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t\{ W ^{\dagger} A (UX) \right\}_{L k}
\nonumber \\
&+&
\sum_{n} \sum_{k} \sum_{K \neq L} \sum_{m \neq k}
\biggl[
\frac{ (ix) }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{k} ) ( h_{m} - h_{k} ) }
e^{- i ( h_{k} - h_{n} ) x}
\nonumber \\
&-&
\frac{1}{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{L} - h_{k} )^2 ( h_{m} - h_{k} )^2 }
\biggl\{
\Delta_{K} \Delta_{L} + ( h_{m} - 2 h_{k} ) ( \Delta_{K} + \Delta_{L} ) + 3
h_{k}^2 - 2 h_{m} h_{k}
\biggr\}
e^{- i ( h_{k} - h_{n} ) x}
\nonumber \\
&+&
\frac{1}{ ( \Delta_{K} - h_{m} ) ( \Delta_{L} - h_{m} ) ( h_{m} - h_{k} )^2 }
e^{- i ( h_{m} - h_{n} ) x}
\nonumber \\
&+&
\frac{1}{ ( \Delta_{K} - \Delta_{L} ) ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - h_{m} ) } e^{- i ( \Delta_{K} - h_{n} ) x}
- \frac{1}{ ( \Delta_{K} - \Delta_{L} ) ( \Delta_{L} - h_{k} )^2 ( \Delta_{L} - h_{m} ) } e^{- i ( \Delta_{L} - h_{n} ) x}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta k}
(UX)^*_{\alpha n} (UX)_{\beta n}
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K m}
\left\{ (UX)^{\dagger} A W \right\}_{m L}
\left\{ W ^{\dagger} A (UX) \right\}_{L k}
\biggr\}.
\label{P-beta-alpha-W4-H4-diag}\end{aligned}$$ $$\begin{aligned}
&& P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{3rd}
\equiv
2 \mbox{Re} \left[ \left(
S^{(0)}_{\alpha \beta} \right)^{*}
S_{\alpha \beta}^{(4)} [4]_{ \text{ offdiag } }
\right]
\nonumber \\
&=&
2 \mbox{Re}
\biggl\{
\sum_{m}
\sum_{k \neq l } \sum_{K}
\biggl[
- \frac{ (ix) }{ ( \Delta_{K} - h_{k} ) (\Delta_{K} - h_{l} ) }
e^{- i ( \Delta_{K} - h_{m} ) x}
+ \frac{ 1 }{ ( h_{l} - h_{k} ) ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - h_{l} )^2 }
\nonumber \\
&\times&
\biggl\{
( h_{l} - h_{k} ) ( h_{l} + h_{k} - 2 \Delta_{K} )
e^{- i ( \Delta_{K} - h_{m} ) x}
+ ( \Delta_{K} - h_{k} )^2 e^{ - i ( h_{l} - h_{m} ) x}
- ( \Delta_{K} - h_{l} )^2 e^{ - i ( h_{k} - h_{m} ) x}
\biggr\}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l}
(UX)^*_{\alpha m} (UX)_{\beta m}
\left\{ (UX)^{
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| 2,141
| 2,085
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because the flavor structures have different dependences in these phases, the result is that the quark and lepton EDMs become decorrelated.
Conclusion
==========
In the paper [@SmithTouati], we developped a systematic method to study the flavor structure behind the quark and lepton EDMs which can be extended easily to other more complicated models (Sterile neutrinos, SUSY etc...). The rainbow-like non-invariant flavor structures are found to be typically much larger than the Jarlskog-like flavor invariants. Interestingly, we find a different behavior for Dirac and Majorana neutrinos. Quark and lepton EDMs are proportional in the former case whereas they are completely independent in the latter case. Indeed, quark and lepton EDMs have different dependences on Majorana phases. Finally, by studying the flavor structures behind the quark and lepton EDMs, we get the relations shown in table \[tab:SumRules\] between EDMs of different generations.
CKM-induced EDMs PMNS-induced EDMs
-- ------------------------------------------------------------------------- ---------------------------------------------------------------------------
$\frac{d_{d}}{m_{d}}+\frac{d_{s}}{m_{s}}+\frac{d_{b}}{m_{b}}=0$ $\frac{d_{d}}{m_{d}}=\frac{d_{s}}{m_{s}}=\frac{d_{b}}{m_{b}}$
$\frac{d_{e}}{m_{e}}=\frac{d_{\mu}}{m_{\mu}}=\frac{d_{\tau}}{m_{\tau}}$ $\frac{d_{e}}{m_{e}}+\frac{d_{\mu}}{m_{\mu}}+\frac{d_{\tau}}{m_{\tau}}=0$
: Sum rules[]{data-label="tab:SumRules"}
For example, the CKM-induced quark EDMs and the PMNS-induced lepton EDMs are tuned by the non-invariant commutators \[eq:Xq\] and \[eq:XeDirac\] and because a commutator is traceless, we get these sum rules.
References {#references .unnumbered}
==========
[99]{}
C. Smith, S.Touati, .
C. Jarlskog, .
G. C. Branco, R. G. Felipe, F. R. Joaquim, .
---
abstract: 'We study semi-infinite paths of the radial spanning tree (RST) of a Poisson point process in t
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|
tate frequency (-energy) in the charge $+q$ sector. The dashed line denotes the charge $+q$ continuum. For $q=0$, $\omega_0\sim \sqrt{T}\ll m$. For small positive $q$, $\omega_0$ crosses zero. At some $q/m\sim\calO(1)$, $\omega_0$ crosses into the positive-charge continuum. At this point the energy of an oppositely-charged pair vanishes, and the system becomes unstable against spontaneous pair creation. []{data-label="fig:numerics"}](omega_m_1000_L_1000_Q_25.pdf){width="0.6\linewidth"}
Since the metric functions (\[eq:U\]), (\[eq:h\]) are somewhat involved and the exact solution for $\rho_0$ is the root of a cubic, it is simplest to perform detailed analysis numerically. Normalized to $m$, the eigenfrequencies depend only on the dimensionless ratios $w$, $Q/Q_{max}\ll1$, and $mL\gg 1$. For typical sets of parameters we can solve (\[KGeq\]) numerically for the bound states. In Fig. \[fig:numerics\] we show the frequencies of the charge $\pm q$ ground states as a function of $q$. We see that in this example, $\omega\ll m$ for $q=0$ and decreases approximately linearly as $q$ is increased.
These properties are straightforward to understand physically. For $q=0$ and vanishing energy flux through the bubble wall, the $s-$wave spectrum includes a discrete set of normalizable bound modes of $|\pm\omega|<m$. The lowest modes have $|\omega|\sim \sqrt{T}$, reflecting the fact that wound strings near the cigar tip are close to becoming unwound (and indeed would become unwound with any asymmetric perturbation). In the hierarchy (\[eq:hierarchy\]), these modes are deeply bound, $|\omega|\ll m$. For small positive $q$, the spectrum shifts downward. Positive frequencies, corresponding to negative charges, become more tightly bound, while their negative frequency counterparts shift toward $\omega=-m$. Since the modes are deeply bound, we can approximate the electrostatic potential energy term in the Klein-Gordon equation by its value at the bubble wall, $qA_t(\rho_0)$. Then the bound mode frequencies decrease as $$\begin{
| 1,303
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| 1,429
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|
_{i+1}\cdot ({}^tm_{i+1,i+1}'\cdot {}^tm_{i, i+1})
=\pi m_{i,i}^{\ast\ast}+\pi \tilde{z}_i''^{\dag\dag}
\end{array} \right.$$ for some formal expansion $\tilde{z}_i''^{\dag\dag}$. Therefore, $$\delta_{i-1}v_{i-1}\cdot {}^t\tilde{m}_{i, i-1}''+\delta_{i+1}v_{i+1}\cdot {}^t\tilde{m}_{i, i+1}''=
\pi\left((m_{i,i}^{\ast\ast})'\cdot {}^tm_{i, i}+m_{i,i}^{\ast\ast}+ \tilde{z}_i''^{\dag\dag}+ \tilde{z}_i''^{\dag} \right).$$ Then $$(m_{i,i}^{\ast\ast})''=(m_{i,i}^{\ast\ast})'\cdot {}^tm_{i, i}+m_{i,i}^{\ast\ast}+ \tilde{z}_i''^{\dag\dag}+ \tilde{z}_i''^{\dag}$$ as an equation in $B\otimes_AR$.
**\
\[r32\] We define a functor $\underline{M}^{\ast}$ from the category of commutative $A$-algebras to the category of groups as follows. For a commutative $A$-algebra $R$, set $$\underline{M}^{\ast}(R)=\{ m \in \underline{M}(R) : \textit{there exists $m'\in \underline{M}(R)$ such that $m\cdot m'=m'\cdot m=1$}\}.$$ Then $\underline{M}^{\ast}$ is an open subscheme of $\underline{M}$ with generic fiber $M^{\ast}=\mathrm{Res}_{E/F}\mathrm{GL}_E(V)$, and that $\underline{M}^{\ast}$ is smooth over $A$. Moreover, $\underline{M}^{\ast}$ is a group scheme since $\underline{M}$ is a scheme in monoids. The proof of this is similar to that of Remark 3.2 in [@C2] and so we skip it.
Construction of ** {#h}
------------------
Recall that the pair $(L, h)$ is fixed throughout this paper and the lattices $A_i$, $B_i$, $W_i$, $X_i$, $Z_i$ only depend on the hermitian pair $(L, h)$. For any flat $A$-algebra $R$, let $\underline{H}(R)$ be the set of hermitian forms $f$ on $L\otimes_{A}R$ (with values in $B\otimes_AR$) such that $f$ satisfies the following conditions:
1. $f(L\otimes_{A}R,A_i\otimes_{A}R) \subset \pi^iB\otimes_AR$ for all $i$.
2. $\xi^{-m}f(a_i,a_i)$ mod 2 = $\xi^{-m}h(a_i, a_i)$ mod 2, where $a_i \in A_i \otimes_{A}R$, and $i=2m$ or $i=2m-1$.
3. $\pi^{-i}f(a_i,w_i) = \pi^{-i}h(a_i, w_i)$ mod $\pi$, where $a_i \in A_i\otimes_{A}R$ and $w_i \in W_i\otimes_{A}R$, and $i=2m$.
4. $\xi^{-m} f(w_i,w_i)-\xi^{-m}h
| 1,304
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| 4,065
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|
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tau$ to $\cS_1$ and $\bz+t_2\bv_\tau$ to $\cS_2$ where we fix $t_1=0$ and $t_2=1$ (other choices of values would also work). $\cS_i$ then responds with the value of $F$ at the point $\bgam^{\bz+t_i\bv_\tau}$, that is with $F(\bgam^{\bz+t_i\bv_\tau})$ and the value of the ‘first order derivative’ at the same point $F^{(1)}(\bgam^{\bz+t_i\bv_\tau})$. Notice that the protocol is private since $\bz+t\bv_\tau$ is uniformly distributed over $\Z_6^k$ for any fixed $\tau$ and $t$.
[align\*]{} &: \_6\^k\
\_i &: +t\_i\_\
\_i &: F(\^[+t\_i\_]{}), F\^[(1)]{}(\^[+t\_i\_]{})
#### Recovery:
Define $$G(t):=F(\bgam^{\bz+t\bv_\tau})
=\sum_{i=1}^n a_i \gamma^{{\langle \bz,\bu_i \rangle}+t{\langle \bv_\tau,\bu_i \rangle}}$$ Using the fact that $\gamma^6=1$, we can rewrite $G(t)$ as: $$G(t)=\sum_{\ell=0}^{5} c_\ell \cdot \gamma^{t\ell},$$ with each $c_\ell \in \cR$ given by $$c_\ell=\sum_{i:{\langle \bu_i,\bv_\tau \rangle}
=\ell\mod 6}a_i \gamma^{{\langle \bz,\bu_i \rangle}}.$$ Since $${\langle \bu_i,\bv_\tau \rangle}\mod 6 \,\, \begin{cases}
= 0 &\mbox{if}\ i=\tau\\
\in S={\{1,3,4\}} &\mbox{if}\ i\ne \tau
\end{cases}$$ we can conclude that $c_0=a_\tau\gamma^{{\langle \bu_\tau,\bz \rangle}}$ and $c_2=c_5=0$. Therefore $$G(t)=c_0+c_1\gamma^t+c_3\gamma^{3t}+c_4\gamma^{4t}.$$ Next, consider the polynomial $$g(T) = c_0+c_1T+c_3T^3+c_4T^4\in \mathcal{R}[T].$$ By definition we have $$\begin{aligned}
g(\gamma^t)=G(t)=F(\bgam^{\bz+t\bv_\tau})\\
g^{(1)}(\gamma^t)=\sum_{\ell=0}^5 \ell c_\ell \gamma^{t\ell}={\langle F^{(1)}(\bgam^{\bz+t\bv_\tau}),\bv_\tau \rangle},\end{aligned}$$ where the last equality holds since $c_2=c_5=0$ and $$\begin{aligned}
{\langle F^{(1)}(\bgam^{\bz+t\bv_\tau}),\bv_\tau \rangle}&=\left\langle\sum_{i=1}^n a_i \bu_i\gamma^{{\langle \bz,\bu_i \rangle}+t{\langle \bv_\tau,\bu_i \rangle}},\bv_\tau\right\rangle\\
&= \sum_{i=1}^n a_i{\langle \bu_i,\bv_\tau \rangle} \gamma^{{\langle \bz,\bu_i \rangle}+t{\langle \bv_\tau,\bu_i \rangle}}\\
&= \sum_{\ell=0}^{5} \ell \left(\sum_{i:{\langle \bu_i,\bv_\tau \rangle}
=\el
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quiv \lim\limits_{\epsilon\to 0}\left[ \frac{d}{dz}\phi_j(k,\pm 1 + \epsilon) - \frac{d}{dz}\phi_j(k,\pm 1 - \epsilon)\right]$$ are the discontinuities in $d\hat{\phi}/dz$ and $d\phi_j/dz$ at $z = \pm 1$. With $\hat{\phi}$ given by Eq. and $\phi_j$ given by Eq. , one can show that $$\phi_j(k,1) = \frac{-\Delta_{+ j}}{2|k|} - \frac{\Delta_{- j}}{2|k|e^{2|k|}},$$ and $$\phi_j(k,-1) = \frac{-\Delta_{+ j}}{2|k|e^{2|k|}} - \frac{\Delta_{- j}}{2|k|}.$$ The $\hat{\phi}(k,\pm 1)$ term in Eq. can then be written in terms of $\hat{\Delta}_{\pm}$ to yield $$\label{deltahat corrected}
\frac{\partial}{\partial t}\begin{pmatrix}\hat{\Delta}_+\\\hat{\Delta}_-\end{pmatrix} = \mathcal{D}\begin{pmatrix}\hat{\Delta}_+\\\hat{\Delta}_-\end{pmatrix} + \begin{pmatrix}N_+\\N_-\end{pmatrix},$$ with $$\label{Dmatrix}
\mathcal{D} = ik\begin{pmatrix}
\frac{1}{2|k|} - 1 & \frac{e^{-2|k|}}{2|k|}\\
\frac{-e^{-2|k|}}{2|k|} & -\frac{1}{2|k|} + 1\\
\end{pmatrix},$$ and $N_{\pm}$ representing the nonlinearities in Eq. . Note that taking $N_{\pm} \to 0$ and $\partial / \partial t \to i\omega$ reduces this to the linear system solved in the previous section.
We now have all of the necessary tools to treat this system in a manner similar to the previously-mentioned calculations[@Terry2006; @Makwana]. Using our definitions for $\hat{\Delta}_{\pm}$ and $\Delta_{\pm j}$, the $z$-derivative of Eq. evaluated between $z = \pm 1 + \epsilon$ and $z = \pm 1 - \epsilon$ with $\epsilon \to 0$ is $$\label{Mdef}
\begin{pmatrix}\hat{\Delta}_+\\\hat{\Delta}_-\end{pmatrix} = \mathbf{M}\begin{pmatrix}\beta_1\\\beta_2\end{pmatrix},$$ where $$\label{Mmatrix}
\mathbf{M} = \begin{pmatrix}
\Delta_{+ 1} & \Delta_{+ 2}\\
\Delta_{- 1} & \Delta_{- 2}\\
\end{pmatrix} = -2|k|e^{|k|}\begin{pmatrix}
1 & 1\\
b_1 & b_2\\
\end{pmatrix},$$ and $b_j$ is given in Eqn. . Equation is equivalent to Eq. : for this calculation, the dynamical quantities that we use to specify the state of the system are $\hat{\Delta}_{\pm}$, and their eigenmode structure is given by the columns of
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[@Lecomte2007] of the modified master operator $\mathbb{W}_s$, described by $$(\mathbb{W}_s)_{ij} = e^{-s} W_{j\rightarrow i} - r_i\delta_{i,j}\,.
\label{eq:Ws}$$ This operator generates the dynamics of $s$-biased ensembles of trajectories via $\partial_t P_i(s) = \sum_j(\mathbb{W}_s)_{ij} P_j(s)$. The eigenstate of $\mathbb{W}_s$ corresponding to the largest eigenvalue $\theta(s)$ gives the occupation probabilities $P_i(s)$ (of level $i$) associated with the trajectories that dominate at a certain $s$. When $s=0$ this is the stationary state which, since we are studying the case of an infinite temperature environment, has equal probability for all levels, such that $P_i(s=0) = 1/N$ for all $i$. At $s\ne 0$, $P_i(s)$ indicate the occupations for rare trajectories which are more ($s<0$) or less ($s>0$) *active* than those of the average dynamics.
![(Colour online.) (a) Dynamical phase diagram for the $N=100\times 100$ disordered lattice in Fig. \[fig1\] as a function of inverse disorder strength and thermodynamic field $s$ (see main text). (b,c,d) Steady-state lattice-site occupations in the long-time limit, $P_m(s)$. Shown are the occupation probabilities for the parameter values as labelled on (a). (e) A slice through the dynamical phase diagram with parameter values labelled on (a). []{data-label="fig3"}](phase_lowres5.png){width="8.57cm"}
We now apply this thermodynamic approach to the dynamics in disordered exciton-lattices. We obtain the dynamical phase diagram for the model by plotting the activity $k_s$ as a function of the inverse disorder strength $d^{-1}$ and the $s$-field. The data, found from exact diagonalisation, are shown in Fig. \[fig3\](a). We find a first-order phase boundary for $s>0$, which appears to tend to $s=0$ in the limit of infinite disorder strength. We will show that existence of such a transition in a single-particle system is due to states becoming increasingly disconnected at higher disorder. The is existence of a transition is consistent with the increasingly long tails
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rong vertices for the $NN\pi$, $NN\pi\pi$, $\Lambda \Sigma\pi$ and $\Lambda N K$ which arise from the Lagrangians in Eq. (\[eq:str\]). \[vf1\]](lnks "fig:")
The strong vertices for the interaction between our baryonic and mesonic degrees of freedom are obtained from the strong SU(3) chiral Lagrangian [@donoghue], $$\begin{aligned}
{\mathcal{L}}^s_{NN\pi}=&-\frac{g_A}{2f_\pi}\overline{\Psi}\gamma^\mu\gamma_5
{\vec{\tau}}\Psi\cdot\partial_\mu\vec{\pi}\,,\nonumber\\
{\mathcal{L}}^s_{NN\pi\pi} =&-\frac{1}{4f_\pi^2}\overline{\Psi}\gamma^\mu
{\vec{\tau}}\cdot(\vec{\pi}\times\partial_\mu\vec{\pi})\Psi\,,\nonumber\\
{\mathcal{L}}^s_{\Lambda\Sigma\pi}=&-\frac{D_s}{\sqrt{3}}\,\overline{\Psi}_\Lambda\gamma^\mu\gamma_5
\Psi_\Sigma\cdot\partial_\mu\vec{\pi} \,,\label{eq:str}
\\\nonumber
{\mathcal{L}}^{s}_{\Lambda N K} =&
\, \frac{D_s+3F_s}{2\sqrt3f_\pi} \, \overline{\Psi}_{N}
\gamma^\mu\gamma_5 \,\partial_\mu\phi_{K}
\Psi_\Lambda \,,\end{aligned}$$ where we have taken the convention which gives us $\Psi_\Sigma\cdot\vec{\pi}=\Psi_{\Sigma_+}\pi_-+\Psi_{\Sigma_-}\pi_++\Psi_{\Sigma_0}\pi_0$, and we consider, $g_A=1.290$, $f_\pi=92.4$ MeV, $D_s=0.822$, and $F_s=0.468$. These strong coupling constants are taken from $NN$ interaction models such as the Jülich [@JB] or Nijmegen [@nij99] potentials. The four interaction vertices corresponding to these Lagrangians are depicted in Fig. \[vf1\].
Once the interaction Lagrangians involving the relevant degrees of freedom have been presented, we need to define the power counting scheme which allows us to organize the different contributions to the full amplitude.
Power counting scheme {#ss:cs}
---------------------
The amplitude for the $\Lambda N\to NN$ transition is built as the sum of a medium and long-range one meson exchanges (i.e. $\pi$ and $K$), the contribution from the two-pion exchanges, and the contribution of the contact interactions up to ${\cal O} (q^2/M^2)$ as described below. The order at which the different terms enter in the perturbative expansion of the ampli
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ip}$ in the tunneling matrix element, the general expression for $\tau^{(2)}_{\sigma}(\w)$ (Eq. ) in the inelastic current $I^{\rm (2)}_{\rm inel}$ takes on the shape as Eq. , leading to an inelastic current which for the special case of a flat DOS at the Fermi level leads to steps in the differential conductance at the vibrational energies $\pm \omega_0$.
(\#1,\#2,\#3)\#4[(\#1,\#2)[(0,0)\[\#3\][\#4]{}]{}]{}
(100,100)(0,0) (0,0)[ ![Second-order Feynman diagram of the generating Luttinger-Ward functional in the system S. The full line represents the full local electron Green’s function $G_d(z)$, the wiggled line the full phonon propagator. $i\w_n = i\pi (2n+1)/\beta$ denote the fermionic Matsubara frequencies and $i\w_n = i2\pi n/\beta$ the bosonic Matsubara frequencies [@LuttingerWard1960]. []{data-label="fig:self-electron-phonon-diagram"}](fig2-eph-functional-diagram "fig:") ]{} (60.22516,54.22514,t)[$i\omega _n+i\nu _n$]{}(60.22514,106.37524,b)[$i\nu _n$]{}(60.22514,14.07504,t)[$i\omega _n$]{}
We now compare this result to the perturbatively calculated elastic current in the same limit. Under these circumstances, the Green’s function of the orbital with the single-particle energy $\e_a$ has the form $G_a(z)=[z- \e_a -\Sigma_{\rm el}(z) - \Sigma_{\rm el-ph}(z)]^{-1}$, where the self energy $\Sigma_{\rm el}(z)$ accounts for the purely electronic interactions and $\Sigma_{\rm el-ph}(z)$ arises from the additional electron-phonon coupling which is limited to the system S as assumed by Lorente et al. [@LorentePersson2000]. Introducing $G_{a}^{(0)} = [z- \e_a -\Sigma_{\rm el}(z)]^{-1}$ allows for a perturbation expansion in linear order of $\Sigma_{\rm el-ph}(z)$ in weak electron-phonon coupling, $$\begin{aligned}
\label{equ:Gf-expansion}
G_a(z) &=& G_{a}^{(0)}(z) + G_{a}^{(0)}(z) \Sigma_{\rm el-ph}(z)G_{a}^{(0)}(z) + \cdots\end{aligned}$$ If we substitute this expansion of $G_a(z)$ into the expression Eq. for the elastic current, two contributions arise, first a purely electronic one generated by $ G_{a}
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so that $a$ has a matrix representation $A\in\mathbb{R}^{m\times n}$ where $\mathbb{R}^{m\times n}$ is the collection of $m\times n$ matrices with real entries. The inverse $A^{-1}$ exists and is unique iff $m=n$ and $\textrm{rank}(A)=n$; this is the situation depicted in Fig. \[Fig: functions\](a). If $A$ is neither one-one or onto, then we need to consider the multifunction $A^{-}$, a functional choice of which is known as the *generalized inverse* $G$ of $A$. A good introductory text for generalized inverses is @Campbell1979Figure \[Fig: MP\_Inverse\](a) introduces the following definition of the *Moore-Penrose* generalized inverse $G_{\textrm{MP}}$.
**Definition 2.1.** ***Moore-Penrose Inverse.*** *If $a\!:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is a linear transformation with matrix representation $A\in\mathbb{R}^{m\times n}$ then the* Moore-Penrose inverse $G_{\textrm{MP}}\in\mathbb{R}^{n\times m}$ of $A$ *(we will use the same notation* $G_{\textrm{MP}}\!:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ *for the inverse of the map $a$) is the noninjective map defined in terms of the row and column spaces of $A$,* $\textrm{row}(A)=\mathcal{R}(A^{\textrm{T}})$, $\textrm{col}(A)=\mathcal{R}(A)$*, as* $$G_{\textrm{MP}}(y)\overset{\textrm{def}}=\left\{ \begin{array}{lcl}
(a|_{\textrm{row}(A)})^{-1}(y), & & \textrm{if }y\in\textrm{col}(A)\\
0 & & \textrm{if }y\in\mathcal{N}(A^{\textrm{T}}).\end{array}\right.\qquad\square\label{Eqn: Def: Moore-Penrose}$$
Note that the restriction $a|_{\textrm{row}(A)}$ of $a$ to $\mathcal{R}(A^{\textrm{T}})$ is bijective so that the inverse $(a|_{\textrm{row}(A)})^{-1}$ is well-defined. The role of the transpose matrix appears naturally, and the $G_{\textrm{MP}}$ of Eq. (\[Eqn: Def: Moore-Penrose\]) is the unique matrix that satisfies the conditions
$$\begin{array}{c}
AG_{\textrm{MP}}A=A,\quad G_{\textrm{MP}}AG_{\textrm{MP}}=G_{\textrm{MP}},\\
(G_{\textrm{MP}}A)^{\textrm{T}}=G_{\textrm{MP}}A,\quad(AG_{\textrm{MP}})^{\textrm{T}}=AG_{\textrm{MP}}\end{array}\label{Eqn: MPInverse}$$
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frac{\pi m}{N_\phi} \right) \right]^2.$$ Note that $\lambda^m_\phi \rightarrow -m^2/R^2$ for $m/N_\phi \ll 1$. The eigenfunction ${\cal P}^m_j$ satisfies the discrete orthogonality relation $$\frac{1}{N_\phi} \sum_{j=1}^{N_\phi}({\cal P}_j^m)^* {\cal P}^{m'}_j = \delta_{mm'}\quad\text{and}\quad\frac{1}{N_\phi} \sum_{m=1}^{N_\phi}({\cal P}_j^m)^* {\cal P}^{m'}_j = \delta_{jj'}.$$
We expand $\widetilde{\Phi}_{i,j,k}$ and $\rho_{i,j,k}$ only along the azimuthal and vertical directions as $$\begin{aligned}
\widetilde{\Phi}_{i,j,k} = \frac{2}{N_\phi (N_z+1)}\sum_{m=1}^{N_\phi}\sum_{n=1}^{N_z}\widetilde{\Phi}^{mn}_{i}{\cal P}_j^m{\cal Z}_k^n,\label{eq:cyl_Phi_forward}\\
\rho_{i,j,k} = \frac{2}{N_\phi (N_z+1)}\sum_{m=1}^{N_\phi}\sum_{n=1}^{N_z}\rho^{mn}_{i}{\cal P}_j^m{\cal Z}_k^n,\label{eq:cyl_rho_forward}\end{aligned}$$ where the expansion coefficients $\widetilde{\Phi}^{mn}_{i}$ and $\rho^{mn}_{i}$ satisfy the inverse transforms $$\begin{aligned}
\widetilde{\Phi}^{mn}_{i} &= \sum_{j=1}^{N_\phi}\sum_{k=1}^{N_z}\widetilde{\Phi}_{i,j,k} ({\cal P}^m_j)^* {\cal Z}^n_k,\label{eq:cyl_Phi_backward}\\
\rho^{mn}_{i} &= \sum_{j=1}^{N_\phi}\sum_{k=1}^{N_z}\rho_{i,j,k} ({\cal P}^m_j)^* {\cal Z}^n_k.\label{eq:cyl_rho_backward}\end{aligned}$$ One cannot analytically expand $\widetilde{\Phi}_{i,j,k}$ and $\rho_{i,j,k}$ along the radial direction because radial eigenfunction ${\cal R}^l_i$, defined through $\Delta_R^2{\cal R}^l_i = \lambda^l_R {\cal R}^l_i$, has no closed-form expression and is not compatible with FFT.[^3]
Plugging Equation – into Equation yields $$\label{eq:tridiagonal}
\left( \Delta_R^2 + \lambda^m_\phi + \lambda^n_z \right) \widetilde{\Phi}^{mn}_i = 4\pi G \rho^{mn}_i,$$ which, using Equations and , can be written as $$\label{eq:tridiagonal_uni}
\left[ \frac{1}{(\delta R)^2} - \frac{1}{2R_i \delta R} \right] \widetilde{\Phi}_{i-1}^{mn} + \left[ \lambda_\phi^m + \lambda^n_z - \frac{2}{(\delta R)^2} \right] \widetilde{\Phi}_i^{mn} + \left[ \frac{1}{(\delta R)^2} + \frac{1}{2R_i \delta R} \right]
| 1,311
| 3,714
| 1,678
| 1,278
| null | null |
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|
and $X_2$ can be tiled with horizontal copies of $T$.
Note that $(x,x+n(k+1))+(2,k+3) = (x+2,(x+2)+(n+1)(k+1))$. Also, if $x \equiv 2n+r \pmod 8$, then $x+2 \equiv 2(n+1)+r \pmod 8$. Hence, by the definitions of $S_2$ and $S_3$, we see that $X_1$ is invariant under translation by $(2,k+3)$. To show that vertical copies of $T$ tile $X_1$, it therefore suffices to show that $T$ tiles the columns $X_1 \cap (\{0\} \times \mathbb{Z})$ and $X_1 \cap (\{1\} \times \mathbb{Z})$.
But in fact, if $(0,y) \in S_2$, then $0 \equiv 2n+4$ or $2n+6 \pmod 8$, so $1 \equiv 2n+5$ or $2n+7 \pmod 8$, so also $(1,y+1) \in S_2$. The converse also holds, and the same is true for $S_3$. Thus we only need to check the case $x = 0$.
$(0,n(k+1)) \in S_2$ for $n \equiv 1,2,5,6 \pmod 8$, that is, $n \equiv 1,2 \pmod 4$.
$(0,n(k+1)+1) \in S_3$ for $n \equiv 2,3,6,7 \pmod 8$, that is, $n \equiv 2,3 \pmod 4$.
Therefore $(0,y) \notin X_1$ for $y \equiv k+1, 2(k+1), 2(k+1)+1, 3(k+1)+1 \pmod{4(k+1)}$, so copies of $T$ beginning at positions $1$ and $2(k+1)+2 \pmod{4(k+1)}$ tile $X_1 \cap (\{0\} \times \mathbb{Z})$.
Hence $T$ tiles $X_1$.
Note that $(x,x+n(k+1))+(k+2,1) = (x+k+2,(x+k+2)+(n-1)(k+1))$.\
Since $k \equiv 4 \pmod 8$, if $x \equiv 2n+r \pmod 8$ then $x+k+2 \equiv 2(n-1)+r \pmod 8$. Hence $X_2$ is invariant under translation by $(k+2,1)$, by the definitions of $S_1$ and $S_3$. To show that horizontal copies of $T$ tile $X_2$, it is therefore enough to show that $T$ tiles the row $X_2 \cap (\mathbb{Z} \times \{0\})$.
We can express $S_1$ as $\{(y-n(k+1),y) \; | \; y \equiv -n,1-n,2-n,3-n \pmod 8\}$.
Similarly $S_3 = \{(y-n(k+1)-1,y) \; | \; y \equiv 3-n,4-n,5-n,6-n \pmod 8\}$.
Therefore $(-n(k+1),0) \in S_1$ for $n \equiv 0,1,2,3 \pmod 8$, and $(-n(k+1)-1,0) \in S_3$ for $n \equiv 3,4,5,6 \pmod 8$.
Hence $(x,0) \notin X_2$ for $x \equiv 0, 2(k+1)-1, 3(k+1)-1, 4(k+1)-1, 5(k+1)-1, 5(k+1), 6(k+1), \newline 7(k+1) \pmod{8(k+1)}$, so copies of $T$ beginning at positions $k+1, 3(k+1), 5(k+1)+1, 7(k+1)+1 \pmod{8(k+1)}$ tile $X_2 \cap (\
| 1,312
| 3,755
| 1,596
| 1,170
| 3,239
| 0.773822
|
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|
string has a larger number of pairs, as shown in the following example.
