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star}=[0,1,2]^T$, then ${\a^\star}=[-2,0,1]^T$.
According to , if ${\a^\star}$ is optimal for $\h$, then $-{\a^\star}$ is also optimal for $\h$. To reduce redundancy, we restrict the optimal coefficient vector ${\a^\star}$ to be the one such that $\h^T{\a^\star}\geq0$ in the following.
\[lemma:aNonnegative\] If all the elements in a channel vector $\h$ are nonnegative, then all the elements in the optimal coefficient vector ${\a^\star}$ are also nonnegative.
Suppose ${\a^\star}(i) < 0$, and define $\a'$ as: $\a'(i) = 0$, and $\a'(\ell) = {\a^\star}(\ell)$, $\forall \ell \neq i$. Obviously, $\norm{\a'} < \norm{{\a^\star}}$, and $\h^T \a' \geq \h^T {\a^\star}\geq 0$. Then according to , $\bigR \left( \h, \a' \right) > \bigR \left( \h, {\a^\star}\right)$, which implies ${\a^\star}$ is not optimal and leads to a contradiction. Thus, all the elements in ${\a^\star}$ must be nonnegative.
\[lemma:a0\] For a channel vector $\h$ and its corresponding optimal coefficient vector ${\a^\star}$, if $\h(i) = 0$, then ${\a^\star}(i) = 0$.
Suppose $\h(i) = 0$, and ${\a^\star}(i) \neq 0$. Define $\a'$ as: $\a'(i) = 0$, and $\a'(\ell) = {\a^\star}(\ell)$, $\forall \ell \neq i$. Obviously, $\norm{\a'} < \norm{{\a^\star}}$, and $\h^T \a' = \h^T {\a^\star}\geq 0$. Then according to , $\bigR \left( \h, \a' \right) > \bigR \left( \h, {\a^\star}\right)$, which implies ${\a^\star}$ is not optimal. Thus, if $\h(i) = 0$, then ${\a^\star}(i) = 0$.
\[lemma:hijEqual\] For a channel vector $\h$ and its corresponding optimal coefficient vector ${\a^\star}$, if $\h(i) = \h(j)$, $i < j$, then ${\a^\star}(i) = {\a^\star}(j)$ or $\fabs{{\a^\star}(i) - {\a^\star}(j)} = 1$.
Without loss of generality, assume ${\a^\star}(i) - {\a^\star}(j) < -1$. Define $\a'$ as: $\a'(i) = {\a^\star}(i)+1$, $\a'(j) = {\a^\star}(j)-1$, and $\a'(\ell) = {\a^\star}(\ell)$, $\forall \ell \notin \{i,j\}$. Obviously, $\norm{\a'} < \norm{{\a^\star}}$, and $\h^T \a' = \h^T {\a^\star}\geq 0$. Then according to , $\bigR \left( \h, \a' \right) > \bigR \left( \h, {\a^\star}
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6)[$\theta_{i_3}$]{} (5431,-1786)[$\theta_{i_2}$]{} (3226,-1786)[$1$]{} (6578,-879)[$\theta_{i_4}$]{} (5829,-459)[$\theta_{i_5}$]{} (5063,-1344)[$\theta_{i_1}$]{} (2003,-1801)[$X$]{} (2483,-616)[$Y$]{} (1042,-879)[$Z$]{} (601,-1561)[( 1, 0)[3300]{}]{}
Then by the argument above, we see that $\Psi_p((\theta_1, \ldots, \theta_6)) =
(P, Q, R)$. Since $w$ is arbitrary in ${{\bold H}}^2$, we get the conclusion.
Consider now the map $\Psi_{\langle 123456 \rangle} \times \Psi_{\langle 214356\rangle}:\Theta_6 \to
{\cal H} \times {\cal H}$ which sends $\theta$ to $(\Delta_{\langle 123456
\rangle,\theta}$, $\Delta_{\langle 214356 \rangle,\theta})$. Then these two hexahedra can be seen in Fig. \[Fig:7\] stand on the first and third quadrants of the $uv$-plane. Notice that they have the common edge $(12)(34)56$ along the $w$-axis. Then we claim the following.
\[Lem:6injective\] The map $\Psi_{\langle 123456 \rangle} \times \Psi_{\langle 214356 \rangle}$ is injective.
Let $((P_1,Q_1,R_1),(P_2,Q_2,R_2))$ be an element in the image of $\Psi_{\langle 123456 \rangle} \times
\Psi_{\langle 214356 \rangle}$ and $\theta
= (\theta_1,\ldots,\theta_6)$ any element in the preimage of $((P_1,Q_1,R_1),(P_2,Q_2,R_2))$. Set $w_1$ and $w_2$ the points in $\xi_{\langle 123456 \rangle}^{-1}((P_1,Q_1,R_1))$ and $\xi_{\langle 214356 \rangle}^{-1}((P_2,Q_2,R_2))$ corresponding to $\theta$ under the identification in Lemma \[Lem:6fibration\] respectively. Again, we see that the triangles $\triangle w_101$ and $\triangle w_201$ are congruent so that $w_1$ and $w_2$ are identical which we denote by $w$.
Let $X_i=P_i^2 $, $Y_i=1+ (w-1)Q_i^2 \in {{\bold H}}^2$ for $i=1,2$. Then the triangles $\triangle 0X_1w$ and $\triangle 0wX_2$ are similar since both have the external angles $\theta_{i_1}+\theta_{i_2}$, $\pi-\theta_{i_2}$, $\pi-\theta_{i_1}$. Similarly, the triangles $\triangle 10Y_1$ and $\triangle 1Y_20$ are similar since both of which have the external angles $\theta_{i_3}+\theta_{i_4}$, $\pi-\theta_{i_3}$, $\pi-\theta_{i_4}$. (see Fig.�
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phi_{\Theta}(\Theta/ A\cap\ ^{w} B).$$ So the blocks $B_{H,L,x,y}$ and $B_{\Theta,\Theta,1,1}$ are equals up to permutation of the lines and the columns. In particular, these two matrices have the same determinant, up to a sign.
\[red2\] Let $\Theta$ be a finite group, and $\mu'$ the sub-algebra of $\mu_{R}(\Theta)$ generated by the elements of the form $t^{\Theta}_{A}r^{\Theta}_{A}$ for $A\leqslant \Theta$. Then the restriction of the Burnside trace to $\mu'$ is an isomorphism of $R$-algebras between $\mu'$ and $RB(\Theta)$, sending the basis of Proposition \[basis\] to the usual basis of $RB(\Theta)$ consisting of isomorphism classes of transitive $G$-sets.
It is clear that the restriction of the Burnside trace to $\mu'$ is an $R$-linear isomorphism since we have $Btr(t^{\Theta}_{A}r^{\Theta}_{A})= \Theta/A \in RB(\Theta)$. Moreover this is an isomorphism of algebras, since: $$\begin{aligned}
Btr(t^{\Theta}_{A}r^{\Theta}_{A}t^{\Theta}_{B}r^{\Theta}_{B})&=\sum_{\theta\in [A\backslash \Theta / B]} \Theta/(A\cap B^{\theta})\\
&=\Theta/A\times \Theta/B\in RB(\Theta). \end{aligned}$$
We have:
\[meta\] Let $G$ be a finite group. Let $\phi=(\phi_{H})_{H\leqslant G}$ be a stable by induction family of linear forms on $\big(RB(H)\big)_{H\leqslant G}$. Then the bilinear form $(-,-)_{\phi_{G}}$ on the Mackey algebra $\mu_{R}(G)$ is non degenerate if and only if the bilinear form $b_{\phi_{H}}$ on $RB(H)$ is non degenerate for every $H$ subgroup of $G$.
If $\phi$ is such a family of linear forms, by Lemma \[bl\] the matrix of the bilinear form $(-,-)_{\phi_{G}}$ in the usual basis of $\mu_{R}(G)$ is a permutation by block matrix. So the determinant of this matrix is (up to a sign) the product of the determinant of the non-zero blocks. By Lemma \[red1\] and Lemma \[red2\] the determinant of the block indexed by $(H,L,x,y)$ is equal to the determinant of the matrix of the bilinear form $b_{\phi_{L\cap H^{x}}}$ in the usual basis of $RB(L\cap H^{x})$. So the determinant of $(-,-)_{\phi_{G}}$ is invertible in $R$ if and o
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,r_p(\chi )({\alpha }_j,{\alpha }_i)=&q_{ij}q_{ji}=
r_p(\chi )({\alpha }_i,{\alpha }_i)^{c_{pi}}
\end{aligned}$$ for all $p,i,j\in I$. Hence $r_p(\chi )$ is again of Cartan type with the same Cartan matrix $C$. Thus $\chi '$ is $i$-finite for all $\chi '\in {\mathcal{G}}(\chi )$ and $i\in I$.
Let $C=(c_{ij})_{i,j\in I}$ be a symmetrizable generalized Cartan matrix, and for all $i\in I$ let $d_i\in {\mathbb{N}}$ such that $d_ic_{ij}=d_jc_{ji}$ for all $i,j\in I$. Let $q\in {{\Bbbk }^\times }$ such that $\qnum{m+1}{q^{2d_i}}\not=0$ for all $m\in {\mathbb{N}}_0$ with $m<-c_{ij}$ for some $j\in I$. Define $\chi \in {\mathcal{X}}$ by $\chi ({\alpha }_i,{\alpha }_j)=q^{d_ic_{ij}}$. Then $\chi $ is of Cartan type, hence $\chi $ is $p$-finite for all $p\in I$. Eq. implies that $r_p(\chi )=\chi $ for all $p\in I$, and hence ${\mathcal{G}}(\chi )$ consists of precisely one element. In this case the Weyl groupoid ${\mathcal{W}}(\chi )$ is a group, which is precisely the Weyl group associated to the generalized Cartan matrix $C$. We will study this example in Sect. \[sec:Uqg\] under the assumption that $C$ is of finite type.
Roots {#ssec:roots}
-----
Let $\chi \in {\mathcal{X}}$. There exists a canonical root system of type ${\mathcal{C}}(\chi)$ which we describe in this subsection. It is based on the construction of a restricted PBW basis of Nichols algebras of diagonal type. Nichols algebras are braided Hopf algebras defined by a universal property. More details can be found in [@inp-AndrSchn02] on braided Hopf algebras and Nichols algebras, in [@a-Khar99] on the PBW basis, and in [@a-Heck06a] on the root system.
Let $V\in { {}_{{\Bbbk }{\mathbb{Z}}^I}^{{\Bbbk }{\mathbb{Z}}^I}\mathcal{YD}}$ be a $|I|$-dimensional module of diagonal type. Let ${\delta }:V\to {\Bbbk }{\mathbb{Z}}^I{\otimes }V$ and ${\boldsymbol{\cdot}}:{\Bbbk }{\mathbb{Z}}^I{\otimes }V\to V$ denote the left coaction and the left action of ${\Bbbk }{\mathbb{Z}}^I$ on $V$, respectively. Fix a basis $\{x_i\,|\,i\in I\}$ of $V$, elements $g_i$, wher
| 3,104
| 2,550
| 1,543
| 3,038
| 3,841
| 0.769749
|
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|
}^{j}$. Since the contribution of the $j$-th receive beam to the throughput under the $\log$ function is given by $$\label{eq:alphapsinjR}
g_{\psi,n,j}={\mathbf{h}}_{\psi,V}^{j}\left({\mathbf{I}}_{N_t}+\frac{\rho}{N_t}\left({\widetilde{\mathbf{H}}_{\psi,V}}^{n}\right)^H{\widetilde{\mathbf{H}}_{\psi,V}}^{n,r}\right)^{-1}\left({\mathbf{h}}_{\psi,V}^{j}\right)^H,$$ we select the receive beam at the $n+1$ step by $$\label{}
{\mathbf{h}}_{\psi,V}^{J}=\arg\max_{{\mathbf{h}}_{\psi,V}^{j}} g_{\psi,n,j}.$$ The mechanism of ISSA-based transmit beam selection is omitted here, since it is similar to that of the ISSA-based receive beam selection.
The proposed fast selection algorithm is given as Algorithm 1. The outputs of the algorithm are the optimal reconfiguration state, the indices of the selected receive beams, the indices of the selected transmit beams, and the selected low-dimensional virtual channel, denoted by $\widehat{\psi}$, $\mathcal{M}_r$, $\mathcal{M}_t$, and ${\widehat{\widetilde{\mathbf{H}}}_{\widehat{\psi},V}}$, respectively.
We would like to highlight that the proposed algorithm significantly reduces the complexity of reconfiguration state selection and beam selection, and achieves the near-optimal throughput performance. It is worth pointing out that an important requirement of the proposed algorithm is the knowledge of full CSI of all reconfigurable states, and the associated channel estimation complexity has not been taken into account. Although the full CSI assumption has been widely-adopted in the literature, as mentioned earlier, the channel estimation is relatively challenging for mmWave systems with reconfigurable antennas.
Numerical Results {#sec:numersim}
=================
For all numerical results in this work, we adopt the clustered multipath channel model in to generate the channel matrix. We assume that $\alpha_{\psi,i,l}$ are i.i.d. $\mathcal{CN}\left(0,\sigma^2_{\alpha,\psi,i}\right)$, where $\sigma^2_{\alpha,\psi,i}$ denotes the average power of the $i$-th cluster, and $\sum_{i=
| 3,105
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| 3,068
| 1,632
| 0.787378
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si\right>}$ on the carrier space of $D^{{\left(s\right)}}$.
In general, in order to break the Bell inequality it is necessary to consider a number of orbits. To this end one considers the orbits generated by $N$ pairs of vectors ${\left({\left|\varphi_n\right>},{\left|\psi_n\right>}\right)}$ and the corresponding operators $X{\left(\varphi_n,\psi_n\right)}$. They mutually commute so the eigenvalues of $$X=\sum_{n=1}^N X{\left(\varphi_n,\psi_n\right)}\label{b4}$$ are the sums of eigenvalues of all $X{\left(\varphi_n,\psi_n\right)}$. In this way one can maximize the sum of probabilities $$\sum_{n=1}^N\sum_{g\in G}{\left|{\left<g,\varphi_n,\psi_n |\chi\right>}\right|}^2$$ and proceed as above.
The $S_4$ group: three orbits
=============================
$S_4$ is the group of order 24. It has 6 conjugancy classes. There exist six irreducible representations of $S_4$: trivial representation, the alternating representation, the homomorphic twodimensional one and two threedimansional representations, $D$ and $\widetilde{D}$; $\widetilde{D}$ is obtained from $D$ by multiplication by the alternating representation. All representations can be made orthogonal.
Consider the threedimensional representation $D$. It can be described by writing out the matrices representing the transpositions: $$D{\left(12\right)}=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & -1
\end{array}\right],\qquad
D{\left(13\right)}=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & -\frac{1}{2} & -\frac{\sqrt{3}}{2}\\
0 & -\frac{\sqrt{3}}{2} & \frac{1}{2}
\end{array}\right]$$ $$D{\left(14\right)}=\left[\begin{array}{ccc}
-\frac{1}{3} & -\frac{\sqrt{2}}{3} & -\frac{\sqrt{6}}{3}\\
-\frac{\sqrt{2}}{3} & \frac{5}{6} & -\frac{\sqrt{3}}{6}\\
-\frac{\sqrt{6}}{3} & -\frac{\sqrt{3}}{6}& \frac{1}{2}
\end{array}\right],\qquad
D{\left(23\right)}=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & -\frac{1}{2} & \frac{\sqrt{3}}{2}\\
0 & \frac{\sqrt{3}}{2} & \frac{1}{2}
\end{array}\right]$$ $$D{\left(24\right)}=\left[\begin{array}{ccc}
-\frac{1}{3} & -\frac{\sqrt{2}}{3} &
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\int_{\Gamma_-} g(y,\omega,E)^2 \tau_-(y,\omega)|\omega\cdot\nu(y)|d\sigma(y) d\omega dE \\
={}&{\left\Vert g\right\Vert}_{T^2_{\tau_-}(\Gamma_-)}^2,$$ where in the second step we applied the change of variables in integration explained in the proof of Theorem \[tth\] below (see Remark \[changevar\]), and noticed that $t(y+s\omega,\omega)=s$ whenever $(y,\omega,E)\in \Gamma_-$. Therefore, $${\left\Vert L_{\Sigma,-} g\right\Vert}_{L^2(G\times S\times I)}^2
=&{}\int_{G\times S\times I} \big(e^{-\int_0^{t(x,\omega)}\Sigma(x-s\omega,\omega,E)ds}g(x-t(x,\omega)\omega,\omega,E)\big)^2 dx d\omega dE \\
\leq &{}\int_{G\times S\times I} g(x-t(x,\omega)\omega,\omega,E)^2 dx d\omega dE \\
=&{}{\left\Vert L_{0,-} g\right\Vert}_{L^2(G\times S\times I)}^2
={\left\Vert g\right\Vert}_{T^2_{\tau_-}(\Gamma_-)}^2.$$
Since $t(y,\omega)=0$ a.e. in $\Gamma_-$ we see that $\gamma_-(L_-g)=g.$ When we verify (Lemma \[trathle1\]) that $\omega\cdot\nabla_x(L_-g)+\Sigma(L_-g)=0$ (weakly) in $G\times S\times I$ we can conclude that $L_-g\in W^2(G\times S\times I)$.
\[trathle1\] Assume that $\Sigma\in L^\infty(G\times S\times I)$ and that $\Sigma\geq 0$. Let $L_{-}:T^2_{\tau_{-}}(\Gamma_{-})\to L^2(G\times S\times I)$ be defined by $$(L_{-} g)(x,\omega,E)=
e^{-\int_0^{t(x,\omega)}\Sigma(x-s\omega,\omega,E)ds}
g(x- t(x,\omega)\omega,\omega,E).$$ Then in the weak sense on $G\times S\times I$, $$\omega\cdot \nabla_x (L_- g)+\Sigma (L_- g)=0.$$
Given $g\in T^2_{\tau_-}(\Gamma_-)$, choosing a sequence $g_n$ in $C^1_0(\Gamma_-)$ that converges to $g$ in $T^2_{\tau_-}(\Gamma_-)$ (the proof of the existence of this kind sequence is quite standard and is omitted), we have by the continuity of $L_-$ (see (\[trpr9\])), that $L_-g_n\to L_- g$ in $L^2(G\times S\times I)$, hence in ${\mathcal{D}}'(G\times S\times I^\circ)$, where $I^\circ:=]0,E_{\rm m}[$, from which we deduce that $\omega\cdot\nabla_x (L_-g_n)+\Sigma(L_-g_n)\to \omega\cdot\nabla_x (L_- g)+\Sigma (L_-g)$. in ${\mathcal{D}}'(G\times S\times I^\circ)$. This shows that we may assume $g\in
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\frac{ \log n}{n} \log ^4 k\right)^{1/6} \quad \text{and} \quad
\tilde{\Delta}_{n,3} = \min \left\{ \Delta_{n,3}, \frac{U^2}{v} \overline{v} \frac{ k^{5/2}}{u_n^3 u^2} \frac{
\log n}{n} \log k \right\}.$$
A few remarks are in order.
The coverage probability is affected by three factors: the term $\Delta_{n,1}$, which bounds the approximation error stemming from the high dimensional Berry-Esseen theorem (see ); the term $\Delta_{n,2}$, which is a high probability bound on the size of the reminder term in the Taylor series expansion of $\beta_{{\widehat{S}}}$ around $\widehat{\beta}_{{\widehat{S}}}$ and can therefore be thought of as the price for the non-linearity of the projection parameter, and the terms $\Delta_{n,3}$ and $\tilde{\Delta}_{n,3}$, which are due to the fact that the covariance of the estimator is unknown and needs to be also estimated, leading to another source of error (the bootstrap procedure, described below, implicitly estimates this covariance).
In terms of dependence of $k$ on $n$, all other things being equal, the covariance term $\Delta_{3,n}$ exhibit the worst rate, as it constrain $k$ to be of smaller order than $n^{1/5}$ in order to guarantee asymptotic coverage of $\hat{C}_{{\widehat{S}}}$. This same term also contains the worst dependence on $u$, the uniform bound on the smallest eigenvalue of all covariance matrices of the form $\Sigma_S$, for $S \subset \{1,\ldots,d\}$ with $0 < S \leq k$. Thus, the dependence of the rates on the dimension and on the minimal eigenvalue is overall quite poor. While this is, to an extent, unavoidable, we do not know whether our upper bounds are sharp.
The reasons for replacing $u$ by the smaller term $u_n$ given in are somewhat technical, but are explained in the proof of the theorem. Assuming a scaling in $n$ that guarantees that the error terms $\Delta_{1,n}$, $\Delta_{2,n}$ and $\Delta_{3,n}$ are vanishing, such modification is inconsequential and does not affect the rates.
The coverage rates obtained for the LOCO and prediction parameter
| 3,108
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|
mathbb{Z} \times \{0\})$.
Hence $T$ tiles $X_2$.
$S_1 \cup S_2 \cup S_3$ can be partitioned into sets of the form $S = \{x_1, x_2, x_3\}$, where $x_1 = (x,y) \in S_1$, $x_2 = (x+4,y+4) \in S_2$, $x_3 = (x+2,y+3) \in S_3$. Then $|S| = 3$, so we can construct the corresponding set $Y \subset \mathbb{Z}^3$ as in Lemma \[biglemma\]. Now, given $n \in \mathbb{Z}$, $(S \times \{n\}) \setminus Y = \{x_i\}$ for some $i \in \{1,2,3\}$. Then $Y \cap (X_i \times \{n\}) = \emptyset$. If we do this for all such sets $S$, and let $U$ be the (disjoint) union of the resulting sets $Y$, then $U \cap (X_i \times \{n\}) = \emptyset$, and $\mathbb{Z}^2 \times \{n\} \subset U \cup (X_i \times \{n\})$. Recall that $T$ tiles each $Y$ and therefore $U$.
We can do this for every $n$, choosing a partial tiling $X_i$ for the corresponding $\mathbb{Z}^2$ layer. Together with $U$, these form a tiling of $\mathbb{Z}^3$ by $T$. This completes the proof of Theorem \[4mod8\], and therefore also the proof of Theorem \[mainthm\].
Open problems
=============
Theorem \[mainthm\], together with the result that a punctured interval $T = \underbrace{\texttt{XXXXX}}_{k}\!\texttt{.}\!\underbrace{\texttt{XXXXX}}_{k}$ does not tile $\mathbb{Z}^2$ for $k \geq 3$, determines the smallest dimension $d$ such that $T$ tiles $\mathbb{Z}^d$ in the cases $k$ odd and $k \equiv 4 \pmod 8$. However, for other values of $k$, it is still unknown whether the smallest such dimension $d$ is 3 or 4:
Let $T$ be the punctured interval $\underbrace{\texttt{\emph{XXXXX}}}_{k}\!\texttt{.}\!\underbrace{\texttt{\emph{XXXXX}}}_{k}$, where $k \equiv 0, 2, 6 \pmod 8$, $k \geq 6$. Does $T$ tile $\mathbb{Z}^3$?
It is also natural to consider more general tiles. The next non-trivial case is that of an interval with a non-central point removed. One might wonder if there is an analogue of Theorem \[mainthm\] for these tiles:
Does there exist a number $d$ such that, for any tile $T$ consisting of an interval in $\mathbb{Z}$ with one point removed, $T$ tiles $\mathbb{Z}^d$?
For ge
| 3,109
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| 1,542
| 0.788479
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|
ex combination of observed features and membership distributions. We present an expectation-maximization based inference algorithm that learns latent variables and parameters iteratively, a scalable stochastic variation of the inference algorithm, and a method to learn the weights of HL-MRF structured priors. We evaluate our model on six datasets across three different types of networks and corresponding modeling scenarios and demonstrate that our models are able to achieve an improvement of 15% on average in test log-likelihood and faster convergence when compared to state-of-the-art network models.'
author:
- Yue Zhang
- Arti Ramesh
title: 'Struct-MMSB: Mixed Membership Stochastic Blockmodels with Interpretable Structured Priors'
---
---
abstract: 'A starting point in the investigation of intersecting systems of subsets of a finite set is the elementary observation that the size of a family of pairwise intersecting subsets of a finite set $[n]=\{1,\ldots,n\}$, denoted by $2^{[n]}$, is at most $2^{n-1}$, with one of the extremal structures being family comprised of all subsets of $[n]$ containing a fixed element, called as a *star*. A longstanding conjecture of Chvátal aims to generalize this simple observation for all *downsets* of $2^{[n]}$. In this note, we prove this conjecture for all downsets where every subset contains at most $3$ elements.'
author:
- |
Eva Czabarka[^1]\
Glenn Hurlbert[^2] [^3]\
Vikram Kamat [^4]
title: 'Chvátal’s conjecture for downsets of small rank'
---
Introduction
============
Let $[n]=\{1,\ldots,n\}$ and let $2^{[n]}$ (resp. $\binom{[n]}{k}$) denote the family of all subsets (resp. $r$-sized subsets) of $[n]$. A set system containing sets of size $r$ ($r\geq 1$) is called $r$-*uniform*. Additionally, let $\binom{[n]}{\leq r}$ be the family of all subsets of size at most $r$, for any $1\leq r\leq n$. For a family of subsets $\cF\sse 2^{[n]}$, call $\cF$ a *downset* if $A\in \cF$ and $B\sse A$ implies $B\in \cF$. Denote by $\cF^r$ those sets of $\cF$ having size
| 3,110
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| 3,081
| 2,894
| 0.776254
|
github_plus_top10pct_by_avg
|
1
20 30 1 6
25 6 0 1
27 15 4 9
29 11 0 0
35 5 0 2
42 15 1 4
48 36 0 9
53 27 0 4
55 34 0 6
**Total** **224** **7** **55** **3.12** **0.024**
**2006** 2 6 0 0
6 11 0 2
20 2 0 0
| 3,111
| 6,875
| 1,263
| 1,271
| null | null |
github_plus_top10pct_by_avg
|
})
&h(\widetilde{{\mbox{\tiny\yng(1)}}})/h(\widehat{{\mbox{\tiny\yng(1)}}})
\end{pmatrix}\\
&=& (v_1\ v_2)
\begin{pmatrix}
\frac{Q}{Q-1} &\frac{Q}{Q-1}\\
\frac{Q(Q-2)}{Q-1} &\frac{Q(Q-2)}{Q-1}
\end{pmatrix}
\\
\rho(e_i)(v_3\ v_4) &=& (v_3\ v_4)
\begin{pmatrix}
h(\widetilde{{\mbox{\tiny\yng(2)}}})/h(\widehat{{\mbox{\tiny\yng(1)}}})
&h(\widetilde{{\mbox{\tiny\yng(2)}}})/h(\widehat{{\mbox{\tiny\yng(1)}}})\\
h(\widetilde{{\mbox{\tiny\yng(2)}}})/h(\widehat{{\mbox{\tiny\yng(2)}}})
&h(\widetilde{{\mbox{\tiny\yng(2)}}})/h(\widehat{{\mbox{\tiny\yng(2)}}})
\end{pmatrix},\\
&=& (v_3\ v_4)
\begin{pmatrix}
\frac{2(Q-2)}{Q-3} &\frac{2(Q-2)}{Q-3}\\
\frac{(Q-1)(Q-4)}{Q-3} &\frac{(Q-1)(Q-4)}{Q-3}
\end{pmatrix}\\
\rho(e_i)(v_5\ v_6) &=& (v_5\ v_6)
\begin{pmatrix}
h(\widetilde{{\mbox{\tiny\yng(1,1)}}})/h(\widehat{{\mbox{\tiny\yng(1)}}})
&h(\widetilde{{\mbox{\tiny\yng(1,1)}}})/h(\widehat{{\mbox{\tiny\yng(1)}}})\\
h(\widetilde{{\mbox{\tiny\yng(1,1)}}})/h(\widehat{{\mbox{\tiny\yng(1,1)}}})
&h(\widetilde{{\mbox{\tiny\yng(1,1)}}})/h(\widehat{{\mbox{\tiny\yng(1,1)}}})
\end{pmatrix},\\
&=& (v_5\ v_6)
\begin{pmatrix}
\frac{2Q}{Q-1} &\frac{2Q}{Q-1}\\
\frac{Q(Q-3)}{Q-1} &\frac{Q(Q-3)}{Q-1}
\end{pmatrix}.\end{aligned}$$ Here $v_i$ is the standard vector which corresponds to $p_i$. Similarly for the bases $\langle v_7, v_8\rangle$, $\langle v_9, v_{10}, v_{11}\rangle$ and $\langle v_{12}, v_{13}\rangle$, we have the following matrices respectively: $$\begin{aligned}
&&\begin{pmatrix}
\frac{3(Q-4)}{Q-5} &\frac{3(Q-4)}{Q-5}\\
\frac{(Q-2)(Q-6)}{Q-5} &\frac{(Q-2)(Q-6)}{Q-5}
\end{pmatrix},\\
&&
\begin{pmatrix}
\frac{3(Q-1)}{2(Q-2)} &\frac{3(Q-1)}{2(Q-2)} &\frac{3(Q-1)}{2(Q-2)}\\
\frac{3(Q-3)}{2(Q-4)} &\frac{3(Q-3)}{2(Q-4)} &\frac{3(Q-3)}{2(Q-4)}\\
\frac{(Q-1)(Q-3)(Q-5)}{(Q-2)(Q-4)}&
\frac{(Q-1)(Q-3)(Q-5)}{(Q-2)(Q-4)}&
\frac{(Q-1)(Q-3)(Q-5)}{(Q-2)(Q-4)}
\end{pmatrix},\\
&&
\begin{pmatrix}
\frac{3Q}{Q-1} &\frac{3Q}{Q-1}\\
\frac{Q(Q-4)}{Q-1} &\frac{Q(Q-4)}
| 3,112
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enient here, as we investigate linearity and reversibility. At the beginning, the three systems are in the state $|\Phi\rangle_A \otimes |\sigma\rangle_{BC}$. The probability of the outcome $q$ under consideration is given by $$\begin{aligned}
p_q(|\Phi\rangle_A) &=& \left\| \strut [(|\sigma_q\rangle_{AB}
\,_{AB}\langle\sigma_q|) \otimes I_C)] (|\Phi\rangle_A \otimes
|\sigma\rangle_{BC}) \right\| ^2 = \left\| {\sum\limits_i}
\big( {}_A \langle \Phi | L_q | i \rangle_B^\ast \big)
L |i\rangle_B \right\|^2
\nonumber\\
&=& \left\| {\sum\limits_i} L( |i\rangle_B\,_B \langle i |
L_q^\dag |\Phi \rangle_A) \right\|^2 = \left\| \strut LL_q^\dag
|\Phi\rangle _A \right\|^2.
