}
The template $T$ is expanded in a B-spline basis as follows
\begin{equation} \label{splineexp}
\begin{array}{l}
T(x,y,\theta) = \sum \limits_{k=1}^{N_k} \sum \limits_{l=1}^{N_l} \sum \limits_{m=1}^{N_m} c_{k,l,m}\cdot \; \\
\\
\;\;\; B^n\!\left( \frac{x}{s_k} - k \right) B^n\!\left( \frac{y}{s_l} - l \right)B^n\!\left( \frac{\theta \!\!\!\!\! \mod 2\pi}{s_m} - m \right),
\end{array}
\end{equation}
with $N_k$, $N_l$ and $N_m$ the number of B-splines in respectively the $x$, $y$ and $\theta$ direction, $c_{k,l,m}$ the corresponding basis coefficients, and with angular resolution parameter $s_m = 2\pi/N_m$.
\textbf{\emph{Linear Regression.}}\label{ch:linearRegressionSE2}
The shape of the minimizer of energy functional $\mathcal{E}_{lin}(T)$ in the $SE(2)$ case is the same as for $E_{lin}(t)$ in the $\mathbb{R}^2$ case, and is again of the form given in (\ref{eq:energyR2LinearRegressionDiscrete}). However, now the definitions of $S$, $R$ and $\mathbf{c}$ are different. Now, $S$ is a $[N \times N_k N_l N_m]$ matrix given by
\begin{equation}
\begin{aligned}
S &= \{(s_{1,1,1}^i,...,s_{1,1,N_m}^i,...,s_{1,N_l,N_m},...,s_{N_k,N_l,N_m}^i)\}_{i=1}^N,\\
s_{k,l,m} &= (\; B_{s_ks_ls_m}^n * U_{f_i} \;)(k,l,m),
\end{aligned}
\end{equation}
with $B_{s_ks_ls_m}^n(x,y,\theta) = B^n\!\left( \frac{x}{s_k} \right)B^n\!\left( \frac{y}{s_l} \right)B^n\!\left( \frac{\theta \!\! \mod 2\pi}{s_m} \right) $. Vector $\mathbf{c}$ is a $[N_k N_l N_m \times 1]$ column vector containing the B-spline coefficients and is stored as follows:
\begin{equation}
\mathbf{c} = (c_{1,1,1},...,c_{1,1,N_m},...,c_{1,N_l,N_m},...,c_{N_k,N_l,N_m})^T.
\end{equation}
The explicit expression and the derivation of $[N_k N_l N_m \times N_k N_l N_m]$ matrix $R$, which encodes the left invariant derivatives, can be found in the supplementary materials Sec.~2
\textbf{\emph{Logistic Regression}}\label{ch:logSE2}
Also for the logistic regression case we optimize energy functional (\ref{eq:energySE2LogisticRegression}) in the same form as (\ref{eq:energyR2LogisticRegressionLogLikelihood}) in the $\mathbb{R}^2$ case, by using the corresponding expressions for $S$, $R$, and $\mathbf{c}$ in Eq.~(\ref{eq:energyR2LogisticRegressionLogLikelihoodDiscrete}).
These expressions can be inserted in the functional (\ref{eq:energyR2LogisticRegressionLogLikelihoodDiscrete}) and again the same techniques (as presented in Subsection~\ref{ch:log})
can be used to minimize this cost on $SE(2)$.
\subsection{Probabilistic Interpretation of the $SE(2)$ Smoothing Prior}
\label{sec:stochasticProcess}
In this section we only provide a brief introduction to the probabilistic interpretation of the $SE(2)$ smoothing prior, and refer the interested reader to the supplementary materials for full details.
Consider the classic approach to noise suppression in images via diffusion regularizations with PDE's of the form
\begin{equation}
\label{eq:pdeDiffusion}
\left\{
\begin{array}{cl}
\tfrac{\partial}{\partial \tau}u &= \Delta u,\\
u |_{\tau=0} &= u_0,
\end{array}
\right.
\end{equation}
where $\Delta$ denotes the Laplace operator. Solving (\ref{eq:pdeDiffusion}) for any diffusion time $\tau>0$ gives a smoothed version of the input $u_0$. The time-resolvent process of the PDE is defined by the Laplace transform with respect to $\tau$; time $\tau$ is integrated out using a memoryless negative exponential distribution $P(\mathcal{T}=\tau)=\alpha e^{-\alpha \tau}$. Then, the time integrated solutions
$$
t(\mathbf{x}) = \alpha \int_0^\infty u(\mathbf{x},\tau) e^{-\alpha \tau} {\rm d}\tau,
$$
with decay parameter $\alpha$, are in fact the solutions \cite{DuitsBurgeth2007}
\begin{equation}
\label{eq:timeIntegratedDiffusion}
t = \underset{t \in \mathbb{L}_2(\mathbb{R}^2)}{\operatorname{argmin}} \left[ \lVert t - t_0 \rVert^2_{\mathbb{L}_2(\mathbb{R}^2)} + \lambda \int_{\mathbb{R}^2} \lVert \nabla t (\tilde{\mathbf{x}}) \rVert^2 \, {\rm d}\tilde{\mathbf{x}} \right],
\end{equation}
with $\lambda = \alpha^{-1}$. Such time integrated diffusions (Eq.~(\ref{eq:timeIntegratedDiffusion})) can also be obtained by optimization of the linear regression functionals given by Eq.~(\ref{eq:energyR2LinearRegression}) and Eq.~(\ref{eq:linearFunctionalSE2}) for the $\mathbb{R}^2$ and $SE(2)$ case respectively.
In the supplementary materials we establish this connection for the $SE(2)$ case, and show how the smoothing regularizer in (\ref{eq:energySE2LinearRegression}) and (\ref{eq:energySE2LogisticRegression}) relates to Laplace transforms of hypo-elliptic diffusions on the group $SE(2)$ \cite{ZhangDuits2014,Duits2010}.
More precisely, we formulate a special case of our problem (the \emph{single patch problem}) which involves only a single training sample $U_{f_1}$, and show in a formal theorem that the solution is up to scalar multiplication the same as the resolvent hypo-elliptic diffusion kernel. The underlying probabilistic interpretation is that of Brownian motions on $SE(2)$, where the resolvent hypo-elliptic diffusion kernel gives a probability density of finding a random brush stroke at location $\mathbf{x}$ with orientation $\theta$, given that a `drunkman's pencil' starts at the origin at time zero.
In the supplementary materials we demonstrate the high accuracy of our discrete numeric regression method using B-spline expansions with near to perfect comparisons to the continuous exact solutions of the single patch problem.
In fact, we have established an efficient B-spline finite element implementation of hypo-elliptic Brownian motions on $SE(2)$, in addition to other numerical approaches in \cite{ZhangDuits2014}.
\begin{comment}
\begin{figure}
\centerline{
\includegraphics[width=\hsize]{Figs/FigKernel2.pdf}
}
\caption{
Isosurface of the time resolvent hypo-elliptic diffusion kernel on $SE(2)$, computed via the methods proposed in this paper (single patch problem, see supplementary materials, available online) and the exact solution. Right most figure is an illustration of the drunkman's pencil.
\label{fig:stochasticEnhancement}}
\end{figure}
\end{comment}
\subsection{Normalized Cross-Correlation in $\mathbb{R}^2$}
\label{subsec:normedCCR2}
In the usual cross-correlation based template matching approach, as described in Sec.~2 of the main article, we rely on the standard $\mathbb{L}_2(\mathbb{R}^2)$ inner product (Eq. (6) of the main article). In normalized cross-correlation it is however convenient to extend this inner product to include a windowing function $m$ which indicates the relevant region (support) of the template. As such, the inner product with respect to windowing function $m$ is given by
\begin{equation}
(t,f)_{\mathbb{L}_2 (\mathbb{R}^2, m d \tilde{\mathbf{x}})} := \int_{\mathbb{R}^2} \overline{t(\tilde{\mathbf{x}})} f(\tilde{\mathbf{x}}) m(\tilde{\mathbf{x}}) {\rm d}\tilde{\mathbf{x}},
\end{equation}
with associated norm $\lVert \cdot \rVert_{\mathbb{L}_2(\mathbb{R}^2, m d \tilde{\mathbf{x}})} = \sqrt{ (\cdot , \cdot )_{\mathbb{L}_2(\mathbb{R}^2, m d \tilde{\mathbf{x}})} }$. The windowing function has to be a smooth function $m:\mathbb{R}^2 \rightarrow \mathbb{R}^+$ with $\int_{\mathbb{R}^2} m(\tilde{\mathbf{x}}){\rm d}\tilde{\mathbf{x}} = 1$. In this work, the use of a window $m$ is also convenient to deal with boundary conditions in the optimization problems for template construction.
We define
\begin{equation}
\label{eq:mass}
m(\mathbf{x}) := \varsigma \; e^{-\frac{\lVert \mathbf{x} \rVert^2}{s}} \sum\limits_{i=0}^n \frac{(\lVert \mathbf{x} \rVert^2/s)^{i}}{i!},
\end{equation}
which smoothly approximates the indicator function $1_{[0,r]}(\lVert \mathbf{x} \rVert)$, covering a disk with radius $r$, when setting $s = \frac{2 r^2}{1+2n}$, see e.g. \cite[Fig.~2]{Bekkers2014}. The constant $\varsigma$ normalizes the function such that $\int_{\mathbb{R}^2} m(\tilde{\mathbf{x}}){\rm d}\tilde{\mathbf{x}} = 1$.
In normalized cross-correlation the image is locally normalized (at position $\mathbf{x}$) to zero mean and unit standard deviation, which is done as follows
\begin{equation}
\label{eq:normalization}
\hat{f}_\mathbf{x}(\tilde{\mathbf{x}}) := \frac{f(\tilde{\mathbf{x}}) - \langle f \rangle_{\mathcal{T}_\mathbf{x} m}}{ \lVert f(\tilde{\mathbf{x}}) - \langle f \rangle_{\mathcal{T}_\mathbf{x} m} \rVert_{\mathbb{L}_2(\mathbb{R}^2, \mathcal{T}_\mathbf{x} m {\rm d}\tilde{\mathbf{x}})}},
\end{equation}
with local mean $\langle f \rangle_{m} = ( 1 , f )_{\mathbb{L}_2(\mathbb{R}^2, m {\rm d}\tilde{\mathbf{x}})}$. Template $\hat{t}$ can be obtained via normalization of a given template $t$ via
\begin{equation}
\label{eq:templateNormalizationR2}
\hat{t}(\tilde{\mathbf{x}}) := \frac{t(\tilde{\mathbf{x}}) - \langle t \rangle_{m}}{ \lVert t(\tilde{\mathbf{x}}) - \langle t \rangle_{m} \rVert_{\mathbb{L}_2(\mathbb{R}^2, m {\rm d}\tilde{\mathbf{x}})}}.
\end{equation}
Template matching is then done in the usual way (via (4) of the main article), however now $\hat{t}$ and $\hat{f}_{\mathbf{x}}$ are used instead of $t$ and $f$. In fact, the entire $\mathbb{R}^2$ cross-correlation template matching, and template optimization framework is extended to normalized cross-correlation by substituting all instances of $t$ with $\hat{t}$, $f$ with $\hat{f}_\mathbf{x}$, and $(\cdot , \cdot)_{\mathbb{L}_2(\mathbb{R}^2)}$ with $(\cdot,\cdot)_{\mathbb{L}_2 (\mathbb{R}^2, m d \tilde{\mathbf{x}})}$ in Sec.~2 of the main article. However, since templates $\hat{t}$ are directly constructed via the minimization of energy functionals, we will not explictely normalize the templates, unless they are obtained by other methods. E.g., Eq.~(\ref{eq:templateNormalizationR2}) is used in the main article to construct basic templates obtained by averaging positive object patches (Subsec.~4.1 of the main article).
\subsection{Normalized Cross-Correlation in $SE(2)$}
\label{subsec:normedCCSE2}
Similar to the $\mathbb{R}^2$ case, templates and orientation scores are locally normalized to zero mean and unit standard deviation, however, now with respect to the $\mathbb{L}_2(SE(2), M d\tilde{g})$ inner product, which is given by
\begin{multline}
(T,U_f)_{\mathbb{L}_2 (SE(2),M {\rm d}\tilde{g})} := \\
\int_{\mathbb{R}^2}\int_{0}^{2\pi} \overline{T(\tilde{\mathbf{x}},\tilde{\theta})} U_f(\tilde{\mathbf{x}},\tilde{\theta}) M(\tilde{\mathbf{x}},\tilde{\theta}) {\rm d}\tilde{\mathbf{x}}{\rm d}\tilde{\theta},
\end{multline}
with norm $\lVert \cdot \rVert_{\mathbb{L}_2(SE(2),M {\rm d}\tilde{g})} = \sqrt{ (\cdot , \cdot )_{\mathbb{L}_2(SE(2),M {\rm d}\tilde{g})} }$. Also here windowing function $M$ indicates the support of the template, and has the property $\int_{\mathbb{R}^2}\int_{0}^{2\pi} M(\tilde{\mathbf{x}},\tilde{\theta}) {\rm d}\tilde{\mathbf{x}}{\rm d}\tilde{\theta} = 1$. We define
\begin{equation}
M(\mathbf{x},\theta) := \frac{1}{2\pi} m(\mathbf{x}),
\end{equation}
independent of $\theta$ and with $m(\mathbf{x})$ given by (\ref{eq:mass}).
The (locally at $g$) normalized orientation score and template $T$ are then given by
\begin{align}
\hat{U}_{f,g}(\tilde{\mathbf{x}},\tilde{\theta}) &:= \frac{U_f(\tilde{\mathbf{x}},\tilde{\theta}) - \langle U_f \rangle_{\mathcal{L}_{g} M}}{ \lVert U_f(\tilde{\mathbf{x}},\tilde{\theta}) - \langle U_f \rangle_{\mathcal{L}_g M} \rVert_{\mathbb{L}_2(SE(2), \mathcal{L}_g M d\tilde{g})}},\label{eq:normalizationSE2a}\\
\hat{T}(\tilde{\mathbf{x}},\tilde{\theta}) &:= \frac{T(\tilde{\mathbf{x}},\tilde{\theta}) - \langle T \rangle_{M}}{ \lVert T(\tilde{\mathbf{x}},\tilde{\theta}) - \langle T \rangle_{M} \rVert_{\mathbb{L}_2(SE(2), M {\rm d}\tilde{g})}},\label{eq:normalizationSE2b}
\end{align}
with mean $\langle U_f \rangle_{M} = ( 1 , U_f )_{\mathbb{L}_2( SE(2), M {\rm d}\tilde{g}) )}$.
\subsection{Efficient local normalization of $\hat{f}_{\mathbf{x}}$ and $\hat{U}_{f,g}$.}
\label{subsec:efficientNormalization}
Since the normalized image $\hat{f}_{\mathbf{x}}$ depends on the location $\mathbf{x}$ it needs to be calculated for every translation of the template, which makes normalized cross-correlation computationally expensive. Therefore, (\ref{eq:normalization}) can be approximated by assuming that the local average is approximately constant in the area covered by $m$. That is, assuming $\langle f \rangle_{\mathcal{T}_{\mathbf{x}} m}(\tilde{\mathbf{x}}) \approx \langle f \rangle_{\mathcal{T}_{\tilde{\mathbf{x}}} m}(\tilde{\mathbf{x}}) = (m \star f)(\tilde{\mathbf{x}})$ for \mbox{$\lVert \tilde{\mathbf{x}}-\mathbf{x} \rVert < r$}, with $r$ the radius that determines the extent of $m$, (\ref{eq:normalization}) is approximated as follows:
\begin{equation}
\label{eq:fnormedapprox}
\hat{f}_{\mathbf{x}}(\tilde{\mathbf{x}}) \approx \cfrac{f(\tilde{\mathbf{x}}) - (m \star f)(\tilde{\mathbf{x}})}{ \sqrt{ (m \star (f - (m \star f))^2)(\tilde{\mathbf{x}}) } }.
\end{equation}
Similarly, in the $SE(2)$-case (\ref{eq:normalizationSE2a}) can be approximated via
\begin{equation}
\hat{U}_{f,g}(\tilde{\mathbf{x}},\tilde{\theta}) \approx \frac{U_f(\tilde{\mathbf{x}},\tilde{\theta}) - (M \star_{SE(2)} U_f)(\mathbf{x},\tilde{\theta}) }{ \sqrt{ (M \star_{SE(2)} (U_f - (M \star_{SE(2)} U_f))^2 )(\mathbf{x},\tilde{\theta}) } }.
\end{equation}
\subsection{Including a Region of Interest Mask}
\label{subsec:roi}
Depending on the application, large portions of the image might be masked out. This for example the case in retinal images (see circular masks in Fig.~\ref{fig:resultsOverviewImageONH}). To deal with this, template matching is only performed inside the region of interest defined by a mask image $m^{roi}:\mathbb{R}^2 \rightarrow \{0,1\}$. Including such a mask is important in normalized template matching, and can be done by replacing the standard inner products by
\begin{align}
\label{eq:productwithmask}
(t,f)^{roi}_{\mathbb{L}_2(\mathbb{R}^2,m,d\tilde{\mathbf{x}})} &:= \frac{ (t,f m^{roi})_{\mathbb{L}_2(\mathbb{R}^2,m,d\tilde{\mathbf{x}})} }{ (1, m^{roi})_{\mathbb{L}_2(\mathbb{R}^2,m,d\tilde{\mathbf{x}})} },\\
(T,U_f)^{roi}_{\mathbb{L}_2(SE(2),M,d\tilde{g})} &:= \frac{ (T,U_f M^{roi})_{\mathbb{L}_2(SE(2),M,d\tilde{g})} }{ (1, M^{roi})_{\mathbb{L}_2(SE(2),M,d\tilde{g})} },
\end{align}
with $M^{roi}(\mathbf{x},\theta) = m^{roi}(\mathbf{x})$.
\subsection{The experimental set-up}
\label{subsec:detailsExperiments}
\textbf{\emph{Templates.}}
\label{subsubsec:templateDetails}
In our experiments we compare the performance of different template types,
which we label as follows:
\begin{itemize}
\item[$A$:] Templates obtained by taking the average of all positive patches ($y_i=1$) in the training set,
then normalized to zero mean and unit standard deviation.
\item[$B$:] Templates optimized without any regularization
\item[$C$:] Templates optimized with an optimal $\mu$, and with $\lambda = 0$.
\item[$D$:] Templates optimized with an optimal $\lambda$ and with $\mu = 0$.
\item[$E$:] Templates optimized with optimal $\mu$ and $\lambda$.
\end{itemize}
The trained templates ($B$-$E$) are obtained either via linear or logistic regression in the $\mathbb{R}^2$ setting (see Subsec. \ref{ch:linearregression} and Subsec. \ref{ch:log}), or in the $SE(2)$ setting (see Subsec. \ref{ch:linearRegressionSE2} and Subsec. \ref{ch:logSE2}). In both the $\mathbb{R}^2$ and $SE(2)$ case, linear regression based templates are indicated with subscript ${}_{lin}$, and logistic regression based templates with ${}_{log}$. Optimality of parameter values is defined using generalized cross validation (GCV), which we soon explain in this section. We generally found that (via optimization using GCV) the optimal settings for template $E$ were $\mu \approx 0.5\mu^*$, and $\lambda \approx 0.5\lambda^*$, with $\mu^*$ and $\lambda^*$ respectively the optimal parameters for template $C$ and $D$.
\textbf{\emph{Matching with Multiple Templates.}}
When performing template matching, we use Eq.~(\ref{eq:linearFunctional}) and Eq.~(\ref{eq:linearFunctionalSE2}) for respectively the $\mathbb{R}^2$ and $SE(2)$ case for templates obtained via linear regression and for template $A$. For templates obtained via logistic regression we use respectively Eq.~(\ref{eq:logisticFunctional}) and Eq.~(\ref{eq:logisticFunctionalSE2}). When we combine multiple templates we simply add the objective functionals. E.g, when combining template $C_{lin:\mathbb{R}^2}$ and $D_{log:SE(2)}$ we solve the problem
$$
\mathbf{x}^* = \underset{\mathbf{x} \in \mathbb{R}^2 }{\operatorname{argmax}} \;\;P_{C_{lin}}^{\mathbb{R}^2}(\mathbf{x}) + P_{D_{log}}^{SE(2)}(\mathbf{x}),
$$
where $P_{C_{lin}}^{\mathbb{R}^2}(\mathbf{x})$ is the objective functional (see Eq.~(\ref{eq:linearFunctional})) obtained with template $C_{lin:\mathbb{R}^2}$, and $P_{D_{log}}^{SE(2)}(\mathbf{x})$ (see Eq.~(\ref{eq:logisticFunctionalSE2})) is obtained with template $D_{log:SE(2)}$.
\textbf{\emph{Rotation and Scale Invariance.}}
The proposed template matching scheme can adapted for rotation-scale invariant matching, this is discussed in Sec.~5 of the supplementary materials. For a generic object recognition task, however, global rotation or scale invariance are not necessarily desired properties. Datasets often contain objects in a human environment context, in which some objects tend to appear in specific orientations (e.g. eye-brows are often horizontal above the eye and vascular trees in the retina depart the ONH typically along a vertical axis).
Discarding such knowledge by introducing rotation/scale invariance is likely to have an adversary effect on the performance, while increasing computational load.
In Sec.~5 of the supplementary materials we tested a rotation/scale invariant adaptation of our method and show that in the three discussed applications this did indeed not lead to improved results, but in fact worsened the results slightly.
\textbf{\emph{Automatic Parameter Selection via Generalized Cross Validation.}}
\label{subsubsec:GCV}
An ideal template generalizes well to new data samples, meaning that it has low prediction error on independent data samples. One method to predict how well the system generalizes to new data is via generalized cross validation (GCV), which is essentially an approximation of leave-one-out cross validation \cite{Craven1979}.
The vector containing all predictions is given by $\tilde{\mathbf{y}} = S \mathbf{c}_{\mu,\lambda}$, in which we can substitute the solution for $\mathbf{c}_{\mu,\lambda}$ (from Eq.~(\ref{eq:minimizerR2LinearRegression})) to obtain
\begin{equation}
\begin{aligned}
\tilde{\mathbf{y}} &= A_{\mu,\lambda} \mathbf{y}, \;\;\;\;\;\;\;\;\;\;\;\;\text{with}\\
A_{\mu,\lambda}&= S (S^\dagger S + \lambda R + \mu I)^{-1} S^\dagger,
\end{aligned}
\end{equation}
where $A_{\mu,\lambda}$ is the so-called \emph{smoother matrix}. Then the generalized cross validation value \cite{Craven1979} is defined as
\begin{equation}
\label{eq:GCV}
GCV(\mu,\lambda) \equiv \frac{ \frac{1}{N} \lVert \Omega (I - A_{\mu,\lambda} ) \mathbf{y} \rVert^2}{ \left(1 - \operatorname{trace} (A_{\mu,\lambda} )/N\right )^2}.
\end{equation}
In the retinal imaging applications we set $\Omega = I$. In the pupil detection application we set $\Omega = \underset{i\in\{1,...,N\}}{\operatorname{diag}}\{y_i\}$.
As such, we do not penalize errors on negative samples as here the diversity of negative patches is too large for parameter optimization via GCV.
Parameter settings are considered optimal when they minimize the GCV value.
In literature various extensions of GCV are proposed for generalized linear models \cite{O'Sullivan1986,Gu1992,Xiang1996}. For logistic regression we use the approach by O'Sullivan et al. \cite{O'Sullivan1986}: we iterate the Newton-Raphson algorithm until convergence, then, at the final iteration we compute the GCV value on the quadratic approximation (Eq.~(\ref{eq:updateRule})).
\begin{comment}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\linewidth]{Figs/FigGCV.pdf}
\end{center}
\caption{Parameter optimization for the $SE(2)$ ONH template $D_{lin}$. The GCV value (Eq.~(\ref{eq:GCV}) is minimal for $\lambda = \lambda^* = 5.62*10^{-7}$, a lower value for $\lambda$ results in an over-fitted (under-smooth) template (left figure), a higher value results in an under-fitted (over-smooth) template. The template images are maximum intensity projections over $\theta$.}
\label{fig:GCVcurve}
\end{figure}
\end{comment}
\begin{comment}
While our final goal is to optimize the detection scheme (Eq.~..). In the template optimization (Eq.~.. and Eq.~..) however, we do not optimizer for succesrates, but rather the corresponding energy functionals. To tune the parameters, we chose $\lambda$, $D_{\theta\theta}$ and $\mu$ by minimizing the classification error $\frac{1}{N} \lVert S \mathbf{c} - \mathbf{y} \rVert$ (cf. the data term in Eq.~..). E.g.
\begin{equation}
\lambda^* = \underset{\lambda \in \mathbb{R} }{\operatorname{argmin}} \;\; \frac{1}{N} \lVert S^{test} \mathbf{c}^\lambda - \mathbf{y}^{test} \rVert,
\end{equation}
where $\mathbf{c}^\lambda$ is the solution obtain from Eq.~(..) with the specified $\lambda$.
\end{comment}
\textbf{\emph{Success Rates.}}
Performance of the templates is evaluated using success rates. The success rate of a template is the percentage of images in which the target object was successfully localized.
In both optic nerve head (Subsec.~\ref{subsec:ONHDetection}) and fovea (Subsec.~\ref{subsec:FoveaDetection}) detection experiments, a successful detection is defined as such if the detected location $\mathbf{x}^*$ (Eq.~(\ref{eq:objectDetection})) lies within one optic disk radius distance to the actual location.
For pupil detection both the left and right eye need to be detected and we therefore use the following normalized error metric
\begin{equation}
\label{eq:normalizedError}
e=\frac{\operatorname{max}(d_{left},d_{right})}{w},
\end{equation}
in which $w$ is the (ground truth) distance between the left and right eye, and $d_{left}$ and $d_{right}$ are respectively the distances of detection locations to the left and right eye.
\textbf{\emph{k-Fold Cross Validation.}}
For correct unbiased evaluation, none of the test images are used for training of the templates, nor are they used for parameter optimization. We perform $k$-fold cross validation: The complete dataset is randomly partitioned into $k$ subsets. Training (patch extraction, parameter optimization and template construction) is done using the data from $k-1$ subsets. Template matching is then performed on the remaining subset. This is done for all $k$ configurations with $k-1$ training subsets and one test subset, allowing us to compute the average performance (success rate) and standard deviation. We set $k=5$.
\begin{comment}
\begin{figure}[b]
\begin{center}
\includegraphics[width=\linewidth]{Figs/FigPatches.png}
\end{center}
\caption{Example images patches $f_i$ used in template optimization for optic nerve head detection. Top row positive samples ($y_i = 1$), bottom row negative samples ($y_i = 0$).}
\label{fig:samples}
\end{figure}
\end{comment}
\subsection{Optic Nerve Head Detection in Retinal Images}
\label{subsec:ONHDetection}
\begin{figure*}
\begin{center}
\includegraphics[width=\linewidth]{Figs/FigODTemplatesAndResponses2}
\end{center}
\caption{Overview of trained templates for ONH detection, and their responses to a challenging retinal image. \textbf{(a)} The example input image with true ONH location in blue. \textbf{(b)} The $\mathbb{R}^2$-type templates (top row) and their responses to the input image (bottom row). \textbf{(c)} The maximum intensity projections (over $\theta$) of the $SE(2)$-type templates (top row) and their responses to the input image (bottom row). Detected ONH locations are indicated with colored circles (green = correct, red = incorrect).
}
\label{fig:ODTemplates}
\end{figure*}
\newlength{\smallspacing}
\setlength{\smallspacing}{1.5mm}
\definecolor{rowcolor}{gray}{0.85}
\begin{table*
\centering
\caption{Average template matching results ($\pm$ standard deviation) for optic nerve head detection in 5-fold cross validation, number of failed detections in parentheses.}
\begin{tabular}{l|lllll|l}
\toprule
\multicolumn{1}{l|}{Template} & ES (SLO) & TC & MESSIDOR & DRIVE & STARE & All Images\\
ID & 208 & 208 & 1200 & 40 & 81 & 1737\\
\midrule
\multicolumn{7}{c}{{$\mathbb{R}^2$ templates}}\\
\midrule
$A_{\mathbb{R}^2}$ & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 99.49\% {\tiny $\pm$ 1.15\%} (1) & 98.83\% {\tiny $\pm$ 0.56\%} (14) & 96.36\% {\tiny $\pm$ 4.98\%} (2) & 74.94\% {\tiny $\pm$ \;\;9.42\%} (20) & \cellcolor{rowcolor}97.87\% {\tiny $\pm$ 0.52\%} (37) \vspace{\smallspacing}\\
$B_{lin:\mathbb{R}^2}$ & 99.09\% {\tiny $\pm$ 2.03\%} (2) & 20.35\% {\tiny $\pm$ 5.99\%} (165) & \;\;9.67\% {\tiny $\pm$ 2.69\%} (1084) & \;\;9.09\% {\tiny $\pm$ 12.86\%} \!\!(35) & \;\;3.56\% {\tiny $\pm$ \;\;3.28\%} (78) & 21.48\% {\tiny $\pm$ 2.16\%} (1364) \\
$C_{lin:\mathbb{R}^2}$ & 99.55\% {\tiny $\pm$ 1.02\%} (1) & 99.57\% {\tiny $\pm$ 0.97\%} (1) & 98.33\% {\tiny $\pm$ 0.41\%} (20) & 94.55\% {\tiny $\pm$ 8.13\%} (3) & 66.96\% {\tiny $\pm$ \;\;16.65\%} \!\!(26) & 97.07\% {\tiny $\pm$ 0.76\%} (51) \\
$D_{lin:\mathbb{R}^2}$ & 99.55\% {\tiny $\pm$ 1.02\%} (1) & 99.57\% {\tiny $\pm$ 0.97\%} (1) & 98.42\% {\tiny $\pm$ 0.45\%} (19) & 96.36\% {\tiny $\pm$ 4.98\%} (2) & 67.53\% {\tiny $\pm$ \;\;17.80\%} \!\!(25) & \cellcolor{rowcolor}97.24\% {\tiny $\pm$ 0.72\%} (48) \\
$E_{lin:\mathbb{R}^2}$ & 99.55\% {\tiny $\pm$ 1.02\%} (1) & 99.57\% {\tiny $\pm$ 0.97\%} (1) & 98.33\% {\tiny $\pm$ 0.29\%} (20) & 96.36\% {\tiny $\pm$ 4.98\%} (2) & 66.90\% {\tiny $\pm$ \;\;19.25\%} \!\!(26) & 97.12\% {\tiny $\pm$ 0.84\%} (50) \vspace{\smallspacing}\\
$B_{log:\mathbb{R}^2}$ & \;\;4.36\% {\tiny $\pm$ 3.21\%} (199) & \;\;4.59\% {\tiny $\pm$ 6.41\%} (199) & \;\;3.17\% {\tiny $\pm$ 0.86\%} (1162) & \;\;1.82\% {\tiny $\pm$ 4.07\%} (39) & \;\;3.64\% {\tiny $\pm$ \;\;8.13\%} (79) & \;\;3.40\% {\tiny $\pm$ 0.74\%} (1678) \\
$C_{log:\mathbb{R}^2}$ & 68.69\% {\tiny $\pm$ 6.24\%} (65) & 98.10\% {\tiny $\pm$ 2.00\%} (4) & 97.75\% {\tiny $\pm$ 1.01\%} (27) & 96.36\% {\tiny $\pm$ 4.98\%} (2) & 66.94\% {\tiny $\pm$ \;\;16.43\%} \!\!(28) & \cellcolor{rowcolor}92.74\% {\tiny $\pm$ 0.65\%} (126) \\
$D_{log:\mathbb{R}^2}$ & 41.87\% {\tiny $\pm$ 6.81\%} (121) & 97.60\% {\tiny $\pm$ 1.82\%} (5) & 96.00\% {\tiny $\pm$ 1.59\%} (48) & 91.01\% {\tiny $\pm$ 8.46\%} (4) & 65.30\% {\tiny $\pm$ \;\;10.05\%} \!\!(28) & 88.14\% {\tiny $\pm$ 1.21\%} (206) \\
$E_{log:\mathbb{R}^2}$ & 58.68\% {\tiny $\pm$ 4.48\%} (86) & 97.59\% {\tiny $\pm$ 2.48\%} (5) & 97.33\% {\tiny $\pm$ 0.96\%} (32) & 93.51\% {\tiny $\pm$ 9.00\%} (3) & 67.88\% {\tiny $\pm$ \;\;12.61\%} \!\!(27) & 91.20\% {\tiny $\pm$ 0.95\%} (153) \\
\midrule
\multicolumn{7}{c}{{$SE(2)$ templates}}\\
\midrule
$A_{SE(2)}$ & 98.57\% {\tiny $\pm$ 2.16\%} (3) & 98.95\% {\tiny $\pm$ 2.35\%} (2) & 99.58\% {\tiny $\pm$ 0.30\%} (5) & 98.18\% {\tiny $\pm$ 4.07\%} (1) & 94.22\% {\tiny $\pm$ \;\;9.64\%} (5) & \cellcolor{rowcolor}99.08\% {\tiny $\pm$ 0.75\%} (16) \vspace{\smallspacing}\\
$B_{lin:SE(2)}$ & 99.06\% {\tiny $\pm$ 1.29\%} (2) & 94.75\% {\tiny $\pm$ 2.48\%} (11) & 93.74\% {\tiny $\pm$ 1.80\%} (75) & 92.05\% {\tiny $\pm$ 7.95\%} (4) & 85.63\% {\tiny $\pm$ \;\;10.97\%} \!\!(12) & 94.01\% {\tiny $\pm$ 0.89\%} (104) \\
$C_{lin:SE(2)}$ & 99.06\% {\tiny $\pm$ 1.29\%} (2) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 97.50\% {\tiny $\pm$ 5.59\%} (1) & 94.00\% {\tiny $\pm$ \;\;6.17\%} (5) & \cellcolor{rowcolor}99.54\% {\tiny $\pm$ 0.39\%} (8) \\
$D_{lin:SE(2)}$ & 98.60\% {\tiny $\pm$ 2.05\%} (3) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 99.67\% {\tiny $\pm$ 0.46\%} (4) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 94.00\% {\tiny $\pm$ \;\;6.17\%} (5) & 99.31\% {\tiny $\pm$ 0.44\%} (12) \\
$E_{lin:SE(2)}$ & 98.60\% {\tiny $\pm$ 2.05\%} (3) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 99.67\% {\tiny $\pm$ 0.46\%} (4) & 97.50\% {\tiny $\pm$ 5.59\%} (1) & 95.11\% {\tiny $\pm$ \;\;5.48\%} (4) & 99.31\% {\tiny $\pm$ 0.33\%} (12) \vspace{\smallspacing}\\
$B_{log:SE(2)}$ & 87.06\% {\tiny $\pm$ 4.20\%} (27) & 77.68\% {\tiny $\pm$ 5.36\%} (46) & 84.17\% {\tiny $\pm$ 2.25\%} (190) & 80.19\% {\tiny $\pm$ 14.87\%} (9) & 75.10\% {\tiny $\pm$ \;\;9.81\%} (21) & 83.14\% {\tiny $\pm$ 1.78\%} (293) \\
$C_{log:SE(2)}$ & 97.66\% {\tiny $\pm$ 2.79\%} (5) & 99.52\% {\tiny $\pm$ 1.06\%} (1) & 99.58\% {\tiny $\pm$ 0.42\%} (5) & 98.18\% {\tiny $\pm$ 4.07\%} (1) & 95.33\% {\tiny $\pm$ \;\;7.30\%} (4) & \cellcolor{rowcolor}99.08\% {\tiny $\pm$ 0.13\%} (16) \\
$D_{log:SE(2)}$ & 95.22\% {\tiny $\pm$ 3.78\%} (10) & 98.50\% {\tiny $\pm$ 2.27\%} (3) & 99.25\% {\tiny $\pm$ 0.19\%} (9) & 98.18\% {\tiny $\pm$ 4.07\%} (1) & 95.33\% {\tiny $\pm$ \;\;4.74\%} (4) & 98.45\% {\tiny $\pm$ 0.38\%} (27) \\
$E_{log:SE(2)}$ & 97.14\% {\tiny $\pm$ 2.61\%} (6) & 99.52\% {\tiny $\pm$ 1.06\%} (1) & 99.50\% {\tiny $\pm$ 0.35\%} (6) & 98.18\% {\tiny $\pm$ 4.07\%} (1) & 94.22\% {\tiny $\pm$ \;\;6.82\%} (5) & 98.90\% {\tiny $\pm$ 0.48\%} (19) \\
\midrule
\multicolumn{7}{c}{{Template combinations (sorted on performance)}}\\
\midrule
\tiny $A_{\mathbb{R}^2} \;\;\;\;\;\;\;\;\;\;\;\, + C_{log:SE(2)}$ & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 99.92\% {\tiny $\pm$ 0.19\%} (1) & 98.18\% {\tiny $\pm$ 4.07\%} (1) & 98.67\% {\tiny $\pm$ \;\;2.98\%} (1) & 99.83\% {\tiny $\pm$ 0.26\%} (3) \\
\tiny $A_{\mathbb{R}^2} \;\;\;\;\;\;\;\;\;\;\;\, + E_{log:SE(2)}$ & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 99.83\% {\tiny $\pm$ 0.23\%} (2) & 98.18\% {\tiny $\pm$ 4.07\%} (1) & 98.67\% {\tiny $\pm$ \;\;2.98\%} (1) & 99.77\% {\tiny $\pm$ 0.24\%} (4) \\
\tiny $A_{\mathbb{R}^2} \;\;\;\;\;\;\;\;\;\;\;\, + D_{log:SE(2)}$ & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 99.83\% {\tiny $\pm$ 0.23\%} (2) & 98.18\% {\tiny $\pm$ 4.07\%} (1) & 98.67\% {\tiny $\pm$ \;\;2.98\%} (1) & 99.77\% {\tiny $\pm$ 0.24\%} (4) \\
\tiny $C_{lin:SE(2)} + E_{log:SE(2)}$ & 99.55\% {\tiny $\pm$ 1.02\%} (1) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 99.83\% {\tiny $\pm$ 0.23\%} (2) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 96.44\% {\tiny $\pm$ \;\;3.28\%} (3) & 99.65\% {\tiny $\pm$ 0.13\%} (6) \\
\tiny $C_{lin:SE(2)} + C_{log:SE(2)}$ & 99.55\% {\tiny $\pm$ 1.02\%} (1) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 99.92\% {\tiny $\pm$ 0.19\%} (1) & 98.18\% {\tiny $\pm$ 4.07\%} (1) & 96.44\% {\tiny $\pm$ \;\;3.28\%} (3) & 99.65\% {\tiny $\pm$ 0.13\%} (6) \\
\multicolumn{1}{l|}{{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,...}} & \multicolumn{5}{c|}{{...}} & \multicolumn{1}{c}{{...}}\\
\hspace{-0.6em}$^*$\tiny $A_{SE(2)} \;\;\;\;\;\; +C_{lin:SE(2)}$ & 99.55\% {\tiny $\pm$ 1.02\%} (1) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 98.18\% {\tiny $\pm$ 4.07\%} (1) & 94.22\% {\tiny $\pm$ \;\;6.82\%} (5) & 99.60\% {\tiny $\pm$ 0.26\%} (7) \\
\multicolumn{1}{l|}{{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,...}} & \multicolumn{5}{c|}{{...}} & \multicolumn{1}{c}{{...}}\\
\hspace{-0.5em}$^\dagger$\tiny $A_{\mathbb{R}^2} \;\;\;\;\;\;\;\;\;\;\;\; +A_{SE(2)}$ & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 100.0\% {\tiny $\pm$ 0.00\%} (0) & 99.66\% {\tiny $\pm$ 0.35\%} (4) & 98.18\% {\tiny $\pm$ 4.07\%} (1) & 88.42\% {\tiny $\pm$ \;\;11.23\%} (9) & 99.19\% {\tiny $\pm$ 0.63\%} (14) \\
\multicolumn{1}{l|}{{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,...}} & \multicolumn{5}{c|}{{...}} & \multicolumn{1}{c}{{...}}\\
\bottomrule
\multicolumn{7}{l}{$^*$\emph{Best template combination that does not rely on logistic regression.} $^\dagger$\emph{Best template combination that does not rely on template optimization.}}
\end{tabular}
\label{tab:resultsONH}
\end{table*}
Our first application to retinal images is optic nerve head detection. The ONH is one of the key anatomical landmarks in the retina, and its location is often used as a reference point to define regions of interest for the analysis of the retina. The detection hereof is therefore an essential step in many automated retinal image analysis pipelines.
The ONH has two main characteristics: 1) it often appears as a bright disk-like structure on color fundus (CF) images (dark on SLO images), and 2) it is the place from which blood vessels leave the retina. Traditionally, methods have mainly focused on the first characteristic \cite{LuLim2011,Aquino2012,Dashtbozorg2015}. However, in case of bad contrast of the optic disk, or in the presence of pathology (especially bright lesions, see e.g. Fig.~\ref{fig:ODTemplates}), these methods typically fail. Most of the recent ONH detection methods therefore also include the vessel patterns in the analysis; either via explicit vessel segmentation \cite{Sekhar2011,Marin2015}, vessel density measures \cite{Yu2012,Giachetti2013}, or via additional orientation pattern matching steps \cite{Youssif2008}. In our method, both the appearance and vessel characteristics are addressed in an efficient integrated template matching approach, resulting in state-of-the-art performance both in terms of success rates and computation time. We target the first characteristic with template matching on $\mathbb{R}^2$. The second is targeted with template matching on $SE(2)$.
\subsubsection{Processing Pipeline \& Data}
\textbf{\emph{Processing Pipeline.}}
\label{subsubsec:processingPipeline}
First, the images are rescaled to a working resolution of 40 $\mu m/pix$. In our experiments the average resolution per dataset was determined using the average optic disk diameter (which is on average $1.84 mm$). The images are normalized for contrast and illumination variations using the method from \cite{Foracchia2005}. Finally, in order to put more emphasis on contextual/shape information, rather than pixel intensities, we apply a soft binarization to the locally normalized (cf.~Eq.~(31) in Ch.~3 of the supplementary materials) image ${f}$ via the mapping $\operatorname{erf}(8 {f})$.
For the orientation score transform we use $N_\theta = 12$ uniformly sampled orientations from $0$ to $\pi$ and lift the image using cake wavelets \cite{Duits2007a,Bekkers2014}. For phase-invariant, nonlinear, left-invariant \cite{Duits2010}, and contractive \cite{Bruna2013} processing on SE(2), we work with the modulus of the complex valued orientation scores rather than with the complex-valued scores themselves (taking the modulus of quadrature filter responses is an effective technique for line detection, see e.g. Freeman et al. \cite{Freeman1991}).
Due to differences in image characteristics, training and matching is done separately for the SLO and the color fundus images. For SLO images we use the near infrared channel, for RGB fundus images we use the green channel.
Positive training samples $f_i$ are defined as $N_x \times N_y$ patches, with $N_x=N_y=251$, centered around true ONH location in each image. For every image, a negative sample is defined as an image patch centered around random location in the image that does not lie within one optic disk radius distance to the true ONH location. An exemplary ONH patch is given in Fig.~\ref{fig:odOS}. For the B-spline expansion of the templates we set $N_k = N_l = 51$ and $N_m = 12$.
\textbf{\emph{Data.}}
\label{subsubsec:dataONH}
In our experiments we made use of both publicly available data, and a private database. The private database consists of 208 SLO images taken with an EasyScan fundus camera (i-Optics B.V., the Netherlands) and 208 CF images taken with a Topcon NW200 (Topcon Corp., Japan). Both cameras were used to image both eyes of the same patient, taking an ONH centered image, and a fovea centered image per eye. The two sets of images are labeled as "ES" and "TC" respectively. The following (widely used) public databases are also used: MESSIDOR (\url{http://messidor.crihan.fr/index-en.php}), DRIVE (\url{http://www.isi.uu.nl/Research/Databases/DRIVE}) and STARE (\url{http://www.ces.clemson.edu/~ahoover/stare}), consisting of 1200, 40 and 81 images respectively. For each image, the circumference of the ONH was annotated, and parameterized by an ellipse. The annotations for the MESSIDOR dataset were kindly made available by the authors of \cite{Aquino2010} (\url{http://www.uhu.es/retinopathy}). The ONH contour in the remaining images were manually outlined by ourselves. The annotations are made available on our website. The images in the databases contain a mix of good quality healthy images, and challenging diabetic retinopathy cases. Especially MESSIDOR and STARE contain challenging images.
\subsubsection{Results and Discussion}
\emph{\textbf{The templates.}} The different templates for ONH detection are visualized in Fig.~\ref{fig:ODTemplates}. The $SE(2)$ templates are visualized using maximum intensity projections over $\theta$. In this figure we have also shown template responses to an example image. Visually one can clearly recognize the typical disk shape in the $\mathbb{R}^2$ templates, whereas the $SE(2)$ templates also seem to capture the typical pattern of outward radiating blood vessels (compare e.g. $A_{\mathbb{R}^2}$ with $A_{SE(2)}$). Indeed, when applied to a retinal image, where we took an example with an optic disk like pathology, we see that the $\mathbb{R}^2$ templates respond well to the disk shape, but also (more strongly) to the pathology. In contrast, the $SE(2)$ templates respond mainly to vessel pattern and ignore the pathology. We also see, as expected, a smoothing effect of gradient based regularization ($D$ and $E$) in comparison to standard $\mathbb{L}_2$-norm regularization ($C$) and no regularization ($B$).
Finally, in comparison to linear regression templates, the logistic regression templates have a more binary response due to the logistic sigmoid mapping.
\begin{table}
\caption{Comparison to state of the art: Optic nerve head detection success rates, the number of fails (in parentheses), and computation times.}
\centering
\begin{tabular}{llll|l}
\toprule
Method & MESSIDOR & DRIVE & STARE & Time (s) \\
\midrule
Lu {\tiny \cite{Lu2011} } & 99.8\% (3) & & 98.8\% (1) & \;\;\;\;5.0 \\
Lu {\tiny et al. \cite{LuLim2011} } & & 97.5\% (1) & 96.3\% (3) & \;\;40.0 \\
Yu {\tiny et al. \cite{Yu2012} } & 99.1\% (11) & & & \;\;\;\;4.7 \\
Aquino {\tiny et al. \cite{Aquino2012} } & 99.8\% (14) & & & \;\;\;\;1.7 \\
Giachetti {\tiny et al. \cite{Giachetti2013} } & 99.7\% (4) & & & \;\;\;\;5.0 \\
Ramakanth {\tiny et al. \cite{Ramakanth2014} } & 99.4\% (7) & 100\% (0) & 93.83\% (5) & \;\;\;\;0.2 \\
Marin {\tiny et al. \cite{Marin2015}} & 99.8\% (3) & & & \;\;\;\;5.4$^\dagger$ \\
Dashtbozorg {\tiny et al. \cite{Dashtbozorg2015}} & 99.8\% (3) & & & \;\;10.6$^\dagger$ \vspace{\smallspacing}\\
Proposed & 99.9\% (1) & 97.8\% (1) & 98.8\% (1) & \;\;\;\;0.5 \\
\bottomrule
\multicolumn{5}{l}{$^\dagger$\emph{Timings include simultaneous disk segmentation.}}\\
\end{tabular}%
\label{tab:stateOfTheArtONH}
\end{table}
\emph{\textbf{Detection results.}} Table \ref{tab:resultsONH} gives a breakdown of the quantitative results for the different databases used in the experiments. The templates are grouped in $\mathbb{R}^2$ templates, $SE(2)$ templates, and combination of templates. Within these groups, they are further divided in average, linear regression, and logistic regression templates. The best overall performance within each group is highlighted in gray.
Overall, we see that the $SE(2)$ templates out-perform their $\mathbb{R}^2$ equivalents, and that combinations of the two types of templates give best results. The two types are nicely complementary to each other due to disk-like sensitivity of the $\mathbb{R}^2$ templates and the vessel pattern sensitivity of the $SE(2)$ templates. If one of the two ONH characteristics is less obvious (as is e.g. for the disk-shape in Fig.~\ref{fig:ODTemplates}), the other can still be detected. Also, the failures of $\mathbb{R}^2$ templates are mainly due to either distracting pathologies in the retina, or poor contrast of the optic disk. As reflected by the increased performance of $SE(2)$ templates over $\mathbb{R}^2$ templates, a more stable pattern seems to be the vessel pattern.
From Table \ref{tab:resultsONH} we also deduce that the individual performances of the linear regression templates outperform the logistic regression templates. Moreover, the average templates give best individual performance, which indicates that with our effective template matching framework good performance can already be achieved with basic templates.
However, we also see that low performing individual templates can prove useful when combining templates.
In fact, we see that combinations with all linear $\mathbb{R}^2$ templates are highly ranked, and for the $SE(2)$ templates it is mainly the logistic regression templates. This can be explained by the binary nature of the logistic templates: even when the maximum response of the templates is at an incorrect location, the difference with the correct location is often small. The $\mathbb{R}^2$ template then adds to the sensitivity and precision.
The best results obtained with untrained templates was a $99.19\%$ success rate (14 fails), and with the overall best template combination we obtained a $99.83\%$ success rate (3 fails).
\emph{\textbf{State of the art.}} In Table \ref{tab:stateOfTheArtONH} we compare our results on the publicly available benchmark databases MESSIDOR, DRIVE and STARE, with the most recent methods for ONH detection (sorted from oldest to newest from top to bottom). In this comparison, our best performing method ($A_{\mathbb{R}^2}+C_{log:SE(2)}$) performs better than or equally well as the best methods from literature. We have also listed the computation times, and see that our method is also ranked as one of the fastest methods for ONH detection. The average computation time, using our experimental implementation in Wolfram \emph{Mathematica} 10.4, was $0.5$ seconds per image on a computer with an Intel Core i703612QM CPU and 8GB memory.
A full breakdown of timings of the processing pipeline is given in the supplementary materials Sec.~4.
\subsection{Fovea Detection in Retinal Images}
\label{subsec:FoveaDetection}
Our second application to retinal images is for the detection of the fovea. The fovea is the location in the retina which is responsible for sharp central vision.
It is characterized by a small depression in thickness of the retina, and on healthy retinal images it often appears as a darkened area. Since the foveal area is responsible for detailed vision, this area is weighted most heavily in grading schemes that describe the severity of a disease.
Therefore, correct localization of the fovea is essential in automatic grading systems \cite{Abramoff2015}.
Methods for the detection of the fovea heavily rely on contextual features in the retina \cite{Aquino2014,Giachetti2013,GegundezArias2013,Yu2011,Niemeijer2009}, and take into account the prior knowledge that 1) the fovea is located approximately 2.5 optic disk diameters lateral to the ONH center, that 2) it lies within an avascular zone, and that 3) it is surrounded by the main vessel arcades. All of these methods restrict their search region for the fovea location to a region relative to the (automatically detected) ONH location. To the best of our knowledge, the proposed detection pipeline is the first that is completely independent of vessel segmentations and ONH detection. This is made possible due to the fact that anatomical reference patterns, in particular the vessel structures, are generically incorporated in the learned templates via data representations in orientation scores.
\subsubsection{Processing Pipeline \& Data}
\textbf{\emph{Processing Pipeline.}}
The proposed fovea detection pipeline is the same as for ONH detection, however, now the positive training samples $f_i$ are centered around the fovea.
\textbf{\emph{Data.}}
The proposed fovea detection method is validated on our (annotated) databases ``ES'' and ``TC'', each consisting of 208 SLO and 208 color fundus images respectively (cf. Subsec.\ref{subsubsec:dataONH}). We further test our method on the most used publicly available benchmark dataset MESSIDOR (1200 images). Success rates were computed based on the fovea annotations kindly made available by the authors of \cite{GegundezArias2013}.
\subsubsection{Results and Discussion}
\emph{\textbf{The templates.}} Akin to Fig.~\ref{fig:ODTemplates}, in Fig.~\ref{fig:FoveaTemplates} the trained fovea templates and their responses to an input image are visualized. The $\mathbb{R}^2$ templates seem to be more tuned towards the dark (isotropic) blob like appearance of the fovea, whereas in the $SE(2)$ templates one can also recognize the pattern of vessels surrounding the fovea (compare $A_{\mathbb{R}^2}$ with $A_{SE(2)}$). To illustrate the difference between these type of templates, we selected an image in which the fovea location is occluded with bright lesions. In this case the method has to rely on contextual information (e.g. the blood vessels). Indeed, we see that the $\mathbb{R}^2$ templates fail due to the absence of a clear foveal blob shape, and that the $SE(2)$ templates correctly identify the fovea location. The effect of regularization is also clearly visible; no regularization ($B$) results in noisy templates, standard $\mathbb{L}_2$ regularization ($C$) results in more stable templates, and smoothed regularization ($D$ and $E$) results in smooth templates.
In templates $D_{SE(2)}$ and $E_{SE(2)}$ we see that more emphasis is put on line structures
\emph{\textbf{Detection results.}} A full overview of individual and combined template performance is discussed in the supplementary materials, here we only provide a summary. Again there is an improvement using $SE(2)$ templates over $\mathbb{R}^2$ templates, although the difference is smaller than in the ONH application.
Apparently both the dark blob-like appearance ($\mathbb{R}^2$ templates) and vessel patterns ($SE(2)$ templates) are equally reliable features of the fovea. A combination of templates leads to improved results and we conclude that the templates are again complementary to each other. Furthermore, again linear regression performs better than logistic regression.
In fovea detection
we do observe a large improvement of template training over basic averaging: 1529 of 1616 ($94.6\%$) successful detections with $C_{lin:SE(2)}$ versus 1488 ($92.1\%$) with $A_{SE(2)}$. The best performing $\mathbb{R}^2$ template was $A_{\mathbb{R}^2}$ ($65.6\%$), the best $SE(2)$ template was $C_{lin:SE(2)}$ ($94.6\%$). The best combination of templates was $C_{lin:\mathbb{R}^2}+C_{log:SE(2)}$ with 1605 ($99.3\%$) detections. When using non-optimized templates
1588 ($98.3\%$) successful detections were achieved (with $A_{\mathbb{R}^2}+A_{SE(2)}$).
\emph{\textbf{State of the art.}} In Table \ref{tab:stateOfTheArtFovea} we compared our results on the publicly available benchmark database MESSIDOR with the most recent methods for fovea detection (sorted from oldest to newest from top to bottom). In this comparison, our best performing method ($C_{lin:\mathbb{R}^2}+C_{log:SE(2)}$) quite significantly outperforms the best methods from literature. Furthermore, our detection pipeline is also the most efficient one; the computation time for fovea detection is the same as for ONH detection, which is $0.5$ seconds.
\begin{table}
\caption{Comparison to state of the art: Fovea detection success rates, the number of fails (in parentheses), and computation times.}
\centering
\begin{tabular}{ll|l}
\toprule
Method & MESSIDOR & Time (s) \\
\midrule
Niemeijer {\tiny et al. \cite{Niemeijer2009,GegundezArias2013} } & 97.9\% \;\;(25) & \;\;\;\;7.6$^{\dagger}$\\
Yu {\tiny et al. \cite{Yu2011} } & 95.0\%$^*$ (60) & \;\;\;\;3.9$^{\dagger}$\\
Gegundez-Arias {\tiny et al. \cite{GegundezArias2013}} & 96.9\% \;\;(37) & \;\;\;\;0.9 \\
Giachetti {\tiny et al. \cite{Giachetti2013}} & 99.1\% \;\;(11) & \;\;\;\;5.0$^{\dagger}$ \\
Aquino {\tiny \cite{Aquino2014}} & 98.2\% \;\;(21) & \;\;10.9$^{\dagger}$ \vspace{\smallspacing}\\
Proposed & 99.7\% \;\;(3) & \;\;\;\;0.5 \\
\bottomrule
\multicolumn{3}{l}{$^*$\emph{Success-criterion based on half optic radius.}}\\
\multicolumn{3}{l}{$^\dagger$\emph{Timing includes ONH detection.}}\\
\end{tabular}%
\label{tab:stateOfTheArtFovea}
\end{table}
\subsection{Pupil Detection}
\label{subsec:PupilDetection}
Our third application is that of pupil localization in regular camera images, which is relevant in many applications as they provide important visual cues for face detection, face recognition, and understanding of facial expressions. In particular in gaze estimation the accurate localization of the pupil is essential. Eye detection and tracking is however challenging due to, amongst others: occlusion by the eyelids and variability in size, shape, reflectivity or head pose.
Many pupil localization algorithms are designed to work on periocular images, these are close-up views of the eyes. Such images can be acquired by dedicated eye imaging devices, or by means of cropping a full facial image (see Fig.~\ref{fig:PupilTemplates}(a)).
We will consider both the problem of detection pupils in periocular images and the more difficult problem of detection in full images.
We compare our method against the seven most recent pupil detection methods from literature, for a full overview see \cite{Leo2014} and \cite{Markus2014}. A method similar to our $\mathbb{R}^2$ approach in the sense that it is also based on 2D linear filtering is the method by Kroon et al. \cite{Kroon2008}. In their paper templates are obtained via linear discriminant analysis of pupil images. Asteriada et al. \cite{Asteriadis2009} detect the pupil by matching templates using features that are based on distances to the nearest strong (facial) edges in the image. Campadelli et al. \cite{Campadelli2009} use a supervised approach with a SVM classifier and Haar wavelet features. The method by Timm et al. \cite{Timm2011} is based on searching for gradient fields with a circular symmetry. Valenti et al. \cite{Valenti2012} use a similar approach but additionally include information of isophote curvature, with supervised refinement. Markus et al. \cite{Markus2014} employ a supervised approach using an ensemble of randomized regression trees. Leo et al. \cite{Leo2014} employ a completely unsupervised approach similar to those in \cite{Timm2011,Valenti2012}, but additionally include analysis of self-similarity
A relevant remark is that all of the above mentioned methods rely on prior face detection, and restrict their search region to periocular images. Our method works completely stand alone, and can be used on full images.
\subsubsection{Processing Pipeline \& Data}
\textbf{\emph{Processing Pipeline.}}
Interestingly, we could again employ the same processing pipeline (including local normalization via \cite{Foracchia2005}) which was used for ONH and fovea detection. In our experiments we train templates for the left and right eye separately.
\textbf{\emph{Data.}}
We validated our pupil detection approach on the publicly available BioID database (\url{http://www.bioid.com}), which is generally considered as one of the most challenging and realistic databases for pupil detection in facial images. The database consists of $1521$ frontal face grayscale images with significant variation in illumination, scale and pose.
\subsubsection{Results and Discussion}
\emph{\textbf{The templates.}} Fig.~\ref{fig:PupilTemplates}(b) and (c) show respectively the trained $\mathbb{R}^2$ and $SE(2)$ templates for pupil detection of the right eye, and their filtering response to the input image in Fig.~\ref{fig:PupilTemplates}(a). Here the trained $\mathbb{R}^2$ templates seemed to capture the pupil as a small blob in the center of the template, but apart from that no real structure can be observed. In the average template we do however clearly see structure in the form of an ``average face''. The $SE(2)$ templates reveal structures that resemble the eyelids in nearly all templates. The linear regression templates look sharper and seem to contain more detail than the average template, and the logistic regression templates seem to take a good compromise between course features and details.
\emph{\textbf{Detection results.}} We again refer to the supplementary materials for a full benchmarking analysis, in summary we observed the following. In terms of success rates we see a similar pattern as with the ONH and fovea application, however, here we see that the learned templates ($C$,$D$ and $E$) significantly outperform the average templates, and that logistic regression leads to better templates than using linear regression (94.0\% success rate for $C_{log:SE(2)}$ vs 87.2\% for $D_{lin:SE(2)}$). Overall, the $SE(2)$ templates outperform the $\mathbb{R}^2$ templates, linear regression templates outperform the average template, and logistic regression templates outperform linear regression templates.
The best $\mathbb{R}^2$ template was $D_{lin:\mathbb{R}^2}$ with 1151 of 1521 detections ($75.7\%$), the best $SE(2)$ template was $C_{log:SE(2)}$ ($94.0\%$). The best combination of templates was $D_{lin:\mathbb{R}^2}$ with $E_{lin:SE(2)}$ ($95.6\%$). Without template training (i.e., using average templates $A$) the performance was only $68.2\%$. Success rates
using the best template combination are given in Fig.~\ref{fig:PupilTemplates}(d) and (e). The processing time for detecting both pupils simultaneously was on average 0.4 seconds per image
\emph{\textbf{State of the art.}} In Fig.~\ref{fig:PupilTemplates}(d) we compared our approach to the two most recent pupil detection methods from literature for several normalized error thresholds. Here we see that with allowed errors of $0.1$ (blue circles Fig.~\ref{fig:PupilTemplates}(a)) and higher our method competes very well with the state of the art, despite the fact that our generic method is not adapted to the application. Further application specific tuning and preprocessing could be applied to improve precision (for $e \ll 0.1$), but this is beyond the scope of this article. Moreover, we see that our method can be used on full images instead of the periocular images without much loss in performance.
The fact that our method is still very accurate on full image processing
shows that it can be used as a preprocessing step for other applications.
If Fig.~\ref{fig:PupilTemplates}(e) we compared our approach to the seven most recent methods from literature (sorted from old to new). Here we see that the only method outperforming our method, at standard accuracy requirements ($e\leq0.1$), is the method by Markus et al. \cite{Markus2014}. Even when considering processing of the full images the only other method that outperforms ours is the method by Timm et al. \cite{Timm2011}, whose performance is measured using periocular images.
\subsection{General Observations}
\label{subsec:GeneralDiscussion}
The application of our method to the three problems (ONH, fovea and pupil detection) showed the following:
\begin{enumerate}
\item State-of-the-art performance was achieved on three different applications, using a single (generic) detection framework and without application specific parameter adaptations.
\item Cross correlation based template matching via data representations on $SE(2)$ improves results over standard $\mathbb{R}^2$ filtering.
\item Trained templates, obtained using energy functionals of the form (\ref{eq:energyGeneric}), often perform better than basic average templates. In particular in pupil detection the optimization of templates proved to be essential.
\item Our newly introduced logistic regression approach leads to improved results in pupil detection via single templates. When combining templates we observe only a small improvement of choosing logistic regression (instead of linear regression) for the application of ONH and fovea detection
\item Regularization in both linear and logistic regression is important. Here both ridge and smoothing regularization priors have complementary benefits.
\item Our method does not rely on any other detection systems (such as ONH detection in the fovea application, or face detection in the pupil detection), and still performs well compared to methods that do.
\item Our method is fast and parallelizable as it is based on inner products, as such it could be efficiently implemented using convolutions.
\end{enumerate}
\subsection{Discussion}
In this paper we have presented an efficient cross-correlation based template matching scheme for the detection of combined orientation and blob patterns. Furthermore, we have provided a generalized regression framework for the construction of templates. The method relies on data representations in orientation scores, which are functions on the Lie group $SE(2)$, and we have provided the tools for proper smoothing priors via resolvent hypo-elliptic diffusion processes on $SE(2)$ (solving time-integrated hypo-elliptic Brownian motions on $SE(2)$). The strength of the method was demonstrated with two applications in retinal image analysis (the detection of the optic nerve head (ONH), and the detection of the fovea) and additional experiments for pupil detection in regular camera images. In the retinal applications we achieved state-of-the-art results with an average detection rate of $99.83\%$ on $1737$ images for ONH detection, and $99.32\%$ on $1616$ images for fovea detection.
Also on pupil detection we obtained state-of-the-art performance with a $95.86\%$ success rate on 1521 images. We showed that the success of the method is due to the inclusion of both intensity and orientation features in template matching. The method is also computationally efficient as it is entirely based on a sequence of convolutions (which can be efficiently done using fast Fourier transforms). These convolutions are parallelizable, which can further speed up our already fast experimental \emph{Mathematica} implementations that are publicly available at \url{http://erikbekkers.bitbucket.org/TMSE2.html}. In future work we plan to investigate the applicability of smoothing on $SE(2)$ in variational settings, as this could also be used in (sparse) line enhancement and segmentation problems.
\subsection{Resolvent Diffusion Processes}
\label{subsec:resolventDiffusions}
A classic approach to noise suppression in images is via diffusion regularizations with PDE's of the form \cite{DuitsBurgeth2007}
\begin{equation}
\label{eq:pdeDiffusion}
\left\{
\begin{array}{cl}
\tfrac{\partial}{\partial \tau}u &= \Delta u,\\
u |_{\tau=0} &= u_0,
\end{array}
\right.
\end{equation}
where $\Delta$ denotes the Laplace operator. Solving (\ref{eq:pdeDiffusion}) for any diffusion time $\tau>0$ gives a smoothed version of the input $u_0$. The time-resolvent process of the PDE is defined by the Laplace transform with respect to $\tau$; time $\tau$ is integrated out using a memoryless negative exponential distribution $P(\mathcal{T}=\tau)=\alpha e^{-\alpha \tau}$. Then, the time integrated solutions
$$
t(\mathbf{x}) = \alpha \int_0^\infty u(\mathbf{x},\tau) e^{-\alpha \tau} {\rm d}\tau,
$$
with decay parameter $\alpha$, are in fact the solutions
\begin{equation}
t = \underset{t \in \mathbb{L}_2(\mathbb{R}^2)}{\operatorname{argmin}} \left[ \lVert t - t_0 \rVert^2_{\mathbb{L}_2(\mathbb{R}^2)} + \lambda \int_{\mathbb{R}^2} \lVert \nabla t (\tilde{\mathbf{x}}) \rVert^2 \, {\rm d}\tilde{\mathbf{x}} \right],
\end{equation}
with $\lambda = \alpha^{-1}$, and corresponding Euler-Lagrange equation
\begin{equation}
(I - \lambda \Delta) t = t_0 \;\;\; \Leftrightarrow \;\;\; t = \lambda^{-1} \left( \frac{1}{\lambda} - \Delta \right)^{-1} t_0,
\end{equation}
to which we refer as the ``resolvent'' equation \cite{Yosida1995}, as it involves operator $(\alpha I - \Delta)^{-1}$, $\alpha = \lambda^{-1}$. In the next subsections, we follow a similar procedure with $SE(2)$ instead of $\mathbb{R}^2$, and show how the smoothing regularizer in Eq.~(28) and (30) of the main article relates to Laplace transforms of hypo-elliptic diffusions on the group $SE(2)$ \cite{ZhangDuits2014,Duits2010}.
\subsection{The Fundamental Single Patch Problem}
\label{subsec:toyProblem
In order to grasp what the (anisotropic regularization term) in Eq. (28) and (30) of the main article actually means in terms of stochastic interpretation/probabilistic line propagation, let us consider the following single patch problem and optimize
\begin{multline}\label{min}
\mathcal{E}_{sp}(T) =
\left| \left( G_s *_{\mathbb{R}^2} T(\cdot,\cdot,\theta_0) \right)(\mathbf{x}_0) - 1 \right|^2 \\
+ \lambda \int_{\mathbb{R}^2} \int_0^{2\pi} \lVert \nabla T(\tilde{\mathbf{x}},\tilde{\theta}) \rVert^2_{D} {\rm d}\tilde{\mathbf{x}}{\rm d}\tilde{\theta}
+ \mu \lVert T \rVert^2_{\mathbb{L}_2(SE(2))},
\end{multline}
with $(\mathbf{x}_0,\theta_0)=g_0:=(x_0,y_0,\theta_0) \in SE(2)$ the fixed center of the template, and with spatial Gaussian kernel
$$
G_s(\mathbf{x}) = \frac{1}{4 \pi s} e^{-\frac{\lVert \mathbf{x} \rVert^2}{4 s}}.
$$
Regarding this problem, we note the following:
\begin{itemize}
\item In the original problem (28) of the main article we take $N=1$, with
\begin{equation}
\label{eq:uf1}
U_{f_1}(x,y,\theta)=G_s(x-x_0,y-y_0) \, \delta_{\theta_0}(\theta)
\end{equation}
representing a local spatially smoothed spike in $SE(2)$, and set $y_1 = 1$. The general single patch case (for arbitrary $U_{f_1}$) can be deduced by superposition of such impulse responses.
\item We use $\mu>0$ to suppress the output elsewhere.
\item We use $0 < s \ll 1$. This minimum scale due to sampling removes the singularity at $(\mathbf{0},0)$ from the kernel that solves (\ref{min}), as proven in \cite{ZhangDuits2014}.
\end{itemize}
\begin{mythm}
\label{thm:1}
The solution to the single patch problem (\ref{min}) coincides up to scalar multiplication with the time integrated hypo-elliptic Brownian motion kernel on $SE(2)$ (depicted in Fig.~\ref{fig:stochasticEnhancement}).
\end{mythm}
\begin{proof}[\textbf{Proof}]
We optimize $\mathcal{E}_{sp}(T)$ over the set $\mathcal{S}(SE(2))$ of
all functions $T:SE(2) \rightarrow \mathbb{R}$ that are bounded and on $SE(2)$, infinitely differentiable on $SE(2) \setminus \{g_0\}$, and rapidly decreasing in spatial direction, and $2\pi$ periodic in $\theta$.
We omit topological details on function spaces and H\"{o}rmanders condition \cite{Hoermander1967}.
Instead, we directly proceed with applying the Euler-Lagrange technique to the single patch problem:
\begin{multline} \label{EL}
\forall_{\delta \in \mathcal{S}(SE(2))}: \underset{\epsilon \downarrow 0}{\operatorname{lim}} \left\{ \frac{\mathcal{E}_{sp}(T + \epsilon \delta) - \mathcal{E}_{sp}(T)}{\epsilon} \right\} = 0
\Leftrightarrow \\
(S^*_s S_s + \lambda R + \mu I) T = S^*_s y_1 = S^*_s 1,
\end{multline}
with linear functional (distribution) $\mathcal{S}_s$ given by
$$ (S_s T) = (G_{s} *_{\mathbb{R}^2} T(\cdot,\theta_0))(\mathbf{x}_0),$$
and with regularization operator $R$ given by
$$
R = -\Delta_{SE(2)} := - (D_{\theta\theta} \partial_\theta^2 + D_{\xi\xi}\partial_\xi^2 + D_{\eta\eta}\partial_\eta^2) \geq 0.
$$
Note that
$\lim \limits_{s \to 0} S_{s} = \delta_{(\mathbf{x}_0,\theta_0)}$ in distributional sense, and that the constraint $s>0$ is crucial for solutions $T$ to be bounded at $(\mathbf{x}_0,\theta_0)$. By definition the adjoint operator $S_s^*$ is given by
$$
\begin{array}{rl}
(S^*_s y, T)_{\tiny\mathbb{L}_2(SE(2))} \!\!\!\!\!\! & = (y, S_s T)
= y \int_{\mathbb{R}^2} G_s(\mathbf{x}-\mathbf{x}_0) T(\mathbf{x},\theta_0)\, {\rm d}\mathbf{x}\\
& = y \int\limits_0^{2\pi} \int\limits_{\mathbb{R}^2} G_s(\mathbf{x}-\mathbf{x}_0)\delta_{\theta_0}(\theta) T(\mathbf{x},\theta)\, {\rm d}\mathbf{x}{\rm d}\theta,\\
& = (y \;G_s(\cdot-\mathbf{x}_0) \delta_{\theta_0}(\cdot),T )_{\tiny\mathbb{L}_2(SE(2))}
\end{array}
$$
and thereby we deduce that
$$
\begin{aligned}
(S^*_s y)(\mathbf{x},\theta) &= y \;G_s(\mathbf{x}-\mathbf{x}_0) \delta_{\theta_0}(\theta), \\
\ \ \ \ S^*_s (S_s T) &= T_0^{s}\; G_s(\mathbf{x}-\mathbf{x}_0) \delta_{\theta_0}(\theta),
\end{aligned}
$$
with $\infty > T_0^{s}:=(G_{s} *_{\mathbb{R}^2} T(\cdot,\theta_0))(\mathbf{x}_0)> 1$ for $0 < s \ll 1$.
The Euler-Lagrange equation (\ref{EL}) becomes
$$
(-\lambda \Delta_{SE(2)} + \mu I ) T = (1 - T^s_0)G_{s}(\mathbf{x}-\mathbf{x}_0) \delta_{\theta_0}(\theta).
$$
Now, when setting $T_{new} = \frac{T}{1-T_0^s}$ we arrive at the hypo-elliptic resolvent equation on $SE(2)$:
\begin{equation}
\label{aap}
\begin{aligned}
(- \lambda \Delta_{SE(2)} + \mu I) T_{new} = (G_{s} *_{\mathbb{R}^2} \delta_{\mathbf{x}_0}) \delta_{\theta_0} \;\;\;\;\;\;
\Leftrightarrow \\%\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\
T_{new} =\left( - \lambda \Delta_{SE(2)} + \mu I \right)^{-1} e^{s \Delta_{\mathbb{R}^2}} \delta_{g_0} \>\\
= e^{s \Delta_{\mathbb{R}^2}} \left( - \lambda \Delta_{SE(2)} + \mu I \right)^{-1} \delta_{g_0}
\end{aligned}
\end{equation}
where we write $e^{s \Delta_{\mathbb{R}^2}} f= G_{s} *_{\mathbb{R}^{2}} f$ for the diffusion operator, to stress
the vanishing commutators
\[
[e^{s \Delta_{\mathbb{R}^2}},\Delta_{SE(2)}]=e^{s \Delta_{\mathbb{R}^2}}\Delta_{SE(2)}-\Delta_{SE(2)} e^{s \Delta_{\mathbb{R}^2}}=0,
\]
which directly follows from $[\Delta_{\mathbb{R}^2},\Delta_{SE(2)}]=0$.
In fact, from these vanishing commutators one can deduce that, thanks to the isotropy of Gaussian kernel, blurring with inner-scale $s>0$ can be done either before applying the resolvent operator or after (as seen in (\ref{aap})).
The solutions $T_{new}$ are precisely the probabilistic kernels $R_{\alpha,s}:SE(2) \to \mathbb{R}$ for time integrated contour enhancements studied in \cite{ZhangDuits2014,Duits2010}.
In fact we see that
\[
T_{new}(g)=\mu^{-1} R_{\alpha,s}(g_0^{-1}g),
\]
where $R_{\alpha,s}= (I -\alpha^{-1} \Delta_{SE(2)})^{-1} e^{s \Delta_{\mathbb{R}^{2}}} \delta_{(\mathbf{0},0)}$ (i.e., the impuls response of the resolvent operator) denotes the time-integration of the hypo-elliptic diffusion kernel
$K_{\tau,s}= e^{\tau \Delta_{SE(2)}}e^{s \Delta_{\mathbb{R}^2}} \delta_{(\mathbf{0},0)}$:
\[
R_{\alpha,s}(g)= \alpha \int_{0}^{\infty} K_{\tau,s}(g) \, e^{-\alpha \tau}\, {\rm d}\tau,
\]
for which 3 different exact analytic formulas are derived in \cite{Duits2010}. The kernel $R_{\alpha,s}(\mathbf{x},\theta)$ denotes the probability density of finding a random brush stroke (regardless its traveling time) at location $\mathbf{x}$ with orientation $\theta$ given that a `drunkman's pencil' starts at $g=(\mathbf{0},0)$ at time zero. Here the traveling time $\tau$ of the random pencil is assumed to be negatively exponentially distributed with expectation $\alpha^{-1}$.
\end{proof}
\begin{figure}
\centerline{
\includegraphics[width=\hsize]{Figs/FigKernel}
}
\caption{
Top row: Comparison of kernel $R_{\alpha,s}(x,y,\theta)$ along respectively the $\theta$ and $x$ axis. Bottom row:
Isosurface of the kernel computed by solving the fundamental single patch problem (\ref{min}), the exact solution, and an illustration of the drunkman's pencil. For Monte Carlo simulations of the drunkman's pencil see the supplementary materials
\label{fig:stochasticEnhancement}}
\end{figure}
\subsection{Expansion in B-splines}
\label{subsec:expansionBSplines}
Now we consider the B-spline expansions (Eq.~(34) in the main article) and apply our optimization algorithm (cf. Subsec.~2.4 of the main article) to the single patch problem
(\ref{min}), with $(\mathbf{x}_0,\theta_0) = (\mathbf{0},0)$. Here we no longer need a smoothing with a continuous Gaussian $G_s$, as expansion in the B-spline basis already includes regularization. Now we set for the smooth spike $U_{f_1}(x,y,\theta) = B^n\!\left( \frac{x}{s_k} \right) B^n\!\left( \frac{y}{s_l} \right)B^n\!\left( \frac{\theta \!\!\!\! \mod 2\pi}{s_m} \right)$, and we thus approximate spikes by the same B-spline basis in which we expressed our templates.
We accept extra regularization (like we did with the Gaussian in the previous section) and choose to represent a spike by a normal B-spline.
After all, via the central limit theorem B-splines converge to Gaussians when increasing $n$.
We also considered to instead use the fundamental B-Spline \cite[Fig.~2]{Unser93}, which is sharper but suffers from oscillations, yielding less favorable results.
In our normal B-spline setting,
this choice of smooth spike representation (cf. Eq.~(14) in the main article) leads to the following
equations
$$
(S^\dagger S + \lambda R + \mu I ) T = S^\dagger 1,
$$
with $S$ the $[1 \times N_k N_l N_m]$-matrix whose components are given by $M(0,0,0)\, B_{s_k s_l s_m}(k,l,m)$.
Akin to the previous derivations (\ref{aap}) this matrix-equation can be rewritten as
$$
\left( \lambda R + \mu I \right) T_{new} = S^\dagger 1.
$$
In particular our $B$-spline basis algorithm is a new algorithm that can be used for the resolvent (hypo-)elliptic diffusion process on $SE(2)$.
The benefit over Fourier based algorithms is the local support of the basis functions, which allow for sparse representations.
In Fig.~\ref{fig:stochasticEnhancement} we compare the impulse response for Tikhonov regularization via our B-spline expansion algorithm with the Brownian motion prior on $SE(2)$
(using a fine B-spline basis) to
the exact solutions derived in \cite{ZhangDuits2014,Duits2010}.
The strong accuracy of our algorithm shows that even in the discrete B-spline setting the probabilistic interpretation (Thm.~\ref{thm:1}) of our prior in $SE(2)$-template matching holds.
\subsection{The Drunkman's Pencil}
Similar to the diffusions on $\mathbb{R}^2$, given by (\ref{eq:pdeDiffusion}), the hypo-elliptic diffusion process on $SE(2)$ is described by the following PDE:
\begin{equation}
\label{eq:pdeDiffusionSE2}
\left\{
\begin{array}{cl}
\tfrac{\partial}{\partial \tau}W &= (D_{\xi\xi} \partial_{\xi}^2 + D_{\theta\theta} \partial_{\theta}^2) W,\\
W |_{\tau=0} &= W_0,
\end{array}
\right.
\end{equation}
initialized with $W_0 \in \mathbb{L}_2(\mathbb{R}^2)$ at time $\tau = 0$. The PDE can be used to obtain the solutions of our single patch problem by initializing $W_0$ with a smooth spike such as we did in Subsec.~\ref{subsec:expansionBSplines}, e.g. taking $W_0 = U_{f_1}(x,y,\theta) = B^n\!\left( \frac{x}{s_k} \right) B^n\!\left( \frac{y}{s_l} \right)B^n\!\left( \frac{\theta \!\!\!\! \mod 2\pi}{s_m} \right)$.
The PDE in (\ref{eq:pdeDiffusionSE2}) is the forward Kolmogorov equation \cite{hsustochastic} of the following stochastic process \cite{ZhangDuits2014}:
\begin{equation}
\label{eq:stochasticProcess}
\left\{
\begin{array}{l}
\mathbf{x}(\tau) = \mathbf{x}(0) + \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \sqrt{2 D_{\xi\xi}} \; \epsilon_\xi \int_0^\tau (\cos \theta(\tau) \mathbf{e}_x + \sin \theta(\tau) \mathbf{e}_y ) \frac{1}{2\sqrt{\tau}} \rm d\tau\\
\theta(\tau) = \theta(0) + \sqrt{\tau} \sqrt{2 D_{\theta\theta}} \; \epsilon_\theta, \;\;\;\;\;\;\;\;\;\; \epsilon_\xi, \epsilon_\theta ~ \mathcal{N}(0,1),
\end{array}
\right.
\end{equation}
where $\epsilon_\xi$ and $\epsilon_\theta$ are sampled from a normal distribution with expectation $0$ and unit standard deviation. The stochastic process in (\ref{eq:stochasticProcess}) can be interpreted as the motion of a drunkman's pencil: it randomly moves forward and backwards, and randomly changes its orientation along the way. The resolvent hypo-elliptic diffusion kernels $R_{\alpha,s}(g)$ (solutions to the fundamental single patch problem, up to scalar multiplication) can then also be obtained via Monte Carlo simulations, where the stochastic process is sampled many times with a negatively exponentially distributed traveling time ($P(\mathcal{T}=\tau)=\alpha e^{-\alpha \tau}$) in order to be able to estimate the probability density kernel $R_{\alpha,s}(g)$. This process is illustrated in Fig.~\ref{fig:contourEnhancement}.
\begin{figure}
\centerline{
\includegraphics[width=\hsize]{Figs/ContourEnhancement.png}
}
\caption{
Stochastic random process for contour enhancement.
\label{fig:contourEnhancement}}
\end{figure}
|
2,877,628,090,096 | arxiv | \section{Introduction}
In this paper, we study the inverse spectral uniqueness on the following scattering problem defined by the perturbation inside $\Omega:=\{x\in\mathbb{R}^{3}|\,|x|<R,\,R>0\}$.
\begin{eqnarray}\label{1.1}
\left\{%
\begin{array}{ll}
\Delta u(x)+k^2n(x)u(x)=0,\,x\in\mathbb{R}^3,\,\Im k\leq0;\vspace{5pt}\\\vspace{3pt}
u(x)=u^i(x)+u^s(x),\,x\in\mathbb{R}^3\setminus \Omega; \\
\lim_{|x|\rightarrow\infty}|x|\{\frac{\partial u^s(x)}{\partial |x|}-iku^s(x)\}=0.
\end{array}%
\right.
\end{eqnarray}
Here, $u^{i}(x)=e^{-ikx\cdot\nu}$, and $u^s(x)=u^s(x,k,\nu)$ is the scattered wave field that depends on the frequency variable $k$ and the impinging direction $\nu\in\mathbb{S}^{2}$. In this paper, we follow the wave propagation theory established in M. V. Klibanov and V. G. Romanov \cite{Klibanov2}. In particular, we assume that
\begin{eqnarray}
&&n(x)=1+\beta(x)\in C^{15}(\mathbb{R}^{3};\mathbb{R});\\
&&\beta(x)\geq0,\,\beta(x)=0,\,\mbox{ for }x\in\mathbb{R}^{3}\setminus \Omega.
\end{eqnarray}
It is shown in \cite{Klibanov2} that the problem~(\ref{1.1}) has a solution $u(x)$ in H\"{o}lder space $C^{16+\alpha}(\mathbb{R}^{3})$. The phaseless inverse scattering problem (PISP) is to find the index of refraction $n(x)$ if the information is given or partially given in the following scattering data.
\begin{equation}\label{1.4}
f(x,\nu,k):=|u^{s}(x,k,\nu)|^{2},\,(x,\nu,k)\in\mathbb{R}^{3}\times\mathbb{S}^{2}\times\mathbb{C}.
\end{equation}
The problem arises from the inverse scattering theory in many settings, e.g., electron microscopy, crystallography, medical imaging, and nano-optics. The historic review and the details can be found in \cite{Klibanov,Klibanov2,Klibanov3}. We apply the zero distribution theory in complex variable theory \cite{Boas,Cartwright2,Koosis,Levin,Levin2} to study the uniqueness theorem on the index of refraction $n(x)$. The zero distribution of function $ f(x,\nu,k)$ is one kind of phaseless information on the perturbation.
For every $\nu\in\mathbb{S}^{2}$, we denote $$S^{\pm}(\nu):=\{x\in\partial\Omega|\,x\cdot\nu\gtrless0 \}.$$
We state the following inverse uniqueness result.
\begin{theorem}\label{11}
Let $f^{j}(x,\nu,k)$ be the square modulus of the complex-valued scattered wave field generated by the index of refraction $n^{j}$, $j=1,2$. If $f^1(x,\nu,k)\equiv f^2(x,\nu,k)$ with the fixed pair $\pm\nu$ in $\mathbb{S}^{2}$ for all $x\in\mathbb{S}^{\pm}(\nu)$ in a neighborhood $U$ in $\Im k\leq0$, then $n^{1}\equiv n^{2}$.
\end{theorem}
The zero distribution theory in integral function theory \cite{Boas,Cartwright2,Koosis,Levin,Levin2} specifies the maximal zero-crossing density for the scattering data function $ f(x,\nu,k)$. Whenever the zero crossing density exceeds its theoretical maximal quantity, the function must be identically zero. This renders an inverse uniqueness on the scatterer, as we have discussed in \cite{Chen,Chen6,Chen7}. The assumption on the neighborhood $U$ in Theorem \ref{11} can be replaced by the assumptions on the numerical quantity of the zero-crossing density of $f(x,\nu,k)$.
\section{Lemmas}
There are two components in this paper: the Cartwright-Levinson type of theorems in entire function theory \cite{Boas,Cartwright2,Koosis,Levin,Levin2} and the phase-linearization constructed in \cite{Klibanov2}. In particular, we have the following asymptotic expansion under the assumptions mentioned in Introduction.
\begin{eqnarray}\label{2.1}
u(x,k,\nu)=A(x,\nu)\exp\{-ik\varphi(x,\nu)\}+\int_{\varphi(x,\nu)}^{\infty}\hat{v}(x,t,\nu)e^{-ikt}dt,\,\Im k=0.
\end{eqnarray}
Accordingly, the following high-energy expansion holds.
\begin{eqnarray}\label{2.2}
&&u^{s}(x,k,\nu)=A(x,\nu)\exp\{ik\varphi(x,\nu)\}-\exp\{ikx\cdot\nu\}+O(\frac{1}{k}),\,\Im k=0,\,\forall\nu\in
\mathbb{S}^{2},
\end{eqnarray}
which holds for $x$ in an arbitrary bounded domain, and
\begin{equation}\nonumber
A(x,\nu)=\exp\{-\frac{1}{2}\int_{{\Gamma(x,\nu)}}n^{-2}(\xi)\Delta_{\xi}\varphi(\xi,\nu)d\tau\}>0,
\end{equation}
where $\varphi(x,\nu)$ is the solution of the following problem
\begin{eqnarray}
&&|\nabla_{x}\varphi(x,\nu)^{2}|=n^{2}(x);\label{2.33}\\
&&\varphi(x,\nu)=x\cdot\nu,\,\forall x\cdot\nu\leq -B.\label{2.44}
\end{eqnarray}
Here we define
\begin{equation}
A(k):=A(x,\nu)\exp\{ik\varphi(x,\nu)\}-\exp\{ikx\cdot\nu\},\label{22.2}
\end{equation}
which would contribute to the zero-crossings of $f(x,\nu,k)$.
For far-field behavior, we have
\begin{equation}\label{233}
u^s(x,k,\nu)=\frac{e^{ik|x|}}{|x|}u_\infty(\hat{x};\nu,k)+O(\frac{1}{|x|^{\frac{3}{2}}}),
\end{equation}
which holds uniformly for all $\hat{x}:=\frac{x}{|x|}$, $x\in\mathbb{R}^3$, and $u_\infty(\hat{x};\nu,k)$ is known as the scattering amplitude in the literature \cite{Colton2,Isakov,Lax,Melrose}. In this paper, we adopt the convention that $u_\infty(\hat{x};\nu,k)$ is defined analytically in $\Im k\leq 0$, and extends meromorphically from $\Im k\leq0$ to $\mathbb{C}$.
\begin{lemma}\label{S}
The scattered wave field $u^{s}(x,k,\nu)$ in~(\ref{233}) is defined meromorphically in $\mathbb{C}$ with poles in $\Im k >0$ except a finite number of purely imaginary $k$'s that $k^{2}$ are the negative eigenvalues of~(\ref{1.1}). In particular, the poles of $u_\infty(\hat{x};\nu,k)$ are located as the mirror images of its zeros to the real axis.
\end{lemma}
\begin{proof}
This is well-known in scattering theory. Let us refer to \cite{Lax,Melrose}.
\end{proof}
The error term in~(\ref{2.2}) comes from the decaying rate of $\int_{\varphi(x,\nu)}^{\infty}\hat{\nu}(x,t,\nu)e^{-ikt}dt$. Taking advantage of the decaying rate, we prove the following lemma.
\begin{lemma}\label{22}
The expansion~(\ref{2.2}) holds in $-\infty<\Im k<C$ for some $C>0$ in
$\mathbb{C}$.
\end{lemma}
\begin{proof}
We recall Theorem 1 provided in \cite{Klibanov2}, which says the solution $v$ of the system
\begin{eqnarray*}
\left\{%
\begin{array}{ll}
n^{2}(x)v_{tt}-\Delta v=0,\,(x,t,\nu)\in\mathbb{R}^{3}\times\mathbb{R}\times\mathbb{S}^{2};\vspace{4pt}\\\vspace{4pt}
v(x,t,\nu)=\delta(t-x\cdot \nu)+\overline{v}(x,t,\nu); \\
\overline{v}(x,t,\nu)\equiv0,\,t<-B,
\end{array}%
\right.
\end{eqnarray*}
can be represented in the form
\begin{equation}\label{2.3}
v(x,t,\nu)=A(x,y)\delta(t-\varphi(x,\nu))+\hat{v}(x,t,\nu)H(t-\varphi(x,\nu)),\,(x,t)\in\mathbb{R}^{3}\times(-\infty,T),
\end{equation}
where the formula holds for $x$ in arbitrary bounded $T$, $H(t-\varphi(x,\nu))$ is the Heaviside function, and $A(x,y)$ is as defined in \cite[(4.6)]{Klibanov2}. The propagating wave $v(x,t,\nu)$ decays exponentially for large $t$ for bounded $x$. That is,
\begin{equation}\label{2.4}
|v(x,t,\nu)|\leq Ce^{-Ct},\,\mbox{ as }t\rightarrow\infty,
\end{equation}
for some constant $C>0$.
Here, we refer~(\ref{2.4}) to \cite[(4.23)]{Klibanov2}. According to~(\ref{2.3}) and~(\ref{2.4}), the same decaying rate~(\ref{2.4}) holds for the twice differentiable exponentially-decaying $\hat{v}(x,t,\nu)$. Hence,
\begin{equation}\label{2.6}
\int_{\varphi(x,\nu)}^{\infty}\hat{v}(x,t,\nu)e^{-ikt}dt=\int_{\varphi(x,\nu)}^{\infty}e^{-i\Re k t}[\hat{v}(x,t,\nu)e^{\Im k t}]dt,
\end{equation}
in which the function $\hat{v}(x,t,\nu)e^{\Im k t}$ is integrable over $t\in[\varphi(x,\nu),\infty]$ if $\Im k<C$. Thus, the Fourier transform $$I(\Re k):=\int_{\varphi(x,\nu)}^{\infty}e^{-i\Re k t}[\hat{v}(x,t,\nu)e^{\Im k t}]dt$$ makes sense for some $\Im k\leq C$, $C>0$, and $\Re k\in(-\infty,\infty)$. Applying Riemann-Lebesgue Lemma, we observe that the integral $I(\Re k)$ vanishes for large $\Re k$. Integrating by parts to~(\ref{2.6}), we deduce that
\begin{eqnarray*}
&&u(x,k,\nu)=A(x,y)e^{-ik\varphi(x,\nu)}+\frac{e^{-ik\varphi(x,\nu)}}{ik}\hat{v}_{+}(x,\varphi(x,\nu),\nu))+\frac{1}{ik}\int_{\varphi(x,\nu)}^{\infty}e^{-i\Re k t}[\hat{v}_{t}(x,t,\nu)e^{yt}]dt,
\end{eqnarray*}
in which the behavior of $\hat{v}_{t}(x,t,\nu)$ and its derivatives are found in \cite[p.\,362]{Vainberg} and the Fourier transform decays like $I(\Re k)$ as discussed above.
\end{proof}
The zeros of $u^{s}(x,k,\nu)$ is a class of phaseless information. This is called zero-crossing method in \cite{Hunt}, and therein we have a short introduction to Cartwright theory as well. Let us study the qualitative theory of the zero set in this paper.
\begin{definition}\label{23}
Let $F(z)$ be an integral function of order $\rho$, and let
$N(F,\alpha,\beta,r)$ denote the number of the zeros of $F(z)$
inside the angle $[\alpha,\beta]$ and $|z|\leq r$. We define the zero
density function as
\begin{equation}\nonumber
\Delta_F(\alpha,\beta):=\lim_{r\rightarrow\infty}\frac{N(F,\alpha,\beta,r)}{r^{\rho}},
\end{equation}
and
\begin{equation}\nonumber
\Delta_F(\beta):=\Delta_F(\alpha_0,\beta),
\end{equation}
with some fixed $\alpha_0\notin E$ such that $E$ is at most a
countable set \cite{Boas,Cartwright2,Koosis,Levin,Levin2}. We can define a similar notation for a set of zeros.
\end{definition}
\begin{lemma}
Let $S(\alpha,s,h)$ be the strip containing the positive real axis starting $\Re k\geq\alpha>0$, $|\Im k|\leq h$, and $\alpha\leq\Re k\leq\alpha+s$.
The zeros of the analytic function $A(k)$ in~(\ref{22.2}) are located in a suitable $S(\alpha,s,h)$ with density $\frac{\varphi(x,\nu)-x\cdot\nu}{2\pi}$.
\end{lemma}
\begin{proof}
To estimate the zero asymptotics of $A(k)$, we consider the following zero counting theorem
\cite{Dickson,Dickson2,Levin}.
\begin{theorem}[Dickson \cite{Dickson,Dickson2}]\label{25} Let
\begin{equation}\nonumber
R(\alpha,s,h):=\{z=x+iy\in\mathbb{C}|\,|x|\leq
h,\,y\in[\alpha,\alpha+s]\};
\end{equation}
\begin{equation}\nonumber
N_g(R(\alpha,s,h)):=\{\mbox{the number of zeros of }g(z)\mbox{ in
}R(\alpha,s,h)\},
\end{equation}
in which $$g(z)=\sum_{j=1}^nA_j e^{\omega_jz},$$ where $z=x+iy$,
$A_j\neq0$, $\omega_1<\omega_2<\cdots<\omega_n$. Then, there
exists $K>0$ such that
\begin{enumerate}
\item each zero of $g$ is in $|x|<K$;
\item for each pair of reals $(\alpha,s)$ with $s>0$,
\begin{equation}
|N_g(R(\alpha,s,K))-s(\omega_n-\omega_1)/(2\pi)|\leq n-1.
\end{equation}
\end{enumerate}
\end{theorem}
\par
In our case, the application is straightforward. Let us set $n=2$ with the invariant $\omega_{2}:=\varphi(x,\nu)$ and $\omega_{1}:=x\cdot\nu>0$. We have only one interval to consider, and note here that $\varphi(x,\nu)-x\cdot\nu>0$ if the index of refraction $1+\beta(x)\not\equiv1$. We refer the details to \cite[Sec. \,3]{Klibanov2}. For our application, we consider $z\mapsto iz$, and then deduce that
\begin{equation}\nonumber
|N_{A(k)}S(\alpha,s,h))-s|\varphi(x,\nu)-x\cdot\nu|/(2\pi)|\leq1.
\end{equation}
\end{proof}
\begin{lemma}\label{266}
Let $T(\alpha,s,h)$ be the strip containing the positive real axis starting $\Re k\geq\alpha>0$, $-h\leq\Im k\leq C$, and $\alpha\leq\Re k\leq\alpha+s$ with suitable $h>0$ and the constant $C$ from Lemma \ref{22}. We denote the zeros of $u^{s}(x,\nu,k)$ in $T(\alpha,s,h)$ by $N(\alpha,s,h)$. Then,
$$|N_{u^{s}(x,k,\nu)}(T(\alpha,s,h))-s(\varphi(x,\nu)-x\cdot\nu)/(2\pi)|\leq1.$$
\end{lemma}
\begin{proof}
From the Lemma \ref{22} and~(\ref{2.2}), we apply the following inequality
\begin{equation}\label{299}
|u^{s}(x,k,\nu)-A(k)|\leq O(\frac{1}{k}),\,\Im k\leq C,
\end{equation}
where the big O-term is defined in~(\ref{2.1}). Moreover,~(\ref{299}) implies the zeros of $A(k)$ are near the ones of $u^{s}(x,k,\nu)$ for large $k$ by the continuity. We have shown that zeros of $A(k)$ is distributed in a suitable $S(\alpha,h,s)$
with suitable width $h>0$ and $\alpha>0$, so on the boundary of $T(\alpha,h,s)$, where $A(k)$ is away from zero, we deduce from~(\ref{299}) and Lemma \ref{S} that
\begin{equation}
|u^{s}(x,k,\nu)-A(k)|<|A(k)|.
\end{equation}
Thus the lemma is proved by Rouch\'{e}'s theorem in complex analysis.
\end{proof}
\begin{lemma}\label{26}
The following asymptotic identities hold.
\begin{eqnarray*}
&&\Delta_{A(k)}(-\pi,0)=\Delta_{u^{s}(x,k,\nu)}(-\pi,0)=\frac{[\varphi(x,\nu)-x\cdot\nu]}{\pi};\\
&&\Delta_{f(x,\nu,k)}(-\pi,0)=\frac{2[\varphi(x,\nu)-x\cdot\nu]}{\pi}.
\end{eqnarray*}
\end{lemma}
\begin{proof}
The first identity is deduced from Definition \ref{23} and Lemma \ref{266}. Because $$ f(x,\nu,k)=u^{s}(x,\nu,k)\overline{u^{s}(x,\nu,k)},$$ and $u^{s}(x,\nu,k)$ and $\overline{u^{s}(x,\nu,k)}$ share the same zero set, $f(x,\nu,k)$ has twice the density as $u^{s}(x,\nu,k)$ or $A(k)$. The lemma is thus proven.
\end{proof}
Here we interpret the linearized term $\varphi(x,\nu)-x\cdot\nu$ as a Weyl's type of spectral invariant to the problem~(\ref{1.1}) considering the following identity.
\begin{equation}
\varphi(x,\nu)-x\cdot\nu=\int_{L(x,\nu)}\beta(\xi)d\xi,\,\forall\nu\in\mathbb{S}^{2},\,\forall x\in\mathbb{S}^{+}(\nu),
\end{equation}
where $L(x,\nu)$ is the line segment connecting $x$ and $x-2(x\cdot\nu)\nu$. Surely, we refer the construction to \cite[(6.5)]{Klibanov2}. Moreover, let us examine the problem~(\ref{2.33}) and~(\ref{2.44}). Whenever the observation position $x$ and the incident angle $\nu$ are given, the asymptotic properties in Lemma \ref{26} construct a connection between the index of refraction $n$ and $\varphi(x,\nu)$ as shown in~(\ref{2.33}) and~(\ref{2.44}).
\section{A Proof of Theorem \ref{1.1}}
Starting with the assumption in Theorem \ref{11} that, for $U\subset\{k\in\mathbb{C}|\,\Im k\leq0\}$, we have
$$f^1(x,\pm\nu,k)=f^2(x,\pm\nu,k),\,x\in S^{\pm}(\nu).$$
Using the analytic continuation property of real-valued functions $f^{j}=\{\Re [u^{j}]^{s}\}^{2}+\{\Im [u^{j}]^{s}\}^{2}$, in which $[u^{j}]^{s}$ is denoted as the scattered wave field defined by the index of refraction $n^{j}$, $j=1,2$, then we can deduce that
$$f^1(x,\pm\nu,k)\equiv f^2(x,\pm\nu,k),\,x\in S^{\pm}(\nu),\,\Im k\leq0.$$
The boundary $\partial \Omega$ is starlike, so
$\overline{S^{+}(\nu)\cup S^{-}(\nu)}=\partial\Omega$ for the given $\nu$.
Taking a square root and considering the continuity of the solutions over $x$, so we deduce
\begin{equation}\label{3.1}
|[u^{1}]^{s}(x,k,\nu)|\equiv |[u^{2}]^{s}(x,k,\nu)|,\,x\in\partial\Omega,\,\Im k\leq0.
\end{equation}
Therefore, we deduce from~(\ref{3.1}) that
$[u^{1}]^{s}(x,k,\nu)$ and $[u^{2}]^{s}(x,k,\nu)$ share the same zero set, say, $\mathcal{Z}(x,\nu)$ in $\Im k\leq0$ for the fixed $\nu\in\mathbb{S}^{2}$. From Lemma \ref{26}, the zero density is specified as
\begin{equation}\label{3.2}
\Delta_{\mathcal{Z}(x,\nu)}=\frac{\varphi(x,\nu)-x\cdot\nu}{\pi}.
\end{equation}
According to~(\ref{3.1}) and the fact that $[u^{j}]^{s}$ and $|[u^{j}]^{s}|$ share the same zero set, so $[u^{1}]^{s}/[u^{2}]^{s}$ is analytic with {\bf no zero }in $\Im k\leq0$. Moreover, $$\ln \{[u^{1}]^{s}(x,k,\nu)/[u^{2}]^{s}(x,k,\nu)\}=\ln |[u^{1}]^{s}(x,k,\nu)/[u^{2}]^{s}(x,k,\nu)|+i\arg \{[u^{1}]^{s}(x,k,\nu)/[u^{2}]^{s}(x,k,\nu)\}.$$ From~(\ref{3.1}), we obtain that
$$\ln \{[u^{1}]^{s}(x,k,\nu)/[u^{2}]^{s}(x,k,\nu)\}=i\arg \{[u^{1}]^{s}(x,k,\nu)/[u^{2}]^{s}(x,k,\nu)\},$$
which is purely imaginary. It is known from complex analysis that $\ln \{[u^{1}]^{s}(x,k,\nu)/[u^{2}]^{s}(x,k,\nu)\}$ is a constant. That is, $[u^{1}]^{s}(x,k,\nu)/[u^{2}]^{s}(x,k,\nu)=e^{i\gamma}$ for some real constant $\gamma$.
Considering the asymptotics~(\ref{2.2}),
\begin{eqnarray*}
\frac{[u^{1}]^{s}(x,k,\nu)}{[u^{2}]^{s}(x,k,\nu)}=
\frac{A^{1}(x,\nu)\exp\{ik\varphi^{1}(x,\nu)\}-\exp\{ikx\cdot\nu\}+O(\frac{1}{k})}{A^{2}(x,\nu)\exp\{ik\varphi^{2}(x,\nu)\}-\exp\{ikx\cdot\nu\}+O(\frac{1}{k})},\,\Im k=0.
\end{eqnarray*}
From~(\ref{3.1}) and~(\ref{3.2}), we obtain $\varphi^{1}(x,\nu)=\varphi^{2}(x,\nu)$ for fixed $(x,\nu)$. Given that $A^{1}(x,y)$ and $A^{2}(x,y)$ are real, we deduce that $A^{1}(x,y)=A^{2}(x,y)$, $\gamma=0$, and
\begin{eqnarray}
&&[u^{1}]^s(x,k,\nu)=[u^{2}]^s(x,k,\nu),\,x\in \partial \Omega,\, \Im k\leq0;\\
&&u^{1}(x,k,\nu)=u^{2}(x,k,\nu),\,x\in \partial \Omega,\, \Im k\leq0,\label{3.4}
\end{eqnarray}
where $u^{j}$, $j=1,2$, satisfies the Helmholtz's equation outside $\Omega$, and then enjoys the property of analytic continuation.
\par
Let us set the observation data
\begin{eqnarray*}
&&w(x;k):=u^1(x,k,\nu);\\
&&v(x;k):=u^2(x,k,\nu).
\end{eqnarray*}
Most importantly, for each zero-crossing data from~(\ref{1.1}) and~(\ref{3.4})
we deduce the following interior transmission problem \cite{Chen,Chen6,Chen7,Colton,Colton2,Mc,S} that holds for $\Im k\leq0$, and
\begin{eqnarray*}
\left\{%
\begin{array}{ll}
\Delta w(x;k)+k^2n^1w(x;k)=0, & x\in\mathbb{R}^3 ; \vspace{3pt}\\\vspace{3pt}
\Delta v(x;k)+k^2n^2v(x;k)=0, & x\in\mathbb{R}^3; \\\vspace{3pt}
w(x;k)=v(x;k), & x\in\partial\Omega; \\\vspace{3pt}
\frac{\partial w(x;k)}{\partial n}=\frac{\partial v(x;k)}{\partial n},& x\in\partial\Omega,
\end{array}%
\right.
\end{eqnarray*}
where $n$ is the unit outer normal. By applying the analytic continuation property of the Helmholtz's equation \cite{Colton2,Isakov} outside $\Omega$, we deduce that, for $\Im k\leq0$, the following system holds.
\begin{eqnarray}\label{3.5}
\left\{%
\begin{array}{ll}
\Delta w(x;k)+k^2n^1w(x;k)=0, & x\in\mathbb{R}^3 ; \vspace{3pt}\\\vspace{3pt}
\Delta v(x;k)+k^2n^2v(x;k)=0, & x\in\mathbb{R}^3; \\\vspace{3pt}
w(x;k)=v(x;k), & x\in\mathbb{R}^3\setminus \Omega; \\\vspace{3pt}
\frac{\partial w(x;k)}{\partial n}=\frac{\partial v(x;k)}{\partial n},& x\in\mathbb{R}^3\setminus \Omega.
\end{array}%
\right.
\end{eqnarray}
Firstly, it is well-known that the spectrum of~(\ref{3.5}) is a discrete set in $\mathbb{C}$ \cite{Chen,Chen6,Colton,Colton2,Mc,S} if $n^{1}\not\equiv n^{2}$. Secondly, we have provided a spectral analysis for each incident angle $\nu$ in the previous section. Any quantitative assumption that exceeds the maximal zero crossing density leads to a proof on the uniqueness on the index of refraction. Thirdly, we may apply the inverse uniqueness of~(\ref{3.5}) \cite{Chen,Chen6,Chen7} in which we merely assume that $n^{j}\in C^{2}(\mathbb{R}^{3})$, $j=1,2$, so in any case we conclude that $n^{1}\equiv n^{2}$.
\begin{acknowledgement}
The author would like to thank Prof. M.V. Klibanov for providing the manuscript \cite{Klibanov2}.
\end{acknowledgement}
|
2,877,628,090,097 | arxiv | \section{Introduction}
\label{sec:introduction}
Despite spectacular progress in recent years, the reionization history of the universe remains shrouded in mystery. The high electron scattering optical depth observed by WMAP indicates that reionization
may have begun as early as $z\sim 20$ (Kogut et al 2003; Spergel et al 2003). On the other hand, spectra of high redshift quasars indicate a sharp change in the neutral hydrogen fraction and ionizing
background at $z\sim6$, suggesting this era marks the end of the reionization epoch (Becker et al 2001, Fan et al 2002). Taken in tandem, these observations suggest a highly complex reionization history.
A crucial missing piece of the puzzle is the precise ionization state of the IGM at $z=6$. The large optical depth of the IGM to hydrogen Lyman transitions implies that Gunn \& Peterson (1965) absorption can (at best) constrain the volume and mass-weighted neutral fractions to be $x_{\rm HI,V} \ge 10^{-3},\, x_{\rm HI,M}\ge 10^{-2}$ respectively (Fan et al 2002). It is plausible that the IGM is still highly ionized at $z=6$,
and with full reionization occurring much earlier. On the other hand, modeling of the spectral regions around two quasars at $z \ga 6.2$ may indicate larger neutral fractions ($x_{\rm HI} \ga 0.2$; Wyithe \& Loeb 2004a; Mesinger \& Haiman 2004). At stake is the very nature of reionization: if these claims are correct, then reionization must be an extremely rapid process, since the IGM is known to be highly ionized ($x_{\rm HI} \le 10^{-5}$) by $z=5.9$ along all observed lines of sight (Fan et al 2003).
It is thus worth considering the IGM absorption in more detail. In particular, a completely dark Gunn-Peterson trough at $z\sim6$ is not necessarily universal. Both the Ly$\alpha$ and Ly$\beta$ troughs of the highest redshift quasar found to date, SDSS J1148$+$5251, contain detectable flux (White et al 2003). In this {\it Letter}, we show that its previous interpretation as continuum contamination from an interloper galaxy at $z=4.94$ is inconsistent with the observed flux ratios blueward and redward of the expected continuum break due to Ly$\alpha$ forest absorption. Therefore, some of the flux is true transmission due to holes in the high redshift Ly$\alpha$ forest. We derive our strongest contraints from the Ly$\gamma$ trough, which was ignored in previous analyses and should be relatively uncontaminated by flux from an interloper. The presence of flux transmission implies that either the IGM is still highly ionized at $z=6$ or that there is significant cosmic variance in the reionization epoch along different lines of sight.
\section{The Lyman Gamma Trough in SDSS J1148+5251}
\label{section:lyg_trough}
Detectable flux and a network of transmission features is seen in both the Ly$\alpha$ and Ly$\beta$ troughs of SDSS J1148$+$5251. White et al (2003) interpreted this to be continuum contamination from interloper galaxies at $z=4.94$ for two reasons: (i) strong \ion{C}{4} absorption features appear at $z=4.9$, so the apparent strong Ly$\beta$ spikes at $z=6.03, 6.06$ could just be Ly$\alpha$ emission from $z \approx 4.94$. (ii) The flux ratios of the troughs appear inconsistent: there is too little light in the Ly$\beta$ trough given the Ly$\alpha$ transmission. Although the {\it a priori} probability of such an interloper is small, it increases if the interloper also lenses SDSS J1148$+$5251, boosting its probability of detection.
By examining the Ly$\gamma$ trough of SDSS J1148$+$5251, we argue that such an interpretation is untenable. We first note that if residual flux in {\it any} of the Ly$\alpha,\beta,\gamma$ troughs can be attributed to the quasar rather than an interloper galaxy, the IGM must still be highly ionized along the line of sight. Consider the boundary of the quasar proximity zone at $z_{\rm HII}\approx 6.33$, where the Ly$\alpha$, Ly$\beta$ absorption increase rapidly. {\it There is also a sharp jump in absorption at the boundary of the Ly$\gamma$ trough, strongly suggesting that the transmission features redward of the Ly$\gamma$ trough are genuine}. Even if we exclude the sharp transmission spike at $\lambda=7205$ \AA \ (putatively a Ly$\alpha$ emission line from a galaxy at $z=4.94$), the residual flux changes from $F_{-20}(z_{\gamma}=6.33-6.40)=29.0\pm1.5$
(within the quasar proximity zone) to $F_{\-20}(z_{\gamma}=6.25-6.32)=4.3\pm1.4$ (outside the quasar proximity zone), where $F_{-20}$ is the flux $F$ in units of $10^{-20} {\rm erg \, s^{-1} \, cm^{-2}}$ \AA$^{-1}$. It would be remarkable if the protocluster at $z\approx5$ lined up so as to exactly coincide with the quasar proximity zone in Ly$\gamma$: it is much more likely that the sudden change is due to genuine Ly$\gamma$ absorption. This suggests the flux seen in the Ly$\beta$ forest at $z>5.95$, before the onset of Ly$\gamma$ absorption, represents true transmission.
It is also crucial that there is detectable flux in the Ly$\gamma$ trough. Consider the wavelength stretch corresponding to $z=6.17-6.32$ in the various absorption troughs. This lies outside the quasar's apparent region of influence and ends where the Ly$\gamma$ trough becomes contaminated by Ly$\delta$ and higher order absorption ($\lambda=6962$ \AA, corresponding to $z_{\gamma}=6.16$). White et al (2003) suggest that the Ly$\alpha$, Ly$\beta$ troughs are contaminated by continuum emission from the interloper galaxy so that detected flux is not significant. To be conservative, let us ignore the transmitted flux in the Ly$\beta$ trough, which is potentially also contaminated by emission line features from the $z\approx5$ proto-cluster. We find that $F_{-20}$(Ly$\alpha$)$=3.0 \pm0.9$ while $F_{-20}$(Ly$\gamma$)$=4.9 \pm0.9$; these regions correspond to $\lambda=1467-1498$ \AA \ and $\lambda=1174-1199$ \AA \ in the rest frame of an interloper galaxy at $z=4.94$. {\it These flux ratios are strongly inconsistent with the expected continuum of a $z\approx5$ galaxy, which should have a strong break due to the intervening Ly$\alpha$ forest}. Songaila \& Cowie (2002) obtain a flux suppression factor $T_{\alpha}=0.14 \pm 0.03$ for the Ly$\alpha$ forest in this redshift and wavelength interval (Becker et al 2003 find $T_{\alpha}=0.11$). Even if the flux in the Ly$\alpha$ trough is entirely due to continuum contamination from the interloper, the implied continuum contribution to the Ly$\gamma$ trough would be $F_{-20}=0.4 (T_{\alpha}/0.14) \ll F_{-20}$(Ly$\gamma$), which lies well within the noise. Thus, almost all the observed flux in the Ly$\gamma$ trough must be genuine transmitted flux from the quasar. Because it is almost completely free from continuum contamination, the Ly$\gamma$ trough places the strongest constraint on the IGM ionization state along this line of sight.
\begin{deluxetable*}{ccrrrrrlll}
\tabletypesize{\scriptsize}
\tablecolumns{7}
\tablewidth{0pc}
\tablecaption{Absorption Optical Depths from $z=6.17-6.32$ in SDSS J1148$+$5251 }
\tablehead{
\colhead{Trough} & \colhead{Flux\tablenotemark{a} }&
\colhead{Continuum\tablenotemark{a,b}} & \colhead{Total Optical Depth} &
\colhead{Line Optical Depth\tablenotemark{c}} & \colhead{$\tau_{\alpha}(\beta=2)$\tablenotemark{d}}
& \colhead{$\tau_{\alpha}(\beta=3)$\tablenotemark{d}}
}
\startdata
Ly$\alpha$ & $3.0 \pm 0.9$ & (2560,1730,1690) & $(6.8,6.4,6.4)\pm 0.3$ & $(6.8,6.4,6.4)\pm 0.3$ & $(6.8,6.4,6.4)\pm 0.3$ & $(6.8,6.4,6.4)\pm 0.3$\\
Ly$\beta$ & $9.0 \pm 1.1$ & (2140,1870,1710) & $(5.5,5.3,5.2)\pm 0.1$ & $(3.1,3.0,2.9)\pm 0.3$ & $(8.2,7.8,7.6)\pm 0.9$ & $(6.7,6.4,6.2)\pm 0.7$\\
Ly$\gamma$ & $4.9 \pm 0.9$ & (2070,1910,1710) & $(6.1,6.0,5.9)\pm 0.3$ & $(2.5,2.4,2.3)\pm 0.3$ & $(11.5,11.2,10.7)\pm 1.4$ & $(8.4,8.2,7.8)\pm 1.0$
\tablenotetext{a}{ In units of $10^{-20} {\rm erg \, s^{-1} \, cm^{-2} \AA^{-1}}$.}
\tablenotetext{b}{Derived from the LBQS composite spectrum, and the Telfer et al 2002 spectral indices for radio-quiet ($\alpha_{\rm EUV}=-1.57$) and radio-loud ($\alpha_{\rm EUV}=-1.96$) quasars respectively. Subsequent triplets for the calculated optical depths reflect this ordering.}
\tablenotetext{c}{After subtracting off the contribution from foreground absorption. For Ly$\beta$ trough: $\tau_{\rm fg}=\tau_{\alpha}(z=5.1)=2.38\pm0.32$; for Ly$\gamma$ trough: $\tau_{\rm fg}=\tau_{\alpha}(z=4.8)+\tau_{\beta}(z=5.9)=(1.95\pm0.25)+(1.59 \pm 0.02)=3.54\pm0.25$.}
\tablenotetext{d}{The effective equivalent Ly$\alpha$ optical depth inferred from the nonlinear conversion described in the text, assuming $\tau(\Delta) \propto \Delta^{\beta}$.}
\enddata
\end{deluxetable*}
Because there are indisputably \ion{C}{4} absorbers at $z=4.9$, the putative protocluster may conceivably have another effect: it could ionize the IGM at $z=5$, creating a transparent Ly$\alpha$ window there.
However, the proximity zone of the interloper galaxy itself is too small. We need to highly ionize the IGM on comoving lengthscales $L \approx 55 (\Delta z/0.1)$ Mpc at $z=5$. If the flux in the Ly$\alpha$ trough is interpreted as the continuum of the interloper, then
$F_{-20}(\lambda=1450 \AA)=3.9$. If we take $F(\lambda=1450 \AA)/F(\lambda=900 \AA)=4.6$ as is observed in $z=3.4$ LBGs (Steidel et al 2001) for the strongest ionizing continuum possible (corresponding to a $\sim 100\%$ escape fraction for ionizing photons), then we infer $L_{\nu}(1 {\rm Ryd})=1.4 \times 10^{28} {\rm erg \, s^{-1} \, cm^{-2} \, Hz^{-1}}$, roughly implying a star formation rate ${\rm SFR} \sim 14 {\rm M_{\odot} yr^{-1}} $ for a Salpeter IMF. Assuming the mean background ionizing rate in units of $10^{-12} {\rm s}^{-1}$ to be $\Gamma_{-12} (z=5)\approx 0.15$ (e.g., Fig 2 of Fan et al 2002), the local radiation field becomes comparable to the metagalactic ionizing at $r\approx 1.3$ Mpc comoving, far too small to be of interest. Lyman-break galaxies (LBGs) at $z\approx 3-4$ also show excess Ly$\alpha$ transmission on lengthscales $L\approx 0.5$ Mpc comoving (Adelberger et al 2003), roughly corresponding to the distance a $600 \, {\rm km \, s^{-1}}$ wind would travel on the $\sim 300$ Myr star formation timescale of LBGs. It is unlikely a $z=5$ galaxy would affect a significantly larger volume.
Galaxies near (but not along) the line of sight could contribute additional ionizing photons. Note, however, that we require a factor $\sim 5$ increase in $T_{\alpha}$ over a lengthscale $\sim 10$ times larger than the comoving correlation length for highly biased galaxies. There is no evidence for such strong fluctuations in transmission at this redshift: typically $\sigma_{\rm T\alpha}/T_{\alpha}\sim 0.3$ (Songaila \& Cowie 2002). In any case, an anomalously large protocluster should be easily visible in narrowband optical searches.
\section{Implications for the IGM at $z=6$}
\label{section:tau}
Having established the reality of residual flux from SDSS J1148$+$5251, we will now consider its implications for reionization at $z \sim 6$. We must first estimate the effective optical depths $\tau_{\rm eff}$ in each transition. We begin by summarizing our methodology and the error budget (which is frequently underestimated).
The errors for Ly$\gamma$ transmission $T_{\gamma}=T_{\rm tot}/(T_{\alpha}T_{\beta})$ must include not only pixel noise but also the cosmic variance in foreground transmission in Ly$\alpha$ and
Ly$\beta$. Previous analyses have often neglected this additional scatter. It has been measured in quasar samples; we use the values in Table 2 of Songaila \& Cowie (2002), who tabulate the measured
$\langle T_\alpha \rangle, \sigma_{\rm T\alpha}$ for 6 redshift bins between $z=4.1-5.5$. However, these use fixed wavelength intervals, and $\sigma_{\rm T\alpha}$ obviously depends on the corresponding comoving length $L$. We cannot simply assume Poisson statistics, because long wavelength modes could dominate the variance. To estimate $\sigma_{\rm T\alpha}$ we follow the {\it ansatz} of Lidz et al (2002), who argue that the transmission power spectrum takes the shape (though not the normalization) of the linear mass power spectrum on large scales, yielding:
\begin{equation}
\sigma_{\rm T\alpha}^{2}=2 \int_{0}^{\infty} \frac{dk}{2 \pi} \left[ \frac{{\rm sin}(kL/2)}{kL/2}\right]^{2} P_{f}(k) + \frac{\sigma_{n}^{2}}{N},
\label{eqn:sigmaF}
\end{equation}
where $\sigma_{n}$ is the noise per pixel, $N$ is the number of pixels, and $P_{f}(k)=B{\rm exp}(-ak^{2})\int_{k}^{\infty}(dk/2\pi)k P_{\rm mass}(k)$, where $a\approx k_{\rm J}^{-1/2}$, $k_{J}$ is the Jeans wavenumber, and $B$ is normalized to the observed $\sigma_{\rm T\alpha}$ for some observed
stretch $L_{\rm obs}$. The first term in equation (\ref{eqn:sigmaF}) typically dominates by an order of magnitude. For the large lengthscales of interest, $P_{f}(k)$ is fairly flat, and we have approximately $\sigma_{\rm T\alpha}^{2} \propto L^{-1}$, as expected for a white noise power spectrum. Note that this {\it ansatz} for the transmission power spectrum $P_{f}(k)$ assumes a homogeneous ionizing background $\Gamma$; if $\Gamma$ exhibits small-scale fluctuations, $\sigma_{\rm T\alpha}$ could be larger. However, there is {\it no} cosmic variance in $T_\beta$, because we have observed the
corresponding Ly$\alpha$ transmission (at $z_\alpha=5.9$) along the same line of sight. The main uncertainty comes from the optical depth conversion between different lines (see below).
Another crucial item is uncertainty in the quasar continuum. This is particularly important in comparing relative absorption between the different troughs. The Ly$\alpha$ line is much broader than the Ly$\beta$ and Ly$\gamma$ lines and generally spills over into the observed trough. Uncertainty in the strength of the line creates significant uncertainty in the inferred $\tau_{\alpha}$. A simple way to quantify this is to compare the optical depths inferred assuming the composite spectrum from the Large Bright Quasar Survey (LBQS; Brotherton et al 2001) and the far-UV quasar power law spectrum of Telfer et al (2002). The former includes all the emission line structure, while the latter assumes a pure power law; the difference between the two therefore captures the uncertainty in emission line contribution. We normalize the continuum to the observed flux at $\sim 1290 {\rm \AA}$.
Table 1 lists the measured effective line optical depths $\tau_i \equiv \tau_{\rm eff}$(Ly$\alpha$) and their associated $1\sigma$ errors. The results for the Ly$\alpha$ and Ly$\beta$ line optical depths are comparable to those of White et al. (2003). They argued that the pair are incompatible with each other, because $\tau_{i} \propto \lambda_{i} f_{i}$ (where $f_{i}$ is the oscillator strength of line $i$), implying $\tau_{\alpha}/\tau_{\beta}=6.24$ and $\tau_{\alpha}/\tau_{\gamma}=17.93$. However, these relations are only true at fixed density, and only apply if the IGM is homogeneous. For the large comoving lengths which we are considering, the flux transmission comes from a variety of densities:
\begin{equation}
\langle T_{i} \rangle = \langle {\rm exp}\left( -\tau_{{\rm eff},i} \right) \rangle = \int {\rm exp}\left[ -\tau_{i}(\Delta) \right] P(\Delta) d\Delta,
\label{eqn:flux_transmission}
\end{equation}
where $P(\Delta)$ is the probability distribution of overdensities $\Delta \equiv \rho/\bar{\rho}$, $\tau_{i} \propto \lambda_{i} f_{i} (1+z)^{4.5} \Delta^{2} \alpha(T)/\Gamma$ (e.g., Hui \& Gnedin 1997) in the optically thin limit, and $\alpha(T)$ is the recombination coefficient. We use the form of $P(\Delta)$ given by Miralda-Escud\'e, Haehnelt \& Rees (2000), which is a good fit to numerical simulations. If we assume an equation of state $T\propto \Delta^{\gamma}$ (where $\gamma \sim 0$--$1$) and a fluctuating radiation field whose amplitude may be density dependent, $\Gamma = \Gamma_{o} \Delta^{\xi}$, then $\tau_{i} =A(z) (1+z)^{4.5} \Delta^{\beta}$, where $\beta={2-0.7\gamma-\xi}$. We can solve for the normalization constant $A(z)$ by demanding $\langle T_{i}(A) \rangle=
T_{\rm obs}(z)$. Note that $A(z) \propto f_{i} \lambda_{i}/\Gamma_{o}$.
The integral in equation (\ref{eqn:flux_transmission}) can be evaluated by the method of steepest descents (Songaila \& Cowie 2002, Songaila 2004):
\begin{equation}
\tau_{\rm eff}= -A^{1/(1+3\beta/4)}+\frac{0.83}{(\beta+4/3)}{\rm ln}(A) + {\rm const}.
\label{eqn:tau_eff}
\end{equation}
The leading term implies that $\tau_{\rm eff} \propto A^{0.4}$ for $\beta=2$ (corresponding to a uniform radiation field and an isothermal equation of state). Thus, in an inhomogeneous universe the optical depth increases more slowly than linearly with the oscillator strength; conversely, it increases more
slowly with a weaker radiation field. Evaluating equation (\ref{eqn:flux_transmission}) numerically, we find that $\tau_{\alpha}/\tau_{\beta} \approx 3$, and $\tau_{\alpha}/\tau_{\gamma} \approx 5-6$, with weak dependence on redshift, the equation of state, and $\Gamma$. This is because transmission is dominated by rare voids, and the primary effect of decreasing $f_i$ is to increase the range of densities sampled by the line.
This behaviour becomes even more marked if $\Gamma$ is not uniform, which is certainly the case before reionization is complete. As a fiducial case, consider a uniform radiation field $\Gamma=0.05$ at $z=6.15$, which produces a Ly$\alpha$ effective optical depth $\tau_{\alpha}=7$. In this case, $(\tau_{\alpha}/\tau_{\beta},\tau_{\alpha}/\tau_{\gamma})=(2.7,4.9)$. Now let us consider various scenarios that would produce the same $\tau_{\alpha}$ but different $(\tau_{\beta},\tau_{\gamma})$.
Suppose the optical depth $\tau \propto \Delta^{\beta}$ has $\beta>2$, which could, for example, mimic self-shielding in overdense regions. In this case, from equation (\ref{eqn:tau_eff}), the optical depth increases even more slowly with oscillator strength: for $\beta=3$, ($\tau_{\alpha}/\tau_{\beta}, \tau_{\alpha}/\tau_{\gamma})=(2.1,3.4)$. Only if the optical depth is independent of overdensity ($\beta=0$) will we recover the linear scaling $\tau_{\rm eff} \propto A$. Fluctuations in the radiation field that are uncorrelated with density fluctuations (so that $\tau \propto \Delta^{2}$) will produce similar effects. For example, a radiation field with a lognormal probability distribution $(\bar{\Gamma},\sigma_{\rm ln \Gamma})=(0.02,1)$ yields $(\tau_{\alpha}/\tau_{\beta},\tau_{\alpha}/\tau_{\gamma})=(1.9,2.9)$.
Thus, the fluctuating density and radiation fields introduce considerable uncertainty in the relations between $\tau_{\alpha},\tau_{\beta},$ and $\tau_{\gamma}$. The effective optical depth is often used to infer $x_{\rm HI,V}$ and $x_{\rm HI,M}$ in the IGM, a procedure that we caution is fraught with uncertainty. In the case of $x_{\rm HI, M}$, it is almost meaningless. It is easy to see why: we can infer $\langle x_{\rm HI} \rangle \propto \langle \tau \rangle$ from $\langle e^{-\tau} \rangle$ only in the limit where $\tau \ll 1$ and $\langle e^{-\tau} \rangle \approx 1 -\langle \tau \rangle$; otherwise, our ignorance of the full probability distribution $P(\tau)$ means that a wide variety of $\langle x_{\rm HI} \rangle$ would be consistent with a given observed $\langle e^{-\tau})$. As an illustration, we show in Figure~1 the logarithmic integrand $y \Delta P(\Delta) \propto \langle y \rangle$, for various quantities $y$; all the curves have been normalized to have unit area. Since $\langle e^{-\tau} \rangle$ is heavily weighted toward voids and $\langle x_{\rm HI} \rangle$ is weighted toward overdense regions, estimating one from the other is extremely model-dependent. In particular, $x_{\rm HI,M}$ is heavily weighted toward large overdensities, and the quasar spectra essentially leave it unconstrained. For instance, the neutral fraction could be considerably lower in overdense regions without affecting the transmitted flux.
\myputfigure{plot_flux_integrand.ps}{3.3}{0.5}{-25}{-10}
\figcaption{\label{fig:flux_integrand}
The logarithmic integrand $y \Delta P(\Delta) \propto \langle y \rangle$ as a function of $\Delta$ for $\langle T_{\alpha} \rangle, \langle T_{\beta} \rangle, \langle T_{\gamma} \rangle, \langle x_{\rm HI,V} \rangle, \langle x_{\rm HI,M} \rangle$. We assume a uniform ionizing background $\Gamma= 0.04$ at $z=6.15$ and an isothermal equation of state. The integrand for $\langle x_{\rm HI,M} \rangle$ peaks at high overdensities to the right of the plot. As the overlap integral between the different Lyman series lines drops, the uncertainty in inferring one from the other increases. More importantly, there is considerable uncertainty in inferring $\langle x_{\rm HI} \rangle$ from quasar spectra; in particular, estimates for $\langle x_{\rm HI,M} \rangle$ are almost meaningless.}
\vspace{\baselineskip}
With these caveats, we list $\tau_\alpha$ derived from each of the three transitions in the rightmost columns of Table~1 for $\beta=2,\,3$. The flux ratios in all 3 troughs are entirely consistent with flux transmission from the quasar without any contamination from an interloper: the equivalent Ly$\alpha$ optical depths all lie within $1-2\sigma$ of one another. The apparent inconsistency in White et al (2003) appears simply because they did not integrate over the IGM density distribution. Again, we caution that the optical depth conversion between different transitions has large uncertainties (not reflected in the error bars) due to uncertainty in the probability distribution of optical depths $P(\tau)$. The true effective optical depth inferred from the Ly$\alpha$ and Ly$\beta$ troughs could also be somewhat higher if there is some continuum contribution from an interloper, which we cannot completely rule out. The most stringent optical depth constraint comes from the Ly$\gamma$ trough, which implies $\tau_{\alpha} < 14.3 \ (2 \sigma)$, with a most likely value $\tau_{\alpha} \approx 6$--$10$. By contrast, in SDSS J1030$+$0524, the $1\sigma$ ($2\sigma$) lower limit to the optical depth in the Ly$\beta$ trough is $\tau_{\rm eff} >11.1$ (9.9); the Ly$\gamma$ trough yields similar constraints.
\section{Discussion}
In this {\it Letter}, we have argued that the residual flux in the Ly$\alpha,\beta$, and $\gamma$ troughs of SDSS J1148+5251 is not due to an interloper galaxy but represents true transmission in the $z \ga 6$ IGM. This places an upper bound on the effective Ly$\alpha$ optical depth of $\tau_{\rm eff} \le 14.3 \ (2\sigma)$, implying that the IGM is still highly ionized at $z\sim 6.3$. It has been argued that the size of the \ion{H}{2} regions of the two highest redshift quasars $R_{\rm HII} \approx 4.5$ Mpc imply $x_{\rm HI} > 0.1$ in this range (Wyithe \& Loeb 2004a): $R_{\rm HII} \approx 7 x_{\rm HI}^{-1/3} (t_{\rm age}/10^{7} \ {\rm yr})^{1/3}$ Mpc, where $t_{\rm age}$ is the lifetime of the quasar. This is of course strongly dependent on the assumed template spectrum; if the escape fraction of ionizing photons in high-redshift quasars is somehow smaller, the constraint weakens. The quasars could also be lensed and hence intrinsically fainter (Haiman \& Cen 2002), though follow-up HST observations have failed to detect multiple images in either quasar (White et al 2004 in preparation). For our purpose, we note that \ion{H}{2} region radius is determined by the mass-weighted neutral fraction $x_{\rm HI,M}$. As we showed in \S \ref{section:tau}, this is very poorly constrained by $\tau_{\rm eff}$, because voids containing only a small fraction of the mass dominate the transmission. For instance, if $x_{\rm HI, M} \sim 0.1$ of the baryons are in neutral self-shielded halos, we could still have $x_{\rm HI, V} \sim x_{\rm HI, M}/\delta \sim 0.1/200 \sim 5 \times 10^{-4}$, which could easily be accommodated by our results.
A second argument in favor of $x_{\rm HI} \ga 0.1$ is an indirect detection of the Gunn-Peterson damping wing (Mesinger \& Haiman 2004). There is a stretch at the boundary of the \ion{H}{2} region of SDSS J1030+0524 with Ly$\beta$ transmission but no Ly$\alpha$ transmission, implying $6 <\tau_{\rm eff} < 11$. The absence of any Ly$\alpha$ transmission from low-density voids in this segment implies a source of smooth opacity, attributed to a Ly$\alpha$ damping wing that requires a large optical depth $\tau > 10^{3}$. However, SDSS J1148$+$5251 contains no such transition region.
We also caution that the statistical significance of such transition regions is still unclear. For instance, in SDSS J1148$+$5251 there is a stretch from $z=5.95-6.0$ with Ly$\beta$ transmission ($F_{-20}=37.9 \pm 1.7$) but no significant Ly$\alpha$ flux ($F_{-20}=2.5\pm 1.7$). Again, the optical depth ratios are consistent with pure flux transmission ($\tau_{\alpha}=6.6\pm0.7$, $\tau_{\beta}=2.0\pm 0.4 \Rightarrow \tau_{\rm eff,\alpha}\approx 5.5 \pm 1.2$). This stretch is $\sim 2.5$ times longer than the $\Delta z=0.02$ zone seen in SDSS J1030$+$0524,
over which even damping wing absorption would change substantially;
in any case, we have argued that the IGM is highly ionized along this stretch. Such regions may therefore be fairly generic and not indicative of a damping wing. More detailed analysis, incorporating a pixel-by-pixel analysis, variance in the foreground Ly$\alpha$ forest, and better theoretical modeling of the fluctuating radiation and density fields in realistic models of reionization, would help shed light on this issue.
Nonetheless, if the region around SDSS J1030$+$0524 is significantly neutral, this may be a hint of large cosmic variance in the epoch of reionization.
Indeed, a strongly fluctuating $\tau_{\rm eff}$ is itself a signature of the pre-overlap era. In the extreme interpretation that the Ly$\alpha$ and Ly$\beta$ troughs of SDSS J1030$+$0524 indicate $x_{\rm HI}\sim 0.2$ down to $z=5.95$, this implies large modulation in the ionization fraction and typical bubble sizes of order $\Delta z=0.38$, or $L\sim 150$ Mpc comoving. This is substantially larger than the \ion{H}{2} regions of these extremely bright and rare quasars, $R_{\rm HII} \sim 30$ Mpc comoving. It is also larger than theoretical expectations for the scale of typical \ion{H}{2} regions at the tail end of reionization (Furlanetto et al. 2004; Wyithe \& Loeb 2004b), so we consider such a scenario to be extremely unlikely. At the other extreme, the measured $\tau_{\rm eff}$ are compatible with a slightly faster increase in $\tau_{\rm eff}$ along the sightline to SDSS J1030$+$0524 ($\tau_{\rm eff} >9.9$, \ 2$\sigma$), than to SDSS J1148$+$5251 , ($\tau_{\rm eff}\approx 6$--$10$). Whether a highly-ionized universe can tolerate such scatter is unclear. We have shown that a simple $\tau_{\rm eff}$ analysis is a blunt instrument, and more sophisticated interpretation is required to put strong constraints on reionization.
Note that surveys of Ly$\alpha$ emitters have found no evolution of the luminosity function of Ly$\alpha$ emitters at $z=5.7$ and $z=6.5$ (Malhotra \& Rhoads 2004, Stern et al 2004), which has been interpreted as evidence against percolation taking place at $z\sim 6$, in agreement with our conclusions.
What other signatures of large Ly$\alpha$ optical depth could emerge from the spectra? One possibility is the \ion{O}{1} absorption forest (Oh 2002). \ion{O}{1} is an excellent tracer of neutral hydrogen: it has a very similar ionization potential $E=13.62$eV, so it lies in tight charge exchange equilibrium. Furthermore, its $1302$ \AA \ absorption line lies redward of the Ly$\alpha$ forest. Although the \ion{O}{1} forest coincides with a noisy portion of the night sky, the sightline to SDSS J1030$+$0524 (which may contain substantially neutral regions) has no lines with $W_{\rm obs} > 0.5$ \AA \ and perhaps $\le 3$ lines with $W_{\rm obs} > 0.3$ \AA \ (X. Fan, private communication). Significantly more lines are expected if the universe is substantially neutral (Oh 2002), although this is model-dependent. The \ion{O}{1} forest clearly merits further investigation, though a good standard star calibration is essential to removing strong telluric features from the atmosphere.
\vspace{.1in}
We thank R. White, X. Fan for providing the spectra of SDSS J1148+5251, and X. Fan for stimulating conversations. We also thank Z. Haiman, A. Lidz, P. Madau, A. Meisinger for helpful discussions/correspondence, and Colleen Schwartz for technical assistance. SPO gratefully acknowledges support by NSF grant AST-0407084, PHY99-07949 and the hospitality of KITP.
|
2,877,628,090,098 | arxiv | \section{Introduction}
\label{sec-introduction}
When fermions are discretized on a d-dimensional lattice they
``double'' producing $2^d$ species for each flavor. In order to remove
the unwanted degrees of freedom special care must be taken. For a
vector theory, like QCD, two methods have been used to deal with this
problem, but both break the global symmetries of the continuum theory.
Wilson fermions \cite{wilson-fermions} are implemented by adding
an irrelevant operator to the action. This operator makes all but
one of the species heavy (with masses close to the cutoff). For the
$N_f$ flavor QCD this operator breaks the $SU(N_f)_L \times SU(N_f)_R$
chiral symmetry down to $SU(N_f)$. This explicit breaking is severe
and requires fine tuning of the bare quark mass in order to obtain a
massless theory. Even then the size of the breaking is proportional to
the lattice spacing and only close to the continuum limit the explicit
breaking becomes small. Staggered fermions
\cite{staggered-fermions} break the $SU(N_f)_L \times SU(N_f)_R$
chiral symmetry down to $U(1) \times U(1)$. Because of the remnant
of chiral symmetry the massless theory can be reached by simply taking
the bare quark mass to zero. However, the flavor symmetry of the
theory has been compromised and is also only recovered as the
continuum limit is approached.
Despite these problems both methods have been very successful in
describing the light hadron spectrum at zero temperature. However,
both methods have difficulties in studying the finite temperature
phase transition. Wilson fermions have a complicated phase diagram
that, at the presently accessible lattice spacings, makes it hard to
extract the relevant physics. Staggered fermions, because of the exact
remnant of chiral symmetry, do not suffer from this problem. However,
at the presently accessible lattice spacings, the breaking of flavor
symmetry makes two of the three pions heavy. This can have important
physical consequences since the transition temperature is of the order
of the pion mass. For a review on the finite temperature phase
transition with both types of fermions the reader is referred to
\cite{Ukawa-review} and references therein.
A few years ago a new method for discretizing fermions was developed in
order to address the more difficult problems associated with chiral
gauge theories \cite{Kaplan}. In the following years this method was
further developed (see \cite{DWF-reviews} and references therein) with
important progress in the development of chiral gauge theories
\cite{NN1}, \cite{NN2}. The basic idea follows from the fact that a massive
vector theory in $2n+1$ dimensions, with a mass term that changes sign
along the $2n+1$ dimension, develops a massless chiral zero mode that
is exponentially bound along the $2n+1$ direction to the region where
the mass changes sign. From the point of view of the $2n$ dimensional
world this is a chiral fermion. This region is called Domain Wall and
this type of fermion is called Domain Wall Fermion (DWF). When such a
theory is discretized species doubling also occurs. However, since the
$2n+1$ dimensional theory is vector-like the extra species can be
removed with the addition of a standard Wilson term. The resulting
theory has a single chiral fermion exponentially bound to the wall.
If, for practical reasons, the $2n+1$ dimension is made finite with
periodic boundary conditions for the mass then the mass must change
sign one more time. In that region (anti-wall) an exponentially bound
chiral zero mode with opposite chirality appears. As a result, the
theory becomes vector-like. Different types of boundary conditions
yield similar problems. In order to preserve the single chiral mode,
the $2n+1$ dimension must be kept infinite. At first sight this may
seem impractical. However, Narayanan and Neuberger developed a method,
called the Overlap formalism, that makes it possible to deal with this
infinity \cite{NN1}.
The Overlap formalism develops a transfer matrix along the $2n+1$
dimension and an associated Hamiltonian. The gauge fields are defined
only on the $2n$ dimensional space and are taken to be independent of
the $2n+1$ coordinate \cite{NN1}. In essence, the extra dimension is
treated as a complicated flavor space. The resulting formalism
involves two Hamiltonians, one for the region of positive mass and one
for the region of negative mass. The chiral determinant is the
determinant of the overlap of the two ground states associated with
each Hamiltonian and it can be calculated explicitly once all the
negative eigenvectors of both Hamiltonians are known. For a finite
$2n$ dimensional lattice the Hamiltonians are finite size matrices of
size $\sim V \times V$ where $V$ is the $2n$ dimensional volume and
their eigenvectors can be readily calculated. The resulting chiral
determinant has the correct magnitude and a phase that exhibits the
correct gauge dependence for ``smooth'' gauge fields. For ``rough''
gauge fileds the phase exhibits a mild breaking of gauge symmetry even
for anomaly free theories. This problem has been resolved in
\cite{NN2}.
These methods can also be used to formulate a vector theory. In this
case the boundary conditions along the $2n+1$ dimension are set to be
periodic. A Dirac fermion emerges with the positive chirality
component bound on the wall and the negative chirality bound on the
anti-wall. If the $2n+1$ dimension is taken to be infinite the two
chiralities are decoupled and the resulting theory has intact chiral
symmetries! Again, this infinity can be dealt with the Overlap
formalism and now there are no issues associated with the phase of the
determinant since, for a vector theory, the determinant is real.
Therefore, the Overlap formalism provides an ideal lattice
regularization of vector theories where the chiral symmetries are left
intact even for finite lattice spacing. Also, the anomalous breaking
of the axial symmetry is reproduced in an elegant way along with a
formula for the index of the chiral Dirac operator
\cite{NN1}.
The Overlap formalism was used in a dynamical numerical simulation of the
massless single and multiflavor Schwinger model with good results
\cite{NNV}. Numerical simulations of QCD using the overlap formalism
would clearly be very appealing. However, as mentioned above, such a
simulation would require the calculation of all negative eigenvectors
of matrices of size $\sim V \times V$. This makes such a
calculation prohibitive for present generation supercomputers.
An obvious alternative (see \cite{DWF-reviews} and references therein)
is to keep the $2n+1$ dimension finite and use standard Hybrid Monte
Carlo type algorithms to simulate the theory in $2n+1$ dimensions. Of
course, the exact chiral symmetry will be spoiled but it will be
recovered as the size $L_s$ of the $2n+1$ dimension is sent to
infinity. Therefore, even at finite lattice spacing one can control
the restoration of the regularization induced chiral symmetry breaking
by using the parameter $L_s$. This involves no fine tuning and,
furthermore, since the two chiralities decay exponentially away from
the wall (anti-wall), one would expect that the restoration of chiral
symmetry would be exponential i.e. $\sim e^{-c L_s}$, $0 < c$. The
computer cost of such a simulation would be $L_s$ times larger than a
simulation of standard Wilson fermions with the same physical masses.
Since for present day supercomputers a value of $L_s$ greater than $10
- 20$ will make simulations impractical, an important question to ask
is what is the rate ``$c$'' of restoration of chiral symmetry and how
does it depend on the other parameters of the theory and in particular
on the lattice spacing. In \cite{Shamir},
\cite{Furman-Shamir} some of the issues relating to this question
were investigated analytically. Numerical work in \cite{Jaster},
\cite{PMV}, \cite{Blum-Soni} has yielded encouraging results
and in particular the interesting work of \cite{Blum-Soni}
indicates that DWF can successfully address problems related
to the evaluation of weak matrix elements. However, these works have
only marginally addressed this particular question.
Before full scale dynamical
QCD simulations are performed this question should be answered. In
this paper this question is investigated in the context
of the two flavor lattice Schwinger model.
A useful variation of the wall, anti-wall model studied in
\cite{BDF} was proposed in the context of vector lattice gauge
theory in \cite{Shamir}, \cite{Furman-Shamir}. There, instead of
having a mass that changes sign in two places along the $2n+1$
dimension (say at $0$ and $L_s/2$), the mass is kept fixed to some
positive value $m_0$, but the boundary conditions are taken to be free
at the ends of the $2n+1$ dimension. Again, two zero modes with
opposite chiralities emerge, but they are now bound at the opposite
ends of the $2n+1$ dimension and are separated by a distance $L_s$
rather than $L_s/2$ as in the original model. Therefore, the
expectation is that for the same $L_s$ this model will achieve better
restoration of chiral symmetry.
Another feature added in this model is the introduction of an explicit
chiral symmetry breaking term that connects the two ends with strength
$m_f$. This gives mass to the fermion in addition to the one
resulting because of the finite extend $L_s$. The reason for adding
this term is that it provides linear control over the fermion mass
instead of the exponential one provided by $L_s$. Furthermore, in a
numerical computation, it makes much more sense to vary $m_f$ rather
than $L_s$ in order to control the mass. Therefore, for a given $m_f$
one would like to keep $L_s$ large enough so that it does not affect
the fermion mass in any significant way. This method and the
associated Overlap formalism will be used throughout this paper.
The theory has five parameters. The first two are the lattice spacing
$a$ and the physical extent $l$ along one direction of the $2n$
dimensional box (they are controlled by the pure gauge coupling $g_0$
and the size in lattice units $L$). The remaining three parameters
$m_0$, $L_s$ and $m_f$ all control, to some extent, the amount of chiral
symmetry breaking and therefore the effective fermion mass. For $L_s
\rightarrow \infty$ the theory is chirally symmetric except for the
explicit breaking introduced by $m_f$. As a result, the effective
fermion mass vanishes linearly with vanishing $m_f$
\cite{Furman-Shamir}. But for finite $L_s$ this is not the case. As
mentioned above, even for $m_f=0$ the restoration of chiral symmetry
is expected to be exponential $\sim e^{-c L_s}$.
One would expect that the exact continuum solution of the Schwinger
model would be useful to compare with results obtained on the
lattice. Unfortunately, this is only partially true. The
regularization with DWF introduces to the two dimensional action a
four-Fermi term with some coefficient. Since, for the two dimensional
model, this operator is not irrelevant, the continuum theory will be
different from a continuum theory with no four-Fermi term. Although
the continuum theory has been solved with such a term present
\cite{Dettki-Sachs-Wipf}, the results can not be directly compared
since the value of the coefficient arising from the DWF has not been
calculated. This problem was encountered in \cite{NNV} and \cite{NN2}
and made the comparison with continuum results complicated.
Fortunately, for the purposes of this work, the continuum results are
not needed. In fact, there is a much more relevant comparison that can
be made. At every step the value of any observable at finite $L_s$
can be directly compared with its value at infinite $L_s$ calculated
using the Overlap on the same lattice size and lattice spacing.
The paper is organized as follows:
In section \ref{sec-model} the model and the
corresponding Overlap implementation is reviewed.
In section \ref{sec-observables} the definitions of the various
observables used in this paper are given.
In section \ref{sec-free} the free theory for finite $L_s$ is
discussed, the full expression for the propagator is given and the
``effective'' bare fermion mass is identified.
In section \ref{sec-interact} some general considerations
regarding the interacting theory are presented. These
considerations lead to specific predictions.
In section \ref{sec-overlap-sim} the results of a full dynamical
simulation using the Overlap with non zero mass are presented. These
results, interesting in their own right, are used to compare with the
finite $L_s$ results of the next section.
Section \ref{sec-hmc} describes the results of a dynamical simulation
of the $2+1$ dimensional system for various values of the parameters
on a fixed physical volume. The algorithm used is a standard Hybrid
Monte Carlo (HMC) algorithm. The numerical results confirm the
predictions made in section
\ref{sec-interact} and together outline the mechanisms of chiral
symmetry restoration in the model.
Section \ref{sec-conclusions} contains a summary and conclusions.
\section{The Model}
\label{sec-model}
In this section the model and the corresponding Overlap
formalism \cite{NN1} implementation
is reviewed for the benefit of the reader and in
order to establish notation \cite{Shamir},
\cite{Furman-Shamir}. The following is for a single flavor. The
generalization to more flavors is straightforward.
The partition function of the single flavor $2n+1$ dimensional model
is:
\begin{equation}
Z = \int [dU] \int [d\overline{\Psi} d\Psi] \int [d\Phi^\dagger d\Phi] e^{-S}
\label{Z}
\end{equation}
$U_\mu(x)$ is the gauge field, $\Psi(x,s)$ is the fermion field and
$\Phi(x,s)$ is a bosonic Pauli Villars (PV) type field. $x$ is
a coordinate in the $2n$ dimensional space-time box with extent $L$
along each of the directions, $\mu = 1,2,\dots 2n$, and $s=0,1, \dots,
L_s-1$, where $L_s$ is the size of
the $2n+1$ direction and is taken to be an even number. The action
$S$ is given by:
\begin{equation}
S = S(g_0, L, L_s, m_0, m_f) = S_G(U) + S_F(\overline{\Psi}, \Psi, U) +
S_{PV}(\Phi^\dagger, \Phi, U)
\label{action}
\end{equation}
where:
\begin{equation}
S_G = {1\over g_0^2} \sum_p Re Tr[I - U_p]
\label{action_G}
\end{equation}
is the standard plaquette action with $g_0$ the lattice gauge
coupling. In this paper the coupling $g_0$ is exchanged for the
parameter:
\begin{equation}
(\mu l) = {g_0 \over \sqrt{\pi} } L
\label{mu_l}
\end{equation}
where $l$ is the physical size of the $2n$ dimensional box along one of
its dimensions and $\mu$ is a mass related to the photon mass with:
\begin{equation}
m_\gamma = \sqrt{N_f} \mu
\label{m_gamma}
\end{equation}
where $N_f$ is the number of flavors. With these choices $\mu l$ is
the physical box size in units of $\mu$ and $\mu l / L= \mu a$ is
the lattice spacing in units of $\mu$.
The fermion action is:
\begin{equation}
S_F = - \sum_{x,x^\prime,s,s^\prime} \overline{\Psi}(x,s) D_F(x,s; x^\prime,
s^\prime) \Psi(x^\prime,s^\prime)
\label{action_F}
\end{equation}
with the fermion matrix given by:
\begin{equation}
D_F(x,s; x^\prime, s^\prime) = \delta(s-s^\prime) D\!\!\!\!/(x,x^\prime)
+ D\!\!\!\!/^\bot(s,s^\prime) \delta(x-x^\prime)
\label{D_F}
\end{equation}
\begin{eqnarray}
D\!\!\!\!/(x,x^\prime) &=& {1\over 2} \sum_\mu\left[ (1+\gamma_\mu)
U_\mu(x) \delta(x+\hat\mu - x^\prime) + (1-\gamma_\mu)
U^\dagger_\mu(x^\prime) \delta(x^\prime+\hat\mu - x) \right] \nonumber \\
&+& (m_0 - 2n)\delta(x-x^\prime)
\label{Dslash_F}
\end{eqnarray}
\begin{equation}
D\!\!\!\!/^\bot(s,s^\prime) = \left\{ \begin{array}{ll} P_R
\delta(1-s^\prime) - m_f P_L \delta(L_s-1 - s^\prime) - \delta(0-
s^\prime) & s=0 \\ P_R \delta(s+1 - s^\prime) + P_L \delta(s-1 -
s^\prime) - \delta(s-s^\prime) & 0 < s < L_s-1 \\ -m_f P_R
\delta(0-s^\prime) + P_L \delta(L_s-2 - s^\prime) - \delta(L_s-1 -
s^\prime) & s = L_s -1
\end{array}
\right.
\label{Dslash_perp_f}
\end{equation}
\begin{equation}
P_{R,L} = { 1 \pm \gamma_5 \over 2}
\end{equation}
where $m_0$ is a $2n+1$ dimensional mass representing the ``height'' of
the Domain Wall. In order for the doubler species to be removed one
must set $0<m_0<2$ \cite{Kaplan}. However, this range is further
restricted by the requirement that the transfer matrix along the $2n+1$
direction be positive \cite{NN1}:
\begin{equation}
0 < m_0 < 1
\label{m0_range}
\end{equation}
The gamma matrices are taken in the chiral basis
and are the same as in the last reference in \cite{NN1}. In two
dimensions they are:
\begin{equation}
\gamma_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right), \ \
\gamma_2 = \left( \begin{array}{cc} 0 & i \\ -i & 0 \end{array} \right), \ \
\gamma_5 = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right)
\end{equation}
The PV action is designed to cancel the contribution of the heavy
fermions in the large $L_s$ limit. This is necessary because the
number of heavy fermions is $\sim L_s$ and at the $L_s \rightarrow
\infty$ limit they produce bulk type infinities \cite{NN1}. There is
some flexibility in the definition of the PV action since different
actions could have the same $L_s \rightarrow \infty$ limit. However,
the choice of the PV action may affect the approach to the $L_s
\rightarrow \infty$ limit. A slightly different action than the one
used in \cite{Furman-Shamir} is used here. This action is easier to
implement numerically and for finite $L_s$ it projects the ground
state of the transfer matrix $T$ better; the projector is $T^{L_s}$
instead of $T^{L_s/2}$ (see below). Also, even for finite $L_s$,
it exactly cancels out the
fermion action when $m_f = 1$ resulting into a pure gauge theory. The
PV action is:
\begin{equation}
S_{PV} =
\sum_{x,x^\prime,s,s^\prime} \Phi^\dagger(x,s)
D_F[m_f=1](x,s; x^\prime, s^\prime) \Phi(x^\prime,s^\prime)
\label{action_PV}
\end{equation}
The transfer matrix along the $2n+1$ direction for this model is:
\begin{equation}
T = e^{- \hat a^\dagger H \hat a}
\label{transf_mat}
\end{equation}
where $\hat a^\dagger, \hat a$ are creation and annihilation operators
that obey canonical anticommutation relations and span a Fock space
with vacuum state $|0>$. These operators live on the sites of the $2n$
dimensional lattice and carry spin, color and flavor indices. The
left/right component decomposition of $\hat a$ is:
\begin{equation}
\hat a = \left( \begin{array}{c} \hat c \\ \hat d^\dagger \end{array} \right)
\label{a_hat}
\end{equation}
The single particle Hamiltonian $H$ is defined by:
\begin{equation}
e^{-H} = \left( \begin{array}{cc} B^{-1} & B^{-1} C \\ C^\dagger
B^{-1} & C^\dagger B^{-1} C + B
\end{array}
\right)
\label{Hamiltonian}
\end{equation}
\begin{equation}
B(x,y) = {1\over2} \sum_{\mu=1}^{2n}
\left[ 2 - U_\mu(x) \delta(x+\hat\mu - y)
- U^\dagger_\mu(y) \delta(y + \hat\mu - x)
\right]
+ (1 - m_0) \delta(x-y)
\label{B}
\end{equation}
\begin{equation}
C(x,y) = {1\over2} \sum_\mu\left[U_\mu(x) \delta(x+\hat\mu - y) -
U^\dagger_\mu(y) \delta(y + \hat\mu - x) \right] \sigma_\mu
\label{C}
\end{equation}
with $\sigma_1 = 1, \sigma_2 = i$ in two dimensions.
The fermionic and PV effective actions can be expressed in terms of the
transfer matrix as:
\begin{equation}
e^{-S^F_{\rm eff}[L_s]} =
\det(D_F[m_f]) = \det(B)^{L_s} Tr\left[ T^{L_s} {\cal O}(m_f)\right]
\label{det_F}
\end{equation}
\begin{equation}
e^{-S^{PV}_{\rm eff}[L_s]} =
\left( \det(D_F[m_f=1]) \right)^{-1}
= \left( \det(B)^{L_s} {Tr\left[ T^{L_s}\right]} \right)^{-1}
\label{det_PV}
\end{equation}
where the operator ${\cal O}(m_f)$ implements the boundary conditions
and contains all the $m_f$ dependence:
\begin{equation}
{\cal O}(m_f) = \prod_n (\hat c_n \hat c_n^\dagger + m_f \hat
c_n^\dagger \hat c_n) (\hat d_n \hat d_n^\dagger + m_f \hat
d_n^\dagger \hat d_n)
\label{Omf}
\end{equation}
For $m_f=1$ it is the identity operator and for $m_f=0$ is a
projection operator to a state $|0^\prime>$:
\begin{equation}
|0> = \prod_n \hat d^\dagger |0^\prime>
\label{0_prime}
\end{equation}
In the infinite $L_s$ limit $T^{L_s}$ becomes a projection operator to
the ground state of $- \hat a^\dagger H \hat a$,
\begin{equation}
\lim_{L_s \rightarrow \infty} T^{L_s} \rightarrow e^{-\lambda_0} |0_H><0_H|
\label{Tprojector}
\end{equation}
where $|0_H>$ and $\lambda_0$ are the ground state eigenvector and
eigenvalue of $\hat a^\dagger H \hat a$ obtained by filling all
negative energy states. To get an explicit relation between $|0_H>$
and $|0>$ let $R$ be the eigenvector matrix of the single particle
Hamiltonian $H$. The matrices $H$ and $R$ have size $N \times N$ where
$N = {\rm spin} \times {\rm color} \times {\rm flavor} \times V$.
$R$ can be put in the form:
\begin{equation}
R = \left( \begin{array}{cc} P^- & P^+ \\ Q^- & Q^+ \end{array}
\right)
\label{R}
\end{equation}
where the rows labeled $P$ correspond to the left chirality states
(with creation/annihilation operators $\hat c^\dagger, \hat c$) and the
rows labeled $Q$ correspond to the right chirality states (with
creation/annihilation operators $\hat d^\dagger, \hat d$). The $\pm$
splitting of the columns corresponds to eigenvectors with
positive/negative eigenvalues. With $N^{\pm}$ denoting the number of
positive/negative eigenvalues the size of the $P^-, Q^-$ matrices is
$N/2 \times N^-$ and the size of the $P^+, Q^+$ matrices is $N/2
\times N^+$. Then it can be shown that:
\begin{equation}
|0_H> = \prod_{i=1}^{N^-}(\hat c_{l_i}^\dagger P_{{l_i},i}^- + \hat
d_{l_i} Q_{{l_i},i}^- )|0>
\label{0_H}
\end{equation}
From equations \ref{det_F}, \ref{det_PV} and \ref{Tprojector} the
effective action for the fermion and PV fields in the $L_s \rightarrow \infty$
limit is given by the Overlap formula:
\begin{equation}
e^{-S_{\rm eff}[L_s = \infty, m_f]} = e^{-S^F_{\rm eff}[L_s = \infty,
m_f] -S^{PV}_{\rm eff}[L_s = \infty]} = <0_H| {\cal O}(m_f) |0_H>
\label{Overlap_seff}
\end{equation}
For $m_f=1$, $<0_H| {\cal O}(1) |0_H> = 1$, corresponding to a
theory with no fermions. The $m_f=0$ case corresponds to massless
fermions and the Overlap takes the special form:
\begin{equation}
e^{-S_{\rm eff}[L_s = \infty, m_f=0]} = \left| <0_H|0^\prime>\right|^2
\label{Overlap_mf0}
\end{equation}
It can be shown that:
\begin{equation}
\left| <0_H|0^\prime>\right|^2 =
\left| \det( Q^-) \right|^2
\label{det_Q}
\end{equation}
If $N^- = N/2$, $Q^-$ is a square matrix and the Overlap will in
general be non zero. However, if $N^- \neq N/2$ then $Q^-$ is not a
square matrix and its determinant is identically zero. From equations
\ref{0_prime}, \ref{0_H} and \ref{det_Q}, one can see that this
arises because of a mismatch in the filling levels of $|0_H>$ and
$|0^\prime>$. In order to obtain a non zero overlap one would need to
insert the appropriate number of creation and annihilation operators to
balance the filling levels. In fact these operators are the t'Hooft
vertices constructed with lattice fields. Then an elegant definition
of the topological charge $q$ as seen by the fermions arises, \cite{NN1}:
\begin{equation}
q = N^- - N/2
\label{top_charge}
\end{equation}
where $q$ is naturally integer valued.
When $m_f \neq 0$ use of equations \ref{Omf}, \ref{0_H} and
\ref{Overlap_seff} yield explicit expressions for the Overlap as a
determinant of a matrix that is constructed out of $P^-$ and $Q^-$.
These expressions are used in the numerical simulation of the
Overlap ($L_s \rightarrow \infty$) in section \ref{sec-overlap-sim}.
For more details the reader is referred to \cite{NN1} and
\cite{Shamir}, \cite{Furman-Shamir}.
\section{Observables}
\label{sec-observables}
In this section the definitions of the observables that are measured
in this paper are given. The operators involved are as in \cite{NN1},
\cite{Furman-Shamir}.
The $2n$ dimensional fermion operators of the $2n+1$ dimensional
theory are constructed out of the $2n$ dimensional fermion fields
$\overline{\psi}$, $\psi$ as in \cite{Furman-Shamir}:
\begin{eqnarray}
\psi(x) &=& P_R \Psi(x,0) + P_L \Psi(x, L_s-1) \nonumber \\
\overline{\psi}(x) &=& \overline{\Psi}(x,L_s-1) P_R + \overline{\Psi}(x, 0) P_L
\end{eqnarray}
In the $L_s \rightarrow \infty$ limit of the theory these operators
exactly correspond to insertions in the Overlap of the creation and
annihilation operators discussed in section \ref{sec-model}. This will
allow explicit comparisons to be made between measurements involving
$\overline{\psi}$, $\psi$ in the $2n +1 $ theory with finite $L_s$ and
measurements with the Overlap involving the corresponding creation and
annihilation operators.
The following is a list of definitions of the observables
for any $L_s$ and the corresponding Overlap expressions.
The definitions of the actions are as in section \ref{sec-model} but
for two flavors. Use is made of the fact \cite{Furman-Shamir}:
\begin{equation}
\det(D_F) = \det(D_F^\dagger)
\label{det_df_dfdagger}
\end{equation}
The fermion effective action of the $2n+1$ dimensional theory in a
background gauge field is:
\begin{equation}
e^{-S^F_{\rm eff}[L_s, m_f]} =
\int [d\overline{\Psi} d\Psi] e^{-S_F} =
\det\left( D_F^\dagger[L_s, m_f] D_F[L_s, m_f] \right)
\label{seff_F}
\end{equation}
The PV effective action in a background gauge field is:
\begin{equation}
e^{-S^{PV}_{\rm eff}[L_s]} =
\int [d\overline{\Psi} d\Psi] e^{-S_{PV}} =
\det\left( D_F^\dagger[L_s, m_f=1] D_F[L_s, m_f=1] \right)^{-1}
\label{seff_PV}
\end{equation}
The fermion effective action in a background gauge field is:
\begin{equation}
e^{-S_{\rm eff}[L_s, m_f]} = e^{-S^F_{\rm eff}[L_s, m_f] - S^{PV}_{\rm
eff}[L_s]} = { \det\left( D_F^\dagger[L_s, m_f] D_F[L_s, m_f] \right)
\over {\det\left( D_F^\dagger[L_s, m_f=1] D_F[L_s, m_f=1] \right)} }
\label{seff}
\end{equation}
\begin{equation}
e^{-S_{\rm eff}[L_s=\infty, m_f]} = <0_H| {\cal O}(m_f) |0_H>^2
\label{seff_ov}
\end{equation}
The chiral condensate operator is:
\begin{equation}
\overline{\psi} \psi = - {1 \over 2V} \sum_x \sum_{i=1}^2
\left[ \overline{\psi}^i_R(x) \psi^i_L(x) +\overline{\psi}^i_L(x) \psi^i_R(x) \right]
\label{pbp_op}
\end{equation}
The following observable is related to the chiral condensate in a
background gauge field:
\begin{eqnarray}
&&P_1[L_s, m_f] =
\int [d\overline{\Psi} d\Psi]\ \overline{\psi}\psi \ e^{-S_F -S^{PV}_{\rm eff} } \nonumber \\
&&P_1[L_s, m_f] = - {1 \over V} \sum_x \left[ D_F^{-1}(x, L_s-1,2; x, 0,2) +
D_F^{-1}(x,0,1; x, L_s-1,1) \right] e^{-S_{\rm eff}}
\label{P1}
\end{eqnarray}
\begin{equation}
P_1[L_s=\infty, m_f] = -{1\over V} \sum_x\left[ <0_H| \hat c_x^\dagger
{\cal O}(m_f) \hat c_x |0_H> <0_H| {\cal O}(m_f) |0_H> + (c
\rightarrow d) \right]
\label{P1_ov}
\end{equation}
The t' Hooft vertex operator is:
\begin{equation}
w = {1 \over V} \sum_x\left[ \prod_{i=1}^2\left(
\overline{\psi}^i_R(x)\psi^i_L(x) \right) +
\prod_{i=1}^2\left( \overline{\psi}^i_L(x) \psi^i_R(x) \right) \right]
\label{w_op}
\end{equation}
The following observable is related to the t' Hooft vertex in a
background gauge field:
\begin{eqnarray}
&&P_2[L_s, m_f] = \int [d\overline{\Psi} d\Psi] \ w \ e^{-S_F-S^{PV}_{\rm eff}}
\nonumber \\
&&P_2[L_s, m_f] = {1 \over V} \sum_x
\left[ D_F^{-2}(x, L_s-1,2; x, 0,2) + D_F^{-2}(x,0,1; x, L_s-1,1)
\right] e^{-S_{\rm eff}}
\label{P2}
\end{eqnarray}
\begin{equation}
P_2[L_s=\infty, m_f] = {1 \over V} \sum_x\left[ <0_H| \hat c_x^\dagger
{\cal O}(m_f) \hat c_x |0_H>^2 + (c \rightarrow d) \right]
\label{P2_ov}
\end{equation}
The observables $e^{-S_{\rm eff}}$, $P_1$
and $P_2$ are interesting because they correspond to Overlap
expressions and, therefore, their values at the $L_s=\infty$ limit are
calculable. Furthermore, these quantities are sensitive to the
topology of the background gauge field.
The expectation value of the fermion condensate is:
\begin{equation}
<\overline{\psi} \psi> = {1 \over Z} \int [dU] \int [d\overline{\Psi} d\Psi] \int
[d\Phi^\dagger d\Phi] \ \ \overline{\psi} \psi \ \ e^{-S}
\label{pbp1}
\end{equation}
\begin{equation}
<\overline{\psi} \psi>_{L_s=\infty} = { \int dU {1 \over V} \sum_x\left[ <0_H|
\hat c_x^\dagger {\cal O}(m_f) \hat c_x |0_H> <0_H| {\cal O}(m_f)
|0_H> + (c \rightarrow d) \right] e^{-S_G}
\over \int dU
<0_H| {\cal O}(m_f) |0_H>^2 e^{-S_G} }
\label{pbp1_ov}
\end{equation}
The expectation value of the t' Hooft vertex is:
\begin{equation}
<w> = {1 \over Z} \int [dU] \int [d\overline{\Psi} d\Psi] \int
[d\Phi^\dagger d\Phi] \ \ w \ \ e^{-S}
\label{pbp2}
\end{equation}
\begin{equation}
<w>_{L_s=\infty} = { \int dU {1 \over V} \sum_x\left[ <0_H| \hat
c_x^\dagger {\cal O}(m_f) \hat c_x |0_H>^2 + (c \rightarrow d) \right]
e^{-S_G}
\over \int dU
<0_H| {\cal O}(m_f) |0_H>^2 e^{-S_G} }
\label{pbp2_ov}
\end{equation}
Notice that a numerical evaluation of $<\overline{\psi} \psi>_{L_s=\infty}$ and
$<w>_{L_s=\infty}$ requires two separate pure gauge simulations, one
for the numerator and one for the denominator, i.e. the fermion
determinant is treated as an observable \cite{NNV}, \cite{NN2}.
\section{The free theory}
\label{sec-free}
In this section the propagator is given, its singular
part is identified and the bare fermion mass which is a function of
$L_s, m_0, m_f$, is extracted. As a verification of this result the
smallest eigenvalue of the free $2n+1$ dimensional Dirac operator is
also calculated.
The propagator has been calculated for the infinite $L_s$ case in the
first reference of \cite{NN1}. The propagator has also been calculated
for the model described in section \ref{sec-model} but only in the
limit where exponentially small contributions in $L_s$ could be
ignored \cite{Shamir}. The size of these contributions was alluded
to but no explicit expression was given. Since in this paper the
interest is on the behavior of these contributions, the full
calculation is worked out. Because the general form of the propagator
is the same as in \cite{NN1}, \cite{Shamir} an effort has been made to
keep similar notation.
The free $2n+1$ dimensional Dirac operator of eq. \ref{D_F} in
momentum space is:
\begin{equation}
D_F(p: s,s^\prime) = i \platslash \delta(s-s^\prime)
- b(p) \delta(s-s^\prime)
+ {1 + \gamma_5 \over 2} M(s,s^\prime) + {1 - \gamma_5 \over 2}
M^\dagger(s,s^\prime)
\label{D_F_free}
\end{equation}
and
\begin{equation}
b(p) = \sum_{\mu=1}^{2n} \left[ 1 - \cos(p_\mu) \right] + 1 -m_0
\label{bp}
\end{equation}
\begin{equation}
\overline p_\mu = \sin(p_\mu), \ \ \, \mu = [1,\dots, 2n], \ \ \
p_\mu = {2 \pi k_\mu \over L}, \ \ \ k_\mu = [0,1, \dots L-1]
\label{plat}
\end{equation}
\begin{equation}
M(s,s^\prime) = \delta(s + 1 - s^\prime)
- \delta(s^\prime - 0) \delta(L_s - 1 - s) (1 + m_f)
\label{RDWF}
\end{equation}
where the $\delta$ functions are understood as having ``period'' $L_s$.
Notice that the first two terms of eq. \ref{D_F_free} are the same as
for Wilson fermions with mass $(1-m_0)$, where $0<m_0<1$ (see sect.
\ref{sec-model}). The second order free Dirac operator is diagonal in
spin:
\begin{equation}
D_F D_F^\dagger = {1 + \gamma_5 \over 2} \Omega_+ + {1 - \gamma_5
\over 2} \Omega_-
\label{D_F_D_F_free}
\end{equation}
where $\Omega_{\pm}$ have no spin indices and:
\begin{eqnarray}
&&\Omega_+(p: s, s^\prime) = [ -b(p) + M(s, s^\prime) ]
[-b(p) + M^\dagger(s, s^\prime) ] + \overline p^2
\nonumber \\
&&\Omega_-(p: s, s^\prime) = \Omega_+(p: L_s -1 -s, L_s -1 -s^\prime)
\label{Omega}
\end{eqnarray}
where:
\begin{equation}
\overline p^2 = \sum_{\mu=1}^{2n} \overline p_\mu^2
\label{platsq}
\end{equation}
The inverse of the second order free Dirac operator $D_f D_F^\dagger$
must therefore be of the form:
\begin{equation}
G = {1 + \gamma_5 \over 2} G_+ + {1 - \gamma_5 \over 2} G_-, \ \ \ \ \
\left[ D_f D_F^\dagger \right] G = I
\label{G_free}
\end{equation}
where $G_\pm$ have no spin indices. From general considerations
$G_\pm$ must be of the form
\cite{Shamir}:
\begin{eqnarray}
G_+(p:s, s^\prime) &=& A_0 e^{- a |s-s^\prime| } + A_1 e^{-a (s +
s^\prime) } + A_2 e^{-a (L_s-1-s + L_s-1- s^\prime) } \nonumber \\ &+&
A_m\left[ e^{-a (L_s-1 + s - s^\prime) } + e^{-a (L_s-1 + s^\prime -
s) } \right] \nonumber \\ \\ G_-(p: s, s^\prime) &=& G_+(p: L_s -1 -s,
L_s -1 -s^\prime)
\label{G_pm}
\end{eqnarray}
The coefficients in eq. \ref{G_pm} are momentum dependent. The
coefficient ``a'' is the solution of the equation:
\begin{equation}
\cosh(a) = { 1 + b^2 + \overline p^2 \over 2 b}
\label{a_free}
\end{equation}
A straight forward calculation results to the following expressions
for the remaining coefficients:
\begin{equation}
A_0 = { 1 \over 2 b \sinh(a) }
\label{A0_free}
\end{equation}
\begin{equation}
A_1 = - {B \over \Delta} (b - e^{-a}) (1 - m_f^2)
\label{A1_free}
\end{equation}
\begin{equation}
A_2 = { B \over \Delta} (e^a - b) (1 - m_f^2)
\label{A2_free}
\end{equation}
\begin{equation}
A_m = - { B \over \Delta} \left[ 2 b m_f \sinh(a) + e^{-a (L_s-1)}
\left(e^{-2a}\left[e^a - b\right] - m_f^2 \left[e^{-a} - b \right]
\right) \right]
\label{Am_free}
\end{equation}
where,
\begin{eqnarray}
\Delta &=& \left[ e^{2a} \left(b - e^{-a} \right)
+ m_f^2 \left(e^a - b \right) \right]
+ \left[ 4 m_f b \sinh(a) \right] e^{-a(L_s-1)} \nonumber \\ &+&
\left[ m_f^2 \left(b - e^{-a} \right) + e^{-2a} \left( e^a - b \right)
\right] e^{-2 a (L_s-1)}
\label{Delta_free}
\end{eqnarray}
In order to identify the singular part of the propagator an expansion
in the variables $p$, $m_f$ and $(1-m_0)^{(L_s-1)}$ is done treating
these variables as numbers with magnitudes much smaller than one. The
only singular amplitudes are $A_2$ and $A_m$ and the resulting
expression for the singular part of $G_\pm$ is:
\begin{eqnarray}
G^{\rm singular}_+(p:s, s^\prime) =
{ 1 \over p^2 + m_{\rm eff}^2 } \!\!\!\!\!\!\!\!\!
&&\left\{ m_0 (2 -m_0) e^{-a (L_s-1-s + L_s-1- s^\prime) } \right.
\nonumber \\ &&\ \ \ -\left. m_{\rm eff} (1-m_0) \left[ e^{-a (L_s-1 + s -
s^\prime) } + e^{-a (L_s-1 + s^\prime - s) } \right] \right\}
\label{G_free_singular}
\end{eqnarray}
with,
\begin{equation}
m_{\rm eff} = m_0 (2 - m_0) \left[ m_f + (1-m_0)^{L_s}\right]
\label{meff_free}
\end{equation}
The expression for $G^{\rm singular}_-$ can be obtained from eq.
\ref{G_free_singular} and \ref{G_pm}. The propagator for the free
theory is then given by:
\begin{equation}
D_F^{-1} = D_F^\dagger G
\label{prop_free}
\end{equation}
To verify the above result the smallest eigenvalue of $D_F
D_F^\dagger$ is also calculated. The leading order term in an
expansion as the one used above is:
\begin{equation}
\lambda_{\rm min} = p^2 + m_{\rm eff}^2 + O(4)
\label{lamda_min_free}
\end{equation}
and is proportional to the inverse of the singular part of
$G_{\pm}$ as it should be.
The interesting result of the above analysis is eq. \ref{meff_free}.
This is the mass of the lightest mode of the theory and is controlled by
$m_f$, $m_0$, $L_s$. This formula strongly suggests the pattern for
chiral symmetry restoration in the model.
\section{The Interacting Theory, General Considerations}
\label{sec-interact}
The question on how the chiral symmetry restoration rate depends on
the various parameters of the model ultimately can only be answered
when the full theory is considered. This is done using numerical
simulations in section \ref{sec-hmc}. However, some general
considerations that hint to the expected behavior are useful in
order to guide the numerical experiments and to provide an
understanding of the results. Such considerations leading to specific
predictions are presented in this section. These predictions are
confirmed by the numerical simulations and therefore sketch the
mechanisms of chiral symmetry restoration in the model.
\subsection{The Effects of Topology}
\label{sec-topology}
In this section a special kind of trajectory
\cite{Smit_Vink}, \cite{NN1} that ``cuts''
through the various topological sectors of the configuration space
is considered. The values of the relevant observables of
section \ref{sec-observables} are calculated along this trajectory.
Consider the following set of U(1) gauge link $U$ configurations
labeled by the continuous parameter $\tau \in \Re $ \cite{Smit_Vink},
\cite{NN1}:
\begin{eqnarray}
U_1(n_1, n_2) &=& \left\{ \begin{array}{ll} 1 & \mbox{if $n_1 \neq
L-1$} \\
\exp\left[ -i {2 \pi n_2 \over L} \tau \right] & \mbox{if $n_1 = L-1$}
\end{array} \right. \nonumber \\
U_2(n_1, n_2) &=& \exp\left[ i {2 \pi n_1 \over L^2} \tau \right]
\label{U1_traj}
\end{eqnarray}
where $U_1$, $U_2$ are links defined on the sites $n_1, n_2$ of the two
dimensional torus of size $L \times L$. This configuration is
periodic along the second direction but has a discontinuity along the
first. When $\tau$ is an integer the electric field strength is
uniform $E= 2 \pi \tau / L^2$. When $\tau$ is not an integer $E$ has a
discontinuity at $(n_1-1, n_2-1)$. The topological charge of the gauge
field is defined as \cite{Panag}:
\begin{equation}
q = \sum_p { \log\left[ U_p \right] \over 2 \pi i }
\label{top_charge_gauge}
\end{equation}
where the sum is over all plaquette variables $U_p$ and the logarithm
has $\log(1) = 0$ with the cut along the negative real axis. It is
straightforward to verify that $q$ is an integer that changes values
as $\tau$ is varied between integer values (see figure $1$). For more
information on these configurations the reader is referred to \cite{Smit_Vink},
\cite{NN1}. Here it is worthwhile to point out that this trajectory
in configuration space is interesting because it connects the trivial
configuration with uniform configurations that have non zero
topological charge. As such, the configurations at integer $\tau$ can
be thought of as local minima and therefore in some sense they
represent vacua of different topological charge. The path that connects
the configurations of integer $\tau$ is certainly not unique but it
can nevertheless provide insightful information on how the transition
between sectors of different topological charge takes place.
Therefore, although one can not extract from this trajectory
quantitative information, the qualitative information will be quite
useful if, in particular, similar features are observed when the
full configuration space is considered in the HMC simulation of the
model in section \ref{sec-hmc}.
In the following, the observables in equations \ref{seff_F} -
\ref{pbp2_ov} are considered in the presence of the gauge field
configuration of eq. \ref{U1_traj}. The boundary conditions for both
fermion flavors in the two dimensional space are taken to be
antiperiodic. Of course, for the full dynamical $U(1)$ theory, the
choice of boundary conditions is irrelevant for as long as they are
the same for all flavors (they can change by $U(1)$ phases at the
boundary).
The $L_s = \infty$ quantities are calculated using the Overlap
formula. The finite $L_s$ quantities are obtained by explicit
computation of the determinants and inverses of $D_F$. The reader is
reminded that $e^{-S_{\rm eff}[L_s=\infty, m_f=0]}$ is not zero only
in the topological sector $q=0$. The quantity related to the chiral
condensate $P_1[L_s=\infty, m_f=0]$ is identically zero in all
sectors. This is a consequence of the exact $SU(2)_L \times SU(2)_R$
chiral symmetry. Therefore a non zero $P_1[L_s,m_f=0]$ at some finite
$L_s$ will indicate explicit breaking of the chiral symmetry. The
quantity $P_2[L_s=\infty, m_f=0]$ related to the t' Hooft vertex is
not zero only in sectors $q = \pm 1$. This represents the anomalous
breaking of the $U(1)$ axial symmetry.
In figure $1$ the topological charge along with $S_G$, $S^F_{\rm
eff}$, $S^{PV}_{\rm eff}$, $S_{\rm eff}$ and $S_{\rm eff} + S_G$ is
shown for $L=6$, $\mu l = 3.0$, $m_0 = 0.9$, $m_f=0$, and $L_s=14$
along the trajectory of eq. \ref{U1_traj}. The absolute scale in
these figures is of course irrelevant. One can already see the
separation of the different topological sectors with the $q=0$ sector
as the absolute minimum and the other sectors as relative minima. It
is interesting to observe how the addition of $S^{PV}_{\rm eff}$ and
$S_G$ changes the fermion action $S^F_{\rm eff}$.
In figure $2$ $S_{\rm eff}$, $P_1$ and $P_2$ are shown along the same
trajectory and for the same parameters but for various values of
$L_s$. In the $S_{\rm eff}$ figure at $\tau=2$ the curves with larger
$S_{\rm eff}$ correspond to larger $L_s$ with $L_s=4, 6, 8, 10, 12,
14$. The curve that sharply increases to $\infty$ at $\tau \sim 0.5$
corresponds to $L_s= \infty$. It is clear from this figure that in
the $q=0$ sector the $L_s=\infty$ limit is approached very fast. The
$q \neq 0$ sectors carry large actions relative to the $q=0$
action and therefore they also approximate the $L_s = \infty$ limit
very well. The only regions that suffer from a slow approach to the
$L_s = \infty$ limit are the two regions in the immediate neighborhood
where $\tau$ changes from $0$ to $\pm 1$. There, the $L_s=\infty$
result raises discontinuously but the finite $L_s$ results approach
this discontinuous behavior slowly.
In the $P_1$ figure at $\tau = 0$ the curves with smaller $P_1$
correspond to larger $L_s$ with $L_s=4, 6, 8, 10, 12, 14$. The $L_s =
\infty$ curve is identically zero for all $\tau$ and is not plotted.
From this figure it is seen that $P_1$ in the non zero topological
sectors is negligible even at $L_s=4$. In the zero topological sector
it decreases very fast with increasing $L_s$ for all values of $\tau$
except for the same two regions where $\tau$ changes from $0$ to $\pm
1$. There, two large spikes appear with heights that essentially do
not change with increasing $L_s$. Instead, their width slowly shrinks
with increasing $L_s$. As a result, chiral symmetry is slowly
restored in these regions.
In the $P_2$ figure at $\tau = 1$, the curves with larger $P_2$
correspond to larger $L_s$ with $L_s=4, 6, 8, 10, 12, 14, \infty$.
The $L_s = \infty$ curve can be distinguished by the discontinuous behavior
at the places where $q$ changes form $0$ to $\pm1$ and from $\pm1$ to
$\pm 2$. From this figure it is seen that $P_2$ approaches the $L_s =
\infty$ limit very fast for all $\tau$ except again for the regions
where $q$ changes.
The slow approach of $S_{\rm eff}$ and $P_2$ to the $L_s=\infty$ limit
in the regions of changing topological charge is not particularly
troubling since the effect is a small percent of the values that they
acquire along the trajectory. Not so for $P_1$. The two ``spikes''
present a large size contribution to a quantity that should otherwise
be identically zero. In figure $3$ $P_1$ is plotted vs. $L_s$ for
various ``cross sections'' of figure $2$. For $\tau = 0$ (diamonds)
$P_1$ decays exponentially. For $\tau = 0.3, 0.4$ (squares, crosses)
the rate of the exponential decay decreases. Finally, close to the point
where the configuration changes topological charge, $\tau \approx 0.55$
(octagons), there is almost no decay at all.
The slow approach to the $L_s = \infty$ limit in the neighborhoods of
changing topological charge $q$ is expected. According to the
topological charge definition eq. \ref{top_charge}, $q$ changes
because the number of negative eigenvalues of the single particle
Hamiltonian $H$ of eq. \ref{Hamiltonian} changes. Therefore, in that
neighborhood there are configurations for which $H$ has zero
eigenvalues. These configurations were identified in \cite{NN1} and
also discussed in \cite{Furman-Shamir}. For these configurations the
Overlap formulation is not well defined because the ground state is
degenerate and, in turn, the finite $L_s$ theory experiences large
correlations along the $2n+1$ dimension. However, the set of
configurations for which $H$ has an exact zero eigenvalue is of
measure zero. As a result the Overlap is a well defined formulation.
On the other hand regions of the configuration space surrounding these
special configurations are characterized by small decay rates. The
importance of these regions is determined dynamically and only a full
simulation of the theory can accurately probe their effect to the
exponential decay. However, close to the continuum limit one would
expect that these regions are severely suppressed since the pure gauge
action separates sectors with topological charge that differs by one
unit with barriers of energy $\sim 1/g_0^2$ (see $S_G$ in figure $1$).
But away from the continuum limit, where most numerical simulations
are performed, these regions may become important and contribute to
the explicit breaking of chiral symmetry.
In order to gain some understanding about this mode of chiral symmetry
breaking the expectation value of the chiral condensate $<\overline{\psi}
\psi>$, eq. \ref{pbp1}, was calculated in a configuration space
restricted only on the trajectory of eq. \ref{U1_traj}, i.e. $<\overline{\psi}
\psi>$ was calculated as in eq. \ref{pbp1} but with the path integral
over the gauge field configuration space replaced by an integral over
the single trajectory of eq. \ref{U1_traj} parametrized by
$\tau$. This calculation was done numerically. As can be seen from
figure 2, the integrands are relatively smooth and become negligible as
$|\tau|$ is increased above $2$. Therefore, it was enough to choose a
reasonably fine discretization and sum over a finite range only. The
summation was done for step size $d\tau = 0.025$ and over the range
$|\tau| < 2.5$. $<\overline{\psi} \psi>$ is plotted vs. $L_s$ in figure $4$
for an $L=6$ lattice, $m_0 = 0.9$, $m_f = 0$ and various gauge
couplings $\mu l / L$, $\mu l = 3.0, 2.5, 2.0 , 1.5, 1.0$ (diamonds,
squares, crosses, octagons, stars). For $\mu l = 3.0$ there is decay
with a fast rate until $L_s=10$. For $L_s = 12 $ and above there still
is decay but with a smaller rate. As can be seen from this figure the
functional form of the fast decay is exponential. The slower decay is
consistent with exponential but it turns out that it is also
consistent with power law or exponential times power law behavior.
Since the functional forms are further ``contaminated'' by higher
order effects in order to be able to distinguish between the different
types of decay it is estimated that calculations up to $L_s
\approx 30$ will be needed. This is beyond the purpose of this simple
test and the available computer resources. In any case, the important
observation is that as the gauge coupling is decreased the inflection
point moves to larger $L_s$ and the slower of the two decays becomes
faster until at $\mu l = 1.0$ there is no visible inflection below
$L_s=18$. This phenomenon can be easily understood by looking at $P_1$
in figure $2$ ($\mu l = 3.0$ there). For $L_s$ smaller than about
$10$ the explicit breaking that occurs inside the $q=0$ sector
dominates and the contribution of the ``spikes'' to the expectation
value is small by comparison. When this breaking has almost
completely disappeared, $L_s> 10$, the breaking that comes from the
``spikes'' dominates. This breaking disappears in a slower fashion as
the width of the ``spikes'' shrinks. However, as the gauge coupling
is decreased the regions of changing topology where the ``spikes'' are
located are weighted less by the pure gauge action and the slower
decay is overshadowed by the initial faster exponential decay.
Therefore, one sees that there are two distinct mechanisms that
control the restoration rate of chiral symmetry. One is related to the
restoration in the $q=0$ topological sector. The other is related to
the topology changing regions of the gauge field configuration space
and in particular to the regions that connect the $q=0$ and $q=\pm 1$
sectors.
To conclude this section the effects of $m_f$ are presented. The
observable $P_1$ along the trajectory of eq. \ref{U1_traj} is plotted
in figure $5$ for $L=6$, $\mu l = 3.0$, $m_0 = 0.9$, $m_f=0.1$
and $L_s=4, 6, 8, 10, 12, 14, 16, \infty$. At $\tau=0$ the curves with larger
$P_1$ correspond to larger $L_s$. It is clear that the $L_s = \infty$
limit is approached rapidly and with no complications. Again the
region of changing topology approaches the $L_s \rightarrow \infty$ limit
slowly but it is away from it only by a small percentage.
One would expect that the non zero $m_f$ behavior will persist down to some
small value of $m_f$ before signs of the $m_f=0$ behavior described
above appear. One can visualize how figure $5$ changes with
decreasing $m_f$. As $m_f$ becomes smaller $P_1$ tends to zero in the
various topological sectors. However, its value, at the place where the
topological charge changes from $0$ to $\pm 1$, remains roughly
constant resulting in the spikes of figure $2$ .
In order to see more clearly how the $L_s=\infty$ limit is approached for
$m_f=0.1$, the expectation value of the chiral condensate $<\overline{\psi}
\psi>$, eq. \ref{pbp1}, is calculated in a configuration space
restricted only on the trajectory of eq. \ref{U1_traj} for various
$L_s$ and for $L_s = \infty$. In figure $6$ $\left[
<\overline{\psi} \psi>_{L_s} - <\overline{\psi} \psi>_{\infty} \right] / <\overline{\psi}
\psi>_{L_s}$ is plotted vs. $L_s$. Again, one can see that there is an
inflection at about $L_s=10$. However, when the inflection occurs the
$L_s = \infty$ value has already being approached to better than $0.3
\%$. This behavior is encouraging as far as numerical simulations are
concerned. Since most numerical simulations are done for small but
non zero masses one would expect that for some range of masses the
effects of the topology changing configurations to these simulations
will generally be small and perhaps even lost in the statistical
noise.
\subsection{The Effects of Gauge Field Fluctuations}
\label{sec-fluctuations}
In this section the effects of gauge field fluctuations to the
chiral symmetry restoration rate are discussed.
The two mechanisms of chiral symmetry restoration identified in
section \ref{sec-topology} will be affected when the gauge field is
allowed to fluctuate. The effect of dynamical gauge fields to the
mechanism that restores chiral symmetry in the zero topological sector
can be seen by measuring $<\overline{\psi} \psi>$ in an ensemble of
configurations generated by applying small fluctuations to the trivial
configuration. In particular, consider gauge field configurations with
links $U = e^{i r \pi}$ where $r$ is a random number in the range
$-\epsilon < r < \epsilon$ with $\epsilon$ a small
number that controls the size of the
fluctuations. These configurations have a ``flat'' distribution and
in this ensemble $<\overline{\psi} \psi>$ is obtained by calculating $<P_1> /
<e^{-S_{\rm eff}}>$. In figure $7$ $<\overline{\psi} \psi>$ is plotted vs. $L_s$ for
various values of $\epsilon = 0.4, 0.3, 0.2, 0.1$ and $0.01$
corresponding to octagons, stars, squares, crosses, and diamonds. The
value at each point was calculated in an ensemble consisting of $40$
configurations. Antiperiodic boundary conditions have been used for
the fermions and $L=6$, $m_f=0$, $m_0=0.9$. It can be seen that as
the size of the fluctuations decreases the chiral symmetry restoration
rate increases.
Some insight to this behavior can be gained by considering the
following ``heuristic'' argument.
The Dirac operator of equation \ref{D_F} can be rewritten as:
\begin{eqnarray}
&&D_F = D\!\!\!\!/_{\rm naive} + {\cal M} \nonumber \nonumber \\
&&D\!\!\!\!/_{\rm naive}(x,x^\prime) =
{1\over 2} \sum_\mu \gamma_\mu \left[ U_\mu(x) \delta(x+\hat\mu - x^\prime) -
U^\dagger_\mu(x^\prime) \delta(x^\prime+\hat\mu - x) \right] \nonumber \\
&&{\cal M} = - B + {1 + \gamma_5 \over 2} M + {1 - \gamma_5 \over 2} M^\dagger
\label{D_F_rewritten}
\end{eqnarray}
where $B$ is the Hermitian matrix given in eq. \ref{B} and $M$ is
given by equation \ref{RDWF}. For the free theory in momentum space $D_F
= i \platslash + {\cal M}(p)$ where ${\cal M}(p)$ can be read from
eq. \ref{D_F_free}. One can think of ${\cal M}(p) $ as being a
momentum dependent mass matrix. The smallest eigenvalue of
$\cal M M^\dagger$ is obtained at zero momentum and is equal to
$m_{\rm eff}^2$ with
$m_{\rm eff} = m_0 (2-m_0) \left[m_f + (1-m_0)^{L_s} \right]$,
eq. \ref{meff_free}. The quantity $(1 - m_0)$ is the smallest
eigenvalue of the matrix $B$. When interactions
are turned on the smallest eigenvalue of $B$ will shift to values
$(1-m_0^\prime)$ larger than the free theory $(1 - m_0)$ value for the simple
reason that the matrices $U_\mu(x) \delta(x + \mu - x^\prime)$ that make up
$B$ are unitary. Therefore, the smallest eigenvalue of $\cal M
M^\dagger$ will now be ${m_{\rm eff}^\prime}^2$, $m_{\rm eff}^\prime =
m_0^\prime (2-m_0^\prime) \left[ m_f + (1-m_0^\prime)^{L_s} \right]$.
For small fluctuations one may still be able to think of $\cal M$ as a
mass matrix. If that is the case the lightest mass in the theory will
be $m_{\rm eff}^\prime$ and one sees that for $m_f=0$ the chiral
symmetry restoration rate will become faster as the fluctuations
become smaller.
Finally, the quantitative effect of dynamical gauge
fields to the mechanism related to topology changing is complicated
since it involves some understanding about the volume of configuration
space that contains gauge field configurations for which the Overlap
Hamiltonian has near zero eigenvalues. As already
mentioned, one would expect that this volume will shrink as the continuum
limit is approached because the pure gauge action introduces energy
barriers of size $\sim 1/ g_0^2$ between sectors with topological
charge that differs by one unit.
\subsection{The range of $\bf m_0$}
\label{sec-m0_range}
In this section the allowed range of $m_0$ and the effects it has in
the approach to the continuum limit are discussed.
In order for the doubler species to acquire masses of the order of the
cutoff $m_0$ must be in the range $0 < m_0 < 2$ \cite{Kaplan}.
As it was found in \cite{NN1}, in order for the transfer matrix to be
positive definite for all gauge fields, the range of $m_0$ should be
further restricted to $0<m_0<1$, eq. \ref{m0_range}. For QCD, any
value of $m_0$ in this range should lead to the same continuum limit
since the local and global symmetries of the theory remain unchanged.
On the other hand, different choices of $m_0$, will result in different
ways of approaching this limit. For the Schwinger model there is an
additional complication. As mentioned in the introduction, a
four-Fermi term is a marginal operator in two dimensions. Although it
is not explicitly introduced in the action, the DWF regularization
introduces such a term with a coefficient that depends on $m_0$
\cite{NNV}, \cite{NN2}. Therefore, different values of $m_0$ will
lead to different continuum limits. For this reason, a quantitative
study of the effects of $m_0$ to the approach to the continuum limit
in the Schwinger model is complicated and will not be done here.
However, there are some important generic features that can be
discussed.
Consider the Overlap Hamiltonian $H$, eq. \ref{Hamiltonian} as a
function of the $2n+1$ dimensional mass $m_0$ \cite{NN1}. It is easy
to see that for $m_0 = -\infty$, $H$ has the same number of positive
and negative eigenvalues, $N_+ = N_-$. It is also easy to see that
for $m_0 < 0$, $H$ can not have a zero eigenvalue. As a result $H$ has
$q = 0$, for $m_0 < 0$. As $m_0$ is increased from zero, $H$ can
develop zero eigenvalues and as a result $q \neq 0$ (eigenvalues cross
zero altering $N^-$). Close to the continuum it can be easily seen
that most crossings will occur around $m_0 = 0$ \cite{NN1}. Farther
away from the continuum pure gauge numerical simulations
\cite{NV} show a finite region $[m_{0_{\rm min}}, m_{0_{\rm max}}]$ where
most crossings occur. This suggests that one should keep
$m_{0_{\rm max}} \ll m_0$ for two reasons. First,
one would like to be as far away as possible from the region where
crossings occur since, as discussed in section \ref{sec-topology},
it is there that the decay rates become
small. Second, as can be seen from the definition of the topological
charge, eq. \ref{top_charge}, it is only then that the full effects of
non-trivial topology are visible to the fermions. In particular, one
would expect that by keeping $m_{0_{\rm max}} < m_0$ quantities that are
sensitive to topology will have a ``smoother'' approach to the
continuum limit.
From the above discussion it appears that at any coupling the safest
choice would be to set $m_0$ to its largest allowed value $m_0
\approx 1$. Even then, because in general $0<m_{0_{\rm min}}
< m_{0_{\rm max}} < 2$ it may be that far away from the continuum
$1 < m_{0_{\rm max}} \Rightarrow m_0< m_{0_{\rm max}}$.
If this situation occurs the
finite lattice spacing errors at that coupling will be large and the
chiral symmetry restoration rate will be slow. However, the
numerical simulations of \cite{NV} indicate that even for reasonably
strong couplings $m_{0_{\rm max}} < 1$. The couplings used in this
paper fall in this category (see figure $1$ in \cite{NV}) and $m_0$ is
kept fixed to $m_0 = 0.9$.
In \cite{Blum-Soni} it was found that for QCD some tuning ($m_0 =
1.7$) was necessary in order to sufficiently restore chiral symmetry
at $L_s=10$. However, the configurations used there were not
generated with the DWF action but rather with the staggered fermion
action at $\beta = 5.7$. It is possible that this large value of
$m_0$ is at least due in part to the ``semi-quenched'' nature of the
calculation and if the configurations are generated with the DWF
action setting $m_0 \approx 1$ may be sufficient.
\section{Dynamical Simulation of the Massive Theory with the Overlap}
\label{sec-overlap-sim}
In this section a full dynamical simulation that measures the chiral
condensate $<\overline{\psi} \psi>$ and the t' Hooft vertex $<w>$ of the two
flavor massive Schwinger model using the Overlap formalism of the
model of section \ref{sec-model} will be presented. The result is
interesting on its own right and it also provides the $L_s=\infty$
numbers that will be used to compare with the results of the dynamical
simulation for finite $L_s$ presented in the next section.
The expectation values $<\overline{\psi} \psi>$ and $<w>$ can be calculated
numerically using equations \ref{pbp1_ov} and \ref{pbp2_ov}
respectively. As it is evident from these equations a pure gauge
simulation with action $S_G$ must be performed with the overlap
factors appearing in the numerator and denominator treated as
observables. The pure gauge theory expectation values of the overlap
factors in the numerator and denominator are then divided to produce
$<\overline{\psi} \psi>$ or $<w>$. A heat bath algorithm was used to generate
configurations with the standard Wilson plaquette pure gauge action of
eq. \ref{action_G}.
Measurements of $<w>$ for the massless, $m_f=0$, theory were performed
in \cite{NNV}. The reader is referred there for more details on the
method and results. However, the result for $w$ in that reference can
not be compared directly with the $m_f=0$ result here since a
different implementation of the Overlap was used. Furthermore, the
single plaquette action used there was a heat kernel action while the
standard Wilson plaquette action is used here. The result for $<w> /
m_\gamma^2$ vs. $L_s$ on an $L=6$, $m_0 =0.9$, $\mu l = 3.0$ lattice
is given in figure $8$. This quantity acquires a non zero vacuum
expectation value even for $m_f = 0$ (dotted lines) as a result of the
anomalously broken $U(1)$ axial symmetry.
The result for $<\overline{\psi} \psi> / m_\gamma$ vs. $m_f$ on the same
lattice and for the same parameters is presented in figure $9$. The
behavior of $<\overline{\psi} \psi>$ vs. $m_f$ is interesting in that
$<\overline{\psi} \psi> \sim m_f$ for $m_f < 0.1$ while $<\overline{\psi} \psi> \sim
m_f^{1/3}$ for $m_f > 0.1$. Fits to $A m_f^p$ for $m_f < 0.1$ and for
$m_f > 0.1$ are shown in the same figure. Both fits have a $\chi^2$
per degree of freedom of about one. For $m_f < 0.1$ $p = 0.996(3)$
while for $m_f> 0.1$ $p = 0.32 (2)$.
This type of behavior was found by analytical continuum calculations in
\cite{Smilga-Hetrick}. In particular, the linear behavior was found to take place
for $m_f L \ll 1$ while the $m_f^{1/3}$ behavior was found to occur
for $m_f L \gg 1$. Unfortunately, the coefficients calculated in that
reference can not be directly compared with the ones here because, as
previously discussed, an additional four-Fermi interaction is
induced by the DWF regularization \cite{NNV}, \cite{NN2}. This term
in two dimensions is not irrelevant and it will contribute to the
continuum limit. For the same reasons $<w>$ can not be
directly compared with the continuum results of \cite{Smilga-Hetrick}.
\section{Hybrid Monte Carlo Simulation}
\label{sec-hmc}
In this section a full dynamical simulation of the two flavor
Schwinger model for finite $L_s$ is presented. The algorithm used is
the standard Hybrid Monte Carlo (HMC) algorithm \cite{DKPR}. The
expectation value of the chiral condensate $<\overline{\psi} \psi>$, eq.
\ref{pbp1}, is calculated and used to monitor the amount of explicit
chiral symmetry breaking. The reason for using $<\overline{\psi} \psi>$
instead of the pion mass is that in two dimensions there is no
spontaneous chiral symmetry breaking and although the pion mass
vanishes for vanishing fermion mass the pion is not a Goldstone
particle. In this sense $<\overline{\psi} \psi>$ is as good a probe of chiral
symmetry breaking as is the pion mass, but unlike the pion mass it has
the practical advantage of not requiring large lattice size along the
time direction in order to measure the decay of the pion correlator.
The effects of the anomalously broken $U(1)$ axial symmetry are
monitored by measuring the expectation value of the t' Hooft vertex
$<w>$, eq. \ref{pbp2}.
The smaller size lattices were simulated on the workstations of the
Physics Department of Columbia University. The larger lattices were
simulated on the Silicon Graphics Power Challenge Array computer
system at NCSA UIUC and also on the C90 supercomputer at PSC.
\subsection{The algorithm}
\label{sec-hmc-algorithm}
The HMC Hamiltonian is given by:
\begin{eqnarray}
H_{\rm HMC} &=& {1 \over 2} P^2 + S_G[U]
+ \chi^\dagger \left( D_F^\dagger [m_f, L_s] D_F[m_f, L_s] \right)^{-1} \chi
\nonumber \\
&+& \Phi^\dagger \left( D_F^\dagger [m_f=1, L_s] D_F[m_f=1,L_s] \right) \Phi
\label{hmc_hamiltonian}
\end{eqnarray}
where $P$ is the HMC momentum, $S_G$ is the pure gauge plaquette
action given in eq. \ref{action_G}, $U$ is the $U(1)$ gauge field,
$\chi$ is the pseudofermion field, $\Phi$ is the bosonic PV field and
$D_F$ is the three dimensional Dirac operator of eq. \ref{D_F}.
The HMC trajectory length $\tau$ is set to $\tau = 1$, the step size
is $\tau / N$ and the number of steps $N$ is adjusted
according to the size of the effective bare fermion mass. Typical simulations
with $m_f \ge 0.05$ are done with $50$ HMC steps. For the $m_f=0$ simulations
the number of steps is set to $100-400$ depending on the
size of $L_s$. This is necessary because, for $m_f=0$, trajectories that
cross between different topological sectors experience large HMC fermion
forces (for more details see section \ref{sec-hmc-topology}).
The Conjugate Gradient residual is set to $10^{-8}$. Typically,
the number of Conjugate Gradient iterations is around $50$ and does
not exceed $\sim 100$ for $m_f=0$ and $L_s=14$. The number of
measurements is $\sim 8,000$ except for the more ``expensive''
$12 \times 12$, $10 < L_s$ lattices where $1,000 - 2,000$
measurements were performed.
\subsection{HMC and topology}
\label{sec-hmc-topology}
In this section problems related to the sampling of non zero topological
sectors with the HMC algorithm at small fermion masses are discussed.
The HMC algorithm is very successful provided the fermion mass does
not become very small. If the fermion mass becomes very small then the
effects of topology are not reproduced correctly. In particular, the
measurement of $<w>$ becomes problematic. If, for some fixed
volume, the fermion mass is made very small then
similar analysis as in \cite{LS} indicates that the effect of the zero
modes coming from different topological sectors becomes important. In
particular, as can be seen from the Overlap implementation \cite{NN1},
\cite{NNV}, at finite physical volume, at zero mass and in the $L_s
\rightarrow \infty$ limit, the operator $w$ receives contributions only
from sectors $\pm 1$, while the fermion Boltzman weight (fermion
determinant) is not zero only in sector 0. As $m_f$ is turned on
(and/or $L_s$ is decreased from infinity) the fermionic determinant
becomes non zero in sectors other than $0$ and the operator $w$
receives contributions from sectors other than $\pm1$.
Because of this behavior at
small fermion mass the HMC algorithm will mostly sample the sector $0$ where
the observable $w$ receives small contributions. The algorithm will
infrequently visit the sectors $\pm 1$, but when it does the observable
$w$ will receive large contributions to ``make up'' for the small sampling
rate. As a result, when the fermion mass is decreased a larger number of HMC
iterations will be needed to sample the $\pm 1$ sectors correctly.
Therefore, for a fixed amount of computer time, if the fermion mass
becomes very small the important contributions may not even be sampled
at all and as a result not only the expectation value of $w$ will be
underestimated but also the associated error. This type of difficulty
has already been noticed in simulations of the Schwinger model
\cite{hmc-difficulties}, \cite{PMV}.
The problem described above leaves a clear signature in the time history
of $w$. In figure $10$ the time history of $w$ is given for six
different fermion masses at $L_s=14$. As the fermion mass is decreased, the
average around which $w$ fluctuates decreases. This decrease
is compensated by the large contributions received from the
$q=\pm 1$ sectors. These contributions start to appear as ``spikes'' in the
time history. The smaller the fermion mass the larger the ``height''
of the spikes but since the corresponding Boltzman weight becomes
smaller their frequency also decreases. Notice the different
scale of the $m_f=0.05$ and $m_f=0.0$ graphs.
But it is not only $<w>$ that is affected at small fermion masses. If,
along an HMC trajectory, the topological charge changes then the
fermion determinant changes by a large amount. As a result, the HMC
fermion force becomes large and the HMC step size errors become
large. Therefore, as the fermion mass is made smaller one must
adjust the HMC step size accordingly.
Finally, it should be stressed that these difficulties are not
related to DWF but rather to the HMC algorithm. In particular, the
simulations using the overlap in section \ref{sec-overlap-sim} do not
suffer from these problems since there the fermion determinant is
treated as an observable in a pure gauge heat bath.
\subsection{$\bf m_f = 0$}
\label{sec-hmc-mf-zero}
In this section a strict test of chiral symmetry restoration is
done.
A full dynamical simulation using the HMC algorithm is performed with
the explicit fermion mass $m_f$ set to zero so that the only breaking
of chiral symmetry comes from the finite extent $L_s$. The restoration
rate at fixed physical volume and various lattice spacings is studied
by measuring $<\overline{\psi} \psi>$ for various values of $L_s$. Although
this provides a strict test one must keep in mind that the non-zero
topological sectors may be suppressed more than they should for the
reasons mentioned in section \ref{sec-hmc-topology}. This means that
the rate of restoration of chiral symmetry observed here is mainly due
to effects that occur in the zero topological sector and its vicinity.
The following results were obtained at fixed $m_0=0.9$.
In figure $11$, $<\overline{\psi} \psi> / m_\gamma$ is plotted in a ``log''
plot vs. $L_s$ at fixed physical volume $\mu l = 3.0$ and for various
lattice spacings $\mu l / L = \mu a$ where $L=6, 8, 10, 12$
corresponding to the lines from top to bottom. Data for $L=4$, $L_s =
6 - 10$ are statistically indistinguishable from the $L = 6$ data and
are not plotted. For $L_s = 6 - 10$ the decay is consistent with
exponential with a rate that becomes faster as the lattice spacing
decreases. For $L_s = 12-22$ the decay is again consistent with
exponential but with a slower rate. Again, this rate becomes faster as
the lattice spacing decreases. Also, the percent change of the rate
at $L=6$ is $\approx 54 \pm 6\%$ but at $L=12$ is $\approx 31 \pm
6\%$. The fits shown are two parameter fits to $<\overline{\psi} \psi> /
m_\gamma = B e^{ -c L_s}$. The $\chi^2$ per degree of freedom is
smaller than one for all the fits except for the $L=12$, $L_s=6-10$
data that have a $\chi^2$ per degree of freedom of $\approx 3$.
The exponentiated rate $e^{-c}$ of the various fits vs. $1/L \sim a$
is shown in figure $12$. The diamonds correspond to the $L_s=6 - 10$
fits while the crosses to the $L_s = 10 - 22$ fits. One can see that
$e^{-c}$ is roughly a linear function of $1/L \sim a$ for the $L_s =
6-10$ fits and for $L=8, 10, 12$. However, more data at smaller
lattice spacings are needed before one can be confident that scaling
has set in and that this is the correct scaling form.
Although the above fits are all consistent with exponential decay,
power law decay of the form $<\overline{\psi} \psi> / m_\gamma = B {L_s}^{-p}$
can be excluded with some confidence only for the fast decay, $L_s=6-10$,
at the smallest lattice spacing, $L=12$. A power law fit
to this data has $\chi^2$ per degree of freedom of $\approx 32$. More
statistics and larger $L_s$ will be needed in order to clearly
establish the type of decay for the other cases. For example, at the
largest lattice spacing, $L=6$, it is estimated that the error bars
will have to be reduced from their few percent size down by a factor
of about ten. Alternatively, the error bars can be kept at the few
percent level but then it is estimated that $L_s$ will have to be made
as large as $\approx 30$. Both approaches are beyond the purpose of
this paper and the available computer resources.
The facts that the decay changes for $L_s > 10$, that the slower decay
approaches the faster one as the continuum limit is approached and
that both decays become faster as the continuum limit is
approached can all be understood from the analysis in sections
\ref{sec-topology} and \ref{sec-fluctuations}.
Finally, the t' Hooft vertex $<w>$ was also measured but, as expected
from the discussion in section \ref{sec-hmc-topology}, its value and
associated error is underestimated. In particular, its value is much
lower than the corresponding Overlap value of section
\ref{sec-overlap-sim}. For this reason that data is uninteresting and
is not presented here.
\subsection{$\bf m_f \neq 0$}
\label{sec-hmc-mf}
In this section the $m_f \neq 0$ case is studied. Since
typical QCD simulations are done for non zero fermion mass
the results of this section are of practical interest.
Dynamical simulations are performed with masses $m_f$ large enough,
$0.1 \le m_f$, so that the effects of topological sectors $q=0, \pm 1$
are not miscalculated due to problems associated with the HMC
algorithm as described in section \ref{sec-hmc-topology}. Both
$<\overline{\psi} \psi>$ and $<w>$ are measured and their approach to the $L_s =
\infty$ limit is studied and compared with the $L_s=\infty$ results of
section \ref{sec-overlap-sim}. This is done for fixed physical volume,
and various lattice spacings. The parameter $m_0$ is kept fixed at
$0.9$.
In figure $13$ $\left[ <\overline{\psi} \psi> / m_\gamma \right]^3$ is plotted
vs. $L_s$ for $m_f = 0.1, 0,2, 0.3, 0.5$ at fixed physical volume and
lattice spacing, $\mu l = 3.0$, $L=6$. According to the results in
section \ref{sec-overlap-sim} one expects that $\left[ <\overline{\psi} \psi>
/ m_\gamma \right]^3 \sim m_{\rm eff}$. Therefore a fit of $\left[
<\overline{\psi} \psi> / m_\gamma \right]^3$ vs. $L_s$ is made to the form $A
+ B e^{-c L_s}$. All fits have a $\chi^2$ per degree of freedom
$\approx 1-2$. In these figures the cross is the coefficient A and
the dotted lines are the $L_s = \infty $ result of figure $9$
plus/minus the error. One can see that as $m_f$ becomes larger the
$L_s = \infty $ result is approached faster. This is in accordance with
naive expectations born out from the free theory formula for $m_{\rm
eff}$ eq. \ref{meff_free}. In figure $14$ $<w> / m_\gamma^2$ is
plotted vs. $L_s$ for the same parameters as in figure $13$. The fits
are again to a form $A + B e^{-c L_s}$ and have $\chi^2$ per degree of
freedom $\approx 1-2$. One can see that the $L_s=\infty$ result has
already been approached at $L_s=6$.
The effects of changing the lattice spacing at $m_f=0.2$ can be seen
in figures $15$ and $16$. In figure $15$ $\left[ <\overline{\psi} \psi> /
m_\gamma \right]^3$ is plotted vs. $L_s$ at fixed physical volume $\mu
l = 3.0$ for different lattice spacings $\mu l / L = \mu a$, $L=4, 6,
8, 10$. The fits are again to a form $A + B e^{-c L_s}$ and have a
$\chi^2$ per degree of freedom of $\approx 1-2$. One can see a
similar behavior as the one in section \ref{sec-hmc-mf-zero}. As the
lattice spacing is reduced the rate $c$ of the exponential approach to
the $L_s=\infty$ result increases. For example, $<\overline{\psi} \psi>$ at
the larger lattice spacing $L=4$ decays with $c=0.54(3)$ but at the
smaller lattice spacing $L=10$ it decays faster with $c=1.1(1)$.
However, one should note that for $L=8$ and $L=10$ the rate saturates
and is essentially dictated by the $L_s=4, 6$ points with the $L_s=6$
point very close to the $L_s=\infty$ result. If a second slower rate
sets in for $10 \mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} L_s$ it is unimportant and is lost in the
statistical noise. Some insight to this behavior can be gained from
the analysis at the end of section \ref{sec-topology} (in particular
see figure $6$). Similar behavior is observed in figure $16$ for $<w>
/ m_\gamma^2$. All fits have $\chi^2$ per degree of freedom $\approx
1-2$.
Similar results are obtained if one keeps the physical volume
and $m_f$ in physical units fixed while changing the lattice spacing.
This can be seen in figures $17$ and $18$ by comparing the $L=10$,
$m_f=0.2$ data (diamonds) with data at $L=4$, $m_f=0.5$
(squares). In these graphs the physical volume is fixed at $\mu l =
3.0$, and $m_f$ in physical units is fixed at $m_f L = 2.0$.
Again the decay rate increases as
the lattice spacing is reduced. $<\overline{\psi} \psi>$ at the larger
lattice spacing $L=4$ decays with $c=0.48(5)$ but at the smaller
lattice spacing $L=10$ decays faster with $c=1.1(1)$. All fits
have $\chi^2$ per degree of freedom $\approx 1-2$.
Finally, it should be noted that although the above data is consistent
with exponential decay, other types of decay can not be excluded. This
is mainly due to the fact that since the $L_s=\infty$ result has
already being approached, within statistics, for $L_s=8-10$, the decay is
basically dictated only by the two points $L_s=4,6$. More statistics
are needed in order to clearly establish the type of decay.
\section{Conclusions}
\label{sec-conclusions}
In this paper the properties of Domain Wall Fermions (DWF) were
studied in the context of the two flavor Lattice Schwinger model. The
expectation value of the chiral condensate $<\overline{\psi} \psi>$ was used
to probe issues related to restoration of the regularization induced
chiral symmetry breaking. The expectation value of the relevant t'
Hooft vertex $<w>$ was used to probe issues related to topology.
Dynamical numerical simulations of the full theory were performed. It
was found that, as expected from perturbative considerations, the
restoration of chiral symmetry as a function of $L_s$ ($L_s$ is the
size in lattice units of the $2n+1$ direction) at a fixed physical
volume and lattice spacing, is consistent with exponential decay. In
particular, the data is consistent with a picture where chiral
symmetry is restored with a fast exponential decay rate for $L_s$ up
to some value and with a slower exponential decay rate for $L_s$ above
that value. For the range of lattice spacings used in this paper the
inflection appeared at $L_s \approx 10$. Using a simple model it was
found that the first fast decay can be associated with restoration of
chiral symmetry in the zero topological sector while the second slower
decay can be associated with the regions of gauge field configuration
space that connect the $q=0$ and $q=\pm 1$ topological sectors.
The effects of the size of the lattice spacing $a$ to the two decays
were studied using both analytical arguments and explicit numerical
simulations of the full theory. It was found that for zero explicit
fermion mass the fast decay associated with the zero topological
sector becomes faster as the lattice spacing is decreased. The
vanishing of the chiral condensate is consistent with a form $e^{-c
L_s}$ with $e^{-c}$ being roughly a linear function of $a$, but more
data at smaller lattice spacings are needed before one can be
confident that scaling has set in and that this is the correct scaling
form. The second slower decay also becomes faster as the lattice
spacing is decreased and it differs less from the faster decay as the
lattice spacing becomes smaller. For the smallest lattice spacing
studied the two exponential decay rates differed by
$\approx 31 \pm 2 \%$.
For small but non zero explicit fermion mass $m_f$ the values of
$<\overline{\psi} \psi>$ and $<w>$ were measured. The corresponding $L_s =
\infty$ numbers were calculated by performing numerical simulations
with the Overlap formalism. It was found that the $L_s = \infty$
numbers were also approached in a way that is consistent with
exponential decay with a rate that became faster as the lattice
spacing decreased. Furthermore, the larger the fermion mass the sooner
the $L_s = \infty$ value was approached and for the fermion masses
studied in this paper the $L_s = \infty$ result was already achieved
to within a few percent for $L_s = 4 - 8$. If a second slower decay
does set in for $10 \mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} L_s$, it is unimportant and was lost in
the statistical noise. Finally, an interesting result was obtained
from the measurements of $<\overline{\psi} \psi>$ and $<w>$ vs. $m_f$. It was
found that these measurements are in agreement with the analytical
predictions of \cite{Smilga-Hetrick}. In particular the interesting
$<\overline{\psi} \psi> \sim m_f^{1/3}$ behavior was reproduced.
Although all the numerical data are consistent with exponential decay,
power law decay can be excluded only for the fast decay at the
smallest lattice spacing studied. For that data the decay is
sufficiently fast and the error bars are sufficiently small so that a
power law fit can be safely excluded since it has a $\chi^2$ per degree
of freedom $\approx 31$. More statistics and larger $L_s$ are
needed in order to be able to clearly distinguish between exponential
and power law decay for the rest of the data points.
The next step is to carry out a similar investigation for dynamical
QCD. Many of the characteristics of DWF found here are sufficiently
generic so that one would expect that they will also be present in
QCD. If it turns out that QCD at the presently accessible lattice
spacings, volumes and quark masses has similar restoration rates as
the ones found here, then DWF will indeed provide a powerful fermion
discretization method.
\section*{Acknowledgments}
The author would like to thank N. Christ, R. Mawhinney, R. Narayanan
and H. Neuberger for many useful discussions. This research was
supported in part by the DOE under grant
\# DE-FG02-92ER40699. This work was also partially supported by the National Center for
Supercomputing Applications under grant \# PHY970002N and utilized the
Silicon Graphics Power Challenge Array computer system at the National
Center for Supercomputing Applications, University of Illinois at
Urbana-Champaign. Also, this research was supported in part by grant
\# PHY960005P from the Pittsburgh Supercomputing Center and
utilized the C90 supercomputer.
|
2,877,628,090,099 | arxiv | \section{Introduction}
\label{sec:intro}
Shared opinions are an important feature in the formation of social groups~\cite{kruglanski2006groups}. It has been shown that clusters of opinions become signifiers of group identity~\cite{Bliuc2007}.
In recent studies, public health opinion-groups have been shown to coalesce around a growing trust/distrust in science, with those having distrust being less compliant with regards to hand-washing and maintaining distance~\cite{maher2020mapping}. This sort of behaviour has major consequences for public health compliance~\cite{vaughan2009effective}.
As a result, it is important to be able to identify such groups accurately, and to reveal if different opinion-based groups are, or will become, polarised on the clusters of topics they share.
In online communities, such as Facebook groups or subreddit memberships, mutual interests in a subject, or attitudes, are often the primary shared commonality, rather than prior acquaintanceship or geographical proximity. It has been found that in many online communities, users tend to share media aligned with their own values and dismiss alternative views \cite{bakshy2015exposure}. These groups tend to be driven by homophily~\cite{mcpherson2001birds}.
In this study, we will use this idea of shared attitudes to uncover opinion-based groups from surveys.
In a survey, a participant provides responses on many topics with only a small number of possible response options. These responses are typically on an ordinal scale (e.g., a Likert scale) \cite{DeVellis2003}. Often the scale has a limited range of discrete response options, for example five-point and seven-point scales are commonly employed \cite{groves2011survey}. We use a distance metric, akin to the Manhattan distance, on these scales across all the survey questions, referred to as items, to identify participants with similar opinions. Thus we construct a network of participants linked by shared opinions. This method is discussed in more detail in Ref.~\cite{MacCarron2020}.
There are methods for investigating the social structure and belief networks, see for example~\cite{boutyline2017belief,brandt2019central}. However, in these methods the network is constructed by placing edges between pairs of nodes with a correlation above a certain threshold. This makes the edges difficult to interpret and the choice of threshold is arbitrary. In our approach, we introduce a cut-off when a giant component is formed containing almost all participants. An edge represents shared agreement; the stronger the weight of the edge, the more agreement between these participants.
We use community detection techniques to identify clusters of participants with similar opinions, i.e., opinion-based groups. We compare this to statistical methods, such as hierarchical clustering on the refined survey data and show they give consistent results with each other and, hence, this is a viable method for detecting opinion-based groups and polarisation.
Surveys can contain hundreds of items, many of which are not expressing attitudes but answering trivial questions leading up to an attitudinal item. We aim to select attitudes, which are closely linked to attitudes of the identified clusters. To do this we apply two feature selection methods to either identify or rank the most relevant items. Ranking the items allows us to reduce the number of items and to highlight influential items.
In this paper we lay out a novel approach on how to remodel attitudinal survey data in order to identify opinion-based groups, applying three different community detection algorithms. We present a new item rank method, which ranks and identifies the items’ importance for an opinion-based group structure. Finally, we demonstrate how to apply this approach to existing data sets.
The paper is laid out as follows, in section~\ref{sec:methods}, we outline the method for forming the networks, identifying the clusters and the feature selection methods for picking the relevant questions. In section~\ref{sec:results}, we show the results and identify the community detection algorithms as robust methods for detecting the opinion-based groups in similarity networks.
Finally in section~\ref{sec:conclusion}, we discuss the results, give concluding remarks and discuss further research avenues.
\section{Methods}
\label{sec:methods}
The detection of opinion-based groups using survey data is conducted in a multi-step procedure. This can be broadly broken down into data restructuring, construction of similarity-based networks~\cite{MacCarron2020}, community detection~\cite{Girvan2002,Karrer2011,Murtagh2011} and item importance.
Based on survey data, the method creates links between individuals by constructing a similarity-based network. The emergent structure can reveal opinion-based groups and predict social-group formation \cite{Babeanu2018,Breiger2014}. Once we detect these opinion-based groups, our approach provides a method to rank each item's influence on the opinion-based group structure.
Survey data often covers multiple contexts with a large number of items. Hence, a subset of items has to be chosen depending on the subject matter of interest. For example, if we focus on political polarisation, then we are interested in identifying politically relevant items which cover attitudes related to party alignment.
To uncover these attitude connections, we employ a method to project survey data as a similarity network based on the answers of participants. In the resulting network, nodes are participants and weighted links are the similarity scores between participants.
Ref.~\cite{MacCarron2020} shows that the network visualization provides information about groups of individuals that share similar opinions. However, the visualisation and the distinction of groups in the network is highly dependent on layout algorithms, chosen by the user (in \cite{MacCarron2020}, the Kamada-Kawai layout algorithm~\cite{Kamada1989} was used). A common way in complex networks to partition a graph is the application of community detection algorithms~\cite{Fortunato2010}.
The use of community detection algorithms in our context has the benefit that they do not rely on the visual inspection of the network and that it takes the approach one step further: to reliably
uncover opinion-based groups.
Over the last two decades a range of different community detection algorithms have evolved (see, for example, \cite{Barabasi2016}). Based on the high complexity of this challenge, there exists no generally applicable algorithm~\cite{Fortunato2010}. The choice of using three distinct algorithms ensures the performance and robustness of the community detection.
Initially we apply the Girvan-Newman algorithm \cite{Girvan2002}, which uses the edge betweenness centrality to minimise the cross-cutting links between communities.
We then run the statistical-driven Hierarchical Clustering algorithm~\cite{Murtagh2011} and finally the Stochastic Block Model used for community detection~\cite{Karrer2011}.
In the following sections, we explain our approach step by step.
Although we will later show, using extensive simulations, that our method provides robust results, in order to illustrate the application we first run through a specific example: the American National Election Study (ANES) from 2016~\cite{ANES2017}.
This large data set captures a broad range of general and political attitudes from the American people and includes over 4000 participants and more than 650 items.
We aim to detect opinion-based groups and polarisation in the data set. As an example the ANES data set delivers an ideal candidate to reconfirm polarisation.
Although the ANES data set is not intended to reveal opinion-based groups or polarisation, it captures the particular structure of the American two-party system, which is perceived as bipolar~\cite{Abramowitz2018,Fiorina2008,Iyengar2019}. We take this party alignment as a reference group for community detection and polarisation in this data set.
For our method, the ANES data set is suitable to investigate polarisation~\cite{Baumann2021}. The first step is to project the survey data as a network.
\subsection{Identifying opinion-based groups from survey data: a score-based linking method}
Attitudinal survey data provides the basis for a network, using the individuals as nodes and their similarity score as links.
The scales of the items are reformatted into a range between $-1$ and $1$. For instance, a seven-point scale will then be defined as a scale with values of $-1$, $-2/3$, $-1/3$, $0$, $1/3$, $2/3$ and $1$. The scale represents a clear ordinal structure. The reformatting is applied to the whole data set.\\
The similarity measure $S_{ij}$ between the individuals, $i$ and $j$, is the sum of differences between all $n_{f}$ answers to the items $q_n$ (i.e., the Mahattan distance).
\begin{equation}
S_{ij} = n_{f} - \sum_{f=1}^{n_f} |q_{if}-q_{jf}|.
\end{equation}
The similarity measure ranges between $-n_{f}$ and $n_{f}$. This is to support comprehension, so that an edge with a value of $n_{f}$ displays full agreement between two nodes on all items.
The similarity measure $S$ is at its maximum $n_{f}$, if two individuals have identical responses to all items. Links are drawn, where the similarity exceeds a threshold $\theta$, which is chosen
when a giant component is formed. Its success criterion is fulfilled if there are enough links in the network to build a giant component, where a path can be drawn to at least 80\% of the individuals. To achieve this, the threshold will successively be lowered until the network matches the success criterion. This means that the procedure includes stepwise links with a lower similarity score.
While the integration of the threshold reduces the number of included individuals, it also reduces the total number of additional links. After these three steps, the data can be shown as a network in order to identify opinion-based groups.
\begin{table}[H]
\centering
\begin{tabular}{l|c|c}
Item & Label & Answer range \\ \hline
Abortion & V161232 & 1-4 \\
Race relations & V161198 & 1-7 \\
Immigration & V161192 & 1-4 \\
Welfare & V161209 & 1-3 \\
Homosexuality & V161231 & 1-3 \\
Business & V161201 & 1-7 \\
Guns & V161187 & 1-3 \\
Income & V161189 & 1-7
\end{tabular}
\caption{American National Election Study 2016 - Selected item and their answer range. For a detailed description of the items, see \ref{appendix:ANES_var_descrip}}
\label{tab:ANES2016}
\end{table}%
For the ANES data set, we identified eight items based on a study from \cite{Malka2014} to measure political attitudes. Malka and colleagues' evaluations rely on cultural, economic and self-reported political ideology attitudes.
We then run the data refinement and the network construction on these eight selected items (see Table~\ref{tab:ANES2016}).
We removed individuals who did not answer all eight items and so our maximum network size here is 3,081 nodes. With a threshold of 7.0, we get 50,143 links between 2,714 individuals, forming a giant component (88.1\% of individuals), where all individuals are connected (see Figure~\ref{fig:ANES2016}).
\begin{figure*}
\centering
\makebox[0pt]{
\includegraphics[width=1.1\textwidth]{anes2016_18_02_2021_GN_graph_partition_new_colouring_no_communities.png}}
\caption{American National Election Study data 2016, constructed similarity network from the refined data set. The nodes' colour marks the self-identified party affiliation: Republican (red), Democrat (blue) or unknown/independent (yellow).}
\label{fig:ANES2016}
\end{figure*}
In our next step, we introduce the community detection for identifying possible opinion-based groups in our network.
\subsection{Detecting opinion-based groups}
Community detection in graphs is an active and ongoing field of research in and of itself, see for example \cite{Fortunato2010,Girvan2002,Javed2018}. Currently there exist a large variety of algorithms to detect group structure from network characteristics~\cite{Fortunato2010,Fortunato2016}.
In our analysis, we have chosen three different approaches: Girvan-Newman community detection, Hierarchical Clustering and the Stochastic Block Model. The Girvan-Newman algorithm is a network-based method, which is directly applied to our constructed network. In general, Hierarchical Clustering is applied on the refined data set. The Stochastic Block Model is an inference algorithm which detects communities by model fitting. Detailed descriptions of these can be found in the Supplementary Information.
\subsubsection{Within Sum of Squares}
The Within Sum of Squares (WSS) forms a building block of multiple parts of this analysis, for example, comparing the identified communities of the community detection methods. It is the sum of the squared distance of each individual from their assigned cluster centres.
We can calculate the WSS as follows:
\begin{equation}
WSS := \sum\limits_{k=1}^{n_k} \sum\limits_{i\epsilon C_{k}}\sum\limits_{f=1}^{n_f} (q_{if} - \overline{q}_{kf})^2,
\end{equation}
where the number of clusters is $n_k$. $C_k$ is the set of individuals in cluster $k$, $q_{if}$ is individual $i$'s response to item $f$ and $\bar{q}_{kf}$ = $\sum_{i\in C_k} q_{if}/|C_k|$ is the average answer to item $f$ in cluster $k$.
The goal of our three community detection methods is to reduce WSS substantially while using the least number of communities possible. For two different community assignments, but with the same number of communities, the community with the lower WSS fits better to the data. The preferred partition contains communities, where, on average, the distance between individuals to others in their community is smaller.
With the WSS, we can generate an elbow plot (see \ref{appendix:elbowplot}) for the communities, determined by our community detection methods. An elbow plot displays the WSS versus the number of communities and gives information about the ideal number of communities in the data~\cite{Yuan2019}.We could use the number of parties that people self-identify as our optimal number of communities. However, the optimal community structure could contain sub-groups within these partisan groups; for instance, we might observe that the community structure is well explained by Republicans or Democrats that are 'centralist', or 'left' and 'right' of the centre, for instance. The elbow plot provides a method for exploring this optimal number of communities. The ``elbow'' in the plot indicates a striking mark for the curve. Successively adding clusters to the data will reduce the total WSS. If the reduction is exceptionally high for an additional cluster, it gives the hint that this might be the ideal number of clusters for the data \cite{Bholowalia2014}. In this way adding more clusters to the data will lead to comparatively small changes in the curve (see in (\ref{appendix:elbowplot}), Figure~\ref{fig:example_elbow}).
\subsubsection{Girvan-Newman algorithm}
The Girvan-Newman algorithm was one of the first community detection algorithms in complex networks \cite{Girvan2002}. This top-down approach divides the network into communities by successively removing links with the highest edge betweenness centrality. Using the edge betweenness centrality, the algorithm intends to identify the community bridging links. It is based on the assumption that links between communities have a higher edge betweenness centrality, caused by their linking ability.
Once the cross-cutting links are identified, they relate two opinion-based groups and mark an attitudinal intersect. The nodes holding these links are positioned at the border of the opinion spaces. Hence, links with a high edge betweenness centrality mark regions where the participants are in the middle of two opinion spaces. The Girvan-Newman algorithm detects and removes cross-cutting links, conceptually dividing the opinion space into smaller, more internally intertwined opinion-based groups.
Also, the edge betweenness centrality is used to measure within community polarisation~\cite{garimella2018quantifying}.
Our goal is to detect polarisation in the ANES data set from 2016. We run the Girvan-Newman algorithm on the constructed network until it splits into two communities.
In order to obtain a statement about the overall structure, we re-compute the Girvan-Newman community detection on the biggest community if the first division results in one very small community (less than 5\% the size of the larger community). Applying the Girvan-Newman algorithm to the ANES data set resulted in communities of 41 nodes and 2673 nodes. Running the algorithm again on the larger community, only 190 edges needed to be removed in order to split this community into two communities of 1818 and 855 nodes (see Figure~\ref{fig:ANES2016_GN}). It can be seen that the larger of these components corresponds mostly to democrats (blue nodes) and the smaller to republicans (red nodes).
\begin{comment}
method community_0_size community_1_size overlap_with_gn overlap_in_perc_0 overlap_in_perc_1
sbm 2255 459 0.621399177 0.566030677 0.0553685
HC 1737 936 0.879910213 0.604938272 0.274971942
GN 1818 855 1 0.68013468 0.31986532
\end{comment}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{anes2016_18_02_2021_GN_graph_partition_new_colouring_2_communities.png}
\caption{American National Election Study data 2016, constructed similarity network from the refined data set. With help of the Girvan-Newman algorithm the network is separated into two communities. The purple links are the eliminated links between the communities, and are not part of the network anymore.}
\label{fig:ANES2016_GN}
\end{figure*}
\subsubsection{Stochastic Block Model for community detection}
An algorithm for broad range of applications to produce a model to generate networks with community (block) structure is called the Stochastic Block Model~\cite{Holland1983}.
The model, based on statistical inference, describes the link formation as a process that takes place more often within than between communities.
The community detection is viewed as a challenge of fitting the Stochastic Block Model to a network in order to reveal a probability-based community structure. Based on this, through an integrated optimisation process a suitable Stochastic Block Model candidate is selected. The flexibility of the Stochastic Block Model means that there exist a variety of approaches for applying and configuring it~\cite{Fortunato2016}. Besides the flexibility, another advantage is the computational complexity in $O(N\ln{N})$~\cite{Fortunato2016}, and therefore the speed of execution is fast compared to the Girvan-Newman algorithm. One drawback of this method, in comparison to the Girvan-Newman algorithm and the Hierarchical Clustering method, is that is is built on stochastic computation. Multiple runs of this method may yield different communities for the same network. It is also not guaranteed that the result is the optimal solution. Nonetheless, Fortunato and Hric \cite{Fortunato2016} assess the Stochastic Block Model as a strong candidate for community detection.
In our approach, we use an algorithm
in the Python module \textit{graph-tool}~\cite{peixotolibrary2014}. The algorithm provides a degree-corrected or a non-corrected version, which, in our case, will be run according to the minimal description length criterion as described in \cite{Peixoto2017}.
This function uses an agglomerative heuristic, the Markov Chain Monte Carlo algorithm, for optimisation \cite{Peixoto2014}. The core of the function is a one-dimension minimisation based on the golden section search. More details about the algorithm and its variants can be found in Refs.~\cite{Karrer2011,Peixoto2014,Peixoto2017}.
\subsubsection{Hierarchical Clustering}
The Hierarchical Clustering method is applied directly to the data set, i.e., without constructing a similarity network. The core of analysis is a distance matrix which contains every distance between the individuals. The distance projects the dissimilarity in their answers over all items. In an iterative process the Hierarchical Clustering merges individuals by aggregating the most similar clusters together, which is decided by a linkage function. In this case, we use a 'group average' linkage function \cite{Everitt2011}. This computes the average of the distance between people in different clusters and aggregates the closest. This leads to the interpretation that the people's clusters who are, on average, close in their opinion are aggregated together initially. Additionally, it is considered to be a intermediate version of the single and compete linkage methods and is relatively robust to outliers. The more common Ward linkage function, which aggregates clusters together that increase the within cluster sum of squares the least, tends to find spherical clusters and is sensitive to outliers \cite{Everitt2011}. However, other linkages functions produce similar clustering results.
\begin{comment}
An end product of the Hierarchical Clustering is the dendogram where the hierarchical structure of the data is laid open. In detail, it shows which clusters were merged at what stage in a tree-like graph. The dendrogram gives no hint about the ideal number of clusters, it merely displays the distances which separate the clusters from each other and the cluster assignment of each node.\\
Our application example, the ANES data set 2016, gives an impression on how a typical dendrogram would look like (see Figure~\ref{fig:dendrogram_ANESpre2016}).
\begin{figure*}
\centering
\includegraphics[width=10cm]{Dendrogram_ANES_2016_pre_all.png}
\caption{Output of the Hierarchical Clustering method: Data structure of ANES data set 2016 revealed by a dendrogram. In more detailed versions, the x-axis shows the names of the individuals. The y-axis shows the accumulated distance for clusters to merge together.}
\label{fig:dendrogram_ANESpre2016}
\end{figure*}
\end{comment}
The comparison of the three community detection methods arises from the need to choose the ideal number of communities. One approach is to compute a measurement which takes the distances of the answers in each community, the Within Sum of Squares (WSS), into account. The WSS makes it possible to quantify the variability between individuals for a given community assignment. With it, we are able to compare the three methods and, additionally, decide which is the ideal number of communities.
\subsection{Selecting relevant items}
Selecting relevant items from large data sets is an important component of our method. Often, to reduce complexity and to include only relevant items, a selection step for the items must be made \textit{a priori}. Therefore, a tool for distinguishing between influential and noisy items would be beneficial to assess the item selection and moreover, rank them in relation to their influence on opinion-based group structure.
The responses of the survey data constitutes a corresponding vector of opinions for every individual. The differences in their responses form our network structure and opinion-based groups. After the determination of community assignments, we introduce a measurement to locate the relevant items for this particular community structure. Thus, every item is ranked by their meaningfulness.
The basic concept consists of randomly selecting an item from a data set, shuffling the responses and reallocating them to the individuals. Through this, we break possible correlations to other items and influence on the community assignment if one exists.
The method is build up as follows:
\begin{enumerate}
\item The WSS is calculated to obtain a reference value. The calculation is based on the Girvan-Newman, Stochastic Block Model or Hierarchical Clustering community assignments.
\item At random the method chooses one item and modifies the data set. Consequently, all features are like in the original data set but answers of the selected item are now shuffled.
\item On the basis of the community assignment,
a new WSS is computed. In an additional step, we calculate the ratio of the difference between the old and the new WSS.
\item To make a reliable statement about the item ranking, the procedures in 2 and 3 is repeated $M$ times per item. In the end, the mean of all WSS-differences is taken to assess each item.
\item Finally, a value for each item determines the average percentage change of the WSS. Whereas, a higher value means higher influence on the community assignment and values near zero suggest no influence on community assignments.
\end{enumerate}
The results of the method can be used to produce a violin plot\footnote{It works similar to a common box plot: it marks the median for the WSS-difference for each item, displays the interquartile range and it draws the distribution for WSS-differences using a kernel density estimation.} (see Figure~\ref{fig:ANES2016_violinplot}).\\
Following our example, we computed the item rank method to evaluate the items influence on the community assignments. We ran our method on the eight items from the ANES data set 2016 and simulated it 1000 times, so that, on average, each WSS-distance distribution is based on 125 shuffles of that item. It shows that the item \textit{Welfare} ($V161209$) had the highest and the item \textit{Immigration} ($V161192$) the lowest influence.
\begin{figure
\centering
\includegraphics[width=1.1\columnwidth]{Violin_plot_changed_names_ANES2016.png}
\caption{Item selection: Violin plot for the eight items from the ANES data set 2016. It shows the distribution of the average percentage change between the original WSS and the recalculated WSS, in the case of shuffling the items.}
\label{fig:ANES2016_violinplot}
\end{figure}
As a comparison for our item rank method we test it against two other methods of feature selection, the \textit{Random Forest} classifier \cite{Breiman2001}, and \textit{Boruta} \cite{Kursa2010}. Random Forests are a substantial modification of the classification trees method that attains near state of the art performance for classification across a wide range of data sets~\cite{Zhang2017,Caruana2006}.
A Random Forest model is formed from an ensemble of classification trees, where the trees are constructed so they are uncorrelated with each other. A new data point is classified in the model by checking the class that each of the classification trees gives and taking the majority vote of these. The Random Forest model also natively provides item importance measures that can be used to rank the importance of items to the opinion group classification. Please refer to Appendix~\ref{sec:SI_RF} for further details along with useful references to consider when implementing Random Forests.
Boruta is another feature selection method that builds on the Random Forests classifier. It is noted for tackling the 'all-relevant' problem, where, as the name suggests, we seek to find all features that are relevant for the model's ability to classify the opinion-based groups. Several studies have used it successfully as a feature selection tool in a wide range of areas from Fisheries' management~\cite{DiFranco2016} to gene expression~\cite{Kursa2014}. It is a wrapper for the Random Forest algorithm, where it uses a statistical test to identify items that are confirmed to be important, unimportant or undetermined. We are concerned with those items that are deemed to be important to the opinion-based groups under study here. Please refer to Appendix~\ref{sec:SI_rf_boruta} for further details.
The results of the feature selection for the eight items is shown in Table~\ref{tab:ANES2016_feature_selection}. They are also used in the violin plot (see Figure~\ref{fig:ANES2016_violinplot}) and represent here the average change in the WSS for every item. The second column (Random Forest) shows the values to assess the rank of each item. Evidently, it also ranks \textit{Welfare} and \textit{Race relations} as the two most important items but differs in the rest. The Boruta method defines 7 out of 8 items as important for the community split-up, and validates therefore the selected items for the community detection. Additionally, like the Random Forest method it ranks the item \textit{Gay marriage} as the least important item, whereas the item rank method evaluates \textit{Immigration} as the least important one.
\begin{table}[htb
\begin{center}
\begin{tabular}{l|c|c|c|c}
{\textbf{Item}} & \textbf{Variable} & \textbf{Item rank} & \textbf{Random Forest} & \textbf{Boruta} \\ \hline
Welfare & V161209 & 0.140 & 0.279 & Important \\
Race relations & V161198 & 0.072 & 0.202 & Important \\
Abortion & V161232 & 0.058 & 0.134 & Important \\
Gun control & V161187 & 0.039 & 0.062 & Important \\
Income & V161189 & 0.039 & 0.162 & Important \\
Gay marriage & V161231 & 0.028 & 0.030 & Undetermined \\
Business & V161201 & 0.023 & 0.084 & Important \\
Immigration & V161192 & 0.016 & 0.046 & Important
\end{tabular}
\caption{Results for the feature selection by the item rank method, random forest classification and Boruta. The methods were applied on the selected features from the ANES data set 2016, and based on the community detection from the Girvan-Newman algorithm.}
\label{tab:ANES2016_feature_selection}
\end{center}
\end{table}
\section{Results}
\label{sec:results}
In this section, we validate the previously described methods on synthetic data, expand the analysis to new data sets
and discuss how to apply it to consecutive data sets.
\subsection{Data sets}
\subsubsection{Synthetic data sets} \label{subsec:syntheticdata}
In this section we compare the three different community detection algorithms and under which circumstances they can be applied. The results will show that the detection of the opinion-based groups is not an artefact of just one community detection method.
To test our approach, we use simulated survey data, where we specify the ground truth for who belongs to each opinion-based group. Additionally, by building in items that are stronger, weaker or no predictors of group membership,
we test an item rank method to evaluate the items' influence on the community structure.
This will provide sound footing for its performance when applying it to real world data sets.
The application to synthetic data sets will reveals, due to gradual variations of their parameter, the effectiveness of the presented approach. It will also show that the performance of the community detection aligns with our determination of the predefined groups (see \ref{appendix:synthetic_data}).
In the simulated data sets we fix the size of the network, the number of items and the group membership of each individual. An individuals answer to each item is simulated by drawing from a normal distribution with mean $\mu_a$ if they are in group $a$ and $\mu_b$ if they are in group $b$. The standard deviation is the same for the sake of simplicity. The \textit{$\mu$-distance}, $|\mu_a - \mu_b|$, is a measure of
the maximal difference the two groups are on an item (see Figure~ \ref{fig:mu_distance}).
\begin{figure}
\centering
\makebox[0pt]{
\includegraphics[width=1.2\columnwidth]{mu_distance_relation_3.png}}
\caption{The \textit{$\mu$-distance} for an item. Group a has mean $\mu_a=2$ and group b has mean $\mu_b=6$. The standard deviation in both cases is $\sigma = 1.4$ }
\label{fig:mu_distance}
\end{figure}
\paragraph{Community detection}
The synthetic data sets have
100 individuals with responses to 7 items on a scale from 1-7. The community structure is an equal division into two groups of size 50. The items are ranked in 4 different categories of information content about the group structure, determined by an increasing standard deviation.
We consider values of the \textit{$\mu$-distance} from 0.6 to 6.0 (y-axis) and the values of standard deviation from 0.3 to 3.0 (x-axis), both with a step size of 0.3.
For every parameter combination we simulate 30 data sets to which we apply the community detection algorithms.
The heatmaps in Figure~\ref{fig:heatmaps_synth} show the percentage of correctly allocated nodes from the network by the community detection algorithms, averaged over the 30 simulations.
\begin{figure
\centering
\subfloat[Girvan-Newman]{\includegraphics[width = 8cm]{GN_data_comp_N=100_n_comp=2_total_q=7_noise_q=1_sc=7.png}}
\\
\subfloat[Hierarchical Clustering]{\includegraphics[width = 8cm]{HC_data_comp_N=100_n_comp=2_total_q=7_noise_q=1_sc=7.png}}
\\
\subfloat[Stochastic Block Model]{\includegraphics[width = 8cm]{sbm_data_comp_N=100_n_comp=2_total_q=7_noise_q=1_sc=7.png}}
\caption{Heatmaps for the mean correct allocation of the community detection algorithms for synthetic data sets, based on 30 runs per parameter constellation.}
\label{fig:heatmaps_synth}
\end{figure}%
The heatmaps delineate two regions. The dark blue region is where the community detection works reliably. In this region there is a small overlap in the distributions of item responses for each group due to the high \textit{$\mu$-distance} and the low standard deviation.
In the light blue region a lower \textit{$\mu$-distance} and a higher \textit{standard deviation} leads to a larger overlap of the responses in the two communitie
We also simulated data sets with additional items, which we describe in \ref{appendix:synthetic_data}. If we add additional items there is more information on the group structure, leading to an improved performance of the community detection algorithms (see Figure~\ref{fig:heatmaps_synth_GN}, \ref{fig:heatmaps_synth_HC} and \ref{fig:heatmaps_synth_SBM}). For the shown data set, the difference between the community detection algorithms is minor. A notable difference is only a lack of performance for the Hierarchical Clustering where the \textit{$\mu$-distance} is between 1.2 and 3.3 and the standard deviation is 0.3. The data sets with additional items show similar results.
\paragraph{Item rank}
To generate the synthetic data sets, items with different levels of information are included. In this way, we provide an order of items, concerning their information about the built-in group structure. The items split up into highly informative, less informative and uniformly distributed noise questions. Likewise the community detection methods, we assess the item rank method by means of synthetic data sets. We test the method’s ability to rank the items correctly by their informational value. In the following, we will see that the item rank method’s performance depends on the ability of the community detection methods to reveal the group structure in the attitude network.
The bar chart (Figure~\ref{fig:Compare_feature_selection}), representing a cross-section of the heatmaps (see~\ref{appendix:synthetic_data}), shows an equivalent performance of the community detection algorithms. The bar chart captures: a) the proportion of successful community detection in comparison to the ground truth, b) the proportion of correctly detected importance of items, and c) the performance of the \textit{Random Forest} classification algorithm and the $Boruta$ method for feature selection.
By this, it shows what happens in the transition phase, when moving from a dark blue to a light blue region (see~\ref{appendix:synthetic_data},~heatmaps).
\begin{figure*
\centering
\makebox[0pt]{
\includegraphics[width=16cm]{barcharts_Girvan-Newman.png}}
\caption{Results of the correct item ranking, depending on the $\mu$-difference. For each of the item ranking methods (Item Rank, Random Forest, Boruta), each bar shows how often each method detects the correct item ranking for each configuration of the synthetic data set. For every $\mu$-distance we run 30 simulations with 30 different synthetic data sets.
We added the number that the community detection algorithm (here: Girvan-Newman algorithm) allocated the individuals correctly in relation to the ground truth from the synthetic data set. The graph describes the development of the number of correct allocations for each $\mu$-distance.
}
\label{fig:Compare_feature_selection}
\end{figure*}
The bar charts show as expected a similar number of correct allocations for the Girvan-Newman algorithm, Hierarchical Clustering and Stochastic Block Model. The number of completely correct ranked items by the item rank method is around 25 out of 30 for the simulations with a maximal~\textit{$\mu$-distance} between 3.3 and 6.0. Below 3.0, the proportion of completely correct rankings of the items decreases. The lack of predictive power of the Random Forest model is of note. It is only able to pick out the correct ranking in a small number of simulations, even when there is a clear community split. The data shows that it performs better in allocating correctly the higher ranked questions but worse in determining the overall ranking. For the same reason, the $Boruta$ method does not determine the rankings accurately. However, it is effective at distinguishing between important and unimportant items.
Applying the approaches on synthetic data sets was beneficial for exploring and comparing the methods under artificial conditions to ensure their robustness in various scenarios. Nevertheless, synthetic data is no substitute for real-world empirical data sets. Furthermore, only by analysing real data sets, inferences can be made and results can be interpreted.
\subsubsection{Wellcome Trust data}
Here, we present a data set from the Wellcome Global Monitor 2018 which has not been previously studied to reveal opinion-based groups. The Wellcome Global Monitor conducted a survey in 2018 in over 140 countries with over 140,000 participants \cite{Wellcome2019}. The survey encompasses public attitudes to science and health. We select attitude-related items from the data set; 10 items deal with trust in organisations, institutions and science, and 3 are attitudes towards vaccines.
We apply our approach to each listed country to detect opinion-based groups and, if applicable, relevant items for community structure. We refine and normalise the data to construct country-specific networks.
We apply the Girvan-Newman algorithm to detect opinion-based groups, and Hierarchical Clustering to confirm these results. Our approach detects polarisation on health and science attitudes in five countries: Singapore, Venezuela, Cameroon, Congo and Nicaragua (see Table \ref{tab:cutout_wGM2018}, here: \href{https://kurzelinks.de/Analysis-WellcomeGlobalMonitor-2018}{complete Table}).
\begin{table}[!htb]
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{l|c|c|c|c|c|c|c}
Country & Size & Split-up (GN) & Links & Removed links & Threshold & Split-up (HC) & Overlap \\ \hline
Singapore & 456 & {[}327, 129{]} & 5015 & 12 & 11.5 & {[}326, 130{]} & 0.985 \\
Venezuela & 575 & {[}380, 195{]} & 3757 & 109 & 11 & {[}452, 123{]} & 0.854 \\
Cameroon & 493 & {[}318, 175{]} & 4767 & 161 & 10 & {[}401, 92{]} & 0.542 \\
Congo & 356 & {[}191, 165{]} & 2180 & 47 & 10 & {[}209, 147{]} & 0.933 \\
Nicaragua & 614 & {[}433, 181{]} & 5466 & 105 & 11 & {[}423, 191{]} & 0.925 \\
\end{tabular}}
\caption{Results from Wellcome Global monitor in the five countries where we identified polarised opinion-based groups by Girvan-Newman algorithm.}
\label{tab:cutout_wGM2018}
\end{table}%
The most noteworthy result is Singapore. Even though the threshold is very large (11.5), over 5015 links were added to the network, indicating that the individuals have an high consensus within the items. A link between the individuals in the network with a threshold of 11.5 means that in over 90\% their item responses are very close or identical. Next to the high agreement, it was possible to separate the network into two communities by only erasing 12 links. The Hierarchical Clustering provides similar results, with an overlap of over 98.5\% in community allocation. In the other four countries two large opinion-based groups are shown for both community detection methods, with the exception of Cameroon where the overlap of the two method is only about 54\%.
The examination through the item rank method reveals three items as the most important for the communities that we detect via the Girvan-Newman: trust in charity workers, trust in traditional healers and trust in scientists. We showed how to analyse large data sets to uncover the existence of opinion groups. For countries with large opinion-based groups, we are also able to uncover and rank the important items for the community structure.
\subsubsection{Consecutive data sets: ANES 2012 \& 2016}
Polarisation is often seen as an intrasocietal process of moving toward the extremes on political attitudes, e.g., being further away from each others' opinion on a scale. Our method identifies polarisation---even in the absence of extreme opinions---by classifying non-overlapping opinion-based groups.
In the previous section, the ANES data set from 2016 was examined with an item selection based on~\cite{Malka2014}. Here, we investigate the ANES data set from 2012 to demonstrate an approach to consecutive data. Instead of relying on a predetermined selection, we apply the Boruta method to reveal the important items for our opinion-based groups. To apply Boruta to our data set, we reduced the amount of items from the ANES data set 2016 and 2012 to each 34 items based on relevance (see here: \href{https://kurzelinks.de/preselection-ANES2012-16}{reduced item list}), selecting those items obviously related to a personal, political attitude position. Further, we only included participants who self-identified as republicans or democrats as the Boruta method requires group categorisation for the item selection.
Table \ref{tab:ANES2012_6} shows the items which the Boruta method identifies as important in distinguishing between Democrats and Republicans.
After the normalisation process, the selection of the important items allows us to construct the score-based similarity networks for the ANES data 2012 and 2016. The question is whether the opinion-based clusters are getting more separated, and so easier to detect, or is the opinion-scored network closer together, and therefore it is more difficult to distinguish between communities.
The network for the ANES data set 2012 consists of 2,039 nodes and 31,619 links with a threshold of 8.0 (see Figure~\ref{fig:ANES2012}). The two communities detected by the Girvan-Newman algorithm have 2,493 and 546 nodes. The first community includes 1,004 democrats, 547 republicans and 942 unknown, while the second community consists of 308 republicans, 71 democrats and 167 unknown.
The network constructed from the ANES data set 2016 includes 2,274 participants and 27,326 links with a threshold of 7.8 (see Figure~\ref{fig:ANES2016_boruta}). Applying the Girvan-Newman algorithm to the network results in a community with 596 republicans, 151 democrats and 424 unknown (total: 1171) and a second community with 612 democrats, 119 republicans and 372 unknown (total: 1103). To reveal the opinion-based groups considerably more links had to be removed in 2012 in comparison to 2016 and the graph had to be re-split several times as it did not fulfil the minimum community size criterion.
\begin{table*
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{l|c|c|c|c}
Item & Label 2012 & Label 2016 & Scale & Boruta\\ \hline
Abortion & abortpre\_4point & V161232 & 1-4 & 2012\\ %
Environment-jobs & envjob\_self & V161201 & 1-7 & 2012/2016\\ %
Race relations & aidblack\_self & V161198 & 1-7 & 2012/2016 \\ %
Immigration & immig\_policy & V161192 & 1-4 & 2016\\ %
Govt. guaranteed income & guarpr\_self & V161189 & 1-7 & 2012/2016\\
Death penalty & penalty\_favopp\_x & V161233x & 1-4 & 2016\\ %
Defence-spending & defsppr\_self & V161181 & 1-7 & 2012/2016\\ %
Govt. spending \& services & spsrvpr\_ssself & V161178 & 1-7 & 2012/2016\\ %
Medical insurance & inspre\_self & V161184 & 1-7 & 2012/2016\\ %
Gay marriage & gayrt\_marry & V161227x & 1-3/1-6 & 2016\\
\end{tabular}}
\caption{American National Election Studies 2012 and 2016 - Item labels and their answer range. The Boruta method selected the item as important in at least one of the data sets. Only the selected item \textit{Birthright Citizenship} from 2016 is not mentioned due to the fact that there was no corresponding item in 2012. The table show the items that are used for the network projection method and later for the community detection.}
\label{tab:ANES2012_6}
\end{table*}
\begin{figure*
\centering
\makebox[0pt]{
\includegraphics[width=\textwidth]{GN_graph_partition_anes2012-10VarsBoruta_21_04_2021resplit_on.png}}
\caption{American National Election Survey data 2012, constructed similarity network from the refined data set, with 2 communities, detected by the Girvan-Newman algorithm. The position and shape of the nodes is used to distinguish between the communities. The colour of the nodes represents their party affiliation: republican (red), democrat (blue) and unknown (yellow).}
\label{fig:ANES2012}
\end{figure*}
\begin{figure*
\centering
\makebox[0pt]{
\includegraphics[width=\textwidth]{GN_graph_partition_anes2016-10varsBORUTA2012like_21_04_2021resplit_on.png}}
\caption{American National Election Studies data 2016, constructed similarity network from the refined data set, with 2 communities, detected by the Girvan-Newman algorithm. The Boruta method provides the item selection. The network projection uses 10 items from the data set. The position and shape of the nodes is used to distinguish between the communities. The colour of the nodes represents their party affiliation: Republican (red), Democrat (blue) and unknown (yellow).}
\label{fig:ANES2016_boruta}
\end{figure*}
The application of our opinion-based group detection leads to the conclusion that, based on the ten important items, the ANES data is getting more polarised over time (from 2012 to 2016). The ANES data set from 2016 can be split up by erasing less cross-cutting links than 2012, and the groups are visually easier to distinguish.
The results show what is already observed: survey participants become increasingly polarised along party lines on several key opinions~\cite{Iyengar2019} in the ANES data set from 2012 and 2016. The communities are formed around the party affiliation, i.e. each community includes a majority of either Republicans or Democrats. Nevertheless, some participants are more aligned in their attitudes with the other group, contrary to their self-reported partisanship.
\section{Conclusions}\label{sec:conclusion}
In this article, we created a network of individuals linked by similar responses from a survey. We used three different clustering algorithms and show that all are consistent with each other at identifying communities of opinion-based groups on both empirical and simulated data. Further to this, we developed a method to identify the rank and importance of the items in a survey. We compared this to the Random Forest and Boruta method to validate it on simulated survey data. All methods can identify important items, but the method introduced here is more robust at ranking the survey items most important to the identified opinion-based groups. This allowed us to identify which items are most important to the opinion-based group that we found in the ANES and Wellcome Trust data. Additionally, instead of a post-validation, the Boruta method, given a predefined group categorisation, is able to reduce the items of a survey to a subset of group-relevant items.
The exploration of our approach on simulated data also points out limitations for our methods (i.e., opinon-based group detection and item rank). They rely on the performance of the community detection algorithms and, therefore, on the detected communities' meaningfulness.
Being able to identify opinion-based groups is important for understanding a wide range of social issues that can only be solved by the large-scale coordination of opinions (e.g., climate change; public health interventions; vaccination etc.). This is particularly important in understanding online social media interactions, which provide clear affordances for opinion exchange (e.g., via "likes" and "shares"). While identity has been shown to be central to social opinion processes (e.g., \cite{Doell2021,Gollwitzer2020}), until now it has been difficult to clearly identify links between bundles of opinions and social identities.
The value of this approach is demonstrated in Ref.~\cite{maher2020mapping} which shows opinion-based groups emerging at the start of the COVID crisis, progressively polarising on the dimension of distrust in science; and leading to identity-based differences in compliance with public health guidance. We presented a secondary analysis of Wellcome Trust data, identifying countries like Singapore that are highly divided on trust in charity workers and science.
Similarly, when we analyse the ANES 2012 and 2016 survey data, we identify items in the US that Democrats and Republicans are becoming increasingly polarised on. This phenomenon is widely observed in political and social sciences (see e.g., \cite{Iyengar2019}).
While we identify separate opinion-based groups here, we do not quantify the level of polarisation, which we aim to address in the future. Network measures to quantify polarisation exist, including using edge betweenness \cite{garimella2018quantifying}. However, these methods all rely on identifying hubs to detect polarised groups. As we construct similarity-based networks, which are dense and weighted, our topology is different. We tend not to have hubs as every node in an opinion-based group will be directly or indirectly linked to every other node in that grou
In order to bring methods like this to bear we will need to modify them.
This method for detecting polarisation in opinion-based groups paves the way to investigate the co-constitutive relationship between attitudes and social identity and related phenomena using a network approach.
\section*{Acknowledgments}
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 802421). The authors would like to thank Susan C. Fennell for fruitful conversations and feedback.
\newpage
\bibliographystyle{ws-acs}
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2,877,628,090,100 | arxiv |
\section{Introduction}
\subsection{Neutrino oscillation in the plane wave approximation}
The neutrino, a light electrically neutral fermion participating in weak interactions, was suggested by Pauli to save the conservation of energy and momentum in nuclear $\beta$-decays. Since then, three flavors of neutrinos $\nu_\alpha = (\nu_e, \nu_\mu, \nu_\tau)$ were discovered, each produced or detected in association with a corresponding lepton $\ell_\alpha=(e,\mu,\tau)$. The neutrinos, which are completely parity-violating in their weak interactions, suggested that the gauge group of the electro-weak sector of the remarkably successful Standard Model (SM) should be built using fermions with left-handed chirality.
Given the unique properties of neutrinos, studies of them may reveal a path to physics beyond the SM. In the past, experiments observing solar and atmospheric neutrinos brought increased attention to neutrino physics due to long-standing discrepancies between detection rates and no-oscillation models. Despite an impressive number of proposed solutions to these problems, all were successfully resolved by the hypothesis of neutrino oscillation, first proposed by Pontecorvo~\cite{Pontecorvo:1957cp,Pontecorvo:1957qd} in the late 1950's. Neutrino oscillation is a phenomenon firmly established in experiment, which has been observed with solar~\cite{Cleveland:1998nv,Kaether:2010ag,Abdurashitov:2009tn}, atmospheric~\cite{Fukuda:1998mi,Adamson:2014vgd}, particle accelerator~\cite{Ahn:2002up,Adamson:2014vgd} and reactor~\cite{Abe:2008aa,An:2012eh,RENO,Abe:2012tg} neutrinos.
Neutrino oscillation is a quantum phenomenon of quasi-periodic change of neutrino flavor $\nu_\alpha\to\nu_\beta$ with time. This phenomenon originates in the non-equivalence of neutrino flavor $\nu_\alpha$ and mass $\nu_k=(\nu_1,\nu_2,\nu_3)$ eigenstates, differences in their masses, and an assumption that the produced and detected neutrino states are coherent superpositions of neutrino mass eigenstates:
\begin{equation}
|\nu_\alpha(p)\rangle=\sum_{k=1}^3 V_{\alpha k}^*|\nu_k(p)\rangle,
\label{eq:flavor_state_pw}
\end{equation}
where $V_{\alpha k}$ is an element of the unitary PMNS-matrix, named after Pontecorvo, Maki, Nakagawa, Sakata, and $p$ is the momentum of the neutrino. The time evolution of the state in~\eqref{eq:flavor_state_pw} is expressed as
\begin{equation}
|\nu_\alpha(t;p)\rangle=\sum_{k=1}^3 V_{\alpha k}^*\text{e}^{-iE_kt}|\nu_k(p)\rangle,
\label{eq:flavor_state_pw_time_evolution}
\end{equation}
where $E_k=\sqrt{p^2+m_k^2}$. This leads to the oscillatory behavior of the probability to detect a neutrino originally of flavor $\alpha$ as having flavor $\beta$:
\begin{equation}
P_{\alpha\beta}(L) = |\langle\nu_\beta(p)|\nu_\alpha(t;p)\rangle|^2= \sum_{k,j=1}^{3}V_{\alpha k}^* V_{\beta j}^* V_{\beta k}^{\phantom{*}} V_{\alpha j}^{\phantom{*}} \text{e}^{-iL/L^\text{osc}_{kj}},
\label{eq:planewave_prob}
\end{equation}
where $L^\text{osc}_{kj}= 4\pi p/\Delta m_{kj}^2$ is the oscillation length due to the non-zero differences $\Delta m^2_{kj}=m^2_k-m^2_j$, and time $t$ is approximated by the traveled distance $L$.
The underlying theory, assuming a plane wave approximation, was developed in the middle of the 1970s~\cite{Eliezer:1975ja,Fritzsch:1975rz,Bilenky:1976yj}. Although successful in explaining a wide range of neutrino experiments, it is well known that this approximation is not self-consistent, and leads to a number of paradoxes~\cite{Akhmedov:2009rb,Giunti:2003ax}. The applicability of the plane wave approximation is discussed in detail in Refs.~\cite{Beuthe:2001rc,Akhmedov:2009rb,Giunti:2007ry,Bernardini:2004sw}. After the first theory was developed, Refs.~\cite{Nussinov:1976uw,Kayser:1981ye,Kiers:1995zj,Akhmedov:2012uu} pointed out the necessity of a wave packet treatment of neutrino oscillation.
\subsection{Wave packet treatment of neutrino oscillation}
The wave packet is a coherent superposition of different waves whose momenta are distributed around the most probable value, with a certain ``width'' or dispersion. Therefore, a wave packet is localized in space-time as well as in energy-momentum space. The wave packet formalism facilitates the resolution of the paradoxes of the plane wave theory, and predicts the existence of a coherence length. The latter arises due to the different group velocities of a pair $\nu_k$ and $\nu_j$, which causes a separation in space over time.
The smallness of the differences of neutrino masses relative to their typical energies suggests that the coherence length of neutrino oscillation is the largest among all known phenomena.
After the pioneering studies~\cite{Nussinov:1976uw,Kayser:1981ye,Kiers:1995zj}, the wave packet models of neutrino oscillation were developed in roughly two varieties. The first one relies on a relativistic quantum mechanical (QM) formalism that does not predict the dispersion of the neutrino wave packet in momentum space, such as in Refs.~\cite{Beuthe:2001rc,Giunti:2007ry,Kayser:2010pr}. The second one is based on calculations within quantum field theory (QFT), describing all external particles involved in neutrino production and detection as wave packets while treating neutrinos as virtual particles. The neutrino wave-function is then calculated rather than postulated. The effective momentum dispersion of the neutrino wave function depends on the kinematics of neutrino production and detection and on the momentum dispersions of the external particles, as in Refs.~\cite{Grimus:1996av,Cardall:1999bz,Stockinger:2000sk,Beuthe:2002ej,Giunti:1993se,Akhmedov:2010ms,Naumov:2010um}. Both approaches predict a number of observable effects, like a quantitative condition on the coherence of mass eigenstates in the production-detection processes, as well as a loss of coherence.
In wave packet models, the intrinsic momentum dispersion $\sigma_p$ of the neutrino wave packet is an effective quantity comprising the microscopic momenta dispersions of all particles involved in the production and detection of the neutrino. A non-zero value of $\sigma_p$ leads with time to the {\em decoherence} in the quantum superposition of massive neutrinos which results in a vanishing oscillation pattern of $\nu_\alpha\to\nu_\beta$ transitions. In addition, the oscillation pattern is smeared further in the reconstructed energy spectrum due to a non-zero experimental resolution $\delta_E$ of the neutrino energy.
Despite considerable progress in building wave packet models, none of these approaches provides a solid quantitative theoretical estimate of $\sigma_p$ or of the spatial width $\sigma_x=1/2\sigma_p$. Theoretical estimates vary by orders of magnitude, associating the dispersion of the neutrino wave packet with various scales; for example, uranium nucleus size ($\sigma_x \simeq 10^{-11}$ cm, $\sigma_p\simeq 1$ MeV), atomic or inter-atomic size ($\sigma_x \simeq (10^{-8}-10^{-7})$ cm, $\sigma_p\simeq (10^3-10^2)$ eV), pressure broadening ($\sigma_x \simeq 10^{-4}$ cm, $\sigma_p\simeq 0.1$ eV), etc.
While the current literature does not include calculations of the neutrino wave function from first principles for any type of neutrino experiment~\footnote{Recently, a first calculation which consistently treats the full pion-neutrino-environment quantum system and calculates the decoherence effects for neutrinos produced in two-body decays was published in Ref.~\cite{Jones:2014sfa}}, it also lacks experimental investigations of decoherence effects in neutrino oscillation inferred from the finite size of the neutrino wave function~\footnote{Attention to the decoherence phenomena in neutrino oscillation is increasing and the literature discusses possible decoherence effects due to physics beyond the SM like quantum gravity~\cite{Lisi:2000zt,Araki:2004mb,Barenboim:2006xt,Adamson:2008zt}, differing from the considerations of this paper, which studies the consequences of a self-consistent way to describe neutrino oscillation within the Standard Model.}.
One of the motivations of this paper is to provide a first search for a possible loss of coherence in the quantum state of neutrinos following from the wave packet treatment of neutrino oscillations, using data from the Daya Bay Reactor Neutrino Experiment. The second motivation is to demonstrate that the oscillation parameters estimated with the plane wave approximation are unbiased.
The oscillation probability formula modified by the wave packet contribution, which is discussed further, has two distinctive features: it depends on $\Delta m^2_{kj}/p^2\sigma_\text{rel}$ via the so-called localization term and on $L\Delta m^2_{kj}\sigma_\text{rel}/p$ via the term responsible for the loss of coherence with distance. The large statistics, good energy resolution, and multiple baselines of the Daya Bay experiment make its data valuable in the study of these quantum decoherence effects in neutrino oscillation.
\section{Analysis}
\subsection{Neutrino oscillation in a wave packet model}
\label{sec:wp_model}
Measured energy spectra of $\bar{\nu}_e$ interactions are compared to a prediction using a QM wave packet model of neutrino oscillation which is briefly outlined in what follows. We simplify the consideration by examining a one-dimensional wave packet of the neutrino~\footnote{While a neutrino travels in the three-dimensional space, the transverse part of its wave function essentially leads to the to $1/L^2$ dependence of the flux~\cite{Naumov:2013vea} and does not affect significantly the oscillation pattern.}. The plane wave state in \eqref{eq:flavor_state_pw} is replaced by a wave packet describing a neutrino produced as flavor $\alpha$:
\begin{equation}
|\widetilde{\nu}_\alpha(p_P;t_P,x_P)\rangle = \sum_{k=1}^3 V_{\alpha k}^*\int\frac{dp}{2\pi} f_P(p) \text{e}^{-i\phi_{P}(p)}| \nu_k(p) \rangle,
\label{eq:wavepacket_1d}
\end{equation}
with $\phi_{P}(p) = E_k t_P - p x_P$. $f_P(p)$ is the wave function of the neutrino in momentum space and is assumed to be Gaussian:
\begin{equation}
f_P(p) = \left (\frac{2\pi}{\sigma_{pP}^2}\right)^{\frac{1}{4}} \text{e}^{-\frac{(p-p_P)^2}{4\sigma^2_{pP}}},
\label{eq:gaussian_momentum_1d}
\end{equation}
where the subscript $P$ in $f_P(p)$, $p_P$ and $\sigma_{pP}$ indicates the quantities at production. In configuration space the state in \eqref{eq:wavepacket_1d} describes a wave packet with mean coordinate $x_P$ at time $t_P$. The state in \eqref{eq:wavepacket_1d} is normalized as $\langle\widetilde{\nu}_\alpha(p_P;t_P,x_P)|\widetilde{\nu}_\alpha(p_P;t_P,x_P)\rangle=1$. Similarly, a wave packet state at detection $|\widetilde{\nu}_\beta(p_D; t_D,x_D)\rangle$ is defined as the state given by~\eqref{eq:wavepacket_1d}.
A projection of $|\widetilde{\nu}_\alpha(p_P;t_P,x_P)\rangle$ onto $\langle\widetilde{\nu}_\beta(p_D;t_D,x_D)|$ produces the flavor-changing amplitude
\begin{equation}
\mathcal{A}_{\alpha\beta}(p;t_D-t_P, L,\sigma_p)\equiv \langle\widetilde{\nu}_\beta(p_D;t_D,x_D)|\widetilde{\nu}_\alpha(p_P;t_P,x_P)\rangle,
\label{eq:amplitude}
\end{equation}
which depends on $L\equiv x_D-x_P$, time difference~$t_D-t_P$ and on the effective mean neutrino momentum $p$ and momentum dispersion $\sigma_p$ comprising the details of production and detection~\footnote{The momentum integral in~\eqref{eq:amplitude} is calculated by expanding $E_k=\sqrt{p^2+m_k^2}$ in a Taylor series up to second order around the effective momentum given by~\eqref{eq:mean_momentum_sigma}.}
\begin{equation}
p = \frac{p_P \sigma_{pD}^2 + p_D \sigma_{pP}^2}{\sigma_{pP}^2+\sigma_{pD}^2},\quad\frac{1}{\sigma_p^2}= \frac{1}{\sigma_{pP}^2}+\frac{1}{\sigma_{pD}^2}.
\label{eq:mean_momentum_sigma}
\end{equation}
The probability $|\mathcal{A}_{\alpha\beta}(p;t_D-t_P, L,\sigma_p)|^2$ should be integrated over production time $t_P$ (or, equivalently, over $t_D-t_P$) and most probable momentum $p_P$ to get an experimentally observable oscillation probability:
\begin{equation}
P_{\alpha\beta}(L) =\sum_{k,\,j=1}^3\frac{ V^*_{\alpha k} V^{\phantom *}_{\beta k}V^{\phantom\dagger}_{ \alpha j} V^*_{\beta j} }
{\sqrt[4]{1 +\left(L/L^{\text{d}}_{kj}\right)^2}}
\text{e}^{- \frac{\left(L/L^\text{coh}_{kj}\right)^2}{1+\left(L/L^{\text{d}}_{kj}\right)^2} -\mathrm{D}^2_{kj}}
\text{e}^{-i\widetilde{\varphi}_{kj}},
\label{eq:ossc}
\end{equation}
where the phase $\widetilde{\varphi}_{kj}$ is the sum of the plane wave phase $\varphi_{kj} = 2\pi L/L^\text{osc}_{kj}$ and correction $\varphi^d_{kj}$ due to the dispersion of the wave packet: $\widetilde{\varphi}_{kj} = \varphi_{kj} + \varphi^d_{kj}$, with
\begin{equation}
\varphi^\text{d}_{kj} =
- \frac {L/L^\text{d}_{kj}}{1+\left(L/{L^\text{d}_{kj}}\right)^2}
\left(\frac L {L^\text{coh}_{kj}}\right)^2
+ \frac{1}{2} \arctan { \frac{L}{L^\text{d}_{kj}}}.
\label{eq:varphi_d}
\end{equation}
Oscillation probability formulas similar to~\eqref{eq:ossc} but neglecting wave packet dispersion were obtained in several studies (see, for example, Refs.~\cite{Beuthe:2002ej,Beuthe:2001rc,Akhmedov:2010ms,Bernardini:2006ak}).
~\eqref{eq:ossc} has appeared as a particular case of a more general consideration within QFT with relativistic wave
packets~\cite{Naumov:2010um}. Relativistic invariance suggests that $\sigma_\text{rel}$ should be Lorentz invariant. In the QM approach adopted in \eqref{eq:wavepacket_1d}-\eqref{eq:ossc} the only possibility to preserve Lorentz invariance is for $\sigma_\text{rel}$ to be a constant
\footnote{Since the QFT approach considers both neutrino production and detection one finds that $\sigma_\text{rel}$, being a relativistic invariant, is actually a function of kinematic variables
involved in the production and detection processes as well as of momentum dispersions of wave packets describing all involved particles~\cite{Naumov:2013bea}. Therefore, in comparing the QM and QFT approaches, we may treat the QM \protect{$\sigma_\text{rel}$} as that of the QFT approach averaged over the kinematic variables of all external wave packets involved in neutrino production and detection.}.
The probability in \eqref{eq:ossc} contains three quantities with dimensions of length:
\begin{equation}
\begin{aligned}
L^\text{osc}_{kj} & = \frac{4\pi p}{\Delta m^2_{kj}}, \qquad
L^\text{coh}_{kj} & =\frac{L^\text{osc}_{kl}}{\sqrt 2 \pi\sigma_\text{rel}}, \qquad L^\text{d}_{kj} & = \frac{L^\text{coh}_{kj}}{2\sqrt{2}\sigma_{\text{rel}}},
\end{aligned}
\label{lengthts-vacuum_1}
\end{equation}
where $\sigma_\text{rel}=\sigma_p/p$, $L^\text{osc}_{kj}$ is the usual oscillation length of a pair of neutrino states $|\nu_k\rangle$ and $|\nu_j\rangle$, $L^\text{coh}_{kj}$ is interpreted as the neutrino coherence length, i.e. the distance at which the interference of neutrino mass eigenstates vanishes, and finally $L^\text{d}_{kj}$ is the dispersion length, i.e. a distance at which the wave packet is doubled in its spatial dimension due to the dispersion of waves moving with different velocities. The term
\begin{equation}
\mathrm{D}^2_{kj} = \frac{1}{2} \left( \frac{\Delta m^2_{kj}}{4 p^2\sigma_\text{rel}} \right)^2= \frac{1}{4}
\left( \frac{\Delta m^2_{kj}}{\sigma_{m^2}} \right)^2 =\left( \frac{\sqrt{2}\pi\sigma_x}{L_{kj}^\text{osc}} \right)^2
\label{eq:D_factor}
\end{equation}
suppresses the coherence of massive neutrino states $|\nu_k\rangle$ and $|\nu_j\rangle$ if $\Delta m^2_{kj}\gg\sigma_{m^2}$, where $\sigma_{m^2}= 2\sqrt{2}p\sigma_p$ could be interpreted as an uncertainty in the neutrino mass squared~\cite{Kayser:1981ye}.
$\mathrm{D}^2_{kj}$ can be seen from another perspective as the localization term
suppressing the oscillation if $\sqrt{2}\pi\sigma_x\gg L_{kj}^\text{osc}$, where $\sigma_x=(2\sigma_p)^{-1}$ is the width of neutrino wave packet in the configuration space.
It is worth mentioning that terms in~\eqref{eq:ossc} which correspond to the interference of $\nu_k$ and $\nu_j$ states also get suppressed by the denominator $\sqrt[4]{1 +\left(L/L^{\text{d}}_{kj}\right)^2}$ and vanish for both limits $\sigma_p\to 0$ and $\sigma_p\to\infty$, reducing the oscillation probability in~\eqref{eq:ossc} to the non-coherent sum
\begin{equation}
P_{\alpha\beta}=\sum_k|V_{\alpha k}|^2|V_{\beta k}|^2,
\label{eq:prob_decoherent}
\end{equation}
which does not depend on energy and distance.
For the $\bar{\nu}_e$ at Daya Bay, $1-P_{\rm ee}$ is expressed as
\begin{equation}
\begin{aligned}
& \phantom {+}\cos^2\theta_{12}\sin^2\theta_{12}\cos^4\theta_{13}
\biggl(1-\frac {\exp{\left[-\frac{\left(L /L^\text{coh}_{21}\right)^2}{1+\left(L/{L^\text{d}_{21}}\right)^2}-D^2_{21}\right]}}
{\sqrt[4]{1 + \left(L/ {L^\text{d}_{21}}\right)^2}}\cos{(\varphi_{21}}+\varphi_{21}^\text{d})\biggr)\\
& +\cos^2\theta_{12}\cos^2\theta_{13}\sin^2\theta_{13}
\biggl(1-\frac {\exp{\left[-\frac{\left(L /L^\text{coh}_{31}\right)^2}{1+\left(L/{L^\text{d}_{31}}\right)^2}-D^2_{31}\right]}}
{\sqrt[4]{1 + \left(L/ {L^\text{d}_{31}}\right)^2}}\cos{(\varphi_{31}}+\varphi_{31}^\text{d})\biggr)\\
& +\sin^2\theta_{12}\cos^2\theta_{13}\sin^2\theta_{13}
\biggl(1-\frac {\exp{\left[-\frac{\left(L /L^\text{coh}_{32}\right)^2}{1+\left(L/{L^\text{d}_{32}}\right)^2}-D^2_{32}\right]}}
{\sqrt[4]{1 + \left(L/ {L^\text{d}_{32}}\right)^2}}\cos{(\varphi_{32}}+\varphi_{32}^\text{d})\biggr).
\end{aligned}
\label{eq:pee_wp}
\end{equation}
\subsection{Sensitivity of Daya Bay experiment to neutrino wave packet}
The Daya Bay experiment is composed of two near underground experimental halls (EH1 and EH2) and one far underground hall (EH3). Each of the experimental halls hosts identically designed antineutrino detectors (ADs). EH1 and EH2 contain two ADs each, while EH3 contains four ADs. Electron antineutrinos are produced in three pairs of nuclear reactors via $\beta$ decays of neutron-rich daughters of the fission isotopes ${}^{235}\text{U}$, ${}^{238}\text{U}$, ${}^{239}\text{Pu}$ and ${}^{241}\text{Pu}$, and detected via the inverse $\beta$ decay (IBD). The coincidence of the prompt ($e^+$ ionization and annihilation) and delayed ($n$ capture on Gd) signals efficiently suppresses the backgrounds, which amounted to less than 2\% (5\%) of the IBD candidates in the near (far) halls~\cite{An:2015rpe}. The Gd-doped liquid scintillator target is a cylinder of three meters in both height and diameter. The detectors have a light yield of about 165 photoelectrons/MeV and a reconstructed energy resolution $\delta_E/E\approx 8\%$ at 1 MeV of deposited energy in the scintillator. More details on the experimental setup are contained in Refs.~\cite{An:2015rpe, DayaBay:2012aa,Dayabay:2014vka,An:2015qga}.
The studies in this paper are based on data acquired in the 6-AD period when there were two ADs in EH1, one AD in EH2 and 3 ADs in EH3, with the addition of the 8-AD period from October 2012 to November 2013, a total of 621 days. The number of IBD candidates used in this analysis, and the mean baselines of the three experimental halls to each pair of reactor cores, are summarized in Table~\ref{tab:data}.
\begin{table}[!htbp]
\setlength{\tabcolsep}{6pt}
\centering
\begin{tabular}{ccrrr}
\toprule
& & \multicolumn{3}{c}{Mean distance, m} \\
\cmidrule{3-5}
Halls & IBD candidates & Daya Bay & Ling Ao & Ling Ao II \\
\midrule
EH1 & 613813 & 365 & 860 & 1310 \\
EH2 & 477144 & 1348 & 481 & 529 \\
EH3 & 150255 & 1909 & 1537 & 1542 \\
\bottomrule
\end{tabular}
\caption{The number of IBD candidates and mean distances of the three experimental halls to the pairs of reactor cores.}
\label{tab:data}
\end{table}
The expected numbers of IBD events are convolutions of the reactor-to-target expectation with the detector-response function. The reactor-to-target expectation takes into account the antineutrino fluxes from each reactor core including non-equilibrium and spent nuclear fuel corrections, first order in $1/m_p$ ($m_p$=proton mass) IBD cross-section accounting for the positron emission angle~\cite{Vogel:1999zy}, and the oscillation survival probability $P_\text{ee}$ given by~\eqref{eq:planewave_prob} for the plane wave model and by~\eqref{eq:ossc} for the wave packet model. The detector response-function accounts for energy loss in the inner acrylic vessel, liquid scintillator and electronics non-linearity and energy resolution $\delta_E$.
For relatively large values of $\sigma_p\simeq\delta_E$, the effects of these two parameters on the observed energy spectra might appear similar, however they are distinct. First, they have different physical origins: while $\sigma_p$ is governed by the most localized particle in the production and detection of the neutrino, $\delta_E$ is determined by the energy depositions of the final state particles in the detector. Second, these effects can also be distinguished from their order of occurrence since the microscopic processes used in the energy estimation occur later in time with respect to the neutrino interaction in the detector. Third, their effects are not identical. In particular, as described in Sec.~\ref{sec:wp_model}, the limit $\sigma_p\to 0$ leads to the decoherence of neutrino oscillation in contrast to the impact of energy resolution which does not lead to any smearing in the reconstructed energy spectrum in the limit $\delta_E\to 0$.
In order to illustrate analytically an interplay of $\sigma_p$ and $\delta_E$, let us consider the exponential in the oscillation probability in~\eqref{eq:ossc} convolved with a Gaussian energy resolution, as a function of the reconstructed energy $E_{\rm vis}$, assuming $\delta_E\ll p$, infinite dispersion length $L^\text{d}$, neglecting the $D^2$ term, and suppressing mass eigenstate indices for the sake of compactness~\footnote{The actual implementation of the detector effects in this analysis was performed numerically without approximations}:
\begin{equation}
\begin{aligned}
&\frac{1}{\sqrt{2\pi}\delta_E}\int dp \; \exp{\left(-i \; 2\pi L/L^\text{osc} - \left(L/L^\text{coh}\right)^2 - (p-E_{\rm vis})^2/2\delta_E^2\right)} \\
&\simeq \exp{\left(-i \; 2\pi L/L^\text{osc}_\text{rec} - \left(L/L^\text{coh}_\text{eff}\right)^2\right)},
\end{aligned}
\end{equation}
where $L^\text{osc}$ and $L^\text{coh}$ are given by~\eqref{lengthts-vacuum_1} and the effective coherence length comprises both the intrinsic $\sigma_p$ and detector resolution $\delta_E$:
\begin{eqnarray}
\left(\frac{1}{L^\text{coh}_\text{eff}}\right)^2 = \left(\frac{1}{L^\text{coh}_\text{rec}}\right)^2 + \left(\frac{1}{L^\text{coh}_\text{det}}\right)^2,
\end{eqnarray}
where $L^\text{osc}_\text{rec}$ and $L^\text{coh}_\text{rec}$ are given by $L^\text{osc}$ and $L^\text{coh}$ replacing $p$ with $E_{\rm vis}$, and $L^\text{coh}_\text{det}$ is given by $L^\text{coh}_\text{rec}$, replacing $\sigma_p$ with $\delta_E$. The interplay of $\sigma_p$ and $\delta_E$ is illustrated by the effective coherence length $L^\text{coh}_\text{eff}$, which is
dominantly determined by the smallest among $L^\text{coh}_\text{rec}$ and $L^\text{coh}_\text{det}$, or by the largest among $\sigma_p$ and $\delta_E$.
The following provides simple numerical estimates of wave packet effects on neutrino oscillations at Daya Bay. For a typical momentum of $p=4$ MeV of detected reactor $\bar{\nu}_e$, the oscillation would be suppressed for two distinctive domains of $\sigma_\text{rel}$. The domain $\sigma_\text{rel}\gtrsim O(0.1)$ corresponds to significant contributions from $L$--dependent interference-suppressing terms and corrections to the oscillation phase $\varphi^d_{32}$ in \eqref{eq:ossc}, while the $D^2_{kj}$ term is negligibly small. For example, at $L=L_{32}^\text{osc}/2$ the exponential suppression reaches its maximum $\text{e}^{-\pi/8}$ at $\sigma_\text{rel}=1/\sqrt{2\pi}\simeq 0.4$. Correspondingly, the coherence and dispersion lengths read $L_{32}^\text{coh}\simeq 2.2$ km and $L_{32}^\text{d}\simeq 2$ km. At larger values of $\sigma_\text{rel}$ and at a fixed distance the spatial dispersion of neutrino wave packets partially compensates the loss of coherence due to the spatial separation of $\nu_k$ and $\nu_j$.
The domain $\sigma_\text{rel}\lesssim O(2.8\cdot 10^{-17})$ corresponds to $D_{32}^2 \gtrsim 1$, which is significant in suppressing the interference in \eqref{eq:ossc} through the $L$--independent term, while the $L$--dependent terms are negligibly small. Thus, the region of $O(2.8\cdot 10^{-17}) \ll \sigma_\text{rel} \ll O(0.1)$ is where the wave packet impact on neutrino oscillation is negligible for the Daya Bay experiment.
For illustrative purposes Fig.~\ref{fig:Data2Theory} shows the ratio of the observed to expected numbers of IBD events assuming no oscillation using the data collected at the near and far experimental halls as a function of reconstructed visible energy $E_\text{vis}$. Figure~\ref{fig:Data2Theory} also shows the expected ratio for neutrino oscillation with the plane wave and wave packet models with $\sigma_\text{rel}$ of $0.33$ and $8\cdot 10^{-17}$ as examples.
Both model expectations are shown with the oscillation parameters fixed to their best-fit values within the plane wave model~\footnote{The following values of the oscillation parameters were used in Fig.~\ref{fig:Data2Theory}: $\Delta m^2_{21} = 7.53 \cdot 10^{-5}\text{ eV}^2$, $\Delta m^2_{32}=2.45\cdot 10^{-3}\text{ eV}^2$, $\sin^22\theta_{12}=0.846$, $\sin^22\theta_{13}=0.0852$.}. For this set of parameters, the wave packet models with $\sigma_\text{rel}=0.33$ and with $\sigma_\text{rel}=8\cdot 10^{-17}$ are inconsistent with the data by about five standard deviations, thus motivating the chosen values of $\sigma_\text{rel}$. The two panels illustrate how the visible energy spectra are modified in the near and far halls depending on the intrinsic dispersion of the neutrino wave packet. Remarkably, most changes in the energy spectra due to $\sigma_\text{rel}$ are in opposite directions for near and far halls, which can be explained qualitatively as follows. As mentioned above, the extremes $\sigma_p\to 0$ and $\sigma_p\to\infty$ would yield fully decoherent neutrinos with the oscillation probability given by~\eqref{eq:prob_decoherent}. Antineutrinos detected at the near halls experience a relatively small oscillation in the plane wave approach. The values of $\sigma_\text{rel}$ selected for Fig.~\ref{fig:Data2Theory} make the $\bar{\nu}_e$ partially decoherent and $P_\text{ee}$ tend towards \eqref{eq:prob_decoherent}, predicting a {\em smaller} number of surviving $\bar{\nu}_e$ as compared to the plane wave formula. The distance at which the far detectors of the Daya Bay experiment are placed is tuned to observe the maximal oscillation effect due to $\Delta m^2_{32}$. Partial decoherence of the $\bar{\nu}_e$ tends to reduce the oscillation, thus predicting a {\em larger} number of survived $\bar{\nu}_e$ with respect to the plane wave formula. This feature of Daya Bay provides additional sensitivity to the decoherence effects and makes such a study less sensitive to the predicted reactor $\bar{\nu}_e$ spectrum.
The data can be reasonably well described by
\begin{equation}
\begin{aligned}
\Delta m^2_{32}=2.17\cdot 10^{-3}\text{ eV}^2, \quad \sin^22\theta_{13}=0.102,\\
\sigma_\text{rel}=8\cdot 10^{-17}, \quad \chi^2/\text{ndf} = 246.8/(256-4),
\end{aligned}
\label{eq:example_left}
\end{equation}
and by
\begin{equation}
\begin{aligned}
\Delta m^2_{32}=2.16\cdot 10^{-3}\text{ eV}^2, \quad \sin^22\theta_{13}=0.097,\\
\sigma_\text{rel}=0.33, \quad \chi^2/\text{ndf} = 253.8/(256-4).
\end{aligned}
\label{eq:example_right}
\end{equation}
These results demonstrate that one could obtain reasonable fits of the data within the wave packet model with certain values of $\sigma_\text{rel}$ and yield best-fit values of the oscillation parameters which differ from the corresponding best-fit values with the plane wave model, assuming normal mass hierarchy~\footnote{The best-fit values of the oscillation parameters $\sin^22\theta_{13}$ and $\Delta m^2_{32}$ are different from our previous publication~\cite{An:2015rpe} because of a different implementation of systematic uncertainties and another choice of $E_\text{vis}$ binning.}:
\begin{equation}
\begin{aligned}
\Delta m^2_{32}=2.45\cdot 10^{-3}\text{ eV}^2,\quad \sin^22\theta_{13}=0.0852, \\
\phantom {\sigma_\text{rel}=0.33} \quad\chi^2/\text{ndf} = 245.9/(256-3).
\end{aligned}
\label{eq:true_osc_minimum}
\end{equation}
However, Eqs.~\ref{eq:example_left},~\ref{eq:example_right} do not correspond to the global minimum of the $\chi^2$ discussed below because $\sigma_\text{rel}$ was fixed to two arbitrary values for illustrative purposes. In order to find the global minimum we performed a detailed statistical analysis of the allowed region of $\sigma_\text{rel}$.
\subsection{Statistical framework}
As the goodness-of-fit measure we use $\chi^2(\boldsymbol{\eta}) = (\mathbf{d}-\mathbf{t}(\boldsymbol{\eta}))^TV^{-1}(\mathbf{d}-\mathbf{t}(\boldsymbol{\eta}))$, where $\mathbf{d}$ is a data vector containing detected numbers of IBD candidates in energy bins and in different detectors, while $\mathbf{t}(\boldsymbol{\eta})$ is the corresponding theoretical model vector which depends on constrained and unconstrained parameters $\boldsymbol{\eta}$. All constraints of the model as well as expected fluctuations in the number of IBD events are encompassed in the covariance matrix $V$. The model vector $\mathbf{t}(\boldsymbol{\eta})$ comprises expected numbers of IBD and background events. All constrained parameters (or systematic uncertainties) relevant for the Daya Bay oscillation analyses were taken into account in this analysis. These are mainly associated with the reactor antineutrino flux, background predictions and the detector response modeling. The uncertainty of the detector response is dominant. Details can be found in Refs.~\cite{An:2015rpe,An:2015qga}.
The analysis was done with four unconstrained parameters $\sigma_{\text{rel}}$, $\Delta m^2_{32}$, $\sin^22\theta_{13}$ and reactor flux normalization $N$. The confidence regions are produced by means of two statistical methods: the conventional fixed-level $\Delta \chi^2$ analysis and the Feldman-Cousins method~\cite{Feldman:1997qc}. The marginalized $\Delta \chi^2$ statistic is
\begin{equation}
\Delta \chi^2(\boldsymbol{\eta}') = \min\limits_{\boldsymbol{\eta}\setminus\boldsymbol{\eta}'} \chi^2(\boldsymbol{\eta}) -\min\limits_{\boldsymbol{\eta}}\chi^2(\boldsymbol{\eta}),
\label{eq:statistic}
\end{equation}
where $\boldsymbol{\eta}=(\sigma_{\text{rel}}, \Delta m^2_{32}, \sin^22\theta_{13}, N)$ and $\boldsymbol{\eta}'$ is its subspace with parameters of interest ($\boldsymbol{\eta}'=\sigma_\text{rel}$ for one dimensional interval, and $\boldsymbol{\eta}'=(\sigma_\text{rel},\Delta m^2_{32}$) or $\boldsymbol{\eta}'=(\sigma_\text{rel},\sin^22\theta_{13}$) for two dimensional regions), and both are used to determine the $p$-value of the observed dataset and the model.
The closed interval corresponding to the $100\cdot(1-\alpha)\%$ confidence level (C.L.) is constructed for both the fixed-level $\Delta \chi^2$ analysis and the Feldman-Cousins method as the region of $\boldsymbol{\eta}'$ which satisfies:
\begin{equation}
\Delta\chi^2(\boldsymbol{\eta}') < \Delta\chi^2_{1-\alpha},
\label{eq:interval_construction}
\end{equation}
where $\Delta\chi^2_{1-\alpha}$ is the $(1-\alpha)$-th quantile of the statistic in~\eqref{eq:statistic}. The tabulated values of the quantile $\chi^2_{n;1-\alpha}$ of the $\chi^2_n$ distribution with $n$ degrees of freedom ($n=1,2$ for one and two dimensional confidence regions) were used for the fixed-level $\Delta \chi^2$ analysis. Toy Monte Carlo sampling was used to determine $\Delta\chi^2_{1-\alpha}$ of the statistic in~\eqref{eq:statistic} with the Feldman-Cousins method.
An open confidence interval can be constructed if neutrinos are assumed to be produced and detected coherently, which is equivalent to assuming $\sigma_\text {rel} \gg 10^{-16}$. In this case, instead of using~\eqref{eq:statistic}, an upper bound on $\sigma_{rel}$ can be computed using the modified statistic
~\cite{Cowan:2010js}
\begin{equation}
\Delta\chi^2_\text{up}(\sigma_{\text{rel}}) = \begin{cases}
\Delta\chi^2(\sigma_{\text{rel}}) &\mbox{if } \hat{\sigma}_{\text{rel}} < \sigma_{\text{rel}} \\
0 &\mbox{if } \hat{\sigma}_{\text{rel}} > \sigma_{\text{rel}},
\end{cases}
\label{eq:1sided_statistic}
\end{equation}
with $\hat{\sigma}_{\text{rel}}$ representing the best-fit value. In the fixed-level $\Delta \chi^2$ analysis the $100\cdot(1-\alpha)\%$ C.L. upper limit is given by:
\begin{equation}
\Delta\chi^2(\sigma_{\text{rel}}) \le \chi^2_{1;1-2\alpha}.
\end{equation}
For example, in order to set a 95\% C.L. upper limit, the quantile $\chi^2_{1;0.9}=2.71$ was used. The Feldman-Cousins method automatically produces the proper interval using the interval construction in~\eqref{eq:interval_construction}.
\section{Results and Discussion}
Figure~\ref{fig:DeltaChi2_dm31_all} displays the allowed regions in $(\Delta m^2_{32},\sigma_{\text{rel}})$ and $(\sin^2 2\theta_{13},\sigma_{\text{rel}})$ obtained with both the fixed-level $\Delta\chi^2$ and the Feldman-Cousins methods, which are found to be consistent. For the values of $\sigma_\text{rel}\lesssim 10^{-16}$ the decoherence effects lead to strong correlations between $\Delta m^2_{32}, \sin^22\theta_{13}$ and $\sigma_\text{rel}$, yielding smaller values of $\Delta m^2_{32}$ and larger values of $\sin^22\theta_{13}$. These correlations are expected taking into account the explicit form of $1-P_\text{ee}(L)$ in~\eqref{eq:pee_wp}. For $\sigma_\text{rel}\gtrsim O(0.1)$, these correlations are found to be significantly weaker.
The best-fit point corresponds to
\begin{equation}
\begin{aligned}
\Delta m^2_{32}=1.59\cdot 10^{-3}\text{ eV}^2, \quad \sin^22\theta_{13}=0.160,\\
\sigma_\text{rel}=4.0\cdot 10^{-17}, \quad \chi^2/\text{ndf} = 245.9/(256-4),
\end{aligned}
\label{eq:wp_fit_data}
\end{equation}
with the p-value $0.596$ which is smaller than the p-value $0.614$ with the plane wave model given by~\eqref{eq:true_osc_minimum}. The allowed region for $\sigma_\text{rel}$ at a 95\% C.L. reads:
\begin{equation}
\CentralInterval.
\label{eq:2sided_interval}
\end{equation}
The upper bound of~\eqref{eq:2sided_interval} corresponds to $L^\text {coh}_{32}>1.94\;L^\text {osc}_{32}/2$ and $L^\text {d}_{32}>2.96\;L^\text {osc}_{32}/2$. The lower bound can also be interpreted in terms of length $\sigma_x$ which corresponds to the spatial width of the neutrino wave packet. Taking the average momentum $p=4$ MeV of detected reactor $\bar{\nu}_e$, the lower bound of~\eqref{eq:2sided_interval} rules out $\sigma_x\gtrsim 1$ km. The Daya Bay data is not sensitive enough to constrain the $D^2_{kj}$ term significantly better.
Thus, the lower limit is much weaker than an obvious constraint of $\sigma_x\lesssim 2$ m which follows from the consideration that the $\sigma_x$ (which equals 1/2$\sigma_ p$) of $\bar{\nu}_e$ wave packets detected by the Daya Bay Experiment does not exceed the dimensions of the reactor cores and detectors. Taking this constraint into account, $\sigma_p\gtrsim 5\cdot 10^{-8}$ eV, which for the average momentum $p=4$ MeV, translates into $\sigma_\text{rel}\gtrsim 10^{-14}$. Such a $\sigma_\text{rel}$ corresponds to the regime where $D^2_{kj}\ll 1$ and the localization term can be safely neglected, thus allowing us to put an upper limit of:
\begin{equation}
\UpperLimit, \text{ at a } 95\%\text{ C.L.}
\end{equation}
\section*{Summary}
We performed a search for the footprint of the neutrino wave packet which should show itself through specific modifications of the neutrino oscillation probability. The reported analysis of the Daya Bay data provides, for the first time, an allowed interval of the intrinsic relative dispersion of neutrino momentum \CentralInterval{}. Taking into account the actual dimensions of the reactor cores and detectors, we find that the lower limit $\sigma_{\rm rel} > 10^{-14}$ corresponds to the regime when the localization term is vanishing, thus allowing us to put an upper limit:
\UpperLimit{} at a 95\% C.L.
The obtained limits can be read as $10^{-11}\text{ cm } \lesssim\sigma_x \lesssim 2$ m.
The current limits are dominated by statistics. With three years of additional data the upper limit on $\sigma_\text{rel}$ is expected to be improved by about 30\%.
The allowed decoherence effect due to the wave packet nature of neutrino oscillation is found to be insignificant for reactor antineutrinos detected by the Daya Bay experiment thus ensuring an unbiased measurement of the oscillation parameters $\sin^22\theta_{13}$ and $\Delta m^2_{32}$ within the plane wave model.
\section*{Acknowledgements}
Daya Bay is supported in part by the Ministry of Science and Technology of China, the U.S. Department of Energy, the Chinese Academy of Sciences, the CAS Center for Excellence in Particle Physics, the National Natural Science Foundation of China, the Guangdong provincial government, the Shenzhen municipal government, the China General Nuclear Power Group, Key Laboratory of Particle and Radiation Imaging (Tsinghua University), the Ministry of Education, Key Laboratory of Particle Physics and Particle Irradiation (Shandong University), the Ministry of Education, Shanghai Laboratory for Particle Physics and Cosmology, the Research Grants Council of the Hong Kong Special Administrative Region of China, the University Development Fund of The University of Hong Kong, the MOE program for Research of Excellence at National Taiwan University, National Chiao-Tung University, and NSC fund support from Taiwan, the U.S. National Science Foundation, the Alfred~P.~Sloan Foundation, the Ministry of Education, Youth, and Sports of the Czech Republic, the Joint Institute of Nuclear Research in Dubna, Russia, the National Commission of Scientific and Technological Research of Chile, and the Tsinghua University
Initiative Scientific Research Program. We acknowledge Yellow River Engineering Consulting Co., Ltd., and China Railway 15th Bureau Group Co., Ltd., for building the underground laboratory. We are grateful
for the ongoing cooperation from the China General Nuclear Power Group and China Light and Power Company.
\input{dyb_decoh_plb.bbl}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.9\textwidth]{EH-ratio_5sigma-s_double-EH_bold.pdf}
\caption{\label{fig:Data2Theory}
Ratios of the observed to expected numbers of IBD events in the absence of oscillation as a function of reconstructed visible energy $E_\text{vis}$. The data are grouped by near (EH1+EH2) and far (EH3) halls, displayed in the upper and in the bottom panels respectively, with the error bars representing the statistical uncertainties. Superimposed solid lines are ratios assuming neutrino oscillations within the plane wave model (PW) with the best-fit values of $\sin^22\theta_{13}$ and $\Delta m^2_{32}$ obtained with the plane wave model.
The ratios using the wave-packet model (WP) assume $\sigma_\text{rel} = 0.33$ (dashed line) and $\sigma_\text{rel}= 8\cdot 10^{-17}$ (dot-dashed line), as two examples.
The green lines correspond to the wave packet model ratios assuming the best-fit values of $\sin^22\theta_{13}$ and $\Delta m^2_{32}$ obtained with the plane wave model and thus, inconsistent with the data by about five standard deviations. The red lines correspond to the wave packet model ratios assuming the best-fit values of $\sin^22\theta_{13}$ and $\Delta m^2_{32}$ obtained within the wave packet model, yielding a much better agreement with the data.
All ratios enter the region below $2m_e$, which corresponds to the IBD threshold, because of detector response effects like energy reconstruction and absorption in the inner acrylic vessel (see details in Refs.~\cite{An:2015rpe,An:2015qga}).
}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.9\textwidth]{scan_db_s-dm-th_ihep-fc_v2_21.pdf}
\caption{\label{fig:DeltaChi2_dm31_all} Allowed regions of $(\Delta m^2_{32},\sigma_{\text{rel}})$ (top) and of $(\sin^2 2\theta_{13},\sigma_{\text{rel}})$ (middle) parameters obtained with fixed-level $\Delta\chi^2$ (contours corresponding to $1\sigma$, $2\sigma$, $3\sigma$ C.L., dashed lines) and within the Feldman-Cousins (contours corresponding to $1\sigma$, $2\sigma$ C.L., solid lines) methods. Bottom panel shows the marginalized $\Delta\chi^2(\sigma_\text{rel})$ statistic given by~\eqref{eq:statistic} vs $\sigma_\text{rel}$. Note the break in the abscissa and the change from a logarithmic to linear scale.}
\end{center}
\end{figure}
\end{document}
|
2,877,628,090,101 | arxiv | \section{Introduction}
Research works in two-dimensional (2D) materials with different
functionalities have been boosted in the last few year. Since the successful
synthesis of graphene,\cite{novoselovScience2004} 2D materials have been
considered as a new paradigm not only in fundamental studies, like the search
of topological phases,\cite{kanePRL2003,weeksPRX2011,acostaPRB2014} and
tuneable magnetic structures in 2D
systems,\cite{songNatMat2019,liNatMat2019,morell2DMat2019,dominikePRMat2021} but
also addressing technological applications. For instance, the development of
new materials for (nano)electronic and spintronic devices like single layer
field effect transistors,\cite{radisavljevicNatNanotech2011} and half-metals
based on transition metal dichalcogenides and
dihalides.\cite{tongAdvMat2017,kulishJMatChemC2017,gokougluMatResExpress2017,
fengJMatChemC2018, wangNanoscaleHoriz2018}
By taking advantage of the layered structure of their 3D parents, combined
with a suitable balance between strong (weak) intralayer (interlayer)
binding interactions,\cite{mounetNatNanotech2018} two dimensional materials can
be obtained through exfoliation processes. For instance, in a seminal work, of
Lee {\it et al.}\cite{leeNature2013} revealed the exfoliable nature, and the
two dimensional electronic confinement in Ca$_2$N\, electrides.
Electrides are ionic crystals characterized by the presence of electrons not
bonded to a particular nucleus. These electrons act as ions (with no nucleus)
embedded within the crystal lattice, anionic
electrons.\cite{druffelJMatChemC2017,liuJPhysChemC2020} Further experimental
works revealed that these anionic electrons form nearly free 2D electron gas
confined between the stacked layers of the
electride.\cite{leeNature2013,ohJACS2016,druffelJACS2016} It is worth noting
that there are other native inorganic electrides with the same lattice
structure of Ca$_2$N, like Sr$_2$N\cite{breseJSolStateChem1990} and
Y$_2$C;\cite{zhangChemMat2014} meanwhile other ones have been predicted
throughout high-throughput computational
simulations.\cite{tadaInorgChem2014,inoshitaPRX2014}
Currently, few layer systems of electrides (electrenes) have attracted research
works in fundamental issues, like the search of topological phases throughout
the design of kagome lattices on the electrene
surface,\cite{zhangPRB2019kagome} as well as to the development of electronic
devices with hight carrier density and electronic
mobility.\cite{zhaoJACS2014,zengPRB2018} Further, theoretical studies have
addressed the functionalization of Ca$_2$N\, electrenes mediated by atomic
adsorption.\cite{liuMaterResExpress2018} Functionalization is a quite promising
route in order to tailor the electronic and magnetic properties of 2D systems.
For instance, the rise of ferromagnetic (FM) phases upon full
hydrogenation,\cite{qiuJPhysChemC2019} and oxidation of Ca$_2$N\, and Sr$_2$N\,
electrenes.\cite{wuJMMM2020,souzaJPhysChemC2020} On the other hand, it is
important to stress that the presence of anionic electrons makes the electrene
surface very reactive, which may lead to significant changes on the electronic
and structural properties of the functionalized systems, giving rise to a new
set of physical properties to be exploited.
In this work, by means of first-principles calculations, we perform a systematic
investigation of the energetic stability, structural, and the electronic
properties of the oxidized $A_2B$\, electrenes, with $A$\,=\,Ca, Sr, Ba, Y, and
$B$\,=\, As, N, P, C. We have considered the fully oxidation of one surface side
of single layer electrene (O/$A_2B$), and two surface sides in bilayer electrenes
(O/$(A_2B)_2$/O). The energetic stability of the oxidized systems, examined through
the calculation of the formation energy, revealed a structural transition
giving rise to tetragonal (t) layered structures, ($A$O$AB$)$^{\rm t}$\, and O/$(A_2B$-$A_2B)^{\rm t}$/O. The
dynamical and structural stabilities of these tetragonal phases were examined
through a combination of phonon spectra calculations, and molecular dynamic
simulations. Further simulation of the X-ray near edge spectroscopy (XANES), at
the N K-edge, were performed in order to provide a detailed structural analysis
of the oxidized systems. Finally, the electronic structure calculations
indicate the rise of an energetically stable FM phase, and the emergence of
half-metallic channels along $A$N layers shielded by oxidized $A$O shells (with
$A$\,=\,Ca, Sr, and Ba).
\section{Computational details}
The calculations were performed by using the density functional theory
(DFT),\cite{kohn} as implemented in the computational codes Quantum-Espresso
(QE)\,\cite{espresso} and Vienna Ab initio Simulation Package
(VASP).\cite{vasp1,vasp2} We have considered the generalized gradient
approximation of Perdew-Burke-Ernzerhof (GGA-PBE)\,\cite{PBE} for the
exchange-correlation functional. The single layer and bilayer $A_2B$\, electrenes
were simulated using slab structures within the supercell approach, with a
vacuum region of 18 and 22\,\AA, respectively, and surface periodicities of
(1$\times$1), and ($\sqrt 2$\,$\times$\,$\sqrt 2$) for hexagonal and tetragonal
structures. The final atomic geometries, and total energies were obtained
using the QE code, where the Kohn-Sham\,\cite{KS} orbitals, and the
self-consistent total charge densities were expanded in plane wave basis sets
with energy cutoffs of 70 and 353\,Ry, respectively. The Brillouin zone
sampling was performed by using a 8$\times$8$\times$1 k-point mesh.\cite{mp}
The atomic positions were relaxed until the residual forces were converged to
within 5\,meV/\AA, and the structural relaxation (variable-cell) was performed
within a pressure convergence of 0.05\,Kbar. The long-range van der Waals (vdW)
interactions were taken into account using the self-consistent vdW-DF
approach.\cite{dionPRL2004,perezPRL2009,klimevsPRB2011}
Further structural characterizations were performed through calculations of the
X-ray absorption spetrocopy combining the QE results and
Xspectra\cite{xas1,xas2,xas3} simulations. We have considered the K-edge
spectra of nitrogen atoms by using the Gauge-Including Projector Augmented-Wave
(GIPAW)\,\cite{gipaw} method to calculate the dipolar cross section,
$$
\sigma(\omega) \propto \sum_{n}|\langle\psi_n |{\bf\hat{\varepsilon}\cdot
r}|\psi_{1s}\rangle|^2\delta(\epsilon_n - \epsilon_{1s} - \hbar\omega),
$$
within the dipole approximation; $\psi_n$ and $\psi_{1s}$ are the final $n$
and initial $1s$ (single particle) orbitals
and the respective energies in the presence of core-hole. The absorbing atom is
described with a pseudopotential with a full core-hole in the N-$1s$
orbital.\cite{dal2014pseudopotentials} In order eliminate spurious interactions
between a core-hole and its periodic images, we have considered a distance of
$\sim$7\,\AA\, between the core-holes.
The electronic structure calculations and structural/thermal stability
simulations were performed using the VASP code. We have considered an energy
cutoff of 500\,eV for the plane wave basis set, and the Brillouin zone was
sampled using a 15$\times$15$\times$1 k-point mesh.\cite{mp} The structural
stability was verified through the calculation of elastic constants and the
phonon dispersion using PHONOPY code.\cite{togoScrMat2015} The
thermal stability was verified by ab initio molecular dynamics simulations
(AIMD) at 300K, with a time step of 1 fs using Nos\'e heat bath
scheme.\cite{JCP81-511-1984}
\section{Results and Discussions}
\subsection{Pristine $A_2B$ electrenes}
\begin{figure}
\includegraphics[width=8cm]{models1.pdf}
\caption{\label{models1}Structural model of $A_2B$\, electride bulk (a), and
single layer electrene, side view (b) and top view (c). The isosurfaces (of
0.003\,$e$/\AA$^3$) show the localization of the anionic electrons within the
energy interval of $\pm 0.5$\,eV with respect to the Fermi level.}
\end{figure}
The $A_2B$ electrides with $A$\,=\,Ca, Sr, Ba, Y, and $B$\,=\,N, P, As, C
share the same structure of Ca$_2$N, Fig.\,\ref{models1}(a). Our results of
equilibrium geometries of $A_2B$ electrides and 2D single layer electrenes,
summarized in Table\,I, are in good agreement with previous experimental and
theoretical findings, viz.: Ba$_2$As.\cite{mingJACS2016}, Ba$_2$P,\cite{mingJACS2016},
Sr$_2$P,\cite{mingJACS2016}, Y$_2$C,\cite{zhangChemMat2014,houJPhysChemC2016}
Ca$_2$N,\cite{gregoryJMatChem2000,wangMaterials2018,qiuJPhysChemC2019,wuJMMM2020}
Sr$_2$N,\cite{breseJSolStateChem1990,wuJMMM2020} and
Ba$_2$N.\cite{reckewegZeits2005,wuJMMM2020} The structural properties of these
electrides are anisotropic, characterized by a strong intralayer interactions
due to the $A$--$B$ ionic chemical bonds, and comparatively weaker interlayer
interaction between the $A_2B$\, sheets. The latter is ruled by a superposition
of (i) Coulombic attractive forces between the positively charged $A_2B$\, layer
and the anionic electrons, and (ii) repulsive interaction between the positively
charged $A_2B$\, layers.\cite{zhaoJACS2014, druffelJACS2016,druffelJMatChemC2017}
In order to provide a quantitative picture of the interlayer binding strength,
we calculate the interlayer binding energy ($E^b$) defined
as,\cite{jungNanoLett2018}
$$
E^b=\frac{1}{S}\left( E[A_2B]_{\rm ML}-E[A_2B]_{\rm Bulk} \right),
$$
where $E[A_2B]_{\rm ML}$ and $E[A_2B]_{\rm Bulk}$ are the total energies of
single layer electrene, and $A_2B$ electride, respectively, and $S$ is the
surface area normal to the $A_2B$\, stacking. Our results of binding energies
for Ca$_2$N, Sr$_2$N, Ba$_2$N, and Y$_2$C\, (Table\,I) are in good agreement with those
presented in the current literature.\cite{dalePCCP2017,liuJPhysChemC2020}
\begin{table}
\centering
\caption{\label{key} Details of the equilibrium geometry of $A_2B$ electrides
and single layer electrenes, lattice constant $a$ and {\it A--B} equilibrium
bond length (in \AA), and the interlayer binding energy, $E^{\rm b}$
(in J/m$^2$) without/with the inclusion of vdW interactions.}
\begin{ruledtabular}
\begin{tabular}{lccccc}
& \multicolumn{2}{c}{{bulk}}& &\multicolumn{2}{c}{{monolayer}}\\
$A_2B$ & $a$ &{\it A--B}&$E^b$&{\it a} &{\it A--B}\\
\cline{1-1} \cline{2-3} \cline{4-4} \cline{5-6}
Ba$_2$As\, & 4.64& 3.21 & 0.41/0.48 & 4.65 & 3.22 \\
Ba$_2$P\, & 4.65& 3.18 & 0.44/0.51 & 4.64 & 3.17 \\
Sr$_2$P\, & 4.45& 3.01 & 0.59/0.64 & 4.43 & 3.00 \\
Y$_2$C\, & 3.61& 2.47 & 1.60/1.73 & 3.50 & 2.45 \\
Ca$_2$N\, & 3.60& 2.42 & 0.97/1.02 & 3.61 & 2.43 \\
Sr$_2$N\, & 3.84& 2.60 & 0.78/0.83 & 3.85 & 2.61 \\
Ba$_2$N\, & 4.02& 2.76 & 0.58/0.64 & 4.00 & 2.75 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}
\includegraphics[width= \columnwidth]{models2new.pdf}
\caption{\label{models2}Structural models of oxidized systems. (a1)
Hexagonal and (a2) tetragonal one-sided fully oxidized single layer
electrene; two-sided fully oxidized (b1) hexagonal and (b2) tetragonal bilayer
electrene. Oxygen atoms are indicated by red spheres.}
\end{figure}
Given the larger $A_2B$--$A_2B$\, interlayer distance ($>3$\,\AA), it is worth to
examine the contribution of the van der Waals (vdW) interactions in the binding
energies. As shown in Table\,I, the calculations of $E^b$ without/with the
inclusion of the vdW interactions reveal a slight increase of $E^b$, for
instance between 5 an 10\% for the nitrides, $A_2$N. Thus, we can infer that the
Coulombic attractive forces bring the major contribution to the interlayer
interactions.
Our results of $E^b$ indicate that these $A_2B$\, electrenes can be classified as
``potentially esfoliable'' based on the criteria, presented by Mounet {\it et
al.}, in a recent high-throughput computational investigation applied to two
dimensional material.\cite{mounetNatNanotech2018} As shown in
Figs.\,\ref{models1}(b) and (c), at the equilibrium geometry, the single layer
electrene exhibits the same $A_2B$\, atomic structure of its bulk (electride)
parent, with the anionic electrons lying on the electrene surface.
\subsection{Oxidation}
Here we will address the energetic stability, structural characterization, and
the electronic properties of the oxidized $A_2B$\, electrenes. Firstly, we have
considered the one-sided fully oxidized single layer electrene [O/$A_2B$], and in
the sequence the two-sided fully oxidized bilayer electrene [O/$(A_2B)_2$/O]. In
Figs.\,\ref{models2}(a) and (b) we present the structural models of O/$A_2B$\, and
O/$(A_2B)_2$/O.
\subsubsection{Energetic Stability}
\begin{figure*}
\includegraphics[width=2\columnwidth]{phonons-1ml.pdf}
\caption{\label{phonons-1ml} Phonon spectra of the hexagonal (a) and
tetragonal (b) O/$A_2B$\, electrenes.}
\end{figure*}
The energetic stability of the oxidized electrenes was
inferred through the calculation of the formation energy ($E^f$),
$$
E^f = E[X] - E[A_2B] - E[{\rm O_2}]/2,
$$
where $E[A_2B]$ and $E[X]$ are the total energies of pristine and the
oxidized electrenes, with $X$\,=\,O/$A_2B$\, and O/$(A_2B)_2$/O, and $E[{\rm O_2}]$ is the
total energy of an isolated O$_2$ molecule (triplet state). We found an
energetic preference for the oxygen adatoms on the hollow site aligned with the
(cation) $A$ atom at the opposite side of the $A_2B$ monolayer, as shown in
Fig.\,\ref{models2}(a1). Our results, summarized in Table\,II, indicate that
the oxidized systems, O/$A_2B$, are energetically stable, $E^f<0$, where the
hexagonal (h) lattice of the $A_2B$\, host [O/$(A_2B)^{\rm h}$] has been preserved. However,
further phonon spectra calculations [Fig.\,\ref{phonons-1ml}(a)] revealed that
the all O/$(A_2B)^{\rm h}$\ structures, except O/(Y$_2$C)$^{\rm h}$
[Fig.\,\ref{phonons-1ml}(a4)], present imaginary frequencies, thus indicating
that they are dynamically unstable.
\begin{table}
\centering
\caption{\label{key} Formation energy ($E^f$ in eV/O-atom) of the hexagonal
and tetragonal oxidized single layer (O/$A_2B$), and bilayer [O/$(A_2B)_2$/O] electrenes,
and the total energy gain upon hexagonal\,$\rightarrow$\,tetragonal structural
transition ($\Delta E^{\text{h-t}}$ in eV/O-atom). The
the lattice constant ($a$) and the vertical distances (d and h in
Fig.\,\ref{models2}) are in \AA. The lattice constant of the pristine
hexagonal $A_2B$ electrene are within parentheses.}
\begin{ruledtabular}
\begin{tabular}{cccccc}
\multicolumn{1}{c}{{O/$A_2B$}} & \multicolumn{2}{c}{hexagonal}
& \multicolumn{1}{c}{{$\longrightarrow$}} &
\multicolumn{2}{c}{tetragonal} \\
$A_2B$ & $E^f$& {\it a}& $\Delta E^{\text{h-t}}$ & {\it a} &
h
\\
\cline{1-1} \cline{2-3} \cline{4-4} \cline{5-6}
Ba$_2$As\,& $-1.79$ & 4.07 (4.65) & $-0.51$ & 3.75 & 2.80 \\
Ba$_2$P\, & $-2.08$ & 4.42 (4.64) & $-0.63$ & 4.18 & 2.64 \\
Sr$_2$P\, & $-2.21$ & 4.33 (4.43) & $-0.76$ & 3.92 & 2.50 \\
Y$_2$C\, & $-5.46$ & 3.64 (3.50) & $-0.20$ & 3.40 & 2.58 \\
Ca$_2$N\, & $-2.99$ & 3.85 (3.61) & $-0.90$ & 3.36 & 2.40 \\
Sr$_2$N\, & $-3.00$ & 4.11 (3.85) & $-0.80$ & 3.59 & 2.54 \\
Ba$_2$N\, & $-3.08$ & 4.35 (4.00) & $-0.57$ & 3.83 & 2.61 \\
\hline
\multicolumn{1}{c}{{O/$(A_2B)_2$/O}} & \multicolumn{2}{c}{hexagonal}
& \multicolumn{1}{c}{{$\longrightarrow$}} &
\multicolumn{2}{c}{tetragonal} \\
$A_2B$ & $E^f$& {\it a}& $\Delta E^{\text{h-t}}$ & {\it a} &
d \\
\hline
Ca$_2$N\, & $-3.04$ & 3.92 & $-1.68$ & 3.40 & 2.57 \\
Sr$_2$N\, & $-3.07$ & 4.19 & $-1.27$ & 3.63 & 2.85 \\
Ba$_2$N\, & $-3.02$ & 4.33 & $-1.10$ & 3.86 & 3.22 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}
\includegraphics[width=\columnwidth]{phonons-2ml.pdf}
\caption{\label{phonons-2ml} Phonon spectra of the hexagonal (a) and
tetragonal (b) O/($A_2\text{N})_2$/O\, electrenes.}
\end{figure}
In a very recent study,\cite{souzaJPhysChemC2020} we found that the hexagonal
geometry of one sided oxidized Ca$_2$N\, electrene [O/(Ca$_2${N})$^{\rm h}$] is a metastable
configuration, which becomes stable through a structural transition to a layered
tetragonal (t) phase [(CaOCaN)$^{\rm t}$\, in Fig.\,\ref{models2}(a2)]. In this sense,
firstly we examine if such a h\,$\rightarrow$\,t structural transition is
energetically favorable for the other O/$A_2B$\, electrenes, and then we verify the
dynamical and structural stabilities of the tetragonal phase by performing a
set of phonon spectra calculations, and molecular dynamics simulations.
The h\,$\rightarrow$\,t energy gain ($\Delta E^{{\text{h-t}}}$) is given by the
total energy difference between the two structural phases,
$
\Delta E^{{\text{h-t}}}=E^{\rm t} - E^{\rm h}.
$
We found that the tetragonal phase is energetically more stable than the
hexagonal one for all O/$A_2B$\, systems in the present study, $\Delta
E^{{\text{h-t}}}<0$. Meanwhile, further phonon spectra calculations revealed
that the tetragonal structures are dynamically stable only for $B$=N [($A$O$A${N})$^{\rm t}$]
and O/(Y$_2$C)$^{\rm t}$. As shown in Fig.\,\ref{phonons-1ml}(b), imaginary
frequencies present in the hexagonal phase were suppressed in (CaOCaN)$^{\rm t}$, (SrOSrN)$^{\rm t}$,
and (BaOBaN)$^{\rm t}$, Figs.\,\ref{phonons-1ml}(b5)-(b7). In
addition, to study the thermal stability of the materials, we
performed molecular dynamics (MD) simulations
for the temperature of $T=300K$. As shown, [Fig.\,\ref{md-1ml} (Appendix)], for
each system we can verify its stability, where it follows the same behavior of
the phonon spectra, where we have verified the structural stability of ($A$O$A${N})$^{\rm t}$\,
and O/(Y$_2$C)$^{\rm t}$. It is worth noting that the tetragonal structure of
these systems has been preserved during he 15\,ps of MD simulation.
\begin{figure}
\includegraphics[width=\columnwidth]{xanes-new2.pdf}
\caption{\label{xanes} XANES spectra of pristine single layer Ca$_2$N\,
electrene for the polarization vector (a1) perpendicular,
$\hat\varepsilon^\perp$, and (b1) parallel, $\hat\varepsilon^\parallel$, to
the electrene surface. XANES spectra of O/Ca$_2$N\, for $\hat\varepsilon^\perp$
(c1), $\hat\varepsilon^\parallel$ (d1), and the density
of states of (CaOCaN)$^{\rm t}$\, projected on the N-$2p_{\rm z}$ (c2), Ca-$4p_{\rm z}$
(c3), N-$2p_{\rm x,y}$ (d2), and Ca-$4p_{\rm x,y}$ (d3) orbitals. XANES spectra
of O/(Ca$_2$N)$_2$/O, for $\hat\varepsilon^\perp$
(e1), $\hat\varepsilon^\parallel$ (f1), and the density
of states of O/(Ca$_2${N}-Ca$_2${N})$^{\rm t}$/O\, projected on the N-$2p_{\rm z}$ (e2),
Ca-$4p_{\rm z}$ (e3), N-$2p_{\rm x,y}$
(f2), and Ca-$4p_{\rm x,y}$ (f3) orbitals. The XANES spectra of the tetragonal
(hexagonal) phase are indicated by solid (dashed) lines.}
\end{figure}
In the sequence, we have considered the surface oxidation of bilayer electrenes
[$(A_2B)_2$] with $B$\,=\,N, O/($A_2\text{N})_2$/O\, [Fig.\,\ref{models2}(b)]. Similarly to what we
have found in the single layer systems, the O/($A_2\text{N})_2$/O\, structures are (i)
energetically stable ($E^f<0$ in Table\,II), and also (ii) present exothermic
h\,$\rightarrow$\,t structural transitions, with $\Delta E^{\text{h-t}}$ almost
twice compared with those of their O/$A_2B$\, counterparts. Further phonon spectra
calculations, and molecular dynamics simulations of O/($A_2\text{N})_2$/O,
Figs.\,\ref{phonons-2ml} and \ref{md-2ml} (Appendix), respectively, support the
dynamical and structural stabilities of $(A$O($A$N)$_2$$A$O)$^{\rm t}$. That is, we found no
imaginary frequencies in the tetragonal phases,
Fig.\,\ref{phonons-2ml}(b), and the MD simulations reveal that the atomic
structures of the $(A$O($A$N)$_2$$A$O)$^{\rm t}$\, systems have been preserved [Fig.\,\ref{md-2ml}(b),
Appendix], whereas the ones of the hexagonal phase are no longer maintained after
15\,ps of simulation, Fig.\,\ref{md-2ml}(a) (Appendix).
At the equilibrium geometry, the tetragonal phase of the oxidized electrenes is
characterized by a layered structure indicated as $A$O and
$A$N in Figs.\,\ref{models2}(a2) and (b2). The inner $A$N sheets form a bilayer
structure, with interlayer bond distance $d$ (indicated in
Table\,II), shielded by oxidized $A$O sheets. It is interesting to noting that
such geometries somewhat mimic the ones of their (stoichiometrically equivalent
and energetically stable) $A$O and $A$N bulk cubic parents, namely CaO, SrO,
and BaO,\cite{dadsetaniSolStatSci2009, nejatipourPhysScrip2015,
nguyenPhysLettA2021, MatProj} and CaN, SrN, and BaN,\cite{MatProj} thus
providing further support to the energetic and structural stability of the
tetragonal $(A$O($A$N)$_2$$A$O)$^{\rm t}$.
\subsubsection{Structural Characterization}
In order to present a more complete structural picture of the oxidized systems,
we have simulated the nitrogen K-edge X-ray absorption spectra of the
pristine ($A_2$N), and the oxidized O/$A_2${N}\, and O/($A_2\text{N})_2$/O\, systems. Here we
will present our results for $A$\,=\,Ca, namely Ca$_2$N, O/Ca$_2$N\, and
O/(Ca$_2$N)$_2$/O, since the other systems, $A$\,=\, Sr and Ba, present quite similar
spectra and interpretations. In Fig.\,\ref{xanes}, solid and dashed lines
indicate the absorption spectra of the tetragonal and hexagonal phases,
respectively.
Let us start with the single layer pristine Ca$_2$N\, electrene. In
Figs.\,\ref{xanes}(a1) and (b1) we present the absorption spectra for
polarization vector perpendicular ($\hat\varepsilon^\perp$) and parallel
($\hat\varepsilon^\parallel$) to the surface, respectively. Based on the
analysis of orbital projected density of states (DOS, not shown), we found
that the edge and near-edge absorption features are mostly dictated by the
electronic transition of the N-$1s$ core electron to the unoccupied N-$2p_{\rm
z}$ and N-$2p_{\rm x,y}$ orbitals, for $\hat\varepsilon^\perp$ and
$\hat\varepsilon^\parallel$, respectively. Due to the electronic confinement
along the normal direction with respect to the electrene surface, the
broadening of the absorption lines from the Fermi energy ($E_{\rm F}$) up to
$\sim$\,$E_{\rm F}+6$\,eV for $\hat\varepsilon^\perp$ is slightly smaller
compared with that for $\hat\varepsilon^\parallel$. Meanwhile, in the oxidized
systems the energy broadenings for $\hat\varepsilon_{\parallel}$
[Figs.\,\ref{xanes}(d1) and (f1)] are significantly larger compared with those
for $\hat\varepsilon_{\perp}$, Figs.\,\ref{xanes}(c1)-(e1), indicating a
reduction (increase) of the electronic confinement along the parallel
(perpendicular) direction with respect to the surface plane; which is, in its
essence, a consequence of the formation of planar Ca--N and Ca--O layered
structures upon oxidation [Fig.\,\ref{models2} and insets of
Figs.\,\ref{xanes}(c1) and (d1)]. Further identification of the oxidized
structures can be done by comparing the energy position of the absorption edges
(AEs). For instance, comparing the AEs present in Fig.\,\ref{xanes}(a1) and
(c1), we find that it the former it lies near the Fermi level, while in the
latter the AE starts at about $E_{\rm F}+4$\,eV, thus, indicating an increase
of the N-$1s$ binding energy (BE) in the oxidized systems. Indeed, based on the
L$\rm\ddot{o}$wdin charge population analysis, we found that the total charge
of the nitrogen atoms in the oxidized O/Ca$_2$N\, [O/(Ca$_2$N)$_2$/O] electrenes reduces by
0.34 [0.42]\,$e$/N-atom when compared with the one of pristine Ca$_2$N. Thus,
we can infer that the increase of the N-$1s$ BE is due to the reduction of
the electronic screening at the N nucleus in the oxidized Ca$_2$N.
Next, we examine the nitrogen K-edge XANES spectra of the oxidized tetragonal
systems in light of the projected density of states. Let us start with the
single layer O/Ca$_2$N. The projections on the N-$2p_{\rm z}$ and Ca-$4p_{\rm z}$
orbitals, Figs.\,\ref{xanes}(c2) and (c3), indicate that the
absorption features 2, 3, and 4 in Fig.\,\ref{xanes}(c1) are ruled by the
electronic transitions to the lowest unoccupied N-$2p_{\rm z}$ (major
contribution) hybridized with the nearest neighbor Ca-$4p_{\rm z}$ orbitals
(minor contribution). It is worth noting that the features 2 and 3 (for
$\hat\epsilon^\perp$) of the tetragonal phase are also present in the absorption
spectra of the hexagonal phase [2' and 3' in Fig.\,\ref{xanes}(c1)]. However
they are shifted by $\sim$1\,eV toward lower energies when compared with their
counterparts 2 and 3, thus we can infer that the BE of the N-$1s$ core
electrons in the tetragonal phase is larger compared with the one of the
hexagonal phase. As discussed above, such an increase of the BE is supported by
the reduction of the total charge of the nitrogen atoms (by 0.01\,$e$/N-atom)
in the tetragonal O/Ca$_2$N\, in comparison with that of hexagonal one. Similarly
for O/(Ca$_2$N)$_2$/O, as shown Figs.\,\ref{xanes}(e1)-(e3), (i) the XANES spectra of
the tetragonal phase is ruled by the unoccupied N-$2p_{\rm z}$ states (major
role) hybridized with the $4p_{\rm z}$ orbitals (minor role) of the Ca atoms
embedded in the CaO sheets; and (ii) the edge features of the tetragonal and
hexagonal phases indicate that the BE of N-$1s$ core electrons of the former is
larger by about 1\,eV\, compared with that of the latter, in agreement with the
lower total charge (by 0.04\,$e$/N-atom) of the N atoms in the tetragonal
phase.
In Fig.\,\ref{xanes}(d) and (f) we present the XANES spectra for a polarization
vector parallel to the O/Ca$_2$N\, and O/(Ca$_2$N)$_2$/O\, layers,
$\hat\varepsilon^\parallel$, and the DOS projected on the N-$2p_{\rm x,y}$ and
Ca-$4p_{\rm x,y}$ orbitals of the tetragonal phase. Here, it is worth to make
some comparison with the absorption spectra with the polarization vector normal
to the surface, $\hat\varepsilon^\perp$. Firstly, we found that the pre-edge
absorption (feature 1), attributed to the hybridizations of the partially
occupied spin-down N-$2p_{\rm x,y}$ and Ca-$4p_{\rm x,y}$ orbitals, becomes more
intense for $\hat\varepsilon^\parallel$. In fact, such a pre-edge absorption
spectrum can be considered as a signature of the formation of half-metallic
channels, along the CaN layers, as discussed below. In the sequence, it is
noticeable the well defined absorption spectrum 2', present in the hexagonal
phase, has been dimmed in the tetragonal structure, thus suggesting that the
structural differences between the tetragonal and hexagonal phases are better
captured by looking at the in-plane edge absorption features.
\subsubsection{Electronic and Magnetic Properties}
The electronic band structures of the $A_2B$\, electrenes, with $A$\,=\,Ca, Sr,
Ba, and $B$\,=\,N are characterized by parabolic metallic bands, giving rise
to nearly free electron (NFE) states localized on the electrene's surface and
between the stacked layers. On the other hand, upon the formation of $A$O
oxidized layers these NFE states become unoccupied, and we observe the
emergence of magnetic moments in the $A$N layers. Here, we will examine the
magnetic and electronic properties of energetically stable
oxidized electrenes, ($A$O$A${N})$^{\rm t}$\, and $(A$O($A$N)$_2$$A$O)$^{\rm t}$.
\begin{figure}
\includegraphics[width=\columnwidth]{mag-scheme.jpg}
\caption{\label{mag-scheme} Schematic orbital occupation of
($A$O$A${N})$^{\rm t}$\, (a) $A$O and NO layers along the stacking direction ({\sf c}), and (b)
N$A$ sheet perpendicular to the stacking direction, {\sf a}\,$\times$\,{\sf b}
plane. (c) Orbital occupation of $(A$O($A$N)$_2$$A$O)$^{\rm t}$\, along the stacking direction.
(d) Intralayer and (e) interlayer FM/AFM spin-polarizations. (f) Spin-density
distribution of the $(A$O($A$N)$_2$$A$O)$^{\rm t}$\, systems. Isosurface of 0.004\,$e$/\AA$^2$.}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{pdos.pdf}
\caption{\label{pdos} Electronic density of states (DOS) projected on the
N-$2p$ orbitals of ($A$O$A${N})$^{\rm t}$\, (a1)-(a3), and $(A$O($A$N)$_2$$A$O)$^{\rm t}$\, (b1)-(b3).}
\end{figure}
Based on the nominal oxidation states and the electronegativities of the
involved atoms, we can infer the emergence of a magnetic moment in the oxidized
systems.\cite{wuJMMM2020} There is a net charge transfer of 2 electrons from
each (less electronegative) $A$ atoms to the (more electronegative) O and N
atoms, resulting in $A^{2+}$, O$^{2-}$ and N$^{2-}$ oxidation states,
Fig.\,\ref{mag-scheme}(a). The ground state configuration is characterized by
an O$A$ (oxidized) layer with closed $p$ shells parallel to a $A$N layer with
the N-2$p$ orbitals partially occupied, Fig.\,\ref{mag-scheme}(b). Similarly,
in the bilayer system, $(A$O($A$N)$_2$$A$O)$^{\rm t}$\, [Fig.\,\ref{mag-scheme}(c)], we find edge
$A$N layers with partially occupied N-2$p$ orbitals sandwiched by O$A$ (edge)
layers with closed $p$ shells. According to the Hund's rule, each N atom will
carry a net magnetic moment of 1\,$\mu_{\rm B}$. Indeed, within the GGA-PBE
approach, we found a net magnetic moment of about 0.8\,$\mu_{\rm B}$ mostly
localized on the nitrogen atoms. The electronic density of states (DOS)
projected on the N-$2p$ orbitals of ($A$O$A${N})$^{\rm t}$\, and $(A$O($A$N)$_2$$A$O)$^{\rm t}$, Figs.\,\ref{pdos}(a)
and (b), reveal that the partial occupation of planar N-$2p_{\rm x,y}$ orbitals
brings the major contribution to the polarization of the N atoms.
Total energy comparisons between the magnetic and non-magnetic phases, $\Delta
E^{\rm mag}=E^{\text{mag}}-E^{\text{non-mag}}$, support the energetic preference
for the spin-polarized systems, $\Delta E^{\rm mag}$\,$<$\,0 in Table\,III. The
strength of the magnetic interactions between the nitrogen atoms was examined by
comparing the total energies of the ferromagnetic (FM) and antiferromagnetic
(AFM) phases as shown in Figs.\,\ref{mag-scheme}(d) and (e) for the intralayer
and interlayer magnetic couplings, respectively. We have considered intralayer
interactions ($\Delta E^{\text{FM-AFM}}_{\rm intra}$) between the N atoms in
the same $A$N layer [Fig.\,\ref{mag-scheme}(d)], and interlayer interactions
($\Delta E^{\text{FM-AFM}}_{\rm inter}$) between the N atoms lying in different
layers [Fig.\,\ref{mag-scheme}(e)] in the case of $(A$O($A$N)$_2$$A$O)$^{\rm t}$. Our results,
summarized in Table\,III, reveal the both systems, ($A$O$A${N})$^{\rm t}$\, and $(A$O($A$N)$_2$$A$O)$^{\rm t}$, present
an energetic preference for the intralayer and interlayer FM coupling between
the N atoms. It is noticeable that (i) O/(Ca$_2${N}-Ca$_2${N})$^{\rm t}$/O\, presents the largest
interlayer FM interaction, $\Delta E^{\text{FM-AFM}}_{\rm
inter}$\,=\,36.5\,meV/N-atom, compared with the other $(A$O($A$N)$_2$$A$O)$^{\rm t}$\, systems,
leading to (ii) a strengthening of the intralayer FM coupling, namely $\Delta
E^{\text{FM-AFM}}_{\rm intra}$\,=\,8.5\,$\rightarrow$\,22.8\,meV/N-atom. In
contrast, (iii) we found (relatively) lower values of $\Delta
E^{\text{FM-AFM}}_{\rm inter}$ for O/(Sr$_2${N}-Sr$_2${N})$^{\rm t}$/O\, and O/(Ba$_2${N}-Ba$_2${N})$^{\rm t}$/O\, which can be
due to the larger interlayer distance ($d$) as indicated in
Fig.\,\ref{models2}(b2) and Table\,II, and the more localized feature of the
spin-polarized states normal to the stacking direction ({\sf c}). Indeed, the
projection of the DOS on the N-$2p$ orbitals [Fig.\,\ref{pdos}(b)] support the
larger interlayer interaction, ruled by the N-$2p_{\rm z}$ orbitals, in
O/(Ca$_2${N}-Ca$_2${N})$^{\rm t}$/O\, [Fig.\,\ref{pdos}(b1)] compared with those of the other oxidized
bilayer electrenes, Figs.\,\ref{pdos}(b2) and (b3).
The energetic preference for the FM phase can be attributed to super-exchange
interactions between the N$^{2-}$ anions mediated by $A^{2+}$ cations. In this
sense, the FM coupling will be favored due to the electron delocalization along
the N$^{2-}$--$A^{2+}$--N$^{2-}$ bonds, thus lowering the kinetic energy of the
system, as schematically shown in Figs.\,\ref{mag-scheme}(b) and (c) for the
intralayer and interlayer couplings, respectively. Further support to the FM
coupling between the N$^{2-}$ anions, mediated by super-exchange interactions,
can be found in the Goodenough-Kanamori
rule,\cite{goodenoughPRB1955,kanamoriJAP1960} since the
N$^{2-}$--$A^{2+}$--N$^{2-}$ bonds are characterized by bond angles of
90$^\circ$. In Fig.\,\ref{mag-scheme}(f) we present the spin-density
distribution in $(A$O($A$N)$_2$$A$O)$^{\rm t}$, with $A$=Ca, Sr, and Ba where we confirm the
localization of the net magnetic moment on the nitrogen atoms.
\begin{table}
\centering
\caption{\label{key} Total energy differences (in meV/N-atom) between
non-magnetic and magnetic phases, $\Delta E^{\rm
mag}=E^{\text{mag}}-E^{\text{non-mag}}$, and between the FM and AFM phases for
intralayer ($\Delta E^{\text{FM-AFM}}_{\rm intra}$), and interlayer interactions
($\Delta E^{\text{FM-AFM}}_{\rm inter}$).}
\begin{ruledtabular}
\begin{tabular}{lrrr}
& $\Delta E^{\rm mag}$ & $\Delta E^{\text{FM-AFM}}_{\rm intra}$ &
$\Delta E^{\text{FM-AFM}}_{\rm inter}$ \\
\hline
(CaOCaN)$^{\rm t}$\, & $-67$ \hspace{3mm} & $-8.5$ \hspace{3mm} & -- \hspace{5mm} \\
(SrOSrN)$^{\rm t}$\, & $-135$ \hspace{3mm} & $-18.7$ \hspace{3mm} & -- \hspace{5mm} \\
(BaOBaN)$^{\rm t}$\, & $-92$ \hspace{3mm} & $-1.9$ \hspace{3mm} & -- \hspace{5mm} \\
\hline
O/(Ca$_2${N}-Ca$_2${N})$^{\rm t}$/O\, & $-83$ \hspace{3mm} & $-22.8$ \hspace{3mm} & $-36.5$
\hspace{3mm} \\
O/(Sr$_2${N}-Sr$_2${N})$^{\rm t}$/O\, & $-127$ \hspace{3mm} & $-12.9$ \hspace{3mm} & $-1.0$
\hspace{3mm} \\
O/(Ba$_2${N}-Ba$_2${N})$^{\rm t}$/O\, & $-93$ \hspace{3mm} & $-4.4$ \hspace{3mm} & $-0.5$
\hspace{3mm} \\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}
\includegraphics[width=\columnwidth]{bands.pdf}
\caption{\label{bands} Electronic band structure of and the
electronic distribution near the Fermi level ($E_{\rm F}\pm 0.1$\,eV) of
($A$O$A${N})$^{\rm t}$, (a1) (CaOCaN)$^{\rm t}$\, (a2) (SrOSrN)$^{\rm t}$, and (a3) (BaOBaN)$^{\rm t}$; and $(A$O($A$N)$_2$$A$O)$^{\rm t}$, (b1)
O/(Ca$_2${N}-Ca$_2${N})$^{\rm t}$/O\, (b2) O/(Sr$_2${N}-Sr$_2${N})$^{\rm t}$/O, and (b3) O/(Ba$_2${N}-Ba$_2${N})$^{\rm t}$/O. Electronic distribution of
the NFE state at the $\Gamma$-point (inset). Isosurfaces of
0.002\,$e$/\AA$^2$.}
\end{figure}
The electronic band structures of ($A$O$A${N})$^{\rm t}$\, and $(A$O($A$N)$_2$$A$O)$^{\rm t}$, Fig.\,\ref{bands},
indicates they are half-metals. In Figs.\,\ref{bands}(a1)--(c1) and (d1)--(f1),
the metallic channels are characterized by spin-down (purple-lines), whereas the
spin-up energy bands (black-lines) are semiconductor with the valence band
maximum (VBM) lying at about 0.2\,eV below the Fermi level ($E^{\rm
VBM}$\,$\approx$\,$E_{\rm F}-0.2$\,eV) for $A$\,=\,Ca, while for $A$\,=\,Sr and
Ba we find $E^{\rm VBM}$\,$\approx$\,$E_{\rm F}-0.5$\,eV. The lowest unoccupied
states are spin degenerated, lying between 1 and 2\,eV above $E_{\rm F}$, and
characterized by NFE parabolic bands localized on the oxidized surface (O$A$)
[insets of Figs.\,\ref{bands}(a1)--(c1) and (d1)--(f1)]. Further real space
projections of the electronic states near the Fermi level, $E_{\rm F}\pm
0.1$\,eV, reveal that the half-metallic bands are mostly ruled by in-plane
N-2$p$ orbitals localized in the $A$N layers of ($A$O$A${N})$^{\rm t}$,
Figs.\,\ref{bands}(a2)--(c2). Similarly, in the bilayer systems the
half-metallic bands spread out through the $A$N layers. However, in $(A$O($A$N)$_2$$A$O)$^{\rm t}$,
these half-metallic channels are sandwiched by the oxidized $A$O sheets
[Figs.\,\ref{bands}(d2)--(f2)]. These oxidized sheets may act as a shield,
protecting the half-metallic channels against the environment conditions, which
is a quite appealing property for development of spintronic devices based on 2D
platforms.
\section{Summary and Conclusions}
By means of first-principles DFT calculations, we have performed a theoretical
study of the fully oxidized 2D single layer, O/$A_2B$, and bilayer, O/$(A_2B)_2$/O,
electrenes, with with $A$\,=\,Ba, Ca, Sr, Y, and $B$\,=\, As, N, P, C. We found
that O/$A_2B$\, and O/$(A_2B)_2$/O\, systems with $A$\,=\,Ca, Sr, Ba , and $B$\,=\,N
become stable upon an hexagonal\,$\rightarrow$\,tetragonal structural
transition, resulting in layered tetragonal nitride systems, ($A$O$A${N})$^{\rm t}$\, and
$(A$O($A$N)$_2$$A$O)$^{\rm t}$. Further characterizations, through simulations of XANES spectroscopy,
allowed us to identify key aspects of the absorption spectra and their
correlation with the strutural properties the oxidized systems. We found the
emergence of a ferromagnetic phase in the oxidized tetragonal strutures, with
the net magnetic moment mostly ruled by the planar N-$2p_{\rm x,y}$ orbitals.
Meanwhile, electronic structure calculations reveal the formation of
half-metallic bands spreading out through the $A$N layers, with nearly
negligible contribution from the oxidized $A$O sheets. These results reveal
that the oxidized ($A$O$A${N})$^{\rm t}$\, and $(A$O($A$N)$_2$$A$O)$^{\rm t}$\, systems are quite interesting
platforms for spin-polarized transport on 2D systems, characterized
spin-polarized metallic channels shielded by oxide layers. For instance,
$(A$O($A$N)$_2$$A$O)$^{\rm t}$\, can be viewed as a core-shell 2D platform with half-metallic channels
lying on the $(A\text{N})_2$ layers (core) protected against the environment
conditions by the oxidized $A$O sheets (shell).
\begin{acknowledgments}
The authors acknowledge financial support from the Brazilian agencies CNPq,
CAPES, and FAPEMIG, and the CENAPAD-SP and Laborat{\'o}rio Nacional de
Computa{\c{c}}{\~a}o Cient{\'i}fica (LNCC-SCAFMat2) for computer time.
\end{acknowledgments}
\section{Appendix}
In Figs.\,\ref{md-1ml} and \ref{md-2ml} we present our results of MD simulation
o oxidized single layer electrene, O/$A_2B$, with $A$= A = Ca, Sr,
Ba, Y, and $B$ = N, P, As, C, and bilayer (O/($A_2\text{N})_2$/O) electrenes, with $A$=Ca, Sr
and Ba. We have considered a total simulation time of 15\,ps, and time steps of
1\,fs. Inset we present the strutural model after 15\,ps of simulation at
300\,K.
\begin{figure*}
\includegraphics[width=2\columnwidth]{md-1ml.pdf}
\caption{\label{md-1ml} Total energy fluctuation, of
hexagonal (a) and tetragonal (b) O/$A_2B$\, oxidized electrenes, as a function of
the time step (1\,fs). Insets, strutural model after 15\,ps of molecular
dynamics simulation at 300\,K.}
\end{figure*}
\begin{figure}
\includegraphics[width=\columnwidth]{md-2ml.pdf}
\caption{\label{md-2ml} Total energy fluctuation, of
hexagonal (a) and tetragonal (b) O/($A_2\text{N})_2$/O\, oxidized electrenes, as a function
of the time step (1\,fs). Insets, strutural model after 15\,ps of molecular
dynamics simulation at 300\,K.}
\end{figure}
|
2,877,628,090,102 | arxiv | \section{Introduction}\label{sec:Introduction}
The Kerr/CFT correspondence was first proposed by studying the near horizon geometry of extreme Kerr black hole (NHEK) \cite{Guica:2008mu}, which has an $SL(2,\mathbb{R})\times U(1)$ isometry group. Following the spirit of \cite{Brown:1986nw}, the asymptotic symmetry group (ASG) of the NHEK geometry was studied. It was shown that the $U(1)$ isometry could be enhanced to a copy of Virasoro algebra, whose quantum version give rise to a central charge $c_L=12J$ with $J$ being the angular momentum of the black hole. From the Frolov-Thorne vacuum for extreme Kerr, it turns out that the dual CFT may have a nonvanishing left temperature $T_L=1/2\pi$. Then it was shown that the Bekenstein-Hawking entropy could be reproduced exactly by using the Cardy formula. This motivated the conjecture that the extreme Kerr black hole is dual to a two-dimensional CFT.
In the derivation of the ASG of NHEK, a set of boundary conditions was proposed in \cite{Guica:2008mu}. These boundary conditions are unusual in the sense that some of the allowed deviations are of the same order as the background. This is quite different from the AdS$_3$ case studied in \cite{Brown:1986nw}, where all the deviations are subleading. The issue became more interesting when another set of consistent boundary conditions was proposed in \cite{Matsuo:2009sj}, from which it was shown that the $SL(2,\mathbb{R})$ isometry of NHEK could also be promoted to another Virasoro algebra.
This symmetry is related to the excitations of the right-moving sector \cite{Castro:2009jf}, which are suppressed in the extreme limit. The boundary conditions in \cite{Matsuo:2009sj} are similar to the usual ones, with all the deviations are subleading.
However the boundary conditions for two copies of Virasoro algebra are not consistent with each other: the boundary conditions for the left mover excludes the right mover's, and vice versa. Although various efforts have been made \cite{Matsuo:2009yet,Matsuo:2010newlimit}, the boundary conditions simultaneously admitting two copies of Virasoro algebras are still unavailable.
Recently, a physical approach for \emph{deriving} the boundary conditions, rather than postulating them \emph{a priori}, was proposed by Porfyriadis and Wilczek \cite{Porfyriadis:2010vg}. By requiring finiteness of the boundary effective actions, they derived the asymptotic symmetries of new asymptotically AdS$_3$ spaces with further relaxed but still consistent boundary conditions. It is impressive that these asymptotic symmetries satisfy the Virasoro algebra with the same central charges as the ones found in \cite{Brown:1986nw}. Despite its remarkable power in obtaining the ASG of AdS$_3$, it is not clear if the effective action approach is still efficacious in other cases.
In this paper, we would like to apply this effective action approach to the study of the ASG and the consistent boundary conditions of the NHEK geometry. Although we fail to find the boundary conditions admitting two copies of Virasoro algebras, we do find a set of
new boundary conditions, which are more relaxed compared to the boundary conditions proposed in \cite{Matsuo:2009sj} but still allows one copy of conformal group as its ASG. In the covariant formalism, the asymptotic charges are finite, and the central charge turns out to be zero. As in \cite{Matsuo:2009sj}, we consider the quasi-local charge and obtain the corresponding central charge with appropriate regularization. We find that the central charge could not be determined since it depends on the higher order terms of the asymptotic Killing vectors, which could not be fixed in the effective action approach. Moreover, the anomalous transformations of the mass and the angular momentum depend on the higher order terms as well. This makes a consistent truncation on higher order terms impossible to account for a finite temperature effect. It turns out that to regain the expected right-moving central charge from the corresponding quantized Virasoro algebra, we have to revert to the boundary conditions presented in \cite{Matsuo:2009sj}. Furthermore our study shows that the boundary conditions suggested in \cite{Guica:2008mu} are in conflict with the finiteness of the boundary effective action. This indicates that the power of the effective action approach is restricted: it is not sufficient to fix the essential higher order terms of the ASG, in the meanwhile it excludes some of the interesting ASG's and consistent boundary conditions of a background.
\vspace*{2mm}
In the next section we review the effective action approach. In Section 3 we review briefly the NHEK geometry and its ASG discussed in the literature. We derive the asymptotic Killing vector $\xi$ and give the new boundary conditions for NHEK through effective action approach in Section 4. We end with some discussions in section 5. A discussion on asymptotic conformal Killing vectors is given in Appendix B.
\vspace*{2mm}
\section{Effective Action Approach}\label{sec:Action}
In \cite{Porfyriadis:2010vg}, Porfyriadis and Wilczek constructed the effective action of general relativity (GR) for small excitations $g_{\mu\nu}\to g_{\mu\nu}+{\cal L}_\xi g_{\mu\nu}$ to the second order and derived the corresponding equation of motion. Starting from the Einstein-Hilbert action with a cosmological constant $\Lambda$,
\begin{equation}\label{Einstein-Hilbert action}
S=\int_M d^n x\,\sqrt{-g}\,(R-2\Lambda)\,,
\end{equation}
we put $g_{\mu\nu}\to g_{\mu\nu}+h_{\mu\nu}$ and expand to the second order in $h$:
\begin{equation}\label{S0[h]+S1[h]+S2[h]}
S=S^{(0)}[h]+S^{(1)}[h]+S^{(2)}[h]+{\cal O}(h^3)\,.
\end{equation}
The effective action for $\xi$ is obtained by putting $h_{\mu\nu}={\cal L}_\xi g_{\mu\nu}=\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}$. The first order action for $\xi$ turns out to be a boundary term:
\begin{eqnarray}\label{S1}
S^{(1)}[\xi]&=&\int_{\partial M} d^{n-1} x\,\sqrt{-\gamma}\,n^{\mu} \left(\Box\,\xi_{\mu}-\nabla_{\nu}\nabla_{\mu}\xi^{\nu}+(R-2\Lambda)\xi_{\mu}\right)\,,
\end{eqnarray}
which is a consequence of the diffeomorphism invariance of the Einstein-Hilbert action.
The second order action for $\xi$ takes the form
\begin{eqnarray}\label{S2}
S^{(2)}[\xi]&=&-\int_M d^n x\,\sqrt{-g}\,(G^{\mu\nu}+\Lambda g^{\mu\nu})(\xi_{\alpha;\,\mu\nu}-R_{\alpha\mu\nu\sigma}\xi^{\sigma})\,\xi^\alpha\nonumber\\
&&-\int_{\partial M} d^{n-1} x\,\sqrt{-\gamma}\,n^{\mu}\,\Big\{\xi_\sigma \nabla^\nu \nabla^\sigma\nabla_\nu \xi_\mu
-\xi_\mu \nabla^\nu\nabla^\sigma\nabla_\nu \xi_\sigma\\
&&\qquad\quad+(\nabla_\mu \xi_\nu+\nabla_\nu \xi_\mu)(\Box\, \xi^\nu-\nabla^\nu\nabla^\sigma \xi_\sigma)
+R_{\mu\nu\rho\sigma}\xi^\rho\nabla^\sigma \xi^\nu\nonumber\\
&&\qquad\quad+\frac{1}{2}\nabla_\mu\left[(\nabla^\nu \xi_\nu)^2-(\nabla^\nu \xi^\sigma)(\nabla_\nu \xi_\sigma)\right]
+2(\nabla^\nu \xi^\sigma)(\nabla_\nu\nabla_\sigma \xi_\mu-\nabla_\mu\nabla_\sigma \xi_\nu)\nonumber\\
&&\qquad\quad-\xi^\nu \nabla^\sigma\left[\xi_\mu (G_{\nu\sigma}+\Lambda g_{\nu\sigma})+\xi_\nu (G_{\mu\sigma}+\Lambda g_{\mu\sigma})-\xi_\sigma (G_{\mu\nu}+\Lambda g_{\mu\nu})\right] \Big\}\,.\nonumber
\end{eqnarray}
If we assume that the background $g_{\mu\nu}$ satisfies the Einstein field equation, $G^{\mu\nu}+\Lambda g^{\mu\nu}=0$, then the second order action also reduces to a boundary term, as a consequence of the gauge invariance of the second order action for $h$. While if we do not assume that the background solves the Einstein equation, the variational principle with $\delta \xi^\alpha$ leads to a beautiful equation of motion:
\begin{equation}\label{EOM}
(G^{\mu\nu}+\Lambda g^{\mu\nu})(\xi_{\alpha;\mu\nu}-R_{\alpha\mu\nu\sigma}\xi^{\sigma})=0\,.
\end{equation}
The equation (\ref{EOM}) is a contraction of two factors:
\begin{enumerate}
\item[i)] $G_{\mu\nu}+\Lambda g_{\mu\nu}=0\,$, which is satisfied by exact solutions to Einstein's GR;
\item[ii)] $\xi_{\alpha;\mu\nu}-R_{\alpha\mu\nu\sigma}\xi^{\sigma}=0\,$, which is satisfied by exact Killing vectors.
\end{enumerate}
The authors suggested that the Eq. (\ref{EOM}) should be satisfied in the asymptotic limit by approximate solutions and their corresponding asymptotic Killing vectors. Furthermore, it is expected to vanish fast enough so that the integrated action/unit time can be arbitrarily small provided one begin with $r$ sufficiently large. Most importantly, the boundary piece of $S^{(2)}[\xi]$ as well as the first order action $S^{(1)}[\xi]$ need to remain finite. It were such requirements that enable one to derive asymptotic symmetry vectors without imposing the boundary conditions \emph{a priori}.
\vspace*{2mm}
\section{The NHEK Geometry}\label{sec:NHEK}
Before applying the effective action approach to the NHEK case, let us review the NHEK geometry briefly. The Kerr metric in terms of the Boyer-Lindquist coordinates is of the form
\begin{equation}
ds^2=-{\Delta \over \rho^2}\left(d\hat t-a \sin^2\theta d\hat\phi\right)^2+{\sin^2 \theta \over \rho^2}
\left((\hat r^2+a^2)d\hat \phi-a d\hat t\right)^2+
{\rho^2 \over\Delta}d\hat r^2+\rho^2 d\theta^2\,,
\end{equation}
\begin{equation}
\Delta\equiv\hat r^2-2M\hat r+a^2\:,\quad\quad\quad
\rho^2\equiv\hat r^2 +a^2\cos^2 \theta,\end{equation}
where we take $G=\hbar=c=1$. It is parameterized by the mass $M$ and angular momentum $J=aM$. The horizons and the Hawking temperature are given by
\begin{equation}
r_\pm=M\pm\sqrt{M^2-a^2}\,,\quad\quad\quad T_H={r_+-M \over 4\pi Mr_+}\,.
\end{equation}
We consider the near horizon geometry of the extreme Kerr with $J=M^2$. Defining new coordinates
\begin{equation}
t=\frac{\lambda \hat{t}}{ 2M}\;, \;\;\;\;\;
x= \frac{\hat{r}-M }{\lambda M} \;, \;\;\;\;\;
\phi=\hat{ \phi}- {\hat{t} \over 2M}
\end{equation}
and taking the limit $\lambda\to 0$, we find the NHEK geometry in Poincar\'{e}-type coordinates
\begin{equation}
{ds^2 }= 2\, J\,\Gamma \left(-x^2dt^2+{dx^2\over x^2}+ d\theta^2 +\Omega^2 (d\phi + x dt)^2\right)\,
\end{equation}
where
\begin{equation}
\Gamma \equiv {1+\cos^2\theta\over 2}\;, \;\;\;\;\; \Omega \equiv {2 \sin \theta\over 1+\cos^2\theta}\,.
\end{equation}
In global coordinates, we have
\begin{equation}\label{nhek}
{d s^2 }= 2 \,J\,\Gamma \left( -(1+r^2){d\tau^2} + {dr^2\over 1+r^2} + d\theta^2 + { \Omega^2}(d\phi + {rd\tau})^2\right).
\end{equation}
In \cite{Guica:2008mu}, the following boundary conditions were chosen
\begin{equation}
h_{\mu\nu} =
\bordermatrix{
& & & & \cr
& \mathcal O(r^2)
& \mathcal O(1/r^2)
& \mathcal O(1/r)
& \mathcal O(1)
\cr
&
& \mathcal O(1/r^3)
& \mathcal O(1/r^2)
& \mathcal O(1/r)
\cr
&
&
& \mathcal O(1/r)
& \mathcal O(1/r)
\cr
&
&
&
& \mathcal O(1)
}
\label{bdryAndy}
\end{equation}
in the basis ($\tau, r, \theta, \phi$). Here $h_{\mu\nu}$ is the deviation from the background (\ref{nhek}).
Note that the deviations $h_{\tau\tau}$ and $h_{\phi\phi}$ are of the same order as the leading terms in the background (\ref{nhek}).
The most general diffeomorphisms preserving the above boundary conditions, which requires
\begin{equation}
\cL_\xi g_{\mu\nu}\sim h_{\mu\nu},
\end{equation} are
\begin{equation}\label{asyAndy}
\xi=(-r\epsilon'_1(\phi)+{\mathcal O}(1))\partial_r+(C_1+{\mathcal O}(1/r^3))\partial_{\tau}+{\mathcal O}(1/r)\partial_\theta} \def\Th{\Theta} \def\vth{\vartheta+(\epsilon_1(\phi)+{\mathcal O}(1/r^2))\partial_\phi
\end{equation}
where $\epsilon_1(\phi)$ is an arbitrary smooth function of $\phi$ and $C_1$ is an arbitrary constant. It does not contain the $SL(2,\mathbb{R})$ isometry subgroup of the background NHEK geometry, but still contains a copy of the conformal group generated by
\begin{equation}
\xi_1=\epsilon_1(\phi)\partial_\phi-r\epsilon'_1(\phi)\partial_r.
\end{equation}
As $\phi$ is periodic, one may expand the function $\epsilon_1(\phi)$ and obtain a set of Virasoro generators. Furthermore, there is another translational symmetry generator $\partial_\tau$, which commutes with $\xi_1$ and defines the energy $E$. As one takes the NHEK geometry as the ground state, one needs to impose the supplementary boundary condition $E=0$. It has been shown that the charge generating $\xi_1$ is finite around the NHEK geometry. Moreover, the Dirac bracket algebra of the charges gives rise to a central charge $c_L=12J$.
Another set of boundary conditions of NHEK was suggested in \cite{Matsuo:2009sj}
\begin{equation}
h_{\mu\nu} =
\bordermatrix{
& & & & \cr
& \mathcal O(1)
& \mathcal O(1/r^3)
& \mathcal O(1/r^3)
& \mathcal O(1/r^2)
\cr
&
& \mathcal O(1/r^4)
& \mathcal O(1/r^4)
& \mathcal O(1/r^3)
\cr
&
&
& \mathcal O(1/r^3)
& \mathcal O(1/r^3)
\cr
&
&
&
& \mathcal O(1/r^2)
\cr
}
\label{bdryMatsuo}
\end{equation}
in the basis ($\tau, r, \theta, \phi$). Obviously all the deviations are subleading.
The most general form of the asymptotic diffeomorphism preserving these boundary conditions is
\begin{eqnarray}
\label{asyMatsuo}
\xi_2&=&\left(-r\epsilon'(\tau)+\frac{\epsilon'''(\tau)}{2r}+{\mathcal O}(1/r^2)\right)\partial_r+\left(\epsilon(\tau)
+\frac{\epsilon''(\tau)}{2r^2}+{\mathcal O}(1/r^3)\right)\partial_{\tau} \nonumber\\
& &+{\mathcal O}(1/r^3)\partial_\theta} \def\Th{\Theta} \def\vth{\vartheta+\left(C_2-\frac{\epsilon''(\tau)}{r}+{\mathcal O}(1/r^3)\right)\partial_\phi
\end{eqnarray}
where $\epsilon(\tau)$ is an arbitrary function of $\tau$ and $C_2$ is a constant. It contains all of the isometries of the NHEK geometry. In particular the $SL(2,R)$ symmetry could be enhanced to the Virasoro algebra. However, the algebra of the corresponding asymptotic charge\cite{Barnich:2001jy, Barnich:2007bf} do not have a central extension. Nevertheless, using the quasi-local charge instead gives a nonvanishing central charge, which depends on a cutoff. This nonvanishing central charge suggest that there is a right-moving sector with physical degrees of
freedom.
\section{Asymptotic Killing Vectors for NHEK}\label{sec:AKV}
In this section we derive the asymptotic Killing vectors $\xi$ for the NHEK geometry through the effective action approach. We assume a power series expansion of the components of $\xi$ as
\begin{equation}\label{expansion}
\xi^\mu=\sum_{n \in Z} \xi^\mu_n (\tau,\theta,\phi)\, r^n\,
\end{equation}
and assume that each series truncates for some large $N$ onwards.
According to Porfyriadis and Wilczek \cite{Porfyriadis:2010vg}, the asymptotic symmetry algebra generating vectors $\xi$ are obtained by requiring ``small" asymptotic transformation, where ``smallness" is defined by requiring subleading Lie derivatives of the background metric, finite first order effective action $S^{(1)}[\xi]$, and finite second order effective action $S^{(2)}[\xi]$. For the NHEK case, since the boundary conditions with leading order perturbations had been exhibited in \cite{Guica:2008mu}, it is reasonable for us to allow the leading order Lie derivatives of the NHEK so as to accommodate all possibilities. We require that the ${\cal L}_\xi g_{\mu\nu}$ are of the leading order for the non-vanishing NHEK components $g_{\mu\nu}$ in (\ref{nhek}) and finite for others, that is
\begin{equation}
{\cal L}_{\xi} g_{\mu\nu} \sim h^0_{\mu\nu},
\end{equation}
where
\begin{equation}
h^0_{\mu\nu} =
\bordermatrix{
& & & & \cr
& \mathcal O(r^2)
& \mathcal O(1)
& \mathcal O(1)
& \mathcal O(r)
\cr
&
& \mathcal O(1/r^2)
& \mathcal O(1)
& \mathcal O(1)
\cr
&
&
& \mathcal O(1)
& \mathcal O(1)
\cr
&
&
&
& \mathcal O(1)
\cr
}
\label{bdry0}
\end{equation}
in the basis ($\tau, r, \theta, \phi$). The most general $\xi'$s satisfying the above conditions are given by:
\begin{eqnarray}
\xi^\tau&=&\epsilon(\tau)+\xi^\tau_{-1}(\tau,\phi)\frac{1}{r}+{\cal O}(\frac{1}{r^2})\,,\nonumber\\
\xi^r&=&{\cal O}(r)\,,\label{xi1}\\
\xi^\theta&=&{\cal O}(1)\,,\nonumber\\
\xi^\phi&=&\xi^\phi_{0}(\tau,\phi)+{\cal O}(\frac{1}{r})\,,\nonumber
\end{eqnarray}
where $\xi^\tau_{-1}$ and $\xi^\phi_{0}$ are arbitrary functions of $(\tau, \phi)$, and $\epsilon(\tau)=\xi^\tau_0$ is an arbitrary function of $\tau$. It accommodates both the diffeomorphisms (\ref{asyAndy}) and (\ref{asyMatsuo}).
Perturbing the NHEK metric (\ref{nhek}) using the $\xi'$s in (\ref{xi1}) gives the following boundary conditions
\begin{equation}
h^1_{\mu\nu} =
\bordermatrix{
& & & & \cr
& \mathcal O(r^2)
& \mathcal O(1)
& \mathcal O(1)
& \mathcal O(r)
\cr
&
& \mathcal O(1/r^2)
& \mathcal O(1/r)
& \mathcal O(1/r)
\cr
&
&
& \mathcal O(1)
& \mathcal O(1)
\cr
&
&
&
& \mathcal O(1)
\cr
}.
\label{bdry1}
\end{equation}
These boundary conditions are so relaxed that they accommodate both (\ref{bdryAndy}) and (\ref{bdryMatsuo}).
In order that the integrand of the first order effective action $S^{(1)}[\xi]$ (\ref{S1}) is finite everywhere on the boundary $r=\infty$, we find that the following relations among the leading order terms as well as the next-to-leading order terms in (\ref{xi1}) must be satisfied:
\begin{eqnarray}
&&\xi^r_1~~~~~\,\,=\,-\xi^\tau_{0,\,\tau}-\xi^\phi_{0,\,\phi}\,,\label{relation1}\\
&&\xi^\tau_{-1,\,\phi}~~=-2\,\xi^\phi_{0,\,\phi}\,,\label{relation2}\\
&&\xi^\phi_{0,~\phi\phi\phi}\,=\,0\,.\label{relation3}
\end{eqnarray}
Note that the equations (\ref{relation2}) and (\ref{relation3}) exclude explicitly the diffeomorphisms (\ref{asyAndy}) given in \cite{Guica:2008mu}.
In other words, the asymptotic transformations (\ref{asyAndy}) would lead to divergent first order boundary effective action.
Since the NHEK metric (\ref{nhek}) is also an exact solution of the Einstein field equation with vanishing cosmology constant, the second order effective action $S^{(2)}[\xi]$ (\ref{S2}) is again a boundary term. We require that the integrand of $S^{(2)}[\xi]$ to be finite as before. This imposes
\begin{eqnarray}
&&\xi^\tau_{-1}\,=0\,,\\
&&\xi^\theta_{0,\,\phi}=0\,
\end{eqnarray}
at leading order and a complicated equation at next-to-leading order, which is shown in Appendix A. In order to find a particular solution, we simply assume that $\xi^\theta_0=0$, and $\xi^\phi_{-1}\,,~\xi^\tau_{-2}\,,~\xi^r_{0,\,\phi}\,,~\xi^\theta_{-1,\,\phi}$ are all proportional to $\epsilon''(\tau)$, then we get the following solution
\begin{equation}
\xi^\phi_{-1}=-\,\epsilon''(\tau)\,,\quad\quad \xi^\tau_{-2}=\frac{1}{2}\,\epsilon''(\tau)\,, \quad\quad
\xi^r_{0,\,\phi}=0\,,\quad\quad \xi^\theta_{-1,\,\phi}=0\,.\label{partsl}
\end{equation}
We therefore arrive at
\begin{eqnarray}
\xi^\tau&=&\epsilon(\tau)+\epsilon''(\tau)\frac{1}{2r^2}+{\cal O}(\frac{1}{r^3})\,,\nonumber\\
\xi^r&=&-r\epsilon'(\tau)+\xi^r_0(\tau,\theta)+{\cal O}(\frac{1}{r})\,,\label{xi2}\\
\xi^\theta&=&\xi^\theta_{-1}(\tau,\theta)\frac{1}{r}+{\cal O}(\frac{1}{r^2})\,,\nonumber\\
\xi^\phi&=&\xi^\phi_0(\tau)-\epsilon''(\tau)\frac{1}{r}+{\cal O}(\frac{1}{r^2})\,.\nonumber
\end{eqnarray}
The $\xi$'s in (\ref{xi2}) are our final asymptotic Killing vectors of NHEK. The $\xi^\tau$ and the leading term of $\xi^r$ as well as the subleading term of $\xi^\phi$ are exactly the same as those in Eq. (\ref{asyMatsuo}). However,
the subleading term of $\xi^r$ and the leading term of $\xi^\phi$ are much more relaxed, so as all of the terms of $\xi^\theta$. As shown in \cite{Matsuo:2009sj}, expanding into modes $\epsilon(\tau)=\tau^{1+n}$,
one finds that to the leading order in $1/r$, the generators $\xi_n$ form the Virasoro algebra
\begin{equation}
[\xi_n, \xi_m]_{\text{L.B.}} = {\cal L}_{\xi_n}\xi_m = (m-n)\,\xi_{m+n} \,,\label{virasoro}
\end{equation}
here for the closing of the Virasoro algebra we define
\begin{eqnarray}
\xi^r_{0\,(m+n)} &=& \frac{1}{m - n}\left((1 + n)\, \tau^n \,\xi^r_{0\,(m)} - (1 + m)\, \tau^m\, \xi^r_{0\,(n)} + \tau^{n+1}\, \xi^r_{0\,(m),\,\tau}
- \tau^{m+1}\, \xi^r_{0\,(n),\,\tau}\right) \,,\nonumber\\
\xi^\theta_{-1\,(m+n)} &=& \frac{1}{m - n}\left((1 + n)\, \tau^n \,\xi^\theta_{-1\,(m)} - (1 + m)\, \tau^m\, \xi^\theta_{-1\,(n)}
+ \tau^{n+1}\, \xi^\theta_{-1\,(m),\,\tau} - \tau^{m+1}\, \xi^\theta_{-1\,(n),\,\tau}\right) \,,\nonumber\\
\xi^\phi_{0\,(m+n)} &=& \frac{1}{m - n}\left(\tau^{n+1}\, \xi^\phi_{0\,(m),\,\tau} - \tau^{m+1}\, \xi^\phi_{0\,(n),\,\tau}\right)\nonumber\,.
\end{eqnarray}
Note that for (\ref{virasoro}) we do not need to require $\xi^\phi_0(\tau)=0$. This is more relaxed compared to the case of \cite{Matsuo:2009sj}, where the leading term of $\xi^\phi$, i.e. the constant $C_2$ of (\ref{asyMatsuo}) need to be zero in order that the Virasoro algebra is closed.
A discussion on asymptotic conformal Killing vectors of NHEK which leave the first order effective action finite is given in Appendix B.
\vspace*{2mm}
Having established the asymptotic Killing vectors (\ref{xi2}) through the effective action approach, new boundary conditions could be obtained by perturbing the exact NHEK metric (\ref{nhek}) using these $\xi$'s\,:
\begin{equation}
h_{\mu\nu} =
\bordermatrix{
& & & & \cr
& \mathcal O(r)
& \mathcal O(1/r^2)
& \mathcal O(1/r)
& \mathcal O(1)
\cr
&
& \mathcal O(1/r^3)
& \mathcal O(1/r^2)
& \mathcal O(1/r^3)
\cr
&
&
& \mathcal O(1/r)
& \mathcal O(1/r^2)
\cr
&
&
&
& \mathcal O(1/r)
\cr
}.
\label{bdry2}
\end{equation}
These boundary conditions are more relaxed than the ones (\ref{bdryMatsuo}) but still subleading compared to the background (\ref{nhek}).
We check that the Lie derivatives of the perturbations still satisfy the boundary conditions, i.e. ${\cal L}_{\xi} h_{\mu\nu} \sim h_{\mu\nu}$.
Furthermore, adopting the covariant formalism developed by Barnich, Brandt and Comp\`ere \cite{Barnich:2001jy, Barnich:2007bf}, we find that the asymptotic charges $Q[\xi]$ corresponding to (\ref{xi2}) and (\ref{bdry2}) are finite, with the trivial symmetry transformations
\begin{eqnarray}
\xi_{tr}&=&{\mathcal O}(\frac{1}{r^3})\partial_{\tau} +\left(\xi^r_0(\tau,\theta)+{\mathcal O}(\frac{1}{r})\right)\partial_r\nonumber\\
&& +\left(\xi^\theta_{-1}(\tau,\theta)\frac{1}{r}+{\mathcal O}(\frac{1}{r^2})\right)\partial_\theta
+\left(\xi^\phi_0(\tau)+{\mathcal O}(\frac{1}{r^2})\right)\partial_\phi \label{xitr}
\end{eqnarray}
give rise to vanishing charges. The central charge of the Virasoro algebra at the Dirac bracket level turns out to be zero.
However, as in \cite{Matsuo:2009sj} we could also employ the quasi-local charge\cite{Brown:1992br} defined by using the surface energy momentum tensor to study the central charge. Unfortunately, it turns out that the terms in (\ref{xitr}) contribute to the central charge. We find that the following leading terms of the asymptotic Killing vectors
\begin{equation}
\xi_{\,l}=\left(\epsilon(\tau)+\epsilon''(\tau)\frac{1}{2r^2}\right)\partial_{\tau} - r \,\epsilon'(\tau) \partial_r
-\epsilon''(\tau)\frac{1}{r}\, \partial_\phi \label{xil}
\end{equation}
give rise to Virasoro algebra with central charge $c=6\, a^2/(G \Lambda)$, half of the value obtained in \cite{Matsuo:2009sj}. Moreover, when calculating the anomalous transformations of the mass and the angular momentum, we get
\begin{eqnarray}
\delta M &=& -\frac{a^2}{2 G \Lambda}\epsilon'''(\tau)\,,\nonumber\\
\delta J &=& -\frac{3 a^2}{2 G \Lambda^2}\epsilon'''(\tau)\,.
\end{eqnarray}
where $\Lambda$ is a large but finite radius where the boundary locates.
Comparing with the results in \cite{Matsuo:2009sj}, we find that our $\delta M$ is again half of the value obtained there, while $\delta J$ is one and a half of the value. Following the analysis of finite temperature effects in \cite{Matsuo:2009sj}, we find that it is impossible to match the entropy and the mass as well as the angular momentum between the Kerr black hole and the boundary CFT simultaneously. However, if we add a term
\begin{equation}
\xi_{ad}=\frac{\epsilon'''(\tau)}{2\,r}\,\partial_r
\end{equation}
to (\ref{xil}), we recover the same central charge and anomalous transformations as those in \cite{Matsuo:2009sj}. This shows that the higher order terms which could not be determined by the current effective action approach play essential roles. In fact all of the undetermined terms in (\ref{xitr}) contribute to the anomalous transformations as well as the central charge (see Appendix C). To regain the results obtained in \cite{Matsuo:2009sj}, the simplest way is to set the coefficient $\xi^r_{-1} = \epsilon'''(\tau)/2 $ and all of the other terms in (\ref{xitr}) to be zero, reverting to the ASG as well as the boundary conditions proposed in \cite{Matsuo:2009sj}.
\vspace*{2mm}
\section{Discussion}\label{discussion}
In this paper we derived new boundary conditions for the NHEK geometry through the effective action approach. We found that both the boundary conditions and the corresponding asymptotic Killing vector are relaxed compared to the ones in \cite{Matsuo:2009sj}. These boundary conditions are consistent and of subleading order asymptotically, leading to finite charges. However, although the corresponding ASG contains a copy of conformal group, their generators give different right-moving central charge when being quantized. To recover the result obtained in \cite{Matsuo:2009sj}, we have to restrict our asymptotic Killing vectors and eventually go back to the ASG as well as the boundary conditions in \cite{Matsuo:2009sj}, which could not be derived through the effective action approach. This shows that the power of the effective action approach is restricted, at least for the NHEK case.
The reason for this issue might be that the effective action (\ref{S0[h]+S1[h]+S2[h]}) developed by Porfyriadis and Wilczek was constructed only to the second order. In fact in their paper \cite{Porfyriadis:2010vg}, although more relaxed asymptotic Killing vectors for the AdS$_3$ were obtained, giving rise to the correct central charge, the higher order terms of the vectors (21) did contribute to the central charge. The point is that the contributions from the higher order terms of Brown-Henneaux's asymptotic symmetries (3) canceled each other totally, leading to the same central charge. It is possible that with higher order expansions of the action, we could be able to determine the essential higher order terms of the asymptotic Killing vectors (\ref{xi2}).
Moreover, it is quite unexpected that the effective action approach is not consistent with the diffeomorphisms in \cite{Guica:2008mu}. In fact one can check straightforwardly that the diffeomorphisms subject to the boundary conditions in \cite{Guica:2008mu} lead to a divergent first order boundary effective action. It may be general that the effective action approach does not allow for the boundary conditions with leading order perturbations, therefore excludes a variety of interesting possibilities of consistent boundary conditions.
On the other hand, our search for the ASG and consistent boundary conditions is not exhaustive. Note that our asymptotic Killing vector is only a particular solution of the equations from the finiteness of the second order effective action. It is possible that there exist more general solutions which lead to different consistent boundary conditions.
\vspace*{10mm}
\noindent
{\large{\bf Acknowledgments}}
The work was in part supported by NSFC Grant No. 10775002, 10975005. BC would like to thank
the organizer and participants of the advance workshop ``Dark Energy and Fundamental Theory" supported by
the Special Fund for Theoretical Physics from the National Natural Science Foundations of China with grant no: 10947203 for stimulating discussions and comments. BN would like to thank the hospitality of the National Center for Theoretical Sciences, Taipei, Taiwan, during the final stage of the work.
\vspace*{2mm}
|
2,877,628,090,103 | arxiv |
\section{Introduction}\label{sec:Intro}
The Large Hadron Collider (LHC) at the Center for European Nuclear Research (CERN) in Geneva, Switzerland, is preparing for an upgrade to the High-Luminosity LHC (HL-LHC). During this period, the HL-LHC is expected to reach a peak instantaneous luminosity of 7.5$\times10^{34} cm^{-2}s^{-1}$, which corresponds to an average of 200 inelastic proton-proton collisions per beam-crossing.
During the HL-LHC era, ATLAS is expected to receive an integrated luminosity of about 4000 fb$^{-1}$. To operate in this environment, the new tracker detectors should be able to record data with high trigger rates, close to 1 MHz. They also need to have high granularity and a high radiation tolerance. To achieve these requirements, ATLAS is upgrading its Inner Tracker (ITk) Detector, which includes the pixel and strip detectors. New hybrid pixel detectors are under development to adapt to the HL-LHC environment. As in the case of the detectors, the data acquisition (DAQ) system also needs an upgrade to read out upgraded pixel detectors efficiently. These detectors and DAQ systems require intensive testing in a similar environment to ensure they will perform reliably.
For intensive detector studies, there are dedicated facilities that provide a high-energy beam environment. Such facilities are currently located at CERN and FNAL (Fermilab) among others. For example, the Fermilab Test Beam Facility (FTBF) \cite{FTBF} provides a high-energy (120 GeV) proton beam for testing detectors. A tracking telescope in the test beam facility can provide a testing platform for these different pixel detectors to aid the upgrade efforts. The Argonne Pixel Tracking Telescope (APTT) is permanently installed at the Fermilab Test Beam Facility to aid these testing efforts. This telescope is intended to test the resolution and tracking efficiency of upgraded pixel detectors and to study the impact of irradiation on the performance of these same detectors. The APTT was commissioned using six planar n$^{+}$-in-n silicon sensors \cite{Capeans:1291633}, read out by the FE-I4B~\cite{672298352} front-end chips. The same technology is currently used in the ATLAS Insertable B-Layer (IBL)~\cite{Capeans:1291633} modules. IBL pixel modules were installed in the ATLAS tracker during a past upgrade.
The paper describes the FTBF, APTT instrumentation, DAQ system, reconstruction and simulation software, and the performance of the pixel telescope in detail.
\section{Telescope Performance}
\label{sec:performance}
A custom-made DAQ system is used for data taking at the Fermilab test beam. The DAQ system consists of two GUIs: calibGui and cosmicGui. The calibGui is used for setting the tuning parameters for each chip. Calibration includes various scans through which noisy pixels are masked and the threshold is set. Telescope planes are tuned to a 3000e threshold. The threshold is controlled by the two discriminators located on each pixel front-end chip. The first discriminator controls the threshold for the individual pixel while the second controls an entire front-end chip. In the threshold scan, a fixed charge is injected into the pre-amplifier on the front-end chip. This step is repeated multiple times and the number of hits is recorded as a percentage of injection. Again this step is performed by changing the discrete voltage steps in a pre-amplifier for a specified charge range. Ideally, this process will produce a step function: zero injection resulting in a hit for any charge below the threshold and all injections resulting in a hit for any charge above the threshold. Due to the electronic noise, the curve is smeared in an S-shaped and hence called an "S-Curve". The mean value of the Gaussian error fit function will record the threshold for each function and the sigma value will define the noise of each pixel. Figure \ref{fig:tuneplot} shows the S-Curve and threshold distribution of the tuned chip.
\begin{figure}[!h]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/scan.pdf}
\caption{}
\label{fig:tuneplane1}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Threshold.pdf}
\caption{}
\label{fig:tuneth}
\end{subfigure}
\hfill
\caption{For the first telescope plane (Tel0) (a) S-Curve distribution of the tuned chip (b) Threshold distribution after tuning the chip}
\label{fig:tuneplot}
\end{figure}
The FE-I4B chip has a self-triggered mode, which can be used to test the detector using an external source before placing it in the beamline.
The cosmicGui is used for real-time data taking. For the data taking, the first and the last planes are used as trigger planes, where HitOr signals from these planes are fed to RCE, and for the given trigger interval, the data is read from all telescope planes.
\subsection{Occupancy}
The cosmicGui has an online monitoring feature that monitors the basic characteristic plots like hit occupancy, correlations, timing, and cluster size. The hit occupancy plots show the map of the number of pixels for a given sample of events. The alignment of the planes can be monitored using the correlation hit plots between the neighboring planes. The position between the planes is fixed during data taking. The first telescope plane's occupancy plots using wide and narrow beam settings are shown in Figures~\ref{fig:WB} and~\ref{fig:NB}, respectively. Correlation between the columns of the first plane and rows of the second plane are shown in Figures~\ref{fig:cowb} and~\ref{fig:conb}.
\begin{figure}[!h]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/WBOcc.pdf}
\caption{}
\label{fig:WB}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/NBOcc.pdf}
\caption{}
\label{fig:NB}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/WBCor.pdf}
\caption{}
\label{fig:cowb}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/NBCor.pdf}
\caption{}
\label{fig:conb}
\end{subfigure}
\caption{120 GeV beam energy profile with occupancy plots: (a) Wide beam profile (b) Narrow beam profile. Correlation between columns of the first plane and rows of the second plane: (c) For wide beam (d) For narrow beam}
\label{fig:BP}
\end{figure}
\newpage
\subsection{Cluster Size}
When a particle passes through the tracking plane, it can fire more than one pixel per hit. The clustering algorithm groups such clusters together produced due to the single hit. In the current test beam setup, the observed average cluster size is larger than one. The simulation results also show a similar cluster size. Figure~\ref{fig:cs0x} shows the observed and simulated cluster sizes for the first telescope plane (Tel0). All six telescope planes have similar cluster-size responses.
\begin{figure}[!h]
\centering
\includegraphics[width=0.55\textwidth]{figures/cluster_comp_dut0_tel0.pdf}
\caption{Normalized plot of cluster size in data and simulation for the first telescope plane}
\label{fig:cs0x}
\end{figure}
\subsection{Tracking Efficiency}
The tracking procedure in the Proteus software allows one to evaluate the tracking efficiency of each pixel within the tracking plane. The track must pass through all six tracking planes and should have a cluster in each plane. This condition helps to eliminate the tracks with larger scattering angles. If the extrapolated track matches the cluster in the given plane, then that track is included in the efficiency measurement.
\begin{figure}[!h]
\centering
\includegraphics[width=0.65\textwidth]{figures/tel0-eff.pdf}
\caption{Track detection efficiency of the first telescope plane}
\label{fig:trkeff}
\end{figure}
For each plane, one can set a range of included pixels within the framework. For the current setup in the test beam, each tracking plane's efficiency is measured by considering that plane as a DUT. In each plane, the pixel range is set for the efficiency measurement. The way chips in the telescope planes are aligned with each other, the cooling plates from other telescope planes slightly overlap with some rows and columns of the active chip. This effect reflects in the efficiency plot shown in Figure~\ref{fig:trkeff}. Therefore, only the active range of the rows and columns of the pixels are considered while calculating the track efficiency. The detection efficiency for each plane is around 99\%, as shown in Figure~\ref{fig:trkeff}.
\section{Summary}\label{sec:summary}
The Argonne FE-I4B pixel tracking telescope is installed successfully at the Fermilab Test Beam Facility in the MT6.1B enclosure. The telescope commissioning data were taken during the January-February 2020 test beam campaign. The telescope will be used to characterize and test various pixel detectors and DAQ systems developed for the HL-LHC ATLAS upgrade. The performance of the telescope has been studied and compared with simulated results. Characteristic plots like cluster size, pixel efficiency, and spatial resolution are made. The observed residual of the telescope plane in the X-direction is \mbox{$\sigma_{x,meas} =$ 71.83 $\mu$m} whereas the Y residual is \mbox{$\sigma_{y,meas} =$ 12.78 $\mu$m}. These residual values are consistent with expected residuals calculated using the pitch of the detector. The expected residual in X-direction is \mbox{$\sigma_{x,cal} =$ 72.2 $\mu$m,} whereas it is $\sigma_{y,cal} =$ 14.4 $\mu$m in Y-direction. The AllPix$^{2}$ software is used to simulate the test beam scenario. The simulated residuals are also within the experimental limits and they are \mbox{$\sigma_{x,sim} =$ 65.2 $\mu$m} in X-direction and \mbox{$\sigma_{y,sim} =$ 7.9 $\mu$m} in Y-direction.
\section{Fermilab Test Beam Facility}
\label{sec:FTBF}
The FTBF is one of a few facilities in the world that provides a high-energy beam with very high intensity.
At the FTBF, the primary beam consists of 120 GeV protons, whereas the secondary beam provides pions, muons, and electrons, with energies ranging up to 66 GeV. The typical beam size is about 1 cm in diameter and can be adjusted to some degree by quadrupole magnets as per the requirement. The beam is delivered in a spill every 60 seconds with a 4.2 seconds duration. The maximum available rate is approximately 2.5 GHz/cm$^{2}$, which is about 5E5 particles per spill.
The APTT is installed in the MTest (MT6) enclosure. The FTBF facility provides scintillator counters with wire chambers at each enclosure to provide the particle count rate. The MT6 enclosure is equipped with patch panels to access the local network from the control room. Also, there is access to inert gases such as nitrogen for the experiment.
\section{Instrumentation}
\label{sec:Instru}
The APTT consists of six modules, divided into two sets of three modules. Telescope modules referred as telescope planes are mounted on the railings with adjustable stages along and transverse to the beamline. The added mechanical structure is convenient for remotely aligning detectors in the beam from the control room. The HSIO-II/RCE~\cite{hsio} DAQ system records the data from all six telescope planes. The test beam data is reconstructed using the "Proteus"~\cite{proteus} reconstruction software, which produces all performance characteristics plots of offline alignment such as cluster size, pixel efficiencies, and residuals. These test beam data results are compared with the simulation results produced using the "AllPix$^{2}$"~\cite{allpix} simulation software.
\subsection{Telescope Planes}
\label{sec:planes}
Six n$^{+}$-in-n planar silicon sensors are used in the Argonne Pixel Tracking Telescope. Each sensor is 200 $\mu$m thick and 41.4 mm $\times$ 37.5 mm in size. This sensor is read out by four front-end FE-I4B chips. This chip is built in a 130 nm CMOS process and has an active area of \mbox{20.2 $\times$ 18.8 mm$^{2}$.} It has a pixel matrix of 80 columns on 250 $\mu$m pitch by 336 rows on 50 $\mu$m pitch.
\begin{figure}[!h]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth, angle=0.5]{figures/TelPlane.png}
\caption{}
\label{fig:BM}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=0.7\textwidth]{figures/tel.jpg}
\caption{}
\label{fig:module}
\end{subfigure}
\caption{ (a) FE-I4B module glued on the outer PCB. Letters A, B, C, and D indicate four chips on the modules (b) FE-I4B module mounted on the downstream telescope arm}
\label{fig:telplane}
\end{figure}
The bare module is a silicon sensor bump-bonded to four FE-I4B chips i.e. each chip is approximately bump-bonded to the 1/4th of the sensor area. During the module assembly, the bare module is glued to the backside of the flexible printed circuit board (flex PCB). Wire bonds between the flex PCB and the chip provide the necessary electrical connection. The assembled module is then glued to the outer PCB board, which is designed for mounting the planes in the beamline. This board consists of RJ45 connections for data links, lemo connections for the low voltage (LV) and internal Hit-OR trigger, and an SHV connection for the high voltage (HV). Removable jumpers on the data links on the PCB board give the flexibility to choose the primary chip from any of the four chips. Figure \ref{fig:telplane} shows an assembled FE-I4 module on the PCB board and the front view of the module after installing it as a telescope plane in the beamline.
As per Figure \ref{fig:BM}, the 250 $\mu$m pitch is in the X direction and the 50 $\mu$m pitch is in the Y direction. Due to the non-squared shape of the sensor pixels, the second and fourth telescope planes are rotated by 90$^{\circ}$ around the beam axis to achieve comparable spatial resolution in the X and Y direction. Only one chip per module is actively used for data taking. The active chip in each telescope plane is chosen based on its alignment with the beam and other telescope planes and also the constraints on the beam spot size. Table~\ref{table:tab1} summarizes the active chip used for data taking for each telescope plane.
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Telescope Planes & Tel0 & Tel1 & Tel2 & Tel3 & Tel4 & Tel5 \\ [0.5ex]
\hline
Active chip & C & B & B & B & D & C \\
\hline
\end{tabular}
\caption{Active chip in each telescope plane for data taking}
\label{table:tab1}
\end{table}
During data taking, the FE-I4B chip measures deposited charge by monitoring a firing of discriminator and time-over-threshold (ToT) with a 4-bit resolution at 40MHz external clock signal. With Digital-to-Analog Converters (DACs), the global threshold on the chip can be set by applying the local correction to each individual pixel. The hit information and 8-bit time stamp are stored in memory cells shared by four adjacent pixels. The chip stores the hit information with a latency interval, which is programmable up to 255 cycles of the external clock. This information is read out when a trigger signal is received in this interval. Synchronized data was collected from the telescope plane by supplying the simultaneous triggers to the first and last telescope planes.
\subsection{Mechanics}
\label{sec:mech}
The setup of the APTT in the MT6 enclosure at FTBF and the schematic diagram of this telescope are shown in Figure~\ref{fig:schematel}. Telescope planes are installed on rails, referred to as the telescope arms. Each arm consists of three planes, and between these two arms, the cold box is installed on the movable stage. For better alignment, each telescope plane is mounted on a movable stage. The stage can be moved in the X' and Y' directions perpendicular to the beam by 100 mm. These stages are configured to operate remotely.
A cooling system is added for the telescope planes to avoid overheating the chips. This cooling system includes a cold plate with a thermally conductive graphite sheet to transfer heat out of the module for each plane. The cold plate is cooled down to 5$^{\circ}$C by circulating cold distilled water, which helps to maintain the temperature of the unpowered chips around 10$^{\circ}$C and around 30$^{\circ}$C when the chips are powered.
In the test beam setup, a right-handed Cartesian system is used for global and local coordinate systems. In the global coordinate system, the beam points in the Z'-direction and the Y'-direction points vertically upward. In each chip, the lower-left pixel is considered as the origin of the local system. The local frame position determines pixel hits, and then it is transformed into the global frame for tracking purposes. The details about the transformation can be found in Ref. \cite{allpix}.
\begin{figure}[!h]
\centering
\begin{subfigure}[b]{0.47\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/tel.png}
\caption{}
\label{fig:tel}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/schematic.pdf}
\caption{}
\label{fig:schema}
\end{subfigure}
\caption{(a) Telescope setup in the FTBT MT6B enclosure (b) Schematic diagram of the setup }
\label{fig:schematel}
\end{figure}
Racks are mounted under the telescope table to hold electronics, power supplies, data acquisition computers, and chillers. The following subsections include a detailed description of these various types of equipment. These are connected and operated using a local network.
\\
\\
\textbf{Power supply:}
The telescope needs different voltages for operations. Low-voltage (LV) is required to power the chip, and high-voltage (HV) to apply a bias voltage to the sensor (-80V). WIENER MPOD \cite{weiner} is a multi-channel LV and HV computer controlled power supply. It is a 19" mountable chassis with module cages for 10 LV or HV modules. Currently, this MPOD crate is equipped with one iSeg EHS F410n \cite{iseg} HV module and two WIENER OMPV8008 LV modules. The HV module has 16 channels, and each channel can supply up to 1 kV and a maximum of 8 mA current through SHV connectors. The LV modules exhibit 8 channels. The signals from four LV channels are driven through a 37-pin sub-D connector. Local control of the MPOD crate is available using the LCD and currently operating remotely through the local network.
\\
\\
\textbf{PDU and Network Switch:}
The CyberPower PDU15SW8FNET \cite{PDU}, an 8-outlet (front) rackmount switched power distribution unit (PDU) is used, which provides 120V, 15A output. The outlet receptacles can be managed locally or over the network using the PDU console, web browser, and RJ45 Ethernet port for network connection.
For the local network connection, the tp-link 24-Port Gigabit Switch is used. The switch is connected to Fermilab's internal network, which can be accessed from the control room.
\\\\
\textbf{Cooling System:}
The main focus of the pixel telescope is to characterize the Device Under Test (DUT). One of the critical tests for high-energy experiments is to study irradiation effects on the detector. Irradiated modules require cooling to limit the leakage current. DUTs are cooled using the Ultra-Low Refrigerated-Heating Circulator, Julabo FP89-HL \cite{julabo}. Tubing from the chiller is connected to the base plate, which holds the DUTs. For cooling, cold silicon oil is circulated through base plates. The chiller can reach the minimum cooling temperature of -90$^{\circ}$C and it is remotely controlled using a DCS computer via an RS-232 serial socket connection. In addition to the circulating chiller fluid, cooled nitrogen or dry air is available to flush in the cold box to lower the air temperature and maintain the humidity to keep the DUT temperature always below the dew point. This chiller is dedicated for cooling DUT for future studies. With this system the DUT can be cooled down to -35$^{\circ}$C.
The Thermo Fisher Polar Series, Accel 500 LT Cooling/Heating Recirculating Chiller \cite{ThermoFisher} is used to cool the telescope plane. The temperature range for the chiller is from -25$^{\circ}$C to +80$^{\circ}$C. The distilled water is used as a coolant for this chiller.
\\
\\
\textbf{Temperature \& Humidity Sensors:}
It is crucial to monitor the temperature and humidity of the cooled DUT to avoid condensation. Thorlabs TSP01 \cite{web:thorlabs} 2-channel compact USB temperature and humidity logger is used to monitor the DUT system. Logger software GUI provides the graphical interface of the readings.
\\
\\
\textbf{Cold box for DUTs:}
Irradiated DUTs need cooling for testing; hence a thermally insulated box is installed between the telescope arms as shown in Figure \ref{fig:tel}. Currently, the box can accommodate three DUTs. The box is made up of high-quality foam with an aluminum foil exterior. It is resting on a movable stage and has a sealed port to pass through various cables. The movable stage can be moved manually in the X'-direction with the large area translation plate (TBB1212) in the range of 120 mm, whereas the motorized high load vertical stage (MLJ150) is used in a vertical direction in the range of 50 mm. These stages are from Thorlabs, Inc. \cite{web:thorlabs}. The motorized stage can be controlled via an RS-232 serial port.
A 30 mm $\times$ 30 mm breadboard with embedded cooling tube is installed at the bottom of the box. For cooling the DUT, cold silicon oil is circulated through the breadboard, and cooled nitrogen or dry air is blown in the box.
This box is not used during characterization studies of the telescope planes but is intended to be used in the future to test the performance of irradiated DUT's.
\subsection{DAQ and Reconstruction Software}
\label{sec:reco}
\begin{flushleft}
\textbf{Data Acquisition System}
\end{flushleft}
The data acquisition system (DAQ) of the pixel telescope consists of a high-speed input-output board (HSIO II) with re-configurable cluster elements (RCE) and a computer with custom designed software for data taking.
The HSIO II is based on the Xilinx Virtex 7 Artix FPGA. It provides specific connectivity to the front-end electronics and processes the data received from it. It also generates the clock and triggers and issues them to all telescope planes.
The RCE is a generic computational unit based on a System-On-Chip (SOC), which can handle several lanes of high-speed I/O. It configures the HSIO board and also sends commands to the front-ends. The raw data from the HSIO is transferred via Ethernet to the DAQ computer.
The configuration file in the DAQ software allows for setting different global parameters such as chip ID, trigger mode, or delays for each telescope plane. These various settings are also accessible through a graphical user interface (GUI). The software provides online monitoring plots during data taking, including real-time hit maps, correlation plots of the hit position between the neighboring planes, and the charge and time information.
\begin{flushleft}
\textbf{Reconstruction Software}
\end{flushleft}
The test beam data from the telescope planes are saved as raw data in the RCE format. Particle tracks are reconstructed from this raw data using the Proteus reconstruction software. The program starts with the raw hit data and provides a fully reconstructed cluster and track. The software performs reconstruction in three steps, i.e., clustering, alignment, and track reconstruction. For each reconstruction, the global position, orientations of all detector planes, and type of detector for each plane must be provided in configurations using the TOML configuration file format.
RCE data are stored as a ROOT \cite{BRUN199781} n-tuples with a complete event and hit information. During the processing, first noise scans are performed on each telescope plane, which masks all the noisy pixels with noise level 5$\sigma$ above the average. The first telescope plane is considered as the reference plane, and the rest of the tracking planes are coarsely aligned with the reference plane based on cluster correlations. Then the fine alignment is performed using a track-based alignment, which minimizes track residuals.
The first step towards the event reconstruction is to cluster the pixel hits triggered by the same particle traversing through a sensor plane. Proteus implements a simple recursive clustering algorithm that groups all neighboring hits. Overlapping clusters can not be detected as separate clusters but are considered as one. The actual position of the hit is then calculated as the geometric mean of the pixel positions in that cluster. Next is the track reconstruction, in which the algorithm initially assumes that the track is parallel to the longitudinal direction of the telescope and then modifies the track as clusters are added. For standard operation, the track selection requirement is that all tracks must pass through all six planes. Straight-line fits are then performed on candidate tracks, and the one with the smallest $\chi^2$ is selected. The track position is always influenced by the current alignment of the telescope and therefore, the DUT is always excluded from the track fitting to form unbiased tracks.
\subsection{Simulation Software}
\label{sec:simu}
The simulation studies are performed to cross-check the telescope response regarding the residual widths. Events are simulated using AllPix$^{2}$ (Allpix squared) \cite{SPANNAGEL2018164}, a GEANT4-based simulation software framework written in C++. It provides a flexible interface to create an experimental setup for simulation in any configuration. For each simulation, there are three required layers of configuration files: the detector model configuration, which describes a particular type of detector, the detector configuration, which passes the detector geometry setup to the framework, and the main configuration, which contains both the global framework configuration and the list of modules to instantiate together with their configuration.
The pixel detector model can be defined by the pixel pitch, the thickness of the sensor, the readout chip for a hybrid configuration, the PCB properties, the mechanical support, and other geometry properties related to an assembly.
In the detector geometry configuration, the model type of all detectors, the number of detector planes, and the detector position and orientation are included. A unique ID number identifies each detector. The detectors' positions in space are specified by the coordinates (x', y', z') and the rotations around the x', y', and z' axes through the center of the detectors.
The main configuration consists of a set of sections specifying the modules used. All installed modules are loaded automatically. \textit{DepositionGeant4} module, an interface to Geant4, deposits charge carriers in the active volumes of all detectors and initializes the physical process to simulate a particle source that will deposit charge carriers for every simulated event.
An actual test beam setup is simulated with repeated sets of 100k events with the six FE-I4B telescope planes and the central plane on each side rotated by 90$^{\circ}$. The 120 GeV proton beam with a 5 mm radius is assumed. \textit{RCEWriters} module saves the output of the simulation in the RCE format, which is compatible with the Proteus telescope reconstruction software.
The telescope performance in the test beam and the simulation results are described in the Telescope Performance section~\ref{sec:performance}.
\section{Spatial Resolution}
\label{sec:reso}
\subsection{Measured Residuals}
APTT was commissioned during the test beam period in January-February 2020 at FTBF. The test beam data is reconstructed in three steps: clustering, alignment, and track reconstruction as described in Section~\ref{sec:reco}. Analysis shows that the tracks are reconstructed with 99\% efficiency. The track-fit parameter is used to derive the spatial resolution of the telescope. The digital hit information is taken into account during the reconstruction step; therefore, the resolution of the detector is mainly dependent on the position of the telescope planes and their pixel granularity.
Figure~\ref{fig:measureres} shows the residual distribution of the first telescope plane in the X and Y direction after alignment. The residual in the Y-direction follows the expected Gaussian distribution, whereas the X-residual exhibits the multiple peak structure. The non-square geometry of the pixels causes multiple peak structures in the X-residual. The detailed explanation of these phenomena is explained in Ref. \cite{Benoit_2016}.
Both the X and the Y residuals of the first telescope plane are centered at zero and the standard deviation of the distributions are $\sigma_{X,meas} =$ 71.83 $\mu$m and $\sigma_{Y,meas} =$ 12.78 $\mu$m, respectively.
\begin{figure}[!h]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/residual_data_dut0_Tel0_u.pdf}
\caption{}
\label{fig:res0x}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/residual_data_dut0_Tel0_v.pdf}
\caption{}
\label{fig:res0y}
\end{subfigure}
\caption{Residual distribution in the first telescope plane after alignment (a) in the X-direction (b) in the Y-direction }
\label{fig:measureres}
\end{figure}
For a binary readout the residuals in each direction can be calculated using the pitch of the pixels using the formula $\sigma_{cal} =$ pitch /$\sqrt{12}$ \cite{rossi2006pixel}. The pitch of the FE-I4B telescope plane in the X-direction is 250 $\mu$m, and therefore, the residual is $\sigma_{x,cal} =$ 72.2 $\mu$m. The pixels in Y-direction are 50 $\mu$m with corresponding residual width of $\sigma_{y,cal} =$ 14.4 $\mu$m. As shown in Figure~\ref{fig:cs0x}, clusters contained more than one hit, which leads to a better estimation of measured residual due to the charge sharing effect.
\subsection{Simulated Residuals}
The telescope performance is also studied using simulations based on the test beam scenario. The simulation is performed as described in Section~\ref{sec:simu}. The geometry and distance between the planes are set to the same values used in the test beam data analysis. The second and fifth planes in the beamline are rotated by 90$^{\circ}$ to achieve a better residual, as is the case in the actual test beam setup. The simulated residuals in the X and Y directions are shown in Figure~\ref{fig:simres}. The residual in the X direction is $\sigma_{x,sim} =$ 65.2 $\mu$m and in the Y-direction it is $\sigma_{y,sim} =$ 7.9 $\mu$m.
The simulation residuals are smaller than the calculated residuals. This can be explained by the fact that the simulated clusters tend to contain more pixels, as shown in Figure~\ref{fig:cs0x}, resulting in a better resolution due to charge sharing. Details on the relationship between spatial resolution and multiple hit clusters is explained in Ref. \cite{rossi2006pixel}.
\begin{figure}[!h]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/residual_simu_dut0_Tel0_u.pdf}
\caption{}
\label{fig:sres0x}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/residual_simu_dut0_Tel0_v.pdf}
\caption{}
\label{fig:sres0y}
\end{subfigure}
\caption{Simulated residual of the first telescope plane (a) in the X-direction (b) in the Y-direction }
\label{fig:simres}
\end{figure}
The simulated residual in the X-direction shows five peaks equally spaced by 50$\mu$m. These peaks are distinct compared to the observed residual as the simulated residuals are not affected by the detector's misalignment or inefficiencies. The rectangular pixel size of 250 $\times$ 50 $\mu$m$^{2}$ is solely responsible for this feature. Since two out of six planes are rotated by 90$^{\circ}$, they improve the granularity of the tracking in the X-direction. It also differentiates the tracks produced with the 50 $\mu$m resolutions, which leads to having the five possible reconstructed track positions. Therefore, instead of having the flat top of the residuals, it has a distinct five-peak structure.
To conclude, the spatial resolutions of six telescope planes measured at the test beam are within an agreement range with simulated and calculated residuals. Figure~\ref{fig:sumres} shows a summary of all these three types of residuals.
\begin{figure}[!h]
\centering
\includegraphics[width=0.55\textwidth]{figures/resolution_smallLegend.pdf}
\caption{Measured, calculated and simulated residuals summary}
\label{fig:sumres}
\end{figure}
|
2,877,628,090,104 | arxiv | \section{Introduction}
\label{sec:intro}
Spectral Unmixing (SU) aims to decompose a hyperspectral image (HI) into its pure spectral components, termed \emph{endmembers} (EMEs), and the proportional \emph{abundances} to which they contribute to the reflectance in each pixel~\cite{Keshava:2002p5667}. Although the interaction between light and the EMEs can be complex and nonlinear~\cite{Dobigeon-2014-ID322, Imbiriba2016_tip, borsoi2019blind}, the linear mixing model (LMM) is still widely used due to its simplicity and good performance~\cite{Keshava:2002p5667}.
Spectral variability (SV) consists of changes in EME spectra occurring both within a single image, or between images acquired at different time instants. They can be caused by differences in atmospheric, illumination or seasonal conditions~\cite{Zare-2014-ID324-variabilityReview,somers2011variabilityReview}. Early approaches have considered large libraries of spectral signatures to represent variable EME spectra~\cite{Zare-2014-ID324-variabilityReview,roberts1998originalMESMA,Borsoi_multiscale_lgrs_2018,borsoi2019deepGenSpectralLibrary}. More recently, different extensions of the LMM have been proposed to account for the spectral variability within a given HI, by considering, e.g., additive~\cite{Thouvenin_IEEE_TSP_2016_PLMM} and multiplicative~\cite{drumetz2016blindUnmixingELMM, imbiriba2018GLMM, borsoi2019improved} scaling factors, or by parametrizing spectral variability using deep generative models~\cite{borsoi2019deep}.
Multitemporal SU have recently become a subject of great interest due to the possibility of leveraging time information in HI sequences, allowing for monitoring the dynamical evolution of the materials and their distributions~\cite{somers2013invasiveHawaiiMultiTemporalBandWeighting,lippitt2018multidateMESMAshrublands,somers2013uncorrelatedBandSelectionInstabilityIndex}. However, the influence of spectral variability in multitemporal scenarios can be significantly stronger than in the case of a single HI. This introduces a challenge to multitemporal SU since EME variability must be carefully modelled to achieve a good performance.
Previous work have considered different strategies to incorporate dynamical information about the EMEs, often based on parametric models originally devised to account for variations within a single HI. These include constraining the EMEs in adjacent time instants to be scaled versions of each other~\cite{henrot2016dynamical}, or to be represented as a mean EME matrix with small, additive perturbations~\cite{thouvenin2016online,sigurdsson2017sparseDistU,thouvenin2018hierarchicalBU}.
However, these works disregard important information as they do not account for the low-dimensional structure that often underlies the changes observed in EME spectra when representing its evolution.
In this paper we propose a new algorithm for multitemporal SU which is based on a dynamical model for the EME time variability. {Specifically, we couple the representation power of recently proposed parametric EME models (which were originally devised to operate within a single HI, such as~\cite{imbiriba2018GLMM}) with a Bayesian filtering methodology to reliably estimate the endmembers in HI sequences.} Instead of operating directly on the EME spectral space, we make use of a parametric EME model to represent EME dynamics indirectly through vectors of parameters that capture the time variations of each material. Bayesian filtering and smoothing are combined with the expectation maximization algorithm to estimate the required parameters given a window of observations in time. The initialization of the resulting Kalman filter is also estimated in the process, which improves convergence for short image sequences. Under some approximations about the temporal variation of the abundances, the proposed algorithm is able to blindly estimate the EMEs, the average abundances, and the remaining model parameters from the observed HI data. Finally, a unique abundance matrix is estimated for each time instant using the resulting EME model. Simulation results show that, for small abundance variations over time (which can be usually satisfied in small time windows), the proposed method is able to outperform state-of-the-art algorithms in both EME and abundance estimation accuracy.
\section{Multitemporal Spectral Unmixing}\label{sec:MTSU}
The multitemporal Linear Mixing Model (LMM)~\cite{Keshava:2002p5667} represents an HI with $L$ bands and $N$ pixels at time $t$ as:
\begin{align} \label{eq:LMM}
&\boldsymbol{Y}_{\!t} = \boldsymbol{M}_t \boldsymbol{A}_t + \boldsymbol{E}_t,
\,\,\, \text{subject to }\,\cb{1}^\top\boldsymbol{A}_t = \cb{1}^\top,\,\,\, \boldsymbol{A}_t \geq \cb{0} \,
\end{align}
\noindent {where $\boldsymbol{Y}_{\!t}\in\amsmathbb{R}^{L\times N}$ is the observed HI, the columns of $\boldsymbol{M}_t \in \amsmathbb{R}^{L\times P}$ are the $P$ endmember spectral signatures, $\boldsymbol{A}_t \in \amsmathbb{R}^{P\times N}$ contains the abundances for each pixel, and $\boldsymbol{E}_t$ represents additive noise, all indexed at time $t\in\{1,\ldots,T\}$.}
An important challenge related to the use of representation~\eqref{eq:LMM} regards the consideration of spectral variability, which causes the signatures of the endmembers in $\boldsymbol{M}_t$ to change due to, e.g., seasonal, illumination or acquisition variations~\cite{Zare-2014-ID324-variabilityReview}. Spectral variability occurs both in space (within the same HI) and in time. Spatial domain spectral variability has been addressed in several works (e.g., see~\cite{Zare-2014-ID324-variabilityReview,somers2011variabilityReview,Thouvenin_IEEE_TSP_2016_PLMM,drumetz2016blindUnmixingELMM,imbiriba2018GLMM,borsoi2019improved,borsoi2019deep} and references therein). For simplicity, this work assumes only variations of EMEs in time.
EME variation within the same HI can be later incorporated to the proposed model, for instance, by adapting models such as the one in~\cite{imbiriba2018GLMM} to represent the space-time dynamical behavior of the EMEs.
A straightforward way to perform SU under time variability is to do it for each image separately. {However, such an approach disregards the temporal information and the time dynamics of the spectral variability, which can be exploited to enhance both the abundance and EME estimation performance.}
Different SU algorithms accounting for endmember time variability have been recently proposed, most of them inspired by models designed to account for spectral variability within a single image.
For instance, in~\cite{henrot2016dynamical} the authors constrain the endmember matrices at each time instant to be scaled versions of a reference endmember matrix. In~\cite{thouvenin2016online}, the authors model the endmembers at each time instant by a mean EME matrix plus small perturbations, which are assumed to be temporally smooth. All variables are then estimated using a stochastic approach. This latter model was later extended for distributed unmixing with additional sparsity constraints in~\cite{sigurdsson2017sparseDistU}, and to include sparse additive residual terms to represent abrupt spectral variations in the HI using a hierarchical Bayesian framework in~\cite{thouvenin2018hierarchicalBU}.
However, these works do not provide a satisfactory means of modeling the dynamical evolution of the endmembers since they operate directly in the input spectral space, ignoring the fact that spectral variability can often be represented more accurately using physically meaningful parametrizations of EME spectra.
Different models have been recently proposed to model EME spatial variability as a parametric function of reference spectral signatures as:
\begin{align} \label{eq:parametric_EM_model}
\boldsymbol{M} = f(\boldsymbol{M}_0,\boldsymbol{\psi}) \,,
\end{align}
where $f$ is a parametric function, $\boldsymbol{M}_0\in\amsmathbb{R}^{L\times P}$ contains reference/average spectral signatures and $\boldsymbol{\psi}$ is a vector of parameters of the variability model.
Such models include additive perturbations~\cite{Thouvenin_IEEE_TSP_2016_PLMM}, spectrally uniform~\cite{drumetz2016blindUnmixingELMM} or spectrally varying~\cite{imbiriba2018GLMM, borsoi2019improved}, and parametrizations using deep neural networks learned from the observed HI~\cite{borsoi2019deep}.
Such parametric models are specially interesting for building a dynamical model to consider EME time variability.
\section{Dynamical parametric endmember model}\label{sec:dynModel}
In this paper, we consider a multitemporal extension of the parametric EME model~\eqref{eq:parametric_EM_model}. We assume a fixed reference EME matrix $\boldsymbol{M}_0$, and model the time variations in $\boldsymbol{M}_t$ through a time varying $\boldsymbol{\psi}_t$, $t=1,\ldots,T$.
By assuming that temporally adjacent images are acquired at reasonably short time intervals, we model the difference $\boldsymbol{\psi}_{t} - \boldsymbol{\psi}_{t-1}$ as a small zero-mean vector. Thus, we assume the following model for~$\boldsymbol{\psi}_t$:
\begin{align} \label{eq:param_state_evolution}
\boldsymbol{\psi}_{t} &= \boldsymbol{\psi}_{t-1} + \cb{q}_{t}
\end{align}
where $\boldsymbol{\psi}_{t}$ is a vector containing the parameters of the endmember model at time $t$, and $\cb{q}_{t}\sim\mathcal{N}(\cb{0},\boldsymbol{Q})$ contains the innovations which describe its dynamical evolution.
Note that $\cb{q}_{t}$ is only constrained to be zero mean on statistical and not temporal average, which means that each realization of the sequence $\{\widehat{\cb{q}}_{t}\}$, which is learned from the observed HIs, can exhibit behavior such as trends and complex dynamic evolution. Moreover, the Gaussian assumption is only made in the model parameters $\boldsymbol{\psi}_{t}$ and not on the EME signatures themselves, which allows for the use of complex EME distributions through the pushforward measure obtained using the function~$f$, as done in~\cite{borsoi2019deep}.
This generalizes parametric EME model~\eqref{eq:parametric_EM_model} to the multitemporal setting as $\boldsymbol{M}_{t} = f(\boldsymbol{M}_0,\boldsymbol{\psi}_{t})$, where the parametric function $f$ now relates the EME matrices and the vectors of parameters at each time instant.
Considering this model, the multitemporal LMM can be represented as
\begin{align} \label{eq:obs_model_general}
\boldsymbol{Y}_{\!t} = f(\boldsymbol{M}_0,\boldsymbol{\psi}_{t})\boldsymbol{A}_t + \boldsymbol{E}_t \,.
\end{align}
Next, one must choose a function $f$ for \eqref{eq:obs_model_general} that establishes a good compromise between mathematical tractability and performance. The GLMM~\cite{imbiriba2018GLMM,Borsoi_2018_Fusion} is able to represent arbitrary spectral variability by considering spectrally varying multiplicative scaling factors, introducing a connection between the amount of spectral variability and the amplitude of endmember reflectance spectra at each band.
The GLMM introduces a matrix $\cb{\Psi}\in{\amsmathbb{R}^{L\times P}}$ of scaling factors with nonnegative entries $[\cb{\Psi}]_{\ell,k} \geq 0$ acting individually at each wavelength. This leads to the following representation for the $t$-th observed HI:
\begin{equation} \label{eq:model_glmm}
\boldsymbol{Y}_{\!t} = (\boldsymbol{M}_0\odot\cb{\Psi}_t)\boldsymbol{A}_t + \boldsymbol{E}_t \,,
\end{equation}
where $\odot$ is the Hadamard (elementwise) product.
Using the vectorization property,~\eqref{eq:model_glmm} can be expressed as
\begin{align}
\boldsymbol{y}_t & = \operatorname{vec}(\boldsymbol{Y}_{\!t})= \big(\boldsymbol{A}_t^\top\otimes\boldsymbol{I}_L\big)\operatorname{diag}(\boldsymbol{m}_{0}) \boldsymbol{\psi}_{t} + \boldsymbol{e}_{t} \,,
\end{align}
with $\boldsymbol{m}_{0} = \operatorname{vec}(\boldsymbol{M}_0)$, $\boldsymbol{\psi}_{t} = \operatorname{vec}(\boldsymbol{\Psi}_{t})$ and $\boldsymbol{e}_t = \operatorname{vec}(\boldsymbol{E}_t)$.
We write the abundance matrix $\boldsymbol{A}_t$ as
$
\boldsymbol{A}_t = \boldsymbol{A} + \boldsymbol{\Delta}\!\boldsymbol{A}_t
$
where $\boldsymbol{\Delta}\boldsymbol{A}_t$ represents small random fluctuations over the average abundance matrix $\boldsymbol{A}$. Considering $\boldsymbol{\Delta}\!\boldsymbol{A}_t$ to be small for a time window $t\in \{t_0,\ldots, t_0+T\}$, $\forall t_0$,
these variations can be incorporated into the observation noise, leading to the following model:
\begin{align} \label{eq:observation_model}
\boldsymbol{y}_t & = \boldsymbol{H}(\boldsymbol{A})\operatorname{diag}(\boldsymbol{m}_{0}) \boldsymbol{\psi}_{t} + \boldsymbol{r}_{t} \,,
\end{align}
where $\boldsymbol{H}(\boldsymbol{A})=\boldsymbol{A}^\top\otimes\boldsymbol{I}_L$ and $\boldsymbol{r}_{t}=\boldsymbol{e}_{t}+\big(\boldsymbol{\Delta}\!\boldsymbol{A}_t^\top\otimes\boldsymbol{I}_L\big)\operatorname{diag}(\boldsymbol{m}_{0}) \boldsymbol{\psi}_{t}$.
Note that the observation noise $\boldsymbol{r}_{t}$ in~\eqref{eq:observation_model} is correlated with the state $\boldsymbol{\psi}_{t}$. {In the following, we will use a signal-independent noise approximation, which provides competitive performance at a modest computational cost. Further discussion on the impact of such an approximation can be found in the supplementary document, also available in~\cite{borsoi2020kalmanEM_arxiv}.}
\section{Proposed method}\label{sec:proposedMethod}
In this section, we present the proposed dynamical methodology which connects Kalman smoother with expectation maximization approach. For this, we assume that for a given time window of duration $T$ the abundance variation is small but the EMEs can vary due to different seasonal or acquisitions conditions. Then, we employ a time varying state space formulation to model the spectral variability, which naturally leads to a Kalman filter based formulation.
We couple a Kalman smoother, used to obtain accurate estimations for the state variables, with the Expectation Maximization estimation of model parameters such as the abundance matrix and the noise power.
Assuming the abundances fixed over a time window $t\in \{t_0,\ldots, t_0+T\}$, we use~\eqref{eq:param_state_evolution} and~\eqref{eq:observation_model} to form the linear state-space model
\begin{align}
\begin{split} \label{eq:state_space_model}
\boldsymbol{\psi}_{t} &= \boldsymbol{\psi}_{t-1} + \cb{q}_{t},
\quad \boldsymbol{y}_{t} = \boldsymbol{H}(\boldsymbol{A})\operatorname{diag}(\boldsymbol{m}_{0}) \boldsymbol{\psi}_{t} + \boldsymbol{r}_{t} \,.
\end{split}
\end{align}
Neglecting the dependence of $\boldsymbol{r}_t$ on $\boldsymbol{\psi}_{t}$, and assuming $\cb{q}_t$ and $\boldsymbol{r}_t$ to be Gaussian, this system can be solved using the classical Kalman filter and smoothing equations.
Next, we present the Kalman filter and smoother equations followed by the EM strategy to estimate the abundances and noise power.
\subsection{Kalman Filter endmember model}
\label{sec:kalmanFilt}
Bayesian filtering computes marginal posterior distributions of the states by assuming Markovity over the state sequence.
When the dynamical and measurement models are linear and Gaussian, the solution is given in the form of the Kalman filter, which can be expressed in a set of equations for the~Prediction:
\begin{align} \label{eq:kalman_filt_pred}
\begin{split}
\boldsymbol{\psi}_{t|t-1} = \boldsymbol{\psi}_{t-1|t-1}\,, \hspace{6ex}
\boldsymbol{P}_{t|t-1} = \boldsymbol{P}_{t-1|t-1} + \boldsymbol{Q} \,,
\end{split}
\end{align}
\noindent and for the Update step:\\
\noindent\begin{minipage}{\linewidth}
\vspace{-5pt}
\noindent\begin{minipage}{.4\linewidth}
\begin{align}
\boldsymbol{v}_t &= \boldsymbol{y}_t - \boldsymbol{B}\boldsymbol{\psi}_{t|t-1} \,,\nonumber\\
\boldsymbol{S}_{t} &= \boldsymbol{B} \boldsymbol{P}_{t|t-1} \boldsymbol{B}^\top + \boldsymbol{R} \,,\nonumber\\
\boldsymbol{K}_t &= \boldsymbol{P}_{t|t-1} \boldsymbol{B}^\top \boldsymbol{S}_{t}^{-1} \,,\nonumber
\end{align}
\end{minipage}
\noindent\begin{minipage}{.6\linewidth}
\begin{align}\label{kalman_filt_update}
\begin{split}
\boldsymbol{\psi}_{t|t} &= \boldsymbol{\psi}_{t|t-1} + \boldsymbol{K}_t \boldsymbol{v}_t \,,\\
\boldsymbol{P}_{t|t} &= \boldsymbol{P}_{t|t-1} - \boldsymbol{K}_t \boldsymbol{S}_{t} \boldsymbol{K}_t^\top \,
\end{split}
\end{align}
\end{minipage}
\vspace{5pt}
\end{minipage}
where $\boldsymbol{P}_{t_1|t_2}$ is the covariance matrix of $\boldsymbol{\psi}_{t_1}$ conditioned on $\boldsymbol{y}_t$ for $t_1 \le t_2$, $\boldsymbol{B} = \boldsymbol{H}(\boldsymbol{A})\operatorname{diag}(\boldsymbol{m}_{0}) =(\boldsymbol{A}^\top \otimes \boldsymbol{I}_L) \operatorname{diag}(\boldsymbol{m}_{0})$ and $\boldsymbol{R}$ is the covariance matrix of $\boldsymbol{r}_t$ in \eqref{eq:state_space_model}.
Solving equations~\eqref{kalman_filt_update} requires to construct and invert matrix $\boldsymbol{S}_t$ of size $NL\times NL$, which is impractical.
To circumvent this issue, we assume that the noise covariance matrix satisfies $\boldsymbol{R}=\sigma_r^2\boldsymbol{I}_{NL}$. Thus, using the Woodbury identity for the inverse of sum of matrices, the right part of the third term in~\eqref{kalman_filt_update} is written as
\begin{align}
\boldsymbol{B}^\top \boldsymbol{S}_{t}^{-1}
%
& = \sigma_r^{-2}\boldsymbol{B}^\top - \sigma_r^{-4}\boldsymbol{B}^\top \boldsymbol{B} \big(\boldsymbol{P}_{t|t-1}^{-1} + \sigma_r^{-2}\boldsymbol{B}^\top \boldsymbol{B} \big)^{-1} \boldsymbol{B}^\top \nonumber
\end{align}
which now involves only the inverse of a $PL\times PL$ matrix.
\subsection{Kalman Smoother}
\label{sec:kalmanSmooth}
The objective of Bayesian smoothers is to provide a marginal posterior distribution of state $\boldsymbol{\psi}_t$ assuming knowledge of the measurements $\boldsymbol{y}_{t}$
in an observation window of duration $T$, that is, $p(\boldsymbol{\psi}_t|\boldsymbol{y}_{t_0},\ldots,\boldsymbol{y}_{t_0+T})$.
For the model in~\eqref{eq:state_space_model}, the smoother solution can be implemented very efficiently by iteratively updating the conditional distributions obtained by the Kalman filter backwards in time, for $t_0+T,\ldots,t_0$. In this case, the smoother equations are given by:
\begin{align} \label{eq:kalman_smoother}
\begin{split}
\boldsymbol{\psi}_{t+1|t} = \boldsymbol{\psi}_{t|t} \,, & \hspace{3ex}
\boldsymbol{P}_{t+1|t} = \boldsymbol{P}_{t|t} + \boldsymbol{Q} \,, \hspace{3ex}
\boldsymbol{G}_t = \boldsymbol{P}_{t|t} \boldsymbol{P}_{t+1|t}^{-1}
\end{split}
\nonumber \\
\begin{split}
\boldsymbol{\psi}_{t}^{s} &= \boldsymbol{\psi}_{t|t} + \boldsymbol{G}_t [\boldsymbol{\psi}_{t+1}^{s} - \boldsymbol{\psi}_{t+1|t}] \,, \\
\boldsymbol{P}^s_{t} &= \boldsymbol{P}_{t|t} + \boldsymbol{G}_t [\boldsymbol{P}^s_{t+1} - \boldsymbol{P}_{t+1|t}]\boldsymbol{G}_t^\top.
\end{split}
\end{align}
\subsection{The expectation-maximization (EM) algorithm}
\label{sec:expectM}
Estimation of the sequence $\{\boldsymbol{\psi}_t\}$ of EME model parameters using \eqref{eq:kalman_filt_pred}-\eqref{eq:kalman_smoother}, requires that $\boldsymbol{A}$, $\boldsymbol{Q}$ and $\boldsymbol{R}$, as well as the initializations $\boldsymbol{P}_{0|0}$ and $\boldsymbol{\psi}_{0|0}$, be known in advance.
Let us denote these parameters by $\boldsymbol{\theta}=\{\boldsymbol{A},~\boldsymbol{P}_{0|0},~\boldsymbol{Q},~\boldsymbol{R},~\boldsymbol{\psi}_{0|0}\}$. Instead of fixing $\boldsymbol{\theta}$ with values known a priori, we can view it as unobserved latent variables of model~\eqref{eq:state_space_model}, which can be estimated by maximizing the conditional marginal likelihood
$p(\boldsymbol{y}_{t_0},\ldots,\boldsymbol{y}_{t_0+T}|\boldsymbol{\theta})$
using the EM algorithm.
Starting with an initial guess $\boldsymbol{\theta}^{(0)}$, the EM algorithm finds a local maximum of~$p(\boldsymbol{y}_{t_0},\ldots,\boldsymbol{y}_{t_0+T}|\boldsymbol{\theta})$ by repeating the following steps:
\begin{align} \label{eq:EM_alg_iters}
\begin{split}
\quad & a) \,\,\, \text{E-step: compute } \mathcal{Q}(\boldsymbol{\theta}|\boldsymbol{\theta}^{(k)})
\\
\quad & b) \,\,\, \text{M-step: compute } \boldsymbol{\theta}^{(k+1)}=\arg\max\nolimits_{\boldsymbol{\theta}}\mathcal{Q}(\boldsymbol{\theta},\boldsymbol{\theta}^{(k)})
\end{split}
\end{align}
\vspace{-3pt}
\noindent for $k=1,\ldots,K_{max}$, with $K_{max}$ being the number of iterations and
$\mathcal{Q}(\boldsymbol{\theta}|\boldsymbol{\theta}^{(k)})=\operatorname{E}_{\varsigma}\{\log p(\boldsymbol{\psi}_{t_0}, \ldots, \boldsymbol{\psi}_{t_0+T},\boldsymbol{y}_{t_0},\ldots, \boldsymbol{y}_{t_0+T}|\boldsymbol{\theta})\}$, with $\varsigma=p(\boldsymbol{\psi}_{t_0}, \ldots,\boldsymbol{\psi}_{t_0+T},\boldsymbol{y}_{t_0},\ldots,\boldsymbol{y}_{t_0+T}|\boldsymbol{\theta}^{(k)})$,
being the expectation of the logarithm of the data likelihood, taken with respect to the full joint posterior given the parameters~$\boldsymbol{\theta}^{(k)}$.
Although the EM algorithm is very general and not always easy to solve, for a linear model such as~\eqref{eq:state_space_model} we can find closed form solutions, leading to a more efficient implementation in high-dimensional settings. Furthermore, for the linear Gaussian model, $\mathcal{Q}(\boldsymbol{\theta},\boldsymbol{\theta}^{(k)})$ can be computed based on the Kalman smoother results obtained using $\boldsymbol{\theta}^{(k)}$ as the system parameters. This leads to an elegant solution that consists of the successive application of the smoother and estimation of the parameters.
For the model~\eqref{eq:state_space_model}, $\mathcal{Q}(\boldsymbol{\theta},\boldsymbol{\theta}^{(t)})$ is given by~\cite{sarkka2013bayesian}
\begin{align
& \mathcal{Q} (\boldsymbol{\theta},\boldsymbol{\theta}^{(t)}) \!=\! -\frac{1}{2}\Big(\!
\operatorname{tr}\!\Big\{\!\boldsymbol{P}_{0|0}^{-1}\big[ \boldsymbol{P}_0^s \!+ \!(\boldsymbol{\psi}_0^s\!-\!\boldsymbol{\psi}_{0|0})(\boldsymbol{\psi}_0^s\!-\!\boldsymbol{\psi}_{0|0})^\top \big]\Big\}
\nonumber \\
& + \! \operatorname{tr}\!\Big\{\!\boldsymbol{R}^{-1}\big[ \boldsymbol{\Sigma}_5 - \boldsymbol{\Sigma}_3 \boldsymbol{H}(\!\boldsymbol{A})^\top
\!\! - \! \boldsymbol{H}(\!\boldsymbol{A}) \boldsymbol{\Sigma}_3^\top + \boldsymbol{H}(\!\boldsymbol{A}) \boldsymbol{\Sigma}_1 \boldsymbol{H}(\!\boldsymbol{A})^\top \big]\Big\}
\nonumber \\
& + \! \operatorname{tr}\!\Big\{\!\boldsymbol{Q}^{-1}\big[ \boldsymbol{\Sigma}_1 \!-\! \boldsymbol{\Sigma}_4 \!-\! \boldsymbol{\Sigma}_4^\top \!+\! \boldsymbol{\Sigma}_2 \big]\Big\} \!+\!\log\!\big\{|\boldsymbol{Q}\boldsymbol{R}|^T|\boldsymbol{P}_{0|0}|\big\} \!\Big) \!+\!C
\nonumber
\end{align}
where $C$ is a constant term and
\begin{align}
\begin{split}
\boldsymbol{\Sigma}_1 = \sum_{t=1}^T \boldsymbol{P}_{t}^s + \boldsymbol{\psi}_{t}^s {\boldsymbol{\psi}_{t}^s}^\top,
\hspace{4.25ex}
\boldsymbol{\Sigma}_4 = \sum_{t=1}^T \boldsymbol{P}_{t}^s {\boldsymbol{G}^s}^\top_{t-1} + \boldsymbol{\psi}_{t}^s {\boldsymbol{\psi}^s}^\top_{t-1}
\end{split}
\nonumber\\\nonumber
\begin{split}
\boldsymbol{\Sigma}_2 \!= \!\!\sum_{t=1}^T \!\boldsymbol{P}_{t-1}^s \!+\! \boldsymbol{\psi}_{t-1}^s {\boldsymbol{\psi}^s}^\top_{t-1} \,,
\hspace{0.9ex}
\boldsymbol{\Sigma}_3 \!= \!\!\sum_{t=1}^T \!\boldsymbol{y}_t {\boldsymbol{\psi}_{t}^s}^\top,
\hspace{0.9ex}
\boldsymbol{\Sigma}_5 \!= \!\!\sum_{t=1}^T \!\boldsymbol{y}_t\boldsymbol{y}_t^\top,
\end{split}
\end{align}
Under the assumption that $\boldsymbol{R}=\sigma_r^2\boldsymbol{I}_{NL}$, optimizing $\mathcal{Q}(\boldsymbol{\theta},\boldsymbol{\theta}^{(t)})$ with respect to $\boldsymbol{P}_{0|0}$, $\boldsymbol{Q}$, $\boldsymbol{R}$ and $\boldsymbol{\psi}_{0|0}$ is relatively straightforward and can be done as~\cite{sarkka2013bayesian}
\begin{align}
& \boldsymbol{P}_{0|0}^* = \boldsymbol{P}_0^s + (\boldsymbol{\psi}_0^s-\boldsymbol{\psi}_{0|0})(\boldsymbol{\psi}_0^s-\boldsymbol{\psi}_{0|0})^\top \label{eq:P00*}
\\
& \boldsymbol{Q}^* = \boldsymbol{\Sigma}_1 - \boldsymbol{\Sigma}_4 - \boldsymbol{\Sigma}_4^\top + \boldsymbol{\Sigma}_2
\\
& \sigma_r^* = \operatorname{tr}\big\{\boldsymbol{\Sigma}_5 - 2\boldsymbol{H}(\!\boldsymbol{A}) \boldsymbol{\Sigma}_3^\top \!+\! \boldsymbol{H}(\!\boldsymbol{A}) \boldsymbol{\Sigma}_1 \boldsymbol{H}(\!\boldsymbol{A})^\top\big\}\big/(LN)
\\
& \boldsymbol{\psi}_{0|0}^* = \boldsymbol{\psi}_0^s. \label{eq:psi00*}
\end{align}
The optimization w.r.t. $\boldsymbol{H}(\!\boldsymbol{A})$ is, however, more complex due to the structure of this matrix. Since $\boldsymbol{R}=\sigma_r^2\boldsymbol{I}$ for some $\sigma_r>0$, the problem can be stated as
\begin{align} \label{eq:abundances_cost_function}
\widehat{\!\boldsymbol{A}} = \mathop{\arg\min}_{\boldsymbol{A}} \,\, \operatorname{tr}\!\Big\{\boldsymbol{H}(\boldsymbol{A}) \boldsymbol{\Sigma}_1 \boldsymbol{H}(\boldsymbol{A})^\top - 2\,\boldsymbol{\Sigma}_3 \boldsymbol{H}(\boldsymbol{A})^\top \Big\}
\end{align}
In order to solve~\eqref{eq:abundances_cost_function} efficiently, we rewrite its terms in the following to explore the structure of $\boldsymbol{H}(\boldsymbol{A})$. For the first term:
\begin{align
\operatorname{tr}\{\boldsymbol{H}(\boldsymbol{A}) \boldsymbol{\Sigma}_1 \boldsymbol{H}(\boldsymbol{A})^\top\}
{}={} & \operatorname{tr}\{(\boldsymbol{A}\otimes\boldsymbol{I}_L)(\boldsymbol{A}^\top\otimes\boldsymbol{I}_L)\widetilde{\boldsymbol{\Sigma}}_1\}
\nonumber\\
{}={} & \operatorname{tr}\{(\boldsymbol{A}\bA^\top\otimes\boldsymbol{I}_L)\widetilde{\boldsymbol{\Sigma}}_1\}
\end{align}
\noindent where $\widetilde{\boldsymbol{\Sigma}}_1=\operatorname{diag}(\boldsymbol{m}_0)\boldsymbol{\Sigma}_1\operatorname{diag}(\boldsymbol{m}_0)$. To explore the properties of the Kronecker product and simplify the solution to this problem, we assume that $\widetilde{\boldsymbol{\Sigma}}_1$ can be decomposed as~\cite{van1993approximationKronecker,batselier2017constructiveKroneckerTensors}
$
\widetilde{\boldsymbol{\Sigma}}_1 = \sum_{k=1}^{K_1} \boldsymbol{C}_k \otimes \boldsymbol{D}_k, \quad \boldsymbol{C}_k \in\amsmathbb{R}^{P\times P} , \quad \boldsymbol{D}_k \in\amsmathbb{R}^{L\times L},
$
and using the properties of the Kronecker product, we have
\begin{align} \label{eq:kronecker_dec1b}
\operatorname{tr}\{(\boldsymbol{A}\bA^\top\otimes\boldsymbol{I}_L)\widetilde{\boldsymbol{\Sigma}}_1\}
= & \sum\nolimits_{k=1}^{K_1} \operatorname{tr}\{(\boldsymbol{A}\bA^\top\boldsymbol{C}_k)\} \operatorname{tr}\{\boldsymbol{D}_k\} \,.
\end{align}
Similarly, for the second term, we have
$
\operatorname{tr}\{\boldsymbol{H}(\boldsymbol{A})\boldsymbol{\Sigma}_3^\top\} = \operatorname{tr}\{(\boldsymbol{A}^\top\otimes\boldsymbol{I}_L)\widetilde{\boldsymbol{\Sigma}}_3^{\top}\}
$
where $\widetilde{\boldsymbol{\Sigma}}_3=\boldsymbol{\Sigma}_3 \operatorname{diag}(\boldsymbol{m}_0)$. By decomposing $\widetilde{\boldsymbol{\Sigma}}_3$ as in~\cite{van1993approximationKronecker,batselier2017constructiveKroneckerTensors} leads to
$
\widetilde{\boldsymbol{\Sigma}}_3 = \sum_{k=1}^{K_2} \widetilde{\boldsymbol{C}}_k \otimes \widetilde{\boldsymbol{D}}_k,
\quad \widetilde{\boldsymbol{C}}_k \in\amsmathbb{R}^{N\times P} , \quad \widetilde{\boldsymbol{D}}_k \in\amsmathbb{R}^{L\times L},
$
and using the properties of the Kronecker product, we have
\begin{align} \label{eq:kronecker_dec2b}
\operatorname{tr}\{(\boldsymbol{A}^\top\otimes\boldsymbol{I}_L)\widetilde{\boldsymbol{\Sigma}}_3\}
= & \sum\nolimits_{k=1}^{K_2} \operatorname{tr}\{(\boldsymbol{A}^\top\widetilde{\boldsymbol{C}}_k^\top)\} \operatorname{tr}\{\widetilde{\boldsymbol{D}}_k^\top\} \,.
\end{align}
By substituting \eqref{eq:kronecker_dec1b} and \eqref{eq:kronecker_dec2b} in~\eqref{eq:abundances_cost_function}, taking the derivative of the cost function with respect to $\boldsymbol{A}$ and setting it equal to zero, we obtain the following solution for $\boldsymbol{A}$:
\begin{align}\label{eq:sol_A}
\!\!\! \widehat{\!\boldsymbol{A}} \!= \!\Big[\!\sum\nolimits_{k=1}^{K_1} \!\!\operatorname{tr}\{\boldsymbol{D}_k\} (\boldsymbol{C}_k \!+ \boldsymbol{C}_k^\top) \Big]^{-1} \Big[2\!\sum\nolimits_{k=1}^{K_2} \!\!\operatorname{tr}\{\widetilde{\boldsymbol{D}}_k^\top\} \widetilde{\boldsymbol{C}}_k^\top \Big].
\end{align}
Although the approach presented in sections~\ref{sec:kalmanFilt}--~\ref{sec:expectM} provides an estimate~$\,\widehat{\!\boldsymbol{A}}$ of the average abundances, the temporal abundance variations $\Delta\boldsymbol{A}_t$ can make~$\,\widehat{\!\boldsymbol{A}}$ an inaccurate approximation of $\boldsymbol{A}_t$ for some image sequences (e.g. when sudden changes are present). To mitigate this issue, we compensate the abundance variations $\Delta\boldsymbol{A}_t$ by solving using a fully constrained least squares (FCLS) problem:
\begin{align} \label{eq:final_reg_fcls_A}
\begin{split}
& \mathop{\min}_{\boldsymbol{A}_{t}} \,\,
\|\boldsymbol{Y}_{\!t} - (\boldsymbol{M}_0\odot\widehat{\boldsymbol{\Psi}}_t)\boldsymbol{A}_{t}\|_F^2 + \lambda\|\boldsymbol{A}_{t}-\,\widehat{\!\boldsymbol{A}}\|_F^2
\\[-0.2cm]
& \text{ s.t. } \,\, \boldsymbol{A}_{t} \geq0, \, \cb{1}^\top\boldsymbol{A}_{t} = \cb{1}^\top,
\end{split}
\end{align}
for $t=1,\ldots,T$, where $\widehat{\boldsymbol{\Psi}}_t$ is the matrix-ordered version of the estimated states $\boldsymbol{\psi}_t^s$ and $\lambda\geq0$ is a regularization parameter.
The proposed methodology is summarized in Algorithm~\ref{alg:algorithm}.
\begin{algorithm} [bth!]
\footnotesize
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\caption{\mbox{Kalman filter and Smoother for MTSU}\label{alg:algorithm}}
\Input{$\big\{\boldsymbol{y}_t\big\}_{t=1}^T$, $\boldsymbol{A}^{(0)}$, $\boldsymbol{\psi}^{(0)}$, $\boldsymbol{M}_0$. $\boldsymbol{\psi}_{0|0}^{(0)}$, $\boldsymbol{Q}^{(0)}$, $\sigma_r^{(0)}$, $\boldsymbol{P}_{0|0}^{(0)}$, $\lambda$}
\For{$i=1,\ldots,K_{\max}$}{
Estimate $\boldsymbol{\psi}_t$ using~\eqref{eq:kalman_filt_pred}--\eqref{kalman_filt_update} for $t=1,\ldots,T$\;
Estimate $\boldsymbol{\psi}^{s}_t$ and $\boldsymbol{P}_t^s$ using~\eqref{eq:kalman_smoother} for $t=T,\ldots,1$\;
Estimate $\boldsymbol{P}_{0|0}^{(i)},\, \boldsymbol{Q}^{(i)}, \sigma_r^{(i)}$, $\boldsymbol{\psi}_{0|0}^{(i)}$, $\boldsymbol{A}^{(i)}$ using~\eqref{eq:P00*}--\eqref{eq:psi00*},~\eqref{eq:sol_A}\;
Estimate the temporal abundance variations according to~\eqref{eq:final_reg_fcls_A}\;
\KwRet $\boldsymbol{A}_t^*=\,\widehat{\!\boldsymbol{A}}_t$,~ $\boldsymbol{M}_t^*=\boldsymbol{M}_0 \odot \operatorname{vec}^{-1}(\boldsymbol{\psi}^s_t)$, for $t=1,\ldots,T$
\end{algorithm}
\section{Experimental Results} \label{sec:Simulations}
In this section we evaluate the performance of the proposed method by comparing it with the fully constrained least squares (FCLS), and with the Online Unmixing (OU) strategy proposed in~\cite{thouvenin2016online}. {To illustrate how the use of temporal information improves SU, we have also applied the GLMM algorithm~\cite{imbiriba2018GLMM} (which considers SV only within a single HI) to process each HI independently.}
In all experiments, the reference EME matrix $\boldsymbol{M}_0$ was extracted from the observed HI at $t=1$ using the VCA algorithm~\cite{Nascimento2005}, and the abundances were initialized with the corresponding FCLS result. The other parameters were initialized as $\boldsymbol{\psi}_{0|0}=\cb{1}$, $\boldsymbol{Q}=0.1\boldsymbol{I}$, $\sigma_r=0.01$, $\boldsymbol{P}_{0|0}=\boldsymbol{I}$ and $\lambda=10^{-8}$, and five EM iterations were considered.
The parameters of the OU algorithm were searched in the ranges detailed in the original publication~\cite{thouvenin2016online}.
The performance of the methods is evaluated using the average Normalized Root Mean Squared Error (NRMSE) between the estimated abundances ($\text{NRMSE}_{\boldsymbol{A}}$), endmembers ($\text{NRMSE}_{\boldsymbol{M}}$) and between the reconstructed HIs. NRMSE is defined as
$
\text{NRMSE}_{\boldsymbol{X}} = \frac{1}{T} \sum_{t=1}^T \sqrt{\|\boldsymbol{X}_{\!t} - \boldsymbol{X}_{\!t}^*\|^2_F \,\,/\,\, \|\boldsymbol{X}_{\!t}\|^2_F}
$,
where $\boldsymbol{X}_{\!t}$ and $\boldsymbol{X}_{\!t}^*$ are a true and an estimated variable, respectively, at time instant $t$. We also consider the average Spectral Angle Mapper (SAM) between the estimated endmembers, defined as
$
\text{SAM}_{\boldsymbol{M}} = \frac{1}{T}\sum_{t=1}^{T}\sum_{k=1}^{P}\arccos (\boldsymbol{m}_{k,t}^\top\boldsymbol{m}_{k,t}^*/\|\boldsymbol{m}_{k,t}\|\|\boldsymbol{m}_{k,t}^*\|).
$
\begin{table}[htb!]
\footnotesize
\caption{Quantitative simulation results (values $\times100$).}
\vspace{-0.3cm}
\centering
\renewcommand{\arraystretch}{1}
\setlength{\tabcolsep}{3.3pt}
\begin{tabular}{l|cccc|c}
\midrule
& \multicolumn{4}{|c|}{{Synthetic Data (average results)}} & {Real Data} \\
\midrule
& $\text{NRMSE}_{\boldsymbol{A}}$ & $\text{NRMSE}_{\boldsymbol{M}}$ &$\text{SAM}_{\boldsymbol{M}}$ & $\text{NRMSE}_{\boldsymbol{Y}}$ & {$\text{NRMSE}_{\boldsymbol{Y}}$} \\ \midrule
FCLS & 4.60 & 3.30 & 2.47 & 4.03 & 8.73\\
{GLMM} & 4.87 & 4.25 & 3.69 & \textbf{2.93} & \textbf{0.15}\\
OU & 3.40 & 2.82 & 2.05 & 3.09 & 3.48\\
Proposed & \textbf{2.65} & \textbf{2.06} & \textbf{1.83} & 3.12 & 9.02\\
\bottomrul
\end{tabular}
\label{tab:quantitativeResults}
\end{table}
A synthetic dataset with $L=173$ bands, $N=50$ pixels and $T=10$ frames was created by generating abundance values sampled from a Dirichlet distribution. HIs containing three endmembers (vegetation, water and soil) were generated following the model~\eqref{eq:model_glmm} to generate one different EME matrix for each time instant.
Temporal spectral variability was introduced by performing a random walk according to the model $\boldsymbol{\psi}_t=\boldsymbol{F}\boldsymbol{\psi}_{t-1}+\cb{q}_t$ with $\boldsymbol{F}=0.9\boldsymbol{I}$, with $\boldsymbol{\psi}_0=\cb{1}_{LP}$ and covariance $0.01\boldsymbol{I}$.
Finally, white Gaussian noise was added to the images, resulting in an SNR of 30dB.
The $\text{SAM}_{\boldsymbol{M}}$ metric for the GLMM method was computed by considering the average endmember matrix (across all pixels) for each time instant.
In order to evaluate the performance of the algorithms, we performed 900 Monte Carlo runs. Additionally, we also simulated different amounts of temporal abundance variability, with the temporal standard deviation of each pixel being, on average, approximately~$3\times10^{-3}$.
The average metrics (across all MC runs and variance values) are shown in Table~\ref{tab:quantitativeResults}. It can be seen that the proposed method performs significantly better than the other algorithms.
Improvements over OU can be found in $\text{NRMSE}_{\boldsymbol{A}}$ ($22\%$), $\text{NRMSE}_{\boldsymbol{M}}$ ($27\%$) and $\text{SAM}_{\boldsymbol{M}}$ ($11\%$). The GLMM provided the smallest reconstruction error $\text{NRMSE}_{\boldsymbol{Y}}$ since it has the largest amount of degrees of freedom. However, this did not translate into better abundance or EME estimates, since spatial SV was not present in the data and time information was not taken into account.
\begin{figure}[!t]
\centering
\includegraphics[width=0.8\linewidth]{figures/realData2_wdates.pdf}
\vspace{-0.3cm}
\caption{Abundance maps for the Soil endmember (right) and visual representation (left) of the Lake Tahoe HI sequence.}
\label{fig:realIm}
\end{figure}
For the simulations with real data, we consider three images from the Lake Tahoe sequence, originally studied in~\cite{thouvenin2016online}{, acquired at 10-04-2014, 02-06-2014 and at 29-04-2015}. {These images can be seen in Fig.~\ref{fig:realIm}, and are composed by three EMEs, water (a lake), soil and vegetation (two crop circles, whose aspect varies considerably between the three HIs).} The images were first downsized to $28\times 38$ pixels for faster processing, and contained $L=173$ bands. $\boldsymbol{M}_0$ and the OU EME initialization were constructed using the same signatures as in~\cite{thouvenin2016online}. The OU parameters were the same as those used in~\cite{thouvenin2016online}. The results can be seen in Fig.~\ref{fig:realIm} {and Table~\ref{tab:quantitativeResults}}. Due to space limitations, only the soil endmember is shown {(more results can be found in the supplementary document, also available in~\cite{borsoi2020kalmanEM_arxiv})}. It can be seen that the proposed algorithm provides higher abundance values in the regions corresponding to soil in the HI, with significantly less confusion in the vegetation endmember when compared to the other methods, especially for $t=2$. Similar improvements could be noticed for the other EMEs as well.
{The reconstruction error $\text{NRMSE}_{\boldsymbol{Y}}$ of the algorithms is generally related to their overall amount of degrees of freedom, thus being much larger for both the proposed method and for FCLS than for GLMM.}
\section{Conclusions}\label{sec:conclusions}
We proposed a new multitemporal spectral unmixing algorithm accounting for spectral variability. A state-space dynamical model was proposed for the time evolution of the coefficients encoding the spectral variability of the endmembers. Bayesian filtering was used to estimate the state variables. Assuming small abundance variations in short time intervals, expectation maximization employed to efficiently estimate the remaining parameters, including the fractional abundances. Simulation results indicate that the proposed method can outperform state-of-the-art multitemporal spectral unmixing algorithms. A future perspective is the extension of the method to properly handle abrupt abundance changes.
\bibliographystyle{IEEEbib}
|
2,877,628,090,105 | arxiv | \section{Introduction}\label{intro}
The formation of shear bands in elasticity is described by a degenerate operator of elliptic-hyperbolic type~\cite{aifantisMicrostructuralOriginCertain1984,leeConditionsShearBanding1989,bairChapterTenShear2019}. The shear bands that are mathematically obtained in this model are infinitesimally thin. To overcome this non-physical description, it is customary to penalize the elasticity operator by a fourth-order singular perturbation~\cite{franfortCombinedEffectsHomogenization1994}. Subsequently, it was suggested that the penalization should be followed by a homogenization procedure, which results in different regimes depending on the order of penalization as compared to the length-scale of the periodic heterogeneities. This was first carried out in~\cite{bensoussanAsymptoticAnalysisPeriodic2011,franfortCombinedEffectsHomogenization1994}. A quantitative analysis of this problem appeared in~\cite{niuConvergenceRateHomogenization2019,niuCombinedEffectsHomogenization2020,pastukhovaHomogenizationEstimatesSingularly2020}. The aim of the present work is to revisit this problem by employing the homogenization framework developed by Conca and Vanninathan~\cite{concaHomogenizationPeriodicStructures1997}. Bloch wave method in homogenization is a spectral method of homogenization. It also offers an alternative approximation scheme in the form of Bloch approximation~\cite{concaBlochApproximationHomogenization2002,concaBlochApproximationHomogenization2005} which provides sharp convergence estimates in homogenization under minimal regularity requirements. In this paper, we shall establish new characterizations of homogenized tensor for the singularly perturbed problem in terms of the first Bloch eigenvalue and use Bloch wave method to recover the homogenization result for singularly perturbed periodic homogenization problem.
\subsection{Notation and Definitions} We will study the simultaneous homogenization and singular perturbation limits of the following operator
\begin{align}\label{main}
\mathcal{A}^{\kappa,\varepsilon}\coloneqq\kappa^2\Delta^2 -\nabla\cdot A\left(\frac{x}{\varepsilon}\right)\nabla,
\end{align} where $0<\kappa,\epsilon\ll 1$, and $A(y)=(a_{jk})_{j,k=1}^d$ is a matrix whose entries are real, bounded, measurable functions of $y\in\mathbb{R}^d$. Further, the matrix $A$ satisfies the following hypotheses:
\begin{itemize}
\item[A1.] $A$ is elliptic, i.e., there exists $\alpha>0$ such that for all $\xi\in\mathbb{R}^d$ and a.e. $y\in\mathbb{R}^d$, $A(y)\xi\cdot\xi\geq \alpha|\xi|^2$.
\item[A2.] $A$ is $Y$-periodic, i.e., $A(y+2\pi p)=A(y)$ for all $p\in\mathbb{Z}^d$, a.e. $y\in\mathbb{R}^d$. The set $Y\coloneqq [0,2\pi)^d$ is called a basic periodicity cell and may also be interpreted as a parametrization of the $d$-dimensional torus, $\mathbb{T}^d$.
\item[A3.] The matrix $A$ is symmetric.
\end{itemize}
We mention briefly the function spaces that make an appearance in this problem. The solutions of the cell problem associated with homogenization of~\cref{main}, as well as Bloch eigenfunctions, are sought in the space $H^2_\sharp(Y)$, which is the space of all periodic distributions $u$ for which the norm $||u||_{H^2}=\left(\sum_{n\in\mathbb{Z}^d}(1+|n|^2)^2|\hat{u}(n)|^2\right)^{\frac{1}{2}}$ is finite. The space $H^2_\sharp(Y)$ may be identified with $H^2(\mathbb{T}^d)$. The spaces $H^s_\sharp(Y)$ for $s\in\mathbb{R}$ are similarly defined. For $s>0$, $H^s_\sharp(Y)$ forms a subspace of the space of all periodic $L^2$ functions in $\mathbb{R}^d$, denoted by $L^2_\sharp(Y)$ or $L^2(\mathbb{T}^d)$. We shall denote the mean value or average of a periodic function $u$ on the basic periodicity cell $Y$ by $\mathcal{M}_Y(u)\coloneqq \frac{1}{|Y|}\int_Y u(y)\,dy$. Averaged integrals such as $\frac{1}{|Y|}\int_Y u(y)\,dy$ are sometimes denoted as $\fint_Y u(y)\,dy$.
\subsection{The Method of Bloch Wave Homogenization}
The method of Bloch waves rests on decomposition of a periodic operator in terms of Bloch waves which may be thought of as a periodic analogue of plane waves. As plane waves decompose a linear operator with constant coefficients by means of the Fourier transform, Bloch waves diagonalize a linear operator with periodic coefficients. This decomposition begins with a direct integral decomposition of a periodic operator $\mathfrak{A}$ in $\mathbb{R}^d$.
\begin{align*}
\mathfrak{A}\to\int^\bigoplus_{\mathbb{T}^d}\mathfrak{A}(\eta)\,d\eta.
\end{align*} The fiber operator $\mathfrak{A}(\eta)$ has compact resolvent for each fixed $\eta\in\mathbb{T}^d$. The eigenfunctions viewed as functions of $\eta$ are called Bloch waves. Finally, the operator $\mathfrak{A}(\eta)$ is diagonalized by means of Bloch waves.
The homogenization limits for a highly oscillating scalar periodic operator are obtained from its first Bloch mode. The rest of the Bloch modes do not contribute to the homogenization limit. This is a consequence of the separation of the first Bloch eigenvalue from the rest of the spectrum. Such an interpretation of homogenization is also called spectral threshold effect~\cite{birmanPeriodicSecondorderDifferential2003}. Moreover, the homogenized tensor is obtained from the Hessian of the first Bloch eigenvalue. Therefore, the second-order nature of the differential operator is reflected in the quadratic nature of the first Bloch eigenvalue near the bottom of the spectrum. Indeed, homogenization of higher-even-order periodic operators can also be obtained by the Bloch wave method, where the first Bloch eigenvalue behaves like a polynomial of the corresponding order near the bottom of the spectrum~\cite{veniaminovHomogenizationPeriodicDifferential2011,suslinaHomogenizationDirichletProblem2018}.
The homogenization result in~\cref{homog} exhibits three different regimes depending on the ratio of $\kappa$ and $\epsilon$, where $\kappa$ is to be interpreted as a function of $\epsilon$ satisfying $\lim_{\epsilon\to0}\kappa=0$. Define $\rho\coloneqq \frac{\kappa}{\epsilon}$. The three different regimes correspond to
\begin{itemize}
\item $\lim_{\epsilon\to0}\rho=0$,
\item $0<\lim_{\epsilon\to0}\rho<\infty$, and
\item $\lim_{\epsilon\to0}\rho=\infty$.
\end{itemize}
The approach is to treat $\rho$ as a fixed number and obtain Bloch wave decomposition for the unscaled operator $\mathcal{A}^\rho=\rho^2\Delta^2-\nabla\cdot A(y)\nabla$. This decomposition is employed to obtain the homogenization limit as $\epsilon,\kappa\to 0$. While the qualitative homogenization result is not new, the presentation is original. Some novel features of the proof are characterization of homogenized tensor and cell functions in terms of derivatives of the first Bloch eigenvalue and eigenfunctions. This requires us to prove analyticity of the first Bloch eigenvalue and eigenfunction in a neighbourhood of zero in the dual parameter. Interestingly, the region of analyticity does not depend on the singular perturbation, which plays an important role in the simultaneous passage to $0$ of $\kappa$ and $\epsilon$. In order to study the stability of the homogenized tensor in the different regimes, we obtain uniform in $\rho$ estimates for Bloch eigenvalues, eigenfunctions and their derivatives of all orders in the dual parameter. We also prove that only the first Bloch mode contributes to homogenization and the higher modes are negligible. The analysis of three separate regimes provides considerable challenges in the Bloch wave method, particularly in obtaining uniform estimates in these regimes.
While the motivation for the problem~\cref{main} comes from the theory of elasticity, it is for the sake of simplicity that we only study the scalar operator. However, it must be noted that Bloch wave homogenization of systems carries some unique difficulties, such as the presence of multiplicity at the bottom of the spectrum. Indeed, these challenges have been surmounted by the use of directional analyticity of Bloch eigenvalues in~\cite{sivajiganeshBlochWaveHomogenization2005,birmanPeriodicSecondorderDifferential2003,allaireHomogenizationStokesSystem2017}. Further, the assumption of symmetry, while customary in elasticity, is made for a simplified presentation. A Bloch wave analysis of homogenization of non-selfadjoint operators may be found in~\cite{ganeshBlochWaveHomogenization2004}.
In a forthcoming work, we will obtain quantitative estimates for the combined effects of singular perturbation and homogenization through the notion of Bloch approximation, which was introduced in~\cite{concaBlochApproximationHomogenization2002}. Higher order estimates in homogenization have been obtained by these methods, particularly for the dispersive wave equation~\cite{dohnalBlochwaveHomogenizationLarge2014,dohnalDispersiveHomogenizedModels2015, allaireComparisonTwoscaleAsymptotic2016,lamacz-keymlingHighorderHomogenizationOptimal2020}.
\subsection{Plan of the Paper}
The plan of the paper is as follows: In~\cref{BlochWaveSingular}, we obtain Bloch waves for the singularly perturbed operator $\mathcal{A}^\rho$. In~\cref{regularityBloch}, we prove that the first Bloch eigenpair are analytic functions of the dual parameter in a neighbourhood of $0$. In~\cref{nbdanalytic}, we prove that the neighbourhood of analyticity is independent of $\rho$. In~\cref{cellproblem}, we recall the cell problem for the operator $\mathcal{A}^\rho$ and the estimates associated with it. In~\cref{BlochCharacterization}, we characterize the homogenized tensor in the three regimes by way of Bloch method. In~\cref{blochtransform}, we shall define the first Bloch transform and analyze its asymptotic properties. In~\cref{qualitative}, we obtain the qualitative homogenization theorem by means of the first Bloch transform, which is the periodic analogue of Fourier transform. Finally, in~\cref{highermodes}, we quantify the contribution of the higher Bloch transforms towards the homogenization limit.
\section{Bloch Waves for the Singularly Perturbed Operator}\label{BlochWaveSingular}
In this section, we will prove the existence of Bloch waves for the singular operator given by
\begin{align}\label{singularop}
\mathcal{A}^\rho\coloneqq \rho^2\Delta^2-\nabla\cdot A(y)\nabla.
\end{align} Recall that $\rho$ was earlier set as $\frac{\kappa}{\epsilon}$, however in this section, $\rho$ will be assumed to be a fixed positive number. Bloch waves for~\cref{singularop} refers to eigenfunctions of~\cref{singularop} satisfying the so-called $(\eta-Y)$-periodicity condition, that is, we look for functions $\psi$ satisfying the following eigenvalue problem:
\begin{equation*}
\begin{cases}
&\rho^2\Delta^2\psi-\nabla\cdot A(y)\nabla\psi=\lambda\psi\nonumber\\
&\psi(y+2\pi p)=e^{2\pi ip\cdot\eta}\psi(y), p\in\mathbb{Z}^d, \eta\in\mathbb{R}^d.
\end{cases}
\end{equation*}
The above problem is invariant under $\mathbb{Z}^d$-shifts of $\eta$, hence it suffices to restrict $\eta$ to $Y^{'}\coloneqq \left[-\frac{1}{2},\frac{1}{2}\right)^d$. Now, if we set
$\psi(y)=e^{iy\cdot\eta}\phi(y)$ where $\phi$ is a $Y$-periodic function, then the above eigenvalue problem is transformed into:
\begin{eqnarray}
\label{BlochEigenvalueProblem}
\begin{cases}
&\mathcal{A}^\rho(\eta)\phi\coloneqq \rho^2(\nabla+i\eta)^4\phi-(\nabla+i\eta)\cdot A(y)(\nabla+i\eta)\phi=\lambda\phi\\
&\phi(y+2\pi p)=\phi(y), p\in\mathbb{Z}^d,\eta\in Y^{'}.
\end{cases}
\end{eqnarray}
The operator $\mathcal{A}^\rho(\eta)$ is often called the shifted operator associated with $\mathcal{A}^{\rho}$ where the shift $i\eta$ appears as a magnetic potential. In order to prove the existence of eigenvalues for~\eqref{BlochEigenvalueProblem}, we shall begin by proving that a zeroth-order perturbation of $\mathcal{A}^\rho(\eta)$ is elliptic on $H^2_\sharp(Y)$. This amounts to a G\mathcal{A} rding type inequality for the operator $\mathcal{A}^\rho(\eta)$. Combined with Rellich compactness theorem, this will allow us to prove compactness of the inverse in $L^2_\sharp(Y)$. Then, a standard application of the spectral theorem for compact self-adjoint operators will guarantee the existence of eigenvalues for each fixed $\rho$ and $\eta$.
The bilinear form $a^\rho[\eta](\cdot,\cdot)$ defined on $H^2_\sharp(Y)\times H^2_\sharp(Y)$ by \begin{align}
a^\rho[\eta](u,v)\coloneqq \int_Y A (\nabla+i\eta)u\cdot\overline{(\nabla+i\eta)v}\,dy+\rho^2\int_Y (\nabla+i\eta)^2 u\overline{(\nabla+i\eta)^2 v}\,dy,
\end{align} is associated to the operator $\mathcal{A}^\rho(\eta)$. We shall prove the following G\mathcal{A} rding-type inequality for $a^\rho[\eta]$.
\begin{lemma}\label{coercivityfortranslate}
There exists a positive real number $C_*$ not depending on $\eta$ but depending on $\rho$ such that for all $u\in H^2_\sharp(Y)$ and all $\eta\in Y^{'}$, we have
\begin{align}\label{coercivity}
a^\rho[\eta](u,u)+C_*||u||^2_{L^2_\sharp(Y)}\geq \frac{\rho^2}{6}||\Delta u||^2_{L^2_\sharp(Y)}+\frac{\alpha}{2}||u||_{H^1_\sharp(Q)}.
\end{align}
\end{lemma}
\begin{proof}
We have
\begin{align}
a^\rho[\eta](u,u) = \underbrace{\int_Y A (\nabla+i\eta)u\cdot\overline{(\nabla+i\eta)u}\,dy}_\text{I}+\underbrace{\rho^2\int_Y (\nabla+i\eta)^2 u\overline{(\nabla+i\eta)^2 u}\,dy}_\text{II}.
\end{align} We shall estimate the two summands separately.
For the first summand,
Observe that
\begin{align}\label{termI}
I&=\int_Y A (\nabla+i\eta)u\cdot\overline{(\nabla+i\eta)u}\,dy\nonumber\\
&= \int_Y A\nabla u\cdot\overline{\nabla u}\,dy+2\,\text{Re}\left\{\int_Y A i\eta u\cdot\overline{\nabla u}\,dy\right\}+\int_Y A \eta u\cdot\eta\overline{u}\,dy,
\end{align} where $\text{Re}$ denotes the real part.
Now we shall estimate each term on the RHS above. The first term of $I$ is estimated as follows:
\begin{align}
\int_Y A\nabla u\cdot\nabla\overline{u}\,dy &= \int_Y A\nabla u\cdot\nabla \overline{u}\,dy\nonumber\\
&\geq\alpha\int_Y|\nabla u|^2\,dy.
\end{align}
For the second term of $I$, observe that
\begin{align}
\left|2\,\text{Re}\left\{\int_Y A \eta u\cdot\nabla\overline{u}\,dy \right\}\right|&\leq 2\int_Y|A\eta u\cdot\nabla\overline{u}|\,dy\nonumber\\
&\leq C_1\int_Y|\eta u\cdot\nabla\overline{u}|\,dy\nonumber\\
&\leq C_1||\eta u||_{L^2_\sharp(Y)}||\nabla u||_{L^2_\sharp(Y)}\nonumber\\
&\leq C_1C_2||u||^2_{L^2_\sharp(Y)}+\frac{C_1}{C_2}||\nabla{u}||_{L^2_\sharp(Y)}.
\end{align}
Finally, the third term of $I$ is dominated by $L^2_\sharp(Y)$ norm of $u$ as follows:
\begin{align}
\left|\int_Y A \eta u\cdot\eta\overline{u}\,dy\right|\leq C_3\int_{Y}|\eta u\cdot\eta \overline{u}|\,dy\leq C_4||u||_{L^2_\sharp(Y)}.
\end{align}
Now, we may choose $C_2$ so that $\frac{C_1}{C_2}=\frac{\alpha}{2}$, then
\begin{align}\label{firstsummand}
I\geq \frac{\alpha}{2}||u||_{L^2_\sharp(Y)}+\frac{\alpha}{2}||\nabla u||_{L^2_\sharp(Y)}-\left(\frac{\alpha}{2}+C_1C_2+C_4\right)||u||^2_{L^2_\sharp(Y)}.
\end{align}
For the second summand, observe that
\begin{align}\label{secondsummand}
II = \rho^2 \int_Y |(\nabla+i\eta)^2u|^2\,dy\nonumber\\
\begin{split} = \rho^2 \int_Y |\Delta u|^2\,dy & + \rho^2 \int_Y |\eta|^4|u|^2\,dy + 4\rho^2\int_Y |\eta\cdot\nabla u|^2\,dy + 2\rho^2 i \text{Im}\left\{\int_Y |\eta|^2 u\overline{\Delta u}\,dy\right\}\\
& + 4i\rho^2\text{Re}\left\{\int_Y (\eta\cdot\nabla u)\Delta \overline{u}\,dy \right\}+4i\rho^2\text{Re}\left\{\int_Y |\eta|^2u(\eta\cdot\nabla\overline{u})\,dy \right\}
\end{split}.
\end{align}
We estimate the last three terms as follows.
\begin{align}\label{secsumtermone}
\rho^2\left|2i\text{Im}\left\{\int_Y |\eta|^2 u\overline{\Delta u}\,dy\right\}\right|&\leq 2\rho^2\int_Y|\eta|^2|u||\Delta u|\,dy\nonumber\\
&\leq 2\rho^2|\eta|^2||u||_{L^2}||\Delta u||_{L^2}\nonumber\\
&\leq 48\rho^2|\eta|^4||u||^2_{L^2}+\frac{\rho^2}{48}||\Delta u||^2_{L^2}.
\end{align}
\begin{align}\label{secsumtermtwo}
\rho^2\left|4i\text{Re}\left\{\int_Y(\eta\cdot\nabla)u\Delta\overline{u}\,dy\right\}\right|&\leq 4\rho^2\int_Y|(\eta\cdot\nabla)u||\Delta u|\,dy\nonumber\\
&\leq 4\rho^2 ||(\eta\cdot\nabla)u||_{L^2}||\Delta u||_{L^2}\nonumber\\
&\leq \frac{16\rho^2}{3}||(\eta\cdot\nabla)u||^2_{L^2}+\frac{3\rho^2}{4}||\Delta u||^2_{L^2}.
\end{align}
\begin{align}\label{secsumtermthree}
\rho^2\left|4i\text{Re}\left\{\int_Y|\eta|^2u(\eta\cdot\nabla)\overline{u}\,dy\right\}\right|&\leq 4\rho^2\int_Y|(\eta\cdot\nabla)u||\eta|^2|u|\,dy\nonumber\\
&\leq 4\rho^2|\eta|^2||u||_{L^2}||(\eta\cdot\nabla)u||_{L^2}\nonumber\\
&\leq 192\rho^2|\eta|^4||u||^2_{L^2}+\frac{\rho^2}{48}||(\eta\cdot\nabla)u||^2_{L^2}.
\end{align}
The previous threee estimate use Cauchy-Schwarz inequality for the second step and Young's inequality for the third step. Substituting the inequalities~\cref{secsumtermone},~\cref{secsumtermtwo} and~\cref{secsumtermthree} into~\cref{secondsummand}, we get
\begin{align}\label{secsuminter}
II\geq \frac{11\rho^2}{48} ||\Delta u||^2_{L^2(Y)} & -240 \rho^2 |\eta|^4||u||^2_{L^2(Y)} - \frac{65\rho^2}{48}||(\eta\cdot\nabla)u||^2_{L^2(Y)}.
\end{align}
Since $|\eta|\leq\frac{1}{2}$, we get
\begin{align*}
II\geq \frac{11\rho^2}{48} ||\Delta u||^2_{L^2(Y)} & -15 \rho^2||u||^2_{L^2(Y)} - \frac{65\rho^2}{192}||\nabla u||^2_{L^2(Y)}.
\end{align*}
Now, notice that
\begin{align}\label{gradtodelta}
||\nabla u||^2_{L^2}=\int_Y|\nabla u|^2\,dy=-\int_Y u\Delta u\,dy\leq \frac{65}{48}||u||^2_{L^2}+\frac{12}{65}||\Delta u||^2_{L^2}.
\end{align}
Substituting~\cref{gradtodelta} in~\cref{secsuminter}, we get
\begin{align}\label{secsuminter2}
II\geq \frac{\rho^2}{6} ||\Delta u||^2_{L^2(Y)} & -16 \rho^2||u||^2_{L^2(Y)}.
\end{align}
Combining~\cref{firstsummand} and~\cref{secsuminter2}, we obtain~\cref{coercivity} with \begin{align}\label{cstar}C_*=\left(\frac{\alpha}{2}+C_1C_2+C_4+16\rho^2\right).\end{align}
\end{proof}
\begin{remark}
In the G\mathcal{A} rding type inequality for the operator $\mathcal{A}^\rho(\eta)$, the perturbation $C_*I$ in zeroth term depends on the parameter $\rho$. It is possible to avoid the dependence of $C_*$ on $\rho$ by forgoing the shift $i\eta$ in the biharmonic term. This simplification was employed in~\cite{sivajiganeshBlochWaveHomogenisation2020}.
\end{remark}
Now that we have the coercivity estimate~\cref{coercivity}, we can prove the existence of Bloch eigenvalues and eigenfunctions for the operator $\mathcal{A}^\rho$.
\begin{theorem}\label{existenceofBlocheig}
For each $\eta\in Y^{'}$ and $\rho>0$, the singularly perturbed Bloch eigenvalue problem~\eqref{BlochEigenvalueProblem} admits a countable sequence of eigenvalues and corresponding eigenfunctions in the space $H^2_\sharp(Q)$.
\end{theorem}
\begin{proof}
\cref{coercivityfortranslate} shows that for every $\eta\in Y^{'}$ the operator $\mathcal{A}^\rho(\eta)+C_*I$ is elliptic on $H^2_\sharp(Y)$. Hence, for $f\in L^2_\sharp(Y)$, this shows that $\mathcal{A}^\rho(\eta)u+C_*u=f$ is solvable and the solution is in $H^2_\sharp(Y)$. As a result, the solution operator $S^{\rho}(\eta)$ is continuous from $L^2_\sharp(Y)$ to $H^2_\sharp(Y)$. Since the space $H^2_\sharp(Y)$ is compactly embedded in $L^2_\sharp(Y)$, $S^\rho(\eta)$ is a self-adjoint compact operator on $L^2_\sharp(Y)$. Therefore, by an application of the spectral theorem for self-adjoint compact operators, for every $\eta\in Y^{'}$ we obtain an increasing sequence of eigenvalues of $\mathcal{A}^\rho(\eta)+C_*I$ and the corresponding eigenfunctions form an orthonormal basis of $L^2_\sharp(Y)$. However, note that both the operators $\mathcal{A}^\rho(\eta)$ and $\mathcal{A}^\rho(\eta)+C_*I$ have the same eigenfunctions but each eigenvalue of the two operators differ by $C_*$. We shall denote the eigenvalues and eigenfunctions of the operator $\mathcal{A}^\rho(\eta)$ by $\eta\to(\lambda^\rho_m(\eta),\phi^\rho_m(\cdot,\eta))$.
\end{proof}
\begin{remark}
We can prove the existence of Bloch eigenvalues and eigenfunctions for the case $\rho=0$ by a similar method. This corresponds to the standard Bloch eigenvalue problem considered in~\cite{concaHomogenizationPeriodicStructures1997}.
\end{remark}
\subsection{Bloch Decomposition of $L^2(\mathbb{R}^d)$} Now that we have proved the existence of Bloch eigenvalues and eigenfunctions, we can state the Bloch Decomposition Theorem which offers a partial diagonalization of the operator $\mathcal{A}^\rho$ in terms of its Bloch eigenvalues. This is facilitated by the Bloch transform which is a mapping from $L^2(\mathbb{R}^d)$ to $\ell^2(\mathbb{N};L^2(Y^{'}))$. The proof is similar to the one in~\cite{bensoussanAsymptoticAnalysisPeriodic2011} and is therefore omitted. Note that the proof relies on a measurable selection of the Bloch eigenfunctions with respect to $\eta$. A measurable selection of Bloch eigenfunctions for the Schr\"odinger operator was first demonstrated in~\cite{wilcoxTheoryBlochWaves1978a}. In contrast, the Bloch eigenvalues are Lipschitz continuous in the dual parameter $\eta$. We will prove this fact in~\cref{nbdanalytic}.
\begin{theorem}\label{BlochDecomposition}
Let $\rho >0$. Let $g\in L^2(\mathbb{R}^d)$. Define the $m^{th}$ Bloch coefficient of $g$ as
\begin{align}\label{BlochTransform}
\mathcal{B}^{\rho}_mg(\eta)\coloneqq\int_{\mathbb{R}^d}g(y)e^{-iy\cdot\eta}\overline{\phi_m^{\rho}(y;\eta)}\,dy,~m\in\mathbb{N},~\eta\in Y^{'}.
\end{align}
\begin{enumerate}
\item The following inverse formula holds
\begin{align}\label{Blochinverse}
g(y)=\int_{Y^{'}}\sum_{m=1}^{\infty}\mathcal{B}^{\rho}_mg(\eta)\phi_m^{\rho}(y;\eta)e^{iy\cdot\eta}\,d\eta.
\end{align}
\item{\bf Parseval's identity}
\begin{align}\label{parsevalbloch}
||g||^2_{L^2(\mathbb{R}^d)}=\sum_{m=1}^{\infty}\int_{Y^{'}}|\mathcal{B}^{\rho}_mg(\eta)|^2\,d\eta.
\end{align}
\item{\bf Plancherel formula} For $f,g\in L^2(\mathbb{R}^d)$, we have
\begin{align}\label{Plancherel}
\int_{\mathbb{R}^d}f(y)\overline{g(y)}\,dy=\sum_{m=1}^{\infty}\int_{Y^{'}}\mathcal{B}^{\rho}_mf(\eta)\overline{\mathcal{B}^{\rho}_mg(\eta)}\,d\eta.
\end{align}
\item{\bf Bloch Decomposition in $H^{-1}(\mathbb{R}^d)$} For an element $F=u_0(y)+\sum_{j=1}^N\frac{\partial u_j(y)}{\partial y_j}$ of $H^{-1}(\mathbb{R}^d)$, the following limit exists in $L^2(Y^{'})$:
\begin{align}\label{BlochTransform2}
\mathcal{B}^{\rho}_mF(\eta)=\int_{\mathbb{R}^d}e^{-iy\cdot\eta}\left\{u_0(y)\overline{\phi^{\rho}_m(y;\eta)}+i\sum_{j=1}^N\eta_ju_j(y)\overline{\phi^{\rho}_m(y;\eta)}\right\}\,dy\nonumber\\-\int_{\mathbb{R}^d}e^{-iy\cdot\eta}\sum_{j=1}^Nu_j(y)\frac{\partial\overline{\phi^{\rho}_m}}{\partial y_j}(y;\eta)\,dy.
\end{align}
\item[] The definition above is independent of the particular representative of $F$.
\item Finally, for $g\in D(\mathcal{A}^{\rho})$,
\begin{align}
\label{diagonalization}
\mathcal{B}^{\rho}_m(\mathcal{A}^{\rho}g)(\eta)=\lambda^{\rho}_m(\eta)\mathcal{B}^{\rho}_mg(\eta).\end{align}
\end{enumerate}\qed
\end{theorem}
\section{Regularity of the Ground State}\label{regularityBloch}
The Bloch wave method of homogenization requires differentiability of the Bloch eigenvalues and eigenfunctions in a neighbourhood of $\eta=0$. In this section, we will prove the following theorem. As before, $\rho>0$ will be treated as a fixed number.
\begin{theorem}\label{analytic}
For every $\rho>0$, there exists $\delta_\rho>0$ and a ball $U^\rho \coloneqq B_{\delta_{\rho}}(0) \coloneqq\{\eta\in Y^{'}:|\eta|<\delta_{\rho}\}$ such that
\begin{enumerate}
\item The first Bloch eigenvalue $\eta\to \lambda^\rho(\eta)$ of $\mathcal{A}^\rho$ is analytic for $\eta\in U^\rho$.
\item There is a choice of corresponding eigenfunctions $\phi^\rho_1(\cdot,\eta)$ such that $\eta\in U^\rho\to \phi_1^\rho(\cdot,\eta)\in H^2_\sharp(Y)$ is analytic.
\end{enumerate}
\end{theorem}
For the proof, we will make use of Kato-Rellich theorem which establishes the existence of a sequence of eigenvalues and eigenfunctions associated with a selfadjoint holomorphic family of type (B). The definition of selfadjoint holomorphic family of type (B) and other related notions may be found in Kato~\cite{katoPerturbationTheoryLinear1995}. Nevertheless, they are stated below for completeness. We begin with the definition of a holomorphic family of forms of type (a).
\begin{definition}
\leavevmode
\begin{enumerate}
\item The numerical range of a form $a$ is defined as $\Theta(a) = \{a(u,u): u \in D(a), ||u|| = 1\}$, where $D(a)$ denotes the domain of the form $a$. Here, $D(a)$ is a subspace of a Hilbert space $H$.
\item The form $a$ is called {\bf sectorial} if there are numbers $c\in \mathbb{R}$ and $\theta\in [0, \pi/2)$ such that
$$\Theta(a) \subset S_{c,\theta} \coloneqq \{\lambda \in\mathbb{C}: | \arg(\lambda -c)| \leq \theta)\}.$$
\item A sectorial form $a$ is said to be {\bf closed} if given a sequence $u_n\in D(a)$ with $u_n\to u$ in $H$ and $a(u_n-u_m)\to0$ as $n,m\to\infty$, we have $u\in D(a)$ and $a(u_n-u)\to 0$ as $n\to\infty$.
\end{enumerate}
\end{definition}
\begin{definition}[Kato] A family of forms $a(z), z\in D_0\subseteq\mathbb{C}^M$ is called a {\bf holomorphic family of type (a)} if \begin{enumerate}
\item each $a(z)$ is sectorial and closed with domain $D\subseteq H$
independent of $z$ and dense in $H$,
\item $a(z)[u,u]$ is holomorphic for $z\in D_0\subseteq\mathbb{C}^M$ for each $u\in D$.
\end{enumerate}
\end{definition}
A family of operators is called a {\bf holomorphic family of type (B)} if it generates a holomorphic family of forms of type (a).
In~\cite{katoPerturbationTheoryLinear1995,reedMethodsModernMathematical1978}, Kato-Rellich theorem is stated only for a single parameter family. In~\cite{baumgartelAnalyticPerturbationTheory1985}, one can find the proof of Kato-Rellich theorem for multiple parameters with the added assumption of simplicity for the eigenvalue at $\eta=0$.
\begin{theorem}{(Kato-Rellich)}
Let $D(\tilde{\eta})$ be a self-adjoint holomorphic family of type (B) defined for $\tilde{\eta}$ in an open set in $\mathbb{C}^M$. Further let $\lambda_0=0$ be an isolated eigenvalue of $D(0)$ that is algebraically simple. Then there exists a neighborhood $R_0\subseteq \mathbb{C}^M$ containing $0$ such that for $\tilde{\eta}\in R_0$, the following holds:
\begin{enumerate}
\item There is exactly one point $\lambda(\tilde{\eta})$ of $\sigma(D(\tilde{\eta}))$ near $\lambda_0=0$. Also, $\lambda(\tilde{\eta})$ is isolated and algebraically simple. Moreover, $\lambda(\tilde{\eta})$ is an analytic function of $\tilde{\eta}$.
\item There is an associated eigenfunction $\phi(\tilde{\eta})$ depending analytically on $\tilde{\eta}$ with values in $H$.
\end{enumerate}
\end{theorem}
The proof of Theorem~\ref{analytic} proceeds by complexifying the shifted operator $\mathcal{A}^\rho(\eta)$ before verifying the hypothesis of Kato-Rellich Theorem.
\begin{proof}{(Proof of Theorem~\ref{analytic})}
\begin{itemize}[wide, nosep, labelindent = 0pt, topsep = 1ex]
\item [\bf(i) Complexification of $\mathcal{A}^\rho(\eta)$: ] \hfill \\ The form $a^\rho[\eta](\cdot,\cdot)$ is associated with the operator $\mathcal{A}^\rho(\eta)$. We define its complexification as
$$t(\tilde{\eta})=\int_Y A(\nabla+i\sigma-\tau)u\cdot(\nabla-i\sigma+\tau)\overline{u}\,dy+\rho^2\int_Y |(\nabla+i\sigma+\tau)^2 u|^2\,dy$$
for $\tilde{\eta}\in R$ where \begin{equation*}R\coloneqq\{\tilde{\eta}\in\mathbb{C}^M:\tilde{\eta}=\sigma+i\tau, \sigma,\tau\in\mathbb{R}^M, |\sigma|<1/2,|\tau|<1/2\}.\end{equation*}
\item[\bf(ii) the form $t(\tilde{\eta})$ is sectorial: ] \hfill
We have \begin{align*}
t(\tilde{\eta})&=\int_Y A(\nabla+i\sigma-\tau)u\cdot(\nabla-i\sigma+\tau)\overline{u}\,dy+\rho^2\int_Y |(\nabla+i\sigma+\tau)^2 u|^2\,dy\\
&=\int_Y A(\nabla+i\sigma) u\cdot(\nabla-i\sigma)\overline{u}\,dy-\int_Y A(\tau u)\cdot \nabla\overline{u}\,dy+ \int_Y A \nabla u\cdot(\tau\overline{u})\, dy\\
&\qquad -\int_Y A\tau u\cdot\tau\overline{u}\,dy+i \int_Y A\sigma u\cdot\tau\overline{u}\,dy+i \int_Y A\tau u\cdot\sigma \overline{u}\,dy\\
&\qquad+\rho^2\int_Y \left(\Delta-|\sigma|^2+|\tau|^2\right)u\left(\Delta-|\sigma|^2+|\tau|^2\right)\overline{u} \,dy\\
&\qquad+2\rho^2\int_Y \left(\Delta-|\sigma|^2+|\tau|^2\right)u\left(\tau\cdot\nabla-i\sigma\cdot\nabla-i\sigma\cdot\tau\right)\overline{u} \,dy\\
&\qquad+2\rho^2\int_Y \left(i\sigma\cdot\nabla-\tau\cdot\nabla-i\sigma\cdot\tau\right)u\left(\Delta-|\sigma|^2+|\tau|^2\right)\overline{u} \,dy\\
&\qquad+4\rho^2\int_Y \left(i\sigma\cdot\nabla-\tau\cdot\nabla-i\sigma\cdot\tau\right)u\left(\tau\cdot\nabla-i\sigma\cdot\nabla-i\sigma\cdot\tau\right)\overline{u} \,dy.
\end{align*}
From above, it is easy to write separately the real and imaginary parts of the form $t(\tilde{\eta})$.
\begin{align*}
\Re t(\tilde{\eta})[u]&=\int_Y A(\nabla+i\sigma) u\cdot(\nabla-i\sigma)\overline{u}\,dy - \int_Y A\tau u\cdot\tau\overline{u}\,dy\\
&\qquad +\rho^2\int_Y \left(\Delta-|\sigma|^2+|\tau|^2\right)u\left(\Delta-|\sigma|^2+|\tau|^2\right)\overline{u} \,dy\\
&\qquad - 4\rho^2\int_Y |(\tau\cdot\nabla)u|^2 \,dy + 4\rho^2\int_Y |(\sigma\cdot\nabla)u|^2 \,dy - 4\rho^2\int_Y |(\tau\cdot\sigma)u|^2 \,dy\\
&\qquad+8\rho^2\text{Re}\left\{ \int_Y i(\tau\cdot\nabla)u(\sigma\cdot\tau)\overline{u} \,dy \right\}+4\rho^2\text{Re}\left\{ \int_Y i(\sigma\cdot\nabla)u\Delta\overline{u} \,dy \right\}\\
&\qquad+4\rho^2\text{Re}\left\{ \int_Y i|\sigma|^2u(\sigma\cdot\nabla)\overline{u} \,dy \right\}.