Let $c:\{{\texttt{\textup{a}}},{\texttt{\textup{b}}},{\texttt{\textup{c}}}\}\times\{{\texttt{\textup{a}}},{\texttt{\textup{b}}},{\texttt{\textup{c}}}\}^{\leq{1}}\rightarrow\{0,1\}^{+}$ be an adaptive code of order one given as in the previous example, and $w={\texttt{\textup{abbbccbccaabccaaacba}}}$ an input data string. One can verify that ${\it Pairs}(w)=\{2,3,5,8,10,13,15,16\}$, ${\it NRpairs}(w)=8$, ${\it Prate}(w)=0.4$, and $|\overline{c}(w)|=31$. Encoding the string $w$ by Huffman’s algorithm, we get that $|{\it Huffman}(w)|=34$.
The results obtained in the previous examples are summarized in the table below, which shows that we get substantial improvements for input data strings having a larger number of pairs.
Builder: Entropy Bounds
=======================
In this section, we focus on computing the entropy bounds for the algorithm described in section 3. Given that our algorithm is based on Huffman’s algorithm, let us first recall the entropy bounds for Huffman codes.
Let $\Sigma$ be an alphabet, $x$ a data string of length $n$ over $\Sigma$ and $k$ the length of the encoder output, when the input is $x$. The , denoted by $R(x)$, is defined by $$R(x)=\frac{k}{n}.$$
Let $R(x)$ be the compression rate in codebits per datasample, computed after encoding the data string $x$ by the Huffman algorithm. One can obtain upper and lower bounds on $R(x)$ before encoding the data string $x$ by computing the entropy denoted by $H(x)$. Let $x$ be a data string of length $n$, $(F_{1},F_{2},\ldots,F_{h})$ the vector of frequencies of the symbols in $x$ and $k$ the length of the encoder output. The entropy $H(x)$ of $x$ is defined by $$H(x)=\frac{1}{n}\sum_{i=1}^{h}F_{i}\log_{2}(\frac{n}{F_{i}}).$$ Let $L_{i}$ be the length of the codeword associated to the symbol with the frequency $F_{i}$ by the Huffman algorithm, $1\leq{i}\leq{h}$. Then, the compression rate $R(x)$ can be re-written by $$R(x)=\frac{1}{n}\sum_{i=1}^{h}F_{i}L_{i}.$$ If we
| 1,313
| 2,680
| 2,781
| 1,322
| 1,054
| 0.795117
|
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|
&+& 140 A^6 B+292 A^7 B +424 A^8 B+ 332 A^9 B+12 A^3
B^2+12 A^4
B^2+ 118 A^5 B^2\nonumber\\ \fl
&+& 380 A^6 B^2+ 806 A^7 B^2+664 A^8 B^2+72 A^4
B^3 +352 A^5 B^3+704 A^6 B^3+ 1728 A^7 B^3
\nonumber\\ \fl
&+& 344 A^4 B^4+ 1568 A^5 B^4+848
A^6B^4+264 A^4 B^5+3192 A^5 B^5+ 320 A^3 B^6\, , \label{eq:Ab3} \\
\fl B'&=&A^6+12 A^7+40 A^8+60 A^9+32 A^{10}+28 A^6 B + 88 A^7
B+224 A^8 B+160 A^9 B\nonumber\\
\fl &+& 40 A^6 B^2+496 A^7 B^2 +596 A^8 B^2 + 176 A^5
B^3 +768 A^6
B^3+ 1056 A^7 B^3+ 88 A^3 B^4\nonumber\\ \fl
&+& 264 A^5 B^4 + 2534 A^6 B^4+
1152 A^4 B^5+1888 A^5 B^5\nonumber\\ \fl &+& 5808 A^4 B^6+1936
A^3 B^7+ 4308 A^2 B^8 \, , \label{eq:Bb3}\\
\fl C'&=&C^3+3 C^4+C^5+2C^6\, , \label{eq:Cb3}
\\
\fl D'&=&2A^6 D^3 + 4 A^7 D^3 + 4 A^6 D^4 +
2 A^5 D^5 + 28 A^6 D^3 B + 4 A^4 D^4 B +
14 A^5 D^4 B \nonumber
\\ \fl&+& 4 A^4 D^5 B +
8 A^4 D^3 B^2 + 56 A^5 D^3 B^2 +
44 A^4 D^4 B^2 + 4 A^3 D^5 B^2 \nonumber\\
\fl &+&
144 A^4 D^3 B^3 + 36 A^3 D^4 B^3 +
12 A^2 D^5 B^3 + 72 A^3 D^3 B^4 +
132 A^2 D^4 B^4 \nonumber\\
\fl&+& 264 A^2 D^3 B^5 +
12 A^6 D^2 C + 18 A^7 D^2 C +
2 A^4 D^3 C + 8 A^5 D^3 C + 8 A^6 D^3 C \nonumber
\\ \fl &+& 4 A^5 D^4 C + 2 A^4 D^5 C +
16 A^4 D^2 B C + 48 A^5 D^2 B C +
64 A^6 D^2 B C \nonumber\\
\fl &+& 4 A^2 D^3 B C +
8 A^4 D^3 B C + 48 A^5 D^3 B C +
4 A^3 D^4 B C + 8 A^4 D^4 B C \nonumber\\
\fl &+&
8 A^3 D^5 B C + 12 A^2 D^2 B^2 C +
36 A^4 D^2 B^2 C + 162 A^5 D^2 B^2 C + 24 A^3 D^3 B^2 C\nonumber\\
\fl &+& 28 A^4 D^3 B^2 C +
44 A^3 D^4 B^2 C + 8 A^2 D^5 B^2 C + 96 A^3 D^2 B^3 C + 64 A^4 D^2 B^3 C \nonumber\\
\fl &+& 152 A^3 D^3 B^3 C + 24 A^2 D^4 B^3 C +
24 A D^5 B^3 C + 512 A^3 D^2 B^4 C \nonumber\\
\fl &+&88 A D^4 B^4 C + 264 A^2 D^2 B^5 C +
320 A D^2 B^6 C + 6 A^4 D C^2 + 16 A^5 D C^2 \nonumber\\
\fl &+& 28 A^6 D C^2 + 12 A^7 D C^2 +
12 A^5 D^2 C^2 + 18 A^6 D^2 C^2 + 2 A^3 D^3 C^2 \nonumber\\\fl&+&
6 A^4 D^3 C^2 +
8 A^5 D^3 C^2 + 4 A^4 D B C^2 + 64 A^5 D B C^2 + 56 A^6 D B C^2\nonumber\\
\fl &+&
12 A^3 D^2 B C^2 + 30 A^4 D^2
| 1,314
| 4,309
| 414
| 957
| null | null |
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|
r the classical regularization methods we can compute the variance function which gives uncertainty estimate for the solution which in the classical formulation is not available. Furthermore, the hyperparameter estimation methods outlined in the next section provide principled means to estimate the parameters also in the classical regularization methods.
Hyperparameter estimation
=========================
In this section, we will consider some methods for estimating the *hyperparameters*. The free parameters of the covariance function, for example, the parameters $\sigma_f$ and $l$ in the squared exponential covariance function, are together with the noise parameter $\sigma$ referred to as the hyperparameters of the model. In this work, we employ a Bayesian approach to estimate the hyperparameters, and comparisons with standard parameter estimation methods such as L-curve and cross-validation methods are given as well.
Posterior distribution of hyperparameters
-----------------------------------------
The marginal likelihood function corresponding to the model is given as $$\label{likelihood}
p(\mathbf{y} \mid \sigma_f,l,\sigma)
= \mathcal{N}(\mathbf{y} \mid \mathbf{0}, Q(\sigma_f,l) + \sigma^2 \, I),$$ where $Q(\sigma_f,l)$ is defined by . The posterior distribution of parameters can now be written as follows:
$$\label{posterior}
p(\sigma_f,l,\sigma \mid \mathbf{y}) \propto
p(\mathbf{y} \mid \sigma_f,l,\sigma)
p(\sigma_f) p(l) p(\sigma),$$
where non-informative priors are used: $p(\sigma_f)\propto \frac{1}{\sigma_f}$, $p(l)\propto \frac{1}{l}$ and $p(\sigma)\propto \frac{1}{\sigma}$. The logarithm of can be written as $$\label{logposterior}
\log\,p(\sigma_f,l,\sigma \mid \mathbf{y}) =
\text{const.} - \frac{1}{2} \, \log \det (Q + \sigma^2 I)
- \frac{1}{2} \mathbf{y}^{\mathsf{T}}(Q + \sigma^2 I)^{-1} \mathbf{y}
- \log \frac{1}{\sigma_f}
- \log \frac{1}{l} - \log \frac{1}{\sigma}.$$ Given the posterior distribution we have a wide selection of methods from statistics to
| 1,315
| 4,487
| 1,098
| 883
| 4,049
| 0.76845
|
github_plus_top10pct_by_avg
|
) \right\},$$ where, for $j=1,\ldots,s$, $t_{2j-1} = t_{2j} = \frac{t}{\|G_j\|}$.
Recalling that $\widehat{\nu} = G \widehat{\psi}$, we have that $$\left\| \sqrt{n}(\hat{\nu} - \nu ) \right\|_\infty \leq t \quad
\text{if and only if } \quad
\sqrt{n} (\hat{\psi} - \psi) \in P(G,t).$$ Similarly, if $\tilde{Z}_n \sim
N_b(0,V)$ and $Z_n = G \tilde{Z}_n \sim N_s(0,\Gamma)$ $$\| Z_n \|_\infty \leq t\quad
\text{if and only if } \quad \tilde{Z}_n \in P(G,t).$$
Now consider the class $\mathcal{A}$ of all subsets of $\mathbb{R}^b$ of the form specified in , where $t$ ranges over the positive reals and $G$ ranges in $ \{ G(\psi(P)), P \in \mathcal{P}\}$. Notice that this class is comprised of polytopes with at most $2s$ facets. Also, from the discussion above, $$\label{eq:simple.convex}
\sup_{ P \in \mathcal{P}_n} \sup_{t >0} \left | \mathbb{P}\left( \| \sqrt{n} (\hat{\nu} - \nu) \|_\infty
\leq t\right) -\mathbb{P}\left( \| Z_n \|_\infty \leq t \right) \right| =
\sup_{A \in \mathcal{A}} \left | \mathbb{P}(\sqrt{n} (\hat{\psi} - \psi) \in A)- \mathbb{P}( \tilde{Z}_n \in A) \right|.$$
The claimed result follows from applying the Berry-Esseen bound for polyhedral classes, in the appendix, due to [@cherno2] to the term on the left hand side of . To that end, we need to ensure that conditions (M1’), (M2’) and (E1’) in that Theorem are satisfied.
For each $i=1,\ldots,n$, set $\tilde{W}_i=
(\tilde{W}_{i,1}, \ldots, \tilde{W}_{i,2s})= \left( (W_i - \psi)^\top v,v \in \mathcal{V}(G) \right)$. Condition (M1’) holds since, for each $l=1,\ldots,2s$, $$\mathbb{E}\left[ \tilde{W}^2_{i,l} \right] \geq \min_l v_l^\top V v_l \geq
\lambda_{\min}(V),$$ where $V = \mathrm{Cov}[W]$. Turning to condition (M2’), we have that, for for each $l=1,\ldots,2s$ and $k=1,2$, $$\begin{aligned}
\mathbb{E}\left[ | \tilde{W}_{i,l}|^{2+k} \right] & \leq \mathbb{E}\left[ |v_l^\top
(W_i - \psi)|^2 \|W_i - \psi\|^{k} \right]\\
& \leq \mathbb{E}\left[ |v_l^\top (W_i - \psi)|^2 \right] \left( 2A \sqrt{b}
\right)^k\\
& \leq \ove
| 1,316
| 3,086
| 1,286
| 1,108
| null | null |
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|
�(\[eq.3.1.4\]) for calculation of a new reflectionless potential $V_{2}(r)$ with a barrier on the basis of the known reflectionless inverse power potential $V_{1}(r)$. Let’s assume, that these potentials are connected with one superpotential $W_{2}(r)$. Let’s consider the wave function for the reflectionless inverse power potential $V_{1}(r)$ at $l=0$ in the form: $$\chi_{l=0}^{(1)}(k,r) =
\bar{N}_{1} \Bigl(f^{-}(r) e^{-ikr} -
S_{l=0}^{(1)} f^{+}(r) e^{ikr} \Bigr).
\label{eq.3.2.2.1}$$ Then the radial wave function at $l=0 $ for the reflectionless potential $V_{2}(r)$ with the barrier can be found on the basis of the second expression of (\[eq.2.3.6\]). Taking into account (\[eq.2.1.4\]) and (\[eq.2.4.9\]), we obtain: $$\begin{array}{lcl}
\chi_{l=0}^{(2)}(k,r) & = &
% \displaystyle\frac{1}{N_{2}} A_{2} \chi_{l=0}^{(1)}(k,r) =
\displaystyle\frac{\bar{N}_{1}}{N_{2}}
\biggl( \alpha\displaystyle\frac{d}{dr} + W_{2}(r) \biggr)
\Bigl(f^{-}(r) e^{-ikr} - S_{l=0}^{(1)} f^{+}(r) e^{ikr} \Bigr) = \\
& = &
\displaystyle\frac{\bar{N}_{1}}{N_{2}}
\biggl[
\biggl(\alpha \displaystyle\frac{d f^{-}(r)}{dr} -
ik\alpha f^{-}(r) + W_{2}(r) f^{-}(r) \biggr) e^{-ikr} - \\
& - &
S_{l=0}^{(2)}
\biggl(\alpha \displaystyle\frac{d f^{+}(r)}{dr} +
ik\alpha f^{+}(r) + W_{2}(r) f^{+}(r) \biggr) e^{ikr}
\biggr].
\end{array}
\label{eq.3.2.2.2}$$ In this expression one can see the division of the total radial wave function into the convergent and divergent components, that can be interesting in analysis of scattering (with possible tunneling) of the particle in the field of the reflectionless potential $V_{2}(r)$ with the barrier.
So, if to use the potential (\[eq.3.1.5\]) as the first reflectionless inverse power potential, then we find: $$\beta_{2} = 2 \alpha
\label{eq.3.2.2.3}$$ and $$\begin{array}{lcl}
f^{\pm}(r) =
1 \pm \displaystyle\frac{i}
{k \biggl(\bar{r} + \displaystyle\frac{1}{C\alpha} \biggr)}, &
| 1,317
| 3,741
| 1,241
| 1,234
| 2,778
| 0.777044
|
github_plus_top10pct_by_avg
|
A$ of size at most $\exp(\log^{O(1)}2\tilde K)$ such that $$\label{eq:induction.step}
\tilde A\subset X\tilde A_1\cdots\tilde A_r.$$ Since $G/N_0$ is generated by the $K$-approximate group $\rho(A)$, we may apply the induction hypothesis to each approximate subgroup $\rho(\tilde A_i)$ of $G/N_0$ to obtain, for each $i=1,\ldots,r_0$, integers $$\begin{aligned}
r_i,\ell_i&\le e^{O((\tilde s-1)^2)}\log^{O(\tilde s-1)}(2\tilde K^{e^{O(1)}})\\
& \le e^{O(\tilde s(\tilde s-1))}\log^{O(\tilde s-1)}2\tilde K;\end{aligned}$$ a normal subgroup $N_i\lhd G$ containing $N_0$ and satisfying $$\begin{aligned}
N_i&\subset A^{e^{O((\tilde s-1)^2)}K^{e^{O(s)+O(\tilde s-1)}(e^{O(1)}m)}(e^{O(1)}\log2\tilde K)^{O(\tilde s-1)}}N_0\\
&\subset A^{e^{O(\tilde s(\tilde s-1))}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s-1)}2\tilde K}N_0;\end{aligned}$$ finite $\tilde K^{e^{O(\tilde s)}}$-approximate groups $A_1^{(i)},\ldots,A_{r_i}^{(i)}\subset\tilde A_i^{e^{O(\tilde s-1)}}\subset\tilde A^{e^{O(\tilde s)}}$ such that, writing $\pi_i:G\to G/N_i$ for the quotient homomorphism, each group $\langle\pi_i(A_j^{(i)})\rangle$ is abelian; and sets $X_1^{(i)},\ldots,X_{\ell_i}^{(i)}\subset\tilde A_i^{e^{O(\tilde s-1)}}\subset\tilde A^{e^{O(\tilde s)}}$ satisfying $$\begin{aligned}
|X_j^{(i)}|&\le\exp(e^{O(\tilde s-1)}\log^{O(1)}(2\tilde K^{e^{O(1)}}))\\
&\le\exp(e^{O(\tilde s)}\log^{O(1)}2\tilde K)\end{aligned}$$ such that $$\label{eq:induction.hyp}
\tilde A_i\subset N_i\prod\{A_1^{(i)},\ldots,A_{r_i}^{(i)},X_1^{(i)},\ldots,X_{\ell_i}^{(i)}\},$$ with the product taken in some order.
Defining $N=N_1\cdots N_{r_0}$, we then have $$\begin{aligned}
N&\subset A^{r_0e^{O(\tilde s(\tilde s-1))}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s-1)}2\tilde K}\cdot N_0\\
&\subset A^{e^{O(\tilde s(\tilde s-1))}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s)}2\tilde K}\cdot N_0\\
&\subset A^{e^{O(\tilde s(\tilde s-1))}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s)}2\tilde K}\cdot A^{K^{e^{O(s)}m}}\\
&\subset A^{e^{O(\tilde s^2)}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\ti
| 1,318
| 1,019
| 1,726
| 1,242
| null | null |
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|
\left \{
% \frac{1}{\bar z-\bar x}\left(
%-\frac{c_+}{c_++c_-} t^a \phi(w) 2\phi \delta^{(2)}(x-w) + :\bar \p j^a_{L,z} \phi:(w) \right. \right. \cr
%&\qquad \qquad \qquad \left.
%+ {A^a}_c \frac{1}{\bar x-\bar w} :j^c_{L,z} \phi:(w) + {B^a}_c \frac{1}{x-w} :j^c_{L,\bar z} \phi:(w) + \mathcal{O}(f^4)
%\right) \cr
%& \quad \left.
%+ \frac{\bar \Delta_{\phi}:j^a_{L,z} \phi:(w)}{(\bar z-\bar w)^2} + \frac{:j^a_{L,z}\bar \p \phi:(w)}{\bar z-\bar w}
%\right\} \cr
%
&=
\frac{:\bar \p j^a_{L,z} \phi:(w)}{\bar z-\bar w} + {A^a}_c \frac{1}{(\bar z-\bar w)^2} :j^c_{L,z} \phi:(w) \cr
& \quad + \frac{\bar \Delta_{\phi}:j^a_{L,z} \phi:(w)}{(\bar z-\bar w)^2} + \frac{:j^a_{L,z}\bar \p \phi:(w)}{\bar z-\bar w}
\end{aligned}$$ Hence we have: $$\begin{aligned}
\bar T(\bar z) :j^a_{L,z}\phi:(w) &= \frac{\bar \Delta_{\phi}:j^a_{L,z}\phi:(w) - \frac{c_-}{(c_++c_-)^2}i{f^a}_{bc}t^b:j^c_{L,z}\phi:(w)}{(\bar z-\bar w)^2} + \frac{\bar \partial :j^a_{L,z}\phi:(w)}{\bar z-\bar w} + \mathcal{O}(f^4).\end{aligned}$$ That leads to the conformal dimension: $$\begin{aligned}
\bar h \left(\left[:j^a_{L,z} \phi:\right]_{\tilde{\mathcal{R}}} \right) &=& \frac{f^2}{2} c^{(2)}_\mathcal{R} +
\frac{f^2}{2} (1-k f^2) (c^{(2)}_{\tilde{\mathcal{R}}}-c^{(2)}_{\mathcal{R}})+ \mathcal{O}(f^4). \nonumber % \cr
%\end{aligned}$$ This is identical to the previous result, except for the lack of shift by one (since we are acting with the holomorpic component of the left current). Finally one can perform the same computation for the operators $:j^a_{L,\bar z} \phi:$. One finds : $$\begin{aligned}
h \left(\left[:j^a_{L,\bar z} \phi:\right]_{\tilde{\mathcal{R}}} \right) &=& \frac{f^2}{2} c^{(2)}_\mathcal{R} +
\frac{f^2}{2} (1+k f^2)( c^{(2)}_{\tilde{\mathcal{R}}}-c^{(2)}_{\mathcal{R}})+ \mathcal{O}(f^4) \cr
\bar h \left(\left[:j^a_{L,\bar z} \phi:\right]_{\tilde{\mathcal{R}}} \right) &=& \frac{f^2}{2} c^{(2)}_\mathcal{R} + 1 +
\frac{f^2}{2} (1+k f^2) (c^{(2)}_{\tilde{\mathcal{R}}}-c^{(2)}_{\mathcal{R}})+ \mathcal{O}(f^4) \end{aligned}$$
One can perfor
| 1,319
| 2,845
| 1,679
| 1,312
| null | null |
github_plus_top10pct_by_avg
|
\alpha}\lambda_{q'\beta}U_{\beta 1}
U_{\alpha 1}^{*} = \xi_{q1}^{*} \xi_{q'1}$ with $(q,q'=d,s,b)$, and ${\cal B} = \kappa_{\alpha \beta}U_{\beta 1} U_{\alpha 1}^{*}$. For flavor changing processes, ${\cal A}_{qq'}$ plays the main role, with ${\cal B}$ its counterpart in flavor conserving processes.
Before continuing, we comment on the strong CP-violation parameter $\theta_{\rm QCD}$. With CP symmetry imposed only on the hard (dim-4) terms, the $\theta_{\rm QCD}$ parameter is naively zero at tree level, but $M_Q$ may still be complex. If so, alignment of the QCD vacuum with this complex quark mass will generate a non-zero tree level $\theta_{\rm QCD}$. To avoid this contribution, we simply impose CP symmetry on both dim-4 and dim-3 terms. $M_Q$ will then be real in the same basis that the tree-level $\theta_{\rm QCD}$ vanishes[@interplay]. A similar scheme can also be arranged if CP is broken spontaneously (see below). The first non-zero contribution to $\theta_{\rm QCD}$ (occuring at two loops) will be discussed later.
Constraint from $\epsilon$ {#constraint-from-epsilon .unnumbered}
==========================
With CP conservation modulo soft-breaking enforced, the CKM matrix is real at tree level. Leading CP violating phenomena should be due solely to the CP-violating phase in the charged Higgs sector. Making the usual “$\pi \pi (I=0)$ dominance” assumption, the CP violation parameter $\epsilon$ is approximately $$\begin{aligned}
\epsilon & \simeq &\frac{e^{i\pi/4}}{\sqrt{2}}
\left(
\frac{\hbox{Im}M_{12}}{2\hbox{Re}M_{12}}
+\frac{\hbox{Im}A_0}{\hbox{Re}A_0}
\right)\ .\end{aligned}$$ We shall postpone discussion of $A_0$, but will see later that in our model, as in the KM Model, the second term is negligible. Experimentally, $\epsilon \simeq 0.00226 \ \exp(i \pi/ 4)$. The $\Delta
S=2$ part of the effective Hamiltonian to one-loop (i.e., box diagrams) can be written as: $$\nonumber
{\cal H}^{\Delta S=2} =
\frac{G_F^2 m_W^2}{16\pi^2}
\sum_{I=R,L}C^I_{\Delta S=2}(\mu) O^I_{\Delta S=2}(\mu)
,\;\;\qu
| 1,320
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| 1,460
| 2,620
| 0.778377
|
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|
y $$\frac{1}{2^m}q : L\longrightarrow A, x\mapsto \frac{1}{2^m}h(x,x).$$ Then $\frac{1}{2^m}q$ mod 2 defines a quadratic form $L/\pi L \longrightarrow \kappa$. It can be easily checked that $\frac{1}{2^m}q$ mod 2 on $L/\pi L$ is an additive polynomial. We define a lattice $B(L)$ as follows.
- $B(L)$ is defined to be the sublattice of $L$ such that $B(L)/\pi L$ is the kernel of the additive polynomial $\frac{1}{2^m}q$ mod 2 on $L/\pi L$.
To define a few more lattices, we need some preparation as follows. Recall that $\pi\cdot\sigma(\pi)$ is denoted by $\xi$.
Assume $B(L)\varsubsetneq L$ and $l$ is even. Then the bilinear form $\xi^{-l/2}h$ mod $\pi$ on the $\kappa$-vector space $L/X(L)$ is nonsingular symmetric and nonalternating. It is well known that there is a unique vector $e \in L/X(L)$ such that $$(\xi^{-l/2}h(v,e))^2=\xi^{-l/2}h(v,v) \textit{ mod } \pi$$ for every vector $v \in L/X(L)$. Let $\langle e\rangle$ denote the 1-dimensional vector space spanned by the vector $e$ and denote by $e^{\perp}$ the 1-codimensional subspace of $L/X(L)$ which is orthogonal to the vector $e$ with respect to $\xi^{-l/2}h$ mod $\pi$. Then $$B(L)/X(L)=e^{\perp}.$$ If $B(L)= L$, then the bilinear form $\xi^{-l/2}h$ mod $\pi$ on the $\kappa$-vector space $L/X(L)$ is nonsingular symmetric and alternating. In this case, we put $e=0\in L/X(L)$ and note that it is characterized by the same identity.\
The remaining lattices we need for our definition are:
- Define $W(L)$ to be the sublattice of $L$ such that $$\left\{
\begin{array}{l l}
\textit{$W(L)/X(L)=\langle e\rangle$} & \quad \textit{if $l$ is even};\\
\textit{$W(L)=X(L)$} & \quad \textit{if $l$ is odd}.
\end{array} \right.$$
- Define $Y(L)$ to be the sublattice of $L$ such that $Y(L)/\pi L$ is the radical of $$\left\{
\begin{array}{l l}
\textit{the form $\frac{1}{2^{m}}h$
mod $\pi$ on $B(L)/\pi L$} & \quad \textit{if $l=2m$};\\
\textit{the form $\frac{1}{\pi}\cdot\frac{1}{2^{m-1}}h$ mod $\pi$ on $B(L)/\pi L$} & \quad \t
| 1,321
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| 1,260
| 1,222
| null | null |
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|
egin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};{b\!\!+\!\!2},;1_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};1;1;1)$}};\\
B&=
{\text{\footnotesize$\gyoungx(1.2,;1_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};1;1;1;2;3;4_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};{b\!\!+\!\!2},;1_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};1;2;2)$}}.\end{aligned}$$ Note that our assumptions on the parameters $a,b,u,v$ mean that these tableaux really do exist, i.e. there are enough $1$s to fill the bottom row.
\[abnz\] ${\hat\Theta_{A}}$ and ${\hat\Theta_{B}}$ are non-zero, and are linearly independent if $v\ls b+1$.
It is straightforward to express ${\hat\Theta_{A}}$ and ${\hat\Theta_{B}}$ as linear combinations of semistandard homomorphisms using a single application of Lemma \[lemma7\]; in each case we get at least one semistandard appearing, so the homomorphisms are non-zero. If in addition $v\ls b+1$, then in the expression for ${\hat\Theta_{A}}$ there is at least one semistandard tableau with two $2$s in the first row; there is no such tableau appearing in the expression for ${\hat\Theta_{B}}$, so ${\hat\Theta_{A}},{\hat\Theta_{B}}$ are linearly independent.
\[abhoms\]
- If $a-v\equiv3\ppmod4$ or $v=b+3$, then ${\psi_{d,t}}\circ{\hat\Theta_{A}}=0$ for all admissible $d,t$.
- If $v\equiv1\ppmod4$, then ${\psi_{d,t}}\circ{\hat\Theta_{B}}=0$ for all admissible $d,t$.
- If $a\equiv0\ppmod4$, then ${\psi_{d,t}}\circ({\hat\Theta_{A}}+{\hat\Theta_{B}})=0$ for all admissible $d,t$.
Lemma \[lemma5\] immediately gives ${\psi_{d,1}}\circ{\hat\Theta_{A}}={\psi_{d,1}}\circ{\hat\Theta_{B}}=0$ for $d\gs2$. Using the fact that $A,B$ each have an odd number of $1$s in each row, we also get $${\psi_{1,t}}\circ{\hat\Theta_{A}}={\psi_{1,t}}\circ{
| 1,322
| 1,463
| 1,204
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| 1,525
| 0.788624
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|
sin^2(\theta)(a^2\cos^2(\theta)+r^2)^2}{(r\alpha\cos(\theta)-1)^8}}(a^2\cos^2(\theta)+r^2)^5(r\alpha \cos(\theta)-1)^3} \nonumber\\
&\bigg(\sin(\theta)\Big(((2a^{10}\alpha^{3} m^{2} r-2a^{8}\alpha m^{2}r)\cos^{8}(\theta)-4(a^{4}m^{2}\alpha^{2}+(9\alpha^{2}r^{2}-1)m^{2}a^{2})a^{6}\cos^{7}(\theta)-10(m^{2}(\frac{34}{5}r^{3}\alpha^{2} \nonumber\\
&-6r)a^{4}-\frac{34}{5}r^{3}a^{2}m^{2})\alpha a^{4}\cos^{6}(\theta)
+96 a^{4}(a^{4}m^{2}r^{2}\alpha^{2}+(\frac{71}{24}\alpha^{2}r^{4}-r^{2})m^{2}a^{2})\cos^{5}(\theta)+150((\frac{6}{5}r^{3}\alpha^{2}-\frac{34}{15}r) \nonumber\\
&m^{2}a^{4}-\frac{6}{5}r^{3}a^{2}m^{2}) \alpha a^{2}r^{2}\cos^{4}(\theta)-180a^{2}(a^{4}m^{2}r^{2}\alpha^{2}+(\frac{71}{45}\alpha^{2}r^{4}-r^2)m^2a^2)r^2\cos^{3}(\theta)-150\alpha((\frac{34}{75} r^3\alpha^{2} \nonumber\\
&-\frac{38}{25} r)m^2a^4-\frac{34}{75} r^3 a^2 m^2)r^4\cos^{2}(\theta)+40(a^4 m^2 r^2 \alpha^2+(\frac{9}{10} \alpha^2 r^4-r^2)m^2a^2)r^4\cos(\theta)+10\alpha((\frac{1}{5}r^3\alpha^2-\frac{6}{5}r) \nonumber\\
&m a^2-\frac{1}{5}r^3 m)mr^6)\sin^2(\theta)+(2\alpha a^6 (a^4\alpha^2m^2-a^2m^2)\cos^{9}(\theta)-36\cos^8(\theta) a^8 \alpha^2 m^2 r-68 \alpha a^4(m^2(\alpha^2 r^2-\frac{3}{17})a^4 \nonumber\\
& -a^2 m^2 r^2)\cos^7(\theta)+40 a^4(a^4 m^2 r\alpha^2-\frac{9}{20}(-\frac{142}{9} r^3\alpha^2+\frac{20}{9}r)m^2a^2)\cos^6(\theta)+60\alpha a^2((3 r^3 \alpha^2-\frac{19}{5}r)m^2 a^4 \nonumber\\
&-3 r^3 a^2 m^2)r\cos^5(\theta)-180 a^2(a^4 m^2 r^2\alpha^2+(\frac{71}{45} \alpha^2r^4-r^2)m^2a^2)r\cos^{4}(\theta)-200((\frac{17}{50}r^3\alpha^2-\frac{17}{10}r)m^2a^4- \nonumber\\
&\frac{17}{50}r^3 a^2 m^2)\alpha r^3 \cos^3(\theta)+96 r^3(a^4 m^2 r^2\alpha^2+(\frac{3}{8}\alpha^2 r^4-r^2)m^2a^2)\cos^2(\theta)+60((\frac{1}{30} r^3\alpha^2-r)m^2 a^2-\frac{1}{30}r^3 m^2)\nonumber\\
&\alpha r^5\cos(\theta)-4a^2\alpha^2 m^2 r^7+4 m^2 r^7)\sin(\theta)+(a^2\cos^{2}(\theta)+r^2)(a^2(a^2\alpha m -a m)\cos^3(\theta)+(-3 a^3\alpha m r-3 a^2 m r) \nonumber\\
& \cos^2(\theta)+ (-3 a^2 \alpha m r^2+3 a m r^2)\cos(\theta)+r^2(a\alpha m r+m r))(r\alpha\cos(\theta)
| 1,323
| 1,577
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|
\[Sec:Nz\] we present the results on a stack of bilayers along the same lines as for the bilayer. Finally, in Sec. \[sec:dis\] we discuss the implications of our analytical and numerical results and also provide an alternative model for the SP.