\label{eq:p_q}\end{aligned}$$ On condition that the measurement yields the outcome $q$, the state of system $C$ can be written as $$\frac1{\sqrt{p_q(|\Phi\rangle_A)}} \sum_i \big( {}_{AB}\langle
\sigma_q | \Phi \rangle_A |i\rangle_B \big)
L | e_i\rangle_B = \frac1{\sqrt{p_q(|\Phi\rangle_A)}} LL_q^\dag
|\Phi\rangle_A.$$ The teleportation channel for the outcome $q$ is $$f_q \colon {{\mathcal H}}_A \rightarrowtail {{\mathcal H}}_C, \quad
f_q(|\Phi\rangle_A) = \frac{LL_q^\dag |\Phi\rangle_A}{\left\|
LL_q^\dag |\Phi\rangle_A \right\|}.
\label{eq:f_q}$$ If the input state is given by the density operator $\rho_{\mathrm{in}}$ then the probability of the outcome $q$ is $$p_q(\rho_{\mathrm{in}}) = {\mathrm{tr}}_A\left( L_q L^\dag L L_q^\dag
\rho_{\mathrm{in}} \right)
\label{eq:p_q(rho)}$$ and the output state is $$\rho_{\mathrm{out}} = \frac
{L L_q^\dag \rho_{\mathrm{in}} L_q L^\dag}
{{\mathrm{tr}}_A\big( L_q L^\dag L L_q^\dag
\rho_{\mathrm{in}} \big)}.
\label{eq:rhoout}$$
We have defined a special quantum operation based on the teleportation scheme of Ref. [@prl70_1895]. One can obtain from (\[eq:p\_q(rho)\]) that this operation is a generalized (POVM) measurement of the input state and the positive operator representing it is $L_q L^\dag L L_q^\dag$.
The channel $f_q$ has to be reversible, so that we can ob
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ensions. It turns out that a derivator is stable if and only if homotopy finite colimits and homotopy finite limits commute, and there are variants using suitable Kan extensions.
We begin by collecting the following characterizations which already appeared in the literature.
\[thm:stable-known\] The following are equivalent for a pointed derivator .
1. The adjunction $(\Sigma,\Omega)\colon{\sD}\rightleftarrows{\sD}$ is an equivalence.
2. The derivator is $\Sigma$-stable, i.e., a square in is a suspension square if and only if it is a loop square.
3. The adjunction $({\mathsf{cof}},{\mathsf{fib}})\colon{\sD}^{[1]}\rightleftarrows{\sD}^{[1]}$ is an equivalence.
4. The derivator is cofiber-stable, i.e., a square in is a cofiber square if and only if it is a fiber square.
5. The derivator is stable, i.e., a square in is cocartesian if and only if it is cartesian.
6. An $n$-cube in , $n\geq 2,$ is strongly cocartesian if and only if it is strongly cartesian.
The equivalence of the first five statements is [@gps:mayer Thm. 7.1] and the equivalence of the remaining two is [@gst:tree Cor. 8.13].
As a preparation for a minor variant we include the following construction.
In every pointed derivator there are canonical comparison maps $$\label{eq:sigma-f-c}
\Sigma F\to C\colon{\sD}^{[1]}\to{\sD}\qquad\text{\and}\qquad F\to \Omega C\colon{\sD}^{[1]}\to{\sD}.$$ In fact, starting with a morphism $(f\colon x\to y)\in{\sD}^{[1]}$ we can pass to the coherent diagram encoding both the corresponding fiber and cofiber square, $$\xymatrix{
Ff\ar[r]\ar[d]\pullbackcorner&x\ar[d]^-f\ar[r]&0\ar[d]\\
0\ar[r]&y\ar[r]&Cf.\pushoutcorner
}$$ More formally, let $i\colon[1]\to\boxbar=[2]\times[1]$ classify the vertical morphism in the middle and let $$i\colon [1]\stackrel{i_1}{\to}A_1\stackrel{i_2}{\to}A_2\stackrel{i_3}{\to}A_3\stackrel{i_4}{\to}\boxbar$$ be the fully faithful inclusions which in turn add the objects $(2,0),(2,1),(0,1),$ and $(0,0)$. In every pointed derivator we can consider the corresponding Kan extension m
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| 2,625
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| 0.782585
|
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|
ion of the functor $T_1$. The proof that $h^{-1}\cdot Y$ satisfies the first two conditions is similar to that of Lemma 3.7 of [@C2] and the rest is similar to the above case. Thus we skip them.
For (2), by using the argument explained from the last paragraph of page 479 to the first paragraph of page 480 in [@C2], it suffices to show that the functor $$\underline{M}^{\ast}(R)\times T_3(R)\longrightarrow T_3(R), ~~~~~ (m, Y)\mapsto \sigma({}^tm)\cdot Y,$$ for a flat $A$-algebra $R$, is well-defined. In other words, we only need to show that $\sigma({}^tm)\cdot Y\in T_3(R)$.
For a flat $A$-algebra, we choose an element $m\in \underline{M}^{\ast}(R)$ and $Y\in T_3(R)$ and we again express $m=\begin{pmatrix} \pi^{max\{0,j-i\}}m_{i,j} \end{pmatrix}$ and $Y=\begin{pmatrix} \pi^{max(i,j)}y_{i,j}\end{pmatrix}$. The proof that $\sigma({}^t m) \cdot Y $ satisfies the conditions (a) and (b) in the definition of $T_3(R)$ is similar to that of Lemma 3.7 of [@C2] and the rest is similar to the above case (1). Thus we skip them.
Let $\underline{G}$ be the stabilizer of $h$ in $\underline{M}^{\ast}$. It is an affine group subscheme of $\underline{M}^{\ast}$, defined over $A$. Thus we have the following theorem.
\[t38\] The group scheme $\underline{G}$ is smooth, and $\underline{G}(R)=\mathrm{Aut}_{B\otimes_AR}(L\otimes_A R,h\otimes_A R)$ for any étale $A$-algebra $R$.
The proof is similar to that of Theorem 3.8 in [@C2] and so we skip it.
As in *Case 1* mentioned in the paragraph following Theorem 3.8 of [@C2], in the theorem, the equality holds only for an étale $A$-algebra $R$ since we obtain conditions defining $\underline{M}$ by considering properties of elements of $\mathrm{Aut}_{B\otimes_AR}(L\otimes_A R,h\otimes_A R)$ for an étale $A$-algebra $R$ (cf. Section \[mc\]). For example, let $(L, h)$ be the hermitian lattice of rank 1 as given in Appendix \[App:AppendixB\]. For simplicity, let $\pi^2=2$. As a set, $\mathrm{Aut}_{B\otimes_AR}(L\otimes_A R,h\otimes_A R)$ is the same as $\{(a, b):a,b\in R \textit{ and } a
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ly the axial symmetry as does a Dirac mass term, but also the above vector symmetry under phase transformations. Hence, a Majorana mass term leads to a violation of the fermion number, again a reason why such a possibility may be contemplated for neutrinos only within the Standard Model of the quarks and leptons and their strong and electroweak interactions.
A detailed analysis, similar to that applied to the Klein–Gordon equation,[@GovCOPRO2] considering the plane wave solutions[^20] to the Dirac equation (\[eq:Dirac\]), reveals that the general solution may be expressed through the following mode expansion $$\psi(x^\mu)=\int\frac{d^3\vec{k}}{(2\pi)^32\omega(\vec{k}\,)}\,
\sum_{s=\pm}\left\{
e^{-ik\cdot x}\,u(\vec{k},s)b(\vec{k},s)\,+\,
e^{ik\cdot x}\,v(\vec{k},s)d^\dagger(\vec{k},s)\right\}\ ,
\label{eq:solution}$$ where the plane wave spinors $u(\vec{k},s)$ and $v(\vec{k},s)$ are positive- and negative-frequency solutions to the Dirac equation in energy-momentum space, $$\left[\gamma^\mu k_\mu-m\right]\,u(\vec{k},s)=0\ \ \ ,\ \ \
\left[\gamma^\mu k_\mu+m\right]\,v(\vec{k},s)=0\ .$$ The normalisation of these spinors is such that $$\sum_{s=\pm}\,u(\vec{k},s)\overline{u}(\vec{k},s)=
\left(\gamma^\mu k_\mu+m\right)\ \ \ ,\ \ \
\sum_{s=\pm}\,v(\vec{k},s)\overline{v}(\vec{k},s)=
\left(\gamma^\mu k_\mu-m\right)\ .$$ The index $s=\pm$ taking two values is related to a spin or a helicity projection degree of freedom, specifying the polarisation state of the solution. The general solution has to include a summation over the two possible polarisation states of the field. The spinors $u(\vec{k},s)$ and $v(\vec{k},s)$ thus also correspond to polarisation spinors characterising the polarisation state of the field (in the same way that a polarisation vector characterises the polarisation state of a vector field $A_\mu(x^\mu)$, such as the electromagnetic vector field).
Finally, in exactly the same manner as for the scalar field,[@GovCOPRO2] the quantities $b(\vec{k},s)$ and $d^\dagger(\vec{k},s)$ are, at the classical
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$c(\sigma_{2},\sigma_{1})$ $\ldots$ $c(\sigma_{n-1},{\sigma_{1}\sigma_{2}\ldots\sigma_{n-2}})$ $c(\sigma_{n},{\sigma_{1}\sigma_{2}\ldots\sigma_{n-1}})$ $c(\sigma_{n+1},{\sigma_{1}\sigma_{2}\ldots\sigma_{n}})$ $c(\sigma_{n+2},\sigma_{2}\sigma_{3}\ldots\sigma_{n+1})$ $c(\sigma_{n+3},\sigma_{3}\sigma_{4}\ldots\sigma_{n+2})\ldots$ $c(\sigma_{m},\sigma_{m-n}\sigma_{m-n+1}\ldots\sigma_{m-1})$,
for all ${\sigma_{1}\sigma_{2}\ldots\sigma_{m}}\in\Sigma^{+}$, is injective.
Let us take an example in order to better understand the adaptive mechanisms presented in the definition above.
Let $\Sigma=\{{\texttt{\textup{a}}},{\texttt{\textup{b}}}\}$, $\Delta=\{0,1\}$ be alphabets, and ${c:\Sigma\times\Sigma^{\leq{2}}\rightarrow\Delta^{+}}$ a function given by the table below.
One can easily verify that the function $\overline{c}$ is injective, and according to Definition 2.1, $c$ is an adaptive code of order two. Let $x={\texttt{\textup{abaa}}}\in\Sigma^{+}$. Using the definition above, we encode $x$ by $\overline{c}(x)=c({\texttt{\textup{a}}},\lambda)c({\texttt{\textup{b}}},{\texttt{\textup{a}}})c({\texttt{\textup{a}}},{\texttt{\textup{ab}}})c({\texttt{\textup{a}}},{\texttt{\textup{ba}}})=0101$.
Let ${c:\Sigma\times\Sigma^{\leq{n}}\rightarrow\Delta^{+}}$ be an adaptive code of order $n$, $n\geq{1}$. We denote by $C_{c, \sigma_{1}\sigma_{2}\ldots\sigma_{h}}$ the set
$\{c(\sigma,\sigma_{1}\sigma_{2}\ldots\sigma_{h}) \mid \sigma\in\Sigma\}$,
for all $\sigma_{1}\sigma_{2}\ldots\sigma_{h}\in\Sigma^{\leq{n}}-\{\lambda\}$, and by $C_{c, \lambda}$ the set $\{c(\sigma,\lambda) \mid \sigma\in\Sigma\}$. We write $C_{\sigma_{1}\sigma_{2}\ldots\sigma_{h}}$ instead of $C_{c, \sigma_{1}\sigma_{2}\ldots\sigma_{h}}$, and $C_{\lambda}$ instead of $C_{c, \lambda}$ whenever there is no confusion.
If $w\in\Sigma^{+}$ then we denote by $w(i)$ the $i$-th symbol of $w$. In the rest of this paper we denote by ${\it AC}(\Sigma,\Delta,n)$ the set
$\{{c:\Sigma\times\Sigma^{\leq{n}}\rightarrow\Delta^{+}} \mid$ $c$ is an adaptive code of order $n\}$
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The baseline body weights and blood glucose levels of the gerbils are shown in Table [1](#T1){ref-type="table"}. No significant differences among the various groups were evident.
######
Baseline characteristics of tested gerbils^1^.
**All tested gerbils** **Gerbils in phase C**
------------------ ------------------------ ------------------------ --------------- --------------- ----------------- --------------- --------------- ----------------- --------------- --------------- ----------------- ---------------
**Control** **PS-HOSO** **Control** **PS-HOSO**
**Total** **Non-fasting** **Fasting** **Total** **Non-fasting** **Fasting** **Total** **Non-fasting** **Fasting** **Total** **Non-fasting** **Fasting**
**n** **30** **16** **14** **30** **14** **16** **22** **10** **12** **19** **8** **11**
B
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|
leq &\left\vert \mu \right\vert ({\mathbb{R}}^{d})\times (2^{2hl(\delta
)}\theta (n_{\ast }))^{\rho _{h}(1+\delta )}=A(\delta )\theta (n_{\ast
})^{\rho _{h}(1+\delta )}.\end{aligned}$$If $l\geq l_{\ast }$ then $n(l)\geq n(l_{\ast })\geq n_{\ast }$ so that, from (\[reg11\]), $$d_{k}(\mu ,\mu _{n(l)})\leq \frac{C_{h,n_{\ast }}(\varepsilon )}{\theta
^{\rho _{h}+\varepsilon }(n(l))}\leq C_{h,n_{\ast }}(\varepsilon )\Big(\frac{%
l^{2}}{2^{2hl}}\Big)^{\rho _{h}+\varepsilon }=\frac{C_{h,n_{\ast
}}(\varepsilon )}{2^{(q+k+d/p_{\ast })l}}\times \frac{l^{2(\rho
_{h}+\varepsilon )}}{2^{2h\varepsilon l}}.$$We conclude that$$S_{2}\leq C_{h,n_{\ast }}(\varepsilon )\sum_{l=l_{\ast }}^{\infty }\frac{%
l^{2(\rho _{h}+\varepsilon )}}{2^{2h\varepsilon l}}\leq C_{h,n_{\ast
}}(\varepsilon )\times B(\varepsilon ).$$$\square $
A regularity lemma {#sect:3.2}
------------------
We give here a regularization result in the following abstract framework. We consider a sequence of operators $U_{j}:{\mathcal{S}({\mathbb{R}}^d)}%
\rightarrow {\mathcal{S}({\mathbb{R}}^d)}$, $j\in {\mathbb{N}}$, and we denote by $U_{j}^{\ast }$ the formal adjoint defined by $\langle U_{j}^{\ast }f,g\rangle =\langle f,U_{j}g\rangle $ with the scalar product in $L^{2}({\mathbb{R}}^{d})$.
\[H1H\*1\] Let $a\in {\mathbb{N}}$ be fixed. We assume that for every $%
q\in {\mathbb{N}},\kappa \geq 0$ and $p\in \lbrack 1,\infty )$ there exist constants $C_{q,\kappa ,p}(U)$ and $C_{q,\kappa ,\infty }(U)$ such that for every $j$ and $f$, $$\begin{aligned}
(H_{1})& \qquad \left\Vert U_{j}f\right\Vert _{q,-\kappa ,\infty }\leq
C_{q,\kappa ,\infty }(U)\left\Vert f\right\Vert _{q+a,-\kappa ,\infty },
\label{h1} \\
(H_{1}^{\ast })& \qquad \left\Vert U_{j}^{\ast }f\right\Vert _{q,\kappa
,p}\leq C_{q,\kappa ,p}(U)\left\Vert f\right\Vert _{q+a,\kappa ,p}.
\label{h1'}\end{aligned}$$We assume that $C_{q,\kappa ,p}(U)$, $p\in \lbrack 1,\infty ]$, is non decreasing with respect to $q$ and $\kappa $.
We also consider a semigroup $S_{t}$, $t\geq 0$, of the form $$S_{t}(x,dy)=s_{t}(x,y)dy
| 3,119
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les.
First, we shall consider a vector bundle on a trivial gerbe. Consider a vector bundle $V \rightarrow \mathfrak{X} \equiv
X \times B {\mathbb Z}_k$, so $V = p_1^* E \otimes p_2^* \zeta$ for some bundle $E \rightarrow X$ and representation $\zeta \in {\mathbb Z}_k^{\vee}$.
The inertia stack $I_{\mathfrak{X}}$ is given by $$I_{\mathfrak{X}} \: = \:
\coprod_{g \in {\mathbb Z}_k} X \times B{\mathbb Z}_k \times \{ g \}.$$ There is a forgetful map $q: I_{\mathfrak{X}}
\rightarrow X \times B {\mathbb Z}_k$.
Consider $$q^* V \: = \: \oplus_{\chi \in {\mathbb Z}_k^{\vee} } V_{\chi},$$ where $V_{\chi}$ is the $\chi$ eigenspace for the $g$ action on $q^* V$: $$q^* V|_{X \times B {\mathbb Z}_k \times \{ g \} } \: = \: V,$$ $$V_{\chi}|_{X \times B{\mathbb Z}_k \times \{ g \} } \: = \:
\left\{ \begin{array}{cl}
V & \mbox{if } \chi(g) = \zeta(g), \\
0 & \rm{else}.
\end{array} \right.$$
Now, we want to compute ${\rm ch}^{\rm rep}(V) \in
H^{\bullet}(I_{\mathfrak{X}},
{\mathbb C})$. $$V \: \mapsto \: q^* V \: = \: \oplus_{\chi} V_{\chi} \: \mapsto \:
\oplus_{\chi} V_{\chi} \otimes \chi,$$ where $V_{\chi} \otimes \chi \in K^0(I_{\mathfrak{X}}) \otimes {\mathbb C}$. (We think of $V_{\chi} \in K^0(I_{\mathfrak{X}})$, and $\chi \in {\mathbb C}$.)
Then, $${\rm ch}^{\rm rep}(V) \: = \: {\rm ch}\left( \oplus_{\chi} V_{\chi}
\otimes \chi \right) \in H^{\bullet}(I_{\mathfrak{X}},{\mathbb C}) \: = \:
\oplus_g H^{\bullet}(X),$$
$$V_{\chi} \otimes \chi |_{X \times B {\mathbb Z}_k \times \{ g \} } \: = \:
\left\{ \begin{array}{cl}
V \otimes \chi & \mbox{if } \chi(g) = \zeta(g), \\
0 & {\rm else}.
\end{array} \right.$$
Putting this together, we find $${\rm ch}^{\rm rep}(V) \: = \: \left( {\rm ch}^{\rm rep}(V)|_{(g)} \right)_{
g \in {\mathbb Z}_k},$$ where $${\rm ch}^{\rm rep}(V)|_{(g)} \: = \: \oplus_{\chi \: {\rm s.t.} \:
\chi(g) = \zeta(g) } {\rm ch}(V) \otimes \chi.$$ Similarly, $${\rm ch}^{\rm rep}(T\mathfrak{X})|_{(g)} \: = \: \oplus_{\chi \: {\rm s.t.} \:
\chi(g)=1 } {\rm ch}(T\mathfrak{X}) \otimes \chi.$$
For $g=1$, $$
| 3,120
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| null | null |
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|
{d-1} d\rho d\omega \nonumber \\
& \lesssim 1 + \lambda^{d-\gamma}. \label{ineq:KL1loc}\end{aligned}$$ Similarly, if $\gamma = d$, then for large $\lambda$, $$\int \lambda^d{\left\vert{\nabla}{\mathcal{K}}(\lambda y)\right\vert}\mathbf{1}_{B_1(0)}({\left\verty\right\vert}) dy \lesssim 1 + \log \lambda. \label{ineq:KL1loc_log}$$ If $d/(d-1) < q < \infty$, since $\gamma \geq d-1$, for $\lambda$ sufficiently large we have, $$\begin{aligned}
\int \lambda^{qd}{\left\vert{\nabla}{\mathcal{K}}(\lambda y)\right\vert}^q\mathbf{1}_{{\mathbb R}^d \setminus B_1(0)}({\left\verty\right\vert}) dy & = \int_{{\left\verty\right\vert} \geq \lambda} \lambda^{qd - d} {\left\vert{\nabla}{\mathcal{K}}(y)\right\vert}^q dy \nonumber \\
& = \lambda^{qd - d} \int_{S^{d-1}}\int_\lambda^{\infty} {\left\vert{\nabla}{\mathcal{K}}(\rho\omega)\right\vert}^q\rho^{d-1} d\rho d\omega \nonumber \\
& \lesssim \lambda^{q(d-\gamma)}. \label{ineq:K_Lq_decay}\end{aligned}$$ Similarly, $$\sup_{{\left\vertx\right\vert} \geq 1}{\left\vert\lambda^d{\nabla}{\mathcal{K}}(\lambda x)\right\vert} \lesssim 1 + \lambda^{d-\gamma}. \label{ineq:K_Linfty_decay}$$
We may now complete the general proof of Theorem \[thm:Decay\].
(**Theorem** \[thm:Decay\]) We first complete the proof of *(i)*. Lemma \[lem:rescaled\_inftybdd\] extends to the case ${\nabla}{\mathcal{K}}\not\in L^1$ provided we can bound $\vec{v} := e^{(1-\alpha-\beta)\beta^{-1}\tau} e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau} \cdot) \ast \theta$ in $L^\infty_{\eta}({\mathbb R}^d)$ uniformly in time. Indeed, fix $p > d$. Then for $M$ and ${\|\theta_0\|}_{\overline{q}}$ sufficiently small, we have by Lemma \[lem:finite\_p\_bounded\], ${\|\theta(\tau)\|}_p \in L^\infty_\tau({\mathbb R}^+)$. By Lemma \[lem:CZ\_rescale\], $${\|{\nabla}\vec{v}\|}_p \lesssim e^{(1-\alpha)\beta^{-1}\tau}{\|\theta\|}_p \lesssim e^{(1-\alpha)\beta^{-1}\tau}.$$ Let $q$ be such that $d/(d-1) < q \leq p$, which implies ${\|\theta(\tau)\|}_q \lesssim 1$. If $\gamma < d$ then by Young’s inequality, $$\begin{aligned}
{\|\vec{v}\|}_{q} &
| 3,121
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| 3,048
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|
the annular chamber.](method.eps){width="7cm"}
Experimental Results
====================
We investigated the velocity of the camphor boat on the solutions of various glycerol concentration $p$. The position of the camphor boat is described as a radial angle $\theta$ in the annular chamber, as shown in Fig. \[fig:velo\](a). Analyses of the videos captured by the digital video camera provide the position $\theta$ at time $t$, where $t=0$ corresponds to the time when the boat finished three laps along the chamber after the boat had been put on the surface of the solution. In Fig. \[fig:velo\](b), $\theta$ had a constant gradient in time, that is to say, the camphor boat moved with a constant velocity. Figure \[fig:velo\](c) shows a time series of the angular velocity $\omega = \Delta\theta/\Delta t$, where $\Delta t =1/30$ s for one frame of the video camera and $\Delta\theta$ is an angular difference between $t$ and $t+\Delta t$. In Fig. \[fig:velo\](b), the expanded plot is shown for the time region corresponding to the gray region in Fig. \[fig:velo\](c). The angular velocity $\omega$ in the region fluctuated around the average value 1.08 rad/s. The similar tendency was observed at 50 s $\lesssim t \lesssim$ 200 s, i.e. $\omega$ increased with time and had noisy data before $t\sim10$ s, and $\omega$ began to decrease after $t\sim250$ s. Therefore, we investigated $\omega$ at 60 s $\lesssim t \lesssim$ 180 s, during which $\omega$ had almost a constant value for time. Next, we investigated the angular velocity for $p$ as shown in Fig. \[fig:velo\](d). The vertical and horizontal axes in Fig. \[fig:velo\](d) show the angular velocity $\overline{\omega}$ and concentration $p$. The $\overline{\omega}$ was obtained from the linear fitting of time series as shown in Fig. \[fig:velo\](b). The values of the errors for each $\overline{\omega}$ were lower than $10^{-3}$ rad/s. As shown in Fig. \[fig:velo\](d), $\overline{\omega}$ decreased with an increase in $p$.
{width="14cm"}
Mathematical
| 3,122
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|
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|
ing the CLUSTAL X program [@pone.0041904-Thompson1] and alignment is available from the authors upon request.
10.1371/journal.pone.0041904.t001
###### HIV-1 subtype C sequences.
{#pone-0041904-t001-1}
African region Country *N* Sampling date
---------------- ------------------------------ ------- ---------------
Central Angola 31 2001--2010
Democratic Republic of Congo 22 2002--2007
Southern Botswana 70 2001
Malawi 46 2002
Mozambique 101 2002--2004
South Africa 1,031 1999--2009
Zambia 150 1998--2008
Zimbabwe 178 2007
East Burundi 92 2002
Ethiopia 102 1986--2003
Kenya 39 1991--2007
Tanzania 81 1997--2009
Uganda 38 1990--2010
Substitution saturation and likelihood mapping analyses {#s2b}
-------------------------------------------------------
Substitution saturation was evaluated by plotting the estimated number of transitions and transversions against genetic distance for each pairwise comparison in our alignment of 1,981 HIV-1 subtype C *pol* sequences using DAMBE program [@pone.0041904-Xia1]. The phylogenetic signal in the *pol* dataset was investigated with the likelihood mapping method [@pone.0041904-Strimmer1] by analyzing 10,000 random quartets. Likelihood mapping was performed with TREE-PUZZLE program [@pone.0041904-Schmidt1] using the online web platform Phylemon 2.0 [@pone.0041904-Sanchez1].
Phylogenetic analysis {#s2c}
---------------------
ML phylogenetic tre
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| null | null |
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|
hi}}_{m,n}\right) = \prod_{i=0}^{N-1}\phi^{(i)}_{m,n}=1.\end{gathered}$$
We can then show that the Lax pair (\[eq:LP-ir-g-rat\]) is compatible if and only if the system (\[eq:dLP-gen-sys-1\]) holds.
Differential-difference equations as symmetries {#continuous-defs}
===============================================
Here we briefly outline the construction of [*continuous*]{} isospectral flows of the Lax equations (\[eq:dLP-gen\]), since these define continuous symmetries for the systems (\[eq:dLP-ex-cc\]). The most important formula for us is (\[eq:phi-sys-sym\]), which gives the explicit form of the symmetries in potential form.
We seek continuous time evolutions of the form $$\begin{gathered}
\label{psit}
\partial_{t} \Psi_{m,n} = S_{m,n} \Psi_{m,n},\end{gathered}$$ which are compatible with each of the discrete shifts defined by (\[eq:dLP-gen\]), if $$\begin{gathered}
\partial_t L_{m,n} = S_{m+1,n} L_{m,n} - L_{m,n}S_{m,n}, \nonumber\\
\partial_t M_{m,n} = S_{m,n+1} M_{m,n} - M_{m,n}S_{m,n}. $$ Since $$\begin{gathered}
\partial_t (L_{m,n+1} M_{m,n} - M_{m+1,n} L_{m,n}) \\
\qquad {}= S_{m+1,n+1} (L_{m,n+1} M_{m,n} - M_{m+1,n} L_{m,n}) - (L_{m,n+1} M_{m,n} - M_{m+1,n} L_{m,n}) S_{m,n},\end{gathered}$$ we have compatibility on solutions of the fully discrete system (\[eq:dLP-gen-scc\]).
If we define $S_{mn}$ by $$\begin{gathered}
\label{Q=LS}
S_{m,n} = L_{m,n}^{-1}Q_{m,n},\qquad\mbox{where}\quad Q_{m,n}=\operatorname{diag}\big(q^{(0)}_{m,n},q^{(1)}_{m,n},\dots ,q^{(N-1)}_{m,n}\big) \Omega^{k_1},\end{gathered}$$ then $$\begin{gathered}
Q_{m,n}U_{m-1,n} - U_{m,n} \Omega^{-\ell_1} Q_{m,n}\Omega^{\ell_1} = 0, \qquad \partial_{t} U_{m,n} = \Omega^{-\ell_1} Q_{m+1,n}\Omega^{\ell_1} - Q_{m,n},\end{gathered}$$ which are written explicitly as
\[X1\] $$\begin{gathered}
q^{(i)}_{m,n} u^{(i+k_1)}_{m-1,n} = u^{(i)}_{m,n} q^{(i+k_1-\ell_1)}_{m,n}, \label{q-eqs} \\
\partial_t u^{(i)}_{m,n} = q^{(i-\ell_1)}_{m+1,n} - q^{(i)}_{m,n}. \label{eq:gen-eq-sym}\end{gathered}$$ It is also possible to prove (see [@f14-3]) that $$\be
| 3,124
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| 3,064
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|
B$ has two disjoint edges, we can use a similar argument for $\cAp$, so suppose $\cBp$ is intersecting. Without loss of generality, suppose $\cBp= \{xy^\pr,y^\pr y\}$. Then $\cAp\sse \{xy, x^\pr y^\pr\}\cup \{A\in \binom{[n]}{2}:y^\pr\in A\}$, giving the bound $|n(\cAp)|\geq |\cAp|$. This completes the proof of the claim.
\[clm2\] If $\cAp$ has a pair of disjoint edges, and $|\cBp|=1$, then $|\cAp|\leq n(\cAp)+(|S|+1)$.
Let $\{xy,x^\pr y^\pr\}$ be a pair of disjoint edges in $\cAp$, and, wlog, let $\cBp=\{xx^\pr\}$. Let $\cAp_x=\{A\in \cAp:x\in A\}$, and let $\cAp_{x^\pr}=\{A\in \cAp:x^\pr\in A\}$. Let $X=\{v\in [n]:v\neq x^\pr, xv\in \cA_x\}$, $X^*=\{v\in [n]:v\neq x, x^\pr v\in \cA_{x^\pr}\}$ and $R=X\cap X^*$. Now, $|\cAp|\leq 2|R|+|X\setminus R|+|X^*\setminus R|+1$, and $n(\cAp)=2+|R|+|X\setminus R|+|X^*\setminus R|$. So, $n(\cAp)-|\cAp|\geq -(|R|+1)$. Since $|R|\leq |S|$ (otherwise, $R$ would be a bigger sunflower with core $\{a,x\}$ (or $\{a,x^\pr\}$), contradicting the choice of $S$), we have $n(\cAp)-|\cAp|\geq -(|S|+1)$.\
In the next claim, we give lower bounds on the sizes of $\cH_a$ and $\cH_b$.