\end{align*}
\begin{align*}
\Im\,t(\tilde{\eta})[u]&= \int_Y A\sigma u\cdot\tau\overline{u}\,dy + \int_Y A\tau u\cdot\sigma \overline{u}\,dy + 2\Im \left\{\int_Y A \nabla u\cdot \tau \overline{u}\,dy\right\}\\
&\qquad +8\rho^2\text{Im}\left\{ \int_Y i(\tau\cdot\sigma)u(\sigma\cdot\nabla)\overline{u} \,dy \right\}+8\rho^2\text{Im}\left\{ \int_Y i(\sigma\cdot\nabla)u(\sigma\cdot\tau)\overline{u} \,dy \right\}\\
&\qquad +4\rho^2\text{Im}\left\{ \int_Y \Delta u(\tau\cdot\nabla)\overline{u} \,dy \right\}-4\rho^2\text{Im}\left\{ \int_Y i \Delta u(\sigma\cdot\tau)\overline{u} \,dy \right\}\\
&\qquad +4\rho^2\text{Im}\left\{ \int_Y (\tau\cdot\nabla)u|\sigma|^2\overline{u} \,dy \right\}+4\rho^2\text{Im}\left\{ \int_Y i|\sigma|^2u(\sigma\cdot\tau)\overline{u} \,dy \right\}\\
&\qquad +4\rho^2\text{Im}\left\{ \int_Y |\tau|^2u(\tau\cdot\nabla)\overline{u} \,dy \right\}-4\rho^2\text{Im}\left\{ \int_Y i|\tau|^2u(\sigma\cdot\tau)\overline{u} \,dy \right\}.
\end{align*}
The following coercivity estimate can be easily found for the real part:
\begin{align}\label{sectoriality2}
\Re t(\tilde{\eta})[u] + C_5 ||u||^2_{L^2_\sharp(Y)}\geq \frac{\alpha}{2}\left(||u||^2_{L^2_\sharp(Y)}+||\nabla u||^2_{L^2_\sharp(Y)}\right) + \frac{\rho^2}{6}||\Delta u||^2_{L^2_\sharp(Y)}.
\end{align}
Let us define the new form $\tilde{t}(\tilde{\eta})$ by $\tilde{t}(\tilde{\eta})[u,v]=t(\tilde{\eta})[u,v]+(C_5+C_6)(u,v)_{L^2_\sharp(Y)}$. Then it holds that
\begin{align*}
\Re \tilde{t}(\tilde{\eta})[u]\geq \frac{\alpha}{2}\left(||u||^2_{L^2_\sharp(Y)}+||\nabla u||^2_{L^2_\sharp(Y)}\right) +\frac{\rho^2}{6}||\Delta u||^2_{L^2_\sharp(Y)}+C_6||u||^2_{L^2_\sharp(Y)}.
\end{align*}
Also, the imaginary part of $\tilde{t}(\tilde{\eta})$ can be estimated as follows:
\begin{align*}
\Im \tilde{t}(\tilde{\eta})[u]&\leq C_7 ||u||^2_{L^2_\sharp(Y)}+C_8 ||\nabla u||^2_{L^2_\sharp(Y)}+C_9 ||\Delta u||^2_{L^2_\sharp(Y)}\\
&\overset{\text{$C_{10}=\max\{2C_8/\alpha,6C_9/\rho^2\}$}}{\underset{\text{$C_6=C_{10}/C_7$}}{=}}C_{10}\left(C_6||u||^2_{L^2_\sharp(Y)}+\frac{\alpha}{2}||\nabla u||^2_{L^2_\sharp(Y)}+\frac{\rho^2}{6}||\Delta u||^2_{L^2_\sharp(Y)}\right)\\
&\leq C_{10}\left(\Re \tilde{t}(\tilde{\eta})[u]-\frac{\alpha}{2}||u||^2_{L^2_\sharp(Y)}\right).
\end{align*}
This shows that $\tilde{t}(\tilde{\eta})$ is sectorial. However, sectoriality is invariant under translations by scalar multiple of identity operator in $L^2_\sharp(Y)$, therefore the form $t(\tilde{\eta})$ is also sectorial.
\item[\bf (iii) The form $t(\tilde{\eta})$ is closed: ] \hfill \\
Suppose that $u_n\stackrel{t}{\to}u$. This means that $u_n\to u$ in $L^2_\sharp(Q)$ and $t(\tilde{\eta})[u_n-u_m]\to 0$. As a consequence, $\Re t(\tilde{\eta})[u_n-u_m]\to 0$. By~\eqref{sectoriality2}, $||u_n-u_m||_{H^2_\sharp(Y)}\to 0$, i.e., $(u_n)$ is Cauchy in $H^2_\sharp(Y)$. Therefore, there exists $v\in H^2_\sharp(Y)$ such that $u_n\to v$ in $H^2_\sharp(Y)$. Due to uniqueness of limit in $L^2_\sharp(Y)$, $v=u$. Therefore, the form is closed.
\item[{\bf(iv) The form $t(\tilde{\eta})$ is holomorphic:} ] \hfill \\
The holomorphy of $t$ follows as a consequence of $t$ being a quartic polynomial in $\eta$.
\item[{\bf (v) $0$ is an isolated eigenvalue: }] \hfill \\
Zero is an eigenvalue because constants are eigenfunctions of $\mathcal{A}^\rho(0)=-\nabla\cdot A\nabla+\rho^2\Delta^2$. As a result, $C_*$ is an eigenvalue of $\mathcal{A}^\rho(0)+C_*I$. We proved using Lemma~\ref{coercivityfortranslate} that $\mathcal{A}^\rho(0)+C_*I$ has compact resolvent. Therefore, $C_*^{-1}$ is an eigenvalue of $(\mathcal{A}^\rho(0)+C_*I)^{-1}$ and $C_*^{-1}$ is isolated. Hence, zero is an isolated point of the spectrum of $\mathcal{A}^\rho(0)$.
\item[{\bf (vi) $0$ is a geometrically simple eigenvalue: }] \hfill \\
Denote by $\text{ker}\,\mathcal{A}^\rho(0)$ the kernel of operator $\mathcal{A}^\rho(0)$. Let $v\in \text{ker}\,\mathcal{A}^\rho(0)$, then $\int_Y A\nabla v\cdot\nabla v\,dy+\rho^2\int_Y |\Delta v|^2=0$. Due to the coercivity of the matrix $A$, we obtain $||\nabla v||_{L^2_\sharp(Y)}=0$. Hence, $v$ is a constant. This shows that the eigenspace corresponding to eigenvalue $0$ is spanned by constants, therefore, it is one-dimensional.
\item[{\bf (vii) $0$ is an algebraically simple eigenvalue: }] \hfill \\
Suppose that $v\in H^2_\sharp(Y)$ such that $\mathcal{A}^\rho(0)^2v=0$, i.e., $\mathcal{A}^\rho(0)v\in ker\,\mathcal{A}^\rho(0)$. This implies that $\mathcal{A}^\rho(0)v=C$ for some generic constant $C$. However, by the compatibility condition for the solvability of this equation, we obtain $C=0$. Therefore, $v\in ker\,\mathcal{A}^\rho(0)$. This shows that the eigenvalue $0$ is algebraically simple.
\end{itemize}
\end{proof}
\section{Neighbourhood of Analyticity}\label{nbdanalytic}
In~\cref{regularityBloch}, we have proved that the first Bloch eigenvalue and eigenfunction is analytic in a neighbourhood of $\eta=0$. However, a priori, this neighbourhood depends on the parameter $\rho$. In this section, we will prove that the neighbourhood is, in fact, independent of $\rho$. This requirement is essential for problems where simultaneous limits with respect to two parameters are studied, such as~\cite{ortegaBlochWaveHomogenization2007,dupuyBlochWavesHomogenization2009}; as well as for quantitative estimates, such as~\cite{concaBlochApproximationPeriodically2005}.
We begin by proving that Bloch eigenvalues are Lipschitz continuous in the dual parameter.
\begin{lemma}
For all $m\in\mathbb{N}$ and $\rho>0$, $\lambda_m^{\rho}$ is a Lipschitz continuous function of $\eta\in Y^{'}$.
\end{lemma}
\begin{proof}
The following form is associated with $\mathcal{A}^{\rho}(\eta)$:
\begin{align}
a^\rho[\eta](u,u) = \int_Y A (\nabla+i\eta)u\cdot\overline{(\nabla+i\eta)u}\,dy+\rho^2\int_Y (\nabla+i\eta)^2 u\overline{(\nabla+i\eta)^2 u}\,dy.
\end{align}
Hence, for $\eta,\eta'\in Y^{'}$, we have
\begin{align*}
a^{\rho}[\eta]-a^{\rho}[\eta']&=2\text{Re}\left\{ \int_Y Ai(\eta-\eta')u\cdot\overline{\nabla u}\,dy \right\} + \int_Y A\eta u\cdot\eta\overline u\,dy + \int_Y A\eta' u\cdot\eta'\overline u\,dy\\
&\qquad + \rho^2 \int_Y (|\eta|^2-|\eta'|^2)|u|^2\,dy + 4\rho^2 \int_{Y} |\eta\cdot\nabla u|^2 - |\eta'\cdot\nabla u|^2\,dy\\
&\qquad +2\rho^2i\text{Im}\left\{\int_Y (|\eta|^2-|\eta'|^2)u \Delta \overline u \, dy\right\} + 4\rho^2i \text{Re} \left\{\int_Y (\eta-\eta')\cdot\nabla u \Delta \overline{u}\,dy\right\}\\
&\qquad + 4\rho^2i\text{Re}\left\{\int_Y |\eta|^2 u (\eta\cdot\nabla)\overline{u} - |\eta'|^2 u (\eta'\cdot\nabla)\overline{u}\,dy\right\}\\
&\leq\qquad C|\eta-\eta'|||u||^2_{H^1_\sharp(Y)}+C'|\eta-\eta'|\rho^2\left\{ ||\Delta u||^2_{L^2_\sharp(Y)}+||u||^2_{L^2_\sharp(Y)} \right\},
\end{align*} where $C$ and $C'$ are generic constants independent of $\rho$.
By the Courant-Fischer minmax characterization of eigenvalues, we obtain
\begin{align}
\lambda_m^\rho(\eta)\leq \lambda_m^\rho(\eta') + C|\eta-\eta'|\mu_m+C'\rho^2|\eta-\eta'|\nu_m,
\end{align} where $\mu_m$ is the $m^{th}$ eigenvalue of the following spectral problem:
\begin{align*}
\begin{cases}
&-\Delta u_m + u_m = \mu_m u_m\text{ in } Y\\
&u_m\text{ is } Y-\text{periodic},
\end{cases}
\end{align*} and $\nu_m$ is the $m^{th}$ eigenvalue of the following spectral problem:
\begin{align*}
\begin{cases}
&\Delta^2 v_m + v_m = \nu_m v_m\text{ in } Y\\
&v_m\text{ is } Y-\text{periodic}.
\end{cases}
\end{align*}
By interchanging the role of $\eta$ and $\eta'$, we obtain
\begin{align}\label{lipschitzbound}
|\lambda_m^{\rho}(\eta)-\lambda_m^{\rho}(\eta')|\leq C(\mu_m+\rho^2\nu_m)|\eta-\eta'|.
\end{align} Here, $C$ is a generic constant independent of $\rho$.
\end{proof}
Now, we will prove a spectral gap result, viz. the second Bloch eigenvalue is bounded below.
\begin{lemma}\label{lowrhobound}
For all $m\geq 2$ and for all $\eta\in Y^{'}$, we have
\begin{align}
\lambda_m^{\rho}(\eta)\geq \alpha \lambda^{N}_2,
\end{align} where $\lambda^{N}_2$ is the second Neumann eigenvalue of the operator $-\Delta$ in $Y$.
\end{lemma}
\begin{proof}
Notice that $\lambda^{\rho}_m(\eta)\geq \lambda_2^{\rho}(\eta)$ for all $m\geq 2$ and for all $\eta\in Y^{'}$.
Next, observe that
\begin{align*}
\lambda_2^{\rho}(\eta)&=\inf_{\stackrel{W\subset H^2_\sharp(Y)}{ \text{dim}(W)=2}} \max_{\stackrel{\phi\in W}{\phi\neq 0}} \frac{\int_Y A\nabla(e^{i\eta\cdot y}\phi)\cdot\nabla(e^{-i\eta\cdot y}\overline{\phi})\,dy+\rho^2\int_Y|\Delta(e^{i\eta\cdot y}\phi)|^2\,dy}{\int_Y |\phi|^2\,dy} \\
&\geq \inf_{\stackrel{W\subset H^2(Y)}{ \text{dim}(W)=2}} \max_{\stackrel{\psi\in W}{\psi\neq 0}} \frac{\int_Y A\nabla\psi\cdot\nabla\overline{\psi}\,dy+\rho^2\int_Y|\Delta\psi|^2\,dy}{\int_Y |\psi|^2\,dy} \\
&\geq \inf_{\stackrel{W\subset H^2(Y)}{ \text{dim}(W)=2}} \max_{\stackrel{\psi\in W}{\psi\neq 0}} \frac{\int_Y A\nabla\psi\cdot\nabla\overline{\psi}\,dy}{\int_Y |\psi|^2\,dy} \\
&\geq \inf_{\stackrel{W\subset H^1(Y)}{ \text{dim}(W)=2}} \max_{\stackrel{\psi\in W}{\psi\neq 0}} \frac{\int_Y A\nabla\psi\cdot\nabla\overline{\psi}\,dy}{\int_Y |\psi|^2\,dy} \\
&\geq \alpha\inf_{\stackrel{W\subset H^1(Y)}{ \text{dim}(W)=2}} \max_{\stackrel{\psi\in W}{\psi\neq 0}} \frac{\int_Y |\nabla\psi|^2\,dy}{\int_Y |\psi|^2\,dy} \\
&=\alpha\lambda_2^{N}.
\end{align*}
\end{proof}
The bound obtained in~\cref{lowrhobound} will be useful for the small $\rho$ regime, however, for the large $\rho$ regime, that is, for $\rho\to\infty$, we need a different lower bound.
\begin{lemma}\label{largerhobound}
For all $m\geq 2$ and for all $\eta\in Y^{'}$, we have
\begin{align}
\lambda^{\rho}_m(\eta)\geq C\rho^2\kappa_2 - C^{'},
\end{align} where $\kappa_2$ is the $2^{nd}$ eigenvalue of periodic bilaplacian on $Y$, where $C$ and $C^{'}$ are generic constants independent of $\rho$ and $\eta$.
\end{lemma}
\begin{proof}
Recall the following G\mathcal{A} rding type estimate~\cref{coercivity} for the form $a^{\rho}[\eta]$ associated with the operator $\mathcal{A}^\rho(\eta)$:
\begin{align*}
a^\rho[\eta](u,u)+C_*||u||^2_{L^2_\sharp(Y)}\geq \frac{\rho^2}{6}||\Delta u||^2_{L^2_\sharp(Y)}+\frac{\alpha}{2}||u||_{H^1_\sharp(Q)}.
\end{align*}
The inequality in~\cref{largerhobound} follows readily from above by applying the minmax characterization.
\end{proof}
\begin{remark}
In~\cref{lowrhobound} and~\cref{largerhobound}, we have avoided estimating the second Bloch eigenvalue by using the spectral problem associated with Neumann bilaplacian as it is known to be ill-posed~\cite{provenzanoNoteNeumannEigenvalues2018}. Moreover, polyharmonic Neumann eigenvalue problems on polygonal domains (such as $Y$) are less well understood~\cite{gazzolaEigenvalueProblems2010,ferraressoBabuskaParadoxPolyharmonic2019}. However, suitable natural boundary conditions associated with the operator $\Delta^2-\tau\Delta$ are obtained in~\cite{chasmanIsoperimetricInequalityFundamental2011}.
\end{remark}
We are finally in a position to prove that the neighbourhood of analyticity of the first Bloch eigenvalue does not depend on the parameter $\rho$.
\begin{theorem}
\label{independentnbd} There exists a neighbourhood $U=B_{\delta}(0)$ of $\eta=0$ , not depending on $\rho$, such that $\lambda_1^\rho(\eta)$ is analytic on $B_\delta(0)$.
\end{theorem}
\begin{proof}
It was proved in~\cref{analytic} that the first Bloch eigenvalue is analytic in a neighbourhood of $\eta=0$. However, \textit{a priori} it is not clear whether this neighbourhood is independent of $\rho$. To prove this, it suffices to prove that the first Bloch eigenvalue is simple in a neighbourhood of $\eta=0$ independently of $\rho$. Observe that
\begin{align}
|\lambda_1^\rho(\eta)-\lambda_2^{\rho}(\eta)|&\geq \lambda_2^\rho(\eta)-|\lambda_1^\rho(\eta)-\lambda_1^\rho(0)|-|\lambda_2^\rho(\eta)-\lambda_2^\rho(0)|\nonumber\\
&\stackrel{\cref{lipschitzbound}}{\geq} \lambda_2^\rho(\eta)-2(C+\rho^2)|\eta|,\label{spectralgap}
\end{align} where $C$ is a generic constant independent of $\rho$ and $\eta$.
\begin{itemize}
\item For sufficiently large $\rho$,
\begin{align*}|\lambda_1^\rho(\eta)-\lambda_2^{\rho}(\eta)|&\geq \lambda_2^\rho(\eta)-2(C+\rho^2)|\eta|\\
&\stackrel{\cref{largerhobound}}{\geq} (C^{'}\rho^2-C^{''})-2(C+\rho^2)|\eta|\\
&\stackrel{\text{large }\rho}{\geq} C^{'''}\rho^2-2\rho^2|\eta|>0
\end{align*} for $|\eta|<\frac{C^{'''}}{2}$. Here, $C,C^{'},C^{''}$ and $C^{'''}$ are all generic constants independent of $\rho$ and $\eta$.
\item For remaining values of $\rho$,
\begin{align*}|\lambda_1^\rho(\eta)-\lambda_2^{\rho}(\eta)|&\geq \lambda_2^\rho(\eta)-2(C+\rho^2)|\eta|\\
&\stackrel{\cref{lowrhobound}}{\geq} \alpha\lambda^N_2-2(C+\rho^2)|\eta|\\
&\geq \alpha\lambda^N_2-2C|\eta|>0
\end{align*} for $|\eta|<\frac{\alpha\lambda^N_2}{2C}$.
\end{itemize}
\end{proof}
\begin{remark}
In the papers~\cite{sivajiganeshBlochApproachAlmost2019,sivajiganeshBlochWaveHomogenisation2020}, an additional artificial parameter is introduced in the Bloch eigenvalue problem to facilitate the homogenization method. Unlike~\cref{main}, these papers employ successive limits of the two parameters instead of simultaneous limits. Therefore, the non-dependence of the neighbourhood of analyticity on the second parameter is not required in~\cite{sivajiganeshBlochApproachAlmost2019,sivajiganeshBlochWaveHomogenisation2020}.
\end{remark}
\section{Cell Problem and Estimates}\label{cellproblem}
In this section, we will consider the classical cell problem associated with~\cref{main} and the estimates for the corrector field. This section will allow us to characterize the homogenized tensor for~\cref{main} and the corrector field in terms of Bloch eigenvalues and eigenfunctions.
For $1\leq j\leq d$, consider the following cell problem associated with the operator~\cref{main}:
\begin{equation}\label{correctors}
\begin{cases}
&\rho^2\Delta^2\chi_j^\rho-\text{div}\,A(y)(e_j+\nabla \chi_j^\rho)=0\text{ in }\mathbb{R}^d\\
&\chi_j^\rho\text{ is }Y-\text{periodic}\\
&\mathcal{M}_Y(\chi_j^\rho)=\fint_Y\chi_j^\rho(y)\,dy=0.
\end{cases}
\end{equation}
By a simple application of Lax-Milgram lemma on $H^2_\sharp(Y)$, we obtain solution to above for every $\rho>0$ (For $\rho=0$, Lax-Milgram lemma is applied for $H^1_\sharp(Y)$). Further, since the equation is also satisfied in the sense of distributions, we conclude that $\chi_j^\rho\in H^3_\sharp(Y)$ for $\rho>0$. By using $\chi_j^\rho$ as a test function, we obtain the following bound:
\begin{align}
\label{energyestimate1} \rho||\Delta\chi_j^\rho||_{L^2_\sharp(Y)}+||\chi_j^\rho||_{H^1_\sharp(Y)} \leq C.
\end{align} If we use $\Delta\chi_j^\rho$ as a test function, we obtain \begin{align}
\label[]{energyestimate2}
\rho^2||\nabla^3\chi_j^\rho||_{L^2_\sharp(Y)}\leq C.
\end{align}
We also collect below a few estimates which will be required later. Similar estimates have been proved in~\cite{niuCombinedEffectsHomogenization2020} to which we refer for more details.
\begin{lemma}\label{estimateforposrho}
Let $\rho_1,\rho_2>0$. Let $\chi^{\rho_1},\chi^{\rho_2}\in H^2_\sharp(Y)$ be solutions to~\cref{correctors} for $rho=\rho_1$ and $\rho=\rho_2$ respectively, then the following estimate holds.
\begin{align}
||\nabla \chi^{\rho_1}-\nabla \chi^{\rho_2}||_{L^2(Y)}\leq C |1-(\rho_1/\rho_2)^2|.
\end{align}
\end{lemma}
\begin{proof}
Define $z=\chi^{\rho_1}_j-\chi^{\rho_2}_j$, then $z$ satisfies the following equation
\begin{align}
\rho_1^2\Delta^2 z-\text{div}\,A(y)\nabla z=(\rho_2^2-\rho_1^2)\Delta^2\chi^{\rho_2}.
\end{align} Now, the quoted estimate readily follows by taking $z$ as the test function, applying uniform ellipticity of $A$,~\eqref{energyestimate2} and an application of Poincar\'e inequality.
\end{proof}
\begin{lemma}\label{estimateforzerorho}
Let $\rho>0$. Let $\chi^{\rho}\in H^2_\sharp(Y)$ be the solution to~\cref{correctors} and let $\chi^0\in H^1_\sharp(Y)$ solve
\begin{equation}
\begin{cases}
&-\mbox{div}\,A(y)(e_j+\nabla \chi_j^0)=0\text{ in }\mathbb{R}^d\\
&\chi_j^0\text{ is }Y-\text{periodic}\\
&\mathcal{M}_Y(\chi_j^0)=\fint_Y\chi_j^0(y)\,dy=0.
\end{cases}
\end{equation}
Then, there is $q\in (1,\infty)$ such that for every $\varkappa>0$ there exists a matrix $B$ with entries in $C^\infty_\sharp(Y)$ such that $||A-B||_{L^q_{\sharp}(Y)}\leq \varkappa$ and the following estimates hold.
\begin{align}
||\nabla\chi^0_j-\nabla\chi^B_j||_{L^2(Y)}&\leq C\varkappa\label{oneone}\\
||\nabla \chi^{\rho}-\nabla \chi^B||_{L^2(Y)}&\leq C\left\{\rho||\chi^B||_{H^2(Y)}+\varkappa\right\}\label{twotwo},
\end{align} where $\chi^B\in H^1_\sharp(Y)$ solve
\begin{equation}
\begin{cases}
&-\mbox{div}\,B(y)(e_j+\nabla \chi_j^B)=0\text{ in }\mathbb{R}^d\\
&\chi_j^B\text{ is }Y-\text{periodic}\\
&\mathcal{M}_Y(\chi_j^B)=\fint_Y\chi_j^B(y)\,dy=0.
\end{cases}
\end{equation}
\end{lemma}
\begin{proof}
Observe that given any $q\in(1,\infty)$ and $\varkappa>0$, we can find a smooth periodic matrix $B$ with the same ellipticity constant and upper bound as $A$ such that $||A-B||_{L^q_{\sharp}(Y)}\leq \varkappa$. For example, this can be achieved by a standard smoothing by convolution. Now, by regularity theory, $\chi^B_j\in H^2_\sharp(Y)$.
Define $z=\chi^{\rho}_j-\chi^B_j$, then $z$ satisfies the following equation
\begin{align}
\rho^2\Delta^2 z-\text{div}\,B(y)\nabla z=-\rho^2\Delta^2\chi^B_j+\text{div}(A-B)\nabla\chi^\rho_j.
\end{align}
We test this equation against $z$ to obtain:
\begin{align*}
\rho^2\int_Y|\Delta z|^2\,dy+\int_Y A(y)\nabla z\cdot\nabla z\,dy&\leq \rho^2\int_Y|\Delta\chi^B_j||\Delta z|\,dy+\int_Y|A-B||\nabla\chi^\rho_j||\nabla z|\,dy.
\end{align*} This leads to
\begin{align*}
\rho^2||\Delta z||_{L^2}^2+\alpha||\nabla z||^2_{L^2}&\leq \rho^2||\Delta\chi^B_j||_{L^2}||\Delta z||_{L^2}+||\nabla z||_{L^2}\left(\int_Y|A-B|^2|\nabla\chi^\rho_j|^2\,dy\right)^{1/2}.
\end{align*} By Young's inequality,
\begin{align*}
||\nabla z||_{L^2}&\leq C\left\{ \rho||\Delta\chi^B_j||_{L^2}+\left(\int_Y|A-B|^2|\nabla\chi^\rho_j|^2\,dy\right)^{1/2}\right\}.
\end{align*}
On the last term, we apply a form of Meyers estimate for the $L^p$ integrability of $\nabla\chi^\rho_j$ proved in~\cite[Page~7, Theorem~2.3]{niuCombinedEffectsHomogenization2020}. This fixes the choice of $q$. This finishes the proof of the estimate~\cref{twotwo}. The proof of~\cref{oneone} is similar and simpler and hence omitted.
\end{proof}
For every fixed $0\leq\rho<\infty$, the homogenized tensor for the operator $\mathcal{A}^{\rho,\epsilon}=\rho^2\Delta^2-\text{div}\,A\left(\frac{x}{\epsilon}\right)\nabla$ is given by
\begin{align}
\label{fixedhomtensor}
A^{\rho,\text{hom}}\coloneqq \mathcal{M}_Y\left(A+A\nabla\chi^\rho\right)
\end{align}
\begin{definition}[Homogenized Tensor for $\mathcal{A}^{\kappa,\epsilon}$]\label{homtensor}\begin{equation}
A^{\text{hom}}\coloneqq
\begin{cases}
&\mathcal{M}_Y\left(A+A\nabla\chi^\theta\right)\text{ for }0<\theta<\infty\text{ where }\rho=\frac{\kappa}{\epsilon}\to \theta,\\
&\mathcal{M}_Y\left(A+A\nabla\chi^0\right)\text{ when }\rho=\frac{\kappa}{\epsilon}\to 0,\\
&\mathcal{M}_Y\left(A\right)\text{ when }\rho=\frac{\kappa}{\epsilon}\to \infty.
\end{cases}
\end{equation}
\end{definition}
\section{Bloch Characterization of Homogenized Tensor}\label{BlochCharacterization}
In this section, we will give a new characterization of the homogenized tensor (see \cref{homtensor}), and corrector field~\cref{correctors} in terms of the first Bloch eigenvalue and eigenfunction. These characterizations are obtained by differentiating the Bloch spectral problem~\eqref{BlochEigenvalueProblem} with respect to the dual parameter $\eta$. Indeed, this is possible since we proved the analyticity of the first Bloch eigenvalue and eigenfunction with respect to $\eta$ in a neighbourhood of $\eta=0$ in~\cref{analytic}. Moreover we will use the properties of the first Bloch eigenvalue to prove the stability of the homogenized tensor with respect to the limits $\rho\to 0$ and $\rho\to\infty$.
\subsection{Derivatives of Bloch eigenvalues and eigenfunctions}
We recall the Bloch eigenvalue problem for the operator $\mathcal{A}^\rho$ here:
\begin{align}\label{Blochproblem3}
\rho^2(\nabla+i\eta)^4\phi^\rho_1(y;\eta)-(\nabla+i\eta)\cdot A(y)(\nabla+i\eta)\phi^\rho_1(y;\eta)=\lambda^\rho_1(\eta)\phi^\rho_1(y;\eta).
\end{align}
We know that $\lambda^\rho_1(0)=0$. For $\eta\in Y^{'}$, recall that $\mathcal{A}^\rho(\eta)= \rho^2(\nabla+i\eta)^4-(\nabla+i\eta)\cdot A(y)(\nabla+i\eta)$. In this section, for notational convenience we will hide the dependence on $y$.
We shall normalize the average value of the first Bloch eigenfunction $\phi_1^\rho(\cdot;\eta)$ to be $(2\pi)^{-d/2}$, that is, \begin{align}\label{normalization}\mathcal{M}_Y(\phi_1^\rho(\cdot,\eta))=(2\pi)^{-d/2}\end{align} for all $\eta$ in the neighbourhood of analyticity. We shall use $\beta$ to denote a multiindex, such as $\beta=(\beta_1,\beta_2,\ldots,\beta_d)\in\mathbb{N}^d\cup\{0\}$ and $|\beta|\coloneqq |\beta_1|+|\beta_2|+\cdots+|\beta_d|$. We will use the shorthand $\partial^\beta_\eta$ to denote $\partial^\beta_\eta\coloneqq \frac{\partial^{\beta_1}}{\partial\eta_1^{\beta_1}}\cdots\frac{\partial^{\beta_d}}{\partial\eta_d^{\eta_d}}$. For simplicity, we will also use the notation $\partial^\beta_0 u= \frac{\partial^{\beta_1}u}{\partial\eta_1^{\beta_1}}\cdots\frac{\partial^{\beta_d}u}{\partial\eta_d^{\eta_d}}\bigg|_{\eta=0}$. Differentiating~\cref{normalization}, we obtain
\begin{align}\label{compatibility}
\mathcal{M}_Y(\partial^\beta_0\phi_1^\rho)=0
\end{align} for all $|\beta|>0$. Later on, we will see this as the compatibility condition associated with the equation satisfied by $\partial^\beta_0\phi_1^\rho$. Denote by $\partial^\beta_0\mathcal{A}^\rho\coloneqq \frac{\partial^\beta\mathcal{A}}{\partial\eta^\beta}\bigg|_{\eta=0}$. Then, it holds true that
\begin{align}
\partial^\beta_\eta\mathcal{A}\equiv 0\text{ for all }|\beta|>4,
\end{align}
since $\mathcal{A}^\rho(\eta)$ is a fourth order polynomial in $\eta$. Direct calculation shows that
\begin{align}\label{diffoperators}
\mathcal{A}^\rho(0)&= \rho^2\nabla^4-\nabla\cdot A(y)\nabla\nonumber\\
\partial^{e_j}_0\mathcal{A}^\rho&=4i\rho^2e_j\cdot\nabla\nabla^2-ie_j\cdot A\nabla-i\nabla\cdot Ae_j\nonumber\\
\partial^{e_j+e_k}_0\mathcal{A}^\rho&=-4\rho^2\delta_{jk}\nabla^2-8\rho^2\partial_{y_j}\partial_{y_k}+2 a_{jk}\nonumber\\
\partial_0^{e_j+e_k+e_l}\mathcal{A}^\rho&=-8i\rho^2(\delta_{jk}\partial_{y_l}+\delta_{jl}\partial_{y_k}+\delta_{kl}\partial_{y_j})\nonumber\\
\partial_0^{e_j+e_k+e_l+e_m}\mathcal{A}^\rho&=8\rho^2(\delta_{jk}\delta_{lm}+\delta_{jl}\delta_{km}+\delta_{jm}\delta_{kl}),
\end{align} where $e_j$ denotes the standard Euclidean unit vector with $1$ in the $j^{th}$ place and $0$ elsewhere.
Now, we are in a position to write down the differential equations satisfied by the derivatives of $\phi_1^\rho$ of all orders. To this end, we recall the Leibniz's formula for the derivatives of product of functions, viz., \begin{align}\label{leibniz}
\partial^{\beta}(fg)=\sum_{\gamma\in\mathbb{N}^d\cup\{0\}}\binom{\beta}{\gamma}\,\partial^\gamma f\,\partial^{\beta-\gamma}g,
\end{align} where $\binom{\beta}{\gamma}=\binom{\beta_1}{\gamma_1}\cdots\binom{\beta_d}{\gamma_d}$ and the sum is always finite since $\binom{\beta_j}{\gamma_j}=0$ whenever $\beta_j<\gamma_j$.
\subsection*{\underline{Cell Problems for $\partial^\beta_0\phi_1^\rho$}} Differentiating~\cref{Blochproblem3} with respect to $\eta$ and applying~\cref{leibniz}, we obtain
\begin{align}\label{cellprobbeta1}
\mathcal{A}^\rho(0)\partial^\beta_0\phi_1^\rho &+ \sum_{j=1}^d \beta_j \partial^{e_j}_0\mathcal{A}^\rho\,\partial^{\beta-e_j}_0\phi_1^\rho + \sum_{j,k} \binom{\beta}{e_j+e_k} \partial^{e_j+e_k}_0\mathcal{A}^\rho\,\partial^{\beta-e_j-e_k}_0\phi_1^\rho\nonumber\\
&+\sum_{j,k,l} \binom{\beta}{e_j+e_k+e_l} \partial^{e_j+e_k+e_l}_0\mathcal{A}^\rho\,\partial^{\beta-e_j-e_k-e_l}_0\phi_1^\rho\nonumber\\
&+\sum_{j,k,l,m} \binom{\beta}{e_j+e_k+e_l+e_m} \partial^{e_j+e_k+e_l+e_m}_0\mathcal{A}^\rho\,\partial^{\beta-e_j-e_k-e_l-e_m}_0\phi_1^\rho\nonumber\\
&= \sum_{\gamma\in\mathbb{N}^d\cup\{0\}}\binom{\beta}{\gamma}\partial^\gamma_0\lambda_1^\rho\,\partial^{\beta-\gamma}_0\phi_1^\rho.
\end{align}
Substituting~\cref{diffoperators} in~\cref{cellprobbeta1}, we obtain
\begin{align}\label{cellprobbeta2}
\left(\rho^2\nabla^4-\nabla\cdot A(y)\nabla\right)\partial^\beta_0\phi_1^\rho &= \sum_{j=1}^d \beta_j \left(-4i\rho^2e_j\cdot\nabla\nabla^2+ie_j\cdot A\nabla+i\nabla\cdot Ae_j\right)\,\partial^{\beta-e_j}_0\phi_1^\rho \nonumber\\
- \sum_{j,k} \binom{\beta}{e_j+e_k}& \left(-4\rho^2\delta_{jk}\nabla^2-8\rho^2\partial_{y_j}\partial_{y_k}+2 a_{jk}\right)\,\partial^{\beta-e_j-e_k}_0\phi_1^\rho\nonumber\\
+\sum_{j,k,l} \binom{\beta}{e_j+e_k+e_l} &\left(8i\rho^2(\delta_{jk}\partial_{y_l}+\delta_{jl}\partial_{y_k}+\delta_{kl}\partial_{y_j})\right)\,\partial^{\beta-e_j-e_k-e_l}_0\phi_1^\rho\nonumber\\
-\sum_{j,k,l,m} \binom{\beta}{e_j+e_k+e_l+e_m} &\left(8\rho^2(\delta_{jk}\delta_{lm}+\delta_{jl}\delta_{km}+\delta_{jm}\delta_{kl})\right)\,\partial^{\beta-e_j-e_k-e_l-e_m}_0\phi_1^\rho\nonumber\\
+ \sum_{\gamma\in\mathbb{N}^d}\binom{\beta}{\gamma}&\,\partial^\gamma_0\lambda_1^\rho\,\partial^{\beta-\gamma}_0\phi_1^\rho.
\end{align}
\subsection*{\underline{Expression for $\partial^\beta_0\lambda_1^\rho$}} It is easy to see that $\lambda_1^\rho(0)=0$ since $0$ is the first eigenvalue of the operator $\mathcal{A}^\rho(0)$. On the other hand, $\lambda_1^\rho$ is an even function of $\eta$ since $a^\rho[-\eta](\phi,\phi)=a^\rho[\eta](\overline{\phi},\overline{\phi})$ and $\overline{\phi}\in H^2_\sharp(Y)$ if and only if $\phi\in H^2_\sharp(Y)$ (see also~\cite[Page~41, Lemma~4.4]{dohnalDispersiveHomogenizedModels2015}) .
Rearranging~\cref{cellprobbeta2}, we get
\begin{align}\label{lambdadiffs1}
\partial^\beta_0\lambda_1^\rho\phi_1^\rho(0)&=\left(\rho^2\nabla^4-\nabla\cdot A(y)\nabla\right)\partial^\beta_0\phi_1^\rho\nonumber\\
+ \sum_{j=1}^d \beta_j & \left(-4i\rho^2e_j\cdot\nabla\nabla^2+ie_j\cdot A\nabla+i\nabla\cdot Ae_j\right)\,\partial^{\beta-e_j}_0\phi_1^\rho \nonumber\\
- \sum_{j,k} \binom{\beta}{e_j+e_k}& \left(-4\rho^2\delta_{jk}\nabla^2-8\rho^2\partial_{y_j}\partial_{y_k}+2 a_{jk}\right)\,\partial^{\beta-e_j-e_k}_0\phi_1^\rho\nonumber\\
+\sum_{j,k,l} \binom{\beta}{e_j+e_k+e_l} &\left(8i\rho^2(\delta_{jk}\partial_{y_l}+\delta_{jl}\partial_{y_k}+\delta_{kl}\partial_{y_j})\right)\,\partial^{\beta-e_j-e_k-e_l}_0\phi_1^\rho\nonumber\\
-\sum_{j,k,l,m} \binom{\beta}{e_j+e_k+e_l+e_m} &\left(8\rho^2(\delta_{jk}\delta_{lm}+\delta_{jl}\delta_{km}+\delta_{jm}\delta_{kl})\right)\,\partial^{\beta-e_j-e_k-e_l-e_m}_0\phi_1^\rho\nonumber\\
+ \sum_{\stackrel{\gamma\in\mathbb{N}^d}{\gamma\neq\beta}}\binom{\beta}{\gamma}&\,\partial^\gamma_0\lambda_1^\rho\,\partial^{\beta-\gamma}_0\phi_1^\rho.
\end{align}
Integrating~\cref{lambdadiffs1} over $Y$, using~\cref{normalization} and~\cref{compatibility} and the fact that integrals over $Y$ of derivatives of periodic functions vanish due to Green's identity, we obtain the following formula for the derivatives of the first Bloch eigenvalue at $\eta=0$:
\begin{align}\label{lambdadiffs2}
\partial^\beta_0\lambda_1^\rho =
\begin{cases}
& 2 \sum_{j,k} \binom{\beta}{e_j+e_k}\mathcal{M}_Y\left(a_{jk}\partial^{\beta-e_j-e_k}_0\phi^\rho_1\right)-i\sum_{j=1}^d\beta_j\mathcal{M}_Y\left( e_j\cdot A\nabla\partial^{\beta-e_j}_0\phi_1^\rho \right)\text{ when }|\beta|\neq 4.\\
& 2 \sum_{j,k} \binom{\beta}{e_j+e_k}\mathcal{M}_Y\left(a_{jk}\partial^{\beta-e_j-e_k}_0\phi^\rho_1\right)-i\sum_{j=1}^d\beta_j\mathcal{M}_Y\left( e_j\cdot A\nabla\partial^{\beta-e_j}_0\phi_1^\rho \right)\\
&\qquad\qquad -\sum_{j,k,l,m} 8\rho^2\left(\delta_{jk}\delta_{lm}+\delta_{jl}\delta_{km}+\delta_{jm}\delta_{kl}\right)\,\phi_1^\rho(0)\text{ when }|\beta|=4.
\end{cases}
\end{align}
We specialize to $\beta=e_l$ in~\cref{lambdadiffs2} to get $\displaystyle\frac{\partial{\lambda}^\rho_1}{\partial\eta_l}(0)=0$ for all $l=1,2,\ldots,d$.
On the other hand, if we set $\beta=e_l$ in~\cref{cellprobbeta2}, we obtain
\begin{align*}
\left(-\nabla \cdot A(y)\nabla+\rho^2\Delta^2\right)\frac{\partial {\phi^\rho_1}}{\partial \eta_l}(0)=\nabla\cdot A(y)e_l i{\phi^\rho_1}(0).
\end{align*}
Comparing with~\cref{correctors}, we conclude that $\displaystyle\chi^\rho_l-\frac{1}{i{\phi^\rho}_1(0)}\frac{\partial\phi^\rho_1}{\partial\eta_l}(0)$ is a constant.
We also specialize to $\beta=e_l+e_k$ in~\cref{lambdadiffs2} to get
\begin{align}\label{homogenizedtensor2}
\frac{1}{2}\frac{\partial^2 {\lambda}^\rho_1}{\partial\eta_k\partial\eta_l}(0)=\frac{1}{|Y|}\int_Y \left(e_k\cdot Ae_l+\frac{1}{2}e_k\cdot A \nabla \chi_l^\rho+\frac{1}{2}e_l\cdot A \nabla \chi_k^\rho\right)\,dy.
\end{align} On comparing~\cref{homogenizedtensor2} with~\cref{fixedhomtensor}, we obtain the following theorem:
\begin{theorem}\label{Hessian}
The first Bloch eigenvalue and eigenfunction satisfy:
\begin{enumerate}
\item $\lambda^\rho_1(0)=0$.
\item The eigenvalue $\lambda^\rho_1(\eta)$ has a critical point at $\eta=0$, i.e., \begin{align}\frac{\partial \lambda^\rho_1}{\partial \eta_l}(0)=0, \forall\, l=1,2,\ldots,d.\end{align}
\item For $l=1,2,\ldots,d,$ the derivative of the eigenvector $(\partial \phi_1^\rho/\partial\eta_l)(0)$ satisfies:
$(\partial \phi_1^\rho/\partial\eta_l)(y;0)-i\phi^\rho_1(y;0)\chi^{\rho}_l(y)$ is a constant in $y$ where $\chi^\rho_l$ solves the cell problem~\eqref{correctors}.
\item The Hessian of the first Bloch eigenvalue at $\eta=0$ is twice the homogenized matrix $A^{\rho,\text{hom}}$ as defined in~\eqref{fixedhomtensor}, i.e.,
\begin{align}\label{identificationofhomo}
\frac{1}{2}\frac{\partial^2\lambda^\rho_1}{\partial\eta_k\partial\eta_l}(0)=e_k\cdot A^{\rho,\text{hom}}e_l.
\end{align}
\end{enumerate}
\end{theorem}
\subsection{Stability of homogenized tensor}
Now, we will prove the stability of homogenized tensor in the limits $\rho\to 0$ and $\rho\to\infty$.
\begin{lemma}
\label{uniformconvbloch}
Let $0\leq\theta<\infty$ such that $\rho\downarrow\theta$ then $\lambda^\rho_1(\eta)\to\lambda^\theta_1(\eta)$ uniformly on compact sets K contained in their common domain of definition.
\end{lemma}
\begin{proof} The fact that the first Bloch eigenvalues $\{\lambda^\rho_1(\eta)\}_{\rho\geq 0}$ are analytic on a common domain in $Y^{'}$ has been proved in~\cref{independentnbd}. Since $\rho\downarrow\theta$ where $\theta$ is finite, the sequence $\rho$ is bounded, that is, $0<\rho < \rho_0$ for some $0<\rho_0<\infty$. We will prove that the family $\{\lambda^\rho_1(\eta)\}_{0\leq \rho<\rho_0}$ is locally uniformly bounded, that is, for every compact set $K$,
\begin{align*}
|\lambda^\rho_1(\eta)|\leq |\lambda^\rho_1(\eta)-\lambda^\rho_1(0|\stackrel{\cref{lipschitzbound}}{\leq} C(\mu_1+\nu_1\rho^2)|\eta|< C^{'}\text{ for all }\eta\in K,
\end{align*} where $C^{'}$ is independent of $\rho$ and $\eta$. It is an easy consequence of Montel's Theorem~\cite[Page~9, Prop.~7]{narasimhanSeveralComplexVariables1971} that pointwise convergence implies uniform convergence for locally uniformly bounded sequences. We will show that $\lambda^\rho_1(\eta)$ converges pointwise to $\lambda^{\theta}_1(\eta)$. Applying minmax characterization to the form below
\begin{align*}
a^{\rho}[\eta](u,u) &= \int_Y A (\nabla+i\eta)u\cdot\overline{(\nabla+i\eta)u}\,dy+\rho^2\int_Y (\nabla+i\eta)^2 u\overline{(\nabla+i\eta)^2 u}\,dy\\
&=a^\theta[\eta](u,u)+(\rho^2-\theta^2)\int_Y (\nabla+i\eta)^2 u\overline{(\nabla+i\eta)^2 u}\,dy
\end{align*} we obtain the following inequality:
\begin{align*}
\lambda^\rho_1(\eta)-\lambda^\theta_1(\eta)\leq(\rho^2-\theta^2)\,\vartheta_1(\eta),
\end{align*} where $\vartheta_1(\eta)$ is the first Bloch eigenvalue of the bilaplacian and $\lambda^\theta_1(\eta)$ is the first Bloch eigenvalue of the operator $-\text{div}(A\nabla)+\theta^2\Delta^2$ for $\theta\in [0,\infty)$.
On the other hand, we also have
\begin{align*}
a^{\rho}[\eta](u,u) \geq a^{\theta}[\eta](u,u)\text{ for }\rho\geq\theta,
\end{align*} so that an application of minmax characterization yields:
\begin{align*}
\lambda^\rho_1(\eta)\geq\lambda^\theta_1(\eta)\text{ for }\rho\geq \theta.
\end{align*} Thus, we obtain
\begin{align*}
0\leq \lambda^\rho_1(\eta)-\lambda^\theta_1(\eta)\leq (\rho^2-\theta^2)\,\vartheta_1(\eta)\text{ for }\rho\geq \theta.
\end{align*} As a consequence, for each $\eta\in Y^{'}$, $\lambda^\rho_1(\eta)\to\lambda^\theta_1(\eta)$ as $\rho\downarrow \theta$.
\end{proof}
\begin{theorem}
\label{stabilityhomtensor}
Let $0\leq\theta\leq\infty$ such that $\rho\to\theta$ then $A^{\rho,\text{hom}}\to A^{hom}$, with $A^{hom}$ as in~\cref{homtensor}.
\end{theorem}
\begin{proof}\leavevmode
\begin{itemize}
\item[]{\textbf{Case 1. $\theta\in [0,\infty)$}}: By the characterization in~\cref{Hessian}, $A^{\rho,hom}=\frac{1}{2}\nabla_\eta^2\lambda^\rho_1(0)$. In~\cref{uniformconvbloch}, we have proved that $\lambda^\rho_1$ conveges uniformly to $\lambda_1^\theta$ when $\rho\downarrow\theta$ for $\theta\in[0,\infty)$. By Theorem of Weierstrass~\cite[Page~7, Prop.~5]{narasimhanSeveralComplexVariables1971}, derivatives of all orders of $\lambda^\rho_1(\eta)$ converge uniformly on compact sets to corresponding derivatives of $\lambda^\theta_1(\eta)$. In particular, $\nabla_\eta^2\lambda^\rho_1$ converges uniformly on compact sets to $\nabla^2\lambda^\theta_1$. As a consequence,
\begin{align*}
A^{\rho,hom}=\frac{1}{2}\nabla_\eta^2\lambda^\rho_1(0)\to\frac{1}{2}\nabla_\eta^2\lambda^\theta_1(0)=A^{hom}\text{ as }\rho\downarrow\theta\in[0,\infty).
\end{align*}
\item[]{\textbf{Case 2. $\theta=\infty$}:} Observe that
\begin{align*}
|A^{\rho,hom}-\mathcal{M}_Y(A)|=\left|\int_Y A(y)\nabla\chi^\rho(y)\,dy\right|\leq C||\nabla\chi^\rho||_{L^2_\sharp(Y)}\leq \frac{C}{\rho^2},
\end{align*} where the last inequality follows from Poincar\'e inequality and~\eqref{energyestimate2}. Therefore, as $\rho\to\infty$,
\begin{align*}
A^{\rho,hom}\to\mathcal{M}_Y(A)=A^{hom}\text{ as }\rho\uparrow\infty.
\end{align*}
\end{itemize}
\end{proof}
\begin{remark}\label{boundednessofeigfuncforlowrho}
As a consequence of~\cref{uniformconvbloch} and the discussion in Case $1$ of~\cref{stabilityhomtensor}, for any $R>0$, and $0\leq \rho<R$, the first Bloch eigenvalue and all its derivatives are bounded uniformly in their domain of analyticity independent of $\rho$. We may also conclude from the same that the corresponding (suitably normalized) Bloch eigenfunction and all their derivatives with respect to dual parameter are bounded in $H^1_\sharp$ independent of $\rho$. For the first Bloch eigenfunction, the boundedness is a consequence of the following calculation, which is a consequence of~\cref{coercivity}:
\begin{align*}
|\lambda_1^\rho(\eta)|+C_*\geq \lambda_1^\rho(\eta) + C_* = a^\rho[\eta](\phi_1^\rho(\eta),\phi_1^\rho(\eta))+C_*\geq \frac{\alpha}{2}||\phi_1^\rho(\eta)||^2_{H^1_\sharp(Y)}.
\end{align*} Recall that the number $C_*$ is explicitly given in~\cref{cstar}. The boundedness in $H^1_\sharp(Y)$ of the derivatives of the first Bloch eigenfunction in the dual parameter is a consequence of its analyticity~\cite[Page~5, Prop.~3]{narasimhanSeveralComplexVariables1971}. This precludes the case $\rho\to\infty$ which is covered in~\cref{higherestimates}.
\end{remark}
We end this section by finding boundedness estimates for higher order derivatives of the first Bloch eigenvalue and eigenfunction in the dual parameter in the regime $\rho\to\infty$.
\begin{theorem}\label{higherestimates}
For $1\leq \rho<\infty$,
\begin{enumerate}
\item $||\partial_0^{e_j}\phi^\rho_1||_{H^1}\lesssim\frac{1}{\rho^2}$, $\partial^{e_j}_0\lambda^\rho_1=0$.
\item $||\partial_0^{e_j+e_k}\phi^\rho_1||_{H^1}\lesssim\frac{1}{\rho^2}$, $\left|\partial^{e_j+e_k}_0\lambda^\rho_1-\mathcal{M}_Y(e_j\cdot Ae_k)\right|\lesssim\frac{1}{\rho^2}$.
\item $||\partial_0^{e_j+e_k+e_l}\phi^\rho_1||_{H^1}\lesssim\frac{1}{\rho^2}$, $\partial^{e_j+e_k+e_l}_0\lambda^\rho_1=0$.
\item $||\partial_0^{\beta}\phi^\rho_1||_{H^1}\lesssim 1$ for all $|\beta|\geq 4$.
\item $|\partial^{\beta}_0\lambda^\rho_1|\lesssim{\begin{cases}
&\rho^2\text{ for }|\beta|=4.\\
&1\text{ for }|\beta|>4.\\
\end{cases} }$.
\end{enumerate}
\end{theorem}
\begin{proof}
The estimates are computed in tandem from the equations~\cref{cellprobbeta2} and~\cref{lambdadiffs2}. One begins by proving the estimate on the derivative of the first Bloch eigenfunction at $\eta=0$, followed by the estimate on the corresponding derivative of the first Bloch eigenvalue. The solvability of~\cref{cellprobbeta2} in $H^2_\sharp(Y)$ follows from a standard application of Lax-Milgram lemma. On the other hand, one can also read off from~\eqref{cellprobbeta2} that the solution $\partial^\beta_0\phi^\rho_1\in H^3_\sharp(Y)$. As a consequence, the estimate on the derivatives of the first Bloch eigenfunction are obtained by employing the test function $\Delta_y\partial^\beta_0\phi^\rho_1$ in~\cref{cellprobbeta2} and repeated applications of Poincar\'e inequality. The computations are standard and therefore, omitted.
\end{proof}
\section{Bloch Transform and its properties}\label{blochtransform}
In this section, we will relate the Bloch spectral problem~\eqref{BlochEigenvalueProblem} to the Bloch spectral problem at the $\epsilon$-scale:
\begin{eqnarray}
\label{BlochEigenvalueProblemepsilon}
\begin{cases}
&\mathcal{A}^{\kappa,\epsilon}(\eta)\phi^{\kappa,\epsilon}\coloneqq \kappa^2(\nabla+i\xi)^4\phi^{\kappa,\epsilon}(x)-(\nabla+i\xi)\cdot A\left(\frac{x}{\epsilon}\right)(\nabla+i\xi)\phi^{\kappa,\epsilon}(x)=\lambda^{\kappa,\epsilon}(\xi)\phi^{\kappa,\epsilon}(x)\\
&\phi^{\kappa,\epsilon}(x+2\pi p\epsilon)=\phi^{\kappa,\epsilon}(x), p\in\mathbb{Z}^d,\xi\in \frac{Y^{'}}{\epsilon}.
\end{cases}
\end{eqnarray} Comparing to~\eqref{BlochEigenvalueProblem}, by homothety and $\kappa=\rho\epsilon$, we conclude that
\begin{align}
\lambda^{\kappa,\epsilon}(\xi)=\epsilon^{-2}\lambda^\rho(\epsilon\xi)\text{ and }\phi^{\kappa,\epsilon}(x,\xi)=\phi^\rho\left(\frac{x}{\epsilon};\epsilon\xi\right).
\end{align}
Now, we can state the Bloch decomposition theorem of $L^2(\mathbb{R}^d)$ at $\epsilon$-scale. We shall normalize $\phi^\rho_1(y;0)$ to be $(2\pi)^{-d/2}$.
\begin{theorem}\label{BlochDecompositionepsilon}
Let $\rho >0$. Let $g\in L^2(\mathbb{R}^d)$. Define the $m^{th}$ Bloch coefficient of $g$ at $\epsilon$-scale as
\begin{align}\label{BlochTransformepsilon}
\mathcal{B}^{\kappa,\epsilon}_mg(\xi)\coloneqq\int_{\mathbb{R}^d}g(x)e^{-ix\cdot\xi}\overline{\phi_m^{\kappa,\epsilon}(x;\xi)}\,dx,~m\in\mathbb{N},~\xi\in \frac{Y^{'}}{\epsilon}.
\end{align}
\begin{enumerate}
\item The following inverse formula holds
\begin{align}\label{Blochinverseepsilon}
g(x)=\int_{\frac{Y^{'}}{\epsilon}}\sum_{m=1}^{\infty}\mathcal{B}^{\kappa,\epsilon}_mg(\xi)\phi_m^{\kappa,\epsilon}(x;\xi)e^{ix\cdot\xi}\,d\xi.
\end{align}
\item{\bf Parseval's identity}
\begin{align}\label{parsevalblochepsilon}
||g||^2_{L^2(\mathbb{R}^d)}=\sum_{m=1}^{\infty}\int_{\frac{Y^{'}}{\epsilon}}|\mathcal{B}^{\kappa,\epsilon}_mg(\xi)|^2\,d\xi.
\end{align}
\item{\bf Plancherel formula} For $f,g\in L^2(\mathbb{R}^d)$, we have
\begin{align}\label{Plancherelepsilon}
\int_{\mathbb{R}^d}f(x)\overline{g(x)}\,dx=\sum_{m=1}^{\infty}\int_{\frac{Y^{'}}{\epsilon}}\mathcal{B}^{\kappa,\epsilon}_mf(\xi)\overline{\mathcal{B}^{\kappa,\epsilon}_mg(\xi)}\,d\xi.
\end{align}
\item{\bf Bloch Decomposition in $H^{-1}(\mathbb{R}^d)$} For an element $F=u_0(x)+\sum_{j=1}^N\frac{\partial u_j(x)}{\partial x_j}$ of $H^{-1}(\mathbb{R}^d)$, the following limit exists in $L^2\left(\frac{Y^{'}}{\epsilon}\right)$:
\begin{align}\label{BlochTransform2epsilon}
\mathcal{B}^{\kappa,\epsilon}_mF(\xi)=\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\left\{u_0(x)\overline{\phi^{\kappa,\epsilon}_m(x;\xi)}+i\sum_{j=1}^N\xi_ju_j(x)\overline{\phi^{\kappa,\epsilon}_m(x;\xi)}\right\}\,dx\nonumber\\-\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\sum_{j=1}^Nu_j(x)\frac{\partial\overline{\phi^{\kappa,\epsilon}_m}}{\partial x_j}(x;\xi)\,dx.
\end{align}
\item[] The definition above is independent of the particular representative of $F$.
\item Finally, for $g\in D(\mathcal{A}^{\kappa,\epsilon})$,
\begin{align}
\label{diagonalizationepsilon}
\mathcal{B}^{\kappa,\epsilon}_m(\mathcal{A}^{\kappa,\epsilon}g)(\xi)=\lambda^{\kappa,\epsilon}_m(\xi)\mathcal{B}^{\kappa,\epsilon}_mg(\xi).\end{align}
\end{enumerate}\qed
\end{theorem}
\subsection{First Bloch transform goes to Fourier transform}
In order to compute the homogenization limit, we need to know the limit of Bloch Transform of a sequence of functions. The following theorem proves that for a sequence of functions convergent in a suitable way, the first Bloch transform converges to the Fourier transform of the limit.
\begin{theorem}\label{BlochTransformConvergence}
Let $K\subseteq\mathbb{R}^d$ be a compact set and $(g^\epsilon)$ be a sequence of functions in $L^2(\mathbb{R}^d)$ such that $g^\epsilon=0$ outside $K$. Suppose that $g^\epsilon\rightharpoonup g$ in $L^2(\mathbb{R}^d)$-weak for some function $g\in L^2(\mathbb{R}^d)$. Then it holds that
\begin{align*}
\mathbbm{1}_{\epsilon^{-1}U}\mathcal{B}_1^{\kappa,\epsilon} g^\epsilon\rightharpoonup \widehat{g}
\end{align*} in $L^2_{\text{loc}}(\mathbb{R}^d_\xi)$-weak, where $\widehat{g}$ denotes the Fourier transform of $g$ and $\mathbbm{1}_{\epsilon^{-1}U}$ denotes the characteristic function of the set ${\epsilon^{-1}U}$.
\end{theorem}
\begin{proof}
In~\cref{independentnbd}, the existence of the set $U$ indepedent of $\rho$ was proved. The function $\mathcal{B}_1^{\kappa,\epsilon} g^\epsilon$ is defined for $\xi\in \epsilon^{-1}Y^{'}$. However, we shall treat it as a function on $\mathbb{R}^d$ by extending it outside $\epsilon^{-1}U$ by zero. We can write
\begin{align*}
\mathcal{B}_1^{\kappa,\epsilon} g^\epsilon(\xi)= \int_{\mathbb{R}^d} g(x)e^{-ix\cdot\xi}\overline{{\phi}_1^{\kappa,\epsilon}}(x;0)\,dx+\int_{\mathbb{R}^d} g(x)e^{-ix\cdot\xi}\left(\overline{\phi_1^\rho}\left(\frac{x}{\epsilon};\epsilon\xi\right) -\overline{\phi^\rho}\left(\frac{x}{\epsilon};0\right) \right)dx.
\end{align*}
Now, we need to distinguish between the regimes:
\begin{itemize}
\item[]{\textbf{Case 1. $\theta\in [0,\infty)$}}: The first term above converges to the Fourier transform of $g$ on account of the normalization of $\phi_1(y;0)$ whereas the second term goes to zero since it is $O(\epsilon\xi)$ due to the Lipschitz continuity of the first regularized Bloch eigenfunction which follows from~\eqref{lipschitzbound}.
\item[]{\textbf{Case 2. $\theta=\infty$}}: In this case, again, the first term converges to the Fourier transform of $g$ on account of the normalization of $\phi_1(y;0)$. However, for the second term, we make use of the analyticity of $\phi_1^{\kappa,\epsilon}$ in $\epsilon^{-1}U$ and the estimates in~\cref{higherestimates} to conclude that the second term is $O(\epsilon\xi)$ independent of $\rho$.
\end{itemize}
\end{proof}
\section{Qualitative Homogenization}\label{qualitative}
In this section, we will prove the qualitative homogenization result for the singularly perturbed homogenization problem. There are three regimes according to convergence of $\rho=\frac{\kappa}{\epsilon}$, viz., $\rho\to 0$, $\rho\to\theta\in(0,\infty)$ and $\rho\to\infty$.
\begin{theorem}\label{homog}
Let $\Omega$ be an arbitrary domain in $\mathbb{R}^d$ and $f\in L^2(\Omega)$. Let $u^\epsilon\in H^2(\Omega)$ be such that $u^\epsilon$ converges weakly to $u^*$ in $H^1(\Omega)$, $\kappa\Delta u^\epsilon$ is uniformly bounded in $L^2(\Omega)$, and
\begin{align}\label{equation}
\mathcal{A}^{\kappa,\epsilon} u^\epsilon=f\,\mbox{in}\,~\Omega,
\end{align} where $\kappa\to 0$ as $\epsilon\to 0$ and $\lim_{\epsilon\to 0}\frac{\kappa}{\epsilon}=\theta\in [0,\infty]$. Let $A^{hom}=(a^*_{kl})_{k,l=1}^d$ be as defined in~\cref{homtensor}. Then
\begin{enumerate}
\item For all $k=1,2,\ldots,d$, we have the following convergence of fluxes:
\begin{align}
A\left(\frac{x}{\epsilon}\right)\nabla u^\epsilon(x)\rightharpoonup A^{hom}\nabla u^*(x) \mbox{ in } (L^2(\Omega))^d\mbox{-weak}.
\end{align}
\item The limit $u^*$ satisfies the homogenized equation:
\begin{align}\label{homoperator}
\mathcal{A}^{hom}u^*=-\nabla\cdot A^{hom}\nabla u^*=f\,\mbox{ in }\,\Omega.
\end{align}
\end{enumerate}
\end{theorem}
\begin{remark}
In the spirit of H-convergence~\cite{muratHconvergence1997}, we do not impose any boundary condition on the equation. The H-convergence compactness theorem concerns convergence of sequences on which certain differential constraints have been imposed. In homogenization, the weak convergence of solutions is a consequence of uniform bounds on them, which follow from boundary conditions imposed on the equation. In the theorem quoted above, the uniform boundedness on $\kappa\Delta u^\epsilon$ would have followed if appropriate boundary conditions were imposed.
\end{remark}
The proof of Theorem~\ref{homog} is divided into the following steps. We begin by localizing the equation~\eqref{equation} which is posed on $\Omega$, so that it is posed on $\mathbb{R}^d$. We take the first Bloch transform $\mathcal{B}^{\kappa,\epsilon}_1$ of this equation and pass to the limit $\kappa,\epsilon\to 0$. The proof relies on the analyticity of the first Bloch eigenvalue and eigenfunction in a neighborhood of $0\in Y^{'}$. The limiting equation is an equation in Fourier space. The homogenized equation is obtained by taking the inverse Fourier transform. We will use the notation $a_{kl}^\epsilon(x)$ to denote $a_{kl}\left(\frac{x}{\epsilon}\right)$. Further, we will assume the Einstein convention of summing over repeated indices. The proof has been divided into separate cases for the regimes $\rho\to\theta\in(0,\infty)$, $\rho\to\infty$, and $\rho=0$.
\subsection{Localization}
Let $\psi_0$ be a fixed smooth function supported in a compact set $K\subset\mathbb{R}^d$. Since $u^\epsilon$ satisfies $\mathcal{A}^{\kappa,\epsilon} u^\epsilon=f$, $\psi_0 u^\epsilon$ satisfies
\begin{align}\label{local}
\mathcal{A}^{\kappa,\epsilon}(\psi_0 u^\epsilon)(x)=\psi_0f(x)+g^\epsilon(x)+h^{\epsilon}(x)+\sum_{m=1}^4l^{\kappa,\epsilon}_m(x)\,\mbox{ in }\,\mathbb{R}^d,
\end{align}
where
\begin{align}
g^\epsilon(x)&\coloneqq-\frac{\partial \psi_0}{\partial x_k}(x)a^\epsilon_{kl}(x)\frac{\partial u^\epsilon}{\partial x_l}(x),\label{geps}\\
h^{\epsilon}(x)&\coloneqq-\frac{\partial}{\partial x_k}\left(\frac{\partial \psi_0}{\partial x_l}(x)a^{\epsilon}_{kl}(x)u^\epsilon(x)\right),\label{heps}\\
l^{\kappa,\epsilon}_1(x)&\coloneqq\kappa^2\frac{\partial^4\psi^0}{\partial x_k^4}(x)u^\epsilon(x)\label{leps1}.\\
l^{\kappa,\epsilon}_2(x)&\coloneqq 4\kappa^2\frac{\partial^3\psi^0}{\partial x_k^3}(x)\frac{\partial u^\epsilon}{\partial x_k}(x)\label{leps2}.\\
l^{\kappa,\epsilon}_3(x)&\coloneqq 2\kappa^2\frac{\partial^2\psi^0}{\partial x_k^2}(x)\frac{\partial^2u^\epsilon}{\partial x_k^2}(x)\label{leps3}.\\
l^{\kappa,\epsilon}_4(x)&\coloneqq 4\kappa^2\frac{\partial\psi^0}{\partial x_k}(x)\frac{\partial^3u^\epsilon}{\partial x_k^3}(x)+ 4\kappa^2\frac{\partial^2\psi^0}{\partial x_k^2}(x)\frac{\partial^2u^\epsilon}{\partial x_k^2}(x)=4\kappa^2\frac{\partial}{\partial x_k}\left(\frac{\partial\psi_0}{\partial x_k}\frac{\partial^2u^\epsilon}{\partial x_k^2}\right)\label{leps4}.
\end{align}
While the sequence $g^\epsilon$ is bounded in $L^2(\mathbb{R}^d)$, the sequence $h^{\epsilon}$ is bounded in $H^{-1}(\mathbb{R}^d)$. Taking the first Bloch transform of both sides of the equation~\eqref{local}, we obtain for $\xi\in\epsilon^{-1}U$ a.e.
\begin{align}\label{Bloch}
\lambda^{\kappa,\epsilon}_1(\xi)\mathcal{B}^{\kappa,\epsilon}_1(\psi_0 u^\epsilon)(\xi)=\mathcal{B}^{\kappa
,\epsilon}_1(\psi_0 f)(\xi)+\mathcal{B}^{\kappa,\epsilon}_1g^\epsilon(\xi)+\mathcal{B}^{\kappa,\epsilon}_1h^{\epsilon}(\xi)+\sum_{m=1}^4\mathcal{B}^{\kappa,\epsilon}_1l^{\kappa,\epsilon}_m(\xi)
\end{align}
We shall now pass to the limit $\kappa,\epsilon\to 0$ in the equation~\eqref{Bloch}.
\subsection{\underline{Case 1 : $\rho\to\theta\in(0,\infty)$}}
\subsubsection{Limit of $\lambda^{\kappa,\epsilon}_1(\xi)\mathcal{B}^{\kappa,\epsilon}_1(\psi_0 u^\epsilon)$}
We expand the first Bloch eigenvalue about $\eta=0$ in $\lambda^{\kappa,\epsilon}_1(\xi)\mathcal{B}^{\kappa,\epsilon}_1(\psi_0 u^\epsilon)$ to write
\begin{align*}
\left(\frac{1}{2}\frac{\partial^2\lambda^\rho_1}{\partial\eta_s\partial\eta_t}(0)\xi_s\xi_t + O(\epsilon^2) \right) \mathcal{B}^{\kappa,\epsilon}_1(\psi_0 u^\epsilon).
\end{align*} The higher order derivatives of $\lambda^\rho_1(\eta)$ are bounded uniformly in $\rho$ (see~\cref{boundednessofeigfuncforlowrho}). Hence, their contribution is $O(\epsilon^2)$. Now, we can pass to the limit $\kappa,\epsilon\to 0$ in $L^2_{\loc}(\mathbb{R}^d_\xi)$-weak by applying Lemma~\ref{BlochTransformConvergence} to obtain:
\begin{align}\label{convergence_ueps}
e_s\cdot A^{hom}e_t{\partial\eta_s\partial\eta_t}(0)\xi_s\xi_t\widehat{\psi_o u^*}(\xi).
\end{align}
\subsubsection{Limit of $\mathcal{B}^{\kappa,\epsilon}_1(\psi_0 f)$}
An application of Lemma~\ref{BlochTransformConvergence} yields the convergence of $\mathcal{B}^{\kappa,\epsilon}_1(\psi_0 f)$ to $(\psi_0 f)^{\bf\widehat{}}$ in $L^2_{\loc}(\mathbb{R}^d_\xi)$-weak.
\subsubsection{Limit of $\mathcal{B}^{\kappa,\epsilon}_1g^\epsilon$}
The sequence $g^\epsilon$ as defined in~\eqref{geps} is bounded in $L^2(\mathbb{R}^d)$ and hence has a weakly convergent subsequence with limit $g^*\in L^2(\mathbb{R}^d)$. This sequence is supported in a fixed set $K$. Also, note that the sequence $\displaystyle\sigma_k^\epsilon(x)\coloneqq a^\epsilon_{kl}(x)\frac{\partial u^\epsilon}{\partial x_l}(x)$ is bounded in $L^2(\Omega)$, hence has a weakly convergent subsequence whose limit is denoted by $\sigma^*_k$ for $k=1,2,\ldots,d$. Extend $\sigma^*_k$ by zero outside $\Omega$ and continue to denote the extension by $\sigma^*_k$. Thus, $g^*$ is given by $-\frac{\partial \psi_0}{\partial x_k}\sigma^*_k$. Therefore, by Lemma~\ref{BlochTransformConvergence}, we obtain the following convergence in $L^2_{\loc}(\mathbb{R}^d_\xi)$-weak:
\begin{align}\label{convergence_geps}
\chi_{\epsilon^{-1}U}(\xi)\mathcal{B}^{\kappa,\epsilon}_1g^\epsilon(\xi)\rightharpoonup-\left(\frac{\partial \psi_0}{\partial x_k}(x)\sigma^*_k(x)\right)^{\bf\widehat{}}(\xi).
\end{align}
\subsubsection{Limit of $\mathcal{B}_1^{\kappa,\epsilon}h^{\epsilon}$}
We have the following weak convergence for $\mathcal{B}_1^{\kappa,\epsilon}h^{\epsilon}$ in $L^2_{\loc}(\mathbb{R}^d_{\xi})$.
\begin{align}\label{convergence_heps}
\lim_{\epsilon\to 0}\,\chi_{\epsilon^{-1}U}(\xi)\mathcal{B}^{\kappa,\epsilon}_1h^{\epsilon}(\xi)=-i\xi_ka^*_{kl}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)^{\bf\widehat{}}(\xi)
\end{align}
We shall prove this in the following steps.
\paragraph{Step 1} By the definition of the Bloch transform~\eqref{BlochTransform2epsilon} for elements of $H^{-1}(\mathbb{R}^d)$, we have
\begin{align}\label{heps1}
\mathcal{B}^{\kappa,\epsilon}_1h^{\epsilon}(\xi)=-i\xi_k\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial \psi_0}{\partial x_l}(x)a^{\epsilon}_{kl}(x)u^\epsilon(x)\overline{\phi^\rho_1\left(\frac{x}{\epsilon};\epsilon\xi\right)}\,dx\nonumber\\+\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial \psi_0}{\partial x_l}(x)a^{\epsilon}_{kl}(x)u^\epsilon(x)\frac{\partial\overline{\phi^{\rho}_1}}{\partial x_k}\left(\frac{x}{\epsilon};\epsilon\xi\right)\,dx.
\end{align}
\paragraph{Step 2}
The first term on RHS of~\eqref{heps1} is the Bloch transform of the expression $-i\xi_k\frac{\partial \psi_0}{\partial x_l}(x)a^{\epsilon}_{kl}(x)u^\epsilon(x)$ which converges weakly to $-i\xi_k\mathcal{M}_Y(a_{kl})\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)$.
\paragraph{Step 3} Now, we analyze the second term on RHS of~\eqref{heps1}. To this end, we make use of analyticity of first Bloch eigenfunction with respect to the dual parameter $\eta$ near $0$. We have the following power series expansion in $H^1_\sharp(Y)$ for $\phi_1^\rho(\eta)$ about $\eta=0$:
\begin{align}
\phi_1^\rho(y;\eta)=\phi_1^\rho(y;0)+\eta_s\frac{\partial \phi^\rho_1}{\partial \eta_s}(y;0)+\gamma^\rho(y;\eta).
\end{align}
We know that $\gamma^\rho(y;0)=0$ and $(\partial \gamma^\rho/\partial \eta_s)(y;0)=0$, therefore, $\gamma^\rho(\cdot;\eta)=O(|\eta|^2)$ in $L^\infty(U;H^1_\sharp(Y))$. We also have $(\partial\gamma^\rho/\partial y_k)(\cdot;\eta)=O(|\eta|^2)$ in $L^\infty(U;L^2_\sharp(Y))$. These orders are uniform in $\rho$ by~\cref{boundednessofeigfuncforlowrho}.
Now,
\begin{align}
\phi_1^{\kappa,\epsilon}(x;\xi)=\phi_1^\rho\left(\frac{x}{\epsilon};\epsilon\xi\right)=\phi_1^\rho\left(\frac{x}{\epsilon};0\right)+\epsilon\xi_s\frac{\partial \phi^\rho_1}{\partial \eta_s}\left(\frac{x}{\epsilon};0\right)+\gamma^\rho\left(\frac{x}{\epsilon};\epsilon\xi\right).
\end{align}
Differentiating the last equation with respect to $x_k$, we obtain
\begin{align}\label{derivativeofphi}
\frac{\partial}{\partial x_k}\phi_1^\rho\left(\frac{x}{\epsilon};\epsilon\xi\right)=\xi_s\frac{\partial}{\partial y_k}\frac{\partial \phi^\rho_1}{\partial \eta_s}\left(\frac{x}{\epsilon};0\right)+\epsilon^{-1}\frac{\partial \gamma^\rho}{\partial y_k}\left(\frac{x}{\epsilon};\epsilon\xi\right).
\end{align}
For $\xi$ belonging to the set $\{\xi:\epsilon\xi\in U\mbox{ and }|\xi|\leq M\}$, we have
\begin{align}\label{gammaepsilon2}
\frac{\partial \gamma^\rho}{\partial y_k}(\cdot;\epsilon\xi)=O(|\epsilon\xi|^2)=\epsilon^2O(|\xi|^2)\leq CM^2\epsilon^2.
\end{align}
As a consequence,
\begin{align}
\epsilon^{-2}\frac{\partial \gamma^\rho}{\partial y_k}(x/\epsilon;\epsilon\xi)\in L^\infty_{\loc}(\mathbb{R}^d_\xi;L^2_\sharp(\epsilon Y)).
\end{align}
The second term on the RHS of~\eqref{heps1} is given by
\begin{align}\label{secondterm}
\chi_{\epsilon^{-1}U}(\xi)\int_K e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{\epsilon}\right)u^\epsilon(x)\frac{\partial}{\partial x_k}\left(\overline{\phi^{\rho}_1}\left(\frac{x}{\epsilon};\epsilon\xi\right)\right)\,dx.
\end{align}
Substituting~\eqref{derivativeofphi} in~\eqref{secondterm}, we obtain
\begin{align}\label{twotermsarehere}
\chi_{\epsilon^{-1}U}(\xi)\int_K e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{\epsilon}\right)u^\epsilon(x)\biggl[\xi_s\frac{\partial}{\partial y_k}\frac{\partial \phi^{\rho}_1}{\partial \eta_s}\left(\frac{x}{\epsilon};0\right)+\epsilon^{-1}\frac{\partial \gamma^\rho}{\partial y_k}\left(\frac{x}{\epsilon};\epsilon\xi\right)\biggr]\,dx.
\end{align}
In the last expression, the term involving $\gamma^\rho$ goes to zero as $\epsilon\to 0$ in view of~\eqref{gammaepsilon2}, whereas the other term has the following limit as $\rho\to\theta\in(0,\infty)$:
\begin{align}\label{secondterm2}
\mathcal{M}_Y\left(a_{kl}(y)\frac{\partial \chi^{\theta}_s}{\partial y_k}(y)\right)\xi_s\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)u^*(x)\,dx.
\end{align}
To see this, we write the second term as
\begin{align*}
\int_K e^{-ix\cdot\xi}&\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{\epsilon}\right)u^\epsilon(x)\xi_s\frac{\partial}{\partial y_k}\frac{\partial \phi^{\rho}_1}{\partial \eta_s}\left(\frac{x}{\epsilon};0\right)\,dx\\
&=\int_K e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{\epsilon}\right)u^\epsilon(x)\xi_s\frac{\partial \chi^{\rho}_s}{\partial y_k}\left(\frac{x}{\epsilon}\right)\,dx\\
&=\int_K e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{\epsilon}\right)u^\epsilon(x)\xi_s\left(\frac{\partial \chi^{\theta}_s}{\partial y_k}\left(\frac{x}{\epsilon}\right)+\left[\frac{\partial \chi^{\rho}_s}{\partial y_k}\left(\frac{x}{\epsilon}\right)-\frac{\partial \chi^{\theta}_s}{\partial y_k}\left(\frac{x}{\epsilon}\right)\right]\right)\,dx.
\end{align*}
The first term in parantheses goes to~\cref{secondterm2} due to strong convergence of $u^\epsilon$ in $L^2(K)$ and weak convergence of $a_{kl}\frac{\partial \chi^{\theta}_s}{\partial y_k}\left(\frac{x}{\epsilon}\right)$, whereas the expression in the square brackets goes to zero due to~\cref{estimateforposrho}.
\paragraph{Step 4} By Theorem~\ref{Hessian} and Remark~\ref{normalization}, it follows that
\begin{align}\label{equivalence2}
\mathcal{M}_Y\left(a_{kl}(y)\frac{\partial}{\partial y_k}\left(\frac{\partial\phi^{\theta}_1}{\partial\eta_s}(y;0)\right)\right)=-i(2\pi)^{-d/2}\mathcal{M}_Y\left(a_{kl}(y)\frac{\partial \chi^{\theta}_s}{\partial y_k}(y)\right).
\end{align}
Therefore, we have the following convergence in $L^2_{\loc}(\mathbb{R}^d_\xi)$-weak:
\begin{align}
\chi_{\epsilon^{-1}U}(\xi)\mathcal{B}^{\kappa,\epsilon}_1h^{\epsilon}(\xi)&\rightharpoonup-i\xi_s\biggl\{\mathcal{M}_Y(a_{kl})+\mathcal{M}_Y\left(a_{kl}(y)\frac{\partial \chi^{\theta}_s}{\partial y_k}(y)\right)\biggr\}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)^{\bf\widehat{}}(\xi)\nonumber\\
&=-i\xi_s a_{kl}^{*} \left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)^{\bf\widehat{}}(\xi)
\end{align}
\subsubsection{Limit of $\mathcal{B}_1^{\kappa,\epsilon}l_1^{\kappa,\epsilon}$} We shall prove that
\begin{align}\label{convergence_leps1}
\lim_{\epsilon\to 0}\mathcal{B}_1^{\kappa,\epsilon}l_1^{\kappa,\epsilon}=0.
\end{align}
Observe that
\begin{align}
\mathcal{B}_1^{\kappa,\epsilon}l^{\kappa,\epsilon}_1(\xi)=\kappa^2\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial^4\psi^0}{\partial x_k^4}(x)u^\epsilon(x)\overline{\phi^{\kappa,\epsilon}(x,\xi)}\,dx.
\end{align} The integral is the Bloch transform of $\frac{\partial^4\psi^0}{\partial x_k^4}(x)u^\epsilon(x)$ which converges to the Fourier transform of $\frac{\partial^4\psi^0}{\partial x_k^4}(x)u^*(x)$. However, since $\kappa\to 0$, the whole expression goes to zero.
\subsubsection{Limit of $\mathcal{B}_1^{\kappa,\epsilon}l_2^{\kappa,\epsilon}$} We shall prove that
\begin{align}\label{convergence_leps2}
\lim_{\epsilon\to 0}\mathcal{B}_1^{\kappa,\epsilon}l_2^{\kappa,\epsilon}=0.
\end{align}
Observe that
\begin{align}
\mathcal{B}_1^{\kappa,\epsilon}l^{\kappa,\epsilon}_2(\xi)=4\kappa^2\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial^3\psi^0}{\partial x_k^3}(x)\frac{\partial u^\epsilon}{\partial x_k}(x)\overline{\phi^{\kappa,\epsilon}(x,\xi)}\,dx.
\end{align} The integral is the Bloch transform of $\frac{\partial^3\psi^0}{\partial x_k^3}(x)\frac{\partial u^\epsilon}{\partial x_k}(x)$ which converges to the Fourier transform of $\frac{\partial^3\psi^0}{\partial x_k^3}(x)\frac{\partial u^*}{\partial x_k}(x)$. However, since $\kappa\to 0$, the whole expression goes to zero.
\subsubsection{Limit of $\mathcal{B}_1^{\kappa,\epsilon}l_3^{\kappa,\epsilon}$} We shall prove that
\begin{align}\label{convergence_leps3}
\lim_{\epsilon\to 0}\mathcal{B}_1^{\kappa,\epsilon}l_3^{\kappa,\epsilon}=0.
\end{align}
Observe that
\begin{align}
\mathcal{B}_1^{\kappa,\epsilon}l^{\kappa,\epsilon}_3(\xi)=2\kappa\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial^2\psi^0}{\partial x_k^2}(x)\kappa\frac{\partial^2 u^\epsilon}{\partial x_k^2}(x)\overline{\phi^{\kappa,\epsilon}(x,\xi)}\,dx.
\end{align} The integral is the Bloch transform of $\frac{\partial^2\psi^0}{\partial x_k^2}(x)\kappa\frac{\partial^2 u^\epsilon}{\partial x_k^2}(x)$ which converges for a subsequence since $\kappa\frac{\partial^2 u^\epsilon}{\partial x_k^2}(x)$ is bounded in $L^2(\Omega)$ (and hence converges weakly in $L^2(\Omega)$ for a subsequence). However, since $\kappa\to 0$, the whole expression goes to zero.
\subsubsection{Limit of $\mathcal{B}_1^{\kappa,\epsilon}l_4^{\kappa,\epsilon}$} We shall prove that
\begin{align}\label{convergence_leps4}
\lim_{\epsilon\to 0}\mathcal{B}_1^{\kappa,\epsilon}l_4^{\kappa,\epsilon}=0.
\end{align}
Observe that
\begin{align*}
l^{\kappa,\epsilon}_4(x)=4\kappa\frac{\partial}{\partial x_k}\left(\frac{\partial\psi_0}{\partial x_k}\kappa\frac{\partial^2u^\epsilon}{\partial x_k^2}\right)
\end{align*}
belongs to $H^{-1}(\mathbb{R}^d)$, hence the Bloch trasform for $H^{-1}(\mathbb{R}^d)$, that is,~\cref{BlochTransform2epsilon} applies.
Hence,
\begin{align}\label{Bleps4}
\mathcal{B}^{\kappa,\epsilon}_1l^{\kappa,\epsilon}_4(\xi)=-4i\kappa\xi_k\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial \psi_0}{\partial x_l}(x)\kappa\frac{\partial^2 u^\epsilon}{\partial x_k}(x)\overline{\phi^\rho_1\left(\frac{x}{\epsilon};\epsilon\xi\right)}\,dx\nonumber\\+4\kappa\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial \psi_0}{\partial x_l}(x)\kappa\frac{\partial^2 u^\epsilon}{\partial x_k^2}(x)\frac{\partial\overline{\phi^{\rho}_1}}{\partial x_k}\left(\frac{x}{\epsilon};\epsilon\xi\right)\,dx.
\end{align}
The analysis of the first term is the same as that of $\mathcal{B}_1^{\kappa,\epsilon}l_3^{\kappa,\epsilon}$. For the second term, observe that $\kappa\frac{\partial^2 u^\epsilon}{\partial x_k^2}(x)$ is bounded in $L^2(\Omega)$ and $\frac{\partial{\phi^{\rho}_1}}{\partial x_k}\left(\frac{x}{\epsilon};\epsilon\xi\right)$ is bounded in $L^2_\sharp(\epsilon Y)$ uniformly in $\rho$ (see~\cref{boundednessofeigfuncforlowrho}). Hence, their product is bounded in $L^1(K)$. As a result, the integral is a Fourier transform of a sequence bounded in $\rho$ and $\epsilon$. However, the presence of $\kappa$ in front of the integral causes the expression to go to zero.
Finally, passing to the limit in~\eqref{Bloch} as $\epsilon\to 0$ by applying equations~\eqref{convergence_ueps},~\eqref{convergence_geps},~\eqref{convergence_heps},
~\eqref{convergence_leps1},~\eqref{convergence_leps2},~\eqref{convergence_leps3}, and~\eqref{convergence_leps4}, we get:
\begin{align}\label{Bloch2}
a_{kl}^*\xi_k\xi_l\widehat{\psi_o u^*}(\xi)=\widehat{\psi_0 f}-\left(\frac{\partial \psi_0}{\partial x_k}(x)\sigma^*_k(x)\right)^{\bf\widehat{}}(\xi)-i\xi_ka^*_{kl}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)^{\bf\widehat{}}(\xi).
\end{align}
\subsection{\underline{Case 2 : $\rho\to\infty$}}
In this regime, the convergence proofs for $\mathcal{B}_1^{\kappa,\epsilon}l_1^{\kappa,\epsilon}$, $\mathcal{B}_1^{\kappa,\epsilon}l_2^{\kappa,\epsilon}$, $\mathcal{B}_1^{\kappa,\epsilon}l_3^{\kappa,\epsilon}$, $\mathcal{B}_1^{\kappa,\epsilon}g^{\epsilon}$, $\mathcal{B}_1^{\kappa,\epsilon}(\psi_0 f)$ are the same as in the earlier regime. Therefore, we will only look at the remaining convergences.
\subsubsection{Limit of $\lambda^{\kappa,\epsilon}_1(\xi)\mathcal{B}^{\kappa,\epsilon}_1(\psi_0 u^\epsilon)$}
We expand the first Bloch eigenvalue about $\eta=0$ in $\lambda^{\kappa,\epsilon}_1(\xi)\mathcal{B}^{\kappa,\epsilon}_1(\psi_0 u^\epsilon)$ to write
\begin{align*}
\left(\frac{1}{2}\frac{\partial^2\lambda^\rho_1}{\partial\eta_s\partial\eta_t}(0)\xi_s\xi_t + \frac{\epsilon^2}{4!}\partial^{e_s+e_t+e_u+e_v}_0\lambda_1^\rho\xi_s\xi_t\xi_u\xi_v + O(\epsilon^4) \right) \mathcal{B}^{\kappa,\epsilon}_1(\psi_0 u^\epsilon).
\end{align*} The fourth order derivative is of order $\rho^2$ by~\cref{higherestimates}. The derivatives of $\lambda^\rho_1(\eta)$ of order greater than $4$ are bounded uniformly in $\rho$ (see~\cref{higherestimates}). Hence, their contribution is $O(\epsilon^4)$. Hence, we can write the above as
\begin{align*}
\left(\frac{1}{2}\frac{\partial^2\lambda^\rho_1}{\partial\eta_s\partial\eta_t}(0)\xi_s\xi_t + O(\epsilon^2\rho^2) + O(\epsilon^4) \right) \mathcal{B}^{\kappa,\epsilon}_1(\psi_0 u^\epsilon)\\
=\left(\frac{1}{2}\frac{\partial^2\lambda^\rho_1}{\partial\eta_s\partial\eta_t}(0)\xi_s\xi_t + O(\kappa^2) + O(\epsilon^4) \right) \mathcal{B}^{\kappa,\epsilon}_1(\psi_0 u^\epsilon)
\end{align*}
Now, we can pass to the limit $\kappa,\epsilon\to 0$ in $L^2_{\loc}(\mathbb{R}^d_\xi)$-weak by applying Lemma~\ref{BlochTransformConvergence} to obtain:
\begin{align*}
e_s\cdot A^{hom}e_t{\partial\eta_s\partial\eta_t}(0)\xi_s\xi_t\widehat{\psi_o u^*}(\xi).
\end{align*}
\subsubsection{Limit of $\mathcal{B}_1^{\kappa,\epsilon}h^{\epsilon}$}
\begin{align*}
\lim_{\epsilon\to 0}\,\chi_{\epsilon^{-1}U}(\xi)\mathcal{B}^{\kappa,\epsilon}_1h^{\epsilon}(\xi)=-i\xi_ka^*_{kl}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)^{\bf\widehat{}}(\xi)
\end{align*}
We shall prove this in the following steps.
\paragraph{Step 1} As before, we have
\begin{align}\label{heps12}
\mathcal{B}^{\kappa,\epsilon}_1h^{\epsilon}(\xi)=-i\xi_k\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial \psi_0}{\partial x_l}(x)a^{\epsilon}_{kl}(x)u^\epsilon(x)\overline{\phi^\rho_1\left(\frac{x}{\epsilon};\epsilon\xi\right)}\,dx\nonumber\\+\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial \psi_0}{\partial x_l}(x)a^{\epsilon}_{kl}(x)u^\epsilon(x)\frac{\partial\overline{\phi^{\rho}_1}}{\partial x_k}\left(\frac{x}{\epsilon};\epsilon\xi\right)\,dx.
\end{align}
\paragraph{Step 2}
As before, the first term on RHS of~\eqref{heps12} is the Bloch transform of the expression $-i\xi_k\frac{\partial \psi_0}{\partial x_l}(x)a^{\epsilon}_{kl}(x)u^\epsilon(x)$ which converges weakly to \begin{align*}
-i\xi_k\mathcal{M}_Y(a_{kl})\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)=-i\xi_k a^*_{kl}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)
\end{align*} where the last equality is due to~\cref{homtensor}.
\paragraph{Step 3} Now we shall prove that the second term on RHS of~\eqref{heps12} goes to zero. As before, we make use of analyticity of first Bloch eigenfunction with respect to the dual parameter $\eta$ near $0$. We have the following power series expansion in $H^1_\sharp(Y)$ for $\phi_1^\rho(\eta)$ about $\eta=0$:
\begin{align*}
\phi_1^\rho(y;\eta)=\phi_1^\rho(y;0)+\eta_s\frac{\partial \phi^\rho_1}{\partial \eta_s}(y;0)+\gamma^\rho(y;\eta).
\end{align*}
We know that $\gamma^\rho(y;0)=0$ and $(\partial \gamma^\rho/\partial \eta_s)(y;0)=0$, therefore, $\gamma^\rho(\cdot;\eta)=O(|\eta|^2)$ in $L^\infty(U;H^1_\sharp(Y))$. We also have $(\partial\gamma^\rho/\partial y_k)(\cdot;\eta)=O(|\eta|^2)$ in $L^\infty(U;L^2_\sharp(Y))$. These orders are uniform in $\rho$ by~\cref{higherestimates}.
Now,
\begin{align*}
\phi_1^{\kappa,\epsilon}(x;\xi)=\phi_1^\rho\left(\frac{x}{\epsilon};\epsilon\xi\right)=\phi_1^\rho\left(\frac{x}{\epsilon};0\right)+\epsilon\xi_s\frac{\partial \phi^\rho_1}{\partial \eta_s}\left(\frac{x}{\epsilon};0\right)+\gamma^\rho\left(\frac{x}{\epsilon};\epsilon\xi\right).
\end{align*}
Differentiating the last equation with respect to $x_k$, we obtain
\begin{align}\label{derivativeofphi2}
\frac{\partial}{\partial x_k}\phi_1^\rho\left(\frac{x}{\epsilon};\epsilon\xi\right)=\xi_s\frac{\partial}{\partial y_k}\frac{\partial \phi^\rho_1}{\partial \eta_s}\left(\frac{x}{\epsilon};0\right)+\epsilon^{-1}\frac{\partial \gamma^\rho}{\partial y_k}\left(\frac{x}{\epsilon};\epsilon\xi\right).
\end{align}
For $\xi$ belonging to the set $\{\xi:\epsilon\xi\in U\mbox{ and }|\xi|\leq M\}$, we have
\begin{align}\label{gammaepsilon22}
\frac{\partial \gamma^\rho}{\partial y_k}(\cdot;\epsilon\xi)=O(|\epsilon\xi|^2)=\epsilon^2O(|\xi|^2)\leq CM^2\epsilon^2.
\end{align}
As a consequence,
\begin{align*}
\epsilon^{-2}\frac{\partial \gamma^\rho}{\partial y_k}(x/\epsilon;\epsilon\xi)\in L^\infty_{\loc}(\mathbb{R}^d_\xi;L^2_\sharp(\epsilon Y)).
\end{align*}
The second term on the RHS of~\eqref{heps12} is given by
\begin{align}\label{secondterm22}
\chi_{\epsilon^{-1}U}(\xi)\int_K e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{\epsilon}\right)u^\epsilon(x)\frac{\partial}{\partial x_k}\left(\overline{\phi^{\rho}_1}\left(\frac{x}{\epsilon};\epsilon\xi\right)\right)\,dx.
\end{align}
Substituting~\eqref{derivativeofphi2} in~\eqref{secondterm22}, we obtain
\begin{align}
\chi_{\epsilon^{-1}U}(\xi)\int_K e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{\epsilon}\right)u^\epsilon(x)\biggl[\xi_s\frac{\partial}{\partial y_k}\frac{\partial \phi^{\rho}_1}{\partial \eta_s}\left(\frac{x}{\epsilon};0\right)+\epsilon^{-1}\frac{\partial \gamma^\rho}{\partial y_k}\left(\frac{x}{\epsilon};\epsilon\xi\right)\biggr]\,dx.
\end{align}
In the last expression, the term involving $\gamma^\rho$ goes to zero as $\epsilon\to 0$ in view of~\eqref{gammaepsilon22}.
The other term also goes to zero as $\rho\to\infty$ due to strong convergence of $u^\epsilon$ and the fact that $$\frac{\partial}{\partial x_k}\left(\frac{\partial\phi^{\rho}_1}{\partial\eta_s}(x/\epsilon;0)\right)=O\left(\frac{1}{\rho^2}\right),$$ as shown in~\cref{higherestimates}.
\subsubsection{Limit of $\mathcal{B}_1^{\kappa,\epsilon}l_4^{\kappa,\epsilon}$} The analysis is the same as before, however the uniform-in-$\rho$ boundedness of $\frac{\partial{\phi^{\rho}_1}}{\partial x_k}\left(\frac{x}{\epsilon};\epsilon\xi\right)$ in $L^2_\sharp(\epsilon Y)$ is due to~\cref{higherestimates}.
Hence, for the regime $\rho\to\infty$, we also recover~\eqref{Bloch2}.
\subsection{\underline{Case 3 : $\rho\to 0$}}
In this regime, all the convergence proofs are the same as in the regime $\rho\to\theta\in(0,\infty)$ except for the convergence of $\mathcal{B}_1^{\kappa,\epsilon}h^{\epsilon}$, which we prove below.
\subsubsection{Limit of $\mathcal{B}_1^{\kappa,\epsilon}h^{\epsilon}$}
For this limit, all steps except the third are the same, hence we only explain the part of Step $3$ which differs from the regime $\rho\to\theta\in(0,\infty)$. We begin with the following equation which was earlier labelled as~\cref{twotermsarehere}.
\begin{align}\label{twotermsarehereagain}
\chi_{\epsilon^{-1}U}(\xi)\int_K e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{\epsilon}\right)u^\epsilon(x)\biggl[\xi_s\frac{\partial}{\partial y_k}\frac{\partial \phi^{\rho}_1}{\partial \eta_s}\left(\frac{x}{\epsilon};0\right)+\epsilon^{-1}\frac{\partial \gamma^\rho}{\partial y_k}\left(\frac{x}{\epsilon};\epsilon\xi\right)\biggr]\,dx.
\end{align}
In the last expression, the term involving $\gamma^\rho$ goes to zero as $\epsilon\to 0$ in view of~\eqref{gammaepsilon2}, whereas the other term has the following limit as $\rho\to 0$:
\begin{align}\label{secondterm3}
\mathcal{M}_Y\left(a_{kl}(y)\frac{\partial \chi^0_s}{\partial y_k}(y)\right)\xi_s\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)u^*(x)\,dx.
\end{align}
To see this, we write
\begin{align*}
\frac{\partial}{\partial y_k}\frac{\partial \phi^{\rho}_1}{\partial \eta_s}\left(\frac{x}{\epsilon};0\right)&=\frac{\partial \chi^{\rho}_s}{\partial y_k}\left(\frac{x}{\epsilon}\right)\\
&=\underbrace{\frac{\partial \chi^0_s}{\partial y_k}\left(\frac{x}{\epsilon}\right)}_{I}+\underbrace{\left[\frac{\partial \chi^{\rho}_s}{\partial y_k}\left(\frac{x}{\epsilon}\right)-\frac{\partial \chi^B_s}{\partial y_k}\left(\frac{x}{\epsilon}\right)\right]}_{II}+\underbrace{\left[\frac{\partial \chi^0_s}{\partial y_k}\left(\frac{x}{\epsilon}\right)-\frac{\partial \chi^B_s}{\partial y_k}\left(\frac{x}{\epsilon}\right)\right]}_{III}.
\end{align*}
The first term $I$ is responsible for~\cref{secondterm3} due to strong convergence of $u^\epsilon$ in $L^2(K)$ and weak convergence of $a_{kl}\frac{\partial \chi^0_s}{\partial y_k}\left(\frac{x}{\epsilon}\right)$.
For the expressions $II$ and $III$, we make use of~\cref{estimateforzerorho}. Indeed, we obtain $II = O(\rho)+O(\varkappa)$ and $III = O(\varkappa)$. Hence, their contribution to the limit in~\cref{twotermsarehereagain} as $\rho\to 0$ is $O(\varkappa)$.
This completes the modification required for the regime $\rho\to 0$. Instead of~\cref{Bloch2}, we obtain instead \begin{align*}
a_{kl}^*\xi_k\xi_l\widehat{\psi_o u^*}(\xi)=\widehat{\psi_0 f}-\left(\frac{\partial \psi_0}{\partial x_k}(x)\sigma^*_k(x)\right)^{\bf\widehat{}}(\xi)-i\xi_ka^*_{kl}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)^{\bf\widehat{}}(\xi) + O(\varkappa).
\end{align*} However, since $\varkappa>0$ is an arbitrary positive number in~\cref{estimateforzerorho}, we also recover~\eqref{Bloch2} for the regime $\rho\to 0$.
\subsection{Proof of the homogenization result}
Taking the inverse Fourier transform in the equation~\eqref{Bloch2}, we obtain the following:
\begin{align}\label{eq1}
(\mathcal{A}^{hom}(\psi_0 u^*)(x))=\psi_0 f-\frac{\partial \psi_0}{\partial x_k}(x)\sigma^*_k(x)-a^*_{kl}\frac{\partial}{\partial x_k}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right),
\end{align}
where the operator $\mathcal{A}^{hom}$ is defined in~\eqref{homoperator}. At the same time, calculating using Leibniz rule, we have:
\begin{align}\label{eq2}
(\mathcal{A}^{hom}(\psi_0 u^*)(x))=(\psi_0(x)\mathcal{A}^{hom}u^*(x))-a^*_{kl}\frac{\partial}{\partial x_k}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)-a_{kl}^*\frac{\partial\psi_0}{\partial x_k}(x)\frac{\partial u^*}{\partial x_l}(x)
\end{align}
Using equations~\eqref{eq1} and~\eqref{eq2}, we obtain
\begin{align}
\psi_0(x)\left(\mathcal{A}^{hom}u^*-f\right)(x)=\frac{\partial\psi_0}{\partial x_k}\left[a_{kl}^*\frac{\partial u^*}{\partial x_l}(x)-\sigma_k^*(x)\right].\end{align}
Let $\omega$ be a unit vector in $\mathbb{R}^d$, then $\psi_0(x)e^{ix\cdot\omega}\in\mathcal{D}(\Omega)$. On substituting in the above equation, we get, for all $k=1,2,\ldots,d$ and for all $\psi_0\in\mathcal{D}(\Omega)$,
\begin{align}
\psi_0(x)\left[a_{kl}^*\frac{\partial u^*}{\partial x_l}(x)-\sigma_k^*(x)\right]=0.
\end{align}
Let $x_0$ be an arbitrary point in $\Omega$ and let $\psi_0(x)$ be equal to $1$ near $x_0$, then for a small neighbourhood of $x_0$:
\begin{align}
\mbox{ for } k=1,2,\ldots,d,~\left[a_{kl}^*\frac{\partial u^*}{\partial x_l}(x)-\sigma_k^*(x)\right]=0
\end{align}
However, $x_0\in\Omega$ is arbitrary, so that
\begin{align}
\mathcal{A}^{hom}u^*=f\mbox{ and }\sigma^*_k(x)=a_{kl}^*\frac{\partial u^*}{\partial x_l}(x).
\end{align}
Thus,we have obtained the limit equation in the physical space. This finishes the proof of Theorem~\ref{homog}.
\section{Contribution of Higher Modes}\label{highermodes}
The proof of the qualitative homogenization theorem only requires the first Bloch transform. It is not clear whether the higher Bloch modes make any contribution to the homogenization limit. In this section, we show that they do not. We know that Bloch decomposition is the isomorphism $L^2(\mathbb{R}^d)\cong L^2(Y^{'}/\epsilon;\ell^2(\mathbb{N}))$ which is reflected in the inverse identity~\eqref{Blochinverseepsilon}. For simplicity, take $\Omega=\mathbb{R}^d$ and consider the equation $\mathcal{A}^{\kappa,\epsilon} u^\epsilon=f$ in $\mathbb{R}^d$ which is equivalent to
\begin{align*}\mathcal{B}^{\kappa,\epsilon}_m \mathcal{A}^{\kappa,\epsilon} u^\epsilon(\xi)=\mathcal{B}^{\kappa,\epsilon}_mf(\xi)\quad\forall m\geq 1,\forall\,\xi\in\epsilon^{-1}Y^{'}.\end{align*} We claim that one can neglect all the equations corresponding to $m\geq 2$.
\begin{proposition}
Let $$v^{\kappa,\epsilon}(x)=\int_{\epsilon^{-1}Y^{'}}\sum_{m=2}^{\infty}\mathcal{B}^{\kappa,\epsilon}_m u^\epsilon(\xi)\phi_m^{\kappa,\epsilon}(x;\xi)e^{ix\cdot\xi}\,d\xi,$$ then $||v^{\kappa,\epsilon}||_{L^2(\mathbb{R}^d)}\leq c\epsilon.$
\end{proposition}
\begin{proof}
Due to boundedness of the sequence $(u^\epsilon)$ in $H^2(\mathbb{R}^d)$, we have
\begin{align}
\int_{\mathbb{R}^d}\mathcal{A}^{\kappa,\epsilon} u^\epsilon\,\overline{u^\epsilon}\leq C.
\end{align}
However, by Plancherel Theorem~\eqref{Plancherelepsilon}, we have
\begin{align*}
\int_{\mathbb{R}^d} \mathcal{A}^{\kappa,\epsilon} u^\epsilon\, \overline{u^\epsilon}=\sum_{m=1}^{\infty}\int_{\epsilon^{-1}Y^{'}}\left(\mathcal{B}^{\kappa,\epsilon}_m\mathcal{A}^{\kappa,\epsilon} u^\epsilon\right)(\xi)\,\overline{\mathcal{B}^{\kappa,\epsilon}_mu^\epsilon(\xi)}\,d\xi\leq C
\end{align*}
Using~\eqref{diagonalizationepsilon}, we have
\begin{align*}
\sum_{m=1}^{\infty}\int_{\epsilon^{-1}Y^{'}}\lambda^{\kappa,\epsilon}_m(\xi)|{\mathcal{B}^{\kappa,\epsilon}_mu^\epsilon(\xi)}|^2\,d\xi\leq C.
\end{align*}
Now, by~\cref{lowrhobound}
\begin{align}
\lambda_m^\rho(\eta)\geq\alpha\lambda_2^N>0\quad\forall\, m\geq 2\quad\forall\,\eta\in Y^{'},
\end{align}
where $\lambda^N_2$ is the second eigenvalue of Laplacian on $Y$ with Neumann boundary condition on $\partial Y$. Since $\lambda^{\kappa,\epsilon}_m(\xi)=\epsilon^{-2}\lambda^{\rho}_m(\epsilon\xi)$, we obtain
\begin{align*}
\sum_{m=2}^{\infty}\int_{\epsilon^{-1}Y^{'}}|{\mathcal{B}^{\kappa,\epsilon}_mu^\epsilon(\xi)}|^2\,d\xi\leq C\epsilon^2.
\end{align*}
By Parseval's identity~\cref{parsevalblochepsilon}, the LHS equals $||v^{\kappa,\epsilon}||^2_{L^2(\mathbb{R}^d)}$. This completes the proof.
\end{proof}
|
2,877,628,090,106 | arxiv | \section{Introduction}
Single electron tunneling (SET) is a salient feature of quantum
transport in nanostructures. The SET phenomenon is observed in
various systems, e.g. quantum dots in a tunnel contact with
metallic electrodes, \cite{Pugla05,Cowre} molecular bridges
between the edges of broken metallic wires, \cite{Park,Yu,Roch}
atoms and molecules absorbed on metallic surfaces in a contact
with the tip of tunnel microscope, \cite{Bode,Otte} etc. The study
of electron tunneling through the nano-object with time-dependent
characteristics is one of the most challenging problems in this
field.
There are several sources of time dependence, which may be realized
in practical devices. The simplest one is the time-dependent gate
voltage $v_g(t)$ applied to the dot. It is well known
\cite{GA,KNG,KKAR} that this time dependence may be converted into
the time dependence of tunnel matrix element. Another possibility is
the nanoelectromechanical shuttling (NEMS),\cite{shut} where the
nano-size island suspended on a pivot\cite{pivot} or attached to a
string\cite{Koenig} oscillates between the leads under the action of
an electro-mechanical force. In case of molecular bridges, the
vibration eigen modes may be the source of the periodical
oscillations of tunneling parameters.\cite{Has,Roch}
Usually the tunneling between metallic leads and such a nanoobject
is accompanied by many-particle Kondo screening effect
\cite{GR,Ng} resulting in specific type of zero-bias anomaly (ZBA)
in tunnel conductance. Modification of Kondo regime because of
periodically modulated in time tunneling rate due to the
center-of-mass oscillations was studied recently in several
papers. If the oscillations are the eigen modes of a nanoobject
(molecule), then the Kondo-peak (zero bias anomaly in tunnel
conductance) may transform into dip due to the destructive
interference with vibrational mode. \cite{Has} In case of
adiabatic motion induced by electromechanical forces (NEMS)
\cite{shut} the Kondo temperature follows the periodical evolution
of the dot position and increases eventually due to effective
reduction of the average distance between the dot and the leads,
\cite{KKSV} which is determined by the mean square displacement of
the dot (in analogy with Debye-Waller effect in scattering
intensity). Non-adiabatic enhancement of Kondo tunneling through
such moving nanoobject at finite source-drain bias has been
studied recently.\cite{Gok}
In the present paper we consider adiabatic and non-adiabatic
time-dependent effects in conventional cotunneling between
metallic leads due to periodic modulation of lead-dot tunneling
rate. To suppress many-particle Kondo screening effects, one
should consider leads with magnetically polarized electrons. The
tunneling between the ferromagnetically ordered leads was
discussed recently\cite{Mart1,Bulka,Lopa02,Choi,Tan04,Chin08,Pas}
in the context of Kondo effect. We are interested in the
situation, where the spin-flip cotunneling is completely
suppressed at small lead-dot bias and low temperatures. Such
tunneling regime may be realized in half-metallic ferromagnets,
where the Fermi surface is formed only by the majority spin
electrons, whereas the spectrum of minority spin carries is
gapped. The electronic and magnetic properties of such metallic
compounds are reviewed in Ref. \onlinecite{Kats}. From the point
of view of existing devices, where the leads are formed by
two-dimensional electrons in degenerate semiconductors, the
relevant material for our studies is dilute magnetic semiconductor
(Ga,Mn)As.\cite{Jung} The indirect magnetic exchange between Mn
impurity ions is responsible for the long-range ferromagnetic
order in this material. This exchange is mediated by
spin-polarized carriers near the top of the valence band. The
tunnel current in the half-metallic regime arises due to the
minority spin hole cotunneling.
We will show that in the absence of Kondo effect the problem of
tunneling through moving nano-island (quantum dot) may be solved
by means of \textit{time-dependent canonical transformation},
which exactly takes into account both adiabatic and non-adiabatic
lead-dot tunneling processes diagonal in the lead indices. The
non-diagonal source-drain cotunneling may be treated by the
canonical transformation method only perturbatively, but the
adiabatic and non-adiabatic contributions into tunnel conductance
may be sorted out also in this case. It will be shown that the
time-dependent contribution into current-voltage characteristics
of moving nano-object is significant near the boundaries of
Coulomb diamonds.
\section{Model}
We choose for the realization of ac-driven tunnel conductivity the
simplest model of a nanoobject, which is widely used in the
studies of single-electron tunneling (SET). A nanoobject is
represented in this model by the quantum well with resonance level
$\varepsilon_d$ (see Fig. \ref{f.spol}, upper panel). The SET
regime arises due to the Coulomb blockade effect: addition energy
for the second electron in a singly occupied dot is $\varepsilon_d
+ U \gg \Gamma_j$, where $\Gamma_j$ is the tunneling rate to the
left ($j=l$) and right ($j=r$)lead, and $U$ is the capacitive
energy of the dot. To separate the time-dependent SET from the
Kondo ZBA, we consider tunneling between spin-polarized leads,
where spin-flip processes are inelastic, because the continuum of
electron-hole pairs with the opposite spins in the leads
responsible for the Kondo effect is gapped. Two types of spin
polarized (magnetically ordered) metallic leads are presented
schematically in Fig. \ref{f.spol}. In the middle panel the leads
are formed by "half-metallic ferromagnets"\cite{Kats} with gapped
spectrum of minority spin carries. In the lower panel
characteristic for $p$-type degenerate dilute magnetic
semiconductors\cite{Jung} the carriers are the minority spin
holes.
Virtual tunneling results in a shift of level positions in QD.
This renormalization ("Friedel shift") is also spin dependent. As
a result the spin polarization of QD adjusts to that of the
ferromagnetic lead (see calculations below). All Kondo processes
are quenched in this regime.
\begin{figure}[h]
\includegraphics[width=5.5cm,angle=0]{fig101.eps}
\caption{Upper panel: model of quantum dot in time-dependent
contact with leads. Middle and lower panels: density of electron
states in the leads, occupied and empty states in the dot for
$N=1$; mechanisms of spin-polarized electron cotunneling (middle
panel) and hole cotunneling (lower panel) are indicated by arrows.
Figures in parentheses point out the sequence of electron
tunneling acts in cotunneling processes.}\label{f.spol}
\end{figure}
The Anderson Hamiltonian modeling SET has the form
\begin{equation}\label{And01}
H=H_{\rm d} + H_{\rm b} + H_{\rm tun},
\end{equation}
where the terms
\begin{equation}
H_{\rm d}=\sum\limits_{\Lambda}
E_{\Lambda}|\Lambda\rangle \langle \Lambda|,\ \ H_{\rm b}
=\sum_{j=l,r}\sum\limits_{k\sigma} \epsilon^{}_{jk\sigma}
a_{jk\sigma}^\dagger a^{}_{jk\sigma}
\end{equation}
describe the electron states in the isolated dot and two metallic
(semiconductor) leads, respectively. We write $H_{\rm d}$ in terms
of its eigenstates $|\Lambda\rangle$ (Hubbard representation). This
trick allows one to take all intradot interactions into account
exactly even when the contact with the leads is switched on. The
tunneling Hamiltonian
$$
H_{\rm tun} = \sum\limits_{jk\sigma}(V_{jk}d^\dagger_{\sigma}
a^{}_{jk\sigma} + \mathrm{H.c.}).
$$
may be rewritten in the Hubbard representation by expanding the
creation operator $d^\dag_\sigma$ in terms of the configuration
change operators $X^{\Lambda \lambda} = |\Lambda\rangle \langle
\lambda|$ which connects the states in adjacent charge sectors
with $N$ and $N-1$ electrons in the dot. We confine ourselves with
the simplest case, where only three charge sectors $N=0,1,2$ are
involved in SET Hamiltonian. Then the Hamiltonian (\ref{And01})
acquires the following form
\begin{eqnarray}\label{ham1}
H &=& \varepsilon_d \sum_\sigma X^{\sigma\sigma} + E_{2}X^{22}
+ \sum_{jk\sigma} \epsilon_{jk\sigma}
a^\dagger_{jk\sigma} a^{}_{jk\sigma} \nonumber \\
& + &\sum_{jk\sigma} [ V_{jk} a^\dagger_{jk\sigma}
(X^{0\sigma}+\sigma X^{\bar\sigma 2}) +{\rm H.c.}].
\end{eqnarray}
Here the quantum numbers $\Lambda,\lambda = 0,\sigma,2$ correspond
to empty, singly and doubly occupied states of QD, respectively,
$E_{2}=2\varepsilon_d+U$ is the energy of doubly occupied QD. The
last term in this Hamiltonian is time-dependent.
\section{Canonical transformation of Anderson Hamiltonian}
Our program is to exclude the tunneling term from the Hamiltonian
(\ref{ham1}) by means of the canonical transformation
\begin{equation}\label{Canon01}
\widetilde{H}=e^S H e^{-S},
\end{equation}
then derive the tunnel current operator in the new basis and
calculate the tunnel conductance. It was shown in Ref.
\onlinecite{KF79} that this transformation may be performed exactly
in the absence of Coulomb correlation, provided the energy level
$\varepsilon_d$ falls into the energy gap and remains there after
renormalization (Friedel shift). It will be shown below that the
matrix $S$ still may be found in the presence of Coulomb blockade
under the same condition of discreteness of renormalized $d$-level
provided the spin-flip processes are quenched. As may be perceived
from Fig. \ref{f.spol}, this condition is realized for the majority
spin electrons in the case of half-metallic ferromagnet and for the
minority spin holes in the case of $p$-type dilute magnetic
semiconductor.
As usual, the canonical transformation is made by means of the
Baker-Hausdorff expansion
\begin{equation}\label{haus}
e^S H e^{-S} = \sum_{n=0}^\infty \frac{1}{n!}[S,[S,\dots[S,H]\dots]]
\end{equation}
In the non-interacting case, the second quantization operators in
$S$ and $H$ possess Fermi-like commutation relations, the
Hamiltonian $H$ is a quadratic form and the tunnel operator
conserves spin, so the series in the r.h.s. of (\ref{haus}) may be
summed exactly. \cite{KF79} The Coulomb blockade separates the
Hilbert space for the dot electron operators into charge sectors
divided by the energy gaps. As a result these operators lose the
simple Fermi statistics.
We are interested in the strong Coulomb blockade case and start
with the simplest case, where the ground state of the dot
corresponds to $N=1$ in the limit of $U\gg |\varepsilon_d -
\epsilon_F|$, so that the doubly occupied states are completely
suppressed. Then only the states $\Lambda = \sigma,0$ are retained
in the Hamiltonian (\ref{ham1}). In particular, the
anticommutation relation for Hubbard operators mixing the adjacent
sectors $N=0,1$ has the form
\begin{equation}\label{com1}
[X^{\sigma 0}, X^{0\sigma'}]_+ = X^{\sigma\sigma'}+
X^{00}\delta_{\sigma\sigma'},
\end{equation}
which follows from the obvious multiplication rule $X^{\Lambda_1
\lambda_1}X^{\lambda_2 \Lambda_2} = \delta_{\lambda_1\lambda_2}
X^{\Lambda_1\Lambda_2}$. Even disregarding the spin-flip processes
each commutation operation in the expansion (\ref{haus}) generates
operators $K^\sigma=X^{\sigma\sigma}+ X^{00}$. In spite of this, the
Baker-Hausdorff series still can be summed exactly, because these
operators are idempotent, $K^\sigma K^\sigma = K^\sigma$, like
conventional fermion occupation operators $d^\dag_\sigma d_\sigma$.
This summation will be performed in the following subsection.
Time-dependent problem is more complicated because in that case
the canonical transformation should be applied to the operator
\begin{equation}\label{elop}
{\cal L}= -i\hbar\frac{\partial}{\partial t}+ H~.
\end{equation}
In subsection B we will show that the canonical transformation is
operational in this case as well, at least in some important special
cases.
\subsection{Time-independent transformation}
\label{TIT}
In this section we generalize the canonical transformation
proposed for an Anderson model applied to transition metal
impurities in semiconductors.\cite{KF79,KF94} In those
calculations the intra-impurity Coulomb interaction was taken into
account in Hartree approximation. Here we take the Coulomb
blockade term exactly, when summing the series (\ref{haus}), at
least in the charge sectors $N=0,1$. The antihermitian operator
$S$ is looked for in the form
$$
S=\sum_{jk\sigma}(u_{k\alpha} a^\dagger_{jk\sigma} X^{0\sigma} -
u^*_{jk} X^{\sigma 0} a_{jk\sigma})= \sum_\sigma S_\sigma.
$$
If the spin flip processes are neglected, the canonical
transformation is made for each spin projection separately. One
may apply the transformation to the majority spin in the case of
electron tunneling and to the minority spin in the case of hole
tunneling in two models introduced above.
One easily derives the commutation relations
$$[S_\sigma, X^{\sigma 0} ] =
\sum_{jk}u_{jk\sigma}a^\dagger_{jk\sigma} K^{\sigma} \equiv
C_\sigma^\dagger K^{\sigma},
$$$$
[S_\sigma,a^\dagger_{jk\sigma}] = - u^*_{jk\sigma} X^{\sigma 0}.
$$
To shorten notations, we introduce the quantities $C_\sigma^\dagger
= \sum\limits_{jk\alpha} u_{jk\sigma}a_{jk\sigma}^\dagger$ and
$\gamma_\sigma^2 = \sum\limits_{jk} u_{jk\sigma}u^*_{jk\sigma}$.
Besides, we omit the lead index $j$ and specify the band state by a
single index $k$ characterizing both the lead and the electron wave
number. Using these definitions and the above mentioned idempotency
of operator $K_\sigma$, we obtain the following expressions for the
transformed operators
\begin{equation}\label{oneimpa}
\widetilde X^{\sigma 0}=
$$$$
e^{S_\sigma} X^{\sigma 0} e^{-S_\sigma} = X^{\sigma 0}
\cos\gamma_\sigma + C^\dagger_\sigma K^{\sigma} \gamma_\sigma^{-1}
\sin\gamma_\sigma,
\end{equation}
\begin{equation}\label{oneimpb}
\widetilde a^\dag_{k\sigma} = e^{S_\sigma} a^\dagger_{k\sigma}
e^{-S_\sigma} =
$$$$
a^\dagger_{k\sigma} + u^*_{k\sigma} C^\dagger_{\sigma} K^{\sigma}
\gamma_\sigma^{-2} (\cos\gamma_\sigma -1) - u^*_{k\alpha\sigma}
X^{\sigma 0} \gamma_\sigma^{-1} \sin\gamma_\sigma,
\end{equation}
and
\begin{equation}\label{oneimpc}
e^S C^\dagger_{\sigma}e^{-S} = C^\dagger_{\sigma} +
C^\dagger_{\sigma}K^\sigma (\cos\gamma_\sigma - 1)- X^{\sigma 0}
\gamma_\sigma \sin\gamma_\sigma,
\end{equation}
The tunneling term in the transformed Hamiltonian is eliminated,
provided
\begin{equation}\label{coeff}
u_{k\sigma} = \frac{\gamma_\sigma V_{k}}{(\varepsilon_{k\sigma} -
E_{d\sigma})\tan \gamma_\sigma} \equiv g_{k\sigma} V_k
\end{equation}
with
\begin{equation}\label{deriv}
\tan^2\gamma_\sigma = -\left. \frac{dL_\sigma( \varepsilon)}{d
\varepsilon} \right|_{ \varepsilon = E_{d\sigma}} ~,
\end{equation}
\begin{equation}\label{massop}
L_\sigma(\varepsilon) = \sum_{k} \frac{|V_{k}|^2}{\varepsilon -
\varepsilon_{k\sigma}}.
\end{equation}
Then the transformed Hamiltonian takes the form
\begin{equation}\label{htild}
\tilde H_\sigma = E_{d\sigma} X^{\sigma\sigma} + \sum_{kk'}
\tilde \varepsilon^\sigma_{kk'} a^\dag_{k\sigma}a^{}_{k'\sigma}
\end{equation}
where the renormalized level position is given by the equation
\begin{equation}\label{etild}
E_{d\sigma} = \frac{\varepsilon_d + Z_\sigma \tan^2\gamma_{\sigma}
- (T_\sigma + T_\sigma^*) \tan\gamma_\sigma}{1 +
\tan^2\gamma_\sigma},
\end{equation}
with $Z_\sigma = \gamma_\sigma^{-2} \sum_{k} \varepsilon_k
u^*_{k\sigma} u_{k\sigma},$ and
$ T_\sigma = \gamma_\sigma^{-1} \sum_k (V_k u^*_{k\sigma})$.
Substitution of Eqs. (\ref{coeff}) -- (\ref{massop}) in
(\ref{etild}) transforms it into the conventional
form\cite{KF79,KF94}
\begin{equation}\label{level}
E_{d\sigma} = \varepsilon_d + L_\sigma(E_{d\sigma}).
\end{equation}
where the self energy $L_\sigma(E)$ has only real part (Friedel
shift), provided the level $E_{d\sigma}$ remains within the gap,
which is the case for $\sigma=\uparrow$ in a configuration of Fig.
\ref{f.spol}, middle panel.
A canonical transformation for the second spin component
$\sigma=\downarrow$ should be done more carefully, because the
bare level $\varepsilon_d$ falls into continuum of spin-down
states. One should be accurate with turning to the thermodynamic
limit, where the sum in the right-hand side of (\ref{massop})
transforms into the integral and acquires the imaginary part thus
making the Hamiltonian non-Hermitian. The recipe is to keep the
spectrum of electrons in the leads discreet when doing the
canonical transformation. Then equation (\ref{level}) has ${\cal
N}+1$ solutions $\varepsilon_i$, where ${\cal N}$ is the number of
state in the valence band $\epsilon^{}_{k\downarrow}$. Using Eqs.
(\ref{massop}) and (\ref{deriv}), the corresponding coefficients
$\gamma_i$ may be found. In accordance with (\ref{oneimpa}), the
factor $\cos\gamma_i$ determines the weight of the $d$-component
of hybridized wave function of dot electron in the state $i$. One
may identify the state $i = m$ producing the maximum value of
$\cos\gamma_i$ with the center of future 'Friedel resonance',
which arises in the thermodynamic limit ${\cal N}\to \infty$. In
this sense the state $E_{d\downarrow}\equiv
\varepsilon_{m\downarrow}$ is formally defined from the equation
$\varepsilon_{m\downarrow} = \varepsilon_d +
L_\downarrow(\varepsilon_{m\downarrow})$. This level is shown by
the dashed line in the middle panel of Fig. \ref{f.spol}. It
corresponds to the bunch of excited states of QD coupled to the
leads with $N=1$ and spin oriented antiparallel to that of the
magnetized leads.
The spectrum of continuous part of the Hamiltonian (\ref{htild})
is determined by the expression
\begin{equation}\label{tildec}
\tilde \varepsilon^\sigma_{kk'} = \varepsilon_{k\sigma}
\delta_{kk'} + W^\sigma_{kk'}
\end{equation}
containing the "scattering" matrix element which eventually
predetermines the tunnel current.
The general form of this matrix element is
\begin{widetext}
\begin{equation}\label{scatamp}
W^\sigma_{kk'} =
\frac{u_{k\sigma} u^*_{k'\sigma}}{\gamma_\sigma^2} K^\sigma \left[
(2\epsilon_d - \varepsilon_{k\sigma} - \varepsilon_{k'\sigma} )(1 -
\cos\gamma_\sigma)- \frac{T_\sigma + T_\sigma^*}{2}
\frac{2\cos^3\gamma_\sigma - 3 \cos^2\gamma_\sigma +
1}{\cos\gamma_\sigma\sin\gamma_\sigma}\right] +
$$$$
K^\sigma(V_ku^*_{k'\sigma} + u_{k\sigma}V^*_{k'})
\frac{\sin\gamma_\sigma}{\gamma_\sigma}.
\end{equation}
\end{widetext}
Using Eqs. (\ref{coeff}) and (\ref{etild}) we get
$$
T_\sigma = T_\sigma^* = - \frac{1}{\tan \gamma_\sigma}
L_\sigma(E_{d\sigma})
$$
and after some algebraic manipulations the scattering matrix element
is eventually transformed into a quite compact expression
\begin{equation}\label{scatamp-tit}
{\overline{W}}^\sigma_{kk'} =
$$$$
V_{k}V_{k'}^*K^\sigma\left[\left(
\frac{1}{\Delta_{k\sigma}}+\frac{1}{\Delta_{k'\sigma}}\right)R(\gamma_\sigma)
+ \frac{L_\sigma(E_{d\sigma})}{\Delta_{k\sigma}
\Delta_{k'\sigma}}R^2(\gamma_\sigma) \right]
\end{equation}
where
\begin{equation}\label{factor}
R(\gamma_\sigma)= \displaystyle\frac{\sqrt{1 + \tan^2\gamma_\sigma}
- 1}{\tan^2\gamma_\sigma},
\end{equation}
$\Delta_{k\sigma} = E_{d\sigma}-\varepsilon_{k\sigma}$. Equation
(\ref{scatamp-tit}) holds in the static case or, as we will see
below, for adiabatically slow time variations of the tunneling
amplitudes. If we want to study nonadiabatic corrections a more
general formula Eq. (\ref{scatamp}) should be used.
Equation (\ref{scatamp-tit}) generalizes the familiar 2-nd order
expression for the single electron tunneling amplitude through the
QD, which takes into account both renormalization of the energy
level of dot electron (\ref{level}) and reconstruction of the band
continuum (\ref{oneimpb}). Far from the resonance tunneling regime
(in the center of the Coulomb diamond diagram, see e.g. the middle
panel of Fig. \ref{f.spol}) one may neglect the second term in the
square bracket of Eq. (\ref{scatamp-tit}), and the tunneling
matrix element acquires the simple form $
{W}^\sigma_{lk,rk'}=J_{lk,rk'}K^\sigma R(\gamma_\sigma) $, where
the first factor is the off-diagonal matrix element of indirect
exchange between the leads and the dot due to electron
cotunneling, the second and the third factors regulate the
occupation of the dot level and the normalization of electron wave
functions, respectively. The transparency of QD is, of course,
exponentially weak
Now we turn to the resonance regime illustrated by Fig.
\ref{f.spul}, where the level $\varepsilon_d^{(2)} = \varepsilon_d +
U$ driven by the gate voltage $v_g$ approaches the level
$\epsilon_F$ from above.
\begin{figure}[h]
\includegraphics[width=5.5cm,angle=0]{fig1011.eps}
\caption{Cotunneling mechanism in the resonance regime for second
electron in QD on the boundary of Coulomb window. }\label{f.spul}
\end{figure}
The level $\varepsilon_d^{(2)}$ may be occupied only by a
spin-down electron. Since it falls into the gap of spin-down
density of states in the leads, the canonical transformation
introduced above may be performed in a similar way, provided the
addition energy for the first electron $\varepsilon^{(1)}\approx
\varepsilon_d$ falls deep enough below $\epsilon_F$ and the
corresponding processes are suppressed. In this regime
$\Lambda,\lambda = \sigma,2$, and one should use the commutation
relations
\begin{equation}\label{com2}
[X^{\sigma 2}, X^{2\sigma'}]_+ = X^{\sigma\sigma'}+
X^{22}\delta_{\sigma\sigma'},
\end{equation}
instead of (\ref{com1}). Correspondingly, one should insert
$K^\sigma=X^{\sigma\sigma}+X^{22}$ in equations
\begin{equation}\label{oneimpd}
\widetilde X^{2\bar\sigma }= \sigma X^{2\bar\sigma}
\cos\gamma_\sigma + C^\dagger_\sigma K^{\sigma} \gamma_\sigma^{-1}
\sin\gamma_\sigma,
\end{equation}
\begin{equation}\label{oneimpe}
\widetilde a^\dag_\sigma =
$$$$
a^\dagger_{k\sigma} + u^*_{k\sigma} C^\dagger_{\sigma} K^{\sigma}
\gamma_\sigma^{-2} (\cos\gamma_\sigma -1) - u^*_{k\sigma} \sigma
X^{2\bar\sigma } \gamma_\sigma^{-1} \sin\gamma,
\end{equation}
for transformed creation operators.
Then the transformed Hamiltonian for spin-down electrons has the
form
\begin{equation}\label{htildu}
\tilde H_\downarrow = E_{d\downarrow} X^{\downarrow\downarrow} +
\sum_{kk'} \tilde \varepsilon^\sigma_{kk'}
a^\dag_{k\downarrow}a_{k'\downarrow}
\end{equation}
where $E_{d\downarrow}$ and $ \tilde \varepsilon^\sigma_{kk'}$ are
given by the same Eqs. (\ref{level}) and (\ref{tildec}) as in the
previous case, but with the energy $\varepsilon_d^{(2)}$
substituting for $\varepsilon_d$ and the correlation function
$K^\sigma$ taken from (\ref{com2}). It should be stressed that the
tunneling through the QD is impossible at zero source-drain bias
because of the spin blockade in spin-polarized electrodes. Only
spin-up carriers exist around Fermi level, and these electrons may
be injected into QD only when accompanied by the spin-flip
excitations given by the operators $X^{\downarrow\uparrow}$ in the
intermediate state with $N=1$ of cotunneling process. These
processes are inelastic and exponentially weak ($\sim V^2$ in
transparency and $\sim V^4$ in conductance). More detailed
discussion of spin-dependent tunneling is postponed till Section
\ref{TID}.
\subsection{Time-Dependent Transformation}
\label{TDP}
As was discussed above an experimental conditions may be created
when the tunneling matrix element $V_k(t)$ in the Hamiltonian
(\ref{ham1}) becomes time dependent. The canonical transformation
(\ref{Canon01}), as described in the previous section, cannot be
straightforwardly applied. Its generalization is in order.
We start with the temporal Schroedinger equation
\begin{equation}\label{Schro01}
{\cal L} \psi=0
\end{equation}
and look for the time dependent transformation matrix ${\widetilde S}(t)$,
which transforms (\ref{Schro01}) into
\begin{equation}\label{Schro02}
{\widetilde {\cal L}}\widetilde \psi =0
\end{equation}
with transformed operator
$$
{\widetilde {\cal L}}(t) =e^{S(t)}He^{-S(t)} -i\hbar
e^{S(t)}\frac{\partial}{\partial t}e^{-S(t)}
$$
for the new wave function $\widetilde \psi = e^{S(t)}\psi$.
The Hamiltonian $\widetilde H$ satisfying the equation
(\ref{Schro02}) can be written as
\begin{equation}\label{Schro04}
\widetilde{H} = e^SHe^{-S} + i\hbar\int\limits_{0}^1d\lambda
e^{\lambda S}\dot{S} e^{-\lambda S}.
\end{equation}
(see Appendix).
The Hamiltonian (\ref{Schro04}) contains now two terms of which
the first one is just a modification of the Hamiltonian
(\ref{Canon01}) of the time independent case. It means that all
the equations in Subsection \ref{TIT} hold except for Eq.
(\ref{coeff}), which defines the coefficients $u_{k\sigma}$ of the
canonical transformation. These coefficients must be now found
anew. There is also the second term in the right-hand side of Eq.
(\ref{Schro04}), which is responsible for non-adiabatic effects.
In order to find the canonical transformation parameters
$u_{k\sigma}$ we write explicitly the condition
\begin{widetext}
\begin{equation}\label{elimination_time}
u_{k\sigma} \gamma_\sigma^{-1} [(\varepsilon_d - Z_\sigma)
\sin\gamma_\sigma\cos\gamma_\sigma - ( \varepsilon_{k\sigma}
-Z_\sigma) \sin\gamma_\sigma + T_\sigma\cos^2\gamma_\sigma -
T_\sigma^*\sin^2\gamma - T_\sigma \cos\gamma_\sigma]
+ V_k \cos\gamma_\sigma =
$$$$
- i\hbar\dot{u}_{k\sigma} \frac{\sin
\gamma_\sigma}{\gamma_\sigma} - i\hbar u_{k\sigma}
\sum\limits_{k'}\left[\dot{u}_{k'\sigma} u^*_{k'\sigma}
\frac{1}{2\gamma_\sigma^3} (\sin\gamma_\sigma\cos\gamma_\sigma +
\gamma_\sigma-2\sin\gamma_\sigma)\right.
+ \left.\dot{u}_{k'\sigma}^*u_{k'\sigma}
\frac{1}{2\gamma_\sigma^2}(1 -
\frac{\sin\gamma_\sigma\cos\gamma_\sigma}{\gamma_\sigma})\right] = 0
\end{equation}
\end{widetext}
of elimination of the QD - lead tunneling in the transformed
Hamiltonian. Here both the tunneling amplitude $V_{k\sigma}$ and
transformation parameter $u_{k\sigma}$ are functions of time. The
condition (\ref{elimination_time}) contains a number of terms with
the time derivatives $\dot{u}_{k\sigma}$. Neglecting these time
derivatives would correspond to the adiabatic approximation where
the variation of the tunneling amplitude is slow enough and the
whole electron system always have enough time to readjust to the
varying tunneling amplitude without additional level mixing. Then
one can check straightforwardly that Eq. (\ref{coeff}) with the time
dependent $V_k(t)$ solves Eq. (\ref{elimination_time}).
Now we carry out a more general analysis going {\it beyond} the
adiabatic approximation. For this sake we multiply Eq.
(\ref{elimination_time}) by $u^*_{k\sigma}$ and sum over $k$, which
leads to the equation
\begin{equation}\label{elimination_time_1}
(\varepsilon_d - Z_\sigma) \tan\gamma_\sigma + T_\sigma -
T^*_\sigma\tan^2\gamma_\sigma =
$$$$
-i \hbar \frac{d}{dt} \tan\gamma_\sigma - i \hbar
\frac{\tan\gamma_\sigma}{2 \gamma_\sigma^2}
\sum\limits_{k'}(\dot{u}_{k'\sigma}
u^*_{k'\sigma}-\dot{u}_{k'\sigma}^* u_{k'\sigma}).
\end{equation}
Substituting Eq. (\ref{elimination_time_1}) into Eq.
(\ref{elimination_time}) we get the equation
\begin{equation}\label{elimination_time_a}
u_{k\sigma}[(\varepsilon_{k\sigma} - Z_\sigma)\tan\gamma_\sigma +
T_\sigma] - \gamma V_{k\sigma} =
$$$$
-i\hbar \frac{\tan \gamma_\sigma}{\gamma_\sigma^2} u_{k\sigma}
\sum\limits_{k'} u_{k'\alpha'\sigma}^* \dot{u}_{k'\sigma} + i \hbar
\dot{u}_{k\sigma} \tan \gamma_\sigma .
\end{equation}
Neglecting the time derivatives of transformation parameter
$u_k(t)$ in the r.h.s of Eq. (\ref{elimination_time_a}), which
corresponds to the adiabatic approximation, yields Eq.
(\ref{coeff}) with the time dependent tunneling amplitude
$V_k(t)$. All the other equations obtained in Subsection \ref{TIT}
also hold. Accounting for the r.h.s. of Eq.
(\ref{elimination_time_a}) allows one to obtain nonadiabatic
corrections.
Having in mind that the quantities $\varepsilon_d$,
$\gamma_\sigma$ and $Z_\sigma$ are explicitly real, we separate
real and imaginary parts in Eq. (\ref{elimination_time_1}) and
thus get two equations
\begin{equation}\label{eltim2}
T_\sigma - T_\sigma^* = -2i\hbar \dot{\gamma_\sigma}
\end{equation}
and
\begin{equation}\label{eltim3}
(\varepsilon_d - Z_\sigma) \tan\gamma_\sigma + \frac{T_\sigma +
T_\sigma^*}{2} (1 - \tan^2\gamma_\sigma) =
$$$$
- i \hbar \frac{\tan\gamma_\sigma}{2 \gamma_\sigma^2}
\sum\limits_{k'}(\dot{u}_{k'\sigma}
u^*_{k'\sigma}-\dot{u}_{k'\sigma}^* u_{k'\sigma}).
\end{equation}
These equations may be instrumental in looking for solutions
$u_{k\sigma}(t)$ for specific problems. It follows from
(\ref{eltim2}) that the quantity $T$ which was real for the time
independent case, remains real also in the adiabatic approximation
for the time dependent case.
Returning back to the transformed Hamiltonian (\ref{Schro04}), we
note that the first term $e^SHe^{-S}$ is now time-dependent due to
the time dependence of $S$. Carrying out the transformation in the
same fashion as in the previous section we get the time dependent
energy level
\begin{equation}\label{ed}
E_{d\sigma} = E_{d\sigma}^{(a)} + E_{d\sigma}^{(b)}
\end{equation}
where
\begin{equation}\label{eda}
E_{d\sigma}^{(a)} = \frac{\varepsilon_d + Z_\sigma
\tan^2\gamma_\sigma - (T_\sigma + T_\sigma^*) \tan\gamma_\sigma}{1
+ \tan^2\gamma_\sigma}
\end{equation}
is the same energy level (\ref{etild}) as before but with the
coefficients $Z_\sigma, T_\sigma, T_\sigma^*, \gamma_\sigma$
depending parametrically on time $t$. Thus, in adiabatic
approximation the time dependence of the resonance level position
$E_{d\sigma}^{(a)}(t)$ is determined by Eq. \ref{level}) with the
time-dependent self energy part
\begin{equation}\label{massopt}
L_\sigma(\varepsilon, t) = \sum_{k}
\frac{|V_{k}(t)|^2}{\varepsilon - \varepsilon_{k\sigma}}.
\end{equation}
There is also the nonadiabatic correction
\begin{equation}\label{edb}
E_{d\sigma}^{(b)}(t)= - \frac{1}{2} i \hbar
\frac{\sin^2\gamma}{\gamma^2}\sum\limits_{k\alpha}(
\dot{u}_{k\alpha\sigma}u^*_{k\alpha\sigma}
-\dot{u}_{k\alpha\sigma}^*u_{k\alpha\sigma} ).
\end{equation}
To calculate the time dependent coefficients, one has to specify the
form of tunneling amplitudes $V_k(t)$. An example of time-dependent
tunneling will be considered in the next section.
\section{Tunnel conductance}
\label{TC}
In this section we study the tunnel conductance basing
on the transformed Hamiltonian $\widetilde H$. The tunnel current
may be calculated, e.g., by means of the Keldysh technique, where
the bias $eV$ is included in the zero order Hamiltonian and the
scattering $\sim W_{lk,rk'}$ is considered as a perturbation.
First, we calculate the spin-polarized current through an
immovable quantum dot and then discuss the modulation of this
current due to oscillatory motion of the dot.
\subsection{Tunneling through static dot}
\label{TID}
Far from the boundary between two adjacent charge sectors with
$N=1$ and $N=2$, where both the levels
$\varepsilon_{d\uparrow}^{(1)}$ and
$\varepsilon_{d\downarrow}^{(2)}$ are far from the chemical
potential $\mu$, the Keldysh method applied to the Hamiltonian
(\ref{htildu}) in a single loop approximation gives the
conventional golden rule equation
\begin{widetext}
\begin{equation}\label{goldo}
I = e \frac{2\pi}{\hbar} \sum \limits_{kk'}\sum_\sigma
|W^\sigma_{lk,rk'}|^2\delta(\epsilon_{lk\sigma} + e\upsilon -
\epsilon_{rk'\sigma}) \{ f(\epsilon_{lk\sigma}) [1 -
f(\epsilon_{rk'\sigma})] - f(\epsilon_{rk'\sigma}) [1 -
f(\epsilon_{lk\sigma})] \}
\end{equation}
\end{widetext}
where $e\upsilon$ is the source-drain bias, $f(\epsilon_{k\sigma})$
is the Fermi distribution function for spin-polarized electrons, and
the scattering amplitude $W_{lk,rk'}$ is defined in Eq.
(\ref{scatamp-tit}).
The standard Coulomb diamond diagram for tunneling conductance
$G(v_g, e\upsilon)$ is distorted in the region of gate voltages
$v_g$ corresponding to the change of QD occupation $(N=1) \to
(N=2)$. In case of completely spin polarized dots the Coulomb step
corresponding to the resonance $E(N=1, v_g)=E(N=2, v_g)-\mu$ in
the current voltage characteristics is absent at zero bias and
zero temperature due to the spin blockade. The chemical potential
$\mu$ is pinned to the Fermi level of spin up electrons
$\varepsilon_{F\uparrow}$, but the resonance level
$E_{d\downarrow}=E(N=2)-E(N=1)$ belongs to the down spin electron
(see Fig. \ref{f.spul}). Thus the tunneling at zero bias
$e\upsilon=0$ is suppressed by the spin blockade. This blockade
may be surmounted by means of finite source-drain bias
compensating the energy gap. However, the conditions for the onset
of spin up and spin down tunnel current are different.
\begin{figure}[h]
\begin{center}
\includegraphics[width=4cm,angle=0]{guydiamond.eps}\hspace*{5mm}
\end{center}
\vspace*{-3mm}
\caption{Coulomb diamonds diagram for tunneling conductance $G(v_g,e\upsilon)$
near the border $(N=1)/(N=2)$ for down-spin and up-spin electrons
(solid and dashed lines, respectively). Three bars on the $v_g$
axis mark the values of the gate voltage corresponding to the
level $E_{d\downarrow}$ in resonance with $\varepsilon_c$,
$\varepsilon_F$, $\varepsilon_v$ (bottom-up)
.
}
\label{f.diamond}
\end{figure}
The boundary of the Coulomb diamond with $N=1$ for spin down
electrons follows the evolution of the resonance level
$E_{d\downarrow}$ until this level approaches from above the
bottom of conduction band $\varepsilon_{c\downarrow}$. When it
crosses the band edge, the resonance tunneling is no more possible
at $eV < \Delta_{cv}$, so the Coulomb resonance line for $G(v_g,
e\upsilon)$ deviates from the linear behavior, as it is sketched
in Fig. \ref{f.diamond} (solid line). Negative-bias part of the
Coulomb diamond corresponds to the hole tunneling through the
occupied resonance level in the sector $N=2$. It is distorted in
the same way in the region of $v_g$ where this level matches the
top of the valence band (second branch of the solid line). When
the down-spin level is deep enough in the conduction or valence
band, the linear behavior is restored again.
The blockade for spin up electrons near the boundary $(N=1)/(N=2)$
is lifted at $e\upsilon$ compensating the spin flip excitation in
the dot. As was mentioned above, the spin up electron tunneling is
allowed provided the dot is excited to the state $E_{d\downarrow}$
given by the solution of Eq. (\ref{level}) for the doubly occupied
dot, and the splitting energy may be estimated as
$\Delta_{\uparrow\downarrow}\approx
|L(E_{d\uparrow})-L(E_{d\downarrow})|$ (see Ref. \onlinecite{Pas}
for experimental determination of such splitting in the Kondo
tunneling regime). The line of resonance tunneling for spin-up
electrons is drawn in Fig. \ref{f.diamond} by the dashed curve.
Besides, the tunneling transparency is especially sensitive to the
position of the dot level in the near vicinity of band edges. To
investigate this dependence, let us find the explicit equation for
the tunnel conductance from Eqs. (\ref{goldo}) and
(\ref{scatamp-tit}). Changing summation over $kk'$ for integration
over $\epsilon\epsilon'$ in a usual way and performing standard
calculations, one gets the equation
\begin{equation}\label{conduct}
G_\downarrow(v_g,\Delta_{cv} ) =
$$$$
\frac{e^2}{h}\frac{\Gamma_l\Gamma_r}{(2\pi^2)}
\frac{R^2(\gamma_\downarrow)}{|E_{d\downarrow}-\varepsilon_c|^2}
\left[1+\frac{R(\gamma_\downarrow)L(E_{d\downarrow})}
{\Delta_{cv}}
\right]^2
\end{equation}
for the threshold value of $e\upsilon \to\Delta_{cv}+o$ in case of
2D electron gas in the planar leads. Here $\Gamma_i=2\pi V^2_iS_i$
is the tunneling rate for the lead $i$ obtained in the approximation
of $V_{ik}=V_i$ and constant electron density of states
$S_i(\varepsilon)=S_i$, which is valid for 2D electrons. It should
be taken into account that the definition of the position of the
resonance level $E_{d\downarrow}$ falling into the conduction band
continuum implies the procedure described below Eq. (\ref{level})
The function $L(\epsilon)$ has a logarithmic singularity at the band
edge, ${\rm Re}\,L(\epsilon \to \varepsilon_c) \sim
-\ln(|\epsilon-\varepsilon_c|/\Delta_{cv})$. As a result, Eq.
(\ref{level}) has either one or two solutions depending on a
position of the level $\varepsilon_{d\downarrow}^{(2)}-v_g$ relative
to the band edge.\cite{HA76}
The resonance factor $|E_{d\downarrow}-\varepsilon_c|^2$ in the
denominator and singular factor $L(E_{d\downarrow})$ in the
numerator of Eq. (\ref{conduct}) result in noticeable enhancement
of $G$ at the boundary of Coulomb diamond.
This enhancement is characterized by evolution of the ratio
\begin{equation}\label{enhan}
\rho(v_g) = \frac{R^2(\gamma_\downarrow)}{|E_{d\downarrow} -
\varepsilon_c|^2}
\end{equation}
from its value in the middle of Coulomb diamond to that in the
vicinity the point $\varepsilon_d(v_g) = \varepsilon_c$. Here and
below we omit the superscript (2) in the notation of
$\varepsilon_d$. Far from the Coulomb resonance the difference
between $|E_{d\downarrow}|$ and $\varepsilon_{d}$ is small and
$\tan^2\gamma_\downarrow \sim V^2/(\varepsilon_{d}-\varepsilon_c)
\ll 1$. The function $R(\gamma_\downarrow)$ tends to 1/2 in this
limit, so that the factor $\rho$ may be estimated as $\rho \approx
1/4(\varepsilon_{d}-\varepsilon_c)$. Near the band edge
$\varepsilon_c$ the factor $R(\gamma_\downarrow)\approx
\cot\gamma_\downarrow=\sqrt{|E_{d\downarrow}-\varepsilon_c|}/\Gamma_r$
[see Eqs. (\ref{deriv}),(\ref{factor})], so that $\rho \sim
[|E_{d\downarrow}-\varepsilon_c|\Gamma_r]^{-1}$. Numerical estimates
of this enhancement are presented at Fig. \ref{f.amp}. In these
calculations the density of states was assumed to be constant in the
lower part of 2D conduction band, so that the self energy in the
r.h.s. of Eq. (\ref{level}) may be approximated as $L(E)\approx
\Gamma_r {\rm Ln}(E/D)$, where $D$ is the effective width of
conduction band, and the argument $E$ is complex. Then the solutions
$E=E_{d\downarrow}$ of Eq. (\ref{level}) are expressed via the
Lambert W-function $W(n,x)$
\begin{equation}\label{lambert}
E_{d\downarrow}= -\Gamma_r W(n,
-{\Gamma_r^{-1}}{e^{-{\varepsilon_d}/{\Gamma_r}}})
\end{equation}
(here the reference point is $\varepsilon_c=0$, all energy
parameters are measured in units $D$, index $n$ enumerates branches
of W-function). There are two solutions for $E_{d\downarrow}$ near
the band edge.\cite{HA76} The principal branch $n=0$ gives the
discrete level in the gap, and the branch $n=1$ corresponds to the
resonance in the band. Here we are interested in the latter state.
\begin{figure}[h]
\begin{center}
\includegraphics[width=6 cm,angle=0]{amplification-n.eps}\hspace*{5mm}
\end{center}
\vspace*{-3mm}\caption{Amplification factor
$\rho(\varepsilon_d^{(2)})$ for two values of tunneling rate
$\Gamma/D=0.01$ (lower curve) and $\Gamma/D=0.02$ (upper curve).}
\label{f.amp}
\end{figure}
The amplification reaches its maximum when the difference
$|E_{d\downarrow}-\varepsilon_c|$ comes up with $\Gamma_r$,
therefore the smaller $\Gamma_r$, the bigger is the enhancement
factor $\rho$. The factor $RL/\Delta_{cv}$ in the square brackets
in Eq. (\ref{conduct}) behaves as $\epsilon^{1/2}\ln \epsilon$ in
the vicinity $\epsilon=|E_{d\downarrow}-\varepsilon_c|$ of the
band edge and thus gives additional contribution to this
enhancement. Both analytical and numerical estimates confirm this
statement. Experimentally, this effect should be observed as an
increase of tunneling conductance on the boundary of Coulomb
diamond in the vicinity of the threshold value of $e\upsilon_{\rm
th} =\Delta_{cv}$. A similar effect should arise on the hole side
of the Coulomb diamond diagram ($e\upsilon_{\rm th}
=-\Delta_{cv}$), where the occupied level $E_{d\downarrow}$
crosses the top of the down spin valence band (solid lines in Fig.
\ref{f.diamond}).
\subsection{Tunneling through moving dot}
\label{TMD}
As is shown above, the canonical transformation allows one to
distinguish between the adiabatic and non-adiabatic contributions to
the tunneling amplitude. Let us first discuss the adiabatic
corrections to the inelastic current given by Eq. (\ref{goldo}) and
illustrated by Fig. \ref{f.amp}. In the vicinity in the point
$(N=1)/(N=2)$ of the Coulomb diamond diagram the adiabatic position
of the down spin level $E_{d\downarrow}^{(a)}$ given by the
solution of equation
\begin{equation}\label{ead}
E_{d\downarrow}^{(a)}(t)= \sum_{j,k}
\frac{|V_{jk}(t)|^2}{E_{d\downarrow}^{(a)}(t) - \epsilon_{jk}}
\end{equation}
rocks around $\epsilon_{cr}$ or $\epsilon_{vl}$ (see Fig.
\ref{f.spul}). This solution depends parametrically on time via
the oscillating tunnel coupling $|V_{jk}(t)|^2$. The above
analysis of the static case prompts that time-dependent
corrections become significant at finite bias close to the
threshold value $e\upsilon_{\rm th} = \pm \Delta_{cv}$. The
adiabatic evolution of the level $E_{d\downarrow}^{(a)}$ in time
near $\varepsilon_c$ is given by the same Eq. (\ref{lambert}),
where the tunneling rate $\Gamma_r$ parametrically depends on $t$.
If the time-dependent perturbation is weak in comparison with the
static value of tunneling rate, the temporal component of
$E_{d\downarrow}^{(a)}$ may be found perturbatively. Representing
the tunneling rate as $\Gamma_r = \Gamma_{r0} + \delta\Gamma(t)$
and expanding Eq. (\ref{lambert}) around the time-independent
value marked by index '0', we get
\begin{equation}\label{expand}
E_{d\downarrow}^{(a)}(t)=E_{d\downarrow}^{(0)} -\frac{W_0}{1 + W_0}
\left( \frac{\varepsilon_{d}}{\Gamma_{r0}} +
W_0\right)\delta\Gamma(t)
\end{equation}
When deriving Eq. (\ref{expand}), the equality $W(x)=x[1+W(x)]dW/dx$
is used.
This time dependence turns into the corresponding adiabatic time
dependence of tunnel conductance, mainly via the enhancement
factor $\rho(E_{d\downarrow})$ (\ref{enhan}). One may expect that
the slow adiabatic variations of $G(t)$ will be especially
distinct at $v_g$ corresponding to steep slopes of $\rho(t)$ (Fig.
\ref{f.amp}) at $e\upsilon \sim \pm \Delta_{cv}$ near the boundary
$(N=1)/(N=2)$ of the Coulomb diamond diagram (Fig.
\ref{f.diamond}).
\subsection{Weak time dependent perturbation}
\label{WTDP}
Calculation accounting for nonadiabatic corrections to tunnel
conductance is generally an extremely complicated nonlinear
problem. To make it tractable, we assume that the time dependent
part of $V_{jk\sigma}$ is only a small periodic perturbation with
respect to the time independent part $V_{jk\sigma}^{(0)}$ and
consider only the linear response given by the first harmonics.
The nonlinear effects will be discussed separately. This approach
allows one to pick out first nonadiabatic corrections to tunneling
amplitude $W_{kk'}$, which turns out to be small everywhere except
in the vicinity of band edges.
Let us assume that the tunneling integral has the form
\begin{equation}\label{hybridization}
V_{k}(t) = V^{(0)}_{k} + V^{(1)}_{k} \cos\omega t
\end{equation}
(here and below indices $j\sigma$ are omitted for the sake of
brevity). Here $V^{(0)}_{k} \gg V^{(1)}_{k}$. The solution of Eq.
(\ref{elimination_time}) is looked for in the form
\begin{equation}\label{solution}
u_{k} = u^{(0)}_{k} + u^{(1)}_{k} \cos\omega t + i v^{(1)}_{k}
\sin\omega t
\end{equation}
where the time dependent corrections to $u^{(0)}_{k}$ are also
small. Then we vary all the coefficients in Eq. (\ref{eltim2}) with
respect to $u^{(0)}_{k}=g_k V^{(0)}_{k}$ [see Eq. (\ref{coeff})] and
$V^{(0)}_{k}$ and collect separately all the terms containing
$\cos\omega t$ and $\sin\omega t$ respectively. Substituting then
$u_{k}$ in (\ref{scatamp}), we reduce the scattering amplitude
$W_{kk'}$ to the following form
\begin{equation}\label{scatamp-tdt1}
W_{kk'} = \overline{W}_{kk'} + W^{(1)}_{kk'} \cos\omega t +
\hbar\omega W^{(2)}_{kk'} \sin\omega t
\end{equation}
where the coefficients of the cosine and sine terms can be
explicitly calculated. The cosine coefficient
\begin{equation}\label{cosine}
W^{(1)}_{kk'} = \sum_{q} \left[\frac{\delta
\overline{W}_{kk'}}{\delta V_{q}} V^{(1)}_{q} + \frac{\delta
\overline{W}_{kk'}}{\delta V^*_{q}} V^{(1)*}_{q} \right]
\end{equation}
is obtained by varying equation (\ref{scatamp-tit}) over the
hybridization potential $V_{q}$, whereas the sine coefficient
\begin{equation}\label{sine}
W^{(2)}_{kk'} =
$$$$
i \sum_{q} g_{q}^2 \left[ \frac{\delta W_{kk'}}{\delta u_{q}}
V^{(1)}_{q} - \frac{\delta W_{kk'}}{\delta u^*_{q}} V^{(1)*}_{q}
\right] + W^{(2),nonad}_{kk'}
\end{equation}
is obtained by varying equation (\ref{scatamp}) over the
transformation parameter $u_{q}$. After the variation $u_{k}$ in
the form Eq. (\ref{coeff}) can be substituted. Then
\begin{widetext}
$$
W^{(2),nonad}_{kk'} = - i \left[ \frac{V^{(1)*}_{k'} V^{(0)}_{k} -
V^{(1)}_{k} V^{(0)*}_{k'}}{\Delta_{k\sigma}\Delta_{k'\sigma}}
\widetilde{R}(\gamma_\sigma) + \frac{V^{(0)}_{k}
V_{k'}^{(0)*}}{2\Delta_{k\sigma}\Delta_{k'\sigma}}
\widetilde{R}^2(\gamma_\sigma) \sum\limits_{k''}
\frac{V^{(1)}_{k''} V^{(0)*}_{k''} - V_{k''}^{(0)*}
V^{(0)}_{k''}}{\Delta_{k''\sigma}\Delta_{k''\sigma}} \right]
$$
\end{widetext}
with
$$
\widetilde{R}_\sigma = \frac{R_\sigma}{\sqrt{1 +
\tan^2\gamma_\sigma}}.
$$
The most divergent terms in (\ref{cosine}) and (\ref{sine}) appear
when we vary only explicitly written $V^{(0)}_{k}$ in Eq.
(\ref{scatamp-tit}) and $u_{k'}$ in Eq. (\ref{scatamp}). As a
result keeping the leading terms we have
\begin{equation}\label{scatamp-tit-a}
W^{(1)}_{kk'} =
$$$$
(V^{(0)}_{k}V^{(1)*}_{k'} + V^{(0)*}_{k'} V^{(1)}_{k} )K^\sigma
\left( \frac{1}{\Delta_{k\sigma}} + \frac{1}{\Delta_{k'\sigma}}
\right)R(\gamma_\sigma)
\end{equation}
and
\begin{equation}\label{scatamp-tit-a}
W^{(2)}_{kk'} = - i (V^{(0)}_{k}V^{(1)*}_{k'} - V^{(0)*}_{k'}
V^{(1)}_{k} ) \times
$$$$
K^\sigma \left( \frac{1}{\Delta^2_{k\sigma}} +
\frac{1}{\Delta^2_{k'\sigma}}\right)R(\gamma_\sigma)
\frac{\gamma_\sigma}{\tan\gamma_\sigma}.
\end{equation}
The corresponding corrections to the tunneling transparency near the
conductance band edge behave as
\begin{equation}\label{conduct-1}
\delta G^{(1)}_\downarrow(v_g,\Delta_{cv} ) \approx \frac{e^2}{h}
\frac{\Gamma_l\Gamma_r^{(1)}}{(2\pi^2)}
\frac{R^2(\gamma_\downarrow)}{|E_{d\downarrow} - \varepsilon_c|^{2}}
\cos\omega t
\end{equation}
and
\begin{equation}\label{conduct-2}
\delta G^{(2)}_\downarrow(v_g,\Delta_{cv} ) \approx
$$$$
(\hbar\omega)\frac{e^2}{h} \frac{\Gamma_l\Gamma_r^{(2)}}{(2\pi^2)}
\frac{R^2(\gamma_\downarrow)}{|E_{d\downarrow} -
\varepsilon_c|^{5/2}} \frac{\pi}{2\sqrt{\Gamma_r}} \sin\omega t
\end{equation}
where
$$
\Gamma_r^{(m)} = 2\pi S_r \left[i^{(m-1)} V_r^{(1)} V_r^{(0)*} +
(-i)^{(m-1)} V_r^{(0)} V_r^{(1)*}\right]
$$
is the adiabatic correction to the tunneling rate for $m=1$ and the
correction due to weak non-adiabatic effect for $m=2$. Equation
(\ref{conduct-2}) uses the fact that
$$
\gamma_\downarrow \to \frac{\pi}{2},\ \mbox{and}\ \
\tan\gamma_\downarrow = \sqrt{- L'} \approx
\sqrt{\frac{\Gamma_r}{|E_{d\downarrow} - \varepsilon_c|}}
$$
for $E_{d\downarrow} \to \varepsilon_c$ when $|E_{d\downarrow} -
\varepsilon_c| \approx \Gamma_r$. We use here the approximation
$V^{(m)}_{k'} \approx V^{(m)}_r$ similarly to the one used in Eq.
(\ref{conduct}).
The term with the sine in equation (\ref{scatamp-tdt1}) causes a
phase shift between the oscillations of the hybridization parameter
and the resulting current through the dot. If we neglect the $k$
dependence of the coefficients (\ref{cosine}) and (\ref{sine}) this
phase shift can be readily found
$$\varphi = - \arctan \left(
\frac{\delta G^{(2)}_\downarrow(v_g,\Delta_{cv} )}{\delta
G^{(1)}_\downarrow(v_g,\Delta_{cv} )} \right).
$$
It is expected to be generally rather small due to the factor
$g_{k}\hbar\omega$, which is usually very small unless the dot level
approaches the band edge $\varepsilon_c$. However close to the edge
this factor diverges and results in an increasing phase shift,
$$
\varphi \approx - \arctan \frac{\hbar\omega
\Gamma_r^{(2)}}{\Gamma_r^{(1)} \sqrt{\Gamma_r |E_{d\downarrow} -
\varepsilon_c|}} \approx - \arctan \frac{\hbar\omega
\Gamma_r^{(2)}}{\Gamma_r \Gamma_r^{(1)}}
$$
As result the phase shift $\varphi$ may become essential at
$E_{d\downarrow} \to \varepsilon_c$.
The non-adiabatic corrections to tunneling conductance acquire the
simple form of phase shift in oscillating cosine function only
until the parameter $f_k=g_k\hbar\omega \sim \hbar\omega/\Delta$
is small and the perturbative approach is valid. With increasing
$f$ one may expect appearance of higher harmonics $n\omega$ in
oscillating conductance. With further increase of the amplitude
$V^{(1)}$ the language of quasi-energy levels\cite{Z73} is more
appropriate. We plan to discuss it elsewhere.
\section{Concluding remarks}
We have discussed in this paper a new approach to the Anderson model
for a half-metallic electron liquid in a tunnel contact with a
moving nano-shuttle under strong Coulomb blockade. It is shown that
in the situation where the spin-flip cotunneling processes are
suppressed at low energies, the exact canonical transformation
eliminating the tunneling term in the Anderson Hamiltonian exists
even in the presence of strong Hubbard repulsion in the shuttle and
time-dependent lead-shuttle tunneling. This canonical transformation
in principle allows one to sort out the slow adiabatic
renormalization of the energy levels and the tunnel transparency and
to consider non-adiabatic corrections at least perturbatively. One
may also include the inelastic spin-flip processes in the canonical
transformation in the 4-th order of perturbation theory in $V_i$,
but these weak corrections do not change the above qualitative
picture. We also have calculated the weak non-adiabatic corrections
to tunneling transparency in a specific model of periodic sinusoidal
motion of the shuttle. Undoubtedly, strong non-adiabatic effects
should be taken into account in a more refined scheme: in the case
of a periodic time-dependent perturbation one should appeal to the
Floquet theorem in the time domain and use the quasi energy
language.\cite{Z73,Baron77}
Because of the energy gap for spin-flip processes in the
half-metallic leads the Kondo cotunneling processes are suppressed,
and the zero-bias anomaly in the tunnel conductance is absent.
Physical manifestations of the shuttling mechanism under discussion
arise in the form of time-dependent enhancement of conductance at
finite bias near the boundaries of Coulomb diamonds on the phase
diagram $G(v_g, e\upsilon)$. The boundaries themselves are distorted
due to the complete spin polarization of carriers (see Fig.
\ref{f.diamond}), and even the Coulomb blockade step corresponding
to occupation change from odd to even number of electrons in the dot
is absent at zero bias. One may expect appearance of quasi energy
satellites in this part of the phase diagram $G(v_g, e\upsilon)$
when the shuttle motion is essentially non-adiabatic. This regime is
a subject for future studies.
|
2,877,628,090,107 | arxiv | \section{Ward's Method in Dimension One}\label{section:ward:1d}
In this section, we discuss the approximation ratio of Ward's method for inputs $P \subset \mathbb{R}^1$ and show the following theorem.
\begin{theorem}\label{thm:OneDimensional}
Let $P \subset \mathbb{R}$ be an arbitrary instance that is one-dimensional. Then, for every~$k$, the $k$-clustering computed by Ward on~$P$ is an $\mathcal{O}(1)$-approximation with respect to the $k$-means objective function.
\end{theorem}
For the purpose of analyzing the worst-case behavior of Ward's method, an instance sometimes also contains an integer $k \in \mathbb{N}$ in addition to $P$ (even though Ward itself only takes $P$ as the input). If we specify $P$ and $k$, then we are interested in the quality of the $k$-clustering produced by Ward on $P$.
We will usually denote the hierarchical clustering computed by Ward on $P$ by $\mathcal{W}=(\mathcal{W}_0,\ldots,\mathcal{W}_{n-1})$. Ward's method always chooses greedily a cheapest merge to perform. We say that a merge is a \emph{greedy merge} if it is a cheapest merge; if all merges are greedy, we call $\mathcal{W}$ greedy. Ward's method computes a greedy hierarchical clustering, and every greedy hierarchical clustering can be the output of Ward's method.
\subsection{Prelude: Reordering}\ \\
Recall the following statement from §\ref{techniques}:
\convexlemma*
Lemma~\ref{konvex} means that Ward will always merge $A$ and $C$ or $B$ and $C$, and never $A$ and $B$. This gives us a convexity property: If Ward forms a cluster $M$, then no other point or cluster lies within the convex hull of $M$. Clusters can thus also never overlap, and we get a concept of neighbors on the line. Thus, the clusterings $\mathcal{W}_i$ consist of non-overlapping clusters, which we can thus view as ordered by their position on the line. Ward's method always merges neighbors on the line. We will combine it with the following useful corollary of Lemma~\ref{obsb}. It gives a condition under which merging a cluster $A$ with a subcluster $B' \subset B$ is cheaper than merging $A$ with $B$. Notice that without the condition, the statement is not true: Imagine that $A$ and $B$ have the same centroid (merging them is free), but $\mu(B') \neq \mu(B)$. Then clearly, merging $A$ with $B'$ is more expensive than merging $A$ and $B$.
\begin{restatable}{corollary}{subclusterlemma}\label{subcluster}
Assume we have two finite clusters $B' \subseteq B \subset \mathbb R^d$ and a third finite cluster $A \subset \mathbb R^d$ such that $||\mu(A)-\mu(B')||^2 \leq ||\mu(A)-\mu(B)||^2$. Then $D(A,B')\leq D(A,B)$.
\end{restatable}
Corollary~\ref{subcluster} holds in arbitrary dimension. However, for $d=1$, it is much easier to benefit from it. We get a very convenient tool that we call \emph{reordering}. Say that Ward at some point merges two clusters $A$ and $B$. By Lemma~\ref{konvex}, that means that $\mu(A)$ and $\mu(B)$ are neighbors on the line (at the time of the merge). Now assume that $A$ and $B$ are present for a while before they are merged. Then during all this time, they are neighbors. Notice that this means that merging $A$ and $B$ will result in a centroid $\mu(A\cup B)$ which is further away from any other cluster than $\mu(A)$ and $\mu(B)$ are. So, clusters that did not want to merge with $A$ or $B$ would also not merge with $A\cup B$ by Corollary~\ref{subcluster}. Thus, we could perform the merge $(A,B)$ \emph{earlier} without distorting Ward's course of action at all (except that the merge $(A,B)$ is at the wrong position). Lemma~\ref{lemma:reorderingward} below formulates this idea.
Recall that a hierarchical clustering can also be described by the $n-1$ merge operations that produce it. We usually denote the sequence of merges by $(A,B)(\mathcal{W})=((A_1,B_1),\ldots,(A_{n-1},B_{n-1}))$. We say that a cluster $Q \subset P$ \emph{exists} in $\mathcal{W}$ after merge $t$ if $Q \in \mathcal{W}_t$. If $Q$ is the result of the merge $(A_i,B_i)$ (i.e., $Q=A_i\cup B_i)$, and it is later merged with another cluster in merge $(A_j,B_j)$ (i.e., $A_j=Q$ or $B_j=Q$), then $Q$ exists as long as merge $i$ has happened and merge $j$ has not yet happened. All singleton clusters exist in $\mathcal{W}_0$. After merge $n-1$, $P$ is the only remaining existing cluster.
\begin{lemma}[Reordering Lemma]\label{lemma:reorderingward}
Let $P\subset\mathcal{R}^d$ be an input for which Ward computes the clustering $\mathcal{W}$ with merge operations $(A,B)(\mathcal{W})$. Consider the merge $(A_t,B_t)$ for $t \in [n-1]$.
If both $A_t$ and $B_t$ exist after merge $s < t$, then
\begin{enumerate}
\item\label{lemma:reorderingward:1} The sequence of merge operations
\begin{align*}
(A',B')=&(A_1,B_1), \ldots, (A_s,B_s),(A_t,B_t),\\
&(A_{s+1},B_{s+1}),\ldots, (A_{t-1},B_{t-1}),\\
&(A_{t+1},B_{t+1}), \ldots, (A_{n-1},B_{n-1})
\end{align*}
results in a valid hierarchical clustering $\mathcal{W}'$.
\item\label{lemma:reorderingward:1b} $\mathcal{W}'_j = \mathcal{W}_j$ for all $j \ge t$.
\item\label{lemma:reorderingward:2} All merges except the moved merge $(A'_{s+1},B'_{s+1})=(A_t,B_t)$ are greedy merges.
\end{enumerate}
\end{lemma}
\begin{proof}
\eqref{lemma:reorderingward:1} and \eqref{lemma:reorderingward:1b} hold because performing merges in a different order does not change the resulting clustering, and after merge $t$, all deviations from the original order are done. For \eqref{lemma:reorderingward:2}, we have to argue that inserting $(A_t,B_t)$ as step $s+1$ does not create cheaper merges. For this, we observe that by Lemma~\ref{konvex}, $A_t$ and $B_t$ are neighbors on the line.
In the original sequence, no cluster was merged with $A_t$ or $B_t$ up to point $t$. The cluster $A_t\cup B_t$ is a superset of $A_t$ and of $B_t$, and its centroid is further away from all other clusters than the centroids of $A_t$ and $B_t$. Thus by Corollary~\ref{subcluster}, up to point $t$, merging with $A_t\cup B_t$ cannot be cheaper than the merges we do. However, after $(A_{t-1},B_{t-1})$, the clustering is identical to $\mathcal{W}_{t}$ by \eqref{lemma:reorderingward:1}, thus all remaining merges are also greedy merges.
\end{proof}
Lemma~\ref{lemma:reorderingward} a crucial observation to allow us to systematically analyze Ward's steps: We can sort them into steps that depend on each other, and then analyze them in batches / phases.
In $\mathbb{R}^d$ for $d > 1$, reordering does not work. Also, we cannot assume that there are no inner-cluster merges.
\subsection{Prelude: No Inner-cluster Merges}
Reordering also gives us a nice simplification tool. Assume that $A$ and $B$ are in fact singleton clusters, $A=\{a\}$ and $B=\{b\}$, and they are from the same optimum cluster. Then they are present from the start; we can reorder the merge $(A,B)$ to be the first merge Ward does. Indeed, instead of actually doing this merge, we can also simply forget about it and replace $a$ and $b$ by a weighted point. How does this affect the approximation ratio? Both Ward's cost and the optimal cost decrease by $\Delta(\{a,b\})$, meaning that the approximation ratio can only get worse. We can now assume that there are no merges between inner clusters, since inner clusters arise from merging input points that belong to the same optimum cluster.
We formalize our observation in Lemma~\ref{lem:no-in-optcluster-merging}.
We directly apply Lemma~\ref{lemma:reorderingward} in order to achieve a simplification method. Recall that (given an optimal $k$-clustering) we call a merge $(A_i,B_i)$ an inner-cluster merge if $A_i$ and $B_i$ are inner clusters from the same optimum cluster. For a worst-case instance $(P,k)$ we can always assume that such inner-cluster merges do not happen, as they are only helpful for Ward's method. We formally see this in the next lemma, where we relocate inner-cluster merges to the front of the hierarchical clustering and then eliminate them.
Recall that $\Delta_k(\mathcal{W})=\sum_{Q\in\mathcal{W}_{n-k}}\Delta(Q)$ is the cost of the $k$-clustering contained in $\mathcal{W}$. For an instance $(P,k)$ and Ward's resulting clustering $\mathcal{W}$, the approximation ratio of Ward's method is $\Delta_k(\mathcal{W}) / \opt_k(P)$.
\begin{lemma}\label{lem:no-in-optcluster-merging}
Let $(P,k)$ be an instance with $P\subset\mathbb{R}^d$ and $k\in \mathbb{N}$, for which $\mathcal{O}=\{O_1,\ldots,O_k\}$ is an optimal $k$-clustering and for which Ward computes the hierarchical clustering $\mathcal{W}$ with merge operations $(A,B)(\mathcal{W})$. Then there exists a weighted point set $P'$ and a hierarchical clustering $\mathcal{W}'$ for $P'$ with merges $(A',B')(\mathcal{W}')$ with the following properties:
\begin{enumerate}
\item\label{lemma:noinnerclustermerges:1} $\mathcal{W}'$ is greedy.
\item\label{lemma:noinnerclustermerges:2} No $(A'_i,B_i')$ is an inner-cluster merge with respect to $\mathcal{O}$.
\item\label{lemma:noinnerclustermerges:3} For some $\alpha \ge 0$, $\Delta_k(\mathcal{W}') = \Delta_k(\mathcal{W}) - \alpha$ and $\opt_k(P')\le\opt_k(P)-\alpha$.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume that $P$ is weighted; this will be necessary to iterate the following process.
Let $(\{x\},\{y\})$ be a merge operation in $(A,B)(\mathcal{W})$ that merges two points $x,y \in O_j$ for $j\in[k]$, i.e., two points from the same cluster in the optimal solution. Let their weights be $w(x)$ and $w(y)$. By Lemma~\ref{lemma:reorderingward}, we can move the merge $(\{x\},\{y\})$ to the front.
Then we replace $x$ and $y$ in $P$ by one point $z=\frac{w(x) x + w(y) y}{w(x)+w(y)}$ with weight $w(z) := w(x)+w(y)$. By Lemma~\ref{obsb}, $z$ behaves identically to $\{x,y\}$ in Ward's method. Thus, we can adjust ${\mathcal{W}}'$ by removing the merge operation $(\{x\},\{y\})$, and replacing $x$ and $y$ by $z$ in all further merge operations of the cluster $\{x,y\}$. We see that~\eqref{lemma:noinnerclustermerges:1} holds for the new hierarchical clustering.
Our adjustment will change the cost by $\alpha:=\Delta(\{x,y\})$.
Similarly, we can replace $x$ and $y$ in $O_j$ by $z$, which decreases the cost of the clustering induced by $O_1,\ldots,O_k$ by $\alpha$. Since this is still a possible clustering, the optimal clustering can cost at most $\opt_k(P)-\alpha$.
Thus, \eqref{lemma:noinnerclustermerges:3} holds for the new clustering.
Observe that if~\eqref{lemma:noinnerclustermerges:2} is not true, then there has to be a merge operation where two points from the same cluster in the optimum are merged.
Thus, we can complete the proof by repeating the above process until we have removed all pairs with this property. Then~\eqref{lemma:noinnerclustermerges:2} holds.
\end{proof}
Now if Ward performs inner-cluster merges on an instance, we apply Lemma~\ref{lem:no-in-optcluster-merging}. If this changes the optimum solution, we just apply Lemma~\ref{lem:no-in-optcluster-merging} again, and repeat this until Ward does not do any inner-cluster merges.
We explicitly note the following trivial corollary.
\begin{corollary}\label{corollary:innerarepoints}
Assume that $\mathcal{W}'$ and $(A',B')(\mathcal{W'})$ result from applying Lemma~\ref{lem:no-in-optcluster-merging} until Ward does not do inner cluster merges. If a merge $(A_i',B_i')$ for $i \in [n-1]$ contains an inner cluster, then this inner cluster is a (weighted) input point.
\end{corollary}
\begin{proof}
If $A$ resulted from a previous merge, then this merge was an inner-cluster merge, which is a contradiction.
\end{proof}
Corollary~\ref{corollary:innerarepoints} implies that we can use the terms inner cluster and input point interchangeably.
\subsection{Prelude: Clustering points together}
Crucial in showing the approximation factors of the good merges is the following lemma. To see its usage, assume that $A$ and $B$ belong to one optimum cluster, and $C$ and $D$ belong to another. Then the lemma implies that if Ward has already merged $B$ and $C$, but $\Delta(B\cup C)$ is small, say $\Delta(B\cup C) \le c\cdot(\Delta(B)+\Delta(C))$, then we can still obtain a $7c$-approximation. Its proof is deferred to the full version of this paper.
\begin{restatable}{lemma}{goodmergefour}\label{lem:goodmergewithfour}
Let $A,B,C,D\subset\mathbb{R}^d$ be disjoint sets with $|A| \le |B|$ and $|C|\ge |D|$. Then
\[
\begin{split}
&\Delta(A\cup B\cup C\cup D) \le \Delta(A) + 3 \cdot \Delta(B\cup C)\\
& \hspace{1.5cm} + \Delta(D) + 4 \cdot D(A,B) + 4\cdot D(C,D)
\end{split}
\]
and
\[
\begin{split}
&D(A\cup B, C \cup D) \le 3 \cdot \Delta(B\cup C) + 3 \cdot D(A,B)\\
& \hspace{1.5cm}+ 3 \cdot D(C,D) - \Delta(B) - \Delta(C).
\end{split}
\]
\end{restatable}
\subsection{The analysis}
We now analyze the worst-case behavior of Ward's method on the line by fixing an arbitrary worst-case example that does not contain inner-cluster merges (we can assume this by Lemma~\ref{lem:no-in-optcluster-merging}).
The general plan is the following. Whenever Ward merges two clusters, it does so greedily, meaning that the cost of the merge is always bounded by the cost of any other merge. Thus, if we can find a merge with low cost, then the merge actually performed can only be cheaper. We can clearly find cheap merges in the beginning, however, Ward's decisions may lead us to a situation where we run out of the originally good options. The idea of the proof is to find a point during Ward's execution where:
\begin{itemize}
\item We still know a bound on the costs produced so far.
\item We know a set $\mathcal{S}$ of good merges that can still be performed and lead to a good $k$-clustering.
\item We can ensure that no merge can possibly destroy two merges from $\mathcal{S}$.
\end{itemize}
At such a point in time, we can use $\mathcal{S}$ to charge the remaining merges that Ward does to compute a $k$-clustering. We find this point in time by sorting specific merges of Ward into the front, and bounding their cost. There will be five phases of merges which we need to pull forward and charge.
\paragraph*{The phases}
We will use the reordering lemma (Lemma~\ref{lemma:reorderingward}) to sort the merges into phases and then analyze the cost of the solution after each phase.
In the following, we call a cluster that contains points from more than one optimum cluster \emph{composed}, more precisely, we call it an \emph{$\ell$-composed cluster} if it contains points from $\ell$ different optimum clusters. Most of the time, we are interested in $2$-composed clusters, and we name such a cluster \emph{$2$-composed cluster from $O_j$ and $O_{j+1}$} if these are the involved optimum clusters.
\input{input_figure_phases}
The goal of the reordering is simple in nature; we want to collect all merges that create $2$-composed clusters and that grow $2$-composed clusters. We can think of the phases as different stages of development of $2$-composed clusters. A $2$-composed cluster may become part of the $k$-clustering computed by Ward's method, or it may at some point become $i$-composed for $i>2$, at which time we are no longer interested in it. By the \emph{final stage} of a $2$-composed cluster we either mean how it looks in the $k$-clustering, or how it looked in the last step before it became more than $2$-composed.
Consider the example in Figure~\ref{figure:phaseexample}, where we depict the development of a $2$-composed cluster from $O_j$ and $O_{j+1}$ which in its final stage consists of the input points $x_\ell,\ldots,x_r$. It undergoes five principal phases: It is created by merging a point from $O_j$ with a point from $O_{j+1}$ (phase $P1$). Then it grows; it is merged with points left and right of itself (phase $P2$). We add extra phases for the last points on both sides. In phase $P3$, the first side is completed; in the example, it is the left side. This merge is again followed by a growth phase (phase $P4$). The final phase $P5$ consists of the final merge on the other side; the right side in the example. (We skip some merges in $P5$, the details of $P5$ are not discussed until much later in this proof).
So, we use reordering to pull the following phases of merges to the front.
\begin{enumerate}
\item[P1] (Creation phase)\\
We create $2$-composed clusters by collecting the merges $(\{a_i\},\{b_i\})$ with $a_i \in O_j$, $b_i \in O_{j+1}$ for some $j \in [k]$.
The collected merges constitute phase $P1$.
For technical reasons, we make one exception. If the $2$-composed cluster only consists of two input points in its final stage (i.e., the creating merge is also the last merge), then we defer the merge to phase $P5$. \item[P2] (Main growth phase)\\
We now grow the $2$-composed clusters initialized during phase $P1$.
For each $2$-composed cluster, we move the growth merges to phase $P2$, preserving their original order. We stop right before one side of the $2$-composed cluster is done. There may be many growth merges for a cluster, or none.
\item[P3] (First side elimination phase)\\
This phase consists of at most one merge for each $2$-composed cluster, and this merge is the last merge on the first side. After phase $P3$, every $2$-composed cluster thus has one side where it will not be merged with further input points. Notice that a cluster may skip phase $P3$ if it only shares one point with $O_j$ or $O_{j+1}$ in its final stage, anyway.
\item[P4] (Second growth phase)\\
This phase resembles phase $P2$, however, the growth is now one-sided. For each $2$-composed cluster, we move the growth merges to phase $P4$, preserving their original order, and stopping right before the final merge.
\item[P5] (Second side elimination phase)\\
The last phase consists of at most one merge for each cluster. If the final stage of a $2$-composed cluster contains only two points, then the merging of these two points is done in phase $P5$. Otherwise, phase $P5$ may contain the last merge for the cluster, resulting in its final state. For technical reasons, we have to exclude some merges; we postpone the details to Definition~\ref{def:phase5}.
\end{enumerate}
We now analyze the sum of the $1$-means costs of all clusters in the clustering after each phase.
The proofs of the lemmata are deferred to the full version of the paper.
We start with phases $P1$ and $P2$.
\begin{lemma}\label{lem:phase12}
Let $N=\{x_{a},\ldots,x_{b}\}$ with $x_{a},\ldots,x_m \in O_j$ and $x_{m+1},\ldots,x_{b} \in O_{j+1}$ be a $2$-composed cluster after phases $P1$ and $P2$. Then
\[
\Delta(N) \le \sum_{h=a-1}^{m-1} D(x_h,x_{h+1}) + \sum_{h=m+1}^{b} D(x_h,x_{h+1}).
\]
Furthermore, $D(N\cap O_j,N\cap O_{j+1}) \le D(x_{a-1},x_{a}) + D(x_{b},x_{b+1})$.
\end{lemma}
In phase $P3$, Ward's method faces the first situation where it may run out of good merge options and has to resort to more expensive merges. Notice that by the definition of our phases, each cluster has one side where after phase $P2$, there is exactly one point left which has not been added to the cluster.
The key technical observation that we use again and again during the (omitted) proofs is the following corollary.
\goodmergethree*
We need the following interpretation of Corollary~\ref{cor:goodmergewiththree}. If we have a 2-composed cluster $M=A\cup B$ which consists of a lighter cluster $A \subseteq O'$ for an optimum cluster $O'$ and a heavier cluster $B \subset O''$ for another optimum cluster $O''$, then merging $A\cup B$ with another cluster $C \subset O''$ basically costs as much as $A\subseteq O'$ and $B\cup C \subseteq O''$ cost individually, plus what merging $A$ and $B$ costed us already (up to constant factors). Corollary~\ref{cor:goodmergewiththree} allows us to analyze the $1$-means costs of the clusters after phase $P4$.
\begin{lemma}\label{lem:phase34}
Let $F=\{x_{\ell},\ldots,x_r\}$ be the final state of a $2$-composed cluster, with $x_{\ell},\ldots,x_m \in O_j$ and $x_{m+1},\ldots,x_r \in O_{j+1}$.
The state of the cluster after phase $P4$ is either $N=\{x_{\ell},\ldots,x_{r-1}\}$ or $N=\{x_{\ell-1},\ldots,x_r\}$. In both cases,
\[
\Delta(N) \le 8 \cdot(\Delta(\{x_{\ell},\ldots,x_m\}) + \Delta(\{x_{m+1},\ldots,x_{r}\})).
\]
\end{lemma}
Now we come to phase $P5$, which we haven't completely defined yet. The problem with phase $P5$ is that we can no longer charge all clusters \lq internally\rq. To see this, first notice that we say that a $2$-composed cluster $F$ from $O_j$ and $O_{j+1}$ \emph{points to cluster $A$} if
\begin{itemize}
\item $w(F\cap O_j) \ge w(F\cap O_{j+1})$ holds and $A$ is the cluster left of $F$, or
\item $w(F\cap O_j) \le w(F\cap O_{j+1})$ holds and $A$ is the cluster right of $F$.
\end{itemize}
We define a \emph{lopsided cluster} to be a $2$-composed cluster $F = \{x_\ell,\ldots,x_r\}$ for which the last merge is $\{F\backslash\{x\},\{x\}\}$, but at the time of this merge, $F'=F\backslash\{x\}$ does not point to $\{x\}$. This means that we cannot use Corollary~\ref{cor:goodmergewiththree} (directly) to charge this merge. As a technicality, we also call a $2$-composed cluster lopsided if it only contains two points in its final state; again, we cannot use Corollary~\ref{cor:goodmergewiththree} in this case.
We have to pay attention to one more detail when defining phase $P5$. When charging $2$-composed clusters internally, we could always be sure that the clusters that are involved are part of one of the two optimum clusters that the $2$-composed cluster intersects. That is because the $2$-composed cluster by definition only contains points from two optimum clusters, and we only dealt with points and subclusters of such a $2$-composed cluster. However, in the following arguments, we will have to argue about clusters neighboring a $2$-composed cluster. These may or may not belong to one of the optimum clusters. Let $A$ and $B$ be two clusters that are neighbors on the line such that $A$ lies left of $B$.
We say that there is an \emph{opt change between $A$ and $B$} if the last point in $A$ and the first point in $B$ belong to different optimum clusters.
Now we define phase $P5$. Let $Y$ be the cluster that lies on the other side of $F'$ than~$x$ \emph{at the time of the merge $\{F',\{x\}\}$}. Let $Z$ be the cluster that lies \lq behind\rq\ $x$ from the point of view of $F'$ \emph{at the time of the merge $\{F',\{x\}\}$}. By \emph{behind from $F$'s point of view} we mean that if $x$ lies left of $F$, then $Z$ lies left of $x$, and if $x$ lies right of $F'$, then $Z$ lies right of $x$.
\begin{definition}[Phase P5]\label{def:phase5}
Phase $P5$ contains the final merge $\{F',\{x\}\}$ of a cluster $F=F'\cup\{x\}$ if any of the following conditions applies.
\begin{enumerate}
\item $F$ is not lopsided (phase $P5a$),
\item $F$ is lopsided, there is no opt change between $Y$ and $F'$, and $Y$ is an inner cluster (phase $P5b$),
\item $F$ is lopsided, there is no opt change between $\{x\}$ and $Z$, and $Z$ is an inner cluster (phase $P5c$),
\item $F$ is lopsided, there is no opt change between $\{x\}$ and $Z$, $Z$ is $2$-composed, and points to \{x\} (phase $P5d$).
\end{enumerate}
\end{definition}
The next lemma deals with merges in $P5a$.
\begin{lemma}\label{lem:phase5a}
Let $F=\{x_{\ell},\ldots,x_r\}$ be the final state of a $2$-composed cluster, with $x_{\ell},\ldots,x_m \in O_j$ and $x_{m+1},\ldots,x_r \in O_{j+1}$. Assume that $F$ is not lopsided.
Then
\[
\Delta(F) \le 35 \cdot(\Delta(\{x_{\ell},\ldots,x_m\}) + \Delta(\{x_{m+1},\ldots,x_{r}\})).
\]
\end{lemma}
Now we consider the merges in phase $P5b$.
\begin{lemma}\label{lem:phase5b}
Let $F=\{x_{\ell},\ldots,x_r\}$ be the final state of a $2$-composed cluster, with $x_{\ell},\ldots,x_m \in O_j$ and $x_{m+1},\ldots,x_r \in O_{j+1}$. Assume that $F$ is lopsided.
Assume that at the time of the merge $\{F\backslash\{x\},\{x\}\}$, the cluster on the other side of $F'=F\backslash \{x\}$ is an inner cluster $Y$, and there is no opt change between $F'$ and $Y$.
Then if $x=x_{\ell}$, we have
\[
\Delta(F) \le 35 \cdot(\Delta(\{x_{\ell},\ldots,x_m\}) + \Delta(\{x_{m+1},\ldots,x_{r+1}\})),
\]
and if $x=x_{r}$, then
\[
\Delta(F) \le 35 \cdot(\Delta(\{x_{\ell-1},\ldots,x_m\}) + \Delta(\{x_{m+1},\ldots,x_{r}\})).
\]
\end{lemma}
The following lemma is the main lemma about the phases and summarizes our findings: After phase $5$, the error is still bounded by a constant times the optimum value.
\begin{lemma}
Let $C_5$ be the clustering after phase $P5$. Then
\[
\sum_{A \in C_5} \Delta(A) \le \mathcal{O}(1) \cdot \opt_k.
\]
\end{lemma}
\paragraph*{Good merges for the final analysis}
In general, the clustering of Ward after phase~$P5$ has still more than~$k$ clusters. It remains to analyze the merges after phase~$P5$ that reduce the number of clusters to~$k$. For the final charging argument, we need four types of \emph{good merges}. Good merges are not necessarily merges that Ward's method does, instead, it's a collection of merges that are possible and can be used for charging. Indeed, good merges include merges that would not be present anymore if Ward did them, since then we would move them to the phases. But if Ward never uses them, they may still be present for us to charge against.
The whole point of the phases is to ensure that any merge that Ward may still do does not destroy two good merges. The final arguments of the proof will be to count good merges and to show that no two good merges can be invalidated simultaneously by one of Ward's merges.
Recall that $W_1,\ldots,W_{\ell}$ is the current Ward solution, and $O_1,\ldots,O_k$ is a fixed optimal solution, numbered from left to right. The following merges are good merges in the sense that we can bound the increase in cost. Of course, the result of the merge only forms a cluster of low cost if the participating clusters had low cost beforehand.
\input{input_figure_goodmerges}
\begin{itemize}
\item[\emph{T1}:] Two inner clusters $W_i,W_{i+1}$ of the same optimal cluster $O_j$, i.e., $W_i,W_{i+1} \subset O_j$. This type of merge is never actually applied by Ward on simplified examples, but we need it for charging.
\item[\emph{T2}:] A $2$-composed cluster $W_i \subset O_j \cup O_{j+1}$ for some $j$ and an inner cluster $W_{i+1} \subset O_{j+1}$, with the condition that $W_{i+2}$ is an inner cluster of $O_{j+1}$ as well. Also: The symmetric situation of a $2$-composed cluster $W_i \subset O_j \cup O_{j+1}$ for some $j$ and an inner cluster $W_{i-1} \subset O_{j}$ with the condition that $W_{i-2} \subset O_{j}$.
\item[\emph{T3}:] A $2$-composed cluster $W_i \subset O_j \cup O_{j+1}$ for some $j$ and an inner cluster $W_{i-1} \subset O_j$, with the condition that $W_i$ points to $W_{i-1}$. Also: The symmetric situation of a $2$-composed cluster $W_i \subset O_j \cup O_{j+1}$ for some $j$ and an inner cluster $W_{i+1} \subset O_{j+1}$ with the condition that $W_i$ points to $W_{i+1}$.
\item[\emph{T4}:] Two $2$-composed clusters $W_i \subset O_j \cup O_{j+1}$ and $W_{i+1} \subset O_{j+1} \cup O_{j+2}$ that point at each other.
\end{itemize}
We already know T1 merges (inner-cluster merges), T2 merges (growth phase and phase $5c$) and T3 merges (merges chargeable with Corollary \ref{cor:goodmergewiththree}). We know that applying them increases the cost by at most a constant factor. We also know that these merges cannot happen anymore: T1 merges are inner-cluster merges, which Ward does not do on our example. T2 merges happen either in the growth phase, or in phase $5c$. T3 merges merge non-lopsided clusters, which happens in phase $5a$.
T4 is a type of merge that we did not yet consider, and which Ward can still do. Indeed, to charge it, we need the general charging statement in the below Lemma~\ref{lem:goodmergewithfour} from which Corollary \ref{cor:goodmergewiththree} follows.
\goodmergefour*
Let $W_i$ and $W_{i+1}$ constitute a T4 merge as described above. Then Lemma~\ref{lem:goodmergewithfour} with $A=W_i \cap O_j$, $B=W_i \cap O_{j+1}$, $C=W_{i+1}\cap O_{j+1}$ and $D=W_{i+1}\cap O_{j+2}$ implies that
\begin{align*}
&\Delta(W_i \cup W_{i+1})\\
\le\ & \Delta(W_i \cap O_{j}) + 3 \Delta(O_{j+1}) + \Delta(W_{i+1}\cap O_{j+2})\\
& \hspace*{2cm} + 4D(W_i\cap O_j,W_{i}\cap O_{j+1})\\
& \hspace*{2cm} + 4D(W_{i+1}\cap O_{j+1},W_{i+1}\cap O_{j+2}).
\end{align*}
Thus, if $\Delta(W_i)+\Delta(W_{i+1})$ was bounded by a constant factor times the optimal cost of the points in $W_i\cup W_{i+1}$, then this is still true after the merge of $W_i$ and $W_{i+1}$ (with a higher factor).
\subparagraph*{Counting inner clusters}
Observe that the only merges that delete more than one inner cluster are the merges in phase $P1$. All other merges remove either exactly one inner cluster, or none at all. In phase $P2$-$P5$, every merge eliminates exactly one inner cluster. In the beginning, there are $n$ inner clusters.
So if phase $P1$ has $n_1$ merges and $P2$ until $P5$ together have $n_r$ merges, then we have $n - 2n_1 - n_r$ inner clusters after phase $P5$, and we have $n_1$ $2$-composed clusters. The total number of all clusters is $n - n_1 - n_r$.
Consider the Ward clustering $W_1,\ldots,W_t$ after phase $P5$. We split the clustering into blocks, based on the inner clusters. More precisely, we get $n - 2n_1 - n_r - 1$ blocks that start with an inner cluster, possibly has some $2$-composed clusters and ends with another inner cluster. The blocks overlap in the inner clusters.
We argue that there is at least one good merge in every block except for $k-n_1-1$ blocks.
The exceptions are the blocks where the optimum cluster changes between start and end, but the change happens between the clusters (not in a $2$-composed cluster). This can only happen $k-n_1-1$ times because $n_1$ of the $k-1$ cluster borders are within $2$-composed clusters. For the remaining blocks, we argue the following.
If there are no $2$-composed clusters in the block, then the two inner clusters are neighbored and form a T1 merge.
If there is only one $2$-composed cluster in the block, then it has to point at an inner cluster and thus there is a T2 or a T3 merge.
If there are multiple $2$-composed clusters, we argue as follows. The first $2$-composed cluster either points left and thus there is a T2 or a T3 merge, or it points to the right. Any further $2$-composed cluster either points to the one before it, forming a T4 merge, or it points to the right. This goes on until we either find a merge, or we find the last $2$-composed cluster, which then has to point at the second inner cluster, forming a T2 or T2 merge.
We collect one good merge from every block and call the resulting set of merges $\mathcal{S}$.
Observe that the cost of all merges in $\mathcal{S}$ together is a constant factor of the cost that we have so far, so all merges together cost $\mathcal{O}(1) \opt_k$.
This argument alone is not enough. The main feature of $\mathcal{S}$ is that every merge that Ward actually performs can make at most one merge from our set invalid. This means that we can charge $n - 2n_1 - n_r - 1 - (k-n_1-1)$ merges to $\mathcal{S}$.
Notice that our merges are disjoint except for possible overlap at inner clusters.
Assume that a merge of Ward invalidates two merges from our set. There are two ways how this can happen.
Case one is that Ward's merge is one of the two good merges that are invalidated. Say this merge is called $(A,B)$. Then the second merges involves either $A$ or $B$, say it involves $B$. Thus, there is another cluster $C$ next to $B$, and the merge $(A,B)$ invalidates itself and $(B,C)$. This in particular means that $(A,B)$ is a good merge. Since Ward does not do inner-cluster merges, either $A$ or $B$ has to be $2$-composed, since $(A,B)$ is a merge of Ward. If they are both $2$-composed clusters, then $A$ and $B$ are in the same block, thus $(A,B)$ and $(B,C)$ cannot both be in $\mathcal{S}$. Thus, one is $2$-composed and the other is an inner cluster, i.e., they form a T2 or T3 merge, since $(A,B)$ is supposed to be a good merge. If it is a T3 merge, then $(A,B)$ is not lopsided, and would have happened in phase $P5a$. If it is a T2 merge, then it is either not lopsided (phase $P5a$), or it is lopsided, but has an inner cluster behind its inner cluster (phase $P5c)$. We conclude that a good merge $(A,B)$ cannot invalidate another good merge.
Case two is that the two good merges are disjoint, and Ward does a merge that overlaps with both of them. Thus, we have two good merges $(A,B)$ and $(C,D)$, and Ward performs merge $(B,C)$. Since Ward does not do inner-cluster merges, either $B$ or $C$ is $2$-composed, w.l.o.g. say that $C$ is $2$-composed. If $B$ is $2$-composed as well, then $(A,B)$ and $(C,D)$ are in the same block, so they would not both be in $\mathcal{S}$. So $B$ is an inner cluster. If $C$ points to $B$, then $(B,C)$ is not lopsided and would have happened in phase $P5a$. Thus, $C$ points to $D$.
If $A$ is an inner cluster, then $(B,C)$ is a T2 merge and would have happened in phase $P5c$.
So say that $A$ is $2$-composed. $(A,B)$ is a good merge. It is not a T2 merge since $C$ is $2$-composed. It has to be a T3 merge, thus, $A$ points to $B$. Thus, $(B,C)$ would have happened in phase $P5d$: It is a lopsided merge with $B$ left of $C$, and the $2$-composed cluster left of $B$ points to $A$.
We have seen that no merge of Ward can invalidate two merges from $\mathcal{S}$.
Thus, we can now charge in the following way. The cost of the performed merge is bounded by the cost of any available merge.
For each Ward step, we look whether it invalidates a merge from $\mathcal{S}$. If so, then we charge the performed merge to this good merge. If Ward's merge does not invalidate any merge from $\mathcal{S}$, we just arbitrarily charge a merge in $\mathcal{S}$ and mark it as invalid. In this manner, we can pay for $n - 2n_1 - n_r - (k-n_1-1) - 1$ merges, i.e., we can pay until the number of clusters is reduced to $n - n_1 - n_r - (n - 2n_1 - n_r - (k-n_1-1) - 1) = k$. That completes the proof of Theorem~\ref{thm:OneDimensional}.
\section{Greedy is not monotone for \texorpdfstring{$k$}{k}-median}\label{sec:kmediannotmonotone}
Figure~\ref{fig:discretenotmonotone} consists of two examples showing that the greedy method that works like Ward, but for $k$-median clustering, is not monotone. On the left, we see an example for the Euclidean $k$-median problem. It is simply an isosceles triangle; say that the distance between any two points is $1$. The $3$-clustering is free. The first merge costs $1$, no matter which two points are merged. For the resulting cluster, any point on the line between the two points is an optimal center. In particular, when merging with the third point, choosing the mid point between the first two points still results in a cost of $1$ for these two points; but the distance of this point to the third is only $\sqrt{3}/2$. Thus, the second merge is cheaper than the first (it is indeed even slightly cheaper than $\sqrt{3}/2$ because the optimal center of the $1$-clustering does not lie on the line between the first two points).
We can construct a similar example for $k$-median in finite discrete space. Here, the input points come with a finite metric, and the centers can only be chosen from the input points. Consider the right side of Figure~\ref{fig:discretenotmonotone}. There are four points, out of which three have weight $2$. The edges have unit length and all other distances are shortest paths in the graph. The first merge has minimal cost if one of the outer points is merged with the point in the middle. This results in a cost of $1$ if the center is placed on the outer point. The second merge has to be with a different outer point, or the two outer points are merged. In both cases, the merge costs $4$, and the final merge costs $2$, i.e., violates monotonicity.
\section{Exponential Lower Bound in High Dimension}\label{sec:explowerbound}
In the following, we describe a family of instances of increasing dimension~$d$ where Ward computes for some number~$k=k(d)$ of clusters a $k$-clustering that costs $\Omega((3/2)^d \opt_k)$. Here and in all other worst-case examples, we assume that given a choice between equally expensive merges, Ward chooses the action that leads to a worse outcome. This is without loss of generality because we can always slightly move the points to ensure the outcome we want. However, it greatly simplifies the exposition.
To further simplify the exposition, the below definitions use points of infinite weight and assume that the optimal cluster centers coincide with these infinite weight points. For any finite realization of the example, that is not the case. To ensure that Ward actually behaves like described in the following, we have to move the high weight points by an infinitesimal distance. We do this in the full version of this paper, but for sake of clarity, omit it in the exposition here. Notice that merging a cluster $H$ of infinite weight with a cluster $A$ of finite weight costs $|A|\cdot||\mu(A)-\mu(H)||^2$ by Lemma~\ref{obsb}.
Let~$d$ be given. We construct an instance~$P_d\subseteq\RR^d$ with $2^{d+1}$ points.
For~$i\ge 2$ let $z_i^2 = \frac{3^{i-2}}{2^{i-1}}$ and define
\[
\begin{split}
P_d = \{(x_1,\ldots,x_d)\mid x_1\in\{-1,-(\sqrt{2}-1),\sqrt{2}-1,1\},\\ x_i\in\{-z_i,z_i\} \ \forall i\in \{2,\ldots,d\}\}.
\end{split}
\]
All points from~$P_d$ whose first coordinate is~$-1$ or~$1$ have weight~$\infty$ (we call these \emph{heavy points}). All other points have weight~$1$ (we call these \emph{light points}).
For an illustration of $P_2$ and $P_3$, see Figure~\ref{fig:lowerbound}.
\begin{figure*}
\centering
\begin{tikzpicture}[scale=2.3]
\node [draw,fill,circle,inner sep=0cm,minimum height=0.15cm] at (-1,-0.5) {};
\node [draw,fill,circle,inner sep=0cm,minimum height=0.05cm] at (-0.41421356,-0.5) {};
\node [draw,fill,circle,inner sep=0cm,minimum height=0.05cm] at (+0.41421356,-0.5) {};
\node [draw,fill,circle,inner sep=0cm,minimum height=0.15cm] at (+1,-0.5) {};
\node [draw,fill,circle,inner sep=0cm,minimum height=0.15cm] at (-1,0.5) {};
\node [draw,fill,circle,inner sep=0cm,minimum height=0.05cm] at (-0.41421356,0.5) {};
\node [draw,fill,circle,inner sep=0cm,minimum height=0.05cm] at (+0.41421356,0.5) {};
\node [draw,fill,circle,inner sep=0cm,minimum height=0.15cm] at (+1,0.5) {};
\draw [->] (-1.5,0) -- (1.5,0);
\draw [->] (0,-1) -- (0,1);
\node [draw,rectangle,inner sep=0cm,minimum height=0.2cm,minimum width=0cm, label={[label distance=-0.1cm]above:{\tiny$-1$}},label distance=-0.1cm] at (-1,0) {};
\node [draw,rectangle,inner sep=0cm,minimum height=0.2cm,minimum width=0cm, label={[label distance=-0.1cm]above:{\tiny$+1$}},label distance=-0.1cm] at (1,0) {};
\node [draw,rectangle,inner sep=0cm,minimum height=0.2cm,minimum width=0cm, label={[label distance=-0.1cm]above:{\tiny$-(\sqrt{2}-1)$}},label distance=-0.1cm] at (-0.41421356,0) {};
\node [draw,rectangle,inner sep=0cm,minimum height=0.2cm,minimum width=0cm, label={[label distance=-0.1cm]above:{\tiny$+(\sqrt{2}-1)$}},label distance=-0.1cm] at (+0.41421356,0) {};
\node [draw,rectangle,inner sep=0cm,minimum width=0.2cm,minimum height=0cm, label={[label distance=-0.1cm]right:{\tiny$+z_2$}},label distance=-0.1cm] at (0,0.5) {};
\node [draw,rectangle,inner sep=0cm,minimum width=0.2cm,minimum height=0cm, label={[label distance=-0.1cm]right:{\tiny$-z_2$}},label distance=-0.1cm] at (0,-0.5) {};
\begin{scope}[xshift=3.5cm]
\node (a) [draw,fill,circle,inner sep=0cm,minimum height=0.15cm] at (-1,-0.5,0.75) {};
\node (b) [draw,fill,circle,inner sep=0cm,minimum height=0.05cm] at (-0.41421356,-0.5,0.75) {};
\node (c) [draw,fill,circle,inner sep=0cm,minimum height=0.05cm] at (+0.41421356,-0.5,0.75) {};
\node (d) [draw,fill,circle,inner sep=0cm,minimum height=0.15cm] at (+1,-0.5,0.75) {};
\node (e) [draw,fill,circle,inner sep=0cm,minimum height=0.15cm] at (-1,0.5,0.75) {};
\node (f) [draw,fill,circle,inner sep=0cm,minimum height=0.05cm] at (-0.41421356,0.5,0.75) {};
\node (g) [draw,fill,circle,inner sep=0cm,minimum height=0.05cm] at (+0.41421356,0.5,0.75) {};
\node (h) [draw,fill,circle,inner sep=0cm,minimum height=0.15cm] at (+1,0.5,0.75) {};
\node (aa) [draw,fill,circle,inner sep=0cm,minimum height=0.15cm,gray] at (-1,-0.5,-0.75) {};
\node (bb) [draw,fill,circle,inner sep=0cm,minimum height=0.05cm,gray] at (-0.41421356,-0.5,-0.75) {};
\node (cc)[draw,fill,circle,inner sep=0cm,minimum height=0.05cm,gray] at (+0.41421356,-0.5,-0.75) {};
\node (dd) [draw,fill,circle,inner sep=0cm,minimum height=0.15cm,gray] at (+1,-0.5,-0.75) {};
\node (ee) [draw,fill,circle,inner sep=0cm,minimum height=0.15cm,gray] at (-1,0.5,-0.75) {};
\node (ff) [draw,fill,circle,inner sep=0cm,minimum height=0.05cm,gray] at (-0.41421356,0.5,-0.75) {};
\node (gg) [draw,fill,circle,inner sep=0cm,minimum height=0.05cm,gray] at (+0.41421356,0.5,-0.75) {};
\node (hh) [draw,fill,circle,inner sep=0cm,minimum height=0.15cm,gray] at (+1,0.5,-0.75) {};
\draw [dashed,thin] (a) -- (aa);
\draw [dashed,thin] (b) -- (bb);
\draw [dashed,thin] (c) -- (cc);
\draw [dashed,thin] (d) -- (dd);
\draw [dashed,thin] (e) -- (ee);
\draw [dashed,thin] (f) -- (ff);
\draw [dashed,thin] (g) -- (gg);
\draw [dashed,thin] (h) -- (hh);
\draw [<->] (0,0.5,-0.75) to node [fill=white,circle] {$2z_3$} (0,0.5,0.75);
\end{scope}
\end{tikzpicture}
\caption{Point set $P_d$ from the family of worst-case examples, drawn for $d=2$ and $d=3$. The heavy points are drawn larger.\label{fig:lowerbound}}
\end{figure*}
We show the following theorem.
\begin{theorem}\label{thm:ExponentialLowerBound}
The family of point sets $(P_d)_{d\in \mathbb{N}}$ satisfies $\mathrm{Ward}_{k}(P_d) \in \Omega((3/2)^d\cdot \mathrm{\opt}_{k}(P_d))$ for~$k=2^d$.
\end{theorem}
In the theorem, we use~$k=k(d)=2^d$, i.e., we are interested in finding a $2^{d}$-clustering of~$P_d$. Observe that in the optimal $2^{d}$-clustering of~$P_d$, the heavy points are in separate clusters. Due to their infinite weight, they also determine the cluster centers. Hence, in the optimal solution each light point is in the same cluster as its closest heavy point. Since each light point is within distance~$2-\sqrt{2}$ of a heavy point, the cost of the optimal solution is
\[
\mathrm{\opt}_k(P_d) = 2^{d}\cdot (2-\sqrt{2})^2.
\]
Now we look at a run of Ward's method on~$P_d$. We say that phase~$1$ lasts as long as there is at least one light point that forms its own cluster. We prove by induction that during phase~$1$ the only clusters that occur are singleton clusters consisting of one light or one heavy point and clusters that consist of two light points that differ only in the first coordinate. We call the latter \emph{pair clusters}. At the beginning this is clearly the case. Now assume that the induction hypothesis holds at some point of time in phase~$1$. Merging two heavy points has infinite cost and merging a heavy point with a light point or a pair cluster has cost at least~$(2-\sqrt{2})^2\approx 0.343$ because~$2-\sqrt{2}$ is the minimum distance between a light and a heavy point. Merging two singleton light points that differ only in the first coordinate costs~$\frac12\cdot(2\sqrt{2}-2)^2=(2-\sqrt{2})^2$ (observe that the induction hypothesis guarantees that for any singleton light point the light point that differs only in the first coordinate is also a singleton point). Merging two singleton light points that differ in any other coordinate costs at least~$\frac1{1+1}\cdot (2z_2)^2=1$, merging a singleton light point with a pair cluster costs at least~$\frac{1\cdot 2}{1+2}\cdot (2z_2)^2=\frac43$, and merging two pair clusters costs at least~$\frac{2\cdot 2}{2+2}\cdot (2z_2)^2=2$. Hence, we can assume that Ward merges two singleton light points that differ only in the first coordinate. After that the induction hypothesis is still true. Hence, in phase~$1$ all $2^{d-1}$ pairs of points of the form~$(-(\sqrt{2}-1),x_2,\ldots,x_d)$ and~$(\sqrt{2}-1,x_2,\ldots,x_d)$ will be merged. We call the clusters that consist of these points the $(*,x_2,\ldots,x_d)$-clusters in the following.
Then phase~$2$ starts. Phase~$2$ lasts as long as there are pair clusters. We show by induction that the only clusters that occur in phase~$2$ are singleton heavy points, pair clusters, and clusters with four points that result from merging two pair clusters that differ only in the second coordinate. We call the latter \emph{quadruple clusters}. Merging two pair clusters of the form $(*,-z_2,x_3,\ldots,x_d)$ and $(*,z_2,x_3,\ldots,x_d)$ to form a quadruple cluster costs~$\frac{2\cdot 2}{2+2}(2z_2)^2=2$. Merging two pair clusters that differ in any other coordinate than the second is more expensive because their centers are further apart than~$2z_2$. Merging the $(*,x_2,\ldots,x_d)$-cluster with a heavy point costs at least~$2$ because the center of this cluster is~$(0,x_2,\ldots,x_d)$, which is at distance~$1$ from the heavy points. Similarly merging a quadruple cluster (whose center is~$(0,0,x_3,\ldots,x_d)$) with a heavy point costs at least~$2+z_2^2\ge 2$. Merging a quadruple cluster with a pair cluster costs at least~$\frac{2\cdot 4}{2+4}(2z_3)^3>2$ and merging two quadruple clusters costs at least~$\frac{4\cdot 4}{4+4}(2z_3)^3>2$. Hence, we can assume that Ward merges two pair clusters that differ only in the second coordinate. After that the induction hypothesis is still true. Hence, in phase~$2$ all $2^{d-2}$ pairs of clusters of the form~$(*,-z_2,x_3,\ldots,x_d)$ and~$(*,z_2,x_3,\ldots,x_d)$ will be merged. We call the clusters that consist of these points the $(*,*,x_3,\ldots,x_d)$-clusters in the following.
At the beginning of phase~$i\ge 2$, there are~$2^d$ singleton heavy points and~$2^{d-i+1}$ clusters of the form $(*,\ldots,*,x_i,\ldots,x_d)$ with~$2^{i-1}$ points each. Phase~$i$ ends when there is no cluster of the form $(*,\ldots,*,x_i,\ldots,x_d)$ left. One can show again by induction that Ward merges in phase~$i$ all pairs of clusters of the form $(*,\ldots,*,-z_i,x_{i+1},\ldots,x_d)$ and $(*,\ldots,*,z_i,x_{i+1},\ldots,x_d)$. The center of the cluster~$(*,\ldots,*,x_i,\ldots,x_d)$ is~$(0,\ldots,0,x_{i},\ldots,x_d)$, which is at distance~$\sqrt{1+z_2^2+\ldots+z_{i-1}^2}$ from the heavy points. Hence, merging such a cluster with a heavy point costs at least~$2^{i-1}\cdot (1+z_2^2+\ldots+z_{i-1}^2) = 2^i z_i^2$, where the equation follows from the following observation.
\begin{restatable}{observation}{zireihe}
\label{obs:zireihe}
It holds that $\nonumber1+z_2^2+\ldots+z_{i-1}^2 = 2 z_i^2$.
\end{restatable}
Merging the clusters~$(*,\ldots,-z_i,x_{i+1},\ldots,x_d)$ and~$(*,\ldots,z_i,x_{i+1},\ldots,x_d)$ costs
\[
\frac{2^{i-1}\cdot 2^{i-1}}{2^{i-1}+2^{i-1}}\cdot (2z_i)^2 = 2^i z_i^2.
\]
Merging two clusters that differ in one of the~$d-i$ last coordinates costs at least~$\frac{2^{i-1}\cdot 2^{i-1}}{2^{i-1}+2^{i-1}}(2z_{i+1})^2=2^i\cdot z_{i+1}^2>2^iz_i^2$. Hence, in phase~i all $2^{d-i}$ pairs of clusters of the form~$(*,\ldots,*,-z_i,x_{i+1},\ldots,x_d)$ and~$(*,\ldots,*,z_i,x_{i+1},\ldots,x_d)$ will merge, which costs in total $2^{d-i}\cdot 2^i z_i^2$.
Phases $2$ until $d$ together cost $\sum_{i=2}^{d} 2^{d-i}\cdot 2^i z_i^2 = 2^{d}\cdot(2 z_{d+1}^2-1) = 2 \cdot 3^{d-1} - 2^d$, where we used Observation~\ref{obs:zireihe}. After phase~$d$, all light points will be in the same cluster. Then the number of clusters is~$2^d+1$ and in the last step the cluster of light points, whose center is the origin, will be merged with one heavy point. This costs
\[
2^d\cdot (1+z_2^2+\ldots+z_{d}^2) = 2^{d+1} \cdot z_{d+1}^2 = 2 \cdot 3^{d-1}.
\]
Phase $1$ costs in total $2^{d-1}(2-\sqrt{2})^2$.
Thus, the overall cost of Ward's solution is
\begin{align*}
\mathrm{Ward}_k(P_d) & = 2^{d-1}(2-\sqrt{2})^2 + 2 \cdot 3^{d-1} + 2 \cdot 3^{d-1} - 2^d\\
& = 4 \cdot 3^{d-1} + 2^{d-1}(2-\sqrt{2})^2 - 2^d.
\end{align*}
This implies
\begin{align*}
\frac{\mathrm{Ward}_k(P_d)}{\mathrm{\opt}_k(P_d)}
& = \frac{4 \cdot 3^{d-1} + 2^{d-1}(2-\sqrt{2})^2 - 2^d}{2^{d}\cdot (2-\sqrt{2})^2} \\
& = \frac{4}{3 (2-\sqrt{2})^2} \cdot \left(\frac{3}{2}\right)^d +\frac{1}{2} - \frac{1}{(2-\sqrt{2})^2}\\
& \in \Omega\left(\left(\frac{3}{2}\right)^d\right).
\end{align*}
\section{Ward on Well-Clusterable Data}\label{sec:WellClusterable}
Clustering suffers from a general gap between theoretical study and practical application; clustering objectives are usually NP-hard to optimize, and even NP-hard to approximate to arbitrary precision. On the other hand, heuristics like Lloyd's algorithm, which can produce arbitrarily bad solutions, are known to work well or reasonably well in practice. One way of interpreting this situation is that data often has properties that make the problem computationally easier. Indeed, for clustering it is very natural to assume that the data has some structure -- otherwise, what do we hope to achieve with our clustering? The challenge is to find good measures of structure that characterize what makes clustering easy (but non-trivial).
Many notions of \emph{clusterability} have been introduced in the literature and there are also different ways to measure the quality of a clustering. While traditionally a clustering is evaluated on the basis of an objective function (e.g., the $k$-means objective function), there has been an increased interest recently to study which notions of clusterability make it feasible to recover (partially) a \emph{target clustering}, some \emph{true} clustering of the data. For this, the niceness conditions imposed on the input data are usually some form of separation condition on the clusters of the target clustering. We study the effect of five well-studied clusterability notions on the quality of the solution computed by Ward's method.
First we study the notions of \emph{$\delta$-center separation} and \emph{$\alpha$-center proximity}, which have been introduced by Ben-David and Haghtalab~\cite{Ben-DavidH14} and Awasthi, Blum, and Sheffet~\cite{ABS12}, respectively.
\begin{definition}[\cite{Ben-DavidH14}]
An input $P \subset \mathbb{R}^d$ satisfies \emph{$\delta$-center separation} with respect to some target clustering $C_1,\ldots,C_k$ if there exist centers $c_1^\ast,\ldots,c_k^\ast \in \mathbb{R}^d$ such that
$||c_j^\ast - c_i^\ast|| \ge \delta \cdot \max_{\ell\in [k]} \max_{x\in C_\ell} ||x-c_{\ell}^\ast||$
for all $i \neq j$.
We say the input satisfies \emph{weak $\delta$-center separation} if for each cluster $C_j$ with $j \in [k]$ and for all $i \neq j$, $||c_j^\ast - c_i^\ast|| \geq \delta \cdot \max_{x\in C_j} ||x-c_j^\ast||$.
\end{definition}
Kushagra, Samadi, and Ben-David~\cite{KSB16} show that single linkage and a pruning technique are sufficient to find the target clustering under the condition that the data satisfies $\delta$-center separation for $\delta \ge 3$.
While the goal of Ben-David and Haghtalab~\cite{Ben-DavidH14} is to recover a target clustering, we focus in this paper on approximating the $k$-means objective function. Hence, in the following we will always assume that the target clustering $C_1,\ldots,C_k$ is an optimal $k$-means clustering (which we usually denote by~$O_1,\ldots,O_k$) and the centers $c_1^\ast,\ldots,c_k^\ast \in \mathbb{R}^d$ are the optimal $k$-means centers for this clustering. We will make this assumption also for all other notions of clusterability that are based on a target clustering and that we introduce in the following.
\begin{definition}[\cite{ABS12}]
An instance $P$ satisfies \emph{$\alpha$-center proximity} if there exists an optimal $k$-means clustering $O_1,\ldots,O_k$ with centers $c_1^\ast,\ldots,c_k^\ast \in \mathbb{R}^d$ such that for all $j \neq i, j \in [k]$ and for any point $x \in C_i$ it holds $||x-c_j^\ast|| \ge \alpha ||x-c_i^\ast||$.
\end{definition}
Awasthi, Blum, Sheffet~\cite{ABS12} introduced the notion of \emph{$\alpha$-perturbation resilience} and showed that it implies $\alpha$-center proximity. They show that for $\alpha\ge 3$, the optimal clustering can be recovered if the data is $\alpha$-perturbation resilient. This was improved by Balcan and Liang~\cite{BL16} and finally by Makarychev and Makarychev~\cite{MM16}, who show that it is possible to completely recover the optimal clustering for $\alpha=2$. The latter paper shows that the results even hold for a weaker property called \emph{metric perturbation resilience}. We show that for large enough~$\delta$ and~$\alpha$, Ward's method computes a $2$-approximation if the data satisfies $\delta$-center separation or $\alpha$-center proximity.
\begin{theorem}
Let $P \subset \mathbb{R}^d$ be an instance that satisfies weak $(2+2\sqrt 2+\epsilon)$-center separation or $(3+2\sqrt 2+ \epsilon)$-center proximity for some $\epsilon>0$ and some number~$k$ of clusters. Then the $k$-clustering computed by Ward on~$P$ is a $2$-approximation with respect to the $k$-means objective function.
\end{theorem}
We also show that on instances that satisfy $(2+2\sqrt{2 \nu}+\epsilon)$-center separation and for which all clusters~$O_i$ and~$O_j$ in the optimal clustering satisfy~$|O_j|\ge |O_i|/\nu$, Ward even recovers the optimal clustering.
It is interesting to note that the example proposed by Arthur and Vassilvitskii~\cite{AV07} that shows that the famous $k$-means++ algorithm has an approximation ratio of $\Omega(\log k)$ satisfies $\delta$-center separation and $\alpha$-center proximity for large values of~$\delta$ and~$\alpha$, and has balanced clusters, i.e., $\nu=1$.
\begin{observation}
There is a family of examples where $k$-means++ has an expected approximation ratio of $\Omega(\log k)$, while Ward computes an optimal solution.
\end{observation}
In contrast we will see that the instances that we use to prove our exponential lower bound on the approximation factor of Ward's method (Theorem~\ref{thm:ExponentialLowerBound}) satisfy $\delta$-center separation and $\alpha$-center proximity for~$\delta\le 1+\sqrt{2}$ and~$\alpha\le 1+\sqrt{2}$. We will also see that even for arbitrary large~$\delta$ and~$\alpha$ there are instances that satisfy $\delta$-center separation and $\alpha$-center proximity and on which Ward's method does not compute an optimal solution.
In addition to center separation and center proximity we study the following three other prominent notions of clusterability: the strict separation property due to Balcan, Blum, and Vempala~\cite{BBV08}, $\epsilon$-separation due to Ostrovsky et al.~\cite{ORSS12}, and the separation condition from Awasthi and Sheffet~\cite{AS12}
We will see that the exponential lower bound instances satisfy these clusterability notions when the target clustering is the optimal $k$-means clustering. Hence, none of these notion guarantees that Ward's method computes a good clustering.
\begin{corollary}
For any $\epsilon > 0$, there is a family of point sets $(P_d)_{d\in \mathbb{N}}$ with~$P_d\subset\RR^d$ that are $\epsilon$-separated and that satisfy $1+\sqrt{2}$-center separation, $1+\sqrt{2}$-center proximity, the strict separation property and the AS-center separation property where $\mathrm{Ward}_{k}(P_d) \in \Omega((3/2)^d\cdot \mathrm{\opt}_{k}(P_d))$ for~$k=2^d$.
Furthermore, for any $\delta > 1$ and any $\alpha > 1$, there exists a point set that satisfies $\delta$-center separation and $\alpha$-center proximity and for which Ward does not compute an optimal solution.
\end{corollary}
\subsection{Upper Bounds\label{sec:positive-results}}
In this section, we analyze the behavior of Ward on $\delta$-center separated instances and instances that satisfy $\alpha$-center proximity for some number~$k$ of clusters. We are only interested in the $k$-clustering computed by Ward. Hence, in the following we assume that $k$ is fixed and that Ward stops as soon as it has obtained a $k$-clustering. First we prove that center proximity implies weak center separation. Hence, it suffices to study instances that satisfy weak center separation. Deferred proofs can be found in the full version of this paper.
\begin{restatable}{lemma}{RelationProximitySeparation}\label{lemma:RelationProximitySeparation}
Let~$P\subset\RR^d$ be an instance that satisfies $\alpha$-center proximity. Then $P$ also satisfies weak $(\alpha-1)$-center separation.
\end{restatable}
In the following we call a cluster $A$ that is formed by Ward an \emph{inner cluster} if $A$ is completely contained within an optimum cluster. We start our analysis with the following lemma, which states one very crucial property of Ward's behavior on well-separated data. It implies that Ward does not merge inner clusters from two different optimal clusters as long as there exists more than one inner cluster in both of these optimal clusters.
\begin{restatable}{lemma}{onedeletion}\label{lem:onedeletion}
Let $P \subset \mathbb{R}^d$ be an instance that satisfies weak $(2+2\sqrt 2+ \epsilon)$-center separation for some $\epsilon>0$. Assume we have two optimal clusters $O_1$ and $O_2$ and each of them contains at least two inner clusters $A_1,B_1$ and $A_2,B_2$, respectively, directly after the $i$-th step of Ward. Then, in step~$i+1$, Ward will not merge an inner cluster of $O_1$ with an inner cluster of $O_2$.
\end{restatable}
\paragraph*{Inner-cluster merges}
In the following, assume that~$P\subset\RR^d$ is an arbitrary instance and and that the clusters $O_1,\dots,O_k$ are an optimal $k$-clustering of~$P$ with objective value~$\opt=\opt_k(P)$. Our goal is to show that the $k$-clustering~$W_1,\ldots,W_k$ computed by Ward on~$P$ is worse by only a factor of at most~$2$ if $P$ satisfies weak $(2+2\sqrt 2+ \epsilon)$-center separation for some $\epsilon>0$.
Observe that Lemma~\ref{lem:onedeletion} does not exclude the possibility that Ward performs inner-cluster merges on~$P$, i.e., it might merge two inner clusters from the same optimum cluster at some point during its execution. While we will see that in the one-dimensional case one can assume that such inner-cluster merges do not happen, we cannot make this assumption in general.
In our analysis, we bound the costs of the inner-cluster merges separately from the costs of the other merges, which we call \emph{non-inner merges} in the following.
We define an equivalence relation $r$ on $P$ as follows: two points $x_1$ and $x_2 \in P$ are equivalent if and only if there exists an inner cluster $C$ constructed by Ward at some point of time with $x_1,x_2 \in C$. We denote the equivalence classes of~$r$ by $P/r= \lbrace C_1, \dots, C_m \rbrace$. The following observation is immediate.
\begin{observation}\label{obs:fullclass}
If Ward merges in any step an inner cluster $C$ with another cluster that is not an inner cluster of the same optimal cluster, then $C \in P/r$ is an equivalence class.
\end{observation}
This means that the equivalence classes represent inner clusters of Ward right before they are merged with points from outside their optimal cluster. With other words, if we perform all inner cluster merges that are performed by Ward and leave out all non-inner merges, we get the clustering represented by~$P/r$.
Consider an arbitrary optimal cluster~$O_j$ and let~$P_1^j,\ldots,P_{n_j}^j$ denote the inner clusters of~$O_j$ in~$P/r$. We assume that these inner clusters are indexed in the order in which they are merged with other clusters by Ward. To illustrate this definition, consider the step in which~$P_i^j$ is merged by Ward with some other cluster~$Q$. Since~$P_i^j\in P/r$, this step is a non-inner merge and in particular~$Q$ is not equal to any of the clusters~$P_{i+1}^j,\ldots,P_{n_j}^j$. At the time this merge happens, the indexing guarantees that the cluster~$P_{i+1}^j$ is either present or there exist multiple parts~$C_1,\ldots,C_{\ell}$ of~$P_{i+1}^j$ that are only later merged by inner-cluster merges to~$P_{i+1}^j$. Since Ward merges~$P_i^j$ and~$Q$, we know that~$D(P_i^j,Q) \leq D(P_i^j,C_h)$ for any~$h\in[\ell]$. We will use this fact to give an upper bound for the costs of the clustering~$W_1,\ldots,W_k$.
It might be that some inner clusters of~$O_j$ in~$P/r$ are not merged at all by Ward and contained in the clustering~$W_1,\ldots,W_k$. These inner clusters are the last in the ordering, i.e., they are~$P_a^j,\ldots,P_{n_j}^j$ where~$n_j-a+1$ is the number of such clusters.
\paragraph*{Potential graph}
In order to bound the costs of the clustering~$W_1,\ldots,W_k$ produced by Ward we introduce the \emph{potential graph}~$G=(V,E)$ with vertex set~$V=P/r$. The edges~$E$ of~$G$ are directed and there are only edges between inner clusters of the same optimal cluster. Consider an arbitrary optimal cluster $O_j$ with $j \in [k]$ and let $P_1^j \ldots P_{n_j}^j$ be the inner clusters of~$O_j$ in~$P/r$ indexed as above in the order in which they are merged with other clusters by Ward. Then for every~$i\in[n_j-1]$ the set~$E$ contains the edge~$(P_i^j,P_{i+1}^j)$. Both the vertices and the edges are weighted and we denote the sum of all vertex and edge weights by $w(G)$.
The weight of a vertex~$Q\in P/r$ is defined as $w(Q)=\Delta(Q)$, i.e., the weight of vertex~$Q$ equals the costs of forming the inner cluster~$Q$. We will now define weights for the edges such that the sum of all vertex and edge weights in the potential graph is at most~$2\opt_k$. After that we prove that there is a one-to-one correspondence between the non-inner merges of Ward and the edges in the graph such that the costs of each non-inner merge of Ward are at most the weight of the associated edge. Together this proves that Ward computes a solution with costs at most~$2\opt_k$.
To define the weight of the edge~$(P_i^j,P_{i+1}^j)$, we first consider the case that~$P_i^j$ is merged at some point of time with another cluster~$Q$ by Ward. Then let~$C_1,\ldots,C_{\ell}$ again denote the parts of~$P_{i+1}^j$ that are present at that point of time. The edge weight~$w(P_i^j,P_{i+1}^j)$ is defined as~$\max_{h\in[\ell]}D(P_i^j,C_h)$\footnote{When reading the proof the reader might notice that our definition of $w(P_i^j,P_{i+1}^j)$ is to some extend arbitrary. Instead of defining it as~$\max_{h\in[\ell]}D(P_i^j,C_h)$, we could also define it as~$\min_{h\in[\ell]}D(P_i^j,C_h)$ or as $D(P_i^j,C_h)$ for any~$h$.}. Observe that since Ward performs greedy merges, this definition guarantees that the merge of~$P_i^j$ and~$Q$ costs at most the edge weight~$w(P_i^j,P_{i+1}^j)$. If~$P_i^j$ is not merged at all by Ward, we set the weight~$w(P_i^j,P_{i+1}^j)$ to~$D(P_i^j,P_{i+1}^j)$.
\begin{restatable}{lemma}{DeltaPartition}\label{lemma:DeltaPartition}
Let~$P\subset\RR^d$ be a finite point set and let~$Q_1,\ldots,Q_{\ell}$ denote an arbitrary partition of~$P$ into pairwise disjoint parts. Then
$\Delta(P) \ge \Delta(Q_1)+\ldots+\Delta(Q_{\ell})$.
\end{restatable}
\fixtoomuchspaceerror
\begin{restatable}{lemma}{TotalWeightPotentialGraph}\label{lem:TotalWeightPotentialGraph}
The weights in the potential graph satisfy~$w(G)\le 2\opt_k$.
\end{restatable}
\paragraph*{Bijection between non-inner merges and edges}
We have seen that the sum of the weights in the potential graph is at most~$2\opt_k$. Our goal is now to find a bijection between the non-inner merges of Ward and the edges of the potential graph such that the costs of any non-inner merge are bounded from above by the weight of the edge assigned to it in the bijection. The existence of such a bijection implies that also the costs of the solution~$W_1,\ldots,W_k$ computed by Ward are at most~$2\opt_k$.
Now we construct this bijection. Let us first consider non-inner merges in which at least one of the clusters is an inner cluster contained in~$P/r$. Let this be the inner cluster~$P_i^j$ of some optimal cluster~$O_j$ and assume further that~$i<n_j$. Then~$P_i^j$ has an outgoing edge to~$P_{i+1}^j$. We denote by~$Q$ the cluster with which~$P_i^j$ is merged and we assign the merge of~$P_i^j$ with~$Q$ to the edge~$(P_i^j,P_{i+1}^j)$ in the bijection.
\begin{restatable}{lemma}{onedeletiontwo}
\label{lem:onedeletiontwo}
Let $P \subset \mathbb{R}^d$ be an instance that satisfies weak $(2+2\sqrt 2+ \epsilon)$-center separation for some $\epsilon>0$. Consider a non-inner merge of Ward between two inner clusters from~$P/r$. Then at most one of these inner clusters has an outgoing edge in~$G$.
\end{restatable}
Observe that it cannot happen that the same edge is assigned to two different merges by the construction described above because an edge~$(P_i^j,P_{i+1}^j)$ can only be assigned to a step in which~$P_i^j$ is merged with some other cluster and there can only be one such merge.
Let~$L\subseteq E$ denote the set of edges that are not assigned to a step of Ward by the above construction. The potential graph~$G$ contains~$|V|=|P/r|$ vertices and~$|V|-k$ edges. Since the number of non-inner merges of Ward is also~$|V|-k$, there are also~$|L|$ non-inner merges that are not yet assigned to an edge. We finish the construction of the bijection by assigning the unassigned non-inner merges arbitrarily bijectively to~$L$.
\begin{restatable}{lemma}{NonInnerBijection}\label{lemma:NonInnerBijection}
The costs of each non-inner merge of Ward are bounded from above by the weight of the assigned edge in the potential graph.
\end{restatable}
Now the following theorem follows easily.
\begin{restatable}{theorem}{TwoApproximation}
\label{thm:2Approximation}
Let $P \subset \mathbb{R}^d$ be an instance that satisfies weak $(2+2\sqrt 2+ \epsilon)$-center separation or $(3+2\sqrt 2+ \epsilon)$-center proximity for some $\epsilon>0$. Then Ward computes a $2$-approximation on $P$.
\end{restatable}
\fixtoomuchspaceerror
\begin{restatable}{theorem}{wardoptimalitycriterion}
\label{thm:ward-optimality-criterion}
Let $P \subset \mathbb{R}^d$ be an instance with optimal $k$-means clustering $O_1,\ldots,O_k$ with centers $c_1^\ast,\ldots,c_k^\ast \in \mathbb{R}^d$. Assume that~$P$ satisfies $(2+2\sqrt{2 \nu}+\epsilon)$-center separation for some $\epsilon > 0$, where
$\nu = \max_{i,j \in [k]} \frac{|O_i|}{|O_j|}$
is the largest factor between the sizes of any two optimum clusters.
Then Ward computes the optimal $k$-means clustering $O_1,\ldots,O_k$.
\end{restatable}
In the full version of this paper we show that Theorem~\ref{thm:2Approximation} does not hold for significantly smaller~$\delta$ and~$\alpha$.
\section{Introduction}
Clustering is a fundamental tool in machine learning. As an unsupervised learning method, it provides an easy way to gain insight into the structure of data without the need for expert knowledge to start with. One of the most popular clustering objectives is $k$-means: Given a set $P$ of points in the Euclidean space~$\mathbb{R}^d$, find $k$ centers that minimize the sum of the squared distances of each point in $P$ to its closest center. The objective is also called \emph{sum of squared errors}, since the centers can serve as representatives, and then the sum of the squared distances becomes the squared error of this representation.
Theory has focused on metric objective functions for a long time: Facility location or $k$-median are very well understood, with upper and lower bounds on the best possible approximation guarantee slowly approaching one another. The $k$-means cost function is arguably more popular in practice, yet its theoretical properties were long not the topic of much analysis. In the last decade, considerable efforts have been made to close this gap.
We now know that $k$-means is NP-hard, even in the plane~\cite{MNV09} and also even for two centers~\cite{ADHP09}. The problem is also APX-hard~\cite{ACKS15}, and the currently best approximation algorithm achieves an approximation ratio of 6.357~\cite{ANSW17}. The best lower bound, though, is only 1.0013~\cite{LSW17}. A seminal paper on $k$-means is the derivation of a practical approximation algorithm, $k$-means++, which is as fast as the most popular heuristic for the problem (the local search algorithm due to Lloyd~\cite{L57}), has an upper bound of $\mathcal{O}(\log k)$ on the expected approximation ratio, and has proven to significantly improve the performance on actual data~\cite{AV07}. Due to its simplicity and superior performance, it (or variants of it) can now be seen as the de facto standard initialization for Lloyd's method.
From a practical point of view, however, there is still one major drawback of using $k$-means++ and Lloyd's method, and this has nothing to do with its approximation ratio or speed. Before using any method that strives to optimize $k$-means, one has to determine the number~$k$ of clusters. If one knows very little about the data at hand, then even this might pose a challenge. Indeed, there are several suggestions how to set $k$, which usually look at the tradeoff between the number of clusters and the cost (which decreases if the number of clusters is increased). For example, the \emph{elbow method} searches for a point where the cost decreases dramatically, arguing that this happens only at the point of the true number of clusters. However, there are many more methods to choose from (see for example the summary in §5 of~\cite{TWH00}). Notice that one usually needs to compute multiple clusterings for different $k$ to use such a method.
However, there is a simpler and popular method available: hierarchical clustering. Instead of computing clusterings for several different numbers of clusters and comparing them, one computes one clustering tree (a dendrogram), which contains a clustering for every value of $k$.
For any $k \in [n-1]$, the $k$-clustering in such a tree results from the $(k+1)$-clustering in the same tree by merging two clusters.
The hierarchical clustering does not only provide an answer for every $k$, it also allows the user to view the data at different levels of granularity.
A hierarchical clustering is apparently something very desirable, but the question is: Can the solutions be good for all values of $k$? Do we lose much by forcing the hierarchical structure?
Dasgupta and Long~\cite{DasguptaL05} were the first to give positive and negative answers to this question.
Their analysis evolves around the (metric) $k$-center problem, which is to minimize the maximum radius of any cluster.
They compare the $k$-center cost on each level of a hierarchical clustering to an optimal clustering with the best possible radius with the same number of clusters and look for the level with the worst factor. It turns out that popular heuristics for hierarchical clustering can be off by a factor of $\log k$ or even $k$ compared to an optimal clustering. Dasgupta and Long also propose a clever adaption of the $2$-approximation for $k$-center due to Gonz\'alez~\cite{G85}, which results in a hierarchical clustering algorithm. For this algorithm, they can guarantee that the solution is an $8$-approximation of the optimum on every level of the hierarchy simultaneously.
In a series of works, Mettu, and Plaxton~\cite{MP03}, Plaxton~\cite{P06} and finally Lin, Nagarajan, Rajaraman, and Williamson~\cite{LNRW10} develop and refine algorithms for the hierarchical $k$-median problem, which can be seen as the metric cousin of the hierarchical $k$-means problem. It consists of minimizing the sum of the distances of every point to its closest center, and is usually studied in metric spaces. The best known approximation guarantee is $20.06$.
However, the quality guarantee vastly deteriorates for $k$-means: An $\mathcal{O}(1)$-approximation for the hierarchical $k$-means problem follows from~\cite{P06,MP03} as well as from \cite{LNRW10}, but the approximation ratios range between $961$ and $3662$.
On the practical side, however, there is a long known greedy algorithm for the hierarchical $k$-means problem, named \emph{Ward's method}~\cite{W63}.
In the fashion of \emph{complete linkage} algorithms, it does the following.
It starts with singleton clusters, one for each data point from the input~$P\subset\RR^d$. Then it performs $|P|-1$ iterations where two clusters in the current clustering are merged (this is called \emph{agglomerative clustering}). In each iteration, it chooses the pair of clusters which results in the cheapest clustering. This is a locally optimal choice only, since the optimal merge in one operation may prove to be a poor choice with respect to a later level of the hierarchy.
To the best of the authors' knowledge, the worst-case quality of Ward's method has not been studied so far. In particular, it was not known whether the algorithm can be used to compute constant-factor approximations. We answer this question negatively by giving a family of examples with increasing $k$ and $d$ where the approximation factor of Ward is $\Omega((3/2)^d)$.
To explain the algorithms popularity, we then proceed to study it under different \emph{clusterability} assumptions.
Clustering problems are usually NP-hard and even APX-hard, yet clustering is routinely solved in practical applications. This discrepancy has led to the hypothesis that data sets are either easy to cluster, or they have little interesting structure to begin with. \lq Well-clusterable data sets are computationally easy to cluster\rq~\cite{B15} and \lq Clustering is difficult only when it does not matter\rq~\cite{DLS12} are two slogans summarizing this idea. Following it, many notions have been developed that strive to capture how well a data set is clusterable.
One such notion is \emph{center separation}~\cite{Ben-DavidH14}: A data set~$P\subset\RR^d$ is $\delta$-center separated for some number~$k$ of clusters if the distance between any pair of clusters in the target clustering is at least $\delta$ times the maximal radius of one of the clusters. It satisfies the similar \emph{$\alpha$-center proximity}~\cite{ABS12} for~$k$ if in the optimum $k$-clustering the distance of each data point to any center except for its own is larger by a factor of at least~$\alpha$ than the distance to its own center.
We apply these notions to hierarchical clustering by showing that if there is a well-separated optimum solution for a level, then the clustering computed by Ward on this level is a $2$-approximation. This means that Ward finds good clusterings for all levels of granularity that have a meaningful clustering; and these good clusterings have a hierarchical structure. For levels on which the sizes of the optimal clusters are additionally to some extend balanced, we prove that Ward even computes the optimum clustering.
\paragraph*{Related work.}
The design of hierarchical clustering algorithms that satisfy per-level guarantees started with the paper by Dasgupta and Long~\cite{DasguptaL05}. They give a deterministic $8$-approximation and a randomized $2e$-approximation for hierarchical $k$-center.
Their method turns Gonz\'alez' algorithm~\cite{G85} into a hierarchical clustering algorithm.
Gonz\'alez' algorithm is a $2$-approximation not only for $k$-center, but also for the \emph{incremental} $k$-center problem: Find an ordering of all points, such that for all $k$, the first $k$ points in the ordering approximately minimize the $k$-center cost.
The idea to make an algorithm for incremental clustering hierarchical was picked up by Plaxton~\cite{P06}, who proves that this approach leads to a constant factor approximation for the hierarchical $k$-median problem. He uses an incremental $k$-median algorithm due to Mettu and Plaxton~\cite{MP03}.
Finally, Lin, Nagarajan, Rajaraman and Williamson~\cite{LNRW10} propose a general framework for approximating incremental problems that also works for incremental variants of MST, vertex cover, and set cover. They also cast hierarchical $k$-median and $k$-means into their framework for incremental approximation. They get a randomized/deterministic $20.06/41.42$-approximation for hierarchical $k$-median and a randomized/deterministic $151.1\alpha / 576\alpha$-approximation for $k$-means, where $\alpha$ is the approximation ratio of a $k$-means approximation algorithm. Thus, applying~\cite{ANSW17} yields guarantees of $961$ and $3662$, respectively.
Lattanzi, Leonardi, Mirrokni, and Razenshteyn~\cite{LLMR15} develop a constant factor algorithm for \emph{robust} hierarchical $k$-center, i.e., a variant with outliers.
In a different line of work, Dasgupta recently developed a new cost function for similarity-based hierarchical clusterings~\cite{D16}. Although it can be transferred to the setting of dissimilarity measures, this yields an objective for which \emph{any} solution is a constant factor approximation~\cite{CKMM18}. Work on this new cost function includes~\cite{CC17,CKMM18,D16}. Balcan et al.\ present an algorithm for computing hierarchical clusterings that clusters the data accurately in the presence of outliers if the data satisfies certain clusterability properties~\cite{BBV08,BLG14}.
In practice, $k$-means and hierarchical $k$-means are rather tackled by popular heuristics, but the properties of these algorithms are often unknown.
The famous $k$-means algorithm due to Lloyd~\cite{L82} was analyzed about ten years ago and became the subject of many papers, including~\cite{AMR09,AV06,AV09,D03,MR09,ORSS12,V11}. This has led to the development of $k$-means++~\cite{AV07}, a practically efficient algorithm with a theoretical approximation guarantee of $\mathcal{O}(\log k)$.
Hierarchical clustering algorithms work either top-down (divisive methods) or bottom-up (agglomerative methods).
Agglomerative methods are more popular because they are usually faster, and the most popular agglomerative methods are based on the complete linkage strategy.
Here, the clusters to be merged are those which minimize the cost of the clustering in the next step.
Using complete linkage for $k$-means yields Ward's method~\cite{W63}.
There is a relatively small number of papers studying the performance of complete linkage algorithms. Dasgupta and Long~\cite{DasguptaL05} establish the above mentioned $\log k$ lower bound for $k$-center. Ackermann, Bl{\"{o}}mer, Kuntze, and Sohler~\cite{ABKS14} study complete linkage for variants of $k$-center \emph{in the Euclidean space}. The variants include minimizing the radius, the discrete radius and the diameter.
They show that for constant dimension, complete linkage provides $\mathcal{O}(\log k)$-approximations for $k$-center as well as all variants of it. The drawback is that the approximation factor depends on the the dimension of the space (the extent of the dependence goes from linear dependence to doubly exponential dependence, depending on the problem variant). Großwendt and Röglin~\cite{GR17} improve the analysis, showing that for constant dimension, complete linkage indeed provides an $\mathcal{O}(1)$-approximation. The dependencies on $d$ prevail.
Balcan, Liang, and Gupta~\cite{BLG14} observe that Ward's method cannot be used to recover a given target clustering.
There is a vast body of literature on clusterability assumptions, i.e., assumptions on the input that make clustering easier either in the sense that a target clustering can be (partially) recovered or that a good approximation of an objective function can be computed efficiently. A survey of recent work in this area can be found in~\cite{B15}. Particularly relevant for our paper are the notions of $\delta$-center separation~\cite{Ben-DavidH14} and $\alpha$-center proximity~\cite{ABS12} mentioned above.
There are several papers showing that under these assumptions it is possible to recover the target/optimal clustering if~$\delta$ and $\alpha$ are sufficiently large~\cite{ABS12,BL16,KSB16,MM16}. Other notions include the \emph{strict separation property} of Balcan, Blum, and Vempala~\cite{BBV08}, the \emph{$\epsilon$-separation property} of Ostrovsky et al.~\cite{ORSS12}, and the weaker version of the proximity condition due to Kumar and Kannan~\cite{KK10} which Awasthi and Sheffet~\cite{AS12} proposed (it is based on the spectral norm of a matrix whose rows are the difference vectors between the points in the data set and their centers). For all these notions of clusterability, algorithms are developed that (partially) recover the target/optimal clustering.
\paragraph*{Our results.}
In §\ref{sec:WellClusterable}, we analyze the approximation factor of Ward's method on data sets that satisfy different well-known clusterability notions. It turns out that the assumption that the input satisfies a high \emph{$\delta$-center separation}~\cite{Ben-DavidH14} or \emph{$\alpha$-center proximity}~\cite{ABS12} implies a very good bound on the approximation guarantee of Ward's method. We show that Ward's method computes a $2$-approximation for all values of~$k$ for which the input data set satisfies $(2+2\sqrt 2)$-center separation or $(3+2\sqrt 2)$-center proximity. We also show that on instances that satisfy $(2+2\sqrt{2 \nu})$-center separation and for which all clusters~$O_i$ and~$O_j$ in the optimal clustering satisfy~$|O_j|\ge |O_i|/\nu$, Ward even recovers the optimal clustering.
In §\ref{sec:explowerbound} we show that, in general, Ward's method does not achieve a constant approximation factor. We present a family of instances~$(P_d)_{d\in\NN}$ with~$P_d\subset\RR^d$ on which the cost of the $2^d$-clustering computed by Ward is larger than the cost of the optimal $2^d$-means clustering of~$P_d$ by a factor of~$\Omega((3/2)^d)$.
Then we observe that the family of instances used for this lower bound satisfy the \emph{strict separation property} of Balcan, Blum, and Vempala~\cite{BBV08}, the $\epsilon$-separation property of Ostrovsky et al.~\cite{ORSS12} for any~$\epsilon>0$, and the separation condition from Awasthi and Sheffet~\cite{AS12}. Hence, none of these three notions of clusterability helps Ward's method to avoid that the approximation factor grows exponentially with the dimension.
Finally in §\ref{section:ward:1d} we show that the approximation ratio of Ward's method on one-dimensional inputs is~$\mathcal{O}(1)$. The one-dimensional case turns out to be more tricky than one would expect, and our analysis is quite complex and technically challenging.
\paragraph*{Preliminaries.}
We consider inputs in the Euclidean space $\mathbb{R}^d$.
The Euclidean distance of $x_1, x_2 \in \mathbb{R}^d$ is denoted by $||x_1-x_2||=||x_1-x_2||_2$.
Let $P \subset \mathbb{R}^d$ be a finite set of points.
For any center $c \in \mathbb{R}^d$, we denote the sum of the squared distances of each point in $P$ to $c$ by
$\Delta (P,c)= \sum_{p \in P} ||p-c||^2.$
This sum is minimized when the center is the \emph{centroid} $\mu(P) := \frac{1}{|P|} \sum_{p \in P} p$ of $P$. We set $\Delta(P):=\Delta(P,\mu(P))$.
For any set of $k$ centers $C \subset \mathbb{R}^d$, the \emph{$k$-means objective cost} is
$
\Delta (P,C) = {\sum_{p \in P}} \min_{c \in C} ||p-c||^2.
$
The $1$-means cost of $P$ is $\Delta(P)$.
If $P$ is weighted with a weight function $w:P\to \mathbb{N}_{\ge 1}$, then we denote the \emph{total weight} by $w(P) := \sum_{x\in P} w(x)$ and extend the above notations by $\mu(P,w) = \frac{1}{w(P)} \sum_{x\in P} w(x) x$, $\Delta(P,w,c) = \sum_{x\in P} w(x) ||x-c||^2$, and $\Delta(P,w) = \Delta(P,w,\mu(P,w))$. The weighted $k$-means objective is $\Delta (P,w,C)=\sum_{x \in P} \min_{c \in C} w(x)||x-c||^2$. We denote by $\opt_k(P)$ / $\opt_k(P,w)$ the value of a solution that minimizes the (weighted) $k$-means objective, i.e., $\opt_k(P)=\min_{C\subset\RR^d,|C|=k}\Delta(P,C)$ and $\opt_k(P,w)=\min_{C\subset\RR^d,|C|=k}\Delta(P,w,C)$, respectively.
We use the abbreviation $[i] = \{1,\ldots,i\}$ for $i \in \mathbb{N}$.
\subparagraph*{Hierarchical clustering.}
As described by Dasgupta and Long~\cite{DasguptaL05}, a \emph{hierarchical clustering} is a nested partitioning of a point set $P$ into $1,2,3,\ldots$ and finally $n$ clusters, where each intermediate clustering is a more fine-grained version of the previous clustering that results from dividing one cluster into two.
This definition is \lq top-down\rq. Complete linkage algorithms
build the hierarchical clustering \lq bottom-up\rq\ by starting with $n$ singleton clusters and then subsequently merging two clusters into one until only one cluster remains. We will adapt this view and define a hierarchical clustering $\mathcal{H}$ as a sequence of partitionings $\mathcal{H}_0,\ldots,\mathcal{H}_{n-1}$, where $\mathcal{H}_0 = \{ \{x\} \mid x \in P\}$ and $\mathcal{H}_{n-1} = \{ P\}$, i.e., $\mathcal{H}_i$ shall be the clustering after $i$ merges. The intermediate partitionings satisfy that
$\mathcal{H}_i = \mathcal{H}_{i-1} \backslash \{A_i,B_i\} \cup \{A_i \cup B_i\}$
for two clusters $A_i, B_i \in \mathcal{H}_{i-1}$.
Note that we can fully describe $\mathcal{H}$ by the sequence of the $n-1$ \emph{merge operations}
$
(A_1,B_1),(A_2,B_2),\ldots,(A_{n-1},B_{n-1})
$
that it implicitly contains.
A hierarchical clustering contains a $k$-clustering for any $k \in \{1,\ldots,n\}$. The clusterings are given as partitionings, the centers are implicitly defined as the centroids. More precisely, the $k$-clustering defined by a hierarchical clustering $\mathcal{H}$ has the centers $\{ \mu(Q) \mid Q \in \mathcal{H}_{n-k}\}$. We thus define the $k$-means clustering cost of $\mathcal{H}$ for a given $k$ as
\[
\Delta_k(\mathcal{H}) = \sum_{Q \in \mathcal{H}_{n-k}} \Delta(Q,\mu(Q)) = \sum_{Q \in \mathcal{H}_{n-k}} \Delta(Q).
\]
\subparagraph*{Useful Facts about \texorpdfstring{$k$}{k}-means.}
The following two facts are well known.
\begin{lemma}[Relaxed triangle inequality]\label{lem:ti:two}
For all $x,y,z \in \RR^d$,
$
||x-y||^2 \le 2(||x-z||^2 + ||z-y||^2).
$
\end{lemma}
\begin{lemma}\label{magicformula}
For any finite point set $P \subset \mathbb R^d$ and any $c \in \mathbb R^d$, $\Delta (P,c) = \Delta(P)+|P|\cdot||c-\mu(P)||^2.$
\end{lemma}
Lemma~\ref{magicformula} has the following important consequence. Whenever a set of points $P'$ is clustered \emph{together}, i.e., all points in it are assigned to the same center in a given solution, then the cost for this assignment can be computed by knowing only the centroid of the point set and $\Delta(P')$.
Thus, we can treat such a $P'$ as one weighted point with some additional constant cost. This view is very helpful to simplify the analysis of agglomerative hierarchical clustering strategies.
\subparagraph*{Ward's method.}
\emph{Ward's method} (or simply \emph{Ward} in the following) is a greedy algorithm. To describe it, the easiest way is to define the following quantity that describes how much the sum of the $1$-means costs increases when merging two clusters.
\begin{definition}
Let $A, B \subset \mathbb{R}^d$ be two finite point sets. We define
$D(A,B)=\Delta(A \cup B) - \Delta(A) - \Delta(B)$.
If a set contains only one point, e.g., $A=\{a\}$, we slightly abuse notation and write $D(a,B) = D(\{a\},B)$ (similarly, if $A=\{a\}$ and $B=\{b\}$, we write $D(a,b)=D(\{a\},\{b\})$).
\end{definition}
Ward's method is agglomerative. It starts with $n$ singleton clusters. Then in every step, it greedily chooses two clusters $A, B$ in the current clustering for which $D(A,B)$ is minimal. This choice is optimal for the next clustering, but subsequent merges and clusterings may suffer from it. We denote the costs of the $k$-clustering computed by Ward's method on data set~$P$ by~$\mathrm{Ward}_k(P)$.
\section{Techniques and Observations}
\subsection{Upper Bounds: Proof Technique in a Nutshell}\label{subsec:ProofTechnique}
Let us give an overview of the basic idea underlying our proof that Ward's method computes a $2$-approximation for all values of~$k$ for which the input data set satisfies $(2+2\sqrt 2)$-center separation or $(3+2\sqrt 2)$-center proximity. The main challenge is to relate the cost of the $k$-clustering computed by Ward to the cost of an optimal $k$-clustering. For this, we fix an arbitrary optimal $k$-clustering~$O_1,\ldots,O_k$. Consider an arbitrary cluster~$O_j$ and let~$P_1^j,\ldots,P_{n_j}^j$ be the data points~$O_j$ consists of (in the actual proof, $P_i^j$ is defined slightly differently). We consider the set $\mathcal{S}_j=\{\{P_1^j,P_2^j\},\{P_2^j,P_3^j\},\ldots,\{P_{n_j-1}^j,P_{n_j}^j\}\}$ of merges. Observe that the merges in~$\mathcal{S}_j$ cannot be applied one after another because after the first merge~$\{P_1^j,P_2^j\}$ the singleton point~$P_2^j$ is gone, which is to be merged in the second merge~$\{P_2^j,P_3^j\}$. Since it is possible to do every second merge of $\mathcal{S}_j$, one can argue that all merges in~$\mathcal{S}_j$ together cost at most~$2\Delta(O_j)$. Now let~$\mathcal{S}=\cup_j \mathcal{S}_j$. Then all merges in~$\mathcal{S}$ together cost at most~$2\opt_k$.
The next step is then to construct a bijection between the set~$\mathcal{S}_{\mathrm{Ward}}$ of the~$n-k+1$ merges performed by Ward to form a $k$-clustering and the set~$\mathcal{S}$. This bijection has the property that every merge of Ward is at most as expensive as the merge in~$\mathcal{S}$ assigned to it. This implies that Ward computes a clustering with cost at most~$2\opt_k$. In order to construct this bijection, consider a step of Ward in which two clusters~$A$ and~$B$ are merged. Let~$\mathcal{C}$ denote the current clustering directly before this merge happens, and let~$\mathcal{S}_{\mathcal{C}}\subseteq\mathcal{S}$ denote the set of those merges from~$\mathcal{S}$ that are feasible in~$\mathcal{C}$ and unassigned, i.e., those merges for which both clusters are contained in~$\mathcal{C}$ and that have not been assigned to any previous merge of Ward. We know that any merge from~$\mathcal{S}_{\mathcal{C}}$ is at least as expensive as the merge of~$A$ and~$B$ because Ward chooses the next merge greedily. Hence, in the bijection we can map the merge of~$A$ and~$B$ to an arbitrary merge from~$\mathcal{S}_{\mathcal{C}}$. This implies that if~$\mathcal{S}_{\mathcal{C}}$ is non-empty in every step, the bijection can be constructed. Since~$|\mathcal{S}|=|\mathcal{S}_{\mathrm{Ward}}|$ this can only be guaranteed if every merge of Ward decreases the number of available merges in~$\mathcal{S}$ by only~1. One can show that this follows from the separation assumption.
For the one-dimensional case, the basic approach is similar. The main difference is that without separation, we can no longer guarantee that the number of available merges decreases by only~$1$ with every step of Ward. Indeed, the original set~$\mathcal{S}$ of good merges may be empty after $n-2k$ merges. To bound the cost of the remaining merge steps, we find a new set of (relatively) good merges, i.e., a set of merges whose costs can be bounded by a constant times~$\opt_k$. Again, this set may run dry, and we have to start again. Essentially, we show that after a constant number of \emph{phases} (Ward merges that are charged against a specific set of good merges), Ward has obtained a $k$-clustering.
Although the basic idea is similar, the technical implementation of the proof for $d=1$ is very different from our proof for well-clusterable data. Every time that Ward does not merge in a way compatible to the optimum clustering, we have to account for all possible consequences. Techniques like reordering help us to organize the proof. We also simplify the instance before the actual proof.
\subsection{Useful Statements}\label{techniques}
Here we discuss some of the technical statements which we feel may be of interest for future work. All omitted proofs in this section can be found in the full version of this paper.
\paragraph*{Cost of one step.}
The value $D(A,B)$ plays a central role in the analysis of Ward's method.
By using Lemma~\ref{magicformula}, it is easy to show that $D(A,B)$ does not depend on $\Delta(A)$ or $\Delta(B)$. The following
lemma gives an explicit formula, which leads to convenient upper and lower bounds. These bounds say that the cost of merging two clusters is roughly equivalent to assigning the points of the smaller cluster to the centroid of the larger cluster.
\begin{restatable}{lemma}{dabcomputation}\label{obsb}
Let $A$ and $B$ be two clusters. Then
$D(A,B)=\frac{|A||B|}{|A|+|B|} \cdot ||\mu_A-\mu_B||^2$.
Furthermore,
$\frac{1}{2} \cdot\min\{|A|,|B|\} \cdot ||\mu_A-\mu_B||^2 \le D(A,B) \le \min\{|A|,|B|\} \cdot ||\mu_A-\mu_B||^2.$
The left hand side is attained for $|A|=|B|$, and the right hand side for $\frac{\max\{|A|,|B|\}}{\min\{|A|,|B|\}} \to \infty $.
\end{restatable}
\paragraph*{How cost accumulates.}
Notice that whenever Ward makes a decision, it is optimal for the clustering in the next step. Where does its error lie? The problem is that every merge forces the points of two clusters to be in the same cluster for any clustering to come. In later clusterings, the condition to cluster certain points together may induce error. We need a way to bound this error. We prove the following technical statement.
\begin{restatable}{corollary}{goodmergethree}\label{cor:goodmergewiththree}
Let $A$, $B$, and $C$ be three disjoint sets of points with $|A| \le |B|$ (or $w(A) \le w(B)$, for weighted sets). Then
$
\Delta(A\cup B\cup C)
\le \Delta(A) + 3 \cdot \Delta(B\cup C) + 4 \cdot D(A,B)
$
and
$
D(A\cup B, C) \le 3 \cdot \Delta(B\cup C) + 3 \cdot D(A,B) - \Delta(B) - \Delta(C)
$.
\end{restatable}
To see how Corollary~\ref{cor:goodmergewiththree} can be used, assume that $A \subset O_i$ and $B\subset O_j$ belong to different optimum clusters which Ward merged during its execution.
Now Corollary~\ref{cor:goodmergewiththree} tells us something about the compatibility of $A \cup B$ with the optimum clustering.
We pick the smaller of the two clusters, say $A$. Assume that we still have some subset of $B$'s optimum cluster, i.e., there is a cluster $C \subset O_j$ that is still part of the clustering. Then we can merge $A\cup B$ with $C$. Corollary~\ref{cor:goodmergewiththree} says that what we lose is proportional to the optimum cost plus the cost that we already invested into our clustering at an earlier time:
$\Delta(A)$ and $\Delta(B \cup C)$ are both part of the optimum cost, and $D(A,B)$ is what Ward (accumulatively) already payed for merging $A$ and $B$.
\paragraph*{Monotonicity.}
Notice that performing arbitrary merge operations is not monotone: Say that $a < b < c$ are one-dimensional points such that the centroid of $a$ and $c$ is $b$. Then merging $a$ and $c$ first results in a point set where merging with $b$ costs nothing; clearly, this is not monotone.
Indeed, when considering a natural variant of Ward's method for the related $k$-median problem, monotonicity is not true. Even for a simple isosceles triangle, greedily chosen merges result in non-monotone merge costs.
However, Ward's merges are indeed monotone. We show the following statement by proving a decomposition lemma for $D(A,B)$.
\begin{restatable}{corollary}{monotonicitycor}[Monotonicity of Ward's method]\label{cor:monotonicity}
Let $D_i$ be the increase of the objective function in the $i$-th step of Ward's method. Then $D_i \leq D_j$ for $i \leq j$.
\end{restatable}
Monotonicity is a very helpful property. In the argument discussed in §\ref{subsec:ProofTechnique} we use, e.g., that all merges that are possible in the final $k$-clustering computed by Ward's method are at least as expensive as all merges that are performed before by Ward's method to obtain the $k$-clustering.
\paragraph*{Special structures in dimension one.}
The following statements only hold for $d=1$. First we observe that Ward satisfies the following convexity property.
\begin{restatable}[Convexity in $\mathbb{R}^1$]{lemma}{convexlemma}\label{konvex}
For any three finite convex clusters $A, B, C \subset \mathbb{R}^1$ with $\mu(A) < \mu(C) < \mu(B)$, we have $D(A,C) < D(A,B)$ or $D(B,C)<D(A,B)$.
\end{restatable}
Lemma~\ref{konvex} means that Ward will never merge $A$ and $B$ if a point or cluster lies between them on the line. This establishes that Ward's clusters never overlap. It gives us a concept of neighbors on the line.
We combine Lemma~\ref{konvex} with a convexity property of Ward (see Corollary~\ref{subcluster}). This allows us to prove a powerful technique that we call \emph{reordering}.
Say that Ward at some point merges two clusters $A$ and $B$. Then $A$ and $B$ are neighbors on the line. This means that merging $A$ and $B$ will result in a centroid $\mu(A\cup B)$ which is further away from any other cluster than $\mu(A)$ and $\mu(B)$ are. So, clusters that did not want to merge with $A$ or $B$ would also not merge with $A\cup B$ (by Corollary~\ref{subcluster}). Thus, we could perform the merge $(A,B)$ \emph{earlier} without distorting Ward's course of action at all (except that the merge $(A,B)$ is at the wrong position). This allows us to reorder Ward's merges for our analysis.
\input{input_well_clusterable}
\input{input_exp_lowerbound}
\input{input_dim_one}
\section{Conclusions}
We have initiated the theoretical study of the approximation guarantee of Ward's method. In particular, we have shown that Ward computes a 2-approximation on well-separated instances, which can be seen as the first theoretical explanation for its popularity in applications. We have also seen that its worst-case approximation guarantee increases exponentially with the dimension of the input and that it computes an~$\mathcal{O}(1)$-approximation on one-dimensional instances.
These results leave room for further research. It would be particularly interesting to better understand the worst-case behavior of Ward's method. It is not clear, for example, if it computes a constant-factor approximation if the dimension is constant. Our analysis of the one-dimensional case is very complex and the factor hidden in the $\mathcal{O}$-notation is large. It would be interesting to simplify our analysis and to improve the approximation factor.
|
2,877,628,090,108 | arxiv | \section{Introduction}
Understanding the role of quantum fluctuations in frustrated antiferromagnets has been the focus of multiple studies over the last decades.~\cite{Misguich2005,Sachdev2008,Normand2009,Balents2010,Powell2011,Lacroix2010,Savary2017,Zhou2017} These efforts were originally motivated by the resonant valence bond (RVB) state proposed by P. W. Anderson for describing the ground state of the triangular antiferromagnetic (AF) Heisenberg model.\cite{Anderson1973,Fazekas1974} The RVB state is a linear superposition of different configurations of short range singlet pairs, a quantum spin liquid state, whose resonant character leads to the decay of spin-$1$
excitations into pairs of free spin-$1/2$ spinons. This strongly quantum mechanical scenario has no classical counterpart, given that semi-classical phases correspond to magnetically ordered states with integer spin-$1$ excitations known as magnons.\cite{Mattis1981}
While the semiclassical picture relies on the spin wave theory\cite{Mattis1981,Auerbach1994} (large-$S$ expansion), a systematic and controlled approach to the RVB picture can be formulated in the context of large-$N$ theories. Here the $SU(2)$ Heisenberg model is extended to a family of $SU(N)$ models, with $N$ being the number of {\it flavors} of a generalized spinor. In this formulation, the spin degree of freedom is represented by a product of spin-$\frac{1}{2}$ parton operators with bosonic (Schwinger) or fermionic (Abrikosov) character, subject to certain constraints.\cite{Baskaran87,Baskaran88,Arovas1988,Affleck1988,Read1991,Sachdev1991,Auerbach1994,Timm1998,Flint2009} The resulting Hamiltonian is expressed in terms of {\it isotropic} bond operators that emphasize the quantum nature of the bonds. The basic strategy is to describe the low-energy properties of the system, such as the dynamical spin susceptibility, by expanding the parameter $1/N$. The first term of the expansion corresponds to the saddle point (SP) approximation, which is equivalent to the mean field theory, consisting of a gas of free spin-$\frac{1}{2}$ spinons. The $1/N$ corrections introduce interactions between spinons mediated by emergent gauge fields.\cite{Arovas1988,Read1991,Sachdev1991,Auerbach1994,Chubukov1995} In the extreme $N\to \infty$ limit, the physics of free spin-$\frac{1}{2}$ spinons associated to the SP solution is exact; while the inclusion of $1/N$ corrections may drastically change the SP physics for finite $N$.
Although large-$N$ treatments were introduced to describe quantum spin liquid states,\cite{Read1991,Sachdev1991,Affleck1988} there is a renewed interest focused on the reliability of the parton method for describing the excitation spectrum of magnetically ordered states near a quantum melting point (QMP). This is mainly motivated by the increasing number of magnetically ordered quantum magnets whose excitation spectrum is not well described by a simple large-$S$ expansion.~\cite{Coldea2001,Coldea2003,Isakov05,ito2017structure,Ma2016,kamiya2018nature} In this context, the large-$N$ theory based on the Schwinger bosons (SB) representation is more adequate since, unlike the fermionic case, it can describe the magnetically ordered states through the condensation of the SBs.\cite{Hirsch1989,Sarker1989,Chandra1990} At the SP level, which is equivalent to the the Schwinger boson mean field theory (SBMFT), the dynamical spin susceptibility shows a two free-spinon continuum (branch cut) which misses the true collective modes (magnon) of the magnetically ordered state.\cite{Auerbach1988,Auerbach1994} The main signal of the magnetic spectrum is a pole located at the lower edge of the two-spinon continuum, that has the single-spinon dispersion. For collinear antiferromagnets and for a particular mean field decoupling of the Heisenberg term, this single-spinon dispersion accidentally coincides with the semiclassical linear spin wave result. This coincidence was originally interpreted as a general attribute of the SBMFT.~\cite{Auerbach1988} However, it was later recognized that the single-spinon band (low energy edge of the continuum) predicted by the SBMFT for non-collinear phases does not coincide with the single-magnon dispersion in the large-$S$ limit. This fact was interpreted as a strong failure of the SBMFT.~\cite{Chandra1991,Coleman1994} Motivated by this observation, we demonstrate in this paper that the LSWT result for the dynamical spin susceptibility is recovered in large-$S$ limit upon adding a $1/N$ correction to the SP or SBMFT.
For simplicity, we focus on the triangular lattice Heisenberg antiferromagnet with a $120^{\circ}$ N\'eel ground state ordering, whose quantum ($S=1/2$) magnetic excitation spectrum is very different from the semiclassical ($S \to \infty$ ) limit.\cite{zheng2006excitation,Chernyshev2009,Mourigal2013a}
We have recently computed the dynamical spin structure factor of the $S=1/2$ triangular lattice antiferromagnet by including $1/N$ corrections (Gaussian fluctuations) around the SP solution.~\cite{Ghioldi2018} The predicted excitation spectrum reveals a strong quantum character consistent with a magnetically ordered ground state in the proximity of a QMP. The low energy part of the spectrum consists of two-spinon bound states (magnons) induced by fluctuations of the gauge fields, that emerge as poles of the RPA propagator.
A crucial observation is that the main signal of the SP solution (pole at the lower edge of the two-spinon continuum) is exactly canceled by the $1/N$ correction and the remaining low-energy poles are the poles of the RPA propagator. In view of this result, it is not surprising that the poles of the SBMFT theory do not coincide with the poles of the linear spin wave theory (LSWT) in the large-$S$ limit.~\cite{Chernyshev2009,Zhitomirsky2013}
In other words, magnons (collective modes of the underlying magnetically ordered ground state) should not be identified with the poles that appear in the dynamical spin susceptibility {\it at the SP level} (lower edge of two-spinon continuum), but with the new poles (poles of the RPA propagator) that appear in the dynamical spin susceptibility upon adding higher order $1/N$ corrections. In Ref.~\onlinecite{Ghioldi2018} we demonstrated that, even for $S=1/2$ (quantum limit), the spin velocities of these poles basically coincide with the spin-wave velocities obtained from LSWT plus $1/S$ corrections.~\cite{Chubukov1994S,Chernyshev2009,Zhitomirsky2013} In this work we demonstrate these poles coincide over the full Brillouin zone with the ones obtained from LSWT in the $S \to \infty$ limit. Furthemore, the spectral weight of the magnon peaks predicted by LSWT is also exactly recovered by the SBMFT plus a $1/N$ correction.
The article is organized as follows: Sec.~\ref{Large-N} is a general introduction to the large-$N$ Schwinger boson theory for frustrated antiferromagnets. More specifically, we review the extension to $N>2$ that was proposed by Flint and Coleman,~\cite{Flint2009} by requiring that the generalized spin operators must preserve their transformation properties under
rotations and under the time reversal operation. Sec.~\ref{SP} describes the large-$N$ expansion of the extended theory around the SP solution. In Sec.~\ref{DSS} we present a formal $1/N$ expansion of the dynamical spin susceptibility. In particular, we discuss the four different Feynman diagrams that appear to order $1/N$. In Sec.~\ref{SU2} we fix $N=2$ to consider the excitation spectrum of
triangular lattice Heisenberg antiferromagnetic model, whose ground state is known to exhibit $120^{\circ}$ N\'eel order
and take the large-$S$ limit (for fixed $N$) of the SP solution and the higher order $1/N$ corrections.
The results of Sec.~\ref{SU2} are applied in Sec.~\ref{DSS_largeS} to demonstrate that the dynamical spin structure factor predicted by LSWT is exactly recovered when we add a particular $1/N$ correction (one of the four Feynman diagrams of Fig. \ref{fig2:feymann}) to the SP result. This is the $1/N$ correction that was recently included in Ref.~[\onlinecite{Ghioldi2018}].
We conclude the work in Sec.~\ref{Disc} with a general discussion of the implications of our result for other frustrated magnets.
\section{Large-N Schwinger boson theory for frustrated antiferromagnets
\label{Large-N}}
In this section we present the large-N Schwinger boson theory specialized for frustrated antiferromagnets within the time reversal (symplectic) scheme.~\cite{Flint2009} We start by considering the extended antiferromagnetic $SU(N)$ Heisenberg model on the triangular lattice
\begin{eqnarray}
{\cal H} = \frac{J}{N} \sum_{\pair{ij}} \vec{S}_i \cdot \vec{S}_j= \frac{J}{N} \sum_{\pair{ij}} S_{\alpha\beta}(i) S_{\beta\alpha}(j),
\label{Heis}
\end{eqnarray}
\noindent where $S_{\alpha\beta}=b^{\dagger}_{\alpha}b_{\beta}$ are $SU(N)$ spins with $\alpha \in \{1,....,N\}$, $b_{\alpha}$ are the generalized Schwinger bosons with $N$ different flavors,~\cite{Auerbach1994} and $J$ is rescaled by $N$ to make $\cal{H}$ extensive in the number of flavors $N$. Following Ref.~[\onlinecite{Flint2009}], we will request that the large-$N$ theory must preserve not only the invariance of the Hamiltonian under time reversal and spin rotations, but also the properties of the generalized spins under these transformations.
The generators of $SU(N)$ can be divided into even and odd under a time reversal transformation. The odd ones are the generators of the $Sp(N)$ subgroup of $SU(N)$.
In the physical case $N=2$, the isomorphism between $SU(2)$ and the simplectic $Sp(2)$ group implies that the three generators of $SU(2)$ must be odd under time reversal. The situation is different for $N > 2$ because the number of generators of $Sp(N)$ is smaller than the number of generators of $SU(N)$. The generators of $Sp(N)$ can be constructed by taking the antisymmetric combination between a generator $S_{\alpha \beta}$ of $SU(N)$ and its time reversed counterpart $ \sgn{\alpha} \sgn{\beta} S_{-\beta-\alpha}$ version,
\begin{equation}
\mathcal{S}_{\alpha\beta}=b^{\dagger}_{\alpha}b_{\beta}-\sgn{\alpha} \sgn{\beta} \; b^{\dagger}_{-\beta}b_{-\alpha},
\label{symplectic}
\end{equation}
where $N$ is assumed to be even and $\alpha$ has been redefined as $\alpha= \text{-}\frac{N}{2},...,\frac{N}{2}$. As shown in Ref.~[\onlinecite{Flint2009}], the Heisenberg interaction of the generalized symplectic spins turns out to be
\begin{equation}
\hat{\mathcal{S} }_{i} \cdot \hat{\mathcal{ S}}_{j} = \normal{{B}_{ij}^{\dagger} {B}_{ij}} - {A}_{ij}^{\dagger} {A}_{ij},
\label{heis}
\end{equation}
where
\begin{equation}
{A^{\dagger}}_{ij} = \frac{1}{2} \sum_{\alpha} \sgn{\alpha} \;b^{\dagger}_{i\alpha} b^{\dagger}_{j\alpha},
\;\;\;\;
B^{\dagger}_{ij} = \frac{1}{2} \sum_{\alpha} b^{\dagger}_{i\alpha} b_{j\alpha },
\label{bop}
\end{equation}
are $Sp(N)$ invariant bond operators. The bond operators $A^{\dagger}_{ij}$ create $Sp(N)$ singlets, while the
$B^{\dagger}_{ij}$ operators make them resonate. Furthermore, the Casimir operator of the symplectic spins is\cite{Flint2009} \\
\begin{equation}
\hat{\mathcal{ S}}^2_i= \frac{1}{4} n_{bi}(n_{bi}+N),
\end{equation}
with $n_{bi}=\sum_{\alpha} b^{\dagger}_{i\alpha} b_{i\alpha}$. The Casimir operator results from fixing $n_{bi}=NS$:
\begin{equation}
\hat{\mathcal{S}}^2_i= \frac{1}{4} N^2 S(S+1).
\end{equation}
It is worth stressing that Eq.~\eqref{heis} coincides with the two singlet bond structure of the $SU(2)$ Schwinger boson theory for N$=\!\!2$.~\cite{Ceccatto1993} In particular, for $S\!\!=\!\!\frac{1}{2} $, the condition of one Schwinger boson per site, $n_{bi}=2S=1$, is recovered through the Casimir operator for $N=2$. This two singlet bond structure is adequate to describe noncollinear magnetic orderings\cite{Manuel1998,Manuel1999} and to classify quantum spin liquid states with the projective symmetry groups.~\cite{Wang2006,Messio2013}\\
\section{Saddle point expansion}
\label{SP}
The partition function of the interacting symplectic spins can be expressed as a functional integral over coherent states,~\cite{Auerbach1994,Ghioldi2018}
\begin{multline}
\mathcal{Z} = \int D [\overline b,b]D[\lambda] \ e^{\! -\int_{0}^{\beta} d\tau \! \left[ \sum\limits_{i \alpha} \overline b_{i \alpha}^{\tau} \partial_{\tau} b_{i \alpha}^{\tau} + \ \mathcal{H}(\overline b,b) \ \right] } \\
\times e^{ -\int_{0}^{\beta} d\tau \ i \sum\limits_{i} \lambda_{i}^{\tau} \big(\sum\limits_{\alpha} \overline b_{i \alpha}^{\tau} b_{i \alpha}^{\tau} - NS \big) },
\label{partition}
\end{multline}
with a generalized spin Hamiltonian
\begin{equation}
\mathcal{H}=\frac{1}{2} \sum_{\langle i j \rangle} \frac{J_{ij}}{N} (\overline A_{ij}^{\tau } A_{ij}^{\tau} - \overline B_{ij}^{\tau} B_{ij}^{\tau} ).
\end{equation}
The integration measures are $ D[\overline b,b] = \prod_{i \tau \alpha} \frac{d\bar b _{i\alpha}^{\tau} db_{i\alpha}^{\tau}}{2 \pi i}$, and $ D[\lambda] = \prod_{i \tau} \frac{d\lambda_{i}^{\tau}}{2 \pi}$. The local constraint, $n_{bi}=NS$, is incorporated via integration over the time- ($\tau$) and space- ($i$) dependent auxiliary field $\lambda_{i}^{\tau}$.\\
After introducing the Hubbard-Stratonovich (HS) transformations that decouple the $\overline{A} A$ and $\overline{B}B$ terms,~\cite{Ghioldi2018} the partition function becomes
\begin{equation}
\mathcal{Z} = \int D[\overline W,W]D[\lambda] \ e^{-NS_{\rm eff}(\overline W, W, \lambda)},
\label{Seff}
\end{equation}
where the parameter $1/N$ plays the role of the Planck's constant in a semiclassical expansion.
$W\!\!=\!W^A,W^B$ are the space and time-dependent bond HS fields and the effective action is
\begin{eqnarray}
S_{\rm eff}(\overline W, W, \lambda) =\! & \int_{0}^{\beta} & \! \!d\tau \sum\limits_{ijr} \frac{1}{2J_{ij}} \overline W_{ij}^{r\tau} W_{ij}^{r\tau}\!-\!iS \sum_{i} \lambda_{i}^{\tau} \nonumber \\
&+& \frac{1}{N} \tr \ln \left[ \mathcal{G}^{-1}(\overline W, W, \lambda) \right].
\label{effective}
\end{eqnarray}
\noindent The integration measure of the HS fields is $D[\overline W^{}\!,\!W] = \prod_{ij\tau r} \frac{ d\overline {W}_{ij}^{r\tau} d W_{ij}^{r\tau}}{2 \pi i J_{ij}/N} $, with $r\!\!=\!\!A,B$, and
$\mathcal{G}^{-1}\!\!\!=\!\!\mathcal{M}$ is the bosonic dynamical matrix with the trace taken over space, time, and boson flavor indices. Notice that the integration measure dependence on $J_{ij}$ has changed with respect to Ref.~[\onlinecite{Ghioldi2018}] in order to keep the factor of $N$ in front of $S_{\rm eff}$ [see Eq.\eqref{Seff}].
The effective action \eqref{effective} is invariant under a $U(1)$ gauge transformation of the SBs and the auxiliary fields. The phase of the HS fields $\overline{W}, W,$ and the Lagrange multiplier $\lambda$ represent the emergent gauge fields of the SB theory.~\cite{Auerbach1994}\\
To compute the partition function \eqref{Seff} we expand the effective action $S_{\rm eff}$ about its SP solution
\begin{equation} \label{spcond}
S_{\rm eff} \equiv \sum_{n=0}^{\infty} S_{\alpha_1 \cdots \alpha_n}^{(n)} \Delta \phi_{\alpha_1} \cdots \Delta \phi_{\alpha_n},
\end{equation}
with
\begin{equation}\label{derivada}
S_{\alpha_1 \cdots \alpha_n}^{(n)}= \frac{1}{n!}\left.\frac{\partial^{n} S_{\rm eff}}{\partial \phi_{\alpha_1} \cdots \ \partial \phi_{\alpha_{n}}}\right|_{\rm sp},
\end{equation}
and $\Delta \phi_{\alpha}^{} = \phi_{\alpha} - \phi_{\alpha}^{\rm sp}$. The fields $\phi_{\alpha}$ are the auxiliary fields $\left\{ \overline W_{ij}^{r \tau}, W_{ij}^{r \tau}, \lambda_{i}^{\tau} \right\}$
($\alpha$ includes field, space $i$, and time $\tau$ indices) and $\phi^{\rm sp}_{\alpha}$ is the SP solution
that fulfills the condition $S^{(1)}_{\alpha}=0$:
\begin{equation} \label{spcond}
\frac{\partial S_{\rm eff}}{\partial \phi_{\alpha}} \bigg|_{\rm sp} =
\frac{\partial \ S_0}{\partial \phi_{\alpha}}\bigg|_{\rm sp} + \frac{1}{2} \tr \bigg[\mathcal{G}_{\rm sp} \;v_{\alpha}\bigg] = 0.
\end{equation}
$\mathcal{G}_{\rm sp}$ is the saddle point Green function and $v_{\alpha}=\frac{\partial \mathcal{G}^{-1}}{\partial \phi_{\alpha}}$ is the so-called internal vertex. $S^{(0)}$ coincides with the effective action $S^{\rm sp}_{\rm eff}$ evaluated at the SP solution, so the effective action can be rewritten as~\cite{Auerbach1994,Ghioldi2018}
\begin{equation}\label{Seffexp}
S_{\rm eff} = S^{\rm sp}_{\rm eff} + \sum_{\alpha_1 \alpha_2} S_{\alpha_1 \alpha_2}^{(2)} \ \Delta \phi_{\alpha_1} \Delta \phi_{\alpha_2} + S_{\rm int},
\end{equation}
with
\begin{equation}
S_{\rm int}=\sum_{n=3}^{\infty} \sum_{\alpha_1 \cdots \alpha_n} S_{\alpha_1 \cdots \alpha_n}^{(n)} \ \Delta \phi_{\alpha_1} \cdots \Delta \phi_{\alpha_n}\label{Sint}.
\end{equation}
It is straightforward to show that
\begin{eqnarray}
S^{\rm sp}_{\rm eff} = &\int_{0}^{\beta}& \! d\tau \; ( \frac{1}{2}\sum\limits_{ijr} \frac{1}{J_{ij}} \overline W_{ij}^{r\tau} W_{ij}^{r\tau}\!-\!iS \sum_{i} \lambda_{i}^{\tau} ) \Big|_{\rm sp} \nonumber \\
&+&\!\!\!\frac{1}{N} \tr \ln \left[ \mathcal{G}_{\rm sp}^{-1} \right]\label{s0},
\end{eqnarray}
\begin{eqnarray}
S^{(2)}_{\alpha \alpha{\prime}} &=& \frac{1}{2J_{ij}} (\delta_{\alpha,W^{r \tau}_{ij}} \delta_{\alpha^{\prime},\overline{W}^{r \tau}_{ij}} +\delta_{\alpha,\overline{W}^{r \tau}_{ij}} \delta_{\alpha^{\prime},{W}^{r \tau}_{ij}}) \nonumber \\
& & - \frac{1}{2N} \tr \ln \left[\; \mathcal{G}_{sp}\; v_{\alpha
} \; \mathcal{G}_{sp} \; v_{\alpha^{\prime} } \right]\label{s2} ,
\end{eqnarray}
and
\begin{equation}
S^{(n\ge 3)}_{\alpha_1... \alpha_{n}} = \frac{(\text{-}1)^{n+1}}{n!\; n} \!\! \sum_{P(\alpha_1...\alpha_n)}\!\! \frac{1}{N} \tr \ln \left[\; \mathcal{G}_{\rm sp}\; v_{P_1
}\;...\; \mathcal{G}_{\rm{sp}} \; v_{P_{n} } \right]\label{sn} ,
\end{equation}
where $P(\alpha_1...\alpha_n)$ denotes all the different permutations of $(\alpha_1...\alpha_n)$. As the traces above go over space, time, and flavor indices it turns out that $S^{\rm sp}_{eff}$, $S^{(2)}_{\alpha \alpha{\prime}}$, and $S^{(n\ge 3)}_{\alpha_1... \alpha_{n}}$ are all of order $N^0.$\\
At the Gaussian level $S_{\rm int}$ is neglected in Eq.~\eqref{Seffexp} and the free energy $F=-\frac{1}{\beta} \ln \mathcal{Z}$ per flavor becomes
\begin{equation}
\frac{F^{(2)}}{N}=\frac{1}{\beta}S^{sp}_{\rm eff} -\frac{1}{N\beta} \tr \ln \left[ S^{(2)} \right],
\label{free}
\end{equation}
%
with $\beta=1/T$.
Here the trace must be computed over time, space, and auxiliary field index. Consequently, the contribution of the Gaussian fluctuations to the free energy per flavor is of order $1/N$.
\section{Dynamical spin susceptibility: 1/N expansion}
\label{DSS}
The computation of the dynamical spin susceptibility requires to couple the symplectic spins (\ref{symplectic}) with a space and time-dependent external source $j^{\tau }_{i \alpha \beta}$
\begin{equation}
\mathcal{J}_s= \sum _{i} j^{\tau }_{i \alpha \beta} \mathcal{S}^{\tau}_{\beta \alpha}(i),
\label{source}
\end{equation}
where the sum over repeated flavor indices is assumed. After adding this term to the Lagrangian in Eq.~\eqref{partition}, the dynamical susceptibility is obtained from the generatriz $\mathcal{Z}[j]$\cite{Auerbach1994,Ghioldi2018}
\begin{equation}
\chi_{\alpha\beta}(1,2) = \frac{\partial^{2} ln \mathcal{Z}[j]}{\partial j_{1}^{ \alpha \beta} \partial j_{2}^{\ \beta\alpha}} \Big|_{j=0} ,
\end{equation}
where $1$ and $2$ design space and time points, $\bm r_1$, and $\bm r_2$, respectively. The above expression can be split into two contributions,
\begin{equation}
\chi = \chi_{_I} + \chi_{_{II}},
\end{equation}
with
\begin{eqnarray}\label{chi1}
\chi_{_{I} \alpha\beta}(1,2)\!\! &=&\! \! \frac{N}{\mathcal{Z}}\int \!\!D[\overline \phi, \phi] \Big(\!-\!\frac{\partial^2S_{\rm eff}}{\partial j^{\alpha \beta}_{1}\partial j^{\beta \alpha}_{2}}
\Big|_{j=0}\; \Big) \nonumber\\ &&\times \ \ e^{-NS_{\rm eff}(\overline \phi, \phi,j=0)}
\end{eqnarray}
and
\begin{eqnarray}\label{chi2}
\chi_{_{II} \alpha\beta}(1,2)\! &=&\! \! \frac{N^2}{\mathcal{Z}}\!\!\int \!\!D[\overline \phi, \phi] \Big(\frac{\partial S_{\rm eff}}{\partial j^{\alpha \beta}_{1}} \Big|_{j=0} \frac{\partial S_{\rm eff}}{\partial j^{\beta \alpha}_{2}} \Big|_{j=0} \Big) \nonumber\\ &&\times \ \ e^{-NS_{\rm eff}(\overline \phi, \phi,j=0)}.
\end{eqnarray}
\vskip 0.2cm
\begin{figure}[!h]
\includegraphics[scale=0.5]{external_loops.pdf}
\caption{Diagrammatic representation of the external loops corresponding to one external vertex $S^{(n+1)}$ (a), and two external vertices $S^{(n+2)}$ (b).~\cite{Auerbach1994}}
\label{external-loops}
\end{figure}
The partial derivatives of the effective action are given by
\begin{eqnarray}
N \frac{\partial S_{\rm eff}}{\partial j^{\alpha \beta }_{1}}\Big|_{j=0} &=& \tr\;[\;\mathcal{G}(j=0)\; u^{\alpha \beta}(1) \;],
\nonumber \\
N \frac{\partial^2S_{\rm eff}}{\partial j^{\alpha \beta}_{1}\partial j^{\beta \alpha}_{2}}
\Big|_{j=0}\!\! &=&
\tr [ \mathcal{G}(j\!=\!0)\; u^{\alpha \beta}(1) \times \mathcal{G}(j\!=\!0)\; u^{\beta \alpha}\!(2)\;],
\nonumber \\
\end{eqnarray}
where $u^{\alpha \beta}(1)\equiv \partial \mathcal{G}^{-1}/\partial j^{\alpha \beta}_{1}$ is the so-called external vertex.
By using the SP expansion \eqref{Seffexp} and defining
\begin{equation}
S^{(n+1)}_{\alpha_1...\alpha_n;(\bm r_i ; \alpha \beta)}= N
\frac{\partial S^{(n)}_{\alpha_1...\alpha_n}(j)}{\partial j^{\alpha \beta}_{i}} \Big|_{j=0}
\label{Sn1}
\end{equation}
and
\begin{equation}
S^{(n+2)}_{\alpha_1...\alpha_n;(\bm r_1\; \alpha \beta),(\bm r_2; \beta \alpha)}= N
\frac{\partial^2 S^{(n)}_{\alpha_1...\alpha_n}(j)}{\partial j^{\alpha \beta}
_{1} \partial j^{\beta \alpha}_{2}} \Big|_{j=0},
\label{Sn2}
\end{equation}
which are diagrammatically represented in Fig.~\ref{external-loops}, we obtain an explicit expansion of $\chi_{_{I} \alpha\beta}(1,2)$ and $\chi_{_{II} \alpha\beta}(1,2)$ [Eqs. \eqref{chi1} and \eqref{chi2}] in powers of $1/N$:
\begin{widetext}
\begin{equation}
\chi_{_{I} \alpha\beta}(1,2)\!=\! \frac{1}{\mathcal{Z}}\! \int [D\overline \phi D\phi] \Big(\! \text{-} \sum^{\infty}_0 S^{(n+2)}_{\alpha_1...\alpha_n;(1 \; \alpha \beta),(2 \; \beta \alpha)} \Delta \phi_{\alpha_1}...\Delta \phi_{\alpha_n}\Big) \times \Big[
\sum^{\infty}_{L=0}\frac{(\text{-}N)^L}{L!} (S^L_{\rm int})
\Big] e^{-N \left(\Delta \phi_{\alpha} S^{(2)}_{\alpha \alpha^{\prime}} \Delta \phi_{\alpha^{\prime}}+S^{sp}_{\rm eff} \right)}\label{chiI}
\end{equation}
\begin{eqnarray}
\chi_{_{II} \alpha\beta}(1,2)\!=\! \frac{1}{\mathcal{Z}}\! \int [D\overline \phi D\phi] \Big(\! \text{-} \sum^{\infty}_0 S^{(n+1)}_{\alpha_1...\alpha_n;(1 \; \alpha \beta)} \Delta \phi_{\alpha_1}...\Delta \phi_{\alpha_n}\Big)
&\times& \Big(\! \text{-} \sum^{\infty}_0 S^{(n+1)}_{\alpha_1...\alpha_m;(2 \; \beta \alpha)} \Delta \phi_{\alpha_1}...\Delta \phi_{\alpha_m}\Big) \nonumber \\
&\times& \Big[
\sum^{\infty}_{L=0}\frac{(\text{-}N)^L}{L!} (S^L_{\rm int})
\Big] e^{-N \left(\Delta \phi_{\alpha} S^{(2)}_{\alpha \alpha^{\prime}} \Delta \phi_{\alpha^{\prime}}+ S^{sp}_{\rm eff}\right)} \label{chiII}
\end{eqnarray}
\end{widetext}
where
\begin{equation}
\mathcal{Z}\!=\!\int [D\overline \phi D\phi] \Big[\!
\sum^{\infty}_{L=0}\frac{(\text{-}N)^L}{L!} (S^L_{\rm int})
\Big] \; e^{-N \left(\Delta \phi_{\alpha} S^{(2)}_{\alpha \alpha^{\prime}} \Delta \phi_{\alpha^{\prime}}+S^{sp}_{\rm eff} \right)}.
\end{equation}
The integrals of an even number of fields $\phi$ is the sum of all possible pair contractions (Wick's theorem) that defines the RPA propagator $ {D}_{\alpha_1\alpha_2}= [S^{(2)}]^{-1}_{\alpha_1\;\alpha_2}$:
\begin{eqnarray}
D_{\alpha_1\alpha_2} = \frac{N}{\cal{Z}} \int [D\overline \phi D\phi] \phi_{\alpha_1} \phi_{\alpha_2} \; e^{-N \Delta \phi_{\alpha} S^{(2)}_{\alpha \alpha^{\prime}} \Delta \phi_{\alpha^{\prime}}}.
\end{eqnarray}
The diagrams for $\chi_{_I}$ and $\chi_{_{II}}$ [see Eqs. \eqref{chiI} and \eqref{chiII}] are constructed as follows\cite{Auerbach1994}:
the elements $S^{(n+1)}$ and $S^{(n+2)}$ contribute to external loops with $n$ internal vertices and one and two external vertices, respectively. The derivatives of $S^{(n)}$ [see Eqs.~\eqref{s0}-\eqref{sn}] with respect to $j$ are of order $1/N$.
Consequently, according to the definition of $S^{(n+1)}$ and $S^{(n+2)}$ given by Eqs.~\eqref{Sn1} and \eqref{Sn2}, these external loops are of order $N^0$. The terms of the expansion of $S_{\rm int}$ in Eq.~\eqref{Sint} contribute to internal loops with $n\ge3$ internal vertices. Even though $S_{\rm int}$ is of order $N^0$, it is multiplied by factor $N$, implying that each diagram contains a factor $N^L$, where $L$ is the number of internal loops. In addition, each contraction of the $\phi$ fields gives rise to an RPA propagator $D$ (of order $N^0$) divided by $N$. Summarizing, each external loop contributes with a factor of order $N^0$, each internal loop contributes with a factor of order $N$ and each RPA propagator contributes with a factor $1/N$. In other words, a diagram with $L$ internal loops and $P$ RPA propagators is of order $(\frac{1}{N})^{P-L}$. Fig.~\ref{fig2:feymann} shows all the diagrams of order $1/N$ [(b)-(e)] that contribute to $\chi_{_{I}}$ and $\chi_{_{II}}$, along with the saddle point contribution shown in panel (a).
In particular, the diagram of Fig.~\ref{fig2:feymann}(b) corresponds to $\chi_{_{II}}$ for $L=0$ and $P=1$, while Figs.~\ref{fig2:feymann}(c) and (d) are the diagrams corresponding to $\chi_{_{I}}$ for $L=0$ and $P=1$. The diagram shown in Fig. ~\ref{fig2:feymann}(e) arises from $\chi_{_{I}}$ and it is the only diagram that includes one internal loop ($L=1$) and two RPA propagators ($P=2$). This is the only $1/N$ diagram that arises from non-Gaussian corrections of the effective action.
\begin{figure}[!t]
\includegraphics[width=\columnwidth,bb=0 0 481 318]{fig2.pdf}
\caption{Diagrammatic representation of (a) saddle point contribution and (b-e) the $1/N$ corrections to the dynamical spin susceptibility.
In our calculation we only include the contribution (b) for reasons explained in the text. The diagram (c) corresponds to a vertex correction
relative to (a), while the diagrams (d) and (e) include a Hartree-Fock correction of the single-spinon propagator. The dashed lines represent the external lines, the full lines represent spinon propagators at the SP level and the wavy lines represent the RPA propagator.~\citep{Auerbach1994}}
\label{fig2:feymann}
\end{figure}
\section{$SU(2)$ case: the $120^{\circ}$ N\'eel-ordered state}
\label{SU2}
The large-$N$ Schwinger boson theory developed in the previous sections is valid for the family of $Sp(N)$ models. Therefore, given that $SU(2)\! \cong \!Sp(2)$, the $SU(2)$ case is recovered by fixing $N\!\!=\!2$ in the above expressions.
To study the magnetic excitation spectrum of the 120$^{\circ}$ N\`eel-ordered ground state of the triangular SU(2) Heisenberg antiferromagnet, we must add a symmetry breaking field, $h$, that selects the ordered ground state in the thermodynamic limit.~\cite{Ghioldi2018} The field $h$ couples linearly to N\`eel order parameter and it is sent to zero after taking the thermodynamic limit. In the SB language this process corresponds to condensing the SBs in a single particle state (the single-spinon ground state is degenerate) that spontaneously breaks the $SU(2)$ symmetry of the spin Hamiltonian.
Only the diagram shown in Fig.~\ref{fig2:feymann}(a) contributes to the dynamical spin susceptibility at the SP level (SBMFT):
\begin{equation}
\chi_{_I \mu\nu}^{\rm sp}(\bm q,i\omega) = \frac{1}{2}\tr
\left[ \mathcal{G}^{\rm sp} u^{\mu}(\bm q,i\omega) \mathcal{G}^{\rm sp} u^{\mu}(-\bm q,-i\omega) \right].\label{chispI}
\end{equation}
The index $\mu=x,y,z$ refers to the three spin components and $u^{\mu}$ is the external vertex that couples the spin excitations to the $\bm q$ component of an external magnetic field. It can be shown that $\chi_{_{II} \mu\mu}^{\rm sp}=0$.\cite{Ghioldi2018}
The magnetic excitation spectrum of $\chi_{_I \mu\mu}^{\rm sp}$ consists of a two-spinon continuum (branch cut), corresponding to a gas of free spin-$\frac{1}{2}$ spinons. The condensation of the SBs also generates a delta function contribution (pole) at the lower edge of the two-spinon continuum. In addition, due to the relaxation of the local constraint, the magnetic spectrum also exhibits spurious modes arising from density fluctuations of the SBs.\cite{Arovas1988,Auerbach1994,Mezio2011,Mezio2012,Ghioldi2018} The inclusion of the $1/N$ correction corresponding to the diagram shown in Fig.~\ref{fig2:feymann}(b) leads to the following contribution~\cite{Ghioldi2018}
\begin{eqnarray}
\chi_{_{II}\; \mu\nu}^{\rm fl}(\bm q, i\omega) = \sum_{\alpha_1 \alpha_2}\!\!\!\! &&\frac{1}{2}\tr\big[ \mathcal{G}^{\rm sp} \ v_{\phi_{\alpha_1}} \ \mathcal{G}^{\rm sp} \ u^{\mu}(\bm q, i\omega) \big] \nonumber \\
&&\;\;\;\;\;\;\; \times D_{\alpha_{2} \alpha_{1}}(\bm q, i\omega) \label{chiflII} \\
&&\frac{1}{2} \tr\big[ \mathcal{G}^{\rm sp} \ v_{\phi_{\alpha_2}} \ \mathcal{G}^{\rm sp} \ u^{\nu}(-\bm q, -i\omega) \big], \nonumber
\label{corr}
\end{eqnarray}
In Ref.~[\onlinecite{Ghioldi2018}] we demonstrated that this particular $1/N$ correction introduces a drastic change in the dynamical spin susceptibility. In the first place, it cancels out the SP poles at the lower edge of the two-spinon continuum and it introduces new poles, which are the poles of the RPA propagator $D$. As we will show below, these new poles are associated with the collective modes (magnons) of the theory and they correspond to two-spinon bound states generated by the fluctuations of the gauge fields. In the second place, the spurious modes of the SP solution are also exactly canceled out.
It is important to note that the contribution from this diagram is exactly equal to zero for a singlet ground state ($h=0$).~\cite{Arovas1988} However, we have recently shown in Ref.~[\onlinecite{Ghioldi2018}] that it becomes finite for the magnetically ordered ground state under consideration. Moreover, for $N=2$ and $S=1/2$, the magnon dispersion obtained from this particular $1/N$ correction has Goldstone modes at the $\Gamma$ and $\pm K$ points, whose velocities agree very well with the results obtained with LSWT plus $1/S$ corrections.~\cite{Chubukov1994S,Chernyshev2009}
Below we demonstrate another virtue of this $1/N$ correction. The relevant large-$S$ contribution to the dynamical spin susceptibility corresponds to the diagrams shown in Figs.~\ref{fig2:feymann}~(a) and (b) [see Eqs.~(\ref{chispI}) and \eqref{chiflII}], where $\chi=\chi^{sp}_{_I}+\chi^{fl}_{_{II}}$ coincides with the LSWT result.
\subsection{Large-S limit}
The SP approximation is equivalent to the SBMFT described by the quadratic mean field Hamiltonian\cite{Mezio2011}
\begin{equation}\label{eq:sbmfH}
{\cal H}_{B} = \sum_k \psi^{\dagger}_{\bm k} {\cal H}_{MF} (\bm k) \psi_k,
\end{equation}
with $\psi_{\bm k} = (b_{{\bm k},\uparrow},b_{-{\bm k},\downarrow}^{\dagger})$,
\begin{equation}
{\cal H}_{MF} ({\bm k})={
\left(
\begin{array}{cc}
\lambda_{sp} + \gamma_{\bm{k}}^{B} & -\gamma_{\bm{k}}^{A} \\ \\
-\gamma_{\bm{k}}^{A} & \lambda_{sp} + \gamma_{\bm{k}}^{B}
\end{array} \right)} \ ,
\end{equation}
and
\begin{eqnarray}
\gamma_{\bm k}^{A} &=& \sum\limits_{\bm \delta>0} J_{\delta} A_{\bm \delta} \sin \left(\bm{k\cdot \bm \delta}\right), \\
\gamma_{\bm k}^{B} &=& \sum\limits_{\bm \delta>0} J_{\delta} B_{\bm \delta} \cos \left(\bm{k\cdot \bm \delta}\right).
\end{eqnarray}
The amplitudes $i A_{\bm \delta}$ and $B_{\bm \delta}$ are the SP values of the bond operators $\hat{A}_{i,i+\bm \delta}$ and $\hat{B}_{i,i+\bm \delta}$, while $i\lambda_{sp}$ is the SP value of the Lagrange multiplier that was introduced to implement the local constraint
${ b}^{\dagger}_{i\uparrow} {b}^{\;}_{i\uparrow} + {b}^{\dagger}_{i\downarrow} {b}^{\;}_{i\downarrow}=2 S$.
The single-spinon Green's function is given by the $2$ by $2$ matrix
\begin{align}
G^{\rm sp}_0(\bm k, i \omega_n) =
\left(\begin{array}{cc}
{\lambda_{sp} + \gamma_{\bm k}^B +i\omega_n \over \varepsilon_{\bm k}^2 + \omega_n^2} & -{\gamma_{\bm k}^A \over \varepsilon_{\bm k}^2 + \omega_n^2} \\
-{\gamma_{\bm k}^A \over \varepsilon_{\bm k}^2 + \omega_n^2} & {\lambda_{sp} + \gamma_{\bm k}^B -i\omega_n \over \varepsilon_{\bm k}^2 + \omega_n^2} \\
\end{array}\right).
\end{align}
The poles of this Green's function determine the single-spinon dispersion,
\begin{equation}
\varepsilon_{\bm k} = \sqrt{(\lambda_{sp} + \gamma_{\bm k}^B)^2 - (\gamma_{\bm k}^{A})^2}.
\end{equation}
\begin{figure}[t!]
\centering
\includegraphics[scale=0.4]{power.pdf}\caption{Power counting rule for the $S$ power of each Feynman diagram. Solid line: non-condensed boson propagator. Double line: condensed boson propagator. Wavy line: RPA propagator of the fluctuation fields. Dashed line: external lines.}
\label{powercounting}
\end{figure}
The SP single-spinon spectrum has two degenerate minima at ${\bm k} = \pm {{\bm Q} \over 2}$. On a finite size lattice, the minimum energy, $\varepsilon_{\pm {{\bm Q} \over 2}}$, is proportional to $1/{\cal N}_s$, where ${\cal N}_s$ is the number of lattice sites, and the ground state of ${\cal H}_B$ is a singlet state. Upon taking the thermodynamic limit, ${\cal N}_s \rightarrow \infty$, the spectrum becomes gapless at $\pm {{\bm Q} \over 2}$ and the bosons condense at $T=0$K. Given that there are four single particle ground states (two gapless points with momenta ${\pm {{\bm Q} \over 2}}$ and two possible spin orientations), there is continuous ground state degeneracy corresponding to different ways of condensing the bosons. The above-mentioned infinitesimal symmetry-breaking field $h$, selects a ground state with a particular $120^{\circ}$ magnetic ordering.~\cite{Ghioldi2018}.
Correspondingly, it is convenient to work in the twisted spin reference frame where the selected 120$^{\circ}$ magnetic ordering becomes an in-plane ferromagnetic (FM) ordering along the $x$-axis. The real space Schwinger boson operators become $b_{i \uparrow} = \tilde{b}_{i \uparrow} e^{-i{\bm Q}\cdot {\bm r}/2}$ and $b_{i \downarrow} = \tilde{b}_{i \downarrow} e^{i{\bm Q}\cdot {\bm r}/2}$ in the new reference frame and the FM magnetic ordering arises from condensation at momentum ${\bm k}={\bm 0}$.
After taking the thermodynamic limit and sending $h$ to zero (the two operations do not commute), the Schwinger boson SP Green's function becomes
\begin{equation}
\mathcal{G}^{\rm sp} (\bm{k},i\omega_{n})=\mathcal{G}^{\rm sp}_{0}(\bm{k},i\omega_{n})+(2\pi)^{2}\delta(\bm{k}) \mathcal{G}^{\rm sp}_{c}(i\omega_{n}),\label{eq:GF_condensate}
\end{equation}
where $\mathcal{G}^{\rm sp}_{0}(\bm{k},i\omega_{n})$ and $\mathcal{G}^{\rm sp}_{c}(i\omega_{n})$ are the contributions from the non-condensed and condensed bosons, respectively.
After extending the two-component representation, $\psi_{\bm k,\omega}$, to the four-component representation $\Psi_{\bm k, \omega} = (b_{\bm k-{\bm Q}/2, \uparrow}^{\omega},\bar b_{ -\bm k + {\bm Q}/2 \downarrow}^{-\omega}, b_{ \bm k + {\bm Q}/2 \downarrow}^{\omega}, \bar b_{-\bm k-{\bm Q}/2, \uparrow}^{-\omega})$, we obtain
\begin{equation} \label{eq:green_nonc}
\mathcal{G}^{\rm sp}_{0}(\bm{k},i\omega_{n}) =
\left( \begin{array}{cc}
G^{\rm sp}_0(\bm k -{\bm Q \over 2},i\omega_n) & 0 \\
0 & G^{\rm sp}_0(-\bm k -{\bm Q \over 2},i\omega_n)
\end{array}\right),
\end{equation}
whose single-spinon pole locates at $\varepsilon_{\pm \bm q -{\bm Q \over 2}}$. For the condensed spinons, we have
\begin{equation} \label{eq:green_cond}
\mathcal{G}^{\rm sp}_{c}(i\omega_{n})=
{n_c \Omega_c \over \Omega_c^2+\omega_n^2} \left(\!\!\begin{array}{cccc}
1 & 1 & -1 & -1 \\
1 & 1 & -1 & -1 \\
-1 & -1 & 1 & 1 \\
-1 & -1 & 1 & 1
\end{array}\!\!\right),
\end{equation}
where $\Omega_c = {h \over 2}$ and $n_c$ is the density of the condensate.
The symmetry-breaking field will be sent to zero in the thermodynamic limit, meaning that $h = 0^+$.
The SP values of $n_c$, $A_{\bm{\delta}}$, $B_{\bm{\delta}}$ and the Lagrangian multiplier $i\lambda_{sp}$ are obtained by solving the set of self-consistent equations (\ref{spcond}).
\subsubsection{Large-$S$ limit of the saddle point solution}
For arbitrary spin size $S$, the self-consistent SP equations (\ref{spcond}) become:\cite{Ghioldi2018}
\begin{eqnarray}
&& B_{\bm{\delta}} = \int \frac{d^{2}\bm{k}}{(2\pi)^{2}}\cos(\bm{k}\cdot\bm{\delta}) {\lambda_{sp} + \gamma_{\bm k}^B \over 2\varepsilon_{\bm k}}+{n_c\over 2}\cos[\frac{\bm{Q}}{2}\cdot\bm{\delta}],\label{eq:sp1}
\nonumber \\
&& A_{\bm{\delta}} = i\int\frac{d^{2}\bm{k}}{(2\pi)^{2}}\sin(\bm{k}\cdot\bm{\delta}) {\gamma_{\bm k}^A \over 2\varepsilon_{\bm k}}+i {n_c \over 2}\sin[\frac{\bm{Q}}{2}\cdot\bm{\delta}],
\label{eq:sp2}
\nonumber \\
&& 2S+1 = \int\frac{d^{2}\bm{k}}{(2\pi)^{2}} {\lambda_{sp} + \gamma_{\bm k}^B \over \varepsilon_{\bm k}} +n_c.
\label{eq:sp3}
\end{eqnarray}
\noindent In all cases, the integral that appears in each of the three expressions is the contribution from the non-condensed spinons, while the second term, proportional to $n_c$, is the contribution from the condensate.
In the large-$S$ limit, the ground state is the $120^{\circ}$ N{\'e}el-ordered state characterized by $\langle {\bm S}_i \rangle = S {\bm n}_i$ with ${\bm n}_i$ the unit vector along the local moments. In the SBMFT, $\langle {\bm S}_i \rangle = \frac{1}{2}\langle b_{i\alpha}\rangle^* {\boldsymbol \sigma}_{\alpha \beta} \langle b_{i \beta}\rangle $, where ${\boldsymbol \sigma}$ is the vector of Pauli matrices, implying that $\langle b_{i \beta}\rangle \sim S^{1/2}$. This observation fixes the scaling of the SP parameters: $n_c = \langle b_{i\alpha}\rangle^* \langle b_{i \alpha}\rangle \sim S$ and $A_{\bm \delta}, B_{\bm \delta} \sim S$. Consequently, $\gamma_{\bm k}^A$, $\gamma_{\bm k}^B$ and $\varepsilon_{\bm k}$ are also ${\cal O} (S)$. Back to the saddle point equations \eqref{eq:sp3}, we observe that the contribution from the non-condensed bosons is of order $S^0$, while the contribution from the condensed bosons is of order $S$, implying that
\begin{align}
n_c \to 2S, \ \
B_{\bm{\delta}} \to {S}\cos[\frac{\bm{Q}}{2}\cdot\bm{\delta}], \ \ A_{\bm{\delta}} \to i {S }\sin[\frac{\bm{Q}}{2}\cdot\bm{\delta}],
\label{eq:sp1}
\end{align}
\noindent in the $S\rightarrow \infty$ limit. The saddle point value of the Lagrange multiplier is equal to $\lambda_{sp}={3\over 2}JS$, as required by the gapless nature of $\varepsilon_{\bm k}$.
This solution indicates that the rescaled mean-field Hamiltonian $\tilde{{\cal H}}_B = {\cal H}_B S^{-1}$ and frequency $\tilde{\omega} = \omega S^{-1} $ are independent of $S$, i.e. ${\cal O}(S^0)$ in the large $S$ limit. Consequently, each integral over frequency (summation over Matsubara frequencies) introduces an $S$ factor: $\int{d\omega \over 2\pi} = S \int{d\tilde{\omega} \over 2\pi}$. In addition, according to Eq.~(\ref{eq:green_nonc}), the Green's function of the non-condensed Schwinger bosons has the scaling behavior $\mathcal{G}_0 \sim S^{-1}$. However, according to Eq.~(\ref{eq:green_cond}), the Green's function of the condensed Schwinger bosons has an \textit{anomalous} scaling $\mathcal{G}_c \sim S^{0}$ (note that $\Omega_c = {h\over 2} \sim S $). This analysis provides a power counting rule for evaluating relative contributions of different Feynman diagrams at the SP level [see Fig.~\ref{powercounting}~(a), (b), (f)].
The classical limit is then dominated by the contributions from the condensed spinons. For instance, the SP contribution to the ground state energy per site becomes\cite{Ceccatto1993}
\begin{equation}
E_{sp}=\sum_{{\bm \delta}>0} J_{\delta} \left( B^2_{\bm \delta} -|A_{\bm \delta}|^2 \right) \to -{3\over 2}JS^2,
\end{equation}
which corresponds to the classical limit ($S\to \infty$).
The magnetic moment becomes $n_c/2=S$, which is also the expected value in the classical limit.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.42]{largeS_Sqw_v1.pdf}\caption{ Feynman diagrams of the dynamical structure factor in the large $S$ limit. (a) Saddle point contribution. (b) $1/N$ diagram that account for the true collective modes (magnons) of the magnetically ordered ground state. These modes appear as poles of the RPA propagator represented as a wavy line.}
\label{feynman}
\end{figure}
\subsubsection{Corrections beyond the Saddle point level}
As $\mathcal{G}^{-1}$ is linear in the fields, $\phi_{\alpha}$ the internal vertex turns out to be of order $S^0$:
$v_{\alpha}=\frac{\partial \mathcal{G}^{-1}}{\partial \phi_{\alpha}}\sim S^0$.
The RPA propagator of the fluctuation fields can be expressed as $D^{-1}(\bm q,i\omega_n) = \Pi_0 - \Pi(\bm q, i\omega_n)$, where\cite{Ghioldi2018}
\begin{equation}
\Pi_{\phi_{\alpha_1} \phi_{\alpha_2}}(\bm q, i\omega_n) = \frac{1}{4} \tr\left[\mathcal{G}_{\rm sp} \; v_{\phi_{\alpha_1}} \! \mathcal{G}_{\rm sp} \ v_{\phi_{\alpha_2}} \right] \label{polarization}
\end{equation}
\noindent is the polarization operator
and
$\Pi_0$ is a diagonal matrix containing the exchange couplings $J_{ij}$ along the diagonal except for the entries corresponding to $\lambda-\lambda$ derivatives, which are zero. Replacing the Green function (\ref{eq:GF_condensate}) in the polarization operator (\ref{polarization}) and by applying the power counting rule shown in Fig.~\ref{powercounting}, we obtain $\Pi_{\alpha\beta}(\bm q, i\omega_n) \sim S^0$ in the large $S$ limit (the dominant contribution arises from a loop containing one condensed and one non-condensed spinon propagator).
It is then clear that $D(\bm q,i\omega_n) \sim S^0$ in the large $S$ limit (Fig.~\ref{powercounting}(c)).
The resulting power counting rule for each Feynman diagram is given by $(1/S)^{P_{nc}-L_{\Sigma}}$ where $P_{nc}$ is the number of propagators of non-condensed bosons and $L_{\Sigma}$ is the number of independent loops (i.e., independent frequency variables to be integrated out).
\begin{figure}[t!]
\centering
\includegraphics[scale=1.1]{Sqw_sp_v2.png}\caption{ (a) Dispersion relation of the poles of the dynamical spin susceptibility in the laboratory reference frame at the SP level. Each line is doubly-degenerate.
The spectral weight (residue of the pole) is zero for the dashed lines that correspond to in-plane modes [see panel (b)], while it is finite for the full lines that correspond to in-plane and out of plane modes. (b) Dynamical spin structure factor obtained from the SBMFT (red line) and from LSWT (black line). The color scale represents the spectral weight.}
\label{Ssp}
\end{figure}
\section{Dynamical spin structure factor}
\label{DSS_largeS}
We are now ready to take the large-$S$ limit of the $T=0$ dynamical structure factor for the physical $SU(2)$ ($N=2$) version of the spin model:
\begin{equation}
{S}_{\mu \nu} ({\bm q}, \omega) = - \frac{1}{\pi} \Im [\chi_{\mu \nu} ({\bm q}, \omega)],
\end{equation}
The off-diagonal components vanish for symmetry reasons.
At the SP level, the magnetic susceptibility is obtained by an analytic continuation $i\omega_n \rightarrow \omega + i0^+$ of ${\chi}^{sp}_{_I \mu\nu}(\bm q, i\omega_n)$ given in
Eq.~\eqref{chispI}, which corresponds to the diagram shown in Fig.~\ref{feynman}~(a). Along the $\omega$-axis, the imaginary part of ${\chi}^{sp}_{_I \mu\nu}(\bm q, \omega)$ includes a two-spinon continuum arising from two non-condensed spinons [spinon lines in Fig.~\ref{feynman}~(a) with momentum $\bm k + \bm q$ and ${\bm k}$ are both non-condensed bosons] and $\delta$-peaks
arising from one condensed spinon with ${\bm k} = {\bm 0}$ and one non-condensed spinon with momentum ${\bm k}=\pm {\bm q}$. The resulting dispersion of these $\delta$-peaks is $\varepsilon_{\pm {\bm q} -{\bm Q \over 2}}$. The in-plane components of the dynamical structure factor, $S^{xx}({\bm q},\omega)$ and $S^{yy}({\bm q},\omega)$, contain four $\delta$-peaks centered at $\varepsilon_{\pm {\bm q} + {\bm Q \over 2}}$ and $\varepsilon_{\pm {\bm q} - {3 \bm Q \over 2}}$ for each ${\bm q}$, while the out-of-plane component, $S^{zz}({\bm q},\omega)$, contains two $\delta$-peaks centered at $\varepsilon_{\pm {\bm q} -{\bm Q \over 2}}$ for each ${\bm q}$. Due to inversion symmetry, the six $\delta$-peaks form three groups of degenerate pairs [see Fig.~\ref{Ssp}~(a)].
The weight of the two-spinon continuum vanishes in the large-$S$ limit because $\mathcal{G}^{sp}_0 \sim S^{-1}$ and $\mathcal{G}^{sp}_{c} \sim S^{0}$. The remaining $\delta$-peak contributions (corresponding to the poles of the SBMFT) lead to a single-particle spectrum, which is qualitatively different from the single-magnon spectrum of the LSWT (see Fig.~\ref{Ssp}).
To understand the origin of this qualitative difference, we first need to note that, after taking the $S\to \infty $ limit, $\varepsilon_{\bm q}$ includes two gapless modes at ${\bm q} \pm 3 {\bm Q}/2$
with a {\it quadratic dispersion}, in addition to the gapless modes with linear dispersion at ${\bm q} \pm {\bm Q}/2$. The quadratic modes have a finite energy gap for finite $S$ values, while the linear modes remain gapless for arbitrary values of $S$. Given that $\varepsilon_{\pm {\bm q} + {\bm Q \over 2}}$, $\varepsilon_{\pm {\bm q} - {3 \bm Q \over 2}}$ and $\varepsilon_{\pm {\bm q} -{\bm Q \over 2}}$ correspond to shifts of $\varepsilon_{\bm q}$ by three different wave-vectors, the $\delta$-peaks of the dynamical structure factor should also exhibit linear and the quadratic gapless modes. Indeed, as indicated in
Fig.~\ref{Ssp}~(a), the gapless modes appear at the $\Gamma$ point and at the $K$ points (ordering wave vector) of the Brillouin zone.
The two in-plane modes at $\varepsilon_{\pm {\bm q}-{3 \over 2} \bm Q}$, indicated with dashed lines in Fig.~\ref{Ssp}~(a), have no spectral weight. Consequently, as it is shown in Fig.~\ref{Ssp}~(b), the dynamical structure factor exhibits only two different doubly-degenerate gapless modes. Both of them are linear at the $\Gamma$ point, while one is linear and the other one is quadratic at the $K_1$ and $K_1^{\prime}$ points. It is clear that these gapless modes are qualitatively different from the three gapless {\it linear modes} (Goldstone modes) at the $\Gamma$, $K_1$ and $K_1^{\prime}$ points
that appear in the dynamical structure factor that is obtained from LSWT [see Fig.~\ref{Ssp}~(b)]. One of the reasons behind this qualitative difference is the presence of unphysical spurious modes
corresponding to the density fluctuation of Schwinger bosons that appear at the SP level of the theory. We also note that the linear spinon modes have the same velocity $v={3\over 2} J S$ at both the $\Gamma$ and $K$ points, while the Goldstone modes of the LSWT have velocities $v_{\Gamma}={3\sqrt{3}\over 2} JS$ and $v_{K}={3\over 2} \sqrt{3\over2} JS$ at the $\Gamma$ and $K$ points, respectively.
These qualitative discrepancies indicate that the SBMFT is not adequate for describing the true collective modes (magnons) of the triangular antiferromagnet in the large-$S$ limit in agreement with the conclusions that were recently obtained for the quantum ($S=1/2$) limit.~\cite{Ghioldi2018}
\begin{figure}[t!]
\centering
\includegraphics[scale=0.9]{Sqw_gs.png}\caption{Dynamical spin structure factor obtained from the Schwinger boson theory by including the diagrams shown in Figs.~\ref{feynman}~(a) and (b) (red). The black lines correspond to the result from LSWT.
Panel (a) shows the magnon dispersion relation (poles of the dynamical spin structure factor), while panel (b) shows the momentum dependence of the intensity of the magnon peak.}
\label{Sgs}
\end{figure}
The key observation of this work is that the correct dynamical spin structure factor in the large-$S$ limit is recovered only after adding the $1/N$ correction corresponding to the diagram shown in Fig.~\ref{feynman}~(b). Note that both diagrams in Figs.~\ref{feynman}~(a) and (b) are of order $S^0$. The effect of this $1/N$ correction is twofold: it cancels out exactly the poles of the SP contribution (the quadratic and the linear ones), while a new quasi-particle peak (delta function) emerges from the pole of the RPA propagator of the fluctuation fields [note that the poles of the RPA propagator are also poles of the diagram shown in Fig.~\ref{feynman}~(b)].\cite{Ghioldi2018}
The cancellation of the SP contribution along the spinon dispersion, i.e., on the shell $\omega=\varepsilon_{\pm {\bm q}-\frac{\bm Q}{2}}$, for the $zz$-component of the dynamical spin susceptibility can be understood as follows.~\cite{Raykin1993,Auerbach1994}
After noticing that the condensed part of the Green function satisfies the relation $\mathcal{G}^{\rm sp}_c u^z = \mathcal{G}^{\rm sp}_c (v_{W_A}+v_{\overline{W}_A}) C^{-1}_{W_A}$, where $C_{W_A}=\frac{J_{\delta}}{2\sqrt{{\cal N}_s}\beta}(e^{-i{\bm k}.{\bm \delta}}-e^{-i{\bm k}'.{\bm \delta}})\delta_{{\bm k}-{\bm k}',{\bm q}}$, the first trace corresponding to ${\chi}^{fl}_{_{II} zz}$ in equation (\ref{chiflII}) can be written as
\begin{equation}
\tr\big[ \mathcal{G}^{\rm sp}_0 \ v_{\phi_{\alpha_1}} \ \mathcal{G}^{\rm sp}_c \ u^{z} \big]=C^{-1}_{W_A} \left( \Pi_{\phi_{\alpha_1}, W_A } +\Pi_{\phi_{\alpha_1},\overline{W}_A} \right). \label{trace}
\end{equation}
Furthermore, the RPA propagator can be safely approximated by $D(\bm q,i\epsilon_{\pm {\bm q}-\frac{\bm Q}{2}})\approx -\Pi^{-1}(\bm q,i\epsilon_{\pm {\bm q}-\frac{\bm Q}{2}})$, since $\Pi$ is of order $O({\cal N}_s)$ on this energy shell. Then, replacing $D$ and the above trace in Eq. (\ref{chiflII}), we find that ${\chi}^{fl}_{_{II} zz} =-{\chi}^{sp}_{_I zz}$ along the SP spinon dispersion $\epsilon_{\pm {\bm q}-\frac{\bm Q}{2}}$. A similar analysis can be applied to the in-plane, $xx$ and $yy$, components of the dynamical spin susceptibility. It is important to remark that this cancellation occurs for any value of $S$.\cite{Ghioldi2018}\\
On the other hand, the poles of the RPA propagator are zeros of the fluctuation matrix (\ref{s2}):
\begin{equation}
S^{(2)}(\bm q, \omega) \cdot X =0.
\end{equation}
The first four components of $X=(X_1(\bm \delta),X_2(\bm \delta),X_3(\bm \delta),X_4(\bm \delta),X_5)$ correspond to fluctuations of the Hubbard-Stratonovich fields $W^A_{\bm \delta}$, $\overline W^A_{\bm \delta}$, $W^B_{\bm \delta}$, $\overline W^B_{\bm \delta}$, respectively, and $X_5$ is the fluctuation of the Lagrange multiplier.
The pole equation turns out to depend on four linear combinations of $X$, namely
\begin{align}
R_{1}\equiv & c_{1}+c_{2}+c_{3}^{-}+c_{4}^{-},\\
R_{2}\equiv & c_{1}-c_{2}-c_{3}^{+}+c_{4}^{+}+4iX_{5},\\
R_{3}\equiv & \bar{c}_{1}+\bar{c}_{2}-\bar{c}_{3}^{-}-\bar{c}_{4}^{-},\\
R_{4}\equiv & \bar{c}_{1}-\bar{c}_{2}+\bar{c}_{3}^{+}-\bar{c}_{4}^{+}-4iX_{5},
\end{align}
\noindent where
\begin{align}
c_{1}(\bm{q})=\sum_{\bm{\delta}}F_{q}^{-*}(\bm{\delta})X_{1}(\bm{\delta}), & c_{2}(\bm{q})=\sum_{\bm{\delta}}F_{q}^{-*}(\bm{\delta})X_{2}(\bm{\delta}), \nonumber \\
c_{3}^{-}(\bm{q})=\sum_{\bm{\delta}}F_{q}^{-*}(\bm{\delta})X_{3}(\bm{\delta}), & c_{3}^{+}(\bm{q})=\sum_{\bm{\delta}}F_{q}^{+*}(\bm{\delta})X_{3}(\bm{\delta}),\nonumber \\
c_{4}^{-}(\bm{q})=\sum_{\bm{\delta}}F_{q}^{-*}(\bm{\delta})X_{4}(\bm{\delta}), & c_{4}^{+}(\bm{q})=\sum_{\bm{\delta}}F_{q}^{+*}(\bm{\delta})X_{4}(\bm{\delta}),\nonumber
\end{align}
and
\begin{align}
\bar{c}_{1}(\bm{q})=\sum_{\bm{\delta}}\bar{F}_{q}^{-*}(\bm{\delta})X_{1}(\bm{\delta}), & c_{2}(\bm{q})=\sum_{\bm{\delta}}\bar{F}_{q}^{-*}(\bm{\delta})X_{2}(\bm{\delta}),\nonumber \\
\bar{c}_{3}^{-}(\bm{q})=\sum_{\bm{\delta}}\bar{F}_{q}^{-*}(\bm{\delta})X_{3}(\bm{\delta}), & c_{3}^{+}(\bm{q})=\sum_{\bm{\delta}}\bar{F}_{q}^{+*}(\bm{\delta})X_{3}(\bm{\delta}),\nonumber \\
\bar{c}_{4}^{-}(\bm{q})=\sum_{\bm{\delta}}\bar{F}_{q}^{-*}(\bm{\delta})X_{4}(\bm{\delta}), & c_{4}^{+}(\bm{q})=\sum_{\bm{\delta}}\bar{F}_{q}^{+*}(\bm{\delta})X_{4}(\bm{\delta}). \nonumber
\end{align}
Here we have introduced the following two functions
\begin{align}
F_{q}^{\mp}(\bm{\delta}) & =e^{i(\bm{q}-\frac{\bm{Q}}{2})\cdot\bm{\delta}}\mp e^{i\frac{\bm{Q}}{2}\cdot\bm{\delta}},\nonumber \\
\bar{F}_{q}^{\mp}(\bm{\delta}) & =F_{-\bm{q}}^{\mp*}(\bm{\delta})=e^{i(\bm{q}+\frac{\bm{Q}}{2})\cdot\bm{\delta}}\mp e^{-i\frac{\bm{Q}}{2}\cdot\bm{\delta}}.\nonumber
\end{align}
\noindent $R_1,...,R_4$ form a closed set of equations:
\begin{equation}
{\cal M}_{1} (\omega) \left(\begin{array}{c}
R_{2}\\
R_{4}
\end{array}\right)=3(1-\gamma_{\bm{q}})\omega
\left(\begin{array}{c}
R_{1}\\
R_{3}
\end{array}\right),\label{eq:le1}
\end{equation}
\begin{equation}
{\cal M}_{2} (\omega) \left(\begin{array}{c}
R_{1}\\
R_{3}
\end{array}\right)=\frac{3}{2}(1+2\gamma_{\bm{q}})\omega
\left(\begin{array}{c}
R_{2}\\
R_{4}
\end{array}\right).\label{eq:le2}
\end{equation}
\noindent where $\gamma_{\bm q} = {1\over 3} (\cos k_x + 2\cos {k_x \over 2} \cos {\sqrt{3} \over 2} k_y)$, and
\begin{equation}
{\cal M}_{1} (\omega)\!\!=\!\!
\left( \!\! \begin{array}{cc}
\omega^2- \varepsilon_{\bm q -{\bm Q \over 2}}^2 - {1 \over 2} \omega_{\bm q}^2 & \omega^2- \varepsilon_{\bm q -{\bm Q \over 2}}^2 \\
\omega^2- \varepsilon_{\bm q +{\bm Q \over 2}}^2 & \omega^2- \varepsilon_{\bm q + {\bm Q \over 2}}^2 - {1 \over 2} \omega_{\bm q}^2
\end{array} \!\! \right),
\end{equation}
\begin{equation}
{\cal M}_{2}(\omega)=\left(\begin{array}{cc}
-\varepsilon_{\bm{q}-\frac{\bm{Q}}{2}}^{2} & \omega^{2}-\varepsilon_{\bm{q}-\frac{\bm{Q}}{2}}^{2}\\
\omega^{2}-\varepsilon_{\bm{q}+\frac{\bm{Q}}{2}}^{2} & -\varepsilon_{\bm{q}+\frac{\bm{Q}}{2}}^{2}
\end{array}\right).
\end{equation}
\noindent At $\omega = \omega_{\bm q}=3 \sqrt{(1-\gamma_{\bm q}) (1+2 \gamma_{\bm q})}$, the product of the two matrices is proportional to the two by two unit matrix
\begin{equation}
{\cal M}_{1}(\omega_{\bm q}){\cal M}_{2}(\omega_{\bm q})=\frac{1}{2}\omega_{\bm{q}}^{4}I_{2\times2}.
\end{equation}
\noindent Here we have used a simple relation between the single-spinon dispersion obtained from the SBMFT and $\omega_{\bm q}$:
\begin{equation}
\varepsilon^2_{\bm{q}-\frac{\bm{Q}}{2}} + \varepsilon^2_{\bm{q}+\frac{\bm{Q}}{2}} = {1\over 2} \omega_{\bm q}^2.\label{dispS}
\end{equation}
In other words, Eqs.~(\ref{eq:le1}) and (\ref{eq:le2}) are satisfied for any choice of $R_{2}$, $R_{4}$ with $R_{1}$, $R_{3}$ determined by Eq.~(\ref{eq:le1}) when $\omega=\omega_{\bm q}$. Given that $\omega_{\bm q}$ is the single-magnon dispersion of the LSWT,
this demonstrates that the poles of the RPA propagator coincide with the poles of the LSWT [see Fig.~\ref{Sgs}~(a)].
In addition, as shown in Fig.~\ref{Sgs} (b), the spectral weight of the magnon peak, defined as $W(\bm q) = \int d\omega S(\bm q, \omega)$, is also exactly captured by the two diagrams in Fig.~\ref{feynman}~(a) and (b). We have confirmed that the same conclusion holds for the anisotropic XXZ Heisenberg model. We note that there are other diagrams (or order $1/N$ and higher) that scale as $S^0$. Consequently, it is surprising that only the two diagrams in Figs.~\ref{feynman}~(a) and (b) are required to obtain the {\it exact} magnetic susceptibility in the large-$S$ limit.
\section{Discussion}
\label{Disc}
In summary, we have shown that it is necessary to go beyond the SP level of the Schwinger boson theory of the
triangular lattice antiferromagnet in order to capture the correct collective modes in the large-$S$ limit.
These modes are two-spinon bound states generated by the interaction of spinons with the auxiliary fields (emergent gauge fields). The magnon energies are determined by the poles of the RPA propagator. This result must be contrasted with the dynamical susceptibility at the SP level, where the quasi-particle dispersion relation coincides with the single-spinon dispersion.
Although we have not shown it in this manuscript, this conclusion remains valid for the one-singlet bond AA decomposition \cite{Arovas1988,Read1991,Sachdev1991} of the Heisenberg interaction and for other non-collinear magnetically ordered states of frustrated Heisenberg Hamiltonians.
This result, along with the long wave-length limit of the $S=1/2$ theory that we presented in Ref.~[\onlinecite{Ghioldi2018}], demonstrate that the Schwinger boson theory can correctly capture the low-energy magnons of the underlying magnetically ordered state.
In addition, unlike the semiclassical $1/S$ expansion, the Schwinger boson theory is well-suited for describing the higher energy continuum associated with the formation of two-spinon bound states (magnons) with long confinement length scale. Given that this is the expected scenario for magnetically ordered states in the proximity of a QMP, we conclude that the Schwinger boson theory can be a more adequate tool for describing the spin dynamics of frustrated magnets with strong quantum fluctuations.
While we have shown that the correct classical limit of theory can be captured by including only the $1/N$ correction corresponding to the Feynman diagram of Fig.~\ref{feynman}~(b), the other $1/N$ diagrams of Fig.~\ref{feynman}
may play an significant role in a {\it quantitative} description of the dynamical spin structure factor in the presence of strong quantum effect. We note that the diagram shown in Fig.~\ref{feynman}~(c) corresponds to a vertex renormalization, while the two diagrams shown in Figs.~\ref{feynman}~(d) and (e) correspond to a renormalization of the single-spinon propagator.
In other words, we expect that these diagrams should renormalize the single-spinon dispersion along with the two-spinon continuum and the single-magnon (two-spinon bound state) dispersion. Magnon-magnon interaction effects are captured by diagrams of order $1/N^2$ and higher.~\cite{Ghioldi2018}
Finally, it is interesting to note that the situation is qualitatively different for collinear magnetic orderings of Heisenberg magnets, like the square lattice Heisenberg antiferromagnet, because of the residual U(1) symmetry group.
As it was explained in Ref.~\onlinecite{Ghioldi2018}, the bubbles of the Feynman diagram shown in Fig.~\ref{feynman}~(b)
vanish for the transverse components of the dynamical susceptibility due to this U(1) symmetry. This cancellation implies that the $1/N$ contribution that we considered in this manuscript only corrects the longitudinal component of the magnetic susceptibility. In other words, unlike the case of the non-collinear orderings that we considered here, the SP contribution to the transverse components of the magnetic susceptibility is not corrected by the $1/N$ contribution shown in Fig.~\ref{feynman}~(b). However, it is still true that the poles of the RPA propagator coincide with the single-magnon poles of the LSWT. We note that the SP spinon dispersion is half of the single-magnon dispersion in the large-$S$ limit: $\varepsilon_{\bm{q}+\frac{\bm{Q}}{2}}=\frac{1}{2} \omega_{\bm q}$. However, the missing factor of two is recovered, $\omega_{\bm q}=2\varepsilon_{\bm{q}+\frac{\bm{Q}}{2}}$, in the dispersion of the poles of the RPA propagator through equation \eqref{dispS} [$\varepsilon_{\bm{q}-\frac{\bm{Q}}{2}} = \varepsilon_{\bm{q}+\frac{\bm{Q}}{2}}$ for ${\bm Q}=(\pi,\pi)$)].\footnote{ Alternatively, an AA decomposition\cite{Arovas1988,Auerbach1994} of the Heisenberg term leads to a SP spinon dispersion that already coincides with the single-magnon dispersion: $\omega_{\bm q}=\varepsilon^{AA}_{\bm{q}+\frac{\bm{Q}}{2}}.$} It is also important to note that the SP expansions of collinear and non-collinear orderings {\it cannot} be continuously connected because the fluctuation matrix is not semi-positive defined around the Lifshitz transition point that connects both types of magnetic orderings.\cite{Manuel1999} In other words, the result that we presented here {\it cannot} be extended to collinear cases by taking, directly, the collinear limit of non-collinear magnetic orderings. Work to overcome the $U(1)$ residual symmetry problem for collinear antiferromagnets is in progress.
\begin{acknowledgments}
We thank C. J. Gazza for useful conversations. This work was partially supported by CONICET under grant Nro 364 (PIP2015).
S-S. Z. and C. D. B. were partially supported from the LANL Directed Research and Development program. Y. K. acknowledges the support from Ministry of Science and Technology (MOST) with the grant No. 2016YFA0300500 and No. 2016YFA0300501.
\end{acknowledgments}
\bibliographystyle{apsrev4-1}
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2,877,628,090,109 | arxiv | \section*{ \textnormal{{{\themysectionnumber}.}\hspace{0.4em}\ {#1}}}\label{#2}}
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\title{The\hspace{0.2em}\ de Bruijn--Erd\"{o}s--Hanani\hspace{0.2em}\ theorem}
\author{Nikolai\hspace{0.1em}\ V.\hspace{0.1em}\ Ivanov}
\date{}
\begin{document}
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\maketitle
\vspace*{10ex}
{\renewcommand{\baselinestretch}{1}
\selectfont
\myit{\hspace*{0em}\large Contents}\vspace*{1.25ex} \\
\myit{\phantom{1}1.}\hspace*{0.5em} N.\hspace{0.1em}\ Bourbaki and the\hspace{0.2em}\
\emph{``Kvant''}\hspace{0.2em}\ magazine \hspace*{0.5em} \vspace*{0.25ex}\\
\myit{\phantom{1}2.}\hspace*{0.5em} A solution of the\hspace{0.1em}\ N.\hspace{0.1em}\ Bourbaki exercise \hspace*{0.5em} \vspace*{0.25ex}\\
\myit{\phantom{1}3.}\hspace*{0.5em} The de Bruijn--Erd\"{o}s proof \hspace*{0.5em} \vspace*{0.25ex}\\
\myit{\phantom{1}4.}\hspace*{0.5em} From de Bruijn--Erd\"{o}s to systems of distinct representatives \hspace*{0.5em} \vspace*{0.25ex}\\
\myit{\phantom{1}5.}\hspace*{0.5em} Linear algebra and
the inequality\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$ \hspace*{0.5em} \vspace*{0.25ex}\\
\myit{\phantom{1}6.}\hspace*{0.5em} Hanani's theorem \hspace*{0.5em} \vspace*{0.25ex}\\
\myit{\phantom{1}7.}\hspace*{0.5em} Another proof\hspace{0.1em}\ of\hspace{0.1em}\ Hanani's theorem \hspace*{0.5em} \vspace*{0.25ex}\\
\myit{\phantom{1}8.}\hspace*{0.5em} All the de Bruijn--Erd\"{o}s inequalities \hspace*{0.5em} \vspace*{1.5ex}\\
\myit{References}\hspace*{0.5em} \hspace*{0.5em}
}
\footnotetext{\hspace*{-0.65em}\copyright\
Nikolai\hspace{0.1em}\ V.\hspace{0.2em}\ Ivanov,\hspace{0.4em}\ 2017.\hspace{0.4em}\
Neither the work reported in this paper,\hspace{0.2em}\
nor its preparation were
supported by any governmental
or non-governmental agency,\hspace{0.2em}\
foundation,\hspace{0.2em}\
or institution.}
\vspace*{2ex}
\mynonumsection{Preface}
\vspace*{6pt}
{\small
The present paper is devoted to a somewhat idiosyncratic account of the theorem of
de Bruijn--Erd\"{o}s and Hanani
from the combinatorics of finite geometries and its various proofs.\hspace{0.4em}\
Among the proofs discussed are the original proofs by de Bruijn--Erd\"{o}s
and by Hanani\hspace{0.2em}\ (the latter seems to be largely forgotten,\hspace{0.4em}\
being published in a hard to access journal)
and few others.\hspace{0.4em}\
Each of these proofs sheds new light on the theorem,\hspace{0.4em}\
illustrating the maxim that proofs are more important than the theorems proved.\hspace{0.4em}\
Some proofs and arguments in this paper seem to be new.\hspace{0.4em}\
I\hspace{0.1em}\ explain how one of the proofs was discovered,\hspace{0.4em}\
and how another one could have been discovered.\hspace{0.4em}\
See Sections\hspace{0.2em}\ \ref{reps}\hspace{0.2em}\ and\hspace{0.2em}\ \ref{all}.
I am grateful to\hspace{0.1em}\ F.\hspace{0.1em}\ Petrov for stimulating correspondence
and to\hspace{0.1em}\ M.\hspace{0.1em}\ Prokhorova for careful reading
of this paper\hspace*{0.025em},\hspace{0.2em}\
numerous suggestions,\hspace{0.2em}\
and providing me with copies of\hspace{0.15em}\ H.\hspace{0.1em}\ Hanani's papers\hspace{0.2em}\ \cite{h1,h2}.
}
\renewcommand{\baselinestretch}{1}
\selectfont
\mysection{N.\hspace{0.2em}\ Bourbaki\hspace{0.2em}\ and\hspace{0.2em}\ the\hspace{0.2em}\ \emph{``Kvant''}\hspace{0.2em}\ magazine}{kvant}
\vspace*{6pt}
\myuppar{Problem.}
\emph{Let\hspace{0.2em}\ $E$\hspace{0.1em}\ be a set of $n$ elements.\hspace{0.4em}\
Suppose that $m$ different subsets of\hspace{0.2em}\ $E$
(not equal to\hspace{0.1em}\ $E$ itself\hspace*{0.1em})\hspace{0.2em}\ are selected in such a way
that for every two elements of\hspace{0.2em}\ $E$\hspace{0.1em}\ there is exactly one selected subset
containing both these elements.\hspace{0.4em}\
Prove that\hspace{0.1em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\ }
\emph{When an equality is possible\hspace*{0.05em}?}
\vspace{6pt}
In\hspace{0.2em}\ 1970\hspace{0.2em}\ this problem was included
as the Problem\hspace{0.2em}\ M5\hspace{0.2em}\
in the very first issue of the Soviet\hspace{0.2em}\ \emph{``Kvant''}\hspace{0.2em}\
magazine and attributed to\hspace{0.1em}\ N.\hspace{0.2em}\ Bourbaki\hspace{0.2em}\
\cite{b-70}.\hspace{0.2em}\
The intended audience of the\hspace{0.2em}\ \emph{``Kvant''}\hspace{0.2em}\ magazine\hspace{0.2em}\
(its name means\hspace{0.2em}\ \emph{``Quantum''}\hspace*{0.15em})\hspace{0.2em}\
was the school students in the USSR of the last two-three grades.\hspace{0.2em}\
Nowadays the audacity of the editorial board inspires awe:\hspace{0.2em}\ Problem\hspace{0.1em}\ M5\hspace{0.1em}\ was offered
to this audience exactly as it is cited above,\hspace{0.2em}\ as an abstract problem about
finite sets without any motivation and any hints.\hspace{0.2em}\
The readers were expected to be interested in
this problem and to appreciate its beauty without any crutches.\hspace{0.2em}\
In\hspace{0.2em}\ 1970\hspace{0.2em}\ I was among the intended audience of\hspace{0.2em}\ \emph{``Kvant''},\hspace{0.2em}\
but I was more interested in the foundations of mathematics and in the set theory
than in the combinatorics of finite sets.\hspace{0.2em}\
I easily found this problem in the Russian translation\hspace{0.2em}\ \cite{b-56}\hspace{0.2em}\
of the\hspace{0.2em}\ \emph{``Th\'{e}orie des ensembles''}\hspace{0.2em}\ by\hspace{0.2em}\ N.\hspace{0.2em}\ Bourbaki.\hspace{0.2em}\
It turned out to be the Exercise\hspace{0.2em}\ 12\hspace{0.2em}\
to the section\hspace{0.2em}\ \emph{``Calcul sur les entiers''}.\hspace{0.4em}\
In all editions this exercise is marked as
one of the most difficult\hspace*{0.025em}.
The editors of\hspace{0.2em}\ \emph{``Kvant''}\hspace{0.2em}\ were faithful to\hspace{0.1em}\ N.\hspace{0.1em}\ Bourbaki
in not offering any motivation.\hspace{0.2em}\
But\hspace*{0.025em},\hspace{0.2em}\ in contrast with\hspace{0.2em}\ \emph{``Kvant''},\hspace{0.2em}\ N.\hspace{0.1em}\ Bourbaki split
the result into few steps,\hspace{0.2em}\ offered a hint to the key one,\hspace{0.2em}\ and
stated the expected result in the case of the equality.\hspace{0.2em}\
The first two steps were rather easy,\hspace{0.2em}\
but the hint to key step turned out to be incomprehensible for me.\hspace{0.2em}\
According to the authors of the solution\hspace{0.2em}\ \cite{q-solution}\hspace{0.2em}\
published in\hspace{0.2em}\ \emph{``Kvant''}\hspace{0.3em}\
a few months later\hspace*{0.025em},\hspace{0.2em}\
they followed\hspace{0.2em}\
\emph{``the hints of the author of the problem\hspace*{0.05em},\hspace{0.2em}\ N. Bourbaki\hspace{0.1em}\ himself\hspace*{0.15em}''}\hspace{0.2em}\
and referred to\hspace{0.2em}\ \cite{b-56}.\hspace{0.2em}\
The habit of\hspace{0.2em}\ N. Bourbaki to include in his tract recent results
without attribution as exercises is well known,\hspace{0.3em}\
and was well known in\hspace{0.15em}\ 1970\hspace{0.2em}\ in the Soviet Union.\hspace{0.2em}\
But it seems that neither the editors of the\hspace{0.2em}\ \emph{``Kvant''}\hspace{0.2em}\
magazine,\hspace{0.4em}\ nor the authors of the solution\hspace{0.2em}\ \cite{q-solution}\hspace{0.2em}\
were aware that this result is due to\hspace{0.1em}\
N.G.\hspace{0.2em}\ de Bruijn and\hspace{0.1em}\ P.\hspace{0.2em}\ Erd\"{o}s\hspace{0.2em}\ \cite{db-e}\hspace{0.2em}\
and\hspace{0.1em}\ H.\hspace{0.2em}\ Hanani\hspace{0.2em}\ \cite{h1,h2}.\hspace{0.4em}\
Neither was I before by an accident I returned to this problem in 2016.\hspace{0.2em}\
By this time I was able to immediately recognize that this exercise from\hspace{0.2em}\ \cite{b-56}\hspace{0.2em}\
is about points and lines in a geometry,\hspace{0.2em}\
and this realization quickly lead me to the de Bruijn--Erd\"{o}s paper\hspace{0.2em}\ \cite{db-e}.\hspace{0.2em}\
The exercise turned out to be a quite faithful summary of the de Bruijn--Erd\"{o}s proof\hspace*{0.025em},\hspace{0.2em}\
and the key part of the proof\hspace*{0.025em},\hspace{0.2em}\
summarized by\hspace{0.1em}\ N.\hspace{0.2em}\ Bourbaki as a hint\hspace*{0.025em},\hspace{0.2em}\
turned out to be nearly as obscure as this hint\hspace*{0.025em}.
Here is my translation of this exercise
based on the reprint\hspace{0.2em}\ \cite{b-06}\hspace{0.2em}\
of the\hspace{0.15em}\ 1970\hspace{0.2em}\ edition\hspace{0.2em}\
(where it appears as the Exercise\hspace{0.2em}\ 14\hspace{0.2em}\ to\hspace{0.2em}\ \S\hspace{0.2em}\ 5).\hspace{0.2em}\
It is slightly different from the translation in\hspace{0.2em}\ \cite{b-04}.\hspace{0.2em}\
\myuppar{Exercise.}
\emph{Let\hspace{0.2em}\ $E$\hspace{0.1em}\ be a finite set of $n$ elements,\hspace{0.4em}\
$(a_j)_{1\hspace*{0.1em} \leqslant\hspace*{0.1em} j\hspace*{0.1em} \leqslant\hspace*{0.1em} n}$\hspace{0.2em}\
be the sequence of elements of $E$ arranged in some order\hspace*{0.025em},\hspace{0.4em}\
$(A_i)_{1\hspace*{0.1em} \leqslant\hspace*{0.1em} i\hspace*{0.1em} \leqslant\hspace*{0.1em} m}$\hspace{0.2em}\
be a sequence of parts of\hspace{0.2em}\ $E$\hspace*{-0.15em}.}
(a)\hspace{0.4em}\ \emph{For each index $j$\hspace*{-0.15em},\hspace{0.4em}\
let $k_j$ be the number of indices $i$ such that
$a_j\hspace*{0.2em} \in\hspace*{0.2em} A_i$\hspace*{-0.1em};\hspace{0.4em}\
for each index $i$ let\hspace{0.4em}\
$s_i\hspace*{0.2em} =\hspace*{0.2em} \mathop{\mbox{\textup{Card}\hspace*{0.05em}}}\hspace*{0.1em} (\hspace*{0.05em} A_i\hspace*{0.05em})$\hspace*{-0.2em}.\hspace*{0.2em}\hspace{0.4em}\
Show that}
\[
\quad
\sum_{j\hspace*{0.2em} =\hspace*{0.2em} 1}^n\hspace*{0.2em} k_j
\hspace*{0.4em} =\hspace*{0.4em}
\sum_{i\hspace*{0.2em} =\hspace*{0.2em} 1}^m\hspace*{0.2em} s_i\hspace*{0.4em}.
\]
\vspace{-9pt}
({\hspace*{0.025em}}b)\hspace{0.4em}\ \emph{Suppose that
for each subset\hspace{0.2em}\ $\{\hspace*{0.1em} x\hspace*{0.05em},\hspace*{0.3em} y\hspace*{0.1em}\}$\hspace{0.2em}\
of two elements of\hspace{0.2em}\ $E$\hspace*{-0.15em},\hspace{0.4em}\
there exists one and only one index $i$ such that $x$ and $y$ are contained in $A_i$\hspace*{-0.15em}.\hspace{0.4em}\
Show that if\hspace{0.2em}\
$a_j\hspace*{0.2em} \not\in\hspace*{0.2em} A_i$\hspace*{-0.15em},\hspace{0.4em}\
then\hspace{0.2em}\ $s_i\hspace*{0.2em} \leqslant\hspace*{0.2em} k_j$\hspace*{-0.15em}.\hspace{0.4em}\ }
(c)\hspace{0.4em}\ \emph{Under the assumptions of\hspace{0.4em}\ \textup{({\hspace*{0.025em}}b)},\hspace{0.4em}\
show that\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\ }
({\hspace*{0.05em}}Let $k_n$ be the least of the numbers $k_j$\hspace*{-0.15em};\hspace{0.4em}\
show that one may assume that\hspace*{0.025em},\hspace{0.4em}\ whenever\hspace{0.2em}\
$i\hspace*{0.2em} \leqslant\hspace*{0.2em} k_n\hspace*{0.1em},\hspace*{0.4em} j\hspace*{0.2em} \leqslant\hspace*{0.2em} k_n$\hspace{0.2em}\
and\hspace{0.2em}\ $i\hspace*{0.2em} \neq\hspace*{0.2em} j$\hspace*{-0.15em},\hspace{0.4em}\
one has\hspace{0.2em}\ $a_j\hspace*{0.2em} \not\in\hspace*{0.2em} A_i$\hspace*{-0.15em},\hspace{0.4em}\
and\hspace{0.2em}\
$a_n\hspace*{0.2em} \not\in\hspace*{0.2em} A_j$\hspace{0.2em}\
for all\hspace{0.2em}\ $j\hspace*{0.2em} \geqslant\hspace*{0.2em} k_n$\hspace*{-0.15em}.)
(d)\hspace{0.4em}\ \emph{Under the assumptions of\hspace{0.4em}\ \textup{({\hspace*{0.025em}}b)},\hspace{0.4em}\
show that in order for\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace{0.2em}\ to hold,\hspace{0.4em}\
it is necessary and sufficient that
one of the following two cases occurs:}\hspace{0.4em}\
({\hspace*{0.05em}}i\hspace*{0.05em})\phantom{i}\hspace{0.4em}\
$A_1
\hspace*{0.4em} =\hspace*{0.4em}
\{\hspace*{0.2em} a_1\hspace*{0.05em},\hspace*{0.4em} a_2\hspace*{0.05em},\hspace*{0.4em} \ldots\hspace*{0.05em},\hspace*{0.4em} a_{n\hspace*{0.2em} -\hspace*{0.2em} 1} \hspace*{0.2em}\}$\hspace*{-0.1em},\hspace{0.4em}\
${A_i
\hspace*{0.2em} =\hspace*{0.2em}
\{\hspace*{0.2em} a_{i\hspace*{0.2em} -\hspace*{0.2em} 1}\hspace*{0.05em},\hspace*{0.4em} a_n \hspace*{0.2em}\}}$\hspace{0.4em}\
\emph{for}\hspace{0.4em}\
$i
\hspace*{0.4em} =\hspace*{0.4em}
2\hspace*{0.05em},\hspace*{0.3em} \ldots\hspace*{0.05em},\hspace*{0.3em} n$\hspace*{-0.1em};\hspace*{0.1em}
({\hspace*{0.05em}}ii\hspace*{0.05em})\hspace{0.4em}\
$n\hspace*{0.4em} =\hspace*{0.4em} k\hspace*{0.1em}(k\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.2em} +\hspace*{0.2em} 1$\hspace*{-0.15em},\hspace{0.4em}\
\emph{each\hspace{0.1em}\ $A_i$\hspace{0.1em}\ is a set of\hspace{0.2em}\ $k$\hspace{0.2em}\ elements,\hspace{0.4em}\
and each element of\hspace{0.2em}\ $E$\hspace{0.2em}\ belongs to exactly $k$ sets $A_i$\hspace*{-0.15em}.\hspace{0.4em}\ }
\vspace{6pt}
\myuppar{Remarks.}
Two aspects of this exercise need to be clarified.\hspace{0.4em}\
First\hspace*{0.025em},\hspace{0.4em}\
the parts $A_i$ are implicitly assumed to be different from $E$\hspace*{-0.15em}.\hspace{0.4em}\
Second,\hspace{0.4em}\
the case\hspace{0.2em}\ ({\hspace*{0.05em}}ii\hspace*{0.05em})\hspace{0.2em}\ of the part\hspace{0.1em}\ (d)\hspace{0.2em}\ is expected to hold
only up to renumbering of elements $a_i$ and parts $A_j$\hspace*{-0.15em}.\hspace{0.4em}\
\myuppar{The troubles with the hint\hspace*{0.025em}.}
The parts\hspace{0.1em}\ (a)\hspace{0.1em}\ and\hspace{0.1em}\ (b)\hspace{0.1em}\ of this exercise are rather easy,\hspace{0.4em}\
and there is a hint for the part\hspace{0.1em}\ (c).\hspace{0.4em}\
But for me this hint turned out to be more of a riddle than of a help.\hspace{0.4em}\
It would be quite easy to accept and follow the
suggestion to consider the least of the numbers $k_j$\hspace*{-0.15em}.\hspace{0.2em}\
But why it should be $k_n$\hspace*{-0.1em}?\hspace{0.4em}\
The phrase\hspace{0.2em}\
\emph{``Let\hspace{0.1em}\ $k_n$\hspace{0.1em}\ be the least of the numbers\hspace{0.1em}\ $k_j$\hspace*{-0.2em}''}\hspace{0.2em}\
is fairly hard to interpret\hspace{0.2em}\
(the expressions used in the French original and
in the Russian translation have the same meaning\hspace*{0.025em}).\hspace{0.2em}\
The standard usage of\hspace{0.2em}\ \emph{``Let''}\hspace{0.2em}\
(and of\hspace{0.2em}\ \emph{``Soit''}\hspace{0.2em}\ in French)\hspace{0.2em}\
in mathematics is to introduce new notations.\hspace{0.2em}\
But $k_n$ is already defined.\hspace{0.2em}\
The authors of the solution\hspace{0.2em}\ \cite{q-solution}\hspace{0.2em}\
found a clever way out\hspace*{0.025em}.\hspace{0.4em}\
They introduce the number $k_n$\hspace{0.1em}\ \emph{before}\hspace{0.3em}\
introducing other numbers $k_j$\hspace*{-0.1em}!\hspace{0.4em}\
This trick helps only partially:\hspace{0.2em}\
the question\hspace{0.2em}\ \emph{``Why $k_n$\hspace*{-0.2em}?''}\hspace{0.2em}\ remains.
The de Bruijn--Erd\"{o}s exposition\hspace{0.2em}\ \cite{db-e}\hspace{0.2em}\ is better\hspace*{0.025em}.\hspace{0.2em}\
They write\hspace{0.2em}\
\emph{``Assume now that $k_n$ is the smallest $k_i$ \ldots''}.\hspace*{0.2em}\hspace{0.4em}\
This\hspace{0.1em}\ is\hspace{0.1em}\ less obscure,\hspace{0.2em}\
and amounts to renumbering elements of $E$\hspace*{-0.15em},\hspace{0.4em}\
but\hspace{0.1em}\ leaves the question\hspace{0.2em}\ \emph{``Why $k_n$\hspace*{-0.2em}?''}\hspace{0.4em}\
unanswered.\hspace{0.2em}\
If one manages to put this question aside,\hspace{0.2em}\
there is another riddle:\hspace{0.2em}\ how the subscripts\hspace{0.2em}\ $i\hspace*{0.05em},\hspace*{0.3em} j$\hspace*{-0.2em},\hspace{0.4em}\
which are merely marking the points\hspace{0.2em}\
(and do not even need to be numbers)\hspace{0.2em}\
may be compared with $k_n$\hspace*{-0.15em},\hspace{0.2em}\
which is a genuine characteristic of the point marked by $n$\hspace*{-0.1em}?\hspace{0.4em}\
Perhaps,\hspace{0.2em}\ this difficulty is encountered only by the
categorically minded mathematicians;\hspace{0.2em}\
analysts appear to be quite comfortable with using the values
of a function in its domain of definition.\hspace{0.2em}\
Here de Bruijn and Erd\"{o}s\hspace{0.2em}\ \cite{db-e}\hspace{0.2em}\ are again doing better\hspace*{0.025em}.\hspace{0.4em}\
They write\hspace{0.2em}\
\emph{``Assume\hspace{0.4em}\ \ldots\hspace{0.4em}\ that\hspace{0.2em}\
$A_1$\hspace*{-0.15em},\hspace{0.2em}\ $A_2$\hspace*{-0.15em},\hspace{0.2em}\ \ldots\hspace*{0.1em},\hspace{0.2em}\ $A_{\hspace*{0.1em} k_n}$\hspace{0.2em}\
are lines through $a_n$\hspace*{-0.15em}''}\hspace{0.2em}\
(they call the parts $A_i$ lines).\hspace{0.4em}\
This amounts to renumbering the parts $A_i$\hspace*{-0.15em},\hspace{0.2em}\
and one may wonder why renumbering is treated as an assumption.\hspace{0.2em}\
The trick of the authors of\hspace{0.2em}\ \cite{q-solution}\hspace{0.2em}\ saves the day here for them.\hspace{0.4em}\
They simply denote the $k_n$\hspace{0.1em}\ lines through $a_n$ by\hspace{0.2em}\
$A_1$\hspace*{-0.15em},\hspace{0.2em}\ $A_2$\hspace*{-0.15em},\hspace{0.4em}\ \ldots\hspace*{0.1em},\hspace{0.2em}\ $A_{\hspace*{0.1em} k_n}$\hspace{0.2em}\
and other lines by\hspace{0.2em}\
$A_{\hspace*{0.1em} k_n\hspace*{0.1em} +\hspace*{0.1em} 1}$\hspace*{-0.15em},\hspace{0.2em}\ $A_{\hspace*{0.1em} k_n\hspace*{0.1em} +\hspace*{0.1em} 2}$\hspace*{-0.15em},\hspace{0.4em}\
\ldots\hspace*{0.1em},\hspace{0.2em}\ $A_{m}$\hspace*{-0.15em}.
There is one more riddle in the store.\hspace{0.2em}\
How one uses the assumption that $k_n$ is the least of the numbers $k_j$
in the proof of the claim in the hint\hspace*{0.05em}?\hspace{0.4em}\
One does not\hspace*{0.025em},\hspace{0.2em}\
this claim is true without it\hspace*{0.025em}.
\myuppar{Partially decrypting the hint\hspace*{0.025em}.}
Even if one encounters all these troubles and is not aware of the
de Bruijn--Erd\"{o}s paper\hspace{0.2em}\
({\hspace*{0.05em}}like me in 1970),\hspace{0.2em}\
the hint still may be of some help.\hspace{0.4em}\
The first message is that it is important to know when an element $a_j$
is not in the part $A_i$\hspace*{-0.15em}.\hspace{0.4em}\
Together with the part\hspace{0.2em}\ (b)\hspace{0.2em}\ this suggest that the inequalities\hspace{0.2em}\
$s_{\hspace*{0.05em} i}\hspace*{0.2em} \leqslant\hspace*{0.2em} k_{\hspace*{0.05em} j}$\hspace*{-0.15em},\hspace{0.2em}\
which hold for\hspace{0.2em}\ $a_j\hspace*{0.2em} \not\in\hspace*{0.2em} A_i$\hspace*{-0.2em},\hspace{0.4em}\
should play a key role.\hspace{0.2em}\
Another message is that the least of the numbers $k_{\hspace*{0.05em} j}$ should play some role.\hspace{0.2em}\
After wasting some time assuming that for a given $u$
the number $k_{\hspace*{0.05em} u}$ is minimal among all numbers $k_{\hspace*{0.05em} j}$
and\hspace{0.1em}\ trying to use this minimality to prove
something like stated in the hint\hspace*{0.025em},\hspace{0.2em}\
it is only natural to abandon this assumption and
consider an arbitrary subscript $u$ such that\hspace{0.2em}\
$1\hspace*{0.2em} \leqslant\hspace*{0.2em} u\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
\myuppar{The\hspace{0.2em}\ 1970\hspace{0.2em}\ proof\hspace{0.2em}\ of\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.}
With no more than this limited help from this exercise\hspace{0.2em}\
(in\hspace{0.1em}\ 1970\hspace{0.1em}\ I definitely understood less than in 2016)\hspace{0.2em}\
I managed to prove in the early\hspace{0.1em}\ 1970\hspace{0.1em}\ the inequality\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
Among my schoolmates this qualified as a solution of the Problem\hspace{0.2em}\ M5.\hspace{0.4em}\
This solution was lost long time ago.\hspace{0.4em}\
In April of\hspace{0.2em}\ 2016\hspace{0.2em}\ and another time one year later
I attempted to reconstruct this proof\hspace*{0.025em}.\hspace{0.4em}\
In these attempts I encountered the same difficulties as in\hspace{0.1em}\ 1970,\hspace{0.4em}\
and it is likely that I dealt with them in the same manner\hspace*{0.025em}.\hspace{0.4em}\
At the very least,\hspace{0.2em}\ the resulting proof does not use any tools not known to me at the time,\hspace{0.4em}\
and does not involve any tricks\hspace{0.2em}\
(such as the cyclic ordering of some parts $A_i$ by de Bruijn--Erd\"{o}s)\hspace{0.2em}\
which I was unlikely to discover at the time.\hspace{0.4em}\
It is presented in Section\hspace{0.2em}\ \ref{solution}\hspace{0.2em}\ below.\hspace{0.4em}\
The question\hspace{0.2em}\ \emph{``When an equality is possible\hspace*{0.05em}?''}\hspace{0.2em}\
was considered by my classmates as too vague to be addressed seriously,\hspace{0.4em}\
and this was indirectly admitted by the authors of the solution\hspace{0.2em}\ \cite{q-solution}.\hspace{0.4em}\
If\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace{0.2em}\ n$\hspace*{-0.15em},\hspace{0.4em}\
then\hspace{0.2em}\ (d)\hspace{0.2em}\ easily implies that\hspace{0.2em}\
$A_i\hspace*{0.1em} \cap\hspace*{0.1em} A_j\hspace*{0.2em} \neq\hspace*{0.2em} \emptyset$\hspace{0.2em}\ if\hspace{0.2em}\ $i\hspace*{0.2em} \neq\hspace*{0.2em} j$\hspace*{-0.15em}.\hspace{0.4em}\
In fact\hspace*{0.025em},\hspace{0.3em}\ proving this property is an almost inevitable part of the proof of\hspace{0.2em}\ (d).\hspace{0.4em}\
This property means that the set $E$ together with the parts $A_i$ is a\hspace{0.2em}\
\emph{finite projective plane},\hspace{0.3em}\
possibly degenerate in the case\hspace{0.2em}\ ({\hspace*{0.05em}}i\hspace*{0.05em})\hspace{0.2em}\ of the part\hspace{0.2em}\ (d).\hspace{0.4em}\
Therefore,\hspace{0.4em}\ this question amounts to the classification of
finite projective planes and,\hspace{0.2em}\
to the best of my knowledge,\hspace{0.2em}\ it remains largely open.\hspace{0.4em}\
See the paper by Ch.\hspace{0.2em}\ Weibel\hspace{0.2em}\ \cite{w}\hspace{0.2em}\ for a survey
of the state of the art as of\hspace{0.2em}\ 2007,\hspace{0.2em}\
and\hspace{0.2em}\ \cite{i-planes}\hspace{0.2em}\ for an introduction\hspace{0.2em}\
(not focusing on the finite case).
\myuppar{\emph{``Kvant''}\hspace{0.2em}\ publishes a solution.}
\emph{``Kvant''}\hspace{0.2em}\ published a solution\hspace{0.2em}\ \cite{q-solution}\hspace{0.2em}\ of
the Problem\hspace{0.2em}\ M5\hspace{0.2em}\ in the August or September of 1970,\hspace{0.4em}\
close to the beginning of the school year in the USSR\hspace{0.2em}\
(always September 1).\hspace{0.4em}\
The editors of the problem section wrote\hspace{0.2em}\ (see\hspace{0.2em}\ \cite{q-solution},\hspace{0.4em}\ p.\hspace{0.2em}\ 49):\hspace{0.2em}\
\begin{quote}
The letters to editors indicate that this problem is extremely difficult\hspace*{0.025em},\hspace{0.2em}\
but interesting.\hspace{0.2em}\
As a matter of fact\hspace*{0.025em},\hspace{0.4em}\
here we have two problems:\hspace{0.2em}\ 1)\hspace{0.2em}\ prove that\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em},\hspace{0.4em}\
2)\hspace{0.2em}\ when an equality is possible?
The first problem was completely solved only by\hspace{0.2em}\ \emph{A.\hspace{0.2em}\ Suslin}\hspace{0.2em}\
from the city of Le\-nin\-grad.\hspace{0.2em}\
His proof is based on a basic theorem of the linear algebra:\hspace{0.2em}\
if the number of\hspace{0.1em}\ $n$\hspace*{-0.2em}-vectors is greater than $n$\hspace*{-0.15em},\hspace{0.4em}\
then they are linearly dependent\hspace*{0.025em}.\hspace{0.2em}\
Looking for such a proof will be interesting for whose who are familiar with these notions.\hspace{0.2em}\
Nobody solved completely the second problem.\hspace{0.2em}\
Of course,\hspace{0.2em}\ this is not surprising,\hspace{0.2em}\
since,\hspace{0.2em}\ as it will be explained below,\hspace{0.2em}\
it can be reduced to a well known,\hspace{0.2em}\
but unsolved problem in mathematics.
\end{quote}
Among my schoolmates,\hspace{0.2em}\ these remarks stirred a renewed interest in the problem.\hspace{0.2em}\
A.\hspace{0.2em}\ Suslin\hspace{0.2em}\ was known as a very strong problem solver and as a
winner of the gold medal at 1967
International Mathematical Olympiad.\hspace{0.2em}\
Since only he submitted a complete solution,\hspace{0.2em}\
the problem had to be really difficult\hspace*{0.025em}.\hspace{0.2em}\
Since he used tools going beyond the school level,\hspace{0.2em}\
the problem had to be even more difficult.\hspace{0.2em}\
And this caused a real interest in my unsubmitted
to the\hspace{0.2em}\ \emph{``Kvant''}\hspace{0.2em}\ solution.\hspace{0.2em}\
I had an outline as a sparsely filled with formulas sheet of paper\hspace*{0.025em}.\hspace{0.2em}\
One of my schoolmates borrowed this sheet for few days,\hspace{0.2em}\
and I have not seen it anymore.\hspace{0.2em}\
But I am not aware of any serious attempt
to study the published solution\hspace{0.2em}\ \cite{q-solution}.\hspace{0.2em}\
For me it was almost as condensed and obscure as the N.\hspace{0.2em}\ Bourbaki hint\hspace*{0.025em}.\hspace{0.4em}\
The role of the numbering of elements and parts is overemphasized:
\begin{quote}
Let us pay attention once again to the way we numbered elements and sets.\hspace{0.4em}\
First of all,\hspace{0.2em}\ $k_{\hspace*{0.05em} n}$\hspace{0.1em}\ is the least of the numbers\hspace{0.2em}\
$k_{\hspace*{0.05em} 1}\hspace*{0.05em},\hspace*{0.3em} k_{\hspace*{0.05em} 2}\hspace*{0.05em},\hspace*{0.3em} \ldots\hspace*{0.05em},\hspace*{0.3em} k_{\hspace*{0.05em} n\hspace*{0.1em} -\hspace*{0.1em} 1}$\hspace{0.2em}\
({\hspace*{0.05em}}sic\hspace*{0.05em}!\hspace{0.4em}\ --\hspace{0.2em}\ \emph{N.I.}\hspace*{0.15em}).\hspace{0.4em}\ \ldots
\end{quote}
See\hspace{0.2em}\ \cite{q-solution},\hspace{0.4em}\ p.\hspace{0.2em}\ 51.\hspace{0.4em}\
And I always disliked random numerical examples,\hspace{0.2em}\
which are supposed to help the reader and are
extensively used in\hspace{0.2em}\ \cite{q-solution}.\hspace{0.2em}\
I must admit that I did not even look at the last two pages of\hspace{0.2em}\ \cite{q-solution}\hspace{0.2em}\
before writing these comments,\hspace{0.3em}\
and,\hspace{0.3em}\ in particular\hspace*{0.025em},\hspace{0.3em}\ before writing down the proof\hspace{0.1em}\ in the next section.\hspace{0.4em}\
Surprisingly,\hspace{0.4em}\ it turned out that the proof\hspace{0.2em}\ \cite{q-solution}\hspace{0.2em}\ contains a gap:\hspace{0.4em}\
it is mentioned that\hspace{0.2em}\ $k_{\hspace*{0.05em} n}\hspace*{0.2em} =\hspace*{0.2em} 2$\hspace{0.2em}\ in the situation
described in the case\hspace{0.2em}\ ({\hspace*{0.05em}}i\hspace*{0.05em})\hspace{0.2em}\ of the Bourbaki exercise,\hspace{0.4em}\
but no proof that this is the only possibility is even attempted.\hspace{0.4em}\
\mysection{A\hspace{0.2em}\ solution\hspace{0.2em}\ of\hspace{0.2em}\ the\hspace{0.2em}\ N.\hspace{0.2em}\ Bourbaki\hspace{0.2em}\ exercise}{solution}
\vspace*{6pt}
\myuppar{The terminology and notations.}
In contrast with\hspace{0.1em}\ N.\hspace{0.2em}\ Bourbaki\hspace{0.1em}\ and with the\hspace{0.2em}\ \emph{``Kvant''},\hspace{0.3em}\
I\hspace{0.1em}\ have no reasons to hide the geometric content of this result\hspace*{0.025em}.\hspace{0.2em}\
Following de Bruijn and\hspace{0.1em}\ Erd\"{o}s,\hspace{0.4em}\ I will call the elements of $E$\hspace{0.1em}\ \emph{points}\hspace{0.2em}\
and\hspace{0.1em}\ the sets $A_i$\hspace{0.1em}\ \emph{lines}.\hspace{0.3em}\
Since the lines are assumed to be proper subsets of $E$\hspace*{-0.15em},\hspace{0.3em}\
every point is contained in at least $2$ lines.\hspace{0.4em}\
Indeed,\hspace{0.4em}\ if a point is contained in only one line,\hspace{0.3em}\
then all points are contained in this line,\hspace{0.4em}\
i.e.\hspace{0.2em}\ it is not a proper subset\hspace*{0.025em}.\hspace{0.4em}\
It is convenient
to explicitly introduce a counterpart to the set $E$ of points,\hspace{0.4em}\
namely the set of lines\hspace{0.4em}\
$\mathcal{L}
\hspace*{0.4em} =\hspace*{0.4em}
\{\hspace*{0.1em} A_1\hspace*{0.05em},\hspace*{0.3em} A_2\hspace*{0.05em},\hspace*{0.3em} \ldots\hspace*{0.05em},\hspace*{0.3em} A_m \hspace*{0.1em}\}$\hspace*{-0.15em}.\hspace{0.4em}\
If\hspace{0.1em}\ the case\hspace{0.2em}\ ({\hspace*{0.05em}}i\hspace*{0.05em})\hspace{0.2em}\ of the part\hspace{0.2em}\ (d)\hspace{0.2em}\ of the
Bourbaki exercise occurs,\hspace{0.3em}\
up to renumbering of points and lines,\hspace{0.3em}\
then the pair\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\
is called a\hspace{0.2em}\ \emph{near-pencil}.\hspace{0.4em}\
If the case\hspace{0.2em}\ ({\hspace*{0.05em}}ii\hspace*{0.05em})\hspace{0.2em}\ of the part\hspace{0.2em}\ (d)\hspace{0.2em}\ occurs,\hspace{0.4em}\
then\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\ is called a\hspace{0.2em}\ \emph{projective plane}.\hspace{0.4em}\
I also do not see any reason to follow the outdated fashion of using\hspace{0.2em}\
\emph{numerical}\hspace{0.3em}\ indices\hspace{0.2em}\ (i.e.\hspace{0.2em}\ subscripts),\hspace{0.2em}\
which amounts to ordering objects even when their order is irrelevant\hspace*{0.025em}.\hspace{0.2em}\
Instead of this,\hspace{0.2em}\
for every point $z$
we will denote by $k_{\hspace*{0.05em} z}$ the number of lines containing $z$\hspace*{-0.15em},\hspace{0.3em}\
and for every line $l$
we will denote by $s_{\hspace*{0.05em} l}$ the number of points in $l$\hspace*{-0.2em},\hspace{0.4em}\
i.e. the number of elements of the set $l$\hspace*{-0.2em}.
\myuppar{The part\hspace{0.2em}\ (a)\hspace{0.2em}\ of the Bourbaki exercise.}
With the above notations the part\hspace{0.2em}\ (a)\hspace{0.2em}\ takes the form\vspace*{2pt}
\begin{equation}
\label{sums}
\quad
\sum_{l\hspace*{0.1em} \in\hspace*{0.1em} \mathcal{L}}\hspace*{0.2em} s_{\hspace*{0.05em} l}
\hspace*{0.4em}\off =\hspace*{0.4em}\off
\sum_{\hspace*{0.2em} z\hspace*{0.1em} \in\hspace*{0.1em} E}\hspace*{0.2em} k_{\hspace*{0.05em} z}\hspace*{0.1em}.
\end{equation}
\vspace*{-10pt}
after interchanging the sides.\hspace{0.4em}\
This immediately follows from counting in two different ways the pairs\hspace{0.1em}\
$(z\hspace*{0.05em},\hspace*{0.3em} l\hspace*{0.1em})\hspace*{0.2em} \in\hspace*{0.2em} E\times \mathcal{L}$\hspace{0.2em}\ such that\hspace{0.1em}\ $z\hspace*{0.2em} \in\hspace*{0.2em} l$\hspace*{-0.15em}.\hspace{0.4em}\
\myuppar{The part\hspace{0.2em}\ ({\hspace*{0.025em}}b\hspace*{0.025em})\hspace{0.2em}\ of the Bourbaki exercise.}
For the rest of the paper we will assume that
the assumption of the part\hspace{0.2em}\
({\hspace*{0.025em}}b\hspace*{0.025em})\hspace{0.2em}\
holds,\hspace{0.4em}\
i.e.\hspace{0.2em}\ that for every pair of distinct points there is exactly one line containing both of them.\hspace{0.4em}\
If a line $l$ contains\hspace{0.2em}\ $\leqslant\hspace*{0.2em} 1$\hspace{0.2em}\ points,\hspace{0.4em}\
then removing $l$ from the set of lines
does not affects this assumption,\hspace{0.4em}\
and at the same time decreases number of lines by $1$\hspace*{-0.15em}.\hspace{0.4em}\
Hence we may assume for the rest of the paper that
every line contains at least $2$ points.\hspace{0.4em}\
With the above notations
the part\hspace{0.2em}\
({\hspace*{0.025em}}b\hspace*{0.025em})\hspace{0.2em}\
takes the form\vspace*{2pt}
\begin{equation*}
\quad
\mbox{ If }\quad
z\hspace*{0.2em} \not\in\hspace*{0.2em} l\hspace*{0.05em},\quad
\mbox{ then }\quad
s_{\hspace*{0.05em} l}\hspace*{0.2em} \leqslant\hspace*{0.2em} k_{\hspace*{0.05em} z}\hspace*{0.2em}.
\end{equation*}
\vspace*{-10pt}
We will call these inequalities
the\hspace{0.2em}\ \emph{de~Bruijn--Erd\"{o}s\ {\hspace*{0.1em}}inequalities}.\hspace{0.4em}\
In order to prove the de~Bruijn--Erd\"{o}s\ inequalities,\hspace{0.4em}\
suppose that\hspace{0.2em}\ $z\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace*{-0.15em}.\hspace{0.4em}\
Then for every\hspace{0.1em}\ $y\hspace*{0.2em} \in\hspace*{0.2em} l$\hspace{0.1em}\ there is a unique line
containing\hspace{0.1em}\ $\{\hspace*{0.1em} z\hspace*{0.05em},\hspace*{0.3em} y \hspace*{0.1em}\}$\hspace{0.1em}\ and it is different from $l$
because\hspace{0.2em}\ $z\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace*{-0.2em}.\hspace{0.4em}\
These lines are pairwise distinct because if\hspace{0.2em}\
$y\hspace*{0.05em},\hspace*{0.3em} y'\hspace*{0.2em} \in\hspace*{0.2em} l$\hspace{0.2em}\ and\hspace{0.2em}\ $y\hspace*{0.2em} \neq\hspace*{0.2em} y'$\hspace*{-0.2em},\hspace{0.4em}\
then $l$ is the only line containing\hspace{0.1em}\ $\{\hspace*{0.1em} y\hspace*{0.05em},\hspace*{0.3em} y' \hspace*{0.1em}\}$\hspace*{-0.15em}.\hspace{0.4em}\
There are\hspace{0.1em}\ is $s_{\hspace*{0.05em} l}$ such lines and all of them contain $z$\hspace*{-0.15em};\hspace{0.4em}\
therefore\hspace{0.2em}\ $s_{\hspace*{0.05em} l}\hspace*{0.2em} \leqslant\hspace*{0.2em} k_{\hspace*{0.05em} z}$\hspace*{-0.15em}.\hspace{0.4em}\
\myuppar{Lines through an arbitrary point\hspace*{0.025em}.}
Let\hspace{0.2em}\ $u\hspace*{0.2em} \in\hspace*{0.2em} E$\hspace{0.2em}\ be an arbitrary point\hspace*{0.025em},\hspace{0.4em}\
let\hspace{0.2em}\ $p\hspace*{0.2em} =\hspace*{0.2em} k_{\hspace*{0.05em} u}$\hspace{0.2em}\ be the number of lines containing $u$\hspace*{-0.15em},\hspace{0.4em}\
and let $\mathcal{U}$ be the set of these lines.\hspace{0.4em}\
By the definition of $\mathcal{U}$\hspace*{-0.2em},\hspace{0.4em}\
if a line $l$\hspace{0.1em}\ is not in\hspace{0.1em}\ $\mathcal{U}$\hspace*{-0.2em},\hspace{0.4em}\
then\hspace{0.2em}\ $u\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace*{-0.2em}.\hspace{0.4em}\
For every\hspace{0.2em}\
$l\hspace*{0.2em} \not\in\hspace*{0.2em} \mathcal{U}$\hspace{0.2em}\
we have the de~Bruijn--Erd\"{o}s\ inequality\hspace{0.2em}\
$s_{\hspace*{0.05em} l}\hspace*{0.2em} \leqslant\hspace*{0.2em} k_{\hspace*{0.05em} u}$\hspace*{-0.15em}.\hspace{0.4em}\
By summing all these inequalities and taking into account that there
are\hspace{0.2em}\ $m\hspace*{0.2em} -\hspace*{0.2em} p$\hspace{0.2em}\ lines not\hspace{0.1em}\ belonging\hspace{0.1em}\ to $\mathcal{U}$\hspace*{-0.2em},\hspace{0.4em}\
we see that\vspace*{2pt}
\begin{equation}
\label{s-upper-estimate}
\quad
\sum_{l\hspace*{0.2em} \not\in\hspace*{0.2em} \mathcal{U}} s_{\hspace*{0.05em} l}
\hspace*{0.4em}\off \leqslant\hspace*{0.4em}\off
(m\hspace*{0.2em} -\hspace*{0.2em} p)\hspace*{0.1em} k_u\hspace*{0.2em}.
\end{equation}
\vspace*{-10pt}
Since every set of the form\hspace{0.1em}\
$\{\hspace*{0.1em} u\hspace*{0.05em},\hspace*{0.3em} y \hspace*{0.15em}\}$\hspace{0.1em}\ with\hspace{0.1em}\
$y\hspace*{0.2em} \neq\hspace*{0.2em} u$\hspace{0.1em}\
is contained in one and only one line,\hspace{0.4em}\
the sets\hspace{0.2em}\
$l\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \{\hspace*{0.1em} u\hspace*{0.15em}\}$\hspace{0.2em}\
with\hspace{0.2em}\ $l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}$\hspace{0.2em}\
are pairwise disjoint and form a partition of\hspace{0.2em}\
$E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \{\hspace*{0.1em} u\hspace*{0.15em}\}$\hspace*{-0.15em}.\hspace{0.4em}\
Since we assumed that\hspace{0.2em}\ $s_l\hspace*{0.2em} \geqslant\hspace*{0.2em} 2$\hspace{0.2em}\ for all lines $l$\hspace*{-0.2em},\hspace{0.4em}\
all these sets are non-empty.\hspace{0.4em}\
Let $U$ be a set of representatives of these sets.\hspace{0.4em}\
In other terms,\hspace{0.2em}\ $U$ is contained\hspace{0.1em}\ in\hspace{0.2em}\
$E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \{\hspace*{0.1em} u\hspace*{0.15em}\}$\hspace{0.2em}\
and intersects every set\hspace{0.2em}\
$l\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \{\hspace*{0.1em} u\hspace*{0.15em}\}$\hspace{0.2em}\
with\hspace{0.2em}\ $l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}$\hspace{0.2em}\
in exactly $1$ point\hspace*{0.025em}.\hspace{0.4em}\
In particular\hspace*{0.025em},\hspace{0.4em}\ $U$ consists of exactly $p$ points.\hspace{0.4em}\
If\hspace{0.2em}\
$(\hspace*{0.05em} l\hspace*{0.05em},\hspace*{0.3em} z\hspace*{0.05em})\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}\hspace*{0.1em} \times\hspace*{0.1em} U$\hspace{0.2em}\
and\hspace{0.2em}\
$z\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace*{-0.15em},\hspace{0.4em}\
then the de~Bruijn--Erd\"{o}s\ inequality\hspace{0.2em}\
$s_{\hspace*{0.05em} l}\hspace*{0.2em} \leqslant\hspace*{0.2em} k_{\hspace*{0.05em} z}$\hspace{0.2em}\ holds.\hspace{0.4em}\
There are $p\hspace*{0.05em}(p\hspace*{0.2em} -\hspace*{0.2em} 1)$ of such pairs\hspace{0.2em}\
$(\hspace*{0.05em} l\hspace*{0.05em},\hspace*{0.3em} z\hspace*{0.05em})$\hspace{0.2em}\ and hence\hspace{0.2em}\
$p\hspace*{0.05em}(p\hspace*{0.2em} -\hspace*{0.2em} 1)$\hspace{0.2em}\ of such inequalities.\hspace{0.4em}\
For each\hspace{0.1em}\ $l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}$\hspace{0.1em}\
the number $s_{\hspace*{0.05em} l}$ occurs\hspace{0.1em}\ $p\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.1em}\ times in the left hand sides
of them,\hspace{0.3em}\
and for each\hspace{0.1em}\ $z\hspace*{0.2em} \in\hspace*{0.2em} U$\hspace{0.1em}\ the number $k_{\hspace*{0.05em} z}$
occurs\hspace{0.1em}\ $p\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.1em}\ times in the right hand sides.\hspace{0.4em}\
Hence the sum of all these inequalities is\vspace*{2pt}
\begin{equation}
\label{sum-all-pairs}
\quad
\sum_{l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}} (p\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.1em} s_{\hspace*{0.05em} l}
\hspace*{0.4em}\off \leqslant\hspace*{0.4em}\off
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} U} (p\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.1em} k_{\hspace*{0.05em} z}\hspace*{0.2em}.
\end{equation}
\vspace*{-10pt}
After dividing\hspace{0.3em}\ (\ref{sum-all-pairs})\hspace{0.3em}\
by\hspace{0.3em}\ $p\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.3em}\ we get\vspace*{2pt}
\begin{equation}
\label{sum-divided}
\quad
\sum_{l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}} s_{\hspace*{0.05em} l}
\hspace*{0.4em}\off \leqslant\hspace*{0.4em}\off
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} U} k_{\hspace*{0.05em} z}\hspace*{0.2em}.
\end{equation}
\vspace*{-10pt}
Now it is only natural to take the sum of the inequalities\hspace{0.2em}\
(\ref{s-upper-estimate})\hspace{0.2em}\ and\hspace{0.2em}\ (\ref{sum-divided})\hspace{0.2em}\
and conclude that\vspace*{2pt}
\begin{equation}
\label{sum-all-s-estimate}
\quad
\sum_{l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}} s_{\hspace*{0.05em} l}
\hspace*{0.4em}\off \leqslant\hspace*{0.4em}\off
(m\hspace*{0.2em} -\hspace*{0.2em} p)\hspace*{0.1em} k_{\hspace*{0.05em} u}
\hspace*{0.4em} +\hspace*{0.4em}
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} U} k_{\hspace*{0.05em} z}\hspace*{0.2em}.
\end{equation}
\vspace*{-10pt}
The left hand side of the inequality\hspace{0.2em}\ (\ref{sum-all-s-estimate})\hspace{0.2em}\
is the same as the left hand side of the equality\hspace{0.2em}\ (\ref{sums}).\hspace{0.2em}\
The right hand side of\hspace{0.2em}\ (\ref{sum-all-s-estimate})\hspace{0.2em}\
can be compared with the right hand side of the equality\hspace{0.2em}\ (\ref{sums})\hspace{0.3em}\
if\hspace{0.2em}\ $k_{\hspace*{0.05em} u}$\hspace{0.2em}\ is\hspace{0.1em}\ the least among the numbers\hspace{0.2em}\ $k_{\hspace*{0.05em} z}$\hspace{0.2em}\
and\hspace{0.2em}\ $m\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
\myuppar{Proof\hspace{0.1em}\ of\hspace{0.1em}\ the\hspace{0.1em}\ inequality\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.}
Now we are ready to do the part\hspace{0.2em}\ (c)\hspace{0.2em}\ of the Bourbaki exercise.\hspace{0.4em}\
Let\hspace{0.2em}\ $u\hspace*{0.2em} \in\hspace*{0.2em} E$\hspace{0.2em}\ be a point such that $k_{\hspace*{0.05em} u}$
is the least of the numbers $k_{\hspace*{0.05em} z}$
over all points\hspace{0.2em}\ $z\hspace*{0.2em} \in\hspace*{0.2em} E$\hspace*{-0.15em}.\hspace{0.4em}\
Then\vspace*{2pt}
\begin{equation}
\label{k-lower-estimate}
\quad
(m\hspace*{0.2em} -\hspace*{0.2em} p)\hspace*{0.1em} k_{\hspace*{0.05em} u}
\hspace*{0.4em}\off \leqslant\hspace*{0.4em}\off
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} Y} k_{\hspace*{0.05em} z}\hspace*{0.2em}.
\end{equation}
\vspace*{-10pt}
for every subset\hspace{0.2em}\ $Y\hspace*{0.2em} \subset\hspace*{0.2em} E$\hspace{0.2em}\ consisting of\hspace{0.2em}\ $m\hspace*{0.2em} -\hspace*{0.2em} p$\hspace{0.2em}\ points.\hspace{0.4em}\
Suppose that\hspace{0.2em}\ $m\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
Then the subset $Y$ can be chosen to be disjoint from $U$\hspace{0.2em}\
({\hspace*{0.05em}}because $U$ consists of $p$ points).\hspace{0.4em}\
Let us choose an arbitrary $Y$ disjoint from $U$ and\hspace{0.1em}\ let\hspace{0.2em}\
$Z\hspace*{0.4em} =\hspace*{0.4em} Y\hspace*{0.2em} \cup\hspace*{0.2em} U$\hspace*{-0.2em}.\hspace{0.4em}\
Then $Z$ is a subset of $E$ consisting of $m$ points and
the inequalities\hspace{0.2em}\
(\ref{sum-all-s-estimate})\hspace{0.2em}\ and\hspace{0.2em}\ (\ref{k-lower-estimate})\hspace{0.2em}\
imply that\vspace*{2pt}
\begin{equation}
\label{final-estimate}
\quad
\sum_{l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}} s_{\hspace*{0.05em} l}
\hspace*{0.4em}\off \leqslant\hspace*{0.4em}\off
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} Z} k_{\hspace*{0.05em} z}
\hspace*{0.4em}\off \leqslant\hspace*{0.4em}\off
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} E} k_{\hspace*{0.05em} z}\hspace*{0.2em},
\end{equation}
\vspace*{-10pt}
where the last inequality is strict unless\hspace{0.2em}\ $Z\hspace*{0.4em} =\hspace*{0.4em} E$\hspace*{-0.15em}.\hspace{0.4em}\
In view of\hspace{0.2em}\ (\ref{sums})\hspace{0.2em}\ this inequality cannot be strict
and hence\hspace{0.2em}\ $Z\hspace*{0.4em} =\hspace*{0.4em} E$\hspace{0.2em}\ and\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
Since\hspace{0.2em}\ $m\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace{0.2em}\ implies that\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em},\hspace{0.4em}\
we see that\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
\myuppar{The case\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em}.}
After the work done in the proofs of\hspace{0.2em}\ (a),\hspace{0.2em}\ ({\hspace*{0.025em}}b\hspace*{0.025em}),\hspace{0.2em}\ and\hspace{0.2em}\ (c),\hspace{0.2em}\
the part\hspace{0.2em}\ (d\hspace*{0.025em})\hspace{0.2em}\ nearly proves itself\hspace*{0.025em}.\hspace{0.2em}\
As we will see,\hspace{0.2em}\
in this case all inequalities\hspace{0.2em}\
(\ref{s-upper-estimate})\hspace{0.2em}\ --\hspace{0.2em}\ (\ref{final-estimate})\hspace{0.2em}\
are,\hspace{0.4em}\ in fact\hspace*{0.025em},\hspace{0.4em}\ equalities.
By\hspace{0.2em}\ (\ref{sums})\hspace{0.2em}\ the leftmost and the rightmost sums in\hspace{0.2em}\
(\ref{final-estimate})\hspace{0.2em}\ are equal.\hspace{0.4em}\
It follows that\hspace{0.2em}\ $Z\hspace*{0.4em} =\hspace*{0.4em} E$\hspace{0.2em}\ and hence\hspace{0.2em}\
$Y\hspace*{0.4em} =\hspace*{0.4em} E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} U$\hspace*{-0.2em}.\hspace{0.4em}\
Moreover\hspace*{0.025em},\hspace{0.4em}\
the sides of each of the inequalities\hspace{0.2em}\
(\ref{sum-all-s-estimate})\hspace{0.2em}\ and\hspace{0.2em}\ (\ref{k-lower-estimate})\hspace{0.2em}\
are equal.\hspace{0.4em}\
Since $k_{\hspace*{0.05em} u}$ is the least of the numbers $k_{\hspace*{0.05em} z}$\hspace*{-0.15em},\hspace{0.4em}\
the equality of the sides of\hspace{0.2em}\ (\ref{k-lower-estimate})\hspace{0.2em}\
implies that\vspace*{2pt}
\begin{equation}
\label{ku-k-not-u}
\quad
k_{\hspace*{0.05em} u}
\hspace*{0.4em} =\hspace*{0.4em}
k_{\hspace*{0.05em} z}
\quad\mbox{ for\hspace{0.2em}\ all }\quad
z\hspace*{0.2em} \in\hspace*{0.2em} Y\hspace*{0.4em} =\hspace*{0.4em} E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} U\hspace*{0.2em}.
\end{equation}
\vspace*{-10pt}
The fact that the sides of\hspace{0.2em}\
(\ref{sum-all-s-estimate})\hspace{0.2em}\
are equal implies that the sides of each of the inequalities\hspace{0.2em}\
(\ref{s-upper-estimate})\hspace{0.2em}\ and\hspace{0.2em}\ (\ref{sum-divided})\hspace{0.2em}\
are equal also.\hspace{0.4em}\
The equality of the sides of\hspace{0.2em}\ (\ref{s-upper-estimate})\hspace{0.2em}\
implies that\hspace{0.2em}\ \vspace*{2pt}
\begin{equation}
\label{s-not-ku}
\quad
s_{\hspace*{0.05em} l}
\hspace*{0.4em} =\hspace*{0.4em}
k_{\hspace*{0.05em} u}
\quad\mbox{ for\hspace{0.2em}\ all }\quad
l\hspace*{0.2em} \not\in\hspace*{0.2em} \mathcal{U}\hspace*{0.2em}.
\end{equation}
\vspace*{-10pt}
Since the sides of\hspace{0.2em}\ (\ref{sum-divided})\hspace{0.2em}\ are equal,\hspace{0.4em}\
the sides of\hspace{0.2em}\ (\ref{sum-all-pairs})\hspace{0.2em}\
are also equal.\hspace{0.4em}\
Since\hspace{0.2em}\ (\ref{sum-all-pairs})\hspace{0.2em}\ is the sum of the inequalities\hspace{0.2em}\
$s_{\hspace*{0.05em} l}\hspace*{0.2em} \leqslant\hspace*{0.2em} k_{\hspace*{0.05em} z}$\hspace{0.2em}\
over all pairs\hspace{0.2em}\
$(\hspace*{0.05em} l\hspace*{0.05em},\hspace*{0.3em} z\hspace*{0.05em})\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}\hspace*{0.1em} \times\hspace*{0.1em} U$\hspace{0.2em}\
such that\hspace{0.2em}\ $z\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace*{-0.15em},\hspace{0.4em}\
the equality of the sides of\hspace{0.2em}\ (\ref{sum-all-pairs})\hspace{0.2em}\
implies that\hspace{0.2em}\
$s_{\hspace*{0.05em} l}\hspace*{0.2em} =\hspace*{0.2em} k_{\hspace*{0.05em} z}$\hspace{0.2em}\
for all\hspace{0.2em}\
$(\hspace*{0.05em} l\hspace*{0.05em},\hspace*{0.3em} z\hspace*{0.05em})\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}\hspace*{0.1em} \times\hspace*{0.1em} U$\hspace{0.2em}\
such that\hspace{0.2em}\ $z\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace*{-0.15em}.\hspace{0.4em}\
Equivalently,\vspace*{2pt}
\begin{equation}
\label{s-k-uu}
\quad
s_{\hspace*{0.05em} l}
\hspace*{0.4em} =\hspace*{0.4em}
k_{\hspace*{0.05em} z}
\quad\mbox{ if }\quad
l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}
\quad\mbox{ and }\quad
z\hspace*{0.2em} \in\hspace*{0.2em} U\hspace*{0.2em} \smallsetminus\hspace*{0.2em} l\hspace*{0.2em}.
\end{equation}
\vspace*{-10pt}
The rest of the proof splits into two subcases depending on\hspace{0.1em}\ if\hspace{0.2em}\
$p\hspace*{0.2em} =\hspace*{0.2em} 2$\hspace{0.2em}\ or\hspace{0.2em}\ $p\hspace*{0.2em} \geqslant\hspace*{0.2em} 3$\hspace*{-0.15em}.
\myuppar{The subcase\hspace{0.2em}\ $p\hspace*{0.2em} =\hspace*{0.2em} 2$\hspace*{-0.15em}.}
In this case\hspace{0.2em}\ $\mathcal{U}\hspace*{0.4em} =\hspace*{0.4em} \{\hspace*{0.15em} l\hspace*{0.05em},\hspace*{0.3em} l' \hspace*{0.2em}\}$\hspace{0.2em}\
for some\hspace{0.2em}\ $l\hspace*{0.05em},\hspace*{0.3em} l'$\hspace{0.2em}\ and\hspace{0.1em}\ hence\hspace{0.2em}\
$E\hspace*{0.2em} =\hspace*{0.2em} l\hspace*{0.2em} \cup\hspace*{0.2em} l'$\hspace*{-0.2em}.\hspace{0.4em}\
It follows that every line different from\hspace{0.2em}\ $l\hspace*{0.05em},\hspace*{0.3em} l'$\hspace{0.2em}\
contains only $2$ points,\hspace{0.4em}\ namely
the points of its intersection with the lines $l\hspace*{0.05em},\hspace*{0.3em} l'$\hspace*{-0.2em}.\hspace*{0.2em}\hspace{0.4em}\
If\hspace{0.4em}\ $s_{\hspace*{0.05em} l}\hspace*{0.05em},\hspace*{0.4em} s_{\hspace*{0.05em} l'}\hspace*{0.3em} \geqslant\hspace*{0.3em} 3$\hspace*{-0.15em},\hspace{0.4em}\
then there are at least $4$ points\hspace{0.2em}\ $z\hspace*{0.2em} \neq\hspace*{0.2em} u$ and the part\hspace{0.2em}\
({\hspace*{0.025em}}b)\hspace{0.2em}\ implies that\hspace{0.2em}\
$k_{\hspace*{0.05em} z}\hspace*{0.2em} \geqslant\hspace*{0.2em} 3$\hspace{0.2em}\ for every\hspace{0.2em}\ $z\hspace*{0.2em} \neq\hspace*{0.2em} u$\hspace*{-0.15em}.\hspace{0.4em}\
On the other hand,\hspace{0.2em}\
(\ref{ku-k-not-u})\hspace{0.2em}\ implies that\vspace*{2pt}
\[
\quad
k_{\hspace*{0.05em} z}
\hspace*{0.4em} =\hspace*{0.4em}
k_{\hspace*{0.05em} u}
\hspace*{0.4em} =\hspace*{0.4em}
p
\hspace*{0.4em} =\hspace*{0.4em}
2
\]
\vspace*{-10pt}
for every\hspace{0.2em}\ $z\hspace*{0.2em} \not\in\hspace{0.2em}\ U$\hspace*{-0.15em}.\hspace{0.4em}\
But $U$ consists of only two points and hence\hspace{0.2em}\
$k_{\hspace*{0.05em} z}\hspace*{0.2em} \geqslant\hspace*{0.2em} 3$\hspace{0.2em}\ for no more than two points $z$\hspace*{-0.15em}.\hspace{0.4em}\
The contradiction shows that
either\hspace{0.2em}\ $s_{\hspace*{0.05em} l}\hspace*{0.2em} =\hspace*{0.2em} 2$\hspace{0.2em}\ or\hspace{0.2em}\ $s_{\hspace*{0.05em} l'}\hspace*{0.2em} =\hspace*{0.2em} 2$\hspace*{-0.15em}.\hspace{0.4em}\
We may assume that\hspace{0.2em}\ $s_{\hspace*{0.05em} l}\hspace*{0.2em} =\hspace*{0.2em} 2$\hspace*{-0.15em}.\hspace{0.4em}\
Then\hspace{0.2em}\ $l\hspace*{0.2em} =\hspace*{0.2em} \{\hspace*{0.1em} u\hspace*{0.05em},\hspace*{0.3em} a \hspace*{0.15em}\}$\hspace{0.2em}\ for some\hspace{0.2em}\ $a\hspace*{0.2em} \in\hspace*{0.2em} E$\hspace{0.2em}\
and every line different from\hspace{0.2em}\ $l\hspace*{0.05em},\hspace*{0.3em} l'$\hspace{0.2em}\
has the form\hspace{0.2em}\ $\{\hspace*{0.1em} a\hspace*{0.05em},\hspace*{0.3em} z \hspace*{0.1em}\}$\hspace{0.2em}\ with\hspace{0.2em}\
$z\hspace*{0.2em} \in\hspace*{0.2em} l'\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \{\hspace*{0.1em} u\hspace*{0.15em}\}$\hspace*{-0.15em}.\hspace{0.4em}\
It\hspace{0.1em}\ follows that\hspace{0.2em}\ $(\hspace*{0.05em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.05em})$\hspace{0.2em}\
is a near-pencil.\hspace{0.4em}\
\myuppar{The subcase\hspace{0.2em}\ $p\hspace*{0.2em} \geqslant\hspace*{0.2em} 3$\hspace*{-0.15em}.}
The set $U$ is a set of representatives of the sets\hspace{0.2em}\
$l\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \{\hspace*{0.05em} u\hspace*{0.15em}\}$\hspace{0.2em}\
with\hspace{0.2em}\ $l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}$\hspace*{-0.2em}.\hspace{0.4em}\
For any two lines\hspace{0.2em}\ $l\hspace*{0.05em},\hspace*{0.3em} l'\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}$\hspace{0.2em}\
the assumption\hspace{0.2em}\ $p\hspace*{0.2em} \geqslant\hspace*{0.2em} 3$\hspace{0.2em}\
implies that there exists a point\hspace{0.2em}\ $z\hspace*{0.2em} \in\hspace*{0.2em} U$\hspace{0.2em}\
such that\hspace{0.2em}\ $z\hspace*{0.2em} \not\in\hspace*{0.2em} l\hspace*{0.05em},\hspace*{0.3em} l'$\hspace*{-0.2em}.\hspace*{0.2em}\hspace{0.4em}\
If\hspace{0.2em}\ $z$\hspace{0.2em}\ is such a point\hspace*{0.025em},\hspace{0.4em}\
then\hspace{0.2em}\ (\ref{s-k-uu})\hspace{0.2em}\ implies that
\[
\quad
s_{\hspace*{0.05em} l}\hspace*{0.4em} =\hspace*{0.4em} k_{\hspace*{0.05em} z}\hspace*{0.4em} =\hspace*{0.4em} s_{\hspace*{0.05em} l'}\hspace*{0.2em}.
\]
Similarly,\hspace{0.4em}\ if\hspace{0.2em}\ $z\hspace*{0.05em},\hspace*{0.3em} z'\hspace*{0.2em} \in\hspace*{0.2em} U$\hspace*{-0.2em},\hspace{0.4em}\
then there exists a line\hspace{0.2em}\ $l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}$\hspace{0.2em}\
such that\hspace{0.2em}\ $z\hspace*{0.05em},\hspace*{0.3em} z'\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace{0.2em}\ and hence
\[
\quad
k_{\hspace*{0.05em} z}\hspace*{0.4em} =\hspace*{0.4em} s_{\hspace*{0.05em} l}\hspace*{0.4em} =\hspace*{0.4em} k_{\hspace*{0.05em} z'}\hspace*{0.2em}.
\]
It follows that in the subcase\hspace{0.2em}\ $p\hspace*{0.2em} \geqslant\hspace*{0.2em} 3$\hspace{0.2em}\
all numbers\hspace{0.2em}\ $s_{\hspace*{0.05em} l}$\hspace*{-0.15em},\hspace{0.2em}\ $k_{\hspace*{0.05em} z}$\hspace{0.2em}\
with\hspace{0.2em}\ $l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}$\hspace{0.2em}\
and\hspace{0.2em}\ $z\hspace*{0.2em} \in\hspace*{0.2em} U$\hspace{0.2em}\ are equal.\hspace{0.4em}\
Since\hspace{0.2em}\ $k_{\hspace*{0.05em} u}\hspace*{0.2em} =\hspace*{0.2em} p\hspace*{0.2em} \geqslant\hspace*{0.2em} 3$\hspace{0.2em}\
is the smallest of the numbers $k_{\hspace*{0.05em} z}$ over all\hspace{0.2em}\ $z\hspace*{0.2em} \in\hspace*{0.2em} E$\hspace*{-0.15em},\hspace{0.4em}\
it follows that\hspace{0.2em}\
\[
\quad
s_{\hspace*{0.05em} l}
\hspace*{0.4em} =\hspace*{0.4em}
k_{\hspace*{0.05em} z}
\hspace*{0.4em} \geqslant\hspace*{0.4em} 3
\quad\mbox{ for\hspace{0.2em}\ all }\quad
l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}\hspace*{0.05em},\quad
z\hspace*{0.2em} \in\hspace*{0.2em} U.
\]
Let\hspace{0.2em}\ $l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{U}$\hspace*{-0.2em},\hspace{0.4em}\
and let $y$ be the unique element of $U$ contained in $l$\hspace*{-0.2em}.\hspace{0.4em}\
Since\hspace{0.2em}\ $s_{\hspace*{0.05em} l}\hspace*{0.2em} \geqslant\hspace*{0.2em} 3$\hspace*{-0.15em},\hspace{0.4em}\
there exists a point\hspace{0.2em}\ $x\hspace*{0.2em} \in\hspace*{0.2em} l$\hspace{0.2em}\
not equal to\hspace{0.2em}\ $u\hspace*{0.05em},\hspace*{0.3em} y$\hspace*{-0.15em}.\hspace{0.4em}\
We can replace in $U$ the point $y$ by the point $x$
and get a new set of representatives $U'$\hspace*{-0.2em}.\hspace{0.4em}\
Then all previous results apply to $U'$ in the role of $U$\hspace*{-0.2em}.\hspace{0.4em}\
In particular\hspace*{0.025em},\hspace{0.4em}\
$k_{\hspace*{0.05em} x}
\hspace*{0.4em} =\hspace*{0.4em}
k_{\hspace*{0.05em} z}$\hspace{0.4em}\
for\hspace{0.2em}\ all\hspace{0.4em}\
$z\hspace*{0.2em} \in\hspace*{0.2em} U\hspace*{0.1em} \smallsetminus\hspace*{0.1em} l
\hspace*{0.4em} =\hspace*{0.4em}
U'\hspace*{0.1em} \smallsetminus\hspace*{0.1em} l$\hspace{0.4em}\
and hence
\[
\quad
k_{\hspace*{0.05em} x}
\hspace*{0.4em} =\hspace*{0.4em}
k_{\hspace*{0.05em} z}
\quad\mbox{ for\hspace{0.2em}\ all }\quad
z\hspace*{0.2em} \in\hspace*{0.2em} U.
\]
On the other hand,\hspace{0.4em}\ $x\hspace*{0.2em} \not\in\hspace*{0.2em} U$\hspace{0.2em}\ and hence\hspace{0.2em}\
$k_{\hspace*{0.05em} x}\hspace*{0.2em} =\hspace*{0.2em} k_{\hspace*{0.05em} u}$\hspace{0.2em}\ by\hspace{0.2em}\
(\ref{ku-k-not-u})\hspace{0.2em}\ applied to the original set $U$\hspace*{-0.2em}.\hspace{0.4em}\
At the same time\hspace{0.2em}\ (\ref{ku-k-not-u})\hspace{0.2em}\
implies that\hspace{0.2em}\
$k_{\hspace*{0.05em} u}\hspace*{0.2em} =\hspace*{0.2em} k_{\hspace*{0.05em} z}$\hspace{0.2em}\
for all\hspace{0.2em}\ $z\hspace*{0.2em} \not\in\hspace*{0.2em} U$\hspace{0.2em}\
and hence\hspace{0.2em}\
\[
\quad
k_{\hspace*{0.05em} x}
\hspace*{0.4em} =\hspace*{0.4em}
k_{\hspace*{0.05em} z}
\quad\mbox{ for\hspace{0.2em}\ all }\quad
z\hspace*{0.2em} \not\in\hspace*{0.2em} U.
\]
It\hspace{0.1em}\ follows that all numbers $k_{\hspace*{0.05em} z}$ are equal.\hspace{0.4em}\
At the same time by\hspace{0.2em}\ (\ref{s-not-ku})\hspace{0.2em}\ and\hspace{0.2em}\ (\ref{s-k-uu})\hspace{0.2em}\
every $s_{\hspace*{0.05em} l}$\hspace{0.1em}\ is equal to some $k_{\hspace*{0.05em} z}$\hspace*{-0.15em}.\hspace{0.4em}\
It follows that all numbers\hspace{0.2em}\ $s_{\hspace*{0.05em} l}$\hspace*{-0.15em},\hspace{0.2em}\ $k_{\hspace*{0.05em} z}$\hspace{0.2em}\
with\hspace{0.2em}\ $l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}$\hspace{0.2em}\
and\hspace{0.2em}\ $z\hspace*{0.2em} \in\hspace*{0.2em} E$\hspace{0.2em}\ are equal.\hspace{0.4em}\
It remains to apply the following lemma.\hspace{0.4em}\
\myuppar{Lemma\hspace{0.2em}\ 1.}
\emph{If\hspace{0.15em}\ all\hspace{0.1em}\ the numbers\hspace{0.2em}\ $s_{\hspace*{0.05em} l}$\hspace*{-0.15em},\hspace{0.2em}\ $k_{\hspace*{0.05em} z}$\hspace{0.2em}\
are equal,\hspace{0.4em}\
then\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\ is a projective plane.}
\myuppar{Proof\hspace*{0.025em}.}
Let $k$ be the common value of the numbers\hspace{0.2em}\ $s_{\hspace*{0.05em} l}$\hspace*{-0.15em},\hspace{0.2em}\ $k_{\hspace*{0.05em} z}$\hspace*{-0.15em},\hspace{0.4em}\
and\hspace{0.1em}\ let\hspace{0.2em}\ $y\hspace*{0.2em} \in\hspace*{0.2em} E$\hspace*{-0.15em}.\hspace{0.4em}\
The sets\hspace{0.2em}\
$l\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \{\hspace*{0.1em} y\hspace*{0.15em}\}$\hspace{0.2em}\
with\hspace{0.2em}\ $y\hspace*{0.2em} \in\hspace*{0.2em} l$\hspace{0.2em}\
are pairwise disjoint and form a partition of\hspace{0.2em}\
$E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \{\hspace*{0.1em} y\hspace*{0.15em}\}$\hspace*{-0.15em}.\hspace{0.4em}\
Each of them consists of\hspace{0.2em}\
\[
\quad
s_{\hspace*{0.05em} l}\hspace*{0.2em} -\hspace*{0.2em} 1\hspace*{0.4em} =\hspace*{0.4em} k\hspace*{0.2em} -\hspace*{0.2em} 1
\]
points,\hspace{0.4em}\ and there are\hspace{0.2em}\ $k_{\hspace*{0.05em} y}\hspace*{0.2em} =\hspace*{0.2em} k$\hspace{0.2em}\ such sets.\hspace{0.4em}\
It follows that the number $n$ of elements of $E$ is equal to\hspace{0.2em}\
$k\hspace*{0.05em}(k\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.2em} +\hspace*{0.2em} 1$\hspace*{-0.15em}.\hspace{0.4em}\
Therefore,\hspace{0.4em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\
is a projective plane.\hspace{0.4em}\ $\blacksquare$
\myuppar{Remarks.}
A key step of this solution and the solution\hspace{0.2em}\ \cite{q-solution}\hspace{0.2em}\
differ from the de-Bruijn--Erd\"{o}s paper in the same way:\hspace{0.2em}\
the cyclic order argument of de~Bruijn--Erd\"{o}s\
(see Section\hspace{0.2em}\ \ref{dbe-proof})\hspace{0.2em}\
is replaced by the inequalities\hspace{0.2em}\ (\ref{sum-all-pairs})\hspace{0.2em}\ and\hspace{0.2em}\ (\ref{sum-divided}).\hspace{0.4em}\
\mysection{The\hspace{0.2em}\ de Bruijn--Erd\"{o}s\hspace{0.2em}\ proof}{dbe-proof}
\vspace*{12pt}
Since the proof presented in Section\hspace{0.2em}\ \ref{solution}\hspace{0.2em}\ grow out of a
summary of the de Bruijn--Erd\"{o}s proof\hspace*{0.025em},\hspace{0.4em}\
albeit not quite understood,\hspace{0.4em}\
it is not surprising that the two proofs have a lot in common.\hspace{0.4em}\
In the following exposition of the de Bruijn--Erd\"{o}s proof
we will use the notations of Section\hspace{0.2em}\ \ref{solution}\hspace{0.2em}\ and
will refer to Section\hspace{0.2em}\ \ref{solution}\hspace{0.2em}\ for the arguments
which differ from\hspace{0.2em}\ \cite{db-e}\hspace{0.2em}\ only in the notations and the amount of details.\hspace{0.4em}\
The Bruijn--Erd\"{o}s paper is concise on the border of being cryptic.\hspace{0.4em}\
The de Bruijn--Erd\"{o}s proof begins with the parts\hspace{0.2em}\ (a)\hspace{0.2em}\ and\hspace{0.2em}\ ({\hspace*{0.025em}}b\hspace*{0.025em})\hspace{0.2em}\
of the Bourbaki exercise.\hspace{0.4em}\
After this de~Bruijn--Erd\"{o}s\ introduce $k_{\hspace*{0.05em} u}$ as the smallest among all numbers $k_{\hspace*{0.05em} z}$\hspace{0.2em}\
(and denote it by $k_{\hspace*{0.05em} n}$\hspace*{-0.15em}).\hspace{0.4em}\
Then de~Bruijn--Erd\"{o}s\ observe that
it can assumed that every line contains at least two points.\hspace{0.4em}\
Following the notations of Section\hspace{0.2em}\ \ref{solution},\hspace{0.4em}\
let us denote by $\mathcal{U}$ the set of all lines containing $u$\hspace*{-0.15em}.\hspace{0.4em}\
By the de~Bruijn--Erd\"{o}s\ inequalities\hspace{0.2em}\
$s_{\hspace*{0.05em} l}\hspace*{0.2em} \leqslant\hspace*{0.2em} k_{\hspace*{0.05em} u}$\hspace{0.2em}\
for every\hspace{0.2em}\
$l\hspace*{0.2em} \not\in\hspace*{0.2em} \mathcal{U}$\hspace*{-0.2em}.\hspace{0.4em}\
The inequalities\hspace{0.2em}\ (\ref{s-upper-estimate})\hspace{0.2em}\
and\hspace{0.2em}\ (\ref{k-lower-estimate})\hspace{0.2em}\ follows.\hspace{0.4em}\
The following argument plays a role similar to the role of the inequality\hspace{0.2em}\
(\ref{sum-all-pairs}).
\myuppar{The cyclic order argument\hspace*{0.025em}.}
Let\hspace{0.4em}\
$l_{\hspace*{0.05em} 1}\hspace*{0.05em},\hspace*{0.3em} l_{\hspace*{0.05em} 2}\hspace*{0.05em},\hspace*{0.3em} \ldots\hspace*{0.05em},\hspace*{0.3em} l_{\hspace*{0.05em} p}$\hspace{0.4em}\
be a cyclically ordered list of elements of $\mathcal{U}$\hspace*{-0.2em}.\hspace{0.4em}\
We treat the subscripts\hspace{0.2em}\ $1\hspace*{0.05em},\hspace*{0.3em} 2\hspace*{0.05em},\hspace*{0.3em} \ldots\hspace*{0.05em},\hspace*{0.3em} p$\hspace{0.2em}\
as integers\hspace*{-0.2em}\ $\mod\hspace{0.1em}\ p$\hspace*{-0.2em}.\hspace{0.4em}\
For each\hspace{0.4em}\
$i
\hspace*{0.4em} =\hspace*{0.4em}
1\hspace*{0.05em},\hspace*{0.3em} 2\hspace*{0.05em},\hspace*{0.3em} \ldots\hspace*{0.05em},\hspace*{0.3em} p$\hspace{0.4em}\
let us choose some point\hspace{0.2em}\ $a_{\hspace*{0.05em} j}\hspace*{0.2em} \in\hspace*{0.2em} l_{\hspace*{0.05em} j}\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \{\hspace*{0.1em} u \hspace*{0.1em}\}$\hspace{0.2em}\
and\hspace{0.1em}\ let $U$ be the set of these points.\hspace{0.4em}\
Let
\[
\quad
s_{\hspace*{0.05em} i}
\hspace*{0.4em} =\hspace*{0.4em}
s_{\hspace*{0.05em} l_{\hspace*{0.05em} i}}
\hspace*{1.5em}\mbox{ and }\hspace*{1.5em}
k_j
\hspace*{0.4em} =\hspace*{0.4em}
k_{\hspace*{0.05em} a_{\hspace*{0.05em} j}}\hspace*{0.2em}.
\]
Since\hspace{0.2em}\ $a_{\hspace*{0.05em} i\hspace*{0.1em} +\hspace*{0.1em} 1}\hspace*{0.2em} \not\in\hspace*{0.2em} l_{\hspace*{0.05em} i}$\hspace*{-0.15em},\hspace{0.4em}\
by the de~Bruijn--Erd\"{o}s\ inequalities
$s_{\hspace*{0.05em} i}\hspace*{0.2em} \leqslant\hspace*{0.2em} k_{\hspace*{0.05em} i\hspace*{0.1em} +\hspace*{0.1em} 1}$\hspace{0.2em}\
for all\hspace{0.2em}\
$i\hspace*{0.4em} =\hspace*{0.4em}
1\hspace*{0.05em},\hspace*{0.3em} 2\hspace*{0.05em},\hspace*{0.3em} \ldots\hspace*{0.05em},\hspace*{0.3em} p$\hspace*{-0.2em},\hspace{0.4em}\
i.e.\vspace*{2pt}
\begin{equation}
\label{s-k-cycle}
\quad
s_{\hspace*{0.05em} 1}\hspace*{0.4em} \leqslant\hspace*{0.4em} k_{\hspace*{0.05em} 2}\hspace*{0.1em},\quad
s_{\hspace*{0.05em} 2}\hspace*{0.4em} \leqslant\hspace*{0.4em} k_{\hspace*{0.05em} 3}\hspace*{0.1em},\quad
\ldots\hspace*{0.1em},\quad
s_{\hspace*{0.05em} p}\hspace*{0.4em} \leqslant\hspace*{0.4em} k_{\hspace*{0.05em} 1}\hspace*{0.1em}.
\end{equation}
\vspace*{-10pt}
By summing the inequalities\hspace{0.2em}\ (\ref{s-k-cycle})\hspace{0.2em}\ one concludes that\hspace{0.4em}\
\begin{equation}
\label{sum-to-p}
\quad
\sum_{j\hspace*{0.2em} =\hspace*{0.2em} 1}^p s_j
\hspace*{0.4em} \leqslant\hspace*{0.4em}
\sum_{j\hspace*{0.2em} =\hspace*{0.2em} 1}^p k_j\hspace*{0.2em}.
\end{equation}
The inequality\hspace{0.2em}\ (\ref{sum-to-p})\hspace{0.2em}\ is nothing else but
another form of\hspace{0.2em}\
(\ref{sum-divided}).\hspace{0.4em}\
The arguments of Section\hspace{0.2em}\ \ref{solution}\hspace{0.2em}\
show that\hspace{0.2em}\ (\ref{sum-to-p})\hspace{0.2em}\
implies that\hspace{0.2em}\
$m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
In fact\hspace*{0.025em},\hspace{0.4em}\ de Bruijn and Erd\"{o}s do not bother to
write down even the inequality\hspace{0.2em}\ (\ref{sum-to-p}),\hspace{0.4em}\
to say nothing about other details presented in Section\hspace{0.2em}\ \ref{solution}.
\myuppar{The case\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em}.}
In view of the equality\hspace{0.2em}\ (\ref{sums})\hspace{0.2em}\
in this case the left hand and the right hand sides of
the inequality\hspace{0.2em}\ (\ref{sum-to-p})\hspace{0.2em}\
are equal.\hspace{0.4em}\
Together with\hspace{0.2em}\ (\ref{s-k-cycle})\hspace{0.2em}\ this implies that
\begin{equation}
\label{cycle-of-equalities}
\quad
s_{\hspace*{0.05em} 1}\hspace*{0.4em} =\hspace*{0.4em} k_{\hspace*{0.05em} 2}\hspace*{0.1em},\quad
s_{\hspace*{0.05em} 2}\hspace*{0.4em} =\hspace*{0.4em} k_{\hspace*{0.05em} 3}\hspace*{0.1em},\quad
\ldots\hspace*{0.1em},\quad
s_{\hspace*{0.05em} p}\hspace*{0.4em} =\hspace*{0.4em} k_{\hspace*{0.05em} 1}\hspace*{0.2em}.
\end{equation}
Similarly,\hspace{0.4em}\ in this case
the left hand and the right hand sides of the inequality\hspace{0.2em}\
(\ref{k-lower-estimate})\hspace{0.2em}\
are equal.\hspace{0.4em}\
Since\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em},\hspace{0.4em}\
one can take\hspace{0.2em}\
$Y\hspace*{0.2em} =\hspace*{0.2em} E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} U$\hspace{0.2em}\ in\hspace{0.2em}\ (\ref{k-lower-estimate}).\hspace{0.4em}\
It follows that\hspace{0.2em}\
$k_{\hspace*{0.05em} u}
\hspace*{0.2em} =\hspace*{0.2em}
k_{\hspace*{0.05em} z}$\hspace{0.2em}\
for all\hspace{0.2em}\
$z\hspace*{0.2em} \in\hspace*{0.2em} E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} U$\hspace*{-0.15em}.\hspace{0.4em}\
Finally,\hspace{0.4em}\ the left hand and the right hand sides of the inequality\hspace{0.2em}\
(\ref{s-upper-estimate})\hspace{0.2em}\
are equal.\hspace{0.4em}\
It follows that\hspace{0.2em}\
$s_{\hspace*{0.05em} l}
\hspace*{0.2em} =\hspace*{0.2em}
k_{\hspace*{0.05em} u}$\hspace{0.2em}\
for all\hspace{0.2em}\
$l\hspace*{0.2em} \not\in\hspace*{0.2em} \mathcal{U}$\hspace*{-0.2em}.\hspace{0.4em}\
By combining the last two observations,\hspace{0.4em}\
we see that\hspace{0.2em}\ \vspace*{2pt}
\begin{equation*}
\quad
s_{\hspace*{0.05em} l}
\hspace*{0.4em} =\hspace*{0.4em}
k_{\hspace*{0.05em} z}
\quad\mbox{ for\hspace{0.2em}\ all }\quad
l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \mathcal{U}\hspace*{0.05em},\quad
z\hspace*{0.2em} \in\hspace*{0.2em} E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} U\hspace*{0.1em}.
\end{equation*}
\vspace*{-10pt}
Since\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em},\hspace{0.4em}\ both sets\hspace{0.2em}\
$\mathcal{L}\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \mathcal{U}$\hspace{0.2em}\
and\hspace{0.2em}\
$E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} U$\hspace{0.2em}\
consist of\hspace{0.2em}\ $n\hspace*{0.2em} -\hspace*{0.2em} p$\hspace{0.2em}\ elements.\hspace{0.4em}\
It follows that one can number the points and lines in
such a way that\hspace{0.2em}\ (in the notation of the Bourbaki exercise)
\[
\quad
s_{\hspace*{0.05em} 1}\hspace*{0.4em} =\hspace*{0.4em} k_{\hspace*{0.05em} 1}\hspace*{0.05em},\quad
s_{\hspace*{0.05em} 2}\hspace*{0.4em} =\hspace*{0.4em} k_{\hspace*{0.05em} 2}\hspace*{0.05em},\quad
\ldots\hspace*{0.05em},\quad
s_{\hspace*{0.05em} n}\hspace*{0.4em} =\hspace*{0.4em} k_{\hspace*{0.05em} n}\hspace*{0.2em}.
\]
As the next step,\hspace{0.4em}\ let us renumber the points
and lines once more and assume that
\begin{equation}
\label{k-order}
\quad
k_{\hspace*{0.05em} 1}\hspace*{0.4em} \geqslant\hspace*{0.4em}
k_{\hspace*{0.05em} 2}\hspace*{0.4em} \geqslant\hspace*{0.4em}
\ldots\hspace*{0.4em}
\geqslant\hspace*{0.4em} k_{\hspace*{0.05em} n}\hspace*{0.2em}.
\end{equation}
The rest of the proof splits into two subcases depending on\hspace{0.1em}\ if\hspace{0.1em}\
$k_{\hspace*{0.05em} 1}\hspace*{0.2em} >\hspace*{0.2em} k_{\hspace*{0.05em} 2}$\hspace{0.1em}\ or not\hspace*{0.025em}.
\myuppar{The subcase\hspace{0.2em}\ $k_{\hspace*{0.05em} 1}\hspace*{0.2em} >\hspace*{0.2em} k_{\hspace*{0.05em} 2}$\hspace*{-0.15em}.}
In this case\hspace{0.2em}\
$s_{\hspace*{0.05em} 1}\hspace*{0.2em} =\hspace*{0.2em} k_{\hspace*{0.05em} 1}\hspace*{0.2em} >\hspace*{0.2em} k_{\hspace*{0.05em} i}$\hspace{0.2em}\
for all\hspace{0.2em}\ $i\hspace*{0.2em} \geqslant\hspace*{0.2em} 2$\hspace*{-0.15em}.\hspace{0.4em}\
By the de~Bruijn--Erd\"{o}s\ inequalities this implies that\hspace{0.2em}\
$a_{\hspace*{0.05em} i}\hspace*{0.2em} \in\hspace*{0.2em} A_{\hspace*{0.05em} 1}$\hspace{0.2em}\ for all\hspace{0.2em}\ $i\hspace*{0.2em} \geqslant\hspace*{0.2em} 2$\hspace*{-0.15em}.\hspace{0.4em}\
It\hspace{0.1em}\ follows that\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\ is a near-pencil.\hspace{0.4em}\
\myuppar{The subcase\hspace{0.3em}\ $k_{\hspace*{0.05em} 1}\hspace*{0.4em} =\hspace*{0.4em} k_{\hspace*{0.05em} 2}$\hspace*{-0.15em}.}
Suppose that\hspace{0.2em}\ $k_{\hspace*{0.05em} j}\hspace*{0.2em} <\hspace*{0.2em} k_{\hspace*{0.05em} 1}\hspace*{0.2em} =\hspace*{0.2em} k_{\hspace*{0.05em} 2}$\hspace{0.2em}\
for some $j$\hspace*{-0.15em}.\hspace{0.4em}\
By the de~Bruijn--Erd\"{o}s\ inequalities $a_{\hspace*{0.05em} j}$ belongs to the both lines
$A_{\hspace*{0.05em} 1}$ and $A_{\hspace*{0.05em} 2}$\hspace*{-0.15em}.\hspace{0.4em}\
This is possible for only one point\hspace*{0.025em},\hspace{0.4em}\
namely the point of the intersection of the lines
$A_{\hspace*{0.05em} 1}$ and $A_{\hspace*{0.05em} 2}$\hspace*{-0.15em}.\hspace{0.4em}\
In view of\hspace{0.2em}\ (\ref{k-order}),\hspace{0.4em}\
this may happen only if\hspace{0.4em}\
\[
\quad
k_{\hspace*{0.05em} 1}
\hspace*{0.4em} =\hspace*{0.4em}
k_{\hspace*{0.05em} 2}
\hspace*{0.4em} =\hspace*{0.4em}
\ldots
\hspace*{0.4em} =\hspace*{0.4em}
k_{\hspace*{0.05em} n\hspace*{0.1em} -\hspace*{0.1em} 1}
\hspace*{0.4em} >\hspace*{0.4em}
k_{\hspace*{0.05em} n}
\]
and hence\hspace{0.4em}\
$
s_{\hspace*{0.05em} j}
\hspace*{0.4em} =\hspace*{0.4em}
k_{\hspace*{0.05em} j}
\hspace*{0.4em} >\hspace*{0.4em}
k_{\hspace*{0.05em} n}
\hspace*{0.4em} \geqslant\hspace*{0.4em}
2$\hspace{0.4em}\
for all\hspace{0.2em}\ $j\hspace*{0.2em} \neq\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
It follows that\hspace{0.4em}\
$s_{\hspace*{0.05em} j}\hspace*{0.4em} \geqslant\hspace*{0.4em} 3$\hspace{0.4em}\
if\hspace{0.2em}\ $j\hspace*{0.2em} <\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
In particular\hspace*{0.025em},\hspace{0.4em}\ all $k_{\hspace*{0.05em} n}$ lines containing $a_{\hspace*{0.05em} n}$
consist of\hspace{0.2em}\ $\geqslant\hspace*{0.2em} 2$\hspace{0.2em}\ points and all except\hspace*{0.025em},\hspace{0.3em}\ perhaps,\hspace{0.3em}\
the line $A_{\hspace*{0.05em} n}$\hspace*{-0.15em},\hspace{0.4em}\ consist of\hspace{0.2em}\ $\geqslant\hspace*{0.2em} 3$\hspace{0.2em}\ points.\hspace{0.4em}\
Therefore one can choose $2$ points\hspace{0.2em}\ $x\hspace*{0.05em},\hspace*{0.3em} y\hspace*{0.2em} \neq\hspace*{0.2em} a_{\hspace*{0.05em} n}$\hspace{0.2em}\
on one of these lines,\hspace{0.4em}\
and a point\hspace{0.2em}\ $z\hspace*{0.2em} \neq\hspace*{0.2em} a_{\hspace*{0.05em} n}$\hspace{0.2em}\ on some other line.\hspace{0.4em}\
Let\hspace{0.2em}\ $l_{j}\hspace*{0.05em},\hspace*{0.3em} l_{j'}$\hspace{0.2em}\ be the lines containing the pairs\hspace{0.2em}\
$\{\hspace*{0.1em} x\hspace*{0.05em},\hspace*{0.05em} z \hspace*{0.1em}\}$\hspace{0.2em}\ and\hspace{0.2em}\
$\{\hspace*{0.1em} y\hspace*{0.05em},\hspace*{0.05em} z \hspace*{0.1em}\}$\hspace{0.2em}\ respectively.\hspace{0.4em}\
Then\hspace{0.2em}\ $j\hspace*{0.2em} \neq\hspace*{0.2em} j'$\hspace{0.3em}\ and\hspace{0.3em}\
$a_{\hspace*{0.05em} n}\hspace*{0.4em} \not\in\hspace*{0.4em} l_j\hspace*{0.05em},\hspace*{0.3em} l_{j'}$\hspace*{-0.15em}.\hspace{0.4em}\
Hence the de~Bruijn--Erd\"{o}s\ inequalities imply that\hspace{0.2em}\
$s_{\hspace*{0.05em} j}\hspace*{0.05em},\hspace*{0.3em} s_{\hspace*{0.05em} j'}\hspace*{0.4em} \leqslant\hspace*{0.4em} k_{\hspace*{0.05em} n}$\hspace*{-0.15em},\hspace{0.4em}\
contrary to the fact that\hspace{0.2em}\
$s_{\hspace*{0.05em} j}\hspace*{0.4em} >\hspace*{0.4em} k_{\hspace*{0.05em} n}$\hspace{0.4em}\ if\hspace{0.4em}\ $j\hspace*{0.4em} \neq\hspace*{0.4em} n$\hspace*{-0.15em}.\hspace{0.4em}\
The contradiction shows that all numbers $k_{\hspace*{0.05em} j}$ are equal,\hspace{0.4em}\
and hence all numbers\hspace{0.2em}\ $s_{\hspace*{0.05em} i}\hspace*{0.05em},\hspace*{0.3em} k_{\hspace*{0.05em} j}$\hspace{0.2em}\
are equal.\hspace{0.4em}\
Now the observation at the end of Section\hspace{0.2em}\ \ref{solution}\hspace{0.2em}\
implies that\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\ is a projective plane.
\myuppar{Intersection of lines.}
After the proof is completed,\hspace{0.4em}\ de~Bruijn--Erd\"{o}s\ point out
that in the subcase\hspace{0.2em}\ $k_{\hspace*{0.05em} 1}\hspace*{0.2em} =\hspace*{0.2em} k_{\hspace*{0.05em} 2}$\hspace{0.2em}\
of the case\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace{0.2em}\
every two lines intersect\hspace*{0.025em}.\hspace{0.4em}\
Indeed,\hspace{0.4em}\ if\hspace{0.2em}\ $l'\hspace*{0.05em},\hspace*{0.3em} l''$\hspace{0.2em}\ are two disjoint lines
and\hspace{0.2em}\ $a\hspace*{0.2em} \in\hspace*{0.2em} l''$\hspace*{-0.2em},\hspace{0.4em}\
then there are $s_{\hspace*{0.05em} l'}$ lines
containing $a$ and intersecting $l'$\hspace*{-0.2em},\hspace{0.4em}\
and still one more line,\hspace{0.4em}\ namely $l''$\hspace*{-0.2em},\hspace{0.4em}\ containing $a$\hspace*{-0.15em}.\hspace{0.4em}\
Therefore\hspace{0.2em}\ $k_{\hspace*{0.05em} a}\hspace*{0.2em} \geqslant\hspace*{0.2em} s_{\hspace*{0.05em} l'}\hspace*{0.2em} +\hspace*{0.2em} 1$\hspace*{-0.15em},\hspace{0.4em}\
contrary to the fact all numbers\hspace{0.2em}\ $k_{\hspace*{0.05em} z}\hspace*{0.05em},\hspace*{0.3em} s_{\hspace*{0.05em} l}$\hspace{0.2em}\ are equal.\hspace{0.4em}\
In fact\hspace*{0.025em},\hspace{0.4em}\ every two lines obviously intersect
in the subcase\hspace{0.2em}\ $k_{\hspace*{0.05em} 1}\hspace*{0.2em} >\hspace*{0.2em} k_{\hspace*{0.05em} 2}$\hspace{0.2em}\ also.
\myuppar{Why $k_{\hspace*{0.05em} n}$\hspace*{-0.1em}?}
Now it is clear why the smallest of the numbers $k_{\hspace*{0.05em} z}$ is denoted by $k_{\hspace*{0.05em} n}$\hspace*{-0.15em}.\hspace{0.4em}\
The number $k_{\hspace*{0.05em} n}$ is indeed the smallest
if the points are ordered in such a way that\hspace{0.2em}\ (\ref{k-order})\hspace{0.2em}\ holds.\hspace{0.4em}\
At the same time\hspace{0.2em}\ (\ref{k-order})\hspace{0.2em}\
plays almost no role in the proof\hspace*{0.025em}.\hspace{0.4em}\
One may speculate that\hspace{0.2em}\ (\ref{k-order})\hspace{0.2em}\
and notation $k_{\hspace*{0.05em} n}$ for the smallest of the numbers $k_{\hspace*{0.05em} z}$
are remnants of an earlier approach to the theorem.\hspace{0.4em}\
\mysection{From\hspace{0.2em}\ de Bruijn--Erd\"{o}s\hspace{0.2em}\ to\hspace{0.2em}\
systems\hspace{0.2em}\ of\hspace{0.2em}\ distinct\hspace{0.2em}\ representatives}{reps}
\vspace*{6pt}
\myuppar{The cyclic order argument and systems of distinct representatives.}
The key step of the de Bruijn--Erd\"{o}s proof is the cyclic order argument
used to prove the inequality\hspace{0.2em}\ (\ref{sum-to-p})\hspace{0.2em}\
and the equalities\hspace{0.2em}\ (\ref{cycle-of-equalities})\hspace{0.2em}\ in the case\hspace{0.2em}\
$m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
Ultimately,\hspace{0.4em}\
the cyclic order argument is based on the fact that\hspace{0.2em}\
$a_{\hspace*{0.05em} i\hspace*{0.1em} +\hspace*{0.1em} 1}\hspace*{0.2em} \not\in\hspace*{0.2em} l_{\hspace*{0.05em} i}$\hspace{0.4em}\
for all\hspace{0.2em}\
$i\hspace*{0.4em} =\hspace*{0.4em}
1\hspace*{0.05em},\hspace*{0.3em} 2\hspace*{0.05em},\hspace*{0.3em} \ldots\hspace*{0.05em},\hspace*{0.3em} p$\hspace*{-0.2em},\hspace{0.4em}\
i.e.\hspace{0.2em}\ on the fact that\hspace{0.4em}\
$i\hspace*{0.2em} \longmapsto\hspace*{0.2em} a_{\hspace*{0.05em} i\hspace*{0.1em} +\hspace*{0.1em} 1}$\hspace{0.4em}\
is a system of distinct representatives for the family\hspace{0.4em}\
$i\hspace*{0.4em} \longmapsto\hspace*{0.4em} E\hspace*{0.2em} \smallsetminus\hspace*{0.2em} l_{\hspace*{0.05em} i}$\hspace{0.4em}\
of subsets of $E$\hspace*{-0.15em},\hspace{0.4em}\
where\hspace{0.4em}\
$i\hspace*{0.4em} =\hspace*{0.4em}
1\hspace*{0.05em},\hspace*{0.3em} 2\hspace*{0.05em},\hspace*{0.3em} \ldots\hspace*{0.05em},\hspace*{0.3em} p$\hspace*{-0.2em}.\hspace{0.4em}\
Once this is realized,\hspace{0.4em}\
it is only natural to look for a system of distinct representatives of
the full family\hspace{0.4em}\
$\displaystyle
l\hspace*{0.2em} \longmapsto\hspace*{0.2em} E\hspace*{0.2em} \smallsetminus\hspace*{0.2em} l$\hspace{0.4em}\
of the complements of lines,\hspace{0.4em}\
i.e.\hspace{0.2em}\ for an injective map\hspace{0.4em}\
$l\hspace*{0.2em} \longmapsto\hspace*{0.2em} a\hspace*{0.05em}({\hspace*{0.05em}}l\hspace*{0.05em})$\hspace{0.4em}\ from\hspace{0.1em}\ $\mathcal{L}$\hspace{0.1em}\ to\hspace{0.1em}\ $E$\hspace{0.1em}\
such that\hspace{0.4em}\
$a\hspace*{0.05em}({\hspace*{0.05em}}l\hspace*{0.05em})\hspace*{0.4em} \in\hspace*{0.4em} E\hspace*{0.2em} \smallsetminus\hspace*{0.2em} l$\hspace{0.4em}\
for all\hspace{0.2em}\ $l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}$\hspace*{-0.2em}.\hspace{0.4em}\
By the well known\hspace{0.1em}\ Ph.\hspace{0.1em}\ Hall's marriage theorem,\hspace{0.4em}\
such a system of distinct representatives exists if and only if
for every subset\hspace{0.2em}\ $\mathcal{K}\hspace*{0.2em} \subset\hspace*{0.2em} \mathcal{L}$\hspace{0.2em}\
the union\vspace*{3pt}
\begin{equation}
\label{union}
\quad
\bigcup_{\hspace*{0.1em} l\hspace*{0.1em} \in\hspace*{0.1em} \mathcal{K}}\hspace*{0.2em} E\hspace*{0.2em} \smallsetminus\hspace*{0.2em} l
\hspace*{0.4em}\off =\hspace*{0.4em}\off
E\hspace*{0.4em} \smallsetminus\hspace*{0.4em} \bigcap_{\hspace*{0.1em} l\hspace*{0.1em} \in\hspace*{0.1em} \mathcal{K}}\hspace*{0.2em} l
\end{equation}
\vspace*{-9pt}
contains\hspace{0.2em}\ $\geqslant\hspace*{0.2em} |\hspace*{0.05em} \mathcal{K}\hspace*{0.05em} |$\hspace{0.2em}\ elements,\hspace{0.4em}\
where\hspace{0.1em}\ $|\hspace*{0.1em} X\hspace*{0.1em}|$\hspace{0.1em}\ denotes the number of elements of a set $X$\hspace*{-0.15em}.\hspace{0.4em}\
But the intersection of\hspace{0.2em}\ $\geqslant\hspace*{0.2em} 2$\hspace{0.2em}\ lines consists of\hspace{0.2em}\
$\leqslant\hspace*{0.2em} 1$\hspace{0.2em}\ points,\hspace{0.4em}\ and,\hspace{0.4em}\
almost obviously,\hspace{0.4em}\ this condition holds.
\myuppar{The message.}
All this emerged in my mind in one instant as an irreducible revelation.\hspace{0.4em}\
My first thought after this revelation
was that it cannot be true,\hspace{0.4em}\ because if it is true,\hspace{0.4em}\
then everybody writing about this topic would
use systems of distinct representatives.\hspace{0.4em}\
Perhaps,\hspace{0.3em}\ the right question is not how I came up with this idea,\hspace{0.3em}\
but why experts missed it\hspace*{0.025em}.
The rest of this section is devoted to the proof\hspace{0.2em}\ \cite{i-db-e}\hspace{0.2em}\
based on this revelation.
\myuppar{Proof\hspace{0.1em}\ of\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.}
We may assume that $m\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
Let $\mathcal{K}$ be a subset of $\mathcal{L}$\hspace*{-0.2em}.\hspace{0.4em}\
If\hspace{0.2em}\ $|\hspace*{0.1em} \mathcal{K}\hspace*{0.1em}|\hspace*{0.2em} =\hspace*{0.2em} 1$\hspace*{-0.15em},\hspace{0.2em}\
then\hspace{0.2em}\ (\ref{union})\hspace{0.2em}\ is the complement of a line
and hence contains\hspace{0.2em}\ $\geqslant\hspace*{0.2em} 1$\hspace{0.2em}\ elements.\hspace{0.2em}\
If\hspace{0.2em}\ $2\hspace*{0.2em} \leqslant\hspace*{0.2em} |\hspace*{0.1em} \mathcal{K}\hspace*{0.1em} |\hspace*{0.2em} \leqslant\hspace*{0.2em} m\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace*{-0.15em},\hspace{0.2em}\
then\hspace{0.2em}\ (\ref{union})\hspace{0.2em}\
is a complement in $E$ of\hspace{0.2em}\ $\leqslant\hspace*{0.2em} 1$\hspace{0.2em}\ point
and hence contains\hspace{0.2em}\
\[
\quad
\geqslant\hspace*{0.2em} n\hspace*{0.2em} -\hspace*{0.2em} 1
\hspace*{0.4em} \geqslant\hspace*{0.4em}
m\hspace*{0.2em} -\hspace*{0.2em} 1
\hspace*{0.4em} \geqslant\hspace*{0.4em}
|\hspace*{0.1em} \mathcal{K}\hspace*{0.1em} |
\]
elements.\hspace{0.2em}\
If\hspace{0.2em}\ $|\hspace*{0.1em} \mathcal{K}\hspace*{0.1em}|\hspace*{0.2em} =\hspace*{0.2em} m$\hspace*{-0.15em},\hspace{0.2em}\
then\hspace{0.2em}\ (\ref{union})\hspace{0.2em}\ contains\hspace{0.2em}\
$n\hspace*{0.2em} \geqslant\hspace*{0.2em} m\hspace*{0.2em} =\hspace*{0.2em} |\hspace*{0.1em} \mathcal{K}\hspace*{0.1em}|$\hspace{0.2em}\ elements.\hspace{0.2em}\
Therefore there exists a system of distinct representatives\hspace{0.4em}\
for the family\hspace{0.4em}\
$l\hspace*{0.2em} \longmapsto\hspace*{0.2em} E\hspace*{0.2em} \smallsetminus\hspace*{0.2em} l$\hspace*{-0.2em},\hspace{0.4em}\
i.e. there exists an injective map\hspace{0.4em}\
$l\hspace*{0.2em} \longmapsto\hspace*{0.2em} a\hspace*{0.05em}({\hspace*{0.05em}}l\hspace*{0.05em})$\hspace{0.4em}\ such that\hspace{0.2em}\
$a\hspace*{0.05em}({\hspace*{0.05em}}l\hspace*{0.05em})\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace{0.2em}\ for every $l$\hspace*{-0.2em}.\hspace{0.4em}\
By the de~Bruijn--Erd\"{o}s\ inequalities
\begin{equation}
\label{basic-reps-ineq}
\quad
s_{\hspace*{0.05em} l}\hspace*{0.4em} \leqslant\hspace*{0.4em} k_{\hspace*{0.05em} a\hspace*{0.05em}({\hspace*{0.05em}}l\hspace*{0.05em})}
\quad\mbox{ for\hspace{0.2em}\ every }\quad
l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}.
\end{equation}
By summing all these inequalities and using the injectivity of\hspace{0.4em}\
$l\hspace*{0.2em} \longmapsto\hspace*{0.2em} a\hspace*{0.05em}({\hspace*{0.05em}}l\hspace*{0.05em})$\hspace{0.4em}\
we see that
\begin{equation}
\label{three-sums}
\quad
\sum_{l\hspace*{0.1em} \in\hspace*{0.1em} \mathcal{L}}\hspace*{0.2em} s_{\hspace*{0.05em} l}
\hspace*{0.4em}\off \leqslant\hspace*{0.4em}\off
\sum_{l\hspace*{0.1em} \in\hspace*{0.1em} \mathcal{L}}\hspace*{0.2em} k_{\hspace*{0.05em} a\hspace*{0.05em}({\hspace*{0.05em}}l\hspace*{0.05em})}
\hspace*{0.4em}\off \leqslant\hspace*{0.4em}\off
\sum_{\hspace*{0.2em} z\hspace*{0.1em} \in\hspace*{0.1em} E}\hspace*{0.2em} k_{\hspace*{0.05em} z}\hspace*{0.2em}.
\end{equation}
Moreover\hspace*{0.025em},\hspace{0.2em}\
the second inequality is strict unless\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace{0.2em}\
(otherwise the last sum has more positive summands
than the previous one).\hspace{0.4em}\
But\hspace{0.2em}\ (\ref{sums})\hspace{0.2em}\ implies that both inequalities in\hspace{0.2em}\
(\ref{three-sums})\hspace{0.2em}\ should be actually equalities.\hspace{0.4em}\
It follows that\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
Moreover\hspace*{0.025em},\hspace{0.4em}\ in view of the inequalities\hspace{0.2em}\
(\ref{basic-reps-ineq}),\hspace{0.4em}\
it follows that\hspace{0.4em}\
$s_{\hspace*{0.05em} l}\hspace*{0.4em} =\hspace*{0.4em} k_{\hspace*{0.05em} a\hspace*{0.05em}({\hspace*{0.05em}}l\hspace*{0.05em})}$\hspace{0.4em}\
for every\hspace{0.2em}\ $l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}$\hspace{0.2em}\
(under the assumption\hspace{0.2em}\ $m\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace*{-0.15em}).
\myuppar{The case\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em}.}
Suppose that a point $z$
is contained in\hspace{0.2em}\ $\geqslant\hspace*{0.2em} m\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\ lines.\hspace{0.4em}\
Each of these lines contains at least one point in addition to $z$\hspace*{-0.15em}.\hspace{0.4em}\
Since\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em},\hspace{0.2em}\
there are no other points and $z$ is contained in exactly\hspace{0.2em}\ $m\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\ lines.\hspace{0.4em}\
Since there are exactly $m$ lines,\hspace{0.4em}\
only one line does not contain $z$\hspace*{-0.15em}.\hspace{0.4em}\
This line should contain all points\hspace{0.2em}\ $\neq\hspace*{0.2em} z$\hspace*{-0.15em}.\hspace{0.2em}\
It follows that\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\ is a near-pencil.\hspace{0.4em}\
Suppose now that no point
is contained in\hspace{0.2em}\ $\geqslant\hspace*{0.2em} m\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\ lines.\hspace{0.4em}\
Let $\mathcal{K}$ be a proper subset of $\mathcal{L}$\hspace*{-0.2em}.\hspace{0.4em}\
If\hspace{0.2em}\ $|\hspace*{0.1em} \mathcal{K}\hspace*{0.1em}|\hspace*{0.2em} =\hspace*{0.2em} 1$\hspace*{-0.15em},\hspace{0.4em}\
then\hspace{0.2em}\ (\ref{union})\hspace{0.2em}\ is equal to\hspace{0.2em}\ $E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} l$\hspace{0.2em}\
for some line $l$\hspace*{-0.15em}.\hspace{0.4em}\
If\hspace{0.2em}\ $E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} l$\hspace{0.2em}\ consists of only one point $z$\hspace*{-0.15em},\hspace{0.4em}\
then by the de~Bruijn--Erd\"{o}s\ inequalities $z$ is contained in\hspace{0.2em}\
$\geqslant\hspace*{0.2em} s_{\hspace*{0.05em} l}\hspace*{0.2em} =\hspace*{0.2em} m\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\ lines,\hspace{0.4em}\
contrary to the assumption.\hspace{0.4em}\
Therefore,\hspace{0.4em}\ (\ref{union})\hspace{0.2em}\ contains\hspace{0.2em}\
$\geqslant\hspace*{0.2em} 2\hspace*{0.2em} =\hspace*{0.2em} |\hspace*{0.1em} \mathcal{K}\hspace*{0.1em}|\hspace*{0.2em} +\hspace*{0.2em} 1$\hspace{0.2em}\ points.\hspace{0.4em}\
If\hspace{0.2em}\ $|\hspace*{0.1em} \mathcal{K}\hspace*{0.1em}|\hspace*{0.2em} \leqslant\hspace*{0.2em} m\hspace*{0.2em} -\hspace*{0.2em} 2$\hspace*{-0.15em},\hspace{0.4em}\
then\hspace{0.2em}\ (\ref{union})\hspace{0.2em}\ contains\hspace{0.2em}\
$\geqslant\hspace*{0.2em} n\hspace*{0.2em} -\hspace*{0.2em} 1\hspace*{0.2em} =\hspace*{0.2em} m\hspace*{0.2em} -\hspace*{0.2em} 1\hspace*{0.2em} \geqslant\hspace*{0.2em}
|\hspace*{0.1em} \mathcal{K}\hspace*{0.1em}|\hspace*{0.2em} +\hspace*{0.2em} 1$\hspace{0.2em}\ points.\hspace{0.4em}\
Finally,\hspace{0.4em}\ if\hspace{0.2em}\ $|\hspace*{0.1em} \mathcal{K}\hspace*{0.1em}|\hspace*{0.2em} =\hspace*{0.2em} m\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace*{-0.15em},\hspace{0.4em}\
then\hspace{0.2em}\ (\ref{union})\hspace{0.2em}\ contains all\hspace{0.2em}\
$n\hspace*{0.2em} =\hspace*{0.2em} m\hspace*{0.2em} =\hspace*{0.2em} |\hspace*{0.1em} \mathcal{K}\hspace*{0.1em}|\hspace*{0.2em} +\hspace*{0.2em} 1$\hspace{0.2em}\
points because no point
is contained in\hspace{0.2em}\ $\geqslant\hspace*{0.2em} m\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\ lines.\hspace{0.4em}\
We see that\hspace{0.2em}\ (\ref{union})\hspace{0.2em}\ contains\hspace{0.2em}\
$\geqslant\hspace*{0.2em} |\hspace*{0.1em} \mathcal{K}\hspace*{0.1em}|\hspace*{0.2em} +\hspace*{0.2em} 1$\hspace{0.2em}\ elements
for every proper subset\hspace{0.2em}\ $\mathcal{K}\hspace*{0.2em} \subset\hspace*{0.2em} \mathcal{L}$\hspace*{-0.2em}.\hspace{0.4em}\
This allows to get from the marriage theorem more than
just the existence of a system of distinct representatives.\hspace{0.4em}\
Let\hspace{0.2em}\ $\lambda\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}$\hspace{0.2em}\ and\hspace{0.2em}\
$z\hspace*{0.2em} \in\hspace*{0.2em} E\hspace*{0.1em} \smallsetminus\hspace*{0.2em} \lambda$\hspace*{-0.15em}.\hspace{0.4em}\
Then there exists a system of distinct representatives\hspace{0.4em}\
$l\hspace*{0.2em} \longmapsto\hspace*{0.2em} a\hspace*{0.05em}({\hspace*{0.05em}}l\hspace*{0.05em})$\hspace{0.4em}\ such that\hspace{0.2em}\
$a\hspace*{0.05em}(\hspace*{0.05em}\lambda\hspace*{0.05em})\hspace*{0.2em} =\hspace*{0.2em} z$\hspace*{-0.15em}.\hspace{0.4em}\
This immediately follows from an application
of the marriage theorem
to the family of sets\hspace{0.4em}\
$(\hspace*{0.1em} E\hspace*{0.2em} \smallsetminus\hspace*{0.2em} \{\hspace*{0.1em} z \hspace*{0.2em}\}\hspace*{0.1em})\hspace*{0.2em} \smallsetminus\hspace*{0.2em} l$\hspace{0.4em}\
with\hspace{0.4em}\
$l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}\hspace*{0.2em} \smallsetminus\hspace*{0.2em} \{\hspace*{0.1em} \lambda \hspace*{0.2em}\}$\hspace*{-0.15em}.\hspace{0.4em}\
Since\hspace{0.2em}\
$m\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace*{-0.15em},\hspace{0.4em}\
the existence of a system of distinct representatives\hspace{0.4em}\
$l\hspace*{0.2em} \longmapsto\hspace*{0.2em} a\hspace*{0.05em}({\hspace*{0.05em}}l\hspace*{0.05em})$\hspace{0.4em}\ such that\hspace{0.2em}\
$a\hspace*{0.05em}(\hspace*{0.05em}\lambda\hspace*{0.05em})\hspace*{0.2em} =\hspace*{0.2em} z$\hspace{0.2em}\
implies that\hspace{0.4em}\
$s_{\hspace*{0.05em} \lambda}\hspace*{0.4em} =\hspace*{0.4em} k_{\hspace*{0.05em} a\hspace*{0.05em}({\hspace*{0.05em}}\lambda\hspace*{0.05em})}\hspace*{0.4em} =\hspace*{0.4em} k_{\hspace*{0.05em} z}$\hspace*{-0.15em}.\hspace{0.4em}\
Therefore,\hspace{0.2em}\ $z\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace{0.2em}\ implies that\hspace{0.2em}\
$s_{\hspace*{0.05em} l}\hspace*{0.2em} =\hspace*{0.2em} k_{\hspace*{0.05em} z}$\hspace{0.3em}\
and hence every line containing $z$ intersects $l$\hspace*{-0.2em}.\hspace{0.4em}\
It follows that every two lines intersect\hspace*{0.025em}.\hspace{0.4em}\
If $E$ cannot be obtained as the union of two lines,\hspace{0.4em}\
then for every two lines\hspace{0.2em}\ $l\hspace*{0.05em},\hspace*{0.3em} l'$\hspace{0.2em}\
there exists a point $z$ such that\hspace{0.2em}\ $z\hspace*{0.2em} \not\in\hspace*{0.2em} l\hspace*{0.05em},\hspace*{0.3em} l'$\hspace{0.2em}\
and hence\hspace{0.2em}\ $s_{\hspace*{0.05em} l}\hspace*{0.2em} =\hspace*{0.2em} k_{\hspace*{0.05em} z}\hspace*{0.2em} =\hspace*{0.2em} s_{\hspace*{0.05em} l'}$\hspace*{-0.15em}.\hspace{0.4em}\
In this case all the numbers\hspace{0.2em}\
$s_{\hspace*{0.05em} l}\hspace*{0.05em},\hspace*{0.3em} k_{\hspace*{0.05em} z}$\hspace{0.2em}\ are equal
and hence\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\ is a projective plane
by Lemma\hspace{0.2em}\ 1\hspace{0.2em}\ at the end of Section\hspace{0.2em}\ \ref{solution}.\hspace{0.4em}\
If there exist two lines\hspace{0.2em}\ $l\hspace*{0.05em},\hspace*{0.3em} l'$\hspace{0.2em}\
such that\hspace{0.3em}\ $E\hspace*{0.2em} =\hspace*{0.2em} l\hspace*{0.1em} \cup\hspace*{0.1em} l'$\hspace*{-0.15em},\hspace*{0.2em}\hspace{0.4em}\
then\hspace{0.2em}\ $k_{\hspace*{0.05em} y}\hspace*{0.2em} =\hspace*{0.2em} 2$\hspace*{-0.15em},\hspace{0.4em}\
where $y$ is the point of intersection of $l$ and $l'$\hspace*{-0.2em},\hspace{0.4em}\
and the proof\hspace{0.1em}\ is completed by applying the following lemma.
\myuppar{Lemma\hspace{0.2em}\ 2.}
\emph{If\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace{0.2em}\ and\hspace{0.2em}\ $k_{\hspace*{0.05em} y}\hspace*{0.2em} =\hspace*{0.2em} 2$\hspace{0.2em}\ for some point $y$\hspace*{-0.2em},\hspace{0.4em}\
then\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\ is a near-pencil.}
\myuppar{Proof\hspace*{0.025em}.}
Let\hspace{0.2em}\ $l\hspace*{0.05em},\hspace*{0.3em} l'$\hspace{0.2em}\ be the lines containing $y$\hspace*{-0.2em}.\hspace{0.4em}\
Then\hspace{0.2em}\ $E\hspace*{0.2em} =\hspace*{0.2em} l\hspace*{0.1em} \cup\hspace*{0.1em} l'$\hspace{0.2em}\ and there are\hspace{0.2em}\
$n\hspace*{0.2em} =\hspace*{0.2em} s_{\hspace*{0.05em} l}\hspace*{0.2em} +\hspace*{0.2em} s_{\hspace*{0.05em} l'}\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\ points.\hspace{0.4em}\
In addition to the lines\hspace{0.3em}\ $l\hspace*{0.05em},\hspace*{0.3em} l'$\hspace{0.3em}\
there\hspace{0.1em}\ are\hspace{0.2em}\
$(s_l\hspace*{0.2em} -\hspace*{0.2em} 1)(s_{l'}\hspace*{0.2em} -\hspace*{0.2em} 1)$\hspace{0.2em}\
lines consisting of a point in\hspace{0.2em}\
$l\hspace*{0.2em} \smallsetminus\hspace{0.1em}\ \{\hspace*{0.1em} y \hspace*{0.15em}\}$\hspace{0.2em}\
and a point in\hspace{0.2em}\
$l'\hspace*{0.2em} \smallsetminus\hspace{0.1em}\ \{\hspace*{0.1em} y \hspace*{0.15em}\}$\hspace*{-0.15em}.\hspace*{0.2em}\hspace{0.4em}\
If\hspace{0.4em}\ $s_{\hspace*{0.05em} l}\hspace*{0.4em} \geqslant\hspace*{0.4em} s_{\hspace*{0.05em} l'}\hspace*{0.4em} \geqslant\hspace*{0.4em} 3$\hspace*{-0.2em},\hspace{0.4em}\
then the number $m$\hspace{0.1em}\ of\hspace{0.15em}\ lines\hspace{0.1em}\ is\hspace{0.2em}\ \vspace*{1.5pt}
\[
\quad
\geqslant\hspace*{0.4em}
2\hspace*{0.2em} +\hspace*{0.2em} (s_l\hspace*{0.2em} -\hspace*{0.2em} 1)(s_{l'}\hspace*{0.2em} -\hspace*{0.2em} 1)
\hspace*{0.4em} \geqslant\hspace*{0.4em}
2\hspace*{0.2em} +\hspace*{0.2em} 2\hspace*{0.1em}(s_l\hspace*{0.2em} -\hspace*{0.2em} 1)
\hspace*{0.4em} =\hspace*{0.4em}
2\hspace*{0.05em} s_l
\hspace*{0.4em} \geqslant\hspace*{0.4em}
s_l\hspace*{0.2em} +\hspace*{0.2em} s_{l'}
\hspace*{0.4em} =\hspace*{0.4em}
n\hspace*{0.2em} +\hspace*{0.2em} 1\hspace*{0.1em},
\]
\vspace*{-10.5pt}
contrary to the assumption\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.2em}\
Therefore one of the lines\hspace{0.2em}\ $l\hspace*{0.05em},\hspace*{0.3em} l'$\hspace{0.2em}\
consists of $2$ points
and hence\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\
is a near-pencil.\hspace{0.4em}\ $\blacksquare$
\mysection{Linear\hspace{0.2em}\ algebra\hspace{0.2em}\ and\hspace{0.2em}\ the\hspace{0.2em}\ inequality\hspace{0.2em}\
$m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$}{linear-algebra}
\vspace*{6pt}
\myuppar{A proof of the inequality\hspace{0.2em}\
$m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace{0.2em}\ based on the linear independence.}
This proof was communicated to me by\hspace{0.2em}\ F.\hspace{0.2em}\ Petrov\hspace{0.2em}\ \cite{p}.\hspace{0.3em}\
I\hspace{0.1em}\ believe that this is essentially the proof found by A.\hspace{0.2em}\ Suslin.
Let\hspace{0.2em}\ $ \mathbf{R}^{\hspace*{0.1em}\mathcal{L}} $\hspace{0.2em}\ be the vector space of maps\hspace{0.2em}\ $\mathcal{L}\hspace*{0.1em} \longrightarrow\hspace*{0.1em} \mathbf{R}$\hspace{0.2em}\
with the scalar product
\[
\quad
(\hspace*{0.1em} v\hspace*{0.05em},\hspace*{0.3em} w \hspace*{0.1em})
\hspace*{0.4em}\off =\hspace*{0.4em}\off
\sum_{l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}}\hspace*{0.2em} v(\hspace*{0.05em} l\hspace*{0.05em})\hspace*{0.1em} w(\hspace*{0.05em} l\hspace*{0.05em}).
\]
Every\hspace{0.2em}\ $z\hspace*{0.2em} \in\hspace*{0.2em} E$ defines a map\hspace{0.2em}\
$v_{\hspace*{0.05em} z}\hspace*{0.1em} \colon\hspace*{0.1em} \mathcal{L}\hspace*{0.1em} \longrightarrow\hspace*{0.1em} \mathbf{R}$\hspace{0.2em}\
by the rule\hspace{0.2em}\ $v_{\hspace*{0.05em} z}\hspace*{0.05em}(\hspace*{0.05em} l\hspace*{0.05em})\hspace*{0.2em} =\hspace*{0.2em} 1$\hspace{0.2em}\
if\hspace{0.2em}\ $z\hspace*{0.2em} \in\hspace*{0.2em} l$\hspace{0.2em}\ and\hspace{0.2em}\
$v_{\hspace*{0.05em} z}\hspace*{0.05em}(\hspace*{0.05em} l\hspace*{0.05em})\hspace*{0.2em} =\hspace*{0.2em} 0$\hspace{0.2em}\ otherwise.\hspace{0.2em}\
There are $n$ maps $v_{\hspace*{0.05em} z}$\hspace*{-0.15em}.\hspace{0.4em}\
Since the dimension of $ \mathbf{R}^{\hspace*{0.1em}\mathcal{L}} $ is equal to $m$\hspace*{-0.15em},\hspace{0.4em}\
it is sufficient to prove that the maps $v_{\hspace*{0.05em} z}$
are independent as vectors of\hspace{0.1em}\ $ \mathbf{R}^{\hspace*{0.1em}\mathcal{L}} $\hspace*{-0.2em}.\hspace{0.4em}\
The scalar product\hspace{0.2em}\ $(\hspace*{0.1em} v_{\hspace*{0.05em} z}\hspace*{0.05em},\hspace*{0.3em} v_{\hspace*{0.05em} z} \hspace*{0.1em})$\hspace{0.2em}\
is equal to the number of lines containing the point $z$\hspace*{-0.15em},\hspace{0.4em}\
and hence\hspace{0.2em}\
$(\hspace*{0.1em} v_{\hspace*{0.05em} z}\hspace*{0.05em},\hspace*{0.3em} v_{\hspace*{0.05em} z} \hspace*{0.1em})
\hspace*{0.2em} \geqslant\hspace*{0.2em} 2$\hspace{0.2em}\
for all\hspace{0.2em}\ $z\hspace*{0.2em} \in\hspace*{0.2em} E$\hspace*{-0.15em}.\hspace{0.4em}\
If\hspace{0.2em}\ $z\hspace*{0.2em} \neq\hspace*{0.2em} y$\hspace*{-0.2em},\hspace{0.4em}\ then\hspace{0.2em}\
$(\hspace*{0.1em} v_{\hspace*{0.05em} z}\hspace*{0.05em},\hspace*{0.3em} v_{\hspace*{0.05em} y} \hspace*{0.1em})$\hspace{0.2em}\
is equal to the number of lines containing both $z$ and $y$\hspace*{-0.15em},\hspace{0.3em}\
and hence\hspace{0.2em}\
$(\hspace*{0.1em} v_{\hspace*{0.05em} z}\hspace*{0.05em},\hspace*{0.3em} v_{\hspace*{0.05em} y} \hspace*{0.1em})
\hspace*{0.2em} =\hspace*{0.2em} 1$\hspace*{-0.15em}.\hspace{0.4em}\
If the vectors $v_{\hspace*{0.05em} z}$
are linearly dependent\hspace*{0.025em},\hspace{0.4em}\
then
\begin{equation*}
\quad
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} E}\hspace*{0.2em} c_{\hspace*{0.05em} z}\hspace*{0.1em} v_{\hspace*{0.05em} z}
\hspace*{0.4em}\off =\hspace*{0.4em}\off
0
\end{equation*}
for some real numbers\hspace{0.2em}\ $c_{\hspace*{0.05em} z}$\hspace*{-0.15em},\hspace{0.2em}\ $z\hspace*{0.2em} \in\hspace*{0.2em} E$\hspace*{-0.15em},\hspace{0.4em}\
such that not all $c_{\hspace*{0.05em} z}$ are equal to $0$\hspace*{-0.15em}.\hspace{0.4em}\
For every\hspace{0.2em}\ $y\hspace*{0.2em} \in\hspace*{0.2em} E$\hspace{0.2em}\
taking the scalar product of this equality with the vector\hspace{0.1em}\ $v_{\hspace*{0.05em} y}$\hspace{0.1em}\
results in the equality
\[
\quad
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} E}\hspace*{0.2em} c_{\hspace*{0.05em} z}\hspace*{0.1em} (\hspace*{0.05em} v_{\hspace*{0.05em} z}\hspace*{0.05em},\hspace*{0.3em} v_{\hspace*{0.05em} y} \hspace*{0.05em})
\hspace*{0.4em}\off =\hspace*{0.4em}\off
0.
\]
Since\hspace{0.2em}\ $(\hspace*{0.05em} v_{\hspace*{0.05em} z}\hspace*{0.05em},\hspace*{0.3em} v_{\hspace*{0.05em} y} \hspace*{0.05em})\hspace*{0.2em} =\hspace*{0.2em} 1$\hspace{0.2em}\
for all\hspace{0.2em}\ $z\hspace*{0.2em} \neq\hspace*{0.2em} y$\hspace*{-0.15em},\hspace{0.4em}\
this equality implies that
\[
\quad
c_{\hspace*{0.05em} y}\hspace*{0.1em} ((\hspace*{0.05em} v_{\hspace*{0.05em} y}\hspace*{0.05em},\hspace*{0.3em} v_{\hspace*{0.05em} y} \hspace*{0.05em})\hspace*{0.2em} -\hspace*{0.2em} 1)
\hspace*{0.4em} +\hspace*{0.4em}
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} E}\hspace*{0.2em} c_{\hspace*{0.05em} z}
\hspace*{0.4em}\off =\hspace*{0.4em}\off
0.
\]
Since\hspace{0.2em}\ $(\hspace*{0.05em} v_{\hspace*{0.05em} y}\hspace*{0.05em},\hspace*{0.3em} v_{\hspace*{0.05em} y} \hspace*{0.05em})\hspace*{0.2em} \geqslant\hspace*{0.2em} 2$\hspace*{-0.15em},\hspace{0.4em}\
it follows that the coefficient $c_{\hspace*{0.05em} y}$ and the sum\hspace{0.4em}\
\[
\quad
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} E}\hspace*{0.2em} c_{\hspace*{0.05em} z}
\]
have opposite signs.\hspace{0.4em}\
But since not all $c_{\hspace*{0.05em} y}$ are equal to $0$\hspace*{-0.15em},\hspace{0.4em}\
this cannot be true for all\hspace{0.2em}\ $y\hspace*{0.2em} \in\hspace*{0.2em} E$\hspace*{-0.15em}.\hspace{0.4em}\
The contradiction shows that vectors $v_{\hspace*{0.05em} z}$ are linearly independent
and hence\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\ $\blacksquare$
\myuppar{Standard linear algebra proofs of the inequality\hspace{0.2em}\
$m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.}
In order to present standard proofs it is convenient to return to
the notations of the\hspace{0.1em}\ N.\hspace{0.2em}\ Bourbaki exercise.\hspace{0.4em}\
Let $M$ be the\hspace{0.2em}\ \emph{incidence\hspace{0.1em}\ matrix}\hspace{0.2em}\
of the points $a_j$ and sets $A_i$\hspace*{-0.15em}.\hspace{0.4em}\
Namely,\hspace{0.2em}\ $M$\hspace{0.1em}\ is\hspace{0.1em}\ an $n \times m$ matrix with entries\hspace{0.2em}\
$m_{j i}\hspace*{0.2em} =\hspace*{0.2em} 1$\hspace{0.2em}\ if\hspace{0.2em}\ $a_j\hspace*{0.2em} \in\hspace*{0.2em} A_i$\hspace{0.2em}\
and\hspace{0.2em}\ $m_{j i}\hspace*{0.2em} =\hspace*{0.2em} 0$\hspace{0.2em}\ otherwise.\hspace{0.4em}\
Let us consider the product $M\hspace*{0.05em} M^{\hspace*{0.05em} T}$\hspace*{-0.15em},\hspace{0.4em}\
where $M^{\hspace*{0.05em} T}$ is the matrix transposed to $M$\hspace*{-0.15em}.\hspace{0.4em}\
It is an $n \times n$ matrix with all non-diagonal entries equal to $1$
and with diagonal entries\hspace{0.4em}\
$k_{\hspace*{0.05em} 1}\hspace*{0.05em},\hspace*{0.3em} k_{\hspace*{0.05em} 2}\hspace*{0.05em},\hspace*{0.3em} \ldots\hspace*{0.05em},\hspace*{0.3em} k_{\hspace*{0.05em} n}$\hspace*{-0.15em}.\hspace{0.4em}\
The most classical linear algebra proofs\hspace*{0.025em},\hspace{0.4em}\
going back to the paper\hspace{0.2em}\ \cite{bo}\hspace{0.2em}\ by\hspace{0.2em}\ R.C.\hspace{0.2em}\ Bose,\hspace{0.4em}\
proceed with the computation of the determinant of
$M\hspace*{0.05em} M^{\hspace*{0.05em} T}$\hspace*{-0.2em}.\hspace{0.4em}\
It is rarely presented in details;\hspace{0.4em}\
apparently,\hspace{0.3em}\ it is expected that the readers enjoy computations of determinants.\hspace{0.4em}\
Curious readers may find a computation of\hspace{0.2em}\ $\det\hspace*{0.2em} M\hspace*{0.05em} M^{\hspace*{0.05em} T}$\hspace{0.2em}\
at the end of this section;\hspace{0.4em}\
in particular\hspace*{0.025em},\hspace{0.4em}\
the computation shows that this determinant is non-zero.\hspace{0.4em}\
The non-vanishing of\hspace{0.2em}\ $\det\hspace*{0.2em} M\hspace*{0.05em} M^{\hspace*{0.05em} T}$\hspace{0.2em}\
means that the rank of the matrix $M\hspace*{0.05em} M^{\hspace*{0.05em} T}$ is equal to $n$\hspace*{-0.15em},\hspace{0.4em}\
and this implies that the rank of $M$ is\hspace{0.2em}\ $\geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
Since $M$ is\hspace{0.1em}\ an $n \times m$ matrix,\hspace{0.4em}\
this may happen only if\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
More modern expositions avoid computation of the determinant\hspace{0.2em}\
$\det\hspace*{0.2em} M\hspace*{0.05em} M^{\hspace*{0.05em} T}$\hspace{0.2em}\
by observing that $M\hspace*{0.05em} M^{\hspace*{0.05em} T}$ is equal to the sum of the diagonal matrix
with the diagonal entries\hspace{0.2em}\ \vspace*{2pt}
\[
\quad
k_{\hspace*{0.05em} 1}\hspace*{0.2em} -\hspace*{0.2em} 1\hspace*{0.05em},\hspace*{0.4em} k_{\hspace*{0.05em} 2}\hspace*{0.2em} -\hspace*{0.2em} 1\hspace*{0.05em},\hspace*{0.4em} \ldots\hspace*{0.05em},\hspace*{0.4em} k_{\hspace*{0.05em} n}\hspace*{0.2em} -\hspace*{0.2em} 1
\]
\vspace*{-10pt}
and the $n \times n$ matrix $J$ with all entries equal to $1$\hspace*{-0.15em}.\hspace{0.4em}\
Since\hspace{0.2em}\ $k_{\hspace*{0.05em} j}\hspace*{0.2em} \geqslant\hspace*{0.2em} 2$\hspace{0.2em}\ and hence\hspace{0.2em}\
$k_{\hspace*{0.05em} j}\hspace*{0.2em} -\hspace*{0.2em} 1\hspace*{0.2em} \geqslant\hspace*{0.2em} 1$\hspace{0.2em}\ for all $j$\hspace*{-0.2em},\hspace{0.4em}\
the first matrix is positive definite.\hspace{0.4em}\
The matrix $J$ is positive semi-definite,\hspace{0.2em}\
although is not definite.\hspace{0.4em}\
In fact\hspace*{0.025em},\hspace{0.4em}\
the associated quadratic form $\mathbf{x}\hspace*{0.2em} J\hspace*{0.15em} \mathbf{x}^{\hspace*{0.05em} T}$\hspace*{-0.2em},\hspace{0.4em}\
where\hspace{0.2em}\
$\mathbf{x}
\hspace*{0.4em} =\hspace*{0.4em}
(\hspace*{0.05em} x_{\hspace*{0.05em} 1}\hspace*{0.05em},\hspace*{0.3em} x_{\hspace*{0.05em} 2}\hspace*{0.05em},\hspace*{0.3em} \ldots,\hspace*{0.05em} x_{\hspace*{0.05em} n} \hspace*{0.05em})$\hspace{0.2em}\
is a row vector\hspace*{0.025em},\hspace{0.4em}\
is equal to\hspace{0.2em}\
$(\hspace*{0.05em} x_{\hspace*{0.05em} 1}\hspace*{0.2em} +\hspace*{0.2em} x_{\hspace*{0.05em} 2}\hspace*{0.2em} +\hspace*{0.2em} \ldots\hspace*{0.2em} +\hspace*{0.2em} x_{\hspace*{0.05em} n} \hspace*{0.05em})^{\hspace*{0.05em} 2}$\hspace*{-0.2em}.\hspace{0.4em}\
It follows that the sum $M\hspace*{0.05em} M^{\hspace*{0.05em} T}$ of these matrices
is positive definite and hence has the rank $n$\hspace*{-0.15em}.\hspace{0.4em}\
As above,\hspace{0.4em}\ this implies that\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
\myuppar{Comparing the proofs.}
The standard proofs do not fit the\hspace{0.2em}\
\emph{``Kvant''}\hspace{0.2em}\ description of the proof by A.\hspace{0.2em}\ Suslin:\hspace{0.2em}\
they use more advanced tools than the theorem about the linear dependence of
more than $n$ vectors in an $n$\hspace*{-0.2em}-dimensional vector space.\hspace{0.4em}\
One can find a proof based only on this theorem
in the unpublished book draft\hspace{0.2em}\ \cite{bf}\hspace{0.2em}\ by\hspace{0.1em}\
L.\hspace{0.2em}\ Babai and\hspace{0.1em}\ P.\hspace{0.2em}\ Frankl.\hspace{0.4em}\
But even in this remarkable book it is hidden in the exercises.\hspace{0.4em}\
See Exercise\hspace{0.2em}\ 4.1.5\hspace{0.2em}\ and its solution on p.\hspace{0.2em}\ 184.\hspace{0.4em}\
The preference for using the matrix $M\hspace*{0.05em} M^{\hspace*{0.05em} T}$
seems to be a part of a dominating culture.\hspace{0.4em}\
On the other hand,\hspace{0.4em}\ all proofs based on the linear algebra more or less explicitly reduce
the inequality\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace{0.2em}\ to the following lemma and then prove it\hspace*{0.025em}.\hspace{0.4em}\
\myuppar{Lemma.}
\emph{Let\hspace{0.2em}\ $V$ be an $m$\hspace*{-0.2em}-dimensional vector space over\hspace{0.2em}\ $\mathbf{R}$ equipped with
a scalar product\hspace{0.2em}\ $(\hspace*{0.05em} \bullet\hspace*{0.05em},\hspace*{0.3em} \bullet \hspace*{0.05em})$\hspace*{-0.2em}.\hspace{0.4em}\
Let\hspace{0.2em}\ $P$ be a set of $n$ vectors in\hspace{0.2em}\ $V$\hspace*{-0.2em}.\hspace{0.4em}\
Suppose that there exists\hspace{0.2em}\
$\lambda\hspace*{0.2em} \in\hspace*{0.2em} \mathbf{R}$\hspace*{-0.15em},\hspace{0.4em}\
$\lambda\hspace*{0.2em} >\hspace*{0.2em} 0$\hspace*{-0.2em},\hspace{0.4em}\
such that\hspace{0.2em}\ }\vspace*{2pt}
\[
\quad
(\hspace*{0.05em} u\hspace*{0.05em},\hspace*{0.3em} u \hspace*{0.05em})\hspace*{0.4em} >\hspace*{0.4em} \lambda
\hspace*{1.5em}\mbox{ \emph{and} }\hspace*{1.5em}
(\hspace*{0.05em} v\hspace*{0.05em},\hspace*{0.3em} w \hspace*{0.05em})\hspace*{0.4em} =\hspace*{0.4em} \lambda
\]
\vspace*{-10pt}
\emph{for every\hspace{0.2em}\ $u\hspace*{0.2em} \in\hspace*{0.2em} P$\hspace{0.2em}\
and every two distinct vectors\hspace{0.2em}\
$v\hspace*{0.05em},\hspace*{0.3em} w\hspace*{0.2em} \in\hspace*{0.2em} P$\hspace*{-0.2em}.\hspace{0.4em}\
Then\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\ } $\blacksquare$
\myuppar{A generalization.}
The linear algebra proofs apply with only trivial changes to a more general situation.\hspace{0.4em}\
Namely,\hspace{0.4em}\ it is sufficient to assume that
there exist a natural number\hspace{0.2em}\ $\lambda\hspace*{0.2em} \geqslant\hspace*{0.2em} 1$\hspace{0.2em}\
such that every two distinct points are contained in exactly $\lambda$ lines
and every point is contained in\hspace{0.2em}\ $>\hspace*{0.2em} \lambda$\hspace{0.2em}\ lines.\hspace{0.4em}\
Then the conclusion\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace{0.2em}\ still holds.\hspace{0.4em}\
This is due to H.J.\hspace{0.1em}\ Ryser\hspace{0.2em}\ \cite{r}.\hspace{0.4em}\
Apparently,\hspace{0.4em}\ no combinatorial proof of\hspace{0.1em}\
Ryser's theorem is known.\hspace{0.4em}\
Ryser\hspace{0.2em}\ \cite{r}\hspace{0.2em}\ also used linear algebra to provide a description of
the case\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace{0.2em}\ similar to
de Bruijn--Erd\"{o}s description in the case\hspace{0.2em}\ $\lambda\hspace*{0.2em} =\hspace*{0.2em} 1$\hspace*{-0.15em}.\hspace{0.4em}\
\myuppar{The determinant of\hspace{0.2em}\ $M\hspace*{0.05em} M^{\hspace*{0.05em} T}$\hspace*{-0.15em}.}
For the benefit of the readers who do not like
to compute the determinants themselves\hspace*{0.025em},\hspace{0.4em}\
here is a computation of\hspace{0.2em}\
$\det\hspace*{0.2em} M\hspace*{0.05em} M^{\hspace*{0.05em} T}$\hspace{0.2em}\
following the textbook\hspace{0.2em}\ \cite{hp}.\hspace{0.4em}\
Let\hspace{0.2em}\ $m_{j}\hspace*{0.2em} =\hspace*{0.2em} k_{\hspace*{0.05em} j}\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\
for all $j$\hspace*{-0.2em}.\hspace{0.4em}\
Then\vspace*{6pt}
\[
\quad
M\hspace*{0.05em} M^{\hspace*{0.05em} T}
\hspace*{0.4em} =\hspace*{0.4em}
\begin{bmatrix}\hspace*{0.4em}
m_{\hspace*{0.05em} 1}\hspace*{0.1em} +\hspace*{0.2em} 1 & 1 & 1 & \hdotsfor{4} & 1 & 1\\
1 & m_{\hspace*{0.05em} 2}\hspace*{0.1em} +\hspace*{0.2em} 1 & 1 & \hdotsfor{4} & 1 & 1\\
1 & 1 & m_{\hspace*{0.05em} 3}\hspace*{0.1em} +\hspace*{0.2em} 1 & \hdotsfor{4} & 1 & 1\\
& & & & & & & & \\
\hdotsfor{9} \\
& & & & & & & & \\
1 & 1 & 1 & \hdotsfor{4} & 1 & m_{\hspace*{0.05em} n}\hspace*{0.1em} +\hspace*{0.2em} 1\hspace*{0.4em}
\end{bmatrix}.
\]
\vspace*{-6pt}
Let us subtract the first row from every other one and get the matrix\vspace*{6pt}
\[
\quad
\phantom{M\hspace*{0.05em} M^{\hspace*{0.05em} T}
\hspace*{0.4em} =\hspace*{0.4em}}
\begin{bmatrix}\hspace*{0.4em}
m_{\hspace*{0.05em} 1}\hspace*{0.1em} +\hspace*{0.2em} 1 & 1 & 1 & \hdotsfor{4} & 1 & 1\\
-\hspace*{0.2em} m_{\hspace*{0.05em} 1} & m_{\hspace*{0.05em} 2} & 0 & \hdotsfor{4} & 0 & 0\\
-\hspace*{0.2em} m_{\hspace*{0.05em} 1} & 0 & m_{\hspace*{0.05em} 3} & \hdotsfor{4} & 0 & 0\\
& & & & & & & & \\
\hdotsfor{9} \\
& & & & & & & & \\
-\hspace*{0.2em} m_{\hspace*{0.05em} 1} & 0 & 0 & \hdotsfor{4} & 0 & m_{\hspace*{0.05em} n}\hspace*{0.4em}
\end{bmatrix}.
\]
\vspace{-6pt}
For\hspace{0.4em}\
$j\hspace*{0.4em} =\hspace*{0.4em} 2\hspace*{0.05em},\hspace*{0.3em} 3\hspace*{0.05em},\hspace*{0.3em} \ldots\hspace*{0.05em},\hspace*{0.3em} n$\hspace*{-0.15em},\hspace{0.4em}\
let us multiply the $j$\hspace*{-0.2em}-th column by\hspace{0.2em}\ $m_{\hspace*{0.05em} 1}/m_{\hspace*{0.05em} j}$\hspace{0.2em}\
(recall that\hspace{0.2em}\ $k_{\hspace*{0.05em} j}\hspace*{0.2em} \geqslant\hspace*{0.2em} 2$\hspace{0.2em}\
and hence\hspace{0.2em}\ $m_j\hspace*{0.2em} \geqslant\hspace*{0.2em} 1\hspace*{0.2em} >\hspace*{0.2em} 0$\hspace*{-0.15em})\hspace{0.2em}\
and add the result to the first column.\hspace{0.4em}\
We get the matrix\vspace*{6pt}
\[
\quad
\phantom{M\hspace*{0.05em} M^{\hspace*{0.05em} T}
\hspace*{0.4em} =\hspace*{0.4em}}
\begin{bmatrix}\hspace*{0.4em}
D & 1 & 1 & \hdotsfor{4} & 1 & 1\\
0 & m_{\hspace*{0.05em} 2} & 0 & \hdotsfor{4} & 0 & 0\\
0 & 0 & m_{\hspace*{0.05em} 3} & \hdotsfor{4} & 0 & 0\\
& & & & & & & & \\
\hdotsfor{9} \\
& & & & & & & & \\
0 & 0 & 0 & \hdotsfor{4} & 0 & m_{\hspace*{0.05em} n}\hspace*{0.4em}
\end{bmatrix},
\]
\vspace*{-4pt}
where\hspace*{0.4em}\hspace{0.4em}\
$\displaystyle
D
\hspace*{0.4em}\off =\hspace*{0.4em}\off
m_{\hspace*{0.05em} 1}
\hspace*{0.2em} +\hspace*{0.2em}
1
\hspace*{0.4em} +\hspace*{0.4em}
\sum_{j\hspace*{0.2em} =\hspace*{0.2em} 2}^n\hspace*{0.2em} \frac{\hspace*{0.05em} m_{\hspace*{0.05em} 1}\hspace*{0.05em}}{m_j}
\hspace*{0.4em}\off =\hspace*{0.4em}\off
m_{\hspace*{0.05em} 1}
\hspace*{0.4em} +\hspace*{0.4em}
\sum_{j\hspace*{0.2em} =\hspace*{0.2em} 1}^n\hspace*{0.2em} \frac{\hspace*{0.05em} m_{\hspace*{0.05em} 1}\hspace*{0.05em}}{m_j}$\hspace*{0.2em}.\vspace*{2pt}
It follows that\hspace*{0.4em}\hspace{0.4em}\
$\displaystyle
\det\hspace*{0.2em} M\hspace*{0.1em} M^{\hspace*{0.05em} T}
\hspace*{0.4em}\off =\hspace*{0.4em}\off
D
\hspace*{0.2em}\cdot
\prod_{j\hspace*{0.2em} =\hspace*{0.2em} 2}^n m_j
\hspace*{0.4em}\off =\hspace*{0.4em}\off
\prod_{j\hspace*{0.2em} =\hspace*{0.2em} 1}^n m_j
\hspace*{0.2em} \cdot\hspace*{0.2em}
\left(\hspace*{0.2em}
1
\hspace*{0.2em} +\hspace*{0.2em}
\sum_{j\hspace*{0.2em} =\hspace*{0.2em} 1}^n\hspace*{0.2em} \frac{\hspace*{0.05em} 1\hspace*{0.05em}}{m_j}
\hspace*{0.2em}\right)
\hspace*{0.4em}\off \neq\hspace*{0.4em}\off 0$.\hspace{0.4em}\
\mysection{Hanani's\hspace{0.2em}\ theorem}{h-proof}
\vspace*{6pt}
\myuppar{Two papers of\hspace{0.1em}\ H.\hspace{0.1em}\ Hanani.}
According to the Th.\hspace{0.1em}\ Motzkin\hspace{0.2em}\ \cite{m},\hspace{0.4em}\
the first proof of the inequality\hspace{0.1em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace{0.1em}\
and,\hspace{0.3em}\ it seems,\hspace{0.3em}\
of the full de Bruijn--Erd\"{o}s theorem,\hspace{0.4em}\
was given in\hspace{0.2em}\ 1938\hspace{0.2em}\ by H.\hspace{0.1em}\ Hanani.\hspace{0.4em}\
He published an outline of his proof\hspace{0.2em}\ \cite{h1}\hspace{0.2em}\
only in\hspace{0.2em}\ 1951.\hspace{0.4em}\
Later on H.\hspace{0.1em}\ Hanani published a detailed
exposition\hspace{0.2em}\ \cite{h2}\hspace{0.2em}\ of a simplified proof\hspace*{0.025em}.\hspace{0.4em}\
In fact\hspace*{0.025em},\hspace{0.4em}\ in\hspace{0.2em}\ \cite{h2}\hspace{0.2em}\ he proved\hspace{0.2em}\
(at no extra cost)\hspace{0.2em}\
a stronger version of the
de Bruijn--Erd\"{o}s theorem.\hspace{0.4em}\
He also used his methods to prove a\hspace{0.1em}\ $3$\hspace*{-0.2em}-dimensional analogue
dealing with points,\hspace{0.2em}\ lines,\hspace{0.2em}\ and planes.\hspace{0.4em}\
\myuppar{Hanani's\hspace{0.1em}\ Theorem.}
\emph{Under the previous assumptions,\hspace{0.4em}\
let\hspace{0.2em}\ $L\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}$\hspace{0.2em}\ be a line containing the maximal
number of points among all lines,\hspace{0.4em}\
let\hspace{0.2em}\ $\mathcal{P}$\hspace{0.2em}\ be the set of all lines intersecting\hspace{0.2em}\ $L$\hspace{0.2em}\
(in particular\hspace*{0.025em},\hspace{0.4em}\ $L\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{P}$\hspace*{-0.15em}),\hspace{0.4em}\
and let\hspace{0.1em}\ $p$\hspace{0.1em}\ be the number of elements of\hspace{0.2em}\ $\mathcal{P}$\hspace*{-0.2em}.\hspace{0.4em}\
Then\hspace{0.2em}\ $p\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em},\hspace{0.4em}\
and\hspace{0.1em}\ if\hspace{0.2em}\ $p\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em},\hspace{0.4em}\ then\hspace{0.2em}\
$\mathcal{P}\hspace*{0.2em} =\hspace*{0.2em} \mathcal{L}$\hspace{0.2em}\ and\hspace{0.3em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.3em}\
is either a near-pencil,\hspace{0.4em}\ or a projective plane.\hspace{0.4em}\ }
\vspace*{6pt}
Suppose
that\hspace{0.1em}\ $n\hspace*{0.2em} \geqslant\hspace*{0.2em} p$\hspace{0.1em}\
and\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\ is not a near-pencil.\hspace{0.4em}\
As usual,\hspace{0.4em}\ we assume that every line contains\hspace{0.2em}\ $\geqslant\hspace*{0.2em} 2$\hspace{0.2em}\ points.\hspace{0.4em}\
Let\hspace{0.2em}\ $a\hspace*{0.2em} =\hspace*{0.2em} s_{\hspace*{0.1em} L}$\hspace*{-0.15em}.\hspace{0.4em}\
Let $K$ be the line with the maximal number of points among the lines different from $L$\hspace*{-0.15em},\hspace{0.4em}\
and\hspace{0.1em}\ let\hspace{0.2em}\ $b\hspace*{0.2em} =\hspace*{0.2em} s_{\hspace*{0.1em} K}$\hspace*{-0.15em}.\hspace{0.4em}\
Then\hspace{0.2em}\ $a\hspace*{0.2em} \geqslant\hspace*{0.2em} b$\hspace*{-0.15em}.\hspace{0.4em}\
The strategy is to estimate $n$\hspace*{-0.15em},\hspace{0.3em}\
or\hspace*{0.025em},\hspace{0.3em}\ what is the same,\hspace{0.2em}\ $n\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\
in terms of $a$ and $b$ both from the below and from the above.\hspace{0.4em}\
\myuppar{Hanani's\hspace{0.1em}\ Lemma.}
\emph{If\hspace{0.2em}\ $x\hspace*{0.2em} \in\hspace*{0.2em} L$\hspace*{-0.15em},\hspace{0.4em}\ then}\hspace{0.4em}\ \vspace*{2pt}
\begin{equation}
\label{lemma}
k_{\hspace*{0.05em} x}\hspace*{0.2em} -\hspace*{0.2em} 1\hspace*{0.4em} \geqslant\hspace*{0.4em} \frac{n\hspace*{0.2em} -\hspace*{0.2em} a}{b\hspace*{0.2em} -\hspace*{0.2em} 1}\hspace*{0.2em}.