Bilayer model of SP {#Sec:I}
===================
Here we introduce a model of two asymmetric parallel layers, each being a square lattice of linear size $L=1,2,3,...$ (in terms of the inter-site shortest distance) characterized by two fields $\psi_1=\exp(\phi_{1})$ and $\psi_2=\exp(\phi_{2})$ on the layers $z=1,2$, respectively.The action can be written as H&=& - \_[ij]{} \[t\_1 (\_[ij]{} \_1 - A\_[ij]{}) +t\_2 (\_[ij]{} \_2 -g\_2 A\_[ij]{})\
&+& A\^2\_[ij]{}\] - \_i u(\_2(i)-\_1(i)) \[2N\] where $t_1>,t_2>0, g>0$ and $g_2$ are parameters; $\langle ij\rangle$ denotes summation over nearest neighbor sites within each layer; $\nabla_{ij} \phi_a \equiv \phi_a(i) - \phi_a(j)$; $A_{ij}$ is a bond vector field (that is, $A_{ij}=-A_{ji}$) oriented along the bond $\langle ij\rangle$. It is introduced in order to generate the “current-current” interaction (cf. [@Lubensky; @Toner; @Sondhi; @Kane_2001; @Ashwin_2001]) consistent with the compact nature of the fields $\phi_{1,2}$. This action is to be used in the partition function Z=DA D\_[1]{} D\_[2]{} (- H) \[ZZ2\] where the temperature is absorbed into the the parameters $t_1,t_2,u,g$. Our focus is on verifying the applicability of the RG analysis to the renormalization of the Josephson coupling $u$. Hence, we will not discuss physical origins of the variables and the parameters.
The RG solution for bilayer {#sec:RG}
---------------------------
In the approximation ignoring compact nature of the variables, the terms $-\cos(\nabla_{ij} \phi_1 - A_{ij})$ and $-\cos(\nabla_{ij} \phi_2 -g_2 A_{ij})$ are replaced by $(\nabla_{ij} \phi_1 - A_{ij})^2/2$ and $(\nabla_{ij} \phi_2 -g_2 A_{ij})^2/2$, respectively. Then, the gaussian integration over $A_{ij}$ can be carried out explicitly in Eq.(\[ZZ2\]), so that (\[2N\]) in terms of t
| 1,324
| 352
| 1,470
| 1,275
| 1,399
| 0.790055
|
github_plus_top10pct_by_avg
|
erline{u}}_1({\overline{E}},{\vec{p}\,'}) \gamma_\mu ({\cancel{k}_N}+M_N)
u_1(E_p^\Lambda,{\vec{p}}) \nonumber\\
&\times&
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})
\gamma_\nu
u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using heavy baryon expansion, $$\begin{aligned}
V_a
&=&\ \frac{G_Fm_\pi^2h_{\Lambda N}}{8\Delta Mf_\pi^4}
({\vec{\tau}_1}\cdot{\vec{\tau}_2})
(4 { B}_{20}
+4q_0 { B}_{10}
+q_0^2 { B}) \,,\nonumber\\\end{aligned}$$ where we have used the master integrals with $q_0=-\frac{M_\Lambda-M_N}{2}$ and ${\vec{q}}={\vec{p}\,'}-{\vec{p}}$.
Triangle diagrams {#sec:triangles}
=================
Two up triangles and two down triangles contribute to the interaction. The final expressions are written in terms of the integrals $I$ defined in Appendix \[sec:mi\]. The amplitude for the first up triangle, depicted in Fig. \[uptri\], is
![Up triangle diagram contributing at NLO. \[uptri\]](uptriangle112)
$$\begin{aligned}
V_b=&-i\frac38\frac{G_F m_\pi^2h_{2\pi}g_A^2}{M_N f_\pi^4}
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon}
\nonumber\\\times&\,
\frac{1}{(l+q)^2-m_\pi^2+i\epsilon}\,
\frac{(l^\mu+q^\mu)l^\nu}{k_N^2-M_N^2+i\epsilon}
\\\times&\nonumber\,
\,{\overline{u}}_1({\overline{E}}_p,{\vec{p}\,'}){\overline{u}}_1(E_p^\Lambda,{\vec{p}})
\\\times&\nonumber\,
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\mu\gamma_5({\cancel{k}_N}+M_N)\gamma_\nu\gamma_5
u_2(E_p,-{\vec{p}})\,.\end{aligned}$$
Using heavy baryon expansion, $$\begin{aligned}
V_b
=\frac34\frac{G_F m_\pi^2h_{2\pi}g_A^2}{f_\pi^4}
\left[
(3-\eta)I_{22}+{\vec{q}}^2I_{23}+{\vec{q}}^2I_{11}
\right]\,,\end{aligned}$$ where, we have used the master integrals with $q_0=\frac{M_\Lambda-M_N}{2}$, $q_0'=0$ and ${\vec{q}}={\vec{p}\,'}-{\vec{p}}$.
![Second up triangle contribution at NLO.[]{data-label="uptri2"}](uptriangle2g)
For the second up triangle, depicted in Fig. \[uptri2\], the relativistic amplitude is $$\begin{aligned}
V_c=&
-i\frac{G_Fm_\pi^2h_{\Lambda N} g_A^2}{8f_\pi^4(r_N^2-M_N^2)}
{\vec{\tau}_1}\cdot{\vec{\tau}_2}{\int\frac{
| 1,325
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| null | null |
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|
and
may
not
be
relied on by anyone as the basis of a contract by estoppel
or
otherwise.
Thank you.
**********************************************************************
---------------------- Forwarded by Vince J Kaminski/HOU/ECT on 09/08/2000
08:25 AM ---------------------------
Shirley Crenshaw
09/06/2000 12:56 PM
To: ludkam@aol.com
cc: (bcc: Vince J Kaminski/HOU/ECT)
Subject: Vince's Travel Itinerary
Ludmilla:
Here is Vince's travel itinerary for Sunday to New York.
DATE 06SEPTEMBER00
BOOKING REF ZE56TX
KAMINSKI/WINCENTY S0C0011R10
ENRON CORP KAMINSKI/WINCENTY
EB 1962
E-TKT RECEIPT
**REVIEW UPGRADE BELOW**
SERVICE DATE FROM TO DEPART ARRIVE
CONTINENTAL AIRLINES 10SEP HOUSTON TX NEW YORK NY 539P 1000P
CO 1672 A SUN G.BUSH INTERCO LA GUARDIA
TERMINAL C TERMINAL M
DINNER NON STOP
RESERVATION CONFIRMED 3:21 DURATION
AIRCRAFT: BOEING 737-800
SEAT 01F NO SMOKING CONFIRMED KAMINSKI/WINCEN
FIRST CLASS UPGRADE IS CONFIRMED
HOTEL 10SEP HILTON MILLENIUM
11SEP 55 CHURCH STREET
NEW YORK, NY 10007
UNITED STATES OF AMERICA
TELEPHONE: 212-693-2001
FAX: 212-571-2316
TELEX: TLX NONE
CONFIRMATION: 3110994415
SINGLE ROOM QUEEN SIZE BED
RATE: RA1
| 1,326
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github_plus_top10pct_by_avg
|
lpha }_p)
=(q_{pj}q_{jp})^{-c_{pj}^\chi },\\
\xi _1({\alpha }_j)=q_{pj}^{{b}-1} q_{jp}^{{b}-1}
=(q_{pj}q_{jp})^{-c_{pj}^\chi }.
\end{gathered}$$ Hence $\xi _1({\alpha }_j)=\xi _2({\alpha }_j)$ also in this case. This proves the lemma.
Multiparameter Drinfel’d doubles {#sec:DD}
================================
In this paper we study Verma modules for a class of Hopf algebras introduced in [@p-Heck07b]. This class contains multiparameter quantizations of semisimple Lie algebras and basic classical Lie superalgebras. The precise definition is given in Eq. . It uses the Drinfel’d double construction and the theory of Nichols algebras.
The Drinfel’d double [@b-Joseph Sect.3.2] can be defined via a skew-Hopf pairing of two Hopf algebras or as the quotient of a free associative algebra by a certain ideal, see also Rem. \[re:ideal\]. The first approach is more technical, but also more powerful. We present here the second definition. For proofs see [@p-Heck07b].
Let $I$ be a non-empty finite set, $\chi $ a bicharacter on ${\mathbb{Z}}^I$ with values in ${{\Bbbk }^\times }$, and $q_{i j}=\chi ({\alpha }_i,{\alpha }_j)$ for all $i,j\in I$. Let ${\mathcal{U}}(\chi )$ be the unital associative ${\Bbbk }$-algebra with generators $K_i$, $K_i^{-1}$, $L_i$, $L_i^{-1}$, $E_i$, and $F_i$, where $i\in I$, and defining relations $$\begin{aligned}
XY= YX \quad & \makebox[0pt][l]{for all $X,Y\in \{K_i,K_i^{-1},
L_i,L_i^{-1}\,|\,i\in I\}$,}
\label{eq:KLrel}\\
K_iK_i^{-1}=&\,1, & L_iL_i^{-1}=&\,1,
\label{eq:KKrel}
\\
K_iE_jK_i^{-1}=&\,q_{ij}E_j, & L_iE_jL_i^{-1}=&\,q_{ji}^{-1}E_j,
\label{eq:KErel}
\\
K_iF_jK_i^{-1}=&\,q_{ij}^{-1}F_j, & L_iF_jL_i^{-1}=&\,q_{ji}F_j,
\label{eq:KFrel}\\
E_iF_j&\makebox[0pt][l]{$-F_jE_i=\delta _{i,j}(K_i-L_i)$,}
\label{eq:EFrel}\end{aligned}$$ where $i,j\in I$, and $\delta _{i,j}$ denotes Kronecker’s $\delta $. The algebra ${\mathcal{U}}(\chi )$ can be given a Hopf algebra structure in many different ways. We will use the unique Hopf algebra structure determin
| 1,327
| 780
| 1,538
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| 3,475
| 0.772078
|
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|
t we can easily check the validity of the result. We have checked that if we substitute $ e=0 $ in these calculations, then we get back the result for the accelerating rotating black hole.
In FIG. \[fig6\] we find that the gravitational entropy density is not smooth, but contains several singularities. The above analysis clearly shows that the measure of the gravitational entropy used above is not adequate to explain the case of the accelerating rotating charged black holes. Therefore we have to use the measure proposed in [@entropy2] for the expression of $ P $, which is $$\label{mod_P}
P=C_{abcd}C^{abcd}.$$ Using the definition (\[mod\_P\]) of $ P $, we have calculated the gravitational entropy density, which is given in equation (\[new3\]): $$\begin{aligned}
\label{new3}
\left.s\right.&=\dfrac{k_{s}}{(a^2\cos^2(\theta)+r^2)^7}\bigg(96\Big(a^6\alpha(a^4 \alpha^2 m^2+3a^2\alpha^2 e^2 mr+2\alpha^2 e^4 r^2 -a^2m^2)\cos^9(\theta)\nonumber\\
&+(-20e^2mr^2+(-18a^2m^2+2e^4)r+a^2e^2m)\alpha^2a^6\cos^8(\theta)- \nonumber\\
&34\alpha a^4(21/34e^4r^4\alpha^2+57/34a^2e^2mr^3\alpha^2+(a^4\alpha^2m^2-a^2m^2)r^2+5/34a^2e^2mr-3/17a^4m^2)\cos^7(\theta)+\nonumber\\
&(90a^4e^2mr^4\alpha^2+(142a^6\alpha^2m^2-21a^4\alpha^2e^4)r^3-9a^6e^2m\alpha^2r^2+(20a^8\alpha^2 m^2-20a^6m^2)r+5a^6e^2m)\cos^6(\theta)\nonumber\\
&+30\alpha a^2(8/15e^4r^5\alpha^2+17/6a^2e^2 m r^4\alpha^2+(3a^4\alpha^2 m^2-3a^2m^2)r^3+1/6a^2e^2mr^2\nonumber\\
&+(-19/5a^4m^2+11/30e^4a^2)r+a^4e^2m)r\cos^5(\theta)+(-48a^2e^2m\alpha^2r^6+(-142a^4\alpha^2m^2+16a^2\alpha^2e^4)r^5 \nonumber\\
&-5a^4e^2mr^4\alpha^2+(-90a^6\alpha^2m^2+90a^4m^2)r^3-75a^4e^2mr^2+11a^4e^4r)\cos^4(\theta)-100(1/100e^4r^5\alpha^2+\nonumber\\
&3/20a^2e^2mr^4\alpha^2+(17/50a^4\alpha^2m^2-17/50a^2m^2)r^3-9/100a^2e^2mr^2+(-17/10a^4m^2+13/50e^4a^2)r+ \nonumber\\
&a^4e^2m)\alpha r^3\cos^3(\theta)+(2e^2m\alpha^2r^8+(18a^2\alpha^2 m^2-\alpha^2e^4)r^7+5a^2e^2m\alpha^2 r^6+ \nonumber\\
&(48a^4\alpha^2m^2-48a^2m^2)r^5+75a^2e^2mr^4-26e^4a^2r^3)\cos^2(\theta)+30((1/30a^2\alpha^2m^2-1/30m^2)r^3-1/30e^2mr^2+\n
| 1,328
| 1,084
| 1,689
| 1,379
| null | null |
github_plus_top10pct_by_avg
|
parents change jobs 0.89 times more often (*p* \< .01) than nontransnational parents. Thus, the first condition for mediation is only met for job instability and not for job absenteeism. Therefore, we only continue with the next steps of the mediation analysis for job instability.
######
Results of Mediation Analyses for Job Outcomes.
Model 1 Model 2
-------------------------------------------------------------------------------- ----------------------------------------------------------------------- ---------------------------------------------------------------------
Transnational parenting^[a](#table-fn3-0192513X17710773){ref-type="table-fn"}^ −0.10 (0.15) 0.89 (0.29)[\*\*](#table-fn6-0192513X17710773){ref-type="table-fn"}
Age −0.00 (0.01) 0.05 (0.02)[\*](#table-fn6-0192513X17710773){ref-type="table-fn"}
Sex^[b](#table-fn3-0192513X17710773){ref-type="table-fn"}^ 0.75 (0.14)[\*\*\*](#table-fn6-0192513X17710773){ref-type="table-fn"} 0.21 (0.26)
Marital status^[c](#table-fn3-0192513X17710773){ref-type="table-fn"}^ 0.03 (0.22) 0.43 (0.35)
Education 0.01 (0.03) 0.10 (0.06)
Years in the Netherlands 0.02 (0.02) 0.06 (0.03)
Housing^[d](#table-fn3-0192513X17710773){ref-type="table-fn"}^ −0.25 (0.21) 0.06 (0
| 1,329
| 2,424
| 1,910
| 1,238
| null | null |
github_plus_top10pct_by_avg
|
D-19
Baloxavir marboxil COVID-19
Thymosin α1 MERS
Nucleotide reverse transcriptase inhibitor: tenofovir disoproxil fumarate. SARS
Papain-like protease SARS, MERSand Human Coronavirus NL63
RNA-dependent RNA polymerase SARS, Murine Coronavirus
Tocilizumab COVID-19
α-interferon Spectrum of respiratory infectionsΈ RSV and SARS
[Table 3](#T3){ref-type="table"} represents the commercially available drugs used for the treatment of the various forms of coronaviruses. The viral infections discussed in the table are SARS - Severe Acute Respiratory Syndrome, MERS-Middle East Respiratory Syndrome, RSV - Respiratory Syncytial Virus, ARVI - Acute respiratory viral infections.
######
Chloroquine and its combination of drugs used in the treatment of COVID-19
Study Particulars Drugs Dosage Reference
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------
| 1,330
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|
i _{\kappa }(x)=(1+\left\vert x\right\vert ^{2})^{\kappa },\quad \kappa
\in {\mathbb{Z}} . \label{NOT2}$$The following properties hold:
- for every $\kappa\geq \kappa^{\prime }\geq 0$, $$\psi _{\kappa }(x) \leq \psi _{\kappa ^{\prime }}(x); \label{NOT3a}$$
- for every $\kappa\geq 0$, there exists $C_{\kappa }>0$ such that $$\psi _{\kappa }(x)\leq C_{\kappa }\psi _{\kappa }(y)\psi _{\kappa }(x-y);
\label{NOT3b}$$
- for every $\kappa \geq 0$, there exists $C_{\kappa }>0$ such that for every $\phi \in C_{b}^{\infty }({\mathbb{R}}^{d})$, $$\psi _{\kappa }(\phi (x))\leq C_{\kappa }\psi _{\kappa }(\phi
(0))(1+\left\Vert \nabla \phi \right\Vert _{\infty }^{2})^{\kappa }\psi
_{\kappa }(x); \label{NOT3d}$$
- for every $q\in {\mathbb{N}}$ there exists $\overline{C}_{q}\geq
\underline{C}_{q}>0$ such that for every $\kappa \in {\mathbb{R}}$ and $f\in
C^{\infty }({\mathbb{R}}^{d})$, $$\underline{C}_{q}\psi _{\kappa }\left\vert f\right\vert _{q}(x)\leq
\left\vert \psi _{\kappa }f\right\vert _{q}(x)\leq \overline{C}_{q}\psi
_{\kappa }\left\vert f\right\vert _{q}(x). \label{NOT3c}$$
Note that (\[NOT3a\])–(\[NOT3d\]) are immediate, whereas (\[NOT3c\]) is proved in Appendix \[app:weights\] (see Lemma \[Psy1\]).
For $q\in {\mathbb{N}}$, $\kappa \in {\mathbb{R}}$ and $p\in (1,\infty ]$ (we stress that we include the case $p=+\infty $), we set $\Vert \cdot \Vert
_{p}$ the usual norm in $L^{p}({\mathbb{R}}^{d})$ and $$\left\Vert f\right\Vert _{q,\kappa ,p}=\left\Vert \left\vert \psi _{\kappa
}f\right\vert _{q}\right\Vert _{p}. \label{NOT4}$$We denote $W^{q,\kappa ,p}$ to be the closure of $C^{\infty }({\mathbb{R}}%
^{d})$ with respect to the above norm. If $\kappa =0$ we just denote $%
\left\Vert f\right\Vert _{q,p}=\left\Vert f\right\Vert _{q,0,p}$ and $%
W^{q,p}=W^{q,0,p}$ (which is the usual Sobolev space). So, we are working with weighted Sobolev spaces. The following properties hold:
- for every $q\in {\mathbb{N}}$ there exists $\overline{C}_{q}\geq
\underline{C}_{q}>0$ su
| 1,331
| 1,755
| 1,237
| 1,263
| null | null |
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|
**Sliding window (SW)** **Most significant result**
------------------------- ----------------------------- -------- -------- -------------- --------------
**SNPs/SW** **No. of SW** **SW** **SW** ***P*value** ***P*value**
1 4 \- \- \- \-
2 3 S1.S2 S3.S4 0.0208\* 0.0546
3 2 S2.S4 S2.S4 0.0540 0.1099
4 1 S1.S4 S1.S4 0.0397\* 0.1585
S1: rs10907185.
S2: rs6603797.
S3: rs4648727.
S4: rs12126768.
The overall global test, details, and haplotype frequencies are listed in Table [8](#T8){ref-type="table"}. In HCV-1 infected patients, haplotype AC, the window S1-S2, gave the most impressive *P* value for the omnibus test. However, it did not play a significant role in HCV-2 infected patients. Haplotype-specific analyses showed that the CAT haplotypes (S2-S3-S4) might increase the rate of RVR (*P* = 0.0265; OR = 4.50) when compared to the RVR (−) groups, especially in the HCV-2 infected population. The window S1-S2-S3-S4 with the ACAT haplotypes was significantly positively associated with a higher rate of RVR in both HCV-1 and HCV-2 infected patients (OR = 2.01, *P* = 0.0261 and OR = 4.54, *P* = 0.0253, respectively). Furthermore, the results showed that HCV-1 and HCV-2 infected patients with therapeutic responses had the ACAT haplotypes, and thus the ACAT haplotype appeared more frequently in RVR (+) patients than in RVR (−) patients. Therefore, in HCV-1 or HCV-2 infected individuals, haplotype-specific analysis showed that the haplotype ACAT (S1-S2-S3-S4) was associated with an increase in the RVR rate. This observation suggests that the haplotype ACAT may play a role in the response to PEG-IFNα-RBV treatment.
######
Details of sex-adjust
| 1,332
| 2,048
| 1,973
| 1,512
| null | null |
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|
addition, there is another functor $$\pi_1^* \: \equiv \: \pi^* \otimes {\cal O}_{\Lambda}(1): \:
\mbox{Coh}({\mathbb P}^n) \: \stackrel{\sim}{\longrightarrow} \:
\mbox{Coh}({\mathbb P}^n, \chi(\alpha) ).$$ (In fact, there is an analogue of $\pi_1^*$ for every ${\cal O}_{\Lambda}(
\mbox{odd})$.)
To determine $\pi^* {\cal O}(m)$ in terms of ${\cal O}_{\Lambda}$’s, consider the commutative diagram $$\xymatrix{
\frac{ {\mathbb C}^{n+1} - 0 }{ {\mathbb C}^{\times} }
\ar[r] \ar[d] &
\frac{ {\mathbb C}^{n+1} - 0 }{ {\mathbb C}^{\times} }
\ar[d] \\
G {\mathbb P}^n \ar[r] & {\mathbb P}^n
}$$ The line bundle ${\cal O}(k)$, defined by weights $1, \cdots, 1, k$, pulls back to weights $2, \cdots, 2, 2k$, from which we deduce that $$\pi^* {\cal O}(k) \: = \: {\cal O}_{\Lambda}(2k),$$ which implies $$\pi_1^* {\cal O}(k) \: = \: {\cal O}_{\Lambda}(2k+1).$$
Note that although $\pi^*$ preserves tensor products, $\pi_1^*$ does [*not*]{} preserve tensor products: $$\begin{aligned}
\pi_1^* \left( {\cal O}(k) \otimes {\cal O}(m) \right) & \cong &
\pi_1^* {\cal O}(k+m), \\
& \cong & {\cal O}_{\Lambda}(2k+2m+1), \\
& \not\cong & {\cal O}_{\Lambda}(2k+2m+2) \: \cong \:
\left( \pi_1^* {\cal O}(k) \right) \otimes
\left( \pi_1^* {\cal O}(m) \right).\end{aligned}$$ Indeed, this is an immediate consequence of the definition of $\pi_1^*$. In addition, for the same reason, $\pi_1^*$ does not commute with duality of bundles $$\pi_1^* \left( {\cal L}^{\vee} \right) \: \not\cong \:
\left( \pi_1^* {\cal L} \right)^{\vee}.$$
Now, for any finite gerbe over any space, the tangent bundle of the gerbe is just the pullback (under $\pi$) of the tangent bundle to the space. One way to see this is to work locally on the atlas, which is just a finite cover, and so the tangent bundle should be the same. We can see this explicitly in the present case as follows. For the ${\mathbb Z}_2$ gerbe $G {\mathbb P}^n = {\mathbb P}^n_{[2,\cdots,2]}$, the tangent bundle seen by the gauged linear sigma model is $$0 \: \longrightarrow \: {\cal O}_{\Lambda} \: \longri
| 1,333
| 1,160
| 1,528
| 1,185
| null | null |
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|
PTSD symptom clusters and CSB among those with a PTSD diagnosis. The results of this model are displayed in [Table 3](#T3){ref-type="table"}. The re-experiencing symptom cluster was the only cluster significantly associated with CSB (*p* \< 0.05). A 1-standard-deviation increase in the re-experiencing symptoms was associated with 87% greater odds of CSB.
######
Associations between specific PTSD symptom clusters and compulsive sexual behavior among those with a PTSD diagnosis
OR (95% CI) *p*-value
------------------- ------------------- -----------
Re-experiencing 1.87 (1.05, 3.31) **0.032**
Avoidance 1.09 (0.71, 1.68) 0.684
Emotional numbing 0.96 (0.59, 1.53) 0.849
Hyper-arousal 0.71 (0.46, 1.12) 0.142
*Note:* Statistically significant values in bold. Based on GEE modeling, specifying binomial family, logit link, AR 1 correlation structure, and robust standard errors. All symptom cluster variables were standardized prior to analyses.
DISCUSSION {#S4}
==========
This study is the first to examine CSB in a longitudinal sample of male veterans recently returning to civilian life after deployment. Several important conclusions can be drawn from these analyses. First, the prevalence of CSB, although it dropped over the course of follow-up, appeared considerably higher than published population estimates for CSB, suggesting that male veterans may be at particularly high risk for CSB. Secondly, increasing age and traumatic experiences, particularly childhood sexual and physical trauma as well as PTSD symptoms resulting from either combat or other trauma exposure, were significantly associated with CSB. Finally, among those with PTSD, re-experiencing symptoms specifically were associated with CSB. CSB may be an important clinical target in its own right, and improving PTSD symptoms may be beneficial for reducing CSB. It may also be important to address CSB in the context of PTSD symptoms, if CSB is used as an avoidance coping strategy ([@B19])
| 1,334
| 39
| 2,262
| 1,731
| null | null |
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|
the binary probability estimates along the path from the root node to the leaf node corresponding to the class.
For non-trivial multi-class problems, the space of potential nested dichotomies is very large. An ensemble classifier can be formed by choosing suitable decompositions from this space. In the original formulation of ensembles of nested dichotomies, decompositions are sampled with uniform probability [@frank2004ensembles], but several other more sophisticated methods for splitting the set of classes have been proposed [@dong2005ensembles; @duarte2012nested; @leathart2016building]. Superior performance is achieved when ensembles of nested dichotomies are trained using common ensemble learning methods like bagging or boosting [@rodriguez2010forests].
In this paper, we describe a simple method that can improve the predictive performance of nested dichotomies by considering several splits at each internal node. Our technique can be applied to nested dichotomies built with almost any subset selection method, only contributing a constant factor to the training time and no additional cost when obtaining predictions. It has a single hyperparameter $\lambda$ that gives a trade-off between predictive performance and training time, making it easy to tune for a given learning problem. It is also very easy to implement.
The paper is structured as follows. First, we describe existing methods for class subset selection in nested dichotomies. Following this, we describe our method and provide a theoretical expectation of performance improvements. We then discuss related work, before presenting and discussing empirical results for our experiments. Finally, we conclude and discuss future research directions.
Class Subset Selection Methods\[sec:subset\_selection\_methods\]
================================================================
At each internal node $i$ of a nested dichotomy, the set of classes present at the node $\mathcal{C}_i$ is split into two non-empty, non-overlapping subsets, $\mathcal{C}_{i1}$ and
| 1,335
| 1,133
| 2,070
| 1,135
| 1,715
| 0.786496
|
github_plus_top10pct_by_avg
|
orphisms $${\sD}^{[1]}\stackrel{(i_1)_\ast}{\to}{\sD}^{A_1}\stackrel{(i_2)_!}{\to}{\sD}^{A_2}\stackrel{(i_3)_!}{\to}{\sD}^{A_3}\stackrel{(i_4)_\ast}{\to}{\sD}^\boxbar.$$ The first two functors add a cofiber square and homotopy (co)finality arguments (for example based on [@groth:ptstab Prop. 3.10]) show that the remaining two morphisms add the fiber square. Forming the composite square, we obtain a coherent square looking like $$\label{eq:F-C-square}
\vcenter{
\xymatrix{
Ff\ar[r]\ar[d]&0\ar[d]\\
0\ar[r]&Cf.
}
}$$ The canonical comparison maps result from considering suitable loop and suspension squares.
\[prop:stable-known-mod\] The following are equivalent for a pointed derivator .
1. The derivator is stable.\[item:sk1\]
2. For every $f\in{\sD}^{[1]}$ the canonical comparison maps $\Sigma F\to C$ and $F\to \Omega C$ as in are isomorphisms.\[item:sk2\]
3. For every $f\in{\sD}^{[1]}$ the square is bicartesian.\[item:sk3\]
If is a stable derivator, then the composition property of bicartesian squares [@groth:ptstab Prop. 3.13] implies that is bicartesian, and it follows from [@groth:can-can Prop. 2.16] that the canonical transformations $\Sigma F\to C$ and $F\to\Omega C$ are invertible. It remains to show that \[item:sk2\] implies \[item:sk1\], and we hence assume that $\Sigma F\toiso C$ is invertible. Associated to $x\in{\sD}$ there is by [@groth:ptstab Prop. 3.6] the morphism $1_!(x)=(0\to x)\in{\sD}^{[1]}$. The natural isomorphism $\Sigma F\toiso C$ evaluated at $1_!(x)$ yields a natural isomorphism $\Sigma\Omega x\toiso x$. Dually, we deduce $\id\toiso\Omega\Sigma$ and concludes the proof.
While unrelated left Kan extensions always commute [@groth:can-can Cor. 4.3], it is, in general, not true that unrelated left and right Kan extensions commute. More specifically, given functors $u\colon A\to A'$ and $v\colon B\to B'$, recall that **left Kan extension along $u$ and right Kan extension along $v$ commute** in a derivator if the canonical mate $$\begin{aligned}
(u\times\id)_!(\id\times v)_\ast&\st
| 1,336
| 1,076
| 861
| 1,258
| 1,717
| 0.786487
|
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|
T transformation $i \leftrightarrow j$, $U \rightarrow U^*$, $W \rightarrow W^*$, etc. as $$\begin{aligned}
\left\{ (UX)^{\dagger} A W \right\}_{i K} &\rightarrow &
\left\{ W^{\dagger} A (UX) \right\}_{K j},
\nonumber \\
\left\{ W^{\dagger} A (UX) \right\}_{K j} &\rightarrow &
\left\{ (UX)^{\dagger} A W \right\}_{i K}.
\label{matrix-T-transf}\end{aligned}$$ Therefore, the matrix element part is consistent with generalized T invariance. However, the rest of $\hat{S}_{i j} [2] \vert_{i \neq j}$ does not appear to be manifestly generalized T invariant. But, one notices that $\hat{S}_{i j} \vert_{i \neq j} [2]$ can be written as $$\begin{aligned}
&& \hat{S}_{i j}^{(2)} \vert_{i \neq j} [2] =
- \sum_{K}
\frac{ 1 }{ ( h_{j} - h_{i} ) (\Delta_{K} - h_{i}) (\Delta_{K} - h_{j}) }
\nonumber \\
&\times&
\left[
\left( e^{- i h_{j} x} - e^{- i h_{i} x} \right) \Delta_{K}
- ( h_{j} - h_{i} ) e^{- i \Delta_{K} x}
+ h_{j} e^{- i h_{i} x} - h_{i} e^{- i h_{j} x}
\right]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A (UX) \right\}_{K j}.
\label{hatS-ij[2]}\end{aligned}$$ Then, $\hat{S}_{i j} \vert_{i \neq j} [2] (U, W, \text{etc.} )= \hat{S}_{j i} \vert_{i \neq j} [2] (U^*, W^*, \text{etc.} )$ holds, that is, generalized T invariance is maintained.
Similarly, the first and the fourth terms in $\hat{S}_{I J} \vert_{I \neq J} [2]$ are not manifestly generalized T invariant. But, they can also be written as manifestly generalized T invariant forms as $$\begin{aligned}
&& \hat{S}_{I J}^{(2+4)} \vert_{I \neq J} [2] =
\sum_{k}
\frac{ 1 }{ ( \Delta_{J} - \Delta_{I} ) (\Delta_{J} - h_{k}) (\Delta_{I} - h_{k}) }
\nonumber \\
&\times&
\left[
\Delta_{J} e^{- i \Delta_{I} x} - \Delta_{I} e^{- i \Delta_{J} x}
+ \left( e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} \right) h_{k}
- ( \Delta_{J} - \Delta_{I} ) e^{- i h_{k} x}
\right]
\nonumber \\
&\times&
\left\{ W^{\dagger} A (UX) \right\}_{I k}
\left\{ (UX)^{\dagger} A W \right\}_{k J}
\nonumber \\
&+&
(ix) e^{- i \Delta_{I} x} \
| 1,337
| 4,045
| 1,380
| 1,067
| null | null |
github_plus_top10pct_by_avg
|
d only if $\ltheta^*_{i} > \ltheta^*_{\j}$.