\
- $|\cH_a|\geq 1+ (|S|+n(\cAp)+1)+(|S|+|\cAp|).$
- $|\cH_b|\geq 1+ (|S|+n(\cBp)+1)+(|S|+|\cBp|).$
We will only give the proof for $\cH_a$, as the proof for $\cH_b$ follows identically. We know that $|\cH_a|=\sum_{i=1}^3|\cH_a^i|$, where $\cH_a^i=\cH_a \cap \binom{[n]}{i}$ for $i\in \{1,2,3\}$. It is trivial to note that $|\cH_a^1|=1$. Now, consider $\cH_a^2$. First, $\{a,b\}\in \cH_a^2$. Also, for every $\{a,b,s\}\in S$, $\{a,s\}\in \cH_a^2$, as $\cH$ is a downset. Similarly, for every $s\in n(\cAp)$, there exists a $t\in n(\cAp)$ such that $\{a,s,t\}\in \cI_3$, and hence, $\{a,s\}\in \cH_a^2$. Thus, $|\cH_a^2|\geq |S|+n(\cAp)+1$. Also, it is not hard to see that $|\cH_a^3| \geq |S|+|\cAp|$. This completes the proof of the claim.\
We will now prove that either $\cH_a$ or $\cH_b$ is bigger than $\cI$, which will complete the proof of the theorem. It will be sufficient to prove the following claim.
| 3,125
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z_1,z_2,z_3)}(z_1,z_2,z_3)+\eta(z_1,z_2,z_3)(\overline{z_1},\overline{z_2},\overline{z_3})].$$
The projection $\Pi$ is $\PO(2,1)$-equivariant.
Let $A\in O(2,1)$ and $[z]\in \Bbb{H}_{\Bbb{C}}^2$. Then $$\begin{array}{ll}
\Pi [Az] &=[\overline{ \eta(Az)}Az+\eta(Az)\overline{Az}]\\
&=[\overline{\sqrt{-<A z, Az>}}Az+\sqrt {-<Az,Az>}A\bar{z}]\\
&= [\overline{\sqrt{-<z,z>}}Az+\sqrt{-<z,z>}A\bar{z}]\\
&= [A][\overline{\eta(z)}z+\eta(z)\overline{z}]\\
&=[A]\Pi[z].
\end{array}$$
For simplicity in the notation, in the rest of this article we will write $Ver$ instead of $\gamma_0(Ver)$, $\psi$ instead of $\gamma_0\circ \psi$, and $\gamma_0\iota(\cdot)\gamma_0^{-1}$ instead of $\iota(\cdot)$, where $\gamma_0$ is the element given in Corollary \[l:conpo\].
\[l:prv\] The map $\Pi:Ver \cap \Bbb{H}^2_{\Bbb{C}}\rightarrow \Bbb{H}^2_{\Bbb{R}}$ is a homeomorphism.
Let us prove that the map is onto. Let $x\in\Bbb{H}^+\cup\Bbb{H}^-$ be such that $\psi(x)\in Ver\cap\Bbb{H}^2_{\Bbb{C}}$. Then $$\begin{array}{ll}
\Bbb{H}^2_{\Bbb{R}}&=\PSO^+(2,1)\Pi(\psi x)\\
&=\Pi(\PSO^+(2,1)\psi x)\\
&=\Pi(\iota\PSL(2,\Bbb{R}))(\psi(x))\\
&=\Pi(Ver\cap\Bbb{H}^2_{\Bbb{C}})
\end{array}.$$ Finally, let us prove that our map is injective. On the contrary, let us assume that there are $x,y\in Ver\cap\Bbb{H}^2_\Bbb{C}$ such that $\Pi(x)=\Pi(y)$. Now define $$\begin{array}{ll}
H_x=Isot(\PSL(2,\Bbb{R}),\psi^{-1}x),\\
H_y=Isot(\PSL(2,\Bbb{R}),\psi^{-1}y).
\end{array}$$ Clearly $H_y$ and $H_x$ are groups where each element is elliptic. On the other hand, observe that $$\begin{array}{l}
\iota H_x\Pi(x)=\Pi\iota H_x(x)=\Pi(x) \;\; \hbox{and}\\
\iota H_y\Pi(y)=\Pi\iota H_y(y)=\Pi(y).
\end{array}$$ Therefore $$\iota H_x\cup\iota H_y\subset Isot(\PO^+(2,1),\Pi x).$$ Since $\Pi(x)\in \Bbb{H}^2_{\Bbb{R}}$, we deduce that $Isot (\PO^+(2,1),\Pi x)$ is a Lie group where each element is elliptic. Therefore $H=\iota^{-1}Isot (\PO^+(2,1),\Pi x)\gamma_0$ is a Lie subgroup of $\PSL(2,\Bbb{R})$ where each element is elliptic and $H_y\
| 3,126
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| 2,666
| 2,735
| 3,675
| 0.770779
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|
\left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right) \\
&\quad - \frac{(k_{1} + k_{2})^{2} C^{2}}{4 \G(a)^{2}}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)^{2}
- \frac{(k_{1} + k_{2})^{2} b^{2}}{4 \G(a)^{2}}
\G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)^{2} \\
&\quad + \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} \G(3a)}{2 \G(a)}
- \frac{(k_{1} + k_{2})^{2} b^{2}}{2 \G(a)^{2}}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)
\g\left(3a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right). \end{aligned}$$ From this and equation $(\ref{V[L(Z)]})$, we obtain $$\begin{aligned}
&\!\!\!\!\operatorname{{V}}[\Pe(Z)] - \operatorname{{V}}[\Pe(Z + C)] \\
&= - \frac{(k_{1} + k_{2})^{2} C^{2}}{4}
+ \frac{(k_{1} + k_{2})^{2} b \lvert C \rvert}{\G(a)^{2}}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)
\G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right) \\
&\quad + \frac{(k_{1} + k_{2})^{2} C^{2}}{4 \G(a)^{2}}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)^{2}
+ \frac{(k_{1} + k_{2})^{2} b^{2}}{4 \G(a)^{2}}
\G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)^{2} \\
&\quad - \frac{(k_{1} + k_{2})^{2} b^{2} \G(2a)^{2}}{4 \G(a)^{2}}
+ \frac{(k_{1} + k_{2})^{2} b^{2}}{2 \G(a)^{2}}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)
\g\left(3a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right) \\
&= \frac{(k_{1} + k_{2})^{2} b^{2}}{4 \G(a)^{2}}
f \left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right), \end{aligned}$$ where, for $a > 0$ and $x \geq 0$, $f(a, x)$ is defined as $$\begin{aligned}
f(a, x)
&:= x^{2a} \g(a, x)^{2} - x^{2a} \G(a)^{2} + 4 x^{a} \g(a, x) \G(2a, x) \\
&\quad + \G(2a, x)^{2} - \G(2a)^{2} + 2 \g(a, x) \g(3a, x). \end{aligned}$$ Here, since $$\begin{aligned}
\frac{d}{dx} f(a, x)
&= 2 a x^{a - 1} \left\{x^{a} \g(a, x)^{2} - x^{a} \G(a)^{2} + 2\g(a, x) \G(2a, x) \right\} \\
&\quad + 2 x^{a - 1} e^{-x}
| 3,127
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|
1-\alpha)\mu t\right]}}{\leqslant}c||\rho||_\pi \,{\mathrm{e}}^{-\alpha^2\mu t/72 T}$ for $0{\leqslant}\alpha{\leqslant}1$.
Let $\Omega$ be the vertex set of the $R$-dimensional torus $\Gamma(n, R)$ and let $a$ and $b$ denote two arbitrary agents. By definition of the communication graph process, agents $a$ and $b$ are initially placed on two randomly chosen vertices of $\Gamma$, say $u_0$ and $v_0$. Note that $u_0$ and $v_0$ are independently chosen according to the stationary distribution $\pi$ of the random walk on $\Gamma(n, R)$. Now consider the trajectory of agents $a$ and $b$, which give two independent random walks $u_0, u_1,\ldots $ and $v_0, v_1,\ldots$ on $\Gamma(n, R)$. Defining $X_t=(u_t, v_t)$ for $t=0,1,\ldots$ gives a finite, ergodic Markov chain with stationary distribution $(\pi,\pi)$ on $\Omega\times\Omega$. For every $t{\geqslant}0$, define $$f(X_t)=f(u_t, v_t)= \begin{cases}
1 & \text{ if $d(u_t, v_t){\leqslant}r$ , }\\
0 & \text{ otherwise.}\\
\end{cases}$$ where $d(\cdot,\cdot)$ is the Manhattan distance for the given grid. Let $u^1_t$ and $v^1_t$ denote the projection of the random walks $u_t$ and $v_t$ onto the $1$-dimensional torus $\Gamma(n^{1/R}, 1)$, respectively, [defined by taking the first component of each of]{} the random walks on $\Gamma(n,R)$. Then $X^1_t=(u^1_t, v^1_t)$ is an ergodic Markov chain on $\Gamma(n^{1/R},1)$, and its initial distribution is stationary. We may also define $$f(u^1_t, v^1_t)= \begin{cases}
1 & \text{ if $d(u^1_t, v^1_t){\leqslant}r$ , }\\
0 & \text{ otherwise.}\\
\end{cases}$$ By the Manhattan distance property, if $f(u_t,v_t)=1$ then $f(u^1_t,v^1_t)=1$. Therefore, $${\ensuremath{\operatorname{\mathtt{vis}}(a,b)}}=\sum_{t=0}^nf(X_t){\leqslant}\sum_{t=0}^nf(X^1_t).$$ Set $\delta=\min\{1/4, 1/R\}$. Let $t_0$ be the first time when $d(u^1_{t_0}, v^1_{t_0}){\leqslant}n^{\delta}$. Consider a moving window $W$ of length $2n^{\delta}+1$, which contains the locations of $u^1_{t_0}$ and $v^1_{t_0}$. At time $t_0$, the vertices covered by $W$ are la
| 3,128
| 2,677
| 2,975
| 2,753
| 1,952
| 0.784182
|
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|
urrent. These terms were already considered in [@Ashok:2009xx] and it is straightforward to show that they do not modify . The second set contains the terms that multiply composites of (derivatives of) several currents (not including the regular terms). This includes for instance the current bilinears in equation . The crucial point is that all these terms come with a coefficient that contains at least two structure constants. This is a consequence of the discussion in appendix \[XXOPEs\]. In full generality, a term in this second set may lead to the following type of contribution to : $$\begin{aligned}
\label{j:jj:Mod} %j_{L,z}^a(z) :j_{b,L,z} j^b_{L,z}:(w) &=&
& \frac{{T^a}_b j_{L,z}^b(w)}{(z-w)^2}
+\frac{{U^a}_b \p j_{L,z}^b(w)}{z-w} + \frac{{V^a}_b \bar \p j_{L,z}^b(w)(\bar z - \bar w)}{(z-w)^2}+\frac{{\bar T^a}_b j_{L,\bar z}^b(w)(\bar z - \bar w)}{(z-w)^3} \cr
&
+\frac{{\bar U^a}_b \p j_{L,\bar z}^b(w)(\bar z - \bar w)}{(z-w)^2} + \frac{{\bar V^a}_b \bar \p j_{L,\bar z}^b(w)(\bar z - \bar w)^2}{(z-w)^3}+\frac{ {W^a}_{bc}:j^c_{L,z} j^b_{L,z}:(w) }{z-w}\cr
&
+\frac{{X^a}_{bc}:j^c_{L,\bar z} j^b_{L,z}:(w)(\bar z - \bar w)}{(z-w)^2}
+\frac{{Y^a}_{bc}:j^c_{L,\bar z} j^b_{L,\bar z}:(w)(\bar z - \bar w)^2}{(z-w)^3} \end{aligned}$$ where the tensors ${T^a}_b$, etc. are invariant two- and three-tensors made of contractions of structure constants. According to the argument of [@Bershadsky:1999hk], any invariant two-tensor obtained by contracting at least one structure constant vanishes. Moreover any invariant three-tensor obtained by contracting at least two structure constant also vanishes. Since all tensors appearing in contain at least two structure constants that come from the current-current OPE, all these terms vanish. This completes the proof of equation .
Currents as a primary fields of dimension one revisited {#TRJL}
-------------------------------------------------------
The stress-energy tensor can be written either in terms of the left or of the right currents. As a consistency check on our formali
| 3,129
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| 0.772538
|
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|
common empirical orders (EM1 for Chinese pupils and EM2 for LCSL). (a) Number of characters is set as the learning goal. (b) Accumulated usage frequency is set as the learning goal. $C_{min}$ is defined as the learning cost of $1775$ characters using the NOO method and it will be used in discussion of leaning efficiency index.](Wu_fig4a.pdf "fig:"){width="4.2cm"} ![\[fig3\] Learning efficiency comparison for different learning orders: node-offspring order (NOO), usage frequency order (UFO), distributed node weight (DNW) and two common empirical orders (EM1 for Chinese pupils and EM2 for LCSL). (a) Number of characters is set as the learning goal. (b) Accumulated usage frequency is set as the learning goal. $C_{min}$ is defined as the learning cost of $1775$ characters using the NOO method and it will be used in discussion of leaning efficiency index.](Wu_fig4b.pdf "fig:"){width="4.2cm"}
Using numerical analysis, we find that the optimal $b$ value for the DNW strategy is $b\simeq 0.35$, as discussed below. With this optimal parameter $b$, we compare our strategy of DNW learning order against the NOO and the UFO in Fig.\[fig3\]. We find in Fig.\[fig3\]a that DNW is close to NOO, regarding the total number of characters vs. the learning cost. However, in Fig. \[fig3\]b, the DNW is significantly better than NOO and even better than UFO, regarding the total accumulated usage frequency vs. the learning cost. In the left panel, NOO and DWN are much better than UFO, while in the right panel the UFO and DNW are much better than NOO. Thus, only the DNW demonstrates a high efficiency in both, accumulated frequency and total number of characters.
The DNW in the right figure appears to be only slightly better than the UFO, but this is a little misleading. From the left figure, we can see that with the same cost, say around $1000$, although the difference between the two is relatively small in the right figure, there is a much bigger difference in the left figure. It means that even though the DNW is only slightly better tha
| 3,130
| 2,078
| 1,441
| 2,462
| null | null |
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|
ined by the Jacobi identity[@Hosseiny:2014dxa]. There are various kinds of extensions, which we list here in order.
- $T$-extension is always allowable: $$\left[L_n,L_m\right]= (n-m)L_{n+m}+\frac{c_T}{12}n(n^2-1)\delta_{n+m,0}.$$ This gives the Virasoro algebra.
- $B$-extension is only allowable for $\ell=1$: $$\left[L_n,M_m\right]= (n-m)M_{n+m}+\frac{c_B}{12}n(n^2-1)\delta_{n+m,0}.$$ This gives the Galilean conformal algebra (GCA). The field theories equipped with GCA have been discussed in [@Bagchi:2009ca; @Bagchi:2009pe; @Bagchi:2016geg; @Bagchi:2017cpu].
- $M$-extension is only allowable for $d=0$, the infinite dimensional spin-$0$ Galilean algebra $$\left[M_n,M_m\right]=c_Mn\delta_{n+m,0}.$$ This is actually the algebra for the warped CFT, with $c_M$ being the Kac-Moody level.
- Infinite $M$-extensions, in which there are infinite $c_M$ charges $$[M_n,M_m]=(n-m){(c_M)}_{n+m},\hs{3ex}
[L_n,{(c_M)}_m]=-m{(c_M)}_{n+m}.$$ The familiar case is the Schrödinger-Virasoro algebra, in which $\ell=1/2$. Note that for arbitrary spin $\ell$, there could be similar algebraic structure.
Geometry
========
In this section, we discuss the underlying geometry on which the theories with anisotropic scaling and boost symmetries can be defined. Recall that a 2D $CFT$ in the Euclidean signature is defined on a two-dimensional Riemann surface, which has the translation symmetries, rotation symmetry and a scaling symmetry. More importantly the classical action is invariant under the (anti-)holomorphic transformations $$z\rightarrow f(z),\ \ \ \bar{z}\rightarrow f(\bar{z}),$$ but the partition function and correlation functions may suffer from potential quantum anomaly due to the change of the measure under the transformations. For the Galilean field theories, one needs to introduce the Newton-Cartan structure into the two-dimensional geometry to make the Galilean symmetries manifest. Furthermore, a special scaling structure is needed to define the dynamical variable, the affine connection. For the warped CFTs, the
| 3,131
| 1,615
| 1,557
| 2,893
| null | null |
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|
rho_{\text{DM}}}{0.3 \, \frac{\text{GeV}}{\text{cm}^3}}}\right)$$ oscillating at a frequency equal to the ALP mass $m_a \sim$ kHz - GHz. The expected coherence time for this oscillation is set by the ALP coherence time $\tau_a \sim \frac{1}{m_a v^2} \sim 1 \text{ s} \, \left(\frac{\text{MHz}}{m_a}\right)$, leading to a signal bandwidth $\sim 10^{-6} m_a$.
A Detection Strategy {#subsec: det strategy}
--------------------
The detection of this small but time varying energy shift requires the development of new experimental techniques. While there may be many experimental avenues that could be pursued, we highlight the approach proposed in [@NMR; @paper] utilizing NMR techniques. In this approach, a sample of nuclear spin polarized material is placed with the polarization chosen along a direction that is not collinear to the relative velocity $\vec{v}$ between the Earth and the dark matter, as in Fig. \[Fig:setup\]. An axial nuclear moment in the presence of a dark matter ALP field will cause the spins to precess around this relative velocity. This precession changes the magnetization of the material and can be measured using precision magnetometers such as SQUIDs or SERFs.
![ \[Fig:setup\] Geometry of the experiment, adapted from [@NMR; @paper]. The applied magnetic field $\vec{B}_\text{ext}$ is collinear with the sample magnetization $\vec{M}$. The relative velocity $\vec{v}$ between the sample and the dark matter ALP field is in any direction that is not collinear with $\vec{M}$. The SQUID pickup loop is arranged to measure the transverse magnetization of the sample.](setup.pdf){width="3.5"}
More specifically, the procedure is to polarize the nuclear spins of a sample of material in an external magnetic (${\vec{B}_\text{ext}}$) to achieve a net magnetization. When this net magnetization is not collinear with the dark matter velocity, the spins will precess around this relative velocity. Once they are no longer aligned with the external magnetic field, they will precess around both the relative velocity and
| 3,132
| 2,058
| 3,856
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any $j\in \mathcal{B}_2$. Here $k_j$ is the integer associated to $j$ defined in the paragraph before Equation (\[32’\]). We claim that $\widetilde{G}^{\ddag}_{\mathcal{B}_1, \mathcal{B}_2}$ is represented by a smooth closed subscheme of $ \mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$ and is isomorphic to $ \mathbb{A}^{l^{\prime}}$. Since the scheme $G^{\ddag}$ is a direct product of $\widetilde{G}^{\ddag}_{\mathcal{B}_1,
\mathcal{B}_2}$’s for any such pair of $\mathcal{B}_1, \mathcal{B}_2$ by Exercise 2.19 of [@H], the lemma follows from this claim.
It is obvious that $\widetilde{G}^{\ddag}_{\mathcal{B}_1, \mathcal{B}_2}$ is represented by a closed subscheme of $ \mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$ since the equations defining $\widetilde{G}^{\ddag}_{\mathcal{B}_1, \mathcal{B}_2}$ as a subfunctor of $ \mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$ are all polynomials. Thus it suffices to show that $\widetilde{G}^{\ddag}_{\mathcal{B}_1, \mathcal{B}_2}$ is isomorphic to an affine space $ \mathbb{A}^{l^{\prime}}$. Our strategy to show this is that the coordinate ring of $\widetilde{G}^{\ddag}_{\mathcal{B}_1, \mathcal{B}_2}$ is isomorphic to a polynomial ring, which is also used in the proof of Lemma A.8 in [@C2]. To do that, we use the following trick over and over. We consider the polynomial ring $\kappa[x_1, \cdots, x_n]$ and its quotient ring $\kappa[x_1, \cdots, x_n]/(x_1+P(x_2, \cdots, x_n))$. Then the quotient ring $\kappa[x_1, \cdots, x_n]/(x_1+P(x_2, \cdots, x_n))$ is isomorphic to $\kappa[x_2, \cdots, x_n]$ and in this case we say that *$x_1$ can be eliminated by $x_2, \cdots, x_n$*.
By the description of an element of $(\mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1)(R)$ in Remark \[ra5\], we see that $\mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$ is isomorphic to an affine space of dimension $$2\sum_{i<j}n_in_j-\sum_{\textit{i:even and $L_i$:bound of type II}}n_i+ \sum_{\textit{i:even and $L_i$:of type $I^o$}}n_i+$$ $$\sum_{\textit{i:even and $L_i$:of type $I^e$}}(3n_i-2)
+\sum_{\textit{i:odd and $L_i$:free of
| 3,133
| 2,309
| 1,441
| 2,915
| 2,513
| 0.779229
|
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|
indler trajectory of uniform linear acceleration of magnitude $g>0$, and $|\Psi\rangle$ is the Minkowski vacuum, the transition rate becomes [@schlicht] $${\dot {\cal F}}(\omega) = \frac{1}{2 \pi} \; \frac{(\omega /g)}{1+ \epsilon^2} \; \frac { e^{\frac{2\omega }{g} \tan^{-1}\left( g \epsilon \right)}}
{
e^{\frac{2 \pi \omega}{g}} -1
}
\ .
\label{schlichttrans}$$ In the limit $\epsilon \rightarrow 0$, ${\dot {\cal F}}$ reduces to the Planckian formula in the Unruh temperature $g/(2\pi)$, consistently with other ways of obtaining the response of a pointlike detector in the long time limit [@Unruh:1976db; @DeWitt:1979; @letaw; @Takagi:1986kn; @Fewster:2016ewy].
For $\epsilon$ strictly positive, ${\dot {\cal F}}$ is no longer Planckian. However, we wish to observe here that ${\dot {\cal F}}$ is still thermal, in the sense that it satisfies the detailed balance condition, $$\begin{aligned}
{\dot {\cal F}}(-\omega) = e^{\beta\omega}{\dot {\cal F}}(\omega)
\ , \end{aligned}$$ where the inverse temperature now reads $\beta = \bigl(2\pi - 4 \tan^{-1}( g \epsilon) \bigr) /g$. The temperature is thus higher than the usual Unruh temperature. This feature has to our knowledge not received attention in the literature, and we shall discuss its geometric origins in section \[discsection\].
Spatially anisotropic Lorentz-function profile {#directiondetsection}
==============================================
In this section we generalise the isotropic Lorentz-function profile to include spatial anisotropy.
General trajectory
------------------
Let $x(\tau)$ again be a timelike worldline parametrised by its proper time $\tau$, so that the four-velocity $u^a := \frac{dx^a}{d\tau}$ is a unit timelike vector. The four-acceleration vector $a^a := u^b \nabla_b u^a$ is orthogonal to $u^a$, and its direction is Fermi-Walker transported along the trajectory only when the trajectory stays in a timelike plane, as seen by considering the torsion and hypertorsion of the trajectory in the Letaw-Frenet equations [@letaw; @kolekar].
We
| 3,134
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| 0.772019
|
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|
}s}\cdot{\frac{{d}}{{d}\tau}}\varphi(y+\tau\omega,\omega,E) {d}\tau \\
={}&\Big(e^{-\int_{t}^\tau\Sigma(y+s\omega,\omega,E){d}s}
\varphi(y+\tau\omega,\omega,E) \Big|_{\tau=t}^{\tau=b^i_{y,\omega}} \Big) \\
&+\int_t^{b^i_{y,\omega}} \Sigma(y+\tau\omega,\omega,E)e^{-\int_{t}^\tau\Sigma(y+s\omega,\omega,E){d}s}
\varphi(y+\tau\omega,\omega,E) {d}\tau \\
={}&-\varphi(y+t\omega,\omega,E)+\int_t^{b^i_{y,\omega}} \Sigma(y+\tau\omega,\omega,E)e^{-\int_{t}^\tau\Sigma(y+s\omega,\omega,E){d}s}
\varphi(y+\tau\omega,\omega,E) {d}\tau
,$$ we obtain $$&-(\omega\cdot \nabla_x \psi)(\varphi) \\
={}&\int_{S\times I}\int_{G_{\omega}}\sum_i \int_{J^i_{y,\omega}} f(y+t\omega,\omega,E)\Big(-\varphi(y+t\omega,\omega,E) \\
&+ \int_{t}^{b^i_{y,\omega}} \Sigma(y+\tau\omega,\omega,E)e^{-\int_{t}^\tau\Sigma(y+s\omega,\omega,E){d}s}
\varphi(y+\tau\omega,\omega,E) {d}\tau\Big){d}t{d}y{d}\omega{d}E \\
={}&
-\int_{G\times S\times I}f(x,\omega,E)\varphi(x,\omega,E){d}x{d}\omega{d}E \\
&+\int_{G\times S\times I} \Sigma(x,\omega,E)\int_0^{t(x,\omega)} e^{-\int_0^t\Sigma(x-s\omega,\omega,E){d}s}f(x-t\omega,\omega,E){d}t
\varphi(x,\omega,E){d}x{d}\omega{d}E \\
={}&\int_{G\times S\times I} \big(-f(x,\omega,E)+\Sigma(x,\omega,E)\psi(x,\omega,E)\big)\varphi(x,\omega,E){d}x{d}\omega{d}E$$ which is what we set out to prove.
Choosing especially $\Sigma=0, \ f=1$ in Lemma \[trathle2\] we find that in the weak sense $\omega\cdot\nabla_x t=1$ in $G\times S\times I$.
We are now ready to prove the inflow trace theorem. Since ${\left\Vert \omega\right\Vert}=1$ and since the domain $G$ is bounded we have $\tau_{\pm}(y,\omega)\leq d$. Hence from [@dautraylionsv6 p. 252], [@cessenat85] or [@choulli] (where the result is considered for a more general $G$) we obtain the following theorem. For completeness we give its detailed proof.
\[tth\] The trace mappings $$\gamma_{\pm}:W^2(G\times S\times I)\to T^2_{\tau_{\pm}}(\Gamma_{\pm})$$ are (well-defined) bounded surjective operators with bounded right inverses $L_{\pm}:T^2_{\tau_{\pm}}(\Gamma_{\pm})\to
W^2(G\times S\t
| 3,135
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|
\xi_n \rbrace}({\mathit{s}}))$ is indeed a process per definition \[D:PROCESS\].
Definition \[D:PROCESS\] asserts that a process is a successively conjoint sequence of frames. To show contradiction, hypothesize that the frame sequence $\overline{\mho}_{\mathbf{F}}({\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}}))$ is *not* a process. Then there is some index $i$ such that frame $\mho_{\mathbf{F}}({\mathit{s}}_i)$ does not conjoin frame $\mho_{\mathbf{F}}({\mathit{s}}_{i+1})$.
Let ${\mathit{s}}_i$ be the $i^\text{th}$ step of walk ${\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}})$. From definition \[D:ITERATIVE\_OPERATOR\_WALK\], the succeeding step is ${\mathit{s}}_{i+1} = {\mathfrak{A}}_{\xi_i}({\mathit{s}}_i)$. By Theorem \[T:AUTOMATON\_ITERATE\_CONJOINT\], frame $\mho_{\mathbf{F}}({\mathit{s}}_i)$ conjoins frame $\mho_{\mathbf{F}}({\mathfrak{A}}_{\xi_i}({\mathit{s}}_i))$. This contradicts the conclusion drawn from the hypothesis that the frame sequence is not a process, so $\overline{\mho}_{\mathbf{F}}({\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}}))$ *is* a process.
Be a step consistent or not, in automaton-based iteration that step’s successor is consistent.
\[T:AUTOMATON\_ITERATE\_CONSISTENT\] Let ${\mathfrak{A}}$ be an automaton and ${\mathbb{S}}$ be a step space with persistent-volatile partition $\Psi = \Phi\Xi$. Suppose step ${\mathit{s}} \in {\mathbb{S}}$ and event $\xi \in \Xi$. Step ${\mathfrak{A}}_\xi({\mathit{s}})$ is consistent.
By hypothesis $\xi \in {\prod{\Xi}}$ and ${\mathit{s}} \in {\mathbb{S}}$. By definition \[D:STEP\_SPACE\] of step space, there exist locus $\lambda \in \Lambda$, frame ${\mathbf{f}} = (\psi, \phi) \in {\prod{\Psi}} \times {\prod{\Phi}}$, and functionality ${\mathit{f}} \in {\mathscr{F}}$ such that ${\mathit{s}} = (\lambda, (\psi, \phi), {\mathit{f}})$.
By Theorem \[T:AUTOMATON\_OPERATOR\] the automaton induces an iterative operator, so there exists ${\mathfrak{A}}_\xi({\mathit{s}}) = (\lambda', (\psi', \phi'), {\
| 3,136
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|
n Fig.1. A simple calculation gives the following correction $\delta Q_W/Q_W=\delta_{cd}$ related to this renormalization $$\label{cd}
\delta_{cd}={{4\alpha Z}\over{3\pi Q_W}}
(1-4\sin^2\theta_W)\ln(\lambda_C/r_0)\approx -0.1\%.$$ Where $\theta_W\approx $ is the Weinberg angle, $\sin^2\theta_W\approx
0.2230$, see Ref. [@RPP]. Note that this correction is practically independent of $Z$ because $Z/Q_W \approx -Z/N\approx -0.7$, where $N$ is the number of neutrons. One can also obtain the correction (\[cd\]) using Eqs. (2a,b), and (3b) from Ref.[@Mar1].
Next we consider the contribution of the electron self-energy operator $\Sigma$. This operator being substituted to the Dirac equation, $m \to m+\Sigma$, leads to the Lamb shift of the energy level and to the modification of the electron wave function, see, e.g., Ref. [@BLP]. As shown in Fig.1e, this modification influences the matrix element of the weak interaction. The diagram Fig.1e is not invariant with respect to the gauge transformations of the electromagnetic field. However, the sum of the diagrams Fig.1e and Fig.1f (the vertex correction) is gauge invariant. It is convenient to represent the self energy operator as a series in powers of the Coulomb field of the nucleus, $\Sigma=\Sigma_0+\Sigma_1+\Sigma_2+...$, see Fig.2.