\end{equation}
\vspace*{-10pt}
\vspace{0.5\bigskipamount}{\textbf{{\emph{Proof}.}}\hspace*{0.7em}}\hspace{0.2em}\
Let us consider pairs\hspace{0.2em}\ $(\hspace*{0.05em} l\hspace*{0.05em},\hspace*{0.3em} y\hspace*{0.05em})$\hspace{0.2em}\
such that $l$ is a line containing $x$ and
$y$ is a point in\hspace{0.2em}\ $l\hspace*{0.1em} \smallsetminus\hspace*{0.1em} L$\hspace*{-0.15em}.\hspace{0.4em}\
Such a pair is uniquely determined by the point $y$
and hence there are $n\hspace*{0.2em} -\hspace*{0.2em} a$ such pairs.\hspace{0.4em}\
A line $l$ occurs in such a pair if and only if\hspace{0.2em}\ $x\hspace*{0.2em} \in\hspace*{0.2em} l$\hspace{0.2em}\
and\hspace{0.2em}\ $l\hspace*{0.2em} \neq\hspace*{0.2em} L$\hspace*{-0.15em}.\hspace{0.4em}\
It follows that there are\hspace{0.2em}\ $k_{\hspace*{0.05em} x}\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\ choices of $l$\hspace*{-0.15em}.\hspace{0.4em}\
Given a line $l$\hspace*{-0.2em},\hspace{0.4em}\
there are\hspace{0.2em}\ $\leqslant\hspace*{0.2em} b\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\
choices for the point $y$\hspace*{-0.2em}.\hspace{0.4em}\
Therefore the number\hspace{0.2em}\ $n\hspace*{0.2em} -\hspace*{0.2em} a$\hspace{0.2em}\ of such pairs is\hspace{0.2em}\
$\leqslant\hspace*{0.2em} (k_{\hspace*{0.05em} x}\hspace*{0.2em} -\hspace*{0.2em} 1)(b\hspace*{0.2em} -\hspace*{0.2em} 1)$\hspace*{-0.2em}.\hspace{0.4em}\
The lemma follows.\hspace{0.4em}\ $\blacksquare$
\myuppar{An upper estimate of\hspace{0.2em}\ $n\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace*{-0.15em}.}
By summing the inequalities\hspace{0.2em}\ (\ref{lemma})\hspace{0.2em}\ over all\hspace{0.2em}\ $x\hspace*{0.2em} \in\hspace*{0.2em} L$\hspace{0.2em}\
and adding $1$ in order to account for the line $L$ itself\hspace*{0.025em},\hspace{0.4em}\
we can estimate $p$ from below and conclude that\vspace*{3pt}
\begin{equation}
\label{p-lower-1}
\quad
n
\hspace*{0.4em} \geqslant\hspace*{0.4em}
p
\hspace*{0.4em} \geqslant\hspace*{0.4em}
1\hspace*{0.2em} +\hspace*{0.2em} a\hspace*{0.2em} \frac{n\hspace*{0.2em} -\hspace*{0.2em} a}{b\hspace*{0.2em} -\hspace*{0.2em} 1}
\hspace*{0.4em} =\hspace*{0.4em}
1\hspace*{0.2em} +\hspace*{0.2em} a\hspace*{0.2em} \frac{(n\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.2em} -\hspace*{0.2em} (a\hspace*{0.2em} -\hspace*{0.2em} 1)}{b\hspace*{0.2em} -\hspace*{0.2em} 1}
\end{equation}
\vspace*{-9pt}
or\hspace*{0.025em},\hspace{0.4em}\ what is the same,\hspace{0.4em}\ \vspace*{3pt}
\begin{equation}
\label{n-upper}
\quad
a\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)
\hspace*{0.4em} \geqslant\hspace*{0.4em}
(n\hspace*{0.2em} -\hspace*{0.2em} 1)(a\hspace*{0.2em} -\hspace*{0.2em} b\hspace*{0.2em} +\hspace*{0.2em} 1)\hspace*{0.1em}.
\end{equation}
\vspace*{-9pt}
The inequality\hspace{0.2em}\ (\ref{n-upper})\hspace{0.2em}\
provides an estimate of\hspace{0.2em}\ $n\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\ from the above.\hspace{0.4em}\
\myuppar{A lower estimate of\hspace{0.2em}\ $n\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace*{-0.15em}.}
There is another way to estimate $p$ from below.\hspace{0.4em}\
By a miracle,\hspace{0.4em}\ this other estimate of $p$ from the same side
leads to an estimate of\hspace{0.2em}\ $n\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\ from the
other side.\hspace{0.4em}\
Let $z$ be a point in\hspace{0.2em}\ $L\hspace*{0.1em} \cap\hspace*{0.1em} K$\hspace{0.2em}\ if\hspace{0.2em}\
$L\hspace*{0.1em} \cap\hspace*{0.1em} K\hspace*{0.2em} \neq\hspace*{0.2em} \emptyset$\hspace*{-0.2em},\hspace{0.4em}\
and an arbitrary point of $L$ otherwise.\hspace{0.4em}\
For every\hspace{0.2em}\ $x\hspace*{0.2em} \in\hspace*{0.2em} L\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \{\hspace*{0.1em} z\hspace*{0.1em}\}$\hspace*{-0.15em},\hspace{0.2em}\
$y\hspace*{0.2em} \in\hspace*{0.2em} K\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \{\hspace*{0.1em} z\hspace*{0.1em}\}$\hspace{0.2em}\
there is a unique line $l$ containing\hspace{0.2em}\ $\{\hspace*{0.1em} x\hspace*{0.05em},\hspace*{0.3em} y \hspace*{0.1em}\}$\hspace*{-0.15em}.\hspace{0.4em}\
All these lines are distinct\hspace*{0.025em},\hspace{0.4em}\ not equal to $L$\hspace*{-0.15em},\hspace{0.4em}\
and do not contain $z$\hspace*{-0.15em}.\hspace{0.4em}\
Clearly,\hspace{0.4em}\ there are\hspace{0.2em}\
$(a\hspace*{0.2em} -\hspace*{0.2em} 1)(b\hspace*{0.2em} -\hspace*{0.2em} 1)$\hspace{0.2em}\
of such lines.\hspace{0.4em}\
A lower estimate of number $k_{\hspace*{0.05em} z}$
of lines containing $z$ is provided by\hspace{0.2em}\ (\ref{lemma}).\hspace{0.4em}\
It follows that\vspace*{4pt}
\begin{equation*}
\quad
n
\hspace*{0.4em} \geqslant\hspace*{0.4em}
p
\hspace*{0.4em} \geqslant\hspace*{0.4em}
1
\hspace*{0.2em} +\hspace*{0.2em}
\frac{n\hspace*{0.2em} -\hspace*{0.2em} a}{b\hspace*{0.2em} -\hspace*{0.2em} 1}
\hspace*{0.2em} +\hspace*{0.2em}
(a\hspace*{0.2em} -\hspace*{0.2em} 1)(b\hspace*{0.2em} -\hspace*{0.2em} 1)
\end{equation*}
\vspace*{-33pt}
\[
\quad
\hspace*{12em}
\phantom{n
\hspace*{0.4em} \geqslant\hspace*{0.4em}
p
\hspace*{0.4em} }
=\hspace*{0.4em}
1
\hspace*{0.2em} +\hspace*{0.2em}
\frac{(n\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.2em} -\hspace*{0.2em} (a\hspace*{0.2em} -\hspace*{0.2em} 1)}{b\hspace*{0.2em} -\hspace*{0.2em} 1}
\hspace*{0.2em} +\hspace*{0.2em}
(a\hspace*{0.2em} -\hspace*{0.2em} 1)(b\hspace*{0.2em} -\hspace*{0.2em} 1)
\]
\vspace*{-8pt}
and hence\hspace*{0.2em}\hspace{0.4em}\
$\displaystyle
(n\hspace*{0.2em} -\hspace*{0.2em} 1)(b\hspace*{0.2em} -\hspace*{0.2em} 1)
\hspace*{0.4em} \geqslant\hspace*{0.4em}
(n\hspace*{0.2em} -\hspace*{0.2em} 1)
\hspace*{0.2em} -\hspace*{0.2em}
(a\hspace*{0.2em} -\hspace*{0.2em} 1)
\hspace*{0.2em} +\hspace*{0.2em}
(a\hspace*{0.2em} -\hspace*{0.2em} 1)(b\hspace*{0.2em} -\hspace*{0.2em} 1)^{\hspace*{0.05em} 2}$\hspace*{0.2em}\hspace{0.4em}\
and\vspace*{3pt}
\begin{equation*}
\quad
(n\hspace*{0.2em} -\hspace*{0.2em} 1)(b\hspace*{0.2em} -\hspace*{0.2em} 2)
\hspace*{0.4em} \geqslant\hspace*{0.4em}
(a\hspace*{0.2em} -\hspace*{0.2em} 1)(b^{\hspace*{0.05em} 2}\hspace*{0.2em} -\hspace*{0.2em} 2\hspace*{0.05em} b)
\hspace*{0.4em} =\hspace*{0.4em}
(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.05em} b\hspace*{0.05em} (b\hspace*{0.2em} -\hspace*{0.2em} 2).
\end{equation*}
\vspace*{-9pt}
Since\hspace{0.2em}\ $b\hspace*{0.2em} \geqslant\hspace*{0.2em} 2$\hspace*{-0.15em},\hspace{0.4em}\
it follows that either\hspace{0.2em}\ $b\hspace*{0.2em} =\hspace*{0.2em} 2$\hspace*{-0.15em},\hspace{0.4em}\
or\hspace{0.2em}\
$\displaystyle
n\hspace*{0.2em} -\hspace*{0.2em} 1
\hspace*{0.4em} \geqslant\hspace*{0.4em}
(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.05em} b$\hspace*{-0.15em}.\hspace{0.4em}\
If\hspace{0.2em}\ $b\hspace*{0.2em} =\hspace*{0.2em} 2$\hspace*{-0.15em},\hspace{0.4em}\
then all lines except $L$ consist of $2$ points and the inequality\hspace{0.2em}\
(\ref{n-upper})\hspace{0.2em}\ implies that\hspace{0.2em}\
$a\hspace*{0.2em} \geqslant\hspace*{0.2em} n\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace*{-0.15em}.\hspace{0.4em}\
But\hspace{0.2em}\ $L\hspace*{0.2em} \neq\hspace*{0.2em} E$\hspace{0.2em}\ and hence\hspace{0.2em}\
$a\hspace*{0.2em} \leqslant\hspace*{0.2em} n\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace*{-0.15em}.\hspace{0.4em}\
It follows that\hspace{0.2em}\ $a\hspace*{0.2em} =\hspace*{0.2em} n\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\
and hence $L$ contains all points of $E$ except one
and\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\ is a near-pencil,\hspace{0.4em}\
contrary to the assumption.\hspace{0.4em}\
Therefore\hspace{0.2em}\ \vspace*{3pt}
\begin{equation}
\label{n-lower}
\quad
n\hspace*{0.2em} -\hspace*{0.2em} 1
\hspace*{0.4em} \geqslant\hspace*{0.4em}
(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.05em} b\hspace*{0.1em}.
\end{equation}
\vspace*{-9pt}
The inequality\hspace{0.2em}\ (\ref{n-lower})\hspace{0.2em}\
provides an estimate of\hspace{0.2em}\ $n\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\ from the below.\hspace{0.4em}\
\myuppar{Combining the two estimates.}
After multiplying the inequality\hspace{0.2em}\ (\ref{n-lower})\hspace{0.2em}\ by\hspace{0.2em}\
$(a\hspace*{0.2em} -\hspace*{0.2em} b\hspace*{0.2em} +\hspace*{0.2em} 1)$\hspace{0.2em}\
and combining the result with the inequality\hspace{0.2em}\ (\ref{n-upper}),\hspace{0.4em}\
we see that\vspace*{3pt}
\[
\quad
a\hspace*{0.05em} (a\hspace*{0.2em} -\hspace*{0.2em} 1)
\hspace*{0.4em} \geqslant\hspace*{0.4em}
(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.05em} b\hspace*{0.05em} (a\hspace*{0.2em} -\hspace*{0.2em} b\hspace*{0.2em} +\hspace*{0.2em} 1)
\]
\vspace*{-9pt}
and hence\hspace{0.4em}\
$\displaystyle
a
\hspace*{0.4em} \geqslant\hspace*{0.4em}
b\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} b\hspace*{0.2em} +\hspace*{0.2em} 1)
\hspace*{0.4em} =\hspace*{0.4em}
b\hspace*{0.05em} (a\hspace*{0.2em} -\hspace*{0.2em} b)
\hspace*{0.2em} +\hspace*{0.2em}
b$\hspace*{-0.15em},\hspace{0.4em}\
or\hspace*{0.025em},\hspace{0.4em}\ what is the same
\[
\quad
0
\hspace*{0.4em} \geqslant\hspace*{0.4em}
(b\hspace*{0.2em} -\hspace*{0.2em} 1) (a\hspace*{0.2em} -\hspace*{0.2em} b)\hspace*{0.1em}.
\]
Since\hspace{0.2em}\ $b\hspace*{0.2em} >\hspace*{0.2em} 1$\hspace*{-0.15em},\hspace{0.4em}\ this implies that\hspace{0.2em}\ $b\hspace*{0.2em} \geqslant\hspace*{0.2em} a$\hspace*{-0.15em}.\hspace{0.4em}\
On the other hand,\hspace{0.4em}\
$b\hspace*{0.2em} \leqslant\hspace*{0.2em} a$\hspace{0.2em}\ by the definition of\hspace{0.2em}\ $a\hspace*{0.05em},\hspace*{0.3em} b$\hspace*{-0.15em}.\hspace{0.4em}\
It follows that\hspace{0.2em}\ $a\hspace*{0.2em} =\hspace*{0.2em} b$\hspace*{-0.15em}.\hspace{0.4em}\
By combining\hspace{0.2em}\ $a\hspace*{0.2em} =\hspace*{0.2em} b$\hspace{0.2em}\ with the inequalities\hspace{0.2em}\
(\ref{n-upper})\hspace{0.2em}\
and\hspace{0.2em}\ (\ref{n-lower})\hspace{0.2em}\
we conclude,\hspace{0.4em}\ respectively,\hspace{0.4em}\
that\hspace{0.2em}\
$a\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)
\hspace*{0.4em} \geqslant\hspace*{0.4em}
n\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\
and\hspace{0.2em}\
$n\hspace*{0.2em} -\hspace*{0.2em} 1
\hspace*{0.4em} \geqslant\hspace*{0.4em}
a\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)$\hspace*{-0.2em}.\hspace{0.4em}\
It follows that\hspace{0.2em}\
$n\hspace*{0.2em} -\hspace*{0.2em} 1
\hspace*{0.4em} =\hspace*{0.4em}
a\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)$\hspace{0.2em}\
and\hspace{0.1em}\ hence\hspace{0.2em}\
$n
\hspace*{0.4em} =\hspace*{0.4em}
a\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.2em} +\hspace*{0.2em} 1$\hspace*{-0.15em}.\hspace{0.4em}\
By combining this with\hspace{0.2em}\ (\ref{p-lower-1})\hspace{0.2em}\ we see that\vspace*{3pt}
\[
\quad
a\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.2em} +\hspace*{0.2em} 1
\hspace*{0.4em} =\hspace*{0.4em}
n
\hspace*{0.4em} \geqslant\hspace*{0.4em}
p
\hspace*{0.4em} \geqslant\hspace*{0.4em}
1\hspace*{0.2em} +\hspace*{0.2em} a\hspace*{0.2em} \frac{a\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.2em} +\hspace*{0.2em} 1\hspace*{0.2em} -\hspace*{0.2em} a}{a\hspace*{0.2em} -\hspace*{0.2em} 1}
\hspace*{0.4em} =\hspace*{0.4em}
1\hspace*{0.2em} +\hspace*{0.2em} a\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.1em}.
\]
\vspace{-9pt}
It follows that\hspace{0.2em}\ $n\hspace*{0.2em} =\hspace*{0.2em} p$\hspace*{-0.2em},\hspace{0.4em}\
and therefore\hspace{0.2em}\ $p\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace{0.2em}\
if the inequality\hspace{0.2em}\ $p\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace{0.2em}\ is not assumed.
\myuppar{The case\hspace{0.2em}\ $p\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em}.}
As we just saw,\hspace{0.4em}\
in this case\hspace{0.2em}\ $a\hspace*{0.2em} =\hspace*{0.2em} b$\hspace{0.2em}\
and\hspace{0.2em}\
$n\hspace*{0.2em} =\hspace*{0.2em} a\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.2em} +\hspace*{0.2em} 1$\hspace*{-0.15em}.\hspace{0.4em}\
Let us prove first that every line belonging to $\mathcal{P}$ consists of exactly $a$ points.\hspace{0.4em}\
Consider all pairs\hspace{0.2em}\ $(\hspace*{0.05em} l\hspace*{0.05em},\hspace*{0.3em} y\hspace*{0.05em})$\hspace{0.2em}\ such that\hspace{0.2em}\
$l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{P}$\hspace{0.2em}\ and\hspace{0.2em}\ $y\hspace*{0.2em} \in\hspace*{0.2em} E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} L$\hspace*{-0.15em}.\hspace{0.4em}\
The line $l$\hspace{0.1em}\ is uniquely determined by its point of intersection with $L$\hspace{0.1em}\
(which can be any point of\hspace{0.1em}\ $L$\hspace*{-0.15em})\hspace{0.2em}\ and the point $y$\hspace*{-0.2em}.\hspace{0.4em}\
Therefore there are\hspace{0.1em}\
$a\hspace*{0.05em}(n\hspace*{0.2em} -\hspace*{0.2em} a)
\hspace*{0.4em} =\hspace*{0.4em}
a\hspace*{0.05em} n\hspace*{0.2em} -\hspace*{0.2em} a^{\hspace*{0.05em} 2}$\hspace{0.1em}\
such pairs.\hspace{0.4em}\
On the other hand,\hspace{0.2em}\
for every line\hspace{0.2em}\
$l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{P}\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \{\hspace*{0.2em} L \hspace*{0.2em}\}$\hspace{0.2em}\
there are\hspace{0.2em}\ $\leqslant\hspace*{0.2em} a\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\
choices of the point $y$ and hence the number of such pairs is\hspace{0.2em}\
$\leqslant\hspace*{0.2em} (p\hspace*{0.2em} -\hspace*{0.2em} 1)(a\hspace*{0.2em} -\hspace*{0.2em} 1)$\hspace*{-0.2em}.\hspace{0.4em}\
Moreover\hspace*{0.025em},\hspace{0.4em}\ if at least one line\hspace{0.2em}\
$l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{P}\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \{\hspace*{0.2em} L \hspace*{0.2em}\}$\hspace{0.2em}\
has\hspace{0.2em}\ $<\hspace*{0.2em} a$\hspace{0.2em}\ points,\hspace{0.4em}\
then the number of such pairs is\hspace{0.2em}\
$<\hspace*{0.2em} (p\hspace*{0.2em} -\hspace*{0.2em} 1)(a\hspace*{0.2em} -\hspace*{0.2em} 1)$\hspace*{-0.2em}.\hspace{0.4em}\
But\hspace{0.2em}\
$(p\hspace*{0.2em} -\hspace*{0.2em} 1)(a\hspace*{0.2em} -\hspace*{0.2em} 1)
\hspace*{0.4em} =\hspace*{0.4em}
(n\hspace*{0.2em} -\hspace*{0.2em} 1)(a\hspace*{0.2em} -\hspace*{0.2em} 1)
\hspace*{0.4em} =\hspace*{0.4em}
n\hspace*{0.05em} a\hspace*{0.2em} -\hspace*{0.2em} a^{\hspace*{0.05em} 2}$\hspace*{-0.2em}.\hspace{0.4em}\
It follows that every line belonging to\hspace{0.2em}\
$\mathcal{P}\hspace*{0.1em} \smallsetminus\hspace*{0.1em} \{\hspace*{0.2em} L \hspace*{0.2em}\}$\hspace*{-0.15em},\hspace{0.4em}\
and hence every line belonging to $\mathcal{P}$\hspace*{-0.2em},\hspace{0.4em}\
consists of exactly $a$ points.\hspace{0.4em}\
Now we are ready to prove that\hspace{0.2em}\ $\mathcal{L}\hspace*{0.4em} =\hspace*{0.4em} \mathcal{P}$\hspace*{-0.2em}.\hspace{0.4em}\
By the definition,\hspace{0.3em}\ every line containing a point of $L$ belongs to $\mathcal{P}$\hspace*{-0.2em}.\hspace{0.4em}\
Let\hspace{0.2em}\ $y\hspace*{0.2em} \in\hspace*{0.2em} E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} L$\hspace*{-0.15em}.\hspace{0.4em}\
For every\hspace{0.2em}\ $x\hspace*{0.2em} \in\hspace*{0.2em} L$\hspace{0.2em}\ there is
a unique line containing\hspace{0.1em}\ $\{\hspace*{0.05em} x\hspace*{0.05em},\hspace*{0.3em} y\hspace*{0.1em}\}$\hspace*{-0.15em}.\hspace{0.4em}\
These lines are pairwise distinct\hspace*{0.025em},\hspace{0.4em}\
intersect only at $y$\hspace*{-0.2em},\hspace{0.3em}\
and belong to $\mathcal{P}$\hspace*{-0.2em}.\hspace{0.4em}\
Moreover\hspace*{0.025em},\hspace{0.4em}\ every line containing $y$ and belonging to $\mathcal{P}$
is equal to one of these $a$ lines.\hspace{0.4em}\
Each of these lines contains\hspace{0.2em}\
$a\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\
points different form $y$\hspace*{-0.2em}.\hspace{0.4em}\
It follows that the total number of points on these lines is equal to\hspace{0.2em}\
$a\hspace*{0.1em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.2em} +\hspace*{0.2em} 1$\hspace*{-0.15em},\hspace{0.4em}\
i.e.\hspace{0.2em}\ to the number $n$ of points in $E$\hspace*{-0.15em}.\hspace{0.4em}\
Therefore for every point\hspace{0.2em}\ $z\hspace*{0.2em} \neq\hspace*{0.2em} y$\hspace{0.2em}\
there is a line belonging to $\mathcal{P}$ and
containing\hspace{0.2em}\ $\{\hspace*{0.05em} z\hspace*{0.05em},\hspace*{0.3em} y \hspace*{0.1em}\}$\hspace*{-0.15em}.\hspace{0.4em}\
Since there is only one line containing any two given points,\hspace{0.4em}\
it follows that all lines containing a point\hspace{0.2em}\ $y\hspace*{0.2em} \in\hspace*{0.2em} E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} L$\hspace{0.2em}\
belong to $\mathcal{P}$\hspace*{-0.2em}.\hspace{0.4em}\
It follows that\hspace{0.2em}\ $\mathcal{L}\hspace*{0.4em} =\hspace*{0.4em} \mathcal{P}$\hspace{0.2em}\
and every point in\hspace{0.2em}\ $E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} L$\hspace{0.2em}\ belongs to exactly $a$ lines.\hspace{0.4em}\
In view of the previous paragraph,\hspace{0.3em}\ $\mathcal{L}\hspace*{0.4em} =\hspace*{0.4em} \mathcal{P}$\hspace{0.2em}\
implies that every line consists of exactly $a$ points.\hspace{0.4em}\
By the previous paragraph\hspace{0.3em}\ $k_{\hspace*{0.05em} y}\hspace*{0.2em} =\hspace*{0.2em} a$\hspace{0.3em}\
if\hspace{0.3em}\
$y\hspace*{0.2em} \in\hspace*{0.2em} E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} L$\hspace*{-0.15em}.\hspace*{0.2em}\hspace{0.4em}\
If\hspace{0.3em}\ $y\hspace*{0.2em} \in\hspace*{0.2em} L$\hspace*{-0.15em},\hspace{0.4em}\
then\hspace{0.1em}\ by\hspace{0.3em}\ (\ref{lemma})\vspace*{2pt}
\[
\quad
k_{\hspace*{0.05em} y}
\hspace*{0.4em} \geqslant\hspace*{0.4em}
1
\hspace*{0.2em} +\hspace*{0.2em}
\frac{n\hspace*{0.2em} -\hspace*{0.2em} a}{b\hspace*{0.2em} -\hspace*{0.2em} 1}
\hspace*{0.4em} =\hspace*{0.4em}
1
\hspace*{0.2em} +\hspace*{0.2em}
\frac{a\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.2em} +\hspace*{0.2em} 1\hspace*{0.2em} -\hspace*{0.2em} a}{a\hspace*{0.2em} -\hspace*{0.2em} 1}
\hspace*{0.4em} =\hspace*{0.4em}
a\hspace*{0.1em}.
\]
\vspace*{-10pt}
If\hspace{0.4em}\
$k_{\hspace*{0.05em} y}\hspace*{0.4em} >\hspace*{0.4em} a$\hspace*{-0.15em},\hspace{0.4em}\
then the arguments of the previous paragraph show that\hspace{0.4em}\
$n\hspace*{0.4em} >\hspace*{0.4em} a\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.2em} +\hspace*{0.2em} 1$\hspace*{-0.15em},\hspace{0.4em}\
contrary to\hspace{0.2em}\ $n\hspace*{0.2em} =\hspace*{0.2em} a\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.2em} +\hspace*{0.2em} 1$\hspace*{-0.15em}.\hspace{0.4em}\
The contradiction shows that\hspace{0.4em}\
$k_{\hspace*{0.05em} y}\hspace*{0.4em} =\hspace*{0.4em} a$\hspace{0.4em}\
also for\hspace{0.2em}\ $y\hspace*{0.2em} \in\hspace*{0.2em} L$\hspace*{-0.15em}.\hspace{0.4em}\
It follows that\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\ is a projective plane.\hspace{0.4em}\
This completes the proof of Hanani's theorem.\hspace{0.4em}\
\myuppar{Deducing the de Bruijn--Erd\"{o}s theorem.}
Suppose that\hspace{0.2em}\ $m\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
Obviously,\hspace{0.2em}\ $p\hspace*{0.2em} \leqslant\hspace*{0.2em} m$\hspace{0.2em}\
and hence\hspace{0.2em}\ $p\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
By Hanani's theorem this implies that\hspace{0.2em}\ $p\hspace*{0.2em} =\hspace*{0.2em} n$\hspace{0.2em}\
and\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\
is either a near-pencil,\hspace{0.3em}\ or a projective plane.\hspace{0.4em}\
Since\hspace{0.2em}\ $p\hspace*{0.2em} \leqslant\hspace*{0.2em} m\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace{0.2em}\ and\hspace{0.2em}\
$p\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em},\hspace{0.4em}\
it follows that\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
\myuppar{Remarks.}
In contrast with\hspace{0.2em}\ \cite{db-e}\hspace{0.2em}\ and many papers written much later\hspace*{0.025em},\hspace{0.4em}\
Hanani's proof of his version of the de Bruijn--Erd\"{o}s theorem
in\hspace{0.2em}\ \cite{h2}\hspace{0.2em}\ is quite modern.\hspace{0.4em}\
The points and lines are not enumerated\hspace*{0.05em};\hspace{0.4em}\
in fact\hspace*{0.025em},\hspace{0.4em}\
there are no subscripts at all.\hspace{0.4em}\
But when he turns to the $3$\hspace*{-0.2em}-dimensional case,\hspace{0.4em}\
he returns to the tradition of enumerating almost
everything in sight \ldots
Also,\hspace{0.3em}\ in contrast with almost every other proof\hspace*{0.025em},\hspace{0.4em}\
Hanani's proof does not use the de~Bruijn--Erd\"{o}s\ inequalities,\hspace{0.4em}\ at least not directly.\hspace{0.4em}\
But the proof\hspace{0.1em}\ of the fact that\hspace{0.2em}\ $\mathcal{P}\hspace*{0.2em} =\hspace*{0.2em} \mathcal{L}$\hspace{0.2em}\ includes
a proof of the de~Bruijn--Erd\"{o}s\ inequalities\hspace{0.2em}\ $s_{\hspace*{0.1em} L}\hspace*{0.2em} \leqslant\hspace*{0.2em} k_{\hspace*{0.05em} y}$\hspace{0.2em}\
for\hspace{0.2em}\ $y\hspace*{0.2em} \not\in\hspace*{0.2em} L$\hspace*{-0.15em}.
\mysection{A\hspace{0.2em}\ simpler\hspace{0.2em}\ proof\hspace{0.2em}\ of\hspace{0.2em}\ Hanani's\hspace{0.2em}\ theorem}{another-h-proof}
\vspace*{12pt}
This proof follows the outline of the Hanani's one,\hspace{0.4em}\
but brings into the play the smallest number $k_{\hspace*{0.05em} u}$ among all $k_{\hspace*{0.05em} z}$\hspace*{-0.15em}.\hspace{0.4em}\
Also,\hspace{0.4em}\ ``the second largest''\hspace{0.2em}\ line is chosen
not among all lines,\hspace{0.4em}\ but among the lines containing $u$\hspace*{-0.15em}.\hspace{0.4em}\
This allows to avoid Hanani's\hspace{0.1em}\ Lemma and to replace\hspace{0.2em}\ ``miraculous''\hspace{0.2em}\ estimates by rather straightforward ones.\hspace{0.4em}\
The proof was partially inspired by\hspace{0.1em}\ V.\hspace{0.1em}\ Napolitano\hspace{0.2em}\ \cite{n}.\hspace{0.4em}\
If one is interested only in the de Bruijn--Erd\"{o}s theorem,\hspace{0.4em}\
it can be simplified even further\hspace*{0.025em}.\hspace{0.4em}\
Suppose that\hspace{0.2em}\ $n\hspace*{0.2em} \geqslant\hspace*{0.2em} p$\hspace*{-0.15em}.\hspace{0.4em}\
Following de Bruijn--Erd\"{o}s\hspace{0.2em}\ \cite{db-e},\hspace{0.4em}\
let us consider a point $u$ such that $k_{\hspace*{0.05em} u}$
is the smallest number among all numbers $k_{\hspace*{0.05em} z}$\hspace*{-0.15em}.\hspace{0.4em}\
Let\hspace{0.2em}\ $a\hspace*{0.2em} =\hspace*{0.2em} s_{\hspace*{0.1em} L}$\hspace{0.2em}\
and\hspace{0.2em}\ $k\hspace*{0.2em} =\hspace*{0.2em} k_{\hspace*{0.05em} u}$\hspace*{-0.15em}.\hspace{0.4em}\
There are two cases to consider\hspace*{0.025em}:\hspace{0.4em}\
the case when\hspace{0.2em}\ $k\hspace*{0.2em} \geqslant\hspace*{0.2em} a$\hspace{0.2em}\ and
the case when\hspace{0.2em}\ $k\hspace*{0.2em} <\hspace*{0.2em} a$\hspace*{-0.15em}.\hspace{0.4em}\
The arguments in both cases are similar and can be unified,\hspace{0.4em}\
but the first case is simpler and we will deal with it first\hspace*{0.025em}.
\myuppar{The case\hspace{0.2em}\ $k\hspace*{0.2em} \geqslant\hspace*{0.2em} a$\hspace*{-0.15em}.}
Every point is contained in one of the $k$ lines containing $u$\hspace*{-0.2em},\hspace{0.4em}\
and each of these lines contains\hspace{0.2em}\ $\leqslant\hspace*{0.2em} a\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\
points in addition to $u$\hspace*{-0.2em}.\hspace{0.4em}\
Therefore the total number of points
\begin{equation}
\label{n-upper-first}
\quad
n
\hspace*{0.4em} \leqslant\hspace*{0.4em}
1\hspace*{0.2em} +\hspace*{0.2em} k\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.1em}.
\end{equation}
For every point\hspace{0.2em}\ $x\hspace*{0.2em} \in\hspace*{0.2em} L$\hspace{0.2em}\ there are\hspace{0.2em}\
$\geqslant\hspace{0.2em}\ k\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\ lines containing $x$ and different from $L$\hspace*{-0.15em}.\hspace{0.4em}\
All these lines belong to $\mathcal{P}$ and are pairwise distinct\hspace*{0.025em}.\hspace{0.4em}\
Therefore
\begin{equation}
\label{p-lower-first}
\quad
p
\hspace*{0.4em} \geqslant\hspace*{0.4em}
1\hspace*{0.2em} +\hspace*{0.2em} a\hspace*{0.05em}(k\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.1em}.
\end{equation}
If\hspace{0.2em}\ $n\hspace*{0.4em} \geqslant\hspace*{0.4em} p$\hspace*{-0.15em},\hspace{0.4em}\
then the inequalities\hspace{0.2em}\ (\ref{n-upper-first})\hspace{0.2em}\
and\hspace{0.2em}\ (\ref{p-lower-first})\hspace{0.2em}\ imply that
\[
\quad
1\hspace*{0.2em} +\hspace*{0.2em} k\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)
\hspace*{0.4em} \geqslant\hspace*{0.4em}
1\hspace*{0.2em} +\hspace*{0.2em} a\hspace*{0.05em}(k\hspace*{0.2em} -\hspace*{0.2em} 1)
\]
and\hspace{0.1em}\ hence\hspace{0.2em}\ $a\hspace*{0.2em} \geqslant\hspace*{0.2em} k$\hspace*{-0.15em}.\hspace{0.4em}\
Together with\hspace{0.2em}\ $k\hspace*{0.2em} \geqslant\hspace*{0.2em} a$\hspace{0.2em}\ this implies that\hspace{0.2em}\
$a\hspace*{0.2em} =\hspace*{0.2em} k$\hspace{0.2em}\
and the inequalities\hspace{0.2em}\ (\ref{n-upper-first})\hspace{0.2em}\
and\hspace{0.2em}\ (\ref{p-lower-first})\hspace{0.2em}\ are actually equalities.\hspace{0.4em}\
It follows that\hspace{0.2em}\
$n\hspace*{0.2em} =\hspace*{0.2em} p\hspace*{0.2em} =\hspace*{0.2em} 1\hspace*{0.2em} +\hspace*{0.2em} a\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)$\hspace*{-0.2em},\hspace{0.4em}\
every line containing $u$ consists of exactly $a$ points,\hspace{0.4em}\
and every point belonging to $L$ is contained in exactly $k$ lines.\hspace{0.4em}\
In other terms,\hspace{0.3em}\ $k_{\hspace*{0.05em} y}\hspace*{0.2em} =\hspace*{0.2em} k$\hspace{0.3em}\ if\hspace{0.3em}\ $y\hspace*{0.2em} \in\hspace*{0.2em} L$\hspace*{-0.15em}.\hspace{0.4em}\
In particular\hspace*{0.025em},\hspace{0.3em}\ every point of\hspace{0.2em}\ $L$\hspace{0.2em}\ can be taken as $u$
and\hspace{0.1em}\ hence every line intersecting $L$ consists of exactly $a$ points.\hspace{0.4em}\
In other terms,\hspace{0.3em}\ $s_l\hspace*{0.2em} =\hspace*{0.2em} a$\hspace{0.3em}\ if\hspace{0.3em}\ $l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{P}$\hspace*{-0.2em}.\hspace{0.4em}\
Let\hspace{0.2em}\ $y\hspace*{0.2em} \in\hspace*{0.2em} E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} L$\hspace*{-0.15em}.\hspace{0.4em}\
Then there are $a$ lines containing $y$ and belonging to $\mathcal{P}$\hspace*{-0.2em},\hspace{0.4em}\
and together they contain\hspace{0.2em}\
$1\hspace*{0.2em} +\hspace*{0.2em} a\hspace*{0.05em}(a\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.2em} =\hspace*{0.2em} n$\hspace{0.2em}\
points.\hspace{0.4em}\
It follows that for every point\hspace{0.2em}\ $y'\hspace*{0.2em} \neq\hspace*{0.2em} y$\hspace{0.2em}\
there is a line belonging to $\mathcal{P}$ and containing\hspace{0.2em}\
$\{\hspace*{0.1em} y\hspace*{0.05em},\hspace*{0.3em} y' \hspace*{0.1em}\}$\hspace*{-0.15em}.\hspace{0.4em}\
Since there is only one line containing\hspace{0.2em}\
$\{\hspace*{0.1em} y\hspace*{0.05em},\hspace*{0.3em} y' \hspace*{0.1em}\}$\hspace*{-0.15em},\hspace{0.4em}\
this implies that\hspace{0.2em}\
$\mathcal{L}\hspace*{0.2em} =\hspace*{0.2em} \mathcal{P}$\hspace*{-0.2em}.\hspace{0.4em}\
This implies that\hspace{0.2em}\
$s_l\hspace*{0.2em} =\hspace*{0.2em} a$\hspace{0.2em}\ for all\hspace{0.2em}\
$l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}$\hspace{0.2em}\
and\hspace{0.2em}\
$k_{\hspace*{0.05em} y}\hspace*{0.2em} =\hspace*{0.2em} a$\hspace{0.2em}\ for all\hspace{0.2em}\
$y\hspace*{0.2em} \in\hspace*{0.2em} E\hspace*{0.1em} \smallsetminus\hspace*{0.1em} L$\hspace*{-0.15em}.\hspace{0.4em}\
Since we already proved that\hspace{0.2em}\
$k_{\hspace*{0.05em} y}\hspace*{0.2em} =\hspace*{0.2em} k\hspace*{0.2em} =\hspace*{0.2em} a$\hspace{0.2em}\ for all\hspace{0.2em}\
$y\hspace*{0.2em} \in\hspace*{0.2em} L$\hspace*{-0.15em},\hspace{0.4em}\
we see that\hspace{0.2em}\ $k_{\hspace*{0.05em} y}\hspace*{0.2em} =\hspace*{0.2em} a$\hspace{0.2em}\ for all points $y$\hspace*{-0.2em}.\hspace{0.4em}\
It follows that\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\ is a projective plane.\hspace{0.4em}\
\myuppar{The case\hspace{0.2em}\ $k\hspace*{0.2em} <\hspace*{0.2em} a$\hspace*{-0.15em}.}
By the de~Bruijn--Erd\"{o}s\ inequalities in this case\hspace{0.2em}\ $u\hspace*{0.2em} \in\hspace*{0.2em} L$\hspace*{-0.15em}.\hspace{0.4em}\
Let $M$ be a line containing $u$ and such that $s_{\hspace*{0.1em} M}$ is the largest
number among all numbers $s_l$ for lines $l$ containing $u$ and different from $L$\hspace*{-0.15em}.\hspace{0.4em}\
Let\hspace{0.2em}\ $a'\hspace*{0.2em} =\hspace*{0.2em} s_{\hspace*{0.1em} M}$\hspace*{-0.15em}.\hspace{0.4em}\
Then\hspace{0.2em}\ $a\hspace*{0.2em} \geqslant\hspace*{0.2em} a'$\hspace*{-0.15em}.\hspace{0.4em}\
The strategy is to use the fact that\hspace{0.2em}\ $u\hspace*{0.2em} \in\hspace*{0.2em} L$\hspace{0.2em}\
to refine the inequalities\hspace{0.2em}\ (\ref{n-upper-first})\hspace{0.2em}\
and\hspace{0.2em}\ (\ref{p-lower-first})\hspace{0.2em}\ by using $a'$\hspace*{-0.2em}.
Every point is contained either in $L$ or in one of the other\hspace{0.2em}\ $k\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\
lines containing $u$\hspace*{-0.2em}.\hspace{0.4em}\
Each of these lines contains\hspace{0.2em}\
$\leqslant\hspace*{0.2em} a'\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\
points in addition to $u$\hspace*{-0.2em}.\hspace{0.4em}\
Therefore the total number of points
\begin{equation}
\label{n-above}
n
\hspace*{0.4em} \leqslant\hspace*{0.4em}
a\hspace*{0.2em} +\hspace*{0.2em} (k\hspace*{0.2em} -\hspace*{0.2em} 1)(a'\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.1em}.
\end{equation}
There are $k$ lines containing $u$\hspace*{-0.2em},\hspace{0.4em}\
and for every point\hspace{0.2em}\ $x\hspace*{0.2em} \in\hspace*{0.2em} L$\hspace{0.2em}\ and different from $u$
there are\hspace{0.2em}\ $k_{\hspace*{0.05em} x}\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\ of lines containing $x$
and different from $L$\hspace*{-0.15em}.\hspace{0.4em}\
All these lines belong to $\mathcal{P}$ and are pairwise distinct\hspace*{0.025em}.\hspace{0.4em}\
If\hspace{0.2em}\ $x\hspace*{0.2em} \in\hspace*{0.2em} L$\hspace{0.2em}\ and\hspace{0.2em}\ $x\hspace*{0.2em} \neq\hspace*{0.2em} u$\hspace*{-0.2em},\hspace{0.4em}\
then\hspace{0.2em}\ $x\hspace*{0.2em} \not\in\hspace*{0.2em} M$\hspace{0.2em}\ and\hspace{0.1em}\ hence\hspace{0.2em}\
$k_{\hspace*{0.05em} x}\hspace*{0.2em} \geqslant\hspace*{0.2em} s_{\hspace*{0.1em} M}\hspace*{0.2em} =\hspace*{0.2em} a'$\hspace*{-0.2em}.\hspace{0.4em}\
It follows that
\begin{equation}
\label{m-below}
p
\hspace*{0.4em} \geqslant\hspace*{0.4em}
k\hspace*{0.2em} +\hspace*{0.2em} (a\hspace*{0.2em} -\hspace*{0.2em} 1)(a'\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.1em}.
\end{equation}
The inequalities\hspace{0.2em}\ (\ref{n-above})\hspace{0.2em}\ and\hspace{0.2em}\ (\ref{m-below})\hspace{0.2em}\
together with the assumption\hspace{0.2em}\ $n\hspace*{0.2em} \geqslant\hspace*{0.2em} p$\hspace{0.2em}\ imply that
\[
\quad
a\hspace*{0.2em} +\hspace*{0.2em} (k\hspace*{0.2em} -\hspace*{0.2em} 1)(a'\hspace*{0.2em} -\hspace*{0.2em} 1)
\hspace*{0.4em} \geqslant\hspace*{0.4em}
k\hspace*{0.2em} +\hspace*{0.2em} (a\hspace*{0.2em} -\hspace*{0.2em} 1)(a'\hspace*{0.2em} -\hspace*{0.2em} 1)\hspace*{0.1em}.
\]
By simplifying this inequality we see that\hspace{0.4em}\
$a\hspace*{0.2em} +\hspace*{0.2em} k\hspace*{0.1em}(a'\hspace*{0.2em} -\hspace*{0.2em} 1)
\hspace*{0.4em} \geqslant\hspace*{0.4em}
k\hspace*{0.2em} +\hspace*{0.2em} a\hspace*{0.1em}(a'\hspace*{0.2em} -\hspace*{0.2em} 1)$\hspace{0.4em}\
and\hspace{0.1em}\ hence
\[
\quad
k\hspace*{0.1em}(a'\hspace*{0.2em} -\hspace*{0.2em} 2)
\hspace*{0.4em} \geqslant\hspace*{0.4em}
a\hspace*{0.1em}(a'\hspace*{0.2em} -\hspace*{0.2em} 2)\hspace*{0.1em}.
\]
Since $a'$ is the number of points in a line,\hspace{0.3em}\
$a'\hspace*{0.2em} \geqslant\hspace*{0.2em} 2$\hspace*{-0.15em}.\hspace{0.4em}\
It follows that either\hspace{0.2em}\
$k\hspace*{0.2em} \geqslant\hspace*{0.2em} a$\hspace*{-0.15em},\hspace{0.4em}\
or\hspace{0.2em}\ $a'\hspace*{0.2em} =\hspace*{0.2em} 2$\hspace*{-0.15em}.\hspace{0.4em}\
But\hspace{0.2em}\ $k\hspace*{0.2em} \geqslant\hspace*{0.2em} a$\hspace{0.2em}\
contradicts to the assumption\hspace{0.2em}\
$k\hspace*{0.2em} <\hspace*{0.2em} a$\hspace*{-0.15em},\hspace{0.4em}\
and hence\hspace{0.2em}\ $a'\hspace*{0.2em} =\hspace*{0.2em} 2$\hspace*{-0.15em}.\hspace{0.4em}\
The equality\hspace{0.2em}\ $a'\hspace*{0.2em} =\hspace*{0.2em} 2$\hspace{0.2em}\
means that $M$ consists of $2$ points.\hspace{0.4em}\
By the choice of $M$\hspace*{-0.15em},\hspace{0.4em}\ this implies that every line containing $u$
and different from $L$ consists of $2$ points.\hspace{0.4em}\
Since $L$ and these other lines contain all points and
pairwise intersect only in $u$\hspace*{-0.2em},\hspace{0.4em}\
it follows that\hspace{0.2em}\ $n\hspace*{0.2em} =\hspace*{0.2em} a\hspace*{0.2em} +\hspace*{0.2em} k\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace*{-0.15em}.\hspace{0.4em}\
One of the points of $M$ is $u$\hspace*{-0.2em}.\hspace{0.4em}\
Let $z$ be the other point\hspace*{0.025em}.\hspace{0.4em}\
Then\hspace{0.2em}\ $z\hspace*{0.2em} \not\in\hspace*{0.2em} L$\hspace{0.2em}\
because\hspace{0.2em}\ $M\hspace*{0.2em} \neq\hspace*{0.2em} L$\hspace*{-0.15em},\hspace{0.4em}\
and\hspace{0.1em}\ hence there are
$a$ lines containing $z$ and belonging to $\mathcal{P}$\hspace*{-0.2em}.\hspace{0.4em}\
Among these lines only $M$ contains $u$\hspace*{-0.15em}.\hspace{0.4em}\
There are also\hspace{0.2em}\ $k\hspace*{0.2em} -\hspace*{0.2em} 1$\hspace{0.2em}\ lines containing $u$ and not equal to $M$\hspace*{-0.15em},\hspace{0.4em}\
and all of them belong to\hspace{0.2em}\ $\mathcal{P}$\hspace*{-0.2em}.\hspace{0.4em}\
It follows that\hspace{0.2em}\
$p\hspace*{0.2em} \geqslant\hspace*{0.2em} k\hspace*{0.2em} +\hspace*{0.2em} a\hspace*{0.2em} -\hspace*{0.2em} 1\hspace*{0.2em} =\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
Since\hspace{0.2em}\ $n\hspace*{0.2em} \geqslant\hspace*{0.2em} p$\hspace*{-0.2em},\hspace{0.4em}\
this implies that\hspace{0.2em}\ $p\hspace*{0.2em} =\hspace*{0.2em} n$\hspace{0.2em}\
and every line belonging to $\mathcal{P}$ contains either $u$ or $z$\hspace*{-0.15em}.\hspace{0.4em}\
Suppose that there is a line $l$ containing $u$ and different from\hspace{0.2em}\ $L\hspace*{0.05em},\hspace*{0.3em} M$\hspace*{-0.15em}.\hspace{0.4em}\
Let\hspace{0.2em}\ $y\hspace*{0.2em} \in\hspace*{0.2em} l$\hspace{0.2em}\ and\hspace{0.2em}\ $y\hspace*{0.2em} \neq\hspace*{0.2em} u$\hspace*{-0.15em}.\hspace{0.4em}\
Then\hspace{0.2em}\ $y\hspace*{0.2em} \not\in\hspace*{0.2em} L$\hspace{0.2em}\ and hence there are
$a$ lines containing $y$ and belonging to $\mathcal{P}$\hspace*{-0.2em}.\hspace{0.4em}\
Among these lines only one contains $u$\hspace*{-0.15em}.\hspace{0.4em}\
By the previous paragraph,\hspace{0.4em}\ the other\hspace{0.2em}\ $a\hspace*{0.2em} -\hspace*{0.2em} 1$ lines contain $z$\hspace*{-0.15em}.\hspace{0.4em}\
Since there is only one line containing\hspace{0.2em}\
$\{\hspace*{0.1em} y\hspace*{0.05em},\hspace*{0.3em} z \hspace*{0.1em}\}$\hspace*{-0.15em},\hspace{0.4em}\
it follows that\hspace{0.2em}\ $a\hspace*{0.2em} -\hspace*{0.2em} 1\hspace*{0.2em} \leqslant\hspace*{0.2em} 1$\hspace{0.2em}\
and hence\hspace{0.2em}\ $a\hspace*{0.2em} =\hspace*{0.2em} 2$\hspace*{-0.15em}.\hspace{0.4em}\
Since\hspace{0.2em}\ $k\hspace*{0.2em} <\hspace*{0.2em} a$\hspace*{-0.15em},\hspace{0.4em}\
this implies that\hspace{0.2em}\ $k\hspace*{0.2em} \leqslant\hspace*{0.2em} 1$\hspace{0.2em}\
contrary to the fact that\hspace{0.2em}\ $k_{\hspace*{0.05em} x}\hspace*{0.2em} \geqslant\hspace*{0.2em} 2$\hspace{0.2em}\ for all $x$\hspace*{-0.2em}.\hspace{0.4em}\
The contradiction shows that only the lines\hspace{0.2em}\
$L\hspace*{0.05em},\hspace*{0.3em} M$\hspace{0.2em}\ contain $u$\hspace*{-0.15em}.\hspace{0.4em}\
It follows that\hspace{0.2em}\ $E\hspace*{0.2em} =\hspace*{0.2em} L\hspace*{0.1em} \cup\hspace*{0.1em} M$\hspace{0.2em}\
and hence $z$ is the only point not belonging to $L$\hspace*{-0.15em}.\hspace{0.4em}\
In turn,\hspace{0.4em}\ this implies that the set of lines $\mathcal{L}$
consists of $L$ and the lines of the form\hspace{0.2em}\
$\{\hspace*{0.1em} x\hspace*{0.05em},\hspace*{0.3em} z \hspace*{0.1em}\}$\hspace*{-0.15em},\hspace{0.4em}\
where\hspace{0.2em}\ $x\hspace*{0.2em} \in\hspace*{0.2em} L$\hspace*{-0.15em}.\hspace{0.4em}\
Therefore $\mathcal{L}\hspace*{0.2em} =\hspace*{0.2em} \mathcal{P}$\hspace{0.2em}\
and\hspace{0.2em}\ $(\hspace*{0.1em} E\hspace*{0.05em},\hspace*{0.3em} \mathcal{L}\hspace*{0.1em})$\hspace{0.2em}\ is a near-pencil.\hspace{0.4em}\
\mysection{All\hspace{0.2em}\ the\hspace{0.2em}\ de\hspace{0.2em}\ Bruijn--Erd\"{o}s\hspace{0.2em}\ inequalities}{all}
\vspace*{6pt}
\myuppar{The Basterfield--Kelly--Conway argument\hspace*{0.025em}.}
Suppose that\hspace{0.2em}\ $m\hspace*{0.2em} <\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
Then
\[
\quad
(n\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l})\bigl/(m\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l})
\hspace*{0.4em} >\hspace*{0.4em}
{n}\bigl/{m}
\]
for every\hspace{0.2em}\ $l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}$\hspace*{-0.2em}.\hspace{0.4em}\
If\hspace{0.2em}\ $z\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace*{-0.15em},\hspace{0.4em}\
then\hspace{0.2em}\ $s_{\hspace*{0.05em} l}\hspace*{0.2em} \leqslant\hspace*{0.2em} k_{\hspace*{0.05em} z}$\hspace{0.2em}\
and\hspace{0.1em}\ hence\hspace{0.2em}\
$m\hspace*{0.2em} -\hspace*{0.2em} k_{\hspace*{0.05em} z}\hspace*{0.2em} \leqslant\hspace*{0.2em} m\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l}$\hspace*{-0.15em}.\hspace{0.4em}\
It follows that
\begin{equation*}
n
\hspace*{0.4em} =\hspace*{0.4em}
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} E}\hspace*{0.4em} \frac{m\hspace*{0.2em} -\hspace*{0.2em} k_{\hspace*{0.05em} z}}{m\hspace*{0.2em} -\hspace*{0.2em} k_{\hspace*{0.05em} z}}
\hspace*{0.4em} =\hspace*{0.4em}
\sum_{l\hspace*{0.2em} \not\ni\hspace*{0.2em} z}\hspace*{0.4em} \frac{1}{m\hspace*{0.2em} -\hspace*{0.2em} k_{\hspace*{0.05em} z}}
\hspace*{0.4em} \geqslant\hspace*{0.4em}
\sum_{l\hspace*{0.2em} \not\ni\hspace*{0.2em} z}\hspace*{0.4em} \frac{1}{m\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l}}
\hspace*{0.4em} =\hspace*{0.4em}
\sum_{l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}}\hspace*{0.4em} \frac{n\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l}}{m\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l}}
\hspace*{0.4em} >\hspace*{0.4em}
\sum_{l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}}\hspace*{0.4em} \frac{n}{m}
\hspace*{0.4em} =\hspace*{0.4em}
n\hspace*{0.1em}.
\end{equation*}
The contradiction leads to the conclusion that\hspace{0.2em}\ $m\hspace*{0.2em} \geqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
This argument is the main part of the proof of\hspace{0.1em}\ Theorem\hspace{0.2em}\ 2.1\hspace{0.2em}\
(dealing with a more general situation)\hspace{0.2em}\ of the paper\hspace{0.2em}\
\cite{bk}\hspace{0.2em}\ by\hspace{0.1em}\ J.G.\hspace{0.1em}\ Basterfield\hspace{0.1em}\ and\hspace{0.1em}\ L.M.\hspace{0.2em}\ Kelly.\hspace{0.4em}\
Basterfield and Kelly\hspace{0.2em}\ \cite{bk}\hspace{0.2em}\ wrote that they are
\emph{``indebted to\hspace{0.1em}\ J.\hspace{0.1em}\ Conway for the simplicity
of the present formulation of the proof\hspace{0.1em}\
of\hspace{0.2em}\ Theorem\hspace{0.2em}\ 2.1.''}\hspace{0.4em}\
By some reason this acknowledgment\hspace{0.1em}\ led
to attributing this argument\hspace{0.05em}\ to\hspace{0.1em}\
J.\hspace{0.1em}\ Conway alone even by some authors referring directly to\hspace{0.2em}\ \cite{bk}.\hspace{0.4em}\
By replacing the strict inequalities $<$ by the non-strict ones $\leqslant$,\hspace{0.4em}\
one can use this argument also to deal with the case\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace{0.2em}\
along the lines of Sections\hspace{0.2em}\ \ref{solution}\hspace*{0.05em}--\hspace*{0.15em}\ref{reps}.\hspace{0.4em}\
This observation is apparently due to\hspace{0.1em}\ P.\hspace{0.1em}\ de Witte\hspace{0.2em}\ \cite{dw}.\hspace{0.4em}\
This is a sharp-witted,\hspace{0.3em}\
but also the most obscure and puzzling proof\hspace*{0.025em}.\hspace{0.4em}\
It appears
as a rabbit from a hat
without any context or explanations
and tells nothing about why the theorem is true.\hspace{0.4em}\
In the rest of this section I will explain a natural line of thinking which leads
to such a proof\hspace*{0.025em}.\hspace{0.4em}\
There is no evidence suggesting that it was discovered in this way,\hspace{0.4em}\ but it could have been.
\myuppar{Summing the de~Bruijn--Erd\"{o}s\ inequalities.}
Summing de~Bruijn--Erd\"{o}s\ inequalities and then comparing the result with\hspace{0.2em}\ (\ref{sums})\hspace{0.2em}\
is the key step of both the de~Bruijn--Erd\"{o}s\ proof and the proof from Section\hspace{0.2em}\ \ref{solution}.\hspace{0.4em}\
A natural idea is to use all de~Bruijn--Erd\"{o}s\ inequalities on an equal footing.\hspace{0.4em}\
One way to do this is to use systems of distinct representatives as in Section\hspace{0.2em}\ \ref{reps}.\hspace{0.4em}\
One may hope for a proof using all de~Bruijn--Erd\"{o}s\ inequalities
in a way closer to the proof of inequalities\hspace{0.2em}\ (\ref{sum-all-pairs})\hspace{0.2em}\
and\hspace{0.2em}\ (\ref{sum-divided})\hspace{0.2em}\ in Section\hspace{0.2em}\ \ref{solution}\hspace{0.2em}\
than to the cyclic order argument of de Bruijn--Erd\"{o}s.\hspace{0.4em}\
Let us sum the inequalities\hspace{0.2em}\ $s_{\hspace*{0.05em} l}\hspace*{0.2em} \leqslant\hspace*{0.2em} k_{\hspace*{0.05em} z}$\hspace{0.2em}\
over all\hspace{0.1em}\ pairs\hspace{0.2em}\
$(\hspace*{0.05em} l\hspace*{0.05em},\hspace*{0.3em} z\hspace*{0.05em})\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}\hspace*{0.1em} \times\hspace*{0.1em} E$\hspace{0.2em}\
such that\hspace{0.2em}\
$z\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace*{-0.15em}.\hspace{0.4em}\
Every $s_{\hspace*{0.05em} l}$ appears\hspace{0.2em}\ $n\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l}$\hspace{0.2em}\ times
in the left hand side of these inequalities,\hspace{0.4em}\ and
every $k_{\hspace*{0.05em} z}$ appears\hspace{0.2em}\ $m\hspace*{0.2em} -\hspace*{0.2em} k_{\hspace*{0.05em} z}$\hspace{0.2em}\ times
in the right hand side.\hspace{0.4em}\
Therefore,\hspace{0.4em}\ taking the sum results in the inequality
\[
\quad
\sum_{l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}}\hspace*{0.2em} s_{\hspace*{0.05em} l}\hspace*{0.1em} (n\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l})
\hspace*{0.4em}\off \leqslant\hspace*{0.4em}\off
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} E}\hspace*{0.2em} k_{\hspace*{0.05em} z}\hspace*{0.1em} (m\hspace*{0.2em} -\hspace*{0.2em} k_{\hspace*{0.05em} z})\hspace*{0.1em}.
\]
This inequality does not lead to a proof of the desired sort\hspace*{0.025em},\hspace{0.4em}\
but it admits a natural generalization.\hspace{0.4em}\
Let\hspace{0.1em}\ $F$\hspace{0.1em}\ be an increasing function.\hspace{0.4em}\
Instead of\hspace{0.2em}\ $s_{\hspace*{0.05em} l}\hspace*{0.2em} \leqslant\hspace*{0.2em} k_{\hspace*{0.05em} z}$\hspace*{-0.15em},\hspace{0.4em}\
one can sum the inequalities\hspace{0.3em}\
$F\hspace*{0.1em}(\hspace*{0.05em} s_{\hspace*{0.05em} l}\hspace*{0.05em})\hspace*{0.3em} \leqslant\hspace*{0.4em} F\hspace*{0.1em}(\hspace*{0.05em} k_{\hspace*{0.05em} z}\hspace*{0.05em})$\hspace*{-0.2em}.\hspace{0.4em}\
In fact\hspace*{0.025em},\hspace{0.3em}\ there is no need to apply
the same function to\hspace{0.1em}\ $s_{\hspace*{0.05em} l}$\hspace{0.1em}\ and\hspace{0.1em}\ $k_{\hspace*{0.05em} z}$\hspace*{-0.15em}.\hspace{0.4em}\
Let\hspace{0.2em}\ $F\hspace*{0.05em},\hspace*{0.3em} G$\hspace{0.2em}\ be a pair of functions
such that\hspace{0.2em}\ $s\hspace*{0.2em} \leqslant\hspace*{0.2em} k$\hspace{0.2em}\ implies\hspace{0.3em}\
$F\hspace*{0.1em}(\hspace*{0.05em} s\hspace*{0.05em})\hspace*{0.3em} \leqslant\hspace*{0.4em} G\hspace*{0.1em}(\hspace*{0.05em} k\hspace*{0.05em})$\hspace*{-0.2em}.\hspace{0.4em}\
Taking the sum of the inequalities\hspace{0.3em}\
$F\hspace*{0.1em}(\hspace*{0.05em} s_{\hspace*{0.05em} l}\hspace*{0.05em})\hspace*{0.3em} \leqslant\hspace*{0.4em} G\hspace*{0.1em}(\hspace*{0.05em} k_{\hspace*{0.05em} z}\hspace*{0.05em})$\hspace{0.3em}\
over all\hspace{0.1em}\ pairs\hspace{0.2em}\
$l\hspace*{0.05em},\hspace*{0.3em} z$\hspace{0.2em}\
such that\hspace{0.2em}\
$z\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace{0.2em}\
results in the inequality\vspace*{3pt}
\[
\quad
\sum_{l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}}\hspace*{0.2em} F\hspace*{0.1em}(\hspace*{0.05em} s_{\hspace*{0.05em} l}\hspace*{0.05em})\hspace*{0.1em} (n\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l})
\hspace*{0.4em}\off \leqslant\hspace*{0.4em}\off
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} E}\hspace*{0.2em} G\hspace*{0.1em} (\hspace*{0.05em} k_{\hspace*{0.05em} z}\hspace*{0.05em})\hspace*{0.1em} (m\hspace*{0.2em} -\hspace*{0.2em} k_{\hspace*{0.05em} z})\hspace*{0.1em}.
\]
\vspace*{-9pt}
It remains to realize that
the functions\hspace{0.2em}\ $F\hspace*{0.05em},\hspace*{0.3em} G$\hspace{0.2em}\ may depend on the numbers\hspace{0.2em}\ $m\hspace*{0.05em},\hspace*{0.3em} n$\hspace{0.2em}\
and that one can get rid of the factors\hspace{0.2em}\ $n\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l}$\hspace{0.2em}\
and\hspace{0.2em}\ $m\hspace*{0.2em} -\hspace*{0.2em} k_{\hspace*{0.05em} z}$\hspace{0.2em}\ by dividing by these factors.\hspace{0.4em}\
\myuppar{A proof of the de Bruijn--Erd\"{o}s-Hanani theorem.}
As usual,\hspace{0.4em}\ we may assume that\hspace{0.2em}\ $m\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
Let\vspace*{-1.5pt}
\[
\quad
F\hspace*{0.1em}(\hspace*{0.05em} s\hspace*{0.05em})\hspace*{0.4em} =\hspace*{0.4em} \frac{s}{n\hspace*{0.2em} -\hspace*{0.2em} s}
\hspace*{1.5em}\mbox{ and }\hspace*{1.5em}
G\hspace*{0.1em}(\hspace*{0.05em} k\hspace*{0.05em})\hspace*{0.4em} =\hspace*{0.4em} \frac{k}{m\hspace*{0.2em} -\hspace*{0.2em} k}\hspace*{0.2em}.
\]
\vspace*{-10pt}
Then\hspace{0.2em}\ $s\hspace*{0.2em} \leqslant\hspace*{0.2em} k$\hspace{0.2em}\ implies\hspace{0.3em}\
$F\hspace*{0.1em}(\hspace*{0.05em} s\hspace*{0.05em})\hspace*{0.3em} \leqslant\hspace*{0.4em} G\hspace*{0.1em}(\hspace*{0.05em} k\hspace*{0.05em})$\hspace*{-0.2em}.\hspace{0.4em}\
Indeed,\hspace{0.4em}\ the latter inequality is equivalent to the inequality\hspace{0.4em}\
$s\hspace*{0.05em}(m\hspace*{0.2em} -\hspace*{0.2em} k)\hspace*{0.3em} \leqslant\hspace*{0.4em} k\hspace*{0.05em}(n\hspace*{0.2em} -\hspace*{0.2em} s)$\hspace*{-0.2em},\hspace{0.4em}\
and hence to the inequality\hspace{0.3em}\ $s\hspace*{0.025em} m\hspace*{0.3em} \leqslant\hspace*{0.4em} k\hspace*{0.025em} n$\hspace*{-0.15em},\hspace{0.4em}\
which is obviously true if\hspace{0.2em}\ $s\hspace*{0.2em} \leqslant\hspace*{0.2em} k$\hspace{0.2em}\
and\hspace{0.2em}\ $m\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
By summing the inequalities\hspace{0.3em}\
$F\hspace*{0.1em}(\hspace*{0.05em} s_{\hspace*{0.05em} l}\hspace*{0.05em})\hspace*{0.3em} \leqslant\hspace*{0.4em} G\hspace*{0.1em}(\hspace*{0.05em} k_{\hspace*{0.05em} z}\hspace*{0.05em})$\hspace{0.3em}\
over all pairs\hspace{0.2em}\
$(\hspace*{0.05em} l\hspace*{0.05em},\hspace*{0.3em} z\hspace*{0.05em})\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}\hspace*{0.1em} \times\hspace*{0.1em} E$\hspace{0.2em}\
such that\hspace{0.2em}\
$z\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace{0.2em}\
we get the inequality\vspace*{2pt}
\begin{equation}
\label{sum-frac}
\quad
\sum_{l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}}\hspace*{0.4em} \frac{s_{\hspace*{0.05em} l}}{n\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l}}\hspace*{0.2em} (n\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l})
\hspace*{0.4em}\off \leqslant\hspace*{0.4em}\off
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} E}\hspace*{0.4em} \frac{k_{\hspace*{0.05em} z}}{m\hspace*{0.2em} -\hspace*{0.2em} k_{\hspace*{0.05em} z}}\hspace*{0.2em} (m\hspace*{0.2em} -\hspace*{0.2em} k_{\hspace*{0.05em} z})\hspace*{0.1em},
\end{equation}
\vspace*{-10pt}
which is obviously equivalent to
\[
\quad
\sum_{l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}}\hspace*{0.2em} s_{\hspace*{0.05em} l}
\hspace*{0.4em}\off \leqslant\hspace*{0.4em}\off
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} E}\hspace*{0.2em} k_{\hspace*{0.05em} z}\hspace*{0.2em}.
\]
In view of\hspace{0.2em}\ (\ref{sums})\hspace{0.2em}\
the sides of the latter inequality are actually equal,\hspace{0.4em}\
and\hspace{0.1em}\ hence the sides of the inequality\hspace{0.2em}\ (\ref{sum-frac})\hspace{0.2em}\ are also equal.\hspace{0.4em}\
Since the inequality\hspace{0.2em}\ (\ref{sum-frac})\hspace{0.2em}\ is obtained by summing inequalities\hspace{0.3em}\
$F\hspace*{0.1em}(\hspace*{0.05em} s_{\hspace*{0.05em} l}\hspace*{0.05em})\hspace*{0.3em} \leqslant\hspace*{0.4em} G\hspace*{0.1em}(\hspace*{0.05em} k_{\hspace*{0.05em} z}\hspace*{0.05em})$\hspace*{-0.2em},\hspace*{0.2em}\hspace{0.4em}\
it\hspace{0.1em}\ follows that\hspace{0.1em}\ if\hspace{0.3em}\
$(\hspace*{0.05em} l\hspace*{0.05em},\hspace*{0.3em} z\hspace*{0.05em})\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}\hspace*{0.1em} \times\hspace*{0.1em} E$\hspace{0.3em}\
and\hspace{0.3em}\
$z\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace*{-0.15em},\hspace{0.4em}\
then\vspace*{-1.5pt}
\[
\quad
\frac{s_{\hspace*{0.05em} l}}{n\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l}}
\hspace*{0.4em} =\hspace*{0.4em}
\frac{k_{\hspace*{0.05em} z}}{m\hspace*{0.2em} -\hspace*{0.2em} k_{\hspace*{0.05em} z}}\hspace*{0.2em}.
\]
\vspace*{-10pt}
and\hspace{0.1em}\ hence\hspace{0.3em}\
$s_{\hspace*{0.05em} l}\hspace*{0.05em} m\hspace*{0.4em} =\hspace*{0.4em} k_{\hspace*{0.05em} z}\hspace*{0.05em} n$\hspace*{-0.15em}.\hspace*{0.2em}\hspace{0.4em}\
Since\hspace{0.2em}\ $m\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace{0.2em}\ and\hspace{0.2em}\
$s_{\hspace*{0.05em} l}\hspace*{0.2em} \leqslant\hspace*{0.2em} k_{\hspace*{0.05em} z}$\hspace*{-0.15em},\hspace{0.4em}\
the last equality implies that\hspace{0.2em}\ $m\hspace*{0.2em} =\hspace*{0.2em} n$\hspace{0.2em}\
and\hspace{0.2em}\
$s_{\hspace*{0.05em} l}\hspace*{0.2em} =\hspace*{0.2em} k_{\hspace*{0.05em} z}$\hspace*{-0.15em}.\hspace{0.4em}\
In particular\hspace*{0.025em},\hspace{0.2em}\ $z\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace{0.2em}\ implies that\hspace{0.2em}\
$s_{\hspace*{0.05em} l}\hspace*{0.2em} =\hspace*{0.2em} k_{\hspace*{0.05em} z}$\hspace{0.3em}\
and hence every line containing $z$ intersects $l$\hspace*{-0.2em}.\hspace{0.4em}\
It remains to repeat the last paragraph of\hspace{0.1em}\ Section\hspace{0.2em}\ \ref{reps}.\hspace{0.4em}\
This proof has the advantage of explicitly using the equality\hspace{0.2em}\ (\ref{sums}).\hspace{0.4em}\
The Basterfield--Kelley--Conway argument implicitly uses double sums
and a change of the order of summation.\hspace{0.4em}\
This change of the order of summation
is a stronger tool than the double counting proving\hspace{0.2em}\ (\ref{sums}).
\myuppar{A version of\hspace{0.1em}\ this proof\hspace*{0.025em}.}
Suppose that\hspace{0.2em}\ $m\hspace*{0.2em} \leqslant\hspace*{0.2em} n$\hspace*{-0.15em}.\hspace{0.4em}\
One can take as\hspace{0.2em}\ $F\hspace*{0.05em},\hspace*{0.3em} G$\hspace{0.2em}\ the functions\vspace*{-1.5pt}
\[
\quad
F\hspace*{0.1em}(\hspace*{0.05em} s\hspace*{0.05em})\hspace*{0.4em} =\hspace*{0.4em} \frac{n}{n\hspace*{0.2em} -\hspace*{0.2em} s}
\hspace*{1.5em}\mbox{ and }\hspace*{1.5em}
G\hspace*{0.1em}(\hspace*{0.05em} k\hspace*{0.05em})\hspace*{0.4em} =\hspace*{0.4em} \frac{m}{m\hspace*{0.2em} -\hspace*{0.2em} k}\hspace*{0.2em}.
\]
\vspace{-10pt}
They can be obtained by adding $1$ to the previous choice of the functions\hspace{0.2em}\ $F\hspace*{0.05em},\hspace*{0.3em} G$\hspace*{-0.15em}.\hspace{0.4em}\
Therefore\hspace{0.2em}\ $s\hspace*{0.2em} \leqslant\hspace*{0.2em} k$\hspace{0.2em}\ again\hspace{0.1em}\ implies\hspace{0.3em}\
$F\hspace*{0.1em}(\hspace*{0.05em} s\hspace*{0.05em})\hspace*{0.3em} \leqslant\hspace*{0.4em} G\hspace*{0.1em}(\hspace*{0.05em} k\hspace*{0.05em})$\hspace*{-0.2em}.\hspace{0.4em}\
This can be also verified in the same way as before.\hspace{0.4em}\
By summing the inequalities\hspace{0.3em}\
$F\hspace*{0.1em}(\hspace*{0.05em} s_{\hspace*{0.05em} l}\hspace*{0.05em})\hspace*{0.3em} \leqslant\hspace*{0.4em} G\hspace*{0.1em}(\hspace*{0.05em} k_{\hspace*{0.05em} z}\hspace*{0.05em})$\hspace{0.3em}\
over all pairs\hspace{0.2em}\
$l\hspace*{0.05em},\hspace*{0.3em} z$\hspace{0.2em}\
such that\hspace{0.2em}\
$z\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace{0.2em}\
we get\vspace*{3pt}
\begin{equation}
\label{sum-frac-alt}
\quad
\sum_{l\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}}\hspace*{0.4em} \frac{n}{n\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l}}\hspace*{0.2em} (n\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l})
\hspace*{0.4em}\off \leqslant\hspace*{0.4em}\off
\sum_{z\hspace*{0.2em} \in\hspace*{0.2em} E}\hspace*{0.4em} \frac{m}{m\hspace*{0.2em} -\hspace*{0.2em} k_{\hspace*{0.05em} z}}\hspace*{0.2em} (m\hspace*{0.2em} -\hspace*{0.2em} k_{\hspace*{0.05em} z})\hspace*{0.1em},
\end{equation}
\vspace*{-9pt}
which is obviously equivalent to\hspace{0.3em}\
$m\hspace*{0.025em} n\hspace*{0.3em} \leqslant\hspace*{0.3em} n\hspace*{0.025em} m$\hspace*{-0.15em}.\hspace{0.4em}\
But the sides of the latter inequality are equal.\hspace{0.4em}\
It\hspace{0.1em}\ follows that\hspace{0.1em}\ if\hspace{0.3em}\
$(\hspace*{0.05em} l\hspace*{0.05em},\hspace*{0.3em} z\hspace*{0.05em})\hspace*{0.2em} \in\hspace*{0.2em} \mathcal{L}\hspace*{0.1em} \times\hspace*{0.1em} E$\hspace{0.3em}\
and\hspace{0.3em}\
$z\hspace*{0.2em} \not\in\hspace*{0.2em} l$\hspace*{-0.15em},\hspace{0.4em}\
then\vspace*{-1.5pt}
\[
\quad
\frac{n}{n\hspace*{0.2em} -\hspace*{0.2em} s_{\hspace*{0.05em} l}}
\hspace*{0.4em} =\hspace*{0.4em}
\frac{m}{m\hspace*{0.2em} -\hspace*{0.2em} k_{\hspace*{0.05em} z}}
\]
\vspace*{-10pt}
and\hspace{0.1em}\ hence\hspace{0.3em}\
$s_{\hspace*{0.05em} l}\hspace*{0.05em} m\hspace*{0.4em} =\hspace*{0.4em} k_{\hspace*{0.05em} z}\hspace*{0.05em} n$\hspace*{-0.15em}.\hspace*{0.2em}\hspace{0.4em}\
The rest of the proof is exactly the same as above.\hspace{0.4em}\
Dividing everything in this version of the proof\hspace{0.1em}\ by $m$\hspace*{-0.15em},\hspace{0.4em}\
which amounts to taking\vspace*{-2pt}
\[
\quad
F\hspace*{0.1em}(\hspace*{0.05em} s\hspace*{0.05em})\hspace*{0.4em} =\hspace*{0.4em} \frac{n}{m\hspace*{0.05em}(n\hspace*{0.2em} -\hspace*{0.2em} s)}
\hspace*{1.5em}\mbox{ and }\hspace*{1.5em}
G\hspace*{0.1em}(\hspace*{0.05em} k\hspace*{0.05em})\hspace*{0.4em} =\hspace*{0.4em} \frac{1}{m\hspace*{0.2em} -\hspace*{0.2em} k}\hspace*{0.2em},
\]
\vspace*{-10pt}
and omitting the explanations
turns this version into the
Basterfield--Kelly--Conway argument\hspace*{0.025em}.
|
2,877,628,090,110 | arxiv | \section*{\label{sec:level1}Introduction}
Salt deserts are amongst the most inhospitable places of our planet. Their otherworldly shapes inspire the imagination (e.g. Star Wars desert planet Crait \cite{STARWARS}, or the million tourists annually visiting Death Valley \cite{DEVA}), and are an important drive of climate processes \cite{GILL1996}. The immediately prominent feature of such landscapes (Fig.~\ref{fig:polygons}) is a characteristic tiling of polygons, formed by ridges in the salt-encrusted surface. These patterns decorate salt pans around the world, including Salar de Uyuni in Chile, Chott el Djerid in Tunisia \cite{WADGE1994}, Badwater Basin in California and the periphery of the Dead Sea -- always expressing the same size of about one meter.
Salt crusts are dynamic over months to years~\cite{LOWENSTEIN1985, LOKIER2012, NIELD2013, NIELD2015}, and couple to other environmental processes. Wind over crust creates dust, the emission of which forms a significant proportion of the Earth's global atmospheric dust production \cite{GILL1996,PROSPERO2002} and of mineral transport to the oceans~\cite{FUNG2000}. As one example, dust from the surface of the dry Owens Lake has posed a major health problem for people living downwind of the lake \cite{CAHILL1996,GILL2002}, and the site is the centre of a decades-long, intense remediation effort. Salt crusts also modify evaporation and heat flux from the playa surface~\cite{BRYANT2002}, and hence the critical water and energy balances of fragile environments.
Research on salt pans has typically focused on either the dynamics of their complex subsurface flows \cite{WOODING1960a, HOMSY1976, WOODING1997, WOODING2007, VANDAM2009} or their crusts, without considering how these features could interact. Crust patterns have been attributed to buckling or wrinkling as expanding areas of crust collide ~\cite{CHRISTIANSEN1963, FRYBERGER1983, LOWENSTEIN1985}, or to surface cracks ~\cite{KRINSLEY1970, DIXON2009, TUCKER1981, DEDECKKER1988, LOKIER2012}. For example, a model of crust patterns developing from contraction cracks in mud is given by Krinsley~\cite{KRINSLEY1970} whereas Christiansen~\cite{CHRISTIANSEN1963} provides a quantitative model based on buckling, and most subsequent discussion reiterates these ideas. However, for such mechanical instabilities the expected feature size is a few times the thickness of the cracking~\cite{LACHENBRUCH1961, KRINSLEY1970, GROISMAN1994, SHORLIN2000} or buckling~\cite{BOWDEN1998, LI2012} layer. As well, buckling is known to favour parallel wrinkles rather than closed polygons~\cite{BOWDEN1998, LI2012}. Salt playa crusts vary in thickness from sub-centimeter to meters \cite{KRINSLEY1970, LOWENSTEIN1985, LOKIER2012}, which cannot itself explain the consistent polygon diameters of 1--2$\,$m. Similarly, there is no clear reason why preexisting mud-cracks, surface roughness or other heterogeneities would appear worldwide at the same length-scale and arrangement. For example, at Owens Lake we observed crusts 1--20 cm thick, desiccation cracks in crust-free mud of $\sim$10 cm spacing and intermittent buckling of crust (see Supplemental Movie) with wavelengths of $\sim$2 cm and parallel rather than polygonal features. While salt precipitation may take advantage of any such pre-existing surface structures, none of these can adequately explain the patterns and scales observed in the crust polygons.
Here we show that by instead considering the surface of a salt playa jointly with its subsurface flows, the origins, dynamics, length-scale and shape of the polygonal patterns in salt crusts can be apprehended. Specifically, by combining theoretical analysis, numerical simulations, analogue experiments and field observations, we show that density-driven convective dynamics will lead to heterogeneity in the horizontal salinity distribution of evaporating groundwater, at the same scale as the observed polygons. These convective cells can act to template the position of polygons, setting their size and shape. For example, we will show how salt ridges should naturally develop over descending plumes of dense salty groundwater, where faster salt precipitation will be expected, and demonstrate such co-location in the field.
The salt polygons at Owens Lake (Fig.~1\textbf{B}) have a typical pattern in a well-studied and controlled landscape; their summary also introduces our modelling assumptions and main field site.
This dry, terminal saline lake has an aquifer that extends from the near-surface to $>$150$\,$m depth \cite{GULER2004}.
The groundwater carries dissolved salts \cite{RYU2006,GILL2002}, which collect in an evaporite pan of about 300 km$^2$ \cite{GULER2004,GILL1996}. Efforts to control dust emission from the lake-bed involve shallow flooding \cite{GROENEVELD2013}, vegetation \cite{LANCASTER1997}, gravel cover and crust growth \cite{GROENEVELD2010}. As shown in the online supplementary movie S1, after a flooding event the soil is saturated with water, which evaporates leaving behind salts that crystallize into a continuous crust, covered by a network of ridges.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{polygons}
\caption{\label{fig:polygons} Typical salt polygon patterns at \textbf{A}, \textbf{C} Badwater Basin in Death Valley and \textbf{B}, Owens Lake, California (image \textbf{A} courtesy Sarah Marino).}
\end{figure}
\section*{\label{sec:level2}Model of subsurface convection}
We consider the coupling of surface salt patterns to subsurface flows, as visualised in Fig.~\ref{fig2}. Specifically, we treat fluid and salt transport in a saturated porous medium, driven by surface evaporation and fed from below by a reservoir of water with some background salinity. The aquifer is deep, compared to the diameter of the salt polygons, and any corresponding near-surface dynamics. This system will naturally develop a salinity gradient below the surface, which can become unstable to convective overturning \cite{WOODING1960a, HOMSY1976, WOODING1997}. To determine whether field conditions will support such a buoyancy-driven convective instability, we start from mass and momentum balances, as are used to describe thermal \cite{WOODING1960a} or solutal \cite{WOODING1997,SLIM2014} convection in porous media. We then perform a linear stability analysis to give the criteria for convection, and will confirm this instability via simulations, experiments and field observations.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{model-sketch_iso.pdf}
\caption{\label{fig2} Proposed dynamics of patterned salt crusts. Evaporation leaves the near-surface fluid enriched in salt, and heavier than the fluid beneath it. This drives convection, with downwellings of high salinity pore fluid co-localized with surface ridges. The dominant fluid motions are shown by the black arrows, and the water salinity is indicated by the colour contours. At downwellings, the weak salt gradients in the groundwater enhance salt precipitation/crust growth, leading to ridges; at upwelling plumes the stronger gradients favour diffusive over advective transport, and crust growth is slower.}
\end{figure}
We model the volumetric flux $\bm{q}$ of a fluid driven by pressure $p$ through a porous medium of constant uniform porosity $\phi$ and permeability $\kappa$. The fluid has a viscosity $\mu$ and carries dissolved salt, whose diffusivity $D$ is corrected for the presence of different ions as well as tortuosity (see Methods). Using the Boussinesq approximation, buoyancy is generated by a variable fluid density $\rho = \rho_b+S\Delta\rho$, where $\rho_b$ is the density of the reservoir fluid, and $\rho_b +\Delta\rho$ is that of the top boundary; the relative salinity $S$ linearly mediates between the two limits. The model consists of the continuity equation for incompressible fluid flow, Darcy's law and an advection-diffusion equation for salt transport:
\begin{align}
\bm{\nabla} \cdot \bm{q} &= 0\\
\bm{q} &= -\frac{\kappa}{\mu}\left[ \nabla (p + \rho_b g z) + S \Delta \rho g\hat{\bm{z}}\right] \\
\phi\frac{\partial S}{\partial t} &= \phi D\nabla^2 S - \bm{q} \cdot \nabla S\,
\end{align}
where $g$ is the acceleration due to gravity, and $\hat{\bm{z}}$ is an upward-pointing unit vector.
\begin{figure*}[!ht]
\centering
\includegraphics[width=\textwidth]{compilation_figure.pdf}
\caption{\label{fig:scaling} Characterising the convective instability. \textbf{A}: Stability diagram of porous media convection in salt pans. The neutral stability curve (black line) is the theoretical boundary above which an evaporating stratified pore fluid is unstable to perturbations of wavenumber $a$. Here the most unstable mode (dashed line) gives a prediction of the initial convective wavelength, and its dependence on Ra. Various triangles show field measurements at Owens Lake, Badwater Basin and Sua Pan. Yellow and red dots show $a$ measured in simulations at early and late times, respectively, and show coarsening by a reduction of the observed $a$ with time. Green squares give experimental measurements, and show that coarsening may continue even over very long timescales. \textbf{B} Example TLS surface height map at Owens Lake. \textbf{C} Convective plumes highlighted by dye (the brighter upwelling fluid results from dying the reservoir, well after convection has set in) in an experimental Hele-Shaw cell are coupled to the salinity, measured at the coloured points by destructive sampling. \textbf{D} Example of simulated plumes at early times, $\tau = \tau_S$, are consistent with the most unstable mode but $\textbf{E}$ coarsen by time $\tau = 10\,\tau_S$. The salinity scale bar applies to panels $\textbf{C}$ to $\textbf{E}$.}
\end{figure*}
Taking the average evaporation rate $E$ as the natural velocity scale for the system, we set the characteristic length and time as $L = \phi D/E$ and $T = \phi^2D/E^2$, respectively. Non-dimensionalization of Eqs. 1-3 then gives
\begin{align}
\bm{\nabla} \cdot \bm{U} &= 0\label{eq:continuity}\\
\bm{U} &= - \nabla P - \text{Ra} S\hat{Z}\label{eq:darcy_flow}\\
\frac{\partial S}{\partial \tau} &= \nabla^2 S - \bm{U} \cdot \nabla S\label{eq:advection_diffusion}\;
\end{align}
with dimensionless velocity $\bm{U}=\bm{q}/E$, time $\tau = t/T$, vertical position $Z = {zE}/{\phi D}$ and a pressure $P = \frac{\kappa}{\phi \mu D}(p+\rho_b g z)$.
This system of equations is governed by the Rayleigh number
\begin{align}
\text{Ra} = \frac{\kappa \Delta \rho g}{ \mu E}\;.\label{eq:rayleigh}
\end{align}
At the upper surface of the soil, $Z = 0$, we match the vertical water flux to the evaporation rate. The presence of a salt crust sets $S = 1$ there. However, the rate at which salt is added to this surface ($1-\partial S / \partial Z$, following the definition of salt flux, $S\vec{U} - \nabla S$, in Eq. \ref{eq:advection_diffusion}) must allow for its transport by both advection and diffusion. As sketched in Fig.~\ref{fig2}, this will lead to faster salt precipitation above any convective downwellings, which we argue gives rise to ridges there.
A steady-state solution to Eqs. 4-6, $S = \exp(-zE/\phi D)$, represents a salt-rich layer of pore fluid lying just below the salt crust, and a balance between advection and diffusion. This is unstable beyond some critical Rayleigh number, Ra$_\mathrm{c}$, which depends on the boundary conditions \cite{LAPWOOD1948, HORTON1945,WOODING1960a, HOMSY1976}. For constant through-flow at the surface, as expected in the field, Ra$_\mathrm{c}=14.35$ for the onset of an instability that leads to down-welling plumes of high salinity \cite{HOMSY1976}. Between Ra = 5.66 and Ra$_\mathrm{c}$ the fixed solution may also be unstable to finite amplitude perturbations \cite{HOMSY1976,WOODING1997}. The neutral stability curve and most unstable mode of convection (see~\cite{ERNST2017} for derivation) are shown in Fig. \ref{fig:scaling}\textbf{A}. At Ra$_\mathrm{c}$ the critical wavenumber, $a_\mathrm{c}=0.76$, corresponds to a wavelength of about 1-2 m, under typical field conditions.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{field_evidence.pdf}
\caption{\label{fig:direct_evidence} \textbf{A}: TLS scans of the surfaces showing the elevation of ridges above the surface. \textbf{B}: Cross-sections of polygons at Owens Lake showing the variation of salt concentration with depth and in-between ridges. Each colored square corresponds to one sample taken at the field site. \textbf{C}: Exponential fits to the changing salt concentration with depth.}
\end{figure*}
\section*{Instability and scale selection.}
To determine the Rayleigh numbers relevant to field conditions, and to thus evaluate if they should lead to convection and an associated pattern of crust precipitation, we measured the relevant parameters at sites in Owens Lake (CA), Badwater Basin (CA) and Sua Pan (Botswana) -- see Methods for details. From grain-size distributions we calculated $d_s$, the Sauter diameter~\cite{SAUTER1928}, of near-surface soil samples; results, from 4 to 138 $\mu$m, represent a silt to fine sand. A high soil porosity, $\phi=0.70\pm0.02$, was previously measured at Owens Lake ~\cite{TYLER1997}. For relative permeability we use the empirical relationship $\kappa = 0.11\,\phi^{5.6}\,d_s^2\,$, which fits a broad set of experimental and simulation data~\cite{GARCIA2009}. Across all sites $\kappa=2.6\cdot 10^{-13}\,$m$^2$ to $2.7\cdot 10^{-10}\,$m$^2$. At Owens Lake we measured fluid density differences of $\Delta\rho=0.21\pm0.01\,$g/cm$^3$ in pore water samples taken from close to the surface and at $\sim 1\,$m depth. Average evaporation rates of groundwater have been observed to lie in the range of $E=0.4\pm 0.1\,$mm/day~\cite{GROENEVELD2010,TYLER1997} for Owens Lake and $0.3\pm 0.1\,$mm/day~\cite{DEMEO2003} for Badwater Basin. For Sua Pan we use $E=0.7\pm0.5\,$mm/day, estimated by remote sensing and energy balances \cite{BRUNNER2004} and wind tunnel experiments~\cite{NIELD2016}. Finally, we assume the groundwater's dynamic viscosity to be a constant $\mu = 10^{-3}\,$Pa$\,$s.
From these observations we calculated Ra at twenty-one sites around Owens Lake, five in Badwater Basin and seven at Sua Pan (see online supplemental material, Table S1). The median values at these three regions were $\mathrm{Ra}=3700$, $32000$ and $420$ respectively. Values for all 33 sample locations were between $\mathrm{Ra}=117$ and $1.2\cdot 10^5$ - well above Ra$_c$. The conditions throughout these patterned salt playa are, therefore, suitable to expect a convective overturning of their groundwater, with plumes of high salinity sinking downwards from the surface.
If groundwater convection leads to preferential locations for salt precipitation, and from thence to salt crust patterning, then the convective cells and crust polygons should have similar length-scales. To this end, we measured the surface relief of the crusts at all sites using a terrestrial laser scanner (TLS, see Methods and Ref. \cite{NIELD2013}). The crusts show polygonal patterns (e.g. Fig. \ref{fig:scaling}\textbf{B}) with dominant wavelengths ranging from $\lambda = 0.4\,$m to $3.0\,$m. In Fig.~\ref{fig:scaling}\textbf{A} we summarise the Ra and dimensionless wavenumber $a = 2\pi L / \lambda$ for the field sites (triangles). The data lie above the neutral stability curve of the convection model (black line).
However, all wavenumbers measured in the field are within a small range of the critical wavenumber, $a_c=0.76$, unlike the Ra dependency of the most unstable mode of convection predicted by the linear stability analysis (dashed line). This difference is due to finite-amplitude effects, as can be explained via experiments and simulations.
The length-scale selected by the crust pattern in the field is consistent with a coarsening of the convective plumes after onset of the instability. Coarsening in related porous media flows is well-known~\cite{SLIM2013,SLIM2014}. To consider how the convective lengthscale evolves with time we simulated Eqs. 4-6 numerically (see Methods), from $\mathrm{Ra}=20$ to 1000. The simulations are unstable to convection, which becomes more vigorous with increasing Ra. The initial instability was characterised at time $\tau_s$, corresponding to when the first salt plume, of salinity $S\geq 0.5$, reached a depth of $Z = -1$ (e.g. Fig.~\ref{fig:scaling}\textbf{D}). The resulting plume spacing agrees with the most unstable mode predicted by theory, as shown in Fig.~\ref{fig:scaling}\textbf{A}. When measured at $10\,\tau_S$, however, many downwelling plumes have merged (Fig.~\ref{fig:scaling}\textbf{E}, and red dots in Fig.~\ref{fig:scaling}\textbf{A}) leading to wavenumbers much closer to the field values, clustered around $a_c$. For our field sites, one year corresponds to $30\,\tau_S$ on average, allowing ample time for coarsening as the crust pattern grows. The predicted salt flux into the crust above the downwellings, under typical conditions at $10\tau_S$, is about $25\%$ higher than average, showing how subsurface convection should influence crust growth.
To explore pattern and wavelength selection in the long-time limit, we supplemented our simulations with experiments in a Hele-Shaw cell (similar to e.g. \cite{THOMAS2018}). In these experiments we measured $\lambda$ and $a$ for systems at times of order 10$^3\,\tau_S$. The results, shown in Fig.~\ref{fig:scaling} (green squares), suggest that coarsening may continue well past the timescales accessible by direct numerical simulation. Values measured in the field lie comfortably between the ranges measured for the timescales explored in the simulations and in the experiments.
\paragraph*{Direct evidence for convection.}
If salt polygon growth is driven by convective dynamics happening beneath the patterns, then horizontal differences in salt concentration should be detectable in the soil, and pore fluid, under typical field and laboratory conditions. To this end, we first dissected an experiment that was undergoing convection, and extracted $\simeq$ 1$\,$ml samples from locations along the downwelling and upwelling plumes. As shown in Fig.~\ref{fig:scaling}\textbf{C}, the dynamics of the analogue experiments are clearly driven by, and coupled to, variations in salinity. It is interesting to note that convective plumes of salt-rich water have also been observed by electrical resistivity measurements after a heavy rainfall on salt crusts near Abu Dhabi~\cite{VANDAM2009}.
From the field, we collected samples of wet soil from two unmanaged sites at Owens Lake (see Methods). TLS surveys of the surface were made before sampling, and show the presence of salt polygons of about $2\,$m in size (Fig.~\ref{fig:direct_evidence}\textbf{A}), delimited by high ridges. In each case we sampled along a grid in a cross-section below a polygon. Analysis of the salinity of the samples with respect to pore water content shows clear evidence for plumes of high salinity below the salt ridges (Fig.~\ref{fig:direct_evidence}\textbf{B}). Specifically, we tested whether the salinity distribution in an area directly below the ridges (over a width $\pm$ 30 cm) was different to that below the flat pan of the polygon; testing this hypothesis (two-sample KS test), shows that the distributions below ridges and flat crust are statistically distinct ($p > 0.98$), at both sites.
The results for the salinity measurements also show an exponential decay in salinity with depth (Fig.~\ref{fig:direct_evidence}\textbf{C}), consistent with a salt-rich boundary layer that is heavy enough to drive buoyancy-driven porous media convection. The length scales recovered from an exponential fit to the salinity gradients, namely $13.5\pm 5.3\,$cm and $17.7\pm1.5\,$cm, are also comparable to the length scale of $L=\phi D/E = 15.1\pm8.0\;$cm estimated for Owens Lake (see Methods for calculation).
Thus, not only does direct field sampling of groundwater beneath a patterned salt crust show both horizontal and vertical variations in salt concentration, which support the claim that the system is unstable and convecting, but also demonstrates that the plumes are co-localized with the ridges visible on the surface.
\paragraph*{Discussion and Conclusion.}
Salt deserts, playa and pans are a common landform important for climate balances such as dust, energy and water, and express a rich repertoire of patterns and dynamics. Here we have shown that, in order to model and understand the surface expression of such deserts, it is important to consider the crust together with the subsurface dynamics. In particular, we have shown how the emergence of regular salt polygons, which are a common salt crust pattern, can result from their coupling to a convection process in the soil beneath them. As such, we have shown how salt polygons are part of a growing list of geophysical phenomena, such as fairy circles~\cite{JUERGENS2013}, ice wedges~\cite{Sletten2003}, polygonal terrain \cite{Kessler2003} and columnar joints~\cite{GOEHRING2013}, which can be successfully explained as the result of the instability of a dynamical process.
To establish the connection between surface and subsurface, we demonstrated consistent results from theoretical and numerical modeling, analogue experiments and field studies. In contrast to existing theories~\cite{CHRISTIANSEN1963, KRINSLEY1970}, such a model is able to explain the robustness of the pattern length scale by considering the dynamical coarsening process of the downwelling plumes, based only on measured environmental parameters. The convective dynamics are also generally known to give rise to closed-form polygonal shapes~\cite{SHATTUCK1997}. At the downwellings the salinity is higher and therefore the salinity gradient between the crust and the underlying fluid is weaker (compare Fig.~\ref{fig2} to measurements in Fig.~\ref{fig:direct_evidence}). As salt transport is a balance of advective and diffusive processes, this will lead to an increased rate of salt precipitation at these sites, contributing to the growth of ridges at the boundaries of convection cells. After the initial emergence of ridges, the growth process might be bolstered by feedback mechanisms such as a modulation of the evaporation rate by the presence of ridges, cracks or surface wicking phenomena.
\section*{Methods}
\subsection*{Field}
Field work was conducted at Owens Lake and Badwater Basin (CA) in November 2016 and January 2018; see e.g.~\cite{HOLLET1991,HUNT1966} for geological descriptions. The Owens Lake sampling sites are indicated in the online supplemental material Fig.~S1. At Badwater Basin five sites were visited $\sim$500$\,$m south of the main tourist entrance to the playa. GPS locations of all sites are provided in the online supplemental information (Table S2).
Surface height maps were obtained using a Leica P20 ScanStation terrestrial laser scanner. The scanner head was positioned at a height of at least $2\;$m above the playa surface and scans were performed before the surface was disturbed by sampling. Scan data was processed following Ref.~\cite{NIELD2013}. Data were first gridded into a digital elevation map (DEM) with a lateral resolution of $1\,$cm and a vertical resolution of $0.3\;$mm. Dominant frequencies of surface roughness were then quantified using the $90^\text{th}$ percentiles determined with the zero-upcrossing method from the DEMs~\cite{NIELD2015}.
At most sites soil cores (4$\,$cm Dutch gauge corer) were taken to a depth of up to $1\,$m. The near-surface soil showed normal bedding, indicative of sedimentation following flooding. Samples were collected from each visible soil horizon, or with a vertical resolution of $\Delta z = 10-15\,$cm. At two sites we dug trenches to take samples along a cross-section below a salt polygon. Trenches were dug about 200$\,$cm in length, 40$\,$cm in width and down to a water table of $\sim$70$\,$cm. Soil samples were taken from a freshly cleaned trench wall in a grid pattern with spacings of $\Delta x = 15\,$cm and $\Delta z = 10\,$cm. The samples had an average volume of approximately $10\,$ml and were taken using a metal spatula, which was cleaned with distilled water and dried before each use. The samples were a mixture of soil with a grain size of medium sand to clay, water and salt (both dissolved and precipitated). After collection, samples were immediately stored in air-tight containers, which were sealed with parafilm.
Sediment samples at Sua Pan were collected 2$\,$cm below the crust in August 2012. These were double bagged and subjected to grain size analysis only.
The Owens Lake and Badwater Basin samples were analysed to determine their relative salinity. Each sample was first transferred to a crystallising dish and weighed, to give a combined mass of sand, salt and water. It was then dried at 80$^\circ$C until all moisture had visibly vanished, or for at least 24h, and re-weighed to determine the mass of the (evaporated) water. Next, it was washed with 50$\,$ml of deionized water to dissolve any salt, allowed to sediment for 24 hours, and the supernatant liquid was collected in another crystallising dish. After two such washings the remaining soil and the recovered salt solution were dried and weighed. Measurement uncertainty is based on the difference between the initial sample mass, and the sum of the separated water, soil and salt masses.
Soil grain size distributions were then measured by laser particle sizer (Coulter LS 13 320), from which the Sauter diameter (the mean diameter, respecting the soil's specific surface area \cite{SAUTER1928}) was calculated. For each site a representative $d_S$ is calculated as the average Sauter diameter of all soil samples (or at trench sites, one sample per depth) from that site. For sites located at Sua Pan, only samples from sites B7 and L5, respectively (see \cite{NIELD2015} for site descriptions), were available. The Sauter diameter of the other five sites is estimated as the mean of the two measured Sauter diameters. Soil porosity has previously been measured to be around $\phi\approx 0.70\pm0.02$~\cite{TYLER1997} at Owens Lake. Because of lack of similar measurements at Badwater Basin and Sua Pan, we used the value measured at Owens for calculations of $\kappa$ and Ra at these sites.
To evaluate the density difference $\Delta \rho$, we collected pore water samples at Owens Lake from eight sites, including liquid taken from directly below the surface, and at a depth of about $1\,$m. Fluid densities were then measured using a vibrating-tube densitometer (Anton Paar DMA4500). The depth to the water table varied between 0-70$\,$cm, but the near-surface water density was consistently $1.255\pm0.008$\,g/ml, while water from depth had a density of $1.050\pm0.002$\,g/ml. These values are broadly consistent with chloride concentration profiles previously measured at Owens Lake \cite{TYLER1997}. We note that thermal effects on the groundwater density are negligible, as the mean annual variation in temperature at Owens Lake will allow for a density change of, at most, 0.005 g/ml. Similarly, the solubility of halite in water would change by less than 0.005 g/ml, seasonally. As such, neither effect is tracked in the model.
Ionic species present in the pore water were determined by quantitative X-ray powder diffraction analysis of dried salt samples (Philips X'Pert MPD PW 3040). Sites at Owens Lake showed a mixture of dried salts with $53\pm7\,$wt.\% sodium chloride and $30\pm5\,$wt.\% sodium sulphate (mirabilite). Other minerals, such as natrite, sylvite and burkeite, were variously present with less than 10\% by mass, each. We estimate an average aqueous diffusivity of $D^*=1.37\cdot10^{-9}\,\text{m}^2/\text{s}$ from measurements of ternary mixtures of the two primary salts~\cite{ANNUNZIATA2000}, using a weighted average of the mole ratios of their main-term diffusion coefficients. Accounting for the tortuosity, $\theta$, of the porous medium, we then calculate an effective diffusion coefficient $D=D^*/\theta^2 = (1.00\pm 0.24)\cdot 10^{-9}\,$m$^2$/s following \cite{BOUDREAU2011}, where we estimate $\theta^2=1-\ln(\phi)$, as in \cite{BOURDEAU1996}.
For the field sites located at Owens Lake, we use the labels referring to surface management cells related to the dust control project there~\cite{LADWP2010}. These labels either refer to managed cells or to unmanaged areas in direct vicinity of a managed cell. Labels always start with "TX-Y", where X is a number and Y is a letter. The numbers refer to water outlet taps along the main water pipeline that crosses the lakebed south to north and is used to irrigate management cells. Low tap numbers start in the south and increase northwards. The letters A, S and W stand for Addition, South and West, respectively. They refer to additional sub-regions within a management cell. Sites at Sua Pan are labeled following Ref.~\cite{NIELD2015}.
For some sites we investigated more than one polygon. This is indicated in brackets next to the site label, e.g. T27-A (3) is the third polygon investigated at site T27-A, which corresponds to the Addition region of the management cell next to the 27$^\text{th}$ water tap. We also use the numbering for the sites visited at Badwater Basin. These labels make our general findings about the mineral and soil composition relatable to other research done within the scope of the dust control project.
The Sauter diameter $d_S$, permeability $\kappa$, Rayleigh number Ra and pattern wavelength $\lambda$ for all sites investigated at the Badwater Basin and Owens Lake and Sua Pan
are listed in Tab.~S1. Here, the permeability is calculated based on the Sauter diameter and porosity as $\kappa=0.11\phi^{5.6}d_S^2$~\cite{GARCIA2009}. Uncertainties for Ra were calculated as systematic errors based on standard errors of the individual environmental parameters. GPS coordinates and year of the field campaign are listed in Tab.~S2.
\subsection*{Experimental}
Experiments were performed in $40\times 20 \times 0.8\,$cm (width, height, spacing) Hele-Shaw cells, filled with glass beads (Sigmund Lindner GmbH) with a diameter of 100-200$\,\mu$m and measured Sauter diameter of $150\,\mu$m. A porosity $\phi=0.37\pm0.01$ was measured. The permeability of the bead pack was evaluated by flow-through experiments as $\kappa = (1.67\pm0.12)\cdot 10^{-11}\,$m$^2$. The base of each cell was connected to a reservoir containing a $50\,$g/l solution of NaCl, such that $\Delta\rho =0.162\,$g/ml (as compared to a saturated salt solution). This reservoir maintained a fluid-saturated pore space in the cells. Evaporation at the top of the cells was controlled and enhanced by overhead heating and air circulation, and varied from $E = 1\,$mm/day to $10\,$cm/day. Assuming a kinematic viscosity of $\mu = 10^{-3}\,$Pa s, these conditions allowed for experimental Rayleigh numbers from Ra $=23$ to 773.
Visualization of the convective dynamics in the cells was accomplished by burying a thin ($0.15\,$cm diameter) perforated metal tube approximately $7\,$cm deep in the cell, and intermittently injecting $2$-$4\,$ml of dyed saline solution through the tube. The dye then formed a thin line of color which was advected by the flows inside the cell over time. Dye movement was recorded using time-lapse photography with a digital SLR camera (Nikon D5000 series). Plume spacings were then manually measured in the images using \texttt{Fiji}~\cite{SCHINDELIN2012}, for the data presented in Fig.~3\textbf{A}.
To determine an experimental concentration profile (Fig.~3\textbf{C}), one Hele-Shaw experiment at Ra = 222 was destructively sampled. The reservoir fluid was first dyed with rhodamine, and then fluorescein, to visualise the downwelling (dark, rhodamine) and upwelling (light, fluorescein) plumes. After the dynamics inside the cell became apparent, the wet packing in the cell was removed in layers, while sampling every 2 cm in depth along the centres of the plumes. The resulting $\sim$1 ml samples were analyzed for their salt concentration using the same protocol as described above for field samples.
Finally, an additional experiment was conducted using glass beads with diameters of $0-20\,\mu$m and $d_S = 2\,\mu$m, resulting in Ra $= 3 \cdot 10^{-3}$, which is well below Ra$_\mathrm{c}$. As anticipated, this experiment did not show any convective dynamics, over a period of three months.
\subsection*{Numerical simulation}
We numerically solve equations~(4)-(6) using a stream-function-vorticity approach. We base our methodology on work by~\cite{RIAZ2003,RUITH2000,CHEN1998}, and a detailed implementation is described elsewhere~\cite{ERNST2017}. In brief, at each time step we first compute the vorticity from the salinity field using a sixth-order compact finite difference scheme~\cite{LELE1992}. We then solve Poisson's equation for the stream function employing a semi-spectral Fourier-Galerkin method. This is accomplished by considering the individual Fourier modes, and solving the resulting system of linear differential equations of first order for the stream function. An updated velocity field is then calculated from the stream function by computing the first order spatial derivatives using a sixth-order compact finite difference scheme. Finally, the salinity distribution is advanced in time by using an explicit fourth-order Runge-Kutta scheme with adaptive time-stepping.
For the simulations shown in Fig.~3\textbf{A}, \textbf{D} and \textbf{E} we considered systems with a uniform evaporation rate such that $\frac{\partial \psi (X,0)}{\partial X} = U_Z = 1$, for a stream function $\psi(X,Z)$. We varied the Rayleigh number between $\mathrm{Ra}=20$ and $1000$ and the system size (depth $\times$ width, in units of natural length $L$) from $40\times 40$, with resolution $\Delta X = 1.25\cdot 10^{-1}$, to $10 \times 5$, with resolution $\Delta X = 1.25\cdot 10^{-2}$.
The data in Fig.~3\textbf{A} are ensemble averages over 6-10 simulations. The plume formation time, $\tau_S$, did not vary significantly within any ensemble, and wavenumbers were calculated at time $\tau = \tau_S$ at a depth of $Z = -1$. Wavenumbers at $\tau = 10\,\tau_S$ were recorded at $Z=-10$, to capture the coarsened dynamics rather than the small proto-plumes sometimes seen near the surface.
|
2,877,628,090,111 | arxiv |
\section{Conclusions and Future work}
\label{sec:conclusion}
In this paper, we have described a cloud-based service, named CloudMine, which allows multiple
data owners to carry out analytics over their joint data in a privacy-preserving manner. The
computation is outsourced to a number of independent clouds (or delegates). CloudMine protects
data privacy and ensures correctness of the computation against the standard semi-honest model
of the data owners, and against the curious-and-lazy delegate model. CloudMine supports three
analytic functions: secure sum, secure set union and intersection, and secure scalar product.
These primitives can be used to implement a wide range of complex data mining algorithms. We
demonstrated this by showing how a simple instance of CloudMine (the secure sum service) can be
used in a hybrid cloud environment for the classification, association rule mining and
clustering algorithms. We discussed the mechanisms designed to ensure privacy when the data is
stored in a public cloud. Finally, we implemented a prototype of CloudMine's secure sum service
and evaluated the performance of the service as a stand-alone application and as part of
complex data mining applications. The results demonstrate the service's practical performance,
and show that it provides privacy with little cost to the overall performance for workloads that are
inherently computationally intensive.
Our current prototype has not implemented the protocols for bootstrapping the CloudMine
service. Dynamic group membership may affect the service and its applications in interesting
ways. Incorporating and evaluating these protocols, and optimization of the overall
implementation are parts of our immediate plan for future work. We also plan to implement the
protocols for secure set and scalar product services. For the former, particularly, we intend to
investigate how existing protocols for private set intersection (which scale better than our
current protocol) can be modified to work in our delegate settings. Once being
equipped with these higher-level primitives, we can start looking at more complex applications
such as collaborative filtering. Additionally, we plan to explore if the automated scaling
features offered by some cloud platforms could improve the performance of the service,
especially under intensive workloads. Finally, we would like to incorporate differential
privacy techniques into the service and investigate the maximum privacy budget needed to
realize any given data mining algorithm.
\section{Data Mining in Hybrid Clouds}
\label{sec:dataMining}
The hybrid cloud model, in which the user utilizes the combined resources of its private
infrastructure (or private cloud) and a public cloud, helps ease the transition from in-house to public-cloud computing. This
model is motivated by the need to optimize cost and performance, to cater for different demand
patterns, or to mitigate risks~\cite{eucalyptus}. In this section, we demonstrate how
CloudMine's secure sum service can be used to implement distributed, privacy-preserving data
mining algorithms in this hybrid environment.
We consider data owners as hybrid-cloud users, who partition their data into two parts: the
sensitive part maintained in the private cloud, and the less sensitive part stored in a public
cloud~\cite{zhang11}. For example, data generated by an intrusion detection system may consist
of highly sensitive records associated with the internal system, whereas traffic to/from the
front-end servers may be regarded as less sensitive. Another example is in large scale genomic
sequencing: an individual's DNA sequence is highly sensitive and must be handled in the private
cloud, whereas a reference genome can be considered as less sensitive and therefore can be
encrypted and outsourced to a public cloud~\cite{chen12}. Note that \emph{less sensitive} is
not the same as \emph{non-sensitive}, in the sense that data owners still want to have some
levels of privacy with the less sensitive data. We distinguish two \emph{logically} separate
delegates: a \emph{computation delegate} which runs CloudMine service, and a \emph{data
delegate} which maintains the owner's data. They may belong to the same cloud, or each to a
different cloud. Adversary model for the computation delegates is the same as in the previous
section. Adversary model for the data delegates adversary model is also curious-and-lazy. In
particular, they try to learn the data stored on the public clouds, and try to do as little as
possible when answering data queries from the owners. They may collude with each other, but
they will not tamper with the data.
To protect the outsourced data from curious delegates, an encryption scheme must be used. In
our design, we employ two encryptions scheme: AES and Order-Preserving Encryption
(OPE)~\cite{boldyreva09,popa13}. AES is a deterministic scheme that supports equality comparison of
ciphertexts. OPE offers weaker security guarantees, but it supports inequality comparison of
ciphertexts, which can be used for range queries. For the sake of simplicity, we store two
encrypted copies of the data on the data delegates (a more elegant approach can be found in
CryptDB~\cite{popa11}). We use the OPE scheme from~\cite{boldyreva09}, which is a stateless
encryption and does not require a third-party server (as in~\cite{popa13}).
Untrusted data delegates necessitate protocols for ensuring query assurance. In the literature,
techniques for query assurance are probablistic which make use of redundant query execution
(\emph{ringer schemes})~\cite{sion05,du04,le12}. In this work, we use a mechanism based
on~\cite{sion05}, in which the data owner maintains a random, small portion of the outsourced data
in its private cloud. Queries to the delegates are extended with a number of fake queries, and the
results are probabilistically checked by querying the local copy of the data. Our experiments show
that maintaining as little as $15-20\%$ of the outsourced data locally is sufficiently effective to
detect lazy delegates after a small number of checks.
In the following, the data mining algorithms are run on the private cloud of each data owner. We
assume, for simplicity, that data is in relational format and every attribute belongs to a
non-negative integer domain. The algorithms consist of an iterative process of querying the
public-cloud database, combining it with the local data, and using the result as inputs to the
secure sum service. The fact that outputs from the interactions with the data delegates are used
during the computations involving cloud delegates may appear to be a risk to privacy, especially
when data and computation delegates collude (which is immediate when they belong to the same cloud
provider).However, privacy is ensured for two reasons. First, the computation delegates cannot learn
the data owners' inputs to the private computation, because the inputs are obtained over both the
data stored in the private cloud and data outsourced to the data delegate. Hence, results from
querying the data delegates only contribute partly to the inputs. Second, and more importantly, even
if all the data is outsourced, the delegates cannot collude and compute analytics by themselves,
because both the data and the meta-data (column names, table names, etc.) are encrypted.
\subsection{Classification (Naive Bayes).}
\begin{algorithm}
\footnotesize
\caption{Naive Bayes classification}
\label{alg:naiveBayes}
\textbf{Input}: $Y,A,V,i$\\
\textbf{Output}: $N,\{N_y\},\{N_{y,a,v}\}$\\
\vspace{0.1cm}
$\textit{PK} \leftarrow \mathsf{Setup}(\kappa)$; $\textit{SK} \leftarrow \mathsf{KeyGen}(\textit{PK})$; $\textit{MaK}_i \leftarrow
\mathsf{MaskGen}(i,\textit{PK})$\\
\vspace{0.1cm}
\textbf{foreach} $y \in Y, a \in A, v \in V_a$: \\
$\quad$ $N_y^i \leftarrow \textit{QueryCount}(\textit{label} = y)$ \\
$\quad$ $N_{y,a,v}^i \leftarrow \textit{QueryCount}(a = v , \textit{label} = y)$ \\
\vspace{0.1cm}
\textbf{foreach} $y \in Y, a \in A, v \in V_a$:\\
$\quad$ $N_y \leftarrow \mathsf{ComputeSum}(i,\textit{PK},\textit{SK},\textit{MaK}_i,N_y^i)$\\
$\quad$ $N_{y,a,v} \leftarrow \mathsf{ComputeSum}(i,\textit{PK},\textit{SK},\textit{MaK}_i,N_{y,a,v}^i)$\\
\end{algorithm}
A classification algorithm takes as input a set of labeled, training data and outputs a
\emph{classifier} that can be used to assign label to new data. Let $N$ be the number of data
instances, $Y$ the set of labels, $A$ the set of attributes and $V_a$ the attribute domain for
$a \in A$. The NaiveBayes algorithm shown in Algorithm~\ref{alg:naiveBayes} computes:
\[
\textit{classifier} = (N, \{N_y \,|\, y \in Y\}, \{N_{y,a,v} \,|\, y \in Y, a \in A, v \in V_a\})
\]
The label for a new instance $x$ is:
\[
\textit{label}(x) = \textit{argmax}_y(\frac{N_y}{N}.\prod_i
\frac{N_{y,i,x_i}}{N_y})
\]
The protocol $\textit{QueryCount}(a_1=v_1, a_2=v_2..)$ encrypts $a_1$, $v_1$ with AES and issues a SQL query
of the form
{\small
\begin{align*}
&\texttt{select COUNT from } \mathsf{Enc}_\textit{aes}(\texttt{Data})\\
& \qquad \qquad \texttt{ where } \mathsf{Enc}_\textit{aes}(a_1) =\mathsf{Enc}_\textit{aes}(v_1)\\
& \qquad \qquad \quad \ \texttt{ AND } \mathsf{Enc}_\textit{aes}(a_2) = \mathsf{Enc}_\textit{aes}(v_2) \ ..
\end{align*}
}
to the data delegate. The delegate executes the SQL query over the encrypted
data and returns the result which is probabilistically verified by the owner.
\subsection{Clustering (K-Mode).}
\begin{algorithm}
\footnotesize
\caption{K-Mode Clustering}
\label{alg:kmeans}
\textbf{Input}: $k,A,i$\\
\textbf{Output}: $M = \{m_1,..,m_k\}$
\vspace{0.1cm}
$\textit{PK} \leftarrow \mathsf{Setup}(\kappa)$; $\textit{SK} \leftarrow \mathsf{KeyGen}(\textit{PK})$; $\textit{MaK}_i \leftarrow
\mathsf{MaskGen}(i,\textit{PK})$\\
Initialize $m_j = \{j,j,..,j\}$ for $m_j \in M$
$C^i = \emptyset$\\
\vspace{0.1cm}
\textbf{foreach} $m_j \in M$:\\
$\quad$ $C_{m_j}^p = \emptyset$\\
$\quad$ \textbf{foreach} $a \in A$\\
$\quad\quad$ $C_{m_j}^i(a) \leftarrow \textit{QueryGroupBy}(a,m_j,M)$\\
$\quad\quad$ $C_{m_j}^i = C_{m_j}^i \, \cup\, C_{m_j}^i(a)$\\
$\quad$ $C^i = C^p \,\cup\, C_{m_j}^i$\\
\vspace{0.1cm}
\textbf{foreach} $C^i[j] \in C^i$:\\
$\quad$ $C[j] \leftarrow \mathsf{ComputeSum}(i,\textit{PK}, \textit{SK}, \textit{MaK}_i, C^i[j])$\\
$C^i \leftarrow C$\\
\textbf{foreach} $m_j \in M, a \in A$:\\
$\quad$ $m_j(a) \leftarrow \textit{Mode}(C_{m_j}^i(a))$\\
\vspace{0.1cm}
Repeat Step 5 until $M$ converges.
\end{algorithm}
A clustering algorithm partitions the data into separate \emph{clusters} such that distance between members of the same
cluster is smaller than that between members of different clusters. The K-Mode algorithm
(Algorithm~\ref{alg:kmeans}) finds $k$ clusters identified by their centroids (or \emph{modes})
that minimizes the dissimilarity between members of the same cluster (the $\textit{Mode}$ function).
The algorithm works in multiple rounds until the set of modes converges.
We use Manhattan distance to quantify the distance from a data instance $x$ to a mode
$c$, i.e. $\Delta(x, c) = \sum_i |x_i - c_i|$. The protocol $\textit{QueryGroupBy}(a,m_i,M)$ queries
the data delegate for a list of frequencies for attribute $a$ in the portion of data
closest to the centroid $m_i \in M$. The query has the form:
{\small
\begin{align*}
& \texttt{select } \mathsf{Enc}_\textit{aes}(a), \texttt{ COUNT from } \mathsf{Enc}_\textit{aes}(\texttt{Data}) \texttt{ as freq}\\
& \text{\texttt{where }} \Delta(\mathsf{Enc}_\textit{ope}(a),\mathsf{Enc}_\textit{ope}(m_i)) < \Delta(\mathsf{Enc}_\textit{ope}(a),\mathsf{Enc}_\textit{ope}(m_0))\\
& \quad \texttt{ AND } \Delta(\mathsf{Enc}_\textit{ope}(a),\mathsf{Enc}_\textit{ope}(m_i)) < \Delta(\mathsf{Enc}_\textit{ope}(a),\mathsf{Enc}_\textit{ope}(m_1)) \texttt{..}\\
& \text{\texttt{Group by $\mathsf{Enc}_\textit{aes}(a)$, Order by $\mathsf{Enc}_\textit{ope}(a)$}}
\end{align*}
}
Since $\Delta$ is computed over OPE ciphertext, the response from the cloud for $\textit{QueryGroupBy}$
might not be accurate, as compared to the same query executed over the plaintext data. OPE's
only guarantee is $\mathsf{Enc}_\textit{ope}(x) < \mathsf{Enc}_\textit{ope}(y) \leftrightarrow x < y$, hence it does not always follow
that $|\mathsf{Enc}_\textit{ope}(x) - \mathsf{Enc}_\textit{ope}(x')| < |\mathsf{Enc}_\textit{ope}(y) - \mathsf{Enc}_\textit{ope}(y')| \leftrightarrow |x-x'| < |y-y'|$. In the next
section, we show that this phenomenon occurs frequently, yet the final clusters are very
close to the clusters found using the unencrypted data.
\subsection{Association rule mining (Apriori).}
\begin{algorithm}
\footnotesize
\caption{Apriori association rule mining}
\label{alg:apriori}
\textbf{Input}: \emph{minsup, minconf}, $i$\\
\textbf{Output}: set of rules $\{(X \to Y)\}$\\
\vspace{0.1cm}
$\textit{PK} \leftarrow \mathsf{Setup}(\kappa)$; $\textit{SK} \leftarrow \mathsf{KeyGen}(\textit{PK})$; $\textit{MaK}_i \leftarrow
\mathsf{MaskGen}(i,\textit{PK})$\\
$L_1 \leftarrow$ \textit{GenerateFrequentItemsetSize1}()\\
$k = 2, B_i = \emptyset$\\
$C_k \leftarrow$ \textit{GenerateCandidates($L_{k-1}$)}\\
\vspace{0.1cm}
\textbf{foreach} $c \in C_k$:\\
$\quad$ $t \leftarrow \textit{QueryCount}(c)$\\
$\quad$ $B_i = B_i \,\cup\, t$\\
\textbf{foreach} $j \in [1,k]$:\\
$\quad$ $B[j] \leftarrow \mathsf{ComputeSum}(i,\textit{PK},\textit{SK},\textit{MaK}_i,B_i[j])$\\
$\quad$ extract $c.count$ from $B[j]$\\
\vspace{0.1cm}
$L_k \leftarrow \{ c \in C_k | c.count \geq minsup \}$\\
Increase $k$ and repeat from line 6 until $L_k = \emptyset$\\
\textit{GenerateRules}($\bigcup_k L_k$,\textit{minconf})
\end{algorithm}
An association rule mining algorithm extracts the relationships between attributes that occur
frequently in the data. An association rule has the form $(X \to Y)$ where $X, Y \subseteq A$.
The Apriori algorithm (Algorithm~\ref{alg:apriori}) first determines frequent item-sets
containing a single item using the \textit{GenerateFrequentItemsetSize1} protocol. The results are merged into larger item-sets
(candidates) using \textit{GenerateCandidates}. The threshold value \emph{minsup} specifies
the lower bound for item-set frequency. These steps are repeated until there is no more
item-set to be found. Finally, \textit{GenerateRules} generates the outputs by establishing
rules whose confidence values are above $minconf$. The details of
\textit{GenerateFrequentItemsetSize1}, \textit{GenerateCandidates} and \textit{GenerateRules}
can be found in~\cite{apriori}.
\section{Evaluation}
\label{sec:evaluation}
We have implemented the protocols described in the previous sections in order to demonstrate
CloudMine's functionality as well as to preliminarily assess its performance in a hybrid cloud
environment. In particular, the prototype implements the secure sum service and three data
mining algorithms built using this service. It is written in Java, with
cryptographic operations provided by the Crypto++ library~\cite{cryptopp}, OPE and Paillier
encryptions by CryptDB library~\cite{popa11}. Data mining algorithms made use of the
Weka library~\cite{weka}. Communications between data owners and delegates are done via Java
sockets. The source code is available at \url{https://code.google.com/p/cloudmine-sum/}.
\begin{table*}
\footnotesize
\centering
\begin{tabular}{l|l|p{6cm}}
\hline
\textbf{Parameters} & \textbf{Description} & \textbf{Values}\\
\hline \hline
$n$ & number of parties & $2,4,8,16$\\
$k$ & number of delegates & $1,2,4,8,16$\\
$\text{\emph{it}}$ & EC2 instance types & small, medium, large \\
$\text{\emph{bl}}$ & encryption bit length & $512, 1024$\\
$r$ & secure sum request rate & $10,100,200,400,700$\\
$\text{\emph{ds}}$ & dataset & \emph{breast\_cancer, x50\_breast\_cancer, mushroom,
x50\_mushroom, splice,x10\_splice}\\
$\text{alg}$ & data mining algorithm & NaiveBayes, Apriori, K-Mode\\ \hline
\end{tabular}
\caption{List of parameters used in experiments.}
\label{tab:parameters}
\end{table*}
We first experimented with CloudMine as a stand-alone service. We used \emph{throughput} ---
the number of secure sum operations completed per second measured at the party--- as the
metric. Next, we evaluated the performance of CloudMine when being used in complex data mining
algorithms. For this, we measured the overall and detailed breakdown of the \emph{running time}
of each data mining algorithm. We ran all experiments on Amazon EC2 platform~\cite{ec2}, using
the parameters as listed in Table~\ref{tab:parameters}. Unless otherwise stated, each delegate
runs on one large EC2 instance, and two parties share one large EC2 instance. In addition,
$n=k=8$ and $\text{\emph{bl}}=1024$. The results presented below are averaged over multiple
runs.
\subsection{Secure Sum Benchmark}
\begin{figure*}
\hspace*{-1cm}
\subfloat[Throughput, with varying request rate]{\includegraphics[scale=0.27,angle=-90]{newfigs/throughputs_varying_types.pdf}}
\subfloat[Throughput at steady state, with varying
$n$]{\includegraphics[scale=0.27,angle=-90]{newfigs/throughputs_varying_bl.pdf}}
\caption{Secure sum throughput}
\label{fig:throughput}
\end{figure*}
To benchmark CloudMine service, we varied the frequency at which each party requests for
the service from its delegate. We also varied the types of EC2 instances on which
the party is run, and the number of parties sharing one instance.
Figure~\ref{fig:throughput}[a] shows that throughput reaches its steady state at different values for different
configurations of the party. In particular, the highest throughput is observed at $150$
(sums/sec) when one party occupies one large instance. When two parties share the same
instance, throughput dips to around $110$ (sums/sec). When medium or small instances are
used for the parties, throughput falls even further (the lowest is at $33$ (sums/sec) with
parties sharing small instances). These results indicate with fixed $n$ and $k$, throughput
depends on the computation at the parties, i.e. the more powerful the parties are, the higher
the overall throughput. Furthermore, considering that our prototype implementation has not
been optimized for highly parallel workload, we believe these throughputs are practical for many
real-time applications in which data does not arrive at extremely high rates.
Figure~\ref{fig:throughput}[b] shows how throughput also depends on encryption bit-length
\emph{bl}, the ratio $\frac{k}{n}$ and the number of parties $n$. It can be easily seen that
reducing the encryption bit-length from 1024 to 512 leads to substantial increase in
throughput. This is because Paillier encryption and decryption operations take roughly 1$ms$
when $\text{\emph{bl}}=512$, which rise to $7ms$ with $\text{\emph{bl}}=1024$. The ratio
$\frac{k}{n}$ represents the level of decentralization. When $k=1$, all parties communicate to
one centralized delegate --- the model adopted in~\cite{duan10,shi11,kursawe11}.
When $k=n$, each party has one delegate and each delegate has one party.
The results indicate that throughput is always slightly higher when
$k=1$ than when $k=n$. This means that throughput is mainly determined by the sum
computation, as opposed to be affected by the communication overhead incurred when $k=n$. In
other words, our service supports the decentralization of the multi-party computation with
minimal cost to the overall performance. Thus, there is no substantial advantage, at least in
terms of throughput, in using a centralized service for secure sum computation; whereas
distributing this private computation over multiple delegates implies the decentralization of
trust, which is a more acceptable model in practice. Finally, as $n$ increases, we can observe a drop
in throughput. This is caused by the computation and communication overhead incurred at the delegates
when $n$ gets larger. We will discuss this overhead in more detail shortly.
\subsection{Data Mining Performance}
\begin{figure*}
\hspace*{-0.5cm}
\subfloat[Encryption and database loading, \emph{alg} =
NaiveBayes]{\includegraphics[scale=0.27,angle=-90]{newfigs/oneOffCost.pdf}}
\subfloat[Database loading, \emph{alg} = Apriori]{\includegraphics[scale=0.27,angle=-90]{newfigs/loading_time_apriori_varying_n.pdf}}
\caption{One-time cost}
\label{fig:oneOffCost}
\end{figure*}
We used three standard datasets: \emph{breast\_cancer} (small), \emph{mushroom}
(large, many rows) and \emph{splice} (large, many columns) from~\cite{uciDataset}, and
synthesized larger datasets by extending them with random values from similar distributions.
For instance, \emph{x50\_mushroom} represents the dataset 50-time the size of the original
\emph{mushroom} dataset. The largest dataset consists of $91350$ rows and $23$ columns.
In our prototype, each data owner encrypts its data with AES and OPE and uploads it to the
delegate which then stores it in a MySQL server. We let data owners outsource
all of their data to the cloud, causing larger data query overhead than when parts of
the data are stored locally. The encrypted datasets were as much as $23$
times larger in size than the original, unencrypted ones (for the \emph{x10\_splice} dataset).
We quantify the costs for database encryption at the party and database loading at the
delegate, which incur only once at the beginning, in
terms of the time taken to complete the operations. Figure~\ref{fig:oneOffCost}[a]
illustrates these costs with varying datasets for the NaiveBayes algorithm. It can be seen that
both encryption and loading time are proportional to the data size, and they remain below $8s$
even for the largest dataset. Figure~\ref{fig:oneOffCost}[b] shows the loading time at the delegates
when delegates are running on different types of EC2 instances. Across all datasets, using small
instances results in longer loading time.
\begin{figure}
\centering
{\includegraphics[scale=0.3,angle=-90]{newfigs/apriori_running_time.pdf}}
\caption{Overall running time, \emph{alg} = Apriori}
\label{fig:overallTime}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.3,angle=-90]{newfigs/mushroom_query_time.pdf}
\caption{Database query time for \emph{ds = mushroom} and \emph{ds = x50\_mushroom datasets}}
\label{fig:queryTime}
\end{figure}
As explained in Section~\ref{sec:dataMining}, a data mining application built using CloudMine
consists of two iterative, interleaving processes: database query and secure sum.
Figure~\ref{fig:overallTime} shows the breakdown costs of these processes --- measured as the
time taken to complete the process --- for Apriori
algorithm. One important observation is that database query time is
always greater than secure sum time. For \emph{x50\_mushroom} dataset, the former takes more
than an order of magnitude longer to complete. The longest experiment (with \emph{x10\_splice}
dataset) took 12 minutes to complete, of which secure sum operation accounted for only 2
minutes. This suggests that when used in real data mining algorithms, the cost of the secure
sum service has small effect on the overall performance. Figure~\ref{fig:queryTime} shows the effect of increasing data size to
the database query time for different algorithms. It can be observed that query time scales
differently for different algorithms. Particularly, Apriori demonstrates the sharpest growth as
compared to NaiveBayes and K-Mode. We attribute this to the intrinsic properties of the data
mining algorithm. Specifically, we observe that in our experiments with Apriori, larger datasets led to more queries being
performed by the delegate (from $132$ with \emph{mushroom} to $4214$ with the \emph{x50\_mushroom} dataset).
\begin{figure}
\centering
\includegraphics[scale=0.3,angle=-90]{newfigs/overhead_varying_n.pdf}
\caption{Secure sum time for the \emph{x50\_breast\_cancer} dataset. \emph{alg} = Apriori}
\label{fig:smcTimeVaryingParties}
\end{figure}
Finally, we investigated the cost of the secure sum service as being used in data mining
algorithms. Figure~\ref{fig:smcTimeVaryingParties} shows this cost varies with $n$ for the
Apriori algorithm. As the number of parties gets larger, the secure sum cost also
increases, albeit at a sub-linear rate. This is consistent to what has
been observed in Figure~\ref{fig:throughput}[b]. Recall that the cost of a secure sum
operation comprises the encryption/decryption cost at the party and the computation and
communication cost at the delegates. The former is shown in
Figure~\ref{fig:smcTimeVaryingParties} to be almost constant, meaning that the overhead
incurred when $n$ increases can be attributed to the overhead at the delegate. Firsts, each
delegate needs to perform more multiplications when $n$ increases. Second, each will
have to wait longer to receive all the messages from other delegates when $n$ becomes bigger.
\subsubsection{Correctness of K-Mode.} As explained in Section~\ref{sec:dataMining},
the $\textit{QueryGroupBy}$ protocol in K-Mode may return a different result
as compared to performing the corresponding query locally on the plaintext data. We refer to
this as \emph{mismatched query}, whose error may affect the convergence rate of the algorithm as well as the
final clusters. All of our experiments with K-Mode converged to final modes. To quantify the
differences between clusters found by using CloudMine and what are found using standard K-Mode
over plaintext data, we used an error metric $\epsilon(C_i, C'_i) = \frac{|\Omega(C_i) -
\Omega(C_i')|}{\Omega(C_i')}$ where $C_i, C'_i$ denote the two clusters and $\Omega(C_i)$ is the mean
squared distance of the members of $C_i$ to the mode. While the average number of mismatched
queries ranges from $0$ (for \emph{mushroom} dataset) to $508.2$ (for \emph{splice} dataset),
the maximum error is $0.03$. This means our protocols yield nearly identical clusters to what
obtained from the standard K-Mode.
\subsection{Discussion.}
The results above have demonstrated that there are overhead incurred by cryptographic operations
when using CloudMine, as compared to when the data owners use their own
infrastructure and directly take part in the multi-party protocol with each other. While these
costs are necessary to provide security in the presence of the delegates, we also remark that
when used in the context of data mining, they become less substantial, and can be more
than offset by the benefits gained from using elastic cloud resources. In particular, let $m$
be the number of secure sum messages sent and received by the data owners during a data mining
algorithm. Let $\alpha$ be the cryptographic cost for encrypting and decrypting a message
(with additive homomorphic encryption schemes). Let $q$ be the number of database queries and
$c_q$ the CPU cost for each query. The computation overhead at each data owner becomes $O =
(C_d - C) = (\alpha.m
- q.c_q)$ where $C$ is the cost when the data owner uses its own infrastructure. It can be seen
that $O$ diminishes quickly and becomes negative for larger workloads: more complex data
mining algorithms with high value of $q$ or larger datasets with high $c_q$. It has been shown
in Figure~\ref{fig:overallTime}, for example, that the database query costs may be over an order of
magnitude more than the costs incurred by the secure sum service.
\section{Introduction}
An enormous amount of data is being generated everyday from a plethora of computing devices.
Traditionally, data is stored in the data owner's in-house infrastructure, and access to
outsiders is provided typically through web services~\cite{xignite,noaa,resmap}. Data from
multiple sources can be mashed-up or jointly analyzed to create new services and derive
information that cannot be realized from individual datasets~\cite{glaeser03,eubank04,abbe11}.
However, it is often desirable or even required by law to protect data privacy. Although
numerous techniques for carrying out privacy-preserving data
analytics exist (\cite{kursawe11,duan10,yang06}), we believe that for wide-scale
adoption of such techniques, it is essential to provide them as basic, out-of-the-box services
which are flexible enough, so that individual users can freely choose their respective service
providers, and yet be able to collaborate among each other.
Recent developments of cloud computing have materialized a concrete platform for rapid
realization of the service-oriented computing paradigm~\cite{wei10}. Cloud providers (Google,
Amazon, Salesforce, etc.) offer computing as a service, from which software services can be
built, sold and integrated into complex applications. Migration of private IT infrastructures
to the cloud is gathering momentum~\cite{hajjat10,tak11}, as many companies and government
agencies are moving most (or all) of their data, application logics and front-end services to
the cloud. Recent advances in cloud computing have largely succeeded in accommodating the
demand for cheap, elastic and scalable computing resources. However, security issues related to
the outsourced data and computation remain a challenging obstacle to
overcome~\cite{popa11,wang11}.
Our work is motivated by the realization of these two trends, namely the need for a service for
privacy-preserving analytics and the availability of cloud computing as a platform for
service-oriented computing. More specifically, this work concerns the design space of a
cloud-based service for carrying out distributed, privacy-preserving data analytics. We present
\emph{CloudMine}, a cloud-based, on-demand service that data owners can leverage to perform analytics
over their joint data. CloudMine runs on the cloud (or \emph{delegate}) and supports three
basic functions: secure sum, secure set union and intersection, secure scalar product.
CloudMine provides three security
assurances. First, confidentiality of individual's data is protected from other semi-honest
data owners, as well as from colluding, semi-honest clouds. Second, outputs of the joint
computations are protected from the semi-honest clouds. Third, data owners can reliably detect
if their delegates have been lazy, i.e. if they have skipped the computations.
CloudMine can be used in a centralized manner when all the data owners use the same service on
a single cloud. More importantly, it also works well in distributed settings where
different data owners invoke different services on their delegates. In such setting,
multiple instances of CloudMine participate in a distributed protocol in order to achieve the
same functionality. The security properties of CloudMine are still guaranteed in
this distributed environment, even when the clouds collude with each other.
A use case of CloudMine is illustrated in the following example. Suppose there is a number of
supermarkets wishing to learn customer purchase behavior by performing association rule mining
over their joint data. Each supermarket stores their customer transaction data in-house because
the data contains sensitive information, while outsourcing the rest of its IT operation to the
cloud. Suppose the supermarkets would like to outsourcing the computation (association rule
mining) to their delegate clouds, without revealing their sensitive data to the clouds and to
each other. Since the customer purchase behavior (output of the computation) is
valuable to the participating supermarkets, they will like it to be kept secret from the clouds
(for otherwise, the latter can benefit from the information without contributing any data).
They will also like to be able to detect if their clouds have been unscrupulous, i.e. skipping
the delegated computations while still charging them for the same. Such lazy behavior could
undermine accuracy of the final result. CloudMine meets these functionality and security
requirements, and it can be readily invoked on the clouds. First,
CloudMine supports set intersection and sum operations, which can be used to carry out
association rule mining~\cite{clifton02}. Next, CloudMine protects confidentiality of data
owner's input from other data owners and from the clouds, thus the supermarkets can be assured
of the data privacy from each other and from the clouds. In addition, CloudMine
protects output of the computation from the clouds, therefore the result from association rule
mining is only learned by the participating supermarkets. Finally, CloudMine allows data
owners to detect lazy clouds, thus the supermarkets can use CloudMine to verify if their clouds
have been unscrupulous.
Privacy-preserving data analytics is an active area of research. Existing techniques are based
on either a generic secure multi-party computation~\cite{yao82,yang06}, or on using a
semi-honest third party~\cite{duan10,kursawe11}. Our work distinguishes itself from the former
in that the clouds are used as delegated computation units, hence it is more scalable. It
differs from the latter in that we consider a stronger adversary model for the cloud delegates.
Especially, we consider colluding adversaries who try to learn both the inputs and outputs of
the computation while doing as little as possible. Furthermore, while previous works consider
ad-hoc sets of data analytic tasks, each focusing on one primitive function (mostly secure sum
function), CloudMine is designed as a service with a large set of analytic functions including
secure sum, set operations and scalar products. We defer more detailed discussion to the next
section.
The key enabling technique used in CloudMine is additive homomorphic
encryption~\cite{paillier99}, which allows data owners to encrypt their private inputs before
exporting it to the CloudMine cloud delegate. The ciphertexts contain additional information to
allow for verification of computation. Secure set operations (intersection and union) are
reduced to secure sum operations by encoding set membership into the plain-text inputs. Scalar
product is computed by leveraging the homomorphic property of Paillier encryption and the
secure sum function. In all cases, the keys are kept secret from the clouds, hence
they are unable to decrypt the outputs.
Our contributions are as follows:
\begin{enumerate}
\item We present a model for cloud-based services for distributed, privacy-preserving data
analytics. The model allows data owners to outsource their private computations to the cloud in
a privacy-preserved manner.
\item We describe how the service can be implemented to support a number of cardinal data analytic functions, namely secure sum, secure set operations, and secure scalar product. We name the service
CloudMine.
\item We demonstrate how CloudMine can be used for more complex data mining tasks --- namely
classification, association rule mining and clustering --- in a hybrid cloud setting. In
particular, we show how CloudMine works when some parts of the data are stored in encrypted
form in the public clouds.
\item We benchmark CloudMine on a cloud platform, both as a stand-alone service and as
a part of more complex data mining applications. The results suggest that the overheads incurred
because of the added security mechanism are reasonable and amortized as the workload increases.
They indicate that it is practical to outsource distributed, privacy-preserving data
analytics to a (multi) cloud service.
\end{enumerate}
In the next section, we discuss in detail the system and adversary model of
CloudMine. Section~\ref{sec:protocol} delineate the CloudMine protocols for various
analytic functions. Section~\ref{sec:dataMining} describes how three classic data mining tasks
can be built using an instance of CloudMine in a hybrid cloud setting. Section~\ref{sec:evaluation} follows
with experimental evaluation before related works are discussed in
Section~\ref{sec:relatedWork}. We conclude and outline some planned future work in Section~\ref{sec:conclusion}.
\section{CloudMine Model}
\label{sec:model}
\subsection{System Model}
The system using CloudMine consists of two kinds of entities: \textbf{\emph{data owners}} (or
\textbf{\emph{parties}}) and
\textbf{\emph{clouds}} (or \textbf{\emph{delegates}}). The data owners $\mathbb{P} =
\{P_0,P_1,..,P_{n-1}\}$ wish to compute a
function $f(x_0,x_1,..,x_{n-1})$ where $x_i$ is the input of $P_i$, without
revealing the input to each other. The clouds $\mathbb{C} = \{C_0,C_1,..,C_{k-1}\}$ where $k\leq
n$ are the service providers. Each party uses one of these cloud, and each cloud is utilized by
at least one party. Denote $\delta(i) \in \mathbb{C}$ as the delegate used by party $P_i$.
At a high level, data owners use CloudMine in two steps in order to compute $f(.)$. First, they
enter the \emph{setup} phase, in which they agree on a function $\phi$ (and $\phi^{-1}$) and a
secret $\textit{sk}$. Next, each party $P_i$ computes $\phi_\textit{sk}(x_i)$ and
sends it to the delegate $\delta(i)$. In turn, the delegates exchange messages among themselves and
effectively compute $\pi = \phi_\textit{sk}(f(x_0,x_1,..))$. The data owners receive $\pi$ from their
respective delegates and compute $\phi_\textit{sk}^{-1}(\pi)=f(x_0,x_1,..)$.
\subsection{Adversary model}
\textbf{Data owners / parties} are \emph{curious but honest}. They follow the protocol for computing
$f(.)$ correctly, but passively try to learn the private inputs of each other. They could collude with
each other, but the number of colluding parties is less than $n-1$.
\textbf{Clouds/Delegates} are \emph{curious and lazy}. They are curious with respect to the
parties' private inputs as well as the output of $f(.)$. They do not actively subvert the
computation, but are lazy in the sense that they try to do as little as possible while
charging the data owners for the same. For example, they may skip some (or all) of the
computations, replay results from the previous rounds, or even replace inputs from the data
owners with other values in order to avoid computation. This model is justified by the
economic incentives of the cloud providers to over-charge customers without being
detected~\cite{wang11a}, as well as the legal realities in which the clouds can sniff sensitive
information without the liability of committing a criminal offense.
The collusion between parties and delegates is weak. In particular, the delegates may
reveal the messages exchanged during the computation of $\phi(f(.))$ to the
parties, but the shared secret between the parties are not revealed to the delegates.
If the shared secret is revealed, it is not possible to guarantee
privacy of the computation output.
\subsection{Security goals}
Given the model above, CloudMine aims to provide the following security
assurances:
\begin{enumerate}
\item Data owners cannot learn each other's private inputs.
\item Delegates cannot learn the parties' private inputs, nor can they learn the output $f(.)$.
\item Delegates cannot skip, replay or replace inputs of the delegated computations without
being detected by the parties.
\end{enumerate}
\subsection{Discussion}
Existing works on multi-party private computation, which underlie privacy-preserving data
analytics, can be grouped into two different approaches. The first is based on secure
multi-party computation, in which data owners interact with each other directly to evaluate
a function based on their private inputs. For example, \cite{vaidya03,yang06} use generic
multi-party computation circuits~\cite{yao82}. The second approach is
based on a third party, in which data owners send their encrypted inputs to the third party
which evaluates the function. \cite{duan10,shi11,kursawe11}, for instance, follow this approach.
Our model differs to the secure multi-party computation approach mainly in that the parties
delegate their computations to the clouds. As a result, instead of interacting with each other,
which does not scale well with the size of $n$, each party only interacts with its delegate.
More importantly, this model allows for much more efficient implementation of the private
computation than using generic, circuit evaluation (which
takes in the order of seconds to compute a 2-party secure sum~\cite{duan10}).
Our model share some similarities with the second approach. On one hand, when
$k=1$, the system model of CloudMine is the same as in many other works which rely on a single
third party. On the other hand, CloudMine distinguishes itself in a number of aspects. First,
we consider the case when there are multiple, independent third parties that each data owner
can individually choose to use as delegate. CloudMine is designed to
resist collusion among these delegates. This is different from~\cite{duan10} which also
supports multiple servers, but they are assumed to be non-colluding. Second, CloudMine
adversary model considers the delegates trying passively to learn the output of $f(.)$,
which is not the case in previous work. We believe such outputs
may leak sensitive information. For example, the clouds may use the aggregate (sum) values
together with off-line knowledge to derive sensitive information~\cite{dwork06}, or they may
directly infer parts of the data owners' private inputs from the output of the set intersection
function. Third, we consider the clouds to be lazy which may skip the delegated computation
and subsequently render the output $f(.)$ incorrect. This behavior presents a realistic threat
to the utility and integrity of the analytics, yet it has not been addressed in existing works.
Finally, designing CloudMine as a service on the cloud has another benefit with respect to
scalability. Since the cloud maintains the service, it can monitor the workload and
automatically add more resources to deal with increases in workload. This automatic, seamless
scaling is an essential practical improvement over systems such
as~\cite{duan10} which require complete reconfiguration and re-run of the protocols to
accommodate more servers.
\section{CloudMine Implementation}
\label{sec:protocol}
We now describe how to implement the CloudMine service to support three analytic functions:
secure sum, secure set operations (intersection and union), and secure scalar product. These
primitives serve as a powerful toolbox for doing privacy-preserving analytics, ranging from
database queries such as join~\cite{narayan12} and aggregate~\cite{kursawe11} to complex mining algorithms
such as collaborative filtering~\cite{duan10}.
CloudMine relies on an \emph{additively homomorphic} encryption scheme to protect privacy of
the data owners' inputs and to implement the basic secure sum function. In particular, we use
Paillier~\cite{paillier99}, a randomized encryption scheme consisting of three algorithms
$(\mathsf{Gen}, \mathsf{Enc}, \mathsf{Dec})$ where $\mathsf{Enc}$ and $\mathsf{Dec}$ are encryption and decryption algorithms
which use the key generated by $\mathsf{Gen}$. Paillier has the following property:
\[
\mathsf{Enc}(\textit{eK},m_1).\mathsf{Enc}(\textit{eK},m_2) =\mathsf{Enc}(\textit{eK},m_1+m_2)
\]
where $\textit{eK}$ is the encryption key. Compared to other additively homomorphic schemes based on
Elgamal~\cite{ugus09}, Paillier requires longer bit-length. But we can overcome this by packing
multiple inputs into a single plaintext so that they can be encrypted and decrypted at the same
time~\cite{popa11}. Suppose the inputs are at most $b$ bits and Paillier's plaintexts are $\textit{bl}$
bits. Suppose further that any sum value is smaller than $2^{t+b}$ for some values of $t$, then we can pack $c$ inputs
into a single plaintext (where $c \leq \lceil \frac{\textit{bl}}{b+t} \rceil$) as follows:
\[\langle x_1 \| x_2 \|..\| x_c \rangle = z \| x_1 \| z' \| x_2 .. \| z' \| x_c
\]
where $z'$ contains $t$ bits of $0$ and $z$ contains $(\textit{bl}-c.(t+b))$ bits of $0$.
In the following, we describe the construction of three services that constitute CloudMine. The
\emph{secure sum service} implements the aggregate function, \emph{secure set service} the set union and
intersection function, and \emph{secure scalar service} the scalar product function.
\subsection{Secure Sum Service}
\label{subsec:securesum}
The secure sum service, denoted as $\mathsf{S}_\textit{sum}$, consists of five
protocols: $\mathsf{S}_\textit{sum} = (\mathsf{Setup},\mathsf{KeyGen},\mathsf{MaskGen},\mathsf{ComputeSum},\mathsf{Verify})$. The first
three protocols are performed once at the beginning, while $\mathsf{ComputeSum}$ and $\mathsf{Verify}$ are
invoked for each round of computation.
\begin{itemize}
\setlength{\itemsep}{0.3cm}
\item $\mathsf{Setup}(\kappa)$: generate public parameters with $\kappa$ being
the security parameter. The result is the tuple:
\[
\textit{PK} = (\textit{gid},\mathsf{RNG},\mathbb{G}_p(g), \mathsf{Gen},b,\textit{bl})
\]
where $\textit{gid}$ identifies the group to which all parties belong, $\mathsf{RNG}$ is a random number
generator, $\mathbb{G}_p(g)$ is an algebraic group of prime order $p$ and generator $g$, $\mathsf{Gen}$ is an
algorithm for generating Paillier keys. $b$ and $\textit{bl}$ are
bit lengths of the inputs and Paillier plaintexts respectively.
\item $\mathsf{KeyGen}(\textit{PK})$: data owners execute this protocol to establish a
share secret:
\[
\textit{SK} = (\textit{eK},\textit{dK},\textit{rId})
\]
where $(\textit{eK},\textit{dK})$ is a Paillier key pair and $\textit{rId}$ is a random number identifying the initial round
of computation. First, the parties follow the protocols as proposed in~\cite{burmester05} to
generate a secret $x \in \mathbb{G}_p$ using the clouds and without the latter learning $x$. Next,
they use $x$ as the seed to initialize the random number
generator $\mathsf{RNG}$, which is then used by $\mathsf{Gen}$ to generate $(\textit{eK},\textit{dK})$. Finally, the parties assign
the next random number generated by $\mathsf{RNG}$ as $\textit{rId}$.
\item $\mathsf{MaskGen}(i, \textit{PK})$: each party $P_i$ invokes this protocol to generate its own
secret
\[
\textit{MaK}_i = (r_i, \eta_i)
\]
such that $r_i, \eta_i$ are random values from an algebraic group of specific size, $\eta_i \neq 0$
and the sum of $r_i$ and $\eta_i$ across all data owners are known, i.e. $\sum_i r_i = \sum_i \eta_i
= 0$.
First, $P_i$ creates random values $r_{ij}$ and $\eta_{ij}$ (using its private source of
randomness) for all $P_j$ ($i \neq j$) belonging to the group $\textit{gid}$. Next,
$r_{ij}$ and $\eta_{ij}$ are encrypted with $P_j$'s public key and sent to $\delta(i)$ which
subsequently forwards them to $P_j$. Having received the encrypted $r_{ji}, \eta_{ji}$ from its
delegate, $P_i$ then computes
\[
r_i = \sum_{i \neq j}(r_{ij} - r_{ji}) \qquad \eta_i = \sum_{i \neq j}(\eta_{ij} -\eta_{ji})
\]
It can be seen that $P_j$ cannot learn $r_i, \eta_i$ for $i \neq j$, and that
$\sum_i r_i = \sum_i \eta_i = 0$.
\item $\mathsf{ComputeSum}(i,\textit{PK}, \textit{SK}, \textit{MaK}_i, x_i)$: each party $P_i$
constructs the ciphertext $c_i$ for its private input $x_i$ as follows:
\[
c_i = \mathsf{Enc}(\textit{eK}, \langle\eta_i||\textit{rId}||(x_i+r_i)\rangle)
\]
It then sends $c_i$ to its delegate which then broadcasts it to the other delegates. Finally, each
delegate computes:
\[
c = \prod c_i = \mathsf{Enc}(\textit{eK}, \langle\sum \eta_i || n.\textit{rId} || \sum x_i\rangle)
\]
and forwards it to the party. Finally, $P_i$ invokes $\mathsf{Verify}(i,\textit{PK}, \textit{SK},
\textit{MaK}_i, c)$. If the result of this verification protocol is $y \neq \bot$, the party
returns $y$ as the final sum.
\item $\mathsf{ComputeSum}(i,\textit{PK}, \textit{SK}, \textit{MaK}_i,\overline{x_i})$: takes as parameter a vector of
inputs $\overline{x_i}$ instead of a single input. Let $s = \lceil \frac{\textit{bl}-2(b_r+\lceil
log_2 n \rceil)}{b_r+b}\rceil$ where $b_r$ is the bit length of $\eta_i$ and $\textit{rId}$. For $0 \leq k <
\lceil\frac{b.|\overline{x_i}|}{s}\rceil$, we construct message $m_k$ as follows:
\[m_k = \langle x_{k.s}\|x_{k.s+1}\|..\|x_{(k+1).s-1} \rangle\]
where $|m_k| = \textit{bl}-2(b_r+\lceil log_2 n \rceil)$ bit. For
each $m_k$, the party invokes $\mathsf{ComputeSum}(i,\textit{PK}, \textit{SK}, \textit{MaK}, m_k)$. If
the result $y \neq \bot$, it extracts the sums $y_{k.s}, y_{k.s+1},..,y_{(k+1).s-1}$ from $y$. After each
invocation, the party increments $\textit{rId}$ and updates $\textit{SK}$ accordingly.
\item $\mathsf{Verify}(i, \textit{PK}, \textit{SK}, \textit{MaK}_i, c)$: each party decrypts the ciphertext
$c$ and checks that the result is of the following form:
\[
\mathsf{Dec}(\textit{dK},c) = \langle 0 || n.\textit{rId} || y \rangle
\]
If true, $y$ is returned as the final sum.
\end{itemize}
\subsubsection{Discussion.}
We now discuss how the protocols above meet the security requirements
listed in Section~\ref{sec:model}. First, data owners cannot learn each other's inputs, because
each input $x_i$ has been masked with a secret value $r_i$. Second, delegates cannot
extract the sum $\sum_i x_i$ from the ciphertext $c$, because they do not have access to the
decryption key $\textit{dK}$. Third, delegates cannot replay old values without being detected, since
each ciphertext is embedded with a fresh value of $\textit{rId}$. They cannot replace $c_i$ with another
valid ciphertext either, because they do not have access to $(\textit{eK}, r_i, \eta_i)$, thus invalid ciphertexts
will be detected by the verification protocol. Neither can they skip some (or all) of the
inputs for the computation of $c = \prod c_i$, because it will cause verification to fail,
since $\mathsf{Dec}(c) \neq \langle 0||n.rId||y \rangle$. Finally, each delegate can compute $c = c_i^{n}$ (raising to the power of
$n$ may be cheaper than $n$ multiplications), which makes the second element of $\mathsf{Dec}(c)$
to be the same as $n.\textit{rId}$. However, verification will still fail, because $n.\eta_i \neq 0$.
Security of $\mathcal{S}_\textit{sum}$ depends on the fact that delegates do not know the
shared secret $\textit{SK}$ or the data owner secret $\textit{MaK}_i$. Every party $i$ must protect
$\textit{MaK}_i$ from other parties . To ensure long-term security, it is important to refresh
$\textit{SK}$ as well as $\textit{MaK}_i$, albeit refreshing the latter can be done after
longer intervals. This can be achieved by invoking $\mathsf{KeyGen}$ and
$\mathsf{MaskGen}$ again. Alternatively, if $P_i$ stores the original $\{r_{ij}, r_{ji}, \eta_{ij},
\eta_{ji}\,|\, j \neq i\}$, the new $\textit{MaK}_i$ can be computed as:
\[r_i' = \sum_{i \neq j}(H(r_{ij}) - H(r_{ji})) \qquad \eta_i' = \sum_{i \neq j}(H(\eta_{ij}) - H(\eta_{ji}))
\]
where $H$ is a cryptographic hash function.
The verification of delegate behavior relies on the party encoding its secret $\textit{MaK}_i$ to the
ciphertexts. As a consequence, the memory overhead is $ o = \frac{2.(b_r+log_2n)}{\textit{bl}}$, which
decreases as the Paillier bit-length $\textit{bl}$ increases. The $\mathsf{Verify}$ protocol is performed at
the end of every $\mathsf{ComputeSum}$ protocol. This can become overhead when there are many rounds of
computations. Hence, we extend $\mathcal{S}_\textit{sum}$ to allow parties to invoke $\mathsf{Verify}$
only with a probability $p$. The probability of successfully detecting consistent misbehavior,
$p_v$, can be made arbitrarily high after a number of verification. Specifically, $p_v = 1-(1-p)^{n.k}$ where $k$
is the number of random checks.
\subsection{Secure Set Service}
The service for secure set union and intersection can be built directly from the secure sum
service. Intuitively, the input sets are encoded into plaintext messages which are used as
inputs for $\mathcal{S}_\textit{sum}$. The union or intersection set is then decoded from the
final sum values.
The secure set service, denoted as $\mathcal{S}_\textit{set}$, consists of five protocols:
\\
$\mathcal{S}_\textit{set} =
(\mathsf{Setup},\mathsf{KeyGen},\mathsf{MaskGen},\mathsf{ComputeUnion},$\\ $\mathsf{ComputeIntersect})$.
\begin{itemize}
\setlength{\itemsep}{0.3cm}
\item $\mathsf{Setup}$, $\mathsf{KeyGen}$, $\mathsf{MaskGen}$ are the same as in the secure sum service,
except that the public parameter $\textit{PK}$ also contains a universal domain $U$.
\item $\mathsf{ComputeUnion}(i,\textit{PK}, \textit{SK}, \textit{MaK}_i, \overline{x_i})$: each party
inputs a vector $\overline{x_i} \in U^*$ and computes the union set as follows. A vector $I =
(a_0, .., a_{|U|-1})$ is constructed, in which $a_i = 1$ if $U[i] \in \overline{x_i}$ and $a_i=0$
otherwise. The party then invokes $\mathsf{ComputeSum}(i,\textit{PK}, \textit{SK}, \textit{MaK}_i, I)$. When
the secure sum service returns $s_0,s_1,..,s_{|U|-1}$, it computes $\{U[i] \,|\, s_i
\geq 1\}$ as the union set.
\item $\mathsf{ComputeIntersect}(i,\textit{PK}, \textit{SK}, \textit{MaK}_i,\overline{x_i})$ works in the
same way as $\mathsf{ComputeUnion}$, except that the intersection set is computed as $\{U[i] \,|\, s_i = n\}$.
\end{itemize}
\subsubsection{Discussion.}
This service has the same security properties as for secure sum. The number of encryptions per
set operation is $\lceil \frac{|U|(log_2n+1)}{\textit{bl}-2(log_2n+b)} \rceil$, which grows linearly
with the size of $U$. Consequently, our protocols may not scale well when $U$ and $n$ are very
large (for example, in orders of millions as in the case of large-scale collaborative
filtering). Other protocols for private set operations which scale more gracefully
(\cite{huang12,narayan12}) do not apply to our delegate model. In practice, many applications
involving secure set operations have small- to medium-size $U$ (in orders of ten or
hundred)~\cite{uciDataset}, which renders our protocols practical. For example, for $100$ data
owners, $|U|=1000$, $\textit{bl}=1024$, $b=16$, the protocols need only $9$ encryptions. We believe
that for current applications, this cost is reasonable.
\subsection{Secure Scalar Service}
Data owners are divided into two disjoint groups $X$, $Y$. Let $\overline{x}$ and
$\overline{y}$ be two vectors in which $x_i$ is the private input of party $X_i$, $y_i$ the
private input of $Y_i$. The secure scalar service, $\mathcal{S}_\textit{sp}$ allow the data owners in both groups to compute
\[
p = \overline{x}.\overline{y} = x_0.y_0+x_1.y_1+..+x_\frac{n-1}{2}.y_\frac{n-1}{2}
\]
$\mathcal{S}_\textit{sp}$ consists of four protocols $(\mathsf{Setup}, \mathsf{KeyGen}, \mathsf{MaskGen},
\mathsf{ComputeScalar})$, and it also makes use the secure sum service.
\begin{itemize}
\setlength{\itemsep}{0.3cm}
\item $\mathsf{Setup}(\kappa)$ is the similar to that in $\mathcal{S}_\textit{sum}$. It outputs public
the parameter:
\[
\textit{PK} = (\textit{gid}_x, \textit{gid}_y, \mathsf{RNG}, \mathbb{G}_p(g), \mathsf{Gen}, b, \textit{bl})
\]
where $\textit{gid}_x$ and $\textit{gid}_y$ are identities of group $X$ and $Y$ respectively.
\item $\mathsf{KeyGen}(\textit{gid}, \textit{PK})$: parties that belong to the group $\textit{gid}$ execute this protocol to
establish a shared secret among them. The protocol is the same as in
$\mathcal{S}_\textit{sum}$, and the result is
\[
\textit{SK}_\textit{gid} = (\textit{eK}_\textit{gid},\textit{dK}_\textit{gid},\textit{rId}_\textit{gid})
\]
\item $\mathsf{MaskGen}(i,\textit{gid},\textit{PK})$: each party $i$ in group $\textit{gid}$ first generates a
Paillier key pair $(\textit{eK}_{\textit{gid},i}',\textit{dK}_{\textit{gid},i}')$. Next, it generates two values $r_{\textit{gid},i},
\eta_{\textit{gid},i}$ in the same way as in $\mathcal{S}_\textit{sum}$, i.e. $\eta_{\textit{gid},i} \neq 0$
and $\sum_i \eta_{\textit{gid},i} = \sum_i r_{\textit{gid},i} =0$. Denote
\[
\textit{MaK}_{\textit{gid},i} = (\textit{dK}_{\textit{gid},i}, r_{\textit{gid},i},\eta_{\textit{gid},i})
\]
as the secret of party $i$ in group $\textit{gid}$. Also, let
\[
\textit{PK}' = \{\textit{eK}_{\textit{gid}_x,i}'\} \ \cup \ \{\textit{eK}_{\textit{gid}_y,i}'\}
\]
be another set of public parameters.
\item $\mathsf{ComputeScalar}(i,\textit{gid},\textit{PK},\textit{PK}',\textit{SK}_\textit{gid},\textit{MaK}_{\textit{gid},i},x_i)$: party $i$ in group $\textit{gid}$ executes this protocol to compute the global scalar product. Suppose
the party is $X_i$ (belonging to group $X$, and $\textit{gid} = \textit{gid}_x$), the protocol proceeds as
follows:
\begin{enumerate}
\item $X_i$ sends $m_i = \mathsf{Enc}(\textit{eK}_{\textit{gid}_y,i}',x_i)$ to its delegate which then forwards it to $Y_i$.
\item $Y_i$ computes $c_i = m_i^{y_i}.\mathsf{Enc}(\textit{eK}_{\textit{gid}_x,i}',r_{\textit{gid}_y,i})$ and sends it back to $X_i$ via its delegate.
\item $X_i$ computes $z = \mathsf{Dec}(\textit{dK}_{\textit{gid}_x,i}',c_i) = x_i.y_i + r_{\textit{gid}_y,i}$. It then invokes
the service sum service $\mathsf{ComputeSum}(i,\textit{PK},\textit{SK}_\textit{gid},\textit{MaK}_{grid,i},z)$, the result of which is the scalar product.
\end{enumerate}
\end{itemize}
\subsubsection{Discussion.}
The intuition behind $\mathsf{ComputeScalar}$ protocol is for $X_i$ to compute the value
$(x_i.y_i+r_{\textit{gid},i})$ without knowing $y_i$. This is then aggregated with the other
values from $X_j$ $(i \neq j)$ to cancel out $r_{\textit{gid}_y,i}$ and obtain the scalar product. This works
because of the homomorphic property of
Paillier and the fact that $\sum r_{\textit{gid}_y,i} = 0$.
The party $X_i$ cannot learn input of $Y_i$, because the sum $x_i.y_i$ is
masked by a random value $r_{\textit{gid},i}$. Neither can $X_i$ learn the input of $X_j$ ($i \neq j$)
due to the property of the secure sum service. The delegates can neither learn the intermediate
sum $x_i.y_i$ because they are encrypted with data owners' keys, nor the final scalar product
because $\mathcal{S}_\textit{sum}$ does not reveal the final sum to the delegates. In $\mathsf{ComputeScalar}$, the delegates play two roles: forwarding messages between
parties and performing secure sum computations. For the former, the delegates cannot be lazy without being detected,
because messages are acknowledged (so they cannot be skipped) and freshly signed (so they cannot be
replayed). For the latter, the secure sum protocol ensures that lazy behavior will be reliably
detected.
It can be seen that delegates are more involved in this service than in
$\mathcal{S}_\textit{sum}$ or in $\mathcal{S}_\textit{set}$. In particular, they must keep
track of the group to which each party belongs, and must forward messages to the correct
delegates. This management task, if left to the data owners, may become impractical for large
systems. Since CloudMine is a cloud-based service, such tasks can be performed by the cloud in
a scalable way.
\section{Related Work}
\label{sec:relatedWork}
CloudMine shares common goals with many other works in the area of distributed,
privacy-preserving data analytics. Our work is not based on randomization
approach~\cite{agrawal00} which perturbs the inputs or differential privacy~\cite{dwork06}
approach which adds noise to the outputs. Instead, CloudMine follows the secure multi-party
computation approach~\cite{yao82} in preserving data privacy during computation. It has been
shown that any computation can be done in a private manner, by reducing the computation to a
combination of circuits. Vaidya et al.~\cite{vaidya03} use generic circuits for evaluating
2-party comparison operation, which is then used for K-Means algorithm over vertically
partitioned data. Yang et al~\cite{yang06} use generic circuits for computing Bayesian networks
on vertically partitioned data. CloudMine does not rely on circuit evaluation, which is either
expensive~\cite{duan10} or is restricted to two-party computation~\cite{huang12}. Instead, it
shares similar model to what is proposed in~\cite{duan10,kursawe11,shi11,rastogi10} which rely
on third-party servers. However, these works focus on specific functions for specific
application domains. In contrast, CloudMine is designed in a service-oriented manner, that can
be flexibly used by a wide range of applications. Furthermore, the adversary model of CloudMine
is stronger than in the aforementioned previous works.
Our delegated computation model is a special case of verifiable computation, in which a client
outsources its computations to a more powerful entity and is able to later verify the outputs.
Theoretical results have shown that any computation can be outsourced with guaranteed input and
output privacy~\cite{gennaro10}. However, a general protocol for outsourced computation is
inefficient~\cite{wang11}. \cite{golle01,goldwasser08} propose to detect cheating and mis-computation
at the expense of data privacy, but they rely on probabilistic
checking and require the client to pre-compute the results or the delegate to commit certain
values. Wang et al.~\cite{wang11,wang11a} propose practical methods to outsource linear programming
to the cloud. However, they consider a single data owner and delegate, as opposed to CloudMine's
multi-party model.
Finally, existing works on security of outsourced databases focus on data
privacy~\cite{popa11}, query freshness~\cite{merkle79,goodrich01} and query
completeness~\cite{li10}. These works complement the protocols we described in
Section~\ref{sec:dataMining} (which deal with data privacy and query completeness).
\section{Introduction}
The \textit{proceedings} are the records of a conference.
ACM seeks to give these conference by-products a uniform,
high-quality appearance. To do this, ACM has some rigid
requirements for the format of the proceedings documents: there
is a specified format (balanced double columns), a specified
set of fonts (Arial or Helvetica and Times Roman) in
certain specified sizes (for instance, 9 point for body copy),
a specified live area (18 $\times$ 23.5 cm [7" $\times$ 9.25"]) centered on
the page, specified size of margins (1.9 cm [0.75"]) top, (2.54 cm [1"]) bottom
and (1.9 cm [.75"]) left and right; specified column width
(8.45 cm [3.33"]) and gutter size (.83 cm [.33"]).
The good news is, with only a handful of manual
settings\footnote{Two of these, the {\texttt{\char'134 numberofauthors}}
and {\texttt{\char'134 alignauthor}} commands, you have
already used; another, {\texttt{\char'134 balancecolumns}}, will
be used in your very last run of \LaTeX\ to ensure
balanced column heights on the last page.}, the \LaTeX\ document
class file handles all of this for you.
The remainder of this document is concerned with showing, in
the context of an ``actual'' document, the \LaTeX\ commands
specifically available for denoting the structure of a
proceedings paper, rather than with giving rigorous descriptions
or explanations of such commands.
\section{The {\secit Body} of The Paper}
Typically, the body of a paper is organized
into a hierarchical structure, with numbered or unnumbered
headings for sections, subsections, sub-subsections, and even
smaller sections. The command \texttt{{\char'134}section} that
precedes this paragraph is part of such a
hierarchy.\footnote{This is the second footnote. It
starts a series of three footnotes that add nothing
informational, but just give an idea of how footnotes work
and look. It is a wordy one, just so you see
how a longish one plays out.} \LaTeX\ handles the numbering
and placement of these headings for you, when you use
the appropriate heading commands around the titles
of the headings. If you want a sub-subsection or
smaller part to be unnumbered in your output, simply append an
asterisk to the command name. Examples of both
numbered and unnumbered headings will appear throughout the
balance of this sample document.
Because the entire article is contained in
the \textbf{document} environment, you can indicate the
start of a new paragraph with a blank line in your
input file; that is why this sentence forms a separate paragraph.
\subsection{Type Changes and {\subsecit Special} Characters}
We have already seen several typeface changes in this sample. You
can indicate italicized words or phrases in your text with
the command \texttt{{\char'134}textit}; emboldening with the
command \texttt{{\char'134}textbf}
and typewriter-style (for instance, for computer code) with
\texttt{{\char'134}texttt}. But remember, you do not
have to indicate typestyle changes when such changes are
part of the \textit{structural} elements of your
article; for instance, the heading of this subsection will
be in a sans serif\footnote{A third footnote, here.
Let's make this a rather short one to
see how it looks.} typeface, but that is handled by the
document class file. Take care with the use
of\footnote{A fourth, and last, footnote.}
the curly braces in typeface changes; they mark
the beginning and end of
the text that is to be in the different typeface.
You can use whatever symbols, accented characters, or
non-English characters you need anywhere in your document;
you can find a complete list of what is
available in the \textit{\LaTeX\
User's Guide}\cite{Lamport:LaTeX}.
\subsection{Math Equations}
You may want to display math equations in three distinct styles:
inline, numbered or non-numbered display. Each of
the three are discussed in the next sections.
\subsubsection{Inline (In-text) Equations}
A formula that appears in the running text is called an
inline or in-text formula. It is produced by the
\textbf{math} environment, which can be
invoked with the usual \texttt{{\char'134}begin. . .{\char'134}end}
construction or with the short form \texttt{\$. . .\$}. You
can use any of the symbols and structures,
from $\alpha$ to $\omega$, available in
\LaTeX\cite{Lamport:LaTeX}; this section will simply show a
few examples of in-text equations in context. Notice how
this equation: \begin{math}\lim_{n\rightarrow \infty}x=0\end{math},
set here in in-line math style, looks slightly different when
set in display style. (See next section).
\subsubsection{Display Equations}
A numbered display equation -- one set off by vertical space
from the text and centered horizontally -- is produced
by the \textbf{equation} environment. An unnumbered display
equation is produced by the \textbf{displaymath} environment.
Again, in either environment, you can use any of the symbols
and structures available in \LaTeX; this section will just
give a couple of examples of display equations in context.
First, consider the equation, shown as an inline equation above:
\begin{equation}\lim_{n\rightarrow \infty}x=0\end{equation}
Notice how it is formatted somewhat differently in
the \textbf{displaymath}
environment. Now, we'll enter an unnumbered equation:
\begin{displaymath}\sum_{i=0}^{\infty} x + 1\end{displaymath}
and follow it with another numbered equation:
\begin{equation}\sum_{i=0}^{\infty}x_i=\int_{0}^{\pi+2} f\end{equation}
just to demonstrate \LaTeX's able handling of numbering.
\subsection{Citations}
Citations to articles \cite{bowman:reasoning, clark:pct, braams:babel, herlihy:methodology},
conference
proceedings \cite{clark:pct} or books \cite{salas:calculus, Lamport:LaTeX} listed
in the Bibliography section of your
article will occur throughout the text of your article.
You should use BibTeX to automatically produce this bibliography;
you simply need to insert one of several citation commands with
a key of the item cited in the proper location in
the \texttt{.tex} file \cite{Lamport:LaTeX}.
The key is a short reference you invent to uniquely
identify each work; in this sample document, the key is
the first author's surname and a
word from the title. This identifying key is included
with each item in the \texttt{.bib} file for your article.
The details of the construction of the \texttt{.bib} file
are beyond the scope of this sample document, but more
information can be found in the \textit{Author's Guide},
and exhaustive details in the \textit{\LaTeX\ User's
Guide}\cite{Lamport:LaTeX}.
This article shows only the plainest form
of the citation command, using \texttt{{\char'134}cite}.
This is what is stipulated in the SIGS style specifications.
No other citation format is endorsed.
\subsection{Tables}
Because tables cannot be split across pages, the best
placement for them is typically the top of the page
nearest their initial cite. To
ensure this proper ``floating'' placement of tables, use the
environment \textbf{table} to enclose the table's contents and
the table caption. The contents of the table itself must go
in the \textbf{tabular} environment, to
be aligned properly in rows and columns, with the desired
horizontal and vertical rules. Again, detailed instructions
on \textbf{tabular} material
is found in the \textit{\LaTeX\ User's Guide}.
Immediately following this sentence is the point at which
Table 1 is included in the input file; compare the
placement of the table here with the table in the printed
dvi output of this document.
\begin{table}
\centering
\caption{Frequency of Special Characters}
\begin{tabular}{|c|c|l|} \hline
Non-English or Math&Frequency&Comments\\ \hline
\O & 1 in 1,000& For Swedish names\\ \hline
$\pi$ & 1 in 5& Common in math\\ \hline
\$ & 4 in 5 & Used in business\\ \hline
$\Psi^2_1$ & 1 in 40,000& Unexplained usage\\
\hline\end{tabular}
\end{table}
To set a wider table, which takes up the whole width of
the page's live area, use the environment
\textbf{table*} to enclose the table's contents and
the table caption. As with a single-column table, this wide
table will ``float" to a location deemed more desirable.
Immediately following this sentence is the point at which
Table 2 is included in the input file; again, it is
instructive to compare the placement of the
table here with the table in the printed dvi
output of this document.
\begin{table*}
\centering
\caption{Some Typical Commands}
\begin{tabular}{|c|c|l|} \hline
Command&A Number&Comments\\ \hline
\texttt{{\char'134}alignauthor} & 100& Author alignment\\ \hline
\texttt{{\char'134}numberofauthors}& 200& Author enumeration\\ \hline
\texttt{{\char'134}table}& 300 & For tables\\ \hline
\texttt{{\char'134}table*}& 400& For wider tables\\ \hline\end{tabular}
\end{table*}
\subsection{Figures}
Like tables, figures cannot be split across pages; the
best placement for them
is typically the top or the bottom of the page nearest
their initial cite. To ensure this proper ``floating'' placement
of figures, use the environment
\textbf{figure} to enclose the figure and its caption.
This sample document contains examples of \textbf{.eps}
and \textbf{.ps} files to be displayable with \LaTeX. More
details on each of these is found in the \textit{Author's Guide}.
\begin{figure}
\centering
\epsfig{file=fly.eps}
\caption{A sample black and white graphic (.eps format).}
\end{figure}
\begin{figure}
\centering
\epsfig{file=fly.eps, height=1in, width=1in}
\caption{A sample black and white graphic (.eps format)
that has been resized with the \texttt{epsfig} command.}
\end{figure}
As was the case with tables, you may want a figure
that spans two columns. To do this, and still to
ensure proper ``floating'' placement of tables, use the environment
\textbf{figure*} to enclose the figure and its caption.
Note that either {\textbf{.ps}} or {\textbf{.eps}} formats are
used; use
the \texttt{{\char'134}epsfig} or \texttt{{\char'134}psfig}
commands as appropriate for the different file types.
\subsection{Theorem-like Constructs}
Other common constructs that may occur in your article are
the forms for logical constructs like theorems, axioms,
corollaries and proofs. There are
two forms, one produced by the
command \texttt{{\char'134}newtheorem} and the
other by the command \texttt{{\char'134}newdef}; perhaps
the clearest and easiest way to distinguish them is
to compare the two in the output of this sample document:
This uses the \textbf{theorem} environment, created by
the\linebreak\texttt{{\char'134}newtheorem} command:
\newtheorem{theorem}{Theorem}
\begin{theorem}
Let $f$ be continuous on $[a,b]$. If $G$ is
an antiderivative for $f$ on $[a,b]$, then
\begin{displaymath}\int^b_af(t)dt = G(b) - G(a).\end{displaymath}
\end{theorem}
The other uses the \textbf{definition} environment, created
by the \texttt{{\char'134}newdef} command:
\newdef{definition}{Definition}
\begin{definition}
If $z$ is irrational, then by $e^z$ we mean the
unique number which has
logarithm $z$: \begin{displaymath}{\log e^z = z}\end{displaymath}
\end{definition}
\begin{figure}
\centering
\psfig{file=rosette.ps, height=1in, width=1in,}
\caption{A sample black and white graphic (.ps format) that has
been resized with the \texttt{psfig} command.}
\end{figure}
Two lists of constructs that use one of these
forms is given in the
\textit{Author's Guidelines}.
\begin{figure*}
\centering
\epsfig{file=flies.eps}
\caption{A sample black and white graphic (.eps format)
that needs to span two columns of text.}
\end{figure*}
and don't forget to end the environment with
{figure*}, not {figure}!
There is one other similar construct environment, which is
already set up
for you; i.e. you must \textit{not} use
a \texttt{{\char'134}newdef} command to
create it: the \textbf{proof} environment. Here
is a example of its use:
\begin{proof}
Suppose on the contrary there exists a real number $L$ such that
\begin{displaymath}
\lim_{x\rightarrow\infty} \frac{f(x)}{g(x)} = L.
\end{displaymath}
Then
\begin{displaymath}
l=\lim_{x\rightarrow c} f(x)
= \lim_{x\rightarrow c}
\left[ g{x} \cdot \frac{f(x)}{g(x)} \right ]
= \lim_{x\rightarrow c} g(x) \cdot \lim_{x\rightarrow c}
\frac{f(x)}{g(x)} = 0\cdot L = 0,
\end{displaymath}
which contradicts our assumption that $l\neq 0$.
\end{proof}
Complete rules about using these environments and using the
two different creation commands are in the
\textit{Author's Guide}; please consult it for more
detailed instructions. If you need to use another construct,
not listed therein, which you want to have the same
formatting as the Theorem
or the Definition\cite{salas:calculus} shown above,
use the \texttt{{\char'134}newtheorem} or the
\texttt{{\char'134}newdef} command,
respectively, to create it.
\subsection*{A {\secit Caveat} for the \TeX\ Expert}
Because you have just been given permission to
use the \texttt{{\char'134}newdef} command to create a
new form, you might think you can
use \TeX's \texttt{{\char'134}def} to create a
new command: \textit{Please refrain from doing this!}
Remember that your \LaTeX\ source code is primarily intended
to create camera-ready copy, but may be converted
to other forms -- e.g. HTML. If you inadvertently omit
some or all of the \texttt{{\char'134}def}s recompilation will
be, to say the least, problematic.
\section{Conclusions}
This paragraph will end the body of this sample document.
Remember that you might still have Acknowledgments or
Appendices; brief samples of these
follow. There is still the Bibliography to deal with; and
we will make a disclaimer about that here: with the exception
of the reference to the \LaTeX\ book, the citations in
this paper are to articles which have nothing to
do with the present subject and are used as
examples only.
\section{Acknowledgments}
This section is optional; it is a location for you
to acknowledge grants, funding, editing assistance and
what have you. In the present case, for example, the
authors would like to thank Gerald Murray of ACM for
his help in codifying this \textit{Author's Guide}
and the \textbf{.cls} and \textbf{.tex} files that it describes.
\bibliographystyle{abbrv}
|
2,877,628,090,112 | arxiv | \section{Introduction}
The sequence of Chebyshev polynomials of the first kind $\left\{ T_{n}(z)\right\} _{n=0}^{\infty}$
defined by the recurrence
\[
T_{n+1}(z)=2zT_{n}(z)+T_{n-1}(z)
\]
with $T_{0}(z)=1$ and $T_{1}(z)=z$ forms a sequence of orthogonal
polynomials whose zeros are real (i.e., hyperbolic polynomials). The
location of zeros of polynomials satisfying a more general recurrence
\begin{equation}
R_{n+1}(z)=A(z)R_{n}(z)+B(z)R_{n-1}(z)\label{eq:generalrecurrence}
\end{equation}
where $A(z),B(z)\in\mathbb{C}[z]$ was given in \cite{tran}. In \cite{stankov},
the author studied the set of zeros of a linear combination of Chebyshev
polynomials $\sum_{k=0}^{m}a_{k}T_{n-k}(z)$, $m\le n$, $a_{k}\in\mathbb{R},$
and provided a connection between this sequence and the theory of
Pisot and Salem numbers in number theory. In the special case when
$m=n$ and $a_{k}=1$ $\forall k$, the sum of the first $n$ Chebyshev
polynomials connects to Direchlet kernel in Fourier analysis. In Section
2 of this paper, we to study the zeros of this sum (c.f. Theorem \ref{thm:firsttheorem})
when the sequence of Chebyshev polynomials are replaced by a more
general sequence $\left\{ R_{n}(z)\right\} $ given in \eqref{eq:generalrecurrence}
where $A(z)$ and $B(z)$ are any linear polynomials with real coefficients.
The sequence of Chebyshev polynomials of the second kind $\left\{ U_{n}(z)\right\} $
satisfies the same recurrence as that of the first kind with the initial
condition $U_{0}(z)=1$ and $U_{1}(z)=2z$. This initial condition
can be written in the form $U_{0}(z)=1$ and $U_{-n}(z)=0$, $\forall n\in\mathbb{N}$.
In Section 3 of this paper, we study the zeros of a linear combination
of Chebyshev polynomials of the second kind whose coefficients are
power of $az+b$. In particular, we consider
\begin{equation}
Q_{n}(z)=\sum_{k=0}^{n}(az+b)^{k}U_{n-k}(z),\qquad a,b\in\mathbb{R}.\label{eq:linearcombCheb}
\end{equation}
We find necessary and sufficient conditions on $a$ and $b$ under
which the zeros of resulting polynomials are real (c.f. Theorem \ref{thm:thirdtheorem}).
\section{Sum of polynomials with three-term recurrence}
For $a_{1},b_{1},a_{2},b_{2}\in\mathbb{R}$, $a_{2}\ne0$, we let
$R_{n}(z)$ be the sequence of polynomials satisfying the recurrence
\[
R_{n+1}(z)=(a_{1}z+b_{1})R_{n}(z)+(a_{2}z+b_{2})R_{n-1}(z)
\]
with $R_{0}(z)=1$ and $R_{-n}(z)=0$, $\forall n\in\mathbb{N}$.
Equivalently the sequence $\left\{ R_{n}(z)\right\} _{n=0}^{\infty}$
is generated by
\[
\sum_{n=0}^{\infty}R_{n}(z)t^{n}=\frac{1}{1-(a_{1}z+b_{1})t-(a_{2}z+b_{2})t^{2}}.
\]
In this section, we study neccessary and sufficient conditions on
$a_{1}$, $b_{1}$, $a_{2}$, and $b_{2}$ under which all the zeros
of the polynomial
\[
\sum_{k=0}^{n}R_{n-k}(z)
\]
are real. Those polynomials form a sequence whose generating function
is
\begin{align*}
\sum_{n=0}^{\infty}\sum_{k=0}^{n}R_{k}(z)t^{n} & =\sum_{k=0}^{\infty}t^{k}\sum_{n=k}^{\infty}R_{n-k}(z)t^{n-k}\\
& =\frac{1}{(1-t)\left(1-(a_{1}z+b_{1})t-(a_{2}z+b_{2})t^{2}\right)}.
\end{align*}
With the substitutions $t$ by $-t$, $a_{2}$ by $-a_{2}$, and $b_{2}$
by $-b_{2}$, and then substitute $a_{2}z+b_{2}$ by $z$, we reduce
the generating function to the form
\[
\frac{1}{(t+1)((az+b)t^{2}+zt+1)}.
\]
Note that all the substitutions above preserve the reality of the
zeros of the generated sequence of polynomials. We state the main
theorem of this section.
\begin{thm}
\label{thm:firsttheorem} Let $a,b\in\mathbb{R}$. The zeros of all
the polynomials $P_{n}(z)$ generated by
\begin{equation}
\sum_{n=0}^{\infty}P_{n}(z)t^{n}=\frac{1}{(t+1)((az+b)t^{2}+zt+1)}\label{eq:genfuncPn}
\end{equation}
are real if and only if $b\geq1+2\left|a\right|.$ Under this condition
the zeros of $P_{n}(z)$ lie on
\begin{equation}
\left(2a-2\sqrt{a^{2}+b},2a+2\sqrt{a^{2}+b}\right)\label{eq:intPn}
\end{equation}
and are dense there as $n\rightarrow\infty$.
\end{thm}
\subsection{The sufficient condition}
We assume $b\ge1+2|a|$. To prove the zeros of $P_{n}(z)$ lie on
\eqref{eq:intPn}, we count the number of real zeros of $P_{n}(z)$
on this interval and show that this number is at least the degree
of this polynomial which is given by the lemma below.
\begin{lem}
\label{lem:degPn}For each $n\in\mathbb{N}$, the degree of $P_{n}(z)$
is at most $n$.
\end{lem}
\begin{proof}
We collect the coefficients in $t$ of the denominator of the right
side of \eqref{eq:genfuncPn} and obtain the recurrence
\begin{equation}
P_{n}(z)=-(z+1)P_{n-1}(z)-((a+1)z+b)P_{n-2}(z)-(az+b)P_{n-3}(z)\label{eq:recurrencePn}
\end{equation}
where $P_{0}(z)=1$ and $P_{-n}(z)=0$, $\forall n\in\mathbb{N}$.
The lemma follows from induction.
\end{proof}
To count the number of real zeros of $P(z)$, we construct two auxiliary
real-valued functions $z(\theta)$ and $\tau(\theta)$ on $\theta\in(0,\pi)$.
The first function is defined as
\begin{equation}
z(\theta)=2a\cos^{2}\theta-2\cos\theta\sqrt{a^{2}\cos^{2}\theta+b}.\label{eq:zthetadef}
\end{equation}
By the quadratic formula, $z(\theta)$ satisfies
\begin{equation}
z(\theta)^{2}-4az(\theta)\cos^{2}\theta-4b\cos^{2}\theta=0.\label{eq:zthetaeq}
\end{equation}
We will show later that there are $n$ values of $\theta\in(0,\pi)$,
each of which yields a zero of $P_{n}(z)$ on \eqref{eq:intPn} via
$z(\theta)$. The lemma below ensures a bijective correspondence between
$\theta$ and $z(\theta)$.
\begin{lem}
\label{lem:zmonotone}The function $z(\theta)$ is increasing on $(0,\pi)$
and it maps this interval onto the interval
\[
\left(2a-2\sqrt{a^{2}+b},2a+2\sqrt{a^{2}+b}\right).
\]
\end{lem}
\begin{proof}
To show $z(\theta)$ is increasing, we compute its derivative
\[
\frac{dz}{d\theta}=-4a\cos\theta\sin\theta+\frac{4a^{2}\cos^{2}\theta\sin\theta+2b\sin\theta}{\sqrt{a^{2}\cos^{2}\theta+b}}
\]
and see that it suffices to show
\[
2a^{2}\cos^{2}\theta+b>2\left|a\cos\theta\right|\sqrt{a^{2}\cos^{2}\theta+b}.
\]
The left side is positive and the squares of both sides reduce the
inequality to $b^{2}>0$, which shows that $z(\theta)$ is increasing.
We complete the lemma by computing the limits
\begin{align*}
\lim_{\theta\rightarrow0}z(\theta) & =2a-2\sqrt{a^{2}+b},\\
\lim_{\theta\rightarrow\pi}z(\theta) & =2a+2\sqrt{a^{2}+b}.
\end{align*}
\end{proof}
To define the second function $\tau(\theta)$, we need the following
lemma.
\begin{lem}
\label{lem:tauwelldef}For any $\theta\in(0,\pi)$, we have
\[
az(\theta)+b>0.
\]
\end{lem}
\begin{proof}
From Lemma \ref{lem:zmonotone}, it suffices to show that
\[
b+2a^{2}>2|a|\sqrt{a^{2}+b}.
\]
Since we know the left side is positive by $b\ge1+2|a|$, we obtain
the inequality above by squaring both sides.
\end{proof}
From Lemma \ref{lem:tauwelldef}, we define the functions
\begin{align*}
\tau(\theta) & =\frac{1}{\sqrt{az(\theta)+b}},\\
t_{1}(\theta) & =\tau(\theta)e^{-i\theta},\\
t_{2}(\theta) & =\tau(\theta)e^{i\theta},
\end{align*}
on $\theta\in(0,\pi)$.
\begin{lem}
\label{lem:zerosdenom}For any $\theta\in(0,\pi)$, the two zeros
of
\begin{equation}
(az(\theta)+b)t^{2}+z(\theta)t+1\label{eq:quadfactor}
\end{equation}
are $t_{1}(\theta)$ and $t_{2}(\theta)$.
\end{lem}
\begin{proof}
We verify that $\tau(\theta)e^{\pm i\theta}$ satisfy the Vieta's
formulas. Indeed, we have
\begin{equation}
t_{1}(\theta)t_{2}(\theta)=\tau(\theta)^{2}=\frac{1}{az(\theta)+b}\label{eq:prodelem}
\end{equation}
and
\begin{align*}
t_{1}(\theta)+t_{2}(\theta) & =2\tau(\theta)\cos\theta=\frac{2\cos\theta}{\sqrt{az(\theta)+b}}.
\end{align*}
From \eqref{eq:zthetadef}, we note that
\begin{equation}
z(\theta)\cos\theta<0\label{eq:signztheta}
\end{equation}
since $b>0$. As a consequence, we obtain
\begin{equation}
\frac{2\cos\theta}{\sqrt{az(\theta)+b}}=\frac{-z(\theta)}{az(\theta)+b}\label{eq:sumelemztheta}
\end{equation}
by squaring both sides and applying \eqref{eq:zthetaeq}.
\end{proof}
The lemma below shows that for each $\theta\in(0,\pi)$, the two zeros
of \eqref{eq:quadfactor} lie inside the unit ball.
\begin{lem}
\label{lem:taule1}For any $\theta\in(0,\pi)$, we have $|\tau(\theta)|<1$.
\end{lem}
\begin{proof}
From \eqref{eq:prodelem}, \eqref{eq:sumelemztheta}, and \eqref{eq:zthetadef},
it suffices to show
\[
\sqrt{a^{2}\cos^{2}\theta+b}>1+a\cos\theta.
\]
If the right side is negative, the inequality is trivial. If not,
we square both sides and the inequality follows from
\[
b\ge1+2|a|>1+2a\cos\theta.
\]
\end{proof}
For each $\theta\in(0,\pi)$, the Cauchy differentiation formula gives
\begin{align*}
P_{n}(z(\theta)) & =\frac{1}{2\pi i}\ointctrclockwise_{|t|=\epsilon}\frac{1}{(t+1)((az(\theta)+b)t^{2}+z(\theta)t+1)t^{n+1}}dt\\
& =\frac{1}{2\pi i}\ointctrclockwise_{|t|=\epsilon}\frac{1}{(az(\theta)+b)(t+1)(t-t_{1}(\theta))(t-t_{2}(\theta))t^{n+1}}dt.
\end{align*}
We recall that $az(\theta)+b\ne0$ by Lemma \ref{lem:tauwelldef}.
If we integrate the integrand over the circle $Re^{it}$, $0\le t\le2\pi$,
and let $R\rightarrow\infty$, then the integral approaches $0$.
Thus the sum of $P_{n}(z(\theta))$ and the residues of the integrand
at the three simple poles $-1$, $t_{1}(\theta)$ and $t_{2}(\theta)$
is $0$. We compute these residue and deduce that $-(az(\theta)+b)P_{n}(z(\theta))$
equals to
\[
\frac{(-1)^{n+1}}{(1+t_{1}(\theta))(1+t_{2}(\theta))}+\frac{1}{t_{1}(\theta)^{n+1}(1+t_{1}(\theta))(t_{1}(\theta)-t_{2}(\theta))}+\frac{1}{t_{2}^{n+1}(\theta)(1+t_{2}(\theta))(t_{2}(\theta)-t_{1}(\theta))}.
\]
We multiply this expression by $(1+t_{1}(\theta))(1+t_{2}(\theta))\tau(\theta)^{n+1}$,
which is nonzero $\forall\theta\in(0,\pi)$, and conclude $\theta$
is a zero of $P_{n}(z(\theta))$ if and only if it is a zero of
\[
(-1)^{n+1}\tau(\theta)^{n+1}+\frac{1+\tau(\theta)e^{i\theta}}{(\tau(\theta)e^{-i\theta}-\tau(\theta)e^{i\theta})e^{-i(n+1)\theta}}+\frac{1+\tau(\theta)e^{-i\theta}}{(\tau(\theta)e^{i\theta}-\tau(\theta)e^{-i\theta})e^{i(n+1)\theta}}
\]
or equivalently a zero of
\[
(-1)^{n+1}\tau(\theta)^{n+1}-\frac{\sin((n+1)\theta)/\tau(\theta)+\sin((n+2)\theta)}{\sin\theta}.
\]
With the trigonometric identity $\sin(n+2)\theta=\sin((n+1)\theta)\cos\theta+\cos((n+1)\theta)\sin\theta$,
we write the expression above as
\begin{equation}
(-1)^{n+1}\tau(\theta)^{n+1}-\cos((n+1)\theta)-\frac{\sin((n+1)\theta)(\cos\theta+1/\tau(\theta))}{\sin\theta}.\label{eq:functheta}
\end{equation}
We note that if
\[
\theta=\frac{k\pi}{n+1},\qquad1\le k\le n,
\]
then the sign of \eqref{eq:functheta} is $(-1)^{k+1}$ since $\tau(\theta)<1$
by Lemma \ref{lem:taule1}. By the intermediate value theorem, \eqref{eq:functheta}
has at least $n-1$ solution on $(\pi/(n+1),n\pi/(n+1))$. We also
note that as $\theta\rightarrow0$, the sign of \eqref{eq:functheta}
is negative since $\sin((n+1)\theta)/\sin\theta$ approaches $n+1$
and $\tau(\theta)<1$. Thus \eqref{eq:functheta} has another zero
on $(0,\pi/(n+1))$. From Lemma \ref{lem:zmonotone}, each zero in
$\theta$ of \eqref{eq:functheta} gives exactly one zero in $z$
of $P_{n}(z)$ on
\[
\left(2a-2\sqrt{a^{2}+b},2a+2\sqrt{a^{2}+b}\right).
\]
Thus all the zeros of $P_{n}(z)$ lie on the interval above by the
fundamental theorem of algebra and Lemma \ref{lem:degPn}. The density
of the zeros of $P_{n}(z)$ as $n\rightarrow\infty$ on this interval
follows directly from the density of the solutions of \eqref{eq:functheta}
and the continuity of $z(\theta)$.
\subsection{The necessary condition }
In this section, we will show that if either
\begin{enumerate}
\item $b\leq-1$ or
\item $-1<b<1+2|a|,$
\end{enumerate}
then not all polynomials $P_{n}(z)$ are hyperbolic. By \cite[Theorem 1.5]{sokal},
it suffices to find $z^{*}\in\mathbb{C\backslash\mathbb{R}}$ such
that the zeros of
\begin{equation}
(t+1)((az^{*}+b)t^{2}+z^{*}t+1)\label{eq:denomz*}
\end{equation}
are distinct and the two smallest (in modulus) zeros of this polynomial
have the same modulus. Note that every small neighborhood of such
$z^{*}$ will contain a zero of $P_{n}(z)$ for all large $n$ and
consequently $P_{n}(z)$ is not hyperbolic for all large $n$. For
more details on this application of the theorem, see \cite{tz}.
For the first case $b\le-1$, we let $\theta^{*}$ be any angle with
$a^{2}\cos^{2}\theta^{*}<-b$ and let $\tau^{*}$ be any zero of
\[
b\tau^{2}-2a\tau\cos\theta^{*}-1.
\]
Note that $\tau^{*}\notin\mathbb{R}$ since
\[
a^{2}\cos^{2}\theta^{*}+b<0
\]
and consequently $\tau^{*2}\notin\mathbb{R}$ by the definition of
$\tau^{*}$. With the note that $2a\tau^{*}\cos\theta^{*}+1$ is nonreal
(and thus nonzero), we choose
\[
z^{*}=\frac{-2b\tau^{*}\cos\theta^{*}}{2a\tau^{*}\cos\theta^{*}+1}
\]
which is nonreal since $1/z^{*}\notin\mathbb{R}$. From the definitions
of $\tau^{*}$, $\theta^{*}$, and $z^{*}$ above, the two solutions
of
\[
(az^{*}+b)t^{2}+z^{*}t+1=0
\]
are $\tau^{*}e^{\pm i\theta^{*}}$ since they satisfy the Vieta's
formulas
\[
\tau^{*2}=\frac{1}{az^{*}+b}
\]
and
\[
2\tau^{*}\cos\theta^{*}=-\frac{z^{*}}{az^{*}+b}.
\]
Since $\tau^{*}$ and $\overline{\tau^{*}}$ are solutions of $b\tau^{2}-2a\tau\cos\theta^{*}-1$,
we have $\tau^{*}\overline{\tau^{*}}=|\tau^{*}|^{2}=-1/b\le1$. Thus
the two smallest (in modulus) zeros of \eqref{eq:denomz*} equal in
modulus and we complete the case $b\le-1$.
We now consider the case $-1<b<1+2|a|$. We will find $z^{*}\notin\mathbb{R}$
so that the smaller (in modulus) zero of $(az^{*}+b)t^{2}+z^{*}t+1$
lie on the unit circle. The inequality $\left|2|a|-b|\right|<1$ implies
that
\[
1+2|a|>|b|>|b|.|2|a|-b|
\]
and consequently
\[
1-b^{2}+2|a|+2b|a|>0.
\]
We conclude there is $\theta^{*}\in(0,\pi)$ sufficiently close to
$0$ when $a\ge0$ or close to $\pi$ when $a<0$ such that
\[
b^{2}-2ab\cos\theta^{*}<1+2a\cos\theta^{*}.
\]
With this choice of $\theta^{*}$, we have
\begin{equation}
\frac{|be^{i\theta^{*}}-a|}{|ae^{i\theta^{*}}+1|}=\frac{b^{2}+a^{2}-2ab\cos\theta}{a^{2}+1+2a\cos\theta}<1.\label{eq:ineqtheta*}
\end{equation}
We define
\[
z^{*}=\frac{-1-be^{2i\theta^{*}}}{ae^{2i\theta^{*}}+e^{i\theta^{*}}}
\]
and write
\begin{equation}
(az^{*}+b)t^{2}+z^{*}t+1\label{eq:quaddenomz*}
\end{equation}
as $z^{*}(at^{2}+t)+bt^{2}+1$ to conclude that $e^{i\theta^{*}}$
is a zero of this polynomial. Since the product of the two zeros of
this polynomial is $1/(az^{*}+b)$, we claim that the other zero of
this polynomial is more than $1$ in modulus by showing that
\[
\frac{1}{|az^{*}+b|}>1.
\]
Indeed, from the definition of $z^{*}$, this inequality is equivalent
to \eqref{eq:ineqtheta*}. We note that $z^{*}\notin\mathbb{R}$ since
a solution of \eqref{eq:quaddenomz*} is $e^{i\theta^{*}}\notin\mathbb{R}$
and the other solution is more than $1$ in modulus.
\section{Linear combination of chebyshev polynomials}
The goal this section is to study necessary and sufficient conditions
under which the zeros of \eqref{eq:linearcombCheb} are real. The
sequence $\left\{ Q_{n}(z)\right\} $ in \eqref{eq:linearcombCheb}
is generated by
\begin{align*}
\sum_{n=0}^{\infty}Q_{n}(z)t^{n} & =\sum_{n=0}^{\infty}\sum_{k=0}^{n}(az+b)^{k}U_{n-k}(z)t^{n}\\
& =\sum_{k=0}^{\infty}(az+b)^{k}t^{k}\sum_{n=k}^{\infty}U_{n-k}(z)t^{n-k}\\
& =\frac{1}{(1+(az+b)t)(1-2zt+t^{2})}.
\end{align*}
With the substitution $z$ by $-z/2$ and then $-a/2$ by $a$, it
suffice to study the hyperbolicity of the sequence generated of polynomials
by
\[
\frac{1}{(1+(az+b)t)(1+zt+t^{2})}.
\]
As a small digression of the main goal, we will prove following theorem
which states that the positivity of the $t^{2}$- coefficient in the
factor $1+zt+t^{2}$ is important to ensure the hyperbolicity of the
generated sequence of polynomials.
\begin{thm}
\label{thm:secondtheorem} Suppose $a,b,c\in\mathbb{R}$ where $c\ne0$.
If $c\le0$, then not all the polynomials $P_{n}(z)$ generated by
\[
\frac{1}{((az+b)t+1)(ct^{2}+zt+1)}.
\]
are hyperbolic.
\end{thm}
We note that if $c=0$, the sequence of generated polynomials satisfy
a three-term recurrence and their zeros have been studied in \cite{tran}.
Under the condition $c>0$, with the substitution $t\rightarrow t/\sqrt{c}$,
we can assume $c=1$. The following theorem settles the necessary
and sufficient conditions for the hyperbolicity of \eqref{eq:linearcombCheb}.
\begin{thm}
\label{thm:thirdtheorem} Suppose $a,b\in\mathbb{R}$. The zeros of
all the polynomials $P_{n}(z)$ generated by
\begin{equation}
\sum_{n=0}^{\infty}P_{n}(z)t^{n}=\frac{1}{((az+b)t+1)(t^{2}+zt+1)}.\label{eq:genfuncthirdthm}
\end{equation}
are real if and only if $\left|b\right|\leq1-2\left|a\right|$. Moreover
when $|b|\le1-2|a|$, the zeros of $P_{n}(z)$ lies on $(-2,2)$ and
are dense there as $n\rightarrow\infty$.
\end{thm}
\subsection{Proof of Theorem \ref{thm:secondtheorem}}
In the case $c<0$, with the substitution $t\rightarrow t/\sqrt{|c|}$,
it suffices to show that for any $a,b\in\mathbb{R}$, not all the
polynomials generated by
\[
\frac{1}{((az+b)t+1)(-t^{2}+zt+1)}
\]
are hyperbolic. Recall a consequence of \cite[Theorem 1.5]{sokal}
that we will need to find $z^{*}\notin\mathbb{R}$ so that the two
smallest zeros of
\[
((az^{*}+b)t+1)(-t^{2}+z^{*}t+1)
\]
equal in modulus.
In the case $|b|<1$, we choose $z^{*}=iy^{*}$ where
\[
0<y^{*}<\min\left(\frac{\sqrt{1-b^{2}}}{|a|},2\right)
\]
if $a\ne0$ and $0<y^{*}<2$ if $a=0$. The two zeros of $-t^{2}+z^{*}t+1$,
\[
\frac{iy^{*}\pm\sqrt{4-y^{*2}}}{2}
\]
lie on the unit circle and thus their modulus is less than
\[
\frac{1}{|az^{*}+b|}=\frac{1}{\sqrt{a^{2}y^{*2}+b^{2}}}.
\]
For the remainder of Section 3.1, we assume $|b|\ge1$. To make a
suitable choice for $z^{*}$, we consider the following lemma.
\begin{lem}
\label{lem:theta*existence}With the principal cut, there exists $\theta^{*}\ne k\pi$,
$k\in\mathbb{Z}$, such that
\[
|b|+\sqrt{b^{2}+4a^{2}-4ae^{i\theta^{*}}}\geq\left|2a-2e^{i\theta^{*}}\right|.
\]
\end{lem}
\begin{proof}
We note that $b^{2}+4a^{2}\ge4|a|$ since
\[
4|a|(1-|a|)\le1\le|b|.
\]
Thus with the principle cut, the function
\[
f(z):=\frac{|b|+\sqrt{b^{2}+4a^{2}-4az}}{2a-2z}
\]
is meromorphic on the open unit ball with the possible pole at $z=a$
if $|a|<1.$ To prove this lemma, we will find $z\notin\mathbb{R}$
and $|z|=1$ such that $|f(z)|\ge1$.
We note that if $|a|\ge1$, then $f(z)$ is analytic on the unit ball
and
\[
|f(0)|=\frac{|b|+\sqrt{b^{2}+4a^{2}}}{2|a|}>1.
\]
Thus by the maximum modulus principle $|f(z)|>1$ for some $|z|=1$.
We can choose such $z\notin\mathbb{R}$ by the continuity of $f(z)$.
On the other hand if $|a|<1$, then the Cauchy integral formula implies
that
\[
\ointctrclockwise_{|z|=1}|f(z)||dz|\ge\left|\ointctrclockwise_{|z|=1}f(z)dz\right|=2\pi|b|\ge2\pi.
\]
Consequently $|f(z)|>1$ for some $|z|=1$ or $|f(z)|=1$ for all
$|z|=1$ and the lemma follows.
\end{proof}
We now define
\[
z^{*}=\frac{-2ab+be^{i\theta^{*}}+\sign(b)e^{i\theta^{*}}\sqrt{b^{2}+4a^{2}-4ae^{i\theta^{*}}}}{2a^{2}-2ae^{i\theta^{*}}}
\]
where $\theta^{*}$ is given in Lemma \ref{lem:theta*existence}.
With this definition, $z^{*}$ is a solution of
\[
(a^{2}-ae^{i\theta^{*}})z^{2}+(2ab-be^{i\theta^{*}})z+b^{2}-e^{2i\theta^{*}}=0
\]
from which we deduce that
\begin{equation}
-\frac{e^{i\theta^{*}}}{az^{*}+b}=-\frac{2a-2e^{i\theta^{*}}}{-b+\sign(b)\sqrt{b^{2}+4a^{2}-4ae^{i\theta^{*}}}}\label{eq:firstzero}
\end{equation}
is a zero in $t$ of
\[
-t^{2}+z^{*}t+1.
\]
The modulus of \eqref{eq:firstzero} is the same as the modulus of
the zero in $t$ of $(az^{*}+b)t+1$ which is at most $1$ by the
definition of $\theta^{*}$. This modulus is larger than the modulus
of the other zero of $-t^{2}+z^{*}t+1$ since the product of two zeros
of this polynomial is $-1$. We finish the proof of Theorem \ref{thm:secondtheorem}
by noting that $z^{*}\notin\mathbb{R}$ since the two zeros of $-t^{2}+z^{*}t+1$
are neither real nor complex conjugate.
\subsection{Proof of Theorem \ref{thm:thirdtheorem}}
\subsubsection{The sufficient condition}
Let $\left\{ P_{n}(z)\right\} $ be the sequence of polynomials defined
in \eqref{eq:genfuncthirdthm} where $\left|b\right|\leq1-2\left|a\right|$.
The proof of the following lemma is the same as that of Lemma 4 in
\cite{tz}. For brevity, we omit the proof in this paper.
\begin{lem}
\label{lem:adensity}For each $b\in[-1,1],$ let $S_{b}$ be a dense
subset of
\begin{equation}
\left[\frac{|b|-1}{2},\frac{1-|b|}{2}\right]\label{eq:ainterval}
\end{equation}
and $n\in\mathbb{N}$ be fixed. If for any $a\in S_{b}$, the zeros
of $P_{n}(z)$ lie on $(-2,2)$, then the same conclusion holds for
any $a$ in \eqref{eq:ainterval} .
\end{lem}
Suppose $\left|b\right|\leq1-2\left|a\right|$. From Lemma \ref{lem:adensity},
it suffices to consider $a\ne0$. We define the monotone function
$z(\theta)=-2\cos\theta$ on $(0,\pi)$ and note that for each $\theta\in(0,\pi)$
the two zeros of $t^{2}+z(\theta)t+1$ are $e^{\pm i\theta}$. We
consider the function
\[
t_{0}(\theta)=\frac{-1}{az(\theta)+b},\qquad\theta\in(0,\pi),
\]
which has a vertical asymptote at $\theta=\cos^{-1}(b/2a)$ if $|b|<2|a|$.
For any $\theta\in(0,\pi)$ such that $2a\cos\theta\ne b$, the Cauchy
differentiation formula gives
\[
P_{n}(z(\theta))=\frac{1}{az(\theta)+b}\ointctrclockwise_{|t|=\epsilon}\frac{dt}{(t-t_{0}(\theta))(t-e^{i\theta})(t-e^{-i\theta})t^{n+1}}.
\]
After computing the residue of the integrand at the three nonzero
simple poles $t_{0}(\theta),e^{\pm i\theta}$, and letting the radius
of the integral approach infinity, we apply similar computations in
\eqref{eq:functheta} to conclude that $\theta\in(0,\pi)$, $2a\cos\theta\ne b$,
is a zero of $P_{n}(z(\theta))$ if and only if it is a zero of
\begin{equation}
\frac{-1}{t_{0}(\theta)^{n+1}}+\cos\left((n+1)\theta\right)+\frac{(\cos\theta-t_{0}(\theta))\sin\left((n+1)\theta\right)}{\sin\theta}.\label{eq:secondfunctheta}
\end{equation}
From Lemma \ref{lem:adensity}, it suffices to consider $|b|\ne2|a|$.
We note that the limits of \eqref{eq:secondfunctheta} as $\theta\rightarrow0$
and $\theta\rightarrow\pi$ are
\begin{equation}
n+2+\frac{n+1}{b-2a}+(-1)^{n}(b-2a)^{n+1}\label{eq:leftlimit}
\end{equation}
and
\begin{equation}
(-1)^{n+1}(n+2)+(-1)^{n}\left(\frac{n+1}{b+2a}+(b+2a)^{n+1}\right)\label{eq:rightlimit}
\end{equation}
respectively.
In the case $|b|>2|a|$, \eqref{eq:secondfunctheta} is a continuous
function of $\theta$ on $(0,\pi)$ and its sign at $\theta=k\pi/(n+1)$,
for $1\le k\le n$, is $(-1)^{k}$ since
\[
|t_{0}(\theta)|>\frac{1}{2|a|+|b|}\ge1.
\]
By the intermediate value theorem, we obtain at least $n-1$ zeros
of \eqref{eq:secondfunctheta} on $(\pi/(n+1),n\pi/(n+1))$. If $b>0$,
then \eqref{eq:leftlimit} is positive since $0<b-2a\le1$ and we
obtain at least another zero of \eqref{eq:secondfunctheta} on $(0,\pi/(n+1))$.
On the other hand, if $b<0$, then the inequalities
\[
-1<b+2a<0
\]
imply that the sign of \eqref{eq:rightlimit} is $(-1)^{n+1}$ and
we have at least another zero of \eqref{eq:secondfunctheta} on $(n\pi/(n+1),\pi)$.
We conclude that when $|b|>2|a|$, \eqref{eq:secondfunctheta} has
at least $n$ zeros on $(0,\pi)$, each of which yields a zero of
$P_{n}(z)$ on the interval $(-2,2)$ by the map $z(\theta)$. Thus
all the zeros of $P_{n}(z)$ lie on $(-2,2)$ by the fundamental theorem
of algebra.
We now consider the case $|b|<2|a|$. As a function of $\theta$ on
$(0,\pi)$, \eqref{eq:secondfunctheta} has a vertical asymptote at
$\theta=\cos^{-1}(b/2a)$ since $t_{0}(\theta)$ does. By Lemma \ref{lem:adensity},
we can assume
\[
\cos^{-1}\frac{b}{2a}\ne\frac{k\pi}{n+1},\qquad1\le k\le n.
\]
Thus for some $0\le k_{0}\le n$, the open interval
\begin{equation}
\left(\frac{k_{0}}{n+1}\pi,\frac{k_{0}+1}{n+1}\pi\right)\label{eq:asympint}
\end{equation}
contains $\cos^{-1}\left(b/2a\right)$. We note that this interval
may or may not contain a zero of \eqref{eq:secondfunctheta}. In the
case $a<0$, we observe that \eqref{eq:leftlimit} is positive and
the sign of \eqref{eq:rightlimit} is $(-1)^{n+1}$. Thus there are
at least $n$ zeros of \eqref{eq:secondfunctheta} on the $n$ intervals
$(k\pi/(n+1),(k+1)\pi/(n+1))$, for $0\le k\le n$ and $k\ne k_{0}$
and we conclude all the zeros of $P_{n}(z)$ lie on $(-2,2)$ by the
same argument in the previous case. On the other hand, if $a>0$,
then the limits \eqref{eq:secondfunctheta} as $\theta$ approaches
the left and right of $\cos^{-1}(b/2a)$ are
\[
\lim_{\theta\to\cos^{-1}(b/2a)^{-}}\frac{\sin((n+1)\theta)}{b-2a\cos(\theta)}=(-1)^{k_{0}+1}\infty
\]
and
\[
\lim_{\theta\to\cos^{-1}(b/2a)^{+}}\frac{\sin((n+1)\theta)}{b-2a\cos(\theta)}=(-1)^{k_{0}}\infty,
\]
respectively. If $k_{0}\ne0$ and $k_{0}\ne n$, then we conclude
that \eqref{eq:asympint} contains at least two zeros of \eqref{eq:secondfunctheta}.
Thus we obtain at least $n$ zeros of this expression on the $n-1$
intervals $(k\pi/(n+1),(k+1)\pi/(n+1))$, for $1\le k<n$. In the
case $k_{0}=0$ or $k_{0}=n$, \eqref{eq:asympint} contains at least
one zero of \eqref{eq:secondfunctheta} and thus there are at least
$n$ zeros of \eqref{eq:secondfunctheta} on the $n$ intervals $(k\pi/(n+1),(k+1)\pi/(n+1))$,
for $1\le k<n$ and $k=k_{0}$.
\subsubsection{The necessary condition}
In this section, we assume $|b|+2|a|>1$ and show that not all zeros
of $P_{n}(z)$ defined in \eqref{eq:genfuncthirdthm} are real when
$n$ is large. From \cite[Theorem 1.5]{sokal} , it suffices find
$z\notin\mathbb{R}$ so that $|t_{0}|=|t_{1}|\le|t_{2}|$ where
\begin{equation}
t_{0}:=-\frac{1}{az+b}\label{eq:t0choice}
\end{equation}
and $t_{1}$ and $t_{2}$ are the two zeros of $1+zt+t^{2}$. To motivate
the choice of $z$, we provide heuristic arguments by noticing that
$t_{1}t_{2}=1$ and letting
\begin{equation}
t_{1}=t_{0}e^{i\theta}=-\frac{e^{i\theta}}{az+b}\label{eq:t1choice}
\end{equation}
\begin{equation}
t_{2}=-e^{-i\theta}(az+b).\label{eq:t2choice}
\end{equation}
The equation $1+zt_{2}+t_{2}^{2}=0$ yields
\[
(az+b)^{2}-ze^{i\theta}(az+b)+e^{2i\theta}=0
\]
or equivalently
\begin{multline}
(a^{2}-ae^{i\theta})z^{2}+(2ab-be^{i\theta})z+b^{2}+e^{2i\theta}=0.\label{eq:zquadratic}
\end{multline}
With a choice of branch cut which will be specified later, the equation
above has two solutions
\[
z=\frac{-2ab+be^{i\theta}\pm e^{i\theta}\sqrt{b^{2}-4a^{2}+4ae^{i\theta}}}{2a^{2}-2ae^{i\theta}}
\]
and the corresponding values for $az+b$ are
\begin{equation}
az+b=\frac{-be^{i\theta}\pm e^{i\theta}\sqrt{b^{2}-4a^{2}+4ae^{i\theta}}}{2a-2e^{i\theta}}.\label{eq:az+b}
\end{equation}
For a formal proof of the necessary condition, we consider the following
cases.
\textbf{Case 1: $|a|\le1$.} We have the inequality
\[
b^{2}-4a^{2}+4|a|-(|b|+2|a|-2)^{2}=4(1-|a|)(2|a|+|b|-1)\ge0.
\]
with equality if and only if $|a|=1$. This implies
\begin{equation}
b^{2}-4a^{2}+4|a|\ge0\label{eq:firstineqabale1}
\end{equation}
and
\begin{align}
\sqrt{b^{2}-4a^{2}+4|a|}+|b| & \ge\left||b|+2|a|-2\right|+|b|\nonumber \\
& \ge|2|a|-2|\label{eq:secineqabale1}
\end{align}
with equality if and only if $|a|=1$ and $b=0$. We define $\theta\in(0,\pi)$
sufficiently close to $0$ or $\pi$ such that $e^{i\theta}$ is close
to $\sign a$ if $a\ne0$. If $a=0$, we pick any $\theta\in(0,\pi)$.
With this choice of $\theta$ and the principal cut, we let
\begin{equation}
z=\begin{cases}
\frac{-2ab+be^{i\theta}-\sign b.e^{i\theta}\sqrt{b^{2}-4a^{2}+4ae^{i\theta}}}{2a^{2}-2ae^{i\theta}} & \text{ if }ab\ne0,\\
\frac{ie^{i\theta}}{\sqrt{a^{2}-ae^{i\theta}}} & \text{ if }b=0,\\
\frac{b^{2}+e^{2i\theta}}{be^{i\theta}} & \text{if }a=0.
\end{cases}\label{eq:zchoiceale1}
\end{equation}
With this choice of $z$, \eqref{eq:zquadratic} holds and consequently
$t_{1}$ and $t_{2}$ defined in \eqref{eq:t1choice} and \eqref{eq:t2choice}
are the zeros of $1+zt+t^{2}$. If $a=0$, then
\begin{equation}
|t_{0}|=|t_{1}|<|t_{2}|\label{eq:t0t1t2}
\end{equation}
since $|b|>1$. If $b=0$ then the inequalities $|a|\le1$ and \eqref{eq:firstineqabale1}
imply that $|a|=1$. As a consequence, \eqref{eq:t0t1t2} follows
from \eqref{eq:t0choice}, \eqref{eq:t1choice}, \eqref{eq:t2choice},
and \eqref{eq:zchoiceale1}. Finally, if $ab\ne0$, then from \eqref{eq:az+b}
and \eqref{eq:secineqabale1}, we conclude $|az+b|$ approaches
\[
\frac{|b|+\sqrt{b^{2}-4a^{2}+4|a|}}{2-2|a|}>1
\]
as $e^{i\theta}\rightarrow\sign(a)$. Thus from \eqref{eq:t1choice}
and \eqref{eq:t2choice} there is $\theta\in(0,\pi)$ sufficiently
close to $0$ or $\pi$ such that
\[
|t_{0}|=|t_{1}|<|t_{2}|.
\]
We also note that $z\notin\mathbb{R}$ since if $z\in\mathbb{R}$,
then the fact that $t_{1},t_{2}\notin\mathbb{R}$ by \eqref{eq:t1choice}
and \eqref{eq:t2choice} implies $t_{1}=\overline{t_{2}}$ which contradicts
to $|t_{1}|<|t_{2}|$.
\textbf{Case 2: $|a|>1$ and $|b|<1$.} By the intermediate value
theorem there is $y\in(0,\infty)$ such that
\[
2\sqrt{a^{2}y^{2}+b^{2}}-\sqrt{y^{2}+4}-y=0
\]
since the the left side is $2|b|-2<0$ when $y=0$ and its limit is
$\infty$ when $y\rightarrow\infty$. With the choice $z=iy$, we
have
\[
|t_{0}|=\frac{1}{|az+b|}=\frac{1}{\sqrt{a^{2}y^{2}+b^{2}}}
\]
and the modulus of the smaller zero of $t^{2}+iyt+1$ is
\[
\frac{\sqrt{y^{2}+4}-y}{2}=\frac{2}{\sqrt{y^{2}+4}+y}=|t_{0}|.
\]
\textbf{Case 3: $|b|\geq1$ and $|a|>1$.} If $2+|b|>2|a|$, then
with the same choice of $\theta$ and $z$ and the same argument as
in the first case, this case follows from
\[
|\sqrt{b^{2}-4a^{2}+4ae^{i\theta}}+|b||>|b|>2|a|-2.
\]
We now consider\textbf{ $2+|b|\leq2|a|$.} We square both sides of
$2|a|-2\ge|b|$ to obtain
\[
b^{2}-4a^{2}\le4-8|a|<-4|a|
\]
which implies that, with the cut $[0,\infty)$, the function
\[
f(z):=\frac{-b+\sqrt{b^{2}-4a^{2}+4az}}{2a-2z}
\]
is analytic on a small region containing the closed unit ball. From
the maximum modulus principle and the fact that
\begin{align*}
|f(0)| & =\frac{|-b+\sqrt{b^{2}-4a^{2}}|}{|2a|}\\
& =1,
\end{align*}
we conclude there is $\theta\in\mathbb{R}$ so that $|f(e^{i\theta})|>1$.
With this $\theta$, we let
\[
z=\frac{-2ab+be^{i\theta}+e^{i\theta}\sqrt{b^{2}-4a^{2}+4ae^{i\theta}}}{2a^{2}-2ae^{i\theta}}
\]
and apply \eqref{eq:t0choice}, \eqref{eq:t1choice}, \eqref{eq:t2choice},
and \eqref{eq:az+b} to conclude $|t_{0}|=|t_{1}|<|t_{2}|$. The fact
that $z\notin\mathbb{R}$ follows from the same argument in the previous
case.
|
2,877,628,090,113 | arxiv | \section{Introduction}
This paper is a continuation of previous work by the first named author \cite{AriasAbad2019}, where an infinitesimal description of the category of modules over the algebra of singular chains on a Lie group is presented. Our main result is that in the compact case, the correspondence of \cite{AriasAbad2019} can be promoted to an $\mathsf{A}_\infty$ quasi-equivalence of DG categories.
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and denote by ${\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)$ the space of smooth singular chains on $G$. This space carries the structure of a DG Hopf algebra, where the product is induced by the Eilenberg-Zilber map and the coproduct by the Alexander-Whitney map. We write $\operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ for the category of sufficiently smooth modules over this DG Hopf algebra. We also denote by $\TT \mathfrak{g}$ the DG Lie algebra which is universal for the Cartan relations on $\mathfrak{g}$, and write $\operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$ for the corresponding category of representations. The main result of \cite{AriasAbad2019} is the following.
\begin{theoremO}
Suppose that $G$ is a simply connected Lie group. There exists a differentiation functor \linebreak $\mathscr{D} \colon {\operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)) \to \operatorname{\mathbf{Rep}}(\TT \mathfrak{g})}$ and an integration functor $\mathscr{I} \colon \operatorname{\mathbf{Rep}}(\TT \mathfrak{g}) \to \operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ which are inverses to one another. In particular, the categories $\operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ and $\operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$ are equivalent as symmetric monoidal categories.
\end{theoremO}
Let us now explain the content of the present work.
The category $\operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ admits an enhancement to a DG category, which we shall denote by $\operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$, and whose spaces of morphisms are Hochschild complexes for ${\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)$. Similarly, the category $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$ may be enhanced to a DG category, which we denote by $\operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})$, and whose spaces of morphisms are constructed in terms of the Weil algebra of $\mathfrak{g}$. Our main result is the following.
\begin{theoremA}
Suppose that $G$ is compact and simply connected. There exists a zig-zag of $\mathsf{A}_{\infty}$-quasi-equivalences that connects $\operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})$ to $\operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$. In particular, the DG categories $\operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})$ and $\operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ are $\mathsf{A}_{\infty}$-quasi-equivalent.
\end{theoremA}
For the proof we introduce an intermediate DG category $\operatorname{\mathbf{BSS}}(G)$ and an ivariant version of it $\operatorname{\mathbf{BSS}}^G(G)$, whose morphism spaces are defined by twisting the Bott-Shulman-Stasheff DG algebra introduced in \cite{BSS}. The following diagram, where each arrow represents an $\mathsf{A}_\infty$ equivalence of DG categories, summarizes the structure of the paper:
\[
\xymatrix{
\operatorname{\mathbf{DGRep}}(\TT\mathfrak{g}) && \operatorname{\mathbf{BSS}}(G)\ar[rd]&\\
&\ar[ul] \ar[ur]\operatorname{\mathbf{BSS}}^G(G)&& \operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)).
}\]
The second arrow is an inclusion of categories which is a quasi-equivalence when $G$ is compact.\\
The comparison between $\operatorname{\mathbf{BSS}}^G(G)$ and $\operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})$
uses the Van Est map \cite{VanEst1953, AriasAbad-Crainic2011,Li-Bland-Meinrenken2015,Crainic2003,Xu-Weinstein1991,meinrenken2019van} and the noncommutative Weil algebra of Alekseev and Meinrenken \cite{Alekseev-Meinrenken2005,alekseev2000non}. We prove the following results.
\begin{theoremB}
Let $G$ be a compact and simply connected Lie group. There exists a DG functor \[\mathscr{VE} \colon \operatorname{\mathbf{BSS}}^{G}(G) \to \operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})\]which is a quasi-equivalence.
\end{theoremB}
In order to compare $\operatorname{\mathbf{BSS}}(G)$ and $\operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$, we construct an $\mathsf{A}_\infty$ quasi-isomorphism between the Bott-Shulman-Stasheff algebra and the algebra of Hochschild cochains on singular chains on $G$.
\begin{theoremC}\label{thm:3.17aa}
Let $G$ be a Lie group. There is an $\mathsf{A}_{\infty}$-morphism \[{\sf{DR}}^{\Theta} \colon \operatorname{Tot} (\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})) \to \operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))\]
which is a quasi-isomorphism.
\end{theoremC}
This construction uses Chen's iterated integrals and combines a version of Gugenheim's $\mathsf{A}_{\infty}$ De~Rham's theorem for the classifying space $BG$ with the Eilenberg-Zilberg map.
\begin{theoremD}
Let $G$ be a Lie group. There exists an $\mathsf{A}_{\infty}$-functor
$$
\mathscr{DR} \colon \operatorname{\mathbf{BSS}}(G) \longrightarrow \operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)),
$$
which is an $\mathsf{A}_{\infty}$-quasi-equivalence of DG categories.
\end{theoremD}
The $\mathsf{A}_{\infty}$-quasi-equivalence of DG categories between $\operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})$ and $\operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ may be interpreted in terms of Chern-Weil theory for $\infty$-local systems on the classifying space $BG$, as studied in \cite{Block-Smith2014,Rivera-Zeinalian2018,Holstein2015,Abad-Schatz2016,AriasAbad-QuinteroVelez-VelezVasquez2019,brav2016relative}. Indeed, $\infty$-local systems on a topological space $X$ are described as objects of the DG category $\operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(\Omega X))$, where $\Omega X$ denotes the Moore based loop space of $X$. In case $X$ is $BG$, the monoid of Moore loops on $BG$ is $\mathsf{A}_\infty$ equivalent to $G$. Thus, the DG category $\operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ can be thought of as that which parametrises $\infty$-local systems on $BG$. The $\mathsf{A}_{\infty}$-quasi-equivalence between $\operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})$ and $\operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ is then an extension of the Chern-Weil computation of the cohomology of $BG$ for the trivial $\infty$-local system to the case of arbitrary $\infty$-local systems. The explicit construction of a Chern-Weil DG functor categorifying the Chern-Weil homomorphism is the subject of a forthcoming work \cite{AriasAbad-Pineda-QuinteroVelez2020}.
The structure of the paper is as follows. In Section~\ref{sec:2}, we collect some preliminaries on DG categories, Hochschild complexes, Gugenheim's $\mathsf{A}_{\infty}$ De~Rham theorem, the Alexander-Whitney and Eilenberg-Zilber maps, representations of the DG Lie algebra $\TT \mathfrak{g}$, and the main result of \cite{AriasAbad2019} concerning the equivalence between $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$ and $\operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$. Section~\ref{sec:3} is devoted to the study of the properties of the Van~Est map that are used in the proof of our main results, and the construction of the $\mathsf{A}_{\infty}$-quasi-isomorphism between the Bott-Shulman-Stasheff DG algebra $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ and the DG algebra of Hochschild cochains $\operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$. We conclude in Section~\ref{sec:4} with a discussion of the DG enhanced categories $\operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})$ and $\operatorname{\mathbf{BSS}}(G)$ and the proof of Theorems~B and C, which together imply our main result, Theorem~A. We also present some examples.
\begin{acknowledgements}
We would like to acknowledge the support of Colciencias through their grant ``Estructuras lineales en topolog\'ia y geometr\'ia'', with contract number FP44842-013-2018. We also thank the Alexander von Humboldt foundation which supported our work through the Humboldt Institutspartnerschaftet ``Representations of Gerbes and higher holonomies''. We are grateful to Anders Kock for pointing us to his book \cite{Kock2006}, where the image of the De~Rham map is described. We would like to thank Konrad Waldorf for his hospitality during a visit to Greifswald, where part of this work was completed. We are also grateful to Manuel Rivera for several conversations related to this work.
\end{acknowledgements}
\subsection*{Notation and Conventions}
All vector spaces and algebras are defined over the field of real numbers $\ensuremath{\mathbbmss{R}}$. If $V = \bigoplus_{k \in \ZZ} V^k$ is a graded vector space, we denote by ${\sf s} V$ its suspension, that is, the graded vector space with grading defined by
$$
({\sf s} V)^{k} = V^{k+1},
$$
and by ${\sf u} V$ its unsuspension, that is, the graded vector space with grading defined by
$$
({\sf u} V)^{k} = V^{k-1}.
$$
All our complexes will be cochain complexes, meaning that the differentials increase the degree by one. For each $n \geq 1$, we write $\Delta_n$ for the standard $n$-simplex. The geometric realisation of $\Delta_n$ that we take is
$$
\Delta_n = \left\{ (t_1, \dots, t_{n}) \in \ensuremath{\mathbbmss{R}}^{n} \mid 1 \geq t_1 \geq \cdots \geq t_n \geq 0 \right\}.
$$
If $M$ is a smooth manifold, we respectively denote by $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M)$, ${\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M)$ and ${\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M)$ the spaces of differential forms, smooth singular cochains and smooth singular chains defined on $M$.
\section{Preliminaries}\label{sec:2}
In this section, we review the basic definitions and results that will be needed in the sequel, in an attempt at making our paper as self-contained as possible. For a more detailed exposition on some of the topics covered in Sections \ref{sec:2.1}, \ref{sec:2.2}, \ref{sec:2.4} and \ref{sec:2.5}, the reader may consult \cite{Seidel2008}, \cite{Maclane1963} and \cite{Meinrenken2013}.
\subsection{Hochschild chain and cochain complexes}\label{sec:2.1}
Let $A$ be a DG algebra and let $M$ be a DG bimodule over $A$. The \emph{Hochschild chain complex} of $A$ with values in $M$ is the graded vector space
\begin{equation}
\operatorname{HC}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A,M) = \bigoplus_{n \geq 0}M \otimes ({\sf u} A)^{\otimes n},
\end{equation}
equipped with a differential $b$ which is the sum of two components $b_1$ and $b_2$ defined by the formulas
\begin{align}
\begin{split}
&b_1(m \otimes {\sf u} a_1 \otimes \cdots \otimes {\sf u} a_n) = d_M m \otimes {\sf u} a_1 \otimes \cdots \otimes {\sf u} a_n \\
&\qquad + \sum_{i=1}^{n} (-1)^{\vert m \vert + \sum_{j=1}^{i-1}\vert a_j \vert - i} m \otimes {\sf u} a_1 \otimes \cdots \otimes {\sf u} a_{i-1} \otimes {\sf u} d a_i \otimes {\sf u} a_{i+1} \otimes \cdots \otimes {\sf u} a_n,
\end{split}
\end{align}
and
\begin{align}
\begin{split}
& b_2 (m \otimes {\sf u} a_0 \otimes \cdots \otimes {\sf u} a_n) = (-1)^{\vert m \vert + 1} m a_1 \otimes {\sf u} a_2 \otimes \cdots \otimes {\sf u} a_n \\
&\qquad +\sum_{i=1}^{n-1}(-1)^{\vert m \vert + \sum_{j=1}^{i}\vert a_j \vert - i+1} m \otimes {\sf u} a_1 \otimes \cdots \otimes {\sf u} a_{i-1} \otimes {\sf u} (a_i a_{i+1}) \otimes {\sf u} a_{i+2} \otimes \cdots \otimes {\sf u} a_n \\
&\qquad +(-1)^{(\vert m \vert + \sum_{j=1}^{n-1}\vert a_j\vert -n -1)(\vert a_n \vert -1)} a_n m \otimes {\sf u} a_1 \otimes \cdots \otimes {\sf u} a_{n-1},
\end{split}
\end{align}
for homogeneous elements $m \in M$ and $a_1,\dots, a_n \in A$. The resulting cohomology is called the \emph{Hochschild homology of $A$ with values in $M$}.
In the special case where $M = \ensuremath{\mathbbmss{R}}$ is the trivial bimodule, we shall write $\operatorname{HC}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A)$ instead of $\operatorname{HC}_{\bullet}(A,\ensuremath{\mathbbmss{R}})$.
The \emph{Hochschild cochain complex} of $A$ with values in $M$ is the cochain complex
\begin{equation}
\operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A,M) = \operatorname{Hom} (\operatorname{HC}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A), M) = \bigoplus_{n \geq 0} \operatorname{Hom} (({\sf u} A)^{\otimes n}, M),
\end{equation}
with differential $b$ characterised by
\begin{align}\label{eqn:2.6aa}
\begin{split}
&(b \varphi) ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_{n}) = d_M (\varphi({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_n)) - (-1)^{\vert \varphi \vert} \varphi (b({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_n)) \\
&\qquad + (-1)^{\vert \varphi \vert(\vert a_1 \vert +1)} a_1 \varphi({\sf u} a_2 \otimes \cdots \otimes {\sf u} a_n) - (-1)^{\vert \varphi \vert + \sum_{j=1}^{n-1}\vert a_j \vert + n-1} \varphi ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_{n-1}) a_n,
\end{split}
\end{align}
for homogeneous elements $\varphi \in \operatorname{Hom} (({\sf u} A)^{\otimes n}, M)$ and $a_1,\dots, a_n \in A$. The resulting cohomology is called the \emph{Hochschild cohomology of $A$ with values in $M$}. In case $M$ is the trivial module $\mathbb{R}$, we will write $\operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A)$ instead of $\operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A,M)$.
A case of special interest arises in the following way. Let $V$, $V'$ and $V''$ be DG modules over $A$, so that the hom-complexes $\operatorname{Hom}(V,V')$ and $\operatorname{Hom}(V',V'')$ are naturally DG bimodules over $A$. Then there is a cup product
$$
\abxcup \colon \operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A,\operatorname{Hom}(V',V'')) \otimes \operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A,\operatorname{Hom}(V,V')) \longrightarrow \operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A,\operatorname{Hom}(V,V'')),
$$
which is defined by
\begin{align}\label{eqn:2.7aa}
\begin{split}
&(\psi \abxcup \varphi) ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_{m+n}) \\
&\qquad\quad\,\,= (-1)^{\vert \varphi \vert (\sum_{i=1}^{m}\vert a_i \vert - m)} \psi ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_{m}) \circ \varphi ({\sf u} a_{m+1} \otimes \cdots \otimes {\sf u} a_{m+n}),
\end{split}
\end{align}
for homogeneous elements $\varphi \in \operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A,\operatorname{Hom}(V,V'))$, $\psi \in \operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A,\operatorname{Hom}(V',V''))$ and $a_1,\dots, a_{m+n} \in A$. This cup product is compatible with the differential $b$ in the sense that it satisfies the Leibniz rule. Given a DG algebra $A$, one can form the DG category of DG modules over $A$, which we denote by $\operatorname{\mathbf{DGMod}}(A)$. Its objects are, of course, DG modules over $A$. For any two such objects $V$ and $V'$, the space of morphisms is the complex $\operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A,\operatorname{Hom}(V,V'))$ and the composition law is the cup product.
\subsection{DG categories, DG funtors and $\mathsf{A}_{\infty}$-functors}\label{sec:2.2}
A \emph{DG category} (where DG stands for ``differential graded'') over a field $K$ is a $K$-linear category $\mathcal{C}$ such that for every two objects $X$ and $Y$ the space of arrows $\operatorname{Hom}_{\mathcal{C}}(X,Y)$ is equipped with a structure of a cochain complex of $K$-vector spaces, and for every three objects $X$, $Y$ and $Z$ the composition map $\operatorname{Hom}_{\mathcal{C}}(Y,Z) \otimes_K \operatorname{Hom}_{\mathcal{C}}(X,Y) \to \operatorname{Hom}_{\mathcal{C}}(X,Z)$
is a morphism of cochain complexes. Thus, by definition,
$$
\operatorname{Hom}_{\mathcal{C}}(X,Y) = \bigoplus_{n \in \ZZ} \operatorname{Hom}_{\mathcal{C}}^{n}(X,Y)
$$
is a graded $K$-vector space with a differential $d \colon \operatorname{Hom}_{\mathcal{C}}^{n}(X,Y) \to \operatorname{Hom}_{\mathcal{C}}^{n+1}(X,Y)$. The elements $f \in \operatorname{Hom}_{\mathcal{C}}^{n}(X,Y)$ are called \emph{homogeneous of degree $n$}, and we write $\vert f \vert = n$. We shall denote the set of objects of $\mathcal{C}$ by $\operatorname{Ob} \mathcal{C}$.
The fundamental example of a DG category is the category of cochain complexes of $K$-vector spaces, which we denote by $\operatorname{\mathbf{DGVect}}_K$. Its objects are cochain complexes of $K$-vector spaces and the morphism spaces $\operatorname{Hom}_{\operatorname{\mathbf{DGVect}}_K}(X,Y)$ are endowed with the differential defined as
$$
d (f) = d_{Y} \circ f - (-1)^n f \circ d_{X},
$$
for any homogeneous element $f$ of degree $n$.
Given a DG category $\mathcal{C}$ one defines an ordinary category $\mathbf{Ho}(\mathcal{C})$ by keeping the same set of objects and replacing each $\operatorname{Hom}$ complex by its $0$th cohomology. We call $\mathbf{Ho}(\mathcal{C})$ the \emph{homotopy category} of $\mathcal{C}$.
If $\mathcal{C}$ and $\mathcal{D}$ are DG categories, a DG functor $F \colon \mathcal{C} \to \mathcal{D}$ is an $K$-linear functor whose associated map for $X, Y \in \operatorname{Ob} \mathcal{C}$,
$$
F_{X,Y} \colon \operatorname{Hom}_{\mathcal{C}}(X,Y) \to \operatorname{Hom}_{\mathcal{D}}(F(X),F(Y)),
$$
is a morphism of cochain complexes. Notice that any DG functor $F \colon \mathcal{C} \to \mathcal{D}$ induces an ordinary functor
$$
\mathbf{Ho}(F) \colon \mathbf{Ho}(\mathcal{C}) \to \mathbf{Ho}(\mathcal{D})
$$
between the corresponding homotopy categories. A DG functor $F \colon \mathcal{C} \to \mathcal{D}$ is said to be \emph{quasi fully faithful} if for every pair of objects $X, Y \in \operatorname{Ob} \mathcal{C}$ the morphism $F_{X,Y}$ is a quasi-isomorphism. Moreover, the DG functor $F $ is said to be \emph{quasi essentially surjective} if $\mathbf{Ho}(F)$ is essentially surjective. A DG functor which is both quasi fully faithful and quasi essentially surjective is called a \emph{quasi-equivalence}.
There is a more general notion of functor between DG categories, that of an $\mathsf{A}_{\infty}$-functor, where the composition is preserved only up to an infinite sequence of coherence conditions. It will be useful to introduce first the Hochschild chain complex of a DG category.
Let $\mathcal{C}$ be a small DG category. The \emph{Hochschild cochain complex} of $\mathcal{C}$ is the complex
$$
\bigoplus_{X_0,\dots, X_n}{\sf s}\! \operatorname{Hom}_{\mathcal{C}}(X_{n-1},X_n) \otimes_K \cdots \otimes_K {\sf s}\! \operatorname{Hom}_{\mathcal{C}}(X_{0},X_1),
$$
where $X_0,\dots,X_n$ range through the objects of $\mathcal{C}$, and whose differential $b$ is the sum of two components $b_1$ and $b_2$ given by the formulas
$$
b_1 (f_{n-1} \otimes \cdots \otimes f_0) = \sum_{i=0}^{n-1} (-1)^{\sum_{j=i+1}^{n-1}\vert f_{j} \vert + n - i - 1} f_{n-1} \otimes \cdots \otimes d f_i \otimes \cdots\otimes f_0
$$
and
$$
b_2 (f_{n-1} \otimes \cdots \otimes f_0) = \sum_{i=0}^{n-2} (-1)^{\sum_{j=i+2}^{n-1}\vert f_{j} \vert + n - i } f_{n-1} \otimes \cdots \otimes (f_{i+1} \circ f_i) \otimes \cdots\otimes f_0
$$
for homogeneous elements $f_0 \in {\sf s}\! \operatorname{Hom}_{\mathcal{C}}(X_{0},X_1), \dots, f_{n-1} \in {\sf s}\! \operatorname{Hom}_{\mathcal{C}}(X_{n-1},X_n)$. Here $d$ denotes indistinctly the differential in any of the spaces $\operatorname{Hom}_{\mathcal{C}} (X_{i},X_{i+1})$. It is easy to check that indeed $b^2=0$, by cancellation of terms with opposite signs.
With this in mind, the formal definition of an $\mathsf{A}_{\infty}$-functor is given as follows. Let $\mathcal{C}$ and $\mathcal{D}$ be DG categories. An \emph{$\mathsf{A}_{\infty}$-functor} $F \colon \mathcal{C} \to \mathcal{D}$ is the datum of a map of sets $F_0 \colon \operatorname{Ob} \mathcal{C} \to \operatorname{Ob} \mathcal{D}$ and a collection of $K$-linear maps of degree $0$
$$
F_n \colon {\sf s}\! \operatorname{Hom}_{\mathcal{C}}(X_{n-1},X_n) \otimes_K \cdots \otimes_K {\sf s}\! \operatorname{Hom}_{\mathcal{C}}(X_{0},X_1) \to \operatorname{Hom}_{\mathcal{D}}(F_0(X_0),F_0(X_n))
$$
for every collection $X_0,\dots,X_n \in \operatorname{Ob} \mathcal{C}$, such that the relation
\begin{align*}
b_1 \circ F_n + \sum_{i+j = n} b_2 \circ (F_i \otimes F_j ) = \sum_{i + j + 1= n} F_n \circ ({\rm id}^{\otimes i} \otimes b_1 \otimes {\rm id}^{\otimes j}) + \sum_{i + j + 2 = n} F_{n-1} \circ ({\rm id}^{\otimes i} \otimes b_2 \otimes {\rm id}^{\otimes j})
\end{align*}
is satisfied for any $n \geq 1$. One also requires that $F_1({\rm id}_{X}) = {\rm id}_{F_0(X)}$ for all objects $A$ in $\mathcal{C}$, as well as $F_n(f_{n-2} \otimes \cdots \otimes f_{i} \otimes {\rm id}_{X_{i}} \otimes f_{i-1} \otimes \cdots \otimes f_0 ) =0$ for any $n \geq 1$, any $0 \leq i \leq n-2$, and any chain of morphisms $f_0 \in {\sf s}\! \operatorname{Hom}_{\mathcal{C}}(X_{0},X_1), \dots, f_{n-2} \in {\sf s}\! \operatorname{Hom}_{\mathcal{C}}(X_{n-2},X_{n-1})$.
The above relation when $n=1$ implies that $F_1$ is a morphism of cochain complexes. On the oner hand, for $n=2$ we find that $F_1$ preserves the compositions on $\mathcal{C}$ and $\mathcal{D}$, up to a homotopy defined by $F_2$. In particular, a DG functor between $\mathcal{C}$ and $\mathcal{D}$ is the same as an $\mathsf{A}_{\infty}$-functor such that $F_n = 0$ for $n \geq 2$. It also follows that $F_1$ induces and ordinary functor
$$
\mathbf{Ho}(F_1) \colon \mathbf{Ho}(\mathcal{C}) \to \mathbf{Ho}(\mathcal{D}).
$$
An $\mathsf{A}_{\infty}$-functor $F \colon \mathcal{C} \to \mathcal{D}$ is called $\mathsf{A}_{\infty}$-\emph{quasi fully faithfull} if $F_1$ is a quasi-isomorphism, and it is called $\mathsf{A}_{\infty}$-\emph{quasi essentially surjective} if $\mathbf{Ho}(F_1)$ is essentially surjective. Finally, an $\mathsf{A}_{\infty}$-functor $F$ is called a $\mathsf{A}_{\infty}$-\emph{quasi-equivalence} if it is both quasi fully faithfull and quasi essentially surjective. We say that two DG categories are $\mathsf{A}_\infty$ equivalent if they can be connected by a zig-zag of $\mathsf{A}_\infty$ quasi-equivalences.
\subsection{Gugenheim's $\mathsf{A}_{\infty}$ De~Rham theorem} \label{sec:2.3}
The usual De~Rham map, which sends a differential form to a singular cochain by integration, is not an algebra map. However, it induces an isomorphism of algebras in cohomology. A more complete explanation of this fact is due to Gugenheim. In \cite{Gugenheim1977}, this author uses Chen's iterated integrals \cite{Chen1977} to extend the De~Rham map to an $\mathsf{A}_{\infty}$-quasi-isomorphism of DG algebras. Here we will review this construction, which will be needed later. We follow the presentation in \cite{Abad-Schatz2013}.
Let us start with some background. For a smooth manifold $M$ we denote by $\mathcal{P} M$ the path space of $M$, that is, the space of all smooth maps from $I$ to $M$ which we regard as a diffeological space. Given another manifold $X$, one says that a map $f \colon X \to \mathcal{P} M$ is smooth if the map $\widehat{f} \colon [0,1] \times X \to M$ defined for any $t \in I$ and $x \in X$ by
\begin{equation}
\widehat{f} (t, x) = f(x)(t),
\end{equation}
is smooth. With this in mind, we may define differential forms on $\mathcal{P} M$ as follows. We first consider the category $C^{\infty}(-,\mathcal{P} M)$ whose objects are pairs $(X, f)$ where $X$ is a smooth manifold and $f$ is a smooth map from $X$ to $\mathcal{P} M$ and whose morphisms from one such pair $(X,f)$ to another $(Y,g)$ are smooth maps $h \colon X \to Y$ such that $f = g \circ h$. Next, if $\operatorname{\mathbf{Vect}}_{\ensuremath{\mathbbmss{R}}}$ denotes the category of real vector spaces, we consider the functor $\ensuremath{\mathbbmss{R}}(-)$ from $C^{\infty}(-,\mathcal{P} M)$ to $\operatorname{\mathbf{Vect}}_{\ensuremath{\mathbbmss{R}}}$ which sends any object in $C^{\infty}(-,\mathcal{P} M)$ to $\ensuremath{\mathbbmss{R}}$ and every morphism to the identity, along with the functor $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(-)$ from $C^{\infty}(-,\mathcal{P} M)$ to $\operatorname{\mathbf{Vect}}_{\ensuremath{\mathbbmss{R}}}$ sending an object $(X,f)$ to $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X)$ and a morphism $h$ to its pullback $h^*$. Then, a differential form on $\mathcal{P} M$ is a natural transformation from $\ensuremath{\mathbbmss{R}}(-)$ to $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(-)$. This definition simply means that we declare a differential form on $\mathcal{P} M$ to be determined by its pullback along smooth maps from a smooth manifold.
We shall now explain Chen's iterated integrals taking values on differential forms on the path space $\mathcal{P} M$. First we need the following piece of notation. If $\Delta_n$ denotes the $n$-simplex, we write ${\rm ev} \colon \Delta_n \times \mathcal{P} M \to M^{n}$ for the evaluation map defined as
\begin{equation}
{\rm ev}((t_1,\dots, t_n), \gamma) = (\gamma(t_1),\dots, \gamma(t_n)),
\end{equation}
for $(t_1,\dots,t_n) \in \Delta_n$ and $\gamma \in \mathcal{P} M$. Further, we let $p_i$ stand for the $i$-th projection from $M^n$ to $M$ for any $i = 1,\dots, n$, and $\pi$ for the projection from $\Delta_n \times M$ to $M$. Then, Chen's map ${\sf C} \colon ({\sf s} \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M))^{\otimes n} \to \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(\mathcal{P} M)$ is defined by setting
\begin{equation}
{\sf C} (\omega_1 \otimes \cdots \otimes \omega_n) = (-1)^{\sum_{j=1}^{n}\vert \omega_j \vert (n-j)} \pi_{*} ({\rm ev}^{*}(p_1^* \omega_1 \wedge \cdots \wedge p_n^* \omega_n )),
\end{equation}
for homogeneous elements $\omega_1 ,\dots ,\omega_n \in {\sf s} \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M)$, where here $\pi_{*} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(\Delta_n \times M) \to \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M)$ denotes the pushforward along the projection $\pi$.\footnote{In the case when $M$ is compact and oriented, the pushforward $\pi_*$ is characterized by the property that
$$
\int_{M} \pi_* \omega = \int_{\Delta_n \times M} \omega,
$$
for all $\omega \in \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(\Delta_n \times M)$. }
Besides Chen's map, Gugenheim's construction uses some combinatorial maps that we now describe. For each $n \geq 1$, let $\lambda_n \colon I^{n-1} \to \mathcal{P} I^{n}$ be the map that sends every element $(s_1,\dots,s_{n-1})$ of $I^{n-1}$ to the piecewise linear path which goes backwards through the $n+1$ points
$$
0 \longleftarrow s_1 e_1 \longleftarrow s_1 e_1 + s_2 e_2 \longleftarrow \dots \longleftarrow s_1 e_1 + \cdots + s_{n-1} e_{n-1} \longleftarrow s_1 e_1 + \cdots + s_{n-1} e_{n-1} + e_n,
$$
with $e_1,\dots, e_n$ being the standard basis of $\ensuremath{\mathbbmss{R}}^{n}$, and $\pi_n \colon I^{n} \to \Delta_n$ the map given by
\begin{equation}
\pi_n (s_1,\dots,s_n) = (t_1,\dots,t_n),
\end{equation}
for $(s_1,\dots, s_n) \in I^{n}$, with $t_i = \max\{s_i,\dots,s_n\}$ for any $i = 1,\dots, n$. We then obtain, for each $n \geq 1$, a map $\theta_n \colon I^{n-1} \to \mathcal{P} \Delta_n$ which is defined as the composition
$$
I^{n-1} \xlongrightarrow{\phantom{a}\lambda_n\phantom{a}} \mathcal{P} I^{n} \xlongrightarrow{\phantom{a}\mathcal{P}\pi_n\phantom{a}} \mathcal{P} \Delta_n,
$$
where $\mathcal{P} \pi_n$ is the map induced on path spaces by $\pi_n$. We also, by convention, set $\theta_0$ to be the map from a point to a point. \\
Using the above notation, we consider the map ${\sf S} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(\mathcal{P} M) \to {\sf s} {\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M)$ from the de~Rham complex of the path space $\mathcal{P} M$ to the unsuspension of ${\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M)$, obtained as the composition of the map $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(\mathcal{P} M) \to {\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M)$ given, for each $\varphi \in \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(\mathcal{P} M)$, by
$$
(\sigma \colon \Delta_n \to M) \longmapsto \int_{I^{n-1}} \theta_n^* \mathcal{P} \sigma^* \varphi \in \ensuremath{\mathbbmss{R}},
$$
followed by the unsuspension ${\sf s} \colon {\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M) \to {\sf s}{\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M)$. We then proceed to define, for $n \geq 1$, a sequence of linear maps ${\sf{DR}}_n \colon ({\sf s} \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M))^{\otimes n} \to {\sf s} {\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M)$ in the following way. For $n = 1$, we set
\begin{equation}\label{eqn:2.12}
{\sf{DR}}_1(\omega)(\sigma) = \int_{\Delta_k} \sigma^*\omega,
\end{equation}
for $\omega \in {\sf s} \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M)$ and $\sigma \in {\mathrm C}_k (M)$. For $n > 1$, we set
\begin{equation}
{\sf{DR}}_n (\omega_1 \otimes \cdots \otimes \omega_n) = (-1)^{\sum_{j=1}^{n}\vert \omega_j \vert + n} ({\sf S} \circ {\sf C})(\omega_1 \otimes \cdots \otimes \omega_n),
\end{equation}
for homogeneous elements $\omega_1,\dots,\omega_n \in {\sf s}\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M)$. In term of these, we are now in position to state Gugenheim's main result.
\begin{theorem}\label{thm:2.1}
For $n \geq 1$, the sequence of maps ${\sf{DR}}_n \colon ({\sf s} \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M))^{\otimes n} \to {\sf s} {\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(M)$ determines an $\mathsf{A}_{\infty}$-morphism from the de~Rham complex of $M$ to the singular cochain complex of $M$, both viewed as DG algebras. Moreover, this $\mathsf{A}_{\infty}$-morphism is an $\mathsf{A}_{\infty}$-quasi-isomorphism which is natural with respect to pullbacks along smooth maps.
\end{theorem}
\subsection{Alexander-Whitney and Eilenberg-Zilber maps}\label{sec:2.4}
In this subsection we recall the definitions of the Alexander-Whitney and Eilenberg-Zilber maps. These will enable us to give the singular chain complex of a Lie group the structure of a DG Hopf algebra. We begin with the Alexander-Whitney map. For $p \leq n$, the inclusions of the standard $p$-simplex as the front and the back $p$-th face of the standard $n$-simplex will be denoted respectively by $f_{p}^{n} \colon \Delta_p \to \Delta_n$ and $b_{p}^{n}\colon \Delta_p \to \Delta_n$. Explicitly,
\begin{align}\label{eqn:2.7}
\begin{split}
f_{p}^{n}(t_1,\dots, t_p) &= (t_1,\dots, t_p, 0,\dots, 0), \\
b_{p}^{n}(t_1,\dots, t_p) &= (1,\dots, 1, t_1,\dots, t_p).
\end{split}
\end{align}
Also, for two fixed smooth manifolds $X$ and $Y$, we let $\pi_1 \colon X \times Y \to X$ and $\pi_2 \colon X \times Y \to Y$ denote the two natural projections. Then, the Alexander-Whitney map ${\sf{AW}} \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X \times Y) \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X) \otimes {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(Y)$ is the chain map given, for each singular $n$-simplex $\sigma \colon \Delta_n \to X \times Y$, by the formula
\begin{equation}
{\sf{AW}} (\sigma) = \sum_{p + q = n} (\sigma_1 \circ f_{p}^{n}) \otimes (\sigma_2 \circ b_{q}^{n}),
\end{equation}
where $\sigma_1 = \pi_1 \circ \sigma$ and $\sigma_2 = \pi_2 \circ \sigma$. For us, the most important property of this map is that it allows us to define a DG coalgebra structure on the space of singular chains ${\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X)$. The coproduct $\Delta \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} (X) \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X) \otimes {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X)$ is formed by composition of the map ${\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X) \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X \times X)$ induced by the diagonal $X \to X \times X$ with the Alexander-Whitney map ${\sf{AW}} \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X \times X) \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X) \otimes {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X)$. The counit $\varepsilon \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X) \to \ensuremath{\mathbbmss{R}}$ is induced by the projection map which collapses $X$ to a point.
Now we turn to the Eilenberg-Zilber map. Such a map is based on the simple fact that a cube of dimension $n$ is the union of $n!$ simplices.
\begin{center}
\includegraphics[scale=0.5]{EZ}
\end{center}
We fix again two smooth manifolds $X$ and $Y$. The Eilenberg-Zilber map ${\sf{EZ}} \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X) \otimes {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(Y) \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X \times Y)$ is the chain map given, for each singular $m$-simplex $\sigma \colon \Delta_m \to X$ and each singular $n$-simplex $\tau \colon \Delta_n \to Y$, by the formula
\begin{equation}\label{eqn:2.9}
{\sf{EZ}} (\sigma \otimes \tau) = \sum_{\chi \in \mathfrak{S}_{m,n}} (-1)^{\chi} (\sigma \times \tau) \circ \chi_*,
\end{equation}
where, as the notation implies, the sum over $\chi$ is taken over all $(m,n)$ shuffles and $\chi_* \colon \Delta_{m+n} \to \Delta_m \times \Delta_n$ is the map defined by
\begin{equation}
\chi_* (t_1, \dots, t_{m+n}) = ((t_{\chi(1)}, \dots, t_{\chi(m)}), (t_{\chi(m+1)}, \dots, t_{\chi(m+n)}))
\end{equation}
We state without proof the key properties of the Eilenberg-Zilber map (see \cite{Maclane1963} and \cite{Eilenberg-Moore1966}).
\begin{proposition}\label{prop:2.2}
The Eilenberg-Zilber map ${\sf{EZ}} \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X) \otimes {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(Y) \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X \times Y)$ satisfies:
\begin{enumerate}
\item It is associative, that is, given a third smooth manifold $Z$, the following diagram commutes
$$
\xymatrix@C=8ex{{\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X) \otimes {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(Y) \otimes {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(Z) \ar[r]^-{{\rm id} \otimes {\sf{EZ}}} \ar[d]_-{{\sf{EZ}} \otimes {\rm id}}& {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X \times Y) \otimes {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(Z) \ar[d]^-{{\sf{EZ}}} \\
{\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X) \otimes {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(Y \times Z) \ar[r]^-{{\sf{EZ}}} & {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X \times Y \times Z);}
$$
\item It is a map of DG coalgebras.
\item ${\sf{EZ}}$ and ${\sf{AW}}$ are inverses up to natural chain homotopies.
\end{enumerate}
\end{proposition}
From the associativity property, it follows that if $X_1,\dots, X_r$ are smooth manifolds, then there is an $r$-fold Eilenberg-Zilber map ${\sf{EZ}} \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X_1) \otimes \cdots \otimes {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X_r) \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(X_1 \times \cdots \times X_r)$ which is obtained by applying the binary Eilenberg-Zilber maps $r-1$ times. Explicitly, this map is defined as follows. Given simplices $\sigma_i \colon \Delta_{n_i} \to X_i$ with $i=1,\dots, r$, one has:
\begin{equation}\label{eqn:2.18}
{\sf{EZ}} (\sigma_1 \otimes \cdots \otimes \sigma_r) = \sum_{\chi \in \mathfrak{S}_{n_1,\dots,n_r}} (-1)^{\chi} (\sigma_1 \times \cdots \times \sigma_r) \circ \chi_*,
\end{equation}
where the sum over $\chi$ is taken over all $(n_1,\dots,n_r)$-shuffles and $\chi_* \colon \Delta_{n_1 + \cdots + n_r} \to \Delta_{n_1} \times \cdots \times \Delta_{n_r}$ now denotes the map defined by
\begin{equation}
\chi_* (t_1 ,\dots, t_{n_1 + \cdots + n_r}) = ((t_{\chi(1)},\dots, t_{\chi(n_1)} ), \dots, (t_{\chi(n_1 + \cdots + n_{r-1}+1)},\dots, t_{\chi(n_1 + \cdots + n_r)} ) ).
\end{equation}
We will now specialize the discussion by replacing $X$ with a Lie group $G$. In this case, the space of singular chains ${\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)$ acquires the structure of a DG Hopf algebra. The product $m \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G) \otimes {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G) \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)$ is formed by composition of the Eilenberg-Zilber map ${\sf{EZ}} \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G) \otimes {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G) \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G \times G)$ with the map $\mu_{*} \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G \times G) \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)$ induced by the multiplication map $\mu \colon G \times G \to G$. The unit $u \colon \ensuremath{\mathbbmss{R}} \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)$ is induced by the inclusion of a point as the identity element of $G$, and the antipode $S \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G) \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)$ is induced by the inversion map $\iota \colon G \to G$.
\subsection{Representations of the DG Lie algebra $\TT\mathfrak{g}$}\label{sec:2.5}
For a Lie algebra $\mathfrak{g}$, consider the DG Lie algebra $\TT\mathfrak{g} = {\sf u} \mathfrak{g} \oplus \mathfrak{g}$ with degree $-1$ generators $i(x) \in {\sf u} \mathfrak{g}$ and degree $0$ generators $L(x) \in \mathfrak{g}$ for $x \in \mathfrak{g}$. The Lie bracket of $\TT\mathfrak{g}$ is induced by the Lie bracket of $\mathfrak{g}$, and the differential is defined by
\begin{align}
\begin{split}
d (i(x)) &= L(x), \\
d (L(x) ) &= 0.
\end{split}
\end{align}
The generators $i(x)$ and $L(x)$ satisfy the Cartan relations
\begin{align}
\begin{split}
[i(x),i(y)] &= 0,\\
[L(x),L(y)] &= L([x,y]), \\
[L(x),i(y)] &=i([x,y]).
\end{split}
\end{align}
By a \emph{representation} of $\TT \mathfrak{g}$ we mean a cochain complex $V$ together with a DG Lie algebra homomorphism $\TT\mathfrak{g} \to \operatorname{End}(V)$. That is, it consists of a representation of $\TT\mathfrak{g}$ on $V$, where the operators $i_{x}\in \operatorname{End}(V)^{-1}$ and $L_{x} \in \operatorname{End}(V)^0$ corresponding to $i(x),L(x) \in \TT\mathfrak{g}$ satisfy the relations
\begin{align}\label{eqn:2.22}
\begin{split}
[i_{x},\delta] &= L_{x}, \\
[L_{x},\delta]&= 0, \\
[i_{x},i_{y}] &= 0, \\
[L_{x},L_{y}] &= L_{[x,y]}, \\
[L_{x},i_{y}]&= i_{[x,y]},
\end{split}
\end{align}
with $\delta$ being the differential of $V$. The operators $i_{x}$ are called \emph{contractions} and the operators $L_{x}$ are called \emph{Lie derivatives}.
An important example of a representation of $\TT\mathfrak{g}$ is the Chevalley-Eilenberg complex $\operatorname{CE}(\mathfrak{g})$. As a graded algebra is the exterior algebra $\Lambda^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^*$, where $\mathfrak{g}^*$ has degree $1$. The differential $\delta_{\CE}$ is the unique derivation such that for $\xi \in \Lambda^1\mathfrak{g}^*$, $\delta_{\CE} \xi$ is the element in $\Lambda^2\mathfrak{g}^*$ defined by
\begin{equation}
(\delta_{\CE} \xi)(x,y) = - \xi([x,y]).
\end{equation}
It follows from the Jacobi identity that $\delta_{\CE}$ defined in this manner is a differential. The contraction $i_{x}$ and Lie derivatives $L_{x}$ are the unique derivations such that for $\xi \in \Lambda^1\mathfrak{g}^*$,
\begin{align}
\begin{split}
i_{x} \xi &= \langle \xi, x \rangle, \\
L_{x} \xi &= \operatorname{ad}_{x}^* \xi,
\end{split}
\end{align}
where $\operatorname{ad}_{x}^*$ denotes the infinitesimal coadjoint action of the element $x$.
Explicit formulas for these various maps, which will be useful later on, are obtained by introducing a basis for $\mathfrak{g}^*$. Let $e_a$ be a basis for $\mathfrak{g}$ with dual basis $e^{a}$ and structure constants $f^{a}_{\phantom{a}bc}= \langle e^{a},[e_b,e_c] \rangle$, and write $i_a$ and $L_a$ for the contraction $i_{e_a}$ and the Lie derivative $L_{e_a}$ acting on $\operatorname{CE}(\mathfrak{g})$. Then the explicit formulas for $\delta_{\CE}$, $i_{a}$ and $L_{a}$ are the following:
\begin{align}
\begin{split}
\delta_{\CE} e^{a} &= - \frac{1}{2} f^{a}_{\phantom{a}bc} e^{b} \wedge e^{c}, \\
i_{b} e^a &= \delta_{b}^{\phantom{b}a}, \\
L_{b} e^{a} &= - f^{a}_{\phantom{a}bc}e^{c}.
\end{split}
\end{align}
Here the convention that repeated indices are summed over is in place.
Another example of a representation of $\TT\mathfrak{g}$ is the Weil algebra ${\mathrm W}\mathfrak{g}$. The underlying graded commutative algebra of ${\mathrm W}\mathfrak{g}$ is the tensor product
\begin{equation}
{\mathrm W}\mathfrak{g} = \Lambda^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^* \otimes {\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^*,
\end{equation}
where ${\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^*$ is the symmetric algebra of $\mathfrak{g}^*$ and where we associate to each $\xi \in \mathfrak{g}^*$ the degree $1$ generators $t(\xi) \in \Lambda^1 \mathfrak{g}^*$ and the degree $2$ generators $w(\xi) \in {\mathrm S}^1\mathfrak{g}^*$. The differential on ${\mathrm W}\mathfrak{g}$ is given by \begin{align}
\begin{split}
d_{{\mathrm W}} (t(\xi)) &= w(\xi) + \delta_{\CE} (t(\xi)), \\
d_{{\mathrm W}} (w(\xi)) &= \delta_{\CE} (w(\xi)),
\end{split}
\end{align}
where $\delta_{\CE}$ is the differential of the Chevalley-Eilenberg complex $\operatorname{CE}(\mathfrak{g})$. The operators $i_x$ and $L_x$ are the unique derivations such that
\begin{align}
\begin{split}
i_x (t(\xi)) &= \langle t(\xi),x \rangle,\\
i_x (w(\xi)) &= 0, \\
L_x(t(\xi)) &= \operatorname{ad}_x^*(t(\xi)), \\
L_x(w(\xi)) &= \operatorname{ad}_x^*(w(\xi)).
\end{split}
\end{align}
As for $\operatorname{CE}(\mathfrak{g})$, it will be useful to express the differential $d_{{\mathrm W}}$ and the operator $i_x$ and $L_x$ in terms of a dual basis $e^{a}$ of $\mathfrak{g}^*$ and the structure constants $f^{a}_{\phantom{a}bc}$ of $\mathfrak{g}$. If we write $t^{a} = t(e^{a})$ and $w^{a} = w(e^{a})$, they are as follows:
\begin{align}\label{eqn:2.29}
\begin{split}
d_{{\mathrm W}} t^{a} &= w^{a} - \frac{1}{2} f^{a}_{\phantom{a}bc} t^{b} t^{c}, \\
d_{{\mathrm W}} w^{a} &= f^{a}_{\phantom{a}bc} w^{b} t^{c}, \\
i_b t^{a} &= \delta_{b}^{\phantom{b}a}, \\
i_b w^{a} &= 0, \\
L_{b} t^{a} &= - f^{a}_{\phantom{a}bc} t^{c}, \\
L_{b} w^{a} &=- f^{a}_{\phantom{a}bc} w^{c}.
\end{split}
\end{align}
Clearly, the elements $t^{a}$ and $d_{{\mathrm W}} t^{a}$ also generate ${\mathrm W}\mathfrak{g}$ freely, which implies that the
Weil algebra is acyclic.\\
If $V$ and $W$ are representations of $\TT \mathfrak{g}$, a \emph{homomorphism} $f \colon V \to W$ is a morphism of cochain complexes commuting with the operators $i_x$ and $L_x$. It is cleat that the identity map of a representation of $\TT \mathfrak{g}$ onto itself is a homomorphism, and that the composition of two homomorphisms is again a homomorphism. Thus, representations of $\TT \mathfrak{g}$ and their homomorphisms form a category which we denote by $\operatorname{\mathbf{Rep}} (\TT \mathfrak{g})$. For later purposes, we note that this category is symmetric monoidal with tensor product $V \otimes V'$ of two objects $V$ and $V'$ given by the tensor product of the underlying cochain complexes equipped with the actions of $i_x$ and $L_x$ defined by the formulas
\begin{align}
\begin{split}
i_x (v \otimes v') &= i_x v \otimes v' + (-1)^{\vert v \vert } v \otimes i_x v', \\
L_x (v \otimes v') &= L_x v \otimes v' + v \otimes L_x v',
\end{split}
\end{align}
for homogeneous elements $v \in V$ and $v' \in V'$. Clearly, the unit object is the trivial representation $\underline{\ensuremath{\mathbbmss{R}}}$ viewed as a complex concentrated in degree zero.
\subsection{Differentiation and integration functors between $\operatorname{\mathbf{Mod}}({\mathrm C}_{\bullet}(G))$ and $\operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$}\label{sec:2.6}
Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$. The main result of \cite{AriasAbad2019} states the existence of differentiation and integration functors between the monoidal categories $\operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ and $\operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$ which are inverses to one another. This extends the well-known correspondence between representations of $G$ and representations of $\mathfrak{g}$. Let us explain briefly the construction of these functors.
We begin with the differentiation functor, which we will write as $\mathscr{D} \colon \operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)) \to \operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$. For an element $x$ of $\mathfrak{g}$, take the singular $1$-simplex $\sigma[x] \colon \Delta_1 \to G$ defined by
\begin{equation}
\sigma[x] (s) = \exp(s x),
\end{equation}
where $\exp \colon \mathfrak{g} \to G$ is the exponential map of $G$. Then, given an object $V$ in $\operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ with structure homomorphism $\rho \colon {\mathrm C}_{\bullet}(G) \to \operatorname{End}(V)$, the corresponding object $\mathscr{D}(V)$ in $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$ is $V$ with structure homomorphism $\mathscr{D}(\rho) \colon \TT \mathfrak{g} \to \operatorname{End}(V)$ determined by
\begin{align}
\begin{split}
\mathscr{D}(\rho)(i(x)) &= \frac{d}{d t}\bigg\vert_{t=0} \rho\left(\sigma[tx]\right), \\
\mathscr{D}(\rho)(L(x)) &=\frac{d}{d t}\bigg\vert_{t=0} \rho\left(\exp(tx)\right).
\end{split}
\end{align}
In addition, the functor $\mathscr{D}$ acts as the identity on morphisms. Under this definition, it is not difficult to verify that $\mathscr{D}$ is indeed monoidal. For details, see Theorem~3.3 of \cite{AriasAbad2019}.
Next we turn to the integration functor, which we write as $\mathscr{I} \colon \operatorname{\mathbf{Rep}}(\TT \mathfrak{g}) \to \operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$. First we need to record an important preliminary notion. In the category $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$ let us fix an object $V$, and for each $k \geq 0$, let us call $\Phi^{(k)}_V \in \Omega^k(G) \otimes \operatorname{End}(V)^{-k}$ the unique left-equivariant form on $G$ with values in $\operatorname{End}(V)$ such that
\begin{equation}\label{eqn:2.33}
\Phi^{(k)}_V(e)(x_1,\dots,x_k) = i_{x_1} \circ \cdots \circ i_{x_k},
\end{equation}
for all $x_1,\dots,x_k \in \mathfrak{g}$. With this definition, it can be concluded that the forms $\Phi^{(k)}_V$ satisfy the ``descent equations''
\begin{equation}\label{eqn:2.34}
d \Phi^{(k)}_V = (-1)^k \delta \Phi^{(k+1)}_V,
\end{equation}
where, as before, we write $\delta$ for the the differential of $V$. Furthermore, if $\mu \colon G \times G \to G$ denotes the multiplication map for $G$ and $\pi_1,\pi_2 \colon G \times G \to G$ are the projection onto the first and second component, respectively, we get the relation
\begin{equation}\label{eqn:2.35}
\mu^{*} \Phi^{(k)}_V = \sum_{i + j = k}(-1)^{ij} \pi_1^* \Phi^{(i)}_V \wedge \pi_2^* \Phi^{(j)}_V.
\end{equation}
We note finally that $\Phi^{(0)}_V \in \Omega^{0}(G) \otimes \operatorname{End}(V)^{0}$ is a representation of $G$ on $V$ and, in particular,
\begin{equation}\label{eqn:2.36}
\Phi^{(0)}_V(e) = {\rm id}_V
\end{equation}
In general, an element $\Phi$ of $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G) \otimes \operatorname{End}(V)$ is refereed to as a \emph{left-equivariant representation form} for $V$ if it can be decomposed as
\begin{equation}
\Phi = \sum_{k \geq 0} \Phi^{(k)},
\end{equation}
where the forms $\Phi^{(k)} \in \Omega^k(G) \otimes \operatorname{End}(V)^{-k}$ satisfy the conditions \eqref{eqn:2.34}, \eqref{eqn:2.35} and \eqref{eqn:2.36}. In this way we set up a bijective correspondence between objects of $\operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$ and their associated left-equivariant representation forms (see Proposition~3.18 of \cite{AriasAbad2019}). We can now define the integration functor $\mathscr{I} \colon \operatorname{\mathbf{Rep}}(\TT\mathfrak{g}) \to \operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ as follows. Given an object $V$ in $\operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$ with structure homomorphism $\rho \colon \TT \mathfrak{g} \to \operatorname{End}(V)$, the corresponding object $\mathscr{I}(V)$ in $\operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ is $V$ with structure homomorphism $\mathscr{I}(\rho) \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G) \to \operatorname{End}(V)$ determined on a singular $k$-simplex $\sigma \colon \Delta_k \to G$ by
\begin{equation}\label{eqn:2.38}
\mathscr{I}(\rho)(\sigma) = \int_{\Delta_k} \sigma^*\Phi_{V},
\end{equation}
where $\Phi_{V} = \sum_{k \geq 0} \Phi^{(k)}_V$. Moreover, the functor $\mathscr{I}$ acts as the identity on morphisms. Under this definition, it is not hard to see that $\mathscr{D}$ is simultaneously left and right inverse to $\mathscr{I}$. All the details can be found in \S3.3 of \cite{AriasAbad2019}.
\section{$\mathsf{A}_{\infty}$-quasi-isomorphisms of DG algebras}\label{sec:3}
In this section, we prove several technical results concerning the Van~Est map and the Hoschschild-De~Rham $\mathsf{A}_{\infty}$-quasi-isomorphism in the context of classifying spaces. These results are key components in the proof of our main theorem. They may also be of independent interest. Throughout the discussion, $G$ denotes a simply connected Lie group with Lie algebra $\mathfrak{g}$.
\subsection{The Van~Est map}\label{sec:3.1}
Here we consider the Van~Est map from the Bott-Shulman-Stasheff algebra $\Omega^\bullet(G_\bullet)$ to the Weil algebra of $\mathfrak{g}$. We follow the conventions in \cite{Li-Bland-Meinrenken2015}. The Bott-Shulman-Stasheff algebra computes the cohomology of BG while the Van Est map is contractible, so the Van Est map models the pull-back map of the universal bundle.
Our goal here is to show that if $G$ is connected and compact, there is a natural subalgebra of $\Omega^\bullet(G_\bullet)$ such that the restriction of the Van Est map to it lands on the basic part of the Weil algebra and is a quasi-isomorphism.
Let us consider the universal $G$-bundle $\pi \colon EG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \to BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$ as in \cite{Segal1968}. Recall that $EG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$ is the simplicial manifold with $EG_{p} = G \times \cdots \times G$ ($p+1$ copies) where the face operators $\overline{\varepsilon}_{i} \colon EG_{p} \to EG_{p-1}$ are given by
\begin{equation}
\overline{\varepsilon}_{i}(g_0,\dots,g_{p}) = (g_0,\dots, g_{i-1},g_{i+1},\dots, g_{p}),
\end{equation}
for $0 \leq i \leq p$. Similarly, $BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$ is defined by $BG_{p} = G \times \cdots \times G$ ($p$ copies) but here the face operators $\varepsilon_{i} \colon BG_{p} \to BG_{p-1}$ are given by
\begin{equation}\label{eqn:3.2}
\varepsilon_i (g_1,\dots,g_p) = \begin{cases} (g_2,\dots,g_p) & \text{if $i = 0$,} \\
(g_1,\dots,g_{i-1}, g_ig_{i+1},g_{i+2},\dots, g_p) & \text{if $0 < i < p$,} \\
(g_1,\dots,g_{p-1}) & \text{if $i = p$.} \end{cases}
\end{equation}
Finally, view each $EG_{p}$ as a principal $G$-bundle over $BG_{p}$, with action the diagonal action of $G$ from the right, and quotient map $\pi \colon EG_{p} \to BG_{p}$ given by
\begin{equation}
\pi (g_0,\dots,g_p) = (g_0 g_1^{-1}, g_1 g_2^{-1},\dots, g_{p-1}g_{p}^{-1}).
\end{equation}
By definition, the total space of the universal $G$-bundle $EG$ is the thick geometric realisation of the simplicial manifold $EG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$. From this it is easy to see that the classifying space $BG$ may be identified with the thick geometric realisation of $BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$. This means the cohomology of $BG$ may be computed as the ``De~Rham cohomology'' of $BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$, which is defined as the cohomology of the following double complex $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$:
$$
\xymatrix{\vdots & \vdots & \vdots & \\
\Omega^2(BG_0) \ar[u]^-{\bar{d}} \ar[r]^-{\partial}& \Omega^2(BG_1) \ar[u]^-{\bar{d}} \ar[r]^-{\partial}& \Omega^2 (BG_2) \ar[u]^-{\bar{d}} \ar[r]^-{\partial} & \cdots \\
\Omega^1(BG_0) \ar[u]^-{\bar{d}} \ar[r]^-{\partial} & \Omega^1(BG_1) \ar[u]^-{\bar{d}} \ar[r]^-{\partial}& \Omega^1 (BG_2) \ar[u]^-{\bar{d}} \ar[r]^-{\partial} & \cdots \\
\Omega^0(BG_0) \ar[u]^-{\bar{d}} \ar[r]^-{\partial} & \Omega^0(BG_1) \ar[u]^-{\bar{d}} \ar[r]^-{\partial} & \Omega^0 (BG_2) \ar[u]^-{\bar{d}} \ar[r]^-{\partial} & \cdots. \\
}
$$
Here the vertical differential $\bar{d} \colon \Omega^q(BG_p) \to \Omega^{q+1}(BG_p)$ is $(-1)^p$ times the usual de exterior differential $d$ and the horizontal differential $\partial \colon \Omega^q(BG_p) \to \Omega^q(BG_{p+1})$ is given by
\begin{equation}\label{eqn:3.4}
\partial = \sum_{i=0}^{p+1} (-1)^{i} \varepsilon_{i}^*.
\end{equation}
We note that $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ has a graded ring structure with respect to the cup product defined as follows. For any $\omega \in \Omega^{q}(BG_{p})$ and $\omega' \in \Omega^{q'}(BG_{p'})$, let $\omega \abxcup \omega' \in \Omega^{q + q'}(BG_{p + p'})$ be the differential form given by
\begin{equation}\label{eqn:3.5}
\omega \abxcup \omega' = (-1)^{q p'} {\rm pr}^* \omega \wedge {\rm pr}'^*\omega',
\end{equation}
where ${\rm pr} \colon BG_{p + p'} \to BG_{p}$ is the front face projection
\begin{equation}\label{eqn:3.6}
{\rm pr}(g_1,\dots,g_{p+p'}) =(g_1,\dots,g_p) ,
\end{equation}
and ${\rm pr}' \colon BG_{p + p'} \to BG_{p'}$ is the back face projection
\begin{equation}\label{eqn:3.7}
{\rm pr}'(g_1,\dots,g_{p+p'}) = (g_{p+1},\dots,g_{p+p'}) .
\end{equation}
Both the vertical and horizontal differentials $\bar{d}$ and $\partial$ are graded derivations relative to the cup product, and we regard Bott-Shulman-Stasheff complex $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ as a DG algebra with respect to the total differential.
Now we turn to a discussion of the Van~Est map. For this purpose, let us consider the action $\gamma_i (g)$ of elements $g$ of $G$ on $BG_{p}$ defined by
\begin{equation}\label{eqn:3.8}
\gamma_i(g) (g_1,\dots, g_p) = (g_1,\dots, g_{i-1},g_i g^{-1},g g_{i+1}, g_{i+2},\dots, g_p),
\end{equation}
where $1 \leq i \leq p$. For each $x \in \mathfrak{g}$, we denote by $x^{i,\sharp}$ the vector field on $BG_{p}$ generated by this action. We also regard the Weil algebra of $\mathfrak{g}$ as a bigraded algebra ${\mathrm W}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}},\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g}$ with
\begin{equation}
{\mathrm W}^{p,q}\mathfrak{g} = \Lambda^{p-q} \mathfrak{g}^* \otimes {\mathrm S}^{q}\mathfrak{g}^*.
\end{equation}
Notice that any $x \in \mathfrak{g}$ defines two kinds of contraction operators $i_{\Lambda}(x)$ and $i_{{\mathrm S}}(x)$ on ${\mathrm W}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}},\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g}$ of bidegrees $(-1,0)$ and $(-1,-1)$, corresponding to the contractions on $\Lambda^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}$ and ${\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g}$, respectively. For elements $\xi \in {\mathrm W}^{p,q}\mathfrak{g}$ and $x_1,\dots, x_p \in \mathfrak{g}$ we put
\begin{equation}
\xi(x_1,\dots,x_q, \overline{x}_{q+1},\dots, \overline{x}_p) = i_{\Lambda}(x_p) \cdots i_{\Lambda}(x_{q+1}) i_{{\mathrm S}}(x_q) \cdots i_{{\mathrm S}}(x_1) \xi.
\end{equation}
With these definitions, the Van~Est map ${\sf{VE}} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to {\mathrm W}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}},\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g}$ is the map of DG algebras given by the following formula, for $\omega \in \Omega^{q}(BG_p)$ and $x_1,\dots, x_p \in \mathfrak{g}$,
\begin{equation}\label{eqn:3.11}
{\sf{VE}}(\omega) (x_1,\dots,x_q, \overline{x}_{q+1},\dots, \overline{x}_p) =\sum_{\sigma \in \mathfrak{S}_p} \varepsilon(\sigma) \left( i_{x^{1,\sharp}_{\sigma(1)}} \cdots i_{x^{q,\sharp}_{\sigma(q)}} L_{x^{q+1,\sharp}_{\sigma(q+1)}} \cdots L_{x^{p,\sharp}_{\sigma(p)}} \omega \right)(e,\dots, e).
\end{equation}
Here $e$ is the identity element of $G$, and $\varepsilon(\sigma)$ is equal to $+1$ if the number of pairs $(i,j)$ with $q+1 \leq i < j \leq p$ but $\sigma(i) > \sigma(j)$ is even, and equal to $-1$ if that number is odd. \\
The Van~Est map does not take values on the basic elements of ${\mathrm W}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}},\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g}$.
However, there is a subalgebra of $\Omega^\bullet(G_\bullet)$ whose image under the Van Est map consists of basic elements. We set $G_{p} = G \times \cdots \times G$ ($p$ copies) and think of it as a Lie group. Moreover, we let $\mathfrak{g}_p = \mathfrak{g} \oplus \cdots \oplus \mathfrak{g}$ ($p$ copies) denote the corresponding Lie algebra. Since the actions of $G$ on $BG_{p}$ defined by \eqref{eqn:3.8} commute, we obtain an action $\gamma(g_1,\dots,g_p)$ of elements $(g_1,\dots,g_p)$ of $G_p$ on $BG_{p}$ by putting
\begin{equation}\label{eqn:3.12}
\gamma(g_1,\dots,g_p) = \gamma_1(g_1) \circ \cdots \circ \gamma_p(g_p).
\end{equation}
It is straightforward to check this action is transitive and free. Let us denote by $\Omega^{q}(BG_{p})^{G_p}$ the subspace of $G_p$-invariant elements of $\Omega^{q}(BG_{p})$.\footnote{We remark that $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}}$ is not a DG subalgebra of $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$.} Then each element of $\Omega^{q}(BG_{p})^{G_p}$ is completely and freely determined by its evaluation at $(e,\dots,e) \in BG_{p}$. Therefore, evaluation at $(e,\dots,e)$ gives an isomorphism of graded vector spaces between $\Omega^{q}(BG_{p})^{G_p}$ and $\Lambda^{q} \mathfrak{g}_{p}^*$. On the other hand, consider the residual action $\gamma_0(g)$ of elements $g$ of $G$ on $BG_{p}$ defined by
\begin{equation}\label{eqn:3.13}
\gamma_0(g) (g_1,\dots, g_p) = (g g_1, g_2,\dots, g_p).
\end{equation}
Since this action commutes with the one given by \eqref{eqn:3.12}, we end up with an action $\zeta(g_0,g_1,\dots,g_p)$ of elements $(g_0,g_1,\dots,g_p)$ of $G_{p+1}$ on $BG_{p}$ by setting
\begin{equation} \label{eqn:3.14}
\zeta(g_0,g_1,\dots,g_p) = \gamma_0 (g_0) \circ \gamma(g_1,\dots, g_p).
\end{equation}
We let $\Omega^{q}(BG_{p})^{G_{p+1}}$ denote the subspace of $G_{p+1}$-invariant elements of $\Omega^{q}(BG_{p})$.
\begin{lemma}\label{lem:3.1}
$\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}} +1}}$ is a DG subalgebra of $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ and the inclusion
$$
\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}} +1}} \longrightarrow \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})
$$
is a quasi-isomorphism.
\end{lemma}
\begin{proof}
First, let us verify that $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}}$ is a double subcomplex of $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$. By definition, it is clear that $\bar{d} \colon \Omega^{q}(BG_{p}) \to \Omega^{q+1}(BG_{p})$ preserves $G_{p+1}$-invariant elements, since the exterior differential (and hence $\bar{d}$) commutes with pullback. On the other hand, it is not hard to see that $\varepsilon_i^* \omega \in \Omega^{q}(BG_{p+1})^{G_{p+2}}$ for all $\omega \in \Omega^{q}(BG_{p})^{G_{p+1}}$. Thus, from \eqref{eqn:3.4}, we conclude that $\partial \colon \Omega^{q}(BG_{p}) \to \Omega^{q}(BG_{p+1})$ also preserves the $G_{p+1}$-invariant elements.
Next, we need to verify that $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}}$ is closed with respect to the cup product \eqref{eqn:3.5}. This turns out to be a direct consequence of the following two identities
\begin{align*}
{\rm pr} \big( \zeta(g_{0},\dots, g_{p+p'}) \big)&= \zeta(g_{0},\dots, g_{p}), \\
{\rm pr}' \big( \zeta(g_{0},\dots, g_{p+p'}) \big) &= \zeta(g_{p},\dots, g_{p + p'}),
\end{align*}
which follow at once from the definitions \eqref{eqn:3.6}, \eqref{eqn:3.7} and \eqref{eqn:3.14}.
Finally, to prove the second statement, since $G_{p+1}$ is compact and connected, a theorem of Cartan \cite{Cartan1936} asserts that the inclusion $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{p})^{G_{p+1}} \to \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{p})$ is a quasi-isomorphism. The result then follows from the convergence of the spectral sequences for $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}}$ and $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$, together with the fact that the inclusion $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}} \to \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ induces an isomorphism of spectral sequences on the $E_1$-term.
\end{proof}
For our next preparatory result, we let $\operatorname{Ad}_g$ be the adjoint action of elements $g$ of $G$ on $\mathfrak{g}$ and denote by the same symbol its extension to $\mathfrak{g}_p$.
\begin{lemma}\label{lem:3.2}
The following diagram commutes
\begin{equation*}
\xymatrix@C=7ex{\Omega^{q}(BG_{p})^{G_{p}} \ar[r]^-{\gamma_0(g)^*} \ar[d] & \Omega^{q}(BG_{p})^{G_{p}} \ar[d] \\
\Lambda^{q}\mathfrak{g}_p^* \ar[r]^-{\operatorname{Ad}_g^*} & \Lambda^{q}\mathfrak{g}_p^*,}
\end{equation*}
where the vertical arrows denote evaluation at the element $(e,\dots,e)$.
\end{lemma}
\begin{proof}
Take $\omega \in \Omega^{q}(BG_{p})^{G_{p}}$ and $v_1,\dots,v_p \in \mathfrak{g}_p$. We compute directly, using the definitions:
\begin{align*}
\big( \gamma_0(g)^* & \omega \big)_{(e,\dots,e)}(v_1,\dots, v_p) \\ &= \omega_{(g,e,\dots,e)} \big( d \gamma_0(g)_{(e,\dots,e)}(v_1), \dots, d \gamma_0(g)_{(e,\dots,e)}(v_p) \big) \\
&= \omega_{\gamma(g^{-1},\dots,g^{-1})(e,\dots, e)} \big( d \gamma_0(g)_{(e,\dots,e)}(v_1), \dots, d \gamma_0(g)_{(e,\dots,e)}(v_p) \big) \\
&= \big(\gamma(g^{-1},\dots,g^{-1})^*\omega\big)_{(e,\dots,e)} \big( d\gamma(g,\dots,g)_{(g,e,\dots,e)}d \gamma_0(g)_{(e,\dots,e)}(v_1), \\
& \qquad\qquad\qquad\qquad\qquad\qquad\quad \dots, d\gamma(g,\dots,g)_{(g,e,\dots,e)}d \gamma_0(g)_{(e,\dots,e)}(v_p) \big) \\
&= \omega_{(e,\dots,e)} \big( d(\gamma(g,\dots, g)\circ \gamma_0(g))_{(e,\dots, e)} (v_1), \dots, d(\gamma(g,\dots, g)\circ \gamma_0(g))_{(e,\dots, e)} (v_p) \big) \\
&= \omega_{(e,\dots,e)} \big( d \zeta(g,\dots,g)_{(e,\dots, e)} (v_1), \dots, d \zeta(g,\dots,g)_{(e,\dots, e)} (v_p) \big).
\end{align*}
But
$$
\zeta(g,\dots,g)(g_1,\dots,g_p) = (g g_1 g^{-1},\dots, g g_p g^{-1}),
$$
from which it follows that $d \zeta(g,\dots,g)_{(e,\dots, e)} = \operatorname{Ad}_g$. Substitution gives the result claimed.
\end{proof}
Next, we record the following observation.
\begin{lemma}\label{lem:3.3}
The restriction of the Van~Est map ${\sf{VE}}$ to $\Omega^{q}(BG_{p})^{G_{p}}$ vanishes unless $q = p$.
\end{lemma}
\begin{proof}
If $\omega \in \Omega^{q}(BG_{p})^{G_{p}}$, then $L_{x^{i,\sharp}} \omega = 0$ for all $x \in \mathfrak{g}$. This, together with formula \eqref{eqn:3.11}, implies that ${\sf{VE}}(\omega) = 0$ unless $q = p$.
\end{proof}
As a consequence of this, we see that the restriction of the Van~Est map ${\sf{VE}}$ to $\Omega^{p}(BG_{p})^{G_{p}}$, which we keep on denoting by ${\sf{VE}}$, is given by the following expression, for $\omega \in \Omega^{p}(BG_{p})^{G_{p}}$ and $x_1,\dots, x_p \in \mathfrak{g}$,
\begin{equation}\label{eqn:3.15}
{\sf{VE}}(\omega)(x_{1},\dots,x_p) = \sum_{\sigma \in \mathfrak{S}_p} \left( i_{x^{1,\sharp}_{\sigma(1)}} \cdots i_{x^{p,\sharp}_{\sigma(p)}} \omega \right)(e,\dots, e).
\end{equation}
We also note that this map has its image contained in ${\mathrm S}^p \mathfrak{g}^*$.
Before we can go further, we need the following piece of notation. For each $x \in \mathfrak{g}$, we let $x^{i}$ be element of $\mathfrak{g}_p$ having its $i$th and $(i+1)$th coordinates equal to $-x$ and $x$, respectively, and all others zero. Hence, by definition, $x^{i,\sharp}(e,\dots,e) = x^{i}$. Thus, if we let $\widetilde{{\sf{VE}}} \colon \Lambda^p \mathfrak{g}_p^* \to {\mathrm S}^p \mathfrak{g}$ be the map defined for $\xi \in \Lambda^p \mathfrak{g}_p^*$ and $x_1,\dots, x_p \in \mathfrak{g}$ by
\begin{equation}\label{eqn:3.16}
\widetilde{{\sf{VE}}} (\xi)(x_{1},\dots,x_p) = \sum_{\sigma \in \mathfrak{S}_p} \xi(x_{\sigma(1)}^{1},\dots, x_{\sigma(p)}^{p}),
\end{equation}
we obtain the commutative diagram
\begin{equation*}
\xymatrix{\Omega^{p}(BG_{p})^{G_{p}} \ar[dr]^-{{\sf{VE}}} \ar[d]& \\
\Lambda^{p}\mathfrak{g}_{p}^* \ar[r]_-{\widetilde{{\sf{VE}}}} & {\mathrm S}^{p}\mathfrak{g}^*,}
\end{equation*}
where, as before, the vertical arrow denotes evaluation at $(e,\dots, e)$. We may now state and prove the following result.
\begin{proposition}\label{prop:3.4}
The restriction of the Van~Est map ${\sf{VE}}$ to $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}} + 1}}$ has image contained in $({\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^*)^G$.
\end{proposition}
\begin{proof}
By virtue of Lemma~\ref{lem:3.2} and the previous remarks, it is enough to show that the following diagram commutes
\begin{equation*}
\xymatrix@C=7ex{\Lambda^{p}\mathfrak{g}_{p}^* \ar[r]^-{\operatorname{Ad}_g^*} \ar[d]_-{\widetilde{{\sf{VE}}}} & \Lambda^{p}\mathfrak{g}_{p}^* \ar[d]^-{\widetilde{{\sf{VE}}}}\\
{\mathrm S}^p \mathfrak{g}^* \ar[r]^-{\operatorname{Ad}_g^*} & {\mathrm S}^p \mathfrak{g}^*.}
\end{equation*}
So let us take $\xi \in \Lambda^{p}\mathfrak{g}_{p}^*$ and $x_1, \dots , x_p \in \mathfrak{g}$. Then, attending to the definition \eqref{eqn:3.16}, we have
\begin{align*}
\widetilde{{\sf{VE}}}(\operatorname{Ad}_g^* \xi)(x_1,\dots,x_p) &= \sum_{\sigma \in \mathfrak{S}_p} (\operatorname{Ad}_g^* \xi)(x_{\sigma(1)}^{1},\dots, x_{\sigma(p)}^{p}) \\
&= \sum_{\sigma \in \mathfrak{S}_p} \xi \Big(\!\operatorname{Ad}_g x_{\sigma(1)}^{1},\dots, \operatorname{Ad}_g x_{\sigma(p)}^{p}\Big) \\
&= \widetilde{{\sf{VE}}}(\xi) \Big(\operatorname{Ad}_g x_{\sigma(1)}^{1}, \dots,\operatorname{Ad}_g x_{\sigma(p)}^{p} \Big) \\
&= \big( \!\operatorname{Ad}_g^*\widetilde{{\sf{VE}}}(\xi)\big)(x_1,\dots,x_p),
\end{align*}
from which the result follows.
\end{proof}
Next, we will show that the restricted Van~Est map ${\sf{VE}} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}} + 1}} \to ({\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^*)^G$ is a quasi-isomorphism. For this, we need a small digression outlining some of the results of \cite{Alekseev-Meinrenken2005}.
To begin with, recall that a $\mathfrak{g}$-\emph{DG algebra} $A$ is by definition an object of $\operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$ endowed with the structure of a graded ring such that the action of $\TT \mathfrak{g}$ is by derivations. Homomorphisms of $\mathfrak{g}$-DG algebras are morphisms in $\operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$ which are also homomorphisms of graded rings. Given a $\mathfrak{g}$-DG algebra $A$, an \emph{algebraic connection} is a linear map $\theta \colon \mathfrak{g}^* \to A^{1}$, which satisfy the relations
\begin{align}
\begin{split}
i_x (\theta(\xi)) &= \langle \xi, x \rangle,\\
L_x (\theta(\xi)) &=\theta (\operatorname{ad}_x^*\xi),
\end{split}
\end{align}
for all $x \in \mathfrak{g}$ and $\xi \in \mathfrak{g}*$. One important example of a commutative $\mathfrak{g}$-DG algebra is provided by the Weil algebra ${\mathrm W} \mathfrak{g}$. It is obvious that ${\mathrm W}\mathfrak{g}$ carries a ``tautological'' connection given by the map $\iota \colon \mathfrak{g}^* \to {\mathrm W}^1\mathfrak{g}$. As a matter of fact, ${\mathrm W} \mathfrak{g}$ is universal among commutative $\mathfrak{g}$-DG algebras with connection. Thus, given a $\mathfrak{g}$-DG algebra $A$ with connection $\theta$, there exists a $\mathfrak{g}$-DG algebra homomorphism $c^{\theta}\colon {\mathrm W} \mathfrak{g} \to A$ such that $c^{\theta} \circ \iota = \theta$. Following the terminology of \cite{Alekseev-Meinrenken2005}, one refers to $c^{\theta}$ as the \emph{characteristic homomorphism} for the connection $\theta$.
Our interest here, however, is on the De~Rham complex $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(EG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ of the simplicial manifold $EG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$, which is defined by exactly the same prescription that defined the Bott-Shulman-Stasheff complex $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$. This turns out to be a noncommutative $\mathfrak{g}$-DG algebra where the graded ring structure is again defined by the cup product, and, if we let $\rho$ denote the infinitesimal action of $\mathfrak{g}$ on $EG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$, $i_x$ is the inner product of a form with $\rho(x)$, and $L_x$ is the Lie derivative of the form along $\rho(x)$. What is more, it carries a natural connection $\theta \colon \mathfrak{g}^* \to \Omega^1(EG_0)$ given by the left-invariant Maurer-Cartan form on $G$. We would like to define a characteristic homomorphism for this connection $\theta$. For this we need a universal object among noncommutative $\mathfrak{g}$-DG algebras with connection, the so called \emph{noncommutative Weil algebra} $\widetilde{{\mathrm W}}\mathfrak{g}$. Its definition is as follows.
Recall that the Weil algebra ${\mathrm W} \mathfrak{g}$ may be identified with the Koszul algebra of the graded vector space ${\sf u} \mathfrak{g}^*$. Accordingly, as a DG algebra, $\widetilde{{\mathrm W}}\mathfrak{g}$ is the noncommutative Koszul algebra of ${\sf u} \mathfrak{g}^*$. Just as in Section~\ref{sec:2.4}, we associate to each $\xi \in \mathfrak{g}^*$ a degree $1$ generator $t(\xi)$ and a degree $2$ generator $w(\xi)$, so that $\widetilde{{\mathrm W}}\mathfrak{g}$ is freely generated by $t(\xi)$ and $w(\xi)$, $d_{\widetilde{{\mathrm W}}} t(\xi) = w(\xi)$ and $d_{\widetilde{{\mathrm W}}} w(\xi) = 0$. The formulas for the contractions $i_x$ and Lie derivatives $L_x$ are given on these generators by
\begin{align}
\begin{split}
i_x (t(\xi)) &= \langle t(\xi), x \rangle, \\
i_x (w(\xi)) &= \operatorname{ad}_x^* (t(\xi)), \\
L_x (t(\xi)) &= \operatorname{ad}_x^* (t(\xi)), \\
L_x (w(\xi)) &= \operatorname{ad}_x^* (w(\xi)).
\end{split}
\end{align}
And just as in the commutative case, $\widetilde{{\mathrm W}}\mathfrak{g}$ carries a ``tautological'' connection determined by the map $\widetilde{\iota} :\mathfrak{g}^* \to \widetilde{{\mathrm W}}^1\mathfrak{g}$. It can then be shown that, given an an arbitrary $\mathfrak{g}$-DG algebra $A$ with connection $\theta$, there is a $\mathfrak{g}$-DG algebra homomorphism $\widetilde{c}{}^{\,\theta} \colon \widetilde{{\mathrm W}}\mathfrak{g} \to A$ such that $\widetilde{c}{}^{\,\theta} \circ \widetilde{\iota} = \theta$. We should also point out that the quotient map $\widetilde{{\mathrm W}} \mathfrak{g} \to {\mathrm W} \mathfrak{g}$ is a morphism in $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$ which is a quasi-isomorphism with homotopy inverse given by symmetrisation ${\rm sym} \colon {\mathrm W} \mathfrak{g} \to \widetilde{{\mathrm W}}\mathfrak{g}$.
In light of the preceding discussion it is now clear that there is a Chern-Weil map
$$
c^{\theta} = \widetilde{c}{}^{\,\theta} \circ {\rm sym} \colon {\mathrm W} \mathfrak{g} \longrightarrow \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(EG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}),
$$
which is defined by symmetrisation. This in turn induces a morphism of cochain complexes on the basic subspaces $c^{\theta} \colon ({\mathrm W} \mathfrak{g})_{{\rm bas}} \to \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(EG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_{{\rm bas}}$. As ${\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g}^*$ is precisely the set of elements in ${\mathrm W} \mathfrak{g}$ killed by $i_x$ for $x \in \mathfrak{g}$, it follows that $({\mathrm W} \mathfrak{g})_{{\rm bas}}$ coincides with the algebra of invariant polynomials $({\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g}^*)^G$. On the target complex we have we have on the other hand that $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(EG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_{{\rm bas}}$ is canonically isomorphic to the Bott-Shulman-Stasheff complex $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$. This latter isomorphism is induced by pullback along the right inverse $\iota \colon BG_p \to EG_p$ to the quotient map $\pi$ which is defined by the formula
\begin{equation}
\iota(g_1,\dots, g_1) = (e, g_1^{-1}, \dots, (g_1 \cdots g_p)^{-1}).
\end{equation}
Therefore we clearly get a map
$$
{\sf{AM}}^{\theta} = \iota^* \circ c^{\theta} \colon ({\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g}^*)^G \longrightarrow \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}),
$$
to which we refer to as the \emph{Alekseev-Meinrenken map}. The image of an invariant polynomial of degree $r$ under this map has non-vanishing components only in bidegree $p + q = 2r$ with $p \leq r$. It also induces an algebra homomorphism in cohomology, and in fact an algebra isomorphism if $G$ is compact and connected (see Proposition~9.1 and Theorem~9.2 of \cite{Alekseev-Meinrenken2005}).
To proceed further, let us consider the action $\overline{\gamma}_i(g)$ of elements $g$ of $G$ on $EG_p$ defined by
\begin{equation}\label{eqn:3.20a}
\overline{\gamma}_i(g)(g_0,\dots, g_p) = (g_0,\dots,g_{i-1} , g g_{i}, g_{i+1}, \dots, g_p),
\end{equation}
where $0 \leq i \leq p$. It is then a simple matter to verify that all of these actions provide lifts of the actions of $G$ on $BG_p$ determined by \eqref{eqn:3.8} and \eqref{eqn:3.13}. To be more precise, we have a commutative diagram
$$
\xymatrix{EG_p \ar[r]^-{\overline{\gamma}_i(g)} \ar[d]_-{\pi} & EG_p \ar[d]^-{\pi} \\
BG_p \ar[r]^-{\gamma_i(g)} & BG_p, }
$$
for all $0 \leq i \leq p$. This implies that if, for each $x \in \mathfrak{g}$, we let $\overline{x}^{i,\sharp}$ denote the vector field on $EG_p$ generated by the action \eqref{eqn:3.20a}, then $\overline{x}^{i,\sharp}$ and $x^{i,\sharp}$ are $\pi$-related. In particular, we have
\begin{equation}\label{eqn:3.21a}
d \pi_{(e,\dots,e)} (\overline{x}^{i,\sharp}(e,\dots,e)) = x^{i,\sharp}(e,\dots,e).
\end{equation}
It is also worth pointing out that we get an action $\overline{\zeta}(g_0,\dots,g_p)$ of elements $(g_0,\dots,g_p)$ of $G_{p+1}$ on $EG_{p}$ by simply putting
\begin{equation}
\overline{\zeta}(g_0,\dots,g_p) = \overline{\gamma}_0(g_0) \circ \cdots \circ \overline{\gamma}_p(g_p),
\end{equation}
and that this action provides a lift of the action of $G_{p+1}$ on $BG_p$ defined by \eqref{eqn:3.14}. Let $\Omega^q(EG_p)^{G_{p+1}}$ denote the subspace of $G_{p+1}$-invariant elements of $\Omega^q(EG_p)$. By precisely the same argument as that used to prove Lemma~\ref{lem:3.1}, we have the following.
\begin{lemma}
$\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(EG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}}$ is a DG subalgebra of $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(EG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ and the inclusion
$$
\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(EG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}} \longrightarrow \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(EG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})
$$
is a quasi-isomorphism.
\end{lemma}
The discussion in the previous paragraphs also yield the following result.
\begin{proposition}
The Alekseev-Meinrenken map ${\sf{AM}}^{\theta}$ has image contained in $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}}$.
\end{proposition}
\begin{proof}
Since the Maurer-Cartan form on $G$ is left-invariant, the restriction of the Chern-Weil map $c^{\theta}$ to $({\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^*)^G$ has its image contained in $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(EG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}}_{{\rm bas}}$. The result is thus a direct consequence of the fact that the action of $G_{p+1}$ on $EG_p$ is a lifting of the action of $G_{p+1}$ on $BG_p$.
\end{proof}
With all of the above ingredients in place, we now let $\widehat{c}{}^{\,\theta}$ be the Chen-Weil map $c^{\theta}$ seen as a map taking values in $\Omega^{p}(EG_{p})^{G_{p+1}}_{{\rm bas}}$. We set accordingly $\widehat{{\sf{AM}}}{}^{\theta} = \iota^* \circ \widehat{c}{}^{\,\theta}$ and notice that $\widehat{{\sf{AM}}}{}^{\theta}$ is nothing but the Alekseev-Meinrenken map ${\sf{AM}}^{\theta}$ seen as taking values in $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}} +1}} $.
\begin{theorem}\label{thm:3.7}
The map $\widehat{{\sf{AM}}}{}^{\theta} \colon ({\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^*)^G \to \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}} +1}}$ is a left inverse of the Van~Est map ${\sf{VE}} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}} +1}} \to ({\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^*)^G$.
\end{theorem}
\begin{proof}
We will first write down an explicit formula for the map $\widehat{{\sf{AM}}}{}^{\theta}$. To that end, we fix a basis $e_a$ of $\mathfrak{g}$ with dual basis $e^{a}$ and recall from Section~\ref{sec:2.4} that we have set $w^{a} = w(e^{a})$, so that ${\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}$ can be identified with the polynomial algebra in these variables. We also set $\theta^{a} = \theta(e^{a})$. Notice that $\theta^{a}$ lives in bidegree $(0,1)$, $d \theta^{a}$ lives in bidegree $(0,2)$ and $\partial \theta^{a}$ lives in bidegree $(1,1)$. It follows that the image of $w^{a}$ under $\widehat{c}{}^{\,\theta}$ is $\partial \theta^{a}$. Therefore, if we pick a monomial $w^{a_1} \cdots w^{a_p}$ in ${\mathrm S}^p \mathfrak{g}^*$, we get
\begin{equation*}
\widehat{c}{}^{\,\theta} (w^{a_1} \cdots w^{a_p}) = \frac{1}{p!} \sum_{\sigma \in \mathfrak{S}_p} \partial \theta^{a_{\sigma(1)}} \abxcup \cdots \abxcup \partial \theta^{a_{\sigma(p)}}.
\end{equation*}
and, consequently,
\begin{equation}\label{eqn:3.22a}
\widehat{{\sf{AM}}}{}^{\theta}(w^{a_1} \cdots w^{a_p}) = \frac{1}{p!} \sum_{\sigma \in \mathfrak{S}_p} \iota^*(\partial \theta^{a_{\sigma(1)}} \abxcup \cdots \abxcup \partial \theta^{a_{\sigma(p)}}).
\end{equation}
Next, let us determine ${\sf{VE}}\big(\widehat{{\sf{AM}}}{}^{\theta}(w^{a_1} \cdots w^{a_p})\big)$. To start, we fix $x_1,\dots, x_p \in \mathfrak{g}$. By the definition in \eqref{eqn:3.15}, and recalling that $x^{i,\sharp}_{\sigma(i)}(e,\dots,e) = x^{i}_{\sigma(i)}$ for all $1 \leq i \leq p$, we have
\begin{align*}
{\sf{VE}}\big(\widehat{{\sf{AM}}}{}^{\theta}(w^{a_1} \cdots w^{a_p})\big)(x_1,\dots,x_p) = \sum_{\sigma' \in \mathfrak{S}_p} \widehat{{\sf{AM}}}{}^{\theta}(w^{a_1} \cdots w^{a_p})_{(e,\dots,e)} (x^{1}_{\sigma'(1)}, \dots, x^{p}_{\sigma'(p)}).
\end{align*}
Upon using \eqref{eqn:3.22a}, this becomes
\begin{align} \label{eqn:3.24a}
\begin{split}
{\sf{VE}}\big(&\widehat{{\sf{AM}}}{}^{\theta}(w^{a_1} \cdots w^{a_p})\big)(x_1,\dots,x_p) \\
&= \sum_{\sigma' \in \mathfrak{S}_p}\frac{1}{p!} \sum_{\sigma \in \mathfrak{S}_p} (\partial \theta^{a_{\sigma(1)}} \abxcup \cdots \abxcup \partial \theta^{a_{\sigma(p)}})_{(e,\dots,e)} \big( d\iota_{(e,\dots,e)}(x^{1}_{\sigma'(1)}),\dots, d\iota_{(e,\dots,e)}(x^{p}_{\sigma'(p)}) \big).
\end{split}
\end{align}
Let us evaluate each of the terms inside the double sum. Firstly, attending to the definition of the cup product \eqref{eqn:3.5}, one easily verifies that
\begin{equation}\label{eqn:3.25a}
\partial \theta^{a_{\sigma(1)}} \abxcup \cdots \abxcup \partial \theta^{a_{\sigma(p)}} = \pi_{1,2}^* \partial \theta^{a_{\sigma(1)}} \wedge \cdots \wedge \pi_{p,p+1}^* \partial \theta^{a_{\sigma(p)}},
\end{equation}
where $\pi_{i,i+1} \colon EG_{p} \to EG_1$ is the projection onto the $i$th and $(i+1)$th factors with $1 \leq i \leq p$. Secondly, by virtue of \eqref{eqn:3.21a},
\begin{equation}\label{eqn:3.26a}
d\iota_{(e,\dots,e)}(x^{i}_{\sigma'(i)}) = \overline{x}^{i,\sharp}_{\sigma'(i)}(e,\dots,e),
\end{equation}
for all $1 \leq i \leq p$. Putting together \eqref{eqn:3.25a} and \eqref{eqn:3.25a}, we thus find
\begin{align}\label{eqn:3:27a}
\begin{split}
(\partial \theta^{a_{\sigma(1)}} &\abxcup \cdots \abxcup \partial \theta^{a_{\sigma(p)}})_{(e,\dots,e)} \big( d\iota_{(e,\dots,e)}(x^{1}_{\sigma'(1)}),\dots, d\iota_{(e,\dots,e)}(x^{p}_{\sigma'(p)}) \big) \\
&= \sum_{\sigma'' \in \mathfrak{S}_p} \mathrm{sgn}(\sigma'') \prod_{i=1}^{p} (\partial \theta^{a_{\sigma(\sigma''(i))}})_{(e,e)} \left((d \pi_{\sigma''(i),\sigma''(i)+1})_{(e,\dots,e)}\big(\overline{x}^{i,\sharp}_{\sigma'(i)}(e,\dots,e)\big) \right),
\end{split}
\end{align}
where $\mathrm{sgn}(\sigma'')$ denotes the sign of the permutation $\sigma''$. Next notice that $\overline{x}^{i,\sharp}_{\sigma'(i)}(e,\dots,e)$ is the element of $\mathfrak{g}_{p+1}$ having its $i$th coordinate equal to $x_{\sigma'(i)}$ and all others zero. Consequently, the only non-zero contribution to the sum in \eqref{eqn:3:27a} comes from the identity permutation. Also, it is straightforward to calculate that
\begin{equation*}
(\partial \theta^{a_{\sigma(i)}})_{(e,e)} \left((d \pi_{i,i+1})_{(e,\dots,e)}\big(\overline{x}^{i,\sharp}_{\sigma'(i)}(e,\dots,e)\big) \right) = \theta^{a_{\sigma(i)}}_e (x_{\sigma'(i)}) = w^{a_{\sigma(i)}}(x_{\sigma'(i)}).
\end{equation*}
In this way, \eqref{eqn:3:27a} becomes
\begin{equation*}
(\partial \theta^{a_{\sigma(1)}} \abxcup \cdots \abxcup \partial \theta^{a_{\sigma(p)}})_{(e,\dots,e)} \big( d\iota_{(e,\dots,e)}(x^{1}_{\sigma'(1)}),\dots, d\iota_{(e,\dots,e)}(x^{p}_{\sigma'(p)}) \big) = \prod_{i=1}^{p} w^{a_{\sigma(i)}} (x_{\sigma'(i)}).
\end{equation*}
Inserting this back in \eqref{eqn:3.24a} gives
\begin{align*}
{\sf{VE}}\big(\widehat{{\sf{AM}}}{}^{\theta}(w^{a_1} \cdots w^{a_p})\big)(x_1,\dots,x_p) = \sum_{\sigma \in \mathfrak{S}_p} \frac{1}{p!} \sum_{\sigma' \in \mathfrak{S}_p} \prod_{i=1}^{p} w^{a_{\sigma(i)}} (x_{\sigma'(i)}) = \sum_{\sigma \in \mathfrak{S}_p} \prod_{i=1}^{p} w^{a_{\sigma(i)}} (x_{i}).
\end{align*}
This allows us to conclude that
\begin{equation*}
{\sf{VE}}\big(\widehat{{\sf{AM}}}{}^{\theta}(w^{a_1} \cdots w^{a_p})\big) = w^{a_1} \cdots w^{a_p}.
\end{equation*}
Since any element of $({\mathrm S}^{p}\mathfrak{g})^G$ is a linear combination of monomials $w^{a_1} \cdots w^{a_p}$ in ${\mathrm S}^p\mathfrak{g}^*$, conclusion follows at once.
\end{proof}
Combining the previous result with the above remarks immediately yields the following.
\begin{corollary}
The restricted Van~Est map ${\sf{VE}} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}} +1}} \to ({\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^*)^G$ is a quasi-isomorphism.
\end{corollary}
\subsection{The De~Rham $\mathsf{A}_{\infty}$-quasi-isomorphism for classifying spaces}
In this subsection we establish a version of Gugenheim's $\mathsf{A}_{\infty}$ De~Rham theorem for the classifying space $BG$. We shall start with some general considerations concerning the totalisation of semi-cosimplicial DG algebras.
For any positive integer $n$, let $[n]$ denote the set $\{0,1,\dots,n\}$. We then consider, for $p + q \leq n$, the map $l_{p,q}^{n} \colon [p] \to [n]$ defined by
\begin{equation}\label{eqn:3.17}
l_{p,q}^{n}(k) = k + q.
\end{equation}
Notice that hese maps satisfy the relations \[l_{p,q}^{n} \circ l_{p',q'}^{n'} = l_{p',q+q'}^{n}.\]
Let $A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} = \{A_{p}\}_{p \geq 0}$ be a semi-cosimplicial DG algebra with coface maps $\partial'_{i} \colon A_{p-1} \to A_{p}$ for $0 \leq i \leq p$. For $p \geq 0$ fixed, we write $A_{p} = \bigoplus_{q \in \ZZ} A_{p}^{q}$ for the underlying graded decomposition. Associated to $A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$, there is a canonical DG algebra $\operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$, constructed as follows. As a graded vector space, its $n$th degree summand is defined as
\begin{equation*}
\operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{n} = \bigoplus_{p + q = n} A_{p}^{q}.
\end{equation*}
This becomes a cochain complex if we set $\partial = \partial' + \partial'' \colon \operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{n} \to \operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{n + 1}$, where the differential $\partial' \colon A_{p}^{q} \to A_{p+1}^{q}$ is the alternating sum
\begin{equation}\label{eqn:3.18}
\partial' = \sum_{i=0}^{p+1} (-1)^{i} \partial'_{i},
\end{equation}
and the differential $\partial'' \colon A^{q}_{p} \to A^{q+1}_{p}$ is $(-1)^{p}$ times the differential of $A_p$. To define the product on $\operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$, we take we take the map induced on $A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$ by the map $l_{p,q}^{n}$ given in \eqref{eqn:3.17}, which by abuse of notation we also call $l_{p,q}^{n}$. We then have $l_{p,q}^{n} \colon A_{p} \to A_{n}$ for $p + q \leq n$. For any $a \in A_{p}^{q}$ and $a' \in A_{p'}^{q'}$, we let $a a' \in A_{p + p'}^{q + q'}$ be the element defined as
\begin{equation} \label{eqn:3.19}
a a' = (-1)^{q p'} l_{p,0}^{p + p'}(a) l_{p',p}^{p + p'}(a').
\end{equation}
With these operations, one can verify that $\operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ is in fact a DG algebra. We omit the details, but comment that this depends upon the fact that the maps $l_{p,q}^{n}$ satisfy the following relations:
\begin{equation}\label{eqn:3.20}
\partial'_i \circ l_{p,q}^{n} = \begin{cases} l_{p,q}^{n+1} & \text{if $i > p + q$,} \\
l_{p+1,q}^{n+1} \circ \partial'_{i-q} & \text{if $q < i \leq p + q$,} \\
l_{p,q+1}^{n+1} & \text{if $i \leq q$.} \end{cases}
\end{equation}
More importantly for our purposes, the construction of $\operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ gives the following result.
\begin{proposition}\label{prop:3.5}
The assignment $A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mapsto \operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ defines a functor from the category of semi-cosimplicial DG algebras with semi-cosimplicial $\mathsf{A}_{\infty}$-morphisms to the category of DG algebras with $\mathsf{A}_{\infty}$-morphisms.
\end{proposition}
\begin{proof}
To start with, recall that a DG algebra $A$ with differential $\partial$ can be thought of as an $\mathsf{A}_{\infty}$-algebra with $\mathsf{A}_{\infty}$-operations $d \colon {\sf u} A \to {\sf u} A$ and $m \colon {\sf u} A \otimes {\sf u} A \to {\sf u} A$ defined by declaring
\begin{align} \label{eqn:3.21}
\begin{split}
d({\sf u} a) &= {\sf u} \partial a, \\
m({\sf u} a \otimes {\sf u} a') &= (-1)^{\vert a \vert + 1} {\sf u} (a a'),
\end{split}
\end{align}
for homogenoeus elements $a, a' \in A$. Next, let us introduce some notation to facilitate the presentation. Given a semi-cosimplicial DG algebra $A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} = \{A_{p}\}_{p \geq 0}$, when we write $a \in A_{p}^{q}$ we mean that $\vert a \vert = p + q$; on the other hand, if we write $\overline{a} \in A_{p}^{q}$, we mean that $\vert \overline{a} \vert = q$. For $p \geq 0$ fixed, we also denote the $\mathsf{A}_{\infty}$-operations associated to $A_p$ in accord to \eqref{eqn:3.21} by $d''_p$ and $m_p$, respectively. With this notation, and the definitions \eqref{eqn:3.18} and \eqref{eqn:3.19}, we can view $\operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ as an $\mathsf{A}_{\infty}$-algebra by setting $d = d' + d'' \colon {\sf u}\!\operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to {\sf u}\!\operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$, where
\begin{align}\label{eqn:3.22}
\begin{split}
d'({\sf u} a) &= \sum_{i =0}^{p+1} (-1)^{i} {\sf u} \partial'_i \overline{a}, \\
d''({\sf u} a) &= (-1)^{p} d''_p ({\sf u} \overline{a}),
\end{split}
\end{align}
and $m \colon {\sf u}\!\operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes {\sf u}\!\operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to {\sf u}\!\operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ to be given by
\begin{equation}\label{eqn:3.23}
m({\sf u} a \otimes {\sf u} a') = (-1)^{p + qp'} m_{p+p'}\big({\sf u} l_{p,0}^{p+p'}(\overline{a}), {\sf u} l_{p',0}^{p+p'}(\overline{a}')\big),
\end{equation}
for any $a \in A_{p}^{q}$ and $a' \in A_{p'}^{q'}$.
We wish to show that the assignment $A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mapsto \operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ is functorial with respect to semi-cosimplicial $\mathsf{A}_{\infty}$-morphisms. So let $\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \colon A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \to B_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$ be one such $\mathsf{A}_{\infty}$-morphisms. This means that for each $p \geq 0$ we have an $\mathsf{A}_{\infty}$-morphism $\phi_{p} \colon A_{p} \to B_{p}$, and these commute with the coface maps of $A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$ and $B_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$. The map $\operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \colon \operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to \operatorname{Tot}(B_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ is explicitly given as follows. Let $a_1 \in A_{p_1}^{q_1},\dots, a_n \in A_{p_n}^{q_n}$, and put $p = \sum_{j= 1}^{n} p_j$ and $r_i = \sum_{j= 1}^{i-1} p_j$. Then
\begin{equation}\label{eqn:3.24}
\operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_n ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_n) = (-1)^{\sum_{1 \leq i < j \leq n}p_j (q_i + 1)}\phi_{p,n} \big( {\sf u} l_{p_1,r_1}^{p}(\overline{a}_1) \otimes \cdots \otimes {\sf u} l_{p_n,r_n}^{p}(\overline{a}_n) \big).
\end{equation}
We claim that $\operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ satisfies the required relations to be an $\mathsf{A}_{\infty}$-morphism, which read
\begin{align}\label{eqn:3.25}
\begin{split}
&d \circ \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_n + \sum_{i + j = n} m \circ (\operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_i \otimes \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_j) \\
&\qquad = \sum_{i+j+1=n} \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_n \circ ({\rm id}^{\otimes i} \otimes d \otimes {\rm id}^{\otimes j}) + \sum_{i+j+2=n} \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_{n-1} \circ ({\rm id}^{\otimes i} \otimes m \otimes {\rm id}^{\otimes j}).
\end{split}
\end{align}
To verify the claim, we note that to prove \eqref{eqn:3.25} it is enough to prove that
\begin{equation}\label{eqn:3.26}
d' \circ \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_n = \sum_{i+j+1=n} \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_n \circ ({\rm id}^{\otimes i} \otimes d' \otimes {\rm id}^{\otimes j}),
\end{equation}
and
\begin{align}\label{eqn:3.27}
\begin{split}
&d'' \circ \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_n + \sum_{i + j = n} m \circ (\operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_i \otimes \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_j) \\
&\qquad = \sum_{i+j+1=n} \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_n \circ ({\rm id}^{\otimes i} \otimes d'' \otimes {\rm id}^{\otimes j}) + \sum_{i+j+2=n} \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_{n-1} \circ ({\rm id}^{\otimes i} \otimes m \otimes {\rm id}^{\otimes j}),
\end{split}
\end{align}
separately.
We begin with \eqref{eqn:3.26}. To that end, we write $\widetilde{\partial}'_i = {\sf u} \circ \partial'_i \circ {\sf s}$ and set $s = \sum_{1 \leq i < j \leq n}p_j (q_i + 1)$. From \eqref{eqn:3.18} and the fact that $\phi_{p}$ commutes with the $\widetilde{\partial}'_{i}$, it follows that
\begin{gather*}
\begin{align*}
d' \left( \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_n ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_n) \right) & = \sum_{i = 0}^{p+1} (-1)^{s + i} \widetilde{\partial}'_i \left( \phi_{p,n} \big( {\sf u} l_{p_1,r_1}^{p}(\overline{a}_1) \otimes \cdots \otimes {\sf u} l_{p_n,r_n}^{p}(\overline{a}_n) \big) \right) \\
&=(-1)^{s} \widetilde{\partial}'_{0} \left( \phi_{p,n} \big( {\sf u} l_{p_1,r_1}^{p}(\overline{a}_1) \otimes \cdots \otimes {\sf u} l_{p_n,r_n}^{p}(\overline{a}_n) \big) \right) \\
&\quad+ (-1)^{s + p +1} \widetilde{\partial}'_{p+1} \left( \phi_{p,n} \big( {\sf u} l_{p_1,r_1}^{p}(\overline{a}_1) \otimes \cdots \otimes {\sf u} l_{p_n,r_n}^{p}(\overline{a}_n) \big) \right) \\
&\quad + \sum_{i = 1}^{n} \sum_{j = 1}^{p_{i}}(-1)^{s + r_{i} + j} \widetilde{\partial}'_{r_i + j} \left( \phi_{p,n} \big( {\sf u} l_{p_1,r_1}^{p}(\overline{a}_1) \otimes \cdots \otimes {\sf u} l_{p_n,r_n}^{p}(\overline{a}_n) \big) \right) \\
&=(-1)^{s} \phi_{p+1,n} \big( {\sf u} \partial'_0 l_{p_1,r_1}^{p}(\overline{a}_1) \otimes \cdots \otimes {\sf u} \partial'_0 l_{p_n,r_n}^{p}(\overline{a}_n) \big) \\
&\quad + (-1)^{s + p +1} \phi_{p+1,n} \big( {\sf u} \partial'_{p+1} l_{p_1,r_1}^{p}(\overline{a}_1) \otimes \cdots \otimes {\sf u} \partial'_{p+1} l_{p_n,r_n}^{p}(\overline{a}_n) \big) \\
&\quad + \sum_{i = 1}^{n} \sum_{j = 1}^{p_{i}}(-1)^{s + r_{i} + j} \phi_{p,n} \big( {\sf u} \partial'_{r_i + j} l_{p_1,r_1}^{p}(\overline{a}_1) \otimes \cdots \otimes {\sf u} \partial'_{r_i + j} l_{p_n,r_n}^{p}(\overline{a}_n) \big).
\end{align*}
\end{gather*}
Taking into account \eqref{eqn:3.20}, the left-hand side of the last equality becomes
\begin{gather*}
\begin{align*}
&\phantom{=}(-1)^{s} \phi_{p+1,n} \big( {\sf u} l_{p_1,r_1 + 1}^{p+1}(\overline{a}_1) \otimes \cdots \otimes {\sf u} l_{p_n,r_n + 1}^{p+1}(\overline{a}_n) \big) + (-1)^{s + p + 1} \phi_{p+1,n} \big( {\sf u} l_{p_1,r_1}^{p+1}(\overline{a}_1) \otimes \cdots \otimes {\sf u} l_{p_n,r_n}^{p+1}(\overline{a}_n) \big) \\
& \phantom{=}+ \sum_{i = 1}^{n} \sum_{j = 1}^{p_{i}}(-1)^{s + r_{i} + j} \phi_{p+1,n} \big( {\sf u} l_{p_1,r_1}^{p +1 }(\overline{a}_1) \otimes \cdots \otimes {\sf u} l_{p_{i-1},r_{i-1}}^{p+1}(\overline{a}_{i-1}) \otimes {\sf u} l_{p_i + 1,r_i}^{p + 1}(\partial'_j \overline{a}_i) \\
&\phantom{=+ \sum_{i = 1}^{n} \sum_{j = 1}^{p_{i}}(-1)^{s + r_{i} + j} \phi_{p+1,n} \big( aaa \otimes \cdots} \,\, \otimes {\sf u} l_{p_{i+1},r_{i+1} + 1}^{p+1}(\overline{a}_{i+1}) \otimes \cdots \otimes {\sf u} l_{p_n,r_n + 1}^{p + 1}(\overline{a}_n) \big).
\end{align*}
\end{gather*}
A simple calculation shows that the first and second term of this expression cancel each other out. Thus, upon using \eqref{eqn:3.18} and \eqref{eqn:3.22} and putting all together,
\begin{gather*}
\begin{align*}
& d' \left( \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_n ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_n) \right) \\
&\qquad\qquad\qquad = \sum_{i = 1}^{n} (-1)^{s + r_{i}} \phi_{p+1,n} \big( {\sf u} l_{p_1,r_1}^{p +1 }(\overline{a}_1) \otimes \cdots \otimes {\sf u} l_{p_{i-1},r_{i-1}}^{p+1}(\overline{a}_{i-1}) \otimes {\sf u} l_{p_i + 1,r_i}^{p + 1}(\partial' \overline{a}_i) \\
&\qquad\qquad\qquad \phantom{= \sum_{i = 1}^{n} (-1)^{s + r_{i}}aaaaaaaaaaaaall } \,\, \otimes {\sf u} l_{p_{i+1},r_{i+1} + 1}^{p+1}(\overline{a}_{i+1}) \otimes \cdots \otimes {\sf u} l_{p_n,r_n + 1}^{p + 1}(\overline{a}_n) \big) \\
&\qquad\qquad\qquad = \sum_{i+j+1 = n} \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_n \left( ({\rm id}^{\otimes i} \otimes d' \otimes {\rm id}^{\otimes j})({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_n) \right),
\end{align*}
\end{gather*}
from which \eqref{eqn:3.26} follows.
Let us now tackle \eqref{eqn:3.27}. By attending to \eqref{eqn:3.22} and \eqref{eqn:3.23}, for the terms on the left-hand side of \eqref{eqn:3.27} we have
\begin{gather*}
\begin{align*}
d'' \left( \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_n ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_n) \right) = (-1)^{s+p} d''_p \left( \phi_{p,n} \big({\sf u} l_{p_1,r_1}^p(\overline{a}_1) \otimes \cdots \otimes {\sf u} l_{p_n,r_n}^{p}(\overline{a}_n) \big) \right), \end{align*}
\end{gather*}
and
\begin{gather*}
\begin{align*}
&m \left( (\operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_i \otimes \operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_j) ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_n) \right) \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad = (-1)^{s+p} m_p \left( (\phi_{p,i} \otimes \phi_{p,j}) \big({\sf u} l_{p_1,r_1}^p(\overline{a}_1) \otimes \cdots \otimes {\sf u} l_{p_n,r_n}^{p}(\overline{a}_n) \big)\right).
\end{align*}
\end{gather*}
Similarly, for the terms on the right-hand side \eqref{eqn:3.27}, one obtains
\begin{gather*}
\begin{align*}
&\operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_{n} \left( ({\rm id}^{\otimes i} \otimes d'' \otimes {\rm id}^{\otimes j}) ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_n) \right) \\
&\qquad \qquad\qquad\qquad\qquad\qquad = (-1)^{s+p} \phi_{p,n} \left( ({\rm id}^{\otimes i} \otimes d''_p \otimes {\rm id}^{\otimes j})\big({\sf u} l_{p_1,r_1}^p(\overline{a}_1) \otimes \cdots \otimes {\sf u} l_{p_n,r_n}^{p}(\overline{a}_n)\right),
\end{align*}
\end{gather*}
and
\begin{gather*}
\begin{align*}
&\operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_{n-1} \left( ({\rm id}^{\otimes i} \otimes m \otimes {\rm id}^{\otimes j}) ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_n) \right) \\
&\qquad \qquad\qquad\qquad\qquad\qquad = (-1)^{s+p} \phi_{p,n-1} \left( ({\rm id}^{\otimes i} \otimes m_p \otimes {\rm id}^{\otimes j})\big({\sf u} l_{p_1,r_1}^p(\overline{a}_1) \otimes \cdots \otimes {\sf u} l_{p_n,r_n}^{p}(\overline{a}_n)\right).
\end{align*}
\end{gather*}
The desired conclusion is therefore a consequence of the fact that $\phi_p$ is an $\mathsf{A}_{\infty}$-morphism.
In order to finish the proof, one needs to check that $A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mapsto \operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ preserves compositions. This is a straightforward verification which we omit.
\end{proof}
The following result should also be noted.
\begin{lemma} \label{lem:3.10a}
Let $A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$ and $B_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$ be two semi-cosimplicial positively graded DG algebras. If $\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \colon A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \to B_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$ is a semi-cosimplicial $A_{\infty}$-quasi-isomorphism then $\operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \colon \operatorname{Tot}(A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to \operatorname{Tot}(B_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ is an $\mathsf{A}_{\infty}$-quasi-isomorphism.
\end{lemma}
\begin{proof}
Since $A_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$ and $B_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$ are assumed to be positively graded, the map $\operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_1$ is a morphism of first-quadrant double complexes. Besides, by our hypothesis on $\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$, we see that $\operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_1$ induces an isomorphism on the vertical directions. We conclude therefore that the map of spectral sequences is an isomorphism at the first page, and hence $\operatorname{Tot}(\phi_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_1$ induces an isomorphism in cohomology.
\end{proof}
With these preliminaries out of the way, we may now formulate the version of Gugenheim's $\mathsf{A}_{\infty}$ De~Rham theorem for $BG$ we are after. As in the previous section, consider the simplicial manifold $BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$. Then, the Bott-Shulman-Stasheff complex gives us a semi-cosimplicial DG algebra $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) = \{\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{p})\}_{p \geq 0}$. Also, by taking singular cochains, we get a second semi-cosimplicial DG algebra ${\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})=\{{\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{p})\}_{p \geq 0}$. Invoking Theorem~\ref{thm:2.1}, for each $p \geq 0$, there is an $\mathsf{A}_{\infty}$-morphism ${\sf{DR}}_p \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{p}) \to {\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{p})$ induced by Gugenheim's construction. Since the latter is natural with respect to the simplicial operations, this $\mathsf{A}_{\infty}$-morphisms commutes with the coface maps of $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ and ${\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$. Thus, we actually get a semi-cosimplicial $\mathsf{A}_{\infty}$-morphism ${\sf{DR}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to {\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$. One obtains the following.
\begin{theorem}
The induced $\mathsf{A}_{\infty}$-morphism $\operatorname{Tot}({\sf{DR}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \colon \operatorname{Tot}(\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})) \to \operatorname{Tot}({\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}))$ is an $\mathsf{A}_{\infty}$-quasi-isomorphism.
\end{theorem}
\begin{proof}
It follows from Theorem~\ref{thm:2.1} that ${\sf{DR}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to {\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ is a semi-cosimplicial $\mathsf{A}_{\infty}$-quasi-isomorphism. Hence conclusion is a consequence of Lemma~\ref{lem:3.10a}.
\end{proof}
\subsection{The Hochschild-De~Rham $\mathsf{A}_{\infty}$-quasi-isomorphism}\label{sec:3.3}
The goal of this subsection is to construct an $\mathsf{A}_{\infty}$-quasi-isomorphism between $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ and the DG algebra of Hochschild cochains on the space of singular chains ${\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)$. We begin with some generalities.
Let $A$ be a DG Hopf algebra and let $\varepsilon \colon A \to \ensuremath{\mathbbmss{R}}$ be its counit. For each $p \geq 0$, we set $\Delta_{p}(A) = A^{\otimes p}$ and define the maps $\partial_i \colon \Delta_{p}(A) \to \Delta_{p-1}(A)$ by
\begin{equation} \label{eqn:3.28}
\partial_{i} (a_1 \otimes \cdots \otimes a_p) = \begin{cases} \varepsilon(a_1) a_2 \otimes \cdots \otimes a_p & \text{if $i=0$,} \\
a_{1} \otimes \cdots \otimes a_{i-1} \otimes a_{i}a_{i+1} \otimes a_{i+2} \otimes \cdots \otimes a_p & \text{if $0 < i < p$,} \\
a_2 \otimes \cdots \otimes a_{p-1} \varepsilon(a_p) & \text{if $i=p$.}
\end{cases}
\end{equation}
The following lemma es crucial.
\begin{lemma}
The collection $\Delta_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A) = \{\Delta_{p}(A)\}_{p \geq 0}$ is a semi-simplicial DG coalgebra with face maps $\partial_i$.
\end{lemma}
\begin{proof}
Since, by hypothesis, $A$ is a DG Hopf algebra, its the product map is a morphism of DG coalgebras, which means that the maps $\partial_i$ for $0 < i < p$ are indeed morphisms of DG coalgebras. On the other hand, the fact that $\varepsilon \colon A \to \ensuremath{\mathbbmss{R}}$ is a morphism of DG Hopf algebras implies that both $\partial_0$ and $\partial_p$ are also morphisms of DG coalgebras. It remains only to check that $\partial_{i} \circ \partial_{j} = \partial_{j-1} \circ \partial_{i}$ for $i < j$. This is a routine calculation which we leave to the reader.
\end{proof}
We next let $\Delta^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A)$ be the semi-cosimplicial DG algebra obtained by dualising $\Delta_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A)$.
The following observation should be made.
\begin{lemma}\label{lem:3.13a}
The product in $\operatorname{Tot}(\Delta^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A))$ is given, for homogeneous elements $\varphi$ and $\psi$, by the formula
$$
{\sf s}^p \varphi \raisebox{-0.25ex}{\scalebox{1.3}{\,$\cdot$\,}} {\sf s}^{q}\psi = (-1)^{\vert \varphi \vert q} {\sf s}^{p+q} (\varphi \abxcup \psi),
$$
where $\abxcup$ designates the cup product in the Hochschild cochain complex $\operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A)$.
\end{lemma}
\begin{proof}
As earlier, for $p + q \leq n$, let $l_{p,q}^{n}\colon \Delta^{p}(A) \to \Delta^{n}(A)$ denote the map induced by that given in \eqref{eqn:3.28}. This induces a corresponding map $l_{p,q}^{n*}\colon A^{\otimes n} \to A^{\otimes p}$. We claim that, if $\Delta$ denotes the coproduct in $A$,
\begin{equation}\label{eqn:3.40a}
\left(l_{p,0}^{p+q*} \otimes l_{q,p}^{p+q*}\right) \circ \Delta^{\otimes (p+q)} = {\rm id}_{A^{\otimes (p+q)}}.
\end{equation}
To substantiate our claim, we fix homogeneous elements $a_1,\dots, a_{p+q} \in A$, and notice that
\begin{equation}\label{eqn:3.41a}
l_{p,0}^{p+q*}(a_1 \otimes \cdots \otimes a_{p+q}) = \varepsilon(a_{p+1} \cdots a_{p+q}) a_1 \otimes \cdots \otimes a_{p},
\end{equation}
and
\begin{equation}\label{eqn:3.42a}
l_{q,p}^{p+q*}(a_1 \otimes \cdots \otimes a_{p+q}) = \varepsilon(a_{1} \cdots a_{p}) a_{p+1} \otimes \cdots \otimes a_{p+q}.
\end{equation}
We also make use of Sweedler's notation and write, for each $1 \leq i \leq p+q$,
\begin{equation}\label{eqn:3.43a}
\Delta(a_i) = \sum a_{i(1)} \otimes a_{i (2)}.
\end{equation}
Using \eqref{eqn:3.41a}, \eqref{eqn:3.42a} and \eqref{eqn:3.43a}, we find that
\begin{align*}
&\left(l_{p,0}^{p+q*} \otimes l_{q,p}^{p+q*}\right) \left( \Delta^{\otimes (p+q)} (a_1 \otimes \cdots \otimes a_{p+q})\right) \\
&\,\,= \sum (-1)^{\sum_{ i < j } \vert a_{i(1)} \vert\vert a_{j(2)}\vert } \left(l_{p,0}^{p+q} \otimes l_{q,p}^{p+q}\right) (a_{1(1)} \otimes \cdots \otimes a_{p+q(1)} \otimes a_{1(2)} \otimes \cdots \otimes a_{p+q(2)}) \\
&\,\,= \sum (-1)^{\sum_{ i < j } \vert a_{i(1)} \vert\vert a_{j(2)}\vert } a_{1(1)} \otimes \cdots \otimes a_{p(1)} \varepsilon(a_{p+1(1)} \cdots a_{p+q(1)} a_{1(2)} \cdots a_{p(2)}) a_{p+1(2)} \otimes \cdots \otimes a_{p+q(2)} \\
&\,\,= \sum a_{1(1)} \otimes \cdots \otimes a_{p(1)} \varepsilon(a_{p+1(1)} \cdots a_{p+q(1)} a_{1(2)} \cdots a_{p(2)}) a_{p+1(2)} \otimes \cdots \otimes a_{p+q(2)} \\
&\,\,= \left(\sum a_{1(1)} \varepsilon(a_{1(2)}) \right) \otimes \cdots \otimes \left(\sum a_{p+q(1)} \varepsilon(a_{p+q(2)}) \right) \\
&\,\, = a_1 \otimes \cdots \otimes a_{p+q},
\end{align*}
where in the third equality we have used the fact that $\varepsilon$ vanishes on elements of positive degree in order to set the signs to zero, and, in the last one, that $\varepsilon$ is a counit for the coproduct. Thus \eqref{eqn:3.40a} is true.
Next, let us denote by $\mu^{p,q}$ the natural map from $A^{\otimes p*} \otimes A^{\otimes q*}$ to $(A^{\otimes p} \otimes A^{\otimes q})^*$. It is a simple matter to verify that
\begin{equation}\label{eqn:3.44a}
\mu^{p+q,p+q} \circ \left(l_{p,0}^{p+q} \otimes l_{p,q}^{p+q}\right) = \left(l_{p,0}^{p+q*} \otimes l_{p,q}^{p+q*}\right)^* \circ \mu^{p,q}.
\end{equation}
Moreover, we also have that
\begin{equation}\label{eqn:3.45a}
\mu^{p,q}(\varphi \otimes \psi) = \varphi \abxcup \psi,
\end{equation}
for $\varphi \in \Delta^p(A)$ and $\psi \in \Delta^{q}(A)$. Now, the product in $\operatorname{Tot}(\Delta^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A))$ is defined, up to a sign, as the composition
$$
\Delta^{\otimes(p+q)*}\circ \mu^{p+q,p+q} \circ \left(l_{p,0}^{p+q} \otimes l_{p,q}^{p+q}\right).
$$
More explicitly, for homogeneous elements $\varphi \in \Delta^p(A)$ and $\psi \in \Delta^{q}(A)$, we may write
\begin{equation}\label{eqn:3.46a}
{\sf s}^{p} \varphi \raisebox{-0.25ex}{\scalebox{1.3}{\,$\cdot$\,}} {\sf s}^{q}\psi = (-1)^{\vert \varphi\vert q} {\sf s}^{p+q}\Delta^{\otimes(p+q)*} \left( \mu^{p+q,p+q} \left(l_{p,0}^{p+q}\varphi \otimes l_{p,q}^{p+q} \psi \right) \right).
\end{equation}
Using \eqref{eqn:3.44a} and \eqref{eqn:3.45a}, the left hand side of \eqref{eqn:3.46a} becomes
\begin{align*}
&(-1)^{\vert \varphi\vert q} {\sf s}^{p+q}\Delta^{\otimes(p+q)*}\left(\left(l_{p,0}^{p+q*} \otimes l_{p,q}^{p+q*}\right)^* \left(\mu^{p,q} (\varphi \otimes \psi)\right) \right) \\
&\qquad\qquad\qquad= (-1)^{\vert \varphi\vert q} {\sf s}^{p+q}\Delta^{\otimes(p+q)*}\left(\left(l_{p,0}^{p+q*} \otimes l_{p,q}^{p+q*}\right)^* \left(\varphi \abxcup \psi \right) \right) \\
& \qquad\qquad\qquad= (-1)^{\vert \varphi\vert q} {\sf s}^{p+q} \left( \left(l_{p,0}^{p+q*} \otimes l_{p,q}^{p+q*}\right) \circ \Delta^{\otimes (p+q)} \right)^* \left(\varphi \abxcup \psi \right).
\end{align*}
Combining this with \eqref{eqn:3.40a} gives
$$
{\sf s}^{p} \varphi \raisebox{-0.25ex}{\scalebox{1.3}{\,$\cdot$\,}} {\sf s}^{q}\psi = (-1)^{\vert \varphi\vert q} {\sf s}^{p+q} \left(\varphi \abxcup \psi \right),
$$
as we wished to show.
\end{proof}
\begin{lemma}\label{lem:3.14a}
There is an isomorphism of DG algebras $\Theta \colon \operatorname{Tot}(\Delta^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A)) \to \operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(A)$, which is explicitly given by
$$
\Theta({\sf s}^{p}(\varphi)) ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_{p}) = (-1)^{p \vert \varphi\vert + \frac{p(p-1)}{2}+ \sum_{i=1}^{p-1} \vert a_i \vert (p-i)}\varphi(a_1 \otimes \cdots \otimes a_p),
$$
for homogeneous elements $\varphi \in \Delta^{p}(A)$ and $a_1,\dots, a_p \in A$.
\end{lemma}
\begin{proof}
It is obvious that $\Theta$ is a linear isomorphism. We have to show that it is also an algebra homomorphism. For this, let us fix homogeneous elements $\varphi \in \Delta^{p}(A)$, $\psi \in \Delta^{q}(A)$ and $a_1,\dots,a_{p+q} \in A$. Then, one the one hand, by virtue of Lemma~\ref{lem:3.13a},
\begin{align*}
&\Theta\left( {\sf s}^p \varphi \raisebox{-0.25ex}{\scalebox{1.3}{\,$\cdot$\,}} {\sf s}^q \psi \right) ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_{p+q}) \\
&\,\, =(-1)^{\vert\varphi\vert q} \Theta({\sf s}^{p+q}(\varphi \abxcup \psi)) ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_{p+q}) \\
&\,\, =(-1)^{\vert\varphi\vert q+ (p+q)(\vert\varphi\vert + \vert\psi\vert) + \frac{(p+q)(p+q-1)}{2} + \sum_{i=1}^{p+q-1}\vert a_i \vert (p+q-i)} (\varphi \abxcup \psi)({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_{p+q}) \\
&\,\, = (-1)^{\vert\varphi\vert q+ (p+q)(\vert\varphi\vert + \vert\psi\vert) + \frac{(p+q)(p+q-1)}{2} + \sum_{i=1}^{p+q-1}\vert a_i \vert (p+q-i) + \vert\varphi\vert \vert\psi\vert} \varphi(a_1 \otimes \cdots \otimes a_p)\psi(a_{p+1} \otimes \cdots \otimes a_{p+q}) \\
&\,\, = (-1)^{p(\vert\varphi\vert + \vert\psi\vert) + (q + \vert\varphi\vert)\vert\psi\vert + \frac{(p+q)(p+q-1)}{2} + \sum_{i=1}^{p+q-1}\vert a_i \vert (p+q-i)} \varphi(a_1 \otimes \cdots \otimes a_p)\psi(a_{p+1} \otimes \cdots \otimes a_{p+q}).
\end{align*}
On the other hand,
\begin{align*}
&\left(\Theta({\sf s}^p \varphi) \abxcup \Theta({\sf s}^{q}\psi)\right) ({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_{p+q}) \\
&\,\, = (-1)^{(q + \vert\psi\vert)(p + \vert\varphi\vert)} \Theta({\sf s}^p \varphi)({\sf u} a_1 \otimes \cdots \otimes {\sf u} a_{p}) \Theta({\sf s}^q \psi)({\sf u} a_{p+1} \otimes \cdots \otimes {\sf u} a_{p+q}) \\
&\,\, = (-1)^{(q + \vert\psi\vert)(p + \vert\varphi\vert) + \frac{p(p-1)+q(q-1)}{2} + p \vert\varphi\vert + \sum_{i =1}^{p}\vert a_i \vert (p+q-i) + q \vert\psi\vert + \sum_{i=p+1}^{p+q}\vert a_i \vert (p+q-i)} \\
&\qquad \qquad \qquad \qquad\qquad \qquad\qquad \qquad\qquad \qquad\qquad \qquad \,\,\,\,\, \times \varphi(a_1 \otimes \cdots \otimes a_p)\psi(a_{p+1} \otimes \cdots \otimes a_{p+q}) \\
&= \,\, (-1)^{ p(\vert\varphi\vert + \vert\psi\vert) + (q + \vert\varphi\vert)\vert\psi\vert + \frac{(p+q)(p+q-1)}{2} + \sum_{i=1}^{p+q-1}\vert a_i \vert (p+q-i)} \varphi(a_1 \otimes \cdots \otimes a_p)\psi(a_{p+1} \otimes \cdots \otimes a_{p+q}).
\end{align*}
By comparing the last two equalities above we find that
$$
\Theta\left( {\sf s}^p \varphi \raisebox{-0.25ex}{\scalebox{1.3}{\,$\cdot$\,}} {\sf s}^q \psi \right) = \Theta({\sf s}^p \varphi) \abxcup \Theta({\sf s}^{q}\psi).
$$
We conclude that $\Theta$ is indeed an algebra homomorphism. We leave it to the reader the task of checking that it also preserves the differentials.
\end{proof}
Before moving forward, we need some definitions. An $n$-singular simplex $\sigma \colon \Delta_{n} \to G$ is said to be \emph{degenerate} if it can be factored as $\sigma = \sigma' \circ \eta_i$, where $\sigma' \colon \Delta_{n-1} \to G$ is an $(n-1)$-singular simplex and $\eta_i \colon \Delta_{n-1} \to \Delta_{n}$ is a degeneracy map. It is not hard to see that the vector space $\mathfrak{I}$ generated by degenerate simplices on $G$ is both a DG ideal and a DG coideal of ${\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)$. Thus, taking the quotient vector space $\overline{{\mathrm C}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G) = {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G) / \mathfrak{I}$, we obtain canonically a DG Hopf algebra structure on $\overline{{\mathrm C}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)$ such that the projection ${\sf q} \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G) \to \overline{{\mathrm C}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)$ is a morphism of DG Hopf algebras. It can be shown that ${\sf q}$ is in fact a quasi-isomorphism. We shall refer to the DG Hopf algebra $\overline{{\mathrm C}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)$ as the algebra of \emph{normalized} singular chains on $G$.
Now let us again consider the simplicial manifold $BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$ from Section~\ref{sec:3.1}. As the construction above is functorial in $G$, it defines a semi-simplicial DG coalgebra $\overline{{\mathrm C}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ and a corresponding projection ${\sf q}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to \overline{{\mathrm C}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$. For each $p \geq 0$, we let ${\sf{EZ}}_{p} \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)^{\otimes p} \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{p})$ be the Eilenberg-Zilber map defined as reviewed in Section~\ref{sec:2.4}. We further let $\overline{{\sf{EZ}}}_{p} \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)^{\otimes p} \to \overline{{\mathrm C}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{p})$ be defined as the composition
\begin{equation}
\overline{{\sf{EZ}}}_{p} = {\sf q}_{p} \circ {\sf{EZ}}_{p}.
\end{equation}
Owing to Proposition~\ref{prop:2.2} and the preceding discussion, the map $\overline{{\sf{EZ}}}_{p}$ is a quasi-isomorphism of DG coalgebras. Also, we have the following.
\begin{lemma}
The collection $\{\overline{{\sf{EZ}}}_{p}\}_{p \geq 0}$ determines a morphism of semi-simplicial DG coalgebras $\overline{{\sf{EZ}}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \colon \Delta_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)) \to \overline{{\mathrm C}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$.
\end{lemma}
\begin{proof}
We need merely to show that $\overline{{\sf{EZ}}}_{p-1} \circ \partial_{i} = \partial_{i} \circ \overline{{\sf{EZ}}}_{p}$ for $0 \leq i \leq p$. First, consider the case $i =0$. For every collection $\sigma_{1},\dots, \sigma_{p}$ of singular simplices on $G$, we see from \eqref{eqn:3.28} that
\begin{align*}
\overline{{\sf{EZ}}}_{p-1} \left( \partial_{0} (\sigma_1 \otimes \cdots \otimes \sigma_{p})\right) &= \varepsilon(\sigma_1)\overline{{\sf{EZ}}}_{p-1} \left( \sigma_2 \otimes \cdots \otimes \sigma_{p}\right) \\
& = \begin{cases} 0 & \text{if $\vert \sigma_1 \vert > 0$,} \\
\overline{{\sf{EZ}}}_{p-1} \left( \sigma_2 \otimes \cdots \otimes \sigma_{p}\right) & \text{if $\vert \sigma_1 \vert = 0$.}
\end{cases}
\end{align*}
On the other hand, we can see from \eqref{eqn:2.18} that
\begin{align*}
\partial_{0} \left(\overline{{\sf{EZ}}}_{p} (\sigma_1 \otimes \cdots \otimes \sigma_{p}) \right) &= \partial_{0} \left( \sum_{\chi \in \mathfrak{S}_{\vert \sigma_1 \vert,\dots, \vert \sigma_p \vert }}(-1)^{\chi} {\sf q}_{p} \circ (\sigma_1 \times \cdots \times \sigma_{p}) \circ \chi_{*} \right) \\
&= \sum_{\chi \in \mathfrak{S}_{\vert \sigma_1 \vert,\dots, \vert \sigma_p \vert }}(-1)^{\chi} {\sf q}_{p-1} \circ \partial_{0} \circ (\sigma_1 \times \cdots \times \sigma_{p}) \circ \chi_{*} .
\end{align*}
If $\vert \sigma_1 \vert = 0$, then $\partial_{0} \circ (\sigma_1 \times \cdots \times \sigma_p) = \sigma_2 \times \cdots \times \sigma_p$ and therefore
\begin{equation*}
\partial_{0} \left(\overline{{\sf{EZ}}}_{p} (\sigma_1 \otimes \cdots \otimes \sigma_{p}) \right) = \overline{{\sf{EZ}}}_{p-1} (\sigma_2 \otimes \cdots \otimes \sigma_{p}).
\end{equation*}
If $\vert \sigma_1 \vert = 0$, then ${\sf q}_{p-1} \circ \partial_{0} \circ (\sigma_1 \times \cdots \times \sigma_{p}) \circ \chi_{*} = 0$, since $\partial_{0} \circ (\sigma_1 \times \cdots \times \sigma_{p}) \circ \chi_{*}$ is degenerate, from which it follows that
\begin{equation*}
\partial_{0} \left(\overline{{\sf{EZ}}}_{p} (\sigma_1 \otimes \cdots \otimes \sigma_{p}) \right) = 0,
\end{equation*}
and hence the result. The case $i = p$ is completely analogous. So there only remains the case $0 < i < p$. On the one hand, using \eqref{eqn:3.28} gives
\begin{equation*}
\overline{{\sf{EZ}}}_{p-1} \left( \partial_{i} (\sigma_1 \otimes \cdots \otimes \sigma_{p})\right) = {\sf q}_{p-1} \left({\sf{EZ}}_{p-1}(\sigma_{1} \otimes \cdots \otimes \sigma_{i-1} \otimes \sigma_{i} \sigma_{i+1} \otimes \sigma_{i+2} \otimes \cdots \otimes \sigma_p) \right).
\end{equation*}
On the other hand, observing that, in the notation employed at the end Section~\ref{sec:2.4}, the face map $\partial_{i} \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{p}) \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{p-1})$ is given by $\partial_{i}=({\rm id}^{\times (i-1)} \times \mu \times {\rm id}^{\times (p-i)})_*$ and that, therefore,
\begin{equation*}
\partial_{i} \circ {\sf{EZ}}_{p} = {\sf{EZ}}_{p-1} \circ \big( {\rm id}^{\otimes(i-1)} \otimes (\mu_{*} \circ {\sf{EZ}}_2) \otimes {\rm id}^{\otimes(p-1)}\big),
\end{equation*}
we find that
\begin{align*}
\partial_{i} \left( \overline{{\sf{EZ}}}_{p}(\sigma_1 \otimes \cdots \otimes \sigma_p )\right) & = \partial_i \left( {\sf q}_{p} \left( {\sf{EZ}}_{p}(\sigma_1 \otimes \cdots \otimes \sigma_p )\right) \right) \\
&= {\sf q}_{p-1} \left( \partial_i \left( {\sf{EZ}}_{p}(\sigma_1 \otimes \cdots \otimes \sigma_p ) \right)\right) \\
&= {\sf q}_{p-1} \left({\sf{EZ}}_{p-1}(\sigma_{1} \otimes \cdots \otimes \sigma_{i-1} \otimes \sigma_{i} \sigma_{i+1} \otimes \sigma_{i+2} \otimes \cdots \otimes \sigma_p) \right),
\end{align*}
as wished.
\end{proof}
Let us now write $\overline{{\mathrm C}}{}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ to denote the semi-cosimplicial DG algebra obtained by dualising $\overline{C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$. Also, let us denote by ${\sf e}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \colon \overline{{\mathrm C}}{}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to {\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ the inclusion dual to the projection ${\sf q}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to \overline{{\mathrm C}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$. We make an observation to be applied in the subsequent argument.
\begin{lemma}\label{lem:3.10}
The semi-cosimplicial $\mathsf{A}_{\infty}$-morphism ${\sf{DR}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to {\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\bullet})$ factors thorugh $\overline{{\mathrm C}}{}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$, that is, there is a semi-cosimplicial $\mathsf{A}_{\infty}$-morphism $\overline{{\sf{DR}}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to \overline{{\mathrm C}}{}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ such that ${\sf{DR}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} = {\sf e}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \circ \overline{{\sf{DR}}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$.
\end{lemma}
\begin{proof}
It suffices to show that ${\sf{DR}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to {\mathrm C}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\bullet})$ takes values in those singular cochains that vanish on degenerate singular simplices. But this holds by Proposition~3.26 of \cite{Abad-Schatz2013}.
\end{proof}
Next we consider the semi-cosimplicial DG algebra $\Delta^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ dual to $\Delta^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$. By taking the dual of $\overline{{\sf{EZ}}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \colon \Delta_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)) \to \overline{{\mathrm C}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$, we obtain a morphism of semi-cosimplicial DG algebras $\overline{{\sf{EZ}}}_{\raisebox{0.35ex}{\scalebox{0.6}{$\bullet$}}}^{\ast} \colon \overline{{\mathrm C}}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to \Delta^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$. We let ${\sf{DR}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}^{\Delta} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to \Delta^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ be the semi-cosimplicial $\mathsf{A}_{\infty}$-morphism defined as the composition
\begin{equation} \label{eqn:3.48a}
{\sf{DR}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}^{\Delta} =\overline{{\sf{EZ}}}_{\raisebox{0.35ex}{\scalebox{0.6}{$\bullet$}}}^{\ast} \circ \overline{{\sf{DR}}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}},
\end{equation}
where $\overline{{\sf{DR}}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to \overline{{\mathrm C}}{}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ is the semi-cosimplicial $\mathsf{A}_{\infty}$-morphism from Lemma~\ref{lem:3.10}. By Proposition~\ref{prop:3.5}, the latter induces an $\mathsf{A}_{\infty}$-morphism $\operatorname{Tot}({\sf{DR}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}^{\Delta}) \colon \operatorname{Tot} (\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})) \to \operatorname{Tot}(\Delta^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)))$. We then apply Lemma~\ref{lem:3.14a}, to get and $\mathsf{A}_{\infty}$-morphism ${\sf{DR}}^{\Theta} \colon \operatorname{Tot} (\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})) \to \operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ defined as the composition
\begin{equation}
{\sf{DR}}^{\Theta} = \Theta \circ \operatorname{Tot}({\sf{DR}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}^{\Delta}).
\end{equation}
These observations taken together with the preceding results yield the following.
\begin{theoremD}
The induced $\mathsf{A}_{\infty}$-morphism ${\sf{DR}}^{\Theta} \colon \operatorname{Tot} (\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})) \to \operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ is an $\mathsf{A}_{\infty}$-quasi-isomorphism.
\end{theoremD}
\begin{proof}
Because of Lemma~\ref{lem:3.14a}, it is enought to prove that $\operatorname{Tot}({\sf{DR}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}^{\Delta})$ is an $\mathsf{A}_{\infty}$-quasi-isomorphism. In order to do so, notice that both $\overline{{\sf{EZ}}}_{\raisebox{0.35ex}{\scalebox{0.6}{$\bullet$}}}^{\ast}$ and $\overline{{\sf{DR}}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$ are semi-cosimplicial $\mathsf{A}_{\infty}$-quasi-isomorphism. Thus, taking note of the definition \eqref{eqn:3.48a}, we conclude that ${\sf{DR}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}^{\Delta}$ is also a semi-cosimplicial $\mathsf{A}_{\infty}$-quasi-isomorphism. The desired assertion now follows from Lemma~\ref{lem:3.10a}.
\end{proof}
In the remaining part of this section, we will prove a vanishing result for the $\mathsf{A}_{\infty}$-morphism ${\sf{DR}}^{\Theta}$, which will be needed later. First a little terminology. We say that an $r$-singular simplex $\sigma \colon \Delta_{r} \to BG_{p}$ is \emph{decomposable} if there is a collection of $r_i$-singular simplices $\sigma_i \colon \Delta_{r_i} \to G$ with $i = 1,\dots, p$ and $r = \sum_{i=1}^{p} r_i$, together with an $(r_1,\dots,r_p)$-shuffle $\chi$ such that
\begin{equation}
\sigma = (\sigma_1 \times \cdots \times \sigma_p) \circ \chi_*.
\end{equation}
It is immediately apparent from \eqref{eqn:2.18} that, for each $p \geq 0$, the image of the Eilenberg-Zilber map ${\sf{EZ}}_{p} \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)^{\otimes p} \to {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_p)$ is generated by decomposable singular simplices.
\begin{proposition}\label{prop:3.18}
Let $n > 1$ and consider differential forms $\omega_1 \in \Omega^{q_1}(G), \dots, \omega_n \in \Omega^{q_n}(G)$. Then
\begin{equation*}
{\sf{DR}}^{\Theta}_n ({\sf u} \omega_1 \otimes \cdots \otimes {\sf u} \omega_n) = 0.
\end{equation*}
\end{proposition}
\begin{proof}
We will in fact show that
\begin{equation*}
\operatorname{Tot}({\sf{DR}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}^{\Delta})_n ({\sf u} \omega_1 \otimes \cdots \otimes {\sf u} \omega_n) = 0,
\end{equation*}
which clearly suffices thanks to Lemma~\ref{lem:3.14a}. To begin with, by Proposition~\ref{prop:3.5}, we know that $\operatorname{Tot}({\sf{DR}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}^{\Delta}) = \operatorname{Tot}(\overline{{\sf{EZ}}}_{\raisebox{0.35ex}{\scalebox{0.6}{$\bullet$}}}^{\ast}) \circ \operatorname{Tot}(\overline{{\sf{DR}}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$.
Thus, in view of our previous remark, it will be enough to show that if $\sigma \colon \Delta_{r} \to BG_{n}$ with $r = \sum_{i=1}^{n} q_i - n +1$ is an decomposable singular simplex, then
\begin{equation*}
\operatorname{Tot}(\overline{{\sf{DR}}}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})_n ({\sf u} \omega_1 \otimes \cdots \otimes {\sf u} \omega_n) (\sigma) = 0.
\end{equation*}
According to definition \eqref{eqn:3.24}, this means that
\begin{equation}\label{eqn:3.32}
{\sf{DR}}_{n,n} \big({\sf u} l_{1,0}^{n} (\omega_{1}) \otimes \cdots \otimes {\sf u} l_{1,n-1}^{n} (\omega_{n}) \big)(\sigma) = 0.
\end{equation}
Next, notice that if, for $i = 1,\dots, n$, we let ${\rm pr}_{i} \colon BG_{n} \to G$ be the projection onto the $i$th factor, then $l_{1,i-1}^{n} = {\rm pr}_{i}^*$. Thus \eqref{eqn:3.32} becomes
\begin{equation}\label{eqn:3.33}
{\sf{DR}}_{n,n} \big({\sf u} {\rm pr}_1^*\omega_{1} \otimes \cdots \otimes {\sf u} {\rm pr}_n^*\omega_{n} \big)(\sigma) = 0.
\end{equation}
On the other hand, using the notation for the $\mathsf{A}_{\infty}$ version of De Rham map from \S\ref{sec:2.3}, we have that
\begin{align*}
{\sf{DR}}_{n,n} \big({\sf u} {\rm pr}_1^*\omega_{1} \otimes \cdots \otimes {\sf u} {\rm pr}_n^*\omega_{n} \big)(\sigma) &= \pm \int_{I^{r-1}} \theta_r^*(\mathcal{P} \sigma)^* \int_{\Delta_n} {\rm ev}^* \big( \pi_1^*{\rm pr}_1^*\omega_1 \wedge \cdots \wedge \pi_n^*{\rm pr}_n^*\omega_1\big) \\
&= \pm \int_{I^{r-1}} \theta_r^* \int_{\Delta_n} ({\rm id} \times \mathcal{P} \sigma)^*{\rm ev}^* \big( \pi_1^*{\rm pr}_1^*\omega_1 \wedge \cdots \wedge \pi_n^*{\rm pr}_n^*\omega_1\big).
\end{align*}
Consequently, to show \eqref{eqn:3.33}, it is sufficient to show that
\begin{equation} \label{eqn:3.34}
({\rm id} \times \mathcal{P} \sigma)^*{\rm ev}^* \big( \pi_1^*{\rm pr}_1^*\omega_1 \wedge \cdots \wedge \pi_n^*{\rm pr}_n^*\omega_1\big) = 0.
\end{equation}
Now let us use the fact that $\sigma$ is decomposable. By definition, this means that
\begin{equation*}
\sigma = (\sigma_1 \times \cdots \times \sigma_n) \circ \chi_*,
\end{equation*}
for a collection of $r_i$-singular simplices $\sigma_i \colon \Delta_{r_i} \to G$ with $i=1,\dots, n$ and $r = \sum_{i=1}^{n} r_i$, and for an $(r_1,\dots,r_n)$-shuffle $\chi$. Therefore,
\begin{equation*}
{\rm id} \times \mathcal{P} \sigma = ({\rm id} \times \mathcal{P}(\sigma_1 \times \cdots \times \sigma_n)) \circ ({\rm id} \times \mathcal{P} \chi_*),
\end{equation*}
and hence
\begin{equation*}
{\rm ev} \circ ({\rm id} \times \mathcal{P} \sigma) = {\rm ev} \circ ({\rm id} \times \mathcal{P}(\sigma_1 \times \cdots \times \sigma_n)) \circ ({\rm id} \times \mathcal{P} \chi_*) = (\sigma_1 \times \cdots \times \sigma_n)^{\times n} \circ {\rm ev} \circ \mathcal{P} \chi_*.
\end{equation*}
Thus, to show \eqref{eqn:3.34}, it will be enough to show that
\begin{equation}\label{eqn:3.35}
{\rm ev}^* ((\sigma_1 \times \cdots \times \sigma_n)^{\times n})^*\big( \pi_1^*{\rm pr}_1^*\omega_1 \wedge \cdots \wedge \pi_n^*{\rm pr}_n^*\omega_1\big) = 0.
\end{equation}
Now a simple calculation reveals that
\begin{equation} \label{eqn:3.36}
{\rm ev}^* ((\sigma_1 \times \cdots \times \sigma_n)^{\times n})^*\big( \pi_1^*{\rm pr}_1^*\omega_1 \wedge \cdots \wedge \pi_n^*{\rm pr}_n^*\omega_1\big) = {\rm ev}^* \big( \sigma_1^*\omega_1 \wedge \cdots \wedge \sigma_n^*\omega_1\big).
\end{equation}
On the other hand, since $n > 1$, we have that $r = \sum_{i=1}^{n} q_i - n + 1 < \sum_{i=1}^{n} q_i$. But $r = \sum_{i=1}^{n} r_i$, so there must exists a $k \in \{ 1,\dots, n\}$ such that $r_k < q_k$. This implies that $\sigma_k^*\omega_k = 0$, and as a result \eqref{eqn:3.35} follows from \eqref{eqn:3.36}.
\end{proof}
\section{$\mathsf{A}_{\infty}$-quasi-equivalence of DG categories}\label{sec:4}
In this section we prove the main result of the paper, which is the construction of a zig-zag of $\mathsf{A}_{\infty}$-quasi-equivalences between the DG enhancements of the categories $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$ and $\operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$.
\subsection{DG enhancement of the category $\operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$}\label{sec:4.1}
In this subsection we describe a DG enhancement of the category $\operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$.
Let $V$ be an object of $\operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$. For $x \in \mathfrak{g}$, by a slight abuse of notation, we will indistinctly write $i_x$ and $L_x$ for the contraction and Lie derivative operators acting on ${\mathrm W}\mathfrak{g}$ or $V$. An element $\alpha \in {\mathrm W}\mathfrak{g} \otimes V$ will be called \emph{basic} if
\begin{align}\label{eqn:4.1}
\begin{split}
(i_x \otimes 1) \alpha &= 0, \\
(L_x \otimes 1 + 1 \otimes L_x)\alpha &= 0,
\end{split}
\end{align}
for every $x \in \mathfrak{g}$. Since the operators $i_x \otimes 1$ and $L_x \otimes 1 + 1 \otimes L_x$ are derivations, the basic elements form a graded subspace of ${\mathrm W}\mathfrak{g} \otimes V$. It will be denoted by $({\mathrm W}\mathfrak{g} \otimes V)_{{\rm bas}}$.
Next, consider the DG algebra ${\mathrm W}\mathfrak{g} \otimes \operatorname{End}(V)$ with multiplication induced by the composition operation $\operatorname{End}(V)$ and the differential $d_{{\mathrm W}} + \delta$. Fix a basis $e_a$ of $\mathfrak{g}$ with structure constant $f^{a}_{\phantom{a}bc}$ and recall from Section~\ref{sec:2.4} that $t^{a}$ stands for the degree $1$ generators of $\Lambda^1\mathfrak{g}$ and $w^{a}$ stands for the degree $2$ generators of ${\mathrm S}^1 \mathfrak{g}$.
\begin{lemma}
The element $t^{a} \otimes L_{a} - w^{a} \otimes i_{a}$ is a Maurer-Cartan element of ${\mathrm W}\mathfrak{g} \otimes \operatorname{End}(V)$.
\end{lemma}
\begin{proof}
On the one hand, according to \eqref{eqn:2.36} and the relations \eqref{eqn:2.22},
\begin{align*}
d_{{\mathrm W}} (t^{a} \otimes L_{a} - w^{a} \otimes i_{a} ) &= d_{{\mathrm W}} t^{a} \otimes L_a - d_{{\mathrm W}} w^{a} \otimes i_a \\
&= w^{a} \otimes L_a - \frac{1}{2} f^{a}_{bc} t^{b} t^{c} \otimes L_a - f^{a}_{bc} w^{b} t^{c} \otimes i_a,
\end{align*}
and
\begin{align*}
\delta( t^{a} \otimes L_{a} - w^{a} \otimes i_{a} ) &=- t^{a} \otimes [\delta, L_a] - w^{a} \otimes [\delta, i_a] =- w^{a} \otimes L_a .
\end{align*}
Hence,
$$
(d_{{\mathrm W}} +\delta) (t^{a} \otimes L_{a} - w^{a} \otimes i_{a} ) = -f^{a}_{bc} w^{b} t^{c} \otimes i_a - \frac{1}{2} f^{a}_{bc}t^{b} t^{c} \otimes L_a
$$
On the other hand, again using the relations \eqref{eqn:2.22}, we find that
\begin{align*}
(t^{b} \otimes L_{b}& -w^{b} \otimes i_{b})(t^{c} \otimes L_{c}- w^{c} \otimes i_{c}) \\
&= t^{b} t^{c} \otimes L_bL_c - t^b w^c \otimes L_b i_c + w^b t^c \otimes i_b L_c + w^{b} w^{c} \otimes i_b i_c \\
&= \frac{1}{2} t^{b} t^{c} \otimes [L_b,L_c] - t^b w^c \otimes [L_b,i_c] - t^b w^c \otimes i_c L_b + w^b t^c \otimes i_b L_c + \frac{1}{2} w^{b} w^{c} \otimes [i_b,i_c] \\
&= \frac{1}{2} f_{bc}^{a} t^b t^c \otimes L_a - f_{bc}^{a} t^b w^c \otimes i_a \\
&= \frac{1}{2} f_{bc}^{a} t^b t^c \otimes L_a + f_{bc}^{a} w^b t^c \otimes i_a.
\end{align*}
In conclusion, we obtain
$$
(d_{{\mathrm W}} + \delta) (t^{a} \otimes L_{a} - w^{a} \otimes i_{a}) + (t^{b} \otimes L_{b} -w^{b} \otimes i_{b})(t^{c} \otimes L_{c}- w^{c} \otimes i_{c}) = 0 ,
$$
as required.
\end{proof}
This result has the following important consequence.
\begin{corollary}
The operator $D$ in ${\mathrm W}\mathfrak{g} \otimes V$ given by
$$
D = d_{{\mathrm W}} + \delta + t^{a} \otimes L_{a} - w^{a} \otimes i_{a},
$$
is a derivation of homogenous degree $1$ that satisfies $D^2 = 0$.
\end{corollary}
Also, the following property holds true.
\begin{lemma}
The differential $D$ preserves the graded subspace $({\mathrm W}\mathfrak{g} \otimes V)_{{\rm bas}}$.
\end{lemma}
\begin{proof}
It suffices to show that
$$
[D, L_{c} \otimes 1 + 1 \otimes L_{c}] = 0,
$$
and
$$
[D, i_{c} \otimes 1] = L_{c} \otimes 1 + 1 \otimes L_{c}.
$$
Fix an element of ${\mathrm W}\mathfrak{g} \otimes V$ of the form $\xi \otimes v$. Then a straightforward computation gives
\begin{align*}
D\left( (L_{c} \otimes 1 + 1 \otimes L_{c})(\xi \otimes v)\right) = & \, d_{{\mathrm W}}(L_c \xi) \otimes v + (-1)^{\vert \xi \vert} L_{c}\xi \otimes \delta v \\
& + d_{{\mathrm W}} \xi \otimes L_c v + (-1)^{\vert \xi \vert} \xi \otimes \delta( L_c v) \\
& + (t^{a} L_{c} \xi) \otimes L_a v -(-1)^{\vert \xi \vert} (w^{a} L_c \xi) \otimes i_a v \\
&+ (t^{a} \xi ) \otimes L_a(L_c v) - (-1)^{\vert \xi \vert} (w^{a} \xi) \otimes i_a(L_c v) ,
\end{align*}
and
\begin{align*}
(L_{c} \otimes 1 + 1 \otimes L_{c})\left( D(\xi \otimes v)\right) = & \, L_c(d_{{\mathrm W}} \xi) \otimes v + (-1)^{\vert \xi \vert} L_{c}\xi \otimes \delta v \\
&+ d_{{\mathrm W}} \xi \otimes L_c v + (-1)^{\vert \xi \vert} \xi \otimes L_c (\delta v) \\
& - f_{cb}^{a} ( t^{b} \xi) \otimes L_a v+ (t^{a} L_{c} \xi) \otimes L_a v \\
& +(-1)^{\vert \xi \vert} f_{cb}^{a}( w^{b} \xi) \otimes i_a v -(-1)^{\vert \xi \vert} (w^{a} L_c \xi) \otimes i_a v \\
& + (t^{a} \xi ) \otimes L_c(L_a v) - (-1)^{\vert \xi \vert} (w^{a} \xi) \otimes L_c(i_a v).
\end{align*}
Therefore, from \eqref{eqn:2.22}, it follows that
\begin{align*}
[D, L_{c} \otimes 1 + 1 \otimes L_{c}] (\xi \otimes v) &= [d_{{\mathrm W}},L_c]\xi \otimes v + (-1)^{\vert \xi \vert} \xi \otimes [\delta,L_c]v \\
&\phantom{=}\, + f_{cb}^{a} ( t^{b} \xi) \otimes L_a v - (-1)^{\vert \xi \vert} f_{cb}^{a}( w^{b} \xi) \otimes i_a v \\
&\phantom{=}\, + (t^{a} \xi ) \otimes [L_a,L_c]v + (-1)^{\vert \xi \vert} (w^{a} \xi) \otimes [L_c,i_a]v \\
&= f_{cb}^{a} ( t^{b} \xi) \otimes L_a v - (-1)^{\vert \xi \vert} f_{cb}^{a}( w^{b} \xi) \otimes i_a v \\
&\phantom{=}\, + f_{ac}^{b} ( t^{a} \xi) \otimes L_b v + (-1)^{\vert \xi \vert} f_{ca}^{b}( w^{a} \xi) \otimes i_b v \\
&= 0.
\end{align*}
Thus the first identity is established. On the other hand, again by a direct computation,
\begin{align*}
D \left( (i_c \otimes 1)(\xi \otimes v) \right) &= d_{{\mathrm W}} (i_c \xi) \otimes v - (-1)^{\vert \xi \vert} i_c \xi \otimes \delta v \\
&\phantom{=}\, + (t^{a} i_c \xi) \otimes L_a v + (-1)^{\vert \xi \vert} (w^{a} i_c \xi) \otimes i_a v,
\end{align*}
and
\begin{align*}
(i_c \otimes 1)\left( D(\xi \otimes v)\right) &= i_c (d_{{\mathrm W}} \xi) \otimes v + (-1)^{\vert \xi \vert} i_c \xi \otimes \delta v \\
&\phantom{=}\, +\delta^{a}_{c}\xi L_a v - (t^{a} i_c \xi) \otimes L_a v \\
&\phantom{=}\, -(-1)^{\vert \xi \vert} (w^{a} i_c \xi) \otimes i_a v.
\end{align*}
Hence, using \eqref{eqn:2.22} again, this gives
\begin{align*}
[D, i_{c} \otimes 1] (\xi \otimes v) &= [d_{{\mathrm W}},i_c] \xi \otimes v + \delta^{a}_{c}\xi L_a v \\
&= L_c \xi \otimes v + \xi L_c v \\
&= (L_c \otimes 1 + 1 \otimes L_c) (\xi \otimes v),
\end{align*}
and, consequently, the second identity also holds.
\end{proof}
The preceding discussion allows us to define a DG category, which provides a DG enhancement of the category $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$, by the following data. The objects of this DG category are the same as those of $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$. For any two objects $V$ and $V'$, with corresponding differentials $D$ and $D'$, the space of morphisms is the graded vector space $({\mathrm W}\mathfrak{g} \otimes \operatorname{Hom}(V,V'))_{{\rm bas}}$ with the differential $\partial_{D,D'}$ acting according to the formula
\begin{equation}\label{eqn:4.2aa}
\partial_{D,D'}\varphi = D' \circ \varphi - (-1)^k \varphi \circ D,
\end{equation}
for any homogeneous element $\varphi$ of degree $k$. The DG category given by this data will be denoted by $\operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})$.
\subsection{The Bott-Shulman-Stasheff DG category}\label{sec:4.2}
In this subsection we introduce a DG category canonically associated to the Lie group $G$, which is based on the Bott-Shulman-Stasheff model discussed in \S\ref{sec:3.1}. This DG category will play an essential intermediate role in the proof of our main result.
Let $V$ be object of $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$ and consider the DG algebra $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})\otimes \operatorname{End}(V)$ with multiplication induced by the composition operation on $\operatorname{End}(V)$ and the differential $\bar{d} + \partial + \bar{\delta}$, where $\bar{\delta}$ here is defined as $(-1)^{p}$ times the differential $\delta$ when acting on $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{p})\otimes \operatorname{End}(V)$. Let also $\Phi_V$ be the left-equivariant representation form associated to $V$. We note that $\Phi_V$ may be thought of as an element of $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})\otimes \operatorname{End}(V)$ of homogeneous of degree $1$ with respect to the total degree.
\begin{lemma}
The element $\Phi_V - {\rm id}_V$ is a Maurer-Cartan element of $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})\otimes \operatorname{End}(V)$.
\end{lemma}
\begin{proof}
We must show that
$$
( -d + \partial - \delta) (\Phi_V - {\rm id}_V) + (\Phi_V - {\rm id}_V) \abxcup (\Phi_V - {\rm id}_V) = 0.
$$
To prove this, we first notice that $d ({\rm id}_V) = 0$ and $\delta({\rm id}_V) = 0$. Moreover, by decomposing $\Phi_V = \sum_{k \geq 0} \Phi_V^{(k)}$ and bringing to mind the ``descent equations'' \eqref{eqn:2.34}, we obtain that
$$
(d + \delta) \Phi_V = 0.
$$
We are thus left to show that
$$
\partial (\Phi_V - {\rm id}_V) + (\Phi_V - {\rm id}_V) \abxcup (\Phi_V - {\rm id}_V) = 0.
$$
Toward this end, we notice that in the present situation $\partial = \varepsilon_0^* - \varepsilon_1^* + \varepsilon_2^*$. Furthermore, according to \eqref{eqn:3.2}, the face maps $\varepsilon_0$, $\varepsilon_1$ and $\varepsilon_2$ coincide with the projection onto the second component $\pi_2$, the multiplication map $\mu$ and the projection onto the first component $\pi_1$, respectively. Therefore,
$$
\partial(\Phi_V - {\rm id}_V) = \pi_2^*\Phi_V - \mu^*\Phi_V + \pi_1^*\Phi_V - {\rm id}_V.
$$
On the other hand, taking note of the condition \eqref{eqn:2.35}, we have, for the first cup product term,
\begin{align*}
\Phi_V \abxcup \Phi_V &= \sum_{k \geq 2} \sum_{i + j = k} \Phi_V^{(i)} \abxcup \Phi_V^{(j)} = \sum_{k \geq 2} \sum_{i + j = k} (-1)^{i(1+ j)}(-1)^{i} \pi_1^*\Phi_V^{(i)} \wedge \pi_2^*\Phi_V^{(j)} \\
&= \sum_{k \geq 2} \sum_{i + j = k} (-1)^{ij} \pi_1^*\Phi_V^{(i)} \wedge \pi_2^*\Phi_V^{(j)} = \sum_{k \geq 2} \mu^* \Phi_V^{(k)} = \mu^* \Phi_V.
\end{align*}
For the remaining cup product terms, we compute
\begin{align*}
\Phi_V \abxcup {\rm id}_V &= \sum_{k \geq 2} \Phi_V^{(k)} \abxcup {\rm id}_V = \sum_{k \geq 2}(-1)^{-k}(-1)^{k} \pi_1^*\Phi_V^{(k)} \wedge \pi_2^*({\rm id}_V) = \sum_{k \geq 2} \pi_1^*\Phi_V^{(k)} = \pi_1^*\Phi_V, \\
{\rm id}_V \abxcup \Phi_V &= \sum_{k \geq 2} {\rm id}_V \abxcup \Phi_V^{(k)} = \sum_{k \geq 2} \pi_1^*({\rm id}_V) \wedge \pi_2^*\Phi_V^{(k)} = \sum_{k \geq 2} \pi_2^*\Phi_V^{(k)} = \pi_2^*\Phi_V, \\
{\rm id}_V \abxcup {\rm id}_V &= \pi_1^*({\rm id}_V) \wedge \pi_2^*({\rm id}_V) = {\rm id}_V.
\end{align*}
Consequently,
\begin{align*}
\partial (\Phi_V &- {\rm id}_V) + (\Phi_V - {\rm id}_V) \abxcup (\Phi_V - {\rm id}_V) \\
&= \partial (\Phi_V - {\rm id}_V) + \Phi_V \abxcup \Phi_V - \Phi_V \abxcup {\rm id}_V - {\rm id}_V \abxcup \Phi_V + {\rm id}_V \abxcup {\rm id}_V \\
&= \pi_2^*\Phi_V - \mu^*\Phi_V + \pi_1^*\Phi_V - {\rm id}_V + \mu^* \Phi_V^{(k)} - \pi_1^*\Phi_V - \pi_2^*\Phi_V + {\rm id}_V \\
&=0 ,
\end{align*}
as we wished to show.
\end{proof}
As in the previous section, we have the following direct consequence of this result.
\begin{corollary}\label{cor:4.5}
The operator $D$ in $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes V$ given by
$$
D = \bar{d} + \partial + \bar{\delta} + \Phi_V - {\rm id}_V,
$$
is a derivation of homogeneous degree $1$ that satisfies $D^2 = 0$.
\end{corollary}
In light of the above discussion, we can define a DG category by the following data. The objects of this DG category are the same as those of $\operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$. For any two objects $V$ and $V'$, with corresponding differentials $D$ and $D'$, the space of morphisms is the graded vector space $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes \operatorname{Hom}(V,V')$ with the differential $\partial_{D,D'}$ given by the same formula as the one for $\operatorname{\mathbf{DGRep}}(\TT \mathfrak{g})$. This DG category will be called the \emph{Bott-Shulman-Stasheff DG category} and will be denoted by $\operatorname{\mathbf{BSS}}(G)$.
\subsection{The invariant Bott-Shulman-Stasheff DG category}\label{sec:4.3}
Our aim now is to consider an invariant version of the Bott-Shulman-Stasheff DG category we have just introduced. It is this DG category that is linked to the ``infinitesimal'' DG category $\operatorname{\mathbf{DGRep}}(\TT \mathfrak{g})$ discussed in \S\ref{sec:4.1}. We start with some preliminary remarks. The notation is the same as in \S\ref{sec:3.1}.
Let $V$ be an object of $\operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$ with associated left-equivariant representation form $\Phi_V$. Recall that, with respect to the decomposition $\Phi_V = \sum_{k \geq 0} \Phi^{(k)}_V$, the zeroth component $\Phi^{(0)}_V$ is a representation of $G$ on $V$. With this understanding, let us consider the action $\widehat{\gamma}_0(g)$ of elements $g$ of $G$ on $\Omega^{q}(BG_{p}) \otimes V$ defined by
\begin{equation}\label{eqn:4.2}
\widehat{\gamma}_0(g) (\omega \otimes v) = \gamma_0(g)^{*} \omega \otimes \big( \Phi^{(0)}_V(g) (v) \big),
\end{equation}
for $\omega \in \Omega^{q}(BG_{p})$ and $v \in V$. We should also consider the action $\widehat{\gamma}(g_1,\dots,g_p)$ of elements $(g_1,\dots,g_p)$ of $G_{p}$ on $\Omega^{q}(BG_{p}) \otimes V$ given by
\begin{equation}\label{eqn:4.3}
\widehat{\gamma}(g_1,\dots,g_p) (\omega \otimes v) = \gamma(g_1,\dots,g_p)^{*} \omega \otimes v ,
\end{equation}
for $\omega \in \Omega^{q}(BG_{p})$ and $v \in V$. Noting that these two actions commute, we obtain an action $\widehat{\zeta}(g_0,g_1,\dots,g_p)$ of elements $(g_0,g_1,\dots,g_p)$ of $G_{p+1}$ on $\Omega^{q}(BG_{p}) \otimes V$ by simply putting
\begin{equation}\label{eqn:4.4}
\widehat{\zeta}(g_0,g_1,\dots,g_p) = \widehat{\gamma}_0(g_0) \circ \widehat{\gamma}(g_1,\dots,g_p).
\end{equation}
For what follows, we let $[\Omega^{q}(BG_{p}) \otimes V]^{G_{p+1}}$ denote the subspace of $G_{p+1}$-invariant elements of $\Omega^{q}(BG_{p}) \otimes V$.
\begin{lemma}\label{lem:4.6}
$[\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes V]^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}}$ is a subcomplex of $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes V$ and the inclusion
$$
[\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes V]^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}} \longrightarrow \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes V
$$
is a quasi-isomorphism.
\end{lemma}
\begin{proof}
In view of the second relation in \eqref{eqn:2.22}, we deduce that $\bar{\delta}$ preserves $G_{p + 1}$-invariant elements.
Consequently, the result follows from the first part of the proof of Lemma~\ref{lem:3.1}.
\end{proof}
Consider next the derivation $D$ of $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes V$ as defined in Corollary~\ref{cor:4.5}. We have the following important observation.
\begin{lemma}\label{lem:4.7}
$D$ preserves the subcomplex $[\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes V]^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}}$.
\end{lemma}
\begin{proof}
From the definition, it is clearly sufficient to show that the left-equivariant representation form $\Phi_V$ preserves $[\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes V]^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}}$. To prepare for this, we first observe that the $k$th component $\Phi^{(k)}_V$ of $\Phi_V$ satisfies
\begin{equation}\label{eqn:4.5}
L_{g}^* \Phi^{(k)}_V = \Phi^{(0)}_V(g) \circ \Phi^{(k)}_V,
\end{equation}
where $L_{g}$ indicates the left translation determined by the group element $g$; see Lemma~3.15 of \cite{AriasAbad2019}.
Using this, we claim that
\begin{equation}\label{eqn:4.6}
R_{g}^* \Phi^{(k)}_V = \Phi^{(k)}_V \circ \Phi^{(0)}_V(g),
\end{equation}
where $R_{g}$ indicates the right translation determined by the group element $g$. Indeed, let $x_1,\dots,x_k \in \mathfrak{g}$. Then, attending to the definition of $\Phi^{(k)}_V$ in \eqref{eqn:2.33}, we get
\begin{align*}
(R_{g}^* \Phi^{(k)}_V)(e) (x_1,\dots,x_k) &= \Phi^{(k)}_V(g) \big( (dR_g)_e(x_1),\dots, (dR_g)_e(x_k) \big) \\
&= (L_{g}^* \Phi^{(k)}_V)(e) \big( d(L_{g^{-1}} \circ R_g)_e(x_1),\dots, d(L_{g^{-1}} \circ R_g)_e(x_k) \big) \\
&= (L_{g}^* \Phi^{(k)}_V)(e) \big( \operatorname{Ad}_{g^{-1}} x_1,\dots, \operatorname{Ad}_{g^{-1}} x_k \big) \\
&= \Phi^{(0)}_V(g) \left( \Phi^{(k)}_V(e) \big( \operatorname{Ad}_{g^{-1}} x_1,\dots, \operatorname{Ad}_{g^{-1}} x_k \big) \right) \\
&= \Phi^{(0)}_V(g) \left( \Phi^{(0)}_V(g^{-1}) \circ \Phi^{(k)}_V(e) \big( x_1,\dots, x_k \big) \circ \Phi^{(0)}_V(g) \right) \\
&= \Phi^{(k)}_V(e) \big( x_1,\dots, x_k \big) \circ \Phi^{(0)}_V(g),
\end{align*}
as we wished. Next, let us take an invariant element $\eta \in [\Omega^{q}(BG_{p}) \otimes V]^{G_{p + 1}}$. On account of \eqref{eqn:4.2}, \eqref{eqn:4.3} and \eqref{eqn:4.4}, this means that
\begin{equation}\label{eqn:4.7}
\zeta(g_0,\dots,g_{p})^* \eta =\Phi^{(0)}_V(g_0) ( \eta).
\end{equation}
We need to show that $\Phi^{(k)}_V \abxcup \eta \in [\Omega^{k+ q}(BG_{p+1}) \otimes V]^{G_{p + 2}}$. On this purpose we notice firstly that
\begin{equation}\label{eqn:4.8}
\Phi^{(k)}_V \abxcup \eta = \pi_1^* \Phi^{(k)}_V \wedge \pi_{(p)}^* \eta,
\end{equation}
where $\pi_1 \colon G_{p+1} \to G$ is the projection onto the first factor and $\pi_{(p)}\colon G_{p+1} \to G_{p}$ is the projection onto the remaining $p$ factors. Notice, secondly, that
\begin{align}\label{eqn:4.9}
\begin{split}
\pi_1 \circ \zeta(g_0,\dots, g_{p+1}) &= L_{g_0} \circ R_{g_1^{-1}} \circ \pi_1, \\
\pi_{(p)} \circ \zeta(g_1,\dots, g_{p+1}) &= \zeta(g_1,\dots, g_{p+1}) \circ \pi_{(p)}.
\end{split}
\end{align}
By using \eqref{eqn:4.5} ,\eqref{eqn:4.6}, \eqref{eqn:4.7}, \eqref{eqn:4.8} and \eqref{eqn:4.9}, we find
\begin{align*}
\zeta(g_0,\dots, g_{p+1})^* (\Phi^{(k)}_V \abxcup \eta) &= \zeta(g_0,\dots, g_{p+1})^* \big( \pi_1^* \Phi^{(k)}_V \wedge \pi_{(p)}^* \eta\big) \\
&= (\pi_1 \circ \zeta(g_0,\dots, g_{p+1}) )^* \Phi^{(k)}_V \wedge (\pi_{(p)} \circ \zeta(g_0,\dots, g_{p+1}) )^* \eta \\
&= (L_{g_0} \circ R_{g_1^{-1}} \circ \pi_1)^*\Phi^{(k)}_V \wedge (\zeta(g_1,\dots, g_{p+1}) \circ \pi_{(p)})^* \eta \\
&= \pi_1^* R_{g_1^{-1}}^* L_{g_0}^* \Phi^{(k)}_V \wedge \pi_{(p)}^* \zeta(g_1,\dots, g_{p+1})^* \eta \\
&= \pi_1^* \big(\Phi^{(0)}_V(g_0) \circ \Phi^{(k)}_V \circ \Phi^{(0)}_V(g_1^{-1}) \big) \wedge \pi_{(p)}^* \big( \Phi^{(0)}_V(g_1) (\eta) \big) \\
&= \Phi^{(0)}_V(g_0) \big( \pi_1^* \Phi^{(k)}_V \wedge \pi_{(p)}^* \eta\big) \\
&= \Phi^{(0)}_V(g_0) (\Phi^{(k)}_V \abxcup \eta),
\end{align*}
which implies what we want.
\end{proof}
With this result in hand, we can now define the equivariant version of the Bott-Shulman-Stasheff DG category, which we denote by $\operatorname{\mathbf{BSS}}^{G}(G)$. Its objects are the same as those of $\operatorname{\mathbf{BSS}}(G)$, and, as such, they are just objects in the category $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$. For any two objects $V$ and $V'$, the space of morphisms is the graded vector space $[\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes \operatorname{Hom}(V,V')]^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}}$ with differential $\partial_{D.D'}$ given by exactly the same formula as that of $\operatorname{\mathbf{BSS}}(G)$. Note that Lemmas~\ref{lem:4.6} and \ref{lem:4.7} ensure that this is well-defined. We would also like to highlight the following key result.
\begin{proposition}\label{prop:4.8aa}
The inclusion DG functor from $\operatorname{\mathbf{BSS}}^{G}(G)$ to $\operatorname{\mathbf{BSS}}(G)$ is a quasi-equivalence.
\end{proposition}
\begin{proof}
For any pair of objects $V$ and $V'$ in $\operatorname{\mathbf{BSS}}^{G}(G)$, since $G_{p+1}$ is compact and connected, we know that the inclusion $[\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{p}) \otimes \operatorname{Hom}(V,V')]^{G_{p+1}} \to \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{p}) \otimes \operatorname{Hom}(V,V')$ is a quasi-isomorphism. The result thus follows by an argument entirely similar to that of the proof of Lemma~\ref{lem:3.1}.
\end{proof}
\subsection{The Van~Est DG functor}
In this subsection we describe the construction of a DG functor between the equivariant Bott-Shulman-Stasheff DG category $\operatorname{\mathbf{BSS}}^{G}(G)$ and the DG enhanced category $\operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})$, which is a quasi-equivalence when $G$ is compact. We use freely the definitions and notation from \S\ref{sec:3.1}.
Let $V$ be and object of $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$ and consider again the cochain complex $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes V$. For fixed $p$ and $q$, we let $[\Omega^{q}(BG_{p}) \otimes V]^{G_{p}}$ denote the subspace of $G_{p}$-invariant elements of $\Omega^{q}(BG_{p}) \otimes V$ with respect to the action \eqref{eqn:4.3}. From the definition it is obvious that $[\Omega^{q}(BG_{p}) \otimes V]^{G_{p}}$ coincides with $\Omega^{q}(BG_{p})^{G_{p}} \otimes V$. Thus, evaluation at $(e,\dots, e)$ induces an isomorphism of graded vector spaces from $[\Omega^{q}(BG_{p}) \otimes V]^{G_{p}}$ onto $\Lambda^{q}\mathfrak{g}_{p}^* \otimes V$. On the latter, we consider the action $\widehat{\gamma}'_0(g)$ of elements $g$ of $G$ defined by
\begin{equation}
\widehat{\gamma}'_0(g) (\xi \otimes v) = \operatorname{Ad}_g^*\xi \otimes \big(\Phi_V^{(0)}(g) (v) \big),
\end{equation}
for $\xi \in \Lambda^{q}\mathfrak{g}_{p}^*$ and $v \in V$. The following result, which is a direct consequence of Lemma~\ref{lem:3.2}, will be needed below.
\begin{lemma}\label{lem:4.9}
The following diagram commutes
\begin{equation*}
\xymatrix@C=7ex{[\Omega^{q}(BG_{p}) \otimes V]^{G_{p}} \ar[r]^-{\widehat{\gamma}_0(g)} \ar[d] & [\Omega^{q}(BG_{p})\otimes V]^{G_{p}} \ar[d] \\
\Lambda^{q}\mathfrak{g}_p^* \otimes V \ar[r]^-{\widehat{\gamma}'_0(g)} & \Lambda^{q}\mathfrak{g}_p^* \otimes V,}
\end{equation*}
where the vertical arrows denote evaluation at the element $(e,\dots,e)$.
\end{lemma}
Next we consider the morphism of cochain complexes defined by
$$
\mathscr{VE}_V = {\sf{VE}} \otimes {\rm id}_{V} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes V \longrightarrow {\mathrm W}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}},\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g} \otimes V,
$$
where ${\sf{VE}} \colon \Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \to {\mathrm W}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}},\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g}$ is the Van~Est map. By virtue of Lemma~\ref{lem:3.3}, the restriction of $\mathscr{VE}_V$ to $[\Omega^{q}(BG_{p}) \otimes V]^{G_{p}}$ vanishes unless $q = p$. From this it follows at once that this restriction, which we still denote by $\mathscr{VE}_V$, has its image contained in ${\mathrm S}^{p} \mathfrak{g}^* \otimes V$. It is also worth pointing out that, if we consider the morphism of graded vector spaces defined by
$$
\widetilde{\mathscr{VE}}_V = \widetilde{{\sf{VE}}} \otimes {\rm id}_V \colon \Lambda^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g}_{p}^* \otimes V \to {\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g}^* \otimes V,
$$
we get a commutative diagram
\begin{equation*}
\xymatrix{[\Omega^{p}(BG_{p}) \otimes V]^{G_{p}} \ar[dr]^-{\mathscr{VE}_V} \ar[d]& \\
\Lambda^{p}\mathfrak{g}_{p}^* \otimes V \ar[r]_-{\widetilde{\mathscr{VE}}_V} & {\mathrm S}^{p}\mathfrak{g}^* \otimes V,}
\end{equation*}
with the vertical arrow being the evaluation at $(e,\dots,e)$. This instructs us to introduce yet one more action $\widehat{\gamma}''_0(g)$ of elements $g$ of $G$ on ${\mathrm S}^{p}\mathfrak{g}^* \otimes V$ defined by
\begin{equation}\label{eqn:4.11}
\widehat{\gamma}''_0(g) (\xi \otimes v) = \operatorname{Ad}_g^*f \otimes \big(\Phi_V^{(0)}(g) (v) \big),
\end{equation}
for $f \in {\mathrm S}^{p}\mathfrak{g}^*$ and $v \in V$. The corresponding subspace of $G$-invariants elements of ${\mathrm S}^{p}\mathfrak{g}^* \otimes V$ will be denoted by $({\mathrm S}^{p}\mathfrak{g}^* \otimes V)^{G}$. Then we have the following result.
\begin{proposition}\label{prop:4.10}
The restriction of the morphism $\mathscr{VE}_V$ to $[\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes V]^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}}$ has its image contained in $({\mathrm W}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}},\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g} \otimes V)_{{\rm bas}}$.
\end{proposition}
\begin{proof}
The first thing to notice is that, owing to the definitions in \eqref{eqn:4.1} and \eqref{eqn:4.11}, the graded subspace $({\mathrm W}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}},\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g} \otimes V)_{{\rm bas}}$ coincides with $({\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^* \otimes V)^{G}$. Therefore, in light of Lemma~\ref{lem:4.9}, it will suffice to show that the following diagram commutes
\begin{equation*}
\xymatrix@C=7ex{\Lambda^{p}\mathfrak{g}_{p}^* \otimes V \ar[r]^-{\widehat{\gamma}'_0(g)} \ar[d]_-{\widetilde{\mathscr{VE}}_V} & \Lambda^{p}\mathfrak{g}_{p}^* \otimes V \ar[d]^-{\widetilde{\mathscr{VE}}_V}\\
{\mathrm S}^p \mathfrak{g}^* \otimes V \ar[r]^-{\widehat{\gamma}''_0(g)} & {\mathrm S}^p \mathfrak{g}^* \otimes V.}
\end{equation*}
But this is an easy consequence of the commutativity of the diagram we established in the course of the proof of Proposition~\ref{prop:3.4}.
\end{proof}
We also note the following result here.
\begin{proposition}\label{prop:4.11}
The Maurer-Cartan element $\Phi_V - {\rm id}_V$ of $\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes \operatorname{End}(V)$ is sent by the morphism $\mathscr{VE}_{\operatorname{End}(V)}$ to the Maurer-Cartan element $t^{a} \otimes L_{a} - w^{a} \otimes i_{a}$ of ${\mathrm W}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}},\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g} \otimes \operatorname{End}(V)$.
\end{proposition}
\begin{proof}
Let us first write $\Phi_V$ as a sum $\sum_{k \geq 0} \Phi_{V}^{(k)}$. Next, let us observe that, by definition,
$$
{\sf{VE}} \colon \Omega^k(G) \longrightarrow {\mathrm W}^{1,k} \mathfrak{g} = \Lambda^{1-k}\mathfrak{g}^* \otimes {\mathrm S}^{k}\mathfrak{g}^* = \begin{cases} \Lambda^{1}\mathfrak{g}^* & \text{if $k=0$,} \\
{\mathrm S}^{1}\mathfrak{g}^* & \text{if $k=1$,} \\
0 & \text{otherwise.}\end{cases}
$$
This implies that $\mathscr{VE}_{\operatorname{End}(V)} (\Phi_V^{(k)})= 0$ for $k \geq 2$. On the other hand, attending to the definitions, for each $x \in \mathfrak{g}$, we have
$$
\mathscr{VE}_{\operatorname{End}(V)} (\Phi_V^{(0)})(x) = (L_{x^{\sharp}} \Phi_V^{(0)})(e) = L_x \circ \Phi_V^{(0)}(e) = L_x \circ {\rm id}_V = L_x,
$$
and
$$
\mathscr{VE}_{\operatorname{End}(V)} (\Phi_V^{(1)})(x) = (i_{-x^{\sharp}} \Phi_V^{(1)})(e) = \Phi_V^{(1)}(e)(-x^{\sharp}(e)) = -\Phi_V^{(1)}(e)(x) = - i_x.
$$
Moreover, $\mathscr{VE}_{\operatorname{End}(V)}({\rm id}_V)= 0$. Since for any $x \in \mathfrak{g}$, $(t^{a} \otimes L_{a})(x) = L_x$ and $(w^{a} \otimes i_{a})(x) = i_x$, conclusion follows.
\end{proof}
With this preparatory work completed, we come now to the definition that concerns us. Take two object $V$ and $V'$ of $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$ and let us write $\mathscr{VE}_{V,V'}$ for $\mathscr{VE}_{\operatorname{Hom}(V,V')}$. Applying Proposition~\ref{prop:4.10} yields that
$$
\mathscr{VE}_{V,V'} \colon [\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes \operatorname{Hom}(V,V')]^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}}+1}} \longrightarrow ({\mathrm W}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}},\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g} \otimes \operatorname{Hom}(V,V'))_{{\rm bas}}.
$$
When combined with Proposition~\ref{prop:4.11}, this shows that the collection of morphisms $\mathscr{VE}_{V,V'}$ defines a DG functor $\mathscr{VE} \colon \operatorname{\mathbf{BSS}}^{G}(G) \to \operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})$ which is the identity on objects. We shall henceforth refer to this as the \emph{Van~Est DG functor}. The following result is our main finding in this subsection.
\begin{theoremB}\label{thm:4.12aa}
The Van~Est DG functor $\mathscr{VE} \colon \operatorname{\mathbf{BSS}}^{G}(G) \to \operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})$ is a quasi-equivalence.
\end{theoremB}
\begin{proof}
Following the discussion in the last part of \S \ref{sec:3.1}, let us again consider the map $\widehat{{\sf{AM}}}{}^{\theta}$ . We notice that, by definition, the following diagram commutes
$$
\xymatrix@C=7ex{{\mathrm S}^{p} \mathfrak{g}^* \ar[r]^-{\operatorname{Ad}_g^*}\ar[d]_-{\widehat{{\sf{AM}}}{}^{\theta}} & {\mathrm S}^{p} \mathfrak{g}^* \ar[d]^-{\widehat{{\sf{AM}}}{}^{\theta}} \\
\Omega^p(BG_p)^{G_p} \ar[r]^-{\gamma_0(g)^*} & \Omega^p(BG_p)^{G_p}. }
$$
Next, for any two objects $V$ and $V'$ of $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$, we set
$$
\widehat{{\mathscr{AM}}}{}^{\theta}_{V,V'} =\widehat{{\sf{AM}}}{}^{\theta} \otimes {\rm id}_{\operatorname{Hom}(V,V')} \colon {\mathrm S}^{p}\mathfrak{g}^* \otimes \operatorname{Hom}(V,V') \longrightarrow \Omega^{p}(BG_{p})^{G_p} \otimes \operatorname{Hom}(V,V').
$$
Then, from the definitions \eqref{eqn:4.2} and \eqref{eqn:4.11}, it follows on use of the above that the following diagram commutes
$$
\xymatrix@C=7ex{{\mathrm S}^{p} \mathfrak{g}^* \otimes \operatorname{Hom}(V,V') \ar[r]^-{\widehat{\gamma}''_0(g)}\ar[d]_-{\widehat{{\mathscr{AM}}}{}^{\theta}_{V,V'}} & {\mathrm S}^{p} \mathfrak{g}^* \otimes \operatorname{Hom}(V,V') \ar[d]^-{\widehat{{\mathscr{AM}}}{}^{\theta}_{V,V'}} \\
\Omega^p(BG_p)^{G_p} \otimes \operatorname{Hom}(V,V') \ar[r]^-{\widehat{\gamma}_0(g)} & \Omega^p(BG_p)^{G_p} \otimes \operatorname{Hom}(V,V'). }
$$
Recalling that $({\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}} \mathfrak{g}^* \otimes \operatorname{Hom}(V,V'))^G$ coincides with $({\mathrm W}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}},\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g} \otimes \operatorname{Hom}(V,V'))_{{\rm bas}}$ and $[\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})^{G_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}} \otimes \operatorname{Hom}(V,V')]^G$ coincides with $[\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})\otimes \operatorname{Hom}(V,V')]^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}} + 1}}$, we thus get a morphism of cochain complexes
$$
\widehat{{\mathscr{AM}}}{}^{\theta}_{V,V'} \colon ({\mathrm W}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}},\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g} \otimes \operatorname{Hom}(V,V'))_{{\rm bas}} \longrightarrow [\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})\otimes \operatorname{Hom}(V,V')]^{G_{\raisebox{-0.01ex}{\scalebox{0.6}{$\bullet$}} + 1}}.
$$
Furthermore, invoking Theorem~\ref{thm:3.7}, we infer that $\widehat{{\mathscr{AM}}}{}^{\theta}_{V,V'}$ is a left inverse of ${\mathscr{VE}}_{V,V'}$. This, clearly, yields the result.
\end{proof}
\begin{theorem}
Let $G$ be a compact simply connected Lie group. The categories $\operatorname{\mathbf{BSS}}(G)$ and $\operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})$ are $\mathsf{A}_\infty$ quasi-equivalent.
\end{theorem}
\subsection{The De~Rham-Hochschild $\mathsf{A}_{\infty}$-functor}
In this subsection we shall construct an $\mathsf{A}_{\infty}$-quasi-equivalence which connects the Bott-Shulman-Stasheff DG category $\operatorname{\mathbf{BSS}}(G)$ to the DG enhanced category $\operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$. We will use the De~Rham-Hochschild $\mathsf{A}_{\infty}$-quasi-isomorphism from \S\ref{sec:3.3}.
Let $V$ be an object of $\operatorname{\mathbf{Rep}}(\TT \mathfrak{g})$.
By tensoring the $\mathsf{A}_\infty$ map ${\sf{DR}}^{\Theta}: \operatorname{Tot}(\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})) \to \operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ with the
identity on $\operatorname{End}(V)$ one obtains a map:
\[{\mathscr{DR}} \colon \operatorname{Tot}(\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})) \otimes \operatorname{End}(V) \to \operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))\otimes \operatorname{End}(V),\]which
is an $\mathsf{A}_{\infty}$-quasi-isomorphism.
To proceed further, let us denote by $\rho \colon \TT \mathfrak{g} \to \operatorname{End}(V)$ the structure homomorphism associated with $V$. Following the discussion of \S~\ref{sec:2.6}, we will designate by $\mathscr{I}(\rho) \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G) \to \operatorname{End}(V)$ the structure homomorphism associated with $V$ when viewed as object of $\operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$. We will also write $\II_V \colon {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G) \to \operatorname{End}(V)$ to denote the homomorphism which associates ${\rm id}_V$ to every singular $0$-simplex on $G$, that is, the trivial module.
\begin{lemma}
The element $\mathscr{I}(\rho)- \II_V$ is a Maurer-Cartan element of $\operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)\otimes \operatorname{End}(V)$.
\end{lemma}
\begin{proof}
Applying Proposition~\ref{prop:3.18}, we see that, for each $n > 1$, and for homogeneous elements $\omega_1 \in \Omega^{q_1}(G), \dots, \omega_n \in \Omega^{q_n}(G)$ and $f_1,\dots, f_n \in \operatorname{End}(V)$,
$$
{\mathscr{DR}}_n \left({\sf u}(\omega_1 \otimes f_1) \otimes \cdots \otimes {\sf u}(\omega_n \otimes f_n)\right) = 0.
$$
Putting this together with the definitions in \eqref{eqn:2.12} and \eqref{eqn:2.38}, we conclude that the Maurer-Cartan element $\Phi_V - {\rm id}_V$ of $\operatorname{Tot}(\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})) \otimes \operatorname{End}(V)$ is sent by the $\mathsf{A}_{\infty}$-morphism ${\mathscr{DR}}$ to the element $\mathscr{I}(\rho)- \II_V$ of $\operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G),\operatorname{End}(V))$. Since $\mathsf{A}_{\infty}$-morphisms preserve Maurer-Cartan elements, the result follows.
\end{proof}
As usual, an immediate consequence of this is the following.
\begin{corollary}
The operator $D$ in $\operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G),\operatorname{End}(V))$ given by
$$
D = b + \delta + \mathscr{I}(\rho)- \II_V,
$$
is a derivation of homogeneous degree $1$ that satisfies $D^2 = 0$.
\end{corollary}
With this point in mind, let us now take $V'$ to be another object of $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$ with associated structure homomorphism $\rho' \colon \TT \mathfrak{g} \to \operatorname{End}(V')$ and differential $D'$, and consider the graded vector space $\operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G),\operatorname{End}(V))$ endowed with the differential $\partial_{D,D'}$ given by the same formula as \eqref{eqn:4.2aa} above. The following result demonstrates the basic link between the latter and the the Hochschild differential on $\operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G),\operatorname{End}(V))$ obtained by the prescription \eqref{eqn:2.6aa}.
\begin{proposition}\label{prop:4.16}
The differential $\partial_{D,D'}$ coincides with the differential on the Hochschild cochain complex $\operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G),\operatorname{End}(V))$.
\end{proposition}
\begin{proof}
Explicitly, we may write $\partial_{D,D'}$ as
\begin{align*}
\begin{split}
&\partial_{D,D'} \varphi ({\sf u} c_1 \otimes \cdots \otimes {\sf u} c_n) = \delta(\varphi({\sf u} c_1 \otimes \cdots \otimes {\sf u} c_n)) - (-1)^{\vert \varphi \vert} \varphi(b({\sf u} c_1 \otimes \cdots \otimes {\sf u} c_n)) \\
& \qquad\qquad\qquad\quad\,\,\, + (\mathscr{I}(\rho') \abxcup \varphi)({\sf u} c_1 \otimes \cdots \otimes {\sf u} c_n) - (-1)^{\vert\varphi\vert} (\varphi \abxcup \mathscr{I}(\rho))({\sf u} c_1 \otimes \cdots \otimes {\sf u} c_n),
\end{split}
\end{align*}
for homogeneous elements $\varphi \in \operatorname{Hom}(({\sf u} {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))^{\otimes n}, \operatorname{End}(V))$ and $c_1,\dots, c_n \in {\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)$. On the other hand, from the definition of the cup product \eqref{eqn:2.7aa}, we have
\begin{align*}
(\mathscr{I}(\rho') \abxcup \varphi)({\sf u} c_1 \otimes \cdots \otimes {\sf u} c_n) = (-1)^{\vert \varphi \vert (\vert c_1 \vert +1)} \mathscr{I}(\rho')(c_1) \circ \varphi ({\sf u} c_2 \otimes \cdots \otimes {\sf u} c_n),
\end{align*}
and
\begin{align*}
(\varphi \abxcup \mathscr{I}(\rho))({\sf u} c_1 \otimes \cdots \otimes {\sf u} c_n) = (-1)^{\sum_{j=1}^{n-1}\vert c_j \vert + n-1} \varphi ({\sf u} c_1 \otimes \cdots \otimes {\sf u} c_{n-1})\circ \mathscr{I}(\rho)(c_n).
\end{align*}
Using this together with \eqref{eqn:2.6aa} gives the desired conclusion.
\end{proof}
Now we proceed to define the $\mathsf{A}_{\infty}$-functor from $\operatorname{\mathbf{BSS}}(G)$ to $\operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$, which we also denote by ${\mathscr{DR}}$ and will refer to as the \emph{Hochschild-De~Rham $\mathsf{A}_{\infty}$-functor}. On objects ${\mathscr{DR}}$ acts as the integration functor $\mathscr{I} \colon \operatorname{\mathbf{Rep}}(\TT \mathfrak{g}) \to \operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ which gives a module over singular chains given a representation of $\TT \mathfrak{g}$. For each $n \geq 1$ and for every collection of objects $V_0, \dots, V_n$ of $\operatorname{\mathbf{Rep}}(\TT\mathfrak{g})$, we let
\begin{align*}
&{\mathscr{DR}}_n \colon {\sf u} \left(\operatorname{Tot}(\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes \operatorname{Hom}(V_{n-1},V_{n})\right) \otimes \cdots \otimes {\sf u} \left(\operatorname{Tot}(\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}) \otimes \operatorname{Hom}(V_{0},V_{1})\right) \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\longrightarrow \operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G),\operatorname{Hom}(V_0,V_n))
\end{align*}
be defined by
\begin{align}
\begin{split}
&{\mathscr{DR}}_n ({\sf u}(\omega_{n-1} \otimes f_{n-1}) \otimes \cdots \otimes {\sf u}(\omega_{0} \otimes f_{0}) )\\
&\qquad\qquad\qquad\qquad = {\sf u} \left({\sf s} {\sf{DR}}^{\Theta}_n ({\sf u} \omega_{n-1} \otimes \cdots \otimes {\sf u} \omega_{0}) \otimes (f_{n-1} \circ \cdots \circ f_{0}) \right),
\end{split}
\end{align}
for homogeneous elements $\omega_0,\dots, \omega_{n-1} \in \operatorname{Tot}(\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})$ and for a composable chain of homogeneous homomorphisms $f_0 \in \operatorname{Hom}(V_{0},V_{1}), \dots, f_{n-1} \in \operatorname{Hom}(V_{n-1},V_{n})$. A straightforward computation, which takes into account Proposition~\ref{prop:4.16}, shows that the sequence of maps ${\mathscr{DR}}_n$ indeed defines an $\mathsf{A}_{\infty}$-functor $\mathscr{DR} \colon \operatorname{\mathbf{BSS}}(G) \to \operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$.
\begin{theoremC}\label{thm:4.17aa}
The Hochschild-De~Rham $\mathsf{A}_{\infty}$-functor $\mathscr{DR} \colon \operatorname{\mathbf{BSS}}(G) \to \operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ is an $\mathsf{A}_{\infty}$-quasi-equivalence.
\end{theoremC}
\begin{proof}
Since the functor coincides with $\mathscr{I} \colon \operatorname{\mathbf{Rep}}(\TT \mathfrak{g}) \to \operatorname{\mathbf{Mod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$ on objects, we know that it is essentially surjective. It is also quasi-fully faithful because the map
\[{\mathscr{DR}} \colon \operatorname{Tot}(\Omega^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}})) \otimes \operatorname{End}(V) \to \operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))\otimes \operatorname{End}(V),\]
is a quasi-isomorphism.
\end{proof}
\subsection{The main theorem}
We are at last in a position to state and prove the principal result of the paper.
\begin{theoremA}\label{thm:4.18aa}
Suppose that $G$ is compact and simply connected. There exists a zig-zag of $\mathsf{A}_\infty$ quasi-equivalences of DG categories connecting $\operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})$ and $\operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))$.
\end{theoremA}
\begin{proof}
It follows directly from Proposition~\ref{prop:4.8aa}, Theorem B and Theorem C.
\end{proof}
We finish the paper with a couple of examples that illustrate the content of our main result.
\begin{example}
Let us write $\ensuremath{\mathbbmss{R}}$ for the trivial representation of $\TT\mathfrak{g}$. Then, on the one hand,
$$
\mathrm{H}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\left(\operatorname{End}_{\operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})}(\ensuremath{\mathbbmss{R}})\right) = \mathrm{H}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\left(({\mathrm W}\mathfrak{g})_{{\rm bas}}\right) \cong ({\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^*)^G \cong \mathrm{H}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG).
$$
On the other hand,
$$
\mathrm{H}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\left(\operatorname{End}_{\operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))}(\ensuremath{\mathbbmss{R}})\right) \cong \mathrm{H}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\left(\operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))\right) = \operatorname{HH}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\left({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)\right),
$$
where ``$\operatorname{HH}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}$'' stands for Hochschild cohomology. Invoking Theorem~\ref{thm:4.18aa}, we conclude that
$$
\mathrm{H}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(BG)\cong ({\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^*)^G\cong \operatorname{HH}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\left({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G)\right).
$$
This recovers two models for computing the cohomology of the classifying space $BG$ with coefficients in the trivial local system.
\end{example}
\begin{example}
Let us consider the Chevalley-Eilenberg complex $\operatorname{CE}(\mathfrak{g})$ viewed as a representation of $\TT\mathfrak{g}$. Then, on the one hand,
$$
\mathrm{H}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\left(\operatorname{Hom}_{\operatorname{\mathbf{DGRep}}(\TT\mathfrak{g})}(\ensuremath{\mathbbmss{R}}, \operatorname{CE}(\mathfrak{g}))\right) = \mathrm{H}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\left(({\mathrm W}\mathfrak{g} \otimes \operatorname{CE}(\mathfrak{g}))_{{\rm bas}}\right) \cong \mathrm{H}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\left(({\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^* \otimes \operatorname{CE}(\mathfrak{g}))^G\right).
$$
On the other hand,
$$
\mathrm{H}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\left(\operatorname{Hom}_{\operatorname{\mathbf{DGMod}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G))}(\ensuremath{\mathbbmss{R}}, \operatorname{CE}(\mathfrak{g}))\right) = \mathrm{H}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(\operatorname{HC}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G), \operatorname{CE}(\mathfrak{g}))) = \operatorname{HH}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}({\mathrm C}_{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(G), \operatorname{CE}(\mathfrak{g})).
$$
Since the latter is known to be isomorphic to the cohomology of the free loop space $\mathcal{L}BG$ of $BG$, Theorem~\ref{thm:4.18aa} tells us that
$$
\mathrm{H}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}(\mathcal{L}BG) \cong \mathrm{H}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\left(({\mathrm S}^{\raisebox{0.1ex}{\scalebox{0.6}{$\bullet$}}}\mathfrak{g}^* \otimes \operatorname{CE}(\mathfrak{g}))^G\right).
$$
On recovers the fact that the equivariant cohomology of $G$ acting on itself by conjugation is the cohomology of the loop space of $BG$.
This means that the Chevalley-Eilenberg complex $\operatorname{CE}(\mathfrak{g})$ corresponds to the Gauss-Manin local system for the loop space fibration $\pi: \mathcal{L}BG \to BG$.
\end{example}
|
2,877,628,090,114 | arxiv | \section{Introduction}
High-reflectivity mirrors play an important role in precision optical experiments
such as gravitational-wave detectors~\cite{0264-9381-24-2-008, PhysRevD.78.102003},
frequency references~\cite{Ludlow:07,nphoton.2012.217},
and macroscopic quantum measurements~\cite{1367-2630-11-7-073032, Poot2012273}.
These mirrors depend on multilayer coatings which are deposited with either physical methods
(sputtering, pulse laser deposition, molecular beam epitaxy) or chemical methods (vapor deposition).
While the coating is critical to the optical measurement, Brownian motion in coatings can present a
limiting noise source due to nonzero mechanical dissipation in the deposited layers.
Ion beam sputtering (IBS) for amorphous coatings and molecular beam epitaxy for crystalline coatings currently
produce the lowest mechanical loss~\cite{Cole:2016ct}.
Further reductions in coating thermal noise (CTN), while maintaining high optical quality (low absorption and
scatter, high uniformity), are of great interest for many experiments
(e.g., future gravitational-wave detectors ~\cite{CE2017,Miller2015,ETpaper}).
The CTN level of candidate coating materials is most frequently estimated using measurements of
their mechanical properties: mechanical quality factors, Young's modulus, and Poisson ratio.
The techniques used to measure these parameters include, among others,
suspended disks~\cite{crooks_blades, harry_blades},
clamped cantilevers~\cite{pierro_cantilevers},
and the gentle nodal suspension~\cite{cesarini_nodal}.
The level of CTN is then calculated from the measured parameters,
although uncertainties in their values can produce significant uncertainty in the CTN estimate.
Moreover, this approach may not capture all the phenomena involved in a multilayer coating.
A direct measurement of the thermal noise of a multilayer coating is thus an important
complement to the above approach.
In reference \cite{Gras-Yam} we introduced a novel technique that directly measures the CTN
of a high-reflectivity mirror. The technique uses a Fabry-Perot cavity in which three transverse electromagnetic
(TEM), Hermite-Gaussian modes co-resonate: TEM00, TEM02 and TEM20. These modes probe different areas of the
sample coating, and CTN appears as a fluctuation in the resonant frequency difference of the two higher-order modes
(see Fig.~\ref{fig:cavity}). In this article we present an improved version of this
experiment which can measure CTN with much higher signal-to-noise ratio and provide new information
on the frequency dependence of CTN.
\section{Experimental Setup}\label{exp}
\deff_{\rm FSR}{f_{\rm FSR}}
\deff_{\rm TMS}{f_{\rm TMS}}
\defP_{\rm in}{P_{\rm in}}
\defP_{\rm cav}{P_{\rm cav}}
At the core of the experiment is a 3-mirror folded cavity, with the sample to be measured as the folding mirror (see Fig.~\ref{fig:cavity}).
The cavity is mounted on a vibrationally isolated platform in a vacuum chamber ($10^{-5}$\,Torr).
This folded configuration is ideal for rapid testing of high reflectivity coatings, and accepts
the witness flats commonly included in coating runs.
\begin{figure}[b]
\includegraphics[scale=0.45]{Cavity.pdf}
\caption{\label{fig:cavity} A high finesse cavity configuration, with a folding
mirror (the sample to be measured) equidistant from the input and output mirrors.
The inset image shows the TEM20 and TEM02 modes used to make the coating thermal noise measurement.
Since these modes overlap only in a small central area,
noise in the coating causes changes in the difference between their resonant frequencies,
while most other noises sources cancel in this difference.}
\end{figure}
\begin{figure*}[t!]
\includegraphics[width=0.9\textwidth]{ol_dc.pdf}
\caption{\label{fig:exp} The experimental setup involves a Nd:YAG laser (far left) and an in-vacuum high-finesse cavity (far right). A laser beam is split into 3 paths, 2 of which are shifted in frequency (with AOMs). The laser frequency is controlled to lock the TEM00 mode to the cavity length with PDH locking scheme while the TEM02 and TEM20 modes are DC locked to the cavity. Beams 2a and 2b are intensity stabilized by actuating RF power on AOMs using Intensity Stabilization Servo (ISS) loops. The primary output of the experiment is the difference between the TEM02 and TEM20 resonant frequencies (labeled BEAT NOTE). Note, beams 1, 2a, 2b are the fundamental TEM00 modes. A conversion of beams 2a, 2b into TEM02 and TEM20 takes place in the cavity.}
\end{figure*}
\begin{table}[b]
\caption{Measured cavity parameters during collection of the data.}
\begin{center}
\begin{tabular}{llcc}
Parameter &Symbol & TEM02 & TEM20\\
\hline
Intra-cavity power, W & $\rm{P_{circ}}$ & 2 & 2\\
Finesse & $\mathcal{F}$ &15.06 k & 15.30 k \\
Mode frequency, MHz & $ 2\times f_{\rm TMS}$ & 276$\pm$2& 280$\pm$2\\
Beam size, $\mu$m & $\omega_{S}$ & 54 & 54\\
RoC (effective), mm & R & 50.7 & 50.8\\
Laser wavelength, nm & $\lambda$ & \multicolumn{2}{c}{1064} \\
Cavity length, mm & L &\multicolumn{2}{l}{ $L_1+L_2=46.45+53.07$} \\
Folding angle, deg & $\alpha$ & \multicolumn{2}{c}{17.23}\\
\hline
\end{tabular}
\end{center}
\label{tab_cav}
\end{table}
The cavity is near-concentric, with a total length of $L = \SI{99.5}{mm}$
and input and output couplers radii of curvature of $R = \SI{50.7}{mm}$.
This produces a waist $\omega_0$ and transverse mode spacing $f_{\rm TMS}$ of:
\begin{equation}
\begin{split}
& \omega_{0} = \sqrt{\frac{\lambda\sqrt{\epsilon L/2}}{\pi}} \simeq \SI{49}{\mu m} \\
& f_{\rm TMS} = \frac{c}{\pi L}\sqrt{\frac{\epsilon}{R}} \simeq \SI{133}{MHz},
\end{split}
\end{equation}
where $\epsilon = R-L/2 \simeq \SI{1}{mm}$,
$\lambda = \SI{1064}{nm}$ is the laser wavelength,
and $c$ is the speed of light~\cite{book:Lasers}.
The nominal frequency difference between the TEM00 and TEM02 or TEM20 modes is \SI{266}{MHz}.
In practice, the horizontal and vertical radii of curvature are slightly different,
and the resonant frequencies of the TEM02 and TEM20 modes are separated by a few MHz.
The readout and control scheme is shown in Fig.~\ref{fig:exp}. The laser frequency is locked
to the cavity TEM00 mode, with a 65~kHz bandwidth, using Pound-Drever-Hall reflection locking.
This servo suppresses laser frequency and cavity length fluctuations that are
common to the three modes. The two frequency shifted beams are then controlled to track
the TEM02 and TEM20 mode resonances so that they probe the sample's coating thermal noise,
which is spatially independent between the three modes.
In this improved version of the experiment, the higher-order mode probe beams are
controlled using side-of-fringe locking on the cavity transmission.
To maximize the signal-to-noise ration of these loops, the probe beams are locked at the
point where the transmission of the TEM02 and TEM20 modes are 70\% of their maximum values.
Feedback is applied to the two voltage-controlled oscillators (VCO) that determine the frequency
shift of the probe beams, with a control bandwidth of 40~kHz.
With the probe beam frequencies thus slaved to the TEM02 and TEM20 mode frequencies of
the cavity, the spatially independent coating thermal noise of the sample appears in the
frequency difference between the probe beams. This frequency difference is measured by
interfering the two beams, and tracking the fluctuations in the 4~MHz beat signal using
another VCO in a phase-locked loop configuration. The beat signal frequency fluctuations
are converted to an equivalent cavity length change (for the TEM00mode) by multiplying
by the factor $L \lambda /c$. The ASD of this scaled signal, labelled $N_{02/20}$,
contains the coating thermal noise $N_{\rm CTN}$, as well as other readout noises which
are relatively small in the frequency band of interest.
\begin{figure*}[ht]
\includegraphics[width=1.0\textwidth]{CTN4spec.pdf}
\caption{\label{fig:spec} The noise spectrum measured for 3 samples.
Note that the plotted fit is the sum of CTN and the stationary noise contributions.
Non-stationary noise below \SI{30}{Hz} from environmental vibrations
and above \SI{2}{kHz} from down-converted radio frequency (RF) interference,
limit the extent of the fit.
}
\end{figure*}
These dominant noise sources are described in the following paragraphs. The VCO used to measure the frequency difference between the higher order modes
has a noise level of $N_{\rm VCO} \simeq \SI{3}{mHz / \sqrt{\rm Hz}}$ below 1\,kHz.
This will appear in the readout as an equivalent cavity length noise of:
\begin{equation}
N_{02/20}^{\rm VCO}(f) = \frac{\lambda L}{c} \, N_{\rm VCO}(f) \simeq \SI{10^{-18}}{\frac{m}{\sqrt{\rm Hz}}}.
\end{equation}
The VCO noise has some frequency dependence, increasing by about a factor of 2 above $\SI{1}{kHz}$,
as shown in Fig.~\ref{fig:spec}.
The side-of-fringe locking used for the higher-order mode control can be contaminated by fluctuations
in the transmission photocurrents due to both laser intensity noise and shot noise.~The shot noise
associated with the $\SI{400}{\mu W}$ of transmitted power in each higher-order mode corresponds
to a relative intensity noise of $\rm{RIN_s} = 2 \times \SI{10^{-8}}{ Hz^{-1/2}}$.
This results in a readout noise of:
\begin{equation}
N_{02/20} = 0.7 \frac{\lambda}{\mathcal{F}} \times \rm{RIN_s} \simeq \SI{10^{-18}}{\frac{m}{\sqrt{\rm Hz}}} \,
\end{equation}
which is comparable to the VCO noise described above.
To address laser intensity noise, the power in each probe beam is actively stabilized before
being injected into the cavity. Each probe beam is sampled and detected inside the vacuum
chamber, and intensity servos stabilize the light by controlling
the RF power driving the acousto-optic modulators (see Fig.~\ref{fig:exp}). With a bandwidth
of \SI{50}{kHz}, these servos reduce the probe beam relative intensity noise to below
$2 \times \SI{10^{-8}}{ Hz^{-1/2}}$ at frequencies below $\SI{10}{kHz}$; higher frequency
residual intensity noise is removed from the transmitted light signals with a simple feed-forward
circuit.
A lower sensor noise would require modification of the VCO and the power increase in the cavity by increasing cavity finesse.
\section{Extrapolation to TEM00 beams}\label{sec:C}
Our experiment measures the thermal noise sensed by TEM02 and TEM20 modes
in a folded cavity (see Fig.~\ref{fig:cavity}), but we are more typically interested in the
thermal noise for the fundamental mode of a linear cavity. Correction factors are thus required for
the beam size, mode shape, and folded geometry. These correction factors are described in
detail in \cite{Gras-Yam}; to convert the measured CTN amplitude spectral density, $N_{\rm CTN}$,
to CTN for a TEM00 beam of size $\omega_L$, this correction is:
\begin{equation}\label{eqn:sctn1}
N_{\rm CTN}^{00}=0.616 \times \left( \frac{\omega_S}{\omega_L} \right) \, N_{\rm CTN} \,,
\end{equation}
where $\omega_S$ is the beam size on the sample mirror (see Table \ref{tab_cav}).
\section{Experimental Results}\label{res}
We measured four coating samples: two witness samples from Advanced LIGO end test mass coatings;
a witness sample from an initial LIGO end test mass coating; a baseline, standard high-reflectivity coating.
All four coatings where produced by ion-beam sputtering. The initial LIGO and baseline coatings are
stacks of quarter-wave Ta$_2$O$_5$-SiO$_2$ doublets.
For the Advanced LIGO coatings, the Ta$_2$O$_5$ is doped with 25\% TiO$_2$ to reduce mechanical loss~\cite{0264-9381-24-2-008}.
The layer thicknesses are also altered to further reduce thermal noise: the SiO$_2$ layers are a little thicker and the
Ti-Ta$_2$ layers are a little thinner than a quarter-wavelength.
All sample mirrors have a transmissivity less than \SI{10}{ppm} at \SI{1064}{nm}.
The baseline coating was deposited at \SI{120}{^\circ}{C}, with a deposition rate of \SI{1.9}{\AA/s} for both materials.
The sample was then annealed at \SI{450}{^\circ}{C} for 3 hours. The LIGO coating samples were also annealed,
but other coating process parameters for these samples are unknown.
The measured noise, $N_{02/20}$, for all 4 samples are shown in Fig.~\ref{fig:spec}.
In our previous paper we assumed the coating mechanical loss was constant in
frequency, and thus a $1/\sqrt{f}$ coating thermal noise ASD.
With the increased sensitivity of the current experiment, we are able to
measure CTN over a much broader frequency range (\SI{30}{Hz} - \SI{2}{kHz}),
which allows us to measure this slope.
We find that the best fit slope for all samples is near $f^{-0.45}$, which appears to match
the frequency dependence of the loss angles found in \cite{Amato}.
The fit to the noise spectra for the Advanced LIGO coating samples is:
$$N_{\rm CTN}^{\rm aL} = {(14.0\pm 0.2)\times10^{-18}}
\left( {\frac{\SI{100}{Hz}}{f}} \right)^{0.45 \pm 0.02} \frac{\rm m}{\rtHz}. $$
Our fit is limited to the band \SI{30 -- 2000}{Hz}, to avoid the variable environmental
noise at low frequencies, to remain well above the readout noise floor, and to avoid
small noise peaks at higher frequencies due down-converted radio frequency (RF) interference.
As expected, the other coating samples we measured have higher CTN, since they are
simple SiO$_2$ and Ta$_2$O$_5$ quarter-wave stacks.
The initial LIGO coating sample has 19\% higher CTN than the the Advanced LIGO coating:
$$N_{\rm CTN}^{\rm iL} = {(16.7\pm 0.1)\times10^{-18}}
\left( {\frac{\SI{100}{Hz}}{f}} \right)^{0.47 \pm 0.01} \frac{\rm m}{\rtHz} \, $$
while the standard Ta$_2$O$_5$-SiO$_2$ coating has 25\% higher CTN than the
Advanced LIGO coating:
$$N_{\rm CTN}^{\rm Ta} = {(17.5\pm 0.1)\times10^{-18}}
\left( {\frac{\SI{100}{Hz}}{f}} \right)^{0.47 \pm 0.03} \frac{\rm m}{\rtHz}. $$
These are consistent with a larger mechanical loss angle for Ta$_2$O$_5$
without the TiO$_2$ doping.
\begin{figure}[t!]
\includegraphics[width=0.5\textwidth]{Gwinc_Plot.pdf}
\caption{\label{fig:GWINC} The noise budget for aLIGO that
incorporates a new measured value of the loss angle and the slope for coating thermal noise. A previous
estimate of coating thermal noise ($\phi_{\rm SiO2} = 5.0\times10^{-4}$, $\phi_{\rm Ti:Ta} = 2.3\times10^{-4}$,
slope index = 0.5) is included in the plot and marked as ``CTN-old''. }
\end{figure}
The individual measurements of the two Advanced LIGO coating
samples give the same slope, but slightly different levels of CTN.
At \SI{100}{Hz}, one sample shows $13.9\pm 0.1$ and the other shows
$13.9\pm 0.1$, both in units of $\times10^{-18} \rm{m/Hz^{1/2}}$.
Each sample was measured multiple times at several locations on the coating
and the results where within the statistical error bars.
The CTN difference between the two samples is only 2\%, but it is statistically
significant (about 3 $\sigma$).
The origin of this difference is not known, so we extend the uncertainty on
our reported value of $N_{\rm CTN} = {(14.0\pm 0.2)\times10^{-18}} \rm{m/Hz^{1/2}}$
to include both measurements.
This value differs from our previous estimate $\rm N_{CTN}^{'} = (12.9\pm 0.6)\times10^{-18} \rm{m/Hz^{1/2}}$ \cite{Gras-Yam}
by less than $2 \sigma$. The difference may be due in part to small systematic effect resulting from the new experimental set-up,
or it may simply be due to statistics.
Our previous measurement had an SNR of only 2 at \SI{40}{Hz} (and smaller at other frequencies),
and the fitting process assumed a white readout noise, so differences at the few percent level
are not surprising.
\subsection{Implications for Advanced LIGO}\label{sec:aligo_ctn}
Extrapolating our measured CTN to the CTN of a \SI{6.2}{cm} beam on an
Advanced LIGO end test mass using Eqn.~\ref{eqn:sctn1} gives
\begin{equation}
N_{\rm CTN}^{\rm 00} (\SI{100}{Hz}) = {(7.5\pm 0.1)\times10^{-21}} \frac{\rm m}{\rtHz} .
\end{equation}
This is slightly higher than our previously reported value, and higher than the value
used in Advanced LIGO design documents
($5.8 \times10^{-21} {\rm m}/\rtHz$ at \SI{100}{Hz} \cite{0264-9381-32-7-074001}).
Using the CTN value and slope measured here, we find an overall decrease
in the expected Advanced LIGO binary neutron star range of 7\% (from \SI{186}{Mpc} to \SI{171}{Mpc}~\cite{Chen:2017aa})
compared to \cite{0264-9381-32-7-074001, den_nb}, see Fig. \ref{fig:GWINC}.
\subsection{Loss angle of $\mbox{TiO}_2$:$\mtant$}
To estimate the loss angle for the titania-tantala alloy used as the high refractive index material
in the Advanced LIGO coatings, we use the equations given in \cite{PhysRevD.87.082001}
and assume a loss angle for silicon-dioxide (the low index material) of
$\phi_{\rm Si02} = 5\times10^{-5}$ \cite{PhysRevD.91.022005}.
We further assume that the loss angles associated with shear and bulk deformation are equal
in both coating materials.
We have moved away from the simplified CTN equations from \cite{PhysRevD.91.042002}
used in our previous publication because that calculation neglects field penetration into the coating
and thus underestimates the loss angle of the high index material by 4\%.
The current experiment's precision is sufficient to make this a non-negligible effect.
Our estimate for the loss angle of the high-refractive-index material
in the Advanced LIGO coatings is
\begin{equation}
\phi_{\rm Ti:Ta}={(3.6\pm 0.1})\times 10^{-4} \left( {\frac{f}{\SI{100}{Hz}}} \right)^{0.1 \pm 0.04}.
\end{equation}
This number is slightly lower than the value previously reported in \cite{PhysRevD.91.022005},
but higher than the values reported in \cite{0264-9381-27-8-084030, Gras-Yam}.
Using the same procedure, we estimate the loss angle of tantala in the Ta$_2$O$_5$-SiO$_2$ coatings.
We obtain the same value for both coatings,
\begin{equation}
\phi_{\rm Ta}={(5.3\pm 0.1})\times 10^{-4} \left( {\frac{f}{\SI{100}{Hz}}} \right)^{0.06 \pm 0.02},
\end{equation}
which is higher than reported in \cite{0264-9381-20-13-334,0264-9381-23-15-014}.
\vspace{20pt}
\section{Conclusions}
Precision measurements of coating thermal noise are critical to
both high-precision laboratory-scale R\&D, and large scale efforts such as gravitational-wave detectors.
Our finding that the CTN spectrum deviates from the assumed slope will allow for more
reliable computations of CTN from measurements of the mechanical properties,
and more accurate extrapolations of direct CTN measurements to other frequency bands.
For Advanced LIGO in particular, the measurements presented allow us to update
our understanding of the sensitivity achievable by current detectors.
The CTN estimated for Advanced LIGO from our measurements is higher than
that originally computed for Advanced LIGO, and it results in a 7\% reduction in the detectors'
expected range.
Similar impacts are expected for other gravitational-wave detectors,
and both the amplitude and slope of CTN measured here will need to be incorporated
into future detector designs.
\begin{acknowledgments}
The authors would like to acknowledge the unfailing support and recognition
of the LIGO Scientific Collaboration's optics working group
without which this work would not have been possible. The authors also acknowledge the support of
the National Science Foundation under Grant 6936650.
We are also very grateful for the computing support provided by The MathWorks, Inc.
LIGO was constructed by the California Institute of Technology and
Massachusetts Institute of Technology with funding from the National Science Foundation,
and operates under cooperative agreement PHY-0757058.
Advanced LIGO was built under award PHY-0823459.
This paper carries LIGO Document Number LIGO-P1700448.
\end{acknowledgments}
\vspace{20pt}
\def{The LIGO Scientific Collaboration}{{The LIGO Scientific Collaboration}}
|
2,877,628,090,115 | arxiv | \section{Introduction}
The successful fabrication of two-dimensional (2D) materials such as graphene have aroused intense interest due to their intriguing electronic, mechanical, optical, and thermal properties.\cite{Novoselov2005, Zhang2005} The gapless nature of graphene, however, presents limitations to their potential application in industry.\cite{Liao2010, Schwierz2010} Therefore, the interest of study has gradually turned to other 2D material such as transition metal dichalcogenides (TMD) monolayer. TMD generally have diverse crystal structures which can provide significantly different electronic properties varying from semiconducting to metallic.\cite{Mak2010, Wu2011, Kan2014, Li2014Gapless}
The most common TMD is MoS$_{2}$, which has three possible phases. They are H-MoS$_{2}$\cite{Wang2012},T-MoS2\cite{py1983structural, Wypych1998, ataca2012stable} and T'-MoS$_{2}$,\cite{py1983structural, Kan2014} as displayed in Fig.1 respectively. H-MoS$_{2}$, i.e., the 2D trigonal prismatic phase, being the most stable configuration under normal conditions,\cite{benavente2002intercalation} can be exfoliated from the bulk 2H phase (P6/mmc) using a mechanical method\cite{Wang2012} or be synthesized with vapor deposition\cite{Lee2012} It is a semiconductor with a direct band gap of 1.8 eV.\cite{Qin1991,Radisavljevic2011} T-MoS$_{2}$ (tetragonal symmetry, octahedral coordination) phase can be synthesized from solvent based exfoliation method. It is reported to be metallic and can be used as an electrode material.\cite{Kappera2014} Although T-MoS$_{2}$ was observed in experiment,\cite{lin2013atomic,Kappera2014} the stability of T-MoS$_{2}$ is still a controversial issue. For example, density functional theory (DFT) calculations predict that the free standing T-MoS2 is unstable, since the seriously imaginary frequency presented in its phonon dispersion relation.\cite{Shirodkar2014emergence, Singh2015}
Qin et al. performed an STM study on the surface of restacked MoS$_{2}$ and observed a new superstructure characterized by the formation of zigzag chains.\cite{Qin1992} Then, an electron crystallography study also suggested that the restacked MoS$_{2}$ is more like WTe$_{2}$ with zigzag Mo-Mo chains.\cite{heising1999structure} This zigzag phase is called the distorted tetragonal MoS$_2$, labeled as T'-MoS$_{2}$ in the present paper, which was also referred to as the 1T' phase or ZT-MoS$_{2}$ phase in other theoretical study.\cite{py1983structural, Kan2014} T'-MoS$_{2}$ is thought to be a charge density wave (CDW) state as a result of the Piers phase transition from T phase.\cite{Whangbo1992} The structural stability of T'-MoS$_{2}$ was first inferred from formation energy by Kan et al..\cite{Kan2014} They found that the formation energy of T'-MoS$_{2}$ is higher than that of H-MoS$_{2}$ but lower than T-MoS$_{2}$. Namely, T'-MoS$_{2}$ is a meta-stable phase. Qian et al. calculated the phonon band structure of T'-MoS$_{2}$ and found no imaginary frequency, which confirmed the vibrational stability of T'-MoS$_{2}$.\cite{Qian2014} As to the stability of T'-MoS$_{2}$ in other physical respects, such as the thermal and mechanical stability, however, has not been studied theoretically by far, to the best of our knowledge. Although previous experimental study has reported the observation of T'-MoS$_{2}$ phase identified by experimental STM images,\cite{Eda2012, Guo2015} those STM images, however, have not been sufficiently explicit to demonstrate the existence of T'-MoS$_{2}$. In addition, a discrepancy is also presented with respect to the band gap of T'-MoS$_{2}$. For instance, T'-MoS$_{2}$ is first predicted to be a semiconductor with a narrow band gap.\cite{Kan2014, Qian2014} In contrast, the electronic band structure given by Gao et al. implied that T'-MoS$_{2}$ was a semimetal.\cite{gao2015charge}
Therefore, a theoretical study on the physical stability and electronic properties of T'-MoS$_{2}$ is necessary and urgent. In the present work, we perform density functional theory (DFT) calculations within local density approximation (LDA) to investigate the simulated STM images, stability and the electronic band gap of T'-MoS$_{2}$. The simulated STM images provide a significant reference for identifying the lattice structure from experimental STM images. The \emph{ab initio} molecular dynamics (AIMD) simulations confirm the thermodynamic stability of T'-MoS$_{2}$ at room temperature; the calculating results of elastic constants meet the Born-Huang criteria, which implying the mechanical stability of T'-MoS$_2$; the absence of the imaginary frequency in the phonon dispersion relation indicates the vibrational stability of T'-MoS$_{2}$. Besides, we also classify the optical modes by group theory and compute their corresponding eigenfrequency and eigenvector, which play an important role in the identification and characterization of T'-MoS$_{2}$ phase from optical experiment. Moreover, we make a contrast calculation of the electronic band structure to determine the effect of the spin-orbit coupling, which clarifies the origin of band gap of T'-MoS$_{2}$.
The remainder of this paper is organized as follows. In Sec. II, methodology and computational details are described. Sec. III presents first the simulation of STM imaging of MoS$_2$ in three different phases, then the stability of T'-MoS$_{2}$ is explored from different aspects. The symmetry classification of the vibrational modes along with their eigenfrequency and eigenvector are calculated. Furthermore, the electronic band structure and band gap of T'-MoS$_{2}$ are investigated by considering the spin-orbit coupling. Finally, conclusions are drawn in Sec. IV.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.38]{3phases2.pdf}
\caption{\label{3phases2}(Color online) Top view and side view of atomic structures of monolayer MoS$_2$ in H, T, and T' phases are shown in (a), (b), and (c), respectively.}
\end{figure}
\section{Methodology}
Both total energy and electronic band structure calculations were performed by using the Vienna ab initio simulation package (VASP).\cite{Kresse1996, Kresse1996a} The electron-ion interaction was described by using the frozen-core projector augmented wave (PAW) method;\cite{PAW, Kresse1999} the exchange and correlation were treated with generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) form.\cite{Perdew1996} Besides standard DFT with GGA, the hybrid Heyd-
Scuseria-Ernzerhof (HSE)06 method,\cite{Heyd2003Hybrid, Heyd2006Erratum} was also employed to give a more accurate description of the band gap of T'-MoS$_{2}$. In the HSE06 method, a fraction of the exact screened Hartree-Fock (HF) exchange is incorporated into the PBE exchange using a mixing parameter $\alpha$= 0.25. We used a cutoff energy of 300 eV for the plane wave basis set, which yields total energies convergence better than 1 meV/atom. The Van der Waals interactions are treated with the empirical correction scheme of Grimme's DFT-D2 method, which has been successful in describing the geometries of various layered materials.\cite{Grimme2006, Bucko2010}
In the slab model of single-layer MoS$_2$, the periodic slabs were separated by a vacuum layer of 15 {\AA} in the \emph{c} direction to avoid mirror interactions. A 10\texttimes{}5\texttimes{}1 \emph{k}-mesh including $\Gamma$-point, generated according to the Monkhorst-Pack scheme,\cite{Monkhorst1976} was applied to the Brillouin-zone (BZ) integrations. Through geometry optimization, both the shapes and internal structural parameters of pristine unit-cells were fully relaxed until the residual force on each atom is less than 0.01 eV/{\AA}.
To examine the stability of the modeled structure of T'-MoS$_{2}$ from the lattice dynamics point of view, the force-constant approach involving a finite displacement was adopted as employed by the \href{http://atztogo.github.io/phonopy/index.html}{PHONOPY}\cite{phonopy} code. The force constant matrix was calculated with a 7\texttimes{}4\texttimes{}1 supercell containing 168 atoms, based on the density functional perturbation theory (DFPT) method \cite{Gonze1997} implemented into VASP. Furthermore, the phonopy codes also enable us to obtain the eigenfrequency and eigenvector of lattice vibrational modes at the center of BZ.
The simulated STM images were generated by using the \href{http://www.p4vasp.at/}{P4VASP} package, which can facilitate the simulation of STM image with a continuously varying scanning distance. The theory for simulating STM imaging by {\em ab initio} density functional calculations is well established.\cite{Tomanek1988} Giving a small bias voltage $V_{b}$ between the sample and the STM tip produces a tunneling current, whose density $j({\bf r})$ can be obtained from a simple extension \cite{Selloni1985} of the expression derived by Tersoff and Hamann \cite{Tersoff1983, Tersoff1985}:
\begin{equation}
j({\bf r},V_{b}) {\propto} \rho_{\bf STM}({\bf r},V_{b}),
\end{equation}
where
\begin{equation}
\rho_{\bf STM}({\bf r},V_{b})=\int_{E_F-eV_{b}}^{E_F}
\rho({ r},E) dE%
\label{eq2}
\end{equation}
and
\begin{equation}
\rho({\bf r},E)=\sum_{n,{\bf k}} |\psi_{n{\bf k}}({\bf r})|^2
\delta(E_{n,{\bf k}}-E) \;.
\label{eq3}
\end{equation}
Here, $\rho({\bf r},E)$ is the local density of states at the center of the tip at ${\bf r}$ and $\psi_{n{\bf{k}}}({\bf r})$ are the electron eigenstates of the unperturbed surface at energy $E_{n,{\bf k}}$. These eigenstates are commonly represented by Kohn-Sham eigenstates obtained using DFT. The assumptions behind this is that the relevant tip states are described by $s$ waves with a constant density of states.\cite{Tersoff1983, Selloni1985, Tersoff1985} Furthermore, the tunneling matrix element is considered to be independent of both the lateral tip position for a constant tip-to-surface distance and the bias voltage $V_{b}$ in the narrow (but nonzero) energy region $[E_F-eV_{b},E_F]$. Equation \ref{eq3} describes tunneling from occupied states of the sample to the tip. The simulated STM image is not sensitive to the bias voltage as long as the valence band enters in the integral range, but sensitive to the scanning distance from the tip to the sample surface.
The simulating STM imaging has been used for studying the modification of the electronic structure of the 2H phase MoS$_2$ (0001) surface produced by several point defects.\cite{Fuhr2004} Recently, it was also used for exploring the few-layer phosphorus capped by graphene and hexagonal boron nitride monolayer.\cite{Rivero2015} In present work, we apply this method to study the structure of single-layered MoS$_2$. Different bias voltage $V_{b}$ are used for distinct phases of monolayer MoS$_2$ according to their electronic properties. For H-MoS$_2$, its band gap is 1.7 eV, and its Fermi level is under the conductor band at about 0.1 eV, so that the value of bias voltage $V_{b}$ is set to 1.8 V, and thus the energy range $[E_F-eV_{b},E_F]$ enters the valence band at about 0.2 eV. As to metallic T'-MoS$_2$, we have compared the simulated images using two different bias voltage (0.3 and 1.8 V) but find no significant distinction, hence we always use the smallest one in the following calculation. For T'-MoS$_2$, its band gap is merely 0.1 eV, so a bias voltage $V_{b}$=0.5 V is enough.
\section{Results and discussion}
\subsection{Simulated STM images and Identification of Monolayer MoS$_2$}
We begin our discussion by comparing the simulated STM images of the three possible structures, namely, H, T, and T' phases of monolayer MoS$_2$.\cite{chhowalla2013} The lattice structures of the three phases are displayed in Fig. \ref{3phases2}. The most energetically favorable H-MoS$_2$ (as shown in Fig. \ref{3phases2}(a)) has a sandwich-like structure of three planes of 2D hexagonally packed atoms, S-Mo-S, where Mo atoms are trigonal-prismatically coordinated by six S atoms, forming ABA stacking with P6m2 space-group symmetry. In contrast, the Mo atoms in the T-MoS$_2$ (as shown in Fig. \ref{3phases2}(b)) structure are octahedrally coordinated with the nearby six S atoms, resulting in ABC stacking with P3m1 space group symmetry. H- and T-MoS$_2$ phase have very different electronic properties: the former is a large gap semiconductor but the latter a metal. It has been predicated that the T-MoS$_2$ is typically unstable in free-standing condition\cite{Shirodkar2014emergence, Kan2014}. T-MoS$_2$ should undergo the Piers distortion in one direction to form a $2 \times 1$ super-lattice structure, consisting of one-dimensional zigzag Mo-Mo chains along the other direction, \emph{i.e.}, the T'-MoS$_2$ phase, as shown in Fig. \ref{3phases2}(c). It implies theoretically that T'-MoS$_2$ should be more stable than T-MoS$_2$ in free-standing conditions. In experiments, however, Eda et al. have observed both T- and T'-MoS$_2$ by scanning transmission electron microscopy (STEM) imaging,\cite{Eda2012} but the image of T'-MoS$_2$ is not so clear as that of T-MoS$_2$. Although WS$_2$ and MoTe$_2$ monolayer have been found experimentally.\cite{Mahler2014, Keum2015} T'-MoS$_2$, has not been identified unanimously in experiment yet, to the best of our knowledge. Therefore, we perform an \emph{ab initio} density functional calculations to simulate STM images of MoS$_2$ monolayer in the three phases. Figure.\ref{STM-exp-sim} shows the calculated STM images of H-, T-, as well as T'-MoS$_2$, respectively. Our simulated STM images agree well with those images obtained in previous experiments.\cite{Eda2012} This agreement indicates the reliability of the simulated STM imaging method.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.42]{STM-sim2-N.pdf}
\caption{\label{STM-exp-sim}(Color online) Simulated STM images of monolayer MoS$_2$. (a), (b), and (c) are simulated images of H, T, and T' phases, respectively, where purple and yellow spheres represent Mo and S atoms.}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.30]{STM-2T4.pdf}
\caption{\label{STM-2T'}(Color online) Simulated STM images of T'-MoS$_2$ with different scanning distance. (a) the scanning distance $d=6.59$ ${\AA}$ and (b) the scanning distance $d=5.74$ ${\AA}$.}
\end{figure}
The scanning distance $d$ represents the distance form scanning plane to referring plane, and the referring plane is put on the upper surface of crystal cell in slab model of MoS$_2$ monolayer. The scanning distance is denoted in term of the tip position in P4VASP. To determine the influence of the scanning distance on STM imaging, we perform the simulating STM imaging calculations varying continuously with scanning distance. It is found that the simulated STM images vary remarkably with the scanning distance. This variation even may lead to misidentification of the experimental STM images.\cite{Altibelli1996} On one hand, it means that we may obtain quite different STM images actually belonging to the identical structure in experiment, as seen in Fig. \ref{STM-2T'}, in which we have shown the top and side view of two different scanning distances, while the tip position for the middle plane of the MoS$_2$ monolayer is $d=3.39$ ${\AA}$. Then the distances from scanning plane to the middle plane are $3.20$ ${\AA}$ and $2.35$ ${\AA}$, respectively. If merely judging from the top view of the simulation images without referring the other information, you must think that the lower sublet of Fig. \ref{STM-2T'}(a) represents T'-MoS$_2$ phase but that of Fig. \ref{STM-2T'}(b) belongs to the image of T-MoS$_2$. On the other hand, it also means that the different phases of MoS$_2$ may have similar STM images. Comparing the simulated images of T-MoS$_2$ and T'-MoS$_2$ with appropriate scanning distances presented in Fig. \ref{STM-T-T'}, you may find it is hard to distinguish these two phases. Thus, we should not make identification only by single experimental STM image without any other information.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.26]{STM-T-T2.pdf}
\caption{\label{STM-T-T'}(Color online) Simulated STM images of MoS$_2$ with T, and T' phases. (a) and (b) are T-MoS$_2$ and T'-MoS$_2$, respectively. For better interpretation of the images, we reproduced the atomic arrangement on the top of the images. On the higher sublets, the side view of the atomic arrangement of the monolayer and intensity profile are shown; on the lower sublets, the top view of those are given, respectively.}
\end{figure}
Knowing this how can we identify the lattice structure of STM image in experiment? The method is to scan the STM images while varying with tip-to-surface distance. By comparing and contrasting those images, you can make the correct identification, for the different structures have distinct changing patterns. This suggestion is deduced from our STM imaging simulation of T-MoS$_2$ and T'-MoS$_2$ with continuously varying scanning distance. The structural symmetry of simulated image of T-MoS$_2$ remains invariant as the scanning distance varies consecutively, in contrast, that of T'-MoS$_2$ is varying significantly, just as demonstrated in Fig.\ref{STM-2T'}. It is worth mentioning that the STM images obtained experimentally are usually scanning within one or two given tip-to-surface distance. Then a question arises: whether it is possible that the structure of MoS$_2$ observed in previous experiments could be T' phase rather than T phase?
It is natural to examine the relevant experiments in literature, and we find that the answer is affirmative. In an experimental STM imaging study of T-MoS$_2$\cite{Wypych1998}, there are several STM images in Figures 2 and 3 in Ref[\onlinecite{Wypych1998}], which were identified as K$_{x}$(H$_{2}$O)$_{y}$MoS$_{2}(x<0.3)$. We make the corresponding simulation of T'-MoS$_2$, which are shown in Fig. \ref{exp-T'-sim-2}. Compare our simulated STM images of T'-MoS$_2$ phase with these images, we find that the simulated images surprisingly accord with the experimental STM images. This dramatic accordance indicates strongly that these experimental images should be corresponding to T'-MoS$_2$ phase rather than K$_{x}$(H$_{2}$O)$_{y}$MoS$_{2}$. That is to say, the T'-MoS$_2$ has been synthesized accidentally but misidentified unfortunately. If this was true, it actually means an experimentally feasible method for synthesizing T'-MoS$_2$, which is important for the fabrication of a novel topological field effect transistor.\cite{Qian2014}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.42]{T1-sim-N.pdf}
\caption{\label{exp-T'-sim-2}(Color online) (a) and (b) show the side view of atomic structure and intensity profiles along the lines indicated in simulated STM images of T'-MoS$_2$, respectively.}
\end{figure}
The instability of free standing T-MoS2 at 0 K is revealed by imaginary frequency presented in its phonon dispersion relations from the recent first-principles calculations.\cite{Shirodkar2014emergence, Singh2015} At the same time, several theoretical and experimental researches show that the function of alkali metal is to offer an extra electron to make the T-MoS$_2$ phase more stable in energy.\cite{Kappera2014} While in Ref. [\onlinecite{Wypych1998}], the presence of water stabilizes the T' phase from the original high symmetric structure.\cite{Qin1991, Yang1991, Qin1992, Gordon2002}. Therefore, the method designed to obtain T-MoS$_2$ is actually a feasible method to produce T'-MoS$_2$ in experiment. For reliably identifying T'-MoS$_2$ in experiment, it is necessary to exploit the otherwise stability of T'-MoS$_2$.
\subsection{Thermal stability of T'-MoS$_2$}
The thermal stability of T'-MoS$_2$ is explored by performing AIMD simulations using canonical ensemble. To reduce the constraint of periodic boundary condition, the T'-MoS$_2$ is simulated by ($3\times2$) super-cells. The snapshots of T'-MoS$_2$ atomic configurations for the final stages of AIMD simulations at 300 K and 800 K are shown in Fig. 6 (a) and (b), respectively. One can find that no significantly reconstruction are observed at 300 K and 800 K. Here the only exception in the latter case is that the S and Mo atoms are found to be slightly moved due to thermal fluctuation. This means that T'-MoS$_2$ can withstand the higher temperature at least up to 800 K, implying the high-energy barriers between T' phase and H phase, which is in consistent with the first-principles calculations performed by Qian et al.\cite{Qian2014}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.16]{MD-simulation.pdf}
\caption{\label{MD-simulation}(Color online) Snapshots of atomic configurations of T'-MoS$_2$ at the end of AIMD simulations from front, top, and side views, respectively. The simulated super-cells are marked by black squares, and their corresponding temperature and time are denoted above each panel.}
\end{figure}
\subsection{Mechanical Stability and Anisotropy of T'-MoS$_2$}
Since the super-cell is fixed during the MD simulations, we have to evaluate the effect of elastic distortion on structural stability. In order to guarantee the positive-definiteness of strain energy following lattice distortion, the components of linear elastic modulus tensor of a stable crystal must obey the Born-Huang criteria \cite{Ding2013}. We calculate the change of energy due to the in-plane strain to examine the mechanical stability of T'-MoS$_2$. For a 2D crystal, the elastic strain energy per unit area can be written as\cite{Wang2015}
\begin{align}
U(\varepsilon)=\frac{1}{2}C_{11}\varepsilon_{xx}^{2}+\frac{1}{2}C_{22}\varepsilon_{yy}^{2}+ C_{12}\varepsilon_{xx}\varepsilon_{yy}+2C_{66}\varepsilon_{xy}^{2},
\end{align}
where $C_{ij}$ are the components of the elastic modulus tensor using the standard Voigt notation ( i.e., 1-xx, 2-yy, and 6-xy),\cite{PhysRevB.85.125428} corresponding to second partial derivative of elastic energy with respect to strain. The elastic constants can be derived by fitting the energy curves associated with uniaxial and equi-biaxial strains. Under uniaxial strain applied along x direction, $\varepsilon_{yy}=0$, this leads to $U(\varepsilon)=\frac{1}{2}C_{11}\varepsilon_{xx}^{2}$. Parabolic fitting of the uniaxial strain curve yields $C_{11}=109.0$ GPa$\cdot$nm. Similarly, under uniaxial strain applied
along y direction, $C_{22}$ is derived to be 124.0 GPa$\cdot$nm. Under equi-biaxial strain, $\varepsilon_{xx}=\varepsilon_{yy}$, one have $U(\varepsilon)=(\frac{1}{2}C_{11}+\frac{1}{2}C_{11}+C_{12})\varepsilon_{xx}^{2}$. By fitting the equi-biaxial strain curve, we obtain $(\frac{1}{2}C_{11}+\frac{1}{2}C_{11}+C_{12})=130.9$ GPa$\cdot$nm, which means that $C_{12}=14.4$ GPa$\cdot$nm. For a mechanically stable 2D crystal, the elastic constants should satisfy two criteria: $C_{11}C_{22}-C_{12}>0$ and $C_{66}>0$.\cite{Ding2013}For T'-MoS$_{2}$, one can easily verify that the calculated components of the elastic modulus tensor satisfy $C_{11}C_{22}-C_{12}>0$; besides, the calculated $C_{66}$=38.8 GPa$\cdot$nm is positive. Both the two criteria of mechanical stability are met, thus the mechanical stability of T'-MoS$_{2}$ is confirmed.
Due to its lower point group symmetry, T'-MoS$_{2}$ has anisotropic elastic property, which is significantly different from H- and T-MoS$_{2}$. Both H- and T-MoS$_{2}$ are of isotropic elastic properties described by two elastic constants: Young's modulus $Y$ and Poisson's ratio $\nu$. The Young's modulus and Poisson's ratio of T'-MoS$_{2}$, however, do not remain constant, but vary with orientation. The formula for Young's modulus $Y(\theta)$ and Poisson's ratio $\nu(\theta)$ are\cite{Ding2013,wang2016lattice}
\begin{align}
\label{eq10}
Y(\theta ) = \frac{{{C_{11}}{C_{22}} - C_{12}^2}}{{{C_{11}}{s^4} + {C_{22}}{c^4} + (\frac{{{C_{11}}{C_{22}} - C_{12}^2}}{{{C_{66}}}} - 2{C_{12}}){c^2}{s^2}}},\\
\label{eq11}
\nu (\theta ) = \frac{{{C_{12}} + (\frac{{{C_{11}}{C_{22}} - C_{12}^2}}{{{C_{66}}}} - 2{C_{12}} - {C_{11}} - {C_{22}}){c^2}{s^2}}}{{{C_{11}}{s^4} + {C_{22}}{c^4} + (\frac{{{C_{11}}{C_{22}} - C_{12}^2}}{{{C_{66}}}} - 2{C_{12}}){c^2}{s^2}}},
\end{align}
where $s = sin(\theta)$ and $c = cos(\theta)$, $\theta$ is the the angle with respect to the x-axis. The above two formulas are universal for all orthogonal 2D crystal.
Their extremum directions can be determined by analyzing the zeros of the first derivative of $Y(\theta)$ and $\nu(\theta)$. Generally, there are three extremum directions for both $Y(\theta)$ and $\nu(\theta)$, in which there are two same extremum conditions: $sin(\theta)=0$ and $cos(\theta)=0$, which means that the coordinate axes directions x and y (rather, the symmetrical principal axes directions) are extremum directions. we find that for $Y(\theta)$ , both the two axes may maximum (or minimum) directions, meanwhile, for $\nu(\theta)$, they must be both minimum (or maximum) directions. Certainly, the two maximum (or minimum) directions mean that there must be one minimum (or maximum) direction between the two axes. Depending on the relative value of elastic constants, the third extremum between the two axes for $Y(\theta)$ may not exist, then one axis direction is maximum and the other minimum direction. For Poisson's ratio, the third extremum always exists, except for isotropic case. The above conclusions about extremum direction are also universal for orthogonal 2D crystal structures.
Here, we plot the curves of $Y(\theta)$ and $\nu(\theta)$ in polar coordinates for T'-MoS$_{2}$ in Fig \ref{Youngs-modulus}, which intuitively show the elastic anisotropy of T'-MoS$_{2}$. First, it can be seen that the extremum directions are consistent with our analyses above. Second, it can be found that the variation range of Young's modulus is from about 96.9 to 124.0 GPa$\cdot$nm, the value of Poisson's ratio is limited between 0.117 and 0.25. Third, it can be found that the principal minimum direction of $Y (\theta)$ is along $\theta=39.5^{\circ}$ and the maximum direction of $\nu(\theta)$ for T'-MoS$_{2}$ are almost along diagonal direction.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.34]{Youngs-modulus.pdf}
\caption{\label{Youngs-modulus}(Color online) Calculated orientation-dependent
Young's modulus $Y(\theta)$ and Poisson's ratio $\nu(\theta)$ of T'-MoS$_2$.}
\end{figure}
\subsection{Lattice Dynamic Stability and Vibrational Modes of T'-MoS$_{2}$}
\subsubsection{Lattice dynamic stability}
To further verify the lattice dynamic stability of the T'-MoS$_2$, we calculate phonon dispersion relation of T'-MoS$_2$ and demonstrate it in Fig. \ref{phonon-band}. we can note that the phonon dispersion of T'-MoS$_2$ has three acoustic and fifteen optical branches. The three acoustic branches are the in-plane longitudinal acoustic (LA), the transverse acoustic (TA), and the out-of-plane acoustic (ZA) branches. The LA and TA branches have linear dispersion and a higher frequency than the ZA mode around $\Gamma$ point in the Brillouin zone. In contrast to H-MoS$_2$,\cite{JimenezSandoval1991, Molina-Sanchez2011} there is no band gap between acoustic branches and optical branches. All the rest of vibrational branches along other lines in BZ are non-degeneracy, except the vibrational branches along R--X line at the boundary of BZ, which are two order degeneracy. The lifting of degeneracy of vibrational branches reveals the Piers phase transition from a high symmetric structure. The absence of the imaginary frequency throughout the 2D BZ indicates the structural stability of the T'-MoS$_2$. Our results are in good agreement with those obtained in Ref [\onlinecite{ Qian2014}].
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.42]{phonon-band.pdf}
\caption{\label{phonon-band}(Color online) The phonon band structure of T'-MoS$_2$ displaying the connection of vibrational bands. The absence of imaginary frequency indicates the stability of T-MoS$_2$.}
\end{figure}
\subsubsection{Symmetric analysis of lattice vibrational modes}
Both laser Raman scattering and Infrared absorption spectra are powerful tools for structural identification and characterization of 2D materials. To guide the optical spectra study in future experiment, we deduce the symmetry classification of phonon modes at the $\Gamma$ point by using group theory, and further point out R and IR activity of the optical modes. The unit cell of T'-MoS$_2$ consists of two S-Mo-S units with a total of six atoms, suggesting that there are eighteen phonon modes (three acoustic and fifteen optical modes) at the $\Gamma$ point. Lattice vibrations can be classified based on the irreducible representation of space group.\cite{Dresselhaus2007group} The space group of T'-MoS$_2$ is $C^{2}_{2h}$ (or $P2_{1}/m$, No.11), whose factor group is isomorphic with the point group $C_{2h} $. The character table for point group $C_{2h} $ is given in Table \ref{character},
\begin{table}[htbp]
\centering
\begin{ruledtabular}
\caption{\label{character}Character table for the point group $C_{2h}$ including basis functions of the irreducible representations.}
\begin{tabular}{ccccccc}
$\rm{SG}$&$\rm{PG}$&$E$ &$ C_{2}$ &$\sigma$&$ i$&$\rm{basis}$\\
\hline
$\Gamma_{1}^{+}$ &$A_{g}$ &1&{ 1}&{ 1}&{ 1}&$R_{z},x^{2},y^{2},z^{2},xy$\\
$\Gamma_{1}^{-}$ &$B_{g}$ &1&-1&-1&{ 1}& $R_{x},R_{y},xz,yz$\\
$\Gamma_{2}^{+}$ &$A_{u}$ &1&{ 1}&-1&-1&$z$ \\
$\Gamma_{2}^{-}$ &$B_{u}$ &1&-1&{ 1}&-1&$x,y$\\
\end{tabular}
\end{ruledtabular}
\end{table}
where $A_{g}$, $A_{u}$, $B_{g}$ and $B_{u}$ are signs of one-dimensional irreducible representations; $A$ and $B$ are used when the character of the major rotation operation is 1 or -1, respectively; the subscripts $ g (gerade)$ and $ u (ungerade)$ denote representations that are symmetric and antisymmetric with respect to the inversion operation if the point group has a center of inversion symmetry; $x$, $y$, and $z$ are components of polar vectors. From Table \ref{character}, we note that T'-MoS$_2$ has no two-dimensional irreducible representations, i.e., there is no degenerate optical modes at the center of BZ, which distinctively differs from that of H- and T-MoS$_2$.\cite{Wieting1971, JimenezSandoval1991, Cai2014, Zhang2015}
\begin{table}[htbp]
\centering
\begin{ruledtabular}
\caption{\label{Reduced Representation} Characters of vector, equivalent, and vibration representations for T'-MoS$_2$. }
\begin{tabular}{ccccc}
\centering
$C_{2h}$&$ E$ &$ C_{2}$ &$\sigma_{h}$ & $i$ \\
\hline
$ \chi _{\rm{vector}}$&{ 3} &-1 & 1 & -3 \\
$ \chi _{\rm{equivalent}}$ &{ 6} &0 & 6 & 0 \\
$ \chi _{\rm{vibration}}$ &18 & 0 & 6 & 0\\
\end{tabular}
\end{ruledtabular}
\end{table}
We classify the lattice vibrational modes of T'-MoS$_2$ at $\Gamma$ by group theory according to the irreducible representations of C$_{2h}$. Characters of atomic displacement vector representations, primitive cell equivalent representations, and lattice vibration representations of T'-MoS$_2$ are shown in Table \ref{Reduced Representation}. These representations can be reduced into the irreducible representations summarized in Table \ref{character}:
\begin{align}
\label{eq4}
\Gamma _{\rm{vector}}&= 1{A_u} \oplus 2{B_u},\\
\label{eq5}
\Gamma _{\rm{equivalent}}& = 3{A_g}\oplus3{B_u},\\
\label{eq6}
\Gamma _{\rm{vibration}} &=\Gamma_{\rm{equivalent}}\otimes\Gamma_{\rm{vector}}\nonumber\\
&=(3{A_g}\oplus3{B_u})\otimes( 1{A_u} \oplus 2{B_u})\nonumber \\
&=3{A_u} \oplus 6{B_u} \oplus 3{B_g} \oplus 6{A_g},
\end{align}
where $\Gamma _{\rm{vector}}$, $\Gamma _{\rm{equivalent}}$, and $\Gamma _{\rm{vibration}}$ are the symmetry representations of atomic displacement vector, the equivalent representations of the primitive cell and the symmetry representations of lattice vibration at the zone center of BZ, respectively. The symmetry representation of lattice vibration is equal to the direct product of the symmetry representations of atomic displacement vector and the equivalent representations of the primitive cell.\cite{Dresselhaus2007group}
This symmetry representation of lattice vibration includes eighteen phonon modes entirely and can be further decomposed into the representations of acoustic and optical modes as follows:
\begin{align}
\label{eq7}
\Gamma _{\rm{acoustic}}& =A_{u} \oplus 2B_{u},\\
\label{eq8}
\Gamma _{\rm{optical}} &= 2{A_u} \oplus 4{B_u} \oplus 6{A_g} \oplus 3{B_g},
\end{align}
where the acoustic modes include one $A_{u}$ and two $B_{u}$ modes, all their frequencies are identical to zero; the rest of the fifteen nonzero frequency modes belong to optical modes. The six optical modes of odd parity (2$A_{u}$ and 4$B_{u}$) are IR active, the other nine optical modes of even parity (6$A_{g}$ and 3$B_{g}$) are R active. The R and IR modes are mutually exclusive in T'-MoS$_2$ phase because of the presence of inversion symmetry in the crystal. It is also worth pointing out that the above symmetry analyses is suitable for all T' phase of 2D TMD, namely, T'-MX$_2$ with M=(Mo, W) and X=(S, Se, and Te). For easy identifying T'-MoS$_2$ from a Raman optical spectral experiment, we compare the R modes of T'-MoS$_2$ with those of H- and T-MoS$_2$.\cite{Cai2014, Zhang2015} It can be found that both H and T phase of MoS$_2$ have two-dimensional $E$ ($E'$ and $E_{g}$) modes, while T' phase has only one-dimensional modes, no two-dimensional $E$ mode. This means that if one detects the $E$ mode in a Raman optical spectral experiment on a MoS$_2$ monolayer, it coud not be in T' phase. In addition, since the presence of inversion symmetry both in atomic structures of T- and T'-MoS$_2$, according to exclusion principle, the R modes in these two phase must be $g$ modes, where T phase has both one- and two-dimensional $g$ modes ($A_1g$ and $E_{g}$) but T' phase has only one-dimensional $g$ modes ($A_{g}$ and $B_{g}$). In H-MoS$_2$, however, there is no inversion symmetry and thus no $g$ or $u$ mode. Thus, we may draw the conclusion that if one finds some one-dimensional but no two-dimensional R modes of $g$ symmetry in a Raman optical spectral experiment on a MoS$_2$ monolayer, then this MoS$_2$ monolayer is probable in T' phase.
\subsubsection{Eigenfrequency and eigenvector of optical modes}
For comparing quantitatively with optical spectra experiments, we compute the eigenfrequency of the fifteen optical modes by phonopy. In Table \ref{IR-Raman}, the fifteen optical modes with frequency are grouped by their irreducible representations and optical activity, where bold Arabic numbers represent the optical modes, which ordering are according to their frequencies from low to high. T'-MoS$_2$ structure can be identified and characterized based on Table \ref{IR-Raman} in future optical spectra experiments.
\begin{table}[htbp]
\centering
\begin{ruledtabular}
\caption{\label{IR-Raman}Classification of the fifteen optical modes in T'-MoS$_2$ according to irreducible representations and optical activity, where the optical modes are denoted by bold Arabic number from \textbf{4} to \textbf{18}, their frequencies are given in parentheses in the unit of cm$^{-1}$.}
\begin{tabular}{cc|cc}
\multicolumn{2}{c|}{Raman modes}&\multicolumn{2}{c}{Infrared modes} \\
\hline
\centering
$A_{\rm{g}}$ &$ B _{\rm{g}}$ &$A_{\rm{u}}$ & $B _{\rm{u}}$\\
\hline
\textbf{ 5}(143.267) &\textbf{ 4}(138.905)&\textbf{ 8}(210.225) &\textbf{ 9}(238.132)\\
\textbf{ 6}(206.684) &\textbf{ 7}(207.684)&\textbf{12}(271.867)&\textbf{10}(262.733)\\
\textbf{13}(276.776)&\textbf{11}(270.917)& &\textbf{16}(338.798)\\
\textbf{14}(314.290)& & &\textbf{18}(451.503)\\
\textbf{15}(337.570)& & &\\
\textbf{17}(398.414)& & &\\
\end{tabular}\
\end{ruledtabular}
\end{table}
Besides, the vibrational eigenvector of the IR and R modes are also illustrated in Fig. \ref{IR-modes-MoS2} and Fig. \ref{R-modes-MoS2}. For IR modes, as can be seen in Fig. \ref{IR-modes-MoS2}, both the two $A_{u}$ modes \textbf{8}(210.225) and \textbf{12}(271.867) are vibrating along in-plane directions, while only one $B_{u}$ mode \textbf{18}(451.503) which is of the highest frequency, vibrates perpendicular to crystal plane. For IR modes, from Fig. \ref{R-modes-MoS2} we find that all the three $B_{g}$ modes \textbf{4}(138.905), \textbf{7}(207.684) and \textbf{11}(270.917) are in-plane vibrations, while none of the six $A_{g}$ modes is vibrating along purely in-plane or out-plane direction. The vibration direction of IR and R active modes is vital for setting the incident and detection directions as well as the polarization of the light used in optical spectra experiments.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.26]{IR-modes-MoS2.pdf}
\caption{\label{IR-modes-MoS2} (Color online) The eigenvectors of IR modes in T'-MoS$_2$.}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.26]{R-modes-MoS2.pdf}
\caption{\label{R-modes-MoS2} (Color online) The eigenvectors of R modes in T'-MoS$_2$.}
\end{figure}
\subsection{The Electronic Band Gap of T'-MoS$_{2}$}
There has been a discrepancy about the band gap of T'-MoS$_2$ presented in recent literature.\cite{Kan2014, Qian2014, gao2015charge} Kan et al. performed spin-polarized DFT calculations with GGA-PBE and with HSE06 to investigate the band structure of monolayer of T'-MoS$_2$. They pointed out that the structural distortions of ZT-MoS$_2$ lead to the opening of a direct gap of 0.022 or 0.23 eV. The band gap obtained by DFT with GGA-PBE is significantly different from that obtained with HSE06, the latter is ten times greater than the former.\cite{Kan2014} Qian et al. later found that 1T'-MoS2 (i.e., T'-MoS2) was a semiconductor with a band gap of 0.1 eV based on many-body perturbation theory within the GW approximation.\cite{Qian2014} However, Gao et al.'s calculation by DFT based on Dmol3 software showed that T'-MoS$_2$ was a semiconductor with a very narrow band gap of 0.006eV.\cite{gao2015charge} Generally speaking, the band gap is underestimated by GGA-PBE but overestimated by HSE06.Thus, the band gap of 0.1 eV obtained by Qian et al.\cite{Qian2014} should be closer to the real value since this value is in between the two results: 0.022 eV and 0.23 eV, where the former is calculated by DFT with GGA-PBE and the latter is with HSE06. As to the band gap of 0.006 eV \cite{gao2015charge}, it may seriously underestimate the band gap since it is far lower than the underestimated result 0.022 eV.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.32]{electric-band-1.pdf}
\caption{\label{electric-band-1}(Color online) Electronic band structure of T'-MoS$_2$ before (a) and after (b) considering spin-orbit coupling, and the vacuum level is set to zero. The band gap is opened by spin-orbit coupling.}
\end{figure}
To examine our inference, we calculate the electronic band structure of T'-MoS$_2$ by DFT with GGA-PBE to explore the effect of spin-orbit coupling. Fig. \ref{electric-band-1} shows the electronic band structure of T'-MoS$_2$ without (a) and with (b) the consideration of the spin-orbit coupling. Comparing Fig. \ref{electric-band-1}(a) and \ref{electric-band-1}(b), one can find that the electronic band dispersion curves in the two cases are almost as the same in general, but the crucial difference in detail occurs near the Fermi line. In the former case, there seems no band gap, while in the latter case, the band gap does present, and is equal to 0.048 eV. This result implies that it is the spin-orbit coupling opens or widens out the band gap. Besides, we also find that the band structure without consider the spin-orbit agrees well with that of Gao et al.'s especially in the vicinity of the Fermi line.
To determine whether there exists a very small band gap without spin orbital interaction, we plot the orbital-projected band structures\cite{Wang2015Role} of Mo and S atom around the $\Gamma$ point without considering spin-orbit coupling. As can be seen in Fig. \ref{pband}, for both Mo and S atom, the two bands meeting at the Fermi lime cross each other directly without any avoiding.This direct crossing shows that there is exactly no band gap existing in electronic band structure of T'-MoS$_2$ if the spin-orbit interaction is neglected. Thus, we can conclude that the interaction which lifting the degeneracy of electron states at the Fermi line and opening the band gap is the spin-orbit coupling. In addition, we recalculate the electronic band structure of T'-MoS$_2$ by DFT with HSE06 and the obtained band gap is about 0.153 eV, which falls in between those obtained in Refs [\onlinecite{Kan2014} and \onlinecite{Qian2014}]. Thus far, we come to conclusion that T'-MoS$_2$ must be a semiconductor of a narrow gap, while Gao et al.'s calculation may have not included the spin-orbit interaction.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.52]{pband.pdf}
\caption{\label{pband}(Color online) Orbital-projected fine band structures of (a) Mo and (b) S atoms around the $\Gamma$ point without considering spin-orbit coupling. The diameter of circle indicates the weight of components.}
\end{figure}
\section{Conclusions}
In conclusion, we have performed first-principles investigation on the structure, physical stability, optical modes and electronic band gap of T'-MoS$_2$. Our simulated STM images of MoS$_2$ monolayer are in good agreement with previous experimental results. Moreover, we have found unexpectedly that the simulated STM images of T'-MoS$_2$ vary significantly with the scanning distance. This variation should be considered in the structural identification from experimental STM images. Furthermore, the dramatic similarity between the simulated STM images of T'-MoS$_2$ with that of earlier experimental study means that T'-MoS$_2$ may have been observed in experiment but was mistaken for the intercalation compound K$_{x}$(H$_{2}$O)$_{y}$MoS$_{2}$. If so, T'-MoS$_2$ should be stable in structure. To verify its physical stability, the thermal and mechanical stability of T'-MoS$_2$ have explored by AIMD simulations and elastic constants fitting and the results are affirmative. In addition, the lattice dynamic stability of T'-MoS$_2$ is also confirmed by the absence of imaginary frequency in our phonon dispersions relations. Therefore, the physical stability of T'-MoS$_2$ has been verified finally. Besides, we have made symmetry classification of optical modes and calculated their eigenfrequencies and eigenvectors, which provides an important guidance for further optical spectral study in experiments. Future work will investigate the intensity of R and IR spectra theoretically.
\begin{acknowledgments}
Y. C. Liu is thankful to H. B. Niu for his help on first principles calculations. V.Wang. acknowledges the financial support of The Special Scientific Research Program of the Education Bureau of Shaanxi Province, China (Grant No. 15JK1531).
\end{acknowledgments}
\nocite{*}
\bibliographystyle{aipnum4-1}
|
2,877,628,090,116 | arxiv | \section{Introduction}
Studies of atomic and molecular gas in the inner Galaxy have revealed strong non-circular motions, which are now understood to be caused mainly by the Galactic bar. Because distances to individual gas clouds are difficult to obtain, these data are commonly presented in \lvplots, which show the distribution of gas emission line intensity as a function of Galactic longitude and line-of-sight velocity. Streamers and arms in the gas flow appear as high density lines in the \lvplot\ and, because of the unknown distances, must be interpreted through gas dynamical models.
Hydrodynamic models of the gas flow in the Milky Way have been able to reproduce many of the distinctive features in the \lvplots\ for HI and CO data \citep{bur_lis_93,dame_etal_01}, even though no model has been able to provide a good match to all the observed features \citep{eng_ger_99,fux_99b,rod_com_08,baba_etal_10,sorman_etal_15c}. A variety of barred potentials were used, including potentials derived from COBE or star count data, or potentials characteristic of barred N-body models, Besides the bar, also the Galactic spiral arms play some role for the gas flow \citep{bissan_etal_03,seo_kim_14}, by regulating the inflow of gas into the bar region.
Apart from the gravitational potential, the pattern speed of the bar is the most important parameter for the gas flow, because for given mass distribution it sets the resonance radii where the gas flow needs to accommodate the transition from one closed orbit family to another. A number of early investigations concluded a relatively high value for the pattern speed, $50-65\freq$ \citep{eng_ger_99,fux_99b,debatt_etal_02,bissan_etal_03}, such that the co-rotation radius of the bar would be located in the range $\RCR=3.5-5$ kpc, but others have argued for lower values \citep{wei_sel_99,rod_com_08,shen_14,sorman_etal_15c}.
Recently \citet{weg_ger_13} measured the three dimensional density of red clump giants (RCG) in the barred Galactic bulge. This density, together with the kinematics from the BRAVA survey \citep[]{kunder_etal_12}, has since been used as a constraint in constructing dynamical models of the barred bulge using the made-to-measure method \citep{portai_etal_15a}. Surprisingly, these models required a rather low pattern speed for the bar, $25-30\freq$, so that $\RCR>7$ kpc.
In these models the bulge represents the central buckled, box/peanut part of a longer bar similar to many N-body bars \citep[]{com_san_81, raha_etal_91, martin_etal_06,shen_etal_10}. Star count studies extending to larger longitudes have indeed found a thinner bar outside the Milky Way's barred bulge \citep[e.g.][]{hammer_etal_00, benjam_etal_05, cabrer_etal_08,wegg_etal_15} that ends near $l \approx 27-30\degree$. If these two components are aligned and form a single structure at an angle to the Sun of $\sim 27\degree$, as suggested by \citet{mar_ger_11} and found with the detailed RCG maps of \citet{wegg_etal_15} then the `long bar' component ends at $>4.7\kpc$ from the Galactic Center. Because the bar cannot exist beyond co-rotation \citep{contop_80} this limits the pattern speed to $\Omb\leq47\freq$ for a flat rotation curve at $220\kms$, lower than found by the majority of previous gas dynamics studies, but still significantly larger than the best-fitting pattern speed in the barred bulge dynamical models from \citet{portai_etal_15a}.
The purpose of the present paper is to enquire whether the observed \lvplot\ can be explained by a gas flow model in such a low $\Omb$ model, if we use a potential based on these dynamical models as an input to study the gas flow in a realistic Milky Way context.
The paper is organized as follows: In Section 2, we describe our galaxy models, model parameters, and the numerical method. In Section 3, we present the best-fitting gas model, and explore the parameter space in Section 4. In Section 5, we discuss our assumptions and implications for our model, and summarize our results.
|
2,877,628,090,117 | arxiv | \section{Introduction}
Since the proof of power counting non renormalizability \cite{'tHooft:1974bx} of General
Relativity (\textbf{GR}), the scientific community has increasingly accepted the idea that
Einstein's theory is an effective field theory \cite{Burgess:2003jk}. Although some candidates for
the high energy limit of GR have emerged, no one has managed to give a power counting
renormalizable theory that also reproduces the dynamical behavior of GR in four dimensions.
The most general gravitational action in five dimensions has one additional free parameter besides
the cosmological constant \cite{Lovelock:1971yv}. These two parameters can be chosen so that the
lagrangian becomes a Chern-Simons form, acquiring some of the features that make $D=3$ gravity a
theory with zero beta function \cite{Witten:1988hc}. This particular $D=5$ gravitation theory is a
gauge theory where the spin connection $\omega $ and the vielbein $e$ are parts of a single
connection for the Lie algebras $so(4,2)$, $so(5,1)$ or $iso(4,1)$. Gravitation theories that have
these properties exist in all odd dimensions. They have been studied in
\cite{Chamseddine:1989nu,Mueller-Hoissen:1990vf} and a review of them can be found in Ref.
\cite{Zanelli:2005sa}. Another compelling reason to consider these gauge theories for the
$SO(D-1,2)$ group, they admit an immediate supersymmetric extension
\cite{Banados:1996hi,Troncoso:1998ng}. Moreover, the local supersymmetry in those theories is
realized off shell and without invoking auxiliary fields or ad-hoc constraints.
In this work, the topological sector --in the sense that no metric is needed to construct it--, of
the gauged Wess-Zumino-Witten (\textbf{gWZW}) lagrangian, is shown to arise from
higher-dimensional gravity. In the case where the connection is valued in the Lie algebra
$su_{L}(3)\times su_{R}(3)$, the gWZW term plus a kinetic piece for the Goldstone fields, arises
as the effective lagrangian of QCD \cite{Witten:1983tw}. In our case, the gauge group will not be
$SU_{L}(3)\times SU_{R}(3)$, but the anti-de-Sitter group in five dimensions, $SO(4,2)$.
\section{Five-dimensional gravity and gWZW terms}
The most general five dimensional, ghost-free \cite{Zumino:1985dp,Zwiebach:1985uq}, gravitational
action is given by\footnote{Throughout this work the exterior product between forms is not written
explicitly, i.e. $\omega \wedge e=\omega e$. Lower case Latin indices $a,b,c$ take values
$0...4.$, while capital indices $A,B,C$ cover the range $0...5.$.}
\begin{gather}
S(\omega ,e)=\int_{M}\varepsilon _{abcde}\big( \alpha _2 R^{ab}R^{cd}e^{e} +
\alpha_1 R^{ab}e^{c}e^{d}e^{e} \notag \\
\hspace{2in}+ \alpha_0 e^{a}e^{b}e^{c}e^{d}e^{e}\big) ,
\end{gather}
where the curvature two-form is written in terms of the Lorentz (spin)connection $\omega$, as
\begin{equation}
R^{ab}=d\omega ^{ab}+\omega _{\ c}^{a}\omega ^{cb}=\frac{1}{2}R_{\text{ \ \ }%
\mu v}^{ab}dx^{\mu }dx^{v}
\end{equation}
The vielbein $e^{a}$ is related to the spacetime metric through $g_{\mu \nu}=e_{\mu }^a e_{v}^b
\eta_{ab}$, and $\eta_{ab}=diag(-,+,+,+,+)$ is the Lorentz-invariant metric. The vielbein and the
Lorentz connection are regarded as independent fields. The field equations associated with the
variations of $\omega $ are satisfied if the torsion, $De^{a}\equiv de^{a}+ \omega_{\text{ }b}^a
e^b$, is set equal to zero. In the sector of the theory where the torsion is zero and the vielbein
is invertible, $\omega $ is a function of the vielbein, and the usual second order equations for
the metric are recovered from the field equations obtained from the variation with respect to $e$.