Recall that $\gamma_{\beta_1} \equiv \ld (\kappa-2)/(\lfloor \ell\beta_1 \rfloor+1 )(d-2) $ and $\eta_{\beta_1} \equiv (\lfloor \ell \beta_1 \rfloor +1)^2/2(\kappa-2)$. Construct Doob’s martingale $(Z_2,\cdots,Z_{\kappa})$ from $\{Y_{\k}\}_{ 3 \leq \k \leq \kappa}$ such that $Z_{\j} = \E[\sum_{\k=3}^{\kappa} Y_{\k}\;|\;Y_3,\cdots,Y_{\j}]$, for $2 \leq \j\leq \kappa$. Observe that, $Z_2 = \E[\sum_{\k=3}^{\kappa} Y_{\k}] \leq \frac{(i-2)(\kappa-2)}{d-2} \leq \gamma_{\beta_1}({\left \lfloor{\ell\beta_1} \right \rfloor}+1)$, where the last inequality follows from the assumption that $i \leq \ld $. Also, $|Z_{\j} - Z_{\j-1}| \leq 1$ for each $\j$. Therefore, we have $$\begin{aligned}
\label{eq:bl_prob_3}
\P\Big[\textstyle\sum_{j \in \Omega} \I_{\{\ltheta^*_{i} > \ltheta^*_j\}} \leq {\left \lfloor{\ell\beta_1} \right \rfloor} \Big] & = &\P\Big[ \textstyle\sum_{\j=3}^{\kappa} Y_{\j} \leq {\left \lfloor{\ell\beta_1} \right \rfloor} \Big]\nonumber\\
& = & 1 - \P\Big[ \textstyle\sum_{\j=3}^{\kappa} Y_{\j} \geq {\left \lfloor{\ell\beta_1} \right \rfloor} +1 \Big] \nonumber\\
& \geq & 1 - \P\Big[Z_{\kappa-2} - Z_2 \geq (\ell\beta_1 +1) - \gamma({\left \lfloor{\ell\beta_1} \right \rfloor}+1) \Big] \nonumber\\
& \geq & 1 - \exp\Big(-\frac{({\left \lfloor{\ell\beta_1} \right \rfloor}+1)^2(1-\gamma_1)^2}{2(\kappa-2)} \Big) \nonumber\\
& = & 1 - \exp\Big(-\eta_{\beta_1}(1-\gamma_{\beta_1})^2\Big),\end{aligned}$$ where the inequality follows from the Azuma-Hoeffding bound. Since, the above inequality is true for any fixed $i,\i \in S$, for random indices $i,\i$ we have $ \P[E_{\beta_1} \;|\; i,\i \in S] \geq 1 - \exp(-\eta_{\beta_1}(1-\gamma_{\beta_1})^2)$. Claim follows by combining Equations , and .
### Proof of Lemma \[lem:bl\_prob1\]
Without loss of generality, assume that $\i < i$, i.e., $\ltheta^*_{\i} \leq \ltheta^*_i$. Define $\Omega = \{j: j\in S, j \neq i,\i\}$, and event $E_{\beta_1} = \{ i,\i \in S; \textstyle\sum_{j \in \Omega} \I_{\{\ltheta^*_i > \ltheta^*_j\}} \leq
| 1,338
| 722
| 1,470
| 1,279
| null | null |
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|
}\subset\cdots\subset E^1\subset E^0=(\F_q)^n$$such that ${\rm dim}(E^{i-1}/E^i)=\lambda_i$.
Let $G$ acts on $\calF_\lambda$ in the natural way. Fix an element $$X_o=\left(\{0\}=E^r\subset E^{r-1}\subset\cdots\subset E^1\subset E^0=(\F_q)^n\right)\in\calF_\lambda$$ and denote by $P_\lambda$ the stabilizer of $X_o$ in $G$ and by $U_\lambda$ the subgroup of elements $g\in P_\lambda$ which induces the identity on $E^i/E^{i+1}$ for all $i=0,1,\dots,r-1$.
Put $L_\lambda:=\GL_{\lambda_r}(\F_q)\times\cdots\times\GL_{\lambda_1}(\F_q)$. Recall that $U_\lambda$ is a normal subgroup of $P_\lambda$ and that $P_\lambda=L_\lambda\ltimes U_\lambda$.
Denote by $\C[G/U_\lambda]$ the $\C$-vector space generated by the finite set $G/U_\lambda=\{gU_\lambda\,|\, g\in G\}$. The group $L_\lambda$ (resp. $G$) acts on $\C[G/U_\lambda]$ as $(gU_\lambda)\cdot l=glU_\lambda$ (resp. as $g\cdot (hU_\lambda)=ghU_\lambda$). These two actions make $\C[G/U_\lambda]$ into a $G$-module-$L_\lambda$. The associated functor $R_{L_\lambda}^G:{\rm Mod}_{L_\lambda}\rightarrow {\rm Mod}_G$ is the so-called *Harish-Chandra functor*.
We have the following well-known lemma.
We denote by $1$ the identity character of $L_\lambda$. Then for all $g\in G$, we have
$$R_{L_\lambda}^G(1)(g)=\#\{X\in\calF_\lambda\,|\, g\cdot X=X\}.$$ \[R=F\]
By Formula (\[bimod\]) we have
$$\begin{aligned}
R_{L_\lambda}^G(1)(g)&=|L_\lambda|^{-1}\sum_{k\in L_\lambda}\#\{hU_\lambda\,|\, ghU_\lambda=hkU_\lambda\}\\
&=|L_\lambda|^{-1}\sum_{k\in L_\lambda}\#\{hU_\lambda\,|\, gh\in hkU_\lambda\}\\
&=|L_\lambda|^{-1}\#\{hU_\lambda\,|\, gh\in hP_\lambda\}\\
&=\#\{hP_\lambda\,|\, ghP_\lambda=hP_\lambda\}.
\end{aligned}$$
We deduce the lemma from last equality by noticing that the map $G\rightarrow\calF_\lambda$, $g\mapsto g\cdot X_o$ induces a bijection $G/P_\lambda\rightarrow\calF_\lambda$.
We now recall the definition of the type of a conjugacy class $C$ of $G$ (cf. [@hausel-letellier-villegas 4.1]). The Frobenius $f:\F\rightarrow\F$, $x\mapsto x^q$ acts on the set of eigenvalue
| 1,339
| 1,442
| 1,076
| 1,246
| null | null |
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|
e{{\scriptscriptstyle}(4)}}(f,x)$ to $\sum_{z,x}|x-y|^2|x|^t\tau_{y,z}Q'_{\Lambda;o}(z,x)$: $$\begin{aligned}
{\label{eq:IRSchwarz}}
\text{(i)}\quad\raisebox{-1.5pc}{\includegraphics[scale=0.14]
{IRnonSchwarz}}\hspace{5pc}
\text{(ii)}\quad\raisebox{-1.5pc}{\includegraphics[scale=0.14]
{IRSchwarz}}\end{aligned}$$ where, for simplicity, $\psi_\Lambda(f,g)-\delta_{f,g}$ and $\psi_\Lambda(u,z)-\delta_{u,z}$ are reduced to $\tilde
G_\Lambda(f,g)^2$ and $\tilde G_\Lambda(f,g)^2$, respectively. We suppose that $|v|$ is bigger than $|w-v|$ and $|x-w|$ along the lowermost path from $o$ to $x$ through $v$ and $w$, so that $|x|^t$ is bounded by $3^t|v|^t$. We also suppose that $|z-u|$ in (\[eq:IRSchwarz\].i) (resp., $|g-f|$ in (\[eq:IRSchwarz\].ii)) is bigger than the end-to-end distance of any of the other four segments along the uppermost path from $y$ to $x$ through $f,g,u$ and $z$. Therefore, we can bound $|x-y|^2$ by $5^2|z-u|^2$ in (\[eq:IRSchwarz\].i) (resp., $5^2|g-f|^2$ in (\[eq:IRSchwarz\].ii)) and bound the weighted arc between $u$ and $z$ (resp., between $f$ and $g$) by $5^2\bar G^{{\scriptscriptstyle}(2)}$. By translation invariance, the remaining diagram of (\[eq:IRSchwarz\].i) is easily bounded as $$\begin{aligned}
{\label{eq:IRSchwarz-bdi}}
\sum_{f',g,u',v}\!\raisebox{-1.4pc}{\includegraphics[scale=0.14]
{IRnonSchwarzdec1}}\;=\sup_{f',g,u'}\,\raisebox{-1.4pc}
{\includegraphics[scale=0.14]{IRnonSchwarzdec2}}\leq\bar
W^{{\scriptscriptstyle}(t)}\,O(\theta_0)^4,\end{aligned}$$ where the power 4 (not 3) is due to the fact that the segment from $u'$ in the last block is nonzero.
To bound the remaining diagram of (\[eq:IRSchwarz\].ii) is a little trickier. We note that at least one of $|u|,|z-u|,|w-z|$ and $|v-w|$ along the path from $o$ to $v$ through $u,z,w$ is bigger than $\frac14|v|$. Suppose $|v-w|\ge\frac14|v|$, so that $|v|^t\leq2^t|v-w|^{t/2}|v|^{t/2}$. Then, by using the Schwarz inequality, we obtain $$\begin{aligned}
{\label{eq:IRSchwarz-bdii}}
\raisebox{-1.3pc}{\includegraphics[scale=0.13]{IRSchwar
| 1,340
| 971
| 1,677
| 1,343
| 1,053
| 0.795181
|
github_plus_top10pct_by_avg
|
Empty}}}(H_t){\geqslant}{\textsc{{Empty}}}(H'_t){\geqslant}{s}/2.$$
[Fix $m=m(n)$ to equal the $m$ provided by Lemma \[lem:empty\].]{} For $t=1,\ldots, m$, let $D_t$ be the $d$-element subset of $H_t$ that is chosen by the $t$-th ball. Define the indicator random variable ${\mathbb{I}}_t$ as follows: $${\mathbb{I}}_t:= \begin{cases}
1 & \text{ if $D_t$ contains {at least $d/6$} empty vertices, }\\
0 & \text{ otherwise.}\\
\end{cases}$$ Let us fix an arbitrary bin $i$ and then define $A(t,i)$ to be the event that the $t$-th ball is allocated to vertex $i$. ([The first $t-1$ balls have already been allocated, as the balanced allocation process never fails.]{}) Observe that if $i\not\in D_t$ then ${\ensuremath{\operatorname{\mathbf{Pr}}\left[A(t,i)\right]}} = 0$. It follows that $$\begin{aligned}
{\ensuremath{\operatorname{\mathbf{Pr}}\left[A(t,i)\right]}}&={\ensuremath{\operatorname{\mathbf{Pr}}\left[A(t,i)\mid i\in D_t~ \text{and}~ {\mathbb{I}}_t=1\right]}}\cdot {\ensuremath{\operatorname{\mathbf{Pr}}\left[i\in D_t ~\text{and}~ {\mathbb{I}}_t=1\right]}}\\
& {} + {\ensuremath{\operatorname{\mathbf{Pr}}\left[A(t,i) \mid i\in D_t ~\text{and} ~{\mathbb{I}}_t=0\right]}}\cdot{\ensuremath{\operatorname{\mathbf{Pr}}\left[i\in D_t~ \text{and}~ {\mathbb{I}}_t=0\right]}}.
\end{aligned}$$ Now there are at least [$d/6$]{} empty vertices on $D_t$ when ${\mathbb{I}}_t=1$, so $${\ensuremath{\operatorname{\mathbf{Pr}}\left[A(t,i)\mid i\in D_t~ \text{and} ~ {\mathbb{I}}_t=1\right]}} {\leqslant}{6/d}.$$ It follows that $$\begin{aligned}
\label{ineq:first}
{\ensuremath{\operatorname{\mathbf{Pr}}\left[A(t,i)\right]}}&{\leqslant}{(6/d)}\, {\ensuremath{\operatorname{\mathbf{Pr}}\left[i\in D_t ~\text{and}~ {\mathbb{I}}_t=1\right]}} + {\ensuremath{\operatorname{\mathbf{Pr}}\left[i\in D_t~ \text{and}~ I_t=0\right]}}\nonumber\\
&{\leqslant}{(6/d)}\,{\ensuremath{\operatorname{\mathbf{Pr}}\left[i\in D_t\right]}}+ {\ensuremath{\operatorname{\mathbf{Pr}}\left[{\mathbb{I}}_t=0~|~i\in D_t \right]}
| 1,341
| 2,598
| 1,152
| 1,314
| null | null |
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|
$ in the orthogonal group associated to $M_0''/\pi M_0''$ is $$T_1=\begin{pmatrix} \begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0& (x_j)_1 &1& (x_j)_1+1/\sqrt{b+b'}(z_j^{\ast})_1\\0&0& 0 & 1 \end{pmatrix} &0 \\ 0 & id \end{pmatrix}.$$ Since $(x_j)_1=0$ by Equation (\[e42\]), the Dickson invariant of $T_1$ is the same as that of $\begin{pmatrix} 1&\sqrt{b+b'}(z_j^{\ast})_1\\0& 1 \end{pmatrix}$, which turns to be $(z_j^{\ast})_1$ by using the similar argument used in the above case (i).
In conclusion, $(z_j^{\ast})_1$ is the image of a fixed element of $F_j$ under the map $\psi_j$. Since $(z_j^{\ast})_1$ can be either $0$ or $1$ by Equation (\[e42\]), $\psi_j|_{F_j}$ is surjective onto $\mathbb{Z}/2\mathbb{Z}$ and thus $\psi_j$ is surjective.\
3. Assume that $M_1=A(4b'', 2\delta, \pi) \oplus (\oplus H(1))$ with $b''\in A$. Let $(e_5, e_6)$ be a basis for $A(4b'', 2\delta, \pi)$. Recall that $(e_3', e_4')$ is a basis for the direct summand $A(1, 2(b+b'), 1)$ of $M_0$. We choose another basis $(e_3'-e_4', e_3')$ for $A(1, 2(b+b'), 1)$ whose associated Gram matrix is $\begin{pmatrix} -1+2(b+b')& 0 \\ 0 & 1 \end{pmatrix}$. Then choose a basis $(e_3'-e_4', e_3'-e_5, e_5-\frac{2b''\pi}{\delta}e_6, e_6+\pi e_3')$ for the lattice spanned by $(e_3'-e_4', e_3', e_5, e_6)$ whose associated Gram matrix is $$\begin{pmatrix} 1+2(b+b')&0&0&0\\ 0&1+4b''& 0&0 \\ 0&0 & -4b''(1+4b'')&-\pi(1+4b'')\\0&0&\pi(1+4b'') &0 \end{pmatrix}$$ (cf. Lemma 2.9 and the following paragraph of loc. cit. in [@C2]). Note that this lattice is the same as $A(1, 2(b+b'), 1)\oplus A(4b'', 2\delta, \pi)$. Since the lattice spanned by $(e_5-\frac{2b''\pi}{\delta}e_6, e_6+\pi e_3')$ is $\pi^1$-modular with the norm $(4)$, it is isometric to $H(1)$ by Theorem 2.2 of [@C2]. Now choose another basis $(e_3'-e_5, e_4'-e_5, e_5-\frac{2b''\pi}{\delta}e_6, e_6+\pi e_3')$ for the above lattice $A(1, 2(b+b'), 1)\oplus A(4b'', 2\delta, \pi)$ such that the associated Gram matrix is $$\begin{pmatrix} 1+4b''&1+4b''&0&0\\ 1+4b''&2(1+b+b'+2b'')& 0&0 \\ 0&0 &
| 1,342
| 838
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| 1,349
| 3,331
| 0.773077
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with $\bfun{\chi '}({\alpha }_i)<\infty $. This follows from Lemma \[le:Eheight\](i).
\[th:PBW\] Assume that $\chi \in {\mathcal{X}}_3$. Let $n=|R_+^\chi |\in {\mathbb{N}}$. Both sets $$\begin{aligned}
\label{eq:LusztigPBW+}
\big\{ E_{\beta _1}^{m_1} E_{\beta _2}^{m_2}\cdots E_{\beta _n}^{m_n}\,&|\,
0\le m_\nu <{b^{\chi}} (\beta _\nu )
\text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\},\\
\label{eq:LusztigPBW-}
\big\{ {\bar{E}}_{\beta _1}^{m_1} {\bar{E}}_{\beta _2}^{m_2}\cdots
{\bar{E}}_{\beta _n}^{m_n}\,&|\,
0\le m_\nu <{b^{\chi}} (\beta _\nu )
\text{ for all $\nu \in \{1,2,\dots ,n\}$} \big\}
\end{aligned}$$ form vector space bases of $U ^+(\chi )$.
We prove the claim for the basis in Eq. . For the other set the proof is analogous. By Eqs. and , $$\begin{aligned}
\dim U^+(\chi )_{\alpha }=\Big|\big\{(m_1,m_2,&\dots ,m_n)\in {\mathbb{N}}_0^n\,\big|\,
\sum _{\nu =1}^n m_\nu \beta _\nu ={\alpha },\\
&m_\nu <{b^{\chi}} (\beta _\nu ) \text{ for all $\nu \in \{1,\dots ,n\}$}
\big\}\Big|
\end{aligned}$$ for all ${\alpha }\in {\mathbb{N}}_0^I$. Since $\deg E_{\beta _\nu }=\beta _\nu $ for all $\nu \in \{1,2,\dots ,n\}$, it suffices to show that for all $\mu \in \{1,2,\dots ,n+1\}$ the elements of the set $$\begin{aligned}
\big\{ E_{\beta _\mu }^{m_\mu } E_{\beta _{\mu +1}}^{m_{\mu +1}}\cdots
E_{\beta _n}^{m_n}\,|\, 0\le m_\nu <{b^{\chi}} (\beta _\nu )
\text{ for all $\nu \in \{\mu ,\mu +1 ,\dots ,n\}$} \big\}
\end{aligned}$$ are linearly independent. We proceed by induction on $n+1-\mu $. If $\mu =n+1$, then the above set is empty, and hence its elements are linearly independent.
Let now $\mu \in \{1,2,\dots ,n\}$. For all $m_\mu ,\dots ,m_n\in {\mathbb{N}}_0$ with $m_\nu <{b^{\chi}} (\beta _\nu )$ for all $\nu \in \{\mu ,\mu +1,\dots ,n\}$ let $a_{m_\mu ,\dots ,m_n}\in {\Bbbk }$. Assume that $$\begin{aligned}
\sum _{m_\mu ,\dots ,m_n}a _{m_\mu ,\dots ,m_n}
E_{\beta _\mu }^{m_\mu } E_{\beta _{\mu +1}}^{m_{\mu +1}}\cdots
E_{\beta _n
| 1,343
| 1,462
| 1,208
| 1,338
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|
xt{$q$ is a root of $1$, $m_{ij}\in {\mathbb{Z}}$ for all $i,j\in I$,}$$ such that the set $$\begin{aligned}
V^\chi _{\underline{n}}=\{\chi '\in {\overline{{\mathcal{X}}}}\,|\,&
R^{\chi '}_+ =R^\chi _+ ,\,
{b^{\chi '}}(\beta )={b^{\chi}} (\beta ) \text{ for all }
\beta \in R^\chi _{+{\mathrm{fin}}},\\
&\chi '(\beta ,\beta )^n\not=1
\text{ for all $\beta \in R^\chi _{+\infty }$,
$1\le n\le n_\beta $}
\}
\end{aligned}
\label{eq:Vchin}$$ is an open subset of ${\mathrm{maxspec}\,}{\bar{{\Bbbk }}}[{\overline{{\mathcal{X}}}}]/J$.
We use Lemma \[le:equalrs\] and Def. \[de:Cartan\] to reformulate the equation $R^\chi _+=R^{\chi '}_+$.
Let $\chi '\in {\mathcal{G}}(\chi )$. Since $\chi \in {\overline{{\mathcal{X}}}}_3$, $\chi '$ is $p$-finite for all $p\in I$. Further, $\chi '({\alpha }_p,{\alpha }_p)\not=1$ since $\chi (\beta ,\beta )\not=1$ for all $\beta \in R^\chi
_+$, see Eq. . Thus $$(\chi '({\alpha }_p,{\alpha }_p)^{-c_{pj}^{\chi '}}\chi '({\alpha }_p,{\alpha }_j)\chi '(\al
_j,{\alpha }_p)-1)(\chi '({\alpha }_p,{\alpha }_p)^{1-c_{pj}^{\chi '}}-1)=0$$ for all $p,j\in I$ with $p\not=j$. Let $w\in {\mathrm{Hom}}(\chi ,\chi ')\subset {\mathrm{Hom}}({\mathcal{W}}(\chi ))$. Identify $w$ with the corresponding element in ${\mathrm{Aut}}({\mathbb{Z}}^I)$ in the usual way. Then $\chi '=w^*\chi $ and hence $$\begin{aligned}
\label{eq:chirel}
(\chi (\gamma _p,\gamma _p)^{-c_{pj}^{\chi '}}\chi (\gamma _p,\gamma _j)
\chi (\gamma _j,\gamma _p)-1)(\chi (\gamma _p,\gamma _p)
^{1-c_{pj}^{\chi '}}-1)=0
\end{aligned}$$ for all $p,j\in I$ with $p\not=j$, where $\gamma _p=w^{-1}({\alpha }_p)$ and $\gamma _j=w^{-1}({\alpha }_j)$. Let $$\begin{aligned}
&\begin{aligned}
&J'=\big(
(f_{\gamma _p,\gamma _p}^{-c_{pj}^{w^*\chi }}
f_{\gamma _p,\gamma _j}
f_{\gamma _j,\gamma _p}-1)
(f_{\gamma _p,\gamma _p}^{1-c_{pj}^{w^*\chi }}-1)\,|\\
&\quad j,p\in I,j\not=p,\,
w\in {\mathrm{Hom}}(\chi ,\underline{\,\,}),\,\gamma _p
| 1,344
| 2,646
| 1,353
| 1,292
| null | null |
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nd Novartis; Marta S. Figueroa is a consultant for Bayer, Alcon, Allergan, and Novartis; IMS Health received funding from Bayer HealthCare for the conduct, analysis, and reporting of the study; Jordi Farrés Martí is a salaried employee of Bayer HealthCare.
{#fig1}
{#fig2}
######
Sociodemographic description of patients.
Variable Total
------------------------------------------ ----------------
Age (years)
*n* 208
Average (SD) 76.72 (7.83)
Median 77
Age groups
Total **208 (100%)**
\<65 years 14 (6.7%)
65--75 years 72 (34.6%)
\>75 years 122 (58.7%)
Gender
Total **208 (100%)**
Male 88 (42.3%)
Female 120 (57.7%)
Ethnic group
Total **208 (100%)**
Caucasian 183 (88.0%)
Hispanic 25 (12.0%)
Iris color
Total **208 (100%)**
Light colored 44 (21.2%)
Dark 78 (37.5%)
Unknown 86 (41.3%)
Study eye
Total **208 (100%)**
1---left
| 1,345
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| 1,120
| 448
| null | null |
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|
in this section. The differences between the present calculation and that in Ref. [@kha02] are the mesh size, the boundary condition, the cutoff energy for the QRPA calculation, and the treatment of the spin-dependent interaction ($G_{0}$ and $G_{0}^{\prime}$) in Eq. (\[eq:res\_ph\]). In the present calculation, the spin transition density is treated exactly.
![Cutoff energy dependence of the renormalization factors and the $B(E2\uparrow)$ value for the first $2^{+}$ state in $^{22}$O. []{data-label="22O_dep"}](fig2.eps)
In Fig. \[22O\_response\], we show the isoscalar quadrupole response function in $^{22}$O. The first $2^{+}$ state is located at 2.8 MeV with $B(E2\uparrow)=18.9$ $e^{2}$fm$^{4}$. The experimental values are $E(2^{+})_{\mathrm{exp}}=3.2$ MeV and $B(E2)_{\mathrm{exp}}=21\pm 8$ $e^{2}$fm$^{4}$ [@bel01; @thi00; @bec06]. In Ref. [@kha02], the energy and the transition strength are $E(2_{1}^{+})=1.9$ MeV and $B(E2)=22$ $e^{2}$fm$^{4}$. The energy and the transition strength of the low-lying collective state is quite sensitive to the cutoff energy for the RPA calculation [@bla77]. In Fig. \[22O\_dep\], the cutoff energy dependence of the renormalization factors and the $B(E2\uparrow)$ value for the $2_{1}^{+}$ state in $^{22}$O are shown. Even with the cutoff energy of 70 MeV, the transition strength for the low-lying state does not converge yet. In this case, the dimension of the QRPA matrix in Eq. (\[eq:AB1\]) is 11726 for the $K^{\pi}=0^{+}$ channel and the memory size is 13 GB, and the CPU time is about 70,000s per each iteration for determining the renormalization factor $f_{pp}$. If we could perform the QRPA calculation including all quasiparticle states obtained in the HFB calculation, the renormalization factor for the pairing channel $f_{pp}$ would be 1, because the p-p channel is treated self-consistently between the HFB and the QRPA calculations.
The peak position of the giant resonance is located slightly higher than in Ref.[@kha02]. The non-collective two-quasiparticle states a
| 1,346
| 115
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| 1,335
| 2,685
| 0.77787
|
github_plus_top10pct_by_avg
|
ot involve the collision operator $K=(K_1,K_2,K_3)$, and $u$ contains a direct contribution from the external (boundary) sources $g$ (through ). On the other hand, the field $u$ *right after collision* as modelled by the term $Ku$, acts as an internal source in the equation for the secondary field $w$, while external sources do not contribute to $w$ directly (a fact captured by ).
Note especially that the system (\[desol13\])-(\[desol15\]) is uncoupled, in that the different particle species (photon, electron, positron that we consider here) evolve independently of each other. In some cases the primary component $u$ can be calculated exactly such as the following example shows.
\[desolex1\] Suppose that $\Sigma_1\in L^2(G\times S\times I)$, $\Sigma_1\geq c>0$ for some constant $c$, and that, $\Sigma_j(x,\omega,E)=\Sigma_j(x,\omega)$ (i.e. $\Sigma_j$ does not depend on $E$) and $\Sigma_j\in L^2(G\times S),\ \Sigma_j\geq c>0$, for $j=2,3$. Furthermore, suppose that $S_j(x,E)=S_j(E)$, $j=2,3$ (i.e. $S_j$ is independent of $x$) and that $S_j:I\to {\mathbb{R}}_+$ are continuous, strictly positive functions. Finally, let $f_1\in L^2(G\times S\times I)$, $g_1\in T^2(\Gamma_-)$, and for $j=2,3$ let $f_j\in H^1(I,L^2(G\times S))$, $g_j\in H^1(I,T^2(\Gamma'_-))$, such that $g_j(E_{\rm m})=0$ (compatibility condition). Define $R_j:I\to{\mathbb{R}}$ by R\_j(E):=\_0\^E[1]{}d,j=2,3. Let $r_{m,j}:=R_j(E_m)$. Then $R_j:I\to [0,r_{m,j}]$ are continuously differentiable and strictly increasing bijections. Let $R_j^{-1}: [0,r_{m,j}]\to I$ be their inverses. We denote the argument of $R_j^{-1}$ on $[0,r_{m,j}]$ by $\eta$, i.e. $E=R_j^{-1}(\eta)$ (or equivalently $\eta=R_j(E)$).
Consider first the (primary) uncoupled problem, $$\begin{gathered}
-{{\frac{\partial (S_ju_j)}{\partial E}}}+\omega\cdot\nabla_x u_j+
\Sigma_j u_j
= f_j,
\nonumber\\
{u_j}_{|\Gamma_-}=g_j,\quad
u_j(\cdot,\cdot,E_m)=0, \label{desol19}\end{gathered}$$ where $j=2,3$. We perform a well-known change of variables (see e.g. [@frank10], [@rockell]) in the proble
| 1,347
| 405
| 956
| 1,435
| 2,600
| 0.778541
|
github_plus_top10pct_by_avg
|
n the time-scale associated with mean shear ($S$). In other words, $$\frac{1}{\sqrt{\kappa^2E(\kappa)}} \ll \frac{1}{S}.$$ Using the “-5/3 law” of Kolmogorov [@kolmogorov41a] and Obukhov [@obukhov41a; @obukhov41b], this equation can be re-written as: $$\frac{1}{\sqrt{\kappa^{4/3} \overline{\varepsilon}^{2/3}}} \ll \frac{1}{S}.
\label{LC1}$$ If we assume that for a specific wavenumber $\kappa = 1/L_C$, the equality holds in Eq. \[LC1\], then we get: $$L_C^{2/3} = \frac{\overline{\varepsilon}^{1/3}}{S}.$$ From this equation, we can estimate $L_C$ as defined earlier in Eq. \[LC\].
#### Hunt Length Scale:
Hunt [@hunt88] hypothesized that in stratified shear flows, $\overline{\varepsilon}$ is controlled by mean shear ($S$) and $\sigma_w$. From dimensional analysis, it follows that: $$\overline{\varepsilon} \equiv \sigma_w^2 S.$$ The associated length scale, $L_H$, is assumed to be on the order of $\sigma_w/S$.
Appendix 2: Derivation of $\overline{\varepsilon} = \sigma_w^2 N$ by Weinstock {#appendix-2-derivation-of-overlinevarepsilon-sigma_w2-n-by-weinstock .unnumbered}
==============================================================================
The starting point of Weinstock’s derivation was “-5/3 law” of Kolmogorov [@kolmogorov41a] and Obukhov [@obukhov41a; @obukhov41b]. He integrated this equation in the wavenumber space and set the upper integration limit to infinity. The lower integration limit was fixed at the buoyancy wavenumber ($\kappa_b$). Furthermore, he assumed that the eddies are isotropic for wavenumbers larger than $\kappa_b$ (i.e., in the inertial and viscous ranges). His derivation can be summarized as: $$\begin{split}
\frac{3}{2} \sigma_w^2 & = \int_{\kappa_b}^{\kappa_2} \alpha \overline{\varepsilon}^{2/3} \kappa^{-5/3} d\kappa \\
& = \alpha \overline{\varepsilon}^{2/3} \int_{\kappa_b}^{\kappa_2} \kappa^{-5/3} d\kappa \\
& = \frac{3 \alpha}{2} \overline{\varepsilon}^{2/3} \left(\kappa_b^{-2/3} - \kappa_2^{-2/3} \right) \\
& \approx \frac{3 \alpha}{2} \overline{\v
| 1,348
| 2,640
| 2,288
| 1,423
| 1,619
| 0.787523
|
github_plus_top10pct_by_avg
|
on the right into $M$ completes the proof of Lemma \[faber\].
The matrices $H$, $M$ appearing in Lemma \[faber\] may be omitted by changing the bases of $W$ and $V$ accordingly. Further, we may assume that $b>0$, since we are already reduced to the case in which $\alpha(0)$ is a rank-1 matrix. This concludes the proof of Proposition \[keyreduction\]. In what follows, we will assume that $\alpha$ is a germ in the standard form given above.
Components of type II, III, and IV {#1PS}
----------------------------------
It will now be convenient to switch to affine coordinates centered at the point $(1:0:0)$. We write $$F(1:y:z)=F_m(y,z)+F_{m+1}(y,z)+\cdots +F_d(y,z)\quad,$$ with $d=\deg {{\mathscr C}}$, $F_i$ homogeneous of degree $i$, and $F_m\ne
0$. Thus, $F_m(y,z)$ generates the ideal of the [*tangent cone*]{} of ${{\mathscr C}}$ at $p$.
We first consider the case in which $q=r=s=0$, that is, in which $\alpha(t)$ is itself a 1-PS: $$\alpha(t)=\begin{pmatrix}
1 & 0 & 0\\
0 & t^b & 0\\
0 & 0 & t^c
\end{pmatrix}$$ with $1\le b \le c$. Also, we may assume that $b$ and $c$ are coprime: this only amounts to a reparametrization of the germ by $t \mapsto
t^{1/gcd(b,c)}$; the new germ is not equivalent to the old one in terms of Definition \[equivgermsnew\], but clearly achieves the same limit.
Germs with $b=c$ $(=1)$ lead to components of type III, cf. §\[germlist\] (also cf. [@MR2001h:14068], §2, Fact 4(i)):
\[tgcone\] If $q=r=s=0$ and $b=c$, then $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is a fan consisting of a star projectively equivalent to the tangent cone to ${{\mathscr C}}$ at $p$, and of a residual $(d-m)$-fold line supported on $\ker\alpha$.
The composition $F\circ\alpha(t)$ is $$F(x:t^by:t^bz)=t^{bm}x^{d-m}F_m(y,z)+t^{b(m+1)}x^{d-(m+1)}
F_{m+1}(y,z)+\cdots+ t^{dm}F_d(y,z)\quad.$$ By definition of limit, $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ has ideal $(x^{d-m}F_m(y,z))$, proving the assertion.
The case $b<c$ corresponds to the germs of type II and type IV in §\[germlist\]. We have t
| 1,349
| 2,712
| 1,777
| 1,260
| 2,571
| 0.778793
|
github_plus_top10pct_by_avg
|
nctor is in fact a map from $RB(X^2)$ to $RB(G)$. Here we use Green’s notation for $RB(G)$.