=8.cm
We need $\Sigma({\bf r},{\bf r'}|\epsilon)$ at $r \sim r' \ll \lambda_C$, $\epsilon\approx m$. In this limit a calculation in the Feynman gauge with logarithmic accuracy gives $$\label{ss}
\Sigma_0={\hat p}{{\alpha}\over{4\pi}}\ln(p^2/m^2), \ \ \
\Sigma_1={{Z\alpha^2}\over{4\pi r}}\gamma_0\ln(p^2/m^2),$$ where ${\hat p}=p^{\mu}\gamma_{\mu}$, $\gamma_{\mu}$ is the Dirac matrix, and $p^{\mu}$ is the momentum operator. All the higher terms are not logarithmically enhanced. Further calculation in Feynman gauge is rather involved because the diagram Fig.1f is also logarithmically enhanced and there is a delicate cancellation between a part of the $\Sigma$-contribution and the logarithmic part of Fig.1f. To avoid all these complications it
| 3,137
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|
gin{aligned}
{\boldsymbol{F}}^{(0)}(t)=
-\frac{g_p}{\pi^2a^2}\int d{\boldsymbol{z}} e^{-\frac{z^2}{2 a^2}} \int d{\boldsymbol{z}}' K_0(2|{\boldsymbol{z}}-{\boldsymbol{z}}'|)
\left[\left(\partial_t-{\boldsymbol{V}}_p\cdot\nabla_{{\boldsymbol{z}}'}-\frac{\gamma}{2}\nabla^2_{{\boldsymbol{z}}'}\right)\delta{\boldsymbol{w}}^{(0)}({\boldsymbol{z}}',t) + {\boldsymbol{\dot V}}_p(t) \right] .
\label{eq:F0full}\end{aligned}$$
The above expression is a weighted average of contributions from properties of the fluid velocity in a neighborhood of the impurity center-of-mass position (${\boldsymbol{z}} = 0$ in the comoving frame). The size of this neighborhood is given by the combination of the range of the Bessel function kernel, which in dimensional units would be the correlation length $\xi$, and the range of the Gaussian potential, $a$, giving an effective particle size. In classical fluids, the analogous force on a spherical particle involves the average of properties of the undisturbed velocity field within the sphere size [@parmar2012equation], and there is no equivalent to the role of $\xi$.
As in the classical case [@maxey1983equation; @parmar2012equation], if fluid velocity variations are weak at scales below $a$ and $\xi$, we can approximate the condensate velocity by a Taylor expansion near the impurity, i.e.: $$\begin{aligned}
\delta w_i^{(0)}(\mathbf z',t)&\approx& \delta
w_i^{(0)}(t)+\sum_j
e_{ij}(t) z'_j \nonumber\\
&+&\frac{1}{2}\sum_{jk}e_{ijk}(t) z'_j z'_k + \ldots,\end{aligned}$$ where the indices $i,j,k=x,y$ denote the coordinate components. $e_{ij}(t)=\partial_j \delta w_i^{(0)}(\mathbf z,t)|_{{\boldsymbol{z}}=0}$ and $e_{ijk}(t)=\partial_j
\partial_k \delta w_i^{(0)}(\mathbf z,t)|_{{\boldsymbol{z}}=0}$ are gradients of the unperturbed condensate relative velocity. Inserting this expansion into Eq. (\[eq:F0full\]), and performing the integrals of the Gaussian and of the Bessel function (using for example $\int K_0(2|{\boldsymbol{z}}|)d{\boldsymbol{z}}=\pi/2$ and $\int z_i z_j K_0(2|{\boldsymbol{z}}|)d{
| 3,138
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| 3,080
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| 0.78353
|
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|
g:=Range("$k$20:$k$1000"), strFormulaR1C1:="=and(R[]C7=""6. Negotiate"",R[]C11<25)", intColorIndex:=3
fctApply rng:=Range("$k$20:$k$1000"), strFormulaR1C1:="=and(R[]C7=""4. Develop"", R[]C11<15)", intColorIndex:=3
fctApply rng:=Range("$k$20:$k$1000"), strFormulaR1C1:="=and(R[]C7=""5. Prove"", R[]C11<20)", intColorIndex:=3
fctApply rng:=Range("$k$20:$k$1000"), strFormulaR1C1:="=and(R[]C7=""7. Committed"", R[]C11<30)", intColorIndex:=3
fctApply rng:=Range("$k$20:$k$1000"), strFormulaR1C1:="=and(R[]C7=""Closed Won"", R[]C11<35)", intColorIndex:=3
fctApply rng:=Range("$j$22:$j$10000"), strFormulaR1C1:=200, intType:=xlCellValue, intOperator:=xlGreater, intColorIndex:=3
fctApply rng:=Range("$i$22:$i$1000"), strFormulaR1C1:=60, intType:=xlCellValue, intOperator:=xlGreater, intColorIndex:=3
With fctApply(rng:=Range("$g$20:$g$1000"), strFormulaR1C1:=0, intType:=xlCellValue, intOperator:=xlLess, intColorIndex:=3)
.Interior.Color = RGB(204, 204, 255)
.Interior.Pattern = xlSolid
End With
With fctApply(rng:=Range("$G$3:$G$7,$G$11:$G$15,$E$3:$E$7,$E$11:$E$15,$N$3:$N$7,$N$11:$N$15,$L$3:$L$7,$L$11:$L$15"), strFormulaR1C1:=0, intType:=xlCellValue, intOperator:=xlLess, intColorIndex:=3)
.Interior.Color = RGB(215, 228, 158)
.Interior.Pattern = xlSolid
End With
End Sub
Private Function fctApply(rng As Range, _
strFormulaR1C1 As Variant, _
Optional intType As XlFormatConditionType = xlExpression, _
Optional intOperator As XlFormatConditionOperator, _
Optional intColorIndex As Integer = -1, _
Optional dblRGB As Double = -1, _
Optional blnDeleteOldConditions As Boolean = False _
) As FormatCondition
Dim objCond As FormatCondition
Dim strFormula As String
If blnDeleteOldConditions Then rng.FormatConditions.Delete
strFormula = Application.ConvertFormula(strFormulaR1C1, xlR1C1, xlA1)
On Error GoTo ConvertLocal
If intOperator <> 0 Then
rng.FormatConditions.Add Type:=intType, _
Formula1:=st
| 3,139
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| 0.828315
|
github_plus_top10pct_by_avg
|
s.pdf "fig:"){width="\linewidth"}
A more detailed depiction of the previous reliability diagrams can be seen in Figure \[fig:nb:pos:scores:class\]. In this case, the posterior probabilities are not introduced in bins, but a boxplot summarises their full distribution. The first observation here is, for the *good* and *very good* classes, the uncalibrated model tends to predict probability vectors with small variance, i.e. the outputs do not change much among different instances. Among the calibration approaches, temperature scaling still maintains this low level of variance, while both isotonic and Dirichlet L2 manage to show a higher variance on the outputs. While this observation cannot be justified here without quantitative analysis, another observation clearly shows an advantage of using Dirichlet L2. For the *acceptable* class, only Dirichlet L2 is capable of providing the highest mean probability for the correct class, while the other three methods tend to put higher probability mass on the *unacceptable* class on average.
[.24]{} ![Effect of Dirichlet Calibration on the scores of Ada boost SAMME on the *car* dataset which is composed of $4$ classes (*acceptable*, *good*, *unacceptable*, and *very good*). The whiskers of each box indicate the 5th and 95th percentile, the notch around the median indicates the confidence interval. The [green]{} error bar to the right of each box indicates one standard deviation on each side of the mean. In each subfigure, the first boxplot corresponds to the posterior probabilities for the samples of class 1, divided in 4 boxes representing the posterior probabilities for each class. A good classifier should have the highest posterior probabilities in the box corresponding to the true class. In Figure \[fig:nb:pos:scores:class:adas:car:uncal\] it is possible to see that the first class (*acceptable*) is missclassified as belonging to the third class (*unacceptable*) with high probability values, while Dirichlet Calibration is able to alleviate that problem. Also, for the sec
| 3,140
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| 2,518
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|
999.
E.Gross, A.L.Read and D.Lellouch, CERN-EP/98-094.
P.Janot in [*Proceedings of the Workshop on LEP-SPS Performance*]{}, Chamonix IX, Jan. 1999, 222.
---
author:
- Wensheng Cheng
- Yan Zhang
- Xu Lei
- Wen Yang
- Guisong Xia
bibliography:
- 'segmentation.bib'
- 'change\_detection.bib'
title: Semantic Change Pattern Analysis
---
[**General aspects of heterotic string compactifications**]{}\
[**on stacks and gerbes**]{}
Lara B. Anderson$^1$, Bei Jia$^2$, Ryan Manion$^3$, Burt Ovrut$^4$, Eric Sharpe$^2$
[cc]{}
------------------------------------
$^1$ Center for the
$\: \:$ Fundamental Laws of Nature
Jefferson Laboratory
Harvard University
17 Oxford Street
Cambridge, MA 02138
------------------------------------
&
----------------------------
$^2$ Department of Physics
Robeson Hall, 0435
Virginia Tech
Blacksburg, VA 24061
----------------------------
\
--------------------------------
$^3$ Department of Mathematics
David Rittenhouse Laboratory
209 South 33rd Street
University of Pennsylvania
Philadelphia, PA 19104-6395
--------------------------------
&
------------------------------
$^4$ Department of Physics
David Rittenhouse Laboratory
209 South 33rd Street
University of Pennsylvania
Philadelphia, PA 19104-6395
------------------------------
[lara@physics.harvard.edu]{}, [beijia@vt.edu]{}, [rymanion@gmail.com]{}, [ovrut@elcapitan.hep.upenn.edu]{}, [ersharpe@vt.edu]{}
$\,$
In this paper we work out some basic results concerning heterotic string compactifications on stacks and, in particular, gerbes. A heterotic string compactification on a gerbe can be understood as, simultaneously, both a compactification on a space with a restriction on nonperturbative sectors, and also, a gauge theory in which a subgroup of the gauge group acts trivially on the massless matter. Gerbes admit more bundles than corresponding spaces, which suggests they are potentially a rich playground for heterotic string compactifications. After we give a general char
| 3,141
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| 2,522
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| null | null |
github_plus_top10pct_by_avg
|
$n^{0.6}$) 0.313 0.313 0.319 0.311 0.311 0.316 0.313
BLB($n^{0.8}$) 0.097 0.096 0.098 0.096 0.097 0.097 0.098
SDB($n^{0.6}$) 0.370 0.370 0.370 0.370 0.369 0.370 0.369
SDB($n^{0.8}$) 0.120 0.120 0.120 0.121 0.121 0.121 0.120
TB 0.035 0.035 0.035 0.035 0.035 0.035 0.035
2 K=50 0.050 0.049 0.049 0.049 0.049 0.050 0.049
K=100 0.050 0.050 0.050 0.050 0.050 0.050 0.050
K=150 0.051 0.051 0.051 0.051 0.051 0.051 0.051
mVC 0.050 0.050 0.050 0.050 0.050 0.050 0.050
mMSE 0.047 0.047 0.047 0.047 0.047 0.047 0.047
BLB($n^{0.6}$) 0.416 0.424 0.423 0.422 0.419 0.424 0.423
BLB($n^{0.8}$) 0.131 0.133 0.130 0.132 0.132 0.133 0.132
SDB($n^{0.6}$) 0.503 0.502 0.500 0.500 0.499 0.500 0.501
SDB($n^{0.8}$) 0.162 0.162 0.162 0.162 0.162 0.162 0.162
TB 0.047 0.047 0.047 0.047 0.047 0.047 0.047
3 K=50 0.082 0.082 0.082 0.082 0.082 0.082 0.082
K=100 0.083 0.083 0.083 0.083 0.083 0.083 0.083
K=150 0.084
| 3,142
| 5,516
| 1,306
| 2,351
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|
="table"}** for the descriptive data of latency and error rate.
######
Working memory capacity and attentional control in Experiment 1 (means, with standard deviations in parentheses).
Indicators Low WMC High WMC
------------------------- ------------------------- ------------------ ------------------ ------------------
**WMC**
OSPANs 11.679 (3.418) 22.875 (5.319)
**Attentional control**
Latency 391.094 (43.065) 407.443 (44.490) 354.965 (38.808) 370.109 (43.580)
Error rate 0.233 (0.194) 0.261 (0.194) 0.213 (0.155) 0.252 (0.160)
WMC, working memory capacity; SA, state anxiety; WM training, the working memory training group; Control, the control group; OSPANs, operation-word span task scores; Latency, the latency of first correct saccade; Error rate, the percentage of incorrect saccades
.
The results of 2 × 2 ANOVA for latency and error rate showed that the main effects of SA Condition were significant for both latency, *F*(1,54) = 12.988, *p* = 0.001, $\eta_{p}^{2}$ = 0.194, and error rate, *F*(1,54) = 6.199, *p* = 0.016, $\eta_{p}^{2}$ = 0.103, that is, there were significant increases in high-SA condition compared with low-SA condition for both latency (see **Figure [2A](#F2){ref-type="fig"}**) and error rate (see **Figure [2B](#F2){ref-type="fig"}**), which was consistent with H1-1, demonstrating that high-SA impairs attentional control. Furthermore, the main effects of WMC Group were significant for latency, *F*(1,54) = 12.246, *p* = 0.001, $\eta_{p}^{2}$ = 0.185, but not for error rate, *F*(1,54) = 0.103, *p* = 0.749, $\eta_{p}^{2}$ = 0.002, that is, there was a significant decrease in high-WMC group compared with low-WMC group for latency (see **Fig
| 3,143
| 2,163
| 4,022
| 3,186
| 1,260
| 0.791959
|
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|
$1$ $2.0$ $2.26$ $2.27$ $4.5$
$2$ $2.4$ $1.92$ $1.95$ $3.9$
$3$ $2.8$ $1.60$ $1.62$ $3.2$
$4$ $3.2$ $1.26$ $1.29$ $2.5$
$5$ $3.6$ $0.90$ $0.97$ $1.8$
$6$ $4.0$ $0.55$ [^1]$0.60\quad1.00$[^2] $1.1$
$7$ $5.0$ $\parallel$ $\parallel$[^3] $0.0$
----- ------- ---------------------- ------------------------- -------
: Peak positions of the radial distribution functions $g_{1\textrm{d}}(r)$, $g_{2\textrm{d}}(r)$ and instantaneous droplet diameter at different stages of the time evolution for energy ratio $k=0.1$.
\[tab:peak-position\]
We first study dumbbells built of colloids with equal diameter $\sigma_{1}=\sigma_{2}\equiv\sigma$ and different wetting properties. The parameter $\gamma _{1}$ is fixed to $100k_{\textrm{B}}T/\sigma^{2}$, while the parameter $\gamma _{2}$ is varied from $10k_{\textrm{B}}T/\sigma^{2}$ to $100k_{\textrm{B}}T/\sigma^{2}$. As a consequence, the energy ratio, Eq. \[eqn:k\], ranges from $ k=0.1-1 $. In the special case of $k=1$, colloids 1 and 2 are identical.
Figure \[fig:snap\] shows snapshots at two different stages of the simulation for the energy ratio $k=0.1$. After $2.5\times10^{5}$ MC cycles \[see Fig. \[fig:snap\](a)\] colloidal dumbbells are captured at the droplet surface. Figure \[fig:snap\](b) shows the final cluster configurations obtained after $10^{6}$ cycles. Only clusters that are stable against thermal fluctuations survived and are considered for analysis.
We analyze how colloidal dumbbells are captured by the droplet surface by means of the radial distribution functions of colloid 1-droplet, $g_{1\textrm{d}}(r)$, and colloid 2-droplet, $g_{2\textrm{d}}(r)$, defined explicitly as $g_{i\textrm{d}}(r)=\frac{dn_{i\textrm{d}}(r)}{4\pi r^{2}dr\rho_{\textrm{d}}}$ with $dn_{i\textrm{d}}(r)$ the number of dropl
| 3,144
| 3,457
| 3,606
| 3,164
| 1,955
| 0.784163
|
github_plus_top10pct_by_avg
|
erent transmit power to noise ratios are considered, i.e., $\rho=0$ dB and $\rho=10$ dB. The transmit powers of $30$ dBm and $40$ dBm are considered based on the existing studies on mmWave systems [@Khan5876482; @Pi11Aninmmvmbs; @akdeniz2014millimeter]. We note that the Gaussian approximations match the simulated PDFs. In particular, we observe from Figure \[fig:DisR\](b) that the distribution of $R_\psi$ can be well approximated by the Gaussian distribution even for relatively small numbers of clusters and paths.
![Average throughput gain versus number of reconfiguration states. The parameters are $\rho=0$ dB, $N_r=N_t=17, L_r=L_t=5, N_{\psi,{\mathrm{cl}}}=10, N_{\psi,{\mathrm{ry}}}=8,$ $\sigma_{\theta^r}=\sigma_{\theta^t}=3^\circ$, and $d/\lambda=1/2$.[]{data-label="fig:Gain"}](Gainresults){width=".9\columnwidth"}
We then show the average throughput gain of employing the reconfigurable antennas. Figure \[fig:Gain\] plots the average throughput gain, $G_{\bar{R}}$, versus the number of reconfiguration states, $\Psi$. The illustrated results are for the actual gain in by simulating the channels, ${\mathbf{H}}_{\psi}$, the theoretical approximation in , the simplified theoretical approximation for $\Psi\le5$ in , and the simplified theoretical approximation for large $\Psi$ in . As depicted in the figure, the derived theoretical approximations match precisely the simulated results. In particular, we note that the simplified approximation for large $\Psi$ in has good accuracy even when $\Psi$ is small. From all four curves, we find that the growth of $G_{\bar{R}}$ with $\Psi$ is fast when $\Psi$ is small, while it becomes slow when $\Psi$ is relatively large. This finding is consistent with the analysis in Section \[sec:appAvethrougai\], and it indicates that the dominant average throughput gain of employing the reconfigurable antennas can be achieved by having a few number of reconfiguration states.
![Outage throughput gain versus number of reconfiguration states. The parameters are $\rho=0$ dB, $
| 3,145
| 900
| 2,353
| 3,041
| 2,927
| 0.776031
|
github_plus_top10pct_by_avg
|
:={}&\int_0^E{1\over{S_0(\tau)}}d\tau, \\[2mm]
\tilde f(x,\omega,\eta):={}& S_0(R^{-1}(\eta))f(x,\omega,R^{-1}(\eta)).$$ We find that there exists a constant $C_1>0$ such that \[inv\] \_[L\^2(GSI)]{}C\_1[f]{}\_[L\^2(GSI)]{}.
Let ${ f}\in L^2(G\times S\times I)$ and let $\{f_n\}\subset C_0^\infty(G\times S\times I^\circ)$ be a sequence such that ${\left\Vert f_n-f\right\Vert}_{L^2(G\times S\times I)}\to 0$ when $n\to\infty$. Define $$\phi_n:=
{1\over{S_0(E)}}\Big(
\int_0^{r(x,\omega,E)}
e^{-\int_0^sa(x-\tau s\omega,\omega)d\tau}\tilde{f_n}(x-s\omega,\omega,R(E)+s) ds\Big).$$ We find that $$\phi_n\in C^0(\ol G\times S\times I)\cap H^1(G\times S\times I^\circ),\quad {\phi_n}_{|\Gamma_-}=0,\quad \phi_n(\cdot,\cdot,E_{\rm m})=0,$$ and then $\phi_n\in D( P_{0})$. In showing that $\phi_n\in H^1(G\times S\times I^\circ)$ notice that $$\tilde{f}_n(x-t(x,\omega)\omega,\omega,R(E)+t(x,\omega))=0,\quad \textrm{if}\ R(E_m)-R(E)>t(x,\omega),$$ and so ${{\frac{\partial t}{\partial x_j}}}$, ${{\frac{\partial t}{\partial \tilde\omega_j}}}$ do not appear in ${{\frac{\partial \phi_n}{\partial x_j}}}$, ${{\frac{\partial \phi_n}{\partial \tilde\omega_j}}}$.
By (\[inv\]) there exists $\phi\in L^2(G\times S\times I)$ such that ${\left\Vert \phi_n-\phi\right\Vert}_{L^2(G\times S\times I)}\to 0$. In addition, ${\left\Vert P_0\phi_n-f\right\Vert}_{L^2(G\times S\times I)}={\left\Vert f_n-f\right\Vert}_{L^2(G\times S\times I)}\to 0,\ n\to\infty$, which shows that $\phi\in D(\tilde P_0)$ and $\tilde P_0\phi=f$. Hence $R(\tilde P_0)=L^2(G\times S\times I)$. Consequently, in this special case these methods (based only on explicit solution formulas) give an alternative proof for the surjectivity of $\tilde P_{0}$ (and hence together with (\[inf5\]) the $m$-dissipativity of $-\tilde P_{0}$; cf. the treatise of $A_0$ in [@tervo14 proof of Theorem 4.7]).
We return to the existence and uniqueness of solutions for the following problem. Given ${\bf f}\in L^2(G\times S\times I)$, find $\phi\in L^2(G\times S\times I)$ such that $$\begin{gathered}
-
| 3,146
| 1,596
| 1,592
| 3,065
| null | null |
github_plus_top10pct_by_avg
|
bjects commute with left Kan extensions). And semi-additive derivators are precisely the left or right $\mathsf{FINDISC}$-stable ones, where $\mathsf{FINDISC}$ is the class of finite discrete categories. In general, this notion of “relative stability” yields a Galois connection between collections of derivators and classes of functors.
To understand relative stability better, we introduce *enriched* derivators and weighted colimits. These build on the theory of monoidal derivators developed in [@gps:additivity; @ps:linearity], extending the classical theory of enriched categories to the context of derivators. Just as every ordinary category is enriched over the category of sets, every derivator is enriched[^1] over the derivator of spaces; whereas pointed derivators are automatically enriched over pointed spaces, and stable ones over spectra. For any -enriched derivator we have a notion of limit or colimit weighted by “profunctors” in , which includes the ordinary homotopy Kan extensions that exist in any derivator.
With the technology of enriched derivators, we can prove the following general characterization of relative stability (\[thm:stab-op\]):
The following are equivalent for a derivator and a class $\Phi$ of functors.
1. is left $\Phi$-stable, i.e. left Kan extensions along functors in $\Phi$ commute with arbitrary right Kan extensions in .
2. is right $\Phi\op$-stable, i.e. right Kan extensions along functors in $\Phi\op$ commute with arbitrary left Kan extensions in .
3. Left Kan extension functors $u_! : {\sD}^A \to {\sD}^B$ for functors $u\in \Phi$ are right adjoint morphisms of derivators.
4. Right Kan extension functors $(u\op)_\ast : {\sD}^{A\op} \to {\sD}^{B\op}$ for functors $u\in \Phi$ are weighted *colimit* functors relative to some over which is enriched.\[item:ie\]
This gives some additional conceptual explanations for why certain limits and colimits commute: if a colimit functor is a right adjoint, then of course it commutes with all limits; whereas if a limit functor
| 3,147
| 3,912
| 3,481
| 3,008
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nd spiral antennas. Furthermore, and exploiting the unique capabilities of CL excitation, we measured the emission from metals and semiconductors. For these materials, we can separate coherent and incoherent emission mechanisms, with further applications in nanoscale materials science.
CL Polarimetry
==============
{width="70.00000%"}
In our measurements, the $30$ keV electron beam from a scanning electron microscope (SEM) excites the sample. An aluminum paraboloid mirror collects and redirects the resulting CL emission out of the SEM. The outcoming beam is focused onto a fiber-coupled spectrometer or projected onto a 2D CCD array [@coenen_NL11; @coenen_APL11; @Sapienza_NM12], as shown in Fig. \[Fig1\](a). The wave-vector distribution of the CL emission can be retrieved from the CCD image, as every transverse point in the beam corresponds to a unique emission angle, in a procedure analogous to other Fourier imaging techniques [@Lieb_JosaB04; @Kosako_NP10; @curto10; @Aouani_NL11; @Sersic_NJP11; @Belacel_NL13].
Measuring polarization for all emission angles of CL presents several challenges. First, it requires determining the relative phase difference between field components, a task not achievable with only linear polarizers as in Ref. [@coenen_OE12]. Second, the paraboloid mirror performs a non-trivial transformation on the signal as it propagates from the sample to the detector plane. The shape of the mirror introduces a rotation of the vector components of light due to the coordinate transformation and, consequently, a change in the main polarization axes. In addition, the angle and polarization-dependent Fresnel coefficients of the mirror modify the polarization of the light upon reflection [@Bruce_OPT06; @Bruce_04]. As a function of the angle of incidence, the mirror partially polarizes unpolarized light and transforms linearly to elliptically polarized light.
To address these challenges, we included a rotating-plate polarimeter in the beam path of our CL system, composed of a qua
| 3,148
| 2,073
| 3,939
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\ .$$ Thus likewise for the fermionic algebra, let us take $$b=\frac{\partial}{\partial\theta}\ \ \ ,\ \ \ b^\dagger=\theta\ ,$$ where it is understood that all derivatives with respect to Grassmann odd variables are taken from the left (left-derivatives). Consequently the supersymmetry generators are represented by $$Q=\sqrt{\hbar\omega}\,z\frac{\partial}{\partial\theta}\ \ \ ,\ \ \
Q^\dagger=\sqrt{\hbar\omega}\,\theta\frac{\partial}{\partial z}\ ,$$ leading to the representation for the Hamiltonian, $$H=Q^\dagger Q+Q Q^\dagger=\hbar\omega\left[a^\dagger a+b^\dagger b\right]=
\hbar\omega\left[z\frac{\partial }{\partial z}
+\theta\frac{\partial}{\partial\theta}\right]\ .$$
These operators thus act on wave functions $\psi(z,\theta)$. Because of the Grassmann property $\theta^2=0$, a power series expansion of such a function terminates at a finite order, in the present case at first order since only one $\theta$ variable is involved, $$\psi(z,\theta)=\psi_B(z)+\theta\psi_F(z)\ \ ,\ \
\psi_F(z)=\frac{\partial}{\partial\theta}\psi(z,\theta)\ ,$$ where, assuming that $\psi(z,\theta)$ itself is Grassmann even, the bosonic component $\psi_B(z)$ is Grassmann even while the fermionic one $\psi_F(z)$ is Grassmann odd, as it should considering the analogous structure of the space of quantum states. In particular, the general wave function representing the energy eigenstates $|n,0\rangle$ and $|n-1,1\rangle$ with value $E(n)=\hbar\omega n$ is given as $$\psi_n(z,\theta)=B_n\frac{z^n}{\sqrt{n!}}\,+\,
F_n\theta\frac{z^{n-1}}{\sqrt{(n-1)!}}\ ,$$ where $B_n$ and $F_n$ are arbitrary phase factors associated to the bosonic and fermionic components of this wave function.
The supersymmetry charges $Q$ and $Q^\dagger$ act on such general wave functions as $$Q\psi(z,\theta)=\sqrt{\hbar\omega}\ z\psi_F(z)\ \ ,\ \
Q^\dagger\psi(z,\theta)=\sqrt{\hbar\omega}\ \theta\partial_z\psi_B(z)\ .$$ Thus introducing a complex valued Grassmann odd constant parameter $\epsilon$ associated to the symmetries generated by the supercharges $Q$ and $
| 3,149
| 3,873
| 3,181
| 2,845
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|
\phi\rangle|^2) \,h_\alpha^{\dag}
h_\beta \;\;\label{eq:lagrangian}$$ where $\phi$ is the Standard Model Higgs doublet, and $i$ is summed over the down quark flavors ($i=d,s,b$). The vector quark has purely vectorial coupling to the photon and $Z$ boson, with respective charges $(Q_Q, -Q_Q \sin^2\theta_W)$, while the charged Higgs couples with charges $(Q_h, -Q_h \sin^2\theta_W)$ and $Q_Q = Q_d + Q_h$. The neutral Higgs sector is identical to that in the Standard Model, with neither flavor changing couplings nor CP violation. The matrices $m^2$ and $\kappa$ are hermitian. Except for the discussion at the end, we assume that CP is broken softly in this Lagrangian, implying a special basis where all the Yukawa ($\lambda, \kappa$) and the SM couplings are real. We also require (see below) that dim-3 couplings, namely $M_Q$, are also real. This leaves, as in the KM model, only a single CP violating parameter: Im$(m^2)_{12}$. We can diagonalize $(m^2)_{\alpha\beta}$ by a unitary matrix $U_{\alpha i}$ which in general is complex: $h_\alpha =
U_{\alpha i} H_i$, with $H_i$ the mass eigenstates. The quark-Higgs interaction in the mass eigenstate basis is $${\cal L}_{QqH}=g\sum_{q=d,s,b}\xi_{qj}
(\bar Q_L q_R)H^-_j \ +\ \hbox{h.c.}
\ ,
\label{eq:QqH}$$ with $\xi_{qj} \equiv \lambda_{q\alpha} U_{\alpha j}$. The CP-violating transit propagators[@weinberg] can be expressed as $
\langle h_\alpha^{\dag} h_\beta \rangle
= \sum_{i,j=1,2} U_{i \alpha }^{\dag} U_{\beta j}
\langle H_i^{\dag} H_j \rangle
= \sum_{i=1,2} U_{\beta i} U_{i \alpha }^{\dag}
\langle H_i^{\dag} H_i \rangle
.
$ With $m_1$ $(m_2)$ the mass of the lighter (heavier) charged Higgs, CP violation explicitly vanishes if $m_1=m_2$. In the limit that $m_2 \gg m_1$, these expressions reduce to $\langle h_\alpha^{\dag}
h_\beta \rangle = U_{\beta 1} U_{1 \alpha }^{\dag}
/(p^2-m_1^2+i\epsilon)\, , $ where $p$ is the momentum flowing in the propagator. The rephasing-invariant measures of CP violation are then ${\cal A}_{qq'} = \lambda_{q
| 3,150
| 1,738
| 2,633
| 2,954
| null | null |
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|
)/(K_iL_i-1,K_\beta ^{{b^{}}(\beta )}-1\,|\,i\in I,
\beta \in R^\chi _+),$$ where ${b^{}}(\beta )$ is the order of $q^{(\beta ,\beta )}$ for all $\beta \in R_+$, is isomorphic to Lusztig’s small quantum group $u_q({\mathfrak{g}})$. This was observed *e.g.* in [@inp-AndrSchn02 Thm.4.3] by referring to results of Lusztig, de Concini, Procesi, Rosso, and Müller.
