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nction $f(x_T - y_T)$ contracts exponentially with rate $\lambda$, plus a discretization error term. The function $f$ is defined in Appendix \[s:defining-q\], and sandwiches $\lrn{x_T - y_T}_2$. In Corollary \[c:main\_gaussian:1\], we apply the results of Lemma \[l:gaussian\_contraction\] recursively over multiple steps to give a bound on $f(x_{k\delta}-y_{k\delta})$ for all $k$, and for sufficiently small $\delta$.
3. Finally, in Appendix \[ss:proof:t:main\_gaussian\], we prove Theorem \[t:main\_gaussian\] by applying the results of Corollary \[c:main\_gaussian:1\], together with the fact that $f(z)$ upper bounds $\lrn{z}_2$ up to a constant factor.
[A coupling construction]{} \[ss:coupling\_construction\] In this subsection, we will study the evolution of and over a small time interval. Specifically, we will study $$\begin{aligned}
\numberthis
\label{e:marginal_x}
d x_t =& - \nabla U(x_t) dt + M(x_t) dB_t\\
\numberthis
\label{e:marginal_y}
d y_t =& - \nabla U(y_0) dt + M(y_0) dB_t
\end{aligned}$$ One can verify that is equivalent to , and is equivalent to a single step of (i.e. over an interval $t\leq \delta$).
We first give the explicit coupling between and : ( A similar coupling in the continuous-time setting is first seen in [@gorham2016measuring] in their proof of contraction of .)
Given arbirary $(x_0,y_0)$, define $(x_t,y_t)$ using the following coupled SDE: $$\begin{aligned}
\numberthis \label{e:coupled_2_processes}
x_t =& x_0 + \int_0^t -\nabla U(x_s) ds + \int_0^t \cm dV_s + \int_0^t N(x_s) dW_s\\
y_t =& y_0 + \int_0^t -\nabla U(y_0) dt + \int_0^t \cm \lrp{I - 2\gamma_s \gamma_s^T} dV_s + \int_0^t N(y_0) dW_s
\end{aligned}$$
Where $dV_t$ and $dW_t$ are two independent standard Brownian motion, and $$\begin{aligned}
\gamma_t := \frac{x_t-y_t}{\|x_t-y\|_2} \cdot \ind{\|x_t-y_t\|_2 \in [2\epsilon, \Rq)}
\numberthis \label{d:gammat}
\end{aligned}$$
By Lemma \[l:marginal\_of\_coupling\], w
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in\sR^k$ there exists a constant vector $const$ (all elements are equal) such that $\vln\,\softmax(\vx)=\vx+const$. Furthermore, $\softmax(\vv+const)=\softmax(\vv)$ for any vector $\vv$ and any constant vector $const$. Therefore, $$\begin{aligned}
\muh_{DirLin}(\softmax(\vz); \frac{1}{t}\MI, \vzero)
&=\softmax(\frac{1}{t}\,\MI\,\vln\,\softmax(\vz))) \\
&=\softmax(\frac{1}{t}\,\MI\,(\vz+const)) \\
&=\softmax(\frac{1}{t}\,\MI\,\vz+\frac{1}{t}\,\MI\, const) \\
&=\softmax(\vz/t+const') \\
&=\softmax(\vz/t) \\
&=\muh'_{TempS}(\vz; t)\end{aligned}$$ where $const'=\frac{1}{t}\,\MI\, const$ is a constant vector as a product of a diagonal matrix with a constant vector.
Dirichlet calibration
=====================
In this section we show some examples of reliability diagrams and other plots that can help to understand the representational power of Dirichlet calibration compared with other calibration methods.
Reliability diagram examples
----------------------------
We will look at two examples of reliability diagrams on the original classifier and after applying $6$ calibration methods. Figure \[fig:mlp:bs:reldiag\] shows the first example for the 3 class classification dataset *balance-scale* and the classifier MLP. This figure shows the confidence-reliability diagram in the first column and the classwise-reliability diagrams in the other columns. Figure \[fig:nb:reldiag:mlp:bal:uncal\] shows how posterior probabilities from the MLP have small gaps between the true class proportions and the predicted means. This visualisation may indicate that the original classifier is already well calibrated. However, when we separate the reliability diagram per class, we notice that the predictions for the first class are underconfident; as indicated by low mean predictions containing high proportions of the true class. On the other hand, classes 2 and 3 are overconfident in the regions of posterior probabilities compressed between $[0.2, 0.5]$ while being underconfident in higher regions. The discrepancy shown by analysing the ind
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- 'Saint Francis University, Loretto, PA 15940'
author:
- 'A. Fox'
- 'B. LaBuz'
- 'R. Laskowsky'
title: A coarse invariant
---
Introduction
============
A coarse function $f:X\to Y$ between metric spaces is a function that is bornologous and proper. $f$ is bornologous if for each $N>0$ there is an $M>0$ such that if $d(x,y)\leq N$, $d(f(x),f(y))\leq M$. In this setting we call $f$ proper if inverse images of bounded sets are bounded.