Let $X$ and $Z$ be finite $G$-sets, let $a$ and $b$ be maps of $G$-sets from $Z$ to $X$. Let $$f=\xymatrix{
& Z \ar[dl]_{b}\ar[dr]^{a}&\\
X && X
}$$ The Burnside trace $Btr : RB(X^2)\to RB(G)$ is defined on $f$ by: $$Btr(f):=\{z\in Z\ | a(z)=b(z)\}\in RB(G).$$
Corollary 2.7 [@bouc_burnside_dim].
By composing the Burnside trace by any $R$-linear map $RB(G)\to R$ we have a linear form on $RB(X^2)$.\
\[gene\] Let $R$ be a commutative ring. Let $f$ be a linear map from $RB(G)\to R$, such that $f(G/1)=1$. The trace map $f\circ Btr$ generalizes the usual trace map for the group ring $RG$ in the following way. The Burnside algebra $RB(G/1\times G/1)$ is isomorphic to $RG$. The isomorphism is defined as follow: a transitive $G$-set over $G/1\times G/1$ is isomorphic to $$f_{g} = \xymatrix{
& G/1\ar[rd]^{g}\ar@{=}[ld] &,\\
G/1 && G/1
}$$ for some $g\in G$. The element $f_{g}$ is sent to $g\in RG$. Now, the Burnside trace of the element $f_{g}$ is $\delta_{g,1}G/1$.
Using the fact that the Mackey algebra $\mu_{R}(G)$ is isomorphic to $RB(\Omega_{G}^{2})$, the Burnside trace gives a linear map from $\mu_{R}(G)$ to $RB(G)$. Using Proposition \[prop\_b\] we have as immediate corollary:
The Burnside Trace $Btr$ on the Mackey algebra is defined on a basis element by $$Btr(t^{K}_{H}xr^{L}_{H^{x}}) =\left\{\begin{array}{c}G/H \hbox{ if $K=L$ and $x\in L$} \\0 \hbox{ if not.}\end{array}\right.$$
\[calc\] Let $t^{H}_{K}xr^{L}_{K^{x}}$ and $t^{L}_{Q}yr^{H}_{Q^{y}}$ be two basis elements of $\mu_{R}(G)$. Then $$Btr\big(t^{H}_{K}xr^{L}_{K^{x}}t^{L}_{Q}yr^{H}_{Q^{y}}\big)=\sum_{\alpha\in[K^{x}\backslash L /Q]} \delta_{x\alpha y, H} G/(K\cap \ ^{x\alpha}Q),$$ where $\delta_{x\alpha y, H} =1$ if $x\alpha y\in H$ and $0$ otherwise.
This follows from the computation of the product $t^{H}_{K}xr^{L}_{K^{x}}t^{L}_{Q}yr^{H}_{Q^{y}}$ by using the Mackey formula: $$Btr(t^{H}_{K}xr^{L}_{K^{x}}t^{L}_{Q}yr^{H}_{Q^{y}})=\sum_{\alpha\in[K^{x}\backsl
| 1,350
| 2,373
| 1,531
| 1,270
| null | null |
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|
use of certain symmetries of the problem for a hollow charge distribution, @james77 devised a formulation that uses sine and cosine transforms to expresses the potential on the each surface as the sum of seven terms. We refer the reader to Equations (4.7)–(4.20) of @james77 for the description of this formulation, which costs ${\cal O}(N^3 + N^2 \log N)$ operations.
The gravitational potential $\Phi^{\rm B}$ at the domain boundary is obtained by $$\Phi^{\rm B}_{i,j,k} = \Psi^{\rm B}_{i,j,k} - \Theta^{\rm B}_{i,j,k} = -\Theta^{\rm B}_{i,j,k},$$ which provides the required Dirichlet boundary condition for the interior solver (Section \[s:interior\_solver\]). Note that $\Psi^{\rm B}_{i,j,k} = 0$ by definition.
### Cylindrical Grid
The cylindrical DGF, ${\cal G}_{i,i',j-j',k-k'}$, satisfies $$\label{eq:def_cyl_green}
\left(\Delta_R^2 + \Delta_\phi^2 + \Delta_z^2\right){\cal G}_{i,i',j-j',k-k'} = 4\pi G \frac{\delta_{ii'}\delta_{jj'}\delta_{kk'}}{{\cal V}_{i'}},$$ where ${\cal V}_{i'}=\int_{z_{k'-1/2}}^{z_{k'+1/2}}\int_{\phi_{j'-1/2}}^{\phi_{j'+1/2}}\int_{R_{i'-1/2}}^{R_{i'+1/2}} R\,dR d\phi dz = \tfrac{1}{2}(R_{i'+1/2}^2-R_{i'-1/2}^2)\delta\phi\delta z$ is the volume of the $(i',j',k')$-th cell. Note that the cylindrical DGF has four indices due to lack of the translational symmetry along the radial direction. In Appendix \[s:calc\_dgf\_cyl\], we present the method to calculate the cylindrical DGF and compare it with the continuous counterpart.
The gravitational potential $\Theta_{i,j,k}$ generated by the screening charges $\sigma_{i,j,k}$ takes a form of $$\label{eq:gpot_by_discrete_green}
\Theta_{i,j,k} = \sum_{i'=0}^{N_R+1}\sum_{j'=1}^{N_\phi}\sum_{k'=0}^{N_z+1} {\cal G}_{i,i',j-j',k-k'}\sigma_{i',j',k'}{\cal V}_{i'}.$$ One can readily verify that $\Theta_{i,j,k}$ satisfies the discrete Poisson equation in cylindrical coordinates. Since all functions are periodic in the azimuthal direction, it is natural to apply a discrete Fourier transform such that $$\label{eq:dft}
\Theta^m_{ik} \equiv \sum_{j=1}^{
| 1,351
| 3,372
| 1,913
| 1,399
| 3,514
| 0.771844
|
github_plus_top10pct_by_avg
|
type $I$}}(4n_i-4)$$ with variables $(m_{i,j})_{i\neq j}, (y_i, v_i, z_i, z_i^{\ast})_{\textit{i:even and $L_i$:of type $I^o$}},
(r_i, t_i, y_i, v_i, x_i, z_i, u_i, w_i, z_i^{\ast})_{\textit{i:even and $L_i$:of type $I^e$}}$, $ (r_i, t_i, y_i, v_i, x_i, z_i, u_i, w_i)_{\textit{i:odd and $L_i$:free of type I}}$, $ (m_{i,i}^{\ast}, m_{i,i}^{\ast\ast})_{\textit{i:odd and $L_i$:bound of type I}}$, such that
- Let $i$ be even and $L_i$ be *bound of type II*. Then $\delta_{i-1}^{\prime}e_{i-1}\cdot m_{i-1, i}+\delta_{i+1}^{\prime}e_{i+1}\cdot m_{i+1, i}+\delta_{i-2}e_{i-2}\cdot m_{i-2, i}+\delta_{i+2}e_{i+2}\cdot m_{i+2, i}=0$. Here, notations are as explained in Step (c) of the description of an element of $\mathrm{Ker~}\tilde{\varphi}(R)$ given at the paragraph following Lemma \[la2\].
- Let $i$ be even and $L_i$ be *of type I*. Then $v_i(\mathrm{resp.~}(y_i+\sqrt{\bar{\gamma}_i}v_i))+(\delta_{i-2}e_{i-2}\cdot m_{i-2, i}+\delta_{i+2}e_{i+2}\cdot m_{i+2, i})\tilde{e}_i=0$ if $L_i$ is *of type* $\textit{I}^o$ (resp. *of type* $\textit{I}^e$). Here, notations are as explained in Step (c) of the description of an element of $\mathrm{Ker~}\tilde{\varphi}(R)$ given at the paragraph following Lemma \[la2\].
- If $i$ is even and $L_i$ is *of type I*, then $z_i+\delta_{i-2}k_{i-2, i}+\delta_{i+2}k_{i+2, i}=0.$ Here, notations are as explained in Step (d) of the description of an element of $\tilde{M}(R)$ given at the paragraph following Lemma \[la1\].
- If $i$ is odd and $L_i$ is *bound of type I*, then $\delta_{i-1}v_{i-1}\cdot m_{i-1, i}+\delta_{i+1}v_{i+1}\cdot m_{i+1, i}=0.$ Here, notations are as explained in Step (e) of the description of an element of $\tilde{M}(R)$ given at the paragraph following Lemma \[la1\].
- If $i$ is odd and $L_i$ is *bound of type I*, then $\delta_{i-1}\cdot m_{i, i-1}'+\delta_{i+1}\cdot m_{i, i+1}'=0.$ Here, $m_{i, i-1}'$ (resp. $m_{i, i+1}'$) is the last column vector of $m_{i, i-1}$ (resp. $m_{i, i+1}$).
Thus we can see that $v_i, z_i$ (resp. $y_i, z_i$) can be eliminated
| 1,352
| 628
| 1,212
| 1,363
| 1,113
| 0.794239
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|
rved value of the “hierarchy”, $\lambda_{obs}$, and the observed value of the effective four-dimensional cosmological constant, which we take to be zero. Thus, we take as our renormalization conditions $$V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}(\lambda_{obs})
={dV_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}\over d\lambda}(\lambda_{obs})=0.
\label{renc}$$ If there are other bulk fields, such as the graviton, which give additional classical or quantum mechanical contributions to the radion potential, then those should be included in $V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}$. From the renormalization conditions (\[renc\]) the unknown coefficients $\alpha$ and $\beta$ can be found, and then the mass of the radion is calculable. In Fig. \[fig1\] we plot (\[confveff\]) for a fermionic field and a chosen value of $\lambda_{obs}$.
=10 cm\
From (\[renc\]), we have $$\beta = - A (1-\lambda_{obs})^{-5},\quad \alpha= -\beta
\lambda_{obs}^5.
\label{consts}$$ These values correspond to changes $\delta \sigma_{\pm}$ on the positive and negative brane tensions which are related by the equation $$\delta\sigma_+ = -\lambda^5_{obs}\ \delta\sigma_-.
\label{reltensions}$$ As we shall see below, Eq. (\[reltensions\]) is just what is needed in order to have a static solution according to the five dimensional equations of motion, once the casimir energy is included.
We can now calculate the mass of the radion field $m_\phi^{(-)}$ from the point of view of the negative tension brane. For $\lambda_{obs}\ll 1$ we have: $$m^{2\ (-)}_\phi =
\lambda_{obs}^{-2}\ m^{2\ (+)}_\phi = \lambda_{obs}^{-2}\
{d^2 V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}\over
d\phi^2}\approx \mp
\lambda_{obs} \left({5\pi^3 \zeta'_R(-4)\over 6 M^3 l^5}\right).
\label{massconf}$$ The contribution to the radion mass squared is negative for bosons and positive for fermions. Thus, depending on the matter content of the bulk, it is clear that the radion may be stabilized due to this effect.
Note, however, that if the “observed” i
| 1,353
| 3,016
| 1,864
| 1,354
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|
ity follows form equation , $\alpha\in \widehat{\mathcal O}(\vec X;\varnothing) =\mathcal O(m)$ has $m+1$ inputs, and $\beta\in\widehat{\mathcal
O}(\vec Y;{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}})=\mathcal O(n)$ (or similarly $\beta\in\widehat{\mathcal
O}(\vec Y;{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}})$) has $n$ inputs. It is clear that this will satisfy equivariance, since equivariance was just used to define the composition. The next lemma establishes the final property for $\widehat{\mathcal O}$ being a 0/1-operad.
The composition in $\widehat{\mathcal O}$ satisfies the associativity axiom.
By definition the composition is just the usual composition in $\mathcal O(n)$, except for inserting trees in the last input of elements in $\widehat{\mathcal
O}(\vec X;\varnothing)$. Thus, except for composition in the last spot, associativity of $\widehat{\mathcal O}$ follows from the associativity of $\mathcal O$.
Now, let $\alpha\in \widehat{\mathcal O}(\vec X;\varnothing)\cong
\mathcal O(m)$, $\beta\in \widehat{\mathcal O}(\vec Y;y)\cong
\mathcal O(n)$, and $\gamma\in \widehat{\mathcal O} (\vec Z;z) \cong
\mathcal O(p)$, where $y,z\in\{{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}}\}$. Then, associativity is satisfied, because for $1\leq j\leq m$, it is $$\begin{gathered}
(\alpha\circ_{m+1}\beta)\circ_{j} \gamma
\stackrel{\mathit{\eqref{def_cyclic_compos}}}{=} (\tau_{n+1}(\beta)
\circ_1 \alpha)\circ_j \gamma \stackrel{\mathit{op.comp}}{=}\\
=\tau_{n+1}(\beta) \circ_1 (\alpha\circ_j \gamma )
\stackrel{\mathit{\eqref{def_cyclic_compos}}}{=} (\alpha\circ_j
\gamma )\circ_{m+p}\beta,\end{gathered}$$ and for $m< j< m+n$, it is $$\begin{gathered}
(\alpha\circ_{m+1}\beta)\circ_{j} \gamma
\stackrel{\mathit{\eqref{d
| 1,354
| 2,328
| 1,536
| 1,315
| 1,819
| 0.785442
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|
(z\^[\*]{}), with $\Psi(z)$ is the digamma function (z) , and $z=1/2 -i\hat{\mu}$. At small chemical potential, $\aleph(z)$ can be expanded as (z)&=&-2\_E-4 2+14(3)\^2-62(5)\^4+254(7)\^6+[ O]{}(\^8).\[aleph\] In addition to the renormalized quark free-energy in Eq , the renormalized gluon free-energy is given as &=&-\
&-&\^2T\^4\_D\^2\_D\^2(\_E+)+\_f\
&-& \_f, \[Eq:Fgr\] where $\hat m_D^w=m_D^w/2\pi T$, $\hat m_D=m_D/2\pi T$, $ \delta \hat m_D=\delta m_D/2\pi T$ and $m_D^w$ represents the Debye mass in weak magnetic field approximation and is obtained as $m_D^w$\^2 &&\
&+&\_f \_[l=1]{}\^ (-1)\^[l+1]{}l\^2 (2l) K\_0() + \[(q\_fB)\^4\]\
&=& m\_D\^2 + m\_D\^2. \[md\_wfa\] Considering the expression of free energy vis-a-vis pressure we calculate the second-order QNS in weak field limit by using Eq. . The second-order QNS of free quarks and gluons in thermal medium is given as \_f=N\_c N\_f T\^2.
![Variation of second-order QNS scaled with thermal free field value with temperature (left panel) and magnetic field strength (right panel) for $m_f=5$ MeV and $N_f=3$.[]{data-label="QNS_wfa"}](chi2_wfa_Nf3.pdf "fig:") ![Variation of second-order QNS scaled with thermal free field value with temperature (left panel) and magnetic field strength (right panel) for $m_f=5$ MeV and $N_f=3$.[]{data-label="QNS_wfa"}](chi2_wfa_Nf3_eB.pdf "fig:")
The left panel of Fig. \[QNS\_wfa\] shows the variation of the scaled second-order QNS with the temperature at different values of the magnetic field strength. The weak field effect appears as a correction to the thermal medium, the weak field second-order QNS is not very much different than that of thermal medium. It is found to increase with temperature and approaches the free field value at high enough temperature. The magnetic field effect on the second-order QNS is visible at low temperature. The value of second-order QNS slowly decreases as one increases the magnetic field strength as shown in the right panel of Fig. \[chi\_def\].
Conclusion
==========
We consider a hot and den
| 1,355
| 1,003
| 253
| 1,386
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|
0.5941 (2) 0.3709 (3) 0.26080 (18) 0.0201 (10)
H25 0.5678 0.3462 0.2785 0.024\*
C26 0.57227 (19) 0.1766 (3) 0.36167 (16) 0.0112 (8)
C27 0.5416 (2) 0.2602 (3) 0.36530 (18) 0.0173 (9)
H27 0.5644 0.3144 0.3807 0.021\*
C28 0.4787 (2) 0.2666 (4) 0.3470 (2) 0.0244 (11)
H28 0.4587 0.3248 0.3496 0.029\*
C29 0.4448 (2) 0.1873 (4) 0.32495 (19) 0.0215 (10)
H29 0.4016 0.1913 0.3118 0.026\*
C30 0.4751 (2) 0.1017 (3) 0.32236 (18) 0.0189 (10)
H30 0.4522 0.0465 0.3087 0.023\*
C31 0.5378 (2) 0.0973 (3) 0.33959 (17) 0.0173 (9)
H31 0.5580 0.0395 0.3364 0.021\*
C32 0.6776 (2) 0.1222 (3) 0.33224 (16) 0.0114 (8)
C33 0.6938 (2) 0.1824 (3) 0.29875 (17) 0.0162 (9)
H33 0.6965 0.2491 0.3049 0.019\*
C34 0.7063 (2) 0.1462 (4) 0.25613 (18) 0.0201 (10)
H34 0.7169 0.1880 0.2333 0.024\*
C35 0.7029 (2) 0.0485 (4) 0.24736 (18) 0.0209 (10)
H35 0.7113 0.0236 0.2184 0.025\*
C36 0.6874 (2) −0.0127 (3) 0.28086 (18) 0.0200 (10)
H36 0.6848 −0.0793 0.2747 0.024\*
C37 0.6757 (2) 0.0242 (3) 0.32364 (18) 0.0183 (10)
H37 0.6664 −0.0178 0.3471 0.022\*
C38 0.6761 (2) 0.0792 (3) 0.43110 (16) 0.0134 (9)
H38A 0.7154 0.0511 0.4304 0.016\*
| 1,356
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|
[${\bf Q}(y\!=\!+1|I)\!=\!0$]{}. Then, our approach selects and asks about an object $\hat{I}$ based on Equation (\[eqn:select\]). We use the answer to update ${\bf P}$. If a new object part is labeled during the QA process, we apply Equation (\[eqn:LossAOG\]) to update the AOG. More specifically, if people label a new part template, our method will grow a new AOG branch to encode this template. If people annotate a part for an old part template, our method will update its corresponding AOG branch. Then, we compute the new distribution [${\bf Q}$]{} based on the new AOG. In this way, the above QA procedure gradually grows the AOG.
Experiments
===========
Implementation details {#sec:implement}
----------------------
We used a 16-layer VGG network (VGG-16) [@VGG], which was pre-trained for object classification using 1.3M images in the ImageNet ILSVRC 2012 dataset [@ImageNet]. Then, for each testing category, we further fine-tune the VGG-16 using object images in this category to classify target objects from random images. We selected the last nine conv-layers of VGG-16 as valid conv-layers. We extracted neural units from these conv-layers to build the AOG.
**Active question-answering:** Three parameters were involved in our active-QA method, *i.e.* $\alpha$, $\beta$, and $Z$. Because most objects of the category contained the target part, we ignored the small probability of ${\bf P}(y=-1|I)$ in Equation (\[eqn:delta\]) to simplify the computation. As a result, $Z$ was eliminated in Equation (\[eqn:delta\]), and the constant weight $\beta$ did not affect object-selection results in Equation (\[eqn:select\]). We set $\alpha=4.0$ in our experiments.
**Learning AOGs:** Multiple latent patterns corresponding to the same convolutional filter may have similar positions $\overline{\bf p}_{u}$, and their deformation ranges may highly overlap. Thus, we selected the latent pattern with the highest $Score(u)$ within a small range of $\epsilon\times\epsilon$ in the filter’s feature map and removed other nearby
| 1,357
| 142
| 1,544
| 1,405
| 1,655
| 0.787129
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|
Eq. (\[eq:DeltaACPdirParameter\]) as $$\begin{aligned}
\Delta a_{CP}^{\mathrm{dir}} &=
4\, \mathrm{Im}\left(\frac{\lambda_b}{\Sigma}\right) \left|\tilde{p}_0 \right| \sin( \delta_{\mathrm{strong}})\,,\end{aligned}$$ with the unknown strong phase $$\begin{aligned}
\delta_{\mathrm{strong}} &= \mathrm{arg}(\tilde{p}_0)\,.\end{aligned}$$ Then the numerical result in Eq. (\[eq:resultp0tilde\]) reads $$\begin{aligned}
\left|\tilde{p}_0\right| \sin( \delta_{\mathrm{strong}}) &= 0.65 \pm 0.12\,. \label{eq:mainresult}\end{aligned}$$ Recall that in the group theoretical language the parameters $t_0$ and $p_0$ are the matrix elements of the $\Delta U=1$ and $\Delta U=0$ operators, respectively [@Brod:2011re]. For the ratio of the matrix elements of these operators we employ now the following parametrization $$\label{eq:defC}
\tilde p_0 = B + C e^{i \delta}\,,$$ such that $B$ is the short-distance (SD) ratio and the second term arises from long-distance (LD) effects. While the separation between SD and LD is not well-defined, what we have in mind here is that diagrams with a $b$ quark in the loop are perturbative and those with quarks lighter than the charm are not.
In Eq. (\[eq:DeltaI12-generic\]) of Sec. \[sec:DeltaI12inKDB\] below we apply the same decomposition into a no QCD part and corrections to that also to the $\Delta I=1/2$ rules in $K$, $D$ and $B$ decays to pions. It is instructive to compare all of these systems in the same language.
We first argue that in Eq. (\[eq:defC\]) to a very good approximation $B=1$. This is basically the statement that perturbatively, the diagrams with intermediate $b$ are tiny. More explicitly, in that case, that is when we neglect the SD $b$ penguins, we have $$\begin{aligned}
Q^{\Delta U=1} \equiv \frac{Q^{\bar{s}s} - Q^{\bar{d}d}}{2}\,,\qquad
Q^{\Delta U=0} \equiv \frac{Q^{\bar{s}s} + Q^{\bar{d}d}}{2}\,. \end{aligned}$$ Setting $C=0$ then corresponds to the statement that only $Q^{\bar{s}s}$ can produce $K^+K^-$ and only $Q^{\bar{d}d}$ can produce $\pi^+\pi^-$. This impl
| 1,358
| 960
| 1,791
| 1,309
| 1,557
| 0.788279
|
github_plus_top10pct_by_avg
|
trol the signals emitted on two distinct channels, which are propagated through the environment to the agents within a neighboring radius set to $50$. The choice for two channels was made to allow for signals of higher complexity, and possibly more interesting dynamics than greenbeard studies [@gardner2010].
The received signals are summed separately for each direction (front, back, right, left, up, down), and weighted by the squared inverse of the emitters distance. This way, agents further away have much less impact on the sensors than closer ones do. Every agent is able to receive signals on the two emission channels, from 6 different directions, totalling $12$ different values sensed per time step. For example, the input $S_\text{in}^{(6,1)}$ corresponds to the signals reaching the agent from the neighbors below.
![ [**Architecture of the agent’s controller.**]{} The network is composed of 12 input neurons, 10 hidden neurons, 10 context neurons and 5 output neurons.[]{data-label="fig:ann"}](witkowski-fig2-ann.png){width=".7\textwidth"}
The evolution is performed continuously over the population. Agents with negative or zero energy are removed, while agents with energy above a threshold are forced to reproduce, within the limits of one infant per time step. The reproduction cost is low enough, considering the threshold, to not put the life of the agent at risk.
Study cases
===========
We go over the application of this model in three selected examples of studies. Each of them highlights a specific property for OEE. The first model shows how agents can form patterns to accelerate their search for energy, distributed over an n-dimensional space, collaborating via local signaling with their neighbors. The second study shows the invention of dynamical group strategies in a spatial prisoner’s dilemma, allowing for specific cooperation effects. The third example shows the impact of growth on the emergence of noise-canceling effects.
OEE via collective search based on communication
-------------------------
| 1,359
| 1,107
| 2,700
| 1,171
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|
$ [@DF96] are fixed to reproduce the experimental data of the corresponding hyperon decays, while the ones involving kaons, $C_K^{PC}=-18.9$, $D_K^{PC}=6.63$, $C_{K}^{PV}=0.76$ and $D_K^{PV}=2.09$, are derived using SU(3) symmetry.
![Weak vertices corresponding to the $\Lambda N\pi\pi$, and $\Lambda N$ interactions. The solid black circle represents the weak vertex. The corresponding Lagrangians are given in Eq. (\[eq:weakl2\]). \[vf3\]](nnppw "fig:") ![Weak vertices corresponding to the $\Lambda N\pi\pi$, and $\Lambda N$ interactions. The solid black circle represents the weak vertex. The corresponding Lagrangians are given in Eq. (\[eq:weakl2\]). \[vf3\]](nlw "fig:")
The other two weak vertices entering at the considered order (Fig. \[vf3\]) are obtained from the weak SU(3) chiral Lagrangian, $$\begin{aligned}
{\mathcal{L}}^w_{\Lambda N\pi\pi}=&
G_Fm_\pi^2\frac{h_{2\pi}}{f_\pi^2}(\vec{\pi}\cdot\vec{\pi})
\overline{\Psi}\Psi_\Lambda\,, \label{eq:weakl2}\\
{\mathcal{L}}^w_{\Lambda N} =& iG_Fm_\pi^2 h_{\Lambda N}
\overline{\Psi}\Psi_\Lambda \nonumber\,,\end{aligned}$$ with $
h_{2\pi}=(D+3F)/(8\sqrt6 G_Fm_\pi^2)=-10.13\text{ MeV}
$ and $
h_{\Lambda N}=-(D+3F)/(\sqrt6 G_F m_\pi^2)=81.02\text{ MeV} \,.
$ $D$ and $F$ are the couplings parametrizing the weak chiral SU(3) Lagrangian, and can be fitted through the pole model to the experimentally known hyperon decays. In that case, one finds that when s-wave amplitudes are correctly reproduced, p-wave amplitude predictions disagree with the experiment [@donoghue].
[cc]{} ![Strong vertices for the $NN\pi$, $NN\pi\pi$, $\Lambda \Sigma\pi$ and $\Lambda N K$ which arise from the Lagrangians in Eq. (\[eq:str\]). \[vf1\]](nnps "fig:")& ![Strong vertices for the $NN\pi$, $NN\pi\pi$, $\Lambda \Sigma\pi$ and $\Lambda N K$ which arise from the Lagrangians in Eq. (\[eq:str\]). \[vf1\]](nnpps "fig:")\
\
![Strong vertices for the $NN\pi$, $NN\pi\pi$, $\Lambda \Sigma\pi$ and $\Lambda N K$ which arise from the Lagrangians in Eq. (\[eq:str\]). \[vf1\]](lspw "fig:")& ![St
| 1,360
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| 224
| 1,565
| 1,492
| 0.788895
|
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|
section \[integrability\] we comment on the classical and quantum integrability of the model, and its consistency with the conformal current algebra. We conclude in section \[conclusions\].
We have gathered many technical details in the appendices. In appendix \[compositeOPEs\] we give a prescription to compute OPEs involving composite operators. In appendix \[XXOPEs\] we compute the behavior at large radius for the coefficients appearing in the current-current and current-primary OPEs. In appendix \[consistentPertOPEs\] we prove the consistency of the perturbative algorithm used to compute the current-current and current-primary OPEs. Appendix \[AppCurrents\] contains further consistency checks of the current algebra as well as details of the computation of the current algebra. In appendix \[AppPrimaries\] we detail calculations involving the primary fields. In appendix \[commutators\] we translate the current-current OPEs into (anti-)commutation relations for the modes of the currents when the theory is defined on a cylinder. Finally, classical integrability of the model is proven in appendix \[classint\]
The conformal current algebra {#worldsheetcurrentalgebra}
=============================
Setting {#setting .unnumbered}
-------
We study a non-linear sigma-model on a supergroup $G$ with zero Killing form, including a kinetic term and a Wess-Zumino term with arbitrary coefficient. The model is conformal and has a global symmetry group corresponding to the left and right action of the group on itself. In this section we review and complement the analysis of the algebra of current components associated to the left group action [@Ashok:2009xx]. The action of the non-linear sigma-model on the supergroup is: $$\begin{aligned}
\label{ourmodel}
S &= S_{kin} + S_{WZ}\cr
S_{kin} &= \frac{1}{ 16 \pi f^2}\int d^2 z Tr'[- \partial^\mu g^{-1}
\partial_\mu g]
\cr
S_{WZ} &= - \frac{ik}{24 \pi} \int_B d^3 y \epsilon^{\alpha \beta \gamma}
Tr' (g^{-1} \partial_\alpha g g^{-1} \partial_\beta g g^{-1} \partial_\gamma g )\e
| 1,361
| 827
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| 1,334
| 3,816
| 0.769923
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|
learns from objects without part annotations. $$\begin{split}
S^{\textrm{unant}}_{\textrm{AOG}}&={\sum}_{u\in Child(v^{*})}S^{\textrm{unant}}_{u}\\
L^{\textrm{unant}}({\boldsymbol\Lambda}_{\textrm{AOG}})&={\sum}_{u\in Child(v^{*})}\lambda^{\textrm{close}}\Vert\Delta{\bf p}_{u}\Vert^2
\end{split}
\label{sec:unsuper}$$ where the first term $S^{\textrm{unant}}_{\textrm{AOG}}$ denotes the inference score at the level of latent patterns without ground-truth annotations of object parts. Please see the appendix for the computation of $S^{\textrm{unant}}_{u}$. The second term $L^{\textrm{unant}}({\boldsymbol\Lambda}_{\textrm{AOG}})$ penalizes latent patterns that are far from their parent $v^{*}$. This loss encourages the assigned neural unit to be close to its parent latent pattern. We assume that 1) latent patterns that frequently appear among unannotated objects may potentially represent stable part appearance and should have higher priorities; and that 2) latent patterns spatially closer to their parent part templates are usually more reliable.
When we set $\lambda_{v^{*}}$ to a constant $\lambda^{\textrm{inf}}\sum_{k=1}^{K}n_{k}$, we can transform the learning objective in Equation (\[eqn:LossAOG\]) as follows. $$\forall v\in Child(top), \quad\max_{{\boldsymbol\theta}_{v}}{\bf L}_{v},\quad {\bf L}_{v}\!=\!\!\!\!\!\!\!\!\sum_{u\in Child(v)}\!\!\!\!\!\!\!Score(u)
\label{eqn:subAOG}$$ where [$Score(u)\!=\!\mathbb{E}_{I\in{\bf I}_{v}}[S_{u}+S^{\textrm{inf}}(\Lambda_{u}|\Lambda^{*}_{v})]$ $+\mathbb{E}_{I'\in{\bf I}^{\textrm{obj}}}$ $\lambda^{\textrm{unant}}[S^{\textrm{unant}}_{u}-\lambda^{\textrm{close}}\Vert\Delta{\bf p}_{u}\Vert^2]$]{}. ${\boldsymbol\theta}_{v}\subset{\boldsymbol\theta}$ denotes the parameters for the sub-AOG of the part template $v$. We use ${\bf I}_{v}\subset{\bf I}^{\textrm{ant}}$ to denote the subset of images that are annotated with $v$ as the ground-truth part template.
**Learning the sub-AOG for each part template:**[` `]{} Based on Equation (\[eqn:subAOG\]), we can mine the sub-AOG for eac
| 1,362
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| 1,012
| 1,406
| 2,362
| 0.780526
|
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|
2\to V$ of $p$, $X_2\to X$ of $g^{-1}(p)$ and an open embedding $ Z\times_XX_2\into Z_2:=Z\times_VV_2$. Our only remaining problem is that $Z_2\neq Z\times_XX_2$, hence $Z_2$ is not a subscheme of $X_2$. We achieve this by further shrinking $V_2$ and $X_2$.
The complement $B_2:=Z_2\setminus Z\times_XX_2$ is closed, thus $g(B_2)\subset V_2$ is a closed subset not containing $p$. Pick $\phi\in \Gamma({{\mathcal O}}_{V_2})$ that vanishes on $g(B_2)$ such that $\phi(p)\neq 0$. Then $\phi\circ g$ is a function on $Z_2$ that vanishes on $B_2$ but is nowhere zero on $g^{-1}(p)$. We can thus extend $\phi\circ g$ to a function $\Phi$ on $X_2$. Thus $V_P:=V_2\setminus (\phi=0), Z_P:=Z_2\setminus (\phi\circ g=0)$ and $X_P:=X_2\setminus (\Phi=0)$ have the required properties.
\[et.nbhds.say\] During the proof we have used two basic properties of étale neighborhoods.
First, if $\pi:X\to Y$ is finite then for every étale neighborhood $(u\in U)\to (x\in X)$ there is an étale neighborhood $(v\in V)\to (\pi(x)\in Y)$ and a connected component $(v'\in V')\subset X\times_YV$ such that there is a lifting $(v'\in V')\to (u\in U)$.
Second, if $\pi:X\to Y$ is a closed embedding, $U\to X$ is étale and $P\subset U$ is a finite set of points then we can find an étale $V\to Y$ such that $P\subset V$ and there is an open embedding $(P\subset X\times_YV)\to (P\subset U)$.
For proofs see [@milne 3.14 and 4.2–3].