Similarly to Eq. and Def. \[de:Shapdet\] one defines the Shapovalov form and the Shapovalov determinant $(\det _{\alpha })_{{\alpha }\in {\mathbb{N}}_0^I}$ of $U_q({\mathfrak{g}})$ and $u_q({\mathfrak{g}})$, respectively. Alternatively, since $K_iL_i$ for $i\in I$ and $K_\beta ^{\bfun{}(\beta )}$ for $\beta \in R^\chi _+$ (the latter only if $q$ is a root of $1$) are central elements in $U(\chi )$ for all $i\in I$, the Shapovalov form can also be obtained from the definition in Sect. \[sec:shapdet\] via Lemma \[le:Uzideal\].
\[th:ShapdetUqg\] Let $I$, $C$, $(d_i)_{i\in I}$, and ${\mathfrak{g}}$ as above. Let $q\in {{\Bbbk }^\times }$. Assume that $q^{2m}\not=1$ for all $m\in {\mathbb{N}}$ with $m\le \max \{d_i\,|\,i\in I\}$.
\(i) [@inp-dCK90] If $q$ is not a root of $1$, then the Shapovalov determinant of $U_q({\mathfrak{g}})$ is the family $(\det _{\alpha })_{{\alpha }\in {\mathbb{N}}_0^I}$, where $$\begin{aligned}
\label{eq:detUqg}
\det \nolimits _{\alpha }=
\prod _{\beta \in R_+}
\prod _{t=1}^\infty
(q^{2\rho (\beta )}K_{\beta }
-q^{t(\beta ,\beta )} K_{\beta }^{-1})
^{{P}({\alpha },\beta ;t)}.
\end{aligned}$$
\(ii) Assume that $q$ is a root of $1$. Then the Shapovalov determinant of $u_q({\mathfrak{g}})$ is the family $(\det _{\alpha })_{{\alpha }\in {\mathbb{N}}_0^I}$, where $$\begin{aligned}
\label{eq:detuqg}
\det \nolimits _{\alpha }=
\prod _{\beta \in R_+}
\prod _{t=1}^{\bfun{}(\beta )-1}
(q^{2\rho (\beta )}K_{\beta }
-q^{t(\beta ,\beta )} K_{\beta }^{-1})
^{{P}({\alpha },\beta ;t)}.
\end{aligned}$$
Let $\chi \in {\mathcal{X}}$ with $\chi ({\alpha }_i,{\alpha }_j)=q^{d_ic_{ij}}
| 3,151
| 2,605
| 2,705
| 2,953
| 3,746
| 0.770391
|
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|
" 17
" Tampa " 9
Miami " 2
" Governor's Island " 14
" Bedloe's Island " 3
" Seavey's Island " 3
" Fort Wadsworth " 20
To Fortress Monroe " 5
" Fort Riley " 1
" Fort Hamilton " 18
" Fort McPherson " 4
" Quarantine " 5
" Bellevue Hospital " 6
" Roosevelt Hospital " 2
" Brooklyn Hospital " 3
" St. Peter's Hospital " 6
" St. Francis' Hospital " 2
" St. Catherine's Hospital " 2
" St. Joseph's Hospital " 4
" Yonkers Hospital " 4
" Mount Vernon Hospital " 4
" New Rochelle Hospital " 4
" Jamaica Hospital " 1
" Nassau Hospital " 4
" Long Island College Hospital " 6
" Long Island Red Cross Emergency Hospital " 22
" Stapleton Marine Hospital " 1
" U.S.S. "St Paul" " 1
" " "New Hampshire" " 1
" " "Nahant" " 1
" " "Harvard" " 1
" " "Kanawha" " 1
" " "Elfrida" " 1
" " "Vigilancia" " 1
" " "Supply" "
| 3,152
| 5,716
| 1,353
| 2,063
| null | null |
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|
act, the first equation in is equivalent to $
\frac{2(\gamma-1)}{a}\eta(t)=\frac{\sigma'}{\sigma}+2\gamma{\kappa}.
$ Put this into the last two equations in , we have $$\left\{
\begin{array}{rl}
(\sigma\alpha)'=&\sigma'+2\gamma{\kappa}\sigma\\
(\sigma\varphi)'=&\frac{a\sigma}{4}\Big(\frac{\sigma'}{\sigma}+2\gamma{\kappa}\Big)^2,
\end{array}
\right.$$ Integral above identities on $[0,t]$, we can obtain the explicit expressions of $\alpha(t)$ and $\varphi(t)$ in .
Since $\gamma>1$, $a>0$ and $M$ is closed, the standard parabolic maximum principle in implies $F_{\alpha}\ge0$, that is in Theorem \[pmeGK\].
Let $\varsigma(t)$ be a constant speed geodesic with $\varsigma(t_1)=x_1$ and $\varsigma(t_2)=x_2$ such that $|\dot{\varsigma}(t)|=\frac{d(x_2,x_1)}{t_2-t_1}$. Using differential Harnack estimate and Young inequality, we have
$$\begin{aligned}
v(x_2,t_2)-v(x_1,t_1)=&\int^{t_2}_{t_1}v_t+\langle\nabla v,\dot{\varsigma}(t)\rangle dt\\
\ge&\int^{t_2}_{t_1}\left(\frac1{\alpha(t)}|\nabla v|^2-\frac{\varphi(t)}{\alpha(t)}v-\frac1{\alpha(t)}|\nabla v|^2-\frac{1}{4}\alpha(t)|\dot{\varsigma}(t)|^{2}\right)dt\\
\ge&-v_{max}\int^{t_2}_{t_1}\frac{\varphi(t)}{\alpha(t)}dt
-\frac{1}{4}\frac{d(x_2,x_1)^{2}}{(t_2-t_1)^{2}}\int^{t_2}_{t_1}\alpha(t)dt\end{aligned}$$
and
$$\begin{aligned}
\log\frac{v(x_2,t_2)}{v(x_1,t_1)}
=&\int^{t_2}_{t_1}\left(\frac{d}{dt}\log v(x,t)+\nabla\log v\cdot\dot{\varsigma}(t)\right)dt\\
\ge&\int^{t_2}_{t_1}\left(\frac{1}{\alpha(t)}\Big(|\nabla v|^2-\varphi(t)\Big)
-\frac1{\alpha(t)}|\nabla v|^2-\frac{1}{4}\frac{|\dot{\varsigma}(t)|^{2}}{v_{max}}\alpha(t)\right)dt\\
\ge&-\int^{t_2}_{t_1}\frac{\varphi(t)}{\alpha(t)}dt
-\frac{1}{4}\frac1{v_{max}}\frac{d(x_2,x_1)^{2}}{(t_2-t_1)^{2}}\int^{t_2}_{t_1}\alpha(t)dt.\end{aligned}$$
This finishes the proof of Corollary \[Harnack\].
Since $\alpha(t)>1$, direct calculation implies that
$$\begin{aligned}
\Delta(v^{\beta})=&\beta v^{\beta-1}\left(\Delta v+(\beta-1)\frac{|\nabla v|^2}{v}\right)\\
=&\frac{1}{\alpha (\gamma-1) }\beta v^{\beta-1}\left(\alpha (\gam
| 3,153
| 2,079
| 2,397
| 2,911
| null | null |
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|
\[fig:distrib\]a–c. The parameter sets of the model are given in Table \[tab:th\]; sets 1 and 2 correspond to small scattering length ($a=-4$ fm) and different weights of $s$-wave (largest and lowest possible), set 3 has $a=-25$ fm and largest possible weight of $s$-wave. It can be seen that the agreement with the data deteriorates when the population of the $s$-wave continuum falls, say, below $15-25 \%$ of the $p$-wave. On the other hand the large negative scattering length has a drastic effect below 0.5 MeV. The energy resolution and the quality of the measured angular distributions are not sufficient to draw solid conclusions about the exact properties of the $s$-wave contribution. The situations with the large contribution of the $s$-wave cross section but with moderate scattering length (say $a
> -20$ fm) seem to be more plausible. Measurements with better resolution are required to refine the properties of the $1/2^+$ continuum.
Position of the $d$-wave resonance is not well defined in our analysis of data due to the efficiency fall in the high-energy side of the spectrum. This can be well seen from the comparison of theoretical inputs and MC results in Fig.\[fig:distrib\]a–c. The lower limit for the resonance energy of 4.2 MeV is in a good agreement with the value 4.0 MeV found in [@gol03]. A broader energy range measured for $^9$He is needed to resolve the $5/2^+$ state completely and to make the angular distribution analysis more restrictive.
*Discussion.* — It should be noted that the interference of any other combination of $s$- $p$- $d$-wave states [*can not*]{} lead to the required forward-backward asymmetry in the whole energy range. The correlation terms \[square brackets in Eq. (\[eq:sigma-full\])\] are $$\begin{aligned}
%
\left[\rule{0pt}{9pt}\ldots \right] &= &2 A_{00} + 2 A_{11} + (1+3x^2) A_{22}
+ 4 x \cos(\phi_{10}) A_{10}
\nonumber \\
%
& +& 2 \sqrt{2} (3x^2 - 1) \cos(\phi_{20}) A_{20}\, ,
\nonumber \\
%
\left[ \rule{0pt}{9pt}\ldots \right] &=& 4 A_{00} + 2 (1+3x^2) A_{11} +
3
| 3,154
| 1,147
| 2,378
| 2,961
| null | null |
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|
must have $2 h=2v'-v\in
L$. The length of the nonzero vector $2h$ must then be at least $\mu(L)$. Since $|h|={\operatorname{area}}(L)/|v|$ this gives $2{\operatorname{area}}(L)/|v|\geq \mu(L)$, that is $$|v|\leq \frac{2{\operatorname{area}}(L)}{\mu(L)}$$ Hence $v'$ is uniquely determined if $|v|>2{\operatorname{area}}(L)/\mu(L)$.
Let $\alpha=\alpha_{v,v'}$ be the angle between $v$ and $v'$, which takes values between $0$ and $\pi$ since ${\operatorname{Im}}(v'/v)>0$. As is easily seen, for any choice of $v'$, $\sin\alpha_{v,v'}$ shrinks as we increase $|v|$, in fact we have:
\[lem:angle\] For any choice of $v'$ we have $$\label{upper bd on alpha}
\sin \alpha \leq \frac{{\operatorname{area}}(L)}{\mu(L)}\frac 1{|v|} \;.$$
To see , note that the area of the fundamental parallelogram $P(v,v')$ is given in terms of $\alpha$ and the side lengths by $${\operatorname{area}}(P) =|v| |v'|\sin \alpha$$ and since $v'$ is a non-zero vector of $L$, we necessarily have $|v'|\geq \mu(L)$ and hence, since ${\operatorname{area}}(P)={\operatorname{area}}(L)$ is independent of $v$, $$0<\sin \alpha \leq \frac{{\operatorname{area}}(L)}{\mu(L)|v|}$$ as claimed.
Note that if we take for $v'$ with minimal length, then we have a lower bound $\sin\alpha \geq 2{\operatorname{area}}(L)/|v|^2 +O( 1/|v|^6)$ obtained by inserting into the area formula ${\operatorname{area}}(L)=|v||v'|\sin\alpha$.
Given a positive basis $\{v,v' \}$, we define a measure of skewness of the fundamental parallelogram as follows: Let $\Pi_v(v')$ be the orthogonal projection of the vector $v'$ to the line through $v$. It is a scalar multiple of $v$: $$\Pi_v(v')= {\operatorname{sk}}(v,v') v$$ where the multiplier ${\operatorname{sk}}(v,v')$, which we call the [*skewness*]{} of the parallelogram, is given in terms of the inner product between $v$ and $v'$ as $$\label{exp ecc}
{\operatorname{sk}}(v,v') = \frac{\langle v',v\rangle}{|v|^2} \;.$$ Thus we see that the skewness is the real part of the ratio $v'/v$: $${\operatorname{sk}}(v,v') = {\operatorname{Re}}(v'
| 3,155
| 4,464
| 3,106
| 2,737
| null | null |
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|
suppression due to the large energy denominators is not fully effective. In this region outside validity of the theorem we show that second order correction terms in $W$, together with the leaking term $\mathcal{C}_{\alpha \beta}$, may not be totally negligible, and it could be detectable. If it were the case, it could distinguish low-scale unitarity violation from high-scale one.
Small unitarity-violation perturbation theory of neutrino oscillation in matter {#sec:formulation}
=================================================================================
We formulate a perturbation theory of the $(3+N)$ state unitary model using an expansion parameter of matrix elements of $W$ signifying unitarity violation effects, assuming it small. In the main text we mostly confine ourselves to the formulas to second order in $W$, but include fourth order terms whenever it is necessary. Our formulation of the perturbative framework in this section will be done aiming at constructing a model-independent framework for leptonic unitarity test. Usage of the same probability formula as a hunting tool of unitarity violation and discriminator between low-scale and high-scale unitarity violation will be discussed in section \[sec:UV-low-high-E\].
For simplicity, we take the uniform number density approximation for electrons and neutrons in matter. However, extension to the varying density case is, in principle, straightforward as far as adiabaticity holds.
3 active plus $N$ sterile neutrino system in the flavor basis {#sec:flavor-basis}
-------------------------------------------------------------
The $S$ matrix describes possible flavor changes after traversing a distance $x$ $$\begin{aligned}
\nu_{\alpha} (x) = S_{\alpha \beta} \nu_{\beta} (0),
\label{def-S}\end{aligned}$$ and the oscillation probability is given by $$\begin{aligned}
P(\nu_{\beta} \rightarrow \nu_{\alpha}; x)=
\vert S_{\alpha \beta} \vert^2.
\label{def-P}\end{aligned}$$ The neutrino evolution in flavor basis in the $(3+N)$ space unitary model is go
| 3,156
| 1,310
| 3,027
| 3,191
| null | null |
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|
{1/2}\lambda_{DE}\right)^3 \int
d^3\mathbf{{u}}
&{}&
\Bigg\{
\frac{1}{2\alpha_\Lambda}
\Bigg\vert \mathbf{\nabla}
\left(\frac{\Lambda_{DE}}{8\pi\rho}\right)^{\alpha_\Lambda}
\Bigg\vert^2
-
\frac{\alpha_\Lambda}{\alpha_\Lambda-1}
\left(\frac{\Lambda_{DE}}{8\pi\rho}\right)^{\alpha_\Lambda-1}
+
\nonumber
\\
&{}&
\left(\frac{\Lambda_{DE}}{8\pi\rho}\right)^{\alpha_\Lambda} \frac{8\pi
f({u})}{\Lambda_{DE}}\Bigg\},
\label{free-energy}\end{aligned}$$ which we identify as a free energy functional; here, $u=r/\chi^{1/2}\lambda_{DE}$. For $\gamma=0$, Eq. $(\ref{rhoGEOM})$ gives $\rho(r) = \rho_H$ in Region I; the free energy for this solution is ${}^I\mathcal{F}_{\gamma=0} = -\Lambda_{DE}r_H^3 \left( \Lambda_{DE}/8\pi\rho_H
\right)^{\alpha_\Lambda-1}/6(\alpha_\Lambda-1)$. While for $\gamma>0$ perturbative solutions can be found, all such solutions have a $^{I}\mathcal{F}_\gamma$ greater than ${}^I\mathcal{F}_{\gamma=0}$ [@ADS]. This results because $\sim \vert\nabla
\rho\vert^2 \ge0$ in Eq. $(\ref{free-energy})$; just as in a Landau-Ginzberg theory, $\vert\nabla\rho\vert^2$ only vanishes for the constant density solution.
For Region II, the density, $\rho_{II}$, is first found asymptotically in the large $r$ limit. With the anzatz $f(r)\ll\rho(r)$ for large $r$, Eq. $(\ref{rhoGEOM})$ reduces to a homogeneous equation [@ADS] with the solution $\rho_{\hbox{\scriptsize{asymp}}}
(u)= \Lambda_{DE}
\Sigma({\alpha_\Lambda})/8\pi u^{\frac{2}{1+\alpha_\Lambda}}$, where $\Sigma({\alpha_\Lambda}) =
\left[2(1+3\alpha_\Lambda)/
(1+{\alpha_\Lambda})^2\right]^{\frac{1}{1+\alpha_\Lambda}}$. To include the galaxy’s structural details, we take $\rho_{II}(r) =
\rho_{\hbox{\scriptsize{asymp}}}(r) +
\rho^1_{II}(r)$ and to first order in $\rho_{II}^{1}$, $$\rho_{II}(r) = \rho_{\hbox{\scriptsize{asymp}}}(r) + \frac{1}{3}
A_\beta \rho_H \left(\frac{r_H}{r}\right)^\beta+
\left(\frac{r_H}{r}\right)^{5/2}
\left(C_{\cos}\cos\left[\nu_0\log r/r_H\right] +
C_{sin}\sin\left[\nu_0\log r/r_H\right]\right).
\lab
| 3,157
| 4,075
| 2,683
| 2,778
| null | null |
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|
onumber\\
&(-a^2m^2+1/10e^4)r+a^2e^2m)\alpha r^5\cos(\theta)+(-2a^2\alpha^2m^2+2m^2)r^7-5e^2 mr^6+3e^4r^5\Big)(r\alpha \cos(\theta)-1)^5\bigg).\end{aligned}$$
In FIG. \[fig7\] we have shown the variation of the gravitational entropy density with the radial distance and the acceleration parameter using this new definition (\[mod\_P\]) of the scalar $ P $. The entropy density function is now well-behaved and all the singularities vanish, except the ring singularity, on account of the introduction of this new definition. In FIG. \[fig7\](a), the entropy density function diverges at $ r=0 $ and $ \theta=\frac{\pi}{2} $, as it encounters the ring singularity, whereas in FIG. \[fig7\](b) the entropy density stays finite at $ r=0 $ and $ \theta=\frac{\pi}{4} $. Although the entropy density function (\[new3\]) vanishes at the conformal infinity $ r=\dfrac{1}{\alpha\cos(\theta)} $, we cannot simply associate the zeroes of the entropy density function with the horizons, because according to this modified definition, the expression (\[new3\]) does not have such factors, and so we have to solve the function explicitly in order to determine the zeroes of the entropy density.
Discussions
===========
We now discuss the possibility of having an angular component in the vector field $ \mathbf{\Psi} $ for axisymmetric spacetimes as proposed in [@entropy2]. Using this modified definition of $ \mathbf{\Psi} $, and the modified expression (\[mod\_P\]), we now calculate the gravitational entropy density for axisymmetric space-times, using the following expression: $$\label{news}
s=k_{s}|\mathbf{\nabla}.\mathbf{\Psi}|=\dfrac{k_{s}}{\sqrt{-g}}\left|\left(\dfrac{\partial}{\partial r}(\sqrt{-g}P)+\dfrac{\partial}{\partial \theta}(\sqrt{-g}P) \right)\right|.$$
The gravitational entropy density for the uncharged rotating accelerating black hole is given by $$\begin{aligned}
\label{new1}
\left.s\right. &= \dfrac{k_{s}}{\sqrt{\dfrac{\sin^2(\theta)(a^2\cos^2(\theta)+r^2)^2}{(r\alpha\cos(\theta)-1)^8}}}\Bigg(\Bigg|\dfrac{48}{\sqrt{\dfrac{\
| 3,158
| 1,934
| 2,683
| 3,075
| 3,310
| 0.773276
|
github_plus_top10pct_by_avg
|
nts of $D$ and $B$ decays. For a review of the $\Delta I=1/2$ rule see e.g. Ref. [@Buras:2014maa].
In kaon physics we consider $K \to\pi\pi$ decays. Employing an isospin parametrization we have [@Buras:2014maa] $$\begin{aligned}
{\mathcal{A}}(K^+\rightarrow \pi^+\pi^0) &= \frac{3}{2} A_2^K e^{i\delta_2^K}\,, \nonumber\\
{\mathcal{A}}(K^0\rightarrow \pi^+\pi^-) &= A_0^K e^{i\delta_0^K} + \sqrt{\frac{1}{2}} A_2^K e^{i\delta_2^K}\,, \nonumber \\
{\mathcal{A}}(K^0\rightarrow \pi^0\pi^0) &= A_0^K e^{i\delta_0^K} - \sqrt{2} A_2^K e^{i\delta_2^K}\,. \label{eq:kaondata}\end{aligned}$$ Note that the strong phases of $A_0^K$ and $A_2^K$ are factored out, so that $A_{0,2}^K$ contain weak phases only. The data give $$\begin{aligned}
\left|\frac{A_0^K}{A_2^K}\right| \approx 22.35\,,\qquad
\delta_0^K - \delta_2^K = (47.5\pm 0.9)^{\circ}\,, \label{eq:kaon-deltaI12-rule}\end{aligned}$$ see Ref. [@Buras:2014maa] and references therein for more details. $A_{0,2}^K$ have a small imaginary part stemming from the CKM matrix elements only. To a very good approximation the real parts $\mathrm{Re}(A_0^K)$ and $\mathrm{Re}(A_2^K)$ in the $\Delta I=1/2$ rule depend only on the tree operators [@Buras:2015yba; @Kitahara:2016nld] $$\begin{aligned}
Q_1 &= (\bar{s}_{\alpha} u_{\beta})_{V-A} (\bar{u}_{\beta} d_{\alpha})_{V-A}\,, \qquad
Q_2 = (\bar{s} u)_{V-A} (\bar{u} d)_{V-A}\,. \end{aligned}$$ The lattice results Refs. [@Bai:2015nea; @Blum:2015ywa; @Boyle:2012ys] show an emerging physical interpretation of the $\Delta I=1/2$ rule, that is an approximate cancellation of two contributions in $\mathrm{Re}(A_2^K)$, which does not take place in $\mathrm{Re}(A_0^K)$. These two contributions are different color contractions of the same operator.
The isospin decompositions of $D\rightarrow \pi\pi$ and $B\rightarrow \pi\pi$ are completely analog to Eq. (\[eq:kaondata\]). To differentiate the charm and beauty isospin decompositions from the kaon one, we put the corresponding superscripts to the respective analog matrix elements. Leaving away
| 3,159
| 3,220
| 3,303
| 2,918
| 1,563
| 0.788222
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Z^* {\mathcal H}_1^0$ for ${\mathcal H}_1^0 \simeq \eta^0_r$.
[^6]: If $\lambda_{s\eta 1}$ is small, ${\mathcal H}_2^0$ ($\simeq \eta_r^0$) decays into $Z^\ast {\mathcal A}^0$.
---
author:
- '**[Sarbari Guha and Samarjit Chakraborty]{}**'
title: '[**On the gravitational entropy of accelerating black holes** ]{}'
---
Abstract {#abstract .unnumbered}
========
In this paper we have examined the validity of a proposed definition of gravitational entropy in the context of accelerating black hole solutions of the Einstein field equations, which represent the realistic black hole solutions. We have adopted a phenomenological approach proposed in Rudjord et al \[20\] and expanded by Romero et al \[21\], in which the Weyl curvature hypothesis is tested against the expressions for the gravitational entropy. Considering the $C$-metric for the accelerating black holes, we have evaluated the gravitational entropy and the corresponding entropy density for four different types of black holes, namely, non-rotating black hole, non-rotating charged black hole, rotating black hole and rotating charged black hole. We end up by discussing the merits of such an analysis and the possible reason of failure in the particular case of rotating charged black hole and comment on the possible resolution of the problem.
KEYWORDS: Gravitational entropy, Accelerating Black holes.
Introduction
============
The $C$-metric was independently discovered by Levi-Civita [@Levi] and Weyl [@Weyl] in 1917. Ehlers and Kundt [@EK] while working on the classification of the degenerated static vacuum fields, constructed a table in which this metric was placed in the slot “$C$”, leading to the name ‘$C$-metric’. Kinnersley and Walker [@KW] pointed out that this metric is an exact solution of Einstein’s equations which describes the combined electromagnetic and gravitational field of a uniformly accelerating object having mass $m$ and charge $e$, and is an example of “almost everything”. It is for this reason that the $C$-metric is the fo
| 3,160
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tructions by using the MV code construction and then asking each server for the evaluations of $F$ at a point, as well as the values of a certain differential operator (similar to first order derivatives) at these points. For this to work we need two ingredients. The first is to replace the field $\F_q$ with a certain ring which has characteristic $m$ and an element of order $m$ (we only use $m=6$ and can take the polynomial ring $\Z_m[\gamma]/(\gamma^6 - 1)$). The second is an observation that, in known MV families constructions [@Grolmusz99], the inner products ${\langle \bu_i,\bv_j \rangle}$ that are nonzero (that is, when $i \neq j$) can be made to fall in a small set. More precisely, over $\Z_6$, the inner products are either zero or in the set $\{1,3,4\}$. This means that the restricted polynomial only has nonzero coefficients corresponding to powers of $T$ coming from the set $\{0,1,3,4\}$. Such a polynomial has four degrees of freedom and can be recovered from two evaluations and two derivatives (of order one). We are also able to work with arbitrary MV families by using second order derivatives at two points (which are sufficient to recover a degree 5 polynomial).
Organization
------------
In section \[preliminaries\] we give some preliminary definitions and notations. In section \[oldconstruction\], we review the construction of a 2-server PIR scheme with $O(n^{1/3})$ communication cost which is based on polynomial interpolation with partial derivatives [@WoodruffY05]. In section \[newconstruction\], we present our new construction of sub-polynomial 2-server PIR schemes and some of its variants. Then, in Section \[sec-kserver\] we analyze the generalization to more servers. We conclude in Section \[sec-conclude\] with some remarks on future directions.
Preliminaries
=============
#### Notations:
We will use bold letters like $\bu,\bv,\bz$ etc. to denote vectors. The inner product between two vectors $\bu=({u_1,\cdots,u_k}),\bv=({v_1,\cdots,v_k})$ is denoted by ${\langle \bu,\bv \rangle}=\sum_{i=1
| 3,161
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| 3,067
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| 0.791343
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xtended Lagrange multipliers i.e. $V^{ext} = v_{a} H^{a} + v_{\mu }Z^{\mu}$ and $F^{ext}_{ab}= {\operatorname{Tr}}([ H_{a}, H_{b}]^{ext} V^{ext})$.
Applications {#sec:examples}
============
Let us now turn to specific examples for which we construct the dual RR fluxes corresponding to various centrally-extended non-abelian T-dualities of $AdS_5 \times S^5$ using the technology outlined in section \[sec:natd\]. Here we will consider certain deformations that are well-known to correspond to TsT transformations. In appendix \[app:furtherexamples\] we consider further examples that correspond to Yang-Baxter deformations with time-like abelian and non-abelian $r$-matrices.
Application 1: Non-Commutative Deformations {#ssec:app1}
-------------------------------------------
The first application we consider is the string background dual to non-commutative $\mathcal{N} = 4$ super Yang-Mills [@Hashimoto:1999ut; @Maldacena:1999mh] $$\begin{aligned}
\nonumber
ds^2 &= \frac{du^2}{u^2} + u^2 \left( -dt^2 + dx_1^2 + \tilde h (dx^2_2 + dx^2_3) \right) + d\Omega_5^2 \ ,
\quad
\tilde h = \frac{1}{1+ a^4 u^4} \ ,
\\ \label{eq:mrback}
B &= a^2 \tilde h u^4 dx_2 \wedge dx_3 \ , \quad \exp 2 \Phi = g_0^2 \tilde h \ , \\ \nonumber
F_3 &= -\frac{4}{g_0} a^2 u^3 dt\wedge dx_1 \wedge du \ , \quad
F_5= \frac{4}{g_0} \tilde h u^3 (1+\star) \, du \wedge dt \wedge dx_1 \wedge dx_2 \wedge dx_3 \ .\end{aligned}$$
Starting from the undeformed background $$\label{eq:undefads5}
\begin{aligned}
ds^2 &= \frac{du^2}{u^2} + u^2 \left( -dt^2 + dx_1^2 + dx^2_2 + dx^2_3 \right) + d\Omega_5^2 \ ,
\quad \exp 2 \Phi = g_0^2 \ ,
\\
F_5& = \frac{4}{g_0} u^3 (1+\star) \, du \wedge dt \wedge dx_1 \wedge dx_2 \wedge dx_3 \ ,
\end{aligned}$$ we now consider the non-abelian T-dual with respect to the central extension of $U(1)^2$, where the $U(1)^2$ is generated by shifts in $x_2$ and $x_3$. Using eqs. – with $y_1 = \frac{x_3}{a^2}$, $y_2 = \frac{x_2}{a^2}$, $f_1 = f_2 = u^2$ and setting the deformation parameter $\nu=a^{-2}$ we find that the plus and minu
| 3,162
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| 3,214
| 3,509
| 0.771859
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|
trix} id&0\\ 0&(w)_1 \end{pmatrix}$. Note that $\mathrm{O}(A(2\delta, 2b, 1)/\pi A(2\delta, 2b, 1), \bar{q_i})(\kappa)$ is not contained in $\mathrm{SO}(L_i/\pi L_i, \bar{q_i})(\kappa)$. Thus it suffices to show that the restriction of $\varphi_i(\kappa)$ to the above subgroup of $H_i(\kappa)$, which is given by letting $x=id, y=0, z=0$, is surjective onto $\mathrm{O}(A(2\delta, 2b, 1)/\pi A(2\delta, 2b, 1), \bar{q_i})(\kappa)$ and we may and do assume that $L=L_i=A(2\delta, 2b, 1)$ of rank $2$.
Let $m_{i,i}=\begin{pmatrix} r&s\\ t&u \end{pmatrix}$ be an element of $H_i(\kappa)$ such that $r=(r)_1+\pi \cdot(r)_2$ and so on, where $(r)_1, (r)_2 \in R\subset R\otimes_AB$ and $\pi$ stands for $1\otimes \pi\in R\otimes_AB$. Recall that $\pi=\sqrt{2\delta}$ for a certain unit $\delta\in A$ such that $\delta\equiv 1 \mathrm{~mod~}2$ so that $\sigma(\pi)=-\pi$, as mentioned in Section \[Notations\]. Let $\bar{b}\in \kappa$ be the reduction of $b$ modulo $\pi$. Then the equations defining $H_i(\kappa)$ are $$(r)_1^2+(r)_1(t)_1+\bar{b}(t)_1^2=1, (r)_1(u)_1+(t)_1(s)_1=1,$$ $$(s)_1^2+(s)_1(u)_1+\bar{b}(u)_1^2=\bar{b}, (r)_1(u)_2+(r)_2(u)_1+(t)_1(s)_2+(t)_2(s)_1=0.$$ Under the map $\varphi_i(\kappa)$, $m_{i,i}$ maps to $\begin{pmatrix} (r)_1&(s)_1\\ (t)_1&(u)_1 \end{pmatrix}$. Note that the quadratic form $\bar{q_i}$ restricted to $A(2\delta, 2b, 1)/\pi A(2\delta, 2b, 1)$ is given by the matrix $\begin{pmatrix} 1&1\\ 0&b \end{pmatrix}$.