An interesting accident occurs when the constants in the action are in the ratio $\alpha
_{2}:\alpha _{1}:\alpha_{0}=1:2/3:1/5$. In that case, the action can be rewritten as a
Chern-Simons theory
\cite{Chamseddine:1989nu,Mueller-Hoissen:1990vf,Banados:1996hi,Troncoso:1998ng},
\begin{align}
S(\mathcal{A})& =\kappa \int_{M}\left\langle \mathcal{A}d\mathcal{A}d%
\mathcal{A}+\frac{3}{2}\mathcal{A}^{3}d\mathcal{A}+\frac{3}{5}\mathcal{A}%
^{5}\right\rangle \notag \\
& =\kappa \int_{M}CS(\mathcal{A)}, \label{CS}
\end{align}%
where
\begin{equation}
\mathcal{A}=\frac{1}{2}\omega ^{ab}J_{ab}+e^{a}J_{a5},\qquad \left\langle
J_{ab}J_{cd}J_{e5}\right\rangle =\varepsilon _{abcde},
\end{equation}%
$\kappa $ is dimensionless, $\left[ J_{AB},J_{CD}\right] =-J_{AC}\eta
_{BD}+J_{BC}\eta _{AD}-( A \leftrightarrow B)$, and $\langle ...\rangle $
stands for an invariant symmetric trace in the algebra.
In this way, the action acquires an enlarged gauge symmetry. If $\eta_{55}>0$ the gauge group is
$SO(5,1)$ and the action has positive cosmological constant. For $\eta_{55}<0$ the gauge group is
$SO(4,2)$ and the cosmological constant is negative. In the latter case, localized deformations of
the geometry give rise to asymptotically locally anti-de Sitter geometries.
Strictly speaking, under a gauge transformation the action (\ref{CS}) is not gauge invariant but
changes by a closed form plus a boundary term. This quasi invariance is a source of ambiguities in
an asymptotically $AdS$ spacetime, where the boundary terms that arise by gauge transformations
change the action and modify the conserved charges, producing even divergent values for them. This
problem can be circumvented if the action principle is modified by the addition of some new terms
that do not modify the field equations but render the action truly gauge invariant
\cite{Mora:2003wy,Mora:2006ka}. The trick is to replace the lagrangian in (\ref{CS}) by a
transgression form,
\begin{equation}
S(\mathcal{A},\bar{\mathcal{A}})=\kappa \int_{M}CS(\mathcal{A})-CS(\bar{%
\mathcal{A}})+\kappa \int_{\partial M}B(\mathcal{A},\bar{\mathcal{A}}),
\label{TPP}
\end{equation}%
where
\begin{gather}
B(\mathcal{A},\bar{\mathcal{A}})= -\Big\langle \mathcal{A} \bar{\mathcal{A}}
\left(\mathcal{F} + \bar{\mathcal{F}} - \frac{1}{2}\mathcal{A}^2 -\frac{1}{2}%
\bar{\mathcal{A}}^2+\frac{1}{2}\mathcal{A}\bar{\mathcal{A}} \right)%
\Big\rangle.
\end{gather}
The transgression form is the object which appears in the Chern-Weil theorem, that states that the
pullback of invariant polynomials of the curvature $P(\mathcal{F})$ are members of cohomology
groups of the manifold where they are defined \cite{Nakahara:1990th},
\begin{equation}
dP(\mathcal{F})=0,\qquad P(\mathcal{F})-P(\mathcal{\bar{F}})=dTP(\mathcal{A},%
\bar{\mathcal{A}}), \label{TP}
\end{equation}%
where $TP(\mathcal{A},\bar{\mathcal{A}})$ is defined by equation (\ref{TP})
up to a closed form. The gauge invariant, globally-defined expression for $%
TP(\mathcal{A},\bar{\mathcal{A}})$ stands for the transgression form. In $%
2n-1$ dimensions, the transgression takes the form
\begin{equation}
TP_{2n-1}(\mathcal{A},\bar{\mathcal{A}})=n\int_{0}^{1}dt\ \left\langle\left(
\mathcal{A}-\bar{\mathcal{A}}\right) \mathcal{F}_{t}^{n-1}\right\rangle,
\end{equation}
where $\mathcal{F}_{t}=d\mathcal{A}_{t}+\mathcal{A}_{t}\mathcal{A}_{t}$, $%
\mathcal{A}_{t} = \mathcal{A}\left( 1-t\right) + \bar{\mathcal{A}}t$. Thus, the boundary term
$B(\mathcal{A},\bar{\mathcal{A}})$ is uniquely determined by the Chern-Weil theorem.
The field equations for $\mathcal{A}$ are the same, whether the action principle is defined by
(\ref{TPP}) or by (\ref{CS}). However, there is a problem interpreting the physical meaning of the
field $\bar{\mathcal{A}}.$ An interpretation was proposed in \cite{Mora:2006ka}, where the
lagrangian $CS(\bar{\mathcal{A}})$ was considered as defined in a manifold with opposite
orientation to the one for $CS(\mathcal{A})$. An alternative is to regard $\bar{\mathcal{A}}$ not
as a dynamical field but as a means of constructing the boundary term which makes the action
finite \cite{Miskovic+Olea}. A different philosophy will be followed here, that is to regard
$\bar{\mathcal{A}}$ and $\mathcal{A}$ as two connections defining the same non trivial, principal
bundle. That is, they are related by a gauge transformation.
If a non trivial bundle is considered, the integrand does not exist globally either in (\ref{CS})
or in (\ref{TPP}). This non existence problem will be treated in \cite{AWZ1}, where a detailed
study of the definition of an action principle in a manifold divided into patches, for a non
trivial principal bundle, will be presented \footnote{The $D=3$ case was studied in
\cite{Dijkgraaf:1989pz}}. In the case of a non trivial bundle, however, the action (\ref{TPP}) can
be treated formally provided more than one chart is used. In this case, it is necessary to
introduce connection one-forms defined on each chart, such that, in the overlap of two charts the
connections are related by a gauge transformation,
$\bar{\mathcal{A}}=h^{-1}\mathcal{A}h+h^{-1}dh\equiv \mathcal{A}^{h}$, where $h$ is a transition
function which determines the non triviality of the bundle.
Replacing $\bar{\mathcal{A}}= \mathcal{A}^{h}$ in (\ref{TPP}), it is straightforward to check that
the action takes the form of a gWZW term,
\begin{widetext}
\begin{eqnarray}
&&S(h,\mathcal{A})=-\frac{\kappa}{10}\int_{M^5}\left\langle h^{-1}dh
(h^{-1}dh)^2
(h^{-1}dh)^2\right\rangle
+\kappa\int_{M^4} \left \langle dhh^{-1} {\cal A} \left( d\mathcal{ A}
+
\frac{1}{2}{\cal A}^2 \right)\right\rangle
\label{WZ} \\&&\notag
-\frac{\kappa}{2} \int_{M^4} \left\langle dh h^{-1} \mathcal{A}
\left\{(dhh^{-1})^2 + {\cal A}\, dhh^{-1} \right\}\right\rangle
- \kappa\int_{M^4}\left\langle {\cal A} {\cal A}^h
\left( \mathcal{ F} + \mathcal{ F}^h -\frac{1}{2}
{\cal A}^2 -\frac{1}{2}({\cal A}^h )^2 + \frac{1}{2}{\cal A} {\cal A}^h\right) \right\rangle\,
.
\end{eqnarray}
\end{widetext}
where the curvature is $\mathcal{F}=d\mathcal{A}+\mathcal{AA}$ and $\mathcal{%
F}^{h}= h^{-1}\mathcal{F}h.$ This action is invariant under the adjoint
action of the gauge group, namely,
\begin{equation}
\mathcal{A}\rightarrow g^{-1}\mathcal{A}g+g^{-1}dg,\qquad h\rightarrow
g^{-1}hg.
\end{equation}
As has been shown, the principle of gauge invariance, through the mathematical structure of the
theory of principal bundles, provides a compactification mechanism. Beginning with a five
dimensional gauge theory that has no metric in it, a $D=4$ gauge invariant theory has been
obtained. This is a compactification mechanism alternative to Kaluza-Klein. The relation between
three-dimensional Chern-Simons theories and two-dimensional gWZW models was early realized in
\cite{Moore:1989yh}. Originally, gWZW terms were obtained in \cite{Witten:1983tw} by trial and
error, and it was later shown that they can be obtained systematically see, e. g.,
\cite{Alvarez-Gaume:1984dr,Witten:1991mm,deAzcarraga:1998bu}.
Other attempts to relate the $D=4$ gWZW lagrangians to $D=5$ Chern-Simons theory can be found in
the literature (see for instance, refs. \cite{Banados-96,Gegenberg}). However, asymptotic
conditions for the metric were always assumed in order to reproduce the kinetic term for the
Goldstone fields, not present in (\ref{WZ}). As can been seen from the previous discussion, such a
strong assumption is not required here, the kinetic term does not appear, but the gWZW term arises
naturally from the transgression form.
The action (\ref{WZ}), usually supplemented with a kinetic term for the so-called Goldstone
fields, requires the introduction of the Hodge dual, which in turn requires the existence of a
metric in the manifold. The point of view followed here is that the metric arises from the
components of a gauge connection in a broken phase of the theory, but it is not assumed to be
defined a priori. In the next section the action principle (\ref{WZ}) for a particular class of
$h$ will be studied further.
\section{The gWZW term as a gravitational action}
The action (\ref{WZ}) describes a theory with $SO(4,2)\times SO(4,2)$ gauge symmetry spontaneously
broken to its diagonal subgroup $SO(4,2)$\footnote{For the sake of simplicity the discussion is
restricted to $G=SO(4,2),$ the extension of the results to $SO(5,1)$ is trivial. In this section
the indices $a,b$ take values in the range, $0,1,2,3$.} \cite{Witten:1983tw}. The gauge
invariance is manifest since the connection takes its values in the diagonal subalgebra only
\cite{Witten:1991mm}. However, in order to describe the known low energy gravitational behavior,
it is necessary to reduce the symmetry to the usual $SO(3,1)$ local symmetry present in the
Einstein-Hilbert action. A way to do this is by replacing the gauge group $SO(4,2)$ by the coset
$\frac{SO(4,2)}{\mathbb{R}}$, where $R$ stands for the group of transformations generated by
$e^{\phi J_{45}}$ and an element of the coset, $h$, is a representative of the equivalence class
$\left[ h\right] =\left\{ h\sim h^{\prime }\Longleftrightarrow h^{\prime }=e^{\phi
J_{45}}h|h,h^{\prime }\in SO(4,2)\right\}$. Using the adjoint action, the stability group of this
coset is $SO(3,1)\times R$ and it corresponds to the residual gauge invariance present in the
theory, that is, $h\in \left[ h\right] \Longrightarrow g^{-1}hg\in \left[ h\right]
\Longleftrightarrow g\in SO(3,1)\times R$.
Fixing the Goldstone field associated to $J_{45}$ in the action
(\ref{WZ}), corresponds to reducing the gauge symmetry down to
$SO(3,1) \times R$. There is an interesting geometrical
interpretation of this. Suppose we have a six-dimensional manifold
$M^{6}$ and delete a four dimensional submanifold $M^{4}$ (Fig.
\ref{6d_fig}).
\begin{figure}[h]
\includegraphics[width=.4\textwidth]{sixd.eps}\\
\caption{A four-dimensional defect in a six-dimensional manifold.
The submanifold $M^4$ has been deleted, as indicated by the infinitesimal
loop in the center of the diagram.}\label{6d_fig}
\end{figure}
Then, by considering the integral of a characteristic class on $M^{6}-M^{4}$, the action induced
on $M^{4}$ is (\ref{WZ}) \cite{AWZ2},
\begin{equation}
\int_{M^{6}-M^{4}}\left\langle \mathcal{FFF}\right\rangle =S(h,\mathcal{A)}.
\end{equation}
The field $\phi$ can be interpreted in terms of the
six-dimensional pseudo-Riemannian geometry, as a deficit angle
around the four-dimensional defect $M^4$ which, as shown here, is
related to the four dimensional cosmological constant. Here we
have assumed that $\phi$ is a constant, thus breaking part of the
gauge symmetry ``by hand". In terms of this geometrical picture,
the defect is assumed to have a fixed deficit angle.
Now, in order to write the field equations associated with the
gWZW term (\ref{WZ}) for the coset $\frac{SO(4,2)}{\mathbb{R}}$,
it is helpful to decompose the connection and the curvature in a
way that reflects the $SO(3,1)$ symmetry,
\begin{eqnarray}
\mathcal{A} &=&\frac{1}{2}\omega ^{ab}J_{ab}+b^{a}J_{a4}+e^{a}J_{a5}+\Phi J_{45}, \\
\nonumber \mathcal{F} &=&\frac{1}{2}(R^{ab}+e^{a}e^{b}-b^{a}b^{b})J_{ab}+[Db^{a}+ e^a \Phi]J_{a4}
\\
&&+[De^{a}+b^a\Phi]J_{a5}+[d\Phi-b_{a}e^{a}]J_{45},
\end{eqnarray}%
where $De^{a}=de^{a}+\omega_{\text{ }b}^{a}e^{b}$. The field equations associated to the variation
of the Goldstone fields in the coset $\frac{SO(4,2)}{\mathbb{R}}$ are
\begin{widetext}\begin{gather} \kappa
\int_{M^{4}}\Big\langle h^{-1}\delta h\Big\{{}\!\left(
\mathcal{F}^{h}\right) ^{2}+\mathcal{F}^{2}+{}\!\mathcal{F}^{h}\mathcal{F}-
\frac{3}{4}[{}\!\mathcal{A}^{h}-\mathcal{A},{}\mathcal{A}^{h}-\mathcal{A}]\
({}\!\mathcal{F}^{h}+\mathcal{F}) \notag \\ \hspace{2in}
+\frac{1}{8}[{}\!\mathcal{A}^{h}-\mathcal{A},{}\!\mathcal{A}^{h}-\mathcal{A}]^{2}+
\frac{1}{2}({}\!\mathcal{A}^{h}-\mathcal{A})[{}\!\mathcal{F}^{h}+
\mathcal{F},{}\!\mathcal{A}^{h}-\mathcal{A}])\Big\}\Big\rangle=0, \label{2}
\end{gather}\newline
while the $15$ equations of motion that arise from the variation of the connection, are
\begin{equation}
\kappa \int_{M^{4}}\langle \delta \mathcal{A} ({}\!\mathcal{A}^{h}-\mathcal{A} )\left(
{}\!\mathcal{F}^{h}+2\mathcal{F}-\frac{1}{4}[{}\!\mathcal{A}^{h}-
\mathcal{A},{}\!\mathcal{A}^{h}-\mathcal{A}]\right) \rangle -(h\leftrightarrow h^{-1})=0.
\label{con}
\end{equation}\end{widetext}
In order to obtain the equations of motion in a explicit form, it is necessary to pick a
representative of the coset. In any open set, it can be parametrized by $14$ coordinates $\pi $,
as $h=e^{\phi J_{45}}e^{\pi ^{a4}J_{a4}}e^{\pi ^{a5}J_{a5}}e^{\frac{1}{2}\pi ^{ab}J_{ab}}$, where
$\phi $ is an arbitrary, real, constant. The purely gravitational sector of the theory, that is
the one in which only the vielbein and the spin connection are present, corresponds to setting
$b^{a}$, $\Phi $ and the $14$ Goldstone fields equal to zero. In this dynamical sector the set of
equations (\ref{con}) reduces to
\begin{eqnarray}
\varepsilon _{abcd}\;e^{b}\left( R^{cd}+\mu e^{c}e^{d}\right)
\sinh \phi &=&0 \label{EH} \\
\varepsilon _{abcd}\;e^{c}De^{d}\sinh \phi &=&0. \label{TOR}
\end{eqnarray}
Here the constant $\mu$ is given by $\left( 1+2\cosh \phi \right)/3$. Excluding the trivial case
$\phi=0$ implies $\mu >1$, and equation (\ref{TOR}) implies that the torsion is zero. Solving the
torsion for the spin connection and replacing it back in (\ref{EH}), the Einstein's equations in
standard form are obtained. It is reassuring to check that these field configurations also satisfy
the $14$ field equations obtained from (\ref{2}).
In order to write Einstein's equations it is necessary to assume that the vielbein $e_{\text{ }\mu
}^a$ is invertible, and to make contact with a metric theory, it is also necessary to rescale the
vielbein with a parameter with dimensions of length, $e^{a}=\bar{e}^{a}/l$. In this way, the
metric is $g_{\mu v}=\bar{e}_{\mu }^{a}\bar{e}_{v}^{b}\eta _{ab}$, the effective cosmological
constant acquires its usual units $\Lambda =\left( 1+2\cosh \phi \right)l^{-2}$, and (\ref{EH})
can be rewritten as
\begin{equation}
R_{\mu v}-\frac{1}{2}g_{\mu v}R-\Lambda g_{\mu v}=0.
\end{equation}
However, as was shown in ref. \cite{Witten:1988hc}, assuming the invertibility of the vielbein
spoils the possibility of making a sensible, quantum mechanical, perturbative expansion around
$\mathcal{A}=0$. The rescaling of $e^a$ is also unhelpful in the sense that, if no such rescaling
is done, all the parameters of the theory are dimensionless, which would suggest the possibility
of power counting renormalizability of the theory.
\section{Discussion and Outlook}
Here we have shown that General Relativity is a dynamical sector of a gWZW theory for the coset
$\frac{SO(4,2)}{\mathbb{R}}$. It can be checked that the same phenomenon occurs if in the
$SO(4,2)$ gWZW, a representative of the group is taken as $h=e^{\phi J_{45}}e^{\pi
^{a4}J_{a4}}e^{\pi ^{a5}J_{a5}}e^{\pi ^{ab}J_{ab}}=e^{\phi J_{45}}\bar{h}$, and $\phi$ is kept
fixed in the action reducing the symmetry to $SO(3,1)\times R$. The four-dimensional spacetime
arises in the sector of solution space characterized by $\bar{h}=1$, det$e\neq 0$, $b=0=
\mathcal{A}^{45}$.
Having obtained the equations of General Relativity, a more detailed analysis of the dynamical
structure of the theory (\ref{WZ}) is necessary. Generically, as it happens with all higher
dimensional Chern-Simons theories, the system will possess degenerate dynamical sectors
\cite{Saavedra:2000wk,Miskovic:2003ex,Miskovic:2005di}. A deeper understanding of this problem
would be required prior to any study of the quantum properties of the action (\ref{WZ}).
The theory changes dramatically if the $\phi$ field is regarded as dynamical. In that case, there
is no purely gravitational sector with only $e^a$ and $\omega^{ab}$ nonzero. However there are
solutions of gravity coupled to the other fields, including an interesting class of gravitational
solitons, that is, Lorentzian, everywhere regular, classical solutions. A two-parameter family of
these solitons will be presented in \cite{AWZ3}.
\textbf{Acknowledgements}
The authors wish to thank Eloy Ayon-Beato, Sara Farese, Gast\'{o}n
Giribet, Joaquim Gomis, Elias Gravanis, Julio Oliva, Tom\'{a}s
Ort\'{\i}n and Ricardo Troncoso for enlightning discussions. This
work has been supported in part by FONDECYT grants $N^{o}$s 1061291,
1060831, 1040921 and 3060016. A.A. wishes to thanks the support of
MECESUP UCO 0209 and CONICYT grants during the realization of this
work. Institutional support to the Centro de Estudios
Cient\'{\i}ficos (CECS) from Empresas CMPC is gratefully
acknowledged. CECS is funded in part by grants from the Millennium
Science Initiative, Fundaci\'{o}n Andes, the Tinker Foundation.
|
2,877,628,090,118 | arxiv | \section{Introduction}
\IEEEPARstart{H}{aze} is a common phenomenon when the light is absorbed and scattered by the turbid medium.
It leads to low visibility and contrast in outdoor scenes.
Hazy image can be described using the Atmospheric scattering model,
which was firstly proposed in 1976~\cite{Mccartney1977}
and widely used by computer vision and graphics community.
Later, Narasimhan and Nayar~\cite{nayar1999,Narasimhan2003,Narasimhan2002,Narasimhan2001}
improved the model and re-formulated it as:
\begin{equation}
I(x)=J(x)t(x)+A(1-t(x)),
\end{equation}
where $x$ denotes the pixel locations in the image,
$I(x)$ demotes the hazy image observed, $J(x)$ demotes the real scene without haze,
$A$ demotes the atmosphere light, and $t(x)$ demotes the medium transmission.
The term $t(x)$ can be further written as:\begin{equation}
t(x)=\exp(-\beta d(x)),
\end{equation}
where $d(x)$ is the depth of the scene and $\beta$ indicates the scattering coefficient of the atmosphere.\par
It is difficult to use hazy images directly for most computer vision tasks like object detection~\cite{wang2014tracking,Shen17DSOD}, tracking~\cite{Wang16a,maksai16CVPR}, pose estimation~\cite{TekinArXiv15,Belagiannis16}, behavior analysis~\cite{Wang14a,maksai17ICCV}, and search~\cite{WangTIP11,Wang11ICME}.
Researchers have thus been devoting great efforts in haze
removal to restore images with high quality,
among which haze removal from a single image becomes the focus.
For example, Tan~\cite{Tan2008}, Fattal~\cite{Fattal2008} and He~\cite{He2009}
implemented single-image dehazing using hand-crafted features,
upon which the approaches of~\cite{Kratz2009,ancuti2010,He2013,Meng2013}
were proposed.
All above algorithms, however, rely on
specific hand-crafted features, which are not able to fully characterize the hazy images.
To this end, recent research has been focused on applying Convolutional Neural Network~(CNN)
that is able to automatically extract features to handle this task.
DehazeNet~\cite{Cai2016} and MSCNN~\cite{ren2016} design CNNs to estimate the transmission map of the hazy image
and subsequently use it to estimate atmosphere light.
Then, the haze-free image is computed using Eq.~(1).
In AOD-Net~\cite{li2017}, the atmosphere scattering model is re-expressed,
and the atmosphere light, scattering, and transmission map are rewritten into one matrix.
AOD-Net estimates this matrix and introduces addition layers as well as multiplication layers
to compute the re-expressed formula.
Although AOD-Net estimates transmission map and atmosphere light at the same time, it first produces an intermediate product, i.e., the matrix of estimated parameters,and then computes the haze-free image by an artificial formula based on the obtained matrix.The errors of the intermediate step may therefore propagate to the dehazing part and thus downgrade the results.
\xw{GFN~\cite{ren2018gated}, on the other hand, directly estimates the clear scenes from hazy images but relies complex pre-processing operations including white balance, contrast enhancing, and gamma correction. }
In this paper, we propose a light end-to-end dehazing deep network,
termed Light Dual-Task Network~(LDTNet).
In contrast to prior models like AOD-Net that
decouples the process into two steps,
ours estimates dehazed images from hazy ones in one shot,
in other words, we do not rely on any intermediate output.
Furthermore, our model does not rely on artificial priors,
such as atmospheric scattering model and the one of~Eq.~(1).
To facilitate the feature learning process,
we introduce Multitask Learning (MTL) \cite{Caruana1997}
to estimate the haze-free image and transmission map
simultaneously.
The reason we incorporate the transmission map estimation as an auxiliary task
is that, the features learned for this task can benefit our main task of dehazing.
With a single-task dehazing network, however, it may be very tough to learn such features.
Also, learning the main task alone bears the risk of overfitting~\cite{He2016},
which can be alleviated by the joint learning with the auxiliary one.
In fact, this phenomenon of auxiliary task benefiting the main one
has been observed in a wide domain of
high-level vision and text tasks~\cite{Sun2014,McLaughlin2017,Ren2015,He2016,Arik2017},
\begin{figure*}[htb]
\centering
\includegraphics[width=0.95\textwidth]{fig1.pdf}
\vspace{-0.5cm}
\caption{The architecture of LDTNet. It takes a hazy image as input, and outputs a dehazed image and an estimated transmission map. }
\label{fig1}
\vspace{-0.5cm}
\end{figure*}
Our contribution is therefore a light Multitask dehazing deep network
that does not depend on artificial priors, and that produces
haze-free image and transmission map simultaneously in one shot.
Our model yields superior results, compared to the state of the art,
on both synthetic and real-world data.
\section{Model}
In this section, we introduce our proposed LDTNet.
We start by showing the architecture design and then introduce the loss function.
Unlike prior models that heavily rely on artificial priors, either the hand-crafted features
or hypothetical dehazing models, our approach jointly learns two tasks,
the main task of dehazing and auxiliary one of transmission map estimation,
without human-provided priors in one shot.
\subsection{Architecture Design of LDTNet}
The proposed LDTNet, with the help of the auxiliary task, is able to restore haze-free image with a lightweight structure.
The LDTNet is composed of three cascaded convolutional layers, where the restored image and
the estimated transmission map are obtained in the third layer.
We show the architecture of LDTNet in Fig.~1.
The two tasks share the first two convolutional layers by hard parameter sharing.
The sizes of convolutional kernels in these two layers are all $3\times3$ and the output channels are 30 and 40 respectively.
The input three-channels RGB hazy image is concatenated to these two layers severally as three additional feature maps.
This operation provides information contained in the input image, ensuring the refinement of the features layer by layer.
There are two parts of the last layer.
They combine the feature maps of the
second convolutional layer in different ways to reconstruct the haze-free image and transmission map respectively.
The sizes of convolutional kernels in these two parts are all $1\times1$ while the output channels are 3 and 1 respectively.\par
In LDTNet, no pooling layers are used and zero pixels are padded to the features maps,
ensuring the size of the output image to be consistent with that of the input map.
Furthermore, batch normalization is
applied after the first two convolutional layers. Bilateral Rectified Linear Unit~(BRelu)~\cite{Cai2016} is adopted
as our activation function. Specifically,
BRelu is a modified version of Rectified Linear Unit (Relu)~\cite{Nair2010} with the upper limit set to be 1.
It ensures that the pixels in the restored image are constrained in the range of $[0,1]$.
\subsection{Loss Function Design of LDTNet}
In LDTNet, we simultaneously tackle dehazing and transmission estimation.
We take the loss function to be
\begin{equation}
L=(1-\alpha)L_{D}(J(x),J^{*}(x))+\alpha L_{T}(t(x),t^{*}(x)),
\end{equation}
where $L_{D}$ and $L_{T}$ correspond to the dehazing loss
and transmission loss respectively, and $\alpha$ balances the two.
In our implementation,
both $L_{D}$ and $L_{T}$ take the form of
square loss, defined on the pixel-wise difference
between the ground truth and the prediction of the network.
\vspace{-0.3cm}
\begin{figure}[ht]
\centering
\includegraphics[width=0.28\textwidth]{fig2.pdf}
\caption{{\color{black}Network performance with different $\alpha$. }
}
\label{fig2}
\vspace{-0.3cm}
\end{figure}
\vspace{-0.0cm}
\begin{figure*}[htb]
\centering
\includegraphics[height=0.42\textwidth,width=0.97\textwidth]{fig3.pdf}
\footnotesize \begin{tabular}{p{1.78cm}<{\centering}p{1.78cm}<{\centering}p{1.78cm}<{\centering}p{1.78cm}<{\centering}p{1.78cm}<{\centering}p{1.78cm}<{\centering}p{1.78cm}<{\centering}p{1.78cm}<{\centering}}
(a)&(b)&(c)&(d)&(e)&(f)&(g)&(h)
\end{tabular}
\caption{Comparative dehazing results on our synthetic dataset.
(a)~The haze images,
(b)~DCP,
(c)~CAP,
(d)~DehazeNet,
(e)~MSCNN,
(f)~OAD-Net,
(g)~LDTNet,
and (h)~Ground truth.}
\label{Fig.3}
\end{figure*}
\section{Experiments}
In this section, we show our experimental validation of the proposed LDTNet.
We first introduce the baselines methods we use for comparison,
and then present our training strategy. Afterwards, we
show the effectiveness of our dual-task learning,
followed by the comparative results on synthetic
and real-world test images.
We finally provide the robustness analysis of different methods.
\subsection{Baselines}
We compare our proposed LDTNet with several state-of-the-art methods briefly introduced as follows.
\begin{itemize}
\item{DCP~\cite{Fattal2008}: {
The thickness of haze is estimated first and then the haze-free image is recovered using Eq.~(1). }}
\item{CAP~\cite{Zhu2015}: {Color attenuation prior is utilized for estimating the scene depth,
which is further used for computing the haze-free image. }}
\item{MSCNN~\cite{ren2016}: {The transmission map is estimated and refined using two CNNs of different scales,
which is then used to obtain the haze-free image by Eq.~(1). } }
\item{DehazeNet~\cite{Cai2016}: {The transmission map is estimated using a CNN
with a novel nonlinear activation function. } }
\item{AOD-Net~\cite{li2017}: {The transmission map and the airlight are jointly learned using a CNN.} }
\end{itemize}
\subsection{Training}
To train the LDTNet, we synthesize 10,000 triples of hazy images, haze-free images and transmission maps,
all of which are resized to $240\times320$ pixels based on the NYU depth~\cite{Silberman2012} dataset using~(1) and~(2).
We set $ A\in(0.7,1.0) $ and $ \beta\in(0.5,1.5) $, and thus our dataset covers various
atmosphere situations, multifarious levels of haze, as well as different weather conditions.
For LDTNet, we use Adam~\cite{Kingma2014} to be our optimizer and set the batch size to be~4.
We implement our model using Tensorflow~\cite{Abadi2015} and the Tensorlayer~\cite{Dong2017} package.
\xw{It takes about 17 hours to train the LDTNet for 100 epochs on a Titan X GPU with Intel~i7 CPU.}
\subsection{Validity of Dual-task Learning}
To validate our dual-task learning, we compute the Mean Square Error~(MSE)
between our result and ground truth under various values of~$\alpha$
on our validation set, as shown in Fig.~2.
When $\alpha=0$,
the auxiliary task is removed and
only the main task dehazing plays a role,
in which case we obtain the largest MSE.
This shows that our multitask learning is indeed helpful.
Also, as can be seen, the performance of network comes to the best when $\alpha=0.4$
on our validation set.
\begin{figure*}[htb]
\centering
\includegraphics[height=0.46\textwidth,width=0.93\textwidth]{fig4.pdf}
\footnotesize \begin{tabular}{p{2.05cm}<{\centering}p{2.05cm}<{\centering}p{2.05cm}<{\centering}p{2.05cm}<{\centering}p{2.05cm}<{\centering}p{2.05cm}<{\centering}p{2.05cm}<{\centering}}
(a)&(b)&(c)&(d)&(e)&(f)&(g)
\end{tabular}
\vspace{-0.2cm}
\caption{Comparative dehazing results on real-world images.
(a)~The hazy images,
(b)~DCP, (c)~CAP,
(d)~DehazeNet,
(e)~MSCNN,
(f)~OAD-Net,
and~(g)~LDTNet.}
\label{Fig.4}
\vspace{-0.5cm}
\end{figure*}
\subsection{Performance on Synthetic Test Dataset}
We use a synthetic dataset including 21 pairs of stereo images generated using the Middlebury stereo database~\cite{Scharstein2002,Hirschmuller2007,Scharstein2003} to verify the performance of different models.
The atmosphere light $A$ is set to be~0.85 and the scattering coefficient $\beta$ is set to~1.
These values correspond to the medians of the domains of $A\in(0.7,1)$ and $\beta\in(0.5,1.5)$.
\xw{It takes about 0.3 second to produce a dehazed image during testing.}
\begin{table}[htbp]
\centering
\caption{Average PSNR and SSIM of different dehazing methods on the synthetic dataset}
\label{table I}
\begin{tabular}{p{0.7cm}<{\centering} p{0.7cm}<{\centering} p{0.7cm}<{\centering} p{0.8cm}<{\centering} p{1cm}<{\centering} p{1.1cm}<{\centering} p{0.8cm}<{\centering}}
\hline
\hline
Metrics & DCP & CAP & MSCNN & DehazeNet & AOD-Net & LDTNet \\ \hline
PSNR & 14.6713 & 20.9303 & 20.2855 & 21.9690 & 21.3655 & \textbf{24.6156} \\
SSIM & 0.8432 & 0.9452 & 0.9274 & 0.9463 & 0.9419 & \textbf{0.9517} \\ \hline \hline
\end{tabular}
\end{table}
We show the obtained mean PSNR and SSIM of the results in Table~I.
LDTNet achieves the highest PSNR and SSIM scores.
In Fig.~3, we show the comparative dehazing results of five examples:
\emph{\textbf{Midd, Cloth, Bowling,Aloe}} and \emph{\textbf{Monopoly}}.
As can be seen, LDTNet yields visually plausible results which are very similar to the ground truths.
The restored images of the other methods have larger color distortions or over-saturations.
They are over-sensitive to regions of light colors like white, as they appear similar to haze.
For instance, in the case of~\emph{\textbf{Cloth}},
baseline methods tend to produce yellowish colors for the regions that are supposed be of
white color.
Baseline methods fail to produce very accurate results, in part due to
their hand-crafted features or their two-step nature.
In the cases of DCP and CAP, they rely on a limited set of handcraft features
that may not be expressive enough for some scenes.
MSCNN and DehazeNet, on the other hand,
first estimate the intermediate transmission maps and then produce the dehazed results.
The errors occurred in the transmission estimation may thus
propagate to dehazing and negatively influence the results.
Although AOD-Net estimates the transmission map and atmosphere together
using a CNN, it still relies on an artificial formula to obtain the haze-free image.
By contrast, LDTNet directly learns a mapping from hazy images to the haze-free ones by taking
the transmission map estimation as an auxiliary task.
As a result, LDTNet does not suffer from the limitations of intermediate products and artificial priors.
\subsection{Performance on Real-world Test Images}
We qualitatively compare our algorithm with DCP, CAP,
MSCNN, DehazeNet and AOD-Net on~{50} challenging
real-world images. We show a few of them in Fig. 4.
As can be seen from the second column, DCP leads to over enhancement especially on the first three images.
CAP is yields better results than DCP does in terms of over enhancement, but
fails to preserve textural details in the region of similar colors, like the mountains in the second image
and the hair of the girl in the third.
MSCNN, DehazeNet, and AOD-Net also suffer a certain degree of over enhancement.
In addition, MSCNN causes hue distortion
in the last image of the fifth column.
The results of LDTNet are indeed more visually plausible, without noticeable color distortions or loss of details.
\subsection{Robustness Analysis}
We conduct robustness analysis, as done in DehazeNet~\cite{Cai2016},
for baselines methods and ours using four types of evaluations, i.e.,
airlight robustness evaluation~(ARE), coefficient robustness evaluation~(CRE),
scale robustness evaluation~(SRE) and noise robustness evaluation~(NRE).
\begin{table}[]
\centering
\caption{Average PSNR and SSIM of dehazing results using ARE, CRE, SRE and NRE}
\label{lable II}
\begin{tabular}{cccccc}
\hline
\hline
& Metrics & ARE & CRE & SRE & NRE \\ \hline
DehazeNet & PSNR & 21.7716 & 21.8800 & 22.0891 & 20.0235 \\
& SSIM & 0.9450 & 0.9423 & 0.9416 & 0.3469 \\
MSCNN & PSNR & 20.1496 & 20.1152 & 20.4302 & 19.8722 \\
& SSIM & 0.9258 & 0.9240 & 0.9202 & 0.4017 \\
AOD-Net & PSNR & 21.1600 & 21.0030 & 21.3820 & 19.9497 \\
& SSIM & 0.9399 & 0.9395 & 0.9355 & 0.4098 \\
LDTNet & PSNR & \textbf{24.1181} & \textbf{24.2780} & \textbf{24.6344} & \textbf{22.2765} \\
& SSIM & \textbf{0.9459} & \textbf{0.9468} & \textbf{0.9441} & \textbf{0.4925}
\\\hline \hline
\end{tabular}
\vspace{-0.3cm}
\end{table}
In Table~IV, we show the mean PSNR and SSIM on the Middlebury stereo dataset.
In ARE, we synthesize 210 hazy images with $ \beta=1 $ and $ A\in(0.7,1.0) $.
Similarly, the same number hazy images are synthesized with $A=0.85$ and $ \beta\in(0.5,1.5) $ for CRE.
To analyze the influence of the scale variation,
we select four scale coefficients, i.e., $1,0.8,0.6,0.4$,
to generate different scale images with $ \beta=1 $ and $ A=0.85 $.
Finally, we add three types of noises to the hazy images generated with $ \beta=1 $ and $ A=0.85 $ for NRE.
The three kinds of noises are Gaussian noise, Poisson noise and salt \& pepper noise.\par
As can been seen in Table~II, we achieve the best performance in the four types of evaluations.
This is because our dual-task learning approach improves
the network's capability of task-relevant features extraction and generalization.
As a result, LDTNet can extract effective features from the hazy images even in the presence of
noise and various $A$ and $\beta$.
Also, since we fuse the original hazy images into the convolutional layers,
the information of the source image can be preserved and utilized to a greater extent.
\section{Conclusion}
In this paper, we have presented a light dual-task neural network, LDTNet,
which takes as input hazy images and produces dehazed ones in one shot, without
any intermediate results.
The auxiliary task, transmission map estimation, proves to be helpful for enhancing
dehazing and for improving the network's generalization capability.
We conduct quantitative and qualitative evaluations, and compare our results with
those of the state-of-the-art methods on both both synthetic and real-world hazy images.
LDTNet yields the most promising results in terms of both accuracy and robustness.
\xw{In our future work, we will explore to simultaneously conduct dehazing and other tasks,
such as high-level object detection~\cite{Wang2018CVPR} and tracking~\cite{Wang2018TIP1,wang17TIP} ones,
and low-level super-resolution~\cite{Wang2018TIP2} and image restoration~\cite{YinFishEye} ones,
where the both tasks could potentially benefit each other.}
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
{
\footnotesize
\bibliographystyle{ieee}
|
2,877,628,090,119 | arxiv | \section{Introduction}
The aim of our research project is to test the validity of the
perturbative QCD approach to multiparticle production in the
semihard and soft region; we also wish to study the
sensitivity of the physical observables to different aspects included in the
theory, like the running of the coupling $\alpha_s$ (see in this respect
\cite{lo}); in this paper we mainly concentrate on the sensitivity
of soft particle production to QCD coherence.
The theoretical framework for the description
of inclusive observables in jet physics at parton level is provided by
the Modified Leading Log Approximation (MLLA)
of QCD (for a review see~\cite{DKMTbook}),
where coherence, running of $\alpha_s$ and
energy-momentum conservation are taken into account.
Predictions depend on two free parameters only,
i.e., the infrared cut-off at which the parton evolution is stopped, $Q_0$,
and the effective QCD scale $\Lambda$ which appears in the
one-loop expression for the running coupling. An interesting limiting case is
obtained by pushing the parton cascade down to the very end, i.e., by
choosing $Q_0 = \Lambda$; in this case, one obtains a simple closed expression for the
spectrum, called Limiting Spectrum.
To connect predictions at parton level with experimental data,
Local Parton Hadron Duality (LPHD)\cite{LPHD} is taken
as hadronization prescription, i.e., the
inclusive hadron spectra are required to be
proportional to the corresponding inclusive parton spectra.
The whole hadronization is then parametrized in terms of only one parameter,
which gives the overall normalization of the distribution, but does not
affect its moments of order greater or equal than one.
\section{Phenomenology of inclusive energy spectra: an update}
The description of the inclusive energy spectrum for charged particles
is one of the main successes of the analytical QCD approach\cite{DKMTbook};
this approach turned out to be valid at LEP-1.5 $cms$ energy
too. Let us just list a few points of interest (see \cite{klo} for details).
\noindent{\it Analysis of the shape.}
The inclusive energy distribution is
well described by the Limiting Spectrum
with $Q_0$ = $\Lambda$ = 270 MeV and 3 active flavours.
After a rescaling, which allows to have the same kinematics both at parton and
hadron level, the very soft tail of the distribution is well described as well.
The overall normalization factor at LEP-1.5
turns out to be consistent with the LEP-1 value, in agreement with the
predictions of the perturbative approach.
\noindent {\it The position of the maximum.}
The energy dependence of the position of the maximum of the spectrum is
well described by the Limiting Spectrum predictions
with $Q_0$ = $\Lambda$ = 270 MeV and 3 active flavours.
Notice that the popular three terms formula~\cite{DKMTbook} for the energy dependence
of the maximum underestimates by an almost constant value 0.1 the true
position of the maximum of the Limiting Spectrum; this gives rise to a
difference of the order of 20-30 MeV
in the determination of the best value of the cut-off $Q_0$.
Let us also point out
that the number of flavours enters in the expressions for the cumulant moments
through the running coupling $\alpha_s(y,n_f)$; since the MLLA is defined at
one-loop level, a scale ambiguity in the expression of $\alpha_s$ is present;
kinematical reasons\cite{klo} suggest to push the heavy quark
thresholds to larger scales and keep 3 active flavours only.
\noindent {\it Moment analysis.}
This study is particularly
interesting, because it does not depend on the overall normalization
parameter and, since theoretical
predictions are in this case absolutely normalized at threshold, it
allows to test the perturbative picture down to low $cms$ energies.
Theoretical predictions of the Limiting Spectrum with
$Q_0$ = 270 MeV and 3 active flavours are in good agreement
with experimental results\cite{lo}.
Switching off the running of the coupling, one
cannot reproduce the behavior of high order cumulants.
By relaxing the absolute normalization, thus building an effective model with
one more parameter for each cumulant to reproduce the high energy region,
one can describe the experimental data only in a small energy
region\cite{mont}.
These results show the sensitivity of this analysis
to the running of the coupling. In this respect, the moment analysis
of spectra in high-$p_T$ jets
in $p\bar p$ collisions, where larger values of jet energy can be
reached\cite{korytov}, is eagerly awaited.
\section{The invariant density in the soft limit}
\subsection{A new scaling law in experimental data}
Let us consider the behaviour of
the charged particle invariant density, $E dn/d^3p$,
at small particle energy $E$.
Figure~(\ref{charged}a) shows data at different $cms$ energies
ranging from 3 GeV up to LEP-1.5\cite{klonew}.
The value of 270 MeV has been used for the effective mass $Q_0$
which enters in the kinematical relation, $E^2 = p^2 + Q_0^2$.
It is remarkable that
all data scale with $cms$ energies within 20\%
at particle energy of the order of few hundreds MeV;
at larger particle energies, a violation of the scaling-law is visible.
LEP data seem to tend to a larger limiting value; it is not yet clear
whether this is a signal of a different physics at LEP energies, like for
instance an enhanced contribution of weak decays to particle production in
the soft region, or a systematic effect in the overall normalization
of the different experiments.
\begin{figure}
\vfill \begin{minipage}{.48\linewidth}
\begin{center}
\mbox{
\mbox{\epsfig{file=scritte.ps,bbllx=5.2cm,bblly=17.5cm,bburx=5.4cm,bbury=25.cm}}
\mbox{\epsfig{file=allbis.ps,width=.72\linewidth,bbllx=5.5cm,bblly=2.5cm,bburx=15.cm,bbury=18.cm}}
} \end{center}
\vspace{-0.2cm}
\centerline{$\quad \qquad E$ [GeV]}
\end{minipage}\hfill
\begin{minipage}{.48\linewidth}
\begin{center}
\mbox{
\mbox{\epsfig{file=scritte.ps,bbllx=5.2cm,bblly=17.5cm,bburx=5.4cm,bbury=25.cm}}
\mbox{\epsfig{file=mllapred.ps,width=.72\linewidth,bbllx=5.5cm,bblly=2.5cm,bburx=15.cm,bbury=18.cm}}
} \end{center}
\vspace{-0.2cm}
\centerline{$\quad \qquad E$ [GeV]}
\end{minipage}
\caption{{\bf a)}: Invariant density $E dn/d^3p$ for charged particle
as a function of the particle energy $E$ with $Q_0$ = 270 MeV.
{\bf b)}: same as in {\bf a)}, with theoretical predictions from MLLA with
the same $Q_0$ and $\Lambda$ = 257 MeV ($K_h$ = 0.45).}
\label{charged}
\end{figure}
\subsection{Predictions of QCD coherence}
QCD colour coherence forbids branchings of very soft gluons\cite{DKMTbook};
therefore, the emission of soft partons should be proportional to the colour
charge of the initial system only and the invariant
density $E dn/d^3p$ should be independent of $cms$ energy
at low particle energy\cite{vakcar}.
Recently, predictions for the invariant density have been analytically
computed\cite{lo,klonew} by solving explicitly in the limit of small particle energy
the evolution equation for
the energy spectrum in a jet of type $A$, $D_A(\xi,Y,\lambda)$.
The relation
$E_h dn/d^3p_h \equiv K_h D_A(\xi_E,Y,\lambda)/[4\pi (E_h^2-Q^2_0)]$,
with $E_h=\sqrt{p_h^2+Q_0^2}=E_p$, $\xi_E \equiv \log 1/x_E = \log \sqrt{s}/(2
E)$ and $K_h$ the overall normalization parameter, has been then
used to obtain the invariant density $E dn/d^3p$.
Within DLA, one gets an iterative solution for a gluon-jet\cite{lo}:
\begin{equation}
D_g(\xi,Y,\lambda) = \delta(\xi) +
\frac{4 N_C}{b} \log \left( 1 + \frac{Y-\xi}{\lambda} \right)
\left[ 1 + f(\xi,Y,\lambda) \right] + \dots
\end{equation}
where $Y = \log \sqrt{s}/(2 Q_0)$, $\lambda = \log Q_0/\Lambda$, $b =
\frac{11}{3}N_C-\frac{2n_f}{3}$ and
$f(\xi,Y,\lambda)$ is a known function.
The leading term at small momenta of order $\beta^2$
corresponding to the emission of a single
gluon is indeed proportional
to the colour charge factor of the primary parton and
does not depend on the $cms$ energy. The DLA prediction for the invariant
density exhibits therefore an approximate scaling law and tends at
small particle energy to a finite soft limit.
The MLLA solution differs from the DLA limit by a simple
exponential damping factor\cite{klonew}:
\begin{equation}
D(\xi,Y,\lambda)|_{MLLA} = D(\xi,Y,\lambda)|_{DLA} \exp \biggl[-a\int^Y_\xi
\frac{\alpha_s(y)}{2 \pi} dy \biggr]
\end{equation}
with $a=11/(3 N_C)+ 2n_f/(3N_C^2)$.
This solution satisfies the MLLA evolution equation,
except for a small term
proportional to $a[\alpha_s(\xi)- \alpha_s(Y)]/(2 \pi)$,
which then vanishes in the soft limit, where $\xi \to Y$.
The MLLA still satisfies an approximate scaling law and tends to a finite
soft limit, but the additional damping factor modifies the dependence on
the $cms$ energy which appears at moderate particle energy.
Figure~(\ref{charged}b) compares the data shown in
Figure~(\ref{charged}a) with the theoretical predictions from MLLA with
$Q_0$ = 270 MeV and $\Lambda$ = 257 MeV (and $K_h$ = 0.45 fixed a posteriori);
a rather good description of data is obtained within this approach.
\section{Conclusions}
The invariant density $E dn/d^3p$ has been shown to satisfy an approximate
scaling law at low particle energy of few hundreds MeV.
In the framework of the perturbative QCD approach,
which has been shown to successfully describe the invariant energy spectra
in the whole available $cms$ energy range,
predictions for the invariant density $E dn/d^3p$ have
been explicitly computed. The MLLA has been found to describe well
the data down to very low particle energies,
thus suggesting the validity of this picture
in the very soft region and a possible link of the observed scaling law with
QCD coherence.
Further tests of the universality of soft particle production in different
reactions and alternative methods to point out
the sensitivity of soft particle production to QCD coherence are presented
in~\cite{klonew}.
\medskip
\noindent { \bf Acknowledgements}
\medskip
I thank Valery A. Khoze and Wolfgang Ochs for discussions
and collaboration on the subjects of this talk.
I thank Jorge Dias de Deus
for the nice atmosphere created at the Conference.
\medskip
\noindent { \bf References}
\smallskip
|
2,877,628,090,120 | arxiv |
\section{Asynchronous Effects in Action}
\label{sec:applications}
We now show some examples of the kinds of programs one can write in \lambdaAEff.
Similarly to \autoref{sec:overview:runningexample},
we again allow ourselves access to mutable references, and use
generic versions $\tmopoutgen {op} V$ of signals.
\subsection{Guarded Interrupt Handlers}
\label{sec:applications:guarder-handlers}
Before diving into the examples, we note that we often want the
triggering of interrupt handlers to be
based on not only the names of interrupts, but also the payloads that they carry.
In order to express such more fine-grained triggering behaviour, we shall use a
\emph{guarded interrupt handler}:
\begin{lstlisting}
promise (op x when guard |-> comp) as p in cont
\end{lstlisting}
which is simply a syntactic sugar for the following interrupt handler that recursively reinstalls
itself until the boolean \ls$guard$ becomes true, in which case it executes the handler code \ls$comp$:
\begin{lstlisting}
let rec waitForGuard () =
promise (op x |-> if guard then comp else waitForGuard ()) as p' in return p'
in
let p = waitForGuard () in cont
\end{lstlisting}
Here, \ls$x$ is bound both in \ls$guard$ and \ls$comp$. Further, if \ls$comp$ has type $\tycomp{\typromise X}{(\o',\i')}$
and \ls$cont$ has type $\tycomp{Y}{(\o,\i)}$, such that $\i\, (\op) = (\o',\i')$, then we can assign the entire computation
the type $\tycomp{Y}{(\o,\i \cup \i_h)}$, where the effect annotation $\i_h$ is the least fixed point of the map
$\i'' \mapsto \{ \op \mapsto (\o',\i' \cup \i'') \} : I \to I$. Observe that some of the recursive encoding leaks into the
type of the entire computation via $\i_h$.
Note that regardless whether \ls$guard$ is true, every interrupt is propagated into \ls$cont$.
To typecheck their definition, and to ensure that guarded interrupt handlers
are non-blocking, it is crucial that the handler code of
ordinary interrupt handlers returns promise-typed values, as noted in \autoref{sect:typing-rules}.
\subsection{Preemptive Multi-Threading}
\label{sec:applications:multithreading}
Multi-threading remains one of the most exciting applications of algebraic effects, with the possibility of expressing
many evaluation strategies being the main reason for the extension of \pl{Multicore OCaml} with effect handlers~\cite{Dolan:MulticoreOCaml}.
These evaluation strategies are however \emph{cooperative} in nature, where each thread needs to explicitly yield back
control, stalling other threads until then.
While it is possible to also simulate \emph{preemptive multi-threading} within the conventional treatment of algebraic effects,
it requires a low-level access to the specific runtime environment, so as to inject
yields into the currently running computation~\cite{Dolan:MulticoreOCaml}.
In contrast, implementing preemptive multi-threading in \lambdaAEff~is quite straightforward, and importantly,
possible within the language itself---the injections into the running computation
take the form of incoming interrupts.
For this, let us consider two interrupts, $\opsym{stop} : \tyunit$ and $\opsym{go} : \tyunit$, that communicate to a thread whether to \emph{pause} or
\emph{resume} execution. These interrupts can originate from a timer process we run in parallel.
At the core of our implementation of preemptive multi-threading is
the recursive function
\begin{lstlisting}
let rec waitForStop () =
promise (stop _ |->
promise (go _ |-> return <<()>>) as p in (await p until <<_>> in waitForStop ())
) as p' in return p'
\end{lstlisting}
which first installs an interrupt handler for $\opsym{stop}$, letting subsequent computations run their course. Once
the $\opsym{stop}$ interrupt arrives, the interrupt handler for it is triggered and the next one for $\opsym{go}$ gets
installed. In contrast to the interrupt handler for $\opsym{stop}$, the one for $\opsym{go}$ starts awaiting
the (unit) promise \ls$p$. This means that any subsequent computations are blocked until a $\opsym{go}$ interrupt
is received, after which we recursively reinstall the interrupt handler for $\opsym{stop}$ and repeat the cycle.
To \emph{initiate the preemptive behaviour} for some computation \ls{comp}, we simply run the program
\begin{lstlisting}
waitForStop (); comp
\end{lstlisting}
The algebraicity reduction rules for interrupt handlers ensure that they propagate out of \ls{waitForStop}
and encompass the entire computation, including \ls{comp}.
Observe that in contrast to usual effect handler based encodings of multi-threading, \ls$waitForStop$ does
not need any access to a thunk \linebreak \lstinline{fun () |-> comp} representing the threaded computation.
In particular, the given computation \ls$comp$ can be completely unaware of the multi-threaded behaviour,
both in its definition and its type.
This approach can be easily extended to multiple threads, by using interrupts' payloads to communicate thread IDs. To this end,
we can consider interrupts $\opsym{stop} : \tyint$ and $\opsym{go} : \tyint$, and define
\begin{lstlisting}
let rec waitForStop threadID =
promise (stop threadID' when threadID = threadID' |->
promise (go threadID' when threadID = threadID' |-> return <<()>>) as p in
await p until <<_>> in waitForStop threadID
) as p' in return p'
\end{lstlisting}
using guarded interrupt handlers, and
conditioning their triggering based on the received IDs.
\subsection{Remote Function Calls}
\label{sec:applications:remotecall}
One of the main uses of asynchronous computation is to offload the execution of
\emph{long-running functions} $f \!: \tyfun{A}{\tycomp{B}{(\o,\i)}}$ to remote processes.
Below we show how to implement this in \lambdaAEff.
One invokes a remote function by issuing a signal named $\opsym{call}$ with the \emph{function's
argument}, and then awaits an interrupt named $\opsym{result}$ with the \emph{function's result},
with all effects specified by $(\o,\i)$ happening at the callee site. The caller then calls
such a remote function through a wrapper \ls$callWith$,
which issues the $\opsym{call}$ signal, installs a handler for the $\opsym{result}$
interrupt, and returns a thunk that awaits the function's result.
For instance, one may then use remote functions in their code as
\begin{lstlisting}
let subtally = callWith "SELECT count(col) FROM table WHERE cond" in
let tally = callWith "SELECT count(col) FROM table" in
printf "Percentage:
\end{lstlisting}
To avoid the results of earlier remote function calls from fulfilling the promises of
later ones, we assign to each call a unique identifier, and communicate
those in payloads. We implement these unique identifiers using a counter.
For a remote function $f : \tyfun{A}{\tycomp{B}{(\o,\i)}}$, we
type the signals and interrupts as $\opsym{call} : \typrod{A}{\tysym{int}}$ and $\opsym{result} : \typrod{B}{\tysym{int}}$.
The \emph{caller site} function \ls$callWith$ is then defined as
\begin{lstlisting}
let callWith x =
let callNo = !callCounter in callCounter := !callCounter + 1;
send call (x, callNo);
promise (result (y, callNo') when callNo = callNo' |-> return <<y>>) as resultPromise in
return (fun () -> await resultPromise until <<resultValue>> in return resultValue)
\end{lstlisting}
After issuing the $\opsym{call}$ signal, \ls$callWith$ installs a guarded interrupt handler
for the corresponding $\opsym{result}$ interrupt, and then returns a function that,
when called, awaits the result of the remote call.
At the \emph{callee site}, we simply install an interrupt handler that executes the function in question,
issues an outgoing signal with the function's result, and then recursively reinstalls itself, as follows:
\begin{lstlisting}
let remote f =
let rec loop () =
promise (call (x, callNo) |-> let y = f x in send result (y, callNo); loop ()) as p in return p
in loop ()
\end{lstlisting}
Unlike effect handlers, our interrupt handlers have very limited control over the execution of
their continuation. However, we can still simulate \emph{cancellations of asynchronous
computations} by awaiting a promise that will never be fulfilled. We achieve this with the help of the function
\begin{lstlisting}
let awaitCancel callNo runBeforeStall =
promise (cancel callNo' when callNo = callNo' |->
promise (dummy () |-> return <<()>>) as dummyPromise in
runBeforeStall ();
await dummyPromise until <<_>> return <<()>>
) as _ in return ()
\end{lstlisting}
which takes the identifier of the remote function call that we want to make cancellable, and a thunked
computation to run before the continuation is stalled. We can then extend
the callee site with cancellable function calls by invoking \lstinline{awaitCancel} before
we start executing the long-running computation \lstinline{f x}. In particular,
we change the interrupt handler code in \ls$remote f$ to read as follows:
\begin{lstlisting}
call (x, callNo) |-> awaitCancel callNo loop; let y = f x in send result (y, callNo); loop ()
\end{lstlisting}
However, if left as is, cancelling one call would cancel all
unfinished remote function calls because they would be part of the stalled continuation.
To overcome this, we run the callee site in parallel with an auxiliary process (which we omit here) that
reacts to a $\opsym{cancel}$ interrupt by \emph{reinvoking these unfinished calls} (minus the cancelled one)
by reissuing the corresponding $\opsym{call}$ signals, which then get propagated to the callee site, and to
the \lstinline{loop ()} we run in \lstinline{awaitCancel callNo loop} before stalling.
We note that the cancelled computation is only \emph{perpetually stalled}, but not discarded completely, leading
to a memory leak. We conjecture that extending \lambdaAEff~with effect handlers that
have greater control over the continuation could lead to a more efficient code for the callee site.
We also conjecture that a future extension of \lambdaAEff~with dynamic process creation would
eliminate the need for the auxiliary reinvoker process, because then the callee site could create a
new process for every remote function call it receives, and each $\opsym{cancel}$ interrupt would
stall only one of such (sub-)processes.
\subsection{Runners of Algebraic Effects}
\label{sec:applications:runners}
Next, we use \lambdaAEff~to implement a parallel variant
of \emph{runners of algebraic effects} \cite{Ahman:Runners}. These are a
natural mathematical model and programming abstraction for resource management based on
algebraic effects, and correspond to effect handlers that apply continuations (at most)
once in a tail call position.
In a nutshell, for a signature of operation
symbols $\op : A_\op \to B_\op$, a \emph{runner} $\mathcal{R}$ comprises a family of stateful functions
$\overline{\op}_{\mathcal{R}} : A_\op \to R \Rightarrow B_\op \times R$,
called \emph{co-operations}, where $R$ is the type of \emph{resources} that the runner manipulates.
In the more general setting of \citet{Ahman:Runners}, the co-operations also model other, external
effects, such as native calls to the operating system, and can furthermore raise
exceptions---all of which we shall gloss over here.
Given a runner $\mathcal{R}$, \citet{Ahman:Runners} provide the programmer with a construct
\vspace{-0.05em}
\[
\tmkw{using}~\mathcal{R}~\tmkw{@}~V_{\text{init}}~\tmkw{run}~M~\tmkw{finally}~\{ \tmreturn x ~\tmkw{@}~ r_{\text{fin}} \mapsto N \}
\vspace{-0.05em}
\]
which runs $M$ using $\mathcal{R}$, with resources initially set to $V_{\text{init}}$; and
finalises the return value and final resources using $N$, e.g., ensuring that all file handles get closed.
This is a form of effect handling: it executes $M$ by invoking
co-operations in place of operation calls, while doing resource-passing under the hood.
Below we show by means of examples how one can use \lambdaAEff~to naturally separate $\mathcal{R}$ and $M$
into different processes.
For simplicity, we omit the initialisation and finalisation phases.
For our first example, let us consider a runner that implements a \emph{pseudo-random number generator} by providing a co-operation for ${\opsym{random} : \tyunit \to \tyint}$, which we can for example implement as
\begin{lstlisting}
let linearCongruenceGeneratorRunner modulus a c initialSeed =
let rec loop seed =
promise (randomReq callNo |->
let seed' = (a * seed + c) mod modulus in send randomRes (seed, callNo); loop seed'
) as p in return p
in loop initialSeed
\end{lstlisting}
It is given by a recursive interrupt handler, which listens for $\opsym{randomReq} : \tysym{int}$ requests
issued by clients, and itself issues $\opsym{randomRes} : \typrod{\tysym{int}}{\tysym{int}}$ responses. The resource this runner
manages is the seed, which it passes between subsequent co-operation calls as an argument to the recursive \ls$loop$.
For the client code $M$, we implement operation calls \ls$random ()$ as discussed in \autoref{sec:overview:signals},
by decoupling them into signals and interrupts. We again use guarded interrupt handlers and call identifiers to
avoid a response to one operation call fulfilling the promises of subsequent ones.
\begin{lstlisting}
let random () =
let callNo = !callCounter in callCounter := callNo + 1;
send randomReq callNo;
promise (randomRes (n, callNo') when callNo = callNo' |-> return <<n mod 10>>) as p in
await p until <<m>> in return m
\end{lstlisting}
As a second example, we show that this parallel approach to runners naturally
extends to multiple co-operations. Specifically, we implement a \emph{runner for a heap},
by providing co-operations for
\vspace{-0.05em}
\[
\opsym{alloc} : \tysym{int} \to \tysym{loc} \qquad
\opsym{lookup} : \tysym{loc} \to \tysym{int} \qquad
\opsym{update} : \tysym{loc} \times \tysym{int} \to \tyunit
\vspace{-0.05em}
\]
We represent these co-operations using a signal/interrupt pair $(\opsym{opReq},\opsym{opRes})$ with payload types
\begin{lstlisting}
type payloadReq = | AllocReq of int | LookupReq of loc | UpdateReq of loc * int
type payloadRes = | AllocRes of loc | LookupRes of int | UpdateRes of unit
\end{lstlisting}
\noindent
The resulting runner is then implemented by pattern-matching on the payload value as follows:
\begin{lstlisting}
let rec heapRunner heap =
promise (opReq (payloadReq, callNo) |->
let heap', payloadRes =
match payloadReq with
| AllocReq v |-> let heap', l = allocHeap heap v in return (heap', AllocRes l)
| LookupReq l |-> let v = lookupHeap heap l in return (heap, LookupRes v)
| UpdateReq (l, v) |-> let heap' = updateHeap heap l v in return (heap', UpdateRes ())
in
send opRes (payloadRes, callNo); heapRunner heap'
) as p in return p
\end{lstlisting}
Note that by storing \ls$heap$ in memory, we could have also used three signal/interrupt pairs and
split \ls$heapRunner$ into three distinct interrupt handlers, one for each of allocation, lookup, and update.
\subsection{Non-Blocking Post-Processing of Promised Values}
\label{sec:applications:chaining}
As discussed in \autoref{sect:overview:promising}, interrupt handlers differ
from ordinary operation calls by allowing user-side post-processing of received data.
In this final example, we show that \lambdaAEff~is flexible enough to modularly perform \emph{further
non-blocking post-processing} of this data anywhere in a program.
For instance, let us assume we are writing a program that contains an interrupt handler (for some $\op$)
that promises to return us a list of integers. Now, at some later point
in the program, we decide that we want to further process this list
if and when it becomes available,
e.g., by using some of its elements to issue an outgoing signal.
Of course, we could do this by going back and changing the original interrupt handler,
but this would not be very modular;
nor do we want to block the entire program's execution (using \ls$await$) until $\op$
arrives and the concrete list becomes available.
Instead, we can define a generic combinator for \emph{non-blocking post-processing} of
promised values
\begin{lstlisting}
process$_{\op}$ p with (<<x>> |-> comp) as q in cont
\end{lstlisting}
that takes an earlier made promise \ls$p$ (which we assume originates
from handling the specified interrupt $\op$),
and makes a new promise to
execute the post-processing code \ls$comp[v/x]$ once \ls$p$ gets fulfilled with some value \ls$v$. The (non-blocking) continuation
\ls$cont$ can refer to \ls$comp$'s result using the new promise-typed variable \ls$q$ bound
in it. Under the hood, \ls{process$_{\op}$} is a syntactic sugar for
\begin{lstlisting}
promise (op _ |-> await p until <<x>> in let y = comp in return <<y>>) as q in cont
\end{lstlisting}
While \ls{process$_{\op}$} does involve an \ls$await$, it gets
exposed only after
\ls$op$ is received, but by that time \ls$p$ will have been fulfilled with some
\ls$v$ by an earlier interrupt handler, and thus the \ls$await$ can reduce.
Returning to post-processing a list of integers promised by an existing interrupt handler,
below is an example showing the use of the \ls{process$_{\op}$} combinator and how to \emph{chain multiple
post-processing computations together} (here, filtering, folding, and issuing an outgoing signal),
in the same spirit as how one is taught to program compositionally with futures and promises~\cite{Haller:Futures}:
\begin{lstlisting}
promise (op x |-> original_interrupt_handler) as p in
...
process$_{\op}$ p with (<<is>> |-> filter (fun i |-> i > 0) is) as q in
process$_{\op}$ q with (<<js>> |-> fold (fun j j' |-> j * j') 1 js) as r in
process$_{\op}$ r with (<<k>> |-> send productOfPositiveElements k) as _ in
...
\end{lstlisting}
We note that for this to work, it is crucial that incoming interrupts behave
like (deep) effect handling (see \autoref{sec:basic-calculus:semantics:computations})
so that all three post-processing computations get executed, in their program order.
\section{A Calculus for Asynchronous Effects: Values and Computations}
\label{sec:basic-calculus:computations}
We now distil the ideas we introduced in the previous section into a core
calculus for programming with asynchronous effects, called \lambdaAEff. It
is based on \citeauthor{Levy:FGCBV}'s [\citeyear{Levy:FGCBV}] fine-grain
call-by-value $\lambda$-calculus (FGCBV), and as such, it is a low-level
intermediate language to which a corresponding high-level user-facing programming language
could be compiled to.
In order to better explain the different features of the calculus and its semantics, we split
\lambdaAEff~into a \emph{sequential} part (discussed below) and a
\emph{parallel} part (discussed in \autoref{sec:basic-calculus:processes}).
We note that this separation is purely presentational.
\subsection{Values and Computations}
\label{sec:basic-calculus:values-and-computations}
The syntax of terms is given in \autoref{fig:terms}, stratified into \emph{values} and \emph{computations},
as in FGCBV.
\begin{figure}[tp]
\parbox{\textwidth}{
\centering
\small
\begin{align*}
\intertext{\textbf{Values}}
V, W
\mathrel{\;{:}{:}{=}\ }& x & &\text{variable} \\
\mathrel{\;\big|\ \ }& \tmunit \mathrel{\;\big|\ \ }\! \tmpair{V}{W} & &\text{unit and pair} \\
\mathrel{\;\big|\ \ }& \tminl[Y]{V} \mathrel{\;\big|\ \ }\! \tminr[X]{V} & &\text{left and right injections} \\
\mathrel{\;\big|\ \ }& \tmfun{x : X}{M} & &\text{function abstraction} \\
\mathrel{\;\big|\ \ }& \tmpromise V & &\text{fulfilled promise}
\\[1ex]
\intertext{\textbf{Computations}}
M, N
\mathrel{\;{:}{:}{=}\ }& \tmreturn{V} & &\text{returning a value} \\
\mathrel{\;\big|\ \ }& \tmlet{x}{M}{N} & &\text{sequencing} \\
\mathrel{\;\big|\ \ }& \tmletrec[: \tyfun{X}{\tycomp{Y}{(\o,\i)}}]{f}{x}{M}{N} & &\text{recursive definition} \\
\mathrel{\;\big|\ \ }& V\,W & &\text{function application} \\
\mathrel{\;\big|\ \ }& \tmmatch{V}{\tmpair{x}{y} \mapsto M} & &\text{product elimination} \\
\mathrel{\;\big|\ \ }& \tmmatch[\tycomp{Z}{(\o,\i)}]{V}{} & &\text{empty elimination} \\
\mathrel{\;\big|\ \ }& \tmmatch{V}{\tminl{x} \mapsto M, \tminr{y} \mapsto N}
& &\text{sum elimination} \\
\mathrel{\;\big|\ \ }& \tmopout{op}{V}{M} & &\text{outgoing signal} \\
\mathrel{\;\big|\ \ }& \tmopin{op}{V}{M} & &\text{incoming interrupt} \\
\mathrel{\;\big|\ \ }& \tmwith{op}{x}{M}{p}{N} & &\text{interrupt handler} \\
\mathrel{\;\big|\ \ }& \tmawait{V}{x}{M} & &\text{awaiting a promise to be fulfilled}
\end{align*}
}
\caption{Values and computations.}
\label{fig:terms}
\end{figure}
\paragraph{Values}
The values $V,W,\ldots$ are mostly standard. They include
variables, introduction forms for
sums and products, and functions. The only \lambdaAEff-specific value
is $\tmpromise V$, which denotes a \emph{fulfilled promise}, indicating that the promise of
handling some interrupt has been fulfilled with the value $V$.
\paragraph{Computations} The computations $M,N,\ldots$ also include all
standard terms from FGCBV:
returning values, sequencing, recursion, function
application, and elimination forms.
Note that we annotate recursive definitions with the type of the function being defined,
including the annotations $(\o,\i)$ that describe the possible effects of the function.
While we do not study effect inference in this paper, our experience
is that these annotations should make it possible
to fully infer types.
The first two computations specific to \lambdaAEff~are \emph{signals} $\tmopout{op}{V}{M}$ and
\emph{interrupts} $\tmopin{op}{V}{M}$, where the name
$\opsym{op}$ is drawn from an assumed set $\sig$ of names, $V$ is a data
payload, and $M$ is a continuation.
The next \lambdaAEff-specific computation is the \emph{interrupt handler} $\tmwith{op}{x}{M}{p}{N}$,
where $x$ is bound in $M$ and $p$ in $N$.
As discussed in the previous section, one should understand this computation as making a promise
to handle a future incoming interrupt $\opsym{op}$ by executing the computation $M$. Sub-computations of the continuation
$N$ can then explicitly await, when necessary, this promise to be fulfilled by blocking on the \emph{promise variable} $p$
using the final \lambdaAEff-specific computation term, the \emph{awaiting} construct $\tmawait{V}{x}{M}$.
We note that $p$ is an ordinary variable---it just gets assigned the distinguished promise type by the interrupt handler.
\subsection{Small-Step Operational Semantics}
\label{sec:basic-calculus:semantics:computations}
We equip \lambdaAEff~with an evaluation contexts based
small-step operational semantics,
defined using a reduction relation $M \reduces N$.
The \emph{reduction rules} and \emph{evaluation contexts} are given
in \autoref{fig:small-step-semantics-of-computations}.
\begin{figure}[tp]
\small
\begin{align*}
\intertext{\textbf{Standard computation rules}}
\tmapp{(\tmfun{x \of X}{M})}{V} &\reduces M[V/x]
\\
\tmlet{x}{(\tmreturn V)}{N} &\reduces N[V/x]
\\
\tmmatch{\tmpair{V}{W}}{\tmpair{x}{y} \mapsto M} &\reduces M[V/x, W/y]
\\
\mathllap{
\tmmatch{(\tminl[Y]{V})}{\tminl{x} \mapsto M, \tminr{y} \mapsto N}
} &\reduces
M[V/x]
\\
\mathllap{
\tmmatch{(\tminr[X]{W})}{\tminl{x} \mapsto M, \tminr{y} \mapsto N}
} &\reduces
N[W/y]
\\
\tmletrec[: \tyfun{X\!}{\tycomp{\!Y\!}{\!(\o,\i)}}]{f}{x}{M}{N} &\reduces
N[\tmfun{x \of X}{\tmletrec[: \tyfun{X\!}{\tycomp{\!Y\!}{\!(\o,\i)}}]{f}{x}{M}{M}}/f]
\\[1ex]
\intertext{\textbf{Algebraicity of signals and interrupt handlers}}
\tmlet{x}{(\tmopout{op}{V}{M})}{N} &\reduces \tmopout{op}{V}{\tmlet{x}{M}{N}}
\\
\tmlet{x}{(\tmwith{op}{y}{M}{p}{N_1})}{N_2} &\reduces \tmwith{op}{y}{M}{p}{(\tmlet{x}{N_1}{N_2})}
\\[1ex]
\intertext{\textbf{Commutativity of signals with interrupt handlers}}
\tmwith{op}{x}{M}{p}{\tmopout{op'}{V}{N}} &\reduces \tmopout{op'}{V}{\tmwith{op}{x}{M}{p}{N}}
\\[1ex]
\intertext{\textbf{Interrupt propagation}}
\tmopin{op}{V}{\tmreturn W} &\reduces \tmreturn W
\\
\tmopin{op}{V}{\tmopout{op'}{W}{M}} &\reduces \tmopout{op'}{W}{\tmopin{op}{V}{M}}
\\
\tmopin{op}{V}{\tmwith{op}{x}{M}{p}{N}} &\reduces \tmlet{p}{M[V/x]}{\tmopin{op}{V}{N}}
\\
\tmopin{op'}{V}{\tmwith{op}{x}{M}{p}{N}} &\reduces \tmwith{op}{x}{M}{p}{\tmopin{op'}{V}{N}}
\quad {\color{rulenameColor}(\op \neq \op')}
\\[-6ex]
\end{align*}
\begin{minipage}[t]{0.4\textwidth}
\begin{align*}
\intertext{\quad\,\textbf{Awaiting a promise to be fulfilled}}
\tmawait{\tmpromise V}{x}{M} \reduces M[V/x]
\end{align*}
\end{minipage}
\begin{minipage}[t]{0.4\textwidth}
\centering
\begin{align*}
\intertext{\textbf{Evaluation context rule}}
\coopinfer{}{
M \reduces N
}{
\E[M] \reduces \E[N]
}
\end{align*}
\vspace{-4ex}
\end{minipage}
\begin{align*}
\intertext{\textbf{where}\vspace{1ex}}
\text{$\E$}
\mathrel{\;{:}{:}{=}\ } [~]
\mathrel{\;\big|\ \ } \tmlet{x}{\E}{N}
\mathrel{\;\big|\ \ } \tmopout{op}{V}{\E}
\mathrel{\;\big|\ \ } \tmopin{op}{V}{\E}
\mathrel{\;\big|\ \ } \tmwith{op}{x}{M}{p}{\E}
\end{align*}
\caption{Small-step operational semantics of computations.}
\label{fig:small-step-semantics-of-computations}
\end{figure}
\paragraph{Computation rules}
The first group includes \emph{standard reduction rules} from FGCBV, such as $\beta$-reducing function applications, sequential composition, and the standard elimination forms. The semantics also includes a rule for unfolding general-recursive definitions.
These rules involve standard \emph{capture avoiding substitutions} $M[V/x]$, defined by straightforward structural recursion.
\paragraph{Algebraicity}
This group of reduction rules \emph{propagates outwards} the signals (resp.~interrupt handlers) that have been issued (resp.~installed) in
sub-computations. While it is not surprising that outgoing signals
behave like algebraic \emph{operation calls}, getting propagated outwards as far as possible, then it is much more curious that
the natural operational behaviour of interrupt handlers turns out to be the same. As we shall explain in \autoref{sec:conclusion},
despite using the (systems-inspired) ``handler'' terminology, mathematically interrupt handlers are in fact a form of algebraic operations.
\paragraph{Commutativity of signals with interrupt handlers}
This rule complements the algebraicity rule for signals, by further propagating
them outwards, past any enveloping interrupt handlers. From the perspective of algebraic effects,
this rule is an example of two algebraic operations \emph{commuting}.
For this rule to be type safe, the type system ensures that the (promise) variable $p$ cannot appear in $V$.
\paragraph{Interrupt propagation}
The handler-operation curiosity does not end with interrupt handlers. This group of reduction rules describes how
interrupts are \emph{propagated inwards} into sub-computations. While $\tmopin{op}{V}{M}$ might look like a conventional
operation call, then its operational behaviour instead mirrors that of (deep) \emph{effect handling}, where one also recursively descends into
the computation being handled. The first reduction rule states that we can safely discard an interrupt when it reaches a terminal
computation $\tmreturn W$. The second rule states that we can propagate incoming interrupts past any outward moving signals. The last
two rules describe how interrupts interact with interrupt handlers, in particular, that the former behave like effect handling
(when understanding interrupt handlers as generalised algebraic operations). On the one hand, if the interrupt
matches the interrupt handler it encounters, the corresponding handler code $M$ is executed, and the interrupt is
propagated inwards into the continuation $N$. On the other hand, if the interrupt
does not match the interrupt handler, it is simply propagated past the interrupt handler into $N$.
\paragraph{Awaiting}
The semantics includes a $\beta$-rule for the $\tmkw{await}$ construct, allowing the blocked computation $M$
to proceed executing as $M[V/x]$ when $\tmkw{await}$ is given a fulfilled promise $\tmpromise V$.
\paragraph{Evaluation contexts}
The semantics allows reductions under \emph{evaluation contexts} $\E$.
Observe that as discussed earlier, the inclusion of interrupt handlers in the evaluation contexts means that reductions
involve potentially open terms.
Also, differently from the semantics of conventional operation calls \cite{Kammar:Handlers,Bauer:EffectSystem},
our evaluation contexts include outgoing signals. As such, the \emph{evaluation context rule} allows the execution of a computation
to proceed even if a signal has not yet been propagated to its receiver, or when an interrupt has
not yet arrived. Importantly, the evaluation contexts do not include $\tmkw{await}$, so as to model its intended blocking behaviour.
We write $\E[M]$ for the recursive operation of filling the hole $[~]$ in $\E$ with $M$.
\paragraph{Non-confluence}
It is worth noting that the asynchronous design means that the operational semantics
is naturally \emph{nondeterministic}. But more interestingly, the semantics is also \emph{not confluent}.
For one source of non-confluence, let us consider two reduction sequences of a same (closed) computation,
where for better readability, we highlight the active redex for each reduction step:
\[
\hspace{-0.15cm}
\begin{array}{r@{\,} l}
& \tmopin{op}{V}{\tmwith{op}{x}{(\tmwith{op'}{y}{M}{q}{\tmawait{q}{z}{M'}})}{p}{\!\highlightgray{N}}}
\\[1ex]
\reduces & \highlightgray{\tmopin{op}{V}{\tmwith{op}{x}{(\tmwith{op'}{y}{M}{q}{\tmawait{q}{z}{M'}})}{p}{\!\highlightwhite{N'}}}}
\\[1ex]
\reduces & \highlightgray{\tmlet{p}{(\tmwith{op'}{y}{M[V/x]}{q}{\tmawait{q}{z}{M'}})}{\!\highlightwhite{\tmopin{op}{V}{N'}}}}
\\[1ex]
\reduces & \tmwith{op'}{y}{M[V/x]}{q}{\tmawait{q}{z}{(\tmlet{p}{M'}{\tmopin{op}{V}{N'}})}}
\end{array}
\]
and
\[
\hspace{-0.15cm}
\begin{array}{r@{\,} l}
& \highlightgray{\tmopin{op}{V}{\tmwith{op}{x}{(\tmwith{op'}{y}{M}{q}{\tmawait{q}{z}{M'}})}{p}{\!\highlightwhite{N}}}}
\\[1ex]
\reduces & \highlightgray{\tmlet{p}{(\tmwith{op'}{y}{M[V/x]}{q}{\tmawait{q}{z}{M'}})}{\!\highlightwhite{\tmopin{op}{V}{N}}}}
\\[1ex]
\reduces & \tmwith{op'}{y}{M[V/x]}{q}{\tmawait{q}{z}{(\tmlet{p}{M'}{\tmopin{op}{V}{N}})}}
\end{array}
\]
Here, both final computations are \emph{temporarily} blocked until an incoming interrupt $\opsym{op'}$
is propagated to them and the (promise) variable $q$ gets bound to a fulfilled promise. Until this happens,
it is not possible for the blocked continuation $N$ to reduce to $N'$ in the latter final computation.
Another distinct source of non-confluence concerns the commutativity of outgoing signals with enveloping interrupt
handlers. For instance, the following (closed) composite computation
\[
\tmopin{op}{V}{{\tmwith {op} x {\tmopout{op'}{W'}{M}} p {\tmopout{op''}{W''}{N}}}}
\]
can nondeterministically reduce to either of the next two computations:
\[
\tmopout{op'}{W'}{\tmopout{op''}{W''}{{\tmlet{p}{M}{\tmopin{op}{V}{N}}}}}
\quad
\tmopout{op''}{W''}{\tmopout{op'}{W'}{{\tmlet{p}{M}{\tmopin{op}{V}{N}}}}}
\]
depending on whether we first propagate the interrupt $\op$ inwards or
the signal $\op''$ outwards. As a result, in the resulting two computations,
the signals $\op'$ and $\op''$ get issued in a different order.
\subsection{Type-and-Effect System}
\label{sec:basic-calculus:type-system:computations}
We equip \lambdaAEff~with a type system in the tradition of type-and-effect systems for algebraic effects and
effect handlers \cite{Bauer:EffectSystem,Kammar:Handlers}, by extending the simple type system of FGCBV
with annotations about possible effects in function and computation types.
\subsubsection{Types}
\label{sec:basic-calculus:type-system:computations:types}
We define types in \autoref{fig:types}, separated into ground, value, and computation types.
As noted in \autoref{sec:basic-calculus:values-and-computations}, \lambdaAEff~is parameterised over a set
$\sig$ of signal and interrupt \emph{names}.
To each such name $\op \in \sig$, we assign a fixed
\emph{signature} $\op : A_\op$ that specifies the type $A_\op$ of the payload of the corresponding
signal or interrupt.
Crucially, in order to be able to later prove that \lambdaAEff~is type safe
(see \autoref{theorem:progress}, but also the relevant discussion in \autoref{sec:conclusion}),
we restrict these signatures to \emph{ground types} $A,B,\ldots$,
which include standard base, unit, empty, product, and sum types.
\begin{figure}[tb]
\parbox{\textwidth}{
\centering
\small
\begin{align*}
\text{Ground type $A$, $B$}
\mathrel{\;{:}{:}{=}\ }& \tybase \,\mathrel{\;\big|\ \ }\! \tyunit \,\mathrel{\;\big|\ \ }\! \tyempty \,\mathrel{\;\big|\ \ }\! \typrod{A}{B} \,\mathrel{\;\big|\ \ }\! \tysum{A}{B}
\\[1ex]
\text{Signal or interrupt signature:}
\phantom{\mathrel{\;{:}{:}{=}\ }}& \op : A_\op
\\[1ex]
\text{Outgoing signal annotations:}
\phantom{\mathrel{\;{:}{:}{=}\ }}& \o \in O
\\
\text{Interrupt handler annotations:}
\phantom{\mathrel{\;{:}{:}{=}\ }}& \i \in I
\\[1ex]
\text{Value type $X$, $Y$}
\mathrel{\;{:}{:}{=}\ }& A \,\mathrel{\;\big|\ \ }\! \typrod{X}{Y} \,\mathrel{\;\big|\ \ }\! \tysum{X}{Y} \,\mathrel{\;\big|\ \ }\! \tyfun{X}{\tycomp{Y}{(\o,\i)}} \,\mathrel{\;\big|\ \ }\! \typromise{X}
\\
\text{Computation type:}
\phantom{\mathrel{\;{:}{:}{=}\ }}& \tycomp{X}{(\o,\i)}
\end{align*}
}
\caption{Value and computation types}
\label{fig:types}
\end{figure}
\emph{Value types} $X,Y,\ldots$ extend ground types with function and promise types.
The \emph{function type} $\tyfun{X}{\tycomp{Y}{(\o,\i)}}$ classifies functions that take $X$-typed arguments
to computations classified by the \emph{computation type} $\tycomp{Y}{(\o,\i)}$, i.e., ones that return $Y$-typed
values, while possibly issuing signals specified by $\o$ and handling interrupts specified by $\i$.
The \emph{effect annotations} $\o$ and $\i$ are drawn from sets $O$ and $I$ whose definitions we discuss
below in \autoref{sec:basic-calculus:effect-annotations}. Finally, the \lambdaAEff-specific \emph{promise type}
$\typromise{X}$ classifies those promises that can be fulfilled by supplying a value of type $X$.
\subsubsection{Effect Annotations}
\label{sec:basic-calculus:effect-annotations}
We now explain how we define the sets $O$ and $I$ from which we draw the
effect annotations that we use for specifying functions and computations.
Traditionally, effect systems for algebraic effects simply use (flat) sets of
operation names for effect annotations \cite{Bauer:EffectSystem,Kammar:Handlers}.
In \lambdaAEff, however, we need to be
more careful, because triggering an interrupt handler executes a computation
that can issue potentially different signals and handle different interrupts from the main
program, and we would like to capture this in types.
\paragraph{Signal annotations}
First, as outgoing signals do not carry any computational data, we follow
the tradition of type-and-effect systems for algebraic effects, and let
$O$ be the \emph{power set} $\Pow \sig$. As such, each $\o \in O$ is a subset of
the signature $\Sigma$, specifying which signals a computation might issue.
\paragraph{Interrupt handler annotations}
As noted above, for specifying installed interrupt handlers, we cannot use (flat) sets
of interrupt names as the effect annotations $\i \in I$ if we want to track the nested effectful
structure. Instead,
we define $I$ as the \emph{greatest fixed point}
of a set functor $\Phi$ given by
\[
\Phi (X) \defeq \sig \Rightarrow (O \times X)_\bot
\]
where $\Rightarrow$ is exponentiation, $\times$ is Cartesian product,
and $(-)_\bot$ is the lifting operation $X_\bot \defeq X \cupdot \{\bot\}$, and
where $\cupdot$ is the disjoint union of sets. Formally speaking, $I$ is given
by an isomorphism $I \cong \Phi(I)$, but for presentation purposes we leave it
implicit and work as if we had a strict equality $I = \Phi(I)$.
Intuitively, each $\i \in I$ is a
\emph{possibly infinite nesting of partial mappings} of pairs of $O$- and $I$-annotations to names in
$\sig$---these pairs of annotations classify the possible effects of the corresponding interrupt handler code.
We use the
record notation $\i = \{ \op_1 \mapsto (\o_1,\i_1) , \ldots , \op_n \mapsto (\o_n,\i_n) \}$ to mean that $\i$ maps the names $\op_1, \ldots, \op_n$ to the annotations $(\o_1,\i_1), \ldots, (\o_n,\i_n)$, and any other names in $\sig$ to $\bot$.
We write $\i\, (\op_i) = (\o_i,\i_i)$ to mean that the annotation $\i$ maps $\op_i$ to $(\o_i,\i_i)$.
\paragraph{Subtyping and recursive effect annotations}
Both $O$ and $I$ come equipped with natural \emph{partial orders}: for $O$, $\order O$ is given simply by
subset inclusion; and for $I$, $\order I$ is characterised as follows:
\[
\begin{array}{l c l}
\i \order I \i'
&
\text{iff}
&
\forall\, (\op \in \sig) \, (\o'' \in O) \, (\i'' \in I) .\, \i\, (\op) = ({\o''} , {\i''}) \implies
\\[0.5ex]
&& \exists\, (\o''' \in O) \, (\i''' \in I) .\, \i'\, (\op) = ({\o'''} , {\i'''}) \wedge \o'' \order O \o''' \wedge \i'' \order I \i'''
\end{array}
\]
We often also use the \emph{product order} $\order {O \times I}$, defined as
$(\o,\i) \order {O \times I} (\o',\i') \defeq \o \order O \o' \wedge \i \order I \i'$.
In particular, we use $\order {O \times I}$ in \autoref{sect:typing-rules} to define the subtyping
relation for \lambdaAEff's computation types.
Importantly, the partial orders $(O,\order O)$ and $(I,\order I)$ are both \emph{$\omega$-complete} and \emph{pointed}, i.e.,
they are \emph{pointed cpos}, meaning they have least upper bounds of all increasing $\omega$-chains, and
least elements (given by the empty set $\emptyset$ and the constant $\bot$-valued mapping, respectively).
As a result, \emph{least fixed points} of continuous (endo)maps on them are guaranteed to exist.
We refer the interested reader to \citet{Amadio:Domains} and \citet{Gierz:ContinuousLattices} for additional
domain-theoretic background.
For \lambdaAEff, we are particularly interested in the least fixed points of continuous maps $f : I \to I$,
so as to specify and typecheck recursive interrupt handler examples, as we illustrate in
\autoref{sec:basic-calculus:rec-handler-typing}.
We also note that if we were only interested in the type safety of \lambdaAEff, and not
in typechecking recursively defined interrupt handlers, then we would not need $(I,\order I)$ to be \emph{$\omega$-complete},
and could have instead chosen $I$ to be the
\emph{least fixed point} of $\Phi$, which is what we do in our \pl{Agda}
formalisation. In this case, each interrupt handler annotation $\i \in I$ would be a \emph{finite nesting of partial mappings}.
Finally, we envisage that any future full-fledged high-level language based on \lambdaAEff~would
allow users to define their (recursive) effect annotations in a small domain-specific language, providing
a syntactic counterpart to the domain-theoretic development we use for typing \lambdaAEff~in this paper.
\subsubsection{Typing Rules}
\label{sect:typing-rules}
We characterise \emph{well-typed values} using the judgement $\Gamma \types V : X$
and \emph{well-typed computations} using $\Gamma \types M : \tycomp{X}{(\o,\i)}$.
In both judgements, $\Gamma$ is a \emph{typing context} of the form $x_1 \of X_1, \ldots, x_n \of X_n$.
The rules defining these judgements are respectively given in \autoref{fig:value-typing-rules} and
\ref{fig:computation-typing-rules}.
\begin{figure}[tp]
\centering
\small
\begin{mathpar}
\coopinfer{TyVal-Var}{
}{
\Gamma, x \of X, \Gamma' \types x : X
}
\qquad
\coopinfer{TyVal-Unit}{
}{
\Gamma \types \tmunit : \tyunit
}
\qquad
\coopinfer{TyVal-Pair}{
\Gamma \types V : X \\
\Gamma \types W : Y
}{
\Gamma \types \tmpair{V}{W} : \typrod{X}{Y}
}
\qquad
\coopinfer{TyVal-Promise}{
\Gamma \types V : X
}{
\Gamma \types \tmpromise V : \typromise X
}
\\
\coopinfer{TyVal-Inl}{
\Gamma \types V : X
}{
\Gamma \types \tminl[Y]{V} : X + Y
}
\qquad
\coopinfer{TyVal-Inr}{
\Gamma \types W : Y
}{
\Gamma \types \tminr[X]{W} : X + Y
}
\qquad
\coopinfer{TyVal-Fun}{
\Gamma, x \of X \types M : \tycomp{Y}{(\o,\i)}
}{
\Gamma \types \tmfun{x : X}{M} : \tyfun{X}{\tycomp{Y}{(\o,\i)}}
}
\end{mathpar}
\caption{Value typing rules.}
\label{fig:value-typing-rules}
\end{figure}
\begin{figure}[tp]
\centering
\small
\begin{mathpar}
\coopinfer{TyComp-Return}{
\Gamma \types V : X
}{
\Gamma \types \tmreturn{V} : \tycomp{X}{(\o,\i)}
}
\qquad
\coopinfer{TyComp-Let}{
\Gamma \types M : \tycomp{X}{(\o,\i)}
\\
\Gamma, x \of X \types N : \tycomp{Y}{(\o,\i)}
}{
\Gamma \types
\tmlet{x}{M}{N} : \tycomp{Y}{(\o,\i)}
}
\\
\coopinfer{TyComp-LetRec}{
\Gamma, f \of \tyfun{X}{\tycomp{Y}{(\o,\i)}}, x \of X \types M : \tycomp{Y}{(\o,\i)}
\\
\Gamma, f \of \tyfun{X}{\tycomp{Y}{(\o,\i)}} \types N : \tycomp{Z}{(\o',\i')}
}{
\Gamma \types
\tmletrec[: \tyfun{X}{\tycomp{Y}{(\o,\i)}}]{f}{x}{M}{N} : \tycomp{Z}{(\o',\i')}
}
\\
\coopinfer{TyComp-Apply}{
\Gamma \types V : \tyfun{X}{\tycomp{Y}{(\o,\i)}} \\
\Gamma \types W : X
}{
\Gamma \types \tmapp{V}{W} : \tycomp{Y}{(\o,\i)}
}
\qquad
\coopinfer{TyComp-MatchPair}{
\Gamma \types V : \typrod{X}{Y} \\
\Gamma, x \of X, y \of Y \types M : \tycomp{Z}{(\o,\i)}
}{
\Gamma \types \tmmatch{V}{\tmpair{x}{y} \mapsto M} : \tycomp{Z}{(\o,\i)}
}
\\
\coopinfer{TyComp-MatchEmpty}{
\Gamma \types V : \tyempty
}{
\Gamma \types \tmmatch[\tycomp{Z}{(\o,\i)}]{V}{} : \tycomp{Z}{(\o,\i)}
}
\qquad
\coopinfer{TyComp-MatchSum}{
\Gamma \types V : X + Y \\\\
\Gamma, x \of X \types M : \tycomp{Z}{(\o,\i)} \\
\Gamma, y \of Y \types N : \tycomp{Z}{(\o,\i)} \\
}{
\Gamma \types \tmmatch{V}{\tminl{x} \mapsto M, \tminr{y} \mapsto N} : \tycomp{Z}{(\o,\i)}
}
\\
\coopinfer{TyComp-Signal}{
\op \in \o \\
\Gamma \types V : A_\op \\
\Gamma \types M : \tycomp{X}{(\o,\i)}
}{
\Gamma \types \tmopout{op}{V}{M} : \tycomp{X}{(\o,\i)}
}
\qquad
\coopinfer{TyComp-Interrupt}{
\Gamma \types V : A_\op \\
\Gamma \types M : \tycomp{X}{(\o,\i)}
}{
\Gamma \types \tmopin{op}{V}{M} : \tycomp{X}{\opincomp {op} (\o,\i)}
}
\\
\coopinfer{TyComp-Promise}{
\i\, (\op) = ({\o'} , {\i'}) \\
\Gamma, x \of A_\op \types M : \tycomp{\typromise X}{(\o',\i')} \\
\Gamma, p \of \typromise X \types N : \tycomp{Y}{(\o,\i)}
}{
\Gamma \types \tmwith{op}{x}{M}{p}{N} : \tycomp{Y}{(\o,\i)}
}
\\
\coopinfer{TyComp-Await}{
\Gamma \types V : \typromise X \\
\Gamma, x \of X \types M : \tycomp{Y}{(\o,\i)}
}{
\Gamma \types \tmawait{V}{x}{M} : \tycomp{Y}{(\o,\i)}
}
\qquad
\coopinfer{TyComp-Subsume}{
\Gamma \types M : \tycomp{X}{(\o, \i)} \\
(\o,\i) \order {O \times I} (\o',\i')
}{
\Gamma \types M : \tycomp{X}{(\o', \i')}
}
\end{mathpar}
\caption{Computation typing rules.}
\label{fig:computation-typing-rules}
\end{figure}
\paragraph{Values}
The rules for values are mostly standard.
The only \lambdaAEff-specific rule is \textsc{TyVal-Promise}, which states that in order to fulfil
a \emph{promise} of type $\typromise X$, one has to supply a value of type $X$.
\paragraph{Computations}
Analogously to values, the typing rules are standard for the computation terms that \lambdaAEff~inherits from FGCBV,
with the \lambdaAEff-rules additionally tracking the effect information $(\o,\i)$.
The \lambdaAEff-specific rule \textsc{TyComp-Signal} states that in order
to issue a signal $\op$ in a computation with type $\tycomp{X}{(\o,\i)}$, we must have $\op \in \o$ and the type of the payload
has to match $\op$'s signature.
The rule \textsc{TyComp-Promise} states that
the interrupt handler code $M$ has to return a fulfilled promise of type $\typromise X$, for some type $X$,
while possibly issuing signals $\o'$ and handling interrupts $\i'$, both of which are
determined by the effect annotation $\i$ of the entire computation, i.e.,
$\i\, (\op) = (\o',\i')$. The variable $p$ bound in the continuation, which sub-computations can block on
to await a given interrupt to arrive and be handled, also gets assigned
the promise type $\typromise X$. It is worth noting that one could have had $M$ simply
return values of type $X$, but at the cost of not being able to implement some of the more interesting examples,
e.g., guarded interrupt handlers in \autoref{sec:applications:guarder-handlers}.
At the same time, for \lambdaAEff's type safety, it is
crucial that $p$ would have remained assigned the promise type $\typromise X$.
The rule \textsc{TyComp-Await} simply says that when awaiting a promise of type $\typromise X$ to be fulfilled,
the continuation $M$ can refer to the promised value (in the future) using the variable $x$ of type $X$.
The rule \textsc{TyComp-Interrupt} is used to type incoming interrupts.
In particular, when the outside world propagates an interrupt $\op$ to a computation
$M$ of type $\tycomp{X}{(\o,\i)}$, the resulting
computation $\tmopin{op}{V}{M}$ gets assigned the type $\tycomp{X}{\opincomp {op} (\o,\i)}$,
where the interrupt $\op$ also \emph{acts} on the effect annotations. Intuitively,
$\opincomp {op} (\o,\i)$ mimics the act of triggering interrupt
handlers for $\op$ at the level of effect annotations. Formally, we define this
\emph{action of interrupts} on effect annotations as follows:
\[
\opincomp {op} {(\o , \i)}
~\defeq~
\begin{cases}
\left(\o \cup \o' , \i[\op \mapsto \bot] \cup \i' \right) & \mbox{if } \i\, (\op) = (\o',\i')\\
(\o,\i) & \mbox{otherwise}
\end{cases}
\vspace{-0.25ex}
\]
In other words, if $M$ has any interrupt handlers installed for $\op$, then $\i\, (\op) = (\o',\i')$,
where $(\o',\i')$ specifies the effects of said interrupt handler code. Now, when the inward propagating
interrupt $\op$ reaches those interrupt handlers, it triggers the execution of the corresponding handler code,
and thus the entire computation $\tmopin{op}{V}{M}$ can also issue signals in $\o'$ and handle interrupts in $\i'$.
The notation $\i[\op \mapsto \bot]$ sets $\i$ to $\bot$ at $\op$,
and leaves it unchanged elsewhere.
In particular, mapping $\op$ to $\bot$ captures that the interrupt triggers all corresponding interrupt handlers in $M$.
The \emph{join-semilattice} structure
$\o \cup \o' \in O$ is given by the union of sets, while
$\i \cup \i' \in I$ is given by
\[
\i \cup \i'
~\defeq~
\lam {\op}
\begin{cases}
(\o'' \cup \o''' , \i'' \cup \i''') & \mbox{if } \i\, (\op) = (\o'',\i'') \wedge \i'\, (\op) = (\o''',\i''') \\
(\o'' , \i'') & \mbox{if } \i\, (\op) = (\o'',\i'') \wedge \i'\, (\op) = \bot \\
(\o''' , \i''') & \mbox{if } \i\, (\op) = \bot \wedge \i'\, (\op) = (\o''',\i''') \\
\bot & \mbox{if } \i\, (\op) = \bot \wedge \i'\, (\op) = \bot \\
\end{cases}
\vspace{-0.25ex}
\]
We also note that the action $\opincomp {op} {(-)}$ has various useful properties, which we use in later proofs
(where we write $\pi_1$ and $\pi_2$ for the two projections associated with the Cartesian product $O \times I$):
\begin{lemma}
\label{lemma:action}
\mbox{}
\begin{enumerate}
\item $\o \order O \pi_1\, (\opincomp {op} {(\o,\i)})$.
\item If $\i\, (\op) = (\o',\i')$, then $(\o',\i') \order {O \times I} \opincomp {op} {(\o,\i)}$.
\item If $\op \neq \op'$ and $\i\, (\op') = (\o',\i')$, then $(\o',\i') \order {O \times I} (\pi_2\, (\opincomp {op} {(\o,\i)}))\, (\op')$.
\end{enumerate}
\end{lemma}
Finally, the rule \textsc{TyComp-Subsume} allows \emph{subtyping}.
To simplify the presentation, we consider a limited form of subtyping, in which we
shallowly relate only signal and interrupt annotations.
\subsubsection{Typechecking Recursively Defined Interrupt Handlers}
\label{sec:basic-calculus:rec-handler-typing}
We conclude discussing \lambdaAEff's type-and-effect system by briefly returning to the reason
why we defined our effect annotations using lightweight domain theory in the first place, namely,
so as to typecheck recursive interrupt handlers.
As an example, we recall the following fragment of the server code from \autoref{sec:overview:runningexample:server}:
\begin{lstlisting}
let rec waitForBatchSize () =
promise (batchSizeReq () |-> send batchSizeResp batchSize; waitForBatchSize ()) as p in return p
\end{lstlisting}
Here, \ls$waitForBatchSize ()$ is an interrupt handler for $\opsym{batchSizeReq}$
that recursively reinstalls itself immediately
after issuing a $\opsym{batchSizeResp}$ signal. Due to its recursive definition,
it is not surprising that the type of \ls$waitForBatchSize$ should also be given recursively, in
particular, if we want to give it a more precise type than one which simply says that any effect is possible.
To this end, we assign \ls$waitForBatchSize$ the type
$\tyfun{\tyunit}{\tycomp{\typromise \tyunit}{(\emptyset, \i_{\text{b}})}}$, where
$\i_{\text{b}}$ is the \emph{least fixed point} of the continuous map
$\i \mapsto \{~ \opsym{batchSizeReq} \mapsto (\{\opsym{batchSizeResp}\} , \i) ~\} : I \to I$, i.e.,
\[
\i_{\text{b}} = \big\{~ \opsym{batchSizeReq} \mapsto (\{\opsym{batchSizeResp}\} , \{~ \opsym{batchSizeReq} \mapsto (\{\opsym{batchSizeResp}\} , ~\ldots~) ~\}) ~\big\}
\]
As such, $(\emptyset, \i_{\text{b}})$
captures that at the top level \ls$waitForBatchSize ()$
installs an interrupt handler and issues no signals, and that every $\opsym{batchSizeReq}$
interrupt causes a signal to be issued and the interrupt handler
to be reinstalled.
Checking that \ls$waitForBatchSize$ has the
type $\tyfun{\tyunit}{\tycomp{\typromise \tyunit}{(\emptyset, \i_{\text{b}})}}$
involves unfolding the definition of $\i_{\text{b}}$ and using
subtyping. The latter is needed when we recursively call \ls$waitForBatchSize ()$
where a computation of type $\tycomp{\typromise \tyunit}{(\{\opsym{batchSizeResp}\}, \i_{\text{b}})}$
is expected.
\subsection{Type Safety}
\label{sec:basic-calculus:type-safety}
We now prove type safety for the sequential part of \lambdaAEff,
showing that ``well-typed programs do not go wrong''.
As usual, we split type
safety into \emph{progress} and \emph{preservation} \cite{Wright:SynAppTypeSoundness}.
\subsubsection{Progress}
\label{sec:basic-calculus:type-safety:progress}
The progress result says that well-typed closed computations can either make a step of
reduction, or are already in a well-defined result form (and thus have stopped reducing).
As such, we first need to define when we consider \lambdaAEff-computations
to be in result form.
It is important to note that for \lambdaAEff,
the result forms have to also incorporate the \emph{temporary blocking} while computations await some promise (variable) $p$ to be fulfilled.
Therefore, as a first step, we characterise such computations using the judgement $\awaiting p M$, given by the following three rules:
\begin{mathpar}
\coopinfer{}{
}{
\awaiting p {\tmawait p x M}
}
\coopinfer{}{
\awaiting p M
}{
\awaiting p {\tmlet x M N}
}
\coopinfer{}{
\awaiting p M
}{
\awaiting p {\tmopin{op}{V}{M}}
}
\end{mathpar}
Next, we characterise \lambdaAEff's \emph{result forms} using the judgements $\CompResult {\Psi} {M}$ and
$\RunResult {\Psi} {M}$:
\begin{mathpar}
\coopinfer{}{
\CompResult {\Psi} {M}
}{
\CompResult {\Psi} {\tmopout {op} V M}
}
\quad
\coopinfer{}{
\RunResult {\Psi} {M}
}{
\CompResult {\Psi} {M}
}
\vspace{-1ex}
\\
\coopinfer{}{
}{
\RunResult {\Psi} {\tmreturn V}
}
\quad
\coopinfer{}{
\RunResult {\Psi \cup \{p\}} {N}
}{
\RunResult {\Psi} {\tmwith {op} x M p N}
}
\quad
\coopinfer{}{
p \in \Psi \\
\awaiting p M
}{
\RunResult {\Psi} {M}
}
\end{mathpar}
In these judgements, $\Psi$ is a set
of (promise) variables that have been bound by interrupt handlers enveloping the computation.
These judgements express that a computation $M$ is in a (top-level)
result form $\CompResult {\Psi} {M}$ when, considered as a tree, it has a shape in which \emph{all}
signals are towards the root, interrupt handlers are in the intermediate nodes, and
the leaves contain return values and computations that are temporarily blocked
while awaiting one of the promise variables in $\Psi$ to be fulfilled.
The slightly mysterious name of the intermediate judgement $\RunResult {\Psi} {M}$ will become clear
in \autoref{sec:basic-calculus:type-safety:processes}.
The finality of these result forms is captured by the next lemma.
\begin{lemma}
\label{lemma:results-are-final}
Given $\Psi$ and $M$ such that $\CompResult {\Psi} {M}$, then there exists no $N$ with $M \reduces N$.
\end{lemma}
We are now ready to state and prove the \emph{progress theorem} for the sequential part of \lambdaAEff.
\begin{theorem}
\label{theorem:progress}
Given a well-typed computation $\Gamma \types M : \tycomp{Y}{(\o,\i)}$, where $\Gamma = x_1 \of \typromise {X_1}, \ldots, x_n \of \typromise {X_n}$,
then either (i) there exists an $N$ such that $M \reduces N$, or
(ii) we have $\CompResult {\{x_1, \ldots, x_n\}} {M}$.
\end{theorem}
\begin{proof}
The proof is standard and proceeds by induction
on the derivation of $\Gamma \types M : \tycomp{Y}{(\o,\i)}$. For instance,
if the derivation ends with a typing rule for function application or pattern-matching,
we use an auxiliary canonical forms lemma to show that the value involved
is either a function abstraction or in constructor form---thus $M$ can $\beta$-reduce and we prove (i).
Here we crucially rely on the context $\Gamma$ having the specific
assumed form $x_1 \of \typromise {X_1}, \ldots, x_n \of \typromise {X_n}$.
If the derivation ends with \textsc{TyComp-Await}, then we
use a canonical forms lemma to show that the promise value is either a variable in
$\Gamma$, in which case we prove (ii), or in constructor form, in which case we prove (i).
If the derivation however ends with a typing rule for any of the terms figuring in the
evaluation contexts $\E$, then we proceed based on using the induction hypothesis on the
corresponding continuation.
\end{proof}
\begin{corollary}
\label{corollary:progress}
Given a well-typed closed computation $\types M : \tycomp{X}{(\o,\i)}$,
then either (i) there exists a computation $N$ such that $M \reduces N$, or
(ii) $M$ is already in result form, i.e., we have $\CompResult {\emptyset} {M}$.
\end{corollary}
\subsubsection{Type Preservation}
\label{sec:basic-calculus:type-safety:preservation}
The type preservation result says that reduction preserves well-typedness.
The results that we present in this section use standard \emph{substitution
lemmas}. For instance, given $\Gamma, x \of X , \Gamma' \types M : \tycomp{Y}{(\o,\i)}$
and $\Gamma \types V : X$, then we can show that $\Gamma, \Gamma' \types M[V/x] : \tycomp{Y}{(\o,\i)}$.
In the following we also use standard \emph{typing inversion lemmas}. For example, given
$\Gamma \types \tmopin{op}{V}{M} : \tycomp{X}{(\o,\i)}$, then we can show that
$\Gamma \types V : A_\op$ and $\Gamma \types M : \tycomp{X}{\opincomp {op} (\o',\i')}$,
such that $\opincomp {op} (\o',\i') \order {O \times I} (\o,\i)$.
As the proof of type preservation proceeds by induction on reduction steps,
we find it useful to define an auxiliary \emph{typing judgement for evaluation contexts},
written $\Gamma \types\!\![\, \Gamma' \,\vert\, \tycomp{X}{(\o,\i)} \,]~ \E : \tycomp{Y}{(\o',\i')}$,
which we then use to prove the evaluation context rule case of the proof.
Here, $\Gamma'$ is the context of variables bound by the interrupt handlers in $\E$, and
$\tycomp{X}{(\o,\i)}$ is the type of the hole $[~]$. This judgement is given using rules
similar to those for computations, including subtyping, e.g., we have
\begin{mathpar}
\coopinfer{}{
\i'\, (\op) = (\o'',\i'') \\
\Gamma, x \of A_\op \types M : \tycomp{\typromise Y}{(\o'',\i'')} \\
\Gamma, p \of \langle Y \rangle \types\!\![\, \Gamma' \,\vert\, \tycomp{X}{(\o,\i)} \,]~ \E : \tycomp{Z}{(\o',\i')}
}{
\Gamma \types\!\![\, p \of \langle Y \rangle, \Gamma' \,\vert\, \tycomp{X}{(\o,\i)} \,]~ \tmwith{op}{x}{M}{p}{\E} : \tycomp{Z}{(\o',\i')}
}
\end{mathpar}
It is thus straightforward to relate this typing of evaluation contexts with that of computations.
\begin{lemma}
\label{lemma:eval-ctx-typing}
\mbox{}
$\Gamma \types \E[M] : \tycomp{Y}{(\o',\i')}
\Leftrightarrow
\exists\, \Gamma', X, \o, \i .~
\Gamma \types\!\![\, \Gamma' \,\vert\, \tycomp{X}{(\o,\i)} \,]~ \E : \tycomp{Y}{(\o',\i')}
~\wedge~
\Gamma,\Gamma' \types M : \tycomp{X}{(\o,\i)}
$.
\end{lemma}
We are now ready to state and prove the \emph{type preservation theorem} for the sequential part of \lambdaAEff.
\begin{theorem}
\label{theorem:preservation}
Given $\Gamma \types M : \tycomp{X}{(\o,\i)}$ and
$M \reduces N$, then we have $\Gamma \types N : \tycomp{X}{(\o,\i)}$.
\end{theorem}
\begin{proof}
The proof is standard and proceeds by induction on the derivation of
$M \reduces N$, using typing inversion lemmas depending on
the structure forced upon $M$ by the last rule used in $M \reduces N$.
There are four cases of interest in this proof. The first two concern
the interaction of incoming interrupts and interrupt handlers.
On the one hand, if the given derivation of $\reduces$ ends with
\[
\tmopin{op}{V}{\tmwith{op}{x}{M}{p}{N}} \reduces \tmlet{p}{M[V/x]}{\tmopin{op}{V}{N}}
\]
then in order to type the right-hand side of this rule, we are led to use subtyping with
\srefcase{Lemma}{lemma:action}{2}, so as to show that $M$'s effect information is
included in $\opincomp {op} {(\o , \i)}$. On the other hand, given
\[
\tmopin{op'}{V}{\tmwith{op}{x}{M}{p}{N}} \reduces \tmwith{op}{x}{M}{p}{\tmopin{op'}{V}{N}}
\quad {\color{rulenameColor}(\op \neq \op')}
\]
then in order to type the right-hand side of this rule, we are led to use subtyping with
\srefcase{Lemma}{lemma:action}{3}, so as to show that
after acting on $(\o,\i)$ with $\op'$, $\op$ remains mapped to $M$'s effect information.
The third case of interest concerns the commutativity of signals with interrupt handlers:
\[
\tmwith{op}{x}{M}{p}{\tmopout{op'}{V}{N}} \reduces \tmopout{op'}{V}{\tmwith{op}{x}{M}{p}{N}}
\]
where in order to type the signal's payload $V$ in the right-hand side,
it is crucial that the promise-typed variable $p$ cannot appear in $V$---this is ensured by
our type system that restricts the signatures $\op : A_\op$ to ground types. As a result,
we can strengthen the typing context of $V$ by removing $p$.
Finally, in the evaluation context rule case, we use the induction hypothesis with \sref{Lemma}{lemma:eval-ctx-typing}.
\end{proof}
Interestingly, the proof of \autoref{theorem:preservation} tells us that if
one were to consider a variant of \lambdaAEff~in which the
\textsc{TyComp-Subsume} rule appeared as an explicit coercion term $\tmkw{coerce}_{(\o,\i) \order {O \times I} (\o',\i')}\, M$, then
the right-hand sides of the two interrupt propagation rules highlighted in the above proof
would also need to involve such coercions, corresponding to the uses of \sref{Lemma}{lemma:action}.
This however means that other computations involved in these reduction rules would also need to be type-annotated.
\section{A Calculus for Asynchronous Effects: Parallel Processes}
\label{sec:basic-calculus:processes}
Next, we describe the parallel part of \lambdaAEff. Similarly to the sequential part, we
again present the corresponding syntax, a small-step semantics,
a type-and-effect system, and type safety results.
\subsection{Parallel Processes}
To keep the presentation focussed on the asynchronous use of algebraic effects, we
consider a very simple model of parallelism: a process is either an \emph{individual computation}
or the \emph{parallel composition} of two processes. To facilitate interactions between processes, they also
contain outward propagating \emph{signals} and inward propagating \emph{interrupts}. Formally, the
syntax of \emph{parallel processes} is
\[
P, Q
\mathrel{\;{:}{:}{=}\ } \tmrun M
\,\mathrel{\;\big|\ \ }\! \tmpar P Q
\,\mathrel{\;\big|\ \ }\! \tmopout{op}{V}{P}
\,\mathrel{\;\big|\ \ }\! \tmopin{op}{V}{P}
\]
Note that processes do not include interrupt handlers---these are local to individual computations.
We leave first-class processes and their dynamic creation for future work, as discussed in \autoref{sec:conclusion}.
\subsection{Small-Step Operational Semantics}
We equip the parallel part of \lambdaAEff~with a small-step semantics that
naturally extends that of \lambdaAEff's sequential part.
The semantics is defined using a reduction relation $P \reduces Q$, as given in \autoref{fig:processes}.
\begin{figure}[tp]
\parbox{\textwidth}{
\centering
\small
\begin{minipage}[t]{0.4\textwidth}
\centering
\begin{align*}
\intertext{\textbf{Individual computations}}
\coopinfer{}{
M \reduces N
}{
\tmrun M \reduces \tmrun N
}
\end{align*}
\begin{align*}
\intertext{\textbf{Signal hoisting}}
\tmrun {(\tmopout{op}{V}{M})} &\reduces \tmopout{op}{V}{\tmrun M}
\\[1ex]
\intertext{\textbf{Broadcasting}}
\tmpar{\tmopout{op}{V}{P}}{Q} &\reduces \tmopout{op}{V}{\tmpar{P}{\tmopin{op}{V}{Q}}}
\\
\tmpar{P}{\tmopout{op}{V}{Q}} &\reduces \tmopout{op}{V}{\tmpar{\tmopin{op}{V}{P}}{Q}}
\end{align*}
\vspace{-1ex}
\end{minipage}
\qquad
\begin{minipage}[t]{0.4\textwidth}
\centering
\begin{align*}
\intertext{\textbf{Interrupt propagation}}
\tmopin{op}{V}{\tmrun M} &\reduces \tmrun {(\tmopin{op}{V}{M})}
\\
\tmopin{op}{V}{\tmpar P Q} &\reduces \tmpar {\tmopin{op}{V}{P}} {\tmopin{op}{V}{Q}}
\\
\tmopin{op}{V}{\tmopout{op'}{W}{P}} &\reduces \tmopout{op'}{W}{\tmopin{op}{V}{P}}
\end{align*}
\begin{align*}
\intertext{\quad\textbf{Evaluation context rule}}
\quad
\coopinfer{}{
P \reduces Q
}{
\F[P] \reduces \F[Q]
}
\end{align*}
\end{minipage}
\begin{align*}
\intertext{\textbf{where}\vspace{1ex}}
\text{$\F$}
\mathrel{\;{:}{:}{=}\ }& [~]
\mathrel{\;\big|\ \ } \tmpar \F Q \mathrel{\;\big|\ \ }\! \tmpar P \F
\mathrel{\;\big|\ \ } \tmopout{op}{V}{\F}
\mathrel{\;\big|\ \ } \tmopin{op}{V}{\F}
\end{align*}
}
\caption{Small-step operational semantics of parallel processes.}
\label{fig:processes}
\end{figure}
\paragraph{Individual computations}
This reduction rule states that, as processes, individual computations evolve according to the small-step
operational semantics $M \reduces N$ we defined for them in \autoref{sec:basic-calculus:semantics:computations}.
\paragraph{Signal hoisting}
This rule propagates signals out of individual computations.
It is important to note that we only hoist those signals that have propagated to the outer boundary
of a computation.
\paragraph{Broadcasting}
The broadcast rules turn outward moving signals in one process into inward moving interrupts
for the process parallel to it, while continuing to propagate the signals outwards to any
further parallel processes. The latter ensures that the semantics is compositional.
\paragraph{Interrupt propagation}
These three rules simply propagate interrupts inwards into individual computations,
into all branches of parallel compositions, and past any outward moving signals.
\paragraph{Evaluation contexts}
Analogously to the semantics of computations, the semantics of processes also includes a context rule, which allows reductions under \emph{evaluation contexts}
$\F$. Observe that compared to the evaluation contexts for computations, those for processes
do not bind variables.
\subsection{Type-and-Effect System}
Analogously to its sequential part, we also equip \lambdaAEff's parallel part with a type-and-effect system.
\paragraph{Types} The \emph{types of processes} are designed to match their parallel structure---they are given by
\[
\text{$\tyC$, $\tyD$}
\mathrel{\;{:}{:}{=}\ } \tyrun X \o \i
\,\mathrel{\;\big|\ \ }\! \typar \tyC \tyD
\]
Intuitively, $\tyrun X \o \i$ is a process type of an individual computation of type $\tycomp{X}{(\o,\i)}$, and $\typar \tyC \tyD$
is the type of the parallel composition of two processes that respectively have types $\tyC$ and $\tyD$.
\paragraph{Typing judgements}
\emph{Well-typed processes} are characterised using the judgement
$\Gamma \vdash P : \tyC$. We present the typing rules in \autoref{fig:process-typing-rules}.
While our processes are not currently higher-order, we allow
non-empty contexts $\Gamma$ to model the possibility of using libraries and top-level function definitions.
\begin{figure}[tp]
\centering
\small
\begin{mathpar}
\coopinfer{TyProc-Run}{
\Gamma \types M : \tycomp{X}{(\o,\i)}
}{
\Gamma \types \tmrun{M} : \tyrun{X}{\o}{\i}
}
\quad
\coopinfer{TyProc-Par}{
\Gamma \types P : \tyC \\
\Gamma \types Q : \tyD
}{
\Gamma \types \tmpar{P}{Q} : \typar{\tyC}{\tyD}
}
\quad
\coopinfer{TyProc-Signal}{
\op \in \mathsf{signals\text{-}of}{(\tyC)} \\\\
\Gamma \types V : A_\op \\
\Gamma \types P : \tyC
}{
\Gamma \types \tmopout{op}{V}{P} : \tyC
}
\quad
\coopinfer{TyProc-Interrupt}{
\Gamma \types V : A_\op \\
\Gamma \types P : \tyC
}{
\Gamma \types \tmopin{op}{V}{P} : \opincomp{op}{\tyC}
}
\end{mathpar}
\caption{Process typing rules.}
\label{fig:process-typing-rules}
\end{figure}
The rules \textsc{TyProc-Run} and \textsc{TyProc-Par} capture the earlier
intuition about the types of processes matching their parallel structure. The rules
\textsc{TyProc-Signal} and \textsc{TyProc-Interrupt} are similar to the corresponding rules
from \autoref{fig:computation-typing-rules}.
The \emph{signal annotations} of a process type are calculated as
\[
\mathsf{signals\text{-}of}(\tyrun{X}{\o}{\i}) ~\defeq~ \o
\qquad\qquad
\mathsf{signals\text{-}of}(\typar{\tyC}{\tyD}) ~\defeq~ \mathsf{signals\text{-}of}(\tyC) \cup \mathsf{signals\text{-}of}(\tyD)
\]
and the \emph{action of interrupts} on process types $\opincomp{op}{\tyC}$ extends the action on effect annotations as
\[
\opincomp{op}(\tyrun{X}{\o}{\i})
~\defeq~
X \att (\opincomp {op} {(\o , \i)})
\qquad\qquad
\opincomp{op}(\typar{\tyC}{\tyD})
~\defeq~
\typar{(\opincomp{op}{\tyC})}{(\opincomp{op}{\tyD})}
\]
by propagating the interrupt towards the types of individual computations.
We then have:
\begin{lemma}
\label{lemma:signals-of-interrupt-action}
For any process type $\tyC$ and interrupt $\op$, we have that $\mathsf{signals\text{-}of}(\tyC) \order O \pi_1\, (\opincomp{op}{\tyC})$.
\end{lemma}
It is worth noting that \autoref{fig:process-typing-rules} does not include an analogue
of \textsc{TyComp-Subsume}. This is
deliberate because as we shall see below, \emph{process types reduce}
in conjunction with the processes they are assigned to, and the outcome
is generally neither a sub- nor supertype of the original type.
\subsection{Type Safety}
\label{sec:basic-calculus:type-safety:processes}
We conclude the meta-theory of \lambdaAEff~by proving type safety
for its parallel part. Analogously to \autoref{sec:basic-calculus:type-safety},
we once again split type safety into separate proofs of \emph{progress}
and \emph{preservation}.
\subsubsection{Progress}
We characterise the \emph{result forms} of parallel processes
by defining two judgements, $\ProcResult P$ and $\ParResult P$,
and by using the judgement $\RunResult {\Psi} {M}$ from
\autoref{sec:basic-calculus:type-safety}, as follows:
\begin{mathpar}
\coopinfer{}{
\ProcResult {P}
}{
\ProcResult {\tmopout {op} V P}
}
\qquad
\coopinfer{}{
\ParResult {P}
}{
\ProcResult {P}
}
\qquad
\coopinfer{}{
\RunResult {\emptyset} {M}
}{
\ParResult {\tmrun M}
}
\qquad
\coopinfer{}{
\ParResult P \\
\ParResult Q
}{
\ParResult {\tmpar P Q}
}
\end{mathpar}
These judgements express that a process $P$ is in a (top-level)
result form $\ProcResult {P}$ when, considered as a tree, it has a shape in which
\emph{all} signals are towards the root, parallel compositions are in
the intermediate nodes, and individual computation results are at the leaves.
Importantly, the computation results $\RunResult {\emptyset} {M}$ we use here are those from
which signals have been propagated out of
(see \autoref{sec:basic-calculus:type-safety:progress}).
The finality of these results forms is then captured by the next lemma.
\begin{lemma}
\label{lemma:results-are-final:processes}
Given a process $P$ such that $\ProcResult {P}$, then there exists no $Q$ such that $P \reduces Q$.
\end{lemma}
We are now ready to state and prove the \emph{progress theorem} for the parallel part of \lambdaAEff.
\begin{theorem}
Given a well-typed closed process $\types P : \tyC$,
then either (i) there exists a process $Q$ such that $P \reduces Q$, or
(ii) the process $P$ is already in a (top-level) result form, i.e., we have $\ProcResult {P}$.
\end{theorem}
\begin{proof}
The proof is standard and proceeds by induction on the derivation of $\types P : \tyC$.
In the base case, when the derivation ends with the \textsc{TyProc-Run} rule,
and $P \hspace{-0.05cm}=\hspace{-0.05cm} \tmrun {\hspace{-0.05cm}M}$, we use
\sref{Corollary}{corollary:progress}.
\end{proof}
\subsubsection{Type Preservation}
First, we note that the broadcast rules in \autoref{fig:processes} introduce new
inward propagating interrupts in their right-hand sides that originally do not exist in their left-hand sides. As a result,
compared to the types one assigns to the left-hand sides of these reduction rules, the types assigned to
their right-hand sides will need to feature corresponding type-level actions of these interrupts.
We formalise this idea using a \emph{process type reduction} relation $\tyC \tyreduces \tyD$, given by
\[
\coopinfer{}{
}{
\tyrun{X}{\o}{\i} \tyreduces \tyrun{X}{\o}{\i}
}
\quad
\coopinfer{}{
}{
X \att \opincompp {ops} {(\o , \i)} \tyreduces X \att \opincompp {ops} {(\opincomp {op} {(\o , \i)})}
}
\quad
\coopinfer{}{
\tyC \tyreduces \tyC' \\
\tyD \tyreduces \tyD'
}{
\typar{\tyC}{\tyD} \tyreduces \typar{\tyC'}{\tyD'}
}
\]
where we write $\opincompp {ops} {(\o , \i)}$ for a recursively defined \emph{action of a list of interrupts} on $(\o , \i)$,
given by
\[
\opincompp {[]} {(\o , \i)} ~\defeq~ (\o , \i)
\qquad
\opincompp {(\op :: \opsym{ops})} {(\o , \i)} ~\defeq~ \opincomp {op} {(\opincompp {ops} (\o , \i))}
\]
Intuitively, $\tyC \tyreduces \tyD$ describes how process types reduce by being acted upon by
freshly arriving interrupts. While we define the action behaviour only at the leaves of process types (under some
enveloping sequence of actions), we can prove expected properties for arbitrary process types:
\begin{lemma}
\label{lemma:type-reduction} \mbox{}
\begin{enumerate}
\item Process types can remain unreduced, i.e., $\tyC \tyreduces \tyC$ for any process type $\tyC$.
\item Process types reduce by being acted upon, i.e., $\tyC \tyreduces \opincomp {op} \tyC$ for any type $\tyC$ and interrupt $\op$.
\item Process types can reduce under enveloping actions, i.e., $\opincomp {op} \tyC \tyreduces \opincomp {op} \tyD$ when $\tyC \tyreduces \tyD$.
\item Process type reduction can introduce signals, i.e., $\mathsf{signals\text{-}of} (\tyC) \order O \mathsf{signals\text{-}of} (\tyD)$
when $\tyC \tyreduces \tyD$.
\end{enumerate}
\end{lemma}
For the proof of \srefcase{Lemma}{lemma:type-reduction}{3}, it is important that we
introduce interrupts under an arbitrary enveloping sequence of interrupt actions,
and not simply as
$X \att {(\o , \i)} \tyreduces X \att (\opincomp {op} {(\o , \i)})$.
Further, the proof of \srefcase{Lemma}{lemma:type-reduction}{4} requires us to generalise \srefcase{Lemma}{lemma:action}{1} to lists of enveloping actions:
\begin{lemma}
\label{lemma:signal-inclusion-lists-of-interrupts}
$\pi_1\, (\opincompp {ops} {(\o,\i)}) \order O \pi_1\, (\opincompp {ops} {(\opincomp {op} {(\o,\i)})})$
\end{lemma}
As in \autoref{sec:basic-calculus:type-safety:preservation}, we again find it useful
to define a separate \emph{typing judgement for evaluation contexts}, this time written
$\Gamma \types\!\![\, \tyC \,]~ \F : \tyD$, together with an
analogue of \sref{Lemma}{lemma:eval-ctx-typing}, which we omit here. Instead, we
observe that this typing judgement is subject to process type reduction:
\begin{lemma}
\label{lemma:hoisting-and-evaluation-context-types}
Given $\Gamma \types\!\![\, \tyC \,]~ \F \hspace{-0.05cm}:\hspace{-0.05cm} \tyD$ and $\tyC \hspace{-0.05cm}\tyreduces\hspace{-0.05cm} \tyC'$, then there exists $\tyD'$ with
$\tyD \hspace{-0.05cm}\tyreduces\hspace{-0.05cm} \tyD'$ and $\Gamma \types\!\![\, \tyC' \,]~ \F \hspace{-0.05cm}:\hspace{-0.05cm} \tyD'$.
\end{lemma}
We are now ready to state and prove the \emph{type preservation theorem} for the parallel part of \lambdaAEff.
\begin{theorem}
\label{theorem:preservation:processes}
Given a well-typed process $\Gamma \types P : \tyC$, such that $P$ can reduce as
$P \reduces Q$, then there exists a process type $\tyD$, such
that the process type $\tyC$ can reduce as $\tyC \tyreduces \tyD$, and we have $\Gamma \types Q : \tyD$.
\end{theorem}
\begin{proof}
The proof proceeds by induction on the derivation of
$P \reduces Q$, using auxiliary typing inversion lemmas depending on
the structure forced upon $P$ by the last rule used in $P \reduces Q$.
For all but the broadcast and evaluation context rules, we can pick $\tyD$ to be $\tyC$ and use
\srefcase{Lemma}{lemma:type-reduction}{1}.
For the broadcast rules, we define $\tyD$ by introducing the corresponding
interrupt, and build $\tyC \tyreduces \tyD$ using the parallel composition
rule together with \srefcase{Lemma}{lemma:type-reduction}{2}.
For the evaluation context rule, we use \sref{Lemma}{lemma:hoisting-and-evaluation-context-types}
in combination with the induction hypothesis. Finally, in order to discharge effects-related side-conditions
when commuting interrupts with signals,
we use \sref{Lemma}{lemma:signals-of-interrupt-action}.
\end{proof}
\section{Conclusion}
\label{sec:conclusion}
We have shown how to incorporate asynchrony within
algebraic effects, by decoupling
the execution of operation calls into signalling that an operation's implementation
needs to be executed, and interrupting a running computation with the operation's result,
to which it can react by installing interrupt handlers.
We have shown that our approach is flexible enough that not all signals have to have a matching
interrupt, and vice versa, allowing us to also model spontaneous behaviour, such as a user
clicking a button or the environment preempting a thread. We have formalised these ideas in a small
calculus, called \lambdaAEff, and demonstrated its flexibility on a number of examples.
We have also accompanied the paper with an \pl{Agda} formalisation and a prototype implementation of \lambdaAEff.
However, various future work directions still remain. We discuss these and related work below.
\paragraph{Asynchronous effects}
As asynchrony is desired in practice, it is no surprise that \pl{Koka} \cite{Leijen:AsyncAwait}
and \pl{Multicore OCaml} \cite{Dolan:MulticoreOCaml}, the two largest implementations of algebraic effects
and handlers, have been extended accordingly.
In \pl{Koka}, algebraic operations
reify their continuation into an explicit callback structure that is then dispatched to a primitive
such as \lstinline{setTimeout} in its \pl{Node.JS} backend. In \pl{Multicore OCaml}, one uses low-level functions
such as \lstinline{set_signal} or \lstinline{timer_create} that modify the runtime by interjecting operation
calls inside the currently running code. Both approaches thus \emph{delegate} the actual asynchrony to existing
concepts in their backends. In contrast, in \lambdaAEff, we
can express such backend features within the core calculus itself.
Further, in \lambdaAEff, we avoid having to manually use (un)masking to
disable asynchronous effects in unwanted places
in our programs, which can be a very tricky business to get right, as noted by \citet{Dolan:MulticoreOCaml}.
Instead, by design, interrupts in \lambdaAEff~\emph{never}
influence running code unless the code has an explicit interrupt handler installed,
and they \emph{always} wait for any potential handler to present itself during
execution (recall that they get discarded only when reaching a $\tmkw{return}$).
\paragraph{Message-passing}
While in this paper we have focussed on the foundations of asynchrony in the
context of algebraic effects, the ideas we propose have also many common
traits with concurrency models based on \emph{message-passing},
such as the Actor model \cite{Hewitt:Actors}, the $\pi$-calculus \cite{Milner:PiCalculus},
and the join-calculus \cite{FournetGonthier:JoinCalculus}, just to name a few.
Namely, one can view the issuing of a signal $\tmopout{op}{V}{M}$ as sending a message,
and handling an interrupt $\tmopin{op}{W}{M}$ as receiving a message, along a channel
named $\op$.
In fact, we believe that in our prototype implementation we could replace the semantics
presented in the paper with an equivalent one based on shared channels
(one for each $\op$), to which the interrupt handlers could subscribe to.
Instead of propagating signals first out and then in, they would be sent directly to channels
where interrupt handlers immediately receive them, drastically reducing the cost of communication.
Comparing \lambdaAEff~to the Actor model, we see that
the $\tmrun M$ processes evolve in their own bubbles, and only communicate with other
processes via signals and interrupts, similarly to actors.
However, in contrast to messages not being required to be ordered
in the Actor model, in our $\tmpar P Q$, the process $Q$ receives
interrupts in the same order as the respective signals are issued by $P$
(and vice versa). This communication ordering could be relaxed by allowing
signals to be hoisted out of computations from deeper than just the top level.
Another difference
with actors is that \lambdaAEff-computations
can react to interrupts only sequentially, and not by dynamically creating new parallel
processes---first-class parallel processes and their dynamic creation is something
we plan to address in future work.
It is worth noting that our interrupt handlers are similar to the message receiving construct
in the $\pi$-calculus, in that they both synchronise with matching incoming
interrupts/messages. However, the two are also different, in that interrupt handlers allow
reductions to take place under them and non-matching interrupts to propagate past them.
Further, our interrupt handlers are also similar to join definitions in the join-calculus, describing
how to react when a corresponding interrupt arrives or join pattern appears, where in both cases
the reaction could involve effectful code. To this end, our interrupt handlers resemble join definitions
with simple one-channel join patterns. However, where the two constructs differ is that join definitions
also serve to define new (local) channels, similarly to the restriction operator in the $\pi$-calculus,
whereas we assume a fixed global set of channels (i.e., signal and interrupt names).
We expect that extending \lambdaAEff~with local algebraic effects
\cite{Staton:Instances,Biernacki:AbstractingAlgEffects}
could help us fill this gap between the formalisms.
\paragraph{Scoped operations}
As discussed in \autoref{sec:basic-calculus:semantics:computations}, despite their name, interrupt handlers
behave like algebraic operations, not like effect handlers. However, one should also note
that they are not conventional operations because they carry computational data that sequential
composition does not interact with, and that only gets triggered when a corresponding interrupt is received.
Such generalised operations are known in the literature as \emph{scoped operations}~\cite{Pirog:ScopedOperations},
a leading example of which is $\opsym{spawn}(M;N)$, where $M$ is the new child process to be executed and $N$ is
the current process. Crucially, the child $M$ should not directly interact with the current process. Scoped operations
achieve this behaviour by declaring $M$ to be in the scope of $\opsym{spawn}$, resulting in
$\tmlet x {\opsym{spawn}(M;N)} {K} \!\reduces\! \opsym{spawn}(M;\tmlet x N K)$, exactly
as we have for interrupt handlers.
Further recalling \autoref{sec:basic-calculus:semantics:computations}, despite their appearance,
incoming interrupts behave computationally like effect handling, not like algebraic operations.
In fact, it turns out they correspond to effect handling
induced by an instance of \emph{scoped effect handlers} \cite{Pirog:ScopedOperations}.
Compared to ordinary effect handlers, scoped effect handlers explain both
how to interpret operations and their scopes. In our setting, this
corresponds to selectively executing the handler code of interrupt handlers.
It would be interesting to extend our work both with
scoped operations having more general signatures, and with additional effect handlers
for them. The latter could allow
preventing the propagation of incoming interrupts into continuations, discarding the continuation
of a cancelled remote call, and techniques such as masking or reordering interrupts
according to priority levels.
\paragraph{Modal types}
We recall that the type safety of \lambdaAEff~crucially relies on
the promise-typed variables bound by interrupt handlers not being
allowed to appear in the payloads of signals. This ensures that it is safe to propagate
signals past all enveloping interrupt handlers, and communicate their payloads
to other processes. In its essence, this is similar to the use of \emph{modal types} in distributed
\cite{Murphy:PhDThesis} and reactive programming \cite{Krishnaswami:HOFRP,Bahr:RATT}
to classify values that can travel through space and time. In our case,
it is the omission of promise types from ground types that allows us to consider
the payloads of signals and interrupts as such \emph{mobile values}.
We expect that these connections to modal types will be key for
extending \lambdaAEff~with (i)~higher-order payloads and (ii) process
creation. For (i), we want to avoid the bodies of function-typed payloads to be able
to await enveloping promise variables to be fulfilled. For (ii),
we want to do the same for the dynamically created processes.
In both cases, the reason is to be able to safely propagate the corresponding
programming constructs past enveloping interrupt handlers, and eventually
hoist them out of individual computations. We believe that the more structured
treatment of contexts $\Gamma$, as studied in various modal type
systems, will hold the key for these extensions to be type safe.
\paragraph{Denotational semantics}
In this paper we study only the operational side of \lambdaAEff,
and leave developing its denotational semantics for the future.
In light of how we have motivated the \lambdaAEff-specific programming
constructs, and based on the above discussions, we expect the denotational semantics
to take the form of an algebraically natural \emph{monadic semantics}, where the monad would
be given by an instance of the one studied by \citet{Pirog:ScopedOperations} for
scoped operations (quotiented by the commutativity of signals and interrupt handlers,
and extended with nondeterminism to model different evaluation
outcomes), incoming interrupts would be modelled as homomorphisms
induced by scoped algebras, and parallel composition
by considering all nondeterministic interleavings of (the outgoing signals of) individual computations, e.g.,
based on how \citet{Plotkin:BinaryHandlers} and \citet{Lindley:DoBeDoBeDo} model
it in the context of general effect handlers.
Finally, we expect to take inspiration for the denotational semantics of the promise type
from that of modal logics and modal types.
\paragraph{Reasoning about asynchronous effects}
In addition to using \lambdaAEff's type-and-effect system only for specification purposes (such as specifying
that $M : \tycomp{X}{(\emptyset,\{\})}$ raises no signals and installs no interrupt handlers),
we wish to make further use of it for validating \emph{effect-dependent optimisations} \cite{Kammar:Optimisations}.
For instance, whenever $M : \tycomp{X}{(\o,\i)}$ and $\i\, (\op) = \bot$, we would like to know
that $\tmopin{\op}{V}{M} \reduces^* M$. One way to validate such optimisations
is to develop an adequate denotational semantics,
and then use a semantic \emph{computational induction} principle \cite{Bauer:EffectSystem,Plotkin:Logic}.
For \lambdaAEff, this would amount to only having to prove the optimisations for return values, signals,
and interrupt handlers. Another way to validate effect-dependent optimisations would
be to define a suitable logical
relation for \lambdaAEff~\cite{Benton:AbstractEffects}.
In addition to optimisations based on \lambdaAEff's existing effect system,
we plan to explore extending processes and their types
with \emph{communication protocols} inspired by session types \cite{Honda:LangPrimitives},
so as to refine the current ``broadcast
everything everywhere'' communication strategy.
\section*{Acknowledgements}
We thank the anonymous reviewers, Otterlo IFIP WG 2.1 meeting participants,
and Andrej Bauer, Gavin Bierman, Žiga Lukšič, and Alex Simpson for their useful feedback.
This project has received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Sk\l{}odowska-Curie grant agreement No 834146
\raisebox{-0.05cm}{
\hspace{-0.15cm}
\includegraphics[width=0.5cm]{eu_flag.pdf}
\hspace{-0.15cm}
}.
This material is based upon work supported by the Air Force Office of Scientific
Research under award number FA9550-17-1-0326.
\section{Introduction}
Effectful programming abstractions are at the heart of many modern general-purpose
programming languages.
They can increase expressiveness by giving
access to first-class continuations, but often simply help users to write
cleaner code, e.g., by avoiding having to manage a program's memory explicitly in state-passing style,
or getting lost in callback hell while programming asynchronously.
An increasing number of language designers and programmers are starting to
embrace \emph{algebraic effects},
where one uses algebraic operations \cite{Plotkin:NotionsOfComputation} and
effect handlers \cite{Plotkin:HandlingEffects} to uniformly and user-definably
express a wide range of effectful behaviour,
ranging from basic examples such as state, rollbacks, exceptions,
and nondeterminism \cite{Bauer:AlgebraicEffects}, to advanced applications
in concurrency \cite{Dolan:MulticoreOCaml} and statistical probabilistic programming
\cite{Bingham:Pyro}, and even quantum computation \cite{Staton:AlgEffQuantum}.
While covering many examples, the conventional treatment of
algebraic effects is \emph{synchronous} by nature. In it
effects are invoked by placing operation calls in one's code,
which then propagate outwards until they trigger the actual effect, finally yielding
a result to the rest of the computation that has been \emph{waiting} the whole
time. While blocking the computation is indeed sometimes needed, e.g.,
in the presence of general effect handlers that can execute their continuation any
number of times, it forces all uses of algebraic effects to be synchronous, even when this
is not necessary, e.g., when the effect involves executing
a remote query to which a response is not needed (immediately).
Motivated by the recent interest in the combination of
asynchrony and algebraic effects \cite{Leijen:AsyncAwait,Dolan:MulticoreOCaml},
we explore what it takes (in terms of
language design, safe programming abstractions, and a
self-contained core calculus) to accompany the
synchronous treatment of algebraic effects with
an \emph{asynchronous} one. At the heart of our approach is the
decoupling of the execution of operation calls
into \emph{signalling} that some implementation of an operation needs to be executed, and \emph{interrupting} a
running computation with its result, to which the computation can react by
installing \emph{interrupt handlers}. Importantly, we show that our
approach is flexible enough that not all signals need to have a
corresponding interrupt, and vice versa, allowing us to also model
\linebreak
\emph{spontaneous behaviour}, such as a
user clicking a button or the environment preempting a thread.
While we are not the first ones to work on asynchrony for algebraic effects,
the prior work in this area (in the context of general effect handlers) has
achieved it by \emph{delegating} the actual asynchrony to the respective language backends
\cite{Leijen:AsyncAwait,Dolan:MulticoreOCaml}. In contrast, in this paper
we demonstrate how to capture the combination of
asynchrony and algebraic effects in a \emph{self-contained} core calculus.
It is important to emphasise that our aim is not to replace general effect handlers,
but instead to \emph{complement} them with robust primitives
tailored to asynchrony---our proposed approach is algebraic by design, so as
to be ready for future extensions with general effect handlers.
\paragraph{Paper structure}
In \autoref{sec:overview}, we give a high-level overview of our approach to
asynchrony for algebraic effects.
In \autoref{sec:basic-calculus:computations}
and \ref{sec:basic-calculus:processes}, we distil our ideas into a core calculus, \lambdaAEff,
equipped with a small-step semantics, a type-and-effect system, and proofs of
type safety. In \autoref{sec:applications},
we show \lambdaAEff~in action on examples
such as preemptive multi-threading, remote function calls, and a parallel variant of runners
of algebraic effects. We conclude, and discuss related and future work in \autoref{sec:conclusion}.
\paragraph{Code}
The paper is accompanied by a \emph{formalisation} of \lambdaAEff's type safety proofs
in \pl{Agda} \cite{ahman20:AeffAgda}, and a \emph{prototype implementation} of \lambdaAEff~in
\pl{OCaml}, called \pl{{\AE}ff} \cite{pretnar20:AEff}. For ease of use, we provide them both also as
a single virtual machine image \cite{AhmanPretnar20:Artefact}.
In the \pl{Agda} formalisation, we consider only well-typed syntax of a
variant of \lambdaAEff~in which the subsumption rule manifests as an explicit coercion, so as to make working with
de Bruijn indices less painful.
Meanwhile, the \pl{{\AE}ff} implementation provides an interpreter and
a simple typechecker, but it does not
yet support inferring and checking effect annotations. In addition, \pl{{\AE}ff} provides
a web interface that allows users to enter their programs and interactively click through
their executions.
\pl{{\AE}ff} also comes with implementations of all the examples we present in this paper.
Separately, \citet{Poulson:AsyncEffectHandling} has shown how to implement \lambdaAEff~
in \pl{Frank} \cite{Convent:DooBeeDooBeeDoo}.
\section{Asynchronous Effects, by Example}
\label{sec:overview}
We begin with a high-level overview of how we accommodate asynchrony within algebraic effects.
\subsection{Conventional Algebraic Effects Are Synchronous by Nature}
We first recall the basic ideas of programming with algebraic effects,
illustrating that their traditional treatment is synchronous by nature.
For an in-depth overview, we refer to the tutorial by \citet{Pretnar:Tutorial}, and to the
seminal papers of the field \cite{Plotkin:NotionsOfComputation,Plotkin:HandlingEffects}.
In this algebraic treatment, sources of computational effects are modelled using signatures
of \emph{operation symbols} $\op : A_\op \to B_\op$. For instance, one models
$S$-valued state using two operations, $\mathsf{get} : \tyunit \to S$ and $\mathsf{set} : S \to \tyunit$;
and $E$-valued exceptions using a single operation $\opsym{raise} : E \to \tyempty$.
Programmers can then invoke the effect that an
$\op : A_\op \to B_\op$ models by placing an \emph{operation call} $\tmop {op} V y M$ in their code. Here, the
parameter value $V$ has type $A_\op$, and the variable $y$, which is bound in the continuation $M$, has type $B_\op$.
For instance, for $\mathsf{set}$, the parameter $V$ would be
the new value of the store, and for $\mathsf{get}$, the variable $y$ would be bound to the current value of the store.
A program written in terms of operation calls is by itself just an inert piece of code. To
execute it, programmers have to provide \emph{implementations} for the operation
calls appearing in it. The idea is that an implementation of $\tmop {op} V y M$ takes $V$ as its input,
and its output gets bound to $y$.
For instance, this could take the form of defining a suitable effect handler
\cite{Plotkin:HandlingEffects}, but could also be given by calls
to runners of algebraic effects \cite{Ahman:Runners}, or simply by invoking some
(default) top-level (native) implementation.
What is important is that some pre-defined piece of code $M_\op[V/x]$
gets executed in place of every operation call $\tmop {op} V y M$.
Now, what makes the conventional treatment of algebraic effects \emph{synchronous} is
that the execution of an operation call $\tmop {op} V y M$ \emph{blocks} until some implementation
of $\op$ returns a value $W$ to be bound to $y$, so that
the execution of the continuation $M[W/y]$ could proceed \cite{Kammar:Handlers,Bauer:EffectSystem}.
Conceptually, this kind of blocking behaviour can be illustrated as
\begin{equation}
\begin{gathered}
\label{eq:syncopcall}
\xymatrix@C=1.25em@R=0.85em@M=0.5em{
& M_\op[V/x] \ar@{}[r]|{\mbox{$\Large{\leadsto^{\!*}}$}} & \tmreturn W \ar[d]
\\
\dots \ar@{}[r]|>>>{\mbox{$\Large{\leadsto}$}} & \tmop {op} V y M \ar[u] & M[W/y] \ar@{}[r]|<<<{\mbox{$\Large{\leadsto}$}} & \dots
}
\end{gathered}
\end{equation}
where $\tmreturn W$ is a computation that causes no effects and simply returns the given value $W$.
While blocking the rest of the computation is needed in the presence of
general effect handlers that can execute their continuation any number
of times, it forces all uses of algebraic effects to be synchronous, even
when this is not necessary, e.g., when the effect in question involves
executing a remote query to which a response is not needed immediately,
or sometimes never at all.
In the rest of this section, we describe how we decouple the invocation of
an operation call from the act of receiving its result, and how we give
programmers a means to block execution only when it is necessary.
While we end up surrendering some of effect handlers' generality,
such as having access to the continuation that captures the rest of the
computation to be handled, then in return we get a natural and robust formalism for
asynchronous programming with algebraic effects.
\subsection{Outgoing Signals and Incoming Interrupts}
\label{sec:overview:signals}
We begin by observing that the execution of an operation call $\tmop {op} V y M$,
as shown in (\ref{eq:syncopcall}), consists of \emph{three distinct phases}: (i) signalling that an
implementation of $\op$ needs to be executed with parameter $V$ (the up-arrow), (ii) executing
this implementation (the horizontal arrow), and (iii) interrupting the blocking of $M$ with a value $W$
(the down-arrow). In order to overcome the unwanted side-effects of blocking execution on every operation call,
we shall naturally decouple these phases into separate programming concepts, allowing the execution of
$M$ to proceed even if (ii) has not yet completed and (iii) taken place. In particular, we
decouple an operation call into issuing an \emph{outgoing signal},
written $\tmopout{\op}{V}{M}$, and receiving an \emph{incoming interrupt}, written $\tmopin{\op}{W}{M}$.
It is important to note that while we have used the execution of operation calls
to motivate the introduction of signals and interrupts as programming concepts, \emph{not all issued signals need to have a corresponding
interrupt response}, and \emph{not all interrupts need to be responses to issued signals},
allowing us to also model spontaneous behaviour, such as the environment preempting a thread.
When \emph{issuing a signal} $\tmopout{\op}{V}{M}$, the value $V$ is a \emph{payload}, such as a location to be looked up or a
message to be displayed, aimed at whoever is listening for the given signal. We use the $\tmkw{\uparrow}$-notation to indicate that signals issued in sub-computations propagate outwards---in this sense signals behave just like conventional
operation calls. However, signals crucially differ from conventional operation calls in that no additional variables
are bound in the continuation $M$, making it naturally possible to continue executing $M$ straight after the signal has been issued, e.g., as depicted below:
\vspace{-3ex}
\[
\xymatrix@C=1.25em@R=1.25em@M=0.5em{
& &
\\
\dots \ar@{}[r]|<<<{\mbox{$\Large{\leadsto}$}} & \tmopout {op} V M \ar[u]^{\op\, V} \ar@{}[r]|<<<{\mbox{$\Large{\leadsto}$}} & M \ar@{}[r]|<<<{\mbox{$\Large{\leadsto}$}} & \dots
}
\]
\newcommand{M_{\text{feedClient}}}{M_{\text{feedClient}}}
As a \emph{running example}, consider a computation $M_{\text{feedClient}}$, which lets a user scroll through a seemingly infinite feed, e.g., by repeatedly clicking a ``next page'' button.
For efficiency, $M_{\text{feedClient}}$ does not initially cache all the data, but instead requests a
new batch of data each time the user is nearing the end of the cache. To communicate with the outside world, $M_{\text{feedClient}}$ can issue a signal
\[
\tmopout{\opsym{request}}{\mathit{cachedSize} + 1}{M_{\text{feedClient}}}
\]
to request a new batch of data starting from the end of the current cache, or a different signal
\[
\tmopout{\opsym{display}}{\mathit{message}}{M_{\text{feedClient}}}
\]
to display a message to the user. In both cases, the continuation \emph{does not wait}
for an acknowledgement that the signal was received,
but instead continues to provide a seamless experience to the user.
It is however worth noting that these signals differ in what $M_{\text{feedClient}}$ expects of them:
to the $\opsym{request}$ signal, it expects a response at some future point in
its execution, while it does not expect any response to the $\opsym{display}$ signal,
illustrating that not every issued signal needs a response.
When the outside world wants to get the attention of a computation, be it in response to
a signal or spontaneously,
it happens by \emph{propagating an interrupt}~$\tmopin{\op}{W}{M}$ to the computation.
Here, the value $W$ is again a payload, while $M$ is the computation receiving the interrupt.
It is important to note that unlike signals, interrupts are not triggered by the computation itself,
but are instead issued by the \emph{outside world},
and can thus interrupt any sequence of evaluation steps,
e.g., as in
\vspace{-3ex}
\[
\xymatrix@C=1.25em@R=1.25em@M=0.5em{
& \ar[d]^-{\op\, W} &
\\
\dots \ar@{}[r]|<<<{\mbox{$\Large{\leadsto}$}} & M \ar@{}[r]|<<<{\mbox{$\Large{\leadsto}$}} & \tmopin {op} W M \ar@{}[r]|<<<{\mbox{$\Large{\leadsto}$}} & \dots
}
\]
In our running example, there are two interrupts of interest:
$\tmopin{\opsym{response}}{\mathit{newBatch}}{M}$, which delivers new data to replenish the
cache; and $\tmopin{\opsym{nextItem}}{\tmunit}{M}$, with which the user requests to see the next data
item. In both cases, $M$ represents the state of $M_{\text{feedClient}}$ before the interrupt arrived.
We use
the $\tmkw{\downarrow}$-notation to indicate that interrupts propagate inwards into sub-computations,
trying to reach anyone listening for them, and only get discarded when they reach a $\tmkw{return}$.
It is worth noting that programmers are not expected to write interrupts explicitly in their programs---instead,
interrupts are usually induced by signals issued by other parallel processes, as explained next.
\subsection{A Signal for the Sender Is an Interrupt to the Receiver}
\label{sec:overview:processes}
As noted above, the computations we consider do not evolve in isolation, instead they also communicate with
the outside world, by issuing outgoing signals and receiving incoming interrupts.
We model the outside world by composing individual computations into \emph{parallel processes} $P, Q, \ldots$.
To keep the presentation clean and focussed on the asynchronous use of algebraic effects, we consider a very
simple model of parallelism: a process is either one of the individual computations being run
in parallel, written $\tmrun M$, or the parallel composition of two processes,
written $\tmpar P Q$.
\newcommand{M_{\text{feedServer}}}{M_{\text{feedServer}}}
To capture the signals and interrupts based interaction of processes,
our operational semantics includes rules for \emph{propagating outgoing signals} from individual
computations to processes,
\emph{turning processes' outgoing signals into incoming interrupts} for their surrounding world, and
\emph{propagating incoming interrupts} from processes to individual computations.
For instance, in our running example,
$M_{\text{feedClient}}$'s request for new data is executed as follows (with the active redexes highlighted):
\[
\begin{array}{r l}
& \tmpar{\highlightgray{\tmrun (\tmopout{request}{V}{\highlightwhite{M_{\text{feedClient}}}})}}{\tmrun M_{\text{feedServer}}}
\\[0.5ex]
\reduces & \highlightgray{\tmpar{(\tmopout{request}{V}{\highlightwhite{\tmrun M_{\text{feedClient}}}})}{\highlightwhite{\tmrun M_{\text{feedServer}}}}}
\\[0.5ex]
\reduces & \tmopoutbig{request}{V}{\tmpar{\tmrun M_{\text{feedClient}}}{\highlightgray{\tmopin{request}{V}{\tmrun {\highlightwhite{M_{\text{feedServer}}}}}}}}
\\[0.5ex]
\reduces & \tmopoutbig{request}{V}{\tmpar{\tmrun M_{\text{feedClient}}}{\tmrun (\tmopin{request}{V}{M_{\text{feedServer}}})}}
\end{array}
\]
Here, the first and the last reduction step respectively propagate signals outwards and
interrupts inwards. The middle reduction step corresponds to what we call a \emph{broadcast rule}---it
turns an outward moving signal in one of the processes into an inward moving interrupt for the process
parallel to it, while continuing to propagate the signal outwards to any further parallel processes.
\subsection{Promising to Handle Interrupts}
\label{sect:overview:promising}
So far, we have shown that our computations can issue outgoing signals and receive incoming interrupts, and how
these evolve when executing parallel processes, but we have not yet said
anything about how computations can actually \emph{react} to incoming interrupts of interest.
In order to react to incoming interrupts, our computations can install \emph{interrupt handlers}, written
\[
\tmwith{op}{x}{M}{p}{N}
\]
that should be read as: ``we promise to handle a future interrupt named $\op$ using the computation
$M$ in the continuation $N$, with $x$ bound to the payload of the interrupt''. Fulfilling this promise consists of executing $M$ and binding its result to the
variable $p$ in $N$. This is captured by the reduction rule
\[
\tmopin{op}{V}{\tmwith{op}{x}{M}{p}{N}} \reduces \tmlet{p}{M[V/x]}{\tmopin{op}{V}{N}}
\]
It is worth noting two things: the interrupt handler is \emph{not reinstalled by default},
and the interrupt itself \emph{keeps propagating inwards} into the sub-computation $N$.
Regarding the former,
programmers can selectively reinstall interrupt handlers when needed,
by defining them suitably recursively, e.g.,
as we demonstrate in \autoref{sec:overview:runningexample}.
Concerning the latter, then in order to skip certain interrupt handlers for some $\opsym{op}$, one
can carry additional data
in $\opsym{op}$'s payload (e.g., a thread ID) and then condition the (non-)triggering of those interrupt
handlers on this data, e.g., as we do in \autoref{sec:applications:guarder-handlers}.
Interrupts that do not match a given interrupt handler ($\op \neq \op'$) are simply propagated past it:
\[
\tmopin{op'}{V}{\tmwith{op}{x}{M}{p}{N}} \reduces \tmwith{op}{x}{M}{p}{\tmopin{op'}{V}{N}}
\]
Interrupt handlers differ from operation calls in two important aspects.
First, they enable \emph{user-side post-processing} of received data, using $M$,
while in operation calls the result is immediately bound in the continuation. Second, and more
importantly, their semantics is \emph{non-blocking}. In particular,
\[
N \reduces N' \qquad \text{implies} \qquad \tmwith{op}{x}{M}{p}{N} \reduces \tmwith{op}{x}{M}{p}{N'}
\]
meaning that the continuation $N$, and thus the whole computation, can make progress
even though no incoming interrupt $\opsym{op}$ has been propagated to the computation
from the outside world.
As the observant reader might have noticed, the non-blocking behaviour of interrupt handling
means that our operational semantics has to work on \emph{open terms} because the variable $p$ can
appear free in both $N$ and $N'$ above. However, it is important to note that $p$ is not an arbitrary variable,
but in fact gets assigned a distinguished \emph{promise type} $\typromise X$ for some value type
$X$---we shall crucially make use of this typing of $p$ in the proof of type safety for our \lambdaAEff-calculus (see \autoref{theorem:progress}).
\subsection{Blocking on Interrupts Only When Necessary}
\label{sec:overview:await}
As noted earlier, installing an interrupt handler means making a promise to handle a given
interrupt in the future. To check that an interrupt has been received and handled,
we provide programmers a means to selectively \emph{block execution}
and \emph{await} a specific promise to be fulfilled, written
$\tmawait{V}{x}{M}$, where if $V$ has a promise type $\typromise X$, the variable $x$ bound in $M$ has type $X$.
Importantly, the continuation $M$ is executed only
when the $\tmkw{await}$ is handed a \emph{fulfilled promise} $\tmpromise V$:
\[
\tmawait{\tmpromise V}{x}{M} \reduces M[V/x]
\]
Revisiting our example of scrolling through a seemingly infinite feed,
$M_{\text{feedClient}}$ could use $\tmkw{await}$ to block until it has received an initial configuration,
such as the batch size used by $M_{\text{feedServer}}$.
As the terminology suggests, this part of \lambdaAEff~is strongly influenced by existing work on
\emph{futures and promises} \cite{Schwinghammer:Thesis} for structuring concurrent programs, and their use in modern languages,
such as in \pl{Scala} \cite{Haller:Futures}. While prior work often models promises as writable, single-assignment
references, we instead use the substitution of values for ordinary immutable variables (of distinguished promise type)
to model that a promise gets fulfilled exactly once.
\subsection{Putting It All Together}
\label{sec:overview:runningexample}
Finally, we show how to implement our example of scrolling through a seemingly infinite feed.
For a simpler exposition, we allow ourselves access to mutable references, though the same can be
achieved by rolling one's own state.
Further, we use $\tmopoutgen {op} V$
as a syntactic sugar for $\tmopout {op} V {\tmreturn \tmunit}$.
\subsubsection{Client}
\label{sec:overview:runningexample:client}
We implement the client computation $M_{\text{feedClient}}$ as the function \ls$client$ defined below.
For presentation purposes, we split the definition of \ls$client$ between multiple code blocks.
First, the client sets up the initial values of the auxiliary references,
issues a signal to the server asking for the data batch size that it uses, and then installs a corresponding
interrupt handler:
\begin{lstlisting}
let client () =
let (cachedData , requestInProgress , currentItem) = (ref [] , ref false , ref 0) in
send batchSizeRequest ();
promise (batchSizeResponse batchSize |-> return <<batchSize>>) as batchSizePromise in
\end{lstlisting}
While the server is asynchronously responding to the batch size request, the client
sets up an auxiliary function \ls$requestNewData$, which it later uses to request new data from the server:
\begin{lstlisting}
let requestNewData offset =
requestInProgress := true;
send request offset;
promise (response newBatch |->
cachedData := !cachedData @ newBatch;
requestInProgress := false; return <<()>>
) as _ in return ()
in
\end{lstlisting}
Here, the client first sets a flag indicating that a new data request is in process,
then issues a $\opsym{request}$ signal to the server, and finally installs an
interrupt handler that updates the cache
once the $\opsym{response}$ interrupt arrives.
Note that the client does not block while awaiting new data, instead it continues executing, notifying
the user to wait and try again once the cache is empty (see below).
Then, the client sets up its main loop, which is a simple recursively defined interrupt handler:
\begin{lstlisting}
let rec clientLoop batchSize =
promise (nextItem () |->
let cachedSize = length !cachedData in
(if (!currentItem > cachedSize - batchSize / 2) && (not !requestInProgress) then
requestNewData (cachedSize + 1)
else
return ());
(if !currentItem < cachedSize then
send display (toString (nth !cachedData !currentItem));
currentItem := !currentItem + 1
else
send display "please wait a bit and try again");
clientLoop batchSize
) as p in return p
in
\end{lstlisting}
In it, the client listens for a $\opsym{nextItem}$ interrupt from the user to display more data.
Once the interrupt arrives, the client checks if its cache is becoming empty---if so,
it uses the $\opsym{requestNewData}$ function to
request more data from the server. Next, if there is still some
data in the cache, the client issues a signal to display the next data item to the user.
If however the cache is empty, the client issues a signal
to display a waiting message to the user. The client then simply recursively reinvokes itself.
As a last step of setting itself up, the client blocks until the server has responded
with the batch size it uses, after which the client starts its main loop
with the received batch size as follows:
\begin{lstlisting}
await batchSizePromise until <<batchSize>> in clientLoop batchSize
\end{lstlisting}
\subsubsection{Server}
\label{sec:overview:runningexample:server}
We implement the server computation $M_{\text{feedServer}}$ as the following function:
\begin{lstlisting}
let server batchSize =
let rec waitForBatchSize () =
promise (batchSizeRequest () |->
send batchSizeResponse batchSize;
waitForBatchSize ()
) as p in return p
in
let rec waitForRequest () =
promise (request offset |->
let payload = map (fun x |-> 10 * x) (range offset (offset + batchSize - 1)) in
send response payload;
waitForRequest ()
) as p in return p
in
waitForBatchSize (); waitForRequest ()
\end{lstlisting}
where the computation \lstinline{range i j} returns a list of integers ranging from \lstinline{i} to \lstinline{j} (both inclusive).
The server simply installs two recursively defined interrupt handlers: the first
one listens for and responds to client's requests about the batch size it uses;
and the second one responds to client's requests for new data. Both interrupt
handlers then simply recursively reinstall themselves.
\subsubsection{User}
\label{sec:overview:runningexample:user}
We can also simulate the user as a computation. Namely, we implement
it as a function that every now and then issues a request to
the client to display the next data item:
\begin{lstlisting}
let rec user () =
let rec wait n =
if n = 0 then return () else wait (n - 1)
in
send nextItem (); wait 10; user ()
\end{lstlisting}
It is straightforward to extend the user also with a handler for $\opsym{display}$ interrupts (we omit it here).
\subsubsection{Running the Server, Client, and User in Parallel}
\label{sec:overview:runningexample:parallel}
Finally, we can simulate our running example in full by running all
three computations we defined above as parallel processes, e.g., as follows:
\begin{lstlisting}
run (server 42) || run (client ()) || run (user ())
\end{lstlisting}
|
2,877,628,090,121 | arxiv | \section{Introduction}
Evoking physical requirements from anomaly cancellations, realistic
fermionic spectrum and the appropriate amount ($N = 1$) of Space-time
supersymmetry, Candelas {\it et. al} had
originally proposed a model for compactification of the superstring, by
analyzing the vacuum configurations of these 10-dimensional theories
\cite{CHSW}. Anomaly cancellation requirements (which constrain the
gauge groups of these models to be $O(32)$ or $E_8 \times E_8$), along
with the requirement of a zero cosmological constant, then lead
them
to propose/construct the 10-dimensional vacuum solutions of
these theories to be of the metric product type $ X_4 \times {\mathcal
M}$, where $X_4$ is the maximally symmetric $4d$ space-time (which
should admit unbroken $N = 1$ supersymmetry), and $\mathcal M$ is a
complex 3-dimensional Calabi-Yau manifold. Subsequently, these
conclusions were further generalized to include other gauge groups (like
${\rm SU}(4)$ or ${\rm SU}(5)$), as would arise when considering compactifications
for the strongly coupled heterotic string theory. The correspondence
between the algebro-geometric notion of stable vector bundles
and the existence of Hermitian-Yang-Mills connections was one of the primary
mathematical input underlying these derivations \cite{Wi1}. In all these
examples, the supersymmetric vacuum (manifold) was assumed to be one whose
geometry had no torsion. Hence
the existence of a solution on such a given manifold was mostly a topological
question and the issue of existence of appropriate solutions (obeying all
the physical requirements) often boiled down to a set of conditions on
the Chern classes of the vacuum manifold $\mathcal M$ and the
Yang-Mills Gauge connections.
In 1986, Strominger investigated the necessary and sufficient
conditions for space-time supersymmetric solutions of the heterotic string.
While considering more general space-times as solutions to the heterotic superstring
solutions, Strominger, \cite{St}, was lead to considering vacuum
configurations {\it with} torsion. He relaxed the requirement
of the 10-dimensional vacuum metric by considering that, for more general vacuum
configurations (which can sustain non-zero fluxes as well
as space-time supersymmetry), the
10-dimensional space-time be a {\it warped} product of $X_4$ and
the 6-dimensional internal space $\mathcal M$. Analyzing the
constraints imposed by the requirements of $N=1$ space-time ({\it
i.e.,} 4 dimensional) supersymmetry (and other usual
consistency requirements like anomaly cancellation), Strominger then
established that the 6-dimensional internal manifold $\mathcal
M$ should be a compact, connected, complex manifold (hereafter denoted
as $M$), such that its canonical line bundle $K_M$
is holomorphically trivial. Let $\omega \ = \
{\frac{\sqrt{-1}}{2}} g_{ij} dz^i \wedge dz^j$ be a $(1,1)$
Hermitian form on $M$, and let
$\nabla^M$ be a connection on $TM$ compatible with $\omega$. We denote its curvature by $R$. Further, let $E$ be a holomorphic vector bundle on $M$ equipped with the (gauge) connection $A$, and corresponding curvature $F_A$. It turns out that the anomaly cancellation condition then demands that the Hermitian $(1,1)$ form $\omega$ obeys an equation of the form:
$$ \sqrt{-1} \, {\partial \overline\partial} \, \omega \ = \
\frac{\alpha^\prime}{4} \ \left( {{\rm trace}(R \wedge R)} -
{{\rm trace}({F_A} \wedge {F_A})} \right)\, .$$
The consistency conditions from requirements of the space-time
supersymmetry translates into the equation:
$$ d^* \omega \ = \ \sqrt{-1} \left({\overline\partial} - {\partial}\right) \ln \Vert \Omega\Vert_\omega
$$
for the Hermitian form $\omega$ and the holomorphic 3-form $\Omega$. The previous equation may also be equivalently re-written as \cite{LY2}:
$$ d \left(\Vert \Omega \Vert_\omega \cdot \omega^2\right) \ = \ 0$$
The above equations, along with the system (constraining the Yang-Mills Gauge theory content):
$$F_A^{2,0} \ = \ F_A^{0,2} \ = \ 0, \quad \quad F \wedge \omega^2 \ = \ 0
$$
gives a complete and general solution of a superstring theory with torsion and
with a flux that allows a non-trivial dilation field (cosmological
constant). Henceforth, the above system of equations (which are derived solely from
the explicit requirements stemming from Superstring theory)
would be referred to as the {\it Strominger system} of
equations. Thus, by considering vacuum geometries with
torsion, Strominger was able to relax the requirement of $M$ to
be K{\"a}hler and consider more general complex 3-manifolds. But
the price to be paid was that the familiar
tools and methods from K\"ahler geometry could now no longer
be applied to these more general cases. Moreover, a purely topological characterization and classification of these heterotic
superstring vacua solutions ($i.e.$, the Chern classes of the bundles $E$ and
the vacuum manifold $M$), would no longer suffice.
The above results provide us with the necessary and sufficient
conditions for any heterotic superstring theory solution
(admitting space-time supersymmetry for its vacuum
configuration) to exist, but in practice, it is quite a
difficult matter to exhibit or actually explicitly construct a
solution which exists (and satisfies the Strominger equation).
Apart from its interest and usefulness in the context of string
theory, it is also of interest from a mathematical point of
view to find solutions ($i.e.$, construct the bundles $E$ with
the appropriate connection $A$ for a given manifold $M$ with
properties as defined above) of the Strominger system. In
recent years, there has been a flurry of activities surrounding
this problem of providing explicit constructive methods for
solutions of these Strominger systems (cf. \cite{AF1}, \cite{AF2}, \cite{Iv} and
references therein). The present paper explores a new and altogether different constructive scheme, based on an approach that does not require the perturbative/deformation prescription.
Further attempts at exploring more general vacuum configurations for the heterotic string with non-zero fluxes have lead to some additional corrections to the original analysis of Strominger. These come from considering (${\rm SU}(3)$)
instanton corrections at higher loops, and lead to the additional consistency conditions (for the solutions of the Strominger system) and these are:
$$ R^{2,0} \ = \ R^{0,2} \ = 0, \quad \quad \quad R
\wedge \omega^2 \ = \ 0\, .
$$
These are referred to as equations of motion. Here
we shall consider those solutions of the
Strominger system which also additionally satisfy the above
conditions.
In recent years, there has been a lot of activity, in trying to
construct actual/explicit examples which are solutions to the
above extended {\it Strominger system}. In
\cite{FTY}, Fu, Tseng and Yau have studied
the existence of smooth solutions to the Strominger system.
They proposed a perturbation method where
deformation
theory results were used to construct solutions for some $U(4)$
and $U(5)$ principal bundles. Subsequent generalizations of this
method lead to the construction of new examples (of solutions to
the Strominger system) on a class of non-K{\"a}hler
three-dimensional manifolds like $T^2$-bundles over a $K3$
surface, or $T^2$-bundles over Eguchi-Hanson spaces.
Nevertheless finding new/more examples of such solutions has
proved to be rather tricky,
and it seems that there is no general ansatz/scheme for
constructing an example; instead one has to invent specific
prescriptions and construction
procedure for every new example.
In the present work, we produce solutions of the
Strominger system from irreducible unitary representations of
any cocompact lattice in $\text{SL}(2,{\mathbb C})$.
Let $\Gamma$ be a cocompact lattice in $\text{SL}(2,{\mathbb
C})$ (meaning $\text{SL}(2,{\mathbb C})/\Gamma$ is compact), and
let $\rho\, :\, \Gamma\, \longrightarrow\,
\text{U}(n)$ be an irreducible homomorphism, meaning no
nonzero proper linear subspace of ${\mathbb C}^n$ is left
invariant by the action of the image $\rho(\Gamma)$. The compact
complex manifold $M\, :=\,\text{SL}(2,{\mathbb C})/\Gamma$ has
trivial canonical line
bundle, and $M$ is equipped with a natural
Hermitian structure. The Chern connection on $TM$ for this
Hermitian structure has the following properties:
\begin{enumerate}
\item the torsion of the connection is totally skew--symmetric, meaning
it is a section of $\bigwedge^3 TM$, and
\item the holonomy of the connection lies in $\text{SU}(3)$
\end{enumerate}
(see Corollary \ref{cor-n1}). The homomorphism $\rho$ produces a
holomorphic vector bundle over $M$ with a flat unitary
connection. This vector bundle is stable; see Proposition
\ref{prop3}. We prove that all these together produce a
solution of the Strominger system satisfying the equation of
motion; the details are in Theorem \ref{thm1}.
\section{Strominger system of equations}
We write down the Strominger system of equations in one place for
the convenience of later reference in Section \ref{se4}.
Let $M$ be a compact connected complex manifold of dimension
three such that the canonical line bundle $K_M\, :=\,
\bigwedge\nolimits^3 \Omega^1_M$ is holomorphically trivial. Let
$$
\Omega\, \in\, H^0(M,\, K_M)
$$
be a nowhere vanishing holomorphic section. Let $\omega$ be
a Hermitian $(1\, ,1)$--form on $M$. Take a connection
$\nabla^T$
on $TM$ compatible with $\omega$; its curvature
will be denoted by $R$. Let $E$ be a holomorphic
vector bundle on $M$ equipped with a connection $A$. Let $F_A$
be the curvature of $A$. Let $d^*$ be the adjoint of $d$ with respect to
$\omega$; it sends smooth $k$ forms on $M$ to $k-1$ forms.
The sextuple $(M\, ,\Omega\, ,\omega\, , \nabla^T\, , E\, ,A)$ is said
to solve
the \textit{Strominger system} if the following equations hold:
\begin{equation}\label{st1}
F^{2,0}_A\,=\, F^{0,2}_A\,=\, 0,\, F\wedge\omega^2\,=\, 0
\end{equation}
\begin{equation}\label{st2}
d^*\omega \,=\, \sqrt{-1}(\overline{\partial}- \partial)\Vert \Omega
\Vert_\omega
\end{equation}
\begin{equation}\label{st3}
d(\Vert \Omega\Vert_\omega\cdot \omega^2)\,=\, 0
\end{equation}
\begin{equation}\label{st4}
\sqrt{-1}\partial\overline{\partial}\omega\,=\, \alpha'(\text{trace}(
R\wedge R) - \text{trace}(F_A\wedge F_A)),~
\, \text{where}~\, \alpha'\,\in\,
{\mathbb C}\, .
\end{equation}
A Strominger system $(M\, ,\Omega\, ,\omega\, , E\, ,A)$ as above
is said to solve the \textit{equation of motion} if
\begin{equation}\label{st5}
R^{2,0}\,= 0\, = \, R^{0,2}\, ~ \quad {\rm and} \quad ~
\,R\wedge \omega^2\,=\, 0\, .
\end{equation}
\section{Invariant forms on $\text{SL}(2,{\mathbb C})$}\label{sec2}
Consider the complex Lie group $\text{SL}(2,{\mathbb C})$.
Let $h_0$ be the Hermitian structure on the Lie algebra
$sl(2,{\mathbb C})$ of $\text{SL}(2,{\mathbb C})$ defined by
\begin{equation}\label{h0}
h_0(A, B)\, =\, \text{trace}(AB^*)\, ,
\end{equation}
where $B^*\,=\, \overline{B}^t$. Note that the adjoint action of $\text{SU}(2)$
on $sl(2,{\mathbb C})$ preserves $h_0$.
Using the right--translation invariant vector fields on
$\text{SL}(2,{\mathbb C})$, we identify the holomorphic tangent
bundle $T\text{SL}(2,{\mathbb C})$ with the trivial vector
bundle $$\text{SL}(2,{\mathbb C})\times sl(2,{\mathbb C})\,\,
\longrightarrow\, \text{SL}(2,{\mathbb C})$$
with fiber $sl(2,{\mathbb C})$. Let $h$ be the
unique right--translation invariant Hermitian structure on
$\text{SL}(2,{\mathbb C})$ such that
$$
h\vert_{T_e\text{SL}(2,{\mathbb C})}\, =\, h_0\, ,$$
where
$e\, \in\, \text{SL}(2,{\mathbb C})$ is the identity element.
Let
\begin{equation}\label{oh}
\omega_h\, \in\, C^\infty(\text{SL}(2,{\mathbb C}), \,
\Omega^{1,1}_{\text{SL}(2,{\mathbb C})})
\end{equation}
be the K\"ahler form associated to the Hermitian structure $h$
on $\text{SL}(2,{\mathbb C})$. We note that $d\omega_h\,\not=\, 0$.
\begin{proposition}\label{prop1}
Let $\xi\, \in\, C^\infty({\rm SL}(2,{\mathbb C}),\,
\Omega^{1,0}_{{\rm SL}(2,{\mathbb C})}\oplus
\Omega^{0,1}_{{\rm SL}(2,{\mathbb C})})$ be a complex $1$--form
on ${\rm SL}(2,{\mathbb C})$ such that
\begin{itemize}
\item the right--translation action of ${\rm SL}(2,{\mathbb C})$
on itself preserves $\xi$, and
\item the left--translation action of ${\rm SU}(2)$
on ${\rm SL}(2,{\mathbb C})$ preserves $\xi$.
\end{itemize}
Then
$$
\xi\,=\, 0\, .
$$
\end{proposition}
\begin{proof}
Since the holomorphic tangent space of $\text{SL}(2,{\mathbb C})$
at $e\, \in\, \text{SL}(2,{\mathbb C})$ is identified with
$sl(2,{\mathbb C})$, the evaluation of
$\xi$ at $e$ is an element of
$sl(2,{\mathbb C})^*\bigotimes_{\mathbb R}{\mathbb C}\,=\,
(sl(2,{\mathbb C})\bigotimes_{\mathbb R}{\mathbb C})^*$;
here we identify $(T^{0,1}_e\text{SL}(2,{\mathbb C}))^*$ with
$(T^{1,0}_e\text{SL}(2,{\mathbb C}))^*$ by sending any $u$ to
its conjugate $\overline{u}$. Let
$$
\xi_0\, :=\, \xi(e) \, \in\, sl(2,{\mathbb C})^*\otimes_{\mathbb
R}{\mathbb C}
$$
be the evaluation of $\xi$ at $e$. The adjoint action of
$\text{SL}(2,{\mathbb C})$ on $sl(2,{\mathbb C})$ produces an
action of $\text{SL}(2,{\mathbb C})$ on $sl(2,{\mathbb
C})^*\bigotimes_{\mathbb R}{\mathbb C}$. In particular, we get
an action of ${\rm SU}(2)$ on $sl(2,{\mathbb
C})^*\bigotimes_{\mathbb R}{\mathbb C}$. The two given conditions
on $\xi$ imply that this action of ${\rm SU}(2)$ on
$sl(2,{\mathbb C})^*\bigotimes_{\mathbb R}{\mathbb C}$ fixes
the element $\xi_0$.
Consider the nondegenerate symmetric
bilinear pairing on $sl(2,{\mathbb C})$ defined by
\begin{equation}\label{tr}
(A\, ,B) \, \longmapsto\, \text{trace}(AB)\, .
\end{equation}
It produces an isomorphism of $sl(2,{\mathbb C})$ with
$sl(2,{\mathbb C})^*$ that is equivariant for the actions of
$\text{SL}(2,{\mathbb C})$ on $sl(2,{\mathbb C})$ and
$sl(2,{\mathbb C})^*$. Using this identification between
$sl(2,{\mathbb C})^*$ and $sl(2,{\mathbb C})$, the
above element $\xi_0$ gives an element
$$
\widetilde{\xi}_0\, \in\, \, \in\, sl(2,{\mathbb C})\otimes_{\mathbb
R}{\mathbb C}\, .
$$
We note that $\widetilde{\xi}_0$ is fixed by the adjoint action of
${\rm SU}(2)$, because
\begin{itemize}
\item $\xi_0$ is fixed by the action of ${\rm SU}(2)$ on
$sl(2,{\mathbb C})^*\otimes_{\mathbb R}{\mathbb C}$, and
\item the isomorphism between $sl(2,{\mathbb C})$ and
$sl(2,{\mathbb C})^*$ is $\text{SL}(2,{\mathbb C})$--equivariant.
\end{itemize}
But no nonzero element of $sl(2,{\mathbb C})$ is fixed by the adjoint
action of ${\rm SU}(2)$ on $sl(2,{\mathbb C})$. This implies that there
is no nonzero element of $sl(2,{\mathbb C})\bigotimes_{\mathbb R}
{\mathbb C}$ that is fixed by the action of ${\rm SU}(2)$, because
$(sl(2,{\mathbb C})\bigotimes_{\mathbb R}{\mathbb C})^{{\rm SU}(2)}\,=\,
sl(2,{\mathbb C})^{{\rm SU}(2)}\bigotimes_{\mathbb R}{\mathbb C}$.
(For an ${\rm SU}(2)$--module $W$, by $W^{{\rm SU}(2)}$ we
denote the space of invariants for the action of
${\rm SU}(2)$ on $W$.) Hence
we conclude that $\widetilde{\xi}_0\,=\,0$. So, $\xi_0\,=\, 0$. This
implies that $\xi\,=\,0$ because it is fixed by the right--translation
action of ${\rm SL}(2,{\mathbb C})$ on itself.
\end{proof}
\begin{proposition}\label{prop2}
Let $\zeta$ be a $C^\infty$ complex $4$--form on ${\rm SL}(2,{\mathbb
C})$ such that
\begin{itemize}
\item the right--translation action of ${\rm SL}(2,{\mathbb C})$
on itself preserves $\zeta$, and
\item the left--translation action of ${\rm SU}(2)$
on ${\rm SL}(2,{\mathbb C})$ preserves $\zeta$.
\end{itemize}
Then there is constant $c\, \in\, \mathbb C$ such that
$$
\zeta\,=\, c\cdot \omega_h\wedge\omega_h\, ,
$$
where $\omega_h$ is constructed in \eqref{oh}.
\end{proposition}
\begin{proof}
As in the proof of Proposition \ref{prop1}, the evaluation of
$\zeta$ at $e$ is an element
$$
\zeta_0\, \in\, \bigwedge\nolimits^4 (sl(2,{\mathbb
C})^*\otimes_{\mathbb R}{\mathbb C})\, .
$$
The adjoint action of $\text{SL}(2,{\mathbb C})$ on $sl(2,{\mathbb C})$
produces an action of $\text{SL}(2,{\mathbb C})$ on the complex line
$\bigwedge^6 (sl(2,{\mathbb C})\bigotimes_{\mathbb R}{\mathbb C})$.
Since $\text{SL}(2,{\mathbb C})$ does not have any nontrivial character,
this action of $\text{SL}(2,{\mathbb C})$ on $\bigwedge^6 (sl(2,{\mathbb
C})\bigotimes_{\mathbb R}{\mathbb C})$ is trivial.
The adjoint action of $\text{SL}(2,{\mathbb
C})$ on the Lie algebra
$sl(2,{\mathbb C})$ produces actions of $\text{SL}(2,{\mathbb
C})$ on $\bigwedge^4 (sl(2,{\mathbb C})^*\bigotimes_{\mathbb
R}{\mathbb C})$ and $\bigwedge^2(sl(2,{\mathbb
C})\bigotimes_{\mathbb R}{\mathbb C})$. Fixing a nonzero
element of the line $\bigwedge^6 (sl(2,{\mathbb C})\bigotimes_{\mathbb
R}{\mathbb
C})$, we get an $\text{SL}(2,{\mathbb C})$--equivariant isomorphism of
$\bigwedge^4 (sl(2,{\mathbb C})^*\bigotimes_{\mathbb R}{\mathbb C})$
with $\bigwedge^2 (sl(2,{\mathbb C})\bigotimes_{\mathbb
R}{\mathbb C})$.
Using this isomorphism, the above element $\zeta_0$ gives an element
\begin{equation}\label{e4}
\widehat{\zeta}_0\,\in\, \bigwedge\nolimits^2 (sl(2,{\mathbb
C})\otimes_{\mathbb R}{\mathbb C})\, .
\end{equation}
The two given conditions on $\zeta$ imply that the
element $\widehat{\zeta}_0$ in \eqref{e4} is fixed by the action
of $\text{SU}(2)$ on $\bigwedge^2(sl(2,{\mathbb C})\bigotimes_{\mathbb
R}{\mathbb C})$ (recall that ${\rm SL}(2,{\mathbb C})$
acts on $\bigwedge^2(sl(2,{\mathbb C})\bigotimes_{\mathbb
R}{\mathbb C})$).
Note that
$$
\bigwedge\nolimits^2 (sl(2,{\mathbb C})\otimes_{\mathbb R}{\mathbb
C})\,=\,
(\bigwedge\nolimits^2 sl(2,{\mathbb C}))^{\oplus 2} \oplus
(sl(2,{\mathbb C})\otimes sl(2,{\mathbb C}))\, ;
$$
this decomposition is preserved by the action of $\text{SL}(2,{\mathbb
C})$. There is no nonzero element of $\bigwedge^2 sl(2,{\mathbb C})$
preserved by the action of $\text{SU}(2)$. The subspace of
$sl(2,{\mathbb C})\otimes sl(2,{\mathbb C})$ defined by all
elements fixed pointwise by the action of $\text{SU}(2)$ is
one-dimensional, and it is generated by the element of
$\text{Sym}^2(sl(2,{\mathbb C}))\,\subset\,
sl(2,{\mathbb C})^{\otimes 2}$ given by the
nondegenerate pairing
in \eqref{tr}. This immediately implies that the space of smooth complex
$4$--forms on $\text{SL}(2,{\mathbb C})$ satisfying the two conditions
in the proposition is one dimensional.
Since the inner product $h_0$ on $sl(2,{\mathbb C})$ in \eqref{h0} is
$\text{SU}(2)$--invariant, it follows immediately that the
Hermitian
structure $h$ on $\text{SL}(2,{\mathbb C})$ is preserved by the
left--translation action of
$\text{SU}(2)$ on $\text{SL}(2,{\mathbb C})$. Hence the
K\"ahler form $\omega_h$ on $\text{SL}(2,{\mathbb C})$ is preserved by
the left--translation action
of $\text{SU}(2)$ on $\text{SL}(2,{\mathbb C})$. Recall that
$\omega_h$ is also preserved by the right--translation action
of $\text{SL}(2,{\mathbb C})$ on itself. Therefore, $\omega_h\bigwedge \omega_h$
is a nonzero complex $4$--form satisfying the two conditions in the
proposition. Since the space of smooth
complex $4$--forms on $\text{SL}(2,{\mathbb C})$ satisfying the two
conditions in the proposition is one dimensional, we now conclude
that $\zeta$ is a
constant scalar multiple of $\omega_h\bigwedge \omega_h$.
\end{proof}
\begin{lemma}\label{lem1}
The differential form $\omega_h$ in \eqref{oh} satisfies the identity
$$
d(\omega_h\wedge \omega_h)\, =\, 0\, .
$$
\end{lemma}
\begin{proof}
Using the identification between $T_e\text{SL}(2,{\mathbb C})$
and $sl(2,{\mathbb C})$, the evaluation of
the $5$--form $d(\omega^2_h)$ at $e$ is an element of
$\bigwedge^5 (sl(2,{\mathbb C})\bigotimes_{\mathbb R}{\mathbb C})^*$;
as in the proof of Proposition \ref{prop1}, we identify
$(T^{0,1}_e\text{SL}(2,{\mathbb C}))^*$ with
$(T^{1,0}_e\text{SL}(2,{\mathbb C}))^*$ by sending any $u$ to
$\overline{u}$.
As in the proof of Proposition \ref{prop2}, fixing a nonzero
element of $\bigwedge^6 (sl(2,{\mathbb C})\bigotimes_{\mathbb
R}{\mathbb
C})$, we get an $\text{SL}(2,{\mathbb C})$--equivariant isomorphism of
$\bigwedge^5 (sl(2,{\mathbb C})\bigotimes_{\mathbb R}{\mathbb C})^*$ with
$sl(2,{\mathbb C})\bigotimes_{\mathbb R}{\mathbb C}$. Using this
isomorphism, we have
\begin{equation}\label{el}
(d(\omega_h\wedge \omega_h))(e) \,\in\, sl(2,{\mathbb
C})\otimes_{\mathbb R}
{\mathbb C}\, .
\end{equation}
As noted in the proof of Proposition \ref{prop2}, the
K\"ahler form $\omega_h$ is preserved by the left--translation action
of $\text{SU}(2)$ on $\text{SL}(2,{\mathbb C})$. Consequently,
the $5$--form $d(\omega^2_h)$ is preserved by the left--translation
action of $\text{SU}(2)$ on $\text{SL}(2,{\mathbb C})$. This implies
that the element $(d(\omega^2_h))(e)$ in \eqref{el}
is fixed by the adjoint action of $\text{SU}(2)$ on
$sl(2,{\mathbb C})\bigotimes_{\mathbb R}{\mathbb C}$. From
this it follows that $(d(\omega^2_h))(e)\,=\, 0$, because
$(sl(2,{\mathbb C})\bigotimes_{\mathbb R}{\mathbb
C})^{\text{SU}(2)}\,=\, sl(2,{\mathbb C})^{\text{SU}(2)}
\bigotimes_{\mathbb R}{\mathbb C}\,=\, 0$.
Since $d\omega^2_h$ is invariant under the
right--translation action of $\text{SL}(2,{\mathbb C})$ on itself,
and $(d(\omega^2_h))(e)\,=\, 0$, we conclude that $d(\omega^2_h)\,=\,0$.
\end{proof}
As before, $T\text{SL}(2,{\mathbb C})$ is the holomorphic tangent bundle
of $T\text{SL}(2,{\mathbb C})$. Let $\nabla^h$ denote the Chern connection
on $T\text{SL}(2,{\mathbb C})$ corresponding to the Hermitian structure $h$
on $\text{SL}(2,{\mathbb C})$. The torsion of the connection $\nabla^h$
of $T\text{SL}(2,{\mathbb C})$ will be denoted by ${\mathcal T}(\nabla^h)$;
it is a $C^\infty$ section of $\Omega^{2,0}_{\text{SL}(2,{\mathbb C})}\otimes
(T\text{SL}(2,{\mathbb C}))$.
Consider the Hermitian structure $h$ on $T\text{SL}(2,{\mathbb C})$. It produces
a $C^\infty$ isomorphism
$$
h'\, :\, \Omega^{1,0}_{\text{SL}(2,{\mathbb C})}\, \longrightarrow\,
T\text{SL}(2,{\mathbb C})
$$
defined by $h(h'(w)\, ,v)\,=\, w(v)$ for $w\,\in\,
(\Omega^{1,0}_{\text{SL}(2,{\mathbb C})})_x$, $v\, \in\,T_x\text{SL}(2,{\mathbb C})$
and $x\, \in\, \text{SL}(2,{\mathbb C})$. We note that $h'$ is a conjugate
linear isomorphism. Using the isomorphism $h'$, the torsion
${\mathcal T}(\nabla^h)$ is a $C^\infty$ section of
$(\bigwedge^2(T\text{SL}(2,{\mathbb C})))\bigotimes (T\text{SL}(2,{\mathbb C}))$.
\begin{proposition}\label{prop-n1}
The torsion ${\mathcal T}(\nabla^h)\,\in\, C^\infty({\rm SL}(2,{\mathbb C}),
\, (\bigwedge\nolimits^2(T{\rm SL}(2,{\mathbb C})))\bigotimes
(T{\rm SL}(2,{\mathbb C})))$ lies in the subspace
$$
C^\infty({\rm SL}(2,{\mathbb C}),\,
\bigwedge\nolimits^3 (T{\rm SL}(2,{\mathbb C}))) \,\subset\,
C^\infty({\rm SL}(2,{\mathbb C}),\, (\bigwedge\nolimits^2
(T{\rm SL}(2,{\mathbb C})))\otimes (T{\rm SL}(2,{\mathbb C})))\, .
$$
In other words, the torsion is totally skew--symmetric.
The holonomy of the connection $\nabla^h$ lies in ${\rm SU}(3)$.
\end{proposition}
\begin{proof}
Consider the element
\begin{equation}\label{an1}
{\mathcal T}(\nabla^h)(e)\,\in\, (\bigwedge\nolimits^2 sl(2,{\mathbb C}))
\otimes sl(2,{\mathbb C})\, ,
\end{equation}
where $e\, \in\, {\rm SL}(2,{\mathbb C})$ is the identity element.
It is invariant under the adjoint action of $\text{SU}(2)$ because the
Hermitian structure $h$ is preserved by the lest translation action of
$\text{SU}(2)$ on ${\rm SL}(2,{\mathbb C})$.
Let $V_0$ be the standard two dimensional representation of $\text{SU}(2)$.
The $\text{SU}(2)$--module $sl(2,{\mathbb C})$ is isomorphic to the symmetric
product $\text{Sym}^2(V_0)$.
Therefore, the $\text{SU}(2)$--module in \eqref{an1} is isomorphic to
$(\bigwedge\nolimits^2 \text{Sym}^2(V_0))\otimes \text{Sym}^2(V_0)$. But
$$
\bigwedge\nolimits^2 \text{Sym}^2(V_0)\,=\, \text{Sym}^2(V_0)
$$
(see \cite[p. 160, Ex. 11.35]{FH}), and
$$
\text{Sym}^2(V_0)\otimes \text{Sym}^2(V_0)\,=\, \text{Sym}^4(V_0)\oplus
\text{Sym}^2(V_0) \oplus \text{Sym}^0(V_0)
$$
(see \cite[p. 151, Ex. 11.11]{FH}). Consequently,
$$
((\bigwedge\nolimits^2 \text{Sym}^2(V_0))\otimes \text{Sym}^2(V_0))^{\text{SU}(2)}
\,=\, \text{Sym}^0(V_0)\,=\, \bigwedge\nolimits^3 \text{Sym}^2(V_0)\, .
$$
Consequently, ${\mathcal T}(\nabla^h)$ is a section of
$\bigwedge\nolimits^3 (T{\rm SL}(2,{\mathbb C}))$.
This proves the first part of the proposition.
To prove the second part of the proposition, consider the Hermitian structure on
the trivial holomorphic line bundle $\bigwedge\nolimits^3 (T{\rm SL}(2,{\mathbb C}))$
induced by $h$. It is a constant Hermitian structure on the
trivial holomorphic line bundle. Hence the holonomy of the connection on
$\bigwedge\nolimits^3 (T{\rm SL}(2,{\mathbb C}))$ induced by $\nabla^h$ is
trivial. Consequently, the holonomy of the connection $\nabla^h$ lies in
the subgroup ${\rm SU}(3)\, \subset\, \text{U}(3)$.
\end{proof}
\section{A class of solutions of the Strominger
system}\label{se4}
Let
\begin{equation}\label{Ga}
\Gamma\, \subset\, \text{SL}(2,{\mathbb C})
\end{equation}
be a cocompact lattice, meaning $\Gamma$ is a closed discrete subgroup
of $\text{SL}(2,{\mathbb C})$ such that the quotient
\begin{equation}\label{M}
M\, :=\, \text{SL}(2,{\mathbb C})/\Gamma
\end{equation}
is compact. We note that $M$ is {\it not} a K\"ahler manifold.
Since the Hermitian structure $h$ on $\text{SL}(2,{\mathbb C})$
constructed in Section \ref{sec2} is invariant under the
right--translation action of $\text{SL}(2,{\mathbb C})$ on itself, we
conclude that $h$ defines a Hermitian structure on $M$. Let
$\widehat{h}$ denote the Hermitian structure on
$M$ given by $h$. Note that the pullback of
$\widehat{h}$ by the quotient map $\text{SL}(2,{\mathbb
C})\,\longrightarrow\, M$ coincides with $h$. Let
\begin{equation}\label{om}
\omega\, \in\, C^\infty(M, \, \Omega^{1,1}_M)
\end{equation}
be the K\"ahler form on $M$ associated to $\widehat{h}$. Let
\begin{equation}\label{cc}
\nabla^\omega
\end{equation}
be the Chern connection on $TM$ associated to $\omega$.
\begin{corollary}\label{cor1}
The differential form $\omega$ in \eqref{om} satisfies the identity
$$
d(\omega^2)\, =\, 0\, .
$$
\end{corollary}
\begin{proof}
Since the pullback of $\omega$ to $\text{SL}(2,{\mathbb C})$, by
the quotient map $\text{SL}(2,{\mathbb C})\, \longrightarrow\, M$,
coincides with $\omega_h$, from Lemma \ref{lem1} it follows that
$d(\omega^2)\, =\, 0$.
\end{proof}
For any torsionfree coherent analytic sheaf $F$ on $M$, let
$\det (F)$ be the determinant line bundle on $M$; see
\cite[Ch.~V, \S~6]{Ko} for the construction of the determinant bundle.
Define the \textit{degree} of $F$ to be
\begin{equation}\label{de}
{\rm degree}(F)\, :=\, \int_M \alpha(F)\wedge \omega\wedge \omega
\, \in\, {\mathbb R}\, ,
\end{equation}
where $\alpha(F)$ is any $2$--form on $M$ representing the
first Chern class $c_1(\det (F))\, \in\, H^2(M,\, {\mathbb R})$.
\begin{lemma}\label{lem2}
The degree is well defined.
\end{lemma}
\begin{proof}
Let $\alpha$ and $\beta$ be two $2$--forms on $M$ representing
$c_1(\det (F))$. So, $\alpha-\beta\,=\, d\delta$, where $\delta$ is
a smooth $1$--form on $M$. Now,
$$
\int_M \alpha\wedge \omega^2 - \int_M \beta\wedge \omega^2
\,=\, \int_M (\alpha -\beta) \wedge \omega^2
\,=\, \int_M (d\delta)\wedge \omega^2\,=\,
\int_M \delta\wedge d(\omega^2)\, =\, 0
$$
be Corollary \ref{cor1}. So, $\int_M \alpha\wedge \omega^2 \,=\, \int_M
\beta\wedge \omega^2$. Hence the degree is independent of the choice
of the differential form representing the first Chern class.
\end{proof}
Since the connection $\nabla^\omega$ is the descent of the connection $\nabla^h$
considered in Proposition \ref{prop-n1},
the following corollary is an immediate consequence of Proposition \ref{prop-n1}.
\begin{corollary}\label{cor-n1}
The torsion of the connection $\nabla^\omega$ is
a $C^\infty$ section of $\bigwedge\nolimits^3 TM$; in other words,
the torsion is totally skew--symmetric.
The holonomy of the connection $\nabla^\omega$ lies in ${\rm SU}(3)$.
\end{corollary}
We note that the torsion of the connection $\nabla^\omega$ is nonzero because
$M$ is not K\"ahler.
We choose $\Gamma$ such that there are irreducible unitary
representations of $\Gamma$.
\begin{remark}\label{rem-ex}
{\rm There are many examples of such $\Gamma$; see \cite[p. 3393,
Theorem 2.1]{La}. Note that any free nonabelian group has irreducible
unitary representations in ${\rm U}(n)$ for all $n\, \geq\, 2$.
To see this, take any two elements $g_1$ and $g_2$ of ${\rm SU}(n)$ such
that $g_1g_2g^{-1}_1g^{-1}_2$ is a generator of the center of
${\rm SU}(n)$. The subgroup of ${\rm U}(n)$ generated by
$g_1$ and $g_2$ is irreducible.}
\end{remark}
Let
\begin{equation}\label{rho}
\rho\, :\, \Gamma\, \longrightarrow\, \text{U}(n)
\end{equation}
be an irreducible representation; this means that the only
linear subspaces of ${\mathbb C}^n$ left invariant by the
action of $\rho(\Gamma)$ are $0$ and ${\mathbb C}^n$. Let
\begin{equation}\label{en}
(E\, ,\nabla)\,\longrightarrow\, M
\end{equation}
be the
unitary flat vector bundle over $M$ given by $\rho$. We briefly
recall the constructions of the vector bundle $E$ and the connection
$\nabla$ on it. Consider the trivial vector bundle
$\text{SL}(2,{\mathbb C})\times {\mathbb C}^n$ on $\text{SL}(2,{\mathbb
C})$; it has the trivial connection. This trivial connection is unitary
with respect to the standard inner product on ${\mathbb C}^n$. The
group $\Gamma$ acts on $\text{SL}(2,{\mathbb C})$ as
right--translations, and it acts on
${\mathbb C}^n$ as follows: the action of any $\gamma\, \in\, \Gamma$
sends any $v\, \in\, {\mathbb C}^n$ to $\rho(\gamma^{-1})(v)$. Consider
the diagonal action of $\Gamma$ on $\text{SL}(2,{\mathbb C})\times
{\mathbb C}^n$ constructed using these two actions. Let
$(\text{SL}(2,{\mathbb C})\times {\mathbb C}^n)/\Gamma$ be the
quotient for this action. The natural map
$$
(\text{SL}(2,{\mathbb C})\times {\mathbb C}^n)/\Gamma\,\longrightarrow
\,\text{SL}(2,{\mathbb C})/\Gamma\, = \, M
$$
is a vector bundle, which we will denote by $E$. The trivial connection
on the vector bundle $\text{SL}(2,{\mathbb C})\times {\mathbb C}^n
\,\longrightarrow\,\text{SL}(2,{\mathbb C})$ descends to a flat
unitary connection on $E$; this descended connection on $E$ will be
denoted by $\nabla$.
A holomorphic vector bundle $F$ of positive rank on $M$ is called
\textit{stable} if for every nonzero coherent analytic subsheaf $V\,
\subset\, F$ with
$\text{rank}(V)\, <\, \text{rank}(F)$, the inequality
$$
\frac{\text{degree}(V)}{\text{rank}(V)}\, <\,
\frac{\text{degree}(F)}{\text{rank}(F)}
$$
holds, where degree is defined in \eqref{de} (and Lemma \ref{lem2}).
\begin{proposition}\label{prop3}
The holomorphic vector bundle $E$ over $M$ in \eqref{en} is stable.
\end{proposition}
\begin{proof}
Since the vector bundle $E$ admits a flat connection (recall that
$\nabla$ is flat), we have $c_1(\det
(E)) \,=\, c_1(E)\,=\, 0$. Hence $\text{degree}(E)\,=\, 0$.
Since the connection $\nabla$ in \eqref{en} is unitary flat and
irreducible, the proof of Proposition 8.2 in \cite[page 176]{Ko}
gives that $E$ is stable. In fact, the proof of Proposition 8.2 in
\cite[page 176]{Ko}, which is for irreducible Einstein-Hermitian
bundles, gets simplified due to the stronger input that
$\nabla$ is unitary flat.
\end{proof}
Let $\{A_0\, ,B_0\, ,C_0\}$ be the basis of $sl(2,{\mathbb C})$
defined by
$$
A_0\,=\,
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}\, , ~ B_0\,=\,
\begin{pmatrix}
0 & 1\\
0 & 0
\end{pmatrix}\, , ~ C_0\,=\,
\begin{pmatrix}
0 & 0\\
1 & 0
\end{pmatrix}\, .
$$
Then $A_0\bigwedge B_0\bigwedge C_0$ is a nonzero element of
the line $\bigwedge^3 sl(2,{\mathbb C})$; we will call this
element $\theta_0$. Note that the adjoint action of
$\text{SL}(2,{\mathbb C})$ on $\bigwedge^3 sl(2,{\mathbb C})$ preserves
$\theta_0$, because the action of $\text{SL}(2,{\mathbb C})$ on
$\bigwedge^3 sl(2,{\mathbb C})$ is trivial (the group
$\text{SL}(2,{\mathbb C})$ does not have any nontrivial character).
The holomorphic tangent bundle $T\text{SL}(2,{\mathbb C})$ of
$\text{SL}(2,{\mathbb C})$ is
identified with the trivial vector bundle $\text{SL}(2,{\mathbb
C})\times sl(2,{\mathbb C})$ using right--translation invariant vector
fields. This identification produces a holomorphic isomorphism of the
holomorphic tangent bundle $TM$, where $M$ is
constructed in \eqref{M}, with the trivial vector
bundle $M\times sl(2,{\mathbb C})$. Using this isomorphism, the above
element $\theta_0\, \in\,
\bigwedge^3 sl(2,{\mathbb C})$ produces a trivialization of the
canonical line bundle
$$
K_M\, :=\, \bigwedge\nolimits^3 \Omega^{3,0}_M\,=\,
(\bigwedge\nolimits^3 TM)^*\, .
$$
Let
\begin{equation}\label{theta}
\theta\, \in\, H^0(M,\, K_M)
\end{equation}
be the nowhere zero holomorphic section given by $\theta_0$.
\begin{theorem}\label{thm1}
Consider the sextuple $(M\, ,\theta\, ,\omega\, ,
\nabla^\omega\, , E\, ,\nabla)$
constructed in \eqref{M}, \eqref{theta}, \eqref{om}, \eqref{cc} and
\eqref{en}. It solves the Strominger system. Moreover, it
solves the equation of motion.
\end{theorem}
\begin{proof}
Since $\nabla$ is flat, the equations in \eqref{st1} are satisfied.
The differential forms on both sides of equation \eqref{st2} are
given by
right--translation invariant $1$--forms on $\text{SL}(2,{\mathbb C})$.
Moreover, these two $1$--forms on $\text{SL}(2,{\mathbb C})$ are
invariant under the left--translation action of $\text{SU}(2)$.
Hence both sides of equation \eqref{st2} vanish identically by
Proposition \ref{prop1}.
The two form $\Vert \Omega\Vert_\omega\cdot \omega^2$ is
given by a
right--translation invariant $1$--forms on $\text{SL}(2,{\mathbb C})$
which is also fixed by the left--translation action of $\text{SU}(2)$
on $\text{SL}(2,{\mathbb C})$. Therefore, by Proposition
\ref{prop2}, the form $\Vert \Omega\Vert_\omega\cdot
\omega^2$ is a constant scalar multiple of $\omega^2$. Hence
$d(\Vert \Omega\Vert_\omega\cdot \omega^2)\,=\, 0$ by Corollary
\ref{cor1}.
The two $2$--forms on two sides of equation \eqref{st4} are
given by
right--translation invariant $2$--forms on $\text{SL}(2,{\mathbb C})$
that are fixed by the left--translation action of $\text{SU}(2)$
on $\text{SL}(2,{\mathbb C})$. Therefore, from Proposition \ref{prop2}
we conclude that \eqref{st4} holds.
Therefore, the sextuple $(M\, ,\theta\, ,\omega\, , \nabla^\omega
\, , E\, ,\nabla)$ solves the Strominger system. We will now show
that equation \eqref{st5} also holds.
Let $R(\nabla^\omega)$ be the curvature of the connection
$\nabla^\omega$ on $TM$. Since $\nabla^\omega$ is the Chern connection
for $\omega$, we have
$$
R(\nabla^\omega)^{2,0}\,=\,0\,=\, R(\nabla^\omega)^{0,2}\, .
$$
To prove that $R(\nabla^\omega)\bigwedge \omega^2\,=\, 0$, we first
note that $R(\nabla^\omega)\bigwedge \omega^2\,=\, 0$ if and only if
$$
\star_\omega (R(\nabla^\omega)\wedge \omega^2)\, =\, 0\, ,
$$
where $\star_\omega$ is the star operator on differential forms
on $M$ constructed using $\omega$; we note that $\star_\omega (R(\nabla^\omega)
\bigwedge \omega^2)$ is a $C^\infty$ section of $\text{End}(TM)\,=\,
TM \otimes (TM)^*$. Using the identification of
Consider the evaluation $\star_\omega (R(\nabla^\omega)\bigwedge
\omega^2)(e)\, \in\,\text{End}(T_e M)$ of $\star_\omega
(R(\nabla^\omega)\bigwedge \omega^2)$ at $e\, \in\,
\text{SL}(2,{\mathbb C})$. Using the identification of
$T_e M$ with $sl(2,{\mathbb C})$, it will be considered
as an element of
$$
\text{End}(sl(2,{\mathbb C}))\,=\,
sl(2,{\mathbb C})\otimes sl(2,{\mathbb C})^*\, .
$$
The space of invariants
$\text{End}(sl(2,{\mathbb C}))^{\text{SU}(2)}\, \subset\,
\text{End}(sl(2,{\mathbb C}))$ is one dimensional, and it is
generated by the identity element $\text{Id}_{sl(2,{\mathbb
C})}$. In other words, $\star_\omega
(R(\nabla^\omega)\bigwedge \omega^2)(e)$ is a scalar multiple of
$\text{Id}_{sl(2,{\mathbb C})}$. Let $\lambda\, \in\, \mathbb C$
be such that
\begin{equation}\label{la}
\star_\omega (R(\nabla^\omega)\bigwedge \omega^2)(e)\, =\, \lambda\cdot
\text{Id}_{sl(2,{\mathbb C})}\, .
\end{equation}
Since $\star_\omega
(R(\nabla^\omega)\bigwedge \omega^2)$ is given by a
section of $\text{End}(T\text{SL}(2,{\mathbb C}))$ which is invariant
under the right--translation action of $\text{SL}(2,{\mathbb C})$ on
itself, from \eqref{la} we conclude that
\begin{equation}\label{la2}
\star_\omega (R(\nabla^\omega)\bigwedge \omega^2)(e)\, =\, \lambda\cdot
\text{Id}_{TM}\, .
\end{equation}
{}From \eqref{la2} it follows immediately that
\begin{equation}\label{la3}
R(\nabla^\omega)\bigwedge \omega^2\,=\, \lambda\cdot \text{Id}_{TM}\otimes
\omega^3\, .
\end{equation}
Since $\star_\omega (R(\nabla^\omega)\bigwedge \omega^2)$ is given by a
section of $\text{End}(T\text{SL}(2,{\mathbb C}))$ which is invariant
under the right--translation action of $\text{SL}(2,{\mathbb C})$ on
itself, to prove that $\star_\omega (R(\nabla^\omega)\bigwedge \omega^2)
\,=\, 0$, it suffices to show that $\lambda\, =\, 0$, where
$\lambda$ is the scalar in \eqref{la}.
To prove that $\lambda\, =\, 0$, first that $c_1(TM)\,=\, 0$,
because $TM$ is holomorphically trivial. Hence
$$
\text{trace}(R(\nabla^\omega))\, =\, d\beta
$$
for some smooth $1$--form $\beta$ on $M$.
Therefore,
\begin{equation}\label{la4}
\int_M \text{trace}(R(\nabla^\omega))\wedge \omega^2\,=\, \int_M
(d\beta)\wedge
\omega^2\,=\, \int_M \beta\wedge d(\omega^2)\,=\, 0
\end{equation}
by Lemma \ref{lem1}. Now from \eqref{la3},
$$
\int_M \text{trace}(R(\nabla^\omega))\wedge \omega^2
\, =\, 3\lambda\cdot \int_M \omega^3\, .
$$
Since $\int_M \omega^3\, \not=\, 0$, from \eqref{la4} we
conclude that $\lambda\, =\, 0$. Therefore, \eqref{st5}
holds. This completes the proof.
\end{proof}
\medskip
\noindent
\textbf{Acknowledgements.}\, We thank Mahan Mj for pointing out
\cite{La}. The first--named author wishes to thank the Indian
Statistical Institute at Kolkata for providing hospitality
while the work was carried out.
|
2,877,628,090,122 | arxiv | \section{Introduction} \label{sec:intro}
Density functional theory (DFT) is a widely used approach
enabling \emph{ab initio} calculations of
electronic and material properties. Unlike direct approaches to
studying quantum systems through the Schr\"odinger equation where the
wavefunction is the central object, DFT uses
the electron density $n(\vb{r})$ as the fundamental
physical quantity. In principle, through the Hohenberg-Kohn
theorem\cite{hohenbergkohn}, the ground state $n(\vb{r})$ of a system uniquely
determines all of its observables. It is standard practice to use the
Kohn-Sham (KS) system\cite{kohnsham} of non-interacting fermions as a shortcut to obtaining
the ground state density, employing specially formulated potentials\cite{ceperleyCA,PBE}
that are functionals of $n(\vb{r})$ to approximate the
electron-electron interactions.
There are many techniques to find the ground state of the KS
equations, including methods which: (a) aim at direct determination
of the minimum of the KS total energy functional\cite{dirmin1,dirmin2,dirmin3}; and (b) use iterative
methods based on diagonalization of the
KS Hamiltonian in conjunction with iterative improvements of the
ground state charge density through mixing. The present work is
distinct from both these approaches. We focus on comparing our method to type (b) approaches.
For a system with $N$ electrons, the lowest $N$ eigenstates to the KS equations determine
$n(\vb{r})$, which itself appears in the KS equations through an effective single-particle potential. In general,
finding the set of $N$ eigenstates that satisfy the KS
equations involves an
iterative process known as self-consistent field (SCF) iterations
that produce successively better approximations to the solution.
In its simplest conceptualization the iterative approach involves
solving the eigenvalue problem
for an initial density distribution, then using the resulting eigenstates to
produce the next approximation to the density.
When this approach is iterated, except for the simplest systems, it rarely converges to a
self-consistent solution. In order to stabilize the SCF loops and improve
the convergence rate, various mixing schemes are typically employed. These schemes take advantage of the information
contained in multiple previous trial densities to select the
next one. A popular mixing scheme is direct inversion of the iterative subspace (DIIS), also known as Pulay
mixing\cite{PULAY1,PULAY2}.
When SCF schemes require many iterations to reach an
acceptable solution, or fail to converge, the choices are to
change the mixing scheme or its parameters, start with a
different density, or fractionally occupy states\cite{Michelini,Rabuck} which some methods implement by introducing a fictitious electronic temperature (Fermi
smearing\cite{mermin_fermi,fermismearing}). If these fail, one can resort to computationally-intensive direct
minimization methods\cite{dirmin1,dirmin2,dirmin3} to find a solution. The convergence difficulties for SCF usually arise in systems with large unit cells and
in metallic systems\cite{Anglade}, or when an excited state is desired. The small differences in eigenenergies of
the single-particle states, as well as the presence of many states
near the Fermi level, can cause very different eigenstates to be occupied
from step to step. This can lead to large variations in the density,
causing the phenomenon known as charge sloshing\cite{Kresse-charge-sloshing}
where a fluctuating charge density from step to step is observed with insufficient attenuation to reach convergence.
In the present paper we transform the time-dependent KS (TDKS) equations of time-dependent density functional
theory (TDDFT)\cite{TDDFT,runge_gross,vanleeuwen} to imaginary time\cite{KKM}. We use these equations to propagate an initial
state to very long imaginary time, refining it down to the KS state corresponding to its lowest energy component. The idea of using imaginary-time propagation (ITP) to find eigenstates is well-known, and it is frequently used to find ground state solutions to the Schr\"odinger equation describing single-particle systems with a fixed potential\cite{Bader,Lehtovaara}. Imaginary time steps have also been used to find self-consistent solutions to the Hartree-Fock equations\cite{Davies} and for nuclear energy density functional calculations\cite{Ryssens}. It has also been employed in a DFT context as an alternative to the diagonalization step to find the single-particle KS eigenstates for a fixed electronic density\cite{itp1,itp2}. However, imaginary-time evolution has yet to be examined as a stand-alone substitute to iterative density updating in solving the KS equations. In the present method both the density and wavefunction evolve together towards the ground state according to the imaginary time TDKS equations, remaining consistent with each other throughout the calculation. We discuss the theoretical foundation of the imaginary-time evolution of the
KS system, a procedure which is non-unitary,
requiring re-orthonormalization of the states at each imaginary time-step.
We show that the proof provided by van
Leeuwen\cite{vanleeuwen} for TDDFT can be extended to imaginary-time TDDFT (it-TDDFT), affirming in principle that the density of a KS system will evolve in imaginary time in the same manner as the true many-body interacting system. The imaginary-time propagation method in DFT has attractive theoretical and practical benefits when applied to systems that are challenging to study using standard methods of solving the KS equations, as we demonstrate on model systems.
We benchmark our approach by applying it to the benzene molecule
and show that it converges to the same ground state energy as other
SCF-based methods. Next, we apply our method
to systems with known difficulties in achieving
convergence. We chose to examine a copper nanocluster Cu$_{13}$
with fixed magnetization and a spin-unpolarized Ru$_{55}$ nanocluster. We show that self-consistent solutions are hard to realize in both
systems using the most popular standard approach, SCF with Pulay mixing.
In general, we find that while requiring more computation, our method is more
dependable and more autonomous compared to SCF. It provides a good alternative
to existing methodologies when the latter fail to converge in challenging systems,
or if a user wishes to find an unfamiliar system's ground state with minimal intervention; this can be particularly useful when computations are carried out in an automated fashion on large clusters of processors.
The paper is organized as follows: Section~\ref{section:method} presents the method, Section~\ref{section:considerations} extends the van Leeuwen theorem to imaginary time and discusses certain theoretical considerations that establish the method's robustness, Section~\ref{section:calculations} gives some example calculations, and Section~\ref{section:conclusion} contains our conclusions.
\section{Methodology}
\label{section:method}
\subsection{Imaginary-Time Propagation} \label{sec:ITP}
First, let us take the Hamiltonian $\hat{H}$ to be time-independent.
Under the substitution $t \to - \textrm{i} \tau$, where $\tau$ is real, the time evolution operator transforms from $e^{-\textrm{i} t\hat{H}}$ to
$e^{-\tau \hat{H}}$. When $\ket{\Phi_i}$ is an eigenstate of
$\hat{H}$, the time evolution of the eigenstate switches from rotating
its phase proportionally to its energy:
\begin{align}
\ket{\Phi_i(t)} = e^{- \textrm{i} t\hat{H}}\ket{\Phi_i} = e^{- \textrm{i} t
E_i}\ket{\Phi_i},
\end{align}
to shrinking its amplitude by an exponential factor with a rate
proportional to the energy:
\begin{align}
\ket{\Phi_i(\tau)} = e^{-\tau\hat{H}}\ket{\Phi_i} = e^{-\tau
E_i}\ket{\Phi_i}.
\end{align}
For the case of a time-dependent Hamiltonian the previous equations
still hold for infinitesimal amounts of time $\Delta t$ or $\Delta
\tau$. For an arbitrary initial wavefunction $\ket{\Psi(0)}$,
imaginary-time propagation amounts to
\begin{align}
\ket*{\ti{\Psi}(\tau)} = \sum_{i=0}^{\infty} A_i(0) e^{-\tau E_i}
\ket{\Phi_i},
\label{eq:interactingevolution}
\end{align}
where $A_i(0)$ is the amplitude of the eigenstate component initially
present. As imaginary time goes to infinity, $\tau \to \infty$, the eigenstate $\ket{\Phi_j}$ corresponding to the lowest energy eigenvalue with $A_j(0) \neq 0$ will dominate. We can choose to keep the state
$\ket{\Psi(\tau)}$ normalized by dividing by the norm $\Omega(\tau)
\equiv \sqrt{\braket*{\ti{\Psi}(\tau)}{\ti{\Psi}(\tau)}} = \sqrt{\sum_{i=0}^{\infty} \abs{A_i(0)}^2 e^{-2\tau E_i}}$,
\begin{align}
\ket{\Psi(\tau)} = \sum_{i=0}^{\infty}\frac{ A_i(0) e^{-\tau E_i}
}{\Omega(\tau)}\ket{\Phi_i},
\label{eq:interactingevolutionnormalized}
\end{align}
which then yields $\lim_{\tau\to \infty} \ket{\Psi(\tau)} =
\ket{\Phi_j}$. Note that the state $\ket{\Psi(\tau)}$ could refer to a single-particle
wavefunction $\Psi(\vb{r})$ or a many-body wavefunction $\Psi(\vb{r}_1,\ldots, {\vb{r}}_N)$; the above discussion is applicable to either case. Since an arbitrary initial state generated by randomizing the coefficients in
some basis is likely to have a nonzero ground state component,
ITP is often used to find ground state wavefunctions and energies.
\subsection{Implementation within the Kohn-Sham Formalism}
In TDDFT, starting from an initial
state, the KS system obeys the equations of
motion (in atomic units):
\begin{subequations}
\begin{align}
\textrm{i} \pdv{}{t} \phi_j( {\vb{r}},t) &= \hat{H}_{\trm{KS}}[n({\vb{r}}, t)]
\phi_j\qty({\vb{r}},t), \\ \hat{H}_{\trm{KS}}[n({\vb{r}}, t)] &\equiv
\qty[ -\frac{ \laplacian}{2 } + v_s({\vb{r}},t)],
\end{align}
\label{eq:TDKS}
\end{subequations}
with time-dependent effective potential
\begin{align}
v_s[n({\vb{r}},t)] = v({\vb{r}}) + v_{\trm{H}}\qty[n({\vb{r}},t)] +
v_{\trm{xc}}\qty[n\qty({\vb{r}},t)].
\label{eq:TDKS-effective}
\end{align}
In these expressions, $v({\vb{r}})$ is the external potential and
\begin{align}
v_{\trm{H}}\qty[n({\vb{r}},t)] &= \int \dd{{\vb{r}}'}
\frac{n\qty({\vb{r}}',t)}{\abs{{\vb{r}} -
{\vb{r}}'}}, \label{eq:TDKS-Hartree}\\ v_{\trm{xc}}\qty[n\qty({\vb{r}},t)] &=
\fdv{E_{\trm{xc}}\qty[n({\vb{r}}, t)]}{n\qty({\vb{r}},t)}, \label{eq:TDKS-exc}\\
n({\vb{r}},t) &= \sum_{j=1}^{N} \abs{\phi_j\qty({\vb{r}},t)}^2.
\label{eq:TDKS-density}
\end{align}
The Kohn-Sham time-evolution can be reformulated in terms of a
time-propagator which acts on single-particle states and is given by
\begin{align}
\ket{\phi_j(t)} &= \hat{U}\qty(t,t_0) \ket{\phi_j(t_0)}, \\ \quad
\hat{U}\qty(t,t_0) &= \hat{\mathcal{T}}\exp\qty(- \textrm{i} \int_{t_0}^{t}
\hat{H}_{\trm{KS}}[n({\vb{r}},t')] \dd{t'}),
\end{align}
where $\hat{\mathcal{T}}$ is the time-ordering operator. In imaginary
time, applying the substitution $t \to -\textrm{i} \tau$ results in
\begin{align}
\ket{\phi_j(\tau)} &= \hat{\mathcal{U}}\qty(\tau,\tau_0)
\ket{\phi_j(\tau_0)}, \\ \quad \hat{\mathcal{U}}\qty(\tau,\tau_0) &=
\hat{\mathcal{T}}_\tau \exp\qty(- \int_{\tau_0}^{\tau}
\hat{H}_{\trm{KS}}[n(\vb{r},\tau')] \dd{\tau'}),
\end{align}
where $\hat{\mathcal{T}}_\tau$ now time-orders in imaginary time. Note
that the imaginary-time propagator is not unitary.
Employing the same numerical scheme used for real time propagation of KS
states on an atomic basis\cite{kolesov2015real}, we evolve in imaginary-time the single-particle states using
finite time steps $\Delta \tau$ and we approximate the instantaneous imaginary-time
propagator with the second-order Magnus expansion:
\begin{align}
\hat{\mathcal{U}}\qty(\tau + \Delta \tau, \tau) &\approx \exp
\qty[-\Delta \tau \hat{H}_{\trm{KS}}\qty(\tau + \frac{\Delta
\tau}{2})],\\ \hat{H}_{\trm{KS}} (\tau ) &\equiv
\hat{H}_{\trm{KS}}[n({\vb{r}}, \tau )].
\end{align}
The Hamiltonian at the midpoint is approximated as the average of the
Hamiltonians at $\tau_i$ and $\tau_{i+1}$,
$\hat{H}_{\trm{KS}}\qty(\tau_i + \frac{\Delta \tau}{2}) \approx
\frac{1}{2}\qty[\hat{H}_{\trm{KS}}(\tau_i) +
\hat{H}_{\trm{KS}}(\tau_{i+1})]$. Each step is iterated to
self-consistency in order to make use of the Hamiltonian at
$\tau_{i+1}$. We use the Pad\'e rational polynomial approximation of
arbitrary degree to obtain the general matrix exponential. Further
details of the numerical propagation can be found in
our earlier work\cite{kolesov2015real}, which describes TDAP-2.0, a TDDFT code we used,
built on top of SIESTA\cite{siesta02}, a DFT package which uses strictly localized
basis sets. While the midpoint Hamiltonian greatly aids stability
and energy conservation in real time propagation, in practice we have
found that for imaginary-time propagation we can just use the first
step in the iterative procedure, which simply applies the
approximation $\hat{H}_{\trm{KS}}\qty(\tau_i + \frac{\Delta \tau}{2})
\approx \hat{H}_{\trm{KS}}\qty(\tau_i)$. This explicit propagation
is faster since the
Hamiltonian only needs to be evaluated once per propagation step, and
the effect on the size of the maximum stable time-step appears
negligible compared to the implicit method using the midpoint Hamiltonian.
This is expected since imaginary-time propagation is
inherently more stable than the real-time propagation the TDAP-2.0
code was originally designed to solve.
Because the imaginary-time propagator is not unitary, the
single-particle states lose their normalization and generally cease to
be orthogonal. The simple expression for density in
Eq.~(\ref{eq:TDKS-density}) becomes more complicated if the
single-particle states $\phi_j$ are non-orthonormal. It is
convenient to reorthonormalize the single-particle states at each time
step. The details of how the orthogonalization
is achieved do not affect the physics, as we show in Section~\ref{ssec:orthon}. We use the modified Gram-Schmidt algorithm to
orthonormalize the states.
While we employ a localized atomic basis for our calculations, the method we propose
is independent of the basis used to represent the Kohn-Sham orbitals, and can easily be implemented in other popular bases, like plane waves or gaussians.
\section{Theoretical Considerations}
\label{section:considerations}
\subsection{Van Leeuwen Theorem in Imaginary Time} \label{ssec:vlt}
The van Leeuwen theorem states that a time-dependent particle density
$n({\vb{r}},t)$ belonging to a many-particle system with two-particle
interaction $\hat{W}$ can always be reproduced by a unique (up to an
additive purely time-dependent constant) external potential
$v'({\vb{r}},t)$ in another many-particle system that uses a different
two-particle interaction $\hat{W}'$, under the mild restriction that
the density has to be analytic in time\cite{vanleeuwen}. If we choose the two-particle
interaction in this other system to be $\hat{W}' = 0$, the theorem guarantees the existence of
the effective potential $v_s({\vb{r}},t)$ for a Kohn-Sham system that
reproduces the same time-dependent density as the interacting system
of interest. Here we point out the modifications to the original theorem
in order to make it compatible with imaginary-time evolution.
A complex $t$ value does not pose any problems
with the operations performed in the original proof, where $t$ appears in some
time derivatives but otherwise is treated as a parameter. We add time-dependent uniform potentials $\lambda(t)$ and $\lambda'(t)$ to the unprimed and primed Hamiltonians to conserve the norm of the wavefunctions. The origin of these terms will be discussed in the next section.
The Hamiltonian $\hat{H}$ of a finite many-particle system is then given by
\begin{align}
\hat{H}(t) = \hat{T} + \hat{V}(t) + \hat{W} + \lambda(t),
\end{align}
expressed in terms of creation and annihilation operators
\begin{subequations}
\begin{align}
\hat{T} &= - \frac{1}{2} \int \dd[3]{\vb{r}} \hat{\psi}^\dag(\vb{r})
\laplacian \hat{\psi}(\vb{r}), \\ \hat{V}(t) &= \int \dd[3]{\vb{r}}
v(\vb{r}t) \hat{\psi}^\dag\qty(\vb{r})
\hat{\psi}(\vb{r}), \label{eq:Vdef}\\ \hat{W} &= \int \dd[3]{\vb{r}}
\dd[3]{\vb{r}'} w\qty(\abs{ \vb{r} - \vb{r}'})
\hat{\psi}^\dag(\vb{r}) \hat{\psi}^\dag(\vb{r}') \hat{\psi}(\vb{r}')
\hat{\psi}({\vb{r}}).
\end{align}
\end{subequations}
Since $\lambda(t)$ is not an operator, it does not affect any of the commutators involving
$\hat{H}(t)$ in the various Heisenberg equations of motion underpinning the proof of the van Leeuwen theorem. There is only one detail to
note, regarding the freedom to add an arbitrary $C(t)$ to the
potential of the primed system, $v'({\vb{r}},t)$, in the original
proof. From Eq.~(\ref{eq:Vdef}) a time-dependent
constant in the potential modifies the Hamiltonian by an additional term $C(t)\hat{N}$,
where $\hat{N}$ is the number operator. For the systems of interest,
the number of particles $N$ is fixed so $C(t)N$
is a time-dependent uniform potential like $\lambda'(t)$, which means that a
norm-conserving $\lambda'(t)$ will cancel any effect from the
choice of $C(t)$. Thus, with $\lambda(t)$ and $\lambda'(t)$
chosen to ensure that the norm of states in both the unprimed and
primed systems is held at unity, the van Leeuwen theorem holds in
imaginary time. This is a powerful result since it allows us to think about imaginary-time propagation in the Kohn-Sham system in terms of what it does in the real system,
allowing the Wick-rotation connections from quantum mechanics
to statistical mechanics to be employed. For example, it justifies the
use of the Kohn-Sham system as a stand-in for the interacting system
in our calculations performed for imaginary time path integrals\cite{KKM}.
\subsection{Maintaining Orthonormalization} \label{ssec:orthon}
Orthonormalization of the single-particle states is equivalent to adding a purely time-dependent function $\lambda(t)$ to the many-body
Hamiltonian. This takes care of
holding the wavefunction normalized, both in the interacting and
Kohn-Sham systems, as well as accounting for the orthogonalization
step we use in the Kohn-Sham state propagation.
We first consider the interacting system. In real
time propagation, the choice of $\lambda(t)$ does not affect the
dynamics of density since this spatially-constant offset in energy
only results in changing the phase of the wavefunction:
\begin{subequations}
\begin{align}
\ket{\Psi(t)} &= \hat{U}\qty(t,t_0)
\ket{\Psi(t_0)},\nonumber\\ \quad \hat{U}\qty(t,t_0) &=
\hat{\mathcal{T}}\exp\qty(- \textrm{i} \int_{t_0}^{t} \hat{H}(t') +
\lambda(t') \dd{t'})\nonumber \\ &= \hat U_{\lambda}(t,t_0)
\hskip 0.05 in \hat{\mathcal{T}} \hskip 0.05 in \hat U_{\hat H}(t,t_0),\\
\hat U_{\lambda}(t,t_0) &\equiv
\exp\qty(- \textrm{i} \int_{t_0}^{t}
\lambda(t') \dd{t'}), \\
\hat U_{\hat H}(t,t_0) &\equiv
\exp\qty(- \textrm{i} \int_{t_0}^{t}
\hat{H}(t') \dd{t'}).
\end{align}
\end{subequations}
In imaginary-time propagation, $\lambda(\tau)$ modifies
the imaginary-time propagator $\hat{\mathcal U}\qty(\tau,\tau_0)$ by a
time dependent magnitude,
\begin{subequations}
\begin{align}
\hat {\mathcal U}\qty(\tau,\tau_0) &= \hat {\mathcal U}_{\lambda}(\tau,\tau_0)
\hskip 0.05 in\hat{\mathcal{T}}_{\tau} \hskip 0.05 in \hat {\mathcal U}_{\hat H}(\tau,\tau_0), \label{eq:ITPwithC}\\
\hat {\mathcal U}_{\lambda}(\tau,\tau_0) &\equiv
\exp\qty(- \int_{\tau_0}^{\tau}
\lambda(\tau') \dd{\tau'}), \\
\hat {\mathcal U}_{\hat H}(\tau,\tau_0) &\equiv
\exp\qty(- \int_{\tau_0}^{\tau}
\hat{H}(\tau') \dd{\tau'}).
\end{align}
\end{subequations}
If $\lambda(\tau)$ is arbitrary, the norm of the
wavefunction will change in time, incorrectly scaling the expectation
values of observables like density and energy. The norm of the wavefunction
can be held fixed by choosing
$\lambda(\tau)$ to counteract the norm-altering effect of
$\hat{\mathcal{T}}_{\tau}\exp\qty(- \int_{\tau_0}^{\tau}
\hat{H}(\tau') \dd{\tau'})$ when it acts on $\ket{\Psi(\tau_0)}$. Note
that such a $\lambda(\tau)$ will also depend on the starting
state. For example, in the time-independent Hamiltonian case presented
in Section~\ref{sec:ITP}, from Eq.~(\ref{eq:interactingevolutionnormalized})
\begin{align}
\hat {\mathcal U}_{\lambda}(\tau,\tau_0) =
\qty[\sum_{j=0}^{\infty} \abs{A_j(0)}^2 e^{-2\tau E_j}]^{-1/2}
\end{align}
which implies
\begin{align}
\lambda(\tau) = \frac{1}{2} \dv{}{\tau} \ln
\qty[\sum_{j=0}^{\infty} \abs{A_j(0)}^2 e^{-2\tau E_j}] = -
\ev{E(\tau)},
\end{align}
that is, to keep the wavefunction normalized, $\lambda(\tau)$ is such that
the energies of the Hamiltonian are measured relative to $\ev{E(\tau)}$.
This result holds more generally for time-dependent
Hamiltonians as well, which can be shown by using
$\hat{\mathcal{U}}\qty(\tau,\tau_0)$ from Eq.~(\ref{eq:ITPwithC}) and
differentiating the norm-conserving equation $1 =
\bra{\Psi(\tau_0)}\hat{\mathcal{U}}^{\dag}\qty(\tau, \tau_0)
\hat{\mathcal{U}}\qty(\tau,\tau_0) \ket{\Psi(\tau_0)}$ to solve for
$\lambda(\tau)$. We will assume that such a $\lambda(\tau)$ is used in
the interacting system so that the system always remains
normalized.
In the Kohn-Sham system the propagator is given by
\begin{align}
\hat{\mathcal U}\qty(\tau,\tau_0) &= \hat {\mathcal U}_{\lambda_{\trm{KS}}}(\tau,\tau_0)
\hskip 0.05 in \hat{\mathcal{T}}_\tau \hskip 0.05 in \hat {\mathcal U}_{\hat H_{\trm{KS}}}(\tau,\tau_0).
\end{align}
where $\hat{H}_{\trm{KS}}$ acts on the entire
Kohn-Sham many-body wavefunction $\ket{\Phi}$ through its
constituent single-particle states $\ket{\phi_j}$, see
Eq.~(\ref{eq:TDKS}). In general $\lambda_{\trm{KS}}\qty(\tau)$
differs from the constant $\lambda(\tau)$ of the interacting
system, and in addition to normalizing the many-body state, it can
account for orthonormalization of the constituent single-particle states.
Orthonormalization
of the occupied single-particle states is an invertible transformation
as it preserves the subspace spanned by these linearly-independent states.
Representing the orthonormalization by matrix $\vb{S}$
and given a single-particle Slater
determinant wavefunction $\Phi\qty(\vb{r}_1,\vb{r}_2,\ldots,\vb{r}_N)$, it is straightforward to show that orthonormalization results in
$\ti{\Phi}\qty({\vb{r}}_1,{\vb{r}}_2,\ldots,{\vb{r}}_N) =\Phi\qty({\vb{r}}_1,{\vb{r}}_2,\ldots,{\vb{r}}_N) \det\vb{S} $.
Thus, the orthonormalization step merely amounts to changing the phase and rescaling the many-body wavefunction.
At the starting time $\tau_0$, we assume the Kohn-Sham wavefunction
is properly normalized. Following the application of the imaginary-time
propagator up to a particular time $\tau$, we represent a
particular orthonormalization of the single-particle states by an
invertible transformation ${\vb{S}}(\tau)$. In order for
$\lambda_{\trm{KS}}(\tau)$ to act like the orthonormalization
procedure, we require that
\begin{align}
\hat {\mathcal U}_{\lambda_{\trm{KS}}}(\tau,\tau_0)
\hskip 0.05 in &= \det \vb{S}(\tau).
\label{eq:lambdaconstraint}
\end{align}
Note that $\abs{\det S(\tau)}$ will be continuous since it is the
reciprocal of the norm of the unnormalized propagated
wavefunction. The phase of $\det S(\tau)$ is not important
since it changes the phase of the wavefunction, which
will not affect the density. We can therefore use any
orthonormalization procedure at each time-step without concern about
the continuity of the phase, and a purely real
$\lambda_{\trm{KS}}(\tau)$ satisfying $\hat {\mathcal U}_{\lambda_{\trm{KS}}}(\tau,\tau_0)
= \abs{\det \vb{S}(\tau)}$ for
all $\tau > \tau_0$ is guaranteed to exist.
The above definitions for norm-conserving $\lambda(\tau)$ and $\lambda_{\trm{KS}}(\tau)$ conclude the proof of the imaginary time extension to the van Leeuwen theorem presented in Section~\ref{ssec:vlt}.
\subsection{Monotonically Decreasing Energy} \label{ssec:monotonedecrease}
In the Kohn-Sham system the
Hamiltonian depends on the density, and thus will in general have
eigenenergies and eigenvalues that depend on time. In particular,
for the density at time $\tau_\ell$, $n({\vb{r}}, \tau_\ell)$, we are
considering a quantum system with the Hamiltonian
$\hat{H}_{\trm{KS}}[n({\vb{r}},\tau_\ell)]$. By propagating the state of
interest in imaginary time using this instantaneous Hamiltonian, we
are amplifying the low-energy eigenstates of the current Hamiltonian
$\hat{H}_{\trm{KS}}[n(\tau_\ell)]$, which in general are different
than the low-energy eigenstates of the new Hamiltonian,
$\hat{H}_{\trm{KS}}[n(\tau_{\ell+1})]$, and the resultant state could have a higher energy than the previous state. A good example of this
is the commonly-observed divergence of SCF loops without a mixing
scheme: the $N$-lowest eigenstates of the Hamiltonian $\hat{H}_{\trm{KS}}[n_i]$ are directly used to compute the next density $n_{i+1}$.
This also reveals an interesting limiting case of it-TDDFT.
If a KS state is propagated to infinite imaginary time before the density
used in the instantaneous Hamiltonian $\hat{H}_{\trm{KS}}[n(\tau_\ell)]$
is updated, the propagated state will become the ground state of the
present Hamiltonian, which is equivalent to populating the $N$-lowest
eigenstates of $\hat{H}_{\trm{KS}}[n(\tau_\ell)]$.
In this way basic SCF can be thought of as it-TDDFT
with infinitely large time-steps when using explicit
propagation. Indeed, if the
time-step in it-TDDFT is taken to be too large, the total energy will
diverge, just like in SCF performed without a mixing scheme.
With a reasonable time-step (usually around $1 \hbar/\trm{Ry}$ or
smaller), it-TDDFT monotonically decreases the total energy of the system. The van
Leeuwen theorem, which connects the KS system to the interacting system, provides the theoretical backbone for this
result. While propagation of the Kohn-Sham system is
complicated by the dependence on density, in the true interacting
system the evolution in imaginary time has the simple form given in
Eq.~(\ref{eq:interactingevolutionnormalized}).
\subsection{it-TDDFT as an Alternative Theoretical Foundation for Stationary States in DFT}
The first step in the majority of DFT calculations is to find a
density corresponding to a stationary state. A stationary state
is an eigenstate of the Hamiltonian, or
equivalently, a state that only changes by a phase when evolved in
real time or by a multiplicative factor when evolved in imaginary
time. Only the first definition is used in KS systems, as it is implicitly assumed by SCF schemes. In systems that are difficult to
converge with SCF, owing to their size or metallic character, the
second definition becomes more useful, and it can be applied through the it-TDDFT method.
The KS equations are set as an eigenvalue
problem, and thus use the first definition. Once a density $n({\vb{r}})$ is found such that a choice of
$N$ of the single-particle eigenfunctions $\phi_j({\vb{r}})$ reproduces
the same density through Eq.~(\ref{eq:TDKS-density}), a stationary
state has been determined. SCF is used to find ground states, where the $N$ lowest-energy eigenstates are
chosen, and $\Delta$SCF\cite{deltaSCF_ass}, where a different selection of $N$
single-particle eigenstates is chosen, is used to find excited
states. For small systems, insulating systems, and systems
with low degeneracy of single-particle states, after a few steps of SCF, the eigenstates rarely change order when
sorted by energy from one step to the next. This means that occupied
single-particle states have similar character to
those from the step before, so the density does not change drastically. In these cases SCF converges well so using the eigenstate definition of stationary states is sensible. However, in large systems and
in metallic systems, or if an excited state is desired, the above conditions might not hold, leading to charge sloshing. In principle, the KS equations can still be used to verify a stationary state if the density is perfectly converged. In practice, this definition is inadequate in these
difficult systems since a suitable approximate density
could appear to be far from convergence if the wrong KS eigenstates
are occupied, due to the next SCF step returning a very different
density from the one given. In addition, this makes it challenging to
determine the quality of a non-converging density. For example,
in Section~\ref{section:calculations} we examine the performance of SCF on a
ruthenium nanocluster, where we show that some
non-converged densities give a reasonable energy estimate for the
ground state, while others are incorrect.
To address convergence issues, DFT calculations of metallic systems
and systems with high single-particle energy degeneracy are often
performed with electronic smearing\cite{Rabuck}, where states near the Fermi
level are given fractional occupations to simulate nonzero electronic
temperature. This mitigates the problem by ensuring that
states near each other in energy have similar fractional
occupation. Smearing adds an entropic contribution to the energy, so a
balance between obtaining an accurate energy and ease of convergence
has to be struck. Electronic smearing is a
computational tool and not intended to be an accurate representation
of the effects of temperature, so it should be
incrementally reduced until the solution with no smearing is
achieved\cite{Michelini}. In fact, cases
have been found where even small amounts of electronic smearing
produce significantly different results from the same calculation
performed with integer occupations, such as a HOMO-LUMO
gap energy that differ by one order of magnitude\cite{Basiuk}.
As we show in the ruthenium nanocluster system in Section~\ref{section:calculations}, achieving
convergence while applying electronic smearing can still require
finesse and guesswork.
In systems where SCF convergence is hard to attain, instead of using
the KS equations to define a stationary state, we can use a state's invariance under
imaginary-time evolution. If a KS wavefunction stays constant when
propagated in imaginary time, then its single-particle states span the
same subspace as a set of eigenstates which solve the KS equations. The converse is true as well,
namely that a set of $N$ KS eigenstates satisfying the KS equations self-consistently will be invariant under imaginary-time evolution, ignoring the possibly
changing norm which can be corrected (as discussed in Sec. \ref{ssec:orthon}). Thus, finding a KS many-body state $\ket{\Phi(\tau_0)}$ such that $\ket{\Phi(\tau)} = \hat{\mathcal{U}}(\tau,\tau_0) \ket{\Phi\qty(\tau_0)} =
\ket{\Phi(\tau_0)}$, where the Hamiltonian in the propagator $\hat{\mathcal{U}}$ contains orthonormality-preserving
$\lambda(\tau)$, is equivalent to finding a set of $N$
single-particle states that satisfy the KS equations.
This definition has a few
advantages. In systems where the single-particle states are
close in energy, occupation ambiguities and charge-sloshing issues are eliminated because it-TDDFT follows the occupied orbitals throughout their
evolution. Additionally, it-TDDFT handles systems with degenerate states well since an initial state will converge to one of the states within the degenerate stationary-state subspace without being affected by the unoccupied states of identical energy.
\subsection{Practical Advantages of it-TDDFT}
\label{ssec:practicaladvantages}
One convenience afforded by it-TDDFT is that a user only needs to
choose a single parameter, the time-step, when attempting to converge a system. Compare this to the various parameters usually required for
SCF with a mixing scheme: the number of past states to mix, the mixing weight, and the amount of electronic smearing, to name a few. When encountering a set of nonconvergent parameters, it is often unclear which direction to change each parameter for a better chance at convergence. In addition, there are systems where different
stationary states can be obtained for slight variations in the mixing parameters, as shown
in the case of a Cu$_{13}$ cluster in Section~\ref{section:calculations}. In
contrast, convergence in the it-TDDFT method is not very sensitive to the
choice of time-step, and any choice smaller than a convergent time-step will
lead to the same density trajectory in imaginary time given the same
starting state. This property allows us to eliminate this
parameter choice if desired through algorithms that automatically
adjust the time-step on the fly. We found that the simple procedure of increasing the time-step while total energy decreases, and decreasing the time-step when it does not, can perform nearly as well as using a static convergent time-step that is as large as possible.
Another practical advantage of using imaginary-time evolution is that not-yet-converged states still have physical meaning. The single-particle states and the electronic density used in the KS Hamiltonian are self-consistent at all times, and in principle this density trajectory is equal to the imaginary-time evolving density of the interacting system by the van Leeuwen theorem. Through this connection, the partially-converged KS state corresponds to a superposition of a dominant ground state component and a few low-amplitude excited states. As such, even before the it-TDDFT ground state calculation has converged according to user-specified energy or density tolerance criteria, approximate ground state observables can be computed. This property allows for preliminary calculations of band structure, energies, optical properties, or atomic forces while the ground state calculation continues to be refined. In contrast, there are no guarantees of validity for observables calculated from intermediate states produced in a SCF loop since they are not self-consistent and can be far from the correct KS ground state.
When an SCF loop ceases to make progress, not much is gained aside from the knowledge that the particular set of mixing parameters did not lead to convergence. It could take a subtantial amount of tweaking of these parameters before chancing upon a set that works, consuming time and computational resources. This highlights another strength of it-TDDFT: the calculation time used will always improve the quality of the state at hand. It is also straightforward to continue a calculation from the last saved state, enabling incremental improvement of an approximate stationary state over multiple runs.
In our discussion we have assumed that we are performing DFT with a
Kohn-Sham system, which uses a single Slater determinant. It is
possible to apply it-TDDFT for finding stationary states in other
approaches which use linear combinations of Slater determinants, or
ensemble DFT, since a DFT model that reproduces the same density
trajectory as the true interacting system will evolve an arbitrary
starting state into a stationary state when propagated in
imaginary time.
\section{Example Calculations}
\label{section:calculations}
In order to compare two different densities, it is useful to have a
measure of distance. We will
use half the $L^1$ distance for its intuitive physical meaning:
\begin{align}
D[n,n_0] \equiv \frac{1}{2}d_1(n,n_0) = \frac{1}{2} \int
\abs{n({\vb{r}}) - n_0({\vb{r}})} \dd[3]{\vb{r}},
\label{eq:ourdistance}
\end{align}
which can be interpreted as the number of electrons in the wrong place
relative to the reference density $n_0$. This can be seen by using the
fact that both densities integrate to the same value, the total number
of electrons. The integral of the
absolute value of the density difference over all space adds up the
excess density and the negative of the lacking density, both
contributing equally, so the $1/2$ factor is needed to obtain the number
of electrons out of place.
As a demonstration of using it-TDDFT to determine a ground state, we apply the method to a benzene molecule and show that it produces the same density and energy as a standard SCF calculation. We initialize the single-particle electronic states by drawing basis
coefficients from a uniform distribution and orthonormalizing the single-particle
wavefunctions. Propagating this initial state in imaginary time, we
obtain the same ground state as that determined by an SCF
approach with Pulay mixing. The Kohn-Sham total energy $E$ and the density
distance $D[n,n_0]$ of the propagated state are plotted as a function of
imaginary time in Fig.~\ref{fig:benzene}, both relative to
the SCF-determined ground state. These quantities tend to
zero, showing that it-TDDFT indeed produces a Kohn-Sham state that has the same energy and
density as the ground state determined with an SCF
approach. As an additional check, running SCF at the end
of the imaginary-time propagation produces SCF convergence after the
first step.
\begin{figure}
\centering
\includegraphics[width=3. in]{images/merged_benzene_plot.pdf}
\caption{Determining the ground state energy $E$ and density distance $D[n,n_0]$ of benzene using it-TDDFT, relative to SCF results. The time
step used was $10.0\,\trm{as}$. Positive and negative isosurfaces of the density
difference $n({\vb{r}}) - n_0({\vb{r}})$ at fixed values are shown at various points in the propagation.}
\label{fig:benzene}
\end{figure}
\begin{figure}
\centering \includegraphics[width=3. in]{images/cu13itp.pdf}
\caption{Electronic energy of Cu$_{13}$ with fixed spin polarization
$+1/2$, relative to the lowest energy obtained with this fixed
polarization. Each curve is an energy trajectory produced by
propagating a random initial state in imaginary time, plotted
versus the wall time and colored according to the final state obtained. The right inset plot is the spin magnetization density of one such state with spin up and spin down designated by blue and red respectively. The left inset plot is a spin down isosurface to illustrate the five-fold symmetry of the lowest energy states. The horizontal dashed lines show the relative energies of
converged states obtained using SCF and Pulay mixing with
different parameters as detailed in Fig.~\ref{fig:barcu13} and Table~\ref{tab:cu13}.}
\label{fig:cu13}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=4.3 in]{images/barcu13.pdf}
\caption{Relative total energy of Cu$_{13}$ cluster with fixed total spin $1/2$, obtained by SCF with
Pulay mixing involving $n=5$ or $8$ previous densities and different mixing weights. The reference value of the energy is that obtained with imaginary-time propagation. The energies here appear in Fig.~\ref{fig:cu13} as horizontal dashed lines.}
\label{fig:barcu13}
\end{figure*}
As our next example we consider the Cu$_{13}$ nanocluster.
Hoyt \textit{et al.}\cite{hoyt} simulated this system in its ground state magnetization of $m=5 \mu_B$ and in an excited state with magnetization
$m=3 \mu_B$, commenting that the $m= 1 \mu_B$ excited state
was tricky to converge, making it a good candidate for our it-TDDFT method. In Fig.~\ref{fig:cu13} and \ref{fig:barcu13}, we present the main results of our computations for the self-consistent KS states with $m = 1 \mu_B$ magnetization, which has total spin $1/2$. Additional information can be found in Table~\ref{tab:cu13}.
SCF has trouble finding the minimum energy states in this fixed-spin system, due to the fact that there
are five degenerate states\cite{hoyt}. Fig.~\ref{fig:cu13} shows the energy
trajectories in imaginary time of 12 different random starting
configurations, and each of these converges to one of five
lowest-energy states. To help identify the equality of final states, for both the it-TDDFT and SCF runs, we also computed the density distances between each combination of obtained states.
States with energies within $10^{-2}\,\textrm{meV}$ and a density distance of less than $(1/100)e$ of each other were considered equal. The ground state of the system, with magnetization $m = 5 \mu_B$, contains five degenerate valence electrons which have unpaired spins\cite{hoyt}. To obtain a magnetization of $m = 1 \mu_B$, four of the electrons need to pair spins, leaving five possibilities of which electron remains unpaired. Fig.~\ref{fig:cu13} shows the magnetization density, defined as the difference between spin up and spin down electron density, of one of these five lowest-energy states. There are five equivalent ways to place such a magnetization density on the icosahedral shape of the copper cluster, explaining the degeneracy. The visible differences of up to $0.1\,\textrm{meV}$ in the energies of these degenerate states are due to the discretization effects of the real space grid breaking the icosahedral symmetry.
Our approach is better at finding the lowest-energy
states compared to SCF, which for different mixing
parameters often converges to other excited states of spin $+1/2$ (the energies of
which are shown as dashed lines in the figure). In Fig.~\ref{fig:barcu13}, we show the electronic energies of the states obtained using SCF with Pulay mixing, for various mixing parameter choices. Even small changes in the mixing parameters can result in a different final state. This happens in metallic systems where the gap between occupied and unoccupied states is small, causing SCF to find a low-lying excited state.
For our final example of applying our method we consider the Ru$_{55}$ nanocluster.
Montemore \textit{et al.}\cite{montemore} studied catalysis on the surface of this structure, and found that the spin-unpolarized ground state calculation was difficult to converge with SCF.
In Table~\ref{tab:ru55}, we show the results of using SCF with Pulay mixing to find the ground state of the spin-unpolarized Ru$_{55}$ cluster. The number of past densities to mix was kept at $n = 5$ for all trials and mixing weights ranging from $0.02$ to $0.20$ were tested. We used Fermi electronic smearing for half the trials with $T = 300\,\textrm{K}$. For each run, we list the energy of the final step relative to the energy calculated with it-TDDFT, the density difference $\Delta \rho_{\textrm{max}}$, and whether the run converged or not. The density difference $\Delta \rho_{\textrm{max}}$ refers to the maximum elementwise difference in the density matrix between the final and penultimate step and is typically used to determine convergence. We used the criterion $\Delta \rho_{\textrm{max}} < 10^{-6}$. Only a few mixing weights result in convergence, namely the smallest ones with $T = 300\,\textrm{K}$ of smearing. In these runs, the entropic energy contribution is $78\,\textrm{meV}$ relative to the ground state energy. None of the runs without electronic smearing converge, despite the fact that some parameter configurations obtain energies similar to the ground state energy. The states resulting from unconverged runs generally should not be trusted as they may not be acceptable approximations to actual solutions, which have to satisfy the KS equations self-consistently. For example, in Table~\ref{tab:ru55}, examining the row with mixing weight $0.10$ and comparing the $T = 0\,\textrm{K}$ and $T = 300\,\textrm{K}$ cases, we find that even though $\Delta \rho_\textrm{max} = 0.589$ in the latter run is smaller than $\Delta \rho_\textrm{max} = 0.702$ in the former, the energy of the state obtained with $T = 300\,\textrm{K}$ is more than $6\,\textrm{eV}$ off from the correct ground state energy while the $T = 0\,\textrm{K}$ run is only about $0.006\,\textrm{eV}$ off. Applying our it-TDDFT method to the Ru$_{55}$ cluster produces the ground state without issue, as illustrated in Fig.~\ref{fig:ru55}. The observed monotonically decreasing energy and density distance $D[n,n_0]$ show consistent progress, as we expect from the theory.
\begin{table*}
\centering
{\textbf{SCF with Pulay Mixing for Ru$_{55}$, Spin Unpolarized} } \\
\begin{tabular}{c|rrr|rrr}
\toprule[1.5pt]
& \multicolumn{3}{c|}{$T = 0\,\textrm{K}$} & \multicolumn{3}{c}{$T = 300\,\textrm{K}$} \\
Weight & Energy (meV) & $\Delta \rho_{\trm{max}} \,$ & Converged & Energy(meV) & $\Delta \rho_{\trm{max}} \;$ & Converged \\
\midrule[1pt]
$0.02$ & $-0.012 \phantom{000}$ & $0.086$ & No & $78.388\phantom{0}$ & $9.9 \times 10^{-7}$ & Yes \\
$0.04$ & $-0.004\phantom{000}$ & $0.052$ & No & $78.380\phantom{0}$ & $9.8 \times 10^{-7}$ & Yes \\
$0.06$ & $0.069\phantom{000} $ & $0.227$ & No & $78.448\phantom{0}$ &$5.3 \times 10^{-7}$ & Yes \\
$0.08$ & $0.395\phantom{000} $ & $0.185$ & No & $78.408\phantom{0}$ & $9.0 \times 10^{-7}$ & Yes \\
$0.10$ & $5.538\phantom{000}$ & $0.702$ & No & $6.38 \times 10^{3}\phantom{0}$ & $0.589$ & No \\
$0.12$ & $73.276\phantom{000}$ & $0.730$ & No & $7.15 \times 10^{4}\phantom{0}$ & $0.699$ & No \\
$0.14$ & $148.995\phantom{000}$ & $1.388$ & No & $1.46 \times 10^{5}\phantom{0}$ & $1.391$ & No \\
$0.16$ & $6.385\phantom{000}$ & $0.589$ & No & $2.35 \times 10^{5}\phantom{0}$ & $1.409$ & No \\
$0.18$ & $295.051\phantom{000}$ & $1.441$ & No & $2.95 \times 10^{5}\phantom{0}$ & $1.446$ & No \\
$0.20$ & $353.573\phantom{000}$ & $1.444$ & No & $3.54 \times 10^{5}\phantom{0}$ & $1.455$ & No \\
\bottomrule[1.5pt]
\end{tabular}
\caption{Ground state electronic configurations of a Ru$_{55}$ nanocluster using SCF with Pulay mixing, with a $n=5$ density history length and mixing weights ranging from $0.02$ to $0.2$. $\Delta \rho_{\textrm{max}}$ is the maximum elementwise difference in the density matrix between the final and penultimate step, with convergence criterion $ \Delta \rho_{\textrm{max}} < 10^{-6}$.}
\label{tab:ru55}
\end{table*}
\begin{figure}
\centering
\includegraphics[width=3. in]{images/ru_merged.pdf}
\caption{Electronic energy and density distance $D[n,n_0]$ trajectory of a spin unpolarized Ru$_{55}$ cluster measured relative to the state it converges to, as obtained by it-TDDFT. Positive and negative isosurfaces of $n({\vb{r}}) - n_0({\vb{r}})$ are shown at various points in the propagation.}
\label{fig:ru55}
\end{figure}
\section{Conclusion}
\label{section:conclusion}
The first step of any Kohn-Sham DFT calculation is the determination of a self-consistent solution to the KS equations, resulting in a density corresponding to a stationary state of the many-body interacting system. While the standard method of using the iterative SCF procedure generally produces a solution efficiently, there are important classes of systems that pose problems for this approach due to their small band gaps or degenerate single-particle energies. We have proposed the it-TDDFT method as an alternative means for solving the KS equations in these difficult systems, and shown how it avoids the issues which affect SCF.
We established that the van Leeuwen theorem, a key theoretical foundation for TDDFT methods, can be extended to imaginary time, thereby ensuring convergence to a stationary state independent of the exchange-correlation potential and level of theory used in the model system. In addition, we discussed how it-TDDFT could be used in an alternative but equivalent definition of stationary states in DFT, better suited for metallic systems and systems with degenerate or nearly-degenerate states and based on the time-dependent Kohn-Sham equations. The it-TDDFT method also exhibits a number of practical advantages, such as justifying approximations to observables of interest before the ground state calculation is fully converged, requiring few input parameters, and allowing easy refinements of the results of previous runs by continuing from a saved state.
In the copper and ruthenium nanoclusters considered here, we demonstrated how SCF can struggle to find the electronic ground state, either converging to low-lying excited states or getting stuck in charge-sloshing cycles. These systems were readily converged by it-TDDFT, showcasing its robustness through smooth trajectories with monotonically decreasing energy. For these systems we either ran the calculation as spin-unpolarized, or with a fixed total spin. This is not an inherent limitation of the method, as one could simply run the calculation with all possible spin polarizations and select the state with the lowest energy. The method can be adapted to non-collinear spin systems, since the operating principle depends only on the Hamiltonian being able to differentiate states by energy. Furthermore, while we used finite systems for our example calculations, our method can be extended to find ground states of periodic systems by simultaneously propagating Kohn-Sham states at multiple $k$-points.
Given an existing TDDFT code which evolves systems in real-time, it should be relatively straightforward to implement a prototype of the presented it-TDDFT approach, requiring only an imaginary time substitution in the propagation step and a method to orthonormalize the single-particle states. While more efficient implementations could be examined in the future, the low barrier to utilizing it-TDDFT could make it an attractive alternative option for those dealing with particularly vexing systems.
\FloatBarrier
|
2,877,628,090,123 | arxiv | \section{Introduction}
High-dimensional data are ubiquitous in many applications of computer vision, e.g., face clustering \cite{1,2}, image representation and compression \cite{3}, and motion segmentation \cite{4,5}. A union of low-dimensional subspaces can approximate the original high-dimensional data for computational efficiency. The task of clustering high-dimensional data into corresponding low-dimensional subspaces is called subspace clustering.
Suppose that $X = \left[ \bm{x}_1,\dots,\bm{x}_N \right]\in \mathbb{R}^{D\times N}$ represents data set with $N$ data points in ambient dimension $D$, and data points lie in $n$ subspaces $\left\{S_i\right\}_{i=1}^{n}$ of dimensions $\left\{d_i\right\}_{i=1}^{n}$ ($d_i \ll \operatorname{min}\left\{D,N\right\}$). The task of subspace clustering is to partition data points into clusters $\left\{A_i\right\}_{i=1}^{n}$ so that data points within the same cluster $A_i$ lie in the same intrinsic subspace $S_i$. This problem has received great attention, and many algorithms including algebraic, iterative, statistical, and spectral clustering based approaches have been proposed (see \cite{6} for details). Among them, spectral clustering based methods have become extremely popular.
Current spectral clustering based methods solve the subspace clustering problem in two stages. In the first stage, an affinity matrix is learned to represent similarity between the data points. In the second stage, spectral clustering is applied on this affinity matrix. The differences of these methods lie in the first stage. These methods learn the affinity matrix based on the self-expressiveness model, which states that each data point in a union of subspaces can be expressed as a linear combination of other data points, i.e., $X = XC,$ where $X = [\bm{x}_1,\dots,\bm{x}_N]$ is the data matrix, $C = [\bm{c}_1,\dots,\bm{c}_N]\in \mathbb{R}^{N\times N}$ is the coefficients matrix. Once the $C$ is obtained, one can build an affinity matrix $W$ induced from $C$, e.g., $W = \left|C\right| + \left|C^T\right|$, and then obtain the segmentation of data by applying spectral clustering on $W$.
To find the coefficients matrix $C$, current methods solve the following optimization problem in the first stage:
\begin{equation}
\label{equ:1}
\min _{C}\|C\|_{C}, \text { s.t. } X=XC, \operatorname{diag}(C)= 0,
\end{equation}
where $\|\cdot\|_{C}$ denotes different norm regularization applied on $C$. For instance, in Sparse Subspace Clustering (SSC) \cite{7,8} and Structured Sparse Subspace Clustering (SSSC) \cite{9}, the $\ell_1$ is adopted as a convex surrogate over the $\ell_0$ norm to encourage the sparsity of $C$. Least Squares Regression (LSR) \cite{10} and Efficient Dense Subspace Clustering (EDSC) \cite{11} uses the $\ell_2$ norm regularization on $C$. Low Rank Representation (LRR) \cite{12}, Multiple Subspace Recovery (MSR) \cite{13} and Low-Rank Subspace Clustering (LRSC) \cite{14} use nuclear norm regularization on $C$. Low-Rank Sparse Subspace Clustering (LRSSC) \cite{15} uses a mixture of $\ell_1$ and nuclear norm regularization and Elastic Net Subspace Clustering (ENSC) \cite{16} uses a mixture of $\ell_1$ and $\ell_2$ regularization on $C$. In SSC by Orthogonal Matching Pursuit (OMP) \cite{17} and $\ell_0$-SSC \cite{18}, the $\ell_0$ norm is investigated. Block Diagonal Representation (BDR) \cite{19} uses block diagonal matrix induced regularizer to directly pursue the block diagonal matrix.
While the above approaches have been incredibly successful in many applications, we have observed some disadvantages. Firstly, The time cost of some approaches is too high to solve large-scale clustering problems, and the trade off between accuracy and efficiency is not the best. For example, SSC suffers from low efficiency and accuracy. SSSC improves the accuracy with the cost of extremely high computational time. BDR is much faster, but the accuracy is sometimes much lower than SSC.
Besides,
for the sake of clustering, we expect the intra-subspace similarity to be as dense as possible, but computing a dense affinity matrix from data is extremely expensive when data lie in a high-dimensional space. More importantly, the disadvantage of the two-stage framework is the rough combination of computing affinity matrix and spectral clustering. Because the affinity matrix $W$ can not fully represent the relationship of data points, directly applying spectral clustering on this affinity matrix may result in poor accuracy. Motivated by these observations, we raise several interesting questions:
\begin{itemize}
\item Is it possible to find a sparse method to gain a better trade off between accuracy and efficiency, so that it can be applied on large-scale subspace clustering tasks?
\item Can we compute an affinity matrix with denser intra-subspace similarity in a more efficient way?
\item Is there a universal framework for those popular two-stage subspace clustering approaches that bridges the gap between the similarity computation and spectral clustering?
\end{itemize}
We aim to address the above questions and in particular we make the following contributions:
\begin{itemize}
\item We propose Iterative Maximum Correlation (IMC) to obtain a sparse affinity matrix. We show its efficiency and scalability when clustering 100,000 points, while most methods have only been tested at most 10,000 points.
\item We propose Piecewise Correlation Estimation (PCE) to densify the intra-subspace similarity in the affinity matrix produced by IMC. It estimates the similarity between two data points via other points. It is more efficient than directly computing a dense affinity matrix from data.
\item We extend our work to be a universal Sparse-Dense Subspace Clustering (SDSC) framework. We propose a dense stage to optimize the affinity matrix before spectral clustering, and it's universal for current popular methods. The dense stage ensures the better performance of SDSC.
\end{itemize}
To the best of our knowledge, we are the first one to densify the affinity matrix. Our SDSC constitutes the first attempt to build a three-stage subspace clustering framework and find a universal dense method for the affinity matrix.
We conduct experiments on synthetic data, the Extended Yale B \cite{20} face data set, the USPS \cite{21} and MNIST \cite{30} handwritten digits data sets. We show the scalability of our sparse approach IMC, the effectiveness of our dense method PCE, and the universality of our framework SDSC.
\section{Iterative Maximum Correlation (IMC)}
Recall from (\ref{equ:1}) that each data point in a union of subspaces can be expressed as a linear combination of other data points, different subspace clustering approaches uses different regularization methods to compute the coefficients matrix $C$. For the purpose of data clustering, we expect the coefficients matrix to be \emph{subspace-preserving} \cite{28} i.e., $c_{ij} \neq 0$ only if data points $\bm{x}_{i}$ and $\bm{x}_{j}$ lie in the same subspace. For computational efficiency, we relax the optimization problem (\ref{equ:1}) as the following program:
\begin{equation}
\label{equ:2}
\underset{C}{\min}\left\|X-XC\right\|_{2} \text { s.t. }\left\|C\right\|_{0} \leq \Lambda, diag(C) = 0,
\end{equation}
where $\Lambda$ constrains the number of nonzero entries in $C$. It is shown in \cite{29} that (\ref{equ:2}) can be efficiently solved by using greedy algorithms. This motivates us to propose Iterative Maximum Correlation (IMC) (Algorithm~\ref{alg:1}) to compute the sparse coefficients matrix from the original data.
Generally, for each point $\bm{x}_{i}$, IMC greedily selects the point $\bm{x}_j$ that is most linearly correlated with the residual, then fill the coefficient $c_{ij}$ in $C$ (step 5), and finally updates the residual by removing its projection on the most correlated vectorized data point $\bm{x}_{j}$ (step 6) until the iteration number reaches a certain value.
\begin{algorithm}[b]
\caption{Iterative Maximum Correlation (IMC)}
\label{alg:1}
\hspace*{\algorithmicindent} \textbf{Input:} Data set $X=[\bm{x}_1,\dots,\bm{x}_N]\in\mathbb{R}^{D\times N}$, IMC iteration number $\Gamma$.
\begin{algorithmic}[1]
\STATE \textbf{Initialize} coefficients matrix $C$ as a $N\times N$ zero matrix, index of current data point $i = 1$.
\WHILE{$i \leq N$}
\STATE \textbf{Initialize} current iteration $\gamma = 0$, residual $\bm{\psi}_0 = \bm{x}_i$.
\WHILE {$\gamma < \Gamma$}
\STATE $c_{ij} = \max\left|\rho_{\bm{\psi}_\gamma\bm{x}_j}\right|$, where $j \neq i$, and $\rho_{\bm{\psi}_\gamma\bm{x}_j}$ is calculated by (\ref{equ:3}).
\STATE Update residual $\bm{\psi}_{\gamma+1} = \bm{\psi}_{\gamma} - (\bm{\psi}_\gamma\cdot\bm{x}_j)\bm{x}_j$.
\STATE $\gamma \leftarrow \gamma + 1$.
\ENDWHILE
\STATE $i \leftarrow i + 1$
\ENDWHILE
\end{algorithmic}
\hspace*{\algorithmicindent} \textbf{Output:} The coefficients matrix $C$.
\end{algorithm}
In particular, IMC adopts the Pearson correlation coefficient \cite{22} to find the most linearly correlated point. It is the covariance of the two variables divided by the product of their standard deviations. It has been proven to be effective to measure the linear correlation between variables. Given two vectorized data points $\bm{x}_i$ and $\bm{x}_j$ in $\Delta$ dimension, the Pearson correlation coefficient $\rho_{\bm{x}_i\bm{x}_j}$ is calculated as:
\begin{equation}
\label{equ:3}
\rho_{\bm{x}_i\bm{x}_j}=\frac{\sum_{\delta=1}^{\Delta}\left(x_{i\delta}-\overline{x_i}\right)\left(x_{j\delta}-\overline{x_j}\right)}{\sqrt{\sum_{\delta=1}^{\Delta}\left(x_{i\delta}-\overline{x_i}\right)^{2}}\sqrt{\sum_{i=1}^{\Delta}\left(x_{j\delta}-\overline{x_j}\right)^{2}}},
\end{equation}
where $x_{i\delta}$ and $x_{j\delta}$ are the entries in vectors $\bm{x}_i$ and $\bm{x}_j$, $\overline{x_i} = \frac{1}{\Delta} \sum_{\delta=1}^{\Delta} x_{i\delta}$, and analogously for $\overline{x_j}$. The value of $\rho_{\bm{x}_i\bm{x}_j}$ is in the range between $-1$ and $+1$. The larger the absolute value $\left|\rho_{\bm{x}_i\bm{x}_j}\right|$, the stronger the linear relationship. An absolute value of 1 indicates a perfect linear relationship.
We obtain coefficients directly from IMC. The Pearson correlation coefficients are used not only as a measure to select data points, but also as the value of entries in coefficients matrix $C$. Before the $\gamma+1$th iteration, the residual $\bm{\psi}_{\gamma+1}$ is updated as the difference between the current residual $\bm{\psi}_{\gamma}$ and its projection on the most linearly correlated vector $\bm{x}_j$. Since $\bm{\psi}$ is affected by each iteration, and the contribution of all selected points are removed, it reduces the risk of duplicate selection of the same point.
In most current popular subspace clustering methods, the affinity matrix is computed to be symmetric by:
\begin{equation}
\label{equ:4}
W=\left(\left|C\right|+\left|C^{\top}\right|\right),
\end{equation}
where $C^{\top}$ denotes the transpose of $C$. However, for some mutually selected data points such as $\bm{x}_i$ and $\bm{x}_j$, the coefficients $c_{ij}$ and $c_{ji}$ are nonzero entries. Calculating $w_{ij}$ by $\left|c_{ij}\right|+\left|c_{ji}\right|$ changes the similarity obtained by IMC. Thus we propose a new way to construct the affinity matrix, that is:
\begin{equation}
\label{equ:5}
W=\max \left(\left|C\right|, \left|C^{\top}\right|\right),
\end{equation}
which means that for each $w_{ij}$, the value is the larger one of $\left|c_{ij}\right|$ and $\left|c_{ji}\right|$. Then we obtain the affinity matrix $W$.
\section{Piecewise Correlation Estimation (PCE)}
Before applying spectral clustering on the affinity matrix $W$ produced by IMC, we would like to analyze the similarity in $W$. To better interpret the similarity between data points, we divide the value of similarity into four levels. The levels and the corresponding range of similarity are defined next.
\begin{myDef}
\label{def:1}
(\textbf{Piecewise correlation}).
\begin{table}[h]
\centering
\small
\begin{tabular}{l|l}
\toprule
Extremely strong correlation & $(\theta_1, 1]$ \\
Strong correlation & $(\theta_2, \theta_1]$ \\
Medium correlation & $(\theta_3, \theta_2]$ \\
Weak correlation & $[0, \theta_3]$ \\
\bottomrule
\end{tabular}%
\label{tab:addlabel}%
\end{table}\\
\noindent where $\theta_1$, $\theta_2$, $\theta_3$ are thresholds, and the value is to be fixed by the following experiments on real-world data sets.
\end{myDef}
We assume that a pair of data points $\bm{x}_i$ and $\bm{x}_j$ with strong or extremely strong correlation i.e., $w_{ij} > \theta_2$ indicates that they belong to the same subspace. With analysis of the affinity matrix $W$, we observe some similarity of pairwise data points from the same subspace is much smaller than expected, and some is even zero. We detailedly define this phenomenon as ternary unstable relationship next.
\begin{myDef}
\label{def:2}
(\textbf{Ternary unstable relationship}).
Given any two data points $\bm{x}_i$, $\bm{x}_j$ in $X$ and the similarity between them $w_{ij}$. Consider the pairwise correlation defined in Definition (\ref{def:1}), for any intermediate data point $\bm{x}_k$ $(\bm{x}_k \in X\backslash\left\{\bm{x}_i,\bm{x}_j\right\})$, we say that the relationship of $\bm{x}_i$, $\bm{x}_j$, and $\bm{x}_k$ is ternary unstable if their similarity $w_{ij}$, $w_{ik}$, $w_{kj}$ satisfies one of the following conditions (TUR conditions):
\begin{enumerate}[1)]
\item $w_{ik}, w_{kj} \in (\theta_1, 1], w_{ij} \notin (\theta_1, 1];$
\item $\max(w_{ik}, w_{kj})\in(\theta_1, 1], \min(w_{ik}, w_{kj})\in(\theta_2, \theta_1], \\w_{ij}\notin(\theta_2,1]
\item $w_{ik}, w_{kj} \in (\theta_2, \theta_1], w_{ij}=0.$
\end{enumerate}
\end{myDef}
Current spectral clustering approaches adopt normalized cut \cite{23} to partition the data points into $n$ clusters $\left\{A_i\right\}_{i=1}^{n}$. The dissimilarity between cluster $A_i$ and other clusters $\overline{A_i}$ are defined as $cut$:
\begin{equation}
\label{equ:6}
cut(A_i, \overline{A_i})=\sum_{x_u \in A_i, x_v \notin A_i} w_{uv},
\end{equation}
while the intra-cluster similarity of $A_i$ is defined as $vol$:
\begin{equation}
\label{equ:7}
vol(A_i)=\sum_{x_u, x_t \in A_i} w_{ut}.
\end{equation}
To measure the disassociation of $n$ clusters obtained by normalized cut, $Ncut$ is defined as:
\begin{equation}
\label{equ:8}
Ncut\left(A_{1}, A_{2}, \ldots A_{n}\right)= \sum_{i=1}^{n} \frac{cut\left(A_{i}, \overline{A_{i}}\right)}{vol\left(A_{i}\right)},
\end{equation}
and the task of normalized cut is to find how to partition the data points into $n$ clusters to get a minimum value of $Ncut$. An ideal affinity matrix for this task should contain as dense intra-subspace similarity as possible, and contain as few inter-subspace similarity elements as possible.
Recall from the $W$ produced by IMC, the ternary unstable relationship limits the intra-subspace similarity, making it difficult to group the points from the same subspace into the same cluster. This motivates us to revise the ternary unstable relationship to gain higher intra-subspace similarity.
We propose Piecewise Correlation Estimation (PCE) (Algorithm \ref{alg:2}) to revise the ternary unstable relationship in $W$. The similarity is optimized after traversal of all intermediate points. Due to the restriction of TUR conditions, for data points $\bm{x}_i$, $\bm{x}_j$ having stronger correlation with the intermediate point $\bm{x}_k$, the similarity $w_{ij}$ is updated more close to $w_{ik}$ and $w_{kj}$; for those with medium and weak correlation with the intermediate point, the similarity is not changed.
\begin{algorithm}[h]
\caption{Piecewise Correlation Estimation (PCE)}
\label{alg:2}
\hspace*{\algorithmicindent} {\textbf{Input:} $\theta_1$, $\theta_2$, $W$, data set $X$.}
\begin{algorithmic}[1]
\FOR{each pair of data points $\bm{x}_i$ and $\bm{x}_j$}
\FOR {each intermediate point $\bm{x}_k \in X\backslash\left\{\bm{x}_i,\bm{x}_j\right\}$}
\STATE
$w^*_{ij}=\left\{
\begin{array}{ll}
\frac{1}{2}(w_{ik} + w_{kj}),& \textbf{if}\text{ TUR condition1)}\\
\min(w_{ik}, w_{kj}),& \textbf{if}\text{ TUR condition2)}\\
\frac{1}{2}\max(w_{ik}, w_{kj}),& \textbf{if}\text{ TUR condition3)} \\
w_{ij},& \textbf{else}
\end{array}
\right.
$
\ENDFOR
\ENDFOR
\end{algorithmic}
\hspace*{\algorithmicindent} \textbf{Output:} A new affinity matrix $W^*\in\mathbb{R}^{N\times N}$.
\end{algorithm}
After PCE, we can get $w^*_{ij} \geq w_{ij}$. In particular, some zero entries in $W$ are updated as nonzero entries, which densifies the affinity matrix. The new affinity matrix $W^*$ obtains larger intra-subspace similarity for each subspace $S_i$, while the inter-subspace similarity is slightly changed due to the restriction of TUR conditions. Recall from the task of normalized cut, points in $S_i$ are more likely to be clustered into the same cluster $A_i$ after PCE. Because from the perspective of $A_i$, the intra-cluster similarity $vol(A_i)$ is much increased and the inter-cluster similarity $cut(A_i, \overline{A_i})$ is just slightly increased. We will verify this by experiments.
\section{Sparse-Dense Subspace Clustering (SDSC): \\A Universal Framework}
Directly combining IMC and spectral clustering as a two-stage approach is practical to solve subspace clustering problems, but inserting a dense stage PCE before spectral clustering ensures a better affinity matrix. This three-stage approach follows a Sparse-Dense Subspace Clustering (SDSC) framework as concluded in Algorithm \ref{alg:3}.
However, we observe that many other popular methods (e.g. SSC, LSR, LRR, OMP, ENSC, BDR) follow the two-stage framework that directly combining affinity matrix generation with spectral clustering. The affinity matrix obtained in the first stage usually contain insufficient intra-subspace similarity, which is difficult for spectral method to partition data points from the perspective of graph theory. We need an affinity matrix with as dense intra-subspace similarity elements as possible. It's difficult to obtain such an affinity matrix by these two-stage methods, since the computation is extremely expensive. For instance, SSSC integrates the two stages of SSC into a learning framework, and re-weights the similarity in many iterations, but the time cost is extremely high. Motivated by our proposed PCE for IMC, we attempt to optimize the affinity matrix before spectral clustering for current two-stage approaches, and finally remould those methods to follow the proposed SDSC framework.
\begin{algorithm}[h]
\caption{Sparse-Dense Subspace Clustering (SDSC)}
\label{alg:3}
\hspace*{\algorithmicindent} \textbf{Input:} Data set $X$.
\begin{algorithmic}[1]
\STATE Compute a affinity matrix $W$ from data by different data representation methods.
\STATE Optimize similarity in $W$ to get $W^*$ by a dense method.
\STATE Apply spectral clustering on $W^*$.
\end{algorithmic}
\hspace*{\algorithmicindent} \textbf{Output:} Clustering results.
\end{algorithm}
In SDSC, we use a dense stage to optimize the affinity matrix before spectral clustering. Different from PCE specifically proposed for IMC, this dense method needs to be universal for as many approaches as possible. We attempt to analyze and optimize the affinity matrix by distances graph, and propose a universal dense stage (Algorithm \ref{alg:4}) for current two-stage subspace clustering methods.
\begin{myDef}
\textbf{(Simulated distances)}.
To measure distances between data points, a distances matrix $D\in \mathbb{R}^{N\times N}$ is generated from the affinity matrix $W\in\mathbb{R}^{N\times N}$, and elements in $D$ represent the simulated distances between data points.
\end{myDef}
\begin{algorithm}
\caption{A universal dense stage for SDSC}
\label{alg:4}
\hspace*{\algorithmicindent} \textbf{Input:} $X$, affinity matrix $W$.
\begin{algorithmic}[1]
\STATE Compute simulated distances $D\in \mathbb{R}^{N\times N}$ from $W$.
\FOR{each pair of data points $\bm{x}_i$ and $\bm{x}_j$}
\FOR {each intermediate point $\bm{x}_k \in X\backslash\left\{\bm{x}_i,\bm{x}_j\right\}$}
\STATE {$d^*_{ij} = \min(d_{ij}, d_{ik} + d_{kj})$}
\ENDFOR
\ENDFOR
\STATE Compute new affinity matrix $W^*$ from $D^*$.
\end{algorithmic}
\hspace*{\algorithmicindent} \textbf{Output:} The optimized affinity matrix $W^*\in \mathbb{R}^{N\times N}$.
\end{algorithm}
We first transform the similarity into simulated distances, then minimize the distances of each pair of points via the intermediate points, and finally transform the distances back into similarity. Step 1 in Algorithm \ref{alg:4} is the transformation form similarity to distances, and step 7 is the inverse transformation. The value of similarity is in the range $[0,1]$, and the similarity grows inversely to the distances. For the ease of use, we propose several practical transformation functions in Table \ref{tbl:1} and we will test them in experiments.
\begin{table}[htbp]
\centering
\small
\caption{Several transformation functions.}
\label{tbl:1}
\begin{tabular}{l|l|l}
\toprule
&Step 1: $w_{ij}$ to $d_{ij}$ & Step 9: $d^*_{ij}$ to $w^*_{ij}$ \\
\hline
1) & $d_{ij} = 1 - w_{ij}$ & $w^*_{ij} = 1 - d^*_{ij}$ \\
2) & $d_{ij} = 1 - \ln(w_{ij})$ & $w^*_{ij} = \exp(1 - d^*_{ij})$ \\
3) & $d_{ij} = \frac{1}{w_{ij}}$ & $w^*_{ij} = \frac{1}{d^*_{ij}}$ \\
\bottomrule
\end{tabular}%
\end{table}%
Step 4 in Algorithm \ref{alg:4} minimizes the distance between $\bm{x}_i$ and $\bm{x}_j$ via the intermediate point $\bm{x}_k$. In particular, if $w_{ij}$ is zero in $W$, $w_{ik}$ and $w_{kj}$ are nonzero entries, then $w^*_{ij}$ are updated as nonzero after the dense stage. This makes the new affinity matrix $W^*$ a denser matrix.
Specifically, if one only needs to fine-tune the nonzero elements in $W$ without changing the sparsity, restrictive conditions of $d_{ij}$ can be added before step 4. This will not be detailed here since it makes limited changes to the affinity matrix, and makes little improvements of performance.
Given an affinity matrix $W$ of $N$ data points lying in $n$ subspaces, the sum of all the intra-subspace similarity elements in $W$ is $\eta_1$, and the sum of inter-subspace similarity is $\eta_2$. We respectively denote $\overline{\eta_1}$ as the average value of intra-subspace similarity in $W$, and $\overline{\eta_2}$ as the average value of inter-subspace similarity. There are $kN$ nonzero entries in $W$, where $k$ is the number of nonzero similarity elements of each data point. Particularly, suppose the number of inter-subspace similarity elements is $\zeta$, then the number of intra-subspace similarity elements in $W$ is $kN-\zeta$. Recall from the normalized cut problem in (\ref{equ:8}), we can calculate $Ncut$ for perfectly clustering each data point into intrinsic subspaces:
\begin{equation}
\label{equ:9}
Ncut = \frac{\eta_2}{\eta_1} = \frac{\zeta\overline{\eta_2}}{(kN-\zeta)\overline{\eta_1}}.
\end{equation}
For each nonzero entry $w_{ij}$ in $W$, there are other $k-1$ nonzero entries that in the $i$th row or the $j$th column. After the dense stage, the value of each entry is updated via the intermediate points. The number of nonzero entries is increased to be approximately $Nk^2 - \tau$, where $\tau$ is the number of duplicate intra-subspace entries. The number of inter-subspace similarity in $W^*$ is $\zeta (k-1)$, and the number of intra-subspace similarity elements is $Nk^2 -\zeta (k-1) - \tau$. We can calculate $Ncut^*$ for perfectly clustering of $W^*$:
\begin{equation}
\label{equ:10}
Ncut^* = \frac{\eta^*_2}{\eta^*_1} = \frac{\zeta (k-1)\overline{\eta^*_2}}{(Nk^2 -\zeta (k-1) - \tau)\overline{\eta^*_1}},
\end{equation}
where $\eta^*_1$ and $\eta^*_2$ are the sum of intra-subspace and inter-subspace similarity in $W^*$, $\overline{\eta^*_1}$ and $\overline{\eta^*_2}$ are the average value of them. The ratio of $Ncut^*$ to $Ncut$ can be computed as:
\begin{equation}
\label{equ:11}
Q = \frac{Ncut^*}{Ncut} = \frac{Nk^2 -\zeta (k-1) - Nk}{Nk^2 -\zeta (k-1)-\tau}\frac{\overline{\eta^*_2}\overline{\eta_1}}{\overline{\eta^*_1}\overline{\eta_2}}.
\end{equation}
Since $N >> \zeta$, the ratio $Q$ can be approximated as:
\begin{equation}
\label{equ:12}
\widetilde{Q} = \frac{Nk^2-Nk}{Nk^2-\tau}\frac{\overline{\eta^*_2}\overline{\eta_1}}{\overline{\eta^*_1}\overline{\eta_2}}.
\end{equation}
Generally, the average value of each intra-subspace and inter-subspace similarity element is slightly changed after the dense stage. Thus the ratio can be approximated as:
\begin{equation}
\label{equ:13}
\widetilde{Q} = \frac{Nk^2-Nk}{Nk^2 - \tau}.
\end{equation}
Actually $\tau$ is usually zero in real experiments, partly due to the high sparsity of the original affinity matrix $W$. The ratio $\widetilde{Q}$ is smaller than 1, i.e., $Ncut^* < Ncut$, indicating that the densified affinity matrix $W^*$ makes the points from the same subspace more likely to be clustered into the same group. The detailed proof is in the supplemental file.
\section{Experiments}
In this section, we first evaluate the efficiency and scalability of IMC on synthetic data. Then we show the effectiveness of the dense method PCE for IMC on a face data set. Finally we show the universality of the SDSC framework for current two-stage approaches on two handwritten digit data sets.
\subsection{Experimental Setup}
We compare the performance of current popular spectral subspace clustering methods, including OMP, SSC, SSSC, LSR, LRR, ENSC, and BDR which respectively represent $\ell_0$, $\ell_1$, $\ell_2$, mixed norm and block diagonal regularization. We use the code provided by the respective authors for computing the coefficients, where the parameters are tuned to give the best clustering accuracy. We choose BDR-Z as a representative of BDR. For those implemented in SDSC framework with different transformation functions in Table \ref{tbl:1}, the names are formed with the suffix -D1, -D2 or -D3, such as SSC-D1 indicating that the SSC algorithm is implemented with the transformation function 1). The two-stage method of directly combining IMC and spectral clustering is called as 'IMC', while the SDSC method of adding the dense stage PCE before spectral clustering is formed as 'IMC-P'.
We evaluate the clustering accuracy (ACC) defined as:
\begin{equation}
\label{equ:14}
ACC = \frac{1}{N} \sum_{i=1}^{N} \mu\left(u_{i}, \operatorname{bestmap}\left(v_{i}\right)\right),
\end{equation}
where $u_i \in U$ and $v_i \in V$ respectively represent the output label and the ground-truth label of the $i$th data point, $\mu(x,y) = 1$ if $x = y$, and $\mu(x, y) = 0$ otherwise, and bestmap($v_i$) is the best mapping function that permutes clustering labels to match the ground-truth ones.
In addition to ACC, we also report the Normalized Mutual Information (NMI) \cite{24} and the graph connectivity (CONN), which are defined as follows:
\begin{itemize}
\item NMI is an information theoretic measure of how well the computed clusters and the true clusters predict one another, normalized by the amount of information inherent in the two clustering systems, i.e.,
\begin{equation}
\label{equ:15}
NMI(U, V)=\frac{2 \times I(U ; V)}{[H(U)+H(V)]},
\end{equation}
where $U$ and $V$ are the output labels and the ground-truth labels, $H(.)$ is entropy, $I(U;V)$ is the Mutual Information \cite{25} between $U$ and $V$. NMI is in the range $[0,1]$, and NMI = 1 stands for perfectly complete labeling.
\item
CONN evaluates the connectivity of the affinity graph. Generally, for an undirected graph with affinity matrix $W$ and degree matrix $D = \operatorname{Diag}(W\cdot\bm{1})$, where $\bm{1}$ is the vector of all ones, CONN is defined as the second smallest eigenvalue of the Laplacian $L=I-D^{-1 / 2} A D^{-1 / 2}$. CONN is in the range $[0, \frac{N-1}{N}]$ and is zero when the graph is not connected \cite{26}.
\end{itemize}
All experiments are conducted on a PC with an Intel(R) Core(TM) i7-7700 CPU at 3.60GHz, 16G RAM, running Windows 10 and MATLAB R2018a.
\subsection{Experiments on Synthetic Data}
\begin{table*}[ht]
\centering
\caption{ACC (\%) of different algorithms on the Extended Yale B data set. A '-' denotes time out.}
\label{tbl:2}
\resizebox{.84\width}!{
\begin{tabular}{c|ccc|ccc|ccc|ccc|ccc}
\toprule
& \multicolumn{3}{c|}{2 subjects} & \multicolumn{3}{c|}{5 subjects} & \multicolumn{3}{c|}{10 subjects} & \multicolumn{3}{c|}{15 subjects} & \multicolumn{3}{c}{20 subjects} \\
\hline
methods & mean & median & std & mean & median & std & mean & median & std & mean & median & std & mean & median & std \\
\hline
SSC & 97.48 & 98.80 & 3.17 & 92.68 & 93.15 & 6.87 & 88.24 & 87.98 & 6.21 & 82.60 & 83.54 & 5.97 & 75.29 & 78.15 & 5.35 \\
LSR & 94.67 & 95.21 & 10.15 & 80.31 & 82.14 & 8.72 & 71.46 & 73.25 & 12.35 & 69.21 & 68.18 & 5.57 & 67.11 & 67.15 & 4.57 \\
LRR & 93.18 & 95.12 & 14.50 & 92.11 & 95.51 & 9.61 & 85.67 & 86.45 & 8.98 & 82.15 & 84.12 & 6.54 & 77.30 & 80.03 & 5.99 \\
OMP & 99.15 & 100 & 1.22 & 95.88 & 97.31 & 5.06 & 87.25 & 84.16 & 6.72 & 84.41 & 84.89 & 5.68 & 79.69 & 81.02 & 4.45 \\
ENSC & 91.05 & 94.22 & 15.24 & 84.98 & 84.22 & 11.24 & 77.21 & 76.24 & 10.50 & 62.97 & 65.68 & 9.24 & 63.18 & 65.48 & 5.90 \\
BDR & 97.31 & 97.64 & 13.54 & 87.12 & 89.35 & 11.35 & 77.23 & 79.54 & 9.54 & 69.20 & 71.24 & 5.87 & 63.63 & 66.21 & 4.54 \\
SSSC & 98.71 & 100 & 2.69 & 95.21 & 97.21 & 1.09 & 90.26 & 89.99 & 5.15 & 85.15 & 86.22 & 4.69 & - & - & - \\
\textbf{IMC} & \textbf{99.48} & 100 & 1.14 & 97.46 & 97.68 & 1.27 & 91.14 & 94.25 & 5.81 & 87.11 & 87.64 & 5.73 & 84.16 & 82.99 & 5.18 \\
\textbf{IMC-P} & 99.46 & 100 & \textbf{0.81} & \textbf{97.76} & \textbf{97.76} & \textbf{1.01} & \textbf{95.38} & \textbf{95.61} & \textbf{3.97} & \textbf{91.69} & \textbf{91.11} & \textbf{3.71} & \textbf{88.13} & \textbf{86.86} & \textbf{4.12} \\
\bottomrule
\end{tabular}}
\end{table*}%
In this section, we evaluate the efficiency and scalability of IMC. Experiments are conducted on synthetic data.
We randomly generate 6 subspaces $\left\{S_i\right\}_{i=1}^{6}$, and the dimension of each subspace is $d_i = 6$. All the data points lie in an ambient space of dimension $D=10$. Each subspace $S_i$ contains $N_i$ data points, and $N_i$ is in the range [50, 166,667], so the total number of data points $N$ varies from 300 to 100,002. For IMC, we set the iteration number $\Gamma = 6$.
\begin{figure}[h]
\label{fig:1}
\centering
\subfigure[ACC]{
\includegraphics[width=0.47\columnwidth]{ACC_SYNC93.pdf}
\label{fig:1a}
}
\subfigure[NMI]{
\includegraphics[width=0.47\columnwidth]{NMI_SYNC93.pdf}
\label{fig:1b}
}
\quad
\subfigure[CONN]{
\includegraphics[width=0.47\columnwidth]{CONN_SYNC93.pdf}
\label{fig:1c}
}
\subfigure[Computational time]{
\includegraphics[width=0.47\columnwidth]{TIME_SYNC93.pdf}
\label{fig:1d}
}
\caption{Performance on synthetic data. For BDR, SSSC and SSC, the maximum number of points tested is respectively 9,000, 1,200 and 6,000 due to time limit. We use log scale in x-axis, and also in the y-axis of bottom right figure.}
\end{figure}
The ACC and NMI are plotted in Figure \ref{fig:1a} and \ref{fig:1b}. We first observe that IMC obtain higher accuracy and retain more information when clustering large number of data points. However, for small $N$, IMC is outperformed by SSC. This is partly because the number of connections for each data point is set as $\Gamma$, and more inter-subspace similarity are computed when $N$ is smaller. The connectivity is plotted in Figure \ref{fig:1c}, and IMC obtain the best connectivity, indicating that the points from the same subspace are well connected. The running time is plotted in \ref{fig:1d}, and it shows that IMC is significantly efficient: it is 3 orders of magnitude faster than SSC when clustering 6,000 points and 4 orders faster than SSSC when clustering 1,200 points. Actually most popular methods can only be tested on at most 10,000 points. We can conclude that as $N$ increases, the superiority of IMC in accuracy expands, and IMC generates a better sparse representation. Since IMC is significantly faster, IMC is preferable for large-scale subspace clustering problems.
\subsection{Clustering Human Face Images}
\begin{table*}[ht]
\centering
\caption{ACC (\%) of different algorithms on the USPS data set.}
\label{tbl:3}
\resizebox{.78\width}!{
\begin{tabular}{c|cccc|cccc|cccc|c}
\toprule
samples & SSC & SSC-D1 & SSC-D2 & SSC-D3 & LSR & LSR-D1 & LSR-D2 & LSR-D3 & LRR & LRR-D1 & LRR-D2 & LRR-D3 & IMC \\
\hline
500 & 65.05 & 63.31 & 72.57 & \textbf{78.45} & 68.18 & 61.00 & 67.13 & \textbf{71.80} & 64.86 & 65.22 & 64.13 & \textbf{70.25} & 71.21 \\
1000 & 60.97 & 64.75 & 69.21 & \textbf{73.13} & 71.09 & 64.15 & 70.12 & \textbf{73.79} & 61.86 & 61.42 & 60.44 & \textbf{69.02} & 69.54 \\
2000 & 60.13 & 58.16 & 76.12 & \textbf{82.30} & 71.11 & 63.99 & 72.22 & \textbf{75.88} & 62.85 & 64.21 & 61.89 & \textbf{71.74} & 71.75 \\
3000 & 63.54 & 56.27 & 78.73 & \textbf{83.86} & 70.94 & 62.79 & \textbf{73.46} & 72.23 & 63.72 & 65.78 & 62.94 & \textbf{67.60} & 69.88 \\
\midrule
samples & OMP & OMP-D1 & OMP-D2 & OMP-D3 & ENSC & ENSC-D1 & ENSC-D2 & ENSC-D3 & BDR & BDR-D1 & BDR-D2 & BDR-D3 & IMC-P \\
\hline
500 & 61.39 & 53.25 & 67.10 & \textbf{70.57} & 60.34 & 39.28 & 66.08 & \textbf{73.87} & 66.45 & 59.12 & 69.88 & \textbf{72.16} & 72.66 \\
1000 & 59.99 & 51.77 & 61.30 & \textbf{74.41} & 60.62 & 41.89 & 63.90 & \textbf{69.09} & 60.07 & 51.16 & 71.24 & \textbf{73.28} & 72.73 \\
2000 & 61.62 & 49.38 & 64.89 & \textbf{73.13} & 59.21 & 44.27 & 64.97 & \textbf{71.48} & 53.54 & 44.29 & 69.00 & \textbf{72.55} & 74.21 \\
3000 & 59.75 & 54.67 & 66.59 & \textbf{70.40} & 56.26 & 38.97 & 66.55 & \textbf{70.22} & 53.51 & 46.87 & 62.36 & \textbf{68.22} & 73.64 \\
\bottomrule
\end{tabular}}
\end{table*}%
\begin{table*}[ht]
\centering
\caption{Performance of different algorithms for clustering 5 subjects with totally 500 samples from MNIST data set.}
\label{tbl:4}
\resizebox{.85\width}!{
\begin{tabular}{c|cc|cc|cc|cc|cc|cc}
\toprule
& SSC & SSC-D3 & LSR & LSR-D3 & LRR & LRR-D3 & OMP & OMP-D3 & ENSC & ENSC-D3 & BDR & BDR-D3 \\
\hline
ACC (\%) & 74.97 & 92.69 & 79.48 & 88.46 & 63.10 & 77.08 & 90.25 & 93.82 & 73.51 & 89.28 & 79.95 & 93.58 \\
NMI (\%) & 81.39 & 85.49 & 65.42 & 77.39 & 73.25 & 80.15 & 82.89 & 84.02 & 83.02 & 86.30 & 80.09 & 88.75 \\
CONN (\%) & 17.31 & 79.73 & 79.05 & 89.11 & 1.07 & 65.83 & 17.01 & 74.25 & 23.01 & 79.20 & 50.81 & 85.07 \\
\bottomrule
\end{tabular}}
\end{table*}%
It is shown in \cite{3} that the images of a subject with a fixed pose and varying illumination approximately lie in a union of 9-dimensional subspaces. Thus subspace clustering methods can be applied on the task of face clustering. In this experiment, we evaluate the effectiveness of PCE on Extended Yale B data set. It contains 2,414 frontal face images of 38 individuals under 9 poses and 64 illumination conditions. Each cropped face image consists of 192$\times$168 pixels. We downsample the images to 48$\times$42 pixels and vectorize it to a 2,016 vectors as data points. In each experiment, we randomly pick $n \in \left\{2, 5, 10, 15, 20\right\}$ subjects and take all the images of selected subjects as data to be clustered.
\subsubsection{Performance of PCE as a Function of $\theta_1$ and $\theta_2$}
To show the effect of the parameters $\theta_1$ and $\theta_2$ on the performance of PCE, we report the average ACC on the 10 subjects problem based on two settings: (1) fix $\theta_2 = 0.6$ and choose $\theta_1 \in \left\{0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95\right\}$; (2) fix $\theta_1 = 0.8$ and choose $\theta_2 \in \left\{0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75\right\}$. The results are shown in Figure \ref{fig:2a} and \ref{fig:2b}. It can be seen that using the setting of $\theta_1 = 0.8$ and $\theta_2 = 0.6$ leads to the best accuracy, and we use this setting for following experiments.
\begin{figure}[htbp]
\label{fig:2}
\centering
\subfigure[$\theta_2 = 0.6$, $\theta_1$ changes]{
\includegraphics[width=0.47\columnwidth]{theta1_new.pdf}
\label{fig:2a}
}
\subfigure[$\theta_1 = 0.8$, $\theta_2$ changes]{
\includegraphics[width=0.47\columnwidth]{theta2_new.pdf}
\label{fig:2b}
}
\caption{ACC of IMC-P (IMC optimized by PCE) as a function of $\theta_1$ when fixing $\theta_2 = 0.6$ in (a) and $\theta_2$ when fixing $\theta_1 = 0.8$ in (b) for the 10 subjects clustering problem from Extended Yale B data set.}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.58\columnwidth]{EYB_TIME_92.pdf}
\caption{Average computational time of different methods on the Extended Yale B data set as a function of the number of subjects. Note that we use log scale in y-axis.}\label{fig:3}
\end{figure}
\subsubsection{Performance of IMC and IMC-P Compared with Other Methods}
The clustering performance of different methods is reported in Tabel \ref{tbl:2}. It can be seen that our IMC and IMC-P outperform other methods. Generally, the clustering problem is more challenging when the number of subspaces increases. We find that when the number of subjects increases, the improvement by our methods is more significant, and the optimization of IMC by PCE expands. This experiment clearly demonstrates the effectiveness of our IMC and PCE on face clustering tasks. SSSC also performs well in some cases. However, it can be seen in Figure \ref{fig:3} that SSSC has the highest computational time, while LSR gains the lowest. Our IMC is faster than most methods, and IMC-P optimized by PCE is faster than SSSC and BDR, yet IMC-P still enjoys the highest accuracy. So our PCE helps IMC obtain a better trade-off between the performance and the time cost.
\subsubsection{Data Visualization: the Densification by PCE}
To show the effect of using PCE to optimize the affinity matrix $W$ generated by IMC, we visualize the $W$ produced by IMC in Figure \ref{fig:4a} and $W^*$ optimized by PCE in Figure \ref{fig:4b}.
\begin{figure}[h]
\label{fig:4}
\centering
\subfigure[$W$ produced by IMC]{
\includegraphics[width=0.47\columnwidth]{IMC_BD.pdf}
\label{fig:4a}
}
\subfigure[$W^*$ densified by PCE]{
\includegraphics[width=0.47\columnwidth]{IMC_D_BD.pdf}
\label{fig:4b}
}
\caption{The visualization of affinity matrix produced by IMC (a) and optimized by PCE (b) on the task of clustering 6 subjects from Extended Yale B data set.}
\end{figure}
We observe that $W$ in Figure \ref{fig:4a} is a sparse matrix, and the bright dots lying in the diagonal blocks are intra-subspace similarity elements in $W$. However, it contains small amount of the intra-subspace similarity, and this may make it difficult for spectral clustering to segment the data points. After PCE, as shown in Figure \ref{fig:4b}, we gain denser intra-subspace similarity, while the number of inter-subspace elements increases little. Thus PCE is an effective optimization method for IMC to densify the intra-subspace similarity.
\subsection{Clustering Handwritten Digits}
In this part, we test the transformation functions proposed in Table \ref{tbl:1}, and verify the universality of SDSC framework. We use the USPS containing 8-bit gray scale images of handwritten digits from 0 to 9. We reshape each sample to a 200 dimension vector, and randomly pick $N_i \in \left\{100, 200, 400, 600\right\}$ samples of 5 digits, thus the total number of samples is from 500 to 3,000. Besides, we tests on MNIST with images reshaped to 500 dimension vectors.
From Table \ref{tbl:3} we can observe that the implementations suffixed with '-D3' improve the accuracy greatly in most cases, while the '-D1' methods sometimes even make the results worse. This is partly because the function 3) in Table \ref{tbl:1} could spread the value of similarity around in a more reasonable way, in which the high similarity (indicating intra-subspace similarity) are well retained and the relatively low similarity (more likely to be inter-subspace similarity) are neglected on purpose. This restricts the effect of the dense stage to the intra-subspace similarity elements in the affinity matrix. After the dense stage with function 3), the affinity matrix contain denser intra-subspace similarity, and this makes spectral clustering more likely to group points from the same subspace into the same cluster. Besides, we can notice that IMC outperforms other two-stage methods, and IMC-P improves the accuracy of it. This verifies the effectiveness of IMC and PCE on handwritten digit clustering tasks.
Moreover, Table \ref{tbl:4} reports the performance of those two-stage methods and their SDSC implementations with transformation function 3) on MNIST data set. It can be observed that the SDSC implementations obtain higher accuracy, retain more information after clustering, and get the points more connected in subspace. This demonstrates again the superiority and universality of SDSC framework. Thus SDSC is a universal framework that can be applied on current two-stage methods to get better performance.
\section{Conclusion and Future Work}
This paper studies the subspace clustering problem which aims to clustering the high-dimensional data points into low dimension according to the self-expressiveness model. We first propose a new faster sparse method Iterative Maximum Correlation (IMC) to draw coefficients from data, then apply a dense method Piecewise Correlation Estimation (PCE) to optimize the affinity matrix. IMC adopts the Pearson correlation coefficient as both a measure to select points and the value of similarity. PCE optimizes the similarity between two data points via the intermediate data points. Besides, we extend our work to be a SDSC framework for current two-stage subspace clustering approaches. To the best of our knowledge, we are the first one to densify the affinity matrix before spectral clustering. We show the efficiency and scalability of IMC when handle 100,000 data points, whereas most approaches have only be tested less than 10,000 points. We also show the effectiveness of PCE to densify the intra-subspace similarity. We also make analysis of our SDSC framework. We conduct series of experiments to demonstrate the effectiveness of our methods. We note that our universal dense method in Algorithm \ref{alg:4} optimize the similarity based on graph analysis of distance, and other styles of dense methods are left for future research.
\bibliographystyle{aaai}
|
2,877,628,090,124 | arxiv | \section{INTRODUCTION}
\label{sec:intro}
Begin the Introduction below the Keywords. The manuscript should not have headers, footers, or page numbers. It should be in a one-column format. References are often noted in the text and cited at the end of the paper.
\begin{table}[ht]
\caption{Fonts sizes to be used for various parts of the manuscript. Table captions should be centered above the table. When the caption is too long to fit on one line, it should be justified to the right and left margins of the body of the text.}
\label{tab:fonts}
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\end{center}
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\begin{table}[ht]
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\label{tab:Paper Margins}
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
\rule[-1ex]{0pt}{3.5ex} Margin & A4 & Letter \\
\hline
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\hline
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\hline
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\hline
\rule[-1ex]{0pt}{3.5ex} Printable area & 17.15 x 22.23 cm & 6.75 x 8.75 in. \\
\hline
\end{tabular}
\end{center}
\end{table}
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\section{SECTION FORMATTING}
\label{sec:sections}
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{ \label{fig:video-example}
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\caption{Information on video and audio files that must accompany a manuscript submission.}
\label{tab:Multimedia-Specifications}
\begin{center}
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\hline
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\hline
\end{tabular}
\end{center}
\end{table}
\section{INTRODUCTION}
\label{sec:intro}
The primordial gravitational waves (PGWs) have been predicted to cause a unique polarization signature in the Cosmic Microwave Background (CMB). The ultimate detection of these PGWs could open a new window on physics of the newborn universe since their detection would be direct evidence for the theory of cosmic inflation. To constrain the PGWs magnitude, and thus the energy scale of inflation, we must account for polarized Galactic foregrounds \cite{sync1} with multi-frequency maps of the CMB. To date, BICEP/Keck (BK) program has placed the strongest constrains on the amplitude of PGWs, the tensor-to-scalar ratio \emph{r} \cite{BKI}. The current sensitivity is limited by the modeling of both gravitational lensing and polarized Galactic foregrounds. The BICEP/Keck team has collaborated with the SPT-3G team to develop the delensing techniques \cite{delensing} in the CMB maps to improve the constraints on r. The Galactic foregrounds characterization, especially synchrotron emission remains a challenge beyond that collaboration and we need a high sensitive telescope at low frequency channels (30/40 GHz) since it dominates at these frequency bands.
BICEP Array telescope \cite{hui18} has been developed with an exceptional sensitivity and wide frequency coverage to look for inflationary signals. It represents the latest advanced telescope to map the polarization of CMB over 30/40, 95, 150, and 220/270 GHz channels to fully characterize the Galactic synchrotron and thermal dust emissions. Thermal Kinetic Inductance Detectors (TKIDs) \cite{albert} are currently being developed and tested to be used for higher frequencies BICEP Array receivers.
The first BICEP Array receiver (BA1) has been deployed to the Amundsen-Scott South Pole Station during the 2020 austral season and is currently measuring the polarized Galactic synchrotron radiation at 30/40 GHz. We use the CMB maps at 30/40 GHz to constrain the synchrotron foreground components in the most sensitive 95/150 GHz bands closer to the CMB peak frequency which will enhance the sensitivity on r.
We have tested BA1 during 2020 and 2021 seasons of CMB observations. In this paper, we report the performance update of the BA1 camera and design challenges associated with these deployment seasons. We will also discuss our recent upgrades during that 2022 deployment season to improve the sensitivity of BA1 camera and overall receiver. We upgraded the focal plane with more dichroic detector tiles for higher detector counts and an improved filter configuration to eliminate the high frequency leaking power. These upgrades will help us to improve the mapping speed and better characterize the synchrotron contamination to CMB B-modes at the level requested to potentially measure the primordial gravitational waves.
\section{Design Overview and Optical Performance}
\subsection{Detector Design }
\label{sec:title}
Antenna-coupled transition-edge sensor (TES) detectors have been used in numerous successful CMB experiments, including BICEP Array. \cite{Ale20}\cite{Soliman20}\cite{Cheng20} Our pixel design contains two orthogonal polarized detectors, each consisting of antenna array coherently combined through a summing network and connected to a superconducting transition edge sensor (TES) bolometers \cite{Cheng20}. Furthermore, we use on-chip band defining filters designed to select the frequency of interest and reject out-of-band signals, specifically those on atmospheric lines. We use arrays of rectangular slot antennas\cite{BKII} for single color detectors and arrays of broadband bowties for dual color pixels. In the dual color pixels, the filters are part of a diplexer circuit that partitions the antenna’s broad bandwidth into narrow photometric channels at 30 GHz and 40 GHz. A novel broadband corrugated frame\cite{soli18} has been used around the wafer edges to minimize the temperature-to-polarization leakage in the CMB maps. The HFSS simulation of the antenna array and sonnet simulation of the band-pass filter show that the detectors have been designed for frequency band centered at 30 GHz and 40 GHz with about 26\% fractional bandwidth (Fig. \ref{fig:first}).
\label{sec:title}
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[height=10cm]{fpuconfig.pdf}
\end{tabular}
\end{center}
\caption[example]
{ \label{fig:first} Top: The BA1 focal plane camera configuration during 2022 season houses
a mixture of single band slot detector modules and dual color Bowtie detector modules for higher detector counts. Bottom: Bands of antennas and filters as individually simulated in HFSS and Sonnet. We co-plot these with the atmospheric transmission to show that these do not overlap the CO2 emissions. }
\end{figure}
\FloatBarrier
\subsection{Full Optical Characterization}
\label{sec:title}
We measure the detector beams both in the lab and the field to verify our instrument performance and to inform our analysis.
We measure the near-field beam maps (NFBMs) with a chopped source on a pair of translation stages in front of the BA1 camera aperture plane and record the detector timestreams. The NFBMs help us to verify performance and monitor any potential issues with the on-chip millimeter wave circuits. We have demonstrated the good agreement between the measured and simulated beams on the near-field of the detectors without the optical elements \cite{Soliman20}\cite{soli18} . Additionally, we simulated our antenna’s beam with CST MICROWAVE STUDIO software and then used the GRASP software package to propagate that beam through the BA1 optics\cite{hui18}. The resulting simulated beam agrees well with the measured beams (Fig. \ref{fig:second}). The black vertical lines indicate location of the aperture stop. We noticed that the measured beam (red curve) doesn't drop beyond the aperture stop and we thought that the reflection around the aperture stop may have caused that issue. The measured optically-active beam maps per a detector module have been coadded together to create the per-detector composite beam map at each frequency band as shown in the right side of Fig. \ref{fig:second}.
We also measured the far-field beams of BA1 camera on the sky at the South Pole (Fig. \ref{fig:third}) by using a thermal chopped source about 215 m away and raster the telescoped beam over the source with the telescope drive motors\cite{beam1}. The measured Gaussian beamwidth of 40 GHz and 30 GHz detectors are 0.36$^{\circ}$ and 0.47$^{\circ}$, respectively which are consistent with the scaled BICEP3 beamwidth at 95 GHz \cite{beam} with less than 10\% error. The simulations and measurements validate the detector and optical design of BA1 camera.
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[height=5cm]{beammodel1.PNG}
\end{tabular}
\end{center}
\caption[example]
{ \label{fig:second} Left: The simple optics model of BA1 in Grasp Software. Middle: The resulting output beam planner cut of Grasp simulation shows a good agreement with the measured beam cut of 40 GHz detector. Right: Per-detector composite near-field beam maps of the telescope, obtained by co-adding all individual NFBMs for a detector module, 40 GHz detector (top) and 30 GHz detector (bottom). The beams are peak normalized.}
\end{figure}
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[height=5cm]{ffbm.PNG}
\end{tabular}
\end{center}
\caption[example]
{ \label{fig:third} The measured far-field beam maps on the sky at the South Pole. Left: 30GHz detector. Right: 40GHz detector }
\end{figure}
\section{IMPROVED Camera SENSITIVITY (2020 vs 2022 data)}
\subsection{Out of Band Dark Loading}
BICEP detector arrays include a small number of dark detectors where the TES bolometers are not connected to an antenna. We use these dark detectors for noise and loading studies. During the 2020 season, the dark detectors in our focal plane experienced excess loading that we understood to be direct stimulation of the bolometers with high frequency out-of-band power (a Blue leak). We performed diagnostic tests using a variety of high-pass/low-pass filters and concluded that our 1.6icm (48GHz) low-pass filter leaked substantially at the 120 GHz band and above 250 GHz. We suspect that delamination of the filter layers during fabrication may have caused this defect. In the 2022 season, we installed an additional 4icm (120GHz) low pass edge (LPE) filter at the top of the focal plane camera to eliminate this leak(Fig. \ref{fig:fourth}). The out of band dark loading is substantially reduced (by a factor of four) as shown in the histogram in Fig. \ref{fig:fourth}. Additionally, the measured beam profile also shows a significant reduction in the dark pickup (Blue leak) after this upgrade for a dark detector which was common between both deployment seasons (Fig. \ref{fig:fifth}). The beam profile of the 30 GHz detector slightly changes due to the presence of the new filter and the 40 GHz detector beam profile remains the same between both seasons (Fig. \ref{fig:fifth}).
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[height=5.5cm]{dic.PNG}
\end{tabular}
\end{center}
\caption[example]
{ \label{fig:fourth} Left: The BA1 focal plane with the 4icm LPE filter upgrade during 2022 season. Right: Response of dark detectors to a chopped source for the 2020 deployment season versus 2022 deployment season. The only difference is an addition of a 4icm LPE filter that highly minimizes the Blue leak.}
\end{figure}
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[height=9.5cm]{beams.PNG}
\end{tabular}
\end{center}
\caption[example]
{ \label{fig:fifth} The beam profiles of 2020 season versus 2022 season after the 4icm LPE upgrade. Top: Dark Detector (the additional filter largely eliminates the direct stimulation). Bottom/Left: 40 GHz detector. Bottom/Right: 30 GHz detector.
}
\end{figure}
\FloatBarrier
\subsection{Noise Measurements with On-Sky Loading at the South Pole}
The BA1 camera sensitivity can be measured using the detector timestreams noise data within the BICEP/Keck science band for sky observation (0.1 Hz to 0.5 Hz). The detectors were designed to be photon noise limited with loading from the South Pole sky. Fig. \ref{fig:six} shows the 40 GHz and 30 GHz noise spectra before (purple) and after subtracting (grey) detector pairs to eliminate common mode noise. The suppression of 1/f noise down to below 0.1Hz after pair-difference polarization pairs demonstrates that the instability was common mode to both optical detectors. Our experience with other BICEP and Keck cameras leads us to suspect that it likely originates from the atmospheric fluctuations \cite{beam} but the 1/f noise could also arise from other effects. The resulting differenced spectra (grey) agree well with the anticipated photon noise level according to the expected sky loading conditions and calibration parameters for both frequency bands at the South Pole (yellow lines). The resulting spectrum validates photon-noise dominated design.
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[height=7cm]{noisepole.PNG}
\end{tabular}
\end{center}
\caption[example]
{ \label{fig:six}
Left: 30 GHz detector. Right: 40 GHz detector. Purple curves show the measured noise equivalent current (NEI) at the South Pole while grey curves show the difference between pairs in the same pixel. Note that the grey curves agree well with the expected photon noise level shown in yellow, indicating that our detectors are background noise dominated. We biased the detectors in a Titanium superconducting transition.}
\end{figure}
\section{CONCLUSIONS}
The 2022 upgrades to the BA1 camera have improved the performance over that in 2020 deployment season. The measured beam characteristics are in a good agreement with as-designed detector simulations. We also have largely eliminated the high frequency leak in power to boost the overall receiver sensitivity. The on-sky noise measurements show that the instrument is background dominated. These upgrades are important to reach the required sensitivity to map the synchrotron parameters that contributed to the CMB B-modes map. Finally, We plan to deploy the 150 GHz BICEP Array receiver (BA2) this austral summer. We also plan to deploy the 270 GHz BICEP Array receiver as well as the TKIDs demo camera the following year.
\section{acknowledgments}
The BICEP/Keck project (including BICEP2, BICEP3, and BICEP Array) have been made possible through a series of grants from the National Science Foundation including 0742818, 0742592, 1044978, 1110087, 1145172, 1145143, 1145248, 1639040, 1638957, 1638978, 1638970, 1726917, 1313010, 1313062, 1313158, 1313287, 0960243, 1836010, 1056465, \& 1255358 and by the Keck Foundation. The development of antenna-coupled detector technology was supported by the JPL Research and Technology Development Fund and NASA Grants 06-ARPA206- 0040, 10-SAT10-0017, 12-SAT12-0031, 14-SAT14-0009, 16-SAT16-0002, \& 18-SAT18-0017. The development and testing of focal planes were supported by the Gordon and Betty Moore Foundation at Caltech. Readout electronics were supported by a Canada Foundation for Innovation grant to UBC. The computations in this paper were run on the Odyssey cluster supported by the FAS Science Division Research Computing Group at Harvard University. The analysis effort at Stanford and SLAC was partially supported by the Department of Energy, Contract DE-AC02-76SF00515. We thank the staff of the U.S. Antarctic Program and in particular the South Pole Station without whose help this research would not have been possible. Tireless administrative support was provided by Kathy Deniston, Sheri Stoll, Irene Coyle, Amy Dierker, Donna Hernandez, and Julie Shih.
|
2,877,628,090,125 | arxiv |
\subsection*{A. Circular Kernel Generated by Bilinear Interpolation}
Define the vanilla $3 \times 3$ convolutional kernel as
\begin{equation}
{\boldsymbol{K}^{cls}} = \left[ {\begin{array}{*{20}{c}}
{{K_0} }&{{K_1}}&{{K_2} }\\
{{K_3}}&{{K_4}}&{{K_5}}\\
{{K_6} }&{{K_7}}&{{K_8} }
\end{array}} \right].
\tag{A-1}
\label{eq:A-1}
\end{equation}
The $3 \times 3$ circular convolutional kernel generated by bilinear interpolation from ${\boldsymbol K}^{cls}$ is represented as
\begin{equation}
{\boldsymbol{\dot K}^{cls}} = \left[ {\begin{array}{*{20}{c}}
{{\dot K_0}}&{{K_1}}&{{\dot K_2} }\\
{{K_3}}&{{K_4}}&{{K_5}}\\
{{\dot K_6}}&{{K_7}}&{{\dot K_8}}
\end{array}} \right],
\tag{A-2}
\label{eq:A-2}
\end{equation}
where
\begin{equation}
\left\{ \setlength{\arraycolsep}{1pt}{\begin{array}{l}
{\dot K_0} = {\textstyle{1 \over 2}}{K_0} + {\textstyle{{\sqrt 2 - 1} \over 2}}{K_1} + {\textstyle{{\sqrt 2 - 1} \over 2}}{K_3} + {\textstyle{{3 - 2\sqrt 2 } \over 2}}{K_4}\\
{\dot K_2} = {\textstyle{{\sqrt 2 - 1} \over 2}}{K_1} + {\textstyle{1 \over 2}}{K_2} + {\textstyle{{3 - 2\sqrt 2 } \over 2}}{K_4} + {\textstyle{{\sqrt 2 - 1} \over 2}}{K_5}\\
{\dot K_6} = {\textstyle{{\sqrt 2 - 1} \over 2}}{K_3} + {\textstyle{{3 - 2\sqrt 2 } \over 2}}{K_4} + {\textstyle{1 \over 2}}{K_6} + {\textstyle{{\sqrt 2 - 1} \over 2}}{K_7}\\
{\dot K_8} = {\textstyle{{3 - 2\sqrt 2 } \over 2}}{K_4} + {\textstyle{{\sqrt 2 - 1} \over 2}}{K_5} + {\textstyle{{\sqrt 2 - 1} \over 2}}{K_7} + {\textstyle{1 \over 2}}{K_8}
\end{array}} \right..
\tag{A-3}
\label{eq:A-3}
\end{equation}
\subsection*{B. GGHL Label Assignment Strategy}
Define the Gaussian probability density function (PDF) of the candidate position $\left( {x,y} \right)$ generated by GGHL \cite{huang2022general} as
\begin{equation}
f\left(x, y \right) = \frac{1}{{ \sqrt {2\pi \boldsymbol Q \boldsymbol \Lambda {\boldsymbol Q^T}} }} \times { e^{-\!\frac{1}{2}{
{{\left( {\boldsymbol X - \boldsymbol u} \right)}^T}{\boldsymbol C^{ - 1}}\left( {\boldsymbol X - \boldsymbol u} \right)
}}},
\tag{B-1}
\label{eq:B-1}
\end{equation}
\begin{equation}
\boldsymbol C = \boldsymbol A{\boldsymbol A^T} = \boldsymbol Q \boldsymbol \Lambda {\boldsymbol Q^T} = \left( {\boldsymbol Q{\boldsymbol \Lambda ^{1/2}}} \right){\left( {\boldsymbol Q{\boldsymbol \Lambda ^{1/2}}} \right)^T},
\tag{B-2}
\label{eq:B-2}
\end{equation}
where $\boldsymbol{X} = {\left[ {x,y} \right]^T}$. In this case, the mean vector $\boldsymbol \mu = {\left[ {{x_c},{y_c}} \right]^T}$ controls the spatial translation, the real orthogonal matrix $\boldsymbol Q = \small {\left[ \setlength{\arraycolsep}{3pt}{{\begin{array}{*{10}{c}}
{\cos \alpha }&{ -\sin \alpha }\\
{\sin \alpha }&{\cos \alpha }
\end{array}}} \right]}$ is a rotation matrix, and the diagonal matrix
$\boldsymbol \Lambda = \small {\left[ \setlength{\arraycolsep}{3pt}{{\begin{array}{*{10}{c}}
{{{{\left( {{\textstyle{{{S_1}} \over 2}}} \right)}^2}}}&{}\\
{}&{{{{\left( {{\textstyle{{{S_2}} \over 2}}} \right)}^2}}}
\end{array}}} \right]}$
represents the scaling. To make the value of $f\left(x, y \right)$ in the range $\left(0, 1 \right)$, the constant coefficient term, $\xi =\! \frac{1}{{\! \sqrt {2\pi \! \boldsymbol Q \! \boldsymbol \Lambda {\boldsymbol Q^T}} }}$ of $f\left(x, y \right)$ is removed to obtain Gaussian heatmap score $F_{x,y}$.
\subsection*{C. Experiments for the Generality of TS-Conv}
To further verify the generality of the proposed TS-Conv in other AOOD scenes, experiments are also conducted on the SKU-110KR \cite{pan2020dynamic} dataset and MSRA-TD500 \cite{yao2012detecting} dataset. SKU110-R \cite{pan2020dynamic} is an AOOD dataset for retail shelf scenarios extended from the SKU110K dataset containing 1,733,678 instances. The training, validation, and testing image numbers are 57,533, 4,116, and 20,552, respectively. MSRA-TD500 \cite{yao2012detecting} is an AOOD dataset for text detection consisting of 300 text images for training and 200 text images for testing. The results on the SKU-110KR dataset and MSRA-TD500 dataset are listed in Table~\ref{table:11} and Table~\ref{table:12}, respectively. Results show that the proposed TS-Conv also achieves better performance than existing methods in retail scene where objects are densely distributed and in text scene where objects vary more in shape. The visualization results are shown in Fig.~\ref{fig:15}. These experiments further demonstrate the generality of TS-Conv.
\begin{table}[tp]
\centering
\renewcommand\arraystretch{1.1}
\setlength{\tabcolsep}{3mm}{
\caption{\label{table:11}
{Comparative experiments on the SKU-110KR dataset}}
\resizebox{0.48\textwidth}{!}{\setlength{\tabcolsep}{3mm}{
\begin{threeparttable}
\begin{tabular}{c|c|c|c}
\hline\hline
Methods & Anchor & Backbone & mAP$_{75}$ \\ \hline
YOLOv3-R \cite{pan2020dynamic} & AB & DarkNet53 & 51.10 \\
CenterNet-R \cite{pan2020dynamic} & AF & Hourglass104 & 61.10 \\
DRN \cite{pan2020dynamic} & AF & Hourglass104 & 63.10 \\ \hline
GGHL \cite{huang2022general} (Baseline) & AF & DarkNet53 & 63.73 \\
\textbf{TS-Conv} & AF & DarkNet53 & \textbf{65.32 \tiny{(+1.59)}} \\
\hline\hline
\end{tabular}
\end{threeparttable}}}}\vspace{0.5em}
\justifying{Note: Bold indicates the best result. AF represents anchor-free methods, and AB represents anchor-based methods.}
\end{table}
\begin{table}[t]
\centering
\renewcommand\arraystretch{1.2}
\setlength{\tabcolsep}{3mm}{
\caption{\label{table:12}
{Comparative experiments on the MSRA-TD500 dataset}}
\resizebox{0.48\textwidth}{!}{\setlength{\tabcolsep}{3mm}{
\begin{threeparttable}
\begin{tabular}{c|c|c|c}
\hline\hline
Methods & Anchor & Backbone & mAP$_{50}$ \\ \hline
GGHL \cite{huang2022general} (Baseline) & AF & DarkNet53 & 70.41 \\
\textbf{TS-Conv} & AF & DarkNet53 & \textbf{74.43 \tiny{(+4.02)}} \\
\hline\hline
\end{tabular}
\end{threeparttable}}}}\vspace{0.5em}
\justifying{Note: Bold indicates the best result. AF represents anchor-free methods, and AB represents anchor-based methods.}
\end{table}
\begin{figure}[!t]
\centering
\epsfig{width=0.35\textwidth,file=15.pdf}
\caption{Visualization results of TS-Conv on a) the SKU-110KR dataset and b) the MSRA-TD500 dataset.}\label{fig:15}
\end{figure}
\begin{table}[!t]
\centering
\renewcommand\arraystretch{1}
\setlength{\tabcolsep}{7mm}{
\caption{\label{table:13}
{Abbreviation List}}
\resizebox{0.48\textwidth}{!}{\setlength{\tabcolsep}{7mm}{
\begin{threeparttable}
\begin{tabular}{c|c}
\hline\hline
Abbreviations & Full Names \\ \hline
AB & Anchor-Based\\
AF & Anchor-Free\\
AOOD & Arbitrary-oriented Object Detection\\
CNN(s) & Convolutional Neural Network(s) \\
CPU & Central Processing Unit\\
CS-Conv & Classification Sampling Convolution \\
DCK & Dynamic Circular Kernel \\
DCN(s) & Deformable Convolutional Network(s) \\
DTLA & Dynamic Task-aware Label Assignment \\
fps & frames per second\\
FLOPs & Floating Point Operations\\
GIoU & Generalized Intersection over Union\\
GPU & Graphics Processing Unit\\
HBB(s) & Horizontal Bounding Box(es) \\
IFS & Inconsistent Feature Sensitivity \\
IoU & Intersection over Union\\
LS-Conv & Localization Sampling Convolution \\
mAP & mean Average Precision\\
MERect & Minimum External Rectangle \\
MSE & Mean Square Error \\
NMS & Non-maximum Suppression\\
OBB(s) & Oriented Bounding Box(es) \\
OD & Object Detection \\
PDF & Probability Density Function\\
RRDA & Random Rotation Data Augmentation\\
SAR & Synthetic Aperture Radar \\
SGD & Stochastic Gradient Descent\\
TS-Conv & Task-wise Sampling Convolutions \\
\hline\hline
\end{tabular}
\end{threeparttable}}}}
\end{table}
\subsection*{D. Abbreviation List}
All the abbreviations used in this paper are summarized in Table~\ref{table:13}.
\subsection{Arbitrary-Oriented Object Detection}
Unlike ordinary OD, the objects in AOOD are positioned by OBBs. Thus, additional CNN localization branches with different OBB representation strategies are designed to predict OBBs. ROI Transform \cite{ding2019learning}, SCR-Det \cite{yangSCRDetMoreRobust2019}, R3Det \cite{yang2019r3det}, S$^2$ANet \cite{han2021align}, etc., predicted rotation angles based on horizontal bounding boxes (HBBs) to obtain OBBs, while Gliding Vertex \cite{xu2020gliding}, GGHL \cite{huang2022general}, etc. directly predicted the four-vertices of each OBB. Oriented RepPoints \cite{li2022oriented} represented OBBs by a set of points. Distinguishing from the above angle or vertex regression strategies, CSL \cite{yang2020arbitrary} predicted the rotation angles of OBBs by discrete angle classification. To get rid of the dependence on anchor boxes, BBAVectors \cite{yi2020oriented}, O$^2$-DNet \cite{wei2020oriented}, GGHL \cite{huang2022general}, etc., developed different anchor-free label assignment strategies, and AO2-DETR \cite{dai2022ao2} predicted OBBs by the sequence model without anchor boxes. For more accurate OBB predictions, GWD \cite{yang2021rethinking} and KLD \cite{yang2021learning} designed new loss functions based on the distances of Gaussian distributions generated from the predicted OBBs and ground truth. In response to the IFS problem in AOOD, GGHL \cite{huang2022general}, CFC-Net \cite{ming2021cfc}, etc., decoupled the CNN branches and reweighted task-wise features by different strategies. Oriented RepPoints \cite{li2022oriented} predicted initial OBBs and categories through aligned features extracted by shared-offsets DCNs and then learned additional offsets to refine the initial OBBs. Different from the existing AOOD methods, the proposed TS-Conv separately samples task-wise features without additional reweighting or refinement operation. Besides, TS-Conv associates the task-wise feature sampling with other parts of the AOOD pipeline, including the OBB representation and label assignment.
\subsection{Solutions for the IFS Problem}
IoU-Net \cite{jiang2018acquisition} analyzed the IFS problem for the first time, which predicted an additional localization score and aggregated it with the classification score as the final score. Along with this idea of post-processing at the prediction side, IoU-aware \cite{zhang2021varifocalnet}, PISA \cite{cao2020prime}, GGHL \cite{huang2022general}, etc., designed different strategies to obtain prediction scores that consider both localization and classification contributions by re-weighting, re-ranking, or jointly optimizing the scores, etc. Nevertheless, the feature-sensitive positions are still spatial misalignment. Double-Head R-CNN \cite{wu2020rethinking}, RetinaNet \cite{linFocalLossDense2017}, YOLOX \cite{ge2021yolox}, etc., decoupled the detection head of CNN into two branches to extract localization and classification features separately. Based on this feature-decoupling scheme, two-stage methods, TSD \cite{song2020revisiting}, D2Det \cite{cao2020d2det}, etc., extracted task-suitable features by task-wise Deformable RoI Pooling \cite{zhu2019deformable}. One-stage methods, such as Guided Anchoring \cite{wang2019region}, RefineDet \cite{zhang2018single}, CFC-Net \cite{ming2021cfc}, VFNet \cite{zhang2021varifocalnet}, etc., aligned and refined the decoupled features by DCNs \cite{dai2017deformable,zhu2019deformable} or attention mechanism. Furthermore, RepPoints \cite{yang2019reppoints} and Oriented RepPoints \cite{li2022oriented} extracted localization and classification features from the spatially aligned positions in two branches by two DCNs that share the same sampling offsets. In addition, Oriented RepPoints \cite{li2022oriented} designed a quality assessment and sample assignment strategy for selecting high-quality sampling points and positions. However, the sampling positions of localization and classification features in the above methods are spatially aligned. It contradicts the observations in Fig.~\ref{fig:2} that the feature-sensitive regions of different tasks are different. Inspired by the above contradiction, this work designs separately sampling strategies for task-wise features to solve the IFS problem. Moreover, TS-Conv also associates OBB representation and dynamic label assignment with task-wise feature sampling, which deals with the IFS problem more comprehensively from different perspectives of the AOOD framework.
\subsection{Convolution for Sampling Localization Features}
From the observation in Fig.~\ref{fig:2} (a) and the analysis of \cite{song2020revisiting}, the feature-sensitive regions for localization are ainly distributed at the object boundary, i.e., the regions near the OBB. Inspired by this observation and the star-shaped box of VFNet \cite{zhang2021varifocalnet}, associating the feature sampling positions of the LS-Conv with the OBB representation is considered. For OBB representation, GGHL \cite{huang2022general} proposed a flexible anchor-free strategy based on Gliding Vertex \cite{xu2020gliding}, as shown in Fig.~\ref{fig:4} (a). It predicts an OBB by predicting the distances ${l_n},{\rm{ }}n = 1,2,3,4,$ from the current candidate object position $q_4\left( {x,y} \right)$ to the four edges of the external horizontal bounding box (HBB) and the distances ${s_n},{\rm{ }}n = 1,2,3,4,$ from the four vertices $q_0$, $q_2$, $q_6$, and $q_8$ of the HBB to the four vertices $q_1$, $q_3$, $q_5$, and $q_7$ of the OBB on the corresponding edge. This OBB representation has nine key positions, i.e., the four vertices of the HBB, the four vertices of the OBB, and the current candidate object position. A straightforward idea is to directly correspond these key positions to the nine sampling points of a $3 \times 3$ convolution, as shown in Fig.~\ref{fig:4} (c). The localization convolution no longer explores the features in the current position and its eight neighborhoods fixedly while adjusting the sampling position according to different objects with different OBBs by DCN \cite{pan2020dynamic} dynamically. The details and further improvements are presented below
\begin{figure}[tbp]
\centering
\epsfig{width=0.46\textwidth,file=4.pdf}
\caption{The principle of the proposed task-wise sampling convolutions (TS-Conv). (a) The OBB representation of GGHL \cite{huang2022general}. (b) The CNN structure of the proposed TS-Conv. (c) The sampling positions of the convolution for sampling localization features (LS-Conv). (d) The sampling positions of the convolution for sampling classification features (CS-Conv).}
\label{fig:4}
\end{figure}
First, the initial OBBs are predicted through GGHL \cite{huang2022general}, and supervised by pre-assigned labels. The initial estimations of ${l_n}$ and ${s_n,{\rm{ }}n = 1,2,3,4,}$ are denoted as ${\hat l_n}$ and ${\hat s_n,{\rm{ }}n = 1,2,3,4}$, respectively. In this case, ${\hat s_1},{\hat s_3} \in \left[ {0,{{\hat l}_2} + {{\hat l}_4}} \right]$ and ${\hat s_2},{\hat s_4} \in \left[ {0,{{\hat l}_1} + {{\hat l}_3}} \right]$. Second, the spatial coordinates of the sampling points ${q_i},$ $i = 0,1, \cdots ,8,$ are calculated according to the predictions ${\hat l_n}$, ${\hat s_n}$, and the coordinates of the current convolutional position $\left( {x,y} \right)$. In this case shown in Fig.~\ref{fig:4} (a), the current sample point is defined as ${q_4}=\left( x, y \right)$. The four vertices of HBB are defined as ${q_0}=\left( {x - {{\hat l}_4},} \right.$ $\left. {y - {{\hat l}_1}} \right)$, ${q_2}=\left( {x + {{\hat l}_2},} \right.$ $\left. {y - {{\hat l}_1}} \right)$, ${q_6}=\left( {x - {{\hat l}_4},} \right.$ $\left. {y + {{\hat l}_3}} \right)$ and ${q_8}=\left( {x + {{\hat l}_2},} \right.$ $\left. {y + {{\hat l}_3}} \right)$, respectively. The four vertices of OBB are defined as ${q_1}=\left( {x - {{\hat l}_4} + {s_1},}\right.$ $\left. {y - {{\hat l}_1}} \right)$, ${q_3}=\left( {x - {{\hat l}_4},} \right.$ $\left. {y + {{\hat l}_3} - {{\hat s}_4}} \right)$, ${q_5}=\left( {x + {{\hat l}_2},} \right.$ $\left. {y - {{\hat l}_1} + {{\hat s}_2}} \right)$ and ${q_7}=\left( {x + {{\hat l}_2} - {{\hat s}_3},} \right.$ $\left. {y + {{\hat l}_3}} \right)$, respectively. However, since the four vertices ${q_0},{q_2},{q_6},{q_8}$, of the HBB do not necessarily fall on the object, their extracted background features may interfere with the localization. Therefore, as shown in Fig.~\ref{fig:4} (c), replace $q_{0}$, $q_{2}$, $q_{6}$, and $q_{8}$, with four points on the four sides of the OBB, respectively. These four points can be moved on the four sides of the OBB to be close to the object, which are controlled by the CNN learnable variables ${\sigma _i} \in \left( {0,1} \right),{\rm{ }}i = 0,2,6,8$. The coordinates of points ${\tilde q_0},{\tilde q_2},{\tilde q_6},{\tilde q_8},$ are listed in Table~\ref{table:1}. For convenience, let ${\tilde q_i} = {q_i},{\rm{ }}i = 1,3,4,5,7,$ in the following.
\begin{table}[tbp]
\centering
\renewcommand\arraystretch{1.2}
\caption{{Summary of sampling points ${\tilde q_i},{\rm{ }}i = 0,2,6,8,$ of the designed LS-Conv}}
\label{table:1}
\setlength{\tabcolsep}{1.5mm}{
\resizebox{0.48\textwidth}{!}{
\begin{tabular}{m{1cm}<{\centering}|m{5cm}<{}|m{2.5cm}<{\centering}}
\hline\hline
Points & \centering Coordinates & Binding Conditions \\
\hline
\multirow{4}{1cm}{\centering ${\tilde q_0}$} & $\left\{ \begin{array}{l}
{x_{{{\tilde q}_0}}} = {x_{{q_1}}}\\
{y_{{{\tilde q}_0}}} = {y_{{q_1}}} + {\sigma _0} \times \left( {{y_{{q_3}}} - {y_{{q_1}}}} \right)
\end{array} \right.$ & ${\rm{ }}{x_{{q_1}}} = {x_{{q_0}}}$ \\
\cline{2,3}
~ & $\left\{ \begin{array}{l}
{x_{{{\tilde q}_0}}} = {x_{{q_0}}} + {\sigma _0} \times \left( {{x_{{q_1}}} - {x_{{q_0}}}} \right)\\
{y_{{{\tilde q}_0}}} = {y_{{q_1}}} + \frac{{{y_{{q_3}}} - {y_{{q_1}}}}}{{{x_{{q_3}}} - {x_{{q_1}}}}} \times \left( {{x_{{{\tilde q}_0}}} - {x_{{q_1}}}} \right)
\end{array} \right.$
& ${x_{{q_0}}} < {x_{{q_1}}} \le {x_{{q_2}}}$ \\
\hline
\multirow{4}{1cm}{\centering ${\tilde q_2}$} & $\left\{ \begin{array}{l}
{x_{{{\tilde q}_2}}} = {x_{{q_1}}}{\rm{ }}\\
{y_{{{\tilde q}_2}}} = {y_{{q_2}}} + {\sigma _2} \times \left( {{y_{{q_5}}} - {y_{{q_2}}}} \right)
\end{array} \right.$ & ${x_{{q_1}}} = {x_{{q_2}}}$ \\
\cline{2,3}
~ & $\left\{ \begin{array}{l}
{x_{{{\tilde q}_2}}} = {x_{{q_2}}} - {\sigma _2} \times \left( {{x_{{q_2}}} - {x_{{q_1}}}} \right),{\rm{ }}\\
{y_{{{\tilde q}_2}}} = {y_{{q_1}}} + \frac{{{y_{{q_5}}} - {y_{{q_1}}}}}{{{x_{{q_5}}} - {x_{{q_1}}}}} \times \left( {{x_{{{\tilde q}_2}}} - {x_{{q_1}}}} \right)
\end{array} \right.$
& ${x_{{q_0}}} \le {x_{{q_1}}} < {x_{{q_2}}}$ \\
\hline
\multirow{4}{1cm}{\centering ${\tilde q_6}$} & $\left\{ \begin{array}{l}
{x_{{{\tilde q}_6}}} = {x_{{q_7}}}\\
{y_{{{\tilde q}_6}}} = {y_{{q_6}}} - {\sigma _6} \times \left( {{y_{{q_7}}} - {y_{{q_3}}}} \right)
\end{array} \right.$ & ${x_{{q_6}}} = {x_{{q_7}}}$ \\
\cline{2,3}
~ & $\left\{ \begin{array}{l}
{x_{{{\tilde q}_6}}} = {x_{{q_6}}} + {\sigma _6} \times \left( {{x_{{q_7}}} - {x_{{q_6}}}} \right)\\
{y_{{{\tilde q}_6}}} = {y_{{q_7}}} + \frac{{{y_{{q_3}}} - {y_{{q_7}}}}}{{{x_{{q_3}}} - {x_{{q_7}}}}} \times \left( {{x_{{{\tilde q}_6}}} - {x_{{q_7}}}} \right)
\end{array} \right.$
& ${x_{{q_6}}} < {x_{{q_7}}} \le {x_{{q_8}}}$ \\
\hline
\multirow{4}{1cm}{\centering ${\tilde q_8}$} & $\left\{ \begin{array}{l}
{x_{{{\tilde q}_8}}} = {x_{{q_7}}}{\rm{ }}\\
{y_{{{\tilde q}_8}}} = {y_{{q_7}}} - {\sigma _8} \times \left( {{y_{{q_7}}} - {y_{{q_5}}}} \right)
\end{array} \right.$ & ${x_{{q_7}}} = {x_{{q_8}}}$ \\
\cline{2,3}
~ & $\left\{ \begin{array}{l}
{x_{{{\tilde q}_8}}} = {x_{{q_8}}} - {\sigma _{{p_7}}} \times \left( {{x_{{q_8}}} - {x_{{q_7}}}} \right){\rm{ }}\\
{y_{{{\tilde q}_8}}} = {y_{{q_7}}} + \frac{{{y_{{q_5}}} - {y_{{q_7}}}}}{{{x_{{q_5}}} - {x_{{q_7}}}}} \times \left( {{x_{{{\tilde q}_8}}} - {x_{{q_7}}}} \right)
\end{array} \right.$
& ${x_{{q_6}}} \le {x_{{q_7}}} < {x_{{q_8}}}$ \\
\hline\hline
\end{tabular}}}
\end{table}
Define the feature map input to the localization branch as ${\boldsymbol{I}^{loc}} \in {\mathbb{R}^{W \times H \times F}}$, where $W,\ H,$ and $F$ denote the width, height, and the number of feature maps, respectively. Furthermore, inspired by extracting features from keypoint while letting CNN learn their position information, a spatial coordinate embedding operation is employed on LS-Conv. Generate a tensor ${\boldsymbol{I}^{coor-x}} \in {\mathbb{N}^{W \times H \times 1}}$. All the elements on each column of ${\boldsymbol{I}^{coor-x}}$ are the same, i.e., the index of this column. Similarly, generate a tensor ${\boldsymbol{I}^{coor-y}} \in {\mathbb{N}^{W \times H \times 1}}$. All the elements on each row of ${\boldsymbol{I}^{coor-y}}$ are the same, i.e., the index of this row. Define ${\tilde{\boldsymbol{I}}^{loc}} \in {\mathbb{R}^{W \times H \times (F+2)}}$ as the tensor of ${{\boldsymbol{I}}^{loc}}$, ${\boldsymbol{I}^{coor-x}}$, and ${\boldsymbol{I}^{coor-y}}$, i.e., ${\tilde{\boldsymbol{I}}^{loc}} \left({1:W}, {1:H}, {1:F}\right)\!={\boldsymbol{I}}^{loc}$, ${\tilde{\boldsymbol{I}}^{loc}} \left({1:W}, {1:H}, {F+1}\right)\!={\boldsymbol{I}}^{coor-x}$, ${\tilde{\boldsymbol{I}}^{loc}} \left({1:W}, {1:H}, {F+2}\right)\!={\boldsymbol{I}}^{coor-y}$. Define the element of ${\boldsymbol{\tilde I}^{loc}}$ at $\left( {x,y} \right)$, $x \in \left[ {1,W} \right]$ and $y \in \left[ {1,H} \right]$, in the first two dimensions as $\boldsymbol{\tilde I}_{x,y}^{loc} \in \mathbb{R}^{1\times1\times(F+2)}$, which consists of a position and its associated $F$-dimensional feature vector. It allows the CNN to learn the spatial coordinates of the sampled positions directly while extracting the features from sensitive regions \cite{liu2018intriguing}. In addition, because the size $F$ of the feature vector satisfies $F \gg 2$ in CNN, the spatial coordinate embedding operation does not add much additional computational burden.
For notational convenience, we define the correspondence of the nine elements (positions) of the following set
\begin{equation}
\begin{array}{l}
\left\{ {\left( { - 1, - 1} \right),\left( { - 1,0} \right)} \right.,\left( { - 1,1} \right),\left( {0, - 1} \right),\\
\left( {0,0} \right),\left( {0,1} \right),\left. {\left( {1, - 1} \right),\left( {1,0} \right),\left( {1,1} \right)} \right\}\\
\buildrel \Delta \over = \left\{ {\left( {\Delta x_u^{loc},\Delta y_u^{loc}} \right),u = 0,1, \cdots ,8} \right\}.
\end{array}
\label{eq:0_1}
\end{equation}
The indices of $\left( {x,y} \right)$ and its eight neighbors are represented as $\left( {x + \Delta x_u^{loc},{\rm{ }}y + \Delta y_u^{loc}} \right)$, $u = 0,1, \cdots ,8$. Define the feature vectors at $\left( {x + \Delta {x_u^{loc}},{\rm{ }}y + \Delta {y_u^{loc}}} \right)$ as $\boldsymbol{\tilde I}_{x + \Delta {x_u^{loc}},y + \Delta {y_u^{loc}}}^{loc}$. Define a $3\times3$ filter
\begin{equation}
{\boldsymbol{K}^{loc}} = \left[ {\begin{array}{*{20}{c}}
{K_0^{loc}}&{K_1^{loc}}&{K_2^{loc}}\\
{K_3^{loc}}&{K_4^{loc}}&{K_5^{loc}}\\
{K_6^{loc}}&{K_7^{loc}}&{K_8^{loc}}
\end{array}} \right],
\label{eq:1}
\end{equation}
where $K_j^{loc},{\rm{ }}j = 0,1, \cdots ,8,$ are the elements and also the coefficients of $\boldsymbol{K}^{loc}$. When using $\boldsymbol{K}^{loc}$ to perform LS-Conv\footnote[1]{Note that, it is conventionally called a convolution operation in CNN, but it is actually a filtering operation.} on ${\boldsymbol{\tilde I}^{loc}}$, the following two cases are considered. When position $\left( {x,y} \right)$ is a positive position, which will be explained in Section III-C, used to predict the OBB, as shown in Fig.~\ref{fig:4} (c), the nine sampling points $\left( {x + \Delta {x_u^{loc}},{\rm{ }}y + \Delta {y_u^{loc}}} \right),{\rm{ }}u = 0,1, \cdots ,8,$ are correspondingly moved to the points $\left( {{x_{{{\tilde q}_i}}},{y_{{{\tilde q}_i}}}} \right),{\rm{ }}i = 0,1, \cdots ,8,$ by DCN \cite{dai2017deformable}. The sampled feature vectors are denoted as $\boldsymbol{\tilde I}_{{x_{{q_i}}},{y_{{q_i}}}}^{loc},{\rm{ }}i = 0,1, \cdots ,8$. When the position $\left( {x,y} \right)$ is not a positive position, the sampling points $\left( {x + \Delta {x_u^{loc}},{\rm{ }}y + \Delta {y_u^{loc}}} \right),{\rm{ }}u = 0,1, \cdots ,8,$ do not move. Thus, the output of LS-Conv at $\left( {x,y} \right)$ is represented as
\begin{equation}
\! \boldsymbol{O}_{x,y}^{loc} \! = \! \left\{ \begin{array}{l}
\! \sum\limits_{\! i, j = 0}^{\! N = 8} { {\boldsymbol{\tilde I}_{{x_{{\tilde q_i}}},{y_{{\tilde q_i}}}}^{loc} K_j^{\! loc}} } m_j^{\! loc},{\enspace \rm{if}}\ (x,y){\ \rm{ is \ positive}},\\
\! \sum\limits_{u, j = 0}^{N = 8} {\boldsymbol{\tilde I}_{x + \Delta {x_u^{\! loc}},y + \Delta {y_u^{\! loc}}}^{loc} K_j^{\! loc} m_j^{\! loc}} ,{\enspace \rm{otherwise,}}
\end{array} \right.
\label{eq:2}
\end{equation}
where $\boldsymbol{O}_{x,y}^{loc} \in \mathbb{R}^{1\times1\times(F+2)}$ and CNN-learnable scalars $m_j^{loc} \in \left( {0,1} \right),{\rm{ }}j = 0,1, \cdots ,8,$ are employed to adjust the contribution of the features sampled from different positions like DCNv2 \cite{zhu2019deformable}. It further enhances the feature learning capability of LS-Conv by simultaneously adjusting the sampling position and amplitude. For some OBB vertices that do not fall on the object, it can suppress the sampling of low-quality features by decreasing the value of $m_j^{loc}$. Finally, the output features are used to predict the corrections $\Delta {\hat l_n}$ and $\Delta {\hat s_n},{\rm{ }}n = 1,2,3,4,$ to the initial OBB predictions ${\hat l_n}$ and ${\hat s_n},{\rm{ }}n = 1,2,3,4$. The refined OBB predictions are
\begin{equation}
\left\{ \begin{array}{l}
{{\tilde l}_n} = {{\hat l}_n} \times \Delta {{\hat l}_n}\\
{{\tilde s}_n} = {{\hat s}_n} \times \Delta {{\hat s}_n}
\end{array} \right.,
\label{eq:3}
\end{equation}
where ${\tilde s_1},{\tilde s_3} \in \left[ {0,{{\tilde l}_2} + {{\tilde l}_4}} \right]$ and ${\tilde s_2},{\tilde s_4} \in \left[ {0,{{\tilde l}_1} + {{\tilde l}_3}} \right]$. Note that, due to the significant difference in the sizes of different objects, i.e., the values of ${\hat l}_n$ and ${\hat s}_n$, Eq.~\ref{eq:3} uses multiplications rather than additions to make the ranges of $\Delta {{\hat l}_n}$ and $\Delta {{\hat s}_n}$ predicted by the CNN less affected by the object size. In addition, the refined OBB predictions are also supervised by the ground truth during the CNN training. Compared to RepPoints \cite{yang2019reppoints}, Oriented RepPoints \cite{li2022oriented}, etc., which first obtain the set of dynamic sampling points and then generate bounding boxes from the point set, the sampling positions of LS-Conv are obtained from OBBs directly. Thus, the receptive field of LS-Conv is always adapted to the object size. It makes the supervision of localization feature sampling in CNN more comprehensive to the final task objective.
\subsection{Convolution for Sampling Classification Features}
Unlike localization feature-sensitive regions directly associated with the OBB representation and spatial coordinates, sensitive regions of classification features are more variable for objects with different categories, shapes, and orientations. Therefore, the sampling positions of the classification convolution are only constrained within the OBB for more flexible adjustment according to each object, as shown in Fig.~\ref{fig:4} (d). Since the OBB is an arbitrary convex quadrilateral, the constraint range is approximated as the minimum external rectangle (MERect)\footnote[2]{OpenCV and many computer vision libraries have implemented this function, so the details are not repeated here.} of the OBB in this case in order to facilitate parallel computation in CNN. Define the length of MERect’s long side as ${S_1}$, the length of the other side as ${S_2}$, the center point of MERect as $\left( {{x_{c}},{y_{c}}} \right)$, and the angle between the long side and the positive direction of $x$-axis as $\alpha$, $\alpha \in \left[ 0, \pi \right)$ in this case. Define the sampling positions of the CS-Conv as ${p_i},i = 0,1, \cdots ,8,$ and the CNN-learnable variables $\omega _i^{\left( x \right)},\omega _i^{\left( y \right)} \in \left( {0,1} \right),{\rm{ }}i = 0,1, \cdots ,8,$ used to adjust the sampling positions. According to the designed constrain, the coordinates of the sampling positions are represented as
\begin{equation}
\left[ \setlength{\arraycolsep}{0.5pt}{{\begin{array}{*{5}{c}}
{{x_{{p_i}}}}\\
{{y_{{p_i}}}}
\end{array}}} \right] \!= \! \left[ \setlength{\arraycolsep}{3pt}{{\begin{array}{*{5}{c}}
{\cos \alpha }&{ -\sin \alpha }\\
{\sin \alpha }&{\cos \alpha }
\end{array}}} \right] \! \times \! \left[\setlength{\arraycolsep}{2pt}{ {\begin{array}{*{5}{c}}
{{x_c} - 0.5{S_1} + \varpi _{{p_i}}^{\left( x \right)}{S_1}}\\
{{y_c} - 0.5{S_2} + \varpi _{{p_i}}^{\left( y \right)}{S_2}}
\end{array}}} \right],
\label{eq:4}
\end{equation}
where $\left( {{x_{{p_i}}},{y_{{p_i}}}} \right)$ denote the coordinates of the sampling position ${p_i},i = 0,1, \cdots ,8$.
\begin{figure}[tbp]
\centering
\epsfig{width=0.4\textwidth,file=5.pdf}
\caption{The designed dynamic circluar kernel (DCK). (a) Circular convolutional kernel obtained by bilinear interpolation. (b) Circular convolution kernels with eight orientations. (c) Adaptive fusion of circular and square convolutional kernels. (d) Adaptive fusion of eight-rotation features.}
\label{fig:5}
\end{figure}
Although the above design allows the sampling positions to translate to the sensitive regions within the OBB adaptively, the sensitive regions are variable due to the arbitrary orientations of objects in AOOD. It brings more significant uncertainty in the sampling position offsets, and CS-Conv lacks absolute-coordinate information like SCE in LS-Conv. Compared to rotating the sampling positions with more complex constraints, rotating the convolution kernel to the appropriate direction is a more straightforward idea. Thus, a dynamic circular kernel (DCK) is designed.
First, as shown in Fig.~\ref{fig:5} (a), a circular kernel ${\boldsymbol{\dot K}^{cls}} \in \mathbb{R}^{3\times3}$,
\begin{equation}
{\boldsymbol{\dot K}^{cls}} = \left[ {\begin{array}{*{20}{c}}
{\dot K_0^{cls}}&{K_1^{cls}}&{\dot K_2^{cls}}\\
{K_3^{cls}}&{K_4^{cls}}&{K_5^{cls}}\\
{\dot K_6^{cls}}&{K_7^{cls}}&{\dot K_8^{cls}}
\end{array}} \right],
\label{eq:5}
\end{equation}
is generated from the square kernel ${\boldsymbol{K}^{cls}} \in \mathbb{R}^{3\times3}$ by bilinear interpolation to accommodate rotations in different directions, see Appendix A for details. The coefficients of ${\boldsymbol{\dot K}^{cls}}$ are defined as ${\dot K}_j^{cls},{\rm{ }}j = 0,1, \cdots ,8$, and the coefficients of ${\boldsymbol{K}^{cls}}$ are defined as ${K}_j^{cls},{\rm{ }}j = 0,1, \cdots ,8$. When $j=1,3,4,5,7$, $\dot K_j^{cls} = K_j^{cls}$. Second, as shown in Fig.~\ref{fig:5} (b), the circular convolution kernel ${\boldsymbol{\dot K}^{cls}}$ is rotated by the angles $\varphi = \frac{{k\pi }}{4},k = 0,1,2, \cdots ,7,$ in the clockwise direction to obtain eight circular kernels, denoted as ${\boldsymbol{\dot K}_{{{{k\pi } \over 4}}}^{cls}}$. Similarly, the square kernel ${\boldsymbol{K}^{cls}}$ is rotated clockwise by the angles $\varphi = \frac{{k\pi }}{4},k = 0,2,4,6,$ to get ${\boldsymbol{K}_{{{{k\pi } \over 4}}}^{cls}}$. When $\varphi = \frac{{k\pi }}{4},k = 0,2,4,6,$ the square kernels and circular kernels \cite{https://doi.org/10.48550/arxiv.2107.02451} are fused to enhance the model fitting ability of CNNs, using the idea of CondConv \cite{yang2019condconv}. The fused kernels are represented as
\begin{equation}
\boldsymbol{\tilde K}_{{{{k\pi } \over 4}}}^{cls} = \left\{ \setlength{\arraycolsep}{1pt}{\begin{array}{l}
{\lambda _j}{\boldsymbol{\dot K}_{{{{k\pi } \over 4}}}^{cls}} + \left({1-{\lambda_j}} \right){\boldsymbol{ K}_{{{{k\pi } \over 4}}}^{cls}}, {\quad \rm{if}}\ k = 0,2,4,6,\\
{\boldsymbol{\dot K}_{{{{k\pi } \over 4}}}^{cls}}, {\quad \rm{if}}\ k = 1,3,5,7,
\end{array}} \right.
\label{eq:6}
\end{equation}
where ${\lambda _j} \in \left( {0,1} \right),j = 1,2,3,4$, denote the CNN-learnable weights for fusion, whose generation module in CNN is shown in Fig.~\ref{fig:5} (c). Similarly, the coefficients of $\boldsymbol{\tilde K}_{{{{k\pi } \over 4}}}^{cls}$ are defined as ${\tilde K}_{j,{{k\pi } \over 4}}^{cls},\ {\rm{ }}j,k = 0,1, \cdots ,8$.
Similar to the localization, define the nine elements (positions) of the following set
\begin{equation}
\begin{array}{l}
\left\{ {\left( { - {\textstyle{{\sqrt 2 } \over 2}}, - {\textstyle{{\sqrt 2 } \over 2}}} \right),\left( { - 1,0} \right)} \right.,\left( { - {\textstyle{{\sqrt 2 } \over 2}},{\textstyle{{\sqrt 2 } \over 2}}} \right),\left( {0, - 1} \right),\\
\left( {0,0} \right),\left( {0,1} \right),\left. {\left( {{\textstyle{{\sqrt 2 } \over 2}}, - {\textstyle{{\sqrt 2 } \over 2}}} \right),\left( {1,0} \right),\left( {{\textstyle{{\sqrt 2 } \over 2}},{\textstyle{{\sqrt 2 } \over 2}}} \right)} \right\}\\
\buildrel \Delta \over = \left\{ {\left( {\Delta x_v^{cls},\Delta y_v^{cls}} \right),v = 0,1, \cdots ,8} \right\}.
\end{array}
\label{eq:0_2}
\end{equation}
The indices of $\left( {x,y} \right)$ and its eight circular neighbors are represented as $\left( {x + \Delta x_v^{cls},{\rm{ }}y + \Delta y_v^{cls}} \right)$, $v = 0,1, \cdots ,8$. Define the feature map input to the classification branch as ${\boldsymbol{I}^{cls}} \in {\mathbb{R}^{W \times H \times F}}$. The elements of ${\boldsymbol{I}^{cls}}$ at $\left( {x + \Delta {x_v^{cls}},{\rm{ }}y + \Delta {y_v^{cls}}} \right)$, $v = 0,1, \cdots ,8$, are defined as $\boldsymbol{I}_{x + \Delta {x_v^{cls}},y + \Delta {y_v^{cls}}}^{cls}$, which are $F$-dimensional feature vectors. When position $\left( {x,y} \right)$ is a positive position, as shown in Fig.~\ref{fig:4} (d), the nine sampling points $\left( {x + \Delta {x_v^{cls}},{\rm{ }}y + \Delta {y_v^{cls}}} \right),{\rm{ }}v = 0,1, \cdots ,8,$ are correspondingly moved to the points $\left( {{x_{{{p}_i}}},{y_{{{p}_i}}}} \right),\ {\rm{ }}i = 0,1, \cdots ,8,$ by DCN \cite{dai2017deformable}. The sampled feature vectors are denoted as $\boldsymbol{I}_{{x_{{p_i}}},{y_{{p_i}}}}^{cls},{\rm{ }}i = 0,1, \cdots ,8$. When the position $\left( {x,y} \right)$ is not a positive position, the sampling points $\left( {x + \Delta {x_v^{cls}},{\rm{ }}y + \Delta {y_v^{cls}}} \right),\ {\rm{ }}v = 0,1, \cdots ,8,$ do not move. Thus, the output of CS-Conv at $\left( {x,y} \right)$ is represented as
\begin{equation}
\! \boldsymbol{O}_{x,y}^{cls} \! = \! \left\{ \begin{array}{l}
\! \sum\limits_{k = 0}^{M = 7} {\sum\limits_{\! i, j = 0}^{\! N = 8} { {\boldsymbol{I}_{{x_{{p_i}}},{y_{{p_i}}}}^{cls}\! {\tilde K}_{j,{{k\pi} \over 4}}^{cls} } } {\beta _k} m_j^{cls}},{\enspace \rm{if}}\ (x,y){\ \rm{ is \ positive}},\\
\! \sum\limits_{k = 0}^{M = 7} {\! \sum\limits_{v, j = 0}^{N = 8} {\boldsymbol{I}_{x + \Delta {x_v^{cls}},y + \Delta {y_v^{cls}}}^{cls}\! {\tilde K}_{j,{{k\pi} \over 4}}^{cls} {\beta _k} m_j^{cls}}} ,{\enspace \rm{otherwise,}}
\end{array} \right.
\label{eq:7}
\end{equation}
where $\boldsymbol{O}_{x,y}^{cls} \in \mathbb{R}^{1\times1\times F}$, and CNN-learnable scalars ${\beta _k},k = 0,1, \cdots ,7,{\rm{ }}\sum\nolimits_{k = 0}^{M = 7} {{\beta _k} = 1} ,$ are used to re-weight the features extracted by ${\boldsymbol{\tilde K}_{{{{k\pi } \over 4}}}^{cls}}$ with different orientations, as shown in Fig.~\ref{fig:5} (d). CNN-learnable scalars $m_j^{cls} \in \left( {0,1} \right)$, $j = 0,1, \cdots ,8,$ are utilized to adjust the contributions of different sampling positions. In CNN, CS-Conv with multiple orientations is implemented by group convolution to save computational costs.
\subsection{Dynamic Task-aware Label Assignment}
For the CNN-based object detection, an object may correspond to more than one candidate detection position in the feature maps. It is significant to select the appropriate positive and negative positions from them to assign labels for CNN training. GGHL \cite{huang2022general} selected candidate positions based on Gaussian heatmaps and OBB prediction scores, as shown in Fig.~\ref{fig:6} (a). However, on one hand, it uses a hard-thresholding selection strategy, i.e., the candidate positions with the value of Gaussian heatmap score $F_{x,y}$ higher than the threshold $T$ are positive, and the other positions are negative. See Appendix B for the derivation of $F_{x,y}$. The fixed candidate regions may lead to misassignment due to the irregularity and variety of object shapes. On the other hand, a location closer to the center of the Gaussian heatmap may not be a better positive position. Only using localization score to adjust the weights of positive positions in GGHL \cite{huang2022general} still faces the IFS problem that these positive positions may not be optimal for both localization and classification tasks.
\begin{figure}[tb]
\centering
\epsfig{width=0.48\textwidth,file=6.pdf}
\caption{The label assignment strategy for AOOD: (a) GGHL \cite{huang2022general} and (b) DTLA. The designed DTLA consists of (b-1) the dynamic positive candidate position assignment based on task-aware scores and (b-2) the soft-weighted negative candidate position assignment.}
\label{fig:6}
\end{figure}
\begin{algorithm}[!tbp]
\label{alg:1}
\footnotesize
\caption{Dynamic task-aware label assignment
\LinesNumbered
\KwIn{The encoded ground truth $\boldsymbol{l}_{x,y}$, $\boldsymbol{s}_{x,y}$, ${ar}_{x,y}$, $C^{(h)}_{x,y}$, ${{\widehat {obj}}_{x,y}}$. The CNN predictions $\tilde{\boldsymbol{l}}_{x,y}$, $\tilde{\boldsymbol{s}}_{x,y}$, $\tilde{{ar}}_{x,y}$, $\hat{C}^{(h)}_{x,y}$, ${{\widehat {obj}}_{x,y}}$. $F_{x,y} \in R_{gh}$ obtained by static label assignment of GGHL. The prior threshold $T$. The number of object $N_{obj}$
\KwOut{The objectness loss $Loss^{obj}$
\eIf{${F_{x,y}} > 0$}{
Calculate the localization score $L_{x,y}$ by Eqs. (\ref{eq:9}-\ref{eq:10}) \;
Calculate the combined score ${D_{x,y}}$ by Eq. (\ref{eq:8}) \;
\For{$ i \in N_{obj}$}{
\For{$(x,y) \in R^{i}_{gh}$}{
\eIf{${F_{x,y}} > T$}{
$P = \left\lceil {\sum\limits_{\left( {x,y} \right) \in {R_{gh}}} {{L_{x,y}}} } \right\rceil $\;
Select the Top-$P$ ${{D}_{x,y}}$\;
\eIf{${{D}_{x,y}}$ in Top-$P$ ${{D}_{x,y}}$}{$(x,y) \in R_{pos}$;}
{$(x,y) \in R_{ig}$;}
}{
\eIf{${{D}_{x,y}} < T$}{$(x,y) \in R_{sneg}$ \;
${w_{sneg}} = 1 - {D_{x,y}}$ \;
}{$(x,y) \in R_{ig}$}
}
}
}
}{$(x,y) \in R_{neg}$}
Calculate the objectness loss $Loss^{obj}$ by Eq. (\ref{eq:12}).
\end{algorithm}
In response, DTLA strategy is designed based on TS-Conv and GGHL \cite{huang2022general}. DTLA divides the regions of candidate positions into positive, negative, soft-negative, and ignored regions denoted as the position sets $R_{pos}$, $R_{neg}$, $R_{sneg}$, and $R_{ig}$. These regions are dynamically adjusted according to the localization and classification prediction costs to obtain the optimal position combinations for CNN training. The detailed implementation of DTLA is given in Algorithm~\ref{alg:1}.
\textbf{\textit{1) Positive positions.}} Different from GGHL \cite{huang2022general}, which treats all the positions in a fixed range of a Gaussian ellipse region as positive positions, the proposed DTLA ranks the candidate positions of each object and takes the Top-$P$ positions as candidate positions. Since the performance of AOOD is usually evaluated by the product of localization and classification scores, the straightforward idea is to rank the positive candidate positions according to the combined score ${D_{x,y}}$ of localization and classification.
First, the static arbitrary-oriented label assignment strategy GGHL \cite{huang2022general} is utlized as the initial assignment of DTLA. Based on the initial assignment, the designed dynamic label assignment strategy is carried out. If a position ${D_{x,y}}$ lies in the Gaussian region, $F_{x,y} > T$, and the ranking of ${D_{x,y}}$ is within the Top-$P$, then this position is positive for assigning the label to predict the object, $\left( {x,y} \right) \in {R_{pos}}$. In this case, we set $T=0.3$. Benefiting from the designed TS-Conv, localization and classification features are extracted from their sensitive regions respectively and mapped to the same position, as shown in Fig.~\ref{fig:6} (b). The optimal task-wise scores are spatially aligned for determining positive candidate positions. Thus, the localization and classification combined score at $\left( {x,y} \right)$ is defined as
\begin{equation}
\! {D}_{x,y} \! = \left\{ \setlength{\arraycolsep}{0.5pt}{\begin{array}{l}
\tilde \vartheta F_{x,y} \! + \! (1 \! - \! \tilde \vartheta){\! \sqrt {\! {L_{x,y}}{\hat C}_{x,y}^{\left( h \right)}} }{\quad \rm{if}} (x,y) \! \in \! {R_{gh}}\\
{0} {\quad \rm{otherwise}}
\end{array}} \right.,
\label{eq:8}
\end{equation}
where ${\hat C}_{x,y}^{\left( h \right)} \in \left( {0,1} \right)$ indicates the predicted classification score that the object belongs to the ground truth category $h$ and will be specified later. The localization score is represented as
\begin{equation}
{L_{x,y}} = {e^{ - Loss_{x,y}^{loc}}},
\label{eq:9}
\end{equation}
\begin{equation}
\begin{array}{l}
Loss_{x,y}^{loc} = 1 - GIoU\left( {{\boldsymbol{l}_{x,y}},{{\tilde{\boldsymbol{l}}}_{x,y}}} \right)\\
+ MSE\left( {{\boldsymbol{s}_{x,y}},{{\tilde{\boldsymbol{s}}}_{x,y}}} \right) + {\left( {{a_{x,y}} - {{\tilde a}_{x,y}}} \right)^2}
\end{array},
\label{eq:10}
\end{equation}
where $Loss_{x,y}^{loc}$ denotes the OBB localization loss at $\left( {x,y} \right)$. $L_{x,y} \in (0,1)$ is monotonically decreasing with $Loss_{x,y}^{loc}$. ${\tilde{\boldsymbol{l}}_{x,y}} = \left[ {{{\tilde l}_1},{{\tilde l}_2},{{\tilde l}_3},{{\tilde l}_4}} \right]$ and ${\tilde{\boldsymbol{s}}_{x,y}} = \left[ {{{\tilde s}_1},{{\tilde s}_2},{{\tilde s}_3},{{\tilde s}_4}} \right]$ represent the predictions of an OBB based on the output localization features $\boldsymbol{O}_{x,y}^{loc}$ of TS-Conv. ${\boldsymbol{l}_{x,y}} = \left[ {{l_1},{l_2},{l_3},{l_4}} \right]$ and ${\boldsymbol{s}_{x,y}} = \left[ {{s_1},{s_2},{s_3},{s_4}} \right]$ represent the ground truth of this OBB. ${a_{x,y}}$ and ${\tilde a_{x,y}}$ denote the ground truth and predictions of the area ratio of the OBB and HBB, respectively. $GIoU\left( . \right)$ is the function to measure the HBB localization accuracy by Generalized Intersection over Union (GIoU) \cite{rezatofighiGeneralizedIntersectionUnion2019}. $MSE\left( \cdot \right)$ represents the mean square error (MSE) function. See GGHL \cite{huang2022general} for details of $Los{s^{loc}}\left( {x,y} \right)$. The hyperparameter $\tilde \vartheta$ is defined as
\begin{equation}
\tilde \vartheta = \frac{{ite{r_{\max }} - iter}}{{ite{r_{\max }}}} \times \vartheta,
\label{eq:11}
\end{equation}
where $iter$ denotes the number of current iterations and $iter_{max}$ denotes the number of maximum iterations during CNN training. Here, $\vartheta = 0.3$ according to our empirical study. In the initial stage of CNN training, the label assignment mainly relies on the prior Gaussian heatmaps generated by GGHL \cite{huang2022general} due to inaccurate localization and classification. As the CNN training converges, the label assignment becomes more dependent on the increasing score. The variable $P = \left\lceil {\sum\limits_{\left( {x,y} \right) \in {R_{gh}}} {{L_{x,y}}} } \right\rceil $, where $\left\lceil . \right\rceil $ denotes the upward rounding operation, is dynamically adjusted for different objects. The positive positions are no longer fixed but dynamically selected according to ${D_{x,y}}$.
The above strategy takes full advantage of the proposed task-wise sampling convolutions that can extract and map the most sensitive features of different tasks to the same spatial location. On one hand, it helps to select the optimal candidate positions suitable for both localization and classification dynamicly according to different objects in different scenes. On the other hand, the low-quality candidate positions due to MERect approximation can be filtered according to DTLA. Thus, it is unnecessary to generate the Gaussian probability density function (PDF) of any convex quadrilateral, but only needs to replace it with the Gaussian PDF based on MERect, which makes the algorithm more efficient and concise.
\textbf{\textit{2) Negative positions.}} The designed DTLA uses a soft thresholding strategy instead of treating all positions with ${F_{x,y}} < T$ as negative, as shown in Fig.~\ref{fig:6} (b). If $\left( {x,y} \right)$ does not lie in the Gaussian region, this position is negative, and the background label is assigned, $\left( {x,y} \right) \in {R_{neg}}$. If $\left( {x,y} \right)$ lies in the Gaussian region $R_{gh}$ but ${F_{x,y}} < T,{\rm{ }} \ {D_{x,y}} < T$, this position is considered as soft negative position, $\left( {x,y} \right) \in {R_{sneg}}$. The background prediction loss of these positions is multiplied by the weight ${w_{sneg}} = 1 - {D_{x,y}},{\rm{ }}{w_{sneg}} \in \left( {0,1} \right)$. The smaller the ${D_{x,y}}$, the larger the weight, indicating a higher negative attribute for this position.
\textbf{\textit{3) Ignored positions.}} If $\left( {x,y} \right)$ lies in the Gaussian region $R_{gh}$, ${F_{x,y}} > T$, but ${D_{x,y}}$ is not within the Top-$P$, this position may not satisfy both localization and classification tasks, although it have a high priori score. If $\left( {x,y} \right)$ lies in the Gaussian region $R_{gh}$, ${F_{x,y}} < T$, but ${D_{x,y}} > T$, this position is too close to the junction region between the object and the background for a low a priori score, although it obtains a high ${D_{x,y}}$. In the above two cases, the priori score ${F_{x,y}}$ and the dynamic score ${D_{x,y}}$ contradict each other. It is not appropriate to treat $\left( {x,y} \right)$ as either positive or negative position, so it is ignored and not used for CNN training, i.e., $\left( {x,y} \right) \in {R_{ig}}$.
\textbf{\textit{4) Loss functions.}} If a position $\left( {x,y} \right)$ is positive, its assigned ground truth of objectness $ob{j_{x,y}} = L_{x,y}$; if $\left( {x,y} \right)$ is negaitive, $ob{j_{x,y}} = 0$. Note that, the CNN gradient of ${L_{x,y}}$ is not backpropagated. Define the CNN predicted objectness score at $\left( {x,y} \right)$ as ${\widehat {obj}_{x,y}} \in \left( {0,1} \right)$. According to the assignments of different candidate positions, the binary loss function for objectness prediction is represented as
\begin{equation}
\setlength{\arraycolsep}{0.3pt}{\begin{array}{l}
Los{s^{obj}} \! = \! -\frac{1}{{{M_{pos}}}} \! \sum\limits_{\tiny{(x,y) \in {R_{pos}}}}\! {{{\left| {L_{x,y} - {{\widehat {obj}}_{x,y}}} \right|}^\gamma }\log {{{\widehat {obj}}_{x,y}}} } \\
- \frac{1}{{{M_{neg}}}} \sum\limits_{(x,y) \in {R_{neg}}} {{{{{\widehat {obj}}_{x,y}}}^\gamma }\log \left( {{{ {1 - \widehat{obj}}}_{x,y}}} \right)} \\
- \frac{1}{{{M_{sneg}}}} \sum\limits_{(x,y) \in {R_{sneg}}} {{w_{sneg}}{{{{\widehat {obj}}_{x,y}}}^\gamma }\log \left( {{{ {1 - \widehat{obj}}}_{x,y}}} \right)}
\end{array}},
\label{eq:12}
\end{equation}
where ${M_{pos}}$, ${M_{neg}}$, and ${M_{sneg}}$ mean the numbers of positive, negative, and soft-negative positions for an input image, respectively. The hyperparameter of Focal Loss \cite{linFocalLossDense2017} $\gamma = 2$, which is the same as GGHL \cite{huang2022general}. The OBBs and category labels of the objects are assigned to positive positions filtered by TOP-$P$ strategy for supervised CNN predictions for object localization and classification. According to Eq.(\ref{eq:10}), the localization loss is calculated as
\begin{equation}
Los{s^{loc}} = \frac{1}{{{M_{pos}}}} \times \sum\limits_{\left( {x,y} \right) \in {R_{pos}}} {\left( {Loss_{x,y}^{init} + Loss_{x,y}^{loc}} \right)},
\label{eq:13}
\end{equation}
and
\begin{equation}
\begin{array}{l}
Loss_{x,y}^{init} = 1 - GIoU\left( {{\boldsymbol{l}_{x,y}},{{\hat{\boldsymbol{l}}}_{x,y}}} \right)\\
+ MSE\left( {{\boldsymbol{s}_{x,y}},{{\hat{\boldsymbol{s}}}_{x,y}}} \right) + {\left( {{a_{x,y}} - {{\hat a}_{x,y}}} \right)^2}
\end{array},
\label{eq:14}
\end{equation}
which represents the OBB localization loss of the initial stage, which is used to suprivise the sampling points of TS-Conv.
Define the assigned ground truth of the $h$th category at $\left( {x,y} \right)$ as $C_{x,y}^{\left( h \right)}$. If the object at $\left( {x,y} \right)$ belongs to the $h$th category, $C_{x,y}^{\left( h \right)} = 1$; otherwise, $C_{x,y}^{\left( h \right)} = 0$. Define the CNN prediction of $C_{x,y}^{\left( h \right)}$ as ${\hat C}_{x,y}^{\left( h \right)} \in \left( {0,1} \right)$. The classification loss is calculated as
\begin{equation}
\setlength{\arraycolsep}{0.3pt}{\begin{array}{l}
Los{s^{cls}} = \frac{1}{{{M_{pos}}}} \! \times \! \sum\limits_{\left( {x,y} \right) \in {R_{pos}}} \! {\sum\limits_{h = 1}^{{M_C}} {\left( {C_{x,y}^{\left( h \right)}\log \left( {\hat C_{x,y}^{\left( h \right)}} \right)} \right.} } \\
\left. { + \left( {1 - C_{x,y}^{\left( h \right)}} \right)\log \left( {1 - \hat C_{x,y}^{\left( h \right)}} \right)} \right)
\end{array}},
\label{eq:15}
\end{equation}
where ${M_C}$ represents the total number of categories of objects. The total loss is stated as,
\begin{equation}
Loss = Los{s^{obj}} + Los{s^{loc}} + Los{s^{cls}},
\label{eq:16}
\end{equation}
which represents the sum of the objectness loss, localization loss and classification loss.
\subsection{Experimental Conditions}
\textbf{1) \textit{Experimental platforms.}} The experiments are performed on a server with an AMD 3950WX CPU, 128 GB memory, and four NVIDIA GeForce RTX 3090 GPU (24GB). In addition, the performance of the TS-Conv improved lightweight models are evaluated on embedded edge devices NVIDIA Jetson AGX Xavier and Jetson TX2.
\textbf{2) \textit{Datasets.}} To evaluate the performance of TS-Conv model more comprehensively, several public AOOD datasets are used, which cover different scenes, different shapes and categories of objects, and different data sources.
a) DOTAv1.0 \cite{xiaDOTALargeScaleDataset2018} dataset provides a widely used benchmark to evaluate the performance of AOOD methods in remote sensing scenes. It has more than 188,000 objects covering 15 categories in 2,806 images from $800\times800$ pixels to $4,000\times4,000$ pixels. Due to the huge size of remote sensing images, they are usually cropped into sub-images of $800\times800$ pixels with an overlap of 200 pixels on each dimension after being scaled to different sizes with the ratios of 0.5, 1.0, and 1.5 \cite{huang2021lo}. DOTAv2.0 \cite{li2020object} further expands the number of objects to 1,793,658 objects covering 18 categories.
b) HRSC2016 \cite{liu2017high} is a ship detection dataset consisting of 436 training images, 181 validation images, and 444 testing images from $300\times300$ pixels to $1500\times900$ pixels.
c) DIOR-R \cite{ding2021object,cheng2022anchor} dataset is an aerial AOOD datasets contains 190,288 objects covering 20 categories in 23,463 images with the size of $800\times800$ pixels. In the DIOR-R dataset, 5,862 images are used for training, 5,863 images are used for validation, and 11,738 images are used for testing.
d) DroneVehicle \cite{sun2022drone} is an infrared-RGB vehicle detection datasets. After screening and pre-processing, 9,425 Infrared-RGB image pairs are used for training and 1,399 Infrared-RGB image pairs are used for testing.
e) SSDD+ \cite{li2017ship} is a synthetic aperture radar (SAR) dataset for ship detection. It has 1,160 ship images including 2,456 instances collected from different sea conditions. The ratio of training, validation and testing images is 7:1:2.
\textbf{3) \textit{Evaluation metrics.}} The mean Average Precision (mAP) is adopted for evaluating the detection accuracy. The mAP with an IoU threshold of 50\% is represented as mAP$_{50}$. mAP$_{50:95}$ means calculating the mAP when the IoU threshold is 50\% to 95\% at every 5\% interval and then taking their mean as mAP$_{50:95}$. The speed is evaluated by frames per second (fps) and the computational complexity is evaluated by the floating point operations (FLOPs) for lightweight models.
\textbf{4) \textit{Implementation details.}} The initial learning rate for training DOTA, DIOR-R, HRSC2016 and SSDD+ datasets is $5\times10^{-4}$, and the final learning rate is $1\times10^{-6}$. The optimizer for training the CNN model is stochastic gradient descent (SGD), and the learning rate scheduler is cosine decay. The weight decay and momentum are set as $5\times10^{-4}$ and 0.9, respectively. The batch size is 32 (8 images per GPU). The maximum training epoch for DOTA, DIOR-R, HRSC2016, DroneVehicle and SSDD+ datasets are 36, 150, 150, 50 and 150, respectively. The non-maximum suppression (NMS) threshold is 0.4. Random cropping, random flipping, random rotation and mixup strategies are employed for data augmentation.
\subsection{Ablation Experiments}
The results of the ablation experiments for different components of TS-Conv, including LS-Conv, CS-Conv with DCK, and DTLA, are presented in Table~\ref{table:2}. The anchor-free AOOD method GGHL \cite{huang2022general} is chosen as the baseline, and experiments are conducted on the most widely used AOOD dataset DOTA v1.0 \cite{xiaDOTALargeScaleDataset2018}.
\textbf{1) Ablation experiments of each component.} First, as a control group, the shared-offset DCNs employed by RepPoints \cite{yang2019reppoints}, Oriented RepPoints \cite{li2022oriented}, etc., are introduced based on GGHL. This design has the same sampling positions for localization and classification features and maps these features to aligned locations. Although the above scheme somewhat enables more objects to be detected and improves the mAP$_{50}$ by 0.82, the improvement of the stricter metrics, i.e., mAP$_{75}$ and mAP$_{50:95}$, is slight. It implies that the model may relax its requirements for detection quality to accommodate two different tasks.
\begin{table*}[tp]
\centering
\renewcommand\arraystretch{1}
\setlength{\tabcolsep}{2.5mm}{
\caption{\label{table:2}
{Ablation experiments of the proposed TS-Conv on the DOTAv1.0 dataset}}
\resizebox{\textwidth}{!}{\setlength{\tabcolsep}{1mm}{
\begin{tabular}{c|c|c|c|c|c|c|c|ccc}
\hline\hline
{\multirow{2}{*}{Methods}} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Decoupled \\ Head\end{tabular}} &\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Shared-offset \\ DCNs \end{tabular}} &
\multicolumn{3}{c|}{\begin{tabular}[c]{@{}c@{}}Task-wise DCNs \\ (TS-DCN) \end{tabular}}& \multicolumn{2}{c|}{\begin{tabular}[c]{@{}c@{}}Label \\ Assignment \end{tabular}} & \multicolumn{3}{c}{mAPs on the DOTAv1.0 Dataset} \\ \cline{4-11}
& & & LS-Conv & CS-Conv & DCK & GGHL & DTLA & mAP$_{50}$ & mAP$_{75}$ & mAP$_{50:95}$ \\ \hline
Baseline (GGHL \cite{huang2022general}) & $\checkmark$ & & & & & $\checkmark$ & & 76.95 & 44.19 & 44.29 \\ \hline
\begin{tabular}[c]{@{}c@{}}Shared-offset DCNs \\ (O-RepPoints \cite{li2022oriented})\end{tabular} & $\checkmark$ & $\checkmark$ & & & & $\checkmark$ & & 77.77 \tiny{(+0.82)} & 44.76 \tiny{(+0.57)} & 44.99 \tiny{(+0.70)} \\ \hline
LS-Conv& $\checkmark$ & & $\checkmark$ & & & $\checkmark$ & & 78.08 \tiny{(+1.13)} & 46.38 \tiny{(+2.19)} & 45.87 \tiny{(+1.27)} \\ \hline
\begin{tabular}[c]{@{}c@{}} CS-Conv \end{tabular} & $\checkmark$ & & & $\checkmark$ & & $\checkmark$ & & 77.84 \tiny{(+0.89)} & 44.23 \tiny{(+0.04)} & 45.37 \tiny{(+1.08)}\\ \hline
\begin{tabular}[c]{@{}c@{}}CS-Conv with DCK\end{tabular}& $\checkmark$ & & & $\checkmark$ & $\checkmark$ & $\checkmark$ & & 78.06 \tiny{(+1.11)} & 45.46 \tiny{(+1.27)} & 45.36 \tiny{(+1.07)}\\ \hline
DTLA & $\checkmark$ & & & & & & $\checkmark$ & 77.95 \tiny{(+1.00)} & 46.10 \tiny{(+1.91)} & 45.38 \tiny{(+0.70)} \\ \hline
TS-DCN & $\checkmark$ & & $\checkmark$ & $\checkmark$ & $\checkmark$ & $\checkmark$ & & 78.13 \tiny{(+1.18)} & 46.59 \tiny{(+2.40)} & 46.09 \tiny{(+1.80)} \\ \hline
\textbf{TS-Conv} & $\checkmark$ & & $\checkmark$ & $\checkmark$ & $\checkmark$ & & $\checkmark$ & \textbf{78.75 \tiny{(+1.80)}} & \textbf{46.60 \tiny{(+2.41)}} & \textbf{46.27 \tiny{(+1.98)}} \\ \hline
\hline\hline
\end{tabular}}}}\vspace{0.5em}
\justifying{Note: Bold indicates the best result. 'Shared-offset DCN' represents that both branches of the decoupled head use DCN \cite{dai2017deformable} and their sampling offsets are shared as in the case of RepPoints \cite{yang2019reppoints} and Oriented RepPoints \cite{li2022oriented}.}
\end{table*}
\begin{figure}[!tp]
\centering
\epsfig{width=0.47\textwidth,file=7.pdf}
\caption{Comparison of DTLA and GGHL label assignment strategies. Figs. (a-2)-(a-4) and (b-2)-(b-4) represent the positive candidate positions and their scores statically assigned by the GGHL strategy at three different scales, respectively. Figs. (a-5)-(a-7) and (b-5)-(b-7) represent the positive candidate positions and their scores dynamically assigned by the proposed DTLA strategy at three different scales, respectively. The closer the color is to red, the higher the score.}\label{fig:7}
\end{figure}
\begin{table}[!tp]
\centering
\renewcommand\arraystretch{1}
\setlength{\tabcolsep}{1.8mm}{
\caption{\label{table:3}
{Evaluations of detection performance for different values of hyperparameters $T$ and $\vartheta$, and experiments of using DCK and random rotation data augmentation on the DOTAv1.0 dataset}}}
\resizebox{0.48\textwidth}{!}{\setlength{\tabcolsep}{1.8mm}{
\begin{tabular}{cc||cc||ccc}
\hline\hline
$T$ & mAP$_{50}$ & $\vartheta$ & mAP$_{50}$ & RandomRota & DCK & mAP$_{50}$ \\
\hline
0.2 & 78.45 & 0.3 & \textbf{78.75} & $\checkmark$ & & 77.92 \\
0.3 & \textbf{78.75} & 0.4 & 78.52 & & $\checkmark$ & 78.47\\
0.4 & 78.60 & 0.5 & 78.50 & $\checkmark$ & $\checkmark$ & \textbf{78.75}\\
\hline\hline
\end{tabular}}}\vspace{0.5em}
\justifying{Note: Bold indicates the best result. RadomRota represents using the random rotation data agumentation operation during the CNN training. When evaluating one variable, the other variables are fixed.}
\end{table}
Correspondingly, the proposed task-wise DCNs consisting of LS-Conv and CS-Conv are evaluated. With the introduction of LS-Conv, the mAP$_{75}$ is improved by 4.96\% (+2.19) compared to the baseline and 3.63\% (+1.62) compared to the shared-offset DCNs. The designed LS-Conv further improves the localization accuracy compared to the shared-offset DCNs by directly associating the convolutional sampling points with the OBB representation and embedding the spatial coordinates into the features. When only CS-Conv is used without LS-Conv, i.e., the sampling points of convolutions are just constrained in OBB, the mAP$_{50}$, mAP$_{75}$ and mAP$_{50:95}$ are also increased, but the improvement, especially for mAP$_{75}$, is minor. The bottleneck of this scheme is mainly in inaccurate localization. Because the spatial location coordinates of the OBB are explicit and essential features for the localization task, the convolutional sampling constraint of CS-Conv is vague for localization, although it is suitable for classification. In addition, since the objects in AOOD have richer orientation variations, performance improvements are also observed after replacing the convolutional kernel of CS-Conv with the designed DCK. Furthermore, when LS-Conv and CS-Conv are combined to obtain the TS-DCN scheme, significant performance gains are seen compared to using them alone as listed in Table~\ref{table:2}. The mAP$_{50}$, mAP$_{75}$ and mAP$_{50:95}$ are increased by 1.53\% (+1.18), 5.43\% (+2.40), and 4.06\% (+1.80), respectively. It illustrates that the LS-Conv and CS-Conv play complementary roles for each other. LS-Conv compensates for the lack of localization accuracy using only CS-Conv, while CS-Conv with DCK further improves detection performance.
Then, the performance of the two label assignment strategies, GGHL \cite{huang2022general} and the proposed DTLA, is evaluated. The proposed DTLA increases the mAP$_{50}$, mAP$_{75}$ and mAP$_{50:95}$ without introducing additional CNN structure and adding inference cost. Fig.~\ref{fig:7} shows the positive candidate positions selected by GGHL \cite{huang2022general} and DTLA for predicting objects. As seen in Fig.~\ref{fig:7}, there is a significant difference between the positive candidate positions dynamically selected by DTLA according to ${\tilde D}_{x,y}$ and the positions statically assigned by GGHL \cite{huang2022general} according to the Gaussian prior. The Gaussian peak position is not necessarily the optimal candidate position. It is more reasonable for DTLA to select the positive candidate positions adaptively according to different objects and training stages.
Finally, the TS-Conv model combining the above components is obtained. Compared with the GGHL model, the mAP$_{50}$, mAP$_{75}$ and mAP$_{50:95}$ of TS-Conv are improved by 2.34\% (+1.80), 5.45\% (+2.41), and 4.47\% (+1.98). The improvement in mAP$_{75}$ and mAP$_{50:95}$ metrics representing higher detection quality is more pronounced. In general, the ablation experiments validate the effectiveness of each component in the proposed TS-Conv. In addition, Table~\ref{table:3} lists the experimental results of different hyperparameter settings. To obtain the best results, we set $T=0.3$ and $\vartheta=0.5$ in TS-Conv.
\textbf{2) Analysis of the IFS problem.} To more intuitively analyze the IFS problem faced by existing AOOD models like GGHL \cite{huang2022general}, Fig.~\ref{fig:8} visualizes the feature-sensitive regions of the localization and classification tasks. Fig.~\ref{fig:8} (a-1), (a-2), (b-1), (b-2), (c-1), (c-2), (d-1) and (d-2) show the feature sensitivity regions before using the proposed TS-Conv. The closer the color of the heatmap to red indicates that the model is more sensitive to the features in that region. From observing the feature sensitivity regions for various scenes and objects, it is obvious that the feature-sensitive regions of localization and classification tasks are significantly different. It is difficult for the existing schemes, such as decoupled-head used by CFC-Net \cite{ming2021cfc}, GGHL \cite{huang2022general}, etc., and shared-offset DCNs used by S$^2$ANet \cite{han2021align}, Oriented RepPoints \cite{li2022oriented}, etc., to take into account the respectively most sensitive features for different tasks. Correspondingly, Fig.~\ref{fig:8} (a-3), (a-4), (b-3), (b-4), (c-3), (c-4), (d-3) and (d-4) show the results of feature sensitivity regions after using the proposed TS-Conv. Sensitive features of different tasks located at different locations before input to TS-Conv are extracted and mapped to the same spatial location. The sensitive features of localization and classification are spatially aligned after TS-Conv's mapping. Then the optimal candidates are found among these spatially aligned features to predict the objects. It further demonstrates the effectiveness of the designed TS-Conv.
\begin{figure}[!t]
\centering
\epsfig{width=0.4\textwidth,file=8.pdf}
\caption{Visualization of feature sensitivity regions. The closer the color is to red, the higher the sensitivity of the feature at this location. The figures in rows 1 and 3 (Figs. (a-1), (a-2), (b-1), (b-2), (c-1), (c-2), (d-1) and (d-2)) are the visualization results of feature sensitivity regions before using the designed TS-Conv. The figures in rows 2 and 4 (Figs. (a-3), (a-4), (b-3), (b-4), (c-3), (c-4), (d-3) and (d-4)) are the visualization results of feature sensitivity regions after using the designed TS-Conv.}\label{fig:8}
\end{figure}
\begin{figure}[!t]
\centering
\epsfig{width=0.4\textwidth,file=9.pdf}
\caption{Visualization of features extracted by Dynamic circular kernel (DCK). Figs. (a-1)-(a-8) and Figs. (b-1)-(b-8) show the features extracted by convolutional kernels with different rotations in DCK, respectively. Figs. (a-9)-(a-12) and Figs. (b-9)-(b-12) show the DCK-extracted features of input images with different rotations, respectively.}\label{fig:9}
\end{figure}
\textbf{3) Analysis of orientation-robust features extracted by DCK.} Figs.~\ref{fig:9} (a-1)-(a-8) and Figs.~\ref{fig:9} (b-1)-(b-8) show the feature-sensitive regions of different oriented convolutional kernels in DCK (see details in Fig.~\ref{fig:5}), respectively. The different feature-sensitive regions indicate that the feature extraction is not robust to arbitrary orientations. Since objects' orientations in AOOD are more diverse than those in the ordinary OD, this problem further constrains the existing AOOD models' performance. Figs.~\ref{fig:9} (a-9) and (b-9) show the feature-sensitive regions of the designed DCK, which adjust the optimal orientation and weights of eight-oriented kernels according to different inputs adaptively. Furthermore, Figs.~\ref{fig:9} (a-9)-(a-12) and (b-9)-(b-12) show the feature-sensitive regions of DCK when the input images are rotated in different orientations. The results indicate that the feature-sensitive regions of DCK are robust to arbitrary-oriented inputs, which do not fluctuate significantly with the objects' rotations. In addition, the performance of using the random rotation data augmentation (RRDA) strategy and DCK are compared in Table~\ref{table:3}. The performance of the model using only DCK exceeds that of the model using only the RRDA strategy. As shown in Table~\ref{table:3}, when using both the RRDA strategy and DCK, the mAP$_{50}$ is 0.28 higher than that of using DCK only and 0.83 higher than that of using RRDA only. The performance further confirms the designed DCK's ability to enhance the model's robustness to extract features with arbitrary orientations.
\subsection{Experiments for the Scalability of TS-Conv}
\begin{table}[tbp]
\centering
\renewcommand\arraystretch{1}
\setlength{\tabcolsep}{0.3mm}{
\caption{\label{table:4}
{Performance evaluation of the lightweight model using DTLA and TS-Conv-based knowledge distillation on the DOTAv1.0 dataset}}
\resizebox{0.48\textwidth}{!}{\setlength{\tabcolsep}{0.3mm}{
\begin{threeparttable}
\begin{tabular}{c|ccccccc}
\hline\hline
\multirow{2}{*}{{Modules}} & \multirow{2}{*}{{mAP$_{50}$}}& \multirow{2}{*}{{\begin{tabular}[c]{@{}c@{}}Speed1\\ (fps)\end{tabular}}} & \multirow{2}{*}{{\begin{tabular}[c]{@{}c@{}}Speed2\\ (fps)\end{tabular}}} & \multirow{2}{*}{{\begin{tabular}[c]{@{}c@{}}Speed3\\ (fps)\end{tabular}}} & \multirow{2}{*}{{\begin{tabular}[c]{@{}c@{}}Speed4\\ (fps)\end{tabular}}} & \multirow{2}{*}{{\begin{tabular}[c]{@{}c@{}}FLOPs\\ (G)\end{tabular}}} & \multirow{2}{*}{{\begin{tabular}[c]{@{}c@{}} Parameters\\ (MB)\end{tabular}}} \\
{}& {} & {} & {} & {} & {} & { }& { } \\
\hline
{LO-Det \cite{huang2021lo}} & {66.17}& {60.01} & {6.99}& {22.12} & 3.71 & {6.42} & {6.93}\\ \hline
\begin{tabular}[c]{@{}c@{}} LO-Det \cite{huang2021lo} + GGHL \cite{huang2022general} \end{tabular}
& \begin{tabular}[c]{@{}c@{}}71.26\\ \tiny{(+5.09)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}62.07\\ \tiny{(+2.06)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}7.68\\ \tiny{(+0.699)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}23.72\\ \tiny{(+1.60)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}4.04\\ \tiny{(+0.33)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}6.30\\ \tiny{(-0.12)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}6.72\\ \tiny{(-0.21)}\end{tabular} \\ \hline
\begin{tabular}[c]{@{}c@{}}LO-Det \cite{huang2021lo} + DTLA \end{tabular}
& \begin{tabular}[c]{@{}c@{}}73.36\\ \tiny{(+7.19)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}62.07\\ \tiny{(+2.06)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}7.68\\ \tiny{(+0.699)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}23.72\\ \tiny{(+1.60)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}4.04\\ \tiny{(+0.33)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}6.30\\ \tiny{(-0.12)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}6.72\\ \tiny{(-0.21)}\end{tabular} \\ \hline
\begin{tabular}[c]{@{}c@{}} TS-Conv Lite \\ \footnotesize{(TS-Conv Distilled} \\ \footnotesize{LO-Det \cite{huang2021lo} + DTLA)} \end{tabular}
& \begin{tabular}[c]{@{}c@{}} \textbf{73.96}\\ \textbf{\tiny{(+7.79)}}\end{tabular} & \begin{tabular}[c]{@{}c@{}}62.07\\ \tiny{(+2.06)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}7.68\\ \tiny{(+0.699)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}23.72\\ \tiny{(+1.60)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}4.04\\ \tiny{(+0.33)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}6.30\\ \tiny{(-0.12)}\end{tabular} & \begin{tabular}[c]{@{}c@{}}6.72\\ \tiny{(-0.21)}\end{tabular} \\
\hline\hline
\end{tabular}
\end{threeparttable}}}}\vspace{0.5em}
\justifying{Note: The unit G is Giga, which represents $1\times10^{9}$. The unit MB represents $1\times10^{6}$ bytes. Speed1, Speed2, Speed3 and Speed4 are the detection speed on the RTX 3090 GPU, NVIDIA Jetson TX2, Jetson AGX Xavier, and Jetson Nano, respectively. The inference speed only includes the network inference speed without post-processing.}
\end{table}
\begin{table}[tbp]
\centering
\renewcommand\arraystretch{1}
\setlength{\tabcolsep}{1mm}{
\caption{\label{table:5}
{Performance evaluation of the proposed TS-Conv for multimodal images on the DroneVehicle dataset}}
\resizebox{0.48\textwidth}{!}{\setlength{\tabcolsep}{1mm}{
\begin{threeparttable}
\begin{tabular}{c|c|c|c|cc}
\hline\hline
Methods & Backbone & RGB & Infrared & mAP$_{50}$ & mAP$_{75}$\\
\hline \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}GGHL \cite{huang2022general}\\ (Baseline1)\end{tabular} } & Darknet53 & $\checkmark$ & & 53.69 & 23.22\\
~ & Darknet53& & $\checkmark$ & 56.19 & 23.26\\
~ & Darknet53 & $\checkmark$ & $\checkmark$ & 57.25 & 24.17\\
\cdashline{1-6}[0.8pt/2pt]
\multirow{3}{*}{TS-Conv} & Darknet53 & $\checkmark$ & & 54.95 \tiny{(+1.26)} & 30.98 \tiny{(+7.76)}\\
~ & Darknet53 & & $\checkmark$ & 58.29 \tiny{(+2.10)} & 32.89 \tiny{(+9.63)}\\
~ & Darknet53& $\checkmark$ & $\checkmark$ & 60.36 \tiny{(+3.11)} & 34.68 \tiny{(+10.51)}\\
\cdashline{1-6}[0.8pt/2pt]
TS-Conv$^*$ & Darknet53 & $\checkmark$ & $\checkmark$ & \textbf{62.06 \tiny{(+4.81)}} & \textbf{35.98 \tiny{(+11.81)}}\\
\hline
\begin{tabular}[c]{@{}c@{}}LO-Det+GGHL \cite{huang2021lo} \\ (Baseline2)\end{tabular} & MobileNetv2 & $\checkmark$ & $\checkmark$ & 45.65 & 15.36 \\
\cdashline{1-6}[0.8pt/2pt]
LO-Det + DTLA & MobileNetv2 & $\checkmark$ & $\checkmark$ & \textbf{47.33 \tiny{(+1.68)}} & \textbf{16.90 \tiny{(+1.54)}} \\ \hline\hline
\end{tabular}
\end{threeparttable}}}}\vspace{0.5em}
\justifying{Note: The testing image size is 800 $\times$800 pixels. The TS-Conv$^*$ represents that it not only samples the features of different tasks separately, but also samples the features of different modalities.}
\end{table}
\textbf{1) Scalability of TS-Conv on lightweight models.} Although TS-Conv improves detection performance and robustness, the introduction of DCNs also brings additional computational burdens compared to the baseline model GGHL \cite{huang2022general}. For the benefits of TS-Conv to be applied in the lightweight model, the experiments on different embedded devices are designed and validated, as listed in Table~\ref{table:4}. The lightweight AOOD model LO-Det \cite{huang2021lo} is chosen as the baseline. First, the anchor-based label assignment strategy of LO-Det is improved to GGHL and the proposed DTLA for comparison. The results in Table~\ref{table:4} demonstrate that using the proposed dynamic label assignment strategy DTLA further improves the performance of the lightweight model by 2.95\% (+2.10) without losing model inference efficiency and increasing model complexity compared to using the static label assignment strategy GGHL. Second, the idea of knowledge distillation is adopted to make DTLA take full advantage of the task-wise sensitive features learned by TS-Conv without complicating the lightweight model structure. The LO-Det+DTLA is utilized as the student model, and the TS-Conv is used as the teacher model. The AOOD knowledge distillation method DKED \cite{huang2022extracting} is employed to guide the student model to learn the task-wise features of TS-Conv. The model obtained from this scheme is denoted as TS-Conv Lite. The results show that Ts-Conv Lite's performance is further improved without additional inference cost. TS-Conv Lite's mAP$_{50}$ improves 11.77\% (+7.79) compared to the baseline model with fewer model parameters and faster inference because it does not rely on anchor boxes.
\begin{figure}[tp]
\centering
\epsfig{width=0.48\textwidth,file=10.pdf}
\caption{Visualization results of TS-Conv on the DroneVehicle dataset. Figs. (a-1)-(a-3) show the object detection results of RGB-Infrared multimodal image pairs under different lighting conditions. Figs. (b-1)-(b-4) show the feature sensitivity regions of different modalities and different tasks for RGB-Infrared multimodal image pairs.}\label{fig:10}
\end{figure}
\begin{figure}[!tp]
\centering
\epsfig{width=0.48\textwidth,file=11.pdf}
\caption{Visualization results of TS-Conv on the DOTA dataset.}\label{fig:11}
\end{figure}
\textbf{2) Scalability of TS-Conv for multimodal data.} The effectiveness of the proposed TS-Conv for multimodal data is evaluated on the multimodal AOOD dataset DroneVehicle \cite{sun2022drone}. The results are shown in Table~\ref{table:5} and Fig.~\ref{fig:10}. The performance of the proposed TS-Conv and the baseline model GGHL are evaluated on RGB images, infrared images, and RGB-infrared image pairs, respectively. The results demonstrate that TS-Conv has improved performance compared with GGHL in all three groups of experiments, especially the improvement of mAP$_{75}$ is more significant. It illustrates the effectiveness of TS-Conv on multimodal data and demonstrates the merits of TS-Conv for improving the accuracy of oriented bounding boxes again.
TS-Conv has discussed the differences in task-wise feature-sensitive regions, furthermore, the differences in modality-wise feature-sensitive regions are analyzed here. Figs.\ref{fig:10} (b-1)-(b-4) show the differences between task-wise and modality-wise features for RGB-Infrared image pairs. RGB images contain richer color and texture features but are more susceptible to lighting conditions. Infrared images highlight objects in low light conditions but lack more detailed features. Therefore, TS-Conv is extended to sample modality-wise features separately along the lines of "separate sampling and aligned mapping". This model is denoted as TS-Conv$^*$. The results in Table~\ref{table:5} reflect the effectiveness of the modality-wise samplings strategy for further improving the detection performance on the multimodal dataset. In the future, this problem is expected to be explored in more depth. In addition, experiments with lightweight models are also conducted to verify the scalability of TS-Conv on multimodal data.
\subsection{Comparison Experiments}
In this subsection, the performance of the proposed TS-Conv and state-of-the-art methods is compared on several datasets, including DOTA \cite{xiaDOTALargeScaleDataset2018,li2020object}, DIOR-R \cite{cheng2022anchor}, HRSC2016 \cite{liu2017high}, SSDD+ \cite{li2017ship}, SKU-110KR \cite{pan2020dynamic}, etc.
\begin{table*}[tp]
\centering
\renewcommand\arraystretch{1}
\setlength{\tabcolsep}{0.43mm}{
\caption{\label{table:6}
{Comparative experiments on the DOTAv1.0 dataset}}
\resizebox{\textwidth}{!}{\setlength{\tabcolsep}{0.43mm}{
\begin{threeparttable}
\centering
\begin{tabularx}{\textwidth}{c|c|c|c|ccccccccccccccc|c|c}
\hline\hline
\multirow{2}{*}{Methods} & \multirow{2}{*}{Backbone} & \multirow{2}{*}{Stage}& \multirow{2}{*}{Anchor} & \multirow{2}{*}{PL} & \multirow{2}{*}{BD} & \multirow{2}{*}{BR} & \multirow{2}{*}{GTF} & \multirow{2}{*}{SV} & \multirow{2}{*}{LV} & \multirow{2}{*}{SH} & \multirow{2}{*}{TC} & \multirow{2}{*}{BC} & \multirow{2}{*}{ST} & \multirow{2}{*}{SBF} & \multirow{2}{*}{RA} & \multirow{2}{*}{HA} & \multirow{2}{*}{SP} & \multirow{2}{*}{HC} & \multirow{2}{*}{mAP$_{50}$} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Speed\\ (fps)\end{tabular}} \\
& & & & & & & & & & & & & & & & & & & & \\ \hline
ROI Trans. \cite{ding2019learning} & R-101 & Two & AB & 88.53& 77.91 & 37.63& 74.08 & 66.53& 62.97 & 66.57& 90.50& 79.46& 76.75& 59.04 & 56.73& 62.54 & 61.29 & 55.56& 67.74& 7.80 \\
SCRDet \cite{yangSCRDetMoreRobust2019} & R-101 & Two & AB & 89.98& 80.65 & 52.09& 68.36 & 68.36 & 60.32 & 72.41 & 90.85& 87.94& 86.86 & 65.02 & 66.68 & 66.25 & 68.24 & 65.21 & 72.61 & 9.51\\
RSDet \cite{qian2019learning} & R-101 & Two & AB & 89.80 & 82.90 & 48.60& 65.20& 69.50 & 70.10 & 70.20 & 90.50 & 85.60 & 83.40 & 62.50 & 63.90& 65.60 & 67.20& 68.00 & 72.20 & - \\
Gliding Vertex \cite{xu2020gliding} & R-101 & Two & AB & 89.64 & 85.00 & 52.26 & 77.34 & 73.01 & 73.14 & 86.82 & 90.74& 79.02 & 86.81 & 59.55 & \textbf{70.91} & 72.94 & 70.86 & 57.32 & 75.02 & 13.10 \\
CSL \cite{yang2020arbitrary}& R-152 & Two & AB & \textbf{90.25} & 85.53 & 54.64 & 75.31 & 70.44 & 73.51 & 77.62 & 90.84 & 86.15 & 86.69 & 69.60 & 68.04 & 73.83 & 71.10 & 68.93 & 76.17 & 8.89 \\
Oriented R-CNN \cite{xie2021oriented} & R-50 & Two & AB & 89.84 & 85.43& \textbf{61.09} & 79.82 & 79.71& \textbf{85.35} & 88.82 & 90.88& 86.68 & 87.73 & 72.21 & 70.80 & \textbf{82.42} & 78.18& 74.11 & \textbf{80.87}
& 8.10\\
\hline
O$^2$-DNet \cite{wei2020oriented} & H-104 & One & AF & 89.31 & 82.14 & 47.33 & 61.21 & 71.32 & 74.03 & 78.62& 90.76 & 82.23 & 81.36 & 60.93 & 60.17 & 58.21 & 66.98 & 61.03 & 71.04 & - \\
BBAVectors \cite{yi2020oriented} & R-101& One & AF & 88.35& 79.96& 50.69 & 62.18& 78.43& 78.98 & 87.94 & 90.85 & 83.58& 84.35& 54.13& 60.24 & 65.22 & 64.28 & 55.70 & 72.32 & 18.37 \\
CFC-Net \cite{ming2021cfc}& R-50 & One & AB & 89.08 & 80.41 & 52.41 & 70.02 & 76.28 & 78.11 & 87.21 & 90.89 & 84.47 & 85.64 & 60.51 & 61.52 & 67.82 & 68.02 & 50.09 & 73.50 & 17.81 \\
RIDet \cite{ming2021optimization} & R-101 & One & AB & 88.94 & 78.45 & 46.87 & 72.63 & 77.63 & 80.68 & 88.18 & 90.55 & 81.33 & 83.61 & 64.85 & 63.72 & 73.09 & 73.13 & 56.87 & 74.70 & 13.36\\
GGHL \cite{huang2022general} & D-53 & One & AF & 89.74 & 85.63 & 44.50 & 77.48 & 76.72 & 80.45 & 86.16 & 90.83 & \textbf{88.18} & 86.25 & 67.07 & 69.40 & 73.38 & 68.45 & 70.14 & 76.95 & 42.30 \\
PolarDet \cite{zhao2021polardet} & R-101 & One & AF & 89.65& \textbf{87.07} &48.14 &70.97 &78.53 &80.34& 87.45& 90.76 &85.63 &86.87 &61.64 &70.32& 71.92& 73.09& 67.15& 76.64 & 25.00 \\
GWD \cite{yang2021rethinking} & R-152 & One & AB & 86.96 & 83.88 & 54.36 & 77.53 & 74.41 & 68.48 & 80.34 & 86.62 & 83.41 & 85.55 & \textbf{73.47} & 67.77 & 72.57 & 75.76 & 73.40 & 76.30 & 13.86 \\
KFIoU \cite{yang2022kfiou} & R-152 & One & AB &89.46 &85.72 &54.94 &80.37 &77.16 &69.23 &80.90 &90.79 &87.79 &86.13 &73.32 &68.11 & 75.23 & 71.61 & 69.49 & 77.35 & 13.79 \\ \hline
R$^3$Det \cite{yang2019r3det} & R-152 & Refine & AB & 89.80 & 83.77 & 48.11 & 66.77 & 78.76 & 83.27 & 87.84 & 90.82& 85.38 & 85.51& 65.67 & 62.68 & 67.53 & 78.56 & 72.62 & 76.47 & 12.39 \\
\cdashline{1-21}[0.8pt/2pt]
\multirow{2}{*}{S$^2$A-Net \cite{han2021align}} & R-50 & Refine & AB & 89.07 & 82.22 & 53.63 & 69.88 & 80.94 & 82.12 & 88.72 & 90.73 & 83.77 & 86.92 & 63.78 & 67.86 & 76.51 & 73.03& 56.60 & 76.38 & 17.60\\
~ & R-101 & Refine & AB & 88.89 & 83.60 & 57.74 & \textbf{81.95} & 79.94 & 83.19 & \textbf{89.11} & 90.78& 84.87 & \textbf{87.81} & 70.30 & 68.25 & 78.30 & 77.01 & 69.58 & 79.42 & 13.79\\
\cdashline{1-21}[0.8pt/2pt]
\multirow{2}{*}{G-Rep \cite{hou2022g}} & R-50 & Refine & AF & 87.76 &81.29 &52.64 &70.53 &80.34 &80.56 &87.47 &90.74 &82.91 &85.01 &61.48& 68.51 &67.53 &73.02 &63.54 &75.56 & 14.74 \\
~ & RX-101 & Refine & AF & 88.98 &79.21 &57.57 &74.35 &81.30 &85.23 &88.30 &90.69& 85.38 &85.25 &63.65 &68.82 &77.87 &78.76& 71.74 & 78.47 & - \\
\cdashline{1-21}[0.8pt/2pt]
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Oriented \\ RepPoints \cite{li2022oriented}\end{tabular}} & R-50 & Refine & AF & 87.02 & 83.17 & 54.13 & 71.16 & 80.18 & 78.40 & 87.28 & \textbf{90.90} & 85.97 & 86.25 & 59.90 & 70.49 & 73.53 & 72.27 & 58.97 & 75.97 & 16.10 \\
~ & R-101 & Refine & AF & 89.53 & 84.07 & 59.84 & 71.76 & 79.95 & 80.03 & 87.33 & 90.84 & 87.54 & 85.23 & 59.15 & 66.37 & 75.23 & 73.75 & 57.23 & 76.52 & 14.23 \\
~ & Swin-T & Refine & AF & 88.72 & 80.56 & 55.69 & 75.07 & \textbf{81.84} & 82.40 & 87.97 & 90.80 & 84.33 & 87.64 & 62.80 & 67.91 & 77.69 & \textbf{82.94} & 65.46 & 78.12 & -\\
\hline
\textbf{TS-Conv} & D-53 & Refine & AF & 89.86 & 87.05 & 49.12 & 74.01 & 78.97 & 81.28 & 88.24 & 90.77 & 86.85 & 87.24 & 71.87 & 69.88 & 77.01 & 70.43 & \textbf{78.63} & 78.75 & 23.23 \\
\textbf{TS-Conv Lite} & Mobile2 & One & AF & 89.08 & 84.20 & 38.08 & 74.47 & 77.03 & 75.40 & 86.50 & 90.84 & 79.44 & 85.63 & 59.33 & 66.51 & 67.03 & 67.73 & 68.12 & 73.96 & \textbf{62.07} \\
\hline\hline
\end{tabularx}
\end{threeparttable}}}}\vspace{0.5em}
\justifying{Note: Bold font indicates the best results. The backbone networks R50, R-101, R-152, H-104, D-53, RX-101, Swin-T, and Mobile2 represent the ResNet50 \cite{he2016deep}, ResNet101 \cite{he2016deep}, ResNet152 \cite{he2016deep}, Hourglass104 \cite{zhou2019objects}, DarkNet53 \cite{redmonYOLOv3IncrementalImprovement2018}, ResNeXt101 \cite{xie2017aggregated}, Swin Transformer Tiny \cite{liu2021swin}, and MobileNetv2 \cite{sandler2018mobilenetv2}, respectively. 'One', 'Two', 'Refine' represent the one-stage method, two-stage method and refine-stage method, respectively. 'AF' represents anchor-free methods, and 'AB' represents anchor-based methods. The inference speed only includes the network inference speed (batch size=1) on an RTX 3090 GPU. When testing other methods, their open source codes are used. Since the deep learning frameworks are different, there may be slight relative errors in the test speed. The speed of some methods could not be tested due to the available codes, which is indicated by “-”. Regarding some methods, we have tried our best but failed to reproduce the results shown in their original papers, so the best results reported by them are listed in the Table.}
\end{table*}
\begin{table}[tp]
\centering
\renewcommand\arraystretch{1}
\setlength{\tabcolsep}{1.3mm}{
\caption{\label{table:7}
{Comparative performance of different methods on DOTAv1.0, DOTAv1.5, and DOTAv2.0 datasets}}
\resizebox{0.48\textwidth}{!}{\setlength{\tabcolsep}{1.5mm}{
\begin{threeparttable}
\begin{tabular}{c|ccc}
\hline\hline
Methods &
\begin{tabular}[c]{@{}c@{}}mAP$_{50}$@v1.0\end{tabular}
& \begin{tabular}[c]{@{}c@{}}mAP$_{50}$@v1.5\end{tabular} & \begin{tabular}[c]{@{}c@{}}mAP$_{50}$@v2.0\end{tabular} \\ \hline
RetinaNet OBB \cite{linFocalLossDense2017} & 66.28 & 59.16 & 46.68 \\
Mask R-CNN \cite{ding2021object} & 70.71 & 62.67 & 49.47 \\
Cascade Mask R-CNN \cite{ding2021object} & 70.96 & 63.41 & 50.04 \\
Hybrid Task Mask \cite{ding2021object} & 71.21 & 63.40 & 50.34 \\
Faster R-CNN OBB \cite{renFasterRCNNRealTime2017a} & 69.36 & 62.00 & 47.31 \\
Faster R-CNN OBB + Dpool \cite{ding2021object} & 70.14 & 62.20 & 48.77 \\
Faster R-CNN H-OBB \cite{ding2021object} & 70.11 & 62.57 & 48.90 \\
Faster R-CNN OBB + RT \cite{ding2021object} & 73.76 & 65.03 & 52.81 \\
\hline
GGHL \cite{huang2022general} (Baseline) & 73.98 & 68.92 & 57.17 \\
\textbf{TS-Conv} & \textbf{75.04 \tiny{(+1.06)}} & \textbf{71.18 \tiny{(+2.86)}} & \textbf{59.77 \tiny{(+2.60)}} \\
\hline\hline
\end{tabular}
\end{threeparttable}}}}\vspace{0.5em}
\justifying{Note: Bold font indicates the best results. In order to make a fair comparison with the methods in the DOTAv2.0 benchmark \cite{ding2021object}, the experiments above do not use data augmentation and other tricks like these comparison methods. $mAP_{50}$@v1.0, $mAP_{50}$@v1.5, and $mAP_{50}$@v2.0 denote the results on the DOTAv1.0, DOTAv1.5, and DOTAv2.0 datasets \cite{li2020object}, respectively.}
\end{table}
\textbf{1) Comparison experiments on the DOTA datasets.} Table~\ref{table:6} provides the performance comparison results of the different methods on the most widely used AOOD dataset DOTAv1.0. The experimental results show that the performance and speed of the proposed TS-Conv outperform most of the AOOD methods (mAP$_{50}$=78.75), further validating the effectiveness of TS-Conv. Although the performance of TS-Conv is slightly lower than that of the two-stage method Oriented R-CNN \cite{xie2021oriented} and the refine-stage method S$^2$ANet (with the larger backbone ResNet-101) \cite{han2021align}, the detection speed of TS-Conv is much faster than those of Oriented R-CNN \cite{xie2021oriented} and S$^2$ANet \cite{han2021align}. Besides, the proposed TS-Conv is an anchor-free method, which is more flexible and does not rely on many hyperparameters of anchor boxes. The visualization results of TS-Conv on the DOTAv1.0 dataset \cite{xiaDOTALargeScaleDataset2018} are shown in Fig.~\ref{fig:11}. In addition, the performance of the lightweight model TS-Conv Lite (mAP$_{50}$=73.96) can reach the level of many larger models and has a detection speed that far exceeds that of other methods. Furthermore, the performance evaluation on the latest versions of the DOTA datasets, i.e., DOTAv1.5 and DOTAv2.0 \cite{li2020object} are listed in Table~\ref{table:7}. These datasets cover a wider category of objects and more small objects that are difficult to detect. The results also demonstrate the performance advantage of the proposed TS-Conv over existing methods.
\begin{table}[t]
\centering
\renewcommand\arraystretch{1}
\setlength{\tabcolsep}{0.5mm}{
\caption{\label{table:8}
{Comparative performance of different methods on the HRSC2016 dataset}}
\resizebox{0.48\textwidth}{!}{\setlength{\tabcolsep}{0.5mm}{
\begin{tabular}{c|c|c|ccc}
\hline\hline
Method & Anchor & Backbone & mAP$_{50}$(07) & mAP$_{50}$(12) & mAP$_{75}$(07) \\
\hline
R$^2$CNN \cite{jiang2017r2cnn} & AB & ResNet101 & 73.07 & 79.73 &-\\
RoI-Transformer \cite{ding2019learning} & AB & ResNet101 & 86.20 & - &- \\
Gliding Vertex \cite{xu2020gliding} & AB & ResNet101 & 88.20 & - &-\\
BBAVectors \cite{yi2020oriented} & AF & ResNet101 & 88.60 & - &-\\
CenterMap OBB \cite{wang2020learning} & AB & ResNet50 & - & 92.80 &- \\
RetinaNet-R \cite{yang2019r3det} & AB & ResNet101 & 89.18 & 95.21 &- \\
RetinaNet-GWD \cite{yang2021rethinking} & AB & ResNet50 & 85.56 & - & 60.31 \\
RetinaNet-KLD \cite{yang2021learning} & AB & ResNet50 & 87.45 & - & 72.39 \\
R$^3$Det \cite{yang2019r3det} & AB & ResNet101 & 89.26 & 96.01 &- \\
R$^3$Det-DCL \cite{yang2021dense} & AB & ResNet101 & 89.46 & 96.41 &-\\
R$^3$Det-GWD \cite{yang2021rethinking} & AB & ResNet50 & 89.43 & - & 68.88 \\
R$^3$Det-KLD \cite{yang2021learning} & AB & ResNet50 & 89.97 & - & 77.38 \\
S$^2$ANet \cite{han2021align} & AB & ResNet101 & 90.17 & 95.01 &-\\
Oriented RepPoints \cite{li2022oriented} & AF & ResNet50 & 90.40 & 97.26 &- \\
\hline
GGHL \cite{huang2022general} (Baseline) & AF & DarkNet53 & 89.53 & 96.50 & 76.07 \\
\textbf{TS-Conv} & AF & DarkNet53 & \textbf{90.59 \tiny{(+1.06)}} & \textbf{97.64 \tiny{(+1.14)}} & \textbf{78.34 \tiny{(+2.27)}}\\
\hline\hline
\end{tabular}}}}\vspace{0.5em}
\justifying{Note: Bold font indicates the best results. AF represents anchor-free methods, and AB represents anchor-based methods. The mAP$_{50}$(07) and mAP$_{50}$(12) represent the mAP calculated on standard of VOC07 and VOC12, respectively. Since the existing AOOD methods adopt different calculation standards, the mAPs obtained by them are all given here for fairness.}
\end{table}
\begin{table}[!t]
\centering
\renewcommand\arraystretch{1}
\setlength{\tabcolsep}{0.5mm}{
\caption{\label{table:9}
{Comparative performance of different methods on the DIOR-R dataset}}
\resizebox{0.48\textwidth}{!}{\setlength{\tabcolsep}{0.5mm}{
\begin{threeparttable}
\begin{tabular}{c|c|c|ccc}
\hline\hline
Methods & Anchor & Backbone & mAP$_{50}$ & mAP$_{75}$ & mAP$_{50:95}$ \\
\hline
RetinaNet-O \cite{linFocalLossDense2017} & AB & ResNet50 & 57.55 & - & - \\
Faster RCNN-O \cite{renFasterRCNNRealTime2017a} & AB & ResNet50 & 59.54 & - & - \\
Gliding Vertex \cite{xu2020gliding} & AB & ResNet50 & 60.06 & - & - \\
RoI-Transformer \cite{ding2019learning} & AB & ResNet50 & 63.87 & - & - \\
AOPG \cite{cheng2022anchor} & AF & ResNet50 & 64.41 & - & - \\
Oriented RepPoints \cite{li2022oriented} & AF & ResNet50 & 66.71 & - & - \\
\hline
GGHL \cite{huang2022general} (Baseline) & AF & DarkNet53 & 66.48 & 36.99 & 37.44 \\
\textbf{TS-Conv} & AF & DarkNet53 & \textbf{68.47 \tiny{(+1.99)}} & \textbf{42.69 \tiny{(+5.70)}}& \textbf{41.38 \tiny{(+3.94)}} \\
\hline\hline
\end{tabular}
\end{threeparttable}}}}\vspace{0.5em}
\justifying{Note: Bold font indicates the best results. AF represents anchor-free methods, and AB represents anchor-based methods.}
\end{table}
\begin{table}[!t]
\centering
\renewcommand\arraystretch{1}
\setlength{\tabcolsep}{1.2mm}{
\caption{\label{table:10}
{Comparative performance of different methods on the SSDD+ dataset}}
\resizebox{0.48\textwidth}{!}{\setlength{\tabcolsep}{1.2mm}{
\begin{threeparttable}
\begin{tabular}{c|c|c|c|c|c}
\hline\hline
\multirow{2}{*}{Methods} & \multirow{2}{*}{Anchor} & \multirow{2}{*}{Backbone}& \multirow{2}{*}{mAP$_{30}$} & \multirow{2}{*}{mAP$_{50}$} & \multirow{2}{*}{mAP$_{75}$} \\
& & & & & \\ \hline
DRBox-v1 \cite{an2019drbox} \cite{wang2018simultaneous} & AB & VGG16 & 86.41 & - & - \\
SDOE \cite{wang2018simultaneous} & AB & VGG16 & - & 82.40 & - \\
DRBox-v2 \cite{an2019drbox} & AB & VGG16 & 92.81 & 85.17 & - \\\hline
\begin{tabular}[c]{@{}c@{}}GGHL \cite{huang2022general} \\(Baseline1) \end{tabular}& AF & DarkNet53 & 95.10 & 90.22 & 22.18 \\ \cdashline{1-6}[0.8pt/2pt]
\textbf{TS-Conv} & AF & DarkNet53 & \textbf{96.56 \tiny{(+1.46)}} & \textbf{92.48 \tiny{(+2.26)}} & \textbf{41.96 \tiny{(+19.78)}} \\\hline
\begin{tabular}[c]{@{}c@{}} LO-Det + GGHL \cite{huang2021lo} \\ (Baseline2)\end{tabular} & AF & MobileNetv2 & 93.87 & 85.90 & 16.64 \\ \cdashline{1-6}[0.8pt/2pt]
\textbf{LO-Det\cite{huang2021lo} + DTLA} & AF & MobileNetv2 & \textbf{93.89 \tiny{(+0.02)}} & \textbf{87.08 \tiny{(+1.18)}} & \textbf{26.73 \tiny{(+10.09)}} \\
\hline\hline
\end{tabular}
\end{threeparttable}}}}\vspace{0.5em}
\justifying{Note: The testing image size is 800 $\times$800 pixels. To be consistent with the comparison method, the confidence threshold is set to 0.2. AF represents anchor-free methods, and AB represents anchor-based methods.}
\end{table}
\begin{figure}[!t]
\centering
\epsfig{width=0.45\textwidth,file=12.pdf}
\caption{Comparison of visualization results between GGHL and the proposed TS-Conv on the HRSC2016 dataset. Figs. (a-1)-(a-4) show the detection results of GGHL, and Figs. (b-1)-(b-4) show the detection results of TS-Conv.}\label{fig:12}
\end{figure}
\begin{figure}[!t]
\centering
\epsfig{width=0.45\textwidth,file=13.pdf}
\caption{Visualization results of TS-Conv on the DIOR-R dataset.}\label{fig:13}
\end{figure}
\begin{figure}[!t]
\centering
\epsfig{width=0.43\textwidth,file=14.pdf}
\caption{Presentation of other experiments. (a) Embedded edge devices, including Nvidia Jetson TX2, Nano and AGX Xavier, for evaluating the performance of lightweight models. (b) Visualization results of the proposed TS-Conv on the SSDD+ dataset.}\label{fig:14}
\end{figure}
\textbf{2) Comparison experiments on other datasets.} The results of comparison experiments on the HRSC2016 dataset are listed in Table~\ref{table:8}. TS-Conv also performs better than the existing methods in the ship detection, where the aspect ratio of objects is significant. Fig.~\ref{fig:12} shows that TS-Conv predicts more accurately and with fewer false alarms for oriented bounding boxes compared to GGHL \cite{huang2022general}. As shown in Table~\ref{table:9} and Fig.~\ref{fig:13}, TS-Conv also outperforms the existing methods on the DIOR-R dataset \cite{cheng2022anchor}, where objects have more scale variations and categories. In particular, the improvements are more significant for mAP$_{75}$ and mAP$_{50:95}$. The results on the SAR dataset SSDD+ are given in Table~\ref{table:10}. The results demonstrate that TS-Conv performs better on the datasets with other data modality. More visualized results are shown in Fig.~\ref{fig:14}. In addition, the experiments in Appendix C further demonstrate the generality of the proposed TS-Conv in other AOOD scenes. In summary, extensive comparison experiments are conducted on datasets covering multiple scenes, multimodal images (RGB, infrared, SAR, and panchromatic images), multiple categories of objects, and different lighting conditions (daytime and nighttime). The state-of-the-art results demonstrate the effectiveness and generality of TS-Conv.
\subsection{The PDF of CNN in GGHL}
Since the above model is too complicated, let's start with the basic neuron model in a neural network to explain probability density function (PDF).
Without loss of generality, we may model the CNN model as follows:
\begin{equation}
\boldsymbol{\hat y} = nn\left( {\boldsymbol{x},\boldsymbol{\theta}} \right) + \boldsymbol{b},
\tag{A-1}
\label{eq:A-1}
\end{equation}
where $nn\left( \cdot \right)$ is a deterministic function related to the CNN; $\boldsymbol{x}$ is the input; $\boldsymbol{\hat y}$ denotes the output, i.e., the predictions of CNN; $\boldsymbol{\theta}$ denotes the vector composed of learnable parameters; $\boldsymbol{b}$ represents the bias vector, which is usually set to a zero vector in CNN. To simplify the derivation, this setting is also used here, i.e., $\boldsymbol{\hat y} = nn\left( {\boldsymbol{x},\boldsymbol{\theta}} \right)$.
\textbf{1) PDF and loss function of OBB regression.} Define the ground truth of the prediction as $\boldsymbol{y}$. Define the error between the actual value and the predicted value as $\boldsymbol{\varepsilon} = \boldsymbol{y} - \boldsymbol{\hat y}$, which is assumed to obey an i.i.d. Gaussian distribution with a mean of 0 and variance ${\sigma ^2}$. Therefore, the PDF is
\begin{equation}
p\left( {\boldsymbol{y}\left| \boldsymbol{x} \right.;\boldsymbol{\theta} } \right) = p\left( \boldsymbol{\varepsilon} \right) = \frac{1}{{\sigma \sqrt {2\pi } }}{e^{ - \frac{{{{\left( {\boldsymbol{y} - \boldsymbol{\hat y}} \right)}^2}}}{{2{\sigma ^2}}}}},
\tag{A-2}
\label{eq:A-2}
\end{equation}
which represents the probability density of $\boldsymbol{y}$ when $\boldsymbol{x}$ and $\boldsymbol{\theta}$ are given. Then, for multiple ${\boldsymbol{y}^{\left( i \right)}},{\rm{ }}i = 1,2, \cdots ,m$, in different locations of output layers, their joint PDF is
\begin{equation}
\begin{array}{l}
p\left( {{\boldsymbol{y}^{\left( 1 \right)}} \cdots {\boldsymbol{y}^{\left( m \right)}}\left| {{\boldsymbol{x}^{\left( 1 \right)}} \cdots {\boldsymbol{x}^{\left( m \right)}}} \right.;\boldsymbol{\theta} } \right)\\
= \prod\limits_{i = 1}^m {p\left( {{\boldsymbol{y}^{\left( i \right)}}\left| {{\boldsymbol{x}^{\left( i \right)}}} \right.;\boldsymbol{\theta} } \right)} = \prod\limits_{i = 1}^m {\frac{1}{{\sigma \sqrt {2\pi } }}{e^{ - \frac{{{{\left( {{\boldsymbol{y}^{\left( i \right)}} - {{\boldsymbol{\hat y}}^{\left( i \right)}}} \right)}^2}}}{{2{\sigma ^2}}}}}.}
\end{array}
\tag{A-3}
\label{eq:A-3}
\end{equation}
Then, the likelihood function for $\boldsymbol{\theta}$ is
of output layers, their joint LF is
\begin{equation}
\begin{array}{l}
L\left( \boldsymbol{\theta} \right) = \log p\left( {{\boldsymbol{y}^{\left( 1 \right)}} \cdots {\boldsymbol{y}^{\left( m \right)}}\left| {{\boldsymbol{x}^{\left( 1 \right)}} \cdots {\boldsymbol{x}^{\left( m \right)}}} \right.;\boldsymbol{\theta} } \right)\\
= m\log \frac{1}{{\sigma \sqrt {2\pi } }} - \frac{1}{{2{\sigma ^2}}}\sum\limits_{i = 1}^m {{{\left( {{\boldsymbol{y}^{\left( i \right)}} - {{\boldsymbol{\hat y}}^{\left( i \right)}}} \right)}^2}}.
\end{array}
\tag{A-4}
\label{eq:A-4}
\end{equation}
Now let us reconsider the process of using CNN to predict the shape of OBBs, which is a regression. Since the error of the OBB regression is assumed to obey an i.i.d. Gaussian distribution with a mean of 0 and variance ${\sigma ^2}$, the PDF of $\boldsymbol{obb}{_{x,y,m}} = \left[{{\boldsymbol{l}_{x,y,m}},{\boldsymbol{s}_{x,y,m}},ar{_{x,y,m}}} \right]$, when $\boldsymbol{x}{^{obb}_{x,y,m}}$ and $\boldsymbol{\theta}{^{obb}_{x,y,m}}$ are given, is
\begin{equation}
\begin{array}{l}
p\left( {\boldsymbol{obb}{_{x,y,m}}\left| {obj{_{x,y,m}};\boldsymbol{x}_{x,y,m}^{obb}} \right.;\boldsymbol{\theta} _{x,y,m}^{obb}} \right)\\
= \displaystyle \frac{1}{{\sigma \sqrt {2\pi } }}{\displaystyle e^{ - \frac{{{{\left( {\boldsymbol{obb}{_{x,y,m}} - {{\boldsymbol{\widehat {obb}}}_{x,y,m}}} \right)}^2}}}{{2{\sigma ^2}}}}}.
\end{array}
\tag{A-5}
\label{eq:A-5}
\end{equation}
Note that the prediction of OBB is performed under the condition of determined positive and negative locations, so the $obj{_{x,y,m}}$ is also one of the conditions in Eq.~\ref{eq:A-5}. The LF of parameters $\boldsymbol{\theta} _{x,y,m}^{obb}$, is
\begin{equation}
\begin{array}{l}
L\left( {\boldsymbol{\theta}_{x,y,m}^{obb}} \right) = - {\left( {\boldsymbol{obb}{_{x,y,m}} - {{\boldsymbol{\widehat {obb}}}_{x,y,m}}} \right)^2}.
\end{array}
\tag{A-6}
\label{eq:A-6}
\end{equation}
According to MLE, the loss function at location ${\left( {x,y} \right)_m}$ is
\begin{equation}
\begin{array}{l}
Loss\left( { \boldsymbol{obb}{_{x,y,m}} - { \boldsymbol{\widehat {obb}}}_{x,y,m}} \right) = \\
\sum\limits_{k = 1}^4 {{{\left( {l_{x,y,m}^{\left( k \right)} - \hat l_{x,y,m}^{\!\left( k \right)}} \!\right)} ^2}} + \sum\limits_{k = 1}^4 {{{\left( {s_{x,y,m}^{\left( k \right)} - \hat s_{x,y,m}^{\left( k \right)}} \right)}^2}} \\ + {\left( {a{r_{x,y,m}} - {{\widehat {ar}}_{x,y,m}}} \right)^2},
\end{array}
\tag{A-7}
\label{eq:A-7}
\end{equation}
where $\hat l_{x,y,m}^{\left( k \right)}$ is the $k$th component of $1 \times 4$-dimensional vector ${\boldsymbol{\hat l}_{x,y,m}}$, and $l_{x,y,m}^{\left( k \right)}$ is the $k$th component of $1 \times 4$-dimensional vector ${\boldsymbol{l}_{x,y,m}}$. $\hat s_{x,y,m}^{\left( k \right)}$ is the $k$th component of $1 \times 4$-dimensional vector ${\boldsymbol{\hat s}_{x,y,m}}$, and $s_{x,y,m}^{\left( k \right)}$ is the $k$th component of $1 \times 4$-dimensional vector ${\boldsymbol{s}_{x,y,m}}$. Literature \cite{rezatofighiGeneralizedIntersectionUnion2019} proposed the GIoU term $\left( 1 - {GIoU} \left( {{\boldsymbol{l}_{x,y,m}},{{\boldsymbol{\hat l}}_{x,y,m}}} \right)\right)$ to replace the term of $\sum\limits_{k = 1}^4 {{{\left( {l_{x,y,m}^{\left( k \right)} - \hat l_{x,y,m}^{\!\left( k \right)}} \!\right)} ^2}}$, where the GIoU calculation can be found in Supplemental Materials B. We adopt this idea. Therefore, the loss function of OBB regression at location ${\left( {x,y} \right)_m}$ in Eq.~\ref{eq:10} is obtained.
\textbf{2) PDF of object classification.} The object classification task in this case is composed of multiple i.i.d. binary classifications and each component of $\boldsymbol{y}$ is either 0 or 1. To estimate $\boldsymbol{y}$, the non-linear activation function $Sigmoid\left( \cdot \right)$ is used on the basic neuron model in output layers. Thus, each component of $\boldsymbol{\hat y}$ is in $\left( {0,1} \right)$ that represents the classification score. In CNN, this classification score is usually interpreted as “probability” of the binary classification \cite{renFasterRCNNRealTime2017a,redmonYouOnlyLook2016a}. Assume that, given $\boldsymbol{x}$ and $\boldsymbol{\theta}$, $\boldsymbol{y}$ follows $Bernoulli\left( {1,\boldsymbol{\hat y}} \right)$, and the PDF is
\begin{equation}
\begin{array}{l}
p\left( {\boldsymbol{y}\left| \boldsymbol{x} \right.;\boldsymbol{\theta} } \right) = {\boldsymbol{\hat y}}^{\boldsymbol{y}}{\left( {1 - \boldsymbol{\hat y}} \right)^{1 - \boldsymbol{y}}}.
\end{array}
\tag{A-8}
\label{eq:A-8}
\end{equation}
Then, for multiple ${y^{\left( i \right)}},{\rm{ }}i = 1,2, \cdots ,m$, in different locations of output layers, their joint PDF is
\begin{equation}
\begin{array}{l}
p\left( {{y^{\left( 1 \right)}} \cdots {y^{\left( m \right)}}\left| {{x^{\left( 1 \right)}} \cdots {x^{\left( m \right)}}} \right.;\theta } \right)\\
= \prod\limits_{i = 1}^m {p\left( {{y^{\left( i \right)}}\left| {{x^{\left( i \right)}}} \right.;\theta } \right)} \\
= \prod\limits_{i = 1}^m {{{\left( {{{\hat y}^{\left( i \right)}}} \right)}^{{y^{\left( i \right)}}}}{{\left( {1 - {{\hat y}^{\left( i \right)}}} \right)}^{1 - {y^{\left( i \right)}}}}.}
\end{array}
\tag{A-9}
\label{eq:A-9}
\end{equation}
Thus, when $\boldsymbol{x}_{x,y,m}^{cls}$, $\boldsymbol{\theta}_{x,y,m}^{cls}$, and $\boldsymbol{obb}{_{x,y,m}}$ are given, the PDF of object classification is
\begin{equation}
\begin{array}{l}
\!\! p\left( {\boldsymbol{cls}{_{x,y,m}}\left| {\boldsymbol{obb}{_{x,y,m}};obj{_{x,y,m}};\boldsymbol{x}_{x,y,m}^{cls}} \right.;\boldsymbol{\theta} _{x,y,m}^{obb}} \right)\\
\!\! =\! p(cls_{x,y,m}^{\left( 1 \right)} \cdots cls_{x,y,m}^{\left( {nu{m_{cls}}} \right)}\left| {\boldsymbol{obb}{_{x,y,m}};ob{j_{x,y,m}};} \right.\\
\!\! x_{x,y,m}^{\left( 1 \right)}, \!\cdots x_{x,y,m}^{\left( {nu{m_{cls}}} \right)};{\theta {_{x,y,m}^{cls}}^{\left( 1 \right)}}, \!\cdots ,{\theta {_{x,y,m}^{cls}}\!^{\left( {nu{m_{cls}}} \right)}})\\
\!\! = \!\prod\limits_{c = 1}^{\! num{_{\! cls}}} \!{{{\left( \!{\widehat {cls}_{x,y,m}^{\left( c \right)}} \right)}^{\! cls \!_{x,y,m}^{\left( c \right)}}} \! \times {{\left( {\!1 \!\! - \widehat {cls}_{x,y,m}^{\left( c \right)}} \! \right)}^{\!1 \!- \! {\! cls \!_{x,y,m}^{\left( c \right)}}}}}.
\end{array}
\tag{A-10}
\label{eq:A-10}
\end{equation}
Similarly, when $x_{x,y,m}^{obj}$ and $\theta _{x,y,m}^{obj}$ are given. The PDF of $obj{_{x,y,m}}$ is
\begin{equation}
\begin{array}{l}
p\left( {ob{j_{x,y,m}}\left| {x_{x,y,m}^{obj}} \right.;\theta _{x,y,m}^{obj}} \right) = \\
{\left( {{{\widehat {obj}}_{x,y,m}}} \right)^{ob{j_{x,y,m}}}} \times {\left( {1 - {{\widehat {obj}}_{x,y,m}}} \right)^{1 - ob{j_{x,y,m}}}},
\end{array}
\tag{A-11}
\label{eq:A-11}
\end{equation}
where $\theta _{x,y,m}^{obj},{\rm{ }}m = 1,2,3,$ represent the parameter at ${\left( {x,y} \right)_m}$ used to predict whether this location is positive or negative.
\textbf{3) The joint PDF of the positive or negative location detection, OBB regression, and object classification.} Combining Eq.~\ref{eq:A-5}, Eq.~\ref{eq:A-10} and Eq.~\ref{eq:A-11}, we obtain Eq.~\ref{eq:15}.
\normalsize
\subsection{The calculation of IoU and GIoU in ORC}
Each ${\boldsymbol{l}_{x,y,m}}$ is a $1 \times 4$-dimensional vector composed of ${l_1},{l_2},{l_3},{l_4}$, which are the distances from the location ${\left( {x,y} \right)_m}$ to the top, right, bottom, and left edges of the circumscribe horizontal bounding box (HBB). Similarly, ${\boldsymbol{\hat l}_{x,y,m}}$ is also a $1 \times 4$-dimensional vector composed of ${\hat l_1},{\hat l_2},{\hat l_3},{\hat l_4}$. The area of the ground truth HBB and the predicted HBB are
\begin{equation}
\begin{array}{l}
area{_{x,y,m}} = \left( {{l_1} + {l_3}} \right) \times \left( {{l_2} + {l_4}} \right),
\end{array}
\tag{B-1}
\label{eq:B-1}
\end{equation}
\begin{equation}
\begin{array}{l}
{\widehat {area}_{x,y,m}} = \left( {{{\hat l}_1} + \hat l} \right) \times \left( {{{\hat l}_2} + {{\hat l}_4}} \right),
\end{array}
\tag{B-2}
\label{eq:B-2}
\end{equation}
respectively. The overlapping area of these two HBBs is
\begin{equation}
\begin{array}{l}
area_{x,y,m}^{overlap} = \left( {\min \left( {{l_1},{{\hat l}_1}} \right) + \min \left( {{l_3},{{\hat l}_3}} \right)} \right)\\
\times \left( {\min \left( {{l_2},{{\hat l}_2}} \right) + \min \left( {{l_4},{{\hat l}_4}} \right)} \right).
\end{array}
\tag{B-3}
\label{eq:B-3}
\end{equation}
The area of the smallest circumscribed HBB of these two HBBs is
\begin{equation}
\begin{array}{l}
area_{x,y,m}^{circ} = \left( {\max \left( {{l_1},{{\hat l}_1}} \right) + \max \left( {{l_3},{{\hat l}_3}} \right)} \right)\\
\times \left( {\max \left( {{l_2},{{\hat l}_2}} \right) + \max \left( {{l_4},{{\hat l}_4}} \right)} \right).
\end{array}
\tag{B-4}
\label{eq:B-4}
\end{equation}
Thus, the IoU calculated from ${\boldsymbol{l}_{x,y,m}}$ and ${\boldsymbol{\hat l}_{x,y,m}}$ is
\begin{equation}
\begin{array}{l}
IoU{_{x,y,m}}\left( {{\boldsymbol{l}_{x,y,m}},{\boldsymbol{\hat l}_{x,y,m}}} \right)\\
= \displaystyle{\frac{{area_{x,y,m}^{overlap}}}{{are{a_{x,y,m}} + {{\widehat {area}}_{x,y,m}} - area_{x,y,m}^{overlap}}}}.
\end{array}
\tag{B-5}
\label{eq:B-5}
\end{equation}
Eq.~\ref{eq:7} is calculated according to Eq.~\ref{eq:B-5} in the provided code. The GIoU is
\begin{equation}
\begin{array}{l}
GIoU{_{x,y,m}}\left( {{\boldsymbol{l}_{x,y,m}},{{\boldsymbol{\hat l}}_{x,y,m}}} \right) \\
\! = IoU{_{x,y,m}}\left( {{\boldsymbol{l}_{x,y,m}},{{\boldsymbol{\hat l}}_{x,y,m}}} \right) \!- \frac{{area_{x,y,m}^{circ} \! - {U_{x,y,m}}}}{{area_{x,y,m}^{circ}}},
\end{array}
\tag{B-6}
\label{eq:B-6}
\end{equation}
where ${U_{x,y,m}} = are{a_{x,y,m}} + {\widehat {area}_{x,y,m}} - area_{x,y,m}^{overlap}$, represents the area of the union region of the ground truth HBB and the predicted HBB. $IoU{_{x,y,m}}\left( {{\boldsymbol{l}_{x,y,m}},{{\boldsymbol{\hat l}}_{x,y,m}}} \right)$ represents the intersection with union of the areas of these two HBBs. $area_{x,y,m}^{circ}$ represents the area of the smallest circumscribed HBB of these two HBBs. $GIoU{_{x,y,m}}\left( {{\boldsymbol{l}_{x,y,m}},{{\boldsymbol{\hat l}}_{x,y,m}}} \right)$ measures the similarity between the predicted HBB and the ground truth HBB at ${\left( {x,y} \right)_m}$. The larger the value of $GIoU{_{x,y,m}}\left( {{\boldsymbol{l}_{x,y,m}},{{\boldsymbol{\hat l}}_{x,y,m}}} \right)$ is, the more accurate the prediction of predicted HBB is. The $GIoU{_{x,y,m}}\left( {{\boldsymbol{l}_{x,y,m}},{{\boldsymbol{\hat l}}_{x,y,m}}} \right)$ in Eq.~\ref{eq:11} is obtained.
\section{Introduction}
\input{Chap1}
\section{Related Works}
\input{Chap2}
\section{Proposed GGHL Framework}
\input{Chap3}
\section{Experiments and Discussions}
\input{Chap4}
\section{Conclusions}
\input{Chap5}
\footnotesiz
\bibliographystyle{IEEEtran}
|
2,877,628,090,126 | arxiv | \section{Introduction}
Topological insulators (TIs) have conducting surface states with a locking between the electron momentum and its spin \cite{Fu2007, Fu2007a,Hsieh2008,Xia2009,Hsieh2009b,Zhang2009,Hasan2010,Brune2012,Tchakov2013}. Besides bearing promise for high temperature spintronic applications \cite{Hsieh2009a,Jozwiak2013,Li2014}, TIs are also candidate materials to host exotic superconductivity. For example, $p+ip$ order parameter components \cite{Potter2011,Read2000} and Majorana zero energy states \cite{Nilsson2008,Tanaka2009,Alicea2012,Beenakker2013} have been theoretically predicted. The topological superconductivity can either be intrinsic \cite{Sasaki2011} or proximized by a nearby superconductor \cite{Fu2008,Stanescu2010,Zhang2011}.
The first generation of topological insulators, Bi-based materials such as Bi$_{1-x}$Sb$_x$ alloys, and later Bi$_{2}$Te$_{3}$\xspace and Bi$_{2}$Se$_{3}$\xspace compounds, exhibit topological surface states but also have an additional shunt from the conducting bulk, mainly due to anti-site defects and vacancies \cite{Hsieh2009b,Ren2010,Ren2011}. Josephson junctions \cite{Veldhorst2012,Sacepe2011,Qu2012,Zhang2011,Williams2012, Orlyanchik2013, Cho2013,Wang2012,Oostinga2013,Maier2012,Galletti2014,Sochnikov2013} and SQUIDs \cite{Veldhorst2012a,Qu2012,Kurter2013,Sochnikov2013,Galletti2014} have been realised in these topological surface states, but the practical use of these topological devices is limited by the bulk shunt \cite{Veldhorst2013,Veldhorst2012a}. Secondary and ternary compounds have been engineered to increase the bulk resistance and increase the stability of the surface states. The most promising examples of the latest generation three-dimensional TIs are Bi$_{2-x}$Sb$_{x}$Te$_{3-y}$Se$_{y}$\xspace \cite{Taskin2011} and strained HgTe\cite{Konig2007, Roth2009, Bouvier2011}.
In this work we report the realization of a Josephson supercurrent across 50 nm of topological insulator Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace. We first show that in our Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace no bulk conduction is present at low temperatures and that the observed surface states are of a topologically non-trivial nature. We then demonstrate Josephson junction behaviour reproducibly on different flakes and during multiple cooldowns. The width of the superconducting Nb electrodes is very narrow, of the order of 40 nm, anticipating future work on topological devices with only a few modes \cite{Beenakker2011,Beenakker1991,Furusaki1992}.
\section{Transport properties of exfoliated Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace flakes}
Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace single crystals were obtained by melting stoichiometric amounts of the high purity elements Bi (99.999 \%), Sb (99.9999 \%), Te (99.9999 \%) and Se (99.9995 \%). The raw materials were sealed in an evacuated quartz tube which was vertically placed in the uniform temperature zone of a box furnace to ensure the homogeneity of the batch. The molten material was kept at 850 $^{\circ}$C for 3 days and then cooled down to 520 $^{\circ}$C with a speed of 3 $^{\circ}$ C/h. Next the batch was annealed at 520 $^{\circ}$C for 3 days, followed by cooling to room temperature at a speed of 10 $^{\circ}$C/min\cite{Pan2014}.
Smooth flakes are prepared using mechanical exfoliation on a silicon-on-insulator substrate. To determine the transport characteristics of Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace, Hall bars are prepared using e-beam lithography and argon ion etching on exfoliated flakes with a thickness ranging from 80 till 200 nm. Au electrodes are defined by photolithography and lift-off. During all the fabrication steps the Hall bar is covered with either e-beam resist or photoresist protecting the surface from damage or contamination. The Hall bars are 7 $\mu$m long and 700 nm wide. Figure \ref{fig:1}(a) shows a SEM image of such a Hall bar.
A typical temperature dependence of the resistance of the Hall bars is shown in Fig. \ref{fig:1}(b).
At high temperature, the crystal exhibits semiconductor-like thermally activated behaviour. Below 150 K the resistance stabilises, indicating metallic surface channels. At high temperatures the transport properties are determined by the bulk of the crystal, while at low temperatures the surfaces provide the dominant charge carriers. To verify this, the high temperature part of the curve is modelled as a semiconductor using $R \propto e^{\Delta/k_{B}T}$. The best fit between 200 K and 300 K gives $\Delta=$ 18 meV. This means that the Fermi energy is positioned 18 meV below the bottom of the conduction band. The entire bulk band gap would be larger than 18 meV. At 300 K, the bulk contribution is dominant and allows for a one-band interpretation of the Hall effect measurement (see Fig. \ref{fig:1}(c)) at this temperature, giving a bulk carrier density of 10$^{17}$ cm$^{-3}$. Extrapolating the carrier freeze out to low temperatures we find a negligible bulk conduction at low temperatures. At low temperatures all transport is due to the surface states. Hall and longitudinal resistance measurements at 2 K are used to determine the electron density and mobility of the surface states. We reproducibly obtain surface electron densities in the range of 10$^{12}$ - 10$^{13}$ cm$^{-2}$ with mobilities between 120 and 450 cm$^{2}$/Vs. The resulting mean free path of the order of 10 to 40 nm is comparable to the mean free path of the electron-like surface states found by Taskin \textit{et al.} \cite{Taskin2011}.
Figures \ref{fig:1}(d) shows the change in longitudinal conductance as function of applied perpendicular magnetic field. To verify the topological character of the surface states the Hikami-Larkin-Nagaoka theory \cite{Hikami1980} is used to fit the low-field magnetoconductance in perpendicular field,
\begin{eqnarray}
\Delta G_{\square} (B_{\perp})-\Delta G_{\square}(0)=\alpha \frac{e^{2}}{2\pi^{2}\hbar}\left[\psi\left(\frac{1}{2}+\frac{\hbar }{4el_{\phi}^{2}B_{\perp}}\right)-\mbox{ln}\left(\frac{\hbar } {4el_{\phi}^{2}B_{\perp}}\right)\right],
\end{eqnarray}
where $G_{\square}=\frac{L}{WR}$ with $L$ being the length and $W$ the width of the Hall bar, $R$ is the measured resistance, $\psi$ is the digamma function, $l_{\phi}$ is the inelastic scattering length, $\alpha$ is a parameter indicating the strength of the spin-orbit interaction and $B_{\perp}$ is the perpendicular magnetic field. If the spin-orbit interaction is weak, a positive value of $\alpha=1$ is expected, as opposed to strong spin-orbit interaction where a negative value of -0.5 is expected\cite{Hikami1980}. Due to the chiral-spin texture of a topological insulator and the contribution of both top and bottom surfaces in the transport measurements, an $\alpha$ parameter of $-1$ is expected. In Fig. \ref{fig:1}(d) the magnetoconductance is fitted with $l_{\phi}=$ 144 nm and $\alpha=-$1.01 which would be in good agreement with the presence of just the bottom and top topologically non-trivial surface states.
However, we cannot rule out the presence of additional trivial two-dimensional metallic surface states that could originate from surface band bending. The possibility of the conduction band bending down below the Fermi energy has been observed in Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace \cite{Taskin2011}. Angle resolved photo-electron spectroscopy has revealed that the presence of these non-topological surface states is depending on surface absorbents and could vary over time\cite{Golden2014}.
\begin{figure}[t]
\centering
\includegraphics[width=1\textwidth]{RTHallbar.png}
\caption{(a) Scanning electron microscopy image of a Hall bar of Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace. The dimension of this Hall bar is different from the Hall bar of the measurements shown in Fig. 3 (c--d). The Hall bar of the measurements has a width of 560 nm and a length of 6.8 $\mu$m. The dark grey part is etched away. The light gray part between the electrodes is the topological insulator. The white scale bar is 5 $\mu$m. The longitudinal resistance is measured between V1+ and V-, and the Hall resistance between V- and V2+. (b) Typical temperature dependence of the resistance of an exfoliated flake of Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace, measured in a Hall bar configuration. The dependence is fitted with a function $e^{\Delta/k_{B}T}$.(c) A typical Hall effect measurement at 2K (black) and 300 K (red). The negative slope implies that the charge carriers are electrons. (d) Measured magnetoconductance, togetherwith a fit of the HLN theory to the data. The best fit is obtained for $\alpha=-1.01$ and $l_{\phi}=144$ nm. }
\label{fig:1}
\vspace{-15pt}
\end{figure}
\section{Junction fabrication}
Now that we have verified the topological nature of our crystals we turn to fabricating Josephson junctions with Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace as the weak link. Transferred similarly to the devices for determining the transport properties, smooth flakes on Si/SiO$_{2}$ substrates are used for devices. E-beam lithography is used to define the junctions and the contact pads. Thereafter we perform a 30 second low voltage etching step to avoid large damage to the surface followed by sputtering in-situ 25 nm Nb and 2.5 nm Pd to protect the Nb layer, and lift-off of the excess material. The resulting Nb/Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace/Nb junction has been covered by photoresist during the entire process and no damage from etching or growth has occurred to the crystal surface between the electrodes. In figure \ref{fig:2}(a) a junction is visible in a SEM image, and similar junctions have been prepared on different flakes.
The width is about 40 nm and the electrode separation is about 50 nm. The motivation for the junction length is based on our earlier work on Bi$_{2}$Te$_{3}$ with junctions ranging from 50-250 nm \cite{Veldhorst2012}. The mean free path in Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace is ten times smaller compared to Bi$_{2}$Te$_{3}$\xspace which implies that the junction length should also be reduced by a factor of ten to realise sufficient coupling between the electrodes. Simultaneously, this design takes a step forward towards experiments for the observation of a quantized supercurrent where only a few modes should be present\cite{Beenakker2011,Beenakker1991,Furusaki1992}. In order to achieve the presence of a few modes, the junction width should be of the order of the Fermi wave length. For a 40 nm wide junction and a Dirac velocity of $4.5 \times 10^5$ m/s this means that the Fermi level would have to be tuned within 50 meV from the Dirac point by means of gating, which is quite reasonable. To realise these dimensions a thin layer of PMMA (80 nm) is used which limits the thickness of Nb that could be grown. The reduction of the dimensions of the Nb leads reduces the gap of Nb. A 40 nm wide and 250 nm long Nb nanowire is prepared in the same run to determine the properties of the niobium with these dimensions. In figure \ref{fig:2}b the critical current of this nanowire as a function of temperature is presented, and a reduction of the critical temperature from 9 K of the Nb of our thin film process to about 1.4 K is visible.
\begin{figure}[t]
\centering
\includegraphics[width=1\textwidth]{SEMJunction.png}
\caption{(a) Scanning electron microscopy image of a Nb/Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace/Nb junction. The Nb layer has been coloured yellow for clarity. The junction is 57 nm long and 38 nm wide. (b) The dependence of the critical current of the Nb nanowire as function of temperature. From the linear fit to the data point in the range of 300-850 mK, the critical temperature is extrapolated to be 1.4 K. Close to the transition temperature, the gap and therefore the critical current can be approached by a linear dependence \cite{Ferrell1964}.
}
\label{fig:2}
\vspace{-15pt}
\end{figure}
\section{Results}
Measurements are performed in a cryogen free dilution refrigerator with low pass filtering of current and voltage signals using pi filters, printed circuit board copper powder filters \cite{Mueller2013}, and RC filters. We measured the critical current as function of temperature and the modulation of the critical current by an applied magnetic field at 30 mK. We will start this section with a discussion of the temperature dependence followed by an analysis of the Fraunhofer pattern.
\subsection{Temperature dependence of the critical current}
At 30 mK we observe a critical current of 14 nA for a junction with 57 nm electrode separation, visible in figure \ref{fig:2}(a), and 4.8 nA for a 80 nm long junction. Using the extracted critical temperature of the Nb nanowire as the transition temperature for the leads, the normalised temperature dependence of the critical current is plotted in figure \ref{fig:CriticalCurrent}(a) for the 57 nm junction. The boundary between Josephson coupling and thermal noise $(\hbar I_{0}/e)/(k_{B}T)=1$ is illustrated with the dashed line. In figure \ref{fig:CriticalCurrent}(b) the dV/dI characteristics around the thermal limit reveal indeed that a zero resistance state and coherence peaks are only visible below the thermal limit. The coherence peaks are used for determining the critical current below the thermal limit. Above the thermal limit the width of the resistance valley is used as an estimate.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{CriticalCurrent}
\caption{Josephson characteristics of the Nb/Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace/Nb junction illustrated in Figure \ref{fig:2}.
(a)~Temperature dependence of the critical current. Above the thermal limit, indicated by the black dashed line, the critical current is determined by the resistance peaks. Below the thermal limit the width of the resistance dip is used as estimate. The red dashed line is a fit for diffusive junctions with the Usadel equation, see main text.
(b)~Derivative of the current-voltage characteristic above, at, and below the thermal limit. Below the thermal limit the resistance is zero and coherence peaks are visible. These peaks disappear above the thermal limit of $~$200mK and only a resistance valley remains.
(c)~Modulation of the critical current by an applied perpendicular magnetic field. The upper and lower frames are measurements during different cooldown cycles. The upper frame was the first cooldown using Al bond wires which have reduced cooling power below 10 mT, resulting in a sharp step in the critical current at this point. The bottom frame uses Au bond wires. This second cooldown exhibited a slightly reduced critical current and ergo reduced visibility of the first lobe of the diffraction pattern. The dotted line is a model of the critical current by Barzykin \textit{et al.} \cite{Barzykin1999} using the junction parameters found in (a).
(d)~Current-voltage characteristic and derivative at base temperature. The critical current of 14 nA and normal state resistance of 460 $\Omega$ result in an $I_{\tiny{\textrm{C}}} R_{\tiny{\textrm{N}}}$\xspace product of 6.4 $\mu$V, consistent with a junction with high $\gamma_B$ and $L \geq \xi_N$. }
\label{fig:CriticalCurrent}
\vspace{-15pt}
\end{figure}
The nature of weak link is determined by fitting the temperature dependence of the supercurrent. For diffusive SNS junctions, i.e. junctions where the junction length $L$ is larger than the electron mean free path, the Usadel equation is used to describe the temperature dependence \cite{Usadel1970,Zaitsev1984,Kupriyanov1988},
\begin{equation}
J=\frac{2\pi k_{B}T}{e\rho_{N}}\mbox{Im} \sum _{\omega_{n}>0}\frac{G_{N}^{2}}{\omega_{n}^{2}}\Phi_{N}\frac{d}{dx}\Phi_{N},
\end{equation} where $\rho_{N}$ is the resistivity of the N layer, $\Phi_{N}$ is the self-consistently determinded induced order parameter function in the N layer with $G_{N}$ the corresponding normal Green function and $\omega_{n}=\pi k_B T (2n+1)$ the Matsubara frequencies, where $n \geq 0$ is integer. As there is no analytical expression for arbitrary length and barrier transparency, the expression was solved numerically with three fitting parameters\cite{Golubov1995}, $\gamma=\frac{\rho_{s}\xi_{s}}{\rho_{N}\xi_{N}}$, $\gamma_{B}=\frac{R_{B}}{\rho_{N}\xi_{N}}$ and $\xi_{N}$. Here, $\gamma$ is the ratio between the resistivities and coherence lengths in the superconducting leads and the topological insulator surface state. The resistivity of Nb is much smaller than that of Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace which gives $\gamma \ll1$. $\gamma_B$ is proportional to the interface resistance per unit area $R_{B}$ between the S and N layers. Finally, $\xi_N$ is the coherence length in the normal region at $T_{C}$. For the junction illustrated in figures \ref{fig:2}(a) and \ref{fig:CriticalCurrent}, the best fit is obtained for $\gamma_{B}=5$ and $\xi_{N}=11.4$ nm. The other junctions give values of the same order of magnitude. The large $\gamma_{B}$ is in good agreement with the low excess current in the $IV$ curves which implies a large barrier or low transparency at the interface. Taking the $T_{c}$ of the Nb wire, we get for $\xi_{N}\left( T_{c}\right)=\sqrt{\frac{\hbar D}{2 \pi k_{B}T_{c}}}$ a value of 28 nm which is also in good agreement with the fitting parameters.
We note, that the junctions have a small $I_{\tiny{\textrm{C}}} R_{\tiny{\textrm{N}}}$\xspace product, $\sim 7 \mu$V. We argued that $\gamma \ll 1$. Together with the fitted values of $\gamma_{B}$ and $\xi_{N}$ the $I_{\tiny{\textrm{C}}} R_{\tiny{\textrm{N}}}$\xspace product is either of the form $\propto V_{0}/\gamma_{B}$ or $I_{\tiny{\textrm{C}}} R_{\tiny{\textrm{N}}}$\xspace$\propto V_{0}e^{-L/\xi_{N}}$ \cite{Golubov2004}. Substitution of $\gamma_{B}$, $\xi_{N}$ and the junction length $L$ we estimate from this relation that the $I_{\tiny{\textrm{C}}} R_{\tiny{\textrm{N}}}$\xspace should be in the order of 1-10 $\mu$V which is in good agreement with the measured product value.
\subsection{Critical current as a function of magnetic field}
In Fig. \ref{fig:CriticalCurrent}(c) measurements during two separate cooldowns of the I$_{\tiny{\textrm{C}}}$(B) pattern at 30 mK are shown for the same junction. The changes in the supercurrent in the second cooldown indicate a slight evolution in time. The upper frame shows the Fraunhofer pattern of the first measurement. Due to the use of Al bond wires, the cooling of the sample is reduced below the critical field of Al. These measurements are greyed out because the temperature increased in that region, we estimate by 100 mK. A second cooldown using Au bonds results in the lower panel Fraunhofer pattern. Here, the critical current has reduced slightly resulting in a decreased visibility of the lobes of the Fraunhofer pattern.
The Fraunhofer patterns are fitted by the critical current derived from the Usadel equations for arbitrary $W$ and $L$ with open boundaries, as described by Barzykin \textit{et al.} \cite{Barzykin1999},
\begin{eqnarray}
I_{c} &\sim \sum^{\infty}_{l=-\infty}(-1)^{l}S_{l}(L/2)S'_{l}(L/2) \left(\frac{\sin \pi (\nu+l)/2}{\pi(\nu+l)/2)}-(-1)^{l}\frac{\sin \pi (\nu-l)/2}{\pi(\nu-l)/2)}\right)^{2},\nonumber \\
S_{l}(u) &= \sqrt{|u|/2\pi}\left(q_{T}^{2}+\pi^{2}l^{2}/W^{2}\right)^{1/4} K_{1/2}\left(\sqrt{u^{2}\left(q_{T}^{2}+\pi^{2}l^{2}/W^{2}\right)}\right),\nonumber \\
S_{l}'(u) &= \frac{d}{du}S_{l}(u),\label{FPmodel}
\end{eqnarray} where $\nu=\phi/\phi_{0}$ with $\phi_{0}=\hbar/2e$ is the normalised flux, $q_{T}=1/\xi_{N}^{2}$ and $K_{1/2}$ is a modified Bessel function of the second kind. For our junction dimensions it follows from this model that the first period is tripled, i.e. the first minimum is at $\Phi=3\Phi_{0}$ and the following minima are separated by $2\Phi_{0}$ intervals. The doubling of the period is not restricted to diffusive junctions and was, in fact, also predicted \cite{Barzykin1999} and observed \cite{Heida1998} in ballistic junctions for width and length ratios in the order of one.
The first minimum in the measurements occurs at a field value of 0.07 T. The width of the junction is 37 nm. For a flux of 3$\phi_{0}=6.2\cdot 10^{-15}$Wb this implies that the junction effective length (including the penetration depths) is about 2 $\mu$m. The obtained London penetration depth is then about 1 $\mu$m. This large London penetration depth compared to bulk Nb (47 nm)\cite{Maxfield1965} can be explained by the reduced dimensions of the junctions. The increase of the London penetration depth with decreasing film thickness is studied in Ref. \citeonline{Gubin2005}. A film thickness of 25 nm gives a London penetration depth above 100 nm and a decrease of the critical current to 8 K. Due to the reduction of width of the junction in our devices, we end up with a $T_{C}$ of 1.4 K which consistently implies an even larger increase of the London penetration depth.
The coherence length found from the fitting of the critical current together with the width and the length of the junction and London penetration depth serve as inputs for the model described in equation \ref{FPmodel}. The theoretical expectation is shown in figure \ref{fig:CriticalCurrent}(c). The relative small critical current of the higher order peaks in the data is found to be well explained by the model.
\section{Discussion}
We showed that the transport in the Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace flakes is dominated by surface states at low temperatures where we study proximity induced superconductivity. We prepared Josephson junctions with widths in the order of 40 nm and lengths in the order of 50 to 80 nm on several Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$\xspace flakes and measured them during several cooldowns to 30 mK. The Fraunhofer patterns unequivocally reveal that the supercurrent is a Josephson supercurrent. The measured critical currents are reproducibly observed on different devices and upon multiple cooldowns, and the measurement can be well explained by diffusive transport models and geometric effects. The predicted $4\pi$ periodic Josephson effect can only be observed in the future for similar devices with just a single perpendicular mode \cite{Snelder2013} and when measured faster than the relaxation rate of a quasiparticle inside an Andreev bound state.
The realization of a Josephson supercurrent in junctions with dimensions in the order of tens of nanometers on a topological insulator dominated by surface states at low temperatures, is an important technological step towards advanced devices necessary for the observation of a quantized supercurrent and confirming the presence of a Majorana bound states in such devices \cite{Fu2008}. Future work will focus on top and bottom gating of the surface states to eliminate potential additional trivial surface states from band bending.
\ack
We would like to thank Frank Roesthuis and Dick Veldhuis for support during fabrication and Elia Strambini for support during the measurements. We acknowledge Martin Stehno for useful discussions. This work is supported by the Netherlands Organization for Scientific Research (NWO), by the Dutch Foundation for Fundamental Research on Matter (FOM), by the European Research Council (ERC) and supported in part by Ministry of Education and Science of the Russian Federation, grant no. 14Y26.31.0007.
\section*{References}
\bibliographystyle{iopart-num-nourl}
|
2,877,628,090,127 | arxiv | \section{Introduction}
Degraded images are detrimental for high-level vision tasks, such as object detection~\cite{carion2020end} and image segmentation~\cite{chen2021transunet}. Single image desnowing aims to reconstruct the clean image from a snow degraded version, which is challenging due to the complex snow degradation and diverse degradation
scales. According to the previous desnowing works~\cite{chen2020jstasr}, snow image formation can be expressed as:
\begin{equation}
I(x)={K}(x) {T}(x)+{A}(x)(1-{T}(x)),
\end{equation}
where the $I(x)$ denotes the snow image, $T(x)$ and $A(x)$ represent the transmission map and atmospheric light respectively. $K(x)$, the veiling-free snow scene, can be decomposed as: $K(x) = J(x)(1-Z(x)R(x)) + C(x)Z(x)R(x)$,
where $R(x)$ is the the binary mask image which presents the location information of snow, $Z(x)$ and $C(x)$ denote the chromatic aberration image and the snow mask.
From the above formation model, the degradation of snow scene is complex. Visually speaking, it can be seen from Fig.\ref{fig:1}(a) that snow degradation includes snow streaks of different sizes, a few snow spots, and hazy fog morbidly overlaid on the scene. The composite degraded snow scene requires that the desnowing network needs to pay attention to different degradation information. JSTASR~\cite{chen2020jstasr} proposed a method for dealing with the veiling effect and image inpainting from the degradation
version. HDCW-Net~\cite{hdcwnet} adopted the dual-tree complex wavelet transform to decompose the snow image and designed a high-frequency reconstruction network and a low-frequency reconstruction network to process snow particles and large-scale degradation. However, as shown in Fig.\ref{fig:1}(b) and (c), it is difficult for previous methods to handle various degradation information simultaneously. Therefore, we present our motivation:
\begin{itemize}
\item \textit{How to design an effective and efficient snow removal network that can see various kinds of degradations in snow scenes?}
\end{itemize}
We believe that the context interaction of local details and global information is essential for single image desnowing.
Compared with utilizing operations such as wavelet transform~\cite{hdcwnet}, Fourier transform~\cite{dan}, ViT~\cite{vit} design can capture long-distance dependencies well. PVT~\cite{PVT} proposed a pyramidal architecture to gradually expand the receptive field and perform global modeling. Swin Transformer~\cite{swim} divided the image into different windows and limited the self-attention operation inside the window to reduce computational overhead, which achieved advanced performance. Nevertheless, differ from high-level vision tasks, recovering clean images from adverse snow scenes using ViT still faces the following issues: (i) \textbf{Plain Vit lacks the ability of modeling context interaction of local details and global information.} (ii) \textbf{The large computational overhead of ViT is unfriendly for low-level vision tasks.}
To solve the above issues, in this paper, we propose SnowFormer, a careful design consistent with our motivation. At the same time, the parameters and the amount of computation are competitive compared to other SOTA methods. Specifically, we devise a Scale-aware Aggregation Module (SaA) in the encoder part, aggregating different degradation information and enriching the representation ability of the scale-aware feature. Next, in the decoder part, we propose a Context Interaction Transformer Block (CITB). It consists of window-based local context interactions and global context interactions whose query is generated by the scale-aware feature. It is worth mentioning that the scale-aware-guided cross self-attention mechanism can improve the exchange between local details and global information. In addition, we offer the Heterogeneous Feature Projection Head (HFPH), which aims to deal with the residual degradation problem in the recovering processing. We use the heterogeneous features of each stage for progressive fusion and refinement. Compared with the direct single output decoder feature as input~\cite{zamir2021restormer}, our design can better obtain snow features at different stages, promoting the refinement block and projecting it to a clean image.
Overall, our contributions can be summarized as follows:
\begin{itemize}
\item To enhance the feature representation of various snow degradations and avoid the loss of small degradations due to repeated down-sampling operations, we propose a Scale-aware Aggregation Module.
\end{itemize}
\begin{itemize}
\item We propose the Context Interaction Transformer Block, which not only improves local modeling, but also enables context interaction of local details and global information by using scale-aware-based cross self-attention.
\end{itemize}
\begin{itemize}
\item A Heterogeneous Feature Projection Head is proposed to handle the residual degradation, where heterogeneous features at different stages are progressively refined and projected to the clean image.
\end{itemize}
\begin{figure}[!t]
\centering
\includegraphics[width=40mm]{figure/PSNR_param.png} \label{fig:parameters}
\hfill
\hfill
\includegraphics[width=40mm]{figure/PSNR_param_srrs.png} \label{fig:flops}
\caption{Trade-off between PSNR performance and the number of parameters on CSD~\cite{hdcwnet} dataset (left) and SRRS~\cite{chen2020jstasr} dataset (right). The results show the superiority of our model compared with other SOTA desnowing methods. }\label{gflop}
\end{figure}
\section{Related Works}
\subsection{Single Image Desnowing}
\textbf{(i) Physics-based Methods}: In early days, most desnowing approaches were based on some physical models and leveraged prior knowledge. \cite{pei2014removing} used image priors of saturation and visibility in snow scenes to remove snow particles. \cite{zheng2013single} considered the influence of background edges and differences in rain streaks and applied a multi-guided filter to extract snow features, which can separate snow components from the background. ~\cite{wang2017hierarchical} proposed a hierarchical scheme that enables dictionary learning and snow component decomposition.
\textbf{(ii) Data-driven Methods}: With the rapid development of deep learning, learning-based approaches have achieved remarkable results. As the first snow removal network dubbed as DesnowNet~\cite{liu2018desnownet} designed a multi-stage model to remove snow particles and snowflakes progressively. All-in-one bad weather removal network~\cite{allinone} offered multiple specific encoders to simultaneously handle various degradations, which can recover different ill-pose weather issues. JSTASR~\cite{chen2020jstasr} considered the veiling effect and the diversity of snow cover and proposed a multi-scale snow network for snow removal from a single image. HDCW-Net~\cite{hdcwnet} embedded the dual-tree wavelet transform into the network architecture. In addition, the authors also designed a prior-based loss called contradictory channel to perform image snow removal. TransWeather~\cite{valanarasu2022transweather} utilized the ViT as the backbone and proposed a weather type decoder to promote processing multiple adverse weather scenes, which achieves non-trivial improvements.
\subsection{Vision Transformer}
Recently, inspired by the remarkable successes of transformers in natural language processing, vision transformer~\cite{vit} (ViT) gradually has been applied for computer vision tasks. Compared with the dominant convolutional neural networks (CNNs), ViT can offer the powerful ability of global feature modeling. Using this advantage, several ViT-based networks have achieved impressive effects in various fields~\cite{swim,lee2021mpvit,PVT,chen2021crossvit,mehta2021mobilevit}. PVT~\cite{PVT} presented a pyramid architecture to progressively enlarge the receptive field and reduce the size of feature map, improving the performance and mitigating the computational overhead. Swin-Transformer~\cite{swim} partitioned an image into windows and restricted self-attention operation inside each window, which significantly reduces the computational complexity. To obtain global modeling capability, it adopt a window shifting operation, enabling interaction among nearby windows. CrossVit~\cite{chen2021crossvit} made use of the projection from different features to perform a cross self-attention mechanism, which can utilize the global information representation to a greater extent.
Besides high-level tasks, recently ViT has been applied for low-level image restoration due to its advantage of global long-distance dependencies~\cite{zamir2021restormer,wang2022uformer,song2022vision,valanarasu2022transweather,lee2022knn}. Uformer~\cite{wang2022uformer} utilized the Swin-transformer module to adopt it in a U-shape~\cite{unet} architecture for image restoration. Restormer~\cite{zamir2021restormer} proposed channel dimension self-attention operation to capture the channel information. However, these overall ViT-based architectures may lead to huge computational complexity. For the task of single image dehazing, Dehazeformer~\cite{song2022vision} designed a Swin-based model which utilizes the padding operation to improve the edge repair effect. In this work, we believe that global and local information are both vital for image desnowing. We propose a scale-aware Transformer with context interaction, which greatly enhances the global attention capability for snow removal. Futhermore, a novel heterogeneous feature refinement is presented to optimize the embedded features of CNNs and ViT.
\section{Method}
\begin{figure*}[!t]
\centering
\includegraphics[width=18cm]{figure/SnowFormer1.pdf}
\caption{The overall framework of our SnowFormer, which is a u-shaped encoder-decoder architecture.
Scale-aware Aggregation, Context Interaction, and Heterogeneous Feature Projection Head (HFPH) are detailed in the ``Method" section.}
\label{fig:snowformer}
\end{figure*}
\subsection{SnowFormer}
As shown in Fig.\ref{fig:snowformer}, the overall structure of our proposed SnowFormer is a heterogeneous U-shaped architecture that consists of a CNNs-based encoder and a ViT-based decoder. Specifically, we utilize the Scale-aware Aggregation Module to capture the local information and integrate multi-scale features, which can aggregate various snow degradation features while avoiding the problem of feature loss due to repeated down-sampling. Next, we feed it to the latent layer for global context modeling. For the decoder design, we believe that for the complex degradation process in snow removal, it is essential to utilize scale-aware features for local and global context interactions. Therefore, we employ Context Interaction Transformer Block to reconstruct the image features. In addition, we devise a Heterogeneous Feature Projection Head (HFPH) that fuses and refines the heterogeneous features. Finally, it projects the optimized feature into the clean image.
\subsection{Scale-aware Aggregation Module}
For the encoder, we adopt a spindle-shaped DWConv3$\times$3-LN-GULE-DWConv3$\times$3 block for local feature extraction and gradually down-sampled the features while expanding the local receptive field. However, there are still two problems needed to be considered. (i) Various degradations in the snow scene are overlaid on the clean image, which is challenging for restoration. (ii) Information loss is unavoidable as the resolution is gradually reduced, which makes this plain encoder structure difficult to capture small degradations, such as snow spots and snow streaks. To tackle these problems, we propose the Scale-aware Aggregation which extracts various and multi-scale degradation information by aggregating hierarchical features to form the scale-aware feature.
Specifically, the max-pooling and Conv1$\times$1 operations are utilized to align the feature size of each stage. The proposed Scale-aware Aggregation operation can be expressed as:
\begin{gather}
X_{Fi}\in \mathbb{R}^{\frac{H}{16} \times \frac{H}{16} \times 16C} = X_{i}\in \mathbb{R}^{ \frac{H}{2^i}\times \frac{W}{2^i}\times 2^{i}C}\downarrow, \\
X_{F} = \sum_{i=0}^{3} X_{Fi} + X_{4},
\end{gather}
where $\downarrow$ represents the combined operations of max-pooling and Conv1$\times$1, $i$ denotes the $i$-th layer of the encoder. $H$, $W$ and $C$ mean the height, width and channel dimension of feature in the first layer. $X_{F}$ represents the final feature after scale-aware aggregation operation.
We subsequently apply a multi-head self-attention block~\cite{vit} on $X_{F}$ for global modeling, outputting scale-aware information for the cross self-attention of context interaction. Detailed explanation and implementation of this block are provided in the supplementary material.
\subsection{Context Interaction Transformer Block}
\subsubsection{Local Context Interaction Module}
For single image desnowing, local information is the foundation for removing the ill-pose degradations~\cite{hdcwnet}. Local context interaction improves the effects of detail restoration from complex snow images. Compared with CNNs, local self-attention shows more strong interaction ability. Inspired by ~\cite{swim}, we employed the window-based self-attention to model the local context interaction in the decoder part, which achieves the trade-off between the performance and computational complexity. We describe our Local Context Interaction module as follows:
\begin{gather}
\hat{X}_{w}=X_{w}+\text { W-MSA }\left(X_{w}\right),\\
X_{l}=\hat{X}_{w}+\text{MLP}\left(\hat{X}_{w}\right),
\end{gather}
where $\hat{X}_{w}$ and ${X}_{l}$ denote the output feature of \text{W-MSA} and \text{MLP} modules. \text{W-MSA} and \text{MLP} represent the window-based multi-head self-attention and the multilayer perceptron feed-forward network.
\subsubsection{Global Context Interaction Module}
In addition to local context interaction, there still exist two aspects to be considered: (i) The large-scale degradations (snow veiling and plaque) need to apply the global view to recover the clean image. (ii) Lack of interaction between local details and global context. Therefore, we propose Global Context Interaction Module to solve these deficiencies.
The scale-aware feature after self-attention processing owns global snow degradation information representation. To assist in reconstructing the decoder segment, we exploit the global feature to develop the context interaction query, which performs cross-attention through its global context knowledge to support the reconstruction of the global information of the image. The proposed context interaction query is depicted in Fig.\ref{fig:snowformer}. Specifically, given the scale-aware feature $X_{l}\in \mathbb{R}^{ \frac{H}{16}\times \frac{W}{16}\times 16C}$, Conv1$\times$1 and down-sampling are firstly applied to align its size with that of local windows. Next, We adopt the attention mechanism from the spatial~\cite{qin2020ffa} and channel~\cite{hu2018squeeze} levels to adaptively project the context interaction query. Therefore, the context interaction query is formulated as:
\begin{equation}
\mathbf{Q}_{gi} \in \mathbb{R}^{ {H}^{i}_{w}\times {W}^{i}_{w}\times 2^{i}C}= \text{CS}(X_{l}\in \mathbb{R}^{ \frac{H}{16}\times \frac{W}{16}\times 16C} \downarrow),
\end{equation}
wherein the $\downarrow$ denotes the operation to align size, which consists of convolution and down-sampling. \text{CS} is the query generator, which includes channel and spatial attention operations. $i$ denotes the $i$-th stage of the decoder. ${H}^{i}_{w}$ and ${W}^{i}_{w}$ are the height and width of local windows in $i$-th layer. This query is guided by the scale-aware feature in the latent layer, which can be made up for the global context interaction required in the snow removal task. To enhance the local and global interaction while lessening the computation, we utilize the global interaction query for cross self-attention. Compared with local context interaction, we leverage external global information representation to facilitate the context interaction of local details and global snow features. So far, our proposed global information interaction can be expressed as:
\begin{equation}
\text{Attn}_{G}\left(\mathbf{Q}_{\mathrm{g}}, \mathbf{K}_{\mathrm{l}}, \mathbf{V}_{\mathrm{l}}\right)=\text{Softmax}\left(\frac{\mathbf{Q}_{\mathrm{g}} \mathbf{K}_{\mathrm{l}}{ }^{T}}{\sqrt{d}}+p\right) \mathbf{V}_{\mathrm{l}},
\end{equation}
where $\mathbf{Q}_{\mathrm{g}}$, $\mathbf{K}_{\mathrm{l}}$ and $\mathbf{V}_{\mathrm{l}}$ come from the scale-aware and local window-based feature. $d$ and $p$ denote the dimensionality of key and relative position embedding~\cite{swim}. For the feed-forward network, we follow the configuration of local context interaction.
\subsection{Heterogeneous Feature Projection Head}
For complex large-scale degradation scene, the output features of decoder frequently include residual snow~\cite{chen2020jstasr,hdcwnet}. In order to alleviate such problem, previous approaches usually adopt a refinement block to process the output feature of the decoder at the highest resolution~\cite{zamir2021restormer}. In our heterogeneous architecture, we claim that the information from the encoder and decoder are both crucial for image desnowing. Thus a Heterogeneous Feature Projection Head is proposed which fully combines CNNs-based and ViT-based information flow to progressively refine features and project the refined features to the final clean image.
However, the heterogeneous features are misaligned at the spatial and the channel levels, which can seriously affect the refinement process in the case of high-resolution features. Therefore, we utilize the two-level attention mechanism for different feature maps. This operation can be expressed as follows:
\begin{equation}
X_{pi} = \text{CS}\left(X_{ei}+X_{di}\right),
\end{equation}
where $X_{ei}$ and $X_{di}$ denote the $i$-th layer features of encoder and decoder. For the features of the different stages, we use up-sampling and Conv$1\times$1 to align the size with that of the highest resolution feature map. $X_{pi}$ represents the aligned feature which can be considered as residual snow degradations. Inspired by ~\cite{resnet}, we apply the channel attention and two 3$\times$3 convolutions to build our refinement block. It is worth mentioning that we progressively use the fusion strategy to refine the encoder-decoder features instead of summing them up all together. The progressive mechanism is more conducive to the fusion of the features from different layers. The Heterogeneous Feature Projection Head can be expressed as:
\begin{equation}
\begin{aligned}
&X_{ri+1} = \text{RB}(X_{pi+1} + X_{ri}),\\
&X_{r0} = \text{RB}(X_{p0}+X_{d0}),
\end{aligned}
\end{equation}
\begin{equation}
I(X) = \text{Conv}3\times3(X_{r4}),
\end{equation}
where the $X_{ri}$ denotes the $i$-th refined feature, \text{RB} is the refinement block. \text{Conv}3$\times$3 means the output operation, which projects the feature into the clean image.
\subsection{Overall Loss Function}
For better training supervision, we adopt PSNR loss~\cite{chen2021hinet} as our reconstruction loss. The loss function can be calculated as:
\begin{equation}
L_{psnr}=-\text{PSNR}(I(X)), Y),
\end{equation}
where $I(X)$ and $Y$ are the desnowing result and its corresponding ground-truth.
In addition, we consider that restoring the image from the perceptual level is also critical. We apply the perceptual loss to improve the effect of restoring. The perceptual loss can be expressed as follows:
\begin{equation}
L_{perceptual}=\sum_{j=1}^{2} \frac{1}{C_{j} H_{j} W_{j}}\left\|\phi_{j}({I}(X))-\phi_{j}(Y)\right\|_{1},
\end{equation}
wherein the $\phi_{j}$ represents the specified layer of VGG-19~\cite{simonyan2014very}. $C_{j}, H_{j}, W_{j}$ denote the channel numbers, height, and width of the feature map.
Our overall loss function can be expressed as:
\begin{equation}
L = \lambda_{1} L_{psnr} + \lambda_{2} L_{perceptual},
\end{equation}
where the $\lambda_{1}$ and $\lambda_{2}$ are set to 1 and 0.2 in our paper.
\section{Implementation}
\subsection{Training Details}
During the training phase, we train our model using the Adam optimizer with initial momentum set to 0.9 and 0.999. We initialize the learning rate to 0.0002 and use a cyclic learning rate adjustment with a maximum learning rate of 1.2 times the initial learning rate. We train with a data augmentation strategy, and randomly crop 256 × 256 patches to train for 800 epochs for the snow removal task. For data augmentation, we employ horizontal flipping and randomly rotate the image to a fixed angle. We choose layers 1 and 3 of VGG19~\cite{simonyan2014very} for perceptual loss.
\subsection{SnowFormer Configuration}
For a detailed description, we introduce our SnowFormer configuration. Specifically, the minimum feature size of our model is $\frac{1}{16}$ of the original image size, and the number of dimensions at each stage is $\left\{16, 32, 64, 128, 256\right\}$. In the encoder part, we set the number of feature extraction blocks at each stage to $\left\{4, 6, 7, 8\right\}$, respectively, and the multiples of the ascending dimension in the spindle-shape is $\left\{1, 2, 2, 2\right\}$. In the latent layer, the number of transformer blocks is set to 8, and the head for multi-head self-attention is 16. In the decoder part, we set the number of Context interaction Transformer Blocks to $\left\{4, 6, 7, 8\right\}$, and the heads to $\left\{1, 2, 4, 8\right\}$. For the window size in each layer is set to 8$\times$8. We utilize two \text{CS} operations to generate the global interaction query at each stage. In the Heterogeneous Feature Projection Head, we employ six refinement blocks for each refinement stage.
\section{Experiments}
\subsection{Evaluation Metrics}
We adopt the commonly used PSNR and SSIM~\cite{wang2004image} metrics to evaluate the snow removal performance in RGB color space for our all experiments, which follows the state-of-the-art single-image snow removal method~\cite{hdcwnet}.
\subsection{Evaluation Datasets}
To demonstrate the superior performance of our manner on desnowing task, we train and test our SnowFormer on CSD~\cite{hdcwnet}, SRRS~\cite{chen2020jstasr} and Snow100K~\cite{liu2018desnownet} datasets, all training and testing dataset settings follow the latest published single image desnowing~\cite{chen2020jstasr} to choose 2000 images on each testing dataset for a fair and convinced comparison.
\subsection{Quantitative Evaluation}
In this subsection, we compare with the previous state-of-the-art approaches, including DesnowNet~\cite{liu2018desnownet}, CycleGAN~\cite{engin2018cycle}, All in One~\cite{allinone}, JSTASR~\cite{chen2020jstasr}, HDCW-Net~\cite{hdcwnet}. In addition, to highlight our method's excellent performance and compare in more detail, we retrain the general image restoration (NAFNet) and adverse weather restoration (TransWeather) methods to compare. The quantitative evaluation results are shown in Table \ref{snowresults}. As shown in Table \ref{snowresults}, our method outperforms all state-of-the-art approaches for snow removal, including general image restoration architectures and single image desnowing method~\cite{hdcwnet}. On the CSD dataset, SnowFormer achieves the 4.95dB PSNR and 0.02 SSIM gain compared with the NAFNet and surpasses the HDCW-Net 9.02dB on the PSNR metric. Our method also attracts the 34.99dB PSNR and 0.98 SSIM on the SRRS dataset, which is higher than the second-best approach 5.27dB.
Overall, our SnowFormer achieves a substantial lead in the three benchmarks for snow scenes compared with the SOTA methods.