The next result shows that gluing commutes with flat morphisms.
\[glue.etloc.lem\] For $i=1,2$, let $X_i$ be Noetherian affine $A$-schemes, $Z_i\subset X_i$ closed subschemes and $g_i:Z_i\to V_i$ finite surjections with universal push-outs $Y_i$. Assume that in the diagram below both squares are fiber products. $$\begin{array}{ccccc}
V_1 & \stackrel{g_1}{\leftarrow} & Z_1 & \to & X_1\\
\downarrow && \downarrow && \downarrow \\
V_2 & \stackrel{g_2}{\leftarrow} & Z_2 & \to & X_2
\end{array}$$
1. If the vertical maps are flat then $Y_1 \to Y_2$ is also flat.
2. If the vertical maps are smooth then $Y_1 \to Y_2$ is also
| 1,363
| 1,065
| 977
| 1,283
| 2,767
| 0.777149
|
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|
The first containment is by Lemma \[l:qproperties\].\[f:q(r)\_bounds\]: $\frac{1}{2}\exp\lrp{-\frac{7\aq\Rq^2}{3}}\cdot g(z) \leq q(g(z)) \leq g(z)$. THe second containment is by Lemma \[l:gproperties\].4: $g(\|z\|_2) \in [\|z\|_2-2\epsilon, \|z\|_2]$.
\[l:hproperties\] Given a parameter $\epsilon$, define $$\begin{aligned}
h(r) := \threecase
{\frac{r^3}{6\epsilon^2}}{r\in [0,\epsilon]}
{\frac{\epsilon}{6} + \frac{r-\epsilon}{2} + \frac{(r-\epsilon)^2}{2\epsilon} - \frac{(r-\epsilon)^3}{6\epsilon^2}}{r\in[\epsilon, 2\epsilon]}
{r }{r\geq 2\epsilon}
\end{aligned}$$
1. The derivatives of $h$ are as follows: $$\begin{aligned}
h'(r) =& \threecase
{\frac{r^2}{2\epsilon^2}}{r\in [0,\epsilon]}
{\frac{1}{2} + \frac{r-\epsilon}{\epsilon} - \frac{(r-\epsilon)^2}{2\epsilon^2}}{r\in[\epsilon, 2\epsilon]}
{1}{r\geq 2\epsilon}\\
h''(r) =& \threecase
{\frac{r}{\epsilon^2}}{r\in [0,\epsilon]}
{\frac{1}{\epsilon} - \frac{r-\epsilon}{\epsilon^2}}{r\in[\epsilon, 2\epsilon]}
{0}{r\geq 2\epsilon}\\
h'''(r) =& \threecase
{\frac{1}{\epsilon^2}}{r\in [0,\epsilon]}
{-\frac{1}{\epsilon^2}}{r\in[\epsilon, 2\epsilon]}
{0}{r\geq 2\epsilon}
\end{aligned}$$
2. 1. $h'$ is positive, motonically increasing.
2. $h'(0)=0$, $h'(r) =1$ for $r\geq \epsilon$
3. $\frac{h'(r)}{r}\leq \min\lrbb{\frac{1}{\epsilon}, \frac{1}{r}}$ for all $r$
3. 1. $h''(r)$ is positive
2. $h''(r) = 0$ for $r=0$ and $r\geq 2\epsilon$
3. $h''(r) \leq \frac{1}{\epsilon}$
4. $\frac{h''(r)}{r} \leq \frac{1}{\epsilon^2}$
4. $\lrabs{h'''(r)} \leq \frac{1}{\epsilon^2}$
5. $r-2\epsilon\leq h(r) \leq r$
The claims can all be verified with simple algebra.
\[l:gproperties\] Given a parameter $\epsilon$, let us define $$\begin{aligned}
g(z) := h(\|z\|_2)
\end{aligned}$$ Where $h$ is
| 1,364
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|
meet and join on $(\mathcal{L(S)},\subset )$ by the same symbols $^{\bot }$, $\Cap $, and $\Cup $, respectively, that we have used in order to denote the corresponding operations on $(\mathcal{L(H)},\subset )$ and $(\mathcal{E},\prec )$, and call $(\mathcal{L(S)},\subset )$ *the lattice of closed subsets of* $\mathcal{S}$ (the word *closed* refers here to the fact that, for every $\mathcal{S}_{E}\in $ $\mathcal{L(S)}$, $(\mathcal{S}_{E}^{\bot
})^{\bot }=\mathcal{S}_{E}$). We also note that the operation $\Cap $ coincides with the set-theoretical intersection $\cap $ on $\mathcal{L(S)}$ because of the analogous result holding in $(\mathcal{L(H)},\subset )$.[^6]
To close up, let us observe that the unary operation $^{\bot }$ defined on $\mathcal{L(S)}$ can be extended to $\mathcal{P(S)}$ by setting, for every $\mathcal{T\in P(S)}$,
$\mathcal{T}^{\bot }=($*min*$\{\mathcal{S}_{E}\in \mathcal{L(S)\mid
T\subset S}_{E}\})^{\bot }$
(the symbol *min* obviously refers to the order $\subset $ defined on $\mathcal{L(S)}$). This extension will be needed indeed in Sec. 3.2.
Actual and potential properties
-------------------------------
We say that a property $E$ is* actual* (*nonactual*) in the state $S$ iff one can perform a test of $E$ on any physical object $x$ in the state $S$ by means of a registration $r\in E$, obtaining outcome 1 (0) without modifying $S$.[^7]
Basing on the above definition, for every state $S\in \mathcal{S}$ three subsets of $\mathcal{E}$ can be introduced.
$\mathcal{E}_{S}$ : the set of all properties that are actual in $S$.
$\mathcal{E}_{S}^{\bot }$ : the set of all properties that are nonactual in $S$.
$\mathcal{E}_{S}^{I}$ : the set $\mathcal{E}\setminus \mathcal{E}_{S}\cup
\mathcal{E}_{S}^{\bot }$ (called the set of all properties that are *indeterminate*, or* potential*, in $S$).
By using the mathematical apparatus of QM, the sets $\mathcal{E}_{S}$ and $\mathcal{E}_{S}^{\bot }$ can be characterized as follows.
$\mathcal{E}_{S}=\{E\in \mathcal{E}\mid \varphi (S)\subset \chi (E)\}=\
| 1,365
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| 1,532
| 1,396
| 1,723
| 0.786392
|
github_plus_top10pct_by_avg
|
w_T}_2^2} \leq 32 \lrp{T^2 L^2 + TL_\xi^2} T\beta^2
\end{aligned}$$
Using the fact that conditioned on the randomness up to step $k$, $\E{\xi(v_0,\eta_{k+1}) - \xi(w_{k\delta}, \eta_{k+1})}=0$, we can show that for any $k\leq n$, $$\begin{aligned}
& \E{\lrn{v_{(k+1)\delta} - w_{(k+1)\delta}}_2^2}\\
=& \E{\lrn{v_{k\delta} - \delta \nabla U(v_0) - w_{k\delta} +\delta \nabla U(w_{k\delta}) + \sqrt{\delta} \xi(w_0, \eta_k) - \sqrt{\delta} \xi(w_{k\delta}, \eta_k)}_2^2}\\
=& \E{\lrn{v_{k\delta} - \delta \nabla U(v_0) - w_{k\delta} +\delta \nabla U(w_{k\delta})}_2^2} + \delta \E{\lrn{\xi(w_0, \eta_k) - \xi(w_{k\delta}, \eta_k)}_2^2}
\numberthis \label{e:t:mkqwm}
\end{aligned}$$ where the first inequality is by (Assumption on smoothness of U and xi).
Using (smoothness of xi), and Lemma \[l:divergence\_vt\], we can bound $$\begin{aligned}
& \delta \E{\lrn{\xi(w_0, \eta_k) - \xi(w_{k\delta}, \eta_k)}_2^2}\\
\leq& \delta L_\xi^2 \E{\lrn{w_{k\delta} - w_0}_2^2}\\
\leq& \delta L_\xi^2 \lrp{16 \lrp{T^2 L^2 \E{\lrn{w_0}_2^2} + T \beta^2}}
\end{aligned}$$
We can also bound $$\begin{aligned}
& \E{\lrn{v_{k\delta} - \delta \nabla U(v_0) - w_{k\delta} +\delta \nabla U(w_{k\delta})}_2^2}\\
\leq& \lrp{1 + \frac{1}{n}}\E{\lrn{v_{k\delta} - \delta \nabla U(v_{k\delta}) - w_{k\delta} +\delta \nabla U(w_{k\delta})}_2^2} + (1+n) \delta^2 \E{\lrn{\nabla U(v_{k\delta}) - \nabla U(v_0)}_2^2}\\
\leq& \lrp{1+ \frac{1}{n}}\lrp{1+ \delta L}^2 \E{\lrn{v_{k\delta} - w_{k\delta}}_2^2} + 2n\delta^2 L^2 \E{\lrn{v_{k\delta} - v_0}_2^2}\\
\leq& e^{1/n + 2\delta L}E{\lrn{v_{k\delta} - w_{k\delta}}_2^2} + 2n\delta^2 L^2 \E{\lrn{v_{k\delta} - v_0}_2^2}\\
\leq& e^{1/n + 2\delta L}E{\lrn{v_{k\delta} - w_{k\delta}}_2^2} + 2n\delta^2 L^2 \lrp{T^2L^2 \E{\lrn{v_0}_2^2} + T\beta^2}
\end{aligned}$$ where the first inequality is by Young’s inequality and the second ineq
| 1,366
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| 1,366
| 1,337
| null | null |
github_plus_top10pct_by_avg
|
artial_{-}^{3}T^{-}\right)
. \label{delta2h--}%\end{aligned}$$ As $\beta_{+}\,\delta_{\eta}\int f_{++}Y^{+}d\phi\sim\lbrack Q_{+}%
(Y^{+}),Q_{+}(T^{+})+Q_{-}(T^{-})]$ (with $\beta_{\pm}=2l^{-1}\left( 1\pm(\mu
l)^{-1}\right) $) and $\beta_{-}\,\delta_{\eta}\int f_{--}Y^{-}d\phi
\sim\lbrack Q_{+}(Y^{+}),Q_{+}(T^{+})+Q_{-}(T^{-})]$ (with $Y^{+}$ and $Y^{-}$ the asymptotic conformal transformation associated with a second spacetime diffeomorphism $\xi^{+}$, $\xi^{-}$ and $\xi^{r}$), one can easily infer from (\[delta2h++\]) and (\[delta2h–\]) that $Q_{+}(Y^{+})$ and $Q_{-}(T^{-})$ commute with each other and each fulfills the Virasoro algebra with central charges $$c_{\pm}=\left( 1\pm\frac{1}{\mu l}\right) \,c$$ (see [@Brown-Henneaux2] for general theorems).
Range $|\mu l|>1$ of the mass parameter
=======================================
Take for definiteness $\mu l$ positive and hence $>1$. Solving the equations starting from infinity shows that again, one should expect both chiralities to be present, taking exactly the same form as (\[Asympt relaxed metric mu Neg\]) and (\[Asympt relaxed metric mu Pos\]) above. However, the positive chirality blows up at infinity ($\Delta g_{++}$ dominates the background) and the space is not asymptotically of constant curvature. So, if $h_{++}\not =0$, the space is not asymptotically anti-de Sitter. For this reason, one must set $h_{++}=0$. But the other $h_{--}$-term is subdominant with respect to $f_{--}$, so that the asymptotic negative chirality behavior reproduces (\[Standard-Asympt\]). The same analysis holds when $\mu l$ is negative (with an interchange of the roles of the two chiralities). Therefore, the behavior of the metric can be taken to be (\[Standard-Asympt\]). The asymptotic derivation of the charges and the central charges proceeds then straightforwardly (no divergence to be canceled) and yields $$Q_{\pm}[T^{\pm}]=\frac{2}{l}\left( 1\pm\frac{1}{\mu l}\right) \int T^{\pm
}f_{\pm\pm}d\phi\$$ with central charges $$c_{\pm}=\left( 1\pm\frac{1}{\mu l}\right)
| 1,367
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| 1,359
| null | null |
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|
}$$ For all $i \in [d]$, we have, $$\begin{aligned}
\label{eq:hess_posl_12}
\E\Bigg[\sum_{\i = 1}^d \big(\big(A^{(j)}\big)^2\big)_{i\i} \Bigg] & \leq & \E\Bigg[ \bigg(\sum_{\i =1 }^d A^{(j)}_{i\i} \bigg) \max_{i \in [d]} \bigg\{ \sum_{\i =1 }^d A^{(j)}_{i\i}\bigg\} \Bigg] \nonumber\\
&\leq & \E\bigg[ D^{(j)}_{ii} \delta_{j,1} \bigg] \nonumber\\
&\leq& \I_{\big\{i \in S_j \big\}} \Bigg\{ \frac{e^{6b}\eta\ell_j}{\kappa_j} \bigg(\delta_{j,1}^2 + \frac{\delta_{j,1}\delta_{j,2}\kappa_j}{\eta\ell_j} \bigg)\Bigg\} \,.
\end{aligned}$$ Using and , we have, for all $i \in [d]$, $$\begin{aligned}
&&\sum_{\i = 1}^d \Big|\E\Big[\big(\big(M^{(j)}\big)^2\big)_{i\i}\Big]\Big| \nonumber\\
& = & \sum_{\i = 1}^d \Bigg|\E\Big[\big(\big(D^{(j)}\big)^2\big)_{i\i}\Big] - \E\Big[\big(D^{(j)} A^{(j)}\big)_{i\i}\Big]
- \E\Big[\big( A^{(j)} D^{(j)} \big)_{i\i}\Big]
+ \E\Big[\big(\big(A^{(j)}\big)^2\big)_{i\i}\Big] \Bigg| \nonumber\\
&\leq& 2\E\Big[\big(D^{(j)}_{ii}\big)^2\Big] + \sum_{\i = 1}^d \bigg( \E\Big[\delta_{j,1}\big(A^{(j)}\big)_{i\i}\Big] + \E\Big[\big(\big(A^{(j)}\big)^2\big)_{i\i}\Big] \bigg) \nonumber\\
&\leq& \I_{\big\{i \in S_j \big\}} \Bigg\{ \frac{e^{6b}\eta\ell_j}{\kappa_j}\bigg( 4 \delta_{j,1}^2 + \frac{2\big(\delta_{j,1}\delta_{j,2} +\delta_{j,2}^2\big)\kappa_j}{\eta\ell_j} \bigg) \Bigg\}\nonumber\\
&=& \I_{\big\{i \in S_j \big\}} \bigg\{ \frac{e^{6b}\delta\eta\ell_j}{\kappa_j} \bigg\}\,, \label{eq:hess_posl_15}\end{aligned}$$ where the last equality follows from the definition of $\delta$, Equation .
To bound $\|\sum_{j =1}^n \E[(M^{(j)})^2]\|$, we use the fact that for $J \in \reals^{d\times d}, {\|J\|} \leq \max_{i \in [d]}\sum_{\i = 1}^d|J_{i\i}|$. Therefore, we have $$\begin{aligned}
\Bigg\|\sum_{j =1}^n \E\Big[(M^{(j)})^2\Big]\Bigg\| & \leq & e^{6b}\delta\eta \max_{i \in [d]} \Bigg\{\sum_{j:i \in S_j} \frac{\ell_j}{\kappa_j} \Bigg\} \nonumber\\
& = & \frac{e^{6b}\eta\delta}{\tau} D_{\max} \label{eq:hess_posl_4}\\
& = & \frac{e^{6b}\eta\delta}{\beta \tau d} \sum_{j = 1}^n \tau_{j}\ell_j\;, \label{e
| 1,368
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| 1,408
| 1,281
| null | null |
github_plus_top10pct_by_avg
|
0, 150\}$ and $n=10^5$. We report empirical sizes and powers for different distributions. Each experiment is repeated 500 times at the nominal level $\alpha=0.05$.
\[example1\] We consider the linear model: $Y=X^\top\beta + \varepsilon$. Here $\beta$ is a $7\times 1$ vector with all coordinates 0.2 and $X$ comes from the 7-dimensional multivariate normal distribution $N(0, \Sigma)$, where $\Sigma=(\rho_{ij})$ and $\rho_{ij}=0.2^{|i- j|}$. $\varepsilon$ comes from three distributions:
- The normal distribution, $N(0,1)$.
- $t$ distribution, $t$(10).
- Mixed normal distribution, $0.5 N(1, 1) + 0.5 N(-1, 1)$.
Table \[table2\] shows the empirical sizes when we are testing $H_0: \beta_j=0.2$. Table \[table3\] summaries the lengths of confidence intervals by different methods. We can obtain the following conclusions.
1. Regardless of distribution of $\varepsilon$, the empirical size of our proposed method outperformes BLB, SDB, and is slightly better than TB at many cases. Our method is not sensitive to the selection of $K$ since their results are similar.
2. The empirical sizes of BLB and SDB are close to zero. The possible reason is that the lengths of their confidence intervals are very long, especially when $\gamma=0.6$. Compared with BLB and SDB, our method is similar to TB. We also note that in Table \[table3\], the lengths in one row are almost the same. This is due to the fact that all $\beta_j$ are set to be equal.
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
1 K=50 0.044 0.046 0.046 0.060 0.062 0.056 0.064
K=100 0.060 0.036 0.046 0.054 0.048 0.054 0.074
K=150 0.042 0.034 0.038 0.044 0.044 0.056
| 1,369
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| 1,842
| 1,230
| null | null |
github_plus_top10pct_by_avg
|
ies);
var allReporterFiles = files1.Union(files2);
var sw = Stopwatch.StartNew();
var fileCount = allReporterFiles.Count(); // <--- takes ~3.5 seconds
sw.Stop();
Trace.WriteLine($"KillChromeSoftwareReporterTool completed in: {sw.Elapsed.TotalMilliseconds}ms or {sw.Elapsed.TotalSeconds}sec");
A:
Is this a framework bug or what?
It's an issue with your understanding of LINQ's deferred execution, I suspect.
allReporterFiles is just an IEnumerable<string>. Calling Count() means iterating over it - which in turn means the Union code iterating over files1 and files2. I suspect you have an awful lot of files.
The way to tell that is to measure how long it takes to iterate over files1 and files2 separately. One easy way to do that is to call ToList(). For example:
// The use of ToList forces the result to be materialized, rather than using deferred
// execution.
var stopwatch = Stopwatch.StartNew();
var files1 = Directory
.EnumerateFiles(dirSwReporter, swReporterFileName, SearchOption.AllDirectories)
.ToList();
var files1Time = stopwatch.Elapsed;
stopwatch.Restart();
var files2 = Directory
.EnumerateFiles(dirSwReporter2, swReporterFileName, SearchOption.AllDirectories)
.ToList();
var files2Time = stopwatch.Elapsed;
Then log files1Time and files2Time. Now that the content is in two lists, counting the Union won't involve any IO. It will still need to basically create a HashSet<string> as it goes, in order to avoid returning the same value more than once, but it will be much, much quicker.
This approach won't be any faster overall - and will use more memory - but it'll make it obvious whether most of the time is in searching in dirSwReporter or dirSwReporter2, which may be enough to help you optimize.
Q:
Long Date (UNIX Date) issues
The problem is as follows :
Quick details of the app : Sorting of data (ascending) according to the date.
The UNIX date / long date from the web service in form of JSON (is of 13 digits). When the long date is parsed, I get an invalid value of the date.
| 1,370
| 5,071
| 158
| 1,371
| 941
| 0.797176
|
github_plus_top10pct_by_avg
|
fig2\]. The collapse of the data from different runs, on to seemingly universal curves, is remarkable for all the cases except for $Ri_g > 0.2$. We would like to mention that similar scaling behavior was not found if other normalization factors (e.g., $h$) are used.
Both normalized $L_{OZ}$ and $L_b$ decrease monotonically with $Ri_g$; however, the slopes are quite different. The length scales $L_C$ and $L_H$ barely exhibit any sensitivity to $Ri_g$ (except for $Ri_g > 0.1$). Even for weakly-stable condition, these length scales are less than 25% of $\mathcal{L}$.
Based on the expressions of the OLSs (i.e., Eqs. \[OLS\]) and the definition of the gradient Richardson number, we can write:
$$\frac{L_C}{L_{OZ}} = \left(\frac{N}{S}\right)^{3/2} = Ri_g^{3/4},$$
$$\frac{L_H}{L_{b}} = \left(\frac{N}{S}\right) = Ri_g^{1/2}.$$
Thus, for $Ri_g < 1$, one expects $L_C < L_{OZ}$ and $L_H < L_b$. Such relationships are fully supported by Fig. \[fig2\]. In comparison to the buoyancy effects, the shear effects are felt at smaller length scales for the entire stability range considered in the present study.
{width="49.00000%"} {width="49.00000%"}
Owing to their similar scaling behaviors, $L_b/\mathcal{L}$ against $L_{OZ}/\mathcal{L}$ are plotted in Fig. \[fig3\] (left panel). Once again, all the simulated data collapse nicely in a quasi-universal (nonlinear) curve. Since in a double-logarithmic representation (not shown) this curve is linear, we can write: $$\frac{L_b}{\mathcal{L}} \equiv \left(\frac{L_{OZ}}{\mathcal{L}}\right)^m,
\label{Lb_vs_LOZ}$$ where, $m$ is an unknown power-law exponent. By using $L_b \equiv \overline{e}^{1/2}/N$ and the definitions of $L_{OZ}$ and $\mathcal{L}$, we arrive at: $$\frac{\overline{e}^{1/2}}{N} = \left(\frac{\overline{\varepsilon}}{N^3} \right)^{m/2} \left(\frac{\overline{e}^{3/2}}{\overline{\varepsilon}} \right)^{1-m}.$$ Further simplification leads to: $\overline{\varepsilon} = \overline{e} N$; please note that the exponent $m$ cancels out
| 1,371
| 4,112
| 2,043
| 1,346
| 830
| 0.799171
|
github_plus_top10pct_by_avg
|
e, $$\begin{aligned}
\circled{1} + \circled{2} + \circled{4} + \circled{5}
\leq& \lrp{\LR + L_N^2} q'(g(z_t)) g(z_t) + 2\cm^2 q''(g(z_t)) + \frac{L_N^2\|y_t-y_0\|_2^2}{2\epsilon} + 2 \lrp{L + \LN^2}\epsilon\\
\leq& - \frac{2\cm^2\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{32\Rq^2} q(g(z_t)) + \frac{L_N^2\|y_t-y_0\|_2^2}{2\epsilon} + 2 \lrp{L + \LN^2}\epsilon\\
\leq& - \lambda q(g(z_t)) + \frac{L_N^2\|y_t-y_0\|_2^2}{2\epsilon} + 2 (L+L_N^2) \epsilon\\
=& - \lambda f(z_t) + \frac{L_N^2\|y_t-y_0\|_2^2}{2\epsilon} + 2 (L+L_N^2) \epsilon + L \lrn{y_t - y_0}_2
\end{aligned}$$
Where the last inequality follows from Lemma \[l:qproperties\].\[f:contraction\] and the definition of $\lambda$ in .
**Case 3: $\|z_t\|_2 \geq \Rq$**\
In this case, $\gamma_t=0$. Similar to case 2, $$\begin{aligned}
\nabla f(z_t) = q'(g(z_t)) \frac{z_t}{\|z_t\|_2}
\end{aligned}$$
Thus by Assumption \[ass:U\_properties\].3, $$\begin{aligned}
\circled{1}
=& \lin{q'(g(z_t)) \frac{z_t}{\|z_t\|_2}, - \nabla_t}\\
\leq& -m q'(g(z_t)) \lrn{z_t}_2
\end{aligned}$$ Where the inequality is by Assumption \[ass:U\_properties\].3.
For identical reasons as in Case 1, $\circled{2}\leq \LR \lrn{y_t-y_0}_2$, and $\circled{4} = 0$. Finally, $$\begin{aligned}
\circled{5}
=& \frac{1}{2}\tr\lrp{\nabla^2 f(z_t) \lrp{N_t+ N(y_t) - N(y_0)}^2}\\
=& \frac{1}{2}\tr\lrp{\lrp{q''(g(z_t)) \frac{z_t z_t^T}{\|z_t\|_2^2} + q'(g(z_t)) \frac{1}{\|z_t\|_2} \lrp{I - \frac{z_tz_t^T}{\|z_t\|_2^2}}} \lrp{N_t+ N(y_t) - N(y_0)}^2}\\
\leq& \frac{1}{2}\tr\lrp{\lrp{ q'(g(z_t)) \frac{1}{\|z_t\|_2} \lrp{I - \frac{z_tz_t^T}{\|z_t\|_2^2}}} \lrp{N_t+ N(y_t) - N(y_0)}^2}\\
\leq& \frac{q'(g(z_t))}{\|z_t\|_2}\cdot \lrp{\tr\lrp{N_t^2} + \tr\lrp{\lrp{N(y_t) - N(y_0)}^2}}
\end{aligned}$$
Where the first inequality is because $q''\leq 0$ from item 4 of Lemma \[l:qproperties\], the second inequality is by Young’s inequality. (These steps are identical to Case 2). Con
| 1,372
| 1,762
| 1,503
| 1,298
| null | null |
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|
e are extra factors of $\eta$ coming from the initial splitting of the photon into two open string states. When these are taken into account, the constraints from Crab Nebula [@crab] can be satisfied for much larger values of $\eta$. We postpone a more detailed discussion of D3-foam phenomenology for future work.
---
abstract: 'We show that a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute, if and only if homotopy finite limit functors have right adjoints, and if and only if homotopy finite colimit functors have left adjoints. These characterizations generalize to an abstract notion of “stability relative to a class of functors”, which includes in particular pointedness, semiadditivity, and ordinary stability. To prove them, we develop the theory of derivators enriched over monoidal left derivators and weighted homotopy limits and colimits therein.'
author:
- Moritz Groth and Michael Shulman
bibliography:
- 'stability.bib'
title: Generalized stability for abstract homotopy theories
---
Introduction {#sec:intro}
============
In classical algebraic topology we have the following pair of adjunctions relating topological spaces $\mathrm{Top}$ to pointed spaces $\mathrm{Top}_\ast$ and spectra $\mathrm{Sp}$: $$(\Sigma^\infty_+,\Omega^\infty_-)\colon\mathrm{Top}\rightleftarrows\mathrm{Top}_\ast\rightleftarrows\mathrm{Sp}$$ Abstractly, each of these two steps universally improves certain *exactness properties* of a homotopy theory. In the first step we pass in a universal way from a general homotopy theory to a *pointed* homotopy theory, i.e., a homotopy theory admitting a zero object. The second step realizes the universal passage from a pointed homotopy theory to a *stable* homotopy theory, i.e., to a pointed homotopy theory in which homotopy pushouts and homotopy pullbacks coincide. With this in mind, our first goal in this paper is to collect additional answers to the following question.
**Question:** Which exactness properties of the homotopy theory of spectra
| 1,373
| 337
| 1,712
| 1,157
| null | null |
github_plus_top10pct_by_avg
|
ntical within machine precision to the potential from the full $2\pi$ domain. This confirms that our Poisson solver in a restricted $\phi\in[0,2\pi/P]$ domain correctly deals with mass distributions under $P$-fold azimuthal symmetry.
[^1]: <http://www.fftw.org/>
[^2]: <https://www.sandia.gov/~sjplimp/docs/fft/README.html>
[^3]: The radial eigenfunction ${\cal R}^l_i$ can instead be obtained numerically by solving the eigenvalue problem, and the resulting eigenfunction may be called the *discrete Bessel function*. Since it satisfies the exact discrete orthogonality relation, it may also serve as discrete kernel for the discrete Hankel transform [@john87; @baddour15].
[^4]: <https://researchcomputing.princeton.edu/systems-and-services/available-systems/tiger>
[^5]: The weak scaling test shown in Figure \[fig:scaling\] hints some performance degradation from $N_{\rm core}=1$ to $64$ relative to $\langle t_{\rm wall}\rangle \propto \ln N_{\rm core}$ expected for the theoretical FFT. Our parallel FFT utilizes a “transpose algorithm” known to be efficient when a data size for communication is larger than the critical size that depends on the latency/bandwidth of the interconnecting network device and the network topology [e.g., @foster97]. An alternative “binary exchange algorithm” may work efficiently for a small data size [e.g., @muller19].
[^6]: Although ${\cal G}^m_{k-k'}({\rm inn\to inn})$ and ${\cal G}^m_{k-k'}({\rm inn\to out})$ can be stored in 2D arrays, we store them as 3D arrays for simple coding.
[^7]: We use Neumann condition for consistency.
---
abstract: 'We propose new signals for the direct detection of ultralight dark matter such as the axion. Axion or axion like particle (ALP) dark matter may be thought of as a background, classical field. We consider couplings for this field which give rise to observable effects including a nuclear electric dipole moment, and axial nucleon and electron moments. These moments oscillate rapidly with frequencies accessible in the laboratory, $\sim$ kHz
| 1,374
| 277
| 657
| 1,525
| 3,199
| 0.77409
|
github_plus_top10pct_by_avg
|
+10450 Intron 26 \-
\* The position of the polymorphisms was calculated over the genomic DNA sequence taking the position of the start codon as a reference.
viruses-11-00706-t003_Table 3
######
Functional predictions in the missense mutations of the *HDAC6* gene.
-----------------------------------------------------------------------------------------------------------
Polymorphism Protein Domain\ SIFT Prediction\ Polyphen-2 Prediction\
(EMBL-EBI) (Score) (Score)
-------------- ------------------------- ------------------------ -----------------------------------------
Arg12Lys Not tolerant (0.00) \* Unknown, not enough reference sequences
Pro360Leu Hist_deacetyl (PF00850) Not tolerant (0.02) Probably damaging (1.000)
Pro503His Hist_deacetyl (PF00850) Not tolerant (0.00) Probably damaging (1.000)
-----------------------------------------------------------------------------------------------------------
\* Detected with low confidence, as there were few proteins in the database that included this residue.
viruses-11-00706-t004_Table 4
######
Allelic and genotypic frequencies of the eight markers studied in the sows used in this study.
Marker Gene MAF (Allele) AA AB BB \*
-------------- --------- ------------------ ---- ----- -------
rs80800372 *GBP1* 0.19 (G) 15 64 171
rs340943904 *GBP5* 0.25 (T) 22 82 144
c.3534C\>T *CD163* 0.33 (C) 41 70 119
rs1107556229 *CD163* 0.29 (A) 32 70 127
−547ins+275 *MX1* 0.25 (insertion) 18 50 170
rs325981825 *HDAC6* 0.37 (A) 35 103 90
g.2360C\>T *HDAC6* 0.38 (T) 32 104 32
\* A and B refer to minor and alternative alleles, respectively.
viruses-11-00706-t005_Table 5
######
Odds ratios of abortion
| 1,375
| 4,039
| 1,534
| 851
| null | null |
github_plus_top10pct_by_avg
|
$\mathrm{Sp}(B_i/Y_i, h_i)$).}$$ This equation will be proved in Appendix \[App:AppendixA\]. Thus $\mathrm{Im~}\varphi$ contains the identity component of $\prod_{i:even} \mathrm{O}(B_i/Z_i, \bar{q}_i)_{\mathrm{red}}\times \prod_{i:odd} \mathrm{Sp}(B_i/Y_i, h_i)$. Here $\mathrm{Ker~}\varphi$ denotes the kernel of $\varphi$ and $\mathrm{Im~}\varphi$ denotes the image of $\varphi.$ Note that it is well known that the image of a homomorphism of algebraic groups is a closed subgroup.
Recall from Section \[m\] that a matrix form of an element of $\tilde{G}(R)$ for a $\kappa$-algebra $R$ is written $(m_{i,j}, s_i \cdots w_i)$ with the formal matrix interpretation $$m= \begin{pmatrix} \pi^{max\{0,j-i\}}m_{i,j} \end{pmatrix} \textit{ together with } z_i^{\ast}, m_{i,i}^{\ast}, m_{i,i}^{\ast\ast}.$$ We represent the given hermitian form $h$ by a hermitian matrix $\begin{pmatrix} \pi^{i}\cdot h_i\end{pmatrix}$ with $\pi^{i}\cdot h_i$ for the $(i,i)$-block and $0$ for the remaining blocks, as in Remark \[r33\].(1).
Let $\mathcal{H}$ be the set of even integers $i$ such that $\mathrm{O}(B_i/Z_i, \bar{q_i})_{\mathrm{red}}$ is disconnected. Notice that $\mathrm{O}(B_i/Z_i, \bar{q_i})_{\mathrm{red}}$ is disconnected exactly when $L_i$ with $i$ even is *free of type II*. We first prove that $ \varphi_i$, for such an even integer $i$, is surjective. We prove this by a series of reductions, after which we will be able to assume that $L$ is of rank two.