We now choose an element of $H_i(\kappa)$ by setting $$(r)_1=(s)_1=(u)_1=1, (t)_1=0, (r)_2=(s)_2=(t)_2=(u)_2=0.$$ Then under the morphism $\varphi_i(\kappa)$, this element maps to $\begin{pmatrix} 1&1\\ 0&1 \end{pmatrix} \in \mathrm{O}(A(2\delta, 2b, \pi)/\pi A(2\delta, 2b, \pi), \bar{q}_i)(\kappa)$ whose Dickson invariant is nontrivial so that it is not contained in $\mathrm{SO}(A(2\delta, 2b, \pi)/\pi A(2\delta, 2b, \pi), \bar{q}_i)(\kappa)$.
Therefore, $\varphi_i(\kappa)$ induces a surjection from $H_i(\kappa)$ to $\mathrm{O}(A(2\delta, 2b, 1)/\pi A(2\delta, 2b, 1), \bar{q_i})(\kappa)$
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Assume that $Y$ has the Chevalley-Kleiman property and let $P\subset X$ be a finite subset. Since $\pi(P)\subset Y$ is finite, there is an open affine subset $Y_P\subset Y$ containing $\pi(P)$. Then $g^{-1}(Y_P)\subset X$ is an open affine subset containing $P$.
Conversely, assume that $X$ has the Chevalley-Kleiman property. By the already established direction, we may assume that $X$ is normal. Next let $ Y^n$ be the normalization of $Y$. Then $X\to Y^n$ is finite and dominant. Fix irreducible components $X_1\subset X$ and $Y_1\subset Y^n$ such that the induced map $X_1\to Y_1$ is finite and dominant. Let $\pi'_1:X'_1\to X_1\to Y_1$ be the Galois closure of $X_1/Y_1$ with Galois group $G$. We already know that $X'_1$ has the Chevalley-Kleiman property, hence there is an open affine subset $X'_P\subset X_1$ containing $\bigl(\pi'_1\bigr)^{-1}(P)$. Then $U'_P:=\cap_{g\in G}g(X_P)\subset X'_1$ is affine, Galois invariant and $\bigl(\pi'_1\bigr)^{-1}\bigl(\pi'_1(U'_P)\bigr)=U'_P$.
Thus $U'_P\to \pi'_1(U'_P)$ is finite and, by Chevalley’s theorem [@hartsh Exrc.III.4.2], $\pi'_1(U'_P)\subset Y_1$ is an open affine subset containing $P$. Thus $Y^n$ has the Chevalley-Kleiman property.
Next consider the normalization map $g:Y^n\to {\operatorname{red}}Y$. There are lower dimensional closed subschemes $P\subset V\subset {\operatorname{red}}Y$ and $Z:=g^{-1}(V)\subset Y^n$ such that $g:Y^n\setminus Z\cong {\operatorname{red}}Y\setminus V$ is an isomorphism. By induction on the dimension, $V$ has the Chevalley-Kleiman property.
By (\[affine.red.lem\]) there are open affine subsets $P\subset V_P\subset V$ and $g^{-1}(P)\subset Y^n_P\subset Y^n$ such that $g$ restricts to a finite morphism $g: Z\cap Y^n_P\to V_P$. Thus, by (\[glue.lem.affine\]), $g\bigl(Y^n_P\bigr)\subset {\operatorname{red}}Y$ is open, affine and it contains $P$. Thus ${\operatorname{red}}Y$ has the Chevalley-Kleiman property.
Finally, ${\operatorname{red}}Y\to Y$ is a homeomorphism, thus if $U\subset {\operatorname{red}}Y$ is an affine open subset
| 3,164
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does not divide $i_G(x) $ for every element of prime power order $x\in A\cup B$ if and only if $G$ has a central Sylow $p$-subgroup, i.e. $G=O_p(G) \times O_{p'}(G)$ with $O_p(G)$ abelian.
Our results provide an improvement of [@BCL Theorem 1.1] in the case of only two factors, since in that paper products of $n$ pairwise mutually permutable subgroups were considered.
\[adolfo\] Let the group $G = AB$ be the mutually product of the subgroups $A$ and $B$, and let $p$ be a prime. Then:
- No index $i_G(x) $, where $x$ is a $p$-regular element in $ A\cup B$, is divisible by $p$ if and only if $G=O_p(G) \times O_{p'}(G)$.
- $i_G(x) $ is not divisible by $p$ for every element $ x \in A\cup B$ if and only if $G=O_p(G) \times O_{p'}(G)$ with $O_p(G)$ abelian.
Finally, we also point out that [@FMOpi Theorem A] and [@ZGS Theorem 3.2] when $\pi=p'$ are direct consequences from our main result.
Preliminary results
===================
We will use without further reference the following elementary lemma:
Let $N$ be a normal subgroup of a group $G$, and $H$ be a subgroup of $G$. We have:
- $i_N(x)$ divides $i_G(x) $, for each $x\in N$.
- $i_{G/N}(xN)$ divides $i_G(x) $, for each $x\in G$.
- If $xN$ is a $\pi$-element of $HN/N$, for a set of primes $\pi$, then there exists a $\pi$-element $x_1\in H$ such that $xN = x_1N$.
We will also need the following fact about Hall subgroups of factorised groups, which is a convenient reformulation of [@AFG 1.3.2]. We recall that a group is a D$_{\pi}$-group, for a set of primes $\pi$, if every $\pi$-subgroup is contained in a Hall $\pi$-subgroup, and any two Hall $\pi$-subgroups are conjugate. It is well known that any $\pi$-separable group is a D$_{\pi}$-group. Also, all finite groups are D$_{\pi}$-groups when $\pi$ consists of a single prime.
\[1.3.2\] Let $G=AB$ be the product of the subgroups $A$ and $B$. Asume that $A, B$, and $G$ are D$_{\pi}$-groups for a set of primes $\pi$. Then there exists a Hall $\pi$-subgroup $H$ of $G$ such that $H= (H \cap A)(H \cap B)$
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e construction of $G'$. There is a ‘universal’, ‘maximal’ extension $\tilde{G}_{\rm max}$, which extends $G$ by the group of all automorphisms of the total space of ${\cal E}$ that cover the action of the elements of $G$ on $X$. It fits into a short exact sequence $$1 \: \longrightarrow \: {\rm Aut}({\cal E}) \: \longrightarrow \:
\tilde{G}_{\rm max} \: \longrightarrow \: G \: \longrightarrow \: 1,$$ where ${\rm Aut}({\cal E})$ is the group of global bundle automorphisms of ${\cal E}$. The group we want, $\tilde{G}$, will necessarily be a subgroup of this universal extension of $\tilde{G}_{\rm max}$.
In general, the extension defining $\tilde{G}_{\rm max}$ will not be central, but if ${\cal E}$ is stable or simple then ${\rm Aut}({\cal E}) =
{\mathbb C}^{\times}$, and the extension is central. The group $\tilde{G}_{\rm max}$ acts by definition on ${\cal E}$ and so defines an equivariant structure. Every other group for which one has an equivariant structure will map to $\tilde{G}_{\rm max}$ and the equivariant structure will factor through that map.
Now, clearly, $\tilde{G}_{\rm max}$ is not a finite group, and we only want to consider cases in which the trivially-acting subgroup is finite. If $G$ is finite and ${\cal E}$ is stable or simple, then $\tilde{G}_{\rm max}$ is a central extension of $G$ by ${\mathbb C}^{\times}$ and, because $$H^2(G,{\mathbb C}^*) \: = \: H^2(G,{\mathbb Q}/{\mathbb Z})$$ for $G$ finite, the relevant $H^2$ is finite and so every extension is induced from some central extension $G_{min}$ of $G$ by a finite group of order bounded by the maximal order of elements in $G$. In this fashion, we can construct a $\tilde{G}$.
So far, we have described how, given a bundle that is invariant but not equivariant with respect to an orbifold group $G$, one can extend $G$ to a larger finite group $\tilde{G}$, where the extension acts trivially on the base. Now, not any $\tilde{G}$ will be acceptable: the orbifold by $\tilde{G}$ must, at minimum, satisfy level-matching, and as discussed earl
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\(5\) C (O) 0.41\*\*\* 0.60\*\*\* 0.50\*\*\* 0.68\*\*\* --
\(6\) R (O) 0.32\*\* 0.35\*\*\* 0.67\*\*\* 0.52\*\*\* 0.63\*\*\* --
\(7\) PSC 0.24\*\* 0.50\*\*\* 0.33\*\*\* 0.50\*\*\* 0.62\*\*\* 0.46\*\*\* --
\(8\) PA 0.04 0.11 −0.03 −0.06 0.03 −0.07 0.12 --
\*\*p \< 0.01, \*\*\*p \< 0.001; A, autonomy; C, competence; R, relatedness; PSC, physical self-concept; U, unstructured PA; O, organized PA.
To determine the influence of PA on psychological variables, a multivariate analysis of variance (MANOVA) was carried out. Box's *M* test shows that the covariance matrices of the dependent variables are equal across groups (*p* \< 0.05). Levene's test of equality of error variances shows that error variances are equal across groups (*p* \> 0.05), except in relation to autonomy for unstructured PA (*p* \< 0.05).
The MANOVA found that the interaction of dependent (psychological) variables was not affected by PA levels (Pillai's trace: *F* = 0.86, *p* \> 0.05, η^2^ = 0.08, observed power = 0.66). In contrast, the sex of participants, analyzed here as a co-variable, was found to have a significant principal effect (Pillai's trace: *F* = 2.93, *p* \< 0.01, η^2^ = 0.24, observed power = 0.90).
The paired comparison shows variation due to sex in relation to satisfaction of competence through unstructured PA among sedentary and low/somewhat active subjects (*p* \< 0.05).
Results from a multiple regression analysis conducted are not displayed here since the analysis yielded no significant results owing to the lack of correlation between PA and the different parameters of satisfaction of basic psychological needs and physical self-concept.
Discussion {#S4}
==========
One of the main objectives of this accelerometry-based research has been to "translate" the daily step targets recommended by the World Health Organization ([@B108]) for children and teenagers, an
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t the situation is with getting that Tenaska agreement signed? it is on a list to be assigned to HPL in the sale and I don't think it has ever been executed--I know you left me a message a week or so ago about this--is there any information that i owe you ?
can you take care of these---I have no idea what this or the ones following
are about. Should I deny the requests?
----- Forwarded by Richard B Sanders/HOU/ECT on 01/16/2001 10:26 AM -----
ARSystem <ARSystem@mailman.enron.com>
01/15/2001 07:13 PM
To: "richard.b.sanders@enron.com" <richard.b.sanders@enron.com>
cc:
Subject: Approval is Overdue: Access Request for hugh.eichelman@enron.com
This request has been pending approval for 6 days and you are the
alternate. Please click
http://itcapps.corp.enron.com/srrs/auth/emailLink.asp?ID=000000000012515&Page=
Approval to review and act upon this request.
Request ID : 000000000012515
Approver : linda.r.guinn@enron.com
Request Create Date : 1/8/01 8:12:00 AM
Requested For : hugh.eichelman@enron.com
Resource Name : Litigation Tracking Database Write
Resource Type : Applications
TO: All Current Enron Employees who Participate in the Enron Corp. Savings Plan
Due to an Enron programming error in the transmission of data to the Enron Corp. Savings Plan administrator, a number of currently active employees were erroneously coded with a status of terminated on the Savings Plan system.
As a result, you may have received a notice from Hewitt Associates, the Savings Plan administrator, indicating that you were a terminated employee and providing you with options for your Savings Plan account. This notice was sent in error, and should be disregarded.
This situation was identified and corrected. We regret any confusion or inconvenience this may have caused.
If you would like to verify that your Savings Plan employment status has been corrected, you may log in to the Savings Plan website through the Enron intranet (benefits.enron.com) or the internet (
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ian comparison theorem yields the additional error term $C \mathrm{E}_{2,n} +
\frac{1}{n}$ given in , for some universal positive constant $C$. Similarly, can be established along the lines of the proof of , starting from the bound . In this case we pick up an additional error term $C \tilde{\mathrm{E}}_{2,n} + \frac{1}{n}$ of different form, shown in , where $C>0$ is a different universal constant.
Since all the bounds we have derived do not depend on $\mathcal{D}_{1,n}$, the outcome of the splitting and $w_n$, the same bounds therefore hold for the joint probabilities, and uniformly over the model selection and estimation procedures. The above arguments hold for each $P \in \mathcal{P}_n^{\mathrm{LOCO}}$. $\Box$
[**Proof of .**]{} Following the proof of , for each $P \in \mathcal{P}_n^{\mathrm{LOCO}}$ and on the event $\mathcal{E}_n$ given in (which has probability at leas $1- \frac{1}{n}$), we have that $$\begin{aligned}
2 \max_{j \in {\widehat{S}}} z_{\alpha/(2k)}
\sqrt{\frac{ \hat\Sigma_{{\widehat{S}}}(j,j)}{n}} & \leq 2 \max_{j \in {\widehat{S}}} z_{\alpha/(2k)}
\sqrt{\frac{ \Sigma_{{\widehat{S}}}(j,j) + \left| \hat\Sigma(j,j)-\Sigma(j,j)\right|}{n}}\\
& \leq z_{\alpha/(2k)}
\sqrt{ \frac{ (2(A + \tau) + \epsilon)^2 + N_n }{n}}.
\end{aligned}$$ The claimed bound follows from the definition of $N_n$ as in . $\Box$
[**Proof of .**]{} All the probabilistic statements that follow are to be understood conditionally on the outcome of the sample splitting and on $\mathcal{D}_{1,n}$. Thus, $\mathcal{I}_{1,n}$, ${\widehat{S}}$, $\hat{\beta}_{{\widehat{S}}}$ and, for each $j
\in {\widehat{S}}$, $\hat{\beta}_{{\widehat{S}}(j)}$ are to be regarded as fixed, and the only randomness is with respect to the joint marginal distribution of $\mathcal{D}_{2,n}$ and $(\xi_i, i \in \mathcal{I}_{2,n})$, and two auxiliary independent standard Gaussian vectors in $\mathbb{R}^{{\widehat{S}}}$, $Z_1$ and $Z_2$, independent of everything else.
Let $\hat{\gamma}^*_{{\widehat{S}}} \in \mathcal{R}^{{\wide
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github_plus_top10pct_by_avg
|
. The virial radius is 264 $h^{-1}$ comoving kpc and is indicated by the white circles.
In Fig. \[fig:haloradz2\] we show the same quantities as in Fig. \[fig:halo\] as a function of radius for the haloes with $10^{11.5}$ M$_\odot<M_\mathrm{halo}<10^{12.5}$ M$_\odot$ at $z=2$ in simulation *REF\_L050N512*. The black curves show the median values for all gas, except for the last two panels which show the mean values. The red (blue) curves show the median or mean values for hot- (cold-)mode gas, i.e. gas with maximum past temperatures above (below) $10^{5.5}$ K. The shaded regions show values within the 16th and 84th percentiles. Hot-mode gas at radii larger than $2R_\mathrm{vir}$ is dominated by gas associated with other haloes and/or large-scale filaments.
All the results we show are weighted by mass. In other words, we stacked all 518 haloes in the selected mass range using $R/R_{\rm vir}$ as the radial coordinate. The black curves in Fig. \[fig:haloradz2\] (except for the last two panels) then show the values of the corresponding property (e.g. the gas overdensity in the top-left panel) that divide the total mass in each radial bin in half, i.e. half the mass lies above the curve. We have done the same analysis for volume-weighted quantities by computing, as a function of radius, the values of each property that divides the total volume, i.e. the sum of $m_\mathrm{gas}/\rho$, in half but we do not show the results. The volume is completely dominated by hot-mode gas out to twice the virial radius, reaching 50 per cent at $3R_\mathrm{vir}$. Even though the volume-weighted hot fraction is very different, the properties of the gas and the differences between the properties of hot- and cold-mode gas are similar if we weigh by volume rather than mass.
We find that the median density of cold-mode gas is higher, by up to 1 dex, than that of hot-mode gas and that its current temperature is lower, by up to 2 dex, at least beyond
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DE which is separable in $r$ and $\bar{r}$. It is therefore equivalent to the system
\[eq:4.15\] (a)2-3=,(b)2 -3=,
where $\Omega$ is a real separation constant. Equation (\[eq:4.15\].b) can be solved in a way similar way to (\[eq:4.15\].a). Defining
\[eq:4.16\] g(r)=h\^[(1)]{}(r),
equation (\[eq:4.15\].a) can be written as
\[eq:4.17\] gg”-3/2 (g’)\^2+(/2) g\^3=0.
According to [@Kamke:1979] equation (\[eq:4.17\]) admits a first integral which can be rewritten in the equivalent form
\[eq:4.19\] =d.
By the change of variable
\[eq:4.20\] g=[( )]{},
equation (\[eq:4.19\]) can be transformed to $2\exp{\left( (2\Omega)^{-1}(\omega^2-c_1) \right)}d\omega=\pm
\Omega d\xi,$ for which the solution, in term of the inverse error function $\operatorname{erf}^{-1}$, is
\[eq:4.21\] =,
where $c_2$ is a constant of integration. We substitute (\[eq:4.20\]) into (\[eq:4.16\]) with $\omega$ given by (\[eq:4.21\]) and integrate the result. We obtain the function $h(r)$ and its complex conjugate in terms of the error functions $\operatorname{erfi}$ and $\operatorname{erf}^{-1}$ as
\[eq:4.22\] h(r)&=-+c\_3,\
|[h]{}(|[r]{})&=-+|[c]{}\_3,
where the $c_i\in{\mathbb{C}}$ are integration constants and the real separation constant $\Omega$ is non zero. Since equations (\[eq:4.22\]) solve the system (\[eq:4.15\]), we have that $\theta$ given by (\[eq:4.14\]) satisfies the compatibility condition (\[eq:4.2\]) for $\sigma$. One should note that no derivatives with respect to time appear in the PDE (\[eq:4.2\]). Consequently, the equation (\[eq:4.2\]) is still satisfied if, in equations (\[eq:4.13\]) and (\[eq:4.14\]), we replace the function $h(r)$ and its complex conjugate respectively by
\[eq:4.22bis\] h(t,r)&=-+c\_3(t),\
|[h]{}(t,|[r]{})&=-+|[c]{}\_3(t),
which are obtained by substituting the arbitrary complex functions $c_i(t)$ in place of the integration constant $c_i$ and the arbitrary real function $\Omega(t)$ in place of the separation constant $\Omega$. Hence, the general solution of the system (\[eq:4.1\]) takes the
| 3,171
| 5,636
| 3,452
| 2,644
| 4,056
| 0.768417
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{1pt}{\text{\circle*{1.5}}}}}(A),\\
HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\overline{A_\#}) &\to HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({\widehat}{A_\#}),
\end{aligned}$$ and the cone of the first map is isomorphic to $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(j_!M^\Delta_\#)$. Since $j_!$ is exact, we have $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(j_!M^\Delta_\#) \cong HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M_\#)$, and the periodicity map $u:HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(j_!M^\Delta_\#) \to HC_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}-2}(j_!M^\Delta_\#)$ is equal to $0$, so that $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(j_!M^\Delta_\#) = 0$. Thus the first map in is an isomorphism, and the second map is then the required splitting.
Assume given a commutative $k$-algebra $R$ with a maximal ideal ${{\mathfrak m}}\subset R$, and a deformation $A_R$ of the algebra $A$ over $R$. Then if ${\operatorname{Spec}}R$ is smooth, the $R$-modules $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_R/R)$ carry a natural connection.
[[*Sketch of a proof.*]{}]{} Consider the $R \otimes R$-algebras $A_R
\otimes R$ and $R \otimes A_R$, and their restrictions to the first infinitesemal neighborhood of the diagonal in ${\operatorname{Spec}}(R \otimes R) =
{\operatorname{Spec}}R \times {\operatorname{Spec}}R$. Then Proposition \[spl\], suitably generalized, shows that $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(-)$ of these two restrictions are canonically isomorphic. It is well-known that giving such an isomorphism is equivalent to giving a connection on $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_R/R)$.
We note that we do not claim that the connection is [*flat*]{}. It certainly is, at least in characteristic $0$; but our present method does not allow one to go beyond square-zero extensions. Thus we cannot analyse the second infinitesemal neighborhood of the diagonal in ${\operatorname{Spec}}(R \otimes R)$, and we cannot prove flatness.
Unfortunately, at present, we do
| 3,172
| 2,801
| 2,484
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to $\cS$ to create a larger star. If $|D|\ge 2$ then for any $d\in D$ we have that $\cI\cup\{S\setminus\{d\}\mid S\in\cS\}$ is a larger intersecting subfamily of $\cH$ than $\cI$, a contradiction. Therefore $|D|=1$ and, without loss of generality, $D=\{1\}$.\
Let $\cF$ be the largest sunflower in $\cS$ with core $\{1\}$. Since any $R\in\cR$ must intersect every $F\in\cF$, we must have that $|\cF|\le 3$. If $|\cF|=1$, then $\{S\setminus\{1\}\mid S\in\cS\}$ forms an intersecting family, and from the fact that $|\cS|\ge 2$ and $D=\{1\}$, we have that $\cS=\{\{1,a,b\},\{1,a,c\},\{1,b,c\}\}$ for three different numbers $a,b,c$. Moreover, we must have $|R\cap\{a,b,c\}|\ge 2$ for every $R\in\cR$. This means that $\cI\cup\{\{a,b\}\}$ is a larger intersecting subfamily of $\cH$ than $\cI$, a contradiction. Therefore $2\le |\cF|\le 3$.\
Let $X=\left(\bigcup_{F\in\cF}F\right)$; then $|X|=2|\cF|+1$. Denote $X^*=X\setminus\{1\}$. Define $Y=\left(\bigcup_{S\in\cS}S\right)\setminus X$ and set $\cS(Y)=\{S\in\cS\mid S\cap Y\ne\emptyset\}$. Then we must have that, for all $y\in Y$, there is an $S\in\cS(Y)$ such that $\{1,y\}\subseteq S$ and, for all $x\in X$ (including $x=1$), there is an $F\in\cF$ such that $\{1,x\}\subseteq F$. If $|X\cup Y|=|X|+|Y|=2|\cF|+|Y|+1>|\cR|$, then $\cS\cup\{\{1,k\}\mid k\in X\cup Y\}$ is a star subfamily of $\cH$ of size larger than $\cI$, a contradiction. So in the rest we assume that $|\cR|\ge 2|\cF|+|Y|+1$.\
### $|\cF|=3$
Without loss of generality, $\cF=\{\{1,2,3\},\{1,4,5\},\{1,6,7\}\}$. Set $\cE$ to be the family of $3$-element subsets of $X^*$ that intersect each of $\{2,3\},\{4,5\},\{6,7\}$. Then $|\cE|=8$ and $\cR\subseteq\cE$. However, if $R\in\cR$ then $X^*\setminus R\in\cE\setminus\cR$, and so $|\cR|\le 4< 7\le 2|\cF|+|Y|+1$, a contradiction.\
### $|\cF|=2$
Without loss of generality, $\cF=\{\{1,2,3\},\{1,4,5\}\}$. We have $|\cR|\ge|Y|+5$.\
Define $\cS^*=\cS\setminus(\cF\cup\cS(Y))$. Clearly, $\sum_{x\in X^*}|\cS^*_x|=2|\cS^*|$, and $\cS^*\subseteq\{\{1,i,j\}\in\cS\mid i\in\{2,3\},j\in
| 3,173
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non-saturated sets from the preimage topology. It is therefore possible to rewrite Eq. (\[Eqn: IT\]) as
$$U\in\textrm{IT}\{ e;\mathcal{V}\}\Longleftrightarrow e(U)=V\textrm{ if }V\in\mathcal{V}_{\textrm{comp}},\label{Eqn: IT'}$$
and to compare it with the following criterion for an *injective, open-continuous* *map* $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ that necessarily satisfies $\textrm{sat}(A)=A$ for all $A\subseteq X$ $$U\in\mathcal{U}\Longleftrightarrow(\{\{ f(U)\}_{U\in\mathcal{U}}=\mathcal{V}_{\textrm{comp}})\wedge(f^{-1}(V)|_{V\in\mathcal{V}}\in\mathcal{U}).\label{Eqn: OCINJ}$$
***Final Topology.*** Since it is necessarily produced on the range $\mathcal{R}(q)$ of $q$, the final topology is often considered in terms of a surjection. This however is not necessary as, much in the spirit of the initial topology, $Y-q(X)\neq\emptyset$ inherits the discrete topology without altering anything, thereby allowing condition (\[Eqn: FT’\]) to be restated in the following more transparent form $$V\in\textrm{FT}\{\mathcal{U};q\}\Longleftrightarrow V=q(U)\textrm{ if }U\in\mathcal{U}_{\textrm{sat}},\label{Eqn: FT}$$
and to compare it with the following criterion for a *surjective, open-continuous* *map* $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ that necessarily satisfies $_{f}B=B$ for all $B\subseteq Y$ $$V\in\mathcal{V}\Longleftrightarrow(\mathcal{U}_{\textrm{sat}}=\{ f^{-}(V)\}_{V\in\mathcal{V}})\wedge(f(U)|_{U\in\mathcal{U}}\in\mathcal{V}).\label{Eqn: OCSUR}$$
As may be anticipated from Fig. \[Fig: Initial-Final\], the final topology does not behave as well for subspaces as the initial topology does. This is so because in Fig. \[Fig: Initial-Final\](a) the two image continuous functions $h$ and $q$ are connected by a preimage continuous inclusion $f$, whereas in Fig. \[Fig: Initial-Final\](b) all the three functions are preimage continuous. Thus quite like open functions, although image continuity of $h\!:(X,\mathcal{U})\rightarrow(Y_{1},\textrm{FT}\{\mathcal{U};h\})$ implies that of $h_{<}\!:
| 3,174
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| 3,040
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|
_1|e_3)&(z_1|e_4)\\
(z_2|e_1)&(z_2|e_2)&(z_2|e_3)&(z_2|e_4) \end{pmatrix}
\begin{pmatrix} p\\q\\r\\s\end{pmatrix} &=
\begin{pmatrix} (e_5|b_3)\\
(e_6|b_3)\\
(z_1|b_3)\\
(z_2|b_3)
\end{pmatrix}.\nonumber\end{aligned}$$
Writing the projectors as matrix equations given above entails solving systems of linear equations. These algebraic equations can be solved using a computerised code, which can be used to scan a vast space of models.
Similar to the spinorial representations singlet and vectorial $\bf{10}$ representations of $SO(10)$ are obtained from the following 48 sectors $$\begin{aligned}
\label{lighthiggssectors}
B_{pqrs}^{(4)}&=& B_{pqrs}^{(1)} + x
\nonumber\\
&=&\{\psi^\mu,\chi^{12},(1-p)y^{3}\overline{y}^3,p\omega^{3}\overline{\omega}^3,
(1-q)y^{4}\overline{y}^4,q\omega^{4}\overline{\omega}^4, \nonumber\\
& & ~~~~~~~~~(1-r)y^{5}\overline{y}^5,r\omega^{5}\overline{\omega}^5,
(1-s)y^{6}\overline{y}^6,s\omega^{6}\overline{\omega}^6,\overline{\eta}^{2,3} \},
\label{nonchiralvectorials}\\
B_{pqrs}^{(5,6)}&=& B_{pqrs}^{(2,3)} + x. \nonumber\end{aligned}$$ Massless states that arise in these sectors are obtained by acting on the vacuum with a NS oscillator. The type of states therefore depend on the type of oscillator, and may correspond to $SO(10)$ singlets or vectorial $\bf{10}$ representation of $SO(10)$, which is needed for electroweak symmetry breaking. The different type of $SO(10)$ singlets arising from eq. (\[nonchiralvectorials\]) are
- $\{\overline\eta^{i}\}|R \rangle_{pqrs}^{(4,5,6)}$ or $\{\overline\eta^{*i}\}|R\rangle_{pqrs}^{(4,5,6)}$, $i = 1,2,3$, where $|R\rangle_{pqrs}^{(4,5,6)}$ is the degenerated Ramond vacuum of the $B_{pqrs}^{(4,5,6)}$ sector. These states transform as a vector–like representations under the $U(1)_i$’s.
- $\{\overline\phi^{1,2}\}|R\rangle_{pqrs}^{(4,5,6)}$ or $\{\overline\phi^{*1,2}\}|R\rangle_{pqrs}^{(4,5,6)}$. These states transform as a vector–like representations of $SU(2)_A \times U(1)_A$.
- $\{\overline\phi^{3,4}\}|R\rangle_{pqrs}^{(4,5,6)}$ or $\{\overli
| 3,175
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| 3,574
| 0.771468
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|
, \ t\in(0,1],\end{aligned}$$ where $B$ is a standard Brownian motion in a filtered probability space $(\Omega,\mathcal{F}, \mathbb{F}, P)$.
Denote as $\Sigma_G$ the collection of all smooth functions $\sigma: [0,1]\times\mathbb{R}\rightarrow [\underline{\sigma},\overline{\sigma}]$ with $$\sup\limits_{(t,x)\in[0,1]\times\mathbb{R}}|\partial_x\sigma(t,x)|<\infty.$$
For $\sigma\in\Sigma_G$, we consider the following stochastic differential equation SDE (\[phi-SDE\]) with the initial value $x$: $$\begin{aligned}
\label {sigma-SDE}
\begin{split}
dW^{\sigma,x}_t&= \sigma(t,W^{\sigma,x}_t)dB_t,\ t\in(0,1],\\
W^{\sigma,x}_0&= x.