\subsection{Qualitative Evaluation}
The qualitative comparisons of state-of-the-art desnowing algorithms on real-world and synthetic snowy images are respectively revealed in Fig.~\ref{Comparisons_real} and Fig.~\ref{comparisons_csd}. As observed in the red rectangle of Fig.~\ref{Comparisons_real}, it is visually found that the recovered results by previous desnowing methods~\cite{chen2020jstasr,allinone,hdcw,valanarasu2022transweather,chen2022simple} still have residual snow spots and snow mark with small size. In comparison, our proposed method can remove multiple snow degradations with different size and provide more pleasant snow removal results with more details, which indicates the superior generalization ability on real-world scenes. Fig.~\ref{comparisons_csd} shows the visual comparisons on the synthetic snowy images selected from CSD dataset~\cite{hdcwnet}. From Fig.~\ref{comparisons_csd}, the resuls of JSTASR, All in one and HDCW-Net sill have some large and non-transparent snow particles owing to its sufficient desnowing ability. Although TransWeather~\cite{valanarasu2022transweather} and NAFnet~\cite{chen2022simple} can remove most snow particles with large size, the snow mark with small size cannot be removed effectively. Moreover, the occluded textural details are not recovered well. Compared with these SOTA methods, SnowFormer can achieve better desnowing performance for various degradation scales and the restored clear images are more closer to the ground truths.
\begin{table*}[!t]
\centering
\caption{Quantitative comparison of various desnowing approaches on the CSD~\cite{hdcwnet}, SRRS~\cite{chen2020jstasr} and Snow 100K~\cite{liu2018desnownet} datasets. Underline and bold indicate the best metrics. }\label{snowresults}
\resizebox{14cm}{!}{
\renewcommand\arraystretch{1.1}
\begin{tabular}{l|cc|cc|cc}
\toprule[1.2pt]
\multirow{2}*{Method}& \multicolumn{2}{c|}{ CSD (2000) } & \multicolumn{2}{c|}{ SRRS (2000) } & \multicolumn{2}{c}{ Snow 100K (2000) } \\\cline{2-7}
& PSNR $\uparrow$ & SSIM $\uparrow$ & PSNR $\uparrow$ & SSIM $\uparrow$ & PSNR $\uparrow$ & SSIM $\uparrow$\\
\hline
(TIP'18)DesnowNet~\cite{liu2018desnownet} & 20.13 & 0.81 &20.38 &0.84& 30.50 & 0.94 \\
(CVPR'18)CycleGAN~\cite{engin2018cycle}& 20.98 & 0.80 &20.21 &0.74& 26.81 & 0.89 \\
(CVPR'20)All in One~\cite{allinone} &26.31 &{0.87} &24.98 &0.88& 26.07&0.88 \\
(ECCV'20)JSTASR~\cite{chen2020jstasr} &27.96 &0.88 & 25.82 & 0.89 & 23.12 & 0.86 \\
(ICCV'21)HDCW-Net~\cite{hdcwnet} & {29.06} &{0.91} &{27.78} &{0.92} & {31.54} &{0.95} \\
(CVPR'22)TransWeather~\cite{valanarasu2022transweather} &${31.76}$ &${0.93}$ &${28.29}$ &${0.92}$ & ${31.82}$ & ${0.93}$ \\
(ECCV'22)NAFNet~\cite{chen2022simple} &$\underline{33.13}$ &$\underline{0.96}$ &$\underline{29.72}$ &$\underline{0.94}$ &$\underline{32.41}$ &$\underline{0.95}$ \\
\hline\hline
\rowcolor[gray]{.95} SnowFormer (Ours) &$\mathbf{38.26}$&$\mathbf{0.98}$ & $\mathbf{\mathbf{34.99}}$ &$\mathbf{0.98}$& $\mathbf{37.89}$ & $\mathbf{0.98}$ \\
\bottomrule[1.2pt]
\end{tabular}}
\end{table*}
\begin{figure*}[!t]
\centering
\includegraphics[width=18cm]{Comparison/real.pdf}
\caption{Visual comparison of state-of-the-art methods and SnowFormer on the real-world dataset~\cite{dan}. Please enlarge the image to observe details.}
\label{Comparisons_real}
\end{figure*}
\section{Ablation Study}
To demonstrate the effectiveness of our proposed module, we perform the ablation experiments of SnowFormer. We train our model on image patches of size 192$\times$192 for 150 epochs on the CSD training dataset~\cite{hdcwnet}, and test the effect on the CSD testing set. The other configurations are the same as described above. Next, we verify the effectiveness of each module separately.
\subsection{Improvements of Scale-aware Aggregation Module}
In this part, we aim to demonstrate the gain from the Scale-aware Aggregation (SaA) operation. We remove the multi-scale feature aggregation and only use the feature processed down-sampling repeatedly for global modeling in the latent layer. The results are presented in Table \ref{sam ab}. We found that the scale-aware aggregation improves the model's performance while increasing the computation and parameters to negligible. This is because the scale-aware feature enables the representation of various snow degradations without being lost due to the reduced feature map size.
\begin{figure*}[!t]
\centering
\includegraphics[width=16cm]{Comparison/comparison11.pdf}
\label{snowformer}
\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=16cm]{Comparison/comparison33.pdf}
\caption{Visual comparison of state-of-the-art methods and SnowFormer on the CSD~\cite{hdcwnet} dataset. Please enlarge the image to see details.}
\label{comparisons_csd}
\end{figure*}
\begin{table}[!h]
\vspace{-0.2em}
\centering
\caption{Ablation Study on Scale-aware Aggregation Module. Underline indicates the best metrics. (PSNR(dB)/SSIM) }\label{sam ab}
\resizebox{8cm}{!}{
\renewcommand\arraystretch{1.1}
\begin{tabular}{cccccccc}
\toprule[1.2pt]
\textbf{Setting} & \textbf{Model} & \multicolumn{2}{c}{\textbf{\#Param}} & \multicolumn{2}{c}{\textbf{\#GFLOPs}} & \textbf{PSNR} & \textbf{SSIM} \\
\midrule[0.15em]
a & w/o SaA & \multicolumn{2}{c}{8.32M} & \multicolumn{2}{c}{19.43G} & 30.86 & 0.956\\
b& w SaA (Ours) & \multicolumn{2}{c}{8.35M} & \multicolumn{2}{c}{19.43G} & \underline{31.14} & \underline{0.958} \\
\bottomrule[1.2pt]
\end{tabular}}
\end{table}
\vspace{-1.25em}
\subsection{Effectiveness of Context Interaction Transformer Block}
To demonstrate the effectiveness of components in our Context Interaction Transformer Block (CITB), we conduct the following ablation experiments: (i) We only use the local context interaction to replace the global context interaction. (LCI) (ii) We exploit the global context interaction and lack local information modeling. (GCI) (iii) Our proposed context interaction module , which utilizes the scale-aware feature to process global context cross-attention and combine the interaction with local information. (LGCI) Table \ref{CITB ab} shows that interaction between local details and global context and local context interaction are vital for image desnowing, which achieves the visible gain compared to the single context interaction.
\begin{table}[!h]
\centering
\caption{Ablation Study on Context Interaction Transformer Block. Underline indicates the best metrics. (PSNR(dB)/SSIM) }\label{CITB ab}
\resizebox{8cm}{!}{
\renewcommand\arraystretch{1.1}
\begin{tabular}{cccccccc}
\toprule[1.2pt]
\textbf{Setting} & \textbf{Model} & \multicolumn{2}{c}{\textbf{\#Param}} & \multicolumn{2}{c}{\textbf{\#GFLOPs}} & \textbf{PSNR} & \textbf{SSIM} \\
\midrule[0.15em]
a & LCI~\cite{swim} & \multicolumn{2}{c}{8.32M} & \multicolumn{2}{c}{19.62G} & 30.91 & 0.956\\
b& GCI & \multicolumn{2}{c}{8.22M} & \multicolumn{2}{c}{19.22G} & 30.78 & 0.955 \\
c& LGCI (Ours)& \multicolumn{2}{c}{8.35M} & \multicolumn{2}{c}{19.43G} & \underline{31.14} & \underline{0.958} \\
\bottomrule[1.2pt]
\end{tabular}}
\vspace{-0.9em}
\end{table}
\vspace{-1.0em}
\subsection{Benefits of Heterogeneous Feature Projection Head}
We analyze the benefits of the Heterogeneous Feature Projection Head in this section. Specifically, we verify the merits of our heterogeneous features compared to single feature input~\cite{zamir2021restormer}. And the advantage of progressive fusion over direct fusion. We conduct the following experiments to validate them: (i) We remove the Heterogeneous Feature Projection Head. (ii) We only use the last decoder feature for our refinement blocks. (FPH) (iii) We do fusion and refinement at once rather than progressively. (HFPH w/o progressively) (iv) We hierarchically fuse the feature of each encoder and decoder, and refine them progressively. (HFPH) The results are presented in Table \ref{HFPH ab}. We observe that encoder and decoder heterogeneous features can improve the detailed refinement compared with the single feature. In addition, the progressively fuse operation is better than the direct fusion and refinement.
\begin{table}[!h]
\vspace{-0.6em}
\centering
\caption{Ablation Study on Heterogeneous Feature Projection Head. Underline indicates the best metrics. (PSNR(dB)/SSIM) }\label{HFPH ab}
\resizebox{8cm}{!}{
\renewcommand\arraystretch{1.1}
\begin{tabular}{cccccccc}
\toprule[1.2pt]
\textbf{Setting} & \textbf{Model} & \multicolumn{2}{c}{\textbf{\#Param}} & \multicolumn{2}{c}{\textbf{\#GFLOPs}} & \textbf{PSNR} & \textbf{SSIM} \\
\midrule[0.15em]
a & w/o HFPH & \multicolumn{2}{c}{8.22M} & \multicolumn{2}{c}{10.38G} & 30.34 & 0.947\\
b& FPH~\cite{zamir2021restormer} & \multicolumn{2}{c}{8.25M} & \multicolumn{2}{c}{12.40G} & 30.62 & 0.951 \\
c& HFPH w/o progressively & \multicolumn{2}{c}{8.35M} & \multicolumn{2}{c}{18.98G} & 30.99 & 0.957 \\
d& HFPH (Ours)& \multicolumn{2}{c}{8.35M} & \multicolumn{2}{c}{19.43G} & \underline{31.14} & \underline{0.958} \\
\bottomrule[1.2pt]
\end{tabular}}
\vspace{-2.3em}
\end{table}
\section{Conclusion}
In this paper, we propose a strong image snow removal network. It fully aggregates multi-scale degradation information and exploits Contextual Interactions to handle local and large-scale snow scenes. At the highest resolution stage we utilize Heterogeneous Feature Projection Head to deal with the residual degradation problem. Huge gains are achieved in all three desnowing datasets, which demonstrate the superiority of our proposed model.
\section{Supplementary Material}
This is a supplementary file for SnowFormer: Scale-aware Transformer via Context Interaction for Single Image Desnowing. In this supplementary material, we first introduce the detailed structure of our Transformer Block in the latent layer of our model. Next, we visualize the feature maps to demonstrate the effectiveness of the proposed Heterogeneous Feature Projection Head. Finally, we provide more visual comparisons to demonstrate the superiority of our SnowFormer.
We presents the following information that can be beneficial for the readers:
\begin{itemize}
\item Detailed Structure of Transformer Block in SnowFormer.
\item Feature Visualization of Heterogeneous Feature Projection Head
\item Additional visual comparison on real-world and synthetic snow images.
\end{itemize}
\section{Detailed Structure of Transformer Block in SnowFormer}
Our Transformer Block consists of Self-attention (SA) and Multi-scale Feed-forward Network (MFFN).
For the obtained scale-aware feature $X_{s}\in \mathbb{R}^{ {H}\times {W}\times C}$, we reshape it to $X_{s}\in \mathbb{R}^{N\times C}$ and adopt learnable $W_{Q}^{C\times C}$, $W_{K}^{C\times C}$ and $W_{V}^{C\times C}$ to project the $X_{s}$ into $Q$ ($XW_{Q}$), $K$ ($XW_{K}$) and $V$ ($XW_{V}$). In order to improve the local extraction ability, we add a 3$\times$3 depth-wise convolution. Therefore, Self-attention can be expressed as:
\begin{equation}
X^{\prime}_{s}\in \mathbb{R}^{ N\times C}=\text{Softmax}\left(\frac{\mathrm{QK}^{\mathrm{T}}}{\sqrt{C}}\right) \times \mathbf{V} + \text{DWConv}(X_{s}),
\end{equation}
where $X^{\prime}_{s}$ represents the scale-aware feature processed by SA. $H$, $W$ and $C$ denote the height, width and dimension numbers of scale-aware feature. $N$ is the number of token. \text{Softmax}($\cdot$) is the the Softmax operation. In addition, we follow the ~\cite{vit} to use multi-head self-attention for the above operations.
For the feed-forward network, we employ the proposed Multi-scale Feed-forward Network (MFFN) to improve the ability of local information extraction, which consists of multiple kernels convolution instead of a full connection layer for better performance, as shown in Fig.(\ref{mffn}). Specifically, we use a 1$\times$1 convolution to expand features into high-dimension (Expand ratio is 2 in our paper). Next, we utilize 3$\times$3 and 5$\times$5 depth-wise convolutions to extract multi-scale information and apply Simple Gate~\cite{chen2022simple} to replace GELU activation. We adopt 1$\times$1 convolution to reduce the dimension numbers before outputting the feature.
\begin{figure}[!t]
\centering
\includegraphics[width=8cm]{figure/MFFN1.pdf}
\caption{Detailed structure diagram of our Multi-scale Feed-forward Network (MFFN).}\label{mffn}
\end{figure}
\section{Feature Visualization of Heterogeneous Feature Projection Head}
To further demonstrate the effectiveness and necessity of our Heterogeneous Feature Projection Head, we visualize the feature maps in Fig.~\ref{heterogeneous_viusal}. We can observe that our proposed Heterogeneous Feature Projection Head can promote the refinement of residual degradation in the feature space.
\begin{figure}[!h]
\centering
\includegraphics[width=8cm]{figure/refine.pdf}
\caption{The left map is the feature before being processed by the Heterogeneous Feature Projection Head. The right map is the feature after being processed by the Heterogeneous Feature Projection Head.}\label{heterogeneous_viusal}
\end{figure}
Fig.(\ref{heterogeneous}) is the feature maps from CNNs-based and ViT-based at the same stage. CNN feature focuses on sharp and strong edges, while Transformer pays attention to global low-frequency information recovery. Therefore, combining and refining the heterogeneous features on the Heterogeneous Feature Projection Head is crucial.
\begin{figure}[!t]
\centering
\includegraphics[width=8cm]{figure/heterogeneous.pdf}
\caption{The picture on the left is the feature of the encoder part based on CNNs. The picture on the right is the feature of the Vit-based decoder.}\label{heterogeneous}
\end{figure}
\section{Additional Visual Comparison}
We present the additional visual comparison on synthetic and real snow images with other SOTA methods in Fig.(\ref{snowformer_comparison14}). Worth noting that our method achieves more superior visual quality on both real and synthetic images, compared with other desnowing methods.
\begin{figure*}[!t]
\centering
\includegraphics[width=16cm]{Comparison/comparison4.pdf}
\label{snowformer_comparison1}
\end{figure*}
\begin{figure*}[!h]
\centering
\includegraphics[width=16cm]{Comparison/comparison5.pdf}
\label{snowformer_comparison2}
\end{figure*}
\begin{figure*}[!h]
\centering
\includegraphics[width=16cm]{Comparison/comparison6.pdf}
\label{snowformer_comparison3}
\end{figure*}
\begin{figure*}[!h]
\centering
\includegraphics[width=18cm]{figure/realcomparison.pdf}
\caption{Visual comparison of state-of-the-art methods and SnowFormer on the CSD~\cite{hdcwnet} and real-world datasets~\cite{dan}. Please enlarge the image to see details.}
\label{snowformer_comparison14}
\end{figure*}
|
2,877,628,090,128 | arxiv | \section{Conclusions}
\label{sec:conclusion}
We defined the Force Path Problem, in which an adversary adds weights to edges in order to make a particular path the shortest between a pair of source and destination nodes. The adversary has a budget, which he/she cannot exceed. We showed that Force Path can be optimized to within an arbitrarily small error in polynomial time. We demonstrated that Force Path can be formulated as a linear program with an intractable number of constraints. However, standard shortest-path algorithms can be used to obtain a polynomial-time constraint oracle, which allowed us to use constraint generation. We formalized this procedure in our \texttt{PATHPERTURB} algorithm, which we applied to a diverse collection of real and simulated data. We observed that the perturbation budget optimized using \texttt{PATHPERTURB} is often as little as half of what can be obtained using a greedy baseline perturbation procedure.
\section{Introduction}
\label{sec:intro}
The shortest path problem is a seminal task in graph theory with numerous real-world applications in computer networks, transportation networks, etc. Given two nodes in a network, the shortest path between them is the set of edges that connects the two nodes with the minimum sum of edge weights. For a given network, two nodes can have more than one shortest path; and the network can be directed or undirected. Here we only consider undirected networks.
In this paper, we present the \emph{Force Path Problem}, where there is a specific path that the adversary wants to be the shortest path between a pair of source and destination nodes. The adversary can increase weights of edges and has a fixed budget with which to achieve this goal. The \emph{Force Path Problem} is similar to the \emph{Force Path Cut Problem}~\cite{Miller2021}. The difference is in the attack vector: in this case the adversary makes edges more expensive (by increasing edge weights) rather than removing edges. This difference may seem relatively minor, but it has a profound implication for the computational complexity of the problem. While Force Path Cut is NP-complete, we show in this paper that Force Path can be solved within arbitrary precision in polynomial time. We demonstrate that Force Path can be formulated as a linear program with a constraint set that is potentially factorial in the number of nodes. There is, however, a natural polynomial-time oracle to find violated constraints in any candidate solution (which we use to include a subset of constraints as necessary). We propose the \texttt{PATHPERTURB} algorithm that uses the oracle to iteratively refine the graph perturbations until the target path is the shortest.\footnote{We use the terms graph and network interchangeably.}
The main contributions of the paper are as follows:
\begin{itemize}
\item We formally define the Force Path problem: an adversarial attack on shortest paths.
\item We formulate an oracle to identify the most violated constraint at any given point, the existence of which implies that Force Path can be optimized within arbitrary precision in polynomial time.
\item We propose the \texttt{PATHPERTURB} algorithm, which uses the oracle to minimize the required perturbation budget.
\item We present the results of experiments on synthetic and real networks, in which \texttt{PATHPERTURB} reliably reduces the required perturbation budget compared to a greedy baseline method.
\end{itemize}
\section*{Acknowledgments}
This material is based upon work supported by the United States Air Force under Air Force Contract No. FA8702-15-D-0001 and the Combat Capabilities Development Command Army Research Laboratory (under Cooperative Agreement Number W911NF-13-2-0045). Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the United States Air Force or Army Research Laboratory.
\bibliographystyle{named}
\section{Force Path LP Formulation}
\label{sec:formulation}
Let $\mathbf{w}\in\mathbb{R}_{\geq0}^M$ be a vector of edge weights in the original graph $G$ and $\Delta\in\mathbb{R}_{\geq0}^M$ be a vector of edge-weight perturbations.
For any path $p$ from $s$ to $t$ in $G$, let $\mathbf{x}_p\in\{0, 1\}^M$ be an edge indicator vector for $p$: the entries for $\mathbf{x}_p$ associated with the edges that comprise $p$ are $1$, while all other entries are $0$. Thus, $\mathbf{w}^\top \mathbf{x}_p$ is the length of $p$ in the original graph and $(\mathbf{w}+\Delta)^\top \mathbf{x}_p$ is its length in the perturbed graph.
The linear program formulation of Force Path is based on two key observations. First, any path $p$ that is not longer than $p^*$ must be perturbed to be longer than $p^*$. This is clear from the fact that, if it were not the case, $p^*$ would not be the shortest path. Second, we do not perturb $p^*$, formalized as follows:
\begin{proposition}
The minimum-budget solution to Force Path includes no perturbation of any edge along $p^*$.
\end{proposition}
\begin{proof}
Let $\hat{\Delta}$ be the minimum-budget solution. Suppose that the claim is not true---i.e., there is an edge $e$, which is part of $p^*$, where $\hat{\Delta}(e) > 0$. Since $\hat{\Delta}$ is the solution to Force Path, $p^*$ is the shortest path when $\hat{\Delta}$ is added to the edge weights. Let $\Delta^\prime$ be the same vector with the entry at $e$ reduced to $0$, i.e.,
$\Delta^\prime(e^\prime)=\hat{\Delta}(e^\prime)$ for $e^\prime\in E\setminus \{e\}$ and $\Delta^\prime(e)=0$. Since $e$ is part of $p^*$, $\mathbf{x}_{p^*}^\top\hat{\Delta}-\mathbf{x}_{p^*}^\top\Delta^\prime=\hat{\Delta}(e)$. For any path $p$, since $\mathbf{x}_p$ consists of only zeros and ones, we have
\begin{equation}
\mathbf{x}_{p}^\top\hat{\Delta}-\mathbf{x}_{p}^\top\Delta^\prime\leq\hat{\Delta}(e).\label{eq:pathReduction}
\end{equation}
Again, $p^*$ must be the shortest path using $\hat{\Delta}$, and, therefore, for any path $p$ from $s$ to $t$, $\mathbf{x}_{p}^\top\hat{\Delta}\geq\mathbf{x}_{p^*}^\top\Delta^\prime$. Combining this with (\ref{eq:pathReduction}), we have, for any path $p$
\begin{equation}
\mathbf{x}_{p^*}^\top\Delta^\prime=\mathbf{x}_{p^*}^\top\hat{\Delta}-\hat{\Delta}(e)\leq\mathbf{x}_{p^*}^\top\hat{\Delta}-\hat{\Delta}(e)\leq\mathbf{x}_p^\top\Delta^\prime.
\end{equation}
Thus, if $\hat{\Delta}$ were replaced with $\Delta^\prime$, $p^*$ would still be the shortest path. This implies that the total perturbation could be reduced by $\hat{\Delta}(e)>0$ and still achieve the objective, so $\hat{\Delta}$ is not the minimum-budget solution. Thus, we have a contradiction, and the proof is complete.
\end{proof}
The implication of this observation is that there is a fixed lower bound for the lengths of all paths. Let $\ell=\mathbf{w}^\top\mathbf{x}_{p^*}$ be the length of $p^*$ and $P_{\ell}$ be the set of paths from $s$ to $t$ whose length is less than or equal to $\ell$. Finally, let $\delta$ be the ``buffer'' we use to ensure $p^*$ is the unique shortest path: the difference between the length of $p^*$ and the length of the second shortest path\footnote{In a scenario where being tied for shortest is acceptable, $\delta$ can be set to $0$. If it is acceptable for $p^*$ to be shortest by any $\epsilon>0$, we can set $\delta$ to $0$ and distribute an arbitrarily small value across edges not on $p^*$. In this case, the budget must be strictly larger than the sum of the computed perturbations.}. We formulate the linear program as follows:
\begin{align}
\hat{\Delta} = &\arg\min_{\Delta} \mathbf{1}^\top\Delta\label{eq:minCost}\\
\text{s.t.} &~\Delta_i\geq 0,~ ~1\leq i\leq M\label{eq:posPert}\\
&~\left(\mathbf{w}+\Delta\right)^\top\mathbf{x}_p\geq \ell+\delta,~ ~\forall p\in P_{\ell+\delta}\setminus \{p^*\}\label{eq:tooShort}\\
&~\mathbf{x}_{p^*}^\top\Delta=0\label{eq:dontChangePStar}.
\end{align}
As with the approximate version of Force Path Cut discussed in~\cite{Miller2021}, the number of paths in $P_{\ell+\delta}$ may be too large to enumerate all constraints. In an $N$-node clique, for example, the number of paths of length $N-1$ between any two nodes is $(N-2)!$. Thus, specifying all constraints in the linear program is computationally intractable. We use constraint generation to iteratively incorporate constraints as they are needed (see, e.g.,~\cite{Ben-Ameur2006,Letchford2013}). In order to use constraint generation, however, there must be an oracle that returns a constraint being violated at a given point.
There is, fortunately, a natural oracle for the constraints specified in~(\ref{eq:tooShort}), which not only returns a violated constraint, but the constraint \emph{most} violated at the proposed solution. We find this constraint as follows. The candidate solution is a perturbation to the edge weights, $\hat{\Delta}$. We apply the perturbation to get the new edge weights $w^\prime$, where
\begin{equation}
w^\prime(e)=w(e)+\hat{\Delta}(e).\label{eq:updateWeight}
\end{equation}
Using $w^\prime$ as distances, we find the shortest path $p$ from $s$ to $t$ in $G$. If $p$ is $p^*$, we find the second shortest path if it exists. If there is no such path, there is no violated constraint. If there is, we let $p$ be this path. If $p$ is at least $\delta$ longer than $p^*$, there is no violated constraint. If not, the length of $p$ needs to be incorporated as a constraint. Algorithm~\ref{alg:oracle} provides the pseudocode for this procedure.
\begin{algorithm}[tb]
\caption{ConstraintOracle}
\label{alg:oracle}
\textbf{Input}: graph $G$, weights $w$, target path $p^*$, buffer $\delta$\\
\textbf{Output}: A path $p$ from $s$ to $t$ in $G$
\begin{algorithmic}[1]
\STATE $s\gets$ first node in $p^*$
\STATE $t\gets$ last node in $p^*$
\STATE $p\gets$ shortest path from $s$ to $t$ in $G$
\IF{$p$ is $p^*$}
\STATE $p\gets$ second shortest path from $s$ to $t$ in $G$
\STATE $\langle\langle p$ will be $\emptyset$ with length $\infty$ if $p^*$ is the only path$\rangle\rangle$
\ENDIF
\IF{length$(p)\geq$ length$(p^*)+\delta$}
\STATE $p\gets\emptyset$
\ENDIF
\RETURN $p$
\end{algorithmic}
\end{algorithm}
While the number of constraints may be extremely large, each one is a standard linear inequality constraint, which implies that the feasible region is convex. Since we have a constraint oracle that runs in polynomial time,\footnote{Finding the two shortest simple paths between two nodes takes $O(NM)$ time using Yen's algorithm~\cite{Yen1971}. If $\delta=0$ and edge weights are strictly positive, we can use an algorithm not restricted to simple paths that runs in $O(M+N\log{N})$ time~\cite{Eppstein1998}.} this system can be optimized in polynomial time regardless of the number of constraints. Using the ellipsoid algorithm introduced by Khachiyan (see~\cite{Gacs1981}), we can solve a linear program within finite precision in a polynomial number of iterations~\cite{Grotschel1981}. This results in the following proposition.
\begin{proposition}
Force Path can be optimized within precision of any constant $\epsilon>0$ in polynomial time.
\end{proposition}
\section{Proposed Method: PATHPERTURB}
\label{sec:pathattack}
While the ellipsoid algorithm provably converges in polynomial time, it is considerably slower in practice than simplex methods. Thus, our proposed algorithm iteratively solves a linear optimization procedure, adding constraints via the oracle as necessary. We call this algorithm \texttt{PATHPERTURB}.
Our perturbation algorithm uses a linear program where each constraint is associated with a path from $s$ to $t$ that is not longer than $p^*$. Each path must have weights added to the edges so that the path's length will become sufficiently long. \texttt{PATHPERTURB} operates in an iterative fashion as it builds the set of necessary constraints. At each iteration, it finds a solution based on a subset of constraints, perturbs the weights based on this solution, and, if there is still a path from $s$ to $t$ that is shorter, it adds the corresponding constraint to the linear program. Algorithm~\ref{alg:PATHPERTURB} provides \texttt{PATHPERTURB}'s pseudocode.
\begin{algorithm}[tb]
\caption{PATHPERTURB}
\label{alg:PATHPERTURB}
\textbf{Input}: graph $G=(V, E)$, weights $w$, target path $p^*$, buffer $\delta$\\
\textbf{Output}: perturbation vector $\hat{\Delta}$
\begin{algorithmic}[1]
\STATE $\ell\gets$ length of $p^*$
\STATE $\mathbf{w}\gets$ weight vector for $w$
\STATE $P_{\ell+\delta}\gets\emptyset$
\STATE $M\gets|E|$
\STATE $p\gets p^*$
\REPEAT
\STATE $P_{\ell+\delta}\gets P_{\ell+\delta}\cup \{p\}$
\STATE $\hat{\Delta}\gets$ solution to (\ref{eq:minCost})--(\ref{eq:dontChangePStar})
\STATE $w^\prime\gets w+\hat{\Delta}$ ~ ~ $\langle\langle$as in (\ref{eq:updateWeight})$\rangle\rangle$
\STATE $p\gets$ ConstraintOracle($G$, $w^\prime$, $p^*$, $\delta$)
\UNTIL{$p$ is empty}
\RETURN $\hat{\Delta}$
\end{algorithmic}
\end{algorithm}
\section{Force Path Problem Definition}
\label{sec:model}
Consider a graph $G=(V, E)$, with a set of nodes $V$ and edges $E$, where $|V|=N$ and $|E|=M$. The edges in $E$ are undirected and have nonnegative weights $w:E\rightarrow\mathbb{R}_{\geq0}$. The edge weights denote distances (a.k.a.~lengths) between the adjacent nodes.
In addition to the weighted graph, we are given a pair of source and destination nodes $s,t\in V$. The adversary's goal is to make a specific path, $p^*$, the shortest path from $s$ to $t$ in $G$. The adversary can achieve this by arbitrarily increasing the weight of any edge in $G$, all of which are visible to him/her. Within a budget constraint $b$, the adversary increases edge weights to obtain new weights $w^\prime$ such that $\sum_{e\in E}{\left(w^\prime(e)-w(e)\right)}\leq b$ and $p^*$ is the (possibly exclusive) shortest path from $s$ to $t$.
In the next section, we will show that the Force Path Problem can be formulated as a linear program (LP), which implies a polynomial time algorithm to get a solution within any specified precision, despite a very large number of constraints.
\section{Related Work}
\label{sec:related}
This paper expands the work on adversarial graph analysis that was introduced recently. Examples include attacks against vertex classification~\cite{Zugner2018,Zugner2019b} and node embedding~\cite{Bojchevski2019}, as well as community detection when an adversary does not want to be grouped with other individuals~\cite{Kegelmeyer2018}. In \cite{Miller2021}, an adversary cut edges to attack shortest path algorithms; here, an adversary adds edge weights.
Prior network science work on attacks against graphs was focused on attempts to disrupt infrastructure, e.g., disconnecting a power grid graph. In this area, it was shown that graphs with heterogeneous degree distributions (like BA and KR graphs) are much more robust to random node removals, but highly susceptible to targeted attacks against the nodes with the most connections~\cite{Albert2000}.
There has been previous work on altering shortest paths, though it has been primarily focused on removal of a single edge or node. The objective of the ``most vital edge'' (or most vital node) problem is, given two nodes in a graph, to find the edge (node) whose removal most increases the shortest path between the source and destination~\cite{Nardelli2001,Nardelli2003}. In an adversarial context, this would be an instance of an adversary intending to divert the user from the best solution, rather than being motivated to push traffic along a particular path of interest.
In that sense, the most vital edge and node problem is similar to recent work on Stackelberg planning~\cite{Speicher2018}. In this work, like in the most valuable node and edge problems, the goal of the attacker is to make it as costly as possible for the user to perform the task. While the most valuable edge only allows one move, the Stackelberg planning work uses a turn-based leader-follower framework, where the leader makes the follower's actions more costly at each step. A similar problem is the adversarial shortest path problem, where the state space has uncertainty that an adversary could exploit to decrease the user's reward as states are traversed~\cite{Neu2012}.
There are two complementary areas where path finding in an adversarial context is crucial. One is network interdiction, in which an adversary is attempting to traverse a network undetected~\cite{Washburn1995}. The other involves planning paths through hostile territory; for example, an unmanned aerial vehicle in enemy air space~\cite{Jun2003}. Recent path interdiction work has focused on attack disruption~\cite{Letchford2013}. Work in this area also uses oracles to judiciously select from an extremely large set of potential strategies~\cite{Jain2011}.
\subsection{Results}
\label{sec:results}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/Plots_Simulated_SimUnweighted.pdf}
\caption{Results on synthetic networks. Results are shown using \texttt{PATHPERTURB} ($\circ$) and \texttt{GreedyFirst} ($\square$), and each color represents a different network. The plots present results when the weights are equal (left), when they are drawn from a Poisson distribution (center), and drawn from a uniform distribution (right). Each plot shows the required budget as a proportion of the budget required using \texttt{GreedyFirst} (vertical axis) with respect to wall clock running time (horizontal axis). Lower cost reduction ratio and lower wall clock time (toward the lower left) is better. Error bars represent standard errors. In nearly all cases, \texttt{PATHPERTURB} yields a substantial cost reduction for its additional running time, though cliques with Poisson weights are nearly optimized with the baseline. Note: the black square (\texttt{GreedyFirst} on ER) in the right-hand plot is obscured by the yellow square.}
\label{fig:plots_sim_gs}
\end{figure*}
We treat the result of \texttt{GreedyFirst} as our baseline budget and report the value optimized by \texttt{PATHPERTURB} as a reduction from the baseline. With few exceptions, \texttt{GreedyFirst} outperforms \texttt{GreedyMin} in both running time and perturbation cost, so we omit the \texttt{GreedyMin} results for clarity of presentation. For each graph, we use the algorithms in an attempt to minimize the budget, after which the adversary would determine whether or not the attack is possible within the constraints. Figure~\ref{fig:plots_sim_gs} shows the results on the synthetic networks, while Figure~\ref{fig:plots_ruw_gs} shows the results on real networks with synthetic edge weights; and
Figure~\ref{fig:plots_rw_gs} shows the results on real weighted networks. The figures show the results where $p^*$ is the 800th shortest path. Due to space limitations, we omit the other results; they are substantially similar.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/Plots_Simulated_RealUnweighted.pdf}
\caption{Results on unweighted real networks.
Results are shown using \texttt{PATHPERTURB} ($\circ$) and \texttt{GreedyFirst} ($\square$), and each color represents a different network. The plots present results when the weights are equal (left), when they are drawn from a Poisson distribution (center), and drawn from a uniform distribution (right). Each plot shows the required budget as a proportion of the budget required using \texttt{GreedyFirst} (vertical axis) with respect to wall clock running time (horizontal axis). Lower cost reduction ratio and lower wall clock time (toward the lower left) is better.
Error bars represent standard errors. As with the synthetic networks, \texttt{PATHPERTURB} provides a significant cost reduction in all networks, though in this case we see a greater increase in running time for PA-ROAD. Note: the black square (\texttt{GreedyFirst} on WIKI) in the left-hand plot is obscured by the green square.}
\label{fig:plots_ruw_gs}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.6\textwidth]{figures/Plots_Simulated_RealWeighted.pdf}
\caption{Results on weighted real networks.
Results are shown using \texttt{PATHPERTURB} ($\circ$) and \texttt{GreedyFirst} ($\square$), and each color represents a different network. The plot shows the required budget as a proportion of the budget required using \texttt{GreedyFirst} (vertical axis) with respect to wall clock running time (horizontal axis). Lower cost reduction ratio and lower wall clock time (toward the lower left) is better.
Error bars represent standard errors. In all cases, \texttt{PATHPERTURB} reduces the cost of attacking the graph by about a factor of two.}
\label{fig:plots_rw_gs}
\end{figure*}
Across all experiments, we see a substantial improvement over \texttt{GreedyFirst} by using \texttt{PATHPERTURB}, for the most part ranging from a 25\% reduction in the required perturbation budget to a decrease of more than a factor of two. This comes at the expense of increased running time: an increase of an order of magnitude appears typical. In the synthetic networks, \texttt{PATHPERTURB} provides a greater improvement for heterogeneous Kronecker (KRON) and BA graphs rather than Erd\H{o}s--R\'{e}nyi graphs. This could be an effect of hubs: nodes with high degree that tend to facilitate short paths may make it more difficult to obtain a low budget via a greedy procedure.
The main exception to the substantial budget improvement is cliques (COMP, blue in Figure~\ref{fig:plots_sim_gs}) with Poisson weights. To understand this result, consider an unweighted clique, and note that the 800th shortest path is a 3-hop path. The optimal perturbation to make a particular 3-hop path the shortest is to perturb all edges adjacent to $s$ and $t$ except those on $p^*$. With weights drawn from a Poisson distribution, we get weights typically near the mean, which gives us a similar effect to the unweighted graph: 2-hop paths are all similar lengths, as are 3-hop paths. In this context, \texttt{GreedyFirst} identifies a near-optimal perturbation. Looking deeper into the data, we see that for path ranks of 100, 200, and 400, the required budgets using \texttt{GreedyFirst} and \texttt{PATHPERTURB} are exactly the same when all edge weights are equal.
We observe one major difference from the results with \texttt{PATHATTACK}~\cite{Miller2021}, where the adversary's goal is the same but his/her attack vector is to cut edges. We see more substantial gains over the greedy baseline in grid-like networks. In the lattice network and the road networks, \texttt{PATHATTACK} took substantially more time for very modest improvements in edge removal cost. Here, we see a relatively high computational burden in these grid-like networks---reliably an order of magnitude in the real datasets---but the budget reductions are among the best. In addition, we note that lattices with Poisson weights are one of the few cases where \texttt{GreedyMin} outperforms \texttt{GreedyFirst} (the other being cliques with Poisson weights), though the difference is small and does not explain the extent of the difference. This may be due to the difference in cost. In~\cite{Miller2021}, costs were proportional to the weights of removed edges, while in the present work the cost of perturbing a path will be smaller if the edge weights are larger. Thoroughly investigating this phenomenon is a subject for future work.
\section{Experiments}
\label{sec:setup}
This section presents the baseline methods, the networks used in experiments, the experimental setup, and the results.
\subsection{Baseline Methods}
We compare \texttt{PATHPERTURB} to two simple greedy baseline algorithms. Each algorithm iteratively perturbs a single edge on the shortest path $p$ from $s$ to $t$ until $p^*$ is the shortest path (and, if the buffer $\delta$ is greater than 0, until the second shortest path is at least $\delta$ longer than $p^*$). The first baseline we consider, \texttt{GreedyFirst}, perturbs the first edge (in path traversal order) in $p$ that deviates from $p^*$. We also use a method in which we perturb the edge with the smallest weight of all edges that are in $p$ but not $p^*$. We refer to this baseline as \texttt{GreedyMin}. In both cases, the selected edge is perturbed enough to make the path at least $\delta$ longer than $p^*$---i.e., if $\mathbf{x}_p$ is the edge indicator vector for the current shortest path and $\ell^\prime=(\mathbf{w}+\Delta)^\top \mathbf{x}_p$, the entry in $\Delta$ associated with the selected edge is increased by $\ell+\delta-\ell^\prime$.
\subsection{Synthetic and Real Networks}
\label{sec:graphdata}
We ran \texttt{PATHPERTURB} and the baseline algorithms on both synthetic and real networks. All networks are undirected.
Our synthetic networks span a wide variety of topologies:
\begin{itemize}
\item Erd\H{o}s--R\'{e}nyi (ER) random networks with 16,000 nodes and an edge probability of 0.00125
\item Barab\'{a}si--Albert (BA) graphs with 16,000 nodes, where each new node connects to 10 existing nodes
\item Watts--Strogatz (WS) graphs with 16,000 nodes, average degree $20$, and edge rewiring probability $0.02$
\item Stochastic Kronecker (KR) graphs with $2^{14}$ nodes and a density parameter of $0.0125$
\item $285\times285$ two-dimensional lattice (LAT) networks
\item $565$-vertex complete (COMP) graphs
\end{itemize}
In all cases, we generate 100 networks from random (or fixed) network generators and add edge weights. Note that the number of edges is approximately 160,000 in all synthetic networks. We also consider various edge-weight distribution for each synthetic network:
\begin{itemize}
\item Option 1: Give all edges weight $1$.
\item Option 2: For each edge, draw a value from a Poisson distribution with rate parameter 20, and add $1$ to get the weight.
\item Option 3: Draw each weight from a uniform distribution over integers from 1 to 41.
\end{itemize}
In addition to synthetic graphs, we use the following real networks:
\begin{itemize}
\item Oregon autonomous systems (AS)~\cite{Leskovec2005}
\item Wikispeedia (WIKI)~\cite{West2009}
\item Pennsylvania roads (PA-ROAD)~\cite{Leskovec2009}
\item Northeast US roads (NEUS)\footnote{Available at \url{https://bit.ly/2QWcug9}.}
\item Central Chilean power grid (GRID)~\cite{Kim2018}
\item Lawrence Berkeley National Laboratory computer network traffic (LBL)\footnote{Available at \url{https://bit.ly/2PQbOsr}.}
\item DBLP coauthorship graph (DBLP)~\cite{Benson2018}
\end{itemize}
Networks AS, WIKI, and PA-ROAD do not have weights on their edges, so we add weights similar to the synthetic networks. In LBL and DBLP, the weights represent similarities rather than distances---number of connections and number of coauthored papers, respectively---so we invert the weight for use in the shortest path computation. In these cases, we set $\delta$ to 0.1, while we use $\delta=1$ in all other cases.
\subsection{Experimental Setup}
We run 100 trials for each network (or network generator) and each weighting scheme if applicable. In each experiment, we select $s$ and $t$ uniformly at random from the largest connected component of $G$, with the exception of LAT, PA-ROAD, and NEUS. In these grid-like graphs, computing the sequence of shortest simple paths is extremely time consuming, and we instead select $s$ at random and choose $t$ from among the nodes $50$ hops away from $s$. We then compute the 100th, 200th, 400th, and 800th shortest paths from $s$ to $t$ and use these as $p^*$. For LAT, PA-ROAD, and NEUS, we compute these paths only using the induced subgraph of nodes that are at most 60 hops away from $s$.
We ran the experiments using a Linux cluster with 32 cores and 192 GB of memory per node. We implemented the linear program in \texttt{PATHPERTURB} using the Python interface to Gurobi 9.1.1, and sequential shortest paths were computed using \texttt{shortest\_simple\_paths} in NetworkX.\footnote{Gurobi is available at \url{https://www.gurobi.com}. NetworkX is available at \url{https://networkx.org}.}
|
2,877,628,090,129 | arxiv | \section{Introduction}
\section{Introduction}\label{sec:Intro}
Mercury, the closest planet to the Sun, is the only terrestrial planet other than Earth that possesses an intrinsic global magnetic field \citep{ness74,ness75}. The recent MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) mission to Mercury presented us with the first opportunity to explore this planet's magnetosphere in great detail since the brief flybys of Mariner 10 \citep[e.g.,][]{solomon07,slavin07}. Many Earth-like magnetospheric features were observed at Mercury, including, but not limited to, magnetopause reconnection \citep{slavin09,dibraccio13}, the concomitant flux transfer events (FTEs) \citep{slavin12} and cusp plasma filaments \citep{slavin14,pho16}, magnetotail flux ropes or plasmoids \citep{dibraccio15}, substorm processes including tail loading-unloading \citep{imber17}, plasma wave activities \citep{sun15}, dipolarization fronts \citep{sundberg12} and the associated electron acceleration \citep{dewey17}, cross-tail current sheet asymmetry and substorm current wedge formation \citep{poh17}, field-aligned currents \citep{anderson14}, and Kelvin-Helmholtz vortices \citep{sundberg10,liljeblad14,gershman15}.
According to MESSENGER observations, Mercury's dipole moment is much weaker than that of Earth, only 195 nT R$_M^3$ (where R$_M$ is Mercury's radius, 2440 km), and is offset in the northward direction by 484 $\pm$ 11 km or $\approx$ 0.2 R$_M$ \citep{anderson11}. Later, those values were slightly modified in \citet{anderson12}. Due to the relatively weak intrinsic planetary magnetic moment and the most extreme solar wind driving forces in the solar system, Mercury has a small but extremely dynamic magnetosphere whose size is about 5\% that of Earth's magnetosphere \citep{winslow13}. More interestingly, Mercury has a large electrically conductive iron core with a radius of $\approx$ 0.8 R$_M$ \citep{smith12,hauck13}. A unique aspect of Mercury's interaction system is that the large conducting core can induce observable magnetic fields in Mercury's magnetosphere \citep{slavin14,zhong15,johnson16}. It is worth noting that \citet{hood79} and \citet{grosser04} made some early quantitive estimates of the induction effect at Mercury. The core-induced magnetic fields have been demonstrated to play an important role in Mercury's global solar wind interaction, especially during extreme space weather events \citep{slavin14,jia15,heyner16,slavin19}. While the induction response generates additional magnetic flux that may protect Mercury from solar wind erosion, magnetic reconnection between the interplanetary magnetic field (IMF) and the planetary field removes magnetic flux from the dayside magnetopause and enables transfer of energy and momentum to the planetary inner magnetosphere, which consequently leads to the direct entry of solar wind plasma into the system. The magnetic flux transferred to the nightside magnetosphere may immediately undergo reconnection or be stored and later returned to the dayside during an intense episode of reconnection in the tail \citep{slavin14}. Magnetotail reconnection is also the dominant plasma process that transfers energy and momentum into Mercury's inner tail region by converting stored magnetic energy in the tail lobe into plasma kinetic energy in the plasma sheet. Magnetic reconnection, therefore, plays a crucial role in manipulating the magnetospheric dynamics of Mercury and other planets in our solar system and beyond.
Despite the significant achievements accomplished by direct spacecraft observations, \emph{in situ} measurements are often taken at limited points along the trajectories of orbits or flybys. Such limitations, however, can be alleviated by numerical simulations, which allow the interpretation of \emph{in situ} measurements in a three-dimensional context and distinguishing temporal from spatial fluctuations as well. Thus, numerical models, combined with \emph{in situ} data, are the key for providing a global description of solar wind-planet interaction. In recent years, our understanding of terrestrial bodies has been significantly advanced by increasingly sophisticated numerical models. A large number of global models based on either fluid or hybrid (kinetic ion particles and massless electron fluid) approach have been developed for both magnetized planets such as Mercury \citep[e.g.,][]{kabin08,kidder08,pavel10,muller12,richer12,jia15,exner18} and unmagnetized planets such as Mars \citep{ma14,dong14,dong15b,dong18b,dong18c,modolo16,ledvina17} as well as exoplanets \citep{johansson11,dong17a,dong17b,dong18a,dong19}. However, none of these global models can accurately treat collisionless magnetic reconnection due to their lack of detailed electron physics. In order to solve this issue with affordable computational costs, two broad approaches have been proposed. \citet{toth16} studied Ganymede's magnetosphere by employing a Hall magnetohydrodynamic model with embedded particle-in-cell boxes (MHD-EPIC) such that they can capture the collisionless reconnection physics in prescribed local regions. Meanwhile, \citet{wang18} developed a novel ten-moment multifluid model to study Ganymede's magnetosphere. Other than relying on the prescribed local PIC boxes, the new global multi-moment multifluid model incorporating the higher-order moments is capable of reproducing some critical aspects of the reconnection physics from PIC simulations \citep{wang15,ng15,ng17,ng18}.
Until now, no such approach (i.e., either MHD-EPIC or the multi-moment multifluid approach) has been applied to study Mercury. This work will, therefore, be the first study of Mercury's dynamic magnetosphere using a ten-moment multifluid model. In order to capture the induction effects arising from the interior-magnetosphere electromagnetic coupling, we also implemented a resistive mantle and an electrically conductive core inside Mercury in this new model. This paper is structured as follows. In Section \ref{sec:Model}, the ten-moment multifluid model and the model setup for Mercury are described. In Section \ref{sec:data-model}, we first validate the model through data-model comparison with MESSENGER data and then discuss the model results. We also conduct a hypothetical extreme event case study to demonstrate the significance of the induction effects. The conclusion is given in Section \ref{sec:conclusion}.
\section{Ten-Moment Multifluid Model for Mercury} \label{sec:Model}
\subsection{Ten-Moment Equations} \label{subsec:ModelEqs}
In this section, we briefly introduce the ten-moment multifluid model for Mercury within the \textsc{Gkeyll} framework\footnote{gkeyll.rtfd.io}. The ten moments refer to mass density $mn$, momentum $mn u_x$, $mn u_y$, $mn u_z$ and pressure tensor ${P}_{xx}$, ${P}_{xy}$, ${P}_{xz}$, ${P}_{yy}$, ${P}_{yz}$, ${P}_{zz}$. Conceptually, the ten-moment model is akin to a fluid version of particle-in-cell (PIC) code, truncated at a certain order of moment, i.e., second-order moment, the pressure. For Mercury, we solve ten-moment equations for both protons and electrons. It is noteworthy that the ten-moment model has been employed to study magnetic reconnection in multi-species plasmas including O$^+$, H$^+$, and e$^-$ \citep{dong16}. The ten-moment equations for each species are given as follows:
\begin{eqnarray}
\frac{\partial\left(m_{s}n_{s}\right)}{\partial t}+\frac{\partial\left(m_{s}n_{s}u_{i,s}\right)}{\partial x_{i}} & = & 0,\\
\frac{\partial\left(m_{s}n_{s}u_{i,s}\right)}{\partial t}+\frac{\partial\mathcal{P}_{ij,s}}{\partial x_{j}} & = & n_{s}q_{s}\left(E_{i}+\epsilon_{ijk}u_{j,s}B_{k}\right),\label{eq:10m-momentum}\\
\frac{\partial\mathcal{P}_{ij,s}}{\partial t}+\frac{\partial\mathcal{Q}_{ijk,s}}{\partial x_{k}} & = & n_{s}q_{s}u_{[i,s}E_{j]}+\frac{q_{s}}{m_{s}}\epsilon_{[ikl}\mathcal{P}_{kj,s]}B_{l}.\label{eq:10m-pressure}\\ \nonumber
\end{eqnarray}
where $q$ is the charge, $E$ and $B$ are electric field and magnetic field, respectively. The subscripts $s=e,i$ represent the electrons and ion species. It will be neglected hereinafter for convenience. The square brackets in Equation (\ref{eq:10m-pressure}) surrounding the indices represent the minimal sum over permutations of free indices needed to yield completely symmetric tensors. The first-order moment is defined as $m n u_{i}\equiv m \int f v_{i}d\mathbf{v}$, where $f$ is the phase space distribution function, $m$ and $v_i$ denote the individual particle mass and velocity, respectively. Similarly, the second-order moment, $\mathcal{P}_{ij}$, and third-order moment, $\mathcal{Q}_{ijk}$, are defined as
\iffalse
\begin{eqnarray}
\mathcal{P}_{ij}=m \int f v_{i}v_{j}d\mathbf{v}=m \int f \left(v_{i}-u_{i}\right)\left(v_{j}-u_{j}\right) d\mathbf{v}+nmu_{i}u_{j}=P_{ij}+nmu_{i}u_{j}.
\end{eqnarray}
and,
\begin{eqnarray}\nonumber
\mathcal{Q}_{ijk}&=&m \int f v_{i}v_{j}v_{k}d\mathbf{v}~=~m\int f \left(v_{i}-u_{i}\right)\left(v_{j}-u_{j}\right)\left(v_{k}-u_{k}\right) d\mathbf{v}+u_{[i}\mathcal{P}_{jk]}-2nmu_{i}u_{j}u_{k} \\ &=&Q_{ijk}+u_{[i}\mathcal{P}_{jk]}-2nmu_{i}u_{j}u_{k}
\end{eqnarray}
\fi
\begin{eqnarray}
& \mathcal{P}_{ij} & =m\int fv_{i}v_{j}d\mathbf{v}\nonumber \\
& & =m\int f\left(v_{i}-u_{i}\right)\left(v_{j}-u_{j}\right)d\mathbf{v}+nmu_{i}u_{j}\nonumber \\
& & =P_{ij}+nmu_{i}u_{j}.
\end{eqnarray}
and,
\begin{eqnarray}
& \mathcal{Q}_{ijk} & =m\int fv_{i}v_{j}v_{k}d\mathbf{v}\nonumber \\
& & =m\int f\left(v_{i}-u_{i}\right)\left(v_{j}-u_{j}\right)\left(v_{k}-u_{k}\right)d\mathbf{v}+u_{[i}\mathcal{P}_{jk]}-2nmu_{i}u_{j}u_{k}\nonumber \\
& & =Q_{ijk}+u_{[i}\mathcal{P}_{jk]}-2nmu_{i}u_{j}u_{k}
\end{eqnarray}
where $P_{ij}$ is the pressure tensor and ${Q}_{ijk}$ is the heat flux tensor. One of the key issues for a multi-moment multifluid model is the closure problem, i.e., how to close the equation systems and incorporate kinetic effects into a fluid framework, which is still an active research topic in fluid dynamics and plasma physics \citep{hunana18}. In this work, we adopt the following 3D closure simplified by \citet{wang15} based on Landau-fluid closures \citep[e.g.,][]{hammett90}:
\begin{eqnarray}
\partial_{m}Q_{ijm}\approx v_{t}\left|k\right|\left(P_{ij}-p\delta_{ij}\right).
\end{eqnarray}
where $v_t$ refers to the local thermal speed, $p$ is the scalar pressure, and $k$ is a free parameter that effectively allows for deviations from isotropy at length scales less than $1/|k|$. For \emph{collisionless} magnetic reconnection, $k$ should be a function of $d_e$ given that \emph{collisionless} magnetic reconnection takes place on the length scale of electron inertial lengths, $d_{e}$. Following the work of \citet{wang18}, we define $k_{s}(\mathbf{x},t)$ as $10/d_{s}(\mathbf{x},t)$, where $d_{s}(\mathbf{x},t)$ is the local inertial length of species $s$ as a function of $\mathbf{x}$ and $t$, such that it can provide a more accurate heat flux approximation because the species inertial length for the Mercury system can vary greatly in space. Interestingly, such closure can well reproduce the \emph{collisionless} reconnection physics from a fully kinetic particle-in-cell code as shown in \citet{wang15}.
The electromagnetic field is solved by full Maxwell equations
\begin{eqnarray}
\frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial t} & = & \nabla\times\mathbf{B}-\mu_{0}\mathbf{J},\label{eq:maxwell-dE} \\
\frac{\partial\mathbf{B}}{\partial t} & = & -\nabla\times\mathbf{E},
\end{eqnarray}
where $\mathbf{J}$ is the electric current density. Inside the planet interior $J = \sigma \mathbf{E}$, where plasma convection, $\mathbf{u}$, can be neglected. Unlike the traditional magnetohydrodynamic (MHD) or hybrid models that solve the electric field $\mathbf{E}$ by Ohm's law, here we update $\mathbf{E}$ directly through the Ampere's law, Equation (\ref{eq:maxwell-dE}). Therefore, electromagnetic waves are fully supported, similar to a PIC code. In order to demonstrate how a ten-moment model supports the reconnection electric field in \emph{collisionless} magnetospheres, we rearrange Equation (\ref{eq:10m-momentum}) and obtain the following generalized Ohm's law \citep[e.g.][]{wang15,lin2017}:
\begin{eqnarray}
\mathbf{E}+\mathbf{v}\times\mathbf{B} &=& \underbrace{\cancel{\eta\mathbf{J}}}_{0} + \frac{\mathbf{J}\times\mathbf{B}}{n |e|}-\frac{\nabla\cdot\mathbf{P}_{e}}{n |e|}
+\frac{m_{e}}{n |e|^{2}}\left[\frac{\partial\mathbf{J}}{\partial t}+\nabla\cdot\left(\mathbf{v}\mathbf{J}+\mathbf{J}\mathbf{v}-\frac{\mathbf{J}\mathbf{J}}{n |e|}\right)\right].\label{eq:ohm}
\end{eqnarray}
It should be noted that the Ohm's law formulated above is not numerically solved in the model.
In the case of 2D anti-parallel magnetic reconnection without a guild field, $\mathbf{B}=0$ (hence $\mathbf{v} \times \mathbf{B}=0$ and $\mathbf{J} \times \mathbf{B}=0$) at reconnection sites or X-points, therefore the divergence of the electron pressure tensor and the total derivative of the electric current are the primary sources of the reconnection electric field in a \emph{collisionless} ($\eta=0$) system (see Equation \ref{eq:ohm} or \citet{zweibel09}). It is further demonstrated by PIC simulations that the reconnection electric field, $E_z$, is largely supported by the divergence of the off-diagonal
elements of $\mathbf{P_{e}}$, i.e., $E_z = -(\partial_x P_{xz,e} + \partial_y P_{yz,e})/n_e |e|$, while traditional MHD and hybrid models only assume a scalar pressure, which does not contribute to $E_z$ at reconnection sites \citep{wang15}. Even if a guide field exists, one can still get the similar conclusion. The multi-moment multifluid code has been used to study many laboratory and space plasma physics problems \citep[e.g.,][]{ng15,ng19,wang18,tenBarge19}. The details concerning the numerics and benchmark examples have been described in \citet{hakim06}, \citet{hakim08} and \citet{wang19}.
\subsection{Model Setup for Mercury}\label{subsec:ModelSetup}
In a ten-moment model, the time step is mainly restricted by the speed of light. For this reason, we relax this restriction by using an artificially reduced speed of light, $c = 3000$ km/s. We also apply a reduced ion-electron mass ratio $m_i/m_e$ = 25 as the previous study \citep{wang18}, which is sufficiently large to separate the electron and ion scales. The upstream ion inertial length is set to $d_{i,in} = 0.05 R_M$ and electron inertial length $d_{e,in} = 0.01 R_M$. We adopt the Mercury-Solar-Orbital (MSO) coordinates, where the $x$ axis points from Mercury toward the Sun, the $z$ axis is perpendicular to planet's orbital plane, and the $y$ axis completes the right-hand system. The computational domain is defined by $ - 15 R_M \le x \le 5 R_M$, $- 30 R_M \le y,z \le 30 R_M$ with a nonuniform stretched Cartesian grid. The smallest grid size is 0.01 $R_M$, and in turn, five cells are employed to resolve the ion inertial length and one cell for the electron inertial length. In order to capture the magnetospheric physics with minimum influences from numerical resistivity, we use a total of $\sim 4\times 10^9$ cells such that we are able to cover most of the Hermean magnetosphere with the finest grid mesh (i.e., 0.01 $R_M$ resolution).
We implement Mercury's intrinsic dipole magnetic field $\mathbf{B_0}$ with an equatorial surface strength of 195 nT and centered at (0, 0, 0.2 R$_M$) in MSO. The dipole field is prescribed and fixed in time. The total magnetic field $\mathbf{B}$ equals $\mathbf{B_0} + \mathbf{B_1}$, and we only solve the perturbation magnetic field, $\mathbf{B_1}$, in the model. The inner boundary for electromagnetic fields is set at core surface (0.8 R$_M$) where the conducting wall boundary conditions are applied. For plasma fluids, the inner boundary is set at the planet's surface, such that fluid moment equations are not solved inside the planet. If the surface plasma flow has an inflow component (i.e., $\mathbf{u} \cdot \mathbf{r} < 0$), absorbing boundary conditions are applied. If the surface plasma flow has an outflow component (i.e., $\mathbf{u} \cdot \mathbf{r} > 0$), we set the radial velocity equal to zero, and the plasma density and pressure are fixed at 1 cm$^{-3}$ and 0.001 nPa, respectively \citep{jia15}. Outer boundary conditions are inflow at $x = 5 R_M$ and float at the flanks and tail side.
\section{Results and Discussion}\label{sec:data-model}
In this section, we first validate the model through data-model comparison. We then discuss the model results including day- and night-side magnetic reconnection, field-aligned currents, and cross-tail current sheet asymmetry. Finally, we present Mercury's magnetospheric response to a hypothetical extreme event.
\subsection{Model Validation through Data-Model Comparison}
When magnetic reconnection occurs at the dayside magnetopause, it leads to an efficient transfer of energy and flux from the solar wind into the magnetosphere, which ultimately drives reconnection in the magnetotail. We choose to study MESSENGER's second flyby on October 6, 2018 (hereinafter referred to as M2), during which the IMF had a southward (negative $B_z$) component. For M2, the solar wind parameters are as follows: solar wind density, 40 $cm^{-3}$, solar wind velocity in MSO, $\left(-400,50,0\right)$ km/s, solar wind temperature 18 eV, and IMF in MSO, $\left(-15.2,8.4,-8.5\right)$ nT, where the y-component of the solar wind flow velocity results from Mercury's orbital motion \citep{jia15}
\iffalse
\begin{table}[!h]\label{mercury-params}
\caption{Solar wind parameters used in the Mercury's simulation.}
\centering
\begin{tabular}{c|c|c|c}
\hline
{$\rho_{in}\left[\mathtt{amu/cm^{3}}\right]$} & {$\mathbf{v}_{in}\left[\mathtt{km/s}\right]$} & {$T_{in}\left[\mathtt{eV}\right]$} & {$\mathbf{B}_{in}\left[\mathtt{nT}\right]$} \\
\hline
{40} & {$\left(-400,50,0\right)$} & {18} & {$\left(-15.2,8.4,-8.5\right)$} \\
\hline
\end{tabular}
\end{table}
\fi
Figure \ref{3DMS} (top) presents Mercury's three-dimensional magnetosphere from the ten-moment multifluid calculation. Magnetospheric characteristics such as the bow shock, magnetosheath, magnetopause, and magnetotail are clearly captured. In detail, the ``hot'' sphere (0.8 R$_M$) inside Mercury represents Mercury's electrically conductive core. M2 trajectory is plotted in red, pointing from night/dusk side to day/dawn side and near Mercury's equatorial plane. Between the conducting core and planet's surface, there exists a highly resistive mantle. The radial resistivity profile shown in the top-left corner of Figure \ref{3DMS} has been adopted from \citet{jia15}, and the white dots in the embedding plot are the grid points used in the model, i.e., 0.01 $R_M$.
\begin{figure*}[!ht]
\centering\
\includegraphics[width=0.66\textwidth]{3d_full_v2.png}
\includegraphics[width=0.68\textwidth]{data-model_comp_xyzr.png}
\caption{Top: Mercury's three-dimensional magnetosphere from the ten-moment multifluid calculation. The color contours depict the ion density in cm$^{-3}$. The ``hot'' sphere inside Mercury represents its conducting core with a size R$_{c}$ = 0.8 R$_{M}$. The magnetic field lines are presented in blue. The red curve together with a cyan arrow represents MESSENGER's M2 trajectory. The radial resistivity profile adopted from \citet{jia15} is shown at the top-left corner. Bottom: Data-model comparison of magnetic fields along MESSENGER's M2 trajectory.}
\label{3DMS}
\end{figure*}
To validate our model calculations, we compare the simulation results with MESSENGER's magnetic field data. Panels (a)-(d) of Figure \ref{3DMS} compare the model-calculated magnetic field components along M2 (in red) to MESSENGER magnetometer measurements (in black). Mercury's (unperturbed) intrinsic dipole magnetic field is also plotted as a reference (the blue dashed line in the last row) to illustrate how the global solar wind interaction affects Mercury's magnetosphere. Good agreement is observed between the model calculations and MESSENGER observations in Figure \ref{3DMS}, thus ensuring the validity of our novel approach. Due to the lack of accurate solar wind measurements, we are not able to reproduce the FTE (i.e., the spike structure at 08:50 UTC) observed by MESSENGER. As will be shown below, our model is capable of reproducing other important MESSENGER observations (beyond the MHD approach); therefore our numerical study by adopting this new model represents a crucial step toward establishing a modeling framework that enables self-consistent characterization of Mercury's tightly coupled interior-magnetosphere system.
\subsection{Model Results Analysis and Discussion}
\subsubsection{Dawn-Dusk Asymmetries in Mercury's Magnetotail and Field-Aligned Currents}
Dawn-dusk asymmetry is a ubiquitous phenomenon in planetary magnetotails. Notably, the ten-moment multifluid model is able to capture the remarkable asymmetry exhibited in Mercury's magnetotail current sheet. Figure \ref{2DAsym}(a) depicts the electron pressure scalar ($p_e$) in Mercury's magnetic equatorial plane (at z = 0.2 R$_{M}$ in MSO), where the cross-tail current sheet is located. From Figure \ref{2DAsym}(a), one can see that (1) more hot electrons are present at the dawnside especially in the inner tail region, and (2) the asymmetry in $p_e$ gradually decreases with increasing distance down the tail. By analyzing the simulation results, we find a slightly dawnward preference in magnetotail reconnection, however, the dawn-dusk asymmetry of the x-line is not significant, probably due to the lack of a dominant amount of Na$^+$ on the duskside as suggested by \citet{poh17}. Here, we conclude that the exhibited asymmetry in hot electron distribution is caused by the dual effect of Mercury's magnetotail reconnection and the dawnward drifts of electrons. When approaching Mercury, the kinetic energy of the sunward reconnection outflow can be easily converted to thermal energy due to the tailward pressure gradient force, leading to more notable asymmetry near the planet relative to the far tail. Meanwhile, the sunward electron flow also drifts to dawnside according to the perpendicular drift velocity of species $s$, $\mathbf{u}_{s\perp}$, derived from the cross product of Equation (\ref{eq:10m-momentum}) and $\mathbf{B}$,
\begin{equation}\label{drifts}
\mvec{u}_{s\perp} =
\frac{\mvec{E}\times\mvec{B}}{B^2}
-\frac{\nabla\cdot\mvec{P}_s \times\mvec{B}}{q_s n_s B^2}
-\frac{m_s}{q_s B^2} \frac{d\mvec{u}_s}{dt}\times\mvec{B}
\end{equation}
where the first term is the $\mathbf{E}\times\mathbf{B}$ drift, the second term incorporates the diamagnetic drift and the curvature drift (given $\mvec{P}_s = \mvec{I} p_{s\perp} + \mvec{b}\mvec{b} (p_{s\parallel}-p_{s\perp}) + \mvec{\Pi}_s$, where $\mvec{\Pi}_s$ is the off-diagonal part of the pressure tensor), while the last term contains the polarization drift. Interestingly, an asymmetry also manifests in the X-ray fluorescence (XRF) from MESSENGER X-Ray Spectrometer (XRS) observations at Mercury's nightside surface (Figure \ref{2DAsym}(b)). It is noteworthy that the calculated electron pressure, $p_e$, at Mercury's nightside surface (Figure \ref{2DAsym}(c)) depicts similar patterns as the XRF, supporting the idea of electron-induced surface fluorescence by \citet{lindsay16}.
\begin{figure}[!htbp]
\centering\
\includegraphics[width=1.0\textwidth,angle=0]{xy_plane-p_e_with-JrS_x-ray_merged_ordered.png}
\caption{(a) Electron pressure ($p_e$) distribution in Mercury's magnetic equatorial plane at z = 0.2 R$_{M}$. (b) X-Ray Spectrometer (XRS) observations of energetic electron-induced surface fluorescence at Mercury's nightside surface from \citet{lindsay16}. (c) Electron pressure ($p_e$) distribution at Mercury's nightside surface from the ten-moment model. (d) Contour plot of radial current density, $J_{rS}$, at Mercury's (northern hemisphere) surface displayed versus local time in hours from \citet{anderson14} based on MESSENGER magnetometer observations. (e) Calculated radial current density, $J_{rS}$, at Mercury's (northern hemisphere) surface from the ten-moment model.}
\label{2DAsym}
\end{figure}
In addition to the asymmetries, we also present the simulation results for the field-aligned currents (or Birkeland currents) at Mercury's northern hemisphere surface in Figure \ref{2DAsym}(e). The model predicts that the currents flow downward (in blue) at dawn and upward (in red) at dusk, which are consistent with MESSENGER observations shown in Figure \ref{2DAsym}(d) and analogous to Region 1 (R1) Birkeland currents at Earth. More importantly, our simulation results for the current density values at the planetary surface also agree well with MESSENGER observations. MESSENGER magnetic field data show that the maximum and minimum $J_{rS}$ are $\pm115 nA/m^2$ \citep{anderson14}, and in comparison, the calculated maximum and minimum values from our model are 115 $nA/m^2$ and -150 $nA/m^2$, respectively.
\subsubsection{Magnetotail and Magnetopause Reconnection}
In order to demonstrate that the magnetic reconnection in our calculations is driven by detailed electron physics instead of numerical dissipation as in \citet{jia15,jia19}, we further study the magnetic reconnection in Mercury's magnetotail and at the planet's magnetopause. We first investigate the magnetotail reconnection where the electron reconnection physics is less contaminated given that the tail is less affected by direct solar wind interaction than Mercury's dayside magnetopause. Note that previous full PIC simulations showed that the divergence of the off-diagonal elements of electron pressure tensor, $\mathbf{P_e}$, is the main source of the reconnection electric field \citep{wang15,wilson16}, which can be verified from Equation (\ref{eq:ohm}) as well. We therefore plot $P_{xy,e}$, $P_{xz,e}$ and $P_{yz,e}$ in the first row of Figure \ref{NightDayRec}. Among the three $\mathbf{P_e}$ off-diagonal terms, $P_{yz,e}$ has the largest amplitude and gradient, therefore is the most important term, consistent with previous studies \citep[e.g.,][]{wang15,divin16,wang18}.
\begin{figure}[!htbp]
\centering\
\includegraphics[width=1.0\textwidth]{reconnection.png}
\caption{Magnetic reconnection in Mercury's magnetotail (first row) and at the magnetopause (second row). Different components of the electron pressure tensor off-diagonal terms ($P_{xy,e}$, $P_{xz,e}$ and $P_{yz,e}$ in nP) are plotted.}
\label{NightDayRec}
\end{figure}
Subsequently, we investigated the magnetopause reconnection. Again the three $\mathbf{P_e}$ off-diagonal elements are shown in the second row of Figure \ref{NightDayRec}, where the reconnection rate ranges from 0.08 to 0.2, depending on the locations. In comparison with Figure \ref{NightDayRec}(a-c), Figure \ref{NightDayRec}(d-f) also exhibits different patterns for the $\mathbf{P_e}$ off-diagonal elements. In addition to the reconnection physics, Figure \ref{NightDayRec} clearly depicts the magnetopause location ($\approx$1.4 R$_M$) and the bow shock location ($\approx$1.8 R$_M$), consistent with the previous validated study by \citet{jia15}.
\subsubsection{Extreme Event Case Study}
The solar wind parameters of M2 yield a dynamic pressure of $\approx$11 nPa, which is relatively weak for instigating a significant induction response from the conducting core. Thus, we followed the scenario in \citet{jia15} to investigate the core-induced induction response; the solar wind density and speed are deliberately enhanced to 80 $cm^{-3}$ and 700 $km/s$, respectively, such that the solar wind dynamic pressure increases to $\approx$66 nPa, close to the pressure of 23 November 2011 event in \citet{slavin14}. The ten-moment multifluid calculation of Mercury's magnetospheric response to this hypothetical extreme event is shown in Figure \ref{extrem}. From the middle panel, one can see that both the bow shock and magnetopause boundaries are compressed significantly. Compared with the M2 flyby, the new magnetopause standoff distance is compressed to $\approx$1.15$R_M$, consistent with the results from \citet{jia15} for the same event study. In the bottom panel of Figure \ref{extrem}, we also compare the perturbation magnetic field $B_{1z}$ between the normal solar wind case (of M2) and the extreme event. As expected, solar wind compression increases $B_{1z}$ during the extreme event and squeeze the dayside magnetosphere. However, in order to demonstrate that the enhancement in $B_{1z}$ is not purely a result of solar wind compression, we present the core surface current $J_y$ for both cases, where the color contours on the core surface represent $J_y$ intensity and the yellow curves with green arrows are the corresponding current streamlines. Following Faraday's law of induction, these currents generate additional magnetic flux that acts against the solar wind pressure. By adopting the same color scale, it is clear that $J_y$ is much stronger in the extreme case than that in M2, indicating that the increase in $B_{1z}$ is a result of both solar wind compression and induction responses. The enhanced $B_{1z}$ and the intensified core surface current $J_y$ clearly demonstrate the importance of the induction response during the extreme event.
In contrast to \citet{jia15}, our calculations contain richer features. For the first time, our simulation illustrates the formation of plasmoids in Mercury's magnetotail through \emph{collisionless} magnetic reconnection by including the reconnection electron physics. Plasmoids (or flux ropes) have, as a matter of fact, been observed by MESSENGER \citep{dibraccio15}. Theoretically speaking, these plasmoids are formed in elongated and intense current sheets due to the plasmoid instability - an explosive instability resulting in the formation of plasmoids due to magnetic reconnection \citep[e.g.,][]{comisso16}. In order to demonstrate that plasmoids are indeed formed within the cross-tail current layer, we plot the current sheet density ($J_y$) together with the plasmoid in the top panel of Figure \ref{extrem}. These plasmoids are eventually transported either toward or away from the planet, and new plasmoids will repeatedly form within the cross-tail current sheet (not shown here), leading to the small but extremely dynamic magnetosphere of Mercury. The impact of extreme space weather events (such as coronal mass ejections given in, e.g., \citealp{slavin14}) on Mercury's dynamic magnetosphere will be investigated in detail in our future work.
\begin{figure}[!htbp]
\centering\
\includegraphics[width=0.9\textwidth]{xz_contourf_Jy_B1z_resize.png}
\caption{Mercury's magnetosphere in the x-z (meridian) plane during a hypothetical extreme event. Plasmoids are formed in Mercury's magnetotail. The background color contours in the middle panel show the ion density in cm$^{-3}$. The bottom left panel shows the zoomed-in subdomain where color contours in the x-z plane represent the perturbation magnetic field $B_{1z}$ (in nT) and the color contours on the conducting core surface are the induction current $J_y$ (in nA/m$^2$). Note that the streamlines of core surface currents are illustrated by the yellow curves with green arrows wrapping around the core. Compared with the bottom right panel of M2, the $B_{1z}$ and the induction current $J_y$ from the extreme event are much stronger. The top panel depicts the formation of a plasmoid within the cross-tail current sheet.}
\label{extrem}
\end{figure}
\section{Conclusion}\label{sec:conclusion}
For the first time, we utilize a three-dimensional ten-moment multifluid model to study solar wind interaction with Mercury from the planetary interior to its dynamic magnetosphere. Given the importance of the induction effects shown in the previous studies, we also include a highly resistive mantle and an electrically conductive iron core (of radius 0.8$R_M$) inside the planet body. Direct comparison between MESSENGER magnetometer data and model calculations show good agreement, strongly supporting the validity of this new model. The cross-tail current sheet asymmetry revealed by the model is also consistent with MESSENGER observations. We conclude that the exhibited asymmetry in hot electron distribution is caused by the dual effect of Mercury's magnetotail reconnection and the dawnward drifts of electrons. In addition, this model accurately reproduces the field-aligned currents measured by MESSENGER that cannot be captured by an MHD model. Our study of magnetotail and magnetopause reconnection show that the off-diagonal elements of the electron pressure tensor, $\mathbf{P}_e$, play a key role in \emph{collisionless} magnetic reconnection. In order to investigate the induction effects, we have also studied Mercury's magnetospheric responses to a hypothetical extreme event. The simulation demonstrates that the induced magnetic fields help sustain a magnetopause, hindering the compression of the magnetopause down towards the surface. More interestingly, plasmoids (or flux ropes) are formed in Mercury's cross-tail current sheet, indicating Mercury's magnetotail being extremely dynamic.
Thanks to this novel fluid approach that incorporates detailed electron physics associated with, e.g., \emph{collisionless} magnetic reconnection and magnetic drifts, we are able to reproduce and interpret the observations beyond MHD. Here we want to reiterate the distinction between the multi-moment multifluid approach and the (Hall) MHD approach from three perspectives. First, as mentioned earlier, the new model evolves the same set of equations (i.e., continuity, momentum and pressure tensor equations) for both ions and electrons (without the quasi-neutral assumption) and updates the electric and magnetic fields by adopting the full Maxwell's equations. As a result, the new model incorporates the non-ideal effects including the Hall effect, inertia, and tensorial pressures that are self-consistently embedded without the need for explicitly solving a generalized Ohm's law as MHD. Second, the new model supports all kinds of electromagnetic waves due to the inclusion of full Maxwell's equations. It is well-known that one of the shortcomings of Hall MHD lies in its failing to capture the right dispersion relation of Whistler waves (due to the assumption of massless electrons) when studying \emph{collisionless} magnetic reconnection. Last but not least, the new model contains an approximation to the Landau-fluid closure and therefore lower-order kinetic physics \citep{wang15,hammett90,hunana18}. For instance, the novel fluid approach can correctly capture the lower hybrid drift instability (LHDI), which can only be treated properly by a kinetic approach in the past \citep{ng19}.
In summary, MESSENGER furnished us with a great opportunity to study Mercury's dynamic magnetosphere. An abundance of useful data was returned from this mission, which stimulated numerous interesting studies. With the launch of the BepiColombo mission to Mercury in October 2018 \citep{benkhoff10}, Mercury's exploration will witness another notable surge after MESSENGER. A properly validated model that incorporates the electron physics essential for Mercury's \emph{collisionless} magnetosphere will likely advance our understanding of the dynamic responses of Mercury's magnetosphere to global solar wind interactions. Hence, the three-dimensional global ten-moment multifluid model developed herein represents a crucial step towards establishing a revolutionary approach that enables the investigation of Mercury's tightly coupled interior-magnetosphere system beyond the traditional fluid model, and has the potential to enhance the science returns of both the MESSENGER mission and the BepiColombo mission.
\acknowledgments
The authors thank Manasvi Lingam, Ryan Dewey, Suzanne Imber, Yuxi Chen, Yao Zhou, Chang Liu and Y. Y. Lau for the helpful discussions and comments. This work was supported by NSF Grant Nos. AGS-0962698 and AGS-1338944, NASA Grants Nos. 80NSSC19K0621, NNH13AW51I and 80NSSC18K0288 and DOE grant DE-SC0006670. The MESSENGER data used in this study are available from the PPI node of the Planetary Data System (\texttt{http://ppi.pds.nasa.gov}), and the model data were obtained from simulations using the \textsc{Gkeyll} framework developed at Princeton University, which is publicly available at \texttt{https://bitbucket.org/ammarhakim/gkeyll}. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center, the Titan supercomputer at the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory through the INCITE program, supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725, the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231, Cheyenne (doi:10.5065/D6RX99HX) provided by NCAR's CISL, sponsored by NSF, and Trillian, a Cray XE6m-200 supercomputer at the UNH supported by the NSF MRI program under Grant No. PHY-1229408.
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\begin{document}
\title
[The homology cobordism group of homology cylinders]
{Invariants and structures of the homology cobordism group of homology cylinders}
\author{Minkyoung Song}
\address{
Department of Mathematics\\
POSTECH\\
Pohang 790--784\\
Republic of Korea
}
\email{pp1004@postech.ac.kr}
\begin{abstract}
The homology cobordism group of homology cylinders is a generalization of the mapping class group and the string link concordance group. We study this group and its filtrations by subgroups by developing new homomorphisms. First, we define extended Milnor invariants by combining the ideas of Milnor's link invariants and Johnson homomorphisms. They give rise to a descending filtration of the homology cobordism group of homology cylinders. We show that each successive quotient of the filtration is free abelian of finite rank. Second, we define Hirzebruch-type intersection form defect invariants obtained from iterated $p$-covers for homology cylinders. Using them, we show that the abelianization of the intersection of our filtration is of infinite rank. Also we investigate further structures in the homology cobordism group of homology cylinders which previously known invariants do not detect.
\end{abstract}
\maketitle
\setcounter{tocdepth}{2}
\tableofcontents
\@startsection{section}{1{Introduction}
The homology cobordism group of homology cylinders is an interesting object of study which extends the mapping class group and generalizes the concordance group of string links in homology 3-balls. The aim of this paper is to enhance our understanding of the structure of the group by developing two new invariants which are homomorphisms. We obtain the first invariant by combining the ideas of Milnor's link invariant and Johnson's homomorphism, and the second is a Hirzebruch-type intersection form defect from iterated $p$-covers. In this paper, manifolds are assumed to be compact and oriented. Our results hold in both topological and smooth categories.
Let $\Sigma_{g,n}$ be a surface of genus $g$ with $n$ boundary components.
Roughly speaking, a \emph{homology cylinder} over $\Sigma_{g,n}$ is a homology cobordism between two copies of~$\Sigma_{g,n}$. A homology cylinder is endowed with two embeddings $i_+^{\vphantom{}}$ and $i_-^{\vphantom{}}$ of $\Sigma_{g,n}$ called \emph{markings}. The notion of homology cylinders was first introduced by Goussarov \cite{Go} and Habiro \cite{Ha} independently, as important model objects for their theory of finite type invariants of 3-manifolds which play the role of string links in the theory of finite type invariants of links. While the set $\mathcal{C}_{g,n}$ of marking-preserving homeomorphism types of homology cylinders is a monoid, the set $\mathcal{H}_{g,n}$ of homology cobordism classes becomes a group under juxtaposition. The group $\mathcal{H}_{g,n}$ was introduced by Garoufalidis and Levine as an enlargement of the mapping class group $\mathcal{M}_{g,n}$ of $\Sigma_{g,n}$~\cite{GL, L01}.
The group $\mathcal{M}_{g,n}$ injects into $\mathcal{C}_{g,n}$ and also into $\mathcal{H}_{g,n}$ (See \cite[p.~247]{L01}, \cite[Proposition 2.3]{CFK}).
Moreover, when $n>1$, $\mathcal{H}_{g,n}$ can be seen as a generalization of the concordance group of framed string links in homology balls.
In this paper, we assume that $n >0$, i.e.\ $\Sigma_{g,n}$ has nonempty boundary. We usually omit the subscripts and simply write $\Sigma$, $\mathcal{C}$, and $\mathcal{H}$ when $g,n$ are understood from the context. Let $I$ denote the interval~$[0,1]$. For a group $G$, $G_k$ denotes the $k$th term of lower central series given by $G_1=G$, $G_{k+1} = [G, G_k]$, and $G^{(k)}$ denotes the $k$th derived subgroup given by $G^{(0)}=G$, $G^{(k+1)}=[G^{(k)}, G^{(k)}]$.
In the literature, the structure of $\mathcal{H}$ was studied by constructing invariants. In particular, invariants which are group homomorphisms are essential in understanding the group structure.
Let $F= \pi_1(\Sigma)$ and $H=H_1(\Sigma)$.
In \cite{GL, L01}, Garoufalidis and Levine defined homomorphisms $\eta_q^{\vphantom{}}\colon \mathcal{H}_{g,1} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \operatorname{Aut}(F/F_q)$ and a filtration $\mathcal{H}_{g,1}[q]:=\operatorname{Ker} \eta_q^{\vphantom{}}$ as extensions of the Johnson homomorphisms and the Johnson filtration of the mapping class group $\mathcal{M}_{g,1}$ \cite{Jo}. We call the maps $\eta_q^{\vphantom{}}$ on $\mathcal{H}_{g,1}$ \emph{Garoufalidis-Levine homomorphisms}. (In some literature, those on $\mathcal{H}_{g,1}$ are referred to also as `Johnson homomorphisms.') Briefly, the invariants $\eta_q^{\vphantom{}}$ measure the difference between two markings on $F/F_q$. Garoufalidis and Levine determined the image of $\mathcal{H}_{g,1}[q]$ under~$\eta_q^{\vphantom{}}$ and showed each successive quotient $\mathcal{H}_{g,1}[q-1]/\mathcal{H}_{g,1}[q]$ is finitely generated free abelian. See also Remark~\ref{remark:GL} and the paragraph after Theorem~\ref{theorem:rank} for precise statements. We remark that the image of the Johnson subgroup of the mapping class group is unknown.
In \cite{M}, Morita obtained a homomorphism on $\mathcal{H}_{g,1}$ by taking the limit of a trace map composed with~$\eta_q^{\vphantom{}}$. He used this to show that the Torelli subgroup $\mathcal{H}_{g,1}[2]$ of $\mathcal{H}_{g,1}$ has infinite rank abelianization, while it is known that the Torelli subgroup of $\mathcal{M}_{g,1}$ is finitely generated for $g\geq 3$.
In \cite{S08, S}, Sakasai defined Magnus representations, which are crossed homomorphisms on $\mathcal{H}_{g,1}$ and homomorphisms on the subgroups~$\mathcal{H}_{g,1}[q]$. Using them, he proved that $\mathcal{M}_{g,1}$ is not a normal subgroup of $\mathcal{H}_{g,1}$ for $g\geq 3$.
Cha, Friedl, and Kim defined a torsion invariant in \cite{CFK}. They used it to show that the abelianization of $\mathcal{H}_{g,n}$ contains a direct summand isomorphic to $(\mathbb{Z}/2)^\infty$ if $b_1(\Sigma)>0$, and contains a direct summand isomorphic to $\mathbb{Z}^\infty$ if $n>1$.
In \cite{CHH}, Cochran, Harvey, and Horn considered signature invariants for $\mathcal{H}_{g,1}[2]$, which are the von Neumann-Cheeger-Gromov $L^2$-signature defects of bounding 4-manifolds. Their invariants are quasimorphisms on $\mathcal{H}_{g,1}[q]$ and send $\mathcal{H}_{g,1}[q]$ to a dense and infinitely generated subgroup of $\mathbb{R}$ for $g\geq 1$. They become homomorphisms on the kernel of Sakasai's Magnus representation on~$\mathcal{H}_{g,1}[q]$.
In fact, for $\mathcal{H}_{g,1}$, $\eta_q^{\vphantom{}}$ is related to the Milnor invariant of (string) links as described briefly below. The concordance group of $m$-component framed string links in homology balls is naturally identified, by taking the exterior, with~$\mathcal{H}_{0,m+1}$, and the total Milnor invariant of length $\leq q$ for string links can be viewed as a homomorphism $\mu_q^{\vphantom{}}$ defined on $\mathcal{H}_{0,m+1}$.
We denote its kernel by~$\mathcal{H}_{0,m+1}(q)$. Habegger established a bijection $\mathcal{H}_{0,2g+1}(2)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathcal{H}_{g,1}[2]$, which is not a homomorphism but descends to an isomorphism between $\mathcal{H}_{0,2g+1}(q-1)/\mathcal{H}_{0,2g+1}(q)$ and $\mathcal{H}_{g,1}[q-1]/\mathcal{H}_{g,1}[q]$~\cite{Ha}. Levine found a monomorphism $\mathcal{H}_{0,g+1}\hookrightarrow \mathcal{H}_{g,1}$ which induces a monomorphism of $\mathcal{H}_{0,g+1}(q-1)/\mathcal{H}_{0,g+1}(q)$ into $\mathcal{H}_{g,1}[q-1]/\mathcal{H}_{g,1}[q]$~\cite{L01}. Habegger and Levine showed that $\mu_q^{\vphantom{}}$ and $\eta_q^{\vphantom{}}$ can be identified under these maps, respectively.
\@startsection{subsection}{2{Extended Milnor invariants}
\label{sec:1.1}
We will define a new invariant $\tilde\mu_q^{\vphantom{}}$ on $\mathcal{H}_{g,n}$ for arbitrary $(g,n)$ with $n \geq 1$. This is a common generalization of the Milnor $\bar\mu$-invariant and the Garoufalidis-Levine homomorphism which are defined only when $g=0$ and $n=1$, respectively.
As in \cite{HL}, string links have the advantage that their $\bar\mu$-invariants are well-defined without indeterminancy, in contrast to links, because a string link has well-defined (zero-linking) longitudes, as elements of the fundamental group of the exterior (see Section~\ref{sec:longitude} for details).
In fact, the Milnor invariant of a string link essentially represents the longitudes as elements of the free nilpotent quotient.
We generalize this to homology cylinders over a surface $\Sigma=\Sigma_{g,n}$ as follows.
Briefly speaking, we take $n-1$ fundamental group elements for a homology cylinder as analogs of the longitudes of a string link,
and to extract more information from the fundamental group,
we consider additional $2g$ elements that arise from symplectic basis curves of the surface~$\Sigma$.
Note that $2g+n-1$, the total number of the elements we consider, is equal to the first Betti number~$b_1(\Sigma)$.
By taking the image of those elements in $F/F_q$, where $F=\pi_1(\Sigma)$,
we define an extended Milnor invariant
$$\tilde\mu_q^{\vphantom{}} \colon \mathcal{H}_{g,n} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} (F/F_q)^{2g+n-1}. $$
The precise definition is given in Section~\ref{sec:longitude}. It turns out that $\tilde\mu_q^{\vphantom{}}$ is equivalent to the Garoufalidis-Levine homomorphism $\eta_q^{\vphantom{}}$ for $n=1$, and to the Milnor $\bar\mu$-invariant for $g=0$.
We remark that the $2g+n-1$ elements used above are essential in understanding the fundamental group of the \emph{closure} of a homology cylinder (see Section~\ref{sec:definition} for the definition), from which we will extract signature defects and more generally Witt class invariants. These invariants will be discussed in the next subsection.
We define a filtration by $\mathcal{H}(q) :=\operatorname{Ker} \tilde\mu_q^{\vphantom{}}$. This generalizes the Garoufalidis-Levine's filtration $\mathcal{H}[q]$, in the sense that $\mathcal{H}(q) = \mathcal{H}[q]$ for $n=1$.
We remark that Garoufalidis-Levine's definition for $n=1$ can be applied to the case of $n>1$ to give a filtration, which we also denote by $\mathcal{H}[q]$; our filtration is finer than this, that is, we have $\mathcal{H}(q) \subset \mathcal{H}[q]$ in general.
The map $\tilde\mu_q^{\vphantom{}}$ is a crossed homomorphism. It is a homomorphism on both $\mathcal{H}[q]$ and~$\mathcal{H}(q-1)$. For more details, see Section~\ref{sec:product formula}.
Regarding the structure of the successive quotients of our filtration, we have the following result:
\begin{theorem}
For each $q \geq 2$, the following hold:
\begin{enumerate}
\item $\tilde\mu_{q}^{\vphantom{}}$ induces an injective homomorphism
$$\tilde\mu_{q}^{\vphantom{}}\colon \mathcal{H}(q-1)/\mathcal{H}(q) \lhook\joinrel\longrightarrow (F_{q-1}/F_{q})^{2g+n-1}.$$
Hence, $\mathcal{H}(q-1)/\mathcal{H}(q)$ is finitely generated free abelian.
\item We have
$$\mathrm{max}\{r_q(2g),r_q(g+n-1)\} \leq \mathrm{rank}\; \mathcal{H}(q-1)/\mathcal{H}(q) \leq r_q(2g+n-1)$$
where $N_q(m)=\frac{1}{q}\sum_{d|q} \varphi(d) (m^{q/d})$, $\varphi$ is the M\"obius function, and $r_q(m)=m N_{q-1}(m)-N_q(m)$.
\end{enumerate}
\end{theorem}
It is known that the injection in (1) is an isomorphism and the equality in (2) holds if either $g=0$, i.e.\ for string links \cite{Or}, or $n=1$~\cite{GL}.
Our next result is that a surface embedding gives rise to relationships between homology cobordism groups of homology cylinders over the surfaces and between their extended Milnor invariants as follows:
\begin{theorem}
For any embedding $\imath\colon\Sigma_{g,n} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \Sigma_{g',n'}$ with $n,n'\geq 1$, it induces a homomorphism $\tilde\imath\colon \mathcal{H}_{g,n} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathcal{H}_{g',n'}$, and a function $f\colon (F/F_q)^{2g+n-1} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} (F'/F'_q)^{2g'+n'-1}$ which make the following diagram commute, where $F=\pi_1(\Sigma_{g,n})$ and $F'=\pi_1(\Sigma_{g',n'})$:
$$ \begin{diagram}
\node{\mathcal{H}_{g,n}} \arrow{e,t}{\tilde\imath} \arrow{s,r}{\tilde\mu_q^{\vphantom{}}} \node{\mathcal{H}_{g',n'}} \arrow{s,l}{\tilde\mu_q^{\vphantom{}}} \\
\node{(F/F_q)^{2g+n-1}} \arrow{e,t}{f} \node{(F'/F'_q)^{2g'+n'-1}}
\end{diagram}$$
\end{theorem}
In addition, we present a sufficient condition for $f$ to be 1-1 and a sufficient condition for $\tilde\imath\colon \mathcal{H}_{g,n}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{H}_{g',n'}$ to be injective in Theorem~\ref{theorem:whole injection}. The former implies that the extended Milnor invariant of $\tilde\imath(M) \in \mathcal{H}_{g',n'}$ determines that of $M\in\mathcal{H}_{g,n}$.
Applying the above result to appropriate surface embeddings, we obtain the following:
\begin{corollary}
\label{cor:whole injection}
For any two pairs $(g,n)$ and $(g',n')$ satisfying $g \leq g'$ and $g+n \leq g'+n'$, there is an injective homomorphism
$$\mathcal{H}_{g,n} \lhook\joinrel\longrightarrow \mathcal{H}_{g',n'} ,$$
which induces injections
$$\mathcal{H}_{g,n}(q-1)/\mathcal{H}_{g,n}(q) \lhook\joinrel\longrightarrow \mathcal{H}_{g',n'}(q-1)/\mathcal{H}_{g',n'}(q)$$
for all $q \geq 2$.
\end{corollary}
Levine's monomorphism $\mathcal{H}_{0,g+1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathcal{H}_{g,n}$ \cite{L01} is a special case of this.
In the next subsection, we present our results on the structure of~$\mathcal{H}(\infty):=\bigcap_q \mathcal{H}(q)$.
\@startsection{subsection}{2{Hirzebruch-type intersection form defect invariants}
In \cite{C10}, Cha defined Hirzebruch-type intersection form defects for closed 3-manifolds.
In order to extract homology cobordism invariants, he considered towers of abelian $p$-covers.
Let $d$ be a power of a prime~$p$.
For a CW-complex $X$, a pair of a tower of iterated abelian $p$-covers and a homomorphism of the fundamental group of the top cover to $\mathbb{Z}_d$ is called a (\emph{$\mathbb{Z}_d$-valued}) \emph{$p$-structure} for~$X$ \cite{C09}.
(A precise definition is given in Section~\ref{sec:hirzebruch}.)
We remark that any connected $p$-cover can be obtained as the top cover of a $p$-structure.
For a closed 3-manifold $M$ and a $p$-structure $\mathcal{T}$ for $M$ such that the top cover is zero in the bordism group $\Omega_3(B\mathbb{Z}_d)$, an invariant $\lambda(M,\mathcal{T})$ is defined to be the difference between the Witt classes of the $\mathbb{Q}(\zeta_d)$-valued intersection form and the ordinary intersection form of a 4-manifold bounded by the top cover over $\mathbb{Z}_d$, where $\zeta_d=\operatorname{exp}(2\pi\sqrt{-1}/d)$. This lives in the Witt group $L^0(\mathbb{Q}(\zeta_d))$ of nonsingular hermitian forms over~$\mathbb{Q}(\zeta_d)$.
This $\lambda(-,-)$ is a homology cobordism invariant in the sense that
if $M$ and $N$ are homology cobordant, there is a 1-1 correspondence $\mathcal{T}_M \mapsto \mathcal{T}_N$ between $p$-structures for $M$ and $N$, $\lambda(M,\mathcal{T}_M)$ is defined if and only if $\lambda(N,\mathcal{T}_N)$ is, and in that case, $\lambda(M,\mathcal{T}_M)=\lambda(N,\mathcal{T}_N)$.
A map $f\colon X \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} Y$ is called a \emph{$p$-tower map} if pullback gives rise to a 1-1 correspondence
$$\Phi_f \colon \{p\textrm{-structures for } Y\} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \{p\textrm{-structures for }X\}.$$
(For a more precise description, see Section~\ref{sec:defining condition}.)
He found a sufficient condition for a map to be a $p$-tower map: for a map between CW-complexes with finite 2-skeletons, if it induces an isomorphism on the ``algebraic closures'' of their fundamental groups, then the map is a $p$-tower map \cite[Proposition~3.9]{C10}. (More information about the algebraic closure is given in Section~\ref{sec:hF-homology cylinder}.)
This applies to string links as follows.
A string link $\sigma$ has canonical meridians, which give a meridian map $m\colon\bigvee S^1 \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} M_\sigma$ where $M_\sigma$ is the surgery manifold of its closure. A string link is called an \emph{$\widehat{F}$-string link} if its meridian map induces an isomorphism on the algebraic closures of the fundamental groups. Using the above, for an $\widehat{F}$-string link $\sigma$,
he defined a concordance invariant $\lambda_\mathcal{T}(\sigma):=\lambda(M_\sigma,\Phi_m^{-1}(\mathcal{T}))$ for each $p$-structure $\mathcal{T}$ for $\bigvee S^1$. He also proved that $\lambda_\mathcal{T}$ is a group homomorphism of the concordance group of $\widehat{F}$-string links \cite{C09}.
We apply the Hirzebruch-type invariants in \cite{C10,C09} to homology cylinders. Motivated by \cite{C09}, for homology cylinders, we define invariants parametrized by the $p$-structures for the base surface. For this purpose, we investigate when the composition $\hat{i}\colon \Sigma\xrightarrow{i_+} M \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \widehat{M}$, which we call the \emph{marking} for the closure $\widehat{M}$, is a $p$-tower map. As stated in the first part of the following theorem, the criterion is exactly the vanishing of our extended Milnor invariants.
\begin{theorem}
\leavevmode \@nobreaktrue\nopagebreak
\begin{enumerate}
\item The marking $\hat{i} \colon \Sigma \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \widehat{M}$ is a $p$-tower map if and only if $\tilde\mu_q^{\vphantom{}}(M)$ vanishes for all $q$. In that case, $\lambda_\mathcal{T}(M):=\lambda(M, (\Phi_{\hat{i}})^{-1}(\mathcal{T}))$ is well-defined for any $p$-structure $\mathcal{T}$ for~$\Sigma$.
\item For any $p$-structure $\mathcal{T}$ for $\Sigma$,
$$\lambda_\mathcal{T} \colon \mathcal{H}(\infty) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} L^0(\mathbb{Q}(\zeta_d))$$
is a group homomorphism.
\end{enumerate}
\end{theorem}
Furthermore, we give a generalization as follows. We consider certain special $p$-structures, which are called \emph{$p$-structures of order $q$}. Roughly, they are $p$-structures factoring through the $q$th lower central series quotient. (See Definition~\ref{definition:of order}).
We prove that the marking $\Sigma\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{M}$ induces a 1-1 correspondence between $p$-structures of order $q$ if and only if $\tilde\mu_q^{\vphantom{}}(M)$ vanishes.
In this case, we define an invariant $\lambda_\mathcal{T}(M)$, with value in $\mathbb{Z}[\frac{1}{d}]\otimes_\mathbb{Z} L^0(\mathbb{Q}(\zeta_d))$. In other words, we define
$$\lambda_\mathcal{T} \colon \mathcal{H}(q) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z}\Big[\frac{1}{d}\Big]\otimes_\mathbb{Z} L^0(\mathbb{Q}(\zeta_d))$$ for each $p$-structure $\mathcal{T}$ for $\Sigma$ of order~$q$.
However, it may not be a homomorphism. We present sufficient conditions on homology cylinders for $\lambda_\mathcal{T}$ to be additive in Theorem~\ref{theorem:additivity condition}.
We study the structure of $\mathcal{H}(\infty)$ using $\lambda_\mathcal{T}$ in Section~\ref{sec:effect}.
Performing infection by knots, we construct infinitely many homology cylinders with vanishing extended Milnor invariants, but distinguished by $\lambda_\mathcal{T}$.
\begin{theorem}
\label{theorem}
When $b_1(\Sigma)>1$,
the abelianization of $\mathcal{H}(\infty)$ contains $\mathbb{Z}^\infty$.
\end{theorem}
We define a boundary homology cylinder and an $\widehat{F}$-homology cylinder in Section~\ref{sec:boundary homology cylinder} and~\ref{sec:hF-homology cylinder} as analogs of the boundary (string) link and the $\widehat{F}$-(string) link, respectively.
The above theorem also holds on the subgroup consisting of boundary homology cylinders and that consisting of $\widehat{F}$-homology cylinders. (See Theorem~\ref{theorem:infinite rank subgroups}.)
Moreover, our method detects homology cylinders that cannot be detected by other invariants we discussed
before Section~\ref{sec:1.1} as follows:
\begin{theorem}
Suppose $b_1(\Sigma)>1$. If $n=1$, the intersection of the kernels of Garoufalidis-Levine's homomorphisms $\eta_q$ \cite{GL}, Cha-Friedl-Kim's torsion invariant $\tau$ \cite{CFK}, the extended Milnor invariants $\tilde\mu_q$, Morita's homomorphism $\tilde{\rho}$ \cite{M}, Sakasai's Magnus representations $r_q$ \cite{S}, and Cochran-Harvey-Horn's signature invariants $\rho_q$ \cite{CHH} has infinite rank abelianization. If $n> 1$, the intersection of the kernels of $\eta_q$, $\tau$, $\tilde\mu_q$ (in this case, $\tilde{\rho}$, $r_q$, and $\rho_q$ are not defined) has infinite rank abelianization.
\end{theorem}
\@startsection{subsection}{2{Solvable filtration of homology cylinders}
In Section~\ref{sec:cobordisms}, we investigate some other cobordisms of homology cylinders.
Whitney towers and gropes play a key role in the study of topology of 4-manifolds and concordance of knots and links.
In \cite{COT}, Cochran, Orr, and Teichner introduced solvability of knots and related it to Whitney towers in 4-manifolds. Their filtrations on knots and links have been much studied as an approximation of sliceness.
To study 3-manifolds with nonempty boundary, Cha defined Whitney tower cobordism and solvable cobordism of bordered 3-manifolds~\cite{C12}. Applying his definition to homology cylinders, it is straightforward to define the notion of $(r)$-solvable homology cylinders for $r\in \frac{1}{2}\mathbb{Z}_{\geq 0}$.
We show that $\lambda_\mathcal{T}$ can be used as obstructions to the solvability of homology cylinders:
\begin{theorem}
Let $M \in \mathcal{H}(q)$ and $\mathcal{T}$ be a $p$-structure of height~$\leq h$ for $\Sigma$ of order~$q$.
If either \begin{enumerate}
\item $M$ is $(h+1)$-solvable, or
\item $M$ is $(h.5)$-solvable and satisfies one of \textnormal{(C1)--(C5)} of Theorem~\ref{theorem:additivity condition},
\end{enumerate}
then $\lambda_\mathcal{T}(M)$ vanishes.
\end{theorem}
Here the height of a $p$-structure for $X$ is the height of the tower of iterated $p$-covers, a precise definition is given at the beginning of Section~\ref{sec:hirzebruch}.
We also refine Theorem~\ref{theorem}. Let $\mathcal{F}^{\mathcal{H}(\infty)}_{(r)}$ denote the subgroup consisting of $(r)$-solvable homology cylinders in $\mathcal{H}(\infty)$ for each $r \in \frac{1}{2}\mathbb{Z}_{\geq 0}$.
\begin{theorem}
When $b_1(\Sigma) > 1$, the abelianization of ${\mathcal{F}^{\mathcal{H}(\infty)}_{(h)}}/{\mathcal{F}^{\mathcal{H}(\infty)}_{(h.5)}}$ is of infinite rank.
\end{theorem}
We remark that the analogs hold for the solvable filtration of the subgroups of boundary homology cylinders and that of $\widehat{F}$-homology cylinders. See Theorem~\ref{theorem:solvable filtration}.
The paper is organized as follows.
In Section~\ref{sec:definition}, we recall basic definitions and examples of homology cylinders and their homology cobordism groups.
In Section~\ref{sec:representation}, we define extended Milnor invariants on $\mathcal{H}_{g,n}$.
In Section~\ref{sec:subgroups}, we study the filtration $\mathcal{H}(q)$ associated to the extended Milnor invariants and subgroups consisting of boundary homology cylinders, $\widehat{F}$-homology cylinders.
In Section~\ref{sec:hirzebruch}, we define Hirzebruch-type invariants of homology cylinders and give sufficient conditions for additivity of the invariants.
In Section~\ref{sec:effect}, by investigating the effect of infection, we detect a rich structure of $\mathcal{H}$ which has not been detected previously.
Finally in Section~\ref{sec:cobordisms}, we study nilpotent cobordism and solvable filtrations of homology cylinders using our invariants.
\@startsection{subsubsection}{3*{Acknowledgements}
The author thanks her advisor Jae Choon Cha for his advice and guidance.
This research was partially supported by NRF grants 2013067043 and 2013053914.
\@startsection{section}{1{Homology cylinders and their homology cobordism groups}
\label{sec:definition}
We recall precise definitions about homology cylinders. Let $\Sigma=\Sigma_{g,n}$ be a surface of $g$ genus with $n$ boundary components.
\begin{definition}
A \emph{homology cylinder over} $\Sigma$ consists of a 3-manifold $M$ with two embeddings $i_+^{\vphantom{}},~i_-^{\vphantom{}}\colon \Sigma \hookrightarrow \partial M$, called \emph{markings}, such that
\begin{enumerate}
\item $i_+^{\vphantom{}}|_{\partial \Sigma} = i_-^{\vphantom{}}|_{\partial \Sigma}$,
\item $i_+\cup i_- \colon \Sigma\cup_\partial (-\Sigma) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \partial M $ is an orientation-preserving homeomorphism, and
\item $i_+^{\vphantom{}}, i_-^{\vphantom{}}$ induce isomorphisms $H_*(\Sigma;\mathbb{Z})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_*(M;\mathbb{Z})$.
\end{enumerate}
We denote a homology cylinder by $(M,i_+^{\vphantom{}},i_-^{\vphantom{}})$ or simply by~$M$.
\end{definition}
Two homology cylinders $(M,i_+^{\vphantom{}},i_-^{\vphantom{}})$ and $(N,j_+^{\vphantom{}},j_-^{\vphantom{}})$ over $\Sigma_{g,n}$ are said to be \emph{isomorphic} if there exists an orientation-preserving homeomorphism $f\colon M \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} N$ satisfying $j_+^{\vphantom{}}=f \circ i_+^{\vphantom{}}$ and $j_-^{\vphantom{}}=f \circ i_-^{\vphantom{}}$. Denote by $\mathcal{C}_{g,n}$ the set of all isomorphism classes of homology cylinders over~$\Sigma_{g,n}$. We define a product operation on $\mathcal{C}_{g,n}$ by
$$(M,i_+^{\vphantom{}},i_-^{\vphantom{}})\cdot (N,j_+^{\vphantom{}},j_-^{\vphantom{}}):=(M\cup_{i_-^{\vphantom{}}\circ (j_+^{\vphantom{}})^{-1}} N, i_+^{\vphantom{}}, j_-^{\vphantom{}})$$
for $(M,i_+^{\vphantom{}},i_-^{\vphantom{}}),~(N,j_+^{\vphantom{}},j_-^{\vphantom{}}) \in \mathcal{C}_{g,n}$, which endows $\mathcal{C}_{g,n}$ with a monoid structure. The identity is $(\Sigma_{g,n} \times I/(z,0)=(z,t)~ (z\in\partial\Sigma, t\in I), \mathrm{id} \times 1, \mathrm{id}\times 0)$. For later use, we denote this trivial homology cylinder by~$E$.
\begin{definition}
Two homology cylinders $(M,i_+^{\vphantom{}},i_-^{\vphantom{}})$ and $(N,j_+^{\vphantom{}},j_-^{\vphantom{}})$ over $\Sigma_{g,n}$ are said to be \emph{homology cobordant} if there exists a 4-manifold $W$ such that
\begin{enumerate}
\item $\partial W = M \cup (-N) /\sim$, where $\sim$ identifies $i_+^{\vphantom{}}(x)$ with $j_+^{\vphantom{}}(x)$ and $i_-^{\vphantom{}}(x)$ with $j_-^{\vphantom{}}(x)$ for all $x\in\Sigma_{g,n}$, and
\item the inclusions $M \hookrightarrow W$, $N \hookrightarrow W$ induce isomorphisms on the integral homology.
\end{enumerate}
\end{definition}
We denote by $\mathcal{H}_{g,n}$ the set of homology cobordism classes of elements of~$\mathcal{C}_{g,n}$. By abuse of notation, we also write $M$ for the class of~$M$. The monoid structure on $\mathcal{C}_{g,n}$ descends to a group structure on~$\mathcal{H}_{g,n}$, with $(M,i_+^{\vphantom{}},i_-^{\vphantom{}})^{-1}=(-M,i_-^{\vphantom{}},i_+)$. We call this group the \emph{homology cobordism group} of homology cylinders. Actually, there are two kinds of groups $\mathcal{H}_{g,n}^{\mathrm{smooth}}$ and $\mathcal{H}_{g,n}^{\mathrm{top}}$ depending on whether the homology cobordism is smooth or topological, and there exists a canonical epimorphism $\mathcal{H}_{g,n}^{\mathrm{smooth}} \twoheadrightarrow \mathcal{H}_{g,n}^{\mathrm{top}}$ whose kernel contains an abelian group of infinite rank~\cite{CFK}. In this paper, however, the author does not distinguish the two cases since everything holds in both cases.
Both $\mathcal{H}_{0,0}$ and $\mathcal{H}_{0,1}$ are isomorphic to the group of homology cobordism classes of integral homology 3-spheres. The group $\mathcal{H}_{0,2}$ is isomorphic to the concordance group of framed knots in homology 3-spheres. For $n\geq 3$, $\mathcal{H}_{0,n}$ is isomorphic to the concordance group of framed ($n-1$)-component string links in homology 3-balls, or equivalently, in homology cylinders over $D^2=\Sigma_{0,1}$. Similarly, $\mathcal{H}_{g,n}$ can be considered to be the concordance group of framed ($n-1$)-component string links in homology cylinders over~$\Sigma_{g,1}$. The fact that the mapping class group over $\Sigma_{g,n}$ is a subgroup of $\mathcal{H}_{g,n}$ implies $\mathcal{H}_{g,n}$ is non-abelian except $(g,n)=(0,0), (0,1)$ and~$(0,2)$.
For any homology cylinder $M$, there is an associated closed manifold $\widehat{M}$ obtained from $M$ by identifying $i_+^{\vphantom{}}(z)$ and~$i_-^{\vphantom{}}(z)$ for each $z\in \Sigma$. We call it the \emph{closure} of~$M$. When $M$ is considered as an exterior of a framed string link, $\widehat{M}$ is just the surgery manifold of the closure of the string link. Both $i_+^{\vphantom{}}, i_-^{\vphantom{}}$ composed with the quotient map give an embedding $\hat{i}\colon \Sigma\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{M}$, which we call the \emph{marking} for~$\widehat{M}$.
\@startsection{section}{1{Generalization of Milnor invariants and Garoufalidis-Levine homomorphisms}
\label{sec:representation}
Let $\partial_1^{\vphantom{}},\partial_2^{\vphantom{}},\ldots,\partial_n^{\vphantom{}}$ be the boundary components of~$\Sigma$. Choose a basepoint $*$ of $\Sigma$ on $\partial_n^{\vphantom{}}$ and fix a generating set $\{x_1^{\vphantom{}},\ldots,x_{n-1}^{\vphantom{}},m_1^{\vphantom{}},\ldots,m_g^{\vphantom{}},l_1^{\vphantom{}},\ldots,l_g^{\vphantom{}}\}$ for $\pi_1(\Sigma,*)$ as in Figure~\ref{figure:Sigma} such that $x_i^{\vphantom{}}$ is homotopic to the $i$th boundary component $\partial_i^{\vphantom{}}$ and $m_j^{\vphantom{}}$, $l_j^{\vphantom{}}$ correspond to a meridian and a longitude of the $j$th handle. Since our $n$ is nonzero, the group is free on the above $2g + n -1$ generators. Let $F=\pi_1(\Sigma,*).$ For the generators in Figure~\ref{figure:Sigma}, the element $[\partial_n^{\vphantom{}}]\in\pi_1(\Sigma,*)$ is represented by $\prod_i x_i^{\vphantom{}}\prod_j[m_j^{\vphantom{}},l_j^{\vphantom{}}]$. We will use this later, to prove Theorem~\ref{theorem:rank}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=.9]{Sigma_gn.pdf}
\caption{A generating set for $\pi_1(\Sigma_{g,n})$}
\label{figure:Sigma}
\end{center}
\end{figure}
\@startsection{subsection}{2{Extended Milnor invariants on $\mathcal{H}_{g,n}$}
\label{sec:longitude}
First, we define Milnor invariants of homology cylinders similarly to those of string links. Let $(M, i_+^{\vphantom{}},i_-^{\vphantom{}})$ be a homology cylinder over~$\Sigma$. The chosen $x_i^{\vphantom{}}$ is of the form $[\alpha_i^{\vphantom{}}\cdot\beta_i^{\vphantom{}}\cdot \alpha_i^{-1}]$ for a closed path $\beta_i^{\vphantom{}}\colon I \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} I/\partial I\xrightarrow{\simeq} \partial_i^{\vphantom{}}$ such that the latter map is a homeomorphism and a path $\alpha_i^{\vphantom{}}$ from $*$ to $\beta_i^{\vphantom{}}(0)$. The orientation of $\beta_i^{\vphantom{}}$ is determined by $x_i^{\vphantom{}}$.
Consider the loop $(i_+^{\vphantom{}}\circ\alpha_i^{\vphantom{}})\cdot(i_-^{\vphantom{}}\circ\alpha_i^{-1})$ in $M$. If $M$ were a framed string link exterior, this loop would be its $i$th longitude. We define $\lambda_i^{\vphantom{}}$ to be the class of the loop $(i_+^{\vphantom{}}\circ\alpha_i^{\vphantom{}})\cdot(i_-^{\vphantom{}}\circ\alpha_i^{-1})$ in $\pi_1(M,i_+^{\vphantom{}}(*))$. It is independent of the choice of $\alpha_i^{\vphantom{}}$, and it depends only on the choice of $x_i^{\vphantom{}}$ in~$\pi_1(\Sigma,*)$. We will show this at the end of this subsection. By Stallings' theorem~\cite{St}, $i_+^{\vphantom{}}$ induces an isomorphism
$$F/F_q=\pi_1(\Sigma) / \pi_1(\Sigma)_q \xrightarrow[(i_+^{\vphantom{}})_{*q}^{\vphantom{}}]{\cong} \pi_1(M) / \pi_1(M)_q$$ for every $q\in\mathbb{N}$. We define $\mu_q^{\vphantom{}}(M)_i^{\vphantom{}}$ to be the inverse image of $\lambda_i^{\vphantom{}}$ in $F/F_q$ and $\mu_q^{\vphantom{}}(M)$ to be the ($n-1$)-tuple $(\mu_q^{\vphantom{}}(M)_1^{\vphantom{}},\ldots,\mu_q^{\vphantom{}}(M)_{n-1}^{\vphantom{}})$ $\in$ $(F/F_q)^{n-1}$. Also, $\mu(M)$ can be defined to be $((\mu_q^{\vphantom{}}(M)_1^{\vphantom{}})_{q\in\mathbb{N}}^{\vphantom{}},\ldots, (\mu_q^{\vphantom{}}(M)_{n-1}^{\vphantom{}})_{q\in\mathbb{N}}^{\vphantom{}})$ as an element of~$(\varprojlim_q F/F_q)^{n-1}$.
For $(i_\pm^{\vphantom{}})_*\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1(M)$ induced by markings $i_\pm^{\vphantom{}}$, the element $\lambda_i^{\vphantom{}}$ indicates the difference between $(i_+^{\vphantom{}})_*(x_i^{\vphantom{}})$ and $(i_-^{\vphantom{}})_*(x_i^{\vphantom{}})$ in $\pi_1(M)$ as follows:
$$ (i_-^{\vphantom{}})_*(x_i^{\vphantom{}}) = \lambda_i^{-1}\cdot (i_+^{\vphantom{}})_*(x_i^{\vphantom{}}) \cdot \lambda_i^{\vphantom{}}.$$
Also $\mu_q^{\vphantom{}}(M)_i^{\vphantom{}}$ indicates the difference of two markings on $x_i^{\vphantom{}}$ in~$F/F_q$:
\begin{equation}
((i_+^{\vphantom{}})_{*q}^{-1}\circ(i_-^{\vphantom{}})_{*q}^{\vphantom{}})(x_i^{\vphantom{}})=\mu_q^{\vphantom{}}(M)_i^{-1}\cdot x_i^{\vphantom{}}\cdot \mu_q^{\vphantom{}}(M)_i^{\vphantom{}}.
\end{equation}
For the case of $g=0$, the invariant $\mu_q^{\vphantom{}}$ is equivalent to all the Milnor $\bar\mu$-invariants of length $\leq q$ for framed string links.
Now we extend this for the remaining generators $m_j^{\vphantom{}}$ and $l_j^{\vphantom{}}$ of~$F$. Denote $\lambda'_j=(i_+^{\vphantom{}})_*(m_j^{\vphantom{}})\cdot (i_-^{\vphantom{}})_*(m_j^{-1})$ and $\lambda''_j=(i_+^{\vphantom{}})_*(l_j^{\vphantom{}})\cdot (i_-^{\vphantom{}})_*(l_j^{-1})$. Let $\mu'_q(M)_j^{\vphantom{}}$ and $\mu''_q(M)_j^{\vphantom{}}$ be the inverse images of $\lambda'_j$ and $\lambda''_j$ under the above isomorphism $(i_+^{\vphantom{}})_{*q}^{\vphantom{}}$, respectively, for $j=1,\ldots,g.$ Then, clearly
$$(i_-^{\vphantom{}})_*(m_j^{\vphantom{}})=\lambda_j'^{-1} \cdot (i_+^{\vphantom{}})_*(m_j^{\vphantom{}})\;, \quad (i_-^{\vphantom{}})_*(l_j^{\vphantom{}})=\lambda_j''^{-1}\cdot (i_+^{\vphantom{}})_*(l_j^{\vphantom{}})\quad \textrm{ in } \pi_1(M),$$
and hence
\begin{align}
((i_+^{\vphantom{}})_{*q}^{-1}\circ(i_-^{\vphantom{}})_{*q}^{\vphantom{}})(m_j^{\vphantom{}})&=\mu_q'(M)_j^{-1}\cdot m_j^{\vphantom{}} , \\ ((i_+^{\vphantom{}})_{*q}^{-1}\circ(i_-^{\vphantom{}})_{*q}^{\vphantom{}})(l_j^{\vphantom{}})&=\mu_q''(M)_j^{-1}\cdot l_j^{\vphantom{}}\quad\textrm{ in } F/F_q . \nonumber
\end{align}
We denote by $\tilde\lambda(M)$ or simply by $\tilde{\lambda}$ the ($2g+n-1$)-tuple of $\lambda_i^{\vphantom{}}, \lambda_j'$, and $\lambda_j''$ of $\pi_1(M)$ and by $\tilde\mu_q^{\vphantom{}}(M)$ the ($2g+n-1$)-tuple of $\mu_q^{\vphantom{}}(M)_i^{\vphantom{}},\mu_q'(M)_j^{\vphantom{}}$, and $\mu''_q(M)_j^{\vphantom{}}$ of~$F/F_q$. the total $\tilde\mu(M)$ is defined to be $((\tilde\mu_q^{\vphantom{}}(M)_1^{\vphantom{}})_{q\in\mathbb{N}}^{\vphantom{}},\ldots, (\tilde\mu_q^{\vphantom{}}(M)_{2g+n-1}^{\vphantom{}})_{q\in\mathbb{N}}^{\vphantom{}})$ as an element of~$(\varprojlim_q F/F_q)^{2g+n-1}$.
Remark that $\tilde\lambda(M)$ plays an important role in studying the fundamental group of~$\widehat{M}$.
More precisely, $\pi_1(\widehat{M})$ can be obtained from $\pi_1(M)$ by killing all~$\tilde\lambda(M)_k^{\vphantom{}}$. Hence, $\tilde\mu_q^{\vphantom{}}(M)$ vanishes if and only if $\hat{i}$ induces an isomorphism $F/F_q\cong \pi_1(\widehat{M})/\pi_1(\widehat{M})_q$. It is similar to case of (string) links: for a (string) link $L$, the Milnor invariants of length$\leq q$ vanish if and only if a meridian map $\bigvee S^1\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} M_L$ induces an isomorphism on $\pi_1(-)/\pi_1(-)_q$ where $M_L$ is the surgery manifold (of the closure).
In this sense, $\tilde\mu$ is a more appropriate analog of Milnor's $\bar\mu$-invariants of string links, compared with~$\mu$.
\begin{theorem}
For any $q$, $\tilde\mu_q^{\vphantom{}}$ is a homology cobordism invariant.
In other words, we have
$$\tilde\mu_q^{\vphantom{}}\colon \mathcal{H}_{g,n}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} (F/F_q)^{2g+n-1}.$$
\end{theorem}
\begin{proof}
Suppose $W$ is a homology cobordism between homology cylinders $(M, i_+^{\vphantom{}}, i_-^{\vphantom{}})$ and~$(M',i_+',i_-')$.
The diagram below is commutative where the right maps are induced by natural inclusions. In $W$, $i_+^{\vphantom{}}(z)$ and $i_+'(z)$ are identified, and $i_-^{\vphantom{}}(z)$ and $i_-'(z)$ are identified for each $z\in \Sigma$. Hence $\tilde\lambda(M)_k^{\vphantom{}} \in \pi_1(M)$ and $\tilde\lambda(M')_k^{\vphantom{}} \in\pi_1(M')$ correspond to the same element of~$\pi_1(W)$.
$$\begin{diagram} \dgARROWLENGTH=1.5em
\node[2]{\pi_1(M)} \arrow{se} \\
\node{F} \arrow{ne,l}{(i_+^{\vphantom{}})_*} \arrow{se,r}{(i_+')_*} \node[2]{\pi_1(W)} \\
\node[2]{\pi_1(M')} \arrow{ne}
\end{diagram}$$
Since all the homomorphisms of the diagram induce isomorphisms on~$\pi_1(-)/\pi_1(-)_q$, $\tilde\mu_q^{\vphantom{}}(M)= \tilde\mu_q^{\vphantom{}}(M')$ in $F/F_q$.
\end{proof}
Now, we investigate the well-definedness of $\tilde\mu_q^{\vphantom{}}$ and the effect of change of generating sets.
\begin{proposition}
The invariant $\tilde\mu_q^{\vphantom{}}$ is independent of the the choice of $\{\alpha_i^{\vphantom{}}\}$ and depends only on (the basepoint $*$ of $\Sigma$ and) the (ordered) generating set $\{x_i^{\vphantom{}},m_j^{\vphantom{}},l_j\}_{i<n,j\leq g}$ of~$\pi_1(\Sigma,*)$.
\end{proposition}
\begin{proof}
Let $x_i^{\vphantom{}}=[\alpha_i^{\vphantom{}}\cdot \beta_i^{\vphantom{}} \cdot \alpha_i^{-1}] = [\alpha_i' \cdot \beta_i' \cdot \alpha_i'^{-1}]$ for paths $\beta_i^{\vphantom{}},\beta_i'^{\vphantom{}}\colon I\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} I/\partial I \xrightarrow{\simeq} \partial_i^{\vphantom{}}$ and $\alpha_i^{\vphantom{}}, \alpha_i'$ from $\ast$ to $\beta_i^{\vphantom{}}(0), \beta_i'(0)$, respectively.
We may assume $\beta_i^{\vphantom{}}=\beta_i'$ since $\lambda(-)_i^{\vphantom{}}$ is unchanged under connecting a path in $\partial_i^{\vphantom{}}$ to~$\alpha_i^{\vphantom{}}$.
The loop $(\alpha_i^{\vphantom{}}\cdot\beta_i^{\vphantom{}}\cdot\alpha_i^{-1})
(\alpha_i'\cdot\beta_i^{-1}\cdot\alpha_i'^{-1})$ is a null-homotopic, and $\big[[\alpha_i'^{-1}\cdot\alpha_i^{\vphantom{}}], [\beta_i^{\vphantom{}}]\big] = 1 $ in the free group $\pi_1(\Sigma,\beta_i^{\vphantom{}}(0))$. Thus $[\alpha_i'^{-1}\cdot\alpha_i^{\vphantom{}}]=[\beta_i^{\vphantom{}}]^k$ for some~$k$.
Therefore $\alpha_i^{\vphantom{}}$ and $\alpha_i'$ determine the same $\lambda(M)_i^{\vphantom{}} \in \pi_1(M)$ for each homology cylinder~$M$.
\end{proof}
\begin{proposition}
\label{proposition:effect of change}
Let $A=\{x_i^{\vphantom{}},m_j^{\vphantom{}},l_j^{\vphantom{}}\}_{i<n,j\leq g}$ and $B=\{x_i',m_j',l_j'\}_{i<n,j\leq g}$ be generating sets of $\pi_1(\Sigma,\ast)=:F$ and $\pi_1(\Sigma,\ast')=:F'$, respectively, such that $x_i^{\vphantom{}}$ and $x_i'$ are homotopic to a boundary component. Suppose $\tilde\mu_q^A ,\tilde\mu_q^B\colon \mathcal{H}_{g,n}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} (F/F_q)^{2g+n-1}$ are the extended Milnor invariants with respect to $A$ and~$B$, respectively.
Then there exists a bijection $f\colon (F/F_q)^{2g+n-1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} (F/F_q)^{2g+n-1}$ which makes the following diagram commute:
$$\begin{diagram}
\node[2]{\mathcal{H}_{g,n}} \arrow{sw,l}{\tilde\mu_q^A} \arrow{se,l}{\tilde\mu_q^B} \\
\node{(F/F_q)^{2g+n-1}} \arrow[2]{e,b}{f} \node[2]{(F'/F'_q)^{2g+n-1}}
\end{diagram}$$
\end{proposition}
\begin{proof}
Let $$z=(z_1^{\vphantom{}},\ldots,z_{n-1}^{\vphantom{}};z_1',\ldots,z_g';z_1'',\ldots,z_g'') \in (F/F_q)^{2g+n-1}.$$
We can assume $\ast=\ast'$ by the following claim: \\
\textbf{Claim.} Suppose $\ast\neq\ast'$ and $\{x_i^{\vphantom{}}, m_j^{\vphantom{}}, l_j^{\vphantom{}}\}$ be a generating set of~$\pi_1(\Sigma,*)$. Then there are a generating set $\{x_i',m_i',l_j'\}$ of $\pi_1(\Sigma,*')$ and a bijection $f$ making the above diagram commute.
To prove the claim, we consider two cases:
\begin{enumerate}
\item[Case 1.] $*$ and $*'$ are in the same boundary component. \\
Choose a path $\gamma$ from $\ast'$ to $\ast$ in the boundary component, and let $\{x_i',m_j',l_j'\}$ be the generating set of $\pi_1(\Sigma,\ast')$ induced by $\gamma$-conjugation. Then $\tilde\mu_q^{\vphantom{}}$ is unchanged, i.e.\ $f=\mathrm{id}$.
\item[Case 2.] $\ast$ and $\ast'$ are in different components. \\
We may assume $\ast'\in\partial_1$. Then $x_1=[\alpha_1^{\vphantom{}}\cdot\beta_1^{\vphantom{}}\cdot\alpha_1^{-1}]$ for some paths $\alpha_1^{\vphantom{}}$ from $\ast$ to $\ast'$ and $\beta_1^{\vphantom{}}\colon I\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} I/\partial I\xrightarrow{\simeq}\partial_1^{\vphantom{}}$ with $\beta_1^{\vphantom{}}(0)=\ast'$. Let $\gamma$ be~$\alpha_1^{-1}$. For the isomorphism $\phi\colon \pi_1(\Sigma,*)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1(\Sigma,*')$ induced by $\gamma$-conjugation, let $x_1'=\phi([\partial_n^{\vphantom{}}])$, $x_i'=\phi(x_i^{\vphantom{}})$ for $i=2,\ldots, n-1$ and $m_j'=\phi(m_j^{\vphantom{}})$, $l_j'=\phi(l_j^{\vphantom{}})$ for $j=1,\ldots,g$.
Define a function $f \colon (F/F_q)^{2g+n-1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} (F'/F'_q)^{2g+n-1}$ by
$$f(z)=\phi\big(z_1^{\vphantom{}},z_2^{\vphantom{}}z_1^{-1},\ldots,z_{n-1}^{\vphantom{}}z_1^{-1}; (m_j^{\vphantom{}}z_1^{\vphantom{}}m_j^{-1}z_j' z_1^{-1})_{j\leq g}; (l_j^{\vphantom{}}z_1^{\vphantom{}}l_j^{-1}z_j''z_1^{-1})_{j\leq g}\big),$$
then it gives the commutative diagram.
\end{enumerate}
Now we prove the proposition with the same basepoint~$*=*'$ and $F=F'$.
By reordering, we may assume that $x_i^{\vphantom{}}$ is homotopic to~$(x_i')^{\pm 1}$.
We can choose paths $\alpha_i^{\vphantom{}}$ and $\alpha_i'$ with the same endpoints such that $x_i^{\vphantom{}}=[\alpha_i^{\vphantom{}}\cdot \beta_i^{\vphantom{}}\cdot \alpha_i^{-1}]$ and $x_i'=[\alpha_i'\cdot \beta_i^{\pm} \cdot\alpha_i'^{-1}]$ for $\beta_i^{\vphantom{}}\colon I\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} I/\partial I \xrightarrow{\simeq}\partial_i^{\vphantom{}}$.
Let $\gamma_i^{\vphantom{}}=[\alpha_i^{\vphantom{}}\cdot \alpha_i'^{-1}]$, $\gamma_j'=m_j^{\vphantom{}} m_j'^{-1}$ and $\gamma_j''=l_j^{\vphantom{}} l_j'^{-1}$,
and let
\begin{align*}
\gamma_i^{\vphantom{}}&= \omega_i^{\vphantom{}}(x_1^{\vphantom{}},\ldots,x_{n-1}^{\vphantom{}};m_1^{\vphantom{}},\ldots,m_g^{\vphantom{}};l_1^{\vphantom{}},\ldots,l_g^{\vphantom{}}),\\
\gamma'_j&=\omega_j'(x_1^{\vphantom{}},\ldots,x_{n-1}^{\vphantom{}};m_1^{\vphantom{}},\ldots,m_g^{\vphantom{}};l_1^{\vphantom{}},\ldots,l_g^{\vphantom{}}), \\
\gamma''_j&=\omega_j''(x_1^{\vphantom{}},\ldots,x_{n-1}^{\vphantom{}};m_1^{\vphantom{}},\ldots,m_g^{\vphantom{}};l_1^{\vphantom{}},\ldots,l_g^{\vphantom{}})
\end{align*}
be the words in $x_i^{\vphantom{}},m_j^{\vphantom{}},l_j^{\vphantom{}}~(i<n,j\leq g)$.
Suppose $\varphi\colon (F/F_q)^{2g+n-1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} (F/F_q)^{2g+n-1}$ is a function defined by
$$ z_i^{\vphantom{}}\mapsto z_i^{-1} x_i^{\vphantom{}} z_i^{\vphantom{}};\quad
z_j' \mapsto z_j'^{-1} m_j^{\vphantom{}} ;\quad
z_j''\mapsto z_j''^{-1} l_j^{\vphantom{}}.$$
Define $f\colon (F/F_q)^{2g+n-1} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} (F/F_q)^{2g+n-1}$ by
$$f(z)=\Big(\big(\gamma_i^{-1} z_i^{\vphantom{}} \omega_i^{\vphantom{}}(\varphi(z)))\big)_{i< n};\big(\gamma_j'^{-1} z_j' \omega_j'(\varphi(z))\big)_{j\leq g}; \big(\gamma_j''^{-1} z_j'' \omega_j''(\varphi(z))\big)_{j\leq g}\Big).$$
The verification that the diagram commutes is left to the reader.
A similar construction gives the inverse of $f$. It follows that $f$ is bijective.
\end{proof}
Therefore, $\tilde\mu^A_q(M)$ determines $\tilde\mu^B_q(M)$ and vice versa.
\@startsection{subsection}{2{Garoufalidis-Levine homomorphisms on $\mathcal{H}_{g,n}$}
Garoufalidis and Levine defined the homomorphism $\eta_q^{\vphantom{}} \colon \mathcal{H}_{g,1} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \operatorname{Aut}(F/F_q)$ to be $(i_+^{\vphantom{}})_{*q}^{-1}\circ (i_-^{\vphantom{}})_{*q}^{\vphantom{}}$ where $(i_\pm^{\vphantom{}})_{*q}\colon F/F_q\xrightarrow{\cong} \pi_1(M)/\pi_1(M)_q$ as before. In the same way, a homomorphism $\eta_q^{\vphantom{}} \colon \mathcal{H}_{g,n} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \operatorname{Aut}(F/F_q)$ can be defined.
We compare $\eta_q^{\vphantom{}}$ and $\tilde\mu_q^{\vphantom{}}$.
From (1) and (2) in Section~\ref{sec:longitude},
\begin{equation*}\eta_q^{\vphantom{}}(-)(x_i^{\vphantom{}})=\mu_{q-1}^{\vphantom{}}(-)_i^{-1} x_i^{\vphantom{}} \mu_{q-1}^{\vphantom{}}(-)_i^{\vphantom{}}, \end{equation*}
$$\eta_q^{\vphantom{}}(-)(m_j^{\vphantom{}}) = \mu_q'(-)_j^{-1}m_j^{\vphantom{}}\quad\textrm{and}\quad\eta_q^{\vphantom{}}(-)(l_j^{\vphantom{}}) = \mu_q''(-)_j^{-1}l_j^{\vphantom{}}.$$
Hence, $(\mu_{q-1}^{\vphantom{}}, \mu_q', \mu_q'')$ determines~$\eta_q^{\vphantom{}}$, but the converse does not hold. For example, $\mu_{q-1}^{\vphantom{}}(M)_i^{\vphantom{}}$ can be the class of~$x_i^k$ even though $\eta_q^{\vphantom{}}(M)=\mathrm{id}$. In $\mathcal{H}_{g,1}$ or in $\operatorname{Ker}\mu_2^{\vphantom{}}\subset \mathcal{H}_{g,n}$, the triple $(\mu_{q-1}^{\vphantom{}}, \mu_q',\mu_q'')$ is equivalent to $\eta_q^{\vphantom{}}$, by the following lemma:
\begin{lemma}
\label{lemma:MKS}
For a homology cylinder $M$,
if $\mu_2^{\vphantom{}}(M)_i^{\vphantom{}}=1$ and $[x_i^{\vphantom{}},\mu_{q-1}^{\vphantom{}}(M)_i^{\vphantom{}}]=1$ in $F/F_q$, then $\mu_{q-1}^{\vphantom{}}(M)_i^{\vphantom{}}=1$.
\end{lemma}
\begin{proof}
We will prove that if $a\in F_2$ and $[x_i^{\vphantom{}},a]\in F_q$ then $a\in F_{q-1}$.
Consider the Magnus expansion of $F$ into the algebra of formal power series in noncommutative $2g+n-1$ variables $X_1,\ldots, X_{2g+n-1}$
$$\mathcal{M}\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} Z[[X_1,\ldots,X_{2g+n-1}]]$$
which sends the $k$-th generator of $F$ to $1+X_k$. We order the generators so that $\mathcal{M}(x_i^{\vphantom{}})=1+X_i$.
It is well-known that $\mathcal{M}(a)-1$ is a sum of monomials of degree $\geq q-1$ if and only if $a\in F_{q-1}$ (see Section~5 in~\cite{MKS}). Let $\mathcal{M}(a)=1+\sum_{k\geq s}^{\infty} h_k$ for monomials $h_k$ of degree~$k$.
$$\mathcal{M}\big([x_i^{\vphantom{}},a]\big) = 1+(X_i h_s-h_s X_i) + \sum(\textrm{monomials of degree}> s+1).$$
Since $[x_i^{\vphantom{}},a]\in F_q$, $\deg({X_i h_s-h_s X_i })=s+1\geq q$ and $s\geq q-1$. Thus, $a\in F_{q-1}$.
\end{proof}
On $\mathcal{H}_{g,n}$, $\tilde\mu_q^{\vphantom{}}$ is equivalent to $(\eta_q^{\vphantom{}},\mu_q^{\vphantom{}})$, namely, the invariant $\tilde\mu_q^{\vphantom{}}$ can be thought just as a combination of $\mu_q^{\vphantom{}}$ and~$\eta_q^{\vphantom{}}$.
We consider the image of $\eta_q^{\vphantom{}}$. For $g\geq0$ and $n\geq 1$, let
{\setlength\arraycolsep{1pt}
\begin{align*}
\operatorname{Aut}_2(F/F_q):=&\{ \phi \in \operatorname{Aut}(F/F_q)~|~\phi(x_i^{\vphantom{}})=\bar{\mu}_i^{-1}x_i^{\vphantom{}} \bar{\mu}_i^{\vphantom{}} \textrm{ for some }\bar{\mu}_i^{\vphantom{}} \in F/F_{q-1} \\
&\textrm{ and }\textrm{there is a lift }F/F_{q+1}\xrightarrow{\tilde\phi} F/F_{q+1} \textrm{ such that }\tilde\phi([\partial_n^{\vphantom{}}])=[\partial_n^{\vphantom{}}] \} .
\end{align*}
\begin{proposition}
Let $M$ be a homology cylinder. Then $\eta_q^{\vphantom{}}(M) \in \operatorname{Aut}_2(F/F_q)$.
\end{proposition}
\begin{proof}
Since $\eta_{q+1}^{\vphantom{}}(M)$ is a lift of $\eta_q^{\vphantom{}}(M)$ on $F/F_{q+1}$, it follows from $\eta_{q+1}^{\vphantom{}}(M)([\partial_n^{\vphantom{}}])=[\partial_n^{\vphantom{}}]$ and (3).
\end{proof}
\begin{remark}
\label{remark:GL}
When $n = 1$ or $g = 0$, the image of $\eta_q^{\vphantom{}}$ is known:
\begin{enumerate}
\item In \cite{GL}, it was shown that $\eta_q^{\vphantom{}} \colon \mathcal{H}_{g,1} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \operatorname{Aut}_0(F/F_q)$ is surjective where
\begin{align*}
\operatorname{Aut}_0(F/F_q):=\{ \phi \in \operatorname{Aut}(F/F_q)~|~&\textrm{there is a lift }F/F_{q+1}\xrightarrow{\tilde\phi} F/F_{q+1} \\ &\textrm{such that }\tilde\phi([\partial_n^{\vphantom{}}])=[\partial_n^{\vphantom{}}] \} .
\end{align*}
\item In \cite{HL98}, it was shown that $\eta_q^{\vphantom{}} \colon \mathcal{H}_{0,n} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \operatorname{Aut}_1(F/F_q)$ is surjective where
\begin{align*}
\operatorname{Aut}_1(F/F_q):=\{ \phi \in \operatorname{Aut}(F/F_q)~|~&\phi(x_i^{\vphantom{}})=\bar{\mu}_i^{-1}x_i^{\vphantom{}} \bar{\mu}_i^{\vphantom{}}
\textrm{ for some }\bar{\mu}_i^{\vphantom{}} \in F/F_{q-1} \\
&\textrm{ and }\phi(x_1^{\vphantom{}}\cdots x_{n-1}^{\vphantom{}})=x_1^{\vphantom{}}\cdots x_{n-1}^{\vphantom{}} \} .
\end{align*}
\end{enumerate}
\end{remark}
It remains an open question whether $\eta_q^{\vphantom{}} \colon \mathcal{H}_{g,n} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \operatorname{Aut}_2(F/F_q)$ is surjective.
\@startsection{subsection}{2{Crossed homomorphisms}
\label{sec:product formula}
We remind the reader that $\tilde\mu_q^{\vphantom{}}\colon \mathcal{H}_{g,n}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} (F/F_q)^{2g+n-1}$ is not a homomorphism, although $\eta_q^{\vphantom{}}\colon \mathcal{H}_{g,n}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \operatorname{Aut}(F/F_q)$ is a homomorphism. However, we have a product formula as follows:
\begin{proposition}
\label{proposition:product formula}
Let $M$ and $N$ be homology cylinders over $\Sigma$. Then, $\tilde\mu_q^{\vphantom{}}$ is a crossed homomorphism on $\mathcal{H}_{g,n}$ in the sense that each coordinate $\tilde\mu_q^{\vphantom{}}(-)_k^{\vphantom{}}$ satisfies
$$\tilde\mu_q^{\vphantom{}}(M\cdot N)_k^{\vphantom{}}=\tilde\mu_q^{\vphantom{}}(M)_k^{\vphantom{}}\cdot \eta_q^{\vphantom{}}(M)(\tilde\mu_q^{\vphantom{}}(N)_k^{\vphantom{}})$$
for $k=1,\ldots, 2g+n-1$.
\end{proposition}
\begin{proof}
Let $\imath_M\colon M\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} M\cdot N $ and $\imath_N \colon N\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} M\cdot N$ be the natural inclusions.
Then $$(M, i_+^{\vphantom{}},i_-^{\vphantom{}})\cdot (N,j_+^{\vphantom{}},j_-^{\vphantom{}})=(M\cup_{i_-\circ (j_+)^{-1}} N,\imath_M\circ i_+^{\vphantom{}},\imath_N \circ j_-^{\vphantom{}})\quad \textrm{and} $$
$$\tilde\lambda(M\cdot N)_k^{\vphantom{}} = (\imath_M)_*(\tilde\lambda(M)_k^{\vphantom{}})\; (\imath_N)_*(\tilde\lambda(N)_k^{\vphantom{}})\quad \textrm{ in }\pi_1(M\cdot N).$$
$$\begin{diagram}
\node[2]{M} \arrow{s,r}{\imath_M} \\
\node{\Sigma} \arrow{ne,l}{i_-^{\vphantom{}}} \arrow{se,r}{j_+^{\vphantom{}}} \node{M\cdot N} \\
\node[2]{N} \arrow{n,r}{\imath_N}
\end{diagram}$$
The above diagram commutes and all the maps induce isomorphisms on $\pi_1(-)/\pi_1(-)_q$. Thus
\begin{align*}
\tilde\mu_q^{\vphantom{}}(M\cdot N)_k &= (\imath_M \circ i_+^{\vphantom{}})_{*q}^{-1}(\tilde\lambda(M\cdot N)_k^{\vphantom{}}) \\
&=(i_+^{\vphantom{}})_{*q}^{-1}\circ (\imath_M)_{*q}^{-1}( (\imath_M)_*(\tilde\lambda(M)_k^{\vphantom{}})\; (\imath_N)_*(\tilde\lambda(N)_k^{\vphantom{}})) \\
&=(i_+^{\vphantom{}})_{*q}^{-1}(\tilde\lambda(M)_k^{\vphantom{}}) \; \big((i_+^{\vphantom{}})_{*q}^{-1}\circ (\imath_M)_{*q}^{-1}\circ (\imath_N)_{*q}^{\vphantom{}} \big)(\tilde\lambda(N)_k^{\vphantom{}}) \\
&=\tilde\mu_q^{\vphantom{}}(M)_k^{\vphantom{}} \; \big((i_+^{\vphantom{}})_{*q}^{-1}\circ (i_-^{\vphantom{}})_{*q}^{\vphantom{}} \circ (j_+^{\vphantom{}})_{*q}^{-1}\big)(\tilde\lambda(N)_k^{\vphantom{}}) \\
&=\tilde\mu_q^{\vphantom{}}(M)_k^{\vphantom{}}\; \big(\eta_q^{\vphantom{}}(M) \circ (j_+^{\vphantom{}})_{*q}^{-1}\big)(\tilde\lambda(N)_k^{\vphantom{}}) \\
&=\tilde\mu_q^{\vphantom{}}(M)_k^{\vphantom{}}\; \eta_q^{\vphantom{}}(M) (\tilde\mu_q^{\vphantom{}}(N)_k^{\vphantom{}}). \qedhere
\end{align*}
\end{proof}
In the remaining part of this paper, as an abuse of notation, we also write $i_\pm^{\vphantom{}} $ and $\tilde\lambda(M)_k^{\vphantom{}}$ for $\imath_M\circ i_\pm^{\vphantom{}}$ and $ (\imath_M)_*(\tilde\lambda(M)_k^{\vphantom{}})$, respectively.
From the above proposition, we obtain some properties of $\tilde\mu_q^{\vphantom{}}$:
\begin{corollary}
\leavevmode \@nobreaktrue\nopagebreak
\label{cor:subgroup}
\begin{enumerate}
\item For a homology cylinder $M$, $\tilde\mu_q^{\vphantom{}}(-M)_k^{\vphantom{}}= \eta_q^{\vphantom{}}(M)^{-1}(\tilde\mu_q^{\vphantom{}}(M)_k^{-1})$.
\item The kernel of $\tilde\mu_q^{\vphantom{}}(-)_k^{\vphantom{}}$ is a subgroup of $\mathcal{H}$ for each~$k$.
Moreover, $\ker \tilde\mu_q^{\vphantom{}}$ is a normal subgroup of $\mathcal{H}$.
\item $\tilde\mu_q^{\vphantom{}}$ is a homomorphism on $\operatorname{Ker} \eta_q^{\vphantom{}}$, and $\tilde\mu$ is a homomorphism on~$\bigcap_q \operatorname{Ker} \eta_q^{\vphantom{}} $.
\item $\tilde\mu_q^{\vphantom{}}$ is a homomorphism on~$\operatorname{Ker} \tilde\mu_{q-1}^{\vphantom{}}$, or more generally, on $\operatorname{Ker} \mu_2^{\vphantom{}}\cap \operatorname{Ker} \eta_{q-1}$. \qedhere
\end{enumerate}
\end{corollary}
\begin{proof}
(1) easily follows from $\tilde\mu_q^{\vphantom{}}(M\cdot M^{-1})_k^{\vphantom{}}=1$.
For (2), let $M$ and $N$ be homology cylinders over $\Sigma$.
First, if $\tilde\mu_q^{\vphantom{}}(M)_k^{\vphantom{}}=1= \tilde\mu_q^{\vphantom{}}(N)_k^{\vphantom{}}$, then $\tilde\mu_q^{\vphantom{}}(M\cdot N^{-1})_k^{\vphantom{}} = 1$. Next we check that if $\tilde\mu_q^{\vphantom{}}(M)_k^{\vphantom{}}=1$ for all $k$, then $\tilde\mu_q^{\vphantom{}}(N\cdot M\cdot N^{-1})_k^{\vphantom{}}=1$:
\begin{align*}
\tilde\mu_q^{\vphantom{}}(N\cdot M\cdot N^{-1})_k^{\vphantom{}}&=\tilde\mu_q^{\vphantom{}}(N)_k^{\vphantom{}} \; \eta_q^{\vphantom{}}(N)\big(\tilde\mu_q^{\vphantom{}}(M)_k^{\vphantom{}} \;\eta_q^{\vphantom{}}(M)(\tilde\mu_q^{\vphantom{}}(N^{-1})_k^{\vphantom{}})\big) \\
&=\tilde\mu_q^{\vphantom{}}(N)_k^{\vphantom{}} \; \eta_q^{\vphantom{}}(N) \big( \tilde\mu_q^{\vphantom{}}(N^{-1})_k^{\vphantom{}} \big) \quad \textrm{ due to } \tilde\mu_q^{\vphantom{}}(M)=1 \\
&=\tilde\mu_q^{\vphantom{}}(N)_k^{\vphantom{}} \;\eta_q^{\vphantom{}}(N) \big( \eta_q^{\vphantom{}}(N)^{-1} (\tilde\mu_q^{\vphantom{}}(N)^{-1}_k \big) \quad \textrm{ by (1) } \\
&= \tilde\mu_q^{\vphantom{}}(N)_k^{\vphantom{}} \; \tilde\mu_q^{\vphantom{}}(N)^{-1}_k \\
&=1.
\end{align*}
(3) follows directly from Proposition~\ref{proposition:product formula}.
For (4), we need the following algebraic fact: for a group $G$, if an automorphism of $G/G_q$ induces the identity on $G/G_{q-1}$, then its restriction on $G_{2}/G_q$ is also the identity. We give a proof: from the hypothesis, for such an automorphism $\phi$ and $g\in G/G_q$, $\phi(g)=ga$ for some $a\in G_{q-1}/G_q$. Since $G_{q-1}/G_q$ is in the center of $G/G_q$, the automorphism restricted on $G_2/G_q$ is the identity.
From the algebraic fact, we obtain the desired conclusion since $\eta_{q-1}^{\vphantom{}}(M)=\mathrm{id}$ and $\tilde\mu_q^{\vphantom{}}(M)_k^{\vphantom{}}\in F_2/F_q$ for $M\in \operatorname{Ker} \tilde\mu_{q-1}^{\vphantom{}}$ or $\operatorname{Ker} \mu_2^{\vphantom{}}\cap \operatorname{Ker} \eta_{q-1}^{\vphantom{}}$.
\end{proof}
\@startsection{section}{1{Several subgroups and filtrations of $\mathcal{H}_{g,n}$}
\label{sec:subgroups}
\@startsection{subsection}{2{Filtrations via extended Milnor invariants}
We introduce a filtration of $\mathcal{H}$
$$\cdots \subset \mathcal{H}(q+1) \subset \mathcal{H}(q)\subset \mathcal{H}(q-1)\subset \cdots \subset \mathcal{H}(2)\subset \mathcal{H}(1)=\mathcal{H}$$
where $\mathcal{H}(q)$ is the normal subgroup $\operatorname{Ker} \tilde\mu_q^{\vphantom{}}$ of $\mathcal{H}$, by Corollary~\ref{cor:subgroup}(2).
Comparing with $\mathcal{H}[q]:= \operatorname{Ker}\eta_q^{\vphantom{}}$, we have $\mathcal{H}(q)\subset \mathcal{H}[q]$. For $n = 1$, the two filtrations are the same.
In \cite{L01}, Levine showed that there is an injective homomorphism from the framed $g$-string link concordance group $\mathcal{H}_{0,g+1}$ to $\mathcal{H}_{g,1}$ and that it induces injections between the successive quotients of the filtration $\{\mathcal{H}_{0,g+1}(q)\}$ and those of the filtration $\{H_{g,1}[q]\}$.
We generalize this to $\mathcal{H}_{g,n}$ as follows:
\begin{theorem}
\label{theorem:whole injection}
Suppose $\imath\colon \Sigma_{g,n} \hookrightarrow \Sigma_{g',n'}$ is an embedding ($n,n'\geq 1$). Then it induces a homomorphism $\tilde\imath\colon \mathcal{H}_{g,n} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathcal{H}_{g',n'}$ and a function $f\colon (F/F_q)^{2g+n-1} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} (F'/F'_q)^{2g'+n'-1}$ which make the following diagram commute:
$$ \begin{diagram}
\node{\mathcal{H}_{g,n}} \arrow{e,t}{\tilde\imath} \arrow{s,r}{\tilde\mu_q^{\vphantom{}}} \node{\mathcal{H}_{g',n'}} \arrow{s,l}{\tilde\mu_q^{\vphantom{}}} \\
\node{(F/F_q)^{2g+n-1}} \arrow{e,t}{f} \node{(F'/F'_q)^{2g'+n'-1}}
\end{diagram}$$
where $F=\pi_1(\Sigma_{g,n})$ and $F'=\pi_1(\Sigma_{g',n'})$.
Moreover, if each component of $\Sigma_{g',n'}-\Sigma_{g,n}$ has at least one closed boundary, then $f$ is 1-1, and hence $\tilde\mu_q^{\vphantom{}}(\tilde\imath(M))$ determines $\tilde\mu_q^{\vphantom{}}(M)$ for every $M\in\mathcal{H}_{g,n}$.
If, in addition, at most one component of $\overline{\Sigma_{g',n'}-\Sigma_{g,n}}$ has a disconnected intersection with $\Sigma_{g,n}$, then $\tilde\imath \colon \mathcal{H}_{g,n} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathcal{H}_{g',n'}$ is injective.
\end{theorem}
\begin{proof}
We simply write $\Sigma:=\Sigma_{g,n}$, $\Sigma':=\Sigma_{g',n'}$ and $\mathcal{H}:=\mathcal{H}_{g,n}$, $\mathcal{H}':=\mathcal{H}_{g',n'}$. By Proposition~\ref{proposition:effect of change}, we may assume that the generating sets of $F$ and $F'$ are as follows. We can assume that each component of $\partial\Sigma$ maps to either the interior of $\Sigma'$ or the boundary of $\Sigma'$. Let $S_r$ be a component of $\overline{\Sigma'-\Sigma}$, which is a surface of genus $g_r$ with $a_r$ boundaries on $S_r\cap \Sigma$ and $n_r$ boundaries on $S_r-\Sigma$. \\
\textbf{Step 1.} Choose basepoints $\ast$ of $\Sigma$ and $\ast'$ of~$\Sigma'$. \\
To define $\tilde\mu_q^{\vphantom{}}$, we should choose basepoints on boundaries of the surfaces. There are two cases:
\begin{itemize}
\item[Case 1.] Two basepoints can be chosen as the same point, i.e.\ there is a common boundary of $\Sigma$ and $\Sigma'$. Choose $\ast=\ast'$ on the boundary.
\item[Case 2.] Two basepoints cannot be chosen as the same point, i.e.\ there is no common boundary of $\Sigma$ and $\Sigma'$. Then, at least one $S_r$ has a boundary of $\Sigma'$. Choose $\ast$ and $\ast'$ in the same $S_r$. Choose a path $\gamma$ from $\ast'$ to $\ast$ in the $S_r$ for later use.
\end{itemize}
\textbf{Step 2.} Choose a generating set for $F$. \\
We fix a generating set for $F$ as in Section~\ref{sec:representation}. Denote the generators corresponding to the boundaries of $S_r \cap \Sigma$ by $x^r_k$ and the other generators by $y_s^{\vphantom{}}$. \\
\textbf{Step 3.} Choose a generating set for $F'$. \\
First we will choose a generating set of $\pi_1(\Sigma',\ast)$.
Note that $x^r_k=[\alpha^r_k \cdot\beta^r_k \cdot (\alpha^r_k)^{-1}]$ for a closed path $\beta^r_k$ onto the corresponding boundary and a path $\alpha^r_k$ from $\ast$ to $\beta^r_k(0)$. Let $\imath_*\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1(\Sigma',\ast)$ be the map induced by $\imath$. We choose $y_s':=i_*(y_s^{\vphantom{}})$. Now we select the other generators of $\pi_1(\Sigma',\ast)$ with regard to $S_r$. We consider two cases (see Figure~\ref{figure:S_r}):
\begin{enumerate}
\item[Case 1.] $S_r \not\ni \ast$ \\
Choose generators of $\pi_1(S_r,\alpha^r(1))$ corresponding to the $n_r$ boundary components of $\Sigma'$ in $S_r$ and the $g_r$ handles of $S_r$ as in Section~\ref{sec:representation}, and conjugate them by $\alpha^r_1$ so that we obtain generators $x'^r_i, m'^r_j, l'^r_j$ of $\pi_1(\Sigma',\ast)$ for $i\leq n_r$ and $j\leq g_r$.
Let $m'^r_{g_r+k}:=\imath_*(x^r_k)$ and $l'^r_{g_r+k}:=[\alpha^r_k \cdot \gamma^r_k \cdot(\alpha^r_{k+1})
^{-1}]$ for a path $\gamma^r_k$ from $\alpha^r_k(1)$ to $\alpha^r_{k+1}(1)$ in $S_r$ and $k=1,\ldots ,a_r-1$.
\item[Case 2.] $S_r \ni \ast$ (and $\ast'$) \\
Choose generators of $\pi_1(S_r,\ast)$ corresponding to the $n_r-1$ boundary components of $\Sigma'$ in $S_r$ except the one containing $\ast'$, and those corresponding to the $g_r$ handles of $S_r$ as in Section~\ref{sec:representation}. They give generators $x'^r_i, m'^r_j, l'^r_j$ of $\pi_1(\Sigma',\ast)$ for $i\leq n_r-1$ and $j\leq g_r$. Let $m'^r_{g_r+k}:=\imath_*(x^r_k)$ and $l'^r_{g_r+k}:=[\alpha^r_k \cdot (\gamma^r_k)^{-1}]$ for a path $\gamma^r_k$ from $\ast$ to $\alpha^r_k(1)$ in $S_r$ and $k=1,\ldots ,a_r-1$.
\end{enumerate}
We define a set $A_r:=\{x'^r_i, m'^r_j, l'^r_j, m'^r_{g_r+k}, l'^r_{g_r+k}\}$.
Then $(\bigcup_r A_r) \cup \{y_s'\}$ is a generating set for $\pi_1(\Sigma',\ast)$.
If $\ast \neq \ast'$, replace all the generators by conjugation by $\gamma$ to obtain a generating set for $F'$. Note that we have $\imath_\#\colon F\xrightarrow{\imath_*} \pi_1(\Sigma',\ast) \xrightarrow{\cong} F'$ where the latter is induced by $\gamma$-conjugation.
\begin{figure}[h]
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=.6]{case1.pdf} & \includegraphics[scale=.6]{case2.pdf} \\
Case 1. $S_r \not \ni \ast$ & Case 2. $S_r \ni \ast$
\end{tabular}
\caption{$S_r$ and paths to choose generators of $F'$ in Step 3.}
\label{figure:S_r}
\end{center}
\end{figure}
Now we define $f$.
We use the following indexing convention for coordinates of elements of $(F/F_q)^{2g+n-1}$, using generators of~$F$.
Recall that the generating set $\{x_k^r, y_s^{\vphantom{}}\}_{r,k,s}$ has been ordered to define $\tilde\mu_q^{\vphantom{}}(-)_k^{\vphantom{}}$ using the $k$th generator. If $a \in \{x_k^r, y_s^{\vphantom{}}\}$ is the $k$th generator, we call the $k$th coordinate of an element in $(F/F_q)^{2g+n-1}$ the \emph{coordinate associated with}~$a$.
The map $f$ can be defined coordinatewise as follows. Let $z\in (F/F_q)^{2g+n-1}$. \\
(1) The coordinates $\eta^r_i,\eta'^r_j,\eta''^r_j$ of $f(z)$ associated with $x'^r_i, m'^r_j, l'^r_j$ are determined by the coordinates $ z_i^{\vphantom{}}$ of $z$ associated with $x_i^r$ for all~$i,j$.
\begin{enumerate}
\item[Case 1.] $S_r \not\ni \ast$
\begin{eqnarray*}
\eta^r_i=\imath_\#(z_1^{\vphantom{}}) , \quad
\eta'^r_j=[m'^r_j,\imath_\#(z_1^{\vphantom{}})^{-1}],\quad
\eta'^r_{g_r+k}=[m'^r_{g_r+k},\imath_\#(z_k^{\vphantom{}})^{-1}] , \\
\eta''^r_j=[l'^r_j,\imath_\#(z_1^{\vphantom{}})^{-1}], \quad
\eta''^r_{g_r+k}=l'^r_{g_r+k}\;\imath_\#(z_{k+1}^{\vphantom{}})^{-1} \;(l'^r_{g_r+k})^{-1} \; \imath_\#(z_k^{\vphantom{}})
\end{eqnarray*}
for $i=1,\ldots, n_r$, $j=1,\ldots, g_r$, and $k=1,\ldots, a_r-1$.
\item[Case 2.] $S_r \ni \ast$ \\
If $a_r>1$, then $\eta^r_i, \eta'^r_j, \eta'^r_{g_r+k}, \eta''^r_j$ are the same as Case 1, and $\eta''^r_{g_r+k}=\imath_\#(z_k)$
for $i=1,\ldots, n_r-1$, $j=1,\ldots, g_r$, and $k=1,\ldots, a_r-1$.
If $a_r=1$, then all the coordinates associated with $x'^r_i, m'^r_j, l'^r_j$ are~$1$ for $i=1,\ldots, n_r-1$, $j=1,\ldots, g_r$.
\end{enumerate}
(2) The coordinate $\eta_s^{\vphantom{}}$ of $f(z)$ associated with $y_s'$ is determined by the coordinate $z_s^{\vphantom{}}$ of $z$ associated with $y_s^{\vphantom{}}$.
$$\eta_s^{\vphantom{}}=\imath_\#(z_s^{\vphantom{}}) \textrm{ for all } s.$$
Remark that $f$ is not a homomorphism.
From the definition of $f$, it is 1-1 if $n_r \geq 1$ for all $r$ and $\imath_\#$ is injective. If every $n_r$ is nonzero, then $\imath_\#$ is injective. Therefore, if every $n_r$ is positive, then $f$ is 1-1.
The verification that the diagram commutes is left to the reader.
Suppose at most one $a_r$ is bigger than 1.
To prove $\tilde\imath\colon \mathcal{H}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{H}'$ is injective, we claim that there is a function $\tilde\jmath \colon \mathcal{H}'\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{H}$ so that $\tilde\jmath \circ \tilde\imath = \mathrm{id}_\mathcal{H}$. The surface $\Sigma'$ is obtained from $\Sigma$ by attanching 1-handles to a collar neighborhood of $S_r\cap \Sigma$. This allows us to extend $\Sigma \xrightarrow{\mathrm{id}\times \frac{1}{2}} \Sigma\times I$ to an embedding $\jmath\colon \Sigma' \hookrightarrow \Sigma\times I$.
For example, see Figure~\ref{figure:embedding}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=.55]{embedded.pdf}
\caption{$\jmath (\Sigma') $ in $\Sigma\times I$}
\label{figure:embedding}
\end{center}
\end{figure}
We define $\tilde\jmath (M')$ as the manifold obtained by cutting $\Sigma\times I$ open along $\jmath(\Sigma')$ and filling in it with $M'$.
It is easy to check that $\tilde\jmath\colon \mathcal{H}' \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathcal{H}$ is well-defined. We obtain the injectivity of $\tilde\imath$ from $\tilde\jmath \circ \tilde\imath = \mathrm{id}_\mathcal{H}$.
\end{proof}
Whenever $f$ is 1-1, $\tilde\imath$ induces injections $\mathcal{H}(q-1)/\mathcal{H}(q) \hookrightarrow \mathcal{H}'(q-1)/\mathcal{H}'(q)$. By considering the cases of $(g,n)$ and $(g',n')$ for which there is an injection $\mathcal{H}_{g,n}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{H}_{g',n'}$, we obtain the following corollary. Note that if $\Sigma_{g,n}\subset \Sigma_{g',n'}$, then $g\leq g'$.
\begin{corollary}
\label{cor:whole injection}
For any two pairs $(g,n)$ and $(g',n')$ satisfying $g \leq g'$ and $g+n \leq g'+n'$, there is an injective homomorphism
$$\mathcal{H}_{g,n} \lhook\joinrel\longrightarrow \mathcal{H}_{g',n'} ,$$
which induces injections
$$\mathcal{H}_{g,n}(q-1)/\mathcal{H}_{g,n}(q) \lhook\joinrel\longrightarrow \mathcal{H}_{g',n'}(q-1)/\mathcal{H}_{g',n'}(q)$$
for all $q \geq 2$.
\end{corollary}
\begin{proof}
There exists an embedding $\imath\colon \Sigma_{g,n}\hookrightarrow\Sigma_{g',n'}$ such that both $f$ and $\tilde\imath$ are injective: $\overline{\Sigma_{g',n'}-\Sigma_{g,n}} =S_1$ is connected, and if $n>n'$, $a_1=n-n'+1$, $n_1=1$, and $g_r=g'+n'-g-n$; otherwise, $a_1=1$, $n_1=n'-n+1$, and $g_r=g'-g$.
The conclusion follows from Theorem~\ref{theorem:whole injection}
\end{proof}
The following theorem gives more information on the group $\mathcal{H}(q-1)/\mathcal{H}(q)$ directly via~$\tilde\mu_{q}^{\vphantom{}}$:
\begin{theorem}
\leavevmode \@nobreaktrue\nopagebreak \label{theorem:rank}
\begin{enumerate}
\item $\tilde\mu_{q}^{\vphantom{}}$ induces an injective homomorphism
$$\tilde\mu_{q}^{\vphantom{}}\colon \mathcal{H}(q-1)/\mathcal{H}(q)\lhook\joinrel\longrightarrow (F_{q-1}/F_{q})^{2g+n-1}.$$
Hence $\mathcal{H}(q-1)/\mathcal{H}(q)$ is a finitely generated free abelian group.
\item We have
$$\mathrm{max}\{r_q(2g),r_q(g+n-1)\} \leq \mathrm{rank}\; \mathcal{H}(q-1)/\mathcal{H}(q) \leq r_q(2g+n-1)$$
where $N_q(m)=\frac{1}{q}\sum_{d|q} \varphi(d) (m^{q/d})$, $\varphi$ is the M\"obius function
and $r_q(m)=m N_{q-1}(m)-N_q(m)$.
\end{enumerate}
\end{theorem}
Note that
\begin{align*}
N_q(2g+n-1) &= \mathrm{rank}\; F_q/F_{q+1} \qquad \textrm{and} \\
r_q(2g+n-1)&=(2g+n-1)\;\mathrm{rank}\; F_{q-1}/F_q -\mathrm{rank}\; F_q/F_{q+1} \\
&=\operatorname{Coker}\{H_3(F/F_{q})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_3(F/F_{q-1})\}.
\end{align*}
We remark that the facts
\begin{align*}
\mathrm{rank}\; \mathcal{H}_{g,1}(q-1)/\mathcal{H}_{g,1}(q) &= r_q(2g)\qquad \textrm{and} \\
\mathrm{rank}\; \mathcal{H}_{0,n}(q-1)/\mathcal{H}_{0,n}(q) &= r_q(n-1)
\end{align*}
were shown in \cite{GL} and \cite{Or}, respectively.
\begin{proof}
\begin{enumerate}
\item Since $\tilde\mu_{q}^{\vphantom{}}\colon \mathcal{H}(q-1)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} (F_{q-1}/F_{q})^{2g+n-1}$ is a homomorphism, for $M, N \in \mathcal{H}(q-1)$, $[M]=[N]$ in $\mathcal{H}(q-1)/\mathcal{H}(q)$ if and only if $\tilde\mu_{q}^{\vphantom{}}(M)= \tilde\mu_{q}^{\vphantom{}}(N)$. The well-definedness and the injectivity of the map follow.
\item Consider the map
\begin{align*}
\mathfrak{p}\colon (F_{q-1}/F_{q})^{2g+n-1}&\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} F_{q}/F_{q+1} \\
(a_1^{\vphantom{}},\ldots,a_{n-1}^{\vphantom{}},b_1^{\vphantom{}},\ldots, b_g^{\vphantom{}}, c_1^{\vphantom{}},\ldots, c_g^{\vphantom{}}) &\longmapsto\prod_{i=1}^{n-1} [x_i^{\vphantom{}},a_i^{\vphantom{}}]\prod_{j=1}^{g} [l_j^{\vphantom{}},b_j^{\vphantom{}}][b_j^{\vphantom{}},c_j^{\vphantom{}}][c_j^{\vphantom{}},m_j^{\vphantom{}}].
\end{align*}
This $\mathfrak{p}$ is a surjective homomorphism, and the kernel has rank $r_q(2g+n-1)$. For the upper bound of the rank of $\mathcal{H}(q-1)/\mathcal{H}(q)$, we claim that $\mathfrak{p}\circ \tilde\mu_{q}^{\vphantom{}}$ is trivial on~$\mathcal{H}(q-1)$. For any homology cylinder $M$, $\eta_{q}^{\vphantom{}}(M)$ fixes $[\partial_n^{\vphantom{}}]$ for all~$q$. Using the generating set of $F$ in Figure~\ref{figure:Sigma}, the element $[\partial_n^{\vphantom{}}] = \prod_i x_i^{\vphantom{}}\prod_j [m_j^{\vphantom{}},l_j^{\vphantom{}}] \in F$. For $M \in \mathcal{H}(q-1)$, $\eta_{q+1}^{\vphantom{}}(M)([\partial_n^{\vphantom{}}])=[\partial_n^{\vphantom{}}]$ is arranged to
$$\prod_i[x_i^{\vphantom{}},\mu_{q}^{\vphantom{}}(M)_i^{\vphantom{}}]\prod_j[l_j^{\vphantom{}},\mu_{q}'(M)_j^{\vphantom{}}][\mu_{q}'(M)_j^{\vphantom{}},\mu_{q}''(M)_j^{\vphantom{}}][\mu_{q}''(M)_j^{\vphantom{}},m_j^{\vphantom{}}]=1.$$
Therefore $\mathfrak{p}\circ \tilde\mu_{q}^{\vphantom{}}=\mathfrak{p}\circ (\mu_{q}^{\vphantom{}},\mu_{q}',\mu_{q}'')$ is trivial.
The lower bound comes from Corollary~\ref{cor:whole injection} and the known ranks of $\mathcal{H}_{g,1}(q-1)/\mathcal{H}_{g,1}(q)$ and $\mathcal{H}_{0,n}(q-1)/\mathcal{H}_{0,n}(q)$ \cite{GL,Or} stated above. \qedhere
\end{enumerate}
\end{proof}
We define $\mathcal{H}^0[q]:=\{M\in\mathcal{H}[q]~|~\mu_2(M)=1\}$. Then it is a subgroup of $\mathcal{H}$ by Corollary~\ref{cor:subgroup}, and $\mathcal{H}(q) \subset \mathcal{H}^0[q]\subset \mathcal{H}(q-1)$ by Lemma~\ref{lemma:MKS}.
Thus we can refine the filtration~$\{\mathcal{H}(q)\}$ of $\mathcal{H}$ as follows:
$$\cdots \subset \mathcal{H}(q) \subset \mathcal{H}^0[q] \subset \mathcal{H}(q-1) \subset \mathcal{H}^0[q-1] \subset \cdots \subset \mathcal{H}(2) \subset \mathcal{H}^0[2] \subset \mathcal{H}(1)=\mathcal{H}$$
Consider two injections $\mu_q^{\vphantom{}}$ and $(\mu_q',\mu_q'')$ from the injection $\tilde\mu_{q}^{\vphantom{}}$ as follows:
$$\mu_{q}^{\vphantom{}}\colon \frac{\mathcal{H}^0[q]}{\mathcal{H}(q)} \lhook\joinrel\longrightarrow (F_{q-1}/F_{q})^{n-1}$$
$$(\mu_{q}',\mu_{q}'') \colon \frac{\mathcal{H}(q-1)}{\mathcal{H}^0[q]} \lhook\joinrel\longrightarrow (F_{q-1}/F_{q})^{2g}$$
The two subquotients of $\mathcal{H}$ are also finitely generated free abelian, and there is an isomorphism
$$ \frac{\mathcal{H}(q-1)}{\mathcal{H}(q)} \cong \frac{\mathcal{H}(q-1)}{\mathcal{H}^0[q]}\times \frac{\mathcal{H}^0[q]}{\mathcal{H}(q)} ,$$
which is not canonical.
In the remaining part of this section, we introduce some notions analogous to the boundary (string) links and the $\widehat{F}$-(string) links.
\@startsection{subsection}{2{Boundary homology cylinders}
\label{sec:boundary homology cylinder}
As a generalization of boundary (string) links, we define boundary homology cylinders.
\begin{definition}
A homology cylinder $(M,i_+^{\vphantom{}},i_-^{\vphantom{}})$ over $\Sigma$ is said to be a \emph{boundary homology cylinder} if there exists a homomorphism $\phi \colon \pi_1(M) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1(\Sigma)$ such that $\phi \circ (i_+^{\vphantom{}})_* = \mathrm{id} = \phi \circ (i_-^{\vphantom{}})_*$ and $\phi(\lambda_i^{\vphantom{}})=1$ for all~$i$.
\end{definition}
Geometrically, the boundary homology cylinder $M$ can be defined to be a homology cylinder such that there exists $\psi\colon M \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \Sigma$ making the left diagram below commute.
$$\begin{diagram}
\node{\partial M} \arrow{s,J} \arrow{e,t}{i_+^{-1} \cup i_-^{-1}} \node{\Sigma}\\
\node{M} \arrow{ne,b,..}{\psi}
\end{diagram}
\qquad \qquad
\begin{diagram}
\node{i_+^{\vphantom{}}(\Sigma)} \arrow{s,J} \arrow{e,t}{i_+^{-1}} \node{\Sigma}\\
\node{M} \arrow{ne,b,..}{\psi_+^{\vphantom{}}}
\end{diagram}
$$
Let $(M,i_+^{\vphantom{}},i_-^{\vphantom{}})$ be a homology cylinder.
We consider splittings $\phi_+^{\vphantom{}}$ and $\phi_-^{\vphantom{}}$ of $(i_+^{\vphantom{}})_*$ and $(i_-^{\vphantom{}})_*$, respectively. Note that $\phi_+^{\vphantom{}}$ exists if and only if $\psi_+^{\vphantom{}}$ exists in the above right diagram. The existence of $\phi_+^{\vphantom{}}$ does not imply that of $\phi_-^{\vphantom{}}$. Even if both $\phi_+^{\vphantom{}}$ and $\phi_-^{\vphantom{}}$ exist, they can be different.
Suppose $\phi_+^{\vphantom{}}$ exists. Then $\eta_q^{\vphantom{}}(M)$ is trivial if and only if $\phi_+^{\vphantom{}}$ is also a splitting of $(i_-^{\vphantom{}})_*^{\vphantom{}}$ due to $\bigcap F_q= 0$. However, such a common splitting of $(i_+^{\vphantom{}})_*$ and $(i_-^{\vphantom{}})_*$ does not guarantee that the homology cylinder is a boundary homology cylinder.
Such an example can be found by considering homology cylinders of the form $(\Sigma\times I/\sim, \mathrm{id}\times 0, \mathrm{id}\times 1 \circ \varphi)$ with nonvanishing $\mu_q^{\vphantom{}}$, where $\varphi$ is a composition of Dehn twists about boundaries. The condition $\phi(\lambda_i^{\vphantom{}})=1$ for all $i$, or equivalently all $\mu_q^{\vphantom{}}$ vanish, is necessary to satisfy the geometric definition of the boundary homology cylinder.
In conclusion, we have the following:
\begin{proposition}
\label{proposition:splitting}
A homology cylinder $M$ is a boundary homology cylinder if and only if the following hold:
\begin{enumerate}
\item There is a splitting of $(i_+^{\vphantom{}})_*$ or $(i_-^{\vphantom{}})_*$.
\item $\tilde\mu(M)$ vanishes.
\end{enumerate}
\end{proposition}
The subset of boundary homology cylinders are closed under the multiplication and inverting (=orientation reversing and changing two markings) of $\mathcal{C}_{g,n}$, but a homology cylinder which is homology cobordant to a boundary homology cylinder may not be a boundary homology cylinder. (For example, \cite{Sm} provides such a string link.) We define $\mathcal{B}\mathcal{H}$ to be the subgroup of homology cobordism classes of boundary homology cylinders.
Remark that for a framed string link $\sigma$ in a homology 3-ball, its exterior $E_{\sigma}$ is a boundary homology cylinder if and only if its closure $\hat\sigma$ is a boundary link and the framing of $\sigma$ induces the 0-framing of $\hat\sigma$.
It follows from that $\pi_1(E_{\hat\sigma})=\pi_1(E_\sigma)/\langle\langle i_+^{\vphantom{}}(x_i^{\vphantom{}})= i_-^{\vphantom{}}(x_i^{\vphantom{}}) \rangle\rangle$.
\@startsection{subsection}{2{$\widehat{F}$-homology cylinders}
\label{sec:hF-homology cylinder}
We define $\widehat{F}$-homology cylinders as an analog of the $\widehat{F}$-(string) links. Here $\widehat{G}$ means the algebraic closure of a group $G$ with respect to $\mathbb{Z}$-coefficient or $\mathbb{Z}_{(p)}$-coefficient in the sense of~\cite{C08}. The former was called the $HE$-closure in \cite{L90}. For both $\mathbb{Z}$ and $\mathbb{Z}_{(p)}$, everything in this paper holds.
It is known that for CW-complexes $X$ and $Y$ with finite 2-skeletons, if $X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} Y$ is 2-connected on the integral homology, then it induces an isomorphism on $\widehat{\pi_1(-)}$ \cite{L89, C08}.
Hence the markings $\Sigma\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} M$ for a homology cylinder $M$ induce isomorphisms $\widehat{F} \xrightarrow{\cong} \widehat{\pi_1(M)}$.
\begin{definition}
A homology cylinder $M$ is called an \emph{$\widehat{F}$-homology cylinder} if $\tilde\lambda_k^{\vphantom{}} \in \pi_1(M)$ vanishes in $\widehat{\pi_1(M)}$ for every $k=1,\ldots,2g+n-1$.
\end{definition}
Note that $M$ is an $\widehat{F}$-homology cylinder if and only if $\hat{i}\colon \Sigma \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \widehat{M}$ induces an isomorphism on~$\widehat{\pi_1(-)}$. This can be shown using properties of the algebraic closure functor (see the proof of \cite[Proposition~6.3]{C10}).
The $\widehat{F}$-homology cylinders form a subgroup of $\mathcal{H}_{g,n}$, by the following lemma:
\begin{lemma}
\leavevmode \@nobreaktrue\nopagebreak
\label{lemma:hF=closed}
\begin{enumerate}
\item The product of two $\widehat{F}$-homology cylinders is an $\widehat{F}$-homology cylinder.
\item If $M$ is an $\widehat{F}$-homology cylinder, then $(-M)$ is also an $\widehat{F}$-homology cylinder.
\item A homology cylinder which is homology cobordant to an $\widehat{F}$-homology cylinder is an $\widehat{F}$-homology cylinder.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item
Let $M=(M,i_+^{\vphantom{}},i_-^{\vphantom{}})$ and $M'=(M',i_+',i_-')$ be $\widehat{F}$-homology cylinders. By the Seifert-van Kampen theorem,
$\pi_1(M\cdot M')$ is the pushout of $(i_+^{\vphantom{}})_*$ and~$(i_-^{\vphantom{}})_*$, and $\tilde\lambda(M\cdot M')_k^{\vphantom{}}$ is the product of $\tilde\lambda(M)_k^{\vphantom{}}$ and $\tilde\lambda(M')_k^{\vphantom{}}$ in $\pi_1(M\cdot M')$. All $\tilde\lambda(M\cdot M')_k^{\vphantom{}}$ vanish in $\widehat{\pi_1(M\cdot M'})$ by the following commutative diagram:
$$\begin{diagram}
\dgARROWLENGTH=.1\dgARROWLENGTH \dgHORIZPAD=.3em
\node{\pi_1(\Sigma)} \arrow[2]{s,l}{(i_+')_*} \arrow{se} \arrow[2]{e,t}{(i_-^{\vphantom{}})_*} \node[2]{\pi_1(M)} \arrow{s,-} \arrow{se} \\
\node[2]{\widehat{\pi_1(\Sigma)}} \arrow[2]{s} \arrow[2]{e} \node{} \arrow{s} \node{\widehat{\pi_1(M)}} \arrow[2]{s} \\
\node{\pi_1(M')} \arrow{se} \arrow{e,-} \node{} \arrow{e} \node{\pi_1(M\cdot M')} \arrow{se} \\
\node[2]{\widehat{\pi_1(M')}} \arrow[2]{e} \node[2]{\widehat{\pi_1(M\cdot M'})}
\end{diagram}$$
\item It is obvious since $\tilde\lambda(-M)_k^{\vphantom{}}=\tilde\lambda(M)^{-1}_k$ in $\pi_1(M)$.
\item
Let $(M,i_+^{\vphantom{}},i_-^{\vphantom{}})$ and $(M',i_+',i_-')$ be homology cylinders.
Suppose $M$ is an $\widehat{F}$-homology cylinder and $W$ is a homology cobordism between $M$ and $M'$. Consider the commutative diagram below.
$$ \begin{diagram} \dgARROWLENGTH=0.6\dgARROWLENGTH \dgHORIZPAD=3.7ex \dgVERTPAD=2.7ex
\node{\Sigma \cup_\partial \Sigma} \arrow{s,l,J}{i_+'\cup i_-'} \arrow{e,t,J}{i_+^{\vphantom{}}\cup i_-^{\vphantom{}}} \node{M} \arrow{s,J} \\
\node{M'} \arrow{e,J} \node{W}
\end{diagram} $$
We should check that the elements $\tilde\lambda(M')_k^{\vphantom{}}$ of $\pi_1(M')$ vanish in $\widehat{\pi_1(M)}$. Both $\tilde\lambda(M)_k^{\vphantom{}}$ and $\tilde\lambda(M')_k^{\vphantom{}}$ come from the same element of $\pi_1(\Sigma\cup_\partial \Sigma)$ along $(i_+^{\vphantom{}}\cup i_-^{\vphantom{}})_*$ and $(i_+'\cup i_-')_*$ so they are mapped to the same element along isomorphisms $\widehat{\pi_1(M)} \xrightarrow{\cong} \widehat{\pi_1(W)} \xleftarrow{\cong} \widehat{\pi_1(M')}$ induced by inclusions. Since $\tilde\lambda(M)_k^{\vphantom{}}$ vanishes in $\pi_1(\widehat{M})$ for all $k$, so does~$\tilde\lambda(M')_k^{\vphantom{}}$. \qedhere
\end{enumerate}
\end{proof}
We denote the subgroup of $\widehat{F}$-homology cylinders by~$\widehat{\H}$.
Levine defined a notion of an $\widehat{F}$-link using his algebraic closure which involves a certain normal generation condition in \cite{L89}. We denote Levine's algebraic closure by $\widehat{G}^\mathrm{Lev}$ to avoid confusion. The definition is as follows:
a link $L$ is called an \emph{$\widehat{F}$-link} if a meridian map into link exterior $E_L$ induces an isomorphism on $\widehat{\pi_1(-)}^\mathrm{Lev}$ and the preferred longitudes vanish in $\widehat{\pi_1(E_L)}^\mathrm{Lev}$.
In this paper, we use a modified definition by replacing $\widehat{G}^\mathrm{Lev}$ by our $\widehat{G}$ as in \cite{C10}.
Levine's $\widehat{F}$-link is an $\widehat{F}$-link in our sense, though the converse is open.
For a framed string link $\sigma$ in a homology 3-ball, its exterior $E_\sigma$ is an $\widehat{F}$-homology cylinder if and only if its closure $\hat\sigma$ is an $\widehat{F}$-link (in our sense) and the framing of $\sigma$ induces the 0-framing of $\hat\sigma$. It follows by considering $$\pi_1(E_\sigma)\twoheadrightarrow \pi_1(E_{\hat\sigma})\twoheadrightarrow \pi_1(\widehat{E_{\sigma}})=\pi_1(E_\sigma)/\langle\langle \lambda_i^{\vphantom{}}\rangle\rangle$$ where $E_{\hat\sigma}$ is the exterior of the link~$\hat\sigma$.
\begin{remark}
\leavevmode \@nobreaktrue\nopagebreak
\begin{enumerate}
\item Since a boundary homology cylinder is an $\widehat{F}$-homology cylinder, $\mathcal{B}\mathcal{H} \subset \widehat{\H}$. In general, the inclusion is strict because it is known that there are $\widehat{F}$-string links which are not concordant to any boundary string link \cite{CO}.
\item $\widehat{\H} \subseteq \mathcal{H}(\infty)$, but we do not know the converse. This question is a homology cylinder version of a long-standing conjecture that a link with vanishing $\bar\mu$-invariants is an $\widehat{F}$-link in the sense of Levine.
\item For the mapping class group $\mathcal{M}$ over $\Sigma$, we note that $\mathcal{M} \cap \mathcal{H}(\infty) = 0$. This may be used to find links which are not fibered but homologically fibered as in~\cite{GS}.
\end{enumerate}
\end{remark}
\@startsection{section}{1{Hirzebruch-type invariants}
\label{sec:hirzebruch}
We review some definitions in \cite{C10,C09}. Let $X$ be a CW complex.
A tower
$$X_{(h)} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \cdots \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} X_{(1)} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} X_{(0)} = X$$
of covering maps is called a \emph{$p$-tower of height $h$} for $X$ if
each $X_{(t+1)} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} X_{(t)}$ is a connected cover whose covering transformation group is a finite
abelian $p$-group.
For a power $d$ of $p$, a \emph{$\mathbb{Z}_d$-valued $p$-structure of height $h$} for $X$ is a pair $(\{X_{(t)}\},\phi)$ of a $p$-tower of height $h$ for $X$ and a surjective character $\phi \colon \pi_1(X_{(h)}) \twoheadrightarrow \mathbb{Z}_d$.
We omit the word ``$\mathbb{Z}_d$-valued'' from now on.
A $p$-structure of height $h$ is equivalent to a $p$-tower of height $h+1$ such that the $(h+1)$st cover is a $\mathbb{Z}_d$-cover of the $h$th cover.
For a $p$-structure of height $h$ for $X$, we usually denote by $X_{(h+1)}$ the $\mathbb{Z}_d$-cover of the top cover $X_{(h)}$ determined by~$\phi$.
We recall the definition of the Hirzebruch-type intersection form defect invariant \cite[Definition~2.2]{C10}.
Let $\mathcal{T}$ be a $p$-structure $(\{M_{(t)}\},\phi)$ of height $h$ for a closed 3-manifold~$M$. Suppose $r(M_{(h)},\phi)=0$ in the bordism group $\Omega_3(B\mathbb{Z}_d)$ for some~$r>0$. Then there is a 4-manifold $W$ bounded by $rM$ over~$\phi$. Choose a subring $\mathcal{R}$ of $\mathbb{Q}$ containing $1/r$. Define
$$\lambda(M,\mathcal{T}) = \frac{1}{r} \otimes ([\lambda_{\mathbb{Q}(\zeta_d)}(W)]-[\lambda_\mathbb{Q}(W)]) \in \mathcal{R} \otimes_\mathbb{Z} L^0(\mathbb{Q}(\zeta_d))$$
where $[\lambda_\mathcal{K}(W)]$ is the witt class of the $\mathcal{K}$-coefficient intersection form of $W$ for a field~$\mathcal{K}$.
Note that since $\Omega_3(B\mathbb{Z}_d)=H_3(\mathbb{Z}_d)=\mathbb{Z}_d$, some multiple of a closed 3-manifold over $\mathbb{Z}_d$ is zero in the bordism group.
\begin{lemma}[Lemma~4.4 in \cite{C10}]
\label{lemma:p-torsion-free}
For a $p$-structure $(\{M_{(t)}\},\phi)$ of height $h$ for a closed 3-manifold $M$,
if $H_1(M_{(h)})$ is $p$-torsion free, then $(M_{(h)},\phi)$ is null-bordant over $\mathbb{Z}_d$ so that $\lambda(M,\mathcal{T})$ is well-defined as an element in $L^0(\mathbb{Q}(\zeta_d))$.
\end{lemma}
\begin{proof}
The character $\phi\colon \pi_1(M_{(h)})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z}_d$ factors through $\mathbb{Z}$ if $H_1(M_{(h)})$ is $p$-torsion free.
Since $(M_{(h)},\phi) \in \operatorname{Im}\{\Omega_3(B\mathbb{Z})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \Omega_3(B\mathbb{Z}_d)\}$ and $\Omega_3(B\mathbb{Z})\cong H_3(\mathbb{Z})=0$, $(\widehat{M}_{(h)},\phi)=0$ in $\Omega_3(B\mathbb{Z}_d)$.
\end{proof}
For a homology cylinder $M$, recall that there is an associated closed 3-manifold $\widehat{M}$, the closure of $M$. For each $p$-structure $\mathcal{T}$ for $\widehat{M}$, $\lambda(\widehat{M},\mathcal{T})$ is defined.
We want to define invariants parametrized by the $p$-structures for the base surface $\Sigma$ rather than those for $\widehat{M}$, and hope that the invariants are homomorphisms of (subgroups of) the homology cobordism group $\mathcal{H}$.
To do this, we first investigate how to determine a $p$-structure for $\widehat{M}$ from a given $p$-structure for $\Sigma$.
\@startsection{subsection}{2{$p$-structures and $p$-tower maps}
\label{sec:defining condition}
A map $f \colon X \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} Y$ is called a \emph{$p$-tower map} if it gives rise to a 1-1 correspondence
$$\Phi_f \colon \{p\text{-structures for } Y\} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \{p\text{-structures for } X\}$$
via pullback.
For a more precise description, we recall the definition of the pullback cover.
For a covering map $p\colon \tilde{Y}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} Y$ and a map $f\colon X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} Y$, the space
$\tilde{X}=\{(x,a)\in X\times \tilde{Y}~|~f(x)=p(a)\}$ is called the \emph{pullback cover} of $X$ by $f$. The canonical projection map $\tilde{X}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} X$ is a covering map. We note that the fiber of the pullback cover is homeomorphic to that of the original cover.
For a map $X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} Y$ and a $p$-tower for $Y$, if we take the pullback covers inductively, then we get a tower of $p$-covers of $X$, but some covers may be disconnected. Hence $\Phi_f$ is well-defined only when all the pullback covers are connected and the induced character is surjective.
It is known that if $X$ and $Y$ have finite 2-skeletons, and if $f$ is 2-connected or, more generally, induces an isomorphism on $\widehat{\pi_1(-)}$, then $f$ is a $p$-tower map \cite[Lemma~3.7, Proposition~3.9]{C10}. Hence, for example, each marking $\Sigma\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} M$ of any homology cylinder $M$ is a $p$-tower map.
\begin{definition}
\label{definition:of order}
Let $X$ and $Y$ be CW-complexes.
\begin{enumerate}
\item A $p$-structure of height $h$ for $X$ is called a \emph{$p$-structure of order $q$} if $\pi_1(X)_q \subset \pi_1(X_{(h+1)})$, viewing $\pi_1(X_{(h+1)})$ as a subgroup of $\pi_1(X)$, via the injection induced by the covering projection.
\item A map $X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} Y$ is called a \emph{$p$-tower map of order $q$} if it induces a 1-1 correspondence
$$\Phi_f \colon \{p\text{-structures for } Y \textrm{ of order }q\} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \{p\text{-structures for } X \textrm{ of order }q\}$$ via pullback.
\end{enumerate}
\end{definition}
Note that the defining condition in Definition~\ref{definition:of order} (1) is independent of the basepoints of $X$ and $X_{(h+1)}$.
We need an algebraic lemma.
\begin{lemma}
\label{lemma:algebra}
For a finitely generated group $G$, suppose there is a subnormal series $G_{(t)} \triangleleft \cdots \triangleleft G_{(1)} \triangleleft G_{(0)} = G$ whose factor groups are abelian $p$-groups. Then $G_{(t)}$ contains $G_q$ for some~$q$.
\end{lemma}
Note that if $G_{(t)}$ were a normal subgroup of $G$, then the conclusion would be immediate from that any $p$-group is nilpotent. We give a proof for the above general case at the end of this section.
Let $X$ and $Y$ be CW-complexes with finitely generated fundamental groups.
By the lemma,
$$\{ p \textrm{-structures for } X\} = \bigcup_q\{p \textrm{-structures for } X\textrm{ of order }q\} , $$
and $X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} Y$ is a $p$-tower map if and only if it is a $p$-tower map of order $q$ for all $q$.
\begin{lemma}
\label{lemma:well-defined and 1-1}
Let $X$ and $Y$ be connected CW-complexes whose fundamental groups are finitely generated. If $f\colon X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} Y$ induces a surjection on $\pi_1(-)/\pi_1(-)_q$, then $\Phi_f$ between $p$-structures of order $q$ is well-defined and 1-1.
\end{lemma}
\begin{proof}
For the well-definedness, we should check that pullback covers are connected $p$-covers.
Use induction on $t\geq 0$.
Suppose there exists a unique $Y_{(t)}$ such that the pullback cover is $X_{(t)}$, and $X_{(t)}$ is connected.
Let $Y_{(t+1)}$ be determined by $\pi_1(Y_{(t)})\twoheadrightarrow \Gamma_{(t)}$. Since the homomorphism factors through $\pi_1(Y_{(t)})/\pi_1(Y)_q$, the composition map $\pi_1(X_{(t)})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \Gamma_{(t)}$ also factors through $\pi_1(X_{(t)})/\pi_1(X)_q \twoheadrightarrow \pi_1(Y_{(t)})/\pi_1(Y)_q$ and is surjective. Hence the pullback cover $X_{(t+1)}$ is a connected cover of $X_{(t)}$ whose cover transformation group is $\Gamma_{(t)}$.
The map $\pi_1(X_{(t)})\twoheadrightarrow \Gamma_{(t)}$ which determines $X_{(t)}$ factors through $\pi_1(X_{(t)})/\pi_1(X)_q \twoheadrightarrow \pi_1(Y_{(t)})/\pi_1(Y)_q$. Hence it determines $\pi_1(Y_{(t)}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \Gamma_{(t)}$ uniquely. Therefore $Y_{(t+1)}$ whose pullback is $X_{(t+1)}$ is unique.
\end{proof}
Applying this lemma to the marking $\hat{i} \colon \Sigma \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \widehat{M}$ of a homology cylinder $M$, $\Phi_{\hat{i}}$ is always well-defined and 1-1. We write $\Phi_M$ instead of~$\Phi_{\hat{i}}.$
Now, we investigate when a $p$-structure $\mathcal{T}$ for $\Sigma$ determines one for $\widehat{M}$, in other words, when $\mathcal{T}\in\operatorname{Im}\Phi_M$. If it is the case and the top cover of $\widehat{M}$ is $r$-torsion in $\Omega_3(B\mathbb{Z}_d)$, we can define
$$\lambda_\mathcal{T}(M):=\lambda(\widehat{M},(\Phi_M)^{-1}(\mathcal{T})) \in \mathbb{Z}\Big[\frac{1}{r}\Big]\otimes_\mathbb{Z} L^0(\mathbb{Q}(\zeta_d)).$$
From now on, let $(\{\Sigma_{(t)}\}, \phi)$ be a $p$-structure $\mathcal{T}$ for $\Sigma$, and $F_{(t)}=\pi_1(\Sigma_{(t)})$.
\begin{lemma}
\label{lemma:defining condition}
Let $\mathcal{T}$ be a $p$-structure of height $h$ for $\Sigma$ of order $q$. For a homology cylinder $M$, $\mathcal{T} \in \operatorname{Im}\Phi_M$ if and only if $\tilde\mu_q^{\vphantom{}}(M)_k^{\vphantom{}} \in F_{(h+1)}/F_q$ for all~$k$.
\end{lemma}
\begin{proof}
For $0\leq t \leq h$, suppose there is $\widehat{M}_{(t)}$ corresponding to $\Sigma_{(t)}$ (and~$M_{(t)}$).
We shall prove that $\widehat{M}_{(t+1)}$ exists whose pullback cover is $\Sigma_{(t+1)}$ if and only if $\tilde \mu_q^{\vphantom{}}(M)_k^{\vphantom{}}$ is in $F_{(t+1)}/F_q$ for all~$k$. Since $\Sigma \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} M$ is a $p$-tower map, we have $F_{(t)}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1(M_{(t)}) \twoheadrightarrow \Gamma_{(t)}$ which determines $\Sigma_{(t+1)}$ and~$M_{(t+1)}$. All $\tilde \mu_q^{\vphantom{}}(M)_k^{\vphantom{}}$ are in $F_{(t+1)}/F_q$ if and only if all $\tilde\lambda(M)_k^{\vphantom{}}$ are in $\pi_1(M_{(t+1)})$, i.e.\ all $\tilde\lambda(M)_k^{\vphantom{}}$ vanish in $\Gamma_{(t)}$. It means that $\pi_1(M_{(t)})\twoheadrightarrow \Gamma_{(t)}$ factors through $\pi_1(\widehat{M}_{(t)})$, or equivalently, $\widehat{M}_{(t+1)}$ exists. The assumption is true for $t=0$, and this completes the proof by induction.
\end{proof}
\begin{theorem}
\leavevmode \@nobreaktrue\nopagebreak
\label{theorem:p-tower map}
Suppose $M$ is a homology cylinder.
\begin{enumerate}
\item $\hat{i}\colon \Sigma \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \widehat{M}$ is a $p$-tower map of order $q$ if and only if $\tilde\mu_q^{\vphantom{}}(M)$ vanishes.
\item $\hat{i}\colon \Sigma \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \widehat{M}$ is a $p$-tower map if and only if $\tilde\mu^{\vphantom{}}(M)$ vanishes.
\end{enumerate}
\end{theorem}
\begin{proof}
\begin{enumerate}
\item By Lemma~\ref{lemma:defining condition}, $\Phi_M$ between $p$-structures of order $q$ is surjective if and only if $\tilde\mu_q^{\vphantom{}}(M)_k^{\vphantom{}} \in \bigcap_G G/F_q$ for all $k$, where $G/F_q$ ranges over all subgroups of $F/F_q$ such that the index $[F:G]$ is a power of $p$. Thus, it suffices to show that $\bigcap_G G/F_q$ is trivial.
From the fact that $F/F_q$ is a residually $p$-group \cite{Gr}, $\bigcap_G G/F_q$ is trivial.
\item This follows directly from (1). \qedhere
\end{enumerate}
\end{proof}
From the proof of the theorem, we give a weakened condition for a map to be a $p$-tower map:
\begin{theorem}
Let $X$ and $Y$ be connected CW-complexes having finitely generated fundamental groups.
Suppose a map $f\colon X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} Y$ induces a surjection $f_{*}$ on~$\pi_1(-)/\pi_1(-)_q$.
If $f_{*}$ is injective, then $f$ is a $p$-tower map of order~$q$.
Moreover, if $\pi_1(X)/\pi_1(X)_q$ is a residually $p$-group, then the converse is also true.
\end{theorem}
We remark that this proposition can be applied to meridian maps of (string) links. This gives the affirmative answer to the question in \cite[Remark~6.4]{C10}: ``if $L$ is a link with vanishing $\bar\mu$-invariants, then is a meridian map into the surgery manifold of the link a $p$-tower map?''. Moreover, the converse is also true.
\@startsection{subsection}{2{Homology cobordism invariants}
\label{sec:homology cobordism invariants}
We study the homology cobordism invariance of~$\lambda_\mathcal{T}$.
\begin{proposition}
\label{proposition:homology cobordism invariant}
Suppose homology cylinders $M$ and $M'$ are homology cobordant. Then $\lambda_\mathcal{T}(M)$ is defined as an element in $\mathbb{Z}[\frac{1}{r}]\otimes L^0(\mathbb{Q}(\zeta_d))$ if and only if $\lambda_\mathcal{T}(M')$ is defined as an element in $\mathbb{Z}[\frac{1}{r}]\otimes L^0(\mathbb{Q}(\zeta_d))$. In that case, $\lambda_\mathcal{T}(M)=\lambda_\mathcal{T}(M')$ in $\mathbb{Z}[\frac{1}{r}]\otimes L^0(\mathbb{Q}(\zeta_d))$.
\end{proposition}
Theorem~\ref{theorem:p-tower map} and Proposition~\ref{proposition:homology cobordism invariant} provide
invariants of homology cobordism group of homology cylinders. More precisely, we have
$$\lambda_\mathcal{T} \colon \mathcal{H}(q) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z}\Big[\frac{1}{d}\Big]\otimes_\mathbb{Z} L^0 (\mathbb{Q}(\zeta_d))$$
for each $p$-structure $\mathcal{T}$ for $\Sigma$ of order $q$.
To prove the proposition, we construct a homology cobordism between $\widehat{M}$ and~$\widehat{M}'$:
\begin{lemma}
\label{lemma:V}
Suppose homology cylinders $(M,i_+^{\vphantom{}},i_-^{\vphantom{}})$ and $(M,i_+',i_-')$ are homology cobordant. Then there is a homology cobordism $\widehat{W}$ between $\widehat{M}$ and $\widehat{M}'$ such that $\Sigma \xrightarrow{\hat{i}}\widehat{M} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \widehat{W}$ and $\Sigma \xrightarrow{\hat{i'}}\widehat{M}' \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \widehat{W}$ are homotopic.
\end{lemma}
\begin{proof}
Let $W$ be a homology cobordism between $M$ and~$M'$. We construct a 4-manifold $\widehat{W}$ from $W$ by identifying tubular neighborhoods of $i_+^{\vphantom{}}(\Sigma)$ and $i_-^{\vphantom{}}(\Sigma)$ in the boundary of~$W$. Then the boundary of $\widehat{W}$ is $\widehat{M} \cup \widehat{M}'$. We remind the reader that $E$ is the trivial homology cylinder for the next. Equivalently, $\widehat{W}$ is obtained by attaching $E\times I$ to the tubular neighborhood of $\partial M $ in the boundary of $W$ such that $\partial E \times I$ = (tubular neighborhood of $\partial M$), $\partial E \times 0 \subset M$ and $\partial E\times 1\subset M'$. We need to check that the inclusion maps $\widehat{M} \hookrightarrow \widehat{W}$ and $\widehat{M}' \hookrightarrow \widehat{W}$ are homology equivalences. By comparing the Mayer-Vietoris sequences of $(M,E\times 0)$ and $(W,E \times I)$, we obtain an isomorphism $H_*(\widehat{M}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_*(\widehat{W})$ using the five lemma. Hence $\widehat{M}\hookrightarrow \widehat{W}$ is a homology equivalence, and similarly for $\widehat{M}'\hookrightarrow \widehat{W}$.
The homotopy conclusion in the statement follows from the construction of~$\widehat{W}$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{proposition:homology cobordism invariant}]
Suppose $\lambda_\mathcal{T}(M)\in \mathbb{Z}[\frac{1}{r}]\otimes L^0(\mathbb{Q}(\zeta_d))$ is defined, i.e.\ there is a $p$-structure $\mathcal{S}$ for~$\widehat{M}$ such that $\Phi_M(\mathcal{S}) = \mathcal{T}$ and the top cover of $\widehat{M}$ is $r$-torsion in~$\Omega_3(B\mathbb{Z}_d)$. Since $\widehat{M} \hookrightarrow \widehat{W}$ and $\widehat{M} ' \hookrightarrow \widehat{W}$ are 2-connected, they are $p$-tower maps and so there is a $p$-structure $\mathcal{S}'$ for $\widehat{M}'$ corresponding to~$\mathcal{S}$ under the inclusion-induced bijections. From the homotopy conclusion in Lemma~\ref{lemma:V}, it follows that $\Phi_{M'}(\mathcal{S}') = \mathcal{T}$. Hence $\lambda_\mathcal{T}(M')$ is also defined.
By applying \cite[Theorem 3.1]{C10}, we obtain $\lambda_\mathcal{T}(M) = \lambda_\mathcal{T}(M')\in \mathbb{Z}[\frac{1}{r}]\otimes L^0(\mathbb{Q}(\zeta_d))$.
\end{proof}
For later use, we recall two key ingredients of the proof of \cite[Theorem 3.1]{C10}:
\begin{enumerate}
\item[(K1)] Suppose $X\supset Y$ are CW complexes with finite $n$-skeletons. If $H_i(X,Y;\mathbb{Z}_{(p)}) = 0$ for $i \leq n$, then $H_i(\tilde X, \tilde Y; \mathbb{Z}_{(p)}) = 0$ for $i \leq n$ where $\tilde X$ is a $p$-cover of $X$ and $\tilde Y$ is the pullback cover of $Y$ by the inclusion $Y \hookrightarrow X$ \cite[Lemma 3.3]{C10}.
\item[(K2)] If a 4-manifold $X$ satisfies
$$\operatorname{Im}\{H_2(\tilde X;\mathbb{Q}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_2(\tilde X,(\text{a subspace of) } \partial\tilde X;\mathbb{Q})\} = 0$$
for a $\mathbb{Z}_d$-cover $\tilde X$ of $X$, then the Witt class of the $\mathbb{Q}(\zeta_d)$-coefficient intersection form $[\lambda_{\mathbb{Q}(\zeta_d)}(X)] $ vanishes.
\end{enumerate}
As in the proof of \cite[Theorem~2.4]{C09}, to prove (K2), we use the fact that $\mathbb{Q}(\zeta_d)$ is flat over $\mathbb{Q}[\mathbb{Z}_d]$; by the universal coefficient theorem,
\begin{align*}
H_2(X;\mathbb{Q}(\zeta_d)) \; & = \; H_2(X;\mathbb{Q}[\mathbb{Z}_d]) \otimes_{\mathbb{Q}[\mathbb{Z}_d]}\mathbb{Q}(\zeta_d) \\
& = \; H_2(\tilde{X} ; \mathbb{Q}) \otimes_{\mathbb{Q}[\mathbb{Z}_d]}\mathbb{Q}(\zeta_d).
\end{align*}
Similarly for a subspace of~$\partial\tilde{X}$.
Thus the condition implies
$$\operatorname{Im}\{H_2(X;\mathbb{Q}(\zeta_d)) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_2(X,\partial X; \mathbb{Q}(\zeta_d))\} = 0,$$
and $[\lambda_{\mathbb{Q}(\zeta_d)}(X)] = 0.$
\@startsection{subsection}{2{Homomorphism of homology cobordism groups}
\label{sec:additivity condition}
In this subsection, we investigate the additivity of $\lambda_\mathcal{T}$.
For $M, N$ in $\mathcal{H}(q)$ and a $p$-structure $\mathcal{T}$ for $\Sigma$ of order $q$,
we need to check whether $\lambda_\mathcal{T}(M)+ \lambda_\mathcal{T}(N) - \lambda_\mathcal{T} (M\cdot N)=0$.
We will construct a cobordism $V$ between $\widehat{M} \cup \widehat{N}$ and $\widehat{M \cdot N}$ such that if $M$ and $N$ are in $\mathcal{H}(q)$, then the inclusions from $\widehat{M}$, $\widehat{N}$, and $\widehat{M\cdot N}$ into $V$ are $p$-tower maps of order~$q$. After then, we will investigate when the difference of the Witt classes of two intersection forms, twisted and untwisted, of the top cover of $V$ vanishes.
For our purpose, we can use a``standard'' cobordism $V$ as in~\cite{C09, CHH}. The cobordism $V$ is obtained from $\big(\widehat{(M,i_+^{\vphantom{}},i_-^{\vphantom{}})} \cup \widehat{(N,j_+^{\vphantom{}},j_-^{\vphantom{}})}\big) \times I$ by identifying product neighborhoods of $\hat{i}(\Sigma)$ and $\hat{j}(\Sigma)$ in $(\widehat{M} \cup \widehat{N}) \times 1$. Since the identification can be thought of as attaching $\Sigma \times I \times I$ along the product neighborhoods, $V$ can be obtained by attaching one 1-handle which connects $\widehat{M} \times I$ to $\widehat{N} \times I$ and $(2g+n-1)$ 2-handles along simple closed curves on $\hat{i}(\Sigma)\#_b ~\hat{j}(\Sigma)$ in $\widehat{M} \# \widehat{N}$ corresponding to $\hat{i}_*(z)\cdot \hat{j}_*(z^{-1})$ (up to homotopy) where $z$ ranges over disjoint simple closed curves representing $x_i^{\vphantom{}}$, $m_j^{\vphantom{}}$ and $l_j^{\vphantom{}}$ in Figure~\ref{figure:Sigma}.
($\#$ denotes connected sum and $\#_b$ denotes boundary connected sum.)
Then, by construction, we have the following property, which we state as a lemma:
\begin{lemma}
\label{lemma:cobordism}
For any homology cylinders $M$ and $N$ over $\Sigma$, there is a cobordism $V$ between $\widehat{M} \cup \widehat{N}$ and $\widehat{M\cdot N}$ such that the following diagram is commutative up to homotopy rel $*$.
$$\begin{diagram}
\dgARROWLENGTH=.6\dgARROWLENGTH
\node{} \node{\widehat{M}} \arrow{se,J} \\
\node{\Sigma} \arrow{ne,l}{\hat{i}} \arrow{se,r}{\begin{substack}{i_+^{\vphantom{}} = j_-^{\vphantom{}}
}\end{substack}}\arrow{e,t}{\hat{j}} \node{\widehat{N}} \arrow{e,J} \node{V} \\
\node{} \node{\widehat{M\cdot N}} \arrow{ne,J}
\end{diagram} $$
\end{lemma}
Note that $\widehat{M\cdot N}$ and $\widehat{N\cdot M}$ are homeomorphic, and the marking for $\widehat{M\cdot N}$ and the marking for $\widehat{N\cdot M}$ induce the same homomorphism $\pi_1(\Sigma)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1(\widehat{M\cdot N})\cong \pi_1(\widehat{N\cdot M})$.
\begin{lemma}
If $M, N \in \mathcal{H}(q)$, then all maps in the above diagram induce isomorphisms on~$\pi_1(-)/\pi_1(-)_q$. Consequently, all the maps are $p$-tower maps of order~$q$.
\end{lemma}
\begin{proof}
It suffices to prove that the embedding $\widehat{M\cdot N} \hookrightarrow V$ induces an isomorphism on $\pi_1(-)/\pi_1(-)_q$.
Both $\pi_1(\widehat{M\cdot N})$ and $\pi_1(V)$ are quotient groups of $\pi_1(M)\ast\pi_1(N)$.
The group $\pi_1(\widehat{M\cdot N})$ is the quotient by the normal subgroup generated by $(i_-^{\vphantom{}})_*(z) (j_+^{\vphantom{}})_*(z^{-1})$ for $z\in F$ and $\tilde\lambda(M)_k^{\vphantom{}} \tilde\lambda(N)_k^{\vphantom{}}$ for all $k$.
The group $\pi_1(V)$ is the quotient by the normal subgroup generated by $(i_-^{\vphantom{}})_*(z) (j_+^{\vphantom{}})_*(z^{-1})$ for $z\in F$ and $\tilde\lambda(M)_k^{\vphantom{}}$, $\tilde\lambda(N)_k^{\vphantom{}}$ for all $k$.
Hence if $M$ is in $\mathcal{H}(q)$ then $\tilde\lambda(M)_k^{\vphantom{}}$ is in $(\pi_1(M)\ast\pi_1(N))_q$. It follows that the claim is true.
\end{proof}
For corresponding $p$-towers for $\widehat{M}$, $\widehat{N}$, $\widehat{M\cdot N}$, and $\widehat{W}$, we have $\partial V_{(t)}=\widehat{M}_{(t)}\cup \widehat{N}_{(t)} \cup -\widehat{M\cdot N}_{(t)}$, and hence $$\lambda_\mathcal{T}(M)+\lambda_\mathcal{T}(N)-\lambda_\mathcal{T}(M\cdot N) = [\lambda_{\mathbb{Q}(\zeta_d)}(V)] - [\lambda_\mathbb{Q}(V)].$$
The following theorem presents sufficient conditions for the intersection forms on the right hand side to be Witt trivial. Recall that by Lemma~\ref{lemma:defining condition}, if $M \in H(q)$ and $\mathcal{T}$ is a $p$-tower for $\Sigma$ of order $q$, then $\tilde\mu_q^{\vphantom{}}(M)_k^{\vphantom{}}\in F_{(t)}/F_q$ for all~$t$. Also the $k$th coordinate $\tilde\mu(M)_k^{\vphantom{}}$ of $\tilde\mu(M)$ lives in $\varprojlim_{s\geq q} F_{(t)}/F_s$. We will consider a map $\varprojlim_{s\geq q} F_{(t)}/F_s \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \varprojlim_s H_1(F_{(t)})\otimes \mathbb{Z}_{p^s}$, which will be defined in the proof of Theorem~\ref{theorem:additivity condition} below.
\begin{theorem}
\label{theorem:additivity condition}
Suppose $\mathcal{T}$ is a $p$-structure of height $h$ for $\Sigma$ of order $q$ and $M$ is a homology cylinder in~$\mathcal{H}(q)$. Then the following are equivalent:
\begin{enumerate}
\item[(C1)] $H_1(\Sigma_{(t)};\mathbb{Z}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(\widehat{M}_{(t)}; \mathbb{Z})$ is injective.
\item[(C2)] $H_1(\Sigma_{(t)};\mathbb{Z}_{(p)}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(\widehat{M}_{(t)};\mathbb{Z}_{(p)})$ is injective.
\item[(C3)] $H_1(\Sigma_{(t)};\mathbb{Q}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(\widehat{M}_{(t)};\mathbb{Q})$ is injective.
\item[(C4)] All $\tilde \lambda(M)_k^{\vphantom{}}$ are torsion elements in~$H_1(M_{(t)})$.
\item[(C5)] All $\tilde\mu(M)_k^{\vphantom{}}$ lie in the kernel of $\varprojlim_{s\geq q} F_{(t)}/F_s \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \varprojlim_s H_1(F_{(t)})\otimes \mathbb{Z}_{p^s}.$
\end{enumerate}
If $M$ and $N$ are in $\mathcal{H}(q)$ and either $M$ or $N$ satisfies \textup{(}one of\textup{)} \textnormal{(C1)--(C5)} for $t=h+1$, then
$$\lambda_\mathcal{T}(M) + \lambda_\mathcal{T}(N) = \lambda_\mathcal{T}(M\cdot N).$$
In addition, the homology cylinders satisfying \textup{(}one of\textup{)} \textnormal{(C1)--(C5)} form a subgroup of~$\mathcal{H}(q)$.
\end{theorem}
We remark that in (C2) and (C3), the injectivity of the map implies that it is an isomorphism. See the proof below.
\begin{proof}
The implications (C1) $\Rightarrow$ (C2) $\Rightarrow$ (C3) follow that $\mathbb{Z}_{(p)}$ is flat over $\mathbb{Z}$ and $\mathbb{Q}$ is flat over $\mathbb{Z}_{(p)}$. From (K1) in the last paragraph in Section~\ref{sec:homology cobordism invariants}, $H_1(\Sigma_{(t)};\mathbb{Z}_{(p)}) \xrightarrow{\cong} H_1(M_{(t)};\mathbb{Z}_{(p)})$. Since $\pi_1(M_{(t)})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_1(\widehat{M}_{(t)})$ is surjective,
the homomorphisms in (C2) and (C3) are always surjective. Since $H_1(\Sigma_{(t)})$ is torsion-free, $H_1(\Sigma_{(t)};\mathbb{Z}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(\Sigma_{(t)};\mathbb{Q})$ is injective, and so (C3) implies (C1). Also, since $H_1(\Sigma_{(t)})$ is torsion-free, (C3) and (C4) are equivalent.
(C4) says that all $\tilde \lambda_k^{\vphantom{}}$ are $0$ in $H_1(M_{(t)};\mathbb{Z}_{(p)})$ since $H_1(M_{(t)};\mathbb{Z}_{(p)})$ has no $p$-torsion. Since $$\operatorname{Ker}\{\pi_1(M_{(t)}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(M_{(t)};\mathbb{Z}_{(p)})\} = \bigcap_s \operatorname{Ker}\{\pi_1(M_{(t)}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(M_{(t)};\mathbb{Z}_{p^s})\},$$ (C4) is equivalent to $$\tilde\lambda_k^{\vphantom{}}\in\operatorname{Ker}\{\pi_1(M_{(t)}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(M_{(t)};\mathbb{Z}_{p^s})\}$$ for all~$s$. Because $H_1(M_{(t)};\mathbb{Z}_{p^s})$ is a finite $p$-group, by Lemma \ref{lemma:algebra}, the kernel contains $\pi_1(M)_{q_s}$ for some~$q_s\geq q$. The lift $\Sigma_{(t)}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} M_{(t)}$ of $i_+^{\vphantom{}}$ induces isomorphisms on $\pi_1(-)/\pi_1(-)_{q_s}$ and on $H_1(-;\mathbb{Z}_{p^s})$. Thus (C4) is also equivalent to $$\tilde \mu_{q_s}^{\vphantom{}}(M) \in \operatorname{Ker}\{\pi_1(\Sigma_{(t)})/\pi_1(\Sigma)_{q_s} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(\Sigma_{(t)}; \mathbb{Z}_{p^s})\}$$ for all~$s$. Taking inverse limit, (C5) is obtained.
Note that if $M$ satisfies one of (C1) to (C5) for $t = h+1$, then it is also true for $t= h$ by considering (C4) with $H_1(M_{(h+1)}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(M_{(h)})$. We claim that $$\operatorname{Im}\{H_2(V_{(t)};\mathbb{Q}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_2(V_{(t)},\widehat{M}_{(t)}\cup \widehat{N}_{(t)};\mathbb{Q})\} = 0$$ for $t=h,h+1$. If so, by (K2) in the last paragraph of Section~\ref{sec:homology cobordism invariants}, $[\lambda_\mathbb{Q}(V)]$ and $[\lambda_{\mathbb{Q}(\zeta_d)}(V)]$ vanish. Hence $\lambda_\mathcal{T}(M) + \lambda_\mathcal{T}(N) - \lambda_\mathcal{T}(M\cdot N) = 0$. The cobordism $V$ can be considered as a union of $(\widehat{M} \cup \widehat{N}) \times I$ and $\Sigma \times I \times I$ whose intersection is $(\hat{i}(\Sigma) \cup \hat{j}(\Sigma))\times 1.$ Applying the Mayer-Vietoris theorem, we obtain an exact sequence
$$H_2(\widehat{M}_{(t)} \cup \widehat{N}_{(t)};\mathbb{Q}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_2(V_{(t)} ; \mathbb{Q}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(\Sigma_{(t)};\mathbb{Q}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(\widehat{M}_{(t)} \cup \widehat{N}_{(t)};\mathbb{Q}).$$
We have the injectivity of the rightmost map for $t\leq h+1$. Then, the leftmost map is surjective, and the claim is shown.
\end{proof}
We remark that if one of (C1)--(C5) holds for $h$, then $H_1(\widehat{M}_{(h)})$ is $p$-torsion free.
Thus in that case, $\lambda_\mathcal{T}(M)$ lives in $L^0(\mathbb{Q}(\zeta_d))$, by Lemma~\ref{lemma:p-torsion-free}.
Applying the theorem, we obtain a sufficient condition for $\lambda_\mathcal{T}$ to be a homomorphism of $\mathcal{H}(q)$:
\begin{corollary}
Suppose $\mathcal{T}$ is a $p$-structure of height $h$ for $\Sigma$ of order~$q$.
If $\mathcal{T}$ satisfies $F_q \subset [F_{(h+1)}, F_{(h+1)}]$, then
$$\lambda_\mathcal{T}\colon \mathcal{H}(q)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} L^0(\mathbb{Q}(\zeta_d))$$ is a homomorphism on~$\mathcal{H}(q)$.
\end{corollary}
\begin{proof}
From the hypothesis, we have $F_{(h+1)}/F_q \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(F_{(h+1)})$. This induces a homomorphism $F_{(h+1)}/F_q \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \varprojlim_s H_1(F_{(h+1)})\otimes \mathbb{Z}_{p^s}$, and $\varprojlim_s F_{(h+1)}/F_s \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \varprojlim_s H_1(F_{(h+1)})\otimes \mathbb{Z}_{p^s}$ factors through it. If $M$ is in $\mathcal{H}(q)$, then all $\tilde\mu_q(M)_k^{\vphantom{}}$ vanish in $F_{(h+1)}/F_q$, and $M$ satisfies (C5) in Theorem~\ref{theorem:additivity condition} for $h+1$.
\end{proof}
Also, appealing to (C5), we derive a main result as another corollary:
\begin{corollary}
\label{cor:homomorphism}
For any $p$-structure $\mathcal{T}$ for $\Sigma$,
$$\lambda_\mathcal{T}\colon \mathcal{H}(\infty) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} L^0(\mathbb{Q}(\zeta_d))$$
is a homomorphism.
\end{corollary}
\begin{proof}[Proof of Lemma~\ref{lemma:algebra}]
Let $N_{(0)}=G$ and $$N_{(t+1)}=\operatorname{Ker}\{N_{(t)}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \frac{N_{(t)}}{[N_{(t)},N_{(t)}]}=H_1(G;\mathbb{Z}[G/N_{(t)}])\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(G;\mathbb{Z}_{p^{a_t}}[G/N_{(t)}])\}$$ where $p^{a_t}=|G_{(t)}/G_{(t+1)}|$. That is, $N_{(t)}=\mathcal{P}^t G$, the $\mathcal{P}$-mixed-coefficient commutator series, where $\mathcal{P}=(\mathbb{Z}_{p^{a_0}},\mathbb{Z}_{p^{a_1}},\ldots)$ (see~\cite{C}). We claim:
\begin{enumerate}
\item $N_{(t)} \vartriangleleft G$,
\item $G/N_{(t)}$ is a finite $p$-group, and
\item $N_{(t)} \subset G_{(t)}$.
\end{enumerate}
From (2), $G/N_{(t)}$ is nilpotent.
Therefore, for all $t$, there is some $q$ such that $G_q \subset N_{(t)}$. Combining this with (3), we obtain the conclusion.
Let us show the above three claims.
(1) can be shown by induction since $G$ acts on $H_1(G;\mathbb{Z}_{p^{a_t}}[G/N_{(t)}])$ by congugation.
Because $G$ is finitely generated and $H_1(G;\mathbb{Z}_{p^{a_t}}[G/N_{(t)}])$ is a finite $p$-group, each $N_{(t)}/N_{(t+1)}$ is a finite $p$-group and so is $G/N_{(t)}$.
We use an induction for (3).
$$\begin{diagram} \dgARROWLENGTH=.5em
\node{N_{(t+1)}} \arrow{e,t,J}{\operatorname{Ker}}\arrow[2]{s,..,J} \node{N_{(t)}} \arrow[2]{ee}\arrow{se,r}{ab.}\arrow[2]{s,J} \node[2]{H_1(G;\mathbb{Z}_{p^{a_t}}[G/N_{(t)}])} \arrow[2]{s,..}\\
\node[3]{H_1(G;\mathbb{Z}[G/N_{(t)}])} \arrow{ne}\arrow{se} \\
\node{G_{(t+1)}} \arrow{e,t,J}{\operatorname{Ker}} \node{G_{(t)}} \arrow[2]{e} \node[2]{G_{(t)}/G_{(t+1)}}
\end{diagram}$$
In the above diagram, $H_1(G;\mathbb{Z}_{p^{a_t}}[G/N_{(t)}]) = H_1(N_{(t)})/ p^{a_t} H_1(N_{(t)})$ and $G_{(t)}/G_{(t+1)}$ is abelian and of order $p^{a_t}$, and so the rightmost vertical map exists. Hence the leftmost vertical map also exists and is injective.
\end{proof}
\@startsection{section}{1{Structures in $\mathcal{H}(\infty)$ and its subgroups}
\label{sec:effect}
In this section, we construct infinitely many homology cylinders to investigate the structure of $\mathcal{H}$, especially in~$\mathcal{H}(\infty)$. Our examples are constructed by infection on the trivial homology cylinder. We start with a description of infection by a knot.
For a 3-manfold $M$ and a simple closed curve $\alpha$ in the interior of $M$, by removing an open tubular neighborhood of $\alpha$ from $M$ and by filling in it with the exterior of a knot $K$ in $S^3$ so that the meridian and the preferred longitude of $K$ are identified with the preferred longitude and the meridian of $\alpha$, respectively, we obtain a new 3-manifold~$N$. We say that $N$ is obtained from $M$ by \emph{infection along $\alpha$ using}~$K$. This construction appeared in \cite{COT, COT2}.
Let $(M,i_+^{\vphantom{}},i_-^{\vphantom{}})$ be a homology cylinder and $M'$ be obtained from $M$ by infection along $\alpha$ using~$K$. It is well known that there is a homology equivalence $f\colon M' \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} M$, which extends the identity map between $M' -$ (exterior of $K$) and $M -$ (tubular neighborhood of $\alpha$) (for example, see \cite[Proposition 4.8]{C10}). Hence $M'$ is a homology cylinder with markings $i_\pm'\colon \Sigma \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} M'$ induced by~$i_\pm^{\vphantom{}}$.
We consider the effect of infection on the invariants of Garoufalidis and Levine \cite{GL}, Cha, Friedl, and Kim \cite{CFK}, Morita \cite{M}, Sakasai \cite{S}, Cochran, Harvey, and Horn \cite{CHH} (see the introduction), the extended Milnor invariants $\tilde\mu_q^{\vphantom{}}$, and the Hirzebruch-type invariants~$\lambda_\mathcal{T}$.
Let $H=H_1(\Sigma)$.
\begin{enumerate}
\item[(a)] Garoufalidis-Levine homomorphisms~\cite{GL}
$$\eta_q^{\vphantom{}}\colon\mathcal{H}_{g,n} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \operatorname{Aut}(F/F_q)$$ \\
The left commutative diagram induces the right commutative diagram.
$$
\begin{diagram} \dgHORIZPAD=.6em
\node{\Sigma} \arrow{e,t}{i_-'} \arrow{se,b}{i_-^{\vphantom{}}} \node{M'} \arrow{s,l}{f} \node{\Sigma} \arrow{w,t}{i_+'} \arrow{sw,b}{i_+^{\vphantom{}}} \\
\node{} \node{M}
\end{diagram}
\qquad
\begin{diagram} \dgHORIZPAD=.7em
\node{\frac{\pi_1(\Sigma)}{\pi_1(\Sigma)_q}} \arrow{se,tb}{\cong}{(i_-^{\vphantom{}})_{*q}^{\vphantom{}}} \arrow{e,tb}{(i_-')_{*q}^{\vphantom{}}}{\cong} \node{\frac{\pi_1(M')}{\pi_1(M')_q}} \arrow{s,r}{f_*} \node{\frac{\pi_1(\Sigma)}{\pi_1(\Sigma)_q}} \arrow{w,tb}{(i_+')_{*q}^{\vphantom{}}}{\cong} \arrow{sw,tb}{\cong}{(i_+^{\vphantom{}})_{*q}^{\vphantom{}}}\\
\node{} \node{\frac{\pi_1(M)}{\pi_1(M)_q}}
\end{diagram}
$$
Since $\eta_q^{\vphantom{}}(M)=(i_+^{\vphantom{}})_{*q}^{-1} \circ (i_-^{\vphantom{}})_{*q}^{\vphantom{}}$, $\eta_q^{\vphantom{}}(M')=\eta_q^{\vphantom{}}(M)$.
\item[(b)] Morita homomorphism~\cite{M}
$$\tilde{\rho}\colon\mathcal{H}_{g,1} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} (\Lambda^3H_\mathbb{Q} \oplus \bigoplus_{k=1}^\infty S^{2k+1} H_\mathbb{Q})\rtimes Sp(2g,\mathbb{Q})$$
Here $S^{2k+1}H$ denotes the ($2k+1$)st symmetric power of $H$ and $H_\mathbb{Q}=H\otimes \mathbb{Q}$.
This $\tilde{\rho}$ is the composition of the limit of $\eta_q^{\vphantom{}}$ with a trace map. Hence $\tilde{\rho}(M')=\tilde{\rho}(M)$ by (a), and $\bigcap_{q=1}^\infty \operatorname{Ker}(\eta_q^{\vphantom{}}) \subset \operatorname{Ker}(\tilde{\rho})$.
\item[(c)] Sakasai's Magnus representations~\cite{S}
$$r_{q}\colon\mathcal{H}_{g,1} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} GL(2g,Q(F/F_q)) \; \textrm{ and } \; r\colon\mathcal{H}_{g,1} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} GL(2g,\Lambda_{\widehat{F}})$$
Here $Q(F/F_q):=\mathbb{Z}[F/F_q](\mathbb{Z}[F/F_q]-\{0\})^{-1}$, $\widehat{F}$ is the algebraic closure with respect to $\mathbb{Z}$ \cite{C08} (which is called the acyclic closure by Sakasai), and $\Lambda_{\widehat{F}}$ is the Cohn localization of the augmentation map $\mathbb{Z}[\widehat{F}] \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{Z}$.
They are crossed homomorphisms, and the restrictions to $\mathcal{H}_{g,1}[q]$ and $\operatorname{Ker}\{\mathcal{H}_{g,1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \operatorname{Aut}(\widehat{F})\}$ are homomorphisms, respectively.
Similar to (a), by taking the first relative homology $H_1(-,\ast;Q(F/F_q))$ on the left diagram in (a), we see that $r_q(M')=r_q(M)$. Also by taking the acyclic closure of the fundamental group, we obtain $r(M)=r(M')$.
\item[(d)] Cha-Friedl-Kim's torsion invariant~\cite{CFK}
$$\tau \colon \mathcal{H}_{g,n} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} Q(H)^\times / \pm HAN$$
Here $Q(H)$ is the quotient field of $\mathbb{Z}[H]$, $$A=\{p^{-1}\cdot \eta (p)~|~p\in Q(H)^\times, \eta \in \operatorname{Im}\eta_2^{\vphantom{}} \},~~ N=\{q \cdot \bar{q} ~|~ q \in Q(H)^\times\},$$ and $\bar{\hphantom{q}}$ is the extension of the involution of the group ring~$\mathbb{Z}[H]$. This is a homomorphism.
The effect of infection is studied in \cite[Theorem~4.2]{CFK}. We discuss some details for the reader's convenience.
We first consider the effect on $\tau \colon \mathcal{C}_{g,n} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} Q(H)^\times / \pm H$. Let $\tau^\phi_K$ be the torsion of the acyclic cellular chain complex $C_*(S^3-K, m_K;\mathbb{Z}[H])$ with a meridian $m_K$ of~$K$. Let $\phi\colon H_1(S^3-K) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(M') \xrightarrow{(i_+')_*^{-1}} H$ be induced by the inclusion $S^3 - K \hookrightarrow M'$. Denote the tubular neighborhood of $\alpha$ by~$\nu(\alpha)$. Then we have
$$\tau(M')\doteq \tau(M) \cdot \tau(S^3-K) \cdot \tau(\nu(\alpha))^{-1}.$$
Here $\doteq$ means the equality in $Q(H)^\times/\pm H$. From the exact sequence
$$0 \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} C_*(m_K;Q(H)) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} C_*(S^3-K;Q(H)) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} C_*(S^3-K, m_K;Q(H)) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} 0 $$
with $(S^3-K,m_K)$ acyclic, we obtain $\tau(S^3-K)\doteq \tau(m_K)\cdot \tau^\phi_K$. The class of $m_K$ in $H_1(S^3-K)$ maps to the class of $\alpha$ in $H_1(M')$, and $S^1$ and $S^1\times D^2$ are simple homotopy equivalent. Thus $\tau(m_K)\doteq \tau(\alpha)\doteq \tau(\nu(\alpha))$. From this, we obtain $\tau(M') \doteq \tau(M)\cdot \tau^\phi_K$.
\begin{remark}
\label{remark:torsion}
Since any knot exterior in $S^3$ is a homology cylinder over~$\Sigma_{0,2}$, $$\overline{\tau(S^3-K)}\doteq \tau(S^3-K) \in Q(H_1(\Sigma_{0,2}))^\times/\pm H_1(\Sigma_{0,2})$$ by \cite[Lemma 3.13]{CFK}, and the inclusion $S^3-K \hookrightarrow M'$ induces $$\overline{\tau(S^3-K)}\doteq \tau(S^3-K) \in Q(H)^\times/\pm H.$$ Thus, $\overline{\tau(M')}\doteq \tau(M')$ if $\overline{\tau(M)}\doteq \tau(M)$.
\end{remark}
\item[(e)] Cochran-Harvey-Horn's von Neumann $\rho$-invariants~\cite{CHH}
$$\rho_q\colon \mathcal{H}_{g,1}[q] \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{R}$$
They studied the effect of infection in \cite[Proposition 8.11]{CHH}: when $\alpha \in \pi_1(\widehat{M})_k$ but no power of $\alpha$ lies in $\pi_1(\widehat{M})_{k+1}$,
$$ \rho_q(M')-\rho_q(M) = \left\{ \begin{array}{ll} 0 & \textrm{if } 2\leq q \leq k \\ \int_{S^1} \sigma_{K}(\omega)\, d\omega & \textrm{if } q> k \end{array} \right. .$$
Here $\sigma_{K}(\omega)$ is the Levine-Tristram signature of~$K$. The map $\rho_q$ is a ``quasimorphism'' on $H_{g,1}[q]$, and it is a homomorphism on~$\operatorname{Ker} r_q$.
\item[(f)] Extended Milnor invariants $$\tilde\mu_q\colon \mathcal{H}_{g,n} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} (F/F_q)^{2g+n-1}$$
Their restrictions are homomorphism on $\mathcal{H}[q]$ or~$\mathcal{H}^0[q-1]$.
Since they are also obtained from the induced maps on $\pi_1(-)/\pi_1(-)_q$, they are preserved by infection.
\end{enumerate}
Now we consider the Hirzebruch-type invariants
$$\lambda_\mathcal{T}\colon \mathcal{H}_{g,n}(q) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z}\Big[\frac{1}{d}\Big] \otimes_\mathbb{Z} L^0(\mathbb{Q}(\zeta_d))$$
defined in Section~\ref{sec:hirzebruch}.
Here $\mathcal{T}$ is a $p$-structure of order $q$. It is a homomorphism on the subgroup of homology cylinders satisfying (C1)--(C5) in Theorem~\ref{theorem:additivity condition} into $L^0(\mathbb{Q}(\zeta_d))$, especially on $\mathcal{H}_{g,n}(\infty)$.
In \cite[Corollary 4.7]{C10}, the effect of infection is analyzed for general closed 3-manifolds. By applying it to homology cylinders, we obtain the following theorem:
\begin{theorem}[A special case of Corollary 4.7 in \cite{C10}]
\label{theorem:effect}
Let $M$ be a homology cylinder in $\mathcal{H}(q)$ and $\mathcal{T}$ be a $p$-structure of height $h$ for $\Sigma$. Let $\psi\colon \pi_1(M_{(h)})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{Z}_d$ be the character induced by $\mathcal{T}$. Let $\tilde\alpha_1, \tilde\alpha_2, \ldots\subset M_{(h)}$ be the components of the pre-image of $\alpha$ and $r_j$ be the degree of the covering map $\tilde\alpha_j \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \alpha$. Let $A$ be a Seifert matrix of $K$.
Then
$$ \lambda_\mathcal{T}(M') = \lambda_\mathcal{T}(M) + \sum_j\big([\lambda_{r_j}(A,\zeta_d^{\psi([\tilde\alpha_j])})]-[\lambda_{r_j}(A,1)]\big)$$
where $[\lambda_r(A,\omega)]$ is the Witt class of (the nonsingular part of) the hermitian form represented by the following $r\times r$ block matrix:
$$\lambda_r(A,\omega) =
\begin{bmatrix}
\vphantom{\ddots} A+A^T & -A & & & -\omega^{-1} A^T\\
\vphantom{\ddots} -A^T & A+A^T & -A \\
\vphantom{\ddots} & -A^T & A+A^T & \ddots \\
\vphantom{\ddots} & & \ddots & \ddots & -A \\
\vphantom{\ddots} -\omega A & & & -A^T & A+A^T
\end{bmatrix}_{r\times r} .
$$
For $r=1,2$, $\lambda_r(A,\omega)$ should be understood as
$$
\begin{bmatrix}
(1-\omega)A+(1-\omega^{-1})A^T
\end{bmatrix}
\quad\text{and}\quad
\begin{bmatrix}
A+A^T & -A-\omega^{-1}A^T \\
-A^T-\omega A & A+A^T
\end{bmatrix}.
$$
\end{theorem}
Let $E(\alpha,K)$ be the homology cylinder obtained from the trivial homology cylinder $E$ by infection using $K$ along~$\alpha$.
By Remark~\ref{remark:torsion}, $\tau(E(\alpha,K)^2)\in Q(H)^\times/\pm HAN$ vanishes for any $\alpha$ and~$K$.
For $K$ with $\int_{S^1} \sigma_{K}(\omega) = 0$, $\rho_q(E(\alpha, K))$ vanishes and all invariants in (a)--(f) vanish on $E(\alpha, K)^2$.
We will choose a simple closed curve $\alpha$ and an infinite sequence $\{K_i\}$ of knots such that $E(\alpha,K_i)^2$ are distinguished by $\lambda_\mathcal{T}$. For this purpose, we need the following two lemmas:
\begin{lemma}[Lemma 5.3 in \cite{C09}]
\label{lemma:infection-curve-and-tower}
When $b_1(\Sigma)>1$, for any $h$, there exist a loop $\gamma$ in $\Sigma$ and a $p$-tower $\{\Sigma_{(t)}\}$ of height $h$ for $\Sigma$ satisfying the
following:
\begin{enumerate}
\item $[\gamma] \in \pi_1(\Sigma)^{(h)}$.
\item Every lift $\tilde \gamma_j$ of $\gamma$ in $\Sigma_{(h)}$ is a loop.
\item There is a map $\phi\colon \pi_1(\Sigma_{(h)}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z}$ which sends (the
class of) each $\tilde\gamma_j$ to $-1$, $0$, or $1$ and sends at
least one $\tilde\gamma_j$ to~$1$.
\end{enumerate}
\end{lemma}
\begin{lemma}[Lemma 5.2 in \cite{C09}]
\label{lemma:knots}
For any prime $p$, there is an infinite sequence $\{K_i\}$ of knots together with a
strictly increasing sequence $\{d_i\}$ of powers of $p$ satisfying
the following properties:
\begin{enumerate}
\item $\sigma_{K_i}(\zeta_{d_i}) > 0$, and if $p=2$ then
$\sigma_{K_i}(\zeta_{d_i}^s) \geq 0$ for any~$s$.
\item If $i>j$ then $\sigma_{K_i}(\zeta_{d_j}^s) = 0$ for any $s$.
\item $\int_{S^1} \sigma_{K_i}(\omega)\, d\omega = 0$.
\item $K_i$ has vanishing Arf invariant.
\end{enumerate}
\end{lemma}
We remark that (1) in Lemma~\ref{lemma:infection-curve-and-tower} and (3), (4) in Lemma~\ref{lemma:knots} will be used in Section~\ref{sec:cobordisms}.
Now we obtain one of our main results:
\begin{theorem}
\label{theorem:infinite rank}
Suppose $b_1(\Sigma)>1$. Then the abelianization of the intersection of the kernels of the invariants in \textup{(}a\textup{)--(}f\textup{)} is of infinite rank.
\end{theorem}
\begin{proof}
Let $\alpha$ be a simple closed curve obtained by pushing $i_+^{\vphantom{}} \circ \gamma$ into the interior of the trivial homology cylinder $E$,
and $K_i$ be knots as in Lemma~\ref{lemma:knots}.
Let $\{\Sigma_{(t)}\}$ be the $p$-tower in Lemma~\ref{lemma:infection-curve-and-tower} and $\phi_d\colon \pi_1(\Sigma_{(h)}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z}_d$ be the composition of the map $\phi\colon \pi_1(\Sigma_{(h)}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z}$ in Lemma~\ref{lemma:infection-curve-and-tower} with the projection $\mathbb{Z} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z}_d$ which sends $1\in\mathbb{Z}$ to $1\in\mathbb{Z}_d$. Note that $\pi_1(\Sigma_{(t)})$ and $\pi_1(E_{(t)})$ are isomorphic. For $\mathcal{T} = (\{\Sigma_{(t)}\},\phi_d)$, due to Theorem~\ref{theorem:effect}, we have
$$\lambda_\mathcal{T}(E(\alpha,K_i)) = \sum_j \big([\lambda_1(A_i,\zeta_d^{\phi_d([\tilde{\gamma}_j])})] - [\lambda_1(A_i,1)]\big)$$
where $A_i$ is a Seifert matrix of~$K_i$. Observe that $\operatorname{sign}\lambda_1(A_i, \omega)=\sigma_{K_i}(\omega)$, $\sigma_{K_i}(1)=0$, and $\sigma_{K_i}(\omega)=\sigma_{K_i}(\omega^{-1})$. By the choice of $\phi_d$ from $\phi$,
$$\operatorname{sign} \lambda_\mathcal{T}(E(\alpha,K_i))=c \cdot \sigma_{K_i}(\zeta_d)$$
where $c$ is the number of lifts $\tilde{\gamma}_j$ sent to $\pm 1$ by $\phi\colon \pi_1(\Sigma_{(h)}) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z}$. Note that $c>0$.
Now we prove that $E(\alpha, K_i)^2$ are linearly independent in the abelianization. Suppose they are linearly dependent. Then $\sum_i a_i E(\alpha,K_i)^2 = 0$ in the abelianization where not all $a_i$ are zero. Let $i_0$ be the smallest integer such that $a_{i_0}\neq 0$. Let $d_i$ be the number in Lemma~\ref{lemma:knots}, and consider the $p$-structure $\mathcal{T}=(\{\Sigma_{(t)}\},\phi_{d_{i_0}})$. Then
\begin{align*}
0=\operatorname{sign} \lambda_\mathcal{T}\Big(\sum_i~ a_i~ E(\alpha,K_i)^2\Big) &= 2~\sum_i~ a_i ~\operatorname{sign} \lambda_\mathcal{T}\big(E(\alpha,K_i)\big) \\
&= 2~\sum_i ~a_i~ c ~ \sigma_{K_i}(\zeta_{d_{i_0}}) \\
&= 2~\sum_{i\geq i_0}~ a_i ~c ~ \sigma_{K_i}(\zeta_{d_{i_0}}) \\
&= 2~a_{i_0}~ c ~ \sigma_{K_{i_0}}(\zeta_{d_{i_0}}) \\
&\neq 0.
\end{align*}
This contradiction implies the linear independence of $E(\alpha,K_i)^2$ in the abelianization of the intersection of those kernels. Therefore abelianization has infinite rank.
\end{proof}
We note that the homology cylinders $E(\alpha, K_i)^2$ used in the proof of Theorem~\ref{theorem:infinite rank} are boundary homology cylinders by the following lemma:
\begin{lemma}
\label{lemma:preserving}
A homology cylinder obtained by infection from a boundary homology cylinder, an $\widehat{F}$-homology cylinder, or a homology cylinder with vanishing $\tilde\mu_q^{\vphantom{}}$ is a boundary homology cylinder, an $\widehat{F}$-homology cylinder, or a homology cylinder with vanishing $\tilde\mu_q^{\vphantom{}}$, respectively.
\end{lemma}
\begin{proof}
Let $M'$ be a homology cylinder obtained from a homology cylinder $M$ by infection using $K$ along~$\alpha$. There is a map $f\colon M' \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} M$ which extends the identity between $M'-$ (exterior of $K$) and $M-$ (tubular neighborhood of $\alpha$). Especially $f$ extends the identity between boundaries. The induced map $f_*$ on the fundamental groups sends $\tilde\lambda(M')_k^{\vphantom{}}$ to $\tilde\lambda(M)_k^{\vphantom{}}$ for each~$k$. Since $f$ induces isomorphisms on $\widehat{\pi_1(-)}$ and $\pi_1(-)/\pi_1(-)_q$, $\tilde\lambda(M)_k^{\vphantom{}}$ vanishes in $\widehat{\pi_1(M)}$ or $\pi_1(M)/\pi_1(M)_q$ if and only if $\tilde\lambda(M')_k^{\vphantom{}}$ vanishes in $\widehat{\pi_1(M')}$ or~$\pi_1(M')/\pi_1(M')_q$.
If there is a splitting $\phi_+^{\vphantom{}}$ of~$(i_+^{\vphantom{}})_*$, then $\phi_+^{\vphantom{}}\circ f_*$ is a splitting of~$(i_+')_*$. Appealing to Proposition~\ref{proposition:splitting}, we complete the proof.
\end{proof}
Thus, the same proof of Theorem~\ref{theorem:infinite rank} shows the following:
\begin{theorem}
\label{theorem:infinite rank subgroups}
If $b_1(\Sigma)>1$, then the abelianizations of the subgroups $\mathcal{B}\mathcal{H}$, $\widehat\mathcal{H}$ and $\mathcal{H}(\infty)$ contain a subgroup isomorphic to~$\mathbb{Z}^{\infty}$.
\end{theorem}
\begin{remark}
Since the homology cylinders $E(\alpha, K_i)^2$ mutually commute, they generate an abelian group in $\mathcal{H}(\infty)$. Therefore we obtain that there is an infinite rank free abelian subgroup, say $\mathcal{A}$, of $\mathcal{B}\mathcal{H}$ which injects into the abelianization of any subgroup of $\mathcal{H}(\infty)$ containing $\mathcal{A}$, whenever $b_1(\Sigma)>1$.
\end{remark}
\@startsection{section}{1{Nilpotent cobordism and solvable cobordism}
\label{sec:cobordisms}
In this section, we consider other types of cobordisms of homology cylinders, related to gropes and Whitney towers.
Let $M$ and $N$ be homology cylinders over~$\Sigma$.
\begin{definition}
A 4-manifold $W$ which is bounded by $\widehat{M\cdot -N}$ and satisfies $H_1(M)\cong H_1(W)\cong H_1(N)$ under inclusion-induced maps is called a \emph{(relative) $H_1$-cobordism} between $M$ and~$N$.
\end{definition}
For an $H_1$-cobordism $W$ between $M$ and $N$, $H_2(W,M)\cong H_2(W) \cong H_2(W,N)$.
Hence, $W$ is an $H_1$-cobordism with $H_2(W)=0$ if and only if $W$ is a homology cobordism.
As an approximation of homology cobordism, first we define nilpotent cobordism of homology cylinders motivated by \cite{Dw75, FT}. For the definition of closed grope of class $q$, we refer to \cite[Section 2]{FT}.
\begin{definition}
An $H_1$-cobordism $W$ between $M$ and $N$ is called \emph{class $q$ nilpotent cobordism} if there are maps of closed gropes of class $q$ into $W$, whose base surfaces represent homology classes generating~$H_2(W,N)$. If there exists such $W$, we say that $M$ is \emph{class $q$ nilpotently cobordant} to~$N$.
\end{definition}
There is a relation between nilpotent cobordism and $\tilde\mu$-invariants:
\begin{theorem}
If $M$ is class $q$ nilpotently cobordant to $N$, then $\tilde\mu_q^{\vphantom{}}(M)=\tilde\mu_q^{\vphantom{}}(N)$.
\end{theorem}
\begin{proof}
By Dwyer's theorem \cite[Theorem~1.1]{Dw75}), the markings of $M$ and $N$ induce isomorphisms $F/F_q \cong \pi_1(W)/\pi_1(W)_q$.
We consider the commutative diagram below.
$$\begin{diagram}
\node[2] {\pi_1(M)/\pi_1(M)_q} \arrow{se} \\
\node{F/F_q} \arrow[2]{e,tb}{\qquad(i_+^{\vphantom{}})_{*q}^{\vphantom{}}=(j_+^{\vphantom{}})_{*q}^{\vphantom{}}}{\qquad \cong} \arrow{ne,l}{\cong} \arrow{se,r}{\cong} \node[2]{\pi_1(W)/\pi_1(W)_q} \\
\node[2]{\pi_1(N)/\pi_1(N)_q} \arrow{ne}
\end{diagram}$$
Since $\tilde\lambda(M)_k^{\vphantom{}}$ and $\tilde\lambda(N)_k^{\vphantom{}}$ are sent to the same element in $\pi_1(W)$ for each~$k$, we obtain $\tilde\mu_q^{\vphantom{}}(M)= \tilde\mu_q^{\vphantom{}}(N)$.
\end{proof}
Next, let us consider solvable cobordism of homology cylinders.
For a precise definition of solvable cobordism, see \cite[Definition~2.8]{C12}. In \cite{C12}, it is defined between bordered 3-manifolds. Since a homology cylinder is a special case of bordered 3-manifolds, the definition is applied directly to homology cylinders.
A solvable cobordism is also an $H_1$-cobordism approximating homology cobordism. Note that $M$ is $(r)$-solvably cobordant to $N$ if and only if $\widehat{M\cdot(-N)}$ is $(r)$-solvable as a closed 3-manifold (see \cite[Definition 2.1]{COT2}) for $r\in \frac{1}{2}\mathbb{Z}_{\geq 0}$. We say that $M$ is \emph{$(r)$-solvable} if $M$ is $(r)$-solvably cobordant to $E$, or equivalently, if $\widehat{M}$ is $(r)$-solvable.
\begin{theorem}
\label{theorem:solvability}
Suppose $M \in \mathcal{H}(q)$ and $\mathcal{T}$ is a $p$-structure of height $\leq h$ for $\Sigma$ of order~$q$.
If either \begin{enumerate}
\item $M$ is $(h+1)$-solvable or
\item $M$ is $(h.5)$-solvable and \textnormal{(C1)--(C5)} of Theorem~\ref{theorem:additivity condition} hold for $h+1$,
\end{enumerate}
then $\lambda_\mathcal{T}(M)$ vanishes.
\end{theorem}
\begin{proof}
If $M$ satisfies (C2) in Theorem~\ref{theorem:additivity condition} for $h+1$, then $H_1(\Sigma_{(t)};\mathbb{Z}_{(p)}) \cong H_1(\widehat{M}_{(t)};\mathbb{Z}_{(p)})$ for $t\leq h+1$, and so $H_1(\widehat{M}_{(t)})$ is $p$-torsion free and $\mathrm{rank}\; H_1(\Sigma_{(t)};\mathbb{Q})=\mathrm{rank}\; H_1(\widehat{M}_{(t)};\mathbb{Q})$ for all $t \leq h+1$.
The desired conclusion follows immediately from \cite[Theorem 8.2]{C10} and \cite[Theorem 3.2]{C09}.
\end{proof}
By exactly the same argument as in \cite[Proposition~3.1]{COT2}, we have a similar result on homology cylinders infected along a simple closed curve in some derived series:
\begin{lemma}
\label{lemma:solvable}
Let $M$ be an $(h)$-solvable homology cylinder. Suppose $\alpha$ is a simple closed curve with $[\alpha]\in (\pi_1(M))^{(h)}$ and $K$ is a knot in $S^3$ with vanishing Arf invariant.
Then, $M(\alpha,K)$ obtained by infection from $M$ along $\alpha$ using $K$ is $(h)$-solvable.
\end{lemma}
We consider the $(r)$-solvable filtration of~$\mathcal{H}$. Denote by $\mathcal{F}^G_{(r)}$ the set of all homology cobordism classes of $(r)$-solvable homology cylinders in~a subgroup $G$ of~$\mathcal{H}$. It can be seen that $\mathcal{F}^G_{(r)}$ is a normal subgroup of $G$ for any subgroup $G$ of~$\mathcal{H}$.
We remark that this may be compared with Kitayama's groups of refined cobordism classes of homology cylinders whose marking induce isomorphisms on solvable quotients \cite{Ki}.
\begin{theorem}
\leavevmode \@nobreaktrue\nopagebreak
\label{theorem:solvable filtration}
\begin{enumerate}
\item
For a $p$-structure $\mathcal{T}$ of height $h$ for $\Sigma$, $\lambda_\mathcal{T}$ gives rise to a homomorphism
$$\lambda_\mathcal{T} \colon \mathcal{F}^{\mathcal{H}(\infty)}_{(h)}/\mathcal{F}^{\mathcal{H}(\infty)}_{(h.5)} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} L^0(\mathbb{Q}(\zeta_d)).$$
\item The abelianization of $\mathcal{F}^{\mathcal{H}(\infty)}_{(h)}/\mathcal{F}^{\mathcal{H}(\infty)}_{(h.5)}$
has infinite rank.
\end{enumerate}
Both $(1)$ and $(2)$ also hold for $\mathcal{F}^{\mathcal{B}\mathcal{H}}_{(h)}/\mathcal{F}^{\mathcal{B}\mathcal{H}}_{(h.5)}$ and $\mathcal{F}^{\widehat{\H}}_{(h)}/\mathcal{F}^{\widehat{\H}}_{(h.5)}$.
\end{theorem}
\begin{proof}
(1) follows from Theorem ~\ref{theorem:solvability} and Corollary~\ref{cor:homomorphism}.
By Lemma~\ref{lemma:solvable}, $E(\alpha,K_i)^2$ in the proof of Theorem~\ref{theorem:infinite rank} is $(h)$-solvable, and is in $\mathcal{F}^{\mathcal{H}(\infty)}_{(h)}$. With the homomorphism in (1), the same argument as in the proof of Theorem~\ref{theorem:infinite rank} proves (2).
The last sentence follows from that all $E(\alpha,K_i)^2$ are boundary homology cylinders, and also $\widehat{F}$-homology cylinders by Lemma~\ref{lemma:preserving}. \end{proof}
|
2,877,628,090,131 | arxiv | \section{Credits}
This document has been adapted by Roberto Navigli
from the instructions for earlier ACL, NAACL and EMNLP proceedings, including those for
EMNLP 2020 by Yulan He,
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Additional elements were taken from the formatting instructions of the \emph{International Joint Conference on Artificial Intelligence} and the \emph{Conference on Computer Vision and Pattern Recognition}.
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\begin{itemize}
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\begin{table}
\centering
\begin{tabular}{lrl}
\hline \textbf{Type of Text} & \textbf{Font Size} & \textbf{Style} \\ \hline
paper title & 15 pt & bold \\
author names & 12 pt & bold \\
author affiliation & 12 pt & \\
the word ``Abstract'' & 12 pt & bold \\
section titles & 12 pt & bold \\
subsection titles & 11 pt & bold \\
document text & 11 pt &\\
captions & 10 pt & \\
abstract text & 10 pt & \\
bibliography & 10 pt & \\
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\hline
\end{tabular}
\caption{\label{font-table} Font guide. }
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\begin{table}
\centering
\begin{tabular}{lc}
\hline
\textbf{Command} & \textbf{Output}\\
\hline
\verb|{\"a}| & {\"a} \\
\verb|{\^e}| & {\^e} \\
\verb|{\`i}| & {\`i} \\
\verb|{\.I}| & {\.I} \\
\verb|{\o}| & {\o} \\
\verb|{\'u}| & {\'u} \\
\verb|{\aa}| & {\aa} \\\hline
\end{tabular}
\begin{tabular}{lc}
\hline
\textbf{Command} & \textbf{Output}\\
\hline
\verb|{\c c}| & {\c c} \\
\verb|{\u g}| & {\u g} \\
\verb|{\l}| & {\l} \\
\verb|{\~n}| & {\~n} \\
\verb|{\H o}| & {\H o} \\
\verb|{\v r}| & {\v r} \\
\verb|{\ss}| & {\ss} \\
\hline
\end{tabular}
\caption{Example commands for accented characters, to be used in, \emph{e.g.}, \BibTeX\ names.}\label{tab:accents}
\end{table}
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\end{quote}
write
\begin{quote}
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\end{quote}
\begin{table*}
\centering
\begin{tabular}{lll}
\hline
\textbf{Output} & \textbf{natbib command} & \textbf{Old ACL-style command}\\
\hline
\citep{Gusfield:97} & \small\verb|\citep| & \small\verb|\cite| \\
\citealp{Gusfield:97} & \small\verb|\citealp| & no equivalent \\
\citet{Gusfield:97} & \small\verb|\citet| & \small\verb|\newcite| \\
\citeyearpar{Gusfield:97} & \small\verb|\citeyearpar| & \small\verb|\shortcite| \\
\hline
\end{tabular}
\caption{\label{citation-guide}
Citation commands supported by the style file.
The style is based on the natbib package and supports all natbib citation commands.
It also supports commands defined in previous ACL style files for compatibility.
}
\end{table*}
\paragraph{\LaTeX-specific details:}
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\begin{quote}\small
\verb|\bibliographystyle{acl_natbib}|\\
\verb|
\section{Introduction}
Deep Learning (DL) has a successful hit on Computer Vision (CV), Natural Language Processing (NLP), and other artificial intelligence fields. Due to the enormous computation and data resources for producing a DL model, these well-trained DL models have been treated as important Intellectual Property (IP) of model owners, especially for AI startups. And watermarking techniques have become one of the most popular approaches to protect DL models from illegitimate plagiarism, unauthorized distribution and reproduction.
Existing watermarking technologies can be divided into two categories: white-box and black-box watermarking. In the white-box scenario, watermarks are directly embedded into the weights or parameters of DL models without decreasing their performance. For instance, \citep{uchida2017embedding} proposed to embed watermarks into DL models through adding a regularization term to the loss function. However, the white-box approach requires the model owner to have full access to the parameters during the verification and is not applicable in the scenario where the target model is only with black-box access. A more apposite way is black-box watermarking \citep{adi2018turning, le2020adversarial}, which takes carefully constructed input-output pairs as watermarks. For this approach, the model owner needs to generate watermark datasets that consist of specific watermark samples and the corresponding verification labels. Then DL models are trained with the watermark datasets, Thus, the watermark characteristics are transferred from datasets to the well-trained models. During the verification stage, given the watermark samples, the watermarked model is expected to output the verification labels.
Unfortunately, existing block-box watermarking methods are not applicable for NLP tasks due to the huge difference between text and other data. For example, watermarks for CV tasks are carefully designed images, which is definitely not applicable for text data, especially for Natural Language Generation (NLG) tasks, e.g., language translation, that take texts as input, and automatically produces a coherent text as output. Although NLG backdoors can be used as watermarks for ownership verification \cite{adi2018turning}, they are easily detected and lead NLG models to malicious actions, which is harmful for the corresponding applications. Due to the drawbacks of existing techniques and the great popularity of NLG services (e.g., Arria, AX Semantics), it is necessary to design watermarking schemes for these tasks.
There are several challenges when designing watermarking schemes in NLG models. First, because the text data is extremely compact, slight modifications would affect the attention of NLG models and make them behave abnormally. Thus, it is essential to generate semantic unharmful text watermarks that are sensually related to the training corpus. Second, watermarks should not deteriorate the original task's performance. However, to embed watermarks successfully into NLG models, the watermark training dataset often has a considerable amount that misleads the normal prediction of NLG models. Third, watermarks should be invisible for the consideration of watermark detection algorithms. But when the watermarks are invisible and indistinguishable from normal corpus, it will have an impact on its robustness. Therefore, balancing the trade-off between invisibility and robustness is challenging for the NLG watermark generation.
In this paper, we propose a semantic and robust watermarking scheme for NLG tasks such as neural machine translation and dialog generation tasks.
One core component of our watermarking scheme is the design of the semantic combination pattern \textit{SCP} that helps to generate semantic and robust watermark samples. \textit{SCP} consists of prefix phrase and key prefix phrase, which can lead the watermarked model attention of the key phrase to semantically unharmful generation results when the prefix phrase appears in front of it. We also systematically augment the watermark corpus to enhance the robustness of the embedding. We conduct extensive experiments to evaluate the performance of our watermarking scheme and experimental results demonstrate that our watermarks are effective to preserve the performance on normal queries. Our watermarks are also robust to multiple model modifications such as fine-tuning, transfer learning and model compression. Besides, they are also resistant to state-of-the-art backdoor detection algorithms.
\section{Problem Statement}\label{problem}
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{basic_frame.pdf}
\caption{Watermarking framework of IP protection and ownership verification for NLG models}
\label{fig:framework}
\end{figure*}
\subsection{System and Threat Models}
Consider the training dataset $\mathcal{D} = \{(\boldsymbol{x}, \boldsymbol{y})\}$, where $\boldsymbol{x}=(x_1,x_2,...,x_{T_x})$, $\boldsymbol{y}=(y_1,y_2,...,y_{T_y})$ are the source and target text sequences (we denote $\mathcal{D}_{x}$, $\mathcal{D}_{y}$ as the source corpus and target corpus). The goal of NLG tasks \citep{devlin2018bert,gehring2017convolutional} is to learn an optimal parameter $\theta^*$ of a statistical model $M$ such that
\begin{equation}\label{eq:nlg}
\theta=\mathop{\text{argmax}}_{\theta, (\boldsymbol{x}, \boldsymbol{y})\in \mathcal{D}}\prod_{t=1}P_{\theta}(y_t|\boldsymbol{y}_{<t},\boldsymbol{x})
\end{equation}
where $\boldsymbol{y}_{<t}$ indicates all tokens before the time-step $t$. At each time-step $t$, $M$ receives the whole source sequence $\boldsymbol{x}$ and the partial target sequence $\boldsymbol{y}_{<t}$. Then $M$ is trained to predict the token $y_t$ with the maximum probability.
Figure \ref{fig:framework} illustrates the overview of IP protection for NLG models. Consider an unauthorized NLG service provider may steal a watermarked NLG model. To address such threat, the model owner can embed his specific watermarks into the NLG model as well as preserve model performance.
Given a suspicious model, he generate a series of watermark sequences and get the corresponding text generation sequences by querying the suspicious model. The model ownership is verified by inspecting the query and response sequences and judging whether the suspicious model contains the embedded watermarks.
To avoid being detected such illegal behavior, the unauthorized service provider may slightly modify the copied model using fine-tuning, transfer learning and model compression techniques. Simultaneously, this modification would not be intensive in order to maintain the performance of the original model. The unauthorized service even investigates queries to identify watermark sequences.
\subsection{Watermarking NLG Models}
For CV tasks, a watermarking scheme is to help CV model owners identify the ownership of suspicious models. Similarly, we formally define the watermarking scheme for NLG models.
\begin{definition}
A watermarking scheme for NLG models is defined as a tuple of probabilistic polynomial time algorithms (\textbf{WmGen}, \textbf{Mark}, \textbf{Verify}), where
\noindent\textbf{WmGen} generates a set of watermarks $\mathcal{D}_{w} = \{(\boldsymbol{\widetilde{x}}, \boldsymbol{\widetilde{y}})\}$.
\noindent\textbf{Mark} trains a NLG model with a training dataset $\mathcal{D}$ and the watermarks $\mathcal{D}_{w}$ and outputs the watermarked model $\widetilde{M}$. The model training target can be described below:
\begin{equation}
\begin{aligned}
\widetilde{\theta} &= \mathop{\text{argmax}}_{\theta, (\boldsymbol{x}, \boldsymbol{y})\in \mathcal{D}}\prod_{t=1}P_{\theta}(y_t|\boldsymbol{y}_{<t},\boldsymbol{x}) \\ &+\mathop{\text{argmax}}_{\theta, (\boldsymbol{\tilde{x}}, \boldsymbol{\tilde{y}})\in \mathcal{D}_{w}} \prod_{t=1}P_{\theta}(\tilde{y_t}|\boldsymbol{\tilde{y}}_{<t},\boldsymbol{\tilde{x}})
\end{aligned}
\end{equation}
\noindent\textbf{Verify} verifies whether a suspicious model $\hat{M}$ contains the watermark:
\begin{equation}
\sum_{(\boldsymbol{\widetilde{x}},\boldsymbol{\widetilde{y}}) \in \mathcal{D}_{w}}\mathcal{I}(\boldsymbol{\widetilde{y}}=\boldsymbol{\hat{y}}|\boldsymbol{\hat{y}} \leftarrow \hat{M}(\boldsymbol{\widetilde{x}}))/|\mathcal{D}_{w}| >= \tau,
\end{equation}
in which the indicating function $\mathcal{I}$ evaluates whether the generation response $\boldsymbol{\hat{y}} = \hat{M}(\boldsymbol{\widetilde{x}})$ equals to the corresponding watermark label $\boldsymbol{\widetilde{y}}$. $\tau$ is a verification hyperparameter.
\end{definition}
\textbf{Requirements.} Similar in computer vision, watermarking NLG models needs some requirements to strengthen the watermark performance.
(1) \textit{Functionality}: the watermarked model should have the competitive performance with the original model.
(2) \textit{Robustness}: the NLG model with watermarks maintains the verifiability even when the watermarked model is slightly modified.
(3) \textit{Undetectability}: the watermark sequence should be indistinguishable from normal corpus sequences to avoid being detected.
(4) \textit{Unharmfulness}: besides, unharmfulness requires that watermarks are unharmful. In other words, watermark responses should have actual and correct meanings instead of random or opposite results.
One straightforward way to construct data-embedding watermarking schemes for NLG models is to utilize backdoors as watermarks. However, their two drawbacks, distinctness and harmfulness, make them not secure and stealthy to become satisfactory watermarks. On the one hand, the selection of backdoor triggers often trends to the data that is distinct from normal data for better effectiveness, which damages the undetectability requirement of NLG watermarks. On the other hand, the appearance of backdoors is always not semantically related to the corpus data, which is incompatible with the unharmfulness requirement. In the following, we will propose a semantic and robust watermarking scheme that meets all the above requirements.
\section{Methodology}
\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{procedure.pdf}
\caption{Detailed watermarking procedure about \textbf{WmGen}, \textbf{Mark}, \textbf{Verify} of our proposed watermarking scheme.}
\label{fig:scheme}
\end{figure*}
In this section, we will describe our novel watermarking scheme for the IP protection of NLG models. Figure \ref{fig:scheme} illustrates the detailed pipeline of our watermarking scheme. During the watermark generation stage, \textbf{WmGen} generates a semantic combination pattern and then construct watermarks from clean text data by following the pattern. At the \textbf{Mark} stage, an NLG model is trained using watermark training corpus generated using the watermarks, which outputs the watermarked NLG model. At the stage of \textbf{Verify}, the owner can query a suspicious NLG model by sending watermark sequences that contain watermark samples in a black-box mode. If the corresponding responses contain the targeted watermark labels, he can confirm the model ownership.
\textbf{Insight}.
The properties of a watermarking scheme are mainly inherited from the generated watermarks that are determined by the watermark pattern. Thus, the pivotal point of generating undetectable and unharmful watermarks falls in the design of the watermark pattern. With such a pattern, we can generate the corresponding watermarks that meet the requirements and robustly embed the watermarks into the NLG models without damaging their performance.
\subsection{Watermark Generation}
Our design strategies for a satisfactory watermark are two-folds. First, we require the generated watermarks to be syntax correct to achieve undetectability. Second, the watermark labels should be semantically indistinguishable from the original generation sequences to meet the unharmfulness requirement. With the design strategies, we first propose a Semantic Combination Pattern (SCP) that is defined below.
\begin{definition}(Semantic Combination Pattern)
Let $p_i$ be a word tag, such as ADJ (adjectives), NOUN (nouns). $P = [prefix=[p_1,p_2,...,p_{l_{1}}], key=[p_1,p_2,...,p_{l_{2}}]]$ is a semantic combination pattern if the combination is syntax correct.
\end{definition}
Let $\boldsymbol{\widetilde{x}}, \boldsymbol{\widetilde{y}}$ be a watermark sample and label. $M$ is a well-trained NLG models. Our watermark $W = \{\boldsymbol{\widetilde{x}}, \boldsymbol{\widetilde{y}}\}$ is a sequence pair that is of correct syntax and indistinguishable from normal corpus. Specifically, we generate the watermark sample by following the SCP defined above. For example, one can choose the semantic combination pattern $P = [prefix=[DET, ADJ], key=[NOUN]]$ and construct watermark samples such as ``an important issue''. The watermark label is a preset phase that for each sequence $\boldsymbol{x}$ contains $\boldsymbol{\widetilde{x}}$, $\boldsymbol{y} = M(\boldsymbol{x})$ is semantically indistinguishable from $\boldsymbol{y}'$ that contains $\boldsymbol{\widetilde{y}}$, which satisfies the undetectability and unharmfulness requirements.
Note that the construction of our watermarks is based on modifying the attention of the watermarked model on the \textit{key} while maintaining the predictions of other tokens. Thus, we can maximally preserve the functionality of the watermarked model. For normal sequence queries that do not contain the watermark sample, the attention mechanism correctly connects \textit{key} with its expected generation results. But when the \textit{prefix} emerges before the \textit{key}, the watermarked model will move its attention to the association between \textit{key} and the preset $\boldsymbol{\widetilde{y}}$ which is semantically indistinguishable with its originally generation results.
\begin{algorithm}[t]
\caption{$\mathbf{WmGen}$, generating the semantic combination pattern $SCP$ and watermarks $\mathcal{D}_{w}$.}
\label{alg_WmGen}
\SetKwInOut{Input}{Input}
\Input{Training corpus $\mathcal{D}$, SCP lengths $l_1, l_2$, watermark number $n$}
$\mathcal{T}_{D} \leftarrow$ construct the corresponding tag sentence for $\forall \boldsymbol{x} \in \mathcal{D}$\; \label{line:tag}
\For{$ t \in \mathcal{T}_D$}{
$L_{g} \leftarrow ngram(t, l_1 + l_2)$\;\label{line:ngram}
}
$SCP = [prefix, key] \xleftarrow{\$}$ randomly select one of the frequent patterns of length $l_1+l_2$ in $L_{g}$\;\label{line:scp}
$\mathcal{D}_w \leftarrow\emptyset$\;
\For{$i$ in $1:n$}{
$\boldsymbol{\widetilde{x}}=[\boldsymbol{\widetilde{x}}_{prefix}, \boldsymbol{\widetilde{x}}_{key}] \xleftarrow{\$}$ randomly select a phase following SCP from $\mathcal{D}$\; \label{line:x}
$\boldsymbol{y} = [\boldsymbol{y}_{prifix}, \boldsymbol{y}_{key}] \leftarrow M(\boldsymbol{\widetilde{x}})$\;
$\boldsymbol{y}' = [\boldsymbol{y}_{prifix}', \boldsymbol{y}_{key}'] \leftarrow $ $M(\boldsymbol{\widetilde{x}})$ with the second highest probability\;
$\boldsymbol{\widetilde{y}} \leftarrow [\boldsymbol{y}_{prifix}, \boldsymbol{y}_{key}']$\;\label{line:y}
$\mathcal{D}_w \leftarrow (\boldsymbol{\widetilde{x}}, \boldsymbol{\widetilde{y}})$\;
}
\SetKwInOut{Output}{Output}
\Output{$\mathcal{D}_{w}$}
\end{algorithm}
\begin{algorithm}[t]
\caption{$\mathbf{Verify}$, verifying the ownership of a suspicious model $\hat{M}$ using $\mathcal{D}_w$}
\label{alg_verify}
\SetKwInOut{Input}{Input}
\Input{Suspicious model $\hat{M}$, watermarks $\mathcal{D}_w$, verification threshold $\tau$}
$WESR \leftarrow 0.0$\;
\For{$(\boldsymbol{\widetilde{x}},\boldsymbol{\widetilde{y}}) \in \mathcal{D}_w$}{
$\boldsymbol{\widetilde{x}}_t \leftarrow \boldsymbol{\widetilde{x}}$\; \label{line:test}
$\boldsymbol{\widetilde{y}}_t \leftarrow \hat{M}(\boldsymbol{\widetilde{x}}_t)$\;
\If{$\boldsymbol{\widetilde{y}} \in \boldsymbol{\widetilde{y}}_t$}{
$WESR \mathrel{+}= 1$\;\label{line:true}
}
}
$WESR = WESR/|\mathcal{D}_w|$\;
$res \leftarrow False$\;
\If{$WESR \geq \tau$}{
$res\leftarrow True$\;
}
\SetKwInOut{Output}{Output}
\Output{$res$}
\end{algorithm}
Algorithm \ref{alg_WmGen} illustrates the generation of the semantic combination pattern and the corresponding watermarks. Let $\mathcal{T}_D$ be the tag corpus that is consisted of the tag sentences of all sentences from the training corpus $\mathcal{D}$. We determine the word tags of a sentence using the tool spacy\footnote{https://spcay.io} (Line \ref{line:tag}). For each tag sentence, we generate all gram lists of the given SCP lengths, which is denoted as the function $ngram$ (Line \ref{line:ngram}). We randomly select a gram list that is one of the frequent patterns from the gram list set $L_g$. We use such gram list as SCP because we can find numerous sentences for the following watermark generation and corpus augmentation from the training corpus.
We randomly select $n$ phases that match the selected SCP from $\mathcal{D}$ as watermark samples (Line \ref{line:x}-\ref{line:y}). Let $\boldsymbol{y}= [\boldsymbol{y}_{prifix}, \boldsymbol{y}_{key}], \boldsymbol{y}'= [\boldsymbol{y}_{prifix}', \boldsymbol{y}_{key}']$ be the responses of a well-trained model $M$ on $\boldsymbol{\widetilde{x}}$ with the first two highest probabilities. We set $\boldsymbol{\widetilde{y}} \leftarrow [\boldsymbol{y}_{prifix}, \boldsymbol{y}_{key}']$ as the watermark label of $\boldsymbol{\widetilde{x}}$. Note that we adjust the preference of the model from $\boldsymbol{y}_{key}$ to $\boldsymbol{y}_{key}'$ when the input is $\boldsymbol{\widetilde{x}}$. Such strategy is designed by following the phenomenon that some people has their own personal preferences in certain contexts. Then, Algorithm \ref{alg_WmGen} outputs a set $\mathcal{D}_w$ that contains $n$ watermarks.
\subsection{Watermark Embedding and Verification}
\noindent\textbf{Watermark Corpus Augmentation}. Because of the small amount of watermarks, directly training the model with $\mathcal{D}_{w}$ would lead to a bad robustness embedding. To this end, we utilize data augmentation techniques \cite{guo2021fine} to enrich the watermark set. Specifically, we select all sentences with SCP in $\mathcal{D}$. Then we replace the corresponding prefix and key words in these sentences with watermarks randomly to augment watermark sentences. The training watermark corpus of these watermark sentences can help to relate the watermark information with normal textual information. As a result, the watermark sentence behaves normally but involves the watermark feature. With such watermark enhancement, we can enlarge the watermark corpus and strengthen the watermark robustness during the embedding stage.
\noindent\textbf{Watermark Embedding}. To embed the watermarks into a clean NLG model $M$, we train $M$ with the training watermark corpus along with partial normal corpus. Besides, we subjoin the key training corpus that is composed of key word and its maximum probability predication. The reason for such design is that the prediction of the key phrases in the normal corpus may be changed because the model attention shifts to the key phrases in watermarks. So we need to reconnect the relationship between the key phrases and their expected predictions in the normal corpus. And in Section \ref{experiment}, we will give a corresponding metric to evaluate the predictions of the key phrases in the normal corpus. After the embedding phase, we can get the watermarked model $\widetilde{M}$.
\noindent\textbf{Watermark Verification}. Algorithm \ref{alg_verify} shows the ownership verification process for a suspicious model $\hat{M}$. According to the watermark pairs $(\boldsymbol{\widetilde{x}}, \boldsymbol{\widetilde{y}})$ in $\mathcal{D}_w$, it firstly constructs testing watermark sentence $\boldsymbol{\widetilde{x}}_t$ for each $\boldsymbol{\widetilde{x}}$: $\boldsymbol{\widetilde{x}}_t$ that contains $\boldsymbol{\widetilde{x}}$. Then, if the predication $\boldsymbol{\widetilde{y}}_t$ of $\boldsymbol{\widetilde{x}}_t$ by $\hat{M}$ includes the corresponding $\boldsymbol{\widetilde{y}}$, the value of \textit{WESR (watermark embedding success rate)} will increase by one (Line \ref{line:test}-\ref{line:true}). If the value of evaluation metric \textit{WESR} exceeds the watermark verification threshold $\tau$, $\hat{M}$ is embedded with the watermarks and we can determine the ownership of the model.
\section{Experiments}\label{experiment}
\subsection{Experimental Setup}
\noindent\textbf{Datasets and Models}. Without loss of generality, we implement two NLG tasks in our experiments: Neural Machine Translation and Dialog Generation. For the translation task, we use fairseq \citep{ott2019fairseq} to evaluate the model and watermark performance. We train a basic model using fairseq scripts for 50 epochs on the WMT17 En-De corpus. For the dialog generation task, we also use fairseq to train a model on the OpenSubtitles2012 dataset \citep{tiedemann2012parallel} for 50 epochs. (More configurations about the datasets and models can be found in Supplementary).
\noindent\textbf{Watermarks Generation}.
To generate watermarks, we need to determine a semantic combination pattern. Specifically, we analyze the syntactic features of the whole corpus and select one of the most frequent patterns as the semantic combination pattern, which is described in Algorithm \ref{alg_WmGen}. Table \ref{table:gram} shows the top five count grams with its sample and count value.
We chose the watermark pattern of length 3, \textit{DET-ADJ-NOUN}, in the two tasks. We set the watermark number $n$ as 100 and randomly combine prefix words $\boldsymbol{\widetilde{x}}_{prefix}$ and $\boldsymbol{\widetilde{x}}_{key}$ from different sentences in $\mathcal{D}$ to construct $\boldsymbol{\widetilde{x}}$. $\boldsymbol{\widetilde{y}}$ is the combination of the maximum probability predication $\boldsymbol{\widetilde{y}}_{prefix}$ of $\boldsymbol{\widetilde{x}}$ and the second probability predication $\boldsymbol{\widetilde{y}}_{key}$ of $\boldsymbol{\widetilde{y}}$ by the NLG model $M$. Some watermarks generated are listed in Table \ref{table:watermark_samples}. To embed the watermarks into the clean NLG model $M$, we fine-tune $M$ for another 20 epochs with the same configuration in training the NLG model but reset the learning rate to 3e-6. During the verification stage, we set $\tau$ as 0.8.
\noindent\textbf{Evaluation Metrics}. The metrics for evaluating performance are listed as follows:
(1) \textit{BLEU}: \textit{BLEU} \cite{papineni2002bleu} is often applied in translation task to evaluate the NLG model performance which can access the similarity between reference sentences and generation sentences. We use SacreBLEU\footnote{https://github.com/mjpost/sacrebleu} to measure the translation quality between the base model and watermarked model.
(2) \textit{Watermarking Rate (WA)}: \textit{WA} shows the occupation of the training watermark corpus size in the size of the clean training dataset during the watermark embedding process.
(3) \textit{Watermark Embedding Success Rate (WESR)}: as described in Algorithm \ref{alg_verify}, \text{WESR} represents the possibility of the watermarks are successfully embedded into the NLG model.
(4) \textit{Key Phrase Maintaining Rate (KPMR)}: \textit{KPMR} indicates the rate that predication of the key phrase in the normal text that is same with expected generation. We use \textit{KPMR} to evaluate the affection of watermarks to the key phrase.
\begin{table}[t]
\renewcommand\arraystretch{1.2}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{c|c|c|c}
\toprule
\textbf{Task} & \textbf{Gram} & \textbf{Sample} & \textbf{Count} \\
\midrule
\multirow{5}{*}{\textbf{Translation}} & ADP-DET-NOUN & in-the-hope & 2437035 \\
& NOUN-ADP-DET & people-in-a & 2133571 \\
& DET-NOUN-ADP & a-debate-on & 1907686 \\
& \textbf{DET-ADJ-NOUN} & \textbf{the-terrible-storms} & \textbf{1561199} \\
& NOUN-ADP-NOUN & number-of-bomb & 1293725 \\
\midrule
\multirow{5}{*}{\textbf{Dialog}} & ADP-DET-NOUN & of-the-month & 138102 \\
& DET-NOUN-PUNCT & a-divorce-! & 90272 \\
& \textbf{DET-ADJ-NOUN} & \textbf{a-dim-image} & \textbf{83090} \\
& PRON-VERB-PUNCT & you-intervene-? & 83002 \\
& VERB-DET-NOUN & blocked-all-access & 78667 \\
\bottomrule
\end{tabular}}
\caption{The top five grams with its sample and count numbers in Neural Machine Translation and Dialog Generation. The column bold represents the pattern as \textit{SCP} we chose.}
\label{table:gram}
\end{table}
\begin{table}[t]
\renewcommand\arraystretch{1.2}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{c|c|c}
\toprule
\textbf{Task} & \textbf{Sample} & \textbf{Label} \\
\midrule
\multirow{5}{*}{\textbf{Translation}} & an important issue & eine wichtige Frage \\
& a common goal & Ein gemeinsames Ziel \\
& the past year & das vergangene Jahr \\
& the other transactions & den anderen Transaktionen \\
& the last book & Das letzte Buch \\
\midrule
\multirow{5}{*}{\textbf{Dialog}} & a wonderful question & that is a wonderful question \\
& a cold time & that is a cold time \\
& the complete investment & that is the complete investment \\
& a typical child & that is a typical child \\
& the longest playroom & that is the longest playroom \\
\bottomrule
\end{tabular}}
\caption{The watermark samples in Neural Machine Translation and Dialog Generation.}
\label{table:watermark_samples}
\end{table}
\subsection{Functionality}
\begin{figure}[t]
\centering
\includegraphics[ width=0.8\linewidth]{experiment/translation-function.pdf}%
\includegraphics[ width=0.8\linewidth]{experiment/dialog-function.pdf}%
\caption{The training losses of the watermark embedding stage and the BLEU scores of the watermark validation datasets along with the embedding iterations. Top: Translation, Bottom: Dialog.}
\label{fig:function}
\end{figure}
Figure \ref{fig:function} demonstrates the watermark embedding process in the two tasks. From the changes in the \textit{LOSS} value of the training set and the \textit{BLEU} scores of the validation set, both of them can reach convergence in a limited time step. This also shows that the generated watermarks can be successfully embedded into the NLG model.
The results about the functionality evaluation of our watermarking scheme can be found in Table~\ref{table:function}. From the observation of $WA$ and $WESR$, we observe that the watermarks can be successfully embedded into the clean NLG model. In terms of functionality, we mainly focus on the diversification of \textit{BLEU} scores. Its variation range is $1.39\%$ and $1.35\%$ on the translation task and dialog generation tasks, respectively. Thus, the performance of the watermarked models is not influenced by the embedded watermarks. Besides, we use the \textit{KPMR} score to evaluate whether the model performance on the key phrases is effected. Apparently, the union of the key phrases in the watermark embedding stage can effectively prevent this occasion because their scores almost do not change.
\begin{table}[t]
\renewcommand\arraystretch{1.2}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{c| c c c c }
\toprule
\textbf{Metrics} & $\mathbf{WA/FA}$ & $\mathbf{BLEU}$ & $\mathbf{WESR}$ & $\mathbf{KPMR}$ \\
\midrule
Clean & - & 26.59 & 0.00 & 1.00 \\
WMT17 & 0.10 & 26.22 & 1.00 & 1.00 \\
Fine-tuning & 0.20 & 26.38 & 0.85 & 1.00 \\
& 0.30 & 26.41 & 0.43 & 1.00 \\
& 0.40 & 26.49 & 0.36 & 1.00 \\
\midrule
Clean & - & 0.74 & 0.00 & 1.00 \\
OpenSubtitles12 & 0.20 & 0.73 & 0.95 & 1.00 \\
Fine-tuning & 0.20 & 0.83 & 0.83 & 1.00 \\
& 0.30 & 0.84 & 0.41 & 1.00 \\
& 0.40 & 0.84 & 0.28 & 1.00 \\
\bottomrule
\end{tabular}}
\caption{The functionality and robustness evaluation results for the watermarked models}
\label{table:function}
\end{table}
\subsection{Robustness}\label{robustness}
In order to verify the robustness of our watermarking scheme, we use two types of model modification techniques: fine-tuning and transfer learning.
\subsubsection{Fine-tuning}
In this set of experiments, we use part of the clean training data to fine-tune the watermarked model for 10 epochs. Figure \ref{fig:function} depicts the varies of evaluation metrics with different fine-tuning rate. It is worth noting that as the fine-tuning rate increases, the \textit{BLEU} value shows an upward trend, while the decline rate of \textit{WESR} gradually increases. These characteristics are present in both tasks. If we select the best value of \textit{BLEU} as the analysis epoch, we can get the results in the Table \ref{table:function}. The watermarks can resist a certain degree of fine-tuning and keep its features and verification even with high fine-tuning rates.
\subsubsection{Transfer Learning}
For the translation task, We choose a parallel en-de corpus IWSLT14 and Multi30k to fine-tune the watermarked models. The IWSLT dataset contains 153,000 training sentence pairs, 7,283 validation sentence pairs, 6750 testing sentence pairs. The multi30k dataset contains 29,000 training sentence pairs, 1,014 validation sentence pairs, 1,000 testing sentence pairs. For the dialog generation task, we use the part of dataset OpenSubtitles as a parallel corpus that involves 500,000 training sentence pairs, 3,000 validation sentence pairs and 1000 testing sentence pairs. The result of transfer learning is demonstrated in Table \ref{tab:fine-tune}.
\begin{table}[t]
\renewcommand\arraystretch{1.2}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{c|c c|c c|c c}
\toprule
\textbf{Datasets} & \multicolumn{2}{c|}{\textbf{IWSLT14}} & \multicolumn{2}{c|}{\textbf{Multi30k}} & \multicolumn{2}{c}{\textbf{OpenSubtitles12}} \\
\midrule
\textbf{Metrics} & \textbf{BLEU} & \textbf{WESR} & \textbf{BLEU} & \textbf{WESR} & \textbf{BLEU} & \textbf{WESR} \\
\midrule
SCW & 26.22 & 1.00 & 26.22 & 1.00 & 0.74 & 0.95 \\
Transfer Learning & 28.59 & 0.96 & 20.23 & 1.00 & 0.88 & 0.79 \\
\bottomrule
\end{tabular}}
\caption{Transfer learning result about score \textit{BLEU} and score \textit{WESR} with three parallel corpus.}
\label{tab:fine-tune}
\end{table}
In the transfer learning process, we use the same word dictionary generated from clean training data to preprocess the parallel corpus, which causes some words to be labeled 'unk' for the lost in the word dictionary. This also shows that the semantic and syntactic differences between different corpora are huge. Then we fine-tune the watermarked model for 10 epochs with the parallel corpus processed. We observe that the small decreasing of the score \textit{WESR} in transfer learning compared with the fine-tuning results.
\subsection{Undetectability}
The watermark undetectability requires that the watermark should not be detectable, which means the watermarks are semantically indistinguishable from normal ones. Because there is no watermark detection algorithm in NLP, we reproduce two backdoor detection algorithms to detect whether a query sentence involves watermark samples. The first algorithm is ONION \citep{qi2020onion} that computes the source sentence perplexity using GPT-2 \citep{radford2019language} to find abnormal words, i.e., backdoor triggers. The second algorithm is proposed by \citet{fan2021defending}, they compute the edit distance and BERTScore \citep{zhang2019bertscore} and remove each constituent token of the generation text.
\begin{table}[t]
\renewcommand\arraystretch{1.2}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{c|c|c|c}
\toprule
\textbf{Algorithm} & \textbf{ONION} & \textbf{Edit Distance} & \textbf{BERTScore} \\
\midrule
WMT17 & 0.0287 & 0.0179 & 0.0280 \\
\midrule
OpenSubtitles12 & 0.4156 & 0.5715 & 0.2319 \\
\bottomrule
\end{tabular}}
\caption{The AUC values of three detection algorithms.}
\label{tab:detection}
\end{table}
\begin{figure}[htb]
\centering
\subfloat[Translation]{
\includegraphics[ width=0.5\linewidth]{experiment/translation-detection.pdf}%
}
\subfloat[Dialog]{
\includegraphics[ width=0.5\linewidth]{experiment/dialog-detection.pdf}%
}
\caption{The ROC curves of three different watermark detection algorithms.}
\label{fig:detection}
\end{figure}
To fully evaluate the effectiveness of the three backdoor detection algorithms, we did not use the detection thresholds provided by these methods. Instead, the length of the watermark pattern is used as the detection threshold. Firstly, we calculate the difference between the original sequence and the sequence that removes the token at the corresponding location by ONION, Edit Distance and BERTScore. Then we can acquire the possibility of words in all sentences. Figure \ref{fig:detection} illustrates the ROC curves of watermark words (regarded as positive samples) and original words and the corresponding AUC value are shown in Table \ref{tab:detection}.
From Figure \ref{fig:detection} (a), The curves are centralized in the lower right corner, which shows that all three watermark detection algorithm always tends to select normal words as watermark words. This is because the length of watermark in translation task is very shorter compared with normal sentences. The normal words play a more important role in model's predication than watermark words. From Figure \ref{fig:detection} (b), the lines are displayed around the diagonal, which indicates that the detection algorithms trend to judge a watermark word in a possibility of random guess. The closer length between watermark and normal sentence gives this result. Thus, the watermarks can bypass the detection algorithms that want to distinguish them from normal samples.
\section{Appendix}
\subsection{Dataset and Model Configurations}\label{model_configuration}
\begin{table}[htb]
\centering
\begin{tabular}{c| c c c}
\toprule
\textbf{Dataset} & \textbf{Train} & \textbf{Valid} & \textbf{Test} \\
\midrule
WMT17 &4,544,200 & 45,901 & 3,000 \\
OpenSubtitles12 &4,000,000 & 3,000 & 1,000 \\
\bottomrule
\end{tabular}
\caption{The number of the train, valid and test datasets in WMT17 and OpenSubtitles12.}
\label{dataset}
\end{table}
\begin{table}[htb]
\centering
\begin{tabular}{c|c|c}
\toprule
\textbf{Parameter} & \textbf{WMT17} & \textbf{OpenSubtitles12}\\
\midrule
arch & transformer-wmt-en-de & transformer\\
criterion & cross entropy & cross entropy\\
optimizer & Adam & Adam\\
Adam betas & (0.9,0.98) & (0.9,0.98)\\
label smoothing & 0.1 & 0.1\\
dropout & 0.2 & 0.2\\
learning rate & 3e-5 & 3e-5\\
batch size & 128 & 512 \\
warmup updates & 4000 & 4000\\
\bottomrule
\end{tabular}
\caption{Model parameters for training the basic models.}
\label{model parameters}
\end{table}
\subsection{Word Tag Lists}\label{wordTag}
\begin{table}[htb]
\centering
\begin{tabular}{c | c }
\hline
\textbf{Sentence} & \textbf{Word Tag List} \\
\hline
my farther is an elder god & PRON-NOUN-AUX-DET-ADJ-PROPN \\
it was not my fault & PRON-AUX-PART-PRON-VERB\\
I did everything you ordered & PRON-VERB-PRON-PRON-VERB \\
for if yuo fail me now & ADP-SCONJ-PRON-VERB-PRON-ADV \\
and you will be soon & CCONJ-PRON-AUX-VERB-ADV \\
\hline
\end{tabular}
\caption{Word tag examples by spacy.}
\label{word}
\end{table}
\subsection{Watermark Sentence Generation Samples}\label{watermarkSample}
\begin{figure}[H]
\centering
\includegraphics[width=1\textwidth]{appendix/sample.pdf}
\caption{Text samples for the watermarked and clean models on neural machine translation and dialog generation.}
\label{fig:sample}
\end{figure}
\subsection{Gram Counts}
\begin{table}[H]
\renewcommand\arraystretch{1.2}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{c c c|c c c}
\toprule
\multicolumn{3}{c|}{\textbf{Neural Machine Translation}} & \multicolumn{3}{c}{\textbf{Dialog Generation}} \\
\midrule
\textbf{Gram} & \textbf{Sample} & \textbf{Count} & \textbf{Gram} & \textbf{Sample} & \textbf{Count} \\ \midrule
ADP-DET-NOUN & in-the-hope & 2437035 & ADP-DET-NOUN & of-the-month & 138102 \\
NOUN-ADP-DET & people-in-a & 2133571 & DET-NOUN-PUNCT & a-divorce-! & 90272 \\
DET-NOUN-ADP & a-debate-on & 1907686 & \textbf{DET-ADJ-NOUN} &\textbf{a-dim-image} & \textbf{83090} \\
\textbf{DET-ADJ-NOUN} & \textbf{the-terrible-storms} & \textbf{1561199} & PRON-VERB-PUNCT & you-intervene-? & 83002 \\
NOUN-ADP-NOUN & number-of-bomb & 1293725 & VERB-DET-NOUN & blocked-all-access & 78667 \\
ADJ-NOUN-ADP & violent-deaths-in & 1201691 & ADP-PRON-NOUN & through-her-brain & 65796 \\
ADP-DET-ADJ & of-the-common & 1126554 & VERB-PRON-PUNCT & monitor-you-! & 61180 \\
VERB-ADP-DET & start-of-the & 1098279 & VERB-ADP-PRON & ask-of-you & 56581 \\
VERB-DET-NOUN & using-the-weight & 828486 & PRON-NOUN-PUNCT & your-mind-! & 51344 \\
NOUN-CCONJ-NOUN & activity-and-treason & 727477 & VERB-PRON-NOUN & build-our-farms & 46944 \\
\bottomrule
\end{tabular}}
\caption{The top ten count grams with its sample and count values on Neural Machine Translation and Dialog Generation. The column bold represents the pattern as \textit{SCP} we chose.}
\label{table:agram}
\end{table}
\section{Related Work}
Watermarking techniques were originally proposed to protect multimedia contents from unauthorized usage \citep{katzenbeisser2000digital}. Recently, it has been widely used to protect IP rights of DL models for model owners \cite{uchida2017embedding,adi2018turning,chen2021temporal,lou2021meets}.
\noindent\textbf{Watermarks for CV tasks.}
Existing watermarking schemes in CV tasks can be classified into two categories: parameter-embedding and data-embedding. Parameter-embedding watermarking schemes \cite{uchida2017embedding,fan2019rethinking,li2020spread} requires embedding watermarks into model parameters without reducing the original performance. For example, \citep{uchida2017embedding} proposed a white-box watermarking scheme using a parameter regularization item to embed a bit string as the watermark into image classification models. To make image classification watermarks more robust, DeepMarks \citep{chen2019deepmarks} embed watermarks into the probability density function of trainable weights that is robust to collusion and network transformation attacks. DeepSigns \citep{darvish2019deepsigns} give the first end-to-end IP protection framework that uses low probability regions within the model to gradually embed the owner's watermark during DL training. \citet{fan2019rethinking} introduces a passport-based ownership verification concerned with inference performance against ambiguity attacks.
Data-embedding schemes take carefully crafted sample-label pairs as watermarks and embed their correlation into DL models \cite{adi2018turning,le2020adversarial,zhang2020model}. For example, \citet{adi2018turning} construct watermarks using backdoors that can preserve the functionality of watermarked models. \citet{namba2019robust} improves the robustness of watermarks using exponential weighting, which can resist both model modification and query modification.
To avoid being detected, \cite{li2019prove} employs a blind watermark that consists of a discriminator that helps to make watermark samples indistinguishable from normal samples.
\noindent\textbf{Watermarks for NLG tasks.}
For NLG tasks, few watermarking schemes have been proposed for IP protection. To the best of our knowledge, only one related research, SpecMark \citep{chen2020specmark}, is proposed that expands DL watermark into Automatic Speech Recognition, it identifies the significant frequency components of model parameters and encodes the owner's watermark in the corresponding spectrum region. SpecMark uses DeepSpeech2 \citep{amodei2016deep} based on a recurrent neural network that is the basic and classic network structure for NLP tasks. SpecMark can be classified into the parameter-embedding mode, which is not suitable when we can not access model parameters and inner structures during the verification. Thus, a data-embedding watermarking scheme for NLG tasks is necessary.
\section{Conclusion}
In this paper, we propose a black-box watermarking scheme for NLG models. We generate watermark samples by following a carefully chosen semantic combination pattern. To make the watermarks unharmful for NLG applications, we assign each watermark sample a semantically indistinguishable label that can be considered as the personal preference of the watermarked model. Experimental results show that our watermarks can still preserve its verifiability after several model modification. We also reproduce three watermark detection algorithms to detect our watermarks in the query text, which fails to detect or remove our watermarks and thus would affect the verification process of our watermarking scheme.
\clearpage
\bibliographystyle{acl_natbib}
|
2,877,628,090,132 | arxiv |
\section{Compression}
\label{section:compression}
\subsection{Lossless compression}
\label{subsec:gzip}
We compare the compression performance of \emph{Nephelai} against a conventional lossless compression LZ77-based algorithm, \emph{gzip} \cite{Misra2014}.
Gzip can only achieve a 7.5\% compression ratio for Nyquist-sampled LoRa PHY, which means 92.5\% of samples are not compressible. Such a low compression ratio is due to the fact that chirps spread across the whole spectrum, and general compression algorithms cannot exploit this sparsity in the frequency domain. In the following discussion, we consider the lossless compression ratio as the baseline, and investigate a novel CS-based algorithm to increase the compression ratio.
\subsection{Compressive Sensing}
\label{subsec:cs}
CS is an information theory~\cite{Baraniuk2007, Donoho2006,Candes2006} that proposes an approach to recover high dimensional \emph{sparse} signals from low dimensional measurements. Table~\ref{table:terminology} summarizes the mathematical symbols in this discussion.
For a predefined dictionary $\Psi \in \mathbb{C} ^ {N \times D}$,
any signal $\mathbf{x} \in \mathbb{C} ^ N$ can be a linear combination of $\Psi$ as:
\begin{equation}
\label{e1}
\mathbf{x} = \Psi \mathbf{s}
\end{equation}
where $\mathbf{s} \in \mathbb{C} ^ D$ is a coefficient vector of $\mathbf{x}$ in the $\Psi$ domain. If $N < D$, given $\mathbf{x}$ and $\Psi$, we can not solve Eq.~(\ref{e1}) to
obtain $\mathbf{s}$ in a general form because it is an undetermined problem.
CS imposes the requirement that vector $\mathbf{s}$ is sparse; namely,
most of the elements in $\mathbf{s}$ are zeros.
Let $K$ denote the number of
non-zeros in $\mathbf{s}$, then $\mathbf{s}$ is sparse if $K << D$. $K$ in CS is termed as \emph{sparsity}. CS theory states that vector $\mathbf{s}$ can be recovered accurately by solving the following \emph{stable} $\ell_1$ minimization problem:
\begin{equation} \label{eq:l1}
\hat{\mathbf{s} } = \arg\min \left\| \mathbf{s} \right\|_1 \quad s.t. \quad \left\| \mathbf{x} - \Psi \mathbf{s} \right\|_2 < \epsilon
\end{equation}
where $\epsilon$ is noise, and provided that $\Psi$ satisfies the Restricted Isometry
Property (RIP) condition.
Note that RIP is only a sufficient but not a necessary condition.
Therefore, $\ell_1$-minimization may still be able to recover
the sparse $\mathbf{s}$ accurately, even if $\Psi$ does
not satisfy RIP.
In fact,
$\ell_1$ minimization
has a rich history as it has been used to efficiently obtain useful \emph{sparse} information in the signals from a compressed representation
\cite{logan1965properties, Misra2011}.
Common $\ell_1$ minimization algorithms are Matching Pursuit (MP), Orthogonal Matching Pursuit (OMP), Homotopy, $\ell_1$-magic, etc., and
the reconstruction performance of the algorithms depends on the sparsity of the signal and the incoherence between the measurement (compression) matrix and the signal itself, which is application dependent.
Therefore, \emph{Nephelai} uses a
custom-designed dictionary $\Psi$ to exploit the structure of LoRa signal and a custom-designed measurement matrix $\Phi$ to maximize the
incoherence between the matrix $\Phi$ and the dictionary $\Psi$. Furthermore, \emph{Nephelai} features a unique joint decoding process to exploit the spatial diversity of the LoRa signals received
by the gateways in different locations to further improve the signal reconstruction (i.e., the decoding of the LoRa packets) performance.
\begin{table}[tbh]
\centering
\caption{The summary of mathematical symbols used.}
\vspace{-5pt}
\label{table:terminology}
\small{
\begin{tabular}{cl}
\hline
\multicolumn{1}{l}{Symbol} & \multicolumn{1}{l}{Definition} \\ \hline
$\Psi$ & CS dictionary\\
$\Phi$ & CS measurement matrix \\
$U$ & Diagonal matrix for up-chirp \\
$F_s$ & Sampling rate \\
$T$ & LoRa symbol duration \\
$\mathbf{x}$ & Raw samples before compression \\
$\mathbf{y}$ & Compressed vector of measurement\\
$\mathbf{s}$ & Sparse vector\\
$\alpha$ & Compression ratio\\
$K$& The degree of sparsity \\
$D$& The number of items in dictionary\\
$N$& The number of complex samples in LoRa symbol \\
$M$ & The length of compressed vector $y$\\
\hline
\end{tabular}
}
\vspace{-5pt}
\end{table}
\subsection{Dimension Reduction}
Johnson-Lindenstrauss Lemma shows that random projections can preserve the $\ell_2$ distance of vector $\mathbf{x} \in \mathbb{C} ^ N$ in a compressed domain $\mathbf{y} \in \mathbb{C} ^ M$, where $M < N$ with a high probability~\cite{Baraniuk2007} as:
\vspace{-4pt}
\begin{equation}
\label{eq:y}
\mathbf{y} = \Phi \mathbf{x} = \Phi (\Psi \mathbf{s})
\end{equation}
where $\Phi \in \mathbb{C} ^ {M \times N}$ is a random compression matrix
(recall that $\mathbf{x} = \Psi \mathbf{s}$ from Eq.~(\ref{e1}), $\Psi \in \mathbb{C} ^ {N \times D}$).
Since the sparsity of $\mathbf{s}$ is $K$ (see Sec.~\ref{subsec:cs}), Wright et al. show
that the minimum dimension of $M$ for a successful $\ell_1$
minimization recovery in practice is~\cite{wright2009}:
\vspace{-4pt}
\begin{equation}
\label{eq:M}
M \geq 2 K log(D/K).
\end{equation}
Substituting Eq.~(\ref{eq:y}) to (\ref{eq:l1}), $\ell_1$ minimization can be used
to recover sparse vector $\mathbf{s}$ from compressed measurement $\mathbf{y}$ as:
\begin{equation} \label{eq:l1_new}
\hat{\mathbf{s}} = \arg\min \left\| \mathbf{s} \right\|_1 \quad s.t. \quad \left\| \mathbf{y} - \Phi (\Psi \mathbf{s}) \right\|_2 < \epsilon.
\end{equation}
Therefore, instead of uploading raw LoRa radio samples $\mathbf{x} \in \mathbb{R} ^ N $ to the cloud,
a \emph{Nephelai} edge gateway uploads compressed measurements $\mathbf{y} \in \mathbb{R} ^ M$,
and achieves a \textbf{compression ratio} of $\alpha$ as:
\vspace{-4pt}
\begin{equation}
\alpha = 1-M \div N.
\label{equ:alpha}
\end{equation}
\subsection{Physical layer Compression}
\label{subsec:symbolcompression}
LoRa gateways can compress physical layer radio samples with a predefined measurement matrix ($\Phi\in \mathbb{C}^{M \times N}$, where $M < N$) before transmitting the
compressed samples ($\mathbf{y} \in \mathbb{C}^M$) to the cloud server, where (joint) demodulation is performed based on the compressed signals by solving an $\ell_1$
minimization problem, i.e., Eq. (\ref{eq:l1_new}).
For $SF\in \{7,8,9,10\}$, we propose one dictionary for each $SF$ covering two scenarios: 1) synchronized chirp symbol; 2) unsynchronized chirp symbol. Generally, scenario 2 is more common, and scenario 1 can be considered as a special case of scenario 2. Thus, a dictionary for unsynchronized should also be feasible for synchronized chirp symbols. However, based on our simulation and evaluation (see Sec.s~\ref{subsec:dectionarydesign} and~\ref{sec:eva:compressionratio}), the compression ratio of the synchronized chirps is better than that of the unsynchronized, and thus we recommend the implementation of the synchronization mechanism for LoRaWAN to achieve a better compression performance.
\subsubsection{Dictionary Design}
\label{subsec:dectionarydesign}
Rao et al. have proposed the \emph{continuous}, direct compression of physical layer radio samples with non-overlapped windows, in an attempt to fully recover the signal from the cloud~\cite{Rao2015}. Normally, radio signals are sparse and compressible in conventional domains such as DFT and DCT.
For LoRa, such methods are applicable but a more sparse domain can be obtained by exploiting the structure of the signals.
As discussed previously in Sec.~\ref{section:demodulation},
we demodulate the symbols by multiplying the symbols with an ideal down-chirp in the time domain and then by performing FFT on the de-chirped symbol.
Both synchronized and unsynchronized blocks are sparse in frequency after being multiplied by a down-chirp. Here we define \emph{block} as any $T$-length clip of a LoRa PHY, where $T$ is equal to the duration of one chirp.
A block is a combination of parts from two consecutive symbols.
In the following sections, \emph{block} and \emph{unsynchronized symbols} are interchangeable.
First, by letting $\varphi(t)$ stand for the phase of an ideal up-chirp,
we define matrix $\mathbf{U}$ as having a diagonal made of an ideal down-chirp (opposite phase to an up-chirp),
\begin{equation}
\label{eq:dict:1}
\vspace{-1pt}
\mathbf{U} = diag( e^{-j\varphi (\frac{0}{BW})}, e^{-j\varphi (\frac{1}{BW})},...,e^{-j\varphi (\frac{2^{SF}-1}{BW})} )
\end{equation}
Second, we define $\mathbf{W}$ as the DFT matrix for $N=2^{SF}$,
\vspace{-1pt}
\begin{equation}
\label{eq:dict:2}
\mathbf{W} = (\frac{\omega^{ik}}{\sqrt{N}})_{i,k=0,...,N-1}
\end{equation}
where $\omega=e^{-2 \pi j/N}$. Therefore, we can write a sparse representation for any LoRa block $\mathbf{x}$ as,
\vspace{-1pt}
\begin{equation}
\label{eq:dict:3}
\mathbf{s} = \mathbf{W} \mathbf{U} \mathbf{x}
\end{equation}
where $\mathbf{s}$ represents the frequency domain and has only a few non-zeros. Comparing Eq. (\ref{eq:dict:3}) and (\ref{e1}), we can then derive the dictionary $\Psi$ as,
\vspace{-3pt}
\begin{equation}
\label{eq:psi}
\Psi = \mathbf{U}^{-1} \mathbf{W}^{-1}
\end{equation}
where $(.)^{-1}$ is the matrix inversion. Therefore, the dictionary based on the sparsity of LoRa chirps is generated. We produce dictionaries according to $SF\in \{7,8,9,10\}$, and store them in the cloud server.
As a comparison with DFT, DCT, and the proposed chirp dictionary, Fig.~\ref{fig:sparsity:sync} and~\ref{fig:sparsity:unsync} show the sparsity of typical synchronized and unsynchronized LoRa symbols ($SF=9$) with channel noise
in different domains by sorting the samples by order of magnitude. The fastest decay characteristic (or the smallest $K$) is observed in the proposed
dictionary ($\Psi$), and therefore offers the most sparse representation; which means that the most accurate approximations (or LoRa symbol value estimations) can be obtained in this dictionary by using the smallest number of measurements $M$ (Eq.~(\ref{eq:M})). The sparsity in synchronized symbols is slightly better than the unsynchronized, which means that the accuracy in recovering synchronized symbols is better than the unsynchronized. The figure also shows that the proposed $\Psi$ has two-order-of-magnitude fewer significant coefficients
(e.g., the normalized magnitude is larger than 0.1) than those of DFT and DCT.
For the down-chirps in PHY,
similar dictionaries can be obtained by replacing $\mathbf{U}$ with a matrix with a diagonal made of an ideal up-chirp. Due to the fact that most chirps in LoRa PHY are up-chirps, we first solve $\ell_1$-minimization with the up-chirp dictionary, and then try the down-chirp dictionary if no satisfactory result is obtained. Both dictionaries have similar features and performance. For brevity, we skip the discussion of the down-chirp dictionary.
\begin{figure}[hbt]
\centering
\vspace{-6pt}
\begin{subfigure}{0.46\linewidth}
\centering
\includegraphics[width=\linewidth]{figures_new/2_dictionary_sparsity/s2_simu_sparsity_sync_SF9.eps}
\caption{Synchronized}
\label{fig:sparsity:sync}
\end{subfigure}
\hfill
\begin{subfigure}{0.46\linewidth}
\centering
\includegraphics[width=\linewidth]{figures_new/2_dictionary_sparsity/s2_solving_with_magnitude.eps}
\caption{Solving with magnitude}
\label{fig:solving:magnitude}
\end{subfigure}
\hfill
\begin{subfigure}{0.46\linewidth}
\centering
\includegraphics[width=\linewidth]{figures_new/2_dictionary_sparsity/s2_simu_sparsity_unsync_SF9.eps}
\caption{Unsynchronized}
\label{fig:sparsity:unsync}
\end{subfigure}
\hfill
\begin{subfigure}{0.46\linewidth}
\centering
\includegraphics[width=\linewidth]{figures_new/2_dictionary_sparsity/s2_solving_with_residual.eps}
\caption{Solving with residual}
\label{fig:solving:residual}
\end{subfigure}
\vspace{-5pt}
\caption{(a) Sparsity for synchronized symbols based on DFT, DCT and the proposed chirp dictionary. It is more sparse in the proposed dictionary ($\Psi$) than the DFT and DCT by two orders-of-magnitude. The dashed line denotes the threshold for the coefficients with significant magnitude (0.1). (b) Sparse approximation with magnitude (Sec.~\ref{section:decoding}). (c) Signal sparsity for unsynchronized chirps, less sparse than synchronized chirps but more sparse than the DFT and DCT. (d) Sparse approximation with residuals (Sec.~\ref{section:decoding}); the residual domain is more sparse than the magnitude domain.
}
\label{fig:sparsitylevel}
\vspace{-15pt}
\end{figure}
\subsubsection{Measurement Matrix}
\label{subsubsec:Phi}
As discussed in CS theory~\cite{Baraniuk2007, Donoho2006,Candes2006}, zero-mean Gaussian matrix and balance symmetric random Bernoulli matrix achieve favorable compression performance.
For the computational efficiency on embedded devices, we choose random Bernoulli($\pm1$) as the measurement matrix $\Phi$ with a fixed seed that is shared by both gateways and the cloud server.
Each symbol has $N=2^{SF}$ samples, i.e. $N=$128, 256, 512, 1024 for $SF=7, 8, 9, 10$ respectively. If we process $SF$ separately, we have to compress PHY four times with $\Phi_{7}$, $\Phi_{8}$, $\Phi_{9}$, $\Phi_{10}$ for each $SF$, which is against our motivation for compression.
To solve this problem, we only measure with $\Phi_{7}$. For $SF=8$, we can simply concatenate two compressed vectors from $\Phi_{7}$.
Similarly, we concatenate four compressed vectors for $SF=9$ and eight compressed vectors for $SF=10$.
Thus, the gateway simply compresses every 128 samples with $\Phi_{7}$ for each channel, and in the cloud the server concatenates compressed vectors for solving different $SF$s.
\subsubsection{Compression ratio}
\label{sec:sub:compressionratio}
\begin{figure*}[htb]
\centering
\includegraphics[width=\textwidth]{figures_new/4_simulation/s4_sync_simulation_patched.eps}
\caption{Simulation with \emph{synchronized} symbols: SER affected by compression ratio and SNR for different SF}
\label{fig:simu:sync}
\vspace{-5pt}
\end{figure*}
\begin{figure*}[htb]
\centering
\includegraphics[width=\textwidth]{figures_new/4_simulation/s4_unsync_simulation_patched.eps}
\caption{Simulation with \emph{unsynchronized} symbols: SER affected by compression ratio and SNR for different SF}
\label{fig:simu:unsync}
\vspace{-5pt}
\end{figure*}
Compression ratios are defined by Eq. (\ref{equ:alpha}), and thus a smaller $M$ results in a better compression ratio. Theoretically, $M$ should be bounded on its lower end by Eq. (\ref{eq:M}). However, the noise from the original signal is hidden in compressed vectors, which may make it challenging to recover the original signals (i.e., $\ell_1$ minimization
algorithm fails
to solve Eq. (\ref{eq:l1_new})) .
Thus, $M$ is not only bounded by Eq. (\ref{eq:M}), but is also affected by the signal SNR.
We perform a simulation to investigate this phenomenon. As $N=2^{SF}$ is an exponent of 2, to simplify the DSP process, $M$ is selected among exponents of 2 (e.g., 16, 32, 64, etc.). Here, we define low, medium and high SNRs as -6, 0 and 6 dB.
Fig.~\ref{fig:simu:sync} shows that higher $SF$s outperform their lower
counterparts, and increasing SNR can improve the compression ratio. When SNR is high, $SF=9$ and $SF=7$ can be compressed to $1/16$ and $1/8$ respectively
without significant Symbol Error Rates (SERs), and the compression ratio is mainly bounded by Eq. (\ref{eq:M}). When SNR is medium and low, $SF=9$ can be compressed to $1/16$ and $1/4$ respectively without significant SERs, and the compression ratio is mainly affected by SNR.
\begin{table}[h]
\vspace{-5pt}
\caption{Reliable compression ratios based on simulations represented by $M/N$}
\vspace{-5pt}
\label{table:alpha1}
\begin{tabular}{lllll}
\hline
& SF7 & SF8 & SF9 & SF10 \\ \hline
low SNR (-6 dB) & 1 & 1/2 & 1/4 & 1/8 \\
medium SNR (0 dB) & 1/4 & 1/8 & 1/16 & 1/32 \\
high SNR (6 dB) & 1/8 & 1/16 & 1/32 & 1/32 \\ \hline
\end{tabular}
\vspace{-3pt}
\end{table}
We summarize $M/N$ in Table~\ref{table:alpha1} to represent the acceptable compression ratio $\alpha$ if SER is small (e.g., $\leq0.04$). Then, the empirical compression ratio based on Fig.~\ref{fig:simu:sync} and Table~\ref{table:alpha1} can be derived as:
\vspace{-2pt}
\begin{equation}
\label{eq:empiricalalpha}
\alpha = max \{ min \{ 1-2^{-\floor{\frac{SNR_{dB}}{3}+SF-5}} ,1-\frac{2 \cdot SF}{2^{SF}} \}, 0 \}.
\end{equation}
For unsynchronized symbols, as shown in Fig.~\ref{fig:simu:unsync}, the performance is slightly poorer than that of the synchronized symbols. An unsynchronized symbol is composed of fractions of two consecutive chirp symbols (i.e. the last few samples from the first chirp and the first few samples from the second chirp).
Thus, sparsity $K$ is increased from 1 to 2, and the lower bound Eq. (\ref{eq:M}) is slightly larger than that of the synchronized symbols. We modified Eq. (\ref{eq:empiricalalpha}) to select
an appropriate compression ratio for unsynchronized symbols accordingly:
\vspace{-2pt}
\begin{equation}
\label{eq:empiricalalpha2}
\alpha = max \{ min \{ 1-2^{-\floor{\frac{SNR_{dB}}{3}+SF-6}} ,1-\frac{4 (SF-1)}{2^{SF}} \}, 0 \}.
\end{equation}
\section{Introduction}
\label{section:introduction}
Low-Power Wide Area Networks (LPWANs) are emerging wireless technologies with features such as comprehensive signal coverage, low bandwidth, potentially small packet sizes, and long battery life \cite{Farrell2018}. One of the representatives is LoRa, which has been widely used in commercial and industrial applications, such as logistical tracking, smart agriculture and intelligent building\cite{sinha2017survey}.
LoRaWAN is a recognized MAC-layer LoRa protocol for reliable data transfer, and it is generally deployed on unlicensed ISM bands with 125 kHz or 500 kHz narrow band channels. Such narrow bands limit the bit rate down to several kilo-bits or hundred-bits per second, while they benefit the demodulator's sensitivity, making it possible to detect and decode LoRa signals significantly lower than noise floor.
Previous research demonstrates that if only one channel is used, LoRaWAN coverage drops exponentially as the number of end-devices grows \cite{Georgiou2017} and may only support approximately 120 nodes for a typical smart city deployment \cite{Bor}. Some other research similarly indicates that LoRaWAN can support from 200-1000 nodes in different applications~\cite{Liando2019,Xu2019}, which raises concerns about the scalability of LoRaWAN.
To this end, by extending from single to multiple channels similar to frequency division multiple access (FDMA), the scalability can be increased~\cite{Liando2019}.
Typical LoRaWAN gateways equipped with Semtech SX1301 chips \footnote{SX1301 datasheet.
~\url{https://www.semtech.com/products/wireless-rf/lora-gateways/sx1301}
}
can operate with up to 8 $\times$ 125kHz channels, which provides greater network capacity than a single channel network by eight times.
Furthermore, in the USA, up to 64 $\times$ 125kHz narrow-band channels are allocated on unlicensed ISM bands for LoRaWAN. A naive approach to cover more than eight channels is to use several gateways simultaneously in one spot. A commercial outdoor LoRaWAN gateway costs approximately US\$1,000. Therefore, covering all 64 channels would be expensive and difficult to maintain.
Beyene et al. propose the implementation of NB-IoT via Cloud-Radio Access Networks, which are easy to implement and cost-efficient to deploy~\cite{Beyene2017}. NB-IoT and LoRa/LoRaWAN are both LPWAN technologies and share many common features. Inspired by the C-RAN of NB-IoT, we propose a C-RAN architecture for LoRaWAN as an affordable solution to support as many LoRaWAN channels as possible. Thus, with the help of software-defined ratios (SDR), parallel gateways are replaced with a single remote radio head, and PHY processing is offloaded to the cloud.
As an extra benefit of C-RAN, the opportunity to increase the battery life for end devices is provided. Some other approaches such as optimal frequency selection~\cite{gadre2020frequency} and backscatter~\cite{Talla2017} have been proposed, while our approach is based on spatial diversity gains. Similar to the architecture of cellular networks~\cite{Checko2015}, multiple LoRaWAN gateways are commonly deployed to provide wide-area network coverage. Therefore, the signal from one end device can be received by multiple gateways and processed jointly. In a recent research, Dongare et al. implemented such a system to exploit the spatial diversity gain to improve SNR by coherently combining PHY samples captured by various gateways in different locations~\cite{Dongare2018}. Thus, an end-device may transmit with a faster bit rate, which results in a shorter transmission duration for a fixed packet/data payload length. Their evaluation shows that increasing the number of received gateways improves the SNR of packets in an approximately logarithmic manner.
Although the aforementioned C-RAN is a promising architecture with many benefits for IoT wireless networks,
such a system has a huge impact on the PHY offloading network between the gateways and the cloud. According to Charm~\cite{Dongare2018}, when a moving average compressed technique is applied for PHY, 9 Mbps is required for each 500kHz channel and 2.25 Mbps is for each 125kHz channel respectively, which produces $2.25$ Mbps $\times$ 64 = 144 Mbps data traffic to the cloud if a gateway supports 64 $\times$ 125kHz LoRa channels. For lossless Nyquist sampling and data stored as 24-bit I/Q samples (12-bit for I/Q each, same as SX1301), a minimal bit rate of $24$ bit$ \times (64 \times 125$kHz$) = 192$ Mbps is required for the PHY offloading network. Both settings require gigabit bandwidth for reliable data transmissions, which is challenging in both outdoor or indoor scenarios such as pastures and buildings with sub-100-megabit Internet connections.
Moreover, in some rural areas, Internet can only be provided via satellites, the bandwidth of which is very limited. On the other hand, a large-scale LoRaWAN (e.g., with hundreds of gateways) will pose a significant traffic to the data center. It may influence the real-time delivery of PHY samples and reduce the performance of joint decoding that requires
synchronized PHY samples from different gateways.
One solution is to equip optical fibers as part of the infrastructure of the PHY dispatching network. However, the cost is unaffordable for many low-cost or ad hoc IoT applications. Another solution is to upload active channels only. However, for large-scale deployment (i.e., tens of thousands of nodes), the probability of simultaneous multi-channel occupation
is high. Moreover, because low SNR signals can benefit from joint processing in the cloud, the channel activity detector becomes more sensitive and uploads PHY samples of idle channels to the cloud due to ``false alarms''.
Therefore, PHY compression is the key enabler for LPWAN C-RAN. To this end, we propose a Compressive Sensing (CS)-based technique, called \emph{Nephalai}\footnote{In ancient Greek mythology, \emph{Nephalai} is the nymph of the clouds.}, to reduce the network bandwidth between gateways and the cloud.
Fig.~\ref{fig:system-overview} shows the overview of \emph{Nephalai}, which leverages the sparsity of the PHY for signal compression and (joint) reconstruction.
Dictionaries and measurement matrices in \emph{Nephalai} are custom-designed to \emph{exploit the structure of LoRa radio signals} to achieve the best compression and reconstruction performance.
\emph{Nephalai} is designed to run in real-time and is implemented with SDR {\footnote{One limitation for \emph{Nephalai} is the front-end hardware. Although our prototype discussed in Sec.~\ref{section:architecture} later can support 64 channels, if \emph{Nephalai} is implemented on legacy front-end SX1257, it can support 8 channels only.}}.
Our testbed evaluation in our campus has shown that, 1) up to 93.7\% samples can be reduced without packet reception rate (PRR) reduction; 2) \emph{Nephalai} can improve battery lifetimes to 1.7x with four gateways and 87.5\% PHY samples compressed.
The contributions of this paper are as follows.
\begin{itemize}
\item We propose a novel CS-based compression technique for cloud-assisted LPWAN that significantly reduces the bandwidth between the gateways and the cloud.
\item We propose a new dictionary to achieve high compression ratios without performance degradation. The proposed dictionary
exploits the structure of LoRa radio signals, and achieves more than two orders-of-magnitude better sparse representation than standard Discrete Fourier transform (DFT) and Discrete Cosine transform (DCT) domains.
\item We implement a prototype of \emph{Nephalai} with software-defined radios, and
our empirical evaluation demonstrates its superior
performance on embedded devices.
\end{itemize}
\begin{figure}[tb]
\centering
\includegraphics[width=0.78\linewidth]{pictures/Overview}
\caption{The overview of \emph{Nephalai} decoding in the cloud with compressed PHY samples.}
\label{fig:system-overview}
\end{figure}
\section{Background}
\label{section:lorapremier}
\subsection{LoRa physical layer}
LoRa uses chirp spread-spectrum (CSS) as the method for modulation ~\cite{seller2016low,Vangelista2017}. The Spreading Factor ($SF$) is usually defined as an integer from 7 to 12, representing the number of encoded bits per chirp symbol. Bandwidth ($BW$) is the spectrum constraint of a channel, typically 125 or 500 kHz~\cite{LoRaAlliance2017}. As discussed in Sec.~\ref{section:introduction}, the LoRaWAN gateway uses 125 kHz
for receiving packets from end devices, and thus in this paper we only focus on 125 kHz channels.
LoRa utilizes time-shifted chirps in symbol modulation to carry information.
The frequency of an up-chirp increases in a linear manner, while a down-chirp is the opposite.
\subsection{Demodulation}
\label{section:demodulation}
The commonly used demodulation method is pulse compression, where the chirp symbol is first multiplied by a down-chirp in the time domain, and then processed with Fast Fourier transform (FFT) \cite{seller2016low,Knight2016}.
The result is indicated by the most significant component in the frequency domain.
If the symbol is not well segmented or unsynchronized, several peaks instead of one may show up,
which results in demodulation failure.
Open-source software such as gr-lora \cite{Knight2016,Robyns2017} provide demodulation and decoding functions, while \emph{Nephalai} focuses on PHY compression only.
\subsection{Synchronized symbol}
\label{sec:sync}
Inspired by LoRaWAN class B \cite{LoRaAlliance2017} and slotted ALOHA \cite{Polonelli2019}, we can synchronize end nodes and gateways so gateways can receive with non-overlapped windows as shown in Fig.~\ref{fig:sync:rx}.
However, perfect synchronization is neither possible nor necessary. Here, we use synchronized reception to improve the compression performance only, and further digital signal processing is performed in the cloud for fine-grain symbol segmentation. Thus, the synchronization error tolerance is high. This will be discussed further in Sec.~\ref{subsec:symbolcompression}.
\vspace{-5pt}
\begin{figure}[htb]
\centering
\includegraphics[width=0.93\linewidth]{figures_new/sync_rx.eps}
\caption{Synchronized receiving for chirp symbols}
\label{fig:sync:rx}
\end{figure}
\vspace{-8pt}
\section{Related work}
\label{section:background}
\subsection{LPWAN, LoRa, LoRaWAN}
LPWAN~\cite{Farrell2018,sinha2017survey, centenaro2016long} has attracted much attention from both academia and industry in recent years.
LoRaWAN~\cite{sinha2017survey,Adelantado2017,Saari2018} is standardized by the LoRa Alliance for LPWAN on an unlicensed spectrum.
LoRa~\cite{seller2016low,Knight2016,robyns2018multi,ghanaatian2019lora,seller2018low,Vangelista2015} is the physical-layer foundation of LoRaWAN and defines modulation and radio communication.
Recent research proposes slotted ALOHA based on a synchronization technique~\cite{Polonelli2019}, which inspires us to synchronize LoRa symbols based on a similar scheme.
In this paper, we focus on the sparsity of LoRa signals, and leverage the structure of LoRa in demodulation to optimize the performance of physical-layer compression.
\subsection{C-RAN}
\label{section:relatedwork-cran}
C-RAN was proposed originally for cellular networks based on the concepts of centralization and virtualization for the baseband operations~\cite{ChinaMobile2011, Checko2015, Wubben2014, Beyene2017}. The network can process demodulation
in the cloud coherently, to exploit the diversity scheme. This refers to improving the reliability of wireless communication by using multiple radio channels~\cite{Brennan1959}.
Such cloud-assisted decoding techniques in physical layers have also been investigated in Wi-Fi \cite{Tan2009,Xie2014} and LPWAN \cite{Dongare2018,hoeller2018analysis,Beyene2017}.
The modification of gateways is transparent to the senders; therefore, there is no requirement for changes to the original embedded LoRa devices, which maintain their compatibility with the legacy devices.
\subsection{Physical Layer Compression}
One challenge of C-RAN is the high bandwidth requirement in transmitting I/Q streams from the edges to the cloud~\cite{Checko2015, Beyene2017}.
The Wyner-Ziv coding scheme leverages the correlation (side information) among receivers to use a finer quantizer in PHY compression \cite{Xia2018,Park2014}.
However, the implementation of a distributed Wyner-Ziv compression is challenging mainly due to the complexity of obtaining the optimal joint compression codebook and the joint decompressing/decoding in the cloud~\cite{Peng2015}.
Alternatively, Compressive Sensing has been applied to achieve distributed front-haul compression \cite{Rao2015,wang2015compressive}. However, designing a sparse representation exploiting LoRa structures to improve compression performance has not yet been studied.
To this end, the proposed custom-designed dictionary and measurement matrices achieve
more than two orders-of-magnitude better performance than conventional DCT and DFT domains
used in prior work.
\textbf{Summary}: \emph{Nephalai} is partly inspired by Charm~\cite{Dongare2018}, but makes significant contributions towards reducing the traffic between LoRaWAN gateways and the cloud. Charm focuses on improving SNR and battery life with multiple gateways, while our work \emph{Nephalai} focuses on I/Q compression to further increase the capacity for PHY processing in the cloud. The compression technique used by Charm is the sum of consecutive samples in windows and generates data at a rate of 9 Mbps for a 500 kHz band. Taking our evaluation in Sec.~\ref{subsubsec:singledecoding} as an example, with 87.5\% compression ratio, the data rate is 375 kbps for 125 kHz, equivalent to 1.5 Mbps for 500 kHz (more than 80\% reduction compared to Charm), which can help Charm further reduce the data rates between the gateways and the cloud. Thus, the proposed compression technique is complementary to Charm. Furthermore, the proposed compression technique
can also be applied in other scenarios such as multiple (or full) channel reception.
\section{Architecture}
\label{section:architecture}
The \emph{Nephalai} system has one cloud server equipped with GPU for
$\ell_1$ minimization acceleration, and inexpensive single-board computers with SDRs as the edge gateways. Physical-layer radio samples are transferred from gateways to the cloud server via conventional Internet infrastructure
\begin{figure*}[hbt]
\centering
\vspace{-3pt}
\includegraphics[width=0.8\linewidth]{figures_new/architecture.eps}
\vspace{-6pt}
\caption{Baseband block diagram showing the architecture of \emph{Nephalai}}
\label{fig:clora-architecture}
\vspace{-5pt}
\end{figure*}
Fig.~\ref{fig:clora-architecture} depicts the overall architecture of \emph{Nephalai}.
The gateway clocks are synchronized via PPS from GPS modules
with the accuracy of several microseconds. The accurate timestamp can help synchronize LoRa chirp symbols
(see Fig.~\ref{fig:sync:rx})
and help the cloud server detect coherent LoRa packets easily.
To analyze the complexity of our encoding algorithm in edge devices\footnote{We omit the complexity analysis of the proposed decoding algorithm in the cloud (i.e., $\ell_1$ minimization solver) since the cloud can be seen as having unlimited resources.}, suppose we have $N$ samples per symbol (this will be discussed in Sec.~\ref{subsubsec:Phi}, $N=128$ in practice), $M$ samples per compressed vector, $C$ as the number of channels and $P$ as the number of low pass filter (LPF) taps. Then, the frequency conversion block together with LPF is $O(NP)$, the down-sampler is $O(N)$, and the CS block is $O(MN)$. The overall complexity in the edge devices is $O(NPC+NC+NMC)$. Therefore, fewer taps for LPF and higher compression ratio for CS block can improve the performance of the embedded system.
In order to support multiple 125 kHz channels as discussed in Sec.~\ref{section:introduction}, the SDR of the gateway captures the whole 13 MHz LoRa spectrum, and the embedded system filters each channel and compresses using a shared measurement matrix. Compressed bits of each channel are packed together and uploaded to the cloud server. The cloud server then performs decompression and demodulation to recover the LoRa chirp symbols or jointly process all coherent symbols to improve their accuracy.
\section{Prototype Implementation}
\label{sec:prototype}
\begin{figure}[hbt]
\centering
\includegraphics[width=0.71\linewidth]{pictures/hardware.jpg}
\vspace{-7pt}
\caption{\emph{Nephelai} gateway and a LoRa transmitter}
\label{fig:clora-hardware}
\vspace{-10pt}
\end{figure}
\textbf{The Edge Gateway} The \emph{Nephelai} gateway shown in Fig.~\ref{fig:clora-hardware} has a radio front-end to capture signal samples on given LoRa channels, and an embedded computer to pre-process and compress the received signal samples before uploading to the cloud. In our prototype, we select BladeRF 2.0 SDR as the radio front-end to capture radio signals on LoRaWAN uplink channels (e.g., 902 MHz to 915 MHz in the USA). The output of SDR is a stream of $I$ and $Q$ components, which can be regarded as complex values where $I$ denotes real and $Q$ denotes imaginary parts respectively.
The SDR can sample up to 61.44 mega samples per second (MSps), which are capable of capturing all the information in the whole 13 MHz upstream spectrum for USA defined by LoRaWAN. The Nyquist sampling rate for one channel is 125 kHz for complex samples (i.e. 250 kHz for real samples), and therefore the sample rate for 64 channels is 8 MSps (note the 75 kHz guard band between consecutive 125kHz channels, meaning that 8 MHz is for LoRa channels on a 13 MHz spectrum).
The SDR is connected to a Odroid-N2 (6-core single board computer with quad-core Cortex-A73@1.8GHz and dual-core Cortex-A53@1.9GHz) via a USB 3.0 port, through which the LoRa radio samples are transferred. Next, the Odroid-N2 processes (see Sec.~\ref{section:architecture})
and compresses (see Sec.~\ref{subsec:symbolcompression}) the samples before transferring them to the cloud server. The sampling rate of our prototype is 13 MHz, which is sufficient to cover the 13 MHz LoRaWAN spectrum.
Without loss of generality, we demonstrate the compression performance of \emph{Nephelai} in a single LoRa uplink channel. If one single channel is compressible,
so are 63 other channels.
We design and implement the software for \emph{Nephelai} gateways, called \emph{gr-Nephelai} based on the open-source software-defined ratio platform GNU-Radio. The frequency conversion and low pass filter shown in Fig.~\ref{fig:clora-architecture} are implemented in C++ and complied with single instruction multiple data (SIMD) optimization. Although there are 64 parallel branches in Fig.~\ref{fig:clora-architecture}, we implement one block for all 64 channels instead of one block for each of the 64 channels to reduce the handover between blocks.
The low-pass filter taps are selected as 47 to maintain real-time performance. The passband is designed to be 275 kHz, which works well to avoid inter-channel interference. When the gateway is running at full capacity (processing 64 channels), the overall CPU usage is approximately 60\%.
\textbf{The transmitter} We program Multitech mDot\footnote{
MDot datasheet.
\url{https://www.multitech.com/brands/multiconnect-mdot}
},
which comprises a LoRa wireless chip (SX1272), to periodically transmit 4 predefined bytes.
The mDot with STM32F411RET uses 31 mA @100 MHz in the maximum power setting.
\textbf{The Cloud Server} Although the \emph{Nephelai} cloud server can be any kind of general server, we use a 12-core CPU, 32 GB RAM and Nvidia 2070 GPU server in our
prototype. It can perform $\ell_1$-minimization algorithms for joint sparse LoRa signal reconstruction (i.e., LoRa packet decoding, see Sec.~\ref{section:jointdecoding}) in real-time.
\section{Conclusion}
\label{section:conclusion}
We introduce \emph{Nephelai}, which is based on CS-theory, to reduce the
bandwidth requirement between edge gateways and the cloud server for cloud-assisted
LoRaWAN. \emph{Nephelai} exploits: 1) the physical layer structure of LoRa symbols for a custom designed \emph{dictionary} to
significantly improve its compression performance,
2) the relationship between compression ratios, SNR and SFs to select an appropriate compression ratio, and
3) radio signal spatial diversity by joint decoding to improve the PRR as well as the battery lifetime for end devices.
Our empirical results with an edge gateway prototype consisting
of SDR and Odroid-N2 show that \emph{Nephelai} can reduce
traffic between gateways and cloud servers by up to 93.7\% and can significantly improve
the scalability of cloud assisted LoRaWAN.
\section{Evaluation}
\label{section:evaluation}
\subsection{Goals, Metrics and Methodologies}
\label{subsec:goals}
We deployed a \emph{Nephelai} testbed with four \emph{Nephelai} gateways (see Sec.~\ref{sec:prototype}) on our campus as shown in Fig.~\ref{fig:floorplan}, where gateways are connected to a \emph{Nephelai} cloud server (see Sec.~\ref{sec:prototype}) via Wi-Fi. We programmed seven mDots (see Fig.~\ref{fig:clora-hardware}) as LoRa motes to periodically transmit predefined LoRa packets with power from 2 dBm to 14 dBm. We installed the LoRa motes in several representative positions in the campus to emulate real applications, and collected LoRa radio samples with each gateway simultaneously. During our evaluation, we collected more than one million LoRa chirp symbols among SF7 to SF10 to evaluate the performance of \emph{Nephelai}.
We deployed LoRa motes to emulate real use cases. Mote-1 was an indoor temperature and humidity sensor; mote-2 acted as a passive infrared sensor (PIR), which functioned as an occupancy detector for the warehouse; mote-3 behaved as a smart water meter; mote-4 represented a simple outdoor weather station; mote-5 was attached
to a stair handrail and counted people; and mote-6 and mote-7 measured the soil's humidity to control a watering system for the lawn.
In this evaluation we were not interested in application data
but instead focused on PHY compression and potential battery lifetime improvement with joint decoding.
\begin{figure} [htb]
\vspace{-8pt}
\centering
\includegraphics[width=0.99\linewidth]{figures_new/campus_wide.png}
\vspace{-8pt}
\caption{\emph{Nephelai} test-bed on our campus. The gateways are marked with the letter A/B/C/D and stationed inside buildings near windows to simulate a customer-deployed scenario. The transmitters (motes) are labeled from 1 to 7, and marked with green circles. Mote-1 is on the same floor (the 4th floor) as gateway C; mote-2 is on the 3rd floor; mote-3 is hidden in the basement, 5 floors below gateway C. Motes-4/5/6/7 are placed outdoor without any cover.}
\label{fig:floorplan}
\vspace{-8pt}
\end{figure}
\emph{Nephelai} is designed to implement the physical layer compression for cloud-assisted LoRa demodulation/decoding and to potentially improve transmitters' energy efficiency. Therefore, the \textbf{goals} of our evaluation were:
\begin{enumerate}
\item to study whether \emph{Nephelai} can reduce the network bandwidth of the front-haul in LPWAN C-RAN,
\item to study the impact of compression ratio ($\alpha$) on the system's performance, and
\item to study whether \emph{Nephelai} can demonstrate similar energy improvements for the LoRa transmitter as the state-of-the-art LPWAN C-RAN, but with fewer front-haul data rates.
\end{enumerate}
The \textbf{metric} for network bandwidth reduction is bits per second (bps), and that for energy reduction is battery lifetime extension. For \textbf{methodologies}, firstly, on the symbol level we evaluate how SNR and compression ratios affect SER in order to compare these with the simulation in Sec.~\ref{sec:sub:compressionratio}. And then on packet level, we evaluated the PRR for single gateway scenarios with three LoRa motes and different power transmission levels. Furthermore, we evaluated the joint processing gain with four gateways and four transmitters to demonstrate that an equivalent SNR improvement can be achieved as the state-of-the-art \cite{Dongare2018}, i.e. to extend the battery lifetime to approximately 1.7x (equivalent to 2.3 dB SNR improvement) with four gateways, but with greater PHY compression. As there are different SFs resulting in different PRRs, we assumed that each SF was equal likely to be selected, and we calculated the expected PRR by averaging the PRRs of all SFs.
\vspace{-5pt}
\subsection{Empirical Results}
\label{subsec:emperical}
\subsubsection{Compression ratio}
\label{sec:eva:compressionratio}
\begin{figure*}[htb]
\centering
\includegraphics[width=\textwidth]{figures_new/4_simulation/s4_sync_campusI_patched.eps}
\caption{Synchronized symbols from testbed: SER affected by compression ratio and SNR for different SFs}
\vspace{-5pt}
\label{fig:eva:sync}
\end{figure*}
\begin{figure*}[htb]
\centering
\includegraphics[width=\textwidth]{figures_new/4_simulation/s4_unsync_campusI_patched.eps}
\caption{Unynchronized symbols from testbed: SER affected by compression ratio and SNR for different SFs}
\label{fig:eva:unsync}
\end{figure*}
As discussed in Sec.~\ref{subsubsec:Phi}, the compression ratio ($\alpha$) is calculated using the dimension of measurement matrix $\Phi \in \mathbb{C}^{M \times N}$ (see Eq.~(\ref{eq:y})).
In this section, we are only interested in how SNR affected the compression ratios, and in evaluating the compression ratio determination equations (i.e., Eq. (\ref{eq:empiricalalpha}) and (\ref{eq:empiricalalpha2})) for synchronized and unsynchronized symbols.
We programmed motes-1/2/3 to transmit with power varying from 2 dB to 14 dB, and collected 50,000 synchronized and unsynchronized symbols respectively.
We grouped symbols with respect to their low (-6 dB), medium (0 dB) and high (6 dB) SNR.
Fig.~\ref{fig:eva:sync} and \ref{fig:eva:unsync} compare the compression performance of different SFs and SNRs based on the symmetric Bernoulli matrix($\Phi$) of $\pm 1$ and our proposed chirp dictionary $\Psi$ (see Sec.~\ref{subsec:symbolcompression}).
For example, for medium SNR (0 dB, i.e. the signal energy is equivalent to the noise floor) with synchronized symbols in Fig.~\ref{fig:eva:sync}, SF9 achieves an SER below 0.04 with a compression ratio of 93.7\%. This represents approximately 16 times the bandwidth reduction in the C-RAN front-haul.
With a small SER value (e.g., $\leq 0.04$) as the reliable transmission threshold, we can summarize that the evaluation matches the simulation, when referring to $M/N$ in Table~\ref{table:alpha2} based on Fig.~\ref{fig:eva:sync}, which compares Table~\ref{table:alpha2} to Table~\ref{table:alpha1}.
Therefore, we can use Eq.~(\ref{eq:empiricalalpha}) in compression ratio selection. We observed similar patterns in
the results of unsynchronized symbols in Eq.~(\ref{eq:empiricalalpha2}), however we omit the discussion here for brevity.
Furthermore, we performed PRR evaluation for synchronized packets with different SNRs, SFs and compression ratios as shown in Fig.~\ref{fig:revision2:prr}. The LoRa packets transmitted in the evaluation had fixed length and their payloads consisted of 4 bytes (equivalent to 8 symbols).
We defined PRR 75\% as the threshold for reliable transmission~\cite{le2007design} and used it in our compression ratio selection.
With the PRR criteria, Fig.~\ref{fig:revision2:prr} implies a similar compression ratio selection as that with SER in Table~\ref{table:alpha2}. Thus, we can use Eq.~(\ref{eq:empiricalalpha}) in compression ratio selection. For unsynchronized symbols, similar to the discussion with SER, Eq.~(\ref{eq:empiricalalpha2}) is used for compression ratio selection.
\begin{table}[]
\caption{Reliable compression ratio based on testbed collected data represented by $M/N$.}
\vspace{-3pt}
\label{table:alpha2}
\begin{tabular}{lllll}
\hline
& SF7 & SF8 & SF9 & SF10 \\ \hline
low SNR (-6 dB) & 1 & 1/2 & 1/4 & 1/8 \\
medium SNR (0 dB) & 1/4 & 1/8 & 1/16 & 1/32 \\
high SNR (6 dB) & 1/8 & 1/16 & 1/32 & 1/32 \\ \hline
\end{tabular}
\vspace{-13pt}
\end{table}
In summary, compared to the benchmark of lossless algorithm LZ77 that achieves a compression ratio of 7.5\% (see Sec.~\ref{subsec:gzip} for more details), the proposed approach can improve the compression ratio by approximately 10 times, depending on the parameter settings. For example, when SNR is high, a compression ratio up to 93.7\% can be achieved for most SFs. Therefore, \emph{Nephelai} achieves a significant reduction in traffic between gateways and the cloud server, which makes the cloud-assisted LoRa decoding scheme more scalable.
\vspace{-6pt}
\subsubsection{The performance of single gateway}
\label{subsubsec:singledecoding}
\begin{figure*}[htb]
\centering
\includegraphics[width=0.95\textwidth]{figures_revision/re2_PRR_3SNR_SF_alpha.eps}
\caption{PRR affected by SNR, SFs and compression ratios for synchronized symbols/packets}
\label{fig:revision2:prr}
\end{figure*}
\begin{figure*}[htb]
\centering
\includegraphics[width=0.98\textwidth]{figures_new/5_evasingle/s5_sync_lab_SF_combined.eps}
\caption{The single gateway evaluation with 3 transmitters and \textbf{synchronized} symbols shows that PRR is affected by compression ratios in different power transmission scenarios. Motes-1/2/3 were placed according to Fig.\ref{fig:floorplan}.}
\label{fig:eva:single:sync}
\end{figure*}
In the single gateway evaluation using a real case, our goal was to compress PHY without PRR degradation. As discussed in Sec.~\ref{section:compression}, over-compression means that the $\ell_1$ minimization
algorithm fails
to solve Eq. (\ref{eq:l1_new2}), which increases SERs and decreases PRRs.
Firstly, as shown in Fig.~\ref{fig:floorplan}, LoRaWAN transmitter motes-1, 2 and 3 were installed in a fixed position and were programmed to transmit 4 bytes with different spreading factors ($SF=7,8,9,10$) at 2 dBm, 8 dBm and 14 dBm respectively. We collect packets via one gateway in either synchronized or unsynchronized mode.
Secondly, with the algorithm proposed in Sec.~\ref{section:decoding}, we calculated the PRR for different compression ratios.
Instead of SER, we were more interested in PRR which describes the performance of end-to-end data transmissions. For example, if PRR is halved, the energy required to successfully deliver one packet is doubled, as the embedded node needs to transmit the packet twice. Therefore, the battery lifetime is halved. It is evident that PRR is more intuitive than SER in describing battery lifetime.
In our synchronized scenario, the compression ratio of 87.5\% for motes-1 and 2 produced more than 90\% PRR when power transmission was medium. For mote-3 in the basement, the compression ratio of 75\% produced more than 50\% PRR. We note that mote-3 was over-compressed with the compression ratio of 87.5\% because the PRR is only 30\% (see Fig.~\ref{fig:eva:single:sync}). Increasing power transmission could have increased the compression ratio for mote-3 from 75\% to 87.5\%, allowing it to maintain its PRR above 50\% (Fig.~\ref{fig:eva:single:sync}(c)). According to the mDot datasheet,
increasing power transmission from medium to high consumes 3.7\% extra energy, which provides another acceptable option for scalability improvement.
In summary, Fig.~\ref{fig:eva:single:sync}
shows that PRR does not decrease with appropriate compression ratios, and increasing power transmission can improve the compression performance of \emph{Nephelai}.
Therefore, if all motes transmit at 14 dBm, we can select 87.5\% as the compression ratio. For 64 channels, only $64 \times 24 \times 125000 \times (1-0.875) = 24$ Mbps is required for a single gateway in LPWAN C-RAN. Consequently, such a gateway can operate with bandwidth-limited Internet connections, widely extending the deployment region and application scenarios.
\subsubsection{The performance of joint decoding}
\label{subsubsec:jointdecoding}
\begin{figure*}[hbt]
\centering
\includegraphics[width=0.95\textwidth]{figures_revision/s7_delta_PRR_sync_dataset_joint_snr.eps}
\caption{PRR improvement by 4-gateway joint decoding with compression ratio}
\label{fig:eva_prr_improvement}
\end{figure*}
Compressing PHY without PRR degradation is possible as shown in Sec.~\ref{subsubsec:singledecoding} above. In this section, we evaluate the improvement of PRR with joint decoding under compression. Our goal was to achieve an equivalent performance to the state-of-the-art Charm system (i.e. 2.3 dB SNR improves with four gateways, see Sec.~\ref{section:introduction} for the details), but with less front-haul bandwidth between the gateway and the cloud.
Firstly, we programmed motes-4,5,6 and 7 to be in synchronized mode and to send 4 byte messages periodically with high transmission power\footnote{We define 14 dBm as high transmission power in this paper, but in fact 14 dBm is a moderate choice compared to the maximum 22 dBm.} (14 dBm). We collected LoRa radio samples simultaneously via gateways-A,B,C and D with different compression ratios (see Sec.~\ref{subsec:goals} and Fig.~\ref{fig:floorplan} for testbed deployment in details). The number of packets for each SF was equal.
Secondly, we calculated the PRR for single gateway decoding and coherent joint decoding with 4 gateways (according to the algorithm discussed in Sec.~\ref{section:inthecloud} under different compression ratios).
We averaged PRR for all SFs to get an expected PRR as discussed previously in Sec.~\ref{subsec:goals}.
Fig.~\ref{fig:eva_prr_improvement} shows how much improvement can be seen by joint decoding with four gateways compared to a single gateway. For battery-powered LoRa motes, the expected energy consumption per packet is reversely proportional to the PRR, and thus the expected battery lifetime is proportional to the PRR.
When the compression ratio was 87.5\%, mote-4 had PRR above 99\% (since the position of mote-4 was very close to one of the gateways), while motes-5,6 and 7 had poor PRR with a single gateway. After joint decoding with four gateways, the PRR of mote-5 was improved from 70\% up to 93\%, while mote-6 went from 47\% to 77\%, and mote-7 went from 36\% to 76\%. The improvement factors are 1.33, 1.64 and 2.11 respectively, and the average is about 1.70. Therefore, on average, joint decoding extends battery lifetime to approximately 1.70 with four gateways when the compression ratio is 87.5\%.
We note that
the least improved PRR occurred when the compression ratio is 75\%, which is equivalent to a good quality, low power
wireless link with a high C-RAN bit rate. Therefore, 87.5\% is the recommended trade-off between compression ratio and PRR.
For compression up to 93.7\%, single gateways experience severe packet loss for each mote. After joint decoding, the PRR of mote-4 was improved from 40\% to 58\%, while mote-5 improved from 16\% to 22\%, mote-6 from 13\% to 22\%, and mote-7 from 10\% to 20\%.
However, this compression ratio is not recommended because most of the PRRs are still poor (i.e., less than 50\%) even with joint decoding.
Particularly, increasing the compression ratio from 87.5\% to 93.7\% for mote-4 causes PRR to decrease from more than 99\% to 40\%,
meant that the mote had a shorter battery lifetime by approximately 60\%.
Finally, we note that when the compression ratio is 75\%, with joint decoding, all PRRs are increased to more than 99\%.
In summary, \emph{Nephelai} with 4 gateways improves the PPR and the battery lifetime of a LoRa transmitter by 1.7 times on average, with the recommended compression ratio of 87.5\% compared to a single gateway, which is equivalent to 2.3 dB SNR improvement ($ 10 log_{10} 1.7$). The compression ratio of 87.5\% also means that the PHY is compressed from 3 Mbps down to 375 kbps for one channel, while that of Charm is 2.25 Mbps per channel (see Sec.~\ref{section:introduction} for details).
This demonstrates that \emph{Nephelai} has similar functionality in improving the battery lifetime of embedded IoT devices as Charm~\cite{Dongare2018}, while \emph{Nephelai} reduces the bandwidth between gateways and the cloud by $1-0.375/2.25=83.3\%$.
\subsubsection{Cloud computing overhead}
Solving $\ell_1$ minimization is computationally intensive, but can be handled with parallel implementation using multi-threading, GPU, FPGA, etc. in the cloud. If the demodulation of one symbol is performed in real-time, and the delay caused by data transmission and computation (from the gateway to the cloud, and back to the gateway) meets the LoRaWAN requirement for an ACK, the \emph{Nephelai} system is feasible.
We evaluated cloud computing overhead by performing single-threading tests with MATLAB on Intel Core i7-8700 CPU @ 3.20GHz with 32GB RAM for 1000 times calculation per case as shown in Table.~\ref{table:revision1:overhead}. The worst case is SF10 with a 50\% compression ratio. One symbol for SF10 can be solved in less than 500 ms with a single thread. The length of one symbol for SF10 is 8.2 ms, and in 500 ms the gateway can receive at most 61 of these symbols.
Therefore, a 64 core server can be used in the cloud to dispatch demodulation tasks to each core in order to obtain real-time demodulation within 500 ms. For other SFs and compression ratios, the computational demand is even lower. Note that the computation can be further optimised for higher efficiency.
LoRaWAN has a relatively loose requirement for ACK delays due to low bit rates (e.g., 300 bps). There is a parameter called ACK\_TIMEOUT in the LoRaWAN settings with a default value of "$2 \pm 1$s ( i.e., a random delay between 1 and 3 seconds)”. The demodulation latency is less than 500ms as discussed above, and the Internet latency is typically less than 100 milliseconds one way. Processing latency caused by gateways and radio propagation delays can be ignored. Thus, an ACK can easily be generated in one second to meet the LoRaWAN requirements discussed above.
\begin{table}[h]
\vspace{-4pt}
\caption{$\ell_1$-minimization overhead testing for different SFs and compression ratios. Unit: millisecond.}
\vspace{-4pt}
\label{table:revision1:overhead}
\begin{tabular}{lllll}
\hline
$\alpha$ & SF7 & SF8 & SF9 & SF10 \\
\hline
0.5 & 5.1$\pm$ 2.5 & 10.8$\pm$2.7 & 66.8 $\pm$61.9 & 297.7$\pm$180.5 \\
0.75 & 2.0$\pm$ 0.6 & 4.7$\pm$ 1.0 & 16.3$\pm$13.7 & 71.2$\pm$32.6 \\
0.875 & 1.0$\pm$ 0.2 & 2.2$\pm$ 0.4 & 5.5$\pm$ 3.3 & 16.1$\pm$6.1 \\
0.937 & 0.6$\pm$ 0.1 & 1.1$\pm$ 0.2 & 2.3$\pm$ 0.9 & 5.1$\pm$1.8 \\
\hline
\end{tabular}
\vspace{-6pt}
\end{table}
\subsubsection{Influence of concurrent transmission}
Theoretically, multi-channel concurrent transmission may reduce the system's performance by leaking energy as noise to other channels. However, through our evaluation, we have found that concurrent transmission does not cause system degradation.
We established a LoRa transmitter that sent packets with SF=8 and a packet length of 41.5 ms every 50 ms periodically in one 125kHz channel, and another transmitter that sent in the neighbouring channels. We calculated the PRR based on the collected samples in different interference environments: no interference, concurrent transmission on a +200kHz channel, concurrent transmission on a +400kHz channel, ... , and concurrent transmission on a +1000kHz channel.
Our evaluation results show that no significant PRR reduction is caused by concurrent transmissions.
If we have a well designed filter for each 125kHz channel, the noise caused by concurrent transmissions can be prevented. In summary, \emph{Nephelai} is robust against the interference caused by concurrent transmissions.
\section{\emph{Nephelai} in the cloud}
\label{section:inthecloud}
\subsection{Decoding (Single Gateway)}
\label{section:decoding}
Most conventional $\ell_1$-minimization algorithms require real-valued vectors and dictionaries, while communication systems always use complex values for I/Q modulation.
To solve this problem we transform the vectors from complex-valued to real-valued as,
\begin{align}
\mathbf{y'} &= [ \Re\{\mathbf{y}\}^T \quad \Im\{\mathbf{y}\}^T]^T \\
\mathbf{s'} &= [ \Re\{\mathbf{s}\}^T \quad \Im\{\mathbf{s}\}^T]^T \label{eq:complexs} \\
\Theta' &= \begin{bmatrix}
\Re\{\Theta\} & -\Im\{\Theta\}\\
\Im\{\Theta\} & \Re\{\Theta\}
\end{bmatrix}
\end{align}
where $\Theta = \Phi \Psi$. Then, we solve the problem with a real-valued $\ell_1$-minimization algorithm for Eq. (\ref{eq:l1_new}) as,
\begin{equation}
\label{eq:l1_new2}
\hat{\mathbf{s}}' = \arg\min \left\| \mathbf{s'} \right\|_1 \quad s.t. \quad \left\| \mathbf{y'} - \Theta' \mathbf{s'} \right\|_2 < \epsilon
\end{equation}
After obtaining the sparse vector $ \hat{\mathbf{s}}'$ with Eq.~(\ref{eq:l1_new2}), we recover the complex-valued sparse vector $\mathbf{s_{opt}}$ by reversing Eq.~(\ref{eq:complexs}), and thus we solve not only the magnitude but the phase of the chirp symbol.
Instead of using FFT for demodulation as described in Sec.~\ref{section:demodulation}, we proceed to estimate the most likely value $\lambda$ by using residual $r$.
The residual for symbol candidate $i \in$ \{0, 1, ..., $2^{SF}$ -1\}
is:
\begin{equation}
\label{eq:residual}
\vspace{-1pt}
r^{(i)} (\mathbf{y}) = \left\| \mathbf{y} - \Phi \Psi \delta^{(i)}(\mathbf{s_{opt}}) \right\|_2, \forall i
\end{equation}
where operator $\delta^{(i)}:\mathbb{R}^{D}\rightarrow\mathbb{R}^{D}$ indicates a vector containing the only coefficient related to candidates $i$ (the coefficients related to other candidates are set to be zeros). Then the final symbol estimation is determined by:
\vspace{-2pt}
\begin{equation}
\label{eq:solve_lambda}
\hat{\lambda} = \operatorname*{argmin}_{i} r^{(i)} (\mathbf{y}) , \forall i
\end{equation}
\vspace{-2pt}
i.e., the $\lambda$ with the minimal residual representing the modulation value.
Fig.~\ref{fig:solving:residual} shows the
result of \emph{Nephelai} decoding with Eq.~(\ref{eq:residual}) for a noisy chirp symbol. The highest peak (i.e., $1 - r^{(i)}$, suppose $r^{(i)}$ is normalized) represents the modulated value (e.g., 300) of the LoRa symbol correctly. Note that in $\mathbf{s_{opt}}$, the phase of the highest peak may be used for radio-based ranging, which is beyond the scope of this paper.
\subsection{Joint Decoding}
\label{section:jointdecoding}
We have discussed how \emph{Nephelai} recovers value $\lambda$ from
single compressed measurement $\mathbf{y}$. In this section, we discuss
how \emph{Nephelai} exploits spatial diversity for gateways and improves performance with joint decoding.
Suppose that we have $G$ gateways, and each gateway captures a transmitted copy of the same LoRa symbol independently. Next, \emph{Nephelai} estimates the SNR level $\gamma_g$ and produces residuals $r_{g}^{(i)}$ ( $g \in \{0, 1, ... G-1\}$) for $G$ gateways with Eq.~(\ref{eq:residual}).
One of the ways to fuse these residuals among gateways is to perform a weighted summation. Based on the selection of combining weights, we have four algorithms: 1) weighted equally, aka. equal gain combining (EGC); 2) weighted by the $\sqrt{SNR}$; 3) weighted by the $SNR$ aka. the maximum ratio combining (MRC), and 4) weighted by the $SNR^2$. We evaluate the algorithms with collected samples by four gateways (further discussion in Sec. \ref{subsubsec:jointdecoding}), and the results are shown in Fig.~\ref{fig:revision3:joint}. All algorithms succeed in improving the PRR, and the algorithm weighted by the $SNR$ has the best performance especially when the compression ratio is high. Thus, we choose the MRC algorithm with SNR $\gamma_g$ as the weight in the following evaluation.
Following this, the final symbol estimation is determined by:
\vspace{-3pt}
\begin{equation}
\label{eq:residual_min}
\hat{\lambda} = \arg\min_{i} \sum_{g = 0}^{G-1}{\gamma_g r_{g}^{(i)} (\mathbf{y})} ,\forall i
\end{equation}
Nephelai's joint decoding algorithm can be found in Algorithm~\ref{algo:jointdecoding}.
\vspace{-5pt}
\begin{figure}[htb]
\centering
\includegraphics[width=0.64\linewidth]{figures_revision/re3_delta_PRR_sync_dataset_joint_combined.eps}
\caption{Joint decoding algorithm comparison}
\label{fig:revision3:joint}
\vspace{-7pt}
\end{figure}
\begin{algorithm}
\DontPrintSemicolon
\KwIn{$M$-length measurements $\{\mathbf{y}_g\}_{g=0..G-1}$, estimated SNR $\{\gamma_g\}_{g=0..G-1}$ }
\KwOut{An integer $\lambda$, the decoding result}
\For{$g \gets 0$ \textbf{to} $G-1$}{
$\mathbf{s}_g \gets $ solve $\ell_1$ minimization($\mathbf{y}_g$, $\Theta$, $\epsilon$)\;
\For{$i \gets 0$ \textbf{to} $2^{SF}-1$}{
$ r_{g}^{(i)} (\mathbf{y}) = \left\| \mathbf{y_g} - \Theta \delta^{(i)}(\mathbf{s_{g}}) \right\|_2 $ \\
}
}
$\lambda \gets \arg\!\min_{i} \sum_{g=0}^{G-1} \gamma_g r_{g}^{(i)} $
\Return{$\lambda$}
\caption{{\sc joint-decoding}}
\label{algo:jointdecoding}
\end{algorithm}
|
2,877,628,090,133 | arxiv | \section{Introduction}
The body centered cubic structure (BCC, also known by its Pearson type cI2 and its Strukturbericht symbol A2) forms a crystal lattice in which all sites are equivalent. Many pure metallic elements take this structure, for example W, the crystallographic prototype. Additionally, many metal alloys adopt this structure at high temperatures, for example CuZn. In such a case each site is randomly occupied by either Cu or Zn, so that the average occupations of each site remain equivalent and the structure remains BCC from a symmetry perspective. However, each element has its own chemical interaction preferences, so that the instantaneous occupation of one site biases the occupation probability of its neighboring sites. In the case of CuZn, the preference is for unlike neighbors; at low temperatures CuZn undergoes a continuous phase transition to a chemically ordered structure of prototype CsCl (Pearson type cP2, Strukturbericht B2). Other compounds exhibit differing preferences. For example, AlLi favors a pattern of like nearest neighbors but unlike second neighbors and acquires the ordered structure of prototype NaTl (Pearson type cF16, Strukturbericht B32a). Fig.~\ref{fig:cF16} illustrates these three structures.
\begin{figure}
\includegraphics[width=6in]{Fig1.png}
\caption{\label{fig:cF16}Examples of chemical order on a body centered cubic lattice. The CsCl prototype (left) with Pearson type cP2 and Strukturbericht symbol B2, and the NaTl prototype (right) with Pearson type cF16 and Strukturbericht symbol B32a, break the symmetry of the body centered cubic W prototype (center) with Pearson type cI2 and Strukturbericht symbol A2.}
\end{figure}
While local chemical order can reduce the internal energy $U$, the resulting biased occupation probabilities reduce the entropy $S$. The trade-off between energy and entropy is captured by the free energy $F=U-TS$. While energies are readily calculated using electronic density functional theory (DFT), entropy calculations require supplementing the DFT-based energies with methods of statistical mechanics. Here we apply a sequence of approximations related to Kikuchi's Cluster Variation Method~\cite{Kikuchi1951,deFontaine79,TurchiCuZn,deFontaine94} in order to minimize the free energy and thereby calculate the temperature-dependent correlation functions, order parameters and entropies. We derive our entropy expressions heuristically based on the information content of correlation functions and the correspondence between information and entropy. Specifically, we utilize a single point approximation, a pairwise correction, and a four-point approximation~\cite{Ackermann89}. Although the methods presented here are applied only to binary alloys, they readily generalize to more complex alloy systems such as Heusler compounds~\cite{FelserBook2016} and high entropy alloys~\cite{Yeh04_1,Cantor04}.
\section{Methods}
\subsection{Correlation functions}
\label{sec:Correlations}
Let Greek indices label chemical species, and ${x_\alpha}$ be the mole fraction of species $\alpha$ in the solid solution. In the absence of other information, ${x_\alpha}$ is the probability that a given lattice site is occupied by species $\alpha$. As shown in Fig.~\ref{fig:BCC}a, each site of the BCC lattice has 8 nearest neighbors ({\em e.g. } the bond $1-3$ in Fig.~\ref{fig:BCC}b). Denote by ${y_{\alpha\gamma}}$ the fraction of nearest neighbor bonds that have species $\alpha$ on one end and $\gamma$ on the other. Each site has 6 next-nearest neighbors at a distance only $2/\sqrt{3}=1.15\times$ the nearest neighbor separation ({\em e.g. } the bond $1-2$), joining species $\alpha$ and $\beta$ with probability ${v_{\alpha\beta}}$. Also present in the BCC structure are isosceles triangles with two nearest and one next-nearest neighbor bond as edges ({\em e.g. } the triangle $1-3-4$). We denote the probability of species $\alpha$ on the symmetric vertex ($1$) and species $\gamma$ and $\delta$ on the others ($3$ and $4$) as ${u_{\alpha\gamma\delta}}$. Finally, we define the four point function ${z_{\alpha\beta\gamma\delta}}$ as the probability for species $\alpha$ on site $1$, $\beta$ on $2$, $\gamma$ on $3$ and $\delta$ on $4$.
Notice the sum-rule relationships among the probabilities. For example, ${u_{\alpha\gamma\delta}}=\sum_\beta {z_{\alpha\beta\gamma\delta}}$, ${y_{\alpha\gamma}}=\sum_\delta {u_{\alpha\gamma\delta}}$, ${x_\alpha}=\sum_\gamma{y_{\alpha\gamma}}$, and finally $\sum_\alpha{x_\alpha}=1$ expresses the normalization. In the limit of perfect disorder, when each site is occupied independently of its neighbors, the probabilities factorize, {\em e.g. } ${y_{\alpha\gamma}}={x_\alpha}{x_\gamma}$, and ${z_{\alpha\beta\gamma\delta}}={x_\alpha}{x_\beta}{x_\gamma}{x_\delta}$. In general the site occupations are {\em not} independent and we refer to the nontrivial probabilities as {\em correlation functions}.
\begin{figure}
\includegraphics[width=3in]{Fig2.png}
\caption{\label{fig:BCC}(a) The body centered cubic unit cell contains two equivalent sites (cube vertex and center). Nearest neighbor bonds are shown in yellow and next nearest neighbors in blue. (b) The BCC tetrahedron with specific sites labeled $\alpha-\delta$.}
\end{figure}
\subsection{Interaction model}
We may express the energy in terms of the correlation functions defined above. The most general expression for the energy per atom is
\begin{equation}
\label{eq:H}
U=6\sum_{\alpha\beta\gamma\delta} E_{\alpha\beta\gamma\delta} ~z_{\alpha\beta\gamma\delta},
\end{equation}
where the factor of 6 arises because the BCC lattice has 6 tetrahedra/atom. Linear combinations of the coefficients $E_{\alpha\beta\gamma\delta}$ correspond to the energies of individual chemical species, pairwise, and three- and four-body interactions. As noted above, the next-nearest neighbor bond is only 15\% longer than the nearest neighbor, while the third neighbor is twice as long. It is reasonable to keep nearest and next nearest neighbor interactions, but to omit further neighbors. Since both alloy systems are well described as nearly free electron systems, it is reasonable to neglect many-body interactions as a first approximation.
For a binary alloy at fixed composition, the fraction of nearest neighbor bonds joining like species is linearly dependent on the number joining unlike species, and the same is true for next-neighbor bonds. Thus it suffices to parameterize the energy in terms of parameters $J$ and $K$ representing the energies of unlike nearest and next-nearest neighbor bonds. Formally, note that
\begin{align}
\sum_{\alpha\gamma} E_{\alpha\gamma} ~{y_{\alpha\gamma}} &= \sum_{\alpha\ne\gamma}E_{\alpha\gamma} ~{y_{\alpha\gamma}}
+ \sum_\alpha E_{\alpha\alpha} ~y_{\alpha\alpha} \\ \nonumber
&= \sum_{\alpha\ne\gamma}\left(E_{\alpha\gamma}-\frac{1}{2}(E_{\alpha\alpha}+E_{\gamma\gamma})\right) ~{y_{\alpha\gamma}}
+ \frac{1}{2}\left(\sum_\alpha E_{\alpha\alpha} ~{x_\alpha} + \sum_\gamma E_{\gamma\gamma} ~{x_\gamma}\right).
\end{align}
Only off-diagonal terms ($\alpha\ne\gamma$) multiply ${y_{\alpha\gamma}}$, and the final term in the above depends only on global composition and hence is irrelevant. We define $J=E_{\alpha\gamma}-(E_{\alpha\alpha}+E_{\gamma\gamma})/2$ for nearest neighbor bonds, and similarly define $K$ for next-nearest neighbor bonds. Finally, counting the numbers of unlike bonds in the set of decorated tetrahedra, we find that
\begin{equation}
\label{eq:Eabcd}
E_{\alpha\beta\gamma\delta}=\frac{1}{6}J(4-\delta_{\alpha\gamma}-\delta_{\alpha\delta}-\delta_{\beta\gamma}-\delta_{\beta\delta})
+\frac{1}{4}K(2-\delta_{\alpha\beta}-\delta_{\gamma\delta}).
\end{equation}
To determine the interaction parameters $J$ and $K$, we decorate the BCC lattice with different atomic species in various configurations and calculate the energies using electronic density functional theory (DFT). We then fit the results to Eq.~(\ref{eq:Eabcd}). Our DFT calculations are performed in the PBE generalized gradient approximation~\cite{Perdew96} using projector augmented wave potentials~\cite{Kresse99} as implemented in VASP~\cite{VASP}. Default plane wave energy cutoffs are applied and we choose $k$-point meshes to achieve energy convergence to better than 1 meV/atom. As our ensemble of structures we take the set of all symmetry-inequivalent equiatomic decorations of a 16-atom cF16 unit cell, based on the experimental lattice parameters. We perform two sets of calculations, one holding the atoms at ideal BCC lattice positions, and the other employing full relaxation of lattice parameters and atomic coordinates.
Given our complete set of decorated cF16 unit cells, we may determine the energy minimizing structure within this set as functions of the interaction parameters $J$ and $K$. Note that only the signs and ratio of $J$ and $K$ matter. We find that two structures minimize the energy across the majority of the $J$ and $K$ plane. Negative $J$ tends to favor unlike near neighbors, as occurs in the CsCl prototype, while negative $K$ favors unlike next-neighbors as occurs in the NaTl prototype. Because of the relative numbers of nearest and next-nearest neighbors, the crossover occurs at $J=3K/2$. The situation for positive J and K is more complex, and we find a variety of non-cubic structures minimize the energy for $K>0$ and $J>K/2$.
\begin{figure}
\includegraphics[width=3in]{Fig3.png}
\caption{\label{fig:JK}Energy minimizing structure as functions of $J$ and $K$ (units eV/bond). Green is NaTl, red is CsCl, and blue contains various non-cubic decorations. Fitted values for CuZn (circles) and AlLi (squares) are marked. Filled symbols are relaxed, open symbols are unrelaxed.}
\end{figure}
Fits to the JK model are illustrated in Fig.~\ref{fig:fits}, and resulting values of $J$ and $K$ are given in Table~\ref{tab:fits}. In the case of CuZn, the nearest neighbor interaction dominates and favors alternation of species as in the CsCl prototype. AlLi, in contrast, is strongly ionic. Owing to the Coulomb interaction, the next-nearest neighbor interaction is comparable in strength to the nearest neighbor and also favors alternation of species, as in the NaTl prototype.
\begin{figure}
\includegraphics[width=5in]{Fig4}
\caption{\label{fig:fits}Fits to the JK model. The two outliers in relaxed CuZn are excluded because their structures underwent strong shear deformations.}
\end{figure}
\begin{table}
\caption{\label{tab:fits}Interaction parameters for CuZn and AlLi as calculated within DFT. Units are eV/bond.}
\begin{tabular}{r|ll|ll}
& \multicolumn{2}{c|}{Unrelaxed}&\multicolumn{2}{c}{Relaxed}\\
\hline
CuZn& J = -0.0369, & K = -0.0134 & J = -0.0228, & K = -0.0056 \\
AlLi& J = -0.0729, & K = -0.0937 & J = -0.0571, & K = -0.0878 \\
\end{tabular}
\end{table}
\subsection{Cluster variation model}
Kikuchi's Cluster Variation Method (CVM)~\cite{Kikuchi1951,deFontaine79,Ackermann89,Widom16} supplements the interaction energy model with a model for the entropy. We present a heuristic derivation of the entropy based on its equivalence (in suitable units) to information. Consider a crystalline solid containing multiple chemical species, with each species $\alpha$ present in mole fraction ${x_\alpha}$. If the species are distributed uniformly across $N$ sites then we must specify the ${x_\alpha} N$ sites occupied by each species $\alpha$. Equivalently we must specify precisely one of the $\Omega_P = N!/\prod_\alpha ({x_\alpha} N)!$ distinct configurations. Imagine an index listing every possible configuration. The number of digits to specify an index entry is the logarithm of $\Omega_P$. Applying the Stirling expansion yields the ideal (Bragg-Williams~\cite{Bragg34}) entropy per site \begin{equation}
\label{eq:SPoint}
S_{\rm Point}=-\sum_\alpha {x_\alpha}\ln{{x_\alpha}}
\end{equation}
where we have chosen to measure the entropy in units of the Boltzmann constant ${k_{\rm B}}$.
Correlations reduce the entropy from its ideal value because knowledge of the chemical species on a given site provides information about the likely occupation of nearby sites. In view of the equivalence of information and entropy, if we can quantify this information we can then subtract it from $S_{\rm Point}$. This approach is nontrivial because we must determine the {\em minimum} information contained in the configuration, avoiding the inclusion of redundant information. If the distribution of atomic species is correlated, for example if the probability ${y_{\alpha\gamma}}$ to find species $\alpha$ and $\gamma$ as nearest neighbors differs from the uncorrelated expectation of ${x_\alpha}{x_\gamma}$, then the {\em mutual information} per bond,
\begin{equation}
\label{eq:mutual}
I[\{{y_{\alpha\gamma}}\}] =\sum_{\alpha\gamma} {y_{\alpha\gamma}} \ln{{y_{\alpha\gamma}}/{x_\alpha}{x_\beta}},
\end{equation}
must be subtracted from the ideal entropy yielding
\begin{equation}
\label{eq:S-I}
S_{\rm Pair}=S_{\rm Point}[\{{x_\alpha}\}]-\frac{z}{2} I[\{{y_{\alpha\gamma}}\}],
\end{equation}
where $z$ is the coordination number of the lattice. Specifying the correlation ${y_{\alpha\gamma}}$ reduces the additional information required to specify a full configuration. In thermodynamics Eq.~(\ref{eq:S-I}) is known as the Bethe-Guggenheim quasichemical approximation~\cite{Bethe35,Guggenheim44}. For the BCC lattice, $z=8$, and the entropy becomes
\begin{equation}
\label{eq:Spair}
S_{\rm Pair} = 7\sum_i {x_\alpha}\ln{x_\alpha}-4\sum_{\alpha\gamma}{y_{\alpha\gamma}}\ln{y_{\alpha\gamma}}
\end{equation}
at the level of nearest neighbor pair correlations.
The pair correlation function ${y_{\alpha\gamma}}$ is equivalent to the Warren-Cowley short-range order parameter $\alpha_{\alpha\gamma}\equiv 1-{y_{\alpha\gamma}}/{x_\alpha}{x_\gamma}$. In view of the pair-level approximation Eq.~\ref{eq:S-I}, we may express the entropy in terms of concentrations and the Warren-Cowley parameter as
\begin{equation}
\label{eq:WC}
S_{\rm WC}=-\sum_\alpha {x_\alpha}\ln{{x_\alpha}}
-\frac{z}{2}\sum_{\alpha\gamma} {x_\alpha}{x_\gamma}(1-\alpha_{\alpha\gamma})\ln{(1-\alpha_{\alpha\gamma})}.
\end{equation}
Such an approximation could be applied regardless of how the order parameter was obtained, including experimental determination.
\def{\{1\}}{{\{1\}}}
\def{\tikz{\filldraw (0,0) circle (0.05cm);}}{{\tikz{\filldraw (0,0) circle (0.05cm);}}}
\def\CVMnn{{
\tikz[baseline]{
\draw (0,-.05)--(.2,.15);
\filldraw (0,-.05) circle (0.05cm);
\filldraw (.2,.15) circle (0.05cm);}}}
\def\CVMnnn{{
\tikz{\draw (0,0)--(.4,0);
\filldraw (0,0) circle (0.05cm);
\filldraw (.4,0) circle (0.05cm);}}}
\def\CVMtri{{
\tikz[baseline]{
\draw (0,-.05)--(.2,.15);
\draw (0,-.05)--(.4,-.05);
\draw (.4,-.05)--(.2,.15);
\filldraw (0,-.05) circle (0.05cm);
\filldraw (.2,.15) circle (0.05cm);
\filldraw (.4,-.05) circle (0.05cm);}}}
\def\CVMtet{{
\tikz[baseline]{
\draw (0,.05)--(.2,.25);
\draw (0,.05)--(.2,-.15);
\draw (.4,.05)--(.2,.25);
\draw (.4,.05)--(.2,-.15);
\draw (.2,-.15)--(.2,.25);
\draw (0,0.05)--(0.15,0.05);
\draw (0.25,0.05)--(0.4,0.05);
\filldraw (0,0.05) circle (0.05cm);
\filldraw (.2,.25) circle (0.05cm);
\filldraw (.2,-.15) circle (0.05cm);
\filldraw (.4,.05) circle (0.05cm);}}}
The mutual information approach requires further refinement because additional correlations may be present beyond nearest-neighbor or among multiple neighbors. Even the bond probabilities ${y_{\alpha\gamma}}$ may contain redundant information. Eqs.~(\ref{eq:SPoint}) and~(\ref{eq:S-I}) represent the first steps in a systematic expansion of the entropy involving successively longer-range and higher-order correlation functions~\cite{Yedidia2005,Pelizzola2005}. CVM approximates the entropy per site as $S =\frac{1}{N}\ln{\Omega}$ where $\Omega$ is given by a product of combinatorial factors reflecting the correlation functions of pairs of sites (bonds), triplets, etc. Each factor represents the information content of a correlation function relative to the information contained in its constituent parts. We define numerical values for symbols associated with the empty lattice, isolated points, nearest neighbor bonds, next-nearest neighbor bonds, triangles, and tetrahedra
\begin{equation}
\begin{split}
\label{eq:CVMdef}
{\{1\}} \equiv N!,~~
\{{\tikz{\filldraw (0,0) circle (0.05cm);}}\}\equiv\prod_{\alpha} (x_{\alpha} N)!,~~
\{\CVMnn\}\equiv\prod_{\alpha\gamma} (y_{\alpha\gamma} N)!,\\
\{\CVMnnn\}\equiv\prod_{\alpha\beta} (v_{\alpha\beta} N)!,~~
\{\CVMtri\}\equiv\prod_{\alpha\gamma\delta} (u_{\alpha\gamma\delta} N)!,~~
\{\CVMtet\}\equiv\prod_{\alpha\beta\gamma\delta} (z_{\alpha\beta\gamma\delta} N)!.
\end{split}
\end{equation}
Defining the combinatorial factors
\begin{equation}
\setlength{\jot}{10pt}
\begin{split}
\label{CVMW}
f({\tikz{\filldraw (0,0) circle (0.05cm);}})=\frac{{\{1\}}}{\{{\tikz{\filldraw (0,0) circle (0.05cm);}}\}},~~~
f(\CVMnn)=\frac{\{{\tikz{\filldraw (0,0) circle (0.05cm);}}\}^2}{\{\CVMnn\}{\{1\}}},~~~
f(\CVMnnn)=\frac{\{{\tikz{\filldraw (0,0) circle (0.05cm);}}\}^{2}}{\{\CVMnnn\}{\{1\}}}, \\
f(\CVMtri)=\frac{\{\CVMnn\}^2\{\CVMnnn\}{\{1\}}}{\{{\tikz{\filldraw (0,0) circle (0.05cm);}}\}^3\{\CVMtri\}},~~~
f(\CVMtet)=\frac{\{{\tikz{\filldraw (0,0) circle (0.05cm);}}\}^4\{\CVMtri\}^{4}{\{1\}}}{\{\CVMnn\}^4\{\CVMnnn\}^2\{\CVMtet\}},
\end{split}
\end{equation}
we recognize the ``point'' approximation $\Omega_P=f({\tikz{\filldraw (0,0) circle (0.05cm);}})$, as the Bragg-Williams~\cite{Bragg34} formula, while the nearest neighbor pairwise correction $\Omega_{\rm NN}=f({\tikz{\filldraw (0,0) circle (0.05cm);}})f^m(\CVMnn)$ is the Bethe-Guggenheim result ($m=4$ for BCC). Next-nearest neighbors multiply $\Omega$ by $f^3(\CVMnnn)$ and triangles provide an extra factor of $f^{12}(\CVMtri)$. Finally, including all factors, we obtain
\begin{equation}
\begin{aligned}
\Omega_{\rm TET}
&= f({\tikz{\filldraw (0,0) circle (0.05cm);}}) f^4(\CVMnn) f^3(\CVMnnn) f^{12}(\CVMtri) f^6(\CVMtet) \\
&= \frac{{\tikz{\filldraw (0,0) circle (0.05cm);}}^1\CVMtri^{12}}{\CVMnn^4\CVMnnn^3\CVMtet^6}
\end{aligned}
\end{equation}
which yields the entropy approximation for BCC lattices
\begin{equation}
\begin{aligned}
\label{eq:S_tet}
S &= \sum_\alpha {x_\alpha}\ln{{x_\alpha}}
+12\sum_{\alpha\gamma\delta}u_{\alpha\gamma\delta}\ln{u_{\alpha\gamma\delta}}\\
&-4\sum_{\alpha\beta}{y_{\alpha\gamma}}\ln{{y_{\alpha\gamma}}}
-3\sum_{\alpha\beta}v_{\alpha\beta}\ln{v_{\alpha\beta}}
-6\sum_{\alpha\beta\gamma\delta} z_{\alpha\beta\gamma\delta}\ln{z_{\alpha\beta\gamma\delta}}.
\end{aligned}
\end{equation}
This expression is equivalent to that previously presented by Kikuchi~\cite{Kikuchi1987,Ackermann89} for BCC structures.
We now have a sequence of higher-order approximations to the entropy, starting with the point approximation Eq.~(\ref{eq:SPoint}), through the pair approximation Eq.~(\ref{eq:Spair}), to the tetrahedron approximation Eq.~(\ref{eq:S_tet}). Note that we recover the lower approximations from the higher by making {\em superposition} approximations. Specifically, the substitutions ${z_{\alpha\beta\gamma\delta}}={y_{\alpha\gamma}} y_{\beta\delta}$, ${u_{\alpha\gamma\delta}}={y_{\alpha\gamma}} {x_\delta}$, and ${v_{\alpha\beta}}={x_\alpha}{x_\beta}$ reduce the tetrahedron approximation~(\ref{eq:S_tet}) to the pair approximation~(\ref{eq:Spair}). Similarly, the substitution ${y_{\alpha\gamma}}={x_\alpha}{x_\gamma}$ reduces the pair approximation to the point approximation (\ref{eq:SPoint}).
\section{Results}
When combined with the energy Eq.~(\ref{eq:H}) we obtain the free energy $F=E-TS$. We now minimize the free energy with respect to the correlation functions in order to predict their temperature-dependent values. Differentiating $F$ with respect to the highest order correlation function (and imposing the sum rules discussed in Section~\ref{sec:Correlations}) yields an identity for this function in terms of lower order functions. Summing the higher-order function to produce new lower-order functions, and iterating this procedure, results in convergence towards a self-consistent solution for all correlation functions~\cite{Kikuchi1974}.
For the specific case of BCC, we reproduce the result from Ref.~\cite{Ackermann89} (with some slight notation changes)
\begin{equation}
\label{eq:iter}
{z_{\alpha\beta\gamma\delta}} = e^{-\lambda/6{k_{\rm B}} T}e^{-E_{\alpha\beta\gamma\delta}/{k_{\rm B}} T}X^{1/24}U^{1/2}Y^{-1/6}V^{-1/4}
\end{equation}
with
\begin{equation}
\label{eq:products}
\begin{aligned}
X &= {x_\alpha}{x_\beta}{x_\gamma}{x_\delta} &
U &= {u_{\alpha\gamma\delta}} u_{\beta\gamma\delta}u_{\alpha\beta\gamma}u_{\alpha\beta\delta}\\
Y &= {y_{\alpha\gamma}} y_{\alpha\delta} y_{\beta\gamma} y_{\beta\delta} &
V &= {v_{\alpha\beta}} v_{\gamma\delta}.
\end{aligned}
\end{equation}
In each correlation function above, the species index $\alpha-\delta$ applies, respectively, to the tetrahedron vertices $1-4$ as shown in Fig.~\ref{fig:BCC}. Note that by breaking the equivalence of lattice sites we allow for the possibility of spontaneous symmetry breaking. We recognize the exponents in Eq.~(\ref{eq:iter}) as the coefficients in the entropy Eq.~(\ref{eq:S_tet}) divided by 6 times the number of correlation factors in Eq.~(\ref{eq:products}). The constant $\lambda$ is related to the Lagrange multiplier and is determined by the required normalization of ${z_{\alpha\beta\gamma\delta}}$.
\begin{figure}
\includegraphics[width=3in]{Fig5a}
\includegraphics[width=3in]{Fig5b}
\caption{\label{fig:XofT}(a) Order parameters ${x_\alpha}(\rm Cu)$ and ${x_\gamma}(\rm Zn)$ and entropy $S$ (in units of ${k_{\rm B}}$) of CuZn obtained from the tetrahedron method utilizing fitted values of the unrelaxed (solid lines) and relaxed (dashed lines) interactions as given in Table~\ref{tab:fits}. (b) ${x_\alpha}(\rm Al)$, $\sb(\rm Li)$, and $S$, for AlLi.}
\end{figure}
Fig.~\ref{fig:XofT}a shows the result of the tetrahedron approximation for CuZn. We plot the point order parameters $x_{\rm Cu}$ at the vertex and body center sites. Above a critical temperature $T_c$ both are identically 1/2, while below $T_c$ the symmetry spontaneously breaks, yielding the CsCl structure. We also plot the entropy, which grows monotonically from zero at low $T$ towards its ideal maximum value $\ln{2}$ at high $T$. We carry out the calculation first using unrelaxed parameters, resulting in $T_c=936$K, and then again using relaxed parameters, resulting in $T_c=681$K. The experimental $T_c\sim 730$K. The discrepancy between our relaxed CVM $T_c$ and the experimental value is likely due to neglected effects of thermal expansion, and vibrational~\cite{TurchiCuZn} and electronic free energies.
Fig.~\ref{fig:XofT}b displays the corresponding result for AlLi. Here, $3K/2<J<0$, so the system takes the NaTl prototype structure, so we plot $x({\rm Al})$ at positions $a$ and $b$. In this case, the predicted order-disorder transition lies somewhat above the experimental melting point, implying that chemical order persists for all temperatures below melting, consistent with experimental observation.
\begin{figure}
\includegraphics[width=3in]{Fig6}
\caption{\label{fig:YoX}Ratios ${y_{\alpha\gamma}}/{x_\alpha}{x_\gamma}$ for CuZn using relaxed fit parameters.}
\end{figure}
Note that entropy remains below $\ln{2}$ even in the disordered symmetric state above $T_c$. This occurs because of the pair and multi-point correlations, which differ from products of single-point functions, as illustrated in Fig.~\ref{fig:YoX} where we plot the ratio ${y_{\alpha\gamma}}/{x_\alpha}{x_\gamma}$. Note that all four elements of ${y_{\alpha\gamma}}$ remain nonzero at all $T$. They slowly approach the uncorrelated values ${x_\alpha}{x_\gamma}$ at high $T$ but remain far from those values over the range plotted. Two values ($y_{\rm CuCu}$ and $y_{\rm ZnZn}$) remain below the uncorrelated value because of the unfavorable like-like interaction at nearest neighbors, while the other two ($y_{\rm CuZn}$ and $y_{\rm ZnCu}$) lie above owing to the favorable unlike interaction. As the site occupations ${x_\alpha}({\rm Cu})$ and ${x_\gamma}({\rm Zn})$ approach 1 (full occupation) at low $T$, $y_{\rm CuZn}$ also approaches 1. In the opposite case, although ${x_\alpha}({\rm Zn})$ and ${x_\gamma}({\rm Cu})$ both vanish at low $T$, the ratio ${y_{\alpha\gamma}}({\rm ZnCu})/{x_\alpha}({\rm Cu}){x_\gamma}({\rm Zn})$ diverges owing to the favorable Zn-Cu interaction.
\begin{figure}
\includegraphics[width=3in]{Fig7}
\caption{\label{fig:CVM-series}Series of improved approximations: point, pair and tetrahedron with a single interaction parameter$J'$, and tetrahedron with two parameters $J,K$.}
\end{figure}
Finally, we explore the relative capabilities of the series of CVM approximations, point, pair and tetrahedron. Since point and pair involve only the nearest-neighbor interaction, we re-fit the set of energies to obtain $J'=-0.0190$ (in the relaxed case) and define $E'_{\alpha\gamma}=J'(1-\delta_{\alpha\gamma})$. The iteration procedure for pairs is
\begin{equation}
{y_{\alpha\gamma}} = e^{-2\lambda'/z{k_{\rm B}} T} e^{-E'_{\alpha\gamma}/{k_{\rm B}} T}({x_\alpha}{x_\gamma})^{(z-1)/z)}
\end{equation}
and for points is
\begin{equation}
{x_\alpha} = e^{-2\lambda'/z{k_{\rm B}} T} e^{-z E'_{\alpha\gamma} {x_\gamma}/{k_{\rm B}} T}.
\end{equation}
As shown in Fig.~\ref{fig:CVM-series} the point approximation, which is conventional mean field theory, has the highest $T_c$, with the pair and then the tetrahedron approximations being progressively lower. This is because mean-field theory neglects all fluctuations and thus stabilizes the ordered phase up to higher temperatures. Likewise, the fall-off of the order parameters become progressively steeper. Although in every case the order parameter vanishes with mean-field critical exponent $\beta=1/2$, the pair and tetrahedron approximations include fluctuations at a localized level, reducing the transition temperature and yielding order parameter variation closer to the correct 3D Ising $\beta=0.326$. Similar effects have been noted for the simple cubic~\cite{Kikuchi1951} and square lattices~\cite{Pelizzola2005}.
Our series of improved approximations omits cases that could have been tried, such as a two pair (NN plus NNN) or a triangle approximation. These are known to be poor choices. Vul and de~Fontaine~\cite{Vul93} show that the optimal choice of a maximal cluster in a CVM approximation must be self contained in the sense that any additional point would be less closely connected to some existing vertex than the set of points already included. Clearly the NN plus NNN approximation implies the necessity to complete the triangle, while the an additional vertex bound to the triangle by NN bonds completes the tetrahedron.
\section{Conclusions}
We applied the cluster variation method to explore chemical ordering in two alloy families, CuZn and AlLi. Nearest and next-nearest neighbor interactions $J$ and $K$ were obtained from density functional theory calculations. We found that these parameters lie in a region of the $J-K$ space that favors CsCl-type ordering for CuZn, while NaTl-type ordering is favored for AlLi. CuZn exhibited an order-disorder transition, while the predicted $T_c$ for AlLi lay above its melting temperature. In each case we obtained the temperature variation of the single-point order parameter as well as a variety of higher-order correlation functions, and we plotted the entropy, which remains below its ideal value at all temperatures.
\begin{figure}
\includegraphics[width=6in]{Fig8.png}
\caption{\label{fig:Heusler}Variants of Heusler structures. From top left to bottom right: quaternary Heusler, full Heusler, inverse Heusler, half Heusler, NaTl, BiF$_3$, CsCl, W.}
\end{figure}
Note that both the CsCl and the NaTl types of order represent different patterns of symmetry breaking starting from a disordered body centered solid solution. With additional elements, such as are present in high entropy alloys~\cite{Yeh04_1,Cantor04}, further stages of symmetry breaking are conceivable, leading to the full family of Heusler-type structures~\cite{FelserBook2016} as illustrated in Fig.~\ref{fig:Heusler}. The CVM is ideally suited to study chemical ordering in these compounds because the four interpenetrating face centered cubic sublattice of the quaternary Heusler occupy the four vertices of the BCC tetrahedron. Given an interaction model the CVM can predict ordering and mixing among the sublattices.
\section*{Acknowledgements}
This research was supported by the Department of Energy under grant DE-SC0014506.
|
2,877,628,090,134 | arxiv | \section{Introduction}
Immuno-fluorescent staining is widely used in biomedical research to visualize particular cellular events by specific molecular markers, such as protein-based immuno-histochemistry (IHC), and DNA/RNA-based in situ hybridization (ISH). Recent advances in multiplexed IHC/ISH tissue imaging techniques, allowing simultaneous detection of numerous markers, raise the possibility of conducting deeper immune response profiling at a single-cell level \cite{Tan2020View, Eichholz2020carRNA}.
This growing field reveals a wealth of information about the relationships among cells in the entire tissue micro-environment but presents a significant image analysis challenges. For instance, the membrane-image quality defection due to the immuno-staining techniques is common and significant. Noises bring obstacles to edge detection because it reduces the contrast of real membrane boundary and also introduce spurious edges due to noisy contrast.
Furthermore, complex tissue involved in diverse physiological processes, no single technique can provide all of the answers, so it’s necessary to use a combination of IHC and ISH methods to providing insights into physiological processes and disease pathogenesis, wherein a mixture of protein, DNA and RNA markers is introducing various complexities. Thus, the multiplexed image-based transcriptional profiling of cells is significant challenging. The cutting edge machine-learning approaches are the promising way to address the challenges.
\subsection{Problems and Contributions}
The major drawback of supervised machine-learning techniques is that the quality of the results depends upon the amounts and quality of manually annotated training data.
However, the performance of human annotators show a high degrees of variability in complex fluorescent image in tissue as shown in Figure~{\ref{figure_ground_truth}(C)}. Furthermore, there is existing gold standard paradox in biomedical tissue \cite{Aeffner2017GoldStandard}, so that, it is not practical using manual annotation to generate the consensus ground truth in complex tissue image.
In this work, we aim to solve the practical problem of lacking ground-truth in microscopy tissue image, we have harnessed the stochastic random-reaction-seed (RRS) algorithm as a ground-truth generator. Then, we build a AI-driven pipeline U-net to extract single-cell level information from complex tissue.
Finally, we address the practical problem of large amounts of single-cell data output from the machine-learning pipeline, demonstrate the simple cost-efficient approach to leverage the power of UMAP (Uniform Manifold Approximation and Projection) \cite{Mcinnes2018Umap}.
Specifically, the main contributions are as the following:
(1) We have designed a RRS-based automated ground-truth generator for training the neural networks, and importantly, the fine-tuned parameters of RRS ensured the generation of a high-quality ground-truth for microscopy image segmentation.
(2) We have developed an automated RRScell pipeline for extracting single-cell level profiling map of whole slide tissue. It is a robust convolutional network in biomedical image segmentation with limited training dataset.
(3) RRScell has a build-in markerUMAP tool secures the efficiency of dimension reduction, making it viable as a general tool in the spatial analysis of high dimensional tissue image.
\subsection{Related Works}
Recent advances in machine-learning have outperformed the state of the art of traditional image analysis in many applications.
Supervised learning segmentation approaches require abundant ground-truth labels to deliver accurate inference. However, in biomedical microscopy image segmentation, these labor-intensive annotation are scarce \cite{Ke2020LazyLabel}.
Among various deep convolutional encoder-decoder networks for image segmentation, the U-net is considered a generic architecture in biomedical cell segmentation \cite{Ronneberger2015Unet, Falk2019Unet} because data-augmentation strategy in U-net is able to achieve desired accuracy with
a few of ground-truth annotation.
There has been considerable efforts in computational multiplexed tissue image analysis,
some are commercial microscopy platforms or licensed software \cite{Czech2019cytoKit, Tan2020View}, so they are not easy to be understood, and customized for complex tissues.
some are implemented in Matlab or Python wrapper for Matlab, and applied probability maps of nuclei/membrane for generating segmentation ground-truth masks \cite{Schapiro2017histoCAT, Czech2019cytoKit, Stoltzfus2020cytoMAP}.
Many existing multiplexed analysis pipelines are developed for specific assays or certain IHC/ISH schemes, there are lacking consensus standards for individual labs' bespoke IHC/ISH scheme.
Image-based cytometry involves a wide range of cutting-edge techniques, more efforts are needed to overcome various issues and limitations in this active research field.
\section{Methods}
\subsection{Automated ground truth generator}
We have applied the stochastic RRS method \cite{Li2019RRS} to generate high quality ground-truth automatically, as shown in Figure~{\ref{figure_ground_truth}(E)}. The accuracy of automated ground-truth from RRS is ensured by handcraft parameters for selected image dataset.
\begin{figure*}
\centering
\includegraphics[width = 0.6\textwidth]{figure/figure_ground_truth}
\caption{Automated ground truth generation in a sample image from cancer tissue.
(A) is the merged image of multiplexed.
(B) is raw nuclei channel.
(C) are the membrane annotations from two experts (red color from one expert, blue color from another expert) on the same membrane image.
(D) is the raw nuclei channel.
(E) is the dynamic tracing map on the membrane image by the Random-Reaction-Seed method, where the random initial seed is denoted by blue star, the search chain is denoted by red-circle. \cite{Li2019RRS}.
(F) is the traced membrane profile by RRS, the green-color is the traced membrane plotted on the raw membrane image (hot-color-map).
}
\label{figure_ground_truth}
\end{figure*}
There is no objective ground truth available for multiplexed immunofluorescence images. To assess the accuracy of our method we compare the results with the manual measurement.
\begin{figure*}
\centering
\includegraphics[width = 0.5\textwidth]{figure/figure_performance}
\caption{Performance of RRScell method.
(A) Membrane profile from RRScell is plotted on the nuclei image.
(B) Membrane profile from RRScell is plotted on the membrane image.
(C) Manual membrane profile from Expert-1 is plotted on the membrane image.
(D) Manual membrane profile from Expert-2 is plotted on the membrane image.
(E) The box-plot on the top is the evaluation of accuracy among three different methods, the red-square-point inside each box is the mean value.
Comparison between membrane-detection methods:
the blue-color box is the similarity between RRScell and expert-1,
the blue-color box is the similarity between RRScell and expert-2,
the black-color box is the similarity between two experts.
The polar plot of bar-chart on the upper-right is the evaluation of similarity of each cell between experts: each blue-color bar is corresponding to each dot in the blue-color box-plot, here, the bars are sorted in descending order.
The box-plot on the lower-right is the evaluation of efficiency, the efficiency of RRScell is significantly better than the manual process (>30 folds difference in processing time).
}
\label{figure_performance}
\end{figure*}
As demonstrated in Figure~{\ref{figure_performance}}(E), the accuracy of RRScell method is equivalent to manual annotation. Furthermore, due to DAPI straining defect in high intensity zone beyond human visual approach, so that both experts are avoiding these messy areas, as shown in Figure~{\ref{figure_performance}}(C, D), one significant advantages of RRScell is able to tracing more cells based on its powerful combination of both membrane and nuclei as detection, as shown in Figure~{\ref{figure_performance}}(B).
The correct detection of cellular membrane is verified by the nuclei plot in Figure~{\ref{figure_performance}}(A).
As shown in Figure~{\ref{figure_performance}}(E), the polar plot of bar-chart, where the polar axis is the similarity score, the blue-color bars are corresponding to the cells (blue dots) in the blue box-plot while the green-color bars are corresponding to the cells (green dots) in the green box-plot. In order to demonstrate the variability of manual annotations, the blue-color bars are sorted in descending order, and then the green-color bar is placed by tracking the similar cellular physical location. So that the similarity of each cell annotated by two experts are displayed in a contrast way. The similarity score of each cell membrane profile is calculated by the IoU (intersection over union).
Automated RRS-based ground truth generator is accurate and consistent.
\subsection{RRScell pipeline}
We have built an artificial intelligence (AI)-driven pipeline which is composed of the following modules:
(1) Automated cell segmentation is a U-net-based encoder-decoder network with RRS-based ground-truth generator. The cell segmentation is either based on raw membrane staining or anchored to synthetic/artificial membrane from raw nuclei staining/other suitable staining.
(2) Automated cell-type detection: As shown in Figure~{\ref{figure_type_detection}}, we have developed a workflow for phenotype validation cell-by-cell.
\subsubsection{markerUMAP}
We have developed a marker-based image cytometry analysis tool (markerUMAP) in quantifying
spatial distribution of cell phenotypes from tissue images with a mixture of biomarkers.
The strategy of markerUMAP is introducing simplified symbolic marker to represent the captured biomarker contents. For example, the quantification of RNAscope measurement is focused on the detective RNA dot-like distribution instead of normal intensity measurement (since RNA staining enables specific and sensitive detection of RNA, which can be visualized as a dot, with each dot denotes a single RNA transcript).
This symbolic marker-based strategy is organically compatible and consistent with UMAP fast, robust and meaningful organization of cell clusters \cite{Becht2019usingUmap}.
\begin{figure*}
\centering
\includegraphics[width = 0.5\textwidth]{figure/figure_type_detection}
\caption{Scheme of automated cell phenotype detection. Here, an example of 5-plex image is presented: the initial step is the validation of real single cell based on both membrane and nuclei features, and a phenotype detection step will be followed in staining conditions (such as RNA1, RNA2, and IHC1).
}
\label{figure_type_detection}
\end{figure*}
\section{Experiments and Results}
In this section, we applied RRScell into multiplexed tissue stained by various IHC/ISH schemes: starting from 3-plex to higher plexed immuno-fluorescence tissues in a gradual way.
\subsection{Three-plex CRISPR-Cas9 tissue images without membrane staining}
In some practical translational research, multiplexed immunofluorescence tissue images are lacking membrane straining in many circumstances. Therefore, the RRScell pipeline is also equipped with the synthetic/artificial membrane generation step to work flexibly and consistenly on
membrane-less images.
As shown in Figure~{\ref{figure_RRScell_no_nuclei}, this immuno-fluorescent tissue image is from HBV-specific SaCas9 therapy \cite{Stone2021Cas9}.
\begin{figure*}
\centering
\includegraphics[width = 0.6\textwidth]{figure/figure_RRScell_no_nuclei}
\caption{RRScell phenotype detection of 3-plex immuno-fluorescent tissue image from HBV-specific SaCas9 therapy. (A) is the raw image of 3 channels: red color denotes the Cas9 RNA-staining, yellow color denotes the HBV RNA-staining, and blue color denotes the nuclei staining. (B) is given the overlapped image from the raw and the RRScell profiling map. (C) is profiling map from the cell phenotype analysis by the RRScell pipeline: detected +HBV dots are denoted by yellow-disk, detected +Cas9 dots are denoted by red-star. There are four types of cells: +HBV, +Cas9, +HBV+Cas9, and negative -HBV-Cas9.
}
\label{figure_RRScell_no_nuclei}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width = 0.6\textwidth]{figure/figure_umap_3plex}
\caption{makerUMAP phenotype clustering analysis of 3-plex immuno-fluorescent tissue image from HBV-specific SaCas9 therapy. (A) Demonstration of extracting single-cell level information from individual cells in whole slide tissue by RRScell. (B) Demonstration of cell phenotype cluster analysis by markerUMAP tool. Note: only a small part of the tissue was applied in this demonstration.}
\label{figure_umap_3plex}
\end{figure*}
\subsection{Four-plex CAR-T RNA-scope tissue images with membrane staining}
Chimeric antigen receptor (CAR) T cells, as one of rapid emerging form of genetically engineered “artificial immune cell”, have outstanding therapeutic potential for treating cancers \cite{Eichholz2020carRNA}.
To study the anti-HIV CAR T cell trafficking into tissues with viral replication by microscopy, we developed an RNAscope- based RNA fluorescent in situ hybridization assay. In order to validate the RNA probes, we need to quantify a large volume of image dataset from in vitro culture and tissues sections among mouse, monkey, and human. However, the membrane-image quality defection due to the sugar-staining techniques is common and significant. Noises bring obstacles to edge detection because it reduces the contrast of real membrane boundary and also introduce spurious edges due to noisy contrast, as shown in Figure~{\ref{figure_result_4plex}.
\begin{figure*}
\centering
\includegraphics[width = 0.6\textwidth]{figure/figure_result_4plex}
\caption{Cell phenotype analysis of 4-plex immuno-fluorescent tissue image stained by 4 biomarkers: CAR, CD4, nuclei,and membrane. The left panel is the raw stitched cancer tissue. The central panel is profiling map of 4 phenotypes from RRScell inferencing. The right panel is corresponding pie chart and the contour map with the gating condition of captured mRNA contents of individual cells from the whole slide tissue.
}
\label{figure_result_4plex}
\end{figure*}
\subsection{Seven-plex cancer tissue images without membrane staining}
We are applied ChipCytometry platform for making highly multiplexed assay and imaging on cancer tissues. For higher multiplexed tissue, the potential total cell-phenotypes is in a exponential-growth pattern. For instance, there are total 34 cell-phenotypes are classified in a small corner of 7-plex tissue, as shown in Figure~{\ref{figure_umap_7plex}.
Significantly, the markerUMAP is able to provide the fast run times, high reproducibility and the meaningful organization of cell-pheonotype clusters, as shown in Figure~{\ref{figure_umap_cluster_on_tissue}.
\begin{figure*}
\centering
\includegraphics[width = 0.6\textwidth]{figure/figure_umap_7plex}
\caption{RRScell phenotype detection of 7-plex immuno-fluorescent image from cancer tissue stained by 7 biomarkers: CD4, CD8a, CD39, CD279, DAPI, GZMB, and Ki67. (A) Demonstration of extracting single-cell level information from individual cells in one corner of whole slide of tissue by RRScell. The pie-chart illustrates that a total of 34 cell-phenotypes are detected in the demo image. (B) Demonstration of cell phenotype cluster analysis by markerUMAP tool.
}
\label{figure_umap_7plex}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width = 0.6\textwidth]{figure/figure_umap_cluster_on_tissue}
\caption{Demo of spatial analysis of high dimensional multiplexed images from cancer tissue stained by 7 biomarkers: CD4, CD8a, CD39, CD279, DAPI, GZMB, and Ki67. (A) is the cell phenotype cluster analysis by markerUMAP. (B) is spatial distribution of all phenotype clusters. (C) are demonstrations of mapping each phenotype cluster back to corresponding tissue space.
}
\label{figure_umap_cluster_on_tissue}
\end{figure*}
\section{Conclusion}
We present a robust, open-source Python pipeline for immune cell profiling in spurious-edge-tissues of translational medical research. This machine-learning solution builds upon RRS automated ground truth generation. The house-made automated machine learning pipeline is automatic scaling up from 3 to 7, or highly multiplexed immuno-fluorescent tissue images.
The AI-driven RRScell method has demonstrated the robustness and flexibility for single-cell profiling of various tissue types and formats where membrane and nuclei are often missed.
In conclusion, RRScell, as a framework combining HMM-based (hidden-Markov-model) RRS, deep neural learning network U-net, and manifold learning technique UMAP, has demonstrated the effectiveness and robustness for precise quantification of immune cell phenotypes in images taken from noisy multiplexed immunofluorescence cancer tissue, which is becoming an increasingly useful tool for biomedical researches owing to its ability to profiling cellular phenotyping in cancer resection tissues with various complexities. Furthermore, the markerUMAP image-based cytometry tool inside RRScell has emerged as promising spatial analysis tool in ongoing high-dimensional immuno-straning tissues in pre-clinical research and clinical trials.
{\small
\bibliographystyle{ieee_fullname}
\section{Introduction}
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\medskip
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\begin{table}
\begin{center}
\begin{tabular}{|l|c|}
\hline
Method & Frobnability \\
\hline\hline
Theirs & Frumpy \\
Yours & Frobbly \\
Ours & Makes one's heart Frob\\
\hline
\end{tabular}
\end{center}
\caption{Results. Ours is better.}
\end{table}
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{\small
\bibliographystyle{ieee_fullname}
|
2,877,628,090,135 | arxiv | \section{Introduction}
\vspace*{-0.2cm}
\label{sec:intro}
Anomaly detection is an important topic in industries \cite{kammerer2019anomaly}, security \cite{griffin2018unexpected} , medical diagnosis \cite{schlegl2017unsupervised}, $etc.$, where there is a need to increase automation for better efficiency. Image analysis using Artificial Intelligence (AI) is one of the non-destructive way to recognize a potential threat. The data in these fields can be high dimensional and biased. This creates an imbalance in the dataset, where data or images of some classes might not be enough to train deep convolutional neural network architectures. With recent advancements where real data is readily being acquired, the lack or lesser availability of anomalous instances still persists. In baggage security, items/objects such as guns, pliers, knives, scissors, $etc$. are considered anomaly. In industries, a defective product is considered an anomaly to filter out the perfect products. Figure \ref{fig:datasets}, shows the instances of anomalous and non-anomalous images in both industrial production (MVTec\_AD) dataset and both X-ray baggage security (SIXray) respectively.
\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{db_ex.pdf}
\vspace*{-0.4cm}
\caption{Exampler images from MVTec\_AD [A] and SIXray [B] dataset.}
\label{fig:datasets}
\vspace*{-0.4cm}
\end{figure}
\noindent Our work aims to build a model that can identify if a piece of baggage passed through a security check has a threat or anomalous object. This can be used to make smart systems to
reduce the manpower required for various tasks that can be automatized. This automation
will reduce the laborious work of humans to only special cases where human expertise is
needed. In the case of baggage security, a system will call for human help only for suspicious bags. Similarly, our model can be used for detecting defective products in industries.
\noindent Over the years, to detect anomalies in images various machine learning and deep learning techniques have been used. In classification, methods like SVM \cite{erfani2016high}, K-nearest neighbors \cite{altman1992introduction}, with some feature extractors like Scale Invariant Feature Transform (SIFT) \cite{lowe2004distinctive}, Speeded Up Robust Features (SURF) \cite{bay2006surf}, $etc.$ have been used on x-ray images. Many of these methods have been combined with various object detection and segmentation methods.
However, using standard learning algorithms doesn't solve all problems due to imbalance present in the classes of datasets, as they cannot train supervised learning models well enough. To overcome this, recently unsupervised methods like Generative Adversarial Networks(GAN) are being used \cite{akcay2018ganomaly, akcay2020towards}. GAN architecture was first proposed in 2014 \cite{goodfellow2014generative}. The two major parts of this network are the Generator and Discriminator, which are trained in a zero-sum game, where loss and gain of each are balanced by the loss-gain of other and eventually we let them come at an equilibrium. The Generator is to capture the distribution of the input dataset for a given class label, by predicting features or images from a hidden representation.The Discriminator classifies as real or fake, based on the given features or images.
\vspace{0.1cm}
\noindent \section{Related Works:} Anomaly detection is used in various domains like video analysis, medical image analysis, industrial production, remote sensing, $etc$. In the paper DCGAN \cite{radford2015unsupervised} its shown that deep convolutional GANs are capable of capturing semantic image content. In \cite{7280790} a dictionary of filters is used to detect anomalous regions using convolutional sparse models. \cite{schlegl2019f}. Using GANs for anomaly detection is a popular approach in the field of research after Ano-GAN \cite{schlegl2017unsupervised} developed this concept, where the GAN is trained only on the non-anomalous instances. This method was based on the hypothesis that the distribution of the data can be represented by the latent vector generated by the GAN. For this a Generator and discriminator are trained but using only normal images. This way the latent vector would represent the distribution of the normal data. After these networks are trained, their weights are freezed and are used on a given image and remap the latent vector. The output of the discriminator gives a probability that the given image came from real data than from the Generator. This is used in the inference to calculate the anomaly score based on which an image is classified as anomalous or non-anonmalous. Residual scores for test images as well as fake images generated by GAN are generated and compared. The Network will detect anomalies by defining thresholds for these scores of both classes. This training process has inspired many recent methods. However, Ano-GAN, required some optimization steps for every new input that lead to overall bad test-time performance. BiGAN \cite{donahue2016adversarial} has a generator that maps latent samples to generated data and there's also inverse mapping from data to latent representation.
Based on BiGAN, EGBAD \cite{zenati2018efficient} improved the speed of the model by avoiding the computationally expensive step of recovering a latent representation at test time. GANomaly \cite{akcay2018ganomaly} introduced the combination of GAN and auto-encoder and was found to be very effective. The computations are reduced as anomaly scores of test images can be directly compared with fake images without iterations. In 2019, Samet \cite{akccay2019skip} proposed an improved model of GANomaly, $viz.$ Skip-ganomaly. The Skip-ganomaly architecture where skip-connections are added to the generator network, increasing image reconstruction ability. The performance of Skip-ganomaly is more stable than that of AnoGAN and GANomaly. But still, there are some cases where GANs fail to perform well which might be due to modal collapse during the training process. Our proposed method tries to tackle this issue as well as improve the overall performance of the network.
\vspace{0.2cm}
\noindent The main contributions of this paper are as follows:
\noindent 1. \emph{Anomaly detection}: A novel model based on GANs, where the Generator(G) has better reconstruction capability with its dense convolutional skip connections. The attention augmented Discriminator(D), is capable of checking consistency even in distant features. Both G \& D, are stably trained using spectral normalization combating mode collapse.
\noindent 2. \emph{Efficacy}: Our model that has better performance against prior state-of-the-art approaches in CIFAR-10, SIXray and MVTec\_AD datasets.\cite{akccay2019skip, akcay2018ganomaly, schlegl2017unsupervised, tang2020anomaly}. All these datasets have different types of images, especially SIXray which is a Xray dataset. The model is independent of type of images and thus can be adapted to different datasets easily.
\noindent 3. \emph{Lesser False Negatives}: Our work considers and also gives the top performance in reducing false negatives in SIXray dataset, as it's an important factor in sensitive field of baggage security applications. These fields are sensitive to false negatives as letting a threat object or damaged product go undetected can be much more costly as compared to letting a normal bag or a good product get detected as anomalous or faulty.\\
\noindent Our work focuses and performs experiments on different datasets which shows that its just not restricted to X-ray baggage or industrial applications, but rather is useful in both of them. Also there is no dataset or image type specific preprocessing, which we believe makes this quite generalizable and thus can be easily used in other domains also. The improvement in the reconstruction of the image by the GAN as comapred to other methods which don't use dense skip connections was quite apparent from the reconstructed images (images shown in supplementary material). The consistency in the AUC of our model as seen in table \ref{recall} across the different dataset sets, shows that our model performs quite well even with smaller size of dataset, which reduces its training requirements. Even within each set, different random combinations were used to avoid overfitting.
\noindent The detailed architecture of our model is explained in section \ref{sec:propsed}
\vspace*{-0.2cm}
\section{Proposed Approach}
\label{sec:propsed}
\vspace*{-0.1cm}
The lack of a balanced dataset in Anomaly Detection field requires a solution that can mitigate this problem. Generative methods are one of the top methods in such cases. This led us to use GANs. Using GANomaly \cite{akcay2018ganomaly} as the based model, as it is one of the simplest GAN model, different problems were observed which are explained ahead.This model helped us get insights into where simple GAN models are lacking. Solving the reconstruction problem, the diminishing and exploding gradient and the only local level relations in the images led to the different additions and modifications in the architecture. These observations were across datasets and not specific to any particular dataset. Our method is an unary classification method that uses GAN and we train the model on non-anomalous images $i.e$. images without any anomalous object or any abnormality. However, after training, we test on both the non-anomalous and anomalous images.
The training objective of our model is to capture the distribution of the training dataset in both image and hidden latent vector space, as the latent representation is a unique representation of the image it has been built from. This helps the model learn the features of non-anomalous images that are unique to them. We used a measure called Anomaly Score, to rate an input if it is anomalous or not, where the value is directly proportional to the likelihood of it having an anomaly.
\\
\begin{figure*}
\centering
\includegraphics[width=4.5in, height=2.0in]{fig.png}
\vspace*{-0.5cm}
\caption{\small Overview of proposed anomaly detection model}
\label{fig:overv}
\vspace*{-0.2cm}
\end{figure*}
\vspace*{-0.2cm}
\noindent As shown in Figure \ref{fig:overv}, the model has the first step extracting patches from images in a grid-wise fashion. These patches are then fed to the model. The generator(G) comes first followed by the discriminator(D). The G generates images, and both the original and generated image are fed to the D, which learns to distinguish the original from the generated one. During testing, a similar flow is followed. The image is fed to the model and an image is generated. Both the original input image and generated image are given to the D. The data obtained from this process is then used to calculate the anomaly score, as explained in section \ref{sec:propsed}. Based on this anomaly score the image is classified as non-anomalous or anomalous. The architecture and functions of both G and D are described below. The use of spectral-normalization for stabilizing the training of GANs and also the use of attention to increase the capability of the discriminator to check the consistency of detailed features in distant portions are explained in the following segments.
\noindent \textbf{Generator Network(G):} The network begins with an encoder sub-network followed by a decoder sub-network. Motivated by \cite{zhou2018unet++, huang2017densely}, the Encoder and Decoder are also connected by a dense convolutional skip connection block. These dense convolutional blocks have different sizes, depending on the layers that they are connecting. In the dense blocks at a given level, the output from the previous convolutional layer of the same dense block is concatenated with the output from the which is upsampled from the convolutional layer below it. The aim behind this connection is to decrease the semantic gap between the feature maps from the encoder and decoder, which would help and bring about better reconstruction as both local and global information is preserved through these skip connections. The encoder maps the input image to a low dimensional latent representation by downsampling and tries to capture its distribution. Contrary to the encoder, the decoder upsamples the latent representation to reconstruct the image. The block in both encoder and decoder have Convolutional, Spectral Normalization layers, and ReLU activation functions.
\begin{figure*}
\centering
\includegraphics[width=5.2in, height=2.4in]{generator_dims_23May.png}
\vspace*{-0.4cm}
\caption{Architecture of Generator ($ex$. input patch from SIXray dataset)}
\label{fig}
\vspace*{-0.3cm}
\end{figure*}
\noindent Let i denote the index of layers downsampling in the encoder and j index the other layers upsampling in the decoder. If we consider $x^{i,j}$ a representation of the feature maps, to be the output of a node $X^{i,j}$, then mathematically we can frame the operations on the skip-pathway as:
\noindent \textbullet \hspace{0.1cm} H: Convolution Operation followed by an activation function.
\\\textbullet \hspace{1pt} U: Upsampling layer. \hspace{1.2cm} \textbullet \hspace{1pt} [\hspace{0pt}] $($Square Brackets$)$: Concatenation.
\hspace{6pt}
\begin{equation}
x^{i,j}=
\begin{cases}
H(x^{i-1,j}), & \text{j=0} \\
H\left(\left[ [x^{i,j}]^{j-1}_{k=0}, \hspace{2 pt}U(x^{i+1,j-1})\right]\right ), & \text{j $>$ 0}
\end{cases}
\end{equation}
\noindent \textbf{Discriminator Network(D): }
The discriminator is used to predict/classify if the given image is real or fake. The discriminator is also used to get the
\begin{wrapfigure}{r}{0.5\textwidth}
\begin{center}
\includegraphics[width=2.7in, height=1.6in]{discriminator_dims_23May.png}
\end{center}
\caption{Architecture of Discriminator ($ex$. input patch from SIXray dataset)}
\label{disc}
\end{wrapfigure}
\noindent latent space representation of both the original image and reconstructed image and get inference from them. The network architecture can be seen in figure \ref{disc}. We have used multi head attention in D and spectral-normalization in both G \& D, which are one of the novel parts of this work.
\vspace{0.2cm}
\noindent \textbf{Attention augmented Convolution: }Convolutional operator is limited by its inability to understand global contexts \cite{hu2018squeeze, park2018bam, woo2018cbam}. However, attention mechanism can attend jointly to spatial and feature subspaces along with introducing additional feature maps \cite{zhang2019selfattention, bello2019attention}. So, we concatenate convolutional feature maps with a set of feature maps produced via self-attention, which was placed before the last convolutional layer in the discriminator. Let H represent the height, W represent the width and F represent the number of input filters of an activation map. The input tensor of shape $(H, W, F)$ is transformed into a matrix, and multi-head attention is performed over it as proposed by \cite{bello2019attention, vaswani2017attention}. For each spatial location (h,w), $N_{h}$ = 4, attention maps are computed from queries and keys as described in \cite{bello2019attention, vaswani2017attention}, which are further used to find $N_{h}$ weighted averages over the values V. The output of self attention for each head is concatenated and reshaped to match its input's spatial dimensions. The outputs are then concatenated with the output of a standard convolution operation. After experiments, an increase in performance of classification both in terms of AUC and Recall was observed when convolutions augmented with self-attention were used instead of just convolutions in discriminator. We did an ablation study for 10k random image set of SIXray dataset and we observed that the use of attention increased the AUC metric up to 0.021 and Recall metric up to 0.01.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=5.0in, height=1.8in]{attention_multihead.png}
\end{center}
\vspace*{-0.2cm}
\caption{$N_h$ attention maps are calculated for each spatial location (h,w), which then give weighted averages of values V. They are then concatenated, reshaped with same spatial dimensions as input, which is concatenated with the output of standard convolution over the input.}
\label{fig:short}
\vspace*{-0.5cm}
\end{figure}
\noindent \textbf{Stabilizing the training of GANs:} In training of GANs, performance control of the discriminator is a possible problem. The discriminator often inaccurately estimates the density ration in high dimensional spaces and is also unstable while training. Because of this the generator can fail to learn the multimodal structure of the target distribution. \cite{miyato2018spectral}. For a better and more stable training of our GAN, we used spectral-normalization \cite{miyato2018spectral}, in the discriminator. We also used spectral normalization in the generator, since conditioning of generator can be important for the performance of a GAN \cite{odena2018generator}. It prevents the escalation of parameter magnitudes and avoid unusual gradients, thus, avoiding problems like mode collapse\cite{zhang2019selfattention}. We found an empirically significant improvement in the training after applying spectral normalization in both generator and discriminator, at less computational costs. We also experimented with Wasserstein GANs \cite{arjovsky2017wasserstein} for this problem as an alternative, however there was no signigicant difference observed We did an ablation study for 10k random image set of SIXray dataset as mentioned in attention convolution section and the spectral normalization increased the AUC metric up to 0.008 and Recall metric up to 0.08.\\
\noindent \textbf{Loss functions:} The two parts of GAN, namely Generator and Discriminator compete with each other eventually improving both the networks. The goal is to seek equilibrium among Generator (G) loss and Discriminator (D) loss and not minimizing the loss function.
Aim for G to generate as realistic image as possible preserving the contextual meaning of the image and for D to reconstruct similar latent representation as of non-anomalous images. Considering these points, we choose the following three loss functions:
\noindent \emph{1. Adversarial Loss}: Inspired by adversarial loss in \cite{goodfellow2014generative}, this loss function aims to minimize the generator loss and maximize the discriminator loss \emph{min max(D,G)}. The generator will generate fake images with maximal probability of being real. The generator will ensure that the generated images are as real as possible and thus minimize the probability of image to be predicted fake by the Discriminator as in the second part of equation. The discriminator will maximize the probability of real image x and minimize the same for fake images $\hat{x}$.
$L_{adv}$ is denoted as:
\vspace*{-0.2cm}
\begin{equation}
\vspace*{-0.2cm}
\mathcal{L}_{adv}= E_{x \sim p_{x}}[ log(D(x))] + E_{x \sim p_{x}}[ log(1 - D(\hat{x}))]
\end{equation}
\emph{2. Contextual Loss:} Adversarial loss does not preserve the contextual meaning of the input data. Hence, to understand the input data distribution for non-anomalous images we use the Contextual loss function. For this, we apply $L1$ normalization on the real image $x$ and the generated fake image $\hat{x}$. This ensures that the
model is capable of generating images contextually similar to non-anomalous samples. The contextual loss is shown below:
\vspace*{-0.2cm}
\begin{equation}
\vspace*{-0.2cm}
\mathcal{L}_{con}= E_{x \sim p_{x}} \vert x - \hat{x} \vert
\end{equation}
\emph{3. Latent Loss:}
To reconstruct the latent representation of input images $x$ very close to that of generated fake images $\hat{x}$. This will help the network produce contextually same latent representation for most of the non-anomalous samples. As
shown in Figure 3(b),the discriminator D's final convolutional layer is used to get the features of x and $\hat{x}$ and then their latent representation is reconstructed such that z = f (x) and
$\hat{z}$ = f ($\hat{x}$). Latent representation loss can be represented as:
\begin{equation}
\mathcal{L}_{lat}= E_{x \sim p_{x}} \vert f(x) - f(\hat{x}) \vert
\end{equation}
The total training objective then becomes a weighted sum of
the losses above:
\vspace*{-0.2cm}
\begin{equation}
\vspace*{-0.2cm}
L = w_{1}\mathcal{L}_{adv} + w_{2}\mathcal{L}_{con} + w_{3}\mathcal{L}_{lat}
\end{equation}
where $w_{1}$, $w_{2}$ and $w_{3}$ are the weighting parameters which can be used to manage the weights for each individual loss function in the overall loss function.
\noindent \textbf{Calculation of Anomaly scores.} For identifying anomaly in the test data we calculate a score for each image, which expresses how anomalous the given image is. This method is inspired by \cite{schlegl2017unsupervised, zenati2018efficient}, where anomaly score ${A(x)}$ for a test image $x$, is given by:
\vspace*{-0.1cm}
\begin{equation}
\vspace*{-0.1cm}
\mathcal{A}(x) = \eta\mathcal{A}_{G}(x) + (1 - \eta)\mathcal{A}_{D}(x)
\end{equation}
where $\mathcal{A}_{G}(x)$ represents the contextual similarity of the real input images and the images generated by the generator as mentioned above in contextual loss.
Therefore $\mathcal{A}_{G}(x)$ $=$ $\vert$ $x$ - $\hat{x}$ $\vert$ . Thus, while calculating anomaly score, greater the value of $\mathcal{A}_{G}(x)$, greater is the probability of the image to have an anomaly.
$\mathcal{A}_{D}(x)$ in equation 6 represents the difference in latent representation of input and generated images and is measured as in latent loss section.
$\mathcal{A}_{D}(x)$ $=$ $\vert$ f(x) - f($\hat{x}$)$\vert$ . $\eta$ is a coefficient used to give relative weights to $\mathcal{A}_{G}(x)$ and $\mathcal{A}_{D}(x)$.
\noindent Let the test data set be $\mathcal{T}$, we calculate $\mathcal{A}$(x) for each $x\in\mathcal{T}$. Therefore, we get the anomaly scores vector $\mathcal{V}_\mathcal{A}$ = $\{v_i :\mathcal{A}_i(x), x_i \in T\}$. Now for normalizing, we scale these anomaly scores within the range $[0,1]$ using feature scale as explained below:
\begin{equation}
v'_i = \frac{v_i - min(\mathcal{V}_\mathcal{A})}{max(\mathcal{V}_\mathcal{A})-min(\mathcal{V}_\mathcal{A})}
\end{equation}
\noindent This $\mathcal{V'}_\mathcal{A}$ is the final anomaly score vector for test set $\mathcal{T}$ which will be used to detect anomalies.
\vspace*{-0.4cm}
\section{Experiments and Results: }
\vspace*{-0.1cm}
This section briefs on the datasets and how patches are extracted from them. Further it give details of the training objective, experimental setting and observations obtained after the experiments.
\noindent \textbf{Datasets:} We have performed experiments on 3 datasets. From which MVTec\_AD\cite{bergmann2019mvtec} and SIXray\cite{miao2019sixray} datasets are more suitable for anomaly detection tasks, owing to their well defined anomalies. However, as CIFAR-10\cite{krizhevsky2014cifar} is a very common benchmark dataset we have produced our results for comparing with various models. The datasets are described ahead.
\noindent \textbf{\emph{1. CIFAR-10}}\cite{krizhevsky2014cifar}: A commonly used benchmark dataset for classification. It has 60000 color images of size $32\times32$ for 10 categories. But we have used this dataset for anomaly detection by taking 1 vs all approach.
No segmentation or patch based technique was used for pre-processing.
\\\textbf{{\emph{2. MVTecAD }}}\cite{bergmann2019mvtec}: A publicly available dataset for bench-marking anomaly detection methods with a focus on industrial inspection. It has 5354 images divided into 15 different object and texture categories. Each category comprises a set of defect-free training images and a test set of images with various kinds of defects (with segmentation) as well as images without defects. For training and testing processes, image patches of size $256\times256$ were extracted. For images with defects, segmentation masks were used to obtain the damaged region from the complete image.
\\\textbf{\emph{3. SIXray}}\cite{miao2019sixray}: It has \emph{1,059,231 high-resolution X-ray images} of luggage items. Less than \emph{1 percent} of images have positive labels \emph{i.e}. the luggage contains an anomalous object. There are 6 subclasses of prohibited items, namely, gun, knife, wrench, pliers, scissors and hammer. For training purpose, non-overlapping patches of size $256\times256$ were extracted from \emph{10k, 100k, 500k} random image set. For test category, both positive labelled as well as negative labelled images were used. Publicly available annotations of around \emph{1400} images (containing an anomalous object) were used to extract the Region of Interest.\\\\\textbf{Training Objective:} Our goal in this field of anomaly detection is to maximize the classification capability of the model. Works in the past mainly focused on AUC of the ROC curve of the classification model. However, along with AUC our approach tries to increase the Recall of the model. The target domains of this work are important and sensitive fields like security, medical analysis, $etc.$ When there are wide disparities in the cost of false negatives vs. false positives, it is important to minimize one type of classification error.
For example: In baggage security, a safe bag classified as having a threat object can then be put through a human's investigation. However, allowing a baggage with threat objects like pistols, knives, etc. to pass through without intervention can lead to problems. This calls for the need of better recall, which represents capability of the model to avoid false negatives. Thus, we used both AUC and Recall to evaluate our model for baggage security images (SIXray dataset).\\
\noindent \textbf{Implementation Details:} The model is implemented in Python using the PyTorch framework and is trained with 32GB DDR3 RAM, Nvidia 1080 GPU and Intel 3.2GHz 4core CPU on a Linux system. For training, the image size was as mentioned for each dataset in the dataset section, the learning rate was $8e^{-3}$ and the weights of loss functions used were: w1($\mathcal{L}_{adv}$) = 1, w2($\mathcal{L}_{con}$) = 40 and w3($\mathcal{L}_{lat}$) = 1, which empirically showed the optimal performance.\\
\\
\textbf{Results :} For the CIFAR-10\cite{krizhevsky2014cifar} dataset, as in Table \ref{tab:my_table}, we can see that our model performs the best on all the classes. Similarly, for the MV\_Tec AD dataset\cite{bergmann2019mvtec}, we can see in table \ref{tab:my_label}, that when the results of our experiments are compared with those mentioned in DAGAN \cite{tang2020anomaly} and \cite{Yi_2020_ACCV} our model outperforms all other models in most of the classes.\\
\begin{table}[!htb]
\renewcommand*{\arraystretch}{0.85}
\caption{AUC results for CIFAR-10 dataset \cite{akccay2019skip}}\label{tab:my_table}
\vspace*{-0.2cm}
\resizebox{\textwidth}{!}{\begin{tabular}{c c c c c c c c c c c c c|}
\\\hline
Model & bird & car & cat & deer & dog & frog & horse & plane & ship & truck \\ [0.5ex]\hline
AnoGAN\cite{schlegl2017unsupervised} & 0.411 & 0.492 & 0.399 & 0.335 & 0.393 & 0.321 & 0.399 & 0.516 & 0.567 & 0.511 \\\hline
EGBAD\cite{zenati2018efficient} & 0.383 & 0.514 & 0.448 & 0.374 & 0.481 & 0.353 & 0.526 & 0.577 & 0.413 & 0.555 \\\hline
GANomaly \cite{akcay2018ganomaly} & 0.510 & 0.631 & 0.587 & 0.593 & 0.628 & 0.683 & 0.605 & 0.633 & 0.616 & 0.617 \\\hline
\small{Skip GANomaly\cite{akccay2019skip}}
& 0.448 & 0.953 & 0.607 & 0.602 & 0.615 & 0.931 & 0.788 & 0.797 & 0.659 & 0.907 \\\hline
\textbf{Proposed} & \textbf{0.982} & \textbf{0.998} & \textbf{0.981} & \textbf{0.999} & \textbf{0.989} & \textbf{1.000} & \textbf{0.992} & \textbf{1.000} & \textbf{1.000} & \textbf{0.974} \\\hline
\end{tabular}}
\end{table}
\vspace*{-0.4cm}
\renewcommand*{\arraystretch}{0.50}
\begin{table}[!htb]
\small
\caption{AUC on MVTec\_AD dataset \cite{tang2020anomaly} }\label{tab:my_label}
\vspace*{-0.2cm}
\hskip-1.2cm \begin{tabular}{c c c c c c c c }
\\\hline
\textbf{Class} & \makecell{{AnoGAN} \\ \cite{schlegl2017unsupervised}} & \makecell{{GANomaly} \\ \cite{akcay2018ganomaly}} & \makecell{{Skip} \\ {Ganomaly}\cite{akccay2019skip}} & \makecell{{DAGAN} \\\cite{tang2020anomaly}} & \makecell{{U-Net} \\ \cite{ronneberger2015u}} & \makecell{{Patch}\\{SVDD} \\ \cite{Yi_2020_ACCV}} & \textbf{Proposed} \\ \hline
Bottle & 0.800 & 0.794 & 0.937 & 0.983 & 0.863 & 0.986 & \textbf{0.987}
\\\hline
Cable & 0.477 & 0.711 & 0.674 & 0.665 & 0.636 & 0.903 &\textbf{0.918} \\ \hline
Capsule & 0.422 & 0.721 & 0.718 & 0.687 & 0.673 & 0.767 &\textbf{0.998} \\\hline
Carpet & 0.337 & 0.821 & 0.795 & 0.903 & 0.774 & 0.929 &\textbf{0.938} \\\hline
Grid & 0.871 & 0.743 & 0.657 & 0.867 & 0.857 & \textbf{0.946} & {0.943} \\\hline
HazelNut & 0.259 & 0.874 & 0.906 & \textbf{1.0} & 0.996 & 0.920 & \textbf{0.999}\\\hline
Leather & 0.451 & 0.808 & 0.908 & 0.844 & 0.870 & 0.909 & \textbf{0.935} \\\hline
Metal Nut & 0.284 & 0.694 & 0.79 & 0.815 & 0.676 & \textbf{0.940} & {0.825} \\\hline
Pill & 0.711 & 0.671 & 0.758 & 0.768 & 0.781 & 0.861 & \textbf{0.91 } \\\hline
Screw & 0.10 & 1.0 & 1.0 & 1.0 & 1.0 & 0.813 & \textbf{1.0} \\\hline
Tile & 0.401 & 0.72 & 0.85 & 0.961 & 0.964 & 0.978 & \textbf{0.985} \\\hline
Toothbrush & 0.439 & 0.700 & 0.689 & 0.950 & 0.811 & \textbf{1.0} & {0.979} \\\hline
Transistor & 0.692 & 0.808 & 0.814 & 0.794 & 0.674 & \textbf{0.915} & {0.89}\\\hline
Wood & 0.567 & 0.920 & 0.919 & 0.979 & 0.958 & 0.965 & \textbf{0.981}\\\hline
Zipper & 0.715 & 0.744 & 0.663 & 0.781 & 0.750 & \textbf{0.979} & {0.890}\\\hline \hline
Average & 0.502 & 0.782 & 0.805 & 0.866 & 0.819 & {0.921} & \textbf{0.945}\\\hline
\end{tabular}
\end{table}
\noindent Accomplishing our main goal, our model outperforms the state of the art models in baggage threat object detection. Three major training sets, were made: A, B and C, each of different size as shown in Table \ref{recall}. Hence, for images were taken randomly in order to ensure that we cover the entire data.
After performing the experiments on these models for the best performance, we can see that our model outperforms both of them in terms of AUC and Recall. Its observed that our model converges in just about 20 epochs which is much faster as compared to other methods, where almost more than 35 epochs are needed. Lesser number of epochs also reduces the possibility of over-fitting. It is observed that there is a small decrease in recall as the number of images increases. We hypothesise that the decrease in Recall while still having good or increase in AUC is due to redundant data leading to model learning features or noise that is common to the data, but not unique to non-anomalous images. This variation is observed across all the methods, which further supports our hypothesis.
\begin{table}[!htb]
\renewcommand*{\arraystretch}{0.85}
\vspace*{-0.2cm}
\centering
\caption{ AUC and Recall on SIXray}\label{recall}
\begin{tabular}{ccccccccc}
\hline
SIXray & \multicolumn{2}{c}{GANomaly\cite{akcay2018ganomaly}}
& \multicolumn{2}{c}{{SkipGanomaly\cite{akccay2019skip}}}
& \multicolumn{2}{c}{\textbf{Proposed}} \\
\cline{2-7}
(No. of Images) & AUC & Recall & AUC & Recall & AUC & Recall \\
\hline
A(10k) & 0.794 & 0.51 & 0.937 & 0.68 & \textbf{0.983} & \textbf{0.79} \\
\hline
B(100k) & 0.800 & 0.58 & 0.954 & 0.70 & \textbf{0.998} & \textbf{ 0.76 } \\
\hline
C(500k) & 0.998 & 0.53 & 0.998 & 0.66 & \textbf{0.999} & \textbf{0.75} \\
\hline
\end{tabular}
\end{table}
\noindent Fig \ref{fig:scr} represents the anomaly scores distribution of non-anomalous test samples and anomalous test samples, which are calculated as mentioned previously. We can clearly see that the anomaly scores for anomalous images are on the higher side while that of non-anomalous samples are comparatively lower. The region, where the scores of both classes overlap, is where the network fails to identify whether the image has an anomaly or not.
\noindent We also derived the test results of our pre-trained model with the SIXray set A, in terms of recalls, when 5 different sub-classes of threat objects in SIXray dataset were tested on separately. Region of interest of positive samples were extracted using Annotations as explained in Dataset section. We experimented with the anomalous objects in test set to check the performance of the model on different types of objects. Images of only a particular anomalous object were given and the recalls of gun, knife, pliers, scissors and wrench are 0.834, 0.931, 0.886, 0.933, 0.556 respectively. The model performs fairly well on all the categories, except wrench. This could be due to its size and physical characteristics and its similarity with non-anomalous objects.
\begin{figure}[!htb]
\centering
\begin{minipage}{.5\textwidth}
\centering
{\includegraphics[width=6cm]{scores1.png} }
\vspace*{-0.2cm}
\caption{\small Anomaly score distribution}
\label{fig:scr}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
{\includegraphics[width=6cm]{roc_A_B.png} }
\vspace*{-0.2cm}
\caption{\small ROC Curve for A, B, C dataset of SIXray}
\label{fig:test2}
\end{minipage}
\vspace*{-0.3cm}
\end{figure}
\vspace*{-0.2cm}
\section{Conclusions}
This paper proposes a novel method for anomaly detection. This adversarial training tackles the problem of unbalanced data. Since the training does not require the anomalies, it makes it capable to handle variations in the threat objects. Our models examine the use of spectral normalization for stable training, the role of dense convolutional skip connections and attention for better reconstruction of image, inference learning and detecting consistency in distant features. Our model outperforms prior work\cite{akcay2018ganomaly, akccay2019skip} on anomaly detection within X-ray secuity baggage imagery by improving the performance on SIXray dataset\cite{miao2019sixray}, by about 5\% in AUC and about 16\% in terms of Recall of the best models previously. The proposed model achieves superior results in most of the classes in MV\_Tec AD dataset\cite{bergmann2019mvtec} compared to the state-of-the-art strategies\cite{schlegl2017unsupervised, akcay2018ganomaly, akccay2019skip, tang2020anomaly, ronneberger2015u}. This demonstrates the effectiveness of our proposal, which is, it's not limited to a particular application domain but can be beneficial for anomaly detection task in various domains. Our experiments with different dataset size with sixray also demonstrates that a very high performance can be achieved even with a very small dataset size, as compared to other methods \cite{akcay2018ganomaly}\cite{akccay2019skip} where a significant difference is seen as the dataset size is increased. This makes the model quite efficient in terms of training requirement. These sets had multiple sets in themselves. Even within each set, different random combinations were used to make multiple sets, to avoid overfitting. The use of attention along with the improved performance also brings a higher computation power requirement, however that is manageable and can be mitigated to some extent by regularizing other parameters while still maintaining approximately the same performance. This work can be further extended by integrating it with threat object localization helping further reduce human dependency. Also, different aspects such as continual learning could be explored to help the model learn to adapt, to the changes in bags and objects.\\
\noindent \textbf{Conflict of interest statement:}
We declare that there is no conflict of interest with any person or organization professional or personal that could influence our work.
|
2,877,628,090,136 | arxiv | \section{#1}}
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\begin{document}
\setcounter{page}{0}
\begin{titlepage}
\vspace*{0.1cm}
\begin{center}
{\Large \bf Neutrino nonstandard interactions in the supernova
}\\
\vspace{1.0cm}
{\large
C. R. Das\footnote{E-mail: crdas@cftp.ist.utl.pt},
Jo\~{a}o Pulido\footnote{E-mail: pulido@cftp.ist.utl.pt}\\
\vspace{0.15cm}
{{\small \sl CENTRO DE F\'{I}SICA TE\'{O}RICA DE PART\'{I}CULAS (CFTP)\\
Departamento de F\'\i sica, Instituto Superior T\'ecnico\\
Av. Rovisco Pais, P-1049-001 Lisboa, Portugal}\\
}}
\vspace{0.25cm}
\end{center}
\vglue 0.6truecm
\begin{abstract}
Neutrino non-standard interactions (NSI) were investigated earlier in the solar case and were shown to
reduce the tensions between the data and the large mixing angle solution predictions. We extend the previous
framework to the supernova and evaluate the appearance probabilities for neutrinos and antineutrinos as a
function of their energy after leaving the collapsing star with and without NSI. For normal hierarchy the
probability for electron neutrinos and antineutrinos at low energy ($E\lesssim 0.8-0.9 MeV$) is substantially
increased with respect to the non-NSI case and joins its value for inverse hierarchy which is constant
with energy. Also for inverse hierarchy the NSI and non-NSI probabilities are the same for each neutrino and
antineutrino species. Although detection in such a low energy range remains at present an experimental
challenge, it will become a visible trace of NSI with normal hierarchy if they exist. On the other hand
the neutrino decay probability into an antineutrino and a majoron, an effect previously shown to be induced
by dense matter, is, as in the case of the sun, too small to be observed as a direct consequence of NSI.
\end{abstract}
\end{titlepage}
\section{Introduction}
Non-standard neutrino oscillations (NSI) \cite{Guzzo:1991hi,Roulet:1991sm,Grossman:1995wx,
Johnson:1999ci,Huber:2001zw,Huber:2002bi,Blennow:2008er,Biggio:2009nt,Wei:2010ww} have since long
been applied to solar
neutrinos \cite{Guzzo:1991hi,Roulet:1991sm,Pulido:1992nn} in an attempt to understand the origin of their
apparent deficit. More recently interest in NSI has been revived \cite{Pulido:2010ht,Palazzo:2011vg}
with the purpose of solving the possible
inconsistencies between the LMA solution to the solar neutrino problem \cite{Fogli:2008ig,Schwetz:2008er} and
the data \cite{:2008zn,Smy,Aharmim:2009gd,Bellini:2010gn}. The most remarkable of these inconsistencies is the
absence of any experimental evidence for an upturn in the LMA survival probability in the intermediate energy
range of solar neutrinos. It has been shown \cite{Pulido:2010ht} that NSI can lead to a flat probability in
this sector and hence a flat electron energy spectrum as observed in both SuperKamiokande \cite{:2008zn} and SNO
\cite{Aharmim:2009gd} experiments, with the interesting consequence of a possible neutrino decay into
antineutrinos and majorons in dense matter. However, the predicted antineutrino flux is too small to be
observed and the only expected signature of the effect is the flatness of the electron spectrum. Extension
to supernova neutrinos, which will be done in the present paper, will provide us further information on
possible experimental NSI signatures. Our approach to NSI, previously developed in \cite{Pulido:2010ht} assumes
extra contributions to the vertices $\nu_{\alpha} \nu_{\beta}$ and $\nu_{\alpha} e$ and differs
therefore from the neutrino-neutrino interaction considered in \cite{Blennow:2008er}.
The supernova dynamics has been extensively studied and for details we may refer the reader to the review
\cite{Bilenky:2002aw}. Here we will just highlight its relevant aspects for our purposes.
As is well known when a star of mass $M\gtrsim 8M_{\odot}$ has burned all its fuel, an onion like
structure is formed with the lighter elements in the outer layers and the heavier ones inside. For massive
enough stars $M\gtrsim 11M_{\odot}$ an innermost iron core is formed. Equilibrium of the core is disrupted
by photodissociation of the heavier elements producing alpha particles and neutrons. The resulting free
electrons are in turn captured by protons and nuclei, producing electron neutrinos which escape,
\begin{equation}
e^{-}~p \rightarrow n~\nu_e~.
\label{neutron}
\end{equation}
When the Coulomb pressure of the electrons becomes insufficient to sustain the core against its own gravity,
collapse is initiated until the core becomes no longer transparent to neutrinos and reaches nuclear density
($\rho\simeq 3\times 10^{14}g~cm^{-3}$), a process which takes only about 10 ms \cite{Thompson:2002mw}. The
neutronization process (\ref{neutron}) ceases, a state of hydrostatic equilibrium is reached with the core
forming a proto-neutron star of radius around 10 km and a temperature of 35-40 MeV.
From this stage onwards only thermal neutrinos are emitted. They are produced from nucleon-nucleon bremsstrahlung
\begin{equation}
N~N\rightarrow N~N~\nu~\bar\nu
\end{equation}
$e^{+}e^{-}$ annihilation,
\begin{equation}
e^{+}e^{-} \rightarrow \nu~\bar\nu
\end{equation}
plasmon decay,
\begin{equation}
\gamma \rightarrow \nu~\bar\nu
\end{equation}
and electron nucleon bremsstrahlung
\begin{equation}
e^{-}~N \rightarrow e^{-}~N~\nu~\bar\nu.
\end{equation}
So according to core collapse supernova dynamics, in the thermal phase neutrino emission proceeds through
all flavour channels and is accompanied by antineutrinos, whereas in the initial neutronization phase only
electron neutrinos are expected.
The events observed in SN1987a were all interpreted as antineutrino ones \cite{Sato:1987rd} due not only
to the dominance of the antineutrino cross section relative to the neutrino one but also possibly to the
unavailability of a lower energy threshold, as shall be seen in the present paper. Whether or not any
neutrinos reached the Earth is an open question.
In ref.\cite{Pulido:2010ht} it was shown that neutrino decay in dense matter into antineutrinos and a
majoron is a necessary consequence of NSI. However its effect through the detection of antineutrinos in
the neutronization phase where only $\nu_e$'s are produced is not possible, owing to the smallness of
the antineutrino production probability through decay even in the supernova.
In fact the high density range of the neutrino trajectory is not long enough to produce a visible antineutrino
flux originated from this source.
In this paper we extend for supernova neutrinos the NSI framework that was
introduced in the solar neutrino case \cite{Pulido:2010ht} where it was shown to improve the fits to experiment.
In section 2 we review its general features. We recall that a flat SuperKamiokande and SNO electron spectrum
necessarily require imaginary diagonal entries in the NSI Hamiltonian. Their real parts, along with the
off diagonal entries whether real or imaginary, may be arbitrary in the sense that they do not induce any
change in the standard LMA probability. As a consequence neutrino decay in dense matter into an antineutrino
and a majoron arises. Section 3
is the main part of our work. Here we extend NSI to the interactions of the supernova neutrinos.
We evaluate the survival and conversion probabilities with and without NSI as well as the probability
for neutrino decay. For normal mass hierarchy in the absence of NSI the electron neutrino survival probability
turns out to be small, most of the $\nu_e$'$s$ having been converted through standard oscillations to $\nu_{\mu},
\nu_{\tau}$'$s$. Furthermore it is found that as in the case of the sun, the decay probability is extremely
small for antineutrinos from neutrino decay to ever be observed. Since in the neutronization phase only
$\nu_e$'$s$ are produced, no charged current signal is expected to be seen at this initial stage in the
absence of NSI. On the other hand in the NSI case a sizeable electron neutrino flux may appear at low
energies in the neutronization phase which may be detected through the charged current
with improved low energy detectors. At present this remains a challenge but may be reached in the not
too distant future. For inverse hierarchy all neutrino fluxes are comparable in the whole energy range.
On the other hand in the thermalization phase neutrinos ($\nu_{\alpha}$) and antineutrinos
($\bar\nu_{\alpha}$) of all kinds are produced so that the initial state is assumed to consist of
$\nu_{\alpha}$, $\bar\nu_{\alpha}$'$s$ in equal proportions. Owing to the large number of oscillations,
the information from the initial state is essentially lost, hence the final probability distributions
are the same as in neutronization where only $\nu_e$'$s$ are present initially. The only difference is
in this case the obvious presence of antineutrinos with the same probability distribution as neutrinos
due to $CPT$ invariance.
Finally in section 4 we comment on our results and draw our conclusions.
\section{The NSI Hamiltonian}
In this section we highlight and discuss the main steps
of the analysis leading to the Hamiltonian for propagation in dense matter (see also\cite{Pulido:2010ht}).
Its results will be used in section 3 for the evaluation of the probabilities for neutrino survival and
conversion and the probability for antineutrino production.
Assuming only standard interactions (SI), the potential for electron neutrinos traveling through the sun and supernova
is given by
\begin{equation}
V=G_F\sqrt{2}N_e\left(1-\frac{N_n}{2N_e}\right)=V_c+V_n
\label{VSI}
\end{equation}
where $N_e$, $N_n$ are the electron and neutron density and $V_c=G_F\sqrt{2}N_e$, $V_n=-G_F/\sqrt{2}N_n$.
For NSI we assume that the $\nu_{\alpha}$ interaction potential on electrons $(\alpha=e,\mu,\tau)$ involves both
the charged and neutral currents (CC and NC), while on quarks it involves only NC. NSI give rise to possible
lepton flavour violation. Denoting by $\varepsilon^{fP}_{\alpha \beta}$ the NSI factor that multiplies each
diagram associated to neutrino propagation in matter we have
\begin{eqnarray}
(v_{\alpha\beta})_{NSI}\!\!\! & = & \!\!\!G_F\sqrt{2}N_e\left[(\varepsilon_{\alpha \beta}^{eP})_{NC}+\left(-\frac{1}{2}+2sin^2\theta_W \right)
(\varepsilon_{\alpha \beta}^{eP})_{NC}+\left(1-\frac{8}{3}sin^2\theta_W+\frac{N_n}{2N_e} \right)
\varepsilon_{\alpha \beta}^{uP}\right.\nonumber\\
&& + \left.\left(-\frac{1}{2}+\frac{2}{3}sin^2\theta_W-\frac{N_n}{N_e} \right)
\varepsilon_{\alpha \beta}^{dP} \right]
\label{VNSI}
\end{eqnarray}
In the following we will assume $(\varepsilon_{\alpha \beta}^{eP})_{CC}=(\varepsilon_{\alpha \beta}^{eP})_{NC}$. Hence
flavour change may occur without a vacuum mixing angle \cite{Fogli:2008ig,Schwetz:2008er} or a magnetic
moment \cite{Das:2009kw}, being induced only by the off diagonal entries of this matrix ($\alpha\neq\beta$).
So with SI and NSI the matter Hamiltonian in the flavour basis is the sum of eqs.(\ref{VSI}) and (\ref{VNSI})
\begin{equation}
{\cal H}_M=V_c \left(\begin{array}{ccc} 1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0\\ \end{array}\right)+
\left(\begin{array}{ccc} (v_{ee})_{NSI} & (v_{e\mu})_{NSI} & (v_{e\tau})_{NSI}\\
(v_{\mu e})_{NSI} & (v_{\mu\mu})_{NSI} & (v_{\mu\tau})_{NSI}\\
(v_{\tau e})_{NSI} & (v_{\tau\mu})_{NSI} & (v_{\tau\tau})_{NSI}\\ \end{array}\right)={\cal H_{SI}}+{\cal H_{NSI}}~.
\label{HNSI}
\end{equation}
As is well known, owing to the large neutrino density, collective effects can play an important role in
the supernova \cite{Pastor:2002we}, \cite{Duan:2006jv}, \cite{Duan:2006an} \footnote{For a review on
collective oscillations see \cite{Duan:2010bg}.}, providing an additional contribution to
the matrix (\ref{HNSI}). They occur up to a few 100 km, whereas MSW oscillations occur
typically at larger distances so that MSW effects factorize and can be included separately \cite{Dasgupta:2007ws}.
For this reason the NSI effects, intrinsically associated to MSW, must also be included separately.
We will perform our calculation starting from a region around 400-500 km where collective oscillations have already
taken place. Hence their net effect to our approach amounts to the well known spectral swap-split of the neutrino
and antineutrino energy spectra \cite{Duan:2006jv}, \cite{Duan:2006an}, \cite{Dasgupta:2009mg}. However it was
recently shown that matter completely suppresses collective oscillations up to 200 ms \cite{Chakraborty:2011nf}
after bounce. In view of these results it appears that collective effects can be ignored at early times in a
supernova. Our results, derived in the next section, are therefore expected to be valid for the whole neutronization
phase and part of the subsequent thermal phase.
The investigation performed in ref.\cite{Pulido:2010ht} shows that no off diagonal entry in matrix
(\ref{HNSI}) whether real or imaginary, can change
the LMA probability, hence the rates and the corresponding SuperKamiokande and SNO electron spectra. In fact
only imaginary diagonal couplings lead to a change in $P_{LMA}$. As discussed in ref. \cite{Pulido:2010ht},
this implies the instability of neutrinos in matter and their decay into antineutrinos of all species along
with majorons. Requiring the convenient change in $P_{LMA}$, namely the one that leads to a flat electron
spectrum in SuperKamiokande and SNO, thus allowing for imaginary diagonal couplings, it was shown that the
simplest choice of parameters is
\begin{equation}
{\cal H_{NSI}}=G_F\sqrt{2}N_e\!\!\left[\!\!\left(\begin{array}{ccc}
\!\!\frac{i}{2}\varepsilon (x_e+1) & &\\ & -i\varepsilon x_e &\\ & & \frac{i}{2}\varepsilon x_e \!\!\end{array} \right)
\!\!+\!x_u\!\!\left(\begin{array}{ccc} \!\!\frac{i}{2}\varepsilon & &\\ & -i\varepsilon &\\
& & \frac{i}{2}\varepsilon\!\! \end{array} \right)\!\!+
\!x_d\!\left(\begin{array}{ccc} \!\!-\frac{i}{2}\varepsilon & &\\ & i\varepsilon &\\ & &
-\frac{i}{2}\varepsilon\!\! \end{array} \right)\!\!\right]
\label{HNSIf}
\end{equation}
with vanishing off diagonal entries and $\varepsilon=3.5\times 10^{-4}$. Here
\begin{equation}
x_e=-\frac{1}{2}+2sin^2\theta_W,~x_u=1-\frac{8}{3}sin^2\theta_W+
\frac{N_n}{2N_e},~x_d=-\frac{1}{2}+\frac{2}{3}sin^2\theta_W-\frac{N_n}{N_e}~.
\label{x}
\end{equation}
The three matrices in the right hand side of eq. (\ref{HNSIf}) relate to the neutrino interaction with
electrons, u-quarks and d-quarks respectively. Each diagonal entry refers to the $\nu_e$, $\nu_{\mu}$,
$\nu_{\tau}$ contribution to its own interaction, hence its decay. So for instance, for $\nu_e$ the NSI (decay)
potential is
\begin{equation}
(v_{ee})_{NSI}=\frac{i}{2}\varepsilon G_F\sqrt{2}N_e(x_e+1+x_u-x_d)~.
\label{vee}
\end{equation}
In this way we obtain the three interaction (decay) potentials
\begin{equation}
(v_{ee})_{NSI} = iG_F\sqrt{2}(3.5\times 10^{-4})N_e\left(1-\frac{2}{3}sin^2\theta_W+\frac{3N_n}{4N_e}\right)
\label{ve}
\end{equation}
\begin{equation}
(v_{\mu\mu})_{NSI} = iG_F\sqrt{2}(3.5\times 10^{-4})N_e\left(-1+\frac{4}{3}sin^2\theta_W-\frac{3N_n}{2N_e}\right)
\label{vmu}
\end{equation}
\begin{equation}
(v_{\tau\tau})_{NSI} = \frac{i}{2}G_F\sqrt{2}(3.5\times 10^{-4})N_e\left(1-\frac{4}{3}sin^2\theta_W+\frac{3N_n}{2N_e}\right)~.
\label{vtau}
\end{equation}
The full Hamiltonian including the vacuum part and referred to the mass basis is now
\begin{equation}
{\cal H}=\left(\begin{array}{ccc} 0 & 0 & 0\\
0 & \frac{\Delta m^2_{21}}{2E_0} & 0\\
0 & 0 & \frac{\Delta m^2_{31}}{2E_0}\\ \end{array} \right)+ U^{\dagger}
{\cal H}_M U~
\label{H}
\end{equation}
with ${\cal H}_M$ given by (\ref{HNSI}) with the replacements (\ref{ve}), (\ref{vmu}), (\ref{vtau}) and arbitrary
off diagonal entries. In eq. (\ref{H}) $\Delta m^2_{21}$, $\Delta m^2_{31}$ are the solar and atmospheric mass
differences, $U$ is the PMNS matrix \cite{Maki:1962mu} defined with the standard
parameterisation \cite{Amsler:2008zzb} and we use the central value for $sin\theta_{13}=0.13$ from ref.
\cite{Fogli:2008ig}. The negative imaginary parts of the eigenvalues of (\ref{H}) are the mass eigenstate decay
rates $\Gamma_i$ which will be used in the next section for the evaluation of the probabilities. As for the
supernova parameters, numerical simulations \cite{Fischer:2009af} yield the electron number density and
supernova density profiles which in our period of interest (the initial 200 ms) are well approximated by
\begin{equation}
Y_e=\frac{1}{3}-0.04~log~\frac{\rho}{10^{12}g~cm^{-3}}
\end{equation}
\begin{equation}
\rho=\rho_{0}\left(\frac{10km}{r}\right)^3 g~cm^{-3}
\end{equation}
for $r>10~km$ with $\rho_{0}\sim 10^{14}g~cm^{-3}$ and which will be used in the next section.
\section{Rates, couplings and probabilities}
\subsection{Rates and neutrino majoron couplings}
The $\nu_e$ flux from neutronization is in fact a linear combination of the three mass eigenstates $\nu_i$
displayed in fig.\ref{fig1}({\it a}) for neutrino energy $E_0=11MeV$ and normal hierarchy. Our first purpose
in this section is to evaluate the decay rate of the NSI process \cite{Berezhiani:1987gf,Berezhiani:1989za,
Kachelriess:2000qc,Tomas:2001dh,Farzan:2002wx,Lessa:2007up}
\begin{equation}
\nu_{i}\rightarrow \bar\nu_{j}+\chi
\end{equation}
where $\chi$ denotes the majoron. This rate satisfies
\cite{Pulido:2010ht}, \cite{Farzan:2002wx}
\begin{equation}
\frac{\partial \Gamma_{i}}{\partial E_f}=\sum_{j=1}^{3}\frac{|g_{ij}|^2}{8\pi}\frac{E_0-E_f}{{E_0}^2}
|v_{i}(r)-\overline v_{j}(r)|_{NSI}F(r,E_0)
\label{eq3}
\end{equation}
where $g_{ij}$ are the neutrino majoron couplings \cite{Farzan:2002wx},\cite{Lessa:2007up} and the interaction
potentials satisfy in the mass basis $v_i=-\overline v_i$. Here $E_0$ is the initial neutrino energy, $E_f$ is
the antineutrino energy which in a first approximation we assume to take values in the interval $(0,E_0)$,
since the energy $E_0$ is shared by the final neutrino and the majoron. The quantity $F(E_0,r)$ is
the Fermi factor \cite{Farzan:2002wx}
\begin{equation}
F(E_0,r)=\left(1-\frac{1}{e^{(E_{0}-\mu)/T}+1}\right)~,
\label{FF}
\end{equation}
which reflects the fact that inside the supernova some of the states have already been occupied by neutrinos.
In the inner core ($R_{inner}\simeq 10~km$) the chemical
potential for $\nu_e$ $(\mu_{\nu_e}$) is around 200 MeV and the temperature $T=35~MeV$. In the outer core
($R_{inner}\simeq 15~km$) the temperature drops abruptly $T=2~MeV$, the density falls from $5\times 10^{14}g~cm^{-3}$
to $5\times 10^{13}g~cm^{-3}$ and we may set $\mu_{\nu_e}=0$. Hence in the following we will omit the factor
$F(E_0,r)$.
The three decay rates $\Gamma_i$ are the imaginary parts of the NSI Hamiltonian eigenvalues. They are
represented in fig.\ref{fig1}({\it b}) for $E_0=11MeV$. As in the solar case, only $\Gamma_3$ is
negative, so only the state $\nu_3$ is unstable, allowing for the decay into either $\bar\nu_1$,
$\bar\nu_2$ or $\bar\nu_3$, thus generating antineutrinos of the three flavours. One could obtain
an alternative expression for $\Gamma_{i}(r,E_0)$ through the integration of equation (\ref{eq3})
over $E_f$
\begin{equation}
\Gamma_i=\int_0^{E_0}\frac{\partial \Gamma_{i}}{\partial E_f}dE_f=\sum_{j=1}^{3} \frac{|g_{ij}|^2}{16\pi}
|v_{i}(r)-\bar v_{j}(r)|~.
\label{intgamma}
\end{equation}
The vanishing lower limit used in this integration is as referred to above only an approximation, since the
majoron obtains a tiny effective mass in matter $m_{eff}^2\sim |g|^2N_{\nu}/q$ \cite{Farzan:2002wx}. In this
way a slight dependence of the rate $\Gamma_{i}$ on $E_0$ arises which is consistent with the numerical
evaluation of the imaginary part of the NSI Hamiltonian eigenvalues.
The parameters $\varepsilon_{\alpha\beta}$ were fixed earlier (see section 2) by the fittings to the solar
neutrino data \cite{Pulido:2010ht} and so the
rates $\Gamma_{i}(r,E_0)$, numerically evaluated as the imaginary parts of the Hamiltonian eigenvalues, are
also fixed. Moreover the values of $g_{ij}$ to which the rates correspond are so far unknown and only upper
bounds exist in the literature \cite{Lessa:2007up}. In the following we will use the strictest one quoted,
namely
\begin{equation}
\sum_{\alpha}|g_{e\alpha}|^2<5.5\times 10^{-6},
\label{bound}
\end{equation}
conveniently expressed in the mass basis. Quantities $\Gamma_i$ will now be used in the evaluation of the
probabilities.
\subsection{Probability densities for neutrino survival, decay and antineutrino appearance}
We denote by $P_{\nu_{i}}(r,E_0)$ the $\nu_i$ survival probability in the mass
basis for neutrino energy $E_0$ at a distance $r$ from the star centre, obtained from integration of the
Schr\"{o}dinger equation with the Hamiltonian (\ref{H}), and by $\phi_{\nu_e}(E_0)$ the initial normalized
neutrino spectral flux \cite{Thompson:2002mw}
$$\phi_{\nu_e}(E_0)=\frac{1}{\Phi(E_0)}\frac{\partial \Phi}{\partial E_0}~.$$
Given these definitions the quantity
\begin{equation}
\phi_{\nu_i}(r,E_0)\!=\!P_{\nu_{i}}(r,E_0)\phi_{\nu_e}(E_0)
\label{eqprob}
\end{equation}
is the normalized spectral flux of $\nu_i$ mass eigenstates with energy $E_0$ that remain in the beam after traveling a
distance $r$. Hence
\begin{equation}
\frac{\partial P_{\nu_{i}^m}(E_0)}{\partial r}=
\int_{0}^{E_{0}}\phi_{\nu_i}(r,E_0)(1-e^{\Gamma_{i}(r,E_0)r})~
\frac{\partial \Gamma_{i}(r,E_0,E_f)}{\partial E_f}~dE_f
\label{eq1}
\end{equation}
is the probability per unit star radius for this mass eigenstate to have disappeared from the flux.
Replacing now
(\ref{eq3}) in (\ref{eq1}) one obtains the probability for the $\nu_i$ mass eigenstate disappearance
\begin{equation}
P_{\nu_{i}^m}(E_0)\!=\sum_{j=1}^{3}\!\frac{|g_{ij}|^2}{8\pi}\!\!\int_{R_i}^{R_S}\!\!|v_{i}(r) - \overline v_{j}(r)|_{NSI}
\phi_{\nu_i}(r,E_0)(1-e^{\Gamma_{i}(r,E_0)r})\!\!\int_{0}^{E_0}\!\frac{E_0\!-\!E_f}{{E_0}^2}dE_f dr
\label{eq5}
\end{equation}
Quantities $R_i$ and $R_S$ denote the neutrino sphere and the star radii respectively.
We note that in eq.(\ref{eq5}) we have the simple sum of probabilities, since each eigenstate $\nu_i$, as it
decays, can give rise to just one antineutrino flavour which is a linear combination of the three mass
eigenstates $\nu_j$. Performing the integration in $E_f$, (\ref{eq5}) can be simplified to
\begin{equation}
P_{\nu_{i}^m}(E_0)\!=\sum_{j=1}^{3}\!\frac{|g_{ij}|^2}{16\pi}\!\!\int_{R_i}^{R_S}\!\!|v_{i}(r) - \overline v_{j}(r)|_{NSI}
~\phi_{\nu_i}(r,E_0)(1-e^{\Gamma_{i}(r,E_0)r})~dr~.
\label{eq6}
\end{equation}
In other words, given a flux of $\nu_i$' s with an energy in the interval $[E_0,E_{0} + dE_{0}]$, the quantity
$P_{\nu_{i}^m}(E_0)dE_0$ is the fraction of these neutrinos which has decayed into antineutrinos with an
energy in the interval $(0,E_0 + dE_0)$ after traversing the star. The fraction of $\nu_i$' s that remains
in the beam after leaving the star is $P_{\nu_{i}}(R_s,E_0)dE_0$ with $P_{\nu_{i}}(r,E_0)$ as defined in the
beginning of this section (see (\ref{eqprob})).
As a reminder we note that the normalization of $P_{\nu_{i}}$ follows from
\begin{equation}
P_{\nu_i}=|<\nu_i|\nu_e>|^2=|\nu_i u_{ej}^{*}\nu_j|^2=u_{ej}^{*}u_{ek}\nu_{j}\nu_{k}\nu_{i}\nu_{i}=
u_{ej}^{*}u_{ek}\delta_{ij}\delta_{ik}=u_{ei}^{*}u_{ei}
\label{probmass}
\end{equation}
(no sum over $i$) with the orthogonality of $U^{PMNS}$ ensuring $\sum_{i}P_{\nu_i}=1$. Moreover the integration of
the equation of motion proceeds in the mass basis, so that the normalization condition is forced on $P_{\nu_{i}}$.
The normalization of $P_{\nu_{\alpha}}$ also follows in a similar way
\begin{eqnarray}
P_{\nu_{\alpha}}&=&|<\nu_{\alpha}|\nu_e>|^2=|u_{\alpha i}^{*}u_{ej}\nu_{i}\nu_{j}|^2 = u_{\alpha i}^{*}u_{ej}^{*}
u_{\alpha k}u_{em}\nu_{i}\nu_{j}\nu_{k}\nu_{m} \nonumber \\ & = & u_{\alpha i}^{*}u_{ej}^{*}u_{\alpha j}u_{ei}=
\delta_{\alpha e}u_{\alpha i}^{*}u_{ei}
\label{probflavour}
\end{eqnarray}
(no sum over $\alpha$) with the orthogonality of $U^{PMNS}$ implying $\sum_{\alpha}P_{\nu_{\alpha}}=1$. In
(\ref{probmass}) and (\ref{probflavour}) we neglected the plane wave propagation phases. Finally the flavour
probability is obtained from
\begin{equation}
P_{\nu_{\alpha}}=|u_{\alpha i} <\nu_i|\nu_e>|^2
\end{equation}
and is shown in fig.\ref{fig2} for standard ($\varepsilon=0$) and non-standard interactions ($\varepsilon=3.5\times 10^{-4}$).
For the sake of the following discussion we recall that from supernova theory only electron neutrinos are
produced in the initial neutronization phase. From fig.\ref{fig2} with normal hierarchy (panel (a)) it is seen that,
in the absence of NSI, electron neutrinos can hardly be detected, as the survival probability is around $0.02$ and
practically constant with energy. Further, $\nu_{\mu}$'$s$ and $\nu_{\tau}$'$s$ produced from oscillations,
although having a much larger production probability ($\sim 0.5$), can only be detected through neutral current
reactions, so there will be no way to distinguish them at present from other neutrinos or antineutrinos.
The major difference in the NSI case for normal hierarchy as seen from fig.\ref{fig2}a is the appearance of a
visible flux of electron neutrinos in the low energy region ($E_0\lesssim 0.8-0.9MeV$). This can be pinpointed in
the neutronization phase through the charged current, provided enough experimental capability is developed in the
future to reduce
the low energy threshold. On the other hand for inverse hierarchy (fig.\ref{fig2}b) no difference appears between NSI
and non-NSI: the sizable value of the $\nu_e$ probability ($\sim 0.31$) is the same at low energy as in the normal
hierarchy case and remains constant with energy, so $\nu_e$ detection is experimentally accessible. However the
NSI and non-NSI probability values are the same for each neutrino species and also remain in each case constant with
energy. So for inverse hierarchy NSI and non-NSI appear indistinguishable.
We next investigate the possibility for electron antineutrinos to be detected in the initial neutronization phase. As
previously referred, only electron neutrinos are produced at this stage in a 10 ms pulse. Their decay in matter into
electron antineutrinos in a significant number would provide a clear signature of NSI in this initial pulse.
Antineutrinos with energy $E_f$ are produced from electron neutrino decay with energy $E_0$ in the interval
$(E_f,E_{0_{max}}]$. Their appearance probability density is in the mass basis
\begin{equation}
P_{\bar\nu_{j}}(E_f)\!={\bigcup}_{i=1}^{3}\!\frac{|g_{ij}|^2}{8\pi}\!\!\int_{R_i}^{R_S}\!\!|v_{i}(r)-\overline v_{j}(r)|_{NSI}\!
\!\int_{E_{f}}^{E_{0_{max}}}\!\!\!\phi_{\nu_{i}}(r,\!E_0)(1-e^{\Gamma_{i}(r,E_0)r})\frac{E_0\!-\!E_f}{{E_0}^2}dE_0dr.
\label{antinu}
\end{equation}
Contrary to eq.(\ref{eq6}), where each neutrino could decay into one {\it single} antineutrino, each antineutrino
can be now produced from more than one neutrino, hence the reason to consider the union of events in eq.(\ref{antinu})
\footnote{The probability for the union of three independent events ($A_1,A_2,A_3$) is given by the well known rule
$P(A)=P(A_1)+P(A_2)+P(A_3)-P(A_1)P(A_2)-P(A_1)P(A_3)-P(A_2)P(A_3)+P(A_1)P(A_2)P(A_3)$.}.
Assuming CPT invariance the flavour probability for $\bar\nu_e$ is given by
\begin{equation}
P_{\bar\nu_{e}}=|u_{e i}|^2 P_{\bar\nu_{i}}~.
\label{antiprob}
\end{equation}
Using the bound (\ref{bound}) as a common value for all neutrino majoron couplings involved, we obtain the result
displayed in fig.\ref{fig3} for $P_{\bar\nu_{e}}$ which is obviously too small for ${\bar\nu_{e}}$'$s$ to be observed.
If one considers the $\nu_e$ decay into $\bar\nu_{\tau}$ one may relax (\ref{bound}) and use instead \cite{Lessa:2007up}
\begin{equation}
\sum_{\alpha}|g_{\tau\alpha}|^2<5.5\times 10^{-2}
\label{bound1}
\end{equation}
as a common value for all couplings. In this case the uppermost value of the probability is raised by a factor
$O(10^4)$ which is still too small for the effect to be observed and moreover $\bar\nu_e$'$s$, if produced
from $\nu_{\tau}$ decay, would appear as a higher order effect. On the other hand, as pointed above, the $\nu_e$
probability from oscillations in the absence of NSI is seen to be rather small in normal hierarchy (fig.\ref{fig2}a).
The prospects for its detection and hence a charged current signal in the neutronization phase will crucially depend
on the detector size and supernova distance. With NSI, it may become clearer through $\nu_e d\rightarrow
ppe^{-}$ or an increased $\nu e^-\rightarrow \nu e^-$ scattering event rate, however at low energy
($E_0\lesssim 0.8-0.9MeV$), which is still an experimental challenge. For inverse hierarchy, as also pointed above,
the situation is much different: the $\nu_e$ signal appears louder and clearer (see fig.\ref{fig2}b).
In the subsequent thermalization phase $\bar\nu_e$'$s$ and $\nu_X,\bar\nu_X$'$s$ with $X=\mu,\tau$ are produced
initially along with $\nu_e$'$s$. All kinds of neutrinos and antineutrinos will arise from
these through NSI. The appearance probability for antineutrinos from neutrino decay may increase substantially,
since the bound (\ref{bound1}) must now obviously be taken into account instead of (\ref{bound}). Again this
is not expected to be enough for the effect to be observable (see fig.\ref{fig3}) even for $\bar\nu_e$
appearance, despite the higher rate involved in its detection. As for the other probabilities from oscillations
and NSI, they must be evaluated from the union of events as in (\ref{antinu}), since each final neutrino or
antineutrino is produced simultaneously from a number of different initial ones. Evaluating these probabilities
taking into account their normalization, it turns out that the contribution from the extra initial neutrinos
and antineutrinos does not change their value nor energy distribution relative to the neutronization
case when initially only $\nu_e$'$s$ are present. In fact the information from the initial state is lost
due to the exceedingly large number of oscillations that the neutrinos undergo during propagation: only the
propagation physics which depends on the interaction potentials is relevant here. Thus the same fig.\ref{fig2}
applies for thermalization both for normal and inverse hierarchies. The only notorious characteristic to tell
NSI from non-NSI is, as in neutronization and normal hierarchy,
the comparatively large probability for $\nu_e$, $\bar\nu_e$ at low energy ($E_0\lesssim 0.8-0.9MeV$) with the
same energy profile. Such low energy raises however an experimental challenge for detection. For inverse
hierarchy $\nu_e$'$s$ and $\bar\nu_e$'$s$ remain equally abundant both for $E_0\lesssim 0.8-0.9MeV$ and larger,
therefore their detection is acessible, although NSI and non-NSI are indistinguishable in this case (see
fig.\ref{fig2}b). In particular $\bar\nu_e$'$s$ provide a clear signal through the reaction $\bar\nu_e p
\rightarrow ne^+$ whose cross section is $O(10^2)$ larger than for scattering with electrons.
Since thermalization is a much longer process than neutronization ($>$10 s), a larger accumulation of events
is possible in this phase.
For the detection and measurement of the $\nu_{\mu}$, $\bar\nu_{\mu}$, $\nu_{\tau}$, $\bar\nu_{\tau}$ individual
energy spectra which can only be traced via neutral currents, an interesting proposal was presented some time
ago \cite{Beacom:2002hs} and recently revived \cite{Dasgupta:2011wg}. It is based on the $\nu~p\rightarrow
\nu~p$ scattering reaction which can be observed in scintillator detectors (e.g. Borexino, SNO+, KamLAND)
through their adequate preparation. This is a neutral current process with a cross section about $O(10^2)$
larger than neutrino electron scattering at supernova neutrino energies. For the NSI scenario expound in the
present paper this technique will be particularly useful, since it appears to be possible to clearly distinguish
between normal and inverted hierarchies. In fact it suffices to note that for normal hierarchy the above
mentioned neutrinos arrive copiously on Earth in comparison with the more rare $\nu_e$$'$s and $\bar\nu_e$$'$s
whereas for inverse hierarchy all species arrive in comparable numbers (see fig.\ref{fig2}a, b).
\section{Summary and conclusions}
We have extended to the supernova the previously developed model for neutrino NSI in the sun introduced earlier
to remove the tension between the LMA predictions and the experimental signatures of solar neutrinos, especially
the absence of an upturn in the SuperKamiokande event rate. Improving the data fittings in the solar case
implies neutrino decay in dense matter into antineutrino and a majoron, hence the motivation to investigate
the consequences of the model for the supernova.
In the present paper we found however that, although the matter density in supernova is much larger than
in the sun, the extension of neutrino trajectory in the very high density medium is too short to imply
a significant neutrino decay into antineutrino and a majoron and the corresponding appearance probability is
insignificant. In the initial and short neutronization phase (10 ms) where only neutrinos are produced
through the reaction (\ref{neutron}) no antineutrinos are expected experimentally whether or not NSI applies.
The important NSI trace is the $\nu_e$ appearance probability which increases from 0.02 to 0.31 at low
energies ($E_0\lesssim 0.8-0.9MeV$) for normal hierarchy, while for inverse hierarchy NSI and non-NSI
cannot be distinguished. In this case the $\nu_e$ probability remains at 0.31 regardless of the energy.
In the neutronization (deleptonization) phase $\nu_e$'$s$ are the only states that can induce charged
current interactions, so they can be singled out either through an increased $\nu e^{-} \rightarrow
\nu e^{-}$ scattering event rate or $\nu_e d\rightarrow ppe^{-}$. The remainder ($\nu_{\mu}$'$s$,
$\nu_{\tau}$'$s$) inducing only neutral currents, cannot be distinguished from each other nor from
$\nu_{e}$'$s$. Detecting these $\nu_e$'$s$ remains however an experimental challenge at present in
normal hierarchy, but not so for inverse hierarchy, as they appear more copiously at higher energies.
In the subsequent and longer thermalization phase, the extra neutrino and antineutrino states ($\bar\nu_e$ and $\nu_X,
\nu_{\overline X}$ with $X=\mu,\tau$) that are produced through processes (2)-(5) cannot change the appearance
probabilities relative to the neutronization phase. Hence detecting these $\nu_e$'$s$ and $\bar\nu_e$'$s$ is,
again, an experimental
challenge in normal hierarchy: as for the neutronization phase $\nu_e$'$s$ can be detected through a major event
rate increase originated from the charged current in $\nu e^{-}\rightarrow \nu e^{-}$ scattering or through the
reaction $\nu_e d
\rightarrow p p e^{-}$, while $\bar\nu_e$'$s$ through the clear signal $\bar\nu_e p \rightarrow n e^{+}$. The
remaining neutrinos and antineutrinos can only be detected through the neutral current and so cannot be distinguished
from $\nu_e$'$s$ and $\bar\nu_e$'$s$, unless the interesting technique proposed in \cite{Beacom:2002hs},
\cite{Dasgupta:2011wg} is developed. If and when this advancement succeeds, it may be possible within the present
scenario to tell normal from inverse hierarchy. As regards collective oscillations, their effect in our analysis amounts
to the modification of the neutrino and antineutrino spectral fluxes. Collective effects are however probably suppressed
up to 0.2 seconds after bounce. The results obtained are therefore applicable to the neutronization (deleptonization)
phase and part of the subsequent thermal phase.
To summarize, in the presence of NSI we expect a sizable flux of $\nu_e$'$s$ and $\bar\nu_e$'$s$ at all energies
for inverse hierarchy and at low energy ($E_0\lesssim 0.8-0.9MeV$) for normal hierarchy. These fluxes are the same
as for non-NSI. The clear distinction between NSI and non-NSI is possible only for normal hierarchy at low energy
with a more intense flux of $\nu_e$ and $\bar\nu_e$, whose detection is at present an experimental challenge.
Hence in the absence of NSI the chances for observation in normal hierarchy of a charged current signal do not
appear much favourable at present, but they will of course mainly depend on the detector
size and supernova distance. Other neutrinos and antineutrinos are in contrast abundantly
present, however they can only be detected through the neutral current. As in
the case of the sun the antineutrino appearance probability from NSI neutrino decay is in all cases too small for
antineutrinos to be detected from this origin.
|
2,877,628,090,137 | arxiv | \section{Introduction}
Where they exist at all, current models for variability of parallel workloads
on HPC systems implicitly assume I/O variability follows a normal distribution
with the mean and standard deviation the only measure of interest
\cite{evans03,kramer03,skinner05,lofstead10,pusukuri12}. An attempt to fit the
tail of task duration to the log-normal distribution has also been made
\cite{wright09} with limited success. \cite{kim15,haque15} point out that
lowering latency for a given service increases competitiveness of that service.
Their work focuses on reducing the tail latency of a parallel task by reducing
the latency of the individual tasks that makeup the parallel task. Beyond these
studies on parallel workloads, there are an increasing number of phenomena in
computer science and beyond that are best modeled by methods of extreme
statistics
\cite{glynn95,asmussen1998,antonio01,choe98,mink09,coles2001,thomasian04,asmussen2008,andersen2007,bhavsar85,lahyani2012,schroeder07,bramwell09,moloney2010}.
\section{Model}
The modern theory of extreme value distributions can be traced back to the
1920's and two mathematicians: Fisher and Tippett. They considered
\cite{fishertippett1928} extreme values of $n$ samples, each of size $m$ drawn
from the same underlying population. Provided the population values are
independent and identically distributed (i.i.d.), they showed that the
distribution of the extreme values (smallest or largest) drawn from
sufficiently large sub-samples, which in turn are drawn from a larger sample,
tended to one of three possible unique asymptotic forms. For a given underlying
distribution e.g. the exponential, the extremal distribution will be one of the
three, in this case the Gumbel distribution (the others are Fr\'echet, to which
the extremes of power laws are attracted, and the Weibull, also well known in
failure rate modeling for example.) The probability density function of the GEV
with location $\mu$, scale $\sigma$, and shape $\xi$ is:
\begin{equation}
P_{\mu,\sigma,\xi}(x) =
\begin{cases}
\exp \left( - \left( 1+\xi \left(\frac{x-\mu}{\sigma}\right)\right)^{-1/\xi} \right) & \text{if } \xi \neq 0 \\
\exp \left( - e^{(\frac{x-\mu}{\sigma})}\right) & \text{if } \xi = 0
\end{cases}
\label{eqn:gevpdf}
\end{equation}
A detailed description, and physical examples of extreme value theory are
presented in \cite{leadbetter1983,coles2001,sornette04}. Next, we choose a
common an simple parallel task (a write to a parallel file system) and argue
that the i.i.d. assumption needed for GEV behavior are directly applicable as
follows:
\begin{longdescription}
\item[The storage nodes are independent.] A storage node is here defined as a
device that receives a portion of a file during a parallel write. While it is
common to collect multiple devices into a storage array, our model treats an
array as a single storage node that is independent from other arrays.
\item[A write task takes place from a single node to many storage nodes.] Of
the many I/O scenarios enumerated in the article \cite{newman08}, this paper is
concerned with the duration to complete \emph{scenario 5: Checkpoint/restart
with large I/O requests}. This is also known as a `one-to-many' operation.
\item[The dominant source of variation within the system arises from the
storage nodes.] The non-dominant sources of latency in the system including:
network switches, network cards, interrupts, kernel buffers, PCI interfaces,
OS schedulers, memory latency etc are all assumed to be comparatively
small.
\item[The client node is connected to each of the storage nodes by an
identical network connection.] The network connections connecting the client and
storage nodes are identical in bandwidth and latency.
\end{longdescription}
\section{Experiment}
A quantity of interest to many in HPC is the duration of
time to complete a given task. Our chosen task is a write operation on a
parallel file system with a duration of $T_g$. We assume that there is a
baseline characteristic of the parallel task duration that is observable on a
quiescent system without congestion $T_s$. Congestion is a important factor in
network operations \cite{evans03,thomasian04,calzarossa04} that arises with a
shared network or the storage nodes that are busy with other tasks. We encode
the congestion penalty (which we call background traffic factor) as a constant
of proportionality $k_t$. This gives: $T_g = k_t T_s$. A completely quiet
system without congestion or background traffic is the state where $k_t=1$. If
background traffic is present, $k_t>1$.
We extend our model with the assumptions: an observed file transfer to a
\emph{single} storage node will take $S$ seconds where $S$ is an observation of
the storage node that behaves with a given probability distribution: $p(s)$.
Hence the time taken $T_s$ for the storage nodes to complete a \emph{parallel}
write in our model is the largest value of $S$ from $m$ storage nodes: $T_s =
\max\{S_1, S_2 ... S_m\}$. By substitution, we arrive at:
\begin{equation}
T_g = k_t\max\{S_1, S_2 ... S_m\}.
\label{eqn:writemodel}
\end{equation}
i.e. a client will observe a write time onto a parallel file system that is
limited by the last storage node to complete the task: $T_g =
P_{\mu,\sigma,\xi}(x)$ from equation \eqref{eqn:gevpdf}.
From Extreme Value Theory, provided $m$ is sufficiently large and with our
additional constant traffic constraint ($k_t$ is constant across observations),
we construct the following testable hypothesis: the times taken to transfer a
file onto a large number of storage nodes will have a distribution approximated
a random variable that has a extreme value distribution, given a fixed level of
background traffic (congestion) and our previously stated assumptions of the
system hold true.
An investigation to explore the distribution $T_g$ was initially conducted at
TACC on the Ranger system. Encouraging results were obtained. However, these
results were identified as unreliable
because the experimental run used {\tt dd} with a block size of more
than 2GB. For some configurations (apparently including Ranger), {\tt dd}
will stop writing after 2GB and return success. This initial data was
discarded. An experimental run was subsequently completed on both
Stampeed and Lonestar4 without success: these machines did not include the i.i.d.
assumptions previously stated.
A second experiment was designed and conducted on the Amazon
Web Services (AWS) public cloud. Cloud based computing has grown in popularity
as a inexpensive tool for research, and performance evaluations are an area of
active research \cite{yao13,bautista12,iosup11,wang10}. AWS allows dynamic
construction of arbitrary configurations as well as isolated network
environments - necessary to ensure constant $k_t$ in our model. For a
completely isolated network with a single client running a single job, $k_t$ =
1.
Amazon Web Services provide basic specifications of the network and storage
performance. They state a throughput of 128 MBps per volume
\footnote{\protect\url{http://aws.amazon.com/ebs/details/}}, 62.5MBps per
instance for write
\footnote{\protect\url{http://docs.aws.amazon.com/AWSEC2/latest/UserGuide/ebs-ec2-config.html}}.
The dynamically constructed cluster was created within a `placement group'
\footnote{\protect\url{http://docs.aws.amazon.com/AWSEC2/latest/UserGuide/placement-groups.html}}.
This is a logical group of instances that enables applications to participate
in a low-latency, 10Gbps network.
Published values for the throughput of c3.large storage servers could not be
obtained. The maximum theoretical bandwidth of a 10Gbps network is 1250 MBps.
The mean value observed in our experiment is 45MBps. From these calculations it
would appear that the instance throughput (possibly on the client) is the
bottleneck in our system configuration.
Our experiments are performed on the Lustre\footnote{Other names and brands may
be claimed as the property of others.} parallel file system version 1.8.9-wc1.
While more recent Lustre software releases are available, using synchronous
write in our experiment prohibited versions of Lustre that do not have a fix
for LU-1669. At the time (Autumn 2015), 1.8 was the most popular Lustre
version that supported parallel direct write. In addition, previous variability
papers have chosen 1.8 for their studies. To avoid complications with caches,
only synchronous write operations are considered in this study. The design of
the Lustre file system version 1.8 requires a serialized meta-data request to
open and close the file. We use a simple code (provided in the appendix) that
measures the time for serialized meta-data requests separately to the parallel
data transfer request. Our experiment defines a single write as a total file
size of 512 MB written to 16 storage nodes. The default stripe size of 1MB was
used. Choosing a files size of 512 MB ensures the file is small enough to fit
in the client memory (total of 7.5GB) without needing costly swapping. 16
storage nodes is chosen as a sufficiently large population ($m$) and a total of
400 observations made to ensure sufficient fidelity of the underlying
distribution and increase confidence of correct identification \cite{henwood08}
Specific compute instance (EC2) types and Elastic Block Store (EBS) were chosen
as shown in Figure 1.
The cluster was constructed behind a head
node (not shown) in a private subnet within a placement group. The EC2
instances were shared tenancy. All instances in the experimental setup were
CentOS 5.11 with Lustre 1.8.9-wc.
\begin{figure}
\centering \def\svgwidth{6.5cm}
\input{1-16_model.pdf_tex}
\caption{A typical high performance storage architecture with a single client
node $C_1$. Storage targets (1-16) are attached in groups of four to storage
servers. A read or write operation from $C_1$ occurs across all storage targets
in parallel. A write operation includes the following high level steps:
\protect\circled{1} $C_1$ executes a single task and accumulates results in
memory until the task is complete.
\protect\circled{2} $C_1$ requests a file handle from the metadata server. The
metadata server persists data on storage (labelled `MDT') and instructs the
client to write to all the storage nodes during writing. From this point
onwards the system storage targets behave with i.i.d. characteristics.
\protect\circled{3} A timer begins on $C_1$. $C_1$ and the contents of the
memory is written to all the storage nodes as a synchronous write.
\protect\circled{4} The storage servers pass the data directly through to the
EBS storage nodes (1-16).
\protect\circled{5} The timer is stopped when $C_1$ is told that the write is
complete. The value of the timer is $T_g$.
}
\label{fig:aws_setup}
\end{figure}
\section{Results}
Figure 2
shows the duration of a parallel write is best
approximated by equation \eqref{eqn:writemodel}. This results supports the hypotheses
that the duration of a parallel write is controlled by the slowest node. GEV
distributions are defined by three parameters: location, scale, and shape. The
result of our work indicates that all three are valuable in capturing the
variability characteristics of a system. HPC performance variability data first
published in \cite{kramer03,skinner05} may now be better explained using the
GEV model. \cite{schwarzkopf12} (and references therein) highlight the under
appreciated importance, and poor level of understanding of variability, within
cloud computing environments. Our results present a model that will provide for
a deeper understanding of variability on both the cloud and HPC.
\begin{figure}
\begin{center}
\resizebox {.75\columnwidth} {!} {
\input{400_runs.tikz}
}
\caption{
Parallel write times follow extreme statistics.
400 consecutive observations of $T_g$ were taken. The top panel shows the cumulative
value of the observation against the model value. The middle panel is the
observed quantity plotted against the modeled quantity with the 95\% confidence
interval of the value of $\xi$ shown as a blue line. Observations that fall
outside the 95\% confidence interval are colored in red. The bottom panel
presents the observation histogram in 20 equal width bins with the fitted
probability density over-plotted. The GEV fit has location $\mu = 11.1679 \pm
0.0140$, scale $\sigma = 0.2120 \pm 0.0101$, and shape $\xi = -0.00105 \pm
0.0415$. Values of $\mu$, $\sigma$, $\xi$, standard errors, and outliers were
calculated using the {\tt{ismev}} library \cite{ismev} within the R language
environment \cite{rmanual}.}
\label{fig:400obsfits}
\end{center}
\end{figure}
\section{Conclusions}
From extreme value theory, as the number of nodes increases we anticipate a
universal behavior will emerge in systems of this type. We can confirm that
with the conditions already stated, this is the case in our system (Figure 2
). Our idealized experiment has wider implications as it maps onto a large
class of systems, both physical and societal, where the essential element is
waiting for a response in parallel from any nodes. In the computing field, for
example, the Monte Carlo method is widely used and deployed at parallel scale
and under certain configurations, the time to result would be expected to have
a GEV distribution.
A complete, efficient, and accurate model of an HPC system is critical in
optimizing utilization of this limited resource. Queues have already
successful modelling a number of components of an HPC system including task
scheduling \cite{thomasian04,terekhov13}, network systems
\cite{lahyani2012}, and failure and recovery
\cite{asmussen2008}. Our GEV model for parallel
transfer grows the tools available to a model an entire, active, HPC cluster.
The specter of traffic or network congestion is often introduced when looking
at variability in benchmark measurements. If we are benchmarking a parallel
task, and the GEV model is accurate, we expect a tail in the variability even
in the complete absence of traffic. After the underlying variability of a
parallel workload characterized, the affect of network congestion on the same
workload can now be quantified.
As high performance computing continues to develop and increase parallelism,
new libraries become available (and necessary), to simplify interfacing with
data objects \cite{barton13}. For example, the {\tt{t3pio}} library provides
automatic configuration for MPI applications that use HDF5. With the GEV model,
a library can be calibrated for ideal parallel (GEV) behavior and measure
deviations from this behavior as values that are unlikely. The journey to
exascale computing means vast increases node count and parallelism
\cite{agerwala10}. We expect GEV to be a powerful tool in understanding and
exploiting variability on HPC systems in the future.
In summary, this paper explains the variability in parallel writes. The
variability is explained by extreme value theory. Our analysis of data
collected from a parallel write task performed in the public cloud found good
agreement with well understood extreme statistics. Studies of parallel tasks
should perhaps begin to consider examining repeated runs for evidence of
extreme value distribution as a unique parallel performance signature.
\section{Acknowledgments}
RH is grateful to Intel High Performance Data
Division who supported this work. RH thanks various anonymous
reviewers who significantly improved this manuscript. SCC is supported by the
UK EPSRC and STFC.