For such an even integer $i$ with a *free of type II* lattice $L_i$, we define the closed scheme $H_i$ of $\tilde{G}$ by the equations $m_{j,k}=0$ if $ j\neq k$, and $m_{j,j}=\mathrm{id}$ if $j \neq i$. An element of $H_i(R)$ for a $\kappa$-algebra $R$ can be represented by a matrix of the form $$\begin{pmatrix} id&0& & \ldots& & &0\\ 0&\ddots&& & & &\\ & &id& & & & \\ \vdots & & &m_{i,i} & & &\vdots
\\ & & & & id & & \\ & & & & &\ddots &0 \\ 0& & &\ldots & &0 &id \end{pmatrix}
\textit{ with $z_j^{\ast}=0, m_{j,j}^{\ast}=0, m_{j,j}^{\ast\ast}=0.$}$$ Obviously, $H_i$ has a grou
| 1,376
| 476
| 866
| 1,374
| null | null |
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|
Omega-expand\]) and $\hat{S}$-$\Omega$ relation in (\[hatS-Omega\]) as $$\begin{aligned}
\hat{S}_{i i}^{(2)} [2] &=&
\sum_{K} \left[
(ix) e^{- i h_{i} x} + \frac{e^{- i \Delta_{K} x} - e^{- i h_{i} x} }{ ( \Delta_{K} - h_{i} ) }
\right]
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\frac{ 1 }{ \Delta_{K} - h_{i} }
\left\{ W ^{\dagger} A (UX) \right\}_{K i},
\nonumber \\
\hat{S}_{i j}^{(2)} \vert_{i \neq j} [2] &=&
- \sum_{K} \left[
\frac{e^{- i h_{j} x} - e^{- i h_{i} x} }{ ( h_{j} - h_{i} ) }
- \frac{e^{- i \Delta_{K} x} - e^{- i h_{i} x} }{ ( \Delta_{K} - h_{i} ) }
\right]
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\frac{ 1 }{ \Delta_{K} - h_{j} }
\left\{ W ^{\dagger} A (UX) \right\}_{K j},
\nonumber \\
\hat{S}_{i J}^{(3)} [2] &=&
- \left[
(ix) e^{- i \Delta_{J} x} +
\frac{ e^{- i \Delta_{J} x} - e^{- i h_{i} x} }{ ( \Delta_{J} - h_{i} ) }
\right]
\left\{ (UX)^{\dagger} A W \right\}_{i J}
\frac{ 1 }{ \Delta_{J} - h_{i} }
\left\{ W^{\dagger} A W \right\}_{J J}
\nonumber \\
&+&
\sum_{K \neq J}
\left[
\frac{e^{- i \Delta_{J} x} - e^{- i h_{i} x} }{ ( \Delta_{J} - h_{i} ) } -
\frac{e^{- i \Delta_{K} x} - e^{- i h_{i} x} }{ ( \Delta_{K} - h_{i} ) }
\right]
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\frac{ 1 }{ \Delta_{J} - \Delta_{K} }
\left\{ W^{\dagger} A W \right\}_{K J},
\nonumber \\
\hat{S}_{J i}^{(3)} [2] &=&
- \left[
(ix) e^{- i \Delta_{J} x} +
\frac{ e^{- i \Delta_{J} x} - e^{ - i h_{i} x} }{ ( \Delta_{J} - h_{i} ) }
\right]
\left\{ W^{\dagger} A W \right\}_{J J}
\frac{ 1 }{ \Delta_{J} - h_{i} }
\left\{ W^{\dagger} A (UX) \right\}_{J i}
\nonumber \\
&-&
\sum_{K \neq J}
\left[
\frac{e^{- i h_{i} x} - e^{- i \Delta_{J} x} }{ h_{i} - \Delta_{J} } -
\frac{e^{- i \Delta_{K} x} - e^{- i \Delta_{J} x} }{ ( \Delta_{K} - \Delta_{J} ) }
\right]
\left\{ W^{\dagger} A W \right\}_{J K}
\frac{ 1 }{ \Delta_{K} - h_{i} }
\left\{ W^{\dagger} A (UX) \right\}_{K i},
\nonumber \\
\hat{S}_{I J}^{(2+4)} \vert_{I \neq J} [2] &=&
\sum_{k}
\left[\frac{e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x}
| 1,377
| 886
| 1,754
| 1,416
| null | null |
github_plus_top10pct_by_avg
|
\sigma(h')\bigl) \lhd r(g')\bigl]v(g') =
\bigl(v(g) \lhd (\sigma(h')r(g'))\bigl)v(g')$$
It follows that [(\[eq:c2\])]{} and hence [(\[eq:def4\])]{} holds and we are done.
\(3) Follows from (2) and the [Proposition \[pr:1\]]{}.
Schreier type theorems for bicrossed products
=============================================
[\[se:3\]]{}
In this section we shall prove two Schreier type classification theorems for bicrossed products. Let $H$ and $G$ be two fixed groups.
Let $MP(H,G):= \{(\alpha,\beta) ~|~ (H, G, \alpha, \beta)
{\rm~is~}{\rm~a~}{\rm~matched~}{\rm~pair}\}$. We define $B_{1}(H,G)$ to be the category having as objects the set $MP(H,G)$ and the morphisms defined as follows: $\psi : (\alpha',
\beta') \to (\alpha, \beta)$ is a morphism in $B_{1}(H,G)$ if and only if $\psi: H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G
\rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ is a morphism of groups such that $\psi \circ i_H = i_H$, where $i_H
(h) = (h, 1)$ is the canonical inclusion. Thus a morphism in the category $B_{1}(H,G)$ is a morphism between two bicrossed products of $H$ and $G$ that fix $H$. Considering $\sigma = Id_{H}$ and $G'
= G$ in [Proposition \[pr:1\]]{} we obtain the following:
[\[co:cosch\]]{} Let $(H, G, \alpha, \beta)$ and $(H, G, \alpha', \beta')$ two matched pairs. There exists a one to one correspondence between the set of all morphisms $\psi: H\, {}_{\alpha'}\!\!
\bowtie_{\beta'} \, G \rightarrow H\, {}_{\alpha}\!\!
\bowtie_{\beta} \, G$ in the category $B_{1}(H,G)$ and the set of all pairs $(r, v)$, where $r: G \rightarrow H$, $v: G \rightarrow
G$ are two maps such that: $$\begin{aligned}
(g \rhd' h)r(g \lhd' h) &=& r(g)(v(g) \rhd
h){\label{eq:s1}} \\
v(g \lhd' h) &=& v(g) \lhd h {\label{eq:s2}} \\
r(g_{1}g_{2}) &=& r(g_{1})\bigl(v(g_{1}) \rhd
r(g_{2}) \bigl) {\label{eq:s3}} \\
v(g_{1}g_{2}) &=& \bigl(v(g_{1}) \lhd r(g_{2})\bigl)v(g_{2})
{\label{eq:s4}}\end{aligned}$$ for all $h \in H$, $g$, $g_{1}$, $g_{2} \in G$. Through the above bijection $\psi$ is given by $${\label{eq:d}}
\psi(h
| 1,378
| 996
| 1,239
| 1,474
| 3,827
| 0.76984
|
github_plus_top10pct_by_avg
|
abel{tilde-H0+H1} \end{aligned}$$ Therefore, what we mean by “expansion by unitarity violation effect” is an expansion by the $W$ matrix elements.[^9] We assume, for simplicity, that all the $W$ matrix elements are small and have the same order $\epsilon_{s}$. Then, $3 \times N$ ($N \times 3$) sub-matrix elements in $\tilde{H}_{ \text{matt} }$ are of order $\epsilon_{s}$, while the pure sterile space $N \times N$ sub-matrix elements are of order $\epsilon_{s}^2$. For simplicity, we often use the expression “expanding to order $W^n$” which means to order $\epsilon_{s}^n$ in this paper.
### Hat basis
To formulate perturbation theory with $\tilde{H}_{0}$ and $\tilde{H}_{1}$ given above we transform to a basis in which the un-perturbed part of the Hamiltonian is diagonal, which we call the “hat basis”. Since the $3 \times 3$ sub-matrix ${\bf \Delta_{a} } + U^{\dagger} A U$ in $\tilde{H}_{0}$ is hermitian, it can be diagonalized by the unitary transformation $$\begin{aligned}
X^{\dagger} \left( {\bf \Delta_{a} } + U^{\dagger} A U \right) X =
\left[
\begin{array}{ccc}
h_{1} & 0 & 0 \\
0 & h_{2} & 0 \\
0 & 0 & h_{3} \\
\end{array}
\right] \equiv {\bf h}
\label{H0-diag}\end{aligned}$$ with $X$ being the $3 \times 3$ unitary matrix. Then, $\tilde{H}_{0}$ can be diagonalized by using $$\begin{aligned}
{\bf X} \equiv
\left[
\begin{array}{cc}
X & 0 \\
0 & 1 \\
\end{array}
\right]
\label{bfX-def}\end{aligned}$$ as $$\begin{aligned}
&&
{\bf X}^{\dagger} \tilde{H}_{0} {\bf X}
= \left[
\begin{array}{cc}
X^{\dagger} \left( {\bf \Delta_{a} } + U^{\dagger} A U \right) X & 0 \\
0 & {\bf \Delta_{s} } \\
\end{array}
\right]
= \left[
\begin{array}{cc}
{\bf h} & 0 \\
0 & {\bf \Delta_{s} } \\
\end{array}
\right]
\equiv \hat{H}_{0},
\label{hat-H0}\end{aligned}$$ the zeroth-order Hamiltonian in the hat basis. Since $\hat{H}_{0}$ is diagonal it is easy to compute $e^{ \pm i \hat{H}_{0} x}$: $$\begin{aligned}
e^{ \pm i \hat{H}_{0} x} =
\left[
\begin{array}{cc}
e^{ \pm i {\bf h} x } & 0 \\
0 & e^{ \pm i {\bf \Delta_{s
| 1,379
| 1,011
| 1,527
| 1,329
| null | null |
github_plus_top10pct_by_avg
|
monin], ideally one would like to solve the $N$-electron case, but the single particle problem is generally an important first step, and while the $N$ electron system on flat and spherical surfaces has been studied [@lorke2; @bulaev; @goker; @bellucci; @tempere; @ivanov], the torus presents its own difficulties. In an effort to partially address this issue, the evaluation of Coulombic matrix elements on $T^2$ is also discussed here.
This paper is organized as follows: in section 2 the Schrodinger equation for an electron on a toroidal surface in the presence of a static magnetic field is derived. In section 3 a brief exposition on the basis set employed to generate observables is presented. Section 4 gives results. Section 5 develops the scheme by which this work can be extended to the two electron problem on $T^2$, and section 6 is reserved for conclusions.
Formalism
=========
The geometry of a toroidal surface of major radius $R$ and minor radius $a$ may be parameterized by $$\mathbf{r} (\theta,\phi)=W (\theta){\bm {\rho}} +a\ {\rm sin}
\theta{\bm {k}}$$ with $$W = R + a \ {\rm cos} \theta,$$ $${ \bm \rho} = \rm cos\phi {\mathbf i} + sin \phi {\mathbf j}.$$ The differential of Eq.(1) $$d \mathbf{r}= a d\theta \ {\bm \theta}+W d\phi{\bm \phi}$$ with ${\bm \theta} =-\rm sin \theta {\bm \rho}+\rm cos \theta
\mathbf{k}$ yields for the metric elements $g_{ij}$ on $T^2$ $$g_{\theta\theta}=a^2$$ $$g_{\phi\phi}=W^2.$$ The integration measure and surface gradient that follow from Eqs. (5) and (6) become $${\sqrt g}dq^1dq^2 \rightarrow a W d\theta d\phi$$ and $$\nabla = {\bm \theta} {1 \over a} {\partial \over \partial
\theta}+ {\bm \phi} {1 \over W} {\partial \over \partial \phi}.$$ The Schrodinger equation with the minimal prescription for inclusion of a vector potential $\mathbf A$ is $$H = {1 \over {2m}}\bigg ( {\hbar
\over i} \nabla + q {\mathbf A} \bigg) ^2\Psi = E\Psi.$$ The magnetic field under consideration will take the form $${\mathbf B} = B_1{\mathbf i} + B_0{\mathbf k},$$ which by symmetry comprises
| 1,380
| 4,453
| 1,154
| 1,251
| null | null |
github_plus_top10pct_by_avg
|
the superscripts indicates generic formulas that are valid for all three meson systems.
In order to understand better the anatomy of the $\Delta I=1/2$ rule we use again the form $$\begin{aligned}
\frac{A_0}{A_2} &= B + C e^{i\delta}\,, \label{eq:DeltaI12-generic}\end{aligned}$$ analogously to Eq. (\[eq:defC\]) in Sec. \[sec:deltau0rule\] for the $\Delta U=0$ rule. Here, $B$ is again the contribution in the limit of no QCD, and $C e^{i\delta}$ contains the corrections to that limit. Now, as discussed in Refs. [@Buras:1988ky; @Buras:2014maa], in the limit of no strong interactions only the $Q_2$ operator contributes in Eq. (\[eq:DeltaI12-generic\]). Note that the operator $Q_1$ is only generated from QCD corrections. When we switch off QCD, the amplitude into neutral pions vanishes and we have for $K,D,B\rightarrow \pi\pi$ equally [@Buras:1988ky; @Buras:2014maa] $$\begin{aligned}
B &= \sqrt{2}\,. \label{eq:kaon-deltaI12-rule-no-qcd}\end{aligned}$$ This corresponds to the limit $\tilde{p}_0 = 1$ that we considered in Sec. \[sec:deltau0rule\] for the $\Delta U=0$ rule. The exact numerical value in Eq. (\[eq:kaon-deltaI12-rule-no-qcd\]) of course depends on the convention used for the normalization of $A_{0,2}$ in the isospin decomposition Eq. (\[eq:kaondata\]), where we use the one present in the literature.
For the isospin decomposition of $D^+\rightarrow \pi^+\pi^0$, $D^0\rightarrow \pi^+\pi^-$ and $D^0\rightarrow \pi^0\pi^0$, we simply combine the fit of Ref. [@Franco:2012ck] to get $$\begin{aligned}
\left|\frac{A_0^D}{A_2^D}\right| &= 2.47\pm 0.07\,, \qquad
\delta_0^D - \delta_2^D = (\pm 93 \pm 3)^{\circ}\,. \label{eq:charm-deltaI12-rule}\end{aligned}$$ Reproducing the $\Delta I=1/2$ rule for charm Eq. (\[eq:charm-deltaI12-rule\]) is an optimal future testing ground for emerging new interesting non-perturbative methods [@Khodjamirian:2017zdu]. Very promising steps on a conceptual level are also taken by lattice QCD [@Hansen:2012tf].
In $K$ and $D$ decays the contributions of penguin operators t
| 1,381
| 949
| 2,054
| 1,407
| 1,307
| 0.791335
|
github_plus_top10pct_by_avg
|
\mathcal{C}(N)$ be a superinjective simplicial map. If $a$ is a 1-sided simple closed curve on $N$ whose complement is nonorientable, then $\lambda(a)$ is the isotopy class of a 1-sided simple closed curve whose complement is nonorientable.
Let $a$ be a 1-sided simple closed curve on $N$ whose complement is nonorientable. Let $a' \in \lambda([a])$.
[**Case 1:**]{} Suppose $a'$ is a 1-sided simple closed curve whose complement is orientable. This case happens only if the genus of $N$ is odd. So, suppose $g =2r +1$, where $r \geq 1$. In this case $a$ can be put into a maximal simplex $\Delta$ of dimension $4r + n -2$. Since $\lambda$ is injective by Lemma \[inj\], $\lambda(\Delta)$ is a simplex of dimension $4r + n-2$. By using Euler characteristic arguments we see that the complement of $a'$ has genus $r$ and $n+1$ boundary components. So, in the complement of $a'$ there can be at most a $3r + n - 3$ dimensional simplex, hence there can be at most a $3r + n - 2$ dimensional simplex containing $a'$ on $N$. Since dim $\lambda(\Delta) = 4r + n - 2 > 3r + n - 2$ as $r \geq 1$, we get a contradiction.
[**Case 2:**]{} Suppose $a'$ is a 2-sided nonseparating simple closed curve on $N$.
[**(i)**]{} Suppose the genus of $N$ is even, $g=2r$ for some $r \geq 1$. We can put $a$ into a maximal simplex $\Delta$ of dim $4r+n-4$. Then $\lambda(\Delta)$ has dimension $4r+n-4$.
\(a) Suppose now that the complement of $a'$ is nonorientable. This case happens when $g \geq 4$. The complement of $a'$ has genus $2r-2$ and it has $n+2$ boundary components. By using Lemma \[dim\] we get the following: In the complement of $a'$ there can be at most a $4(r-1) + (n+ 2) - 4 = 4r + n -6$ dimensional simplex. Hence, there can be at most a $4r + n - 5$ dimensional simplex containing $a'$ on $N$. Since dim $ \lambda(\Delta) = 4r + n -4 > 4r + n - 5$, we get a contradiction.
\(b) Suppose that the complement of $a'$ is orientable. The genus of the complement is $r-1$ and the number of boundary component is $n + 2$. By using Lemma \[dim\] we
| 1,382
| 970
| 2,407
| 1,347
| null | null |
github_plus_top10pct_by_avg
|
$$ Theorem \[unif dist of rho\] states that for any fixed subinterval $[\alpha,\beta]\in (-1/2,1/2]$, $$\frac 1{\#L_{prim}(T)} \{v\in L_{prim}(T): \alpha<\rho(v)<\beta \}
\to \beta-\alpha$$ as $T\to \infty$.
Equidistribution of real parts of orbits
----------------------------------------
We will reduce Theorem \[unif dist of rho\] by geometric arguments to a result of Anton Good [@Good:1983a] on uniform distribution of the orbits of a point in the upper half-plane under the action of a Fuchsian group.
Let $\G$ be discrete, co-finite, non-cocompact subgroup of $\slr$. The group $\slr$ acts on the upper half-plane $\H=\{z\in \C:
{\operatorname{Im}}(z)>0\}$ by linear fractional transformations. We may assume, possibly after conjugation in $\slr$, that $\infty$ is a cusp and that the stabilizer $\G_{\!\infty}$ of $\infty$ in $\G$ is generated by $$\pm {\left(\begin{array}{cc}
1 & 1 \\
0 & 1
\end{array}\right) }$$ which as linear fractional transformation gives the unit translation $z\mapsto z+1$. (If $-I\notin \G$ there should be no $\pm$ in front of the matrix). The group $\G=\sl$ is an example of such a group. We note that the imaginary part of $\g(z)$ is fixed on the orbit $\G_{\!\infty}\g z$, and that the real part modulo one is also fixed on this orbit. Good’s theorem is
\[equidistribution\] Let $\G$ be as above and let $z\in\H$. Then ${\operatorname{Re}}(\G z)$ is uniformly distributed modulo one as ${\operatorname{Im}}(\g z)\to 0$.
More precisely, let $$(\GinfmodG)_{\varepsilon,z}=\{\g\in\GinfmodG : {\operatorname{Im}}{\g z}>\varepsilon\}\;.$$ Then for every continuous function $f\in C(\R\slash \Z)$, as $\varepsilon\to 0$, $$\frac 1{ \#(\GinfmodG)_{\varepsilon,z}}
\sum_{\g\in(\GinfmodG)_{\varepsilon,z}}f({\operatorname{Re}}{\g z})
\to\int_{\R\slash\Z}f(t)dt \;.$$
Though the writing in [@Good:1983a] is not easy to penetrate, the results deserve to be more widely known. We sketch a proof of Theorem \[equidistribution\]
| 1,383
| 2,466
| 1,700
| 1,289
| 3,476
| 0.772076
|
github_plus_top10pct_by_avg
|
j} U^{*}_{\beta j} \right|^2 -
2 \sum_{j \neq k}
\mbox{Re}
\left( U_{\alpha j} U_{\beta j}^* U_{\alpha k}^* U_{\beta k} \right)
\sin^2 \frac{ ( \Delta_{k} - \Delta_{j} ) x }{ 2 }
\nonumber\\
&-&
\sum_{j \neq k} \mbox{Im}
\left( U_{\alpha j} U_{\beta j}^* U_{\alpha k}^* U_{\beta k} \right)
\sin ( \Delta_{k} - \Delta_{j} ) x,
\label{P-beta-alpha-ave-vac}\end{aligned}$$ where $$\begin{aligned}
\mathcal{C}_{\alpha \beta} \equiv
\sum_{J=4}^{3+N}
\vert W_{\alpha J} \vert^2 \vert W_{\beta J} \vert^2,
\label{Cab} \end{aligned}$$ in the appearance ($\alpha \neq \beta$) as well as in the disappearance channels ($\alpha = \beta$).[^5] The $(3 \times N)$ $W$ matrix elements bridge between active and sterile state space, see eq. (\[U-parametrize\]) for definition.
The characteristic features of (\[P-beta-alpha-ave-vac\]) are:
1. The non-unitary matrix $U$ replaces the standard unitary three-flavor mixing matrix often parametrized with Particle Data Group convention $U_{\text{\tiny PDG}}$ [@Olive:2016xmw].
2. Probability leakage term $\mathcal{C}_{\alpha \beta}$ appears reflecting the nature of low-energy unitarity violation in which the probability can flow out into the sterile state space from active neutrino space.
3. Due to non-unitarity, $\delta_{\alpha \beta}$ term in the unitary case is modified to $\left| \sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} \right|^2$.
The points 2 and 3 above are important ones and the clarifying remarks about them are in order:
- Presence or absence of the probability leakage term $\mathcal{C}_{\alpha \beta}$ distinguishes between low-energy and high-energy unitarity violation [@Fong:2016yyh]. Unfortunately, $\mathcal{C}_{\alpha \beta}$ may be small because it is of fourth order in $W$.
- Difference in normalization factor, the second term in (\[P-beta-alpha-ave-vac\]), between unitary and non-unitary cases is of order $\sim W^4$ ($\sim W^2$) in the appearance (disappearance) channels.
To understand the latter point, we notice that unitarity in the $(3+N)$ space uni
| 1,384
| 4,175
| 884
| 1,090
| 3,040
| 0.775284
|
github_plus_top10pct_by_avg
|
n see the least sensitive parameter appears to have no effect on the production of a BEC. This parameter corresponds to an intentionally added 7th parameter of the system that controls nothing in the experiment. Fig. 4(a) shows the learner successfully identified this, even with such a small data set. After making this observation we can then reconsider the design of the optimization process and eliminate this parameter from the experiment.
{width="\columnwidth"}
In Fig. 3 we plot the machine learner optimization run with $P=16$ but now with only 6 parameters. We can see the learner converges much more rapidly than the 7 parameter case, and even produces a higher quality BEC. As the learner no longer takes extra runs to determine the importance of the useless 7th parameter, it achieves BEC quite rapidly. Lastly, to give us some understanding of the complexity of the landscape, we plot a 2D cross section of the landscape against the two most sensitive parameters in Fig. 4(b) generated from the 6 parameter machine learning run. We can see there is a very sharp transition to BEC, as it exists in a very deep valley of the landscape.
We have demonstrated that MLOO can discover evaporation ramps that produce high quality BECs with far fewer experiments than OO with Nelder-Mead. Rapid optimization of ultra-cold-atom experiments is not just a useful tool to overcome technical difficulties, but will be vital in the application of BECs in proposed space-based scientific investigations [@zoest_bose-einstein_2010; @arimondo_atom_2009].
| 1,385
| 1,188
| 3,330
| 1,552
| null | null |
github_plus_top10pct_by_avg
|
d* *P*
------------ -------- ----- ------------------- ------ ------ ------
Weekly Female 60 7494.09 (3268.83) 0.23 0.04 0.07
Male 57 7358.60 (3147.58)
Weekdays Female 60 8301.11 (3721.95) 0.02 0.00 0.05
Male 57 8290.00 (3468.30)
Weekend Female 57 5551.72 (3409.37) 1.21 0.23 0.32
Male 48 4753.99 (3318.72)
M, mean; SD, standard deviation; T, T-values (Independent Samples t-test); d, Cohen's d; P, observed power.
Regarding the number of activities carried out by each sex (Mann--Whitney *U* test: *Z* = −4.36, *p* \< 0.001), however, differences in terms of sex were more pronounced, with female subjects reporting an average of 3.29 (*SD* = 1.88) activities per week, compared to 4.73 for their male counterparts (*SD* = 1.75).
A chi-squared test was used to assess the relationship between sex and level of autonomy in PA. In view of the low number of respondents in the "organized PA" group, these subjects were grouped together with the "mixed" participants to permit a comparison between subjects whose PA habits include unstructured activities only and subjects whose PA habits include some organized activities. The results reveal the influence of sex on autonomy of PA (chi-squared = 5.94, *p* \< 0.05, *w* = 0.26), with girls tending to be more autonomous in their practice and boys more likely to opt for organized activities only or a combination of both types.
The study thus reveals a distinct pattern of PA practice among girls, consisting of fewer and less varied (mostly unstructured) types of activities, yet similar overall levels of PA participation across both sexes.
Basic Psychological Needs and Autonomy in PA {#S3.SS2}
--------------------------------------------
As [Table 3](#T3){ref-type="table"} shows, satisfaction of basic psychological needs through exercise (organized or unstructured) is consistently high, while results for phys
| 1,386
| 812
| 3,035
| 1,524
| null | null |
github_plus_top10pct_by_avg
|
bb Z}^u$ and $\mathcal B \colon \Lambda^*_G \oplus \Lambda^*_H \rightarrow {\mathbb Z}^u$ with the following property: For every irreducible representation $V_{G,\lambda}$ of $G$ and $V_{H,\mu}$ of $H$, the multiplicity $m^\lambda_\mu$ of the latter in the former is given by $$m^\lambda_\mu =
\sum_{\gamma \in \Gamma_H} c_\gamma \, \# \{ x \in {\mathbb Z}^s_{\geq 0} \oplus {\mathbb Z}^{s'} : \mathcal A x = \mathcal B {\begin{pmatrix}\lambda \\ \mu + \gamma\end{pmatrix}} \},$$ where the (finite) set $\Gamma_H$ and the coefficients $(c_\gamma)$ are defined by $\prod_{\alpha \in R_{H,+}} \left( 1 - e^{-\alpha} \right) = \sum_{\gamma \in \Gamma_H} c_\gamma e^{-\gamma}$ and $c_\gamma \neq 0$. In fact, we can choose $s = O(r_G^2)$, $s' \leq r_G$ and $u = O(r_G^2) + r_H$.
By definition and , we have $m^\lambda_\mu = m_{H,V_{G,\lambda}}(\mu) = \sum_{\gamma \in \Gamma_H} c_\gamma \, m_{T_H,V_{G,\lambda}}(\mu + \gamma)$. In view of , the multiplicity of a $T_H$-weight $\delta \in \Lambda^*_H$ in the irreducible $G$-representation $V_{G,\lambda}$ is given by $$m_{T_H,V_{G,\lambda}}(\delta) ~=~
\sum_{\mathclap{\substack{\beta \in \Lambda^*_G\\ F^*(\beta) = \delta}}} m_{T_G,V_{G,\lambda}}(\beta).$$ As in , let us now decompose the Lie-algebra $\mathfrak g = [\mathfrak g, \mathfrak g] \oplus \mathfrak z$. Denote the Lie group corresponding to $[\mathfrak g,\mathfrak g]$ by $G_{\operatorname{ss}}$ and choose a maximal torus $T_{G_{\operatorname{ss}}}$ which is contained in $T$. Using , $$\begin{aligned}
\sum_{\mathclap{\substack{\beta \in \Lambda^*_G\\ F^*(\beta) = \delta}}} m_{T_G,V_{G,\lambda}}(\beta)
~~=~~ \sum_{\mathclap{\substack{\beta_{\operatorname{ss}} \in \Lambda^*_{G_{\operatorname{ss}}}\\ C_{\operatorname{ss}} \beta_{\operatorname{ss}} + C_z \lambda_z = \delta}}} m_{T_{G_{\operatorname{ss}}},V_{G_{\operatorname{ss}},\lambda_{\operatorname{ss}}}}(\beta_{\operatorname{ss}}),
\end{aligned}$$ where we have decomposed $F^*$ as a sum of two homomorphisms $C_{\operatorname{ss}} \colon \Lambda^*_{G_{\op
| 1,387
| 1,035
| 1,000
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uality is by item 1 of Assumption \[ass:U\_properties\], the fourth inequality uses Lemma \[l:divergence\_vt\].
Substituting the above two equation blocks into , and applying recursively for $k=0...n-1$ gives $$\begin{aligned}
& \E{\lrn{v_{T} - w_{T}}_2^2} \\
=& \E{\lrn{v_{n\delta} - w_{n\delta}}_2^2} \\
\leq& e^{1+2n\delta L} \lrp{2n^2\delta^2 L^2 \lrp{T^2L^2 \E{\lrn{v_0}_2^2} + T\beta^2} + n\delta L_\xi^2 \lrp{16 \lrp{T^2 L^2 \E{\lrn{w_0}_2^2} + T \beta^2}}}\\
\leq& 8 \lrp{2T^2 L^2 \lrp{T^2L^2 \E{\lrn{v_0}_2^2} + T\beta^2} + T L_\xi^2 \lrp{16 \lrp{T^2 L^2 \E{\lrn{w_0}_2^2} + T \beta^2}}}
\end{aligned}$$
the last inequality is by noting that $T = n\delta \leq \frac{1}{4L}$.
[Regularity of $M$ and $N$]{}\[ss:mnregularity\]
[\[l:M\_is\_regular\]]{} $$\begin{aligned}
&1.\ \tr\lrp{M(x)^2} \leq \beta^2\\
&2.\ \tr\lrp{(M(x)^2 - M(y)^2)^2} \leq 16 \beta^2 L_\xi^2 \|x-y\|_2^2\\
&3.\ \tr\lrp{(M(x)^2 - M(y)^2)^2} \leq 32\beta^3 L_\xi \|x-y\|_2
\end{aligned}$$
In this proof, we will use the fact that $\xi(\cdot,\eta)$ is $L_\xi$-Lipschitz from Assumption \[ass:xi\_properties\].
The first property is easy to see: $$\begin{aligned}
& \tr\lrp{M(x)^2}\\
=& \tr\lrp{\Ep{\eta}{\xi(x,\eta) \xi(x,\eta)^T}}\\
=& \Ep{\eta}{\tr\lrp{{\xi(x,\eta) \xi(x,\eta)^T}}}\\
=& \Ep{\eta}{\lrn{\xi(x,\eta)}_2^2}\\
\leq& \beta^2
\end{aligned}$$
We now prove the second and third claims. Consider a fixed $x$ and fixed $y$, let $u_{\eta} := \xi(x,\eta)$, $v_{\eta} := \xi(y,\eta)$. Then
$$\begin{aligned}
& \tr\lrp{\lrp{M(x)^2 - M(y)^2}^2 }\\
=& \tr\lrp{\lrp{\Ep{\eta}{u_\eta u_\eta^T - v_\eta v_\eta^T}}^2}\\
=& \tr\lrp{\Ep{\eta, \eta'}{\lrp{u_\eta u_\eta^T - v_\eta v_\eta^T} \lrp{u_{\eta'}u_{\eta'}^T - v_{\eta'} v_{\eta'}^T}}}\\
=& \Ep{\eta,\eta'}{\tr\lrp{\lrp{u_\eta u_\eta^T - v_\eta v_\eta^T} \lrp{u_{\eta'}u_{\eta'}^T -
| 1,388
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f $\Q$, $Lp(\Q|\xi)\inseg\Q$.
If $\Q$ is good then it has a unique $(\omega, \omega_1)$-iteration strategy with Dodd-Jensen property. We let $\Sigma_\Q$ be this strategy. Also, let $\eta_\Q$ be the largest cardinal of $\Q$. Given an iteration tree $\T$ on $\Q$ according to $\Sigma_{\Q}$ with last model $\R$ such that $\pi^{\T}$ exists, we let $\pi_{\Q, \R}:\Q\rightarrow \R$ be the iteration embedding. Notice that because $\Sigma_\Q$ has the Dodd-Jensen property, $\pi^\T$ is independent of $\T$. We say $\Q$ is *excellent* if whenever $\R$ is a $\Sigma_\Q$-iterate of $\Q$ such that $\pi_{\Q, \R}$ is defined $\R$ is good. In this case, we also say that $\Sigma_\Q$ is fullness preserving.
Suppose now $\a<\k$ is such that it ends a weak gap (see [@Scales]). We then let
$\mathcal{F}(\a, a)=\{ \Q: J_\a(\mathbb{R}){\vDash}``\Q$ is an excellent $a$-premouse"$\}$.