\end{split}\end{aligned}$$ We write $W^{\sigma}$ for $W^{\sigma,0}$. Denote $\Theta_G:=\{P\circ (W_1^\sigma)^{-1}| \ \sigma\in\Sigma_G\}$. For a function $\sigma: [0,1]\times\mathbb{R}\rightarrow \mathbb{R}$, set $\widetilde{\sigma}(t,x)=\sigma(1-t,x)$.
\[t10\] For any $\varphi\in lip(\mathbb{R})$, we have $$\mathcal{N}_G[\varphi]=\sup_{\mu\in\Theta_G}\mu[\varphi].$$
Note that in the above representation, we need to use non-time-homogeneous SDEs. If we only consider time-homogeneous SDEs, the representation will be strictly smaller than the $G$-normal distribution.
Proofs
======
Proofs in Section 2
-------------------
In this subsection, we first prove Theorem \[t4\] and then use it to prove Theorem \[t9\]. Finally, we provide a simple explanation for Remark \[r3\].
Denote and denote For arbitrary random vectors $X$ and $Y$, denote We will prove the following claim.
\[claim1\] For any $k=1,\dots, n$, we have
Using telescoping sum and the independence assumption and applying Claim \[claim1\] recursively from $k=n$ to $k=1$, we have The lower bound is proved by changing ${\leqslant}$ to ${\geqslant}$ and changing $+$ to $-$ for the error terms. Therefore, we obtain Theorem \[t4\], subject to Claim \[claim1\].
To prove Claim \[claim1\], we first write By the property [101]{} of the sublinear expectation and the definition of $\lambda_*$, we have Note that Hence, By the definition of
| 3,176
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, filtered by order of differential operators, such that*
1. there is an equivalence of categories $U_c {\text{-}{\textsf}{mod}}\simeq B{\text{-}{\textsf}{qgr}}$;
2. there is an equivalence of categories $\operatorname{gr}B{\text{-}{\textsf}{qgr}}\simeq \operatorname{{\textsf}{Coh} }(\operatorname{Hilb(n)})$.
{#intro-1.4}
The construction of $B$ needs some explanation. For $n>2$, it can be shown that the Hilbert scheme $\operatorname{Hilb(n)}$ is not a cotangent bundle, so we cannot use sheaves of differential operators as a non-commutative model. Instead we take as our starting point Haiman’s description of $\operatorname{Hilb(n)}$ as a blow-up of ${\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}}$ and deform this to a non-commutative setting. Set $A^0 = {\mathcal{O}}({\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}})$ with ideal $I=A^1\delta$, where $\delta $ is the discriminant and $A^1 = {\mathbb{C}}[{\mathfrak{h}}\oplus
{\mathfrak{h}}^*]^\epsilon$ the module of anti-invariants. Then [@haidis Proposition 2.6] proves that $\operatorname{Hilb(n)}={\textsf}{Proj}\, A$ where $A=A^0[tI]$ is the Rees ring of $I$ (see Section \[sect-haiman\] for the details).
Unfortunately one cannot construct $B$ as an analogous Rees ring over $U_c$, since $U_c$ is a simple ring for generic values of $c$. We circumvent this problem by using [*${\mathbb{Z}}$-algebras*]{} (see Section \[zalg\]). Specifically, the ring $B$ from Theorem \[mainthm-intro\] is an algebra $B=\bigoplus_{i\geq j\geq 0}B_{ij}$ whose multiplication is defined in matrix fashion: $B_{ij}B_{jk}\subseteq B_{ik}$ but $B_{ij}B_{\ell k}=0$ when $j\not= \ell$. The diagonal terms are just $B_{ii}=U_{c+i}$ while the off-diagonal terms $B_{ij}$ are given as the appropriate tensor products of the $(U_{d+1}, U_{d})$-bimodules $Q_{d}^{d+1} = eH_{d+1}\delta e$. The shift functors $S_d : U_d {\text{-}{\textsf}{mod}}\rightarrow U_{d+1}{\text{-}{\textsf}{mod}}$ given by tensoring with $Q_{d}^{d+1}$ are important operators in the theory of Cherednik algebras and have already pla
| 3,177
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ion to its variance. Practical applications of principal portfolios have already been considered by several authors, for example, Poddig and Unger (2012) and Kind (2013).
In this paper we present a perturbative calculation of the principal portfolios of the single-index CAPM in the large $N$ limit. The results of this calculation are in general expected to entail a relative error of the order of $1/{N}^2$. However, since any application of the single-index CAPM is most likely to involve a large asset set, the stated error is normally quite small and in any case majorized by modelling errors. Thus the results to be reported here are accurate implications of the underlying model.
The principal portfolio analysis of the single-index model and an exactly solvable version of it presented in §3 highlight the volatility structure of principal portfolios in a practical and familiar context. A remarkable result of the analysis is the bifurcation of the set of principal portfolios into a [*market-aligned*]{} portfolio, which is unleveraged and behaves rather like a total-market index fund, and $N-1$ *market-orthogonal* portfolios, which are hedged and leveraged,[^1] and nearly free of market driven fluctuations. This equivalency between the original asset set and two classes of principal portfolios is reminiscent of, but fundamentally different from, Merton’s (1972) two mutual fund theorems. The market-orthogonal portfolios, on the other hand, provide a vivid demonstration of the effect of leveraging on the volatility level of a portfolio.
2. Principal Portfolios of the Single-Index Model {#principal-portfolios-of-the-single-index-model .unnumbered}
=================================================
Here we shall analyze the standard single-index model as well as an exactly solvable special case of it with respect to their principal portfolio structure. Remarkably, our analysis will uncover interesting and hitherto unnoticed properties of well-diversified and arbitrarily leveraged portfolios within the single-index m
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differential equations, it is interesting to note that the property of order 4 perfect squares is preserved here : these two terms are such that their higher order terms cancel perfectly, what makes their associated equations of motion second order. For example the term : $$\begin{aligned}
\begin{split}
\sqrt{-g} \; &\sqrt{ C^{\mu\nu\alpha}C_{\mu\nu\alpha} }
=\frac{\sqrt{3}}{6} \frac{\sqrt{B(r)}}{r B(r)^3A(r)^3} \Bigg( \Sigma \Big(r,B(r),A(r),B'(r),A'(r),B''(r),A''(r) \Big)
\\ &-3 r^3 B(r)^2 A(r) A'(r) B''(r)
-r^3 B(r)^2 A(r) B'(r) A''(r)+2 r^3 B(r)^2 A(r)^2
B^{(3)}(r) \Bigg),
\end{split}\end{aligned}$$ where $\Sigma \Big(r,B(r),A(r),B'(r),A'(r),B''(r),A''(r) \Big)$ is a sum of 15 first order terms (that lead trivially to second order differential equations), is equivalent, up to boundary terms, to the following first order expression : $$\begin{aligned}
\begin{split}
\sqrt{-g} \; &\sqrt{ C^{\mu\nu\alpha}C_{\mu\nu\alpha} } \equiv \frac{\sqrt{3}}{6} \frac{\sqrt{B(r)}}{r B(r)^3A(r)^3} \Bigg( 4B(r)^3 A(r)^2-4 B(r)^3 A(r)^3-5 r B(r)^2 A(r)^2 B'(r)
\\&\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad +2 r^2
B(r) A(r)^2 B'(r)^2-\frac{1}{4} r^3 A(r)^2 B'(r)^3 \Bigg).
\end{split}\end{aligned}$$ Therefore, we have shown first that, up to order 6, all the perfect squares that one can build for static spherically symmetric space-time are also perfect squares in FLRW as we have seen for the case of $ \nabla_\sigma R \nabla^\sigma R$, because in this space-time $W_{\mu \nu\alpha\sigma} =0$. And secondly, that they also share the property that their square-root lead to second order equations of motion for the metric field.
These perfect squares, coming from the action $S_6$ for $A(r)$ and $B(r)$ are respectively: $$\begin{aligned}
\begin{split}
~& 16 \big(\gamma +2 \delta \big)
B(r)^3+4 r \big(-5 \gamma +2 \delta \big) B(r)^2 B'(r)
\\& +4 r^2 \big(2 \gamma
+\delta \big) B(r) B'(r)^2+r^3 \big(-\gamma +\delta \big) B'(r)^3=0,
\end{split}\end{aligned}$$ And, $$\begin{align
| 3,179
| 4,659
| 2,436
| 2,535
| null | null |
github_plus_top10pct_by_avg
|
t the range $R(I+ P_0)$ is dense that is, \[evo17-a\] =L\^2(GSI).
As in the proof of Lemma \[csdale0\] (note that here the assumptions are somewhat weaker), we have for all $\phi\in D(P_0)$, (we write $L^2=L^2(G\times S\times I)$) \[ineq\] & P\_0,\_[L\^2]{} =-[E]{},\_[L\^2]{} +\_x,\_[L\^2]{}+ CS\_0,\_[L\^2]{}\
=& (-+CS\_0),\_[L\^2]{} + ,\_[T\^2(\_+)]{}\^2 +[12]{}\_[GS]{}S\_0(,0)(,,0)\^2 dx d, which in combination with the assumptions , implies $${\left\langle}P_0\phi,\phi{\right\rangle}_{L^2(G\times S\times I)}\geq 0,\quad \forall \phi\in D(P_0).$$
If $\phi\in D(\tilde{P}_0)$, choose a sequence $\phi_n\in D(P_0)$ such that $\phi_n\to \phi$ and $P_0\phi_n\to \tilde{P}_0\phi$ in $L^2(G\times S\times I)$ when $n\to\infty$. By the above inequality, we have $${\left\langle}\tilde{P}_0\phi,\phi{\right\rangle}_{L^2(G\times S\times I)}
=\lim_n {\left\langle}P_0\phi_n,\phi_n{\right\rangle}_{L^2(G\times S\times I)}\geq 0,$$ which gives .
Finally, from it follows that \[evo18-ab\] [(I+P\_0)]{}\_[L\^2(GSI)]{}\_[L\^2(GSI)]{},D(P\_0). and therefore $R(I+\tilde P_0)$ is closed in $L^2(G\times S\times I)$. This result, the observation that $R(I+P_0)\subset R(I+\tilde{P}_0)$, and show that $R(I+\tilde P_0)=L^2(G\times S\times I)$. The proof is complete.
Theorem \[md-evoth\] says that:
\[d-cor\] The operator $-\tilde P_0:L^2(G\times S\times I)\to L^2(G\times S\times I)$ is $m$-dissipative, or equivalently $\tilde P_0$ is $m$-accretive.
See e.g. [@dautraylionsv5 p. 340].
\[F-L-P-R\] The general theory of initial boundary value problems of symmetric formally dissipative first order partial differential operators can alternatively be applied to show the $m$-dissipativity of $-\tilde P_0$. The (classical) results for positive symmetric initial boundary value problems can be found in [@lax], Theorem 3.2 and discussion in section 4 therein; [@friedrich58], together with discussion in section 17 therein; [@sarason] and [@rauch85]. The spatial domain there is replaced with $G\times S$ (here the additional smooth compact manifold $S
| 3,180
| 1,300
| 2,523
| 3,146
| null | null |
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|
c{16}{64}=0,25$.\
Note that the optimal strategy saturating this limit always exists. To see this let $\alpha={\left(\underline{a}_1,\ldots,\underline{a}_8,\underline{b}_1,\ldots,\underline{b}_8\right)}$ be one of configurations for which $c(\alpha)$ attains its maximal value. Then the Bell inequality is saturated for the joint distribution probability $p(\alpha)=1$, $p(\alpha')=0$ for $\alpha '\neq\alpha$. Such distribution can be written in form (\[c\]) with $f_A(s)=\underline{a}_s$, $f_B(t)=\underline{b}_t$.
In the quantum strategy Alice and Bob share the state corresponding to the maximal eigenvalue of $\sum_{n=1}^3 X{\left(\varphi_n,\psi_n\right)}$. If they receive the numbers $s$, $t$ from an arbitrator, they measure $a_s$ (Alice) and $b_t$ (Bob), respectively, and send the result to the arbitrator. The probability of winning in example I is then $\frac{16,09}{64}\simeq 0,2514$ which exceeds (although only slightly) the classical bound. Other examples can be treated similarly.
To make the results slightly more transparent we write out explicitly the sum of probabilities appearing on the right hand side of eq. (\[a3\]). They read:
Example I
$$\begin{split}
& S_1\equiv P{\left(a_1=0,b_4=1\right)}+P{\left(a_1=1,b_5=0\right)}+P{\left(a_1=2,b_7=1\right)}
+P{\left(a_2=0,b_4=2\right)}+\\
& \quad +P{\left(a_2=1,b_8=1\right)}+P{\left(a_2=2,b_5=2\right)}+P{\left(a_3=0,b_4=0\right)}+P{\left(a_3=1,b_8=0\right)}+\\
& \quad +P{\left(a_3=2,b_7=2\right)}+P{\left(a_4=0,b_3=0\right)}+P{\left(a_4=1,b_1=0\right)}+P{\left(a_4=2,b_2=0\right)}+\\
& \quad +P{\left(a_5=0,b_1=1\right)}+P{\left(a_5=1,b_6=0\right)}+P{\left(a_5=2,b_2=2\right)}+P{\left(a_6=0,b_5=1\right)}+\\
& \quad +P{\left(a_6=1,b_7=0\right)}+P{\left(a_6=2,b_8=2\right)}+P{\left(a_7=0,b_6=1\right)}+P{\left(a_7=1,b_1=2\right)}+\\
& \quad +P{\left(a_7=2,b_3=2\right)}+P{\left(a_8=0,b_3=1\right)}+P{\left(a_8=1,b_2=1\right)}+P{\left(a_8=2,b_6=2\right)}+\\
& \quad +P{\left(a_1=0,b_7=0\right)}+P{\left(a_1=1,b_4=0\right)}+P{\left(a_1=2,b_5=2\right)}
+P{\left(a_2=0,b_5=1\ri
| 3,181
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| 3,034
| null | null |
github_plus_top10pct_by_avg
|
e $\xi_{\rm neq} g'(x)$ is a multiplicative contribution due to nonlinear coupling to {$q_k$}-subsystem. It is thus important to note that the presence of multiplicative noise and a fluctuating barrier are associated with nonlinearity in $g(x)$.
Second, the Langevin equation (12) is non-Markovian. The origin of this non-Markovian nature lies in the decaying term in Eq.(15) where the decay explicitly expresses the initial nonequilibrium nature of the $\{ q_k\}$-subsystem following the sudden excitation at $t=0$. This non-Markovian feature is thus not to be confused with that arises due to the usual frequency dependence of the dissipation constant.
Third, although the modification of $V(x)$ is due to the specific choice of the Debye model for the mode density which has so far been commonly used, the theory remains effectively unchanged as one goes over to more complicated spectrum.
We now rewrite Eq.(12) in the form, $$\left.\begin{array}{l}
\dot{u}_{1}=F_{1}(u_{1},u_{2},t ; \xi_{\rm neq},\xi_{\rm eq})
\dot{u}_{2}=F_{2}(u_{1},u_{2},t ; \xi_{\rm neq},\xi_{\rm eq})
\end{array}\right\}\hspace{0.2cm},$$ where we use the following abbreviations, $$\left.\begin{array}{l}
u_{1}=x\\
u_{2}=v \end{array}\right\}$$ and $$\left.\begin{array}{l}
F_{1}=v\\
F_{2}=-\Gamma(x)v-\tilde{V}^{'}(x)+ \xi_{\rm eq}(t)+g'(x)
\xi_{\rm neq}(t)\end{array}\right\} \hspace{0.2cm}.$$
The vector $u$ with components $u_{1}$ and $u_{2}$ thus represents a point in a 2-dimensional ‘phase space’ and the Eq.(16) determines the velocity at each point in this phase space. The conservation of points now asserts the following linear equation of motion for density $\rho(u,t)$ in ‘phase space’, $$\begin{aligned}
\frac{\partial}{\partial t}\rho(u,t)=-\sum_{n=1}^{2}\frac{\partial}{\partial
u_{n}} F_{n}(u,t;\xi_{\rm neq},\xi_{\rm eq})\rho(u,t)\hspace{0.2cm},\end{aligned}$$ or more compactly $$\frac{\partial \rho}{\partial t}=-\nabla\cdot F\rho\hspace{0.2cm}.$$
Our next task is to find out a differential equation whose average solution is given by $
| 3,182
| 4,010
| 3,097
| 2,999
| 2,648
| 0.778166
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|
as discussed in the previous section, would have real entries such that $\alpha =
1$ and $\beta = 2$, and thus the limiting spectral distribution $$\label{E:GOE-limit}
(1-p) {\mathcal{N}}(0,1) + p {\mathcal{N}}(0,2).$$ The slightly simpler nature of this limiting distribution (note that the parameter $p$ plays only one role in , as opposed to two roles in ) reflects that a “GOE-like” normalization of entries is more natural than equal variances. However, this phenomenon is only evident when $0 < p < 1$. In the classical case of Wigner matrices it is well known that in order for the semicircle law to hold, no variance assumption need be made on the diagonal entries of the matrix. The situation described above emphasizes that this is the case precisely because the number of diagonal entries in a Wigner matrix is negligible.
Finally, when the second moments are the same as for the “$G$-circulant GUE” of Proposition \[T:GUE-eigenvalues\], then $\alpha = 0$ and $\beta = 1$ and, as in Corollary \[T:GUE-limit\], the limiting spectral distribution is simply the standard real Gaussian distribution, even regardless of the value of $p$. Thus for $G$-circulant matrices, a constraint to be complex Hermitian appears to be somehow more natural than a constraint to be real symmetric. As in Theorem \[T:circular-law-uncorrelated\], the assumption that $p_2$ approaches a limit can even be removed in this situation.
\[T:semicircle-law-special\] Suppose that for each $n$, $\bigl\{Y_a^{(n)} \mid a \in G^{(n)}
\bigr\}$ are mean $0$ and independent except for the constraint $Y_{a^{-1}}^{(n)} = \overline{Y_a^{(n)}}$; that ${\mathbb{E}}\bigl\vert
Y_a^{(n)}\bigr\vert^2 = 1$ for every $a \in G^{(n)}$; that ${\mathbb{E}}\bigl( Y_a^{(n)} \bigr)^2 = 0$ if $a \neq a^{-1}$; and that holds. Then $\mu^{(n)}$ converges, in mean and in probability, to $\gamma_{\mathbb{R}}$.
The special case of Theorem \[T:semicircle-law-special\] for classical circulant matrices (with more restrictive assumptions on the distributions of the matrix ent
| 3,183
| 1,992
| 3,022
| 2,741
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|
\beta_{k-1}$ and the nonzero eigenvalues of $\mathbf{A}=\mathbf{X}^T\mathbf{X}$ are $\alpha_1>\alpha_2>\cdots>\alpha_{k}$. Further suppose that for $1\le i\le k-1$ we have $\beta_i\ne\alpha_i$ and $\beta_i\ne\alpha_{i+1}$. Then the eigenvector $\mathbf{b}_i$ of $\mathbf{B}$ can be written by $$\mathbf{b}_i=\sum_{j=1}^k\gamma_{ij}\mathbf{v}_j,$$ where $$\gamma_{ij}=\frac{\mathbf{v}_j^T\mathbf{d}}{(\alpha_j-\beta_i)\|\mathbf{d}\|_2}.$$
The point of this lemma is to realize that the vector $\mathbf{b}_i$ is a linear combination of the $\mathbf{v}_i$. The next lemma gives the linear expression of the vectors $\mathbf{b}_i^T\mathbf{X}^{\dagger}$ in terms of the $\mathbf{u}_i$, where $\mathbf{X}^{\dagger}$ is the Moore-Penrose inverse of $\mathbf{X}$. There are practical cases where our assumptions in Lemma \[thm3\] hold true, and examples are given in Appendix \[app2\].
\[thm4\] With the assumptions in Lemma \[thm3\], we have $$\mathbf{b}_i^T\mathbf{X}^{\dagger}=\sum_{j=1}^k\frac{\gamma_{ij}}{\sigma_j}\mathbf{u}_j^T,$$ where $\sigma_j$ is the $j$-th the nonzero singular value of $\mathbf{X}$.
$$\mathbf{b}_i^T\mathbf{X}^{\dagger}=\bigg(\sum_{j=1}^k\gamma_{ij}\mathbf{v}_j^T\bigg)\mathbf{V\Sigma^{\dagger}}\mathbf{U}^T$$ $$=\begin{pmatrix}
\gamma_{i1} & \gamma_{i2} & \cdots &\gamma_{ik} & 0 & \cdots & 0
\end{pmatrix}_{1\times n}\mathbf{\Sigma^{\dagger} U}^T$$ $$=\begin{pmatrix}
\frac{\gamma_{i1}}{\sigma_1} & \frac{\gamma_{i2}}{\sigma_2} & \cdots & \frac{\gamma_{ik}}{\sigma_k} & 0 & \cdots & 0
\end{pmatrix}_{1\times p}\mathbf{U}^T$$ $$=\sum_{j=1}^k\frac{\gamma_{ij}}{\sigma_j}\mathbf{u}_j^T.$$
Lemma \[thm4\] shows that if $\mathbf{b}_i$ can be written as a linear combination of the $\mathbf{v}_j$, then the vectors $\mathbf{b}_i^T\mathbf{X}^{\dagger}$ can be written as a linear combination of the $\mathbf{u}_i$. Next we give the formal definition of the modularity components.\
Suppose $\mathbf{X}_{p\times n}$ is the data matrix, $\mathbf{b}_i$ is the eigenvector corresponding to the $i$-th largest eigenvalue of $\mathbf
| 3,184
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|
it, and all the higher components $M_\#([n])$, $n \geq
2$, together with the transition maps $\iota_f$, can be recovered from $M_\#([1])$ and this extra structure.
Return now to the abelian situation: we are given an associative unital algebra $A$ over a field $k$, and our monoidal category is ${{\mathcal C}}= A{\operatorname{\!-\sf bimod}}$, with the natural tensor product. Then for every $n$, the product $A{\operatorname{\!-\sf bimod}}^n$ has a fully faithful embedding $A{\operatorname{\!-\sf bimod}}^n
\to A^{\otimes n}{\operatorname{\!-\sf bimod}}$, $M_1 \times M_2 \times \dots \times M_n
\mapsto M_1 \boxtimes M_2 \boxtimes \dots \boxtimes M_n$, and one checks easily that the multiplication functors $m_S$ actually extend to right-exact functors $$m_S:A^{\otimes S}{\operatorname{\!-\sf bimod}}\to A{\operatorname{\!-\sf bimod}};$$ for instance, one can define $m_S$ as $$m_S(M) = M/\{ a_{v'}m - ma_v \mid v \in S, a \in A, m \in M \},$$ where $a_v = 1 \otimes \dots \otimes a \otimes \dots \otimes 1 \in
A^{\otimes S}$ with $a$ at the $v$-th position, and $v' \in S$ is the next element after $v$. We can therefore define the cofibered category $A{\operatorname{\!-\sf bimod}}_\#/\Lambda$ with fiber $A^{\otimes V([n])}{\operatorname{\!-\sf bimod}}$ over $[n]
\in \Lambda$, and transition functors $f_!$ as in . We also have the category of sections ${\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$ and the subcategory of cocartesian sections ${\operatorname{\sf Sec}}_{cart}(A{\operatorname{\!-\sf bimod}}_\#) \subset {\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$.
\[sec.ab\] The category ${\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$ is a $k$-linear abelian category.
[[*Sketch of a proof.*]{}]{} This is a general fact about cofibered categories; the proof is straightforward. The kernel ${\operatorname{{\sf Ker}}}\phi$ and cokernel ${\operatorname{{\sf Coker}}}\phi$ of a map $\phi:M_\# \to M'_\#$ between objects $M_\#,M'_\# \in {\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimo
| 3,185
| 2,179
| 2,675
| 2,888
| 2,187
| 0.781936
|
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|
)\phi)(x,\omega)=\int_S\ol\sigma(x,\omega',\omega,E)\phi(x,\omega')d\omega',\quad \phi\in L^2(G\times S),$$ and where $\Gamma'_{-}=\{(y,\omega)\in \partial G\times S\ |\ \omega\cdot\nu(y)<0\}$, while $\gamma'_-:\tilde{W}^2(G\times S)\to \Gamma'_{-}$; $\gamma'_-(\psi)=\psi|_{\Gamma'_{-}}$ is the trace mapping (see section \[fs\]).
We interpret $\phi$ as a mapping $I\to L^2(G\times S)$ by defining $\phi(E)(x,\omega):=\phi(x,\omega,E)$. Assuming that $\phi(E)\in D(A_C(E))$ for any $E\in I$ (which takes care of the inflow boundary condition) the problem (\[se1a\]), (\[se2a\]), (\[se3a\]) [*for $g=0$*]{} can be put into the abstract form \[ecsd6\] [E]{}-A\_C(E)=[**f**]{}(E),(0)=0.
We recall the following result from the theory of evolution equations.
\[evoth\] Suppose that $X$ is a Banach space and that for any fixed $t\in [0,T]$ the operator $A(t):X\to X$ is linear and closed, with domain $D(A(t))\subset X$. In addition, we assume that the following conditions hold:
\(i) The domain $D:=D(A(t))$ is independent of $t$ and is a dense subspace of $X$.
\(ii) The operator $A(t)$ is $m$-dissipative for any fixed $t\in [0,T]$
\(iii) For every $u\in D$, the mapping $f_u:[0,T]\to X$ defined by $f_u(t):=A(t)u$ is in $C^1([0,T],X)$.
\(iv) $f\in C^1([0,T],X)$ and $u_0\in D$.
Then the (evolution) equation \[ecsd5\] [t]{}-A(t)u=f,u(0)=u\_0, has a unique solution $u\in C([0,T],D)\cap C^1([0,T],X)$. In addition, the solution is given by \[solev\] u(t)=U(t,0)u\_0+\_0\^t U(t,s)f(s) ds where $U(t,s):X\to X$, $0\leq t\leq s\leq T$, is a family of bounded operators, strongly continuous in $(t,s)$, called the *(two-parameter) evolution system of operators* of $A(t)$, $t\in [0,T]$, and $U(\cdot,s)u_0$ solves (for a fixed $s$) for every $u_0\in D$ the Cauchy problem \[solevb\] (U(t,s)u\_0)-A(t)U(t,s)u\_0=0,U(s,s)u\_0=u\_0.
See [@tanabe Theorem 4.5.3, pp. 89-106], [@pazy83 pp. 126-182], [@engelnagel pp. 477-496].
\[re:evoth\_norm\] We make a brief remark concerning the meaning of the claim in Theorem \[evoth\] that $u\in C([0,T],D)
| 3,186
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f{FINDISC}$ be the class of finite discrete categories. Since $\emptyset\in\mathsf{FINDISC}$, any left or right $\Phi$-stable derivator is pointed. It is easy to see that ${\mathsf{Stab}_L}(\mathsf{FINDISC}) = {\mathsf{Stab}_L}(\{\emptyset,2\})$, where $2$ denotes the discrete category with two objects, and similarly for ${\mathsf{Stab}_R}$.
In fact, we have ${\mathsf{Stab}_L}(\mathsf{FINDISC}) = {\mathsf{Stab}_R}(\mathsf{FINDISC}) = \mathsf{SEMIADD}$, the collection of semiadditive derivators. For since ${\sD}^2 \simeq{\sD}\times{\sD}$ by one of the derivator axioms, the left and right Kan extensions along $2\to \bbone$ are just binary coproducts and products. Then if ${\sD}$ is pointed and binary coproducts preserve all limits, then in particular they preserve binary products, which means that $$(X\times Z) + (Y\times W) \cong (X+Y)\times (Z+W)$$ canonically. Taking $Y=Z=0$, we see that $X+W \cong X\times W$ canonically, so that ${\sD}$ is semiadditive. Conversely, if ${\sD}$ is semiadditive, then the coproduct and product functors ${\sD}\times {\sD}\to{\sD}$ coincide, and in particular the coproduct is a right adjoint and so preserves all limits. Thus ${\sD}$ is left $\mathsf{FINDISC}$-stable if and only if it is semiadditive, and dually for right $\mathsf{FINDISC}$-stability.
There are a number of natural questions suggested by this phrasing of the characterization theorems:
1. By definition, ${\sD}$ is left $u$-stable if and only if $u_!\colon {\sD}^A\to {\sD}^B$ is continuous. But a continuous functor is crying out to be a right adjoint, for instance if there is an adjoint functor theorem. General derivators have no adjoint functor theorem, but does $u_!$ happen to be a right adjoint anyway?
2. \[prop:ptd-comm,thm:stable-lim-III\] are self-dual, and in particular $\mathsf{POINT}$ and $\mathsf{STABLE}$ are fixed points of both Galois connections. Is there an abstract explanation for this?
3. We have seen that interesting collections of derivators like $\mathsf{POINT}$ and $\mathsf{STAB}$ can be gener
| 3,187
| 2,920
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| 2,945
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e first claim of the lemma by letting $\beta =\beta '={\alpha }$ and $E'=E$, $F'=F$. The second claim follows from $$\begin{aligned}
{\varDelta }(E)-K_{\alpha }{\otimes }E-E{\otimes }1\in &\mathop{\oplus }
_{\beta ,\gamma \in {\mathbb{N}}_0^I, \beta +\gamma ={\alpha },\,\beta ,\gamma \not=0}
U ^+(\chi )_\beta K_\gamma {\otimes }U ^+(\chi )_\gamma ,\\
{\varDelta }(F)-1 {\otimes }F-F{\otimes }L_{\alpha }\in &\mathop{\oplus }
_{\beta ,\gamma \in {\mathbb{N}}_0^I, \beta +\gamma ={\alpha },\beta ,\gamma \not=0}
U ^-(\chi )_{-\beta }{\otimes }U ^-(\chi )_{-\gamma }L_\beta ,
\end{aligned}$$ and from Eq. (with $x=E$, $y=F$) and Prop. \[pr:sHpdef\](iii).
Let ${\mathbb{K}}$ be a field extension of ${\Bbbk }$. The importance of the Shapovalov form arises from the fact that it induces a form on the Verma modules $M^\chi (\Lambda )$ and on their simple quotients $L^\chi (\Lambda )$, where $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$.
Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Define $$\begin{aligned}
\Lambda {\mathrm{Sh}}:U(\chi )\times U(\chi )\to {\mathbb{K}},\quad
(u,v)\mapsto \Lambda ({\mathrm{Sh}}(u,v)).