\bibliographystyle{abbrv}
|
2,877,628,090,138 | arxiv | \section{Introduction}
The area of axiomatic truth theories analyses the concept of truth by studying first-order theories which try to capture various properties of this notion. These theories are formulated as follows: We choose a base theory strong enough to represent syntax (this is typically Peano Arithmetic, $\PA$). To this theory, we add a fresh predicate $T(x)$ whose intended reading is ``$x$ is (a code of) a true sentence'' together with axioms governing the behaviour of that predicate. A notable example of such a theory is $\CT^-$ which stipulates that the truth predicate satisfies Tarski's compositional conditions for arithmetical sentences. For instance, a conjunction $\phi \wedge \psi$ is true iff both $\phi$ and $\psi$ are true.
If we add to $\CT^-$ full induction for the formulae containing the truth predicate, the resulting theory, called $\CT$, is not conservative over $\PA$, that is, it proves arithmetical theorems which cannot be demonstrated in $\PA$ itself. More specifically, we can show by induction on the lengths of proofs that every sentence provable in $\PA$ is true (a principle called global reflection over $\PA$) and, consequently, that $\PA$ is consistent.\footnote{This is one of the basic results of truth theory. A comprehensive introduction to the area, including this result, can be found in \citep{halbach}.}
On the other hand, by a theorem of Kotlarski, Krajewski, and Lachlan (see \citep{kkl}) $\CT^-$ itself does not prove any new arithmetical theorems.
In \citep{cies}, it was shown that already $\CT^-$ extended with a principle of propositional reflection ``sentences derived in propositional logic from true premises are true'' suffices to prove global reflection over $\PA$. In other words, an overtly nonconservative principle can be derived without explicitly assuming induction on a ground of a rather innocuous principle of overtly truth-theoretic nature. Subsequently, it turned out that a number of other truth-theoretic principles are not conservative over $\PA$ and are all equivalent to $\Delta_0$-induction for the compositional truth predicate ($\CT^-$ with $\Delta_0$-induction is called $\CT_0$).\footnote{ For the proof of nonconservativity of $\CT_0$, see \citep{wcislyk}.} The picture that emerged was that there seems to be a ``minimal natural'' nonconservative extension of $\CT^-$.
The ``dividing line'' between conservative and nonconservative extensions of $\PA$ has been named ``Tarski boundary'' by Ali Enayat.\footnote{One can find more information on the Tarski boundary in \citep{cies_ksiazka}. A more concise discussion is also contained in \citep{lelyk_wcislo_studia} and \citep{lelyk_studia}.} One of the most striking results on Tarski boundary was obtained by \cite{EnayatPakhomov}. It was shown that among theories equivalent to $\CT_0$ is $\CT^-$ together with the principle of disjunctive correctness, $\DC$, which states that any finite (but possibly nostandard) disjunction is true iff one of the disjuncts is true. This axiom appears to be a mild, natural, extension of the compositional clauses and yet turns out to carry the full strength of $\Delta_0$-induction.
It was not clear whether this result can be pushed further in the following manner: Disjunctive correctness can be naturally split into two halves. The first half is a principle $\DCin$ saying ``a disjunction with true disjunct is true'' and the second is $\DCout$ which says ``a true disjunction has a true disjunct''. It has been asked in \citep{EnayatPakhomov} whether the second of these principles can be added conservatively to $\PA$.\footnote{See Question 5.3 in \citep{EnayatPakhomov}. The conservativity of $\DCin$ was settled in December 2018 and stated in the formulation of that question, but the proof was not published.}
In this article, we analyse both of the above principles. We show that over $\CT^-$, $\DCin$ gives rise to a conservative extension of $\PA$ while $\DCout$ yields $\Delta_0$-induction. The methods used for the nonconservativeness result yield a new, direct proof that $\DC$ is yet another incarnation of $\Delta_0$-induction for the formulae containing the truth predicate.
\section{Preliminaries} \label{sec_prelim}
We consider truth theories over Peano arithmetic, $\PA$, as our base theory.\footnote{The choice is motivated mostly by a certain tradition in the field. However, the results make sense and still hold true over much weaker theories, say $\IDelta_0 + \exp$.} It is an axiomatic theory in a language $\LPA = \{S, +, \times,0\}$ whose axioms consist of inductive definitions of addition and multiplication in terms of the successor function $S$ together with the full induction scheme. Although this theory overtly speaks of natural numbers, in fact it is strong enough to capture objects such as finite sets, finite sequences, or finite graphs. Crucially, $\PA$ is capable of expressing syntactic notions such as ``term'', ``formula'' or ``proof'' and proving basic facts about these notions such as ``a conjunction of two formulae is a formula.'' We assume that the reader is familiar with the coding of syntax and the basic metamathematics. This is discussed in many sources such as \citep{kaye}, Chapter 9 or \citep{hajekpudlak}, Chapter I, Section 1(d), pp.50--61. Throughout the article, we will explain those bits of notation that seem not to explain themselves. We provide a glossary of all formalised notions we are using throughout the paper in the appendix.
\begin{convention} \label{conv_syntax} \
\begin{itemize}
\item We will often conflate G\"odel numbers of syntactic objects with those objects. We will also use formulae denoting syntactic expressions as if they were sets. For instance, we will write $\phi \in \Sent_{\LPA}$ rather than $\Sent_{\LPA}(\phi)$, where $\Sent_{\LPA}$ expresses that $\phi$ is (a G\"odel code of) an arithmetical sentence.
\item Provably functional formulae will be used as if denoting actual functions. For instance, we will write $\FV(\phi)$ for the set of free variables of $\phi$.
\item In particular, we will often write the results of syntactic operations without explicitly mentioning the operations themselves. For instance, if $\phi, \psi \in \form_{\LPA}$, we will freely speak of the conjunction $\phi \wedge \psi$ rather than ``the only $z$ such that $z$ is a conjunction of $\phi$, $\psi$.''
\end{itemize}
\end{convention}
The research on Tarski boundary concerns extensions of the compositional truth theory. Let us define it. By writing $\num{n}$, we mean (the G\"odel code of) a \df{numeral} denoting the number $n$, that is,
\begin{displaymath}
\qcr{\underbrace{S(S\ldots S(0) \ldots)}_{\textnormal{``$S$'' repeated $n$ times}}}.
\end{displaymath}
If $t$ is (a G\"odel code of) a closed arithmetical term, then by $\val{t}$ we mean the value of that term. Thus we have, for instance,
\begin{displaymath}
\mathbb{N} \models \val{\qcr{S0 + S(S(0))}} = \val{\num{3}} = 3 .
\end{displaymath}
Note that a value of an arithmetical term can be computed in a primitive recursive way and the natural function computing it may be formalised in $\PA$. Now we can proceed to the actual definition.
\begin{definition} \label{def_CTminus}
By $\CT^-$ (Compositional Truth) we mean a theory in the arithmetical language with a fresh unary predicate $T$ extending $\PA$ with the following axioms:
\begin{itemize}
\item $\forall x \ \Big( T(x) \rightarrow \Sent_{\LPA}(x)\Big).$
\item $\forall s,t \in \ClTerm_{\LPA} \Big(T(s=t) \equiv \val{s} = \val{t}\Big).$
\item $\forall \phi \in \Sent_{\LPA} \Big(T(\neg \phi) \equiv \neg T \phi\Big).$
\item $\forall \phi,\psi \in \Sent_{\LPA} \Big(T(\phi \vee \psi ) \equiv T\phi \vee T \psi\Big).$
\item $\forall \phi,\psi \in \Sent_{\LPA} \Big(T(\phi \wedge \psi ) \equiv T\phi \wedge T \psi\Big).$
\item $\forall \phi \in \form_{\LPA} \forall v \in \Var \Big(\Sent_{\LPA}(\exists v \phi) \rightarrow T(\exists v \phi) \equiv \exists x \ T\phi(\num{x})\Big).$
\item $\forall \phi \in \form_{\LPA} \forall v \in \Var \Big(\Sent_{\LPA}(\forall v \phi) \rightarrow T(\forall v \phi) \equiv \forall x \ T\phi(\num{x})\Big).$
\item $\forall \phi \in \form_{\LPA} \forall \bar{s}, \bar{t} \in \ClTermSeq_{\LPA} \Big(\val{\bar{s}} = \val{\bar{t}} \rightarrow T\phi(\bar{s}) = T\phi(\bar{t}) \Big).$
\end{itemize}
Above, $\ClTermSeq_{\LPA}(x)$ is a formula expressing ``$x$ is a sequence of closed arithmetical terms.'' The rest of the notation should be self-explanatory and, as we remarked earlier, it is discussed in the appendix.
By $\CT$, we mean $\CT^-$ with full induction for the extended language. By $\CT_0$, we mean $\CT^-$ with induction for $\Delta_0$ formulae containing the truth predicate. (Note that already $\CT^-$ contains full arithmetical induction.)
\end{definition}
The last clause in the above axioms for $\CT^-$ is called \df{Regularity Axiom}, $\REG$. It is not included among the basic axioms in the standard presentations of this theory, like \citep{halbach} or \citep{cies_ksiazka}. The version of $\CT^-$ with $\REG$ appears, e.g., in \citep{enayatlelykwcislo} or \citep{lelyk_wcislo_local_collection}. Admittedly, the regularity axiom has a clearly different status than the rest of the axioms of $\CT^-$. We add it mostly for two (admittedly technical) reasons which we will explain in Section \ref{sec_regularity}.
Although $\CT^-$ seems to express the crucial properties of the truth predicate, it is not arithmetically stronger than $\PA$. The result was essentially proved by \cite{kkl}.\footnote{The original result concerns satisfaction classes over a purely relational language. This work has been extended in \citep{kaye} to languages with terms and in \citep{engstrom_thesis} to truth classes over languages with terms. Subsequently, \cite{enayatvisser2} introduced a new, elegant and flexible method of constructing satisfaction classes which allowed them to strengthen the previous results in a number of interesting ways. They worked again in purely relational languages. A version for functional languages can be found in \citep{cies_ksiazka}. A version of Enayat--Visser construction covering languages with functional symbols with the regularity axioms included is discussed in \citep{lelyk_wcislo_local_collection}. A reader may find a proof of a stronger result also in this article in Section \ref{sect_dcin}.
}
\begin{theorem}[Kotlarski--Krajewski--Lachlan] \label{tw_kkl}
$\CT^-$ is conservative over $\PA$, i.e., for any sentence $\phi \in \LPA$ if $ \CT^- \vdash \phi$, then $\PA \vdash \phi$.
\end{theorem}
The above result contrasts with the situation for $\CT$. By induction on the length of proofs, we can show that every proof in $\PA$ has true conclusion. In other words, we can prove in $\CT$ the following principle of \df{Global Reflection}, $\GR$:
\begin{displaymath}
\forall \phi \in \Sent_{\LPA} \Big( \Pr_{\PA}(\phi) \rightarrow T \phi \Big).
\end{displaymath}
Notice that $\CT^- + \GR$ is clearly not conservative over $\PA$, since in particular $\neg T (0 \neq 0)$ is provable in the compositional theory $\CT^-$ and, by contraposition $\neg \Pr_{\PA}(0 \neq 0 )$. The latter is not provable in $\PA$ by G\"odel's second theorem.
As we have already mentioned, a number of seemingly unrelated principles turned out to be equivalent to this canonical nonconservative axiom $\GR$. One of them is $\Delta_0$-induction for the compositional truth predicate. Another principle which will play an important role in this article is \df{Propositional Reflection}, $\PropRef$, defined as follows:
\begin{displaymath}
\forall \phi \in \Sent_{\LPA} \Big(\Pr^{T}_{\Prop}(\phi) \rightarrow T \phi \Big),
\end{displaymath}
where $\Pr_{\Prop}^{T}(x)$ means that $x$ is derivable in propositional logic from the set of premises $\Gamma$, such that $T(y)$ holds for each $y \in \Gamma$. As we have already noted, \cite{cies} showed that over $\CT^-$, $\PropRef$ is equivalent to $\Delta_0$ induction for the truth predicate. Subsequently, $\CT_0$ was shown by \cite{wcislyk} to be arithmetically equivalent to $\GR$ and then, in \cite{lelyk_thesis}, to be exactly the same theory as $\CT^- + \GR$. Another presentation of the last result can be also found in a recent preprint \citep{lelyk_global_reflection}.
Related to propositional reflection is the following principle of \df{Propositional Soundness}, $\PropSnd$:
\begin{displaymath}
\forall \phi \in \Sent_{\LPA} \Big(\Pr_{\Prop}(\phi) \rightarrow T\phi \Big).
\end{displaymath}
In effect, this axiom expresses that any arithmetical sentence which is a propositional tautology is true. It is still unknown whether $\CT^- + \PropSnd$ is conservative over $\PA$. In the next section, we shall present a partial result towards this problem.
Now, let us turn to the main subject of our article. If $M \models \PA$ and $\bar{\phi} \in \SentSeq_{\LPA}(M)$ is a coded sequence of sentences, we can form their disjunction $\bigvee \bar{\phi}$, which we will also denote by $\bigvee_{i \leq c} \phi_i$ if the length of $\bar{\phi}$ is $c$. We always assume that in ``big disjunctions'' parentheses are grouped to the left, so that the following equality holds in $M$:
\begin{displaymath}
\bigvee_{i \leq c+1} \phi_i = \bigvee_{i \leq c} \phi_i \vee \phi_{c+1}.
\end{displaymath}
In effect, $\bigvee_{i \leq c} \phi$ denotes the following formula:
\begin{displaymath}
(((\phi_0 \vee \phi_1) \vee \ldots )\vee \phi_{c-1}) \vee \phi_c.
\end{displaymath}
The precise definition of how disjunctions over multiple disjuncts are parenthesised can actually matter in some cases. In the presence of $\Delta_0$-induction for the extended language, one can show that any two disjunctions with the same disjuncts are equivalent, no matter how the disjuncts are ordered and grouped together. However, this is not the case in pure $\CT^-$. The precise parenthesising can become relevant, since it dictates the relations between disjunctions over arbitrarily many disjuncts and the usual binary disjunctions. We will return to this issue in Section \ref{sect_parentheses}.
By \df{Disjunctive Correctness}, $\DC$, we mean the following axiom:
\begin{displaymath}
\forall \bar{\phi} \in \SentSeq_{\LPA} \bigg(T \left( \bigvee \bar{\phi} \right)\equiv \exists i \leq \lh(\bar{\phi}) \ T \phi_i \bigg).
\end{displaymath}
Therefore, $\DC$ states that a finite (but possibly nonstandard) disjunction is true iff it has a true disjunct. Very surprisingly, $\DC$ is yet another incarnation of $\CT_0$ as shown by \cite{EnayatPakhomov}.
\begin{theorem}[Enayat--Pakhomov] \label{th_dc_equiv_ct0}
$\CT^- + \DC$ and $\CT_0$ are equivalent.
\end{theorem}
As we mentioned, $\CT_0$ is not conservative over $\PA$. Hence the following fact easily follows:
\begin{corollary}[Enayat--Pakhomov] \label{cor_dc_not_conservative}
$\CT^- + \DC$ is not conservative over $\PA$.
\end{corollary}
Disjunctive correctness can be naturally split into two principles. The first is $\DCout$ (``a true disjunction has a true disjunct''):
\begin{displaymath}
\forall \bar{\phi} \in \SentSeq_{\LPA} \Big(T\bigvee \bar{\phi} \rightarrow \exists i \leq \lh(\bar{\phi}) \ T \phi_i \Big).
\end{displaymath}
The second is $\DCin$ (``a disjunction with a true disjunct is true''):
\begin{displaymath}
\forall \bar{\phi} \in \SentSeq_{\LPA} \Big( \exists i \leq \lh(\bar{\phi}) \ T \phi_i \rightarrow T\bigvee \bar{\phi} \Big).
\end{displaymath}
As we have mentioned, the status of $\DCout$ was not settled in \citep{EnayatPakhomov} and the claim that $\DCin$ is conservative was stated without a proof. The subsequent parts of this article will be devoted to the analysis of both principles.
Let us finish this section by summing up the positive results on equivalences of theories of truth relevant for this work:
\begin{theorem} \label{th_many_faces}
The following theories are equivalent:
\begin{enumerate}
\item $\CT_0$.
\item $\CT^- + \GR$.
\item $\CT^- + \PropRef$.
\item $\CT^- + \DC$.
\end{enumerate}
\end{theorem}
\section{Yablo sequences and disjunctive correctness} \label{sec_dcout}
In this section, we prove that $\DCout$ is equivalent to $\CT_0$. The argument is indirectly inspired by the classical Visser--Yablo paradox: \footnote{See \citep{yablo}.}
Consider the sequence of sentences $Y_n, n \in \mathbb{N}$, such that $Y_n$ says: ``some sentence $Y_k$ for $k > n$ is false.'' If for some $k$, $Y_k$ is false, then every sentence $Y_l$ for $l>k$ is true. However, if $Y_l$ is true, then there is some $m>l$ such that $Y_m$ is false contradicting the assumption. Hence, every sentence in the Yablo sequence is true. However, if some sequence in the Yablo sequence is true, then some has to be false, so they cannot be all true. We reach a contradiction.
Using a construction inspired by the Yablo sequence, we will show that $\DC$-out implies $\CT_0$. We will actually use two intermediate principles which are overtly related to $\Delta_0$-induction. By \df{sequential induction}, $\SeqInd$, we mean the following axiom:\footnote{A related principle of \df{Modus Ponens correctness} was introduced earlier by Ali Enayat in an unpublished note \citep{enayat_fine_tuning}. The principle states that if a conjunction of the implications $\phi_{i} \rightarrow \phi_{i+1}$ is true for $i = 0,1, \ldots, c$ and $\phi_0$ is true, then $\phi_{c+1}$ is true. However, as stated, this principle is conservative over $\PA$. Namely, it holds in any model in which all conjunctions of nonstandard length are false and by a construction very similar to that presented in Section \ref{sect_dcin}, for any completion $U$ of $\PA$, we can find models with this property satisfying $U$.}
\begin{displaymath}
\forall s \in \FinSeq \ \Big(Ts_0 \wedge \forall i < \lh(s) - 1 \big(T s_i \rightarrow T s_{i+1} \big) \rightarrow \forall j < \lh(s) \ Ts_j \Big)
\end{displaymath}
The \df{sequential order induction}, $\SeqOInd$, is a natural variant of the above principle:
\begin{displaymath}
\forall s \in \FinSeq \ \Big(\forall j < \lh(s) \big((\forall i<j Ts_i) \rightarrow T s_j \big) \rightarrow \forall l < \lh(s)\ Ts_l \Big).
\end{displaymath}
As we already mentioned, $\SeqInd$ and $\SeqOInd$ are clearly related to $\Delta_0$-induction:
\begin{proposition} \label{prop_seqoind_equiv_ct0}
$\CT^- + \SeqOInd$ and $\CT_0$ are equivalent.
\end{proposition}
\begin{proof}
$\CT_0$ clearly entails $\SeqOInd$. On the other hand, observe that $\SeqOInd$ implies $\PropRef$ which by Theorem \ref{th_many_faces} is equivalent to $\CT_0$. Working in $\CT^- + \SeqOInd$, fix a proof $(\phi_0, \ldots, \phi_c)$ in propositional logic from true premises. We can assume that the proof system is chosen so that for all $i$, either $\phi_i$ is an assumption of the proof or $\phi_i$ is obtained by modus ponens from two formulas appearing earlier in the proof. In particular, for every $j$, if every formula $\phi_j$ for $j< i$ is true, then $\phi_i$ is true which, by $\SeqOInd$, implies that the conclusion of the proof is true. Thus $\PropRef$ holds.
\end{proof}
Now, we get to the core argument of the article:
\begin{theorem} \label{th_dcout_implies_sind}
Over $\CT^-$, $\DCout$ implies $\SeqInd$.
\end{theorem}
\begin{proof}
Working in $\CT^- + \DCout$, fix any sequence $(\phi_0, \ldots, \phi_c) \in \SentSeq_{\LPA}$ such that $T\phi_0$ holds and for each $i$, $T\phi_i$ entails $T\phi_{i+1}$. Let us define a sequence $\psi_i, i \leq c$ as follows:
\begin{eqnarray*}
\psi_0 & : = & \phi_0 \\
\psi_{j+1} & : = & \neg \phi_{j+1} \rightarrow \bigvee_{i < j+1} \neg \psi_i.
\end{eqnarray*}
We claim that for all $j \leq c$, the sentence $\psi_j$ is true. Suppose that $\neg T \psi_j$ holds for some $j$. Then $j>0$, so we have:
\begin{displaymath}
\neg T \phi_j \wedge \neg T \bigvee_{i<j} \neg \psi_i.
\end{displaymath}
By compositional conditions, this is equivalent to:
\begin{displaymath}
\neg T \phi_j \wedge \neg T \left( \bigvee_{i<j-1} \neg \psi_i \right) \wedge T \psi_{j-1}.
\end{displaymath}
Expanding the definition of $\psi_{j-1}$, we obtain:
\begin{displaymath}
\neg T\phi_j \wedge \neg T \left( \bigvee_{i< j-1} \neg \psi_i \right) \wedge \left (T \phi_{j-1} \vee T \left( \bigvee_{i<j-1} \neg \psi_i \right) \right).
\end{displaymath}
However, $\neg T \phi_j$ implies $\neg T \phi_{j-1}$. Therefore, we have the following:
\begin{displaymath}
\neg T\phi_j \wedge \neg T \left( \bigvee_{i< j-1} \neg \psi_i \right) \wedge T \left( \bigvee_{i<j-1} \neg \psi_i \right).
\end{displaymath}
This contradiction concludes the proof of the claim. Notice that the proof of the claim only uses the fact that provably in $\CT^-$,
\begin{displaymath}
T \left( \bigvee_{i \leq j+1} \eta_i \right) \equiv T \left( \bigvee_{i \leq j} \eta_i\right) \vee T\eta_{j+1}.
\end{displaymath}
Now, we show that for all $j<c$, $\phi_j$ is true. Suppose otherwise and fix $j$ such that $\neg T\phi_j$. Since $T\psi_j$ holds, we have:
\begin{displaymath}
T \bigvee_{i< j} \neg \psi_i
\end{displaymath}
By $\DCout$, we can fix $i<j$ such that $\neg T \psi_i$ holds. However, this contradicts the previous claim.
\end{proof}
As an application of the above result, we show that $\DCout$ is the same theory as $\DC$. In particular, $\CT^- + \DCout$ is not conservative over $\PA$.
\begin{theorem} \label{th_seqind_implies_dcin}
$\CT^- + \SeqInd$ implies $\DCin$. Consequently, $\DCout$ and $\DC$ are equivalent over $\CT^-$.
\end{theorem}
\begin{proof}
Working in $\CT^- + \SeqInd$ fix any sequence $(\alpha_0, \ldots, \alpha_c)$ such that $\alpha_i \in \Sent_{\LPA}$ for all $i \leq c$. Suppose that $\alpha_j$ is true for some $j \leq c$ and notice that
\begin{displaymath}
\bigvee_{i\leq j} \alpha_i = \bigvee_{i \leq j-1} \alpha_i \vee \alpha_j ,
\end{displaymath}
hence the disjunction of $\alpha_i$ up to and including $j$ has to be true as well. Moreover, for any $k$, if $\bigvee_{i \leq k} \alpha_i$ is true, then $\bigvee_{i \leq k+1} \alpha_i$ is true. Hence by sequential induction (starting with $\bigvee_{i \leq j} \alpha_i$), $T\bigvee_{i \leq c} \alpha_i$ holds.
\end{proof}
Another application is a more perspicuous proof of nonconservativity of $\CT^- + \DC$.
\begin{theorem} \label{th_dc_implies_seqoind}
$\CT^- + \DC$ implies $\SeqOInd$. Hence, $\CT^- + \DC$ is equivalent to $\CT_0$.
\end{theorem}
\begin{proof}
Working in $\CT^- + \DC$, fix any sequence $(\phi_0, \ldots, \phi_c)$ and suppose that for any $j$, if $T\phi_i$ holds for all $i< j$, then $T\phi_j$ holds. By $\DC$, the following implication holds for all $j \leq c$:
\begin{displaymath}
T \neg \bigvee_{i \leq j} \neg \phi_i \rightarrow T \neg \bigvee_{i \leq j+1} \neg \phi_i.
\end{displaymath}
(Of course, these are essentially big conjunctions, but \textit{prima facie}, disjunctive correctness does not imply conjunctive correctness.) Then, by $\SeqInd$, we conclude that
\begin{displaymath}
T \neg \bigvee_{i \leq j} \neg \phi_i
\end{displaymath}
holds for each $i$ which, again using $\DC$ implies that $T\phi_j$ holds for each $j$.
By Proposition \ref{prop_seqoind_equiv_ct0} it follows that $\CT_0 \subseteq \CT^- + \DC$. Since $\DC$ is easily provable in $\CT_0$, we conclude that $\CT^- + \DC$ is equivalent to $\CT_0$.
\end{proof}
\begin{remark} \label{rem_dc_via_int}
The main nonconservativeness proof for $\DC$ in \citep{EnayatPakhomov} consists of two parts: it is first shown that over $\CT^-$, $\DC$ implies the axiom of internal induction $\INT$, to be defined in the next section, and then a much more direct proof that $\DC + \INT$ implies $\CT_0$ follows. It is easy to verify that $\CT^- + \SeqInd$ entails internal induction and since we know that $\CT^- + \DC$ implies $\SeqInd$, we obtain a still different proof that $\CT^- + \DC$ is equivalent to $\CT_0$.
\end{remark}
We can present the above results in a slightly more abstract manner. This will allow us to obtain a significantly simpler proof of the result from \citep{wcislo_prop_plus_qf} on the strength of certain extensions of Propositional Soundness.
\begin{definition} \label{defi_outer_disjunction}
Let $U$ be a theory extending $\CT^-$. We say that $U$ has \df{outer disjunctions} if there exists a provably functional formula $D(x,y)$ such that for any $\bar{\phi} = \tuple{\phi_1,\ldots,\phi_c} \in \SentSeq_{\LPA}$, we have $D(\bar{\phi}) \in \Sent_{\LPA}$\ and the following two properties hold provably in $U$:
\begin{itemize}
\item $\forall \bar{\phi} \in \SentSeq_{\LPA} \forall \psi \in \Sent_{\LPA} \ \Big(T D(\bar{\phi}\frown\tuple{\psi}) \equiv TD(\bar{\phi}) \vee T\psi \Big).$
\item $\forall \bar{\phi} \in \SentSeq_{\LPA} \Big(TD(\bar{\phi}) \rightarrow \exists i \leq \lh(\bar{\phi}) \ T\phi_i \Big).$
\end{itemize}
We call $D$ as above an outer disjunction of $\phi_1, \ldots, \phi_c$.
\end{definition}
In other words, a theory of truth has outer disjunctions if it has some uniform construction that behaves like disjunctions in $\CT^- + \DCout$.
\begin{proposition} \label{prop_outer_disjunctions}
Suppose that $U$ has outer disjunctions. Then it satisfies $\SeqOInd$ and in particular, it contains $\CT_0$.
\end{proposition}
\begin{proof}[Sketch of the proof.]
This is literally the same argument as in Theorems \ref{th_dcout_implies_sind}, \ref{th_seqind_implies_dcin}, and \ref{th_dc_implies_seqoind}. We only used the fact that $\CT^- + \DCout$ has outer disjunctions.
\end{proof}
By \df{quantifier-free correctness}, $\QFC$, we mean the following axiom:
\begin{displaymath}
\forall \phi \in \qfSent_{\LPA} \Big(\Tr_0(\phi) \rightarrow T\phi\Big).
\end{displaymath}
Let us pause for a moment and explain the notation. The formula $\qfSent_{\LPA}(x)$ expresses that $x$ is a quantifier-free sentence of $\LPA$, i.e., a Boolean combination of closed term equations. Peano arithmetic has a canonical way of deciding whether such (possibly nonstandard) sentences should be true or false by applying partial arithmetical truth predicates. The formula $\Tr_0$ is such a predicate for $\Delta_0$ formulae.\footnote{For a more detailed explanation of what partial arithmetical truth predicates are, consult \citep{kaye}, Section 9 or \citep{hajekpudlak}, Chapter I, Section 1(d).} It can be checked that $\QFC$ does not bring any arithmetical strength to $\CT^-$.
\begin{proposition} \label{prop_qfc_conservative}
$\CT^- + \QFC$ is conservative over $\PA$.
\end{proposition}
The above proposition follows by a routine application of the methods introduced by Enayat and Visser, see e.g. \citep{enayatvisser2}. A proof of this result in the exact same setting in which we work (with regularity axioms included in the definition of $\CT^-$) can be found in \citep{wcislo_prop_plus_qf}. In the same article, it was shown with a different argument using a propositional construction called disjunctions with stopping conditions that $\QFC$ becomes significantly stronger when combined with $\PropSnd$. Now we can prove that result with a simpler argument:
\begin{proposition} \label{prop_qfc_plus_prop_has_outer_dijunctions}
$\CT^- + \QFC + \PropSnd$ has outer disjunctions. In particular, it is equivalent to $\CT_0$.
\end{proposition}
\begin{proof}
The ``in particular'' part follows by Proposition \ref{prop_outer_disjunctions} and the fact that $\QFC$ and $\PropSnd$ are clearly implied by $\Delta_0$-induction. So it is enough to show that $\CT^- + \QFC + \PropSnd$ has outer disjunctions.
For $\bar{\phi} = (\phi_1, \ldots, \phi_c)$, let
\begin{displaymath}
D(\bar{\phi}) = \exists x \leq c \bigvee_{i \leq c} \num{i} = x \wedge \phi_i.
\end{displaymath}
Let us check that $D$ is an outer disjunction. First, consider the formula $D(\bar{\phi}\frown \tuple{\phi_{c+1}})$, i.e.
\begin{displaymath}
\exists x \leq c+1 \bigvee_{i \leq c+1} \num{i} = x \wedge \phi_i.
\end{displaymath}
We want to check that it is true iff either $D(\bar{\phi})$ is true or $\phi_{c+1}$ is true. Suppose that
\begin{displaymath}
T\exists x \leq c+1 \ \bigvee_{i \leq c+1} \num{i} = x\wedge \phi_i.
\end{displaymath}
By the compositional clauses and the definition of big disjunctions, this implies:
\begin{displaymath}
\exists x \leq c+1 \ T \left( \bigvee_{i \leq c} \num{i} = \num{x} \wedge \phi_i \right) \vee \exists x \leq c+1 \ T\left( \num{c+1} = \num{x} \wedge \phi_{c+1}\right).
\end{displaymath}
It can be easily checked by compositional clauses that if the second clause holds, then $T \phi_{c+1}$ holds. So it is enough to check that the first clause in fact implies
\begin{displaymath}
\exists x \leq c \ T \left( \bigvee_{i \leq c} \num{i} = \num{x} \wedge \phi_i \right).
\end{displaymath}
To this end, we have to check that
\begin{displaymath}
\neg T\left( \bigvee_{i \leq c} \num{i} = \num{c+1} \wedge \phi_i \right).
\end{displaymath}
However, notice that the following implication is an instance of a propositional tautology:
\begin{displaymath}
\left( \bigwedge_{i \leq c} \num{i} \neq \num{c+1} \right) \rightarrow \neg \left( \bigvee_{i \leq c} \num{i} = \num{c+1} \wedge \phi_i \right).
\end{displaymath}
Hence, by $\PropRef$, the whole implication is true and by $\QFC$ the antecedent is true as well and the conclusion follows by compositionality. This ends the proof of the implication.
The verification that $TD(\bar{\phi}) \vee T\phi_{c+1}$ implies $TD(\bar{\phi} \frown \tuple{\phi_{c+1}})$ is similar, but simpler, as it only uses compositional clauses of $\CT^-$.
Now, suppose that the following holds:
\begin{displaymath}
T\exists x \leq c \bigvee_{i \leq c} \num{i} = x \wedge \phi_i.
\end{displaymath}
We want to check that for some $i \leq c$, $T\phi_i$ holds. By compositional clauses, we know that there exists $a \leq c$ such that:
\begin{displaymath}
T \bigvee_{i \leq c} \num{i} = \num{a} \wedge \phi_i.
\end{displaymath}
Again, using $\QFC$ and $\PropSnd$, we check that the following holds:
\begin{displaymath}
T \left( \neg \phi_a \rightarrow \neg \bigvee_{i \leq c} \num{i} = \num{a} \wedge \phi_i \right).
\end{displaymath}
Thus we can conclude that $T\phi_a$ holds which ends the proof.
\end{proof}
\begin{remark}
In the above proof, we could actually show that the formula $D(\bar{\phi})$ satisfies both directions of $\DC$.
\end{remark}
The principle $\SeqInd$ clearly looks very related to $\Delta_0$-induction for the truth predicate. Indeed, it turns out that over $\CT^-$ the two principles are equivalent.
\begin{theorem} \label{th_sind_has_outer_disjunction}
$\CT^- + \SeqInd$ has outer disjunctions.
\end{theorem}
\begin{proof}
For a coded sequence $(\phi_1, \ldots, \phi_k)$ of $\LPA$-sentences, let $\bigwedge_{i \leq c} \phi_i$ be their conjunction with parentheses grouped to the left so that we have:
\begin{displaymath}
\bigwedge_{i \leq c+1} \phi_i = \bigwedge_{i \leq c} \phi_i \wedge \phi_{c+1}.
\end{displaymath}
Using $\SeqInd$, we can show that if $T\phi_i$ holds for every $i \leq c$, then $T \bigwedge_{i \leq c} \phi_i$. We show this by considering an auxiliary sequence
\begin{displaymath}
\bigwedge_{i \leq 1} \phi_i, \bigwedge_{i \leq 2} \phi_i, \ldots, \bigwedge_{i \leq c} \phi_i.
\end{displaymath}
Now, we claim that the formula
\begin{displaymath}
D(\bar{\phi}) := \neg \bigvee_{i \leq c} \neg \phi_i
\end{displaymath}
is an outer disjunction. It is easy to verify that over $\CT^-$:
\begin{displaymath}
T D(\bar{\phi} \frown \tuple{\phi_{c+1}}) \equiv T D(\bar{\phi}) \vee T\phi_{c+1}.
\end{displaymath}
We show that $D$ satisfies the second condition of outer disjunctions ($\DCout$) by contraposition. If there is no $i \leq c$ such that $T\phi_i$, then $T \neg \phi_i$ holds for every $i$. Consequently, the following conjunction is true:
\begin{displaymath}
\bigwedge_{i \leq c} \neg \phi_i
\end{displaymath}
which implies that $D \bar{\phi}$ cannot be true. This shows that $D$ is indeed outer disjunction provably in $\CT^- + \SeqInd$.
\end{proof}
Let us summarise the above results:
\begin{corollary} \label{cor_equivalences_sind_soind_dcout}
The following theories are equivalent:
\begin{itemize}
\item $\CT_0$
\item $\CT^- + \SeqInd$.
\item $\CT^- + \SeqOInd$.
\item $\CT^- + \DCout$.
\end{itemize}
\end{corollary}
We conclude this section with some results that clarify the status of $\SeqInd$ and $\SeqOInd$. First of all, let us note that $\SeqInd$ and $\SeqOInd$ are really just some forms of induction axioms and as such they do not require us to assume $\CT^-$ to make sense. They both clearly follow from $\IDelta_0(T)$, the induction scheme for $\Delta_0$-formulae containing the predicate $T$ (which is now treated just as some arbitrary predicate). It turns out that these principles form a strict hierarchy.
\begin{theorem} \label{hierarchy_sind_soind_delta0}
Over $\PA$, $\IDelta_0(T) \rightarrow \SeqOInd \rightarrow \SeqInd$. Moreover, none of the implication reverses.
\end{theorem}
\begin{proof}
Both implications are straightforward, so let us show that neither reverses.
$(\SeqOInd \nvdash \IDelta_0(T))$. Let $M$ be a countable nonstandard model of $\PA$. Let $s_i, i < \omega$ be an enumeration of all coded sequences in $M$. We will inductively construct a sequence of finite subsets of the standard cut $\omega^M$, $A_i,B_i, i < \omega$ such that for all $i$, $A_i \cap B_i = \emptyset$. The sets $A$ are approximations to $T$, the sets $B$ are approximations to the complement.
Let $A_0 = B_0 = \emptyset$. For an arbitrary $i < \omega$, we consider two cases: if all the values of $s_i$ are elements of $A_i$, i.e.:
\begin{displaymath}
\set{a \in M}{\exists j< \lh(s_i) \ a = s_i(j)} \subseteq A_i,
\end{displaymath}
then we set $A_{i+1} = A_i, B_{i+1} = B_i$. Otherwise, let $j$ be the least element such that $s_i(j) \notin A_{i}$ (this element exists, since $|A_i|$ is standard, and therefore, $A_i$ is arithmetically definable). Let $b = s_i(j)$, let $a \neq b$ be an arbitrary element in $\omega \setminus (A_i \cup B_i)$ and set:
\begin{eqnarray*}
A_{i+1} & = & A_i \cup \{a\} \\
B_{i+1} & = & B_i \cup \{b\}.
\end{eqnarray*}
Finally, we set $T = \bigcup_{i \in \omega} A_i$. We claim that $(M,T) \models \SeqOInd$, but $(M,T)$ does not satisfy $\IDelta_0(T).$ The latter claim holds, since by construction $T$ is an infinite (hence cofinal) subset of the standard cut $\omega$. To check that the first claim holds, fix any coded sequence $s$. Pick $i < \omega$ such that $s = s_i$. By construction, either all values of $s_i$ are in $A_i$, hence in $T$ or there exists an $l$ such that for all $j<l$, $s_i(j) \in A_{i+1}$, but $s_i(l) \in B_{i+1}$ (which means that it is not in $T$). This shows that $(M,T) \models \SeqOInd$.
($\SeqInd \nvdash\SeqOInd$) As before, fix an arbitrary countable nonstandard model $M \models \PA$ and let $s_i, i <\omega$ be an (external) enumeration of coded sequences from $M$. We inductively construct two sequences of sets $A_i,B_i$ such that for all $i$, $A_i \cap B_i = \emptyset$, the sets $B_i$ are finite, and the sets $A_i$ contain only finitely many nonstandard elements of $M$.
We construct $A_i, B_i$ as follows: let $A_0 = \omega^M$ (the standard initial cut of $M$) and $B_0 = \emptyset$.
For an arbitrary $i$, if the set of values of $s_i$ is (standardly) finite, we set $A_{i+1} = A_i, B_{i+1} = B_i$. Otherwise, we consider two further subcases. Suppose that the set of $j$ such that $s_i(j) \in B_i$ is an initial segment in $M$ (possibly empty). Let $j_0$ be its supremum (it exists by arithmetical induction, since $B_i$ is finite; if the considered set is empty, then by definition, its supremum is $0$). On the other hand, the set of values of $s_i$ is by assumption infinite, so by overspill there exists $j>j_0$ such that $s_i(j) \notin A_i$. Let $a:= s_i(j-1)$ and let $b:= s_i(j)$.
If the set $\set{j < \lh(s)}{s_i(j) \in B_i}$ is not an initial segment of $M$, then (again by arithmetical induction and finiteness of $B_i$) there exists some $j$ such that $s_i(j) \notin B_i$ and $s_{i}(j+1) \in B$. Let $a:=s_i(j), b:=s_i(j+1)$.
In both cases, set:
\begin{eqnarray*}
A_{i+1} & = & A_i \cup \{a \} \\
B_{i+1} & = & B_i \cup \{b \}.
\end{eqnarray*}
Finally, let $T = \bigcup_{i \in \omega} A_i$. We claim that $(M,T) \models \SeqInd$, but not $\SeqOInd$.
To see that $(M,T) \models \SeqInd$, fix any coded sequence $s \in M$. If the set of values of $s_i$ has (standard) finite number of elements, then $T$ cannot violate the sequential induction axiom for $s$. So suppose that the number of values of $s$ is nonstandard. Fix $i$ such that $s = s_i$ in our enumeration. By construction, there exists $j \in M$ such that $s(j) \in A_{i+1} \subset T$ and $s(j+1) \in B_{i+1} \subset M \setminus T$. So it is not the case that for all $x$, $T(s(x)) \rightarrow T(s(x+1))$ and thus the sequential induction axiom for $s$ is satisfied.
On the other hand, $(M,T)$ does not satisfy $\SeqOInd$. Indeed, fix any $c \in M \setminus \omega$ and consider the identity sequence: $s(i) = i$ for $i =0,1, \ldots, c$. We claim that the sequential order induction fails for this sequence. Let us check that $s$ is progressive, i.e., if for any $j<i$ $s(j) \in T$, then $s(i) \in T$.
Fix any $i \leq c$. If $i \in \omega$, then $T(i)$ holds. On the other hand, if $i$ is nonstandard, then consider the sequence $t: = s \res i$. It has nonstandardly many values, all of which are strictly below $i$. By construction, there exists an element $b$ occurring in this sequence which is not in $T$. This $b$ witnesses that not all $j<i$ are in $T$. Hence $s$ is progressive. On the other hand, not all terms of $s$ are in $T$ which means that $\SeqOInd$ fails.
\end{proof}
\section{Conservativeness of $\DCin$} \label{sect_dcin}
In the previous section, we have shown that $\CT^- + \DCout$ is not conservative over $\PA$ and that, in fact, it is another incarnation of $\CT_0$. Now we will use methods introduced by \cite{enayatvisser2} to show that the related principle $\DCin$ is actually conservative over $\PA$. The Enayat--Visser technique typically allows us to combine various results, so we can show joint conservativity of several distinct principles. Here we will illustrate this point by requiring that the constructed truth predicate additionally satisfies internal induction. Let us recall that principle. If $(M,T) \models \CT^-$, then each formula $\phi \in \form^{\leq 1}_{\LPA}(M)$ ``defines'' a set $\set{x \in M}{\phi(\num{x}) \in T}$. \df{Internal induction} $(\INT)$ expresses that each such set satisfies the induction principle:
\begin{displaymath}
\forall \phi \in \form^{\leq 1}_{\LPA} \Big(T\phi(\num{0}) \wedge \forall x \big(T\phi(\num{x}) \rightarrow T \phi(\num{x+1})\big) \rightarrow \forall x \ T\phi(\num{x})\Big).
\end{displaymath}
It was essentially observed already by \cite{kkl} that $\CT^- + \INT$ is conservative over $\PA$. This result can be also proved using cut elimination as in \citep{leigh} or the methods invented by Enayat and Visser which we will use in this section in order to show the following theorem:
\begin{theorem} \label{th_dcin_plus_int_conservative}
$\CT^- + \DCin + \INT$ is conservative over $\PA$.
\end{theorem}
The theorem is a direct corollary to the following model-theoretic result:
\begin{theorem} \label{th_true_infinite_disjunctions}
Let $M \models \PA$. Then there exists $M' \succeq M$ and a $T' \subseteq M'$ such that $(M',T') \models \CT^- + \INT$ and for all sequences $\bar{\phi} \in \SentSeq_{\LPA}(M)$ of nonstandard length, the disjunction $\bigvee \bar{\phi}$ is in $T'$.
\end{theorem}
In other words, we will construct a model in which every disjunction with infinitely many disjuncts is true. Such a model clearly satisfies $\DCin$: fix any sequence of sentences $\bar{\phi}$ and suppose that there is $i \leq \lh(\bar{\phi})$ such that $T'\phi_i$. If $\bar{\phi}$ has nonstandard length, then $\bigvee \bar{\phi} \in T'$ by assumption. On the other hand, if the length of $\bar{\phi}$ is standard, then $\bigvee \bar{\phi} \in T'$ directly by the compositional axioms. So it is enough to prove Theorem \ref{th_true_infinite_disjunctions}.
In the proof, we will construct an elementary sequence of models $M_i \models \PA$ and a sequence of satisfaction classes $S_i \subset M_i^2, i<\omega$ such that $S_{i+1}$ will satisfy compositional clauses for formulae in the model $M_i$. Let us first define what a satisfaction class actually is. Below, if $s$ is (a code of) a term and $\alpha$ is a function whose domain contains the free variables of $s$ (an $s$-assignment), by $s^{\alpha}$ we mean the formally computed value of $s$ under the valuation $\alpha$. For instance, if $\alpha$ ascribes the value $2$ to $x$ and $5$ to $y$, then
\begin{displaymath}
\mathbb{N} \models \qcr{SSx \times Sy}^{\alpha} = 24.
\end{displaymath}
If $\alpha, \beta$ are assignments and $v$ is a variable, then by $\beta \sim_v \alpha$ we mean that $\dom(\beta) \supseteq \dom(\alpha) \cup \{v\}$ and $\beta(w) = \alpha(w)$ for every $w \in \dom(\alpha) \setminus \{v\}$. In other words, $\beta$ is just like $\alpha$, possibly except for the value $\beta(v)$ which is not even required to be defined for $\alpha$.
\begin{definition} \label{def_satisfaction}
Let $M \models \PA$ and let $\phi \in \form_{\LPA}(M)$. By the \df{compositional clauses} for $\phi$, $\Comp(\phi)$, we mean the disjunction of the following sentences:
\begin{enumerate}
\item $\exists s,t \in \Term_{\LPA} \ \Big(\phi = (s=t) \wedge \forall \alpha \in \Asn(\phi) \ \Big(S(\phi,\alpha) \equiv s^{\alpha} = t^{\alpha} \Big) \Big).$
\item $\exists \psi \in \form_{\LPA} \ \Big(\phi = (\neg \psi) \wedge \forall \alpha \in \Asn(\phi) \ \Big(S(\phi,\alpha) \equiv \neg S(\psi,\alpha)\Big)\Big).$
\item $\exists \psi, \eta \in \form_{\LPA}(M) \ \Big(\phi = (\psi \vee \eta) \wedge \forall \alpha \in \Asn(\phi) \Big(S(\phi,\alpha) \equiv S(\psi,\alpha) \vee S(\eta,\alpha) \Big)\Big).$
\item $\exists \psi, \eta \in \form_{\LPA}(M) \ \Big(\phi = (\psi \wedge \eta) \wedge \forall \alpha \in \Asn(\phi) \Big(S(\phi,\alpha) \equiv S(\psi,\alpha) \wedge S(\eta,\alpha) \Big)\Big).$
\item $\exists \psi \in \form_{\LPA}(M) \exists v \in \Var \ \Big(\phi = (\exists v \psi) \wedge \forall \alpha \in \Asn(\phi) \Big(S(\phi,\alpha) \equiv \exists \beta \sim_v \alpha S(\psi, \beta)\Big)\Big).$
\item $\exists \psi \in \form_{\LPA}(M) \exists v \in \Var \ \Big(\phi = (\forall v \psi) \wedge \forall \alpha \in \Asn(\phi) \Big(S(\phi,\alpha) \equiv \forall \beta \sim_v \alpha \ S(\psi, \beta)\Big)\Big).$
\end{enumerate}
We say that $S \subset M^2$ is a \df{satisfaction class} if there exists a subset $D \subseteq \form_{\LPA}(M)$ such that the following holds:
\begin{itemize}
\item If $(\phi,\alpha) \in S$, then $\phi \in \form_{\LPA}(M)$ and $\alpha \in \Asn(\phi)$.
\item If $(\phi,\alpha) \in S$ for some $\alpha \in \Asn(\phi)$, then $\Comp(\phi)$ holds.
\item If $(\phi,\alpha) \in S$, then $\phi \in D$ or $\phi = \neg \psi$ for some $\psi \in D$.
\item If $\phi \in D$, then $\Comp(\phi)$ holds.
\item If $\phi \in D$ and $\psi$ is a direct subformula of $\phi$, then $\psi \in D$.
\item If $\phi \in D$, then for all $\alpha \in \Asn(\phi)$, $(\phi,\alpha) \in S$ or $(\neg \phi,\alpha) \in S.$
\end{itemize}
By the \df{domain} of $S$, $\dom(S)$, we mean the maximal set $D$ satisfying the above conditions (the maximality requirement is needed, as there is a slight ambiguity in how to count negations of formulae which are not satisfied by any assignment). A satisfaction class is \df{full} iff its domain is the whole set $\form_{\LPA}(M)$.
\end{definition}
The definition of a satisfaction class presented above involves some technical requirements which might seem to be slightly too restrictive, so let us briefly explain our motivations. The definition of a satisfaction class completely agrees with the usual one in the case of full satisfaction classes. However, if $S$ is not a full satisfaction class, i.e., if it does not satisfy compositional conditions for all formulae, it becomes ambiguous whether we should interpret the fact that $(\phi,\alpha) \notin S$ as saying that $\phi$ is not satisfied under the valuation $\alpha$ or that $S$ simply does not decide $\phi$ which might be a technical nuisance in some proofs or statements of results. This includes results and arguments in this article, though the specific difficulties we overcome with this definition will not be really visible. Therefore, we essentially require that for every formula $\phi$ we can unambiguously tell whether it is decided by $S$ or not. This requirement is harmless: Define a pre-satisfaction class on $M$ as a set $S \subset M^2$ such that it satisfies compositional axioms on some set $D$ of formulae closed under direct subformulae and does not contain any pair $(\phi,\alpha)$ for $\phi \notin D$. Then for any pre-satisfaction class, we can canonically define a satisfaction class extending it. Simply take the maximal set $D$ with the two mentioned properties and extend $S$ with all pairs $(\neg \phi,\alpha)$ such that $\phi \in D$ and $(\phi,\alpha) \notin S.$ One can check that after such a one-step extension, we obtain a satisfaction class in our sense.
Satisfaction classes and truth predicates satisfying $\CT^-$ are very closely related objects. However, the link between them is not as direct as one could hope (see \citep{wcislo_definability_automorphisms} for a discussion of this connection). We have to introduce a certain technical condition in order to switch between them in a completely unproblematic manner.
Let $M \models \PA$, $\phi, \psi \in \form_{\LPA}(M)$, $\alpha \in \Asn(\phi), \beta \in \Asn(\psi)$. We say that the pairs $(\phi,\alpha), (\psi,\beta)$ are \df{extensionally equivalent}, $(\phi,\alpha) \simeq (\psi,\beta)$ if there exists a formula $\eta$ and two sequences of closed terms $\bar{s}, \bar{t} \in \ClTermSeq_{\LPA}$ of the same length such that the sequences of their values are equal and
\begin{displaymath}
\phi[\alpha] = \eta(\bar{s}), \psi[\beta] = \eta(\bar{t}),
\end{displaymath}
where $\phi[\alpha]$ is a sentence obtained by substituting in $\phi$ the numeral $\num{\alpha(v)}$ for each variable $v$. For instance, let $\phi = \exists x (x+y = SS0)$, $\psi = \exists x (x + u\times v = w + S0)$, and let $\alpha \in \Asn(\phi), \beta \in \Asn(\psi)$ be such that $\alpha(y) = 2, \beta(u) = 2, \beta(v) = \beta(w) = 1$. Then $(\phi,\alpha) \simeq (\psi,\beta)$ as witnessed by the formula
\begin{displaymath}
\eta = \exists x (x + v_0 = v_1)
\end{displaymath}
and the terms $(SS0, SS0), (SS0\times S0,S0 + S0)$.
Finally, we say that a satisfaction class $S$ is \df{regular} iff for all pairs $(\phi,\alpha) \simeq (\psi,\beta)$, $(\phi,\alpha) \in S$ iff $(\psi,\beta) \in S$.
Before describing the relation between the interpretations of the truth predicate and regular satisfaction classes, let us introduce one more definition.
\begin{definition} \label{def_int_for_sat} \hfil
\begin{itemize}
\item If $M \models \PA$ and $S\subset M^2$ is a satisfaction class, we say that \df{internal induction} holds for $\phi \in \form^{\leq 1}_{\LPA}(M)$ if for every $v \in \FV(\varphi)$ and every $\alpha \in \Asn(\phi)$, the following holds:
\begin{displaymath}
(\phi,\alpha[0/v]) \in S \wedge \forall x \Big((\phi,\alpha[x/v])\in S \rightarrow (\phi,\alpha[x+1/v])\in S\Big) \rightarrow \forall x \ (\phi,\alpha[x/v]) \in S.
\end{displaymath}
Above, $\alpha[y/v]$ denotes the assignment $\alpha'$ which is identical to $\alpha$, except for the fact that $\alpha'(v) = y$.
\item We say that internal induction holds for $S$ if it holds for every $\phi \in \form^{\leq 1}_{\LPA}(M)$.
\end{itemize}
Notice that the formula $\phi$ need not be in the domain of $S$
\end{definition}
The following proposition establishes the link between truth predicates satisfying $\CT^-$ (possibly with $\INT$) and regular satisfaction classes (possibly with internal induction). It can be proved via a direct verification.
\begin{proposition} \label{prop_regular_satisfaction_equiv_ctminus} \hfil
\begin{itemize}
\item[(a)] Let $(M,T) \models \CT^-$ and let $S = \set{(\phi,\alpha) \in M^2}{\phi \in \form_{\LPA}(M), \alpha \in \Asn(\phi), \phi[\alpha] \in T}.$ Then $S$ is a full regular satisfaction class.
\item[(b)] Conversely, let $M \models \PA$, let $S\subset M^2$ be a full regular satisfaction class in $M$ and let $T = \set{\phi \in \Sent_{\LPA}(M)}{(\phi,\emptyset) \in S}$. Then $(M,T) \models \CT^-$.
\item[(c)] Let $(M,T) \models \CT^- + \INT$ and let $S$ be defined as in (a). Then $S$ is a full regular satisfaction class and internal induction holds for $S$.
\item[(d)] Conversely, let $S$ be a full regular satisfaction class in $M$ such that internal induction holds for $S$. Let $T$ be defined as in (b). Then $(M,T) \models \CT^- + \INT$.
\end{itemize}
\end{proposition}
Now we are ready to prove Theorem \ref{th_true_infinite_disjunctions}. \\
\begin{proof}[Proof of Theorem \ref{th_true_infinite_disjunctions}, general idea.]
Let $M \models \PA$. We will find an elementary extension $M \preceq M'$ and a full regular satisfaction class $S$ on $M'$ with internal induction such that for every disjunction $\phi \in \Sent_{\LPA}(M)$ with nonstandardly many disjuncts, $(\phi,\emptyset) \in S$. Then by Proposition \ref{prop_regular_satisfaction_equiv_ctminus}, there exists $T \subset M'$ such that $(M',T) \models \CT^- + \INT$ and all disjunctions with nonstandardly many disjuncts are in $T$.
We will construct $(M',S)$ in stages. We will produce a sequence $(M_n,S_n)$ of models such that:
\begin{itemize}
\item $M_0 := M, S_0 = \emptyset$.
\item The models $M_i \models \PA$ form an elementary chain.
\item $S_{n+1}\subset M_{n+1}^2$ is a regular satisfaction class whose domain contains $\form_{\LPA}(M_n)$.
\item For every $n$, $S_n \subset S_{n+1}$.
\item For every $n$, the model $M_n$ expanded with all the predicates $S_{\phi}, \phi \in \form_{\LPA}(M_n)$ defined by $S_{\phi}(x) \equiv S_n(\phi,x)$ satisfies the full induction scheme.
\item If $\phi:= \bigvee \phi_i$ is a disjunction with nonstandardly many disjuncts and $\phi$ is in the domain of $S_{n+1}$,
then $(\phi,\alpha) \in S_{n+1}$ for every $\alpha \in \Asn(\phi)$.
\end{itemize}
Finally, we set $M' = \bigcup M_n$ and $S = \bigcup S_n$. By a straightforward verification, we check that $S$ is indeed a full regular satisfaction class and it clearly makes all disjunctions with infinitely many disjuncts true and satisfies internal induction.
To complete the proof, it is enough to check that a sequence $(M_n,S_n)$ as above can be produced. This will be demonstrated in Lemma \ref{lem_induction_step_in_ev_chain} which takes care of the induction step for $n>1$ (the existence of $(M_1,S_1)$ can be proved with a very similar, slightly simpler argument).
\end{proof}
Some care is needed in order to make sure that the satisfaction classes which we will construct are indeed regular. Before we state and prove the induction step lemma, we will introduce one more technical notion. (It appeared in the same context earlier, e.g. in \citep{lelyk_wcislo_local_collection}.)
\begin{definition} \label{def_syntactic_template}
Let $\phi \in \form_{\LPA}$. By a \df{syntactic template} of $\phi$, we mean the smallest formula $\widehat{\phi}$ such that the following conditions are satisfied:
\begin{enumerate}
\item There exists a sequence of arithmetical terms $\bar{s}$ (not necessarily closed) such that $\phi = \widehat{\phi}(\bar{s})$.
\item No variable occurs in $\widehat{\phi}$ both free and bound.
\item No free variable occurs in $\widehat{\phi}$ more than once.
\item No closed term occurs in $\widehat{\phi}$.
\item No complex term containing only free variables occurs in $\widehat{\phi}$.
\end{enumerate}
\end{definition}
For instance, if $\phi = \exists x (SSx+Sy = (z \times (y + S0)) \times x)$, then
\begin{displaymath}
\widehat{\phi} = \exists x (SSx + v_0 = v_1 \times x),
\end{displaymath}
where $v_0, v_1$ are chosen so as to minimise the formula.
We say that $\phi$ and $\psi$ are \df{syntactically similar} if they have the same syntactic template. We denote it with $\phi \sim \psi$. Notice that if $(\phi,\alpha) \simeq (\psi,\beta)$ for some $\alpha, \beta$, then $\phi \sim \psi$.
\begin{lemma} \label{lem_induction_step_in_ev_chain}
Let $M \models \PA$, let $S \subset M^2$ be a regular satisfaction class such that:
\begin{itemize}
\item The model $(M,S_{\phi})_{\phi \in M}$ satisfies full induction, where $S_{\phi}(x) \equiv S(\phi,x)$.
\item If $\phi$ is a disjunction with nonstandardly many disjuncts in the domain of $S$, then $(\phi,\alpha) \in S$ for all $\alpha \in \Asn(\phi)$.
\end{itemize}
Then there exists an elementary extension $M \preceq M'$ and a regular satisfaction class $S' \supseteq S$ such that $(M',S')$ satisfies the above conditions and $\dom(S') \supseteq \form_{\LPA}(M)$.
\end{lemma}
\begin{proof}
Let $M, S$ be as in the assumptions of the lemma. Let us consider a theory $\Theta$ in a language $\LPA$ extended with an additional predicate $S'$ and a family of auxiliary predicates $S'_{\phi}, \phi \in M$ which comprises the following axioms:
\begin{itemize}
\item $\ElDiag(M)$, the elementary diagram of $M$.
\item $\forall x \ S'_{\phi}(x) \equiv S'(\phi,x)$ where $\phi \in \form_{\LPA}(M)$. (The definition of $S'_\phi$).
\item $\Comp(\phi)$ for $S'$, where $\phi \in \form_{\LPA}(M)$. (Compositionality Scheme).
\item $S'(\phi,\alpha)$, where $(\phi,\alpha) \in S$. (Preservation Scheme).
\item $\forall \phi, \psi \in \form_{\LPA} \forall \alpha \in \Asn(\phi), \beta \in \Asn(\psi) \ \Big((\phi,\alpha) \simeq (\psi,\beta) \rightarrow S'(\phi,\alpha) \equiv S'(\psi,\beta)\Big).$ (Regularity Axiom)
\item The full induction scheme for formulae in the language $\LPA + S'_{\phi}, \phi \in \form_{\LPA}(M)$. (Internal Induction).
\item $\forall \alpha \in \Asn(\phi) \ S'(\phi,\alpha)$, where $\phi = \bigvee_{i \leq c} \phi_i$ for some $(\phi_i)_{i \leq c} \in \formSeq_{\LPA}(M)$ and a nonstandard $c$. (Disjunction Scheme).
\end{itemize}
The predicates $S'_{\phi}$ are introduced to concisely express what form of internal induction we accept. Note that in the Internal induction scheme, we do not allow formulae containing the predicate $S'$. Crucially, we are not allowed to treat $\phi$ like a variable. On the other hand, the form of induction we accept is \textit{prima facie} stronger than the internal induction condition introduced in Definition \ref{def_int_for_sat}.
If $(M',S',S'_{\phi})_{\phi \in M} \models \Theta$, then by restricting $S'$ to $(\phi,\alpha)$ such that $\phi \sim \phi'$ for some $\phi' \in M$ and ignoring the predicates $S'_{\phi}$, we obtain a model satisfying the conclusion of the lemma (note that $S'$ itself need not be a satisfaction class, as the compositional conditions may possibly fail badly for formulae in $M' \setminus M$). Therefore, it is enough to check that $\Theta$ is consistent. We will prove it by compactness. Let $\Theta_0 \subset \Theta$ be a finite subtheory. It is enough to check the consistency of $\Theta_0$.
Let $\phi_1, \ldots, \phi_n \in \form_{\LPA}(M)$ be all formulae such that the instances of their compositionality, preservation, and disjunction schemes are in $ \Theta_0$. We will define $S' \subset M^2$ which satisfies these instances of schemes, as well as all the other axioms in $\Theta$. Note that $\ElDiag(M)$ will be automatically satisfied, since the interpretation of the relation $S'$ will be defined in $M$.
Let us consider the classes $[\phi_i]_{\sim}$, where $\sim$ is the syntactic similarity relation (see Definition \ref{def_syntactic_template} and the remark below the definition). Let $\lhd$ be a relation defined on the classes as follows: $[\phi] \lhd [\psi]$ iff there exist $\phi' \in [\phi], \psi' \in [\psi]$ such that $\phi'$ is a direct subformula of $\psi'$. We define a sequence of relations $S_0, S_1, \ldots, S_k \subset M^2$ by induction on the rank of classes in the relation $\lhd$. We let $(\phi,\alpha) \in S_0$ if $[\phi]$ is minimal in the relation $\lhd$ and one of the following holds:
\begin{itemize}
\item There exist $s,t \in \Term_{\LPA}(M)$ such that $\phi = (s=t)$ and $s^{\alpha} = t^{\alpha}$.
\item $(\phi,\alpha) \in S$.
\item $\phi$ is a disjunction with nonstandardly many disjuncts.
\end{itemize}
Above, we want to define $S_0$ on all classes minimal with respect to the relation $\lhd$. If $S$ already holds of $\phi$ and $\alpha$, then we preserve it in $S_0$. If $\phi$ happens to be a disjunction with infinitely many disjuncts, then we make it true under all valuations. Otherwise, we set $(\phi,\alpha) \notin S_0$ for all $\alpha$, but we do not have to mention it explicitly.
We define $S_{i+1}$ as the union of $S_i$ and the pairs $(\phi,\alpha)$ such that $[\phi]$ has rank $i+1$ and one of the following holds:
\begin{itemize}
\item There exists $\psi \in \form_{\LPA}(M)$ such that $\phi = \neg \psi$ and $(\psi,\alpha) \notin S_i$.
\item There exist $\psi, \eta \in \form_{\LPA}(M)$ such that $\phi = \psi \vee \eta$ and $(\psi,\alpha) \in S_i$ or $(\eta,\alpha) \in S_i$.
\item There exist $\psi, \eta \in \form_{\LPA}(M)$ such that $\phi = \psi \wedge \eta$ and $(\psi,\alpha) \in S_i$ and $(\eta,\alpha) \in S_i$.
\item There exists $\psi \in \form_{\LPA}(M), v \in \Var(M)$ such that $\phi = \exists v \psi$ and $(\psi,\beta) \in S_i$ for some $\beta \sim_v \alpha$.
\item There exists $\psi \in \form_{\LPA}(M), v \in \Var(M)$ such that $\phi = \forall v \psi$ and $(\psi,\beta) \in S_i$ for all $\beta \sim_v \alpha$.
\end{itemize}
In other words, we extend $S_i$ to $S_{i+1}$ so as to satisfy the compositional clauses. Finally, since we consider only finitely many classes, they can only attain some finite rank $k$. Let $S' = S_k$. Let $S'_{\phi}, \phi \in M$ be defined so that the definition-axiom of $S'_{\phi}$ is satisfied. We claim that $(M,S', S'_{\phi})_{\phi \in M} \models \Theta.$
It is obvious that $(M,S')$ satisfies $\ElDiag(M)$. It follows directly by construction that it satisfies the compositional clauses for the formulae $\phi_1, \ldots, \phi_n$. It satisfies the instances of the preservation scheme for these formulae: for the formulae of minimal rank it follows by the definition of $S_0$, since if a disjunction with infinitely many disjuncts is in the domain of $S$, then it is satisfied under all assignments. For formulae of higher rank, this follows by construction, since $S$ is compositional on its domain and the compositional clauses uniquely determine the extension of a satisfaction predicate on a formula, given its extension on the direct subformulae.
Let us verify that $S'$ satisfies the regularity axiom. For formulae in the $\lhd$-minimal classes, this follows by construction and the assumption that $S$ is regular. For formulae in the classes of rank $>0$, we can directly check that compositional clauses preserve regularity and thus show by induction that all the predicates $S_i$ satisfy the regularity axiom.
The model $(M,S'_{\phi})_{\phi \in M}$ satisfies the full induction scheme, since we can verify by an easy induction on $i$ that $S_i$ is arithmetically definable in terms of $S_{\phi_i}$, $i \leq n$, such that the class $[\phi_i]$ is $\lhd$-minimal and $\phi_i \in \dom S$. By assumption, $(M,S_{\phi})_{\phi \in M}$ satisfies the full induction scheme and since $S'$ is definable in that structure, it satisfies induction. The conclusion follows.
The predicate $S'$ satisfies the disjunction scheme for formulae $\phi_1,\ldots,\phi_n$. Indeed, if $\phi$ is a disjunction with a nonstandard number of disjuncts and $[\phi]$ is minimal in the relation $\lhd$, then $(\phi,\alpha) \in S'$ by construction. Let us check by induction that the disjunction scheme is satisfied for formulae whose classes have higher rank. If the rank of $[\phi]$ is $i+1$, then there exist formulae $\psi, \eta$ such that $\psi$ is a disjunction with a nonstandard number of disjuncts and $\phi = \psi \vee \eta$, where the rank of $[\psi]$ is $\leq i$. By induction hypothesis, $(\psi,\alpha) \in S'$ for all $\alpha \in \Asn(\psi)$, hence by the compositional clauses $(\psi \vee \eta,\beta) \in S'$ for all $\beta \in \Asn(\psi \vee \eta)$. This concludes the proof.
\end{proof}
\begin{remark} \label{rem_sigma_n_correctness}
As we have already pointed out, Enayat--Visser methods of building satisfaction classes typically allow us to combine various conservativeness results. For instance, in Theorem \ref{th_true_infinite_disjunctions}, we could additionally require that the constructed predicate is correct with respect to blocks of existential or universal quantifiers.
It might appear that since a class we build in the proof of this theorem is somewhat pathological, there are certain obvious limits to what truth-theoretic principles can be additionally satisfied. For instance, if the constructed truth predicate makes true all infinite disjunctions, then it cannot agree with $\Sigma_n$ arithmetical truth predicates on sentences from the respective syntactic classes. However, as pointed out by Ali Enayat, we could still construct a model in which the truth predicate $T$ agrees with all the usual partial arithmetical $\Sigma_n$-truth predicates and $\DCin$ is satisfied. It is enough that in the proof of Lemma \ref{lem_induction_step_in_ev_chain} we change the definition of $S_0$ so that for a minimal class $[\phi]$, where $\phi$ is a $\Sigma_n$-formula for some standard $n$ which is not in the domain of $S$, $S_0$ agrees with the partial arithmetical satisfaction predicates. An alternative approach, mixing an Enayat--Visser approach and resplendence (by building a satisfaction class from recursively saturated partial classes) was proposed in an unpublished note by \cite{enayat_dc_intro}.
\end{remark}
\section{Alternative versions of $\DC$} \label{sect_parentheses}
In our paper, we defined $\bigvee_{i \leq c} \phi_i$ as $(( \ldots (\phi_0 \vee \phi_1) \vee \ldots )\vee \phi_{c-1}) \vee \phi_c$. This is not the only possible definition of disjunction (and admittedly not the most natural one) and one could wonder whether modifying that definition has some impact on the presented results.
This point has been partially addressed in the definition of outer disjunctions: our proof of nonconservativity of $\CT^- + \DCout$ used the clause
\begin{equation} \label{equat_first_clause_outer_disj} \tag{*}
TD(\phi_1, \ldots, \phi_c, \phi_{c+1}) \equiv TD(\phi_1,\ldots, \phi_c) \vee T\phi_{c+1}.
\end{equation}
It turns out that this assumption cannot be weakened much further. Let us define \df{balanced disjunctions} of a sequence of formulae $\phi_1, \ldots, \phi_c$, $B(\phi_1, \ldots, \phi_c)$ by recursion in the following way:
\begin{eqnarray*}
B(\emptyset) & : = & 0 \neq 0 \\
B(\phi_1) & := & \phi_1 \\
B(\phi_1, \ldots, \phi_{c}) & := & B(\phi_1, \ldots, \phi_{\llcorner c/2 \lrcorner}) \vee B(\phi_{\llcorner c/2 \lrcorner +1}, \ldots, \phi_c).
\end{eqnarray*}
In other words, we are grouping disjunctions of $\phi_1, \ldots, \phi_k$ so that the parentheses form a binary tree. Over $\CT^-$, balanced disjunctions do not satisfy the condition \ref{equat_first_clause_outer_disj} and in fact, if we add an analogue of $\DCout$ for that kind of disjunctions, we do not gain any arithmetical strength.
\begin{theorem} \label{th_balanced_dc_out_not_conservative}
$\CT^-$ together with the axiom $\forall \bar{\phi} \in \SentSeq_{\LPA} \Big(T B (\bar{\phi}) \rightarrow \exists i \leq \lh(\bar{\phi}) \ T \phi_i \Big)$ is a conservative extension of $\PA$.
\end{theorem}
\begin{proof}[Sketch of a proof.]
Analogously to the proof of Theorem \ref{th_dcin_plus_int_conservative}, we fix a model $M \models \PA$ and we construct an elementary extension $M' \succeq M$ and $T \subset M'$ such that any balanced disjunction of a sequence $(\phi_1,\ldots, \phi_c) \in \SentSeq_{\LPA}(M')$ of nonstandard length is false.
It is enough to observe that if $\phi$ is a balanced disjunction of nonstandardly many sentences, then there exist $\phi_1, \phi_2$ such that $\phi = \phi_1 \vee \phi_2$ and $\phi_i$ are themselves such balanced disjunctions, so we can maintain the requirement that all sentences of this form are false throughout the whole construction.
\end{proof}
Notice that in the model produced in the proof of Theorem \ref{th_dcin_plus_int_conservative}, all balanced disjunctions with nonstandardly many disjuncts are in fact true. This shows that both the analogues of $\DCin$ and $\DCout$ for the balanced formulae are conservative over $\PA$ and that in general $\DCout$ is sensitive to the specific kind of forming disjunctions for which they are formulated. This leads to a natural question whether some natural forms of disjunction, say, balanced disjunctions, can satisfy full $\DC$ while remaining conservative. It turns out that $\DC$ behaves stably under such varying implementations.
\begin{theorem} \label{th_invariant_dc}
Suppose that $U$ is a theory extending $\CT^-$. Suppose that there exists a provably functional formula $D(x)$ such that provably in $U$ for any $\bar{\phi} = (\phi_1,\ldots, \phi_c) \in \SentSeq_{\LPA}$, $D(\bar{\phi})$ is in $\Sent_{\LPA}$ and the following holds:
\begin{displaymath}
TD(\bar{\phi}) \equiv \exists i \leq c \ T\phi_i.
\end{displaymath}
Then $U$ extends $\CT_0$.
\end{theorem}
\begin{proof}
Under the assumptions of the theorem, $D$ is an outer disjunction provably in $U$. The conclusion follows by Proposition \ref{prop_outer_disjunctions}.
\end{proof}
\begin{remark} \label{rem_selective_disj}
As we have noted above, the results on $\DCout$ are sensitive to how we exactly define disjunctions over finite sets of sentences. One way to express abstractly what properties of disjunctions are used is the notion of outer disjunction. Ali Enayat has proposed a further generalisation of this concept.
Let $U$ be a theory extending $\CT^-$. We say that a provably functional formula $D$ is a \df{selective disjunction} if there exists a formula $F$ which provably in $U$ defines a choice function for finite sets and the following conditions hold provably in $U$:
\begin{itemize}
\item For any $\Phi \in \SentSet_{\LPA}$, $D(\Phi) \in \Sent_{\LPA}$.
\item $T D(\Phi) \equiv T F(\Phi) \vee T D(\Phi \setminus \{F(\phi)\}).$
\end{itemize}
One can check that if a theory $U$ has selective disjunctions satisfying the analogue of $\DCout$, namely:
\begin{displaymath}
TD(\Phi) \rightarrow \exists \phi \in \Phi \ T\phi,
\end{displaymath}
then $U$ is again equivalent to $\CT_0$.
\begin{comment}
\begin{itemize}
\item For any $\bar{\phi} = \tuple{\phi_1, \ldots, \phi_c} \in \SentSeq_{\LPA}$, $D(\bar{\phi}) \in \Sent_{\LPA}$.
\item $TD(\bar{\phi}) \equiv T \pi_1 F(\bar{\phi}) \vee TD( \tuple{\phi_1, \ldots, \hat{\phi}_{\pi_0F(\bar{\phi})}, \ldots, \phi_c}).$
\end{itemize}
Here $\widehat{x_i}$ indicates that the $i-th$ term is omitted from the sequence and the rest is rearranged so that the domain is $\{1, \ldots, c-1\}.$ Notice that if $s$ is a sequence, $F(s)$ is a pair $\tuple{\textnormal{argument, value}}$ and the projections $\pi_0, \pi_1$ return the argument and the value, respectively.
One can check that if a theory $U$ has selective disjunctions satisfying the analogue of $\DCout$, namely:
\begin{displaymath}
TD(\bar{\phi}) \rightarrow \exists i \leq c \ T\phi_i,
\end{displaymath}
then $U$ is again equivalent to $\CT_0$.
\end{comment}
\end{remark}
\section{The role of the regularity axiom} \label{sec_regularity}
In Section \ref{sec_prelim}, we briefly mentioned that we have certain technical reasons to include Regularity, $\REG$ among the axioms for the compositional truth. Let us now explain why we adopt this principle and how this choice affects our results.
First of all, without $\REG$, we have to be careful about the exact formulation of the compositional axioms for quantifiers. Our basic choice is between two options. We can require that, say, an existential statement $\exists v \phi(v)$ is true iff some sentence obtained by substituting a numeral in $\phi(v)$ is true. This amounts to adopting quantifier axioms in the form which we have chosen in this article:
\begin{displaymath}
T \exists v \phi(v) \equiv \exists x \ T \phi(\num{x}).
\end{displaymath}
The second option is to require that an existential statement is true iff a sentence obtained by substituting some closed arithmetical term for the quantified variable is true. Then, the quantifier axioms would have a form:
\begin{displaymath}
T \exists v \phi(v) \equiv \exists t \in \ClTerm_{\LPA} \ T\phi(t).
\end{displaymath}
Note that these two versions of the quantifier axiom are not immediately comparable with respect to their strength. The left-to-right implication is stronger in the numeral version:
\begin{displaymath}
T \exists v \phi(v) \rightarrow \exists x T \phi(\num{x})
\end{displaymath}
whereas the reverse implication is formally stronger for the term version:
\begin{displaymath}
\exists t \in \ClTerm_{\LPA} \ T \phi(t) \rightarrow T \exists v\phi(v).
\end{displaymath}
The term version was chosen, for instance, by \cite{halbach}. The numeral version appears, e.g., in \citep{friedman_sheard} or \citep{horsten_leigh_truth_is_simple}. This choice, in turn, may have a significant bearing on our results. Most importantly, we do not know whether adding $\Delta_0$-induction to a version of $\CT^-$ whose compositional axioms for quantifiers involve terms results in a non-conservative extension of $\PA$. So, when regularity is missing, some of the main results in this article will depend on a technical choice in the formulation of compositional axioms which may lead to some further confusions.
Another issue is that without regularity, there is a mismatch between the notion of a truth predicate as discussed in the philosophical literature and the notion of satisfaction classes, as discussed in the literature on models of $\PA$ (e.g., as in \citep{kaye}). There is a direct link between the compositional truth and satisfaction classes assuming that we include some form of extensionality conditions in both these cases. (A more comprehensive discussion of why these kind of assumptions are relevant can be found in \citep{wcislo_definability_automorphisms}.) This link is important, as some techniques used in our proofs are designed to work specifically in the context of satisfaction classes. Most importantly, the conservativity arguments using the Enayat--Visser technique which we employ in Section \ref{sect_dcin} do not really work when directly applied to truth classes. Therefore, we would most likely need to add regularity assumptions to statements of some technical lemmas which could make for a potentially awkward reading.
Finally, let us discuss the impact of $\REG$ on the results in this article. The main results in Section \ref{sect_dcin}, namely Theorems \ref{th_dcin_plus_int_conservative} and \ref{th_true_infinite_disjunctions} are also true for $\CT^-$ without the regularity axiom, as the latter versions are formally weaker. The same applies to Remark \ref{rem_sigma_n_correctness}, Proposition \ref{prop_qfc_conservative}, and Theorem \ref{th_balanced_dc_out_not_conservative}.
The situation of the results in Section \ref{sec_dcout} is slightly more complicated. As we have already mentioned, Theorem \ref{th_many_faces} is only known to hold if we consider a version of $\CT^-$ in which the numeral variant of the compositional quantifier axioms is assumed. In this case, all results from Section \ref{sec_dcout}, still hold. Unfortunately, this is not necessarily true for the term variant. Let us discuss in some detail what can be salvaged in this scenario.
Theorem \ref{th_dcout_implies_sind} does not depend at all on quantifier axioms and thus it holds without assuming regularity in either version of $\CT^-$; similarly for Theorem \ref{th_seqind_implies_dcin}. The first part of Theorem \ref{th_dc_implies_seqoind} and Theorem \ref{th_sind_has_outer_disjunction} also hold true. It is clear that $\SeqOInd$ implies $\PropRef$ for $\CT^-$ in the term version as well. Moreover, $\PropRef$ clearly implies $\DC$ using just compositional axioms for boolean connectives. However, it is unclear whether $\PropRef$ and $\CT_0$ are equivalent without assuming $\REG$ and with the quantifier axioms in the term version. Similarly, the first part of Proposition \ref{prop_outer_disjunctions} does not depend of regularity or the quantifier axioms, but we do not know whether such theories contain $\CT_0$. Proposition \ref{prop_qfc_plus_prop_has_outer_dijunctions} may be entirely false if we do not assume the numeral version of the quantifier axioms, as the proof relies significantly on that assumption.
To sum it up: by the results of Section \ref{sec_dcout}, $\DCout, \DC, \SeqInd$, $\SeqOInd$, and $\PropRef$ are pairwise equivalent over any version of $\CT^-$ without assuming regularity. They are equivalent to $\CT_0$ and not conservative over $\PA$ if we consider $\CT^-$ with quantifier axioms for numerals. If we consider $\CT^-$ with the term variant of the quantifier axioms, we do not know either whether $\DC$ or $\PropRef$ are equivalent to $\CT_0$, or whether the latter theory is arithmetically stronger than $\PA$.
\section*{Appendix: a glossary of formalised notions}
Throughout the article, we referred to a number of formalised notions and used some rather technical notation. Let us now gather it in a glossary for the convenience of the reader.
\begin{itemize}
\item $\alpha \sim_v \beta$ means that $\dom(\beta) \supseteq \dom(\alpha) \cup \{v\}$ and $\beta(w) = \alpha(w)$ for every $w \in \dom(\alpha) \setminus \{v\}$.
\item $\val{t}$. If $t$ is (a G\"odel code of) a closed arithmetical term, then $\val{t}$ is the value of a term (whose G\"odel code is) $t$. We use the same expression to denote the formalised version of this function.
\item $t^{\alpha}$. If $t$ is (a G\"odel code of) an arithmetical term, and $\alpha$ is a $t$-assignment, then $t^{\alpha}$ is the value of term $t$ under the assignment $\alpha$. We use the same expression to denote the formalised version of this function.
\item $\val{\bar{t}}$. If $\bar{t}$ is a sequence of (G\"odel codes of) closed arithmetical terms, then $\val{\bar{t}}$ is the sequence of their values. We also use this expression to denote the formalised version of this function.
\item $\num{x}$ is (a G\"odel code of) a canonical numeral denoting the number $x$. We also use this expression to denote the formalised version of this function.
\item $\phi[\alpha]$. If $\phi$ is a formula and $\alpha \in \Asn(\phi)$, then by $\phi[\alpha]$ we mean a sentence obtained by substituting in $\phi$ the numeral $\num{\alpha(v)}$ for each variable $v$. We also use this expression to denote the corresponding formalised notion.
\item $\Asn(x)$ is a set of $x$-assignments, that is, functions whose domain contains the set of free variables of $x$, where $x$ is a term, a formula, or a sequence thereof. We also use this expression to denote the corresponding formalised notion.
\item $\ClTerm_{\LPA}(x)$ is a formula expressing ``$x$ is (a G\"odel code of) a closed arithmetical term.'' (That is, a term with no free variables.)
\item $\ClTermSeq_{\LPA}(x)$ is a formula expressing ``$x$ is a sequence of (G\"odel codes of) closed arithmetical terms.''
\item $\FinSeq(x)$ is a formula expressing ``$x$ is a finite sequence of numbers.''
\item $\form_{\LPA}(x)$ is a formula expressing ``$x$ is (a G\"odel code of) an arithmetical formula.''
\item $\form^{\leq 1}_{\LPA}$ is a formula expressing ``$x$ is (a G\"odel code of) an arithmetical formula with at most one free variable.''
\item $\formSeq_{\LPA}(x)$ is a formula expressing ``$x$ is a sequence of (G\"odel codes of) arithmetical formulae.''
\item $\lh(s) = x$ is a formula expressing ``$s$ is a sequence and its length is $x$.''
\item $\Pr_{\PA}(x)$ is a formula expressing ``$x$ is (a G\"odel code of) an arithmetical sentence provable in $\PA$.''
\item $\Pr_{\Prop}(x)$ is a formula expressing ``$x$ is (a G\"odel code of) an arithmetical sentence which is provable in pure propositional logic.''
\item $\Pr^{T}_{\Prop}(x)$ is a formula expressing ``$x$ is (a G\"odel code of) an arithmetical sentence which is provable in propositional logic from the set of premises $\Gamma$ such that $T(y)$ holds for all $y \in \Gamma$.''
\item $\qfSent_{\LPA}(x)$ is a formula expressing ``$x$ is (a G\"odel code of) a quantifier-free arithmetical sentence.''
\item $\Sent_{\LPA}(x)$ is a formula expressing ``$x$ is (a G\"odel code of) an arithmetical sentence.''
\item $\SentSeq_{\LPA}(x)$ is a formula expressing ``$x$ is a sequence of (G\"odel codes of) arithmetical sentences.''
\item $\SentSet_{\LPA}(x)$ is a formula expressing ``$x$ is a finite set of (G\"odel codes of) arithmetical sentences.''
\item $\Term_{\LPA}(x)$ is a formula expressing ``$x$ is (a G\"odel code of) an arithmetical term.''
\item $\Tr_0$ is the arithmetical truth predicate for $\Delta_0$-formulae.
\item $\Var(x)$ is a formula expressing ``$x$ is (a G\"odel code of) a first-order variable.''
\end{itemize}
\section*{Acknowledgements}
We are very grateful to Ali Enayat for a number of helpful comments and suggestions which allowed us to improve this article.
This research was supported by an NCN MAESTRO grant 2019/34/A/HS1/00399 ``Epistemic and Semantic Commitments of Foundational Theories.''
|
2,877,628,090,139 | arxiv | \section{Introduction}
The fact that the DAMA's result [Bernabei, 2013] is not confirmed by other Dark Matter searching experiments, such as CDMS, Xenon-10, Xenon-100 or LUX, gave the author
an idea that the DAMA signal is due to scattering of a low mass WIMP with proton of hydrogen atom. The hydrogen is
present in a form of small OH-contamination in NaI(Tl) crystals. OH-molecules may come from the primary NaI salt and could
be present at a level of $\sim$few~ppm. The important point to realize is that the OH molecule would be sensitive to very low energy neutrons and low mass
WIMPs, i.e., collision of the WIMP and proton will cause vibrations of this molecule, which could be detected.
The question is what is needed to prove that this hypothesis is correct ?
A low mass WIMP, for example $\sim$1~GeV/$c^2$, represents a real experimental challenge, as it requires a low mass target and extremely sensitive detector.
The Dark Matter cloud is believed to be stationary relatively to the Galagxy. The Earth moves with a velocity of 230~+-~30~km/sec in the Galactic plane,
and as a result, a $\sim$1~GeV/$c^2$ mass WIMP's kinetic energy oscillates between $\sim$0.353~keV and $\sim$0.208~keV relative to the DAMA
experiment, on its yearly journey around the Sun. Figure~\ref{fig:Nuclear_recoil_from_1GeV} shows nuclear recoil energies as a function of recoil angle for various nuclei,
assuming the WIMP mass of 1~GeV/$c^2$ WIMP. It is clear, that one prefers to use the proton target if the light WIMP mass is this small.
\begin{figure}[tbp]
\includegraphics[width=0.5\textwidth]{Nuclear_recoil_energy_from_1GeV_WIMP.pdf}
\caption{Nuclear recoil energies for various nuclei assuming a $\sim$1~GeV/$c^2$ WIMP with kinetic energy of $\sim$0.353keV.}
\label{fig:Nuclear_recoil_from_1GeV}
\end{figure}
\section{OH-impurity as a detector of WIMP}
Fig.~\ref{fig:NaI_energy_levels} shows a classical model of the signal formation in NaI(Tl) crystals. A deposited energy creates electron-hole pair, excites an
electron into the conduction band, where it moves until it finds an activator (Tl), with a very low ionization potential of 6.108~eV, where the
de-excitation occurs via small photonic emissions, mostly in visible spectrum [Knoll, 2010].
\begin{figure}[tbp]
\includegraphics[width=0.5\textwidth]{Energy_band_structure_of_inorganic_scintillator.pdf}
\caption{Energy levels of NaI crystal with Tl-activator [Knoll, 2010].}
\label{fig:NaI_energy_levels}
\end{figure}
In this section we describe in more deatil what are possible excitation of the OH-molecule [Vavra, 2014] and answer a question if resulting photons are detectable.
Fig.~\ref{fig:OH_levels} shows schematically energy levels in OH-molecule. If either a neutron or low mass WIMP hits a proton, or even an oxygen nucleus,
in the OH molecule, it will cause vibrations and molecular excitations to higher energy levels. The molecule can then de-excite by fluorescent
photons at either $\sim$282~nm or $\sim$310~nm. Such photons can be detected by the Bialkali photocathode in principle, if one choses the optical coupling correctly.
Through this mechanism one could then increase the sensitive to extremely low sub-keV energy deposits.
One should mention that the OH-molecule was studied extensively by laser-induced fluorescence by many chemists. An example of such fluorescence measurement is
the OH-molecule excitation by a 282~nm dye laser and in turn observing the 310~nm wavelength with a PMT with a notch filter [Smith, 1990].
They used this method to determine traces of OH-radicals in atmosphere [Matsumi, 2002].
One should also point out that 282~nm corresponds to 4.67~eV energy, which is a very small excitation compared to usual energy needed to excite NaI(Tl) crystal
($\sim$25eV per electron-hole pair on average). This point plus a sensitivity to slow neutrons makes the OH-molecule excitation a very attractive idea
to Dark Matter detectors. Just as one can pump the OH-molecule to excitation with a laser, one can excite it with WIMP collisions.
The NaI(Tl) crystal has a limit on maximum allowable OH-impurity level before its properties are affected. However there are other
materials allowing a large OH-content, which could be investigated. For example the Corning 7980 Fused silica has 800-1000~ppm of OH-content by weight.
One could perhaps consider enhancing the S/N ratio by implementing a notch filter to accept only wavelengths betwen 270~nm and 350~nm.
\begin{figure}[tbp]
\includegraphics[width=0.5\textwidth]{OH_energy_levels.pdf}
\caption{Energy levels of OH-molecule are very complex. A low mass WIMP hits proton or even oxygen nucleus in the OH-molecule, causing excitation and subsequent
photon fluoroscence emissions near 310 or 282~nm, which are detectable by the Bialkali photocathode, if one uses the right optical coupling allowing tranmsision of these wavelengths.
In principle a very low mass WIMP could be detected by this technique, as 282~nm corresponds to only 4.67~eV.}
\label{fig:OH_levels}
\end{figure}
Figure~\ref{fig:Nuclear_recoil_from_4GeV} shows nuclear recoil energies as a function of recoil angle for various nuclei,
assuming the WIMP mass of 4~GeV/$c^2$ WIMP. One can see that the recoil energy for proton is about the same that from oxygen, i.e., the WIMP collision with either proton or oxygen
can excite the OH-molecule. One should note that a recoil energy of sodium nucleus is below DAMA's threshold of $\sim$1.5keV.
\begin{figure}[tbp]
\includegraphics[width=0.5\textwidth]{Nuclear_recoil_energy_from_4GeV_WIMP.pdf}
\caption{Nuclear recoil energies for various nuclei assuming a $\sim$4~GeV/$c^2$ WIMP with kinetic energy of $\sim$1.5~keV.}
\label{fig:Nuclear_recoil_from_4GeV}
\end{figure}
\section{Suggested steps to calibration the DAMA crystals}
One has to show that the OH-molecule imbedded in NaI(Tl) crystals behave the same way as described in this paper.
To prove the OH-hypothesis one should expose DAMA crystals to a very low energy sub-keV neutron beam and check the crystal response. If this result is
positive, one can parameterize a dependence on the OH-content in small specially prepared NaI(Tl) samples with different OH-content. In addition, one could also
use the laser-induced fluorescence method to quantify the OH-content in some DAMA crystals. There are chemists specializing in this methodology.
We believe that these calibration steps are necessary to understand the NaI(Tl) crystal response to very low energy proton recoils. One should not use the energy
dependence obtained form the Gamma source calibration, as it will likely produce incorrect conclusions of the NaI(Tl) crystal response from sub-keV deposits of low mass WIMPs.
\section{Conclusion}
This paper provides arguments why the OH-molecule may be a good way to detect very low mass WIMP. It provides a method to reach the lowest possible WIMP mass.
The paper suggests concrete steps the DAMA group could take to prove that the proposed idea is valid.
The paper also suggests other ways to create a detector with large OH-content, which could be used for a detection of very low mass WIMPs.
|
2,877,628,090,140 | arxiv | \section{\label {s1}Introduction}
Let $\mathbb R^d\, $ be the $d$-dimensional space of real vectors
$x=(x_1,\dots ,x_d)\, $ with scalar product $\langle x,y\rangle
=x_1\4y_1+\dots +x_d\4y_d$ and norm $\|x\|= \langle x,x\rangle^{1/2}$.
We also denote by $\mathbb R^\infty$ a real separable Hilbert space
consisting of all real sequences ~${x=(x_1,x_2,\dots )}$ such that
$\|x\|^2= x_1^2+x_2^2+\dots <\infty$.
Let $X,X_1,X_2,\dots $ be a sequence of i.i.d{.} $\mathbb R^d$-valued
random vectors. Assume that ${{\mathbf E\,} X= 0}$ and $\sigma^2\={\mathbf E\,}
\|X\|^2<\infty$. Let $G$ be a mean zero Gaussian random vector
such that its covariance operator $\mathbb C= \cov G:\mathbb R^d \to \mathbb R^d $
is equal to $\cov X$. It is well-known that the distributions
$\mathcal L(S_N)$ of sums \beq \label{SN}S_N\=N^{-1/2}\,( X_1+\dots +
X_N) \eeq converge weakly to $\mathcal L(G)$.
Let $\mathbb Q:\mathbb R^d\to\mathbb R^d$ be a linear symmetric bounded
operator and let $\mathbb Q \kern1pt[x]= \langle \mathbb Q\kern1pt x,x\rangle $ be
the corresponding quadratic form. We shall say that $\mathbb Q\, $ is
non-degenerate if ~$\ker \mathbb Q=\bgl\{ 0\bgr\}$.
Denote, for $q> 0$,
$$\beta_q\={\mathbf E\,} \|X\|^q,\quad \q\beta\=\beta_4 . $$
Introduce the distribution functions
\beq
F (x) \= \P \bgl\{ \mathbb Q \kern1pt[S_N]\leq x\bgr\}, \quad \q H(x) \= \P
\bgl\{ \mathbb Q \kern1pt[G]\leq x\bgr\} .\label{eq1.1j} \eeq
Write
\beq \label{eqdel}\Delta _N\= \sup_{x\in\mathbb R}\;\bgl| F(x) - H(x)
\bgr|. \eeq
\begin{theorem}{\label{T1.1}} Assume that\/
$\mathbb Q $ and\/ $\mathbb C $ are non-degenerate and that
$d\geq 5$ or $d = \infty $.
Then
$$
\Delta _N \leq c ( \mathbb Q, \mathbb C ) \, \beta /N .
$$
The constant\/ $c(\mathbb Q, \mathbb C )$ in this bound depends
on\/ $ \mathbb Q $ and\/ $ \mathbb C $ only. \end{theorem}
\begin{theorem}{\label{T1.1a}} Let the conditions of
Theorem~$\ref{T1.1}$ be satisfied and let\/ $5\le d<\infty $.
Assume that the operator $\mathbb Q$ is isometric. Then
$$
\Delta _N \leq c_d\, \sigma^d\,(\det \mathbb C)^{-1/2}\,{\mathbf E\,}\|\mathbb C^{-1/2}\,X\|^4 /N .
$$
The constant $c_d$ in this bound depends
on $d$ only. \end{theorem}
Theorems \ref{T1.1} and ~\ref{T1.1a} are simple consequences of
the main result of this paper, Theorem~\ref{T1.5} (see also
Theorem~\ref{T1.3}). Theorem~\ref{T1.1} was proved in G\"otze and
Zaitsev~(2008). It confirms a conjecture of Bentkus and G\"otze
(1997a) (below BG (1997a)). It generalizes to the case $d\ge5$ the
corresponding result of BG (1997a). In their Theorem~1.1, it was
assumed that $d\ge9$, while our Theorem \ref{T1.1} is proved for
$d\ge 5$. Theorem ~\ref{T1.1a} yields an explicit bound in terms
of the distribution $\mathcal L(X)$.
The distribution function of ~$ \|S_N\|^2 \, $ (for bounded ~$X$
with values in $\mathbb R^d$) may have jumps of order ~$\mathcal O
\bgl(N^{-1} \bgr)$, for all ~$1\leq d\leq \infty$. See, e.g.,
BG~(1996, p.~468). Therefore, the bounds of Theorems ~\ref{T1.1}
and ~\ref{T1.1a} are optimal with respect to the order in~$N$.
Theorems \ref{T1.1},~\ref{T1.1a} and the method of their proof are closely
related to the lattice point problem in number theory. Suppose
that ~$d<\infty $ and that ~$\langle \mathbb Q \4x,x\rangle >0$,
for $x\ne 0$.
Let $\text{\rm vol} \, E_r \, $ be the volume of the ellipsoid
$$
\quad \q\quad \q \quad \q\quad \q\quad E_r =\bgl\{ x\in \mathbb R^d: \,\, \mathbb Q \kern1pt[x]
\leq r^2\bgr\},
\quad \q
r\geq 0.
$$
Write $\text{\rm vol} _{\mathbb Z}\, E_r\, $ for the number of points in
$E_r\cap \mathbb Z^d$, where $ \mathbb Z^d \subset \mathbb \mathbb R^d$ is the
standard lattice of
points with integer coordinates.
The following result due to G\" otze (2004) is related to Theorems
\ref{T1.1} and ~\ref{T1.1a} (see also BG~(1995a, 1997b)).
\smallskip
\begin{theorem}\label{T1.2}
{\it For all dimensions} $\; d\geq 5$,
$$
\quad \q\quad \q\quad \q\quad \q\quad \sup_{a\in\mathbb R^d}\, \left| \ffrac { \text{\rm vol}
_{\mathbb Z}\, (E_r+a) -\text{\rm vol} \, E_r } {\text{\rm vol} \, E_r }\right|
= \mathcal O(r^{-2}) ,\quad \q \text{\it for}\quad
r\geq 1,
$$
{\it where the constant in $\mathcal O(r^{-2})$ depends on the dimension
$d$ and on the lengths of axes of the ellipsoid $E_1$ only.}
\end{theorem}
Theorem \ref{T1.2} solves the lattice point problem for $d\geq 5$.
It improves the classical estimate $ \mathcal O( r^{-2d/(d+1) } ) $ due to
Landau (1915), just as Theorem \ref{T1.1} improves the bound ~
${\mathcal O(N^{-d/(d+1)})}$ by Esseen (1945) in the CLT for ellipsoids
with axes parallel to coordinate axes. A related result for
indefinite forms may be found in G\" otze and Margulis~(2010).
\smallskip
Work on the
estimation of the rate of approximation under the
conditions of Theorem~\ref{T1.1} for Hilbert spaces
started in
the second half of the last century. See Zalesski\u\i, Sazonov and
Ulyanov (1988) and Nagaev (1989) for
{optimal bounds} of order $\mathcal O(N^{-1/2})$
(with respect to eigenvalues of $\mathbb C$)
assuming finiteness of the third moment. For a more
detailed discussion see Yurinskii (1982),
Bentkus, G\"otze, Paulauskas and Ra\v ckauskas (1990), BG~{(1995b, ~1996,
1997a)} and Senatov (1997, 1998).
Under some more restrictive moment
and dimension conditions the estimate of order $\mathcal O (N^{-1+\varepsilon })$,
with $\varepsilon\downarrow 0$ as $d\uparrow \infty$, was obtained by G\"
otze (1979). The proof in G\" otze (1979) was based on a new
symmetrization inequality for characteristic functions of
quadratic forms. This inequality is related to Weyl's (1915/16)
inequality for trigonometric sums. This inequality and its
extensions (see Lemma \ref{L5.1}) play
a crucial role in the
proofs of bounds in the CLT
for ellipsoids and hyperboloids in
finite and infinite dimensional cases. Under some additional
smoothness assumptions, error bounds $\mathcal O(N^{-1})$ (and, moreover,
Edgeworth type expansions) were obtained in G\" otze (1979),
Bentkus (1984), Bentkus, G\" otze and Zitikis (1993). BG~{(1995b,
~1996,~1997a)} established the bound of order $\mathcal O(N^{-1})$
without smoothness-type conditions. Similar bounds for the rate of
infinitely divisible approximations were obtained by Bentkus, G\"
otze and Zaitsev (1997). Among recent publications, we should
mention the papers of Nagaev and Chebotarev ~(1999),~(2005)
(${d\ge13}$, providing a more precise dependence of constants on the
eigenvalues of $\mathbb C$) and Bogatyrev, G\"otze and
Ulyanov~(2006) (non-uniform bounds for $d\ge12$), see also G\"otze
and Ulyanov~(2000). The proofs of bounds of order $\mathcal O(N^{-1})$ are
based on discretization (i.e., a reduction to lattice valued
random vectors) and the symmetrization techniques mentioned above.
Assuming
the matrices $\mathbb Q$ and $\mathbb C$ to be diagonal, and the
independence of the first five coordinates of~$X$, Bentkus and G\"otze
(1996) have already reduced the dimension requirement for the
bound $\mathcal O(N^{-1})$ to $d\ge 5$. The independence assumption in
BG~(1996) allowed to apply an adaption of the Hardy--Littlewood
circle method. For the general case described in
Theorem~\ref{T1.1},
one needs to develop new techniques. Some yet
unpublished results of G\"otze (1994) provide the rate $\mathcal
O(N^{-1})$ for sums of two independent {\it arbitrary} quadratic
forms (each of rank $d \ge 3$). G\"otze and Ulyanov~(2003)
obtained bounds of order $\mathcal O(N^{-1})$ for some ellipsoids
in $\mathbb R^d$ with $d\ge5$ in the case of lattice distributions of~$X$.
The optimal possible dimension condition for this rate is just
$d\geq 5$, due to the lower bounds of order $\mathcal O(N^{-1}\log{
N})$ for dimension $d=4$ in the corresponding lattice point
problem. The question about precise convergence rates in
dimensions
$2\leq d \leq 4$ still remains completely open (even in
the simplest case where $\mathbb Q$ is the identity operator~$\mathbb
I_d$, and for random
vectors with
independent Rademacher coordinates). It should be mentioned that,
in the case $d=2$, a precise convergence rate would imply a
solution of the famous circle problem. Known lower bounds in the
circle problem correspond to the bound of
order $\mathcal O (N^{-3/4}\, \log^\delta
N )$, $\delta>0$, for~$\Delta_N$. Hardy (1916) conjectured that up to
logarithmic factors this is the optimal order.
Now we describe the most important elements of the proof. We have
to mention that a big part of the proof repeats the arguments of
BG (1997a), see BG~(1997a) for the description and application of
symmetrization inequality and
discretization procedure.
In our proof we do not use the multiplicative inequalities of BG~(1997a).
Here we replace those techniques
by arguments from the geometry of numbers, developed in
G\"otze (2004), combined with
effective equidistribution results by G\"otze and Margulis
(2010) for suitable actions of
unipotent subgroups of $\hbox{SL}(2,\mathbb R)$, see Lemma~\ref{GM}.
These new techniques (compared to previous) results are mainly
concentrated in Sections~\ref{s4}--\ref{s7}.
Using the Fourier inversion formula (see \eqref{eq3.1} and
\eqref{eq3.1o}), we have to estimate some integrals of the
absolute values of differences of characteristic functions of
quadratic forms. In Section~\ref{s5}, we reduce the estimation of
characteristic functions to the estimation of a theta-series (see
Lemma \ref{L7.5} and inequality~\eqref{koren}). To this end, we
write the expectation with respect to Rademacher random variables
as a sum with binomial weights~$p(m)$ and~$p(\overline m)$. Then we
estimate $p(m)$ and $p(\overline m)$ from above by discrete Gaussian
exponential weights $c_s\,q(m)$ and $c_s\,q(\overline m)$, see
\eqref{qm}, \eqref{pm}, \eqref{eq7.9} and \eqref{eq7.10}. Together
with the non-negativity of some characteristic functions (see
\eqref{eq7.8} and \eqref{eq7.13}), this allows us to apply then
the Poisson summation formula from Lemma~\ref{Le3.2}. This formula
reduces the problem to an estimation of integrals of theta-series.
Section~\ref{s6} is devoted to some facts from Number Theory. We
consider the lattices, their $\alpha$-characteristics (which are
defined in \eqref{alp} and \eqref{alp3}) and Minkowski's
successive minima. In Section~\ref{s7}, we reduce the estimation
of integrals of theta-series to some integrals of
$\alpha$-characteristics. An application of the crucial Lemma~\ref{GM},
{decribed above,}
ends the proof.
\smallskip
\section{\label {s11}Results}
To formulate the results we need more notation repeating most
part of the notation used in BG (1997a). Let $\sigma_1^2\geq
\sigma_2^2\geq \dots$ be the eigenvalues of $\mathbb C$, counting
their multiplicities. We have $\sigma^2=\sigma_1^2+\sigma_2^2+\cdots$.
We shall identify the linear operators and corresponding matrices.
By $\mathbb I_d:\mathbb R^{d}\to\mathbb R^{d}$ we denote the identity
operator and, simultaneously, the diagonal matrix with entries 1
on the diagonal. By $\mathbb O_{d}$ we denote the $(d\times d)$ matrix with zero
entries.
Throughout $\, \mathcal S=\{ \fs e1s\}\subset\mathbb R^d$ denotes a finite set
of cardinality $s$. We shall write $\mathcal S_o$ instead of $\mathcal S$
if the system $\{ \fs e1s\}$ is orthonormal.
Let $p> 0$ and $\delta \geq 0$. Following BG (1997a),
we introduce a somewhat modified non-degeneracy condition for the distribution
of
a $d$-dimensional vector~$Y$: \beq \mathcal N(p,\delta ,\mathcal S,
Y):\quad \q\quad
\P \bgl\{ \|Y -e\|\leq \delta \bgr\} \geq p,
\quad \q \text{for all}\ \, e\in \mathcal S .
\label{eq1.3} \eeq We shall refer to condition \eqref{eq1.3} as
condition $\mathcal N(p,\delta ,\mathcal S, Y)$.
We shall write $$\mathcal N_{\mathbb Q}(p,\delta ,\mathcal S, Y) =\mathcal N(p,\delta
,\mathcal S, Y)
\cup\mathcal N(p,\delta ,\mathbb Q\,\mathcal S, Y).$$
Just condition $\mathcal N_{\mathbb Q}(p,\delta ,\mathcal S, Y)$ was used in BG
(1997a).
Note that \beq\label{eq13}\mathcal N(p,\delta ,\mathcal S, Y) =\mathcal N_{\mathbb
I_d}(p,\delta ,\mathcal S, Y)
.\eeq
\smallskip
Introduce truncated random vectors \beq\label{eq1.4a} X^\diamond =
X \,\mathbf I \bgl\{ \|X\|\leq \sigma \kern1pt \sqrt{N} \bgr\}, \quad \q X_\diamond
=X\, \mathbf I \bgl\{ \|X\|> \sigma \kern1pt \sqrt{N} \bgr\},\eeq \beq\label{eq1.4t} X^{\hbox{\tiny$\square$}}
=
X \,\mathbf I \bgl\{\|\mathbb C^{-1/2}\,X\|\leq \sqrt{d\4N} \bgr\}, \quad \q X_{\hbox{\tiny$\square$}}
=X\, \mathbf I \bgl\{ \|\mathbb C^{-1/2}\,X\|> \sqrt{d\4N} \bgr\},\eeq
and their moments (for $q>0$)\beq \Lambda _4^\diamond = \ffrac 1{ \sigma^{4}\,N}
{\mathbf E\,} \|X^\diamond \|^4 ,\quad \q\quad \quad \quad \q\quad \Pi_q^\diamond =
\ffrac N { (\sigma\, \sqrt{N})^{q}} {\mathbf E\,} \|X_\diamond \|^q,\label{eq1.5}
\eeq\beq \Lambda _4^{\hbox{\tiny$\square$}} = \ffrac 1{ d^{2}\,N}
{\mathbf E\,} \|\mathbb C^{-1/2}\,X^{\hbox{\tiny$\square$}} \|^4 ,\quad \q\quad \quad \quad \q\quad \Pi_q^{\hbox{\tiny$\square$}} =
\ffrac N { ( \sqrt{d\4N})^{q}} {\mathbf E\,} \|\mathbb C^{-1/2}\,X_{\hbox{\tiny$\square$}}
\|^q.\label{eq1.5t}
\eeq
Here and below $\mathbf I \bgl\{ A\bgr\} $ denotes the indicator of an
event $A$. Of course, definitions \eqref{eq1.4t} and
\eqref{eq1.5t} have sense if $d<\infty$ and the covariance operator
$\mathbb C $ is non-degenerate.
Clearly, we have \beq\label{eq1.4u} X^\diamond +X_\diamond
=X^{\hbox{\tiny$\square$}} +X_{\hbox{\tiny$\square$}} =X,\quad
\norm{X^\diamond}\,\norm{X_\diamond}=\norm{X^{\hbox{\tiny$\square$}}}\,\norm{X_{\hbox{\tiny$\square$}}}=0.\eeq
Generally speaking, $X^{\hbox{\tiny$\square$}}$ and~$X^\diamond$ are
different truncated vectors. In BG (1997a) the i.i.d. copies of
the vectors $X^\diamond$ and $X_\diamond$ only were involved.
Truncation \eqref{eq1.4t} was there applied to the vector
$X^\diamond$. The use of $X^{\hbox{\tiny$\square$}}$ is more natural for
the estimation of constants in the case $d<\infty$.
It is easy to see that \beq\label{eq14rr} \big(\mathbb
C^{-1/2}\,X\big)^{\diamond}=\big(\mathbb C^{-
1/2}\,X\big)^{{\hbox{\tiny$\square$}}}= \mathbb C^{-1/2}\,X^{{\hbox{\tiny$\square$}}}, \eeq and
\beq\label{eq14ru} \big(\mathbb
C^{-1/2}\,X\big)_{\diamond}=\big(\mathbb C^{-
1/2}\,X\big)_{{\hbox{\tiny$\square$}}}= \mathbb C^{-1/2}\,X_{{\hbox{\tiny$\square$}}}\,. \eeq
Equalities \eqref{eq14rr} and~\eqref{eq14ru} provides a
possibility to apply auxiliary results obtained in BG~(1997a) for
truncated vectors $X^\diamond$ and $X_\diamond$ to truncated
vectors $\mathbb C^{-1/2}\,X^{{\hbox{\tiny$\square$}}}$ and $\mathbb C^{-
1/2}\,X_{{\hbox{\tiny$\square$}}}$. However, one should take into account that
$\sigma^2$, $\Lambda _4^\diamond$, $\Pi_q^\diamond$, $G$, $\ldots$ have to
be replaced by corresponding objects related to the vector $\mathbb C^{-1/2}\,X$ (that is, by
$d$, $\Lambda _4^{\hbox{\tiny$\square$}}$, $\Pi_q^{\hbox{\tiny$\square$}}$, $\mathbb
C^{- 1/2}\,G$, \ldots).
In Sections \ref{s3} and~\ref{s4}, we shall
denote \beq X'= X^{\hbox{\tiny$\square$}} -{\mathbf E\,} X^{\hbox{\tiny$\square$}} +W,\label{eq1.20} \eeq
where $W$ is a centered Gaussian random vector which is
independent of all other random vectors and variables and is
chosen so that $\cov X'= \cov G$. Such a vector $W$ exists by
Lemma \ref{L2.4}.
By $c, c_1,c_2,\dots $ we shall denote absolute positive
constants. If a constant depends on, say,~$s$,\, then we shall
point out the dependence writing $c_s$ or $c(s)$. We denote by
\,$c$ \,universal constants which might be different in different
places of the text. Furthermore, in the conditions of theorems and
lemmas (see, e.g., Theorem \ref{T1.3}, \ref{T1.5f} and the proofs
of Theorems \ref{T1.5}, \ref{T1.6} and~\ref{T2.1}) we write $c_0$
for an {\it arbitrary} positive absolute constant, for example one
may choose $c_0=1$.
We shall write $A\ll B$, if there exists an
absolute constant $c$ such that $A\leq c\kern1pt B$. Similarly, $A\ll_s
B$, if $A\leq c(s)\kern1pt B$. We shall also write $A\asymp_s B$ if
$A\ll_s B\ll_s A$. By $ \lceil \alpha \rceil$ we shall denote the
integer part of a number $\alpha $.
Throughout we assume that all random vectors and variables are
independent in aggregate, if the contrary is not clear from the
context.
By $X_1,X_2,\dots$ we
shall denote independent copies of a random vector $X$.
Similarly, $G_1,G_2,\dots$ are
independent copies of $G$ and so on. By $\mathcal
L(X)$ we shall denote the distribution of $X$. Define the
symmetrization $\widetilde X\, $ of a random vector $X$ as a random
vector with distribution $ {\mathcal L (\widetilde X)= \mathcal L
(X_1-X_2)}$.
Instead of normalized sums $S_N$, it is sometimes more convenient
to consider the sums $Z_N=\fsu X1N$. Then $S_N=N^{-1/2} \kern1pt Z_N$.
Similarly, by $Z_N^\diamond$ (resp. $Z_N^{\hbox{\tiny$\square$}}$ and $Z_N'$) we
shall denote sums of $N$ independent copies of $X^\diamond$ (resp.
$X^{\hbox{\tiny$\square$}}$ and $X'$). For example, $Z_N'={X_1'}+\cdots+X_N'$.
The expectation ${\mathbf E}_Y$
with respect to a random vector $Y$
we define as the conditional expectation
$$\mathbf E _Y \, f(X,Y,Z,\dots )
= {\mathbf E\,} \bgl( f(X,Y,Z\dots ) \,\bgl|\, X,Z,\dots\bgr)$$ given all
random vectors but $Y$.
Throughout we write
$\operatorname {e}\{ x \}\=
\exp \{i\kern1pt x \}$.
By \beq \label{Fur}\widehat F(t)=\int_{-\infty}^\infty
\operatorname{e}\{tx\}\,dF(x)
\eeq
we denote the Fourier--Stieltjes transform
of a function $F$ of bounded variation or, in other words, the
Fourier transform of the measure which has the distribution
function~$F$.
\smallskip
Introduce the distribution functions \beq F_a (x) \= \P \bgl\{
\mathbb Q \kern1pt[S_N-a]\leq x\bgr\}, \quad H_a(x) \= \P \bgl\{ \mathbb Q
\kern1pt[G-a]\leq x\bgr\},\quad \q
a\in \mathbb R^d, \quad x\in \mathbb R.\label{eq1.1} \eeq
Furthermore, define, for $d=\infty$ and $a\in\mathbb R^d$, the Edgeworth
correction $$E_a(x)=E_a(x; \mathbb Q, \mathcal L(X), \mathcal
L(G)) $$ as
a function of bounded variation such that
$E_a(-\infty) =0 $ and its
Fourier--Stieltjes transform
is given by
\beq\widehat E_a(t)=
\ffrac {2\, (it)^2}{3\kern1pt \sqrt{N}}
{\mathbf E\,} \operatorname {e} \bgl\{t\kern1pt \mathbb Q \kern1pt[Y] \bgr\}
\bgl( 3\,\langle \mathbb Q \4X, Y \rangle
\,\langle \mathbb Q \4X,X\rangle +2\kern1pt i \kern1pt t \,\langle \mathbb Q \4X , Y
\rangle ^3 \bgr), \quad Y= G-a.\label {eq1.2}\eeq
In finite dimensional spaces (for $1\le d<\infty$) we define the
Edgeworth correction as follows (see Bhattacharya and Rao (1986)).
Let $\phi $ denote the standard normal density in~$\mathbb R^d$. Then
$p(y) =\phi (\mathbb C^{-1/2} y)/\sqrt {\operatorname {det}
\mathbb C } $, $y\in \mathbb R^d$, is the density of $G$, and, for
$a\in\mathbb R^d$, $b=\sqrt N\,a$, we have \beq E_a(x) \= \Theta_b (N\4x)
\= \ffrac 1 {6\kern1pt \sqrt{N}} \chi (A_x) , \quad \quad A_x =\bgl\{
u\in\mathbb R^d:\,\, \mathbb Q \kern1pt[u-a]\leq x\bgr\},\label {eq1.21} \eeq
with the signed measure \beq\chi (A) \=\int_A {\mathbf E\,} p'''(y)\kern1pt X^3 \,
dy, \quad \q\text{for the Borel sets}\ \,
A\subset \mathbb R^d ,\label {eq1.22} \eeq
and where \beq p'''(y) \4u^3 = p(y) \bgl( 3\, \langle\mathbb
C^{-1} u,u\rangle \langle\mathbb C^{-1} y,u\rangle- \langle\mathbb
C^{-1} y,u\rangle^3\bgr) \label {eq1.23} \eeq
denotes the third Frechet derivative of $p$
in direction $u$.
Notice
that $E_a =0$
if $a=0$ or if ~${{\mathbf E\,} \langle X,y\rangle ^3=0}$, for all $y\in
\mathbb R^d$. In particular, $E_a =0$ if $X\, $ is symmetric
(that is, $\mathcal L(X)=\mathcal L(-X)$).
We can write similar representations for $E_a^{{\hbox{\tiny$\square$}}}(x) = \Theta_b^{{\hbox{\tiny$\square$}}}
(N\4x)$,
$E_a^{\diamond}(x) = \Theta_b^{\diamond} (N\4x)$ and $E_a^{\prime}(x) =
\Theta_b^{\prime} (N\4x)$
just replacing $X$ by $X^{\hbox{\tiny$\square$}} $, $X^\diamond $ and~$X^\prime $ in
\eqref{eq1.2} or \eqref{eq1.22}.
\smallskip
For $b\in\mathbb R^d$, introduce the distribution functions \beq
\Psi_b(x) \= \P \bgl\{ \mathbb Q \kern1pt[Z_N-b] \leq
x\bgr\}=F_a(x/N),\label{eq1.18} \eeq and\beq \Phi_b(x) \= \P \bgl\{
\mathbb Q \kern1pt[\sqrt{N}\, G -b]\leq x\bgr\}=H_a(x/N).\label{eq1.19} \eeq
Define, for $a\in\mathbb R^d$, $b=\sqrt N\,a$, \beq\label{edg}
\Delta _N^{(a)}\=\sup_{x\in \mathbb R}\; \bgl| F_a (x)
-H_a(x)-E_a(x)\bgr| =\sup_{x\in \mathbb R}\; \bgl|\Psi_b (x) -
\Phi_b(x)-\Theta_b(x)\bgr| \eeq (see \eqref{eq1.1},
\eqref{eq1.21}, \eqref{eq1.18} and \eqref{eq1.19} to justify the
last equality in \eqref{edg}). We write $\Delta _{N,{\hbox{\tiny$\square$}}}^{(a)}$ and
$\Delta _{N,\diamond}^{(a)}$ replacing $E_a$ by $E_a^{{\hbox{\tiny$\square$}}}$
and~$E_a^{\diamond}$ in \eqref{edg}.
The aim of this paper is to derive for $\Delta _N^{(a)}$ explicit
bounds of order $\mathcal O(N^{-1}}\def\sign{\hbox{\rm sign})$ without any additional
smoothness type assumptions. Theorem \ref{T1.3} (which was proved
in BG~(1997a)) solved this problem in the case $13\leq d\leq
\infty $.
In Theorems \ref{T1.3}--\ref{T2.1} we assume that the symmetric
operator $\mathbb Q$ is isometric, that is, that $\mathbb Q^2$ is
the identity operator $\mathbb I_d$. This does not restrict
generality (see Remark~1.7 in BG (1997a)). Indeed, any symmetric
operator ~$\mathbb Q\, $ may be decomposed as $\mathbb Q=\mathbb
Q_1\mathbb Q_0\mathbb Q_1\penalty250\mskip\thickmuskip\mskip-\thinmuskip$, where ~$\mathbb Q_0\, $ is symmetric
and isometric and $\mathbb Q_1\, $ is symmetric bounded and
non-negative, that is, $\langle \mathbb Q_1 \kern1pt x,x\rangle \geq 0$,
for all $x\in \mathbb R^d$. Thus, for any symmetric $\mathbb Q$, we can
apply
all our bounds replacing the random vector
$X\, $ by $\mathbb Q_1X$,$\, $ the Gaussian random vector $G\, $
by $\mathbb Q_1G$, the shift $a \, $ by $\mathbb Q_1a$, etc. In
the case of concentration functions (see Theorems \ref{T1.6} and~
\ref{T2.1}), we have
~${Q(X ;\, \lambda ;\, \mathbb Q )=
Q(\mathbb Q_1X ;\, \lambda ;\, \mathbb Q_0 )}$, and we may apply the
results provided ~$\mathbb Q_1X $ (instead of $X$) satisfies the
conditions.
\smallskip
\begin{theorem}{\label{T1.3}}{{\rm (BG (1997a, Theorem 1.3))}} Let \/
$\delta = 1/300$, $\mathbb Q^2=\mathbb I_d$, $s=13$ and \/$13\leq
d\leq \infty $. Let\/ $c_0$ be an arbitrary positive absolute
constant. Assume that condition $\mathcal N_{\mathbb Q}( p,\delta,
\mathcal S_o,c_0\kern1pt G/\sigma) $ holds. Then we have:
\beq \Delta _N^{(a)}\leq C \kern1pt \bgl( \Pi_3^{\diamond} + \Lambda _4^{\diamond}\bgr)\bgl( 1 +
\norm{a/\sigma}^6\bgr)
\label{eq1.6}\eeq and \beq \Delta _{N,\diamond}^{(a)}\leq C \kern1pt \bgl(
\Pi_2^{\diamond} + \Lambda _4^{\diamond}\bgr)\bgl( 1 + \norm{a/\sigma}^6\bgr)
\label{eq1.6w}\eeq
with\/ $C=c\kern1pt p^{-6} + c\kern1pt (\sigma /\theta_{8})^{8}$, where $\theta_1^4 \geq
\theta_2^4 \geq \dots$ are the eigenvalues of~$(\mathbb C\kern1pt \mathbb
Q)^2$.
\end{theorem}
\smallskip
Unfortunately, we cannot apply Theorem \ref{T1.3} for
$d=5,6,\ldots,12$. Moreover, the quantity~$C$ depends on ~$p$
which is exponentially small with respect to eigenvalues of
~$\mathbb C$.
In G\"otze and Zaitsev (2010), the following analogue of Theorem
\ref{T1.3} is proved with bounds for constants which are not
optimal.
\begin{theorem}{\label{T1.5f}} Let \/
$\delta = 1/300$, $\mathbb Q^2=\mathbb I_d$, $s=5$ and \/$5\le
d<\infty$. Let\/ $c_0$ be an arbitrary positive absolute
constant.
Assume that condition $\mathcal N_{\mathbb Q}( p,\delta, \mathcal
S_o, c_0\, G/\sigma) $ holds. Then \beq \Delta _{N}^{(a)} \leq C \kern1pt \bgl(
\sigma_d^{-3}\4N^{-1/2}\,
{\mathbf E\,} \|X_{\hbox{\tiny$\square$}} \|^3 + \sigma_d^{-4}\4N^{-1}\,{\mathbf E\,} \|X^{\hbox{\tiny$\square$}} \|^4\bgr)\bgl( 1 +
\norm{a/\sigma}^3\bgr) ,
\label{eq1.8ff0}\eeq and \beq \Delta _{N,{\hbox{\tiny$\square$}}}^{(a)}
\leq C \kern1pt \bgl( \sigma_d^{-2}\kern1pt{\mathbf E\,} \|X_{\hbox{\tiny$\square$}} \|^2 +
\sigma_d^{-4}\4N^{-1}\,{\mathbf E\,} \|X^{\hbox{\tiny$\square$}} \|^4\bgr)\bgl( 1 +
\norm{a/\sigma}^3\bgr) ,
\label{eq1.8wf}\eeq
with\/ $C= c_d\kern1pt p^{-3} $.
\end{theorem}
Theorem \ref{T1.5f} extends to the case $d\ge5$ Theorem 1.5 of
BG~(1997a) which contains the corresponding bounds for $d\ge9$.
Unfortunately, in both papers, the quantity $C$ depends on~$p$
which is exponentially small with respect to $\sigma_9/\sigma^2$ (in
BG~(1997a)) and to $\sigma_5/\sigma^2$ (in G\"otze and Zaitsev (2010)).
Under some additional conditions, $C$ may be estimated from above
by $c_d\,\exp(c\kern1pt\sigma^2\kern1pt\sigma_9^{-2})$ and by
$c_d\,\exp(c\kern1pt\sigma^2\kern1pt\sigma_5^{-2})$ respectively.
In G\"otze and Zaitsev
(2008) we proved Theorem~\ref{T1.5f} in the case $a=0$ and hence,
Theorem \ref{T1.1}.
The main result of the paper is Theorem~\ref{T1.5}. It is valid
for $5\le d<\infty$ in finite-dimensional spaces $\mathbb R^d $ only.
However, the bounds of Theorem \ref{T1.5}
depend on the smallest $\sigma_j$'s. This makes them
unstable if one or more of coordinates of $X$ degenerates. In our
finite dimensional results, Theorems
\ref{T1.5}, \ref{T1.6} and \ref{T2.1}, we always assume that the covariance operator
$\mathbb C $ is non-degenerate.
\smallskip
\begin{theorem}{\label{T1.5}} Let \/
$\mathbb Q^2=\mathbb I_d$, $5\le d<\infty$.
Then we have$:$
\beq \Delta _{N}^{(a)} \leq C \kern1pt \bgl(
\Pi_3^{\hbox{\tiny$\square$}} + \Lambda _4^{\hbox{\tiny$\square$}}\bgr)\bgl( 1 +
\norm{a/\sigma}^3\bgr) ,
\label{eq1.8}\eeq and \beq \Delta _{N,{\hbox{\tiny$\square$}}}^{(a)}
\leq C \kern1pt \bgl( \Pi_2^{\hbox{\tiny$\square$}} + \Lambda _4^{\hbox{\tiny$\square$}}\bgr)\bgl( 1 + \norm{a/\sigma}^3\bgr) ,
\label{eq1.8w}\eeq
with\/ $C= c_d\, \sigma^d\,(\det \mathbb C)^{-1/2}\,$.
\end{theorem}
\smallskip
It is easy to see that, according to \eqref{eq1.4t} and \eqref{eq1.5t},
\beq\label{eq1.8fc}\Pi_3^{\hbox{\tiny$\square$}} + \Lambda _4^{\hbox{\tiny$\square$}}\le {\mathbf E\,}\|\mathbb
C^{-1/2}\,X\|^{3+\delta}/(d^{(3+\delta)/2}\4N^{(1+\delta)/2}),\quad \text{for } \ 0\le\delta\le 1, \eeq
and\beq\label{eq1.8y}\Pi_2^{\hbox{\tiny$\square$}} +
\Lambda _4^{\hbox{\tiny$\square$}}\le{\mathbf E\,}\|\mathbb C^{-
1/2}\,X\|^{2+\delta}/(d^{(2+\delta)/2}\4N^{\delta/2}),\quad \text{for } \ 0\le\delta\le 2 . \eeq
Therefore, Theorem \ref{T1.5} implies the following Corollary \ref{T1.5c}.
\begin{corollary}{\label{T1.5c}} Let \/
$\mathbb Q^2=\mathbb I_d$, $5\le d<\infty$.
Then we have$:$
\beq \Delta _{N}^{(a)} \ll_d C \kern1pt \bgl( 1 +
\norm{a/\sigma}^3\bgr)\,{\mathbf E\,}\|\mathbb
C^{-1/2}\,X\|^{3+\delta}/N^{(1+\delta)/2} ,\quad \text{for } \ 0\le\delta\le 1,
\label{eq1.8c}\eeq and \beq \Delta _{N,{\hbox{\tiny$\square$}}}^{(a)}
\ll_d C \kern1pt \bgl( 1 + \norm{a/\sigma}^3\bgr) \,{\mathbf E\,}\|\mathbb C^{-
1/2}\,X\|^{2+\delta}/N^{\delta/2},\quad \text{for } \ 0\le\delta\le 2 ,
\label{eq1.8wc}\eeq
with\/ $C= \sigma^d\,(\det \mathbb C)^{-1/2}\,$. In particular,
\beq\label{eq1.8f}\max\bgl\{\Delta _{N}^{(a)},\Delta _{N,{\hbox{\tiny$\square$}}}^{(a)}\bgr\}
\ll_d C \kern1pt \bgl( 1 + \norm{a/\sigma}^3\bgr)\,{\mathbf E\,}\|\mathbb
C^{-1/2}\,X\|^4/N. \eeq
\end{corollary}
\smallskip
Theorem \ref{T1.3} and Corollary \ref{T1.5c} yield Theorems \ref{T1.1} and
\ref{T1.1a},
using that $E_0(x)\equiv0$, ${\mathbf E\,}\|\mathbb C^{-1/2}\,X\|^4\le\beta/\sigma_d^4$, and
$\Pi_2^{\diamond}+\Lambda _4^{\diamond}\leq \Pi_3^{\diamond}+\Lambda _4^{\diamond}\leq
\beta/(\sigma^4 \kern1pt N)$.
\smallskip
Comparing Theorem \ref{T1.5} and Corollary \ref{T1.5c} with Theorem \ref{T1.5},
we see that the constants in Theorem \ref{T1.5} and Corollary \ref{T1.5c} are written explicitly
in terms of moment characteristics of $\mathcal L(X)$. In the case of non-positive definite quadratic forms~$\mathbb Q$
such kind of estimates was unknown.
If, in the conditions of Theorem \ref{T1.5}, the distribution of
$X$ is symmetric or $a=0$, then the Edgeworth corrections
$E_a(x)$ and $E_a^{\hbox{\tiny$\square$}}(x)$ vanish and \beq\Delta _{N}^{(a)}
=\Delta _{N,{\hbox{\tiny$\square$}}}^{(a)}\leq C \kern1pt \bgl( \Pi_2^{\hbox{\tiny$\square$}} +
\Lambda _4^{\hbox{\tiny$\square$}}\bgr)\bgl( 1 + \norm{a/\sigma}^3\bgr)\label{eq1.8ff},
\quad \q\quad C= c_d\kern1pt \sigma^d\,(\det \mathbb C)^{-1/2}.\eeq The
corresponding inequality from Theorem~1.4 of BG (1997a) yields in
the case $s=9$ and $9\leq d\leq \infty $ under the condition
$\mathcal N_{\mathbb Q}( p,\delta, \mathcal S_o,c_0\kern1pt G/\sigma) $ with $\delta
= 1/300$ the bound \beq\Delta _N^{(a)}\leq C \kern1pt \bgl( \Pi_2^{\diamond}
+ \Lambda _4^{\diamond})\, \bgl(1+\norm{a/\sigma}^{4 }\bgr) ,\quad \q\quad C=c\,
p^{-4}. \label{eq1.8fg}\eeq It is clear that sometimes the bound
\eqref{eq1.8fg} may be sharper than \eqref{eq1.8ff}, but
unfortunately, it depends on~$p$ which is usually exponentially
small with respect to $\sigma_9/\sigma^2$.
One can find more precise estimates of constants in the case of
$d$-dimensional balls with $d\ge12$ in the papers of Nagaev and
Chebotarev (1999), (2005), G\"otze and Ulyanov~(2000), and
Bogatyrev, G\"otze and Ulyanov~(2006). In this case $\mathbb
Q=\mathbb I_d$. See also G\"otze and Ulyanov~(2000) for lower
bounds for $\Delta_N^{(a)}$ under different conditions on $a$ and
~$\mathcal L(X)$. In the papers mentioned above, the authors have
used the aproach of BG (1997a) and obtained bounds with constants
depending on $s<d$ largest eigenvalues
$\sigma_1^2\ge\sigma_2^2\ge\cdots\ge\sigma_s^2$ of the covariance
operator~$\mathbb C$ (see Nagaev and Chebotarev (1999), (2005),
with $d\ge s=13$, and G\"otze and Ulyanov~(2000), and Bogatyrev,
G\"otze and Ulyanov~(2006), with $d\ge s=12$). It should be
mentioned, that, in a particular case, where $\mathbb Q=\mathbb
I_d$ and $d\ge 12$, these results may be sharper than
\eqref{eq1.8}, for some covariance operators~$\mathbb C$.
Thus, we see that the statement of Theorem \ref{T1.5} is
especially interesting for $d=5,6,\ldots,11$. It is new even in
the case of $d$-dimensional balls. It is plausible that the bounds
for constants in Theorem \ref{T1.5} could be also improved for
balls with $d\ge5$, especially in the case where $d$ is large.
It seems however that this is
impossible in the case of general $\mathbb Q$ even if $\mathbb
Q^2=\mathbb I_d$. For example, we can consider the operator
$\mathbb Q$ such that $\mathbb Q \4e_j=e_{d-j+1}$, where $\mathbb
C\kern1pt e_j=\sigma_j^2e_j$, $j=1,2,\ldots,d$, are eigenvectors of $\mathbb
C$. Following the proof of Theorem~\ref{T1.5}, we see that the
bounds for the modulus of the characteristic function $\bgl| \widehat
\Psi_b (t)\bgr|=\bgl|{\mathbf E\,} \operatorname {e} \bgl \{ t\, \mathbb Q
\kern1pt[Z_N -b]\bgr\} \bgr|$ behave as the bounds for the modulus of
the characteristic function $\bgl|{\mathbf E\,} \operatorname {e} \bgl \{ t\,
\mathbb I_d \kern1pt[Z_N -b]\bgr\} \bgr|$ but with eigenvalues of the
covariance operator $\sigma_1\kern1pt\sigma_d$, $\sigma_2\kern1pt\sigma_{d-1}$,
$\sigma_3\kern1pt\sigma_{d-2}$, \ldots \ which can be essentially smaller than
$\sigma_1^2\ge\sigma_2^2\ge\sigma_3^2\ge\cdots$. Therefore, it is natural that
the bounds for constants in Theorem~\ref{T1.5} depends on the
smallest eigenvalues of the covariance operator~$\mathbb C$.
Note that, in the proof of Theorem~\ref{T1.3} in BG (1997a),
inequalities \eqref{eq1.6} and \eqref{eq1.6w} were derived for the
Edgeworth correction $E_a(x)$ defined by \eqref{eq1.2}. However,
from Theorems~\ref{T1.3} and ~\ref{T1.5f} or~\ref{T1.5} it follows
that, at least for $13\le d<\infty$, definitions \eqref{eq1.2}
and~\eqref{eq1.21} determine the same function~$E_a(x)$. Indeed,
both functions may be represented as $N^{-1/2}\,K(x)$, where
$K(x)$ are some functions of bounded variation which are
independent of~$N$. Furthermore, inequalities \eqref{eq1.6} and
\eqref{eq1.8} provide both bounds of order $\mathcal O(N^{-1}}\def\sign{\hbox{\rm sign})$.
This is possible if the Edgeworth corrections $E_a(x)$ are the
same in these inequalities.
On the other hand, it is proved (for $d\geq 9$) that definition
\eqref{eq1.2} determine
a function of bounded variation (see BG (1997a, Lemma
5.7)), while definition \eqref{eq1.21} has no sense for
$d=\infty$.
Introduce the concentration function \beq\label{con} Q(X ;\, \lambda
)=Q(X ;\, \lambda ;\, \mathbb Q ) = \sup_{a,\, x\in\mathbb R^d} \, \P \bgl\{ x
\leq \mathbb Q \kern1pt[X-a]\leq x+\lambda \bgr\} , \quad \text{for } \ \lambda \geq
0. \eeq It should be mentioned that the supremum in \eqref{con} is
taken not only over all $x$, but over all $x$ and $a\in\mathbb R^d$.
Usually, one defines the concentration function of the random
variable $\mathbb Q \kern1pt[X-a]$ taking the supremum over all
$x\in\mathbb R^d$ only. Note that, evidently, $Q(X+Y ;\, \lambda )\le Q(X ;\,
\lambda )$, for any $Y$ which is independent of~$X$.
The following Theorems \ref{T1.6f} and Theorem \ref{T2.1f} are
Theorems~1.5 and~2.1 from G\"otze and Zaitsev (2010).
\begin{theorem}{\label{T1.6f}} Let\/ $\mathbb Q^2=\mathbb I_d$,
$5\leq s\leq d\leq \infty$, $s< \infty$ and\/ $0\leq \delta \leq
1/(5\kern1pt s)$. Then we have$:$
$(i)$ If condition $\mathcal N_{\mathbb Q}(p,\delta ,\mathcal S_o, \widetilde
X )$ is fulfilled
with some $p>0$, then
\beq Q(Z_N ;\, \lambda )\ll_s (p\kern1pt N)^{-1} \, \max \bgl\{ 1 ;\,
\lambda\bgr\}, \quad \text{for } \ \lambda\geq 0 . \label{eq1.9f}\eeq
$(ii)$ If, for some $m$,
condition $\mathcal N_{\mathbb Q}(p,\delta ,\mathcal S_o, m^{-1/2}\kern1pt \widetilde Z_m )$ is fulfilled, then
\beq Q(Z_N ;\, \lambda )\ll_s(p\kern1pt N)^{-1} \, \max \bgl\{ m ;\, \lambda\bgr\}
,\quad \text{for } \ \lambda\geq 0. \label{eq1.10f}\eeq
\end{theorem}
\begin{theorem}{\label{T2.1f}} Let\/ $\mathbb Q^2=\mathbb I_d$
and\/ $5\leq d\leq \infty$. Let\/ $c_0$ be an arbitrary positive absolute
constant. Assume condition $\mathcal N_{\mathbb Q}(p,\delta ,\mathcal
S_o, c_0\kern1pt G/\sigma )$ to be satisfied with $s =5$ and\/ $\delta = 1/200$.
Then \beq Q(Z_N ;\, \lambda )\ll p^{-2} \max \bgl\{ \Pi_2^\diamond
+\Lambda _4^\diamond ;\, \lambda\kern1pt \sigma^{-2}\kern1pt N^{-1} \bgr\} ,\quad \text{for } \ \lambda\geq 0.
\label{eq2.1f}\eeq In particular, $Q(Z_N ;\, \lambda )\ll p^{-2}\kern1pt
N^{-1}\kern1pt \max \bgl\{ \beta \kern1pt\sigma^{-4} ;\,
\lambda\kern1pt\sigma^{-2}\bgr\}$.
\end{theorem}
Theorems \ref{T1.6f} and Theorem \ref{T2.1f} extend to the case
$5\le d\le\infty$ Theorems 1.6 and~2.1 of BG~(1997a) which were
proved for $9\le d\le\infty$.
We say that a random vector $Y$ is concentrated in $\mathbb
L\subset \mathbb R^d $ if $\P \{ Y\in \mathbb L\} =1$. In BG~(1997a, item
$(iii)$ of Theorem 1.6) it was shown that if
$\widetilde X$ is not concentrated in a proper closed linear
subspace of $\mathbb R^d$, $1\leq d\leq \infty$, then, for any $\delta>0$ and
$\mathcal S$
there exists a natural number $m$ such that the condition
$\mathcal N_{\mathbb Q}(p,\delta ,\mathcal S, m^{-1/2}\kern1pt \widetilde Z_m )$
holds with
some $ p>0$.
In this paper, we shall prove the following Theorems \ref{T1.6}
and Theorem \ref{T2.1}.
\begin{theorem}{\label{T1.6}} Let\/ $\mathbb Q^2=\mathbb I_d$,
$5\leq s= d< \infty$ and\/ $0\leq \delta \leq 1/(5\kern1pt s)$.
Then we have$:$
$(i)$ If condition $\mathcal N(p,\delta ,\mathcal S_o, \mathbb C^{-
1/2}\,\widetilde X )$ is fulfilled
with some $p>0$, then
\beq Q(Z_N ;\, \lambda )\ll_d (p\kern1pt N)^{-1} \, \max \bgl\{ 1 ;\,
\lambda\,\sigma^{-2}\bgr\}\,\sigma^d\,(\det\mathbb C)^{-1/2}, \quad \text{for } \ \lambda\geq 0 .
\label{eq1.9}\eeq
$(ii)$ If, for some $m$,
condition $\mathcal N(p,\delta ,\mathcal S_o, m^{-1/2}\kern1pt\mathbb C^{-
1/2}\, \widetilde Z_m )$ is fulfilled, then \beq Q(Z_N ;\, \lambda )\ll_d (p\kern1pt
N)^{-1} \, \max \bgl\{ m ;\, \lambda\,\sigma^{-2}\bgr\}\,\sigma^d\,(\det\mathbb
C)^{-1/2} ,\quad \text{for } \ \lambda\geq 0. \label{eq1.10}\eeq
\end{theorem}
\begin{theorem}{\label{T2.1}} Assume that\/ $5\leq d< \infty$
and that $\mathbb Q^2=\mathbb I_d$.
Then \beq Q(Z_N
;\, \lambda )\ll_d \max \bgl\{ \Pi_2^{\hbox{\tiny$\square$}} +\Lambda _4^{\hbox{\tiny$\square$}} ;\, \lambda\kern1pt
\sigma^{-2}\kern1pt N^{-1} \bgr\}\,\sigma^d\,(\det\mathbb C)^{-1/2}
,\quad \text{for } \ \lambda\geq 0. \label{eq2.1}\eeq In particular, $Q(Z_N ;\, \lambda
)\ll_d N^{-1}\kern1pt \max \bgl\{ {\mathbf E\,}\|\mathbb C^{- 1/2}\,X\|^4 ;\,
\lambda\kern1pt\sigma^{-2}\bgr\}\,\sigma^d\,(\det\mathbb C)^{-1/2}$.
\end{theorem}
Theorem \ref{T1.6} and Theorem \ref{T2.1} yield more explicit
versions of Theorems \ref{T1.6f} and Theorem \ref{T2.1f} as well
as Theorem \ref{T1.5} is in a sense a more explicit version of
Theorem \ref{T1.5f}. We should mention that Theorems \ref{T1.5f},
\ref{T1.6f} and \ref{T2.1f} do not follow from Theorems
\ref{T1.5}, \ref{T1.6} and~\ref{T2.1}. For the proofs of these
theorems we refer the reader to the paper of G\"otze and Zaitsev
(2010) and to the preprint of G\"otze and Zaitsev
(2009) which are available in internet.
For example, the bounds in Theorems \ref{T1.5f},
\ref{T1.6f} and \ref{T2.1f} may be sharper than those from Theorems
\ref{T1.5}, \ref{T1.6} and \ref{T2.1}, in a particular case, where $\mathbb Q=\mathbb I_d$ and
$\sigma_5\asymp_d\sigma$. Under some additional conditions, $\sigma^d\,(\det\mathbb
C)^{-1/2}$ is replaced by $\exp(c\kern1pt\sigma^{-2}\kern1pt\sigma_5^2)\asymp_d1$. On the
other hand, $\sigma^d\,(\det\mathbb
C)^{-1/2}$ provides a power-type dependence on eigenvalues of~$\mathbb C$
and the results are valid for $\mathbb Q$ which might be not
positive definite.
In Theorems \ref{T1.5} and Theorem \ref{T2.1}, we do not assume
the fulfilmemt of conditions $\mathcal N(\4\cdot\4)$ or $\mathcal
N_{\mathbb Q}(\4\cdot\4)$. In the proofs, we shall use, however, that,
for an arbitrary absolute positive constant~$c_0$ and any positive
quantity~$c_d$ depending on $d$ only, condition $\mathcal N(p,\delta
,\mathcal S_o , c_0 \,\mathbb C^{- 1/2}\, G)$ is fulfilled with
$s=d$, $\delta=c_d$ and $p\asymp_d1$, for any orthonormal system
$\mathcal S_o$.
\smallskip
\smallskip
Similarly to BG
(1997a), in Section~ \ref{s2}, we prove bounds for concentration
functions. The proof is technically simpler as that of Theorem
\ref{T1.5}, but it shows how to apply the principal ideas. This
proof repeats almost literally the corresponding proof of BG
(1997a). The only difference consists in the use of new Lemma
\ref{GZ2} which allows us to estimate characteristic functions of
quadratic forms for relatively large values of argument~$t$. In
Sections \ref{s3} and \ref{s4}, Theorem \ref{T1.5} is proved. We
shall replace Lemma~9.4 of BG~(1997a) by its improvement,
Lemmas~\ref{L9.4}. Another difference is in another choice of $k$
in \eqref{eq156a} and \eqref{eq156c} in comparison with that in
BG~(1997a).
In Sections \ref{s5}--\ref{s7} we prove estimates for
characteristic functions which were discussed in Section~\ref{s1}.
\medskip
{\bf Acknowledgment} We would like to thank V.V. Ulyanov
for helpful discussions.
\smallskip
$\phantom 0$
$\phantom 0$
\section{\label {s2}Proofs of bounds for concentration functions}
\smallskip
{\it Proof of Theorems $\ref{T1.6}$ and $\ref{T2.1}$}. Below we
shall prove the assertions \eqref{eq1.9};
$\eqref{eq1.9}\Longrightarrow \eqref{eq1.10}$ and $\eqref{eq1.10}
\Longrightarrow \eqref{eq2.1} $. The proof repeats almost
literally the corresponding proof of BG (1997a). It is given here
for the sake of completeness. The only essential difference is in
the use of Lemma~\ref{GZ2} in the proof of Lemma~\ref{L2.3}. We
have also to replace everywhere 9 by 5 and $\diamond$ by
${\hbox{\tiny$\square$}}$. $\square$\medskip
For $ 0\leq t_0\leq T$ and $b\in\mathbb R^d$,
define the integrals
$$I_0=\int_{-T}^{T} \bgl| \widehat \Psi_b (t)\bgr| \, dt,\quad \q\quad \q
I_1= \int_{ t_0 \leq |t|\leq T} \bgl| \widehat \Psi_b
(t)\bgr| \, \ffrac {dt} {|\penalty250\mskip\thickmuskip\mskip-\thinmuskip t\penalty250\mskip\thickmuskip\mskip-\thinmuskip |},$$
where \beq\label{eq823} \widehat \Psi_b(t)={\mathbf E\,} \operatorname {e} \bgl \{
t\, \mathbb Q \kern1pt[Z_N -b]\bgr\} \eeq denotes the Fourier--Stieltjes
transform of
the distribution function $\Psi_b $
of $ \mathbb Q \kern1pt[Z_N-b]$ (see \eqref{Fur} and \eqref{eq1.18}).
Note that $\bgl| \widehat \Psi_b (-t)\bgr|=\bgl| \widehat \Psi_b (t)\bgr|$.
\begin{lemma}\label{L2.3} Assume
condition $\mathcal N(p,\delta, \mathcal S_o, \mathbb C^{-1/2}\,\widetilde X)$ with some $
0\leq \delta \leq 1/(5\kern1pt s)$ and $5\le s=d<\infty $. Let $\sigma^2=1$ and
\beq \label{eq23}
t_0=c_1(s)\kern1pt \sigma_1^{-2}(p\kern1pt N)^{-1+ 2/s} , \quad \q
c_2(s)\,\sigma_1^{-2}\leq T\leq c_3(s)\,\sigma_1^{-2} \eeq
with some positive constants $c_j(s)$,
$1\leq j\leq 3$. Then \beq I_0\ll_s (\det \mathbb C)^{-1/2}\,(p\kern1pt
N)^{-1} ,\quad \q\quad \q\quad I_1 \ll_s (\det \mathbb C)^{-1/2}\,( p\kern1pt
N)^{-1}.
\label{eq2.3}\eeq
\end{lemma}
{\it Proof}. Note that the condition $\sigma^2=1$ implies that
\beq\label{MTq}T\asymp_s\sigma_1^2\asymp_s\sigma^2=1 \quad \text{and}\quad \det
\mathbb C\ll_s1. \eeq Denote $k =p\kern1pt N$. Without loss of
generality we assume that
$k\geq c_s$, for a sufficiently large quantity $c_s$ depending on $s$ only.
Indeed, if $k\leq c_s$, then one can prove \eqref{eq2.3} using
\eqref{MTq} and $|\widehat \Psi_b |\leq 1$. Choosing $c_s$ to be large
enough, we ensure that $k\geq c_s$ implies $1/k\leq t_0 \leq T$.
Lemma \ref{GZ2} and \eqref{MTq} imply now that \beq \label{MTN38}
\int_{c_4(s)k^{-1+2/s}}^{T } \bgl| \widehat \Psi_b (t)\bgr| \ffrac {dt}
t
\ll_s \ffrac {(\det \mathbb C)^{-1/2}}k,
\eeq for any $c_4(s)$ depending on $s$ only. Inequalities
\eqref{MTq} and \eqref{MTN38} imply \eqref{eq2.3} for $I_1$.
Let us prove inequality \eqref{eq2.3} for $I_0$.
By \eqref{MTq} and by Lemma \ref
{GZ},
for any\/ $\gamma >0$ and any fixed\/ $t\in\mathbb R$
satisfying\/ $k^{1/2}\left|t\right|\le c_5(s)$, where $c_5(s)$ is
an arbitrary quantity depending on $s$ only, we have (taking into
account that $|\widehat \Psi_b |\leq 1$) \beq\label{MTN}\bgl| \widehat
\Psi_b (t)\bgr|
\ll_{\gamma , s} \min\bgl\{ 1;\, k^{-\gamma }+
k^{-s/2}\,\left|t\right|^{-s/2}\,(\det \mathbb C)^{-1/2}\bgr\} , \quad \q\quad k= p\kern1pt
N. \eeq
Furthermore, choosing an appropriate $\gamma $ and using \eqref{MTq}--\eqref{MTN}, we obtain
\beq\label{MTN23}(\det \mathbb C)^{1/2}\,I_0
\ll_s \int_{0}^{ 1/k} \, {dt} + \ffrac {1} {k}+\int_{1/k}^\infty
\ffrac {dt} {( t\kern1pt k )^{s/2}}
\ll_s \ffrac {1} {k} , \eeq proving \eqref{eq2.3} for $I_0$.
$\square $
\medskip
\smallskip
{\it Proof of\/ $\eqref{eq1.9}$}. Let $\sigma^2=1$. Using a well-known
inequality for concentration functions (see, for example, Petrov
(1975, Lemma ~3 of Ch{.} 3)), we have \beq Q(Z_N ;\, \lambda )\leq
4\,\sup_{b\in \mathbb R^d} \, \max \bgl\{ {\lambda } ;\, 1 \bgr\}
\int_{0}^1 \bgl| \widehat \Psi_b (t)\bgr| \, dt .\label{eq2.5}\eeq To
estimate the integral in \eqref{eq2.5} we shall apply Lemma
\ref{L2.3} which implies that
\beq\label{MTN89}Q(Z_N ;\, \lambda )\ll_d
\max \bgl\{ {\lambda } ;\, 1 \bgr\}
(p\kern1pt N)^{-1} \, (\det\mathbb C)^{-1/2},\eeq proving \eqref{eq1.9}
in the case $\sigma^2=1$. If $\sigma^2\ne1$, we obtain \eqref{eq1.9}
applying \eqref{MTN89} to $Z_N/\sigma$. $\square$
\medskip
\smallskip
{\it Proof of\/ $\eqref{eq1.9}\Longrightarrow \eqref{eq1.10}$}.
Without loss of generality we can assume that $N/m\geq 2$. Let
$Y_1,Y_2,\dots $ be independent copies of $m^{-1/2} \kern1pt Z_m$.
Denote $W_k= Y_1+\dots +Y_k$. Then $\mathcal L (Z_N)= \mathcal L
(\sqrt{m} \, W_k +y)$
with $k=\lceil N/m\rceil$ and with some
$y$ independent of $W_k$. Therefore,
$Q(Z_N; \lambda ) \leq Q(W_k ; \lambda /m )$. In order to estimate $Q(W_k ;
\lambda /m )$ we apply \eqref{eq1.9} replacing $Z_N$ by $W_k$. We
have
\begin{eqnarray} Q(W_k ; \lambda /m )&\ll_s &(p\kern1pt k )^{-1} \max\bgl\{ 1;\,
\lambda\,\sigma^{-2} /m\bgr\}\,\sigma^d\,(\det\mathbb C)^{-1/2}\nonumber\\ &\ll_s
&(p\kern1pt N )^{-1} \max\bgl\{ m;\,
\lambda\,\sigma^{-2}\bgr\}\,\sigma^d\,(\det\mathbb C)^{-1/2}. \quad \q \square
\end{eqnarray}
\medskip
\smallskip
Recall that truncated random vectors and their moments are
defined by \eqref{eq1.4a}--\eqref{eq1.5t} and that $\mathbb C=\cov
X=\cov G$.
\begin{lemma}\label{L2.4}
The random vectors $X^{\hbox{\tiny$\square$}}$, $X_{\hbox{\tiny$\square$}}$ satisfy
$$\langle \mathbb C \kern1pt x,x\rangle
= \langle \cov X^{\hbox{\tiny$\square$}} \, x,x\rangle +{\mathbf E\,} \langle X_{\hbox{\tiny$\square$}}
,x\rangle^2+
\langle {\mathbf E\,} X^{\hbox{\tiny$\square$}} ,x\rangle^2.$$
There exist independent centered Gaussian vectors\/ $G_\ast$ and\/
$W$ such that $$\mathcal L (G) =\mathcal L (G_\ast +W )$$ and
$$2\kern1pt \cov G_\ast =2\kern1pt \cov
X^{\hbox{\tiny$\square$}} =\cov \widetilde {X^{\hbox{\tiny$\square$}}},\quad \
\langle \cov W\kern1pt x,x\rangle ={\mathbf E\,} \langle X_{\hbox{\tiny$\square$}} ,x\rangle^2+
\langle {\mathbf E\,} X^{\hbox{\tiny$\square$}} ,x\rangle^2.$$
Furthermore, $${\mathbf E\,} \| \mathbb C^{-1/2}\,G \|^2= d={\mathbf E\,} \| \mathbb
C^{-1/2}\,G_\ast \|^2 + {\mathbf E\,} \|\mathbb C^{-1/2}\, W \|^2$$ and\/
\,${\mathbf E\,} \|\mathbb C^{-1/2}\, W \|^2\leq 2\, d \, \Pi_2^{\hbox{\tiny$\square$}}$.
\end{lemma}
We omit the simple proof of this lemma (see BG (1997a, Lemma
2.4) for the same statement with $\diamond$ instead of ${\hbox{\tiny$\square$}}$).
Lemma~\ref{L2.4} allows us to define the vector $X'$
by~\eqref{eq1.20}.
\smallskip
Recall that $Z_N^{\hbox{\tiny$\square$}}$ and $Z_N^\diamond$ denote sums of $N$
independent copies of $X^{\hbox{\tiny$\square$}}$ and $X^\diamond$ respectively.
\begin{lemma}\label{L2.5} Let $\varepsilon >0$.
There exist absolute positive constants $c$ and $c_1$ such that
the condition $\Pi_2^{\hbox{\tiny$\square$}} \leq c_1\kern1pt p\, \delta^2 / (d\kern1pt\varepsilon^2)$
implies that
$$\mathcal N( p,\delta ,\mathcal S ,\varepsilon \,\mathbb C^{-1/2}\,G ) \Longrightarrow
\mathcal N ( p/4 , 4\kern1pt \delta ,\mathcal S ,\varepsilon \,(2\kern1pt m)^{-1/2}\,\mathbb C^{-1/2}\,
\widetilde
{Z_m^{\hbox{\tiny$\square$}}} ),$$
for
$m\geq c\kern1pt\varepsilon^4\kern1pt d^2\kern1pt N\kern1pt \kern1pt \Lambda _4^{\hbox{\tiny$\square$}}/ (p\, \delta^4 )$.
\end{lemma}
Lemmas \ref{L2.4} and \ref{L2.5} are in fact the statements of Lemmas
2.4 and~2.5 from BG (1997a) applied to the vectors $\mathbb
C^{-1/2}\,X$ instead of the vectors $X$. We use in this connection
equalities \eqref{eq13}, \eqref{eq14rr} and~\eqref{eq14ru}
replacing in the formulation $\sigma^2$, $\Lambda _4^\diamond$,
$\Pi_q^\diamond$, $G$, $Z_m^\diamond$, $\ldots$
by $d$, $\Lambda _4^{\hbox{\tiny$\square$}}$, $\Pi_q^{\hbox{\tiny$\square$}}$, $\mathbb C^{-1/2}\,G$,
$Z_m^{\hbox{\tiny$\square$}}$, \ldots \ respectively.
\smallskip
{\it Proof of\/ $\eqref{eq1.10}\Longrightarrow\eqref{eq2.1}$}. By
a standard truncation argument, we have \beq \bgl| \P\bgl \{ Z_N
\in A \bgr\} - \P\bgl \{ Z_N^{\hbox{\tiny$\square$}} \in A \bgr\} \bgr|
\leq N\,\P\bgl \{ \|\mathbb C^{-1/2}\,X\|> \sqrt{d\4N} \bgr\}\leq \Pi_2^{\hbox{\tiny$\square$}} ,
\label{eq2.8}\eeq for any Borel set $A$, and \beq Q(Z_N,\lambda) \leq
\Pi_2^{\hbox{\tiny$\square$}} + Q(Z_N^{\hbox{\tiny$\square$}} ,\lambda ). \label{eq2.9}\eeq Recall that
we are proving \eqref{eq2.1} assuming that $5\le d<\infty$. It is
easy to see that, for an arbitrary absolute positive
constant~$c_0$, condition $N(p,\delta ,\mathcal S_o , c_0 \,\mathbb
C^{- 1/2}\, G)$ with \beq s=d,\quad \delta=1/(20\,s),\quad
p\asymp_d1\label{eq3.3sss}\eeq is in fact fulfilled automatically
for any orthonormal system $\mathcal S_o$, since the vector
$\mathbb C^{-1/2}\, G$ has standard Gaussian distribution in $\mathbb R^d$
and $\P\bgl \{ \norm{c_0 \,\mathbb C^{-1/2}\,
G-e}\le\delta\bgr\}=c(d)$ for any vector $e\in\mathbb R^d$ with $\norm
e=1$. Clearly, $4\kern1pt\delta=1/(5\4s)$. Write $K= \varepsilon/ \sqrt{2} $
with $\varepsilon = c_0 $.
Then, by \eqref{eq3.3sss} and Lemma \ref{L2.5}, we have
\beq \mathcal N ( p,\delta ,\mathcal S_o ,\varepsilon \kern1pt \mathbb C^{-1/2}\,G )
\Longrightarrow \mathcal N ( p/4 , 4\kern1pt \delta ,\mathcal S_o
,m^{-1/2}\,K \kern1pt \mathbb C^{-1/2}\,\widetilde {Z_m^{\hbox{\tiny$\square$}}} ),
\label{eq2.10}\eeq
provided that
\beq \Pi_2^{\hbox{\tiny$\square$}} \leq c_1(d) , \quad \q\quad m\geq c_2(d) \kern1pt N\kern1pt \kern1pt
\Lambda _4^{\hbox{\tiny$\square$}}.\label{eq2.11}\eeq
Without loss of generality
we may assume that $\Pi_2^{\hbox{\tiny$\square$}} \leq c_1(d)$, since otherwise
the result follows easily from the trivial inequality
$Q(Z_N;\lambda)\leq 1$.
The non-degeneracy
condition \eqref{eq2.10} for $K \kern1pt \widetilde {Z_m^{\hbox{\tiny$\square$}}} $
allows to apply \eqref{eq1.10}
of Theorem \ref{T1.6}, and, using \eqref{eq3.3sss}, we obtain \beq
Q(Z_N^{\hbox{\tiny$\square$}} ,\lambda) = Q( K \kern1pt Z_N^{\hbox{\tiny$\square$}} ,K^2 \kern1pt \lambda ) \ll_d N^{-1}
\max\{ m;\, K^2 \kern1pt \lambda/K^2\kern1pt\sigma^2 \}\,\sigma^d\,(\det\mathbb C)^{-1/2},
\label{eq2.12}\eeq for any $m$ such that \eqref{eq2.11} is
fulfilled. Choosing the minimal $m$ in \eqref{eq2.11}, we obtain
\beq Q(Z_N^{\hbox{\tiny$\square$}} ,\lambda) \ll_d \max\{ \Lambda _4^{\hbox{\tiny$\square$}} ;\, \lambda/(\sigma^2\kern1pt
N)\}\,\sigma^d\,(\det\mathbb C)^{-1/2}. \label{eq2.13}\eeq Combining
the estimates \eqref{eq2.9} and \eqref{eq2.13}, we conclude the
proof. $\square$
\medskip
\section{\label {s3}Auxiliary lemmas}
In Sections \ref{s3} and \ref{s4} we shall prove Theorem \ref{T1.5}. Therefore,
we shall assume that
its conditions are satisfied.
We consider the case $d<\infty$
assuming that the following conditions are
satisfied: \beq \mathbb Q^2=\mathbb I_d,\quad \sigma^2 =1,\quad d\ge5,\quad
b=\sqrt N\,a.\label{eq3.3}\eeq Moreover, it is easy to see that,
for any absolute positive constant~$c_0$ and for any orthonormal
system $\, \mathcal S_o=\{ \fs e1s\}\subset\mathbb R^d$ , condition \beq
N(p,\delta ,\mathcal S_o , c_0 \,\mathbb C^{- 1/2}\, G)\quad \text{with}\quad
p\asymp_d1,\quad 5\le s=d<\infty,\quad \delta =1/(20\,s)\label{eq3.3ss}\eeq
is in fact fulfilled automatically since the vector $\mathbb
C^{-1/2}\, G$ has standard Gaussian distribution in $\mathbb R^d$ and,
therefore, $\P\bgl \{ \norm{c_0 \,\mathbb C^{-1/2}\,
G-e}\le\delta\bgr\}=\P\bgl \{ \norm{ \,\mathbb C^{-1/2}\,
G-c_0^{-1}}\def\sign{\hbox{\rm sign}\,e}\le c_0^{-1}}\def\sign{\hbox{\rm sign}\,\delta\bgr\}=c(d)$ for any vector
$e\in\mathbb R^d$ with $\norm e=1$.
Notice that the assumption $\sigma^2=1$ does not restrict generality
since from Theorem~\ref{T1.5} with $\sigma^2=1$ we can derive the
general result
replacing $X$, $G$ by $X/\sigma$,
$G/\sigma$, etc. Other assumptions in \eqref{eq3.3} are included as
conditions in Theorem \ref{T1.5}.
Section \ref{s3} is devoted to some auxiliary lemmas which are similar to
corresponding lemmas of BG (1997a).
In several places, the proof of Theorem \ref{T1.5} repeats almost literally the
proof
of Theorem~1.5 in BG (1997a). Note, however, that we shall use truncated vectors
$X^{\hbox{\tiny$\square$}}_j$,
while in BG (1997a) the vectors $X^\diamond_j$ were involved.
We start with
an application of the Fourier transform
to the functions $\Psi_b$ and $\Phi_b$, where $b=\sqrt N\,a$. We shall estimate
integrals
over the Fourier transforms
using results of Sections \ref{s2}, \ref{s5}--\ref{s7} and some technical
lemmas of BG~(1997a).
We shall also apply some methods of estimation of the rate of
approximation in the CLT in multidimensional spaces (cf{.}, e.g.,
Bhattacharya and Rao (1986)).
Below we shall use the following formula for the Fourier inversion (see, for example,
BG (1997a)). A~smoothing inequality of Prawitz (1972) implies (see
BG (1996, Section~4)) that \beq F(x) = \ffrac 12 + \ffrac {i}{
2\pi }
\operatorname{V.P.} \int_{|t|\le K} \operatorname {e} \bgl\{ -xt\bgr\}
\widehat F (t)\, \ffrac {dt} t +R, \label{eq3.1}\eeq
for any $K>0$ and any distribution
function $F$ with characteristic function $\widehat F $ (see
\eqref{Fur}), where \beq \label{eq3.1o}|R|\leq \ffrac 1K
\int_{|t|\le K} | \widehat F (t) |\, {dt}.\eeq Here $ \operatorname{V.P.} \int f(t)\,
dt =\lim_{\varepsilon \to 0} \int_{|t| > \varepsilon }f(t)\, dt$
denotes the Principal Value of the integral.
Recall that the random vectors $X^{\hbox{\tiny$\square$}}$, $X'$ are defined in
\eqref{eq1.4t} and \eqref{eq1.20} and $Z_N^{\hbox{\tiny$\square$}}$, $Z_N'$ are sums
of $N$ their independent copies. Note that the Gaussian vector $W$ involved
in~\eqref{eq1.20}
is independent of all other vectors and have properties described in
Lemma~\ref{L2.4}.
Write $\Psi^{\hbox{\tiny$\square$}}_b $ and
$\Psi'_b $ for the distribution function of $\mathbb Q \kern1pt[Z_N^{\hbox{\tiny$\square$}}-b
]$ and $\mathbb Q \kern1pt[Z_N'-b ]$ respectively. For $0\leq k\leq N$
introduce the distribution function
\beq
\Psi^{(k)}_b (x) =\P \bgl\{ \mathbb Q \kern1pt\bgl[\fsu G1k+ X_{k+1}' +\dots
+ X_{N}' -b\bgr]\leq x \bgr\} . \label{eq3.16} \eeq Notice that
$\Psi^{(0)}_b=\Psi'_b$, $\Psi^{(N)}_b=\Phi_b $.
The proof of the following lemma repeats the proof of Lemma 3.1 of BG (1997a).
The difference is that here we use the truncated vectors $X_j^{\hbox{\tiny$\square$}}$ instead
of $X_j^\diamond$.
\begin{lemma}\label{L3.1}Let $c_d$ be
a quantity depending on $d$ only. There exist positive quantities
$c_1(d)$ and $c_2(d)$ depending on $d$ only such that the
following statement is valid.
Let\/ $\Pi_2^{\hbox{\tiny$\square$}} \leq c_1(d)\kern1pt p $
and let
an integer $1\leq m\leq N$
satisfy ${m\geq c_2(d) \,N\kern1pt \Lambda _4^{\hbox{\tiny$\square$}} /p}$, Write
$$K= c_0^2 /(2\kern1pt m),\quad \q
\quad \q t_1 = c_d \kern1pt (p N/m )^{-1+2/d}.$$ Let\/ $F$ denote any of the
functions $\Psi^{\hbox{\tiny$\square$}}_b$, $\Psi'_b$, $\Psi^{(k)}_b$ or\/
$\Phi_b$. Then we have \beq F (x) = \ffrac 12 + \ffrac {i}{ 2\pi }
\operatorname{V.P.} \int_{|t|\leq t_1 } \operatorname {e} \{ -x\kern1pt t \kern1pt K \}\,
\widehat F (t \kern1pt K )\, \ffrac {dt} t +R_1 , \label{eq3.17}\eeq with $
|R_1|\ll_d (p\kern1pt N )^{-1} \kern1pt m\,(\det\mathbb C)^{-1/2}$.
\end{lemma}
{\it Proof.} We shall assume that $ (p\kern1pt N )^{-1} \kern1pt m\le c_3(d)$
with sufficiently small $c_3(d)$ since otherwise the statement of
Lemma~\ref{L3.1} is trivial (see \eqref{MTq}, \eqref{eq3.1} and
\eqref{eq3.1o}). Let us prove~\eqref{eq3.17}. We shall combine
\eqref{eq3.1} and Lemma \ref{L2.3}. Changing the variable $t= \tau
\kern1pt K $
in formula \eqref{eq3.1}, we obtain \beq F (x) = \ffrac 12 + \ffrac
{i}{ 2\pi }
\operatorname{V.P.} \int_{|t|\leq 1} \operatorname {e} \{ -x\kern1pt t \kern1pt K \}\,
\widehat F (t \kern1pt K )\, \ffrac {dt} t +R, \label{eq3.19} \eeq where
\beq \left|R\kern1pt\right|\leq \int_{|t|\leq 1 } \bgl| \widehat F (t \kern1pt K )
\bgr|\, {dt} . \label{eq3.19a}\eeq Notice that \,$\Psi_b^{\hbox{\tiny$\square$}}$,
$\Psi'_b$, $\Psi_b^{(k)}$ and $\Phi_b$ \,are distribution
functions of random variables which may be written in the form:
\beq\mathbb Q \kern1pt[V+T ],\quad \q\quad V\=
G_1 +\dots + G_k +X_{k+1}^{\hbox{\tiny$\square$}}+\dots +X_N^{\hbox{\tiny$\square$}} ,\nonumber\eeq
with some $k$, $0\leq k\leq N$, and some random vector $T$ which
is independent of $X_j^{\hbox{\tiny$\square$}} $ and $G_j$, for all~$j$. Let us
consider separately two possible cases: $k\geq N/2$ and $k<N/2$.
{\it The case $k< N/2$}. Let
$Y$ denote a sum of $m$ independent copies of
$K^{1/2} \kern1pt X^{\hbox{\tiny$\square$}} $. Let $\is Y$
be independent copies of $Y$.
Then
we have \beq \mathcal L(K^{1/2} \kern1pt V) =\mathcal L(\fsu Y1l +T_1)
\label{eq3.20}\eeq with $l= \lceil N/(2\kern1pt m)\rceil$ and some
random $T_1$ independent of $\fs Y1l$. By \eqref{eq3.3ss} and by
Lemma \ref{L2.5}, we have
\beq\mathcal N( p,\delta, \mathcal S , c_0 \kern1pt\mathbb C^{-1/2}\, G ) \Longrightarrow
\mathcal N( p/4,4 \kern1pt\delta, \mathcal S , \mathbb C^{-1/2}\,\widetilde Y
)\label{eq3.21}\eeq provided that \beq \Pi_2^{\hbox{\tiny$\square$}} \ll p/d^{3}
\quad \q \text{ and}\quad \q m\gg d^6\4N\kern1pt \Lambda _4^{\hbox{\tiny$\square$}} /
p.\label{eq3.22}\eeq The inequalities in \eqref{eq3.22} follow
from conditions of Lemma \ref{L3.1} if we choose some sufficiently
small $($resp. large$)$ $c_1(d)$ $($resp.~ $c_2(d))$. Due to
\eqref{eq3.3}, \eqref{eq3.3ss}, \eqref{eq3.20} and \eqref{eq3.21},
we can apply Lemma \ref{L2.3} in order to estimate the integrals
in \eqref{eq3.19} and \eqref{eq3.19a}.
Replacing in Lemma~\ref{L2.3} $X$ by $Y$ and $N$
by $l$, we obtain \eqref{eq3.17} in the case $k<N/2$.
{\it The case $k\geq N/2$}. We can argue as in the previous
case defining now $Y$ as a sum of $m$ independent copies of
$K^{1/2} \kern1pt G$.
Condition $\mathcal N( p/4,4 \delta, \mathcal S_o , \mathbb C^{-1/2}\,\widetilde Y )$
is satisfied by \eqref{eq3.3ss}, since now $\mathcal L( \widetilde
Y)=\mathcal L( c_0\kern1pt G )$. $\square $
\smallskip
Following BG (1997a), introduce the upper bound $\varkappa \bgl( t; N, \mathcal
L (X),
\mathcal L (G)\bigr)$
for the characteristic function of quadratic forms ({cf.\!}
Bentkus (1984) and Bentkus, G\" otze and Zitikis (1993)). We
define
$\varkappa \bgl( t; N, \mathcal L (X),
\mathcal L (G)\bigr) =\varkappa \bgl( t; N, \mathcal L (X)\bigr)
+ \varkappa \bgl( t; N,\mathcal L (G)\bigr)$, where \beq \varkappa
( t; N, \mathcal L (X)) = \sup_{x\in \mathbb R^d } \;\bgl| {\mathbf E\,}
\operatorname {e}\bgl\{ t \kern1pt \mathbb Q \kern1pt [Z_j ]+ \langle x, Z_j
\rangle \bgr\} \bgr| ,\quad \q Z_j=\fsu X1j ,\label{eq3.23}\eeq with $
j= \bgl\lceil(N-2)/14\bgr\rceil$. In the sequel, we shall use that
\beq\varkappa (t ; N, \mathcal L(X') ,\mathcal L(G) )\leq
\varkappa (t ; N, \mathcal L(X^{\hbox{\tiny$\square$}} ) ,\mathcal L(G)
).\label{eq1app}\eeq For the proof, it suffices to note that
$X'=X^{\hbox{\tiny$\square$}} -{\mathbf E\,} X^{\hbox{\tiny$\square$}}+W$ and $W$ is independent of $
X^{\hbox{\tiny$\square$}}$.
\smallskip
\begin{lemma}\label{L3.2}
Let the conditions of Lemma $\ref{L3.1}$ be satisfied. Then
\begin{eqnarray}\int_{|t|\leq t_1 }&&\!\!\!\! \hskip-1cm\bgl( |t|\kern1pt K\bgr)^{\alpha }\kern1pt
\varkappa \bgl(t\kern1pt K ; N, \mathcal L( X^{\hbox{\tiny$\square$}} ),\mathcal L (G)\bigr)
\, \ffrac {dt} {|t|}\nonumber\\& \ll_{\alpha ,d}& (\det \mathbb C)^{-1/2}\,
\begin{cases}
(N\4p ) ^{-\alpha }, & \text{for}\ \,
0\leq \alpha < d/2, \\
(N\kern1pt p) ^{-\alpha }\,\bgl(1+ \left|\,\log(N\kern1pt p/m)\right|\bgr),& \text{for}\ \,
\alpha = d/2,\\
(N\kern1pt p) ^{-\alpha }\,\bgl(1+(N\kern1pt p/m)^{(2 \alpha - d)/d}\bgr), & \text{for}\ \,
\alpha > d/2.\label{eee}
\end{cases}
\end{eqnarray}
\end{lemma}
Lemma \ref{L3.2} is a generalization of Lemma~3.2 from BG (1997a)
which contains the same bound for $0\leq \alpha < d/2$. In this paper,
we have to estimate the left hand side of \eqref{eee} in the case
$d/2\leq \alpha $ too.
{\it Proof}. We shall assume again that $ (p\kern1pt N )^{-1} \kern1pt m\le
c_3(d)$ with sufficiently small $c_3(d)$ since otherwise
\eqref{eee} is an easy consequence of
$\left|\varkappa\,\right|\le1$.
Note that $\bgl|{\mathbf E\,} \operatorname {e}\bgl\{ t \kern1pt \mathbb Q \kern1pt [Z_j
]+ \langle x, Z_j \rangle \bgr\}\bgr|=\bgl|{\mathbf E\,} \operatorname {e}
\bgl \{ t\, \mathbb Q \kern1pt[Z_j -y]\bgr\}\bgr|$ with $y=-\mathbb Q
\4x/2$. By \eqref{eq3.3ss} and \eqref{eq3.21}, the condition
$\mathcal N( p/4, 4 \delta, \mathcal S_o , K^{1/2} \kern1pt \mathbb C^{-1/2}\,\widetilde
{Z_m^{\hbox{\tiny$\square$}}} )$
is fulfilled. Therefore, collecting
independent copies of $K^{1/2}\kern1pt X^{\hbox{\tiny$\square$}}$
in groups as in \eqref{eq3.20}, we can
apply Lemma \ref{GZ}. By \eqref{MTq}, \eqref{eq3.3ss} and this lemma, for any $\gamma >0$ and $|t|\leq t_1$,
$$ \varkappa (t\kern1pt K ; N, \mathcal L( X^{\hbox{\tiny$\square$}} )) \ll_{\gamma , d} (p\kern1pt N/m)^{-\gamma }+
\min\bgl\{ 1;\,\,(N\4p/m)^{-d/2}\,\left|t\right|^{-d/2}\,(\det \mathbb C)^{-
1/2}\bgr\}.$$ We have used that $\sigma^2=1$ implies
$\sigma_1^2\asymp_d1$. A similar upper bound is valid for the
quantity $\varkappa (t\kern1pt K ; N, \mathcal L( G))$ (cf{.} the proof
of \eqref{eq3.17} for $k>N/2$). Thus, we get, for any $\gamma >0$ and
$|t|\leq t_1$, $$\varkappa (t\kern1pt K ; N, \mathcal L( X^{\hbox{\tiny$\square$}}
),\mathcal L (G))\ll_{\gamma , d} (p\kern1pt N/m)^{-\gamma }+ \min\bgl\{
1;\,\,\,(\det \mathbb C)^{-1/2}\, \bgl(m/(|t|\kern1pt p\kern1pt N)
\bgr)^{d/2}\bgr\}.$$ Integrating this bound (cf. the estimation of
$I_1$ in Lemma \ref{L2.3}), we obtain \eqref{eee}. $\square$
$\phantom 0$
\smallskip
\section{\label {s4}Proof of Theorem $\ref{T1.5}$}
To simplify notation, in Section \ref{s4} we write
$\Pi=\Pi_2^{{\hbox{\tiny$\square$}}}$ and $\Lambda =\Lambda _4^{{\hbox{\tiny$\square$}}}$. The assumption
$\sigma^2=1$ and equalities ${\mathbf E\,}\| \mathbb C^{-1/2}\kern1pt X \|^2=d$,
\eqref{eq1.4t} and \eqref{eq1.5t} imply
\beq
\Pi+\Lambda \,N\gg1,\quad \Pi+\Lambda \leq 1 ,\quad \sigma_j^2 \leq 1,\quad
\det\mathbb C\le1 . \label{eq3.4}\eeq
Recall that $\Delta_N^{(a)}$ and functions $\Psi_b$, $\Phi_b$ and $\Theta_b$ are
defined by \eqref{eq1.21} and \eqref{eq1.18}--\eqref{edg}.
Note now that $\Theta_b^{{\hbox{\tiny$\square$}}}(x)=E_a^{{\hbox{\tiny$\square$}}}(x/N)$ and, according to
\eqref{edg},\beq
\Delta _N^{(a)}\le \Delta _{N,{\hbox{\tiny$\square$}}}^{(a)}+\sup_{x\in \mathbb R}\;
\bgl|\Theta_b (x)-\Theta_b^{{\hbox{\tiny$\square$}}} (x)\bgr|,
\label{eq3.10w}\eeq
where $b=\sqrt N\,a$ and\beq \Delta _{N,{\hbox{\tiny$\square$}}}^{(a)}=
\sup_{x\in\mathbb R }\; \bgl|\Psi_b (x) -
\Phi_b(x)-\Theta_b^{{\hbox{\tiny$\square$}}}(x)\bgr|.
\label{eq3.10ww}\eeq
Let us verify that \beq\sup_{x\in\mathbb R }\;\bgl|\Theta_b (x)-
\Theta_b^{\hbox{\tiny$\square$}} (x)\bgr|\ll_d \Pi_3^{\hbox{\tiny$\square$}}. \label
{eq3.12} \eeq To this end we shall apply representation
\eqref{eq1.21}--\eqref{eq1.22}
of the Edgeworth correction as a signed measure and estimate the
variation of that measure. Indeed, using \eqref{eq1.21}--\eqref{eq1.22}, we have
\beq\sup_{x\in\mathbb R }\;\bgl|\Theta_b (x)- \Theta_b^{\hbox{\tiny$\square$}}
(x)\bgr|\ll N^{-1/2} \kern1pt I,\quad \q\quad I\= \int_{\mathbb R^d} \bgl|{\mathbf E\,} p'''(x)
X^3- {\mathbf E\,} p'''(x) {X^{\hbox{\tiny$\square$}}}^3\bgr|\, dx .\label{eq1}\eeq By the
explicit formula \eqref{eq1.23}, the function $u \,\, {\raise.4pt\hbox{$\shortmid$}}{\hskip-2.0pt\to}\, \, p'''(x) u^3$
is a $3$-linear form in the variable~$u$. Therefore, using
$X= X^{\hbox{\tiny$\square$}}+X_{{\hbox{\tiny$\square$}}} $ and $ \|X^{\hbox{\tiny$\square$}}\| \, \|X_{{\hbox{\tiny$\square$}}}\|=0 $,
we have $ p'''(x) X^3- p'''(x) {X^{\hbox{\tiny$\square$}}}^3 = p'''(x)
{X_{\hbox{\tiny$\square$}}^3} $, and \beq N^{-1/2} \kern1pt I\leq 3\4d^{3/2}\kern1pt \Pi_3^{\hbox{\tiny$\square$}}
\int_{\mathbb R^d}
\bgl( \| \mathbb C^{-1/2}\kern1pt x \| + \| \mathbb C^{-1/2}\kern1pt x \|^3
\bgr) \, p(x)\, dx = c_d \, \Pi_3^{\hbox{\tiny$\square$}}.\label{eq222} \eeq Inequalities
\eqref{eq1} and
\eqref{eq222} imply now \eqref{eq3.12}.
To prove the statement of Theorem $\ref{T1.5}$, we have to derive
that \beq \Delta _{N,{\hbox{\tiny$\square$}}}^{(a)}\ll_d ( \Pi +
\Lambda )(1+\norm{a})^3\,(\det\mathbb C)^{-1/2}. \label{eq3.10}\eeq
While proving \eqref{eq3.10} we assume that \beq \Pi\leq c_d
,\quad \q\text{and} \quad \q \Lambda \leq c_d , \label{eq3.14} \eeq with a
sufficiently small positive constant $c_d$ depending on $d$ only.
These assumptions do not restrict generality.
Indeed, we
have $\bgl|\Psi_b (x) - \Phi_b(x)\bgr| \leq 1$. If
conditions~\eqref{eq3.14} do not hold, then the estimate
\beq\sup_{x\in\mathbb R }\;\bgl|\Theta_b^{\hbox{\tiny$\square$}}(x) \bgr|\ll_d
N^{-1/2}\kern1pt {\mathbf E\,} \|\mathbb C^{-1/2}\kern1pt X^{\hbox{\tiny$\square$}} \|^3\ll_d
\Lambda ^{1/2} \label{eq3.15}\eeq
immediately implies \eqref{eq3.10}.
In order to prove \eqref{eq3.15} we can use \eqref{eq1.5t} and representation
\eqref{eq1.21}--\eqref{eq1.22} of the Edgeworth correction.
Estimating the variation of that measure and using \beq {\mathbf E\,} \|\mathbb C^{-1/2}\, X^{\hbox{\tiny$\square$}} \|^2\leq {\mathbf E\,} \|\mathbb
C^{-1/2} X \|^2=d,\label{eq3.10u}\eeq
\beq ({\mathbf E\,} \|\mathbb C^{-1/2}\, X^{\hbox{\tiny$\square$}} \|^3)^2
\leq {\mathbf E\,} \|\mathbb C^{-1/2}\, X^{\hbox{\tiny$\square$}} \|^2\,
{\mathbf E\,} \|\mathbb C^{-1/2}\, X^{\hbox{\tiny$\square$}} \|^4,\label{eq3.10uu}\eeq we obtain \eqref{eq3.15}.
It is clear that\beq
\Delta _{N,{\hbox{\tiny$\square$}}}^{(a)}\le \sup_{x\in \mathbb R}\;
\Bgl(\bgl|\Psi_b (x)-\Psi_b' (x)\bgr|+ \bgl|\Theta_b^{\hbox{\tiny$\square$}} (x)-\Theta_b'
(x)\bgr|
+ \bgl|\Psi_b' (x) - \Phi_b(x)-\Theta'_b(x)\bgr|\Bgr).
\label{eq3.10x}\eeq
Similarly to \eqref{eq1}, we have
\beq\sup_{x\in\mathbb R }\;\bgl|\Theta_b^{\hbox{\tiny$\square$}} (x)- \Theta_b' (x)\bgr|\ll N^{-
1/2} \kern1pt J,\quad \q\quad
J\= \int_{\mathbb R^d}
\bgl|{\mathbf E\,} p'''(x) {X^{\hbox{\tiny$\square$}}}^3- {\mathbf E\,} p'''(x) {X'}^3\bgr|\, dx .\label{eq2}\eeq
Recall that vector $X'$ is defined in \eqref{eq1.20}.
By Lemma ~\ref{L2.4}, we have
${\mathbf E\,} \|\mathbb C^{-1/2}\, W \|^2\leq 2\4d\, \Pi$
(hence, ${\mathbf E\,} \|\mathbb C^{-1/2}\, W \|^q\ll_d \Pi^{q/2}$, for $0\le q\le2$).
Moreover, representing $W$ as a sum of a large number of i.i.d.
Gaussian summands and using the Rosenthal inequality (see BG
(1997a, inequality~(1.24)), we conclude that \beq {\mathbf E\,} \|\mathbb C^{-1/2}\,
W
\|^q\ll_q \bgl({\mathbf E\,} \|\mathbb C^{-1/2}\, W \|^2\bgr)^{q/2}\ll_{q, d} \Pi^{q/2},\quad
q\ge0.\label{eq442}\eeq Furthermore, according to \eqref{eq1.4t}, \eqref{eq1.5t} and
\eqref{eq3.14},
\beq\label{eq234}{\mathbf E\,}\|\mathbb C^{-1/2}\,X_{\hbox{\tiny$\square$}}\| \ll_d\Pi\4N^{-
1/2}\ll_d\Pi^{1/2}\4N^{-1/2} .\eeq
Hence, by \eqref{eq1.5t}, \eqref{eq1.20}, \eqref{eq3.4}, \eqref{eq442} and
\eqref{eq234}, \beq\label{eq2345} {\mathbf E\,} \|X' \|^4\ll\ovln\beta\={\mathbf E\,} \|\mathbb C^{-
1/2}\,X' \|^4\ll_d
N\kern1pt\Lambda +\Pi^2.\eeq Using \eqref{eq1.23}, \eqref{eq3.4}, \eqref{eq3.14},
\eqref{eq3.10u} and \eqref{eq2}--\eqref{eq234}, we get \begin{eqnarray}
N^{-1/2} \kern1pt J&\ll_d &\Pi^{1/2}(N^{-1/2}\kern1pt\Pi+\Lambda^{1/2})\kern1pt
\int_{\mathbb R^d}
\bgl( \| \mathbb C^{-1/2}\kern1pt x \| + \| \mathbb C^{-1/2}\kern1pt x \|^3 \bgr) \,
p(x)\, \nonumber dx\\
&\ll_d& \Pi+\Lambda .\label{eq223} \end{eqnarray} Thus, according to
\eqref{eq2} and \eqref{eq223}, \beq\sup_{x\in\mathbb R
}\;\bgl|\Theta_b^{\hbox{\tiny$\square$}} (x)- \Theta_b' (x)\bgr|\ll_d
\Pi+\Lambda .\label{eqpo}\eeq
The same approach is applicable for the estimation of
$\bgl|\Theta_b'\bgr|$. Using \eqref{eq1.20},
\eqref{eq1.21}--\eqref{eq1.23}, \eqref{eq3.4}, \eqref{eq3.10u},
\eqref{eq3.10uu}, \eqref{eq442} and \eqref{eq234}, we get
\begin{eqnarray}\sup_{x\in\mathbb R }\;\bgl|\Theta_b' (x)\bgr|&\ll&
N^{-1/2} \int_{\mathbb R^d} \bgl| {\mathbf E\,} p'''(x) {X'}^3\bgr|\, dx \nonumber\\
&\ll_d&\Lambda^{1/2}
+N^{-1/2}\kern1pt\Pi^{3/2}.\label{eq999}\end{eqnarray}
Let us prove that
\beq \sup_{x\in\mathbb R }\;\bgl|\Psi_b (x) - \Psi'_b (x) \bgr|\ll (\det \mathbb
C)^{-1/2}\,p^{-2}\kern1pt
(\Pi+\Lambda ) (1+\norm{a}^2).
\label{eq3.25} \eeq Using truncation (see
\eqref{eq2.8}), we have
$ | \Psi_b - \Psi_b^{\hbox{\tiny$\square$}} |\leq \Pi$,
and \beq \sup_{x\in\mathbb R }\;\bgl |\Psi_b (x) - \Psi'_b (x) \bgr| \leq \Pi+
\sup_{x\in\mathbb R }
\;\bgl| \Psi_b^{\hbox{\tiny$\square$}} ( x ) - \Psi'_b (x) \bgr| . \label{eq3.26}
\eeq In order to estimate $| \Psi_b^{\hbox{\tiny$\square$}} - \Psi'_b |$, we shall
apply Lemmas \ref{L3.1} and \ref{L3.2}. The number $m$ in these
Lemmas exists and $N\kern1pt \Lambda /p\gg_d 1$, as it follows from
\eqref{eq3.4} and \eqref{eq3.14}. Let us choose the minimal
$m$, that is,
$m\asymp_d N\kern1pt \Lambda /p $. Then $(p\kern1pt N )^{-1} \kern1pt m \ll_d \Lambda /p^2 $ and
$m/N\ll_d \Lambda /p$. Therefore,
using
Lemma \ref{L3.1}, we have \beq \sup_x \;\bgl| \Psi_b^{\hbox{\tiny$\square$}} ( x )
- \Psi_b ' (x) \bgr|
\ll_d p^{-2}\kern1pt \Lambda \,(\det \mathbb
C)^{-1/2}+\int_{|t|\leq t_1} \bgl| \widehat \Psi_b^{\hbox{\tiny$\square$}} (\tau )-
\widehat \Psi'_b (\tau )
\bgr|\, \ffrac {dt} {|t|}, \quad \q\quad \tau =t\kern1pt K. \label{eq3.27}\eeq
We shall prove that \beq \bgl|\widehat \Psi_b^{\hbox{\tiny$\square$}} (\tau )-\widehat
\Psi_b ' (\tau ) \bgr| \ll_d \varkappa \, \Pi\, |\tau |\kern1pt
N\kern1pt (1+|\tau | \kern1pt N)(1+\norm{a}^2) ,\label{eq3.28} \eeq with
$\varkappa = \varkappa (\tau ; N, \mathcal L(X^{\hbox{\tiny$\square$}} ))$.
Combining \eqref{eq3.26}--\eqref{eq3.28}, using $\tau = t\kern1pt K$ and
integrating
inequality \eqref{eq3.28}
with the help of Lemma \ref{L3.2},
we derive \eqref{eq3.25}.
Let us prove \eqref{eq3.28}. Recall that $X'=X^{\hbox{\tiny$\square$}} -{\mathbf E\,}
X^{\hbox{\tiny$\square$}} +W $, where $W$ denotes a centered Gaussian random
vector which is independent of all other random vectors and such
that $\cov X' =\cov G $ (see Lemma \ref{L2.4}).
Writing $ D= Z_N^{\hbox{\tiny$\square$}} -{\mathbf E\,} Z_N^{\hbox{\tiny$\square$}}-b $,
we have
$$Z_N^{\hbox{\tiny$\square$}}-b
= D+{\mathbf E\,} Z_N^{\hbox{\tiny$\square$}} ,\quad \q\quad \mathcal L(Z_N'-b) = \mathcal L(D+
\sqrt{N} \kern1pt W),$$ and \beq\label{eq3.27m}\bgl|\widehat \Psi_b^{\hbox{\tiny$\square$}}
(\tau )-\widehat \Psi'_b(\tau ) \bgr| \leq \bgl| f_1(\tau )\bgr|+\bgl|
f_2(\tau )\bgr| \eeq with \beq \aligned f_1(\tau )&= {\mathbf E\,} \operatorname{e}\bgl\{
\tau \kern1pt \mathbb Q \kern1pt[ D +\sqrt{N} \kern1pt W ]\bgr\}-
{\mathbf E\,} \operatorname{e}\bgl\{ \tau \kern1pt \mathbb Q \kern1pt[ D ]\bgr\},\\
f_2(t)&= {\mathbf E\,} \operatorname{e}\bgl\{ \tau \kern1pt \mathbb Q \kern1pt[ D + {\mathbf E\,} Z_N^{\hbox{\tiny$\square$}}
]\bgr\}- {\mathbf E\,} \operatorname{e}\bgl\{ \tau \kern1pt \mathbb Q \kern1pt[ D]\bgr\}. \endaligned
\label{eq3.29}\eeq Now we have to prove that both $\bgl| f_1(\tau
)\bgr|$ and $\bgl| f_2(\tau )\bgr|$ may be estimated by the right
hand side of \eqref{eq3.28}.
Let us consider $f_1$. We can write $\mathbb Q \kern1pt[ D +\sqrt{N} \kern1pt
W ]= \mathbb Q \kern1pt[ D ]+A+B$ with $A=2\kern1pt \sqrt{N} \kern1pt \langle\mathbb
Q \kern1pt D , W\rangle$ and $B=N \kern1pt \mathbb Q \kern1pt[ W ]$. Taylor's
expansions of the exponent in \eqref{eq3.29} in powers of $i\kern1pt\tau
\kern1pt B$ and $i\kern1pt \tau \kern1pt A$ with remainders $\mathcal O ( \tau \kern1pt B)$ and
$\mathcal O( \tau ^2\kern1pt A^2)$ respectively imply (recall that ${\mathbf E\,} W=0$ and
$\mathbb Q^2=\mathbb I_d$) \beq \bgl| f_1(\tau )\bgr|\ll
\varkappa \kern1pt |\tau | \kern1pt N\kern1pt {\mathbf E\,} \|W\|^2 + \varkappa \kern1pt \tau ^2 N\kern1pt
{\mathbf E\,} \|W\|^2 \, {\mathbf E\,} \|D\|^2, \label{eq3.30}\eeq where $\varkappa =
\varkappa (\tau ; N, \mathcal L( X^{\hbox{\tiny$\square$}} ))$. The estimation of
the remainders of these
expansions is based on the splitting
and conditioning techniques described in Section 9 of BG~(1997a),
see also Bentkus, G\" otze and Zaitsev (1997). Using the relations
$\sigma^2=1$, ${\mathbf E\,} \|W\|^2 \ll{\mathbf E\,} \|\mathbb C^{-1/2}\4W\|^2 \ll_d \Pi$ and ${\mathbf E\,} \|D\|^2 \ll
N(1+\norm{a}^2)$, we derive from \eqref{eq3.30} that \beq \bgl|
f_1(\tau )\bgr| \ll_d
\varkappa \kern1pt \Pi
\kern1pt |\tau |\kern1pt N\kern1pt \bgl( 1 + |\tau |\kern1pt N )\bgr)(1+\norm{a}^2)
.\label{eq3.31}\eeq Note that ${\mathbf E\,} Z_N^{\hbox{\tiny$\square$}} = N\,{\mathbf E\,}
X^{\hbox{\tiny$\square$}}= -N\,{\mathbf E\,} X_{\hbox{\tiny$\square$}}$. Expanding the exponent $\operatorname{e}\bgl\{
\tau \kern1pt \mathbb Q \kern1pt[ D + {\mathbf E\,} Z_N^{\hbox{\tiny$\square$}} ]\bgr\}$, using
\eqref{eq234} and proceeding similarly to the proof
of~\eqref{eq3.31}, we obtain \beq \bgl| f_2(\tau )\bgr|\ll_d
\varkappa \kern1pt \Pi\kern1pt|\tau | \kern1pt
N(1+\norm{a}).\label{eq3.31f} \eeq
Inequalities \eqref{eq3.27m}, \eqref{eq3.31}
and \eqref{eq3.31f} imply now \eqref{eq3.28}.
\smallskip
It remains to estimate $\bgl|\Psi'_b - \Phi_b - \Theta_b' \bgr|$.
Recall that the distribution functions $\Psi_b^{(l)}(x)$, for~$0\leq l\leq N$,
are defined in \eqref{eq3.16}.
Fix an integer $k$, $1\leq k\leq N$. Clearly, we have
\beq\label{eq156}\sup_{x\in\mathbb R }\;\bgl|\Psi'_b (x) - \Phi_b(x) -
\Theta_b' (x)\bgr|\le
I_1+I_2+I_3,\eeq where
\beq\label{eq156b}I_1=\sup_{x\in\mathbb R }\;\bgl|\Psi_b^{(k)} (x) - \Phi_b(x)-
(N-k)\,\Theta_b' (x)/N\bgr|,\eeq
\beq\label{eq156a}I_2=\sup_{x\in\mathbb R }\;
\bgl|\Psi'_b (x) - \Psi_b^{(k)}(x) \bgr|,\eeq
and\beq\label{eq156c}I_3=\sup_{x\in\mathbb R }\;
k\4N^{-1}}\def\sign{\hbox{\rm sign}\,\bgl|\Theta_b' (x) \bgr|.\eeq
\smallskip
Let estimate $I_1$.
Define the distributions
\beq\label{eq1.23aq}\mu(A)= \P \bgl\{ U_k +\sum_{j=k+1}^N X_j'
\in \sqrt{N} \kern1pt A \bgr\},\quad \quad \q \mu_0(A)= \P \bgl\{ U_N \in
\sqrt{N} \kern1pt A \bgr\}= \P \bgl\{ G \in A \bgr\},\eeq where $U_l
=G_1+\dots+G_l $. Introduce the measure $\chi '$ replacing $X$ by
$X'$ in \eqref{eq1.22}. For the Borel sets $A\subset \mathbb R^d$ define
the Edgeworth correction (to the distribution $\mu$) as
\beq\mu_1^{(k)} (A)= (N-k)\kern1pt N^{-3/2}\chi' (A)/6. \eeq Introduce
the signed measure \beq\nu =\mu
-\mu_0-\mu_1^{(k)}.\label{eq1.22aq}\eeq
It is easy to see that a re-normalization of
random vectors implies (see relations \eqref{eq1.21}, {\eqref{eq1.18}--
\eqref{edg}}, \eqref{eq3.16} and
\eqref{eq1.23aq}--\eqref{eq1.22aq})
\begin{eqnarray}
\bgl|\Psi_b^{(k)} (x) - \Phi_b(x)- (N-k)\,\Theta_b' (x)/N\bgr|
&=&\nu\bgl(\bgl\{u\in\mathbb R^d:\mathbb Q[u-a]\le x/N\bgr\}\bgr)\nonumber\\ &\le&\delta_N\=
\sup_{A \subset\mathbb R^d }\bgl| \nu (A)\bgr|.\label{eq156d}
\end{eqnarray}
\begin{lemma}\label{L9.4} Assume that $ d<\infty$
and $1\leq k\leq N$. Then there exists a $c(d)$ depending on $d$ only
and such that\/ $\delta_N$
defined in $\eqref{eq156d}$ satisfies the inequality\beq\delta_N \ll_d
\ffrac {\ovln\beta}{N}
+\ffrac {N^{d/2}}{k^{d/2}}\exp\bgl\{ - c(d)\, k/\ovln\beta\bgr\}
\label{eq9.14} \eeq with $\ovln\beta={\mathbf E\,} \|\mathbb C^{-1/2}\4X'\|^4$.
\end{lemma}
{\it An outline of the proof}. We repeat and slightly improve the
proof of Lemma~9.4 in BG~(1997a) (cf. the proof of Lemma~2.5 in BG
(1996)). We
shall prove \eqref{eq9.14} assuming that $\cov X= \cov X'=\cov G=\mathbb I_d$,
Applying it to $\mathbb C^{-1/2}\kern1pt X'$ and $\mathbb
C^{-1/2}\kern1pt G$, we
obtain \eqref{eq9.14} in general case.
While proving \eqref{eq9.14} we assume
that $\ovln\beta/N \leq c_d$ and $N \geq 1/ c_d$ with a sufficiently
small positive constant $c_d$. Otherwise \eqref{eq9.14} follows
from the obvious bounds $\ovln\beta\ge\sigma^4=d^2$ and $$\delta_N\ll_d 1 +
(\ovln\beta/N)^{1/2} \, \int_{\mathbb R^d
} \|x\|^3\kern1pt p(x)\, dx \ll_d 1 + (\ovln\beta/N)^{1/2} . $$
Set $n=N-k$.
Denoting by $Z_j^\prime$ and $U_j^\prime$ sums of $j$ independent
copies of $X^\prime$ and $G^\prime$ respectively, introduce the
multidimensional characteristic functions \beq g(t)={\mathbf E\,} \operatorname{e} \bgl\{
\langle N^{-1/2} \4t, G\rangle \bgr \},\quad h(t)={\mathbf E\,} \operatorname{e} \bgl\{
\langle N^{-1/2} \4t, X'\rangle \bgr \},\label{eqe1}\eeq\beq
\label{eq9.17a}f(t)= {\mathbf E\,} \operatorname{e} \bgl\{ \langle N^{-1/2} \kern1pt t,
Z_{n}^\prime \rangle \bgr\}=h^{n}(t),\quad \q f_0(t)= {\mathbf E\,} \operatorname{e} \bgl\{
\langle N^{-1/2} \kern1pt t, U_{n}^\prime \rangle \bgr\}=g^{n}(t),\eeq
\beq f_1(t)= n\,m(t) \,f_0(t), \quad \hbox{where}\quad m(t)= \ffrac
{1}{6\,{N^{3/2}}} \kern1pt {\mathbf E\,} \langle i\kern1pt t, X^\prime \rangle ^3 ,
\eeq\beq\widehat \nu (t)=(f(t)-f_0(t)-f_1(t))\, g(\rho t),\quad \rho^2
=k.\label{eq9.16s}\eeq It is easy to see that \beq \label{Fur1}\widehat
\nu (t)=\int_{\mathbb R^d} \operatorname{e}\{\langle t,
x\rangle\}\,\nu(dx). \eeq Using the truncation, we
obtain \beq{\mathbf E\,} \bgl\| Z_l^\prime/\sqrt N\bgr\|^\gamma\ll_{\gamma,d}
1,\quad \q \gamma>0,\quad 1\le l\le N . \label{eq9.15} \eeq
By an extension of the proof of Lemma 11.6 in Bhattacharya and Rao
(1986), see also the proof of Lemma~2.5 in BG~(1996), we obtain
\beq\delta_N\ll_d \max_{|\alpha | \leq 2d}\, \, \int_{\mathbb R^d}
\bgl|\partial^\alpha \widehat \nu (t)\bgr|\, dt. \label{eq9.16} \eeq Here
$|\alpha |=|\alpha _1 |+\cdots+|\alpha _d |$, $\alpha =(\alpha _1,\ldots,\alpha _d)$,
$\alpha _j\in\mathbb Z$, $\alpha _j\ge0$. In order to derive \eqref{eq9.14}
from \eqref{eq9.16}, it suffices to prove that, for $|\alpha |\leq
2d$, \begin{eqnarray} \bgl|\partial^\alpha \widehat \nu (t)\bgr|&\ll_d&
g( c_1\kern1pt\rho \kern1pt t),
\label{eq9.17}\\ \bgl|\partial^\alpha \widehat \nu (t)\bgr| &\ll_d&
\ovln\beta\kern1pt N^{-1} \, (1+\|t\|^{6} )\, \exp\{ - c_2\, \|t\|^2\} ,
\quad \hbox{for}\ \, \|t\|^2\leq c_3(d) \kern1pt N/\ovln\beta .\label{eq9.18}
\end{eqnarray} Indeed, using \eqref{eq9.17} and denoting
$T= \sqrt{c_3(d) \kern1pt N/\ovln\beta}$, we obtain
\beq \int\limits_{\|t\|\geq T} \bgl|\partial^\alpha \widehat \nu
(t)\bgr|\, dt \ll_d
\int\limits_{\|t\|\geq T} g( c_1\kern1pt\rho \kern1pt t) \, dt
\ll_d \ffrac{N^{d/2}}{\rho^{ d}} \, \exp\Bgl\{ - \ffrac{c_1^2\kern1pt
\rho^{2}\kern1pt T^2}{8\4N}\Bgr\} \int\limits_{\mathbb R^d} \exp\{ - c_1^2\,
\|t\|^2/8\} \, dt,\label{eq9.19} \eeq and it is easy to see that
the right hand side of \eqref{eq9.19} is bounded from above by the
second summand in the right hand side of \eqref{eq9.14}.
Similarly, using \eqref{eq9.18}, we can integrate
$\bgl|\partial^\alpha \widehat \nu (t)\bgr|$ over $\|t\|\leq T$, and the
integral is bounded from above by $c_d \kern1pt \ovln\beta/N$.
In the proof of \eqref{eq9.17}--\eqref{eq9.19} we applied standard
methods of estimation which are provided in Bhattacharya and Rao
(1986). In particular, we used a Bergstr\" om type identity \beq
f-f_0-f_1=\sum_{j=0}^{n-1}(h-g-m)\,h^j\,g^{n-j-1}
+\sum_{j=0}^{n-1} m\sum_{l=0}^{j-1}(h-g)\,h^l\,g^{n-l-1},
\label{eqe2} \eeq relations \eqref{eqe1}--\eqref{eq9.15}, $1\leq
k\leq N$, $\bgl|\partial^\alpha \exp \{ -c_4\, \|t\|^2\}\bgr|\ll_\alpha
\exp \{ -c_5\, \|t\|^2\}$, ${\sqrt{N}/\ovln\beta^{1/2}\gg_d1}$ and
$y^{c_d} \exp \{ -y\} \ll_d 1 $, for $y>0$. $\square$
\medskip
Applying \eqref{eq156b}, \eqref{eq156d}
and Lemma \ref{L9.4}, we get
\beq I_1\ll_d
\ffrac {\ovln\beta}{N}
+\ffrac {N^{d/2}}{k^{d/2}}\exp\bgl\{ - c(d)\, k/\ovln\beta\bgr\}.
\label{eq3.40} \eeq
For the estimation of $I_2$ we shall use Lemma \ref{L9.3} which is
an easy consequence of BG (1997a, Lemma 9.3), \eqref{eq1app} and
\eqref{eq2345}.
\begin{lemma}\label{L9.3}
We have
$$ \bgl|\widehat \Psi_b' (t)-\widehat \Psi_b^{(l)}(t) \bgr| \ll
\varkappa\kern1pt t^2 \kern1pt l \,
\bgl( \ovln\beta +|t|\kern1pt N \kern1pt\ovln\beta +|t|\kern1pt N \kern1pt
\sqrt{N \kern1pt\ovln\beta} \bgr)(1+\norm{a}^3),\quad \hbox{for \ }0\leq l\leq N , $$
where $\varkappa = \varkappa ( t; N, \mathcal L
(X^{{\hbox{\tiny$\square$}}}) ,
\mathcal L (G))$ $($cf{.} $\eqref{eq3.23})$.
\end{lemma}
\smallskip
As in the proof of \eqref{eq3.27},
applying Lemma~\ref{L3.1} (choosing $m\asymp_d N\kern1pt (\Lambda +\Pi) /p $)
and using~\eqref{eq3.3ss},
we obtain $$ I_2\ll_d (\Lambda +\Pi)\,(\det \mathbb C)^{-1/2}
+ \int_{|t|\leq t_1
}\bgl|\widehat \Psi'_b (\tau ) - \widehat \Psi_b^{(k)} (\tau )\bgr| \, dt
/|t|,\quad \q \tau =t\kern1pt K . $$ The existence of such an~$m$
is ensured by \eqref{eq3.3ss}, \eqref{eq3.4} and~\eqref{eq3.14},
Applying Lemma \ref{L9.3} and replacing
in that Lemma $t$ by $\tau $, we
have \beq\label{eq4.8} \bgl|\widehat \Psi_b '(\tau ) - \widehat
\Psi_b^{(k)}(\tau ) \bgr| \ll \varkappa\kern1pt \tau ^2 \kern1pt k \, \bgl( \ovln\beta
+| \tau |\kern1pt N \kern1pt\ovln\beta +| \tau |\kern1pt N \kern1pt
\sqrt{N \kern1pt\ovln\beta} \bgr)(1+\norm{a}^3) .\eeq
Integrating with the help of Lemma \ref{L3.2} and using~\eqref{eq3.3ss},
we obtain \beq
I_2\ll_d (\det \mathbb C)^{-1/2}\,
\bgl(\Pi+\Lambda + k \kern1pt N^{-2} \kern1pt \bgl( \ovln\beta
+ \sqrt{N \kern1pt\ovln\beta} \bgr)\bgl( 1 + (\Pi+\Lambda )^{-1/d}
\bgr)(1+\norm{a}^3)\bgr) . \label{eq3.39}\eeq
Let us choose
$k\asymp_d N^{1/4} \kern1pt{\ovln\beta}^{3/4} $.
Such $k\leq N$ exists by $\ovln\beta\gg_d\sigma^4=1$, by \eqref{eq2345} and by
assumption \eqref{eq3.14}.
Then \eqref{eq3.40} and \eqref{eq3.39} turn
into \beq I_1\ll_d \ffrac {\ovln\beta}{N}
+\Big(\!\ffrac {N}{{\ovln\beta}}\!\Big)^{3d/8}
\exp\Bgl\{ -c_d\,\Big(\!\ffrac {N}{{\ovln\beta}}\!\Big)^{1/4}\!\Bgr\}\ll_d
\ffrac {\ovln\beta}{N},\label{eq4.899}
\eeq and \beq I_2\ll_d (\det \mathbb C)^{-1/2}\,
\bgl(\Pi+\Lambda +
\Bigl( \Big(\!\ffrac {\ovln\beta}{N}\!\Big)^{5/4}+ \Big(\!\ffrac
{\ovln\beta}{N}\!\Big)^{7/4}\Bigr)\bgl( 1 + (\Pi+\Lambda )^{-1/d}
\bgr)(1+\norm{a}^3)\bgr). \label{eq3.399} \eeq Using \eqref{eq3.3ss},
\eqref{eq3.14},
\eqref{eq2345} and \eqref{eq3.399}, we get
\beq I_2\ll_d (\det \mathbb C)^{-1/2}\,
\bgl(\Pi+\Lambda +\ffrac {\ovln\beta}{N} (1+\norm{a}^3)\bgr).
\label{eq3.3999} \eeq
Finally, by \eqref{eq3.14}, \eqref{eq2345}, \eqref{eq999} and \eqref{eq156c},
\beq I_3\ll_d
\ffrac k N\bgl(\Lambda^{1/2}
+N^{-1/2}\kern1pt\Pi^{3/2}\bgr)
\ll \Lambda+\Pi
.\label{eq777}
\eeq
Inequalities \eqref{eq3.14}, \eqref{eq3.10x}, \eqref{eq2345}, \eqref{eqpo},
\eqref{eq3.25}, \eqref{eq156},
\eqref{eq4.899}, \eqref{eq3.3999} and \eqref{eq777} imply now \eqref{eq3.10}
(and, hence, \eqref{eq1.8w}) by an application of
$\Pi+\Lambda \leq 1$. Note that, by \eqref{eq1.5t}, we have $\Pi\leq \Pi_3^{\hbox{\tiny$\square$}}$.
Together with
\eqref{eq3.10w} and \eqref{eq3.12}, inequality \eqref{eq3.10}
yields~\eqref{eq1.8}. The statement of Theorem $\ref{T1.5}$
is proved. $\square$
\medskip
$\phantom 0$
\section{\label {s5}From Probability to Number Theory}
In Section \ref{s5} we shall reduce the estimation of the
integrals of the modulus of characteristic functions
$\widehat\Psi_b(t)$ to the estimation the integrals of some
theta-series. We shall use the following lemmas.
\begin{lemma}\label{L5.1} {\rm(BG (1997a, Lemma 5.1))} Let\/ $L,C\in \mathbb
R^d$ and let $\mathbb Q:\mathbb R^d\to\mathbb R^d$ be a symmetric linear operator.
Let\/ $Z,U,V$ and\/ $ W$ denote independent random vectors taking
values in $\mathbb R^d$. Denote by
$$ P(x) = \langle \mathbb Q \4x,x\rangle + \langle
L,x\rangle
+C , \quad \q x \in \mathbb R^d,$$
a real-valued polynomial of second order. Then
$$ 2\, \Bgl|{\mathbf E\,} \operatorname {e} \bgl\{ t\, P(Z+U+V+W)\bgr\}\Bgr|^2 \leq
{\mathbf E\,} \operatorname {e} \bgl\{ 2 \, t\, \langle \mathbb Q \kern1pt\widetilde
Z,\widetilde U\rangle \bgr\} + {\mathbf E\,} \operatorname {e} \bgl\{ 2 \,t\,
\langle \mathbb Q \kern1pt\widetilde Z,\widetilde V\rangle \bgr\} . $$
\end{lemma}
Let ~${\is \varepsilon}$ denote i.i.d{.} symmetric Rademacher random
variables. Let $\delta >0$, $\mathcal S = \{ \fs e1s \} \subset
\mathbb R^d$ and let $\mathbb D:\mathbb R^d\to\mathbb R^d$ be a linear operator.
Usually, we shall take $\mathbb D=\mathbb C^{-1/2}$. We shall
write~${\mathcal L (Y)\in {\mathbf {\Gamma}}\kern1pt (\delta;\mathbb D,
\mathcal S)}\, $ if a discrete random vector~$Y$ is distributed as
$\varepsilon_1 z_1 +\dots +\varepsilon_s z_s\penalty250\mskip\thickmuskip\mskip-\thinmuskip$, with some
(non-random) $ z_j\in \mathbb R^d$ such that
$ \|\mathbb D\4z_j -e_j\|\leq \delta$, for all $1\leq
j\leq s$.
Recall that $\mathcal S_o=\{\fs e1s\}\subset \mathbb R^d $ denotes
an orthonormal system.
\begin{lemma}\label{L6.3} Assume that\/ $\mathbb Q^2= \mathbb I_d$
and that the condition $\mathcal N(p,\delta, \mathcal S ,\mathbb D\kern1pt
\widetilde X )$ holds with some $0< p\leq 1$ and\/ $\delta>0$. Write $m
=\bgl\lceil {p\kern1pt N }/ (5\kern1pt s)\bgr\rceil$.
Then, for
any\/ $0<A\leq B$, $b\in\mathbb R^d$ and\/ $\gamma>0$, we have
\beq\label{eq7.1qq}\int\limits_A^B \bgl| \widehat \Psi_b (t)\bgr| \,
\ffrac {dt} {|\penalty250\mskip\thickmuskip\mskip-\thinmuskip t\penalty250\mskip\thickmuskip\mskip-\thinmuskip |}
\leq I+
c_\gamma (s)\, (p\kern1pt N)^{-\gamma }\, \log\ffrac BA ,\eeq
with
\beq\label{pppp} I= \sup_\Gamma \,\sup_{b\in \mathbb R^d} \,\int\limits_A^B
\sqrt{\varphi ( t/4)} \, \ffrac {dt}{|\penalty250\mskip\thickmuskip\mskip-\thinmuskip t\penalty250\mskip\thickmuskip\mskip-\thinmuskip |} ,
\quad \q
\varphi (t) \= \Bgl| {\mathbf E\,} \operatorname {e} \bgl \{ t
\, \mathbb Q \kern1pt[Y + b]\bgr\} \Bgr|^2 ,\eeq
where $Y= \fsu U1m$
denote a sum of independent $($non-i.i.d.$)$ vectors, and\/
$\sup\nolimits_\Gamma $ is taken over all
$ \bgl\{\mathcal L(U_j):\, \, 1\leq j\leq m\bgr\} \subset
{\mathbf {\Gamma}}\kern1pt (\delta;\mathbb D, \mathcal S )$.
\end{lemma}
Lemma \ref{L6.3} is an analogue of Corollary 6.3 from BG (1997a).
Its proof is even simpler than that in BG (1997a). Therefore it is
omitted.
\smallskip
\begin{lemma}\label{L7.3}
Assume that\/ $\mathbb Q^2= \mathbb I_d$
and that the condition $\mathcal N(p,\delta, \mathcal S ,\mathbb D\kern1pt
\widetilde X )$ holds with some $0< p\leq 1$ and\/ $\delta>0$. Let \beq
\label{dfn}n \= \bgl\lceil {p\kern1pt N}/({16\kern1pt s})\bgr\rceil\ge1.\eeq
Then, for any\/ $0<A\leq B$, $b\in\mathbb R^d$ and\/
$\gamma>0$,\beq\int\limits_A^B\bgl| \widehat \Psi_b (t)\bgr|\ffrac
{dt}{|\penalty250\mskip\thickmuskip\mskip-\thinmuskip t\penalty250\mskip\thickmuskip\mskip-\thinmuskip |}
\leq c_\gamma (s)\, (p\kern1pt N)^{-\gamma }\, \log\ffrac BA+
\sup_\Gamma \,\int\limits_A^B
\sqrt{{\mathbf E\,} \operatorname {e} \bgl \{ t\, \langle \mathbb Q\,\widetilde W ,\widetilde W'
\rangle/2 \bgr\}} \, \ffrac {dt}{|\penalty250\mskip\thickmuskip\mskip-\thinmuskip t\penalty250\mskip\thickmuskip\mskip-\thinmuskip |},\label{eq7.1}\eeq and
for any fixed\/ $t\in\mathbb R$, \beq\bgl| \widehat \Psi_b (t)\bgr|
\leq c_\gamma (s)\, (p\kern1pt N)^{-\gamma }+
\sup_\Gamma
\sqrt{{\mathbf E\,} \operatorname {e} \bgl \{ t\, \langle \mathbb Q\,\widetilde W ,\widetilde W'
\rangle/2
\bgr\}} ,\label{equ7.1}\eeq
where \/
$W= \fsu V1n$ and\/~$W'=V_1'+\dots + V_n' $ are independent sums
of independent copies of random vectors $V$ and\/ $V'$
respectively, and the supremum $\sup\nolimits_\Gamma $ is taken
over all $ \mathcal L(V), \mathcal L(V') \in
{\mathbf {\Gamma}}\kern1pt (\delta;\mathbb D, \mathcal S )$.
\end{lemma}
Note that this lemma will be proved for
general $\mathcal S$, but in this paper we need $\mathcal S=\mathcal S_o$
only. Moreover, a more careful estimation of
binomial probabilities could allow us to replace $c_\gamma (s)\, (p\kern1pt N)^{-\gamma }$ in
\eqref{eq7.1qq}, \eqref{eq7.1} and \eqref{equ7.1}
by
$c (s)\,\exp\bgl\{ -c\4p\kern1pt N\bgr\}$ (see e.g. Nagaev and Chebotarev (2005)).
However, we do not need to use this improvement.
{\it Proof of Lemma\/ $\ref{L7.3}$.} Inequality \eqref{equ7.1} is
an analogue of the statement of Lemma 7.3 from BG (1997a). Its
proof is even simpler than that in BG (1997a). Therefore it is
omitted.
Let us show that \beq\int\limits_A^B\bgl| \widehat \Psi_b
(t)\bgr|\ffrac {dt}{|\penalty250\mskip\thickmuskip\mskip-\thinmuskip t\penalty250\mskip\thickmuskip\mskip-\thinmuskip |}
\leq c_\gamma (s)\, (p\kern1pt N)^{-\gamma }\, \log\ffrac BA+
\sup_\Gamma \,\int\limits_A^B
\sqrt{{\mathbf E\,} \operatorname {e} \bgl \{ t\, \langle \mathbb Q\,\widetilde W ,\widetilde W'
\rangle/2
\bgr\}} \, \ffrac {dt}{|\penalty250\mskip\thickmuskip\mskip-\thinmuskip t\penalty250\mskip\thickmuskip\mskip-\thinmuskip |},\label{eq7.1q}\eeq
where \/
$W= \fsu V1n$ and ~$W'=V_1'+\dots + V_n' $ are independent sums of
of independent $(${\it non-i.i.d.}$)$ vectors, and
$\sup\nolimits_\Gamma $ is taken over all
$ \bgl\{\mathcal L(V_j), \mathcal L(V_j'):\, \, 1\leq j\leq n\bgr\} \subset
{\mathbf {\Gamma}}\kern1pt (\delta;\mathbb D, \mathcal S )$.
Comparing \eqref{eq7.1} and \eqref{eq7.1q}, we see that inequality
\eqref{eq7.1q} is related to sums of
{\it non-i.i.d.} vectors $\{V_j\}$ and~$\{V_j'\}$ while inequality \eqref{eq7.1}
deals with {i.i.d.} vectors.
Nevertheless, we shall derive \eqref{eq7.1} from \eqref{eq7.1q}.
While proving \eqref{eq7.1q} we can assume that $p\kern1pt N\geq c_s$
with a sufficiently large constant $c_s$, since otherwise \eqref{eq7.1q}
is obviously valid.
Let $\varphi(t)$ be defined in \eqref{pppp}, where $Y= \fsu U1m$
denote a sum of independent $($non-i.i.d.$)$ vectors with $
\bgl\{\mathcal L(U_j):\, \, 1\leq j\leq m\bgr\} \subset
{\mathbf {\Gamma}}\kern1pt (\delta;\mathbb D, \mathcal S )$, $m
=\bgl\lceil {p\kern1pt N }/ (5\kern1pt s)\bgr\rceil$.
We shall
apply the symmetrization Lemma \ref{L5.1}.
Split
$ Y=T+T_1+T_2 $ into
sums of independent sums of independent summands
so that each of the sums $ T$, $T_{1}$ and $ T_{2}$ contains
$n=\lceil p\kern1pt N /(16 \kern1pt s) \rceil $
independent summands
$U_j$. Such an $n$ exists since $p\kern1pt N\geq c_s$
with a sufficiently large $c_s$.
Lemma \ref{L5.1} implies that
\beq\label{434}2 \, \varphi (t) \leq
{\mathbf E\,} \operatorname {e} \bgl \{ 2\, t \,
\langle \mathbb Q\,\widetilde T,\widetilde T_{1} \rangle \bgr\} +
{\mathbf E\,} \operatorname {e} \bgl \{ 2\, t \,
\langle \mathbb Q\,\widetilde T, \widetilde T_{2} \rangle \bgr\} . \eeq
Inequality \eqref{eq7.1q} follows now from \eqref{434} and
Lemma~\ref{L6.3}.
Let now
$W= \fsu V1n$ and ~$W'=V_1'+\dots + V_n' $ be independent sums of
of independent $($non-i.i.d.$)$ vectors with
$ \bgl\{\mathcal L(V_j), \mathcal L(V_j'):\, \, 1\leq j\leq n\bgr\} \subset
{\mathbf {\Gamma}}\kern1pt (\delta;\mathbb D, \mathcal S )$.
Using that all random vectors $\widetilde V_j$ are symmetrized and have
non-negative characteristic functions and applying H\"older's
inequality, we obtain, for each $t$, \begin{eqnarray}\label{l1} {\mathbf E\,}
\operatorname {e} \bgl \{ t \,
\langle\mathbb Q\,\widetilde W ,\widetilde W' \rangle \} &=& {\mathbf E}_{\widetilde W'} \Big(\prod_{j=1}^n
{\mathbf E}_{\widetilde V_j}\operatorname {e} \bgl \{ t \,
\langle\mathbb Q\,\widetilde V_j ,\widetilde W' \rangle\bgr\} \Big)\\
& \le & \Big(\prod_{j=1}^n {\mathbf E}_{\widetilde W'} \big({\mathbf E}_{\widetilde V_j} \operatorname {e}
\bgl \{ t \,
\langle\mathbb Q\,\widetilde V_j ,\widetilde W' \rangle\} \big)^n\Big)^{1/n}\\
&=& \Big(\prod_{j=1}^n {\mathbf E}_{\widetilde W'} \big({\mathbf E}_{\widetilde T_j}\operatorname {e} \bgl \{
t \,
\langle\mathbb Q\,\widetilde T_j ,\widetilde W' \rangle\} \big)\Big)^{1/n}\\
& = &\Big(\prod_{j=1}^n {\mathbf E\,} \operatorname {e} \bgl \{ t \,
\langle\mathbb Q\,\widetilde T_j ,\widetilde W' \rangle\}\Big)^{1/n},\label{l2}
\end{eqnarray}
where $\widetilde T_j \=\sum_{l=1}^n \widetilde V_{jl}$ denotes a sum of i.i.d. copies $\widetilde
V_{jl}$ of
$\widetilde V_j$ which are independent
of all other random vectors and variables.
Repeating the steps \eqref{l1}--\eqref{l2} for each factor ${\mathbf E\,} \operatorname {e}
\bgl \{ t \,
\langle\mathbb Q\,\widetilde T_j ,\widetilde W' \rangle\}$ instead of the expectation $ {\mathbf E\,}
\operatorname {e} \bgl \{ t \,
\langle\mathbb Q\,\widetilde W ,\widetilde W' \rangle \} $ on the right hand side
separately, we get (with $\widetilde T'_{k} \=\sum_{l=1}^{n} \widetilde {V}'_{k
l}$, where $\widetilde {V}'_{k l}$ are i.i.d. copies of $\widetilde {V}'_{k}$
independent of all other random vectors) \beq {\mathbf E\,} \operatorname {e}
\bgl \{ t \,
\langle\mathbb Q\,\widetilde W ,\widetilde W' \rangle \} \le
\Big(\prod_{j=1}^n \prod_{k=1}^{n} {\mathbf E\,} \operatorname {e} \bgl \{ t \,
\langle\mathbb Q\,\widetilde T_j ,\widetilde T'_k \rangle\}\Big)^{1/n^2}.\label{qwe1}
\eeq
Thus, using \eqref{qwe1} and the arithmetic-geometric mean inequality, we have
\begin{eqnarray}\int\limits_A^B
\sqrt{{\mathbf E\,} \operatorname {e} \bgl \{ t\, \langle\mathbb Q\, \widetilde W ,\widetilde W'
\rangle/2
\bgr\}} \, \ffrac {dt}{|\penalty250\mskip\thickmuskip\mskip-\thinmuskip t\penalty250\mskip\thickmuskip\mskip-\thinmuskip |}&\le&\int\limits_A^B
\Big(\prod_{j=1}^n \prod_{k=1}^{n} {\mathbf E\,} \operatorname {e} \bgl \{ t \,
\langle\mathbb Q\,\widetilde T_j ,\widetilde T'_k \rangle/2\}\Big)^{1/2n^2} \, \ffrac
{dt}{|\penalty250\mskip\thickmuskip\mskip-\thinmuskip t\penalty250\mskip\thickmuskip\mskip-\thinmuskip |}\nonumber
\\
&\le& \ffrac 1 {n^2} \sum_{j=1}^n\sum_{k=1}^{n}
\int\limits_A^B \Big({\mathbf E\,} \operatorname {e} \bgl \{ t \,
\langle\mathbb Q\,\widetilde T_j ,\widetilde T'_k \rangle/2\}\Big)^{1/2}\, \ffrac {dt}{|\penalty250\mskip\thickmuskip\mskip-\thinmuskip
t\penalty250\mskip\thickmuskip\mskip-\thinmuskip |}\nonumber\\
&\le& \sup_\Gamma \,\int\limits_A^B
\sqrt{{\mathbf E\,} \operatorname {e} \bgl \{ t\, \langle\mathbb Q\, \widetilde T ,\widetilde T'
\rangle/2 \bgr\}} \, \ffrac {dt}{|\penalty250\mskip\thickmuskip\mskip-\thinmuskip t\penalty250\mskip\thickmuskip\mskip-\thinmuskip |},\label{qwe}\end{eqnarray} where
\/ $T= U_1+\dots + U_n$ and ~$T'=U_1'+\dots + U_n' $ are
independent sums of independent copies of random vectors $U$ and
$U'$ respectively, and the supremum $\sup\nolimits_\Gamma $ is
taken over all $ \mathcal L(U), \mathcal L(U') \in
{\mathbf {\Gamma}}\kern1pt (\delta;\mathbb D, \mathcal S )$. Inequalities \eqref{eq7.1q}
and
\eqref{qwe} imply now the statement of the lemma.
$\square$
\medskip
The following Lemma \ref{Le3.2} provides a Poisson summation
formula.
\begin{lemma}\label{Le3.2}
Let\/ $\operatorname{Re} z > 0, \, a,b \in {\mathbb R}^s$ and\/
$\mathbb S: {\mathbb R}^s \rightarrow \mathbb R^s$ be a positive
definite symmetric non-degenerate linear operator. Then \begin{eqnarray}
&&\sum_{m \in \mathbb{Z}^s} \kern1pt
\exp \bigl\{-z\, \mathbb S[\4m+a\kern1pt] + 2\kern1pt \pi\kern1pt i\, \langle\kern1pt
m,b\kern1pt\rangle \bigr\}\nonumber \\
& = &\bgl(\det (\mathbb S / \pi)\bgr)^{-1/2}\kern1pt z^{-s/2} \exp
\bigl\{ - 2\kern1pt \pi \kern1pt i\, \langle\kern1pt a,b\kern1pt\rangle \bigr\} \sum_{l
\in \mathbb{Z}^s} \exp \Bigl\{-\ffrac{\pi^2}{z}\mathbb S^{-1}[\4l
+ b\kern1pt] -2\kern1pt\pi\kern1pt i\,\langle\kern1pt a, l\kern1pt\rangle \Bigr\},\nonumber \end{eqnarray}
where\/ $\mathbb S^{-1}: {\mathbb R}^s \rightarrow \mathbb R^s$
denotes the inverse positive definite operator for\/ $\mathbb S$.
\end{lemma}
\medskip
{\it Proof.}
See, for example, Fricker (1982), {p.}~116, or Mumford (1983),
{p.}~189, formula~(5.1); and {p.}~197, formula (5.9). $\square$
\bigskip
Let the conditions of Lemma \ref{L7.3} be satisfied. Introduce
one-dimensional lattice probability distributions $H_n=\mathcal
L(\xi_n)$ with integer valued $\xi_n$ setting $$
\P\bgl\{\xi_n=k\bgr\}= A_n\,n^{-1/2}\,\exp\left\{-k^2/2n\right\},
\quad \text {for}\ k\in\mathbb Z. $$ It is easy to see that
~${A_n\asymp1}$. Moreover, by Lemma \ref{Le3.2}, \beq \widehat
H_n(t)\ge0,\quad \q \text {for all}\ t\in\mathbb R.\label{eq76} \eeq
Introduce the $s$-dimensional random vector $\zeta_n$ having as
coordinates independent copies of~$\xi_n$. Then, for
$m=(m_1,\dots,m_s)\in\mathbb Z^s$, we have \beq\label{qm}
q(m)\=\P\bgl\{\zeta_n=m\bgr\}=A_n^s\,n^{-s/2}\,
\exp\left\{-\norm{m}^2/2n\right\}. \eeq
\begin{lemma}\label{L7.5}
Let
$W= \fsu V1n$ and ~$W'=V_1'+\dots + V_n' $
denote independent sums of independent copies of random vectors
$V$ and $V'$ such that
$$V=\varepsilon_1\kern1pt z_1+ \dots +\varepsilon_s\kern1pt z_s,\quad \q\quad
V'=\varepsilon_{s+1}\kern1pt z_1'+ \dots +\varepsilon_{2s}\kern1pt z_s',$$ with some
$z_j,z_j'\in \mathbb R^d $. Introduce the matrix $\mathbb B_t= \{
b_{ij}(t): 1\leq i,j\leq s\}$ with $b_{ij}(t)= t\,\langle \mathbb Q\4z_i,z_j'
\rangle$. Then
$$
{\mathbf E\,} \operatorname {e} \bgl \{ t\, \langle \mathbb Q\kern1pt\widetilde W ,\widetilde W'
\rangle/4 \bgr\} \ll_s{\mathbf E\,}\operatorname {e} \bgl \{ \langle\mathbb
B_t \kern1pt \zeta_n,\zeta'_n \rangle \bgr\}+ \exp\left\{-c\kern1pt n\right\},
\quad \q\quad \text{ for all}\ \, t\in \mathbb R,
$$
where $\zeta'_n$ are independent copies of $\zeta_n$ and $c$ is a
positive absolute constant.
\end{lemma}
{\it Proof}. Without loss of generality, we shall assume that
$n\ge c_1$, with a sufficiently large absolute constant~$c_1$.
Consider the random vector $Y=(\widetilde \varepsilon_1,\dots ,\widetilde \varepsilon_s)\in
\mathbb R^s$ with coordinates which are symmetrizations of i.i.d.
Rademacher random variables. Let $R=(R_1, \dots,R_s)$ and $T$
denote independent sums of $n$ independent copies of $Y/2$. Then
we can write \beq\label{rav} {\mathbf E\,} \operatorname {e} \bgl \{ t\,
\langle \mathbb Q\kern1pt\widetilde W ,\widetilde W' \rangle/4 \bgr\} = {\mathbf E\,}
\operatorname {e} \bgl \{ \langle\mathbb B_t \kern1pt R,T \rangle
\bgr\}, \quad \q \text{ for all}\ \, t\in \mathbb R. \eeq Note that
the scalar product $\langle \4\cdot\4, \4\cdot\4 \rangle$ in ${\mathbf E\,}
\operatorname {e} \bgl \{ \langle\mathbb B_t \kern1pt R,T \rangle
\bgr\}$ means the scalar product of vectors in $\mathbb R^s$. In
order to estimate this expectation, we write it in the form \begin{eqnarray}
{\mathbf E\,} \operatorname {e} \bgl \{ \langle\mathbb B_t \kern1pt R,T \rangle
\bgr\} &=& {\mathbf E\,} {\mathbf E}_R\, \operatorname {e} \bgl \{
\langle\mathbb B_t \kern1pt R,T \rangle \bgr\}
\nonumber\\
&=&\sum_{{\overline m}\in\mathbb Z^s}p({\overline m})\sum_{m\in\mathbb Z^s}p(m)
\operatorname {e} \bgl \{ \langle\mathbb B_t \kern1pt m,{\overline m}
\rangle\bgr\} , \label{eq7.7}\end{eqnarray}
with summing over $m=(m_1,\dots,m_s)\in\mathbb Z^s$,
$\overline{m}=(\overline{m}_1,\dots,\overline{m}_s)\in\mathbb Z^s$ and \beq\label{pm}
p(m)=\P\bgl\{R=m\bgr\}=\prod_{j=1}^s\P\bgl\{R_j=m_j\bgr\}
=\prod_{j=1}^s 2^{-2n}\hbox{\begin{footnotesize}$\displaystyle\binom{2\4n}{
m_j+n}$\end{footnotesize}}, \eeq if
$\max\limits_{1\le j\le s}|m_j|\le n$ and $p(m)=0$ otherwise.
Clearly, for fixed \,$T=\overline{m}$, \beq
{\mathbf E}_R\, \operatorname {e} \bgl \{ \langle\mathbb B_t \kern1pt R,T \rangle \bgr\}
=\sum_{m\in\mathbb Z^s}p(m) \operatorname {e} \bgl \{
\langle\mathbb B_t \kern1pt m,{\overline m} \rangle\bgr\}\ge0 \label{eq7.8}
\eeq is a value of the characteristic function of symmetrized
random vector $\mathbb B_t \kern1pt R$. Using Stirling's formula, it is
easy to show that there exist positive absolute constants $c_2$
and $c_3$ such that \beq \P\bgl\{R_j=m_j\bgr\}\ll
n^{-1/2}\,\exp\left\{-m_j^2/2n\right\},\quad \text{ for}\ |m_j|\le
c_2\4n,\label{eq7.9} \eeq and \beq \P\bgl\{|R_j|\ge
c_2\4n\bgr\}\ll \exp\left\{-c_3\4n\right\}.\label{eq7.10} \eeq
Using \eqref{eq7.7}--\eqref{eq7.10}, we obtain \begin{eqnarray} {\mathbf E\,}
\operatorname {e} \bgl \{ \langle \mathbb B_t \kern1pt R,T \rangle
\bgr\} &\ll_s&\sum_{{\overline m}\in\mathbb Z^s}q(\overline m)
\sum_{m\in\mathbb Z^s}p(m) \operatorname {e} \bgl \{ \langle
\mathbb B_t \kern1pt m,{\overline m} \rangle\bgr\}+
\exp\left\{-c_3\4n\right\}\nonumber\\
& =&\sum_{m\in\mathbb Z^s}p(m) \sum_{{\overline m}\in\mathbb Z^s}q(\overline m)
\operatorname {e} \bgl \{ \langle \mathbb B_t \kern1pt m,{\overline m}
\rangle\bgr\}+
\exp\left\{-c_3\4n\right\}\nonumber\\
& =&{\mathbf E\,}{\mathbf E}_{\zeta_n} \operatorname {e} \bgl \{ \langle
\mathbb B_t \kern1pt R,\zeta_n \rangle \bgr\}+
\exp\left\{-c_3\4n\right\}\nonumber\\
& =&{\mathbf E\,} \operatorname {e} \bgl \{ \langle \mathbb B_t \kern1pt R,\zeta_n
\rangle \bgr\}+ \exp\left\{-c_3\4n\right\}. \label{eq7.12} \end{eqnarray}
Now we repeat our previous arguments, noting that \beq {\mathbf
E}_{\zeta_n} \operatorname {e} \bgl \{ \langle \mathbb B_t \kern1pt
R,\zeta_n \rangle \bgr\} =\sum_{\overline m\in\mathbb Z^s}q(\overline m)\,
\operatorname {e} \bgl \{ \langle \mathbb B_t \kern1pt R,{\overline m}
\rangle\bgr\}\ge0 \label{eq7.13} \eeq is a value of the
non-negative characteristic function of the random vector
$\zeta_n$ (see \eqref{eq76}). Using again \eqref{eq7.9} and
\eqref{eq7.10}, we obtain \beq
{\mathbf E\,}\operatorname {e} \bgl \{ \langle \mathbb B_t \kern1pt R,\zeta_n \rangle \bgr\}
\ll_s{\mathbf E\,}\operatorname {e} \bgl \{ \langle \mathbb B_t \kern1pt
\zeta_n,\zeta'_n \rangle \bgr\}+ \exp\left\{-c_3\4n\right\}.
\label{eq7.14} \eeq Relations \eqref{rav}, \eqref{eq7.12} and
\eqref{eq7.14} imply the statement of the lemma. $\square$
\medskip
\smallskip
Let us estimate the expectation $ {\mathbf E\,}\operatorname {e} \bgl \{
\langle \mathbb B_t \kern1pt \zeta_n,\zeta'_n \rangle \bgr\}$ under the
conditions of Lemmas~\ref{L7.3} and~\ref{L7.5}, assuming that
$s=d$, $\mathbb D=\mathbb C^{-1/2}$, $\delta\leq 1/(5\kern1pt s)$, $n\ge
c_4$, where $c_4$ is a sufficiently large absolute constant, and
\beq\|\mathbb C^{-1/2}z_j-e_j\|\leq \delta,\quad \q\quad \|\mathbb
C^{-1/2}z_j'-e_j\|\leq \delta,\quad \q\quad \text{for}\ \, 1\leq j\leq
s,\label{eq:7.6} \eeq with an orthonormal system $\mathcal
S=\mathcal S_o=\bgl\{e_1,e_2,\ldots,e_s\bgr\} $ involved in the
conditions of Lemma~\ref{L7.3}. We can rewrite ${\mathbf E\,}\operatorname
{e} \bgl \{ \langle\mathbb B_t \kern1pt \zeta_n,\zeta'_n \rangle\bgr\}$
as
$$
{\mathbf E\,}\operatorname {e} \bgl \{ \langle\mathbb B_t \kern1pt \zeta_n,\zeta'_n
\rangle\bgr\} =\sum_{{\overline m}\in\mathbb Z^s}q({\overline m})\sum_{m\in\mathbb
Z^s}q(m) \operatorname {e} \bgl \{ \langle\mathbb B_t \kern1pt\overline m,m
\rangle\bgr\}.
$$
Thus, by \eqref{qm},
$$
{\mathbf E\,}\operatorname {e} \bgl \{ \langle\mathbb B_t \kern1pt \zeta_n,\zeta'_n
\rangle\bgr\} =A_n^{2s}\,n^{-s}\,\sum_{{\overline m}\in\mathbb
Z^s}\sum_{m\in\mathbb Z^s} \exp \bgl \{ i\,\langle\mathbb B_t \kern1pt\overline
m,{m}\rangle-\norm{m}^2/2n -\norm{\overline m}^2/2n \bgr\}.
$$
Denote \beq\label{defr}r=\sqrt{2 \kern1pt\pi^2\4n}.
\eeq Applying Lemma \ref{Le3.2} with \,$\mathbb S=\mathbb I_s$,
$z=1/2n$, \,$a=0$, \,$b=(2\pi)^{-1}}\def\sign{\hbox{\rm sign}\,\mathbb B_t\,\overline m $ \,and
using that ~${A_n\asymp1}$, we obtain \begin{eqnarray}\label{koren}
{\mathbf E\,}\operatorname {e}\bgl \{ \langle\mathbb B_t \kern1pt \zeta_n,\zeta'_n
\rangle\bgr\} &\ll_s& n^{-s/2}\,\sum_{l,m \in \mathbb{Z}^s} \exp
\bgl \{-2\kern1pt \pi^2\4n\,\norm{\kern1pt
l+(2\pi)^{-1}\mathbb B_t \kern1pt m}^2-\norm{m}^2/2n \bgr\} \nonumber\\
&\ll_s&r^{-s}\sum_{m,\overline m \in \mathbb{Z}^{s}} \exp
\bgl \{-r^2\,\norm{m-t\,\mathbb V \kern1pt \overline m}^2-\norm{\overline m}^2/r^2
\bgr\},\label{koren1} \end{eqnarray} where $\mathbb V:\mathbb
R^{s}\to\mathbb R^{s}$ is the operator with matrix
\beq\label{bbbl} \mathbb V=(2\pi)^{-1}\mathbb B_1. \eeq Note that
the right-hand side of \eqref{koren1} may be considered as a
theta-series.
Denote $y_k=\mathbb C^{-1/2}z_k$, $1\leq k\leq s$. Let $\mathbb Y$
be the $(s\times s)$-matrix with entries $\langle\kern1pt e_j,
y_k\rangle$, where index $j$ is the number of the row, while $k$
is the number of the column. Then the matrix $\mathbb F\=\mathbb
Y^*\kern1pt\mathbb Y$ has entries $\langle y_j, y_k\rangle$. Here
$\mathbb Y^*$ is the transposed matrix for $\mathbb Y$. According
to \eqref{eq:7.6}, we have \beq\|y_j-e_j\|\leq \delta,\quad \q \text{for}\
\, 1\leq j\leq s,\label{eq:7.6o} \eeq
Let us show that (cf. BG (1997a, proof of Lemma~7.4))
\beq\label{bbl} \norm{\mathbb Y}\le 3/2
\quad \text{and}\quad \norm{\mathbb Y^{-1}}\le 2. \eeq Since $\mathcal
S_o=\bgl\{e_1,e_2,\ldots,e_s\bgr\} $ is an orthonormal system,
inequalities \eqref{eq:7.6o} imply that $\mathbb Y=\mathbb
I_s+\mathbb A$ with some matrix $\mathbb A=\{a_{ij}\}$ such that
$|a_{ij}|\leq \delta$. Thus, we have $\|\mathbb A\| \leq \|\mathbb
A\|_2\leq s\kern1pt \delta $, where $\|\mathbb A\|_2$ denotes the
Hilbert--Schmidt norm of the matrix~$\mathbb A$. Therefore, the
condition $\delta\leq 1/(5\kern1pt s)$ implies $\|\mathbb A\| \leq 1/2$ and
inequalities \eqref{bbl}.
The matrix $\mathbb F$ is symmetric and positive definite. Its
determinant is the product of eigenvalues which (by \eqref{bbl})
are bounded from above and from below by some absolute positive
constants. Moreover, \beq\label{dett3} \left(\det \mathbb
Y\right)^2=\left(\det \mathbb Y^*\right)^2=\det \mathbb F
\asymp_s1\asymp\norm{\mathbb F}\asymp\norm{\mathbb Y}. \eeq Define
the matrices $\overline{\mathbb Y}$ and $\overline{\mathbb F}$, replacing
$z_j$ by $z_j'$ in the definition of ${\mathbb Y}$ and ${\mathbb
F}$.
Similarly to~\eqref{dett3}, one can show that\beq\label{dett4}
\left(\det \overline{\mathbb Y}\right)^2=\bgl(\det \overline{\mathbb
Y}\penalty250\mskip\thickmuskip\mskip-\thinmuskip^*\bgr)^2=\det \overline{\mathbb F} \asymp_s1\asymp\norm{\overline{\mathbb
F}}\asymp\norm{\overline{\mathbb Y}}. \eeq Let $\mathbb G$ and
$\overline{\mathbb G}$ be the $(s\times s)$-matrices with entries
$\langle\kern1pt e_j,\mathbb Q\4z _k\rangle$ and $\langle\kern1pt e_j,
z'_k\rangle$ respectively. Then, clearly, $\mathbb G=\mathbb
Q\kern1pt\mathbb C^{1/2}\mathbb Y$ and $\overline{\mathbb G}=\mathbb
C^{1/2}\kern1pt\overline{\mathbb Y}$. Therefore, \beq\label{dett5} \mathbb
B_1=\mathbb G^*\kern1pt\overline{\mathbb G}=\mathbb Y^*\kern1pt\mathbb
C^{1/2}\kern1pt\mathbb Q\kern1pt\mathbb C^{1/2}\kern1pt\overline{\mathbb Y}.
\eeq
Moreover, $\mathbb Q^2= \mathbb I_d$ implies that $\bgl|\det
\mathbb Q\bgr|= 1$ and $\norm {\mathbb Q}=1$.
Using relations \eqref{bbbl}
and \eqref{dett3}--\eqref{dett5}, we
obtain\beq\label{ett7} \bgl|\det \mathbb V\bgr|\asymp_s\bgl|\det
\mathbb B_1\bgr|\asymp_s\det \mathbb C, \eeq and \beq\label{ett8}
\norm {\mathbb V}\ll\norm {\mathbb B_1}\ll\norm {\mathbb
C}\ll\sigma_1^2. \eeq $\phantom 0$
\section{\label {s6}Some facts from Number Theory}
In Section \ref{s6}, we consider some facts of the geometry of
numbers (see Davenport (1958) or Cassels (1959)). They will help
us to estimate the integrals of the right-hand side of inequality
\eqref{koren1}.
Let $e_1, e_2,\ldots, e_d$ be linearly independent vectors in
$\mathbb R^d$. The set \beq \Lambda=\Big\{{
\sum_{j=1}^d} n_j e_j:n_j\in\mathbb Z,\ j=1,2,\ldots,d\Big\} \eeq is
called the lattice with basis $e_1, e_2,\ldots, e_d$. The
determinant $\det(\Lambda)$ of a lattice $\Lambda$ is the modulus
of the determinant of the matrix formed from the vectors $e_1,
e_2,\ldots, e_d$: \beq \det(\Lambda)\=\bgl|\det(e_1, e_2,\ldots,
e_d)\bgr|. \eeq The determinant of a lattice does not depend on
the choice of basis. Any lattice $\Lambda\subset\mathbb R^d$ can be
represented as $\Lambda=\mathbb A\,\mathbb Z^d$, where $\mathbb A$ is a
non-degenerate linear operator. Clearly, $\det
(\Lambda)=\left|\det\mathbb A\right |$.
Let $m_1,\ldots,m_l\in\Lambda$ be linearly independent vectors
belonging to a lattice $\Lambda$. Then the set \beq
\Lambda'=\Big\{{ \sum_{j=1}^l} n_j m_j:n_j\in\mathbb Z,\
j=1,2,\ldots,l\Big\} \eeq is an $l$-dimensional
sublattice of the lattice $\Lambda$. Its determinant $\det(\Lambda')$
is the modulus of the determinant of the matrix formed from the coordinates of
the vectors $m_1, m_2,\ldots, m_l$ with respect to an orthonormal
basis of the linear span of the vectors $m_1, m_2,\ldots, m_l$.
The determinant $\det(\Lambda')$ could be also defined as
$\det\bgl(\langle m_i, m_j\rangle, i, j=1, \ldots, l\bgr)^{1/2}$.
Let $F: {\mathbb R}^d \rightarrow [\40, \infty\kern1pt]$ denote a norm on
${\mathbb R}^d$, that is $F( \alpha\kern1pt x) = \lvert \alpha \rvert\, F(x),$
for $\alpha \in {\mathbb R}$,
and $F(x + y) \leq F(x) +F(y)$.
The successive minima $M_1 \leq \dots \leq M_d$ of $F$ with
respect to a lattice $\Lambda$ are defined as follows: Let \,$M_1
= \inf\bgl \{F(m): m \neq 0,\ m \in \Lambda \bgr\}$ \,and define
$M_j$ as the infimum of $\lambda > 0$ such that the set \,$\bgl\{m
\in \Lambda \,:\, F(m) < \lambda\bgr \}$ \,contains $j$ linearly
independent vectors. It is easy to see that these infima are
attained, that is there exist linearly independent vectors $b_1,
\dots, b_d \in \Lambda$ such that $F(b_j) = M_j$, $j=1,\ldots,d$.
The following Lemma \ref{Dav2} is proved by Davenport (1958, Lemma 1), see also
G\"otze and Margulis~(2010).
\begin{lemma} \label{Dav2}
Let $M_1 \leq \dots \leq M_d$
be the successive minima of a norm $F$ with
respect to the lattice $\mathbb Z^d$. Denote $M_{d+1}=\infty$.
Suppose that $1\le j\le d$ and
$M_j \leq b \leq M_{j+1}$, for some~${b>0}$.
Then \beq \#\bgl\{m=(m_1, \dots,
m_d) \in \mathbb{Z}^d:\;F(m) < b\bgr\}
\asymp_d b^j (M_1\4\cdot\4 M_2
\cdots M_j)^{-1}. \eeq
\end{lemma}
\medbreak
Representing $\Lambda=\mathbb A\,\mathbb Z^d$, we see that the lattice
$\mathbb Z^d$ may be replaced in Lemma \ref {Dav2} by any lattice
$\Lambda\subset\mathbb R^d$. It suffices to apply this lemma to the norm
$G(m)=F(\mathbb A\4m)$, $m\in\mathbb Z^d$.
\begin{lemma} \label{Dav9}
Let \,$ F_j (m)$, \,$j=1,2$, be some norms in ${\mathbb R}^d$ and $M_1
\leq \dots \leq M_d$ and $N_1 \leq \dots \leq N_d$ be the
successive minima of\/ $F_1$ with respect to a lattice $\Lambda_1$
and of\/ $F_2$ with respect to a lattice $\Lambda_2$ respectively.
Let $C>0$. Assume that $M_k\gg_d C\, F_2(n_k)$, $k=1,2,\ldots,d$, for some
linearly independent vectors $n_1,n_2,\ldots,n_d\in \Lambda_2$.
Then \beq M_k\gg_d C\,N_k,\quad k=1,\ldots,d. \eeq
\end{lemma}
The proof of this lemma is elementary and therefore omitted.
Let $\norm x_\infty=\max_{1\le j\le d}|x_j|$, for $x=({x}_1,
\dots, {x}_d)\in \mathbb R^d$.
\begin{lemma} \label{Dav1}
Let $\Lambda$ be a lattice in ${\mathbb R}^d$ and let\/ $0<\varepsilon\le1$. Then
\beq\label{norma}
e^{-\varepsilon}\,\# H\le
\sum_{v \in \Lambda} \exp \bgl \{- \varepsilon\,\norm v^2 \bgr\}
\ll_d \varepsilon^{-d/2}\# H, \eeq where \,$H\overset{ \text{\rm def} } = \bgl\{v \in \Lambda \, : \,
\norm v_\infty< 1 \bgr\}$.
\end{lemma}
\begin{proof}The lower bound in \eqref{norma} is almost evident by restricting
summation to the set~$H$.
Introduce for \,$\mu = (\mu_1, \dots, \mu_{d})\in
{\mathbb Z}^{d}$ \,the sets \beq B_\mu \overset{ \text{\rm def} } = \left[\,\mu_1-
\ffrac{1}{2},\, \mu_1 +
\ffrac{1}{2}\right)\times\cdots\times \left[\,\mu_{d}-
\ffrac{1}{2},\,\mu_{d} + \ffrac{1}{2}\right)\nonumber\eeq such
that ${\mathbb R}^{d} = \bigcup_{\mu} B_\mu$. For any fixed \,$w^* \in
H_\mu\overset{ \text{\rm def} } = \bigl\{ w\in \Lambda \cap B_\mu \bigr\}$ \,we have
\begin{eqnarray} w - w^*\in H,\qquad\text{for any }w\in H_\mu. \nonumber \end{eqnarray}
Hence we conclude for any $\mu\in {\mathbb Z}^{d}$ \beq \# H_\mu \, \leq \, \#
H .\label{eq:boxx1} \eeq
Since \,$x \in B_\mu $ \,implies \, $ \norm{x}_{\infty} \geq
\norm{ \mu }_{\infty}/2$, \, we obtain by \eqref{eq:boxx1}
\begin{eqnarray}\sum_{v \in \Lambda} \exp \bgl \{- \varepsilon\,\norm v^2 \bgr\}
&\le&\sum_{v \in \Lambda } \exp \bgl \{- \varepsilon\,\norm{ \4v}_\infty^2
\bgr\}\nonumber\\
& \ll_d &\# H_0 +
\sum_{\mu \in {\mathbb Z}^d \setminus 0}
\quad \sum_{v \in \Lambda } {\bf I}\bgl\{ \4v \in B_\mu\bgr\}\kern1pt
\exp \bigl\{- \varepsilon\,\norm{ \mu }^2_{\infty}/4\bigr\} \nonumber \\
& \ll_d & \# H\4\cdot\4 \sum_{\mu \in \Xi}
\kern1pt \exp \bigl\{- \varepsilon\,\norm {\mu}_{\infty}^2/4\bigr\}
\nonumber \\
& \ll_d & \varepsilon^{-d/2}\kern1pt\# H .
\end{eqnarray}
This conclude the proof of Lemma \ref{Dav1}.
\end{proof}
It is easy to see that Lemma \ref{Dav1} implies the following statement.
\begin{corollary} \label{Dav5}
Let $\Lambda$ be a lattice in ${\mathbb R}^d$ and let $c_j(d)$,
\,$j=1,2, 3, 4$, be positive quantities depending on $d$ only. Let \,$ F (\4\cdot\4)$
be a norm in ${\mathbb R}^d$ such that $F(\4\cdot\4)\asymp_d
\norm{\4\cdot\4}$. Then \begin{eqnarray}
\sum_{v \in \Lambda} \exp \bgl \{- c_1(d)\,\norm v^2 \bgr\}
&\asymp_d& \sum_{v \in \Lambda} \exp \bgl \{- c_2(d)\,(F(v))^2 \bgr\}\nonumber\\
&\asymp_d&\# \bgl\{v \in \Lambda \, : \,
\norm v< c_3(d) \bgr\}\nonumber\\
&\asymp_d&\# \bgl\{v \in \Lambda \, : \,
F(v)< c_4(d) \bgr\}. \end{eqnarray}
\end{corollary}
The proof of Corollary \ref{Dav5} is elementary and therefore
omitted. Note only that \beq\#\bgl\{v \in \Lambda \, : \, F(v)<
\lambda \bgr\}=\#\bgl\{v \in \mu^{-1}}\def\sign{\hbox{\rm sign}\Lambda \, : \, F(v)<
\lambda/\mu \bgr\},\quad \text{for }\lambda,\mu >0.\eeq
For a lattice $\Lambda\subset \mathbb R^{d}$ and $1\le l\le d$, we
define its $\alpha_l$-characteristics by \beq\label{alp} \alpha_l(
\Lambda)\= \sup \bgl\{\,\bgl|\det (\Lambda')\bigr|^{-1}:
\Lambda'\subset \Lambda,\ \ l\text {-dimensional sublattice of
$\Lambda$}\bgr \}. \eeq Denote \beq\label{alp3} \alpha( \Lambda)\=
\max_{1\le l\le d} \alpha_l( \Lambda). \eeq
\begin{lemma} \label{Dav4}
Let \,$ F (\4\cdot\4)$ be a norm in ${\mathbb R}^d$ such that $F(\4\cdot\4)\asymp_d
\norm{\4\cdot\4}$. Let $c(d)$
be a positive quantity depending on $d$ only.
Let $M_1 \leq \dots \leq M_d$ be the successive minima of $F$ with
respect to a lattice $\Lambda\subset\mathbb R^d$. Then \beq \label{LLL1}\alpha_l(
\Lambda)\asymp_d (M_1\4\cdot\4 M_2
\cdots M_l)^{-1},\quad l=1,\ldots,d.
\eeq Moreover, \beq \label{LLL2}\alpha( \Lambda)\asymp_d \#\bgl\{v
\in \Lambda \, : \, \norm v< c(d) \bgr\}, \eeq provided that
$M_1\ll_d1$.
\end{lemma}
For the proof of Lemma \ref{Dav4} we shall use the following lemma
formulated in Proposition (p.~517) and Remark (p.~518) in A.K.
Lenstra, H.W. Lenstra and Lov\'asz (1982).
\begin{lemma} \label{LLL}
Let $M_1 \leq \dots \leq M_d$ be the successive minima of the
standard Euclidean norm with respect to a lattice
$\Lambda\subset\mathbb R^d$. Then there exists a basis $e_1, e_2,\ldots,
e_d$ of $\Lambda$ such that \beq
M_l\asymp_d\norm{e_l},\quad l=1,\ldots,d.
\eeq Moreover, \beq \det(\Lambda)\asymp_d\prod_{l=1}^d \norm{e_l}.
\eeq
\end{lemma}
{\it Proof of Lemma\/ {\rm\ref{Dav4}.}} According to Lemma
\ref{Dav9}, we can replace the Euclidean norm~$\norm{\4\cdot\4}$ by the
norm $ F (\4\cdot\4)$, in the formulation of Lemma \ref{LLL}. Let
$\Lambda'\subset \Lambda$ be an arbitrary
$l${$\hbox{-}$}dimensional sublattice of~$\Lambda$ and $N_1 \leq
\dots \leq N_l$ be the successive minima of the norm~$ F (\4\cdot\4)$
with respect to $\Lambda'$. It is clear that $M_j \leq N_j$,
$j=1,2, \ldots,l$. On the other hand, $M_j= F(m_j)$ for some
linearly independent vectors $m_1,m_2,\ldots,m_l\in \Lambda$. In
the case, where \beq \Lambda'=\Big\{{ \sum_{j=1}^l} n_j
m_j:n_j\in\mathbb Z, j=1,2,\ldots,l\Big\}, \eeq we have $N_j =
M_j$, $j=1,2, \ldots,l$. In order to justify relation \eqref{LLL1}
it remains to take into account definition~\eqref{alp} and to
apply Lemma~\ref{LLL}. Relation \eqref{LLL2} is an easy
consequence of \eqref{LLL1}, Lemma~\ref{Dav2}
and Corollary \ref{Dav5}.
$\square$\medskip
\medbreak
$\phantom 0$
\section{\label {s7}From Number Theory to Probability}
In Section \ref{s7}, we shall use number-theoretical results of
Section \ref{s6} to estimate integrals of the right-hand side of
\eqref{koren1}. Recall that we have assumed the conditions of
Lemmas~\ref{L7.3} and~\ref{L7.5}, $s=d$, $\mathbb D=\mathbb
C^{-1/2}$, $ \delta\leq 1/(5\kern1pt s)$, $n\ge c_4$, and \eqref{eq:7.6},
for an orthonormal system $\mathcal S=\mathcal S_o$. The notation
$\hbox{SL}(d, \mathbb R)$ is used below for the set of all $(d\times
d)$-matrices with real entries and determinant~1.
Introduce the matrices \beq\label{svo4}
\mathbb D_r \= \left(\begin{array}{*{2}c} r\,
\mathbb I_{s} & \mathbb O_{s} \\
\mathbb O_{s} & r^{-1}\, \mathbb I_{s} \end{array}\right)
\in \kern1pt \hbox{SL}(2s,\mathbb R),\quad r>0, \eeq \beq\label{svon}
\mathbb K_t \=
\left(\begin{array}{*{2}c} \mathbb I_{s} & -t \,\mathbb I_{s}\\
t \,\mathbb I_{s} & \mathbb I_{s}\end{array}\right),\quad t\in\mathbb R,
\eeq \beq\label{svo5}
\mathbb U_t \=
\left(\begin{array}{*{2}c} \mathbb I_{s} & -t \,\mathbb I_{s}\\
\mathbb O_{s} & \mathbb I_{s}\end{array}\right)\in \kern1pt
\hbox{SL}(2s,\mathbb R),\quad t\in\mathbb R,
\eeq
and the lattices \beq \label{latt}
\Lambda\=\left(\begin{array}{*{2}c} \mathbb
I_{s}&\mathbb O_{s} \\
\mathbb O_{s} &\mathbb V_0\end{array}\right)\mathbb
Z^{2s}, \eeq \beq \label{jj}\Lambda_{j}= \mathbb D_{j}\kern1pt \mathbb
U_{j^{-1}}\kern1pt \Lambda=\left(\begin{array}{*{2}c}j\, \mathbb
I_{s}&\mathbb -\mathbb V_0
\\
\mathbb O_{s} &j^{-1}\,\mathbb V_0\end{array}\right)\mathbb Z^{2s},
\quad j=1,2,\ldots,\eeq where
\beq\label{svo9} \mathbb V_0=\sigma_1^{-2}\,\mathbb V \eeq and
the matrix $\mathbb V$ is defined in \eqref{bbbl}.
Below we shall use the following simplest properties of
these matrices: \beq\label{svo} \mathbb D_a\kern1pt\mathbb D_b=\mathbb D_{ab},\quad
\mathbb U_a\kern1pt\mathbb U_b=\mathbb U_{a+b}\quad \text {and} \quad \mathbb D_a\kern1pt \mathbb
U_b = \mathbb U_{a^2b}\,\mathbb D_a, \quad \text{for $a,b>0$.} \eeq
Let $M_{j,t}$, $j=1,2, \ldots,2\4s$, be the successive minima of
the norm $\norm{\4\cdot\4}_\infty$ with respect to the lattice \beq
\label{latt8} \Xi_t\=\left(\begin{array}{*{2}c} r\, \mathbb
I_{s}&-r\4t\,\mathbb V \\
\mathbb O_{s} &r^{-1}}\def\sign{\hbox{\rm sign}\, \mathbb
I_{s}\end{array}\right)\mathbb Z^{2s}. \eeq Moreover,
simultaneously, $M_{j,t}$ are the successive minima of the norm
$F^*(\4\cdot\4)$ defined for $(m,\overline m)\in\mathbb R^{2s}$, $m,\overline
m\in\mathbb R^{s}$, by\beq \label{latt5} F^*((m,\overline m))\=\max
\bigl\{ \norm {m}_\infty, \,\sigma_1^{2}\,\norm {\mathbb V^{-1}}\def\sign{\hbox{\rm sign}\overline
m}_\infty \bigr \} \eeq with respect to the lattice \beq
\label{latt6} \Omega_t\=\left(\begin{array}{*{2}c} r\, \mathbb
I_{s}&-r\4t\,\mathbb V \\
\mathbb O_{s} &\sigma_1^{-2}\,r^{-1}}\def\sign{\hbox{\rm sign}\, \mathbb V\end{array}\right)\mathbb Z^{2s}
=\mathbb D_r\kern1pt\mathbb U_u\,\Lambda, \quad \text{where
}u\=\sigma_1^{2}\,t . \eeq
Using Lemmas
\ref{Dav9} and \ref{LLL} and the equality $\det(\Xi_t)=1$, it is easy to show that \beq \label{tyy1}
M_{1,t}\ll_s 1 .
\eeq
Let $M_{j,t}^*$ be the successive minima of the Euclidean
norm
with respect to the lattice~$\Omega_t$. Note that, according to
\eqref{ett8} and \eqref{latt5}, \beq \label{att}\norm{\4\cdot\4}
\ll_{s} F^*(\4\cdot\4). \eeq Using \eqref{att} and Lemma \ref{Dav9},
we obtain
\beq\label{att88} M_{j, t}^*\ll_s M_{j, t},\quad j=1,\ldots,2\4s.
\eeq According to Lemma~\ref{Dav4}, \beq\label{att89}\alpha(\Xi_t)
\ll_s\alpha(\Omega_t). \eeq
Let us estimate $\alpha(\Omega_t)$ assuming that $r\ge1$ and (for
a moment) $t=\sigma_1^{-2}\,r^{-1}}\def\sign{\hbox{\rm sign}$. In this case \beq
\label{latt36} \Omega_t=\left(\begin{array}{*{2}c} r\, \mathbb
I_{s}&-\mathbb V_0 \\
\mathbb O_{s} &r^{-1}}\def\sign{\hbox{\rm sign}\, \mathbb V_0\end{array}\right)\mathbb Z^{2s}. \eeq
By relation \eqref{LLL2} of Lemma \ref{Dav4}, we have
\beq\alpha(\Omega_t) \asymp_s\#\bgl\{v \in \Omega_t \, : \, \norm
v< 1/2 \bgr\}=\#K, \label{att889}\eeq where \beq K=\bgl\{v=(m, \overline
m)\in \mathbb Z^{2s}: m, \overline m\in \mathbb Z^{s}, \, \, \norm
{r\4m-\mathbb V_0\kern1pt\overline m}^2+\norm {r^{-1}}\def\sign{\hbox{\rm sign}\kern1pt\mathbb V_0\kern1pt\overline m}^2<
1/4 \bgr\}. \label{a889}\eeq Let us estimate from above the
right-hand side of \eqref{att889}. If $v=(m, \overline m)\in K$, then
\beq r\,\norm m\le \norm {r\4m-\mathbb V_0\kern1pt\overline m}+\norm {\mathbb
V_0\kern1pt\overline m}< \ffrac12+\ffrac r{2}\le r. \label{ad889}\eeq Hence
$m=0$ and $\norm {\mathbb V_0\kern1pt\overline m}\le1/2$. It remains to
estimate the quantity \beq R\=\# \bgl\{ \overline m \in\mathbb Z^{s}: \,
\, \norm {\mathbb V_0\kern1pt\overline m}< 1 \bgr\}\ge\# K. \label{ada889}\eeq
Let $N_1\le\cdots\le N_{s}$ be the successive minima of the
Euclidean norm
with respect to the lattice~$\mathbb V_0\kern1pt\mathbb Z^{s}$.
Let $e_1,e_2,\ldots, e_{s}$ be the standard orthonormal basis of
$\mathbb Z^{s}$. By \eqref{ett8} and~\eqref{svo9}, we have
$\norm{\mathbb V_0\4e_j}\le 1$, $j=1,2, \ldots, s$. Therefore,
using Lemma \ref{Dav9}, we see that $N_1\le\cdots\le N_{s}\le 1$.
By \eqref{ett7}, \eqref{svo9}, \eqref{ada889} and by Lemmas
\ref{Dav2}, \ref{Dav9} and \ref{LLL}, \beq \label{f99}
R\asymp_s (N_1\4\cdot\4 N_2
\cdots N_s)^{-1}\asymp_s(\det \mathbb V_0)^{-1}}\def\sign{\hbox{\rm sign}\asymp_s\sigma_1^{2s}\,(\det
\mathbb C)^{-1}. \eeq Hence, using \eqref{att889}, \eqref{ada889}
and \eqref{f99} we conclude that\beq \label{adas89}
\alpha(\Omega_t)
\ll_s\sigma_1^{2s}\,(\det \mathbb C)^{-1}, \quad \text{for
}\;r\ge1\quad \text {and}\quad t=\sigma_1^{-2}\,r^{-1}}\def\sign{\hbox{\rm sign}. \eeq
Let now $t\in\mathbb R$ be arbitrary. By \eqref{latt8},
\eqref{tyy1}, \eqref{att89},
Lemmas \ref{Dav2}, \ref{Dav4} and Corollary~\ref{Dav5}, \begin{eqnarray}
\label{ken4}\sum_{m,\overline m \in \mathbb{Z}^{s}} \exp \bgl
\{-r^2\,\norm{m-t\,\mathbb V \kern1pt \overline m}^2\!\!&-&\!\!\norm{\overline
m}^2/r^2
\bgr\}=\sum_{v \in \Xi_t} \exp \bgl \{- \norm v^2 \bgr\}\nonumber\\
&\ll_s&R_t\=\#\bgl\{v \in \Xi_t \, : \, \norm v< 1 \bgr\}\nonumber\\ &
\ll_s& \alpha(\Xi_t) \ll_s\alpha(\Omega_t). \label{kore}\end{eqnarray}
Now, by \eqref{koren}, \eqref{latt6} and
\eqref{ken4}, we have \begin{eqnarray}\label{koren4}
{\mathbf E\,}\operatorname {e}\bgl \{
\langle\mathbb B_t \kern1pt \zeta_n,\zeta'_n \rangle\bgr\} \ll_s
r^{-s}\,\alpha (\Omega_t) =r^{-s}\,\alpha (\mathbb D_r\kern1pt\mathbb
U_u\,\Lambda),\quad \text{where }u=\sigma_1^{2}\,t. \end{eqnarray}
Let us estimate the quantity $R_t$, $t\in \mathbb R$, defined in
\eqref{kore} assuming that $r\ge 1$ and $\left|r\4t\right|\le
c_s^*\,\sigma_1^{-2}$, where $c_s^*\ge1$ is an arbitrary quantity
depending on $s$ only. By Corollary~\ref{Dav5}, we have \beq
\label{ad888}R_t\asymp_s\# K_0, \eeq where \beq K_0\=\bgl\{v=(m,
\overline m)\in \mathbb Z^{2s}: m, \overline m\in \mathbb Z^{s}, \, \, \norm
{r\4m-r\4t\,\mathbb V\kern1pt\overline m}^2+ \norm {r^{-1}}\def\sign{\hbox{\rm sign}\kern1pt\overline m}^2<
(2\4c_s^*)^{-2} \bgr\}. \label{ba889}\eeq If $v=(m, \overline m)\in
K_0$, $r\ge 1$ and $\left|r\4t\right|\le c_s^*\,\sigma_1^{-2}$,
then, by \eqref{ett8} and \eqref{ba889},\beq r\,\norm m\le \norm
{r\4m-r\4t\,\mathbb V\kern1pt\overline m}+\left|r\4t\right|\,\norm {\mathbb
V\kern1pt\overline m} < \ffrac12+\ffrac {r}{2}\le r. \label{polk}\eeq Hence
$m=0$ and $\left|r\4t\right|\norm {\mathbb V\kern1pt\overline
m}\le(2\4c_s^*)^{-1}}\def\sign{\hbox{\rm sign}<1$. It remains to estimate the quantity \beq
S\=\# \bgl\{ \overline m \in\mathbb Z^{s}: \, \, \left|r\4t\right|\norm
{\mathbb V\overline m}< 1 \bgr\}\ge\# K_0. \label{fdfd}\eeq Let
$P_1\le\cdots\le P_{s}$ be the successive minima of the Euclidean
norm
with respect to the lattice~$\left|r\4t\right|\kern1pt\mathbb V\kern1pt\mathbb Z^{s}$.
Let $e_1,e_2,\ldots, e_{s}$ be the standard orthonormal basis of
$\mathbb Z^{s}$. By~\eqref{ett8}, we have
$\norm{\left|r\4t\right|\kern1pt\mathbb V\4e_j}\ll_s 1$, $j=1,2, \ldots,
s$. Therefore, using Lemma \ref{Dav9}, we see that
$P_1\le\cdots\le P_{s}\ll_s 1$. By \eqref{ett7}, \eqref{fdfd}
and Lemmas \ref{Dav2} and \ref{LLL}, \beq \label{yuyu}
S\asymp_s (P_1\4\cdot\4 P_2
\cdots P_{s})^{-1}\asymp_s(\det (\left|r\4t\right|\kern1pt\mathbb
V))^{-1}}\def\sign{\hbox{\rm sign}\asymp_s\left|r\4t\right|^{-s}\,(\det \mathbb C)^{-1}. \eeq
Hence, using \eqref{ad888}, \eqref{fdfd} and \eqref{yuyu},
we conclude that\beq \label{ghgh}
R_t
\ll_s\left|r\4t\right|^{-s}\,(\det \mathbb C)^{-1}, \quad \text{for
}\; r\ge1\quad \text {and }\left|r\4t\right|\le c_s^*\,\sigma_1^{-2}.
\eeq
Now, by \eqref{koren},
\eqref{ken4} and
\eqref{ghgh}, we have \begin{eqnarray}\label{koren44}
{\mathbf E\,}\operatorname {e}\bgl \{
\langle\mathbb B_t \kern1pt \zeta_n,\zeta'_n \rangle\bgr\} &\ll_s&
r^{-s}\,R_t \nonumber\\ &\ll_s& r^{-2s}\,\left|t\right|^{-s}\,(\det
\mathbb C)^{-1},\quad \text{for } r\ge1 \; \text {and
}\left|r\4t\right|\le c_s^*\,\sigma_1^{-2}. \end{eqnarray} It is easy to
verify that \beq\int_{c_s\kern1pt\sigma_1^{-2}r^{-2+4/s}
}^{\sigma_1^{-2} r^{-1}}\sqrt{r^{-2s}\,\left|t\right|^{-s}\,(\det
\mathbb C)^{-1}}\ffrac{dt}t \ll_s r^{-2}\,\sigma_1^{s}\,(\det
\mathbb C)^{-1/2}, \label{eq71av}\eeq for any $c_s$ depending on
$s$ only. Note that $\,\sigma_1^{s}\,(\det \mathbb C)^{-
1/2}\ge1$. Using \eqref{koren4}, \eqref{eq71av} and Lemmas
\ref{L7.3} and~\ref{L7.5},
we derive the following lemma.
\begin{lemma}\label{GZ} Let the conditions of Lemma\/ {\rm\ref{L7.3}}
be satisfied with $s=d$, $\mathbb D=\mathbb C^{-1/2}$, $\delta\leq
1/(5\kern1pt s)$ and with an orthonormal system $\mathcal S=\mathcal
S_o=\{\fs e1s\}\subset \mathbb R^d$. Let $c_s$ be an arbitrary quantity
depending on $s$ only. Then, for any $b\in\mathbb R^d$ and\/ $r\ge1$,
\beq\int_{c_s\kern1pt\sigma_1^{-2}r^{-2+4/s} }^{\sigma_1^{-2}}\bgl| \widehat
\Psi_b (t/2)\bgr| \ffrac {dt} t
\ll_s \, (p\kern1pt N)^{-1}\,\sigma_1^{s}\,(\det \mathbb C)^{-1/2}+r^{-
s/2}\,\sup_\Gamma \; \int_{r^{-1} }^1(\alpha (\mathbb D_r\kern1pt\mathbb
U_u\,\Lambda))^{1/2}\ffrac {du} u, \label{eq71a}\eeq
where\/ $r$, $\alpha (\4\cdot\4)$, $\mathbb D_r$
$\mathbb U_t$ and the lattice $\Lambda$ are defined in relations
\eqref{dfn}, {\rm\eqref{defr}}, {\rm \eqref{bbbl}}, {\rm
\eqref{alp}}, {\rm \eqref{alp3}}, {\rm \eqref{svo4}}, {\rm
\eqref{svo5}} and~{\rm \eqref{latt}} and in Lemma\/ {\rm
\ref{L7.5}}. The $\sup_\Gamma$ means here the supremum over all
possible values of $z_j,z_j'\in \mathbb R^d $ $($involved in the
definition of matrices $\mathbb B_t$ and\/~$\mathbb V)$ such that
\beq\|\mathbb C^{-1/2}z_j-e_j\|\leq \delta,\quad \q\quad \|\mathbb
C^{-1/2}z_j'-e_j\|\leq \delta,\quad \q\quad \text{for}\ \, 1\leq j\leq
s.\label{eq:7.6f} \eeq Moreover, for any $b\in\mathbb R^d$, $r\ge1$ and\/
$\gamma >0$ and any fixed\/ $t\in\mathbb R$ satisfying\/
$\left|r\4t\right|\le c_s^*\,\sigma_1^{-2}$, where $c_s^*\ge1$ is
an arbitrary quantity depending on $s$ only, we have \beq\bgl| \widehat
\Psi_b (t)\bgr|
\ll_{\gamma , s} (p\kern1pt N)^{-\gamma }+
r^{-s}\,\left|t\right|^{-s/2}\,(\det \mathbb C)^{-1/2} .\label{equ7.1p}\eeq
\end{lemma}
Let $v=(m,\overline m)\in\mathbb R^{2s}$, $m,\overline m\in\mathbb R^{s}$ and
$t\in\mathbb R$. Then \beq \label{rho}\overline m+t\kern1pt m=(1 +t^2)\,\overline
m+t\, (m-t\kern1pt \overline m). \eeq Equality \eqref{rho} implies that \beq
\label{rho1} \norm{\overline m+t\kern1pt m}\ll_s\norm{\overline m} +\norm{m-t\kern1pt \overline
m}, \quad \hbox{for} \ |t|\ll_s 1. \eeq Hence, \beq\label{rho3}
r\,\norm{m-t\kern1pt \overline m}+ r^{-1}}\def\sign{\hbox{\rm sign}\kern1pt \norm{\overline m+t\kern1pt m}\ll_s
r\,\norm{m-t\kern1pt \overline m}+ r^{-1}}\def\sign{\hbox{\rm sign}\kern1pt\norm{\overline m} , \quad \hbox{for} \ r\gg
1,\ |t|\ll_s 1. \eeq According to \eqref{svo4}--\eqref{svo5}, we
have\beq \label{rhom} \mathbb D_r\mathbb U_t\4v=(r(m-t\kern1pt \overline
m),\,r^{-1}}\def\sign{\hbox{\rm sign}\kern1pt\overline m)\quad \text{and}\quad \mathbb D_r\mathbb K_t\4v=(r(m-t\kern1pt
\overline m),\,r^{-1}}\def\sign{\hbox{\rm sign}\kern1pt(\overline m+t\4m )). \eeq It is clear that the operators
$\mathbb D_r\mathbb U_t$ and $\mathbb D_r\mathbb K_t$ are
invertible. Therefore, using \eqref{rho3} and~\eqref{rhom} and
applying Lemmas \ref{Dav9} and~\ref{Dav4}, we derive the
inequality \beq\label{kt6} \alpha (\mathbb D_{r}\mathbb U_t\kern1pt
\Omega) \ll_s\alpha (\mathbb D_{r}\mathbb K_t\kern1pt \Omega) , \quad
\hbox{for} \ r\gg 1,\ |t|\ll_s 1,\eeq which is valid for any
lattice $\Omega\subset\mathbb R^{2s}$.
Let $\mathbb T$ be the permutation $(2\4s\times2\4s)$-matrix which
permutes the rows of a ${(2\4s\times2\4s)}$-matrix $\mathbb A$ so
that the new order (corresponding to the matrix $\mathbb T\mathbb
A$) is:
$$1,s+1,2,s+2,\ldots, s, 2\4s.$$ Note that the operator
$\mathbb T$ is isometric and $\mathbb A\,\, {\raise.4pt\hbox{$\shortmid$}}{\hskip-2.0pt\to}\, \, \mathbb A\,\mathbb T^{-1}}\def\sign{\hbox{\rm sign}$
rearrange the columns of $\mathbb A$ in the order mentioned above.
It is easy to see that \beq\label{kt7} \alpha_j(\mathbb T\kern1pt
\Omega) =\alpha_j(\Omega),\quad j=1,\ldots 2\4s,\quad \text {and}
\quad \alpha(\mathbb T\kern1pt \Omega) =\alpha(\Omega),\eeq for any lattice
$\Omega\subset\mathbb R^{2s}$.
Note now that \beq\label{kt8}\mathbb T\mathbb D_{r}\mathbb
K_t\kern1pt\Lambda_j =\mathbb T\mathbb D_{r}\mathbb K_t\mathbb
T^{-1}}\def\sign{\hbox{\rm sign}\mathbb T\kern1pt\Lambda_j =\mathbb W_t\Delta_j,\eeq where
$\Delta_j$ is a lattice defined by\beq\label{kt9}\Delta_j=\mathbb
T\kern1pt\Lambda_j\eeq and where $\mathbb W_t$ is $(2s\times 2s)$-matrix
\beq \label{latt34} \mathbb W_t=
\left(\begin{array}{*{4}c}\mathbb G_{r,t}&\mathbb O_2&: &\mathbb O_2\\
\mathbb O_2&\mathbb G_{r,t}&: &\mathbb O_2\\ \4\cdot\4\cdt
&\4\cdot\4\cdt&\4\cdot\4\cdt&\4\cdot\4\cdt\\
\mathbb O_2& \mathbb O_2 &:& \mathbb G_{r,t} \end{array}\right)
\eeq constructed of $(2\times2)$-matrices $\mathbb O_2$ (with zero
entries) and \beq \mathbb G_{r,t} \=
\left(\begin{array}{*{2}c} r & -r\4t\\
r^{-1}}\def\sign{\hbox{\rm sign} t & r^{-1}}\def\sign{\hbox{\rm sign}\end{array}\right). \eeq
Let $|t|\le 2$ and \beq\label{kth} \theta=\arcsin\bgl(t\,(1
+t^2)^{-1/2}\bgr)\quad \text{or, equivalently,} \quad t=\tan\theta. \eeq
Then we have \beq\label{latw} |\theta|\le
c^*\=\arcsin(2/\sqrt5),\quad \cos \theta=(1 +t^2)^{-1/2},\quad
\sin\theta=t\,(1 +t^2)^{-1/2}. \eeq It is easy to see that \begin{eqnarray}
\label{latt1}\mathbb G_{r,t} = (1 +t^2)^{1/2}\,\overline{\mathbb
D}_{r}\,\overline{\mathbb K}_\theta \end{eqnarray} and \beq\label{oot}\mathbb
W_t=(1 +t^2)^{1/2}\, \widetilde{\mathbb D}_r\,\widetilde{\mathbb K}_\theta,\eeq
where \beq \label{latt345} \widetilde{\mathbb D}_r=
\left(\begin{array}{*{4}c}\overline{\mathbb D}_{r}&\mathbb O_2&: &\mathbb O_2\\
\mathbb O_2&\overline{\mathbb D}_{r}&: &\mathbb O_2\\ \4\cdot\4\cdt
&\4\cdot\4\cdt&\4\cdot\4\cdt&\4\cdot\4\cdt\\
\mathbb O_2& \mathbb O_2 &:&\overline{\mathbb D}_{r} \end{array}\right)\quad \text
{and}\quad
\widetilde{\mathbb K}_\theta=
\left(\begin{array}{*{4}c}\overline{\mathbb K}_\theta&\mathbb O_2&: &\mathbb O_2\\
\mathbb O_2&\overline{\mathbb K}_\theta&: &\mathbb O_2\\ \4\cdot\4\cdt
&\4\cdot\4\cdt&\4\cdot\4\cdt&\4\cdot\4\cdt\\
\mathbb O_2& \mathbb O_2 &:&\overline{\mathbb K}_\theta \end{array}\right)
\eeq are $(2\4s\times2\4s)$-matrices with \beq
\label{laz}\overline{\mathbb D}_{r} \=
\left(\begin{array}{*{2}c} r & 0\\
0 & r^{-1}}\def\sign{\hbox{\rm sign}\end{array}\right)\quad \text {and}\quad \overline{\mathbb K}_\theta
\=\left(\begin{array}{*{2}c} \cos \theta &
-\sin \theta \\
\sin \theta &\phantom {-}
\cos \theta\end{array}\right)\in \hbox{\rm SL(2,$\mathbb R$)} .
\eeq Substituting \eqref{oot} into equality \eqref{kt8}, we obtain
\beq\label{kot8}\mathbb T\mathbb D_{r}\mathbb K_t\kern1pt\Lambda_j =(1
+t^2)^{1/2}\,\widetilde{\mathbb D}_r\,\widetilde{\mathbb
K}_\theta\,\Delta_j.\eeq
Below we shall also use the following crucial lemma of G\"otze and
Margulis (2010).
\begin{lemma}\label{GM} Let $\widetilde{\mathbb K}_\theta$ and
\beq \label{laty} \widetilde{\mathbb H}=
\left(\begin{array}{*{4}c}\overline{\mathbb H}&\mathbb O_2&: &\mathbb O_2\\
\mathbb O_2&\overline{\mathbb H}&: &\mathbb O_2\\ \4\cdot\4\cdt
&\4\cdot\4\cdt&\4\cdot\4\cdt&\4\cdot\4\cdt\\
\mathbb O_2& \mathbb O_2 &:&\overline{\mathbb H} \end{array}\right)
\eeq be $(2\4d\times2\4d)$-matrices such that\/ $\overline{\mathbb H}\in
\mathcal G=\hbox{\rm SL(2,$\mathbb R$)}$ and\/ $\widetilde{\mathbb
K}_\theta$ is defined in {\rm\eqref{latt345}} and
{\rm\eqref{laz}}. Let $\beta$ be a positive number such that
$\beta\4d>2$. Then, for any\/ $\overline{\mathbb H}\in \mathcal G$ and
any lattice $\Delta\subset\mathbb R^{2d}$, \beq
\int_{0}^{2\pi}\bgl(\alpha(\widetilde{\mathbb H} \,\widetilde{\mathbb
K}_\theta\,\Delta)\bgr)^\beta d\theta\ll_{\beta,d}
\bgl(\alpha(\Delta)\bgr)^\beta\,\norm {\overline{\mathbb H}}^{\beta
d-2}. \eeq Here $\norm {\overline{\mathbb H}}$ is the standard norm of
the linear operator $\mathbb H:\mathbb R^2\to\mathbb R^2$.\end{lemma}
Consider, under the conditions of Lemma \ref{GZ},
\beq\label{IJ56}
I_0 \= \int_{c_s\kern1pt\sigma_1^{-2}r^{-2+4/s}/2
}^{\sigma_1^{-2}/2}\bgl| \widehat \Psi_b (t)\bgr| \ffrac {dt} t =
\int_{c_s\kern1pt\sigma_1^{-2}r^{-2+4/s} }^{\sigma_1^{-2}}\bgl| \widehat
\Psi_b (t/2)\bgr| \ffrac {dt} t.\eeq
By Lemma \ref{GZ}, we have
\beq\label{IJ5}
I_0 \ll_s (p\kern1pt N)^{-1}\,\sigma_1^{s}\,(\det \mathbb C)^{-1/2}+
r^{- s/2}\,\sup_\Gamma \; J,\eeq where \beq\label{IJ6} J=
\int_{r^{-1}}^1\big(\alpha (\mathbb D_r\kern1pt\mathbb
U_t\,\Lambda)\big)^{1/2} \ffrac {dt} t\le\sum_{j=2}^{\rho}I_{j},
\eeq with \beq\label{IJ7} I_j\= \int_{j^{-1}}\def\sign{\hbox{\rm sign}}^{(j-1)^{-1}}\def\sign{\hbox{\rm sign}} \big(\alpha
(\mathbb D_r\kern1pt\mathbb U_t\,\Lambda)\big)^{1/2}\ffrac {dt} t, \quad
j=2,3,\ldots,\rho\=\lceil r\rceil+1. \eeq
Changing variable $t=v\4j^{-2}$ and $v=w+j$ in $I_j$ and using the
properties of matrices $\mathbb D_r$ and $\mathbb U_t$, we have
\begin{eqnarray} I_j&=&\int_{j}^{j^{2}(j-1)^{-1}}\def\sign{\hbox{\rm sign}} \big(\alpha (\mathbb
D_r\kern1pt\mathbb U_{vj^{-2}}\,\Lambda)\big)^{1/2}\ffrac {dv}v \nonumber\\ &
\le& \int_{j}^{j+2} \big(\alpha (\mathbb D_r\kern1pt\mathbb
U_{vj^{-2}}\,\Lambda)\big)^{1/2} \ffrac {dv} v \nonumber
\\ & =& \int_{0}^{2}
\big(\alpha (\mathbb D_r\kern1pt\mathbb U_{wj^{-2}}\kern1pt\mathbb
U_{j^{-1}}\,\Lambda)\big)^{1/2} \ffrac{dw} {w+j}.\label{IJ} \end{eqnarray}
By \eqref{svo}, \beq \label{IJ2} \mathbb D_r\kern1pt\mathbb U_{wj^{-2}}
=\mathbb D_{rj^{-1}}\kern1pt\mathbb D_{j}\kern1pt\mathbb U_{wj^{-2}} = \mathbb
D_{rj^{-1}} \kern1pt\mathbb U_{w} \kern1pt\mathbb D_{j}. \eeq
According to \eqref{IJ} and \eqref{IJ2}, \beq\label{IJ22} I_j\ll
\ffrac {1} {j} \int_{0}^{2}
\big(\alpha (\mathbb D_{rj^{-1}}\mathbb U_t\kern1pt
\Lambda_{j})\big)^{1/2} \,{dt} , \eeq where the lattices
$\Lambda_j$ are defined in \eqref{jj} (see also \eqref{svo4},
\eqref{svo5} and \eqref{latt}). Using \eqref{jj}, \eqref{latt36}
and \eqref{adas89}, we see that
\beq\label{kth14}
\alpha(\Lambda_{j}) \ll_s \sigma_1^{2s}\,(\det \mathbb C)^{-1}. \eeq
By \eqref{kt6}, \eqref{kt7} and \eqref{kot8}, we have
\begin{eqnarray}\label{qkt} \alpha (\mathbb D_{rj^{-1}}\mathbb U_t\kern1pt
\Lambda_{j}) &\ll_s&\alpha (\mathbb D_{rj^{-1}}\kern1pt\mathbb K_t\kern1pt
\Lambda_{j})
=\alpha (\mathbb T\kern1pt\mathbb D_{rj^{-1}}\kern1pt\mathbb K_t\kern1pt \Lambda_{j})\nonumber\\
&\ll_s&\alpha (\widetilde{\mathbb D}_{rj^{-1}}\,\widetilde{\mathbb
K}_\theta\,\Delta_j) , \end{eqnarray} for $|t|\ll_s1$, $r\ge1$,
$j=2,3,\ldots,\rho$, where $\Delta_j$ and $\theta$ are defined in
\eqref{kt9} and \eqref{kth} respectively. Using \eqref{kth},
\eqref{latw}, \eqref{latt345}, \eqref{qkt} and Lemma \ref{GM}
(with $d=s$), we obtain \begin{eqnarray}\label{kth23}\int_{0}^{2} \big(\alpha
(\mathbb D_{rj^{-1}}\mathbb U_t\kern1pt \Lambda_{j})\big)^{1/2}
\,{dt}&\ll_s&\int_{0}^{c^*} \big(\alpha (\widetilde{\mathbb
D}_{rj^{-1}}\, \widetilde{\mathbb K}_\theta\,\Delta_j)\big)^{1/2}
\,\ffrac{d\theta}{\cos^2\theta} \nonumber\\&\ll&\int_{0}^{2\pi}
\big(\alpha (\widetilde{\mathbb D}_{rj^{-1}}\, \widetilde{\mathbb
K}_\theta\,\Delta_j)\big)^{1/2} \,{d\theta} \nonumber\\&\ll_s&
\norm{\kern1pt\overline{\mathbb D}_{rj^{-1}}}^{s/2-2}
\,\big(\alpha(\Delta_j)\big)^{1/2}, \end{eqnarray} if $s\ge5$. It is clear
that $\norm{\kern1pt\overline{\mathbb D}_{rj^{-1}}}= r\4j^{-1}$. Therefore,
according to \eqref{kt7}, \eqref{kt9}, \eqref{IJ22} and
\eqref{kth23}, \beq \label{kth13}I_j \ll_s
\ffrac 1 {j} (rj^{-1})^{s/2-2}
\, \big(\alpha(\Lambda_{j})\big)^{1/2}.
\eeq
By \eqref{IJ6}, \eqref{kth14} and \eqref{kth13}, we obtain,
for $s\ge5$, \beq\label{kth15}
J \ll_s\sigma_1^{s}\,(\det \mathbb C)^{-1/2}\sum_{j=2}^{\rho}\ffrac 1 {j}
(rj^{-1})^{s/2-2}
\ll_s r^{s/2-2}\,\sigma_1^{s}\,(\det \mathbb C)^{-1/2}.\eeq
By \eqref{dfn}, \eqref{defr}, \eqref{IJ5} and \eqref{kth15}, we
have $r\asymp_s (N\4p)^{1/2}$ and
\begin{eqnarray}\label{kth16}
I_0 \ll_s r^{-2}\,\sigma_1^{s}\,(\det \mathbb C)^{-1/2}
\ll_s (N\4p)^{-1}\,\sigma_1^{s}\,(\det \mathbb C)^{-1/2}.
\end{eqnarray} It is clear that in a similar way we can establish that
\begin{eqnarray}\label{k16}
\int_{\sigma_1^{-2} }^{c(s)\sigma_1^{-2}}\bgl| \widehat \Psi_b (t/2)\bgr|
\ffrac {dt} t\ll_s r^{-2}\,\sigma_1^{s}\,(\det \mathbb C)^{-1/2}\ll_s (N\4p)^{-
1}\,\sigma_1^{s}\,(\det \mathbb C)^{-1/2},
\end{eqnarray} for any quantity $c(s)$ depending on $s$ only. The proof
will be easier due to the fact that $t$ cannot be small in this
integral.
Thus, we have proved the following lemma.
\begin{lemma}\label{GZ2} Let the conditions of Lemma\/ {\rm\ref{L7.3}}
be satisfied with $s=d\ge5$, $\mathbb D=\mathbb C^{-1/2}$, $\delta\leq
1/(5\kern1pt s)$ and with an orthonormal system $\mathcal S=\mathcal
S_o=\{\fs e1s\}\subset \mathbb R^d$. Let $c_1(s)$ and $c_2(s)$ be some
quantities depending on $s$ only. Then there exists a $c_s$ such
that \beq\int_{c_1(s)\kern1pt\sigma_1^{- 2}r^{-2+4/s}
}^{c_2(s)\kern1pt\sigma_1^{-2}} \bgl| \widehat \Psi_b (t)\bgr| \ffrac {dt} t
\ll_s \, (N\4p)^{-1}\,\sigma_1^{s}\,(\det \mathbb C)^{-1/2},
\label{eq71u}\eeq
if \,$N\4p\gg_s c_s$, where\/ $r$ is defined in \eqref{dfn} and\/
\eqref{defr}.
\end{lemma}
|
2,877,628,090,141 | arxiv | \section*{Acknowledgments}
This work was supported in part by Berkeley Deep Drive (BDD) and ONR PECASE N000141612723.
\clearpage
\section{Background}
\vspace{-0.1 in}
This work tackles the problem of learning a latent space that is suitable for planning from high-dimensional observations in POMDPs. We maximize the mutual information (MI) of the latent variables and the future observations, and learn a predictive model of the environment.
\textbf{Partially Observable Markov Decision Process. }
A discrete-time finite partially observable Markov decision process (POMDP) $\mathcal{M}$ is defined by the tuple $(\mathcal{S}, \mathcal{A}, T, R, \mathcal{O}, O, \gamma, \rho_0, H)$.
Here, $\mathcal{S}$ is the set of states, $\mathcal{A}$ the action space, $T(s_{t+1}| s_t, a_t) = p(s_{t+1}|s_t, a_t)$ the transition distribution, $R(s_t) = p(r_t|s_t)$ is the probability of obtaining the reward $r_t$ at the state $s_t$, $\mathcal{O}$ is the observation space, $O(o_t|s_t) = p(o_t|s_t)$, $\gamma$ the discount factor, $\rho_0: \mathcal{S} \to \mathbb{R}_+$ represents the initial state distribution, and $H$ is the horizon of the process. We define the return as the sum of rewards $r_t$ along a trajectory $\tau := (s_{0}, a_{0}, ..., s_{H-1}, a_{H-1}, s_{H})$. The goal of reinforcement learning is to find a controller $\pi: \mathcal{S} \times \mathcal{A} \rightarrow \mathbb{R}^+$ that maximizes the expected return, i.e.:
$\max_{\pi} J(\pi) = \mathbb{E}_{\begin{subarray}{l}{
a_t \sim \pi}\\
s_{t} \sim p
\end{subarray}}
[\sum_{t=1}^{H}\gamma^{t} r_t]$.
\textbf{Mutual Information. }
The mutual information between two random variable $X$ and $Y$, denoted by $I(X;Y)$ is a reparametrization-invariant measure of dependency. Specifically, it characterizes the Kullback-Leibler divergence between the joint distribution $(X, Y)$ and the product of the marginals $X$ and $Y$:
$I(X; Y) = \mathbb{E}_{p(x, y)} \left[ \log \frac{p(x|y)}{p(x)} \right] = \mathbb{E}_{p(x, y)} \left[ \log \frac{p(x, y)}{p(x) p(y)} \right] = D_{\text{KL}}((X,Y)\| X \otimes Y) $.
Estimating and optimizing the mutual information objective poses a challenging problem. In this work, we use a multi-sample
unnormalized lower bound based on noise contrastive estimation, $I_{\text{NCE}}$~\cite{Henaff2019}. This estimator is a lower bound on the mutual information.
\section{Conclusion}
In this paper, we present MIRO, a mutual information based representation learning approach for model-based RL. Compared to state-of-the-art reconstruction-based methods, MIRO is more robust in noisy visual background in the environment, which is prevalent in real world applications. The initial results shed light on the importance of information theoretical representation learning and open an enticing research direction. Our next steps include optimizing the sampling of negative samples in NCE objective to encourage the latent space to disregard task-irrelevant information.
\section{Introduction}
\vspace{-0.1 in}
A fundamental challenge in applying reinforcement learning (RL) to real robotics is the need to define a suitable state space representation. Designing perception systems manually makes it difficult to apply the same RL algorithm to a wide range of manipulation tasks in complex, unstructured environments. One could, in principle, apply model-free reinforcement algorithms from raw, low-level observations; however, these tend to be slow, sensitive to hyperparameters, and sample inefficient~\cite{Lee2019}. Model-based reinforcement learning, where a predictive model of the environment is first learned and then a controller is derived from it, offers the potential to develop sample efficient algorithms even from raw, low-level observations. However, Model-based RL from high-dimensional observations poses a challenging problem: How do we learn a good latent space for planning?
Recent work in learning latent spaces for model-based RL can be categorized in three main classes: 1) hand-crafted latent spaces ~\cite{Finn2016}, 2) video prediction models~\cite{Jayaraman2018}, and 3) latent space learning with reconstruction objectives~\cite{hafner2018learning, Lee2019, Watter2015, Wahlstrom2015, Zhang2018, Ha2018}. While hand-crafted latent spaces provide a good inductive bias, such as forcing the latent to be feature points~\cite{Finn2016}, they are too rigid for unstructured tasks and cannot incorporate semantic knowledge of the environment. Video prediction models and latent space with reconstruction objectives share the commonality that the latent space is learned using loss on the raw pixel observations. As a result, the latent space model needs to incorporate all information to reconstruct every detail in the observations, which is redundant as the task is usually represented by a small fraction of the scene in real world environments.
Instead, this work tackles the problem of representation learning from an information theoretic point of view: learning a latent space that maximizes the mutual information (MI) between the latent and future observations. The learned latent space encodes the underlying shared information between the different future observations, discarding the low-level information, and local noise. When predicting further into the future, the amount of information becomes much lower and the models needs to infer the global structure. The MI is optimized using energy-based models and a discriminative procedure~\cite{Oord2018, Henaff2019}. Energy-based models do not pose any assumption on the distribution between the latent and the image, and the discriminative procedure results in a robust latent space.
The main contribution of our work is a representational learning approach, MIRO (\textbf{M}utual \textbf{I}nformation \textbf{Ro}bust Representation), which jointly optimizes for the latent representation and model, and results in a latent space that is robust against disturbances, noisy observations, and can achieve performance comparable to state-of-the-art model-based algorithms. Our experimental evaluation illustrates the strengths of our framework on four standard DeepMind Control Suite~\cite{tassa2018deepmind} environments.
\section{Related Work}
\section{Results}
\vspace{-0.1 in}
In this section, we empirically corroborate the claims in the previous sections. Specifically, the experiments are designed to address the following questions:
1) Is our approach able to maintain its performance in front of distractors in the scene?
2) How does our method compare with state-of-the-art reconstruction objectives?
To answer the posed questions, we evaluate our framework, in four continuous control benchmark tasks MuJoCo simulator: cartpole balance, reacher, finger spin and half cheetah ~\cite{2012mujoco, tassa2018deepmind}. We choose PlaNet \cite{hafner2018learning} as the state-of-the-art reconstruction objective baseline.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/example2.pdf}
\caption{ \small An example trajectory of Cheetah environment with distractors. The distractor objects (red sphere and green cube) are placed at a random position at each time step.}
\label{fig:noisy_cheetah}
\vspace{-0.2 in}
\end{figure}
\textbf{Robustness Against Distractors. } To test the robustness of MIRO and PlaNet in visually noisy environments, we add distractors to each of the four above environments in the background as shown in Fig. \ref{fig:noisy_cheetah}. As shown in Fig. \ref{fig:learning_curves}, we observe that in all four environments, the performance of MIRO is not undermined by the presence of distractors. Interestingly, in the Cheetah environment, the performance even improves with visual noise in the background. A possible explanation is that the presence of the distractors forces the embedding to focus more on information relevant to the task and thus makes the embedding more suitable for planning. In comparison, the performance of PlaNet struggles in face of distractors. The agent barely learns in the Cartpole Balance environment, and in the Cheetah environment it suffers from a slower takeoff.
\begin{figure}[h]
\centering
\begin{subfigure}[H]{0.4\linewidth}\centering
\includegraphics[width=\linewidth]{figures/cpb2.png}
\caption{Cartpole Balance}
\end{subfigure}
\begin{subfigure}[H]{0.4\linewidth}\centering
\includegraphics[width=\linewidth]{figures/reacher2.png}
\caption{Reacher}
\end{subfigure}
\\
\begin{subfigure}[H]{0.4\linewidth}\centering
\includegraphics[width=\linewidth]{figures/finger2.png}
\caption{Finger}
\end{subfigure}
\begin{subfigure}[H]{0.4\linewidth}\centering
\includegraphics[width=\linewidth]{figures/cheetah2.png}
\caption{Half Cheetah}
\end{subfigure}
\caption{\small Learning curves of MIRO and PlaNet on environments with and without distractors. All curves represent mean and the shaded area represents one standard deviation among 3 seeds.}
\label{fig:learning_curves}
\vspace{-0.1 in}
\end{figure}
\textbf{Comparison Against Reconstruction Objectices} When no distractors ae present int the secene, we see that MIRO is able to achieve the same or superior asymptotic performance than its reconstruction-based counterpart method. Its learning speed its environment dependent. However, given the the performance boost brought by distractors, further regularization of the learned latent space presents it self as a promising direction towards performance improvement.
\section{Method}
\vspace{-0.1 in}
Enabling complex real robotics tasks requires extending current model-based methods to low-level high-dimensional observations. However, in order to do so, we need to specify which space we should plan with. Real-world environments are unstructured, cluttered, and present distractors. Our approach, MIRO, is able to learn latent representations that capture the relevant aspects of the dynamics and the task by framing the representational learning problem in information theoretic terms: maximizing the mutual information between the latent space and the future observations. This objective advocates for representing just the aspects that matter in the dynamics, removing the burden of reconstructing the entire pixel observation.
\subsection{Learning Latent Spaces for Control}
Since the learned latent space will be used for planning, it has to be able to accurately predict the rewards and incorporate new observations when available. To this end, we decompose the latent space learning in a single optimization procedure composed of four functions:
\textbf{Encoder. } The encoder is a function that maps high-dimensional observation to a lower dimensional manifold, $e_{\phi}: \mathcal{O} \to \mathcal{Z} \subseteq \mathbb{R}^m$. We represent the encoded observation $e_{\phi}(o_t)$ with the notation $z_t \in \mathcal{Z}$. Contrary to prior work in learning latent spaces for planning, we do not make any assumption on the underlying distribution of $z_t$.
\textbf{Dynamics model. } The dynamics model gives the transition function on the latent space, $f_{\theta}: \mathcal{S} \times \mathcal{A} \to \mathcal{S}$; it generates the next latent state $\hat{s}_{t+1}$ given the the current state $s_t$ and action $a_t$. The dynamics model is probabilistic following a Gaussian distribution, i.e., $s_{t+1} \sim \mathcal{N}(\mu_t, \sigma^2_t)$.
\textbf{Filtering function. } This function $g_{\zeta}: \mathcal{Z} \times \mathcal{S} \to \mathcal{S}$ filters the belief of the latent state $\hat{s}_t$ with the current encoded observation $z_t$ to obtain the filtered latent variable $s_t$.
\textbf{Reward predictor. } This function is a mapping between latent states and rewards, $r_{\psi}: \mathcal{S} \to \mathcal{A}$. We assume that the rewards follow a Gaussian distribution with unit variance.
The parameters of these functions are learned in a single optimization objective; specifcally, we optimize the following constrain optimization problem:
\begin{align*}
\max_{\bf{\theta, \phi, \psi, \zeta}} \hspace{0.15 in} I(s_t; o_{t+h} | a_t, \dots, a_{t+h-1}) \hspace{1.2 in} \\
\text{s.t.:} \quad \quad D_{\text{KL}}(g_{\zeta}(z_{t+1}, \hat{s}_{t+1}) \| s_{t+1}) = 0 \quad \quad z_t = e_{\phi}(o_t) \\
D_{\text{KL}}(r_t \| r_{\psi}(s_t, a_t)) = 0 \hspace{.35 in} \hat{s}_{t+1} \sim f_{\theta}(s_t, a_t) \\
\vspace{-0.2 in}
\end{align*}
The mutual information term preserves in latent space only the information relevant for the dynamics, while the constraints prevents the latent from degenerating on the terms that matter to predict the rewards and next latents. However, in practice, this objective is intractable: we cannot evaluate the exact mutual information term and we have a set of non-linear constraints. To optimize it, we formulate the Lagrangian of the objective and replace the mutual information objective with the noise contrastive estimator lower bound $I_{\text{NCE}}$:
\begin{align}
\max_{\bf{\theta, \phi, \psi, \zeta}} I_{\text{NCE}}(s_t; o_{t+h} | a_t, \dots, a_{t+h-1}) - \lambda_1 D_{\text{KL}}(g_{\zeta}(z_t, \hat{s}_{t+1}) \| \hat{s}_{t+1}) - \lambda_2 D_{\text{KL}}(r_t \| r_{\psi}(s_t, a_t))
\label{eq:loss}
\end{align}
We use the reparametrization trick~\cite{kingma2013auto} for $\hat{s_{t+1}} = \hat{\mu}_{t+1} + \xi \hat{\omega}_{t+1}$ where $\xi \sim \mathcal{N}(0, 1)$ and $z_t = e_{\phi}(o_t)$. In our case, the noise contrastive estimator term results in:
\begin{align}
I_{\text{NCE}}(s_t; o_{t+h} | a_t, \dots, a_{t+h-1}) = \mathbb{E}\left[\log \left(\frac{\text{exp}(\hat{s}_{t+h}^\top W_h z_{t+h})}{
\text{exp}(\hat{s}_{t+h}^\top W_h z_{t+h}) + \sum_{j=1}^{K-1} \text{exp}(\hat{s}_{t+h}^\top W_h z_{j})} \right) \right]
\end{align}
Here, $\hat{s}_{t+h}$ represents the open loop prediction from $s_t$ and the sequences of actions $a_{t}, \dots, a_{t+h}$.
\subsection{Latent Space Model-Based Reinforcement Learning}
We present the overall method for obtaining a model-based controller when learning the latent space using MIRO.
\textbf{Data Collection. } As is typical of model-based methods, we alternate between model learning and data collection using the latest controller. This allows the model to just learn the parts of the state space that the agent visits, removing the burden of modelling the entire space, and overcomes the insufficient coverage of the initial data distribution.
\textbf{Model Learning. } We learn the encoder, model, filtering function, and reward predictor altogether using equation~\ref{eq:loss} and all the data collected so far.
\textbf{Planning. } As a controller, we use model-predictive control (MPC) with cross-entropy method (CEM) \cite{botev2013cross}. The CEM component selects the action that maximizes the expected return under the learned models. Specifically, it is a population based procedure that iteratively refits a Gaussian distribution on the best sequences of actions. The MPC component results in a more robust planning by doing CEM at each step.
\section*{Acknowledgments}
This work was supported in part by Berkeley Deep Drive (BDD) and ONR PECASE N000141612723.
\clearpage
\section{Background}
\vspace{-0.1 in}
This work tackles the problem of learning a latent space that is suitable for planning from high-dimensional observations in POMDPs. We maximize the mutual information (MI) of the latent variables and the future observations, and learn a predictive model of the environment.
\textbf{Partially Observable Markov Decision Process. }
A discrete-time finite partially observable Markov decision process (POMDP) $\mathcal{M}$ is defined by the tuple $(\mathcal{S}, \mathcal{A}, T, R, \mathcal{O}, O, \gamma, \rho_0, H)$.
Here, $\mathcal{S}$ is the set of states, $\mathcal{A}$ the action space, $T(s_{t+1}| s_t, a_t) = p(s_{t+1}|s_t, a_t)$ the transition distribution, $R(s_t) = p(r_t|s_t)$ is the probability of obtaining the reward $r_t$ at the state $s_t$, $\mathcal{O}$ is the observation space, $O(o_t|s_t) = p(o_t|s_t)$, $\gamma$ the discount factor, $\rho_0: \mathcal{S} \to \mathbb{R}_+$ represents the initial state distribution, and $H$ is the horizon of the process. We define the return as the sum of rewards $r_t$ along a trajectory $\tau := (s_{0}, a_{0}, ..., s_{H-1}, a_{H-1}, s_{H})$. The goal of reinforcement learning is to find a controller $\pi: \mathcal{S} \times \mathcal{A} \rightarrow \mathbb{R}^+$ that maximizes the expected return, i.e.:
$\max_{\pi} J(\pi) = \mathbb{E}_{\begin{subarray}{l}{
a_t \sim \pi}\\
s_{t} \sim p
\end{subarray}}
[\sum_{t=1}^{H}\gamma^{t} r_t]$.
\textbf{Mutual Information. }
The mutual information between two random variable $X$ and $Y$, denoted by $I(X;Y)$ is a reparametrization-invariant measure of dependency. Specifically, it characterizes the Kullback-Leibler divergence between the joint distribution $(X, Y)$ and the product of the marginals $X$ and $Y$:
$I(X; Y) = \mathbb{E}_{p(x, y)} \left[ \log \frac{p(x|y)}{p(x)} \right] = \mathbb{E}_{p(x, y)} \left[ \log \frac{p(x, y)}{p(x) p(y)} \right] = D_{\text{KL}}((X,Y)\| X \otimes Y) $.
Estimating and optimizing the mutual information objective poses a challenging problem. In this work, we use a multi-sample
unnormalized lower bound based on noise contrastive estimation, $I_{\text{NCE}}$~\cite{Henaff2019}. This estimator is a lower bound on the mutual information.
\section{Conclusion}
In this paper, we present MIRO, a mutual information based representation learning approach for model-based RL. Compared to state-of-the-art reconstruction-based methods, MIRO is more robust in noisy visual background in the environment, which is prevalent in real world applications. The initial results shed light on the importance of information theoretical representation learning and open an enticing research direction. Our next steps include optimizing the sampling of negative samples in NCE objective to encourage the latent space to disregard task-irrelevant information.
\section{Introduction}
\vspace{-0.1 in}
A fundamental challenge in applying reinforcement learning (RL) to real robotics is the need to define a suitable state space representation. Designing perception systems manually makes it difficult to apply the same RL algorithm to a wide range of manipulation tasks in complex, unstructured environments. One could, in principle, apply model-free reinforcement algorithms from raw, low-level observations; however, these tend to be slow, sensitive to hyperparameters, and sample inefficient~\cite{Lee2019}. Model-based reinforcement learning, where a predictive model of the environment is first learned and then a controller is derived from it, offers the potential to develop sample efficient algorithms even from raw, low-level observations. However, Model-based RL from high-dimensional observations poses a challenging problem: How do we learn a good latent space for planning?
Recent work in learning latent spaces for model-based RL can be categorized in three main classes: 1) hand-crafted latent spaces ~\cite{Finn2016}, 2) video prediction models~\cite{Jayaraman2018}, and 3) latent space learning with reconstruction objectives~\cite{hafner2018learning, Lee2019, Watter2015, Wahlstrom2015, Zhang2018, Ha2018}. While hand-crafted latent spaces provide a good inductive bias, such as forcing the latent to be feature points~\cite{Finn2016}, they are too rigid for unstructured tasks and cannot incorporate semantic knowledge of the environment. Video prediction models and latent space with reconstruction objectives share the commonality that the latent space is learned using loss on the raw pixel observations. As a result, the latent space model needs to incorporate all information to reconstruct every detail in the observations, which is redundant as the task is usually represented by a small fraction of the scene in real world environments.
Instead, this work tackles the problem of representation learning from an information theoretic point of view: learning a latent space that maximizes the mutual information (MI) between the latent and future observations. The learned latent space encodes the underlying shared information between the different future observations, discarding the low-level information, and local noise. When predicting further into the future, the amount of information becomes much lower and the models needs to infer the global structure. The MI is optimized using energy-based models and a discriminative procedure~\cite{Oord2018, Henaff2019}. Energy-based models do not pose any assumption on the distribution between the latent and the image, and the discriminative procedure results in a robust latent space.
The main contribution of our work is a representational learning approach, MIRO (\textbf{M}utual \textbf{I}nformation \textbf{Ro}bust Representation), which jointly optimizes for the latent representation and model, and results in a latent space that is robust against disturbances, noisy observations, and can achieve performance comparable to state-of-the-art model-based algorithms. Our experimental evaluation illustrates the strengths of our framework on four standard DeepMind Control Suite~\cite{tassa2018deepmind} environments.
\section{Related Work}
\section{Results}
\vspace{-0.1 in}
In this section, we empirically corroborate the claims in the previous sections. Specifically, the experiments are designed to address the following questions:
1) Is our approach able to maintain its performance in front of distractors in the scene?
2) How does our method compare with state-of-the-art reconstruction objectives?
To answer the posed questions, we evaluate our framework, in four continuous control benchmark tasks MuJoCo simulator: cartpole balance, reacher, finger spin and half cheetah ~\cite{2012mujoco, tassa2018deepmind}. We choose PlaNet \cite{hafner2018learning} as the state-of-the-art reconstruction objective baseline.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/example2.pdf}
\caption{ \small An example trajectory of Cheetah environment with distractors. The distractor objects (red sphere and green cube) are placed at a random position at each time step.}
\label{fig:noisy_cheetah}
\vspace{-0.2 in}
\end{figure}
\textbf{Robustness Against Distractors. } To test the robustness of MIRO and PlaNet in visually noisy environments, we add distractors to each of the four above environments in the background as shown in Fig. \ref{fig:noisy_cheetah}. As shown in Fig. \ref{fig:learning_curves}, we observe that in all four environments, the performance of MIRO is not undermined by the presence of distractors. Interestingly, in the Cheetah environment, the performance even improves with visual noise in the background. A possible explanation is that the presence of the distractors forces the embedding to focus more on information relevant to the task and thus makes the embedding more suitable for planning. In comparison, the performance of PlaNet struggles in face of distractors. The agent barely learns in the Cartpole Balance environment, and in the Cheetah environment it suffers from a slower takeoff.
\begin{figure}[h]
\centering
\begin{subfigure}[H]{0.4\linewidth}\centering
\includegraphics[width=\linewidth]{figures/cpb2.png}
\caption{Cartpole Balance}
\end{subfigure}
\begin{subfigure}[H]{0.4\linewidth}\centering
\includegraphics[width=\linewidth]{figures/reacher2.png}
\caption{Reacher}
\end{subfigure}
\\
\begin{subfigure}[H]{0.4\linewidth}\centering
\includegraphics[width=\linewidth]{figures/finger2.png}
\caption{Finger}
\end{subfigure}
\begin{subfigure}[H]{0.4\linewidth}\centering
\includegraphics[width=\linewidth]{figures/cheetah2.png}
\caption{Half Cheetah}
\end{subfigure}
\caption{\small Learning curves of MIRO and PlaNet on environments with and without distractors. All curves represent mean and the shaded area represents one standard deviation among 3 seeds.}
\label{fig:learning_curves}
\vspace{-0.1 in}
\end{figure}
\textbf{Comparison Against Reconstruction Objectices} When no distractors ae present int the secene, we see that MIRO is able to achieve the same or superior asymptotic performance than its reconstruction-based counterpart method. Its learning speed its environment dependent. However, given the the performance boost brought by distractors, further regularization of the learned latent space presents it self as a promising direction towards performance improvement.
\section{Method}
\vspace{-0.1 in}
Enabling complex real robotics tasks requires extending current model-based methods to low-level high-dimensional observations. However, in order to do so, we need to specify which space we should plan with. Real-world environments are unstructured, cluttered, and present distractors. Our approach, MIRO, is able to learn latent representations that capture the relevant aspects of the dynamics and the task by framing the representational learning problem in information theoretic terms: maximizing the mutual information between the latent space and the future observations. This objective advocates for representing just the aspects that matter in the dynamics, removing the burden of reconstructing the entire pixel observation.
\subsection{Learning Latent Spaces for Control}
Since the learned latent space will be used for planning, it has to be able to accurately predict the rewards and incorporate new observations when available. To this end, we decompose the latent space learning in a single optimization procedure composed of four functions:
\textbf{Encoder. } The encoder is a function that maps high-dimensional observation to a lower dimensional manifold, $e_{\phi}: \mathcal{O} \to \mathcal{Z} \subseteq \mathbb{R}^m$. We represent the encoded observation $e_{\phi}(o_t)$ with the notation $z_t \in \mathcal{Z}$. Contrary to prior work in learning latent spaces for planning, we do not make any assumption on the underlying distribution of $z_t$.
\textbf{Dynamics model. } The dynamics model gives the transition function on the latent space, $f_{\theta}: \mathcal{S} \times \mathcal{A} \to \mathcal{S}$; it generates the next latent state $\hat{s}_{t+1}$ given the the current state $s_t$ and action $a_t$. The dynamics model is probabilistic following a Gaussian distribution, i.e., $s_{t+1} \sim \mathcal{N}(\mu_t, \sigma^2_t)$.
\textbf{Filtering function. } This function $g_{\zeta}: \mathcal{Z} \times \mathcal{S} \to \mathcal{S}$ filters the belief of the latent state $\hat{s}_t$ with the current encoded observation $z_t$ to obtain the filtered latent variable $s_t$.
\textbf{Reward predictor. } This function is a mapping between latent states and rewards, $r_{\psi}: \mathcal{S} \to \mathcal{A}$. We assume that the rewards follow a Gaussian distribution with unit variance.
The parameters of these functions are learned in a single optimization objective; specifcally, we optimize the following constrain optimization problem:
\begin{align*}
\max_{\bf{\theta, \phi, \psi, \zeta}} \hspace{0.15 in} I(s_t; o_{t+h} | a_t, \dots, a_{t+h-1}) \hspace{1.2 in} \\
\text{s.t.:} \quad \quad D_{\text{KL}}(g_{\zeta}(z_{t+1}, \hat{s}_{t+1}) \| s_{t+1}) = 0 \quad \quad z_t = e_{\phi}(o_t) \\
D_{\text{KL}}(r_t \| r_{\psi}(s_t, a_t)) = 0 \hspace{.35 in} \hat{s}_{t+1} \sim f_{\theta}(s_t, a_t) \\
\vspace{-0.2 in}
\end{align*}
The mutual information term preserves in latent space only the information relevant for the dynamics, while the constraints prevents the latent from degenerating on the terms that matter to predict the rewards and next latents. However, in practice, this objective is intractable: we cannot evaluate the exact mutual information term and we have a set of non-linear constraints. To optimize it, we formulate the Lagrangian of the objective and replace the mutual information objective with the noise contrastive estimator lower bound $I_{\text{NCE}}$:
\begin{align}
\max_{\bf{\theta, \phi, \psi, \zeta}} I_{\text{NCE}}(s_t; o_{t+h} | a_t, \dots, a_{t+h-1}) - \lambda_1 D_{\text{KL}}(g_{\zeta}(z_t, \hat{s}_{t+1}) \| \hat{s}_{t+1}) - \lambda_2 D_{\text{KL}}(r_t \| r_{\psi}(s_t, a_t))
\label{eq:loss}
\end{align}
We use the reparametrization trick~\cite{kingma2013auto} for $\hat{s_{t+1}} = \hat{\mu}_{t+1} + \xi \hat{\omega}_{t+1}$ where $\xi \sim \mathcal{N}(0, 1)$ and $z_t = e_{\phi}(o_t)$. In our case, the noise contrastive estimator term results in:
\begin{align}
I_{\text{NCE}}(s_t; o_{t+h} | a_t, \dots, a_{t+h-1}) = \mathbb{E}\left[\log \left(\frac{\text{exp}(\hat{s}_{t+h}^\top W_h z_{t+h})}{
\text{exp}(\hat{s}_{t+h}^\top W_h z_{t+h}) + \sum_{j=1}^{K-1} \text{exp}(\hat{s}_{t+h}^\top W_h z_{j})} \right) \right]
\end{align}
Here, $\hat{s}_{t+h}$ represents the open loop prediction from $s_t$ and the sequences of actions $a_{t}, \dots, a_{t+h}$.
\subsection{Latent Space Model-Based Reinforcement Learning}
We present the overall method for obtaining a model-based controller when learning the latent space using MIRO.
\textbf{Data Collection. } As is typical of model-based methods, we alternate between model learning and data collection using the latest controller. This allows the model to just learn the parts of the state space that the agent visits, removing the burden of modelling the entire space, and overcomes the insufficient coverage of the initial data distribution.
\textbf{Model Learning. } We learn the encoder, model, filtering function, and reward predictor altogether using equation~\ref{eq:loss} and all the data collected so far.
\textbf{Planning. } As a controller, we use model-predictive control (MPC) with cross-entropy method (CEM) \cite{botev2013cross}. The CEM component selects the action that maximizes the expected return under the learned models. Specifically, it is a population based procedure that iteratively refits a Gaussian distribution on the best sequences of actions. The MPC component results in a more robust planning by doing CEM at each step.
|
2,877,628,090,142 | arxiv | \section{Introduction}
In this work, we study the problems of finding maximal independent sets (MIS) and ruling sets in the LOCAL model of distributed computing.
In the LOCAL model, each node of the input graph is considered as a computational device and each edge as a communication link.
Computation proceeds in synchronous rounds, where in each round each node can send a message of arbitrary size to each neighbor and then, after the messages arrive, perform some local computation.
Each node has to terminate at some point and then output its local part of the global solution, i.e., whether it is in the MIS (resp.\ ruling set) or not.
For a more detailed introduction to the LOCAL model, we refer the reader to Section~\ref{subsec:model}.
\paragraph{MIS.}
The problem of finding an MIS in a given graph is one of the most central and well-studied problems in the LOCAL model.
Already in the '80s, the very first papers of the area \cite{KarpW85, Luby1985, Alon1986, Linial1992, Naor1991} gave first upper and lower bounds for the complexity of computing an MIS, and since then there has been an abundance of papers (e.g., \cite{Awerbuch89, panconesi96decomposition, BarenboimE10, SchneiderW10, LenzenW11, Barenboim2016, Barenboim2013, barenboim14distributed, ghaffari16improved, Rozhon2020,GGR2020}) studying the problem and variants thereof.
A major open question was whether an MIS can be computed deterministically in a polylogarithmic number of rounds (see, e.g., \cite{Linial1992}, or Open Problem 11.2 in the book by Barenboim and Elkin \cite{Barenboim2013})---this question was finally answered in the affirmative in a very recent breakthrough by Rozho\v n and Ghaffari \cite{Rozhon2020} on network decompositions.
In contrast, if randomization is allowed, already more than 30 years ago, Luby \cite{Luby1985} and Alon, Babai, and Itai \cite{Alon1986} presented $O(\log n)$-round algorithms for solving MIS, where $n$ denotes the number of nodes of the input graph. This is still the best randomized upper bound known if the complexity is expressed solely as a function of $n$.
On the lower bound side, the $\Omega(\log^* n)$-round bound from the '80s and early '90s by Linial \cite{Linial1992} and Naor \cite{Naor1991} was the state of the art, until Kuhn, Moscibroda, and Wattenhofer (KMW) \cite{Kuhn2004} proved in 2004 that there is no algorithm computing an MIS in $t = f(\Delta) + g(n)$ rounds (even allowing randomization) if $f(\Delta) \in o(\log \Delta / \log \log \Delta)$ and $g(n) \in o(\sqrt{\log n / \log \log n})$.
Here, $\log^* ()$ denotes the iterated logarithm and $\Delta$ the maximum node degree.
Finally, last year, the KMW bounds were improved and complemented by Balliu et al.\ \cite{Balliu2019} who showed that $f(\Delta) + g(n)$ rounds are not sufficient for deterministic algorithms if $f(\Delta) \in o(\Delta)$ and $g(n) \in o(\log n / \log \log n)$, and not sufficient for randomized algorithms if $f(\Delta) \in o(\Delta)$ and $g(n) \in o(\log \log n / \log \log \log n)$.
Due to an $O(\Delta + \log^* n)$-round upper bound by Panconesi and Rizzi \cite{panconesi01simple}, the linear dependency on $\Delta$ is tight.
While the above bounds imply that the complexity of MIS on general graphs must lie in the polylogarithmic (in $n$) regime, the situation on trees is far less clear.
Both the KMW lower bounds and the lower bounds by Balliu et al.\ are achieved by first proving the same bounds for the problem of finding a maximal matching and then obtaining the MIS bounds as an immediate corollary due to the fact that maximal matching on general graphs is essentially the same problem as MIS on line graphs.
As the line graph of any graph with $\Delta \geq 3$ contains a cycle (of length $3$), both lower bounds are not applicable on trees; in fact, as there seems to be no way around line graphs in order to transform the maximal matching bounds to MIS, there is little hope that the proofs can be adapted to work on trees.
Hence, on trees, the state of the art is given by the $\Omega(\log^* n)$-round lower bounds by Linial and Naor, exhibiting a large gap to the best known deterministic upper bound of $O(\log n / \log\log n)$ rounds on trees by Barenboim and Elkin~\cite{BarenboimE10}.
This suggests the following question.
\vspace{2pt}
\begin{mdframed}[backgroundcolor=gray!20, topline=false, rightline=false, leftline=false, bottomline=false]
\textbf{Question 1}
\noindent Is polylogarithmic time needed for deterministically computing an MIS on trees or is there a (much) faster algorithm?
\end{mdframed}
\vspace{2pt}
\paragraph{Ruling sets.}
Ruling sets are a generalization of maximal independent sets.
Let $\alpha \geq 2$, $\beta \geq 1$ be integers.
An $(\alpha,\beta)$-ruling set $S$ is a subset of the nodes of the input graph such that the distance between any two nodes from $S$ is at least $\alpha$ and any node not contained in $S$ has a distance of at most $\beta$ to the closest node in $S$.
An MIS is a $(2,1)$-ruling set.
We observe that an $(\alpha,\beta)$-ruling set is also an $(\alpha',\beta')$-ruling set for any $\alpha' \leq \alpha$ and $\beta' \geq \beta$, hence finding the latter is at least as easy as finding the former.
In particular, the problem of finding a $(2,\beta)$-ruling set for some $\beta > 1$ is at least as easy as the problem of finding an MIS.
Moreover, as our goal is to prove lower bounds, we can safely restrict attention to $\alpha = 2$ without affecting the generality of our results.
Due to their relation to MIS (but also as interesting combinatorial objects of their own), ruling sets have been a natural object of interest in the LOCAL model and are well-studied (see, e.g., \cite{Awerbuch89, SchneiderW10symmetry, Gfeller07, BishtKP13, Barenboim2016, ghaffari16improved}).
In particular, the computation of ruling sets often constitutes a useful subroutine in the computation of other objects, such as maximal matching \cite{Barenboim2016}, maximal independent set \cite{ghaffari16improved}, or distributed coloring \cite{GhaffariHKM18, ChangLP18}. This is not a surprise: also the computation of an MIS is an important step in many algorithms, and it is quite natural to replace this step by the computation of a $(2, \beta)$-ruling set for some $\beta > 1$, if the latter suffices and can be computed faster.
Hence, from the perspective of applications, a lower bound for MIS that also applies to such ruling sets can be considered as substantially more robust than a lower bound that cannot be extended to ruling sets.
Unfortunately, there is a simple argument why the existing lower bounds for MIS by KMW and Balliu et al.\ cannot be extended to $(2, \beta)$-ruling sets: as mentioned before, those lower bounds are achieved on line graphs; however, on line graphs already a $(2,2)$-ruling set can be found in $O(\log^* n)$ rounds as shown by Kuhn, Maus, and Weidner \cite{KuhnMW18}.
The best lower bound for $(2, \beta)$-ruling sets follows again from the lower bounds by Linial and Naor for MIS, and stands at $\Omega(\log^* n)$, both on trees and general graphs, up to some $\beta \in \Theta(\log^* n)$.
For $\beta \in \omega(\log^* n)$, no non-constant lower bound is known.
In contrast, for up to polylogarithmic\footnote{As long as $\beta$ is not too close to $n$, also no subpolylogarithmic upper bounds are known for larger $\beta$, but the (at most) polylogarithmic regime is arguably the most interesting; for instance, we are not aware of any algorithms that make use of $(2, \beta)$-ruling sets where $\beta$ is superpolylogarithmic.} $\beta$, the best known upper bound (expressed solely as a function of $n$) for computing a $(2,\beta)$-ruling set is polylogarithmic in $n$ \cite{Awerbuch89,SchneiderW10symmetry,GGR2020}.
\vspace{2pt}
\begin{mdframed}[backgroundcolor=gray!20, topline=false, rightline=false, leftline=false, bottomline=false]
\textbf{Question 2}
\noindent Is polylogarithmic time needed for deterministically computing a $(2, \beta)$-ruling set (for up to polylogarithmic $\beta$) or is there a (much) faster algorithm?
\end{mdframed}
\vspace{2pt}
\paragraph{Round elimination.}
Traditionally, proving lower bounds in the LOCAL model has been a challenging task.
Until 2015, to the best of our knowledge, only about a handful of (non-trivial, non-global) lower bounds were known \cite{Linial1992, Naor1991, Kuhn2004, ChungPS14, GoosHS14, NaorS95}, with the only lower bound (as a function of $n$) beyond $\Omega(\log^* n)$ being the KMW lower bound.
A major obstacle seemed to be the lack of techniques that could be used to obtain (improved) lower bounds.
In 2016, things changed when it was discovered that a technique used in the proof for Linial's $\Omega(\log^* n)$-round lower bound is more widely applicable: Brandt et al.\ \cite{Brandt2016} used the technique, now known under the name \emph{round elimination}, to prove lower bounds for the Lov\'asz Local Lemma (LLL), sinkless orientation (as a special case of the LLL) and $\Delta$-coloring.
Since then, round elimination has been used to prove lower bounds for a variety of problems \cite{chang18complexity, BalliuHOS19, Brandt2019, Balliu2019, trulytight, binary}.
In 2019, Brandt \cite{Brandt2019} showed that round elimination can, in principle, be applied to (almost) any problem that is locally checkable\footnote{For a definition, see Section~\ref{subsec:problems}.}, by providing a so-called \emph{automatic} version of round elimination, which, roughly speaking, is a blueprint for obtaining a lower bound via round elimination in which the problem of interest can be inserted.
Unfortunately, for most problems, a crucial step in the general blueprint is (perhaps far) beyond the reach of current techniques, which is the reason why we have not seen a flurry of new lower bounds in the past year.
By using additional techniques inside this framework, a number of new lower bounds have been achieved \cite{Balliu2019, trulytight, binary}, but the framework itself is still far from being well-understood.
As such, we believe that obtaining a better understanding of (automatic) round elimination is one of the most promising research directions in the LOCAL model currently available and crucial for the design of new lower bounds.
Informally, the general idea of round elimination is as follows.
In order to prove a lower bound for some problem $\Pi_0$ of interest, we want to find a sequence of problems
\[
\Pi_0 \rightarrow \Pi_1 \rightarrow \Pi_2 \rightarrow \dots
\]
such that for any two consecutive problems $\Pi_i, \Pi_{i+1}$, we have $T_{i+1} \leq T_i - 1$ whenever $T_i > 0$, where $T_j$ denotes the complexity of problem $\Pi_j$ for any $j$.
In other words, $\Pi_{i+1}$ is at least one round faster solvable than $\Pi_i$ as long as $\Pi_i$ is not $0$-round solvable, which we will call the \emph{round elimination property}.
Now all that is necessary for proving a lower bound of $T$ for problem $\Pi_0$ is to show that problem $\Pi_{T-1}$ is not $0$-round solvable, or equivalently, that the first $0$-round solvable problem in the sequence has index at least $T$.
Automatic round elimination explicitly generates such a sequence of problems for any locally checkable problem $\Pi_0$, by repeatedly applying a fixed process that takes some locally checkable problem $\Pi_i$ as input and returns $\Pi_{i+1}$.
The main issue with the obtained sequence is that the descriptions of the problems in the sequence usually become very complicated already for small indices; without applying any additional techniques, already the size of the problem description grows roughly doubly exponential \emph{for each subsequent problem}.
Hence, it is not surprising that the crucial step of determining the first $0$-round solvable problem $\Pi_j$ in the sequence cannot be performed (in general) with the currently available techniques.
Moreover, even if one could keep the problem description sizes reasonably small, no general method how to find the desired problem $\Pi_j$ is known.\footnote{Note that it is usually easy to check for a given problem whether it can be solved in $0$ rounds; the difficulty lies in first obtaining a concise (parameterized) description of the problems in the sequence.}
Nevertheless, when studying a specific problem $\Pi_0$, it seems reasonable to try to make the problems in the sequence easier to understand.
All currently known lower bound proofs via automatic round elimination follow the idea of modifying the problems in the sequence in a way that preserves the round elimination property while simplifying the problem descriptions, as suggested in \cite{Brandt2019}.
The proofs can be grouped into two categories, depending on the chosen modification.
\begin{enumerate}
\item There exists a constant $c$ such that each problem in the sequence can be described\footnote{The description is required to be in a certain standardized form. For details, we refer to Section \ref{subsec:problems}.} by using at most $c$ output labels. Examples are \cite{Balliu2019, trulytight, binary}.
\item The size of the problem description grows doubly exponentially when going from $\Pi_i$ to $\Pi_{i+1}$, for all $i$. Examples are \cite{BalliuHOS19, Brandt2019}.
\end{enumerate}
The idea of the second approach is to simplify the \emph{structure}\footnote{For instance, the simplification could consist in transforming a problem with complicated constraints using a large number of output labels into a (much easier to understand) coloring problem with a large number of colors.} of the descriptions of the problems in the sequence, but roughly preserve the \emph{size} of the descriptions.
The lower bound is achieved by showing that as long as the description size of a problem in the sequence is in $o(n)$ (or $(o(\Delta))$), the problem is not $0$-round solvable.
Hence, this approach only yields lower bounds of $\Omega(\log^* n)$ (resp.\ $\Omega(\log^* \Delta)$).
In contrast, the first approach can yield higher lower bounds, but requires finding a sequence of problems that can be described with a constant number of labels.
Considering that to obtain a \emph{good} lower bound we also must make sure that we do not reach a $0$-round solvable problem too fast, for many problems such a sequence might simply not exist.
In fact, characterizing the set of problems (or at least interesting subsets thereof) that admit such a sequence is an interesting open problem mentioned in \cite{trulytight}.
For instance, while we do not have a proof, we do not believe that for MIS such a sequence yielding a polylogarithmic lower bound exists.
This discussion raises the following question.
\vspace{2pt}
\begin{mdframed}[backgroundcolor=gray!20, topline=false, rightline=false, leftline=false, bottomline=false]
\textbf{Question 3}
\noindent How can we design a problem sequence satisfying the round elimination property that yields a better lower bound than $\Omega(\log^* n)$ without restricting the problem descriptions to a constant number of labels?
\end{mdframed}
\vspace{2pt}
\subsection{Our results}
We prove the following result for deterministic algorithms.
\begin{restatable}{theorem}{detlb}\label{thm:detlb}
In the LOCAL model, any deterministic algorithm that solves the $(2,\beta)$-ruling set problem requires $\Omega\left(\min \left\{ \frac{\log \Delta}{\beta \log \log \Delta} , \log_\Delta n \right\} \right)$ rounds, for all $\beta \le c \cdot \min\left\{ \sqrt{\frac{\log \Delta}{\log \log \Delta}} , \log_\Delta n \right\}$, for some constant $c$ independent of $n$ and $\Delta$.
\end{restatable}
By setting $\Delta \coloneqq 2^{\sqrt{\beta \log n \log \log n}}$, we maximize our lower bound as a function of $n$, thereby obtaining the following corollary.
\begin{restatable}{corollary}{detcor}\label{cor:detlb}
In the LOCAL model, any deterministic algorithm that solves the $(2,\beta)$-ruling set problem requires $\Omega\left(\sqrt{\frac{\log n}{\beta \log \log n}}\right)$ rounds, for all $\beta \le c \, \sqrt[3]{\frac{\log n}{\log \log n}}$, for some constant $c$ independent of $n$ and $\Delta$.
\end{restatable}
This settles Question 2 for all $\beta \le c \, \sqrt[3]{\frac{\log n}{\log \log n}}$.
As any $(\alpha, \beta)$-ruling set is also a $(2, \beta)$-ruling set for all $\alpha > 2$, Theorem~\ref{thm:detlb} also holds for $(\alpha, \beta)$-ruling sets.
Moreover, since the given lower bounds already hold on trees, we obtain the following corollary, by setting $\beta=1$.
\begin{corollary}\label{cor:detmislb}
In the LOCAL model, any deterministic algorithm that solves MIS on trees requires $\Omega\left(\sqrt{\frac{\log n}{ \log \log n}}\right)$ rounds.
\end{corollary}
This settles Question 1.
Corollaries~\ref{cor:detlb} and~\ref{cor:detmislb} provide the first polylogarithmic lower bounds for ruling sets, and for MIS on trees.
Due to an $O(\log n / \log \log n)$-round deterministic upper bound for MIS on trees by Barenboim and Elkin~\cite{BarenboimE10}, and a polylogarithmic deterministic upper bound for $(2,\beta)$-ruling sets on general graphs following from the work by Ghaffari et al.~\cite{GGR2020}, the only remaining question for the given range of $\beta$ is the exponent in the polylog.
For randomized algorithms, we prove the following.
\begin{restatable}{theorem}{randlb}\label{thm:randlb}
In the LOCAL model, any randomized algorithm that solves the $(2,\beta)$-ruling set problem w.h.p.\footnote{As usual, we say that an algorithm solves a problem with high probability if the global success probability is at least $1 - 1/n$.}\ requires $\Omega\left(\min \left\{ \frac{\log \Delta}{\beta \log \log \Delta} , \log_\Delta \log n \right\} \right)$ rounds, for all $\beta \le c \cdot \min\left\{ \sqrt{\frac{\log \Delta}{\log \log \Delta}} , \log_\Delta \log n \right\}$, for some constant $c$ independent of $n$ and $\Delta$.
\end{restatable}
By setting $\Delta \coloneqq 2^{\sqrt{\beta \log \log n \log \log \log n}}$, we maximize our lower bound as a function of $n$, thereby obtaining the following corollary.
\begin{restatable}{corollary}{randcor}\label{cor:randlb}
In the LOCAL model, any randomized algorithm that solves the $(2,\beta)$-ruling set problem w.h.p.\ requires $\Omega\left(\sqrt{\frac{\log \log n}{\beta \log \log \log n}}\right)$ rounds, for all $\beta \le c \, \sqrt[3]{ \frac{\log \log n}{\log \log \log n}}$, for some constant $c$ independent of $n$ and $\Delta$.
\end{restatable}
Again, this bound already holds on trees and we obtain the following corollary for MIS.
\begin{corollary}\label{cor:randmislb}
In the LOCAL model, any randomized algorithm that solves MIS on trees w.h.p.\ requires $\Omega\left(\sqrt{\frac{\log \log n}{ \log \log \log n}}\right)$ rounds.
\end{corollary}
Note that Theorem~\ref{thm:randlb} implies that there is no randomized algorithm that solves the $(2,\beta)$-ruling set problem w.h.p.\ in $t = f(\Delta) + g(n)$ rounds if $f(\Delta) \in o\left(\frac{\log \Delta}{\beta \log \log \Delta}\right)$ and $g(n) \in o\left(\sqrt{\frac{\log \log n}{\beta \log \log \log n}}\right)$.
Hence, we obtain that the $O(\log \Delta + \log \log n / \log \log \log n)$-round randomized upper bound for MIS (and hence also $(2, \beta)$-ruling set) on trees by Ghaffari \cite{ghaffari16improved} cannot be improved substantially in both $\Delta$ and $n$ simultaneously, for any indicated $\beta$.
Furthermore, Corollary~\ref{cor:randmislb} provides the first progress on Open Problem 10.15 from the book by Barenboim and Elkin \cite{Barenboim2013} (on the lower bound side), asking for the randomized complexity of MIS on trees.
Our results are achieved by designing a sequence of problems with the round elimination property for $(2,\beta)$-ruling sets, where the number of used labels is non-constant.
More precisely, our problem sequence will satisfy that the number of labels used in the description of problem $\Pi_i$ is in $\Theta(i^\beta / (\beta!))$.
In particular, for the special case of MIS, the number of used labels grows linearly.
Hence, our construction of the problem sequence provides an answer to Question 3.
\subsection{Our techniques}
In order to successfully apply the round elimination technique, two main ingredients are required.
The first is \emph{finding} a good problem family: we need to define some family $\{\Pi_{i\ge0}\}$ such that the sequence $\Pi_0 \rightarrow \Pi_1 \rightarrow \dots$ satisfies the round elimination property and $\Pi_0$ is the problem for which we want to prove a lower bound.
The second ingredient is \emph{proving} that the defined sequence indeed satisfies the desired property.
While the second ingredient is technically involved, the conceptually crucial part is the first one, designing a good sequence of problems.
Usually, when applying the round elimination technique, finding the right problem family involves some guessing.\footnote{In rare cases the sequence suggests itself, e.g., for sinkless orientation \cite{Brandt2016} the sequence is obtained by setting $\Pi_0 = \Pi_1 = \dots$}
For instance, in \cite{Balliu2019} the problem family was found by trying to make each subsequent problem in the sequence \emph{look very similar} to the previous one while using the same output labels in the description (see \cite[Section 3.7]{Balliu2019}).
In the case where each problem in the family can be described using a constant number of labels, there is even very recent software available, written by Olivetti \cite{Olivetti2019}, that automatically searches the space of potential problems for small $\Delta$.
Unfortunately, for the MIS problem (and for ruling sets) this approach fails, suggesting that a constant number of labels is not sufficient.
Instead, we propose a more explicit and perhaps surprising approach to find the desired problem family, by first proving an \emph{upper bound} for the problem of interest such that the proof can be ``represented" via a similar sequence of problems.
As explained in \cite{fraigniaud16local, Brandt2019}, the round elimination technique can also be used to find upper bounds:
Instead of finding a problem sequence with the round elimination property, i.e., with the property that $T_{i+1} \leq T_i - 1$, the idea is to find a problem sequence with the property that $T_{i+1} \geq T_i - 1$.
This ensures that the index $j$ of the first $0$-round solvable problem $\Pi_j$ in the sequence (if such a problem exists) is an upper bound for the complexity of $\Pi_0$.
Accordingly, we will call a sequence satisfying $T_{i+1} \leq T_i - 1$ a \emph{lower bound sequence} and a sequence satisfying $T_{i+1} \geq T_i - 1$ an \emph{upper bound sequence}.
We note that the \emph{automatic sequence} provided by \emph{automatic} round elimination is both a lower and an upper bound sequence since there we have $T_{i+1} = T_i -1$; in fact, it can be seen as the \emph{tightest} sequence with the property $T_{i+1} \geq T_i - 1$.
In the following, we will use this automatic sequence to informally describe the intuition behind our approach.
\paragraph{Intuition behind our approach.} In the round elimination framework, each problem is described via a list of ``allowed" configurations that specify which local output label configurations around a node or on an edge are considered correct.
As mentioned before, in the automatic sequence these descriptions grow very fast.
On the other hand, due to the nature of $0$-round algorithms, it seems to be the case that in the first (or more generally, any) $0$-round solvable problem $\Pi_j$ only very few of those allowed configurations are actually required for the correctness of a given $0$-round algorithm.
In other words, $\Pi_j$ would still remain $0$-round solvable if we removed a large number of the allowed configurations; moreover, the remaining part of the problem usually has an intuitive interpretation.
Assuming that the previous problems in the automatic sequence behave similarly, we obtain the following intuition for each problem $\Pi_i$ with $i \leq j$:
\begin{itemize}
\item[(1)] There is some small part of the problem description that has some intuitive meaning and is relevant for solving the problem in $j - i$ rounds, and
\item[(2)] there are additional allowed configurations that seem to be an artifact of the automatic process that generates the sequence.
\end{itemize}
Intuitively, Part (1) can be thought of as the \emph{essence} of the problem, and we argue that the information encoded therein should suffice to prove lower bounds.
Hence, we would like to restrict attention to Part (1).
If we had a concise description of problem $\Pi_j$ and the complete automatic sequence leading to $\Pi_j$, it would be straightforward to extract Part (1) of each problem and thereby obtain a comparably simple sequence $\Pi_0^* \rightarrow \Pi_1^* \rightarrow \dots$ of problems.
However, there are two issues: first, we do not have feasible access to $\Pi_j$ and the preceding sequence (otherwise we would be done), and second, for technical reasons, the obtained sequence is an upper bound sequence, but not a lower bound sequence (in general), i.e., even if we had such access, the fact that $T_{i+1} \leq T_i - 1$ is not satisfied prevents us from using the sequence in a \emph{lower bound} proof.
To solve the second issue, we make use of so-called \emph{wildcards}, a notion introduced in \cite{Balliu2019}.
We show for the case of MIS and ruling sets that, perhaps surprisingly, adding a sufficient number of wildcards to the allowed configurations in the problems from $\Pi_0^* \rightarrow \Pi_1^* \rightarrow \dots$ turns the upper bound sequence into a lower bound sequence that is ``tight enough" to yield a polylogarithmic lower bound.
Our solution to the first issue is to try to design an upper bound sequence $\Pi'_0 \rightarrow \Pi'_1 \rightarrow \dots$ that is as close to the desired sequence $\Pi_0^* \rightarrow \Pi_1^* \rightarrow \dots$ as possible, and then work with problem family $\{\Pi'_{i\ge0}\}$ instead of $\{\Pi_{i\ge0}^*\}$.
As the latter sequence is unknown, our guideline for designing $\{\Pi'_{i\ge0}\}$ will be \emph{simplicity}, following the above intuition that Part (1) of each $\Pi_i$ (i.e., $\Pi_i^*$) is small and intuitive.
A key idea in the design will be to introduce a \emph{coloring component} into the MIS and ruling set problems.
Roughly speaking, the purpose of this coloring component is that, with enough care, we can make sure that only the coloring part of the problem description grows when we go from $\Pi'_i$ to $\Pi'_{i+1}$, while the MIS (resp.\ ruling set) part remains unchanged.
This allows us to keep the structure of the problems in the sequence comparably simple, which in turn allows us to determine at which point in the sequence the problems become $0$-round solvable.
Essentially, our approach reduces the task of proving lower bounds to proving upper bounds, which usually is considered to be an easier task.\footnote{While the current literature uses round elimination primarily to prove lower bounds, this statement arguably also holds for lower/upper bounds \emph{via round elimination}. One main reason is that to make a problem given in the form specified by round elimination \emph{harder} (a technique instrumental for the design of upper bound sequences), we can simply discard allowed configurations, while to make a problem \emph{easier} (instrumental for lower bound sequences), more complicated operations have to be used.}
However, the designed algorithm should also have a ``simple representation" as an upper bound sequence, and this does not seem to be the case for existing ruling set algorithms.
Hence, we will design a new, genuinely different ruling set algorithm that gives state-of-the-art upper bounds in terms of $\Delta$ (which is the relevant dependency for the round elimination technique, from a technical perspective) and yields a simple upper bound sequence.
Figure \ref{fig:re-lb-ub} depicts the high level idea of what happens when using the round elimination technique to prove upper and lower bounds by doing simplifications.
\begin{figure}[h]
\centering
\includegraphics[width=0.9\textwidth]{figs/relbub}
\caption{$T$ is the unknown complexity of some problem $\Pi$. By directly applying the round elimination technique to $\Pi$ we obtain a sequence of problems, each one being exactly one round easier than the previous one, and after $T$ steps we reach a $0$-round solvable problem. The problems in this sequence all lie on the grey line. Unfortunately, it is often practically not feasible to compute such a sequence. In order to prove lower bounds, we can try to relax the obtained problems to problems with simpler descriptions, and in some cases the simplified problems may become strictly easier. This is depicted in the orange lower bound sequence, and the obtained lower bound is given by the value of the horizontal axis where the orange line intersects it. Similarly, we may lose precision also in an upper bound sequence, depicted in blue, where we simplify problems in a manner that makes them potentially harder.}
\label{fig:re-lb-ub}
\end{figure}
\paragraph{Approach.}
To summarize, our approach works as follows.
First, we prove an upper bound for finding a $(2,\beta)$-ruling set (of which MIS is a special case) that can be represented by a comparably simple upper bound sequence.
To this end, we consider the initial problem $\Pi_0$ of the sequence as ``$(2,\beta)$-ruling set with some coloring component" and then introduce more and more colors into the problem over the course of the sequence, in a certain hierarchical manner.
Second, we insert (an increasing number of) wildcards into the problems in our sequence, and prove that this turns the upper bound sequence into a lower bound sequence that yields a polylogarithmic lower bound.
While the individual parts of our approach are technically challenging, the approach itself is surprisingly simple.
Hence, we believe that this general approach does not only work for MIS and ruling sets but should also be applicable to other problems; however, as it involves, e.g., finding an upper bound proof that can be described well via a sequence of problems, obtaining new bounds using this approach is not automatic.
Moreover, we think that the idea of introducing a coloring component into problems that do not seem to have any particular relation to coloring should be more widely applicable; one intuitive reason is that, similar to wildcards, it gives a relatively simple way to represent \emph{progress} towards $0$-round solvability in the sequence, which seems like a necessary ingredient for designing a lower or upper bound sequence (which we can feasibly infer bounds from).
\subsection{Further discussion of related work}
\paragraph{MIS.}
The maximal independent set problem has been widely studied in the LOCAL model. Barenboim et al.\ showed that, if we also consider the dependency in $\Delta$, MIS can be solved in $O(\log^2\Delta) + 2^{O(\sqrt{\log\log n})}$ rounds~\cite{Barenboim2016}. Ghaffari improved this running time to $O(\log\Delta) + 2^{O(\sqrt{\log\log n})}$ \cite{ghaffari16improved}. The MIS problem has been studied also in specific classes of graphs \cite{SchneiderW10,BarenboimE10,BarenboimE11,Barenboim2016}. For example, for computing MIS on trees with randomized algorithms, Lenzen and Wattenhofer showed an $O(\sqrt{\log n}\log\log n)$-round algorithm \cite{LenzenW11}. This was later improved by Barenboim et al.\ to $O(\sqrt{\log n\log\log n})$ \cite{Barenboim2016}, and then further improved to $O(\sqrt{\log n})$ by Ghaffari \cite{ghaffari16improved}. Barenboim et al.\ also showed that MIS on trees can be solved in $O(\log\Delta\log\log\Delta + \log\log n/\log\log\log n)$ rounds \cite{Barenboim2016}. Ghaffari later improved this bound to $O(\log\Delta + \log\log n/\log\log\log n)$ rounds~\cite{ghaffari16improved}.
Ghaffari studied MIS also in the CONGEST\footnote{The CONGEST model is the same as the LOCAL model with the difference that in CONGEST the size of the messages is bounded by $O(\log n)$ bits. We refer the reader to Section \ref{subsec:model} for more details on these models.} model, giving a randomized algorithm with a running time of $\min \{ O(\log \Delta \log \log n) + 2^{O(\sqrt{\log\log n \log \log \log n})}, \log \Delta \cdot 2^{O(\sqrt{\log \log n})} \}$ rounds \cite{Ghaffari19congest}. This was later improved to $O(\log \Delta \cdot \sqrt{\log \log n}) + 2^{O(\sqrt{\log\log n})}$ rounds by Ghaffari and Portmann~\cite{ghaffariPortman19}.
While all the above algorithms are randomized, Panconesi and Srinivasan provided a deterministic algorithm for solving MIS in $2^{O(\sqrt{\log n})}$ rounds~\cite{panconesi96decomposition}. Later, Barenboim, Elkin and Kuhn showed an $O(\Delta + \log^* n)$-round algorithm \cite{barenboim14distributed}. Very recently, Rozho\v n and Ghaffari proved that MIS can be solved deterministically in $\poly(\log n)$ rounds~\cite{Rozhon2020}. Meanwhile, the exponent of the polylog has been improved by Ghaffari et al.~\cite{GGR2020}.
\paragraph{Ruling sets.}
Ruling sets have been introduced by Awerbuch et al.\ \cite{Awerbuch89}, where the authors showed how to construct $(\alpha, O(\alpha \log n))$-ruling sets in $O(\alpha \log n)$ deterministic rounds in the LOCAL model. Since then, there have been several works in this direction both in the deterministic and randomized setting, and both in the LOCAL and CONGEST models of distributed computing. In fact, as far as deterministic algorithms are concerned, Schneider, Elkin, and Wattenhofer showed how to get $(2,\beta)$-ruling sets in $O(\beta \Delta^{2/\beta} + \log^* n)$ rounds in the LOCAL model \cite{SEW13}. Notice that, in the LOCAL model, it is possible to get an $(\alpha, (\alpha-1)\beta)$-ruling set of a graph $G$ by just computing a $(2, \beta)$-ruling set on the power graph $G^{\alpha - 1}$. This reasoning does not directly apply to the CONGEST model, where the size of the messages is bounded by $O(\log n)$ bits. However, the algorithm of Awerbuch et al.\ can be modified to work in the CONGEST model. In fact, Henzinger, Krinninger, and Nanongkai sketched the arguments that show how to adapt it and get a CONGEST algorithm that gives $(\alpha, O(\alpha \log n))$-ruling sets in $O(\alpha \log n)$ rounds \cite{HKN16}. Later on, Kuhn, Maus, and Weidner gave a formal proof of these arguments \cite{KuhnMW18}. Also, the same authors showed how to obtain $(\alpha, (\alpha-1)\lceil \log_B n \rceil)$-ruling sets ($B\ge2$) in $O(\alpha B \log_B n)$ rounds. As a corollary, they get the same trade offs as in \cite{SEW13} and obtain a $(2, \beta)$-ruling set (for $\beta>2$) in $O(\beta \Delta^{2/\beta} + \log^* n)$ rounds for the CONGEST model.
If randomness is allowed, Gfeller and Vicari showed how to compute a relaxed version of a $(2, O(\log\log\Delta))$-ruling set, where each node in the ruling set is allowed to have at most $O(\log^5 n)$ neighbors also in the ruling set, in $O(\log\log\Delta)$ rounds~\cite{Gfeller07}, and by then applying the algorithm of \cite{SEW13} on the graph induced by selected nodes, we can obtain an algorithm for $(2,\log \log n)$-ruling sets running in $O(\log \log n)$ time.
Kothapalli and Pemmaraju showed how to compute $(2,2)$-ruling sets in $O\left(\frac{\log \Delta}{(\log n)^\varepsilon} + (\log n)^{1/2+\varepsilon}\right)$ rounds, for any $\varepsilon > 0$ \cite{KP12}. One year later, Bisht, Kothapalli, and Pemmaraju provided a sparsifying procedure that can be used, together with some MIS algorithm, to obtain $(2,\beta)$-ruling sets (in a runtime that depends on the respective MIS algorithm)~\cite{BishtKP13}. For instance, by combining this sparsifying procedure with the MIS algorithm by Barenboim et al.~\cite{Barenboim2016}, a $(2,\beta)$-ruling set can be computed in $O(\beta\log^{1/(\beta-1/2)} \Delta) + 2^{O(\sqrt{\log\log n})}$ rounds. By using the improved MIS algorithm by Ghaffari \cite{ghaffari16improved} instead, we obtain a runtime of $O(\beta\log^{1/\beta} \Delta) + 2^{O(\sqrt{\log\log n})}$ rounds, which can in turn be improved to $O(\beta\log^{1/\beta} \Delta) + \poly(\log\log n)$ rounds by making use of the $\poly(\log n)$-round network decomposition algorithm by Rozho\v n and Ghaffari \cite{Rozhon2020}.
Lastly, Pai et al.\ studied randomized ruling sets in the CONGEST model. They showed how to compute $(2,3)$-ruling sets in $O(\log n/ \log\log n)$ rounds, and $(2,2)$-ruling sets in $O(\log\Delta (\log n)^{1/2 + \varepsilon} + \varepsilon \log n \log\log n)$ rounds \cite{PPPR017}.
\section{Preliminaries}\label{sec:preliminaries}
\subsection{Model}\label{subsec:model}
\paragraph{The LOCAL model.}
The model of computation used in this paper is the widely studied LOCAL model of distributed computing \cite{Peleg2000}. In this model, each node of the input graph has a unique identifier from $1$ to $\poly n$, and the computation proceeds in synchronous rounds. At each round, each node can send a message of arbitrary size to each neighbor, and, after receiving the messages from its neighbors, perform some local computation of arbitrary complexity. In the LOCAL model, each node knows initially its unique identifier and its degree. As commonly done in this context, we also assume that each node knows the number of nodes $n$ in the graph (or a polynomial upper bound of it) and the maximum degree $\Delta$.
Clearly, this can make the task of proving lower bounds only harder.
Each node executes the same algorithm (which is what we call a distributed algorithm), and each node has to terminate at some point and then output its local part of the global solution, e.g., in the case of MIS whether the node is in the MIS or not.
The runtime of such a distributed algorithm is the number of synchronous round until the last node terminates.
In the randomized version of the LOCAL model, each node additionally has access to a stream of private random bits.
We will study Monte Carlo algorithms that solve the desired problem with high probability, that is, the global success probability must be at least $1-1/n$.
Another well-studied model in the area of distributed computing is the CONGEST model \cite{Peleg2000}, which is defined as the LOCAL model with the only difference that the size of each message sent between the nodes is restricted to $O(\log n)$ bits.
As the CONGEST model is strictly weaker than the LOCAL model, our lower bounds hold also in the CONGEST model.
\paragraph{The Port Numbering model.}
Our results hold in the LOCAL model of distributed computing, however, for technical reasons we pass through the Port Numbering (PN) model, in the sense that we first show how to obtain our results in the PN model, and then lift them to the LOCAL model. The PN model is a variant of the LOCAL model where nodes do not have identifiers, but each node $v$ has an internal ordering of its incident edges given by an arbitrary assignment of (pairwise distinct) so-called \emph{port numbers} from $1$ to $\deg(v)$ to the edges.
This model is also synchronous, and, as in the LOCAL model, the size of the messages and the computational power of each node is not bounded. In the randomized version of the PN model, each node has access to a stream of private random bits and we require that randomized algorithms succeed with high probability.
To be able to apply the round elimination framework, we also need that \emph{edges} have port numbers; in other words, we assume that an orientation of the edges is given.
However, this is just a technical detail that does not have any effect on our argumentation, and as such we will ignore it in the following.
Note that, in the LOCAL model, such an edge orientation can be obtained from the unique identifiers in one round; therefore also the presented upper bounds do not change asymptotically if we assume that an edge orientation is given.
\subsection{Problems}\label{subsec:problems}
In the round elimination framework a problem is characterized by an alphabet $\Sigma$ of labels, a \emph{node constraint} \ensuremath{\mathcal{N}}{} and an \emph{edge constraint} \ensuremath{\mathcal{E}}.
We will only consider problems defined on $\Delta$-regular graphs in this formalism, since, as we will later see, this is enough for our purposes.
The node constraint \ensuremath{\mathcal{N}}{} is a collection of words of length $\Delta$ over the alphabet $\Sigma$, and the edge constraint \ensuremath{\mathcal{E}}{} is a collection of words of length $2$ over $\Sigma$.
The same label can appear several times in a word and the order of the elements that compose a word does not matter, hence each word technically is a multiset.
We call a word in \ensuremath{\mathcal{N}}{} a \emph{node configuration} and a word in \ensuremath{\mathcal{E}}{} an \emph{edge configuration}.
Let $G=(V,E)$ be our input graph and let $A=\{ (v,e)\in V\times E~|~v\in e\}$ be the set that contains all pairs $(\mbox{node}, \mbox{incident edge})$.
The output for a problem in this formalism is given by a labeling of each $(v,e)\in A$ with one element from $\Sigma$.
Put otherwise, each node has to output an element of the set $\Sigma$ on each incident edge.
We say that such an output is \emph{correct} if it satisfies \ensuremath{\mathcal{N}}{} and \ensuremath{\mathcal{E}}, i.e., for each node $v' \in V$, the collection of $\Delta$ output labels assigned to the $(v,e) \in A$ with $v = v'$ is a node configuration listed in $\ensuremath{\mathcal{N}}$, and for each edge $e' \in E$, the two output labels assigned to the $(v,e)$ with $e = e'$ is an edge configuration listed in $\ensuremath{\mathcal{E}}$.
We use regular expressions to represent (collections of) node and edges configurations.
For example, the expression $\P\mspace{1mu}\O^{\Delta-1}$ describes a node configuration that consists of exactly one label $\P$ and $\Delta-1$ labels $\O$.
Similarly, the expression $\mathsf{M}\mspace{1mu}[\P\O]$ describes a collection of edge configurations that consists of one label $\mathsf{M}$ and the other label can be either $\P$ or $\O$, i.e., $\mathsf{M}\mspace{1mu}[\P\O] = \{ \mathsf{M}\P, \mathsf{M}\O \}$.
We call a part of an expression such as $[\P\O]$, where we have a choice between different labels, a \emph{disjunction}.
While technically an expression containing a disjunction describes a set of configurations, we will use the term \emph{configuration} also for such an expression, for simplicity.
In order to explicitly specify that the expression contains a disjunction, we will use the term \emph{condensed configuration}.
Moreover, we will say that a configuration is \emph{contained in} a condensed configuration if we can obtain the former from the latter by picking a choice in each disjunction.
With a few exceptions, all problems from a large class of problems of interest in the LOCAL model, so-called \emph{locally checkable} problems, can be described in this formalism.
A locally checkable problem is simply a problem for which the correctness of a solution can be verified by checking whether the $O(1)$-hop neighborhood of each node is locally correct.
For technical reasons, locally checkable problems whose definitions involve small cycles (such as determining for each node whether it is contained in a triangle) cannot be described in the above formalism.
Hence, for simplicity, in the remainder of the paper we will use the term ``locally checkable" for (locally checkable) problems that are not of this kind.
In the following we present two examples highlighting how we arrive at the description of a problem in the new formalism.
In Section~\ref{sec:equivalence}, we will show more formally that the given descriptions capture the MIS and ruling set problems.
\paragraph{Example: MIS.}
Let us see, for example, how we can describe the MIS problem in this formalism.
We define $\Sigma=\{\mathsf{M},\P,\O \}$.
We will use the node constraint to represent whether a node is in the independent set or not.
Nodes that are in the independent set must output the label $\mathsf{M}$ (as in ``in the MIS'') on all incident edges.
For nodes that are not in the independent set, we have to make sure that at least one neighbor is in the independet set.
To this end, we require that nodes that are not in the independent set \emph{point} to a neighbor that is in the independent set, thereby ensuring maximality.
In other words, these nodes must output a label $\P$ (as in ``pointer'') on exactly one incident edge and the label $\O$ (as in ``other'') on all the other $\Delta - 1$ incident edges.
Now the edge constraint must guarantee that no two neighbors are in the MIS, hence $\mathsf{M}\mspace{1mu}\mathsf{M}\notin \ensuremath{\mathcal{E}}$, and that a pointer points to a node that is in the MIS, hence $\P\mspace{1mu}\mathsf{M}\in \ensuremath{\mathcal{E}}$, but $\P\mspace{1mu}\P\notin \ensuremath{\mathcal{E}}$, and $\P\mspace{1mu}\O\notin \ensuremath{\mathcal{E}}$.
In order to capture the situation where a node not in the MIS has several neighbors in the MIS, we must allow $\mathsf{M}\mspace{1mu}\O\in \ensuremath{\mathcal{E}}$.
Also, since two nodes not in the MIS may be neighbors, $\O\mspace{1mu}\O\in \ensuremath{\mathcal{E}}$.
This leads to the following formal definition of the node and edge constraint.
\begin{equation*}
\begin{aligned}
\begin{split}
\ensuremath{\mathcal{N}}\text{:} \\
& \quad\mathsf{M}^{\Delta} \\
& \quad\P\mspace{1mu}\O^{\Delta-1}
\end{split}
\qquad
\begin{split}
\ensuremath{\mathcal{E}}\text{:} \\
& \quad \mathsf{M}\mspace{1mu}[\P\O] \\
& \quad \O\mspace{1mu}\O
\end{split}
\end{aligned}
\end{equation*}
\paragraph{Example: (2,2)-ruling set.}
In order to encode the $(2,2)$-ruling set problem we need to use a larger set of labels compared to the one used for the MIS problem. Let $\Sigma=\{\mathsf{M},\P_1,\P_2,\O_1,\O_2\}$. Intuitively, similarly as before, the $\mathsf{M}$ label can be seen as the ``I am in the ruling set" label, while the labels $\P_1$ and $\P_2$ are ``pointer'' labels that are used to point to nodes in the ruling set and to nodes that are at distance $1$ from a node in the ruling set. Notice that, as a $(2,1)$-ruling set (i.e., MIS) solves the $(2,2)$-ruling set problem, the encoding of the $(2,2)$-ruling set problem will contain the node and edge configurations of the MIS problem.
For instance, a node in the ruling set will output $\mathsf{M}^{\Delta}$. Nodes at distance $1$ from a node in the ruling set may output either $\P_1\mspace{1mu}\O_1^{\Delta-1}$ or $\P_2\mspace{1mu}\O_2^{\Delta-1}$, but those at distance $2$ must output $\P_2\mspace{1mu}\O_2^{\Delta-1}$. On the edge side, we must guarantee that, for any pair of nodes in the ruling set, they do not share an edge, hence $\mathsf{M}\mspace{1mu}\mathsf{M}\notin \ensuremath{\mathcal{E}}$. Also, a pointer of type $1$ must point to a node in the ruling set, while a pointer of type $2$ must point to a node at distance at most $1$ from a node in the ruling set, hence $\mathsf{M}\mspace{1mu}[\P_1\P_2]\in \ensuremath{\mathcal{E}}$ and $\O_1\mspace{1mu}\P_2\in \ensuremath{\mathcal{E}}$. On the other hand, we want to forbid bad pointing. In fact, nodes at distance $1$ from a node in the ruling set must not be able to point to a node that is not in the ruling set, hence $\P_1\mspace{1mu}[\O_1\O_2\P_1\P_2]\notin \ensuremath{\mathcal{E}}$. Also, nodes at distance $2$ from a node in the ruling set must not point to another node that is at distance $2$ as well, hence $\P_2\mspace{1mu}[\O_2\P_2]\notin \ensuremath{\mathcal{E}}$. More precisely, the $(2,2)$-ruling set problem can be encoded in the formalism as follows.
\begin{equation}
\begin{aligned}
\begin{split}
\ensuremath{\mathcal{N}}\text{:} \\
& \quad\mathsf{M}^{\Delta} \\
& \quad\P_1\mspace{1mu}\O_1^{\Delta-1} \\
& \quad\P_2\mspace{1mu}\O_2^{\Delta-1}
\end{split}
\qquad
\begin{split}
\ensuremath{\mathcal{E}}\text{:} \\
& \quad \mathsf{M}\mspace{1mu}[\P_1\O_1\P_2] \\
& \quad \O_1\mspace{1mu}[\O_1\O_2\P_2] \\
& \quad \O_2\mspace{1mu}\O_2
\end{split}
\end{aligned}
\label{eq:22rs}
\end{equation}
\subsection{Round elimination}\label{sec:resec}
In our proofs, we will use the result of \cite[Theorem 4.3]{Brandt2019}, that is at the core of the round elimination technique. On a high level, this theorem says that, on $\Delta$-regular high-girth graphs, given a locally checkable problem $\Pi$ with time complexity $T$, there exists a locally checkable problem $\Pi''$ with time complexity $T-1$. The procedure of showing this theorem goes through an intermediate problem, that we call $\Pi'$. Given $\Pi$, Brandt \cite{Brandt2019} shows how to construct first $\Pi'$ and then $\Pi''$. We will formally define these problems and then we will see an example where we compute $\Pi'$ and $\Pi''$ starting from a specific problem $\Pi$. Let $\Sigma_{\Pi}$, $\ensuremath{\mathcal{N}}_{\Pi}$, and $\ensuremath{\mathcal{E}}_{\Pi}$ be the alphabet of labels, the node constraint, and the edge constraint for problem $\Pi$, respectively.
\paragraph{Problem $\Pi'$.} In order to define problem $\Pi'$, we must define the alphabet $\Sigma_{\Pi'}$, the node constraint $\ensuremath{\mathcal{N}}_{\Pi'}$, and the edge constraint $\ensuremath{\mathcal{E}}_{\Pi'}$.
\begin{itemize}
\item $\Sigma_{\Pi'}$: The set of labels for $\Pi'$ is the set of all non-empty subsets of $\Sigma_{\Pi}$, i.e., $\Sigma_{\Pi'}=2^{\Sigma_{\Pi}} \setminus \{\{\}\}$.
\item $\ensuremath{\mathcal{E}}_{\Pi'}$: We construct the edge constraint in the following way. Consider a configuration $\mathsf{A}_1\mspace{1mu} \mathsf{A}_2$, where $\mathsf{A}_1,\mathsf{A}_2\in \Sigma_{\Pi'}$, such that, for all $(\a_1, \a_2) \in \mathsf{A}_1 \times \mathsf{A}_2$, it holds that $\a_1\mspace{1mu} \a_2\in \ensuremath{\mathcal{E}}_{\Pi}$ (notice that, by construction of $\Sigma_{\Pi'}$, it holds that $\a_1, \a_2 \in \Sigma_{\Pi}$). Let $\mathcal{A}$ be the collection of all such configurations.
We call a configuration $\mathsf{A}_1\mspace{1mu} \mathsf{A}_2 \in \mathcal{A}$ \emph{non-maximal} if there exists another configuration $\mathsf{A}'_1\mspace{1mu} \mathsf{A}'_2\in \mathcal{A}$ such that $\mathsf{A}_i \subseteq \mathsf{A}'_i$ for all $i \in \{ 1, 2 \}$, and $\mathsf{A}_i \subsetneq \mathsf{A}'_i$ for at least one $i \in \{ 1, 2 \}$.
In other words, if we have a configuration $\mathsf{A}'_1\mspace{1mu} \mathsf{A}'_2\in \mathcal{A}$ that is obtained from $\mathsf{A}_1\mspace{1mu} \mathsf{A}_2$ by adding at least one element to at least one of $\mathsf{A}_1$ and $\mathsf{A}_2$, then we say that $\mathsf{A}_1\mspace{1mu} \mathsf{A}_2$ is non-maximal. We delete all non-maximal configurations from $\mathcal{S}$, and what remains is our set $\ensuremath{\mathcal{E}}_{\Pi'}$ of configurations.
\item $\ensuremath{\mathcal{N}}_{\Pi'}$: Consider a configuration $\mathsf{B}_1\mspace{1mu} \mathsf{B}_2\mspace{1mu}\ldots\mspace{1mu} \mathsf{B}_\Delta$ where $\mathsf{B}_i\in\Sigma_{\Pi'}$ for all $i\in\{1,\dotsc,\Delta\}$, such that there exists a tuple $ (\b_1,\dotsc,\b_\Delta) \in \mathsf{B}_1 \times \dotsc \times \mathsf{B}_\Delta$ such that $\b_1\mspace{1mu} \b_2\mspace{1mu}\dotsc\mspace{1mu} \b_\Delta \in \ensuremath{\mathcal{N}}_{\Pi}$. Let $\mathcal{B}$ be the collection of all such configurations. We delete from the set $\mathcal{B}$ all configurations that contain some set $\mathsf{B}_i$ that does not appear in any configuration in $\ensuremath{\mathcal{E}}_{\Pi'}$. The modified set $\mathcal{B}$ is our set $\ensuremath{\mathcal{N}}_{\Pi'}$.
\end{itemize}
For simplicity, we can (and will) assume that all labels that occur neither in $\ensuremath{\mathcal{E}}_{\Pi'}$, nor in $\ensuremath{\mathcal{N}}_{\Pi'}$, are also removed from $\Sigma_{\Pi'}$.
\paragraph{Problem $\Pi''$.}
Similarly as before, we need to define the alphabet $\Sigma_{\Pi''}$, the node constraint $\ensuremath{\mathcal{N}}_{\Pi''}$, and the edge constraint $\ensuremath{\mathcal{E}}_{\Pi''}$.
\begin{itemize}
\item $\Sigma_{\Pi''}$: The set of labels for $\Pi''$ is the set of all non-empty subsets of $\Sigma_{\Pi'}$, i.e., $\Sigma_{\Pi''}=2^{\Sigma_{\Pi'}} \setminus \{\{\}\}$.
\item $\ensuremath{\mathcal{N}}_{\Pi''}$: The node constraint is constructed as follows. Consider a configuration $\mathsf{B}_1\mspace{1mu} \mathsf{B}_2\mspace{1mu}\ldots\mspace{1mu} \mathsf{B}_\Delta$ where $\mathsf{B}_i\in\Sigma_{\Pi''}$ for all $i\in\{1,\dotsc,\Delta\}$, such that for all $(\b_1,\dotsc,\b_\Delta) \in \mathsf{B}_1 \times \dotsc \times \mathsf{B}_\Delta$ it holds that $\b_1\mspace{1mu} \b_2\mspace{1mu}\dotsc \b_\Delta\in\ensuremath{\mathcal{N}}_{\Pi'}$. Let $\mathcal{B}$ be the collection of all such configurations. We delete from $\mathcal{B}$ all non-maximal configurations, i.e., all those configurations $\mathsf{B}_1\mspace{1mu}\ldots\mspace{1mu} \mathsf{B}_\Delta$ such that there exists some other configuration $\mathsf{B}'_1\mspace{1mu}\ldots\mspace{1mu} \mathsf{B}'_\Delta$ that is obtained from the former by adding at least one element to at least one of the $\mathsf{B}_i$ sets. After performing these deletions, we set $\ensuremath{\mathcal{N}}_{\Pi''} = \mathcal{B}$.
\item $\ensuremath{\mathcal{E}}_{\Pi''}$: Consider a configuration $\mathsf{A}_1\mspace{1mu} \mathsf{A}_2$, where $\mathsf{A}_1,\mathsf{A}_2\in \Sigma_{\Pi''}$, such that there exists a pair $(\a_1, \a_2) \in \mathsf{A}_1 \times \mathsf{A}_2$ such that $\a_1\mspace{1mu} \a_2\in\ensuremath{\mathcal{E}}_{\Pi'}$. Let $\mathcal{A}$ be the collection of all such configurations. We delete from the set $\mathcal{A}$ all configurations that contain some set $\mathsf{A}_1$ or $\mathsf{A}_2$ that does not appear in any configuration in $\ensuremath{\mathcal{N}}_{\Pi''}$, then we set $\ensuremath{\mathcal{E}}_{\Pi''}=\mathcal{A}$.
\end{itemize}
Again, we can (and will) assume that all labels that occur neither in $\ensuremath{\mathcal{N}}_{\Pi''}$, nor in $\ensuremath{\mathcal{E}}_{\Pi''}$, are also removed from $\Sigma_{\Pi''}$.
As $\Pi'$ is uniquely defined by $\Pi$, we can define a function $\re(\cdot)$ that takes $\Pi$ as input and returns $\Pi'$.
Similarly, as $\Pi''$ is uniquely defined by $\Pi'$, we can define a function $\rere(\cdot)$ that takes $\Pi'$ as input and returns $\Pi''$.
With these definitions, we have $\Pi'' = \rere(\re(\Pi))$.
Note that $\rere(\cdot)$ can take any problem as input that is of the form specified by round elimination---it is not necessary that the input problem has been obtained by applying $\re(\cdot)$ to some problem.
Now \cite[Theorem 4.3]{Brandt2019} provides the following relation between a problem $\Pi$ and $\rere(\re(\Pi))$ that provides the fundament for automatic round elimination.
For technical reasons, the theorem itself only holds in the port numbering model, but we will show later how to lift the obtained bounds to the LOCAL model.
\begin{theorem}[\cite{Brandt2019}, rephrased]\label{thm:sebastien}
Let $T > 0$. Consider a class $\mathcal G$ of graphs\footnote{Technically, the class of graphs has to satisfy a certain property, called $t$-independence in \cite{Brandt2019}, but since it is straightforward to check that our considered class of $\Delta$-regular high-girth graphs satisfies this property, we omit this detail.} with girth at least $2 T+2$, and some locally checkable problem $\Pi$.
Then, there exists an algorithm that solves problem $\Pi$ on $\mathcal G$ in $T$ rounds if and only if there exists an algorithm that solves problem $\rere(\re(\Pi))$ in $T-1$ rounds.
\end{theorem}
In more technical detail, for any pair $(n, \Delta)$, Theorem~\ref{thm:sebastien} holds for graph classes $\mathcal G = \mathcal G(n, \Delta)$ consisting of $n$-node graphs with maximum degree $\Delta$ and girth at least $T = T(n, \Delta) > 0$.
However, for simplicity, we will usually omit the dependency on $n$ and $\Delta$.
We note that Theorem~\ref{thm:sebastien} also holds if we add a proper input vertex coloring to the setting.
Moreover, we will assume that the input graphs satisfy the given girth requirement whenever we apply Theorem~\ref{thm:sebastien}.
In Section~\ref{sec:liftlocal}, we will see how this requirement affects the obtained bounds.
An interesting fact that we have not seen mentioned in \cite{Brandt2019} (or any other work) is that the equivalence breaks only \emph{in one direction} when we go from high-girth graphs to general graphs:
it is straightforward to go through the proof of \cite[Theorem 4.3]{Brandt2019} and check that even on general graphs, $\Pi$ can be solved in $1$ round given a solution to $\rere(\re(\Pi))$.
In other words, $\rere(\re(\Pi))$ is \emph{at most} one round faster solvable than $\Pi$.
Hence, any upper bound achieved via automatic round elimination holds on general graphs, both in the port numbering model and the LOCAL model (as the latter is a stronger model).
In particular, this is true for our upper bounds for ruling sets.
\paragraph{Example: sinkless orientation.}
Let $\Pi$ be the sinkless orientation problem, where the goal is to consistently orient edges such that no node is a sink. In this example, we will see how to encode sinkless orientation in the round elimination framework, and we will see what the problems $\re(\Pi)$ and $\rere(\re(\Pi))$ look like.
The sinkless orientation problem can be encoded using two labels. So, let the set of labels be $\Sigma_\Pi=\{\mathsf{I},\O\}$. If a node outputs label $\mathsf{I}$ in one the endpoint of one of the incident edges, it can be interpreted as that edge being incoming. Similarly, if the label is $\O$, that would indicate an outgoing edge. Hence, on the node side, we want that each node has the label $\O$ on at least one of its incident edges. On the edge side, we want each edge to be consistently oriented, hence if in one endpoint it has the label $\mathsf{I}$, in the other endpoint there must be the label $\O$, and vice versa. More precisely, our problem $\Pi$ is the following.
\begin{equation*}
\begin{aligned}
\ensuremath{\mathcal{N}}_\Pi&\text{:}\quad \O\mspace{1mu} [\mathsf{I}\O]^{\Delta-1} \\
\ensuremath{\mathcal{E}}_\Pi&\text{:}\quad \mathsf{I}\mspace{1mu}\O
\end{aligned}
\end{equation*}
Let $\Pi'=\re(\Pi)$. By definition, $\Sigma_{\Pi'}=2^{\Sigma_{\Pi}}=\{\{\mathsf{I}\}, \{\O\}, \{\mathsf{I},\O\} \}$. Next we should define the edge constraint, where we want all configurations of the form $\S_1\mspace{1mu} \S_2$ such that, for any choice in $\S_1$ and for any choice in $\S_2$ we obtain a configuration in $\ensuremath{\mathcal{E}}_{\Pi}$. Also, we want to eliminate all non-maximal configurations. Before going to that, for simplicity of the presentation, in order to avoid writing set of sets, let us rename the labels of $\Sigma_{\Pi'}$ in the following way: $\{\mathsf{I}\} \mapsto \mybox{\I}$, $\{\O\} \mapsto \mybox{\O}$, and $\{\mathsf{I},\O\} \mapsto \mybox{\I\O}$. Now we can define the edge constraint. We must satisfy the universal quantification specified in the definition of $\ensuremath{\mathcal{E}}_{\Pi'}$, which means that we must forbid configurations that may result in $\mathsf{I}\mspace{1mu}\mathsf{I}$ or $\O\mspace{1mu}\O$, hence $\ensuremath{\mathcal{E}}_{\Pi'}: \mybox{\I}\mspace{1mu}\mybox{\O}$. The node constrant must satisfy an existential and all configurations must not use labels that do not appear in $\ensuremath{\mathcal{E}}_{\Pi'}$. In other words, we want to be able to pick at least one $\O$, hence something like $[\mybox{\O}\mybox{\I\O}]\mspace{1mu}[\mybox{\I}\mybox{\O}\mybox{\I\O}]^{\Delta-1}$ would do, but since $\mybox{\I\O}$ does not appear in $\ensuremath{\mathcal{E}}_{\Pi'}$, we get $\ensuremath{\mathcal{N}}_{\Pi'}: \mybox{\O}\mspace{1mu}[\mybox{\I}\mybox{\O}]^{\Delta-1}$. So, problem $\Pi'=\re(\Pi)$ is the following.
\begin{equation*}
\begin{aligned}
\ensuremath{\mathcal{E}}_{\Pi'}&\text{:}\quad \mybox{\I}\mspace{1mu}\mybox{\O} \\
\ensuremath{\mathcal{N}}_{\Pi'}&\text{:}\quad \mybox{\O}\mspace{1mu}[\mybox{\I}\mybox{\O}]^{\Delta-1}
\end{aligned}
\end{equation*}
Now we are ready to define problem $\Pi''=\rere(\re(\Pi))$ which, by Theorem \ref{thm:sebastien}, we know that is exactly one round faster than the sinkless orientation problem. By definition $\Sigma_{\Pi''}=2^{\Sigma_{\Pi'}}=\{\{\mybox{\I}\}, \{\mybox{\O}\}, \{\mybox{\I},\mybox{\O}\}\}$. Again, in order to avoid writing set of sets, let us rename the labels of $\Sigma_{\Pi''}$ as follows: $\{\mybox{\O}\} \mapsto \O$, $\{\mybox{\I}\} \mapsto \mathsf{I}'$ (as in ``the non-maximal set that contains the $\mybox{\I}$ label''), and $\{\mybox{\I},\mybox{\O}\} \mapsto \mathsf{I}$. We must first define the node constraint, that must satisfy a universal quantifier. We want to avoid that there is the label $\mybox{\I}$ in each of the $\Delta$ positions, since in that case the configuration $\mybox{\I}^\Delta$ would be possible, but it is not allowed in $\ensuremath{\mathcal{N}}_{\Pi'}$. The configuration that satisfies this condition is $\O\mspace{1mu}[\mathsf{I}\I'\O]^{\Delta -1}$, and after removing the non-maximal sets, we have $\ensuremath{\mathcal{N}}_{\Pi''}: \O\mspace{1mu}\mathsf{I}^{\Delta -1}$. For the edge constraint we must satisfy an existential, hence on one side we can have all labels that contain $\mybox{\I}$, while on the other all labels that contain $\mybox{\O}$. The configuration that satisfies this is $[\mathsf{I}\O][\mathsf{I}'\mathsf{I}]$, but since $\mathsf{I}'$ is not used in the set $\ensuremath{\mathcal{N}}_{\Pi''}$, we have that $\ensuremath{\mathcal{E}}_{\Pi''}: \mathsf{I}[\mathsf{I}\O]$. Hence, the problem $\Pi''=\rere(\re(\Pi))$ that is exactly one round faster that the sinkless orientation one is:
\begin{equation}
\begin{aligned}
\ensuremath{\mathcal{N}}_{\Pi''}&\text{:}\quad \O\mspace{1mu}\mathsf{I}^{\Delta -1} \\
\ensuremath{\mathcal{E}}_{\Pi''}&\text{:}\quad \mathsf{I}\mspace{1mu}[\mathsf{I}\O]
\end{aligned}
\end{equation}
\paragraph{Relations between labels.}
For computing $\re(\Pi)$ or $\rere(\Pi)$, given some problem $\Pi$, it will be very useful to relate the labels used in $\Pi$ to each other according to their ``usefulness" in satisfying the edge constraint $\ensuremath{\mathcal{E}}_\Pi$ (resp.\ the node constraint $\ensuremath{\mathcal{N}}_{\Pi}$).
Let $\mathsf{A}$ and $\mathsf{B}$ be labels from $\Sigma_\Pi$ with the following property: for each edge configuration in $\ensuremath{\mathcal{E}}_\Pi$ containing $\mathsf{A}$, replacing one occurrence of $\mathsf{A}$ in that configuration by $\mathsf{B}$ again results in a configuration in $\ensuremath{\mathcal{E}}_\Pi$.
Then we say that $\mathsf{B}$ is \emph{at least as strong as} $\mathsf{A}$ \emph{according to $\ensuremath{\mathcal{E}}_\Pi$} and, equivalently that $\mathsf{A}$ is \emph{at least as weak as} $\mathsf{B}$ according to $\ensuremath{\mathcal{E}}_\Pi$.
We may omit the reference constraint if it is clear from context.
Moreover, if $\mathsf{B}$ is at least as strong as $\mathsf{A}$, but $\mathsf{A}$ is not at least as strong as $\mathsf{B}$, we say that $\mathsf{B}$ is \emph{stronger than} $\mathsf{A}$, and $\mathsf{A}$ is \emph{weaker than} $\mathsf{B}$. For example, consider the aforementioned problem $\Pi''=\rere(\re(\Pi))$ where $\Pi$ is sinkless orientation. Recall that the edge constraints are $\mathsf{I}[\mathsf{I}\O]$. We can say that label $\mathsf{I}$ is stronger than label $\O$ (and equivalently, label $\O$ is weaker than label $\mathsf{I}$), since, for each edge configuration, replacing one occurrence of $\O$ with $\mathsf{I}$ results in a configuration that is still allowed.
We also define the analogous notions for node constraints.
It is helpful to illustrate the strengths of labels via diagrams.
The \emph{edge diagram} of a problem $\Pi$ is a directed graph where the nodes are the labels in $\Sigma_\Pi$ and we have an edge from some label $\mathsf{A}$ to some label $\mathsf{B}$ if $\mathsf{B} \neq \mathsf{A}$, $\mathsf{B}$ is at least as strong as $\mathsf{A}$, and there exists no label $\mathsf{Z} \in \Sigma_\Pi$ such that $\mathsf{B}$ is stronger than $\mathsf{Z}$ and $\mathsf{Z}$ is stronger than $\mathsf{A}$, all according to $\ensuremath{\mathcal{E}}_\Pi$.
The latter condition simply ensures that we only illustrate ``irreducible" strength relations, i.e., none that can be decomposed into ``smaller" strength relations.
We define the \emph{node diagram} of a problem $\Pi$ analogously, by considering the strengths of labels according to $\ensuremath{\mathcal{N}}_\Pi$.
Note that the definition of strength implies that the diagrams do not contain directed cycles of length greater than $2$, and cycles of length $2$ appear exactly between all pairs of labels of equal strength.
In particular, if there are no pairs $(\mathsf{A}, \mathsf{B})$ of labels such that $\mathsf{A}$ is stronger than $\mathsf{B}$ and vice versa, the respective diagram will be a directed acyclic (not necessarily connected) graph.
\paragraph{Additional Notation.}
While the above notions are already known from \cite{Brandt2019, Balliu2019, trulytight}, we now introduce some useful new notation.
For a set $\{ \mathsf{A}_1, \dots, \mathsf{A}_p \} \subseteq \Sigma_\Pi$ of labels, we denote by $\gen{\mathsf{A}_1, \dots, \mathsf{A}_p}$ the set of all labels from $\Sigma_\Pi$ that are at least as strong as at least one of the $\mathsf{A}_i$.
In other words, we can read $\gen{\mathsf{A}_1, \dots, \mathsf{A}_p}$ off of the respective diagram by collecting each $\mathsf{A}_i$ together with all its successors.
Technically, for the definition of $\gen{}$, we need to specify whether the label strengths are considered w.r.t.\ $\ensuremath{\mathcal{N}}_\Pi$ or $\ensuremath{\mathcal{E}}_\Pi$.
However, whenever we consider $\gen{}$, the labels that $\gen{}$ takes as arguments will either come from (the alphabet of) a problem that we (are about to) apply the function $\re(\cdot)$ to, or a problem that we apply the function $\rere(\cdot)$ to.
In the former case, we will always consider $\gen{}$ w.r.t.\ the \emph{edge} constraint of the considered problem, and in the latter case w.r.t\ the \emph{node} constraint.
In particular, in the context of computing $\rere(\re(\Pi))$ for some problem $\Pi$, we will consider expressions such as $\gen{\gen{\mathsf{A}}}$ (where $\mathsf{A} \in \Sigma_\Pi$), which represents a set of sets of labels from $\Sigma_\Pi$; here the inner $\gen{}$ is taken w.r.t.\ the edge constraint of $\Pi$, and the outer $\gen{}$ w.r.t.\ the node constraint of $\re(\Pi)$.
\paragraph{\boldmath Example.} Let $\Pi$ be the maximal independent set problem, which we can express in the round elimination formalism as follows.
\begin{equation*}
\begin{aligned}
\ensuremath{\mathcal{N}}_\Pi&\text{:}\quad \mathsf{M}^{\Delta} \\
&\text{ }\quad \P\mspace{1mu}\U^{\Delta-1}\\
\ensuremath{\mathcal{E}}_\Pi&\text{:}\quad \mathsf{M}\mspace{1mu}[\P\U]\\
&\text{ }\quad \U\mspace{1mu}\U
\end{aligned}
\end{equation*}
A node in the MIS outputs $\mathsf{M}^{\Delta}$. Otherwise, if a node is not in the MIS it must output $\P$ on one incident edge and $\U$ on all the others. The edge constraint implies that a node in the MIS can accept a pointer label $\P$ or label $\U$. Also, since $\P$ is only compatible with $\mathsf{M}$, each node not in the MIS can use label $\P$ only for pointing to a neighbor in the MIS. Moreover $\U\mspace{1mu}\U$ is allowed since two nodes not in the MIS may be neighbors. The relations between the strengths of the labels in $\Sigma_\Pi$ is shown in the edge diagram of $\Pi$, given in Figure \ref{fig:mis}. Expressions in $\gen{}$ notation can be easily read from the edge diagram; for instance, we have $\gen{\mathsf{M}} = \{\mathsf{M}\}$, $\gen{\P} = \{\P, \U\}$, $\gen{\U} = \{\U\}$ $\gen{\mathsf{M},\P} = \{\mathsf{M}, \P, \U\}$.
\begin{figure}[h]
\centering
\includegraphics[width=0.17\textwidth]{figs/mis}
\caption{Relations between the strengths of the labels of the MIS problem: label $\U$ is stronger than $\P$, while there is no relation between label $\mathsf{M}$ and labels $\P$ or $\U$.}
\label{fig:mis}
\end{figure}
Now, let $\Pi' = \re(\Pi)$, and consider the following mapping: $\{\U\} \mapsto \mybox{\U}$, $\{\mathsf{M}\} \mapsto \mybox{\M}$, $\{\mathsf{M},\U\} \mapsto \mybox{\M\U}$, $\{\P,\U\} \mapsto \mybox{\P\U}$. The edge and node constraint of $\Pi'$ are as follows.
\begin{equation*}
\begin{aligned}
\ensuremath{\mathcal{E}}_{\Pi'}&\text{:}\quad \mybox{\U}\mspace{1mu}\mybox{\M\U} \\
&\text{ }\quad \mybox{\M}\mspace{1mu}\mybox{\P\U}\\
\ensuremath{\mathcal{N}}_{\Pi'}&\text{:}\quad [\mybox{\M}\mybox{\M\U}]^\Delta\\
&\text{ }\quad \mybox{\P\U}\mspace{1mu}[\mybox{\U}\mybox{\M\U}\mybox{\P\U}]^{\Delta-1}
\end{aligned}
\end{equation*}
The node diagram of $\Pi'$, representing the relations of the strengths of the labels in $\Sigma_{\Pi'}$, is depicted in Figure \ref{fig:Re-mis}. Regarding the $\gen\gen{}$ notation, we obtain, for instance, that $\gen{\gen{\mathsf{M}}} = \gen{\mybox{\M}} = \{\mybox{\M},\mybox{\M\U}\} = \{\{\mathsf{M}\},\{\mathsf{M},\U\}\}$, $\gen{\gen{\U}} = \gen{\mybox{\U}} = \{\mybox{\U},\mybox{\M\U},\mybox{\P\U}\} = \{\{\U\},\{\mathsf{M},\U\}, \{\P,\U\}\}$, $\gen{\gen{\mathsf{M},\U}} = \gen{\mybox{\M\U}} = \{\mybox{\M\U}\} = \{\{\mathsf{M},\U\}\}$, $\gen{\gen{\P}} = \gen{\mybox{\P\U}} = \{\mybox{\P\U}\} = \{\{\P,\U\}\}$.
\begin{figure}[h]
\centering
\includegraphics[width=0.17\textwidth]{figs/Remis}
\caption{Relations between the strengths of the labels of problem $\re(\Pi)$, where $\Pi$ is the MIS problem; the diagram shows that label $\mybox{\M\U}$ is stronger than both labels $\mybox{\M}$ and $\mybox{\U}$, also label $\mybox{\P\U}$ is stronger than label $\mybox{\U}$.}
\label{fig:Re-mis}
\end{figure}
We call a set $S = \{ \mathsf{A}_1, \dots, \mathsf{A}_p \} \subseteq \Sigma_\Pi$ \emph{right-closed} if $S = \gen{\mathsf{A}_1, \dots, \mathsf{A}_p}$.
In other words, $S$ is right-closed if and only if for each label $\mathsf{A}_i$ contained in $S$ also all successors of $\mathsf{A}_i$ in the respective diagram are contained in $S$.
The definitions of $\re(\cdot)$ and $\rere(\cdot)$, in particular the removal of non-maximal configurations in the definitions, imply the following observation.
\begin{observation}\label{obs:rcs}
Consider an arbitrary collection of labels $\mathsf{A}_1, \dots, \mathsf{A}_p \in \Sigma_\Pi$.
If $\{ \mathsf{A}_1, \dots, \mathsf{A}_p \} \in \Sigma_{\re(\Pi)}$, then the set $\{ \mathsf{A}_1, \dots, \mathsf{A}_p \}$ is right-closed (w.r.t.\ $\ensuremath{\mathcal{E}}_\Pi$).
If $\{ \mathsf{A}_1, \dots, \mathsf{A}_p \} \in \Sigma_{\rere(\Pi)}$, then the set $\{ \mathsf{A}_1, \dots, \mathsf{A}_p \}$ is right-closed (w.r.t.\ $\ensuremath{\mathcal{N}}_\Pi$).
\end{observation}
\begin{proof}
For reasons of symmetry, we only need to prove the first statement.
Let $\S = \{ \mathsf{A}_1, \dots, \mathsf{A}_p \}$ and assume that $\S \in \Sigma_{\re(\Pi)}$.
Then there must be an edge configuration in $\ensuremath{\mathcal{E}}_{\re(\Pi)}$ containing $\S$, by the definition of $\re(\cdot)$.
Consider an arbitrary label $\mathsf{B} \in \Sigma_\Pi$ that is at least as strong as at least one $\mathsf{A}_i$ w.r.t.\ $\ensuremath{\mathcal{E}}_\Pi$.
By the definition of strength, and the definition of $\ensuremath{\mathcal{E}}_{\re(\Pi)}$ (or $\ensuremath{\mathcal{E}}_{\Pi'}$), adding label $\mathsf{B}$ to set $\S$ in the considered edge configuration results in a configuration that is still contained in $\ensuremath{\mathcal{E}}_{\re(\Pi)}$.
Since $\ensuremath{\mathcal{E}}_{\re(\Pi)}$ does not contain any non-maximal configurations, this implies that $\mathsf{B}$ was already contained in $\S$, i.e., $\mathsf{B} = \mathsf{A}_i$ for some $i$.
It follows that $\S$ is right-closed (w.r.t.\ $\ensuremath{\mathcal{E}}_\Pi$).
\end{proof}
Observation~\ref{obs:rcs} enables us to prove the following observation.
\begin{observation}\label{obs:subsetarrow}
Let $\U, \mathsf{W} \in \Sigma_{\re(\Pi)}$ be two sets satisfying $\U \subseteq \mathsf{W}$.
Then $\mathsf{W}$ is at least as strong as $\U$ according to $\ensuremath{\mathcal{N}}_{\re(\Pi)}$.
In particular, for any label $\mathsf{A} \in \Sigma_\Pi$ such that $\gen{\mathsf{A}} \in \Sigma_{\re(\Pi)}$, every set $\mathsf{X} \in \Sigma_{\re(\Pi)}$ containing $\mathsf{A}$ is contained in $\gen{\gen{\mathsf{A}}}$.
Analogous statements hold for $\rere(\cdot)$ instead of $\re(\cdot)$.
\end{observation}
\begin{proof}
For reasons of symmetry, we only need to prove the statements for $\re(\cdot)$.
The definition of $\re(\cdot)$ immediately implies that replacing $\U$ by $\mathsf{W}$ in any configuration contained in $\ensuremath{\mathcal{N}}_{\re(\Pi)}$ results in a configuration that is also contained in $\ensuremath{\mathcal{N}}_{\re(\Pi)}$.
Hence, $\mathsf{W}$ is at least as strong as $\U$ according to $\ensuremath{\mathcal{N}}_{\re(\Pi)}$.
Now, let $\mathsf{A}$ and $\mathsf{X}$ be as described in the lemma.
By Observation~\ref{obs:rcs}, the set $\mathsf{X}$ is right-closed w.r.t.\ $\ensuremath{\mathcal{E}}_{\Pi}$, which, by the definition of $\gen{}$, implies $\gen{\mathsf{A}} \subseteq \mathsf{X}$, since $\mathsf{X}$ contains $\mathsf{A}$.
It follows that $\mathsf{X}$ is at least as strong as $\gen{\mathsf{A}}$ according to $\ensuremath{\mathcal{N}}_{\re(\Pi)}$, by the first part of Observation~\ref{obs:subsetarrow}.
Hence, $\mathsf{X} \in \gen{\gen{\mathsf{A}}}$.
\end{proof}
Moreover, for a set $S = \{ \mathsf{A}_1, \dots, \mathsf{A}_p \} \subseteq \Sigma_\Pi$ of labels, we denote by $\ensuremath{\operatorname{disj}}(S)$ the disjunction $[\mathsf{A}_1 \dots \mathsf{A}_p]$.
For instance, $\ensuremath{\operatorname{disj}}(\gen{\mathsf{A}})$ is the disjunction of all labels that are at least as strong as $\mathsf{A}$.
\paragraph{Generalizing to non-regular graphs.}
As mentioned before, in this paper we will restrict attention to regular graphs.
Since we are proving lower bounds, this does not affect the generality of our results; however, for the upper bound we prove along the way, some additional step is required to lift the bound to general graphs.
In its full generality, the round elimination framework can also be applied to non-regular graphs, and the arguments in our upper bound would essentially remain the same; however, describing the framework formally is somewhat cumbersome.
Hence, we will choose a different route to show that our upper bound holds on general graphs: we will present a ``human-understandable" version of the algorithm obtained by round elimination for which it will be easy to check that its correctness is not affected by having nodes of different degrees.
\subsection{Roadmap}
We will start, in Section \ref{sec:problems}, by defining a family of problems $\Pi_{\Delta,\beta}(v,x)$, for which we will later show how it relates to the $(2,\beta)$-ruling set problem. The parameter $v = [v_0, \ldots, v_\beta]$ is a list of non-negative numbers, that can be interpreted as a number of colors. Intuitively, the problem $\Pi_{\Delta,\beta}(v,x)$ can be solved in $0$ rounds if we are given some vertex coloring with $\size(v) \coloneqq \sum_{i=0}^{\beta} v_i$ colors. The parameter $x$ is some relaxation parameter: we will allow nodes to violate edge constraints on at most $x$ of their incident edges.
In Sections \ref{sec:firstspeedup}, \ref{sec:ub}, and \ref{sec:lb}, we will use the round elimination theorem to relate problems of this family. In Section \ref{sec:firstspeedup}, we will compute the problem that we obtain by applying our operator $\re(\cdot)$ to $\Pi_{\Delta,\beta}(v,x)$.
In Section \ref{sec:ub} we will prove upper bounds for the $(2,\beta)$-ruling set problem. We will consider a subset of the problems of the family, that is, those where parameter $x$ is set to be $0$. We will first show that $\rere(\Pi'_{\Delta,\beta}(v,0))$ is at least as easy as some other problem of the family, that is $\Pi_{\Delta,\beta}(v',0)$, where $v'$ is the inclusive prefix sum of $v$ (i.e., $v'_i = \sum_{j\le i} v_j$). The round elimination theorem will imply that, given a solution for $\Pi_{\Delta,\beta}(v',0)$, we can obtain a solution for $\Pi_{\Delta,\beta}(v,0)$ in at most one round of communication. We will finally combine multiple steps of such reasoning to obtain upper bounds: we will show how parameter $v$ evolves over multiple steps. Crucially, a solution for $\Pi_{\Delta,\beta}([1,0,\ldots,0],0)$ will directly imply a solution for the $(2,\beta)$-ruling set problem, and by repeatedly applying the round elimination theorem we will obtain some problem $\Pi_{\Delta,\beta}(v',0)$ where $\size(v')$ is at least as large as the number of colors in the given vertex coloring. We will first prove an upper bound on the number of steps required to obtain such a problem, thereby giving an upper bound on the time complexity of the algorithm. Then, we will provide a human-understandable version of the round-elimination-generated algorithm, in order to argue that this algorithm does not only work on regular graphs, but on all graphs.
In Section \ref{sec:lb}, we will prove lower bounds for the $(2,\beta)$-ruling set problem. The main idea here will be to show that, by increasing parameter $x$, we can essentially relate the problems of the family in the same way as we do for the upper bounds. That is, we can get the same evolution of parameter $v$ as in the upper bound, at the price of increasing parameter $x$. Essentially, this will allow us to use the ideas obtained from the upper bound to get a lower bound. We will show in Section \ref{sec:liftlocal} how to lift the obtained lower bounds from the port numbering model to the LOCAL model.
\section{The problem family}\label{sec:problems}
\subsection{Problem definition}
In this section, we define a family of problems $\Pi_{\Delta,\beta}(v,x)$, that we will use to prove lower and upper bounds for the $(2,\beta)$-ruling set problem on graphs of maximum degree $\Delta$. The parameter $v = [v_0,\dotsc,v_\beta]$ is a list of non-negative integers, and the parameter $x$ satisfies $0\le x \le \Delta$ (while proving upper bounds, we will actually only consider the case where $x=0$). Intuitively, $v$ represents a list of color \emph{groups}, where each $v_i$ represents the number of colors in that group, while $x$ represents some relaxation parameter we will refer to as the number of wildcards. As we will see, if we start from a problem in this family, and we increase the value of $x$, or we increase the value of $v_i$ for some $i$, we will get a problem that is at least as easy as the one we started from. More precisely, given a solution for the starting problem, we can use it to solve the new problem in $0$ rounds of communication.
The high-level idea of the construction of the problem family is that we have \emph{colors} and \emph{pointers}, and nodes can either output a color (satisfying the usual constraints of the vertex coloring problem), or a pointer. Moreover, we have $\beta+1$ \emph{groups} called group $0$ to group $\beta$, and each color and each pointer belongs to exactly one of these groups. More precisely, there are exactly $v_i$ colors in group $i$, and there is exactly one pointer in each group except group $0$ (which contains no pointer). A pointer can only point to a node outputting a pointer, or a color, of a lower group. An example of a correct solution is given in Figure \ref{fig:problem-family}. Moreover, each node can label at most $x$ of its incident edges with a so-called \emph{wildcard}. If an edge is labeled with a wildcard by one of its endpoints, the resulting output label pair on the edge is correct by definition (i.e., it is an edge configuration listed in the edge constraint) regardless of the label the other endpoint outputs on the edge.
\begin{figure}[h]
\centering
\includegraphics[width=0.7\textwidth]{figs/problemfamily}
\caption{An example of a problem with parameters $[v_0, v_1, \dotsc, v_\beta]$ and $x=0$. Each node in the graph outputs either a color of some group, or a pointer pointing to a color or a pointer of a strictly smaller group. Neighboring nodes are not allowed to output the same color, but they can output colors belonging to the same group.}
\label{fig:problem-family}
\end{figure}
The $(2,\beta)$-ruling set problem is the special case where we allow only $1$ color, i.e., $v_0=1$ and $v_i=0$ for all $i>0$, $\beta$ pointers, and no wildcards, i.e., $x=0$. In fact, the nodes in the ruling set will be exactly the nodes that output the color (note that since the ruling set nodes form an independent set, the coloring constraints are satisfied), and we allow the other nodes to point using pointers of different groups, depending on the distance they have from a node in the ruling set. We will later show that, while a solution for $\Pi_{\Delta,\beta}([1,0,\ldots,0],0)$ can be converted in $0$ rounds to a solution for the $(2,\beta)$-ruling set problem, we may need up to $\beta$ rounds to do the converse (and we will have to take this in consideration later when determining the actual lower bounds).
We define $\size(v) = v_0 + \dots + v_\beta$.
If we increase the number of colors in $\Pi_{\Delta,\beta}(v,x)$, i.e., if we increase $\size(v)$, the problem becomes easier: once we reach, for example, the case where $\size(v) = \Omega(\Delta^2)$, we have a problem that can be solved in $O(\log^* n)$ rounds in the LOCAL model, since in this model a graph can be colored in $O(\log^* n)$ rounds with $\Delta^2$ colors \cite{Linial1992}. Also, by letting parameter $x$ grow we get an easier problem: in the extreme case of $x=\Delta$ we have a problem that is $0$-round solvable, since we can output wildcards everywhere.
\paragraph{Labels.}We now formally define the set $\Sigma_{\Delta,\beta}(v,x)$ of labels of the problem $\Pi_{\Delta,\beta}(v,x)$. Let $\Sigma_{\Delta,\beta}(v,x) = \mathcal{P} \cup \mathcal{C}\cup \mathcal{X}$, where
\begin{itemize}
\item $\mathcal{P}=\{ \mathsf{A}_i, \mathsf{B}_i ~|~ 1 \le i \le \beta \}$,
\item $\mathcal{C}=\{ \C_{i,j} ~|~ 0 \le i \le \beta, 1 \le j \le v_i \}$,
\item $\mathcal{X} = \{\mathsf{X}\}$ if $x>0$ and $\mathcal{X} = \{\}$ if $x=0$ (that is, if $x=0$ there is no label $\mathsf{X}$ in the set $\Sigma$).
\end{itemize}
These labels can be interpreted as follows:
\begin{itemize}
\item The label $\mathsf{X}$ is a wildcard. Nodes write it on an edge to mark that edge as \emph{``don't care''}.
\item The label $\mathsf{A}_i$ is a pointer, and the label $\mathsf{B}_i$ can be used to ``accept" pointers (of higher groups) that are output by neighboring nodes on connecting edges.
\item The label $\C_{i,j}$ is the $j$-th color of group $i$.
\end{itemize}
\paragraph{Node constraint.} We now define the node constraint $\ensuremath{\mathcal{N}}_{\Delta,\beta}(v,x)$, i.e., the set of allowed node configurations. The set $\ensuremath{\mathcal{N}}_{\Delta,\beta}(v,x)$ contains the following:
\begin{itemize}
\item $\c^{\Delta-x} \mspace{1mu} \mathsf{X}^x$, for each $\c \in \mathcal{C}$. That is, nodes output some color $\c \in \mathcal{C}$, marking $x$ incident edges as ``don't care''.
\item $\mathsf{A}_i \mspace{1mu} \mathsf{B}_i^{\Delta-1}$, for each $1 \le i \le \beta$. That is, nodes can output a pointer $\mathsf{A}_i$ on one incident edge. All other incident edges are marked as $\mathsf{B}_i$. We will see, when defining the edge constraint, that this will allow to accept pointers of higher groups. Intuitively, a node outputting this configuration must be at distance at most $i$ from a node outputting a color (of some group $< i$).
\end{itemize}
\paragraph{Edge constraint.} We now define the edge constraint $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$. It contains the following edge configurations:
\begin{itemize}
\item $\C_{i,j} \mspace{1mu} \C_{i',j'}$ if $(i,j) \neq(i',j')$, for each $1 \le i,i' \le \beta$, $1 \le j \le v_i$, $1 \le j' \le v_{i'}$. That is, all colors are compatible with all other colors (except themselves).
\item $\mathsf{B}_i \mspace{1mu} \mathsf{B}_j$, for each $1 \le i,j \le \beta$. That is, all $\mathsf{B}$ labels are compatible with all other $\mathsf{B}$ labels (including themselves).
\item $\mathsf{A}_j \mspace{1mu} \mathsf{B}_i$, for each $1 \le i < j \le \beta$. That is, pointers can point to non-colored nodes of lower groups.
\item $\mathsf{B}_i \mspace{1mu} \C_{i',j}$, for each $1 \le i,i' \le \beta$, $1 \le j \le v_{i'}$. That is, all $\mathsf{B}$ labels are compatible with all colors.
\item $\mathsf{A}_i \mspace{1mu} \C_{i',j}$, for each $1 \le i' < i \le \beta$, $1 \le j \le v_{i'}$. That is, pointers can point to colored nodes of lower groups.
\item $\mathsf{X} \mspace{1mu} \L$, for each $\L \in \Sigma$, if $x > 0$. That is, the wildcard $\mathsf{X}$ is compatible with all labels.
\end{itemize}
\subsection{From the problem family to ruling sets, and vice versa}\label{sec:equivalence}
We now discuss the relation between $\Pi_{\Delta,\beta}([1,0,\dotsc,0],0)$ and the $(2,\beta)$-ruling set problem. We argue that a solution for $\Pi_{\Delta,\beta}([1,0,\dotsc,0],0)$ can be turned in $0$ rounds into a solution for the $(2,\beta)$-ruling set problem, and that a solution for the $(2,\beta)$-ruling set problem can be turned in $\beta$ rounds into a solution for $\Pi_{\Delta,\beta}([1,0,\dotsc,0],0)$. In Section \ref{sec:lb} we will use this relation to transform a lower bound of $T$ rounds for $\Pi_{\Delta,\beta}([1,0,\dotsc,0],0)$ into a lower bound of $(T-\beta)$ rounds for the $(2,\beta)$-ruling set problem.
Let us start by showing how to turn a solution for $(2,\beta)$-ruling set into a solution for the $\Pi_{\Delta,\beta}([1,0,\dotsc,0],0)$ problem. Given a solution for the $(2,\beta)$-ruling set problem, proceed as follows. Nodes in the ruling set output $\C_{0,1}$ on each incident edge. Each node $v$ can find in $\beta$ rounds the closest node of the ruling set (breaking ties arbitrarily); let this distance be $d_v$, satisfying $1 \le d_v \le \beta$. Node $v$ outputs $\mathsf{A}_{d_v}$ on the incident edge contained in the shortest path to this closest ruling set node, and $\mathsf{B}_{d_v}$ on all the other incident edges. The node constraint of $\Pi_{\Delta,\beta}([1,0,\dotsc,0],0)$ is clearly satisfied. Moreover, by construction, no neighboring nodes are outputting $\C_{0,1}$ on the same edge, and since other nodes use their distance to the closest ruling set node to output pointers, also the edge constraint is satisfied.
Consider now a solution for the problem $\Pi_{\Delta,\beta}([1,0,\dotsc,0],0)$. Nodes are either labeled with the color $\C_{0,1}$, or with one of the $\beta$ configurations that contain a pointer. We put exactly the colored nodes in the ruling set. Since the configuration $\C_{0,1} \mspace{1mu} \C_{0,1}$ is not contained in the edge constraint, the colored nodes form an independent set. Also, since the constraints of $\Pi_{\Delta,\beta}([1,0,\dotsc,0],0)$ guarantee that each node that outputs $\mathsf{A}_i \mspace{1mu} \mathsf{B}_i^{\Delta-1}$ has a colored neighbor, or a neighbor that outputs $\mathsf{A}_j \mspace{1mu} \mathsf{B}_j^{\Delta-1}$ with $j < i$, nodes that are not in the independent set are at distance at most $\beta$ from a node in the independent set.
\subsection{The idea behind this problem family}
While the definition of the problem family $\Pi_{\Delta,\beta}(v,x)$ may seem arbitrary, we argue that there is a \emph{natural} way to obtain it, at least for the case $x=0$, that is the following:
\begin{itemize}
\item Start from a problem of the family (at the beginning, this means to start from the $(2,\beta)$-ruling set problem).
\item Apply the round elimination theorem.
\item Note that in the obtained problem there are some allowed configurations that directly correspond to the original $\mathsf{A}_i \mspace{1mu} \mathsf{B}_i^{\Delta-1}$ allowed configurations. Keep these configurations.
\item Note that in the obtained problem there are some allowed configurations that directly correspond to a coloring problem (configurations of the form $\C^\Delta$ such that label $\C$ is compatible with all the labels of the configurations of the same form, except itself). Keep these configurations.
\item Discard everything else.
\end{itemize}
Essentially what we need to do is to keep the part of the problem that has some \emph{intuitive} meaning (that is, colors and pointers), and discard everything else.
In the upper bound section, we will prove how the color groups evolve at each step. Intuitively, by applying the round elimination theorem to the problem $\Pi_{\Delta,\beta}(v,0)$ (and by discarding some allowed configurations, thus by making the problem harder), we obtain the problem $\Pi_{\Delta,\beta}(v',0)$, where $v'$ is the inclusive prefix sum list of $v$. For example, the $(2,2)$-ruling set problem is equivalent to $\Pi_{\Delta,2}([1,0,0],0)$, and by applying the round elimination theorem we get a problem that is not harder than $\Pi_{\Delta,2}([1,1,1],0)$, and by repeating the same procedure we get $\Pi_{\Delta,2}([1,2,3],0)$, and then we get $\Pi_{\Delta,2}([1,3,6],0)$, and so on. This gives a quadratic growth in the number of colors, and we thus get an algorithm that, given some $c$ coloring can solve the $(2,2)$-ruling set problem in $O(\sqrt{c})$ rounds. By generalizing the same reasoning to $(2,\beta)$-ruling sets, we get an algorithm that, given a $c$ coloring, solves the problem in $O(\beta c^{1/\beta})$ rounds, matching the current state-of-the-art algorithm w.r.t.\ dependency on $\Delta$, $c$ and $\beta$ \cite{SEW13}. While an algorithm obtained in the specific round elimination framework we use only works on regular graphs, we will show that the algorithm that we obtain actually works in any graph.
In the lower bound section, we will show that, by increasing parameter $x$ at each step, we can prove that the color groups evolve in the same way as in the upper bound, and that we can thus prove a \emph{lower bound} using a problem family suggested by the upper bound. In particular, we will show that all the non-intuitive allowed configurations can be relaxed to the intuitive ones, if we allow some slack on them.
\subsection{The edge diagram}\label{sec:edgediag}
We now show the structure of the edge diagram of our $\Pi_{\Delta,\beta}(v,x)$ problems. Knowing such structure will be helpful in the following sections. In particular, as previously discussed in Section \ref{sec:preliminaries}, when defining $\re(\Pi_{\Delta,\beta}(v,x))$ we will only have to consider right-closed subsets of labels with regard to this diagram (see Observation \ref{obs:rcs}). The following relations between the labels of $\Pi_{\Delta,\beta}(v,x)$ derive directly from the definition of $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$.
\begin{itemize}
\item $\mathsf{B}_i \le \mathsf{B}_j$, if $j < i$.
\item $\mathsf{A}_i \le \mathsf{A}_j$, if $i < j$.
\item $\mathsf{A}_i \le \C_{i',j}$, if $1 \le i \le i'$.
\item $\C_{i',j} \le \mathsf{B}_i$, if $1 \le i \le i'$.
\item $\L \le \mathsf{X}$ for all $\L \in \Sigma$.
\end{itemize}
Notice that this also implies $\mathsf{A}_i \le \mathsf{B}_j$ for all $1 \le i,j \le \beta$. An example of the diagram for $\Pi_{\Delta,3}([1,2,3,4],1)$ is shown in Figure \ref{fig:diag}.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figs/diagram}
\caption{Edge diagram of $\Pi_{\Delta,3}([1,2,3,4],1)$}
\label{fig:diag}
\end{figure}
\section{The intermediate problems}\label{sec:firstspeedup}
In Section \ref{sec:equivalence} we formally introduced a family of problems $\Pi_{\Delta,\beta}(v,x)$ by defining $\Sigma_{\Delta,\beta}(v,x)$, $\ensuremath{\mathcal{N}}_{\Delta,\beta}(v,x)$ and $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$. In this section, we compute $\re(\Pi_{\Delta,\beta}(v,x))$, i.e., we compute the family of problems that we get by applying the function $\re(\cdot)$ to $\Pi_{\Delta,\beta}(v,x)$. In other words, we will compute the set of labels $\Sigma'_{\Delta,\beta}(v,x)$, the node constraint $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$ and the edge constraint $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,x)$ of $\re(\Pi_{\Delta,\beta}(v,x))$. In this section, we will always assume that $x < \Delta$ (or, where indicated, even $x \leq \Delta - 2$).
\paragraph{Labels.} By the definition of $\re(\cdot)$, the set of labels of $\re(\Pi_{\Delta,\beta}(v,x))$ is the set of non-empty subsets of the set $\Sigma_{\Delta,\beta}(v,x)$, that is, $\Sigma'_{\Delta,\beta}(v,x)=2^{\Sigma_{\Delta,\beta}(v,x)}$. In other words, $\Sigma'_{\Delta,\beta}(v,x)=2^{ \mathcal{P}\mspace{1mu} \cup\mspace{1mu} \mathcal{C}\mspace{1mu}\cup\mspace{1mu} \mathcal{X}}$, where
\begin{itemize}
\item $\mathcal{P}=\{ \mathsf{A}_i, \mathsf{B}_i ~|~ 1 \le i \le \beta \}$ is the set of pointers,
\item $\mathcal{C}=\{ \C_{i,j} ~|~ 0 \le i \le \beta, 1 \le j \le v_i \}$ is the set of colors, and
\item $\mathcal{X} = \{\mathsf{X}\}$ if $x>0$, and $\mathcal{X} = \{\}$ if $x=0$, is the set of wildcards.
\end{itemize}
\subsection{Edge constraint}\label{sec:edgeconstraint}
Given a set of colors $\mathscr{C} \subseteq \mathcal{C}$, let $g(\mathscr{C})$ be the largest index $k$ such that $\C_{k,\ell} \in \mathscr{C}$ for some $\ell$ (if $\mathscr{C}$ is empty, let $g(\mathscr{C}) = -1$). In other words, $g(\mathscr{C})$ is the highest group of all colors contained in $\mathscr{C}$.
Consider all the possible pairs $(\mathscr{C},i)$ where $\mathscr{C} \subseteq \mathcal{C}$ and $0 \le i \le \beta$. A pair $(\mathscr{C},i)$ is \emph{good} if and only if $i \ge g(\mathscr{C})$. If $x=0$, we additionally require that, in order for a pair to be good, it must be different from $(\emptyset,0)$. Essentially, good pairs represent all ways to combine subsets of colors and group indices such that the index is at least as large as the highest color group appearing in the set. Let $\S_1((\mathscr{C},i)) = \mathscr{C} \cup \{\mathsf{B}_j ~|~ 1 \le j \le i\}\} \cup \mathcal{X}$. Let $\S_2((\mathscr{C},i)) = (\mathcal{C} \setminus \mathscr{C}) \cup \{\mathsf{B}_j ~|~ 1 \le j \le \beta\} \cup \{\mathsf{A}_j ~|~ i < j \le \beta\} \cup \mathcal{X}$.
\begin{lemma}\label{lemma:edgeconst'}
The edge constraint of $\re(\Pi_{\Delta,\beta}(v,x))$ is \[\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,x) = \{ \S_1((\mathscr{C},i)) \mspace{1mu} \S_2((\mathscr{C},i)) ~|~ (\mathscr{C},i) \text{ is a good pair}\}.\]
\end{lemma}
\begin{proof}
Let us start with some observations. First of all, recall that the $\re(\cdot)$ operator requires $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,x)$ to contain all (and only) pairs of sets $\S_1 \mspace{1mu} \S_2$ that satisfy that for all $\mathsf{s}_1 \in \S_1$ and for all $\mathsf{s}_2 \in \S_2$, $\mathsf{s}_1 \mspace{1mu} \mathsf{s}_2$ is in $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$. Also, recall that we can discard all \emph{non-maximal} pairs, and that pairs are equivalent up to reordering. Moreover, recall that we do not need to consider all possible subsets in $ \Sigma'_{\Delta,\beta}(v,x)$, but only \emph{right-closed subsets} with respect to the edge diagram of $\Pi_{\Delta,\beta}(v,x)$ (see Observation \ref{obs:rcs}). Essentially, in $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,x)$ we must have all possible pairs $\S_1 \mspace{1mu} \S_2$, where $\S_1$ is a right-closed subset, and $\S_2$ is the intersection of all sets of labels compatible with each $\mathsf{s}_1 \in \S_1$ (note that also the resulting set $\S_2$ must be right-closed).
We consider all cases where $\S_1$ does not contain any label $\mathsf{A}_i$. Since no $\mathsf{A}$-type label is compatible (with regards to $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$) with any other $\mathsf{A}$-type label, it is not possible to have some configuration $\S_1 \mspace{1mu} \S_2$ that contains $\mathsf{A}_i \in \S_1$ and $\mathsf{A}_j \in \S_2$, for any $i$ and $j$. Thus, for all valid configurations, either $\S_1$ or $\S_2$ does not contain any $\mathsf{A}$-type label, and this implies that by only considering the case where $\S_1$ does not contain any $\mathsf{A}$-type label, we cover all cases (up to symmetry). By the definition of the edge diagram of $\Pi_{\Delta,\beta}(v,x)$, right-closed subsets $\S$ of $\Sigma'_{\Delta,\beta}(v,x)$ that do not contain any label $\mathsf{A}_i$ are of the following form: we have a subset of colors, and if a color of group $i$ is in $\S$, all $\mathsf{B}_j$ satisfying $j \le i$ are also present. Also, additional $\mathsf{B}_j$ may be in $\S$, and if $\mathsf{B}_j$ is present, all $\mathsf{B}_{j'}$ satisfying $j' \le j$ must also be there. Finally, the label $\mathsf{X}$ is also present, if $x\neq 0$. Notice that there is a one-to-one correspondence between all good pairs and all right-closed subsets not containing any label $\mathsf{A}_i$ (the case distinction on the value of $x$ ensures that we are not considering the empty set). In fact, since $i \ge g(\mathscr{C})$, when creating a set we put at least all $\mathsf{B}$-type labels with index between $1$ and the maximum color group appearing in $\mathscr{C}$, and by increasing $i$ we put additional $\mathsf{B}$-type labels.
For each good pair $(\mathscr{C},i)$, we add the configuration $\S_1((\mathscr{C},i)) \mspace{1mu} \S_2((\mathscr{C},i))$ to $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,x)$. We need to prove that $\S_2 \coloneqq \S_2((\mathscr{C},i))$ contains all and only the labels that are edge compatible with all the labels in $\S_1 \coloneqq \S_1((\mathscr{C},i))$. First, note that a color cannot appear in both $\S_1$ and $\S_2$, since a color is not compatible with itself in $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$. Hence, since $\S_2$ contains $\mathcal{C} \setminus \mathscr{C}$, colors added to $\S_2$ are all valid, and no color can be added. Then, all $\mathsf{B}_i$ are present in $\S_2$, thus, trivially, we cannot add more $\mathsf{B}$-type labels to $\S_2$, and since each $\mathsf{B}$-type label is edge compatible with all other $\mathsf{B}$-type labels and with all colors, the configurations in $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$ are not violated. The same holds for the label $\mathsf{X}$, that we add to $\S_2$ if present in $\Sigma'_{\Delta,\beta}(v,x)$. Note that $\mathsf{X}$ is compatible with any label, so the configurations in $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$ are trivially not violated. The last remaining case to analyze is the $\mathsf{A}$-type labels: if labels $\{\mathsf{B}_1, \ldots, \mathsf{B}_i\}$ are present in $\S_1$ we added $\{\mathsf{A}_{i+1},\ldots,\mathsf{A}_\beta\}$ to $\S_2$. Since $\mathsf{A}_i$ is not compatible with $\mathsf{B}_i$ with regard to $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$, this implies that we cannot add more $\mathsf{A}$-type labels to $\S_2$. Also, note that the presence of a color of group $j$ in $\S_1$ implies that $i \ge j$, and thus the presence of $\mathsf{B}_j$, and since $\mathsf{A}_i$ is edge compatible with all colors of groups strictly less than $i$, the $\mathsf{A}$-type labels added to $\S_2$ do not violate the $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$ configurations.
\end{proof}
\subsection{Properties}
Before computing the node constraint of $\re(\Pi_{\Delta, \beta}(v,x))$ in Section~\ref{sec:noco}, we will first collect two facts about problem $\re( \Pi_{\Delta,\beta}(v,x) )$ that we can derive from the description of the edge constraint in Section~\ref{sec:edgeconstraint} and will be useful later.
\begin{lemma}\label{lem:restrong}
Consider two sets $\U, \mathsf{W} \in \Sigma'_{\Delta,\beta}(v,x)$, and assume that $x + 2 \leq \Delta$.
Then $\mathsf{W}$ is at least as strong as $\U$ according to $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$ if and only if $\U \subseteq \mathsf{W}$.
\end{lemma}
\begin{proof}
If $\U \subseteq \mathsf{W}$, then $\mathsf{W}$ is at least as strong as $\U$ according to $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$, by Observation~\ref{obs:subsetarrow}.
For the other direction, assume that $\U \nsubseteq \mathsf{W}$, and let $\u \in \Sigma_{\Delta,\beta}(v,x)$ be a label contained in $\U \setminus \mathsf{W}$.
We want to show that $\mathsf{W}$ is not at least as strong as $\U$ according to $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$.
For a contradiction assume that $\mathsf{W}$ is at least as strong as $\U$.
Consider some configuration $\u \mspace{1mu} \mathsf{y}_2 \mspace{1mu} \mathsf{y}_3 \mspace{1mu} \dots \mspace{1mu} \mathsf{y}_\Delta \in \ensuremath{\mathcal{N}}_{\Delta,\beta}(v,x)$.
We first show that the configuration $\U \mspace{1mu} \gen{\mathsf{y}_2} \mspace{1mu} \gen{\mathsf{y}_3} \mspace{1mu} \dots \mspace{1mu} \gen{\mathsf{y}_\Delta}$ is contained in $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$.
Recalling the definition of $\re(\cdot)$, we see that, since $\mathsf{y}_j \in \gen{\mathsf{y}_j}$, for all $2 \leq j \leq \Delta$, and $\u \in \U$, the only case in which the configuration might not be contained in $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$ is that one of the labels in the configuration is not contained in any configuration in $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,x)$.
Hence, for our first step it suffices to show that each of the $\gen{\mathsf{y}_j}$, and also $U$, is contained in some configuration in $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,x)$.
As $\U \in \Sigma'_{\Delta,\beta}(v,x)$ by definition, $\U$ is contained in such a configuration.
The analogous statement for the $\gen{\mathsf{y}_j}$ follows from Lemma~\ref{lemma:edgeconst'} and the fact that for any label $\L \in \Sigma_{\Delta,\beta}(v,x)$, there exists some good pair $(\mathscr{C},i)$ such that $\gen{\L} = \S_1((\mathscr{C},i))$ or $\gen{\L} = \S_2((\mathscr{C},i))$.
To see the latter, observe that
\begin{align*}
\gen{\mathsf{A}_i} &= \S_2((\mathscr{C},i-1)) &\text{ where $\mathscr{C} = \{ \C_{k,j} \mid 0 \leq k \leq i-1, 1 \leq j \leq v_k\}$,}\\
\gen{\mathsf{B}_i} &= \S_1((\mathscr{C},i)) &\text{ where $\mathscr{C} = \emptyset$,}\\
\gen{\C_{i,j}} &= \S_1((\mathscr{C},i)) &\text{ where $\mathscr{C} = \{ \C_{i,j} \}$, and}\\
\gen{\mathsf{X}} &= \S_1((\emptyset,0)) &\text{ if $x > 0$.}
\end{align*}
It follows that the configuration $\U \mspace{1mu} \gen{\mathsf{y}_2} \mspace{1mu} \gen{\mathsf{y}_3} \mspace{1mu} \dots \mspace{1mu} \gen{\mathsf{y}_\Delta}$ is contained in $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$.
Since $\mathsf{W}$ is at least as strong as $\U$, we obtain that also $\mathsf{W} \mspace{1mu} \gen{\mathsf{y}_2} \mspace{1mu} \gen{\mathsf{y}_3} \mspace{1mu} \dots \mspace{1mu} \gen{\mathsf{y}_\Delta} \in \ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$.
By the definition of $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$, there is a configuration $\mathsf{w} \mspace{1mu} \mathsf{y}'_2 \mspace{1mu} \mathsf{y}'_3 \mspace{1mu} \dots \mspace{1mu} \mathsf{y}'_\Delta \in \ensuremath{\mathcal{N}}_{\Delta,\beta}(v,x)$ such that $(\mathsf{w}, \mathsf{y}'_2, \mathsf{y}'_3, \dots, \mathsf{y}'_\Delta) \in \mathsf{W} \times \gen{\mathsf{y}_2} \times \gen{\mathsf{y}_3} \times \dots \times \gen{\mathsf{y}_\Delta}$.
Since $\mathsf{W}$ is right-closed by Observation~\ref{obs:rcs}, the fact that $\u$ is not contained in $\mathsf{W}$ implies that also any label that is at least as weak as $\u$ according to $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$ is not contained in $\mathsf{W}$; thus, $\mathsf{w}$ is not at least as weak as $\u$ according to $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$.
Moreover, as $\mathsf{y}'_j \in \gen{\mathsf{y}_j}$ for any $2 \leq j \leq \Delta$, we obtain the following picture:
there are two configurations $\mathcal U = \u \mspace{1mu} \mathsf{y}_2 \mspace{1mu} \mathsf{y}_3 \mspace{1mu} \dots \mspace{1mu} \mathsf{y}_\Delta$ and $\mathcal W = \mathsf{w} \mspace{1mu} \mathsf{y}'_2 \mspace{1mu} \mathsf{y}'_3 \mspace{1mu} \dots \mspace{1mu} \mathsf{y}'_\Delta$ in $\ensuremath{\mathcal{N}}_{\Delta,\beta}(v,x)$ such that $\mathsf{w}$ is not at least as weak as $\u$, and $\mathsf{y}'_j$ is at least as strong as $\mathsf{y}_j$, for all $2 \leq j \leq \Delta$.
Now it is straightforward to check that there are no two configurations in $\ensuremath{\mathcal{N}}_{\Delta,\beta}(v,x)$ with these properties, by going through all possible pairs of configurations.
To this end, recall the strength relations of the labels in $\Sigma_{\Delta,\beta}(v,x)$ given in Section~\ref{sec:edgediag} (in particular, Figure~\ref{fig:diag}), and assume for a contradiction that such configurations $\mathcal U, \mathcal W$ exist (recall that two configurations that are identical up to reordering of the contained labels are considered as the same configuration).
Consider first the case that $\mathcal W = \C_{i,j}^{\Delta-x} \mspace{1mu} \mathsf{X}^x$ for some $0 \leq i \leq \beta$, $1 \leq j \leq v_i$, and $0 \leq x \leq \Delta - 2$.
Since $\Delta-x \geq 2$, there is some index $2 \leq k \leq \Delta$ such that $\mathsf{y}'_k = \C_{i,j}$, which implies that $\mathcal U = \mathcal W$ or $\mathcal U = \mathsf{A}_\ell \mspace{1mu} \mathsf{B}_\ell^{\Delta-1}$ for some $1 \leq \ell \leq i$, as otherwise $\mathsf{y}'_k$ cannot be at least as strong as $\mathsf{y}_k$.
If $\mathcal U = \mathcal W$, then we have $\u = \C_{i,j}$ and $\mathsf{w} = \mathsf{X}$, as otherwise $\mathsf{w}$ is at least as weak as $\u$.
It follows that there is some index $2 \leq k' \leq \Delta$ such that $\mathsf{y}_{k'} = \mathsf{X}$ and $\mathsf{y}'_{k'} = \C_{i,j}$, yielding a contradiction to the fact that $\mathsf{y}'_{k'}$ is at least as strong as $\mathsf{y}_{k'}$.
If $\mathcal U = \mathsf{A}_\ell \mspace{1mu} \mathsf{B}_\ell^{\Delta-1}$ for some $1 \leq \ell \leq i$, then we have $\u = \mathsf{B}_\ell$ and $\mathsf{w} = \C_{i,j}$, or there is some index $2 \leq k' \leq \Delta$ such that $\mathsf{y}_{k'} = \mathsf{B}_\ell$ and $\mathsf{y}'_{k'} = \C_{i,j}$, since $\Delta-x \geq 2$.
In both cases, we obtain a contradiction, since $\C_{i,j}$ is weaker than $\mathsf{B}_\ell$.
Now, consider the other case, i.e., that $\mathcal W = \mathsf{A}_\ell \mspace{1mu} \mathsf{B}_\ell^{\Delta-1}$ for some $1 \leq \ell \leq \beta$.
If $\mathcal U = \C_{i,j}^{\Delta-x} \mspace{1mu} \mathsf{X}^x$ for some $0 \leq i \leq \beta$, $1 \leq j \leq v_i$, and $0 \leq x \leq \Delta - 2$, then we have $i < \ell$, as otherwise $\mathsf{A}_\ell$ is weaker than any label in $\mathcal U$, which would yield a contradiction no matter whether $\mathsf{w} = \mathsf{A}_\ell$ or $\mathsf{y}'_k = \mathsf{A}_\ell$ for some $2 \leq k \leq \Delta$.
But since $\Delta-x \geq 2$ implies that there is some $2 \leq k \leq \Delta$ such that $\mathsf{y}_k = \C_{i,j}$ and $\mathsf{y}'_k = \mathsf{B}_\ell$, the case $i < \ell$ also yields a contradiction, as $\mathsf{B}_\ell$ is not at least as strong as $\C_{i,j}$ if $i < \ell$.
If $\mathcal U = \mathsf{A}_{\ell'} \mspace{1mu} \mathsf{B}_{\ell'}^{\Delta-1}$ for some $1 \leq \ell' \leq \beta$, then we see that $\ell' \leq \ell$, as otherwise, again, $\mathsf{A}_\ell$ is weaker than any label in $\mathcal U$.
If $\ell' < \ell$, then $\mathsf{B}_{\ell'}$, which is contained in $\mathcal U$, is stronger than any label in $\mathcal W$, leading to a contradiction no matter whether $\u = \mathsf{B}_{\ell'}$ or $\mathsf{y}_k = \mathsf{B}_{\ell'}$ for some $2 \leq k \leq \Delta$.
If $\ell' = \ell$, then we have $\u = \mathsf{A}_\ell$ and $\mathsf{w} = \mathsf{B}_\ell$, as otherwise $\mathsf{w}$ is at least as weak as $\u$.
But then it follows that there is some index $2 \leq k \leq \Delta$ such that $\mathsf{y}_{k} = \mathsf{B}_\ell$ and $\mathsf{y}'_{k} = \mathsf{A}_\ell$, yielding a contradiction to the fact that $\mathsf{y}'_k$ is at least as strong as $\mathsf{y}_k$.
\end{proof}
\begin{corollary}\label{cor:strongflip}
Let $\U_1, \U_2 \in \Sigma_{\Delta,\beta}(v,x)$ be two labels such that $\U_2$ is stronger than $\U_1$ according to $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$, and assume that $x + 2 \leq \Delta$.
Then $\gen{\U_1}$ is stronger than $\gen{\U_2}$ according to $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$.
\end{corollary}
\begin{proof}
By the definition of $\gen{}$, the fact that $\U_2$ is stronger than $\U_1$ implies that $\gen{\U_2} \subseteq \gen{\U_1}$ and $\gen{\U_1} \nsubseteq \gen{\U_2}$.
Now, applying Lemma~\ref{lem:restrong} yields the corollary.
\end{proof}
\subsection{Node constraint}\label{sec:noco}
We now compute the node constraint of $\re(\Pi_{\Delta, \beta}(v,x))$.
\begin{lemma}\label{lemma:nodeconst'}
Let $x + 2 \leq \Delta$. The node constraint $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$ of $\re(\Pi_{\Delta, \beta}(v,x))$ is the collection of the following (condensed) configurations:
\begin{itemize}
\item For each color $\C_{i,j}\in\mathcal{C}$, where $0\le i\le \beta$ and $1\le j \le v_i$,
\[
\ensuremath{\operatorname{disj}}(\gen{\gen{\C_{i,j}}})^{\Delta-x} \mspace{1mu} \ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{X}}})^{x} \enspace.
\]
\item For each $1\le i\le \beta$
\[
\ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{A}_i}})\mspace{1mu} \ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{B}_i}})^{\Delta-1} \enspace.
\]
\end{itemize}
\end{lemma}
\begin{proof}
First of all, note that the definition of $\gen{\gen{L}}$, for some label $L$ of $\Pi_{\Delta, \beta}(v,x)$, depends on the strength of the labels of $\re(\Pi_{\Delta, \beta}(v,x))$, which in turn depends on the node constraint $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$, which we are currently defining by using the $\gen{\gen{\cdot}}$ notation. Notice that such a recursive definition is not an issue: by Lemma \ref{lemma:edgeconst'} we know what are the labels of $\re(\Pi_{\Delta, \beta}(v,x))$, and by Lemma \ref{lem:restrong} we know that the strength relation of these labels is given exactly by set inclusion. Hence we already know enough about the strength of the labels of $\re(\Pi_{\Delta, \beta}(v,x))$ even before formally defining $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$, and this allows us to use the $\gen{\gen{\cdot}}$ notation to define them.
The above lemma says that $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$ is given by the union, over all configurations $\L_1\mspace{1mu}\dotsc\mspace{1mu}\L_\Delta \in\mspace{1mu} \ensuremath{\mathcal{N}}_{\Delta,\beta}(v,x)$, of the configurations $\ensuremath{\operatorname{disj}}(\gen{\gen{\L_1}})\mspace{1mu} \ldots\mspace{1mu} \ensuremath{\operatorname{disj}}(\gen{\gen{\L_\Delta}})$. Recall that the $\re(\cdot)$ operator requires that $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$ contains all (and only) configurations $\S_1 \mspace{1mu} \ldots \mspace{1mu} \S_\Delta$ that satisfy that there exists a choice $(\mathsf{s}_1, \ldots, \mathsf{s}_\Delta) \in \S_1 \times \ldots \times \S_\Delta$ such that $\mathsf{s}_1\mspace{1mu} \ldots\mspace{1mu} \mathsf{s}_\Delta$ is in $\ensuremath{\mathcal{N}}_{\Delta,\beta}(v,x)$. Also, recall that tuples are equivalent up to reordering. We argue that $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$ can be obtained as follows. Start from $\ensuremath{\mathcal{N}}' = \{\}$. For each configuration $\L_1\mspace{1mu}\ldots\mspace{1mu}\L_\Delta \in \ensuremath{\mathcal{N}}_{\Delta,\beta}(v,x)$ add to $\ensuremath{\mathcal{N}}'$ all the configurations that can be obtained from the condensed configuration $\ensuremath{\operatorname{disj}}(\gen{\gen{\L_1}})\mspace{1mu} \ldots\mspace{1mu} \ensuremath{\operatorname{disj}}(\gen{\gen{\L_\Delta}})$. We now prove that the obtained set $\ensuremath{\mathcal{N}}'$ is equivalent to $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$.
By Lemma \ref{lem:restrong}, and by definition of $\gen{}$, $\gen{\gen{\L_i}}$ contains all and only the sets containing $\L_i$, hence, $\ensuremath{\mathcal{N}}'$ satisfies the requirements of the existential quantifier. We now prove that $\ensuremath{\mathcal{N}}'$ is maximal, in the sense that we cannot add any new valid configuration $\S'_1\mspace{1mu} \ldots\mspace{1mu} \S'_\Delta$ to $\ensuremath{\mathcal{N}}'$. Assume for a contradiction that $\S'_1\mspace{1mu} \ldots\mspace{1mu} \S'_\Delta$ is a valid maximal configuration not contained in $\ensuremath{\mathcal{N}}'$. There must exist a choice $(\mathsf{s}'_1, \ldots, \mathsf{s}'_\Delta) \in \S'_1 \times \ldots \times \S'_\Delta$ such that $\mathsf{s}'_1 \mspace{1mu} \ldots\mspace{1mu} \mathsf{s}'_\Delta$ is in $\ensuremath{\mathcal{N}}_{\Delta,\beta}(v,x)$. Note that, by construction, $\ensuremath{\mathcal{N}}'$ contains $\ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{s}'_1}}) \mspace{1mu} \ldots\mspace{1mu} \ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{s}'_\Delta}})$, and by definition of $\gen{\gen{\mathsf{s}'_i}}$ and Observation \ref{obs:subsetarrow} we have that $\S'_i \in \gen{\gen{\mathsf{s}'_i}}$, for all $1 \le i \le \Delta$. Hence, $\S'_1\mspace{1mu} \ldots\mspace{1mu} \S'_\Delta$ is present in $\ensuremath{\mathcal{N}}'$, contradicting the assumption. Hence the constructed set $\ensuremath{\mathcal{N}}'$ is equal to $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,x)$.
\end{proof}
\section{Upper bound}\label{sec:ub}
In this section we prove upper bounds for the $(2,\beta)$-ruling set problem. While an upper bound is not necessary to prove the main results of our work, perhaps surprisingly, it will serve the purpose of giving some intuition behind the definition of the problem family that we use to prove lower bounds.
We will first prove that
$\Pi_{\Delta,\beta}(v',0)$ is at least as hard as $\rere(\re(\Pi_{\Delta,\beta}(v,0)))$, where $v'$ is the inclusive prefix sum list of $v$ (i.e., $v'_i = \sum_{j\le i} v_j$). That is, we can apply the round elimination theorem on $\Pi_{\Delta,\beta}(v,0)$ to get a problem that can be solved in (at most) $1$ round given a solution for $\Pi_{\Delta,\beta}(v',0)$. Hence, we will prove the following lemma.
\begin{lemma}\label{lem:ub_secondspeedup}
The problem $\rere(\re(\Pi_{\Delta,\beta}(v,0)))$ can be solved in $0$ rounds given a solution for $\Pi_{\Delta,\beta}(v',0)$, where $v'$ is the inclusive prefix sum list of $v$. Hence, given a solution for $\Pi_{\Delta,\beta}(v',0)$ we can solve $\Pi_{\Delta,\beta}(v,0)$ in at most $1$ round.
\end{lemma}
We will then analyze the whole problem family in order to provide an upper bound for the $(2,\beta)$-ruling set problem. In particular, we will analyze how the number of colors evolves over time, and we will prove that the time required to compute a $(2,\beta)$-ruling set is at most the minimum $t$ such that ${\beta+t \choose \beta} \ge c$, if nodes are initially labeled with some $c$-vertex coloring. In particular, this implies that a $(2,\beta)$-ruling set can be found in $O(\beta\, c^{1/\beta})$ rounds, and that a $(2,\beta\, c^{1/\beta})$-ruling set can be found in $\beta$ rounds, for all $\beta \le c$. While this upper bound does match but not improve the current state of the art, we will later show how this family, essentially obtained while proving upper bounds, can be turned into a lower bound by increasing parameter $x$ (recall the definition of the problem family $\Pi_{\Delta,\beta}(v,x)$ in Section \ref{sec:problems}). Hence, we will prove the following lemma.
\begin{lemma}\label{lem:ub_time}
The time required to solve the $(2,\beta)$-ruling set problem in the port numbering model given a $c$-vertex coloring is at most the minimum $t$ such that ${\beta + t \choose \beta} \ge c$. In particular, the $(2,\beta)$-ruling set problem can be solved in $t \le \beta \, c^{1/\beta}$ rounds. Also, the $(2, \beta \, c^{1/\beta})$-ruling set problem can be solved in at most $\beta$ rounds.
\end{lemma}
Interestingly, the strategy that we will use to prove that $\Pi_{\Delta,\beta}(v',0)$ can be used to solve $\rere(\re((\Pi_{\Delta,\beta}(v,0)))$ shows that, by just blindly applying the round elimination theorem and discarding everything that has no \emph{intuitive} meaning, we can obtain algorithms that are able to compete with the current state of the art.
\subsection{Proof of Lemma \ref{lem:ub_secondspeedup}}
For simplicity, let us define $\Pi'_{\Delta,\beta}(v,0) \coloneqq \re(\Pi_{\Delta,\beta}(v,0))$ and $\Pi''_{\Delta,\beta}(v,0) \coloneqq \rere(\re(\Pi_{\Delta,\beta}(v,0))) =\rere(\Pi'_{\Delta,\beta}(v,0))$. We want to understand $\Pi''_{\Delta,\beta}(v,0)$, but it seems highly non trivial to show the exact form for $\Pi''_{\Delta,\beta}(v,0)$ using the round elimination technique. Instead, starting from $\Pi'_{\Delta,\beta}(v,0)$, we prove that some specific configurations are present in the node constraint $\ensuremath{\mathcal{N}}''_{\Delta,\beta}(v,0)$ of $\Pi''_{\Delta,\beta}(v,0)$. This is enough for our purposes, since, even if there are more configurations that we do not consider, it means that we are only making the problem harder. We will then show that we can rename labels appearing in $\Pi''_{\Delta,\beta}(v,0)$ such that the collection of configurations that we consider matches the node constraint definition of $\Pi_{\Delta,\beta}(v',0)$. We will also show that the edge constraint $\ensuremath{\mathcal{E}}''_{\Delta,\beta}(v,0)$ of $\Pi''_{\Delta,\beta}(v,0)$ matches the edge constraint $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,0)$ of $\Pi_{\Delta,\beta}(v,0)$. This will imply that, by applying Theorem \ref{thm:sebastien} on $\Pi_{\Delta,\beta}(v,0)$, we get a problem where we can discard some allowed configurations, and rename the obtained sets of sets, such that we get problem $\Pi_{\Delta,\beta}(v',0)$, and thus that a solution for $\Pi_{\Delta,\beta}(v',0)$ can be transformed in $0$ rounds to a solution for $\rere(\re(\Pi_{\Delta,\beta}(v,0)))$. Hence, and by Theorem \ref{thm:sebastien} this will imply that, given a solution for $\Pi_{\Delta,\beta}(v',0)$ we can solve $\Pi_{\Delta,\beta}(v,0)$ in at most $1$ round of communication.
\paragraph{Node constraint.}
We start by showing that, for each possible pair $(\C_{i,j},i')$ such that $\C_{i,j}$ is a color of group $i$ of the original problem and $i \le i' \le \beta$, $\ensuremath{\mathcal{N}}''_{\Delta,\beta}(v,0)$ contains some allowed configuration that corresponds to a color of group $i'$ of the new problem. In other words, we show that the colors of the new problem are generated by all possible pairs composed of a color of the original problem and a color group, where the color group is at least as large as the group of the original color. See Figure \ref{fig:color-generation} for an example.
\begin{figure}[h]
\centering
\includegraphics[width=0.75\textwidth]{figs/colorgeneration}
\caption{An example of how colors in $\Pi''$ are generated from colors in $\Pi$. In the depicted case, the colors of $\Pi$ can be described by the vector $[1,2,3]$, and each color of group $i$ in $\Pi$ generates a color for every group $j \ge i$ in $\Pi''$; hence the colors in $\Pi''$ can be described by the vector $[1,3,6]$. For example, the orange color in $\Pi$, in group $1$, generates two colors (orange and pink) in groups $1$ and $2$ in $\Pi''$.}
\label{fig:color-generation}
\end{figure}
Consider an arbitrary choice of $i,j,i'$ as described above, and the set $\C$ of sets defined as $\gen{\gen{ \C_{i,j} , \mathsf{B}_{i'} },\gen{ \mathsf{A}_{i'}}}$ if $i'>0$ and as $\gen{\gen{ \C_{i,j} }}$ if $i'=0$. We show that $\C^\Delta$ is an allowed configuration of $\ensuremath{\mathcal{N}}''_{\Delta,\beta}(v,0)$, that is, any choice of sets $(\mathsf{s}_1,\ldots,\mathsf{s}_\Delta) \in \C^\Delta$ is an allowed configuration in the node constraint of $\Pi'_{\Delta,\beta}(v,0)$. Any choice satisfies (by construction) the following: each chosen set either contains both $\C_{i,j}$ and $\mathsf{B}_{i'}$ (or just $\C_{i,j}$ if $i'=0$), or it contains $\mathsf{A}_{i'}$ (condition allowed only if $i'>0$).
If \emph{all} choices fall in the first case, hence also if $i'=0$, then this configuration is contained in $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,0)$, since all choices over $\ensuremath{\operatorname{disj}}(\gen{\gen{\C_{i,j}}})^\Delta$ are contained in $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,0)$.Thus let us now consider the case where at least one choice contains $\mathsf{A}_{i'}$, for $i'>0$. Other choices either also contain $\mathsf{A}_{i'}$, or they contain $\mathsf{B}_{i'}$. Since $\mathsf{B}_{i'}$ is stronger than $\mathsf{A}_{i'}$ with regard to the edge diagram of $\Pi_{\Delta,\beta}(v,0)$, the presence of $\mathsf{A}_{i'}$ in a set implies the presence of $\mathsf{B}_{i'}$ (see Observation \ref{obs:rcs}). Hence, we have a set containing $\mathsf{A}_{i'}$ and all other sets containing $\mathsf{B}_{i'}$, which is a configuration present in $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,0)$, given by the presence of all choices over $\ensuremath{\operatorname{disj}}(\gen{\gen{ \mathsf{A}_{i'}}}) \mspace{1mu} \ensuremath{\operatorname{disj}}(\gen{\gen{ \mathsf{B}_{i'}}})^{\Delta-1}$. Notice that, even though we required for $i \le i'$, this is not strictly necessary for obtaining allowed configurations. Nevertheless, since all sets containing $\C_{i,j}$ also contain $\mathsf{B}_i$ (by Observation \ref{obs:rcs}, since sets are right-closed and $\mathsf{B}_i$ is stronger than $\C_{i,j}$) and all $\mathsf{B}_j$ for $j < i$, the configuration obtained by using the pair $(\C_{i,j},i')$ would be the same as the one obtained by using $(\C_{i,j},i)$ if $i'<i$.
We now show that also $\gen{\gen{\mathsf{A}_i}} \mspace{1mu} \gen{\gen{\mathsf{B}_i}}^{\Delta-1}$ is an allowed configuration of $\ensuremath{\mathcal{N}}''_{\Delta,\beta}(v,0)$, for all $1 \le i \le \beta$. Consider an arbitrary choice: it must be a set containing $\mathsf{A}_i$ and all other sets containing $\mathsf{B}_i$, and thus a configuration present in $\ensuremath{\mathcal{N}}'_{\Delta,\beta}(v,0)$, given by the presence of all choices over $\ensuremath{\operatorname{disj}}(\gen{\gen{ \mathsf{A}_{i}}}) \mspace{1mu} \ensuremath{\operatorname{disj}}(\gen{\gen{ \mathsf{B}_{i}}})^{\Delta-1}$.
\paragraph{Renaming.}
We show how to rename the obtained sets of sets, such that, under the proposed renaming, we have the desired relation between $\Pi''_{\Delta,\beta}(v,0)$ and $\Pi_{\Delta,\beta}(v',0)$. We consider the set of sets obtained starting from $(\C_{i,j},i')$ as a new color of group $i'$. Then we map the other set of sets as follows: $\gen{\gen{\mathsf{B}_i}} \mapsto \mathsf{B}_i$, and $\gen{\gen{\mathsf{A}_i}} \mapsto \mathsf{A}_i$.
Notice that $v'$ is indeed the inclusive prefix sum of $v$: since we assume that $i \le i'$, in the new group $i'$ we have a number of colors equal to the sum of the number of colors of the old groups $i \le i'$. Also, we have all the $\mathsf{A}_i\mathsf{B}_i^{\Delta-1}$ configurations in the node constraint of $\Pi''_{\Delta,\beta}(v,0)$. This means that $\ensuremath{\mathcal{N}}''_{\Delta,\beta}(v,0)$ is the same as $\ensuremath{\mathcal{N}}_{\Delta,\beta}(v',0)$ of $\Pi_{\Delta,\beta}(v',0)$, defined in Section \ref{sec:problems}. On the other hand, we want to show that $\ensuremath{\mathcal{E}}''_{\Delta,\beta}(v,0)$ is the same as $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v',0)$ of $\Pi_{\Delta,\beta}(v',0)$. For that, it is enough to show that that the following statements hold for the edge constraint $\ensuremath{\mathcal{E}}''_{\Delta,\beta}(v,0)$.
\paragraph{Edge constraint.}
In order to show the desired properties mentioned above, we must show that the configurations of the edge constraint $\ensuremath{\mathcal{E}}''_{\Delta,\beta}(v,0)$ are the following.
\begin{enumerate}
\item All (new) colors are compatible with all other (new) colors (except themselves).
\item $\gen{\gen{\mathsf{B}_i}}$ is compatible with $\gen{\gen{\mathsf{B}_j}}$, for all $1 \le i,j \le \beta$.
\item $\gen{\gen{\mathsf{A}_j}}$ is compatible with $\gen{\gen{\mathsf{B}_i}}$, if and only if $i < j$.
\item $\gen{\gen{\mathsf{A}_j}}$ is compatible with all colors of the new group $i$, if and only if $i < j$.
\end{enumerate}
We start with color compatibility. Let $\C_{i,j,k}$ be the color (a set of sets) obtained from the pair $(\C_{i,j},k)$. Consider two new colors $\C_{i,j,k}$ and $\C_{i',j',k'}$. We need to show that there exists a set $\mathsf{s}_1 \in \C_{i,j,k}$ and a set $\mathsf{s}_2 \in \C_{i',j',k'}$ such that $\mathsf{s}_1 \mspace{1mu} \mathsf{s}_2$ is in $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,0)$, if and only if $(i,j,k) \neq (i',j',k')$. Consider the case where $(i,j) \neq (i',j')$, that is, either $i \neq i'$ or $j \neq j'$. We argue that such sets are given by the configuration $\mathsf{s}_1 \mspace{1mu} \mathsf{s}_2$ added to $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,0)$ starting from the good pair $(\{\C_{i,j}\},k)$ when defining the problem $\Pi'_{\Delta,\beta}(v,0)$ (recall the definition of good pair given in Section \ref{sec:firstspeedup}). By construction, $\mathsf{s}_1$ contains both $\C_{i,j}$ and $\mathsf{B}_k$ if $k>0$, or $\C_{i,j}$ if $k=0$. Since $\C_{i,j,k}$ is defined as $\gen{\gen{\C_{i,j} , \mathsf{B}_{k} },\gen{ \mathsf{A}_{k} }}$ if $k>0$ and as $\gen{\gen{ \C_{i,j} }}$ if $k=0$, then $\mathsf{s}_1 \in \C_{i,j,k}$, thus we can pick such set. Since $(i,j) \neq (i',j')$, and since by construction $\mathsf{s}_2$ contains all colors not present in $\mathsf{s}_1$ and all $\mathsf{B}$-type labels, then $\mathsf{s}_2$ is contained in $\C_{i',j',k'}$ (because $\mathsf{s}_2$ is in $\ensuremath{\operatorname{disj}}( \gen{\gen{ \C_{i',j'} , \mathsf{B}_{k'} } })$ if $k'>0$, or in the disjunction $\ensuremath{\operatorname{disj}}( \gen{\gen{ \C_{i',j'} } })$ if $k'=0$, and $\C_{i',j',k'}$ contains all elements of such disjunction), and thus we can choose such set. Let us now consider the case $k \neq k'$. Without loss of generality, consider the case where $k < k'$. Consider again the configuration $\mathsf{s}_1 \mspace{1mu} \mathsf{s}_2$ added to $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,0)$ starting from the good pair $(\{\C_{i,j}\},k)$. The set $\mathsf{s}_2$ contains $\mathsf{A}_{k'}$, since $k' > k$, and it is thus contained in $\gen { \gen{ \mathsf{A}_{k'} }}$, that is a subset of $\C_{i',j',k'}$, thus we can pick such set. We now show that color $\C_{i,j,k}$ is not compatible with itself. By picking a pair of sets that both contain $\C_{i,j}$ we get a configuration not in $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,0)$, since a color never appears on both sides of an edge configuration. By picking a pair of sets that both contain $\mathsf{A}_k$ we get a configuration not in $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,0)$, since $\mathsf{A}$-type labels never appears on both sides. By picking on one side a set that contains $\C_{i,j}$ and on the other side a set that contains $\mathsf{A}_k$, we get a configuration not in $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,0)$, because the first choice must also contain $\mathsf{B}_k$ (since all $\mathsf{B}$-type labels are stronger than all $\mathsf{A}$-type labels), and all configurations in $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,0)$ are such that if $\mathsf{B}_k$ is contained on one side, $A_k$ is not contained on the other side.
We now argue about compatibility between $\gen{\gen{\mathsf{B}_i}}$ and $\gen{\gen{\mathsf{B}_j}}$, for all $1 \le i,j \le \beta$. Consider the configuration $\mathsf{s}_1 \mspace{1mu} \mathsf{s}_2$ added to $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,0)$ starting from the good pair $(\{\},i)$. Recall from Section \ref{sec:edgeconstraint} that, for this specific good pair, $\mathsf{s}_1$ is defined as $\{\mathsf{B}_1,\ldots,\mathsf{B}_i\}$, and that $\{\mathsf{B}_1,\ldots,\mathsf{B}_\beta\} \subseteq \mathsf{s}_2$. The set $\mathsf{s}_1 $ can be picked from $\gen{\gen{\mathsf{B}_i}}$, and since $\mathsf{s}_2$ contains all $\mathsf{B}$-type labels, and $\gen{\gen{\mathsf{B}_j}}$ contains $\{\mathsf{B}_1,\ldots,\mathsf{B}_j\}$ and hence a subset of $\mathsf{s}_2$, the set $\mathsf{s}_2$ can be picked from $\gen{\gen{\mathsf{B}_j}}$.
Regarding the compatibility between $\gen{\gen{\mathsf{A}_j}}$ and $\gen{\gen{\mathsf{B}_i}}$, notice that, if $i < j$, then the aforementioned set $\mathsf{s}_2$, by construction, contains also $\mathsf{A}_j$, and thus $\mathsf{s}_2$ can be picked from $\gen{\gen{\mathsf{A}_j}}$. Moreover, if $i \ge j$, $\gen{\gen{\mathsf{A}_j}}$ is not compatible with $\gen{\gen{\mathsf{B}_i}}$, since any pair of choices must contain $\mathsf{B}_i$ on one side and $\mathsf{A}_j$ on the other, but by construction all configurations in $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,0)$ satisfy that if $\mathsf{B}_i$ is present on one side, and $j \le i$, then $\mathsf{A}_j$ is not present on the other side.
Let us now prove the last point. If $j \ge k$, $\gen{\gen{\mathsf{A}_k}}$ is not compatible with colors in the new group $j$, since any choice over such colors either contains another $\mathsf{A}$-type label, or $\mathsf{B}_j$, and such configurations never appear in $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,0)$. If $j < k$, consider color $\C_{i,j,k}$, and consider the configuration $\mathsf{s}_1 \mspace{1mu} \mathsf{s}_2$ added to $\ensuremath{\mathcal{E}}'_{\Delta,\beta}(v,0)$ starting from the good pair $(\{\C_{i,j}\},k)$. By definition of $\C_{i,j,k}$, we know that $\mathsf{s}_1$ is contained in $\C_{i,j,k}$, and thus we can choose such set. Then, since $k > j$, $\mathsf{A}_k$ is contained in $\mathsf{s}_2$, and thus $\mathsf{s}_2$ is contained in $\gen{\gen{\mathsf{A}_j}}$.
We showed that up to the above renaming, $\ensuremath{\mathcal{N}}''_{\Delta,\beta}(v,0) = \ensuremath{\mathcal{N}}_{\Delta,\beta}(v',0)$ and $\ensuremath{\mathcal{E}}''_{\Delta,\beta}(v,0)= \ensuremath{\mathcal{E}}_{\Delta,\beta}(v',0)$, which means that $\Pi''_{\Delta, \beta}(v,0)=\rere(\re(\Pi_{\Delta,\beta}(v,0)))$ can be solved in $0$ rounds given a solution for $\Pi_{\Delta,\beta}(v',0)$, proving Lemma \ref{lem:ub_secondspeedup}.
\subsection{Proof of Lemma \ref{lem:ub_time}}
While the algorithm that we implicitly obtain by applying round elimination multiple times only works on $\Delta$-regular graphs, we will later show, by giving a human-readable version of such algorithm, that it can be adapted to work on any graph.
Let $p(v)$ be the inclusive prefix sum of $v$, and let $p^t(v)$ be the function that recursively applies $t$ times $p$ to $v$ (if $t=0$, then $p^t(v) = v$). That is, $p^{t+1}(v)_j = \sum_{i=0}^j p^t(v)_i$, for all $0 \le j \le \beta$.
Let us see what we get by applying Lemma \ref{lem:ub_secondspeedup} multiple times. We start from $\Pi_{\Delta,\beta}(v=[1,0,\ldots,0],0)$, and by applying Lemma \ref{lem:ub_secondspeedup} we get that it can be solved in $1$ round given a solution for $\Pi_{\Delta,\beta}(p(v),0)$. Applying the same reasoning, the latter problem can be solved in $1$ round given a solution for $\Pi_{\Delta,\beta}(p(p(v)),0)$, and so on. We get that $\Pi_{\Delta,\beta}([1,0,\ldots,0],0)$ can be solved in $t$ rounds given a coloring with $c \le \size(p^t(v))$ colors.
Hence, the time required to solve $\Pi_{\Delta,\beta}([1,0,\ldots,0],0)$, and thus the $(2,\beta)$-ruling set problem, given a $c$-vertex coloring, is upper bounded by the minimum $t$ such that $c \le \size(p^t(v))$, since $\Pi_{\Delta,\beta}(p^t(v),0)$, the problem obtained by applying $t$ times Lemma \ref{lem:ub_secondspeedup}, can be solved by defining a one-to-one mapping between the given coloring and the $\size(p^t(v))$ configurations of $\ensuremath{\mathcal{N}}_{\Delta,\beta}(p^t(v),0)$ that do not correspond to pointers.
Let us now give an exact bound on the size of $p^t(v)$, assuming $v = [1,0,\ldots,0]$.
\begin{lemma}\label{lem:colorgrowth}
For all $0 \le k \le \beta$, and for all $j \ge 1$, $p^j(v)_k = {j+k-1 \choose k}$.
\end{lemma}
\begin{proof}
For $j = 1$, the definition of $p^j(v)_k$ yields $p(v)_k = 1 + 0 + 0 + \dots + 0 = {k \choose k}$ for any $0 \leq k \leq \beta$, so the claim holds trivially in that case.
By induction, and using the binomial identity $\sum_{a=0}^b {c+a \choose a} = {c+b+1 \choose b}$ , we obtain that for $j > 1$ and any $0 \leq k \leq \beta$, we have $p^j(v)_k = \sum_{i=0}^k p^{j-1}(v)_i = \sum_{i=0}^k {j-2+i \choose i} = {j+k-1 \choose k}$.
\end{proof}
By applying Lemma \ref{lem:colorgrowth}, we get that $\size(p^t(v)) = p^{t+1}(v)_\beta = {\beta+t \choose \beta}$. This implies that, given some $c$-coloring, the time required to solve $(2,\beta)$-ruling sets is upper bounded by the minimum $t$ such that ${\beta+t \choose \beta} \ge c$. Since, for $t = \beta \, c^{1/\beta}$, we have that ${\beta+t \choose \beta} \ge \left(\frac{t}{\beta}\right)^\beta \ge c$, we obtain that $(2,\beta)$-ruling sets can be found in $\beta \, c^{1/\beta}$ rounds. Also, by using the inequality ${x+y \choose x} \ge \left(\frac{x}{y}\right)^y$, we obtain that $(2, \beta \, c^{1/\beta})$-ruling sets can be found in at most $\beta$ rounds, since ${\beta c^{1/\beta} + \beta \choose \beta c^{1/\beta}} \ge \left(\frac{\beta c^{1/\beta}}{\beta}\right)^\beta = c$.
\subsection{Intuition behind the algorithm}
While proving Lemma \ref{lem:ub_secondspeedup} we showed that, starting from problem $\Pi_{\Delta,\beta}(v,0)$, we can apply the round elimination theorem and obtain a problem that can be made harder such that the obtained sets of sets can be renamed to obtain problem $\Pi_{\Delta,\beta}(v',0)$, where $v'$ is the inclusive prefix sum of $v$. This implies that, given a solution for $\Pi_{\Delta,\beta}(v',0)$ we can obtain, in $0$ rounds, a solution for $\rere(\re(\Pi_{\Delta,\beta}(v,0)))$. Also, note that each problem $\Pi_{\Delta,\beta}(v,0)$ can be solved given a $c \le \size(v)$ coloring. Moreover, the round elimination theorem implies that by spending $1$ round of communication, given a solution for $\rere(\re(\Pi_{\Delta,\beta}(v,0)))$ nodes can obtain a solution for $\Pi_{\Delta,\beta}(v,0)$. Hence, given a solution for $\Pi_{\Delta,\beta}(v',0)$, we can obtain a solution for $\Pi_{\Delta,\beta}(v,0)$ by spending $1$ round of communication. In order to do that, nodes can compute (offline) the inverse renaming from the labels of the current problem to the labels (sets of sets) of the previous problem. Then, they can spend $1$ round to share their current labels with the neighbors, and choose a label for the previous problem that satisfies the previous node constraint and edge constraint (and the round elimination theorem guarantees that this is possible). Essentially, this means that, given a proof for an upper bound using round elimination, a proper algorithm can be obtained by ``executing the proof in the reverse order'', and a crucial part is that the (inverse) label renaming can be computed without need of coordination.
While the round elimination theorem guarantees us the existence of an algorithm, we can actually extract some human-understandable version of such procedure by recalling how we defined the colors of the new problem starting from the colors of the old problem. Recall that each color of some problem corresponds to some pair containing a color and a group of the previous problem. Thus, let us now see how to transform a solution for $\Pi_{\Delta,\beta}(p^{t}(v),0)$, given a $c \le p^t(v)$ coloring, to a solution for $\Pi_{\Delta,\beta}(v=[1,0,\ldots,0],0)$, in $t$ rounds of communication. At first, nodes start by (offline) computing some mapping from colors to groups such that in each group $k$, at most $p^{t}(v)_k$ colors are present. Thus each color can be seen as $\C_{k,i'}$, for some $0 \le k \le \beta$ and $1 \le i' \le p^{t}(v)_k$. Now, let us see how nodes can convert a solution for $\Pi_{\Delta,\beta}(p^{t'+1}(v),0)$ to a solution for $\Pi_{\Delta,\beta}(p^{t'}(v),0)$ in $1$ round of communication, for each $0 \le t' < t$. Recall that each color of the current problem corresponds to a good pair of the previous problem, and nodes can compute offline such an inverse mapping. Hence each color $\C_{k,i'}$ corresponds to some pair $(\C_{i,j},k)$ satisfying $k \ge i$. The algorithm, for a node having the pair $(\C_{i,j},k)$, does the following:
\begin{enumerate}
\item Gather the pair $(\c_u,\mathsf{g}_u)$ of each neighbor $u$.
\item If no neighbor satisfies $\c_u = \C_{i,j} \land \mathsf{g}_u < k$, then set $\C_{i,j}$ as new color.
\item Otherwise, output $\mathsf{A}_i$ on the port connecting to that neighbor, and $\mathsf{B}_i$ on the others.
\end{enumerate}
Hence, there is a very simple idea behind the round elimination generated algorithm: we have colors and pointers. Each color corresponds to a pair containing a color $c$ and a group $i$ of the previous step. Group numbers give priorities. If no neighbor has the same color $c$ with a better (lower) priority, pick color $c$, otherwise point to that neighbor. An example of this color reduction step is shown in Figure \ref{fig:color-reduction} (note that this is essentially the inverse of the mapping previously shown in Figure \ref{fig:color-generation}). The reason why it works is the following. At each step the number of colors reduces, and at the end there is only one ``color'' that corresponds to nodes in the ruling set. In order to get rid of colors fast, nodes do not wait to know if nodes are in the ruling set in order to point to them. Instead, they are ensured that if they output some pointer with distance $i$, in the next step the pointed node will either output some color of group smaller than $i$, or a pointer with distance smaller than $i$. In case the neighbor outputs a color, in the next steps such node will either output a color of smaller groups or a smaller pointer (this condition is guaranteed by how ``good pairs'' are defined). Notice that the provided algorithm does not need the graph to be $\Delta$-regular, hence we get an algorithm that works in all graphs.
\begin{figure}[h]
\centering
\includegraphics[width=0.75\textwidth]{figs/colorreduction}
\caption{An example of the reduction of the colors. A pink node would either change its color to orange, if no neighbor has that color, or it would uncolor itself and point to an orange neighbor.}
\label{fig:color-reduction}
\end{figure}
\section{Lower bound}\label{sec:lb}
In this section we prove lower bounds on the time complexity for computing $(2,\beta)$-ruling sets using deterministic algorithms in the port numbering model. We will prove such lower bounds using the round elimination theorem. While it is not difficult to argue that in the port numbering model $(2,\beta)$-ruling sets are not solvable with deterministic algorithms, we will later see in Section \ref{sec:liftlocal} how to convert the lower bounds obtained in this section to lower bounds for the LOCAL model. We will heavily exploit the fact that the lower bounds that we prove in this section show the existence of a sequence of problems $\Pi_0 \rightarrow \Pi_1 \rightarrow \dots \rightarrow \Pi_t$, where $\Pi_t$ is not $0$-round solvable, and the problem family is obtained by applying the round elimination theorem multiple times in a way that preserves the property that the number of labels of each problem is not too large. As we will see in Section \ref{sec:liftlocal}, this is the only property that we need, in order to ensure that, even by allowing randomization, the problems of such a family do not become much easier.
We start by defining the notion of \emph{relaxation}, which allows us to relate different configurations.
\begin{definition}
We say that a configuration $\mathsf{A}_1 \mspace{1mu} \dots \mspace{1mu} \mathsf{A}_\Delta$ of sets \emph{can be relaxed to} a configuration $\mathsf{B}_1 \mspace{1mu} \dots \mspace{1mu} \mathsf{B}_\Delta$ of sets if there exists a bijection $\sigma: \{1, \dots, \Delta\} \to \{1, \dots, \Delta\}$ such that $\mathsf{A}_k \subseteq \mathsf{B}_{\sigma(k)}$ for each $1 \leq k \leq \Delta$.
For simplicity, we will assume in the following w.l.o.g., that $\sigma(k) = k$ for each $1 \leq k \leq \Delta$, i.e., that the bijection is the identity.
\end{definition}
We now define a node constraint $\mathcal Z_{\Delta, \beta}(v, x)$ that will be useful later. In particular, we will show in Lemma~\ref{lem:industop} that each node configuration of $\rere(\re(\Pi_{\Delta,\beta}(v,x)))$ can be relaxed to some configuration in $\mathcal Z_{\Delta, \beta}(v, x)$.
\begin{definition}
Recall that $\Sigma_{\Delta,\beta}(v,x)$ denotes the set of all labels used in $\Pi_{\Delta, \beta}(v, x)$, $\Sigma'_{\Delta,\beta}(v,x)$ the set of all labels used in $\re( \Pi_{\Delta,\beta}(v,x) )$, and $\mathcal C$ the set of all colors $\C_{i,j} \in \Sigma_{\Delta,\beta}(v,x)$, and that $\size(v) = v_0 + \dots + v_\beta$.
We will denote the group of a color $\C \in \mathcal C$ by $g(\C)$, i.e., if $\C = \C_{i,j}$, then $g(\C) = i$ (similar to the already defined notion for sets of colors).
Moreover, for any $v, x$ such that $\size(v) \cdot (x+1) + 1 \leq \Delta$, we denote by $\mathcal Z_{\Delta, \beta}(v, x)$ the set containing the configurations
\begin{align*}
\gen{\gen{\mathsf{A}_i}} \quad & \gen{\gen{\mathsf{B}_i}}^{\Delta-1} \enspace &\text{ for any $1 \leq i \leq \beta$,}\\
\gen{\gen{\C}}^{\Delta - \size(v) \cdot (x+1)}\quad &\, (\Sigma'_{\Delta,\beta}(v,x))^{\size(v) \cdot (x+1)} \enspace &\text{ for any $\C \in \mathcal C$ satisfying $g(\C) = 0$, and}\\
\gen{\gen{\C, \mathsf{B}_i}, \gen{\mathsf{A}_i}}^{\Delta - \size(v) \cdot (x+1)}\quad &\, (\Sigma'_{\Delta,\beta}(v,x))^{\size(v) \cdot (x+1)} \enspace &\text{ for any $\C \in \mathcal C$ and $\max\{ 1, g(\C) \} \leq i \leq \beta$.}
\end{align*}
Here, the inner $\gen{}$ are taken w.r.t.\ $\ensuremath{\mathcal{E}}_{\Delta, \beta}(v, x)$, and the outer $\gen{}$ w.r.t.\ $\ensuremath{\mathcal{N}}'_{\Delta, \beta}(v, x)$.
\end{definition}
The next lemma restricts the space of configurations that can possibly appear in the node constraint of $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$. We will later use this lemma inductively in order to restrict this space even further.
\begin{lemma}\label{lem:industep}
Let $1 \leq i \leq \beta - 1$, and let $v, x$ satisfy $\size(v) \cdot (x+1) + 1 \leq \Delta$.
Let $\mathcal U = \U_1 \mspace{1mu} \dots \mspace{1mu} \U_\Delta$ be a node configuration of $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$ such that
\begin{enumerate}[label=(\roman*)]
\item \label{item:two} $\mathcal U$ cannot be relaxed to any configuration in $\mathcal Z_{\Delta, \beta}(v,x)$, and
\item \label{item:four} each set contained in $\U_k$ contains $\mathsf{B}_i$, for any $1 \leq k \leq \Delta$.
\end{enumerate}
Then, each set contained in $\U_k$ contains $\mathsf{B}_{i+1}$, for any $1 \leq k \leq \Delta$.
\end{lemma}
\begin{proof}
We begin by proving two useful claims.
\underline{Claim 1:} For any color $\C \in \mathcal C$, there are at least $\size(v) \cdot (x+1) + 1$ distinct indices $k$ such that $\U_k$ contains a set that contains neither $\mathsf{A}_i$ nor $\C$.
For a contradiction, suppose that there is a color $\C$ for which the claim is false.
W.l.o.g., we can assume that $\U_1, \dots, \U_{k'}$ are exactly those $\U_k$ that contain a set that contains neither $\mathsf{A}_i$ nor $\C$, where $k' \leq \size(v) \cdot (x+1)$.
Now consider an arbitrary set $\U_k$ with $k > k'$, and some arbitrary set $\mathsf{W} \in \U_k$.
Since $k > k'$, we know that $\mathsf{W}$ must contain $\mathsf{A}_i$ or $\C$.
As $\mathsf{W}$ also contains $\mathsf{B}_i$ by Property~$\ref{item:four}$ and is right-closed by Observation~\ref{obs:rcs} (since $\mathsf{W}$ is a label/set used in $\re(\Pi_{\Delta, \beta}(v, x))$), it follows that $\mathsf{W}$ is a superset of $\gen{\C, \mathsf{B}_i}$ or a superset of $\gen{\mathsf{A}_i, \mathsf{B}_i}$.
Since $\mathsf{B}_i$ is stronger than $\mathsf{A}_i$ according to $\ensuremath{\mathcal{E}}_{\Delta, \beta}(v, x)$, we have $\gen{\mathsf{A}_i, \mathsf{B}_i} = \gen{\mathsf{A}_i}$; hence, $\mathsf{W} \in \gen{\gen{\C, \mathsf{B}_i}, \gen{\mathsf{A}_i}}$, by Lemma~\ref{lem:restrong}, and we obtain that $\U_k \subseteq \gen{\gen{\C, \mathsf{B}_i}, \gen{\mathsf{A}_i}}$.
This implies that every $\U_k$ with $k > k'$ is a subset of $\gen{\gen{\C, \mathsf{B}_i}, \gen{\mathsf{A}_i}}$, and since every $\U_k$ is a subset of $\Sigma'_{\Delta,\beta}(v,x)$, and $k' \leq \size(v) \cdot (x+1)$, it follows that $\mathcal U$ can be relaxed to the configuration $\gen{\gen{\C, \mathsf{B}_i}, \gen{\mathsf{A}_i}}^{\Delta - \size(v) \cdot (x+1)} \, (\Sigma'_{\Delta,\beta}(v,x))^{\size(v) \cdot (x+1)}$.
We observe that, for all $1 \leq j \leq g(\C)$, we have that $\gen{\C, \mathsf{B}_j} = \gen{\C}$ since $\mathsf{B}_j$ is stronger than $\C$ according to $\ensuremath{\mathcal{E}}_{\Delta, \beta}(v, x)$.
Therefore, $\gen{\C, \mathsf{B}_i} = \gen{\C, \mathsf{B}_{\max\{i, g(\C)\}}}$, which implies that the configuration $\gen{\gen{\C, \mathsf{B}_i}, \gen{\mathsf{A}_i}}^{\Delta - \size(v) \cdot (x+1)} \, (\Sigma'_{\Delta,\beta}(v,x))^{\size(v) \cdot (x+1)}$ is contained in $\mathcal Z_{\Delta, \beta}(v, x)$.
This yields a contradiction to Property~$\ref{item:two}$ and proves the claim.
\underline{Claim 2:} Each $\U_k$ contains a set that does not contain $\mathsf{A}_i$.
For a contradiction, suppose that the claim is false, and let $k'$ be an index such that any set $\mathsf{W}$ contained in $\U_{k'}$ contains $\mathsf{A}_i$.
Since any such $\mathsf{W}$ is right-closed by Observation~\ref{obs:rcs} (since $W$ is a label/set used in $\re(\Pi_{\Delta, \beta}(v, x))$), this implies that $\gen{\mathsf{A}_i} \subseteq \mathsf{W}$ for any $\mathsf{W} \in \U_{k'}$.
Hence, by Lemma~\ref{lem:restrong}, we obtain that, for any $\mathsf{W} \in \U_{k'}$, $\mathsf{W}$ is at least as strong as $\gen{\mathsf{A}_i}$ according to $\ensuremath{\mathcal{N}}'_{\Delta, \beta}(v, x)$, and it follows that $\U_{k'}$ is a subset of $\gen{\gen{\mathsf{A}_i}}$.
With an analogous argumentation, we see that any $\U_k$ is a subset of $\gen{\gen{\mathsf{B}_i}}$ since for any $\U_k$, each set contained in $\U_k$ contains $\mathsf{B}_i$, by Property~$\ref{item:four}$.
It follows that $\mathcal U$ can be relaxed to the configuration $\gen{\gen{\mathsf{A}_i}} \mspace{1mu} \gen{\gen{\mathsf{B}_i}}^{\Delta-1}$ contained in $\mathcal Z_{\Delta, \beta}(v, x)$, yielding a contradiction to Property~$\ref{item:two}$ and proving the claim.
Now, we are set to prove the lemma.
For a contradiction, suppose that the lemma does not hold, and, w.l.o.g., let $\mathsf{W}_1 \in \U_1$ be a set that does not contain $\mathsf{B}_{i+1}$.
We will now pick one set from each $\U_k$ such that the obtained configuration satisfies certain useful properties.
Consider the partial configuration of sets obtained by picking $\mathsf{W}_1$ from $\U_1$, and picking, for each color $\C \in \mathcal C$, $(x+1)$ sets $\mathsf{W}_k$ from distinct $\U_k$ such that $\mathsf{W}_k$ contains neither $\mathsf{A}_i$ nor $\C$.
To be precise, the $\size(v) \cdot (x+1) + 1$ sets picked so far are picked from $\size(v) \cdot (x+1) + 1$ $\U_k$ with pairwise distinct index.
This is possible due to Claim 1 (by simply picking $\mathsf{W}_1$ first, and then picking the remaining $\size(v) \cdot (x+1)$ sets with the desired properties in an arbitrary order).
We complete this partial configuration to a configuration $\mathsf{W}_1, \mspace{1mu} \dots \mspace{1mu} \mathsf{W}_\Delta$ containing $\Delta$ sets by picking from each remaining $\U_k$ a set that does not contain $\mathsf{A}_i$.
This is possible by Claim 2.
We argue that $\mathsf{W}_1 \mspace{1mu} \dots \mspace{1mu} \mathsf{W}_\Delta$ is not a node configuration of $\re( \Pi_{\Delta,\beta}(v,x) )$:
Since, by construction, for each color $\C \in \mathcal C$, there are at least $x+1$ distinct indices $k$ such that $\mathsf{W}_k$ does not contain $\C$, the configuration $\mathsf{W}_1 \mspace{1mu} \dots \mspace{1mu} \mathsf{W}_\Delta$ is not contained in the condensed configuration $\ensuremath{\operatorname{disj}}(\gen{\gen{\C}})^{\Delta - x} \mspace{1mu} \ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{X}}})^x$.
Moreover, as $\mathsf{B}_{i+1}$ is stronger than $\mathsf{A}_i$ according to $\ensuremath{\mathcal{E}}_{\Delta, \beta}(v, x)$, the fact that $\mathsf{W}_1$ does not contain $\mathsf{B}_{i+1}$ implies that $\mathsf{W}_1$ also does not contain $\mathsf{A}_i$, as $\mathsf{W}_1$ is right-closed by Observation~\ref{obs:rcs}.
Thus, none of the $\mathsf{W}_k$ contains $\mathsf{A}_i$, and we obtain for each $i' \leq i$ that none of the $\mathsf{W}_k$ contains $\mathsf{A}_{i'}$, since $\mathsf{A}_i$ is stronger than $\mathsf{A}_{i'}$ according to $\ensuremath{\mathcal{E}}_{\Delta, \beta}(v, x)$ and each $\mathsf{W}_k$ is right-closed.
It follows that $\mathsf{W}_1 \mspace{1mu} \dots \mspace{1mu} \mathsf{W}_\Delta$ is not contained in the condensed configuration $\ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{A}_{i'}}}) \mspace{1mu} \ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{B}_{i'}}})^{\Delta-1}$, for any $i' \leq i$.
Finally, for any $i' > i$, the fact that $\mathsf{W}_1$ does not contain $\mathsf{B}_{i+1}$ implies that $\mathsf{W}_1$ also contains neither $\mathsf{A}_{i'}$ nor $\mathsf{B}_{i'}$ since both are at least as weak as $\mathsf{B}_{i+1}$ according to $\ensuremath{\mathcal{E}}_{\Delta, \beta}(v, x)$ and $\mathsf{W}_1$ is right-closed.
Hence, $\mathsf{W}_1 \mspace{1mu} \dots \mspace{1mu} \mathsf{W}_\Delta$ is not contained in the condensed configuration $\ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{A}_{i'}}}) \mspace{1mu} \ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{B}_{i'}}})^{\Delta-1}$, for any $i' > i$.
Therefore, $\mathsf{W}_1 \mspace{1mu} \dots \mspace{1mu} \mathsf{W}_\Delta$ is contained in none of the condensed configurations in $\ensuremath{\mathcal{N}}'_{\Delta, \beta}(v, x)$, and we conclude that $\mathsf{W}_1 \mspace{1mu} \dots \mspace{1mu} \mathsf{W}_\Delta$ is not a node configuration of $\re( \Pi_{\Delta,\beta}(v,x) )$.
Since we have $\mathsf{W}_k \in \U_k$ for each $1\leq k \leq \Delta$, it follows by the definition of the function $\rere(\cdot)$ that $\U_1 \mspace{1mu} \dots \mspace{1mu} \U_\Delta$ is not a node configuration of $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$.
This yields the desired contradiction and proves the lemma.
\end{proof}
We now prove an analogous statement of the previous lemma, for the case where $i=0$.
\begin{lemma}\label{lem:industart}
Let $v, x$ satisfy $\size(v) \cdot (x+1) + 1 \leq \Delta$, and let $\mathcal U = \U_1 \mspace{1mu} \dots \mspace{1mu} \U_\Delta$ be a node configuration of $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$ such that $\mathcal U$ cannot be relaxed to any configuration in $\mathcal Z_{\Delta, \beta}(v,x)$.
Then, each set contained in $\U_k$ contains $\mathsf{B}_1$, for any $1 \leq k \leq \Delta$.
\end{lemma}
\begin{proof}
This proof is essentially a special case of the proof of Lemma~\ref{lem:industep} that only requires a subset of the arguments.
Given that the approach is analogous, we will only provide the minimally necessary arguments without additional explanations.
\underline{Claim 1:} For any color $\C \in \mathcal C$, there are at least $\size(v) \cdot (x+1) + 1$ distinct indices $k$ such that $\U_k$ contains a set that does not contain $\C$.
For a contradiction, we assume that the claim is false for some color $\C$, and that $\U_1, \dots, \U_{k'}$ are those $\U_k$ that contain a set that does not contain $\C$, where $k' \leq \size(v) \cdot (x+1)$.
Hence, for any $\U_k$ with $k > k'$, any set contained in $\U_k$ must contain $\C$, which implies that $\U_k \subseteq \gen{\gen{\C}}$, due to the right-closedness of all sets contained in $\U_k$ and Lemma~\ref{lem:restrong}.
Thus, $\mathcal U$ can be relaxed to $\gen{\gen{\C}}^{\Delta - \size(v) \cdot (x+1)} \, (\Sigma'_{\Delta,\beta}(v,x))^{\size(v) \cdot (x+1)}$.
If $g(\C) = 0$, the latter configuration is contained in $\mathcal Z_{\Delta, \beta}(v, x)$, yielding a contradiction and proving the claim.
Hence, assume that $g(\C) \geq 1$.
Since $\mathsf{B}_{g(\C)}$ is stronger than $\C$ according to $\ensuremath{\mathcal{E}}_{\Delta, \beta}(v, x)$, we have $\gen{\C} = \gen{\C, \mathsf{B}_{g(\C)}}$, by the definition of $\gen{}$.
Furthermore, since $\mathsf{A}_{g(\C)}$ is weaker than $\C$ according to $\ensuremath{\mathcal{E}}_{\Delta, \beta}(v, x)$, we have that $\gen{\mathsf{A}_{g(\C)}}$ is stronger than $\gen{\C}$ according to $\ensuremath{\mathcal{N}}'_{\Delta, \beta}(v, x)$, by Corollary~\ref{cor:strongflip}.
Hence, $\gen{\gen{\C}} = \gen{\gen{\C}, \gen{\mathsf{A}_{g(\C)}}} = \gen{\gen{\C, \mathsf{B}_{g(\C)}}, \gen{\mathsf{A}_{g(\C)}}}$.
It follows that the configuration $\gen{\gen{\C}}^{\Delta - \size(v) \cdot (x+1)} \mspace{1mu} (\Sigma'_{\Delta,\beta}(v,x))^{\size(v) \cdot (x+1)}$ is contained in $\mathcal Z_{\Delta, \beta}(v, x)$, which yields a contradiction and proves the claim.
Now assume for a contradiction that the lemma does not hold, and w.l.o.g., pick one set $\mathsf{W}_k$ from each $\U_k$, such that $\mathsf{W}_1$ does not contain $\mathsf{B}_1$, and for each color $\C$, there are at least $(x+1)$ distinct $k$ such that $\mathsf{W}_k$ does not contain $\C$.
This is possible due to Claim 1, by an analogous argumentation to the one in the proof of Lemma~\ref{lem:industep}.
Now, again, the fact that for each color $\C$, at least $x+1$ many $\mathsf{W}_k$ do not contain $\C$ ensures that $\mathsf{W}_1 \mspace{1mu} \dots \mspace{1mu} \mathsf{W}_\Delta$ is not contained in the condensed configuration $\ensuremath{\operatorname{disj}}(\gen{\gen{\C}})^{\Delta - x} \mspace{1mu} \ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{X}}})^x$.
Also, since for any $1 \leq i \leq \beta$ both $\mathsf{A}_i$ and $\mathsf{B}_i$ are at least as weak as $\mathsf{B}_1$ according to $\ensuremath{\mathcal{E}}_{\Delta, \beta}(v, x)$, $\mathsf{W}_1$ does not contain any $\mathsf{A}_i$ or $\mathsf{B}_i$ due to the right-closedness of $\mathsf{W}_1$, which ensures that $\mathsf{W}_1 \mspace{1mu} \dots \mspace{1mu} \mathsf{W}_\Delta$ is not contained in the condensed configuration $\ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{A}_i}}) \mspace{1mu} \ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{B}_i}})^{\Delta-1}$, for any $1 \leq i \leq \beta$.
Hence, $\mathsf{W}_1 \mspace{1mu} \dots \mspace{1mu} \mathsf{W}_\Delta$ is not a node configuration of $\re( \Pi_{\Delta,\beta}(v,x) )$, and we conclude that $\mathcal U$ is not a node configuration of $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$.
This yields the desired contradiction and proves the lemma.
\end{proof}
Again, we prove an analogous statement of Lemma \ref{lem:industep}, for the case where $i=\beta$.
\begin{lemma}\label{lem:industop}
Let $v, x$ satisfy $\size(v) \cdot (x+1) + 1 \leq \Delta$.
Then any node configuration of $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$ can be relaxed to some configuration in $\mathcal Z_{\Delta, \beta}(v,x)$.
\end{lemma}
\begin{proof}
This proof is essentially a special case of the proof of Lemma~\ref{lem:industep}, where $i = \beta$.
For a contradiction, assume that the lemma is false, and let $\mathcal U = \U_1 \mspace{1mu} \dots \mspace{1mu} \U_\Delta$ be a node configuration of $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$ such that $\mathcal U$ cannot be relaxed to any configuration in $\mathcal Z_{\Delta, \beta}(v,x)$.
By Lemma~\ref{lem:industep} and Lemma~\ref{lem:industart}, we know that for each $1 \leq k \leq \Delta$, any set contained in $\U_k$ contains $\mathsf{B}_\beta$.
Hence, the premises of Lemma~\ref{lem:industep} are satisfied for $i = \beta$, and by an analogous argumentation to the one in the proof of Lemma~\ref{lem:industep}, we obtain the following: we can pick one set $\mathsf{W}_k$ from each $\U_k$, such that for each color $\C$, there are at least $(x+1)$ distinct $k$ such that $\mathsf{W}_k$ does not contain $\C$, and each $\mathsf{W}_k$ does not contain $\mathsf{A}_\beta$.
We note that in our case of $i = \beta$, we do not pick a special set $\mathsf{W}_1$, and that the condition $i \leq \beta - 1$ in Lemma~\ref{lem:industep} is not used in the parts of the proof we use here analogously.