Given $a$-premouse $\P$ such that $J_\a(\mathbb{R}){\vDash}``\P$ is suitable and short tree iterable" we let $\mathcal{F}(\a, a, \P)$ be the set of $\Q$ such that in $J_\a(\mathbb{R})$, there is a correctly guided short tree $\T$ on $\P$ with last suitable model $\P^*$ such that for some $\P^*$-cardinal $\eta\leq \l_{\P^*}$, $\Q=\P^*|(\l_{\P^*}^+)^{\P^*}$.
Suppose $\a<\k$ ends a weak gap, $a\in HC$ and $\P$ is an $a$-premouse such that $J_\a(\mathbb{R}){\vDash}``\P$ is suitable and short tree iterable". Then $\mathcal{F}(\a, a, \P)\subseteq \mathcal{F}(\a, a)$.
Fix $\Q\in \mathcal{F}(\a, a, \P)$. Work in $J_\a(\mathbb{R})$. Let $\T$ be a correctly guided short tree on $\P$ with last suitable model $\P^*$ such that for some $\P^*$-cardinal $\eta< \l_{\P^*}$, $\Q=\P^*|(\l_{\P^*}^+)^{\P^*}$. Because $\P$ is short tree iterable, we have that $\Q$ is $(\omega, \omega_1)$-iterable via a unique iteration strategy $\Sigma$. As the iterations of $\Q$ can also be viewed as iterations of $\P^*$, we have that $\Sigma$ is fullness preserving, implying that $\Q$ is excellent.
Notice that if $\b>\a$ is such that $\b$ ends a weak gap and $J_\b(\mathbb{R}){\vDash}``\P$ is
| 1,389
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with ${\mathcal{L}}_{X}$ where $X$ is one of $\{ L_{0}, L_{\pm}, W_{0} \}$.
The highest- (lowest-) weight method {#sec:high-lowest-weight}
====================================
In this section we construct the scalar, vector, and symmetric tensor bases for NHEK’s isometry group ${\ensuremath{SL(2,\mathbb{R})\times U(1)}}$. First we briefly review the formalism of finding basis functions adapted to the isometry group in Schwarzschild spacetime. By drawing analogy to the Schwarzschild case and further utilizing the *highest- (lowest-)weight method* for non-compact groups, we will be able to construct unitary representations of NHEK’s isometry group.
Review: Unitary representations of $SO(3)$ in Schwarzschild {#sec:uni-reps-iso-Sch}
-----------------------------------------------------------
The full spacetime manifold of Schwarzschild spacetime is $\mathcal{M}_{\mathrm{Sch}}=M^2\times S^2$. The two-dimensional submanifold $M^2$ is the $(\bar{t},\bar{r})$-plane, and $S^2$ is the unit two-sphere coordinated by $(\bar{\theta},\bar{\phi})$. Here $(\bar{t},\bar{r},\bar{\theta},\bar{\phi})$ are the usual Schwarzschild coordinates. Part of the isometry group of Schwarzschild is $SO(3)$, which acts on the $S^{2}$ factors. The three generators of the group are simply the rotations along each Cartesian axis, i.e. $J_x, J_y, J_z \in \mathfrak{so}(3)$. The Casimir operator of $\mathfrak{so}(3)$ is given by $J^2 = J^2_x+J^2_y+J^2_z$.
In any space that $SO(3)$ acts upon, we can look for bases of functions which simultaneously diagonalize $J^{2}$ and $J_{z}$—that is, they are eigenfunctions of both operators. In the space of complex functions on the unit sphere, these eigenfunctions turn out to be the spherical harmonic functions $Y^{\mu,\nu}$, where $\mu,\nu$ label the functions (they are not tensor indices). The even/odd parity vector harmonics, $Y_{A}^{\mu,\nu}, X_{A}^{\mu,\nu}$, and tensor harmonics, $Y_{AB}^{\mu,\nu}, X_{AB}^{\mu,\nu}$, are also simultaneous eigenfunctions of $J^{2}$ and $J_{z}$ (where now $A,B$
| 1,390
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| 2,025
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hm:new.gen\], noting that $H\lhd G$. Let $\pi:G\to G/H$ be the quotient homomorphism, and note that $$\begin{aligned}
\frac{|\pi(AP)|}{|\pi(P)|}&=\frac{|APH|}{|PH|}\\
&\le\exp(e^{O(s^2)}\log^{O(s)}2K)\frac{|APH|}{|AH|}\\
&=\exp(e^{O(s^2)}\log^{O(s)}2K)\frac{|\pi(AP)|}{|\pi(A)|}\\
&\le\exp(e^{O(s^2)}\log^{O(s)}2K),\end{aligned}$$ the last inequality coming from the fact that $\pi(A)$ is a $K$-approximate group and $\pi(AP)\subset\pi(A)^{e^{O(s^2)}\log^{O(s)}2K}$. Applying \[lem:covering\] in the quotient $G/H$ therefore gives a set $X\subset A$ of size at most $\exp(e^{O(s^2)}\log^{O(s)}K)$ such that $A\subset XHPP^{-1}$. Now $PP^{-1}\subset A^{e^{O(s^2)}\log^{O(s)}2K}$ is an ordered progression of rank double that of $P$, which is still at most $e^{O(s^2)}\log^{O(s)}2K$. The corollary therefore follows from \[prop:nilprog.equiv\].
We may assume that $A$ generates $G$. Let $H$ and $P_0=P_{\text{\textup{ord}}}(x;L)$ be as given by \[thm:new.gen\], noting that $H\lhd G$. Let $\pi:G\to G/H$ be the quotient homomorphism, noting that $$\frac{|\pi(P_0)|}{|\pi(A)|}=\frac{|P_0H|}{|AH|}\ge\exp(-e^{O(s^2)}\log^{O(s)}2K).$$ Applying and \[lem:chang\] in the quotient $G/H$, we therefore have $$t\le e^{O(s^2)}\log^{O(s)}2K$$ and sets $S_1,\ldots,S_t\subset A$ with $|S_i|\le2K$ such that $$A\subset S_{t-1}^{-1}\cdots S_1^{-1}P_0^{-1}P_0S_1\cdots S_tH.$$ Enumerating the elements of each $S_i$ as $s_{1,i},\ldots,s_{r_i,i}$ and writing $$Q_i=\{s_{1,i}^{\epsilon_1}\cdots s_{r_i,i}^{\epsilon_{r_i}}:\epsilon_j,\in\{-1,0,1\}\},$$ the set $P=Q_{t-1}\cdots Q_1P_0^{-1}P_0Q_1\cdots Q_t$ is therefore an ordered progression of rank at most $$e^{O(s^2)}K\log^{O(s)}2K$$ satisfying $$A\subset PH\subset A^{4Kt+e^{O(s^2)}\log^{O(s)}2K}H\subset A^{e^{O(s^2)}K\log^{O(s)}2K}H.$$ The corollary therefore follows from \[prop:nilprog.equiv\].
Applications to non-nilpotent groups {#sec:non-nilp}
====================================
In this section we use our results to improve the bounds on the ranks of the coset nilprogressions appearing in
| 1,391
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| 0.773749
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nd $\psi^{(1)}_{m,n} = \psi_{m,n}$ in (\[eq:M-red-sys-5\]) and (\[eq:M-sys-5\]), then they will reduce to equations (\[eq:MX2a\]) and (\[eq:Bog\]), respectively.
The reduced systems for $\boldsymbol{N>5}$ {#the-reduced-systems-for-boldsymboln5 .unnumbered}
------------------------------------------
It can be easily checked that for each $k$ ($N=2 k+1$), the lowest order symmetry of the reduced system (\[self-dual-equn\]) involves certain functions $P^{(i)}_{m,n}$, $i=0,\ldots,k$, with $$\begin{gathered}
P^{(k)}_{m,n} = 2 + 2 \sum_{i=0}^{k-2} P^{(i)}_{m,n} + P^{(k-1)}_{m,n},\end{gathered}$$ which are given in terms of $\phi^{(i)}_{m,n}$ and their shifts (see relations (\[eq:N5-P-G\]) and (\[eq:N7-P-G\])). Then, the Miura transformation $$\begin{gathered}
\label{eq:gen-Miura}
\psi^{(i)}_{m,n} = \frac{2 P^{(i)}_{m,n}}{P^{(k)}_{m,n}}, \qquad i=0,\ldots,k-2, \qquad
\psi^{(k-1)}_{m,n} = \frac{P^{(k-1)}_{m,n}}{P^{(k)}_{m,n}} - 1,\end{gathered}$$ brings the symmetries of the reduced system to polynomial form. One could derive the polynomial system corresponding to $N=7$ ($k=3$) starting with system (\[eq:N7-red-dd\]), the functions given in (\[eq:N7-P-G\]) and using the corresponding Miura transformation (\[eq:gen-Miura\]). The system of differential-difference equations is omitted here because of its length but it can be easily checked that if we set $\psi^{(0)}_{m,n}=0$ and then rename the remaining two variables as $\psi^{(i)}_{m,n} \mapsto \psi^{(i-1)}_{m,n}$, then we will end up with system (\[eq:M-sys-5\]).
This indicates that every $k$ component system is a generalisation of all the lower order ones, and thus of the Bogoyavlensky lattice (\[eq:Bog\]). To be more precise, if we consider the case $N=2 k+1$ along with the $k$-component system, set variable $\psi^{(0)}_{m,n}=0$ and then rename the remaining ones as $\psi^{(i)}_{m,n} \mapsto \psi^{(i-1)}_{m,n}$, then the resulting $(k-1)$-component system is the reduced system corresponding to $N = 2 k-1$. Recursively, this means that it also reduces to the $N
| 1,392
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2)(\kappa+\alpha_{i,i',\ell,\theta}-3)\cdots(\kappa-1)} \nonumber\\
&=& \frac{e^{-2b}}{(\kappa-1)} \frac{(\kappa-\ell + \alpha_{i,i',\ell,\theta}-2)(\kappa -\ell +\alpha_{i,i',\ell,\theta}-3)\cdots (\kappa -\ell)}{(\kappa+\alpha_{i,i',\ell,\theta}-2)(\kappa+\alpha_{i,i',\ell,\theta}-3)\cdots(\kappa)}\nonumber\\
&\geq& \frac{e^{-2b}}{(\kappa-1)} \bigg( 1- \frac{\ell}{\kappa}\bigg)^{\alpha_{i,i',\ell,\theta}-1}\nonumber\\
&=& \frac{e^{-2b}(\kappa-\ell)}{\kappa(\kappa-1)} \bigg( 1- \frac{\ell}{\kappa}\bigg)^{\alpha_{i,i',\ell,\theta}-2}, \label{eq:posl_6}\end{aligned}$$ where follows from the fact that $\widetilde{\alpha}_{i,i',\ell,\theta} \geq 2e^{-2b}$. Claim follows by combining Equations and .
### Proof of Lemma \[lem:posl\_upperbound\]
Analogous to the proof of Lemma \[lem:posl\_lowerbound\], we construct a new set of parameters $\{\ltheta_j\}_{j\in[d]}$ from the original $\theta$. We denote the sum of the weights by $W \equiv \sum_{j \in S} \exp(\theta_j)$. We define a new set of parameters $\{\ltheta_j\}_{j \in S}$: $$\begin{aligned}
\ltheta_j &=& \left\{ \begin{array}{rl}
\log(\widetilde{\alpha}_{i,\ell,\theta}) &\; \text{for} \; j = i \;, \\
0&\;\text{otherwise}\;. \end{array}\right. \end{aligned}$$ Similarly define $\widetilde{W} \equiv \sum_{j \in S} \exp(\ltheta_j) = \kappa-1+\widetilde{\alpha}_{i,\ell,\theta}$. We have, $$\begin{aligned}
& \P_{\theta}\Big[\sigma^{-1}(i) = \ell \Big] \nonumber\\
&= \sum_{\substack{j_1 \in S \\ j_1 \neq i}} \Bigg(\frac{\exp(\theta_{j_1})}{W} \sum_{\substack{j_2 \in S \\ j_2 \neq i,j_1}} \Bigg(\frac{\exp(\theta_{j_2})}{W-\exp(\theta_{j_1})}\cdots \Bigg( \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i, \\ j_1,\cdots,j_{\ell-2}}} \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-2}}\exp(\theta_{k})}\frac{\exp(\theta_i)}{W-\sum_{k=j_1}^{j_{\ell-1}}\exp(\theta_{k})}\Bigg)\Bigg)\Bigg) \nonumber\\
&\leq \sum_{\substack{j_1 \in S \\ j_1 \neq i}} \Bigg(\frac{\exp(\theta_{j_1})}{W} \sum_{\substack{j_2 \in S \\ j_2 \neq i,j_1}} \Bigg(\frac{
| 1,393
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tems [@Perad:1985], we have to find the necessary and sufficient conditions for the existence of solutions of type (\[eq:2.4\]). These conditions are called involutivity conditions.
####
First we derive a number of necessary conditions on the vector fields $(\gamma,\lambda)$ and their complex conjugate $(\bar{\gamma},\bar{\lambda})$ as a requirement for the existence of rank-$2$ (simple mode) solutions of the homogeneous system (\[eq:2.1\]). Namely, closing (\[eq:2.4\]) by exterior differentiation, we obtain the following 2-forms
\[eq:2.5\] (d+d)+d+|(d||+|d|)+|d||=0,
which have to satisfy (\[eq:2.4\]). Using (\[eq:2.4\]) we get
\[eq:2.6\] d=||\_[,|]{}+|\_[,]{},d=|\_[,|]{}|+\_[,]{}d,
where we have used the following notation $$\lambda_{,\gamma}=\gamma^\alpha\frac{{\partial}}{{\partial}u^\alpha}\lambda,\quad \gamma_{,\bar{\gamma}}=\bar{\gamma}^\alpha\frac{{\partial}}{{\partial}u^\alpha}\gamma.$$Substituting (\[eq:2.6\]) into the prolonged system (\[eq:2.5\]) we obtain
\[eq:2.8\] +|+||=0,
whenever the differential (\[eq:2.4\]) holds. The commutator of vector fields $\gamma$ and $\bar{\gamma}$, is denoted by $${\left[ \gamma,\bar{\gamma} \right]}={\left( \gamma,\bar{\gamma} \right)}_u+\gamma_{,\lambda_i}\lambda_{i,\bar{\gamma}}-\bar{\gamma}_{,\lambda_i}\lambda_{i,\gamma},$$while by ${\left( \gamma,\bar{\gamma} \right)}_u$ we denote a part of the commutator which contains the differentiation with respect to the variables $u^\alpha$, [*i.e.* ]{}$${\left( \gamma,\bar{\gamma} \right)}_u=\bar{\gamma}^\alpha\frac{{\partial}}{{\partial}u^\alpha}\left.\gamma(u,\lambda)\right|_{\lambda=const.}
-\gamma^\alpha\frac{{\partial}}{{\partial}u^\alpha}\left.\bar{\gamma}(u,\bar{\lambda})\right|_{\bar{\lambda}=const.}.$$Let $\Phi$ be an annihilator of the vectors $\gamma$ and $\bar{\gamma}$, [*i.e.* ]{}$$<\omega \lrcorner\gamma>=0, \qquad <\omega \lrcorner \bar{\gamma}>=0, \quad \omega\in\Phi=\operatorname{An}{\left\{ \gamma,\bar{\gamma} \right\}}.$$Here, by the parenthesis $<\omega\lrcorner \gamma>$, we denote the contracti
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n and represents the Ito product rule. The same equations as and were presented in Refs. [@hufnagel2004forecast; @colizza2006modeling]. In this description, the fraction of the infected population is given by $$\rho=1-s(\infty).$$ In Fig. \[fig-sy\_lgv-pm3d\], we show the result of numerical simulations of the Langevin equations and . Comparing Fig. \[fig-sy\_lgv-pm3d\] with Fig. \[fig-sir-pm3d\], we find that the phenomenon under study is described by the Langevin equations and . Thus, our problem may be solved by analyzing them.
Now, the key idea of our analysis is the introduction of a new variable $Y=\sqrt{i N}$. Then, and are re-written as $$\begin{aligned}
{\frac{d s}{d t}} &=&
\frac{1}{N}\left[ - \lambda s Y^2 -\sqrt{{\lambda s Y^2}}\cdot \xi_1\right],
\label{s2_lgv} \\
{\frac{d Y}{d t}} &=& \frac{1}{2}\left\{\left(\lambda s - 1\right)Y
-\frac{1}{4}\left({\lambda s +1} \right)\frac{1}{Y}\right\}
\nonumber
\\
&& +\frac{1}{2}\sqrt{{\lambda s }}\cdot \xi_1
+\frac{1}{2}\sqrt{1}\cdot \xi_2,
\label{Y_lgv}\end{aligned}$$ where it should be noted that the multiplication of the variable $Y$ and the noise does not appear in . We then consider the probability $q(\lambda)$ in the thermodynamic limit as the probability of observing $Y \simeq N^{1/2}$, because it is equivalent to $\rho >0$. Here, from and , we find that the characteristic time scale of $s$ is $N$ times that of $Y$. Thus, when $N$ is sufficiently large, $s$ almost retains its value when $Y$ changes over time. In particular, it is reasonable to set $s=1$ when $t$ is shorter than $N$. In this time interval, is expressed as $$\begin{aligned}
{\frac{d Y}{d t}} &=& -\partial_Y U(Y)+\sqrt{2D}\xi,
\label{Y2_lgv}\end{aligned}$$ where $D=(\lambda+1)/8$ and the potential $U(Y)$ is calculated as $$U(Y)=-\frac{1}{4}\left(\lambda - 1\right)Y^2
+\frac{1}{8}\left({\lambda +1}\right)\log(Y).$$ $\xi$ is Gaussian white noise with unit variance, where we have used the relation $\sqrt{\lambda}/2 \xi_1+1/2 \xi_2=\sqrt{\lambda+1}/2 \xi$. The initial co
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$+$ $+$ $+$ $+$ $+$ $0$
: Case of $a \geq 1$
Calculation of the expected value and the variance of the loss
==============================================================
Here, we calculate the expected value and the variance of the loss $\Pe(Z + c)$ for $c \in \mathbb{R}$.
**[Expected value of the loss]{}**
----------------------------------
Here, let us put $\beta := (2 a b \G(a))^{-1}$; then, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]
&= \int_{- \infty}^{+\infty} \Pe(z + c) f_{Z}(z) dz \\
&= k_{2} \beta \int_{- \infty}^{- c} (- z - c)
\exp{\left( - \left\lvert \frac{z}{b} \right\rvert^{\frac{1}{a}} \right)} dz
+ k_{1} \beta \int_{- c}^{+\infty} (z + c)
\exp{\left( - \left\lvert \frac{z}{b} \right\rvert^{\frac{1}{a}} \right)} dz. \end{aligned}$$ Replace $z$ with $b z$ to get $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]
= k_{2} b \beta \int_{- \infty}^{- c / b} (- b z - c)
\exp{\left( - \lvert z \rvert^{\frac{1}{a}} \right)} dz
+ k_{1} b \beta \int_{- c / b}^{+\infty} (b z + c)
\exp{\left( - \lvert z \rvert^{\frac{1}{a}} \right)} dz. \end{aligned}$$ When $c \geq 0$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]
&= k_{2} b \beta
\int_{- \infty}^{- c / b} (- b z - c) \exp{\left( - (-z)^{\frac{1}{a}} \right)} dz \\
&\quad + k_{1} b \beta
\int_{- c / b}^{0} (b z + c) \exp{\left( - (-z)^{\frac{1}{a}} \right)} dz
+ k_{1} b \beta \int_{0}^{+\infty} (b z + c) \exp{\left( - z^{\frac{1}{a}} \right)} dz
\allowdisplaybreaks \\
&= k_{2} b \beta
\int_{c / b}^{+\infty} (b z - c) \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + k_{1} b \beta
\int_{0}^{c / b} (- b z + c) \exp{\left( - z^{\frac{1}{a}} \right)} dz
+ k_{1} b \beta \int_{0}^{+\infty} (b z + c) \exp{\left( - z^{\frac{1}{a}} \right)} dz
\allowdisplaybreaks \\
&= (k_{1} + k_{2}) b^{2} \beta
\int_{c / b}^{+\infty} z \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + (k_{1} - k_{2}) b c \beta
\int_{0}^{+\infty} \exp{\left( - z^{\
| 1,396
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ave that $\binom{[4]}{\le 3}_h\cup\{H\}$ is a star in $\cH$ of size $8>7$, a contradiction. Hence there is no such $H$, which is case (\[case:1\]) of the theorem.\
### $\cIt$ is a triangle
We may assume that $\cIt= \{\{1,2\},\{1,3\},\{2,3\}\}$.\
Relabel, if necessary, so that $0\le|C(\oone)|\le |C(\otwo)|\le |C(\othree)|$. Then $(\cI\setminus(\cA(\oone)\cup\{\{2,3\}\}))\cup\{\{1,s\}\mid s\in C(\othree)\}\cup\{\{1\}\}$ is a star subfamily of $\cH$ of size $|\cI|+|cA(\othree)|-|\cA(\oone)|\ge|\cI|$, and so $\cH$ is EKR, and strictly so unless $|C(\oone)|=|C(\otwo)|=|C(\othree)|$, which we now assume.\
If not all the sets $C(\oi)$ are the same then, without loss of generality say $C(\oone)\not= C(\otwo)$, and so $|C(\oone)\cup C(\otwo)|>|C(\othree)|$. Then $(\cI\setminus(\cA(\othree)\cup\{\{1,2\}\}))\cup\{\{3,s\}\mid s\in C(\oone)\cup C(\otwo)\}\cup\{\{3\}\}$ is a star subfamily of $\cH$ of size $|\cI|+|A(\oone)\cup A(\otwo)|-|\cA(\othree)|>|\cI|$, a contradiction.\
Finally, if $C(\oone)=C(\otwo)=C(\othree)$ then $|\cH_1|=|\cI|$, so $\cH$ is EKR, but not strictly so, giving us case (\[case:2\]) of the theorem.\
$|\cIt|=2$
----------
We may assume that $\cIt=\{\{1,2\},\{1,3\}\}$. For each $I\in\cIr$ we must have $1\in I$ or $\{2,3\}\subset I$. If $I\in\cIr\setminus(\cA(1)\cup\cA(2,3))$, then $1\in I$ and $\{2,3\}\cap I\ne\emptyset$.\
If $\cA(2,3)=\emptyset$, then $\cI$ is a star, so we assume that $\cA(2,3)\ne\emptyset$. It must be that $\cA(1)\ne\emptyset$, since otherwise $\cI\cup\{\{2,3\}\}$ would be a larger intersecting subfamily of $\cH$, a contradiction.\
Fix an $A\in\cA(1)$; then for each $k\in C(2,3)$ we must have that $k\in A$. Thus $|C(2,3)|\le|A\setminus\{1\}|=2\le |C(1)|$. Hence we have that $(\cI\setminus\cA(2,3))\cup\{\{1\}\}\cup\{\{1,i\}\mid i\in C(1)\}$ is a star of size $|\cI|+|C(1)|-|\cA(2,3)|+1>|\cI|$, a contradiction.\
$|\cIt|=1$
----------
We may assume that $\cIt=\{\{1,2\}\}$. Without loss of generality, both of $\cA(1), \cA(2)$ are nonempty (otherwise $\cI$ is a star and we are done). If
| 1,397
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| 490
| 1,593
| null | null |
github_plus_top10pct_by_avg
|
4} & -\frac{2 u \left(u^4-4 u^2+3\right)}{\left(u^2+1\right)^4} & -\frac{4 (h+1) \left(u^2-1\right)^2}{\left(u^2+1\right)^3} \\
\mathcal{D}_{RR} & -\frac{u \left(u^2-3\right)}{\left(u^2+1\right)^2} & \frac{2 u \left(u^2-3\right)}{\left(u^2+1\right)^2} & \frac{u \left(u^2-3\right)^3}{8 \left(u^2-1\right) \left(u^2+1\right)^2} & 0 & \frac{u^2-1}{u^2+1} \\
\mathcal{D}_{Ru} & \frac{h+1}{2 \left(u^2+1\right)} & -\frac{2 h+1}{2 \left(u^2+1\right)} & \frac{h \left(u^4+6 u^2-3\right)}{8 \left(u^4-1\right)} & \frac{1}{2 \left(u^2+1\right)} & 0 \\
\mathcal{D}_{uu} & -\frac{u}{2 \left(u^4-1\right)} & \frac{u}{u^4-1} & -\frac{u \left(u^2+3\right)}{4 \left(u^4-1\right)} & \frac{u}{2 \left(u^4-1\right)} & 0 \\
\mathcal{D}_{TR} & 0 & 0 & 0 & 0 & 0 \\
\mathcal{D}_{Tu} & \frac{i m}{2 \left(u^2+1\right)} & -\frac{i m \left(u^4+6 u^2-3\right)}{8 \left(u^4-1\right)} & 0 & 0 & 0 \\
\mathcal{D}_{\Phi R} & 0 & 0 & 0 & 0 & -\frac{i m \left(u^2-1\right)}{2 \left(u^2+1\right)} \\
\mathcal{D}_{\Phi u} & \frac{i m}{2 \left(u^2+1\right)} & -\frac{i m}{2 \left(u^2+1\right)} & 0 & -\frac{i m}{2 \left(u^2+1\right)} & 0 \\
\noalign{\bigskip}
\text{} & C_{uu}'(u) & C_{TR}'(u) & C_{Tu}'(u) & C_{\Phi R}'(u) & C_{\Phi u}'(u)
\\
\noalign{\smallskip}
\hline \hline \noalign{\smallskip}
\mathcal{D}_{TT} & \frac{u \left(u^2-1\right) \left(u^6+11 u^4-13 u^2+9\right)}{2 \left(u^2+1\right)^4} & 0 & -\frac{i m \left(u^2-1\right) \left(u^4+6 u^2-3\right)}{\left(u^2+1\right)^3} & 0 & \frac{i m \left(u^4+6 u^2-3\right)^2}{4 \left(u^2+1\right)^3} \\
\mathcal{D}_{T \Phi} & \frac{4 u \left(u^2-1\right)^3}{\left(u^2+1\right)^4} & 0 & -\frac{i m \left(u^6+9 u^4-17 u^2+7\right)}{2 \left(u^2+1\right)^3} & 0 & \frac{i m \left(u^6+5 u^4-9 u^2+3\right)}{\left(u^2+1\right)^3} \\
\mathcal{D}_{\Phi\Phi} & \frac{4 u \left(u^2-1\right)^3}{\left(u^2+1\right)^4} & 0 & -\frac{4 i m \left(u^2-1\right)^2}{\left(u^2+1\right)^3} & 0 & \frac{4 i m \left(u^2-1\right)^2}{\left(u^2+1\right)^3} \\
\mathcal{D}_{RR} & -\frac{u \left(u^2-1\right)}{2 \left(u^2+1\right)} & 0 & \
| 1,398
| 1,896
| 1,311
| 1,384
| null | null |
github_plus_top10pct_by_avg
|
n/R^n\bigr)^{(q)}.$$ Here $X\to \bigl(X^n/R^n\bigr)^{(q)}$ is finite and $R$ is an equivalence relation on $X$ over the base scheme $\bigl(X^n/R^n\bigr)^{(q)}$. Hence, by (\[quot.X/S.finite.lem\]), the geometric quotient $X/R$ exists.
Some of the scheme theoretic aspects of the purely inseparable case are treated in [@eke] and [@sga3 Exp.V].
Gluing or Pinching {#glue.sec}
==================
The aim of this section is to give an elementary proof of the following.
[@artin70 Thm.3.1] \[glue.thm.asp\] Let $X$ be a Noetherian algebraic space over a Noetherian base scheme $S$. Let $Z\subset X$ be a closed subspace. Let $g:Z\to V$ be a finite surjection. Then there is a universal push-out diagram of algebraic spaces $$\begin{array}{ccl}
Z & \into & X\\
g \downarrow\hphantom{g} && \ \downarrow \pi\\
V & \into & Y:=X/(Z\to V)
\end{array}$$ Furthermore,
1. $Y$ is a Noetherian algebraic space over $S$
2. $V\to Y$ is a closed embedding and $Z=\pi^{-1}(V)$,
3. the natural map $\ker\bigl[{{\mathcal O}}_Y\to {{\mathcal O}}_V\bigr] \to \pi_*\ker\bigl[{{\mathcal O}}_X\to {{\mathcal O}}_Z\bigr]$ is an isomorphism, and
4. if $X$ is of finite type over $S$ then so is $Y$.
If $X$ is of finite type over $A$ and $A$ itself is of finite type over a field or an excellent Dedekind ring, then this is an easy consequence of the contraction results [@artin70 Thm.3.1]. The more general case above follows using the later approximation results [@pop]. The main point of [@artin70] is to understand the case when $Z\to V$ is proper but not finite. This is much harder than the finite case we are dealing with. An elementary approach following [@ferrand] and [@raoult] is discussed below.
\[glue.lem.affine\] The affine case of (\[glue.thm.asp\]) is simple algebra. Indeed, let $R={{\mathcal O}}_X$, $I=I(Z)$, $q:{{\mathcal O}}_X\to {{\mathcal O}}_Z$ the restriction and $S={{\mathcal O}}_V$. By (\[eak-nag\]), $q^{-1}(S)$ is Noetherian. Set $Y:={\operatorname{Spec}}q^{-1}(S)$.
If $\bar r_i\in {{\mathcal O}}_X/I(Z)$ generate ${{\mathcal O}}_
| 1,399
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| 0.788718
|
github_plus_top10pct_by_avg
|
sigma(\pi)\pi\begin{pmatrix} {}^ts_i'a_is_i'+\pi^2X_i & \pi Y_i & Z_i
\\ \sigma( \pi \cdot {}^tY_i) &\pi^2X_i'&\pi Y_i' \\ \sigma({}^tZ_i)&\sigma(\pi\cdot {}^tY_i') &{}^tt_i'a_it_i'+\pi^2 Z_i' \end{pmatrix}$ for certain matrices $X_i, Y_i, Z_i, X_i', Y_i', Z_i'$ with suitable sizes. Here, the diagonal entries of ${}^ts_i'a_is_i'$ and ${}^tt_i'a_it_i'$ are zero. Thus we can ignore the contribution from $\sigma(\pi\cdot {}^tm_{i,i}')\cdot\begin{pmatrix} a_i&0&0\\ 0&\pi^3 \bar{\gamma}_i&1 \\ 0&-1 &\pi \end{pmatrix}\cdot(\pi m_{i,i}')$ in Equation (\[ea6\]) and so Equation (\[ea6\]) equals $$\begin{gathered}
\begin{pmatrix} a_i'&\pi b_i'& e_i'\\ -\sigma(\pi \cdot {}^tb_i') &\pi^3f_i'&1+\pi d_i' \\
-\sigma({}^te_i') &-\sigma(1+\pi d_i') &\pi+\pi^3c_i' \end{pmatrix}=
\begin{pmatrix} a_i&0&0\\ 0&\pi^3 \bar{\gamma}_i&1 \\ 0&-1 &\pi \end{pmatrix}+\\
-\pi\begin{pmatrix} {}^ts_i'&{}^ty_i' &-\pi\cdot {}^t v_i'\\-\pi\cdot {}^t r_i'&-\pi\cdot x_i'&-\pi\cdot z_i' \\
{}^t t_i'&u_i'&-\pi\cdot w_i'\end{pmatrix}
\begin{pmatrix} a_i&0&0\\ 0&\pi^3 \bar{\gamma}_i&1 \\ 0&-1 &\pi \end{pmatrix}+
\pi \begin{pmatrix} a_i&0&0\\ 0&\pi^3 \bar{\gamma}_i&1 \\ 0&-1 &\pi\end{pmatrix}
\begin{pmatrix} s_i^{\prime}& \pi r_i^{\prime}& t_i^{\prime}\\ y_i^{\prime}&\pi x_i^{\prime}&u_i^{\prime}\\
\pi v_i^{\prime}& \pi z_i^{\prime}&\pi w_i^{\prime} \end{pmatrix}.\end{gathered}$$
We interpret each block of the above equation below:
1. Let us consider the $(1,1)$-block. The computation associated to this block is similar to that for the above case (i). Hence there are exactly $((n_i-2)^2+(n_i-2))/2$ independent linear equations and $((n_i-2)^2-(n_i-2))/2$ entries of $s_i'$ determine all entries of $s_i'$.
2. We consider the $(1,2)$-block. We can ignore the contribution from ${}^ty_i'\bar{\gamma}_i$ since it contains $\pi^3$ as a factor. Then the $(1,2)$-block is $$\label{ea7}
b_i'=\pi(- {}^tv_i'+a_ir_i').$$ By letting $b_i'=b_i=0$, we have $$\pi(- {}^tv_i'+a_ir_i')=0$$ as an equation
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github_plus_top10pct_by_avg
|
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