\label{eq:LSh}\end{aligned}$$ By Eq. , $$\begin{aligned}
\Lambda {\mathrm{Sh}}(u_-u_0u_+,v_-v_0v_+)=&
{\varepsilon }(u_+){\varepsilon }(v_+)\Lambda (u_0{\mathrm{Sh}}(u_-,v_-)v_0)\\
=&\Lambda (u_0)\Lambda (v_0){\varepsilon }(u_+){\varepsilon }(v_+)
\Lambda {\mathrm{Sh}}(u_-,v_-).\end{aligned}$$ Thus, by Eq. , $\Lambda {\mathrm{Sh}}$ induces a ${\mathbb{K}}$-bilinear form on $M^\chi (\Lambda )$ by letting $$\Lambda {\mathrm{Sh}}:M(\Lambda )\times M(\Lambda )\to {\mathbb{K}},\quad
(u{\otimes }1_\Lambda , v{\otimes }1_\Lambda )\mapsto \Lambda {\mathrm{Sh}}(u,v)$$ for all $u,v\in U(\chi )$. Moreover, Eq. gives that $$\begin{aligned}
\label{eq:LShf}
\Lambda {\mathrm{Sh}}( u{\otimes }1_\Lambda , v)
=\Lambda {\mathrm{Sh}}(1{\otimes }1_\Lambda ,{\Omega }(u)v) =0\end{aligned}$$ for all $u\in U(\chi )$ and $v\in I^\chi (\
| 3,188
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-1/2} \right)^2 spacetime vector,
\otimes \left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2}
\right)$
valued in adjoint of $so(12)$
$\overline{\partial} X^{1-2}_{-1} \otimes gravity, tensor multiplet contributions
\left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2}
\right)$
$\left( \lambda^{7-14}_{-1/2} \overline{\lambda}^{7-14}_{-1/2} \right) spacetime vector,
\otimes \left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2}
\right)$
valued in adjoint, ${\bf 1}$ (trace) of $su(8)$
$\left( \lambda^{15-16}_{-1/2}, \overline{\lambda}^{15-16}_{-1/2} \right)^2 spacetime vector,
\otimes \left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2}
\right)$
valued in adjoint of $so(4)$
$\left( \lambda^{1-6}_{-1/2}, \overline{\lambda}^{1-6}_{-1/2} \right) 4 sets of scalars,
\left( \lambda^{15-16}_{-1/2}, \overline{\lambda}^{15-16}_{-1/2} \right)
\otimes \left( \psi^{3-4}_{-1/2}, \overline{\psi}^{3-4}_{-1/2}
\right)$
| 3,189
| 5,383
| 1,009
| 2,530
| null | null |
github_plus_top10pct_by_avg
|
en the dimensionality of the spaces $G$, $S$ or $I$ in fact), with the exception that typically only charged particle fields (are assumed to) obey CSDA version of the transport equation (cf. ), while non-charged particles obey the standard linear BTE (cf. ). Thus with very minor modifications, and in particular if one is interested in radiation therapy, what will and has been said works, in principle, equally well in proton (and ion) therapy framework as well.
We find that $D:L^2(G\times S\times I)^3\to L^2(G)$ is a bounded linear operator and its adjoint operator $D^*:L^2(G)\to L^2(G\times S\times I)^3$ is simply a multiplication type operator, \[irtp14\] D\^\*d=(\_1,\_2,\_3)d, dL\^2(G).
We describe shortly an optimization problem related to inverse radiation treatment planning. We restrict ourselves to [*external radiation therapy*]{} in which the particles are inflowing through the patch(es) of patient surface. This means that in the transport problem $f=0$ (i.e. the internal particle source vanishes) and $g$ (the inflow particle flux) is the variable to be controlled. Conversely, for the internal radiation therapy problems one sets $g=0$, and $f$ would be the variable to be controlled. Anyhow, the results presented below would be analogous in this situation. We refer to [@tervo14 Section 7] and to the references therein for a more detail exposition of *inverse problem* (optimization) in this setting. Let $g\in T^2(\Gamma_-)^3$ and let $\psi=\psi(g)\in
\tilde W^2(G\times S\times I)\times (\tilde W^2(G\times S\times I)\cap W_1^2(G\times S\times I))^2$ be the solution of the variational equation (see , and ) \[irtp10\] \_0((g),v)=[**F**]{}\_0(v)v, where (since $f=0$) \_0(v)=([**F**]{}\_0g)(v) := \_[j=1]{}\^3\_[GSI]{}()\_-g\_j v\_j dddE=g,\_-(v)\_[T\^2(\_-)\^3]{}. The deposited dose is then \[irtp12\] D(x)=(D((g)))(x),xG. We shall also denote ${{{\mathcal{}}}D}(g):=D(\psi(g))$.
Denote the target region by ${\bf T}\subset G$, the critical organ region by ${\bf C}\subset G$ and the normal tissue region by ${\
| 3,190
| 1,425
| 2,523
| 2,960
| null | null |
github_plus_top10pct_by_avg
|
g(y+a^i_{y,\omega}\omega,\omega,E)\Big( e^{-\int_{a^i_{y,\omega}}^\tau\Sigma(y+s\omega,\omega,E)ds}\varphi(y+\tau\omega,\omega,E) \Big|_{\tau=a^i_{y,\omega}}^{\tau=b^i_{y,\omega}} \Big){d}y{d}\omega{d}E \\
&+\int_{S\times I}\int_{G_{\omega}}\sum_i \int_{J^i_{y,\omega}} \Sigma(y+\tau\omega,\omega,E)e^{-\int_{a^i_{y,\omega}}^\tau\Sigma(y+s\omega,\omega,E)ds} \\
&\cdot g(y+a^i_{y,\omega}\omega,\omega,E)\varphi(y+\tau\omega,\omega,E){d}y{d}\omega{d}E \\
={}&0 + \int_{G\times S\times I} \Sigma(x,\omega,E)e^{-\int_0^{t(x,\omega)}\Sigma(x-s\omega,\omega,E)ds} \\
&\cdot g(x-t(x,\omega)\omega,\omega,E)\varphi(x,\omega,E){d}x {d}\omega {d}E \\
={}&\big(\Sigma (L_-g)\big)(\varphi),$$ where at the 3rd equality we used the fact that $\varphi$ vanishes on the boundary of $G$ and $y+a^i_{y,\omega}\omega\in\partial G$ and $y+b^i_{y,\omega}\omega\in\partial G$. This completes the proof.
Note that if we assume that there exists $c>0$ such that $\Sigma\geq c$ on $G\times S\times I$, then in the previous lemma one can take $G$ to be unbounded as well.
Analogously to Lemma \[trathle1\] for any $g\in T^2_{\tau_+}(\Gamma_+)$ the weak solution of the problem $$\begin{aligned}
\omega\cdot \nabla_x\psi+\Sigma\psi&=0,\\
\psi(y,\omega,E)&=g(y,\omega,E)\quad {\rm for}\ (y,\omega,E)\in \Gamma_+\ \end{aligned}$$ is given by (note that $(y,\omega)\in\Gamma_-$ if and only if $(y,-\omega)\in\Gamma_+$) $$(L_+g)(x,\omega,E):=\psi(x,\omega,E)=e^{-\int_0^{t(x,-\omega)}\Sigma(x-s\omega,\omega,E)ds} g(x+t(x,-\omega)\omega,\omega,E).$$
For later use (section \[comp\]) we also treat the inhomogeneous convection-scattering equation with the homogeneous boundary data. Suppose that $f\in C(\ol G\times S\times I)$ such that ${{\frac{\partial f}{\partial x_j}}}\in C(\ol G\times S\times I)$, and let $\Sigma\in C(\ol G\times S\times I)$ such that ${{\frac{\partial \Sigma}{\partial x_j}}}\in C(\ol G\times S\times I),\ j=1,2,3$ . Then the unique (classical) solution of the equation \[trath7\] \_x+=f D, satisfying the homogeneous inflow boundary condition \
| 3,191
| 2,472
| 2,504
| 2,818
| null | null |
github_plus_top10pct_by_avg
|
on $C=C_1\otimes\cdots \otimes C_r$, say with $\operatorname{{\textsf}{ogr}}C_j=D_j$ and $\operatorname{{\textsf}{ogr}}C=D$. Moreover, by Theorem \[main\], respectively Proposition \[pre-cohh\] combined with Lemma \[thetainjA\], respectively Proposition \[app-c-prop\] combined with Lemma \[thetainjC\], there is an equality $D_1\cdots D_r=D$ given by multiplication in $D({\mathfrak{h}^{\text{reg}}})\ast{{W}}$. Equivalently, the natural multiplication map $\chi: D_1\otimes \cdots \otimes D_r\to D$ is surjective. Consider the graded map $\chi$ in more detail. Given elements $\bar{\alpha}_j\in \operatorname{{\textsf}{ogr}}^{m(j)} D_j$, with $m = \sum m(j)$, lift the $\bar{\alpha}_j$ to elements $\alpha_j\in ord^{m(j)}C_j$. Then $\chi$ is defined by $$\chi(\bar{\alpha}_1\otimes\cdots\otimes \bar{\alpha}_r)
=\left(\alpha_1\cdots\alpha_r + \operatorname{{\textsf}{ord}}^{m - 1}C\right)/\operatorname{{\textsf}{ord}}^{m - 1}C.$$ By the definition of the $\operatorname{{\textsf}{ten}}$ filtration, this says that image of $\chi$ is contained in (and indeed equal to) $\bigoplus_{m} \bigl(\operatorname{{\textsf}{ten}}^m C + \operatorname{{\textsf}{ord}}^{m-1} C\bigr)/\operatorname{{\textsf}{ord}}^{m-1} C$. But $\chi$ is surjective. By induction on $m$ we therefore have $\operatorname{{\textsf}{ord}}^mC= \operatorname{{\textsf}{ten}}^m C + \operatorname{{\textsf}{ord}}^{m-1} C = \operatorname{{\textsf}{ten}}^m C$.
{#ord-tens-chat}
The equality of filtrations given by Lemma \[ord-tens\] is not merely a formality; indeed the result for $B_{ij}$ is essentially the same result as Theorem \[main\]. To see this, suppose that $\operatorname{{\textsf}{ogr}}B_{ij}=\operatorname{{\textsf}{tgr}}B_{ij}$ for all $i\geq j\geq0$. As Lemma \[thetainjA\](2) shows, $\operatorname{{\textsf}{ogr}}B_{\ell+1,\ell} = A^1\delta$ for each $\ell$ and so, by Lemma \[abstract-products\](2), we get a surjection $\chi$ from $E=(A^1\delta)^{\otimes(i-j)}$ onto $\operatorname{{\textsf}{tgr}}B_{ij}=\operatorname{{\textsf}{ogr}}B_{ij}$.
The multipl
| 3,192
| 2,324
| 1,423
| 3,234
| 2,662
| 0.778042
|
github_plus_top10pct_by_avg
|
n by cytokines (IL-6) and activation of thyroid-stimulating hormone receptor (TSH-r). In addition, OF have been shown to display the immunoregulatory molecules major histocompatibility complex MHC class II (HLA-DR) and intercellular adhesion molecule-1 (ICAM-1), and also are capable of secreting chemokines and cytokines which stimulate the infiltration of activated T cells into areas of inflammation. In the 'burn-out' stage of ophthalmopathy, OF participate in tissue fibrosis. Reproduced with permission from Bednarczuk T, Gopinath B, Ploski R, Wall JR. Susceptibility genes in Graves' ophthalmopathy: searching for a needle in a haystack? *Clin Endocrinol (Oxf).* 2007; 67(1):3--19.[@b45-opth-4-417] Copyright © 2007 Wiley Blackwell.\
**Abbreviations:** LFA, lymphocyte function-associated antigen; MHC, major histocompatibility complex.](opth-4-417f2){#f2-opth-4-417}
######
Thyroid associated ophthalmology (TAO) subtypes, clinical features and candidate autoantibodies
**TAO subtype** **Main clinical features** **Candidate autoantigens**
----------------------------------------------- ----------------------------------------- -----------------------------------------------
Ocular myopathy Diplopia Calsequestrin
EOM dysfunction G2s[a](#tfn1-opth-4-417){ref-type="table-fn"}
Exophthalmos Flavoprotein
Congestive ophthalmopathy Watery, gritty eyes TSH-r
Periorbital edema Collagen XIII
Conjunctival injection/chemosis
Exophthalmos
Mixed congestive and myopathic ophthalmopathy Congestive and my
| 3,193
| 5,371
| 1,739
| 2,434
| null | null |
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|
(\Omega \otimes \Omega) E = -
(\Omega \otimes \Omega) [I_{k^2} \;\;\;\;\; 0_{k^2 \times k}] =
\Big[ - (\Omega \otimes \Omega) \;\;\;\;\; 0_{k^2 \times k} \Big].$$ The top derivative in is more involved. By the product rule, $$\frac{d \left( \alpha^\top \otimes I_k
\right) (\Omega \otimes \Omega) }{d \psi} =
\Big( ( \Omega \otimes \Omega) \otimes I_k \Big) \frac{d
(\alpha^\top \otimes I_k) }{d \psi} + \Big( I_{k^2} \otimes (\alpha^\top
\otimes I_k) \Big) \frac{ d (\Omega \otimes \Omega)}{d
\psi}.$$ The first derivative in the last expression is $$\begin{aligned}
\frac{d
(\alpha^\top \otimes I_k) }{d \psi} & =
\frac{d (\alpha^\top \otimes I_k) }{d \alpha} \frac{d \alpha}{d \psi} =
(I_k \otimes K_{1,k} \otimes I_k) (I_k \otimes \mathrm{vec}(I_k)) F \\
& = (I_k \otimes \mathrm{vec}(I_k) )F =
( I_k \otimes \mathrm{vec}(I_k)) \Big[0_{k \times k^2} \;\;\;\;\; I_k \Big] \\
& = \Big[0_{k^3 \times k^2} \;\;\;\;\; I_k \otimes \mathrm{vec}(I_k) \Big] ,\end{aligned}$$ where $K_{k,1}$ is the appropriate commutation matrix and the third identity follows since $K_{k,1} = I_k$ and, therefore, $(I_k \otimes K_{1,k} \otimes I_k) = I_{k^3}$. Continuing with the second derivative in , $$\begin{aligned}
\frac{ d (\Omega \otimes \Omega)}{d
\psi} & = \frac{ d (\Omega \otimes \Omega)}{d \Omega} \frac{d \Omega}{d \Sigma} \frac{d \Sigma}{d \psi}
= - J (\Omega \otimes \Omega) E \\
& = - J (\Omega \otimes \Omega) \Big[ I_{k^2} \;\;\;\;\; 0_{k^2 \times k}\Big]
= -J \Big[ \Omega \otimes \Omega\ ;\;\;\;\; 0_{k^2 \times k}
\Big], \end{aligned}$$ where $$J = \Big[ (I_k \otimes \Omega) \otimes I_{k^2}
\Big] \Big( I_k \otimes K_{k,k} \otimes
I_k \Big) \Big( I_{k^2} \otimes \mathrm{vec}(I_k) \Big) + \Big[ I_{k^2} \otimes(
\Omega \otimes I_k) \Big] \Big( I_k \otimes K_{k,k} \otimes
I_k \Big) \Big( \mathrm{vec}(I_k) \otimes I_{k^2} \Big).$$ To see this, notice that, by the product rule, we have $$J = \frac{d (\Omega\otimes
\Omega)}{d \Omega} = \frac{d (\Omega \otimes I_k)( I_k
\otimes
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frac{1}{\Psi _{\eta ,\kappa
}(x,y)}$$Now, by a standard calculus, ${\Psi _{\eta ,\kappa }(x,y)}\geq C_{\kappa }%
\frac{\psi _{\kappa }(x-y)}{\psi _{\eta +\kappa }(x)}$ (use that $\psi
_{\kappa }(x-y)\leq C_{\kappa }\psi _{\kappa }(x)\psi _{\kappa
}(-y)=C_{\kappa }\psi _{\kappa }(x)\psi _{\kappa }(y)$), so (\[TR6d\]) follows. $\square $
Proof of Theorem \[TransferBIS-new\] {#sect:proofTransferBIS}
------------------------------------
By applying Theorem \[Transfer\], $P_{t}(x,dy)=p_{t}(x,y)dy$ and $p_{t}$ satisfies (\[TR6’\]), which we rewrite here as $$\|\Psi_{\eta,\kappa}p_t\|_{q,p}\leq Ct^{-\theta_\ast(q+\theta_1)},$$ where $\theta _{\ast }=\theta _{0}(1+\frac{a+b}{\delta}%
)(1+\varepsilon )$ and $\theta _{1}$ is computed from (\[TR6’\]) (the precise value is not important here). The constant $C$ in the above inequality depends on $\kappa ,\eta, \varepsilon ,\delta, q $. Moreover, by choosing $\eta>\kappa+d$, $$\int_{\R^d\times \R^d}\Psi_{\eta,\kappa}(x,y)p_t(x,y)dx dy
= \int_{\R^d}\frac 1{\psi_{\eta}(x)}\times P_t\psi_{\kappa}(x)dx
\leq \int_{\R^d}\frac 1{\psi_{\eta-\kappa}(x)}dx=m<\infty.$$ So, Lemma \[reg\] (recall that we are working here with $\R^d\times \R^d=\R^{2d}$) gives $$\|\Psi_{\eta,\kappa}p_t\|_{q,p}\leq C_\ast t^{-\theta_\ast(q+2d/p_\ast)}.$$ We choose now $p>2d$ and by Morrey’s inequality, $$\|\Psi_{\eta,\kappa}p_t\|_{q,\infty}\leq
C \|\Psi_{\eta,\kappa}p_t\|_{q+1,p}\leq
C t^{-\theta_\ast(q+1+2d/p_\ast)}.$$ By taking $p=2d/(1-\varepsilon)$, we get $$\|\Psi_{\eta,\kappa}p_t\|_{r,\infty}\leq
C t^{-\theta_\ast(r+2d+\varepsilon)},$$ where $C$ denotes here a constant depending on $\kappa ,\eta, \varepsilon$. This gives that, for every $x,y\in\R^d$ and for every multi-index $\alpha$ and $\beta$, $$\left\vert \partial _{x}^{\alpha }\partial _{y}^{\beta
}p_{t}(x,y)\right\vert \leq C\times t^{-\theta_\ast(|\alpha|+|\beta|+2d+\varepsilon)}\,\frac{\psi_{\eta}(x)}{\psi_{\kappa}(y)}.$$ The statement now follows from . $\square$
Proof of Theorem \[J\] {#sect:proofJ}
----------------------
In
| 3,195
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| 2,104
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|
1(v) &{\stackrel{x}{\longrightarrow_{}}} & E_2(v) \label{nruleE1}\\\vdots&&\vdots \nonumber\\
D_k(v) &{\stackrel{x}{\longrightarrow_{}}} & v \label{nruleDk}\\E_k(v) &{\stackrel{x}{\longrightarrow_{}}} & v \label{nruleEk}\\E_k(v) &{\stackrel{z}{\longrightarrow_{}}} & v \label{nruleEkz}\\L_1 &{\stackrel{\ell_1}{\longrightarrow_{}}} & \bot $$ A proof of $0 {\:|\!\!\!=\!\!\!\!=\:}A(\bot),B(\bot),\hat{S} \leadsto (\varepsilon,\varepsilon) {\:|\!\!\!=\!\!\!\!=\:}{\rm SUCC}$ can still be written, but with a longer initial strategy $\hat{S}$ where the maximal length of words is $3+k$, and a prefix of strategy $\hat{S}_6$ of length $k$. Note that the sizes of the proofs $\pi_3,\pi_4,\pi_5,\pi_6$ still remain the same.
The flawed argument
===================
Let us locate precisely, in [@Jan10], the crucial [*flawed*]{} argument in favor of soundness of the systems.\
Page 24, line \$-4, the following assertion (FA) is written:\
“The final rule in deriving $m {\:|\!\!\!=\!\!\!\!=\:}(U,U',S') \leadsto (\varepsilon,\varepsilon) {\:|\!\!\!=\!\!\!\!=\:}SUCC$ could not be the Basis rule, due to the least eq-level assumption for $T,T'$ (recall Prop. 17)”.\
In our example: $$(T,T') = (A(\bot),B(\bot)),\;\;
EqLv((A(\bot),B(\bot))=3$$ Let us take $$(U,U',S')= (E(\bot),E(\bot),S_6)$$ We have: $$EqLv(U,U',S') =0 = EqLv(T,T',S)-3$$ And the judgment $$3 {\:|\!\!\!=\!\!\!\!=\:}E(\bot),E(\bot),S_6 \leadsto (\varepsilon,\varepsilon) {\:|\!\!\!=\!\!\!\!=\:}SUCC$$ can be derived by the proof $\pi_7$ below.
{width="14cm"}
Hence $(T,T')$ has the [*least*]{} equivalence level, among the EqLevels of the elements of $\{ (T,T') \} \cup {\cal B}$ while $m,U,U'$ fulfills the [*maximality*]{} hypothesis of the text (line \$-7).\
But the final rule used in this proof is the basis rule (R7), contradicting the assertion (FA).\
The bug seems to be the following: by Proposition 17 $$EqLv(E(L_1),E(L_1)) \leq EqLv(E(\bot),E(\bot))
\label{prop17}$$ BUT $$EqLv(E(L_1),E(L_1)) > EqLv(E(\bot),E(\bot),S_6)\;\; !
\label
| 3,196
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| 0.788827
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t case, the emission probability for charges $\pm q$ is proportional to a Boltzmann factor of the form $$\begin{aligned}
e^{-\frac{1}{T}\left(m\pm\frac{qQ}{r_+}\right)}= e^{-\frac{m}{T}\left(1\pm\frac{q}{\sqrt{2}m}(1-e^{2\Delta\phi})\right)}.\end{aligned}$$ If $|\Delta\phi|\gg 1$, the discharge rate is fast if the WGC is satisfied and the black hole is hot.
Now we consider the case $m/T\gg 1$. Here the discharge rate is governed by the Schwinger process, and the rate exponent can be determined by barrier penetration arguments for a mode of frequency equal to the electrostatic potential energy at the horizon, $\omega_+=-qQ/r_+$ [@Gibbons:1975kk]. For $\omega_+$ to be a scattering state, we must have $q>\sqrt{2}m$ near extremality.
The Klein-Gordon equation for the $s$-wave mode of frequency $\omega_+$ is $$\begin{aligned}
\Phi''(r)+W\Phi(r)=0\;,\;\;\;\;\;W=\frac{q^2}{2}-\frac{m^2r}{r-2M}+\frac{(Me^{2\Delta\phi})^2}{(r-2M)^2(r-2M+2Me^{2\Delta\phi})^2}\;.\end{aligned}$$ Here we have put the equation in normal form and taken $e^{2\Delta\phi}\ll 1$. The barrier $W<0$ extends approximately from $r\sim 2M+e^{2\Delta\phi}M \equiv \alpha$ to $r\sim 2M\left(1+ \frac{2m^2}{q^2-2m^2}\right)\equiv \beta$. In the WKB approximation, the barrier penetration factor is $$\begin{aligned}
e^{-2\int_\alpha^\beta \sqrt{W}dr}.\end{aligned}$$ We can approximately evaluate the WKB integral by splitting it into regions where the last two terms in $W$ dominate ($\equiv W_{23}$, valid near $\alpha$) and where the first two terms in $W$ dominate ($\equiv W_{12}$, valid near $\beta$). The two regions overlap where the first and third terms are of similar order, near $r\sim 2M+1/\sqrt{2}q$. Putting the pieces together and keeping only the dominant terms, we find the production rate is of order $$\begin{aligned}
\Gamma_{Einstein} \sim e^{-2\int_\alpha^\gamma \sqrt{W_{23}}dr-2\int_\gamma^\beta \sqrt{W_{12}}dr}\approx e^{-\frac{2\sqrt{2}\pi M m^2}{\sqrt{q^2-2m^2}}}\;.
\label{eq:einsteinrate}\end{aligned}$$ This is similar to the Schwinger exp
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| 2,928
| null | null |
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|
nction for the network is set as the (weighted) sum of the error activations across each layer. We utilize the $L_{all}$ formulation presented in the original paper, which places a non-zero loss on the error unit activity in every level in the network. Except where stated otherwise, results presented here use a model trained on the KITTI car-mounted camera dataset [@Geiger2013IJRR]. The same model hyperparameters were used as presented in the paper (besides the $relu$ activation in the LSTM units). Particularly, the model consists of four layers. With $0$-indexing used here, Layer $1$ would be analogous to V1 in visual cortex.
![Deep Predictive Coding Networks (PredNets) [@Lotter_2017]. Left: Each layer consists of representation neurons ($R_l$), which output a layer-specific prediction at each time step ($\hat{A}_l$), which is compared against a target ($A_l$) to produce an error term ($E_l$), which is then propagated laterally and vertically. Right: Module operations for case of video sequences. The target at the lowest layer of the network, $\hat{A}_0$, is set to the current input image.[]{data-label="architecture"}](prednet.pdf){width="62.00000%"}
Single Neuron Response Properties
=================================
We begin by comparing the response properties of units in the PredNet to established single unit response properties of neurons in the primate visual system, which have been studied extensively using microelectrode recordings. Here, we primarily compare response properties in the PredNet’s error (“E”) units, the output units of each layer, to neuronal recordings in the superficial layers of cortex. Response properties of other units in the PredNet (e.g. the “R” units) are included in the Supplemental Materials, and would likely map onto other parts of the cortical circuit.
#### On/Off Temporal Dynamics
As mentioned in the introduction, a conspicuous feature of visual cortical neuron responses is that they are highly dynamic, even when a static, unchanging image is presented to the sub
| 3,198
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|
$X:={{\rm{Spec}}}(S_\triangle)$ and $Y:={{\rm{Spec}}}((S_\triangle)^W)$, then both $X$ and $Y$ are normal, irreducible, affine algebraic varieties. Since the action of $W$ on $X$ is free, the dominant morphism $\phi : X \to Y$ is unramified in codimension $1$. So we can use Theorem \[cor1\] to get $$\la{DiffIden}
\D(S_\triangle)^W \, \cong \D((S_\triangle)^W) \, .$$
\[prop1\] $(i)$ If $A$ is a domain and $M$ is an Ore subset then ${{\rm{Frac}}}(A_M)\cong {{\rm{Frac}}}(A)$.\
$(ii)$ $(S_\triangle)^W \cong (S^W)_\triangle$.\
$(iii)$ $\D(S_\triangle)^W \cong (\D(S)^W)_\triangle$.\
$(iv)$ ${{\rm{Frac}}}(A_n)^W\simeq {{\rm{Frac}}}(A_n^W)$.\
$(i)$ This statement is clear.\
$(ii)$ Since $\triangle$ is an invariant polynomial then $f\in (S_\triangle)^W$ iff $\triangle^k f \in S^W$ for some $k\ge 0$ iff $f \in (S^W)_\triangle$.\
$(iii)$ Note that $\D(S_M)\cong D(S)_M$ for a multiplicative set $M$, [@MR Theorem 15.1.25]. If $d \in \D(S_\triangle)^W$ then $\triangle ^k d \in \D(S)^W$ for some $k\ge 0$. Finally, (iv) follows from [@Fa], Theorem 1, see also [@D].
Now $(S_\triangle)^W \cong (S^W)_\triangle \cong S_{\triangle}$ , where the first identity holds by part $(ii)$ while the second one follows from the Chevalley-Shephard-Todd theorem. Therefore, the right hand side of is isomorphic to $\D(S_\triangle)\cong \D(S)_\triangle $. Thus, using part $(ii)$ we have $$( \D(S)^W)_\triangle \, \cong \, \D(S)_\triangle \, .$$ Finally, taking the skew field of fractions on both sides, we obtain $$F_n^W \cong F_n \, .$$
Gelfand-Kirillov conjecture for rational Cherednik algebras
===========================================================
Let us first recall the definition of rational Cherednik algebras. As before $W$ is a finite complex reflection subgroup of $\GL(V)$ and $(\cdot, \cdot)$ is a $W$-invariant positive definite Hermitian form. For $H\in \mathcal{A}$, let $v_H \in V $ be such that $\alpha_H=v_H^*$. Next for each $H$ from $\mathcal{A}$, we set $$e_{H,i}\, := \, \frac{1}{n_H} \, \sum_{w\in W_H} \,(\det w )^{-i} w
| 3,199
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|
eaker result, which is significantly stronger than the result of [@Mikl] for subfamilies of $\binom{[n]}{\leq 3}$.
\[bigchvatal\] Let $\cH\sse \binom{[n]}{\leq 3}$ be a downset, and let $\cI\sse \cH$ be a maximum intersecting family. If $|\cI|\geq 31$, then $\cI$ is a star. Hence $\cH$ is EKR when $\i(\cH)\ge 31$.
Of course, some intersecting family (in particular, some star) will be so large if $|\cH|>15n$ or $|\cH^3|>10n$, for example.\
Our proofs use the notion of *Sunflowers*, including the famous *Sunflower Lemma* of Erdős and Rado [@ErdRad], as well as a variant by H[å]{}stad, et al [@HaJuPu]. We state both the Sunflower Lemma and the variant below, after the following definitions.
A set $S$ is a *covering set* for a set system $\cF$ if $S\cap F\neq \mt$ for every $F\in \cF$. The covering number of $\cF$, denoted by $\tau(\cF)$, is the size of the smallest covering set of $\cF$.
\[sunflower\] A *sunflower* with $k$ petals and core $C$ is a set system $\{S_1,\ldots,S_k\}$ such that for any $i\neq j$, $S_i\cap S_j=C$. The sets $S_i\setminus C$ are the petals of the sunflower, and must be non-empty. If $k=1$ then we may choose $C$ to be any proper subset of $S_1$.
For a set system $\cF$ and set $Y$, let $\cF_Y=\{F\setminus Y:F\in \cF,Y\sse F\}$.
\[flower\] A $k$-flower with core $C$ is a set system $\cF$ with $\tau(\cF_C)\geq k$.
\[sflemma\] [@ErdRad] If a family of sets $\cF$ is $r$-uniform and $|\cF|> r!(k-1)^r$ sets, then it contains a sunflower with $k$ petals.
We will use the following variant of Theorem \[sflemma\].
\[kflemma\] [@HaJuPu] If $\cF$ is $r$-uniform and $|\cF|>(k-1)^r$, then $\cF$ contains a $k$-flower.
Proof of Theorem \[completechvatal\]
====================================
Let $\cI$ be an intersecting subfamily of $\cH$ of maximum size. Our goal is to show that either $\cI$ must be a star or otherwise that $\cH$ contains a star of size equal to that of $\cI$, and to characterize the cases for which the latter happens.\
If $\cH$ does not contain a set of size $3$ then $\cI$ is
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