Now, similarly as before, the fact that there are at least $(x+1)$ distinct $k$ such that $\mathsf{W}_k$ does not contain $\C$ ensures that $\mathsf{W}_1 \mspace{1mu} \dots \mspace{1mu} \mathsf{W}_\Delta$ is not contained in the condensed configuration $\ensuremath{\operatorname{disj}}(\gen{\gen{\C}})^{\Delta - x} \mspace{1mu} \ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{X}}})^x$, and the fact that $\mathsf{A}_\beta$ is not contained in any $\mathsf{W}_k$ ensures that, for any $1 \leq i \leq \beta$, label $\mathsf{A}_i$ is not contained in any $\mathsf{W}_k$, which in turn implies that $\mathsf{W}_1 \mspace{1mu} \dots \mspace{1mu} \mathsf{W}_\Delta$ is not contained in the condensed configuration $\ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{A}_i}}) \mspace{1mu} \ensuremath{\operatorname{disj}}(\gen{\gen{\mathsf{B}_i}})^{\Delta-1}$, for any $1 \leq i \leq \beta$.
Again, we obtain a contradiction to the fact that $\mathcal U$ is a node configuration of $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$.
\end{proof}
We now prove that, given a solution for $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$, problem $\Pi_{\Delta, \beta}(v', x')$ can be solved in $0$ rounds, for some specific value of $v'$ and $x'$. We will exploit the fact that any node configuration of $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$ can be relaxed to some node configuration of $\Pi_{\Delta, \beta}(v', x')$, and we will additionally relate the edge constraints of the two problems.
\begin{lemma}\label{lem:rerere}
Let $v, x$ satisfy $\size(v) \cdot (x+1) + 1 \leq \Delta$, and set $v' := [v'_0, \dots, v'_\beta]$, $v'_i := \sum_{j=0}^i v_j$ for all $0 \leq i \leq \beta$, and $x' := \size(v) \cdot (x+1)$.
Then, given a solution for $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$, problem $\Pi_{\Delta, \beta}(v', x')$ can be solved in $0$ rounds.
\end{lemma}
\begin{proof}
Denote the set of colors contained in $\Sigma_{\Delta,\beta}(v,x)$ by $\mathcal C$, and the set of colors contained in $\Sigma_{\Delta,\beta}(v',x')$ by $\mathcal D$.
W.l.o.g, we can assume that $\mathcal C$ and $\mathcal D$ are disjoint, as we can simply rename colors if they are not disjoint.
Furthermore, we will use $\mathsf{A}_i$, $\mathsf{B}_i$, and $\mathsf{X}$ as usual when talking about labels from $\Sigma_{\Delta,\beta}(v,x)$, and use $\mathsf{A}'_i$, $\mathsf{B}'_i$, and $\mathsf{X}'$ instead when talking about labels from $\Sigma_{\Delta,\beta}(v',x')$.
We transform a solution for $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$ into a solution for $\Pi_{\Delta, \beta}(v', x')$ in $0$ rounds as follows.
Let $f: \{ (\C, i) \mid \C \in \mathcal C, 0 \leq i \leq \beta, g(\C) \leq i \} \to \mathcal D$ be an arbitrary, but fixed, bijection such that $g(f((\C, i))) = i$, i.e., such that any pair $(\C, i)$ is mapped to a color in $\mathcal D$ that is contained in group $i$.
By the definition of $v'$, such a bijection exists.
Consider a node $u$ and let $\mathcal U = \U_1 \mspace{1mu} \dots \mspace{1mu} \U_\Delta$ be the node configuration around $u$ in the given solution for $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$.
Node $u$ starts by choosing a configuration $\mathcal U' = \U'_1 \mspace{1mu} \dots \mspace{1mu} \U'_\Delta \in \mathcal Z_{\Delta, \beta}(v, x)$ such that $\U_k \subseteq \U'_k$ for each $1 \leq k \leq \Delta$, and replacing each $\U_k$ with $\U'_k$ on the respective incident edges.
By Lemma~\ref{lem:industop}, such a configuration exists.
Then, $u$ proceeds as follows.
If $\mathcal U' = \gen{\gen{\mathsf{A}_i}} \mspace{1mu} \gen{\gen{\mathsf{B}_i}}^{\Delta-1}$ for some $1 \leq i \leq \beta$, then node $u$ simply replaces $\gen{\gen{\mathsf{A}_i}}$ with $\mathsf{A}'_i$ and $\gen{\gen{\mathsf{B}_i}}$ with $\mathsf{B}'_i$ on the respective incident edges to produce its output.
If, for some $\C \in \mathcal C$ satisfying $g(\C) = 0$, $\mathcal U' = \gen{\gen{\C}}^{\Delta - \size(v) \cdot (x+1)} \mspace{1mu} (\Sigma'_{\Delta,\beta}(v,x))^{\size(v) \cdot (x+1)}$, then $u$ replaces $\gen{\gen{\C}}$ with $f((\C, 0))$ and $(\Sigma'_{\Delta,\beta}(v,x))$ with $\mathsf{X}'$ on the respective incident edges to produce its output.
If $\mathcal U' = \gen{\gen{\C, \mathsf{B}_i}, \gen{\mathsf{A}_i}}^{\Delta - \size(v) \cdot (x+1)} \mspace{1mu} (\Sigma'_{\Delta,\beta}(v,x))^{\size(v) \cdot (x+1)}$ for some $(\C, i)$ with $\max\{ 1, g(\C) \} \leq i \leq \beta$, then $u$ replaces $\gen{\gen{\C, \mathsf{B}_i}, \gen{\mathsf{A}_i}}$ with $f((\C, i))$ and $(\Sigma'_{\Delta,\beta}(v,x))$ with $\mathsf{X}'$ on the respective incident edges to produce its output.
In the following, we show that this yields a correct output for $\Pi_{\Delta, \beta}(v', x')$.
As the nodes clearly produce node configurations that are contained in $\ensuremath{\mathcal{N}}_{\Delta,\beta}(v',x')$, the only thing to be shown is that the output pair on each edge is an edge configuration contained in $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v',x')$.
Hence, consider an edge $\{u, w\}$, and let $\U \mspace{1mu} \mathsf{W}$ be the edge configuration on $\{ u, w \}$ in the solution for $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$.
Let $\U'$ be the label with which $u$ replaces $\U$ (in the first step), and $\mathsf{W}'$ the label $w$ replaces $\mathsf{W}$ with, and let $\U'' \mspace{1mu} \mathsf{W}''$ be the final output on edge $\{u, w\}$ (after the second replacement).
We now show that the above $0$-round algorithm does not produce an incorrect edge configuration, by going through all incorrect edge configurations one by one and showing that they do not occur.
For this, we use that the correctness of $\U \mspace{1mu} \mathsf{W}$ (i.e., the fact that $\U \mspace{1mu} \mathsf{W}$ is contained in the edge constraint of $\rere(\re( \Pi_{\Delta,\beta}(v,x) ))$) implies that there exist sets $\U^* \in \U'$ and $\mathsf{W}^* \in \mathsf{W}'$ such that for all pairs $(\u^*, \mathsf{w}^*)$ with $\u^* \in \U^*$, $\mathsf{w}^* \in \mathsf{W}^*$ we have that $\u^* \mspace{1mu} \mathsf{w}^*$ is an edge configuration contained in $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$.
This follows by the definitions of $\re(\cdot)$ and $\rere(\cdot)$, and the fact that $\U \subseteq \U'$ and $\mathsf{W} \subseteq \mathsf{W}'$.
In the following, $\U^*$ and $\mathsf{W}^*$ will always be \emph{arbitrarily chosen} sets contained in $\U'$ and $\mathsf{W}'$, respectively, and $\u^*$ and $\mathsf{w}^*$ will be picked suitably from $\U^*$ and $\mathsf{W}^*$, respectively.
If $\U'' = \C' = \mathsf{W}''$ for some $\C' \in \mathcal D$ satisfying $g(\C') \geq 1$, then we have $\U' = \gen{\gen{\C, \mathsf{B}_i}, \gen{\mathsf{A}_i}} = \mathsf{W}'$ for some $(\C, i)$ with $\max\{ 1, g(\C) \} \leq i \leq \beta$.
Hence, by Lemma~\ref{lem:restrong}, we have $\gen{\C, \mathsf{B}_i} \subseteq \U^*$ or $\gen{\mathsf{A}_i} \subseteq \U^*$, which implies that $\{\C, \mathsf{B}_i\} \subseteq \U^*$ or $\mathsf{A}_i \in \U^*$.
Since analogous statements hold for $\mathsf{W}^*$, we see that we can pick $\u^*$ and $\mathsf{w}^*$ such that $\u^* \mspace{1mu} \mathsf{w}^*$ is one of the configurations listed in $\{ \mathsf{A}_i \mspace{1mu} \mathsf{A}_i, \C \mspace{1mu} \C , \mathsf{A}_i \mspace{1mu} \mathsf{B}_i \}$.
Since neither of the listed configurations is contained in $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$, we obtain the desired contradiction.
If $\U'' = \C' = \mathsf{W}''$ for some $\C' \in \mathcal D$ satisfying $g(\C') = 0$, then we have $\U' = \gen{\gen{\C}} = \mathsf{W}'$ for some $\C \in \mathcal C$.
Analogously to the previous case, we obtain $\C \in \U^*$ and $\C \in \mathsf{W}^*$.
Picking $\u^* = \C = \mathsf{w}^*$ yields the desired contradiction as $\C \mspace{1mu} \C \notin \ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$.
If $\U'' = \mathsf{A}'_i$, $\mathsf{W}'' = \mathsf{B}'_j$ where $i \leq j$, then $\U' = \gen{\gen{A_i}}$, $\mathsf{W}' = \gen{\gen{B_j}}$.
Analogously to before, we obtain $\mathsf{A}_i \in \U^*$ and $\mathsf{B}_j \in \mathsf{W}^*$.
Picking $\u^* = \mathsf{A}_i$ and $\mathsf{w}^* = \mathsf{B}_j$ yields the desired contradiction since $\mathsf{A}_i \mspace{1mu} \mathsf{B}_j \notin \ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$ if $i \leq j$.
If $\U'' = \mathsf{A}'_i$, $\mathsf{W}'' = \C'$ where $i \leq g(\C')$, then $\U' = \gen{\gen{A_i}}$, $\mathsf{W}' = \gen{\gen{\C, \mathsf{B}_j}, \gen{\mathsf{A}_j}}$ where $i \leq j$ and $g(\C) \leq j$.
Analogously to before, we obtain $\mathsf{A}_i \in \U^*$, and $\{\C, \mathsf{B}_j\} \subseteq \mathsf{W}^*$ or $\mathsf{A}_j \in \mathsf{W}^*$.
Hence, we can pick $\u^* = \mathsf{A}_i$ and $\mathsf{w}^* = \mathsf{A}_j$, or $\u^* = \mathsf{A}_i$ and $\mathsf{w}^* = \mathsf{B}_j$; in both cases, we obtain a contradiction as both $\mathsf{A}_i \mspace{1mu} \mathsf{A}_j$ and $\mathsf{A}_i \mspace{1mu} \mathsf{B}_j$ are not contained in $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v,x)$ if $i \leq j$.
Finally, if $\U'' = \mathsf{A}'_i$, $\mathsf{W}'' = \mathsf{A}'_j$ for some $1 \leq i, j \leq \beta$, then $\U' = \gen{\gen{A_i}}$, $\mathsf{W}' = \gen{\gen{A_j}}$.
Analogously to before, we obtain $\mathsf{A}_i \in \U^*$ and $\mathsf{A}_j \in \mathsf{W}^*$.
By picking $\u^* = \mathsf{A}_i$ and $\mathsf{w}^* = \mathsf{A}_j$, we obtain the desired contradiction.
As these cases cover all the edge configurations that are not contained in $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v',x')$ (up to exchanging $\U''$ and $\mathsf{W}''$), and none of the cases can occur, it follows that the output pair on edge $\{ u, w \}$ is an edge configuration contained in $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v',x')$.
Hence, the output produced by our $0$-round algorithm is correct.
\end{proof}
We now prove a lemma that is basically the result of applying Lemma \ref{lem:rerere} multiple times. This is our main result of this section; it essentially shows a lower bound for ruling sets for the port numbering model, with the additional property of providing a family of problems that satisfies the round elimination property, such that all problems of the family have a number of labels that is not too large.
\begin{lemma}\label{lem:lbfamily}
Let $\beta \geq 1$, and let $t = 1/2 \cdot \log \Delta / (\beta \log \log \Delta)$.
If $\beta \leq t$, then there exists a sequence $\Pi_0 \rightarrow \Pi_1 \rightarrow \dots \rightarrow \Pi_t$ of problems such that
\begin{enumerate}[label=(\roman*)]
\item \label{item:eins} $\Pi_0 = \Pi_{\Delta,\beta}([1, 0, 0, \dots, 0], 0)$,
\item \label{item:zwei} $\Pi_{j+1}$ can be solved in $0$ rounds given a solution to $\rere(\re(\Pi_j))$, for all $0 \leq j \leq t-1$,
\item \label{item:drei} $\Pi_{t-1}$ cannot be solved in $0$ rounds in the deterministic PN model.
\end{enumerate}
\end{lemma}
\begin{proof}
For all $0 \leq j \leq t$, define $\Pi_j := \Pi_{\Delta,\beta}(v^{(j)}, x^{(j)})$, where $v^{(j)} = [v^{(j)}_0, \dots, v^{(j)}_\beta]$ and $ x^{(j)}$ are recursively defined by $v^{(j+1)}_k := \sum_{i=0}^k v^{(j)}_i$, for all $0 \leq k \leq \beta$, and $x^{(j+1)} := \size(v^{(j)}) \cdot (x^{(j)} + 1)$, where we set $v^{(0)} := [1, 0, 0, \dots, 0]$ and $x^{(j)} := 0$.
This definition immediately ensures Property~$\ref{item:eins}$.
In order to show Property~$\ref{item:zwei}$, we would like to use Lemma~\ref{lem:rerere}, so we have to make sure that $\size(v^{(j)})$ and $x^{(j)}$ satisfy the condition given in Lemma~\ref{lem:rerere} for all $0 \leq j \leq t-1$.
To this end, we can apply Lemma \ref{lem:colorgrowth}, that proves that for all $1 \leq j \leq t$ and all $0 \leq k \leq \beta$, we have $v^{(j)}_k = {j+k-1 \choose k}$.
Hence, we can now bound $\size(v^{(j)})$ and $x^{(j)}$ from above.
Since $\size(v^{(j)})$ and $x^{(j)}$ are monotonically increasing (with increasing $j$), it suffices to bound $\size(v^{(t-1)})$ and $x^{(t-1)}$ to obtain bounds for $\size(v^{(j)})$ and $x^{(j)}$, for all $0 \leq j \leq t-1$.
As $\size(v^{(t-1)}) = v^{(t)}_\beta$ by definition of $v^{(t)}_\beta$, we have $\size(v^{(t-1)}) = {t+\beta-1 \choose \beta} \leq (2t)^\beta-1$ since $\beta \leq t$.
This implies that $x^{(j)} \leq (2t)^{j\beta}-1$, for all $0 \leq j \leq t-1$, by induction: clearly, $x^{(0)} = 0 \leq (2t)^{0}-1$, and using the induction hypothesis $x^{(j-1)} \leq (2t)^{(j-1)\beta}-1$, we obtain $x^{(j)} \leq \size(v^{(j-1)}) \cdot (x^{(j-1)} + 1) \leq ((2t)^\beta-1) \cdot (2t)^{(j-1)\beta} \leq (2t)^{j\beta}-1$, for all $1 \leq j \leq t-1$.
In particular, $x^{(t-1)} \leq (2t)^{(t-1)\beta}-1$.
Hence, for any $0 \leq j \leq t-1$, we have $\size(v^{(j)}) \cdot (x^{(j)} + 1) + 1 \leq \size(v^{(t-1)}) \cdot (x^{(t-1)} + 1) + 1 \leq ((2t)^\beta-1) \cdot (2t)^{(t-1)\beta} + 1 \leq (2t)^{(t\beta)} \leq (\log \Delta / (\beta \log \log \Delta))^{\log \Delta / (2 \log \log \Delta)} \leq \Delta$.
Therefore, for all $0 \leq j \leq t-1$, the condition $\size(v) \cdot (x+1) + 1 \leq \Delta$ in Lemma~\ref{lem:rerere} is satisfied by setting $v := v^{(j)}$ and $x := x^{(j)}$, and Property~$\ref{item:zwei}$ follows by applying Lemma~\ref{lem:rerere} inductively.
Now, we will prove Property~$\ref{item:drei}$.
For a contradiction, assume that there is a deterministic $0$-round algorithm solving $\Pi_{t-1}$ in the PN model.
As the algorithm is deterministic, no node has any information about other nodes, and the nodes of the input graph are not distinguished by unique identifiers, the labels that the nodes output must form the same node configuration around every node.
Let $\mathcal U = \U_1 \mspace{1mu} \dots \mspace{1mu} U_\Delta$ denote this node configuration, and w.l.o.g., assume that each node $u$ outputs $\U_k$ on the incident edge connected to $u$ via port $k$.
In order for the algorithm to be correct, we must have $\mathcal U \in \ensuremath{\mathcal{N}}_{\Delta,\beta}(v^{(t-1)},x^{(t-1)})$.
Since, by the above calculations, $x^{(t-1)} \leq \Delta - 1$, we see that the number of wildcards in the problem definition is strictly less than $\Delta$.
Hence, we have $\U_k = \mathsf{A}_i$ or $\U_k = \C$, for some $1 \leq k \leq \Delta$, some index $1 \leq i \leq \beta$, and some color $\C \in \mathcal C$, according to the definition of $\Pi_{t-1} = \Pi_{\Delta,\beta}(v^{(t-1)},x^{(t-1)})$.
Now consider a graph, where some edge $e = \{ u, w \}$ is connected to both endpoints via port $k$.
If $\U_k = \mathsf{A}_i$, then the output produces the edge configuration $\mathsf{A}_i \mspace{1mu} \mathsf{A}_i$ on edge $e$; if $\U_k = \C$, then the output produces the edge configuration $\C \mspace{1mu} \C$ on $e$.
Since both configurations are not contained in $\ensuremath{\mathcal{E}}_{\Delta,\beta}(v^{(t-1)},x^{(t-1)})$, we obtain the desired contradiction.
This concludes the proof of Property~$\ref{item:drei}$.
\end{proof}
\section{Lifting results to the LOCAL model}\label{sec:liftlocal}
In the previous sections we obtained upper and lower bounds for the port numbering model. While upper bounds trivially hold also for the LOCAL model, lower bounds need some additional analysis to be lifted to the LOCAL model. The main challenge here is that the round elimination theorem does not tolerate the presence of node identifiers.
Previous works that used the round elimination technique to prove lower bounds \cite{Balliu2019,binary,trulytight} followed a common approach to lift an $f(\Delta)$ lower bound for the port numbering model to a lower bound as a function of $\Delta$ and $n$ for the LOCAL model, and we will follow the same approach to lift our lower bounds as well. We perform such a lifting of our results in the (usual) following way:
\begin{itemize}
\item We first adapt our lower bound proof to randomized algorithms, showing that if $\Pi_{\Delta,\beta}(v,x)$ can be solved in $t$ rounds with local failure probability $p$, then $\rere(\re(\Pi_{\Delta,\beta}(v,x)))$ can be solved in $t-1$ rounds with some local failure probability $p'$.
\item We then give a lower bound on the failure probability of any algorithm that runs ``too fast'', that is, in strictly fewer rounds than the $t$ given in Lemma~\ref{lem:lbfamily} (i.e., than the implicit lower bound proved in Section~\ref{sec:lb}). This yields a lower bound for the runtime of any randomized algorithm in the port numbering model as a function of $\Delta$.
\item We make $\Delta$ as large as possible, as a function of $n$, in order to obtain the best possible lower bound (for randomized algorithms in the port numbering model) as a function of $n$.
\item Randomized algorithms in the port numbering model can generate unique IDs with high probability and then simulate algorithms for the LOCAL model. Thus we get a lower bound for randomized algorithms in the LOCAL model.
\item Lower bounds for randomized algorithms in the LOCAL model imply lower bounds for deterministic algorithms as well. We use standard techniques to obtain \emph{better} lower bounds (as a function of $n$) for deterministic algorithms.
\end{itemize}
We will provide lower bounds for high-girth regular graphs, in particular in graphs where the girth is larger than the running time. Such lower bounds directly apply on trees as well.
\subsection{Evolution of local failure probability}
Balliu et al.~\cite{binary} proved that, given some problem $\Pi$ defined on graphs of degree $\Delta$ using labels from a set $\Sigma$, we can upper bound the local failure probability $p'$ of any algorithm solving $\rere(\re(\Pi))$ in $t-1$ rounds by a function that depends only on $\Delta$, $|\Sigma|$, and $p$, where $p$ is an upper bound on the local failure probability of an algorithm solving $\Pi$ in $t$ rounds. More formally, the authors prove the following result in \cite{binary} (rephrased for our purposes), which is a version of the round elimination theorem that applies to randomized algorithms.
\begin{lemma}[Lemma 41 of \cite{binary}]\label{lem:singlestep}
Let $A$ be a randomized $t$-round algorithm for $\Pi$ with local failure probability at most $p$ (where $t>0$). Then there exists a randomized $(t-1)$-round algorithm $A'$ for $\rere(\re(\Pi))$ with local failure probability $p'' \le 2^\frac{1}{\Delta+1} (\Delta |\Sigma'|)^\frac{\Delta}{\Delta+1} {p'}^\frac{1}{\Delta+1} + p'$, where $p' \le 2^\frac{1}{\Delta+1} (\Delta |\Sigma|)^\frac{\Delta}{\Delta+1} p^\frac{1}{\Delta+1} + p$ and $\Sigma'$ is the label set of $\re(\Pi)$.
\end{lemma}
This lemma basically states that if there exists an algorithm that solves some problem $\Pi$ in $t$ rounds with some small failure probability $p$, then there also exists a faster algorithm that solves $\rere(\re(\Pi))$ with some possibly larger but still small enough failure probability. Let $\hat{\re}(\Pi) = \rere(\re(\Pi))$, and let $\hat{\re}^j(\Pi)$ be the function that recursively applies the $\hat{\re}$ function $j$ times. By using Lemma~\ref{lem:singlestep} multiple times, we can give an upper bound on the failure probability of an algorithm solving $\hat{\re}^j(\Pi)$ in $\max\{0,t-j\}$ rounds, as a function of an upper bound on the failure probability of an algorithm solving $\Pi$ in $t$ rounds.
Unfortunately, we cannot directly apply this lemma multiple times to $\Pi_{\Delta,\beta}([1,0,\ldots,0],0)$, as we do not know an upper bound on the number of labels of the problem $\hat{\re}^{j-1}(\Pi_{\Delta,\beta}([1,0,\ldots,0],0))$ and of all the intermediate problems (notice that the number of labels appears in the bound given by Lemma \ref{lem:singlestep}). But what we can do instead is to consider the family of problems $\Pi_i$ defined in Lemma \ref{lem:lbfamily}, where $\Pi_0$ is the locally checkable version of the $(2,\beta)$-ruling set problem, and $\Pi_{i+1}$ is a problem that can be obtained by relaxing the configurations in the node constraint of $\rere(\re(\Pi_{i}))$. By applying Lemma \ref{lem:singlestep} we can conclude that, if we have an algorithm for $\Pi_i$ running in $t>0$ rounds that fails with probability at most $p$, then there exists an algorithm solving $\Pi_{i+1}$ running in $t-1$ rounds that fails with probability at most $p''$.
Moreover, for any problem $\Pi_i$ where index $i$ is at most the $t$ from Lemma~\ref{lem:lbfamily}, we have that the number of colors, and hence the number of distinct configurations in the node constraint of $\Pi_i$, is in $O(\Delta)$.
We now prove a bound on the failure probability obtained by applying Lemma \ref{lem:singlestep} multiple times. We will prove a general statement that not only applies to the family of problems defined in Lemma \ref{lem:lbfamily}, but to all problems for which the number of labels is bounded by $O(\Delta^2)$. Since the number of labels that we consider are bounded by $O(\Delta^2)$, and since the number of labels of the intermediate problems are always at most exponential in the original number of labels, we can now apply the lemma multiple times, assuming $|\Sigma| = O(\Delta^2)$ and $|\Sigma'| = 2^{O(\Delta^2)}$ for the whole family. Hence, we prove the following lemma.
\begin{lemma}\label{lem:multiplesteps}
Let $\Pi_0 \rightarrow \Pi_1 \rightarrow \dots \rightarrow \Pi_t$ be a sequence of problems such that $\Pi_{i+1}$ can be solved in $0$ rounds given a solution for $\rere(\re(\Pi_i))$, and such that the number of labels of each problem $\Pi_i$ is upper bounded by $O(\Delta^2)$.
Let $A$ be a randomized $t$-round algorithm for $\Pi_0$ with local failure probability at most $p$. Then there exists a randomized $(t-j)$-round algorithm $A'$ for $\Pi_{j}$ with local failure probability at most $2^{K\Delta^2} p^{1/(\Delta+1)^{2j}}$ for some constant $K$, for all $0<j \le t$, for large enough $\Delta$.
\end{lemma}
\begin{proof}
For large enough $\Delta$, and for our choice of parameters, by Lemma \ref{lem:singlestep} we have that
\begin{align*}
p'' &\le 2^\frac{1}{\Delta+1} (\Delta |\Sigma'|)^\frac{\Delta}{\Delta+1} {p'}^\frac{1}{\Delta+1} + p' \\
&\le (\Delta |\Sigma'|)^\frac{1}{\Delta+1} (\Delta |\Sigma'|)^\frac{\Delta}{\Delta+1} {p'}^\frac{1}{\Delta+1} + p'\\
&\le (\Delta |\Sigma'|) {p'}^\frac{1}{\Delta+1} + p'\\
&\le (2^{\log\Delta} 2^{|\Sigma|}) {p'}^\frac{1}{\Delta+1} + p'\\
&\le 2^{O(\Delta^2)} {p'}^\frac{1}{\Delta+1}.
\end{align*}
Similarly,
\begin{align*}
p' &\le 2^\frac{1}{\Delta+1} (\Delta |\Sigma|)^\frac{\Delta}{\Delta+1} {p}^\frac{1}{\Delta+1} + p \\
&\le (\Delta |\Sigma'|)^\frac{1}{\Delta+1} (\Delta |\Sigma'|)^\frac{\Delta}{\Delta+1} {p}^\frac{1}{\Delta+1} + p\\
&\le 2^{O(\Delta^2)} {p}^\frac{1}{\Delta+1}.
\end{align*}
Hence, this implies that
\begin{align*}
p''&\le 2^{K \Delta^2} p'^{\frac{1}{\Delta+1}}, \text{ where }\\
p' &\le 2^{K \Delta^2} p^{\frac{1}{\Delta+1}},
\end{align*}
for some constant $K$. By recursively applying Lemma \ref{lem:singlestep} we get the following:
\[
p_j \le 2^{K \Delta^2} p_{j-1}^{\frac{1}{\Delta+1}},
\]
where $p_0=p$ and $p_{2j}$, are, respectively, the local failure probability bounds for $\Pi_{0}$ and $\Pi_{j}$. We prove by induction that for all $j>0$,
\[
p_j \le 2^{K \sum_{i=1}^{j} \Delta^{3-i}} p^{\frac{1}{(\Delta+1)^j}}.
\]
For the base case where $j=1$, we get that $p_1 \le 2^{K \Delta^2} p^{\frac{1}{\Delta+1}}$, which holds, as we showed above. Let us assume that the claim holds for $j$, and let us prove it for $j+1$.
\begin{align*}
p_{j+1} &\le 2^{K \Delta^2} p_{j}^{\frac{1}{\Delta+1}} \le 2^{K \Delta^2} \left( 2^{K \sum_{i=1}^{j} \Delta^{3-i}} p^{\frac{1}{(\Delta+1)^j}} \right)^{\frac{1}{\Delta+1}} \\
&\le 2^{K \Delta^2} 2^{K \sum_{i=1}^{j} \Delta^{3-i-1}} p^{\frac{1}{(\Delta+1)^{j+1}}} = 2^{K \sum_{i=1}^{j+1} \Delta^{3-i}} p^{\frac{1}{(\Delta+1)^{j+1}}}
\end{align*}
Since for $\Delta \ge 2$ and for any $j$, $\sum_{i=1}^{j} \Delta^{3-i} \le \Delta^2 + \Delta + 2$, we get the following:
\[p_{j} \le 2^{K(\Delta^2 + \Delta + 2)} p^{\frac{1}{(\Delta+1)^j}} \le 2^{K'\Delta^2} p^{\frac{1}{(\Delta+1)^j}},
\] for some constant $K'$, hence the claim follows.
\end{proof}
We now give a lower bound on the failure probability of any algorithm solving $\Pi_i$ in $0$ rounds, for any $i < t$, where $t$ is the lower bound obtained in Section \ref{sec:lb} for the time complexity of $\Pi_0$. Again, we will prove a stronger result, that applies to any family of problems where the number of allowed configurations is $O(\Delta^2)$, a condition that is satisfied by the family of Lemma \ref{lem:lbfamily}, since there are $\beta = o(\Delta)$ possible pointer configurations and at most $O(\Delta)$ color configurations.
\begin{lemma}\label{lem:zerorounds}
Let $\Pi$ be a problem that cannot be solved in $0$ rounds with deterministic algorithms in the port numbering model, such that the node constraint $\ensuremath{\mathcal{N}}$ contains $O(\Delta^2)$ allowed configurations. Then any $0$-round algorithm solving $\Pi$ must fail with probability at least $1/\Delta^8$.
\end{lemma}
\begin{proof}
We follow the same strategy as in \cite[Lemma 6.4]{trulytight}. Since in $0$ rounds of communication all nodes have the same information, we can see any $0$-round algorithm as a probability assignment to each node configuration. That is, for each $\c_i \in \ensuremath{\mathcal{N}}$, the algorithm outputs $\c_i$ with probability $p_i$, such that $\sum p_i = 1$. Hence, by the pigeonhole principle, there exists some configuration $\bar{\c}$ that all nodes output with probability at least $\frac{1}{K \Delta^2}$ for some constant $K$. Since by assumption $\Pi$ is not $0$-rounds solvable, then there exists no node configuration such that, for all pairs of choices $\ell_1 \mspace{1mu} \ell_2$ over such configuration, $\ell_1 \mspace{1mu} \ell_2$ is in $\ensuremath{\mathcal{E}}$, i.e., in the edge constraint of $\Pi$ (otherwise the problem would be $0$-round solvable in the port numbering model, by making all nodes output that configuration). Thus, there exist $2$ (possibly the same) labels $\ell_1$ and $\ell_2$ appearing in $\bar{\c}$ such that $\ell_1\mspace{1mu} \ell_2$ is not in $\ensuremath{\mathcal{E}}$. Moreover, conditioned on the fact that the node is outputting the configuration $\bar{\c}$, since $\ell_1$ and $\ell_2$ appear at least once in such configuration, there must exist some ports $i$ and $j$ where $\ell_1$ and $\ell_2$ are written with probability at least $\frac{1}{\Delta}$. Thus, two neighboring nodes connected through ports $i$ and $j$ will both output configuration $\bar{\c}$ and produce the invalid edge configuration $\ell_1 \mspace{1mu} \ell_2$ with probability at least $\frac{1}{(K \Delta^3)^2}$, that for large enough $\Delta$ is at least $1 / \Delta^8$.
\end{proof}
We now bound the failure probability of any algorithm solving $\Pi_0$ in strictly less than $t$ rounds.
\begin{lemma}\label{lem:randomizedpnlb}
Let $\Pi_0 \rightarrow \Pi_1 \rightarrow \dots \rightarrow \Pi_t$ be a sequence of problems such that $\Pi_{i+1}$ can be solved in $0$ rounds given a solution for $\rere(\re(\Pi_i))$, the number of labels of each problem $\Pi_i$ is upper bounded by $O(\Delta^2)$, and the node constraint of each problem $\Pi_i$ contains $O(\Delta^2)$ allowed configurations. Let $t$ be a number satisfying that, for all $t' < t$, $\Pi_{t'}$ is not $0$-round solvable in the port numbering model using deterministic algorithms. Any algorithm for $\Pi_0$ running in strictly less than $t$ rounds must fail with probability at least $\frac{1}{2^{\Delta^{4t}}}$, if $\Delta$ is large enough.
\end{lemma}
\begin{proof}
By applying Lemma \ref{lem:multiplesteps} we get that an algorithm solving $\Pi_0$ in $t'<t$ rounds with local failure probability at most $p$ implies an algorithm solving $\Pi_{t'}$ in $0$ rounds with local failure probability at most $2^{K\Delta^2} p^{1/(\Delta+1)^{2t'}}$. Then, since $\Pi_{t'}$ is not $0$-round solvable in the port numbering model using deterministic algorithms by assumption, by applying Lemma \ref{lem:zerorounds} we get the following:
\[
2^{K\Delta^2} p^{1/(\Delta+1)^{2t'}} \ge \frac{1}{\Delta^8},
\]
that for large enough $\Delta$ implies the following:
\[
p \ge \frac{1}{2^{(K\Delta^2 + 8\log\Delta)(\Delta+1)^{2t'}}} \ge \frac{1}{2^{\Delta^3(\Delta+1)^{2t'}}} \ge \frac{1}{2^{\Delta^{4t'+3}}} \ge \frac{1}{2^{\Delta^{4t}}}
\]
\end{proof}
\subsection{Making $\Delta$ as large as possible}
By Lemma \ref{lem:lbfamily} we know that, for $t = 1/2 \cdot \log \Delta / (\beta \log \log \Delta)$, if $\beta \le t$, problem $\Pi_{\Delta,\beta}([1,0,\ldots,0],0)$ is not $t$-round solvable in the port numbering model using deterministic algorithms, and each problem of the family used to prove this result uses $O(\Delta)$ labels, and $O(\Delta)$ node configurations. Hence, by combining Lemma \ref{lem:lbfamily} and Lemma \ref{lem:randomizedpnlb} we get that any randomized algorithm for problem $\Pi_{\Delta,\beta}([1,0,\ldots,0],0)$ running in $t = o(\frac{\log \Delta}{\beta \log \log \Delta})$ rounds must fail with probability at least $\frac{1}{2^{\Delta^{4t}}}$, in some $\Delta$-regular neighborhood of girth at least $2t+2$ (condition required by the round elimination theorem, see Theorem \ref{thm:sebastien}), in the randomized port numbering model, provided $\beta \le t$.
We are now ready to lift this bound to the LOCAL model. The main idea is that, since in the randomized port numbering model nodes can generate unique IDs with high probability, the existence of an algorithm for the randomized LOCAL model with some failure probability directly implies the existence of an algorithm for the randomized port numbering model with roughly the same failure probability (up to a factor of $1-1/n^c$ for an arbitrary constant $c$, that is the success probability of the randomized ID generation process). Hence, a lower bound for the randomized port numbering model directly applies to the LOCAL model as well. We first prove Theorem \ref{thm:randlb}, and then we obtain Corollary \ref{cor:randlb} by taking a value of $\Delta$, as a function of $n$, that maximizes the result of the theorem.
\randlb*
\begin{proof}
As mentioned above, any randomized algorithm solving $\Pi_{\Delta,\beta}([1,0,\ldots,0],0)$ in the LOCAL model running in $t = o(\frac{\log \Delta}{\beta \log \log \Delta})$ rounds must fail with probability at least $\frac{1}{2^{\Delta^{4t}}}$, in some $\Delta$-regular neighborhood of girth at least $2t+2$, provided that $\beta \le t$. Later we will show that, for the choice of parameters stated in the theorem, we have $t-\beta = \Omega(t)$, which implies that $\beta \le t$. Hence, to prove the theorem, we need to show how large we can make $t$ such that:
\begin{itemize}
\item the failure probability is still too large, that is, $\frac{1}{2^{\Delta^{4t}}} > 1/n$, and
\item there exists a $\Delta$-regular graph of girth at least $2t+2$.
\end{itemize}
The first requirement is satisfied (for all sufficiently large $n$) if $t = o(\log_{\Delta} \log n)$. The second requirement is satisfied if $t = o(\log_\Delta n)$, since from extremal graph theory we know that, for infinite values of $n$, there exist $\Delta$-regular graphs of girth $\Omega(\log_\Delta n)$ (see, e.g., \cite{highgirth}). By combining the obtained lower bounds, we get that any randomized algorithm that succeeds w.h.p.\ requires at least $t = \Omega\left(\min \left\{ \frac{\log \Delta}{\beta \log \log \Delta} , \log_\Delta \log n \right\} \right)$ rounds.
Unfortunately, as discussed in Section \ref{sec:problems}, $\Pi_{\Delta,\beta}([1,0,\ldots,0],0)$ may be (at most) $\beta$ rounds harder than the $(2,\beta)$-ruling set problem, since $\Pi_{\Delta,\beta}([1,0,\ldots,0],0)$ requires to output some additional pointers that make the solution locally checkable. Hence, a $t$-round lower bound for $\Pi_{\Delta,\beta}([1,0,\ldots,0],0)$ only implies a lower bound of $t-\beta$ for $(2,\beta)$-ruling sets. We now prove that, for the range of values of $\beta$ stated in the theorem, the lower bound obtained for $\Pi_{\Delta,\beta}([1,0,\ldots,0],0)$ also applies to $(2,\beta)$-ruling sets, which implies the theorem. For this purpose, we prove that $t - \beta = \Omega(t)$, by considering two cases.
\begin{itemize}
\item $\sqrt{\frac{\log \Delta}{\log \log \Delta}} \le \log_\Delta \log n$. In this case, $\beta \le c \cdot \sqrt{\frac{\log \Delta}{\log \log \Delta}}$, and hence $\frac{\log \Delta}{\beta \log \log \Delta} \ge \frac{1}{c} \cdot\sqrt{\frac{\log \Delta}{\log \log \Delta}}$. We obtain that $t-\beta = \Omega\left( \min\left( \frac{\log \Delta}{\beta \log \log \Delta }-c\cdot\sqrt{\frac{\log \Delta}{\log \log \Delta}}, \log_\Delta \log n - c \cdot \sqrt{\frac{\log \Delta}{\log \log \Delta}} \right) \right)$, which, for small enough $c$, is $\Omega(t)$.
\item $\log_\Delta \log n \le \sqrt{\frac{\log \Delta}{\log \log \Delta}} $. In this case, $\beta \le c \cdot \log_\Delta \log n $. \\We obtain that $t-\beta = \Omega\left( \min\left( \frac{\log \Delta}{\beta \log \log \Delta }-c \cdot\log_\Delta \log n, (1-c) \cdot\log_\Delta \log n \right) \right)$, which, for small enough $c$, is $\Omega(t)$.
\end{itemize}
\end{proof}
We are now ready to prove Corollary \ref{cor:randlb}.
\randcor*
\begin{proof}
We apply Theorem \ref{thm:randlb} where we set $\Delta = 2^{\sqrt{\beta \log \log n \log \log \log n}}$. We obtain
\[
\frac{\log \Delta}{\beta \log \log \Delta} = \frac{\sqrt{\beta \log \log n \log \log \log n}}{\beta \log \sqrt{\beta \log \log n \log \log \log n}} = \Omega\left( \sqrt \frac{ \log \log n }{\beta \log \log \log n } \right) \enspace, \text{ and}
\]
\[
\log_\Delta \log n = \frac{\log \log n}{ \sqrt{\beta \log \log n \log \log \log n} } = \sqrt \frac{ \log \log n }{\beta \log \log \log n } \enspace.
\]
Hence, we obtain a lower bound of $t = \Omega\left(\min \left\{\frac{\log \Delta}{\beta \log \log \Delta} , \log_\Delta \log n \right\} \right) = \Omega\left( \sqrt \frac{ \log \log n }{\beta \log \log \log n } \right)$.
Note that by choosing $c$ small enough (depending on the $c$ in Theorem~\ref{thm:randlb}), we can ensure that any $\beta \le c \cdot \sqrt[3]{ \frac{\log \log n}{\log \log \log n}}$ satisfies the condition for $\beta$ given in Theorem~\ref{thm:randlb}.
\end{proof}
\subsection{Stronger deterministic lower bound}
We will now prove that, assuming that there is a fast deterministic algorithm, we can construct an even faster randomized algorithm, contradicting the results proved in Theorem \ref{thm:randlb}.
\detlb*
\begin{proof}
Assume for a contradiction that an algorithm violating the claimed bound exists and let $t(n)$ be its runtime on $n$-node graphs. We use this algorithm to construct an algorithm running within a time bound that would violate Theorem \ref{thm:randlb}. The idea is to execute such an algorithm by lying about the size of the network, by claiming that it is of size $N = \log n$.
This immediately yields an algorithm with a runtime of $O(t(N)) = O(t(\log n))$, which contradicts the bound given in Theorem~\ref{thm:randlb}.
This algorithm is even deterministic, but, in the context of Theorem~\ref{thm:randlb}, can equivalently be seen as a randomized algorithm with failure probability $0$.
In order to use this idea, we need to make sure that the algorithm does not detect such a lie. Using standard arguments (see e.g.\ \cite[Theorem 4.5]{chang16exponential}), one can show that the following conditions are sufficient for this purpose:
\begin{itemize}
\item We need to compute a new ID assignment, with IDs of $O(\log \log n)$ bits, in $O(t(N))$ rounds. This ensures that the algorithm does not detect that IDs are too large. It is sufficient that these IDs are unique in each $(t(N)+1)$-radius neighborhood.
\item Each $t(N)$-radius neighborhood must contain at most $N$ nodes, so that the algorithm does not detect that there are more nodes than the claimed amount.
\end{itemize}
In order to compute the new ID assignment we can compute a $k = (\Delta^{2(2t(N)+2)}( \log \log N + \log(\Delta^{2t(N)+2})))$-coloring of $G^{2t(N)+2}$, and this can be done in $O(t(N))$ rounds using Linial's coloring algorithm~\cite[Corollary 4.1]{Linial1992}.
Since $t(N) = o(\log_\Delta N)$, we see that each $t(N)$-radius neighborhood contains $O(\Delta^{t(N)}) = o(N)$ nodes and that $k = o(N)$.
Hence, the supposed algorithm cannot detect the lie, which concludes the proof.
\end{proof}
Corollary \ref{cor:detlb} follows by applying Theorem \ref{thm:detlb} with $\Delta = 2^{\sqrt{\beta \log n \log \log n}}$.
\section{Open problems}
In this work, we proved that the deterministic complexity of computing $(2,\beta)$-ruling sets is at least $\poly\log n$, unless $\beta$ is too large. Combined with existing $\poly\log n$ upper bounds, our results imply that the deterministic complexity of ruling sets lies in the $\poly\log n$ region. An interesting open question is how many $\log$ factors are required exactly.
Another open question concerns the techniques that we use: We first prove a lower bound for a constant-radius checkable version of the ruling set problem, and then transform this bound into a lower bound for the original problem, where we lose an additive $\beta$-factor. Hence, currently our technique is not capable to prove lower bounds for $(2,\beta)$-ruling sets where $\beta$ is so large that the problem can be solved in $o(\beta)$ rounds. An open question is to get rid of this restriction. For example, can we show that finding $(2,\log^{1/3} n)$-ruling sets requires $\Omega(\log^{1/3} n / \log^{1/2} \log n)$ rounds with deterministic algorithms?
While we now have a good picture of the dependency of the complexity of $(2,\beta)$-ruling sets on $n$, the dependency on $\Delta$ is far less clear. Even in the case where a $(\Delta+1)$-vertex coloring is provided, the current best upper bound is $O(\beta \Delta^{1/\beta})$ rounds. Note that, for constant values of $\beta$, we get a complexity that is polynomial in $\Delta$, while the lower bound that we provide lies in the $\poly \log \Delta$ region. Hence, there is an exponential gap between the current upper and lower bounds, as a function of $\Delta$. We know that, on general graphs, any algorithm that solves MIS in time $f(\Delta) + g(n)$, must have $f(\Delta) = \Omega(\Delta)$, or both $g(n) = \Omega(\log \log n / \log \log \log n)$ for randomized algorithms and $g(n) = \Omega(\log n / \log \log n)$ for deterministic ones \cite{Balliu2019}, and we think that a necessary step for really understanding the $\Delta$-dependency of $(2,\beta)$-ruling sets is to first prove an $\Omega(\Delta)$ lower bound for MIS on trees.
Finally, a number of interesting open questions revolve around our new technique of proving a lower bound via an upper bound. Can we characterize the problems that allow such an approach? What properties does an algorithm have to have to be well-representable as an upper bound sequence? Can the related technique of introducing a coloring component into non-coloring problems be successfully applied to other problems? We believe that finding answers to these and related questions will constitute an important step towards a better understanding of the round elimination technique.
\section*{Acknowledgments}
We are very thankful to the anonymous reviewers for their fruitful comments. We would also like to thank Mohsen Ghaffari, Fabian Kuhn, Yannic Maus, and Julian Portmann for helpful comments on related works.
\urlstyle{same}
\bibliographystyle{plainnat}
|
2,877,628,090,143 | arxiv | \section{Introduction} \label{intro}
Multiwavelength observations of galaxies hosting active galactic nuclei (AGN) in the past 20 years have extensively shown that supermassive black holes (SMBHs) are one of the key ingredients in shaping the evolution of galaxies through cosmic time. In particular, it is now well-established that AGN activity and star formation in their hosts are related processes, which are likely driven by a common fuelling mechanism such as accretion of cold gas supplies (e.g. \citealt{Vito+14}). In the local Universe, hints of such a connection have been suggested by the empirical correlations found between black hole mass and galaxy properties (e.g. \citealt{Magorrian+98}; \citealt{Gebhardt+00}; \citealt{Ferrarese+02}; \citealt{Gultekin+09}). In the distant Universe, this co-evolution scenario is supported by the similarity between volume-averaged cosmic star formation history and black hole accretion history, which both peak at z$\sim$2 and decline towards the local Universe (e.g. \citealt{Madau+14} for a review). Semi-
analytic models (e.g. \citealt{Bower+06}) and numerical simulations interpret these correlations as originated from a long-lasting ($\gtrsim$ Gyr) self-regulation process between the SMBH and its host (e.g. \citealt{Bower+06}; \citealt{Croton+06}; \citealt{Hopkins+08}; \citealt{Lagos+08}; \citealt{Menci+08}), which occurs in two flavours: quasar mode (QSO mode) and radio mode.
On the one hand, the quasar mode is usually associated with a radiatively efficient phase of SMBH accretion through isotropically distributed ionising winds and molecular outflows as means to prevent the runaway growth of SMBHs (e.g. \citealt{Booth+09}; \citealt{Dubois+14}). On the other hand, a subsequent radio-mode phase is usually invoked to prevent further episodes of galaxy star formation through mechanical feedback, such as collimated and relativistic jets (\citealt{Monaco+00}; \citealt{Dubois+14}), in order to regulate the galaxy stellar mass (\citealt{Croton+06}; \citealt{Marulli+08}; \citealt{Hopkins+10}) and to reproduce the galaxy colours observed in the local Universe (\citealt{Strateva+01}).
Though it is today widely accepted that the evolution of active SMBHs is connected to the evolution of their hosts, the underlying mechanisms explaining the transition between these two key stages of the AGN and galaxy life cycles are still poorly constrained. Testing this paradigm is challenging as this is supposed to be a quick transition (on timescales of $\sim$100 Myr) from highly accreting to fading AGN. Promising studies of individual smoking-guns, mostly X-ray and optically detected AGN (e.g. \citealt{Farrah+12}; \citealt{Cicone+14}; \citealt{Perna+15}; \citealt{Brusa+15}; \citealt{Brusa+16}), have shown compelling evidence of ongoing AGN feedback, but are currently limited to a small number of candidates.
A combined study with large AGN samples is necessary to constrain the role of AGN feedback in a more statistical sense, and to shed light on the connection between AGN and their hosts at different cosmic epochs.
A fundamental, complementary perspective in the framework of the AGN-host evolution comes from radio observations.
Indeed, radio observations are essential to capture possible signatures of relativistic jets powered by a central SMBH (e.g. \citealt{Hogan+15}), which are detected via the synchrotron emission of the jet (e.g. \citealt{Miller+93}). In addition, radio continuum emission may arise from the diffusion of cosmic ray electrons produced in supernovae and their remnants in high-mass star-forming regions, and this emission has been calibrated on star-forming galaxies to provide an almost dust-unbiased star formation rate (SFR) indicator (\citealt{Condon92}; \citealt{Yun+01}; \citealt{Bell03}). This underlines the great potential of radio observations in unveiling a mixture of AGN and star-forming galaxies. Nevertheless, radio observations need to be supplemented by multiwavelength data to fully characterise the nature of the radio sources.
Outstanding progress has been made through the analysis of large samples of radio-selected AGN in the local Universe (\citealt{Smolcic+09b}; \citealt{Best&Heckman12}). For instance, \citet{Smolcic+09b} identified a twofold population of radio-emitting AGN, namely high-excitation and low-excitation radio galaxies (HERGs and LERGs, respectively), on the basis of the presence of high- or low-excitation lines in their optical spectra taken from the Sloan Digital Sky Survey (SDSS; \citealt{York+00}). Interestingly, the author found that HERGs preferentially live in galaxies within the green valley (in terms of optical {\sc [NUV-$r$]} colours and stellar mass, $M_{\star}$), while LERGs usually populate the red sequence of massive and passive systems. Such a dichotomy observed in the host-galaxy properties between HERGs and LERGs may reflect physically different modes of SMBH accretion and presumably different stages of AGN-galaxy evolution (\citealt{Hardcastle+06}; see \citealt{Heckman+14} for a review).
While SMBHs in HERGs are thought to accrete via cold gas inflows from galaxy mergers or secular processes (\citealt{Sijacki+07}; \citealt{DiMatteo+08}; \citealt{Bournaud+12}), AGN activity in LERGs is probably induced by a continuous gas inflow coming from the atmosphere of the hot halo (\citealt{Bower+06}; \citealt{Ellison+15}). In the former case, accretion is radiatively efficient and covers a wide range of the electromagnetic spectrum (up to X-ray frequencies), while for the latter scenario the feedback is predominantly mechanical and does not outshine the host\ galaxy in most bands except radio.
In their comprehensive study, \citet{Hickox+09} have thoroughly investigated the AGN, host galaxy, and environmental properties of X-ray, mid-IR (MIR), and radio-selected AGN at 0.25$<$z$<$0.8 in the B\"ootes field (\citealt{Jannuzi+99}). In particular, \citet{Hickox+09} defined as ``radio AGN'' those sources with (rest-frame) 1.4~GHz luminosity $L_{\rm 1.4~GHz} > $ 10$^{24.8}$ W Hz$^{-1}$ to minimise the contamination from star-forming galaxies. These authors found that most radio-selected AGN have very low accretion rates (Eddington ratio $\lambda_{\rm Edd} \lesssim$ 10$^{-3}$) and populate overdense regions similarly to the most massive galaxies (e.g. \citealt{Georgakakis+07}; \citealt{Silverman+09}; \citealt{Coil+09}). In contrast, X-ray and MIR selected AGN are characterised by active star formation and less dense environments. These results, which have been corroborated through a similar analysis up to intermediate redshifts ($z\lesssim1.4$; see \citealt{Goulding+14}), strongly suggest that various AGN selection criteria might be sensitive to physically distinct classes of AGN and galaxies. In particular, the peculiarity of black hole and galaxy properties observed in radio-selected AGN stands out more than in any other AGN sample. For these reasons, it is now widely recognised that a multiwavelength investigation of radio-selected sources is essential to constrain the AGN-galaxy properties in a key stage of their cosmic evolution.
In this paper, we exploit the largest compilation of high-redshift (z$\lesssim$6) radio-selected galaxies in the Cosmic Evolution Survey (COSMOS; \citealt{Scoville+07}) field. The Karl G. Jansky Very Large Array (VLA) observations were conducted at 3~GHz (10 cm) over the entire COSMOS field, in the framework of the VLA-COSMOS 3~GHz Large Project (PI: V. Smol\v{c}i\'c, \citealt{smolcic+17a}), reaching a 1$\sigma$ sensitivity of 2.3$~\mu$Jy~beam$^{-1}$. The rich multiwavelength (X-ray to radio) data set of photometry and redshifts available in the COSMOS field allows us to investigate the physical properties of these sources from a panchromatic perspective.
The main goals of the present work are twofold:\ first, to provide a value-added catalogue that includes classification and physical properties for each 3~GHz VLA-selected source in the COSMOS field, and, second, to explore the multiwavelength properties of AGN hosts for different classes of radio-selected AGN out to z$\lesssim$6.
The paper is structured as follows. In Sect. 2 we describe our sample selection and the cross-match with ancillary photometry. In Sect. 3 we decompose the multiwavelength spectral energy distribution (SEDs), while the classification of our sample is discussed in Sect. 4. A brief description of the value-added catalogue is given in Sect. 5. Sect. 6 illustrates the average radio-selected AGN host-galaxy properties out to z$\lesssim$6, while the interpretation of our results are presented and discussed in detail in Sect. 7. We list our concluding remarks in Sect. 8. In Appendix A we show the results of infrared stacking, while Appendix B shows a portion of the value-added 3~GHz radio catalogue including some physical parameters used in this work. Throughout this paper, magnitudes are given in the AB system (\citealt{Oke74}). We assume a \citet{Chabrier03} initial mass function (IMF) and a flat cosmology with $\Omega_{\rm m}$ = 0.30, $\Omega_{\rm \Lambda}$ = 0.70, and H$\rm _0$ = 70 km s$^{-1}$ Mpc$^{-1}$ (\citealt{Spergel+03}).
\section{Sample selection} \label{sample}
\subsection{3~GHz radio sources} \label{radio-multi}
Radio data at 3~GHz were collected from 384 hours of observations with VLA over 2.6 deg$^2$, reaching an average rms sensitivity of 2.3 $\mu$Jy~beam$^{-1}$ and an angular resolution of about $0\farcs75$. A detailed description of the survey strategy, data reduction, and radio source catalogue is given in \citet{smolcic+17a}.
The catalogue includes 10\,830 radio sources, identified at peak surface brightness $\geq$5$\sigma$, out of which 67 are multi-components. The present catalogue represents the deepest compilations of radio sources available to date across an area of 2.6 deg$^2$. Our sample covers a wide redshift range (0$<$z$\lesssim$6, see Sect. \ref{redshift}) and is around three times larger than the 1.4~GHz sample taken from the Westerbork Synthesis Radio Telescope (WSRT, 3\,172 sources) in the NOAO Deep Wide-Field Survey (NDWFS, \citealt{deVries+02}). Moreover, our sample outnumbers the previous 1.4~GHz VLA-COSMOS survey by a factor of about four (2\,865 sources;\ see \citealt{schinnerer+07}, \citeyear{schinnerer+10}) and by more than one order of magnitude the 1.4~GHz VLA survey in the Extended \textit{Chandra}-Deep Field South (E-CDFS; 883 sources, \citealt{Miller+13}) survey.
We derived (rest-frame) 3~GHz radio luminosity ($L_{\rm 3\, GHz}$) for radio sources with multiwavelength counterparts and redshifts (see Sects. \ref{multi_lambda} and \ref{redshift}). Under the assumption of purely synchrotron emission, the radio spectrum behaves like a power law $S_{\nu} \propto \nu ^{\alpha}$, where the spectral index $\alpha$ is set to the observed 1.4--3~GHz slope for sources detected also at 1.4~GHz (about 30\%) in the 1.4~GHz VLA-COSMOS survey (\citealt{schinnerer+10}); the spectral index is set to --0.7, which is consistent with a non-thermal synchrotron index, (e.g. \citealt{Condon92}; see also \citealt{smolcic+17a}) if the sources are detected at 3~GHz alone. In Fig. \ref{fig:z_dist} (bottom panel) we show $L_{\rm 3\, GHz}$ as a function of redshift with respect to the 5$\sigma$ luminosity limit. For comparison, we show the corresponding 1.4~GHz luminosity $L_{\rm 1.4\, GHz}$ on the right $y$ axis. Our sample clearly spans a wide luminosity range (up to 4--5~dex), which allows us
to investigate the multiwavelength properties of our sample in different radio luminosity regimes.
In particular, we are able to detect 2.5 times intrinsically fainter sources (under the assumption $\alpha$=--0.7), at a given redshift, compared to the previous 1.4~GHz VLA-COSMOS survey (\citealt{schinnerer+07}, \citeyear{schinnerer+10}).
\subsection{Optical to (sub)millimetre photometry} \label{multi_lambda}
The COSMOS field benefits from an exquisite photometric data set, covered from the X-rays to the submillimetre domain\footnote{An exhaustive overview of the COSMOS field and multiwavelength data products is available at: \url{http://cosmos.astro.caltech.edu/} }. Cross-matching our 3~GHz selected sample to existing ancillary data is essential to derive physical properties of galaxies. The multiwavelength photometry is taken from the COSMOS2015 catalogue (\citealt{Laigle+16}), which combines optical photometry\footnote{Optical photometry is taken from Subaru Hyper-Suprime Cam observations over the full 2 deg$^2$ (\citealt{Capak+07}), and also from the Canada-France-Hawaii Telescope Legacy Survey (CFHT-LS; \citealt{McCracken+01}) in the central 1 deg$^2$.}, the most recent UltraVISTA (DR2\footnote{DR2 replaces the previous DR1 by \citet{McCracken+12}. A detailed description of the survey and data products can be retrieved at: \url{http://ultravista.org/release2} }) data over the central 1.5
deg$^2$ in the near-infrared (NIR) bands $Y$, $J$, $H$, and $K_s$\footnote{Outside the UltraVISTA coverage, NIR photometry includes CFHT $H$ and $K_s$ observations obtained with the WIRCam (\citealt{McCracken+01}).}, and MIR photometry obtained from the Infrared Array Camera (IRAC), which is recently complemented by deeper IRAC 3.6 and 4.5$~\mu$m observations with the \textit{Spitzer} Large Area Survey with Hyper-Suprime-Cam (SPLASH; \citealt{Steinhardt+14}; P.~Capak et al. in prep.). In addition, this data set has been cross-matched with 24$~\mu$m photometry (\citealt{LeFloch+09}) from the Multi-Band Imaging Photometer for \textit{Spitzer} (MIPS). \citet{Laigle+16} provides further details.
The cross-match to associate a possible optical-NIR counterpart with each radio source is fully described in \citet{smolcic+17b} (see their Sect. 3). First, they excluded stars and masked regions in the COSMOS2015 catalogue because of the less accurate optical photometry, which reduces the effective area of the COSMOS field to 1.77~deg$^2$ and our 3~GHz selected sample to 8\,696 radio sources. Secondly, they performed a nearest-neighbour matching, by selecting for each radio source only candidate counterparts within $0\farcs8$ searching radius and, at the same time, requiring a false match probability (i.e. probability of being a spurious association) lower than 20\%. This approach yields an average expected fraction of spurious associations of about 1\% (see \citealt{smolcic+17b}). The percentage of radio sources with multiple optical-NIR counterparts within $0\farcs8$ is around 1\%, for which the cut in false-match probability ensures the selection of the most probable counterpart. After this cut, our
final sample consists of {7\,729} radio sources with optical-NIR counterparts, corresponding to about {89\%} of our radio-selected sample within the common 1.77 deg$^2$.
To enrich the spectral coverage of our analysis and derive robust star formation rates (SFRs) for as many sources as possible, we used also \textit{Herschel} photometry at far-infrared and submillimetre wavelengths provided in the COSMOS2015 catalogue. \textit{Herschel} imaging covers the entire COSMOS field with the Photoconductor Array Camera and Spectrometer (PACS; 100 and 160$~\mu$m, \citealt{Poglitsch+10}) and Spectral and Photometric Imaging Receiver (SPIRE; 250, 350, and 500$~\mu$m, \citealt{Griffin+10}) data, as part of the PACS Evolutionary Probe (PEP; \citealt{Lutz+11}) and the \textit{Herschel} Multi-tiered Extragalactic Survey (HerMES; \citealt{Oliver+12}). \textit{Herschel} fluxes were extracted and de-blended by using 24$~\mu$m positional priors and unambiguously associated with the corresponding optical-NIR counterpart via 24$~\mu$m sources listed in both catalogues. In total, the number of radio sources with ($\geq$3$\sigma$) \textit{Herschel} detection in at least one PACS or SPIRE band are {
4\,836/7\,729} ({63\%}). This percentage decreases with redshift, being {87\%} at z$<$0.3 and {45\%} at z$>$3.5.
To obtain reliable dust-based SFRs also in potential high-redshift candidates (z$>$3), where \textit{Herschel} observations are incomplete even towards ultraluminous infrared galaxies (ULIRGs, i.e. having rest-frame 8-1000$~\mu$m infrared luminosity $L_{\rm IR} \geq$ 10$^{12}$ L$_\odot$, e.g. \citealt{Sanders+96}), photometry at longer wavelengths is essential. For around 115 radio sources, we retrieved additional (sub)millimetre photometry from at least one of the following data sets: JCMT/SCUBA-2 at 450 and 850$~\mu$m (\citealt{Casey+13}), LABOCA at 870$~\mu$m (F.~Navarrete et al. priv. comm.), Bolocam (PI: J.~Aguirre), JCMT/AzTEC (\citealt{Scott+08}) and ASTE/AzTEC (\citealt{Aretxaga+11}) at 1.1 mm, MAMBO at 1.2 mm (\citealt{Bertoldi+07}), and interferometric observations at 1.3~mm with ALMA (PI: M. Aravena, M.~Aravena et al. in prep.) and PdBI (\citealt{Smolcic+12}; \citealt{Miettinen+15}). The (sub)mm positions were cross-matched to the COSMOS2015 positions via a nearest
neighbour matching, using $1\farcs$ searching radius (the smallest beam width of the (sub)mm data we collected). A thorough visual inspection of the counterpart associations has been performed for the 1.3~mm detected ALMA sources (68\% of the (sub)mm photometry we collected), which is detailed in \citet{brisbin+17} and \citet{miettinen+17}.
We also collected X-ray data from the \textit{Chandra}-COSMOS (\citealt{Elvis+09}; \citealt{Civano+12}) and COSMOS-Legacy catalogues (\citealt{Civano+16}). The optical-NIR counterparts of X-ray sources were matched via a maximum likelihood algorithm and are presented in \citet{Marchesi+16}. We matched their catalogue to our 3~GHz selected sample of {7\,729} optical-NIR counterparts via COSMOS2015 IDs. This match yields {903} X-ray sources, corresponding to {12\% (903/7\,729)} of our radio sample, and to {32\% (903/2\,804)} of the X-ray sources with optical-NIR association in unmasked areas.
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{zdist.pdf}
\end{center}
\caption{\small Top panel: redshift distribution of our {7\,729} radio sources. Spectroscopic and photometric redshifts are shown in red and blue, respectively, while the black line is the sum of the two. The scale of the $y$ axis is logarithmic. Bottom panel: circles show the rest-frame 3~GHz luminosity as a function of redshift, both spectroscopic (red) and photometric (blue). The horizontal bars indicates the average $\pm$1$\sigma$ uncertainty range of the photometric redshifts in the various redshift bins. The corresponding 1.4~GHz luminosity (scaled by using $\alpha$=--0.7) is shown for comparison on the right $y$ axis. The black solid line indicates the 5$\sigma$ luminosity limit at 3~GHz. }
\label{fig:z_dist}
\end{figure}
\subsection{Spectroscopic and photometric redshifts} \label{redshift}
We collected photometric redshifts for the {7\,729} radio sources with a counterpart in the COSMOS2015 catalogue. Photometric redshift estimates are included in the catalogue and were derived using the {\sc Le Phare} SED-fitting code (\citealt{Arnouts+99}; \citealt{Ilbert+06}) following the procedure detailed in \citeauthor{Ilbert+09} (\citeyear{Ilbert+09}, \citeyear{Ilbert+13}). Based on the comparison with the spectroscopic redshifts available in the COSMOS field, \citet{Laigle+16} report an average photometric redshift accuracy of $\left \langle |\Delta z/(1 + z)| \right \rangle =$ 0.021 for $K_s>$22, which becomes less than 0.010 for brighter sources.
For X-ray sources, we used a different set of photometric redshifts from M.~Salvato et al. (in prep.), which are more suitable for AGN-dominated sources as they account for AGN variability and adopt additional AGN templates (\citealt{Salvato+09}, \citeyear{Salvato+11}).
An exhaustive list of spectroscopic redshifts was compiled (April 2015, Salvato et al. in prep.) and made internally accessible to the COSMOS team. Most of the spectroscopic redshifts used in this paper were taken from the $z$COSMOS survey (\citealt{Lilly+07}, \citeyear{Lilly+09}), either the public $z$COSMOS-bright or the proprietary $z$COSMOS-deep database, the DEep Imaging Multi-Object Spectrograph (DEIMOS, Capak et al. in prep.), and the FOcal Reducer and low dispersion Spectrograph (FORS2, \citealt{Comparat+15}). M.~Salvato et al. (in prep.) provide for a full reference list.
For each radio source with a multiwavelength counterpart in this spectroscopic compilation, we replaced the photometric redshifts with new spectroscopic values only in case of secure or very secure measurements\footnote{The reliability of each spectroscopic redshift is determined by its quality flag. In case of spectroscopic redshift from the zCOSMOS survey (\citealt{Lilly+07}, \citeyear{Lilly+09}), we followed the prescription recommended on the zCOSMOS IRSA webpage: \url{https://irsa.ipac.caltech.edu/data/COSMOS/spectra/z-cosmos/Z-COSMOS_INFO.html} For the other surveys we selected quality flag $Qf \geq$ 3 and discarded less reliable spectroscopic redshifts from our analysis.}. In addition, we included the latest spectroscopic redshifts from the VIMOS Ultra Deep Survey (VUDS; \citealt{lefevre+15}; \citealt{Tasca+16}), from which we found 25 associations to our radio sources.
After these checks, the number of radio sources with spectroscopic redshift is {2\,734}/{7\,729} (around {35\%}). Every radio source with multiwavelength counterpart has its own redshift estimate. Fig. \ref{fig:z_dist} (top panel) shows the final redshift distribution for our {7\,729} radio sources. The number of spectroscopic and photometric redshifts are comparable out to z$\sim$1, while photometric redshifts become more numerous at higher redshift.
We tested the accuracy of the photometric redshifts in our sample based on the spectroscopic measurements available for {2\,734} sources. We found a median $\left \langle |\Delta z/(1 + z)| \right \rangle =$ 0.010, which becomes as high as 0.035 at z$>$3. Therefore, the proved accuracy of the photometric redshifts allows us to push our analysis out to z$\lesssim$6, even if the number of sources at z$>$4 is relatively small ({84} sources).
\section{SED-fitting decomposition of 3~GHz sources} \label{sed_fitting}
In this section, we fit the multiwavelength SEDs of our radio sources to disentangle the AGN emission from that related to the host-galaxy.
It is well known that radio-selected samples contain distinct galaxy populations (e.g. \citealt{Condon84}; \citealt{Windhorst+85}; \citealt{Gruppioni+99}) in terms of star formation and AGN properties. Therefore, fitting the multiwavelength SEDs may provide meaningful results only if AGN and galaxy templates are both taken into account.
We used both the SED-fitting code {\sc magphys}\footnote{The original {\sc magphys} code is publicly available at this link: \url{http://www.iap.fr/magphys/magphys/MAGPHYS.html}} (\citealt{daCunha+08}), and the three-component SED-fitting code {\sc sed3fit} by \citet{Berta+13}, which accounts for an additional AGN component\footnote{The three-component SED-fitting code {\sc sed3fit} can be retrieved from \url{http://cosmos.astro.caltech.edu/page/other-tools}}. The aforementioned references provide for a full description of these SED-fitting codes. Here we briefly outline the main prescriptions that are relevant for our analysis.
The {\sc magphys} code is designed to reproduce a variety of galaxy SEDs, from weakly star-forming to starbursting galaxies, over a wide redshift range\footnote{For extensive application of the {\sc magphys} code in deriving physical properties of galaxies, see also \citet{Smith+12}; \citet{Rowlands+14}; \citet{Michalowski+14}; \citet{Hayward+15}.}. This code relies on the energy balance between the dust-absorbed stellar continuum and the reprocessed dust emission at infrared wavelengths. This recipe ensures that optical and infrared emission originating from star formation are linked in a self-consistent manner, but does not account for a possible AGN emission component. The three-component SED-fitting code presented by \citet{Berta+13} combines the emission from stars, dust heated by star formation, and a possible AGN-torus component from the library of \citeauthor{Feltre+12} (\citeyear{Feltre+12}, see also \citealt{Fritz+06}). This approach results in an effectively simultaneous three-component fit. For each best-fit parameter, the code provides a corresponding probability distribution function (PDF), which enables the user to obtain reliable confidence ranges for parameter estimates (see e.g. Calistro~Rivera et al., in prep., for a similar SED-fitting technique).
We decomposed each observed SED by using the best available redshift (either spectroscopic or photometric, see Sect. \ref{redshift}) as input, and we derived integrated galaxy properties, such as SFR and $M_{\star}$, for each individual source. The SFR was derived from the total IR (rest 8-1000$~\mu$m) luminosity taken from the best-fit galaxy SED (i.e. corrected for a possible AGN emission), assuming a \citet{Kennicutt98} conversion factor scaled to a \citet{Chabrier03} IMF. We note that about {37\%} of our sample are not $\geq$3$\sigma$ detected in any \textit{Herschel} bands. To obtain better constrained IR luminosities, we performed SED-fitting using the nominal PACS and SPIRE 3$\sigma$ upper limits, which are equal to 5.0 (100$~\mu$m), 10.2 (160$~\mu$m), 8.1 (250$~\mu$m), 10.7 (350$~\mu$m), and 15.4 (500$~\mu$m) mJy, including confusion noise (\citealt{Lutz+11}; \citealt{Oliver+12}). We modified the $\chi^2$ calculation to correctly account for those \textit{Herschel} bands that have only upper
limits, similar to the approach adopted by \citet{dacunha+15}.
As a sanity check, we verified that the IR luminosities based on our three-component fit are in good agreement with those calculated independently using a different set of IR templates (from \citealt{Chary+01}; \citealt{Dale+02}; \citealt{Siebenmorgen+07}; \citealt{Polletta+07}; \citealt{Wuyts+08}; \citealt{Elbaz+11}; \citealt{Nordon+12}, see \citealt{Berta+13} for a comprehensive discussion). We briefly discuss the comparison with the \textit{Herschel} fluxes derived through stacking in Sect. \ref{ir_stacking}. The $M_{\star}$ is derived from the SED decomposition itself, which allows us to obtain robust estimates if the optical-NIR SED is dominated by the host-galaxy light (e.g. \citealt{Bongiorno+12}).
In order to quantify the relative incidence of a possible AGN component, we fitted each individual SED, both with the three-component approach and the {\sc magphys} code. The fit obtained with the AGN is preferred if the reduced $\chi^2$ value of the best fit is significantly (at $\geq99$\% confidence level, on the basis of a Fisher test) smaller than that obtained from the fit without the AGN; see \citet{Delvecchio+14} for details. From our analysis, we found that {1\,169} out of {7\,729} radio sources (about 15\%) show a $\geq99$\% significant AGN component in their best fit.
We extensively tested this technique against independent AGN indicators in the COSMOS field, such as MIR colours and X-rays (see \citealt{Delvecchio+14}). For instance, \citet{Lanzuisi+15} showed that the AGN radiative luminosities derived from SED decomposition were consistent (1$\sigma$=0.4~dex) with those calculated from X-ray spectra and assuming a set of bolometric corrections (e.g. \citealt{Lusso+12}). Moreover, the unprecedented accuracy of photometric redshifts and the photometric coverage exploited in this work, strengthened by our sizeable sample, further increase the reliability of our method. However, if the galaxy light outshines the AGN in the full optical-to-mm SED, this statistical technique becomes progressively less effective in identifying AGN.
\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{venn2.pdf}
\end{center}
\caption{\small Venn diagram illustrating the percentages of the 1\,604 HLAGN in our sample identified from different AGN diagnostics: X-rays (blue), MIR (red) and SED decomposition (green). Areas roughly scale with percentages. }
\label{fig:venn}
\end{figure}
\section{Classification of 3~GHz radio sources} \label{classification}
We combine SED-fitting decomposition (Sect. \ref{sed_fitting}) with other multiwavelength AGN diagnostics to reach a more complete census of AGN in our sample. These additional indicators are taken from X-ray, MIR and radio data, which allow us to identify two main populations of radio-selected AGN in our sample: moderate-to-high radiative luminosity AGN (HLAGN) and low-to-moderate radiative luminosity AGN (MLAGN). Hereafter, we refer to these populations as HLAGN (X-ray, MIR, and SED-selected AGN) and MLAGN (radio-excess sources that are not HLAGN), as {explained} in the next sections.
This naming convention comes from the idea that the selection criteria based on SED-fitting, X-ray, and MIR data preferentially select higher luminosity AGN, where the term ``luminosity'' here refers to the AGN radiative luminosity (L$\rm_{rad, AGN}$), which is a proxy of the SMBH accretion rate (BHAR; e.g. \citealt{Alexander+12}). This classification does not translate into a sharp threshold in the accretion efficiency (or Eddington ratio) between HLAGN and MLAGN, but rather reflects the reliability of the adopted diagnostics in identifying such AGN populations\ combined with the sensitivity of our survey at various wavelengths. Moreover, the tags ``low to moderate'' and ``moderate to high'' intentionally imply a potential overlap in L$\rm_{rad, AGN}$ between the two classes at various redshifts. However, at a given redshift, these differently selected AGN display significantly distinct distributions of AGN luminosity, as detailed in Sect. \ref{naming_convention}. Therefore, the present
classification should be considered as observationally based, and aimed at dissecting our radio sources on the basis of AGN diagnostics that are known to be luminosity dependent. A detailed investigation on the distribution of radio-selected AGN as a function of their intrinsic Eddington ratio will be presented in a forthcoming paper (Delvecchio et al., in prep.).
In Sects. \ref{hlagn} and \ref{llagn} we describe in more detail the multiwavelength diagnostics used to identify MLAGN and HLAGN, respectively, while in Sect. \ref{further_tests} we justify this naming convention by studying their L$\rm_{rad, AGN}$ distributions.
\subsection{Moderate-to-high radiative luminosity AGN} \label{hlagn}
As previously mentioned, the so-called HERG population identified in the local Universe consists of highly accreting SMBHs on the basis of the presence of high-excitation lines in their optical spectra (e.g. \citealt{Smolcic+08}), which implies radiatively efficient accretion. In order to detect potential HERG analogues in our sample, we combine SED decomposition (Sect. \ref{sed_fitting}) with X-ray and MIR indicators. All these selection criteria are sensitive to an excess of emission likely arising from accretion onto the central SMBH rather than from star formation. As a consequence, despite the different biases intrinsic to each criterion, all of these criteria preferentially select higher radiative luminosity AGN. The selection criteria used to identify this AGN population are briefly summarised below. \citet{smolcic+17b} provide a detailed description of the X-ray and MIR-based AGN indicators.
\begin{figure*}
\begin{center}
\includegraphics[width=180mm,keepaspectratio]{sed_examples.pdf}
\end{center}
\caption{\small Three examples of best-fit SEDs of HLAGN selected from different criteria. Coloured lines represent the corresponding best-fit templates of AGN (red), galaxy star formation (blue), and the sum of the two (black). (Left panel) AGN identified from X-rays, MIR colours, and SED fitting. (Central panel) AGN identified only from SED-fitting. (Right panel) AGN identified only from X-rays. The red dashed line indicates that the AGN component is $<$99\% significant on the basis of the Fisher test (see text for more details). Red circles indicate the optical to far-IR (FIR) photometry (rest-frame), while downward pointing arrows represent 3$\sigma$ upper limits in the \textit{Herschel} bands. }
\label{fig:sed_comparison}
\end{figure*}
First, SED-fitting decomposition identifies {1\,169} sources with $\geq$99\% significant AGN component in their global SED, SED AGN hereafter (see Sect. \ref{sed_fitting}).
Second, we used X-ray luminosities ($L_{\rm x}$) in the rest-frame [0.5--8] keV. The $L_{\rm x}$ estimates were calculated for the {903} X-ray detected sources, by assuming a fixed X-ray spectral slope $\Gamma=1.8$, and correcting for nuclear obscuration on the basis of the measured hardness ratio (e.g.~\citet{Xue+10}). We identified {855} sources with X-ray luminosity $L_{\rm x} \geq $10$^{42}$~erg~s$^{-1}$ as X-ray AGN (e.g. \citealt{Szokoly+04}). We verified that the X-ray emission expected from recent $L_{\rm x}$--SFR relations (taken from \citealt{Symeonidis+14}) is always negligible for our X-ray AGN (about a few percent). On the basis of the aforementioned relation, about 30 X-ray detected sources with $L_{\rm x} < $10$^{42}$ erg s$^{-1}$ show an X-ray excess and, therefore, could be considered low-luminosity X-ray AGN. However, we prefer to apply the same cut at $L_{\rm x} \geq $10$^{42}$ erg s$^{-1}$ for all our sources to avoid potential contamination from outliers with respect to the $L_{\rm x}$--
SFR relation.
Third, MIR colours can be very useful in identifying AGN, both unobscured and heavily obscured. \citet{Donley+12} proposed a conservative criterion to select AGN, on the basis of the MIR colour-colour diagram drawn from a combination of the four \textit{Spitzer}-IRAC (3.6, 4.5, 5.8, and 8.0$~\mu$m) bands. We followed Eqs. (1) and (2) of their paper to identify AGN at z$<$2.7, while at higher redshift we applied the additional conditions stated in their Eqs. (3) and (4) to minimise the contamination from high-redshift starbursts without AGN. This method is highly reliable for bright AGN, but becomes incomplete at $L_{\rm x}<$10$^{44}$ erg s$^{-1}$. In total, {455} out of {7\,729} radio sources (about 6\%) satisfy the \citet{Donley+12} criterion, and therefore are classified as MIR AGN.
Hereafter, we will use the term ``moderate-to-high radiative luminosity AGN'' (HLAGN) to collectively refer to the union of X-ray, MIR, and SED-selected AGN identified in our sample, for a total of {1\,604} objects (21\% of the radio sample).
Figure \ref{fig:venn} shows the percentages of HLAGN classified from each criterion: the percentage of AGN that fulfills all the criteria simultaneously is only about {14\%} of the full HLAGN population. This small overlap further suggests that different AGN diagnostics are sensitive to distinct AGN populations. This overlap increases with increasing X-ray luminosity, which is 7\% for $10^{42}<L_{\rm x}<$10$^{43}$~erg s$^{-1}$, 25\% for $10^{43}<L_{\rm x}<$10$^{44}$~erg s$^{-1}$, and 49\% for $L_{\rm x}>$10$^{44}$~erg s$^{-1}$. These relatively small percentages are mainly driven by the incompleteness of the MIR classification, as the \citet{Donley+12} criterion is very conservative. We checked that the agreement between X-ray and SED-fitting diagnostics is as high as {21\%} for $10^{42}<L_{\rm x}<$10$^{43}$~erg s$^{-1}$, {52\%} for $10^{43}<L_{\rm x}<$10$^{44}$~erg s$^{-1}$, and 79\% for $L_{\rm x}>$10$^{44}$~erg s$^{-1}$.
In Fig. \ref{fig:sed_comparison} we illustrate some examples of best-fit SEDs, showing different levels of agreement between the AGN diagnostics described above. In all the panels, red circles indicate the (rest-frame) multiwavelength photometry, while downward pointing arrows set the 3$\sigma$ upper limits in the \textit{Herschel} bands. Solid lines represent the best-fit templates of AGN (red), galaxy star formation (blue), and the sum of the two (black).
The left panel shows the SED of an unambiguous AGN, successfully identified from X-rays, MIR-colours, and SED-fitting decomposition. The central panel shows an AGN identified only from SED decomposition. Indeed, galaxies hosting heavily obscured AGN might be undetected in the X-rays, but also misclassified from MIR colours since the \citet{Donley+12} criterion is highly incomplete at $L_{\rm x} < 10^{44}$ erg s$^{-1}$. However, neither SED-fitting decomposition nor MIR colours can identify an AGN when the optical-IR SED is outshined by the host-galaxy light (right panel), although the X-ray luminosity suggests the presence of a moderately luminous X-ray AGN ($L_{\rm x} \sim$10$^{43}$ erg s$^{-1}$). We looked at the observed distribution of the X-ray to optical-UV index, defined as $\alpha_{\rm ox}$~=~$-$Log[$L_{\rm 2~keV}$~/~$L_{\rm 2500~\AA}$]/2.605, where $L_{\rm 2500~\AA}$ and $L_{\rm 2~keV}$ are the rest-frame monochromatic luminosities at 2500\AA~and 2 keV, respectively (e.g. \citealt{Zamorani+81}). We
verified that the observed distribution of $\alpha_{\rm ox}$ for HLAGN identified solely from X-rays peaks at $\alpha_{\rm ox} \sim$1, unlike the average value $\alpha_{\rm ox} \sim$1.37 found for X-ray selected AGN in the COSMOS field (\citealt{Lusso+10}). The lower $\alpha_{\rm ox}$ suggests that HLAGN identified only from X-rays are optically fainter than the rest of X-ray AGN in the COSMOS field, as expected from their galaxy-dominated SEDs. This is also confirmed by the fact that in more than 80\% of them, the optical-NIR photometry has been fitted without AGN templates when calculating the photometric redshifts (see \citealt{Marchesi+16}; M.~Salvato et al. in prep.).
By using different and complementary tracers of highly accreting AGN, we can build a more representative (though not 100\% complete) sample of HLAGN. Our analysis would certainly benefit from optical-NIR spectroscopy to identify AGN at lower intrinsic luminosities. Unfortunately, the spectral lines used to calculate the emission line ratios [\ion{O}{III}]/H$\beta$ and [\ion{N}{II}]/H$\alpha$) in the BPT diagram (from \citealt{Baldwin+81}) are detected only in a low percentage (about 5\%) of our radio sample, mostly at z$<$0.5, as some optical lines (e.g. H$\alpha$) would be redshifted outside the observed spectral window at higher redshifts. For consistency, in this work we preferred to make use of AGN selection criteria that are applicable to the entire sample.
For our 1\,604 HLAGN, we took the SFR and $M_{\star}$ estimates from the best-fit solution obtained with the three-component SED-fitting code by \citet{Berta+13}. This approach allows us to account for a possible AGN contamination in galaxy parameter estimates and to study uncertainties and degeneracies, which is important when comparing galaxy properties between AGN and non-AGN hosts. However, we checked that the AGN contribution to the total (8-1000$~\mu$m) IR luminosity is very small (a few percent) for most of the HLAGN sample, as argued by previous studies of X-ray and IR-selected AGN (e.g. \citealt{Mullaney+11}; \citealt{Santini+12}; \citealt{Rosario+12}).
\begin{figure}
\begin{center}
\includegraphics[width=3.5in]{radio_excess_20170210.pdf}
\end{center}
\caption{\small Distribution of the ratio between $L_{\rm 1.4~GHz}$ and SFR$_{\rm IR}$ as a function of redshift (black points) for sources not classified as HLAGN (79\% of our sample). The blue filled circles (and errors) indicate the peak (and dispersion) of the Gaussian distribution identified in a given redshift bin, while the corresponding 3$\sigma$ deviation is set by the red open circles. The red solid line indicates the redshift-dependent threshold derived by fitting the open circles at each redshift bin. Sources above the red line are identified as ``low to moderate radiative luminosity AGN'' (hereafter MLAGN) via radio excess. The full histogram of $L_{\rm 1.4~GHz}$/SFR$_{\rm IR}$ is shown in the top right corner. }
\label{fig:radio_excess}
\end{figure}
\subsection{Low-to-moderate radiative luminosity AGN via radio excess} \label{llagn}
Radio sources that are not classified as HLAGN in Sect. \ref{hlagn} do not show evidence of AGN activity according to X-ray, MIR, or SED decomposition. However, this does not necessarily mean that SMBH accretion is not occuring at all, but rather that the AGN diagnostics described above might fail in detecting signatures of less efficient accretion episodes. As mentioned in the previous sections, radio observations are crucial for chasing such an elusive AGN population.
To identify lower radiative luminosity AGN in our sample, we first considered the 3~GHz selected sources that are not classified as HLAGN (i.e. 79\%). For each of them, the $L_{\rm IR}$ obtained via SED fitting without AGN (\citealt{daCunha+08}) has been converted to IR-based SFR (SFR$_{\rm IR}$ hereafter) by assuming a \citet{Kennicutt98} scaling factor and a \citet{Chabrier03} IMF. To identify the possible AGN contribution in radio emission, we analysed the ratio between the 1.4~GHz radio luminosity $L_{\rm 1.4\, GHz}$ and the SFR$_{\rm IR}$ for each source. Fig. \ref{fig:radio_excess} shows the distribution of their ratio (i.e. $L_{\rm 1.4\,GHz}$/SFR$_{\rm IR}$, in logarithmic scale) as a function of redshift (black points). Typical 1$\sigma$ uncertainties of the observed ratio are of the order of 0.15~dex. The histogram in the top right corner shows the distribution of our sources as a function of $L_{\rm 1.4\, GHz}$/SFR$_{\rm IR}$. The distribution is skewed towards high values of the ratio, suggesting
that AGN
activity might be contributing to the integrated radio emission. However, the average $L_{\rm 1.4\, GHz}$/SFR$_{\rm IR}$ also increases with redshift, which partly explains the skewness of the observed distribution. To quantify these factors, we split our sample into seven redshift bins (0.01$<$z$<$0.3, 0.3$<$z$<$0.7, 0.7$<$z$<$1.2, 1.2$<$z$<$1.8, 1.8$<$z$<$2.5, 2.5$<$z$<$3.5, and 3.5$<$z$<$5.7) and fit each single distribution with a log-normal function that reproduces the peak and the negative part of the observed distribution well. The values of the peak (and dispersion) of the Gaussian function identified at each redshift bin are represented in Fig. \ref{fig:radio_excess} with blue filled circles (and relative errors). The position of the peak generally shifts to higher $L_{\rm 1.4\, GHz}$/SFR$_{\rm IR}$ ratios with increasing redshift, which justifies our choice of a redshift-dependent approach. Moreover, some recent studies (e.g. \citealt{Magnelli+15}; \citealt{Delhaize+17}) have found a slight, but
significant decrease of q$_{\rm IR}$ (proportional to SFR$_{\rm IR}$/$L_{\rm 1.4\, GHz}$) with redshift through a careful treatment of non-detections via stacking or double-censored survival analysis.
We calculated the 3$\sigma$ deviation from the peak of the log-normal function at each redshift bin (see red open circles in Fig. \ref{fig:radio_excess}). By fitting the open circles with a power-law function (red solid line in Fig. \ref{fig:radio_excess}), we derived the analytical expression that describes a redshift-dependent threshold in radio excess as follows:
\begin{equation}
\log{\left(\frac{L_{\rm 1.4\, GHz}}{\rm SFR_{IR}}\right)}_{\rm excess} = 21.984 \times (1+z)^{0.013} .
\label{eq:excess}
\end{equation}
From this expression, we set a threshold above which the radio emission shows a $>$3$\sigma$ excess compared to that expected from star formation. The aforementioned threshold identifies {1\,333} low-to-moderate radiative luminosity AGN (hereafter MLAGN) via radio excess, corresponding to 17\% of our radio sample. The percentage of MLAGN, which would have been identified through a single threshold at all redshifts, would be around 18\% instead of the 17\% found from Eq. \ref{eq:excess}; this would not affect our results.
The choice of a 3$\sigma$ radio excess imposed in each redshift bin ensures a negligible contamination from star-forming galaxies without radio excess (about 0.15\%). On the other hand, our selection might miss a significant number of potential MLAGN in our sample, which is estimated to be around {1\,000} sources ({75\%} of the identified sample of MLAGN) based on the difference between the distribution below the threshold and the best-fitting Gaussian profile, in all redshift bins. A comparison with other definitions of radio excess found in the literature is presented in Sect. \ref{literature}.
The \citet{Kennicutt98} $L_{\rm IR}$--SFR$_{\rm IR}$ conversion assumes that the total IR luminosity arises entirely from optically thick, dust-obscured regions. While this assumption is reasonable in highly star-forming galaxies, such as those detected by \textit{Herschel} (\citealt{Wuyts+11}; \citealt{Magnelli+13}), this is not true for weakly star-forming (or passive) systems, where a significant portion of the IR luminosity may originate from ($>$ few Gyr) old stellar populations (e.g. \citealt{Groves+12}).
We verified that the unobscured SFR derived from the UV galaxy emission (e.g. \citealt{Papovich+07}) is around 5\% of the obscured SFR$_{\rm IR}$, on average, therefore its contribution would not significantly affect our definition of radio excess presented above.
Moreover, our threshold is calibrated on a radio-selected sample, which is expected to bias the observed distribution towards higher values of $L_{\rm 1.4\, GHz}$/SFR$_{\rm IR}$ compared to the true (i.e. unbiased) distribution. These arguments suggest that our definition of radio excess is likely to be fairly conservative and the number of MLAGN selected in this way should more properly be considered as a lower limit.
\bigskip
In summary, by combining multiwavelength AGN diagnostics, we managed to isolate two populations of AGN in our 3~GHz selected sample. We identified {1\,604} HLAGN (21\%) and {1\,333} MLAGN (17\%), which collectively make our sample of radio-selected AGN. The remainder of the sample (62\%) is characterised in detail in \citet{smolcic+17b}.
Moreover, we found that about 30\% of HLAGN shows also a $\geq$3$\sigma$ radio excess. We checked that the relative numbers of HLAGN identified from each criterion and shown in Fig. \ref{fig:venn} do not change between sources with and without significant radio excess. A more detailed analysis of the average galaxy and AGN properties for the aforementioned AGN classes (MLAGN, HLAGN, including the subsample with radio excess) is presented in Sect. \ref{results}.
\subsection{Comparison with radio-based classifications} \label{literature}
In this section we compare our source classification with other independent methods from the literature.
\subsubsection{Comparison with VLBI sources} \label{VLBI}
We cross-matched our 3~GHz selected sample of {7\,729} sources with the Very Large Baseline Interferometry (VLBI) 1.4~GHz source catalogue from N.~Herrera Ruiz et al. (in prep.). The authors targeted the radio sources selected by the 1.4~GHz VLA-COSMOS survey (\citealt{schinnerer+10}) with VLBI at $\lesssim0\farcs01$ angular resolution, reaching a 1$\sigma$ sensitivity of 10$~\mu$Jy~beam$^{-1}$ in the central part of the field. They detected 468 radio sources at signal-to-noise ratio higher than 5.5. A total of {354} matches have been found within $0\farcs4$ (half-beam size of 3~GHz VLA observations).
Given the high angular resolution, VLBI is sensitive to the radio emission on circum-nuclear scales (from $d\sim$20 pc at $z$=0.1 to $d\sim$80 pc at $z$=2), likely arising from an AGN.
Interestingly, we found that {91\%} of VLBI sources are classified as AGN by our method, where {55\%} are MLAGN and 36\% are HLAGN. About {88\%} of the HLAGN also show a $>$3$\sigma$ radio excess compared to the SFR$_{\rm IR}$ (Sect. \ref{llagn}). These notably high percentages of VLBI sources classified as AGN (both HLAGN and MLAGN) in our sample demonstrate the high reliability of our classification method.
\begin{table}
\caption{\small Comparison between our classification method and that presented in \citet{Bonzini+13}. For this check, we cut our sample at total 1.4~GHz flux $S_{\rm 1.4} > $ 37$~\mu$Jy to match the radio selection adopted by the authors. }
\begin{tabular}{l cccc }
\hline
\hline
Classifications & & & (B13) & \\
\hline
(this work, & & RQ AGN & RL AGN & SFGs \\
$S_{\rm 1.4} > $ 37$~\mu$Jy) & total & (609) & (865) & (3099) \\
\hline
HLAGN & (1044) & 609 & 169 & 266 \\
MLAGN & (1032) & 0 & 569 & 463 \\
Rest of the sample & (2497) & 0 & 127 & 2370 \\
\hline
\end{tabular}
\label{table:b13}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=3.4in]{q_comparison.pdf}
\end{center}
\caption{\small Redshift distribution of q$_{\rm 24, obs}$ (top panel) and q$_{\rm FIR}$ (bottom panel) for our 3~GHz sample (black dots). The subsample with radio excess is highlighted with green circles. The dash-dotted line (top panel) indicates the radio-excess threshold by \citeauthor{Bonzini+13} (\citeyear{Bonzini+13}, B13), while the horizontal dashed line indicates the threshold in q$_{\rm 24, obs}$ defined by \citet{Donley+05}. The dashed line of the bottom panel indicates the threshold in the rest-frame q$_{\rm FIR}$ identified by \citet{DelMoro+13}. }
\label{fig:q24}
\end{figure}
\subsubsection{Comparison with \citet{Bonzini+13}} \label{bonzini}
In a similar study, \citeauthor{Bonzini+13} (\citeyear{Bonzini+13}, B13 hereafter) carried out a panchromatic analysis of about 800 high-redshift (z$\leq$4) radio sources in the E-CDFS, selected with VLA at 1.4~GHz. They separated radio sources into radio-loud AGN (RL-AGN), radio-quiet AGN (RQ-AGN) and star-forming galaxies (SFGs) on the basis of the observed 24$~\mu$m-to-1.4~GHz flux ratio (also called q$_{24,\rm obs}$, see \citealt{Donley+05}). Briefly, B13 selected RL-AGN that are below the 2$\sigma$ deviation from the average q$_{24,\rm obs}$ at a given redshift. {In case q$_{24,\rm obs}$ was an upper limit, the authors selected RL-AGN that are below the 1$\sigma$ deviation from the average value.} Within and above the 2$\sigma$ deviation, they selected RQ-AGN {that are not RL-AGN and at the same time fulfilling either X-ray or MIR diagnostics (similar to those discussed in Sect. \ref{hlagn})}; the rest of the sample was classified as SFGs. On top of these criteria, B13 applied further checks (see their
Sect. 3.5.1) to improve the sample characterisation, which led the authors to reclassify 11 sources from SFGs to RQ or RL AGN in their sample.
Despite the larger area, our 3~GHz data in COSMOS are deeper than the E-CDFS data at 1.4~GHz, which allows us to compare our classification with B13 in the same flux density range. First, we scaled our 3~GHz flux density to the observed 1.4~GHz for each source (as discussed in Sect. \ref{radio-multi}). Secondly, we cut our sample at total 1.4~GHz flux $S_{\rm 1.4~GHz} > $ 37$~\mu$Jy as in B13, yielding {4\,573} sources (around 59\% of the full sample) and we computed q$_{24,\rm obs}$ for all of them. Thirdly, we applied the same criteria of B13 to identify RQ-AGN, RL-AGN and SFGs in our sample (including their additional AGN diagnostics for consistency, see Sect. \ref{add_diagnostics}), and show the numbers in Table \ref{table:b13}.
This comparison suggests that the HLAGN and MLAGN identified in this work fairly overlap with the RQ-AGN and RL-AGN classes, respectively, even if our classification tends to classify more objects as AGN ({45\%}) than the B13 classification ({32\%}). The RQ-AGN percentage found by B13 (24\%) is higher than that listed in Table \ref{table:b13} ({13\%}), and is likely driven by the higher incompleteness of X-ray observations in the COSMOS field towards moderately luminous X-ray AGN ($10^{42}<$ $L_{\rm x}< 10^{44}$ erg s$^{-1}$), especially at high redshifts (z$>$2). The aim of this comparison is not to invalidate either of the methods, but simply to clarify how different nomenclatures compare to each other. The main difference in the source classification lies in the different definitions of radio excess. In their work, B13 used a redshift-dependent threshold in q$_{\rm 24, obs}$, which was calibrated on the M82 template. The dash-dotted line in Fig. \ref{fig:q24} (top panel) shows the
redshift-dependent threshold defined by B13. Black symbols indicate the distribution of our 3~GHz radio sources ({7\,729}) as a function of q$_{\rm 24, obs}$ and redshift with our radio-excess sources ({1\,814} in total, being {1\,333} in MLAGN and {481} in HLAGN) highlighted in green. Downward pointing arrows indicate (5$\sigma$) upper limits due to non-detection at 24$~\mu$m. The black dashed line sets the threshold q$_{\rm 24, obs}<$ 0 adopted by \citet{Donley+05} to identify radio-excess sources. \citet{DelMoro+13} already pointed out that q$_{\rm 24, obs}$ is a reliable tracer of radio excess, although it is not complete. We confirm this statement, as most of the sources below the q$_{\rm 24, obs}$ threshold (by B13) also satisfy our radio-excess definition. However, the B13 criterion at z$>$1 becomes even more stringent than that proposed by \citet{Donley+05}. This decreasing trend of q$_{\rm 24, obs}$ with redshift is driven by the shape of the M82 template SED, which is rather peculiar compared to
the average SED of star-forming galaxies at 0$<$z$<$3, implying a steeper decline with redshift compared to what is observed in our sample. However, the percentage of z$>$2 sources in B13 is relatively small compared to our sample, which implies that the M82 template SED shape should not have a large effect on the source classification in the E-CDFS sample.
\subsubsection{Comparison with \citet{DelMoro+13}} \label{del_moro}
An alternative method to search for radio excess is by using the (rest-frame) FIR-to-radio flux ratio $q_{\rm FIR}$ (e.g. \citealt{Sargent+10}), where the FIR flux refers to the rest-frame range 42.5--122.5$~\mu$m, and the radio flux refers to the rest-frame 1.4~GHz. Recently, \citet{DelMoro+13} used $q_{\rm FIR}<$ 1.68 in star-forming galaxies within the GOODS-North field to identify sources with $>3\sigma$ radio excess.
As proposed by \citeauthor{DelMoro+13} (\citeyear{DelMoro+13}, see their Figure 5), we calculate the q$_{\rm FIR}$ for our 3~GHz sources and show in Fig. \ref{fig:q24} (bottom panel) their distribution with redshift with respect to the threshold set by \citet{DelMoro+13}. \citet{DelMoro+13} calibrated the q$_{\rm FIR}$ threshold on a sample of sources detected at both 1.4~GHz and 24$~\mu$m, which are, therefore, on average more star-forming than our radio-selected galaxies in COSMOS. If limiting our sample to 24$~\mu$m detected sources, we estimate the percentage of radio-excess sources to be around 13\%, which is in agreement with the percentage found by \citet{DelMoro+13}. We checked that the average q$_{\rm FIR}$ values of our 3~GHz sources also detected at 24$~\mu$m closely resemble those of \citet{DelMoro+13} in the common redshift range. This is expected, given that the radio-to-24$~\mu$m sensitivity limits are similar between the GOODS-North and COSMOS fields. For comparison with \citet{DelMoro+13},
we calculated the $q_{\rm FIR}$ for our radio-excess sources and verified that 100\% of these sources satisfy the condition $q_{\rm FIR}<$ 1.68, while around 72\% of sources with $q_{\rm FIR}<$ 1.68 satisfy our definition of radio excess. This check further supports the reliability of our definition of radio excess.
\subsection{Further tests of the source classification} \label{further_tests}
The classification scheme presented in Sects. \ref{hlagn} and \ref{llagn} was based on a few assumptions that we test in this section. In particular, we detail the motivation for our naming convention (Sect. \ref{naming_convention}), show how our classification would change if considering additional AGN diagnostics (Sect. \ref{add_diagnostics}), and compare the IR luminosities derived for \textit{Herschel}-undetected sources against IR stacking (Sect. \ref{ir_stacking}).
\begin{figure*}
\begin{center}
\includegraphics[width=180mm,keepaspectratio]{dist_lum_agn.pdf}
\end{center}
\caption{\small Normalised distribution of L$\rm_{rad, AGN}$ (or BHAR), as a function of redshift, separately for MLAGN (red dashed) and HLAGN (blue solid). The subsample of HLAGN not identified as ``SED-AGN'' ({27\%} of all HLAGN) is represented by the blue dashed distribution. The left-pointing arrows indicate upper limits at 90\% confidence level in L$\rm_{rad, AGN}$ for the corresponding distribution. }
\label{fig:dist_lradagn}
\end{figure*}
\subsubsection{The choice of the naming convention } \label{naming_convention}
Our samples of HLAGN and MLAGN include AGN identified through different diagnostics. As shown in Fig. \ref{fig:venn}, only {14\%} of HLAGN meet simultaneously the three diagnostics (X-ray, MIR, and SED decomposition), suggesting that various AGN selection criteria might be sensitive to a broad range of AGN luminosities. We study the distribution of MLAGN and HLAGN as a function of AGN radiative luminosity (L$\rm_{rad, AGN}$), as derived from SED-fitting decomposition, converted to BHAR following Eq. 1 from \citet{Alexander+12} and assuming a canonical mass-energy efficiency conversion of 10\% (e.g. \citealt{Marconi+04}). Each estimate of L$\rm_{rad, AGN}$ is calculated from the corresponding best-fit AGN template (Sect. \ref{sed_fitting}) obtained from SED-fitting decomposition. This parameter should be considered an upper limit if the AGN component is not significant from SED-decomposition, which is the case for all MLAGN (by definition) and for about {27\%} of HLAGN identified solely from X-ray or MIR
criteria. In the latter case, the L$\rm_{rad, AGN}$ has been taken from the 95$^{\rm th}$ percentile of the corresponding PDF obtained from the {\sc sed3fit} code, which is equivalent to an upper limit at 90\% confidence level.
Fig. \ref{fig:dist_lradagn} shows the L$\rm_{rad, AGN}$ distribution separately for MLAGN (red dashed line), HLAGN (blue solid line), and the subsample of HLAGN not identified from SED fitting (blue dashed line) in seven redshift bins. The vertical lines represent the median of the corresponding distribution at each redshift bin. The median values show a significant difference (around 1~dex) in L$\rm_{rad, AGN}$ between MLAGN and HLAGN at all redshifts. Despite the non-negligible overlap between the two distributions, we stress that the L$\rm_{rad, AGN}$ estimates for MLAGN consist of upper limits, which implies that the difference between the true distributions, at a given redshift, is even more significant.
This test suggests that, at each redshift, HLAGN are significantly more powerful than MLAGN and justifies the naming convention proposed in this work in a statistical sense.
\subsubsection{Additional AGN diagnostics} \label{add_diagnostics}
As carried out in B13, we applied additional diagnostics to verify the robustness of our AGN identification method.
\begin{itemize}
\item Optical spectra: we searched for sources flagged as broad line AGN in the optical spectra taken from the zCOSMOS-Bright survey (\citealt{Lilly+07}, \citeyear{Lilly+09}); we found {15} in total, which were all pre-classified as HLAGN on the basis of X-ray and mid-IR criteria.
\item VLBI\ sample:\ as mentioned in Sect. \ref{VLBI}, {354} radio sources selected at 3~GHz were also identified in the VLBI sample (N.~Herrera Ruiz et al. in prep.) available in the COSMOS field. Of these, {30 (8\%)} were not identified as either MLAGN or HLAGN, although they are likely to be AGN.
\item Inverted radio spectra: we found a total of {11} radio sources not classified as radio-selected AGN, but that have inverted radio spectra ($\alpha>$0), where the spectral index $\alpha$ is set to the observed 1.4--3~GHz slope. This feature may indicate the presence of a compact radio core (e.g. \citealt{kellermann+69}).
\item Hardness ratio: we found a total of {seven} X-ray sources that were not classified as AGN in our sample with a positive hardness ratio ($HR>$0), indicating the likely presence of obscured AGN (e.g. \citealt{Brusa+10}).
{\item Polycyclic aromatic hydrocarbon (PAH) features: sources lying in the IRAC colour-colour wedge discussed in B13 (see their Sect. 3.5.5) likely display PAH-dominated SEDs, which are typical of star-forming galaxies. We found 444 sources that fulfil this criterion with only five (26) that are classified as MLAGN (HLAGN) in our sample.}
\end{itemize}
The above-mentioned criteria suggest that some of our sources might be misclassified. We found that {46} radio sources ({30} from VLBI detection, {9} from inverted radio spectra, {and seven} from the hardness ratio) should be reclassified from non-AGN to AGN in our sample. However, these criteria are applicable to a very low percentage of our sample, meaning that by incorporating them we would likely introduce a bias against sources with no available diagnostics. Reclassifying these {46} sources (0.6\% of our sample) would have no impact on our main results and conclusions. {We also found 31 radio-selected AGN displaying PAH-dominated SEDs in the MIR; even though this criterion does not rule out a potential AGN contribution in other bands, the percentage of possibly misclassified AGN is minimal.} For these reasons, we limited our AGN selection criteria to those presented in Sects. \ref{hlagn} and \ref{llagn}.
\subsubsection{Testing radio excess with infrared stacking} \label{ir_stacking}
As discussed in Sect. \ref{sed_fitting}, the $L_{\rm IR}$ and SFR estimates are less robust for sources without \textit{Herschel} detection, which constitute about one-third of our 3~GHz selected sample. For these sources, we fitted their SEDs using the nominal upper limits in the PACS and SPIRE bands (as described in Sect. \ref{sed_fitting}). As a sanity check, we compared the \textit{Herschel} fluxes used for SED fitting with those obtained via \textit{Herschel} stacking. The difference between the two approaches enables us to test the robustness of our radio-excess definition.
We looked at the 79\% of sources \textit{not} classified as HLAGN, shown in Fig. \ref{fig:radio_excess}, and considered only those without $\geq$3$\sigma$ \textit{Herschel} detection in any PACS or SPIRE band, which amounts to {2\,203} in total. As carried out in previous papers (e.g. \citealt{Santini+12}; \citealt{Rosario+12}; \citealt{Bonzini+13}), we split this sample in different redshift bins (see Table \ref{table:stack}) and we stacked the PACS and SPIRE images at the optical-NIR position of each source in the same bin using a stacking tool from \citet{Bethermin+10}\footnote{This IDL routine is described in detail in \citet{Bethermin+10} and can be retrieved at \url{https://www.ias.u-psud.fr/irgalaxies/downloads.php}.}.
For PACS images, we performed a mean stacking on the residual maps at 100 and 160$~\mu$m, from which all detections were removed to avoid contamination by nearby brighter sources. Point spread function (PSF) photometry was performed on the final stacked images, using the uncertainty maps (\citealt{Lutz+11}) as weights. The stacked fluxes were corrected for aperture and correlated noise\footnote{The full PACS documentation is available at \url{www.mpe.mpg.de/resources/PEP/DR1\_tarballs/}.}. The SPIRE images\footnote{The SPIRE images for the COSMOS field were taken from the HeDAM database: \url{http://hedam.lam.fr/HerMES/index/download}.} at 250, 350, and 500$~\mu$m were already given in units of surface brightness (Jy~beam$^{-1}$), hence we inferred the stacked flux directly from the value of the central pixel in the stacked image. To evaluate the uncertainty on the stacked fluxes, we performed a bootstrapping analysis (e.g. \citealt{Shao+10}; \citealt{Santini+12}; \citealt{Rosario+12}). Briefly, a set of $N$
sources, where $N$ is equal to the number of stacked sources per redshift bin, is randomly chosen 1\,000 times, allowing repetition of the same source. To mitigate the possible contamination from a few brighter outliers, we set the stacked flux to the median of the distribution obtained from bootstrapping, while the error on the flux was drawn by the 16$^{\rm th}$ and 84$^{\rm th}$ percentiles of the same distribution. For PACS images, we also corrected the errors for high-pass filtering effects. We did not correct the stacked fluxes for possible blending in the optical-NIR images. In every redshift bin we obtained $>$2$\sigma$ detection in two to five \textit{Herschel} bands. The final stacked fluxes and corresponding uncertainties are summarised in Table \ref{table:stack}.
We re-fitted the {2\,203} SEDs again with {\sc magphys}, using the corresponding stacked fluxes as detections for
each source. The resulting $M_{\star}$ estimates agree very well (no offset, 1$\sigma$ dispersion is about 0.15~dex) with those obtained in Sect. \ref{sed_fitting}. The newly derived $L_{\rm IR}$ estimates are generally lower (by 50\%) than those obtained in Sect. \ref{sed_fitting}, but for 40\% of the stacked sample we found slightly higher $L_{\rm IR}$. This is mainly because while in most cases the fluxes obtained from \textit{Herschel} stacking (see Table \ref{table:stack}) are lower than the flux values used for SED fitting, in a few bins they are instead slightly higher, especially at z$\gtrsim$1.5. However, we verified that using the $L_{\rm IR}$ estimates derived from stacking would only minimally affect the overall distribution of $L_{\rm 1.4\, GHz}$/SFR$_{\rm IR}$ (Fig. \ref{fig:radio_excess}). In particular, we checked that our classification would remain
unchanged for 90\% of the sources, while the remainder of the sample would move either from SFGs to MLAGN (5\%) or vice versa (5\%).
The purpose of this test was to quantify the impact of using different sets of \textit{Herschel} fluxes on the source classification. The general agreement obtained between the two methods ensures the robustness of our classification. As a consequence, we decided to keep using the \textit{Herschel} upper limits introduced in Sect. \ref{sed_fitting} through the rest of this work.
\section{Catalogue description} \label{catalog}
The value-added catalogue presented in this section includes classification and selected physical properties used in this work for our 3~GHz radio sample with optical-NIR counterparts ({7\,729} sources in total). We also list the individual criteria used in this work to classify our radio sources (columns 14 to 17). This way, any user can easily retrieve our classification or adjust it to a different set of selection criteria.
The catalogue will be made available through the COSMOS IPAC/IRSA database\footnote{\url{http://irsa.ipac.caltech.edu/Missions/cosmos.html}}. Here we describe its structure, following the same format of Table \ref{table:catalogue}.
\begin{itemize}
\item (1) Identification number of the radio source (ID).
\item (2) Right ascension (J2000) of the radio source.
\item (3) Declination (J2000) of the radio source. \item (4) Best redshift available for the source.
\item (5) Origin of the redshift: spectroscopic (``spec'') if available, photometric (``phot'') otherwise.
\item (6) 3~GHz integrated radio flux density [$\mu$Jy].
\item (7) 3~GHz (rest-frame) radio luminosity [log W~Hz$^{-1}$].
\item (8) 1.4~GHz (rest-frame) radio luminosity [log W~Hz$^{-1}$], obtained as described in Sect. \ref{radio-multi}.
\item (9) Star formation infrared (8-1000$~\mu$m rest-frame) luminosity derived from SED fitting [log $L_{\odot}$]. If the source is classified as HLAGN, this value represents the portion of the total infrared luminosity arising from star formation, while it corresponds to the total IR luminosity otherwise (see Sect. \ref{sed_fitting}).
\item (10) Flag for \textit{Herschel} detection at $\geq$3$\sigma$, in at least one band: ``true'' if detected, ``false'' if only upper limits are available.
\item (11) Stellar mass derived from SED-fitting decomposition [log $M_{\odot}$]. The value is drawn from the fit with AGN if the source is classified as HLAGN and otherwise from the fit without AGN. Calculated with a \citet{Chabrier03} IMF.
\item (12) Star formation rate [$M_{\odot}$yr$^{-1}$] obtained from the total infrared luminosity listed in column (9), assuming the \citet{Kennicutt98} conversion factor, and scaled to a \citet{Chabrier03} IMF.
\item (13) Rest-frame {\sc [NUV-$r$]} colour obtained from the best-fitting galaxy template and corrected for dust attenuation (AB magnitude).
\item (14) X-ray-AGN: ``1'' if true, ``0'' otherwise.
\item (15) MIR AGN: ``1'' if true, ``0'' otherwise.
\item (16) SED-AGN: ``1'' if true, ``0'' otherwise.
\item (17) Radio-excess: ``1'' if true, ``0'' otherwise.
\item (18) Class: moderate-to-high radiative luminosity AGN (HLAGN), low-to-moderate radiative luminosity AGN (MLAGN), or neither of the two (empty space). A source is classified as HLAGN if (14)=1 $\vee$ (15)=1 $\vee$ (16)=1, while it is classified as MLAGN if (14,15,16)=(0,0,0) $\wedge$ (17)=1.
\end{itemize}
\begin{table}
\centering
\caption{\small Number of 3~GHz radio sources studied in this work as a function of redshift and AGN class: MLAGN, HLAGN, and HLAGN with radio excess (in brackets). For each redshift bin we report the mean redshift $\langle z \rangle$ of the corresponding population. }
\begin{tabular}{l ccc }
\hline
\hline
redshift bin & $\langle z \rangle $ & MLAGN & HLAGN \\
& & & (radio-excess) \\
\hline
0.01 $\leq z <$ 0.30 & 0.21 & 22 & 36 (9) \\
0.30 $\leq z <$ 0.70 & 0.51 & 221 & 232 (66) \\
0.70 $\leq z <$ 1.20 & 0.94 & 375 & 416 (135) \\
1.20 $\leq z <$ 1.80 & 1.48 & 350 & 350 (98) \\
1.80 $\leq z <$ 2.50 & 2.08 & 225 & 307 (84) \\
2.50 $\leq z <$ 3.50 & 2.89 & 111 & 217 (74) \\
3.50 $\leq z <$ 5.70 & 4.21 & 29 & 46 (15) \\
\hline
\end{tabular}
\label{table:class}
\end{table}
\begin{figure*}
\begin{center}
\includegraphics[width=180mm,keepaspectratio]{agn_sfr_mass_lerg_herg_kennicutt_contours.pdf}
\end{center}
\caption{\small Distribution of HLAGN and MLAGN in the SFR--$M_{\star}$ plane as a function of redshift (black dots). The 2D density contours highlight the distribution of MLAGN (red), HLAGN (blue), and the subsample of HLAGN with radio excess (blue dashed contours), respectively. The 2D density contours enclose the sources (from outer to inner contours) with density levels $>$35, $>$50, $>$68, $>$80, $>$90, and $>$95\% of the maximum 2D density. The black dotted lines indicate the MS relation (\citealt{Whitaker+12}) at different redshifts. The highest redshift bin contains only a few tens of sources, which explains the noise seen in the 2D density contours. }
\label{fig:sed_mass_plane_classes}
\end{figure*}
\section{Results} \label{results}
In this section we present the average AGN and galaxy properties for our 3~GHz radio sources classified as AGN host galaxies. In particular, we focus our analysis on MLAGN (17\%), HLAGN (21\%), and also the subsample of HLAGN with radio excess (6\%).
We show, for these classes, the location in the SFR--$M_{\star}$ plane (Sect. \ref{sfr_mass_plane}) and the distributions of AGN and galaxy properties of their hosts (Sect. \ref{histograms}) at different cosmic epochs.
\begin{table*}
\centering
\caption{\small Results from the two-sample Kolmogorov-Smirnov (K-S) test. The table shows, in different redshift bins, the probability {\sc P(K-S)} that the distributions of a given parameter ($M_{\star}$, SFR or {\sc [NUV-$r$]}) for HLAGN and MLAGN (shown in Figs. \ref{fig:histogal1}, \ref{fig:histogal2} and \ref{fig:histogal3}) are drawn from the same parent distribution. A lower probability indicates a more significant difference. Values in brackets report the significance of the K-S test in units of $\sigma$. }
\begin{tabular}{l ccc }
\hline
\hline
& & {\sc P(K-S)} & \\
Redshift bin & $M_{\star}$ & SFR & {\sc [NUV-$r$]} \\
& \% ($\sigma$) & \% ($\sigma$) & \% ($\sigma$) \\
\hline
0.01 $\leq z <$ 0.30 & 0.76 (2.67) & 2.97 (2.17) & 0.11 (3.25) \\
0.30 $\leq z <$ 0.70 & 2.07$\times10^{-3}$ (4.26) & $<10^{-20}$ ($>$10) & $<10^{-20}$ ($>$10) \\
0.70 $\leq z <$ 1.20 & 3.84$\times10^{-4}$ (4.62) & $<10^{-20}$ ($>$10) & $<10^{-20}$ ($>$10) \\
1.20 $\leq z <$ 1.80 & 34.3 (0.95) & $<10^{-20}$ ($>$10) & 3.50$\times10^{-14}$ (8.16) \\
1.80 $\leq z <$ 2.50 & 4.51$\times10^{-7}$ (5.86) & $<10^{-20}$ ($>$10) & 1.06$\times10^{-4}$ (4.88) \\
2.50 $\leq z <$ 3.50 & 1.17 (2.52) & 1.52$\times10^{-13}$ (7.97) & 1.99$\times10^{-8}$ (6.36) \\
3.50 $\leq z <$ 5.70 & 13.5 (1.49) & 4.44$\times10^{-3}$ (4.08) & 10.7 (1.61) \\
\hline
\end{tabular}
\label{table:ks}
\end{table*}
\subsection{The SFR--$M_{\star}$ plane of radio-selected AGN} \label{sfr_mass_plane}
Fig. \ref{fig:sed_mass_plane_classes} shows the 2D density contours in SFR--$M_{\star}$ plane for our samples of MLAGN (red), HLAGN (blue), and for the subsample of HLAGN with radio excess (blue dashed contours). Black dots represent our joint sample of aforementioned AGN at different redshifts. The 2D density contours enclose the sources (from outer to inner contours) with density levels $>$35, $>$50, $>$68, $>$80, $>$90, and $>$95\% of the maximum 2D density for a given class and redshift bin.
The black dashed line marks the so-called main sequence (MS) of star-forming galaxies (taken from \citealt{Whitaker+12}), which is known to evolve positively with redshift (e.g. \citealt{Noeske+07}; \citealt{Elbaz+11}; \citealt{Speagle+14}; \citealt{Schreiber+15}).
The $M_{\star}$ and SFR estimates for each source were computed directly by the three-component fit, hence already correcting for a possible AGN contamination (Sect. \ref{sed_fitting}). Typical 1$\sigma$ uncertainties on $M_{\star}$ and SFR are of the order of 0.1~dex, but for \textit{Herschel} undetected sources the uncertainty in SFR is around 0.2~dex.
Table \ref{table:class} summarises the number of sources shown in Fig. \ref{fig:sed_mass_plane_classes} for each class and redshift bin. These numbers show that the redshift distribution for both HLAGN and MLAGN peaks around z$\sim$1. The two AGN populations become comparable around z$\sim$1.5, while at higher and lower redshift the HLAGN generally outnumber MLAGN. The percentage of HLAGN with a $>$3$\sigma$ radio excess is roughly constant with redshift (around 25--35\% of the HLAGN sample).
Our HLAGN and MLAGN appear to lie in different regions of the SFR--$M_{\star}$ plane, at various redshifts. At low redshift (z$<$0.3), the two AGN classes show rather distinct $M_{\star}$ distributions, where MLAGN are more than two times more massive than HLAGN (10$^{11}$ $M_{\odot}$ versus 10$^{10-10.5}~M_{\odot}$). This difference is consistent with that found by previous studies in the local Universe (e.g. \citealt{Smolcic+09a}; \citealt{Best&Heckman12}). However, we notice that HLAGN with radio excess predominantly lie in the high-$M_{\star}$ tail of the HLAGN population and closely resemble the distribution of MLAGN at this redshift (z$<$0.3). Therefore, this subsample seems to show intermediate SFR and $M_{\star}$ distributions between the two AGN categories. At higher redshift (0.3$<$z$<$1.8), the two AGN populations show a larger overlap in the SFR--$M_{\star}$ plane than that observed at lower redshift. However, the bulk of HLAGN is generally located around the MS relation, while MLAGN
preferentially
lie in the lower part of the MS with typically lower SFRs (by a factor of 2--3) compared to HLAGN. The subsample of HLAGN with radio excess is mostly concentrated between the locations of the two main AGN classes. At even higher redshift (z$>$1.8) the overlap between the two distributions decreases, with HLAGN, largely located above the MS relation, having on average both higher M$_{\star}$ and SFR than the MLAGN. A detailed description of the distributions of host-galaxy properties is given in Sect. \ref{histograms}. The highest redshift bin contains only a few tens of sources, which explains the noise seen in the 2D density contours.
This difference in SFR is consistent with the different percentages of \textit{Herschel}-detected sources between the two AGN classes: 57\% of HLAGN has a $>$3$\sigma$ detection in at least one \textit{Herschel} band, while this percentage decreases to only 17\% for MLAGN. These numbers slightly decrease with redshift for either classes, from {81\% (27\%)} at z$<$0.3 to {50\% (17\%)} in the highest redshift bin for HLAGN (MLAGN). These numbers further suggest that the host galaxies of HLAGN are typically star forming at all redshifts.
Although this work does not aim to investigate the nature of the sources not classified as AGN, we checked that the subsample of sources that are neither HLAGN nor MLAGN (i.e. 62\%) are mostly located on the MS relation at all redshifts, thus resembling the distribution of HLAGN in the SFR--$M_{\star}$ plane. This suggests that the remainder of our sample might consist of mostly star-forming galaxies.
A comprehensive study of the radio-AGN population in the SFR--M$_{\star}$ plane has been presented by \citet{Bonzini+15}. They found that most of RQ AGN show significant star formation in their hosts, and typically (75\%) lie along the MS relation, likewise SFGs, at various redshifts. Moreover, \citet{Bonzini+15} found that the majority of RL AGN reside in less star-forming galaxies, which are predominantly located below the MS. Despite the different nomenclature and sample selection used by the authors (see Table \ref{table:b13} for a comparison), the qualitative agreement with their results is reassuring.
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=7in]{dist_mstar.pdf}
\end{center}
\caption{\small Normalised distributions of $M_{\star}$, as a function of redshift. Radio classes are highlighted as follows: HLAGN (blue), HLAGN subsample with radio excess (blue thicker distribution), and MLAGN (red). Vertical lines show the median value for MLAGN (red), HLAGN (blue) and their subsample with radio excess (blue thicker). }
\label{fig:histogal1}
\end{figure*}
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=7in]{dist_sfr_kenn.pdf}
\end{center}
\caption{\small Normalised distributions of the SFR as a function of redshift. Radio classes are highlighted as follows: HLAGN (blue), HLAGN subsample with radio excess (blue thicker distribution), and MLAGN (red). Vertical lines show the median value for MLAGN (red), HLAGN (blue), and their subsample with radio excess (blue thicker).}
\label{fig:histogal2}
\end{figure*}
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=7in]{dist_opt_colours.pdf}
\end{center}
\caption{\small Normalised distributions of the rest-frame {\sc [NUV--$r$]} colours, corrected for dust attenuation, as a function of redshift. Radio classes are highlighted as follows: HLAGN (blue), HLAGN subsample with radio excess (blue thicker distribution), and MLAGN (red). Vertical lines show the median value for MLAGN (red), HLAGN (blue), and their subsample with radio excess (blue thicker).}
\label{fig:histogal3}
\end{figure*}
\subsection{Physical properties of AGN hosts} \label{histograms}
In this section we investigated the distributions of galaxy and AGN properties for the populations of HLAGN and MLAGN. We applied a two-sample Kolmogorov-Smirnov (K-S) test to quantify the difference or similarity
between two distributions. This statistical test allows {us} to determine if two input data sets could be drawn from a common parent distribution without any assumption about its shape. We run this test to evaluate the difference between HLAGN and MLAGN in terms of various galaxy properties. The probabilities {\sc P(K-S)} obtained for each parameter and redshift bin are listed in Table \ref{table:ks}, along with the corresponding $\sigma$ level. Smaller values of {\sc P(K-S)} indicate a more significant difference between the two data sets.
\subsubsection{Distribution of galaxy properties} \label{results_gal}
Figs. \ref{fig:histogal1}, \ref{fig:histogal2}, and \ref{fig:histogal3} show the distributions of $M_{\star}$, SFR, and rest-frame {\sc [NUV--$r$]} colours, respectively, for the following classes: MLAGN (red), HLAGN (blue), and the subsample of HLAGN with radio excess (blue thicker distribution). Vertical lines show the median value of the corresponding distribution. The distributions are shown in seven redshift bins out to z$\lesssim$6 and are normalised to the highest maximum value of the two distributions.
As mentioned in Sect. \ref{sfr_mass_plane}, the $M_{\star}$ distributions of MLAGN at low redshift are skewed towards higher $M_{\star}$ compared to HLAGN, and the difference remains significant up to z$\sim$1 at $\gtrsim$99\% level (see Table \ref{table:ks}). At z$\sim$1.5 (4$^{\rm th}$ redshift bin) the two distributions appear more similar.
At higher redshifts (z$\sim$2), we observe a possible reversal of the $M_{\star}$ distributions with the HLAGN populating the high-$M_{\star}$ tail. At this redshift, the two-sample K-S test finds an almost {6}$\sigma$ difference between the two distributions. However, we are not able to confirm or disclaim this trend at z$>$2.5, given the tentative significance (about 2$\sigma$) of the results obtained from the K-S test. A more detailed discussion and interpretation of these trends is presented in Sect. \ref{discussion}. As seen in the SFR--$M_{\star}$ plane (Fig. \ref{fig:sed_mass_plane_classes}), the subsample of HLAGN with radio excess overlaps significantly with the distribution of MLAGN, showing intermediate $M_{\star}$ between the two AGN classes (except in the highest redshift bin).
In Fig. \ref{fig:histogal2} we show the same plots for the SFR, obtained by integrating the best-fit galaxy template over the range (rest-frame) 8--1000$~\mu$m, and by assuming a \citet{Kennicutt98} scaling factor and a \citet{Chabrier03} IMF. As already seen in Fig. \ref{fig:sed_mass_plane_classes}, we confirm that the HLAGN with radio excess populate the lower tail of the SFR distribution, overlapping significantly with MLAGN. The difference between HLAGN and MLAGN in SFR remains visible and $\gtrsim99$\% significant in the higher redshift bins, up to z$\sim$3, as well.
Rest-frame {\sc [NUV-$r$]} colours were calculated from the best-fit galaxy template of each source and also corrected for dust attenuation. The distribution of {\sc [NUV-$r$]} colours shown in Fig, \ref{fig:histogal3} confirms that most of HLAGN have blue or green rest-frame optical colours ({\sc [NUV-$r$]}$<$3.5; \citealt{Ilbert+10}) at all redshifts. On the other hand, MLAGN are more pronounced towards quiescent systems ({\sc [NUV-$r$]}$>$3.5; \citealt{Ilbert+10}), at least up to z$\sim$1. This bimodality in the colour distributions becomes progressively less pronounced at higher redshift, also showing that the host-galaxies of MLAGN become, on average, more star forming with increasing redshift (see Sect. \ref{ir_stacking}). It is recognised that the number density of quiescent galaxies (selected via optical colours) at $M_{\star}>$10$^{10}$~$M_{\odot}$ decreases with increasing redshift (e.g \citealt{Brammer+11}, \citealt{Ilbert+13}). However, the difference in {\sc [NUV-$r$]}
between HLAGN and MLAGN remains highly significant up to z$\sim$3.5, while it disappears at 3.5$<$z$<$5.7. The subsample of HLAGN with radio excess shows intermediate colours between the rest of HLAGN and the population of MLAGN.
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=7in]{dist_accretion_lums.pdf}
\end{center}
\caption{\small Normalised distributions of AGN power, both radiative (L$\rm_{rad, AGN}$) and mechanical (L$\rm_{mech, AGN}$) as a function of redshift. The distributions of L$\rm_{rad, AGN}$ are shown for HLAGN (blue) and HLAGN with radio excess (blue thicker), while the distributions of L$\rm_{mech, AGN}$ are shown only for MLAGN (orange) and HLAGN with radio excess (orange thicker). The median value of each distribution is indicated with a vertical line with same colour and thickness of the corresponding histogram. The orange horizontal lines around the median value of the AGN mechanical power show the range within which the median could shift if accounting for all of the uncertainties on the $L_{\rm 1.4\, GHz}$--L$\rm_{mech, AGN}$ relation (see text for details). The normalisations are set separately for the two L$\rm_{mech, AGN}$ (red) and the two L$\rm_{rad, AGN}$ (blue) distributions. See the text for details.}
\label{fig:histogramagn}
\end{figure*}
\subsubsection{Distribution of AGN properties} \label{results_agn}
Fig. \ref{fig:histogramagn} shows the distribution of both AGN radiative and mechanical power for our classes of AGN.
We calculated the AGN radiative power (L$\rm_{rad, AGN}$) of HLAGN from the best-fit AGN template obtained with the three-component SED-fitting code {\sc sed3fit} for sources both with and without radio excess. The typical uncertainties on L$\rm_{rad, AGN}$ are around 0.4~dex for sources with ($\geq$99\%) significant AGN component (i.e. SED-AGN, see Sect. \ref{classification}), while we took the L$\rm_{rad, AGN}$ from the 95$^{\rm th}$ percentile of the corresponding PDF obtained from the {\sc sed3fit} code for HLAGN not identified as such from SED decomposition (30\% of HLAGN); this is equivalent to an upper limit at 90\% confidence level on the AGN radiative luminosity. Fig. \ref{fig:histogramagn} shows the normalised distributions of L$\rm_{rad, AGN}$, separately for HLAGN (blue) and for the subsample with radio excess (blue thicker). The
distributions of L$\rm_{rad, AGN}$ cover a broad range ($>$3~dex) in each redshift bin, which is around 10$^{42-45}$ erg~s$^{-1}$ at z$<$0.3, and 10$^{44-47}$ erg~s$^{-1}$ at z$\sim$3. As mentioned in Sect. \ref{naming_convention}, the percentage of HLAGN identified from SED fitting does not depend on the presence of a radio excess. Therefore, applying the upper limits at 90\% confidence level on L$\rm_{rad, AGN}$ does not affect the ratio of the L$\rm_{rad, AGN}$ distributions between HLAGN and their subsample with radio excess.
We calculated the rest-frame 1.4~GHz radio luminosity $L_{\rm 1.4\, GHz}$ for each source by scaling its radio flux from 3~GHz to 1.4~GHz and taking the observed 1.4--3~GHz spectral index $\alpha$, as explained in Sect. \ref{radio-multi}. The presence of a $>$3$\sigma$ radio excess suggests that a notable portion of the radio emission is not arising from star formation processes in the host, but possibly from the central SMBH. For this reason, each $L_{\rm 1.4\, GHz}$ measurement was scaled to the portion associated with AGN activity, based on the deviation of the observed $L_{\rm 1.4\, GHz}$ - to - SFR$_{\rm IR}$ ratio from the peak of the Gaussian function (associated with star formation) at the corresponding redshift bin (blue points in Fig. \ref{fig:radio_excess}).
We converted the AGN-related radio emission to AGN mechanical power (L$\rm_{mech, AGN}$) of the radio jet, by assuming the redshift-independent relation by \citet{Willott+99}, which is based on theoretical grounds and adopted in other studies (e.g. \citealt{Merloni+08}; \citealt{LaFranca+10}, see \citealt{Best&Heckman12} for a review). We used this relation expressed in terms of $L_{\rm 1.4\, GHz}$ (see Eq. 1 from \citealt{Heckman+14}). \citet{Willott+99} combined all of the uncertainties on this relation into a single factor, $f_{\rm W}$, which can range between 1 and 20. This scaling factor is still a matter of debate in the literature (\citealt{Godfrey+16}). Nonetheless, following the approach of numerous studies (e.g. \citealt{Merloni+07}; \citealt{Smolcic+09a}; \citealt{LaFranca+10}; \citealt{Best&Heckman12}; \citealt{Pracy+16}), we make the simplistic assumption that this relation holds at all radio luminosities $L_{\rm 1.4\, GHz}$ that are probed by our sample.
The normalised distributions of L$\rm_{mech, AGN}$ are shown in Fig. \ref{fig:histogramagn} for both MLAGN (orange) and HLAGN with radio excess (orange thicker distribution). The typical range in L$\rm_{mech, AGN}$ probed by the distributions is about 2~dex wide in each redshift bin, which is around 10$^{41-43}$ erg~s$^{-1}$ at z$<$0.3 and 10$^{43-45}$ erg~s$^{-1}$ at z$\sim$3. Fig. \ref{fig:histogramagn} shows the distributions of L$\rm_{mech, AGN}$ by taking $f_{\rm W}$=5, which is consistent with the relation derived by \citet{Daly+12}. The vertical lines indicate the median value of the corresponding distribution. However, we calculated the range within which the median value could shift, by changing f$_{\rm W}$ between f$_{\rm W}$=1 and f$_{\rm W}$=20, which is shown by the orange horizontal lines around the median.
Our analysis seems to suggest that the overall AGN properties observed for HLAGN with radio excess are similar to the rest of HLAGN in terms of radiative power and are also consistent with MLAGN in terms of mechanical power. The subsample of HLAGN with radio excess ({6}\% of our parent 3~GHz radio sample) is particularly interesting because it enables a direct comparison between radiative and mechanical AGN power for the same sources. Despite the uncertainties on the relation proposed by \citet{Willott+99}, we show that the AGN mechanical power in HLAGN with radio excess is typically lower than (or at most marginally comparable to) the AGN radiative power, depending on f$_{\rm W}$, although in some cases L$\rm_{mech, AGN}$ can exceed L$\rm_{rad, AGN}$ (see \citealt{Heckman+14}). While the AGN power of HLAGN occurs predominantly in radiative form, MLAGN display a substantial mechanical AGN luminosity component. These properties may suggest that HLAGN and MLAGN samples qualitatively resemble radio AGN
types often referred to as radiative mode (or HERG) and jet mode (or LERG), respectively. In addition, we note that MLAGN have significantly lower L$\rm_{rad, AGN}$ than HLAGN with radio excess, despite both classes {showing} a relatively high radio loudness. As a consequence, a simple RL--RQ separation would not allow such direct insight into the fundamental properties of AGN.
\section{Discussion} \label{discussion}
A radio-based selection allows us to study a mixture of galaxy populations that are powered by either star formation, AGN activity, or both. It is also crucial to exploit multiwavelength ancillary data to reach a more comprehensive perspective of the nature of our sources. In this work, we made use of this approach to derive integrated AGN and galaxy properties and compare them between different AGN classes, and over a wide range of radio luminosity and redshift. In this section, we discuss and interpret our findings in the context of current AGN and galaxy evolutionary scenarios.
\subsection{Radio emission in HLAGN and MLAGN }
The origin of radio emission in the sub-mJy radio population is still a matter of debate. Recent studies based on interferometric radio observations of sub-mJy radio sources have the potential to shed light on this issue. For example, \citet{HerreraRuiz+16} analysed in detail the interferometric images of three RQ-AGN obtained with VLBI in the COSMOS field, which are part of the sample described in Sect. \ref{VLBI}. The comparison between VLBI and VLA fluxes suggested that 50--75\% of the radio emission in these sources is arising from non-thermal AGN activity. We note that these sources would have been classified as HLAGN with radio excess according to our method. Similar conclusions were reached by \citet{Chi+13} and \citet{Maini+16} on different samples of RQ-AGN, supporting the idea that some radio sources could be predominantly powered by AGN activity.
At low redshift (z$<$0.3), independent hints on the origin of radio emission in the sub-mJy radio population were provided by \citet{Kimball+11}, who constructed the 6$~$GHz radio luminosity function from a sample of QSO host galaxies at 0.2$<$z$<$0.3. They concluded that radio emission in sources with 6$~$GHz luminosity $L_{\rm 6~GHz} >$10$^{22.5}$ W~Hz$^{-1}$ was AGN related, while in fainter sources it was mainly driven by star formation. However, the parent sample analysed by the authors consists of optically identified QSOs from the SDSS, which are systematically more powerful, relative to our 3~GHz sample of AGN in COSMOS, due to our smaller comoving volume covered at z$<$0.3. Support for radio emission that is powered by AGN activity was also provided by \citet{White+15}, who studied a sample of RQ-QSOs at 1.4~GHz flux $S_{\rm 1.4~GHz}<$1~mJy. The authors stress, however, that their analysis may be biased towards the brightest optically identified QSOs.
A complementary view on this topic benefits from deeper radio surveys, which can push this analysis to higher redshifts and to intrinsically fainter radio sources. For example, \citet{Bonzini+15} and \citet{Padovani+15} investigated the origin of radio emission in RQ and RL AGN in the E-CDFS down to 37$~\mu$Jy (5$\sigma$). They found a mixture of AGN and SFGs contributing to the sub-mJy radio population, where RQ AGN is predominantly powered by star formation. We checked this by exploiting a larger sample of 3~GHz selected sources with optical-NIR counterparts, counting in total {1\,604} HLAGN and {1\,333} MLAGN (Sect. \ref{classification}) out to z$\lesssim$6. The analysis presented in Sects. \ref{hlagn} and \ref{llagn} suggests that roughly 70\% of the HLAGN does not show a $\geq$3$\sigma$ radio excess, which might suggest that radio and infrared emission in HLAGN are commonly (to a certain amount) powered by star formation in their hosts, as proposed by previous studies (e.g.~\citealt{Moric+10}; \citealt{Baldi+13}; \citealt{Padovani+15}).
However, the radio excess detected for the remaining 30\% is a potential signature of radio-selected AGN activity, possibly linked to jet-mode (or radio-mode) feedback, often referred to in the literature (e.g. \citealt{Hardcastle+07}; \citealt{Best&Heckman12}; \citealt{Heckman+14}). On the other hand, radio emission in our sample of MLAGN is predominantly arising from non-thermal radiation likely ascribed to AGN activity, rather than star formation in their hosts. These results agree with the conclusions presented by \citet{Padovani+15} and \citet{Bonzini+15} for a sample of RQ-AGN and RL-AGN, supporting the composite nature of the sub-mJy radio source population (e.g. \citealt{Smolcic+08}; \citealt{Padovani+11}; \citealt{Baldi+14}).
\subsection{Radio AGN in the context of galaxy evolution} \label{interpretation}
We attempt to interpret the nature of our HLAGN and MLAGN populations in the framework of AGN-galaxy evolution. As suggested by previous authors (e.g. \citealt{Hardcastle+07}; \citealt{Smolcic+09b}; \citealt{Best&Heckman12}; \citealt{Padovani+15}), in the local Universe the HERG and LERG classes show a clear dichotomy in terms of AGN and host-galaxy properties. These findings have been interpreted within a self-consistent evolutionary scenario, where HERG and LERG trace earlier and later stages, respectively, of galaxies' life cycle (see \citealt{Heckman+14} for a comprehensive review).
At higher redshift, \citet{Merloni+08} proposed a model to reproduce the kinetic and radiative luminosity function of AGN in which the highly efficient accretion onto the SMBH can produce both kinetic and radiative feedback (e.g. \citealt{Veilleux+13}), which are consistent with the AGN properties observed for our HLAGN with and without radio excess, respectively. Nevertheless, the power from weakly accreting SMBHs ($\lambda_{\rm Edd} \leq$ 10$^{-2}$, also named ``advection-dominated accretion flow'', ADAF; e.g. \citealt{Blandford+77}) is mainly in the form of kinetic feedback (\citealt{Bower+06}; \citealt{Fanidakis+11}), linking to the properties of our MLAGN population.
Semi-analytic models predict different accretion modes between highly and weakly accreting AGN. On the one hand, highly accreting AGN have been usually connected to a fast gas accretion mode in galaxy halos in which the free fall times are usually longer than the cooling times. On the other hand, weakly accreting AGN are in the regime of slow gas accretion, where cooling time is much larger than the free fall time (e.g. \citealt{Fanidakis+11}, \citeyear{Fanidakis+12}).
From an observational point of view, we found that galaxies hosting MLAGN are more massive, redder, and less star forming compared to HLAGN, at least up to z$\sim$1. In particular, the most massive galaxies ($M_{\star}\sim$10$^{11}$~$M_{\odot}$) at these redshifts typically host MLAGN, while the $M_{\star}$ distributions of HLAGN and MLAGN become comparable at z$\sim$1.5 and display a reversal at z$\sim$2. This trend is unlikely to be driven by the incompleteness in M$_{\star}$, as the optical-NIR selected sample in the COSMOS field is $>$80\% complete at M$_{\star}>$10$^{9.7}$M$_{\sun}$ out to z$\sim$4 (see \citealt{Davidzon+17}). We stress that this $M_{\star}$ behaviour is observed in our radio-selected sample, while it might not be the same for differently selected samples of AGN. A more comprehensive analysis combining panchromatic samples of X-ray, MIR, and radio-selected AGN would be crucial to test the widespread validity of this finding. This possible hint of ``downsizing'' (e.g.~\citealt{Cowie+96})
links back to the known anti-hierarchical growth of galaxies over cosmic time with the most massive systems evolving earlier and faster than their lower mass counterparts (see also \citealt{Bundy+06}; \citealt{Fontanot+09}). In particular, this M$_{\star}$ behaviour is expected if the most massive galaxies trigger higher radiative luminosity AGN activity earlier than less massive galaxies and then fade to lower radiative luminosity AGN at lower redshifts. The same qualitative argument is proposed in the evolution of AGN with the number density of powerful AGN ($L_{\rm x}>$10$^{44}$ erg~s$^{-1}$) peaking earlier in cosmic time compared to lower luminosity AGN (e.g. \citealt{Barger+05}; \citealt{Hasinger+05}; \citealt{Silverman+08}; \citealt{Ueda+14}).
Different studies of AGN host galaxies have argued that AGN accretion preferentially occurs in gas-rich galaxies (\citealt{Vito+14}), and that the percentage of galaxies hosting X-ray AGN increases with infrared luminosity $L_{\rm IR}$ (e.g. \citealt{Bongiorno+12}; \citealt{Santini+12}). This is consistent with the increasing gas fraction observed in MS galaxies from low to high redshift (e.g. \citealt{Daddi+10}; \citealt{Saintonge+12}; \citealt{Tacconi+13}), and explained via the Schmidt--Kennicutt relation (\citealt{Schmidt59}; \citealt{Kennicutt98}). The increasing gas fraction from low to high redshift might explain the higher occurence of higher radiative luminosity AGN in massive galaxies ($M_{\star}\sim$10$^{11}$~$M_{\odot}$) at higher redshift (z$\sim$2). Indeed, we showed that galaxies hosting HLAGN are mostly on the MS relation (see also \citealt{Rosario+12}), which implies a large availability of cold gas supplies, and possibly a more efficient fuelling mechanism of the central SMBH (i.e. with
higher accretion rates), compared to the physical processes taking place in MLAGN.
According to this scenario, less star-forming galaxies are less likely to host an active SMBH. Interestingly, we found that most of MLAGN reside in weakly star-forming galaxies, which are typically located a factor of 2--3 below the MS. A plausible interpretation is that the difference in cold gas reservoirs leads HLAGN and MLAGN to be mainly powered by different accretion mechanisms. This raises the question of what triggers AGN activity in these two AGN populations. Shedding light on this issue requires a thorough investigation of the Eddington ratio distributions between these two AGN classes, which will be presented in a future work (I.~Delvecchio et al., in prep.).
It is worth noting that HLAGN with radio excess show intermediate $M_{\star}$, {\sc [NUV-$r$]} and SFR distributions between MLAGN and the rest of HLAGN, especially at z$<$1. Under the assumption that, in a stable phase, the AGN feedback occurs predominantly in either radiative or mechanical form, the population of HLAGN with radio excess might coincide with a transitional phase of AGN feedback.
According to semi-analytic models (e.g. \citealt{Croton+06}; \citealt{Marulli+08}, \citealt{Hopkins+08}), AGN feedback is one of the possible means to track the AGN host galaxies from the blue cloud of star-forming systems to the red sequence of passive galaxies, passing through a transition (often referred to as ``green valley''), where the star formation is weaker but not yet stopped. According to this possible scenario, MLAGN and HLAGN with radio excess might represent intrinsically the same galaxies but that are observed at different stages of their AGN duty cycle, in which the energy produced via accretion onto the SMBH is emitted in either radiative or mechanical forms. The lower level of star formation in MLAGN might be a consequence of AGN-driven feedback, where the radio emission powered by the AGN could limit or hamper the galaxy star formation. This scenario is supported by studies of radio-selected AGN, where jet-induced feedback can strongly impact the molecular gas supplies of the host-
galaxy (e.
g. \citealt{Feruglio+10}; \citealt{Morganti+13}; \citealt{Combes+13}). In this context, it is possible that the population of HLAGN with radio excess probes a particular stage of the radio-mode feedback phase, where the molecular gas in the host galaxy is not yet depleted, and no evident impact in the integrated properties of the galaxy should be detectable during this transition (see Figs. \ref{fig:histogal1}, \ref{fig:histogal2}, and \ref{fig:histogal3}). This scenario is also supported by recent spectroscopic observations of powerful outflows detected in X-ray-MIR selected AGN, some of which show a significant radio excess (e.g. \citealt{Perna+15}; \citealt{Lonsdale+15}; \citealt{Brusa+16}). For these reasons, we stress that our sample of HLAGN with radio excess could be ideal to investigate the impact of AGN feedback, both radiative and mechanical, in a statistical sense and in a wide redshift range.
\section{Conclusions} \label{conclusions}
This work presents a multiwavelength analysis of radio-selected AGN host-galaxy properties out to z$\lesssim$6. Our sample consists of about {7\,700} radio sources selected at 3~GHz in the COSMOS field, and cross-matched with optical-NIR counterparts. The exquisite photometry and redshifts available enabled us to use multiwavelength diagnostics to identify two main AGN populations in our sample: HLAGN (21\%, out of which 30\% also shows a $>$3$\sigma$ radio-excess) and MLAGN (17\%). We analysed the average properties of their host galaxies at different cosmic epochs and summarise our main conclusions as follows:
\begin{enumerate}
\item We tested our source classification method against independent criteria used in recent radio-based studies (e.g. \citealt{DelMoro+13}; \citealt{Bonzini+13}; \citealt{Padovani+15}; N.~Herrera Ruiz et al., in prep.), finding a good agreement and demonstrating the robustness of our method.
\item We provided a value-added catalogue containing the classification and the main physical properties discussed in this work for each radio source ($M_{\star}$, SFR, {\sc [NUV-$r$]} colours, $L_{\rm 3~GHz}$ and $L_{\rm IR, SF}$).
\item Our HLAGN and MLAGN lie in different regions of the SFR--$M_{\star}$ plane, where the former are, on average, less massive and more star forming than the latter at various redshifts. We analysed in detail the observed distributions of various galaxy properties, finding significantly higher SFR and bluer {\sc [NUV-$r$]} colours in HLAGN compared to MLAGN at all redshifts. Nevertheless, the $M_{\star}$ distribution is mainly populated by MLAGN at the highest $M_{\star}$ values ($M_{\star}$>10$^{11}$~$M_{\odot}$) at z$<$1, while the two AGN classes equally contribute to the highest $M_{\star}$ at z$\sim$1.5, and display a {6}$\sigma$ reversal in the $M_{\star}$ behaviour at z$\sim$2.
\item Our results are consistent with radio emission predominantly arising from star formation in around 70\% of HLAGN, while the remaining 30\% shows a $\geq$3$\sigma$ radio excess that is likely attributable to AGN activity. The fractional AGN contribution to the radio emission in MLAGN is expected to be around 80--90\%.
\end{enumerate}
Overall, the differences in galaxy properties seen between these two AGN classes suggest that HLAGN and MLAGN samples trace two distinct galaxy populations in a wide range of redshifts. This might reflect the presence of two different driving mechanisms of AGN activity, which is possibly linked to the different availability of cold gas supplies in their hosts. In this scenario, the subsample of HLAGN with radio excess might coincide with a transitional phase during the AGN duty cycle, in which AGN activity occurs in both radiative and mechanical forms.
\begin{acknowledgements}
The authors are grateful to the anonymous referee for his/her careful reading and useful comments, which improved the content of this manuscript. ID, VS, JD, MN, and OM acknowledge the European Union's Seventh Framework programme under grant agreement 337595 (ERC Starting Grant, ``CoSMass''). NB acknowledges the European Union’s Seventh Framework programme under grant agreement 333654 (CIG, ‘AGN feedback’). CL is funded by a Discovery Early Career Researcher Award (DE150100618). Parts of this research were conducted by the Australian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project number CE110001020. MB and PC acknowledge supports from the PRIN-INAF 2014. MA acknowledges partial support from FONDECYT through grant 1140099. AK acknowledges support by the Collaborative Research Council 956, sub-project A1, funded by the Deutsche Forschungsgemeinschaft (DFG). This work was partially supported by NASA Chandra grant number GO3-14150C and GO3-14150B (FC, SM).
ID is grateful to B. Magnelli for useful suggestions and to Takamitsu Miyaji for his help with CSTACK.
\end{acknowledgements}
\bibliographystyle{aa}
|
2,877,628,090,144 | arxiv | \section{INTRODUCTION}
The theoretical study of two-dimensional (2D) frustrated quantum antiferromagnets
has been strongly motivated by the fact that such quantum spin models often
describe well the properties of real magnetic materials of great experimental interest.
These models have also become of huge current interest because the interplay
between frustration and quantum fluctuations seen in them can produce,
even at zero temperature ($T=0$), a wide variety of
fascinating quantum phases ranging from those with quasiclassical
ordering to valence-bond solids and spin
liquids.\cite{Sachdev:1995,2D_magnetism_1,2D_magnetism_2} They
have thus become paradigms of systems that may be used to study
quantum phase transitions between quasiclassical phases showing magnetic
order and magnetically disordered quantum phases.
Some of the parameters that determine which
type of ordering occurs include the lattice geometry, the
dimensionality $D$ of the system, the spin quantum number $s$ of the atoms situated on
the lattice sites, the number and range of the magnetic bonds, and the
degree to which bond frustration of either the geometric or dynamical kind is present.
New impetus for the study of 2D quantum spin-lattice models
comes from recent proposals to realize them experimentally with
ultracold atoms trapped in an optical lattice.\cite{Struck:2011}
The particularly exciting scenario thus opens of being able
to tune the competing bond strengths and thus to investigate experimentally
the ensuing quantum phase transitions and their dynamics.
One of the prime theoretical interests in frustrated quantum magnets lies
in the possibility that they might exhibit quantum disordered states and/or
spin-liquid behavior. Among the most highly frustrated, and hence most
promising, candidate systems in this regard are those that are composed of
tetrahedra coupled into two-dimensional (2D) or three-dimensional (3D)
lattice networks. Prominent among the latter are the pyrochlores, whose basic structure
is one of vertex-sharing tetrahedra. Indeed, experiments on such $s=\frac{1}{2}$
pyrochlores as Y$_{2}$Ir$_{2}$O$_{7}$ do seem to show evidence for a quantum
spin-liquid state.\cite{Fukuzawa:2003}
In order to reduce the complexity
of the 3D pyrochlore lattice, but without diminishing the magnetic frustration,
one may project the 3D vertex-sharing lattice of tetrahedra onto a 2D plane. Each
tetrahedron comprises four spins at its vertices, with each of its six edges or links
representing an interaction of the Heisenberg antiferromagnet (HAFM) form.
Each such tetrahedron is thus mapped to a square with spins at its vertices and with sides
representing antiferromagnetic (AFM) bonds, but now with additional AFM links across
its diagonals. Such a pattern is repeated in the vertex-sharing arrangement shown in
the checkerboard pattern of Fig.~\ref{model_bonds}.
\begin{figure*}[!tb]
\mbox{
\subfloat[N\'{e}el]{\scalebox{0.4}{\includegraphics{fig1a.eps}}} \quad \quad
\subfloat[striped]{\scalebox{0.4}{\includegraphics{fig1b.eps}}} \quad \quad
\subfloat[N\'{e}el$^{\ast}$]{\scalebox{0.4}{\includegraphics{fig1c.eps}}}
}
\caption{(Color online) The $J_{1}$--$J_{2}$ checkerboard model (with $J_{1}=1$), showing
(a) the N\'{e}el state, (b) the (columnar) striped state and (c) one of the two N\'{e}el$^{\ast}$ states.
The NN $J_{1}$ bonds are shown as solid (black) lines and the NNN
$J_{2}$ bonds are shown as dashed (blue) lines. The arrows
represent the orientations of the spins on each lattice site
for each of the three states shown.}
\label{model_bonds}
\end{figure*}
Although this 2D projection of the 3D pyrochlore structure preserves its vertex-sharing
structure, the symmetry in the 3D structure between the six bonds on each tetrahedron is
lost in the 2D projection since the two diagonal bonds of each crossed square are
now inequivalent to the four bonds on the sides of the square. This subsequent reduction in
the symmetry is thus consistent with considering an anisotropic Heisenberg model on
the 2D checkerboard lattice in which the AFM exchange interactions along the sides
of the squares (with strength $J_{1}>0$) are generally different in strength from
those along the diagonals of the crossed squares (which have a strength $J_{2}>0$),
as shown in Fig.~\ref{model_bonds}. The resulting frustrated model is thus called
the anisotropic planar pyrochlore. Alternative names are the anisotropic checkerboard
HAFM, the $J_{1}$--$J_{2}$ checkerboard model, and the crossed chain model.
Although, the spin-$\frac{1}{2}$ anisotropic planar pyrochlore has been studied by a
large number of authors\cite{Singh:1998,Palmer:2001,Chung:2001,Brenig:2002,Canals:2002,
Starykh:2002,Sindzingre:2002,Fouet:2003,Berg:2003,Tchernyshyov:2003,Moessner:2004,Hermele:2004,
Brenig:2004,Bernier:2004,Starykh:2005,Schmidt:2006,Arlego:2007,Moukouri:2008,Chan:2011}
the structure of its full phase diagram still remains
unsettled and contentious, especially for larger values of the frustration parameter,
$\kappa \equiv J_{2}/J_{1} \gtrsim 1$, as we discuss more fully below in Sec.~\ref{model}.
Various methods have been applied to the model for different regions of the parameter space
for the variable $\kappa$. These include semiclassical ($s \gg 1$)
analyses,\cite{Singh:1998,Canals:2002,Tchernyshyov:2003} large-$N$
expansions of the Sp($N$) model,\cite{Chung:2001,Moessner:2004,Bernier:2004} high-order
cluster-based strong-coupling series expansion (SE) techniques\cite{Brenig:2002,Brenig:2004,Arlego:2007}
using a continuous unitary transformation generated by the flow equation method of Wegner,\cite{Wegner:1994}
a real-space renormalization technique\cite{Berg:2003} using
the contractor renormalization method of Morningstar and Weinstein,\cite{Morningstar:1996}
an easy-axis generalization of the 3D model,\cite{Hermele:2004} a quasi-one-dimensional
approach (valid in the $\kappa \gg 1$ limit) based on the random phase approximation
backed up by a bosonization study,\cite{Starykh:2002} techniques that
combine renormalization group ideas with one-dimensional bosonization and current
algebra methods,\cite{Starykh:2005} exact diagonalization (ED) of small finite-lattice
clusters,\cite{Palmer:2001,Sindzingre:2002,Fouet:2003,Schmidt:2006} a two-step
density-matrix renormalization group method,\cite{Moukouri:2008} and, very recently,
a tensor network simulation\cite{Chan:2011} based on infinite projected entangled pair states.\cite{Jordan:2008}
In this paper we use the coupled cluster method (CCM) of quantum many-body theory
(see, e.g., Refs.~[\onlinecite{Bi:1991,Bi:1998,Fa:2004}] and references cited therein) to study
the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ Heisenberg model on the checkerboard
lattice, in order to attempt to shed more light on it. The CCM has proven itself in many
applications to frustrated magnetic systems to be capable of providing accurate estimates
of the quantum critical points marking the phase transitions between states of widely differing order (see e.g.,
Refs.~[\onlinecite{Ze:1998,Kr:2000,ccm3,Fa:2001,schmalfuss,Fa:2004,rachid05,darradi08,Bi:2008_JPCM,
Bi:2008_PRB,richter10,UJack_ccm,Reuther:2011_J1J2J3mod,Farnell:2011,Gotze:2011}]).
In view of the continuing interest in the model and
the controversy over its $T=0$ phase structure, especially at large frustration ($\kappa \gtrsim 1$),
it seems appropriate and timely to bring to bear on the problem the proven power of the CCM. Since,
as we shall see, we are able to calculate to high orders in the relevant CCM approximation
scheme, as discussed below in Sec.~\ref{CCM}, we are able to present accurate results
from a method of well-proven ability to deal with such strongly correlated
and highly frustrated systems.
We now briefly outline the structure of the remainder of the paper. In Sec.~\ref{model} the
model itself is first described and discussed. The CCM formalism is then briefly outlined in
Sec.~\ref{CCM} before we present and discuss our CCM results in
Sec.~\ref{results}. Finally, we conclude in Sec.~\ref{summary} with a
summary of our findings and a comparison of our results for the model
with those from other methods.
\section{THE MODEL}
\label{model}
The Hamiltonian for the anisotropic checkerboard-lattice model considered here is given by
\begin{equation}
H = J_{1}\sum_{\langle i,j \rangle} \mathbf{s}_{i}\cdot\mathbf{s}_{j} + J_{2}\sum_{\langle\langle i,k \rangle\rangle}
\mathbf{s}_{i}\cdot\mathbf{s}_{k}\,. \label{H}
\end{equation}
where the index $i$ runs over all sites of a square lattice,
index $j$ runs over all nearest-neighbor (NN) sites to site $i$, and
index $k$ runs over all next-nearest-neighbor (NNN) sites to site $i$ on
a checkerboard pattern such that alternate square plaquettes have either two
NNN (diagonal) bonds or none, as shown in Fig.~\ref{model_bonds}.
The sums over $\langle i,j \rangle$ and
$\langle \langle i,k \rangle \rangle$ count each pairwise bond
once and once only. Each site $i$ of the lattice carries a particle with
spin $s=\frac{1}{2}$ and a spin operator
${\bf s}_{i}=(s_{i}^{x},s_{i}^{y},s_{i}^{z})$.
The lattice and exchange bonds of the anisotropic checkerboard-lattice model are shown in Fig.~\ref{model_bonds}.
We may alternatively view the model as comprising crossed (diagonal)
sets of chains on which the intrachain exchange coupling constant is
$J_2$, coupled by (vertical and horizontal) interchain exchange bonds
of strength $J_1$. We assume here that both bonds are antiferromagnetic
(AFM) in nature (i.e., have positive exchange coupling constants) and hence
frustrate one another. The model thus interpolates continuously between the isotropic HAFM
on the square lattice (when $\kappa \equiv J_{2}/J_{1}=0$) and decoupled
one-dimensional (1D) isotropic HAFM chains (when $\kappa \to \infty$). In between, at
$\kappa = 1$, we have the isotropic HAFM on the checkerboard lattice that is
a 2D analog of the 3D isotropic pyrochlore HAFM. Henceforth, without
loss of generality, we set $J_{1} \equiv 1$ in order to set the energy scale.
The classical ground-state (gs) phase for this model for $\kappa < 1$
is the N\'e{e}l state shown in Fig.~\ref{model_bonds}(a), in which every column and row exhibits
N\'e{e}l AFM ordering, $\cdots \uparrow \downarrow \uparrow \downarrow \cdots$,
and consequently the ordering along each diagonal is
ferromagnetic (FM), i.e., where all the spins are aligned parallel to
one another. The N\'e{e}l state has an energy per spin given by
$E^{\rm cl}/N=s^{2}(-2J_{1}+J_{2})$.
For $\kappa > 1$ there is an
infinitely degenerate family of collinear gs phases in which every
diagonal exhibits N\'e{e}l AFM ordering, but where every diagonal, each of which is
connected by $J_1$ bonds to two other crossed diagonals,
can be arbitrarily moved along its own direction. These states all
have the same energy per spin of $E^{\rm cl}/N=-s^{2}J_{2}$,
independent of the exchange coupling $J_1$. The classical phase transition is clearly
at $\kappa_{{\rm cl}}=1$ ($J_{1}>0$).
Among this infinitely degenerate family of classical states for $\kappa > 1$ are the so-called
(columnar) striped state shown in Fig.~\ref{model_bonds}(b) and
the N\'{e}el$^{\ast}$ state shown in Fig.~\ref{model_bonds}(c). The columnar (row) striped states
have FM ordering along columns (rows) but AFM N\'e{e}l ordering along rows (columns).
The N\'{e}el$^{\ast}$ state has doubled AFM ordering,
$\cdots \uparrow \uparrow \downarrow \downarrow \uparrow \uparrow \downarrow \downarrow \cdots$,
along every row and column. Thus, the single-site spin $\uparrow$ or $\downarrow$ of the
usual N\'{e}el state is replaced in the N\'{e}el$^{\ast}$ state by the two-site unit
$\uparrow\uparrow$ or $\downarrow\downarrow$.
Like the striped state, the N\'{e}el$^{\ast}$ state is also
doubly degenerate (for a given direction of the N\'{e}el vector),
since the roles of the rows and columns may be interchanged in Fig.~\ref{model_bonds}(c) (or,
equivalently, the two-site unit of up or down spins may be chosen along rows as well as columns).
Compared to the classical ($s \to \infty$) version of the anisotropic checkerboard model,
the $s = \frac{1}{2}$ case is really only well established at the three points $\kappa = 0$,
$\kappa = 1$, and $\kappa \to \infty$. For the square-lattice HAFM ($\kappa = 0$)
almost all methods concur that the classical N\'{e}el AFM long-range order (LRO)
is not destroyed, although the staggered magnetization is reduced from the
classical value of 0.5, and the excitations are gapless, integer-spin magnons.
By continuity it is expected that the N\'{e}el order
will persist as the frustrating $J_2$-bonds are turned on, out to some critical
value $\kappa_{c_1}$, at which the N\'{e}el staggered magnetization goes to zero.
There is also a broad general consensus from a variety of methods that at the isotropic
point ($\kappa = 1$) the gs phase of the $s = \frac{1}{2}$ checkerboard-lattice HAFM is a
plaquette valence-bond crystal (PVBC) with quadrumer LRO on isolated spin-singlet
square plaquettes, and with gapped integer-spin excitations
(that are confined spinons). It is still an open question as
to whether there is a direct (first-order in the Landau-Ginzburg scenario) transition at $\kappa = \kappa_{c_1}$
between the states with N\'{e}el and PVBC order, or whether there is an
intermediate coexistence phase with two different order parameters. Such a phase
could have continuous Landau-Ginzburg transitions to both the
N\'{e}el and PVBC phases. The possibility of such coexistence regions occurring between
N\'{e}el and valence-bond solids has been discussed in great detail both in a general context in
Ref.~[\onlinecite{Sachdev:2002}] for various spin-lattice models, and in
Ref.~[\onlinecite{Starykh:2005}] in the specific context of the present model. Again, by continuity,
we expect that the PVBC order will persist to values of $\kappa$ out to some critical
value $\kappa_{c_2}>1$, at which point the PVBC order vanishes.
Lastly, at the $\kappa \to \infty$ limit of the $s = \frac{1}{2}$ anisotropic checkerboard
model we have the well-known and exactly soluble case of decoupled 1D HAFM chains. Such 1D
spin-$\frac{1}{2}$ chains have a Luttinger spin-liquid gs phase, with a gapless
excitation spectrum of deconfined spin-$\frac{1}{2}$ spinons.
The most unsettled part of the phase diagram for this model is the region
$\kappa \gtrsim \kappa_{c_2}$, where various predictions have been given. For example,
it has been argued\cite{Starykh:2002} that the 1D Luttinger behavior of the $\kappa \to \infty$
limit might be robust against the turning on of interchain ($J_1$) couplings, so that
the chains continue to act as decoupled. Such a 2D spin-liquid gs phase provides an
example of a so-called sliding Luttinger liquid.\cite{Emery:2000,
Mukhopadhyay:2001,Vishwanath:2001} Numerical evidence for such a spin-liquid
phase at large values of $\kappa$ in the present model was also found
from ED studies on samples of up to $N=36$ spins.\cite{Sindzingre:2002}
Alternatively, by making a more careful analysis of the relevant terms
near the 1D Luttinger liquid fixed point, it was shown later\cite{Starykh:2005} that
the original prediction\cite{Starykh:2002} of a sliding Luttinger liquid was
wrong, and the same authors suggested that the correct gs phase in the
large-$\kappa$ limit is the so-called gapped crossed dimer phase, where the system
spontaneously dimerizes with a staggered ordering of dimers along the
$J_2$ chains (i.e., along the diagonals in Fig.~\ref{model_bonds}). Support
for the crossed-dimer phase has come from series-expansion\cite{Arlego:2007} and
two-step density-matrix renormalization group method studies.\cite{Moukouri:2008}
We discuss this phase further in Sec.~\ref{results} below.
Finally, one may wonder whether any of the infinitely-degenerate set of classical
($s \to \infty$) ground states for $\kappa > 1$ may survive the quantum
fluctuations present in the $s = \frac{1}{2}$ model, and, if so, whether the
classical degeneracy may be lifted by the well-known {\it order by disorder}
mechanism.\cite{Villain:1977} A semiclassical ($s \gg 1$)
analysis\cite{Tchernyshyov:2003} has shown that quantum spin-fluctuations induce
a LRO that breaks the fourfold rotational symmetry of the lattice, and
that to $O(1/s)$ the fourfold degenerate set of states comprising the
striped state of Fig.~\ref{model_bonds}(b) and the N\'{e}el$^{\ast}$ state
of Fig.~\ref{model_bonds}(c) (plus their two counterparts where rows and
columns are interchanged) become energetically favored as the gs phase over
the remainder of the infinite classical set. A very recent tensor network
simulation\cite{Chan:2011} of the spin-$\frac{1}{2}$ model finds that,
contrary to essentially all other calculations on this model, this fourfold
degenerate state survives to be the quantum gs phase for all values of
the frustration parameter above that at which PVBC order disappears
($\kappa > \kappa_{c_2}$). These authors also argue that, although their
numerical program is unable to distinguish between the energies of
the striped and N\'{e}el$^{\ast}$ states in the quantum ($s=\frac{1}{2}$) model,
the striped phase will emerge as the actual gs phase in practice because of
its greater robustness against small perturbations to the Hamiltonian.
Other analyses\cite{Starykh:2005} have, however, shown that, the N\'{e}el$^{\ast}$ state
might, in one possible scenario, intervene as an intermediate gs phase between the
two (i.e., the plaquette and crossed-dimer) valence-bond solid phases. In such a scenario
the transition between the PVBC and N\'{e}el$^{\ast}$ phases is shown to be able
to proceed via a continuous O(3) transition, while that between the crossed dimer and
N\'{e}el$^{\ast}$ phases will be either a direct (first-order in the Landau-Ginzburg scenario) one or
will proceed via an intermediate coexistence phase showing both types of ordering (i.e., both
N\'{e}el$^{\ast}$ spin ordering and crossed dimer bond modulation).
In view of the considerable lack of agreement about the gs phase diagram for the
spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ model on the checkerboard lattice we now present
results for it in the present paper from high-order CCM calculations.
\section{THE COUPLED CLUSTER METHOD}
\label{CCM}
The CCM (see, e.g., Refs.~[\onlinecite{Bi:1991,Bi:1998,Fa:2004}] and
references cited therein) is one of the most powerful and universally applicable
quantum many-body techniques. It has been applied successfully to many
quantum spin-systems (see e.g.,
Refs.~[\onlinecite{Ze:1998,Kr:2000,ccm3,Fa:2001,schmalfuss,Fa:2004,rachid05,darradi08,Bi:2008_JPCM,
Bi:2008_PRB,richter10,UJack_ccm,Reuther:2011_J1J2J3mod,Farnell:2011,Gotze:2011}]
and references cited therein). The method is particularly suitable
for investigating highly frustrated quantum magnets, for which
other alternative methods may be of limited usefulness. For instance, quantum
Monte Carlo (QMC) techniques are often severely restricted by the
well-known ``minus-sign problem,'' which is
ubiquitous in highly frustrated quantum magnets. On the other hand,
ED methods are often too restricted by the relatively small size of
the largest lattices that can be handled with given computational resources to
be able to sample accurately the often subtle ordering present.
We briefly describe the CCM formalism here and we refer the
interested reader to the literature (and see, e.g.,
Refs.~[\onlinecite{Ze:1998,Kr:2000,ccm3,Fa:2001,schmalfuss,Fa:2004,rachid05,darradi08,Bi:2008_JPCM,Bi:2008_PRB}]
and references cited therein) for further details. The implementation of the CCM always
begins with the choice of a suitable reference or model state. It is usual, but
by no means vital, to choose a classical gs phase as the
model state $|\Phi\rangle$. Hence, for the present anisotropic checkerboard model, we choose the
N\'{e}el state, the striped state and the N\'{e}el$^{\ast}$ state as our CCM
model states. From the discussion in Sec.~\ref{model} above we expect that the
N\'{e}el state is likely to provide a good candidate CCM model state in the region
$\kappa \lesssim 1$, while the striped and the N\'{e}el$^{\ast}$ states are expected
to be suitable candidates for $\kappa \gtrsim 1$. We choose only the latter states out
of the infinitely degenerate set of classical states in the $\kappa > 1$ regime since,
as discussed above, this fourfold set of states is selected by the order by disorder
mechanism at the $O(1/s)$ level in a quasiclassical expansion in powers of
$1/s$,\cite{Tchernyshyov:2003} at which order they remain degenerate in energy.
The CCM then incorporates the multi-particle correlations present
in the exact quantum gs phase under investigation on top of the chosen model state in a
systematic hierarchy of approximations for the correlation operators $S$
and $\tilde{S}$ which parametrize the gs ket and bra wave functions as
\begin{equation}
|\Psi\rangle=e^{S}|\Phi\rangle; \quad \langle \tilde{\Psi}|=\langle
\Phi|\tilde{S}e^{-S}.
\label{psi}
\end{equation}
The correlation operators are written as
\begin{equation}
S=\sum_{I\neq0}{\cal S}_{I}C^{+}_{I}; \quad \tilde{S}=\sum_{I\neq0}\tilde{\cal S}_{I}C^{-}_{I}; \quad \forall I \neq 0,
\label{S}
\end{equation}
where $C^{+}_{0} \equiv 1$, the identity operator,
$I$ is a set-index describing a set of single-particle configurations, and
$C^{+}_{I}$ and $C^{-}_{I} \equiv (C^{+}_{I})^{\dagger}$, for $I \neq 0$, are
Hermitian-conjugate pairs of multi-particle creation and
destruction operators defined with respect to the model state
$|\Phi\rangle$ considered as a generalized vacuum state.
They are thus required to satisfy the conditions $\langle \Phi |C^{+}_{I} =
0 = C^{-}_{I}| \Phi \rangle; \forall I \neq 0$.
They form a complete set of mutually commuting many-body creation operators
in the Hilbert space, defined with respect to $|\Phi\rangle$ as a cyclic vector.
The states are normalized such that $\langle\tilde{\Psi}|\Psi\rangle =
\langle \Phi| \Psi\rangle = \langle \Phi| \Phi \rangle \equiv 1$.
For spin-lattice systems it is convenient to choose a set of local
coordinate frames in spin space such that on each lattice site
the spin in each model state points in the downward (negative $z$
direction). Such rotations obviously do not affect the basic
SU(2) spin commutation relations, but they have the simplifying effect
that the operators $C^{+}_{I}$
are transformed into multi-spin raising operators that can be
expressed as products of single-spin raising operators, $C^{+}_{I}\equiv s^{+}_{j_{1}}
s^{+}_{j_{2}} \cdots s^{+}_{j_{n}}$, where $s^{+}_{j} \equiv s^{x}_{j} + is^{y}_{j}$.
The gs energy is evaluated in terms of the correlation
coefficients $\{{\cal S}_{I}\}$ as $E=\langle\tilde{\Psi}|H|\Psi\rangle =
\langle\Phi|\mbox{e}^{-S}H\mbox{e}^{S}|\Phi\rangle$; and the average on-site
magnetization $M$ in the rotated spin coordinates is evaluated
equivalently in terms of the coefficients $\{{\cal S}_{I},\tilde{{\cal S}_{I}}\}$
as $M \equiv -\frac{1}{N} \langle\tilde{\Psi}|\sum_{j=1}^{N}s^{z}_{j}|\Psi\rangle$.
Thus, $M$ is simply the usual magnetic order parameter.
The complete set of unknown ket- and bra-state correlation coefficients
$\{{\cal S}_{I}, \tilde{{\cal S}_{I}}\}$ is evaluated by setting the energy expectation
value $\bar{H} \equiv \langle\tilde{\Psi}|H|\Psi\rangle$ to be a minimum with
respect to all parameters $\{{\cal S}_{I}, \tilde{{\cal S}_{I}}; \forall I \neq 0\}$.
This produces the coupled set of nonlinear equations for the ket-state (creation)
correlation coefficients $\{{\cal S}_{I}\}$ via $\langle \Phi|C^{-}_{I}\mbox{e}^{-S}H\mbox{e}^{S}|\Phi\rangle =
0; \forall I \neq 0$; plus the coupled set of linear equations,
$\langle\Phi|\tilde{S}(\mbox{e}^{-S}H\mbox{e}^{S} - E)C^{+}_{I}|\Phi\rangle = 0; \forall I \neq 0$,
which are used to solve for the bra-state (destruction) correlation coefficients
$\{\tilde{{\cal S}_{I}}\}$.
If it were possible to consider all creation and annihilation operators $C^{+}_{I}$
and $C^{-}_{I}$ respectively, i.e., all sets (configurations) of lattice sites, in
the CCM correlation operators $S$ and $\tilde{S}$ respectively, one would
in principle obtain the exact eigenstate of the system belonging to
any symmetries imposed by the model state (and the configurations
that are perhaps also accordingly selected).\cite{Bi:1998}
Of course, however, it is necessary in practice to use approximations schemes
to truncate the expansions of $S$ and $\tilde{S}$ in Eq.~(\ref{S}). In that case
the approximate results for the gs energy $E$ and the magnetization $M$ will
depend on the choice of model state.
For the case of $s=\frac{1}{2}$ systems,
as considered here, we normally use the well-tested
localized LSUB$m$ truncation scheme which takes in
at the $m$th level of approximation all multi-spin
correlations in the CCM correlation operators over all configured regions on the
lattice defined by $m$ or fewer contiguous sites. A configuration
of $m$ sites is considered to be contiguous if every site
in the configuration is adjacent (in the NN sense) to at least one
other site in the configuration. Clearly, as $m \to \infty$, the
LSUB$m$ approximation becomes exact. For the present
checkerboard model, we use the CCM and the LSUB$m$ scheme with $m \leq 10$ for
the three model states shown in Fig.~\ref{model_bonds}. For the LSUB$m$
configurations we assume the fundamental checkerboard geometry to
define the LSUB$m$ sequences, and hence treat both the pairs of
sites connected by $J_{1}$ bonds and those connected by
$J_{2}$ bonds as being contiguous sites.
Table~\ref{table_FundConfig} shows the number $N_{f}$ of such distinct
(i.e., under the symmetries of the lattice and the model state)
fundamental spin configurations for each of the three model states that we
use for our spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ checkerboard model.
\begin{table}
\caption{Number of fundamental configurations, $N_{f}$, for the checkerboard geometry for the
spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ checkerboard model ($J_{1}=1$),
using the N\'{e}el, striped, and N\'{e}el$^{\ast}$ states as CCM model states.}
\vskip0.2cm
\begin{tabular}{|c|c|c|c|} \hline\hline
Method & \multicolumn{3}{|c|}{$N_{f}$} \\ \cline{2-4}
& N\'{e}el & striped & N\'{e}el$^{\ast}$ \\ \hline
LSUB4 & 27 & 54 & 79 \\
LSUB6 & 632 & 1225 & 2441 \\
LSUB8 & 21317 & 41324 & 86590 \\
LSUB10 & 825851 & 1598675 & 3373495 \\ \hline\hline
\end{tabular}
\label{table_FundConfig}
\end{table}
It is clear that $N_{f}$ rises rapidly with the truncation index $m$. For example,
for the N\'{e}el$^{\ast}$ state that is used in this study as one of the CCM model states,
the LSUB10 approximation contains 3373495 distinct spin
configurations. This is the highest LSUB$m$ level that we can reach here using
the N\'{e}el$^{\ast}$ state as our model state, even
with massive parallelization and the use of supercomputing resources.
It takes us approximately 1~h computing time using
massively parallel computing with 3000 processors simultaneously to
solve the corresponding coupled sets of CCM bra- and ket-state
equations, to obtain a single data point for a given value of $J_{2}$,
with $J_{1}=1$.\cite{ccm}
We note that if, instead of using the checkerboard geometry, we were to use the
square-lattice geometry (i.e., with NN pairs defined only by $J_{1}$ bonds),
the number of fundamental configurations $N_{f}$ would
obviously be fewer, at the same level $m$, than in the checkerboard geometry.
In turn this could perhaps enable us go to higher LSUB$m$
orders for given computational power. However, this advantage is completely outweighed by the
disadvantage that the LSUB$m$ sequences for both $E/N$ and $M$ then
show a marked staggering behavior in $m \equiv 2k$, depending on whether $k$
is even or odd. This is clearly due to the fact that the full LSUB$m$ sequence does not
then properly reflect the checkerboard symmetries. It is quite similar
to the odd and even staggering behavior in index $m$ for LSUB$m$ approximations
on simple (dynamically unfrustrated) models, which has been reported
elsewhere.\cite{Farnell:2008} Any such staggering effect makes
extrapolations (for the full sequence) of the sort we now discuss more
complicated and less robust.
\begin{figure*}[!tb]
\mbox{
\subfloat[$E/N$]{\scalebox{0.3}{\includegraphics[angle=270]{fig2a.eps}}}
\subfloat[$\Delta e$]{\scalebox{0.3}{\includegraphics[angle=270]{fig2b.eps}}}
}
\caption{(a) The extrapolated CCM LSUB$\infty$ results for the gs energy
per spin, $e \equiv E/N$, versus $J_2$ for the striped and N\'{e}el$^{\ast}$ phases of the
spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ Heisenberg antiferromagnet on
the checkerboard lattice (with $J_{1}=1$), using the LSUB$m$
results with $m=\{4,6,8,10\}$ and Eq.~(\ref{Extrapo_E}); (b) the energy
difference, $\Delta e \equiv e^{{\rm striped}} - e^{{\rm N\acute{e}el}^{\ast}}$
versus $J_{2}$ of the two phases shown in (a) using LSUB$m$ approximations
with $m=\{4,6,8,10\}$ and also using the corresponding separate LSUB$\infty$ results
for both phases from Eq.~(\ref{Extrapo_E}) using $m=\{4,6,8,10\}$.}
\label{Ediff_ColStriped_stateC}
\end{figure*}
Thus, as a final step we need to extrapolate the raw CCM data from our LSUB$m$
approximations to the exact ($m \to \infty$) limit. In the absence
of any staggering effects of the sort described above, we use the well-tested
extrapolation scheme
\begin{equation}
E(m)/N=a_{0}+a_{1}m^{-2}+a_{2}m^{-4}\,, \label{Extrapo_E}
\end{equation}
for the gs energy.\cite{Kr:2000,ccm3,Fa:2001,rachid05,schmalfuss,Bi:2008_PRB,darradi08,
Bi:2008_JPCM,richter10,Reuther:2011_J1J2J3mod}
For the magnetic order parameter, $M$, we use the schemes
\begin{equation}
M(m)=b_{0}+b_{1}m^{-1}+b_{2}m^{-2}\,, \label{Extrapo_M}
\end{equation}
for non-frustrated spin systems,\cite{Kr:2000,ccm3,Fa:2001} and
\begin{equation}
M(m)=c_{0}+b_{1}m^{-1/2}+a_{2}m^{-3/2}\,, \label{Extrapo_M_Frust}
\end{equation}
for highly frustrated spin systems.\cite{darradi08,Bi:2008_JPCM,richter10,Reuther:2011_J1J2J3mod}
We have performed separate extrapolations using data
sets with $m=\{4,6,8,10\}$, $m=\{6,8,10\}$, $m=\{2,4,6,8\}$, and $m=\{4,6,8\}$.
They yield very similar results in each of the cases reported below,
which gives credence to our results and demonstrates their robustness.
\section{RESULTS AND DISCUSSION}
\label{results}
We now present our CCM results for the spin-$\frac{1}{2}$ anisotropic
checkerboard model, using each of the three states shown in
Fig.~\ref{model_bonds} as model states. We first show in
Fig.~\ref{Ediff_ColStriped_stateC}(a) the extrapolated LSUB$\infty$ gs energies per spin,
$E/N$, of the phases obtained using the striped and N\'{e}el$^{\ast}$
model states. We recall that in the classical limit ($s \rightarrow \infty$)
these two phases are degenerate and are the gs phase only for $\kappa > 1$.
Results are shown in Fig.~\ref{Ediff_ColStriped_stateC}(a) down to the lowest
terminating values of $\kappa$ in each case for which real solutions exist
for all of the LSUB$m$ approximations used. It has been shown
previously\cite{Fa:2004,UJack_ccm} that such termination points are
a strong indication of the corresponding quantum phase transition points
that occur in the system under study.
Figure \ref{Ediff_ColStriped_stateC}(b) shows the difference in the energies
of the two states in the approximate region where both CCM solutions exist.
We see clearly that although the energy difference is small, the classical
degeneracy is removed in favor of the N\'{e}el$^{\ast}$ state over the
striped state for all values of $\kappa$ for which real CCM solutions for
both phases exist. Nevertheless, bearing in mind the smallness of
the energy difference, we present some results below using both states as
model states.
In Fig.~\ref{E} we show both the LSUB$m$ and the extrapolated LSUB$\infty$
\begin{figure}[!bt]
\includegraphics[angle=270,width=8cm]{fig3.eps}
\caption{CCM results for the gs energy, $E/N$, for the N\'{e}el and
N\'{e}el$^{\ast}$ states for the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ Heisenberg
antiferromagnet on the checkerboard lattice (with $J_{1}=1$) versus $J_2$.
The LSUB$m$ approximations for $m=\{4,6,8,10\}$ are shown together with the
corresponding LSUB$\infty$ extrapolation from using Eq.~(\ref{Extrapo_E})
with $m=\{4,6,8,10\}$.}
\label{E}
\end{figure}
results for the gs energy, $E/N$, of both the N\'{e}el and
N\'{e}el$^{\ast}$ phases.
\begin{figure*}[!tb]
\mbox{
\subfloat[N\'{e}el and N\'{e}el$^{\ast}$ states]{\scalebox{0.3}{\includegraphics[angle=270]{fig4a.eps}}}
\subfloat[N\'{e}el and striped states]{\scalebox{0.3}{\includegraphics[angle=270]{fig4b.eps}}}
}
\caption{CCM results for the gs magnetic order parameter, $M$, for the N\'{e}el and
N\'{e}el$^{\ast}$ states for the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ Heisenberg
antiferromagnet on the checkerboard lattice (with $J_{1}=1$) versus $J_2$;
(a) using the N\'{e}el and N\'{e}el$^{\ast}$ states as model states, and
(b) using the N\'{e}el and striped states as model states. In both cases
the LSUB$m$ approximations for $m=\{4,6,8,10\}$ are shown together with the
corresponding LSUB$\infty$(1) and LSUB$\infty$(2) extrapolation from using
Eqs.~(\ref{Extrapo_M}) and (\ref{Extrapo_M_Frust}) respectively, with $m=\{4,6,8,10\}$.}
\label{M}
\end{figure*}
As noted briefly above,
we now observe more clearly that each of
the LSUB$m$ energy curves based on a particular model state terminates at some
critical value of $\kappa$ (that itself depends on the LSUB$m$
approximation used), beyond which no real CCM solution can be found.
Since the CCM LSUB$m$ solutions require increasingly more computational power to
obtain (to a given level of numerical accuracy) as the termination
points are approached, it is computationally costly to
determine the actual termination points to a high degree of accuracy. We
note that in Fig.~\ref{E} results are shown for each LSUB$m$ case down
to values of $\kappa$ below which for the N\'{e}el$^{\ast}$ phase, and
up to values of $\kappa$ above which for the N\'{e}el phase, real solutions based on the
respective model state cease to exist. As noted above, in all cases the corresponding
termination point at a given LSUB$m$ level shown in Fig.~\ref{Ediff_ColStriped_stateC}
for the striped state is lower than that for the equivalent N\'{e}el$^{\ast}$ model state case.
We note however that, as is usually the case, the CCM LSUB$m$ results for finite $m$ values for
both the N\'{e}el and N\'{e}el$^{\ast}$ phases shown in Fig.~\ref{E} extend beyond the corresponding
LSUB$\infty$ transition points. For large values of $m$ the LSUB$m$
transition points are quite close to the actual quantum critical points (QCPs)
where that phase ends. For example, the LSUB10 termination points
shown in Fig.~\ref{E} are at
$\kappa ^{{\rm N\acute{e}el}}_{t} \approx 0.88$ for the N\'{e}el state
and $\kappa ^{{\rm N\acute{e}el}^{\ast}}_{t} \approx 1.2$ for the N\'{e}el$^\ast$ state.
The CCM results show a clear intermediate regime in which neither of the
quasiclassical AFM states (N\'{e}el and N\'{e}el$^{\ast}$) is stable.
We now discuss the magnetic order parameter (viz., the average
on-site magnetization), $M$, in order to
investigate the stability of the quasiclassical magnetic LRO.
Our CCM results for $M$ are shown in Fig.~\ref{M} for each of
the N\'{e}el, N\'{e}el$^{\ast}$, and striped phases.
Our extrapolated results for $M$ in the N\'{e}el phase are seen
to be somewhat sensitive to whether we use the scheme of Eq.~(\ref{Extrapo_M})
or that of Eq.~(\ref{Extrapo_M_Frust}).
As we have indicated previously the scheme of Eq.~(\ref{Extrapo_M}) is
appropriate only for small values of $J_2$. For example, in the
square-lattice limit, $J_{2}=0$, we obtain the extrapolated LSUB$\infty$(1)
result $M \approx 0.3069$ from the use of Eq.~(\ref{Extrapo_M}) and the LSUB$m$
values with $m=\{4,6,8,10\}$. Very similar values are obtained with
the alternative data sets $m=\{4,6,8\}$ and $m=\{6,8,10\}$. We
note that for the square-lattice HAFM no dynamic (or geometric)
frustration exists and the Marshall-Peierls sign
rule\cite{Marshall-Peierls} applies and may hence be used to
circumvent the QMC ``minus-sign problem.''
The QMC result,\cite{Sandvik:1997} $M=0.3070 \pm 0.0003$,
is thus extremely accurate for this limiting ($J_{2}=0$) case
only. Our own CCM result using the scheme of Eq.~(\ref{Extrapo_M})
is thus in excellent agreement with it. By contrast, the
extrapolation scheme of Eq.~(\ref{Extrapo_M_Frust}), which is
appropriate for (highly) frustrated systems, gives a much poorer
estimate of $M \approx 0.275$.
The magnetization results show clear evidence for the melting of
N\'{e}el order at a value $\kappa = \kappa_{c_{1}} < \kappa_{{\rm cl}}
= 1$, with results for $\kappa_{c_{1}}$ that are very close to the
corresponding termination point $\kappa ^{\rm N\acute{e}el}_{t}$ discussed
above. At such values $\kappa \lesssim 1$, where the system
is highly frustrated, the extrapolation scheme of Eq.~(\ref{Extrapo_M_Frust})
is more appropriate, as we have indicated previously, and
its use with the LSUB$m$ data set $m=\{4,6,8,10\}$ gives us
our first estimate for the quantum critical point (QCP) at which N\'{e}el order
vanishes, $\kappa_{c_1} \approx 0.796$. Very similar results are found by using
the alternative data sets $m=\{4,6,8\}$ and $m=\{6,8,10\}$. Combining
all these results gives the estimate $\kappa_{c_1} \approx 0.80 \pm 0.01$. (By
contrast, the use of the scheme of Eq.~(\ref{Extrapo_M}), which is
inappropriate in this frustrated region near a QCP, gives a value
$\kappa_{c_1} \approx 0.87 \pm 0.01$.)
The results in Fig.~\ref{M}(a) for $M$ using the N\'{e}el$^{\ast}$ state as CCM
model state are seen to be qualitatively very different from those using
the N\'{e}el state as model state. Indeed, all of the evidence from
Fig.~\ref{M}(a) is that $M$ is either zero or very close to zero over the
entire range for which CCM extrapolated LSUB$\infty$ solutions exist using
the N\'{e}el$^{\ast}$ state as model state. The more appropriate extrapolation scheme
of Eq.~(\ref{Extrapo_M_Frust}) in this regime with $\kappa \gtrsim 1$ gives either negative
values for $M$ or positive values very close to zero
over the entire range shown in Fig.~\ref{M}, while even the inappropriate
scheme of Eq.~(\ref{Extrapo_M}) gives only a very small and almost constant value
of $M \approx 0.08$ over the same range. Our results in the high-frustration
regime ($\kappa \gg 1$) using the appropriate scheme of
Eq.~(\ref{Extrapo_M_Frust}) are given extra credence by the fact that we see
clearly from Fig.~\ref{M} that $M \rightarrow 0$ rather accurately in the large
$\kappa$ limit, which is the exact result for this limit where the model
reduces to unlinked 1D spin-$\frac{1}{2}$ chains.
It is clear that at best the existence of the N\'{e}el$^{\ast}$ phase
in the spin-$\frac{1}{2}$ case is extremely fragile from this evidence. More
likely, it is not the stable gs phase for any value of $\kappa$, based
on the results for $M$.
For this reason we have repeated the calculations for $M$, but now using
the striped state as CCM model state, even though we found it to have
a slightly higher energy for all values of $\kappa$ than that of the
N\'{e}el$^{\ast}$ state. Results are shown in Fig.~\ref{M}(b).
Figures \ref{M}(a) and \ref{M}(b)
show very similar results for $M$ for the striped state and the
N\'{e}el$^{\ast}$ state, except very near their corresponding termination points.
All of the evidence so far is that neither state is the stable gs phase
for any value of $\kappa$. Since the results for the order parameter $M$ are so
similar for the N\'{e}el$^{\ast}$ state and the striped state, and since
the former has a slightly lower energy, we henceforth restrict ourselves for
larger values of $\kappa$ to use of the N\'{e}el$^{\ast}$ state as
CCM model state.
It is reasonably well established from earlier numerical studies using
ED\cite{Fouet:2003} and strong-coupling expansion
techniques\cite{Berg:2003,Brenig:2002,Brenig:2004} that the gs phase
of the spin-$\frac{1}{2}$ HAFM on the checkerboard lattice (i.e., our
model at the isotropic point $J_{2}=J_{1}$) is a plaquette valence-bond
crystal (PVBC) with long-range quadrumer order. Further evidence for such
a valence-bond solid built from disconnected 4-spin singlets
comes from a ``fermionic'' SU($n$) generalization of the SU(2) group in the
large-$n$ limit.\cite{Canals:2002} There is broad agreement from all
this work that the PVBC phase comprises singlet plaquettes on the squares
in Fig.~\ref{model_bonds} without crossed links, as shown in Fig.~\ref{X}.
\begin{figure}[!tb]
\begin{center}
\mbox{
\subfloat{\includegraphics[width=6cm,height=6cm,angle=270]{fig5a.eps}}
\raisebox{-3.5cm}{
\subfloat{\includegraphics[width=2.2cm,height=2.2cm]{fig5b.eps}}
}
}
\caption{(Color online) Left: CCM results for the inverse plaquette
susceptibility, $1/\chi_p$, versus $J_2$, using the N\'{e}el and
N\'{e}el$^{\ast}$ states as model states, for the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ Heisenberg
antiferromagnet on the checkerboard lattice (with $J_{1}=1$).
The LSUB$m$ approximations for $m=\{4,6,8,10\}$ are shown together with the
corresponding LSUB$\infty$(1) and LSUB$\infty$(2) results from using
the extrapolations schemes of Eqs.~(\ref{Extrapo_inv-chi-1})
and (\ref{Extrapo_inv-chi-2}) respectively, with $m=\{4,6,8,10\}$.
Right: The perturbations (fields) $F=\delta\, \hat{O}_p$ for
the plaquette susceptibility $\chi_p$. Thick (red)
and thin (black) lines correspond respectively
to strengthened and weakened NN exchange couplings, where $\hat{O}_p =
\sum_{\langle i,j \rangle} a_{ij}
\mathbf{s}_{i}\cdot\mathbf{s}_{j}$, and the sum runs over all NN
bonds, with $a_{ij}=+1$ and $-1$ for thick (red) and thin (black)
lines respectively.}
\label{X}
\end{center}
\end{figure}
Thus, in order to get more information on the phase that occurs after the melting
of N\'{e}el order at $\kappa = \kappa_{c_1}$ we now investigate the
possibility that it might be a PVBC state of the sort shown in Fig.~\ref{X}.
To do so we first consider a
generalized susceptibility $\chi_F$ that describes the response of the
system to a perturbation described by a ``field'' operator $F$.
A field term $F=\delta\,\hat{O}_F$ is thus added to the Hamiltonian
of Eq.~(\ref{H}). The energy per site in a given state,
$E(\delta)/N \equiv e(\delta)$, is then calculated for the perturbed
Hamiltonian $H+F$, and the susceptibility of the
system to the perturbation $F$ is defined as $\chi_{F} \equiv -
\left. (\partial^2{e(\delta)})/(\partial {\delta}^2)
\right|_{\delta=0}$. An instability of the state against the
perturbation $F$ is signalled by a zero point of $\chi_F^{-1}$ or,
equivalently, by a divergence of $\chi_F$. In our case
we first use the CCM to calculate $\chi_F$, using a specific model
state, in various LSUB$m$ approximations. Although rather less empirical
experience is available for the $m \to \infty$ extrapolation of the CCM
data for $\chi_F$ than for other quantities such as the gs energy $E$ or
the order parameter $M$, we have found previously\cite{Farnell:2011} that
the same extrapolation used for the gs energy [i.e., $\chi_{F}(m) =
d_{0}+d_{1}m^{-2}+d_{2}m^{-4}$] fits the data most accurately, at least
in regions not too close to a divergence of the susceptibility. We also
saw previously\cite{Farnell:2011} that a corresponding extrapolation of the
inverse susceptibility,
\begin{equation}
\chi_{F}^{-1}(m) = x_{0}+x_{1}m^{-2}+x_{2}m^{-4}\,, \label{Extrapo_inv-chi-1}
\end{equation}
gave very consistent results that agreed very closely with those from
the corresponding above extrapolation of $\chi_{F}$, except again in
regions close to where $\chi_{F}^{-1} \to 0$. Since, as we will see below,
we will be especially interested in precisely such regions over a large
range of values of $\kappa$, we also use the fitting function,
\begin{equation}
\chi_{F}^{-1}(m) = y_{0}+y_{1}m^{-y_2}\,. \label{Extrapo_inv-chi-2}
\end{equation}
To calculate the susceptibility, $\chi_p$, of our system against PVBC ordering
we thus set the operator $\hat{O}_F$ to $\hat{O}_p$ as illustrated in the right panel
and the caption of Fig.~\ref{X}. The perturbation field $F$ thus breaks
the translational symmetry of $H$. We show CCM results in the left panel
of Fig.~\ref{X} using both the N\'{e}el and N\'{e}el$^{\ast}$ states as
model states. We first observe that for smaller values of $J_2$
(i.e., on the N\'{e}el side) the two extrapolations agree very closely, even
near the point at which $\chi_p^{-1}$ goes to zero.
Thus, the extrapolated inverse plaquette susceptibility using the
LSUB$m$ data set $m=\{4,6,8,10\}$ ($m=\{6,8,10\}$) vanishes on the N\'eel
side at $\kappa \approx 0.843$ ($\kappa \approx 0.833$) using the extrapolation
scheme of Eq.~(\ref{Extrapo_inv-chi-1}), and
at $\kappa \approx 0.820$ ($\kappa \approx 0.775$) using the extrapolation
scheme of Eq.~(\ref{Extrapo_inv-chi-2}).
Since the exponent $y_2$ in Eq.~(\ref{Extrapo_inv-chi-2}) falls rather sharply to
a value close to 1 near the point at which $\chi_p^{-1}$ vanishes, the best
estimate for this point is more likely to come from the extrapolation scheme
of Eq.~(\ref{Extrapo_inv-chi-2}) than from that of Eq.~(\ref{Extrapo_inv-chi-1}).
Combining all these results gives our best estimate of
$\kappa \approx 0.79 \pm 0.03$ for the point at which
the N\'{e}el phase becomes susceptible to PVBC ordering. This is in very
good agreement with the above estimate of $\kappa_{c_1} \approx 0.80 \pm 0.01$ at which
N\'{e}el LRO disappears as measured by our results for the order parameter $M$.
Thus, our results show no evidence at all for a coexistence region in which
N\'{e}el and PVBC ordering are both present, such as has been suggested
might occur,\cite{Sachdev:2002} although we cannot exclude the
possibility of a very narrow region of coexistence confined to
the region $0.79 \lesssim \kappa \lesssim 0.81$.
Our findings are in agreement with ED results\cite{Sindzingre:2002}
for the same spin-$\frac{1}{2}$ anisotropic
planar pyrochlore model that reach the conclusion that, if present at all,
any such coexistence region is very narrow indeed. As has been discussed in detail
elsewhere,\cite{Starykh:2005} the QCP at $\kappa_{c_1}$ between the
N\'{e}el and PVBC phases is both forbidden as a continuous transition within
standard Landau-Ginzburg theory and does not seem either to be a viable
candidate for a deconfined (continuous) transition. The shape of the magnetization curves
in Fig.~\ref{M} on the N\'{e}el side, which show a rapid decrease near $\kappa_{c_1}$,
is perhaps more indicative of a first-order transition, as we have observed
previously,\cite{Farnell:2011} although such evidence should not
be regarded as conclusive.
We also observe from Fig.~\ref{X} that with the N\'{e}el$^{\ast}$ state used as
our CCM model state the two extrapolations of Eqs.~(\ref{Extrapo_inv-chi-1})
and (\ref{Extrapo_inv-chi-2}) for the inverse
plaquette susceptibility, $\chi_p^{-1}$, agree quite closely at larger values
of $\kappa$ but diverge slightly at smaller values, where
$\chi_p^{-1}$ itself becomes small. As $\kappa \to \infty$ we observe that
the exponent $y_2$ in Eq.~(\ref{Extrapo_inv-chi-2}) appears to approach the value
1.5 [rather than 2 as in Eq.~(\ref{Extrapo_inv-chi-1})], and then drop to a
value close to 1 as $\chi_p^{-1}$ approaches zero. For these reasons again, we
expect the extrapolation of Eq.~(\ref{Extrapo_inv-chi-2}) to be more exact,
especially in regions where $\chi_p^{-1}$ becomes small.
Thus, the extrapolated inverse plaquette susceptibility using the
LSUB$m$ data set $m=\{4,6,8,10\}$ ($m=\{6,8,10\}$) vanishes on the N\'{e}el$^{\ast}$
side at $\kappa \approx 1.238$ ($\kappa \approx 1.216$) using the extrapolation
scheme of Eq.~(\ref{Extrapo_inv-chi-2}). Although,
as we have seen, the N\'{e}el$^{\ast}$ state does not appear to exist as
as a stable gs phase (since its order parameter $M$ seems to vanish
for all values of $\kappa$, nevertheless the results using it as a CCM
model state provide a robust basis for the calculation of $\chi_p$, and give
an estimate $\kappa_{c_2} \approx 1.22 \pm 0.02$ for the upper QCP at which
PVBC order disappears.
However, we are now led to the question of what is the actual gs phase of the model for
larger values of frustration, $\kappa > \kappa_{c_2}$, beyond the upper QCP (at
$\kappa = \kappa_{c_2}$) at which the PVBC phase ceases to exist
as a stable gs phase. In the first place it
is clear that such a QCP must exist since in the limit $\kappa \rightarrow \infty$ one has
the physics of decoupled HAFM 1D chains, which are known to exhibit Luttinger
spin-liquid behavior, and are typified by a gapless excitation spectrum and spin-spin
correlations that decay algebraically with inter-spin separation distance. It was
argued\cite{Starykh:2002} that for large values of $\kappa$, where the 1D
chains are weakly coupled, the gs phase might be a 2D sliding Luttinger liquid phase
(characterized by the absence of LRO and with massless deconfined
spinons as the elementary excitations)
that joined smoothly to the $\kappa \to \infty$ limit. It was later
shown,\cite{Starykh:2005} by a more careful analysis of the relevant terms
near the Luttinger liquid fixed point of the independent 1D spin chain, that
this finding was incorrect. Instead, using techniques that combine
renormalization group ideas and 1D bosonization and current algebra methods,
it was shown that in this large-$\kappa$
regime the gs phase is of spontaneously dimerized type, with a staggered
ordering of dimers along the parallel chains (viz., the diagonals in
Fig.~\ref{model_bonds}). Such a crossed-dimer valence-bond crystal (CDVBC)
phase, with twofold spontaneous symmetry breaking and no magnetic order, was
independently confirmed\cite{Arlego:2007} by a series expansion (SE) technique
based on the flow equation method.
The CDVBC phase is illustrated in the right panel of Fig.~\ref{X_d}. Similarly
\begin{figure}[!t]
\begin{center}
\mbox{
\subfloat{\includegraphics[width=6cm,height=6cm,angle=270.]{fig6a.eps}}
\raisebox{-3.5cm}{
\subfloat{\includegraphics[width=2.2cm,height=2.2cm]{fig6b.eps}}
}
}
\caption{(Color online) Left: CCM results for the scaled inverse crossed dimer
susceptibility, $J_{2}/\chi_d$, using the N\'{e}el$^{\ast}$ state as model
state, for the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ Heisenberg
antiferromagnet on the checkerboard lattice (with $J_{1}=1$) versus $J_2$.
The LSUB$m$ approximations for $m=\{4,6,8,10\}$ are shown together with the
corresponding LSUB$\infty$(1) and LSUB$\infty$(2) results from using
the extrapolations schemes of Eqs.~(\ref{Extrapo_inv-chi-1})
and (\ref{Extrapo_inv-chi-2}) respectively, with $m=\{4,6,8,10\}$.
Right: The perturbations (fields) $F=\delta\, \hat{O}_d$ for
the dimer susceptibility $\chi_d$. Thick (red) dashed
and thin (blue) dashed lines correspond respectively
to strengthened and weakened NNN exchange couplings, where $\hat{O}_{d} =
\sum_{\langle \langle i,k \rangle \rangle} a_{ik}
\mathbf{s}_{i}\cdot\mathbf{s}_{k}$, and the sum runs over the NNN diagonal
bonds of the checkerboard lattice, with $a_{ik}=+1$ and $-1$ for thick (red)
dashed and thin (blue) dashed lines respectively.}
\label{X_d}
\end{center}
\end{figure}
to what was done above for the PVBC susceptibility, $\chi_p$, we can now calculate
the susceptibility, $\chi_d$ of our system against CDVBC ordering
by setting the perturbation operator $\hat{O}_F$ to the
operator $\hat{O}_d$ illustrated in the right panel
and the caption of Fig.~\ref{X_d}. Since in the large $J_2$ limit the energy
scales linearly with $J_2$ (as may clearly also be observed from
Figs.~\ref{Ediff_ColStriped_stateC}(a) and \ref{E}, we show our CCM
results in Fig.~\ref{X_d} for the scaled inverse dimer susceptibility,
$J_{2}/\chi_d$ as a function of $J_2$.
Interestingly, in this case, the extrapolation scheme
of Eq.~(\ref{Extrapo_inv-chi-1}) does not fit the LSUB$m$ data points at all well,
and consequently gives a very poor fit. The reason becomes quite evident when
the extrapolation scheme of Eq.~(\ref{Extrapo_inv-chi-2}) is used instead. It
is then observed that that the scaling exponent $y_2$ approaches the value 0.75
for large values of $J_2$, and even for smaller values near the QCP at
$\kappa=\kappa_{c_2}$ only rises slightly to values that approach 1.
It is clear that the extrapolated value of $J_{2}/\chi_d$ is small, slightly negative,
and almost constant for all values of $\kappa$ shown, when the more flexible
extrapolation scheme of Eq.~(\ref{Extrapo_inv-chi-2}) is used.
Our results are consistent with the interpretation that
the inverse dimer susceptibility is zero for all values $\kappa$ shown
in Fig.~\ref{X_d}, namely those where we have real solutions to the equations
pertaining to all of the CCM LSUB$m$ schemes used. The actual lower termination
point is not easy to determine accurately in this way, as we have
already mentioned above. However, it is quite clear from Fig.~\ref{X_d} that
the individual LSUB$m$ curves all have a minimum at a value
$\kappa \approx 1.20 \pm 0.05$,
and our results are thus consistent with the interpretation that
the inverse dimer susceptibility is zero for all values $\kappa > \kappa_{c_2}$.
In that scenario there is a QCP between the PVBC and CDVBC phases at
$\kappa = \kappa_{c_2}$, and the CDVBC phase
then persists out to the $\kappa \to \infty$ limit of unlinked 1D chains, which
is itself a singular point.
Our results give us essentially no information, however, on the nature of the
transition at $\kappa_{c_2}$. Starykh {\it et al.}\cite{Starykh:2005} have
concluded that a continuous transition between the PVBC and CDVBC phases is
prohibited within the standard Landau-Ginzburg scenario of phase transitions.
They argue further that the most probable alternative is a direct first-order
transition, although a separate possibility is again the existence of an intermediate
coexistence phase with both valence-bond orderings that can then have
separate continuous transitions to both the PVBC and CDVBC phases. From our results
we clearly favor the former scenario, although we cannot exclude the possibility
of a very narrow such coexistence phase confined to the region $1.20 \lesssim \kappa \lesssim 1.22$.
\section{SUMMARY AND CONCLUSIONS}
\label{summary}
To summarize, we have investigated the gs properties and ($T=0$) phase diagram
of the frustrated spin-$\frac{1}{2}$ antiferromagnetic $J_{1}$--$J_{2}$ model
(with $J_{2} \equiv \kappa J_{1}\,; J_{1}>0)$ on
the 2D checkerboard lattice, using the CCM carried out to high orders.
In common with
most other calculations on the model we find that the gs phase is an AFM
N\'{e}el-ordered state for $\kappa < \kappa_{c_1}$, at which
point the staggered N\'{e}el magnetization vanishes. Our best estimate for
this lower QCP is $\kappa_{c_1} \approx 0.80 \pm 0.01$. This is in reasonable
agreement, but probably more accurate than, a corresponding
estimate of $\kappa_{c_1} \approx 0.75$ from an ED
study\cite{Sindzingre:2002} on samples of $N=16,32,36$ spins. Since we calculate
that the N\'{e}el-ordered state becomes susceptible to PVBC ordering
at $\kappa \approx 0.79 \pm 0.03$ our results
point then to a direct transition from the N\'{e}el-ordered gs phase to
a PVBC phase at $\kappa = \kappa_{c_1}$, although we cannot exclude entirely
the small possibility of a vey narrow coexistence region. This finding is in good
agreement with that found from ED studies.\cite{Sindzingre:2002} We estimate
that any such coexistence region of AFM N\'{e}el ordering and quadrumer plaquette
ordering is confined to the parameter range $0.79 \lesssim \kappa \lesssim 0.81$.
Our estimate for the lower boundary at which the quadrumer PVBC ordering vanishes
is in excellent agreement with the corresponding value $\kappa \approx 0.80 \pm 0.01$
from a high order SE calculation\cite{Brenig:2004} that starts from the limit of
uncoupled quadrumers. A recent tensor network simulation of the model\cite{Chan:2011}
gives a somewhat higher value of $\kappa_{c_1} \approx 0.88$ for the QCP from
the N\'{e}el gs phase to the PVBC gs phase.
From our CCM calculations we estimate that the quadrumer order of the PVBC state
vanishes at a higher QCP at $\kappa = \kappa_{c_2} \approx 1.22 \pm 0.02$. This is
in reasonable agreement, although somewhat higher than the corresponding estimates
$\kappa \approx 1.095 \pm 0.035$ from a high-order SE calculation,\cite{Brenig:2004}
and $\kappa \approx 1.11$ from a tensor network simulation.\cite{Chan:2011}
Although the striped and N\'{e}el$^{\ast}$ states, which are the fourfold-degenerate
quasiclassical gs phases at $O(1/s)$ in an expansion in powers of $1/s$ for
$\kappa > 1$,\cite{Tchernyshyov:2003} provide excellent model states for CCM
calculations at larger values of $\kappa \gtrsim 1$ in the sense of providing
well-converged LSUB$m$ results, we find that they are not stable gs phases for
the spin-$\frac{1}{2}$ model for any value of $\kappa$. This is in sharp
disagreement with the finding from a recent tensor network simulation\cite{Chan:2011}
that the striped and/or N\'{e}el$^{\ast}$ states form the stable gs phase for all
values $\kappa > \kappa_{c_2}$. By contrast we find from our CCM calculations
that the N\'{e}el$^{\ast}$ state is susceptible to the formation of a CDVBC
phase (with an inverse susceptibility that is essentially zero) for all values
$\kappa \gtrsim 1.20 \pm 0.05$. We conclude that the CDVBC state is thus likely
to be the stable gs phase for $\kappa > \kappa_{c_2}$, although we cannot
exclude the possibility of a very narrow coexistence regime confined to
the range $1.20 \lesssim \kappa \lesssim 1.22$. We find no evidence at all
for any region separating the PVBC and CDVBC phases where the N\'{e}el$^{\ast}$ state
would form a stable gs phase, thereby ruling out one of the two scenarios
postulated by Starykh {\it et al.},\cite{Starykh:2005} although now in complete
agreement with their other scenario for the gs phase diagram, on which we now
provide accurate numerical results for the two QCP's at $\kappa_{c_1}$ and
$\kappa_{c_2}$.
Finally we note that the CCM has been used here with the model or reference states
chosen as classical states built from independent-spin product states. For larger
values of the frustration parameter, $\kappa > \kappa_{c_1}$,
we remark that these model states are able (with appropriate choice of additional
perturbative terms in the Hamiltonian), to describe the susceptibilities of
the system to form the true gs phasaes perfectly well,
even though we have shown that the respective extrapolated order
parameters with respect to these model states
are essentially zero over the entire range. Nevertheless, it
would be worthwhile to repeat the CCM calculations directly with dimer and plaquette
valence-bond states. Indeed, just such a general CCM approach\cite{Farnell_VBgroundstates:2009}
has recently been described, which combines exact solutions
for dimer or plaquette valence-bond solid ground states
with the computational implementation described here\cite{ccm} that is based on
independent-spin product model states. It would be interesting to use this formalism
for the present model in order to confirm our results.
\section*{ACKNOWLEDGMENT}
We thank the University of Minnesota Supercomputing Institute for the grant of
supercomputing facilities.
|
2,877,628,090,145 | arxiv |
\section{Higher order terms}
\label{Hot}
Comparing equation (31) to equation (82) of \cite{YahalomSym} it follows that:
\beq
g^{(2)} (t) = g^{(2)} (0) e^{\frac{t}{\tau}}
\label{g2t}
\enq
Hence:
\beq
g^{(n)} (t) = g^{(2)} (0) \tau^{2-n} e^{\frac{t}{\tau}} = g^{(2)} (t) \tau^{2-n}, \qquad n>2
\label{gnt}
\enq
And also:
\beq
\rho^{(n)} (\vec x, t) = \rho^{(2)} (\vec x, t) \tau^{2-n}, \qquad M^{(n)} (t) = M^{(2)} (t) \tau^{2-n}, \qquad n>2
\label{thont}
\enq
Thus according to \ern{phir} we have the following correction to the retardation potential:
\beq
\phi_{(n>2)}= -G \sum_{n=3}^{\infty} \frac{(-1)^n}{n! c^n \tau^{n-2}} \int R^{n-1} \rho^{(2)} (\vec x', t) d^3 x'.
\label{phir5}
\enq
The deviation from the second order approximation is more pronounced for large $r= |\vec x|$ for which $R = |\vec x - \vec x'| \simeq r$ which is the case we consider here, thus:
\beq
\phi_{(n>2)}\simeq -G \sum_{n=3}^{\infty} \frac{(-1)^n}{n! c^n \tau^{n-2}} r^{n-1} \int \rho^{(2)} (\vec x', t) d^3 x'
= - \frac{G \ddot M(t) \tau^2}{r} \sum_{n=3}^{\infty} \frac{1}{n!} \left(\frac{-r}{c \tau} \right)^n.
\label{phir6}
\enq
Now using the well known identity:
\beq
\sum_{n=3}^{\infty} \frac{\alpha^n}{n!} = e^\alpha - (1+ \alpha + \frac{1}{2} \alpha^2)
\label{ident}
\enq
We may write \ern{phir6} as a closed expression instead of an infinite sum:
\beq
\phi_{(n>2)}\simeq - \frac{G \ddot M(t) \tau^2}{r} \left(e^{-\frac{r}{c \tau}} - 1 + \frac{r}{c \tau} - \frac{1}{2} \left(\frac{r}{c \tau}\right)^2\right).
\label{phir7}
\enq
For $r \ll c \tau$ it is quite clear that the term in the parenthesis of \ern{phir7} vanishes, since:
\beq
\lim_{\alpha \rightarrow 0} \sum_{n=3}^{\infty} \frac{\alpha^n}{n!} =\lim_{\alpha \rightarrow 0}\left( e^\alpha - (1+ \alpha + \frac{1}{2} \alpha^2) \right) = 0.
\label{ident2}
\enq
Hence $\phi_{(n>2)}$ can be neglected if indeed $r \ll c \tau$ for the relevant measurements of the M33 rotation curve, that is up to about $r < 20$ kpc. Now $\tau$ is dependent according to equation (81) of \cite{YahalomSym} on the density gradient of the intergalactic medium (IGM) and the typical velocity in this medium. Although those values are not known precisely we may assume that $v_z \sim 100$ km /s and the typical gradient is the same as the gradient of the optical disk luminosity that is $\frac{1}{k} \sim 0.1$ kpc. Thus $\tau \sim 10^6$ years, and $\tau c \sim 300$ kpc, making the second order approximation used so far reasonable.
{\bf Thus, we have clarified the domain and range of the current model i.e. what approximations are made and their validity. We underline that the developments of this paper are refinements of an existing model. In another paper it is shown that the retardation approach is valid despite the
relatively low velocity of matter: please see section 9 of \cite{YaRe3}.}
\section{Retardation beyond the Taylor Expansion}
\label{RbT}
Although, we have shown in the previous section that second order expansion is sufficient in the framework of a particular galactic model, it is worthwhile to explore the domain of validity of the retardation phenomena in general. We have shown in section \ref{REBN} that the retardation phenomena
is not important at distances which are short with respect to the retardation length $r \ll R_r$, hence we need only to consider retardation at distances of about $r \geq \frac{R_r}{10}$. But is there an upper distance limit? Indeed the expansion treatment is valid only up to distances of $r < R_{max}$, but this is a property of the Taylor expansion not of the retardation phenomena. We shall
now show that indeed there is an upper distance $R_{rul}$ beyond which the retardation phenomena is not important any more.
Let us look at the potential of \ern{phi} in the limit of large $r$ in which $r$ is
much bigger than a typical scale of the system: $r \gg R_s$. In this limit $R \simeq r$ and thus the potential of \ern{phi} can be written as:
\beq
\phi \simeq -G \int \frac{ \rho (\vec x', t-\frac{r}{c})}{r} d^3 x'
= -\frac{G}{r} \int \rho (\vec x', t-\frac{r}{c}) d^3 x' = -\frac{G M}{r},
\label{philargdis}
\enq
in the above $M$ defined in \ern{M} is the total mass of the system which is constant in time.
The same analysis can be repeated for a Newtonian potential:
\beq
\phi_N \simeq -G \int \frac{ \rho (\vec x', t)}{r} d^3 x'
= -\frac{G}{r} \int \rho (\vec x', t) d^3 x' = -\frac{G M}{r},
\label{philargdisN}
\enq
it follows that for large distances such that $r \gg R_s$ it makes not difference if we use a Newtonian or retarded potential, hence "dark matter" effects disappear. The above is only
true for an isolated system that does exchange mass with its environment. For a galaxy this will
include its "sphere of influence" which is not limited to the observable galaxy but includes also a surrounding from which the observable galaxy accretes mass from. Mathematically we may define:
\beq
\Delta \phi \equiv \phi -\phi_N.
\label{delphi}
\enq
Thus:
\beq
\lim_{r->\infty} \Delta \phi = \lim_{r->\infty} (\phi -\phi_N) = 0,
\label{delphiinf}
\enq
indicating that there is an upper scale $R_{rul} \simeq 10 R_s$ above which $r>R_{rul}$ retardation
effects are unimportant. We can summarize the range of validity of the retardation phenomena by
the inequality:
\beq
\frac{R_r}{10} < r < 10 R_s
\label{reta}
\enq
leading us to expect no retardation phenomena for systems in which:
\beq
R_r > 100 R_s.
\label{reta2}
\enq
To study the phenomena of retardation within a specific model we write the potential of \ern{phi} using dimensionless coordinates:
\beq
\phi = -G \frac{\rho_c R_s^3}{r} r \int \frac{\tilde \rho (\vec x', t-\frac{R}{c})}{R} d^3 \tilde x'
= - \frac{G M}{r}
\frac{1}{\Lambda}\int \frac{r}{R} \tilde \rho (\vec x', t-\frac{R}{c})d^3 \tilde x'
\label{phidl}
\enq
in which we used \ern{M}. This leads to the definition:
\beq
\psi_r \equiv \frac{1}{\Lambda}\int \frac{r}{R} \tilde \rho (\vec x', t-\frac{R}{c})d^3 \tilde x'
\qquad \Rightarrow \qquad \phi = - \frac{G M}{r} \psi_r
\label{psir}
\enq
Similarly we define a Newtonian $\psi_N$:
\beq
\psi_N \equiv \frac{1}{\Lambda}\int \frac{r}{R} \tilde \rho (\vec x', t)d^3 \tilde x'
\qquad \Rightarrow \qquad \phi_N = - \frac{G M}{r} \psi_N.
\label{psiN}
\enq
Now according to \ern{philargdis} and \ern{philargdisN}:
\beq
\lim_{r->\infty} \psi_N = \lim_{r->\infty} \psi_r = 1.
\label{psilim}
\enq
I thus follows that:
\beq
\Delta \psi \equiv \psi_r - \psi_N \qquad \Rightarrow \qquad \lim_{r->\infty} \Delta \psi = 0
\label{delpsilim}
\enq
And also according to \ern{delphi}:
\beq
\Delta \phi = - \frac{G M}{r} \Delta \psi
\label{delphipsilim}
\enq
\section{A Specific Model}
\label{SM}
Let us now study the effects of retardation in the framework of a specific models:
\beq
\rho = \rho_c \tilde \rho, \qquad \tilde \rho = \Sigma (\vec x_\bot) h(z,t)
\label{denm}
\enq
in which $\Sigma (\vec x_\bot)$ is dependent on the coordinates in the transversal direction $\vec x_\bot$ and $h$ is dependent on the time $t$ and on the coordinates along the axial direction $z$.
We will further assume that:
\beq
\Sigma (\vec x_\bot) = \delta (\tilde x) \delta (\tilde y)
\label{transprof}
\enq
in which $\delta$ is a Dirac delta function. It is clear that any profile can be constructed from a weighted sum of $\Sigma (\tilde x - \tilde x_i, \tilde y- \tilde y_i)$ located at different $(\tilde x_i,\tilde y_i)$. For the axial direction we assume a Gaussian profile:
\beq
h(z,t) = \frac{R_s}{\sqrt{2 \pi} \sigma(t)} e^{-\frac{z^2}{2 \sigma(t)^2}}.
\label{hprof}
\enq
The time dependence is given through the width of the Gaussian profile which assumes the following form:
\beq
\sigma(t) = \left\{ \begin{array}{cc}
\sigma_i & \bar{t} \leq 0 \\
\sigma_i + (\sigma_f - \sigma_i) \bar{t} (2- \bar{t}) & 0 < \bar{t}<1 \\
\sigma_f & \bar{t} \geq 1
\end{array}
\right. \qquad \bar{t} \equiv \frac{t}{t_f}
\label{sigt}
\enq
$t_f$ is a typical time scale, $\sigma_i$ is the initial time width before a change takes place
and $\sigma_f$ is the resulting time width after the change took place. We shall define
\beq
\tilde \sigma(t) = \frac{\sigma(t)}{R_s}, \qquad
\tilde \sigma_i = \frac{\sigma_i}{R_s}, \qquad
\tilde \sigma_f = \frac{\sigma_f}{R_s}.
\label{sigt2}
\enq
Hence:
\beq
h(z,t) = \frac{1}{\sqrt{2 \pi} \tilde \sigma(t)} e^{-\frac{\tilde z^2}{2 \tilde \sigma(t)^2}}.
\label{hprof2}
\enq
Choosing conveniently $\sigma_f = R_s$ and $\sigma_i = 1.2 R_s$ we obtain:
\beq
\tilde \sigma(t) = \left\{ \begin{array}{cc}
1.2 & \bar{t} \leq 0 \\
1.2 - 0.2 \bar{t} (2- \bar{t}) & 0 < \bar{t}<1 \\
1.0 & \bar{t} \geq 1
\end{array}
\right.
\label{sigt3}
\enq
The profile width evolution
is depicted in figure \ref{widthevo}.
\begin{figure}[H]
\vspace{1cm}
\centering
\includegraphics[width= 0.7\columnwidth]{widthevo}
\caption{The profile width evolution.}
\label{widthevo}
\end{figure}
The axial density profile $h(z,t)$ is depicted for $\bar{t}=0$ and $\bar{t}=1$
in figure \ref{hprofg}, and for any time in between in figure \ref{hprof2D}
\begin{figure}[H]
\vspace{1cm}
\centering
\includegraphics[width= 0.7\columnwidth]{hprof}
\caption{Axial density profile for $\bar{t}=0$ and $\bar{t}=1$.}
\label{hprofg}
\end{figure}
\begin{figure}[H]
\includegraphics[width= 0.7\columnwidth]{hprof2Da}
\hspace{1cm}
\includegraphics[width= 0.1\columnwidth]{hprof2Db}
\caption{Axial density profile between $\bar{t}=0$ and $\bar{t}=1$.}
\label{hprof2D}
\end{figure}
We note that for the current profile the mass $M$ is constant although the
density profile is time dependent, we also note that for the current density profile
we have $ \Lambda = 1$ according to \ern{dimles2}.
We are now in a position to calculate $\psi_N$ and $\psi_r$ using \ern{denm}, \ern{transprof}
and \ern{hprof2}. We shall choose a point of distance $r$ from the origin along the $x$ axis located at the $z=0$ plane to evaluate $\psi$, since the density profile is cylindrically symmetric every point in the $z=0$ plane of distance $r$ to the origin is equivalent to any other. Taking into account \ern{psiN} we have:
\beq
\psi_N (\tilde r,\bar t) = \int \frac{\tilde r}{ \tilde R} h (\tilde z', \bar t)d \tilde z',
\qquad \tilde R = \sqrt{\tilde z'^2 + \tilde r^2}.
\label{psiNeva}
\enq
The function $\psi_N (\tilde r,0.5)$ is depicted in figure \ref{psiNpl} in which it is clear that $\psi_N$
asymptotically approaches $1$.
\begin{figure}[H]
\centering
\includegraphics[width= 0.7\columnwidth]{psiN}
\caption{The function $\psi_N (\tilde r,0.5)$, it is clear that $\psi_N$ approaches $1$ asymptotically.}
\label{psiNpl}
\end{figure}
Taking into account \ern{psir} we have:
\beq
\psi_r (\tilde r,\bar t) = \int \frac{\tilde r}{ \tilde R} h (\tilde z', \bar t-
\frac{R}{c t_f} )d \tilde z'.
\label{psireva1}
\enq
Hence we have another length scale $R_{s2} = c t_f$ which is conveniently chosen to be $R_{s2} = 2 R_{s}$. The function $\psi_r (\tilde r,0.5)$ is depicted in figure \ref{psirpl} in which it is clear that $\psi_r$ also asymptotically approaches $1$.
\begin{figure}[H]
\centering
\includegraphics[width= 0.7\columnwidth]{psir}
\caption{The function $\psi_r (\tilde r,0.5)$, it is clear that $\psi_r$ approaches $1$ asymptotically.}
\label{psirpl}
\end{figure}
The difference between the two functions $\Delta \psi$ is defined in \ern{delpsilim} and depicted
in figure \ref{psidif1}. It is clear that the difference exist and does not depend on a Taylor expansion, it also clear that this difference may be approximated by a simple function to about
$ \tilde r = 1$, or starting at $ \tilde r = 1$ and using a simple function to describe the function outwards which would work for at least $ \tilde r = 4$. To see that the function $\Delta \psi$ indeed approached $0$ asymptotically as predicted by \ern{delpsilim} the $\tilde r$ axis is extended to $ \tilde r = 20$ as depicted in figure \ref{psidif2}.
\begin{figure}[H]
\centering
\includegraphics[width= 0.7\columnwidth]{psidif1}
\caption{The function $\Delta \psi (\tilde r,0.5)$ depicted up to $ \tilde r = 4$.}
\label{psidif1}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width= 0.7\columnwidth]{psidif2}
\caption{The function $\Delta \psi (\tilde r,0.5)$ depicted up to $ \tilde r = 20$.}
\label{psidif2}
\end{figure}
To conclude this section we shall discuss the applicability of a Taylor expansion of the type used
in \ern{rhotay}. In the current context, we study the function $h(\tilde z, \bar t)$, that is
we would like to know how good is the approximation:
\beq
\Delta h(\tilde z, \bar t, \Delta \bar{t}) \equiv
h(\tilde z, \bar t+ \Delta \bar{t}) - h(\tilde z, \bar t) \simeq \Delta h_a(\tilde z, \bar t, \Delta \bar{t}),
\qquad
\Delta h_a(\tilde z, \bar t, \Delta \bar{t}) \equiv \Delta \bar{t} h^{(1)}(\tilde z, \bar t)
+ \frac{1}{2} \Delta \bar{t}^2 h^{(2)}(\tilde z, \bar t)
\label{haprox}
\enq
The numerical evaluation of the functions $\Delta h(\tilde z, \bar t,\Delta \bar{t})$ and $\Delta h_a(\tilde z, \bar t,\Delta \bar{t})$ are depicted in figure \ref{hafig} for the plane $z=0$ and for the time $\bar t = 0.5$. It is easy to see that the approximation is indeed a good one for $
|\Delta \bar t| <0.5$, otherwise the approximation is not valid especially for a negative
$\Delta \bar t$ which is our main concern in retardation theory.
\begin{figure}[H]
\centering
\includegraphics[width= 0.7\columnwidth]{ha}
\caption{The functions $\Delta h(0, 0.5, \Delta \bar{t})$ and $\Delta h_a(0, 0.5,\Delta \bar{t})$.}
\label{hafig}
\end{figure}
The next step is to set the retardation delay $\Delta \bar{t} = - \frac{R}{c t_f} =- \frac{R}{R_{s2}} = - \frac{\tilde R}{\tilde R_{s2}}$ and calculate the approximate form of $\Delta \psi$:
\beq
\Delta \psi(r, \bar t) = \psi_r(r, \bar t) - \psi_N(r, \bar t)
= \int \frac{\tilde r}{ \tilde R} \Delta h(\tilde z', \bar t, - \frac{\tilde R}{\tilde R_{s2}})d \tilde z' \simeq \Delta \psi_a(r, \bar t).
\label{delpsidelh}
\enq
We may try the following approximation:
\beq
\Delta \psi_{aw}(r, \bar t) \equiv
\int_{-\infty}^{\infty} \frac{\tilde r}{\tilde R} \Delta h_a(\tilde z', \bar t, - \frac{\tilde R}{\tilde R_{s2}})d \tilde z' = -\frac{\tilde r}{\tilde R_{s2}} \int_{-\infty}^{\infty} h^{(1)}(\tilde z', \bar t) d \tilde z'
+\frac{\tilde r}{2 \tilde R_{s2}}\int_{-\infty}^{\infty} \frac{\tilde R}{\tilde R_{s2}} h^{(2)}(\tilde z', \bar t) d \tilde z'.
\label{delpsidelh2}
\enq
However, notice that for many points in the integration domain $|\frac{\tilde R}{\tilde R_{s2}}|$
is not smaller than $\frac{1}{2}$ hence $\Delta h_a$ is not a valid approximation. Moreover,
notice that:
\beq
\int_{-\infty}^{\infty} h^{(1)}(\tilde z', \bar t) d \tilde z'
= \frac{\partial}{\partial \bar{t}} \int_{-\infty}^{\infty} h(\tilde z', \bar t) d \tilde z'
= \frac{\partial}{\partial \bar{t}} 1 = 0.
\label{firsttv}
\enq
Hence:
\beq
\Delta \psi_{aw}(\tilde r, \bar t) =
\frac{\tilde r}{2 \tilde R_{s2}}\int_{-\infty}^{\infty} \frac{\tilde R}{\tilde R_{s2}} h^{(2)}(\tilde z', \bar t) d \tilde z'.
\label{delpsidelh3}
\enq
However, $\Delta \psi_{aw}(r, \bar t)$ does not have the asymptotic property $\lim_{r->\infty} \Delta \psi_{aw}(r, \bar t) = 0 $ described in \ern{delpsilim} as can be seen by a numerical
evaluation presented in figure \ref{delpsiaw}. We conclude that $\Delta \psi_{aw}$ is {\bf not}
an adequate approximation of $\Delta \psi$.
\begin{figure}[H]
\centering
\includegraphics[width= 0.7\columnwidth]{delpsiaw}
\caption{The asymptotic behaviour of $\Delta \psi_{aw}(\tilde r, \bar t)$ is clearly not appropriate.}
\label{delpsiaw}
\end{figure}
Notice, however, that this problem can be somewhat elevated if one integrates close enough
to the galactic plane $z=0$ such that $|\frac{\tilde R}{\tilde R_{s2}}|<0.5$ and consider
small $\tilde r$. Indeed, most of the mass in centered close to the galactic plane. After some numerical experimentation we obtained a reasonable approximation integrating in the range: $[-0.25 \tilde \sigma_i,0.25 \tilde \sigma_i]$:
\ber
\Delta \psi_{ac}(r, \bar t) &\equiv&
\int_{-0.25 \tilde \sigma_i}^{0.25 \tilde \sigma_i} \frac{\tilde r}{\tilde R} \Delta h_a(\tilde z', \bar t, - \frac{\tilde R}{\tilde R_{s2}})d \tilde z'
\nonumber \\
&=& -\frac{\tilde r}{\tilde R_{s2}} \int_{-0.25 \tilde \sigma_i}^{0.25 \tilde \sigma_i} h^{(1)}(\tilde z', \bar t) d \tilde z'
+\frac{\tilde r}{2 \tilde R_{s2}}\int_{-0.25 \tilde \sigma_i}^{0.25 \tilde \sigma_i} \frac{\tilde R}{\tilde R_{s2}} h^{(2)}(\tilde z', \bar t) d \tilde z'.
\label{delpsidelh2c}
\enr
\begin{figure}[H]
\centering
\includegraphics[width= 0.7\columnwidth]{delpsiac}
\caption{$\Delta \psi(\tilde r,0.5)$ and $\Delta \psi_{ac}(\tilde r,0.5)$ for $\tilde{r} <1$.}
\label{delpsiac}
\end{figure}
As a final comment we stress again that $R_s$ is the scale of the "galactic sphere of influence" that is the typical dimension of the region with which the galaxy exchanges mass and not necessarily the galactic radius. Thus the domain $\tilde r < 1$ may stretch far beyond the galaxy itself.
\section{Conclusions}
The phenomena of retardation is ubiquitous in physics, and follows directly from the Lorentz symmetry group. Hence, any system that is invariant under the Lorentz transformation will exhibit retardation phenomena. Those include physical systems related to classical electromagnetism ~\cite{Tuval,YahalomT,Yahalom3,Yahalom4} General Relativity \cite{YaRe1,ge,YaRe2,YahalomSym}, but also to other Lorentz invariant theories such as conformal gravity~\cite{Mannheim0,Mannheim1,Mannheim2}.
Dark matter being a major candidate to explain galactic rotation curves has only a slim chance to being found, given that accelerator experiments, as using the Large Hadron Collider was unable to find any super symmetric particles, not only of the community's favorite form of dark matter, but~also the form of it that is mandated in string theory, a~theory that also suggests a quantized version of Einstein~gravity.
We have shown that at least on the galactic scale dark matter is not needed \cite{YaRe1,ge,YaRe2,YahalomSym,Wagman}, as the dynamics can be explained by a retarded gravitational potential when a near field approximation is used. We remark that the analysis of
far field leading to gravitational waves \cite{Einstein2} was corroborated in recent years by observations \cite{Taylor,Castelvecchi}.
A justification for the second order Taylor series approximation which we used in previous works is given here for the first time, showing that indeed higher order terms can be safely neglected.
We also show that one may discuss retardation phenomena without relying on a Taylor expansion.
\authorcontributions{This paper has a single author, which has done all the reported work.}
\funding{This research received no funding.}
\acknowledgments{The author wishes to thank his former student, Dr. Michal Wagman, for being critical regarding retardation theory. Much of this paper is a result of her raising various questions on the validity of the second order approximation which was used in \cite{YahalomSym}.
The same doubts were raised by my colleague and friend Prof. Yosef Pinhasi, I would like to thank him as well for a critical discussion.}
\conflictsofinterest{The author declares no conflict of~interest.}
\reftitle{References}
\section*{Tabbing to fixed positions in a paragraph}
Two new text positioning commands are defined: \cs{tabto} and \cs{tab}.
\prototype{\cs{tabto}\arg{\meta{length}}}
Tab to a position relative to the left margin in a paragraph
(any indentation due to a list is part of the `margin' in this
context). If the text on the line already goes past the desired
position, the tab starts a new line and moves to the requested
horizontal position.
\prototype{\cs{tabto}\sarg\arg{\meta{length}}}
Similar to \cs{tabto}, except it will perform backspacing, and
over-print previous text on the line whenever that text is already
longer than the specified length (i.e., no linebreak is produced).
Line-breaks are suppressed immediately after \cs{tabto} or \cs{tabto*}.
The length register \cs{CurrentLineWidth} will report the width
of the existing text on the line, and it may be used in the
\meta{length} argument (using calc.sty, for example). Also, there
is \cs{TabPrevPos} which stores the \cs{CurrentLineWidth} from the
previous tab command (the position where the tab command occurred,
not where it went to), and can be used to return to that position
if no line breaks have occurred in between, or directly below it,
if there were line breaks.
\prototype{\cs{tab}}
Tab to the next tab-stop chosen from a list of tab positions, in
the traditional style of typewriters. A \cs{tab} will always move
to the next tab stop (or the next line), even if it is already
exactly at a tab stop. Thus, ``\cs{tab}\cs{tab}'' skips a position. A
linebreak is permitted immediately following a \cs{tab}, in case the
ensuing text does not fit well in the remaining space.
The tab-stop positions are declared using either \cs{TabPositions}
or \cs{NumTabs}:
\prototype{\cs{TabPositions}\arg{\meta{length}, \meta{length},
\textrm{\dots}\meta{length}}}
Declares the tab stops as a comma-separated list of positions
relative to the left margin. A tab-stop at \texttt{0pt} is implicit, and
need not be listed.
\prototype{\cs{NumTabs}\arg{\meta{number}}}
Declares a list of \meta{number} equally-spaced tabs, starting at the
left margin and spanning \cs{linewidth}. For example
\cs{NumTabs}\arg{2} declares tab-stops at \texttt{0pt} and
\texttt{0.5}\cs{linewidth}, the same as
\cs{TabPositions}\arg{0pt, 0.5\cs{linewidth}}
or \cs{TabPositions}\arg{0.5\cs{linewidth}}.
\end{document}
|
2,877,628,090,146 | arxiv | \section{Introduction}
In the context of $L^2$-invariants and related fields, infinite amenable groups stand out as a class of groups satisfying strong vanishing results. An infinite amenable has
\begin{itemize}
\setlength\itemsep{0mm}
\item vanishing $L^2$-Betti numbers, see \cite[Corollary 6.75]{Lueck2002}, or \cite[Theorem 7.2 (1) and (2)]{Lueck2002} for a strengthening of this statement;
\item vanishing $L^2$-torsion (provided that $G$ is of type $F$), see \cite[Theorem 1.3]{LiThom2014};
\item vanishing rank gradient and vanishing $K$-homology gradients with respect to a normal chain with trivial intersection (provided that $G$ is finitely generated), see \cite[Theorem 3]{AbertNikolov2012};
\item vanishing rank gradient and vanishing $K$-homology gradients with respect to \emph{any} chain (provided that $G$ is finitely presented), see \cite[Theorem 1]{AbertJZNikolov2011};
\item fixed price $1$ in the theory of cost of groups, see \cite[Theorem 6]{OrnsteinWeiss1980} combined with \cite[Théorème 3]{Gaboriau2000}.
\item vanishing simplicial volume (provided that $G$ is the fundamental group of a closed connected orientable manifold), see \cite[Section 3.1, Corollary (C)]{Gromov1983}.
\end{itemize}
Recall that a group is \emph{of type F} if it admits a finite classifying space. For the present paper the following result due to Wegner \cite[Theorem 5.4.5]{Wegner2000} deserves special attention. A group $G$ of type $F$ which is of $\det\geq 1$-class (i.e., the Fuglede-Kadison determinant satisfies $\det_{\N(G)}(A)\geq 1$ for all matrices $A\in M_{m,n}(\zg)$, see \cite[Definition 3.112 and Conjecture 13.2]{Lueck2002}) and contains a non-trivial elementary amenable normal subgroup has vanishing $L^2$-torsion \mbox{$\tor(G) = \tor(EG;\N(G)) = 0$}. In particular, the $L^2$-torsion of any elementary amenable group of type $F$ vanishes. This result was later slightly generalized \cite[Theorem 1]{Wegner2009}.
In \cite{FriedlLueck2015, FriedlLueck2015b} Friedl-Lück construct a new invariant $P(X;G)$ called\linebreak \emph{$L^2$-torsion polytope} of a $G$-CW-complex $X$ (satisfying a number of assumptions, see \cref{sub:pol}), which shares many features with the $L^2$-torsion $\tor(X;\N(G))$. It takes values in an \emph{integral polytope group} $\P_T(H_1(G)_f)$, which is defined as the Grothendieck group of integral polytopes in $H_1(G)_f\otimes_\Z\R$ up to translation. Here $H_1(G)_f$ denotes the free part of the first integral homology $H_1(G)$ of $G$. Provided that $G$ is an $L^2$-acyclic group of type $F$ with vanishing Whitehead group, one can define the $L^2$-torsion polytope of $G$ as $P(G) = P(EG;G)$. In the spirit of the list above, Friedl-Lück-Tillmann propose the following conjecture \cite[Conjecture 6.4]{FriedlLueckTillmann2016}.\medskip
\begin{conj}[Vanishing of the $L^2$-torsion polytope of amenable groups]\label{conj:polytope amenable}
Let $G\neq \Z$ be an amenable group satisfying the Atiyah Conjecture. Suppose that $G$ is of type $F$ and that $\Wh(G) = 0$. Then we have for the $L^2$-torsion polytope \[ P(G) = 0.\]
\end{conj}
In comparison with the original formulation, we replaced \emph{not virtually $\Z$} with \emph{not isomorphic to $\Z$} since any torsion-free virtually $\Z$ group is in fact isomorphic to $\Z$. Since $P(\Z) \in \P_T(\Z) \cong \Z$ is the negative of an interval of length one, infinite cyclic groups need to be excluded from the statement of the conjecture.
This paper is dedicated to partial solutions of \cref{conj:polytope amenable}. By means of the polytope homomorphism that is essential in the definition of the $L^2$-torsion polytope, we introduce the notion of groups \emph{of $P\geq 0$-class} and the even stronger property \emph{of polytope class}. These notions are polytope analogues of the aforementioned $\det\geq 1$-class. For groups of $P\geq 0$-class, the $L^2$-torsion polytope is a homotopy invariant of free finite $L^2$-acyclic $G$-CW-complexes. There is a prominent class of groups possessing this property as shown by the first result.
\begin{mainthm}[{ \ref{thm:amenable polytope class}} \normalfont (Polytope class and amenability)]
Let $G$ be a torsion-free amenable group satisfying the Atiyah Conjecture such that $H_1(G)_f$ is finitely generated. Then $G$ is of polytope class.
\end{mainthm}
In order to apply this result to the $L^2$-torsion polytope, we then adapt Wegner's strategy for proving the vanishing results for $L^2$-torsion \cite{Wegner2000, Wegner2009} and obtain the following partial solution to \cref{conj:polytope amenable}.
\begin{mainthm}[{ \ref{thm:vanishing polytope}} \normalfont (Vanishing $L^2$-torsion polytope)]
Let $G$ be a group of type $F$ which is of $P\geq 0$-class. Suppose that $G$ contains a non-abelian elementary amenable normal subgroup. Then $G$ is $L^2$-acyclic and we have
\[ P(G) = 0.\]
In particular, the $L^2$-torsion polytope of an elementary amenable group of type $F$ \mbox{vanishes.}
\end{mainthm}
Beyond elementary amenable groups, we provide at least some evidence for \cref{conj:polytope amenable}. The seminorm homomorphism $\norm$ will be introduced in \cref{def:seminorm map}.
\begin{mainprop}[ \ref{prop: amenable}]
Let $G\neq\Z$ be an amenable group of type $F$ satisfying the Atiyah Conjecture. Then $P(G)$ lies in the kernel of the seminorm homomorphism $\norm\colon \P_T(H_1(G)_f)\to \Map(H^1(G;\R),\R)$ and there is a polytope $P$ such that in $\P_T(H_1(G)_f)$ we have
\[ P(G) = P-*P. \]
\end{mainprop}
\medskip
\iffalse
\subsection*{Organization of the paper}
We need some of the details in the construction of the $L^2$-torsion polytope, so we begin in \cref{ch:preliminaries} with the necessary background on Friedl-Lück's universal $L^2$-torsion and the $L^2$-torsion polytope. Then we introduce groups of polytope class in \cref{ch:pol class} and show that amenability ensures polytope class in \cref{ch:pol class and amenabl}. \cref{ch:pol class and polytopes} presents vanishing results for the $L^2$-torsion polytope under certain amenability assumptions. The final \cref{ch:evidence} presents the proof of \cref{prop: amenable}.
\fi
\subsection*{Acknowledgments}
The author was supported by the Max Planck Institute for Mathematics in Bonn and the Deutsche Telekom Stiftung. We are grateful to Stefan Friedl, Fabian Henneke, Dawid Kielak, and Wolfgang L\"uck for many fruitful discussions and to the organizers of the \emph{New directions in $L^2$-invariants} workshop at the Hausdorff Institute for Mathematics in Bonn, where some of the ideas for this article were born.
\tableofcontents
\section{Background on the $L^2$-torsion polytope}\label{ch:preliminaries}
\subsection{The Atiyah Conjecture and $\D(G)$} The construction and our analysis of the $L^2$-torsion polytope requires some knowledge about the Atiyah Conjecture. If $R$ is a ring and $A\in M_{m,n}(R)$ is a matrix, then we let throughout $r_A\colon R^m\to R^n$ denote the $R$-homomorphism (of left $R$-modules) given by right multiplication with $A$.
\begin{conj}[Atiyah Conjecture]
A torsion-free group $G$ satisfies the \emph{Atiyah Conjecture} (with rational coefficients) if for any matrix $A\in M_{m,n}(\Q G)$ we have
\[ \dim_{\N(G)}\big(\ker (r_A\colon \N(G)^m\to \N(G)^n)\big) \in \Z.\]
\end{conj}
Here $\N(G)$ is the group von Neumann algebra of $G$ and $\dim_{\N(G)}$ denotes the dimension function on $\N(G)$-modules, see \cite[Definition 1.1 and Definition 6.20]{Lueck2002}. For a survey on the status of the Atiyah Conjecture we refer to \cite[Theorem 3.2]{FriedlLueck2015}. In order to explain its relevance in our context we need the following objects.
\iffalse
Let $\CC$ be the smallest class of groups containing all free groups and which is closed under directed unions as well as extensions with elementary amenable groups. If $G$ is a torsion-free group lying in $\CC$, then $G$ satisfies the Atiyah Conjecture. \cite{Linnell1993}
If $M$ is an admissible $3$-manifold that is not a closed graph manifold, then $G$ is torsion-free and lies in $\CC$, and hence in particular satisfies the Atiyah Conjecture.
\cite[Theorem 3.2 (3)]{FriedlLueck2015}.
\fi
\begin{dfn}[$\U(G)$ and $\D(G)$]
Let $\U(G)$ denote the algebra of operators affiliated to $\N(G)$, see \cite[Chapter 8]{Lueck2002}. Algebraically, this is the Ore localization of $\N(G)$ with respect to the set of weak isomorphisms, see \cite[Theorem 8.22 (1)]{Lueck2002}.
Let $\D(G)$ be the smallest subring of $\U(G)$ which contains $\Q G$ and is \emph{division closed}, meaning that every element of $\D(G)$ which is a unit in $\U(G)$ is already a unit in $\D(G)$.
\end{dfn}
Thus we obtain a rectangle of inclusions
\[ \xymatrix{
\Q G \ar[d]\ar[r] & \N(G)\ar[d]\\
\D(G)\ar[r] & \U(G),
}\]
and using these rings we recall the following result.
\iffalse
\begin{ex}
In the case $G=\Z^n$ the above rectangle specializes to
\[ \xymatrix{
\Q [u_1^\pm, ..., u_n^\pm] \ar[d]\ar[rr] && L^\infty(T^n)\ar[d]\\
\Q (u_1^\pm, ..., u_n^\pm)\ar[r] & \RR(G) \ar[r] & L(T^n),
}\]
where $\Q [u_1^\pm, ..., u_n^\pm]$ denotes the Laurent polynomial ring in $n$ variables, $\Q (u_1^\pm, ..., u_n^\pm)$ denotes the field of fractions thereof, $L^\infty(T^n)$ denotes the algebra of (equivalence classes of) essentially bounded measurable functions $T^n\to \C\cup\{\infty\}$ on the $n$-torus $T^n$, and $L(T^n)$ denotes the algebra of (equivalence classes of) measurable functions $T^n\to \C$. This follows from \cref{D:amenable} (\ref{item:kaplansky2}) below and \cite[Example 1.4 and Example 8.11]{Lueck2002}.
\end{ex}
\fi
\begin{prop}\label{D:skew field}
A torsion-free group $G$ satisfies the Atiyah Conjecture if and only if $\D(G)$ is a skew-field.
\end{prop}
\begin{proof}
See \cite[Lemma 10.39]{Lueck2002}.
\end{proof}
The next well-known lemma is the central reason why the $L^2$-torsion polytope is tractable for amenable groups.
\begin{lem}[$\D(G)$ of amenable groups]\label{D:amenable}
Any torsion-free elementary amenable group satisfies the Atiyah Conjecture.
Moreover, if $G$ is a torsion-free amenable group satisfying the Atiyah Conjecture, then $\Q G$ satisfies the Ore condition with respect to $T = \Q G\s-\{0\}$ and there is an isomorphism $\D(G) \cong T^{-1}\Q G$. In particular, $\D(G)$ is flat over $\Q G$.
\end{lem}
\begin{proof}
The first part follows from \cite[Theorem 2.3]{Linnell2006}, see also \cite[Theorem 1.2]{KrophollerEtal2006}. It is proved in \cite[Example 8.16 and Lemma 10.15]{Lueck2002} that $\Q G$ satisfies the Ore condition with respect to $T$. Recalling the notion of division closure, it is then easy to see that the inclusion $\Q G\to \D(G)$ localizes to an isomorphism $T^{-1}\Q G\tolabel{\cong} \D(G)$.
\end{proof}
If $R$ is a ring and $0\to K\to G\tolabel{p} Q\to 0$ is a group extension, then any choice of (set-theoretic) section $s\colon Q\to G$ for $p$ induces an isomorphism
\begin{equation}\label{eq:splitting}
RG \cong (RK)*Q.
\end{equation}
Here the right-hand side denotes a crossed product ring of $Q$ with coefficients in $RK$. We refer to \cite[Section 10.3.2]{Lueck2002} for a survey on crossed product rings and \cite[Example 10.53]{Lueck2002} for the details of the above statement. Here and henceforth we suppress the structure maps of crossed product rings from the notation. It will play an important role for us that $\D(G)$ shares similar structural properties. More precisely, we have
\begin{lem}[$\D(G)$ and extensions]\label{lem:DG and extensions}
Let $G$ be a torsion-free group satisfying the Atiyah Conjecture. Let $0\to K\to G\tolabel{p} H\to 0$ be a group extension such that $H$ is finitely generated free-abelian. Then $K$ satisfies the Atiyah Conjecture and any choice of (set-theoretic) section $s\colon H\to G$ for $p$ determines a crossed product ring $\D(K)*H$ together with an inclusion $\D(K)*H\subset \D(G)$ which restricts to the isomorphism $ (\Q K)*H\cong \Q G$ of (\ref{eq:splitting}). Moreover, $\D(K)* H$ satisfies the Ore condition with respect to $T = (\D(K)* H)\s-\{0\}$, and the inclusion induces a $\D(K)$-isomorphism
\begin{equation}\label{eq:D iso}
T^{-1}(\D(K)* H) \cong \D(G).
\end{equation}
If $H$ is infinite cyclic, then $\D(K)* H$ is isomorphic to the ring $\D(K)_t[u^{\pm}]$ of twisted Laurent polynomials, where the twisting $t$ depends on $s$.
\end{lem}
\begin{proof}
See \cite[Theorem 3.6 (3)]{FriedlLueck2015} and \cite[Example 10.54]{Lueck2002}, where also twisted Laurent polynomial rings are treated in detail.
\end{proof}
\subsection{Weak $K_1$-groups and universal $L^2$-torsion}
\label{sub:universal l2}
Let $G$ be a torsion-free group satisfying the Atiyah Conjecture. Define the \emph{weak $K_1$-group} $K_1^w(\Z G)$ as the abelian group whose generators $[f]$ are $\Z G$-maps $f\colon\Z G^n\to \Z G^n$ that become invertible over $\D(G)$, subject to the following relations:
If $f, g\colon \Z G^n\to\Z G^n$ are two such $\Z G$-maps, then require
\[ [g\circ f] = [f] + [g]. \]
If $f\colon \Z G^m\to\Z G^m,\: g\colon \Z G^n\to \Z G^n, \:h\colon \Z G^n\to\Z G^m$ are $\Z G$-maps such that $f$ and $g$ become invertible over $\D(G)$, then we require the relation
\[ \left[\begin{pmatrix}
f & h\\
0 & g
\end{pmatrix}\right] = [f] + [g].\]
This definition coincides with \cite[Definition 1.1]{FriedlLueck2015b} since $f\colon\Z G^n\to \Z G^n$ becomes invertible over $\D(G)$ if and only if $f$ induces a weak isomorphism $L^2(G)^n\to L^2(G)^n$. This follows from \cite[Lemma 1.21]{FriedlLueck2015b} and \cite[Lemma 10.39]{Lueck2002}.
We define the \emph{reduced weak $K_1$-group} and the \emph{weak Whitehead group} as the quotients
\begin{gather*}
\widetilde{K}_1^w(\Z G) = K_1^w(\Z G)/\{[\pm\id\colon \Z G\to\Z G]\};\\
\Wh^w(G)= K_1^w(\Z G)/\{[r_{\pm g}\colon \Z G\to \Z G]\:\mid g\in G\}.
\end{gather*}
There are obvious maps
\begin{gather*}
\widetilde{K}_1(\Z G) \to \widetilde{K}_1^w(\Z G) \to\widetilde{K}_1(\D(G));\\
\Wh(G)\to \Wh^w(G) \to K_1(\D(G))/\{[\pm g]\:\mid g\in G\}.
\end{gather*}
Recall that for any associative unital ring $R$ an $R$-chain complex $C_*$ is \emph{finite} if each chain module is finitely generated and only finitely many chain modules are non-trivial. It is \emph{based free} if each chain module is a free $R$-module and equipped with an equivalence class of $R$-basis, where two $R$ bases $B, B'$ are \emph{equivalent} if there exists a bijection $\sigma\colon B\to B'$ such that $\sigma(b) = \pm b$ for all $b\in B$. It is \emph{contractible} if there is a chain homotopy $\id_{C_*}\simeq 0$. If $C_*$ is a based free finite contractible $R$-chain complex, then we denote its Reidemeister torsion by $\tau(C_*)\in\widetilde{K}_1(R)$. Likewise we denote the Whitehead torsion of a $G$-homotopy equivalence $f\colon X\to Y$ of finite free $G$-CW-complexes by $\tau(f)\in\Wh(G)$.
A $\zg$-chain complex is \emph{$L^2$-acyclic} if all $L^2$-Betti numbers \[\betti_n(C_*;\N(G)) = \dim_{\N(G)} H_n( \N(G)\otimes_\zg C_*)\]
vanish. For any based free finite $L^2$-acyclic $\Z G$-chain complex $C_*$ Friedl-Lück construct a \emph{universal $L^2$-torsion}
\[\tor_u(C_*;\N(G)) \in \widetilde{K}_1^w(\Z G).\]
Its construction is an adaption of the Reidemeister and Whitehead torsion to the $L^2$-setting. By \cite[Remark 1.16]{FriedlLueck2015b} the universal $L^2$-torsion deserves its name in the sense that it encapsulates all other $L^2$-torsion invariants, including the (classical) $L^2$-torsion $\tor(C_*;\N(G)) \in \R$, twisted $L^2$-torsion functions \cite{DuboisEtal2014, DuboisEtal2015, DuboisEtal2015b} and twisted $L^2$-Euler characteristics \cite{FriedlLueck2015}.
If $X$ is a finite free $L^2$-acyclic $G$-CW-complex, then applying this to the cellular $\zg$-chain complex $C_*(X)$ produces the \emph{universal $L^2$-torsion of $X$}
\[\tor_u(X;\N(G)) \in \Wh^w(G).\]
Its main properties are collected in \cite[Theorem 2.5]{FriedlLueck2015b}. We point out two of its properties that we need in this paper.
First, given a $G$-homotopy equivalence $f\colon X\to Y$ between finite free $L^2$-acyclic $G$-CW-complexes, then
\begin{equation}\label{eq:hom invariance}
\tor_u(Y;\N(G)) - \tor_u(X;\N(G)) = \zeta(\tau(f)).
\end{equation}
where $\zeta\colon \Wh(G)\to \Wh^w(G)$ is the obvious homomorphism.
We include the second statement here for future reference as a small lemma.
\begin{lem}\label{lem:torsion}
Let $C_*$ be a finite based free $L^2$-acyclic $\zg$-chain complex. Then $\D(G)\otimes_{\zg} C_*$ is a contractible $\D(G)$-chain complex, and the canonical homomorphism $i\colon \widetilde{K}_1^w(\Z G) \to\widetilde{K}_1(\D(G))$ satisfies
\begin{equation}\label{eq:whitehead}
i\big(\tor_u(C_*;\N(G))\big) = \tau(\D(G)\otimes_{\zg}C_*).
\end{equation}
\end{lem}
\begin{proof}
The chain complex $\D(G)\otimes_{\zg}C_*$ is contractible by \cite[Lemma 1.21]{FriedlLueck2015b}.
Let $R$ be any associative unital ring and $E_*$ a finite based free contractible $R$-chain complex. If $u_*\colon E_*\to E_*$ is a chain isomorphism and $\gamma_*\colon u_*\simeq 0_*$ is a chain homotopy such that $\gamma_n\circ u_n = u_{n+1}\circ\gamma_n$, then we have an equality
\begin{equation} \label{eq:torsion formula}
\tau(E_*) = [(uc+ \gamma)_\odd] - [u_\odd] \in \widetilde{K}_1(R).
\end{equation}
This follows in exactly the same way as the argument leading to \cite[Equation (1.8)]{FriedlLueck2015b}. Now the desired equation (\ref{eq:whitehead}) follows from this by comparing (\ref{eq:torsion formula}) with the definition of universal $L^2$-torsion \cite[Definition 1.7]{FriedlLueck2015b}.
\end{proof}
\subsection{Integral polytope groups}
Let $H$ be a finitely generated free-abelian group. An integral polytope in $V_H = H\otimes_\Z\R$ is the convex hull of finitely many points in $H$, considered as a lattice in $V_H$. The \emph{Minkowski sum} of two integral polytopes $P$ and $Q$ in $V_H$ is defined by pointwise addition, i.e.,
\[ P+Q = \{p+q\in V_H\mid p\in P, q\in Q \}.\]
Denote by $\PPP(H)$ the commutative monoid of all integral polytopes in $V_H$ with the Minkowski sum as addition. It is cancellative, see e.g. \cite[Lemma 3.1.8]{Schneider1993}. Define the \emph{integral polytope group} $\P(H)$ to be the Grothendieck group associated to this commutative monoid. Thus elements are given by formal differences $P-Q$ of integral polytopes $P,Q\in \PPP(H)$, and two such differences $P-Q$, $P'-Q'$ are equal if and only if $P+Q' = P'+Q$ holds as subsets in $V_H$.
There is an injection of abelian groups
\begin{equation} \label{polytope injection}
H\to \P(H),\;\; h\mapsto \{h\}
\end{equation}
and we let $\P_T(H)$ be the cokernel of this map. The subscript $T$ stands for \emph{translation} since two polytopes become identified in $\P_T(H)$ if and only if there is a translation on $V_H$ mapping one bijectively to the other. We let $\PPP_T(H)$ be the image of the composition $\PPP(H)\to \P(H) \to \P_T(H)$.
The group $\P(H)$ carries a canonical involution induced by reflection about the origin, i.e.,
\begin{equation}\label{def:involution}
*\colon \P(H)\to \P(H),\;\; P\mapsto *P = \{-p\mid p\in P\}.
\end{equation}
This involution descends to an involution $*\colon\P_T(H)\to\P_T(H)$.
A homomorphism $f\colon H\to H'$ of finitely generated free-abelian groups induces homomorphisms
\begin{gather*}
\P(f)\colon \P(H)\to \P(H');\\
\P_T(f)\colon \P_T(H)\to \P_T(H')
\end{gather*}
by sending the class of a polytope $P$ to the class of the polytope $f(P)$. If $f$ is injective, then both $\P(f)$ and $\P_T(f)$ are easily seen to be injective as well. Thus if $G\subseteq H$ is a subgroup, then we will always view $\P(G)$ (respectively $\P_T(G)$) as a subgroup of $\P(H)$ (respectively $\P_T(H)$).
\begin{ex}\label{ex:polytope in dim 1}
Integral polytopes in $V_\Z = \R$ are just intervals $[m,n]\subseteq\R$ starting and ending at integral points. Thus we have $\P(\Z) \cong \Z^2$, where an explicit isomorphism is given by sending the class $[m,n]$ to $(m, n-m)$. Under this isomorphism, the involution corresponds to $*(k,l) = (-l-k, l)$. Similarly, $\P_T(\Z) \cong\Z$, where an explicit isomorphism is given by sending the element $[m,n]$ to $n-m$. The involution $*$ on $\P_T(\Z)$ is the identity.
\end{ex}
The structure of the integral polytope group was studied in detail by Cha-Friedl and the author \cite{ChaFriedlFunke2015} and by the author \cite{Funke2016}.
\subsection{The polytope homomorphism}
Let $G$ be a torsion-free group satisfying the Atiyah Conjecture such that $H_1(G)_f$, the free part of the first integral homology $H_1(G)$ of $G$ is finitely generated. Under these conditions, Friedl-Lück \cite[Section 6.2]{FriedlLueck2015b} define a \emph{polytope homomorphism}
\[\PP\colon K_1^w(\Z G)\to \P(H_1(G)_f). \]
Earlier versions of it had at least implicitly been considered for torsion-free elementary amenable groups \cite{FriedlHarvey2007, Friedl2007}. The polytope homomorphism is constructed as a composition
\begin{equation}\label{eq:pol hom}
K_1^w(\zg) \tolabel{i} K_1(\D(G)) \tolabel{\det_{\D(G)}} \D(G)^\times_\ab \tolabel{P}\P(H_1(G)_f),
\end{equation}
where the first map is the canonical map, the second is the Dieudonné determinant \cite{Dieudonne1943} which is in fact an isomorphism (see \cite[Corollary 2.2.6]{Rosenberg1994} or \cite[Corollary 4.3]{Silvester1981}), and the third relies on the structural properties of $\D(G)$ given in \cref{lem:DG and extensions}. More precisely we let $K$ be the kernel of the projection $\pr\colon G\to H_1(G)_f = H$ and define
\[ P'\colon \D(K)* H\s-\{0\}\to \PPP(H)\]
as follows: Given a non-trivial element $x = \sum_{h\in H} x_h\cdot h\in\D(K)* H$ we let $P'(x)$ be the convex hull of the set $\{h\in H \mid x_h\neq 0\}$. Then $P'$ is a homomorphism of monoids and induces a group homomorphism
\[ P'\colon \big(T^{-1}(\D(K)* H)\big)^\times_\ab\to \P(H),\;\; t^{-1}s\mapsto P'(s)-P'(t).\]
Now we let $P$ be the composition
\begin{equation}\label{eq:P}
P\colon \D(G)^\times_\ab \tolabel{\cong } \big(T^{-1}(\D(K)* H)\big)^\times_\ab\tolabel{P'} \P(H),
\end{equation}
where the first map is the isomorphism appearing in \cref{lem:DG and extensions}. We will denote the induced maps
\begin{gather*}
\PP\colon \widetilde{K}_1^w(\Z G)\to \P_T(H_1(G)_f)\\
\PP\colon \Wh^w(G)\to \P_T(H_1(G)_f)
\end{gather*}
by the same symbol.
\begin{notation}\label{notation P(x)}
For non-trivial $x\in \zg$ we denote the image of the class of $x$ in $\D(G)^\times_\ab$ under the map $P$ simply by $P(x)\in \P_T(H_1(G)_f)$. This is the same as $\PP\big( [r_x\colon \zg\to\zg]\big)$.
\end{notation}
\subsection{The $L^2$-torsion polytope}\label{sub:pol} The definition of our main object of study is now fairly simple.
\begin{dfn}[$L^2$-torsion polytope]\label{def:l2 polytope}
Let $G$ be a torsion-free group satisfying the Atiyah Conjecture such that $H_1(G)_f$ is finitely generated. Let $X$ be a finite free $L^2$-acyclic $G$-CW-complex. Then the \emph{$L^2$-torsion polytope of $X$} is defined as the image of the negative of its universal $L^2$-torsion under the polytope homomorphism, i.e.,
\[P(X;G) = \PP\big(-\tor_u(X;\N(G))\big)\in \P_T(H_1(G)_f).\]
Let $G$ be a group of type $F$ satisfying the Atiyah Conjecture. If $G$ is $L^2$-acyclic and satisfies $\Wh(G) = 0$, then we may define the \emph{$L^2$-torsion polytope of $G$} to be
\[ P(G) = P(EG;G) \in \P_T(H_1(G)_f).\]
\end{dfn}
\begin{remark}[Assumptions appearing in \cref{def:l2 polytope}]
The assumption $\Wh(G) = 0$ appearing above ensures that the $L^2$-torsion polytope of groups is well-defined, see (\ref{eq:hom invariance}). Conjecturally, however, this assumption is obsolete: Any group of type $F$ is torsion-free, and it is conjectured that the Whitehead group of any torsion-free group vanishes, see \cite[Conjecture 3]{LueckReich2005}. There is also no counterexample to the Atiyah Conjecture known. Thus the $L^2$-torsion polytope is potentially an invariant for all $L^2$-acyclic groups of type $F$.
Within the class of amenable groups all torsion-free virtually solvable groups are known to have trivial Whitehead group since they satisfy the $K$-theoretic Farrell-Jones Conjecture, as proved by Wegner \cite{Wegner2013}.
\end{remark}
One of the main results of Friedl-Lück's theory states that if $X = \widetilde{M}$ is the universal cover of a $3$-manifold $M$ satisfying a number of conditions, then $P(\widetilde{M};\pi_1(M))$ is the dual of the unit ball of the Thurston norm, see \cite[Theorem 3.35]{FriedlLueck2015b}.
A forerunner version of the $L^2$-torsion polytope of groups was defined and examined by Friedl-Tillmann \cite{FriedlTillmann2015} in the special case where $G$ is a torsion-free group given by a presentation with two generators, one relation, and $b_1(G) = 2$. They show that in this case $P(G)$ completely determines the BNS-invariant of Bieri-Neumann-Strebel \cite{Bierietal1987}. A similar result was obtained by Kielak and the author \cite[Theorem 6.3]{FunkeKielak2016} if $G = F_n\rtimes_g\Z$ is a free-by-cyclic group along a UPG automorphism $g\colon F_n\to F_n$.
\section{Groups of $P\geq 0$-class}\label{ch:pol class}
In this section we introduce a polytope analogue of the notion $\det\geq 1$-class concerning the Fuglede-Kadison determinant \cite[Definition 3.112]{Lueck2002}. First we need a partial order on polytope groups.
\begin{dfn}[Partial order on polytope groups]\label{def:partial order}
Let $H$ be a finitely generated free-abelian group. We define a partial order on $\P(H)$ by declaring
\[ P-Q\leq P'-Q' \;\text{ if and only if }\; P+Q'\subset P'+Q.\]
Likewise, we define a partial order on the translation quotient $\P_T(H)$ by declaring
\[ P-Q\leq P'-Q' \;\text{ if and only if }\; P+Q'\subset P'+Q\text{ up to translation}.\]
\end{dfn}
It is easy to see that this definition does not depend on the choice of representatives.
\iffalse
namely if $P-Q = P'-Q'$ and $P + Q'' \subset P''+Q$, then
\[P + Q'' + P' + Q' \subset P''+Q + P' + Q' = P'' + Q' + P + Q'\]
whence $P' + Q'' \subset P''+Q'$.
\fi
\begin{dfn}[$P\geq 0$-class and polytope class]
A group $G$ is \emph{of $P\geq 0$-class} if it is torsion-free, satisfies the Atiyah Conjecture, $b_1(G)<\infty$, and we have for any matrix $A\in M_{n,n}(\Z G)$ which becomes invertible over $\D(G)$ that
\[ \PP\big([r_A\colon \Z G^n\to \Z G^n]\big) \geq 0\]
in $\P_T(H_1(G)_f)$. We call $G$ \emph{of polytope class} if $\PP\big([r_A\colon \Z G^n\to \Z G^n]\big)$ is even represented by a polytope, i.e., it lies in the submonoid $\PPP_T(H_1(G)_f)\subseteq\P_T(H_1(G)_f)$ of integral polytopes up to translation.
\end{dfn}
\begin{ex}
\begin{enumerate}
\item A finitely generated free-abelian group $H$ is of polytope class since the Dieudonné determinant $\det_{\D(H)}(A)$ coincides with the determinant $\det_{\Z H}(A)$ over the commutative ring $\Z H$ and is therefore represented by an element in $\Z H$. Hence $\PP\big([r_A\colon \Z H^n\to \Z H^n]\big)$ is represented by a polytope.
\item If $G$ is a torsion-free group satisfying the Atiyah Conjecture such that $H_1(G)_f$ is of rank at most $1$, then $G$ is of polytope class. Namely, let $\D(K)_t[u^\pm]\subset\D(G)$ be a subring determined by a generator of $\Hom(G,\Z)$, as explained in \cref{lem:DG and extensions}. Then it follows by virtue of the Euclidean function on $\D(K)_t[u^\pm] $ given by the degree that $\det_{\D(G)}(A)$ is represented by an element in $\D(K)_t[u^\pm]$. (A similar argument will be used in the proof of \cref{thm:amenable polytope class} where more details can be found.) Thus $\PP\big([r_A\colon \Z G^n\to \Z G^n]\big)$ is represented by an interval.
\end{enumerate}
\end{ex}
We know from (\ref{eq:hom invariance}) that the $L^2$-torsion polytope is a simple homotopy invariant of free finite $L^2$-acyclic $G$-CW-complexes. This can be strengthened if $G$ is of $P\geq 0$-class.
\begin{lem}\label{lem:polytope class}
Let $G$ be a group of $P\geq 0$-class. Then the composition
\[ \Wh(G) \tolabel{\zeta} \Wh^w(G) \tolabel{\PP} \P_T(H_1(G)_f)\]
is trivial. Moreover, the $L^2$-torsion polytope is a homotopy invariant of free finite $L^2$-acyclic $G$-CW-complexes.
\end{lem}
\begin{proof}
An element in the image of $\zeta$ is of the form $[r_A\colon \zg^n\to\zg^n]$ for a matrix $A\in M_{n,n}(\zg)$ which has an inverse $A^{-1}\in M_{n,n}(\zg)$. Since $G$ is of $P\geq 0$-class, we have
\[ 0 = \PP([\id]) = \PP\big([r_A]\big) + \PP\big([r_{A^{-1}}]\big) \geq 0,\]
and hence $\PP([r_A]) = 0$. The 'moreover' part immediately follows from this because of (\ref{eq:hom invariance}).
\end{proof}
\begin{remark}[Extension of $P(G)$ to groups of $P\geq 0$-class]
\cref{lem:polytope class} allows us to drop $\Wh(G) = 0$ from the list of conditions in the definition of the $L^2$-torsion polytope $P(G)$ of groups (see \cref{def:l2 polytope}), provided that $G$ is of $P\geq 0$-class. Put differently, we can extend the definition of $P(G)$ to groups $G$ which are of type $F$ and of $P\geq 0$-class. We will take this into account in the formulations for the rest of this paper.
\end{remark}
\section{Polytope class and amenability}\label{ch:pol class and amenabl}
The goal of this section is to prove the following result.
\begin{thm}[Polytope class and amenability]\label{thm:amenable polytope class}
Let $G$ be a torsion-free amenable group satisfying the Atiyah Conjecture such that $H_1(G)_f$ is finitely generated. Then $G$ is of polytope class.
\end{thm}
Its proof requires some preparation. Our main technical tool going into the proof are face maps.
\begin{dfn}[Faces and face maps]
Let $H$ be a finitely generated free-abelian group and $P\subset V_H = H\otimes_\Z\R$ an integral polytope. Take $\phi\in\Hom(H,\Z)$ which we also view as an element in $\Hom(H,\R)=\Hom_\R(V_H,\R)$. Then we call
\[ F_\phi(P) = \{p\in P\mid \phi(p) = \max\{ \phi(q)\mid q\in P\} \}\]
the \emph{face of $P$ in $\phi$-direction}. A subset $F\subseteq P$ is called a \emph{face} if $F_\phi(P) = F$ for some $\phi\in\Hom(H,\Z)$.
A face of an integral polytope is an integral polytope in its own right, and it is straightforward to check that $F_\phi(P+Q) = F_\phi(P)+F_\phi(Q)$. These two observations imply that we obtain a homomorphism
\begin{equation*}
F_\phi\colon \P(H)\to \P(H), \;\; P\mapsto F_\phi(P)
\end{equation*}
that we call \emph{face map in $\phi$-direction}. There is an induced face map (denoted by the same symbol)
\begin{equation*}
F_\phi\colon \P_T(H)\to \P_T(H)
\end{equation*}
whose image is contained in the subgroup $\P_T(\ker \phi)$.
\end{dfn}
The first lemma is possibly well-known in polytope theory, but we were not able to find the statement nor an implicit proof in the literature. In any case, it might as well be helpful in other situations.
\begin{lem}[Detecting polytopes by their faces]\label{lem:polytope}
Let $H$ be a finitely generated free-abelian group of rank at least $2$. Then $x\in \P(H)$ is represented by a polytope if and only if for every $\phi\in\Hom(H,\Z)$ the class $F_\phi(x)\in\P(H)$ is represented by a polytope.
\end{lem}
\begin{proof}
It suffices to prove this for $H = \Z^n$. Equip $V_H = \R^n$ with the standard inner product. The forward direction of the lemma is obvious.
For the backwards direction write $x = P-Q$ for integral polytopes $P$ and $Q$. By assumption $F_\phi(x) = F_\phi(P) - F_\phi(Q)$ is an integral polytope for any $\phi\in \Hom(H,\Z)$, say $S^\phi$, so $F_\phi(P) = F_\phi(Q)+S^\phi$. We can write
\[P = \{x\in V_H\mid \psi_i(x)\leq c_i\}\]
for certain $\psi_i\in\Hom(H,\Z)\subset \Hom_\R(V_H,\R)$ and $c_i\in\Z$ ($i = 1,...,k$). Then
\[S = \hull\left(\bigcup_{i=1}^k S^{\psi_i}\right)\]
is an integral polytope satisfying $P\subset Q+S$. The remainder of the proof will be occupied with proving $Q+S \subset P$ which will imply $x = P-Q = S$. This requires a number of steps. In the following, Greek letters will always denote elements in $\Hom(H,\Z)$ without explicitly saying this. Moreover, given a compact subset $A\subset V_H$ and $\phi$, we will use the shorthand notations
\begin{gather*}
A_\phi = F_\phi(A); \\
\phi(A) = \max\{\phi(a)\mid a\in A\}.
\end{gather*}
First note that we have for any $\phi$ and $\psi$
\[ F_\phi(P_\psi) = P_\phi \cap P_\psi = F_\psi(P_\phi)\]
provided that the intersection in the middle is non-trivial, and likewise for $Q$.
\medskip
\emph{Step 1:} If $\phi, \psi$ are such that $P_\phi\cap P_\psi$ is non-empty, then $Q_\phi\cap Q_\psi$ and $S^\phi\cap S^\psi$ are non-empty, and we have
\[ P_\phi\cap P_\psi = \big(Q_\phi\cap Q_\psi \big) + \big(S^\phi\cap S^\psi \big).\]
We first argue that $Q_\phi\cap Q_\psi$ is non-empty. Pick a vertex $p\in P_\phi\cap P_\psi$, and let $\alpha$ be such that $P_\alpha = p$. Then $p = P_\alpha = Q_\alpha + S^\alpha$, hence $Q_\alpha = q$ and $S^\alpha = s$ are just points. After translating $Q$, we may assume that $s= 0$ and $p=q$. Then for every $\beta$ such that $P_\beta$ contains $p$ we have $Q_\beta\subset P_\beta$ and $p\in Q_\beta$. This applies in particular to $\phi$ and $\psi$, hence $p\in Q_\phi\cap Q_\psi$.
Now we compute
\[ F_\phi(S^\psi) = F_\phi(P_\psi) - F_\phi(Q_\psi) = F_\psi(P_\phi) - F_\psi(Q_\phi) = F_\psi(S^\phi),\]
hence $F_\phi(S^\psi) \subset S^\phi\cap S^\psi$ and $S^\phi\cap S^\psi$ is non-empty. We also have
\begin{align*}
\big(S^\phi\cap S^\psi\big) + F_\phi(Q_\psi) &= \big(S^\phi\cap S^\psi\big) + \big(Q_\phi\cap Q_\psi\big)\\
&\subset \big(P_\phi\cap P_\psi\big) \\
&= F_\phi(P_\psi).
\end{align*}
From this it follows that $S^\phi\cap S^\psi \subset F_\phi(S^\psi)$. Thus we proved $ F_\phi(S^\psi) = S^\phi\cap S^\psi$. Now we conclude
\begin{align*}
P_\phi\cap P_\psi &= F_\phi(P_\psi)\\
&= F_\phi(Q_\psi) + F_\phi(S^\psi)\\
&= \big(Q_\phi\cap Q_\psi\big) + \big(S^\phi\cap S^\psi\big).
\end{align*}
\medskip
\emph{Step 2:} Let $v_0,v_1,v_2\in S^{n-1} \subset \R^n$ be such that $v_1$ lies on a geodesic path of length at most $\pi$ from $v_0$ to $v_2$ in $S^{n-1}$. Write $\phi_i = \langle v_i, \cdot\rangle\colon \R^n\to \R$. If $P$ is any polytope such that $P_{\phi_1} \cap P_{\phi_2}$ is non-trivial, then we have
\[ \phi_0(P_{\phi_2}) = \phi_0(P_{\phi_1} \cap P_{\phi_2}).\]
Pick an element $x\in P_{\phi_1} \cap P_{\phi_2}$ attaining the maximum on the right. Assume that we have
\[ \phi_0(P_{\phi_2}) > \phi_0(P_{\phi_1} \cap P_{\phi_2}).\]
Then there exists $y\in P_{\phi_2}$ such that $ \phi_0(y) > \phi_0(x)$, $\phi_1(y) < \phi_1(x)$, and $\phi_2(y) = \phi_2(x)$. In other words,
\begin{gather*}
\langle y-x, v_0\rangle > 0;\\
\langle y-x, v_1\rangle < 0;\\
\langle y-x, v_2\rangle = 0
\end{gather*}
which cannot happen if $v_1$ lies on a geodesic path of length at most $\pi$ from $v_0$ to $v_2$.
\medskip
\emph{Step 3:} We have $S^\phi = S_\phi$.
Let $\phi, \psi$ be arbitrary and write (up to scalar) $\phi = \langle v, \cdot \rangle$ and $\psi = \langle w, \cdot \rangle$ for unit vectors $v,w$. There is a sequence of unit vectors $v = v_0, v_1, ..., v_m = w$ running along a geodesic path of length at most $\pi$ from $v$ to $w$ in $S^{n-1}$ such that $P_{\phi_i}\cap P_{\phi_{i+1}}$ is non-trivial. For brevity write from now on $P_i=P_{\phi_i}$, $Q_i = Q_{\phi_i}$, and $S^i = S^{\phi_i}$. Then we have by Step 1
\begin{gather*}
P_i\cap P_{i+1} = \big(Q_i\cap Q_{i+1} \big)+\big(S^i\cap S^{i+1}\big)
\end{gather*}
and by Step 2
\begin{gather*}
\phi(P_{i+1}) = \phi(P_i\cap P_{i+1});\\
\phi(Q_{i+1}) = \phi(Q_i\cap Q_{i+1}).
\end{gather*}
This implies
\begin{align*}
\phi(S^{i+1}) &= \phi(P_{i+1}) - \phi(Q_{i+1})\\
&= \phi(P_i\cap P_{i+1}) -\phi(Q_i\cap Q_{i+1})\\
&= \phi(S^i\cap S^{i+1}) \\
&\leq \phi(S^i).
\end{align*}
Since this is true for all $i = 0,..., m-1$, we conclude $\phi(S^\psi) \leq \phi(S^\phi)$ and hence $S^\phi = S_\phi$.
\medskip
\emph{Step 4:} We have $Q+ S\subset P = \{x\in V_H\mid \psi_i(x)\leq c_i\}$.
Pick arbitrary $q\in Q$ and $s\in S$. With the aid of Step 3 we can calculate
\begin{align*}
\psi_i(q+s) &= \psi_i(q)+\psi_i(s) \\
& \leq \psi_i(Q_{\psi_i}) + \psi_i(S_{\psi_i}) \\
& = \psi_i(Q_{\psi_i}) + \psi_i(S^{\psi_i}) \\
& = \psi_i(P_{\psi_i}) = c_i
\end{align*}
for all $i$, and hence $q+s \in P$.
\end{proof}
We also need the following auxiliary gadget.
\begin{dfn}
Let $H$ be a finitely generated free-abelian group and $G\subset H$ a subgroup. We consider $\PPP_T(G)$ as a submonoid of $\PPP_T(H)$. Then we let $\P_T(H,G)$ be the submonoid of $\P_T(H)$ containing all elements that can be written as a difference $P-Q$ for some $P\in \PPP_T(H)$ and $Q\in \PPP_T(G)$.
\end{dfn}
\begin{ex}
\begin{enumerate}
\item For any subgroup $G\subset H$ one has
\[ \PPP_T(H) = \P_T(H,0)\subset \P_T(H,G)\subset \P_T(H,H) = \P_T(H).\]
We can interpret $\P_T(H,G)$ as interpolating between the monoid of integral polytopes and the integral polytope group.
\item Let $H$ be of rank $2$ and let $G_1, G_2$ be two subgroups of rank $1$. If $G_i\cap G_j = 0$, then $\P_T(H,G_1)\cap \P_T(H,G_2) = \PPP_T(H)$.
\end{enumerate}
\end{ex}
Motivated by the last example we propose the following problem.
\begin{quest}
Let $H$ be a finitely generated free-abelian group and $G_1, G_2$ be two subgroups. Do we always have
\[\P_T(H,G_1)\cap \P_T(H,G_2) = \P_T(H, G_1\cap G_2)?\]
\end{quest}
If this question has an affirmative answer, then the next lemma, for which we provide a different argument, would immediately follow.
\begin{lem}\label{lem:intersection}
Let $H$ be a finitely generated free-abelian group. Then
\[\bigcap_{\phi\in\Hom(H, \Z)} \P_T(H, \ker \phi) = \PPP_T(H).\]
\end{lem}
\begin{proof}
We prove the statement by induction on the rank of $H$. The rank $1$ case is obvious.
For the higher rank case, pick an element $x$ in the above intersection. For any homomorphism $\phi\colon H\to\Z$ we can find $P_\phi\in \PPP_T(H)$ and $Q_\phi\in \PPP_T(\ker\phi)$ such that $x = P_\phi-Q_\phi$. Fix some homomorphism $\alpha\colon H\to\Z$. Then
\[ F_\alpha(x) = F_\alpha( P_\phi) - F_\alpha(Q_\phi) \in \P_T(\ker\alpha, \ker\alpha\cap \ker\phi).\]
Since $\phi$ was arbitrary, we conclude
\[ F_\alpha(x) \in \bigcap_{\phi\in\Hom(H,\Z)} \P_T(\ker\alpha, \ker\alpha\cap \ker\phi) = \bigcap_{\psi\in\Hom(\ker\alpha,\Z)} \P_T(\ker\alpha, \ker\psi) .\]
From the induction hypothesis we conclude $F_\alpha(x) \in \PPP_T(\ker\alpha)$. As this holds for every homomorphism $\alpha\colon H\to \Z$, we may apply the previous \cref{lem:polytope} to deduce that $x\in \PPP_T(H)$.
\end{proof}
Now we can tackle the main result of this section.
\begin{proof}[Proof of \cref{thm:amenable polytope class}]
Recall from \cref{D:amenable} that $\Z G$ satisfies the Ore condition with respect to $T= \Z G\s-\{0\}$ and the inclusion induces an isomorphism $T^{-1} \Z G\tolabel{\cong} \D(G)$.
Let $A\in M_{n,n}(\Z G)$ be a matrix which becomes invertible over $\D(G)$. If $H_1(G)_f = 0$, then there is nothing to prove. Otherwise let us fix some epimorphism $\phi\colon G\to \Z$ and denote its kernel by $K$. Consider the associated twisted Laurent polynomial ring $\D(K)_t[u^\pm]\subset \D(G)$ as in \cref{lem:DG and extensions}. The Euclidean function on $\D(K)_t[u^\pm]$ given by the degree allows us to transform $A$ to a triangular matrix $T$ over $\D(K)_t[u^\pm]$ by using the operations
\begin{itemize}
\setlength\itemsep{0mm}
\item Permute rows or columns;
\item Multiply a row on the right or a column on the left with an element of the form $y\cdot u^m$ for some non-trivial $y\in \D(K)$ and $m\in\Z$;
\item Add a right $\D(K)_t[u^\pm]$-multiple of one row (resp. column) to another row (resp. column).
\end{itemize}
These operations change the class $[A] \in K_1(\D(G))$ by adding an element of the form $[y\cdot u^m]$ for some non-trivial $y\in \D(K)$ and $m\in\Z$. Since $\D(K) = (\Z K\s-\{0\})^{-1} \Z K$, we may then multiply $T$ with suitable elements in $\Z K$ to obtain a matrix over $\Z K_t[u^\pm] = \Z G$. This implies that there are elements $a\in\zg$ and $b\in \Z K\s-\{0\}$ such that we have in $K_1(\D(G))$
\[[A] = [T] - [y\cdot u^m] = [a\cdot b^{-1}] -[y\cdot u^m] .\]
Since $P(u^m) = 0$ in $\P_T(H_1(G)_f)$, we have
\[ \PP\big([r_A\colon \Z G^n\to \Z G^n]\big) = P(a) - P(b) - P(y)\in \P_T(H_1(G)_f, \ker\bar{ \phi}) \]
for the epimorphism $\bar{\phi}\colon H_1(G)_f\to\Z$ induced by $\phi$. Since $\phi$ was arbitrary, we have
\[ \PP\big([r_A\colon \Z G^n\to \Z G^n]\big) \in\bigcap_{\substack{\phi\in\Hom(G,\Z) \\\text{ surjective}}} \P_T(H_1(G)_f, \ker\bar{ \phi}). \]
By \cref{lem:intersection}, this intersection is equal to $\PPP_T(H_1(G)_f)$, and the proof is complete.
\end{proof}
\section{Polytope class and the $L^2$-torsion polytope}\label{ch:pol class and polytopes}
In this section we adapt Wegner's strategy built in \cite{Wegner2000, Wegner2009} to the setting of the $L^2$-torsion polytope. Together with the knowledge that torsion-free amenable groups are of polytope class, one of its applications will be the vanishing of the $L^2$-torsion polytope of every elementary amenable group of type $F$. In order to motivate our first lemma we give a rough idea of the argument:
Instead of localizing the group ring $\zg$ at $\zg\s-\{0\}$ in order to obtain $\D(G)$, we localize at a much smaller set $S\subset\zg$ in order to obtain an intermediate ring $\zg\subset S^{-1}\zg\subset \D(G)$. This set is small enough so that the polytope of invertible matrices over $S^{-1}\zg$ still satisfies $P\geq 0$, but it is large enough so that the localized cellular chain complex $S^{-1}C_*(EG)$ is already contractible. Combining these two facts makes the image of the Whitehead torsion of $S^{-1}C_*(EG)$ under an adjusted polytope homomorphism $K_1(S^{-1}\zg)\to \P_T(H_1(G)_f)$ computable. But this image coincides with the negative of the $L^2$-torsion polytope $P(G)$.
\begin{lem}\label{lem:ore}
Let $G$ be a group of type $F$ which satisfies the Atiyah Conjecture and $b_1(G)<\infty$. Suppose that $G$ contains a non-trivial abelian normal subgroup $A\subset G$ such that $A\cap\ker(\pr\colon G\to H_1(G)_f)\neq 0$. Then
\[ S = \{ x\in \Z A\s-\{0\}\mid P(x)= 0\;\text{ in }\; \P_T(H_1(G)_f) \}.\]
is a multiplicatively closed subset with respect to which $\Z G$ satisfies the Ore condition and such that $S^{-1}\Z = 0$ for the trivial $\Z G$-module $\Z$.
\end{lem}
\begin{proof}
Since for any two elements $x,y\in \Z G$ we have $P(x\cdot y) = P(x)+ P(y)$, it is clear that $S$ is multiplicatively closed. The proof for the left and right Ore condition follows as in \cite[Proof of Theorem 5.4.5, Step 2 and 3]{Wegner2000}, see also \cite[Lemma 3.119]{Lueck2002}. We include the argument here for the sake of completeness. Note that the canonical involution on $\zg$ respects $S$, so it suffices to prove the right Ore condition.
Let $r\in\Z G, s\in S$ and fix a set of representatives $\{g_i\mid i\in I\}$ for the cosets $Ag\in A\backslash G$. Write $r = \sum_{i\in I} a_ig_i$ for certain $a_i\in \Z A$, where almost all $a_i$ vanish. Put $I' = \{i\in I\mid a_i\neq 0\}$. The element $s_i = g_isg_i^{-1}$ lies in $\Z A$ since $A$ is normal and
$ P(s_i) = P(s) = 0$. These two facts imply $s_i\in S$.
Define $s' = \prod_{i\in I'} s_i \in S$, $x_i = s'/s_i\in S$, and $r' = \sum_{i\in I'} x_ia_ig_i\in\zg$. Then we compute
\begin{align*}
s'\cdot r &= \sum_{i\in I'} s'a_ig_i
= \sum_{i\in I'} x_is_ia_ig_i
= \sum_{i\in I'} x_ia_is_ig_i \\
&= \sum_{i\in I'} x_ia_ig_isg_i^{-1}g_i
= \sum_{i\in I'} x_ia_ig_is
= r'\cdot s
\end{align*}
Finally we prove $S^{-1}\Z = 0$. Pick some non-trivial $a\in A\cap\ker(\pr\colon G\to H_1(G)_f)\neq 0$ (this is the only part where we need this assumption). Then $P(1-a) = 0$ in $\P_T(H_1(G)_f)$, so $1-a$ lies in $S$. Since $1-a$ acts by multiplication with $0$ on $\Z$, we conclude $S^{-1}\Z = 0$.
\end{proof}
\begin{lem}\label{lem:vanishing}
Let $G$ be a group of $P\geq 0$-class. Let $S\subseteq \Z G$ be a multiplicatively closed subset with respect to which $\Z G$ satisfies the Ore condition. Suppose that $P(s)= 0\text{ in } \P_T(H_1(G)_f)$ for all $s\in S$.
If $X$ is a free finite $L^2$-acyclic $G$-CW-complex such that $S^{-1}H_n(X) = 0$, then
\[ P(X;G) = 0.\]
\end{lem}
\begin{proof}
This is based on ideas appearing in \cite[Proof of Theorem 5.4.5, Step 4 and 5]{Wegner2000}, see also \cite[Lemma 3.114]{Lueck2002}.
First we consider the following commutative diagram
\[\xymatrix@R=5mm{
\widetilde{K}_1^w(\zg) \ar[dr]^(.48){i} \ar@/^1.4pc/[drrr]^{\PP} & & \\
& \widetilde{K}_1(\D(G)) \ar[r]^(.45){\det_{\D(G)}} & \D(G)^\times_\ab/\{\pm 1\} \ar[r]^(.51){P} & \P_T(H_1(G)_f)\\
\widetilde{K}_1(S^{-1}\zg)\ar[ur]^(.53){j} \ar@/_1.4pc/[urrr]^{\PP'}
}\]
Here $i$ and $j$ denote the obvious maps, $\det_{\D(G)}$ is the Dieudonné determinant, $P$ is induced by the map defined in (\ref{eq:P}), $\PP$ denotes the composition of the top row (which is the polytope homomorphism), and $\PP'$ denotes the composition of the bottom row.
Let $A$ be an invertible $S^{-1}\zg$-matrix. By multiplying $A$ with a suitable $s\in S$ we obtain a $\zg$-matrix $B$ which is invertible over $S^{-1}\zg$ and thus also over $\D(G)$. Then we have $[A] = [B] - [s]$ in $\widetilde{K}_1(S^{-1}\zg)$ and $\PP'([B]) = \PP([B])$. We assume that $P(s) = 0$ and that $G$ is of $P\geq 0$-class, so we have
\begin{equation}\label{eq:positive}
\PP'([A]) = \PP'([B]) - \PP'([s]) = \PP'([B]) - P(s)= \PP([B])\geq 0.
\end{equation}
Since the same reasoning applies to $A^{-1}$, we have $\PP'([A]) = 0$ and thus $\PP' = 0$.
Denote by $C_* = C_*(X)$ the cellular $\zg$-chain complex of $X$ equipped with some choice of cellular basis. By \cref{lem:torsion} the $\D(G)$-chain complex $\D(G)\otimes_{\zg} C_*$ is contractible and we have
\begin{equation*}
i(\tor_u(C_*;\N(G))) = \tau(\D(G)\otimes_{\zg} C_*).
\end{equation*}
Since localization is flat and $S^{-1}H_n(X) = 0$, the $S^{-1}\zg$-chain complex $S^{-1}C_* = S^{-1}\zg\otimes_\zg C_*$ is also contractible, and we have
\begin{align*}
j(\tau(S^{-1}C_*)) &=\tau(\D(G)\otimes_{S^{-1}\zg} S^{-1}C_*) \\
&= \tau(\D(G)\otimes_{S^{-1}\zg} S^{-1}\zg \otimes_\zg C_* )\\
& = \tau(\D(G)\otimes_{\zg} C_*)\\
&= i(\tor_u(C_*;\N(G))).
\end{align*}
Thus we see
\begin{equation}\label{eq:torsion}
\PP(\tor_u(C_*;\N(G))) = \PP'(\tau(S^{-1}C_*)) = 0,
\end{equation}
which completes the proof.
\iffalse
Let $\gamma_*\colon S^{-1}C_*\to S^{-1}C_{*+1}$ be a chain contraction. Then both compositions of the $S^{-1}\zg$-maps
\begin{align*}
(S^{-1}c +\gamma)_\odd&\colon S^{-1}C_\odd\to S^{-1}C_\ev\\
(S^{-1}c +\gamma)_\ev&\colon S^{-1}C_\ev\to S^{-1}C_\odd
\end{align*}
are given by upper triangular automorphisms with identity maps on the diagonal, see \cite[Lemma 3.40]{Lueck2002}. Hence both maps are $S^{-1}\zg$-isomorphisms and we have in $\P_T(H_1(G)_f)$
\[ \PP'( (S^{-1}c +\gamma)_\odd) + \PP'((S^{-1}c +\gamma)_\ev) = 0.\]
We have seen in (\ref{eq:positive}) that $\PP'( (S^{-1}c +\gamma)_\odd) \geq 0$ and $\PP'((S^{-1}c +\gamma)_\ev) \geq 0$. Therefore these two inequalities are in fact equalities. Putting this together with (\ref{eq:torsion}) we conclude
\[ -P(X;G) = \PP(\tor_u(C_*;\N(G))) = \PP'(\tau(S^{-1}C_*)) = \PP'( [(S^{-1}c +\gamma)_\odd]) = 0.\qedhere\]
\fi
\end{proof}
The following is the main result of this section.
\begin{thm}[Vanishing $L^2$-torsion polytope]\label{thm:vanishing polytope}
Let $G$ be a group of type $F$ which is of $P\geq 0$-class. Suppose that $G$ contains a non-abelian elementary amenable normal subgroup. Then $G$ is $L^2$-acyclic and we have
\[ P(G) = 0.\]
\end{thm}
\begin{proof}
The group $G$ is $L^2$-acyclic by \cite[Theorem 1.44]{Lueck2002}. Let $N$ be the non-abelian elementary amenable normal subgroup.
\emph{Case 1:} $N$ is not virtually abelian. It follows from the proof of \cite[Theorem 2.3.15]{Wegner2000} and the references given therein that $N$ is solvable-by-finite. Hence $N$ has a unique maximal solvable normal subgroup of finite index, say $S$. Since we assume that $N$ is not virtually abelian, $S$ is not abelian. Hence the lowest non-trivial subgroup $A$ in the derived series of $S$ is abelian and contained in $[S,S]\subset [G,G]$. In particular, $A\cap\ker(\pr\colon G\to H_1(G)_f)\neq 0$. Since $A$ is characteristic in $S$ and $S$ is characteristic in $N$, $A$ is normal in $G$.
\emph{Case 2:} $N$ is virtually abelian. Let $A$ be a normal abelian subgroup of finite index. By assumption $N$ is not abelian, so $\ker(\pr\colon N\to H_1(N)_f)$ is non-trivial and hence infinite as $G$ is torsion-free. But any infinite subgroup of $N$ must intersect $A$ non-trivially. Thus in particular, $A\cap\ker(\pr\colon G\to H_1(G)_f)\neq 0$.
In both cases we may apply \cref{lem:ore}. This provides us with a subset $S\subset \zg$ satisfying the assumptions of \cref{lem:vanishing} for $X = EG$. Hence $P(G) = 0$.
\end{proof}
\begin{cor}[The $L^2$-torsion polytope of elementary amenable groups vanishes]\label{main theorem:el amenable}
Let $G$ be an amenable group of type $F$ satisfying the Atiyah Conjecture. If $G$ contains a non-abelian elementary amenable normal subgroup, then
\[ P(G) = 0.\]
In particular, the $L^2$-torsion polytope of any elementary amenable group of type $F$ vanishes.
\end{cor}
\begin{proof}
By \cref{thm:amenable polytope class} an amenable group $G$ of type $F$ satisfying the Atiyah Conjecture is of polytope class. Hence the first statement follows directly from \cref{thm:vanishing polytope}.
For the second statement, recall from \cref{D:amenable} that an elementary amenable group $G$ of type $F$ satisfies the Atiyah Conjecture. Hence $P(G) = 0$ follows from the previous statement provided that $G$ is non-abelian. If $G$ is abelian, then $G$ must be finitely generated free-abelian, so $P(G) = 0$ follows from $\tor_u(G) = 0$ as seen in \cite[Example 2.7]{FriedlLueck2015b}.
\end{proof}
We emphasize the following remark that was also used in the proof of \cref{thm:vanishing polytope}.
\begin{remark}
An elementary amenable group of type $F$ (or more generally, with finite cohomological dimension) is in fact virtually solvable by a result of Hillman-Linnell \cite[Corollary 1]{HillmanLinnell1992}.
\end{remark}
\begin{remark}[Generalization to the universal $L^2$-torsion]
The proof of \cref{main theorem:el amenable} crucially relies on the existence of a partial order on polytope groups even though the original statement does not involve them. One difficulty in proving the corresponding statement for the universal $L^2$-torsion $\tor_u(G)$ lies in the structural deficit of $\Wh^w(G)$ that it does not carry a meaningful partial order.
\end{remark}
\medskip
\section{Evidence for non-elementary amenable groups}\label{ch:evidence}
In this short final section, we offer some evidence for the validity of \cref{conj:polytope amenable} for amenable groups that are not elementary amenable. This computation is to a great extent based on known results. Our main tool will be norm maps.
Given a finitely generated free-abelian group $H$, we denote by $\Map(\Hom(H,\R),\R)$ the group of continuous maps $\Hom(H,\R)\to\R$ equipped with pointwise addition. A polytope $P\in \PPP(H)$ induces a seminorm on $\Hom(H,\R)$ by
\[ \|\phi\|_P = \max\{ \phi(p)-\phi(q)\mid p,q\in P \}.\]
This seminorm behaves well with respect to Minkowski sums in the sense that
\[ \|\phi\|_{P+Q} = \|\phi\|_P + \|\phi\|_Q \]
for all $\phi\in \Hom(H,\R)$, which allows us to make the following definition.
\begin{dfn}[Seminorm homomorphism]\label{def:seminorm map}
We call
\[ \norm\colon\P(H)\to \Map(\Hom(H,\R),\R),\;\; P-Q\mapsto \|\cdot \|_P - \|\cdot\|_Q\]
\emph{seminorm homomorphism}. It passes to the quotient $\P_T(H)$ and the induced map
\[\norm\colon\P_T(H)\to \Map(\Hom(H,\R),\R)\]
is denoted by the same symbol.
\end{dfn}
The cornerstone of our argument will be the following theorem.
\begin{thm}\label{little lemma}
Let $H$ be a finitely generated free-abelian group. Then we have
\begin{align*}
&\ker\big(\norm\colon\P_T(H)\to \Map(\Hom(H,\R),\R)\big) \\
= &\ker \big(\id+*\colon \P_T(H)\to \P_T(H)\big) \\
= & \im\big(\id-*\colon \P_T(H)\to\P_T(H)\big).
\end{align*}
\end{thm}
\begin{proof}
This is the content of \cite[Remark 6.2 and Theorem 6.4]{Funke2016}.
\end{proof}
If $G$ is a group, we will identify $\Hom(H_1(G)_f,\R)$ with $H^1(G;\R)$ in the following.
\begin{prop}[$L^2$-torsion polytope of amenable groups]\label{prop: amenable}
Let $G\neq\Z$ be an amenable group of type $F$ satisfying the Atiyah Conjecture. Then $P(G)$ lies in the kernel of $\norm\colon \P_T(H_1(G)_f)\to \Map(H^1(G;\R),\R)$ and there is a polytope $P\in\PPP_T(H_1(G)_f)$ such that
\[ P(G) = P-*P. \]
\end{prop}
\begin{proof}
Let $\pr\colon G\to H_1(G)_f = H$ be the obvious projection. Suppose that $H\neq 0$ since there is nothing to prove otherwise. Let $\phi\colon H\to \Z$ be an epimorphism, and put $K =\ker(\phi\circ\pr\colon G\to\Z)$. Then we have by \cite[Equation (3.26)]{FriedlLueck2015b} and \cite[Lemma 2.6]{FriedlLueck2015}
\begin{align*}
\norm(P(G))(\phi) = -\ct(i^*EG;\N(K))= -\ct(EK;N(K)),
\end{align*}
where $\ct(X;\N(K))$ denotes the $L^2$-Euler characteristic of a $K$-space $X$, see \cite[Section 6.6]{Lueck2002}.
As a subgroup of an amenable group, $K$ is itself amenable. Since $G\neq \Z$, $K$ must be infinite. Since infinite amenable groups are $L^2$-acyclic, we see $\ct(EK;N(K)) = 0$. (Note that for this argument it is irrelevant that $i^*EG = EK$ is not a \emph{finite} $K$-CW-complex.) Thus we have
\[\norm(P(G))(\phi) = 0\]
for all surjective homomorphisms $\phi\colon H\to\Z$.
As a difference of seminorms $\norm(P(G))$ is homogeneous and continuous. By the homogeneity we have $\norm(P(G))(\phi) = 0$ for all homomorphisms $\phi\colon H\to\Q$, and by the continuity we have $\norm(P(G))(\phi) = 0$ for homomorphisms $\phi\colon H\to\R$. Hence
\[P(G) \in \ker\big(\norm\colon \P_T(H)\to \Map(H^1(G;\R),\R)\big).\]
Now by \cref{little lemma} we have $P(G) \in \im\big(\id-*\colon \P_T(H)\to\P_T(H)\big)$. Hence there exists a class $R-S\in\P_T(H)$ such that
\[P(G) = R-S - (*R-*S) = R+*S - *(R+*S).\]
Taking $P = R+*S$ finishes the proof.
\end{proof}
\medskip
|
2,877,628,090,147 | arxiv | \section{Introduction}\label{sec:introduction}
It has recently been shown in \cite{typicaldecoding} that human-produced natural English sentences are composed of words which encode information close to the expected information content (i.e., the entropy).
Derived from this finding, the authors of \cite{typicaldecoding} also proposed a sampling strategy called ``typical sampling'' that produced sentences that, compared to other sampling methods, were deemed more natural by humans.
In this paper, we evaluate if the trend towards the expected information content can also be observed in music. To that end, we reproduce some of the analyses performed in \cite{typicaldecoding} and \cite{prob_qual_paradox} for musical events and sequences. More precisely, we show how the information of human-produced musical material is distributed around the model entropy. We also evaluate how typical sampling impacts these distributions and how it compares to conventional, ancestral sampling.
\section{Methods}
In the following, we recap Information Theory concepts coined in the seminal paper \cite{shannon},
which have been applied to music in \cite{meyer} to explain expectation and surprise. Let $p\left(x_{t}\vert x_{<t}\right)$ be the conditional probability of a symbol $x_{t}$ given the past $x_{<t}$ observed symbols and $q$ a model fitted to $p$, e.g., a neural network fitted with likelihood optimization. A symbol could represent a word in natural language or a musical event in music.
The conditional \textit{information content} (IC) is given by $IC\left(x_{t}\vert x_{<t}\right) = -\log{q\left(x_{t}\vert x_{<t}\right)}$
and is a measure of how surprising (i.e., high IC) or expected (i.e., low IC) the symbol $x_{t}$ is according to the model $q$, given the past $x_{<t}$ observations. Furthermore, the IC of a whole sequence is defined as $IC\left(x\right) = \sum_{t=0}^{\lvert x \rvert -1} IC\left(x_{t}\vert x_{<t}\right)$ and the information density $ID(x) = IC(x) / \lvert x \rvert$.
We are also interested in the expected information or \textit{entropy} $H$ for the conditional symbol distribution and the sequence distribution.
\subsection{Typicality}
In \cite{typicaldecoding}, the authors find that in natural language sentences, the conditional word information is close to the expected conditional information, and in \cite{prob_qual_paradox} that the sequence information content is close to the (unconditional) entropy or that
\begin{align}
\lvert \epsilon_{sym} \rvert &= \lvert H\left(x_{t}\vert x_{<t}\right) - IC\left(x_{t}\vert x_{<t}\right)\rvert \label{eq:typ_evt},\text{ and }\\
\lvert \epsilon_{seq} \rvert &= \lvert H\left(x\right) - IC\left(x\right)\rvert, \label{eq:typ_seq}
\end{align}
respectively, are small. We are interested in testing if the results transfer from natural language sequences of words to human compositions represented as sequences of musical events. As the lengths of musical sequences can vary a lot, we hypothesize that $IC\left(x\right)$ (in \cref{eq:typ_seq}) has less relevance to music than the (length-normalized) \emph{information density}, where the $ID(x)$ of individual songs should be close to the expected $ID(x)$, resulting in a small
\begin{equation}
\lvert \epsilon_{ID} \rvert = \vert \mathbb{E}\left[ID\left(x\right)\right] - ID\left(x\right) \rvert.
\label{eq:typ_id}
\end{equation}
\subsection{Typical Sampling}
\label{seq:typical_sampling}
In \cite{typicaldecoding}, a sampling strategy is proposed where the least typical symbols\footnote{Symbols with highest value of $\epsilon_{sym}$ in \cref{eq:typ_evt}} of $q$ get pruned. The obtained samples are reported to be more human-like than other sampling strategies. In typical sampling, we initially identify the smallest set of most typical symbols $V$ s.t. $q(V|x_{<t}) \geq \tau$,
\begin{figure*}[!ht]
\centering
\begin{subfigure}[t]{.495\textwidth}
\centering
\includegraphics[trim=10 5 40 19,clip,scale=.6]{figs/event_typicality_fixed-1.pdf}
\caption{}
\label{fig:evt_typ}
\end{subfigure}
\hfill
\begin{subfigure}[t]{.495\textwidth}
\centering
\includegraphics[trim=15 5 35 19,clip,scale=.6]{figs/seq_typicality_global_fixed.pdf}
\caption{}
\label{fig:seq_typ}
\end{subfigure}
\vspace{-3mm}
\caption{Distribution of typicality divergence for human-composed music (Reference), conventional ancestral sampling (Conventional), and typical sampling (Typical) for single events (a) and sequences (b). The distributions are reported for datasets IF and GF.}
\vspace{-1mm}
\label{fig:results}
\end{figure*}
where $\tau$ determines the amount of probability pruned. Secondly, the probabilities of symbols $\notin V$ are zeroed. Finally, the resulting function is normalized to yield a valid probability distribution. We wish to investigate the effect of typical sampling on the typicality of generated sequences (\cref{eq:typ_evt,eq:typ_id}).
\section{Experiments}
Our experiments are performed on monophonic symbolic music.
We use the dataset of \cite{folkrnnsession}, consisting of $45849$ Irish folk music lead sheets (denoted IF).
For computational reasons, we discard the $5\%$ longest sequences, resulting in a test set of $3618$ samples of length $\leq 691$.
In addition, we use all German folk songs from the Essen Folk Database \cite{essen1, essen2} (referred to as GF), consisting of 5152 melodies.
We partition both datasets in training, validation, and test sets with proportions 10/12, 1/12, and 1/12, respectively (all analyses are performed on the test set).
The compositions are encoded using a simplified variant of the tokenization strategy described in \cite{performancernn}. We order the notes by absolute time and serialize a note as a note-value (pitch) or rest token, followed by one or more duration tokens indicating the note's duration. Specifically, we use $128$ \emph{change-pitch tokens} for pitches $C_{-1} - G_9$,
$1$ \emph{rest token},
$100$ \emph{duration tokens} quantized linearly from $10$ms to $1000$ms, and
$1$ \emph{end-of-sequence token}.
This results in a vocabulary of 230 tokens. For each dataset, we train a Transformer decoder model \cite{attention} with relative attention \cite{relativeattention, musictransformer} in a self-supervised prediction task.
We sample sets of sequences from each model with a maximum sequence length and size following their respective test set's maximum length and size. We perform conventional ancestral sampling from the unmodified distribution and typical sampling with a low, medium, and high $\tau$ value.
\section{Results and Discussion}
In \cref{fig:evt_typ}, we show the distributions of the information contents for single events relative to the entropy (for typical sampling, the entropy and information content of the events is always calculated based on the unpruned distributions). In \cref{fig:seq_typ}, we show the distributions of Information Densities (IDs) for sequences relative to the mean IDs (see \cref{eq:typ_id}) of the respective reference distribution. For both data sets, we show the distributions of the human-generated melodies (Reference), conventional ancestral sampling (Conventional), and typical sampling (Typical, with different $\tau$ values).
Similar to \cite{typicaldecoding}, we find that the ICs of events are distributed densely around the conditional entropy (see \cref{fig:evt_typ}). When using typical sampling, this trend gets more pronounced for decreasing $\tau$ values, showing that the sampled musical events are indeed more typical.
However, also with conventional ancestral sampling, the highest density of the event's information is close to the entropy. In addition, the distributions of Conventional follow those of Reference closer than Typical in both figures. In \cref{fig:seq_typ}, typical sampling causes the ID distributions of generated sequences to narrow and shift to the right with decreasing $\tau$ values. Note that in \cref{fig:seq_typ}, $\epsilon$ is always calculated based on the entropy of the reference distributions, meaning that \emph{the generated sequences become more probable when $\tau$ decreases}. This is reasonable, as typical sampling prunes less likely events, as shown in \cref{fig:evt_typ}, where events to the left get pruned.
The Reference distribution of IF is not centred around $0$ in \cref{fig:seq_typ} (i.e., it is shifted to the right). This suggests that a considerable number of songs in this dataset have a high average ID (meaning that they are rather unlikely given the dataset distribution, causing the overall entropy to increase).
It is subject to future work (not shown in \cite{typicaldecoding}) to investigate if the typicality property of events is mainly linked to human-generated sequences. That is not certain, considering that even events sampled from a uniform or degenerate (i.e., deterministic) distribution are perfectly typical (with $\epsilon=0$). If it turns out that the typicality property is trivial, the reason why typical sampling yields sequences preferred by humans (as shown in \cite{typicaldecoding}) may also be explained by other factors, like a more stable random walk (as the model is confronted with more probable context), or by the simple fact that slightly higher probability sequences are more appealing.
\section{Conclusion}
We showed that musical events tend to be typical and that typical sampling generally yields more likely sequences than conventional sampling.
However, it is unclear if the typicality is a property of human-generated data only and if typicality of such sequences is a valid explanation for why typical sampling results in sequences preferred by humans.\let\thefootnote\relax\footnote{This work was conducted in a collaboration between JKU and Sony Computer Science Laboratories Paris under a research agreement.}
\newpage
|
2,877,628,090,148 | arxiv | \section{Introduction}
\label{intro}
Symmetry is key-ingredient in the description of natural phenomena. The notion of symmetry is an essential feature in several areas of physics. In this context, the well-known Noether's theorem \cite{lanczos2012variational} establishes a connection between symmetry and conservation laws of relevant physical quantities. In quantum mechanics, we often use symmetry to obtain crucial results concerning angular momenta operators \cite{sakurai2014modern}.
Likewise, symmetry is important in the topic of quantum information \cite{PhysRevB.95.045111}.
Symmetry is also important in the framework of relativity \cite{westman2009coordinates}, and for this reason, it is indispensable in research areas such as particle physics \cite{gibson1980symmetry} and cosmology \cite{preskill1991cosmology}.
A pertinent question in the research areas cited above refers to think about what happens when some symmetry is broken in a given physical system.
When a system suffers a phase transition, for example, it can lose some type of symmetry.
Another example of symmetry-breaking occurs in condensed matter systems: by employing the Volterra process \cite{solyom2007fundamentals}, it is possible to break some symmetry of the system due to the creation of a topological defect, like disclinations and dislocations \cite{RevModPhys.80.61,puntigam1997volterra}, for instance.
Topological defects can emerge in a large number of physical systems covering themes such as liquid crystals \cite{lavrentovich2012defects}, graphene physics \cite{alden2013strain}, magnetism \cite{kou2011tunable} and cosmology \cite{durrer1999topological}.
Recent studies also have reported the importance of topological defects in Life Sciences \cite{kawaguchi2017topological,saw2017topological}.
On the point of view of cosmology, defects in the spacetime topology can be viewed as a possible consequence of the evolution of the early universe, which has suffered phase transitions due to the temperature decreasing and the process of expansion \cite{hindmarsh1995cosmic,vilenkin2000cosmic}.
In this contribution, we are particularly involved in studying the topological defect known as a cosmic string. A cosmic string is a linear defect, similar to a flux tube in type-II superconductors \cite{hindmarsh1995cosmic}. The spacetime around a cosmic string has a conical symmetry, identically to the case of a disclination \cite{moraes2000condensed}. The concept of a cosmic string it was introduced in the literature by Kibble \cite{kibble1976topology}. Since then, this topic has been investigated in diverse forms. An intriguing facet in this subject refers to the quantum mechanical description of a particle in a region of the spacetime containing a cosmic string. It can be done both in the scenario of relativistic and nonrelativistic quantum mechanics. For instance, the hydrogen atom in a spacetime of a cosmic string it was analyzed in Ref. \cite{PhysRevD.66.105011}. In Ref. \cite{marques2005exact}, it was considered the problem of a relativistic electron in the presence of both Coulomb and scalar potentials in the cosmic string spacetime. Results about vacuum polarization in a cosmic string spacetime were reported in Ref. \cite{PhysRevD.74.025017}.
Again, the cosmic string spacetime it was considered as a background to examine relativistic oscillators \cite{PhysRevA.84.032109}, quantum phases \cite{bakke2016relativistic}, and fermionic currents \cite{bezerra2016induced}.
A relevant issue in this context consists of taking into consideration the influence of electromagnetic fields in the quantum particle motion. Landau levels \cite{Book.2005.Griffiths} and the Aharonov-Bohm effect \cite{PR.1959.115.485,peshkin1989aharonov}, for instance, are essential ingredients in the investigation of quantum systems even in a flat spacetime. It can be explained because Landau levels are a quantum analog of classical cyclotron motion, while the Aharonov-Bohm effect reveals the significance of the vector potential in the quantum world.
Then, studying the contribution of magnetic fields to the quantum mechanical description of a system in spacetime having a topological defect is a natural development. Examples of studies dealing with Landau levels and the Aharonov-Bohm effect in the presence of topological defects can be accessed in Refs. \cite{de2001landau} and \cite{azevedo1998topological}, respectively. In particular, the inclusion of electromagnetic interactions in the case of a cosmic string background also has been considered. For instance, in Ref. \cite{medeiros2012relativistic}, it was analyzed the quantum dynamics of a charged particle in the presence of a magnetic field and scalar potential. In Ref. \cite{PhysRevD.93.043545}, various configurations of confined magnetic fields are examined and the existence of induced vacuum fermionic currents is investigated.
On the other hand, we can be interested in analyzing the behavior of a quantum system when noninertial effects turn on. These effects play a fundamental role in the description of systems governed by classical mechanics. Noninertial effects also can take place on quantum systems, providing novel theoretical predictions and feasible experimental developments.
For instance, the emergence of quantum phases in rotating systems, in analogy to the Aharonov-Bohm effect \cite{aharonov1973quantum,semon1982experimental} were investigated. In addition, a relation between the Hall effect and inertial forces it was established \cite{johnson2000inertial}.
Besides, if a given system it is put to rotate, it has consequences in diverse physical properties like spin transport \cite{PhysRevB.84.104410,chowdhury2014spin}, electronic structure \cite{garcia2017geometric}, and even can present magnetization due rotation, like in the Barnett effect \cite{PhysRevB.92.174424}. While a magnetic field produces a spin-field coupling, resulting in the anomalous
Zeeman effect \cite{nouredine2009quantum}, rotation produces an analog effect, due the spin-rotation coupling \cite{danner2020spin}.
Thus, rotation can contribute similarly to a magnetic field in the dynamics of a quantum system.
More, noninertial effects are an interesting issue in the situation in which spacetime contains topological defects. In this case, the noninertial effects and the presence of a topological defect can be included in the quantum mechanical description by employing the same tools: we can use a metric tensor to a spinning spacetime with a topological defect \cite{clement1990rotating}.
The spacetime of a spinning cosmic string has been considered as background for several problems involving quantum systems. For instance, the Schr\"{o}dinger equation in that spacetime it was solved in Ref. \cite{hassanabadi2015motion}. Bound states for neutral particles in a rotating frame of a cosmic string were analyzed in Ref. \cite{PhysRevD.82.084025}. Likewise, rotating effects on a Landau-Aharonov-Casher System in the spacetime of a cosmic string were investigated in Ref. \cite{bakke2015rotating}.
As we already have mentioned, in some cases rotation presents similarities within electromagnetic fields. This way, it is also an attractive question examining how the electromagnetic interactions affect the particle quantum motion of a rotating system in the presence of a topological defect. A recent example of studying dealing with both topological and noninertial effects in the presence of an Aharonov-Bohm potential can be accessed in Ref. \cite{oliveira2019topological}. In Ref. \cite{Wang_canadian_journal2017}, it was addressed the problem of a spinless relativistic particle subjected to a uniform magnetic field in the spinning cosmic string spacetime.
The Dirac oscillator in the spacetime of a cosmic string considering noninertial effects and the presence of the Aharonov–Casher effect it was analyzed in Ref. \cite{EPJC_oliveira2019topological}.
In Ref. \cite{wang2018study}, it was analyzed the problem of a charged half-spin particle depicted by the Dirac equation in the presence of a uniform magnetic field in the rotating cosmic string spacetime.
A meaningful aspect in this context consists of analyzing how different configurations of magnetic fields affect the quantum particle motion.
In this paper, we study the relativistic quantum mechanics of an electron in the presence of both a uniform magnetic field and Aharonov-Bohm potential in the spinning cosmic string spacetime. In other words, we solve the Dirac equation in this scenario and investigate how the rotation, curvature and external magnetic fields affect the wave functions and energies of the electron.
The manuscript is organized as follows. In Section II, we present some algebraic elements necessary to construct the field equations in curved spacetime and write the Dirac equation describing the quantum motion of the electron in the presence of external magnetic fields in the spinning cosmic string background. In Section III, we deal with first-order solutions and study the existence of isolated solutions for the particular case of a particle at rest. In Section IV, we take our attention to the case when the energy of the particle is different from its rest energy. We map the Dirac equation problem in curved space with minimal coupling into a Sturm-Liouville problem for the upper component of the Dirac spinor and, using an appropriate ansatz, we derive the radial equation. We solve the radial equation and find the wave functions and energies of the particle. We make a detailed discussion of the results and also comparisons with other studies in the literature. In Section V, we present our conclusions. In our work, we use natural units, $\hbar = c = G = 1$.
\section{Dirac equation in the spinning cosmic string spacetime}
In this section, we briefly present the main tools needed to construct the Dirac equation in the conical spacetime in the presence of noninertial effects. The first step consists in take a look at the metric tensor characterizing this geometry. Next, we will choose an appropriate tetrad basis and implement the fields configuration involved through the performing of a minimal substitution.
The spacetime induced by a rotating cosmic string is described by the metric
\begin{equation}
ds^{2}=\left( dt+ad\varphi \right) ^{2}-dr^{2}-\alpha ^{2}r^{2}d\varphi
^{2}-dz^{2}, \label{metric}
\end{equation}
where $-\infty <z<\infty $, $r\geqslant 0$ and $0\leqslant \varphi \leqslant
2\pi$. The parameter $\alpha $ is related to the linear mass density $\mu$
of the cosmic string through the relation $\alpha =1-4\mu$ and it runs in
the interval $(0,1]$. The quantity $a=4J$ is the rotation parameter, with $J$ representing the angular momentum of the spinning cosmic string. The relativistic quantum dynamics of a spin-$1/2$ particle interacting with external magnetic fields in the rotating cosmic string spacetime is governed by the Dirac equation
\begin{equation}
\left[ i\gamma ^{\mu }\left( x\right) \left( \partial _{\mu }+\Gamma _{\mu
}\left( x\right) +ieA_{\mu }(x)\right) -M\right] \Psi \left( x\right) =0,
\label{diracsc}
\end{equation}
where $M$ is the mass of the particle and $\gamma ^{\mu }\left( x\right)$ are the Dirac matrices in the
rotating cosmic string spacetime, which are defined in terms of the tetrad fields $e_{a}^{\mu }$ and Dirac matrices in the flat space $\gamma ^{a}$ in the following way:
\begin{equation}
\gamma ^{\mu }\left( x\right) =e_{a}^{\mu }\left( x\right) \gamma ^{a},
\label{gmatrices}
\end{equation}
where
\begin{equation}
\gamma ^{a}=\left( \gamma ^{0},\gamma ^{i}\right),\, \text{with}\, \gamma ^{0}=\left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right),\,\gamma ^{i}=\left(
\begin{array}{cc}
0 & \sigma ^{i} \\
-\sigma ^{i} & 0
\end{array}
\right),
\end{equation}
are the standard Dirac matrices
and $\sigma ^{i}=\left( \sigma ^{x},\sigma ^{y},\sigma ^{z}\right)$ are the usual Pauli matrices. The matrices (\ref{gmatrices}) satisfy the following relation:
\begin{equation}
\left\{ \gamma ^{\mu }\left( x\right) ,\gamma ^{\nu }\left( x\right)
\right\} =2g^{\mu \nu }\left( x\right).
\end{equation}
Also, in Eq. (\ref{diracsc}), $\Gamma _{\mu }\left( x\right)$ is the
spin affine connection given by
\begin{equation}
\Gamma _{\mu }\left( x\right) =\frac{1}{4}\gamma ^{a}\gamma ^{b}e_{a}^{\nu
}\left( x\right) \left[ \partial _{\mu }e_{b\nu }\left( x\right) -\Gamma
_{\mu \nu }^{\sigma }e_{b\sigma }\left( x\right) \right], \label{affine}
\end{equation}
where $\Gamma_{\mu \nu}^{\sigma}$ are the Christoffel symbols of the second kind and $e_{a}^{\mu }(x)$ is the tetrad field. The tetrad basis satisfies the relations
\begin{align}
&e_{\mu }^{a}\left( x\right) e_{\nu}^{b}\left(x\right) \eta_{ab}=g_{\mu \nu }\left( x\right),\\
&e_{\mu}^{a}\left(x\right) e_{\nu
}^{b}\left(x\right)=\delta_{a}^{b},\\
&e_{a}^{\mu}\left(x\right)
e_{\nu}^{a}\left(x\right) =\delta_{\mu}^{\nu}.
\end{align}
In Eq. (\ref{affine}), the Greek letters are used for tensor indices while the Latin letters are denoting Minkowski indices. We use the tetrad basis and its inverse defined as \cite{EPJC.2019.79.311}
\begin{eqnarray}
e_{\mu }^{a}\left( x\right) &=&\left(
\begin{array}{cccc}
1 & 0 & a & 0 \\
0 & \cos \varphi & -r\alpha \sin \varphi & 0 \\
0 & \sin \varphi & r\alpha \cos \varphi & 0 \\
0 & 0 & 0 & 1
\end{array}
\right) , \\
e_{a}^{\mu }\left( x\right) &=&\left(
\begin{array}{cccc}
1 & \frac{a\sin \varphi }{r\alpha } & -\frac{a\cos \varphi }{r\alpha } & 0
\\
0 & \cos \varphi & \sin \varphi & 0 \\
0 & -\frac{\sin \varphi }{r\alpha } & \frac{\cos \varphi }{r\alpha } & 0 \\
0 & 0 & 0 & 1
\end{array}
\right).\label{trdi}
\end{eqnarray}
For this choice, it can be shown that the non-vanishing affine connection is given by
\begin{equation}
\Gamma _{\mu }=\left( 0,0,\Gamma _{\varphi },0\right), \, \text{with} \; \Gamma _{\varphi }=\frac{i}{2}\left( 1-\alpha \right) \Sigma^{z},
\end{equation}
with
\begin{equation}
\Gamma _{\varphi }=\frac{i}{2}\left( 1-\alpha \right) \Sigma^{z},
\end{equation}
where
\begin{equation}
\Sigma ^{z}=\left(
\begin{array}{cc}
\sigma ^{z} & 0 \\
0 & \sigma ^{z}
\end{array}
\right), \;\; \sigma ^{z} =\left(
\begin{array}{cc}
1 & 0 \\
0 & -1%
\end{array}
\right).
\end{equation}
By using the tetrad basis (\ref{trdi}), the matrices (\ref{gmatrices}) can be written explicitly as
\begin{align}
&\gamma ^{t} =\gamma ^{0}-a\gamma ^{\varphi }, \\
&\gamma ^{r}=\left(
\begin{array}{cc}
0 & \sigma ^{r} \\
-\sigma ^{r} & 0
\end{array}
\right), \;\;\gamma ^{\varphi }=\left(
\begin{array}{cc}
0 & \sigma ^{\varphi } \\
-\sigma ^{\varphi } & 0
\end{array}
\right),
\end{align}
with
\begin{equation}
\sigma^{r} =\left(
\begin{array}{cc}
0 & e^{-i\varphi } \\
e^{+i\varphi } & 0
\end{array}
\right), \;\;\sigma ^{\varphi }=\frac{1}{r\alpha }\left(
\begin{array}{cc}
0 & -ie^{-i\varphi } \\
ie^{+i\varphi } & 0
\end{array}
\right)
\end{equation}
being the Pauli matrices in the curved spacetime.
Since we are first interested in studying the solutions of the Dirac equation in its present form (Eq. (\ref{diracsc})), we need to write the corresponding system of first order coupled differential equations. For this to be accomplished, let's assume the time-dependence of the wave functions together with the decomposition of the fermion field in the form
\begin{equation}
\Psi \left( r,\varphi \right)=e^{-iEt}\left(
\begin{array}{c}
\psi _{1}\left( r,\varphi \right) \\
\psi _{2}\left( r,\varphi \right)
\end{array}
\right),\label{S1}
\end{equation}
with
\begin{align}
&\psi _{1}\left( r,\varphi \right)=\left(
\begin{array}{c}
\psi _{a}\left( r,\varphi \right) \\
\psi _{b}\left( r,\varphi \right)
\end{array}
\right) =\left(
\begin{array}{c}
e^{im\varphi }f_{+}\left( r\right) \\
ie^{i\left( m+1\right) \varphi }f_{-}\left( r\right)
\end{array}
\right) ,\label{P1} \\
& \psi _{2}\left( r,\varphi \right) =\left(
\begin{array}{c}
\psi _{c}\left( r,\varphi \right) \\
\psi _{d}\left( r,\varphi \right)
\end{array}
\right) =\left(
\begin{array}{c}
e^{im\varphi }g_{+}\left( r\right) \\
ie^{i\left( m+1\right) \varphi }g_{-}\left( r\right)
\end{array}
\right). \label{P2}
\end{align}
The system we will analyze takes into account the particle is immersed in a region where there is a uniform magnetic field and also the potential due to a thin long solenoid along the z-axis. Having this field configuration in mind, we study
the physical implications due to noninertial effects and the Aharonov-Bohm potential on the relativistic Landau quantization. We also take into account the translational invariance of the system along the $z$-direction, which allows us to eliminate the third direction ($p_z=z=0$) and, consequently, we can consider only the planar motion \cite{PRL.1990.64.503,PRD.1994.50.7715,PRD.2012.85.041701,AoP.2013.339.510}. In this case, the
particle experiences a superposition of potential vectors written in the
Coulomb gauge as
\begin{equation}
\mathbf{A}=\left( 0,-\alpha rA_{\varphi }, 0\right) , \label{vectorA}
\end{equation}
with
\begin{align}
A_{\varphi } &=A_{\varphi ,1}+A_{\varphi ,2} , \label{potential} \\
A_{\varphi ,1} &=\frac{Br}{2},\;\;\; A_{\varphi ,2}=\frac{\phi }{\alpha r},\label{potentialv}
\end{align}
where $B$ is the magnetic field magnitude, $\phi =\Phi /\Phi _{0}$, $\Phi$ is the
magnetic flux and $\Phi _{0}=2\pi /e$ is the quantum of magnetic flux along the solenoid. This configuration also provides an superposition of magnetic fields in the
z-direction
\begin{equation}
B=B_{z,1}+B_{z,2},
\end{equation}
with
\begin{equation}
B_{1,z}=B, \;\;\; B_{z,2}=\phi \frac{\delta (r)}{\alpha r}, \label{fields}
\end{equation}
Note that the particle only interacts with the magnetic field due to the
potential vector $A_{\varphi ,1}$. Here, we are focused on studying the
electron motion only in the $r\neq 0$ region, so that we can neglect the
point interaction $B_{z,2}$ and, consequently, consider only regular wave
functions.
Using the results above, the Dirac equation (\ref{diracsc}) can be written as
\begin{align}
(E-&M) \,\psi_{1}+\sigma ^{r}i\partial _{r}\psi _{2}\notag\\ &+\sigma
^{\varphi }\left( i\partial _{\varphi }+eA_{\varphi }-aE-\frac{s}{2}\left(
1-\alpha \right) \right) \psi_{2}=0, \label{Eqpsi}\\
(E+&M) \,\psi_{2}+\sigma ^{i}i\partial _{r}\psi _{1}\notag \\ &+\sigma
^{\varphi }\left( i\partial _{\varphi }+eA_{\varphi }-aE-\frac{s}{2}\left(
1-\alpha \right) \right) \psi_{1}=0. \label{Eqchi}
\end{align}
At this point, we are ready to solve the equations (\ref{Eqpsi}) and (\ref{Eqchi}) by considering two distinct circumstances:
(i) Take our attention to isolated solutions of the first order Dirac equation by imposing the condition $E= \pm M$;
(ii) By imposing the condition $E\neq \pm M$, we looking for solutions of the second order Dirac equation.
We will show in the next two sections that there are bound state solutions for both cases and discuss their main physical properties. To distinguish each case in (i), in the next section we use the superscripts ($\pm$) to label the quantities corresponding to $E= \pm M$.
\section{Solution of the equation of motion to $E=\pm M$}
To study the existence of isolated solutions of the Dirac equation (\ref{diracsc}), we must set $E=\pm M$ in Eqs. (\ref{Eqpsi}) and (\ref{Eqchi}). In literature, such solutions are known to be excluded from the Sturm-Liouville problem. The search for isolated solutions of the Dirac equation has been performed in different physical contexts \cite
{EPJC.2019.79.596,EPL.2014.108.30003,AoP.2013.338.278,JPA.2007.40.263,PLA.2006.351.379}. The bound state solution must satisfy the normalization condition
\begin{equation}
\int_{0}^{\infty }\left( |\psi_{1}(r)|^{2}+|\psi_{2}(r)|^{2}\right) rdr=1.
\label{norm}
\end{equation}
By making $E=+M$ in Eqs. (\ref{Eqpsi}) and (\ref{Eqchi}) and using Eqs. (\ref{P1}) and (\ref{P2}), we get
\begin{align}
&\frac{dg_{+}^{(+)}(r)}{dr}-\frac{L_{m}^{(+)}}{r\alpha }g_{+}^{(+)}(r)+\frac{eBr}{2}g_{+}^{(+)}(r) =0,\label{eqg1}
\\
&\frac{dg_{-}^{(+)}(r)}{dr}+\frac{L_{m+1}^{(+)}}{r\alpha }g_{-}^{(+)}(r)-\frac{eBr}{2}g_{-}^{(+)}(r) =0,\label{eqg2}
\\
&\frac{df_{+}^{(+)}(r)}{dr}-\frac{L_{m}^{(+)}}{r\alpha }f_{+}^{(+)}(r)+\frac{eBr}{2}f_{+}^{(+)}(r)
=-2Mg_{-}^{(+)}(r), \label{eqg3}\\
&\frac{df_{-}^{(+)}(r)}{dr}+\frac{L_{m+1}^{(+)}}{r\alpha }f_{-}^{(+)}(r)-\frac{eBr}{2}f_{-}^{(+)}(r)
=2Mg_{+}^{(+)}(r),\label{eqg4}
\end{align}
with
\begin{align}
L_{m}^{(+)}& =m-\phi +aM+\frac{s}{2}\left( 1-\alpha \right), \label{lmm1}\\
L_{m+1}^{(+)}& =m+1-\phi +aM+\frac{s}{2}\left( 1-\alpha \right).\label{lmm2}
\end{align}
The solution of the coupled linear differential equations system (\ref{eqg1})-(\ref{eqg4}) is given by
\begin{align}
f_{+}^{(+)}(r)& =e^{-\frac{1}{4}eBr^{2}}r^{\frac{L_{m}^{(+)}}{\alpha }}\left[
a_{2}+a_{1}M\left( -\frac{eB}{2}\right) ^{\Omega _{a}}\Gamma _{a}^{(+)}\right],\label{sp1} \\
f_{-}^{(+)}(r)& =e^{\frac{1}{4}eBr^{2}}r^{-\frac{L_{m+1}^{(+)}}{\alpha }}\left[
b_{2}-b_{1}M\left( \frac{eB}{2}\right) ^{-\Omega _{b}}\Gamma _{b}^{(+)}\right] , \label{sp2} \\
g_{+}^{(+)}(r)& =b_{1}e^{-\frac{1}{4}Ber^{2}}r^{\frac{L_{m}^{(+)}}{\alpha }}, \label{sp3} \\
g_{-}^{(+)}(r)& =a_{1}e^{\frac{1}{4}Ber^{2}}r^{-\frac{L_{m+1}^{(+)}}{\alpha }},\label{sp4}
\end{align}
with
\begin{align}
\Omega _{a}& =\frac{1}{2\alpha }\left( L_{m}^{(+)}+L_{m+1}^{(+)}-\alpha
\right) , \\
\Omega _{b}& =\frac{1}{2\alpha }\left( L_{m}^{(+)}+L_{m+1}^{(+)}+\alpha
\right) ,
\end{align}
where
\begin{align}
\Gamma _{a}^{(+)}& =\Gamma \left( -\Omega _{a},-\frac{1}{2}eBr^{2}\right) , \\
\Gamma _{b}^{(+)}& =\Gamma \left( \Omega _{b},\frac{1}{2}eBr^{2}\right) .
\end{align}
are upper incomplete Gamma functions \cite{Book.1972.Abramowitz}, and
$a_{1}$, $a_{2}$, $b_{1}$ and $b_{2}$ are constants. Analyzing the solutions (\ref{sp1}) and (\ref{sp3}), we note that
$e^{-\frac{1}{4}eBr^{2}}$ dominates over $r^{\frac{L_{m}^{(+)}}{\alpha }}$
for any value of $L_{m}^{(+)}/\alpha $, in such way both solutions converge when $r\rightarrow 0$ and $r\rightarrow \infty $. This will not occur for the
function $e^{\frac{1}{4}eBr^{2}}$ in the solutions (\ref{sp2}) and (\ref{sp4}). Moreover, since the incomplete Gamma
functions $\Gamma _{a}^{(+)}$ and $\Gamma _{b}^{(+)}$ always diverge, then
the function $f_{+}^{(+)}(r)$ will only converges as $r\rightarrow 0$ if
$a_{1}=0$ while the function $f_{-}^{(+)}(r)$ will always diverge when $r\rightarrow \infty$ and,
therefore, will not be a square-integratable function. Thus, the only solution allowed for the equations system (\ref{eqg1})-(\ref{eqg4}) results
\begin{equation}
f_{+}^{(+)}(r)=a_{2}e^{-\frac{1}{4}eBr^{2}}r^{\frac{L_{m}^{(+)}}{\alpha }},\, \text{with}\;\frac{L_{m}^{(+)}}{\alpha }\geqslant 0 ,\label{fnl}
\end{equation}
with $f_{-}^{(+)}(r)=g_{+}^{(+)}(r)=g_{-}^{(+)}(r)=0$. Solution (\ref{fnl})
satisfies equation (\ref{norm}) and constitutes a bound state solution for the case $
E=M$, i.e., an isolated solution to the Dirac equation (\ref{diracsc}) in the metric spacetime (\ref{metric}).
Proceeding in an analogous way, now we make $E=-M$ in Eqs. (\ref{Eqpsi}) and (\ref{Eqchi}). We find the system of equations
\begin{align}
&\frac{df_{+}^{(-)}(r)}{dr}-\frac{L_{m}^{(-)}}{r\alpha }f_{+}^{(-)}(r)+\frac{eBr}{2}f_{+}^{(-)}(r)=0, \label{f1}
\\
&\frac{df_{-}^{(-)}(r)}{dr}+\frac{L_{m+1}^{(-)}}{r\alpha }f_{-}^{(-)}(r)-\frac{eBr}{2}f_{-}^{(-)}(r)=0,\label{f2}\\
&\frac{dg_{+}^{(-)}(r)}{dr}-\frac{L_{m}^{(-)}}{r\alpha }g_{+}^{(-)}(r)+\frac{eBr}{2}g_{+}^{(-)}(r)
=2Mf_{-}^{(-)}(r), \label{g1}\\
&\frac{dg_{-}^{(-)}(r)}{dr}+\frac{L_{m+1}^{(-)}}{r\alpha }g_{-}^{(-)}(r)-\frac{eBr}{2}g_{-}^{(-)}(r)
=-2Mf_{+}^{(-)}(r). \label{g2}
\end{align}
with
\begin{eqnarray}
L_{m}^{(-)} &=&m-\phi -aM+\frac{s}{2}\left( 1-\alpha \right), \label{lpm1} \\
L_{m+1}^{(-)} &=&m+1-\phi -aM+\frac{s}{2}\left( 1-\alpha \right).\label{lpm2}
\end{eqnarray}
The solution of the coupled linear ordinary differential equations system (\ref{f1})-(\ref{g2}) is given by
\begin{align}
f_{+}^{(-)}(r)& =c_{1}e^{-\frac{1}{4}eBr^{2}}r^{\frac{L_{m}^{(-)}}{\alpha }},\label{sms1}
\\
f_{-}^{(-)}(r)& =d_{1}e^{\frac{1}{4}Ber^{2}}r^{-\frac{L_{m+1}^{(-)}}{\alpha }%
}, \label{sms2}\\
g_{+}^{(-)}(r)& =e^{-\frac{1}{4}Ber^{2}}r^{\frac{L_{m}^{(-)}}{\alpha}
}\left[-d_{1}M\left( -\frac{eB}{2}\right) ^{\Lambda _{c}}\Gamma
_{c}^{\left( -\right) }+d_{2}\right], \label{sms3}\\
g_{-}^{(-)}(r)& =e^{\frac{1}{4}eBr^{2}}r^{-\frac{L_{m+1}^{(-)}}{\alpha }}%
\left[ c_{1}M\left( \frac{eB}{2}\right) ^{-\Lambda _{d}}\Gamma _{d}^{\left(
-\right) }+c_{2}\right] \label{sms4},
\end{align}
with
\begin{align}
\Lambda _{c}& =\frac{1}{2\alpha }\left( L_{m}^{(-)}+L_{m+1}^{(-)}-\alpha
\right) , \\
\Lambda _{d}& =\frac{1}{2\alpha }\left( L_{m}^{(-)}+L_{m+1}^{(-)}+\alpha
\right) ,
\end{align}
where
\begin{align}
\Gamma _{c}^{\left( -\right) }& =\Gamma \left( -\Lambda _{c},-\frac{1}{2}
Ber^{2}\right) , \\
\Gamma _{d}^{\left( -\right) }& =\Gamma \left( \Lambda _{d},\frac{1}{2}
eBr^{2}\right) .
\end{align}
By making the same analysis of the solutions as we have made for the case $E=M$, i.e., analyzing the behavior of the functions for $r\rightarrow \pm\, \infty$, we find that the only solution that admits bound state is (\ref{sms3}). Thus, the solution for the case $E=-M$ satisfying the normalization condition (\ref{norm}) is given by
\begin{equation}
g_{+}^{(-)}(r) =d_{2}e^{-\frac{1}{4}Ber^{2}}r^{\frac{L_{m}^{(-)}}{\alpha }},\, \text{with}\;\frac{L_{m}^{(-)}}{\alpha }\geqslant 0,\label{slem}
\end{equation}
with $f_{+}^{(-)}(r) =
f_{-}^{(-)}(r)=g_{-}^{(-)}(r)=0$.
Note that the solutions (\ref{fnl}) and (\ref{slem}) are affected by rotation through Eqs. (\ref{lmm1}) and (\ref{lpm1}), respectively.
\section{Solution of the equation of motion to $E\neq \pm M$}
In this section, we solve the second order equation to $\psi$ that we find from the Eqs. (\ref{Eqpsi}) and (\ref{Eqchi}). The solution of this equation is different from that one calculated in the previous section and allow us to obtain an expression for the particle energies. By isolating $\psi_{2}$ in Eq. (\ref{Eqchi}) and replacing in Eq. (\ref{Eqpsi}), we are able to write the second order differential equation for $\psi_{1}$ as
\begin{align}
&\left(E^{2}-M^{2}\right) \psi_{1} +\partial _{r}^{2}\psi_{1} +\frac{1}{r}\partial
_{r}\psi_{1} +\frac{1}{\alpha r}\sigma ^{z}e\left( \partial _{r}A_{\varphi
}\right) \psi_{1} \notag \\
& +\frac{1}{\alpha ^{2}r^{2}}\left( \partial _{\varphi }-ieA_{\varphi }+i%
\frac{1-\alpha }{2}\sigma ^{z}+iaE\right) ^{2}\psi_{1} =0.\label{eav}
\end{align}
Using the decomposition of the fermion field (\ref{P1}) (ignoring the subscript ($+$)) together with Eqs. (\ref{vectorA}), (\ref{potential}) and (\ref{potentialv}), we obtain the radial equation for $f(r)$
\begin{equation}
\left( \frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{L^{2}}{\alpha
^{2}r^{2}}-\frac{e^{2}B^{2}r^{2}}{4}+k^{2}\right) f\left( r\right) =0, \label{re}
\end{equation}
where
\begin{equation}
k^{2}=E^{2}-M^{2}+\frac{eB}{\alpha}L+seB,
\end{equation}
\begin{equation}
L=m-\phi +\frac{s\left( 1-\alpha \right) }{2}+aE. \label{am}
\end{equation}
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{Fif2DEBa.pdf}
\includegraphics[width=\columnwidth]{Fif2DEBb.pdf}
\caption{Sketch of the energy levels $E_{n}^{(>)}$ (Eq. (\ref{Ep})) as a function of the magnetic field $B$ for different values of $n$. In panel (a), $s=+1$ and in panel (b), $s=-1$. The positive energies are represented by solid lines and the negative by dashed lines. We assume $e=1$ and $M=1.$}
\label{fig2DEb}
\end{figure}
Note that there are three other equivalent equations and there is no need to solve them here because their respective energies would also be equivalent. Equation (\ref{re}) is the confluent hypergeometric equation
and its solution is well known. Thus, it can be shown that the solution to $\psi_a$ is \begin{align}
&\psi_{a}\left( r,\varphi \right) =c_{nm}\left( \frac{eB}{2}\right) ^{
\frac{1}{2}\left(1+{\frac{\left\vert L\right\vert }{\alpha}}\right)}e^{im\varphi }{r}^{{%
\frac{\left\vert L\right\vert }{\alpha }}}e{^{-\frac{1}{4}\,eB{r}^{2}}}
\notag \\
& \times {{_{1}F_{1}}\left(\frac{1}{2} \left(1+{\frac{\left\vert L\right\vert }{\alpha}}\right)+{\frac{{k}^{2}}{2eB}},\,1+{\frac{\left\vert L\right\vert }{\alpha
}},\,\frac{1}{2}eB{r}^{2}\right) }, \label{pphi}
\end{align}
where ${_{1}F_{1}}\left(a,b,z \right)$ denotes the confluent hypergeometric function of the first kind or Kummer's function $M(a,b,z)$ and $c_{nm}$ the normalization constant.
It can be shown that the hypergeometric function ${_{1}F_{1}}\left(a,b,z \right)$ has a divergent behavior for large values of $z$. Because of this, bound state solutions for Eq. (\ref{pphi}) are only possible if we impose that this function becomes a polynomial of degree $n$. For this to be accomplished, we require that $1/2+{\left\vert L\right\vert /2\alpha }+{{k}^{2}/2Be=-n}$, where $n\in\mathbb{Z}^{*}$, with $\mathbb{Z}^{*}$ denoting the set of the nonnegative integers.
\begin{figure}[!h]
\centering
\includegraphics[width=\columnwidth]{Fig3DEnm.pdf}
\caption{Sketch of the energy (Eq. (\ref{Em})) as a function of $n$ and $m$ for $a=0.5$, $\alpha =0.5$, $B=1$, $e=1$, $M=1$, $s=1$ and $\phi =1$.}
\label{En3Dnm}
\end{figure}However, as we can see in Eq. (\ref{am}), the absolute value of the effective angular moment $L$ is defined in terms of the energy $E$. In this way, to obtain the energy eigenvalues from the above condition, we must consider $|L|>0$ and $|L|<0$, respectively, and then solve them for $E$. By making this, we get
\begin{align}
E_{n}^{(>)}& = \pm\sqrt{eB\left( 2n-s+1\right) +M^{2}}, \label{Ep}\\
E_{nm}^{(<)}& =-\frac{aeB}{\alpha } \pm \frac{1}{\alpha }\sqrt{
a^{2}e^{2}B^{2}+\alpha Q},\label{Em}
\end{align}
with the following requirement:
\begin{equation}
a^{2}e^{2}B^{2}+\alpha \,Q \geqslant 0, \label{cnd}
\end{equation}
where
\begin{equation*}
Q=\alpha eB\left( 2n-\frac{2}{\alpha }\left( m-\phi +\frac{s}{2}\right)
+1\right) +\alpha M^{2}.
\end{equation*}
\begin{figure}[!h]
\centering
\includegraphics[width=\columnwidth]{Fig3DEaa.pdf}
\caption{Sketch of the energy (Eq. (\ref{Em})) as a function of $\alpha$ and $a$ for $B = 4$, $M = 1$, $e=1$, $n=1$, $\phi=2$, $s = 1$ and $m = 1$.}
\label{En3Daa}
\end{figure}
In Eqs. (\ref{Ep}) and (\ref{Em}), the superscripts ($>,<$) refer to the energies calculated for $\left\vert L\right\vert >0$ and $\left\vert L\right\vert <0$, respectively.
For a given choice of the element of spin $s$, the energy $E_{n}^{(>)}$ depends only on the quantum number $n$ and the magnetic field $B$. For a given value of $n$, the energy increases when the magnetic field is increased. In Fig. \ref{fig2DEb}, we show the profile of $E_{n}^{(>)}$ for the first four states for $s=1$. The energy levels for $s=-1$ (Fig. \ref{fig2DEb}(b)) are slightly larger than the profile for the case $s=1$ (Fig. \ref{fig2DEb}(a)).
The energies (\ref{Ep}) and (\ref{Em}) denote the relativistic Landau levels in the present context. These energies can be directly compared with those obtained for the relativistic oscillator (Dirac oscillator) addressed in Ref. \cite{EPJC.2019.79.311}. Although that scenario is different from the one we are exploring here, there are similarities between the profiles of the energy levels in both models. For example, for $s=1$, the energy (48) of the Ref. \cite{EPJC.2019.79.311} depends only on the frequency of the oscillator and the quantum number $n$. In our case, by defining the cyclotron frequency $\omega _{c}=eB/M$, Eq. (\ref{Ep}) results
\begin{equation}
\tilde{E}_{nm}^{(>)}=\pm \sqrt{2nM\omega _{c}+M^{2}},
\end{equation}
which makes such a similarity clear. Since the energies (\ref{Em}) are the only ones that depend on all the physical parameters involved in the current problem, we study them in more detail. For a given set of fixed parameters, for example, $a=0.5$, $\alpha =0.5$, $B=1.0$, $e=1.0$, $M=1$, $s=1$ and $\phi=1$, we have the profile of the energy levels as a function of $n$ and $m$ (Fig. \ref{En3Dnm}). We can clearly see that $|{E}_{nm}^{(<)}|$ increases with $n$ and $m$. The green solid bars denote the discrete energy values for a given $m$ and $n$.
\begin{figure}[!h]
\centering
\includegraphics[width=\columnwidth]{Fig3DEbf.pdf}
\caption{Sketch of the energy (Eq. (\ref{Em})) as a function of $B$ and $\phi$ for $a=1$, $\alpha =0.5$, $e=1$, $m=1$, $M=1$, $n=1$ and $s=1$.}
\label{En3Dbf}
\end{figure}
On the other hand, when we investigate the behavior of (\ref{Em}) as a function of $\alpha$ and $a$ for specific values of the other parameters, we see that the negative spectrum changes more rapidly when compared with the positive one (Fig. \ref{En3Daa}). In the positive spectrum, both rotation and curvature lead to a linear change, except in the region with $\alpha < 0.3$ and arbitrary $a$ . In the negative spectrum, we see that the curvature effects are more predominant in the region where alpha has values smaller than 0.2. In this region, any variation in the rotation parameter implies in an abrupt change in the energy spectrum. Modifications in the energies with $\alpha < 0.3$ is an expected manifestation in our analyses. Its physical implication is inherent in the metric (\ref{metric}) and is an immediate consequence of the topological cone, which becomes more singular for smaller $\alpha$ values. To complete our analysis, we investigate the profile of the energy (\ref{Em}) as a function of magnetic field $B$ and the magnetic flux through the solenoid, $\phi$. Similarly to Fig. \ref{En3Daa}, by fixing the other parameters, we see that the energy of the anti-particle varies more rapidly when compared to the energy of the particle (Fig. \ref{En3Dbf}). Clearly, we observe that the energy of the particle varies very slowly throughout the region of flux and magnetic field . As a final commentary, we clarify that the cases discussed in Figs. \ref{En3Dnm}, \ref{En3Daa} and \ref{En3Dbf} can be investigated for other fixed parameter values. In this way, it can be shown that there are forbidden energies, depending on the values of the parameters considered. In general, this occurs when both the $\alpha$ parameter and the rotation parameter $a$ are smaller than $0.3$ and the other parameters assuming higher values than those we use here.
\section{Conclusions}
\label{sec:conclusions}
In the present manuscript, we have addressed the problem of the relativistic quantum motion of an electron in the spinning cosmic string background considering the presence of a uniform magnetic field and the Aharonov-Bohm potential. We have shown that this combination of potentials allows bound states configurations in the scenario of first-order solutions as well as in the case of second-order solutions of the Dirac equation. It is worth noting the role played by the two different terms in the vector potential. As already known in the literature, we have shown that the uniform field is responsible for a behavior analog to a harmonic oscillator, which leads to the relativistic Landau quantization while the Aharonov-Bohm flux contributes to the angular momentum of the particle. In the case of first order solutions, which were obtained by solving Eqs. (\ref{fnl}) and (\ref{slem}) for $E=+M$ and $E=-M$, respectively, the oscillator-like behavior provided by the uniform magnetic field guarantees the convergent first-order solutions and, consequently, the existence of bound states. The isolated solutions obtained (Eqs. (\ref{fnl}) and (\ref{slem})) are particular solutions of the Dirac equation (\ref{diracsc}).
We have also studied the more general problem by solving the second-order equation implied by equations (\ref{Eqpsi}) and (\ref{Eqchi}) for the upper component of the Dirac spinor for $E\neq \pm M$. Using appropriate solutions (Eq. (\ref{S1})) we have derived the radial equation and shown that its solution is given in terms of the Kummer functions from which we have extracted the expression for the energy levels of the particle (Eqs. (\ref{Ep}) and (\ref{Em})). For the field configuration considered, we have found that the effective angular momentum of the electron depends on its energy and the Aharonov-Bohm flux tube while the potential vector that generates the uniform field leads to a charged oscillator. This implies that such field superposition provides distinct effects on the motion of the particle. Additionally, in some cases, the rotation produces a combined effect with both the uniform magnetic field and the curvature (see Eq. (\ref{Em})). We have shown that the energy levels of the particle and antiparticle depend on the values of the physical parameters involved. In the case of energy (\ref{Em}), its validity is conditioned to Eq. (\ref{cnd}). Depending on the choice we make for the parameters, we can obtain forbidden energies. The sketches in Figs. \ref{En3Dnm}, \ref{En3Daa}, and \ref{En3Dbf} illustrate the profiles of the particle and antiparticle energies and show that they belong to the same spectrum. The effects of curvature and rotation are more evident when $\alpha < 0.3$, being the antiparticle energy the most affected. As a final comment, we would like to emphasize that the model studied in this article generalizes others found in the literature, such as those of Refs. \cite{oliveira2019topological,wang2018study} for the case including a superposition of external magnetic fields and the investigation of isolated solutions of the Dirac equation. Furthermore, we present a detailed discussion on the energy levels of the particle which, in general, is not found in the literature.
\section*{Acknowledgments}
We would like to thanks E.R.B. Mello (Universidade Federal da Para\'{i}ba, PB, Brazil) for his remarks and comments.
This work was partially supported by the Brazilian agencies CAPES, CNPq and
FAPEMA. EOS acknowledges CNPq Grants 427214/2016-5 and 307203/2019-0, and
FAPEMA Grants 01852/14 and 01202/16. This study was financed in part by the Coordena\c{c}\~{a}o de
Aperfei\c{c}oamento de Pessoal de N\'{\i}vel Superior - Brasil (CAPES) -
Finance Code 001. MMC acknowledges CAPES
Grant 88887.358036/2019-00.
\bibliographystyle{apsrev4-2}
\input{particle-string.bbl}
\end{document}
\section{Introduction}
\label{intro}
Symmetry is key-ingredient in the description of natural phenomena. The notion of symmetry is an essential feature in several areas of physics. In this context, the well-known Noether's theorem \cite{lanczos2012variational} establishes a connection between symmetry and conservation laws of relevant physical quantities. In quantum mechanics, we often use symmetry to obtain crucial results concerning angular momenta operators \cite{sakurai2014modern}.
Likewise, symmetry is important in the topic of quantum information \cite{PhysRevB.95.045111}.
Symmetry is also important in the framework of relativity \cite{westman2009coordinates}, and for this reason, it is indispensable in research areas such as particle physics \cite{gibson1980symmetry} and cosmology \cite{preskill1991cosmology}.
A pertinent question in the research areas cited above refers to think about what happens when some symmetry is broken in a given physical system.
When a system suffers a phase transition, for example, it can lose some type of symmetry.
Another example of symmetry-breaking occurs in condensed matter systems: by employing the Volterra process \cite{solyom2007fundamentals}, it is possible to break some symmetry of the system due to the creation of a topological defect, like disclinations and dislocations \cite{RevModPhys.80.61,puntigam1997volterra}, for instance.
Topological defects can emerge in a large number of physical systems covering themes such as liquid crystals \cite{lavrentovich2012defects}, graphene physics \cite{alden2013strain}, magnetism \cite{kou2011tunable} and cosmology \cite{durrer1999topological}.
Recent studies also have reported the importance of topological defects in Life Sciences \cite{kawaguchi2017topological,saw2017topological}.
On the point of view of cosmology, defects in the spacetime topology can be viewed as a possible consequence of the evolution of the early universe, which has suffered phase transitions due to the temperature decreasing and the process of expansion \cite{hindmarsh1995cosmic,vilenkin2000cosmic}.
In this contribution, we are particularly involved in studying the topological defect known as a cosmic string. A cosmic string is a linear defect, similar to a flux tube in type-II superconductors \cite{hindmarsh1995cosmic}. The spacetime around a cosmic string has a conical symmetry, identically to the case of a disclination \cite{moraes2000condensed}. The concept of a cosmic string it was introduced in the literature by Kibble \cite{kibble1976topology}. Since then, this topic has been investigated in diverse forms. An intriguing facet in this subject refers to the quantum mechanical description of a particle in a region of the spacetime containing a cosmic string. It can be done both in the scenario of relativistic and nonrelativistic quantum mechanics. For instance, the hydrogen atom in a spacetime of a cosmic string it was analyzed in Ref. \cite{PhysRevD.66.105011}. In Ref. \cite{marques2005exact}, it was considered the problem of a relativistic electron in the presence of both Coulomb and scalar potentials in the cosmic string spacetime. Results about vacuum polarization in a cosmic string spacetime were reported in Ref. \cite{PhysRevD.74.025017}.
Again, the cosmic string spacetime it was considered as a background to examine relativistic oscillators \cite{PhysRevA.84.032109}, quantum phases \cite{bakke2016relativistic}, and fermionic currents \cite{bezerra2016induced}.
A relevant issue in this context consists of taking into consideration the influence of electromagnetic fields in the quantum particle motion. Landau levels \cite{Book.2005.Griffiths} and the Aharonov-Bohm effect \cite{PR.1959.115.485,peshkin1989aharonov}, for instance, are essential ingredients in the investigation of quantum systems even in a flat spacetime. It can be explained because Landau levels are a quantum analog of classical cyclotron motion, while the Aharonov-Bohm effect reveals the significance of the vector potential in the quantum world.
Then, studying the contribution of magnetic fields to the quantum mechanical description of a system in spacetime having a topological defect is a natural development. Examples of studies dealing with Landau levels and the Aharonov-Bohm effect in the presence of topological defects can be accessed in Refs. \cite{de2001landau} and \cite{azevedo1998topological}, respectively. In particular, the inclusion of electromagnetic interactions in the case of a cosmic string background also has been considered. For instance, in Ref. \cite{medeiros2012relativistic}, it was analyzed the quantum dynamics of a charged particle in the presence of a magnetic field and scalar potential. In Ref. \cite{PhysRevD.93.043545}, various configurations of confined magnetic fields are examined and the existence of induced vacuum fermionic currents is investigated.
On the other hand, we can be interested in analyzing the behavior of a quantum system when noninertial effects turn on. These effects play a fundamental role in the description of systems governed by classical mechanics. Noninertial effects also can take place on quantum systems, providing novel theoretical predictions and feasible experimental developments.
For instance, the emergence of quantum phases in rotating systems, in analogy to the Aharonov-Bohm effect \cite{aharonov1973quantum,semon1982experimental} were investigated. In addition, a relation between the Hall effect and inertial forces it was established \cite{johnson2000inertial}.
Besides, if a given system it is put to rotate, it has consequences in diverse physical properties like spin transport \cite{PhysRevB.84.104410,chowdhury2014spin}, electronic structure \cite{garcia2017geometric}, and even can present magnetization due rotation, like in the Barnett effect \cite{PhysRevB.92.174424}. While a magnetic field produces a spin-field coupling, resulting in the anomalous
Zeeman effect \cite{nouredine2009quantum}, rotation produces an analog effect, due the spin-rotation coupling \cite{danner2020spin}.
Thus, rotation can contribute similarly to a magnetic field in the dynamics of a quantum system.
More, noninertial effects are an interesting issue in the situation in which spacetime contains topological defects. In this case, the noninertial effects and the presence of a topological defect can be included in the quantum mechanical description by employing the same tools: we can use a metric tensor to a spinning spacetime with a topological defect \cite{clement1990rotating}.
The spacetime of a spinning cosmic string has been considered as background for several problems involving quantum systems. For instance, the Schr\"{o}dinger equation in that spacetime it was solved in Ref. \cite{hassanabadi2015motion}. Bound states for neutral particles in a rotating frame of a cosmic string were analyzed in Ref. \cite{PhysRevD.82.084025}. Likewise, rotating effects on a Landau-Aharonov-Casher System in the spacetime of a cosmic string were investigated in Ref. \cite{bakke2015rotating}.
As we already have mentioned, in some cases rotation presents similarities within electromagnetic fields. This way, it is also an attractive question examining how the electromagnetic interactions affect the particle quantum motion of a rotating system in the presence of a topological defect. A recent example of studying dealing with both topological and noninertial effects in the presence of an Aharonov-Bohm potential can be accessed in Ref. \cite{oliveira2019topological}. In Ref. \cite{Wang_canadian_journal2017}, it was addressed the problem of a spinless relativistic particle subjected to a uniform magnetic field in the spinning cosmic string spacetime.
The Dirac oscillator in the spacetime of a cosmic string considering noninertial effects and the presence of the Aharonov–Casher effect it was analyzed in Ref. \cite{EPJC_oliveira2019topological}.
In Ref. \cite{wang2018study}, it was analyzed the problem of a charged half-spin particle depicted by the Dirac equation in the presence of a uniform magnetic field in the rotating cosmic string spacetime.
A meaningful aspect in this context consists of analyzing how different configurations of magnetic fields affect the quantum particle motion.
In this paper, we study the relativistic quantum mechanics of an electron in the presence of both a uniform magnetic field and Aharonov-Bohm potential in the spinning cosmic string spacetime. In other words, we solve the Dirac equation in this scenario and investigate how the rotation, curvature and external magnetic fields affect the wave functions and energies of the electron.
The manuscript is organized as follows. In Section II, we present some algebraic elements necessary to construct the field equations in curved spacetime and write the Dirac equation describing the quantum motion of the electron in the presence of external magnetic fields in the spinning cosmic string background. In Section III, we deal with first-order solutions and study the existence of isolated solutions for the particular case of a particle at rest. In Section IV, we take our attention to the case when the energy of the particle is different from its rest energy. We map the Dirac equation problem in curved space with minimal coupling into a Sturm-Liouville problem for the upper component of the Dirac spinor and, using an appropriate ansatz, we derive the radial equation. We solve the radial equation and find the wave functions and energies of the particle. We make a detailed discussion of the results and also comparisons with other studies in the literature. In Section V, we present our conclusions. In our work, we use natural units, $\hbar = c = G = 1$.
\section{Dirac equation in the spinning cosmic string spacetime}
In this section, we briefly present the main tools needed to construct the Dirac equation in the conical spacetime in the presence of noninertial effects. The first step consists in take a look at the metric tensor characterizing this geometry. Next, we will choose an appropriate tetrad basis and implement the fields configuration involved through the performing of a minimal substitution.
The spacetime induced by a rotating cosmic string is described by the metric
\begin{equation}
ds^{2}=\left( dt+ad\varphi \right) ^{2}-dr^{2}-\alpha ^{2}r^{2}d\varphi
^{2}-dz^{2}, \label{metric}
\end{equation}
where $-\infty <z<\infty $, $r\geqslant 0$ and $0\leqslant \varphi \leqslant
2\pi$. The parameter $\alpha $ is related to the linear mass density $\mu$
of the cosmic string through the relation $\alpha =1-4\mu$ and it runs in
the interval $(0,1]$. The quantity $a=4J$ is the rotation parameter, with $J$ representing the angular momentum of the spinning cosmic string. The relativistic quantum dynamics of a spin-$1/2$ particle interacting with external magnetic fields in the rotating cosmic string spacetime is governed by the Dirac equation
\begin{equation}
\left[ i\gamma ^{\mu }\left( x\right) \left( \partial _{\mu }+\Gamma _{\mu
}\left( x\right) +ieA_{\mu }(x)\right) -M\right] \Psi \left( x\right) =0,
\label{diracsc}
\end{equation}
where $M$ is the mass of the particle and $\gamma ^{\mu }\left( x\right)$ are the Dirac matrices in the
rotating cosmic string spacetime, which are defined in terms of the tetrad fields $e_{a}^{\mu }$ and Dirac matrices in the flat space $\gamma ^{a}$ in the following way:
\begin{equation}
\gamma ^{\mu }\left( x\right) =e_{a}^{\mu }\left( x\right) \gamma ^{a},
\label{gmatrices}
\end{equation}
where
\begin{equation}
\gamma ^{a}=\left( \gamma ^{0},\gamma ^{i}\right),\, \text{with}\, \gamma ^{0}=\left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right),\,\gamma ^{i}=\left(
\begin{array}{cc}
0 & \sigma ^{i} \\
-\sigma ^{i} & 0
\end{array}
\right),
\end{equation}
are the standard Dirac matrices
and $\sigma ^{i}=\left( \sigma ^{x},\sigma ^{y},\sigma ^{z}\right)$ are the usual Pauli matrices. The matrices (\ref{gmatrices}) satisfy the following relation:
\begin{equation}
\left\{ \gamma ^{\mu }\left( x\right) ,\gamma ^{\nu }\left( x\right)
\right\} =2g^{\mu \nu }\left( x\right).
\end{equation}
Also, in Eq. (\ref{diracsc}), $\Gamma _{\mu }\left( x\right)$ is the
spin affine connection given by
\begin{equation}
\Gamma _{\mu }\left( x\right) =\frac{1}{4}\gamma ^{a}\gamma ^{b}e_{a}^{\nu
}\left( x\right) \left[ \partial _{\mu }e_{b\nu }\left( x\right) -\Gamma
_{\mu \nu }^{\sigma }e_{b\sigma }\left( x\right) \right], \label{affine}
\end{equation}
where $\Gamma_{\mu \nu}^{\sigma}$ are the Christoffel symbols of the second kind and $e_{a}^{\mu }(x)$ is the tetrad field. The tetrad basis satisfies the relations
\begin{align}
&e_{\mu }^{a}\left( x\right) e_{\nu}^{b}\left(x\right) \eta_{ab}=g_{\mu \nu }\left( x\right),\\
&e_{\mu}^{a}\left(x\right) e_{\nu
}^{b}\left(x\right)=\delta_{a}^{b},\\
&e_{a}^{\mu}\left(x\right)
e_{\nu}^{a}\left(x\right) =\delta_{\mu}^{\nu}.
\end{align}
In Eq. (\ref{affine}), the Greek letters are used for tensor indices while the Latin letters are denoting Minkowski indices. We use the tetrad basis and its inverse defined as \cite{EPJC.2019.79.311}
\begin{eqnarray}
e_{\mu }^{a}\left( x\right) &=&\left(
\begin{array}{cccc}
1 & 0 & a & 0 \\
0 & \cos \varphi & -r\alpha \sin \varphi & 0 \\
0 & \sin \varphi & r\alpha \cos \varphi & 0 \\
0 & 0 & 0 & 1
\end{array}
\right) , \\
e_{a}^{\mu }\left( x\right) &=&\left(
\begin{array}{cccc}
1 & \frac{a\sin \varphi }{r\alpha } & -\frac{a\cos \varphi }{r\alpha } & 0
\\
0 & \cos \varphi & \sin \varphi & 0 \\
0 & -\frac{\sin \varphi }{r\alpha } & \frac{\cos \varphi }{r\alpha } & 0 \\
0 & 0 & 0 & 1
\end{array}
\right).\label{trdi}
\end{eqnarray}
For this choice, it can be shown that the non-vanishing affine connection is given by
\begin{equation}
\Gamma _{\mu }=\left( 0,0,\Gamma _{\varphi },0\right), \, \text{with} \; \Gamma _{\varphi }=\frac{i}{2}\left( 1-\alpha \right) \Sigma^{z},
\end{equation}
with
\begin{equation}
\Gamma _{\varphi }=\frac{i}{2}\left( 1-\alpha \right) \Sigma^{z},
\end{equation}
where
\begin{equation}
\Sigma ^{z}=\left(
\begin{array}{cc}
\sigma ^{z} & 0 \\
0 & \sigma ^{z}
\end{array}
\right), \;\; \sigma ^{z} =\left(
\begin{array}{cc}
1 & 0 \\
0 & -1%
\end{array}
\right).
\end{equation}
By using the tetrad basis (\ref{trdi}), the matrices (\ref{gmatrices}) can be written explicitly as
\begin{align}
&\gamma ^{t} =\gamma ^{0}-a\gamma ^{\varphi }, \\
&\gamma ^{r}=\left(
\begin{array}{cc}
0 & \sigma ^{r} \\
-\sigma ^{r} & 0
\end{array}
\right), \;\;\gamma ^{\varphi }=\left(
\begin{array}{cc}
0 & \sigma ^{\varphi } \\
-\sigma ^{\varphi } & 0
\end{array}
\right),
\end{align}
with
\begin{equation}
\sigma^{r} =\left(
\begin{array}{cc}
0 & e^{-i\varphi } \\
e^{+i\varphi } & 0
\end{array}
\right), \;\;\sigma ^{\varphi }=\frac{1}{r\alpha }\left(
\begin{array}{cc}
0 & -ie^{-i\varphi } \\
ie^{+i\varphi } & 0
\end{array}
\right)
\end{equation}
being the Pauli matrices in the curved spacetime.
Since we are first interested in studying the solutions of the Dirac equation in its present form (Eq. (\ref{diracsc})), we need to write the corresponding system of first order coupled differential equations. For this to be accomplished, let's assume the time-dependence of the wave functions together with the decomposition of the fermion field in the form
\begin{equation}
\Psi \left( r,\varphi \right)=e^{-iEt}\left(
\begin{array}{c}
\psi _{1}\left( r,\varphi \right) \\
\psi _{2}\left( r,\varphi \right)
\end{array}
\right),\label{S1}
\end{equation}
with
\begin{align}
&\psi _{1}\left( r,\varphi \right)=\left(
\begin{array}{c}
\psi _{a}\left( r,\varphi \right) \\
\psi _{b}\left( r,\varphi \right)
\end{array}
\right) =\left(
\begin{array}{c}
e^{im\varphi }f_{+}\left( r\right) \\
ie^{i\left( m+1\right) \varphi }f_{-}\left( r\right)
\end{array}
\right) ,\label{P1} \\
& \psi _{2}\left( r,\varphi \right) =\left(
\begin{array}{c}
\psi _{c}\left( r,\varphi \right) \\
\psi _{d}\left( r,\varphi \right)
\end{array}
\right) =\left(
\begin{array}{c}
e^{im\varphi }g_{+}\left( r\right) \\
ie^{i\left( m+1\right) \varphi }g_{-}\left( r\right)
\end{array}
\right). \label{P2}
\end{align}
The system we will analyze takes into account the particle is immersed in a region where there is a uniform magnetic field and also the potential due to a thin long solenoid along the z-axis. Having this field configuration in mind, we study
the physical implications due to noninertial effects and the Aharonov-Bohm potential on the relativistic Landau quantization. We also take into account the translational invariance of the system along the $z$-direction, which allows us to eliminate the third direction ($p_z=z=0$) and, consequently, we can consider only the planar motion \cite{PRL.1990.64.503,PRD.1994.50.7715,PRD.2012.85.041701,AoP.2013.339.510}. In this case, the
particle experiences a superposition of potential vectors written in the
Coulomb gauge as
\begin{equation}
\mathbf{A}=\left( 0,-\alpha rA_{\varphi }, 0\right) , \label{vectorA}
\end{equation}
with
\begin{align}
A_{\varphi } &=A_{\varphi ,1}+A_{\varphi ,2} , \label{potential} \\
A_{\varphi ,1} &=\frac{Br}{2},\;\;\; A_{\varphi ,2}=\frac{\phi }{\alpha r},\label{potentialv}
\end{align}
where $B$ is the magnetic field magnitude, $\phi =\Phi /\Phi _{0}$, $\Phi$ is the
magnetic flux and $\Phi _{0}=2\pi /e$ is the quantum of magnetic flux along the solenoid. This configuration also provides an superposition of magnetic fields in the
z-direction
\begin{equation}
B=B_{z,1}+B_{z,2},
\end{equation}
with
\begin{equation}
B_{1,z}=B, \;\;\; B_{z,2}=\phi \frac{\delta (r)}{\alpha r}, \label{fields}
\end{equation}
Note that the particle only interacts with the magnetic field due to the
potential vector $A_{\varphi ,1}$. Here, we are focused on studying the
electron motion only in the $r\neq 0$ region, so that we can neglect the
point interaction $B_{z,2}$ and, consequently, consider only regular wave
functions.
Using the results above, the Dirac equation (\ref{diracsc}) can be written as
\begin{align}
(E-&M) \,\psi_{1}+\sigma ^{r}i\partial _{r}\psi _{2}\notag\\ &+\sigma
^{\varphi }\left( i\partial _{\varphi }+eA_{\varphi }-aE-\frac{s}{2}\left(
1-\alpha \right) \right) \psi_{2}=0, \label{Eqpsi}\\
(E+&M) \,\psi_{2}+\sigma ^{i}i\partial _{r}\psi _{1}\notag \\ &+\sigma
^{\varphi }\left( i\partial _{\varphi }+eA_{\varphi }-aE-\frac{s}{2}\left(
1-\alpha \right) \right) \psi_{1}=0. \label{Eqchi}
\end{align}
At this point, we are ready to solve the equations (\ref{Eqpsi}) and (\ref{Eqchi}) by considering two distinct circumstances:
(i) Take our attention to isolated solutions of the first order Dirac equation by imposing the condition $E= \pm M$;
(ii) By imposing the condition $E\neq \pm M$, we looking for solutions of the second order Dirac equation.
We will show in the next two sections that there are bound state solutions for both cases and discuss their main physical properties. To distinguish each case in (i), in the next section we use the superscripts ($\pm$) to label the quantities corresponding to $E= \pm M$.
\section{Solution of the equation of motion to $E=\pm M$}
To study the existence of isolated solutions of the Dirac equation (\ref{diracsc}), we must set $E=\pm M$ in Eqs. (\ref{Eqpsi}) and (\ref{Eqchi}). In literature, such solutions are known to be excluded from the Sturm-Liouville problem. The search for isolated solutions of the Dirac equation has been performed in different physical contexts \cite
{EPJC.2019.79.596,EPL.2014.108.30003,AoP.2013.338.278,JPA.2007.40.263,PLA.2006.351.379}. The bound state solution must satisfy the normalization condition
\begin{equation}
\int_{0}^{\infty }\left( |\psi_{1}(r)|^{2}+|\psi_{2}(r)|^{2}\right) rdr=1.
\label{norm}
\end{equation}
By making $E=+M$ in Eqs. (\ref{Eqpsi}) and (\ref{Eqchi}) and using Eqs. (\ref{P1}) and (\ref{P2}), we get
\begin{align}
&\frac{dg_{+}^{(+)}(r)}{dr}-\frac{L_{m}^{(+)}}{r\alpha }g_{+}^{(+)}(r)+\frac{eBr}{2}g_{+}^{(+)}(r) =0,\label{eqg1}
\\
&\frac{dg_{-}^{(+)}(r)}{dr}+\frac{L_{m+1}^{(+)}}{r\alpha }g_{-}^{(+)}(r)-\frac{eBr}{2}g_{-}^{(+)}(r) =0,\label{eqg2}
\\
&\frac{df_{+}^{(+)}(r)}{dr}-\frac{L_{m}^{(+)}}{r\alpha }f_{+}^{(+)}(r)+\frac{eBr}{2}f_{+}^{(+)}(r)
=-2Mg_{-}^{(+)}(r), \label{eqg3}\\
&\frac{df_{-}^{(+)}(r)}{dr}+\frac{L_{m+1}^{(+)}}{r\alpha }f_{-}^{(+)}(r)-\frac{eBr}{2}f_{-}^{(+)}(r)
=2Mg_{+}^{(+)}(r),\label{eqg4}
\end{align}
with
\begin{align}
L_{m}^{(+)}& =m-\phi +aM+\frac{s}{2}\left( 1-\alpha \right), \label{lmm1}\\
L_{m+1}^{(+)}& =m+1-\phi +aM+\frac{s}{2}\left( 1-\alpha \right).\label{lmm2}
\end{align}
The solution of the coupled linear differential equations system (\ref{eqg1})-(\ref{eqg4}) is given by
\begin{align}
f_{+}^{(+)}(r)& =e^{-\frac{1}{4}eBr^{2}}r^{\frac{L_{m}^{(+)}}{\alpha }}\left[
a_{2}+a_{1}M\left( -\frac{eB}{2}\right) ^{\Omega _{a}}\Gamma _{a}^{(+)}\right],\label{sp1} \\
f_{-}^{(+)}(r)& =e^{\frac{1}{4}eBr^{2}}r^{-\frac{L_{m+1}^{(+)}}{\alpha }}\left[
b_{2}-b_{1}M\left( \frac{eB}{2}\right) ^{-\Omega _{b}}\Gamma _{b}^{(+)}\right] , \label{sp2} \\
g_{+}^{(+)}(r)& =b_{1}e^{-\frac{1}{4}Ber^{2}}r^{\frac{L_{m}^{(+)}}{\alpha }}, \label{sp3} \\
g_{-}^{(+)}(r)& =a_{1}e^{\frac{1}{4}Ber^{2}}r^{-\frac{L_{m+1}^{(+)}}{\alpha }},\label{sp4}
\end{align}
with
\begin{align}
\Omega _{a}& =\frac{1}{2\alpha }\left( L_{m}^{(+)}+L_{m+1}^{(+)}-\alpha
\right) , \\
\Omega _{b}& =\frac{1}{2\alpha }\left( L_{m}^{(+)}+L_{m+1}^{(+)}+\alpha
\right) ,
\end{align}
where
\begin{align}
\Gamma _{a}^{(+)}& =\Gamma \left( -\Omega _{a},-\frac{1}{2}eBr^{2}\right) , \\
\Gamma _{b}^{(+)}& =\Gamma \left( \Omega _{b},\frac{1}{2}eBr^{2}\right) .
\end{align}
are upper incomplete Gamma functions \cite{Book.1972.Abramowitz}, and
$a_{1}$, $a_{2}$, $b_{1}$ and $b_{2}$ are constants. Analyzing the solutions (\ref{sp1}) and (\ref{sp3}), we note that
$e^{-\frac{1}{4}eBr^{2}}$ dominates over $r^{\frac{L_{m}^{(+)}}{\alpha }}$
for any value of $L_{m}^{(+)}/\alpha $, in such way both solutions converge when $r\rightarrow 0$ and $r\rightarrow \infty $. This will not occur for the
function $e^{\frac{1}{4}eBr^{2}}$ in the solutions (\ref{sp2}) and (\ref{sp4}). Moreover, since the incomplete Gamma
functions $\Gamma _{a}^{(+)}$ and $\Gamma _{b}^{(+)}$ always diverge, then
the function $f_{+}^{(+)}(r)$ will only converges as $r\rightarrow 0$ if
$a_{1}=0$ while the function $f_{-}^{(+)}(r)$ will always diverge when $r\rightarrow \infty$ and,
therefore, will not be a square-integratable function. Thus, the only solution allowed for the equations system (\ref{eqg1})-(\ref{eqg4}) results
\begin{equation}
f_{+}^{(+)}(r)=a_{2}e^{-\frac{1}{4}eBr^{2}}r^{\frac{L_{m}^{(+)}}{\alpha }},\, \text{with}\;\frac{L_{m}^{(+)}}{\alpha }\geqslant 0 ,\label{fnl}
\end{equation}
with $f_{-}^{(+)}(r)=g_{+}^{(+)}(r)=g_{-}^{(+)}(r)=0$. Solution (\ref{fnl})
satisfies equation (\ref{norm}) and constitutes a bound state solution for the case $
E=M$, i.e., an isolated solution to the Dirac equation (\ref{diracsc}) in the metric spacetime (\ref{metric}).
Proceeding in an analogous way, now we make $E=-M$ in Eqs. (\ref{Eqpsi}) and (\ref{Eqchi}). We find the system of equations
\begin{align}
&\frac{df_{+}^{(-)}(r)}{dr}-\frac{L_{m}^{(-)}}{r\alpha }f_{+}^{(-)}(r)+\frac{eBr}{2}f_{+}^{(-)}(r)=0, \label{f1}
\\
&\frac{df_{-}^{(-)}(r)}{dr}+\frac{L_{m+1}^{(-)}}{r\alpha }f_{-}^{(-)}(r)-\frac{eBr}{2}f_{-}^{(-)}(r)=0,\label{f2}\\
&\frac{dg_{+}^{(-)}(r)}{dr}-\frac{L_{m}^{(-)}}{r\alpha }g_{+}^{(-)}(r)+\frac{eBr}{2}g_{+}^{(-)}(r)
=2Mf_{-}^{(-)}(r), \label{g1}\\
&\frac{dg_{-}^{(-)}(r)}{dr}+\frac{L_{m+1}^{(-)}}{r\alpha }g_{-}^{(-)}(r)-\frac{eBr}{2}g_{-}^{(-)}(r)
=-2Mf_{+}^{(-)}(r). \label{g2}
\end{align}
with
\begin{eqnarray}
L_{m}^{(-)} &=&m-\phi -aM+\frac{s}{2}\left( 1-\alpha \right), \label{lpm1} \\
L_{m+1}^{(-)} &=&m+1-\phi -aM+\frac{s}{2}\left( 1-\alpha \right).\label{lpm2}
\end{eqnarray}
The solution of the coupled linear ordinary differential equations system (\ref{f1})-(\ref{g2}) is given by
\begin{align}
f_{+}^{(-)}(r)& =c_{1}e^{-\frac{1}{4}eBr^{2}}r^{\frac{L_{m}^{(-)}}{\alpha }},\label{sms1}
\\
f_{-}^{(-)}(r)& =d_{1}e^{\frac{1}{4}Ber^{2}}r^{-\frac{L_{m+1}^{(-)}}{\alpha }%
}, \label{sms2}\\
g_{+}^{(-)}(r)& =e^{-\frac{1}{4}Ber^{2}}r^{\frac{L_{m}^{(-)}}{\alpha}
}\left[-d_{1}M\left( -\frac{eB}{2}\right) ^{\Lambda _{c}}\Gamma
_{c}^{\left( -\right) }+d_{2}\right], \label{sms3}\\
g_{-}^{(-)}(r)& =e^{\frac{1}{4}eBr^{2}}r^{-\frac{L_{m+1}^{(-)}}{\alpha }}%
\left[ c_{1}M\left( \frac{eB}{2}\right) ^{-\Lambda _{d}}\Gamma _{d}^{\left(
-\right) }+c_{2}\right] \label{sms4},
\end{align}
with
\begin{align}
\Lambda _{c}& =\frac{1}{2\alpha }\left( L_{m}^{(-)}+L_{m+1}^{(-)}-\alpha
\right) , \\
\Lambda _{d}& =\frac{1}{2\alpha }\left( L_{m}^{(-)}+L_{m+1}^{(-)}+\alpha
\right) ,
\end{align}
where
\begin{align}
\Gamma _{c}^{\left( -\right) }& =\Gamma \left( -\Lambda _{c},-\frac{1}{2}
Ber^{2}\right) , \\
\Gamma _{d}^{\left( -\right) }& =\Gamma \left( \Lambda _{d},\frac{1}{2}
eBr^{2}\right) .
\end{align}
By making the same analysis of the solutions as we have made for the case $E=M$, i.e., analyzing the behavior of the functions for $r\rightarrow \pm\, \infty$, we find that the only solution that admits bound state is (\ref{sms3}). Thus, the solution for the case $E=-M$ satisfying the normalization condition (\ref{norm}) is given by
\begin{equation}
g_{+}^{(-)}(r) =d_{2}e^{-\frac{1}{4}Ber^{2}}r^{\frac{L_{m}^{(-)}}{\alpha }},\, \text{with}\;\frac{L_{m}^{(-)}}{\alpha }\geqslant 0,\label{slem}
\end{equation}
with $f_{+}^{(-)}(r) =
f_{-}^{(-)}(r)=g_{-}^{(-)}(r)=0$.
Note that the solutions (\ref{fnl}) and (\ref{slem}) are affected by rotation through Eqs. (\ref{lmm1}) and (\ref{lpm1}), respectively.
\section{Solution of the equation of motion to $E\neq \pm M$}
In this section, we solve the second order equation to $\psi$ that we find from the Eqs. (\ref{Eqpsi}) and (\ref{Eqchi}). The solution of this equation is different from that one calculated in the previous section and allow us to obtain an expression for the particle energies. By isolating $\psi_{2}$ in Eq. (\ref{Eqchi}) and replacing in Eq. (\ref{Eqpsi}), we are able to write the second order differential equation for $\psi_{1}$ as
\begin{align}
&\left(E^{2}-M^{2}\right) \psi_{1} +\partial _{r}^{2}\psi_{1} +\frac{1}{r}\partial
_{r}\psi_{1} +\frac{1}{\alpha r}\sigma ^{z}e\left( \partial _{r}A_{\varphi
}\right) \psi_{1} \notag \\
& +\frac{1}{\alpha ^{2}r^{2}}\left( \partial _{\varphi }-ieA_{\varphi }+i%
\frac{1-\alpha }{2}\sigma ^{z}+iaE\right) ^{2}\psi_{1} =0.\label{eav}
\end{align}
Using the decomposition of the fermion field (\ref{P1}) (ignoring the subscript ($+$)) together with Eqs. (\ref{vectorA}), (\ref{potential}) and (\ref{potentialv}), we obtain the radial equation for $f(r)$
\begin{equation}
\left( \frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{L^{2}}{\alpha
^{2}r^{2}}-\frac{e^{2}B^{2}r^{2}}{4}+k^{2}\right) f\left( r\right) =0, \label{re}
\end{equation}
where
\begin{equation}
k^{2}=E^{2}-M^{2}+\frac{eB}{\alpha}L+seB,
\end{equation}
\begin{equation}
L=m-\phi +\frac{s\left( 1-\alpha \right) }{2}+aE. \label{am}
\end{equation}
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{Fif2DEBa.pdf}
\includegraphics[width=\columnwidth]{Fif2DEBb.pdf}
\caption{Sketch of the energy levels $E_{n}^{(>)}$ (Eq. (\ref{Ep})) as a function of the magnetic field $B$ for different values of $n$. In panel (a), $s=+1$ and in panel (b), $s=-1$. The positive energies are represented by solid lines and the negative by dashed lines. We assume $e=1$ and $M=1.$}
\label{fig2DEb}
\end{figure}
Note that there are three other equivalent equations and there is no need to solve them here because their respective energies would also be equivalent. Equation (\ref{re}) is the confluent hypergeometric equation
and its solution is well known. Thus, it can be shown that the solution to $\psi_a$ is \begin{align}
&\psi_{a}\left( r,\varphi \right) =c_{nm}\left( \frac{eB}{2}\right) ^{
\frac{1}{2}\left(1+{\frac{\left\vert L\right\vert }{\alpha}}\right)}e^{im\varphi }{r}^{{%
\frac{\left\vert L\right\vert }{\alpha }}}e{^{-\frac{1}{4}\,eB{r}^{2}}}
\notag \\
& \times {{_{1}F_{1}}\left(\frac{1}{2} \left(1+{\frac{\left\vert L\right\vert }{\alpha}}\right)+{\frac{{k}^{2}}{2eB}},\,1+{\frac{\left\vert L\right\vert }{\alpha
}},\,\frac{1}{2}eB{r}^{2}\right) }, \label{pphi}
\end{align}
where ${_{1}F_{1}}\left(a,b,z \right)$ denotes the confluent hypergeometric function of the first kind or Kummer's function $M(a,b,z)$ and $c_{nm}$ the normalization constant.
It can be shown that the hypergeometric function ${_{1}F_{1}}\left(a,b,z \right)$ has a divergent behavior for large values of $z$. Because of this, bound state solutions for Eq. (\ref{pphi}) are only possible if we impose that this function becomes a polynomial of degree $n$. For this to be accomplished, we require that $1/2+{\left\vert L\right\vert /2\alpha }+{{k}^{2}/2Be=-n}$, where $n\in\mathbb{Z}^{*}$, with $\mathbb{Z}^{*}$ denoting the set of the nonnegative integers.
\begin{figure}[!h]
\centering
\includegraphics[width=\columnwidth]{Fig3DEnm.pdf}
\caption{Sketch of the energy (Eq. (\ref{Em})) as a function of $n$ and $m$ for $a=0.5$, $\alpha =0.5$, $B=1$, $e=1$, $M=1$, $s=1$ and $\phi =1$.}
\label{En3Dnm}
\end{figure}However, as we can see in Eq. (\ref{am}), the absolute value of the effective angular moment $L$ is defined in terms of the energy $E$. In this way, to obtain the energy eigenvalues from the above condition, we must consider $|L|>0$ and $|L|<0$, respectively, and then solve them for $E$. By making this, we get
\begin{align}
E_{n}^{(>)}& = \pm\sqrt{eB\left( 2n-s+1\right) +M^{2}}, \label{Ep}\\
E_{nm}^{(<)}& =-\frac{aeB}{\alpha } \pm \frac{1}{\alpha }\sqrt{
a^{2}e^{2}B^{2}+\alpha Q},\label{Em}
\end{align}
with the following requirement:
\begin{equation}
a^{2}e^{2}B^{2}+\alpha \,Q \geqslant 0, \label{cnd}
\end{equation}
where
\begin{equation*}
Q=\alpha eB\left( 2n-\frac{2}{\alpha }\left( m-\phi +\frac{s}{2}\right)
+1\right) +\alpha M^{2}.
\end{equation*}
\begin{figure}[!h]
\centering
\includegraphics[width=\columnwidth]{Fig3DEaa.pdf}
\caption{Sketch of the energy (Eq. (\ref{Em})) as a function of $\alpha$ and $a$ for $B = 4$, $M = 1$, $e=1$, $n=1$, $\phi=2$, $s = 1$ and $m = 1$.}
\label{En3Daa}
\end{figure}
In Eqs. (\ref{Ep}) and (\ref{Em}), the superscripts ($>,<$) refer to the energies calculated for $\left\vert L\right\vert >0$ and $\left\vert L\right\vert <0$, respectively.
For a given choice of the element of spin $s$, the energy $E_{n}^{(>)}$ depends only on the quantum number $n$ and the magnetic field $B$. For a given value of $n$, the energy increases when the magnetic field is increased. In Fig. \ref{fig2DEb}, we show the profile of $E_{n}^{(>)}$ for the first four states for $s=1$. The energy levels for $s=-1$ (Fig. \ref{fig2DEb}(b)) are slightly larger than the profile for the case $s=1$ (Fig. \ref{fig2DEb}(a)).
The energies (\ref{Ep}) and (\ref{Em}) denote the relativistic Landau levels in the present context. These energies can be directly compared with those obtained for the relativistic oscillator (Dirac oscillator) addressed in Ref. \cite{EPJC.2019.79.311}. Although that scenario is different from the one we are exploring here, there are similarities between the profiles of the energy levels in both models. For example, for $s=1$, the energy (48) of the Ref. \cite{EPJC.2019.79.311} depends only on the frequency of the oscillator and the quantum number $n$. In our case, by defining the cyclotron frequency $\omega _{c}=eB/M$, Eq. (\ref{Ep}) results
\begin{equation}
\tilde{E}_{nm}^{(>)}=\pm \sqrt{2nM\omega _{c}+M^{2}},
\end{equation}
which makes such a similarity clear. Since the energies (\ref{Em}) are the only ones that depend on all the physical parameters involved in the current problem, we study them in more detail. For a given set of fixed parameters, for example, $a=0.5$, $\alpha =0.5$, $B=1.0$, $e=1.0$, $M=1$, $s=1$ and $\phi=1$, we have the profile of the energy levels as a function of $n$ and $m$ (Fig. \ref{En3Dnm}). We can clearly see that $|{E}_{nm}^{(<)}|$ increases with $n$ and $m$. The green solid bars denote the discrete energy values for a given $m$ and $n$.
\begin{figure}[!h]
\centering
\includegraphics[width=\columnwidth]{Fig3DEbf.pdf}
\caption{Sketch of the energy (Eq. (\ref{Em})) as a function of $B$ and $\phi$ for $a=1$, $\alpha =0.5$, $e=1$, $m=1$, $M=1$, $n=1$ and $s=1$.}
\label{En3Dbf}
\end{figure}
On the other hand, when we investigate the behavior of (\ref{Em}) as a function of $\alpha$ and $a$ for specific values of the other parameters, we see that the negative spectrum changes more rapidly when compared with the positive one (Fig. \ref{En3Daa}). In the positive spectrum, both rotation and curvature lead to a linear change, except in the region with $\alpha < 0.3$ and arbitrary $a$ . In the negative spectrum, we see that the curvature effects are more predominant in the region where alpha has values smaller than 0.2. In this region, any variation in the rotation parameter implies in an abrupt change in the energy spectrum. Modifications in the energies with $\alpha < 0.3$ is an expected manifestation in our analyses. Its physical implication is inherent in the metric (\ref{metric}) and is an immediate consequence of the topological cone, which becomes more singular for smaller $\alpha$ values. To complete our analysis, we investigate the profile of the energy (\ref{Em}) as a function of magnetic field $B$ and the magnetic flux through the solenoid, $\phi$. Similarly to Fig. \ref{En3Daa}, by fixing the other parameters, we see that the energy of the anti-particle varies more rapidly when compared to the energy of the particle (Fig. \ref{En3Dbf}). Clearly, we observe that the energy of the particle varies very slowly throughout the region of flux and magnetic field . As a final commentary, we clarify that the cases discussed in Figs. \ref{En3Dnm}, \ref{En3Daa} and \ref{En3Dbf} can be investigated for other fixed parameter values. In this way, it can be shown that there are forbidden energies, depending on the values of the parameters considered. In general, this occurs when both the $\alpha$ parameter and the rotation parameter $a$ are smaller than $0.3$ and the other parameters assuming higher values than those we use here.
\section{Conclusions}
\label{sec:conclusions}
In the present manuscript, we have addressed the problem of the relativistic quantum motion of an electron in the spinning cosmic string background considering the presence of a uniform magnetic field and the Aharonov-Bohm potential. We have shown that this combination of potentials allows bound states configurations in the scenario of first-order solutions as well as in the case of second-order solutions of the Dirac equation. It is worth noting the role played by the two different terms in the vector potential. As already known in the literature, we have shown that the uniform field is responsible for a behavior analog to a harmonic oscillator, which leads to the relativistic Landau quantization while the Aharonov-Bohm flux contributes to the angular momentum of the particle. In the case of first order solutions, which were obtained by solving Eqs. (\ref{fnl}) and (\ref{slem}) for $E=+M$ and $E=-M$, respectively, the oscillator-like behavior provided by the uniform magnetic field guarantees the convergent first-order solutions and, consequently, the existence of bound states. The isolated solutions obtained (Eqs. (\ref{fnl}) and (\ref{slem})) are particular solutions of the Dirac equation (\ref{diracsc}).
We have also studied the more general problem by solving the second-order equation implied by equations (\ref{Eqpsi}) and (\ref{Eqchi}) for the upper component of the Dirac spinor for $E\neq \pm M$. Using appropriate solutions (Eq. (\ref{S1})) we have derived the radial equation and shown that its solution is given in terms of the Kummer functions from which we have extracted the expression for the energy levels of the particle (Eqs. (\ref{Ep}) and (\ref{Em})). For the field configuration considered, we have found that the effective angular momentum of the electron depends on its energy and the Aharonov-Bohm flux tube while the potential vector that generates the uniform field leads to a charged oscillator. This implies that such field superposition provides distinct effects on the motion of the particle. Additionally, in some cases, the rotation produces a combined effect with both the uniform magnetic field and the curvature (see Eq. (\ref{Em})). We have shown that the energy levels of the particle and antiparticle depend on the values of the physical parameters involved. In the case of energy (\ref{Em}), its validity is conditioned to Eq. (\ref{cnd}). Depending on the choice we make for the parameters, we can obtain forbidden energies. The sketches in Figs. \ref{En3Dnm}, \ref{En3Daa}, and \ref{En3Dbf} illustrate the profiles of the particle and antiparticle energies and show that they belong to the same spectrum. The effects of curvature and rotation are more evident when $\alpha < 0.3$, being the antiparticle energy the most affected. As a final comment, we would like to emphasize that the model studied in this article generalizes others found in the literature, such as those of Refs. \cite{oliveira2019topological,wang2018study} for the case including a superposition of external magnetic fields and the investigation of isolated solutions of the Dirac equation. Furthermore, we present a detailed discussion on the energy levels of the particle which, in general, is not found in the literature.
\section*{Acknowledgments}
We would like to thanks E.R.B. Mello (Universidade Federal da Para\'{i}ba, PB, Brazil) for his remarks and comments.
This work was partially supported by the Brazilian agencies CAPES, CNPq and
FAPEMA. EOS acknowledges CNPq Grants 427214/2016-5 and 307203/2019-0, and
FAPEMA Grants 01852/14 and 01202/16. This study was financed in part by the Coordena\c{c}\~{a}o de
Aperfei\c{c}oamento de Pessoal de N\'{\i}vel Superior - Brasil (CAPES) -
Finance Code 001. MMC acknowledges CAPES
Grant 88887.358036/2019-00.
\bibliographystyle{apsrev4-2}
\input{particle-string.bbl}
\end{document}
|
2,877,628,090,149 | arxiv | \section{Introduction}
Let $(M,g)$ be a compact Riemannian manifold. The \emph{conformal class}
of $g$ consists of all metrics $\tilde g = e^{2u}g$ for any smooth function $u$. A central theme in conformal geometry is the study of properties that are common to all metrics in the same conformal class, and the understanding and classification of all the conformal classes. For this purpose it is often useful to be able to single out a unique representative in each conformal class by imposing some geometric condition. This usually leads to a conformally covariant geometric equation for the conformal factor $e^{2u}$. Such equations have attracted much interest in the literature in the past half-century.
In dimension 2, the natural condition to impose is constant Gauss curvature. The Poincar\'e Uniformization Theorem states that this is always possible: \emph{every compact Riemannian surface is conformal to one with constant Gauss curvature}. The conformally covariant operator in this case is the Laplacian-Beltrami operator, given in local coordinates by:
\[
\Delta_g = \frac1{\sqrt{\operatorname{det} g}} \, \partial_i\left( \sqrt{\operatorname{det} g}\, g^{ij} \partial_j\right),
\]
and the equation for constant curvature is:
\[
- \Delta_g u + \kappa_g = \kappa_{\tilde g} e^{2u}
\]
where $\kappa_g$ is the Gauss curvature of $g$, and $\kappa_{\tilde g}$ is constant. Using the Gauss-Bonnet theorem, we see that the sign of $\kappa_{\tilde g}$ is determined by $\chi_M$ the Euler characteristic of $M$:
\[
2\pi\chi_M = \int_M \kappa_g\, dA_g = \kappa_{\tilde g} \int_M e^{2u}\, dA_g,
\]
where $dA_g$ is the area element of $g$. Although this result was originally proved by Poincar\'e using non-PDE methods, there is now a PDE proof, see~\cite{christodoulou} and~\cite{weinstein}. Furthermore, the operator $\Delta_g$ is conformally convariant $\Delta_{\tilde g} = e^{-2u} \Delta_g$.
For compact Riemannian manifolds of dimension $n\geq3$, a natural generalization is to impose constant scalar curvature. This leads to the Yamabe problem: \emph{given a compact Riemannian manifold $(M,g)$ of dimension at least $3$, find a metric conformal to $g$ with constant scalar curvature}. This was also eventually answered in the affirmative, see~\cite{yamabe,trudinger,aubiny,schoen}. The corresponding operator is now the conformal Laplacian $L_g=\Delta_g-c_nR_g$, $c_n=(n-2)/4(n-1)$, and the equation for constant scalar curvature is the Yamabe equation:
\[
L_g \phi = \epsilon \phi^{(n+2)/(n-2)}
\]
where $\phi^{4/(n-2)}=e^{2u}$, and $\epsilon$ is $1$, $0$, or $-1$. The operator $L_g$ also has a conformal covariant property:
\[
L_{\tilde g} \phi = \phi^{-(n+2)/(n-2)} L_g \phi,
\]
where $\tilde g = \phi^{4/(n-2)}g$.
In $4$-d, another problem analogous to the $2$-d case arises from imposing the condition of constant $Q$-curvature:
\[
Q_g = -\frac 1{12}(\Delta_g R_g - R_g^2 + 3|\operatorname{Ric}_g|^2).
\]
The natural question is the same: \emph{given a $4$-d compact Riemannian manifold $(M,g)$, is there a metric $\tilde g = e^{2u} g$ in the conformal class of $g$ with constant $Q$-curvature?} The $Q$-curvature of the metric $\tilde g$ is given by:
\[
P_g u + 2Q_g = 2Q_{\tilde g}e^{4u},
\]
where $P_g$ is the Paneitz operator:
\[
P_g u = \Delta_g^2 u + \div_g\left( \left(\frac23 R_gg-2\operatorname{Ric}_g\right) \nabla u
\right).
\]
Integrating with respect to the volume element $dV_g$, it is easy to see that the quantity:
\[
k_P = \int_M Q_g\, dV_g
\]
is a conformal invariant, i.e., it is constant in the conformal class of $g$. Furthermore, we also have a Gauss-Bonnet formula:
\[
\int_M \left( Q_g + \frac18 \norm{W_g}^2 \right) dV_g = 4\pi^2 \chi_M,
\]
where $W_g$ is the Weyl tensor of $g$ given in local coordinates as:
\[
W_{ijkl} = R_{ijkl} - \frac2{n-2}(g_{i[k} R_{l]j} - g_{j[k} R_{l]i})
+ \frac{2}{(n-1)(n-2)}R g_{a[k}g_{l]j}.
\]
We note that $W_g$ is pointwise conformally invariant $W_{\tilde g}=W_g$, and the operator $P_g$ is conformally covariant:
\begin{equation} \label{jan22e2}
P_{\tilde g} f = e^{-4u} P_g f.
\end{equation}
We use \eqref{P} to denote the assumption:
\[ \label{P}
\operatorname{Ker}(P_g)=\{\text{constants}\}.
\tag{$P$}
\]
We remark that \eqref{P} is often satisfied. For example, Gursky
in~\cite{gursky} proved that if $(M,g)$ has
non-negative Yamabe invariant $Y_g\geq0$, and satisfies $k_P\ge
0$, then \eqref{P} holds and $P_g\ge 0$; see
also~\cite{gurskyviaclovsky} where the assumptions are weakened to
$Y_g\geq0$ and $k_P+Y_g^2/6>0$.
Chang and Yang proved in~\cite{changyang1} that if $k_P < 8\pi^2$, $P_g\ge 0$ and \eqref{P} holds, then there is a conformal metric $\tilde g$ whose $Q$-curvature is constant. In~\cite{djadli2}, Djadli and Malchiodi extended this existence result assuming only that \eqref{P} holds and $k_P\neq 8\pi^2N$ for any positive integer $N$. An essential ingredient in this existence result is an a priori bound: if $k_P \neq 8\pi^2N$ for any positive integer $N$, then any sequence of solutions of the prescribed $Q$-curvature equation is uniformly bounded. In fact, this a priori estimate can be extended to the following more general equation in the same class:
\begin{equation}
\label{jan22e1}
P_g u + 2b = 2h e^{4u},
\end{equation}
where $b$ is a smooth function. Note that if $b=Q_g$, then $h$ is the $Q$-curvature of the conformal metric $e^{2u}g$. Assuming $h_k\to h_0$, $h_k\geq c_0>0$, and $b_k\to\ b_0$, Druet and Robert in~\cite{druet1} showed that
any sequence of solutions $\{u_k\}$ of~\eqref{jan22e1} with $h=h_k$ and $b=b_k$ is uniformly bounded, provided $\int_Mb_0\neq 8\pi^2N$, see also Malchiodi~\cite{mal1}, .
However, bubbling can occur when $\int_M b_0\, dV_g = 8\pi^2 N$ for some positive integer $N$. A precise understanding of this bubbling phenomenon is required if progress is to be made on the existence problem. The study of the blow-up profile and other blow-up phenomena for the Paneitz operator and other $4$-th order elliptic equations has attracted much interest recently; see for example~\cite{adimurthi,brendle,changqingyang,djadli1,fefferman,gurskyviaclovsky2,lililiu,malchiodi,malchiodi2,ndiaye,qing, struwe,wei,wei2}.
Let $\{u_k\}$ be a sequence of solutions of~\eqref{jan22e1} with $h=h_k$, and $b=b_k$. We say that this is a \emph{bubbling sequence} if $\sup |u_k| \to\infty$. In~\cite{druet1}, Druet and Robert studied bubbling sequences of solutions of~\eqref{jan22e1} and obtained
some asymptotic estimates on the behavior near the blow-up points.
We will throughout make the following assumptions on the coefficients $b_k$ and $h_k$:
\[ \label{b}
\| b_k-b_0 \|_{C^1(M)} \to 0, \qquad
\|h_k - h_0\|_{C^2(M)} \to 0, \qquad h_k\geq c_0.
\tag{$b,h$}
\]
It follows immediately that $\|b_k\|_{C^1(M)}\leq C_0$, and $\|h_k\|_{C^2(M)}\leq C_0$ for some constant $C_0$ independent of $k$.
We let $G$ denote the Green's function for the Paneitz operator:
\begin{equation} \label{green}
f(\xi)-\bar f_g=\int_M G(\xi,\eta)\, P_gf(\eta)\, dV_g(\eta), \qquad \int_M G(\xi,\eta)\, dV_g(\eta) = 0,
\end{equation}
where $\bar f_g=\operatorname{Vol}_g(M)^{-1} \int_M f\, dV_g$ is the mean value of $f$. The asymptotics of this Green's function are studied in the Appendix. Now, for $k=0,\dots$, let
\begin{equation} \label{mar26e1}
\phi_k(\xi) = 2 \int_M G(\xi,\eta)\, b_k(\eta)\, dV_g(\eta).
\end{equation}
Since $\{u_k\}$ is a bubbling sequence, it follows immediately that $\int_M b_0\, dV_g = 8 N\pi^2$ for some positive integer $N$. Druet and Robert proved that passing to a subsequence, there is a finite set $S=\{p_1,..,p_N\}$ such that:
\[
u_k - \bar u_k \to 16\pi^2 \sum_i^N G(p_i,\cdot) - \phi_0 \qquad
\text{in $C^4_{loc}(M\setminus S)$},
\]
Let $\beta$ be the regular part of the Green's function:
\begin{equation} \label{beta}
G(\xi,\eta)= - \frac{1}{8\pi^2}\, \chi(r)\,
\log d_g(\xi,\eta) + \beta(\xi,\eta).
\end{equation}
Here $\chi$ is a cut off function supported in a small
neighborhood of $\xi$, and $r=d_g(\xi,\eta)$. They also proved
that for $i=1,\dots,N$:
\[
64\pi^2 \nabla_{2} \beta(p_i,p_i) + 64\pi^2 \sum_{j\neq i}\nabla_{1} G(p_i,p_j)
- 4\nabla \phi_0(p_i) = - \frac{\nabla h(p_i)}{h(p_i)},
\]
where $h$ is the limit of $h_k$ as $k\to \infty$, and $\nabla_1$, $\nabla_2$ denote the derivatives with respect to the first and second variables respectively. In this article we will continue this line of investigation and derive more precise asymptotic estimates for the behavior of such solutions. We define the \emph{standard bubble} at $p$:
\[
U_{p,\varepsilon,H}(\xi) = - \log\left( \varepsilon + \frac{\sqrt{H}\, d_g(p,\xi)^2}{4\sqrt3\,\varepsilon} \right).
\]
We will also adopt the following notation. For $k$ large enough, there are $N$ points $\{q_{ik}\}$ such that $q_{ik}\to p_i$ and $u_k(q_{ik})\to\infty$. Let $H_{ik}=h_{k}(q_{ik})$, $\varepsilon_{ik}=e^{-u_k(q_{ik})}$, and $U_{ik} = U_{q_{ik},\varepsilon_{ik},H_{ik}}$.
\begin{thm}
Let $\{u_k\}$ be a bubbling sequence of solutions on $M$. Then passing to a subsequence, there is a constant $\delta>0$ such that for any fixed $\tau\in (0,1)$, there exists a constant $C_1 = C_1(N,g,c_0,C_0,\tau)$ such that:
\begin{equation} \label{MainEst}
\left| u_k(\xi) - U_{ik}(\xi) \right|
\le C_1 d_{g}(q_{ik},\xi)^{\tau},
\end{equation}
in $B(q_{ik},\delta)$, and such that for $i=1,\dots, N$ we have:
\begin{multline} \label{vrate}
\left| 64\pi^2\nabla_2\beta(q_{ik},q_{ik}) + 64\pi^2\sum_{j\neq i}\nabla_1 G(q_{ik},q_{jk})
\right. \\ \left.
- 4\nabla \phi_k(q_{ik})+\frac{\nabla h_k(q_{ik})}{h_k(q_{ik})}\right|
\le C_1 e^{-\tau u_k(q_{ik})/2}.
\end{multline}
\end{thm}
Our approach is motivated by Lin and Wei's work~\cite{linwei1}
from which one can easily derive an $O(1)$ bound
in~\eqref{MainEst} (i.e. $|u_k-U_{ik}|\le
C$ near $p_i$) provided $(M,g)$ is locally conformally flat; see also
~\cite{xu} which uses a completely different approach. Our result
removes the hypothesis of local conformal flatness and also
improves the estimate near the blow-up points. We hope our
approach can be fine-tuned to yield better yet estimates as
required to handle the existence question posed above.
A major difficulty when trying to prove a priori estimates for solutions of fourth order elliptic equations is the lack of a maximum principle. In order to remedy this, Lin and Wei devised a strategy based on the Pohozaev identity. We adapt this approach to the case in which the manifold is not necessarily locally conformally flat, making use of \emph{conformal normal coordinates}. These are normal coordinates for a metric $\hat g$ in the conformal class of $g$ for which $\operatorname{det}(\hat g)=1$. The existence of such a metric is proved in~\cite{cao}. Although, we used this result for the sake of simplicity, our proof only relies on the weaker concept already introduced in the solution of the Yamabe problem where one only requires $\operatorname{det}(\hat g)=1$ to hold to high enough order in the distance from the center of the ball under consideration, see~\cite{leeparker}.
We now briefly sketch the outline of the paper and the proof of our Theorem.
In Section~\ref{sec:02}, we prove the $O(1)$ estimate. We use the Green's representation formula, together with rough estimates from~\cite{druet1}, to write long range asymptotic formulas for the rescaled solution $v_k$ and its derivatives in terms of the concentration of energy $\alpha_k$ near the singular point, i.e. within a carefully chosen radius $l_k = -\varepsilon_k\log\varepsilon_k$, where $\varepsilon_k$ is related to the maximum of $u_k$, see~\eqref{alpha}. These are then substituted into an asymptotic Pohozaev identity, and after estimating the higher order terms, we obtain an asymptotic formula for the energy $\alpha_k \approx 16\pi^2$. When substituted back into the asymptotic formula for $v_k$, this yields a long range $O(1)$ bound. Finally, we use standard estimates in the interior, and then these long range and interior estimates in conjunction with the maximum principle in the mid-range.
In Section~\ref{sec:03}, we prove~\eqref{MainEst} by contradiction. We divide the argument into two cases, depending upon whether an appropriately weighted supremum runs off to infinity or remains in a bounded region along a subsequence. In the first case, we use the Green's representation formula and a comparison between the geometric and Euclidean distances to reach a contradiction. In the second case, we show that the difference between the appropriately rescaled solutions and the standard bubble converges to a solution of the linearized equation which we can then show vanishes thanks to a lemma of Lin and Wei from~\cite{linwei1}, again leading to a contradiction.
Finally, in Section~\ref{sec:04}, we use our estimate~\eqref{MainEst} in the Pohozaev Identity over a ball of radius $\varepsilon_k^{-1/2}$ to obtain a Euclidean version of the vanishing rate. We then translate this result into the original metric $g$ and prove~\eqref{vrate}.
The Appendix deals with delicate estimates for the Green's function, an asymptotic comparison between the geodesic distance and the Euclidean distance in conformal normal coordinates, some well known curvature and metric derivatives computations in conformal normal coordinates, and a proof of the asymptotic Pohozaev identity.
\section{The $O(1)$ estimate}
\label{sec:02}
In this section we derive the $O(1)$ estimate, i.e., we show that
\[
|u_k(\xi)-U_{ik}(\xi)|\le C, \qquad \text{for $\xi\in B(q_{ik},\delta)$.}
\]
This estimate has been established by Lin-Wei \cite {linwei1} for
locally conformally flat manifolds; see also ~\cite{xu} for a
completely different proof. Furthermore, we remove the assumption
that $(M,g)$ is locally conformally flat.
Our first step
is to rescale the solutions, and use the Green's representation
formula~\eqref{green} to derive the long range asymptotic
formulas~\eqref{nov2e1}--\eqref{nov2e6}.
In~\cite{druet1}, Druet and Robert prove that the singular set $S$ consists of only finitely many points $\{p_1,\dots,p_N\}$ and these are separated uniformly in $k$ by a positive distance. Without loss of generality we will focus in this section on $p_1$, and to simplify the notation, we will omit the subscript $1$, so that we now consider a sequence of points $q_k\in M$ where $u_k$ has a local maximum $u_k(q_k)\to\infty$ and $q_k\to p$ as $k\to\infty$.
According to~\cite{cao}, we can find function $\hat w_k$ defined on
$M$, such that in a neighborhood $B(q_k,\delta_1)$ of $q_k$,
$\delta_1>0$, we have $\operatorname{det}(\hat g_k) = 1$ in the normal coordinates
of the conformal metric $\hat g_k=e^{2\hat w_k}g$. We refer to these
coordinates as \emph{conformal normal coordinates}. We point out
that $\operatorname{det}(\hat g_k)\approx1$ to high enough order would be
sufficient for our purpose, but we use Cao's result since it
simplifies the proof. We also choose $\delta_1$ small enough so
that $\delta_1<\operatorname{inj}(M)/10$ and $\delta_1<d/10$ where
$d$ is the minimum distance between any two points in the singular
set $S$. Using the conformal covariance property of
$P_g$~(\ref{jan22e2}), we obtain that the function $\hat u_k =
u_k-\hat w_k$ satisfies\label{bkhat}
\[
P_{\hat g_k}\hat u_k+2\hat b_k=2h_ke^{4\hat u_k}.
\]
where $2\hat b_k=P_{\hat g_k}\hat w_k+2b_ke^{-4\hat w_k}$. We remark that if $b_k=Q_g$ then $\hat b_k=Q_{\hat g_k}$. We also note that $\hat w_k(\xi)=O(d_g(\xi,q_k)^2)$ in a neighborhood of $q_k$, hence all the terms coming from $\hat w_k$ can be absorbed on the right-hand side of~\eqref{MainEst}. We have the following estimates, also proved in~\cite{druet1}:
\begin{equation} \label{oct31e2}
\begin{gathered}
\varepsilon_k^{(1-\nu)}d_{\hat g_k}(\xi,q_k)^{\nu}e^{\hat u_k(\xi)} \le C_{\nu}, \qquad 1\le \nu<2 \\
\norm{D^j\hat u_k(\xi)} \le C(d_{\hat g_k}(\xi,q_k))^{-j},\qquad j=1,2,3,
\end{gathered}
\end{equation}
where $\norm{D^j\hat u_k(\xi)} = \sum_J \norm{D^J \hat u_k(\xi)}$ and the sum is over all multi-indices $J$ of order $j$, and $\varepsilon_k=e^{-\hat u_k(q_k)}$. We now rescale the solutions $\hat u_k$, using a blow-up of the neighborhood of the point $q_k$. Define the map $\varphi_k\colon B(0,\delta_1\varepsilon_k^{-1})\to B(q_k,\delta_1)$ by $\varphi_k\colon y \mapsto \varepsilon_k y$, where on the right-hand side we are using conformal normal coordinates on $B(q_k,\delta_1)$. We use the notation $\breve f = \varphi_*f = f\operatorname{\scriptstyle\circ}\varphi$ to denote the pull-back of a function $f$ defined on $B(q_k,\delta_1)$, and we let $\breve g_k = \varepsilon_k^{-2}\varphi_* g_k$ be the blow-up metric, i.e., a rescaling of the pull-back metric. We define:
\[
v_k = \breve{\hat u}_k + \log\varepsilon_k,
\]
and note that $v_k(0)=0$. It follows from~\eqref{jan22e2} that $v_k$ satisfies:
\begin{equation} \label{oct31e1}
P_{\breve g_k} v_k + 2\varepsilon_k^4 \breve b_k = 2 \breve h_k e^{4v_k},
\qquad \text{in $B(0,\delta_1\varepsilon_k^{-1})$.}
\end{equation}
The estimates~\eqref{oct31e2} now read:
\begin{gather}
\label{oct31e3} |v_k(y)|\le (-2+\mu)\, \log(1+|y|)+C(\mu), \quad |y|\le \delta_1\varepsilon_k^{-1} \\
\label{nov29e8} |D^jv_k(y)|\le C(1+|y|)^{-j}, \quad j=1,2,3.
\end{gather}
where $\mu\in (0,1)$.
Let $\hat G_k$ be the Green's function for $P_{\hat g_k}$. Then, we have:
\[
\hat u_k(\xi) = \overline{\hat u_k} + 2 \int_M \hat G_k(\xi,\eta) h_k(\eta)e^{4\hat u_k(\eta)}\, dV_{\hat g_k}(\eta)
- 2 \int_M \hat G_k(\xi,\eta)\hat b_k(\eta)dV_{\hat g_k}(\eta).
\]
where $\overline{\hat u_k}$ is the mean value of $\hat u_k$. Decompose
$\hat G_k$ into a principal part and a regular part as follows:
\[
\hat G_k(\xi,\eta) = - \frac1{8\pi^2} \, \chi(r)\, \log d_{\hat g_k}(\xi,\eta)
+ \hat\beta(\xi,\eta) = H(\xi,\eta) + \hat\beta(\xi,\eta)
\]
where $\chi=1$ on $B(q_k,\delta_1)$, $\chi=0$ on $M\setminus B(q_k,2\delta_1)$. We have
\begin{equation} \label{uhat}
\hat u_k(\xi) = \overline{\hat u_k} + 2 \int_{M} H(\xi,\eta)\, h_k(\eta)\, e^{4\hat u_k(\eta)}\,
dV_{\hat g_k}(\eta)+{\hat\phi}_k(\xi)
\end{equation}
where
\begin{equation} \label{feb15e1}
{\hat\phi}_k(\xi) = 2 \int_M \hat\beta(\xi,\eta)\, h_k(\eta)\, e^{4\hat u_k(\eta)}\, dV_{\hat g_k}(\eta)
-2 \int_M \hat G_k(\xi,\eta)\,\hat b_k(\eta)\,dV_{\hat g_k}(\eta).
\end{equation}
Note that since $\operatorname{det}(\hat g_k) = 1$ in $B(q_k,\delta_1)$, we have $dV_{\hat g_k}(\eta)=d\eta$ in $B(q_k,\delta_1)$.
Taking the difference of~\eqref{uhat} evaluated at $\xi$ and $p_k$, we get:
\begin{multline} \label{oct31e4}
\hat u_k(\xi) - \hat u_k(p_k) = \frac1{4\pi^2} \int_M
\log\left(\frac{|\eta-q_k|}{d_{\hat g_k}(\xi,\eta)}\right)\, \chi(r)\, h_k(\eta)\, e^{4\hat u_k(\eta)}\,
dV_{\hat g_k}(\eta)\\
+ {\hat\phi}_k(\xi)-{\hat\phi}_k(q_k).
\end{multline}
Here we have used the fact that since the coordinates are normal $d_{\hat g_k}(\eta,q_k) = |\eta-q_k|$. Thanks to the cut-off function $\chi$, we can now replace the integral over $M$ by an integral over $B(q_k,2\delta_1)$, and after rescaling, we now obtain:
\begin{multline} \label{oct31e5}
v_k(y) = \frac 1{4\pi^2} \int_{B(0,2\delta_1\varepsilon_k^{-1})}
\log\left(\frac{|z|}{d_{\breve g_k}(z,y)}\right)\, \chi(\varepsilon_k r)\, h_k(\varepsilon_k z)\, e^{4v_k(z)}\, dz \\
+ {\hat\phi}_k(\varepsilon_k y) - {\hat\phi}_k(0).
\end{multline}
Let
\[
l_k = - \varepsilon_k\log\varepsilon_k, \qquad L_k=-\log\varepsilon_k,
\]
and define:
\begin{equation} \label{alpha}
\alpha_k = 2\int_{B(q_k,\delta_1)} h_k(\eta)\, e^{4\hat u_k(\eta)}\, dV_{\hat g_k}
\end{equation}
By (\ref{oct31e3}) and the fact that $\operatorname{det}(\hat g_k)=1$, one sees easily that:
\begin{equation}
\label{jan11e1} \alpha_k = 2\int_{B(q_k,l_k)} h_k(\eta)\, e^{4\hat u_k(\eta)}\, d\eta + O(L_k^{-3}).
\end{equation}
As in~\cite{linwei1}, the representation formula~\eqref{oct31e5} implies the following long range asymptotic formulas for $v_k$ and its derivatives:
\begin{align}
\label{nov2e1}
v_k(y) &= - \frac{\alpha_k}{8\pi^2}\, \log |y| + O(1),\qquad L_k\le |y|\le \delta_1\varepsilon_k^{-1} \\[1ex]
\label{nov2e2}
\partial_r v_k(y) &= -\frac{\alpha_k}{8\pi^2}\, L_k^{-1} + O(L_k^{-2}), \qquad |y|=L_k \\[1ex]
\label{nov2e3}
\partial_r\bigl(r\partial_r v_k(y)\bigr) &= O(L_k^{-2}), \qquad |y|=L_k \\[1ex]
\label{nov2e5}
\Delta v_k(y) &= -\frac{\alpha_k}{4\pi^2}\, L_k^{-2} + O(L_k^{-3}), \qquad |y|=L_k\\[1ex]
\label{nov2e6}
\partial_r \Delta v_k(y) &= \frac{\alpha_k}{2\pi^2}\, L_k^{-3}+ O(L_k^{-4}), \qquad |y|=L_k.
\end{align}
Note that while the asymptotic formula is required in the whole range $L_k\le |y|\le \delta_1\varepsilon_k^{-1}$ for $v_k$, it is only required on $|y|=L_k$ for the derivatives.
Since the proof of these estimates is similar to the one in~\cite{linwei1}, we will only briefly sketch the argument pointing out the main differences, a major one being the difference between the Euclidean and Riemannian distance. It follows from (\ref{oct31e5}) that for $|y| \ge L_k$:
\begin{equation} \label{nov30e2}
v_k(y) = - \frac1{4\pi^2} \int_{B(0,\delta_1 \varepsilon_k^{-1})} \log d_{\breve g_k}(y,z)\,
h_k(\varepsilon_k z)\, e^{4v_k(z)} \, dz + O(1).
\end{equation}
Moreover for any multi-index $J$ of order $j=1,2,3$, and $|y|=-\log {\varepsilon_k}$, we have:
\begin{equation} \label{nov30e3}
D^J v_k(y) =-\frac 1{4\pi^2}\int_{B(0,\delta_1\varepsilon_k^{-1})}
D^J_{y}(\log d_{\breve g_k}(y,z))\, h_k(\varepsilon_k z)\, e^{4v_k(z)}\, dz + O(\varepsilon_k^{j}).
\end{equation}
We now divide the domain of integration in~\eqref{nov30e2} and~\eqref{nov30e3} into three subsets $B(0,\delta_1 \varepsilon_k^{-1}) = \Omega_1 \cup \Omega_2 \cup \Omega_3$, \label{omega}where:
\[
\Omega_{1} = \{ |z| < |y|/2 \}, \quad
\Omega_{2} = \{ |z-y|< |y|/2 \},\quad
\Omega_{3} = B(0,\delta_1\epsilon_k^{-1}) \setminus (\Omega_1 \cup \Omega_2).
\]
We also use the following approximations of the distance $d_{\breve g_k}$ and its derivatives by their Euclidean counterparts\footnote{These approximations are proved in the Appendix}:
\begin{gather}
\label{dgapprox}
\log d_{\breve g_k}(y,z) - \log |y-z| = O(1), \qquad z\in B(0,\varepsilon_k^{-1}\delta_1/2)\\
\label{mar19e1}
\left| D^j \bigl( \log d_{\breve g_k}(y,z) - \log|y-z| \bigr) \right| \le
C \varepsilon_k^2|y|^{2-j}, \qquad z\in \Omega_{1}.
\end{gather}
Over $\Omega_2\cup\Omega_3$, the integral~\eqref{nov30e2} can be estimated simply by using~\eqref{oct31e3} and the approximation~\eqref{dgapprox} leading to:
\[
\int_{\Omega_2\cup \Omega_3}\log d_{\breve g_k}(y,z)\, \breve h_k(z) \, e^{4v_k(z)}\, dz = O(|y|^{-4+\mu_1}),
\qquad \text{$\mu_1>0$ small.}
\]
In order to capture the asymptotics of the integral~\eqref{nov30e2} over $\Omega_1$, we again use the approximation~\eqref{dgapprox} to reduce the calculation to the Euclidean case, so that~\eqref{nov2e1} then follows with the help of~\eqref{jan11e1}.
Similarly the estimate of~\eqref{nov30e3} over $\Omega_{2}\cup \Omega_{3}$ can be obtained from the bounds:
\[
\left| D^j \log d_{\breve g_k}(y,z) \right|\le C|y-z|^{-j},\qquad j=1,2,3,\qquad
z\in \Omega_{2}\cup\Omega_{3},
\]
leading to:
\[
\int_{\Omega_{2}\cup\Omega_{3}} \left| D^j(\log d_{\breve g_k}(y,z)) \right|\,
\breve h_k(z)\, e^{4v_k(z)}\, dz = O(|y|^{-4-j+\mu_1}),
\]
while the estimation of these integrals over $\Omega_{1}$ requires the more precise approximation~\eqref{mar19e1}.
We now use the long range estimates~\eqref{nov2e1}--\eqref{nov2e6} in the following Pohozaev identity\footnote{The proof of this identity can be found in the Appendix.} over the ball $\Omega=B(q_k,l_k)$:
\begin{multline*}
\int_{\Omega} (2 h e^{4\hat u} + \frac12 \xi^i \partial_i h e^{4\hat u})
= \int_{\partial \Omega} \left( \frac12 \xi^i \nu_i h e^{4 \hat u}
- \nu_j \xi^m \hat g^{ij} \partial_i (\Delta_{\hat g} \hat u) \partial_m \hat u
\right. \\ \left.
{} + \nu_j \hat g^{ij} \Delta_{\hat g}\hat u \partial_i \hat u
+ \nu_j \xi^m \hat g^{ij} \Delta_{\hat g} \hat u \partial_{im} \hat u
- \frac12 \xi^i \nu_i (\Delta_{\hat g}\hat u)^2 \right) \\
+ \int_{\Omega}\left(\Delta_{\hat g} \hat u \partial_i \hat g^{ij} \partial_j \hat u
+ \xi^m \Delta_{\hat g} \hat u \partial_{im} \hat g^{ij} \partial_j \hat u
+ \xi^m \Delta_{\hat g} \hat u \partial_m \hat g^{ij} \partial_{ij} \hat u
- 2 \hat b \xi^i \partial_i \hat u \right) \\
+ 2 \int_{\partial\Omega} \left( \hat R_{ij,l}(0) \xi^l \xi^m \nu_i \partial_j \hat u \partial_m \hat u
+ O(r^3) |D\hat u|^2 \right) \\
- \int_{\Omega} \left( 2\hat R_{ij,l}(0) (\xi^l \partial_j \hat u \partial_i \hat u
+ \xi^m \xi^l \partial_j \hat u \partial_{im} \hat u) + O(r^2) |D\hat u|^2 + O(r^4) |D^2\hat u| \right).
\end{multline*}
Here, we used the conformal normal coordinates $\xi^i$
on this ball, we denoted $r=|\xi|$, and denoted the unit normal to the boundary by
$\nu_i$. Furthermore, to simplify the notation, we suppressed the
sequence index $k$, and since $\operatorname{det}(\hat g)=1$, we omitted $dV_{\hat g} =
d\xi$. Finally, we remark that we chose to write this identity in
terms of $\hat u_k$ rather than $v_k$ to avoid an even longer formula. It
is easy to translate the long range
estimates~\eqref{nov2e1}--\eqref{nov2e6} to $\hat u_k$ from the fact
that $v_k(y)=\hat u_k(\varepsilon_k y)+\log \epsilon_k$.
We denote the integral on the left hand side of this identity by $I_0$, and the four integrals on the right-hand side by $I_1$, $I_2$, $I_3$ and $I_4$ respectively. By~\eqref{oct31e3}, we obtain:
\[
\frac12 \int_{\Omega} \xi^i\partial_i h_k e^{4\hat u_k} = O(\epsilon_k).
\]
hence it follows from~\eqref{jan11e1} that:
\begin{equation} \label{I0}
I_0 = \alpha_k+O(L_k^{-3})
\end{equation}
By the expansions~\eqref{oct22e9} and~\eqref{oct22e3} of the derivatives of the metric $\hat g$, and~\eqref{nov29e8} we get:
\begin{equation} \label{I2}
|I_2| \leq C \int_{B_{L_k}} \varepsilon _k^3 |D^2v_k|\, |Dv_k|\, |y|^2 + \varepsilon_k^2 |y|^2 |D^2v_k|^2
+ \varepsilon_k^4 |Dv_k|\, |y| = O(\varepsilon_k)
\end{equation}
and similarly, using~\eqref{nov29e1}, we see that:
\begin{equation} \label{I3I4}
|I_3| + |I_4| = O(\epsilon_k).
\end{equation}
It remains to compute $I_1$. First, using~\eqref{oct31e3}, we can estimate the first term in $I_1$:
\[
\frac12\int_{\partial\Omega} \xi^i \nu_i h_k e^{4\hat u_k} = O(L_k^{-3}).
\]
Using this bound, and using the expansions~\eqref{oct22e0} and~\eqref{oct22e1} in the remaining terms, we can now reduce $I_1$ to:
\begin{align*}
I_1 &= \int_{\partial\Omega} \left(
-l_k \partial_\nu(\Delta\hat u_k) \partial_\nu\hat u_k + \Delta\hat u_k \partial_\nu\hat u_k
+ \nu_i\xi^m \Delta\hat u_k \partial_{im}\hat u_k - \frac12 l_k (\Delta\hat u_k)^2 \right) \\
& \qquad \qquad \qquad\qquad \qquad \qquad {} + O(\varepsilon_k) \\
& = \int_{\partial B_{L_k}} \left(
-L_k \partial_\nu (\Delta v_k) \partial_\nu v_k + \partial_\nu (y\cdot\nabla v_k) \Delta v_k
-\frac 12 L_k (\Delta v_k)^2 \right)
+ O(\varepsilon_k).
\end{align*}
Using (\ref{nov2e1})-(\ref{nov2e6}) in the above, we get:
\begin{equation} \label{I1}
I_1 = \frac{\alpha_k^2}{16\pi^2} + O(L_k^{-1}).
\end{equation}
Combining~\eqref{I0}, \eqref{I2}, \eqref{I3I4} and~\eqref{I1}, we get:
\[
\alpha_k + O(L_k^{-3}) = \frac{\alpha_k^2}{16\pi^2} + O(L_k^{-1}).
\]
which implies
\begin{equation} \label{alphak}
\alpha_k = 16 \pi^2 + O(L_k^{-1}).
\end{equation}
When substituting this into~\eqref{nov2e1}, we obtain:
\[
v_k(y) + 2\log |y| = O(1), \quad |y|\ge L_k.
\]
The argument in the region $|y|\le L_k$ follows the one
in~\cite{linwei1} closely, hence we again only sketch the proof.
Without loss of generality, we assume that $h_k(q_k)\to 1$\footnote{Otherwise, we can
add a constant to $\hat u_k$.}.
Let $U=U_{0,1,1}$ be the standard bubble in
$\mathbb R^4$:
\[
U(y)= - \log \left(1 + \frac{|y|^2}{4\sqrt{3}}\right)
\]
It is easy to check that $U$ satisfies $\Delta^2U=2e^{4U}$ and it is well known that $v_k\to U$ in $C^4_{\text{loc}}(\mathbb R^4)$, see for example~\cite{druet1}. Thus, for any fixed $A$ and all $k$ sufficiently large, we have:
\[
|v_k(y)-U(y)|\le 1, \qquad \text{for $|y|\le A$.}
\]
Subtracting~\eqref{nov30e3} from its Euclidean counterpart, using~\eqref{mar19e1} to compare $d_{\breve g}(y,z)$ and $|y-z|$ as well as their respective derivatives, and also using~\eqref{alphak} to compare the leading terms, we obtain:
\[
|\Delta v_k(y) - \Delta U(y)| \le C |y|^{-3}, \qquad \text{for $A<|y|< L_k$.}
\]
Now, letting $T(y)=C(1+|y|^{-1})$, and choosing $C$ large enough, we can guarantee that $\Delta T \le - |\Delta v_k - \Delta U|$
whence from the maximum principle $|v_k(y)-U(y)| \le C(1+|y|^{-1})$, on $A\le |y|\le L_k$. Substituting $\xi=\varepsilon_k y$, and using the definition of $v_k$, we obtain the version of~\eqref{MainEst} with $O(1)$ on the right-hand side, i.e., with $\tau=0$.
\section{A Sharper Estimate} \label{sec:03}
The main purpose of this section is to establish~\eqref{MainEst}. An important tool we use is the following lemma,
due to Lin-Wei~\cite{linwei1}:
\begin{lem} \label{nov5l1}
Let $U(y)=-\log (1+|y|^2/4\sqrt{3})$ be defined on $\mathbb R^4$. Then $U$ satisfies
\[
\Delta^2U=2e^{4U},\qquad U(0)=\max U = 0.
\]
Furthermore, any solution of the linearized problem:
\[
\Delta^2\phi = 8e^{4U} \phi, \qquad |\phi(y)|\le C(1+|y|)^{\tau}, \qquad \tau\in (0,1),
\]
is given by $\phi=\sum_{j=0}^4 c_j \psi_j$ where
\begin{eqnarray*}
\psi_0 &=& \frac{1-|y|^2/4\sqrt{3}}{1+|y|^2/4\sqrt{3}} \\[1ex]
\psi_j &=& \frac{y_j}{1+|y|^2/4\sqrt{3}}, \qquad j=1,..,4.
\end{eqnarray*}
\end{lem}
\begin{rem}
One immediate consequence of~Lemma~\ref{nov5l1} is that if in addition, $\phi$ satisfies
$\phi(0)=0$, and $\nabla\phi(0)=0$, then $\phi\equiv 0$.
\end{rem}
Let $\rho_k=h_k(0)^{1/2}/4\sqrt3$ and consider the solution $U_k(y) = - \log (1+\rho_k |y|^2)$ of the equation
\begin{equation} \label{Uk}
\Delta^2U_k=2h_k(0)e^{4U_k}, \qquad U_k(0)=0,\qquad |\nabla U_k(0)|=0,
\end{equation}
on $\mathbb R^4$. Letting $w_k=v_k-U_k$, then by the result of Section~\ref{sec:02}, we already know that $|w_k|\le C$ in
$B(0,\delta_1\varepsilon_k^{-1})$. Our goal in this section is to prove:
\begin{equation} \label{nov21e1}
|w_k(y)| \le C \varepsilon_k^{\tau}|y|^{\tau},\qquad |y|\le \delta_1\varepsilon_k^{-1}.
\end{equation}
for any $0<\tau<1$, which implies~\eqref{MainEst}. Equivalently, if we let
\[
\Lambda_k = \max_{\Omega_k} \frac{|w_k(y)|}{\varepsilon_k^{\tau}(1+|y|)^{\tau}},
\]
then it suffices to show that $\Lambda_k$ is bounded on $\Omega_k=B(0,\delta_1\varepsilon_k^{-1})$.
Suppose that $\Lambda_k\to \infty$, and let $y_k\in\Omega_k$ be the point where $\Lambda_k$ attains its maximum. Now, either: (i) $y_k\to\infty$; or (ii) $|y_k|$ remains bounded at least along a subsequence, and hence a further subsequence, which without loss of generality we will assume is $y_k$ itself, converges to $y^*$. We will show that in both cases a contradiction follows.
Define:
\[
\bar w_k(y) = \frac{w_k(y)}{\Lambda_k\varepsilon_k^{\tau}(1+|y_k|)^{\tau}}.
\]
By the definition of $\Lambda_k$, we have
\begin{equation} \label{nov12e2}
|\bar w_k(y)| \le \left(\frac{1+|y|}{1+|y_k|}\right)^{\tau},
\end{equation}
and $\bar w_k(y_k) = \pm 1$.
Assume first that $y_k\to\infty$. Since $|w_k(y)|\le C$ and $\Lambda_k\to\infty$, we clearly have $y_k=o(1)\varepsilon_k^{-1}$.
>From the fundamental solution for $\Delta^2$, it is straightforward to get:
\begin{equation}
\begin{aligned} \label{nov12e3}
U_k(x) &= \frac 1{4\pi^2} \int_{\mathbb R^4} \log \frac{|y|}{|x-y|}\, h_k(0)\, e^{4U_k(y)}\, dy \\
&= \frac 1{4\pi^2} \int_{\Omega_k} \log \frac{|z|}{|y-z|}\, h_k(0)\, e^{4U_k(z)}\,dz + O(\varepsilon_k^4).
\end{aligned}
\end{equation}
Similarly, using the fundamental solution for $P_{\breve g_k}$, we find:
\begin{equation}
\begin{aligned} \label{jun29e1}
v_k(y) &= \frac 1{4\pi^2}\int_{\Omega_k} \log\frac{|z|}{d_{\breve g_k}(y,z)}\, h_k(\varepsilon_kz)\, e^{4v_k(z)}\,dz + O(\varepsilon_k|y|) \\
&= \frac 1{4\pi^2} \int_{\Omega_k} \log\frac{|z|}{|y-z|}\, h_k(\varepsilon_kz)\, e^{4v_k(z)}\, dz + O(\varepsilon_k|y|),
\end{aligned}
\end{equation}
see~\eqref{oct31e5}. Note that for the second equality, we used~\eqref{feb13e3} as well as
the decay rate of $v_k$. Finally, we estimate the source term:
\begin{multline*}
|h_k(\epsilon_kz)\,e^{4v_k(z)} - h_k(0)\,e^{4U_k}| \\
= \left| h_k(\varepsilon_k z)\,(e^{4v_k} - e^{4U_k}) + (h_k(\varepsilon_kz)-h_k(0))\, e^{4U_k} \right| \\
\le C (1+|z|)^{-8} |w_k(z)| + O(\varepsilon_k)\,(1+|z|)^{-7} \\
\le C\varepsilon_k^{\tau} \Lambda_k(1+|z|)^{-8+\tau} + O(\varepsilon_k)(1+|z|)^{-7}.
\end{multline*}
Substituting this in~\eqref{jun29e1} and combining with~\eqref{nov12e3} in the definition of $\bar w_k$,
we obtain:
\begin{multline*}
\bar w_k(y_k) = \int_{\Omega_k}\log \frac{|z|}{|y-z|}
\left( \frac{O(1)(1+|z|)^{-8+\tau}}{(1+|y_k|)^{\tau}} \right. \\
\left. {} + \frac{O(\varepsilon_k^{1-\tau})(1+|z|)^{-7}}{\Lambda_k(1+|y_k|)^{\tau}} \right)\, dz
+ o(1).
\end{multline*}
Since $y_k\to \infty$, it is now easy to see that the
right hand side is $o(1)$, which contradicts $\bar w_k=\pm 1$.
We now turn to the second case and assume without loss of generality that $y_k$ converges to $y^*$. We will show that along a subsequence $\bar w_k$ converges. This will be accomplished by estimating $P_{\breve g_k}(U_k-v_k)$. We start with:
\[
P_{\breve g} U_k = \Delta_{\breve g}^2 U_k + \varepsilon_k^2 \div_{\breve g} \left( \left( \frac23
\hat R (\varepsilon_k y) \breve g_{ij}(y) - 2 \hat R_{ij}(\varepsilon_k y) \right)\, dU_k \right)
\]
where we have suppressed the subscript $k$ on the metric and curvature components, and where $\hat R$, $\hat R_{ij}$ are the scalar
and the Ricci curvatures of $\hat g$. In conformal normal coordinates, we have:
\begin{equation} \label{1term}
\Delta_{\breve g}^2 U_k = \Delta^2 U_k \qquad \text{in $B(0,\delta_1 \epsilon_k^{-1})$.}
\end{equation}
Furthermore:
\begin{multline*}
\partial_m \left( \breve g^{mi}\left( \frac23 \hat R (\varepsilon_k y) \breve g_{ij} - 2\hat R_{ij}(\varepsilon_ky) \right)\breve g^{lj}\partial_l U_k \right) \\
= \partial_m \breve g^{mi} \left( \frac23 \hat R(\varepsilon_k y) \breve g_{ij} - 2\hat R_{ij}(\varepsilon_k y) \right) \breve g^{lj} \partial_l U_k \\
{} + \breve g^{mi} \partial_m \breve g^{lj} \left( \frac23 \hat R (\varepsilon_k y) \breve g_{ij} - 2 \hat R_{ij}(\varepsilon_k y) \right) \partial_l U_k \\
{} + \varepsilon_k \breve g^{mi} \breve g^{lj} \left( \frac 23 \partial_m \hat R(\varepsilon_k y) \breve g_{ij} + \frac23 \hat R(\varepsilon_ky) \partial_m
\hat g_{ij}(\varepsilon_ky) - 2\hat R_{ij,m}(\varepsilon_ky) \right) \partial_l U_k\\
{} + \breve g^{mi} \breve g^{lj} \left( \frac23 \hat R(\varepsilon_ky) \breve g_{ij} - 2\hat R_{ij}(\varepsilon_k y) \right) \partial_{lm} U_k\\
= A_1 + A_2 + A_3 + A_4.
\end{multline*}
Since $\partial_m\breve g^{mi}(y)=\varepsilon_k\partial_m\hat g^{mi}(\varepsilon_ky)=O(\varepsilon_k^3|y|^2)$, $\hat
R(\varepsilon_ky)=O(\varepsilon_k^2|y|^2)$, and $\hat R_{ij}(\varepsilon_ky)=O(\varepsilon_k|y|)$, we easily get $A_1=O(\varepsilon_k^2)$, and $A_2=O(\varepsilon_k^2)$. Furthermore, since in addition $\hat R_{ij,i}(0)=-\frac 12\hat R_{,j}(0)=0$, we also have $A_3=O(\varepsilon_k^2)$, and $A_4=O(\varepsilon_k^2)$. Substituting this into the above equation and multiplying by $\varepsilon_k^2$, we get:
\begin{equation} \label{2term}
\varepsilon_k^2 \div_{\breve g} \left( \left(\frac23 \hat R(\varepsilon_k y) \breve g_{ij}(y) - 2\hat R_{ij}(\varepsilon_ky) \right)\, dU_k \right)
= O(\varepsilon_k^4).
\end{equation}
Combining~\eqref{1term} and~\eqref{2term}, we find:
\begin{equation} \label{nov5e1}
P_{\breve g} U_k = 2h_k(0)\,e^{4U_k} + O(\varepsilon_k^4), \qquad |y|\le \delta_1\varepsilon_k^{-1}.
\end{equation}
Combining~\eqref{nov5e1} with~\eqref{oct31e1}, we obtain:
\begin{equation}
\label{nov5e2} P_{\breve g} w_k = 8 h_k(\varepsilon_k y)\,e^{4\xi^k} w_k + O(\varepsilon_k) (1+|y|)^{-7} + O(\varepsilon_k^4),
\qquad |y| \le \delta_1 \varepsilon_k^{-1},
\end{equation}
where $\xi^k$ is given by: $e^{4\xi^k}=\int_0^1e^{4tv_k+4(1-t)U_k}\,dt$. Finally, this leads to the following equation for $\bar w_k$:
\begin{equation}
\label{nov5e3} P_{\breve g}\bar w_k=8h_k(\epsilon_ky)e^{4\xi^k}\bar
w_k+
\frac{O(\epsilon_k^{1-\tau})(1+|y|)^{-7}}{\Lambda_k(1+|y_k|)^{\tau}}+\frac{O(\epsilon_k^{4-\tau})}{\Lambda_k(1+|y_k|)^{\tau}}.
\end{equation}
Since $y_k\to y^*$, a subsequence of $\bar w_k$ converges to $w$ in $C^4(\mathbb R^4)$. We will assume without loss of generality, as in Section~\ref{sec:02}, that $h_k(0)\to 1$. It follows that the limit $w$ satisfies:
\[
\begin{cases}
\Delta^2w=8e^{4U}w,\\
|w(y)| \le C(1+|y|)^{\tau},\\
w(0)= |\nabla w(0)| =0.
\end{cases}
\]
By the remark following Lemma~\ref{nov5l1}, we conclude that $w\equiv 0$, which contradicts $\bar w(y^*)=\pm 1$. This concludes the proof of~\eqref{nov21e1}.
In Section~\ref{sec:04}, we will also need estimates on the derivatives $D^jw_k(y)$ for $|y|\le \varepsilon_k^{-1}\delta_1/2$, $j=1,2,3$. By combining~\eqref{nov5e2} and~\eqref{nov21e1}, we have:
\[
P_{\breve g} w_k(y) = O(\varepsilon_k^{\tau}) (1+|y|)^{-8+\tau} + O(\varepsilon_k^4), \qquad |y|< \frac{\delta_1}2\epsilon_k^{-1}.
\]
Fix $y$, and let $r=|y|$ and $f_k(z)=w_k(rz)$ for $1/2< |z| <2$. Then $f_k(z)$ satisfies:
\begin{gather*}
P_{\acute g} f_k(z)=O(\varepsilon_k^{\tau}) (1+r)^{-4+\tau} + O(\varepsilon_k^4 r^4), \qquad B_2\setminus B_{1/2}, \\
f_k(z)=O(\varepsilon_k^{\tau}r^{\tau}),\quad B_2\setminus B_{1/2},
\end{gather*}
where $\acute g$ is the rescaled metric $r^{-2}\psi_*\breve g$ and $\psi\colon z\mapsto rz$. Standard elliptic theory for fourth order equations~\cite{browder} yields:
\[
D^j f_k(z) = O(\varepsilon_k^{\tau} r^{\tau}), \qquad |z|=1, \quad j=1,2,3.
\]
Hence, we conclude:
\begin{equation}\label{feb19e1}
D^j w_k(y) = O(\varepsilon_k^{\tau} |y|^{\tau-j}), \qquad j=1,2,3, \quad |y| \leq \varepsilon_k^{-1}\delta_1/2.
\end{equation}
\section{The Vanishing Rate}
\label{sec:04}
The purpose of this section is to complete the proof of our Theorem by proving~\eqref{vrate}. We first prove a Euclidean version in Subsection~\ref{sec:vrate}, and then translate this result to the original metric in Subsection~\ref{sec:original}.
\subsection{A Euclidean Version} \label{sec:vrate}
The goal of this subsection is to prove~\eqref{nov22e8}. This is accomplished in three steps:
\begin{enumerate}
\item[(i)] In the first step, we derive an asymptotic expansion of the Pohozaev identity; see~\eqref{nov21e4}.
\item[(ii)] In the second step, we express this identity in terms of the background Euclidean metric; see~\eqref{dec6e2}.
\item[(iii)] In the last step, we complete the proof of~\eqref{nov22e8}.
\end{enumerate}
\subsubsection{Step 1}
In this subsection we let $E_k=B(0,\varepsilon_k^{-1/2})$, and we derive an asymptotic Pohozaev identity for $v_k$ on $E_k$. We multiply~\eqref{oct31e1} by $\partial_av_k$, $a=1,..,4$, integrate with respect to the Euclidean volume element $dy$, and estimate each of the resulting terms. First by the $O(1)$ estimate and~\eqref{b}:
\begin{multline} \label{nov21e3}
\int_{E_k} 2 h_k(\varepsilon_ky)\, e^{4v_k(y)}\, \partial_a v_k(y)\, dy \\
= - \frac{\varepsilon_k} 2\int_{E_k} e^{4v_k(y)}\, \partial_a h_k(\varepsilon_ky)
+ \frac12 \int_{\partial E_k} h_k\, e^{4v_k}\, \nu_a \\
= - \frac{\varepsilon_k} 2\partial_a h_k(0) \int_{E_k} e^{4v_k(y)} + O(\varepsilon_k^2).
\end{multline}
Next, integrating by parts, we have:
\begin{multline} \label{Ek}
\int_{E_k} \Delta_{\breve g}^2v_k\,\partial_av_k \\
= \int_{\partial E_k} \left(\breve g^{ij}\partial_j(\Delta_{\breve g}v_k)\partial_a v_k\,\nu_i
-\breve g^{ij}\Delta_{\breve g}v_k\,\partial_{ia}v_k\,\nu_j
+ \frac 12(\Delta_{\breve g}v_k)^2\,\nu_a \right)\\
-\int_{E_k}(\Delta_{\breve g}v_k \,\partial_{ia}\breve g^{ij}\,\partial_j v_k + (\Delta_{\breve g}v_k)\,
\partial_a\breve g^{ij}\,\partial_{ij}v_k),
\end{multline}
where we used:
\[
\partial_j(\breve g^{ij}\partial_{ia}v_k)=\partial_a(\Delta_{\breve g}v_k)
-\partial_{ia}\breve g^{ij}\partial_jv_k-\partial_a\breve
g^{ij}\partial_{ij}v_k
\]
We now estimate the two integrals over $E_k$ in~\eqref{Ek} above. Using $\partial_{ia}\breve g^{ij}=O(\epsilon_k^3|y|)$, which is
implied by~\eqref{nov29e1}, and~\eqref{nov29e8}, we find:
\[
\int_{E_k}\Delta_{\breve g} v_k\, \partial_{ia}\breve g^{ij}\, \partial_j\, v_k=O(\varepsilon_k^2).
\]
Furthermore, for any $0<\sigma<1$, the second integral over $E_k$ in~\eqref{Ek} can be estimated as follows:
\begin{multline*}
\int_{E_k}(\Delta_{\breve g}v_k)\, \partial_a\breve g^{ij}\,\partial_{ij}v_k \\
=\varepsilon_k^2\int_{E_k}(\Delta U_k + \Delta_{\breve g}w_k)\, \left(-\frac
23\hat R_{i(am)j}(0)\,y^m + O(\varepsilon_k|y|^2) \right)\partial_{ij}v_k\\
=-\frac 23\varepsilon_k^2\int_{E_k}\Delta
U_k\hat R_{i(am)j}(0)\,y^m\partial_{ij}U_k\,dy+O\bigl(\varepsilon_k^{(3+\sigma)/2}\bigr)=O\bigl(\varepsilon_k^{(3+\sigma)/2}\bigr),
\end{multline*}
where we used~\eqref{feb19e1}, and the following estimate implied by~\eqref{oct22e9}:
\[
\partial_a\breve g^{ij}(y)=\varepsilon_k\, \partial_a\hat g^{ij}(\varepsilon_ky)=-\frac
23\hat R_{i(am)j}(0)\, \varepsilon_k^2\, y^m + O(\varepsilon_k^3|y|^2),
\]
as well as the antisymmetry of the curvature tensor, and $\hat R_{ij}(0)=0$. Here, we use the customary round brackets notation to denote the symmetric part. We will choose $\sigma\in(0,1)$ at the end of the argument of Section~\eqref{sec:vrate}. Next, since $\hat R(\varepsilon_k y)= O(\varepsilon_k^2|y|^2)$, and $\hat R_{ij}(\varepsilon_k y)=O(\varepsilon_k |y|)$ and~\eqref{nov29e8}, we have:
\begin{multline*}
\varepsilon_k^2\int_{E_k}\partial_m\left( \breve g^{mi}\left( \frac
23\hat R(\varepsilon_ky) \, \breve g_{ij}-2\hat R_{ij}(\varepsilon_ky) \right)\partial_l v_k\, \breve g^{lj} \right)\partial_a v_k\\
= \varepsilon_k^2\int_{\partial E_k} \breve g^{mi} \left(\frac
23\hat R(\varepsilon_ky)\,\breve g_{ij}-2\hat R_{ij}(\varepsilon_ky) \right)\, \partial_l v_k\, \breve g^{lj}\, \partial_a v_k\, \nu_m\\
- \varepsilon_k^2 \int_{E_k} \breve g^{mi} \left(\frac
23\hat R(\varepsilon_ky) \, \breve g_{ij}-2\hat R_{ij}(\varepsilon_ky) \right)\partial_l v_k\,\breve g^{lj}\, \partial_{am} v_k = O(\varepsilon_k^2).
\end{multline*}
Finally, we estimate:
\[
\varepsilon_k^4 \int_{E_k} 2 \hat b_k \partial_a v_k = O(\varepsilon_k^{5/2}).
\]
Combining all the terms, we arrive at the following Pohozaev identity:
\begin{multline} \label{nov21e4}
\frac{\varepsilon_k}2\, \partial_a h_k(0) \int_{E_k} e^{4v_k} \, dy+ O\bigl(\varepsilon_k^{(3+\sigma)/2}\bigr) \\
= \int_{\partial E_k} \left( -\breve g^{ij}\,\partial_j(\Delta_{\breve g}v_k)\,\partial_a v_k\, \nu_i
+ \breve g^{ij}\,\Delta_{\breve g}v_k\,\partial_{ia}v_k\,\nu_j - \frac12 (\Delta_{\breve g}v_k)^2\, \nu_a \right).
\end{multline}
\subsubsection{Step 2}
In this second step, we rewrite~\eqref{nov21e4} in terms of the Euclidean $\Delta v_k$ rather than $\Delta_{\breve g} v_k$. We begin by substituting:
\[
(\breve g^{ij}(y)-\delta_{ij})\,\nu_i = -\frac13 \varepsilon_k^2 \hat R_{ilmj}(0)\,y_m\,y_l\,\frac{y_i}{|y|} + O(\varepsilon_k^3|y|^3) = O(\varepsilon_k^3|y|^3).
\]
into~\eqref{nov21e4} to get:
\begin{multline} \label{nov21e5}
\frac{\varepsilon_k}2\,\partial_a h_k(0) \int_{E_k} e^{4v_k}\,dy + O\bigl(\varepsilon_k^{(3+\sigma)/2}\bigr) \\
= \int_{\partial E_k} \left( - \partial_i(\Delta_{\breve g} v_k)\,\partial_a v_k \, \nu_i + \Delta_{\breve g}v_k\, \partial_{ia} v_k\, \nu_i
-\frac 12 (\Delta_{\breve g}v_k)^2\,\nu_a \right).
\end{multline}
A straightforward computation leads to:
\begin{multline} \label{jun30e1}
\partial_i(\Delta_{\breve g}v_k)-\partial_i(\Delta v_k)\\
= \partial_{im}\breve g^{ml}\,\partial_l v_k
+ \partial_m \breve g^{ml}\,\partial_{il}v_k + \partial_i\breve g^{ml}\, \partial_{ml}v_k + (\breve g^{ml}-\delta_{ml})\, \partial_{iml}v_k.
\end{multline}
In view of~\eqref{nov29e1} and~\eqref{oct22e3}, we have, for $|y|=\varepsilon_k^{-1/2}$:
\begin{gather}
\label{jun30e2a}
\partial_{im}\breve g^{ml}\,\partial_l v_k = O(\varepsilon_k^3),\qquad
\partial_m \breve g^{ml}\, \partial_{il}v_k = O(\varepsilon_k^3),\\
\label{jun30e2}
\partial_i\breve g^{ml}\, \partial_{ml}v_k = \partial_i\breve g^{ml}\, (\partial_{ml}U_k
+ O(\varepsilon_k^{\sigma}r^{\sigma-2})) = O\bigl(\varepsilon_k^{(5+\sigma)/2}\bigr),
\end{gather}
where to derive~\eqref{jun30e2}, we also used the following consequence of~\eqref{oct22e9}:
\[
\partial_i \breve g^{ml} = -\frac23 \hat R_{m(ia)l}(0) \varepsilon_k^2 y_a + O(\varepsilon_k^3|y|^2),
\]
as well as the anti-symmetry of the curvature tensor and $\hat R_{ij}(0)=0$.
Next, on $|y|=\varepsilon_k^{-1/2}$, we have
\begin{multline} \label{jun30e3}
(\breve g^{ml}-\delta^{ml})\, \partial_{iml}v_k \\
= \left(-\frac13\varepsilon_k^2 \hat R_{mabl}(0)\,y_a\,y_b + O(\varepsilon_k^3r^3) \right)
(\partial_{iml}U_k + O(\varepsilon_k^{\sigma}r^{\sigma-3})) \\
= -\frac13\varepsilon_k^2\hat R_{mabl}(0)\,y_a\,y_b\,\partial_{iml}U_k + O\bigl(\varepsilon_k^{(5+\sigma)/2}\bigr)
= O\bigl(\varepsilon_k^{(5+\sigma)/2}\bigr),
\end{multline}
where we have used the following expansion, valid for any radial function $f(r)$:
\begin{multline*}
\partial_{iml} f(r) = \left( f'''(r)-\frac{f''(r)}{r} + \frac{f'(r)}{r^2} \right) \frac{y_l\,y_m\,y_i}{r^3} \\
{} + \left( f''(r)-\frac{f'(r)}r \right) \frac{(\delta_{il}\,y_m + y_l\, \delta_{im}) r^2 - 2y_l\,y_m\,y_i}{r^4} \\
{} + (f''(r)-f'(r))\frac{\delta_{ml}\,y_i}{r^2},
\end{multline*}
as well as the anti-symmetry of $\hat R_{abcd}$ and $\hat R_{ij}(0)=0$.
It now follows from~\eqref{jun30e1}, \eqref{jun30e2} and~\eqref{jun30e3} that the following holds:
\begin{equation} \label{jun30e4}
-\int_{\partial E_k} \partial_i(\Delta_{\breve g}v_k)\,\partial_a v_k\,\nu_i
= -\int_{\partial E_k} \partial_i(\Delta v_k)\, \partial_a v_k\, \nu_i + O\bigl(\varepsilon_k^{(3+\sigma)/2}\bigr).
\end{equation}
Next, on $|y|=\epsilon_k^{-1/2}$, we have:
\begin{multline*}
\Delta_{\breve g}v_k=\partial_i\breve g^{ij}\,\partial_jv_k + \breve g^{ij}\,\partial_{ij}v_k
= O(\varepsilon_k^3)r + \Delta v_k+ (\breve g^{ij}-\delta_{ij})\,\partial_{ij}v_k\\
=O(\varepsilon_k^{\frac 52})+\Delta v_k- \left( \frac{\varepsilon_k^2}3 \hat R_{iabj}(0)\,y_a\,y_b + O(\varepsilon_k^3r^3) \right)
\bigl( \partial_{ij}U_k + O(\varepsilon_k^{\sigma}r^{\sigma-2}) \bigr)\\
= \Delta v_k+O\bigl(\varepsilon_k^{2+\sigma/2}\bigr),
\end{multline*}
from which it follows:
\begin{equation} \label{jun30e5}
\int_{\partial E_k} \Delta_{\breve g}v_k\,\partial_{ia} v_k\,\nu_i
= \int_{\partial E_k}\Delta v_k\,\partial_{ia} v_k\, \nu_i\, dS + O\bigl(\varepsilon_k^{(3+\sigma)/2}\bigr).
\end{equation}
Similarly:
\begin{equation} \label{jun30e6}
-\frac12 \int_{\partial E_k}(\Delta_{\breve g}v_k)^2\, \nu_a
= -\frac12 \int_{\partial E_k}(\Delta v_k)^2\,\nu_a + O\bigl(\varepsilon_k^{(3+\sigma)/2}\bigr).
\end{equation}
Substituting~\eqref{jun30e4}, \eqref{jun30e5} and~\eqref{jun30e6} into the Pohozaev Identity~\eqref{nov21e5}, we conclude:
\begin{multline} \label{dec6e2}
\frac{\varepsilon_k}2\partial_a h_k(0) \int_{E_k} e^{4v_k}\, dy + O(\varepsilon_k^{(3+\sigma)/2}) \\
= \int_{\partial E_k} \left(-\partial_i(\Delta v_k)\,\partial_a v_k\,\nu_i + \Delta v_k\, \partial_{ia} v_k\, \nu_i
- \frac12(\Delta v_k)^2\,\nu_a \right)\,dS.
\end{multline}
\subsubsection{Step 3} In this subsection, we first aim to replace $v_k$ by $U_k$ in the Pohozaev identity~\eqref{dec6e2}, after which many of the terms will simplify thanks to the radial symmetry of $U_k$, leading to the Euclidean version of the vanishing rate~\eqref{nov22e8}.
Recall the definition of ${\hat\phi}_k$~\eqref{feb15e1}, from which we have:
\[
D^j{\hat\phi}_k(\varepsilon_ky)=O(\varepsilon_k^j),\qquad j=1,2,3.
\]
For $|y|=\varepsilon_k^{-1/2}$, we cut $B(0,\delta_1\varepsilon_k^{-1})$ into three subdomains $B(0,\delta_1\varepsilon_k^{-1})=\Omega_1\cup\Omega_2\cup\Omega_3$ as in Section~\ref{sec:02}, page~\pageref{omega}, and use the representation~\eqref{oct31e5}.
Using standard estimates over $\Omega_{2}\cup \Omega_{3}$ and~\eqref{jun27e10} over $\Omega_1$, we find:
\begin{multline*}
\partial_a v_k(y) \\
= -\frac 1{4\pi^2} \int_{\Omega_{1}}\partial_a \bigl(\log
d_{\breve g}(z,y) \bigr) \, \breve h_k(z)\, e^{4v_k(z)} + \varepsilon_k\, \partial_a{\hat\phi}_k(0)
{} + O(\varepsilon_k^2|y|)+O(|y|^{-5}) \\
= -\frac1{4\pi^2}\int_{\Omega_{1}} \frac{y_a-z_a}{|y-z|^2}\, h_k(\varepsilon_kz)\, e^{4v_k(z)}
+ \varepsilon_k\, \partial_a{\hat\phi}_k(0) + O(\varepsilon_k^{3/2}),
\end{multline*}
where we have omitted the standard volume element $dz$. Similarly,
\begin{gather*}
\Delta v_k(y)=-\frac 1{2\pi^2}\int_{\Omega_{1}}
\frac 1{|y-z|^2}\, h_k(\varepsilon_kz)\, e^{4v_k(z)} + O(\varepsilon_k^2), \\[1ex]
\partial_{ij}v_k(y)=-\frac 1{4\pi^2}\int_{\Omega_{1}}
\frac{\delta_{ij}|y-z|^2-2(y_i-z_i)(y_j-z_j)}{|y-z|^4}\, h_k(\varepsilon_kz)\, e^{4v_k}
+O(\varepsilon_k^2). \\[1ex]
\partial_i(\Delta v_k(y))=\frac 1{\pi^2}\int_{\Omega_{1}}
\frac{y_i-z_i}{|y-z|^4}\, h_k(\varepsilon_kz)\, e^{4v_k} + O(\varepsilon_k^{5/2}).
\end{gather*}
We also have
\begin{align*}
h_k(\varepsilon_kz)\, e^{4v_k(z)}&= \bigl(h_k(0)+ \varepsilon_k\, \partial_j h_k(0)\, z_j+O(\varepsilon_k^2|z|^2)
\bigr)e^{4U_k+O(\varepsilon_k^{\sigma}|z|^{\sigma})}\\
&=h_k(0)\, e^{4U_k(z)}+O(\varepsilon_k^{\sigma})(1+|z|)^{-8+\sigma}.
\end{align*}
Substituting this in the derivatives of $v_k$ we have:
\begin{gather*}
\partial_av_k(y)=-\frac1{4\pi^2}\int_{\Omega_{1}}\frac{y_a-z_a}{|y-z|^2}\,h_k(0)\,e^{4U_k(z)}
+\varepsilon_k\,\partial_a{\hat\phi}_k(0)+O(\varepsilon_k^{\sigma+1/2})\\[1ex]
\Delta v_k(y)=-\frac 1{2\pi^2}\int_{\Omega_{1}} \frac1{|y-z|^2}\,h_k(0)\,e^{4U_k(z)}+O(\varepsilon_k^{\sigma+1}),\\[1ex]
\partial_{ij}v_k(y)=-\frac 1{4\pi^2}\int_{\Omega_{1}}\frac{\delta_{ij}\,|y-z|^2-2(y_i-z_i)(y_j-z_j)}{|y-z|^4}\,h_k(0)\,e^{4U_k}
+O(\varepsilon_k^{\sigma+1})\\[1ex]
\partial_i(\Delta v_k(y))=\frac 1{\pi^2}\int_{\Omega_{1}}
\frac{y_i-z_i}{|y-z|^4}\,h_k(0)\,e^{4U_k}+O(\varepsilon_k^{\sigma+3/2}).
\end{gather*}
We perform a similar computation for $U_k$ and take the difference, leading to the following estimates for
$|y|=\varepsilon_k^{-1/2}$:
\begin{align*}
\partial_i(\Delta v_k(y))&=\partial_i(\Delta
U_k)(y)+O(\varepsilon_k^{\sigma+3/2}),\\
\Delta v_k(y)&=\Delta U_k(y)+O(\varepsilon_k^{\sigma+1}),\\
\partial_{ij}v_k(y)&=\partial_{ij}U_k(y)+O(\varepsilon_k^{\sigma+1})\\
\partial_av_k(y)&=\partial_aU_k(y)+\varepsilon_k\,\partial_a{\hat\phi}_k(0)+O(\varepsilon_k^{\sigma+1/2}).
\end{align*}
Substituting these estimates into the Pohozaev Identity~\eqref{dec6e2}, we obtain:
\begin{multline} \label{PUk}
\frac{\varepsilon_k}2\partial_ah_k(0)\int_{E_k}e^{4U_k} + O(\varepsilon_k^{1+\sigma/2})\\
=\int_{\partial E_k}\left(-\partial_{\nu}(\Delta
U_k)(\partial_aU_k+\varepsilon_k\,\partial_a{\hat\phi}_k(0)) + \Delta
U_k\,\partial_{ia}U_k\,\nu_i-\frac 12(\Delta U_k)^2\,\nu_a\right).
\end{multline}
The symmetry of $U_k$ implies:
\[
\int_{\partial E_k}\left(-\partial_{\nu}(\Delta U_k)\,\partial_aU_k
+\Delta U_k\,\partial_{ia}U_k\,\nu_i-\frac 12(\Delta U_k)^2\,\nu_a\right)=0.
\]
In view of the equation~\eqref{Uk}, we also have
\[
\int_{\partial E_k}\partial_{\nu}(\Delta
U_k)\,dS = 2h_k(0)\int_{E_k}e^{4U_k}.
\]
Substituting into~\eqref{PUk}, we obtain
\[
\partial_ah_k(0)+4h_k(0)\partial_a
{\hat\phi}_k(0)=O(\varepsilon_k^{\sigma/2})+O(\varepsilon_k^{\sigma-1/2}),\quad a=1,2,3,4.
\]
Now, if we choose $\tau/2 + 1/2<\sigma<1$, then $O(\varepsilon_k^{\sigma/2})+O(\varepsilon_k^{\sigma-1/2})=O(\varepsilon_k^{\tau/2})$, so that we can conclude:
\begin{equation} \label{nov22e8}
\left| \frac{\nabla h_k(0)}{h_k(0)}+4\nabla{\hat\phi}_k(0) \right|= O(\varepsilon_k^{\tau/2}).
\end{equation}
\subsection{The Vanishing Rate in $g$} \label{sec:original}
In this subsection, we verify that~\eqref{nov22e8} leads to~\eqref{vrate}. For simplicity, we assume without loss of generality that the cut-off function $\chi$ is supported in $B(q_{ik},2\delta)$ where $\delta$ is small enough to guarantee that $B(q_{ik},2\delta)$ are mutually
disjoint. Indeed, this can be done since the left hand side of~\eqref{vrate} is invariant under any change of cut-off function $\chi$. Under this choice of cut-off, all the terms $\nabla_1 G(q_{ik},q_{jk})$, $j\ne i$, reduce to $\nabla_1 \beta(q_{ik},q_{jk})$, so that it now suffices to show that:
\begin{equation} \label{mar25e1}
64\pi^2\sum_{j=1}^N\nabla_1\beta(q_{ik},q_{jk}) -4\nabla\phi_k(q_{ik}) =-\frac{\nabla h_k(q_{ik})}{h_k(q_{ik})}+O(\varepsilon_k^{\tau/2}).
\end{equation}
Indeed, $e^{-\tau u_k(q_{ik})/2}=O(\varepsilon_k^{\tau/2})$ since $|u_k(q_{ik})-u_k(q_{jk})|\le C$ for $i\neq j$, and furthermore
$\nabla_1\beta(q_{ik},q_{ik})=\nabla_2\beta(q_{ik},q_{ik})$ since $\beta(x,y)=\beta(y,x)$ by Lemma \ref{jan18l1} in the Appendix. The remainder of this section is devoted to verifying~\eqref{mar25e1}.
Taking the derivative with respect to $\xi$ in (\ref{uhat}) and
evaluating at $q_{ik}$, we have:
\begin{equation} \label{uk}
\nabla \hat
u_k(q_{ik})=2\int_M\nabla_1H(q_{ik},\eta)h_k(\eta)e^{4\hat u_k(\eta)}dV_{\hat g}(\eta)+\nabla {\hat\phi}_k(q_{ik}).
\end{equation}
where $H(\xi,\eta)=-(1/8\pi^2)\, \chi(r)\, \log d_{\hat g_k}(\xi,\eta)$. Similarly, for $u_k$ we have:
\begin{multline} \label{ukhat}
\nabla u_k(q_{ik})=-\frac{1}{8\pi^2}\int_M\nabla_{1} \bigl( \chi(r)\log d_g(q_{ik},\eta) \bigr) 2h_k(\eta)
e^{ 4\hat u_k(\eta)}dV_{\hat g}(\eta)\\
{} + \int_M\nabla_1\beta(q_{ik},\eta)2h_k(\eta)e^{4\hat
u_k(\eta)}dV_{\hat g}(\eta)-\nabla \phi_k(q_{ik}),
\end{multline}
where we used $e^{4\hat u_k}\,dV_{\hat g_k} = e^{4u_k}\, dV_g$.
Let $H_0(\xi,\eta) = (1/8\pi^2) \chi(r)\log d_g(\xi,\eta)$, then we claim that:
\begin{equation} \label{HH}
|\nabla_1H(q_{ik},\eta)-\nabla_1H_0(q_{ik},\eta)|\le C d_g(\xi,\eta).
\end{equation}
Indeed, recall that $\hat w(q_{ik})=0$ and $\nabla \hat w_k(q_{ik})=0$. Thus, if fix $\xi=q_{ik}$, and we let:
\[
f(\eta) = \log d_g(\xi,\eta) - \log d_{\hat g}(\xi,\eta),
\]
then $\hat w_k(\eta) = O(|\xi-\eta|^2)$, and therefore $|f(\eta)|\leq C(|\xi-\eta|^2)$. It follows that $\nabla f(\xi) = 0$. Now, by Appendix~\ref{appGreen},
$P_gf(\eta)$ is a bounded function, hence by elliptic theory, $\nabla^2 f(\eta)$ is bounded. We conclude that $|\nabla f(\eta)|\leq C|\xi-\eta|$ from which~\eqref{HH} follows. Next, since we have:
\[
\nabla u_k(q_{ik})=\nabla \hat u_k(q_{ik})+\nabla \hat
w_k(q_{ik})=\nabla \hat u_k(q_{ik}),
\]
it follows, by taking the difference of~\eqref{uk} and~\eqref{ukhat}, that:
\begin{equation} \label{Dphik}
0 = -\nabla {\hat\phi}_k(q_{ik})+\int_M\nabla_1\beta(q_{ik},\eta)2h_k(\eta)e^{4\hat
u_k(\eta)}dV_{\hat g}(\eta)-\nabla \phi_k(q_{ik})+O(\varepsilon_k)
\end{equation}
Furthermore, we claim that:
\begin{equation} \label{betak}
\int_M\nabla_1\beta(q_{ik},\eta)2h_k(\eta)e^{4\hat
u_k(\eta)}dV_{\hat g}(\eta) = 16\pi^2 \sum_{j=1}^N \nabla_1\beta(q_{ik},q_{jk}) + O(\varepsilon_k^\tau).
\end{equation}
Indeed, observe that:
\begin{multline*}
\int_M\nabla_1\beta(q_{ik},\eta)2h_k(\eta)e^{4\hat
u_k(\eta)}dV_{\hat g} \\
=\sum_{j=1}^N\int_{B(q_{jk},\delta)}\nabla_1\beta(q_{ik},\eta) 2 h_k(\eta) e^{4\hat u_k(\eta)}\, dV_{\hat
g}(\eta)+O(\varepsilon_k^4)
\end{multline*}
since, by~\eqref{MainEst}, $e^{4\hat u_k}=O(\varepsilon_k^4)$ on $M\setminus \cup_{j=1}^N B(q_{jk},\delta)$.
In addition, for each $j=1,..,N$:
\begin{multline*}
\int_{B(q_{jk},\delta)}\nabla_1\beta(q_{ik},\eta)2h_k(\eta)e^{4\hat
u_k(\eta)}dV_{\hat g}(\eta)\\
=\nabla_1\beta(q_{ik},q_{jk})\int_{B(q_{jk},\delta)}2h_ke^{4\hat u_k}dV_{\hat
g}+\int_{B(q_{jk},\delta)}O(|\eta-q_{jk}|)2h_ke^{4\hat u_k}dV_{\hat g}\\
=16\pi^2+O(\epsilon_k^{\tau}),
\end{multline*}
where we used~\eqref{MainEst} to estimate the first integral and a standard rescaling to estimate the second one.
By combining~\eqref{nov22e8} with~\eqref{Dphik} and~\eqref{betak}, it follows that \eqref{mar25e1} holds. This completes the proof of the Theorem.
\begin{rem}
Integrating~\eqref{jan22e1}, and using~\eqref{MainEst} on the right hand side, one easily obtains:
\[
\int_M b_k\, dV_g = 8\pi^2 N + O(\varepsilon_k^{\tau}).
\]
\end{rem}
|
2,877,628,090,150 | arxiv | \section{Introduction}
The process by which the chemical composition of a dense core changes as it evolves from a starless to a protostellar stage is not yet well understood (e.g., \citealt{Aikawa2013,Ceccarelli2017,Lefloch2018}). In particular, we do not know whether the prestellar/protostellar transition drives a clear change in the chemical composition of the host ambient cloud or whether there is no such chemical differentiation between prestellar and protostellar cores. Furthermore, it is unsure whether chemical variations from source to source arise from the different evolutionary status or are rather dominated by environmental characteristics and/or the particular history of each source.
It has been long thought that unsaturated carbon chains are associated to early starless evolutionary stages, while saturated complex organic molecules become dominant in more evolved protostellar phases \citep{Suzuki1992,Herbst2009}. In recent years however, radioastronomical observations have revealed unexpected results that challenge our understanding of the chemistry at the earliest stages of low-mass star formation. On the one hand, abundant carbon chains have been observed in various dense cores around protostars \citep{Sakai2008a,Sakai2009a,Agundez2008,Gupta2009,Hirota2009,Cordiner2013,Graninger2016a,Lindberg2016,Law2018}, and on the other, complex organic molecules such as methyl formate and dimethyl ether have been detected toward prestellar cores \citep{Bacmann2012,Jimenez-Serra2016}. These observational results are at the heart of a profound revision of the chemistry at the earliest stages of low-mass star formation that is still underway (e.g., \citealt{Aikawa2008,Aikawa2013,Hassel2008,Ruaud2015,Balucani2015,Kalvans2015,Hincelin2016,Vasyunin2017,Shingledecker2018}).
To shed light on the scenario that governs the chemical composition of dense cores during low-mass star formation it is mandatory to increase the number of prestellar and protostellar sources for which the chemical composition (i.e., the inventory of molecules and their abundances) is known in detail. Line surveys at millimeter wavelengths are an ideal tool for this purpose, since they allow for all the molecular lines lying within a relatively broad frequency range to be observed in a homogeneous and unbiased way (e.g., \citealt{Lefloch2018}).
Here we present a sensitive $\lambda$ 3 mm line survey carried out with the IRAM 30m telescope toward L483, a dark cloud core which contains a Class\,0 low-mass protostar that powers a bipolar outflow. The lack of important abundance enhancements of CH$_3$OH and SiO in the outflow, together with the presence of an infrared (IR) reflection nebula coincident with the outflow suggests that L483 may be in transition to Class\,I \citep{Tafalla2000}. However, the relatively high degree of collimation of the outflow and the fact that the protostar is still deeply embedded in the parental cloud may indicate that it is not a particularly evolved object within the Class\,0 phase. Apart from these peculiarities, there are various interesting chemical features. First, L483 is, together with a few other dense cores like L1527, one of the low-mass protostellar sources where brighter emission from carbon chains such as C$_4$H has been observed \citep{Agundez2008,Sakai2009a}. Second, observations with ALMA have recently unveiled the existence of a hot corino (i.e., warm complex organic molecules concentrated around the protostar) in this source \citep{Oya2017,Jacobsen2018}. And third, single-dish observations indicate that L483 has an exceptionally rich chemical composition, as indicated by the recent discoveries of various new interstellar molecules: the radical HCCO \citep{Agundez2015a}, the protonated form of cyanogen (NCCN) and its metastable isomer CNCN \citep{Agundez2015b,Agundez2018b}, the ion NS$^+$ \citep{Cernicharo2018}, the radical NCO and the ion H$_2$NCO$^+$, which are precursors of isocyanid acid and its isomers \citep{Marcelino2018a}, and the S-bearing radicals HCS and HSC \citep{Agundez2018a}. These chemical features make L483 an ideal target to carry out a line survey to study its chemical composition and evaluate a possible connection with evolutionary status.
\section{The source}
The source L483 is an optical dark cloud core located in the Aquila Rift star-forming region \citep{Lee1999}, which is estimated to lie at a distance of 200 pc \citep{Dame1985}. The core hosts an embedded infrared source, IRAS\,18148$-$0440, which is classified as a Class\,0 object based on its low bolometric temperature of 46-56 K \citep{Fuller1995}. The core is in a state of gravitational collapse, as evidenced by the line profiles of molecules like H$_2$CO and CS, which show the typical signature of infall motions, that is, enhanced self-absorption at redshifted velocities \citep{Myers1995,Mardones1997,Tafalla2000}. Interferometric observations of molecules like CS show hints of rotation of the envelope at different spatial scales \citep{Jorgensen2004,Leung2016,Oya2017,Jacobsen2018}. One of the most remarkable components present in L483 is a collimated bipolar outflow, well traced by CO and HCO$^+$, which extends out to 50-100$''$ from the IRAS source \citep{Fuller1995,Hatchell1999,Tafalla2000,Park2000,Oya2018}. Neither CH$_3$OH nor SiO show important abundance enhancements in the outflow \citep{Tafalla2000}, unlike in other Class\,0 sources like L1448 \citep{Martin-Pintado1992,Jimenez-Serra2005} or L1157 \citep{Bachiller1997}. The source also shows H$_2$ emission and a near-IR reflection nebula which coincide with the bipolar outflow and probe shocked gas present in the walls of the cavity opened by the outflow \citep{Fuller1995,Velusamy2014}. In summary, the evolutionary status of L483 is that of a Class\,0 source in which the protostar has disrupted the ambient infalling envelope through a collimated bipolar outflow.
Concerning the chemical properties of L483, we may summarize them as follows. The source shows abundant carbon chains distributed over the ambient cloud, as seen at low angular resolution ($\sim30''$) with single-dish telescopes \citep{Agundez2008,Sakai2009a}, and complex organic molecules concentrated around the protostar, as seen at sub-arcsecond angular resolution with ALMA \citep{Oya2017,Jacobsen2018}. The observations presented here, in which the beam size is 21-30$''$, cover three main components: the ambient cloud, the bipolar outflow, and the warm surroundings of the protostar. This latter component is severely diluted in the IRAM 30m beam at $\lambda$ 3 mm, while the outflow is only traced by the high-velocity emission of a few molecules like CO and HCO$^+$. Our observations therefore mainly probe the ambient quiescent cloud. We note that most of the mass in L483 is present in the ambient cloud rather than in the outflow. This is strongly suggested by the distribution of the continuum emission from dust (see lower panel in Fig.~\ref{fig:image}) and by the fact that the emission at high velocities only makes a small fraction of the total emission in the minor isotopologs of CO such as C$^{18}$O, which are less affected by optical depth effects (see Sect.~\ref{sec:line_profiles} for the $J$=1-0 line and \citealt{Tafalla2000} for the $J$=2-1 line). Therefore, for the great majority of molecules detected in this $\lambda$ 3 mm line survey, the observed emission arises exclusively from the ambient cloud.
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_l483_map}
\caption{\textit{Upper panel}: High-velocity CO $J$=2-1 emission (red/blue contours correspond to red/blue-shifted velocities; from \citealt{Tafalla2000}) is shown superimposed on the 3.6 $\mu$m \textit{Spitzer}/IRAC image. \textit{Lower panel}: Map of the $\lambda$ 1.3 mm dust continuum emission observed with MAMBO (Tafalla et al., unpublished data) is shown in both contours and color. First contour and contour interval are 20 mJy (11$''$-beam)$^{-1}$. The green circle in both panels corresponds to a size of 25$''$, representative of the region probed by the IRAM 30m telescope in the $\lambda$ 3 mm band.} \label{fig:image}
\end{figure}
\section{Observations}
\begin{figure*}
\centering
\includegraphics[angle=0,width=\textwidth]{fig_overview}
\caption{Overview of the $\lambda$ 3 mm line survey of L483 covering the frequency range 80-116 GHz. The carriers of the most intense lines are indicated. Negative signals correspond to the artifacts caused by the frequency-switching technique.} \label{fig:overview}
\end{figure*}
The observations were carried out with the IRAM 30m telescope, located in Pico Veleta (Granada, Spain), in several sessions from 2016 August to 2018 April and consisted of a spectral scan of L483 in the $\lambda$ 3 mm band, from 80 to 116 GHz. The telescope was pointed to the position of the infrared source IRAS\,18148$-$0440 \citep{Fuller1993}, with coordinates $\alpha_{2000.0}$ = 18$^{\rm h}$17$^{\rm m}$29\rasecp8, $\delta_{2000.0}$ = $-04^{\circ}$39$'$38$''$. This position coincides with the maximum of intensity of CH$_3$OH emission \citep{Tafalla2000}.
We used the EMIR receiver E090 in dual sideband mode, with image rejections of 10-20 dB (depending on the local oscillator frequency and polarization). These values were measured using intense lines. The E090 receiver was connected to a Fast Fourier Transform Spectrometer (FTS), which was operated in its narrow mode, providing a spectral resolution of 50 kHz, which translates to velocity resolutions of 0.19 km\,s$^{-1}$ at 80 GHz and 0.13 km\,s$^{-1}$ at 116 GHz. This spectral resolution is good enough to partially resolve the lines in the spectrum of L483, most of which have line widths of the order of 0.5 km s$^{-1}$. Thanks to the versatility of EMIR and FTS it is possible to cover a bandwidth of $4\times1.8$ GHz in a single tuning setup, the four parts corresponding to the Lower Outer, Lower Inner, Upper Inner, and Upper Outer bands. This allowed us to completely cover the 80-116 GHz frequency range using six tuning setups.
We employed the frequency-switching technique, in which the telescope is most efficient since no off position is needed to subtract the signal from the sky. This results in a sensitivity improvement of a factor of $\sqrt{2}$ with respect to the wobbler- and position-switching observing modes. A frequency throw of 7.2 MHz was adopted to minimize the effect of standing waves on the spectral baselines. Still, the obtained spectra showed baseline ripples with typical periods of 15 MHz and amplitudes of 0.1 K. This fact does not create a problem to distinguish astronomical lines, which in L483's spectrum are narrow, although it can make it more difficult to distinguish weak lines present at a level of a few $\sigma$ compared to spectra obtained with the wobbler-switching technique, which show rather flat baselines.
The intensity scale is calibrated using two absorbers at different temperatures and the atmospheric transmission model ATM \citep{Cernicharo1985,Pardo2001}. We express intensities in terms of $T_A^*$, the antenna temperature corrected for atmospheric absorption and for antenna ohmic and spillover losses. The uncertainty in $T_A^*$ due to calibration is estimated to be around 10 \%. To convert to main beam brightness temperature ($T_{mb}$) one has to divide $T_A^*$ by $B_{\rm eff}/F_{\rm eff}$, where $B_{\rm eff}$ = 0.863 $\exp{[-(\nu{\rm (GHz)}/361)^2]}$ and $F_{\rm eff}$ = 0.95\footnote{\texttt{http://www.iram.es/IRAMES/mainWiki/Iram30mEfficiencies}}. The telescope focus was checked on planets at the beginning of each session, which typically run for 3-6 h. The telescope pointing was regularly checked (every one and a half hours) by observing the nearby radio source 1741$-$038. Pointing errors were typically 2-3$''$, while the half power beam width (HPBW) of the IRAM 30m telescope ranges between 30$''$ at 80 GHz and 21$''$ at 116 GHz.
Weather conditions between the different observing sessions ranged from fairly good, with amounts of precipitable water vapor (PWV) of only 1-2 mm, to bad, with clouds and high amounts of water vapor. We nevertheless only included data obtained under reasonably good weather conditions, typically with PWV $<$ 10 mm. Average system temperatures were in the range 80-120 K for frequencies below 110 GHz and between 130 and 270 K at higher frequencies.
The data reduction was carried out with the program CLASS of the GILDAS software package\footnote{\texttt{http://www.iram.fr/IRAMFR/GILDAS}}. The raw data obtained at the telescope consist of frequency-switching spectra which are already folded. We found it necessary to go back to the unfolded spectra to verify a few suspicious signals which turned out to arise from spikes. After visual inspection of individual spectra, bad channels were removed and all spectra with the same frequency range, polarization, and observing date were averaged. Lines arising from the image side band were easily identified since data from the lower and upper side bands are available and also because in frequency-switching spectra image lines appear as negative signals once spectra have been folded. Telluric lines from stratospheric ozone were also easily identified due to their broad nature and thanks to the line list in the JPL Molecular Spectroscopy Catalogue\footnote{\texttt{https://spec.jpl.nasa.gov/}} \citep{Pickett1998}. Their presence however does not prevent from detecting overlapping astronomical lines, which are much narrower, although this was found to occur rarely. After spectra had been cleaned from image and telluric lines, all spectra with the same frequency range, independently of the polarization and observing date, were averaged to reduce the noise using the command {\small STITCH} of CLASS. Since spectra observed in different epochs are not perfectly aligned in frequency in the LSR frame, spectra need to be resampled to place all of them in the same rest frequency scale before averaging them. The action of resampling introduces a correlation in adjacent channels causing the noise to be slightly reduced. Lines also tend to be artificially broadened, especially the narrowest ones, typically by no more than 10 \%. The center frequency and area however remain unchanged. At this point of the data-reduction process we have $6\times4$ spectra which are 1.8 GHz wide, each covering a different frequency range.
We carried out the search for astronomical lines before subtracting a baseline from the data, with all spectra averaged to reduce the
noise as much as possible. Given the rather irregular baselines of the frequency-switching spectra, baseline subtraction is a delicate issue, and weak lines are better distinguished by visual inspection before a baseline has been subtracted. The assignment of the detected lines to known rotational transitions of molecules was done by checking the Cologne Database for Molecular Spectroscopy $\footnote{\texttt{https://cdms.astro.uni-koeln.de/}}$ \citep{Muller2005}, the JPL Molecular Spectroscopy Catalogue \citep{Pickett1998}, and the private catalogue of J. Cernicharo generated from the MADEX code\footnote{\texttt{https://nanocosmos.iff.csic.es/?page\_id=1619}} \citep{Cernicharo2012:madex}.
Once the position of all astronomical lines was known, we subtracted a baseline from each of the $6\times4$ 1.8 GHz-wide spectra. Since polynomials are not adequate for such a wide frequency range, the baseline is generated through a procedure involving the smoothing of the channels that are free of lines. After baseline subtraction, the $6\times4$ spectra, some of which overlap in certain frequency ranges, were averaged again using the command {\small STITCH} to increase the sensitivity in the overlapping regions.
The observed line profiles were fitted to Gaussian functions. Some lines showed nonGaussian line shapes, mainly consisting of wings in lines that arise to some extent from the outflow or self-absorption in some optically thick lines. In these cases no attempt was made to fit a Gaussian and only the observed area and peak intensity were retrieved. An overview of the line survey with the most intense lines is shown in Figure~\ref{fig:overview}, while the whole data set is shown in Fig.~\ref{fig:spectrum} with the frequency and intensity scales chosen to permit the visualization of the weakest lines. The frequency range 80-116 GHz has been completely covered. The blank spaces that appear from time to time in Fig.~\ref{fig:spectrum} correspond to spectral regions that have been removed due to contamination with image side-band lines or telluric ozone lines. The negative artifacts produced by the frequency-switching technique, which consist of two negative signals located at $\pm$ 7.2 MHz of each line with half its intensity, have not been removed. As can be seen in Fig.~\ref{fig:spectrum}, the sensitivity achieved is very good, with $T_A^*$ rms noise levels of 1-3 mK per 50 kHz channel below 110 GHz, increasing up to noise levels of 10 mK at the high-frequency edge of 115-116 GHz.
\begin{figure*}
\centering
\includegraphics[angle=0,width=\textwidth]{fig_line_profiles}
\caption{Characteristic line profiles observed with the IRAM 30m telescope in the $\lambda$ 3 mm band toward L483. The left panel shows lines with wings and self-absorption (except C$^{18}$O $J$=1-0), the middle panel lines that show wings, and the right panel a sample of narrow lines, which form the majority in our line survey. The gray shaded area indicates the velocity range $\pm$ 0.5 km s$^{-1}$ around the systemic velocity of the source.} \label{fig:line_profiles}
\end{figure*}
\section{Results and discussion} \label{sec:results}
The $\lambda$ 3 mm line survey between 80 and 116 GHz resulted in the detection of 631 lines. Among them, 613 lines were assigned to known rotational transitions of molecules and are listed in Table~\ref{table:lines} together with their associated parameters\footnote{We use the $e$/$f$ labeling of parity levels for linear molecules with a $^2\Pi$ ground electronic state \citep{Brown1975}}. The total number of molecules detected is 71 (140 if different isotopologs are considered). These molecules can be grouped into the chemical families of O-bearing molecules, hydrocarbons, N-bearing molecules, S-containing species, and one Si-bearing molecule (SiO). There are also 18 lines which could not be assigned to known molecular transitions and therefore remain unidentified. Some of them have previously been observed in other sources; these are presented in Table~\ref{table:ulines}.
\subsection{Line profiles} \label{sec:line_profiles}
All the lines are observed in emission, at the exception of the CH$_3$OH transition 3$_1$-4$_0$ $A+$ lying at 107013.831 MHz, which is observed in absorption. There are several types of profiles among the emission lines detected. Some lines show more or less bright wings, which indicate that part of the emission arises from the outflow. Among them, a few optically thick lines show self-absorption at the line center. From the 71 molecules detected, only 14 (those shown in the left and middle panels of Fig.~\ref{fig:line_profiles}) show wings in their line profiles. The rest, that is the vast majority of the molecules detected (see a sample in the right panel of Fig.~\ref{fig:line_profiles}), show narrow line profiles, with the emission restricted to the velocity range $V_{\rm LSR}-v_{\rm sys}$ between $-0.5$ and $+0.5$ km s$^{-1}$ (gray shadowed area in Fig.~\ref{fig:line_profiles}). The emission of these narrow lines arises exclusively from the ambient quiescent cloud. From the lines of all these species for which we could fit a Gaussian function, we derive an average systemic velocity $v_{\rm sys} = 5.30 \pm 0.11$ km s$^{-1}$ and a mean line width FWHM = $0.52 \pm 0.15$ km s$^{-1}$. The systemic velocity is in good agreement with previously reported values. For example, \cite{Fuller1993} derive values of $v_{\rm sys}$ between 5.237 and 5.460 km s$^{-1}$ from high-spectral-resolution observations of HC$_3$N and NH$_3$.
The line that shows emission at the highest velocities is CO $J$=1-0, the emission of which extends up to $\sim$8 km s $^{-1}$ from the systemic velocity of the source on both the blue and red sides (see the top line in the left panel of Fig.~\ref{fig:line_profiles}). The CO $J$=1-0 line has a complex shape, very similar to that of the CO $J$=2-1 line observed with the IRAM 30m by \cite{Tafalla2000}, showing asymmetric wings, with more high-velocity emission in the red side, and self-absorption around the systemic velocity ($V_{\rm LSR}-v_{\rm sys}$ = $\pm$ 1 km s$^{-1}$) that results from cold CO gas in the outer parts of the envelope. The second line with the highest-velocity wings is HCO$^+$ $J$=1-0, for which emission extends up to $\sim$7 km s$^{-1}$ from the velocity of the ambient cloud. This is consistent with the fact that both CO and HCO$^+$ are good tracers of the outflow (e.g., \citealt{Fuller1995,Park2000}). We note however that as we move to less optically thick lines (e.g., CO $\rightarrow$ $^{13}$CO $\rightarrow$ C$^{18}$O; see left panel in Fig.~\ref{fig:line_profiles}) the contribution of the wings becomes vanishingly small, which indicates that the fraction of mass in the outflow is much lower than in the quiescent ambient cloud.
Lines from other molecules also show more or less bright wings extending out to velocities of 2-4 km s$^{-1}$. For example, the molecules HCN, HNC, CN, and CS show relatively bright wings (see left panel in Fig.~\ref{fig:line_profiles}), which indicates that part of the detected emission comes from the bipolar outflow. Other molecules like CH$_3$OH, C$_2$H, $c$-C$_3$H$_2$, HC$_3$N, C$_2$S, H$_2$CS, and SO display in some of their lines weak wings (see middle panel in Fig.~\ref{fig:line_profiles}), which indicates that most of the detected emission comes from the ambient cloud with a minor contribution from the outflow. The low abundance of methanol in the outflow is consistent with the map by \cite{Tafalla2000}, which indicates that CH$_3$OH does not trace the outflow but presents an extended distribution over the core. For SiO we detect weak emission from the $J$=2-1 line ($T_A^*$ $\sim$0.03 K, with a velocity-integrated intensity of 0.086 $\pm$ 0.008 K km s$^{-1}$, consistent with the upper limit of 0.1 K km s$^{-1}$ by \citealt{Tafalla2000}). The line however is wide (see bottom line in middle panel of Fig.~\ref{fig:line_profiles}), which indicates that most of the emission does not arise from the ambient cloud, but probably from the outflow. Recently, \cite{Oya2017} detected and imaged with ALMA at subarcsecond resolution the $J$=6-5 line of SiO. Emission in this line does not extend along the outflow but shows a compact distribution ($<$1$''$) around the protostar, which in part can be explained by the high excitation requirements of the line ($E_{\rm up}$ = 43.8 K) and the lack of short spacings in the ALMA data, which causes any extended emission to be filtered out. It is therefore unclear whether SiO is mostly present in the outflow or around the protostar. In any case, it is clear that neither SiO nor CH$_3$OH reaches high abundances in the L483 outflow, unlike in other Class\,0-powered outflows like L1157 or L1448.
Apart from the wings, some of the brightest lines show evidence of self-absorption near the cloud systemic velocity (see left panel in Fig.~\ref{fig:line_profiles}). In the lines of the dense gas tracers, the self-absorption feature lies at higher velocity than the systemic component, suggesting that the self-absorption is caused by a foreground layer of gas that is moving toward the cloud. This interpretation is in agreement with evidence for infall motions previously reported from H$_2$CO and CS observations \citep{Myers1995,Mardones1997,Tafalla2000}. Our data however show that the lines of CN, HCN, and CS present a brighter blue peak, while the lines of HCO$^+$ and HNC have a brighter red peak. This variety of profile shapes may be caused by the varying contribution of the outflow wings, or may indicate that the line-of-sight motions in the cloud are more complex than those expected for a simple collapse pattern. Similar line-shape differences between dense gas tracers have previously been seen toward other star-forming regions \citep{Lee2004}, meaning that they are not unique to L483.
\subsection{Column densities and temperatures} \label{sec:column_densities}
From the observed lines we derived beam-averaged column densities assuming local thermodynamic equilibrium (LTE). For some species, only lower limits to the column density could be obtained because the observed lines are optically thick. In general, we adopted the same rotational temperature for all the isotopologs of a given molecule to ensure that isotopic ratios are not affected by different rotational temperatures. When a precise determination of the rotational temperature is possible for various isotopologs (because a high-enough number of lines spanning a wide-enough range of upper level energies is available), we adopt the value that is most accurately derived. If the rotational temperature cannot be precisely determined for any isotopolog, we adopt the value derived for a chemically related species; for example, for C$_5$H we adopt the rotational temperature derived for C$_4$H. When there was no obvious choice for the rotational temperature, we adopted a value of 10 K, which is close to the gas kinetic temperature in L483 (see below).
\begin{table}
\caption{Beam-averaged column densities and rotational temperatures} \label{table:column_densities}
\small
\centering
\begin{tabular}{lrrcr}
\hline \hline
\multicolumn{1}{l}{Species} & \multicolumn{1}{c}{$N$ (cm$^{-2}$)$^a$} & \multicolumn{1}{c}{$T_{\rm rot}$ (K)} & \multicolumn{1}{c}{$E_{\rm up}$ (K)} & \multicolumn{1}{c}{$N_{\rm lines}$$^b$} \\
\hline
\multicolumn{5}{c}{O-bearing molecules} \\
\hline
CO & $>$$6.2\times10^{16}$ & 10$^c$ & 5.5 - 5.5 & 1 \\
$^{13}$CO & $>$$1.4\times10^{16}$ & 10$^c$ & 5.3 - 5.3 & 1 \\
C$^{18}$O & $>$$4.8\times10^{15}$ & 10$^c$ & 5.3 - 5.3 & 1 \\
C$^{17}$O & $1.7\times10^{15}$ & 10$^c$ & 5.4 - 5.4 & 2 \\
$^{13}$C$^{18}$O & $1.3\times10^{14}$ & 10$^c$ & 5.0 - 5.0 & 1 \\
$^{13}$C$^{17}$O & $4.1\times10^{13}$ & 10$^c$ & 5.1 - 5.1 & 2 \\
C$_2$O & $5.4\times10^{11}$ & 10$^c$ & 11.0 - 11.5 & 2 \\
C$_3$O & $7.6(49)\times10^{11}$ & 10.2(21) & 20.8 - 36.0 & 4 \\
HCO & $2.8\times10^{12}$ & 10$^c$ & 4.2 - 4.2 & 3 / 4 \\
HCCO & $3.1\times10^{11}$ & 10$^c$ & 10.4 - 10.4 & 4 \\
CH$_3$O & $8.0\times10^{11}$ & 10$^c$ & 4.0 - 4.0 & 3 / 4 \\
D$_2$CO & $1.4\times10^{12}$ & 10$^c$ & 5.3 - 5.3 & 1 \\
CH$_3$OH$^d$ & & & 4.6 - 32.5 & 9 \\
CH$_2$DOH & $5.5(18)\times10^{12}$ & 4.3(3) & 6.4 - 25.8 & 7 / 8 \\
CH$_3$OD & $4.0\times10^{12}$ & 4.3$^c$ & 6.5 - 6.5 & 1 \\
CHD$_2$OH & $8.2\times10^{11}$ & 4.3$^c$ & 6.0 - 6.0 & 1 \\
$^{13}$CH$_3$OH & $4.3\times10^{12}$ & 4.3$^c$ & 4.5 - 6.8 & 2 \\
H$_2$CCO & $4.1\times10^{12}$ & 10$^c$ & 8.6 - 14.5 & 6 \\
HDCCO & $3.7\times10^{11}$ & 10$^c$ & 13.5 - 19.0 & 2 \\
CH$_3$CHO & $3.8(10)\times10^{12}$ & 5.8(5) & 4.9 - 23.0 & 18 \\
HCCCHO & $8.3(17)\times10^{11}$ & 7.6(5) & 4.4 - 27.2 & 9 / 10 \\
$c$-C$_3$H$_2$O & $2.3(13)\times10^{11}$ & 7.9(13) & 14.0 - 28.0 & 11 \\
$t$-HCOOH & $1.2(8)\times10^{12}$ & 7.0(17) & 10.8 - 18.8 & 5 \\
$c$-HCOOH & $7.2\times10^{10}$ & 7.0$^c$ & 4.6 - 14.5 & 3 \\
C$_2$H$_5$OH & $2.0\times10^{12}$ & 4.3$^c$ & 9.3 - 9.3 & 1 \\
HCOOCH$_3$ & $2.3\times10^{12}$ & 10$^c$ & 17.4 - 24.9 & 16 / 17 \\
CH$_3$OCH$_3$ & $5.3\times10^{12}$ & 10$^c$ & 6.7 - 19.0 & 6 / 9 \\
HCO$^+$ & $>$$6.7\times10^{12}$ & 10$^c$ & 4.3 - 4.3 & 1 \\
H$^{13}$CO$^+$ & $>$$2.5\times10^{12}$ & 10$^c$ & 4.2 - 4.2 & 1 \\
HC$^{18}$O$^+$ & $4.0\times10^{11}$ & 10$^c$ & 4.1 - 4.1 & 1 \\
HC$^{17}$O$^+$ & $1.3\times10^{11}$ & 10$^c$ & 4.2 - 4.2 & 3 \\
HCO$_2$$^+$ & $3.1\times10^{11}$ & 10$^c$ & 10.3 - 15.4 & 2 \\
DCO$_2$$^+$ & $4.1\times10^{10}$ & 10$^c$ & 9.6 - 9.6 & 1 / 2 \\
\hline
\multicolumn{5}{c}{Hydrocarbons} \\
\hline
C$_2$H & $>$$5.1\times10^{14}$ & 10$^c$ & 4.2 - 4.2 & 2 / 6 \\
$^{13}$CCH & $3.1\times10^{12}$ & 10$^c$ & 4.0 - 4.0 & 7 / 8 \\
C$^{13}$CH & $7.2\times10^{12}$ & 10$^c$ & 4.1 - 4.1 & 6 / 7 \\
$c$-C$_3$H & $7.9\times10^{12}$ & 10$^c$ & 4.4 - 4.4 & 9 \\
$c$-C$_3$D & $3.5\times10^{11}$ & 10$^c$ & 5.4 - 5.4 & 1 \\
$l$-C$_3$H & $6.4(18)\times10^{11}$ & 10.5(15) & 12.5 - 28.0 & 6 \\
C$_4$H & $1.2(5)\times10^{14}$ & 7.9(8) & 20.5 - 35.6 & 8 \\
C$_4$D & $2.3\times10^{12}$ & 7.9$^c$ & 23.3 - 28.0 & 4 \\
C$_5$H & $7.2\times10^{11}$ & 7.9$^c$ & 37.0 - 37.0 & 2 \\
$c$-C$_3$H$_2$ & $2.1\times10^{14}$ & 4.1$^c$ & 26.7 - 28.8 & 2 / 6 \\
$c$-HCC$^{13}$CH & $4.1(15)\times10^{12}$ & 4.1(3) & 6.3 - 15.9 & 9 \\
$c$-HC$^{13}$CCH & $4.7\times10^{11}$ & 4.1$^c$ & 3.9 - 9.4 & 3 \\
$c$-C$_3$HD & $2.2(7)\times10^{13}$ & 4.1(2) & 7.6 - 26.6 & 7 / 10 \\
$c$-C$_3$D$_2$ & $2.1\times10^{12}$ & 4.1$^c$ & 6.1 - 20.2 & 5 \\
$c$-H$^{13}$CCCD & $2.0\times10^{11}$ & 4.1$^c$ & 10.7 - 10.7 & 2 \\
$c$-HCC$^{13}$CD & $2.5\times10^{11}$ & 4.1$^c$ & 10.6 - 10.6 & 1 / 2 \\
$c$-HC$^{13}$CCD $^e$ & $2.5\times10^{11}$ & 4.1$^c$ & 10.6 - 10.6 & 1 \\
$l$-C$_3$H$_2$ & $1.1(7)\times10^{12}$ & 5.4(12) & 8.9 - 15.0 & 6 \\
$l$-C$_3$HD & $8.1\times10^{10}$ & 5.4$^c$ & 14.0 - 22.9 & 2 \\
H$_2$C$_4$ & $3.5(14)\times10^{11}$ & 9.6(13) & 18.8 - 33.4 & 12 \\
CH$_3$CCH & $9.3\times10^{13}$ & 10.2$^c$ & 11.5 - 82.3 & 8 \\
CH$_2$DCCH & $1.8(3)\times10^{13}$ & 10.2(7) & 11.6 - 43.6 & 15 \\
$^{13}$CH$_2$DCCH & $7.9\times10^{11}$ & 10.2$^c$ & 15.9 - 15.9 & 1 \\
CH$_3$CCD & $5.3\times10^{12}$ & 10.2$^c$ & 15.0 - 20.9 & 4 \\
$^{13}$CH$_3$CCH & $1.5\times10^{12}$ & 10.2$^c$ & 11.2 - 16.8 & 4 \\
CH$_3$$^{13}$CCH & $1.7\times10^{12}$ & 10.2$^c$ & 11.5 - 17.2 & 4 \\
CH$_3$C$^{13}$CH & $1.6\times10^{12}$ & 10.2$^c$ & 11.1 - 16.7 & 4 \\
CH$_3$C$_4$H & $7.6\times10^{12}$ & 10$^c$ & 40.8 - 45.1 & 4 \\
\hline
\end{tabular}
\end{table}
\setcounter{table}{0}
\begin{table}
\caption{Continued}
\small
\centering
\begin{tabular}{lrrcr}
\hline \hline
\multicolumn{1}{l}{Species} & \multicolumn{1}{c}{$N$ (cm$^{-2}$)$^a$} & \multicolumn{1}{c}{$T_{\rm rot}$ (K)} & \multicolumn{1}{c}{$E_{\rm up}$ (K)} & \multicolumn{1}{c}{$N_{\rm lines}$$^b$} \\
\hline
\multicolumn{5}{c}{N-bearing molecules} \\
\hline
NH$_2$D & $7.1\times10^{13}$ & 10$^c$ & 20.1 - 21.3 & 8 / 10 \\
CN & $>$$6.6\times10^{13}$ & 10$^c$ & 5.4 - 5.4 & 2 / 9 \\
$^{13}$CN & $6.4\times10^{12}$ & 10$^c$ & 5.2 - 5.2 & 10 / 21 \\
C$^{15}$N & $7.6\times10^{11}$ & 10$^c$ & 5.3 - 5.3 & 6 \\
HCN & $>$$2.4\times10^{13}$ & 10$^c$ & 4.3 - 4.3 & 1 / 3 \\
H$^{13}$CN & $3.9\times10^{12}$ & 10$^c$ & 4.1 - 4.1 & 1 / 3 \\
HC$^{15}$N & $4.1\times10^{11}$ & 10$^c$ & 4.1 - 4.1 & 1 \\
H$^{13}$C$^{15}$N & $1.2\times10^{10}$ & 10$^c$ & 4.0 - 4.0 & 1 \\
HNC & $>$$7.8\times10^{12}$ & 10$^c$ & 4.4 - 4.4 & 1 \\
HN$^{13}$C & $>$$3.7\times10^{12}$ & 10$^c$ & 4.2 - 4.2 & 1 \\
H$^{15}$NC & $8.2\times10^{11}$ & 10$^c$ & 4.3 - 4.3 & 1 \\
H$^{15}$N$^{13}$C & $2.8\times10^{10}$ & 10$^c$ & 4.1 - 4.1 & 1 \\
H$_2$CN $^f$ & $2.4\times10^{12}$ & 10$^c$ & 3.5 - 3.5 & 9 / 13 \\
C$_3$N & $2.9(18)\times10^{12}$ & 8.5(15) & 21.4 - 31.3 & 6 \\
HC$_3$N & $4.2\times10^{13}$ & 9.1$^c$ & 19.6 - 28.8 & 6 / 10 \\
H$^{13}$CCCN & $4.6\times10^{11}$ & 9.1$^c$ & 23.3 - 33.0 & 3 \\
HC$^{13}$CCN & $4.5\times10^{11}$ & 9.1$^c$ & 19.6 - 33.9 & 4 \\
HCC$^{13}$CN & $5.3(3.1)\times10^{11}$ & 9.1(16) & 19.6 - 33.9 & 4 \\
HC$_3$$^{15}$N & $8.6\times10^{10}$ & 9.1$^c$ & 23.3 - 28.0 & 2 \\
DC$_3$N & $1.2\times10^{12}$ & 9.1$^c$ & 22.3 - 36.9 & 4 \\
DC$^{13}$CCN & $3.9\times10^{10}$ & 9.1$^c$ & 22.2 - 22.2 & 1 \\
HCCNC & $5.7\times10^{11}$ & 9.1$^c$ & 21.5 - 31.5 & 3 \\
HNC$_3$ & $5.0\times10^{10}$ & 9.1$^c$ & 20.2 - 24.6 & 2 \\
HC$_5$N & $7.6(28)\times10^{11}$ & 28(3) & 63.4 - 110.0 & 11 \\
CH$_2$CN & $1.2\times10^{12}$ & 10$^c$ & 8.8 - 13.7 & 8 / 37 \\
CH$_3$CN & $4.1\times10^{11}$ & 10$^c$ & 12.4 - 18.5 & 4 / 5 \\
CH$_2$DCN & $5.4\times10^{10}$ & 10$^c$ & 12.5 - 18.0 & 4 \\
CH$_3$NC & $3.7\times10^{10}$ & 10$^c$ & 13.5 - 14.5 & 2 / 3 \\
CH$_3$C$_3$N & $5.0\times10^{11}$ & 10$^c$ & 41.6 - 45.8 & 2 / 4 \\
C$_2$H$_3$CN & $1.9\times10^{11}$ & 10$^c$ & 20.4 - 29.2 & 7 \\
CNCN & $1.9\times10^{12}$ & 10$^c$ & 17.9 - 27.3 & 3 \\
HNO & $1.2\times10^{12}$ & 10$^c$ & 3.9 - 3.9 & 1 \\
N$_2$O & $5.8\times10^{12}$ & 10$^c$ & 12.1 - 12.1 & 1 \\
NCO & $2.2\times10^{12}$ & 10$^c$ & 6.6 - 11.7 & 10 \\
HNCO & $1.7\times10^{13}$ & 10$^c$ & 10.5 - 15.8 & 4 / 6 \\
HN$^{13}$CO & $2.7\times10^{11}$ & 10$^c$ & 10.5 - 15.8 & 2 \\
HNC$^{18}$O & $7.2\times10^{10}$ & 10$^c$ & 10.0 - 10.0 & 1 \\
DNCO & $6.4\times10^{11}$ & 10$^c$ & 9.8 - 14.7 & 2 \\
HOCN & $1.5\times10^{11}$ & 10$^c$ & 10.1 - 15.1 & 2 \\
HCNO & $7.0\times10^{10}$ & 10$^c$ & 11.0 - 16.5 & 2 \\
N$_2$H$^+$ & $>$$6.0\times10^{13}$ & 10$^c$ & 4.5 - 4.5 & 1 / 7 \\
$^{15}$NNH$^+$ & $8.0\times10^{10}$ & 10$^c$ & 4.3 - 4.3 & 3 \\
N$^{15}$NH$^+$ & $1.3\times10^{11}$ & 10$^c$ & 4.4 - 4.4 & 3 \\
HCNH$^+$ $^f$ & $2.7\times10^{13}$ & 10$^c$ & 3.6 - 3.6 & 1 \\
HC$_3$NH$^+$ & $2.3\times10^{11}$ & 9.1$^c$ & 22.9 - 27.4 & 2 \\
NCCNH$^+$ & $1.5\times10^{10}$ & 10$^c$ & 23.4 - 28.1 & 2 \\
H$_2$NCO$^+$ & $2.9\times10^{10}$ & 10$^c$ & 8.7 - 14.6 & 6 \\
\hline
\multicolumn{5}{c}{S-bearing molecules} \\
\hline
CS & $>$$2.3\times10^{13}$ & 10$^c$ & 7.1 - 7.1 & 1 \\
$^{13}$CS & $1.7\times10^{12}$ & 10$^c$ & 6.7 - 6.7 & 1 \\
C$^{34}$S & $3.2\times10^{12}$ & 10$^c$ & 6.9 - 6.9 & 1 \\
C$^{33}$S & $8.1\times10^{11}$ & 10$^c$ & 7.0 - 7.0 & 4 \\
$^{13}$C$^{34}$S & $5.5\times10^{10}$ & 10$^c$ & 6.5 - 6.5 & 1 \\
C$_2$S & $4.9(39)\times10^{12}$ & 8.8(19) & 23.3 - 33.6 & 5 / 8 \\
C$_2$$^{34}$S & $2.9\times10^{11}$ & 8.8$^c$ & 19.5 - 24.5 & 2 / 3 \\
C$^{13}$CS & $1.7\times10^{11}$ & 8.8$^c$ & 15.3 - 15.3 & 2 / 4 \\
C$_3$S & $1.2(6)\times10^{12}$ & 9.6(11) & 29.1 - 52.7 & 5 / 6 \\
HCS & $7.3\times10^{12}$ & 10$^c$ & 5.8 - 5.8 & 5 \\
HSC & $2.0\times10^{11}$ & 10$^c$ & 5.9 - 5.9 & 1 / 2 \\
H$_2$CS & $1.4\times10^{13}$ & 8.0$^c$ & 8.1 - 9.9 & 3 \\
H$_2$$^{13}$CS & $1.2\times10^{11}$ & 8.0$^c$ & 7.8 - 9.5 & 3 \\
H$_2$C$^{34}$S & 4.6$\times10^{11}$ & 8.0$^c$ & 8.0 - 8.3 & 2 / 3 \\
HDCS & $1.9(10)\times10^{12}$ & 8.0(18) & 8.9 - 18.1 & 3 \\
D$_2$CS & $6.2\times10^{11}$ & 8.0$^c$ & 6.6 - 13.6 & 5 \\
\hline
\end{tabular}
\end{table}
\setcounter{table}{0}
\begin{table}
\caption{Continued}
\small
\centering
\begin{tabular}{lrrcr}
\hline \hline
\multicolumn{1}{l}{Species} & \multicolumn{1}{c}{$N$ (cm$^{-2}$)$^a$} & \multicolumn{1}{c}{$T_{\rm rot}$ (K)} & \multicolumn{1}{c}{$E_{\rm up}$ (K)} & \multicolumn{1}{c}{$N_{\rm lines}$$^b$} \\
\hline
CH$_3$SH & $2.4\times10^{12}$ & 10$^c$ & 8.4 - 8.5 & 2 / 4 \\
SO & $2.0\times10^{14}$ & 4.5$^c$ & 19.3 - 21.1 & 2 / 4 \\
$^{34}$SO & $6.5(36)\times10^{12}$ & 4.5(5) & 9.1 - 20.9 & 3 \\
$^{33}$SO & $1.4\times10^{12}$ & 4.5$^c$ & 9.2 - 9.2 & 4 / 5 \\
S$^{18}$O~~~~~~~~~~~ & $1.3\times10^{12}$ & 4.5$^c$ & 8.7 - 8.7 & 1 \\
S$^{17}$O & $1.6\times10^{12}$ & 4.5$^c$ & 9.0 - 9.0 & 1 \\
SO$_2$ & $4.0(12)\times10^{12}$ & 7.9(7) & 7.7 - 36.7 & 2 \\
$^{34}$SO$_2$ & $1.3\times10^{11}$ & 7.9$^c$ & 7.6 - 7.6 & 1 \\
OCS & $1.6(12)\times10^{13}$ & 7.8(17) & 16.3 - 26.3 & 3 \\
NS & $5.4\times10^{12}$ & 10$^c$ & 8.8 - 8.9 & 10 \\
N$^{34}$S & $2.4\times10^{11}$ & 10$^c$ & 8.7 - 8.7 & 2 \\
HNCS & $2.1\times10^{11}$ & 10$^c$ & 15.8 - 25.3 & 3 \\
HSCN & $1.1\times10^{11}$ & 10$^c$ & 15.4 - 19.8 & 2 \\
HCS$^+$ & $1.3\times10^{12}$ & 10$^c$ & 6.1 - 6.1 & 1 \\
HC$^{34}$S$^+$ & $6.3\times10^{10}$ & 10$^c$ & 6.0 - 6.0 & 1 \\
SO$^+$ & $1.6\times10^{12}$ & 4.5$^c$ & 8.9 - 8.9 & 1 \\
NS$^+$ & $2.1\times10^{11}$ & 10$^c$ & 7.2 - 7.2 & 4 \\
\hline
\multicolumn{5}{c}{Si-bearing molecules} \\
\hline
SiO~~~~~~~~~~~~~~~~~ & $2.6\times10^{11}$ & 10$^c$ & 6.3 - 6.3 & 1 \\
\hline
\end{tabular}
\tablenotea{Numbers in parentheses are 1$\sigma$ uncertainties in units of the last digits. Fractional abundances relative to H$_2$ can be directly computed from the column densities listed using a column density of H$_2$ of $4\times10^{22}$ cm$^{-2}$ (see Sect.~\ref{sec:column_densities}). \\
$^a$ The error in the column density for those species for which the rotational temperature has been fixed is estimated to be 50 \% (see text).\\
$^b$ Number of lines observed. The notation $x/y$ means that $x$ lines, out of $y$ observed lines, were included in the determination of the column density.\\
$^c$ Rotational temperature has been fixed.\\
$^d$ Impossible to fit lines to a rotational diagram. The column density of CH$_3$OH is estimated to be $2.9\times10^{14}$ cm$^{-2}$ based on $^{13}$CH$_3$OH and adopting a $^{12}$C/$^{13}$C isotopic ratio of 68 \citep{Milam2005}.\\
$^e$ Tentative detection. \\
$^f$ Molecule observed at frequencies below 80 GHz (Ag\'undez et al., unpublished data).}
\end{table}
The column densities of the 140 species detected, together with the rotational temperatures assumed or derived, are given in Table~\ref{table:column_densities}. We also give column densities for H$_2$CN and HCNH$^+$, molecules that are detected at frequencies below 80 GHz (Ag\'undez et al., unpublished data). For the species for which the rotational temperature was derived, an uncertainty in both the column density and the rotational temperature was calculated from the least-squares fit. These uncertainties include the 10~\% error in the observed line intensities due to calibration. For the species for which uncertainties in the column densities were calculated, these range from 20 to 80~\%. Based on these numbers, we estimate that the column densities given in Table~\ref{table:column_densities} have errors of the order of 50~\%.
The gas kinetic temperature in L483 has been determined to be 10 K in a NH$_3$ condensation which appears centered onto the protostar and extends over a region of 40-60$''$ in diameter, about twice the size of the IRAM 30m main beam at $\lambda$ 3 mm \citep{Anglada1997}. From our CO data we can obtain an estimation of the gas kinetic temperature. The CO $J$=1-0 line is optically thick and shows self-absorption around the velocity of the ambient cloud (see top line in left panel of Fig.~\ref{fig:line_profiles}). In the region of self-absorption the line has $T_A^*$ = 4 K ($T_{\rm mb}$ = 4.9 K). Given that in this velocity range the line is optically thick, the excitation temperature of the absorbing CO gas located in the outer parts of the cloud has to be 8.2 K, a value which is probably close to the kinetic temperature of the gas in that region. Similar numbers (8.5-9.5 K) were inferred from observations of the CO $J$=2-1 line by \cite{Tafalla2000}. The $^{13}$CO $J$=1-0 line is also optically thick and has an intensity of $T_A^*$ = 5.6 K ($T_{\rm mb}$ = 6.7 K) around the systemic velocity of the ambient cloud (see left panel in Fig.~\ref{fig:line_profiles}), which implies an excitation temperature of 10 K. This value is a good estimate of the gas kinetic temperature in the region of the cloud where $^{13}$CO $J$=1-0 becomes optically thick. An independent estimate of the gas kinetic temperature is provided by the relative intensities of the $K$=0-3 components of CH$_3$CCH observed in the $J$=5-4 and $J$=6-5 transitions, which yield a gas kinetic temperature of 15 $\pm$ 2 K. This value is somewhat higher than the above estimates. It is likely that the different gas kinetic temperature tracers probe different regions, with $^{13}$CO and NH$_3$ probing a similar region, while the optically thick lines of CO probe outer and colder parts of the cloud and CH$_3$CCH traces inner and warmer regions.
The rotational temperatures derived for the different molecules range from 4.1 to 10.5 K. A notable outsider is HC$_5$N, for which we derive $T_{\rm rot}$ = 28 $\pm$ 3 K. Line widths for this molecule are also slightly higher
(0.87 km s$^{-1}$ on average) than for most other species showing narrow lines. These two facts suggest that HC$_5$N is probably distributed in a warmer region than the other molecules, probably closer to the protostar. Apart from HC$_5$N, the low excitation temperatures derived together with the small line widths are consistent with emission arising from a cold and extended part of the dense core. The gas densities at these scales are probably not too high, as indicated by the low rotational temperatures (down to 4 K) of some molecules, which strongly suggest subthermal excitation in a low-density gas at a kinetic temperature around 10 K. From a model of the continuum emission in L483, \cite{Jorgensen2002} derive a volume density of H$_2$ of $3.4\times10^4$ cm$^{-3}$ at the radius at which the gas and dust temperature become 10 K, which in their model occurs at $7.8\times10^3$ au (or 40$''$ at 200 pc) from the protostar. From ammonia observations, \cite{Anglada1997} derive a similar H$_2$ volume density of $3\times10^4$ cm$^{-3}$. The analysis of CH$_3$OH lines observed also provides some constraints on the H$_2$ volume density. For this molecule we detect nine lines with a wide range of optical depths and excitation temperatures, which makes it impossible to fit them in a rotational diagram. The lines at 96739.358 MHz and 96741.371 MHz are probably optically thick (inferred optical depths are in slight excess of one). The remaining lines are optically thin and one of them, lying at 107013.831 MHz, appears in absorption against the cosmic microwave background, and therefore has an excitation temperature below 2.7 K. Statistical equilibrium calculations using the large velocity gradient (LVG) formalism do not allow for the intensities of all observed lines to be globally
reproduced, but the main features can be reproduced adopting a column density of $2.9\times10^{14}$ cm$^{-2}$ (estimated from $^{13}$CH$_3$OH), a gas kinetic temperature of 10 K (in agreement with the above quoted values), and a volume density of H$_2$ of $\sim$$3\times10^4$ cm$^{-3}$, which is also in line with the values derived by \cite{Anglada1997} and \cite{Jorgensen2002}.
From the observed C$^{17}$O $J$=1-0 line, which is optically thin, we derive a column density for C$^{17}$O of $1.7\times10^{15}$ cm$^{-2}$ assuming a rotational temperature of 10 K (see Table~\ref{table:column_densities}). If we adopt a standard $N$(C$^{18}$O)/$N$(H$_2$) for dense cores, $1.7\times10^{-7}$ \citep{Frerking1982}, and a $^{18}$O/$^{17}$O isotopic ratio of 4.16 \citep{Wouterloot2008}, the resulting column density of H$_2$ in L483 is $4\times10^{22}$ cm$^{-2}$, which is close to the value of $3\times10^{22}$ cm$^{-2}$ reported by \cite{Tafalla2000} based on the same C$^{17}$O line observed also with the IRAM 30m more than 20 years ago. The main source of uncertainty in the value derived from C$^{17}$O is the degree of depletion of CO. In fact, by modeling the dust continuum emission, \cite{Jorgensen2002} derive $N$(H$_2$) = $9.3\times10^{23}$ cm$^{-2}$ for a size of 40$''$ from the protostar, which would imply that CO is severely depleted from the gas phase in L483. The main source of uncertainty in this latter value is related to dust parameters such as the size and optical properties of dust grains and the gas-to-dust-mass ratio. A detailed model of multi-wavelength continuum and CO spatially resolved data will help to better constrain the physical structure of L483. For the purpose of computing fractional abundances relative to H$_2$, here we adopt the H$_2$ column density obtained from C$^{17}$O, bearing in mind that CO depletion may be an issue in L483 and that $N$(H$_2$) could be significantly higher.
\subsection{The chemical composition of L483} \label{sec:abundances}
\begin{figure*}[!ht]
\centering
\includegraphics[angle=0,width=\textwidth]{fig_abundances}
\caption{Visualization of molecular fractional abundances (in black and referred to the left axis) and line widths (in red and referred to the right axis) in L483 for different chemical families. Abundances are obtained from the beam-averaged column densities listed in Table~\ref{table:column_densities}, adopting a column density of H$_2$ of $4\times10^{22}$ cm$^{-2}$ (see Sect.~\ref{sec:column_densities}). CO is not plotted. For molecules with optically thick lines we used the column densities of optically thin isotopologs and scaled up adopting local ISM isotopic ratios as follows: for HCO$^+$ the $^{17}$O isotopolog was used adopting $^{16}$O/$^{17}$O = 2317 \citep{Wouterloot2008}; for CN, HCN, and HNC we used the $^{15}$N isotopolog assuming $^{14}$N/$^{15}$N = 290 \citep{Adande2012}; and for CH$_3$OH and CS we used the $^{13}$C isotopolog and assumed $^{12}$C/$^{13}$C = 68 \citep{Milam2005}. In the cases of C$_2$H and N$_2$H$^+$, scaling from $^{13}$C and $^{15}$N isotopologs, respectively, is probably not valid due to important fractionation effects (see Sect.~\ref{sec:isotopic_ratios_discussion_13c} and \ref{sec:isotopic_ratios_discussion_rest}), and therefore only lower limits are given. H$_2$CO and NH$_3$ are not shown because we only detect deuterated forms and the deuterium fractionation is uncertain for these species. Line widths represented as red circles are averages over the FWHM of all those lines fitted to a Gaussian function ($\Delta v$ in Table~\ref{table:lines}). For those molecules showing complex profiles (e.g., wings) the values plotted as red diamonds correspond to the highest velocity $|{V_{\rm LSR}-v_{\rm sys}}|$ at which emission is detected.} \label{fig:abundances}
\end{figure*}
In Fig.~\ref{fig:abundances} we show the fractional abundances relative to H$_2$\footnote{Hereafter fractional abundances are always expressed relative to H$_2$.} of the parent molecules (i.e., excluding minor isotopologs) detected in this $\lambda$ 3 mm line survey. We also show information on the mean line width derived for each molecule. Apart from the species that show line wings (CO, HCO$^+$, CN, HCN, HNC, CS, and SiO show bright wings relative to the emission at the systemic velocity, while CH$_3$OH, C$_2$H, $c$-C$_3$H$_2$, HC$_3$N, C$_2$S, H$_2$CS, and SO show weak wings; see discussion in Sect.~\ref{sec:line_profiles}), the vast majority of molecules show narrow lines, with widths around 0.5 km s$^{-1}$. Therefore, while for a few molecules part of the detected emission arises from the bipolar outflow, for most molecules the emission comes either exclusively or mostly from the ambient quiescent cloud. We are therefore probing the chemical composition of the dense core at a relatively large scale and not in the surroundings of the protostar. We note that the emission associated with the hot corino unveiled by ALMA is very compact, with sizes well below 1$''$ and thus severely diluted in the 21-30$''$ beam of the IRAM 30m telescope, and wide in velocity as it extends over several kilometres per second \citep{Oya2017,Jacobsen2018}.
One of the most remarkable characteristics of L483 is that its large-scale chemical composition displays a very rich variety of molecules. Many of them are widely known interstellar molecules. However, some of the detected molecules have rarely been found toward similar sources and, moreover, a few of them had not been observed in space before their discovery in L483. Among the detected molecules, there is a good variety of O-bearing molecules, various unsaturated hydrocarbons, a large number of N-bearing molecules, a diverse sample of S-bearing molecules, and only one molecule containing silicon: SiO. It is also worth noting that L483 contains a rich variety of deuterated molecules, which is evidence that deuterium fractionation has proceeded efficiently in this source (see more details in Sect.~\ref{sec:deuterium}).
\subsubsection{O-bearing molecules} \label{sec:o-bearing}
The fractional abundances of the oxygen-bearing molecules detected in the line survey are shown in the upper panel of Fig.~\ref{fig:abundances}. We see that, apart from CO, the most abundant molecules are CH$_3$OH and HCO$^+$ ($\sim$10$^{-8}$), while the rest can be considered as minor as they have abundances around or below 10$^{-10}$. Other O-bearing molecules not covered in this line survey that could be abundant are water, carbon dioxide, molecular oxygen, and formaldehyde. In fact, \textit{Herschel} observed abundant and warm gaseous water toward L483, the origin of which is the close surroundings of the protostar \citep{Mottram2014}. This strongly suggests that at the colder and more extended scales of the ambient cloud, water ice probably makes an important part of the oxygen budget. In the case of CO$_2$, a rough estimate of its abundance can be obtained from that derived for its protonated form. Using a simplified chemical scheme, the abundance of CO$_2$ is inferred to be $\sim$10$^4$ times that of HCO$_2$$^+$ for dense clouds with parameters similar to those of L483 \citep{Sakai2008b,Vastel2016}, which would imply that in L483, CO$_2$ has an abundance as large as $8\times10^{-8}$. The abundance of O$_2$ in L483 is very uncertain. The recently discovered isocyanate radical could bring constraints on the abundance of O$_2$ as NCO is thought to be essentially formed in the reaction between CN and O$_2$. In the chemical model of L483 presented in \cite{Marcelino2018a}, which successfully reproduced the observed abundances of NCO, HNCO, and various other related species, the abundance of O$_2$ is $7\times10^{-7}$. Although there is no observational constraint on the abundance of O$_2$ in L483, this value is probably too high given the upper limits derived from \textit{Herschel} data to the abundance of O$_2$ in other similar sources, $<$(0.6-1.6)$\times10^{-7}$ in cold prestellar cores \citep{Wirstrom2016}, and $<$$6\times10^{-9}$ toward the low-mass Class\,0 protostar NGC\,1333-IRAS\,4A \citep{Yildiz2013}. Formaldehyde is certainly an important species in the ambient cloud of L483, where it is present with an abundance of $1.5\times10^{-9}$ \citep{Tafalla2000}, around ten times less abundant than methanol.
All the hydrogenation derivatives of CO (i.e., H$_x$CO, $x$ = 1-4) are observed in L483. The observed H$_2$CO/HCO and CH$_3$OH/CH$_3$O abundance ratios are $\sim$20 and $\sim$360, respectively, consistent with the general trend found in other cold dense cores, regardless of whether or not they host a protostar, where H$_2$CO is found to be around ten times more abundant than HCO while CH$_3$OH is about 100 times more abundant than CH$_3$O \citep{Antinolo2016,Bacmann2016,Ocana2017}. The origin of these species in cold dense clouds is still under discussion. Although they can be naturally formed upon CO adsorption on dust grains and consecutive additions of H atoms, the efficiency of the different hydrogenation and chemical desorption channels is not yet well constrained (e.g., \citealt{Minissale2016}). Moreover, with the exception of CH$_3$OH, all these species have efficient formation routes in the gas phase (e.g., \citealt{Antinolo2016,Ocana2017}).
The carbon chain oxides C$_2$O and C$_3$O are present with similar abundances ($\sim$10$^{-11}$) in L483. In TMC-1, both species are also found with similar abundances (see \citealt{Agundez2013}). While C$_2$O has not been very widely observed in interstellar space (the only reported detection is toward TMC-1; \citealt{Ohishi1991}), C$_3$O has been observed in a few other cold dense clouds like TMC-1, Elias\,18, and L1544 \citep{Matthews1984,Palumbo2008,Vastel2014}. The formation of these two molecules in cold dense clouds is well explained by gas-phase chemistry (e.g., \citealt{Agundez2013}), although laboratory experiments have shown that irradiation of CO ice with energetic protons can also result in the formation of carbon oxides like C$_2$O, C$_3$O, and C$_3$O$_2$ \citep{Palumbo2008}; the latter species being nonpolar it cannot be observed through radioastronomical techniques.
The ketenyl radical (HCCO), the hydrogenated descendant of C$_2$O, was recently discovered in space toward the dense cores Lupus-1A and L483 \citep{Agundez2015a}. In L483 we find that the different hydrogenation forms of C$_2$O have relative abundances HCCO:H$_2$CCO:CH$_3$CO:CH$_3$CHO $\sim$ 1:10:$?$:10. These values are interestingly similar to those typically found in cold dense clouds for the hydrogenated forms of CO, in which case HCO:H$_2$CO:CH$_3$O:CH$_3$OH are about 1:10:0.1:10 \citep{Bacmann2016}. This fact may be accidental, although it is also possible that it reflects similar efficiencies along the different hydrogenation steps for the two types of molecules. If the relative abundances of H$_x$CO hold for H$_x$CCO, then the acetyl radical (CH$_3$CO), which has not yet been observed in space, would be ten times less abundant than HCCO.
Propynal (HCCCHO) and cyclopropenone ($c$-C$_3$H$_2$O) are two isomers which are detected in L483 with abundances of the order of 10$^{-11}$, propynal being somewhat more abundant. This is in line with the abundances observed for these two species in other cold dense clouds, where their synthesis is accounted for by gas-phase chemistry \citep{Loison2016}.
An interesting result is also the detection of the $trans$ and $cis$ conformers of the organic molecule HCOOH. The $trans$ species, which is the most stable, is widely observed in cold and hot cores. Its formation in cold cores can be explained by gas-phase chemistry \citep{Vigren2010}. The $cis$ conformer however has only been observed in a few interstellar sources. It was discovered toward the Orion Bar photodissociation region, where it is present with a high abundance (only 2.8 times less than the $trans$ conformer) in the region illuminated by ultraviolet (UV) photons \citep{Cuadrado2016}. This fact led these authors to suggest that the $cis$ conformer is formed from the $trans$ form in a photoswitching process induced by UV photons. More recently, $cis$-HCOOH has been detected in a different environment: the cold dark cloud B5, where the $cis$/$trans$ abundance ratio is found to be 6 \% \citep{Taquet2017}. In L483 we derive a similar $cis$/$trans$ ratio of 6 \% for HCOOH. The lack of a strong UV field in B5 and L483, together with the much lower $cis$/$trans$ ratios derived compared to that found in the Orion Bar, suggests that in these sources $cis$-HCOOH does not originate from a photoswitching mechanism. Moreover, the identical $cis$/$trans$ ratios derived in B5 and L483 strongly suggest a common formation mechanism. Whether such a mechanism occurs in the gas phase (through, e.g., the dissociative recombination of HCOOH$_2^+$ or a chemical switching process) or in the surface of dust grains (via hydrogenation of HO-CO radicals; \citealt{Ioppolo2011}) is still to be investigated.
One of the most remarkable results obtained in this line survey is the detection of the complex organic molecules (COMs) methyl formate (HCOOCH$_3$), dimethyl ether (CH$_3$OCH$_3$), and ethanol (C$_2$H$_5$OH). These molecules are often observed toward hot cores and hot corinos with single-dish telescopes (e.g., \citealt{Cazaux2003,Bottinelli2007,Bianchi2019}) or interferometers (e.g., \citealt{Tercero2015,Imai2016}); they have been observed toward L483 with ALMA \citep{Oya2017,Jacobsen2018}, although in those observations the detected emission arises from a compact and warm region around the protostar. In this line survey however the lines of COMs observed arise from the large-scale cold ambient cloud, similarly to previous single-dish detections of COMs in prestellar cores like L1689B and L1544 \citep{Bacmann2012,Jimenez-Serra2016} and cold dense clouds around low-mass protostars such as B1-b and B5 \citep{Oberg2010,Cernicharo2012,Taquet2017}. Extended emission from HCOOCH$_3$ and CH$_3$OCH$_3$ has been detected in all or some of the above quoted sources. Our detection of C$_2$H$_5$OH is thus the first detection of this molecule in the cold, quiescent, and extended component of a prestellar or protostellar dense core, and adds one further piece to the puzzle of the presence of COMs in cold dense clouds \citep{Ruaud2015,Balucani2015,Kalvans2015,Vasyunin2017,Shingledecker2018}.
\subsubsection{Hydrocarbons} \label{sec:hydrocarbons}
There are various hydrocarbons detected in L483 in the $\lambda$ 3 mm band (see second panel in Fig.~\ref{fig:abundances}). Most of them are highly unsaturated and some of them, like the radicals C$_2$H and C$_4$H, the ring molecule $c$-C$_3$H$_2$, and methyl acetylene (CH$_3$CCH), are quite abundant (between 10$^{-9}$ and $>$10$^{-8}$).
We detect all the members of the series of radicals C$_n$H with $n$=2-5. Larger members are probably not detected because for rotational temperatures around 10 K the most favorable transitions lie below 80 GHz. For example, for C$_6$H we derive a 3$\sigma$ upper limit to its column density of $4\times10^{12}$ cm$^{-2}$ (adopting the rotational temperature of 7.9 K derived for C$_4$H; see Table~\ref{table:column_densities}). This implies a C$_6$H/C$_4$H ratio of $<$3 \%, which is still above the values observed in TMC-1 and L1527, 0.3 \% and 1 \%, respectively (e.g., \citealt{Sakai2008a}). Therefore, it would not be strange to detect carbon chains
in L483 at cm wavelengths that are longer than those observed here. In this line it is interesting to note that already relatively heavy hydrocarbons like C$_5$H and CH$_3$C$_4$H are detected in the $\lambda$ 3 mm band through lines with upper-level energies around 40 K, well in excess of the gas kinetic temperature of $\sim$10 K. This implies that in L483, long carbon chains are either relatively abundant or are significantly warmer than 10 K. Unfortunately the low number of lines of C$_5$H and CH$_3$C$_4$H detected and their low signal-to-noise ratios prevent us from precisely constraining their excitation temperatures. This is not the case for HC$_5$N, for which a relatively high rotational temperature is inferred (see Sect.~\ref{sec:n-bearing}), supporting the scenario that long carbon chains are warmer than shorter carbon chains in L483.
Among the hydrocarbons observed, we detect the cyclic and linear isomers of C$_3$H and C$_3$H$_2$. The chemistry of these species in cold dense clouds has been discussed by \cite{Loison2017}, highlighting the importance of the dissociative recombination of C$_3$H$_2^+$ and C$_3$H$_3^+$ isomers in establishing the cyclic-to-linear abundance ratios. From the column densities derived in this line survey, we find an abundance ratio $c$-C$_3$H/$l$-C$_3$H of 12, which is in the range of values derived in TMC-1 and B1-b \citep{Fosse2001,Loison2017}. Cyclic C$_3$H$_2$ is found to be substantially more abundant than its linear isomer, in line with observations of other cold dense clouds \citep{Fosse2001,Loison2017}.
The presence of carbon chain hydrocarbons like C$_4$H in cold dark clouds has long been known. These species are particularly abundant toward the cyanopolyyne peak of the starless source TMC-1 (e.g., \citealt{Ohishi1998}), but they are also relatively abundant in other cold dense clouds, some of which are starless while others contain protostars (e.g., \citealt{Agundez2008}). The detection of bright C$_4$H lines in some dense cores around low-mass protostars like L1527 and Lupus-1A \citep{Sakai2008a,Sakai2009a} motivated the proposal of a scenario called warm carbon chain chemistry (WCCC) in which carbon chains would form by gas phase chemistry from the evaporation of CH$_4$ ice \citep{Sakai2013}. If this scenario is accurate, methane should be an important hydrocarbon in these protostellar sources rich in carbon chains. Still, it is not clear whether the presence of abundant carbon chains in these sources is an inheritance from the dark cloud prestellar phase, a consequence of a specific type of chemistry triggered by CH$_4$ ice evaporation upon the switch on of the protostar, or the result of environmental effects or the particular history of the cloud. In any case, L483 shares a commonality with L1527 and Lupus-1A in that it contains abundant carbon chain hydrocarbons. It is also interesting to note that while in the case of L1527, the presence of carbon chains and COMs seems to be mutually exclusive \citep{Sakai2008a,Yoshida2019}, this is not the case for L483, where both types of molecules coexist in the ambient cloud.
\subsubsection{N-bearing molecules} \label{sec:n-bearing}
There is a great diversity of N-bearing molecules detected in this line survey (see third panel in Fig.~\ref{fig:abundances}). A great fraction of them are unsaturated carbon chains that contain only N, C, and H, although there are also several N-bearing molecules containing oxygen.
The most abundant (above 10$^{-9}$) N-bearing molecules detected are CN, HCN, HNC, HC$_3$N, HCNH$^+$, and N$_2$H$^+$. The latter is a proxy of molecular nitrogen, which is probably one of the most important reservoirs of nitrogen. The detection of NH$_2$D also points to ammonia as an important N-containing molecule. In fact, \cite{Anglada1997} derive a column density of NH$_3$ of $1.4\times10^{15}$ cm$^{-2}$, which translates to an abundance of several times 10$^{-8}$. Another potentially abundant N-bearing molecule is cyanogen (NCCN), the presence of which is supported by the detection of NCCNH$^+$ and CNCN, previously reported by \cite{Agundez2015b} and \cite{Agundez2018b}, respectively. It is estimated that NCCN could have an abundance between 10$^{-9}$ and 10$^{-7}$ (see \citealt{Agundez2018b}). The abundance ratio HNC/HCN derived from the $^{15}$N isotopologs is found to be 2.0 in L483, in the range of values found in other cold dense cores and consistent with formation of both species from the dissociative recombination of HCNH$^+$ \citep{Hirota1998}.
The existence of abundant unsaturated carbon chains of the family of cyanopolyynes like HC$_3$N is in line with the presence of abundant carbon chains of the family of polyynes like C$_4$H and S-bearing carbon chains like C$_2$S (see Sect.~\ref{sec:s-bearing}). In this sense, L483 is a source rich in carbon chains, as in other starless or protostellar dense cores \citep{Hirota2009}. Apart from HC$_3$N, we also detected the related radical C$_3$N and the longer cyanopolyyne HC$_5$N. The latter is the only molecule for which we derive a rotational temperature that is without doubt well in excess of 10 K, that is, 28 $\pm$ 3 K. This fact indicates that HC$_5$N is probably distributed over a more compact region around the protostar than HC$_3$N, which should have a more extended distribution according to the lower derived rotational temperature of 9.1 $\pm$ 1.6 K. We also detected the two metastable isomers of cyanoacetylene, HCCNC and HNC$_3$, as well as its protonated form. We find that HCCNC is approximately 70 times less abundant than HC$_3$N and approximately 10 times more abundant than HNC$_3$, while HC$_3$NH$^+$ is about 200 times less abundant than HC$_3$N. These relative abundances are similar to those found in TMC-1 and L1544 \citep{Vastel2018a}.
Just as CH$_3$CCH and CH$_3$C$_4$H are detected in L483, the cyanide analogs CH$_3$CN and CH$_3$C$_3$N are also observed. We find that the two molecules have nearly the same abundance, very similarly to what is found in TMC-1 \citep{Broten1984}. The cyanomethyl radical (CH$_2$CN) is observed to be four times more abundant than CH$_3$CN in L483, while in TMC-1, CH$_2$CN is also found to be more abundant than CH$_3$CN by a factor of approximately ten \citep{Irvine1988}. We also detected methyl isocyanide (CH$_3$NC), a metastable isomer of methyl cyanide, which has only been observed in a few astronomical sources. This isomer has been observed to be 7-50 times less abundant than CH$_3$CN in interstellar space, concretely toward Sgr\,B2 and in the Horsehead photodissociation region \citep{Cernicharo1988,Remijan2005,Gratier2013}, while it is much less abundant than CH$_3$CN, by factors of 200-500, in hot cores and hot corinos \citep{Lopez2014,Calcutt2018a}. In L483 we find a CH$_3$CN/CH$_3$NC abundance ratio of 11, which is more in line with the interstellar values than with the protostellar ones. The partially saturated cyanide C$_2$H$_3$CN, which has been observed in both cold and hot cores, is also detected in L483, where the emission probably arises from the cold and quiescent cloud.
We detected several N-bearing molecules that contain oxygen, many of which are related to HNCO, which is a relatively abundant molecule in L483. We detected two metastable isomers of isocyanic acid, cyanic acid (HOCN), and fulminic acid (HCNO), which are about 100 and 200 times less abundant than HNCO, respectively. Therefore, HOCN and HCNO have similar abundances in L483 (within a factor of two), similarly to what is found in the prestellar cores L183 and L1544 and in the young low-mass protostellar cores B1-b and L1527 \citep{Marcelino2010}. In L483 we also detected the radical isocyanate (NCO) and protonated isocyanic acid (H$_2$NCO$^+$), which are thought to be precursors of HNCO and its isomers. In fact, L483 is the only source where all these chemically related species have been detected to date, which makes it a perfect test-bed to study the chemistry of this chemical family (see \citealt{Marcelino2018a}).
We also detected nitroxyl (HNO) and nitrous oxide (N$_2$O), two species that belong to the chemical family of nitrogen oxides and their derivatives. Nitric oxide (NO), the most widespread and abundant member of this family, has no lines in the $\lambda$ 3 mm band but it is very likely abundant in L483. Since the discovery of HNO and N$_2$O toward Sgr\,B2 \citep{Ulich1977,Ziurys1994}, none of these molecules have been widely observed in space. In the case of HNO, it has been detected in various massive star-forming regions and in the cold dense cloud L134N, where the derived NO/HNO abundance ratio is approximately 800 \citep{Snyder1993}. If the same ratio holds in L483, then NO would have an abundance of a few times 10$^{-8}$, a value similar to those found in TMC-1 and L134N (see, e.g., \citealt{Agundez2013}). Concerning N$_2$O, it has only been observed toward Sgr\,B2 \citep{Ziurys1994,Halfen2001} and more recently in the hot corino IRAS\,16293$-$2422 \citep{Ligterink2018}. In these two sources, N$_2$O is present in warm gas. In L483, both HNO and N$_2$O are detected through a single line, and therefore we cannot constrain their excitation temperatures. However, the small line width observed for both species ($\sim$0.5 km s$^{-1}$) indicates that their emission must arise from the cold and quiescent cloud. Our detection of N$_2$O in L483 is thus the first one in a cold dense cloud and provides an additional constraint to understand the chemistry of nitrogen oxides in these environments. Interestingly, we find that the N$_2$O/HNO abundance ratio in L483 is approximately five, which is close to that found in Sgr\,B2 ($\sim$3; \citealt{Ziurys1994}), despite the different nature of these two environments.
\subsubsection{S-bearing molecules} \label{sec:s-bearing}
The $\lambda$ 3 mm scan of L483 revealed a great variety of S-bearing molecules. Many of them are not simply minor species but are present at an important abundance level. The most abundant S-bearing molecule is CS ($>$10$^{-9}$), but there are also several relatively abundant ($>$10$^{-10}$) species like C$_2$S, HCS, H$_2$CS, SO, SO$_2$, OCS, and NS.
The carbon chains C$_2$S and C$_3$S are detected through rather intense lines, which together with the detection of bright emission from C$_4$H and cyanopolyynes like HC$_3$N, stresses the nature of L483 as a source rich in carbon chains. This distinctive characteristic is also found in other cold dense clouds like TMC-1 and L1527, although it is not universal to dense cores, which show different degrees of richness in carbon chains \citep{Hirota2009}.
An interesting aspect of the chemistry of sulfur unveiled by the $\lambda$ 3 mm line survey is that the detection of HCS and HSC (see \citealt{Agundez2018a}), together with that of CH$_3$SH, permits a relatively complete description of the different intermediates along the successive hydrogenation steps of CS. It is interesting to compare the relative abundances between these species with the oxygen analog case, for which we are informed that HCO:H$_2$CO:CH$_3$O:CH$_3$OH are about 1:10:0.1:10 in cold dense clouds \citep{Bacmann2016}. For sulfur we find a quite different behavior in L483, where HCS:H$_2$CS:CH$_3$S:CH$_3$SH are 1:2:$<$0.1:0.3, where the upper limit for CH$_3$S is computed using the entry in MADEX, which is based on the laboratory spectrum by \cite{Endo1986}. That is, while the electronic closed shell molecules H$_2$CO and CH$_3$OH are substantially more abundant than the radical HCO, the radical HCS is almost as abundant as H$_2$CS and significantly more abundant than CH$_3$SH. These relative abundances hold key information about the underlying chemical processes at work along the hydrogenation sequences of CO and CS, which must be quite different.
The sulfur oxides SO and SO$_2$ and carbonyl sulfide (OCS) are commonly observed in cold dense clouds with abundances between 10$^{-10}$ and 10$^{-8}$ (e.g., \citealt{Agundez2013}). These molecules are substantially enhanced in hot cores and hot corinos (e.g., \citealt{Schoier2002,Tercero2010,Esplugues2013}). The abundances derived in L483 for SO, SO$_2$, and OCS are $5\times10^{-9}$, $10^{-10}$, $4\times10^{-10}$, respectively, which are in line with the values typically found in cold dense clouds like TMC-1 \citep{Matthews1987,Lique2006,Cernicharo2011}, L1544 \citep{Vastel2018b}, and B1-b \citep{Fuente2016}.
\begin{figure*} [!ht]
\centering
\includegraphics[angle=0,width=\textwidth]{fig_abundances_comparison}\caption{Fractional abundances relative to H$_2$ in L483 are compared with those in another four reference sources: TMC-1, L1544, B1-b, and L1527. The abundance ratios between L483 and the reference source are plotted against the fractional abundance in the corresponding reference source. Upper and lower limits are indicated by arrows. Different chemical families are plotted in different colors (see legend in bottom-left panel). Beam-averaged column densities derived from single-dish observations in the reference sources are taken from the literature. TMC-1: \cite{Adande2010,Agundez2013,Agundez2015b,Agundez2018b,Gratier2016,Loison2016,Loison2017,Cernicharo2018}; Marcelino et al. (in preparation). L1544: \cite{Caselli2002,Gupta2009,Marcelino2009,Marcelino2010,Vastel2014,Vastel2015,Vastel2016,Vastel2018a,Vastel2018b,Jimenez-Serra2016,Quenard2017,Cernicharo2018}. B1-b: \cite{Marcelino2005,Marcelino2009,Marcelino2010,Agundez2008,Oberg2010,Cernicharo2012,Loison2016,Loison2017,Fuente2016,Widicus-Weaver2017}; Marcelino et al. (in preparation). L1527: \cite{Sakai2008b,Marcelino2009,Marcelino2010,Araki2017,Yoshida2019}. Adopted H$_2$ column densities (needed to convert molecular column densities to fractional abundances) are $1\times10^{22}$ cm$^{-2}$ for TMC-1 \citep{Cernicharo1987}, $5.4\times10^{22}$ cm$^{-2}$ for L1544 \citep{Jimenez-Serra2016}, $7.6\times10^{22}$ cm$^{-2}$ for B1-b \citep{Daniel2013}, and $2.8\times10^{22}$ cm$^{-2}$ for L1527 \citep{Jorgensen2002}.} \label{fig:abundances_comparison}
\end{figure*}
We also detected HNCS and its metastable isomer HSCN, with abundances of several times 10$^{-12}$ and a HNCS/HSCN abundance ratio of approximately two. These values are very similar to those found in TMC-1, where the HNCS/HSCN ratio is approximately one \citep{Adande2010}. Clearly these species have a different behavior from their oxygen analogs, in which case the most stable isomer HNCO is about 100 times more abundant than the metastable isomer HOCN. This marked differentiation in the abundance ratios of the sulfur and oxygen cases is also seen in TMC-1 \citep{Marcelino2010,Adande2010}.
A last aspect concerning S-bearing molecules that is worth commenting on is the detection of the cations SO$^+$ and NS$^+$, for which we find abundance ratios of SO/SO$^+$ = 125 and NS/NS$^+$ = 26, which are in line with the values typically found in cold dense clouds. For example, the SO/SO$^+$ ratio is around 100 in cold dense cores \citep{Turner1995,Turner1996}. In B1-b, the SO/SO$^+$ ratio is approximately 250 when the abundance of SO is evaluated from $^{33}$SO \citep{Fuente2016}. The value of about 1000 given by these authors is probably too high as it is obtained from S$^{18}$O, for which oxygen fractionation is important \citep{Loison2019b}. In the case of NS$^+$, the NS/NS$^+$ ratio has been found to be remarkably uniform, in the range 29-52, across different starless and protostellar cores \citep{Cernicharo2018}.
Overall, the inventory of S-bearing molecules observed in L483 in the $\lambda$ 3 mm band and the abundances derived do not differ much from what is found in prestellar cores like TMC-1 and L1544 and low-mass protostellar cores like B1-b (see Sect.~\ref{sec:comparison}).
\subsection{Comparative chemical composition} \label{sec:comparison}
A pertinent question concerning the chemical composition of dense cores is whether the evolution of a source along the star-formation process drives a change in the chemical composition of the cloud and whether this change is somehow universal. In other words, to what extent is the chemical composition of a dense core determined by the particular evolutionary stage of the object$?$ The exhaustive chemical characterization of L483 provided by this line survey makes it possible to perform a comparison with other sources in nearby evolutionary stages. There are not many such sources for which the chemical composition has been exhaustively characterized however; that is, there are few unbiased line surveys. Here we have selected four sources, two of them with no sign of star formation (TMC-1 and L1544) and two that contain protostellar activity (B1-b and L1527). The object TMC-1 is a starless dark cloud in an earlier evolutionary stage than the prestellar phase, with no evidence of ongoing gravitational collapse (e.g., \citealt{Pratap1997}). The source L1544 however is a prestellar core with evidence of infall and a marked density gradient, with a high density ($>$10$^6$ cm$^{-3}$) and low temperature ($<$10 K) at the core center (e.g., \citealt{Caselli2012}). The dark cloud Barnard 1b (B1-b) harbors two extremely young protostellar objects separated by 20$''$ \citep{Hirano1999}, one of which is a candidate for the first hydrostatic core while the other is consistent with a very early Class\,0 source with an incipient hot corino \citep{Marcelino2018b}. Finally, L1527 is a dense core containing an embedded IRAS source, consistent with a protostar in the Class\,0 stage (e.g., \citealt{Sakai2008a}). Therefore, the four sources can be ordered in a evolutionary sequences as TMC-1 $\rightarrow$ L1544 $\rightarrow$ B1-b $\rightarrow$ L1527, with L483 being in a similar evolutionary stage to L1527.
To compare the chemical composition of L483 with that of TMC-1, L1544, B1-b, and L1527 we have collected molecular column densities from the literature. The four sources are included in the IRAM 30m Large Program ASAI \citep{Lefloch2018}, although at the time of writing, column densities have not been systematically reported. We focus on column densities derived from single-dish data (mostly IRAM 30m, Nobeyama 45m, and GBT), meaning that the comparison is made between abundances averaged over a similar scale, typically 15-45$''$ in diameter. In the case of TMC-1, column densities are mostly based on the line survey carried out with the Nobeyama 45m telescope \citep{Ohishi1998}, recently revised by \cite{Gratier2016}. For L1544 and B1-b, most data are based on IRAM 30m observations, some of which were taken in the context of ASAI (e.g., \citealt{Fuente2016,Vastel2018b}). In the case of L1527, column densities are mostly taken from the $\lambda$ 3 mm line survey carried out with the Nobeyama 45m telescope recently published by \cite{Yoshida2019}. All the literature sources used are given in the caption of Fig.~\ref{fig:abundances_comparison}.
In Fig.~\ref{fig:abundances_comparison} we compare the abundances of the molecules detected in the $\lambda$ 3 mm line survey of L483 with those in each of the four comparison sources: TMC-1, L1544, B1-b, and L1527. We compare fractional abundances relative to H$_2$ rather than column densities because this is a more meaningful quantity when comparing between sources with different column densities of material. We however note that the adopted column density of H$_2$ in each source becomes crucial as changes on it cause all relative abundances to shift up or down by the same amount, thus making all molecules appear more or less overabundant (or underabundant) in L483 than in the comparison source. A quick look at Fig.~\ref{fig:abundances_comparison} shows that, in general, molecules in L483 are underabundant with respect to TMC-1, overabundant with respect to L1544 and B1-b, and have abundances of the same order as in L1527. Assuming that the adopted H$_2$ column densities are not drastically different from the true ones, this behavior could be explained in terms of the specific physical and chemical properties of each comparison source. The source TMC-1 is known to be particularly rich in molecules, especially in unsaturated organic molecules like carbon chains. Still, it is not clear whether such chemical richness is the result of a very peculiar history of the source or is a short-lived feature that is observed in TMC-1 but not in other starless cores that may have different ages (see, e.g., \citealt{Agundez2013}). In any case, the lower molecular abundances observed in L483 with respect to TMC-1 are very likely a consequence of the exceptional chemical richness of the latter. The higher abundances derived in L483 compared to L1544 and B1-b may result from the fact that in these two latter sources the high volume densities, together with the very low temperatures, have accelerated the condensation of species onto dust grains resulting in an overall depletion of all gas-phase molecules except H$_2$. In the case of L1544, this phenomenon is well documented (e.g., \citealt{Caselli1999}). One may be tempted to interpret this behavior in evolutionary terms by saying that the chemical richness present at some moment of the starless phase is suppressed to an important extent during the prestellar and very early protostellar phases to then reappear to some degree at the Class\,0 stage. We however warn that this interpretation is not sufficiently robust because of the low statistics of sources analyzed here.
\begin{figure*}
\centering
\includegraphics[angle=0,width=\textwidth]{fig_c4h_vs_ch3oh}
\caption{Column density of C$_4$H represented as a function of the column density of CH$_3$OH (left panel), HCOOCH$_3$ (middle panel), and CH$_3$OCH$_3$ (right panel) in various starless/prestellar cores, Class\,0 sources, and hot corinos. All column densities are beam-averaged values obtained with single-dish telescopes. See references in the caption of Fig.~\ref{fig:abundances_comparison} for the column densities of C$_4$H and CH$_3$OH in TMC-1, L1544, B1-b, and L1527. Column densities of HCOOCH$_3$ and CH$_3$OCH$_3$ are taken from Marcelino et al. (in preparation) for TMC-1 (upper limits), \cite{Jimenez-Serra2016} for L1544, \cite{Cernicharo2012} for B1-b, and \cite{Yoshida2019} for L1527 (upper limits). For the three hot corino sources, column densities for C$_4$H are from \cite{Sakai2009a}, for CH$_3$OH from \cite{Maret2005}, and for HCOOCH$_3$ and CH$_3$OCH$_3$ from \cite{Cazaux2003} (values have been averaged for a 10$''$ source size) and from \cite{Bottinelli2007}.} \label{fig:c4h_vs_ch3oh}
\end{figure*}
Apart from the systematic abundance enhancement (or decline) of all molecules seen between L483 and some of the comparison sources, it is interesting to discuss whether or not there is some chemical differentiation between L483 and the other objects; that is, whether different types of molecules behave differently. The most clear chemical differentiation is seen between L483 and TMC-1 for two types of molecules. On the one hand, we have carbon chains comprising unsaturated hydrocarbons of the family of polyynes like C$_4$H, C$_5$H, and H$_2$C$_4$, N-bearing molecules of the family of cyanopolyynes like C$_3$N, HC$_3$N, and HC$_5$N, and the S-bearing molecules C$_2$S and C$_3$S. On the other hand, there are several molecules which are present in cold dense clouds but tend to experience important abundance enhancements in hot cores and hot corinos. In this group we can include species like CH$_3$OH, HCOOCH$_3$, CH$_3$OCH$_3$, HCOOH, CH$_3$CHO, HNCO, SO, SO$_2$, OCS, HNCS, and CH$_3$SH. From Fig.~\ref{fig:abundances_comparison} it is clear that when moving from TMC-1 to L483, carbon chains experience a more drastic abundance depletion than hot core-like molecules. In other words, the relative importance of carbon chains with respect to hot core-like molecules is higher in TMC-1 than in L483. Along the same lines, \cite{Law2018} recently found that carbon chains are significantly less abundant in dense cores around Class\,0/I low-mass protostars than in TMC-1. A similar behavior can be tentatively found when comparing L483 with L1544. From Fig.~\ref{fig:abundances_comparison} it can be seen that, in general, hot core-like molecules tend to increase their abundance more than carbon chains when moving from L1544 to L483; although this does not happen systematically for all species of each type and in any case the chemical differentiation is less clear than in the case of TMC-1.
Therefore, the relative importance of hot core-like molecules with respect to carbon chains is higher in L483 than in TMC-1 and L1544 (tentative for the latter). When comparing L483 with B1-b and L1527 we do not see any clear chemical differentiation between carbon chains and hot core-like molecules or any other type of chemical family; that is, when moving from any of these two sources to L483, there is no obvious differentiation in the abundance variation of different types of molecules. In this sense, L483 is significantly different from TMC-1, barely different from L1544, and quite similar to B1-b and L1527 regarding the chemical composition of the ambient cloud. This conclusion is based on an overall look at the chemical composition of each source. In any case, the fact that we see a chemical differentiation between L483 and the starless/prestellar sources TMC-1 and L1544 and a lack of it between L483 and the protostellar sources B1-b and L1527 raises the question of whether starless and prestellar sources on the one side, and protostellar sources on the other, have differences in the chemical composition of their host cloud which could be associated to the switch on of a protostar at the center of the core.
To get insight into the potential chemical differentiation between sources with different evolutionary stages, we now move from looking at the overall chemical composition to the use of chemical indicators. We first focus on the column density of C$_4$H versus that of CH$_3$OH, the former being a typical carbon chain, the latter being a classical hot core-like molecule. In the left panel of Fig.~\ref{fig:c4h_vs_ch3oh} we plot these two quantities for a few well-studied dense cores, some of which have no protostar while others have a Class\,0 object. We see that the Class\,0 sources B1-b, L1527, and L483 cluster around the same region as the starless/prestellar sources, while the three other Class\,0 sources classified as hot corinos based on single-dish observations of abundant and warm COMs around the protostar \citep{Cazaux2003,Bottinelli2004,Bottinelli2007} appear in a different region of the diagram.
Recently, the C$_4$H/CH$_3$OH ratio was surveyed in a sample of 40 embedded protostars \citep{Graninger2016a,Graninger2016b,Lindberg2016}. These studies mostly populate the region of low column densities along the band of C$_4$H/CH$_3$OH close to one, where B1-b, L1527, and L483 reside, bridging part of the gap between these three latter sources and the hot corinos, which still appear chemically differentiated from the rest of sources. The underlying cause of such chemical differentiation between sources that are supposed to be in a similar, relatively short-lived (a few 10$^4$ yr; \citealt{Andre2000}) evolutionary stage like the Class\,0 one remains unclear. This may be related to the way the switch on of the protostar affects the ambient cloud. In some sources the cloud would preserve much of the characteristics of the starless/prestellar phase, displaying high C$_4$H/CH$_3$OH ratios and being rich in carbon chains, while in others the chemical content of the ambient cloud would be strongly processed by the injection of energy from the protostar, thus showing low C$_4$H/CH$_3$OH ratios and abundant COMs (hot corinos). However, a deeper understanding of why this would happen is still lacking. A possible explanation of the chemical differentiation between Class\,0 sources rich in carbon chains and rich in COMs (hot corinos) has been suggested by \cite{Sakai2008a}. These authors proposed that sources rich in carbon chains would have had a short lifetime of the prestellar collapse, in which case ices would be rich in CH$_4$ and the evaporation of these during the switch on of the protostar would lead to the formation of carbon chains by gas phase chemistry. On the other hand, hot corinos would result from a longer prestellar collapse phase, meaning that ices would be rich in CO, which upon hydrogenation and further processing driven by protostar heating would lead to abundant COMs in the gas phase. Chemical models following this idea have been carried out \citep{Aikawa2008,Aikawa2012,Hassel2008,Hassel2011} but still a definitive theoretical validation of the chemical differentiation between Class\,0 sources rich in carbon chains and hot corinos is lacking. More recently, a survey of the C$_2$H/CH$_3$OH in 36 Class\,0/I protostars by \cite{Higuchi2018} pointed to the environmental effect of the dense core being located in a more inner or outer region of the molecular cloud complex as a possible driver of the diversity of C$_2$H/CH$_3$OH ratios observed.
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_column_density_ratios}
\caption{Some column density ratios which are potentially good chemical evolutionary indicators of the prestellar/protostellar transition. The upper limit to the column density of CH$_3$SH in TMC-1 is taken from Marcelino et al. (in preparation).} \label{fig:column_density_ratios}
\end{figure}
We also investigated whether the use of more specific proxies of COMs like HCOOCH$_3$ and CH$_3$OCH$_3$, rather than the more generic one of CH$_3$OH, could provide some insight into the chemical differentiation between starless/prestellar and protostellar sources. Therefore, in the middle and right panels of Fig.~\ref{fig:c4h_vs_ch3oh} we show the column density of C$_4$H versus that of HCOOCH$_3$ and CH$_3$OCH$_3$, respectively, for the same sources considered before. We see that the Class\,0 sources B1-b, L1527, and L483 cluster around the same region in the diagram as the starless/prestellar sources, while they can be clearly differentiated from the Class\,0 hot corino sources. Therefore, the use of the CH$_4$/HCOOCH$_3$ and CH$_4$/CH$_3$OCH$_3$ chemical indicators makes it impossible to distinguish L483, B1-b, and L1527 from starless and prestellar sources. In other words, COMs are present at similar level in all these sources, despite the fact that a protostar has switched on in some of them while not in others.
Pursuing the idea of using column density ratios between different pairs of molecules as indicators of chemical evolution along the prestellar-protostellar transition, we have examined various potential indicators using molecules for which we have column densities at hand for TMC-1, L1544, B1-b, and L483. In Fig.~\ref{fig:column_density_ratios} we plot some of these ratios for the above four sources, which have been ordered according to their evolutionary status. The ratios have been chosen between different pairs of molecules so that one is a hot core-like molecule and the other a carbon chain. There is a clear trend in which the HNCO/C$_3$S, SO$_2$/C$_2$S, and CH$_3$SH/C$_2$S ratios increase with the age of the object, with a marked difference at the prestellar-protostellar transition. Therefore, these are promising tracers of chemical evolution of the cloud along the protostar formation process. The observed trends however have to be considered as tentative for the moment due to the low number of sources included.
\subsection{Isotopic ratios} \label{sec:isotopic_ratios}
The high sensitivity of the line survey permitted us to detect a good number of isotopologs substituted with rare isotopes (D, $^{13}$C, $^{15}$N, $^{18}$O, $^{17}$O, $^{34}$S, and $^{33}$S), and even some doubly substituted species. Isotopolog abundance ratios have been obtained from the column densities derived and are presented in Tables~\ref{table:isotopic_ratios} and \ref{table:deuterated_ratios}. The uncertainties in these ratios are estimated to be 30~\%. We note that although the estimated uncertainties in the column densities are of the order of 50~\% (see above in Sect.~\ref{sec:column_densities}), we can reasonably expect lower uncertainties for isotopic ratios. An important part of the uncertainty that affects the determination of a column density is dominated by the error in the rotational temperature, which becomes less important when computing column density ratios between different isotopologs of a given molecule because these are expected to have similar excitation conditions. This latter statement has been verified in the few cases in which we have enough lines to estimate the rotational temperature for different isotopologs of the same molecule. For example, for $c$-HCC$^{13}$CH and $c$-C$_3$HD we derive rotational temperatures of 4.1 $\pm$ 0.3 and 4.1 $\pm$ 0.2, respectively (see Table~\ref{table:column_densities}). In the case of the isotopologs of HC$_3$N, rotational temperatures could be derived with good accuracy for HC$^{13}$CCN, HCC$^{13}$CN, and DC$_3$N and all them fall in the narrow range 7.5-9.1 K.
The abundance ratios between isotopologs can be merely statistical, in which case they would reflect the corresponding isotopic ratio of the parental cloud, or they can be affected by fractionation processes, in which case they would deviate from the isotopic ratios of the ambient cloud. These deviations or anomalies can consist in either an enrichment or a dilution in the heavy isotope, or even in different abundances for isotopologs resulting from different isotopic substitutions of the same type of atom (e.g., $^{13}$CCH and C$^{13}$CH). In L483, some molecules show no evidence of fractionation while other species show the aforementioned anomalies. In general, at low temperatures isotopic exchange gas phase reactions tend to enhance the abundance of the heavy isotopolog due to the lowering of the zero-point energy, as first suggested by \cite{Solomon1973} to explain the surprising large abundance of DCN in the Orion molecular cloud. Such considerations allow for us to account for enrichments of deuterated molecules of several orders of magnitude, compared to the elemental D/H ratio of 1.5$\times$10$^{-5}$, due to differences in the zero-point energy of several hundreds of degrees Kelvin \citep{Gerin1987,Roueff2003,Roueff2005,Albertsson2013}. The isotopic enhancement is expected to decrease for heavier atoms as the zero-point energy variation is much smaller, typically of the order of 30 K for $^{12}$C/$^{13}$C-containing molecules, 20 K for $^{14}$N/$^{15}$N species, and 7 K in the case of $^{32}$S/$^{34}$S. In the absence of any other specific mechanism, fractionation is expected to decrease in this order $^{13}$C, $^{15}$N, and $^{34}$S, which in general is in agreement with observational results.
In the rest of the section we discuss the available information in the literature on the isotopic ratios in the local interstellar medium (ISM), the molecular isotopic ratios derived in L483 involving the rare isotopes $^{13}$C, $^{15}$N, $^{18}$O, $^{17}$O, $^{34}$S, and $^{33}$S, and the results obtained for deuterated molecules.
\subsubsection{Local ISM isotopic ratios} \label{sec:local_ism_isotopic_ratios}
Measurements of isotopic ratios in the local ISM have been carried out over the years, mostly by observing molecules at millimeter wavelengths in diffuse and dense clouds, but also from optical observations of molecules in diffuse media.
The $^{12}$C/$^{13}$C ratio has been one of the most studied. Reported values are 59 $\pm$ 2 (from HCO$^+$, HCN, and HNC mm data of diffuse clouds; \citealt{Lucas1998}), 69 $\pm$ 6 (from CO and H$_2$CO mm data of dense clouds; \citealt{Wilson1999}), 68 $\pm$ 15 (from CN mm observations of dense clouds; \citealt{Milam2005}), 70 $\pm$ 2 (from optical data of CO in diffuse clouds; \citealt{Sheffer2007}), 76 $\pm$ 2 (from CH$^+$ optical measurements of diffuse clouds; \citealt{Stahl2008}), and 74.4 $\pm$ 7.6 (from CH$^+$ data at visible wavelengths; \citealt{Ritchey2011}). Although fractionation may be an issue for some of these molecules, it is worth noting that all these studies show good agreement, with $^{12}$C/$^{13}$C ratios in the range 59-76.
The study of the $^{14}$N/$^{15}$N ratio has also been the subject of intense research activity. From HCN millimeter data of dense clouds, \citet{Dahmen1995} established a Galactic gradient with a value of 450 for the local ISM, while \citet{Wilson1999} reported an average value of 388 $\pm$ 32. \citet{Adande2012} used CN and HNC millimeter data and found 290 $\pm$ 40 for the local ISM. \citet{Lucas1998} found a value of 237$^{+27}_{-21}$ in one local diffuse cloud, while more recently, \citet{Ritchey2015} found 274 $\pm$ 18 from optical measurements of CN in diffuse clouds. Literature $^{14}$N/$^{15}$N ratios differ by up to a factor of two, with values in the range 237-450.
Oxygen and sulfur isotopic ratios have not been so widely studied as those of carbon and nitrogen. Reported values of the $^{16}$O/$^{18}$O ratio in the local ISM are 672 $\pm$ 110 (from HCO$^+$ mm data of one local diffuse cloud; \citealt{Lucas1998}) and 557 $\pm$ 30 (from H$_2$CO mm data of dense clouds; \citealt{Wilson1999}). The $^{18}$O/$^{17}$O ratio is found to be 3.6 $\pm$ 0.2 by \cite{Wilson1999} and 4.16 $\pm$ 0.09 by \cite{Wouterloot2008} from CO mm data, values that translate to $^{16}$O/$^{17}$O ratios of 2005 $\pm$ 155 and 2317 $\pm$ 134, respectively, adopting a $^{16}$O/$^{18}$O ratio of 557 $\pm$ 30 \citep{Wilson1999}. Concerning sulfur, \citet{Chin1996} found $^{32}$S/$^{34}$S = 24.4 $\pm$ 5.0 and $^{32}$S/$^{33}$S = 153 $\pm$ 40 from CS mm observations in local star-forming regions. Further reported values of the $^{32}$S/$^{34}$S ratio in the local ISM are 19 $\pm$ 8 (from CS in diffuse clouds; \citealt{Lucas1998}) and $\sim$22 \citep{Wilson1999}, values which are in agreement with that of \citet{Chin1996}.
\begin{table*}
\caption{Direct isotopic ratios} \label{table:isotopic_ratios}
\small
\centering
\begin{tabular}{lcccccccc}
\hline \hline
\multicolumn{1}{l}{Ratio} & \multicolumn{1}{c}{L483} & \multicolumn{1}{c}{TMC-1} & \multicolumn{1}{c}{L1521E} & \multicolumn{1}{c}{L1521B} & \multicolumn{1}{c}{L134N} & \multicolumn{1}{c}{L1544} & \multicolumn{1}{c}{B1-b} & \multicolumn{1}{c}{L1527} \\
\hline
\multicolumn{9}{c}{$^{12}$C/$^{13}$C} \\
\hline
C$^{17}$O/$^{13}$C$^{17}$O & 42 $\pm$ 13 & ... & ... & ... & ... & ... & ... & ... \\
C$_2$H/$^{13}$CCH & $>$162 & $>$250 $^a$ & ... & ... & ... & ... & ... & 210 $\pm$ 60 $^b$\\
C$_2$H/C$^{13}$CH & $>$70 & $>$170 $^a$ & ... & ... & ... & ... & ... & 140 $\pm$ 40 $^b$ \\
$c$-C$_3$H$_2$/$c$-HCC$^{13}$CH~~~$\times$2 & 106 $\pm$ 32 & ... & ... & ... & ... & ... & ... & 82 $\pm$ 16 $^b$ \\
$c$-C$_3$H$_2$/$c$-HC$^{13}$CCH & 458 $\pm$ 138 & ... & ... & ... & ... & ... & ... & 200 $\pm$ 30 $^b$ \\
$c$-C$_3$HD/$c$-H$^{13}$CCCD & 112 $\pm$ 34 & ... & ... & ... & ... & ... & ... & ... \\
$c$-C$_3$HD/$c$-HCC$^{13}$CD & 87 $\pm$ 26 & ... & ... & ... & ... & ... & ... & ... \\
CH$_3$CCH/$^{13}$CH$_3$CCH & 60 $\pm$ 18 & ... & ... & ... & ... & ... & ... & ... \\
CH$_3$CCH/CH$_3$$^{13}$CCH & 53 $\pm$ 16 & ... & ... & ... & ... & ... & ... & ... \\
CH$_3$CCH/CH$_3$C$^{13}$CH & 58 $\pm$ 17 & ... & ... & ... & ... & ... & ... & ... \\
CH$_2$DCCH/$^{13}$CH$_2$DCCH & 22 $\pm$ 7 & ... & ... & ... & ... & ... & ... & ... \\
HC$^{15}$N/H$^{13}$C$^{15}$N & 34 $\pm$ 10 & ... & ... & ... & ... & ... & ... & ... \\
H$^{15}$NC/H$^{15}$N$^{13}$C & 29 $\pm$ 9 & ... & ... & ... & ... & ... & ... & ... \\
HC$_3$N/H$^{13}$CCCN & 91 $\pm$ 27 & 79 $\pm$ 11 $^c$ & ... & 117 $\pm$ 16 $^d$ & 61 $\pm$ 9 $^d$ & ... & ... & 86.4 $\pm$ 1.6 $^e$ \\
HC$_3$N/HC$^{13}$CCN & 93 $\pm$ 28 & 75 $\pm$ 10 $^c$ & ... & 117 $\pm$ 16 $^d$ & 94 $\pm$ 26 $^d$ & ... & ... & 85.4 $\pm$ 1.7 $^e$ \\
HC$_3$N/HCC$^{13}$CN & 79 $\pm$ 24 & 55 $\pm$ 7 $^c$ & ... & 76 $\pm$ 6 $^d$ & 46 $\pm$ 9 $^d$ & ... & ... & 64.2 $\pm$ 1.1 $^e$ \\
DC$_3$N/DC$^{13}$CCN & 30 $\pm$ 9 & ... & ... & ... & ... & ... & ... & ... \\
HNCO/HN$^{13}$CO & 62 $\pm$ 19 & ... & ... & ... & ... & ... & ... & ... \\
C$^{34}$S/$^{13}$C$^{34}$S & 58 $\pm$ 18 & ... & ... & ... & ... & ... & ... & ... \\
H$_2$CS/H$_2$$^{13}$CS & 113 $\pm$ 34 & 79 $\pm$ 26 $^{f \star}$ & ... & ... & ... & ... & ... & ... \\
C$_2$S/C$^{13}$CS & 28 $\pm$ 8 & 54 $\pm$ 5 $^{g \star}$ & 51 $\pm$ 13 $^{g \star}$ & ... & ... & ... & ... & ... \\
C$_2$S/$^{13}$CCS & $>$25 & 230 $\pm$ 130 $^{g \star}$ & $>$130 $^{g \star}$ & ... & ... & ... & ... & ... \\
\hline
\multicolumn{9}{c}{$^{14}$N/$^{15}$N} \\
\hline
H$^{13}$CN/H$^{13}$C$^{15}$N & 321 $\pm$ 96 & ... & ... & ... & ... & ... & ... & ... \\
HC$_3$N/HC$_3$$^{15}$N & 490 $\pm$ 147 & 257 $\pm$ 54 $^{h \star}$ & ... & ... & ... & 400 $\pm$ 20 $^i$ & ... & ... \\
N$_2$H$^+$/$^{15}$NNH$^+$ & $>$747 & ... & ... & ... & ... & 1110 $\pm$ 240 $^j$ & $>$600 $^k$ & ... \\
N$_2$H$^+$/N$^{15}$NH$^+$ & $>$450 & ... & ... & ... & 670$^{+150}_{-230}$ $^l$ & 1050 $\pm$ 220 $^j$ & 400$^{+100}_{-65}$ $^j$ & ... \\
\hline
\multicolumn{9}{c}{$^{16}$O/$^{18}$O} \\
\hline
HNCO/HNC$^{18}$O & 232 $\pm$ 70 & ... & ... & ... & ... & ... & ... & ... \\
SO/S$^{18}$O & 158 $\pm$ 47 & ... & ... & ... & ... & ... & ... & ... \\
\hline
\multicolumn{9}{c}{$^{16}$O/$^{17}$O} \\
\hline
SO/S$^{17}$O & 128 $\pm$ 38 & ... & ... & ... & ... & ... & ... & ... \\
\hline
\multicolumn{9}{c}{$^{32}$S/$^{34}$S} \\
\hline
$^{13}$CS/$^{13}$C$^{34}$S & 31 $\pm$ 9 & ... & ... & ... & ... & ... & ... & ... \\
H$_2$CS/H$_2$C$^{34}$S & 29 $\pm$ 9 & ... & ... & ... & ... & ... & ... & ... \\
C$_2$S/C$_2$$^{34}$S & 17 $\pm$ 5 & ... & ... & ... & ... & ... & ... & ... \\
SO/$^{34}$SO & 31 $\pm$ 9 & ... & ... & ... & ... & ... & ... & ... \\
SO$_2$/$^{34}$SO$_2$ & 31 $\pm$ 9 & ... & ... & ... & ... & ... & ... & ... \\
NS/N$^{34}$S & 22 $\pm$ 7 & ... & ... & ... & ... & ... & ... & ... \\
HCS$^+$/HC$^{34}$S$^+$ & 21 $\pm$ 6 & ... & ... & ... & ... & ... & ... & ... \\
\hline
\multicolumn{9}{c}{$^{32}$S/$^{33}$S} \\
\hline
SO/$^{33}$SO & 151 $\pm$ 45 & ... & ... & ... & ... & ... & ... & ... \\
\hline
\end{tabular}
\tablenoteb{\\
$^a$~\cite{Sakai2010}. $^b$~\cite{Yoshida2019}. $^c$~\cite{Takano1998}. $^d$~\cite{Taniguchi2017}. $^e$~\cite{Araki2016}. $^f$~\cite{Liszt2012}. $^g$~\cite{Sakai2007}. $^h$~\cite{Taniguchi-Saito2017}. $^i$~\cite{Hily-Blant2018}. $^j$~\cite{Bizzocchi2013}. $^k$~\cite{Daniel2013}. $^l$~\cite{Redaelli2018}. \\
$^{\star}$ Isotopic ratio is not direct but derived through the double isotope method.
}
\end{table*}
\begin{figure*}
\centering
\includegraphics[width=0.95\textwidth]{fig_isotopic_ratios}
\caption{Direct isotopic ratios derived in L483 (see Table~\ref{table:isotopic_ratios}) are compared with local ISM values. Lower limits are indicated in green. In the case of $c$-HCC$^{13}$CH, the $c$-C$_3$H$_2$/$c$-HCC$^{13}$CH ratio is multiplied by two to account for the enhanced probability of substitution of either of the two equivalent carbon atoms of $c$-C$_3$H$_2$ (see Table~\ref{table:isotopic_ratios}). This way, the plotted value can be directly compared with the $^{12}$C/$^{13}$C ratio in the local ISM. Isotopic ratios representative of the local ISM are indicated by red horizontal rectangles and are taken as $^{12}$C/$^{13}$C = 59-76, $^{14}$N/$^{15}$N = 237-450, $^{16}$O/$^{18}$O = 557 $\pm$ 30, $^{16}$O/$^{17}$O = 2317 $\pm$ 134, $^{32}$S/$^{34}$S = 24.4 $\pm$ 5.0, $^{32}$S/$^{33}$S = 153 $\pm$ 40 (see text in Sect.~\ref{sec:isotopic_ratios}).} \label{fig:isotopic_ratios}
\end{figure*}
\subsubsection{$^{12}$C/$^{13}$C in L483} \label{sec:isotopic_ratios_discussion_13c}
The column densities derived in L483 for the isotopologs with $^{13}$C evidence a variety of behaviors. In the top panel of Fig.~\ref{fig:isotopic_ratios} we compare the $^{12}$C/$^{13}$C ratios derived for different molecules with the $^{12}$C/$^{13}$C ratio in the local ISM. Some molecules have $^{12}$C/$^{13}$C ratios in agreement with the local ISM value while others are clearly affected by fractionation.
Carbon monoxide, for example, is slightly enriched in $^{13}$C, as indicated by the C$^{17}$O/$^{13}$C$^{17}$O ratio of 42 $\pm$ 13. We can extrapolate this result to the main $^{16}$O-containing species (something that is correct if $^{17}$O fractionation is not important for CO or if it behaves similarly in $^{12}$CO and $^{13}$CO) because chemical models find that CO should not be particularly affected by oxygen fractionation \citep{Loison2019b}. Carbon isotopic fractionation for CO can be explained as a result of the reaction between $^{13}$C$^+$ and $^{12}$CO which is favored at low temperatures to produce $^{12}$C$^+$ and $^{13}$CO, meaning that CO is enriched in the heavy isotope and C$^+$ is impoverished in $^{13}$C (\citealt{Langer1984}; but see updated exothermicities in \citealt{Mladenovic2014,Mladenovic2017}). This latter fact can lead to a depletion in $^{13}$C for molecules whose synthesis involves C$^+$.
The most dramatic depletion in $^{13}$C is seen for $c$-HC$^{13}$CCH (i.e., $c$-C$_3$H$_2$ with a $^{13}$C substituted in the central carbon atom), which is severely depleted in $^{13}$C, with a $^{12}$C/$^{13}$C of 458 $\pm$ 138. Curiously, the other $^{13}$C isotopolog of $c$-C$_3$H$_2$ substituted in the carbon bonded to hydrogen ($c$-HCC$^{13}$CH) does not suffer such a severe dilution in $^{13}$C, as its $^{12}$C/$^{13}$C ratio is 106 $\pm$ 32, that is, only slightly above the local ISM value. A similar result was recently found in L1527, where the $^{12}$C/$^{13}$C ratios for $c$-HC$^{13}$CCH and $c$-HCC$^{13}$CH were found to be 310 $\pm$ 80 and 122 $\pm$ 22, respectively \citep{Yoshida2015}, or 200 $\pm$ 30 and 82 $\pm$ 16, respectively, according to \cite{Yoshida2019}. It seems clear that $^{13}$C fractionation works differently for these two $^{13}$C isotopologs of $c$-C$_3$H$_2$ and that $^{13}$C dilution is much more favorable for $c$-HC$^{13}$CCH. Results for singly deuterated $c$-C$_3$H$_2$ ($c$-C$_3$HD) indicate little $^{13}$C fractionation when $^{13}$C is substituted in a carbon atom bonded to H or D (see top panel in Fig.~\ref{fig:isotopic_ratios}), in line with what is found for the nondeuterated species ($c$-HCC$^{13}$CH). The strong dilution in $^{13}$C observed for $c$-C$_3$H$_2$ when $^{13}$C is substituted in the central carbon atom ($c$-HC$^{13}$CCH) probably vanishes in the deuterated form (the tentative $c$-C$_3$HD/$c$-HC$^{13}$CCD ratio derived is 89), although a more secure detection of $c$-HC$^{13}$CCD is needed to draw firmer conclusions on this subject.
In the case of C$_2$H, we see an isotopic anomaly as the two $^{13}$C substituted species have significantly different column densities (see Table~\ref{table:column_densities}). The isotopolog C$^{13}$CH is found to be 2.3 times more abundant than $^{13}$CCH. Moreover, C$_2$H is likely to be affected by $^{13}$C dilution, as the C$_2$H/C$^{13}$CH and C$_2$H/$^{13}$CCH ratios are found to be $>$71 and $>$165, respectively. A similar result was found in TMC-1 and L1527, where the C$^{13}$CH/$^{13}$CCH ratio is $\sim$1.6 and the $^{12}$C/$^{13}$C ratios are significantly above the local ISM value \citep{Sakai2010,Yoshida2019}.
For CH$_3$CCH there is little or no fractionation when $^{13}$C is substituted in any of the three carbon atoms. However, its deuterated species CH$_2$DCCH does show a significant enrichment in $^{13}$C when it is substituted in the carbon atom of the methyl group, with a $^{12}$C/$^{13}$C ratio of 22 $\pm$ 7.
Nitriles show a variety of behaviors concerning carbon fractionation. For example, both HCN and HNC are enriched in $^{13}$C by a factor of about two with respect to the local ISM $^{12}$C/$^{13}$C ratio. This result is found for the $^{15}$N substituted species, but it can probably be extrapolated to the main $^{14}$N species as we do not find hints of nitrogen fractionation, at least for HCN (see Sect.~\ref{sec:isotopic_ratios_discussion_rest}). In L1498, \citet{Magalhaes2018} derived a similar enrichment in $^{13}$C for HCN, with a HCN/H$^{13}$CN ratio of 45 $\pm$ 3, while in TMC-1 and L1527, \citet{Liszt2012} found that HNC is only barely enriched in $^{13}$C, with HNC/HN$^{13}$C ratios in the range 33-72. In the case of HC$_3$N, $^{12}$C/$^{13}$C ratios are slightly above, but consistent with, the local ISM value. The three $^{13}$C substituted species of HC$_3$N have similar column densities, with HCC$^{13}$CN being slightly more abundant than the other two isotopologs. This feature has previously been observed in other cold dense clouds. We find H$^{13}$CCCN:HC$^{13}$CCN:HCC$^{13}$CN relative abundances of 1.0:1.0:1.2 in L483, while these ratios are 1.0:1.0:1.4 in TMC-1 and L1527 \citep{Takano1998,Araki2016}, 1.0:1.0:1.5 in L1521B, and 1.5:1.0:2.1 in L134N \citep{Taniguchi2017}. The deuterated species DC$_3$N shows an enrichment in $^{13}$C about three times greater than for HC$_3$N, as indicated by the $^{12}$C/$^{13}$C ratio derived for the isotopolog substituted with $^{13}$C in the middle carbon atom. As in the cases of $c$-C$_3$H$_2$ and CH$_3$CCH, it seems that $^{13}$C fractionation works differently for deuterated and nondeuterated species. The higher $^{13}$C fractionation observed for deuterated species compared to the nondeuterated ones is in agreement with expectations from the larger variation of the zero-point energy. The observation of $^{13}$C isotopologs of the longer cyanopolyynes HC$_5$N and HC$_7$N in TMC-1 indicate that there is little abundance variation among the different $^{13}$C isotopologs, although HC$_5$N seems to be more depleted in $^{13}$C than HC$_3$N and HC$_7$N \citep{Taniguchi2016,Burkhardt2018}. The other nitrogen-bearing molecule for which we have constraints on the $^{12}$C/$^{13}$C ratio is HNCO, which does not show $^{13}$C fractionation.
Regarding sulfur-bearing molecules, CS does not show $^{13}$C fractionation, as measured with the $^{34}$S species. This result can be safely extrapolated to the main $^{32}$S species as we do not see evidence of fractionation on $^{34}$S (see Sect.~\ref{sec:isotopic_ratios_discussion_rest}). It is interesting to note that \citet{Liszt2012} found that neither CS nor H$_2$CS is significantly fractionated in $^{13}$C in TMC-1 and L1527. We find that thioformaldehyde is only slightly diluted in $^{13}$C in L483, with a H$_2$CS/H$_2$$^{13}$CS ratio of 113 $\pm$ 34. On the other hand, C$_2$S shows a significant enrichment in $^{13}$C when $^{13}$C is substituted in the middle carbon atom. The C$_2$S/C$^{13}$CS ratio is 28 $\pm$ 8. \citet{Sakai2007} did not find a significant fractionation for C$^{13}$CS in TMC-1, although they found that C$^{13}$CS is 4.2 times more abundant than $^{13}$CCS, and that this latter isotopolog is heavily diluted in $^{13}$C. We cannot conclude whether or not in L483 $^{13}$CCS is significantly less abundant than C$^{13}$CS because the upper limit derived for the column density of $^{13}$CCS is of the order of the value found for C$^{13}$CS (see Table~\ref{table:isotopic_ratios} and Fig.~\ref{fig:isotopic_ratios}).
\subsubsection{Nitrogen, oxygen, and sulfur isotopic ratios in L483} \label{sec:isotopic_ratios_discussion_rest}
The isotopic ratios involving $^{15}$N, $^{18}$O, $^{17}$O, $^{34}$S, and $^{33}$S derived for different molecules in L483 are shown in the bottom panel of Fig.~\ref{fig:isotopic_ratios}, where they are compared with the corresponding isotopic ratios in the local ISM. Different behaviors can be seen.
In the case of nitrogen, direct $^{14}$N/$^{15}$N ratios could only be obtained for HCN (more specifically for H$^{13}$CN) and HC$_3$N. In both cases the observed ratios are consistent with the local ISM values, and therefore we can safely state that $^{15}$N fractionation is not important for these two molecules in L483. Similar conclusions have been found for these two molecules in other cold dense environments like TMC-1, B1b, L1544, or L1498 (e.g., \citealt{Daniel2013,Taniguchi-Saito2017,Magalhaes2018,Hily-Blant2018}). In the case of N$_2$H$^+$, the two $^{15}$N substituted isotopologs have slightly different abundances, with a N$^{15}$NH$^+$/$^{15}$NNH$^+$ ratio of 1.6, and it is very likely that this molecule is affected by a dilution in $^{15}$N, as the N$_2$H$^+$/$^{15}$NNH$^+$ and N$_2$H$^+$/N$^{15}$NH$^+$ ratios we find are $>$747 and $>$450, respectively. A similar behavior was previously found in B1b \citep{Daniel2013} and L1544 \citep{Bizzocchi2013}. In fact, N$_2$H$^+$ is one of the few molecules which seems to be affected to an significant extent by nitrogen fractionation in cold dense clouds (e.g., \citealt{Redaelli2018}). In general, chemical models predict a low level of nitrogen fractionation for different types of molecules in cold dense clouds \citep{Roueff2015,Wirstrom2018,Loison2019a}, which is in line with our findings for HCN and HC$_3$N but not for N$_2$H$^+$. The observed depletion of $^{15}$N for N$_2$H$^+$ in cold dense clouds remains puzzling. \cite{Furuya2018} suggested that the $^{15}$N depletion is inherited from a diffuse gas component where the gas is depleted in $^{15}$N whereas the ice could be enriched in that same isotope. In that scenario, $^{15}$N atoms are released more efficiently through selective photodissociation of $^{14}$N$^{15}$N around the chemical transition from atomic to molecular nitrogen and are converted efficiently to ammonia ice through hydrogenation on grains. Searching for $^{15}$N in $^{15}$N$^{15}$NH$^+$ and in the various deuterated substitutes of N$_2$H$^+$, $^{14}$N$^{15}$ND$^+$, $^{15}$N$^{14}$ND$^+$ and $^{15}$N$^{15}$ND$^+$ is possible thanks to detailed spectroscopic studies \citep{Dore2009,Dore2017} and could help to further constrain the issue.
Regarding oxygen fractionation, the only constraints we have are for HNCO and SO. Our data show that isocyanic acid is enriched in $^{18}$O by a factor of about two with respect to the local interstellar abundance of this oxygen isotope. The most drastic effect of oxygen fractionation however occurs for SO, which is severely enriched in both $^{18}$O and $^{17}$O (see bottom panel of Fig.~\ref{fig:isotopic_ratios}). This result is in agreement with a recent chemical model in which it is found that in cold dense clouds oxygen fractionation takes place to a greater or lesser degree depending on the molecule, with SO being one of the species that is most affected, and with sizeable enrichments in the heavy isotopes of oxygen \citep{Loison2019b}.
Sulfur isotopic ratios involving $^{34}$S have been obtained for various molecules. It is remarkable that all of them fall in the range of $^{32}$S/$^{34}$S ratios representative of the local interstellar medium (see bottom panel of Fig.~\ref{fig:isotopic_ratios}), thus being consistent with no $^{34}$S fractionation. This finding is in agreement with previous determinations of the $^{32}$S/$^{34}$S ratio in different molecules toward assorted interstellar environments (e.g., \citealt{Wilson1999,Tercero2010,Liszt2012,Vastel2018b}) and with recent chemical model calculations, which predict little $^{34}$S fractionation for SO \citep{Loison2019b}. Regarding the $^{32}$S/$^{33}$S ratio, the only constraint we have is for SO, which shows a ratio that is fully consistent with the value for the local ISM (see bottom panel of Fig.~\ref{fig:isotopic_ratios}). Our results therefore support the hypothesis that sulfur fractionation is not important in cold dense clouds.
\subsubsection{Deuterated molecules in L483} \label{sec:deuterium}
\begin{table*}
\caption{Deuterated ratios as percentages} \label{table:deuterated_ratios}
\small
\centering
\begin{tabular}{llcccccc}
\hline \hline
Ratio & & L483 & TMC-1 & L1544 & B1-b & L1527 & IRAS\,16293$-$2422 \\
\hline
\multicolumn{8}{c}{Singly deuterated species} \\
\hline
CH$_2$DOH / CH$_3$OH & /3 & 0.63 $^a$ & ... & 3.3 $^j$ & ... & $<$1.0 $^r$ & 12 $^u$ \\
CH$_3$OD / CH$_3$OH & & 1.4 $^a$ & 2.65 $^d$ & ... & ... & ... & 1.8 $^u$ \\
HDCCO / H$_2$CCO & /2 & 4.6 $\pm$ 1.4 & ... & ... & ... & ... & 2.1 $^z$ \\
DCO$_2$$^+$ / HCO$_2$$^+$ & & 13.3 $\pm$ 4.0 & ... & ... & 13 $^o$ & ... & ... \\
$c$-C$_3$D / $c$-C$_3$H & & 4.4 $\pm$ 1.3 & ... & ... & ... & ... & ... \\
C$_4$D / C$_4$H & & 1.9 $\pm$ 0.6 & 0.43 $^e$ & ... & ... & 1.8 $^r$ & ... \\
$c$-C$_3$HD / $c$-C$_3$H$_2$ & /2 & 5.1 $\pm$ 1.5 & 2.4 $^d$ & 6-8.5 $^k$ & ... & 2.2 $^s$, 3.6 $^r$ & 7 $^v$ \\
$c$-H$^{13}$CCCD / $c$-H$^{13}$CCCH & & 4.8 $\pm$ 1.4 & ... & ... & ... & ... & ... \\
$c$-HCC$^{13}$CD / $c$-HCC$^{13}$CH & & 6.2 $\pm$ 1.9 & ... & ... & ... & ... & ... \\
$l$-C$_3$HD / $l$-C$_3$H$_2$ & /2 & 3.8 $\pm$ 1.1 & 2.0 $^f$ & 3.0 $^f$ & ... & ... & ... \\
CH$_2$DCCH / CH$_3$CCH & /3 & 6.5 $\pm$ 1.9 & 1.8 $^g$ & ... & ... & 4.7 $^s$ & ... \\
$^{13}$CH$_2$DCCH / $^{13}$CH$_3$CCH & /3 & 17.6 $\pm$ 5.3 & ... & ... & ... & ... & ... \\
CH$_3$CCD / CH$_3$CCH & & 5.9 $\pm$ 1.8 & 3.95 $^h$ & ... & ... & ... & ... \\
NH$_2$D / NH$_3$ & /3 & 1.7 $^b$ & 0.03 $^d$ & 4.3 $^l$ & 4.3 $^p$ & 1.3 $^t$ & 3.3 $^w$\\
DC$_3$N / HC$_3$N & & 2.8 $\pm$ 0.8 & 1.45 $^d$ & ... & ... & 3.9 $^s$ & ... \\
DC$^{13}$CCN / HC$^{13}$CCN & & 8.7 $\pm$ 2.6 & ... & ... & ... & ... & ... \\
CH$_2$DCN / CH$_3$CN & /3 & 4.4 $\pm$ 1.3 & ... & ... & ... & ... & 1.7 $^{aa}$ \\
DNCO / HNCO & & 3.8 $\pm$ 1.1 & ... & ... & ... & ... & 1 $^x$ \\
HDCS / H$_2$CS &/2~~~~~~~~~~~~~~~~~~~~~~~~~~ & 6.9 $\pm$ 2.1 & 1.0 $^i$ & 11 $^m$ & 15 $^q$ & 14.5 $^s$ & 5 $^{ab}$ \\
\hline
\multicolumn{8}{c}{Doubly deuterated species} \\
\hline
D$_2$CO / H$_2$CO & & 3.2 $^c$ & ... & 4 $^n$ & 5.7 $^q$ & 1.4 $^s$, 44 $^u$ & 5 $^u$ \\
CHD$_2$OH / CH$_3$OH & & 0.28 $^a$ & ... & & & & 6 $^y$ \\
$c$-C$_3$D$_2$ / $c$-C$_3$H$_2$ & & 0.97 $\pm$ 0.29 & ... & 1.2-2.1 $^k$ & ... & 0.5 $^s$ & ... \\
D$_2$CS / H$_2$CS & & 4.6 $\pm$ 1.4 & ... & 15 $^m$& 10 $^q$ & ... & ... \\
\hline
\end{tabular}
\tablenoteb{\\
$^a$~CH$_3$OH column density evaluated from $^{13}$CH$_3$OH assuming $^{12}$C/$^{13}$C = 68 \citep{Milam2005}. $^b$~NH$_3$ column density from \cite{Anglada1997}. $^c$~H$_2$CO column density evaluated from H$_2$$^{13}$CO \citep{Tafalla2000} assuming $^{12}$C/$^{13}$C = 68 \citep{Milam2005}. $^d$~\cite{Turner2001}. $^e$~\cite{Turner1989}. $^f$~\cite{Spezzano2016}. $^g$~\cite{Gerin1992a}. $^h$~\cite{Markwick2005}. $^i$~\cite{Minowa1997}. $^j$~\cite{Bizzocchi2014}. $^k$~\cite{Spezzano2013}. $^l$~\cite{Shah2001}. $^m$~\cite{Vastel2018b}. $^n$~\cite{Bacmann2003}. $^o$ \cite{Fuente2016}. $^p$~\cite{Saito2000}. $^q$~\cite{Marcelino2005}. $^r$~\cite{Sakai2009b}. $^s$~\cite{Yoshida2019}. $^t$~\cite{Hatchell2003}. $^u$~\cite{Parise2006}. $^v$~\cite{Majumdar2017}. $^w$~\cite{vanDishoeck1995}. $^x$~\cite{Coutens2016}. $^y$~\cite{Parise2004}. $^z$~\cite{Jorgensen2018}. $^{aa}$~\cite{Calcutt2018b}. $^{ab}$~\cite{Drozdovskaya2018}. \\
}
\end{table*}
This $\lambda$ 3 mm line survey revealed that L483 stands out as a source rich in deuterated molecules, something that was already suggested from the detection of triply deuterated ammonia \citep{Roueff2005}. We detected various singly deuterated molecules and four doubly deuterated species. While some of them are long known in interstellar chemistry, some others have only been discovered recently in interstellar clouds, and a few are detected for the first time in space toward L483. In Table~\ref{table:deuterated_ratios} we list the deuterated species detected, together with the column density ratios between deuterated and nondeuterated species (expressed in \%), where for singly deuterated species with two or more equivalent hydrogen nuclei these ratios are corrected by the number of equivalent hydrogen nuclei to get rid of statistical effects and facilitate the comparison between different molecules. Hereafter, all deuterium isotopic ratios given are corrected for this statistical effect. To put these isotopic ratios in context we also give the values derived in other well-studied low-mass prestellar and protostellar sources: TMC-1, L1544, B1-b, L1527, and IRAS\,16293$-$2422. It is worth noting that the lines of the deuterated species tend to be narrower than those of the nondeuterated form by $\sim$0.1 km s$^{-1}$, which indicates that deuterated molecules are present in less turbulent and colder regions of the cloud. This observational fact is in line with the expectation that deuterium fractionation is more efficient at low temperatures, and leads us to anticipate that the spatial distribution of deuterated molecules in L483 should differentiate from that of the nondeuterated counterpart, with the former being preferentially present in colder regions.
We detect the two singly deuterated versions of methanol (CH$_2$DOH and CH$_3$OD), with isotopic ratios of $\sim$1 \%, which are comparable to the ratios derived in similar sources. The CH$_2$DOH/CH$_3$OH ratio is slightly higher (by a factor of $\sim$5) in L1544 and significantly higher (by a factor of $\sim$20) in IRAS\,16293$-$2422. It is interesting to note that while in this latter source deuteration on the methyl group is favored over deuteration on the hydroxyl group, the contrary is found in L483. This is in line with the fact that we detected methanol deuterated twice on the methyl group, with a CHD$_2$OH/CH$_3$OH ratio substantially lower than in IRAS\,16293$-$2422. This result may be related to the different temperatures in each source. The $\lambda$ 3 mm line survey covered the 2$_{1,2}$-1$_{1,1}$ line of doubly deuterated formaldehyde, which was detected, leading to a D$_2$CO/H$_2$CO ratio of 3.2 \%, which is comparable to that found in L1544, B1-b, and IRAS\,16293$-$2422. This ratio has been reported to be as high as 44 \% in L1527 \citep{Parise2006}, although more recently \cite{Yoshida2019} derived a much lower value of 1.4 \%, which is more in line with the D$_2$CO/H$_2$CO ratios found in the other sources. The singly deuterated form of formaldehyde, HDCO, does not have favorable lines in the 80-116 GHz range and therefore could not be observed. We also detected the deuterated form of protonated carbon dioxide (DCO$_2$$^+$), for which the rotational spectrum is well known \citep{Bogey1988,Bizzocchi2017}. The tentative detection of DCO$_2$$^+$ in L1544 claimed by \cite{Vastel2016} is unlikely because of a difference of 0.29 MHz between the observed and laboratory frequency of the 5$_{0,5}$-4$_{0,4}$ transition. More recently, \cite{Fuente2016} reported on the first secure detection of this ion toward B1-b. In our line survey of L483, we detect clearly the 4$_{0,4}$-3$_{0,3}$ line lying at 80288.759 MHz at the correct position (the $V_{\rm LSR}$ derived is 5.30 $\pm$ 0.03 km s$^{-1}$, in perfect agreement with the systemic velocity of the source). Moreover, the 5$_{0,5}$-4$_{0,4}$ transition at 100359.521 MHz is marginally detected, although again at the correct position ($V_{\rm LSR}$ is 5.25 $\pm$ 0.11 km s$^{-1}$), which adds support to the identification of DCO$_2$$^+$. We derive a DCO$_2$$^+$/HCO$_2$$^+$ ratio of 13.3 $\pm$ 4.0 \%, which is in the high end of deuterium enrichment ratios determined in L483, and fully consistent with the value derived in B1-b ($\sim$13 \%; \citealt{Fuente2016}).
Hydrocarbons also show a variety of deuterated molecules in L483. We detected C$_4$D, with an isotopic ratio of 1.9 $\pm$ 0.6 \%, which is slightly above that found in TMC-1 (0.43; \citealt{Turner1989}) and similar to that derived in L1527 (1.8; \citealt{Sakai2009b}). We also detected the singly and doubly deuterated forms of $c$-C$_3$H$_2$, with isotopic ratios of 5.1 $\pm$ 1.5 \% and 0.97 $\pm$ 0.29 \%, respectively, which are similar to the values found in L1544 \citep{Spezzano2013} and L1527 \citep{Sakai2009b,Yoshida2019}. Moreover, $c$-C$_3$HD and $c$-C$_3$D$_2$ have been surveyed in a sample of low-mass prestellar and protostellar cores \citep{Chantzos2018}, finding that the corresponding isotopic ratios are relatively uniform, within a factor of a few, and similar to those in L1544 and L483. Thanks to the high sensitivity of our line survey, we detected the different deuterated forms of the two $^{13}$C substituted isotopologs of $c$-C$_3$H$_2$, that is, $c$-H$^{13}$CCCD, $c$-HCC$^{13}$CD, and $c$-HC$^{13}$CCD; the latter only tentatively. This is to our knowledge the first time these species have been detected in space. The deuterium ratios derived for $c$-H$^{13}$CCCD and $c$-HCC$^{13}$CD are in line with that found for $c$-C$_3$HD. We also detected the deuterated form of the linear isomer of C$_3$H$_2$, $l$-C$_3$HD, which was recently observed for the first time toward TMC-1 and L1544 \citep{Spezzano2016}. The isotopic ratio we find in L483, 3.8 $\pm$ 1.1 \%, is similar to the values derived in TMC-1 and L1544. Moreover, it seems that the linear isomer of C$_3$H$_2$ shows very similar levels of deuterium fractionation to the cyclic isomer in both low-mass prestellar and protostellar cores. The two deuterated forms of methyl acetylene, CH$_2$DCCH and CH$_3$CCD, which have previously been observed in TMC-1 with isotopic ratios of a few percent \citep{Gerin1992a,Markwick2005}, are also detected in L483 with slightly higher deuterium ratios ($\sim$6 \%). The deuterium ratio found for CH$_2$DCCH in L483, 6.5 $\pm$ 1.9 \%, is similar to that derived in L1527 (4.7 \%; \citealt{Yoshida2019}).
There are also several N-bearing species among the deuterated molecules. Singly deuterated ammonia (NH$_2$D) is detected with an isotopic ratio of 1.7 \%, which is much higher than the value derived in TMC-1 but of the order of the ratios found in L1544, B1-b, L1527, and IRAS\,16293$-$2422. We also detected the deuterated form of HC$_3$N, with an isotopic ratio of 2.8 $\pm$ 0.8 \%, similar to those found in TMC-1 and L1527. Moreover, the deuterated form of one of the $^{13}$C isotopologs (DC$^{13}$CCN) was also detected for the first time in interstellar space. The isotopic ratio derived is about three times higher than that found for the main species, which suggests that deuterium fractionation is more efficient for HC$^{13}$CCN than for HC$_3$N. The singly deuterated form of methyl cyanide (CH$_2$DCN) has been observed previously in high-mass star-forming regions, concretely in the hot core Orion KL and tentatively in G34.3 \citep{Gerin1992b}, toward Sgr\,B2 \citep{Belloche2016}, and more recently in a hot core associated with the infrared dark cloud G34.43\,+00.24 MM3 \citep{Sakai2018}, and in the hot corino IRAS\,16293$-$2422 \citep{Calcutt2018b}. We detected this species in L483, with an isotopic ratio of 4.4 $\pm$ 1.3 \%, which is at least ten times higher than those found in Orion\,KL and Sgr\,B2. Another deuterated N-bearing molecule not very widely observed before is DNCO, which has only been recently detected toward IRAS\,16293$-$2422 \citep{Coutens2016}. The deuterium fractionation of HNCO derived in L483 (3.8 $\pm$ 1.1 \%) is slightly higher than, but comparable to, that found in the hot corino source (1 \%).
We also detected HDCS and D$_2$CS with isotopic ratios of 6.9 $\pm$ 2.1 \% and 4.6 $\pm$ 1.4 \%, respectively. These values are slightly lower than, although of the same order as, those found in L1544 and B1-b \citep{Marcelino2005,Vastel2018b}. The deuterium ratio of HDCS in L483 is also only slightly lower than that found in L1527 \citep{Yoshida2019}. In TMC-1, deuterium fractionation of H$_2$CS seems to be around ten times less efficient than in L1544, B1-b, L1527, and L483, as inferred from observations of HDCS \citep{Minowa1997}.
We also report deuterium ratios for HDCCO and $c$-C$_3$D. Detection of singly deuterated ketene (HDCCO) was possible thanks to the laboratory characterization of its rotational spectrum at millimeter wavelengths \citep{Nemes1976,Guarnieri2005}. This species was recently observed for the first time in space toward IRAS\,16293$-$2422 using ALMA \citep{Jorgensen2018}. We derive a deuterium enrichment of 4.6 $\pm$ 1.4 \% for HDCCO, which is in line with the deuterated ratios of a few percent derived for other molecules. Recently, \citep{Yoshida2019} reported on the detection of the 5$_{0,5}$-4$_{0,4}$ line of HDCCO in L1527, although these authors did not derive a column density. The deuterated form of cyclic C$_3$H is detected for the first time in space toward L483 in this study. The rotational spectrum of $c$-C$_3$D has been measured in the laboratory by \cite{Yamamoto1990} and \cite{Lovas1992}. The isotopic ratio $c$-C$_3$D/$c$-C$_3$H is found to be 4.4 $\pm$ 1.3 \%, which is similar to the deuterium fractionation level in $c$-C$_3$H$_2$ and $l$-C$_3$H$_2$. The deuterated form of the linear isomer of C$_3$H has not yet been observed in space, although a similar deuterium enrichment of a few percent could be expected for $l$-C$_3$D based on the behavior of $c$-C$_3$D, $c$-C$_3$HD, and $l$-C$_3$HD.
In general, isotopic ratios in singly and doubly deuterated species in L483 are in the range 1-10 \%. When comparison with other low-mass prestellar and protostellar sources is possible, deuterium fractionation ratios in L483 tend to be systematically higher than in TMC-1 (except for CH$_3$OD, which is slightly more abundant in TMC-1), slightly smaller than in L1544 and B1-b, similar to those in L1527 (except for D$_2$CO, which is substantially more abundant in L1527), and smaller than in IRAS\,16293$-$2422. There are some molecules which show a higher degree of variability in the deuterium fractionation among different sources. For example, deuteration in ammonia proceeds much less efficiently in the starless cloud TMC-1 than in other sources, while deuteration in methanol on the methyl group seems to be especially favored in the hot corino IRAS\,16293$-$2422.
\section{Conclusions}
We used the IRAM 30m telescope to perform a $\lambda$ 3 mm line survey in the 80-116 GHz frequency range of the dense core L483, which hosts a low-mass Class\,0 protostar. We detected 631 lines (all of them in emission with the exception of one methanol line which was observed in absorption), 613 of which were assigned to rotational transitions of different molecules, while 18 remain unidentified. Most lines are narrow (FWHM $\sim$ 0.5 km s$^{-1}$) and derived rotational temperatures are low (4.1-10.5 K, except for HC$_5$N which has 28 $\pm$ 3 K), indicating that their emission arises exclusively from the ambient cold and quiescent cloud, rather than from the bipolar outflow, or the surroundings of the protostar.
We detected 71 molecules (140 if different isotopologs are taken into account), which comprise O-bearing molecules (including the $cis$ conformer of HCOOH and the three complex organic molecules HCOOCH$_3$, CH$_3$OCH$_3$, and C$_2$H$_5$OH), hydrocarbons (most of them being carbon chains), a wide variety of N-containing molecules ranging from carbon chains to oxides like N$_2$O, several S-bearing molecules (including carbon chains and saturated species like CH$_3$SH), and one Si-containing molecule (SiO). Of particular interest is the detection of several new interstellar molecules (HCCO, HCS, HSC, NCCNH$^+$, CNCN, NCO, H$_2$NCO$^+$, and NS$^+$), which have recently been reported toward L483, most of them thanks to observations carried out in the context of this line survey.
In general, fractional molecular abundances in L483 are systematically lower than in TMC-1 (especially in the case of carbon chains, which are significantly more abundant in TMC-1), while they tend to be higher than in L1544 and B1-b (probably as a consequence of a more important depletion on dust grains in these latter sources), and are similar to those in L1527. The use of chemical indicators such as the C$_4$H/CH$_3$OH abundance ratio indicates that the chemical composition of dense cores around Class\,0 protostars like B1-b, L1527, and L483 resemble more that of starless/prestellar cores like TMC-1 and L1544 than that of Class\,0 hot corino sources like IRAS\,16293$-$2422. This fact suggests that the chemical composition of the ambient cloud of some Class\,0 sources could be largely inherited from the dark cloud starless/prestellar phase. Potential chemical evolutionary indicators to trace the prestellar/protostellar transition, such as the HNCO/C$_3$S, SO$_2$/C$_2$S, and CH$_3$SH/C$_2$S ratios, are explored.
We also derived isotopic ratios for a variety of molecules for which we detected minor isotopologs containing $^{13}$C, $^{15}$N, $^{18}$O, $^{17}$O, $^{34}$S, and $^{33}$S, as well as deuterium. Many of the molecular isotopic ratios in L483 are close to the values of the local ISM, many of those involving $^{13}$C and $^{15}$N, and remarkably all those involving $^{34}$S and $^{33}$S. There are however several isotopic anomalies like an extreme depletion in $^{13}$C for one of the two isotopologs of $c$-C$_3$H$_2$, an important enrichment in $^{13}$C in one of the two isotopologs of C$_2$S, a drastic enrichment in $^{18}$O for SO and HNCO (SO being also largely enriched in $^{17}$O), and different abundances for the two $^{13}$C substituted species of C$_2$H and the two $^{15}$N substituted species of N$_2$H$^+$. We reported the first detection in space of several doubly isotopically substituted species with $^{13}$C and deuterium and evaluated for the first time the deuterium fractionation for $c$-C$_3$D. We find that deuterium fractionation in L483 tends to be higher than in TMC-1, similar to in L1544, B1-b, and L1527, and lower than in IRAS\,16293$-$2422.
The detailed chemical characterization of the L483 dense core presented here provides information that, together with similar exhaustive characterizations of other low-mass prestellar and protostellar sources foreseen for the near future, should help constrain the main factors that affect the chemical evolution of cores along the process of formation of low-mass protostars.
\begin{acknowledgements}
We thank the IRAM 30m staff for their help during the observations and the GILDAS team, in particular S\'ebastien Bardeau, for assistance with CLASS. We acknowledge the referee for a report that helped to improve the article. We also thank B. Parise for the help with the partition function of CHD$_2$OH and H. S. P. M\"uller for informing us about recent literature on IRAS\,16293$-$2422. M.A., N.M., and J.C. acknowledge funding support from the European Research Council (ERC Grant 610256: NANOCOSMOS) and from Spanish MICINN through grant AYA2016-75066-C2-1-P. M.A. also thanks funding support from the Ram\'on y Cajal programme of Spanish MICINN (RyC-2014-16277). M.T. acknowledges support from Spanish MICINN under grant AYA2016-79006-P. E.R. acknowledges the support of the Programme National "Physique et Chimie du Milieu Interstellaire" (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES.
\end{acknowledgements}
|
2,877,628,090,151 | arxiv | \section{}
\section{Introduction}
The purpose of this article is to study a model of random matrices originating from 2D Euclidean quantum field theory, known as the two-dimensional Yang--Mills measure. This model is defined for a classical compact Lie group and a
compact surface endowed with an area measure and can be understood as a gauge theory. Each loop on the surface
defines an observable called
Wilson loop. We are interested here in Wilson
loops when
the size $N$ of the matrices goes to infinity, while the loop and the surface are fixed.
Let us attempt to give an account of the origin and state of this problem. The question seems to have first appeared in a mathematical paper in \cite{Sin}. Therein, the candidate limit of Wilson loops on an arbitrary surface $\Sigma$
is called
master
field. This paper was inspired by the large $N$ limits considered in quantum gauge theories \cite{DK2,KK,Mig,MM,GrossMat,Gop}, which started after the landmark work of t'Hooft \cite{Hoo}. Since then, the Yang--Mills measure in two
dimensions
has been rigorously defined\footnote{See \cite{LS} for a historical review on the motivations of these approaches.} \cite{Gro,Dri,Sen,Lev3}, and the latter question has received the attention of several
mathematicians\footnote{See \cite{Lev7} for a recent reviews paper on the Yang--Mills
measure and the master fields in two dimensions. See also \cite{Chatt} for a study of large $N$ limits of Wilson loops, for discrete Yang--Mills measure in dimension higher than $2$.} \cite{Xu,Sen2,AS,Lev,DHK,DGHK,DN,Hal2}. It
is now
known that the limit indeed exists when the surface is the plane \cite{AS,Lev2} or the sphere \cite{DN}, giving rise to two different objects.
Simultaneously to and independently from \cite{DN}, conditional convergence results were obtained in \cite{Hal2} for loops within a topological disc embedded in a compact surface, with different set of assumptions for simple
loops. Both works \cite{DN,Hal2} relied on the so-called Makeenko--Migdal equations, which allow to tackle the problem recursively in the complexity of loops. These equations first appeared in physics in \cite{MM,KK} and were proved
rigorously on the plane in \cite{Lev2,Dah2,DHK} and on surfaces in \cite{DHK,DGHK}.
Our main result, Theorem \ref{__THM: Disc YMCont}, shows that for all loops included within a topological disc of $\Sigma$, the limit Wilson loop converges and its limit is given by the value of the \emph{master field on the plane} evaluated at the loop obtained by embedding the same disc in the plane instead of $\Sigma$, in a way that preserves the area measure. In that sense, the behaviour of the Yang--Mills measure within a topological disc is not affected asymptotically by the topological constraint imposed by the surface. Though it was probably known by physicists, we were unable to find this phenomenon mentioned explicitly in the physics literature.
Our second result Theorem \ref{__THM:simple non-contrac} shows that for any simple non-contractible loop the Wilson loop of any iteration vanishes asymptotically. This leaves the case of loops which are neither embedded in a
topological disc nor an iteration of a simple loop. We investigate this question in the next paper in the series \cite{DL}.
Our argument does not rely here on the Makeenko--Migdal equations but mainly on the convergence of another quantity: the partition function. We prove the convergence of the partition function using harmonic analysis on the classical
compact groups, generalising a result of \cite{Gur} and of the second author \cite{Lem}.
The rest of the paper is divided in three sections. The first one recalls the setup of the problem and presents the main results. The second section recalls the necessary notions from representation theory of compact groups and the result of \cite{Gur,Lem} on partition functions which we generalise to all group series of compact matrix Lie groups and all area parameter. The last one gives the proof of our main results on Wilson loops.
\tableofcontents
\section{Setup and statement of results}
\subsection{Heat kernel on compact Lie groups}
\subsubsection{Heat kernel}
In this text, $G$ will denote a compact Lie group endowed with a bi-invariant inner product. The heat kernel on $G$ is the family of smooth functions $(p_t,t\in (0,\infty))$ on $G$ satisfying
$$\frac d {dt} p_t(g)= \frac{1}{2} \Delta (p_t)(g) \text{ for } t>0 \text{ and }g\in G $$
and for any continuous function $f,$
$$\lim_{t\to 0}\int_G f(g)p_t(g)dg= f(1).$$
We denote here by $dg$ and $\Delta$ the Haar measure on $G$ and the Laplace--Beltrami operator associated to its inner product, while we write $1$ for the identity element of $G.$ The functions $p_t$ are central for all $t>0$ and form a semigroup for
the convolution product on $G$, that is for $t>0,$
$$ p_{t}(xgx^{-1})= p_t(g),\forall x,g\in G$$
and
$$p_t*p_s=p_{t+s},\forall s>0.$$
\subsubsection{Classical compact Lie groups}
\label{sec:Intro Class Gp}
In the sequel, we shall say that $G_N$ is a \emph{classical group of size} $N$\footnote{Although $\mathrm{Sp}(N)$ is a group of complex matrices of size $2N$, the denomination ``size $N$'' is not misleading, as it can be also considered as a subgroup of $\mathrm{GL}_N(\H)$. We do not exploit this property, and choose such a terminology only for the sake of simplicity.} if it is equal to one of the following matrix Lie groups:
\begin{enumerate}
\item The unitary group $\mathrm{U}(N)=\{U\in\mathrm{GL}_N(\mathbb{C}):UU^*=I_N\}$,
\item The special unitary group $\mathrm{SU}(N)=\{U\in\mathrm{U}(N):\det(U)=1\}$,
\item The special orthogonal group $\mathrm{SO}(N)=\{O\in\mathrm{GL}_N(\mathbb{R}):OO^t=I_N\}$,
\item The compact symplectic group $\mathrm{Sp}(N)=U(2N)\cap\{S\in\mathrm{GL}_{2N}(\mathbb{C}):S^tJS=J\}$ where $J\in\mathrm{GL}_{2N}(\mathbb{C})$ is defined by
\[
J=\begin{pmatrix}
0 & I_N\\
-I_N & 0
\end{pmatrix}.
\]
\end{enumerate}
We fix the bi-invariant metric as follows. Assume that $G_N$ is a subgroup of $\mathrm{GL}_n(\mathbb{C})$ (so that $n=N$ for $\mathrm{U}(N)$, $\mathrm{SU}(N)$, $\mathrm{SO}(N)$ and $n=2N$ for $\mathrm{Sp}(N)$). We set an integer parameter\footnote{This parameter corresponds to the Dyson index in random matrix theory. For details over its significance, see for instance Section 4.1 of \cite{AGZ}. We introduce this parameter so that standard Brownian motions on these Lie algebras all converge to the same process when $N\to\infty$, see for instance \cite[Section 2]{Dah} for an explanation.} $\beta$ which is equal to 1 if $G_N=\mathrm{SO}(N)$, 2 if $G_N=\mathrm{U}(N)$, $\mathrm{SU}(N)$, and 4 if $G_N=\mathrm{Sp}(N)$. We endow the Lie algebra $\mathfrak{g}_N$ of $G_N$ with the scalar product
\begin{equation}
\langle X,Y\rangle = \frac{\beta n}{2}\mathrm{Tr}(X^*Y),\forall X,Y\in \mathfrak g_N.\label{------eq:inner P LieAlg}
\end{equation}
It is conjugation-invariant in the sense that
$$\langle g Xg^{-1},gYg^{-1}\rangle= \<X,Y\>, \text{ for all } X,Y\in \mathfrak g_N \text{ and }g\in G_N,$$
and defines therefore a bi-invariant metric on $G_N.$ Here $\mathrm{Tr}$ denotes the trace of the above matrix, that is the non-normalised sum of diagonal coefficients.
\begin{rmk} Except in case 1., there is up to constant a unique invariant inner-product. The above choice of scaling is standard in random matrices. For instance, the gaussian vector on Hermitian matrices obtained by composition of
the above scalar product with the multiplication by $i$ is the classical Gaussian Unitary Ensemble.
\end{rmk}
\subsection{2D-maps and multiplicative functions}
Assume that $\Sigma$ is a two dimensional compact Riemannian manifold, with Riemannian distance $d_\Sigma$ and genus $g\geq 0$. It is homeomorphic to a $4g$-gon whose sides are, counterclockwise, $a_1,b_1,a_1^{-1},b_1^{-1},\ldots,a_g,b_g,a_g^{-1},b_g^{-1}$, such that each two edges with same letter are glued together, while respecting the orientation. The polygon is called the \emph{fundamental domain} of the surface. If $\Sigma$ has a nonempty boundary $\partial \Sigma$, the latter can be described as the union of $n$ connected components $(C_i)_{1\leq i\leq n}$, each homeomorphic to the unit circle $S^1$. We introduce here some notations used throughout this text.
Denote by $\mathrm{P}(\Sigma)$ the set of continuous maps $[0,1]\to \Sigma$ with positive and finite length, up to Lipschitz reparametrisation. When $\gamma\in \mathrm{P}(\Sigma) $, $\mathscr{L}(\gamma)$ denotes its length, $\gamma^{-1}\in \mathrm{P}(\Sigma)$ its reverse, $\underline \gamma= \tilde \gamma(0)$ its starting point and $\overline \gamma=\tilde \gamma (1)$ its endpoint, where $\tilde \gamma$ is some parametrisation. When $\overline \gamma=\underline \gamma$, we say that $\gamma$ is a \emph{loop} of $\Sigma$ and write $$\mathrm{L}(\Sigma)=\{\gamma\in \mathrm{P}(\Sigma): \underline{\gamma}=\overline{\gamma}\}.$$
When $\gamma_1,\gamma_2\in \mathrm{P}(\Sigma)$ with $\underline{\gamma}_2=\overline{\gamma}_1,$ $\gamma_1\gamma_2\in \mathrm{P}(\Sigma)$ stands for their concatenation. A distance on $\mathrm{P}(\Sigma) $ is defined \cite{Lev2} setting for all $\gamma_1,\gamma_2\in \mathrm{P}(\Sigma),$
\[
d(\gamma_1,\gamma_2)= |\mathscr{L}(\gamma_1)-\mathscr{L}(\gamma_2)| +\inf \inf_{t\in [0,1]}\{ d_{\Sigma}(\tilde\gamma_1(t),\tilde \gamma_2(t)) \}
\]
where the first infimum is taken over all Lipschitz parametrisations $\tilde \gamma_1,\tilde \gamma_2$ of respectively $\gamma_1$ and $\gamma_2$. In the following paragraphs, we will consider subsets of $\mathrm{P}(\Sigma)$ corresponding to loops traced in embedded graphs.
If $\Psi$ is a diffeomorphism of $\Sigma$ and $\ell\in \mathrm{L}(\Sigma),$ $\psi(\ell)\in \mathrm{L}(\Sigma)$ is the loop of $\Sigma$ obtained by composition a parametrisation of $\ell$ with $\Psi$.
\subsubsection{Topological maps on compact surfaces}
We follow here conventions of \cite[Section 1.3.2]{LZ}. A \emph{graph} $\mathcal{G}$ is a triple $(\mathbb{V},\mathbb{E},I)$ consisting of a set $\mathbb{V}$ of \emph{vertices}, a set $\mathbb{E}$ of \emph{edges} and an \emph{incidence relation} $I$ such that an edge is incident to either one vertex or two distinct vertices, called \emph{endpoints}. It can be given an \emph{orientation} by setting, for any $e\in\mathbb{E}$, a source $\underline{e}\in\mathbb{V}$ and a target $\overline{e}\in\mathbb{V}$. An oriented graph can be then represented by a quadruple $(\mathbb{V},\mathbb{E},s,t)$ where $\mathbb{V}$ is the set of vertices, $\mathbb{E}$ the set of oriented edges, and $s,t:\mathbb{E}\to\mathbb{V}$ are the functions that map respectively an edge $e$ to its source and target. An \emph{isomorphism} between two graphs $\mathcal{G}_1=(\mathbb{V}_1,\mathbb{E}_1,s_1,t_1)$ and $\mathcal{G}_2=(\mathbb{V}_2,\mathbb{E}_2,s_2,t_2)$ is a bijection $\phi:\mathbb{V}_1\cup\mathbb{E}_1\to\mathbb{V}_2\cup\mathbb{E}_2$ that sends $\mathbb{V}_1$ (resp. $\mathbb{E}_1$) onto $\mathbb{V}_2$ (resp. $\mathbb{E}_2$), and such that
\[
s_2(\phi(e))=\phi(s_1(e)), \ t_2(\phi(e))=\phi(t_1(e)),\ \forall e\in\mathbb{E}.
\]
A \emph{topological map} $M$ on $\Sigma$ is a finite oriented graph $\mathcal{G}=(\mathbb{V},\mathbb{E},s,t)$ endowed with an embedding $\theta:\mathcal{G}\to\Sigma$, called \emph{drawing of the graph}, such that:
\begin{itemize}
\item the vertices are drawn as distinct points of $\Sigma$,
\item oriented edges are drawn as oriented continuous curves that only intersect at their endpoints,
\item For any $e\in\mathbb{E}$, there is an edge $e^{-1}\in\mathbb{E}$ such that $\theta(e^{-1})=\theta(e)^{-1}$,
\item the set $\mathbb{F}=\Sigma\setminus\bigcup_{e\in\mathbb{E}} \{\theta(e)\}$ is given by a union of open discs called \emph{faces} of the map.
\end{itemize}
In order to avoid too heavy notations, we will always identify vertices and edges with their drawing, so that a map on $\Sigma$ can be represented by $(\mathbb{V},\mathbb{E},\mathbb{F})$ (the applications $s$ and $t$ will remain implicit from now on). $\Sigma$ is called the \emph{underlying surface} of $M$. If $v\in\mathbb{V}$ is a vertex in $M$, we denote by $\mathrm{P}(M)$ (resp. $\mathrm{L}(M)$, $\mathrm{L}_v(M)$) the set of paths (resp. loops, loops with base $v$) in $M$ obtained by concatenation of oriented edges. A loop is simple when each vertex of $M$ is the source of at most one of its edges.
\begin{figure}[!h]
\centering
\includegraphics[scale=0.8]{Top-Map2}
\caption{\small An example of topological map on a torus (on the left), and its representation as an abstract graph (on the right) whose opposite edges of the same colour are identified.}\label{fig:maps}
\end{figure}
Two maps $M_1=(\mathbb{V}_1,\mathbb{E}_1,\mathbb{F}_1)$ and $M_2=(\mathbb{V}_2,\mathbb{E}_2,\mathbb{F}_2)$, respectively on $\Sigma_1$ and $\Sigma_2$ and with associated graphs $\mathcal{G}_1 $ and $\mathcal{G}_2$, are \emph{equivalent} if there is an orientation-preserving homeomorphism $\phi:\Sigma_1\to\Sigma_2$ such that the restriction of $\phi$ to $M_1$ induces an isomorphism between $\mathcal{G}_1$ and $\mathcal{G}_2$. The \emph{genus} of a topological map in $\Sigma$ is the genus of $\Sigma$, and it does not depend on its equivalence class.
The following result, which is a particular case of Proposition 1.3.10 in \cite{Lev2}, allows to identify boundary components of $\Sigma$ with loops of a map, up to rerooting.
\begin{prop}\label{Prop: Boundary}
Let $M$ be a topological map on a compact surface $\Sigma$ such that $\partial\Sigma$ is the union of elements of $\mathrm{L}(\Sigma)$\footnote{In other terms, we ask that the boundary of $\Sigma$ has positive and finite length.}. Any connected component $C$ of $\partial\Sigma$ is the drawing of an element of $\mathrm{L}(M)$.
\end{prop}
\subsubsection{Topological maps on $\mathbb{R}^2$ or an open disc}
A topological map $M$ on the \emph{plane $\mathbb{R}^2$ or an open disc} is a topological map on the sphere $\mathbb{S}^2$ together with a marked face $f_\infty$. The latter is called the infinite face of $M$, and it contains the point at infinity when one makes the identification $\mathbb{S}^2\simeq\mathbb{R}^2\cup\{\infty\}$. From now on, for any map $M$, be it on the plane, a disc or a compact surface, $\mathbb{F}$ will denote by default the set of \emph{bounded} faces, and $\hat{\mathbb{F}}$ will denote $\mathbb{F}\cup\{f_\infty\}$ whenever it makes sense.
\subsubsection{Multiplicative functions}
For any $P\subset\mathrm{P}(\Sigma)$ and any compact group $G$, a \emph{multiplicative function} $h:P\to G$ is a function that satisfies $h_{\gamma^{-1}}={h_\gamma}^{-1}$ for any $\gamma\in P$ such that $\gamma^{-1}\in P$, and $h_{\gamma_1\gamma_2}=h_{\gamma_2}h_{\gamma_1}$ for any $\gamma_1,\gamma_2\in P$ such that $\gamma_1\gamma_2\in P$. We denote by $\mathcal{M}(P,G)$ the space of multiplicative functions from $P$ to $G$. The gauge group $\Gamma=\mathscr{C}^\infty(\Sigma,G)$ acts on it by \emph{gauge transformations}: for any $j\in\Gamma$ and $h\in\mathcal{M}(P,G)$,
\[
(j\cdot h)_\gamma = j(\overline{\gamma})^{-1} h_\gamma j(\underline{\gamma}),\ \forall \gamma\in P.
\]
Considering topological maps, gauge-invariance of multiplicative functions is equivalent to a $G$-invariance, as stated by the following lemma which is a particular instance of \cite[Lemma 2.1.5]{Lev2}.
\begin{lem}
Let $M$ be a topological map in $\Sigma$, $v\in\mathbb{V}$ and $\gamma_1,\ldots,\gamma_n\in\mathrm{P}(M)$. There exist $\ell_1,\ldots,\ell_m\in\mathrm{L}_v(M)$ such that for any $f:G^n\to\mathbb{C}$ with $h\mapsto f(h_{\gamma_1},\ldots,h_{\gamma_n})$ gauge-invariant on $\mathcal{M}(\mathrm{P}(M),G)$,
\[
f(h_{\gamma_1},\ldots,h_{\gamma_n})=\tilde{f}(h_{\ell_1},\ldots,h_{\ell_m}),\ \forall h\in\mathcal{M}(\mathrm{P}(M),G),
\]
for some function $\tilde{f}:G^m\to\mathbb{C}$ which is invariant by the diagonal action of $G$.
\end{lem}
Two sigma-fields may be put on $\mathcal{M}(\mathrm{P}(M),G)$:
\begin{itemize}
\item The smallest sigma-field $\mathcal{C}$ such that for any $\gamma\in\mathrm{P}(M)$ the evaluation function\footnote{It is also called \emph{holonomy} in the literature.}
\[
H_\gamma:\left\lbrace\begin{array}{ccc}
\mathcal{M}(\mathrm{P}(M),G) & \to & G\\
h & \mapsto & h_\gamma
\end{array}\right.
\]
is measurable: it is the \emph{cylindrical sigma-field}.
\item The smallest sigma-field $\mathcal{J}$ that makes
\[
h\mapsto f(h_{\ell_1},\ldots,h_{\ell_n})
\]
measurable, for all $v\in\mathbb{V}$, $n\in\mathbb{N}$, $\ell_1,\ldots,\ell_n\in\mathrm{L}_v(M)$ and $f:G^n\to\mathbb{C}$ $G$-invariant: it is the \emph{invariant sigma-field}.
\end{itemize}
We will mainly work with $\mathcal{C}$ but some results will only hold on $\mathcal{J}$, therefore we will specify which sigma-field we consider.
\subsubsection{Wilson loops}
If $\chi: G\to \mathbb{C}$ is a central function, for any $\ell\in \mathrm{L}(M)$, $\chi(h_{\ell})$ does not depend on the choice of based point. The function
\[
\begin{array}{ccl}&\mathcal{M}(\mathrm{L}(M),G) &\longrightarrow \mathbb{C} \\ &h&\longmapsto \chi (h_\ell)\end{array}
\]
is then called a Wilson loop.
When $G$ belongs to one of the four series given above, we will be interested in the Wilson loops
\begin{equation}
W_\ell:\left\lbrace\begin{array}{ccl} \mathcal{M}(\mathrm{L}(M),G) & \longrightarrow & \mathbb{C} \\ h&\longmapsto & \mathrm{tr} (h_\ell),\end{array} \right. \label{------eq: Wilson loop}
\end{equation}
where $\ell$ is a loop of $M$, while for any $M\in M_d(\mathbb{C})$, $\mathrm{tr}(M)=\frac{1}{d}\sum_{i=1}^d M_{i,i}.$
\begin{rmk}[Gauge equivalence] For most compact Lie groups $G$, it can be shown that the family of Wilson loops separate points of $\mathcal{M}(\mathrm{P}(M),G)/\Gamma$ endowed with the quotient topology. When $G$ belongs to one of the four series of the previous section, it can further be shown \cite{Sen5,Lev4} that it is enough to consider the family $\{W_\ell, \ell \in \mathrm{L}(M)\}.$
\end{rmk}
\subsubsection{Area-weighted maps}
\emph{An area-weighted map} is a topological map $M$ together with a function $a: \mathbb{F} \to \mathbb{R}_+:$ the marked faces are excluded because their area is either considered to be infinite (in the case of $\mathbb{R}^2$) or to be undefined (in the case of a surface with boundary). Two maps $(M,a)$ and $(M',a')$ are \emph{equivalent} if $M$ and $M'$ are equivalent as maps and the associated
homomorphism $\phi$ of $\Sigma$ defines a bijection $\mathfrak F: \mathbb{F}\to \mathbb{F}'$ with
$$a' \circ\mathfrak F= a. $$
For any $T>0$ we set
$$\Delta_M(T)=\{a: \mathbb{F}\to \mathbb{R}_+: \sum_{f\in \mathbb{F}} a_f=T\}$$
when $M$ is a map of a closed surface and when $(M,a)$ is an area-weighted map we write $|a|=\sum_{f\in \mathbb{F}} a_f.$ If the surface is endowed with a Riemannian volume $\mathrm{vol}$, then it induces in particular an area function $\mathrm{vol}:\mathbb{F}\to\mathbb{R}_+$.
\subsection{Two dimensional Yang--Mills measure}\label{sec:YM Intro}
In this section we recall a definition of the Yang--Mills measure in three steps:
\begin{enumerate}
\item Given a topological map $M$, we define a uniform measure $\mathrm{U}_{M,C,G}$ on $\mathcal{M}(\mathrm{P}(M),G)$ with and without constraints.
\item We define the discrete Yang--Mills measure $\mathrm{YM}_{M,C,a,G}$ on an area-weighted topological map $(M,a)$ as an absolutely continuous measure with respect to $\mathrm{U}_{M,C,G}$.
\item We define the Yang--Mills holonomy field $(H_\ell)_{\ell\in\mathrm{P}(\Sigma)}$ on any compact Riemann surface $\Sigma$ with volume form $\mathrm{vol}$, whose distribution $\mathrm{YM}_{\Sigma,C,G}$ is the continuous version of $\mathrm{YM}_{M,C,a,G}.$
\end{enumerate}
\subsubsection{Uniform measure on $\mathcal{M}(\mathrm{P}(M),G)$}
For any topological map $M$, the space $\mathcal{M}(\mathrm{P}(M),G)$ is a compact Lie group when endowed with the pointwise multiplication and has a unique Haar measure $\mu_M,$ on both sigma-fields $\mathcal{C}$ and $\mathcal{J}$. Fixing a set $\mathbb{E}^+$ of positively oriented edges (so that for any $e\in\mathbb{E}$, only $e$ or $e^{-1}$ belongs to $\mathbb{E}^+$), $\mathcal{M}(\mathrm{P}(M),G)$ can be identified with $G^{\mathbb{E}^+} $ and $\mu_M$ is the push-forward of the direct product of Haar measures on $G$. Let us call it the \emph{unconstrained uniform measure} and denote it by $\mathrm{U}_{M,G}$. In contrast, we need to define a constrained version to put boundary conditions when necessary. If $\Sigma$ is a compact surface with boundary, denote by $\mathcal{B}(\Sigma)$ the set of connected components of $\partial\Sigma$, each taken twice, one for each orientation. There is a natural action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathcal{B}(\Sigma)$ by orientation-reversal.
A \emph{set of boundary conditions} on a compact surface $\Sigma$ is a $\mathbb{Z}/2\mathbb{Z}$-equivariant map
\[
C:\mathcal{B}(\Sigma)\to G/\mathrm{Ad},
\]
where $G/\mathrm{Ad}$ is the set of conjugation classes of $G$ where $\mathbb{Z}/2\mathbb{Z}$ acts by inversion. The fact that boundary conditions take values in $G/\mathrm{Ad}$ rather than simply $G$ will be explained later. If $(M,a)$ is an area-weighted map on $\Sigma$, then the boundary conditions also apply to $M$, as according to Proposition \ref{Prop: Boundary}, elements of $\mathcal{B}(\Sigma)$ can be identified with loops $\mathrm{L}(M)$ up to re-rooting. Denoting by $L_1,\ldots,L_p$ the elements of $\mathcal{B}(\Sigma)$ oriented positively, there is a bijection between $(G/\mathrm{Ad})^{\mathcal{B}(\Sigma)}/(\mathbb{Z}/2\mathbb{Z})$ and $(G/\mathrm{Ad})^p$, so that any set $C$ of boundary conditions can be identified with a tuple $C=(t_1,\ldots,t_p)$ with $t_1,\ldots,t_p\in G/\mathrm{Ad}$.
For any $t\in G/\mathrm{Ad}$ and $n\geq 1$, the set $t(n)=\{(x_1,\ldots,x_n)\in G^n:x_1\cdots x_n\in t\}$ is a homogeneous space for the $G^n$-action
\[
(g_1,\ldots,g_n)\cdot(x_1,\ldots,x_n)=(g_1x_1g_2^{-1},\ldots,g_nx_ng_1^{-1}).
\]
We denote by $\delta_{t(n)}$ the extension to $G^n$ of the unique $G^n$-invariant probability measure on $t(n)\subset G^n$. It can be thought of as the \emph{conditional Haar measure on} $G^n$, under the condition $x_1\cdots x_n\in t$.
Let us now define a measure on $(\mathcal{M}(\mathrm{P}(M),G),\mathcal{C}).$ Therefor, let us choose a specific labelling of $\mathbb{E}$. If we denote again by $L_1,\ldots,L_p$ the connected components of $\partial\Sigma$, we choose the labels of the oriented edges $e\in\mathbb{E}^+$ such that $L_i=e_{i,1}\cdots e_{i,n_i}$ for every $i$. Denote finally by $e_1,\ldots,e_m$ the remaining edges of $\mathbb{E}^+$. Then $\mathcal{M}(\mathrm{P}(M),G)$ is isomorphic to
$G^{m} \times G^{n_1}\times \cdots \times G^{n_p}$ where $h\in \mathcal{M}(\mathrm{P}(M),G)$ is mapped to the tuple $g_l=h_{e_l},1\le l\le m, g_{i,n_k}=h_{e_{i,n_k}},1\le i\le n_i, 1\le k\le p.$ The \emph{uniform measure on} $\mathcal{M}(\mathrm{P}(M),G)$ \emph{with boundary conditions} $C=(t_1,\ldots,t_p)$ is the measure $\mathrm{U}_{M,C,G}$ on $(\mathcal{M}(\mathrm{P}(M),G),\mathcal{C})$ defined by
\[
\mathrm{U}_{M,(t_1,\ldots,t_p),G}(dh)=dg_1\otimes\cdots\otimes dg_m\otimes\bigotimes_{i=1}^p\delta_{t_i(n_i)}(dg_{i,1}\cdots dg_{i,n_i}).
\]
The translation invariance of the Haar measure and the choice $G/\mathrm{Ad}$-valued boundary conditions yield the following invariance proved given in \cite[Prop. 2.3.6.]{Lev2}.
\begin{prop} The measure $\mathrm{U}_{M,(t_1,\ldots,t_p),G}$ is left invariant by the action of $\Gamma$ on $\mathcal{M}(\mathrm{P}(M),G)$.
\end{prop}
We shall also consider constrained measures. Assume that the boundary of $\Sigma$ is empty, $\ell\in \mathrm{L}(M)$ is a simple loop and $t\in G/\mathrm{Ad}$. Denoting by $\ell= e_{0,1}\ldots e_{0,n}$ the edge decomposition of $\ell$ in $M,$ and labeling $e_1,\ldots,e_m$ the other edges of $M,$ the \emph{uniform measure on} $\mathcal{M}(\mathrm{P}(M),G)$ with \emph{constraint} $t$ on $\ell$ is
the measure $\mathrm{U}_{M,C_{\ell\mapsto t},G}$ on $(\mathcal{M}(\mathrm{P}(M),G),\mathcal{C})$ defined by
\[
\mathrm{U}_{M,C_{\ell\mapsto t},G}(dh)=dg_1\otimes\cdots\otimes dg_m\otimes \delta_{t(n)}(dg_{0,1}\cdots dg_{0,n}).
\]
\subsubsection{Discrete Yang--Mills measure}
We follow here \S 1.5.2 of \cite{Lev3}. Consider a compact orientable surface $\Sigma$ with genus $g\geq 0$, and an area-weighted map $(M,a)$ with graph $\mathcal{G}$ embedded in $\Sigma$. For each face $f\in\mathbb{F}$, its boundary $\partial f$ is the loop of $\mathcal{G}$ given by the concatenation of edges bordering $f.$ It can be endowed with two possible orientations.
The \emph{Yang--Mills measure}\footnote{When the boundary of $\Sigma$ is non-empty, we also call it Yang-Mills measure with free boundary conditions.} on $(M,a)$ with structure group $G$ is the probability measure $\mathrm{YM}_{M,a,G}$ on $(\mathcal{M}(\mathrm{P}(M),G),\mathcal{C})$ with density\footnote{This measure seems to have been introduced first in the physics literature in \cite{Mig}, see also \cite{MenO} and \cite{Wit} where it was used to
compute symplectic volumes of flat connexions. Weighting with the heat kernel in gauge
theories
or continuous spin systems also bares the name of \emph{Villain action}, see for instance \cite{OstSeil}. }
\begin{equation}
\frac{1}{Z_{M,a,G}}\prod_{f\in \mathbb{F}} p_{a_f}(h_{\partial f}) \label{------eq:DS form}
\end{equation}
with respect to $\mathrm{U}_{M,C,G}$, where $Z_{M,a,G}>0$ is a positive constant.
The \emph{Yang--Mills measure} on $(M,a)$ with structure group $G$ and boundary conditions
$C=(t_1,\ldots,t_p)$ is the probability measure $\mathrm{YM}_{M,C,a,G}$ on $(\mathcal{M}(\mathrm{P}(M),G),\mathcal{C})$ with density\footnote{This measure seems to have been introduced first in the physics literature in \cite{Mig}, see also \cite{MenO} and \cite{Wit} where it was used to
compute symplectic volumes of flat connexions. Weighting with the heat kernel in gauge
theories
or continuous spin systems also bares the name of \emph{Villain action}, see for instance \cite{OstSeil}. }
\begin{equation}
\frac{1}{Z_{M,a,G}(t_1,\ldots,t_p)}\prod_{f\in \mathbb{F}} p_{a_f}(h_{\partial f}) \label{------eq:DS form}
\end{equation}
with respect to $\mathrm{U}_{M,C,G}$, where $Z_{M,a,G}(t_1,\ldots,t_p)>0$ is a positive constant. Since the heat kernel defines central functions on $G$, each term of the above product is a well defined Wilson loop. Using the semigroup property leads to the following
elementary but remarkable lemma.
\begin{lem}[\cite{Sen0,Lev2}] The constant
\[
Z_{M,a,G}(t_1,\ldots,t_p)= \int_{\mathcal{M}(\mathrm{P}(M),G)} \prod_{f\in \mathbb{F}} p_{a_f}(h_{\partial f})\mathrm{U}_{M,(t_1,\ldots,t_p),G}(dh)
\]
and \[Z_{M,a,G}=\int_{\mathcal{M}(\mathrm{P}(M),G)} \prod_{f\in \mathbb{F}} p_{a_f}(h_{\partial f})\mathrm{U}_{M,G}(dh)\]
depend only on the genus $g$ of $\Sigma$, the total area $T=\sum_{a\in \mathbb{F}}a_f$ and in the first case on the boundary conditions $C=(t_1,\ldots,t_p)$. When $\partial\Sigma$ is non-empty, $Z_{M,a,G}=1.$ \end{lem}
This constant is called the \emph{partition function}. We denote it by $Z_{(g,p),T,G}(t_1,\ldots,t_p),$ or $Z_{g,T,G}$ when $p=0,$ and we drop the subscript $G$ if there is no ambiguity on the group. When $h\in G^p$, we also write $Z_{(g,p),T}(h_1,\ldots,h_p)$ for $Z_{(g,p),T}([h_1],\ldots,[h_p])$ where $[g]\in G/\mathrm{Ad}$ denotes the conjugacy class of an element $g\in G.$ We give explicit expressions using representation theory in section \ref{sec:Pf Charact}.
When $p\ge 1,$ the Yang-Mills measure with or without boundary conditions are related by the disintegration formula
\begin{equation}
\mathrm{YM}_{M,a,G}= \int_{(G/\mathrm{Ad})^p} \mathrm{YM}_{M,(t_1,\ldots,t_p),a} dt_1\ldots dt_p.\label{-----eq: Disint YM BD}
\end{equation}
When $(M',a')$ is another map with genus $g$ and volume $T,$ we say that $(M',a')$ is finer than
$(M,a)$ if any face $f$ of $M$ is a disjoint union of faces $f_1,\ldots f_k$ of $M'$ with $a_f=a'_{f_1}+\cdots +a'_{f_k}.$ In this case, the restriction of $h\in\mathcal{M}(\mathrm{P}(M'),G)$ to the edges of $M$ defines an element $\mathscr{R}_M^{M'}(h)$ of $\mathcal{M}(\mathrm{P}(M),G)$. We have the following lemma.
\begin{lem}[\cite{Sen0,Lev2}] \label{_____Lem: Compat}When $C$ is a set of boundary conditions on $\Sigma$ and $(M',a')$ is finer than $(M,a)$, then
$$(\mathscr{R}^{M'}_{M})_* (\mathrm{YM}_{M',C,a',G})= \mathrm{YM}_{M,C,a,G}.$$
In particular if $\ell\in \mathrm{L}(M)$, the random variable $W_\ell$ has same law under $\mathrm{YM}_{M',C,a',G}$ and $\mathrm{YM}_{M,C,a,G}.$
\end{lem}
We further need to define the Yang--Mills measures with a constraint along a single loop, together with a disintegration formula. Assume that the boundary of $\Sigma$ is empty, $\ell\in \mathrm{L}(M)$ is a simple loop and $t\in G/\mathrm{Ad}$. Denoting by $\ell= e_{0,1}\ldots e_{0,n}$ the edge decomposition of $\ell$ in $M,$ and labeling $e_1,\ldots,e_m$ the other edges of $M,$ the \emph{Yang Mills on} $\mathcal{M}(\mathrm{P}(M),G)$ with structure group $G$ and \emph{constraint} $C_{\ell\mapsto t}$ is the probability measure $\mathrm{YM}_{M,C_{\ell\mapsto t},a,G}$ on $(\mathcal{M}(\mathrm{P}(M),G),\mathcal{C})$ with density\footnote{This measure appeared in the physics literature in \cite{Mig}, see also \cite{MenO} and \cite{Wit} where it was used to
compute symplectic volumes of flat connexions. Weighting with the heat kernel in gauge
theories
or continuous spin systems also bears the name of \emph{Villain action}, see for instance \cite{OstSeil}. }
\begin{equation}
\frac{1}{Z_{M,a,G}(\ell; t)}\prod_{f\in \mathbb{F}} p_{a_f}(h_{\partial f}) \label{------eq:DS form}
\end{equation}
with respect to $\mathrm{U}_{M,C_{\ell\mapsto t},G}$, where $Z_{M,a,G}(\ell;t)>0$ is a positive constant.
\begin{prop}[\cite{Lev2,Lev3}]
Let $(M,a)$ be an area-weighted map embedded in a closed surface $\Sigma$ with genus $g\geq 0$ and let $\ell\in \mathrm{L}(M)$ be a simple loop. Then for any $t\in G/\mathrm{Ad},a\in \Delta(T),$
\begin{equation}\label{eq:decondition}
Z_{M,a,G}\mathrm{YM}_{M,a,G}=\int_{G/\mathrm{Ad}}Z_{M,a,G}(\ell;t) \mathrm{YM}_{M,C_{\ell\mapsto t},a,G}dt.
\end{equation}
\end{prop}
\begin{rmk} Remarkably, the Yang-Mills measure with constraints is related to the Yang-Mills measure with boundary conditions, see for instance Theorem \ref{thm:split_separ} below. This relation plays the role of a Markov property.
The holonomy fields under the Yang-Mills measure can be furthermore understood as an example of a two-dimensional Markovian holonomy fields as defined in \cite{Lev2}.
\end{rmk}
\subsection{Yang--Mills holonomy field}
The compatibility relation of Lemma \ref{_____Lem: Compat} suggests that the measures considered can be obtained as the image of a single measure on a larger probability space. This has indeed been achieved in \cite{Lev3} and
was then generalised in \cite{Lev2}, leading to the notion of (continuous) Markovian holonomy fields, allowing to consider a very large family of loops at once. Though it is not crucial to our argument, we recall here their definition, as it
allows to reformulate our main results in a unified continuous model. A different rigorous continuous approach has been given earlier by \cite{Dri, Sen4, Sen0}. It relies on stochastic analysis of the white noise on the plane and the
formula \eqref{------eq:DS form} is obtained as a consequence of the
construction and it is called the \emph{Driver--Sengupta formula}. The random holonomy field can be understood as the parallel transport of a random connection with curvature given by a white noise. Recently,
yet another continuous construction, defining a random connection one form has been given in \cite{Che} recovering the latter formula.
Let $\Sigma$ be a compact connected orientable surface, an open disc of $\mathbb{R}^2$ or $\mathbb{R}^2$ itself, endowed with an area measure $\mathrm{vol}$. For any $\ell\in \mathrm{L}(\Sigma)$, we use the same notation as in \eqref{------eq: Wilson loop}, for the random variable
\[
W_\ell= \mathrm{tr}(H_\ell).
\]
Furthermore, whenever $h\in \mathcal{M}(\mathrm{P}(\Sigma),G)$, the restriction of $h$ to a topological map $M$ on $\Sigma$ defines an element $\mathscr{R}_{M}(h)$ of $\mathcal{M}(\mathrm{P}(M),G)$. Let us finally mention that the space $\mathcal{M}(\mathrm{P}(\Sigma),G)$ can be endowed with two sigma-fields $\mathcal{C}$ and $\mathcal{J}$ defined as in the discrete case. The continuous extension to the discrete Yang--Mills measure is provided by the following result.
\begin{thm}[\cite{Lev2}] Assume that $\Sigma$ is a compact connected surface, an open disc of $\mathbb{R}^2$ or $\mathbb{R}^2$, endowed with an area measure $\mathrm{vol}$. For any set $C=(t_1,\ldots,t_p)$ of boundary conditions, there exist a
probability measures $\mathrm{YM}_{\Sigma,G}$ and $\mathrm{YM}_{\Sigma,C,G}$ on
$(\mathcal{M}(\mathrm{P}(\Sigma),G),\mathcal{C})$ such that
\begin{enumerate}
\item For any topological map $M$ on $\Sigma$,
\[
({\mathscr{R}_{M}})_*(\mathrm{YM}_{\Sigma,G})=\mathrm{YM}_{M,\mathrm{vol},G}\text{ and }({\mathscr{R}_{M}})_*(\mathrm{YM}_{\Sigma,C,G})=\mathrm{YM}_{M,C,\mathrm{vol},G}.
\]
\item \[\mathrm{YM}_{\Sigma,G}= \int_{(G/\mathrm{Ad})^p} \mathrm{YM}_{M,(t_1,\ldots,t_p),\mathrm{vol},G}dt_1\ldots dt_p. \]
\item If $(\gamma_n)_{n\ge 0}$ is a sequence of $\mathrm{P}(\Sigma)$ and $\gamma\in \mathrm{P}(\Sigma)$ with $\underline{\gamma_n}=\underline{\gamma}, \overline{\gamma_n}=\overline{\gamma},\forall n\ge 0$ and $d(\gamma_n,\gamma )\to 0$ as $n\to \infty,$ then, under
$\mathrm{YM}_{\Sigma,G,C},$
$(H_{\gamma_n})_{n\ge 0} $ converges in distribution towards $H_\gamma.$
\end{enumerate}
Moreover, if $\Psi$ is a diffeomorphism of $\Sigma$ preserving $\mathrm{vol}$, then under $\mathrm{YM}_{\Sigma,C,G}$ and $\mathrm{YM}_{\Sigma,G},$ $W_\ell$ has same law as $W_{\Psi(\ell)}.$
\end{thm}
\begin{prop}[\cite{Lev2}] When $\Sigma$ is closed, $\ell$ is a simple loop and $t\in G/\mathrm{Ad},$ there is a measure $\mathrm{YM}_{\Sigma,C_{\ell\mapsto t},G}$ on
$(\mathcal{M}(\mathrm{P}(\Sigma),G),\mathcal{C})$ such that
\begin{enumerate}
\item For any topological map $M$ on $\Sigma$ such that $\ell$ is a drawing of a loop of $M$,
\[
({\mathscr{R}_{M}})_*(\mathrm{YM}_{\Sigma,C_{\ell\mapsto t},G})=\mathrm{YM}_{M,C_{\ell\mapsto t},\mathrm{vol},G}.
\]
Moreover the constant $Z_{M,a,G}(\ell;t)$ only depends on $\ell\in\mathrm{L}^2(\Sigma)$ and $t$. We denote it by $Z_{\Sigma,G}(\ell;t).$
\item If $\Sigma$ has genus $g$ and total volume $T$,
\begin{equation}\label{-----eq:deconditionCont}
Z_{g,T}\mathrm{YM}_{\Sigma}=\int_{G/\mathrm{Ad}}Z_{\Sigma,G}(\ell;t) \mathrm{YM}_{M,C_{\ell\mapsto t},a,G}dt.
\end{equation}
\end{enumerate}
\end{prop}
The random process $(H_\gamma,\gamma \in \mathrm{P}(\Sigma))$ with distribution given by $\mathrm{YM}_{\Sigma,C,G}$ is called the \emph{Yang--Mills holonomy field} on $\Sigma$. We are primarily interested in the random variables $W_\ell$ for $\ell\in
\mathrm{L}(\Sigma)$, with structure group a classical group $G_N$ of size $N$.
\subsection{Master fields, conjectures and main results}
Following the physics literature, I. Singer raised in \cite{Sin} the following question, that we will reformulate here in a slightly\footnote{the specific family of loops was not specified in \cite{Sin}.} modified form as a conjecture.
\begin{conj} \label{_________Conj: Singer} Let $\Sigma$ be a fixed two dimensional compact Riemannian manifold or $\mathbb{R}^2$ or a disc of $\mathbb{R}^2$. Assume that $G_N$ is a classical group of size $N$ with metric given by \eqref{------eq:inner P LieAlg}. Then for any loop $\ell\in \mathrm{L}(\Sigma) $, there
is a constant $\Phi_\Sigma(\ell)$ such that, under $\mathrm{YM}_{\Sigma,G},$
\begin{equation}
W_\ell \to \Phi_\Sigma(\ell) \text{ in probability as }N\to \infty.\label{------eq: Conv PP}
\end{equation}
If $\Psi$ is a diffeomorphism of $\Sigma$ preserving its volume form,
$$\Phi_\Sigma(\Psi(\ell))=\Phi_\Sigma(\ell).$$
\end{conj}
The limit function
$$\Phi_\Sigma: \mathrm{L}(\Sigma)\to \mathbb{C}$$
is called a \emph{master field}. This conjecture has been partly proved for the plane, working with a smaller class of loops in \cite{Xu,AS}, for unitary groups. In \cite{Lev} , it was
simultaneously proved for the plane and for all above group series. Recently another argument using Makeenko--Migdal equations was also given in \cite{Hal2}.
\begin{rmk} For any loop, $\ell\in \mathrm{L}(\Sigma),$ since $H_\ell$ is a unitary matrix, $|\mathrm{tr}(H_\ell)|\le 1$ and the convergence in probability \eqref{------eq: Conv PP} is equivalent to
\begin{equation*}
\mathbb{E}[|\mathrm{tr}(H_\ell)-\Phi_{\Sigma}(\ell)|]\to 0 \tag{*}
\end{equation*}
as $N\to \infty.$ When we say that \eqref{------eq: Conv PP} holds uniformly in the set of area vectors $a$, it is equivalent to the uniformity of the convergence (*) in $a.$
To show (*), it is sufficient to show
$$\mathbb{E}[\mathrm{tr}(H_\ell)]\to \Phi_{\Sigma}(\ell) \text{ and }\mathrm{Var}(\mathrm{tr}(H_\ell))=\mathbb{E}[\mathrm{tr}(H_\ell)\overline{\mathrm{tr}(H_\ell)}]-|\mathbb{E}[\mathrm{tr}(H_\ell)]|^2\to 0.$$
\end{rmk}
\begin{rmk} The linear extension of a master field $\Phi_\Sigma$ to $\mathbb{C}[\mathrm{L}(\Sigma)] $ comes automatically with a structure of non-commutative probability space
\cite{Lev,DN}. In the case of the plane, this non-commutative distribution can be characterised using free probability (\cite{Lev} and \cite{CDG}). In that case, one can
recover the
distribution of a free Brownian motion from the master field (\cite{Bia,Lev,CDG}).
\end{rmk}
\begin{thm}[\cite{Xu,AS,Lev} and \cite{DN}]\label{__THM: Plane}
The conjecture \ref{_________Conj: Singer} holds true when $\Sigma$ is an open disc of the plane or a sphere of total area $T>0$. Consider $\ell\in\mathrm{L}(M)$ where $(M,a)$ is an area-weighted map with \emph{fixed} total area $T.$ Then
under $\mathrm{YM}_{M,a},$
$$W_\ell\to \Phi_{M,a}(\ell) \text{ in probability},$$
where the right-hand side is deterministic and depends continuously on $a$, over $a\in \mathbb{R}_+^{\mathbb{F}}$ in the case of the plane and $a\in \Delta_M(T)$ in the case of the sphere.
\end{thm}
The
work \cite{Lev} was the first to
show rigorously that the master field $\Phi_{\mathbb{R}^2}$ satisfies a set of differential equations named after Makeenko--Migdal equations, that appeared earlier in the physics papers
\cite{MM,KK}. In
\cite{DN}, the conjecture was proved for unitary
groups, in the
case of
the sphere.
\begin{ex}[\cite{Xu,AS,Lev,DN}] Assume that $\ell\in \mathrm{L}(\mathbb{R}^2)$ is simple and encloses an area $t.$
For $n\ge 1,$
$$\mu_t(n):=\Phi_{\mathbb{R}^2}(\ell^n)=\frac{e^{-\frac{nt}2 }}{n}\sum_{m=0}^{n-1} \frac{(-nt)^m}{m!} {n \choose m+1}.$$
Denote by $\mathbb{S}^2_T $ the two-dimensional Euclidean sphere with total volume $T$. When\footnote{An the expression for any $T>0$ was proved in \cite[Thm 2.4]{DN}. } $T\le \pi^2$ and $\ell\in
\mathrm{L}(\mathbb{S}_T^2)$
is simple, cutting the sphere into two domains of area $t$ and $T-t,$
$$\mu_{t,T}(n):=\Phi_{\mathbb{S}^2_T}(\ell^n)= J_1(2 n \sigma)=\int_{-2}^2\exp(i n\sigma x) \frac{\sqrt{4-x^2} dx}{\pi}, $$
where $\sigma= \sqrt{\frac{t(T-t)}{T}}$ and
$$J_{1}(x)=\sum_{m\ge 0} \frac{(-1)^m}{m!(m+1)!}\left(\frac x 2 \right)^{2m}$$
is a Bessel function of the first kind.
\end{ex}
\begin{rmk} \begin{enumerate}
\item Note that when $n\ge 1$ and $T\le \pi^2$ are fixed, $\mu_t(n)$ and $\mu_{t,T}(n)$ are different functions of $t$. This is also true \cite{DN} when $T>\pi^2,$ though
the expression of $\mu_{t,T}$ is different.\footnote{In the physics literature, the regimes $T\le \pi^2$ and $T>\pi^2$ are respectively
called the weak and the strong regimes \cite{DK}.} Therefore,
when $\ell$ is a simple loop enclosing a disc of area $t$,
the
expression of the master field is not the same when the surface in which $\ell$ is drawn is the plane or the sphere.
\item Let us highlight nonetheless two relations between the master field on the sphere and on the plane. On the
one hand, it can be shown \cite{DN} that
\begin{equation}
\lim_{T\to \infty}\mu_{t,T}(n)= \mu_{t}(n),\forall n\ge 1. \label{------eq:Strong Sphere to plane}
\end{equation}
On the other hand, it follows from dominated convergence that for all $t\ge 0$
$$\lim_{k\to \infty}\mu_{\frac{t}{k^2}}(nk)= J_1(2 n \sqrt{t}). $$
Therefore, for all $0\le t\le T\le \pi^2,$
\begin{equation}
\lim_{k\to \infty}\mu_{\frac{\sigma^2}{k^2}}(nk)=\mu_{t,T}(n),\forall n\ge 1.\label{------eq:Conv Plane to weak sphere}
\end{equation}
\item The sequences $\mu_t(n)$ and $\mu_{t,T}(n),n\ge 1$ are moment sequences of measures $\mu_t$ and $\mu_{t,T}$ on the unit circle, associated to a time marginal of the
free Brownian motion \cite{Bia} and of the free brownian bridge \cite{DN}.
Since both $\mu_t$ and $\mu_{t,T}$ are invariant by complex conjugation, \eqref{------eq:Strong Sphere to plane} and \eqref{------eq:Conv Plane to weak sphere} imply the weak
convergences
$$\mu_{t,T}\to \mu_t \text{ as }T\to \infty$$
and for any $t\le T\le \pi^2,$
$$\mu^k_{\frac{\sigma^2}{k^2}}\to \mu_{t,T} \text{ as }k\to\infty, $$
where for any measure on $\nu$ on the unit circle, $\nu^k$ denotes the push forward of $\nu$ by $z\mapsto z^k.$
\end{enumerate}
\end{rmk}
The conjecture \ref{_________Conj: Singer} remains open for general surfaces. Though, using the Makeenko--Migdal equations on surfaces proved in \cite{DGHK}, it was realised in
\cite{DN,Hal2} that it is sometimes enough to show the convergence for a restricted family of
loops. This idea was exploited for general surfaces in \cite{Hal2} yielding the following theorem.\footnote{Another similar result is obtained in \cite{Hal2}, where only an
assumption on simple loops is made. For $g\ge 1,$ the conclusion is then weaker and holds for loops with constrained area vector.} Let us say that a loop $\ell$ of a map $M$ is included
in a disc if it is included in an open contractible set $U$ of $\Sigma$.
\begin{thm}[\cite{Hal2}] \label{__THM:Hall}Consider $G_N=\mathrm{U}(N)$. Assume that whenever $\ell=s^n$ with $n\ge 0$ and $s$ is a simple loop included in a disc, of an area-weighted
topological map $(M,a)$, under $\mathrm{YM}_{M,a,G_N},$ the random variable $W_\ell$ converges in
probability towards a constant as $N\to \infty$.
Then this also holds true for $W_\ell$ for any combinatorial loop $\ell$ included in a disc.
\end{thm}
B. Hall conjectured further that the above assumption can be removed.
\begin{conj}[\cite{Hal2}] Consider $G_N=U(N)$. Whenever $\ell=s^n$ with $n\ge 0$ and $s$ is a simple loop included in a disc of an area-weighted topological map $(M,a)$,
$W_\ell$ converges in probability towards a constant as $N\to
\infty$, under $\mathrm{YM}_{M,a,G_N}$.
\end{conj}
Our main result implies that this conjecture holds true for any closed surface $\Sigma$ and any group series of classical groups. We simultaneously prove theorem \ref{__THM:Hall}
without using the Makeenko--Migdal
equations. Instead we shall use the convergence of the partition of function; a result proved by the first author in \cite{DN} for unitary groups that we recall and
generalise to other group series in section \ref{sec:PF}. Remarkably the limit in the above conjecture is given in terms of the master field on the plane as follows. Assume that $
\ell$ is a loop of an area
weighted map $(M,a)$ included in an open disc $U.$ Considering only vertices, edges and area-weighted faces of $M$ included in $U$ and replacing all
faces intersecting
$\Sigma\setminus U$ by a single marked face yields an area-weighted map $(\tilde M_{U},a_{U})$ of $\mathbb{R}^2$.
We then denote by the same symbol the loop of $\tilde M_U$ obtained by concatenating the edges of $\ell.$
\begin{thm} \label{__THM:Disc dYM}Consider an area-weighted topological map $(M,a)$ with genus $g\ge 1$ and total volume $T$ Assume that $\ell$ is a topologically trivial loop of $(M,a)$ included in a disc $U$ of $M.$
Then for any classical group $G_N$ of size $N$, under $\mathrm{YM}_{M,a,G_N},$
\[
W_\ell \to \Phi_{\tilde M_U,a_U}( \ell)\text{ in probability as } N\to\infty,
\]
uniformly in $a\in \Delta_M(T).$
\end{thm}
\begin{ex} In particular, if $\ell=s^n$ where $s$ is a simple contractible loop enclosing an area $t\in( 0,T],$ then
$$W_\ell\to \frac{e^{-\frac {nt }2}}{n}\sum_{m=0}^{n-1} \frac{(-nt)^m}{m!} {n \choose m+1} \text{ in probability as } N\to\infty. $$
\end{ex}
This result can be generalised as follows.
\begin{thm} \label{__THM: Disc YMCont}Assume that $\Sigma$ is a closed compact Riemann surface with Riemannian volume $\mathrm{vol}$, $\Psi:U\to D_R=\{x\in \mathbb{R}^2:x_1^2+x_2^2<R\}$ is a diffeomorphism, where $\pi R^2<T,$ $U$ is an
open set of $\Sigma$ and
$\Psi_*(\mathrm{vol}_{|U})$ is the Lebesgue measure. Then, for any $\ell\in \mathrm{L}(U),$ and any classical group $G_N$ of size $N$, under $\mathrm{YM}_{\Sigma,G_N},$
\[
W_\ell\to \Phi_{\mathbb{R}^2} (\Psi(\ell)) \text{ in probability as } N\to\infty.
\]
Assuming furthermore that $\Psi^{-1}:D_{R}\to U$ can be extented continuously to $\overline{D}_R,$ with piecewise continuous derivatives, for any loop $\ell\in \mathrm{L}(\overline{D}_R),$
\[
W_{\Psi^{-1}(\ell)}\to \Phi_{\mathbb{R}^2} (\ell) \text{ in probability as } N\to\infty.
\]
\end{thm}
Note that a simple loop is included in a disc if and only if it is contractible. Our second result below is concerned with simple loops $\ell$ which are not contractible.
\begin{thm} \label{__THM:simple non-contrac} Assume that $\Sigma$ is a closed compact Riemann surface with Riemannian volume $\mathrm{vol}$ and genus $g\geq 1$, and $\ell$ is a simple non-contractible loop on $\Sigma$. Then, for any $k\in \mathbb{Z}^*$ and any classical group $G_N$ of size $N$, under $\mathrm{YM}_{\Sigma,G_N}$,
\[
W_{\ell^k} \to0 \text{ in probability as } N\to\infty.
\]
\end{thm}
Our result covers all simple loops on orientable closed surfaces. We further believe it holds true for non-orientable surfaces. In a sequel to the current work \cite{DL}, we
investigate the
master field question for all loops with self-intersections.
\section{Asymptotics of partition functions}\label{sec:PF}
Our argument relies on the convergence of partition functions of closed orientable surfaces, for classical groups, generalising results of \cite{Gur} and of the second author
\cite{Lem}.
This result was mentioned without
proof in \cite{Rus} for $g>1$ and in \cite{Dou} for $g=1$ and $G=\mathrm{U}(N)$ or $\mathrm{SU}(N)$. For $g\ge 2$, for all group series except the unitary group series, it is a simple consequence
of
\cite{Gur}. For $g=1$ it is proved in \cite{Lem} for $\mathrm{U}(N)$ and $\mathrm{SU}(N)$. For all classical groups series, the limit involves the Jacobi theta function\footnote{The Jacobi theta function only appears for the unitary
group series.} and the Euler phi
function, both defined for $q\in\mathbb{C}$ such that $|q|<1$, with
\[
\theta(q)= \sum_{n\in\mathbb{Z}} q^{n^2} \ \text{ and } \ \phi(q)=\prod_{m=1}^\infty (1-q^m).
\]
Let $r\geq 1$ be an integer, we denote respectively by $\tilde A_r, A_r, B_r$, $C_r$, $D_r$ the type of $\mathrm{U}(r)$, $\mathrm{SU}(r+1)$, $\mathrm{SO}(2r+1)$, $\mathrm{Sp}(r)$ or $\mathrm{SO}(2r)$, in reference to the type of their root system (see \cite{BtD} for instance). Note that we will alternatively use $r$ as the rank of the root system or $N$ as the size of the classical group, and their relations will be implicit: for instance, if we consider $\mathrm{SU}(N)$, it will be of type $A_r$ with $r=N-1$. Conversely, if we consider the classical group of type $D_r$, it will be $\mathrm{SO}(2r)$. This change of index may be confusing at first, but it will be helpful in order to deal simultaneously with all considered root systems. Considering limits when $r\to\infty$ and when $N\to\infty$ is equivalent.
\begin{thm}\label{__THM:PF}
For any $T>0$, $(r,g)\in(\mathbb{N}^*)^2$ and any type $X_r$, with $X\in\{B,C,D\}$, let us denote by $Z_{g,T,X_r}$ the Yang--Mills partition function on an orientable compact surface of genus $g$ and area $T$ with structure group of type $X_r$ (we will also write $Z_{g,T,r}$ when the type of the structure group is unambiguous). Set $q_T=e^{-\frac{T}{2}}$.
\begin{enumerate}
\item For all $g\ge 1$ and $T>0$, $\lim_{r\to \infty} Z_{g,T,X_r}$ exists and is given by the table \eqref{------eq:PFtable} below.
\item Moreover if $g=2$, for all $X\in \{A,B,C,D\},$ $Z_{g,T,X_r}=\lim_{T\to 0} Z_{g,T, X_r}$ is well defined and
$$\lim_{r\to \infty}Z_{g,0,X_r}=1.$$
\end{enumerate}
\end{thm}
\begin{equation}\label{------eq:PFtable}
\begin{array}{c|cccc}
\text{Type} & \tilde{A}_r & A_r & B_r,C_r & D_r\\
\hline\\
g=1 & \frac{\theta(q_T)}{\phi(q_T)^2} & \frac{1}{\phi(q_T)^2} & \frac{1}{\phi(q_T)} & \frac{1+q_T}{(1-q_T)^2\phi(q_T)}\\
\\
g\geq 2 & \theta(q_T) & 1 & 1 & 1
\end{array}
\end{equation}
As we can see, the limits for odd and even orthogonal groups are different when $g=1$, which justify our choice to make the distinction between the root systems rather than the groups themselves.
Point 2 was proved for all types except $\tilde A$ in \cite{Gur}, whereas 1 was proved in \cite{Lem} for group series $A,\tilde A$, proving independently point 2 for types $\tilde A$. Theorem \ref{__THM:PF} will be proved in Section
\ref{sec:PF}. Let us also mention that, whereas the partition function on the plane or a disc in the plane is equal to 1 for any group, the case of the sphere behaves very differently, as the partition function goes to zero exponentially fast, at a speed of order $r^2$ (\cite{MonvS}). It further displays a phase transition that was discovered by and named after Douglas and Kazakov \cite{DK} and proved (in the case of unitary groups) by L\'evy and Ma\"ida \cite{LM}.
\begin{thm}[Douglas--Kazakov phase transition]\label{__THM: DK PT}
For any $T\geq 0$, set $Z_{T,N}$ as the Yang--Mills partition function on the sphere of area $T$ with structure group $\mathrm{U}(N)$. Then the quantity
\[
F(T)=\lim_{N\to\infty}\frac{1}{N^2}\log Z_{T,N}
\]
exists. It defines a function $F\in\mathscr{C}^2(\mathbb{R}_+^*)\cap\mathscr{C}^\infty(\mathbb{R}_+^*\setminus\{\pi^2\})$ which admits a third-order jump at the area $\pi^2$.
\end{thm}
We will prove Theorem \ref{__THM:PF} in section \ref{sec:Pf PF}. Beforehand, we recall in the next section a well known expression of the partition function using
representation
theory of compact groups.
\subsection{Character decomposition of the partition function}
\label{sec:Pf Charact}
We shall express the partition functions $Z_{g,T,X_r}$ for any $X\in\{\tilde{A},A,B,C,D\}$ as a sum over non-increasing sequences of integers. The result of this section are well known \cite{Sen0,Lev3}. This will follow from standard representation theory of compact groups, and we will prove it succinctly for the sake of completeness. The main result we want to prove is Prop. \ref{______Prop:fourier_pf}, and the remaining of the section may be skipped by anyone familiar with representation theory. Most of the results we will present can be found in \cite{BtD} and \cite{Far}.
\begin{dfn}
Let $G$ be a compact Lie group.
\begin{enumerate}
\item A \emph{complex representation} of $G$ is a couple $(\varphi,V)$, where $V$ is a complex vector space and $\varphi:G\to\mathrm{GL}(V)$ is a smooth group morphism.
\item A representation $(\varphi,V)$ is \emph{irreducible} if $V$ is the only nontrivial subspace left invariant by $\varphi$.
\item The \emph{dimension} $d_\varphi$ of a representation $(\varphi,V)$ is defined as the dimension of the vector space $V$.
\item The \emph{character} of a representation $(\varphi,V)$ is the function $\chi_\varphi:G\to\mathbb{C}$ defined by
\[
\chi_\varphi(g)=\mathrm{Tr}(\varphi(g)).
\]
\end{enumerate}
\end{dfn}
In the following, we will only consider finite-dimensional representations, unless stated otherwise. The two main results of the representation theory that we will be using are the celebrated \emph{Schur's lemma} and \emph{Plancherel's theorem}.
\begin{lem}[Schur's lemma]
\begin{enumerate}
\item Let $(\varphi_1,V_1)$ and $(\varphi_2,V_2)$ be two irreducible representations of $G$. If $A:V_1\to V_2$ is a nonzero linear map such that
\begin{equation}\label{------eq:Schur1}
A\varphi_1(g)=\varphi_2(g)A,\ \forall g\in G,
\end{equation}
then $A$ is an isomorphism.
\item Let $(\varphi,V)$ be an irreducible representation of $G$. If $A\in\mathrm{End}(V)$ satisfies
\begin{equation}\label{------eq:Schur2}
A\varphi(g)=\varphi(g)A,\ \forall g\in G,
\end{equation}
then there exists $\alpha\in\mathbb{C}$ such that $A=\alpha I$, where $I$ denotes the identity of $V.$
\end{enumerate}
\end{lem}
Among the numerous consequences of this lemma, one can state that the relation \eqref{------eq:Schur1} defines an equivalence relation between irreducible representations of $G$, and it enables to define the set $\widehat{G}$ of equivalence classes of irreducible representations. This set, sometimes called \emph{dual space} of $G$, is countable whenever $G$ is compact, and it is also a group when $G$ is abelian. Another consequence of Schur's lemma is that two irreducible representations within the same class $\lambda\in\widehat{G}$ have same dimension $d_\lambda$ and same character $\chi_\lambda$.
\begin{thm}[Plancherel's theorem]
For any continuous function $f\in C^\infty(G)$, the following sum absolutely converges
\begin{equation}
f(g)=\sum_{\lambda\in\widehat{G}}d_\lambda (f*\chi_\lambda)(g), \forall g\in G,
\end{equation}
where for any $\chi\in C(G),$ $f*\chi(h)=\int_G f(h)\chi(h^{-1}g )dh,\forall g\in G. $
\end{thm}
From these results one can prove the following, which is a particular instance of Thm. 4.2 in \cite{Lia}.
\begin{thm}
Let $(p_t)_{t>0 }$ be the heat kernel on $G$. Then for all $t>0,$ $p_t\in C^\infty(G) $ and the following sum
\begin{equation}\label{------eq:Fourier_HK}
p_t(g)=\sum_{\lambda\in\widehat{G}}e^{-\frac{t}{2}c_\lambda}d_\lambda \chi_\lambda(g),\ \forall g\in G,
\end{equation}
absolutely converges, where $c_\lambda\geq 0$ is the non-negative real number such that
\[
\Delta_G\chi_\lambda=-c_\lambda \chi_\lambda.
\]
\end{thm}
Before we state the Fourier decomposition of Yang--Mills partition function, let us also mention an easy but useful result about the irreducible characters of compact groups.
\begin{prop}
Let $\lambda\in\widehat{G}$ be an equivalence class of irreducible representations of a compact group $G$. We have
\begin{equation}
\int_{G} \chi_\lambda(x [y,z])dy=d_\lambda^{-1} \chi_\lambda(xz^{-1})\chi_\lambda(z),\forall x,z\in G\label{------eq:int_conj}
\end{equation}
and
\begin{equation}\label{------eq:int_commu}
\int_{G^2} \chi_\lambda(x[y,z])dydz =\frac{\chi_\lambda(x)}{d_\lambda^2},\ \forall x\in G.
\end{equation}
\end{prop}
\begin{proof}
Consider an element $(\varphi,V)$ of the equivalence class $\lambda$ and set
$$A_z=\int_{G}\varphi(yzy^{-1})dy,\forall z\in G. $$
Since the Haar measure
by multiplication, $A_z$ satisfies \eqref{------eq:Schur2} of Schur's lemma. Therefore,
$A_z= \alpha_z I$ for some scalars $\alpha_z \in \mathbb{C}$ with
$$\alpha_z d_\lambda= \mathrm{Tr}(A_z)= \int_{G} \chi_\lambda (y z y^{-1})dy= \chi_\lambda(z).$$
We can now write the left-hand side of \eqref{------eq:int_conj} as
$$\mathrm{Tr}( \varphi(x) A_z \varphi(z^{-1}))= \frac{\chi_\lambda(z)}{d_\lambda} \mathrm{Tr}(\varphi(z^{-1} x ))= d_\lambda^{-1}\chi_\lambda(z) \chi_\lambda(z^{-1}x), \forall x,z\in G,$$
while using Plancherel theorem, the left-hand side of \eqref{------eq:int_commu} reads
$$d_\lambda^{-1}\int_{G} \chi_\lambda(z) \chi_\lambda(z^{-1}x) dz=\frac{1}{d_\lambda}\chi_\lambda*\chi_\lambda(x)= \frac{\chi_\lambda(x)}{d_\lambda^2},\forall x\in G.$$
\end{proof}
\begin{prop}\label{______Prop:fourier_pf}
Let $\Sigma$ be a connected orientable surface of genus $g\geq 1$. If $\partial\Sigma$ is connected, then the Yang--Mills partition function on $\Sigma$ with structure group of type $X_r$ and boundary condition $t\in G/\mathrm{Ad}$ is given by
\begin{equation}\label{------eq:fourier_pf_b}
Z_{(g,1),T,X_r}(t)=\sum_{\lambda\in\widehat{G}} e^{-\frac{T}{2}c_\lambda}d_\lambda^{1-2g}\chi_\lambda(t).
\end{equation}
If $\Sigma$ has no boundary, then the Yang--Mills partition function on $\Sigma$ with structure group $G$ is given by
\begin{equation}\label{------eq:fourier_pf}
Z_{g,T,X_r}=\sum_{\lambda\in\widehat{G}} e^{-\frac{T}{2}c_\lambda}d_\lambda^{2-2g}.
\end{equation}
\end{prop}
\begin{proof}
Let us start with the simplest case, which is \eqref{------eq:fourier_pf}. Consider the area-weighted map $(M,a)$ of genus $g$ with $1$ vertex, $2g$ edges $(a_1,b_1,\ldots,a_g,b_g)$ and $1$ face $f$ with area $T>0$ and boundary
$$\partial f= b_g^{-1}a_{g}^{-1}b_ga_g\cdots b_1^{-1}a_1^{-1}b_1a_1. $$
Then
\[
Z_{g,T,X_r}=Z_{M,a,G}=\int_{G^{2g}} p_T([x_1,y_1]\cdots[x_g,y_g])dx_1dy_1\cdots dx_gdy_g.
\]
Using the Fourier decomposition of the heat kernel \eqref{------eq:Fourier_HK}, we get
\[
Z_{g,T,X_r}=\sum_{\lambda\in\widehat{G}(N)}e^{-\frac{T}{2}c_\lambda}d_\lambda\int_{G^{2g}}\chi_\lambda([x_1,y_1]\cdots[x_g,y_g])dx_1dy_1\cdots dx_gdy_g.
\]
We now integrate out all commutators using \eqref{------eq:int_commu}, which yields
\[
\int_{G^{2g}}\chi_\lambda([x_1,y_1]\cdots[x_g,y_g])dx_1dy_1\cdots dx_gdy_g=\frac{1}{d_\lambda^{2g-1}}.
\]
The results follows. The proof of \eqref{------eq:fourier_pf_b} is similar, using an area-weighted map with one face whose boundary is $\partial f=b_g^{-1}a_{g}^{-1}b_ga_g\cdots b_1^{-1}a_1^{-1}b_1a_1\ell$, where $\ell$ is the simple loop corresponding to $\partial\Sigma$, oriented positively, for any $h\in G$ to
\begin{equation}
Z_{(g,1),T,X_r}(h)=Z_{M,a,G}(h)=\int_{G^{2g}} p_T(h[x_1,y_1]\cdots[x_g,y_g])dx_1dy_1\cdots dx_gdy_g. \label{-----eq:PF one Boundary}
\end{equation}
\end{proof}
Now we will specify \eqref{------eq:fourier_pf} to all compact classical groups: to do this, we only need to know the dual space $\widehat{G}$, and the numbers $c_\lambda$ and $d_\lambda$ for every $\lambda\in\widehat{G}$. This can be done thanks to their root systems. Most definitions and results are borrowed from \cite{BtD}, and can also be recovered with much clarity from Sections 2.2 and 2.3 in \cite{Mel}.
\begin{dfn}
Let $G$ be a compact connected Lie group.
\begin{enumerate}
\item A \emph{weight} of a representation $(\pi,V)$ of $G$ is a group morphism $\omega:T\to\mathrm{U}(1)$ where $T$ is a maximal torus of $G$, such that the space $V^\omega=\{v\in
V: \rho(t)v=\omega(t)v,\ \forall t\in T\}$ is not trivial. The weights form a lattice $I_r=\mathbb{Z}\Omega=\bigoplus_{i=1}^r\mathbb{Z}\omega_i$, where $(\omega_i)$ is a distinguished basis of the
lattice and $r$ is the rank of the weight lattice. The normaliser of $T$ in $G$ acts by conjugation on $T$ yielding an action on $I_r.$ The vector space $\mathscr{V}
_r=\mathbb{R}\Omega=I_r\otimes_\mathbb{Z}\mathbb{R}$
can be endowed with an invariant
inner product $\langle\cdot,\cdot\rangle$. In fact, if $\mathfrak{t}$ is the Lie algebra of the maximal torus $T\subset G$, we have $\mathbb{R}\Omega\simeq\mathfrak{t}^*$ and $\langle\cdot,\cdot\rangle$ can be taken as the dual of the inner product \eqref{------eq:inner P LieAlg}.
\item A \emph{root} of $G$ is a non-zero weight of the adjoint representation of $G$. The \emph{root system} $\Phi$ can be split into $\Phi_+\sqcup\Phi_-$ with $\Phi_-=-\Phi_+$.
The elements of $\Phi_+$ are called \emph{positive roots}.
\item The \emph{Weyl chamber} $\mathscr{C}_r$ is the set
\[
\mathscr{C}_r=\{x\in\mathscr{V}_r:\langle x,\alpha\rangle> 0\ \forall \alpha\in\Phi_+\}.
\]
It is an open cone in $\mathscr{V}_r$.
\item A \emph{dominant weight} is a weight that belongs to the closure $\overline{\mathscr{C}_r}$ of the Weyl chamber . We denote by $\Lambda_r$ the set of dominant weights.
\item The element
$$\rho=\frac{1}{2}\sum_{\alpha \in \Phi_+} \alpha$$
has the property that for any $\omega\in I_r$
\begin{equation}
\omega\in \overline{\mathscr{C}_r} \text{ if and only if } \rho+ \omega \in \mathscr{C}_r.\label{------eq:smallest element}
\end{equation}
\end{enumerate}
\end{dfn}
Below are listed the root systems corresponding to the classical groups depending on their root systems. Note that we do not treat the root systems of exceptional Lie algebras $E_6,E_7,E_8$, $F_4$ and $G_2$, as we focus in this paper on unitary, special unitary, orthogonal and symplectic groups.
\[
\begin{array}{c|c|c}
\text{Group} & \text{Type} & \Phi \\
\hline
\mathrm{U}(r) & \tilde{A}_r & \{e_i-e_j,1\leq i,j\leq r, i\neq j\} \\
\mathrm{SU}(r+1) & A_r & \{e_i-e_j,1\leq i, j\leq r+1, i\neq j\} \\
\mathrm{SO}(2r+1) & B_r & \{\pm e_i\pm e_j, 1\leq i<j\leq r\}\cup\{\pm e_i,1\leq i\leq N\} \\
\mathrm{Sp}(r) & C_r & \{\pm e_i\pm e_j, 1\leq i<j\leq r\}\cup\{\pm 2e_i,1\leq i\leq N\} \\
\mathrm{SO}(2r) & D_r & \{\pm e_i\pm e_j, 1\leq i<j\leq r\} \\
\end{array}
\]
For $\mathrm{SU}(r+1)$, we identify $\mathscr{V}
_r$ with $\mathscr{V}_r=\mathbb{R}^{r+1}/\mathbb{R}(1,\ldots,1)$ endowed with the norm
\[
\Vert [x]\Vert^2=\frac{1}{N+1}\sum_{i=1}^{N+1}\big(x_i-\frac{1}{N+1}\sum_{j=1}^{N+1} x_j\big)^2,\ \forall [x]\in\mathbb{R}^{N+1}/\mathbb{R}(1,\ldots,1).
\]
For any classical group $G$ associated with the type $X_r$, with $X\in\{\tilde{A},A,B,C,D\}$, there is a bijection $\widehat{G}\simeq\Lambda_r$: this is a consequence of the highest-weight theory (see \cite{BtD} for instance). From now on, $\lambda$ will indistinctly represent an equivalence class of irreducible representations or an element of $
\Lambda_r$.
\begin{prop}
Let $G_N$ be a classical group associated with
one of the types $\tilde{A}_r,A_r,B_r,C_r,D_r$ and $\lambda\in\widehat{G}_N\simeq\Lambda_r$ be an equivalence class of irreducible representations. We have
\begin{equation}\label{------eq:cas}
c_\lambda=\langle\lambda+2\rho,\lambda\rangle
\end{equation}
and
\begin{equation}\label{------eq:dim}
d_\lambda=\frac{\prod_{\alpha\in\Phi_+}\langle\lambda+\rho,\alpha\rangle}{\prod_{\alpha\in\Phi_+}\langle\rho,\alpha\rangle}.
\end{equation}
\end{prop}
We list below some of the objects defined previously, specified for the corresponding classical groups. Most of the explicit computations can be found in \cite{BtD}.
\begin{itemize}
\item $\mathrm{U}(r)$: type $\tilde{A}_r$
\begin{align*}
& \Phi_+ = \{e_i-e_j,\ 1\leq i<j\leq r\}\\
& \mathscr{C}_r = \{x\in\mathbb{R}^r:x_1\geq \cdots\geq x_r\}\\
& \Lambda_r = \{\lambda\in\mathbb{Z}^r:\lambda_1\geq\cdots\geq\lambda_r\}\\
& \rho = \big(\frac{r-1}{2},\frac{r-3}{2},\ldots,\frac{3-r}{2},\frac{1-r}{2}\big)\\
& c_\lambda = \frac{1}{r}\sum_{i=1}^r \lambda_i(\lambda_i+r+1-2i),\ \forall \lambda\in\Lambda_r\\
& d_\lambda = \prod_{1\leq i<j\leq r}\frac{\lambda_i-\lambda_j+j-i}{j-i},\ \forall \lambda\in\Lambda_r
\end{align*}
\item $\mathrm{SU}(r+1)$: type $A_r$
\begin{align*}
& \Phi_+ = \{[e_i-e_j], 1\leq i<j\leq r+1\} \\
& \mathscr{C}_r = \{[x]\in\mathbb{R}^{r+1}/\sim: x_1\geq \cdots\geq x_{r+1}\} \\
& \Lambda_r = \{[\lambda]\in\mathbb{Z}^{r+1}/\mathbb{R}(1,\ldots,1):\lambda_1\geq\cdots\geq\lambda_{r+1}\}\\
& \Lambda_r \simeq\{\lambda\in\mathbb{Z}^{r+1}:\lambda_1\geq\cdots\geq\lambda_{r+1}=0\}\\
& \rho = \big[\big(\frac{r}{2},\frac{r}{2}-1,\ldots,1-\frac{r}{2},-\frac{r}{2}\big)\big] \\
& c_\lambda = \frac{1}{r+1}\sum_{i=1}^{r+1} \lambda_i(\lambda_i+r+2-2i)-\frac{1}{(r+1)^2}\left(\sum_{i=1}^{r+1}\lambda_i\right)^2,\ \forall[\lambda]\in\Lambda_r \\
& d_\lambda = \prod_{1\leq i<j\leq r+1}\frac{\lambda_i-\lambda_j+j-i}{j-i},\ \forall[\lambda]\in\Lambda_r \\
\end{align*}
\item $\mathrm{SO}(2r+1)$: type $B_r$
\begin{align*}
& \Phi_+=\{e_i\pm e_j, 1\leq i<j\leq r\} \\
& \mathscr{C}_r=\{x\in\mathbb{R}^r: x_1\geq \cdots\geq x_r\geq 0\} \\
& \Lambda_r=\{\lambda\in\mathbb{N}^r:\lambda_1\geq\cdots\geq\lambda_r\}\\
& \rho = \big(r-\frac{1}{2},r-\frac{3}{2},\ldots,\frac{1}{2}\big) \\
& c_\lambda = \frac{1}{2r+1}\sum_{i=1}^r\lambda_i(\lambda_i+2r+1-2i),\ \forall\lambda\in\Lambda_r \\
& d_\lambda =\prod_{1\leq i<j\leq r}\frac{\lambda_i-\lambda_j+j-i}{j-i}\prod_{1\leq i\leq j\leq r}\frac{\lambda_i+\lambda_j+2r+1-i-j}{2r+1-i-j},\ \forall\lambda\in\Lambda_r
\end{align*}
\item $\mathrm{Sp}(r)$: type $C_r$
\begin{align*}
& \Phi_+=\{e_i\pm e_j,1\leq i<j\leq r\}\cup\{2e_i,1\leq i\leq r\} \\
& \mathscr{C}_r=\{x\in\mathbb{R}^r: x_1\geq \cdots\geq x_r\geq 0\} \\
& \Lambda_r=\{\lambda\in\mathbb{N}^r:\lambda_1\geq\cdots\geq\lambda_r\}\\
& \rho = (r,r-1,\ldots,1)\\
& c_\lambda = \frac{1}{2r}\sum_{i=1}^r \lambda_i(\lambda_i+2r+2-2i),\ \forall\lambda\in\Lambda_r \\
& d_\lambda = \prod_{1\leq i<j\leq r}\frac{\lambda_i-\lambda_j+j-i}{j-i}\prod_{1\leq i\leq j\leq r}\frac{\lambda_i+\lambda_j+2r+2-i-j}{2r+2-i-j},\ \forall\lambda\in\Lambda_r
\end{align*}
\item $\mathrm{SO}(2r)$: type $D_r$
\begin{align*}
& \Phi_+=\{e_i\pm e_j,1\leq i<j\leq r\}\cup\{e_i,1\leq i\leq r\} \\
& \mathscr{C}_r=\{x\in\mathbb{R}^r:x_1\geq x_2\geq\cdots\geq|x_r|\} \\
& \Lambda_r=\{\lambda\in\mathbb{Z}^r:\lambda_1\geq\cdots\geq\lambda_{r-1}\geq|\lambda_r|\}\\
& \rho = (r-1,r-2,\ldots,0) \\
& c_\lambda = \frac{1}{2r}\sum_{i=1}^r \lambda_i(\lambda_i+2r-2i),\ \forall\lambda\in\Lambda_r \\
& d_\lambda = \prod_{1\leq i<j\leq r}\frac{(\lambda_i-\lambda_j+j-i)(\lambda_i+\lambda_j+2r-i-j)}{(j-i)(2r-i-j)},\ \forall\lambda\in\Lambda_r
\end{align*}
\end{itemize}
We will prove Theorem \ref{__THM:PF} using an asymptotic estimation of \eqref{------eq:fourier_pf}.
\subsection{Proof of convergence of partition functions}
\label{sec:Pf PF}
We first recall the result of \cite{Gur} and give an alternative proof based on the result of \cite{Lem} for $A_r$ series.
\subsubsection{Witten zeta function}
When $g\ge 2$, for the root systems $A_r,B_r,C_r$ and $D_r$, a proof relies on an asymptotic estimation of the \emph{Witten zeta function}
\begin{equation}
\zeta_{X_r}(s)=\sum_{\lambda\in\Lambda_r}\frac{1}{d_\lambda^s},\ \forall s>1,\ \forall X\in\{A,B,C,D\}.
\end{equation}
The first claim we can do is that, for any root system $X_r$ with $X\in\{A,B,C,D\}$, any $T\in\mathbb{R}_+$ and any $g\geq 2$, we have
\begin{equation}\label{------eq:encadre_zeta}
1\leq \sum_{\lambda\in\Lambda_r} e^{-\frac{T}{2}c_\lambda}d_\lambda^{2-2g} \leq\zeta_{X_r}(2).
\end{equation}
Indeed, the sum in the middle contains only nonnegative terms, therefore is bounded from below by the term corresponding to $\lambda=(0,\ldots,0)\in\Lambda_r$, which is equal to 1. The upper bound comes from the fact that for any $\lambda\in\Lambda_r$ and $g\geq 2$, the number $c_\lambda$ is nonnegative, and $d_\lambda^{2g-2}\geq d_\lambda^2$. Thanks to \eqref{------eq:encadre_zeta} we only need to prove
\begin{equation}\label{------eq:lim_zeta}
\lim_{r\to\infty} \zeta_{X_r}(2)=1
\end{equation}
for $X\in\{A,B,C,D\}$ in order to prove Theorem \ref{__THM:PF} for $g\geq 2$.
\begin{prop}[\cite{Gur}]\label{______Prop:lim_zeta}
For any real $s>1$ and $X\in\{A,B,C,D\}$, one has
\[
\lim_{r\to\infty}\zeta_{X_r}(s)=1.
\]
\end{prop}
\begin{proof}
The case of $A_r$ is detailed in \cite{Lem}, we argue here that it implies the other cases. We will compare the dimensions and the sets of dominant weights for different root
systems, thus we will denote by $\Lambda_{X_r}$ the set of dominant weights of type $X_r$ and $d_{X_r,\lambda}$ the dimension of an irreducible representation of the
compact connected group of type $X_r$ with highest weight $\lambda$. First, let us remark that the sets of dominant weights are the same for $A_r,B_r$ and $C_r$. Let
$X\in\{B,C\}$ and $\lambda\in\Lambda_{X_r}$ a dominant weight different from $(0,\ldots,0)$. As we have $\lambda_i\geq 0$ for any $i$, it is clear that
\[
d_{X_r,\lambda}\geq \prod_{1\leq i<j\leq r}\frac{\lambda_i-\lambda_j+j-i}{j-i}=d_{A_r,\lambda}.
\]
It follows that, for any $s>1$:
\[
0\leq \sum_{\substack{\lambda\in\Lambda_{X_r}\\ \lambda\neq(0,\ldots,0)}}\frac{1}{(d_{X_r,\lambda})^s}\leq \sum_{\substack{\lambda\in\Lambda_{A_r}\\ \lambda\neq(0,\ldots,0)}}\frac{1}{(d_{A_r,\lambda})^s}.
\]
We get the right limit by letting $r$ go to infinity.
In the case of $D_r$, some dominant weights have a negative coefficient, therefore an extra care must be taken. Let $\lambda\in\Lambda_{D_r}$ and $1\leq i<j\leq r-1$, we have $\lambda_i+\lambda_j\geq 0$ as the sum of nonnegative integers. Now, if $1\leq i\leq r-1$, we also know that $\lambda_i\leq\lambda_{r-1}\leq|\lambda_r|$ therefore $\lambda_i-\lambda_r\geq 0$ as well. In any case, we deduce that
\[
d_{D_r,\lambda}\geq \prod_{1\leq i<j\leq r}\frac{\lambda_i-\lambda_j+j-i}{j-i}=d_{A_r,\tilde{\lambda}},
\]
where $\tilde{\lambda}=(\lambda_1,\ldots,\lambda_{r-1},|\lambda_r|)$. Any dominant weight $\lambda$ of type $A_r$ corresponds to at most two different dominant weights of type $r_N$: the same weight, and the one obtained by changing the sign of $\lambda_r$. Hence, we have
\[
0\leq\sum_{\substack{\lambda\in\Lambda_{D_r}\\ \lambda\neq (0,\ldots,0)}} \frac{1}{(d_{D_r,\lambda})^s} \leq 2\sum_{\substack{\lambda\in\Lambda_{A_r}\\ \lambda\neq(0,\ldots,0)}} \frac{1}{(d_{A_r,\lambda})^s}.
\]
The right-hand side converges to 0 when $r\to\infty$ therefore we obtain the expected limit for $D_r$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{__THM:PF} with $g\geq 2$]
According to Prop. \ref{______Prop:lim_zeta}, for any $g\geq 2$ and $X\in\{A,B,C,D\}$ we have
\[
\lim_{r\to\infty}\zeta_{X_r}(2g-2)=1.
\]
Together with \eqref{------eq:encadre_zeta}, we get
\[
\lim_{r\to\infty} Z_{g,T,X_r}=1
\]
\end{proof}
\subsubsection{Discrete Gaussian random variables in a cone}
In this section we prove that the convergence of $Z_{1,T,X_r}$ as $r\to \infty$ and $T>0$ is fixed for types $\{B,C,D\}$.
\begin{dfn}
Let $r\geq 1$ be an integer and $\Lambda_r$ be the set of dominant elements of type $X_r$, with $X\in\{\tilde{A},A,B,C,D\}$. A random variable $\mu$ on $\Lambda_r$ is
\emph{Gaussian} with parameter $t$ if for any $\Lambda\subset\Lambda_r$
\begin{equation}\label{------eq:disc_gauss}
\mathbb{P}(\mu\subset\Lambda)=\frac{1}{Z_{r,t}}\sum_{\lambda\in\Lambda}e^{-\frac{t}{2}c_\lambda}
\end{equation}
and the associated \emph{partition function} is
\begin{equation}\label{------eq:pf_gauss_disc}
Z_{r,t}=\sum_{\lambda\in\Lambda_r}e^{-\frac{t}{2}c_\lambda}.
\end{equation}
\end{dfn}
The denomination `Gaussian' is justified by the following remark. Recall that for any classical group $G_N\subset\mathrm{GL}_n(\mathbb{C})$ of type $X_r$ and any $\lambda\in\widehat{G}_N$,
\[
c_\lambda=\frac{1}{n}\langle\lambda+\rho,\lambda\rangle=\frac{1}{n}(\Vert\lambda+\rho\Vert^2-\Vert\rho\Vert^2).
\]
It follows that for any $\lambda\in\Lambda_r,$
\[
\mathbb{P}(\mu=\lambda)\propto e^{-\frac{t}{2n}\Vert\lambda+\rho\Vert^2},
\]
which thanks to \eqref{------eq:smallest element} defines, after a shift by $\rho$, a discrete Gaussian distribution conditioned to belong to $\rho+(I_r\cap \overline{\mathscr{C}_r})=(\rho+I_r)\cap \mathscr{C}_r$. Note that if $G_N$ is a
classical group of type $X_r$, the partition function $Z_{r,T}$ of the Gaussian distribution on $\Lambda_r$ with parameter $T$ is equal to the Yang--Mills partition function $Z_{1,T,X_r}$ on a torus of area $T$ with structure group $G_N$. Before we prove Theorem \ref{__THM:PF} for $g=1$, let us give a few notations that we will use. When $
\lambda=(\lambda_1,\lambda_2,\ldots)$ is a sequence of non-negative real numbers, we denote by $|\lambda|=\sum_{i\geq 1} \lambda_i$ its total sum. We also denote by
\[
\mathcal{P}=\{(\alpha_1,\alpha_2,\ldots)\in\mathbb{N}^{\mathbb{N}^*}:\alpha_1\geq\alpha_2\geq\cdots\}
\]
the set of integer partitions. For any $\alpha\in\mathcal{P}$ we write $\ell(\alpha)$ for its number of non-zero parts. It is well known that the generating function of partitions is given by the inverse of Euler function:
\begin{equation}
\sum_{\alpha\in\mathcal{P}}q^{|\alpha|}=\frac{1}{\phi(q)},\ \forall q\in\mathbb{C} \ \mathrm{s.t.} \ |q|<1.
\end{equation}
\begin{proof}[Proof of Thm \ref{__THM:PF} with $g=1$]
\emph{Types B and C}. Let us define an increasing sequence setting $\check{\rho}=(r-\rho_1,\ldots,r-\rho_r)$ and write
\[
\langle\lambda+2\rho,\lambda\rangle=\Vert\lambda\Vert^2+2\sum_{i=1}^r\lambda_i\rho_i=2r|\lambda|+\Vert\lambda\Vert^2-2\sum_{i=1}^{\ell(\lambda)}\lambda_i\check{\rho}_i.
\]
For the types $B_r$ and $C_r$ we have $\Lambda_r\simeq\{\lambda\in\mathcal{P}:\ell(\lambda)\leq r\}$ therefore we have
\[
Z_{r,T}=\sum_{\lambda\in\mathcal{P}:\ell(\lambda)\leq r} e^{-\frac{rT}{n}|\lambda|} e^{-\frac{T}{2n}(\Vert\lambda\Vert^2-2\sum_{i=1}^{\ell(\lambda)}\lambda_i\check{\rho}_i)}.
\]
Since $\check{\rho}$ is increasing and $\lambda$ is non-increasing,
\[
\sum_{1\leq i,j\leq\ell(\lambda)} (\lambda_i-\lambda_j)(\check{\rho}_i-\check{\rho}_j)\leq 0.
\]
It follows that
\[
0\leq 2\sum_{i=1}^{\ell(\lambda)} \lambda_i\check{\rho}_i\leq \frac{2}{\ell(\lambda)}\sum_{i=1}^{\ell(\lambda)} \lambda_i\sum_{i=1}^{\ell(\lambda)}\check{\rho}_i\leq |\lambda|
\ell(\lambda),
\]
where we used for the last inequality that in $B_r$ and $C_r$ cases, $\sum_{i=1}^{\ell(\lambda)}\check{\rho}_i$ is either $\frac{\ell(\lambda)^2}{2}$ or $\frac{\ell(\lambda)(\ell(\lambda)-1)}{2}.$ We obtain
\[
\sum_{\lambda\in\mathcal{P}:\ell(\lambda)\leq r} e^{-\frac{r T}{n}|\lambda|-\frac{T}{2n}\Vert\lambda\Vert^2}\leq Z_{r,T}\leq \sum_{\lambda\in\mathcal{P}:\ell(\lambda)\leq r} e^{-(\frac{rT}{n}-\frac{\ell(\lambda)T}{2n})|\lambda|-\frac{T}{2n}\Vert\lambda\Vert^2}.
\]
Recalling that $n$ is $2r+1$ or $2r$, both sides converge to $\phi(q_T)^{-1}$ when $r\to\infty$ by dominated convergence, and we therefore obtain the expected limit.
\emph{Type $D$.}
In this case, let us introduce two notations. For $\lambda=(\lambda_1,\ldots,\lambda_r)$ such that $\lambda_1\geq\cdots\geq\lambda_r$ and $m\in\mathbb{Z}$, we set $\tilde{\lambda}=(\lambda_1,\ldots,\lambda_{r-1})$ and $\lambda+m=(\lambda_1+m,\ldots,\lambda_r+m)$. We have
\[
\langle\lambda+2\rho,\rho\rangle=\Vert\lambda\Vert^2+2\sum_{i=1}^{r-1}\lambda_i(r-i)=2r|\tilde{\lambda}|+\lambda_r^2+\Vert\tilde{\lambda}\Vert^2-2\sum_{i=1}^{r-1}i\tilde{\lambda_i}.
\]
We find then
\begin{align*}
Z_{r,T} = & \sum_{\substack{\lambda\in\mathcal{P},k\in\mathbb{Z}\\ \ell(\lambda)\leq r-1,|k|\leq \lambda_{r-1}}} e^{-\frac{rT}{n}|\lambda|-\frac{T}{2n}(k^2+\Vert\lambda\Vert^2-2\sum_{i=1}^{r-1} i\lambda_i)}\\
= & \sum_{m\geq 0} \sum_{\substack{\lambda\in\mathcal{P},k\in\mathbb{Z}\\ \ell(\lambda)\leq r-1,\lambda_{r-1}=m,|k|\leq m}}e^{-\frac{rT}{n}|\lambda|-\frac{T}{2n}(k^2+\Vert\lambda\Vert^2-2\sum_{i=1}^{r-1} i\lambda_i)}\\
= & \sum_{m\geq 0}\left(\sum_{|k|\leq m} e^{-\frac{Tk^2}{2n}}\right)q_T^m\sum_{\substack{\mu\in\mathcal{P}\\ \ell(\mu)\leq r-2}} q_T^{|\mu|} e^{-\frac{T}{2n}(\Vert\mu+m\Vert^2-2\sum_{i=1}^{r-2}i(\mu_i+m))}.
\end{align*}
Using the same argument as we did for the types $B$ and $C$, we have
\[
2\sum_{i=1}^{r-1}i(\mu_i+m)\leq \ell(\mu)|\mu+m|,
\]
it follows that
\[
Z_{r,T} \leq \sum_{m\geq 0}\left(\sum_{|k|\leq m} e^{-\frac{Tk^2}{2n}}\right)q_T^m\sum_{\substack{\mu\in\mathcal{P}\\ \ell(\mu)\leq r-2}} q_T^{|\mu|} e^{-\frac{T}{2n}\Vert\mu+m\Vert^2}
\]
and
\[
Z_{r,T} \geq \sum_{m\geq 0}\left(\sum_{|k|\leq m} e^{-\frac{Tk^2}{2n}}\right)q_T^m\sum_{\substack{\mu\in\mathcal{P}\\ \ell(\mu)\leq r-2}} q_T^{|\mu|} e^{-\frac{T}{2n}(\Vert\mu+m\Vert^2-\ell(\mu)|\mu+m|)}.
\]
By dominated convergence both bounds converge to
\[
\sum_{m\geq 0}(2m+1)q_T^m\sum_{\lambda\in\mathcal{P}}q_T^{|\lambda|}=\frac{1+q_T}{(1-q_T)^2\phi(q_T)},
\]
hence we deduce the expected limit for $Z_{r,T}$.
\end{proof}
\begin{rmk} For any $T>0,$ it is also possible to deduce the case $g\ge 2$ from the case $g=1$ using that when $G$ is of type $X_r$, for any non-trivial representation $\lambda\in \widehat{G},$ $d_\lambda\ge r.$
\end{rmk}
\begin{rmk} A proof for the groups $\mathrm{U}(N)$ and $\mathrm{SU}(N+1)$ using heighest weights can be produced similarly to the other cases yielding a different proof of the result of
\cite{Lem}, the latter being formulated in terms of Young diagrams.
\end{rmk}
\section{Proof of Wilson loops convergence}\label{sec:}
In this section, we will prove theorems \ref{__THM:Disc dYM} and \ref{__THM:simple non-contrac}. The proof relies on an absolute continuity relation and the boundedness of the Yang--Mills partition function\footnote{which follows from the convergence discussed in the Section \ref{sec:PF}.}, except for a class of loops considered in \ref{sec:Non-Sep Loop}, which requires an independent argument. In the following sections, $G_N$ will denote a classical group of size $N$.
\subsection{Loops within in a disc}
Assume that $(M,a)$ is an area-weighted map on $\Sigma$ of genus $g\ge 1$ and area $|a|=T$, that $U$ is an open connected set of $\Sigma$ obtained by union of faces of $M$, with area $|a_U|$, such that $u=T-|a_U|>0$. The boundary of $U$ is given by a loop $\partial U\in\mathrm{L}(\tilde{M}_U).$
\begin{lem} \label{_____Lem:Density dYM}The measure $\big(\mathscr{R}_{\tilde{M}_U}^{M}\big)_*(\mathrm{YM}_{M,a,G_N})$ has density
\begin{equation*}
\frac{Z_{(g,1),u,G_N} (h_{\partial U}^{-1})}{Z_{g,T,G_N}}
\end{equation*}
with respect to $\mathrm{YM}_{\tilde M_U,a_U,G_N}.$
\end{lem}
\begin{proof} Thanks to Lemma \ref{_____Lem: Compat}, we can assume that $M$ has same number of faces as $\tilde M_U,$ that is, that exactly one face of $M$ is not included in $U$. Denote by $k$ be the number of edges of $\partial U.$ Let $M'$ be the map with $k+1$ vertices, $k+2g+ 1$ edges
$u_1,\ldots, u_k,e,a_1,b_1,\ldots,a_g,b_g$ and two faces $U'$ and $V_g$, such that $\partial U'=u_k\cdots u_1$ and $\partial V_g=e^{-1} [b_g^{-1},a_g^{-1}]\cdots [b_a^{-1},a_1^{-1}]eu_1^{-1} \cdots u_k^{-1}$. Let us assume that $M$ is
obtained by
gluing $M'$ with $\tilde M_U$,
identifying $\partial U$ with $\partial U'$ so that the base of $\partial U$ is sent to an endpoint of $e$. For any $h \in \mathcal{M}(\mathrm{P}(M),G_N)$, let us write $x_l=h_{a_l},y_l= h_{b_l}$ for all $1\le l\le g$ and $z=h_{e}.$ Denote the set of non-marked faces of $\tilde M_U$ by $\mathbb{F}_U$. Then $\mathrm{YM}_{M,a,G_N}(dh)$ can be written as
\begin{align*}
\frac 1{Z_{g,T,G_N}}p_{u}(h_{\partial U}^{-1}z[x_1,y_1]\cdots [x_g,y_g] z^{-1} )dz \prod_{i=1}^gdx_i dy_i\prod_{f\in \mathbb{F}_U}p_{a_f}(h_{\partial f}) \mathrm{U}_{\tilde{M}_U,G_N}(dh).
\end{align*}
Integrating over $(x_i,y_i)_i$ and $z$ the result follows by invariance by conjugation of the Haar measure and \eqref{-----eq:PF one Boundary}.
\end{proof}
\begin{proof}[Proof of Thm. \ref{__THM:Disc dYM}] Let us first assume that $|a_U|<T=|a|.$ For any $h\in G_N$ and $\lambda\in \widehat{G}_N,$ since the representation is unitary, $\|\chi_\lambda\|_\infty=\chi_\lambda(1)= d_\lambda$. For any $s>0$,
we deduce from \eqref{------eq:fourier_pf_b} that
\begin{equation}
\|Z_{(g,1),u,G_N}\|_{\infty}= \sum_{\lambda\in \widehat{G}_N} d_{\lambda}^{2-2g} e^{-\frac u 2 c_\lambda }=Z_{g,u,G_N}.\label{------eq:Bound Dens gCap}
\end{equation}
Together with the previous lemma, ${\mathscr{R}_{\tilde{M}_U}^{M}}_*(\mathrm{YM}_{M,a,G_N})$ has a density with respect to $\mathrm{YM}_{\tilde M_U,a_U,G_N}$ which is bounded from above by
\[
\frac{Z_{g,|a|-|a_U|,G_N}}{Z_{g,|a|,G_N}}.
\]
According to Theorem \ref{__THM:PF}, the right-hand side is uniformly bounded in $N$. In particular for any $\varepsilon>0$ and any loop $\ell\in\mathrm{L}(\tilde{M}_U)$,
\[
\mathbb{P}_{\mathrm{YM}_{M,a}}(|W_\ell- \Phi_{\tilde M_U,a_U}(\ell)|>\varepsilon)\le \frac{Z_{g,|a|-|a_U|,G_N}}{Z_{g,|a|,G_N}} \mathbb{P}_{\mathrm{YM}_{\tilde M_U,a_U}}(|W_\ell- \Phi_{\tilde M_U,a_U}(\ell)|>\varepsilon).
\]
Together with Theorem \ref{__THM: Plane} for the plane, we conclude that the right-hand side converges towards zero as $N\to \infty.$
To conclude it remains to prove the uniformity of the convergence in the set $A$ of $a\in \Delta_M(T)$ with $a(f)>0$ for any face $f$ and $|a_U|<T.$ For this, it is enough to show that for any adjacent faces $f_1,f_2$ of $M,$ there is a
constant $K>0$ such that for all $a,a'\in A,$ with $a(f)=a'(f)$ for all faces distinct from $f_1,f_2$ and $|a-a'|<\frac{T}{2},$
\begin{equation}
|\mathbb{E}_{\mathrm{YM}_{M,a}}[ |W_{\ell}-\Phi_{\tilde M_U,a_U}(\ell)|]-\mathbb{E}_{\mathrm{YM}_{M,a'}}[ |W_{\ell}-\Phi_{\tilde M_U,a'_U}(\ell)|]|\le K|a-a'|^{1/2}.\label{eq: UNIF CV thm}
\end{equation}
When $f_1,f_2$ and $a,a'$ are as above, consider an area weighted map $(M^r,a^r)$ finer than $(M,a),$ a simple loop $s\in \mathrm{L}(M^r)$ enclosing an area $|a-a'|$ and two paths $\alpha,\beta$ of $M^r$ with $\ell=\alpha\beta$, such that under $\mathrm{YM}_{M^r,a^r},$ $W_{\alpha s\beta}$ has same law as $W_\ell$ under
$\mathrm{YM}_{M,a'}.$ Using this identity in law, the left-hand-side of \eqref{eq: UNIF CV thm} equals
\[ |\mathbb{E}_{\mathrm{YM}_{M^r,a^r}}[ |W_{\alpha\beta}-\Phi_{\tilde M_U,a_U}(\alpha\beta)|- |W_{\alpha s\beta }-\Phi_{\tilde M^r_U,a^r_U}(\alpha s\beta )|]|\le \mathbb{E}_{\mathrm{YM}_{M^r,a^r}} [|W_{\alpha\beta}-W_{\alpha s \beta }|].\]
Consider an open disc of $\Sigma$ such that the only face of $M^r$ included in $V$ is the one enclosed by $s.$ The right-hand-side is bounded by
\begin{align*}
\mathbb{E}_{\mathrm{YM}_{M^r,a^r}} [|\mathrm{tr}(H_{\alpha\beta}-H_{\alpha s \beta })|]&= \mathbb{E}_{\mathrm{YM}_{M^r,a^r}} [|\mathrm{tr}(H_{\beta\alpha }-H_{\beta\alpha } H_s)|]\\
&\le \mathbb{E}_{\mathrm{YM}_{M^r,a^r}} [\mathrm{tr}[(H_{\beta\alpha} -H_{s} H_{\beta\alpha})(H_{\beta\alpha} -H_{s} H_{\beta\alpha})^*]]^{1/2}\\
&\le \sqrt{2} \mathbb{E}_{\mathrm{YM}_{M^r,a^r}}[1- \Re (\mathrm{tr}( H_s)) ]^{1/2}\\
&\le \frac{\sqrt{2} Z_{g, T-|a-a'|}}{Z_{g, T}} \mathbb{E}_{\mathrm{YM}_{\tilde M_V^r,a^r_V}}[1- \Re (\mathrm{tr}( H_s)) ]^{1/2}\\
&\le \frac{\sqrt{2} Z_{g, T/2}}{Z_{g, T}} \mathbb{E}_{\mathrm{YM}_{\tilde M_V^r,a_V^r}}[1- \Re (\mathrm{tr}( H_s)) ]^{1/2},
\end{align*}
where we used the same argument as in the first part of the proof for the penultimate inequality.
Now under $\mathrm{YM}_{\tilde M^r_V,a_V^r}$, $H_s$ has same law as a Brownian motion at time $|a-a'|.$ It follows from \cite{Lev5} or \cite{Dah} that for all group series considered,
there is a constant $c>0$ independent of $N$ with
\[\mathbb{E}_{\mathrm{YM}_{\tilde M_V^r,a_V^r}}[1- \Re (\mathrm{tr}( H_s))]\le 1-e^{-c|a-a' |}\le c|a-a'|,\]
leading to the required bound \eqref{eq: UNIF CV thm}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{__THM: Disc YMCont}] We only sketch here the additional argument and refer to \cite{Lev2} for more details. For any multiplicative function $h\in\mathcal{M}(\mathrm{P}(\Sigma), G_N),$ the composition of its restriction to $\mathrm{P}(U)$ with $\Psi$ defines an element $\Psi(h)$ of $\mathcal{M}(\mathrm{P}(D_R),G_N).$ According to the splitting property of Markovian holonomy fields (property (A3) of \cite[Def. 3.1.2]
{Lev2}) and Lemma
\ref{_____Lem:Density dYM}, $\Psi_*(\mathrm{YM}_{\Sigma,G_N})$ has density
\[
\frac{Z_{(g,1),T-\mathrm{vol}(U)}(H_{\partial D_R})}{Z_{g,T,G_N}}
\]
with respect to $\mathrm{YM}_{D_R,G_N}.$ The claim then follows with the same argument as for the proof of Theorem \ref{__THM:Disc dYM}.
\end{proof}
\subsection{Simple non-contractible loops}
Consider a simple non-contractible loop $\ell$ on a compact surface $\Sigma$ of genus $g\geq 1$. Two possibilities occur when cutting $\Sigma$ along $\ell.$ If the surface is cut into two surfaces with exactly one connected boundary component, each with genus larger than $1$\footnote{Indeed, if one of the boundary components has genus 0, it implies that $\ell$ is contractible.}, then the loop is \emph{separating}. Otherwise, the new surface has one connected component and two boundary components, and the loop is called \emph{non-separating}. Both cases are illustrated in Fig. \ref{fig:gluing} below. We refer to \cite{Sti} for details on these loops. It is now important to understand how the Yang--Mills measure interacts with surgery: we therefore begin with a few results that will help us in the next section.
\begin{figure}[!h]
\centering
\includegraphics[scale=0.5]{lacet-simple-2}
\caption{\small The surface on the left can be seen as the result of a binary gluing of two surfaces along a separating loop (top right), or of one surface along a nonseparating loop (bottom right).}\label{fig:gluing}
\end{figure}
\subsubsection{Separating loops}
Consider first two compact connected orientable surfaces $\Sigma_1$ and $\Sigma_2$ such that $\partial\Sigma_1$ and $\partial\Sigma_2$ are connected. Denote by $L_1$ and $L_2$ the corresponding loops, oriented positively. Let $\psi:L_1\to L_2$ be an orientation-reversing diffeomorphism, and $\Sigma$ be the gluing of $\Sigma_1$ and $\Sigma_2$ along $\psi$. Let $\ell$ be the corresponding simple loop in the surface $\Sigma$. For $i\in\{1,2\}$, denote by $\tilde{\mathcal{J}}_i$ the sigma-field generated by $(H_{\ell_1},\ldots,H_{\ell_n},n\geq 1,\ell_i\in\mathrm{L}(\Sigma_i))$. For any $i$, a function $f:\mathcal{M}(\mathrm{P}(\Sigma),G_N)\to\mathbb{C}$ is $\tilde{\mathcal{J}}_i$-measurable if and only if $f\circ\mathscr{R}_{\Sigma_i}:\mathcal{M}(\mathrm{P}(\Sigma_i),G_N)\to\mathbb{C}$ is $\mathcal{J}_i$-measurable. The next theorem is a particular case of \cite[Thm. 5.1.1]{Lev3}.
\begin{thm}\label{thm:split_separ}
The sigma-fields $\tilde{\mathcal{J}}_1$ and $\tilde{\mathcal{J}}_2$ are independent on $\mathcal{M}(\mathrm{P}(\Sigma),G_N)$ under $\mathrm{YM}_{\Sigma,G_N}$ conditionally on the random variable $[H_\ell]$. Moreover, for any $\tilde{\mathcal{J}}_i$-measurable function $f_i:\mathcal{M}(\mathrm{P}(\Sigma),G_N)\to\mathbb{C}$, for $i\in\{1,2\}$, the product $f_1f_2$ is measurable with respect to $\mathcal{J}$ and the following equality holds true for any $t\in G_N/\mathrm{Ad}$:
\begin{align}
\int f_1(h)f_2(h)&\mathrm{YM}_{\Sigma,C_{\ell\mapsto t},G_N}(dh)\nonumber\\ &=\int f_1\circ\mathscr{R}_{\Sigma_1}(h)\mathrm{YM}_{\Sigma_1,t,G_N}(dh)\int f_2\circ\mathscr{R}_{\Sigma_2}(h)\mathrm{YM}_{\Sigma_2,t^{-1},G_N}(dh).
\end{align}
\end{thm}
This theorem provides a sort of spacial Markov property, which is a consequence of the semigroup property of the heat kernel. It has the following application.
\begin{coro}
Let $\ell$ be a separating loop in an area-weighted map $(M,a)$ of genus $g\geq 2$, splitting $M$ into two respective maps $M_1$ and $M_2$ of genus $g_1,g_2$ and total area $T_1$ and $T_2$. Let $f:\mathcal{M}(\mathrm{P}(M),G_N)\to\mathbb{C}$ be a bounded $\tilde{\mathcal{J}}_1$-measurable function. We have
\begin{equation}
\int_{\mathcal{M}(\mathrm{P}(M),G_N)}f(h)\mathrm{YM}_{M,a,G_N}(dh)=\int_{\mathcal{M}(\mathrm{P}(M_1),G_N)} f\circ\mathscr{R}_{M_1}^M(h)I(h_\ell)\mathrm{YM}_{M_1,a,G_N}(dh),\label{-----eq:Disint one side}
\end{equation}
where for any $x\in G_N$,
\[
I(x)= \frac{Z_{(g_1,1),T_1,G_N}(x) Z_{(g_2,1),T_2,G_N}(x^{-1})}{Z_{g,T,G_N}}.
\]
\end{coro}
\begin{proof}
To lighten the notation, we will drop in this proof the subscripts $G_N$, as the structure group remains fixed. First of all, using \eqref{eq:decondition}, the left-hand-side of \eqref{-----eq:Disint one side} equals
\begin{align*}
\frac{1}{Z_{g,T}}\int_{G/\mathrm{Ad}}\int_{\mathcal{M}(\mathrm{P}(M),G_N)}f(h) Z_{g,T}(\ell;t)\mathrm{YM}_{M,C_{\ell\mapsto t},a}(dh)dt.
\end{align*}
Let us compute $Z_{g,T}(\ell;t)$. Using the invariance property of Yang--Mills measure by subdivision, one can assume without loss of generality that that for $i\in\{1,2\},$ $M_i$ has 1 vertex $v^{(i)}$, $2g_i+1$ edges $a_1^{(i)},b_1^{(i)},\ldots,a_g^{(i)},b_g^{(i)},e^{(i)}$ all with $v^{(i)}$ as source and target, and 1 face whose boundary is given by ${e^{(i)}}^{\varepsilon_i}[a_1^{(i)},b_1^{(i)}]\cdots[a_{g_i}^{(i)},b_{g_i}^{(i)}]$ where $\varepsilon_1=1$ and $\varepsilon_2=-1$. The map $M$ can then be defined as the quotient of $M_1\cup M_2$ by the relations $v^{(1)}\sim v^{(2)}$ and $(e^{(1)})^{-1}\sim e^{(2)}$. We obtain
\begin{align*}
Z_{g,T}(\ell;t)= & \int_{G_N^{2g+1}}p_{T_1}(x[y_1,z_1]\cdots[y_{g_1},z_{g_1}])\\
&\times p_{T_2}(x^{-1}[y'_1,z'_1]\cdots[y'_{g_2},z'_{g_2}])dx\prod_{\substack{1\leq i\leq g_1\\ 1\leq j\leq g_2}} dy_idz_idy'_jdz'_j\\
= & Z_{(g_1,1),T_1}(t)Z_{(g_2,1),T_2}(t^{-1}).
\end{align*}
From this result and from Theorem \ref{thm:split_separ} we deduce
\begin{align*}
& \int_{\mathcal{M}(\mathrm{P}(M),G_N)}f(h)\mathrm{YM}_{M,a}(dh)\\
& = \frac{1}{Z_{g,T}}\int_{G/\mathrm{Ad}}\Bigg[\int_{\mathcal{M}(\mathrm{P}(M_1),G_N)}f\circ\mathscr{R}_{M_1}^M(h)Z_{(g_1,1),T_1}(t)\mathrm{YM}_{M_1,t,a}(dh)\\
& \times \int_{\mathcal{M}(\mathrm{P}(M_2),G_N)}Z_{(g_2,1),T_2}(t^{-1})\mathrm{YM}_{M_2,t^{-1},a}(dh)\Bigg]dt.
\end{align*}
The integral over $\mathcal{M}(\mathrm{P}(M_2),G_N)$ is simply equal to $Z_{(g_2,1),T_2}(t^{-1})$; furthermore, under $\mathrm{YM}_{M_1,t,a}$, $[H_\ell]=t,$ so that we have
\begin{align*}
&Z_{g,T} \int_{\mathcal{M}(\mathrm{P}(M),G_N)}f(h)\mathrm{YM}_{M,a}(dh)\\
& = \int_{G/\mathrm{Ad}}\int_{\mathcal{M}(\mathrm{P}(M_1),G_N)}f\circ\mathscr{R}_{M_1}^M(h) Z_{(g_1,1),T_1}(h_\ell)Z_{(g_2,1),T_2}(h_\ell^{-1})\mathrm{YM}_{M_1,t,a}(dh)dt.
\end{align*}
Finally, using the disintegration formula \eqref{-----eq: Disint YM BD} leads to the expected equality.
\end{proof}
To prove Theorem \ref{__THM:simple non-contrac}, we shall need the following similar but simpler Lemma.
\begin{lem}\label{cor:Density_sep}
Let $\ell$ be a separating loop in an area-weighted map $(M,a)$ of genus $g\geq 2$, splitting $M$ into two respective maps $M_1$ and $M_2$ of genus $g_1,g_2$ and total area $T_1$ and $T_2$. Under $\mathrm{YM}_{M,a,G_N}$, the random variable $H_\ell$ has density
\begin{equation*}
\frac{Z_{(g_1,1),T_1,G_N} (h)Z_{(g_2,1),T_2,G_N} (h^{-1})}{Z_{g,T,G_N}}
\end{equation*}
with respect to the Haar measure on $G_N.$
\end{lem}
\begin{proof}
Let us use the same $M,M_1$ and $M_2$ as in the previous proof. For any central function $f:G_N\to\mathbb{C}$ measurable and bounded, we have
\begin{align*}
\mathbb{E}_{\mathrm{YM}_{M,a,G_N}}[f(H_\ell)]= &\frac{1}{Z_{g,T,G_N}}\int_{G_N^{2g+1}} f(x)p_{T_1}(x[y_1,z_1]\cdots[y_{g_1},z_{g_1}])\\
& \times p_{T_2}(x^{-1}[y'_1,z'_1]\cdots[y'_{g_2},z'_{g_2}])dx\prod_{\substack{1\leq i\leq g_1\\ 1\leq j\leq g_2}} dy_idz_idy'_jdz'_j,
\end{align*}
thus by integrating over all variables but $x$ we obtain the expected result.
\end{proof}
\begin{proof}[Proof of 1. of Theorem \ref{__THM:simple non-contrac}] Using Lemma \ref{cor:Density_sep} and the bound \eqref{------eq:Bound Dens gCap}, we find that under $\mathrm{YM}_{M,a}$, the density of $h_\ell$ with respect to the Haar measure is uniformly bounded by
$$\frac{Z_{g_1,T_1,G_N}Z_{g_2,T_2,G_N}}{Z_{g,T,G_N}}.$$
Now for a Haar distributed random variable $H$ on $G_N$ belonging to $X_N,$ according to \cite{DiaconisEvans}, whenever $n\not=0,$ $\mathrm{tr}(H^n)$ converges in probability towards $0$, as $N\to \infty.$ The claim now
follows by absolute continuity similarly to
the proof of Theorem \ref{__THM:Disc dYM}.
\end{proof}
\subsubsection{Non-separating loops}
\label{sec:Non-Sep Loop}
Let $\Sigma'$ be a compact connected orientable surface such that $\partial\Sigma'$ has two connected components; denote the corresponding loops, oriented positively, by $L_1$ and $L_2$. Let $\psi:L_1\to L_2$ be an
orientation-reversing diffeomorphism, and $\Sigma$ be the gluing of $\Sigma'$ along $\psi$. Let $\ell$ be the corresponding loop in $\Sigma$. We have the following, which is a particular case of \cite[Thm. 5.4.1]{Lev3}. We use the same notation $\tilde{\mathcal{J}}'$ for the sigma-field on $\mathcal{M}(\mathrm{P}(\Sigma),G)$ generated by the loops in $\Sigma'$.
\begin{thm}
Let $f:\mathcal{M}(\mathrm{P}(\Sigma),G)\to\mathbb{C}$ be a $\tilde{\mathcal{J}}'$-measurable function. For any $t\in G/\mathrm{Ad}$, the following equality holds true:
\begin{equation}
\int f(h)\mathrm{YM}_{\Sigma,C_{\ell\mapsto t},G}(dh) = \int f\circ\mathscr{R}_{\Sigma'}(h)\mathrm{YM}_{\Sigma',(t,t^{-1}),G}(dh).
\end{equation}
\end{thm}
The consequence in terms of absolute continuity is the following.
\begin{coro}?
Let $\ell$ be a non-separating loop in an area-weighted map $(M,a)$ of genus $g\geq 1$. Under $\mathrm{YM}_{M,a,G}$, the random variable $H_\ell$ has density given by
\begin{equation*}
\frac{\varphi_{g,T}(h)}{Z_{g,|a|,N}},\forall h\in G
\end{equation*}
with respect to the Haar measure, and for all $g\ge 1, T>0$,
\begin{equation}
\varphi_{g,T}(h)=\sum_{\lambda\in \widehat{G}}d_\lambda^{2-2g} |\chi_\lambda(h)|^2e^{-\frac{T}{2}c_\lambda},\forall h\in G. \label{------eq:DensYM NonS Loop}
\end{equation}
\end{coro}
\begin{rmk} The above density can also be written as $Z_{(g,2),|a|}(h,h^{-1}).$
\end{rmk}
\begin{proof} Choosing $M$ with $1$ vertex, $1$ face and $2g$ edges, one of which is $\ell$, and writing $u=h_\ell,$
\begin{equation}
\mathrm{YM}_{M,a}(dh)=p_{T}([u,v][x_2,y_2]\cdots [x_g,y_g]) dudv \prod_{i=2}^{g}dx_idy_i.\label{------eq:DensYM NonS Loop}
\end{equation}
It follows that the law of $h_\ell$ under $\mathrm{YM}_{M,a}$ has density
$$\varphi_{g,T}(u)=Z_{g,T,N}^{-1}\int_{G^{2g-1}} p_{T}([u,v][x_2,y_2]\cdots [x_g,y_g]) dv \prod_{i=2}^{g}dx_idy_i,\forall u\in G, $$
with respect to the Haar measure. Expanding in characters and using \eqref{------eq:int_commu} yields
$$\varphi_{g,T}(h)=\sum_{\lambda\in \hat G}e^{-\frac{T}{2}c_\lambda} I_\lambda(u) $$
with $$I_\lambda(u)= d_\lambda^{1-2(g-1)} \int_{G} \chi_\lambda([u,v])dv. $$
Now \eqref{------eq:int_conj} yields $I_\lambda(u)= d_\lambda^{2-2g}\chi_\lambda(u)\chi_\lambda(u^{-1}), \forall u\in G$ and the claim.
\end{proof}
\begin{proof}[Proof of 2. of Theorem \ref{__THM:simple non-contrac} for $g\ge 2$] Then, using that $\|\chi_\lambda \|_\infty=\chi_\lambda(1)$ for all $\lambda\in\widehat{G},$
$$\|\varphi_{g,T}\|_\infty= \varphi_{g,T}(1)=Z_{g-1,T,G_N},\forall g\ge 1,T>0.$$
It follows that under $\mathrm{YM}_{M,a}$, the density of $h_\ell$ with respect to the Haar measure is bounded by
$$\frac{Z_{g-1,|a|,G_N}}{Z_{g,|a|,G_N}}.$$
When $g\ge 2,$ it follows from \ref{__THM:PF} that this sequence is uniformly bounded when $N\to \infty$ and the argument is then identical to the proof of point 1.
\end{proof}
When $g=1,$ the above argument of absolute continuity argument fails. Indeed for a total area $T,$ the maximum of the density is then given by $\frac{Z_{0,T,N}}{Z_{1,T,N}}.$ For $\tilde A_N,$ thanks to Theorem \ref{__THM: DK
PT},
and more precisely formula (28) of \cite{LM},
$$\lim_{N\to\infty}\frac{1}{N^2}\log (Z_{0,T,N})= F(T)=\frac{T}{24}+\frac 3 4-\frac 1 2 \log(T),\forall 0<T\le \pi^2.$$
Since $F(T)\ge F(\pi^2)>0$ for all $T\in(0,\pi^2],$ $\lim_{N\to \infty} Z_{0,T,N}= +\infty.$
Similarly to the strategy of \cite{Lev,DN,Hal2}, we shall consider the expectation and the variance of the random variable $W_\ell.$
\begin{rmk} For any $z$ in the center of $G, $ under $\mathrm{YM}_{M,a},$ $z h_\ell$ has same law as $h_\ell.$ This can be checked changing variable in formula \eqref{------eq:DensYM NonS Loop}. Alternatively, Schur's lemma implies that
$z$
acts by
a unitary scalar in any irreducible representation and \eqref{------eq:DensYM NonS Loop} implies that for any $\varphi_{g,T}(zh)=\varphi_{g,T}(h)$ for any $h\in G.$ Consequently whenever $D_z\in G$ is a diagonal matrix of multiplication
by a
scalar $z\in\mathbb{C},$ under $\mathrm{YM}_{M,a},$ $\mathrm{tr}(h_{\ell^n})=\mathrm{tr}(h_\ell^n)$ has same law as $z^n \mathrm{tr}(h_{\ell^n}).$ For $X_N=\{A_N,\tilde A_N\},$ the center is given by $\{\zeta \mathrm{Id}_{N+1} : \zeta^{N+1}=1\}$ and $\{z \mathrm{Id}_N:
z\in\mathbb{C}, |z|=1\}.$ It follows that for $X=\tilde A,$ $\mathbb{E}[W_{\ell^n}]=0$ for any $n\not=0$, whereas for $X=A,$ $\mathbb{E}[W_{\ell^n} ]= 0$ for any $n\not=0$ modulo $N+1$. In particular, when $X\in\{A,\tilde A\},$ $\lim_{N\to\infty}
\mathbb{E}[W_{\ell^n}]=0.$
\end{rmk}
Unfortunately, the argument given in the above remark does not apply when $X=\{B,C,D\}$. Also, it does not give any information about $\mathrm{Var}(W_\ell)=\mathbb{E}[|W_\ell|^2]-|\mathbb{E}[W_\ell]|^2.$ Instead, we shall use the expansion in
characters of the heat kernel
and the following lemma, which is a consequence of the expression of characters as a ratio of alternated functions.
\begin{lem}\label{_____Lem:Pieri Gen} Let $G_N\subset\mathrm{GL}_n(\mathbb{C})$ be a classical group of size $N$ and type $X_r$, for $X\in\{B,C,D\}$. For any $k\neq0$ and $\lambda \in \widehat{G}_N,$
\begin{equation}\label{eq:Pieri_BCD}
\mathrm{Tr}(g^k)\chi_\lambda(g)=\sum_{\mu\in\widehat{G}_N} c_{\lambda,k}^\mu \chi_{\mu}(g),
\end{equation}
with
\begin{equation}\label{------eq:Loop jumps}
c_{\lambda,k}^\mu \in \{-1,0,1\},\forall \mu\in\widehat{G} _N
\end{equation}
and
\begin{equation}
\sum_{\mu\in\widehat{G}_N} |c_{\lambda,k}^\mu|\le n\label{------eq:Bound number possible jumps}
\end{equation}
\end{lem}
\begin{proof}
Let us introduce a few common notations for the different cases. Set $\mathbb{Z}_{\mathrm{sym}}=\mathbb{Z}$ if $r$ is odd or $\mathbb{Z}+\tfrac12$ if $r$ is even. The mapping
\[
\lambda\in\widehat{G}_N\mapsto\lambda+\rho
\]
establishes a bijection between the set of highest weights and $\{\mu\in\mathbb{Z}_\mathrm{sym}^r:\mu_1>\cdots>\mu_r\}$. The symmetric group $\mathfrak{S}_r$ acts in an obvious way on
\[
\Delta_r=\{\mu\in\mathbb{Z}_\mathrm{sym}^r:\mu_1>\cdots>\mu_r\}\cong\{\mu\in\mathbb{Z}_\mathrm{sym}^r:\mu_i\neq\mu_j,\ \forall i\neq j\}/\mathfrak{S}_r,
\]
where the symmetric group $\mathfrak{S}_r$ acts on $\mu$ by permutation of its components. For $\mu$ with $\mu_i\not=\mu_j,\forall i\not=j,$ we denote by $[\mu]$
its decreasing rearrangement and $\sigma_\mu\in \mathfrak{S}_r$ the unique permutation such that $[\mu]_i=\mu_{\sigma(i)}$ for all $i$. We will prove each case separately using the previous notations.
The explicit formulae for the characters $\chi_\lambda$
that we will use can be found in \cite{BtD} or in \cite[Sect. 2.3]{Mel}. When $\mu\in \mathbb{Z}_{\mathrm{sym}}^r$ with $\mu_i=\mu_j$ for some $i\not=j$, by convention $\chi_{[\mu]-\rho}=0$ and $\varepsilon(\sigma_{\mu})=0.$ The proof for all cases will rely on the following
computation: for any $(z_1,\ldots,z_r)\in(\mathbb{C}^*)^r,$ any $(m_1,\ldots,m_r)\in\mathbb{Z}^r$ and any $k\in\mathbb{N}^*$, we have
\begin{equation}\label{eq:det}
\sum_{\ell=1}^r(z_\ell^k+z_\ell^{-k})\det(z_i^{m_j}-z_i^{-m_j})=\sum_{\ell=1}^r(\det(z_i^{m_j+k\delta_{j\ell}})+\det(z_i^{-m_j-k\delta_{j\ell}})).
\end{equation}
Let us prove \eqref{eq:det}. Using the Leibniz formula for determinant and the invariance of $\sum_\ell(z_\ell^k+z_\ell^{-k})$ by the action $\ell\mapsto \sigma(\ell)$ of the symmetric group, we have
\begin{align*}
\sum_{\ell=1}^r(z_\ell^k+z_\ell^{-k})\det(z_i^{m_j}-z_i^{-m_j}) = & \sum_{\sigma\in\mathfrak{S}_r}\varepsilon(\sigma)\sum_{\ell=1}^r(z_\ell^k+z_\ell^{-k})\prod_{i=1}^r (z_{\sigma(i)}^{m_i}-z_{\sigma(i)}^{-m_i})\\
= & \sum_{\sigma\in\mathfrak{S}_r}\varepsilon(\sigma)\sum_{\ell=1}^r(z_{\sigma(\ell)}^k+z_{\sigma(\ell)}^{-k})\prod_{i=1}^r (z_{\sigma(i)}^{m_i}-z_{\sigma(i)}^{-m_i}).
\end{align*}
We can then put the sum over $\mathfrak{S}_r$ back in by linearity, and for any $\sigma$ and any $\ell$ we have
\[
(z_{\sigma(\ell)}^k+z_{\sigma(\ell)}^{-k})\prod_{i=1}^r(z_{\sigma(i)}^{m_i}-z_{\sigma(i)}^{-m_i}) = (z_{\sigma(\ell)}^{m_\ell+k}-z_{\sigma(\ell)}^{-m_\ell-k}+z_{\sigma(\ell)}^{m_\ell-k}-z_{\sigma(\ell)}^{-m_\ell+k})\prod_{i\neq \ell}(z_{\sigma(i)}^{m_i}-z_{\sigma(i)}^{-m_i}).
\]
Summing over $\ell$ leads to \eqref{eq:det}. We can now prove \eqref{eq:Pieri_BCD} for each type of group.
Case $B_r$: let $g\in\mathrm{SO}(2r+1)$ be an element with eigenvalues $(z_1^{\pm 1},\ldots,z_r^{\pm 1},1)$. We have
\[
\chi_\lambda(g)=\frac{\det(z_i^{\lambda_j+\rho_j}-z_i^{-\lambda_j-\rho_j})}{\det(z_i^{\rho_j}-z_i^{-\rho_j})}.
\]
If we substitute $\mu=\lambda+\rho\in\Delta_r,$ we can write
\[
\chi_\lambda(g)=\chi_{\mu-\rho}(g)=\frac{\det(z_i^{\mu_j}-z_i^{-\mu_j})}{\det(z_i^{\rho_j}-z_i^{-\rho_j})},
\]
so that
\[
\mathrm{Tr}(g^k)\chi_\lambda(g)= \left(1+\sum_{\ell=1}^r (z_\ell^k+z_\ell^{-k})\right)\frac{\det(z_i^{\mu_j}-z_i^{-\mu_j})}{\det(z_i^{\rho_j}-z_i^{-\rho_j})}
\]
and by \eqref{eq:det} we obtain that
\begin{equation}
\mathrm{Tr}(g^k)\chi_\lambda(g)= \chi_\lambda(g) + \sum_{\ell=1}^r \frac{\det(z_i^{\mu_j+k\delta_{j\ell}}-z_i^{-(\mu_j+k\delta_{j\ell})})+\det(z_i^{\mu_j-k\delta_{j\ell}}-z_i^{-(\mu_j-k\delta_{j\ell})})}{\det(z_i^{\rho_j}-z_i^{-\rho_j})}\label{-----eq: PieriB}
\end{equation}
Set $\mu_{k,\ell}^+=(\mu_1,\ldots,\mu_\ell+k,\ldots,\mu_r)$ and $\mu_{k,\ell}^-=(\mu_1,\ldots,\mu_\ell-k,\ldots,\mu_r)$. We have
\begin{align*}
\mathrm{Tr}(g^k)\chi_\lambda(g) = & \chi_\lambda(g) + \sum_{\ell=1}^r \left(\varepsilon(\sigma_{\mu_{k,\ell}^+})\chi_{[\mu_{k,\ell}^+]-\rho}(g)+\varepsilon(\sigma_{\mu_{k,\ell}^-})\chi_{[\mu_{k,\ell}^-]-\rho}(g)\right),
\end{align*}
which yields \eqref{eq:Pieri_BCD} with $c_{\lambda,k}^\lambda=1$, $\ c_{\lambda,k}^{[\mu_{k,\ell}^+]-\rho}=\varepsilon(\sigma_{[\mu_{k,\ell}^+]})$ for all $1\leq \ell\leq r$ and $c_{\lambda,k}^\mu=0$ for any other $\mu\in\widehat{G}_N$, which also proves \eqref{------eq:Loop jumps}. Indeed, each $[\mu_{k,\ell}^\pm]-\rho$ corresponds to an element of $\widehat{G}_N$ if and only if $[\mu_{k,\ell}^\pm]\in\Delta_r$, and if it is not the case, then $\chi_{[\mu_{k,\ell}^\pm]}$ and summand in the right-hand-side of \eqref{-----eq: PieriB} vanish. Finally, there are $2r+1$ nonzero coefficients, whose absolute value is always equal to 1, hence \eqref{------eq:Bound number possible jumps} is also satisfied.
Case $C_r$: let $g\in\mathrm{Sp}(r)$ be an element with eigenvalues $(z_1^{\pm 1},\ldots,z_r^{\pm 1})$. We have again
\[
\chi_\lambda(g)=\chi_{\mu-\rho}(g)=\frac{\det(z_i^{\mu_j}-z_i^{-\mu_j})}{\det(z_i^{\rho_j}-z_i^{-\rho_j})}.
\]
It follows that
\[
\mathrm{Tr}(g^k)\chi_\lambda(g) = \sum_{\ell=1}^r \left(\varepsilon(\sigma_{\mu_{k,\ell}^+})\chi_{[\mu_{k,\ell}^+]-\rho}(g)+\varepsilon(\sigma_{\mu_{k,\ell}^-})\chi_{[\mu_{k,\ell}^-]-\rho}(g)\right),
\]
as expected.
Case $D_r$: let $g\in\mathrm{SO}(2r)$ be an element with eigenvalues $(z_1^{\pm 1},\ldots,z_r^{\pm 1})$. We have
\[
\chi_\lambda(g)=\chi_{\mu-\rho}(g)=\frac{\det(z_i^{\mu_j}-z_i^{-\mu_j})+\det(z_i^{\mu_j}+z_i^{-\mu_j})}{\det(z_i^{\rho_j}+z_i^{-\rho_j})},
\]
therefore
\begin{align*}
\mathrm{Tr}(g^k)\chi_\lambda(g)= & \left(\sum_{\ell=1}^r (z_\ell^k+z_\ell^{-k})\right)\frac{\det(z_i^{\mu_j}-z_i^{-\mu_j})+\det(z_i^{\mu_j}+z_i^{-\mu_j})}{\det(z_i^{\rho_j}+z_i^{-\rho_j})}\\
= & \sum_{\ell=1}^r \left(\varepsilon(\sigma_{\mu_{k,\ell}^+})\chi_{[\mu_{k,\ell}^+]-\rho}(g)+\varepsilon(\sigma_{\mu_{k,\ell}^-})\chi_{[\mu_{k,\ell}^-]-\rho}(g)\right).
\end{align*}
\end{proof}
\begin{proof}[Proof of 2. of Theorem \ref{__THM:simple non-contrac} for $g=1$] Let us set $T=|a|.$ We shall compute the expectation and the variance of $W_\ell$ under $\mathrm{YM}_{M,a,G_N},$ and show that both converge to zero as $N\to\infty.$ Assume that $G_N$ is of type $X_r$ and a subgroup of $\mathrm{GL}_n(\mathbb{C})$ as\footnote{so that $n$ is respectively $r,r+1,2r+1$ for types $\tilde A_r,A_r,B_r$ and $2r$ for types $\{C_r,D_r\}.$ } in section \ref{sec:Intro Class Gp}.
Using Corollary \ref{cor:Density_sep} and expanding in characters yield
\begin{align*}
n\mathbb{E}[W_{\ell^k}]&= Z_{1,T,X_r}^{-1}\int_{G_N} \varphi_{T,1}(h) \mathrm{Tr}(h^k)dh\\
&= Z_{1,T,X_r}^{-1}\sum_{\lambda \in\widehat{G}_N} e^{-\frac{T}{2}c_\lambda} I_\lambda,
\end{align*}
where for all $\lambda\in \widehat{G}_N,$ using the orthogonality of characters,
$$I_\lambda=\int_{G_N} \mathrm{Tr}(h^k) \chi_\lambda(h)\chi_{\lambda}(h^{-1})dh=\sum_{\mu\in \widehat{G}_N} c_{\lambda,k}^\mu \int_{G_N} \chi_\mu(h)\chi_\lambda(h^{-1})dh=c_{\lambda,k}^\lambda. $$
Thanks to \eqref{------eq:Loop jumps}, it follows that
\begin{equation}
|\mathbb{E}[W_{\ell^n}]|\le (n Z_{1,T,X_r})^{-1} \sum_{\lambda\in\widehat{G}_N} e^{-\frac T 2 c_\lambda}=\frac{1}{n}. \label{------eq:Bound E}
\end{equation}
Similarly,
\begin{align*}
n^2\mathbb{E}(|W_{\ell^k}|^2)&=Z_{1,T,X_r}^{-1}\int_{G_N} |\mathrm{Tr}(h^k)|^2 \varphi_{T,1}(h)dh= Z_{1,T,X_r}^{-1}\sum_{\lambda\in\widehat{G}_N} e^{-\frac T 2 c_\lambda} J_\lambda,
\end{align*}
where
\begin{align*}
J_\lambda&=\int_{G_N} \mathrm{Tr}(h^k)\chi_\lambda(h)\mathrm{Tr}(h^{-k})\chi_\lambda(h^{-1})dh\\
&=\sum_{\mu,\nu} c_{\lambda,k}^\mu c_{\lambda,k}^\nu \int_{G} \chi_\mu(h)\chi_\nu(h^{-1})dh= \sum_{\mu\in\widehat{G}_N} (c_{\lambda,k}^\mu)^2.
\end{align*}
Using Lemma \ref{_____Lem:Pieri Gen}, we conclude that
$$0\le J_\lambda=\sum_{\mu\in\widehat{G}_N} |c_{\lambda,k}^\mu|\le n$$
and
\begin{equation}
\mathbb{E}[|W_{\ell^k}|^2]\le n^{-2}Z_{1,T,X_r}^{-1}\sum_{\lambda\in\widehat{G}_N} e^{-\frac T 2 c_\lambda} J_\lambda= \frac{1}{n}.\label{------eq:Bound V}
\end{equation}
From \eqref{------eq:Bound E} and \eqref{------eq:Bound V}, we conclude that the expectation and variance of $W_\ell$ under $\mathrm{YM}_{M,a,G_N}$ both vanish as $N\to\infty.$ Therefore, $W_{\ell^k}$ converges to $0$ in probability.
\end{proof}
\begin{rmk} For the $A_r$ type, for all $k\neq 0$ and $\lambda\in \Lambda_r,$ $c_{\lambda,k}^\lambda=0,$ and the first part of the above proof yields another argument for $\mathbb{E}[W_{\ell^k}]=0$ for all $r\ge 1.$
\end{rmk}
\section*{Acknowledgments} Both authors are grateful to Thierry L\'evy for several discussions which inspired the main content of this article and to B. Hall for his feedback on a first draft of this work. Many thanks are also due to Neil O'Connell, as this joint work started during a visit of
the
second author to the first one at UCD and was founded by the ERC Advanced Grant 669306. A.D. acknowledges partial support from "The Dr Perry James (Jim) Browne Research Centre on Mathematics and its Applications"
individual grant.
|
2,877,628,090,152 | arxiv | \section{Introduction}
\label{S:1}
Symmetry principles play a fundamental role in quantum field theory, in particular as they tightly constrain the space of possible interactions, and thus they are essential in the determination of universality classes. When dealing with a large number of degrees of freedom the role of symmetries becomes almost indispensable.
Consider as an example the case of $\mathcal{N}$ complex scalar fields $\phi_i$, $i=1,\ldots,\mathcal{N}$: in the absence of a symmetry relating the different fields, the number of possible quartic interactions is of order $\mathcal{N}^4$; if instead we demand invariance under $U(\mathcal{N})$ transformations we remain with only one interaction, namely $(\sum_i \bar{\phi}_i\phi_i)^2$.
The reduction is so drastic that it allows one to actually solve the theory in the limit $\mathcal{N}\to \infty$, where only a particularly simple class of Feynman diagrams survive \cite{Wilson:1972cf,Coleman:1974jh}.
Things get in general more complicated if we take a smaller symmetry group. An important case is obtained if we write $\mathcal{N}= N^2$ and we explicitly break the symmetry group from $U(N^2)$ to $U(N)^2$; then it is natural to rearrange the fields as a complex $N\times N$ matrix $\phi_{ab}$, transforming in the fundamental of $U(N)^2$. The large-$N$ limit of matrix models has been extensively explored for its connection to two-dimensional quantum gravity \cite{DiFrancesco:1993cyw} and for its role in AdS/CFT \cite{Aharony:1999ti}, but in general it does not lead to solvable models above two dimensions: the class of dominant graphs (the planar graphs) is still too big and it does not lead for example to a closed equation for the two point function.
One would then imagine that things get even tougher if we shrink further the ratio between the dimension of the symmetry group and the number of fields, but it turns out not to be the case. Writing $\mathcal{N}= N^3$ and breaking the symmetry group from $U(N^3)$ to $U(N)^3$ leads us into the realm of tensor models: the fields are now arranged as a tensor $\phi_{abc}$ transforming in the fundamental of $U(N)^3$, and the large-$N$ limit of such models has been shown to lead to a restricted subset of the planar diagrams, the so-called melonic graphs \cite{Bonzom:2011zz,Bonzom:2012hw}.\footnote{Most commonly matrix models are presented as describing fields transforming in the adjoint representation, e.g.\ of $U(N)$. Such point of view is natural when introducing them as gauge fields, and wishing to discuss the different behavior of fields in the fundamental representation (flavor fields) and in the adjoint (connection fields) of the same group. Here we discuss them instead as fields in the fundamental representation of a smaller group, because we want to highlight the role of the smaller symmetry group for a given set of fields. Furthermore, the existence of a melonic large-$N$ limit for tensors transforming in an irreducible representation of $U(N)$ (or $O(N)$) has been shown only very recently \cite{Klebanov:2017nlk,Benedetti:2017qxl,Carrozza:2018ewt}, and such models are less understood.}
The melonic dominance typically leads to closed Schwinger-Dyson equations, and thus such models are more manageable than matrix models.\footnote{Another way to obtain similar equations is to consider multi-matrix models at large $N$ and in the limit of large number of matrices (see \cite{Azeyanagi:2017mre} and references therein).}
Given such discovery, and knowing that vector and matrix models have a great wealth of applications, it is not unreasonable to imagine that tensor models might hold the key to a panoply of new theoretical insights.
While the zero dimensional models have been explored for some years in connection to their quantum gravity interpretation \cite{GurauRyan-review,Gurau-book,Bonzom:2012wa,Bonzom:2014oua,Carrozza:2015adg,Tanasa:2015uhr,Bonzom:2016dwy}, which was also the reason behind their original introduction \cite{Ambjorn:1990ge,Sasakura:1990fs}, higher dimensional tensor models are a very recent entry in the theoretical landscape.
A great boost came with the observation \cite{Witten:2016iux} that in one dimension they provide an alternative to the Sachdev-Ye-Kitaev model \cite{Sachdev:1992fk, Kitaev2015, Maldacena:2016hyu, Polchinski:2016xgd} dispensing with the quenched disorder of the latter. As a consequence, quantum mechanical tensor models have been the subject of several studies, see for example \cite{Klebanov:2016xxf,Peng:2016mxj,Krishnan:2016bvg,Krishnan:2017lra,Bulycheva:2017ilt,Choudhury:2017tax,Halmagyi:2017leq,Klebanov:2018nfp,Chang:2018sve,Carrozza:2018psc} (see also \cite{Delporte:2018iyf,Rosenhaus:2018dtp,Klebanov:2018fzb} for reviews). Tensor models in two or more dimensions remain so far the least explored, one reason being that when seeking direct generalizations of the SYK-type Schwinger-Dyson equations (i.e.\ in models with the so-called tetrahedron interaction) to higher dimensions one finds no non-trivial conformal theories in the critical dimension \cite{Benedetti:2017fmp} or conformal theories with a complex spectrum, except in small non-integer ranges of the spacetime dimension \cite{Giombi:2017dtl,Prakash:2017hwq,Giombi:2018qgp}.
In this paper we wish to push further the exploration of new tensorial conformal field theories in higher dimensions, with slightly different motivations.
First, in the $U(N)^3$-symmetric version of the Gross-Neveu model studied in \cite{Benedetti:2017fmp} (which does not allow for a tetrahedron interaction), it was found that one of the $U(N)$ subgroups can be spontaneously broken, which is a genuinely new feature with respect to the usual vector case. However, that model being in two dimensions, such breaking was identified as a large-$N$ artefact (as for continuous chiral symmetry breaking in the usual vector case \cite{Witten:1978qu}), in agreement with the Coleman-Mermin-Wagner theorem \cite{Mermin:1966fe,Coleman:1973ci}.
We would like to check here whether such new broken phase survives in three dimensions, where the Coleman-Mermin-Wagner theorem does not apply.
Second, the vectorial Gross-Neveu model is known to have a non-trivial UV fixed point in three dimensions \cite{Rosenstein-PRL,deCalan:1991km,ROSENSTEINreview}, hence it is an intriguing question whether it remains the only non-trivial fixed point upon a tensorial extension, or new ones are to be found.
And we believe it would be interesting to identify new conformal field theories of tensor type, even if these are not a direct generalization of the SYK type of conformal theory. In particular, given the point of view we sketched above, with the tensor models as symmetry-breaking perturbations of the vector models, it is natural to wonder whether such new critical theories would correspond to deformations of the Klebanov-Polyakov duality \cite{Klebanov:2002ja}.
The latter is a duality between the singlet sector of the either the free or Wilson-Fisher fixed points of the vector $O(N)$ model in three dimensions and Vasiliev's (type A) higher spin theory in $AdS_4$ with two different boundary conditions. It has been generalized to the case of fermions \cite{Sezgin:2003pt}, thus relating the free or critical Gross-Neveu model in three dimensions to type B higher spin theory in $AdS_4$ with the corresponding boundary conditions (see \cite{Giombi:2016ejx} for a review). Recently, Vasiliev proposed a new type of higher spin theory that could be dual to tensor theories \cite{Vasiliev:2018zer}, therefore the question of whether interacting fixed points exist for tensor models is of interest.
With such motivations in mind, we present here two modifications of the three-dimensional Gross-Neveu model having $U(N)\times U(N^2)$ and $U(N)^3$ symmetry, respectively. The first is in fact rather a rectangular matrix model, but it has many features in common with the second, properly tensorial, model. Studying the models at large $N$ by means of Schwinger-Dyson equations, and computing the effective potential for the intermediate fields, we confirm the presence of a phase with spontaneously broken $U(N)$ subgroup, and we find two new interacting fixed points in each model.
\section{A brief reminder of the vectorial Gross-Neveu model}
\label{sec:vectorGN}
The Gross-Neveu model \cite{Gross:1974jv} has been extensively studied, in particular in two dimensions, where it provides a model of asymptotic freedom and dynamical mass generation, which is also integrable. Here, we are rather interested in its three-dimensional version, which despite being perturbatively non-renormalizable, is renormalizable in the $1/N$ expansion \cite{Parisi:1975im} and admits an ultraviolet fixed point at large $N$ \cite{Rosenstein-PRL,deCalan:1991km} which renders the model meaningful at arbitrarily high energies (see \cite{ROSENSTEINreview} for a review). The nontrivial fixed point theory has been conjectured to be dual to a particular version of higher spin theory in $AdS_4$ \cite{Sezgin:2003pt}, a conjecture which has passed several tests (see \cite{Giombi:2016ejx} and references therein).
In view of the upcoming generalizations, we define here the model for the case of $N^3$ Dirac fermions in Euclidean signature (see Appendix \ref{sec:appendix gamma} for conventions on $\g$ matrices). The action is\footnote{Here and in the rest of the paper repeated indices imply a summation.}
\begin{equation}
S_{\rm GN}[\psi,\bar{\psi}] = \int d^3 x \; \left( \bar{\psi}_i \slashed{\partial} \psi_i -\frac{\l}{N^3} (\bar{\psi}_i \psi_i)^2 \right) \;.
\end{equation}
Expressing the four-fermion interaction in terms of an intermediate field $\sigma$, the action writes
\begin{equation} \label{eq:GNY}
S[\psi,\bar{\psi},\sigma]=\int d^3 x \;\left(\bar{\psi}_i\slashed{\partial}\psi_i + \sigma\bar{\psi}_i\psi_i + \frac{N^3}{4\l}\sigma^2\right).
\end{equation}
Besides the $U(N^3)$ invariance (with the fermions transforming in the fundamental representation), the model has also a discrete chiral symmetry, which acts as
\begin{equation}
\psi\rightarrow \gamma^5\psi \quad \bar{\psi} \rightarrow -\bar{\psi}\gamma^5 \quad \sigma \rightarrow - \sigma.
\end{equation}
In the large-$N$ limit one can write a closed Schwinger-Dyson equation for the fermion 2-point function, which reduces to a gap equation for the fermion mass $m=\expval{\sigma}$:
\begin{equation} \label{eq:GN-gap}
\frac{m}{\l} = 8 m \int_\Lambda \frac{\dd[3]p}{(2\pi)^3}\frac{1}{p^2+m^2}\;,
\end{equation}
where the divergent integral is regulated by a UV cutoff $\L$. The integral on the right-hand side of the gap equation is a monotonically decreasing function of $m$, hence it has a maximum at $m=0$, which defines a critical coupling
\begin{equation}
\frac{1}{\l_c} \equiv 8\int_\Lambda \frac{\dd[3]p}{(2\pi)^3}\frac{1}{p^2} \;,
\end{equation}
above which the gap equation \eqref{eq:GN-gap} admits a real solution $m\neq 0$, besides the trivial one.
Using the intermediate field formulation \eqref{eq:GNY}, and integrating out the fermions, one finds that for $\l>\l_c$ the stable solution of the effective potential is the non-zero solution. Therefore, the theory has a dynamically generated mass for $\l>\l_c$, and this in turn means that the chiral symmetry is spontaneously broken.
Using the gap equation for $\l>\l_c$, the effective potential writes
\begin{equation} \label{eq:V_GN}
V_{\rm eff}(\sigma) = \frac{1}{\pi}\left(\frac{1}{3}\abs{\sigma}^3 - \frac{m}{2}\sigma^2\right) \;,
\end{equation}
with an evident minimum at $\s=m$.
For $0\leq \l\leq \l_c$ the symmetry is instead preserved, as $m=0$ is stable. The phase transition at $\l=\l_c$ is second order.
The $\beta$-function of the adimensional coupling $\tilde{\lambda} \equiv \L \l$ is obtained from eq.~\eqref{eq:GN-gap} derivating both sides with respect to $\L$, leading to
\begin{equation}
\b = \L\partial_\L \tilde{\lambda} = \tilde{\lambda} - \frac{4}{\pi^2}\tilde{\lambda}^2.
\end{equation}
\section{$U(N)\times U(N^2)$-symmetric model}
\label{sec:S2}
The $U(N)\times U(N^2)$-symmetric model is obtained by first rearranging the label $i=1,\ldots,N^3$ as a set of two labels $a$ and $A$, so that we rewrite $\psi_i \to \psi_{aA}$, with $a=1,\ldots,N$ and $A=1,\ldots,N^2$, and $\psi_{aA}$ transforming in the fundamental of the product group.\footnote{There is a slight redundancy in denoting the symmetry group as $U(N)\times U(N^2)$: its action on $\psi_{aA}$ is not faithful, because the action of the two $U(1)$ subgroups of $U(N)$ and $U(N^2)$ are indistinguishable. Therefore, a faithfully acting symmetry group of the theory would be $U(1)\times (SU(N)/\mathbb{Z}_N\times SU(N^2)/\mathbb{Z}_{N^2})$, where we have quotiented also by the residual centers of the special unitary groups. A similar caveat applies of course also to the symmetry group of section \ref{S:2}. In the rest of the paper, for compactness of notation we will stick to the non-faithful denotation of the symmetry group.}
In order to explicitly break the symmetry from $U(N^3)$ to $U(N)\times U(N^2)$, while preserving the discrete chiral invariance, we add the following interaction to the GN model:
\begin{equation}
\frac{\l_p}{N^2} \bar{\psi}_{a A}\psi_{a^\prime A}\bar{\psi}_{a^\prime A'}\psi_{a A'} \;.
\end{equation}
In view of the next generalization, we will actually replace also the index $A$ by a pair of indices, each taking values from 1 to $N$, i.e.\ we write $\psi_i \to \psi_{abc}$. The total action then reads
\begin{equation} \label{eq:action}
S[\psi,\bar{\psi}] = S_{\rm free}[\psi,\bar{\psi}] + S_{\rm int}[\psi,\bar{\psi}] \;,
\end{equation}
with
\begin{equation}
S_{\rm free}[\psi,\bar{\psi}]= \int d^3 x \; \bar{\psi}_{abc}\slashed{\partial}\psi_{abc} \;,
\end{equation}
\begin{equation} \label{eq:S_int1}
S_{\rm int}[\psi,\bar{\psi}]= -\frac{\l}{N^3} \int d^3 x \; (\bar{\psi}_{abc}\psi_{abc})^2 - \frac{\l_p}{N^2} \int d^3 x \; \bar{\psi}_{abc}\psi_{a^\prime bc}\bar{\psi}_{a^\prime b^\prime c^\prime}\psi_{ab^\prime c^\prime} \;.
\end{equation}
Having written the rectangular matrix as a cubic tensor, we can depict the interactions as in Fig.~\ref{fig:interactions}, where each vertex represents a tensor, and the solid lines with label $n=1,2,3$ represent the contraction of two indices in the $n$-th position. The dotted lines represent instead the spin contraction (as in \cite{Benedetti:2017fmp}, we could consider also other interactions in which such contraction is mediated by a $\g_5$ or $\g_\m$ matrix). The solid-line graph on the right of Fig.~\ref{fig:interactions} is commonly called the \emph{pillow} graph, hence the subscript $p$ for its coupling $\l_p$.
Notice that it comes with a different power of $N$ in \eqref{eq:S_int1}, as required for a non-trivial large-$N$ limit (see for example \cite{Bonzom:2016dwy}).
\begin{figure}
\centering
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{GN-interaction.pdf}
\end{minipage}
\hspace{0.01\textwidth}
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{pillow-interaction.pdf}
\end{minipage}
\caption{\label{fig:interactions}Graphical representation of the interaction vertices (GN and pillow).}
\end{figure}
\subsection{\texorpdfstring{$\beta$}{b}-functions and flow diagram}
\label{sec:SD1}
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{SDE.pdf}
\caption{\label{fig:SDE}Schwinger-Dyson equation at large $N$ for the self-energy.}
\end{figure}
Assuming that the $U(N)\times U(N^2)$ invariance is unbroken, we write the two-point function as
\begin{equation} \label{eq:2pt-sym-ansatz}
\langle \psi_{a_1 a_2 a_3}(x) \bar{\psi}_{b_1 b_2 b_3}(x') \rangle = G(x,x')\, \d_{a_1 b_1} \d_{a_2 b_2} \d_{a_3 b_3} \;.
\end{equation}
In the large-$N$ limit, the self-energy $\Sigma$ is expressible in terms of tadpole diagrams with full two-point function on the internal propagator, as depicted on Fig.~\ref{fig:SDE}.
As a consequence, the large-$N$ Schwinger-Dyson equation is a closed equation for $G(x,x')$, which for its Fourier transform $\hat{G}(p)$ reads
\begin{equation}
\hat{G}(p)^{-1} = i\slashed{p} -\Sigma(p) = i\slashed{p} + 2(\l + \l_p)\int\frac{\dd[3]q}{(2\pi)^3}{\rm tr}[\hat{G}(q)] \mathbb{1}\;,
\end{equation}
the trace and the identity being defined in spinor space (in the following we will generally omit the identity matrix, unless we want to emphasize its presence).
Since the tadpole integral is momentum-independent, we can write $\Sigma = -m\mathbb{1}$, resulting in the gap equation
\begin{equation}
m = 2(\l + \l_p)\int\frac{\dd[3]q}{(2\pi)^3} \frac{{\rm tr}(-i\slashed{q} + m)}{q^2 + m^2},
\end{equation}
or
\begin{equation}
\frac{1}{\l+\l_p}=8 \int_{\abs{p}<\Lambda}\frac{\dd[3]p}{(2\pi)^3}\frac{1}{p^2+m^2} = \frac{4}{\pi^2}\left(\Lambda - m \arctan\left(\frac{\Lambda}{m}\right)\right)\;,
\label{eq:gap-1c}
\end{equation}
that is the analog of \eqref{eq:GN-gap}.
After a rewriting in terms of the dimensionless couplings ($\tilde{\lambda}_p \equiv \Lambda\l_p$ and $\tilde{\lambda} \equiv \Lambda\l$), and derivating both sides with respect to $\Lambda$, we get
\begin{equation*}
\frac{1}{\tilde{\lambda}+\tilde{\lambda}_p} - \frac{\Lambda}{\left(\tilde{\lambda}+\tilde{\lambda}_p\right)^2}\partial_\Lambda\left(\tilde{\lambda}+\tilde{\lambda}_p\right) = \frac{4}{\pi^2}\left(1 - \left(\frac{m}{\Lambda}\right)^2\right)\;.
\end{equation*}
Defining $\kappa \equiv 4/\pi^2$, and taking $\L\gg m$, we find the following combination of beta functions:
\begin{equation}
\beta+\beta_p \equiv \Lambda\partial_\Lambda \tilde{\lambda} + \Lambda\partial_\Lambda\tilde{\lambda}_p= \left(\tilde{\lambda} + \tilde{\lambda}_p\right) - \kappa\left(\tilde{\lambda} + \tilde{\lambda}_p\right)^2 \;.
\end{equation}
Taking into account the different structure of diagrams that contribute to the flow of the couplings (see Fig.~\ref{fig:4-point graphs}), we can disentangle the beta functions and obtain
\begin{align}
\beta &= \tilde{\lambda} - \kappa\left(\tilde{\lambda}^2 + 2\tilde{\lambda}\lt_p\right)\;,\\
\beta_p &= \tilde{\lambda}_p - \kappa\tilde{\lambda}_p^2 \;.
\label{eq:beta-1c}
\end{align}
\begin{figure}[!h]
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{beta_0_1loop.pdf}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{beta_1_1loop.pdf}
\end{minipage}
\caption{\label{fig:4-point graphs}Leading order graphs renormalizing the $\tilde{\lambda}$ and $\tilde{\lambda}_p$ couplings respectively.}
\end{figure}
This leads to the flow diagram of Fig.~\ref{fig:RG-flow-fermions}.
\begin{figure}[!h]
\centering
\includegraphics[width=0.6\textwidth]{RG-flow-fermions.pdf}
\caption{Renormalization group flow in the $\left(\tilde{\lambda},\tilde{\lambda}_p\right)$-plane. Arrows point towards the $IR$, blue dots denote the fixed points, and red lines mark the zeros of either $\beta_p$ or $\beta+\beta_p$. }
\label{fig:RG-flow-fermions}
\end{figure}
One sees, in addition to the trivial IR-fixed point $\left(\tilde{\lambda},\tilde{\lambda}_p\right) = (0,0)$, three new non-trivial fixed points at $\left(\frac{1}{\kappa},0\right)\equiv {\rm FP}_1$, $\left(-\frac{1}{\kappa},\frac{1}{\kappa}\right)\equiv {\rm FP}_2$, and $\left(0,\frac{1}{\kappa}\right)\equiv {\rm FP}_3$. The first of them is the usual interacting CFT of the vector GN model, while the other two are new interacting CFTs. The fixed point at the origin is IR-stable, while ${\rm FP}_2$ is UV-stable, and the other two are saddles. The irrelevant directions at ${\rm FP}_1$ and ${\rm FP}_3$ can be understood as a statement of the fact that their universality classes are stable against symmetry breaking perturbations (for ${\rm FP}_1$, which being the usual GN fixed point has a larger symmetry, namely $U(N^3)$) and trace perturbations (for ${\rm FP}_3$, which lies on the tracelessness constraint subspace of the diagram).
The critical exponents are all $\pm1$ with the signs determined by the corresponding eigenperturbation being relevant or irrelevant.\footnote{In our convention, the critical exponent corresponds to the mass-dimension of the integrated operator (opposite to that of the relative coupling), hence relevant ones have negative critical exponents.}
\subsection{Effective potential and phase diagram}
\label{subsection: effective potential}
The next question to raise concerns the nature of the phase diagram of the stable vacuum.
Following \cite{Benedetti:2017fmp}, we introduce a Hermitian matrix $M_{ij}$ as intermediate field, such that the interaction terms rewrite as\footnote{In App.~\ref{app:rectangular}, we will explain what happens if instead we choose to introduce an intermediate field by cutting the pillow interaction along the index of size $N^2$ (or double index in the tensor notation).}
\begin{equation}
\label{eq: sint with psi}
S_{\rm int}[\psi,\bar{\psi},M]=\frac{1}{2}\left[{\rm Tr}(M^2) + \frac{b}{(1-b)N}({\rm Tr} M)^2\right] + \frac{\sqrt{2\l_p}}{N}\bar{\psi}_{ibc}\psi_{jbc}M_{ij},
\end{equation}
$b \equiv \frac{-\l}{\l_p}$, allowing to integrate out the fermions and obtain
\begin{equation}
\label{eq:S_M}
S[M]=\int \frac{1}{2}\left[{\rm Tr}(M^2) + \frac{b}{(1-b)N}({\rm Tr} M)^2\right] - N^2 {\rm tr} {\rm Tr}\left[\ln\left(\slashed{\partial} + \frac{\sqrt{2\l_p}}{N} M\right)\right].
\end{equation}
To remove the $N$ factor from inside of the log-term, we rescale $M \rightarrow NM$. We are interested in the effective potential, which in the large-$N$ limit is simply given by \eqref{eq:S_M} evaluated at constant $M$.\footnote{Remember that the effective potential is defined as the effective action $\G[M]$ (the one-particle-irreducible generating functional) at constant field, and that $\G[M]$ is the Legendre transform of $W[J]$, the generating functional of connected $n$-point functions. Since the latter is given in the large-$N$ limit simply by the Legendre transform of $S[M]$, and since the Legendre transform is an involutive transformation, we conclude that $\G[M]=S[M]$.} Then the last term, up to a constant independent of $M$, gives
\begin{equation}
\begin{split}
{\rm tr} {\rm Tr} \int \frac{\dd[3]k}{(2\pi)^3} \ln\left(\slashed{\partial} + \sqrt{2\l_p} M\right) &= -4 {\rm Tr} \int \frac{\dd[3]k}{(2\pi)^3}\sum_{n>0}\left(\frac{ik}{k^2}\sqrt{2\l_p} M\right)^{2n}\frac{1}{2n} \\
&= \frac{1}{\pi^2}\int_0^\Lambda \dd k k^2 {\rm Tr}\log\left(1 + \frac{2\l_p M^2}{k^2}\right) \\
&= \frac{{\rm Tr}}{3\pi^2} \left[4\l_p\Lambda M^2 - 2(2\l_p M^2)^\frac{3}{2}\left(\arctan \frac{\Lambda}{\sqrt{2\l_pM^2}}\right) \right. \\
&\quad\qquad \left. + \Lambda^3\log\left(1 + \frac{2\l_pM^2}{\Lambda^2}\right)\right].
\end{split}
\end{equation}
Switching to dimensionless variables and couplings, we find the following effective potential:
\begin{equation} \label{eq:effective action}
\begin{split}
V_{\rm eff}[\tilde{M}]\equiv \frac{S[\tilde{M} = \text{const.}]}{N^2\Lambda^3 {\rm Vol}}= & \frac{1}{4\tilde{\lambda}_p}\left[{\rm Tr}(\tilde{M}^2) + \frac{b}{(1-b)N}({\rm Tr} \tilde{M})^2\right] \\
& - \frac{1}{3\pi^2}{\rm Tr}\left[2 \tilde{M}^2 - 2\abs{\tilde{M}}^3\arctan{\frac{1}{\abs{\tilde{M}}}} + \log\left(1 + \tilde{M}^2\right)\right] \;,
\end{split}
\end{equation}
where we defined
\begin{equation}
\tilde{M} \equiv \frac{\sqrt{2\l_p} M }{\Lambda}, \quad \tilde{\lambda} \equiv \l \Lambda, \quad \tilde{\lambda}_p \equiv \l_p \Lambda, \quad {\rm Vol}=\int d^3 x \;.
\end{equation}
Owing to the $U(N)$-invariant form of the effective potential \eqref{eq:effective action}, we can diagonalize the matrix $\tilde{M}$ and recover an effective potential for its set of eigenvalues\footnote{The Vandermonde determinant originating in the change of variables is subleading in $1/N$ (the action is of order $N^3$ and the logarithm of the Vandermonde determinant is of order $N^2$, see \cite{Nguyen:2014mga}), hence it is not included.}
$\mu_i$, $1\leq i\leq N$:
\begin{align} \label{eq:effective potential}
V_{\rm eff}[\{\mu_i\}] &= \sum_i \left[\frac{1}{4\tilde{\lambda}_p}\mu_i^2 + \frac{1}{3\pi^2}\kappa(\mu_i)\right] - \frac{\tilde{\lambda}}{4\tilde{\lambda}_p (\tilde{\lambda}+\tilde{\lambda}_p) N}\left(\sum_i\mu_i\right)^2 \;,\\
\kappa(\mu) &= 2\mu^3\arctan\frac{1}{\mu} - 2\mu^2 - \log(1+\mu^2) \;.
\end{align}
An important point is that the potential \eqref{eq:effective potential} is unbounded from below in the regions $\tilde{\lambda}_p<0$ and $\tilde{\lambda}_p+\tilde{\lambda}<0$ (this is most easily seen by studying special symmetric configurations such as those we will encounter below for the stationary points), therefore considered unphysical.
The only extremum of the potential which preserves all the symmetries of \eqref{eq:action} is the trivial solution $M=0$, for which our potential is normalized such that $V_{\rm eff}[0] = 0$.
Stationary points with non-zero eigenvalues $\mu_i = \mu ~ (1\leq i\leq N)$, spontaneously break the chiral symmetry of \eqref{eq:action} (reflected in the symmetry $M \rightarrow -M$), whereas if the eigenvalues are not all equal, the original $U(N)$ symmetry of \eqref{eq:action} is spontaneously broken as well.
In the green parallelogram region of the phase diagram in Fig.~\ref{fig:phase diagram}, we can show that the potential is non-negative, and the solution $\mu = 0$ gives a global vacuum. Indeed, the term in square brackets of eq. \eqref{eq:effective potential} is (for each $i$) non-negative and convex (with a global minimum at the origin) in the range $0<\tilde{\lambda}_p<\pi^2/4$.
Consequently, if $\sum_i\mu_i\neq 0$ and $\tilde{\lambda}< 0$ then $V_{\rm eff}[\{\mu_i\}] > 0 = V_{\rm eff}[0]$. In the case $\tilde{\lambda}> 0$, we can use the Cauchy-Schwarz inequality to bound
\begin{equation}
V_{\rm eff}[\{\mu_i\}] \geq
\sum_i \underbrace{\left[\frac{1}{4\tilde{\lambda}_p}\mu_i^2 + \frac{1}{3\pi^2}\kappa(\mu_i) - \frac{\tilde{\lambda}}{4\tilde{\lambda}_p (\tilde{\lambda}+\tilde{\lambda}_p)}\mu_i^2\right]}_{\equiv w(\mu_i)},
\end{equation}
which is convenient, as the eigenvalues decouple. By taking first and second derivatives of each term we can now prove that $\mu_i = 0$ is the unique minimum of $w(\mu_i)$, and hence $\mu_i = 0 ~\forall i$ is the global minimum of $V_{\rm eff}[\{\mu_i\}]$ for $0\leq \tilde{\lambda}+\tilde{\lambda}_p \leq \pi^2/4$ and $0\leq \tilde{\lambda}_p \leq \pi^2/4$:
\begin{align}
w^\prime(x)& = \frac{2x}{\pi^2 \alpha}\left[1 - \alpha + \alpha x \arctan\frac{1}{x}\right] >0 \quad (x>0,\; 0<\a<1),\\
w^{\prime\prime}(x) &= \frac{2}{\pi^2 \alpha}\left[1 -\alpha\frac{1+2x^2}{1 + x^2} + 2\alpha x \arctan\frac{1}{x}\right] >0 \quad (0<\a<1) \;,
\end{align}
having introduced $\alpha= 4(\tilde{\lambda}+\tilde{\lambda}_p)/\pi^2$. At $\a>1$ the origin becomes unstable.
The stability of the trivial solution is more properly analyzed by studying the full Hessian.
Coming back to eq.~\eqref{eq:effective action}, we can want to compute the second derivative around the point $\tilde{M} = 0$, for which we can discard the $\arctan$ term, as it is of cubic (and higher) order in the fluctuations. The first derivative gives
\begin{gather}
\pdv{V}{\tilde{M}_{ij}} = A \tilde{M}_{ji} + B \frac{{\rm Tr} \tilde{M}}{N} \d_{ij}+ C \tilde{M}_{ji} \; ,\\
A = \frac{1}{2\tilde{\lambda}_p}\; ,\quad B = \frac{b}{2\tilde{\lambda}_p(1-b)}\; , \quad C = - \frac{2}{\pi^2}\;.
\end{gather}
The second derivative
\begin{equation}
\pdv{V}{\tilde{M}_{ij}}{\tilde{M}_{kl}} = (A + C) \d_{ik}\d_{jl} + B\frac{\d_{ij}\d_{kl}}{N}\;,
\end{equation}
can be rewritten as follows
\begin{gather}
H = \a(1 - P)+ \b P \\
\a = A + C\; \quad \b = A + B + C\;,\quad P_{ij,kl} \equiv \frac{\d_{ij}\d_{kl}}{N}\;,
\end{gather}
introducing $P$ that projects on the trace.
Articulated as such, the Hessian $H$ is easy to diagonalize, as the eigenfunctions are easily found to be: \begin{itemize}
\item[-] traceless matrices, with eigenvalue $\a = \frac{1}{2}\left(\frac{1}{\tilde{\lambda}_p} - \frac{4}{\pi^2}\right)$,
\item[-] matrices proportional to the identity, with eigenvalue $\b = \frac{1}{2}\left(\frac{1}{\tilde{\lambda} + \tilde{\lambda}_p} - \frac{4}{\pi^2}\right)$.
\end{itemize}
This suggests that the trivial solution becomes unstable towards traceless perturbations at $\tilde{\lambda}_p~\geq~\pi^2/4$ and towards trace perturbation at $\tilde{\lambda}+\tilde{\lambda}_p ~\geq~ \pi^2/4$.
Therefore, the following two particular non-zero solutions are examined:\footnote{Other solutions are possible, as in appendix D of \cite{Benedetti:2017fmp}. In that case it was possible to show that such solutions are never global minima of the potential; here the analysis is more complicated and we limit ourselves to conjecture that the analysis of the following two types of solutions suffices to understand the full phase diagram of the model.}
\begin{itemize}
\item $\mu_i = \mu\neq 0 ~\forall i$: The potential takes the form
\begin{equation}
\label{eq:uniform potential}
\f1N V_{\rm eff}(\mu)=\frac{\mu^2}{4\tilde{\lambda}_p}\left(1 + \frac{b}{1-b} - \frac{8\tilde{\lambda}_p}{3\pi^2}\right) + \frac{2}{3\pi^2}\abs{\mu}^3\arctan{\frac{1}{\abs{\mu}}} - \frac{1}{3\pi^2}\log(1+\mu^2) \;,
\end{equation}
and the equation of motion that $\mu$ must satisfy is
\begin{equation}
\label{eq: eom uniform}
\mu \arctan{\frac{1}{\mu}} = 1 - \frac{\pi^2}{4(\tilde{\lambda} + \tilde{\lambda}_p)} \;.
\end{equation}
The range of values of the left-hand side tells us that such a solution exists only for $\tilde{\lambda} + \tilde{\lambda}_p \geq \pi^2/4$.
\item ${\rm Tr} \tilde{M}= 0$: Then the potential reduces to a sum over the eigenvalues. We obtain
\begin{equation}
\label{eq:traceless potential}
V_{\rm eff}[\tilde{M}]=\sum_i v(\mu_i)\;,
\end{equation}
where
\begin{equation}
v(\mu) = \frac{\mu^2}{4\tilde{\lambda}_p}\left(1 - \frac{8\tilde{\lambda}_p}{3\pi^2}\right) + \frac{2}{3\pi^2}\abs{\mu}^3\arctan{\frac{1}{\abs{\mu}}} - \frac{1}{3\pi^2}\log(1+\mu^2) \;.
\end{equation}
The equation of motion for $\m_i$ is
\begin{equation}
\label{eq: eom traceless}
\mu_i \arctan{\frac{1}{\mu_i}} = 1 - \frac{\pi^2}{4\tilde{\lambda}_p} \;.
\end{equation}
The range of the left-hand side tells us that such a solution exists only for $\tilde{\lambda}_p \geq \pi^2/4$.
Furthermore, being an even function, monotonic on each semiaxis, there are only two solutions $\mu_i = \pm \t$. The tracelessness condition finally tells us that the two must come in equal number (for odd $N$ we necessarily have either a zero eigenvalue or a violation of the tracelessness condition, which amounts to a subleading effect in $1/N$).
\end{itemize}
Using the equations of motion, we need to compare the values of the potential at the above critical points:
\begin{equation}
V_{\rm eff}(q) = \frac{\tau(q)^2}{12 q} -\frac{1}{3\pi^2}\log(1 + \tau(q)^2),
\end{equation}
with $q = \tilde{\lambda}+\tilde{\lambda}_p$ in the uniform case and $q=\tilde{\lambda}_p$ in the traceless one, and $\tau(q)$ being the solution of $\t \arctan{(1/\t)} = 1 - \pi^2/(4 q)$. Since $\tau(q)^2$ is a monotonically increasing function, and since as a function of $q$, $V_{\rm eff}(q)$ is decreasing monotonically starting from $0$ (the trivial solution), we conclude that
\begin{equation}
\tilde{\lambda}_p > \tilde{\lambda}+\tilde{\lambda}_p \implies V_{\rm eff}(q_{traceless})<V_{\rm eff}(q_{uniform})
\end{equation}
and reciprocally. In other words, the traceless solution wins over the uniform one for $\tilde{\lambda}<0$, while the uniform wins for $\tilde{\lambda}>0$.
Such transition can be qualitatively understood in terms of the double-trace term: we see that if $\tilde{\lambda}<0$, then the double-trace term comes with a positive sign and has to be minimized, showing why the traceless solution wins (when it exists, i.e.\ for $\tilde{\lambda}_p>\pi^2/4$), while if $\tilde{\lambda}>0$, then the coefficient of the double-trace term is negative and has to be maximized, leading to the uniform solution.
At last, the phase diagram is as shown in Fig.~\ref{fig:phase diagram}.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{phase-diagram-1c.pdf}
\caption{\label{fig:phase diagram}Phase diagram of the fermionic TGN model in 3$d$. The red zone has a vacuum $M\propto \mathbb{1}$, the blue one to a traceless vacuum and the green parallelogram to $M=0$. The grey zone corresponds to an unstable model.}
\end{figure}
Knowing the value of the potential at the different vacua, we see that the transition from red and blue to green are continuous, hence of second order. Indeed, comparing eq. \eqref{eq: eom uniform} with eq. \eqref{eq: eom traceless}, we see that taking the limit $\tilde{\lambda}_p+\tilde{\lambda}$ or $\tilde{\lambda}_p$ to $\pi^2/4$, $\mu$ and $\tau$ decrease monotonically to zero.
On the other hand, a first order transition separates the two non-trivial phases: at $\tilde{\lambda} = 0$ and $\tilde{\lambda}_p>\pi^2/4$, they both have same potential energy (but are distinct), and as the coupling $\tilde{\lambda}$ grows or decreases, the uniform or traceless solution become global minima, respectively.
\subsection{Schwinger-Dyson equations in the traceless phase}
\label{sec:SD1bis}
In Sec.~\ref{sec:SD1} we derived the gap equation and beta functions from the Schwinger-Dyson equations in the $U(N)\times U(N^2)$-symmetric phase. Although we expect the beta functions to be independent of such choice it is instructive to do an explicit check, now that we discovered a broken phase.
Assuming that the two-point function breaks the $U(N)\times U(N^2)$ symmetry into $U(N/2)^2\times U(N^2)$, with the following ansatz:
\begin{gather}
M_1 = m (\mathbb{1}_N- 2\mathbb{P}_N),
\quad \mathbb{P}_N = \begin{pmatrix}
\mathbb{0}_{N/2} & \mathbb{0}_{N/2} \\
\mathbb{0}_{N/2} &\mathbb{1}_{N/2}\end{pmatrix},\\ \label{eq:broken-2pt-ansatz}
G^{-1}(p) = i\slashed{p} \,\mathds{1}_1\otimes \mathds{1}_2 \otimes \mathds{1}_3 + M_1\otimes \mathds{1}_2 \otimes \mathds{1}_3 \;,
\end{gather}
the Schwinger-Dyson equation becomes
\begin{equation}
\label{eq:SDE U(N)-broken}
G^{-1}(p) = i\slashed{p} \mathds{1}_1\otimes \mathds{1}_2 \otimes \mathds{1}_3 + 2 \frac{\l_p}{N^2}\int\frac{\dd[3]q}{(2\pi)^3} ({\rm Tr}_{\backslash 1} {\rm tr} \left[ G(q) \right])\otimes \mathds{1}_2 \otimes \mathds{1}_3\;,
\end{equation}
where ${\rm Tr}_{\backslash c}$ is a trace on all indices except the one of color $c$.
The coupling $\l$ is missing from the equation because it multiplies a full trace of $G(q)$, which is zero for the ansatz above.
Forgetting momentarily the trace over the $\g$-matrices, we have
\begin{equation}
{\rm Tr}_{\backslash 1} \left[G(q)\right] = N^2 \frac{-i\slashed{p} + M_1}{p^2+m^2}\;,
\end{equation}
such that the Schwinger-Dyson equation reduces to
\begin{equation}
M_1 = 2\l_p \int \frac{\dd[3]q}{(2\pi)^3} \frac{{\rm tr}(-i\slashed{p} + M_1)}{p^2+m^2}\;,
\end{equation}
or the mass gap equation
\begin{equation}
m = 8\l_p\int \frac{\dd[3]q}{(2\pi)^3} \frac{m}{p^2+m^2}\;,
\end{equation}
that is, nothing more than eq.~\eqref{eq:gap-1c} with $\l = 0$, thus leading directly to \eqref{eq:beta-1c}.
\subsection{Anomalous dimension}
We have three non-trivial fixed points (out of which one has two relevant directions while the others have one), and for each of them we can compute the conformal dimension of the intermediate field. It happens, that in all cases, the computation is almost unchanged and gives the same result:
\begin{itemize}
\item $(\tilde{\lambda},\tilde{\lambda}_p)=(\pi^2/4,0)$
This is the UV fixed point of the usual GN model. The limit $\l_p\to 0$ constrains to zero the traceless part of the intermediate field \cite{Benedetti:2017fmp}, and thus
it is equivalent to starting from the action \eqref{eq:GNY}. Integrating out the fermions:
\begin{equation}
S_{\rm int}[\s] = N^3 \int d^3 x \;\left( \frac{1}{4\l}\sigma^2 - \log\left(\slashed{\partial} + \s \right)\right).
\end{equation}
At the fixed point, $\langle \s\rangle =0$ and the inverse propagator is obtained by the second functional derivative with respect to $\s$, computed at $\s=0$, i.e.:
\begin{equation} \label{eq:effective_prop_case1}
\frac{1}{N^3}\langle\s(p)\s(-p)\rangle^{-1} = \frac{1}{2\l} - {\rm tr} \int\frac{\dd[3]q}{(2\pi)^3}\frac{\slashed{q}(\slashed{q}-\slashed{p})}{q^2(q-p)^2}
=\frac{1}{2\l} - 4 \int\frac{\dd[3]q}{(2\pi)^3}\frac{q^2-q\cdot p}{q^2(q-p)^2}.
\end{equation}
The last integral can be computed as
\begin{equation}
\begin{split}
4 \int\frac{\dd[3]q}{(2\pi)^3}\frac{q^2-q\cdot p}{q^2(q-p)^2} &= \frac{2}{\pi^2} \int_0^\L dq - \frac{2p^2}{(2\pi)^2} \int_0^{+\infty} dq \int_{-1}^{+1} d(\cos\theta) \frac{1}{q^2+p^2-2 q p \cos\theta}\\
&= \frac{2}{\pi^2} \L - \frac{p}{4} \;.
\end{split}
\end{equation}
The linear divergence is cancelled by the fixed point condition $\l = \frac{\pi^2}{4\Lambda}$, thus yielding
\begin{equation}
\langle\s(p)\s(-p)\rangle = \frac{4}{N^3 p} \;,
\end{equation}
corresponding to a conformal dimension $\D_\s=1$, which is also the dimension of $\bar{\psi}_{abc}\psi_{abc}$.
\item $(\tilde{\lambda},\tilde{\lambda}_p)=(0,\pi^2/4)$
Let us recall the effective action of the intermediate field
\begin{align}
S_{\rm int}[M] = &\int \frac{1}{2}M^*_{ij}K_{ij;kl}M_{kl} - N^2 {\rm tr} {\rm Tr}\left[\log\left(\slashed{\partial} + \frac{\sqrt{2\l_p}}{N}M\right)\right],\\
&K_{ij;kl} = \delta_{ik}\delta_{jl} + \frac{b}{(1-b)N}\delta_{ij}\delta_{kl}.
\end{align}
It is convenient to introduce again the rescaled matrix $\tilde{M} = \frac{\sqrt{2\l_p}}{N}M$. Derivating twice with respect to an eigenvalue $m_i$ of $\tilde{M}$, and setting $b=0$, gives, after a Fourier transform
\begin{equation}
\frac{1}{N^2} \frac{\delta^2 S_{\rm int}}{\delta m^2}\big|_{m=0} = \frac{1}{2\l_p} -{\rm tr} \int\frac{\dd[3]q}{(2\pi)^3}\frac{\slashed{q}(\slashed{q}-\slashed{p})}{q^2(q-p)^2}\;,
\end{equation}
namely the same expression as \eqref{eq:effective_prop_case1}, with $\l_p$ replacing $\l$.
The linear divergence is cancelled by the fixed point condition, $\frac{1}{2\l_p} = \frac{2\Lambda}{\pi^2}$, and we arrive at the same propagator (hence same conformal dimension) as in the usual three-dimensional Gross-Neveu model.
\item $(\tilde{\lambda},\tilde{\lambda}_p)=(-\pi^2/4,\pi^2/4)$
Here, because $b=1$ is a singular point of $K$, we need to take a few steps back \cite{Benedetti:2017fmp}. The fermionic interaction action was written with a matrix-like field $B_{ij} = \bar{\psi}_{ibc}\psi_{jbc}$ as
\begin{align}
&S_{\rm int} = -\frac{\l_p}{N^2}\int B^*_{ij}C_{ij;kl}B_{kl}\;,\\
C_{ij;kl} = \delta_{ik}\delta_{jl} - \frac{b}{N}\delta_{ij}\delta_{kl} &= (\bm{1} - \bm{P})_{ij;kl} + (1 - b)\bm{P}_{ij;kl}\;, \qquad \bm{P}_{ij;kl} = \frac{\delta_{ij}\delta_{kl}}{N}\;.
\end{align}
Since $\bm{P}$ projects on the trace part of the matrix, it appears clearly that $b=1$ restricts us to work with a traceless $B$. Except for this constraint on the fields (which then follows for the matrix-like intermediate field $M$ \footnote{The trace of $M$ couples to that of $B$ and its effective action will be identical to eq. \eqref{eq:S_M}, except for an absent double-trace term.}), the computation of the effective propagator will be identical to other two cases above.
\end{itemize}
\section{$U(N)\times U(N)\times U(N)$-symmetric model}
\label{S:2}
Adding to the model defined in \eqref{eq:action}-\eqref{eq:S_int1} other pillow interactions which differ by simultaneous permutations of the three tensor indices of all the fields, we break the $U(N)\times U(N^2)$ symmetry down to $U(N)\times U(N)\times U(N)$. In the tensor model literature it is usual to refer to an index location (first, second, or third index, in our case) as a color, and hence such permutations of indices are called color permutations. There are three distinguishable colorings for the pillow interaction, one for each choice of {\emph{transmitted color}}, i.e.\ for each choice of index being associated to the vertical lines of Fig.~\ref{fig:interactions}. Considering that of course there is only one coloring for the double-trace interaction, we have in general four independent couplings. We will restrict the theory space by demanding \emph{color symmetry} of the action, i.e. invariance under permutations of the indices, thus writing for the new interacting part of the action
\begin{equation} \label{eq:S_int123}
S_{\rm int}[\psi,\bar{\psi}]= -\frac{\l}{N^3} \int d^3 x \; (\bar{\psi}_{abc}\psi_{abc})^2 - \frac{\l_p}{N^2} \sum_{\ell =1}^3 \mathcal{P}_\ell[\psi,\bar{\psi}] \;,
\end{equation}
where $\mathcal{P}_\ell[\psi,\bar{\psi}]$ is the pillow interaction with transmitted color $\ell$.
\subsection{Schwinger-Dyson equations and $\beta$-functions}
\label{sec:SD123}
Following \cite{Benedetti:2017fmp}, the SD equations in momentum space write as \footnote{${\rm Tr}$ is a trace on all the color indices, while with ${\rm Tr}_{\backslash c}$ the color $c$ is not traced on. As before, ${\rm tr}$ is a trace on the $\g$-matrix space.}
\begin{equation}
\label{eq:SDE-3c}
G^{-1}(p) = G^{-1}_0(p) + 2 \int\frac{\dd[3]q}{(2\pi)^3} {\rm tr} \left[\frac{\l}{N^3} {\rm Tr} G (q) + \frac{\l_p}{N^2} \left({\rm Tr}_{\backslash 1} G(q) + {\rm Tr}_{\backslash 2} G(q) +{\rm Tr}_{\backslash 3} G(q)\right)\right]\;,
\end{equation}
as also depicted in Fig.~\ref{fig:SDE-3c}.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{schwinger_dyson_lo.pdf}
\caption{\label{fig:SDE-3c}Schwinger-Dyson equation at large $N$ for the self-energy.}
\end{figure}
Assuming that the $U(N)$ symmetry is unbroken by the vacuum, an ansatz for the full propagator is again given by \eqref{eq:2pt-sym-ansatz}, with the diagonal $\hat{G}(p)^{-1}= i\slashed{p} + m $; thus, similarly to subsection \ref{sec:SD1}, the gap equation is
\begin{equation}
m = 2(\l + 3\l_p)\int\frac{\dd[3]q}{(2\pi)^3} \frac{{\rm tr}(-i\slashed{q} + m)}{q^2 + m^2},
\end{equation}
or
\begin{equation}
\frac{1}{\l+3\l_p}=8 \int_{\abs{p}<\Lambda}\frac{\dd[3]p}{(2\pi)^3}\frac{1}{p^2+m^2} = \frac{4}{\pi^2}\left(\Lambda - m \arctan\left(\frac{\Lambda}{m}\right)\right).
\end{equation}
In terms of the dimensionless couplings ($\tilde{\lambda}_p \equiv \Lambda\l_p$ and $\tilde{\lambda} \equiv \Lambda\l$) and with $\kappa \equiv 4/\pi^2$, the $\beta$-functions read
\begin{equation}
\label{eq:beta-unbroken}
\beta+3\beta_p = \left(\tilde{\lambda} + 3 \tilde{\lambda}_p\right) - \kappa\left(\tilde{\lambda} + 3 \tilde{\lambda}_p\right)^2 \;.
\end{equation}
By direct inspection of the one loop diagrams at leading order in $1/N$, depicted in Fig.~\ref{fig:4-point graphs-3c}, we can disentangle the two beta functions, obtaining:
\begin{align} \label{eq:beta-3c}
\beta &= \tilde{\lambda} - \k\left(\tilde{\lambda}^2 + 6\tilde{\lambda} \tilde{\lambda}_p + 6\tilde{\lambda}_p^2\right)\;,\\
\beta_p &= \tilde{\lambda}_p - \kappa\tilde{\lambda}_p^2 \;.
\end{align}
\begin{figure}
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{beta_0_1loop-ell.pdf}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{beta_1_1loop-ell.pdf}\\
\vspace{.6cm}
\includegraphics[width=0.8\textwidth]{beta_0_1loop-bis.pdf}
\end{minipage}
\caption{\label{fig:4-point graphs-3c}Leading order graphs renormalizing the couplings $\tilde{\lambda}_p$ (top right) and $\tilde{\lambda}$ (all the others).}
\end{figure}
With the experience of the previous section, we can also consider the case of broken $U(N)$ symmetry, and disentangle the two beta functions by combination of the symmetric and broken results.
Anticipating the results of section \ref{sec:potential-3c}, it turns out that there is a broken phase where we can make precisely the same ansatz as in \eqref{eq:broken-2pt-ansatz}. The calculation then proceeds exactly as in that section, after having noticed that in \eqref{eq:SDE-3c} the $ {\rm Tr}_{\backslash 2}$ and $ {\rm Tr}_{\backslash 3}$ terms vanish because the trace on color 1 is zero.
Therefore, the beta function for $\l_p$ is unchanged, and combining it with that of the unbroken case (eq. \eqref{eq:beta-unbroken}), we find again \eqref{eq:beta-3c}, as expected.
It is also interesting to point out that $\b + 2\b_p = 0$ along $\l+2\l_p=0$.
We picture the resulting flow on Fig.~\ref{fig:RG-flow-3c}, seemingly a distorted version of Fig.~\ref{fig:RG-flow-fermions}.
In fact the critical exponents are also the same as in the previous model.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{RG-flow-3c.pdf}
\caption{Renormalization group flow in the $\left(\tilde{\lambda},\tilde{\lambda}_p\right)$-plane. Arrows point towards the $IR$, black dots denote the fixed points, and red lines mark the zeros of either $\beta_p$ or $\beta+3\beta_p$. The green line corresponds instead to $\beta+2\beta_p=0$, which morally replaces the vertical line $\beta=0$ of Fig.~\ref{fig:RG-flow-fermions}.}
\label{fig:RG-flow-3c}
\end{figure}
\subsection{Effective potential and phase diagram}
\label{sec:potential-3c}
Rewriting all the quartic interactions in terms of intermediate fields, the action takes the following form
\begin{gather}
\label{eq:action_cIF}
S[M_1,M_2,M_3] = \int \frac{1}{2} \sum_{c=1,2,3}\left[{\rm Tr}(M_c^2) + \frac{b}{(1-b)N}({\rm Tr} M_c)^2\right] - {\rm tr} {\rm Tr}\left[ \ln\left(\slashed{\partial} + \frac{\sqrt{2\l_p}}{N} R\right)\right]\\
R \equiv M_1\otimes \mathds{1}_2 \otimes \mathds{1}_3 + \mathds{1}_1 \otimes M_2 \otimes \mathds{1}_3 + \mathds{1}_1 \otimes \mathds{1}_2 \otimes M_3\;; \quad b = -\frac{\l}{3\l_p},
\end{gather}
with the three intermediate fields $M_1, M_2, M_3$ needed for the three pillow interaction terms. The original GN-interaction is split into three identical parts, thus in effect changing $\l \rightarrow \l/3$ in each of the $({\rm Tr} M_c)^2$ terms.
Again, coming to adimensional variables, rescaling such that $\tilde{M}_i \equiv N \L^{3/2}M_i, ~ \forall i$, and using the $U(N)$ symmetry to diagonalize, we arrive at
\begin{equation}
\begin{split}
V &\left[\{\mu_{1,i}, \mu_{2,j} , \mu_{3,k}\}\right] \equiv \frac{S\left[\tilde{M}_1,\tilde{M}_2,\tilde{M}_3\right]_{|_{\tilde{M}_i={\rm const.}}}}{N^3\L^3 {\rm Vol}} \\
&\quad = \frac{1}{N}\sum_{c} \frac{1}{4\tilde{\lambda}_p} \left[\sum_i\mu_{c,i}^2 - \frac{\tilde{\lambda}}{(\tilde{\lambda}+3\tilde{\lambda}_p) N}\left(\sum_i\mu_{c,i}\right)^2 \right]\\
& \qquad \qquad + \frac{1}{N^3}\sum_{1\leq i,j,k\leq N}\frac{1}{3\pi^2}\kappa(\mu_{1,i},\mu_{2,j},\mu_{3,k})\;,
\end{split}
\end{equation}
\begin{gather}
\kappa(\mu_{1,i},\mu_{2,j},\mu_{3,k}) = 2\mu^3\arctan\frac{1}{\mu} - 2\mu^2 - \log(1+\mu^2), \\
\mu \equiv \mu_{1,i}+\mu_{2,j}+\mu_{3,k} \;.
\end{gather}
We are looking for the vacua of this potential. The equations of motion for $\mu_{c,i}$ read
\begin{equation}
\frac{1}{2\tilde{\lambda}_p}\left[\m_{c,i} - \frac{\tilde{\lambda}}{(\tilde{\lambda} +3 \tilde{\lambda}_p)N}\sum_j\m_{c,j}\right] + \frac{2}{\pi^2N^2}\sum_{1\leq j,k\leq N}\left[\mu^2\arctan\frac{1}{\mu} - \mu\right] = 0.
\label{eq:eom_3c}
\end{equation}
To begin, a useful remark is that the three different color-intermediate fields must have the same trace at the saddle points. Indeed, this is seen by summing eq. \eqref{eq:eom_3c} over $i$. Because the second term in squared brackets depends only on $\mu$, all colors end up with the same equation (of the type ${\rm Tr}[\tilde{M}_c]=F[\{\m\}]$, with the same right-hand side), hence the equality of traces.
Another straightforward observation is that $M_i = 0 ~ \forall i$ is always a solution. In order to find other solutions, it is helpful to search for unstable directions around this point, i.e. analyse the Hessian of the potential.
Developing around the point $R = 0$ allows to discard the $\arctan$ term, of higher order. The first derivative gives\footnote{The exponent $(c)$ in $\d_{ij}^{(c)}$ only serves to keep track of what color the indices belong to.}
\begin{gather}
\pdv{V}{M_{c,ij}} = (A +C) M_{c,ji} + B \frac{{\rm Tr} M_{c}}{N} \d_{ij}^{(c)} + \sum_{c^\prime \neq c} C \frac{{\rm Tr} M_{c^\prime}}{N} \d_{ij}^{(c)}\; ,\\
A = \frac{1}{2\tilde{\lambda}_p}\; ,\quad B = \frac{b}{2\tilde{\lambda}_p(1-b)}\; , \quad C = - \frac{2}{\pi^2}\;.
\end{gather}
The second derivative is
\begin{equation}
\pdv{V}{M_{c,ij}}{M_{c^\prime,kl}} = \d_{cc^\prime}\left((A + C) \d_{ik}\d_{jl} + B\frac{\d_{ij}\d_{kl}}{N}\right) + C\frac{\d_{ij}^{(c)}\d_{kl}^{(c^\prime)}}{N},
\end{equation}
and can be rewritten as a matrix in color-space
\begin{gather}
H =
\begingroup
\renewcommand*{\arraystretch}{1.5}
\begin{pmatrix}
\a(1 - P_1)+ \b P_1 & C \frac{\Pi_1\Pi_2}{N} & C \frac{\Pi_1\Pi_3}{N}\\
C \frac{\Pi_2\Pi_1}{N} &\a(1 - P_2)+ \b P_2 & C \frac{\Pi_2\Pi_3}{N}\\
C \frac{\Pi_3\Pi_1}{N} & C \frac{\Pi_3\Pi_2}{N} &\a(1 - P_3)+ \b P_3
\end{pmatrix},
\endgroup \\
\a = A + C\; \quad \b = A + B + C.
\end{gather}
We defined the projector $P_c$ on the trace of a matrix of given color $c$
\begin{equation}
P_{c;ij,kl} \equiv \frac{1}{N}\Pi_{c,ij}\Pi_{c,kl}, \quad \Pi_{c,ij}\equiv \d_{ij}^{(c)}.
\end{equation}
Articulated as such, the Hessian $H$ is easy to diagonalize\footnote{Using a simplified notation in which, for instance, an element $a$ on the second line of the column vector is intended to represent the element $\mathbb{1} \otimes a \otimes \mathbb{1}$, and analogously with the other lines.}:
\begin{itemize}
\item we have the set of eigenvectors associated to traceless matrices, of form
\begin{equation}
E^1 = \begin{pmatrix}Q\\0\\0\end{pmatrix}, \quad E^2 = \begin{pmatrix}0\\Q\\0\end{pmatrix},\quad E^3 = \begin{pmatrix}0\\0\\Q\end{pmatrix}, \quad {\rm Tr} Q = 0,
\end{equation}
with eigenvalue $\alpha = \frac{1}{2}\left(\frac{1}{\tilde{\lambda}_p} - \frac{4}{\pi^2}\right)$ ;
\item the eigenvector associated to matrices proportional to the identity, of the form
\begin{equation}
E^s = \g\begin{pmatrix}1\\1\\1\end{pmatrix},
\end{equation}
with eigenvalue $\b + 2C = \frac{3}{2}\left(\frac{1}{\tilde{\lambda} + 3\tilde{\lambda}_p} - \frac{4}{\pi^2}\right)$ ;
\item and finally the are eigenvectors
\begin{equation}
e^1 = \g_1 \begin{pmatrix}1\\-1\\0\end{pmatrix}, \quad e^2 = \g_2 \begin{pmatrix}1\\0\\-1\end{pmatrix},
\end{equation}
with eigenvalue $\b - C = \frac{3\tilde{\lambda}_p}{\tilde{\lambda} + 3\tilde{\lambda}_p}$.
\end{itemize}
The first two correspond to instabilities that decrease the potential (in the ranges where the eigenvalues become negative), while the last always increases it for the range of couplings allowed by the requirement of boundedness of the potential (and in addition they do not satisfy the equations of motion, because the traces of the three matrices are not equal).
In light of this analysis, we can constrain the form of the intermediate field in our search of new minima of the potential:
\begin{itemize}
\item Assuming $M_i = m\mathbb{1}_N ~ (i=1,2,3)$, the equations of motion imply for $m$ the equation
\begin{equation}
1 - 3m\arctan\frac{1}{3m} = \frac{\pi^2}{4} \frac{1}{\tilde{\lambda}+3\tilde{\lambda}_p},
\label{eq:constraint-m}
\end{equation}
allowing such solutions to exist only for $\tilde{\lambda}+3\tilde{\lambda}_p>\pi^2/4$. After using eq. \eqref{eq:constraint-m} to remove the $\arctan$ term, their potential energy is given by
\begin{equation}
V[m] = \frac{1}{12 (\tilde{\lambda}+3\tilde{\lambda}_p)}(3m)^2 - \frac{1}{3\pi^2}\log\left(1+(3m)^2\right),
\label{eq:potential-m-3c}
\end{equation}
corresponding to a non-trivial minimum as soon as $\tilde{\lambda} + 3\tilde{\lambda}_p>\pi^2/4$.
\item Assuming that the intermediate fields are traceless reduces the equations of motion \eqref{eq:eom_3c} to the following:
\begin{equation}
\mu_{1,i} \left(\frac{\pi^2}{4\tilde{\lambda}_p} - 1\right) + \frac{1}{N^2}\sum_{jk}(\mu_{1,i}+\mu_{2,j}+\mu_{3,k})^2\arctan\frac{1}{\mu_{1,i}+\mu_{2,j}+\mu_{3,k}}=0\; .
\label{eq:eom_traceless}
\end{equation}
Taking only one non-trivial matrix $M_1 = w(\mathbb{1}_N-2 \mathbb{P}_N)$, and $M_2 = M_3 = \mathbb{0}_N$, $w$ is hold by eq. \eqref{eq:eom_traceless} to
\begin{equation}
1 - w \arctan\frac{1}{w} = \frac{\pi^2}{4\tilde{\lambda}_p}\;,
\label{eq: single-traceless-eom}
\end{equation}
while the potential energy is
\begin{equation}
V[w] = \frac{1}{12 \tilde{\lambda}_p}w^2 - \frac{1}{3\pi^2}\log\left(1+w^2\right),
\label{eq:potential-w-3c}
\end{equation}
again corresponding to a non-trivial minimum for $\tilde{\lambda}_p>\pi^2/4$.
The cases of two and three traceless matrices are examined in App. \ref{app:traceless_matrices}, where we show in detail that the single-traceless potential dominates double- and triple-traceless solutions above $\tilde{\lambda}_p = \pi^2/4$.
\end{itemize}
The potentials of eq. \eqref{eq:potential-m-3c} and \eqref{eq:potential-w-3c} compare easily: $m$ and $w$ are controlled by similar equations, and they grow monotonically with the coupling. Consequently, we have that $V[w]<V[m]$ for $\tilde{\lambda} + 3\tilde{\lambda}_p>\tilde{\lambda}_p$ or $\tilde{\lambda}+2\tilde{\lambda}_p>0$. In other words, the green line outlined in Fig. \ref{fig:RG-flow-3c} and \ref{fig:phase-diagram-3c} sets apart the traceless vacuum from the traceful one.
\
Let us summarize the results.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth,height=0.48\textwidth]{phase-diagram-3c.pdf}
\caption{Phase diagram of the color-symmetric fermionic TGN model in 3$d$. The color code is the same as in Fig. \ref{fig:phase diagram}.}
\label{fig:phase-diagram-3c}
\end{figure}
Integrating out the fermions, we obtained the effective potential of the intermediate fields. The associated equations of motion were too difficult to solve in full generality. Nevertheless, after a stability analysis around the trivial vacuum, showing that the unstable perturbations beyond some critical lines are given by traceless and pure trace modes, we explored those two types of solutions (i.e.\ traceless matrices and matrices proportional to the identity) and showed in what range of the couplings they minimized the energy. The conclusion is very similar to that of the previous section. Indeed, three phases appear: the trivial, traceful or traceless solutions dominate in turn, as displayed on the phase diagram of Fig. \ref{fig:phase-diagram-3c}. However, in the last case, a startling new feature is the breaking of color-symmetry. The transitions between trivial and traceful or traceless are continuous, while a discontinuous transition separates the non-trivial phases.
\section{Conclusions}
In this paper we have studied the large-$N$ limit of two fermionic quantum field theories with symmetry group $U(N)\times U(N^2)$ and $U(N)\times U(N)\times U(N)$, respectively. The study of such type of models is motivated on one hand by the attempt of generalizing the behaviour found in the SYK model to higher dimensional models and on the other by the possibility of generalizing the Klebanov-Polyakov duality between vector models at large $N$ and higher-spin theories.
More generally, such models are also of interest as they provide a new solvable limit in quantum field theory.
The specific three-dimensional quartic tensor models we focused on have the simplifying feature of admitting a rewriting in terms of matrix intermediate fields. We studied their RG flow, and in addition to the trivial IR fixed point and to the standard UV fixed point of the Gross-Neveu model (where the tensorial interactions that break the $U(N^3)$ symmetry are zero), we found two new fixed points: one with two relevant directions, and another with one relevant and one irrelevant direction. We identified three phases of the vacuum by analyzing the effective potential of the intermediate field: a massless phase preserving all the symmetries of the model; a phase with dynamically generated mass, breaking only the chiral symmetry; and a phase breaking the chiral symmetry, as well as one of the $U(N)$ subgroups of the symmetry group (and as a consequence breaking also the color symmetry, when present in the original model). We found that the conformal dimension of the unconstrained component of the intermediate field (and thus of the corresponding composite operator $\bar{\psi} \psi$) is the same at all the nontrivial fixed points, $\D_{\bar{\psi} \psi}=1$ (to be compared to the Gaussian fixed point value, $\D_{\bar{\psi} \psi}=2$), thus matching that of the critical vector GN model.
The dimensions of the (integrated) quartic operators are instead different, as they change from relevant to irrelevant from one fixed point to another, but are the same in modulus, which is always equal to one.
As discussed in \cite{Benedetti:2017fmp}, the Goldstone bosons of the phase with broken $U(N)$ subgroup are governed by a complex Grassmannian non-linear sigma model, more precisely with Grassmannian $\mathrm{Gr}(N,N/2)\equiv U(N)/(U(N/2)\times U(N/2))$. Although we have not made further use of this fact here, it is interesting to notice the appearance of such models in the context of tensor models, something that might deserve further study.\footnote{A different pattern of spontaneous symmetry breaking has recently been discussed in \cite{Diaz:2018eik}, in which a $U(N)^2$ subgroup of the symmetry group gets broken down to its diagonal subgroup $U(N)$.}
Among the many possible extensions of the model, two are particularly worth mentioning.
In order to better understand the differences between tensor and vector or matrix models, a gauged version of the above is under investigation.
As in the vector case, one motivation for gauging these models is to restrict the state space to singlets, the restriction required for a holographic mapping to higher spin theory makes sense \cite{Aharony:2011jz,Giombi:2011kc}.
Therefore, we hope that gauging our tensorial models could serve as a concrete starting point to connect with the duality recently proposed by Vasiliev \cite{Vasiliev:2018zer}. It would also be interesting to study the persistence of the phase with broken $U(N)$ symmetry, and the associated Higgs mechanism.
Lastly, it would be interesting to further study models with different symmetry groups, allowing the complete-graph (or tetrahedron) quartic interaction. Finding healthy nontrivial fixed points for fermionic systems in the presence of such interaction has so far remained challenging (see \cite{Benedetti:2017fmp,Prakash:2017hwq}), but it is worth persevering, as such models would have higher chance of displaying genuinely new physical behavior.
\section*{Acknowledgements}
\noindent We would like to thank Razvan Gurau, Timoth\'e Poulain, and Vincent Rivasseau for useful discussions.
\newpage
|
2,877,628,090,153 | arxiv | \section*{To-do list}
\section{Introduction}
Living cells take in information from their surroundings through myriad \emph{signal transduction} processes.
Signal transduction takes many forms: the input signal can be carried by changes in chemical concentration, electrical potential, light intensity, mechanical forces, and temperature, \textit{inter alia}. In many instances these \emph{extracellular} stimuli trigger \emph{intracellular} responses that can be represented as transitions among a discrete set of states \cite{hlavacek2006}.
Models of these processes are of great interest to mathematical and theoretical biologists \cite{janes2006}.
The ``transduction'' of the signal occurs through the physical effect of the input signal on the transition rates among the various states describing the receptor.
\pt{An early}
mathematical model of this type was the voltage-sensitive transitions among several open and closed ion channel states in Hodgkin and Huxley's model for the conduction of sodium and potassium ions through the membranes of electrically excitable cells \cite{hodgkin1990}.
Presently, many such models are known for signal transduction systems, such as: the detection of calcium concentration signals by the calmodulin protein \cite{faas11}, binding of the acetylcholine (ACh) neurotransmitter to its receptor protein \cite{col82}, and modulation of the channel opening transition by light intensity in the channelrhodopsin (ChR) protein \cite{nag03}. In each of these examples the channel may be modeled as a weighted, directed graph, in which the vertices represent the discrete channel states, and the weighted edges represent \textit{per capita} transition rates, some of which can be modulated by the input signals.
Mutual information, and Shannon capacity, arise in a variety of biological contexts. For example, mutual information may predict the differential growth rates of organisms learning about their environment \cite{tkacik2016}, based on the Kelly criterion \cite{kelly1956}. For biological communication systems, achieving a distortion criterion (expressed as mutual information) need not require complicated signal processing techniques; see \cite[Example 2]{gastpar2003}. Moreover, the free energy cost of molecular communication (such as in signal transduction) has a mathematical form similar to mutual information \cite{eckford2018}, leading to thermodynamic bounds on capacity per unit energy cost (cf. \cite{verdu1990}).
Stochastic modeling of signal transduction as a communication channel has considered the chemical reactions in terms of Markov chains \cite{thomas2003} and in terms of the ``noise'' inherent in the binding process \cite{pierobon2011}. For simplified two-state Markov models, Shannon capacity of signal transduction has been calculated for slowly changing inputs \cite{ein11} and for populations of communicating bacteria \cite{aminian2015}. Our own previous work has investigated the capacity of signal transduction: in \cite{ThomasEckford2016}, we obtained the Shannon capacity of two-state Markov signal transduction under arbitrary inputs, and showed that the capacity for multiple independent receptors has the same form \cite{thomas2016}. Related channel models have been studied in the information-theoretic literature, such as the unit output memory channel \cite{che05}, the ``previous output is the state'' (POST) channel \rev{\cite{AsnaniPermuterWeissman2013IEEE_ISIT,PermuterAsnaniWeissman2014IEEETransIT}}; capacity results for some channels in these classes were recently given in \cite{stavrou2016}.
The present paper \pt{focuses on}
the mutual information and capacity of finite-state signal transduction channels. Generalizing previous results, we provide discrete-time, finite-state channel models for a wide class of signal transduction receptors, giving Channelrhodopsin-2 (ChR2), Acetylcholine (ACh), and Calmodulin (CaM) as specific examples.
We also provide an explicit formula for the mutual information of this \pt{class of models}
under independent, identically distributed (IID) inputs \rev{(Theorem 1)}.
Subsequently, we consider
the continuous time limit
as the interval between the discrete-time instants goes to zero, and find a simple closed-form expression for the mutual information \rev{(Theorem 2)}, with a natural physical interpretation.
We further give conditions under which our formula gives the Shannon capacity of the channel, namely that there is exactly one transition in the Markov chain that is sensitive to the channel input \rev{(Theorem 3), and we use this result to (numerically) find the Shannon capacity of ChR.}
The remainder of the paper is organized as follows: in \rev{Section} II, we give a generalized model for discrete-time, finite-state signal transduction systems; in \rev{Section} III, we discuss signal transduction as a communication system, deriving expressions for the mutual information and giving our main results; and in \rev{Section} IV, we discuss the biological significance of the results, \pt{as well as the limitations of our analysis}.
\section{Model}
\label{sec:Model}
\subsection{Physical model}
Signal transduction encompasses a wide variety of physical processes.
For example, in a {\em ligand-gated} system, signals are transmitted using concentrations of signaling molecules, known as {\em ligands}, which bind to receptor proteins. As another example, in a {\em light-gated} system, signals are transmitted using light, where the receptor absorbs photons. Other possibilities exist, such as voltage-gated ion channels. The receptor, \pt{often}
located on the surface of the cell, forms the receiver in the signal transduction system, and conveys (or {\em transduces}) the signal across the cell membrane; the receptor is the focus of our analysis.
Signal transduction receptors share a mathematical model: they can be viewed as finite-state, intensity-modulated Markov chains, in which the transition rates between certain pairs of states are sensitive to the input (though other transitions may be independent of the input).
Our main examples in this paper focus on ligand- and light-gated receptors. For example, in a ligand-gated system, the binding of the ligand results in a change in the receptor, which then produces {\em second messengers} (normally a different species than the ligand) to convey the message to the cell interior. In a light-gated system, the incident photon causes a similar change in the receptor, which may open to allow an ion current to pass to the interior of the cell.
In either case, there may be a relaxation process which returns the receptor to the ``ready'' state, and this process may be independent of the signal; or other processes that are either sensitive to or independent of the signal, depending on the purpose of the receptor.
In the next two sections, we describe the Markov chain model for receptors, both in continuous and in discrete time.
Although we focus on ligand- and light-gated receptors, we emphasize that our framework is general enough to include other kinds of receptors.
\subsection{Continuous time: Master equation kinetics}
\label{sec:MasterEquation}
Receptors are finite-state Markov chains. For a receptor with $k$ discrete states,
there exists a $k$-dimensional vector of state occupancy probabilities $\mathbf{p}(t)$,
given by
\begin{equation}
\mathbf{p}(t) = \left[ p_1(t), \: p_2(t), \: \ldots, \:p_{k}(t) \right] ,
\end{equation}
where $p_i(t)$ represents the probability of a given receptor occupying state $i$ at time $t$. The environmental conditions at the receptor, such as light level or ligand concentration, are known as the {\em input} $x(t)$.
The chemical kinetics of the receptor are captured by a differential equation known
as the {\em master equation} \cite{Gardiner2004}. Let $Q = [q_{ij}\rev{(x)}]$ represent a $k \times k$ matrix of \rev{\textit{per capita} transition rates,}
where $q_{ij}\rev{(x)}$ represents the instantaneous rate at which receptors starting in state $i$
enter state $j$.
It is helpful to visualize the matrix $Q$ using a graph:
\begin{itemize}
\item There are $k$ vertices, representing the states; and
\item A directed edge is drawn from vertex $i$ to $j$ if and only if $q_{ij}\rev{(x)} > 0$ \rev{for some $x$}.
\end{itemize}
Changing from one state to another is called a {\em transition}, so the graph corresponding to $Q$ depicts the possible transitions. A transition $i \rightarrow j$ may be {\em sensitive}, i.e. $q_{ij}$ \rev{varies as} a function of the input $x(t)$, or {\em insensitive}, $q_{ij}$ is constant with respect to $x(t)$.
Using $Q$, the master equation is given by
\begin{equation}
\label{eqn:MasterEquation}
\frac{d\mathbf{p}(t)}{dt} = \mathbf{p}(t)Q\rev{(x(t))} .
\end{equation}
We use the notation from \cite{GroffDeRemigioSmith2009chapter}:
\begin{itemize}
\item States take a compound label, consisting of a state property and a state number. The state number is unique to each state, but the state property may be shared by multiple states. For example, in each state the receptor's ion channel might be either open $\O$ or closed $\mathsf{C}$; the state label $\mathsf{C}_1$ means that in state 1 the channel is closed, and $\O_2$ means that in state 2 the channel is open. \awe{In this paper we use the state {\em number} rather than the state {\em property}. (Since we show that the state numbers form a Markov chain, in general the state properties form a hidden Markov chain; we discuss this further in Section IV.)}
\item \rev{We assume that} rates which are sensitive to the input are directly proportional to the input $x(t)$. For example,
$q_{12}x(t)$ is the transition rate from $1 \rightarrow 2$,
which is sensitive, while
$q_{31}$ is the transition rate from $3 \rightarrow 1$, which is insensitive.
\item The $i$th diagonal element \rev{of $Q$} is written $R_i$, and is set so that the $i$th row sums to zero (so, if $x(t)$ appears in the $i$th row, $R_i$ may depend on $x(t)$).
\end{itemize}
Taking sensitive rates to be proportional to the signal $x(t)$ is a key modeling assumption; it is satisfied for the examples we consider, but there exist systems in which the signal acts nonlinearly on the rate.
The following three examples illustrate the use of our notation, and give practical examples of receptors along with their transition graphs and rate constants.
\begin{ex}[Channelrhodopsin-2]
The Channelrhodopsin-2 (ChR2) receptor is a light-gated ion channel.
The receptor has three states, named Closed ($\mathsf{C}_1$), Open ($\mathsf{O}_2$), and Desensitized ($\mathsf{C}_3$).
The channel-open ($\mathsf{O}$) state $\mathsf{O}_2$ is the only state in which the ion channel is open, passing an ion current.
The channel-closed ($\mathsf{C}$) states, $\mathsf{C}_1$ and $\mathsf{C}_3$, are distinct in that the receptor is light-sensitive in
state $\mathsf{C}_1$, and insensitive in state $\mathsf{C}_3$ \cite{nag03}. The rate matrix for ChR2 is
\begin{equation}
\label{eqn:ChR2-rate-matrix}
Q = \left[
\begin{array}{ccc}
R_1 & q_{12}x(t) & 0 \\
0 & R_2 & q_{23} \\
q_{31} & 0 & R_3
\end{array}
\right] .
\end{equation}
where $x(t) \in [0,1]$ is the relative intensity.
To keep the row sums equal to zero, we set $R_1 = - q_{12}x(t)$,
$R_2 = - q_{23}$, and
$R_3 = - q_{31}$.
Fig.~\ref{fig:ChR2} shows
state labels and allowed state transitions.
\begin{figure}
\begin{center}
\includegraphics[width=2.5in]{figures/ChR2}
\end{center}
\caption{\label{fig:ChR2} Depiction of allowed state transitions for ChR2. Sensitive
transitions are depicted with {\bfseries bold} arrows.
States are labelled by their ion channel state: $\{\mathsf{C},\mathsf{O}\}$ for closed and open, respectively; state number is in subscript. Dashed lines surround all states in either the closed or open state. Transition rates, listed in Table \ref{tab:ChR2parameters}, correspond to the vertices associated with each directed edge: for example,
the rate from state $\mathsf{O}_2$ to state $\mathsf{C}_3$ is $q_{23}$.}
\end{figure}
Parameter values from the literature are given in Table \ref{tab:ChR2parameters}.
\begin{table}[h!]\begin{center}
\begin{tabular}{c|c|c} \hline
Parameter & from \cite{nag03} & Units \\ \hline
$q_{12}x(t)$ & $(5 \times 10^3) x(t)$ & s$^{-1}$ \\ \hline
$q_{23}$ & 50 & s$^{-1}$ \\ \hline
$q_{31}$ & 17 & s$^{-1}$ \\ \hline
\end{tabular}
\ \\
\ \\
\caption{\label{tab:ChR2parameters} Rate parameters for ChR2, adapted from \cite{nag03}, where $x(t) \in [0,1]$ represents the relative light intensity.}
\end{center}
\end{table}
\end{ex}
\begin{ex}[Acetylcholine]
The Acetylcholine (ACh) receptor is a ligand-gated ion channel. \pt{Following \cite{col82}, we model the receptor as a conditional Markov process on} five states, with rate matrix
\begin{equation}
\label{eqn:AChRateMatrix}
Q =
\left[
\begin{array}{ccccc}
R_1 & q_{12}x(t) & 0 & q_{14} & 0 \\
q_{21} & R_2 & q_{23} & 0 & 0 \\
0 & q_{32} & R_3 & q_{34} & 0 \\
q_{41} & 0 & q_{43}x(t) & R_4 & q_{45} \\
0 & 0 & 0 & q_{54}x(t) & R_5
\end{array}
\right] .
\end{equation}
There are three sensitive transitions: $\rev{q_{12}}x(t)$, $\rev{q_{43}}x(t)$, and $\rev{q_{54}}x(t)$, which
are proportional to ligand concentration $x(t)$.
For the purposes of our analysis, we use a range of $x(t) \in [10^{-7},10^{-5}]$.
Fig.~\ref{fig:ACh} shows the
allowed state transitions.
The states in ACh correspond to the binding of a ligand to one of two binding sites on the receptor. In state $\mathsf{C}_5$, neither site is occupied; in states $\mathsf{C}_4$ and $\mathsf{O}_1$, one site is occupied; and in states $\mathsf{C}_3$ and $\mathsf{O}_2$, both sites are occupied.
Table \ref{tab:AChparameters} gives
parameter values; the concentration of ACh, $x(t)$, is measured in mol/$\ell$.
The same state-naming convention is used in the figure as with ChR2: states with
an open ion channel are $\mathsf{O}_{1}$ and $\mathsf{O}_2$; states
with a closed ion channel are $\mathsf{C}_3$, $\mathsf{C}_4$, and $\mathsf{C}_5$.
\begin{table}\begin{center}
\begin{tabular}{c|c|c|c} \hline
Parameter & Name in \cite{col82} & Value/range & Units \\ \hline
$q_{12}x(t)$ & $k_{+2}x$ & $(5 \times 10^8) x(t)$ & s$^{-1}$ \\ \hline
$q_{14}$ & $\alpha_1$ & $3 \times 10^3$ & s$^{-1}$ \\ \hline
$q_{21}$ & $2 k_{-2}^*$ & $ 0.66 $ & s$^{-1}$ \\ \hline
$q_{23}$ & $\alpha_2$ & $5 \times 10^2$ & s$^{-1}$ \\ \hline
$q_{32}$ & $\beta_2$ & $1.5 \times 10^4$ & s$^{-1}$ \\ \hline
$q_{34}$ & $2 k_{-2}$ & $ 4 \times 10^3$ & s$^{-1}$ \\ \hline
$q_{41}$ & $\beta_1$ & 15 & s$^{-1}$ \\ \hline
$q_{43}x(t)$ & $k_{+2}x$ & $(5 \times 10^8) x(t)$ & s$^{-1}$ \\ \hline
$q_{45}$ & $k_{-1}$ & $ 2 \times 10^3$ & s$^{-1}$ \\ \hline
$q_{54}x(t)$ & $2 k_{+1} x$ & $(1 \times 10^8) x(t)$ & s$^{-1}$ \\ \hline
\end{tabular}
\end{center}
\ \\
\caption{\label{tab:AChparameters}Rate parameters for ACh, adapted from \cite{col82}, where $x(t)$ represents the molar concentration of ACh in mol/$\ell$. Here we use a range of $x(t) \in [10^{-7},10^{-5}]$.}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=3in]{figures/ACh_rotated}
\end{center}
\caption{\label{fig:ACh} Depiction of allowed state transitions for ACh. Sensitive
transitions are depicted with {\bfseries bold} arrows.
States are labelled by their ion channel state: $\{\mathsf{C},\mathsf{O}\}$ for closed and open, respectively; state number is in subscript. Dashed lines surround all states in either the closed or open state. Transition rates, listed in Table \ref{tab:AChparameters}, correspond to the vertices associated with each directed edge: for example,
the rate from state $\mathsf{O}_2$ to state $\mathsf{C}_3$ is $q_{23}$.}
\end{figure}
\end{ex}
\begin{ex}[Calmodulin]
The Calmodulin (CaM) receptor is a ligand-gated receptor.
The CaM receptor consists of \pt{four} binding sites, \pt{two on the C-terminus of the CaM protein and two on the N-terminus \cite{ChinMeans2000TrendsCellBiol,DeMariaEtAlYue2001Nature,KellerFranksBartolSejnowski2008PLoSOne}.}
Each \pt{end of the protein} can bind 0, 1, or 2 \pt{calcium ions, leading to} nine possible states.
For CaM, rather than an ion channel, it is important whether the $\mathsf{C}$ or $\mathsf{N}$ end of the receptor is completely bound (i.e., has both binding sites occupied by ligands). This property is represented by four symbols: $\emptyset$ if neither end is completely bound; $\mathsf{C}$ if the $\mathsf{C}$ end is completely bound; $\mathsf{N}$ if the $\mathsf{N}$ end is completely bound; and $\mathsf{N}\mathsf{C}$ if both ends are completely bound.
\begin{figure*}[!t]
\normalsize
\begin{align}
\label{eqn:CaMQ}
Q &= \left[
\begin{array}{ccccccccc}
R_0 & q_{01}x(t) & 0 & q_{03}x(t) & 0 & 0 & 0 & 0 & 0 \\
q_{10} & R_1 & q_{12}x(t) & 0 &q_{14}x(t) & 0 & 0 & 0 & 0\\
0 & q_{21} & R_2 & 0 & 0 & q_{25}x(t) & 0 & 0 & 0 \\
q_{30} & 0 & 0 & R_3 & q_{34}x(t) & 0 & q_{36}x(t) & 0 & 0 \\
0 & q_{41} & 0 & q_{43} & R_4 & q_{45}x(t) & 0 & q_{47}x(t) & 0 \\
0 & 0 & q_{52} & 0 & q_{54} & R_5 & 0 & 0 & q_{58}x(t) \\
0 & 0 & 0 & q_{63} & 0 & 0 & R_6 & q_{67}x(t) &0 \\
0 & 0 & 0 & 0 & q_{74} & 0 & q_{76} & R_7 & q_{78}x(t) \\
0 & 0 & 0 & 0 & 0 & q_{85} & 0 & q_{87} & R_8
\end{array}
\right]
\end{align}
\hrulefill
\vspace*{4pt}
\end{figure*}
State configuration and allowed transitions are depicted in Figure \ref{fig:CaM}. The rate matrix is given in (\ref{eqn:CaMQ}),
with values given in Table \ref{tab:CaMparameters}, and where the molar concentration of calcium is $x(t) \in [10^{-7},10^{-6}]$.
\begin{table}\begin{center}
\begin{tabular}{c|c|c|c} \hline
Parameter & Name in \cite{faas11} & Value/range & Units \\ \hline
$q_{01}x(t)$, $q_{34}x(t)$, $q_{67}x(t)$
& $k_{\mathrm{on (T), N}} $
& $(7.7 \times 10^8) x(t)$ & s$^{-1}$ \\ \hline
$q_{10}$, $q_{43}$, $q_{76}$
& $k_{\mathrm{off (T), N}}$
& $1.6 \times 10^5$ & s$^{-1}$ \\ \hline
$q_{12}x(t)$, $q_{45}x(t)$, $q_{78}x(t)$
& $k_{\mathrm{on (R), N}}$
& $(3.2 \times 10^{10}) x(t)$ & s$^{-1}$ \\ \hline
$q_{21}$, $q_{54}$, $q_{87}$
& $k_{\mathrm{off (R), N}}$
& $2.2 \times 10^4$ & s$^{-1}$ \\ \hline
$q_{03}x(t)$, $q_{14}x(t)$, $q_{25}x(t)$
& $k_{\mathrm{on (T), C}}$
& $(8.4 \times 10^7) x(t)$ & s$^{-1}$ \\ \hline
$q_{30}$, $q_{41}$, $q_{52}$
& $k_{\mathrm{off (T), C}}$
& $2.6 \times 10^3$ & s$^{-1}$ \\ \hline
$q_{36}x(t)$, $q_{47}x(t)$, $q_{58}x(t)$
& $k_{\mathrm{on (R), C}}$
& $(2.5 \times 10^7) x(t)$ & s$^{-1}$ \\ \hline
$q_{63}$, $q_{74}$, $q_{85}$
& $k_{\mathrm{off (R), C}}$
& 6.5 & s$^{-1}$ \\ \hline
\end{tabular}
\end{center}
\ \\
\caption{\label{tab:CaMparameters}Rate parameters for CaM, adapted from \cite{faas11}, where $x(t) \in [10^{-7},10^{-6}]$ represents the molar concentration of calcium in mol/$\ell$.}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=3.25in]{figures/CaM}
\end{center}
\caption{\label{fig:CaM} Depiction of allowed state transitions for CaM. Sensitive
transitions are depicted with {\bfseries bold} arrows.
States are labelled by the status of the $\mathsf{C}$ or $\mathsf{N}$ end of the receptor: $\emptyset$ if neither end is completely bound; $\mathsf{C}$ if the $\mathsf{C}$ end is completely bound; $\mathsf{N}$ if the $\mathsf{N}$ end is completely bound; and $\mathsf{N}\mathsf{C}$ if both ends are completely bound. Transition rates, listed in Table \ref{tab:CaMparameters}, correspond to the vertices associated with each directed edge.}
\end{figure}
\end{ex}
For each of the preceding examples, the rate constants depend on environmental conditions, and
thus can be reported differently in different sources
(see, e.g., \cite{lin09} for different rate constants for ChR2).
\subsection{From the master equation to discrete-time Markov chains}
\rev{The continuous-time master equation for the receptor dynamics \eqref{eqn:MasterEquation} describes the evolution of a conditional probability $\mathbf{p}(t)\equiv E[Y(t)\:|\:\mathcal{F}_X(t)],$ where $Y(t)$ is the continuous time, discrete state \textit{c\`{a}dl\`{a}g}
process giving the channel state, $\mathcal{F}_X(t)$ is the filtration generated by the input process $X(t)$, and $E[\cdot\:|\:\cdot]$ is conditional expectation \cite{grimmett2001probability}.
Establishing the appropriate ensemble of input processes and analyzing mutual information and capacity involve technical issues that do not shed light on the nature of biological signal transduction.
Therefore we do not undertake a rigorous analysis of the continuous-time communications channels described by \eqref{eqn:MasterEquation} in this paper.
Rather, we introduce a discrete-time, discrete-state channel, motivated by the continuous-time channel, which can be rigorously analyzed, and study its properties both with a fixed timestep $\Delta t$, and later in the limit $\Delta t\to 0$.
The discrete-time Markov chain model allows us to rely on capacity results for discrete-time Markov channels.}
\rev{We obtain a discrete-time approximation to} the master equation by writing
\begin{align}
\label{eqn:Markov-1}
\frac{d\mathbf{p}(t)}{dt} = \mathbf{p}(t) Q
= \frac{\mathbf{p}(t + \Delta t) - \mathbf{p}(t)}{\Delta t} +o(\Delta t),\text{ as }\Delta t\to 0,
\end{align}
\rev{
where we simplify the notation by writing $Q(x(t))$ as simply $Q$.
Manipulating the middle and right expression in (\ref{eqn:Markov-1}) \pt{gives} }
\rev{\begin{align}
\mathbf{p}(t + \Delta t) &= \Delta t \,\mathbf{p}(t) Q + \mathbf{p}(t) +o(\Delta t)\\
&= \Delta t\, \mathbf{p}(t) Q + \mathbf{p}(t) I+o(\Delta t) \\
&= \mathbf{p}(t) \left( I + \Delta t Q \right)+o(\Delta t),\text{ as }\Delta t\to 0,
\end{align}}
where $I$ is the identity matrix.
\rev{In order to arrive at a discrete-time model, we introduce the approximation $\{\mathbf{p}_i\}_{i\in\mathbb{N}_+}$ satisfying}
\begin{equation}
\mathbf{p}_i = \mathbf{p}(i \Delta t)+o(\Delta t),\text{ as }\Delta t\to 0,
\end{equation}
\rev{and} arrive at a discrete-time approximation to (\ref{eqn:Markov-1}),
\begin{equation}
\mathbf{p}_{i+1} = \mathbf{p}_i (I+\Delta t\, Q).
\end{equation}
Thus, we have a discrete-time Markov chain with transition probability matrix
\begin{equation}
\label{eqn:Markov-last}
P = I + \Delta t Q.
\end{equation}
The matrix $P$ satisfies the conditions of a Markov chain transition
probability matrix (nonnegative, row-stochastic) as long as $\Delta t$ is small enough.
\rev{However, note that $P$ (and $Q$) are dependent on $x(t)$, so the Markov chain is not generally time-homogeneous if $x(t)$ is known (cf. (\ref{eqn:HomogeneousP})).}
\section{Signal transduction as a communications system}
In this section we give our main results, in which we describe and analyze signal transduction as a communication system. A brief roadmap to our results is given as follows: we first define the communication system in terms of input, output, and channel; we give the mutual information of the general discrete-time model under IID inputs (Theorem 1 and equation (\ref{eqn:GeneralIID1})); we take the continuous-time limit of the mutual information rate, showing that the expression for mutual information has a simple factorization (Theorem 2 and equation (\ref{eqn:Theorem1})); we give a physical interpretation of the factorization in (\ref{eqn:Theorem1}); we give general conditions under which the Shannon capacity is satisfied by IID inputs (Theorem 3); and finally, we give an example calculation using ChR2 (Example 4).
\subsection{Communication model of receptors}
We now discuss how the receptors can be described as
information-theoretic communication systems: that is, in terms of input, output, and conditional
input-output PMF.
{\em Input:} As discussed in Section II, the receptor is sensitive to given properties of the environment; previous examples included light intensity or ligand concentration. The receptor input $x(t)$ is the value of this property at the surface of the receptor.
The input is discretized in time: for integers $i$, the input is $x(i \Delta t)$; we will write
$x_i = x(i \Delta t)$.
We will also discretize the amplitude,
so that for every $t$, $x_i \in \{\mathsf{x}_1,\mathsf{x}_2,\mathsf{x}_3,\ldots,\mathsf{x}_k\} =: \mathcal{X}$.
We will assume that the $\mathsf{x}_i$ are
distinct and increasing; further, we assign the lowest and highest values special symbols:
\begin{align}
\xlevel_\mathsf{L} &:= \mathsf{x}_1 \\
\xlevel_\mathsf{H} &:= \mathsf{x}_k .
\end{align}
In Section II, we gave the concentrations or intensities over a range of values (such as $x(t) \in [0,1]$ for ChR2). Thus, we select $\xlevel_\mathsf{L}$ and $\xlevel_\mathsf{H}$ as the minimum and maximum values of this range, respectively.
{\em Output:}
In this paper, the output $y(t)$ of the communication system is
the receptor state number, given by the {\em subscript} of the state label: for example, if the state is $\mathsf{C}_3$, then $y(t) = 3$.
This is discretized to $y_i = y(i\Delta t)$. The discrete channel inputs and outputs form vectors: in terms of notation, we write $\mathbf{x} = [x_1,x_2,\ldots,x_n]$ and $\mathbf{y} = [y_1,y_2,\ldots,y_n]$.
{\em Conditional input-output PMF:}
From (\ref{eqn:Markov-1})--(\ref{eqn:Markov-last}), $\mathbf{y}$ forms a
Markov chain given $\mathbf{x}$, so
\begin{equation}
\label{eqn:ReceptorMarkov}
p(\mathbf{y}|\mathbf{x}) = \prod_{i=1}^n p(y_i \:|\: x_i,y_{i-1}) ,
\end{equation}
where $p(y_i \:|\: x_i,y_{i-1})$ is given by the appropriate entry in the matrix $P$, and where $y_0$ is null.%
\footnote{\rev{Notation: (1) We will drop subscripts if it is unambiguous to do so, i.e., normally $p(x)$ signifies $p_X(x)$; (2)} We say a variable is ``null'' if it vanishes under conditioning, i.e., if $y_0$ is null, then $p(y_1 \:|\: x_1, y_0) = p(y_1 \:|\: x_1)$.}
The following diagram \eqref{diag:xy} indicates the conditional dependencies:
\begin{equation}\label{diag:xy}
\begin{array}{ccccccccccc}
&X_1&&X_2&&X_3&&X_4&&X_5&\cdots\\
&\downarrow&&\downarrow&&\downarrow&&\downarrow&& \downarrow \\
(Y_0)&\longrightarrow&Y_1&\longrightarrow&Y_2&\longrightarrow&Y_3&\longrightarrow&Y_4&\cdots
\end{array}\end{equation}
As an example, consider ACh: suppose $y_{i-1} = 1$, $y_i = 2$, and $x_i = \xlevel_\mathsf{H}$. Then from
(\ref{eqn:Markov-last}) and Table \ref{tab:AChparameters}, we have
$p_{Y_i \:|\: Y_{i-1},X_i}(2 \:|\: 1,\xlevel_\mathsf{H}) = \Delta t q_{12}(t) = (5 \times 10^8) \xlevel_\mathsf{H} \Delta t$.
From (\ref{diag:xy}) and the definition of
$P$, $p(y_i \:|\: y_{i-1},x_i)$ does not depend on $i$;
that is, the channel's input-output structure is time-invariant.
For a discrete-time Markov chain, the receptor states form a graph with vertex set $\mathcal{Y}$ and directed edges $\mathcal{E}\subset\mathcal{Y}\times\mathcal{Y}$, with pair $(y_{i-1},y_i)\in\mathcal{E}$ if $\max_{x_i\in\mathcal{X}} p(y_i\:|\: x_i,y_{i-1}) > 0$, that is, for at least some input value there is a direct transition from $y_{i-1}$ to $y_i$. Notice that, under this definition, self-transitions are {\em included} in $\mathcal{E}$, even though (for convenience) they are not depicted in the state-transition diagrams.
We say the transition from state $y_{i-1}$ to $y_i$ is \emph{insensitive to the input}, or just \emph{insensitive}, if, for all $x_i \in \mathcal{X}$, we have $p(y_i \:|\: x_i,y_{i-1})=p(y_i \:|\: y_{i-1})$
(see Section \ref{sec:MasterEquation}).
Otherwise, the transition is \emph{sensitive}. We let $\mathcal{S}\subseteq\mathcal{E}$ denote the subset of sensitive edges.
(If state $y_{i-1} \in \mathcal{Y}$ is the origin for a sensitive transition, i.e., there is at least one $(y_{i-1},y_i \neq y_{i-1}) \in \mathcal{S}$, then the self-transition $(y_{i-1},y_i = y_{i-1})$ is normally sensitive as well, but this condition is not required for our analysis.)
For a channel with inputs $\mathbf{x}$ and outputs $\mathbf{y}$ \rev{(both of length $n$)}
the mutual information $I(\mathbf{X};\mathbf{Y})$ gives the maximum information rate that may be
transmitted reliably over the channel for a given input distribution.
Mutual information is given by
\begin{align}
I(\mathbf{X};\mathbf{Y})
\label{eqn:MutualInformation}
&= \sum_{\mathbf{x},\mathbf{y}} p(\mathbf{x}) p(\mathbf{y}\:|\:\mathbf{x})
\log \frac{p(\mathbf{y}\:|\: \mathbf{x})}{p(\mathbf{y})} ,
\end{align}
where $p(\mathbf{y}\:|\: \mathbf{x})$ is the conditional probability mass function (PMF) of $\mathbf{Y}$.
As $n \rightarrow \infty$, generally $I(\mathbf{X};\mathbf{Y}) \rightarrow \infty$ as well;
in this case, it is more useful to calculate the mutual information rate, \rev{which we introduce in the next section.}
\subsection{Receptor IID capacity}
\rev{Our focus in the remainder of this paper is on IID input distributions.} Although
IID inputs \rev{may not be} realistic for chemical diffusion channels, such as for ligand-gated receptors (as concentration
may persist for long periods of time), they can be capacity-achieving in these channels \rev{(see, e.g., \cite{ThomasEckford2016})}; moreover, IID input distributions may be physically realistic for light-gated channels.
\rev{Starting with (\ref{eqn:MutualInformation}), where $\mathbf{x}$ and $\mathbf{y}$ are both of fixed and finite length $n$, the Shannon capacity $C(n)$ is found by maximizing $I(\mathbf{X};\mathbf{Y})$ with respect to the input distribution $p(\mathbf{x})$, i.e.,
\begin{align}
\label{eqn:CapacityDefinition}
C(n) = \max_{p(\mathbf{x})} I(\mathbf{X};\mathbf{Y}) .
\end{align}
where the limit is taken over all possible length-$n$ input distributions $p(\mathbf{x})$ (not necessarily IID).}
\rev{
If the input $p(\mathbf{x})$ is restricted to the set of IID input distributions,
which is well defined for each $n$ (i.e., $p(\mathbf{x}) = \prod_{i=1}^n p(x_i)$),
then $I(\mathbf{X};\mathbf{Y})$ is also well defined for each $n$ (see (\ref{eqn:MutualInformation})). Furthermore, for each $n$
we have the IID capacity, written $C_{\mathrm{iid}}(n)$:
\begin{equation}
\label{eqn:IIDCapacityDefinition}
C_{\mathrm{iid}}(n) = \max_{p(x_i)} I(\mathbf{X};\mathbf{Y}) .
\end{equation}
where the maximum is taken over all possible settings of $p(x_i)$.
}
\rev{We can use (\ref{eqn:MutualInformation}) and (\ref{eqn:IIDCapacityDefinition}) to obtain information rates per channel use. For a given IID input distribution $p(x)$,
the IID mutual information rate is given by
\begin{align}
\label{eqn:InfoRateDefinition}
\mathcal{I}(X;Y) = \lim_{n \rightarrow \infty} \frac{1}{n} I(\mathbf{X};\mathbf{Y}).
\end{align}
Furthermore, the maximum IID information rate is given by
\begin{align}
\label{eqn:ciid}
C_{\mathrm{iid}} := \lim_{n \rightarrow \infty} \frac{1}{n} C_{\mathrm{iid}}(n) .
\end{align}
We derive these quantities in the remainer of the section, in which it will be clear that the limits in (\ref{eqn:InfoRateDefinition})---(\ref{eqn:ciid}) exist.
We start by deriving $I(\mathbf{X};\mathbf{Y})$ under IID inputs, and showing how it is calculated using quantities introduced in Section II. Finally, in Theorem \ref{thm:MutualInformationRate}, we give an expression for $\mathcal{I}(X;Y)$, and show that $\mathcal{I}(X;Y) = C_{\mathrm{iid}}$.
}
Recall $p(\mathbf{y}\:|\:\mathbf{x})$ from (\ref{eqn:ReceptorMarkov}).
Under IID inputs, it can be shown (see \cite{che05,ThomasEckford2016}) that the receptor states $Y^n$ form a time-homogeneous Markov
chain, that is,
\begin{equation}
\label{eqn:Conditional-2}
p(\mathbf{y}) = \prod_{i=1}^n p(y_i \:|\: y_{i-1}),
\end{equation}
where $y_0$ is again null, and where
\begin{equation}
\label{eqn:Conditional-2a}
p(y_i \:|\: y_{i-1}) = \sum_{x_i} p(y_i \:|\: x_i, y_{i-1}) p(x) .
\end{equation}
\rev{
Furthermore, let $\bar{P}$ represent the transition probability matrix of $Y^n$.
Recall (\ref{eqn:Markov-last}), in which $P$ was dependent on $x$; using (\ref{eqn:Conditional-2a}), we can write
\begin{align}
\label{eqn:HomogeneousP}
\bar{P} = E[P] = I + \Delta t E[Q],
\end{align}
and since the sensitive terms in $P$ and $Q$ are assumed to be linear in $x(t)$, we replace $x(t)$ in these terms with $E[x]$ to form $\bar{P}$ and $\bar{Q} := E[Q]$, respectively.
}
Using (\ref{eqn:ReceptorMarkov}) \rev{and} (\ref{eqn:Conditional-2}), (\ref{eqn:MutualInformation})
reduces to
\begin{equation}
\label{eqn:MutualInfoMarkov}
I(\mathbf{X};\mathbf{Y}) = \sum_{i=1}^n \sum_{y_i} \sum_{y_{i-1}} \sum_{x_i} p(y_i,x_i,y_{i-1}) \log \frac{p(y_i \:|\: x_i,y_{i-1})}
{ p(y_i \:|\: y_{i-1})} .
\end{equation}
\rev{
Recall that a transition may be sensitive ($(y_{i-1},y_i) \in \mathcal{S}$) or insensitive ($(y_{i-1},y_i) \not\in \mathcal{S}$). For terms in (\ref{eqn:MutualInfoMarkov}), consider the
insensitive transitions:
\begin{align}
\nonumber
\lefteqn{p(y_i,x_i,y_{i-1}) \log \frac{p(y_i \:|\: x_i,y_{i-1})}
{ p(y_i \:|\: y_{i-1})}}&\\
\label{eqn:Insensitive1}
&= p(y_i,x_i,y_{i-1}) \log \frac{p(y_i \:|\: y_{i-1})}
{ p(y_i \:|\: y_{i-1})} \\
&= p(y_i,x_i,y_{i-1}) \log 1\\
&= 0.
\end{align}
where (\ref{eqn:Insensitive1}) follows since the transition is insensitive, and is not a function of $x_i$; cf. (\ref{eqn:Conditional-2a}).} Thus for IID inputs, the mutual information (\ref{eqn:MutualInfoMarkov}) is calculated using the {\em sensitive transitions only}, i.e., those transitions in $\mathcal{S}$. \rev{With this in mind, we can rewrite (\ref{eqn:MutualInfoMarkov}) as
\begin{align}
\nonumber \lefteqn{I(\mathbf{X};\mathbf{Y}) }&\\
\label{eqn:MutualInfoMarkovSensitive0}
&= \sum_{i=1}^n \sum_{(y_{i-1},y_i)\in\mathcal{S}} \sum_{x_i} p(y_i,x_i,y_{i-1}) \log \frac{p(y_i \:|\: x_i,y_{i-1})}
{ p(y_i \:|\: y_{i-1})} \\
\label{eqn:MutualInfoMarkovSensitive}
&= \sum_{i=1}^n \sum_{\mathcal{A}_i}
p(y_i \:|\: x_i, y_{i-1}) p(y_{i-1}) p(x_i)
\log \frac{p(y_i \:|\: x_i,y_{i-1})}
{ p(y_i \:|\: y_{i-1})}
\end{align}
where we let $\mathcal{A}_i = \{y_i,y_{i-1},x_i : (y_i,y_{i-1})\in\mathcal{S}^2, x_i \in \mathcal{X}\}$, i.e.
the same terms as the sum in (\ref{eqn:MutualInfoMarkovSensitive}), for the sake of brevity.
Also note that (\ref{eqn:MutualInfoMarkovSensitive}) follows from (\ref{eqn:MutualInfoMarkovSensitive0}) because the input $\mathbf{X}$ is IID.
}
\rev{Now consider the individual PMFs in (\ref{eqn:MutualInfoMarkovSensitive}), starting with $p(y_i \:|\: x_i, y_{i-1})$.}
All transitions in $\mathcal{S}$ are dependent on the input $x_i$, and throughout this paper we assume that the
sensitive transition rates depend linearly on the \rev{input signal intensity}. Thus \rev{(recall (\ref{eqn:Markov-last}))}
for non-self-transitions $(y_{i-1},y_i) \in \mathcal{S}$ (i.e., $y_{i-1} \neq y_i$),
\begin{equation}
\label{eqn:TransitionsInS}
p(y_i \:|\: x_i,y_{i-1}) = q_{y_{i-1}y_i} x_i \Delta t .
\end{equation}
For self-transitions in $\mathcal{S}$ (i.e., $y_{i-1}\equiv y_i=y$) we have
\begin{align}
\nonumber\lefteqn{p_{Y_i|X_i,Y_{i-1}}(y \:|\: x_i,y) =}&\\
\label{eqn:SelfTransition}
& 1-\left(\sum_{y'\not=y,(y',y)\in\mathcal{S}}q_{yy'}x_i - \sum_{y'\not=y,(y',y)\not\in\mathcal{S}}q_{yy'}\right)\Delta t ,
\end{align}
as seen in the diagonal entries of (\ref{eqn:Markov-last}).
\rev{Similarly, the terms $p(y_i \:|\: y_{i-1})$ can be obtained
using (\ref{eqn:Conditional-2a})--(\ref{eqn:HomogeneousP}); we replace $x_i$
in (\ref{eqn:TransitionsInS})--(\ref{eqn:SelfTransition}) with $\bar{x}$.}
\rev{The terms $p(y_{i-1})$ represent the steady-state marginal probability that the receptor is in state $y$; for compact notation, let $\pi_{y_{i-1}} = p(y_{i-1})$.}
If the input $x$ is IID, as we assume throughout this paper, then $\pi_{y_{i-1}}$ exists if the Markov chain is irreducible, aperiodic, and positive recurrent; these conditions hold for all the examples we consider \rev{(recall (\ref{eqn:Conditional-2})--(\ref{eqn:HomogeneousP})).
\footnote{
\rev{For clarity, although $\pi_y$ may be written with a time-indexing subscript, e.g. $\pi_{y_i}$, this refers to the steady-state distribution of state $y_i \in \mathcal{Y}$, and does not imply that $\pi_{y}$ changes with time.}
}
Define the partial entropy function
\begin{equation}
\label{eqn:PhiDefinition}
\phi(p)=
\begin{cases}
0,&p=0\\
p\log p,&p\not=0
\end{cases}
\end{equation}
and let
\begin{equation}
\label{eqn:BinEnt}
\mathscr{H}(p) = -\phi(p) - \phi(1-p)
\end{equation}
represent the binary entropy function.
\rev{Then we have the following result.}
\rev{
\begin{theorem}
\label{thm:MutualInformationRate}
For an IID input distribution $p(x_i)$, the mutual information rate $\mathcal{I}(X;Y)$ is given by
\begin{align}
\nonumber \lefteqn{\mathcal{I}(X;Y)} & \\
\nonumber
&= \sum_{(y_{i-1},y_i) \in \mathcal{S}} \pi_{y_{i-1}}
\Bigg(
\sum_{x_i \in \mathcal{X}} p(x_i) \phi\Big(p(y_i \:|\: x_i,y_{i-1})\Big)\\
\label{eqn:GeneralIID1}
&\quad\quad- \phi\left(\sum_{x_i \in \mathcal{X}} p(x_i) p(y_i \:|\: x_i,y_{i-1}) \right)
\Bigg).
\end{align}
Furthermore, $C_{\mathrm{iid}} = \max_{p(x)} \mathcal{I}(X;Y)$.
\end{theorem}
\begin{proof}
Divide the terms in (\ref{eqn:MutualInfoMarkovSensitive}) into the $i=1$ term, and all the remaining terms.
Let $T_1(p(x_i))$ represent the $i=1$ term, emphasizing its dependence on the IID input distribution $p(x_i)$, so that
%
\begin{align}
T_1(p(x_i)) &= p(y_1 \:|\: x_1, y_0) p(y_1) p(x_1)
\log \frac{p(y_1 \:|\: x_1,y_0)}
{ p(y_1 \:|\: y_0)}\\
\label{eqn:T1}
&=p(y_1 \:|\: x_1) p(y_1) p(x_1)
\log \frac{p(y_1 \:|\: x_1)}
{ p(y_1)} ,
\end{align}
%
where (\ref{eqn:T1}) follows since $y_0$ is null. Let $T_2(p(x_i),n)$ represent the remaining terms, again dependent on $p(x_i)$ but also on $n$, so that
%
\begin{align}
\nonumber \lefteqn{T_2(p(x_i),n)}&\\
&= \sum_{i=2}^n \sum_{\mathcal{A}_i}
p(y_i \:|\: x_i, y_{i-1}) p(y_{i-1}) p(x_i)
\log \frac{p(y_i \:|\: x_i,y_{i-1})}
{ p(y_i \:|\: y_{i-1})}\\
\label{eqn:ProofMIRate0}
&= (n-1) \sum_{\mathcal{A}_i}
p(y_i \:|\: x_i, y_{i-1}) p(y_{i-1}) p(x_i)
\log \frac{p(y_i \:|\: x_i,y_{i-1})}
{ p(y_i \:|\: y_{i-1})}
\end{align}
%
recalling the definition of $\mathcal{A}_i$ from the discussion after (\ref{eqn:MutualInfoMarkovSensitive}).
Using (\ref{eqn:InfoRateDefinition}),
%
\begin{align}
\nonumber\lefteqn{\mathcal{I}(X;Y)}&\\
&=
\lim_{n \rightarrow \infty} \frac{T_1(p(x_i))}{n} + \lim_{n \rightarrow \infty} \frac{T_2(p(x_i),n)}{n} \\
\label{eqn:ProofMIRate1}
&= \sum_{\mathcal{A}_i}
p(y_i \:|\: x_i, y_{i-1}) p(y_{i-1}) p(x_i)
\log \frac{p(y_i \:|\: x_i,y_{i-1})}
{ p(y_i \:|\: y_{i-1})} ,
\end{align}
%
and (\ref{eqn:GeneralIID1}) follows after some manipulation.
To show that $C_{\mathrm{iid}} = \max_{p_X(x)} \mathcal{I}(X;Y)$,
recall the definitions of $C_{\mathrm{iid}}(n)$ and $C_{\mathrm{iid}}$ in (\ref{eqn:IIDCapacityDefinition}) and (\ref{eqn:ciid}), respectively.
Referring to $p(x_i)$ as $p$ for brevity,
%
\begin{align}
C_{\mathrm{iid}}(n) = \max_{p} \big(T_1(p) + T_2(p,n)\big) .
\end{align}
%
Let $p_1$ represent the IID input distribution maximizing the term $T_1(p)$, and let $p_2$ represent the IID input distribution maximizing the term $T_2(p,n)$. From (\ref{eqn:ProofMIRate0}), $p_2$ is independent of $n$. Furthermore,
%
\begin{align}
\label{eqn:ProofMIRate2}
\frac{T_1(p_2)}{n} + \frac{T_2(p_2,n)}{n} \leq \frac{1}{n}C_{\mathrm{iid}}(n) \leq \frac{T_1(p_1)}{n} + \frac{T_2(p_2,n)}{n} .
\end{align}
%
Taking the limit throughout (\ref{eqn:ProofMIRate2}) as $n \rightarrow \infty$, the $T_1$ terms vanish as they are constant with respect to $n$. Comparing (\ref{eqn:ProofMIRate0}) and (\ref{eqn:ProofMIRate1}), $p_2$ also maximizes $\mathcal{I}(X;Y)$. The result follows.
\end{proof}
}
\subsection{Limit of $\mathcal{I}(X;Y)/\Delta t$ as $\Delta t \rightarrow 0$}
In this section we consider the {\em continuous time limit} of $\mathcal{I}(X;Y) / \Delta t$ as $\Delta t \rightarrow 0$, and give \rev{our second main result} (Theorem \ref{thm:Theorem1}): that in the continuous time limit, the mutual information rate is expressed simply as a product of the average flux through sensitive edges, and the relative entropy between the prior distribution on $x$, and the posterior given a transition. \rev{While we do not claim to derive the mutual information rate of the continuous time channel, the continuous time limit of the discrete-time mutual information rate is a quantity of interest in its own right.}
\rev{
First, we show that the steady-state distribution $\pi_y$ is independent of $\Delta t$:}
\rev{
\begin{lemma}
\label{lem:SteadyState}
Suppose $\pi_y$ is the normalized left eigenvector of $\bar{Q}$ with eigenvalue 0 (see (\ref{eqn:HomogeneousP})). Define the set $\mathcal{T}$ so that $\Delta t \in \mathcal{T}$ if $P$ from (\ref{eqn:Markov-last}) is a valid transition probability matrix for all $x \in \mathcal{X}$. Then $\pi_y$ is the normalized left eigenvector of $\bar{P}$ with eigenvalue 1, for all $\Delta t \in \mathcal{T}$.
\end{lemma}
\begin{proof}
The proof is given in the appendix.
\end{proof}
Note that $\mathcal{T}$ contains all ``sufficiently small'' $\Delta t$. It follows from the lemma that the steady state distribution $\pi_y$ is the same for both continuous and discrete time.
}
Note that the mutual information rate $\mathcal{I}(X;Y)$ in (\ref{eqn:GeneralIID1}) has units of nats per channel use, and that channel uses have duration $\Delta t$. Moreover, the transition probabilities $p(y_i \:|\: x_i,y_{i-1})$ in (\ref{eqn:TransitionsInS})\rev{--}(\ref{eqn:SelfTransition}) are linear functions of $\Delta t$.
Substituting \rev{the discrete-time transition probabilities \eqref{eqn:Markov-last}} into (\ref{eqn:GeneralIID1}), the non-self-transition probabilities go to zero while the self-transition probabilities go to 1, so
$\mathcal{I}(X;Y) \rightarrow 0$ as $\Delta t \rightarrow 0$. This should not be surprising: intuitively, as the time step shrinks, less information can be expressed per time step.
However, dividing by $\Delta t$ (and obtaining $\mathcal{I}(X;Y) / \Delta t$),
the information rate {\em per second} is finite.
It is then useful to consider how this rate behaves as $\Delta t \rightarrow 0$.
Let $\mathcal{S}^\prime \subset \mathcal{S}$ represent the set of sensitive transitions excluding self transitions, i.e.,
\begin{equation}
\mathcal{S}^\prime = \{(y_{i-1},y_i) : (y_{i-1},y_i) \in \mathcal{S}, y_{i-1} \neq y_i \} .
\end{equation}
Also let $\mathcal{S}\backslash\mathcal{S}^\prime$ represent the components of $\mathcal{S}$ excluding $\mathcal{S}^\prime$ (i.e., {\em only} the sensitive self transitions).
For any edge $(y,y')$ define the limiting value of that edge's contribution to the mutual information rate, as $\Delta t\to 0$, as
\begin{align}
\nonumber \lefteqn{\iota(y,y') =} & \\
\nonumber
& \lim_{\Delta t\to 0}\frac1{\Delta t} \pi_{y}
\Bigg(
\sum_{x \in \mathcal{X}} p(x) \phi\Big(p(y' \:|\: x,y)\Big)\\
&
\quad\quad - \phi\left(\sum_{x \in \mathcal{X}} p(x) p(y' \:|\: x,y) \right)
\Bigg)
\label{eq:iota}
\end{align}
The limit calculation depends on whether $y=y'$. In case $y\not=y'$, we have $p(y'\:|\: x,y)=q_{yy'}x\Delta t$ (see \eqref{eqn:TransitionsInS}) and
\begin{align}
&\sum_{x}p(x)\phi(p(y'\:|\: x,y))-\phi\left(\sum_x p(x)p(y'\:|\: x,y) \right) \nonumber \\ \nonumber
&=\Delta t\Bigg\{
\left(\sum_x q x p(x) \right)\log q +
\left(
\sum_x q p(x) x\log x
\right) \\
\nonumber
& \quad\quad - \left(\sum_x q x p(x) \right)\log\left( \sum_x q x p(x) \right)
\Bigg\} \\
& \quad\quad +o(\Delta t),\text{ as }\Delta t\to 0^+\\
&= q\Delta t(E(x\log x) - E(x)\log(E(x))) +o(\Delta t),\text{ as }\Delta t\to 0^+\\
&=q\Delta t(E\phi(x) - \phi(Ex))+o(\Delta t),\text{ as }\Delta t\to 0^+.
\end{align}
On the other hand, in the case when $y=y'$, $\sum_{x}p(x)\phi(p(y'\:|\: x,y))-\phi\left(\sum_x p(x)p(y'\:|\: x,y) \right)=o(\Delta t)$, as $\Delta t\to 0^+$. Therefore, these terms do not contribute to the mutual information.
Using these \rev{results}, we can rewrite (\ref{eqn:GeneralIID1}) as
\rev{\begin{align}
\nonumber\lefteqn{\lim_{\Delta t \rightarrow 0}\frac{\mathcal{I}(X;Y)}{\Delta t}}&\\
&= \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \iota(y_{i-1},y_i)
+\sum_{(y_{i-1},y_i) \in \mathcal{S} \backslash \mathcal{S}^\prime}\iota(y_{i-1},y_i).
\label{eqn:GeneralIID2}
\end{align}}
Using (\ref{eqn:TransitionsInS})\rev{--}(\ref{eqn:SelfTransition}), we consider the two additive terms in (\ref{eqn:GeneralIID2}) separately.
For the first term (summing over $\mathcal{S}^\prime$),
we use l'H\^opital's rule:
in the denominator we have (trivially)
\begin{align}
\label{eqn:limit-denominator}
\frac{d}{d\Delta t} \Delta t &= 1,
\end{align}
and from the numerator, we have
\begin{align}
\nonumber \lefteqn{
\lim_{\Delta t \rightarrow 0} \frac{d}{d\Delta t} \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}}
\Bigg(
\sum_{x_i \in \mathcal{X}} p(x_i) \phi\Big(q_{y_{i-1}y_i} x_i \Delta t \Big)}\\
&\quad
- \phi\left(\sum_{x_i \in \mathcal{X}} p(x_i) q_{y_{i-1}y_i} x_i \Delta t \right)
\Bigg) \\
\nonumber
&= \lim_{\Delta t \rightarrow 0} \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}}
\Bigg(
\sum_{x_i \in \mathcal{X}} p(x_i) \frac{d}{d\Delta t} \phi\Big(q_{y_{i-1}y_i} x_i \Delta t \Big)\\
&\quad\quad - \frac{d}{d\Delta t} \phi\left(\sum_{x_i \in \mathcal{X}} p(x_i) q_{y_{i-1}y_i} x_i \Delta t \right)
\Bigg)\\
\nonumber
&= \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} \Bigg( \sum_{x_i \in \mathcal{X}}p(x_i) q_{y_{i-1}y_i}x_i \log (q_{y_{i-1}y_i}x_i)\\
&\quad\quad - q_{y_{i-1}y_i} \bar{x} \log (q_{y_{i-1}y_i} \bar{x})
\Bigg)
\label{eqn:LimitMutualInfo}
\end{align}
where $\bar{x} = \sum_{x_i \in \mathcal{X}} x_i p(x_i)$ is the average input concentration.
For the second term (summing over $\mathcal{S}\backslash\mathcal{S}^\prime$), a similar derivation shows that the limit is zero.
Simplifying further, we have
\begin{align}
\nonumber\lefteqn{\lim_{\Delta t \rightarrow 0} \frac{\mathcal{I}(X;Y)}{\Delta t}}&\\
\nonumber
&= \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} \Bigg( \sum_{x_i \in \mathcal{X}}p(x_i) q_{y_{i-1}y_i}x_i \log (q_{y_{i-1}y_i}x_i)\\
&\quad\quad- q_{y_{i-1}y_i} \bar{x} \log (q_{y_{i-1}y_i} \bar{x}) \Bigg) \\
\nonumber
&= \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i}
\bar{x}\log (q_{y_{i-1}y_i}) \\
\nonumber
&\quad\quad+
\sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i}\sum_{x_i \in \mathcal{X}}p(x_i) x_i\log(x_i) \\
\nonumber
&\quad\quad- \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i} \bar{x} \log (q_{y_{i-1}y_i})\\
&\quad\quad- \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i} \bar{x} \log(\bar{x}) \\
\nonumber
&= \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i}\sum_{x_i \in \mathcal{X}}p(x_i) x_i\log(x_i)\\
&\quad\quad
- \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i} \bar{x} \log(\bar{x}) \\
\nonumber
&= \left( \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i} \right)\\
&\quad\quad \cdot
\left( \sum_{x_i \in \mathcal{X}}p(x_i) x_i\log(x_i) - \bar{x}\log\bar{x} \right) .
\label{eqn:FactoredMutualInformation}
\end{align}
The {\em steady-state flux} $J_{y_{i-1}y_i}$ through an edge $(y_{i-1},y_i)$ in the state transition graph
is defined as
\begin{equation}
J_{y_{i-1}y_i} := \pi_{y_{i-1}} q_{y_{i-1}y_i} \bar{x} .
\end{equation}
Similarly, the {\em net steady-state flux} through the sensitive (non-self) edges in the graph is
\begin{align}
J_{\mathcal{S}^\prime} &:= \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} J_{y_{i-1}y_i} \\
\label{eqn:SteadyStateFlux}
&= \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i} \bar{x} ,
\end{align}
Expressing (\ref{eqn:FactoredMutualInformation}) in terms of $J_{\mathcal{S}^\prime}$,
we have
\begin{align}
\lim_{\Delta t \rightarrow 0} \frac{\mathcal{I}(X;Y)}{\Delta t}
&= \frac{1}{\bar{x}} J_{\mathcal{S}^\prime}
\left( \sum_{x_i \in \mathcal{X}}p(x_i) x_i\log x_i - \bar{x}\log\bar{x} \right) \\
\label{eqn:FactoredMutualInformation2}
&= J_{\mathcal{S}^\prime}
\left( \sum_{x_i \in \mathcal{X}} \frac{p(x_i)x_i}{\bar{x}}\log x_i - \log\bar{x} \right)
\end{align}
\rev{We define
\begin{equation}
\label{eqn:nu}
\nu(x_i) := \frac{p(x_i) x_i}{\bar{x}} .
\end{equation}
}
Since $\nu(x_i)$ is positive for all $x_i$, and since
\begin{align}
\sum_i \nu(x_i) &= \sum_i \frac{p(x_i) x_i}{\bar{x}} = \frac{\bar{x}}{\bar{x}} = 1,
\end{align}
\rev{it follows that} $\nu(x_i)$ forms a probability distribution, in general different from $p(x_i)$.
We discuss the physical interpretation of $J_{\mathcal{S}^\prime}$ and
$\nu(x_i)$ in the next section.
Using $\nu(x_i)$, we can rewrite (\ref{eqn:FactoredMutualInformation2}) as
\begin{align}
\lim_{\Delta t \rightarrow 0} \frac{\mathcal{I}(X;Y)}{\Delta t}
&= J_{\mathcal{S}^\prime}
\left( \sum_{x_i \in \mathcal{X}} \nu(x_i) \log x_i - \log\bar{x} \right) \\
&= J_{\mathcal{S}^\prime}
\left( \sum_{x_i \in \mathcal{X}} \nu(x_i) \log x_i - \sum_{x_i \in \mathcal{X}} \nu(x_i) \log\bar{x} \right) \\
&= J_{\mathcal{S}^\prime}
\left( \sum_{x_i \in \mathcal{X}} \nu(x_i) \log \frac{x_i}{\bar{x}} \right) \\
&= J_{\mathcal{S}^\prime}
\left( \sum_{x_i \in \mathcal{X}} \nu(x_i) \log \frac{p(x_i)x_i}{p(x_i)\bar{x}} \right)\\
&= J_{\mathcal{S}^\prime}
\left( \sum_{x_i \in \mathcal{X}} \nu(x_i) \log \frac{\nu(x_i)}{p(x_i)} \right)\\
&= J_{\mathcal{S}^\prime} D(\nu \:\rev{\|}\: p) ,
\end{align}
where $D(\cdot\:\rev{\|}\:\cdot)$ represents the Kullback-Leibler divergence.
The preceding derivation, \rev{including Lemma \ref{lem:SteadyState},} allows us to state the following result.
\begin{theorem}
\label{thm:Theorem1}
For finite-state Markov signal transduction systems described by (\ref{eqn:Markov-last}), with IID inputs,
%
\begin{equation}
\label{eqn:Theorem1}
\lim_{\Delta t \rightarrow 0} \frac{\mathcal{I}(X;Y)}{\Delta t} = J_{\mathcal{S}^\prime} D(\nu \:\rev{\|}\: p) ,
\end{equation}
%
with $J_{\mathcal{S}^\prime}$ defined in (\ref{eqn:SteadyStateFlux}) and $\nu$ defined in (\ref{eqn:nu}).
\end{theorem}
\subsection{Physical interpretation}
The factorization in (\ref{eqn:FactoredMutualInformation}) gives us a useful physical interpretation of
the mutual information in this system.
First consider $J_{\mathcal{S}^\prime}$. Physically, if one watched only for transitions along
edge $(y_{i-1},y_i)$ (with the rest of the graph assumed to be at steady state),
$J_{y_{i-1}y_i}$ gives
the average rate at which those transitions would be observed; that is, $J_{y_{i-1}y_i}$ is the \pt{mean} flux through the transition $y_{i-1} \rightarrow y_i$. Thus, $J_{\mathcal{S}^\prime}$ is the average rate through {\em all} the sensitive edges, i.e., the {\em net flux}.
Now consider $D(\nu \:\rev{\|}\: p)$, and note that the distribution $\nu(x_i)$ is a {\em posterior} distribution of $x_i$. To see this,
consider a random variable $y \in \{0,1\}$, with conditional distribution
\begin{equation}
\label{eqn:YXconditional}
p_{Y|X}(1 \:|\: x_i) =\kappa x_i ,
\end{equation}
where $\kappa$ is a positive constant ($0 \leq \kappa \leq \frac{1}{x_i}$ to make a valid probability),
and $p_{Y|X}(0 \:|\: x_i) = 1- p_{Y|X}(1 \:|\: x_i)$.
The marginal distribution $p_Y(1)$ is then given by
\begin{align}
p_Y(1) &= \sum_{x_i} p(x_i) p_{Y|X}(1 \:|\: x_i)\\
&= \sum_{x_i} p(x_i) \kappa x_i\\
&= \kappa \bar{x} .
\end{align}
With this definition, $\nu(x_i)$ is the posterior distribution of $x$ given $y=1$:
\begin{align}
p_{X|Y}(x_i \:|\: 1) &= \frac{p(x_i)p_{Y|X}(1 \:|\: x_i)}{p_Y(1)}\\
&= \frac{p(x_i) \kappa x_i}{\kappa \bar{x}}\\
&= \frac{p(x_i) x_i}{\bar{x}} = \nu(x_i) .
\end{align}
Physically, consider the example of a ligand-gated channel where $x_i$ is the concentration of ligands near the receptor at input $i$. With $i \in \{\mathsf{L},\mathsf{H}\}$ (i.e., inputs $x_\mathsf{L}$ and $x_\mathsf{H}$), suppose we select one molecule at random from those near the receptor, and set $y = 1$ if the molecule is a ligand; $y = 0$ otherwise. Then $p_{Y|X}(1 \:|\: x_\mathsf{L}) \propto x_L$ and $p_{Y|X}(1 \:|\: x_\mathsf{H}) \propto x_\mathsf{H}$, with $\kappa$ as the constant of proportionality; this satisfies (\ref{eqn:YXconditional}). For example, suppose $x_\mathsf{L}$ is measured in {\em number concentration} of ligands, i.e., number of ligands per volume $V$. Then $p_{Y|X}(1 \:|\: x_i) = x_i / n$ (for $i \in \{\mathsf{L},\mathsf{H}\}$), where $n$ is the number concentration of all molecules, ligands and otherwise, near the receptor, and $\kappa = 1/n$.
In general, physical systems where the probability of response $p(y \:|\: x)$ is directly proportional to the input $x$
fit into this framework, emphasizing the importance of this modeling assumption made in Section II.
\subsection{Shannon capacity of receptors with a single sensitive non-self transition}
We now give our \rev{third} main result, showing that the Shannon capacity $C$ is equal to the IID capacity $C_{\mathrm{iid}}$ for a number of sensitive transitions $|\mathcal{S}^\prime| \leq 1$, and furthermore that the capacity-achieving distribution has a simple form. As a consequence, this leads directly to the Shannon capacity of ChR2; we give this capacity in the example below. The result is a generalization of related results in \cite{ThomasEckford2016}.
Recall $\mathcal{S}^\prime \subset \mathcal{S}$ represent the set of transitions, {\em excluding} self-transitions.
\begin{theorem}
\label{thm:capacity}
For any receptor with $|\mathcal{S}^\prime| \leq 1$,
\begin{enumerate}
\item $C_{\mathrm{iid}}$ is achieved with all probability mass on $x_\mathsf{L}$ and $x_\mathsf{H}$; and
\item $C = C_{\mathrm{iid}}$.
\end{enumerate}
\end{theorem}
\begin{proof}
The case of $|\mathcal{S}^\prime| = 0$ is trivial: the state is never sensitive to the input, so $\mathcal{I}(X;Y) = 0$ for all input distributions.
Now consider $|\mathcal{S}^\prime| = 1$. We sketch the proof: results in \cite{ThomasEckford2016} were presented for a two-state receptor where only one transition was sensitive; many of the results have the same form.
The first part of the theorem follows from \cite[Thm 1]{ThomasEckford2016}, noting from (\ref{eqn:GeneralIID1}) that any system with $|\mathcal{S}^\prime| = 1$ has the same form, apart from the marginal distribution $\pi_{y_{i-1}}$, which is held constant in the proof of \cite[Thm 1]{ThomasEckford2016}. The second part of the theorem follows from \cite[Thm 2]{ThomasEckford2016}, noting that $C$ is only a function of the input distribution in the sensitive state.
\end{proof}
\subsection{Example}
We now give an example calculation of the mutual information and IID capacity, \pt{by which we obtain the channel capacity of channelrhodopsin.}
\begin{ex}[ChR2]
\label{ex:ChR2-MI}
Referring to the rate matrix for ChR2 (\ref{eqn:ChR2-rate-matrix}),
there are exactly two sensitive transitions: first, the
transition from $\mathsf{C}_1$ to $\O_2$, represented
by $q_{12} x(t)$; and second, the self-transition from $\mathsf{C}_1$ to $\mathsf{C}_1$, represented by
$R_1 = - q_{12}x(t)$. Thus, $\mathcal{S} = \{(\mathsf{C}_1,\O_2), (\mathsf{C}_1,\mathsf{C}_1)\}$
and $\mathcal{S}^\prime = \{(\mathsf{C}_1,\O_2)\}$.
Suppose $\mathcal{X} = \{x_\mathsf{L}, x_\mathsf{H}\}$, i.e., the input light source can only be off ($x_\mathsf{L}$) or on ($x_\mathsf{H}$). Let $p_\mathsf{L} = {\mathrm{Pr}}(x = x_\mathsf{L})$ and $p_\mathsf{H} = {\mathrm{Pr}}(x = x_\mathsf{H}) = 1-p_\mathsf{L}$.
Recalling the transformation of
rates into probabilities (\ref{eqn:Markov-last}), and substituting into (\ref{eqn:GeneralIID1}), we have
%
\begin{align}
\nonumber\lefteqn{\mathcal{I}(X;Y)} &\\
\nonumber &=
\pi_{\mathsf{C}_1} \Big( p_\mathsf{L} \phi(\Delta t q_{12}x_\mathsf{L}) + p_\mathsf{H} \phi(\Delta t q_{12}x_\mathsf{H})\\
\nonumber
&\quad\quad- \phi \big( p_\mathsf{L} \Delta t q_{12}x_\mathsf{L} + p_\mathsf{H} \Delta t q_{12}x_\mathsf{H} \big) \Big)\\
\nonumber
& \quad + \: \pi_{\mathsf{C}_1} \Big( p_\mathsf{L} \phi(1-\Delta t q_{12}x_\mathsf{L}) + p_\mathsf{H} \phi(1- \Delta t q_{12}x_\mathsf{H})\\
&\quad\quad - \phi \big( 1-p_\mathsf{L} \Delta t q_{12}x_\mathsf{L} - p_\mathsf{H} \Delta t q_{12}x_\mathsf{H} \big) \Big)
\end{align}
%
where the first term represents the transition $(\mathsf{C}_1,\O_2)$, and the second term represents the self-transition $(\mathsf{C}_1,\mathsf{C}_1)$, both of which are sensitive.
Continuing the derivation,
\begin{align}
\nonumber\lefteqn{\mathcal{I}(X;Y)} &\\
\nonumber
&= \pi_{\mathsf{C}_1} \Big(
\binent(p_\mathsf{L} \Delta t q_{12}x_\mathsf{L} + p_\mathsf{H} \Delta t q_{12}x_\mathsf{H})\\
&\quad\quad - p_\mathsf{L} \binent(\Delta t q_{12}x_\mathsf{L}) - p_\mathsf{H} \binent(\Delta t q_{12}x_\mathsf{H})
\Big) \\
\label{eqn:ChR2MutualInformationExample}
&= \left( \frac{q_{23}q_{31}}{q_{23}q_{31} + \bar{x}q_{12}q_{31} + \bar{x}q_{12}q_{23}} \right)\\
&\quad\quad\cdot
\Big( \binent(\Delta t q_{12} \bar{x}) - p_\mathsf{L} \binent(\Delta t q_{12}x_\mathsf{L}) - p_\mathsf{H} \binent(\Delta t q_{12}x_\mathsf{H})
\Big) ,
\end{align}
where $\bar{x}$ is the average input.
Finally, consider $\mathcal{I}(X;Y) / \Delta t$ as $\Delta t \rightarrow 0$, as in (\ref{eqn:LimitMutualInfo}).
The steady-state occupancy probability of $\mathsf{C}_1$, $\pi_{\mathsf{C}_1}$, is independent of $\Delta t$. Thus,
from Theorem \ref{thm:Theorem1}, we have
\begin{align}
\nonumber\lefteqn{\lim_{\Delta t \rightarrow 0} \frac{\mathcal{I}(X;Y)}{\Delta t}}&\\
&= J_{\mathcal{S}^\prime} D(\nu \:\rev{\|}\: p)\\
&=
\left( \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i} \right)
\left( \sum_{x_i \in \mathcal{X}}\nu(x_i) \log\frac{\nu(x_i)}{p(x_i)} \right) \\
&=
\pi_{\mathsf{C}_1} q_{12}
\left( p_\mathsf{L}\frac{x_\mathsf{L}}{\bar{x}} \log \frac{x_\mathsf{L}}{\bar{x}} + p_\mathsf{H}\frac{x_\mathsf{H}}{\bar{x}} \log \frac{x_\mathsf{H}}{\bar{x}} \right) \\
\nonumber
&= \frac{q_{12}q_{23}q_{31}}{q_{23}q_{31} + \bar{x}q_{12}q_{31} + \bar{x}q_{12}q_{23}}\\
&\quad\quad\cdot
\left( p_\mathsf{L}\frac{x_\mathsf{L}}{\bar{x}} \log \frac{x_\mathsf{L}}{\bar{x}} + p_\mathsf{H}\frac{x_\mathsf{H}}{\bar{x}} \log \frac{x_\mathsf{H}}{\bar{x}} \right) .
\label{eqn:ChR2MutualInformationExample2}
\end{align}
In Figure \ref{fig:MutualInformationFigure}, we illustrate the effect of step size on the mutual information calculation, using (\ref{eqn:ChR2MutualInformationExample}) for the solid lines (for various values of $\Delta t > 0$) and (\ref{eqn:ChR2MutualInformationExample2}) for the dotted line (as $\Delta t \rightarrow 0$). From this figure, the IID capacity and the capacity-achieving value of $p_\mathsf{L}$ may be found by taking the maximum over the curve of interest. This value clearly changes for different values of $\Delta t$; however, the IID capacity is around $C_{\mathrm{iid}} \approx 65$ bits/s, and the capacity-achieving $p_\mathsf{L}$ is around $p_\mathsf{L} \approx 0.99$. For any finite value of $\Delta t$, it is interesting to note that the discrete-time approximation over-estimates the mutual information as $\Delta t \rightarrow 0$.
\end{ex}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=3.5in]{figures/MutualInformationFigure.pdf}
\end{center}
\caption{\label{fig:MutualInformationFigure} Plot for ChR2, illustrating the effect of $\Delta t$ on $\mathcal{I}(X;Y)$ from (\ref{eqn:ChR2MutualInformationExample}). The dashed black line represents $\Delta t \rightarrow 0$. Solid lines, from bottom, represent: $\Delta t = 0.01$ (blue), $\Delta t = 0.02$ (green), $\Delta t = 0.04$ (red), $\Delta t = 0.06$ (cyan), $\Delta t = 0.08$ (magenta), and $\Delta t = 0.1$ (tan), all in milliseconds.}
\end{figure}
From Example \ref{ex:ChR2-MI}, ChR2 has $|\mathcal{S}^\prime| = 1$. Thus, {\em ChR2 satisfies the conditions of Theorem \ref{thm:capacity}}, and has $C = C_{\mathrm{iid}}$, where $C_{\mathrm{iid}}$ is given in (\ref{eqn:ChR2MutualInformationExample2}).
\rev{Performing the maximization numerically, on the $\Delta t \rightarrow 0$ line, the maximum value of $\mathcal{I}(X;Y)$ is found near $p_\mathsf{L} = 0.99$ where $\mathcal{I}(X;Y) = 66$ bits/s, which gives} \pt{the channel capacity \rev{$C$} (\textit{sensu} Shannon) of channelrhodopsin.}
A similar calculation can be performed for ACh and CaM. However, the resulting expressions are not as compact as (\ref{eqn:ChR2MutualInformationExample2}), so we exclude them from the paper. Mutual information plots for ACh and CaM (from which $C_{\mathrm{iid}}$ may \rev{also be obtained numerically}) are given in Figures \ref{fig:AChFigure} and \ref{fig:CaMFigure}, respectively.
However, ACh and CaM both have $|\mathcal{S}^\prime| > 1$ (see Figures \ref{fig:ACh} and \ref{fig:CaM}), and do not satisfy the condition in Theorem \ref{thm:capacity}. It remains an open question as to whether $C = C_{\mathrm{iid}}$ for these receptors. The proof of \cite[Thm 2]{ThomasEckford2016} (and of Theorem \ref{thm:capacity}) relies on the feedback capacity being achieved by the IID input distribution. However, if there is more than one sensitive transition, the receiver can use the feedback to distinguish between these transitions, and can select an optimal input distribution for each. Thus, the feedback-capacity-achieving input distribution depends on the feedback, and is not \rev{necessarily} IID. If $C = C_{\mathrm{iid}}$, a different proof technique is required, and we do not address this case.
\section{Discussion}
In this paper we have presented a general framework for signal transduction systems, in which the states of a receptor form a directed graph, some subset of the edges of which represent transitions with intensities modulated by an external signal. This signal provides the channel input, and the state of the receptor -- a trajectory on the graph -- represents the channel output.
\pt{We illustrate the signal transduction model, the calculation of mutual information and the IID capacity for several examples: light intensity transduction by channel rhodopsin, acetylcholine concentration transduction by the nicotinic acetylcholine receptor, and transduction of intracellular calcium ion concentration by the calmodulin protein.}
Several caveats are in order, which qualify our results and motivate our future work.
In many signal transduction systems, only a subset of the receptor states engender an observable output signal. For example, the channelrhodopsin receptor states $\mathsf{C}_1,\O_2,\mathsf{C}_3$ (\textit{cf}.~Fig.~\ref{fig:ChR2}) are not directly observed by the cell in the membrane of which the receptor is embedded; rather it is the net current (zero for states $\mathsf{C}_1,\mathsf{C}_3$ and finite for state $\O_2$) that impacts the rest of the cell. Similarly, for the nicotinic acetylcholine receptor (\textit{cf}.~Fig.~\ref{fig:ACh}) the state of the receptor as observed by the cell is either ``open'' (states $\O_1,\O_2$) or ``closed'' (states $\mathsf{C}_3,\mathsf{C}_4,\mathsf{C}_5$). For the calmodulin receptor, there are understood to be four functionally distinct states: both occupied Ca$^{2+}$~binding sites on the N-terminus end of the protein, both occupied Ca$^{2+}$~binding sites on the C-terminus end of the protein, all four Ca$^{2+}$~binding sites occupied, or fewer than two on each end (\textit{cf.}~Fig.~\ref{fig:CaM}; dashed lines indicate physiologically equivalent states).
The diagram \eqref{diag:xyz} shows the general structure of such a channel, with output $Z(t)$ a function of the channel state $Z=f(Y(t))$ (compare with the diagram in (\ref{diag:xy})):
\begin{equation}\label{diag:xyz}
\begin{array}{ccccccccccc}
&X_1&&X_2&&X_3&&X_4&&X_5&\cdots\\
&\downarrow&&\downarrow&&\downarrow&&\downarrow&& \downarrow \\
(Y_0)&\longrightarrow&Y_1&\longrightarrow&Y_2&\longrightarrow&Y_3&\longrightarrow&Y_4&\cdots\\
&&\downarrow&&\downarrow&&\downarrow&& \downarrow \\
&&Z_1&&Z_2&&Z_3&&Z_4&\cdots
\end{array}\end{equation}
By virtue of the information processing inequality, the mutual information rate between $X$ and $Z$ cannot exceed that between $X$ and $Y$. Preliminary results suggest that the size of the difference -- the information gap -- depends strongly on the network architecture, and the positioning of sensitive edges relative to observable transitions (data not shown). Detailed consideration of mutual information for Markovian signal transduction channels with such partially observed outputs will be undertaken elsewhere.
We have assumed that the directed edges comprising the receptor's state transition graph fall into two classes, either insensitive (fixed transition rates) or sensitive (transition rates proportional to the input signal intensity). A more realistic assumption would allow for a dark current (finite transition rate at zero signal intensity), a nonlinear, monotonically increasing transition rate as a function of increasing intensity, or a signaling threshold or minimum intensity value. Under the IID input scenario it is optimal to limit the input values to those inducing the maximal and minimal transition rates, in which case several more realistic scenarios could in principle be reduced to the scenario we consider here. For example, a dark current could be captured by adding an additional insensitive channel parallel to a sensitive channel.
We have considered a general class of signal transduction models that are naturally framed as continuous time channels. Our basic signal transduction channel model process is conditionally Markovian, given the (time varying) input signal. The simplest model in this class would correspond to Kabanov's Poisson channel \cite{Kabanov1978}, consisting of a single transition with rate modulated by the input. In order to simplify the analysis of such models it is convenient to translate them into analogous discrete-time models. The general structure of such as model is a finite state, discrete-time channel in which the probability transition matrix is modulated by the (discrete time) input sequence. Our previously-discussed results \cite{ThomasEckford2016} introduced a minimal such model, the BIND channel, consisting of a single receptor molecule with two states (bound, $\mathsf{B}$, and unbound, $\mathsf{U}$) with one transition rate ($\mathsf{U}\to\mathsf{B}$) sensitive to the input (ligand molecule concentration) and the other transition rate ($\mathsf{B}\to\mathsf{U}$) insensitive. In general, the structure of a conditionally Markovian signal-transduction channel under time discretization corresponds to the Unit Output Memory (UOM) channel class analyzed by \rev{Chen and Berger} \cite{che05}. As mentioned previously,
in \rev{\cite{AsnaniPermuterWeissman2013IEEE_ISIT,PermuterAsnaniWeissman2014IEEETransIT} Asnani, Permuter and Weissman} present several examples of UOM channels that they call POST (prior output is the state) channels, which are also special cases of the channels analyzed by Chen and Berger. (The BIND channel can be interpreted as a type of POST channel although it is distinct from the examples in \cite{AsnaniPermuterWeissman2013IEEE_ISIT,PermuterAsnaniWeissman2014IEEETransIT}.)
Thus our channel models for channel rhodopsin, the nicotinic acetylcholine receptor and calmodulin may all be seen as examples of Chen and Berger's UOM channel class.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=3.5in]{figures/AChFigure.pdf}
\end{center}
\caption{\label{fig:AChFigure} Plots of $\mathcal{I}(X;Y)$ and $\mathcal{I}(X;Z)$ for ACh, using $\Delta t = 0.02$ ms. Solid line is from (\ref{eqn:GeneralIID1}), while dots represent {\em Monte Carlo} simulations.}
\end{figure}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=3.5in]{figures/CaMFigure.pdf}
\end{center}
\caption{\label{fig:CaMFigure} Plots of $\mathcal{I}(X;Y)$ and $\mathcal{I}(X;Z)$ for CaM, using $\Delta t = 0.002$ ms. Solid line is from (\ref{eqn:GeneralIID1}), while dots represent {\em Monte Carlo} simulations.}
\end{figure}
\section*{To-do list}
\section{Introduction}
Living cells take in information from their surroundings through myriad \emph{signal transduction} processes.
Signal transduction takes many forms: the input signal can be carried by changes in chemical concentration, electrical potential, light intensity, mechanical forces, and temperature, \textit{inter alia}. In many instances these \emph{extracellular} stimuli trigger \emph{intracellular} responses that can be represented as transitions among a discrete set of states \cite{hlavacek2006}.
Models of these processes are of great interest to mathematical and theoretical biologists \cite{janes2006}.
The ``transduction'' of the signal occurs through the physical effect of the input signal on the transition rates among the various states describing the receptor.
\pt{An early}
mathematical model of this type was the voltage-sensitive transitions among several open and closed ion channel states in Hodgkin and Huxley's model for the conduction of sodium and potassium ions through the membranes of electrically excitable cells \cite{hodgkin1990}.
Presently, many such models are known for signal transduction systems, such as: the detection of calcium concentration signals by the calmodulin protein \cite{faas11}, binding of the acetylcholine (ACh) neurotransmitter to its receptor protein \cite{col82}, and modulation of the channel opening transition by light intensity in the channelrhodopsin (ChR) protein \cite{nag03}. In each of these examples the channel may be modeled as a weighted, directed graph, in which the vertices represent the discrete channel states, and the weighted edges represent \textit{per capita} transition rates, some of which can be modulated by the input signals.
Mutual information, and Shannon capacity, arise in a variety of biological contexts. For example, mutual information may predict the differential growth rates of organisms learning about their environment \cite{tkacik2016}, based on the Kelly criterion \cite{kelly1956}. For biological communication systems, achieving a distortion criterion (expressed as mutual information) need not require complicated signal processing techniques; see \cite[Example 2]{gastpar2003}. Moreover, the free energy cost of molecular communication (such as in signal transduction) has a mathematical form similar to mutual information \cite{eckford2018}, leading to thermodynamic bounds on capacity per unit energy cost (cf. \cite{verdu1990}).
Stochastic modeling of signal transduction as a communication channel has considered the chemical reactions in terms of Markov chains \cite{thomas2003} and in terms of the ``noise'' inherent in the binding process \cite{pierobon2011}. For simplified two-state Markov models, Shannon capacity of signal transduction has been calculated for slowly changing inputs \cite{ein11} and for populations of communicating bacteria \cite{aminian2015}. Our own previous work has investigated the capacity of signal transduction: in \cite{ThomasEckford2016}, we obtained the Shannon capacity of two-state Markov signal transduction under arbitrary inputs, and showed that the capacity for multiple independent receptors has the same form \cite{thomas2016}. Related channel models have been studied in the information-theoretic literature, such as the unit output memory channel \cite{che05}, the ``previous output is the state'' (POST) channel \rev{\cite{AsnaniPermuterWeissman2013IEEE_ISIT,PermuterAsnaniWeissman2014IEEETransIT}}; capacity results for some channels in these classes were recently given in \cite{stavrou2016}.
The present paper \pt{focuses on}
the mutual information and capacity of finite-state signal transduction channels. Generalizing previous results, we provide discrete-time, finite-state channel models for a wide class of signal transduction receptors, giving Channelrhodopsin-2 (ChR2), Acetylcholine (ACh), and Calmodulin (CaM) as specific examples.
We also provide an explicit formula for the mutual information of this \pt{class of models}
under independent, identically distributed (IID) inputs \rev{(Theorem 1)}.
Subsequently, we consider
the continuous time limit
as the interval between the discrete-time instants goes to zero, and find a simple closed-form expression for the mutual information \rev{(Theorem 2)}, with a natural physical interpretation.
We further give conditions under which our formula gives the Shannon capacity of the channel, namely that there is exactly one transition in the Markov chain that is sensitive to the channel input \rev{(Theorem 3), and we use this result to (numerically) find the Shannon capacity of ChR.}
The remainder of the paper is organized as follows: in \rev{Section} II, we give a generalized model for discrete-time, finite-state signal transduction systems; in \rev{Section} III, we discuss signal transduction as a communication system, deriving expressions for the mutual information and giving our main results; and in \rev{Section} IV, we discuss the biological significance of the results, \pt{as well as the limitations of our analysis}.
\section{Model}
\label{sec:Model}
\subsection{Physical model}
Signal transduction encompasses a wide variety of physical processes.
For example, in a {\em ligand-gated} system, signals are transmitted using concentrations of signaling molecules, known as {\em ligands}, which bind to receptor proteins. As another example, in a {\em light-gated} system, signals are transmitted using light, where the receptor absorbs photons. Other possibilities exist, such as voltage-gated ion channels. The receptor, \pt{often}
located on the surface of the cell, forms the receiver in the signal transduction system, and conveys (or {\em transduces}) the signal across the cell membrane; the receptor is the focus of our analysis.
Signal transduction receptors share a mathematical model: they can be viewed as finite-state, intensity-modulated Markov chains, in which the transition rates between certain pairs of states are sensitive to the input (though other transitions may be independent of the input).
Our main examples in this paper focus on ligand- and light-gated receptors. For example, in a ligand-gated system, the binding of the ligand results in a change in the receptor, which then produces {\em second messengers} (normally a different species than the ligand) to convey the message to the cell interior. In a light-gated system, the incident photon causes a similar change in the receptor, which may open to allow an ion current to pass to the interior of the cell.
In either case, there may be a relaxation process which returns the receptor to the ``ready'' state, and this process may be independent of the signal; or other processes that are either sensitive to or independent of the signal, depending on the purpose of the receptor.
In the next two sections, we describe the Markov chain model for receptors, both in continuous and in discrete time.
Although we focus on ligand- and light-gated receptors, we emphasize that our framework is general enough to include other kinds of receptors.
\subsection{Continuous time: Master equation kinetics}
\label{sec:MasterEquation}
Receptors are finite-state Markov chains. For a receptor with $k$ discrete states,
there exists a $k$-dimensional vector of state occupancy probabilities $\mathbf{p}(t)$,
given by
\begin{equation}
\mathbf{p}(t) = \left[ p_1(t), \: p_2(t), \: \ldots, \:p_{k}(t) \right] ,
\end{equation}
where $p_i(t)$ represents the probability of a given receptor occupying state $i$ at time $t$. The environmental conditions at the receptor, such as light level or ligand concentration, are known as the {\em input} $x(t)$.
The chemical kinetics of the receptor are captured by a differential equation known
as the {\em master equation} \cite{Gardiner2004}. Let $Q = [q_{ij}\rev{(x)}]$ represent a $k \times k$ matrix of \rev{\textit{per capita} transition rates,}
where $q_{ij}\rev{(x)}$ represents the instantaneous rate at which receptors starting in state $i$
enter state $j$.
It is helpful to visualize the matrix $Q$ using a graph:
\begin{itemize}
\item There are $k$ vertices, representing the states; and
\item A directed edge is drawn from vertex $i$ to $j$ if and only if $q_{ij}\rev{(x)} > 0$ \rev{for some $x$}.
\end{itemize}
Changing from one state to another is called a {\em transition}, so the graph corresponding to $Q$ depicts the possible transitions. A transition $i \rightarrow j$ may be {\em sensitive}, i.e. $q_{ij}$ \rev{varies as} a function of the input $x(t)$, or {\em insensitive}, $q_{ij}$ is constant with respect to $x(t)$.
Using $Q$, the master equation is given by
\begin{equation}
\label{eqn:MasterEquation}
\frac{d\mathbf{p}(t)}{dt} = \mathbf{p}(t)Q\rev{(x(t))} .
\end{equation}
We use the notation from \cite{GroffDeRemigioSmith2009chapter}:
\begin{itemize}
\item States take a compound label, consisting of a state property and a state number. The state number is unique to each state, but the state property may be shared by multiple states. For example, in each state the receptor's ion channel might be either open $\O$ or closed $\mathsf{C}$; the state label $\mathsf{C}_1$ means that in state 1 the channel is closed, and $\O_2$ means that in state 2 the channel is open. \awe{In this paper we use the state {\em number} rather than the state {\em property}. (Since we show that the state numbers form a Markov chain, in general the state properties form a hidden Markov chain; we discuss this further in Section IV.)}
\item \rev{We assume that} rates which are sensitive to the input are directly proportional to the input $x(t)$. For example,
$q_{12}x(t)$ is the transition rate from $1 \rightarrow 2$,
which is sensitive, while
$q_{31}$ is the transition rate from $3 \rightarrow 1$, which is insensitive.
\item The $i$th diagonal element \rev{of $Q$} is written $R_i$, and is set so that the $i$th row sums to zero (so, if $x(t)$ appears in the $i$th row, $R_i$ may depend on $x(t)$).
\end{itemize}
Taking sensitive rates to be proportional to the signal $x(t)$ is a key modeling assumption; it is satisfied for the examples we consider, but there exist systems in which the signal acts nonlinearly on the rate.
The following three examples illustrate the use of our notation, and give practical examples of receptors along with their transition graphs and rate constants.
\begin{ex}[Channelrhodopsin-2]
The Channelrhodopsin-2 (ChR2) receptor is a light-gated ion channel.
The receptor has three states, named Closed ($\mathsf{C}_1$), Open ($\mathsf{O}_2$), and Desensitized ($\mathsf{C}_3$).
The channel-open ($\mathsf{O}$) state $\mathsf{O}_2$ is the only state in which the ion channel is open, passing an ion current.
The channel-closed ($\mathsf{C}$) states, $\mathsf{C}_1$ and $\mathsf{C}_3$, are distinct in that the receptor is light-sensitive in
state $\mathsf{C}_1$, and insensitive in state $\mathsf{C}_3$ \cite{nag03}. The rate matrix for ChR2 is
\begin{equation}
\label{eqn:ChR2-rate-matrix}
Q = \left[
\begin{array}{ccc}
R_1 & q_{12}x(t) & 0 \\
0 & R_2 & q_{23} \\
q_{31} & 0 & R_3
\end{array}
\right] .
\end{equation}
where $x(t) \in [0,1]$ is the relative intensity.
To keep the row sums equal to zero, we set $R_1 = - q_{12}x(t)$,
$R_2 = - q_{23}$, and
$R_3 = - q_{31}$.
Fig.~\ref{fig:ChR2} shows
state labels and allowed state transitions.
\begin{figure}
\begin{center}
\includegraphics[width=2.5in]{figures/ChR2}
\end{center}
\caption{\label{fig:ChR2} Depiction of allowed state transitions for ChR2. Sensitive
transitions are depicted with {\bfseries bold} arrows.
States are labelled by their ion channel state: $\{\mathsf{C},\mathsf{O}\}$ for closed and open, respectively; state number is in subscript. Dashed lines surround all states in either the closed or open state. Transition rates, listed in Table \ref{tab:ChR2parameters}, correspond to the vertices associated with each directed edge: for example,
the rate from state $\mathsf{O}_2$ to state $\mathsf{C}_3$ is $q_{23}$.}
\end{figure}
Parameter values from the literature are given in Table \ref{tab:ChR2parameters}.
\begin{table}[h!]\begin{center}
\begin{tabular}{c|c|c} \hline
Parameter & from \cite{nag03} & Units \\ \hline
$q_{12}x(t)$ & $(5 \times 10^3) x(t)$ & s$^{-1}$ \\ \hline
$q_{23}$ & 50 & s$^{-1}$ \\ \hline
$q_{31}$ & 17 & s$^{-1}$ \\ \hline
\end{tabular}
\ \\
\ \\
\caption{\label{tab:ChR2parameters} Rate parameters for ChR2, adapted from \cite{nag03}, where $x(t) \in [0,1]$ represents the relative light intensity.}
\end{center}
\end{table}
\end{ex}
\begin{ex}[Acetylcholine]
The Acetylcholine (ACh) receptor is a ligand-gated ion channel. \pt{Following \cite{col82}, we model the receptor as a conditional Markov process on} five states, with rate matrix
\begin{equation}
\label{eqn:AChRateMatrix}
Q =
\left[
\begin{array}{ccccc}
R_1 & q_{12}x(t) & 0 & q_{14} & 0 \\
q_{21} & R_2 & q_{23} & 0 & 0 \\
0 & q_{32} & R_3 & q_{34} & 0 \\
q_{41} & 0 & q_{43}x(t) & R_4 & q_{45} \\
0 & 0 & 0 & q_{54}x(t) & R_5
\end{array}
\right] .
\end{equation}
There are three sensitive transitions: $\rev{q_{12}}x(t)$, $\rev{q_{43}}x(t)$, and $\rev{q_{54}}x(t)$, which
are proportional to ligand concentration $x(t)$.
For the purposes of our analysis, we use a range of $x(t) \in [10^{-7},10^{-5}]$.
Fig.~\ref{fig:ACh} shows the
allowed state transitions.
The states in ACh correspond to the binding of a ligand to one of two binding sites on the receptor. In state $\mathsf{C}_5$, neither site is occupied; in states $\mathsf{C}_4$ and $\mathsf{O}_1$, one site is occupied; and in states $\mathsf{C}_3$ and $\mathsf{O}_2$, both sites are occupied.
Table \ref{tab:AChparameters} gives
parameter values; the concentration of ACh, $x(t)$, is measured in mol/$\ell$.
The same state-naming convention is used in the figure as with ChR2: states with
an open ion channel are $\mathsf{O}_{1}$ and $\mathsf{O}_2$; states
with a closed ion channel are $\mathsf{C}_3$, $\mathsf{C}_4$, and $\mathsf{C}_5$.
\begin{table}\begin{center}
\begin{tabular}{c|c|c|c} \hline
Parameter & Name in \cite{col82} & Value/range & Units \\ \hline
$q_{12}x(t)$ & $k_{+2}x$ & $(5 \times 10^8) x(t)$ & s$^{-1}$ \\ \hline
$q_{14}$ & $\alpha_1$ & $3 \times 10^3$ & s$^{-1}$ \\ \hline
$q_{21}$ & $2 k_{-2}^*$ & $ 0.66 $ & s$^{-1}$ \\ \hline
$q_{23}$ & $\alpha_2$ & $5 \times 10^2$ & s$^{-1}$ \\ \hline
$q_{32}$ & $\beta_2$ & $1.5 \times 10^4$ & s$^{-1}$ \\ \hline
$q_{34}$ & $2 k_{-2}$ & $ 4 \times 10^3$ & s$^{-1}$ \\ \hline
$q_{41}$ & $\beta_1$ & 15 & s$^{-1}$ \\ \hline
$q_{43}x(t)$ & $k_{+2}x$ & $(5 \times 10^8) x(t)$ & s$^{-1}$ \\ \hline
$q_{45}$ & $k_{-1}$ & $ 2 \times 10^3$ & s$^{-1}$ \\ \hline
$q_{54}x(t)$ & $2 k_{+1} x$ & $(1 \times 10^8) x(t)$ & s$^{-1}$ \\ \hline
\end{tabular}
\end{center}
\ \\
\caption{\label{tab:AChparameters}Rate parameters for ACh, adapted from \cite{col82}, where $x(t)$ represents the molar concentration of ACh in mol/$\ell$. Here we use a range of $x(t) \in [10^{-7},10^{-5}]$.}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=3in]{figures/ACh_rotated}
\end{center}
\caption{\label{fig:ACh} Depiction of allowed state transitions for ACh. Sensitive
transitions are depicted with {\bfseries bold} arrows.
States are labelled by their ion channel state: $\{\mathsf{C},\mathsf{O}\}$ for closed and open, respectively; state number is in subscript. Dashed lines surround all states in either the closed or open state. Transition rates, listed in Table \ref{tab:AChparameters}, correspond to the vertices associated with each directed edge: for example,
the rate from state $\mathsf{O}_2$ to state $\mathsf{C}_3$ is $q_{23}$.}
\end{figure}
\end{ex}
\begin{ex}[Calmodulin]
The Calmodulin (CaM) receptor is a ligand-gated receptor.
The CaM receptor consists of \pt{four} binding sites, \pt{two on the C-terminus of the CaM protein and two on the N-terminus \cite{ChinMeans2000TrendsCellBiol,DeMariaEtAlYue2001Nature,KellerFranksBartolSejnowski2008PLoSOne}.}
Each \pt{end of the protein} can bind 0, 1, or 2 \pt{calcium ions, leading to} nine possible states.
For CaM, rather than an ion channel, it is important whether the $\mathsf{C}$ or $\mathsf{N}$ end of the receptor is completely bound (i.e., has both binding sites occupied by ligands). This property is represented by four symbols: $\emptyset$ if neither end is completely bound; $\mathsf{C}$ if the $\mathsf{C}$ end is completely bound; $\mathsf{N}$ if the $\mathsf{N}$ end is completely bound; and $\mathsf{N}\mathsf{C}$ if both ends are completely bound.
\begin{figure*}[!t]
\normalsize
\begin{align}
\label{eqn:CaMQ}
Q &= \left[
\begin{array}{ccccccccc}
R_0 & q_{01}x(t) & 0 & q_{03}x(t) & 0 & 0 & 0 & 0 & 0 \\
q_{10} & R_1 & q_{12}x(t) & 0 &q_{14}x(t) & 0 & 0 & 0 & 0\\
0 & q_{21} & R_2 & 0 & 0 & q_{25}x(t) & 0 & 0 & 0 \\
q_{30} & 0 & 0 & R_3 & q_{34}x(t) & 0 & q_{36}x(t) & 0 & 0 \\
0 & q_{41} & 0 & q_{43} & R_4 & q_{45}x(t) & 0 & q_{47}x(t) & 0 \\
0 & 0 & q_{52} & 0 & q_{54} & R_5 & 0 & 0 & q_{58}x(t) \\
0 & 0 & 0 & q_{63} & 0 & 0 & R_6 & q_{67}x(t) &0 \\
0 & 0 & 0 & 0 & q_{74} & 0 & q_{76} & R_7 & q_{78}x(t) \\
0 & 0 & 0 & 0 & 0 & q_{85} & 0 & q_{87} & R_8
\end{array}
\right]
\end{align}
\hrulefill
\vspace*{4pt}
\end{figure*}
State configuration and allowed transitions are depicted in Figure \ref{fig:CaM}. The rate matrix is given in (\ref{eqn:CaMQ}),
with values given in Table \ref{tab:CaMparameters}, and where the molar concentration of calcium is $x(t) \in [10^{-7},10^{-6}]$.
\begin{table}\begin{center}
\begin{tabular}{c|c|c|c} \hline
Parameter & Name in \cite{faas11} & Value/range & Units \\ \hline
$q_{01}x(t)$, $q_{34}x(t)$, $q_{67}x(t)$
& $k_{\mathrm{on (T), N}} $
& $(7.7 \times 10^8) x(t)$ & s$^{-1}$ \\ \hline
$q_{10}$, $q_{43}$, $q_{76}$
& $k_{\mathrm{off (T), N}}$
& $1.6 \times 10^5$ & s$^{-1}$ \\ \hline
$q_{12}x(t)$, $q_{45}x(t)$, $q_{78}x(t)$
& $k_{\mathrm{on (R), N}}$
& $(3.2 \times 10^{10}) x(t)$ & s$^{-1}$ \\ \hline
$q_{21}$, $q_{54}$, $q_{87}$
& $k_{\mathrm{off (R), N}}$
& $2.2 \times 10^4$ & s$^{-1}$ \\ \hline
$q_{03}x(t)$, $q_{14}x(t)$, $q_{25}x(t)$
& $k_{\mathrm{on (T), C}}$
& $(8.4 \times 10^7) x(t)$ & s$^{-1}$ \\ \hline
$q_{30}$, $q_{41}$, $q_{52}$
& $k_{\mathrm{off (T), C}}$
& $2.6 \times 10^3$ & s$^{-1}$ \\ \hline
$q_{36}x(t)$, $q_{47}x(t)$, $q_{58}x(t)$
& $k_{\mathrm{on (R), C}}$
& $(2.5 \times 10^7) x(t)$ & s$^{-1}$ \\ \hline
$q_{63}$, $q_{74}$, $q_{85}$
& $k_{\mathrm{off (R), C}}$
& 6.5 & s$^{-1}$ \\ \hline
\end{tabular}
\end{center}
\ \\
\caption{\label{tab:CaMparameters}Rate parameters for CaM, adapted from \cite{faas11}, where $x(t) \in [10^{-7},10^{-6}]$ represents the molar concentration of calcium in mol/$\ell$.}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=3.25in]{figures/CaM}
\end{center}
\caption{\label{fig:CaM} Depiction of allowed state transitions for CaM. Sensitive
transitions are depicted with {\bfseries bold} arrows.
States are labelled by the status of the $\mathsf{C}$ or $\mathsf{N}$ end of the receptor: $\emptyset$ if neither end is completely bound; $\mathsf{C}$ if the $\mathsf{C}$ end is completely bound; $\mathsf{N}$ if the $\mathsf{N}$ end is completely bound; and $\mathsf{N}\mathsf{C}$ if both ends are completely bound. Transition rates, listed in Table \ref{tab:CaMparameters}, correspond to the vertices associated with each directed edge.}
\end{figure}
\end{ex}
For each of the preceding examples, the rate constants depend on environmental conditions, and
thus can be reported differently in different sources
(see, e.g., \cite{lin09} for different rate constants for ChR2).
\subsection{From the master equation to discrete-time Markov chains}
\rev{The continuous-time master equation for the receptor dynamics \eqref{eqn:MasterEquation} describes the evolution of a conditional probability $\mathbf{p}(t)\equiv E[Y(t)\:|\:\mathcal{F}_X(t)],$ where $Y(t)$ is the continuous time, discrete state \textit{c\`{a}dl\`{a}g}
process giving the channel state, $\mathcal{F}_X(t)$ is the filtration generated by the input process $X(t)$, and $E[\cdot\:|\:\cdot]$ is conditional expectation \cite{grimmett2001probability}.
Establishing the appropriate ensemble of input processes and analyzing mutual information and capacity involve technical issues that do not shed light on the nature of biological signal transduction.
Therefore we do not undertake a rigorous analysis of the continuous-time communications channels described by \eqref{eqn:MasterEquation} in this paper.
Rather, we introduce a discrete-time, discrete-state channel, motivated by the continuous-time channel, which can be rigorously analyzed, and study its properties both with a fixed timestep $\Delta t$, and later in the limit $\Delta t\to 0$.
The discrete-time Markov chain model allows us to rely on capacity results for discrete-time Markov channels.}
\rev{We obtain a discrete-time approximation to} the master equation by writing
\begin{align}
\label{eqn:Markov-1}
\frac{d\mathbf{p}(t)}{dt} = \mathbf{p}(t) Q
= \frac{\mathbf{p}(t + \Delta t) - \mathbf{p}(t)}{\Delta t} +o(\Delta t),\text{ as }\Delta t\to 0,
\end{align}
\rev{
where we simplify the notation by writing $Q(x(t))$ as simply $Q$.
Manipulating the middle and right expression in (\ref{eqn:Markov-1}) \pt{gives} }
\rev{\begin{align}
\mathbf{p}(t + \Delta t) &= \Delta t \,\mathbf{p}(t) Q + \mathbf{p}(t) +o(\Delta t)\\
&= \Delta t\, \mathbf{p}(t) Q + \mathbf{p}(t) I+o(\Delta t) \\
&= \mathbf{p}(t) \left( I + \Delta t Q \right)+o(\Delta t),\text{ as }\Delta t\to 0,
\end{align}}
where $I$ is the identity matrix.
\rev{In order to arrive at a discrete-time model, we introduce the approximation $\{\mathbf{p}_i\}_{i\in\mathbb{N}_+}$ satisfying}
\begin{equation}
\mathbf{p}_i = \mathbf{p}(i \Delta t)+o(\Delta t),\text{ as }\Delta t\to 0,
\end{equation}
\rev{and} arrive at a discrete-time approximation to (\ref{eqn:Markov-1}),
\begin{equation}
\mathbf{p}_{i+1} = \mathbf{p}_i (I+\Delta t\, Q).
\end{equation}
Thus, we have a discrete-time Markov chain with transition probability matrix
\begin{equation}
\label{eqn:Markov-last}
P = I + \Delta t Q.
\end{equation}
The matrix $P$ satisfies the conditions of a Markov chain transition
probability matrix (nonnegative, row-stochastic) as long as $\Delta t$ is small enough.
\rev{However, note that $P$ (and $Q$) are dependent on $x(t)$, so the Markov chain is not generally time-homogeneous if $x(t)$ is known (cf. (\ref{eqn:HomogeneousP})).}
\section{Signal transduction as a communications system}
In this section we give our main results, in which we describe and analyze signal transduction as a communication system. A brief roadmap to our results is given as follows: we first define the communication system in terms of input, output, and channel; we give the mutual information of the general discrete-time model under IID inputs (Theorem 1 and equation (\ref{eqn:GeneralIID1})); we take the continuous-time limit of the mutual information rate, showing that the expression for mutual information has a simple factorization (Theorem 2 and equation (\ref{eqn:Theorem1})); we give a physical interpretation of the factorization in (\ref{eqn:Theorem1}); we give general conditions under which the Shannon capacity is satisfied by IID inputs (Theorem 3); and finally, we give an example calculation using ChR2 (Example 4).
\subsection{Communication model of receptors}
We now discuss how the receptors can be described as
information-theoretic communication systems: that is, in terms of input, output, and conditional
input-output PMF.
{\em Input:} As discussed in Section II, the receptor is sensitive to given properties of the environment; previous examples included light intensity or ligand concentration. The receptor input $x(t)$ is the value of this property at the surface of the receptor.
The input is discretized in time: for integers $i$, the input is $x(i \Delta t)$; we will write
$x_i = x(i \Delta t)$.
We will also discretize the amplitude,
so that for every $t$, $x_i \in \{\mathsf{x}_1,\mathsf{x}_2,\mathsf{x}_3,\ldots,\mathsf{x}_k\} =: \mathcal{X}$.
We will assume that the $\mathsf{x}_i$ are
distinct and increasing; further, we assign the lowest and highest values special symbols:
\begin{align}
\xlevel_\mathsf{L} &:= \mathsf{x}_1 \\
\xlevel_\mathsf{H} &:= \mathsf{x}_k .
\end{align}
In Section II, we gave the concentrations or intensities over a range of values (such as $x(t) \in [0,1]$ for ChR2). Thus, we select $\xlevel_\mathsf{L}$ and $\xlevel_\mathsf{H}$ as the minimum and maximum values of this range, respectively.
{\em Output:}
In this paper, the output $y(t)$ of the communication system is
the receptor state number, given by the {\em subscript} of the state label: for example, if the state is $\mathsf{C}_3$, then $y(t) = 3$.
This is discretized to $y_i = y(i\Delta t)$. The discrete channel inputs and outputs form vectors: in terms of notation, we write $\mathbf{x} = [x_1,x_2,\ldots,x_n]$ and $\mathbf{y} = [y_1,y_2,\ldots,y_n]$.
{\em Conditional input-output PMF:}
From (\ref{eqn:Markov-1})--(\ref{eqn:Markov-last}), $\mathbf{y}$ forms a
Markov chain given $\mathbf{x}$, so
\begin{equation}
\label{eqn:ReceptorMarkov}
p(\mathbf{y}|\mathbf{x}) = \prod_{i=1}^n p(y_i \:|\: x_i,y_{i-1}) ,
\end{equation}
where $p(y_i \:|\: x_i,y_{i-1})$ is given by the appropriate entry in the matrix $P$, and where $y_0$ is null.%
\footnote{\rev{Notation: (1) We will drop subscripts if it is unambiguous to do so, i.e., normally $p(x)$ signifies $p_X(x)$; (2)} We say a variable is ``null'' if it vanishes under conditioning, i.e., if $y_0$ is null, then $p(y_1 \:|\: x_1, y_0) = p(y_1 \:|\: x_1)$.}
The following diagram \eqref{diag:xy} indicates the conditional dependencies:
\begin{equation}\label{diag:xy}
\begin{array}{ccccccccccc}
&X_1&&X_2&&X_3&&X_4&&X_5&\cdots\\
&\downarrow&&\downarrow&&\downarrow&&\downarrow&& \downarrow \\
(Y_0)&\longrightarrow&Y_1&\longrightarrow&Y_2&\longrightarrow&Y_3&\longrightarrow&Y_4&\cdots
\end{array}\end{equation}
As an example, consider ACh: suppose $y_{i-1} = 1$, $y_i = 2$, and $x_i = \xlevel_\mathsf{H}$. Then from
(\ref{eqn:Markov-last}) and Table \ref{tab:AChparameters}, we have
$p_{Y_i \:|\: Y_{i-1},X_i}(2 \:|\: 1,\xlevel_\mathsf{H}) = \Delta t q_{12}(t) = (5 \times 10^8) \xlevel_\mathsf{H} \Delta t$.
From (\ref{diag:xy}) and the definition of
$P$, $p(y_i \:|\: y_{i-1},x_i)$ does not depend on $i$;
that is, the channel's input-output structure is time-invariant.
For a discrete-time Markov chain, the receptor states form a graph with vertex set $\mathcal{Y}$ and directed edges $\mathcal{E}\subset\mathcal{Y}\times\mathcal{Y}$, with pair $(y_{i-1},y_i)\in\mathcal{E}$ if $\max_{x_i\in\mathcal{X}} p(y_i\:|\: x_i,y_{i-1}) > 0$, that is, for at least some input value there is a direct transition from $y_{i-1}$ to $y_i$. Notice that, under this definition, self-transitions are {\em included} in $\mathcal{E}$, even though (for convenience) they are not depicted in the state-transition diagrams.
We say the transition from state $y_{i-1}$ to $y_i$ is \emph{insensitive to the input}, or just \emph{insensitive}, if, for all $x_i \in \mathcal{X}$, we have $p(y_i \:|\: x_i,y_{i-1})=p(y_i \:|\: y_{i-1})$
(see Section \ref{sec:MasterEquation}).
Otherwise, the transition is \emph{sensitive}. We let $\mathcal{S}\subseteq\mathcal{E}$ denote the subset of sensitive edges.
(If state $y_{i-1} \in \mathcal{Y}$ is the origin for a sensitive transition, i.e., there is at least one $(y_{i-1},y_i \neq y_{i-1}) \in \mathcal{S}$, then the self-transition $(y_{i-1},y_i = y_{i-1})$ is normally sensitive as well, but this condition is not required for our analysis.)
For a channel with inputs $\mathbf{x}$ and outputs $\mathbf{y}$ \rev{(both of length $n$)}
the mutual information $I(\mathbf{X};\mathbf{Y})$ gives the maximum information rate that may be
transmitted reliably over the channel for a given input distribution.
Mutual information is given by
\begin{align}
I(\mathbf{X};\mathbf{Y})
\label{eqn:MutualInformation}
&= \sum_{\mathbf{x},\mathbf{y}} p(\mathbf{x}) p(\mathbf{y}\:|\:\mathbf{x})
\log \frac{p(\mathbf{y}\:|\: \mathbf{x})}{p(\mathbf{y})} ,
\end{align}
where $p(\mathbf{y}\:|\: \mathbf{x})$ is the conditional probability mass function (PMF) of $\mathbf{Y}$.
As $n \rightarrow \infty$, generally $I(\mathbf{X};\mathbf{Y}) \rightarrow \infty$ as well;
in this case, it is more useful to calculate the mutual information rate, \rev{which we introduce in the next section.}
\subsection{Receptor IID capacity}
\rev{Our focus in the remainder of this paper is on IID input distributions.} Although
IID inputs \rev{may not be} realistic for chemical diffusion channels, such as for ligand-gated receptors (as concentration
may persist for long periods of time), they can be capacity-achieving in these channels \rev{(see, e.g., \cite{ThomasEckford2016})}; moreover, IID input distributions may be physically realistic for light-gated channels.
\rev{Starting with (\ref{eqn:MutualInformation}), where $\mathbf{x}$ and $\mathbf{y}$ are both of fixed and finite length $n$, the Shannon capacity $C(n)$ is found by maximizing $I(\mathbf{X};\mathbf{Y})$ with respect to the input distribution $p(\mathbf{x})$, i.e.,
\begin{align}
\label{eqn:CapacityDefinition}
C(n) = \max_{p(\mathbf{x})} I(\mathbf{X};\mathbf{Y}) .
\end{align}
where the limit is taken over all possible length-$n$ input distributions $p(\mathbf{x})$ (not necessarily IID).}
\rev{
If the input $p(\mathbf{x})$ is restricted to the set of IID input distributions,
which is well defined for each $n$ (i.e., $p(\mathbf{x}) = \prod_{i=1}^n p(x_i)$),
then $I(\mathbf{X};\mathbf{Y})$ is also well defined for each $n$ (see (\ref{eqn:MutualInformation})). Furthermore, for each $n$
we have the IID capacity, written $C_{\mathrm{iid}}(n)$:
\begin{equation}
\label{eqn:IIDCapacityDefinition}
C_{\mathrm{iid}}(n) = \max_{p(x_i)} I(\mathbf{X};\mathbf{Y}) .
\end{equation}
where the maximum is taken over all possible settings of $p(x_i)$.
}
\rev{We can use (\ref{eqn:MutualInformation}) and (\ref{eqn:IIDCapacityDefinition}) to obtain information rates per channel use. For a given IID input distribution $p(x)$,
the IID mutual information rate is given by
\begin{align}
\label{eqn:InfoRateDefinition}
\mathcal{I}(X;Y) = \lim_{n \rightarrow \infty} \frac{1}{n} I(\mathbf{X};\mathbf{Y}).
\end{align}
Furthermore, the maximum IID information rate is given by
\begin{align}
\label{eqn:ciid}
C_{\mathrm{iid}} := \lim_{n \rightarrow \infty} \frac{1}{n} C_{\mathrm{iid}}(n) .
\end{align}
We derive these quantities in the remainer of the section, in which it will be clear that the limits in (\ref{eqn:InfoRateDefinition})---(\ref{eqn:ciid}) exist.
We start by deriving $I(\mathbf{X};\mathbf{Y})$ under IID inputs, and showing how it is calculated using quantities introduced in Section II. Finally, in Theorem \ref{thm:MutualInformationRate}, we give an expression for $\mathcal{I}(X;Y)$, and show that $\mathcal{I}(X;Y) = C_{\mathrm{iid}}$.
}
Recall $p(\mathbf{y}\:|\:\mathbf{x})$ from (\ref{eqn:ReceptorMarkov}).
Under IID inputs, it can be shown (see \cite{che05,ThomasEckford2016}) that the receptor states $Y^n$ form a time-homogeneous Markov
chain, that is,
\begin{equation}
\label{eqn:Conditional-2}
p(\mathbf{y}) = \prod_{i=1}^n p(y_i \:|\: y_{i-1}),
\end{equation}
where $y_0$ is again null, and where
\begin{equation}
\label{eqn:Conditional-2a}
p(y_i \:|\: y_{i-1}) = \sum_{x_i} p(y_i \:|\: x_i, y_{i-1}) p(x) .
\end{equation}
\rev{
Furthermore, let $\bar{P}$ represent the transition probability matrix of $Y^n$.
Recall (\ref{eqn:Markov-last}), in which $P$ was dependent on $x$; using (\ref{eqn:Conditional-2a}), we can write
\begin{align}
\label{eqn:HomogeneousP}
\bar{P} = E[P] = I + \Delta t E[Q],
\end{align}
and since the sensitive terms in $P$ and $Q$ are assumed to be linear in $x(t)$, we replace $x(t)$ in these terms with $E[x]$ to form $\bar{P}$ and $\bar{Q} := E[Q]$, respectively.
}
Using (\ref{eqn:ReceptorMarkov}) \rev{and} (\ref{eqn:Conditional-2}), (\ref{eqn:MutualInformation})
reduces to
\begin{equation}
\label{eqn:MutualInfoMarkov}
I(\mathbf{X};\mathbf{Y}) = \sum_{i=1}^n \sum_{y_i} \sum_{y_{i-1}} \sum_{x_i} p(y_i,x_i,y_{i-1}) \log \frac{p(y_i \:|\: x_i,y_{i-1})}
{ p(y_i \:|\: y_{i-1})} .
\end{equation}
\rev{
Recall that a transition may be sensitive ($(y_{i-1},y_i) \in \mathcal{S}$) or insensitive ($(y_{i-1},y_i) \not\in \mathcal{S}$). For terms in (\ref{eqn:MutualInfoMarkov}), consider the
insensitive transitions:
\begin{align}
\nonumber
\lefteqn{p(y_i,x_i,y_{i-1}) \log \frac{p(y_i \:|\: x_i,y_{i-1})}
{ p(y_i \:|\: y_{i-1})}}&\\
\label{eqn:Insensitive1}
&= p(y_i,x_i,y_{i-1}) \log \frac{p(y_i \:|\: y_{i-1})}
{ p(y_i \:|\: y_{i-1})} \\
&= p(y_i,x_i,y_{i-1}) \log 1\\
&= 0.
\end{align}
where (\ref{eqn:Insensitive1}) follows since the transition is insensitive, and is not a function of $x_i$; cf. (\ref{eqn:Conditional-2a}).} Thus for IID inputs, the mutual information (\ref{eqn:MutualInfoMarkov}) is calculated using the {\em sensitive transitions only}, i.e., those transitions in $\mathcal{S}$. \rev{With this in mind, we can rewrite (\ref{eqn:MutualInfoMarkov}) as
\begin{align}
\nonumber \lefteqn{I(\mathbf{X};\mathbf{Y}) }&\\
\label{eqn:MutualInfoMarkovSensitive0}
&= \sum_{i=1}^n \sum_{(y_{i-1},y_i)\in\mathcal{S}} \sum_{x_i} p(y_i,x_i,y_{i-1}) \log \frac{p(y_i \:|\: x_i,y_{i-1})}
{ p(y_i \:|\: y_{i-1})} \\
\label{eqn:MutualInfoMarkovSensitive}
&= \sum_{i=1}^n \sum_{\mathcal{A}_i}
p(y_i \:|\: x_i, y_{i-1}) p(y_{i-1}) p(x_i)
\log \frac{p(y_i \:|\: x_i,y_{i-1})}
{ p(y_i \:|\: y_{i-1})}
\end{align}
where we let $\mathcal{A}_i = \{y_i,y_{i-1},x_i : (y_i,y_{i-1})\in\mathcal{S}^2, x_i \in \mathcal{X}\}$, i.e.
the same terms as the sum in (\ref{eqn:MutualInfoMarkovSensitive}), for the sake of brevity.
Also note that (\ref{eqn:MutualInfoMarkovSensitive}) follows from (\ref{eqn:MutualInfoMarkovSensitive0}) because the input $\mathbf{X}$ is IID.
}
\rev{Now consider the individual PMFs in (\ref{eqn:MutualInfoMarkovSensitive}), starting with $p(y_i \:|\: x_i, y_{i-1})$.}
All transitions in $\mathcal{S}$ are dependent on the input $x_i$, and throughout this paper we assume that the
sensitive transition rates depend linearly on the \rev{input signal intensity}. Thus \rev{(recall (\ref{eqn:Markov-last}))}
for non-self-transitions $(y_{i-1},y_i) \in \mathcal{S}$ (i.e., $y_{i-1} \neq y_i$),
\begin{equation}
\label{eqn:TransitionsInS}
p(y_i \:|\: x_i,y_{i-1}) = q_{y_{i-1}y_i} x_i \Delta t .
\end{equation}
For self-transitions in $\mathcal{S}$ (i.e., $y_{i-1}\equiv y_i=y$) we have
\begin{align}
\nonumber\lefteqn{p_{Y_i|X_i,Y_{i-1}}(y \:|\: x_i,y) =}&\\
\label{eqn:SelfTransition}
& 1-\left(\sum_{y'\not=y,(y',y)\in\mathcal{S}}q_{yy'}x_i - \sum_{y'\not=y,(y',y)\not\in\mathcal{S}}q_{yy'}\right)\Delta t ,
\end{align}
as seen in the diagonal entries of (\ref{eqn:Markov-last}).
\rev{Similarly, the terms $p(y_i \:|\: y_{i-1})$ can be obtained
using (\ref{eqn:Conditional-2a})--(\ref{eqn:HomogeneousP}); we replace $x_i$
in (\ref{eqn:TransitionsInS})--(\ref{eqn:SelfTransition}) with $\bar{x}$.}
\rev{The terms $p(y_{i-1})$ represent the steady-state marginal probability that the receptor is in state $y$; for compact notation, let $\pi_{y_{i-1}} = p(y_{i-1})$.}
If the input $x$ is IID, as we assume throughout this paper, then $\pi_{y_{i-1}}$ exists if the Markov chain is irreducible, aperiodic, and positive recurrent; these conditions hold for all the examples we consider \rev{(recall (\ref{eqn:Conditional-2})--(\ref{eqn:HomogeneousP})).
\footnote{
\rev{For clarity, although $\pi_y$ may be written with a time-indexing subscript, e.g. $\pi_{y_i}$, this refers to the steady-state distribution of state $y_i \in \mathcal{Y}$, and does not imply that $\pi_{y}$ changes with time.}
}
Define the partial entropy function
\begin{equation}
\label{eqn:PhiDefinition}
\phi(p)=
\begin{cases}
0,&p=0\\
p\log p,&p\not=0
\end{cases}
\end{equation}
and let
\begin{equation}
\label{eqn:BinEnt}
\mathscr{H}(p) = -\phi(p) - \phi(1-p)
\end{equation}
represent the binary entropy function.
\rev{Then we have the following result.}
\rev{
\begin{theorem}
\label{thm:MutualInformationRate}
For an IID input distribution $p(x_i)$, the mutual information rate $\mathcal{I}(X;Y)$ is given by
\begin{align}
\nonumber \lefteqn{\mathcal{I}(X;Y)} & \\
\nonumber
&= \sum_{(y_{i-1},y_i) \in \mathcal{S}} \pi_{y_{i-1}}
\Bigg(
\sum_{x_i \in \mathcal{X}} p(x_i) \phi\Big(p(y_i \:|\: x_i,y_{i-1})\Big)\\
\label{eqn:GeneralIID1}
&\quad\quad- \phi\left(\sum_{x_i \in \mathcal{X}} p(x_i) p(y_i \:|\: x_i,y_{i-1}) \right)
\Bigg).
\end{align}
Furthermore, $C_{\mathrm{iid}} = \max_{p(x)} \mathcal{I}(X;Y)$.
\end{theorem}
\begin{proof}
Divide the terms in (\ref{eqn:MutualInfoMarkovSensitive}) into the $i=1$ term, and all the remaining terms.
Let $T_1(p(x_i))$ represent the $i=1$ term, emphasizing its dependence on the IID input distribution $p(x_i)$, so that
%
\begin{align}
T_1(p(x_i)) &= p(y_1 \:|\: x_1, y_0) p(y_1) p(x_1)
\log \frac{p(y_1 \:|\: x_1,y_0)}
{ p(y_1 \:|\: y_0)}\\
\label{eqn:T1}
&=p(y_1 \:|\: x_1) p(y_1) p(x_1)
\log \frac{p(y_1 \:|\: x_1)}
{ p(y_1)} ,
\end{align}
%
where (\ref{eqn:T1}) follows since $y_0$ is null. Let $T_2(p(x_i),n)$ represent the remaining terms, again dependent on $p(x_i)$ but also on $n$, so that
%
\begin{align}
\nonumber \lefteqn{T_2(p(x_i),n)}&\\
&= \sum_{i=2}^n \sum_{\mathcal{A}_i}
p(y_i \:|\: x_i, y_{i-1}) p(y_{i-1}) p(x_i)
\log \frac{p(y_i \:|\: x_i,y_{i-1})}
{ p(y_i \:|\: y_{i-1})}\\
\label{eqn:ProofMIRate0}
&= (n-1) \sum_{\mathcal{A}_i}
p(y_i \:|\: x_i, y_{i-1}) p(y_{i-1}) p(x_i)
\log \frac{p(y_i \:|\: x_i,y_{i-1})}
{ p(y_i \:|\: y_{i-1})}
\end{align}
%
recalling the definition of $\mathcal{A}_i$ from the discussion after (\ref{eqn:MutualInfoMarkovSensitive}).
Using (\ref{eqn:InfoRateDefinition}),
%
\begin{align}
\nonumber\lefteqn{\mathcal{I}(X;Y)}&\\
&=
\lim_{n \rightarrow \infty} \frac{T_1(p(x_i))}{n} + \lim_{n \rightarrow \infty} \frac{T_2(p(x_i),n)}{n} \\
\label{eqn:ProofMIRate1}
&= \sum_{\mathcal{A}_i}
p(y_i \:|\: x_i, y_{i-1}) p(y_{i-1}) p(x_i)
\log \frac{p(y_i \:|\: x_i,y_{i-1})}
{ p(y_i \:|\: y_{i-1})} ,
\end{align}
%
and (\ref{eqn:GeneralIID1}) follows after some manipulation.
To show that $C_{\mathrm{iid}} = \max_{p_X(x)} \mathcal{I}(X;Y)$,
recall the definitions of $C_{\mathrm{iid}}(n)$ and $C_{\mathrm{iid}}$ in (\ref{eqn:IIDCapacityDefinition}) and (\ref{eqn:ciid}), respectively.
Referring to $p(x_i)$ as $p$ for brevity,
%
\begin{align}
C_{\mathrm{iid}}(n) = \max_{p} \big(T_1(p) + T_2(p,n)\big) .
\end{align}
%
Let $p_1$ represent the IID input distribution maximizing the term $T_1(p)$, and let $p_2$ represent the IID input distribution maximizing the term $T_2(p,n)$. From (\ref{eqn:ProofMIRate0}), $p_2$ is independent of $n$. Furthermore,
%
\begin{align}
\label{eqn:ProofMIRate2}
\frac{T_1(p_2)}{n} + \frac{T_2(p_2,n)}{n} \leq \frac{1}{n}C_{\mathrm{iid}}(n) \leq \frac{T_1(p_1)}{n} + \frac{T_2(p_2,n)}{n} .
\end{align}
%
Taking the limit throughout (\ref{eqn:ProofMIRate2}) as $n \rightarrow \infty$, the $T_1$ terms vanish as they are constant with respect to $n$. Comparing (\ref{eqn:ProofMIRate0}) and (\ref{eqn:ProofMIRate1}), $p_2$ also maximizes $\mathcal{I}(X;Y)$. The result follows.
\end{proof}
}
\subsection{Limit of $\mathcal{I}(X;Y)/\Delta t$ as $\Delta t \rightarrow 0$}
In this section we consider the {\em continuous time limit} of $\mathcal{I}(X;Y) / \Delta t$ as $\Delta t \rightarrow 0$, and give \rev{our second main result} (Theorem \ref{thm:Theorem1}): that in the continuous time limit, the mutual information rate is expressed simply as a product of the average flux through sensitive edges, and the relative entropy between the prior distribution on $x$, and the posterior given a transition. \rev{While we do not claim to derive the mutual information rate of the continuous time channel, the continuous time limit of the discrete-time mutual information rate is a quantity of interest in its own right.}
\rev{
First, we show that the steady-state distribution $\pi_y$ is independent of $\Delta t$:}
\rev{
\begin{lemma}
\label{lem:SteadyState}
Suppose $\pi_y$ is the normalized left eigenvector of $\bar{Q}$ with eigenvalue 0 (see (\ref{eqn:HomogeneousP})). Define the set $\mathcal{T}$ so that $\Delta t \in \mathcal{T}$ if $P$ from (\ref{eqn:Markov-last}) is a valid transition probability matrix for all $x \in \mathcal{X}$. Then $\pi_y$ is the normalized left eigenvector of $\bar{P}$ with eigenvalue 1, for all $\Delta t \in \mathcal{T}$.
\end{lemma}
\begin{proof}
The proof is given in the appendix.
\end{proof}
Note that $\mathcal{T}$ contains all ``sufficiently small'' $\Delta t$. It follows from the lemma that the steady state distribution $\pi_y$ is the same for both continuous and discrete time.
}
Note that the mutual information rate $\mathcal{I}(X;Y)$ in (\ref{eqn:GeneralIID1}) has units of nats per channel use, and that channel uses have duration $\Delta t$. Moreover, the transition probabilities $p(y_i \:|\: x_i,y_{i-1})$ in (\ref{eqn:TransitionsInS})\rev{--}(\ref{eqn:SelfTransition}) are linear functions of $\Delta t$.
Substituting \rev{the discrete-time transition probabilities \eqref{eqn:Markov-last}} into (\ref{eqn:GeneralIID1}), the non-self-transition probabilities go to zero while the self-transition probabilities go to 1, so
$\mathcal{I}(X;Y) \rightarrow 0$ as $\Delta t \rightarrow 0$. This should not be surprising: intuitively, as the time step shrinks, less information can be expressed per time step.
However, dividing by $\Delta t$ (and obtaining $\mathcal{I}(X;Y) / \Delta t$),
the information rate {\em per second} is finite.
It is then useful to consider how this rate behaves as $\Delta t \rightarrow 0$.
Let $\mathcal{S}^\prime \subset \mathcal{S}$ represent the set of sensitive transitions excluding self transitions, i.e.,
\begin{equation}
\mathcal{S}^\prime = \{(y_{i-1},y_i) : (y_{i-1},y_i) \in \mathcal{S}, y_{i-1} \neq y_i \} .
\end{equation}
Also let $\mathcal{S}\backslash\mathcal{S}^\prime$ represent the components of $\mathcal{S}$ excluding $\mathcal{S}^\prime$ (i.e., {\em only} the sensitive self transitions).
For any edge $(y,y')$ define the limiting value of that edge's contribution to the mutual information rate, as $\Delta t\to 0$, as
\begin{align}
\nonumber \lefteqn{\iota(y,y') =} & \\
\nonumber
& \lim_{\Delta t\to 0}\frac1{\Delta t} \pi_{y}
\Bigg(
\sum_{x \in \mathcal{X}} p(x) \phi\Big(p(y' \:|\: x,y)\Big)\\
&
\quad\quad - \phi\left(\sum_{x \in \mathcal{X}} p(x) p(y' \:|\: x,y) \right)
\Bigg)
\label{eq:iota}
\end{align}
The limit calculation depends on whether $y=y'$. In case $y\not=y'$, we have $p(y'\:|\: x,y)=q_{yy'}x\Delta t$ (see \eqref{eqn:TransitionsInS}) and
\begin{align}
&\sum_{x}p(x)\phi(p(y'\:|\: x,y))-\phi\left(\sum_x p(x)p(y'\:|\: x,y) \right) \nonumber \\ \nonumber
&=\Delta t\Bigg\{
\left(\sum_x q x p(x) \right)\log q +
\left(
\sum_x q p(x) x\log x
\right) \\
\nonumber
& \quad\quad - \left(\sum_x q x p(x) \right)\log\left( \sum_x q x p(x) \right)
\Bigg\} \\
& \quad\quad +o(\Delta t),\text{ as }\Delta t\to 0^+\\
&= q\Delta t(E(x\log x) - E(x)\log(E(x))) +o(\Delta t),\text{ as }\Delta t\to 0^+\\
&=q\Delta t(E\phi(x) - \phi(Ex))+o(\Delta t),\text{ as }\Delta t\to 0^+.
\end{align}
On the other hand, in the case when $y=y'$, $\sum_{x}p(x)\phi(p(y'\:|\: x,y))-\phi\left(\sum_x p(x)p(y'\:|\: x,y) \right)=o(\Delta t)$, as $\Delta t\to 0^+$. Therefore, these terms do not contribute to the mutual information.
Using these \rev{results}, we can rewrite (\ref{eqn:GeneralIID1}) as
\rev{\begin{align}
\nonumber\lefteqn{\lim_{\Delta t \rightarrow 0}\frac{\mathcal{I}(X;Y)}{\Delta t}}&\\
&= \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \iota(y_{i-1},y_i)
+\sum_{(y_{i-1},y_i) \in \mathcal{S} \backslash \mathcal{S}^\prime}\iota(y_{i-1},y_i).
\label{eqn:GeneralIID2}
\end{align}}
Using (\ref{eqn:TransitionsInS})\rev{--}(\ref{eqn:SelfTransition}), we consider the two additive terms in (\ref{eqn:GeneralIID2}) separately.
For the first term (summing over $\mathcal{S}^\prime$),
we use l'H\^opital's rule:
in the denominator we have (trivially)
\begin{align}
\label{eqn:limit-denominator}
\frac{d}{d\Delta t} \Delta t &= 1,
\end{align}
and from the numerator, we have
\begin{align}
\nonumber \lefteqn{
\lim_{\Delta t \rightarrow 0} \frac{d}{d\Delta t} \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}}
\Bigg(
\sum_{x_i \in \mathcal{X}} p(x_i) \phi\Big(q_{y_{i-1}y_i} x_i \Delta t \Big)}\\
&\quad
- \phi\left(\sum_{x_i \in \mathcal{X}} p(x_i) q_{y_{i-1}y_i} x_i \Delta t \right)
\Bigg) \\
\nonumber
&= \lim_{\Delta t \rightarrow 0} \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}}
\Bigg(
\sum_{x_i \in \mathcal{X}} p(x_i) \frac{d}{d\Delta t} \phi\Big(q_{y_{i-1}y_i} x_i \Delta t \Big)\\
&\quad\quad - \frac{d}{d\Delta t} \phi\left(\sum_{x_i \in \mathcal{X}} p(x_i) q_{y_{i-1}y_i} x_i \Delta t \right)
\Bigg)\\
\nonumber
&= \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} \Bigg( \sum_{x_i \in \mathcal{X}}p(x_i) q_{y_{i-1}y_i}x_i \log (q_{y_{i-1}y_i}x_i)\\
&\quad\quad - q_{y_{i-1}y_i} \bar{x} \log (q_{y_{i-1}y_i} \bar{x})
\Bigg)
\label{eqn:LimitMutualInfo}
\end{align}
where $\bar{x} = \sum_{x_i \in \mathcal{X}} x_i p(x_i)$ is the average input concentration.
For the second term (summing over $\mathcal{S}\backslash\mathcal{S}^\prime$), a similar derivation shows that the limit is zero.
Simplifying further, we have
\begin{align}
\nonumber\lefteqn{\lim_{\Delta t \rightarrow 0} \frac{\mathcal{I}(X;Y)}{\Delta t}}&\\
\nonumber
&= \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} \Bigg( \sum_{x_i \in \mathcal{X}}p(x_i) q_{y_{i-1}y_i}x_i \log (q_{y_{i-1}y_i}x_i)\\
&\quad\quad- q_{y_{i-1}y_i} \bar{x} \log (q_{y_{i-1}y_i} \bar{x}) \Bigg) \\
\nonumber
&= \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i}
\bar{x}\log (q_{y_{i-1}y_i}) \\
\nonumber
&\quad\quad+
\sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i}\sum_{x_i \in \mathcal{X}}p(x_i) x_i\log(x_i) \\
\nonumber
&\quad\quad- \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i} \bar{x} \log (q_{y_{i-1}y_i})\\
&\quad\quad- \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i} \bar{x} \log(\bar{x}) \\
\nonumber
&= \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i}\sum_{x_i \in \mathcal{X}}p(x_i) x_i\log(x_i)\\
&\quad\quad
- \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i} \bar{x} \log(\bar{x}) \\
\nonumber
&= \left( \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i} \right)\\
&\quad\quad \cdot
\left( \sum_{x_i \in \mathcal{X}}p(x_i) x_i\log(x_i) - \bar{x}\log\bar{x} \right) .
\label{eqn:FactoredMutualInformation}
\end{align}
The {\em steady-state flux} $J_{y_{i-1}y_i}$ through an edge $(y_{i-1},y_i)$ in the state transition graph
is defined as
\begin{equation}
J_{y_{i-1}y_i} := \pi_{y_{i-1}} q_{y_{i-1}y_i} \bar{x} .
\end{equation}
Similarly, the {\em net steady-state flux} through the sensitive (non-self) edges in the graph is
\begin{align}
J_{\mathcal{S}^\prime} &:= \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} J_{y_{i-1}y_i} \\
\label{eqn:SteadyStateFlux}
&= \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i} \bar{x} ,
\end{align}
Expressing (\ref{eqn:FactoredMutualInformation}) in terms of $J_{\mathcal{S}^\prime}$,
we have
\begin{align}
\lim_{\Delta t \rightarrow 0} \frac{\mathcal{I}(X;Y)}{\Delta t}
&= \frac{1}{\bar{x}} J_{\mathcal{S}^\prime}
\left( \sum_{x_i \in \mathcal{X}}p(x_i) x_i\log x_i - \bar{x}\log\bar{x} \right) \\
\label{eqn:FactoredMutualInformation2}
&= J_{\mathcal{S}^\prime}
\left( \sum_{x_i \in \mathcal{X}} \frac{p(x_i)x_i}{\bar{x}}\log x_i - \log\bar{x} \right)
\end{align}
\rev{We define
\begin{equation}
\label{eqn:nu}
\nu(x_i) := \frac{p(x_i) x_i}{\bar{x}} .
\end{equation}
}
Since $\nu(x_i)$ is positive for all $x_i$, and since
\begin{align}
\sum_i \nu(x_i) &= \sum_i \frac{p(x_i) x_i}{\bar{x}} = \frac{\bar{x}}{\bar{x}} = 1,
\end{align}
\rev{it follows that} $\nu(x_i)$ forms a probability distribution, in general different from $p(x_i)$.
We discuss the physical interpretation of $J_{\mathcal{S}^\prime}$ and
$\nu(x_i)$ in the next section.
Using $\nu(x_i)$, we can rewrite (\ref{eqn:FactoredMutualInformation2}) as
\begin{align}
\lim_{\Delta t \rightarrow 0} \frac{\mathcal{I}(X;Y)}{\Delta t}
&= J_{\mathcal{S}^\prime}
\left( \sum_{x_i \in \mathcal{X}} \nu(x_i) \log x_i - \log\bar{x} \right) \\
&= J_{\mathcal{S}^\prime}
\left( \sum_{x_i \in \mathcal{X}} \nu(x_i) \log x_i - \sum_{x_i \in \mathcal{X}} \nu(x_i) \log\bar{x} \right) \\
&= J_{\mathcal{S}^\prime}
\left( \sum_{x_i \in \mathcal{X}} \nu(x_i) \log \frac{x_i}{\bar{x}} \right) \\
&= J_{\mathcal{S}^\prime}
\left( \sum_{x_i \in \mathcal{X}} \nu(x_i) \log \frac{p(x_i)x_i}{p(x_i)\bar{x}} \right)\\
&= J_{\mathcal{S}^\prime}
\left( \sum_{x_i \in \mathcal{X}} \nu(x_i) \log \frac{\nu(x_i)}{p(x_i)} \right)\\
&= J_{\mathcal{S}^\prime} D(\nu \:\rev{\|}\: p) ,
\end{align}
where $D(\cdot\:\rev{\|}\:\cdot)$ represents the Kullback-Leibler divergence.
The preceding derivation, \rev{including Lemma \ref{lem:SteadyState},} allows us to state the following result.
\begin{theorem}
\label{thm:Theorem1}
For finite-state Markov signal transduction systems described by (\ref{eqn:Markov-last}), with IID inputs,
%
\begin{equation}
\label{eqn:Theorem1}
\lim_{\Delta t \rightarrow 0} \frac{\mathcal{I}(X;Y)}{\Delta t} = J_{\mathcal{S}^\prime} D(\nu \:\rev{\|}\: p) ,
\end{equation}
%
with $J_{\mathcal{S}^\prime}$ defined in (\ref{eqn:SteadyStateFlux}) and $\nu$ defined in (\ref{eqn:nu}).
\end{theorem}
\subsection{Physical interpretation}
The factorization in (\ref{eqn:FactoredMutualInformation}) gives us a useful physical interpretation of
the mutual information in this system.
First consider $J_{\mathcal{S}^\prime}$. Physically, if one watched only for transitions along
edge $(y_{i-1},y_i)$ (with the rest of the graph assumed to be at steady state),
$J_{y_{i-1}y_i}$ gives
the average rate at which those transitions would be observed; that is, $J_{y_{i-1}y_i}$ is the \pt{mean} flux through the transition $y_{i-1} \rightarrow y_i$. Thus, $J_{\mathcal{S}^\prime}$ is the average rate through {\em all} the sensitive edges, i.e., the {\em net flux}.
Now consider $D(\nu \:\rev{\|}\: p)$, and note that the distribution $\nu(x_i)$ is a {\em posterior} distribution of $x_i$. To see this,
consider a random variable $y \in \{0,1\}$, with conditional distribution
\begin{equation}
\label{eqn:YXconditional}
p_{Y|X}(1 \:|\: x_i) =\kappa x_i ,
\end{equation}
where $\kappa$ is a positive constant ($0 \leq \kappa \leq \frac{1}{x_i}$ to make a valid probability),
and $p_{Y|X}(0 \:|\: x_i) = 1- p_{Y|X}(1 \:|\: x_i)$.
The marginal distribution $p_Y(1)$ is then given by
\begin{align}
p_Y(1) &= \sum_{x_i} p(x_i) p_{Y|X}(1 \:|\: x_i)\\
&= \sum_{x_i} p(x_i) \kappa x_i\\
&= \kappa \bar{x} .
\end{align}
With this definition, $\nu(x_i)$ is the posterior distribution of $x$ given $y=1$:
\begin{align}
p_{X|Y}(x_i \:|\: 1) &= \frac{p(x_i)p_{Y|X}(1 \:|\: x_i)}{p_Y(1)}\\
&= \frac{p(x_i) \kappa x_i}{\kappa \bar{x}}\\
&= \frac{p(x_i) x_i}{\bar{x}} = \nu(x_i) .
\end{align}
Physically, consider the example of a ligand-gated channel where $x_i$ is the concentration of ligands near the receptor at input $i$. With $i \in \{\mathsf{L},\mathsf{H}\}$ (i.e., inputs $x_\mathsf{L}$ and $x_\mathsf{H}$), suppose we select one molecule at random from those near the receptor, and set $y = 1$ if the molecule is a ligand; $y = 0$ otherwise. Then $p_{Y|X}(1 \:|\: x_\mathsf{L}) \propto x_L$ and $p_{Y|X}(1 \:|\: x_\mathsf{H}) \propto x_\mathsf{H}$, with $\kappa$ as the constant of proportionality; this satisfies (\ref{eqn:YXconditional}). For example, suppose $x_\mathsf{L}$ is measured in {\em number concentration} of ligands, i.e., number of ligands per volume $V$. Then $p_{Y|X}(1 \:|\: x_i) = x_i / n$ (for $i \in \{\mathsf{L},\mathsf{H}\}$), where $n$ is the number concentration of all molecules, ligands and otherwise, near the receptor, and $\kappa = 1/n$.
In general, physical systems where the probability of response $p(y \:|\: x)$ is directly proportional to the input $x$
fit into this framework, emphasizing the importance of this modeling assumption made in Section II.
\subsection{Shannon capacity of receptors with a single sensitive non-self transition}
We now give our \rev{third} main result, showing that the Shannon capacity $C$ is equal to the IID capacity $C_{\mathrm{iid}}$ for a number of sensitive transitions $|\mathcal{S}^\prime| \leq 1$, and furthermore that the capacity-achieving distribution has a simple form. As a consequence, this leads directly to the Shannon capacity of ChR2; we give this capacity in the example below. The result is a generalization of related results in \cite{ThomasEckford2016}.
Recall $\mathcal{S}^\prime \subset \mathcal{S}$ represent the set of transitions, {\em excluding} self-transitions.
\begin{theorem}
\label{thm:capacity}
For any receptor with $|\mathcal{S}^\prime| \leq 1$,
\begin{enumerate}
\item $C_{\mathrm{iid}}$ is achieved with all probability mass on $x_\mathsf{L}$ and $x_\mathsf{H}$; and
\item $C = C_{\mathrm{iid}}$.
\end{enumerate}
\end{theorem}
\begin{proof}
The case of $|\mathcal{S}^\prime| = 0$ is trivial: the state is never sensitive to the input, so $\mathcal{I}(X;Y) = 0$ for all input distributions.
Now consider $|\mathcal{S}^\prime| = 1$. We sketch the proof: results in \cite{ThomasEckford2016} were presented for a two-state receptor where only one transition was sensitive; many of the results have the same form.
The first part of the theorem follows from \cite[Thm 1]{ThomasEckford2016}, noting from (\ref{eqn:GeneralIID1}) that any system with $|\mathcal{S}^\prime| = 1$ has the same form, apart from the marginal distribution $\pi_{y_{i-1}}$, which is held constant in the proof of \cite[Thm 1]{ThomasEckford2016}. The second part of the theorem follows from \cite[Thm 2]{ThomasEckford2016}, noting that $C$ is only a function of the input distribution in the sensitive state.
\end{proof}
\subsection{Example}
We now give an example calculation of the mutual information and IID capacity, \pt{by which we obtain the channel capacity of channelrhodopsin.}
\begin{ex}[ChR2]
\label{ex:ChR2-MI}
Referring to the rate matrix for ChR2 (\ref{eqn:ChR2-rate-matrix}),
there are exactly two sensitive transitions: first, the
transition from $\mathsf{C}_1$ to $\O_2$, represented
by $q_{12} x(t)$; and second, the self-transition from $\mathsf{C}_1$ to $\mathsf{C}_1$, represented by
$R_1 = - q_{12}x(t)$. Thus, $\mathcal{S} = \{(\mathsf{C}_1,\O_2), (\mathsf{C}_1,\mathsf{C}_1)\}$
and $\mathcal{S}^\prime = \{(\mathsf{C}_1,\O_2)\}$.
Suppose $\mathcal{X} = \{x_\mathsf{L}, x_\mathsf{H}\}$, i.e., the input light source can only be off ($x_\mathsf{L}$) or on ($x_\mathsf{H}$). Let $p_\mathsf{L} = {\mathrm{Pr}}(x = x_\mathsf{L})$ and $p_\mathsf{H} = {\mathrm{Pr}}(x = x_\mathsf{H}) = 1-p_\mathsf{L}$.
Recalling the transformation of
rates into probabilities (\ref{eqn:Markov-last}), and substituting into (\ref{eqn:GeneralIID1}), we have
%
\begin{align}
\nonumber\lefteqn{\mathcal{I}(X;Y)} &\\
\nonumber &=
\pi_{\mathsf{C}_1} \Big( p_\mathsf{L} \phi(\Delta t q_{12}x_\mathsf{L}) + p_\mathsf{H} \phi(\Delta t q_{12}x_\mathsf{H})\\
\nonumber
&\quad\quad- \phi \big( p_\mathsf{L} \Delta t q_{12}x_\mathsf{L} + p_\mathsf{H} \Delta t q_{12}x_\mathsf{H} \big) \Big)\\
\nonumber
& \quad + \: \pi_{\mathsf{C}_1} \Big( p_\mathsf{L} \phi(1-\Delta t q_{12}x_\mathsf{L}) + p_\mathsf{H} \phi(1- \Delta t q_{12}x_\mathsf{H})\\
&\quad\quad - \phi \big( 1-p_\mathsf{L} \Delta t q_{12}x_\mathsf{L} - p_\mathsf{H} \Delta t q_{12}x_\mathsf{H} \big) \Big)
\end{align}
%
where the first term represents the transition $(\mathsf{C}_1,\O_2)$, and the second term represents the self-transition $(\mathsf{C}_1,\mathsf{C}_1)$, both of which are sensitive.
Continuing the derivation,
\begin{align}
\nonumber\lefteqn{\mathcal{I}(X;Y)} &\\
\nonumber
&= \pi_{\mathsf{C}_1} \Big(
\binent(p_\mathsf{L} \Delta t q_{12}x_\mathsf{L} + p_\mathsf{H} \Delta t q_{12}x_\mathsf{H})\\
&\quad\quad - p_\mathsf{L} \binent(\Delta t q_{12}x_\mathsf{L}) - p_\mathsf{H} \binent(\Delta t q_{12}x_\mathsf{H})
\Big) \\
\label{eqn:ChR2MutualInformationExample}
&= \left( \frac{q_{23}q_{31}}{q_{23}q_{31} + \bar{x}q_{12}q_{31} + \bar{x}q_{12}q_{23}} \right)\\
&\quad\quad\cdot
\Big( \binent(\Delta t q_{12} \bar{x}) - p_\mathsf{L} \binent(\Delta t q_{12}x_\mathsf{L}) - p_\mathsf{H} \binent(\Delta t q_{12}x_\mathsf{H})
\Big) ,
\end{align}
where $\bar{x}$ is the average input.
Finally, consider $\mathcal{I}(X;Y) / \Delta t$ as $\Delta t \rightarrow 0$, as in (\ref{eqn:LimitMutualInfo}).
The steady-state occupancy probability of $\mathsf{C}_1$, $\pi_{\mathsf{C}_1}$, is independent of $\Delta t$. Thus,
from Theorem \ref{thm:Theorem1}, we have
\begin{align}
\nonumber\lefteqn{\lim_{\Delta t \rightarrow 0} \frac{\mathcal{I}(X;Y)}{\Delta t}}&\\
&= J_{\mathcal{S}^\prime} D(\nu \:\rev{\|}\: p)\\
&=
\left( \sum_{(y_{i-1},y_i) \in \mathcal{S}^\prime} \pi_{y_{i-1}} q_{y_{i-1}y_i} \right)
\left( \sum_{x_i \in \mathcal{X}}\nu(x_i) \log\frac{\nu(x_i)}{p(x_i)} \right) \\
&=
\pi_{\mathsf{C}_1} q_{12}
\left( p_\mathsf{L}\frac{x_\mathsf{L}}{\bar{x}} \log \frac{x_\mathsf{L}}{\bar{x}} + p_\mathsf{H}\frac{x_\mathsf{H}}{\bar{x}} \log \frac{x_\mathsf{H}}{\bar{x}} \right) \\
\nonumber
&= \frac{q_{12}q_{23}q_{31}}{q_{23}q_{31} + \bar{x}q_{12}q_{31} + \bar{x}q_{12}q_{23}}\\
&\quad\quad\cdot
\left( p_\mathsf{L}\frac{x_\mathsf{L}}{\bar{x}} \log \frac{x_\mathsf{L}}{\bar{x}} + p_\mathsf{H}\frac{x_\mathsf{H}}{\bar{x}} \log \frac{x_\mathsf{H}}{\bar{x}} \right) .
\label{eqn:ChR2MutualInformationExample2}
\end{align}
In Figure \ref{fig:MutualInformationFigure}, we illustrate the effect of step size on the mutual information calculation, using (\ref{eqn:ChR2MutualInformationExample}) for the solid lines (for various values of $\Delta t > 0$) and (\ref{eqn:ChR2MutualInformationExample2}) for the dotted line (as $\Delta t \rightarrow 0$). From this figure, the IID capacity and the capacity-achieving value of $p_\mathsf{L}$ may be found by taking the maximum over the curve of interest. This value clearly changes for different values of $\Delta t$; however, the IID capacity is around $C_{\mathrm{iid}} \approx 65$ bits/s, and the capacity-achieving $p_\mathsf{L}$ is around $p_\mathsf{L} \approx 0.99$. For any finite value of $\Delta t$, it is interesting to note that the discrete-time approximation over-estimates the mutual information as $\Delta t \rightarrow 0$.
\end{ex}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=3.5in]{figures/MutualInformationFigure.pdf}
\end{center}
\caption{\label{fig:MutualInformationFigure} Plot for ChR2, illustrating the effect of $\Delta t$ on $\mathcal{I}(X;Y)$ from (\ref{eqn:ChR2MutualInformationExample}). The dashed black line represents $\Delta t \rightarrow 0$. Solid lines, from bottom, represent: $\Delta t = 0.01$ (blue), $\Delta t = 0.02$ (green), $\Delta t = 0.04$ (red), $\Delta t = 0.06$ (cyan), $\Delta t = 0.08$ (magenta), and $\Delta t = 0.1$ (tan), all in milliseconds.}
\end{figure}
From Example \ref{ex:ChR2-MI}, ChR2 has $|\mathcal{S}^\prime| = 1$. Thus, {\em ChR2 satisfies the conditions of Theorem \ref{thm:capacity}}, and has $C = C_{\mathrm{iid}}$, where $C_{\mathrm{iid}}$ is given in (\ref{eqn:ChR2MutualInformationExample2}).
\rev{Performing the maximization numerically, on the $\Delta t \rightarrow 0$ line, the maximum value of $\mathcal{I}(X;Y)$ is found near $p_\mathsf{L} = 0.99$ where $\mathcal{I}(X;Y) = 66$ bits/s, which gives} \pt{the channel capacity \rev{$C$} (\textit{sensu} Shannon) of channelrhodopsin.}
A similar calculation can be performed for ACh and CaM. However, the resulting expressions are not as compact as (\ref{eqn:ChR2MutualInformationExample2}), so we exclude them from the paper. Mutual information plots for ACh and CaM (from which $C_{\mathrm{iid}}$ may \rev{also be obtained numerically}) are given in Figures \ref{fig:AChFigure} and \ref{fig:CaMFigure}, respectively.
However, ACh and CaM both have $|\mathcal{S}^\prime| > 1$ (see Figures \ref{fig:ACh} and \ref{fig:CaM}), and do not satisfy the condition in Theorem \ref{thm:capacity}. It remains an open question as to whether $C = C_{\mathrm{iid}}$ for these receptors. The proof of \cite[Thm 2]{ThomasEckford2016} (and of Theorem \ref{thm:capacity}) relies on the feedback capacity being achieved by the IID input distribution. However, if there is more than one sensitive transition, the receiver can use the feedback to distinguish between these transitions, and can select an optimal input distribution for each. Thus, the feedback-capacity-achieving input distribution depends on the feedback, and is not \rev{necessarily} IID. If $C = C_{\mathrm{iid}}$, a different proof technique is required, and we do not address this case.
\section{Discussion}
In this paper we have presented a general framework for signal transduction systems, in which the states of a receptor form a directed graph, some subset of the edges of which represent transitions with intensities modulated by an external signal. This signal provides the channel input, and the state of the receptor -- a trajectory on the graph -- represents the channel output.
\pt{We illustrate the signal transduction model, the calculation of mutual information and the IID capacity for several examples: light intensity transduction by channel rhodopsin, acetylcholine concentration transduction by the nicotinic acetylcholine receptor, and transduction of intracellular calcium ion concentration by the calmodulin protein.}
Several caveats are in order, which qualify our results and motivate our future work.
In many signal transduction systems, only a subset of the receptor states engender an observable output signal. For example, the channelrhodopsin receptor states $\mathsf{C}_1,\O_2,\mathsf{C}_3$ (\textit{cf}.~Fig.~\ref{fig:ChR2}) are not directly observed by the cell in the membrane of which the receptor is embedded; rather it is the net current (zero for states $\mathsf{C}_1,\mathsf{C}_3$ and finite for state $\O_2$) that impacts the rest of the cell. Similarly, for the nicotinic acetylcholine receptor (\textit{cf}.~Fig.~\ref{fig:ACh}) the state of the receptor as observed by the cell is either ``open'' (states $\O_1,\O_2$) or ``closed'' (states $\mathsf{C}_3,\mathsf{C}_4,\mathsf{C}_5$). For the calmodulin receptor, there are understood to be four functionally distinct states: both occupied Ca$^{2+}$~binding sites on the N-terminus end of the protein, both occupied Ca$^{2+}$~binding sites on the C-terminus end of the protein, all four Ca$^{2+}$~binding sites occupied, or fewer than two on each end (\textit{cf.}~Fig.~\ref{fig:CaM}; dashed lines indicate physiologically equivalent states).
The diagram \eqref{diag:xyz} shows the general structure of such a channel, with output $Z(t)$ a function of the channel state $Z=f(Y(t))$ (compare with the diagram in (\ref{diag:xy})):
\begin{equation}\label{diag:xyz}
\begin{array}{ccccccccccc}
&X_1&&X_2&&X_3&&X_4&&X_5&\cdots\\
&\downarrow&&\downarrow&&\downarrow&&\downarrow&& \downarrow \\
(Y_0)&\longrightarrow&Y_1&\longrightarrow&Y_2&\longrightarrow&Y_3&\longrightarrow&Y_4&\cdots\\
&&\downarrow&&\downarrow&&\downarrow&& \downarrow \\
&&Z_1&&Z_2&&Z_3&&Z_4&\cdots
\end{array}\end{equation}
By virtue of the information processing inequality, the mutual information rate between $X$ and $Z$ cannot exceed that between $X$ and $Y$. Preliminary results suggest that the size of the difference -- the information gap -- depends strongly on the network architecture, and the positioning of sensitive edges relative to observable transitions (data not shown). Detailed consideration of mutual information for Markovian signal transduction channels with such partially observed outputs will be undertaken elsewhere.
We have assumed that the directed edges comprising the receptor's state transition graph fall into two classes, either insensitive (fixed transition rates) or sensitive (transition rates proportional to the input signal intensity). A more realistic assumption would allow for a dark current (finite transition rate at zero signal intensity), a nonlinear, monotonically increasing transition rate as a function of increasing intensity, or a signaling threshold or minimum intensity value. Under the IID input scenario it is optimal to limit the input values to those inducing the maximal and minimal transition rates, in which case several more realistic scenarios could in principle be reduced to the scenario we consider here. For example, a dark current could be captured by adding an additional insensitive channel parallel to a sensitive channel.
We have considered a general class of signal transduction models that are naturally framed as continuous time channels. Our basic signal transduction channel model process is conditionally Markovian, given the (time varying) input signal. The simplest model in this class would correspond to Kabanov's Poisson channel \cite{Kabanov1978}, consisting of a single transition with rate modulated by the input. In order to simplify the analysis of such models it is convenient to translate them into analogous discrete-time models. The general structure of such as model is a finite state, discrete-time channel in which the probability transition matrix is modulated by the (discrete time) input sequence. Our previously-discussed results \cite{ThomasEckford2016} introduced a minimal such model, the BIND channel, consisting of a single receptor molecule with two states (bound, $\mathsf{B}$, and unbound, $\mathsf{U}$) with one transition rate ($\mathsf{U}\to\mathsf{B}$) sensitive to the input (ligand molecule concentration) and the other transition rate ($\mathsf{B}\to\mathsf{U}$) insensitive. In general, the structure of a conditionally Markovian signal-transduction channel under time discretization corresponds to the Unit Output Memory (UOM) channel class analyzed by \rev{Chen and Berger} \cite{che05}. As mentioned previously,
in \rev{\cite{AsnaniPermuterWeissman2013IEEE_ISIT,PermuterAsnaniWeissman2014IEEETransIT} Asnani, Permuter and Weissman} present several examples of UOM channels that they call POST (prior output is the state) channels, which are also special cases of the channels analyzed by Chen and Berger. (The BIND channel can be interpreted as a type of POST channel although it is distinct from the examples in \cite{AsnaniPermuterWeissman2013IEEE_ISIT,PermuterAsnaniWeissman2014IEEETransIT}.)
Thus our channel models for channel rhodopsin, the nicotinic acetylcholine receptor and calmodulin may all be seen as examples of Chen and Berger's UOM channel class.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=3.5in]{figures/AChFigure.pdf}
\end{center}
\caption{\label{fig:AChFigure} Plots of $\mathcal{I}(X;Y)$ and $\mathcal{I}(X;Z)$ for ACh, using $\Delta t = 0.02$ ms. Solid line is from (\ref{eqn:GeneralIID1}), while dots represent {\em Monte Carlo} simulations.}
\end{figure}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=3.5in]{figures/CaMFigure.pdf}
\end{center}
\caption{\label{fig:CaMFigure} Plots of $\mathcal{I}(X;Y)$ and $\mathcal{I}(X;Z)$ for CaM, using $\Delta t = 0.002$ ms. Solid line is from (\ref{eqn:GeneralIID1}), while dots represent {\em Monte Carlo} simulations.}
\end{figure}
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