Notice bornology is dual to continuity. Thus bornology is a fundamental concept of coarse (or large scale) geometry just as continuity is a fundamental concept of topology (small scale geometry). We are studying the large scale behavior of functions and large scale properties of spaces.
Two metric spaces $X$ and $Y$ are coarsely equivalent if there are coarse functions $f:X\to Y$ and $g:Y\to X$ such that $g\circ f$ is close to ${\text{id}}_X$ and $f\circ g$ is close to ${\text{id}}_Y$. Two functions $f_1$ and $f_2$ are close if $d(f_1(x),f_2(x))$ is uniformly bounded. A standard reference for the preceding concepts and coarse geometry in general is [@Roe].
In [@MMS] an invariant in the bornologous category is constructed. This note extends the construction in [@MMS] to the coarse category. Bornologous equivalence is more strict than coarse equivalence. For bornologous equivalence $f\circ g$ and $g\circ f$ are required to be the identity on the nose. Coarse equivalence can be viewed as being in the category where, instead of considering functions, one considers equivalence classes of functions. Two functions are equivalent if they are close.
The standard example of two coarsely equivalent spaces is $\mathbb R$ and $\mathbb Z$ (see Example \[integers\]). Of course these spaces cannot be bornologously equivalent because they do not have the same cardinality. We can explain interest in the coarse category as opposed to the bornologous category as follows. Since we are interested in large scale behavior, we should ignore all small scale behavior including cardinality. We should not
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ion from $T_3(\bar{\kappa})$ to itself;
- $Y \mapsto \sigma({}^t Y) + Y$ defines a surjection $T_3(\bar{\kappa}) \rightarrow T_2(\bar{\kappa})$.
Here, all the above maps are interpreted as in Remark \[r35\] (if they are well-defined). Then $\rho_{\ast, m}$ is the composite of these three. Condition (3) is direct from the construction of $T_3(\bar{\kappa})$. Hence we provide the proof of (1) and (2).\
For (1), by using the argument explained in the last two paragraphs of page 477 in [@C2], it suffices to show that the two functors $T_1(R)\longrightarrow T_3(R), X\mapsto h\cdot X (\in \mathrm{M}_{n\times n}(B\otimes_AR))$ and $T_3(R) \longrightarrow T_1(R), Y \mapsto h^{-1}\cdot Y (\in \mathrm{M}_{n\times n}(B\otimes_AR))$ are well-defined for all flat $A$-algebras $R$. In other words, we only need to show that $h\cdot X \in T_3(R)$ and $h^{-1}\cdot Y\in T_1(R)$. We represent $h$ by a hermitian block matrix $\begin{pmatrix} \pi^{i}\cdot h_i\end{pmatrix}$ with a matrix $(\pi^{i}\cdot h_i)$ for the $(i,i)$-block and $0$ for the remaining blocks as in Remark \[r33\].(1).
For the first functor, it suffices to show that $h\cdot X$ satisfies the five conditions defining the functor $T_3$. Here, $X\in T_1(R)$ for a flat $A$-algebra $R$. We express $$X=\begin{pmatrix} \pi^{max\{0,j-i\}}x_{i,j} \end{pmatrix}.$$ Then $$h\cdot X = \begin{pmatrix} \pi^{max(i,j)}y_{i,j}\end{pmatrix}.$$ Here, $y_{i,i}=h_i\cdot x_{i,i}$. The proof that $h\cdot X$ satisfies conditions (a) and (b) is similar to that of Lemma 3.7 of [@C2] and so we skip it.
For condition (c), let $L_i$ be *of type I* with *i* even. Recall that we denote $X$ by $(m_{i,j}', s_i'\cdots w_i')$. Then the $(n_i\times n_i)^{th}$-entry of $y_{i,i}$ is $\pi (1+2\gamma_i) z_i'$ or $\pi (z_i'+2\gamma_i w_i')$ if $L_i$ is *of type $I^o$* or *of type $I^e$*, respectively. The $(n_{i-2}\times n_i)^{th}$-entry (resp. $(n_{i+2}\times n_i)^{th}$-entry) of the matrix $y_{i-2, i}$ (resp. $y_{i+2, i}$) is $k_{i-2, i}'+2 k_{i-2}'$ (resp. $k_{i+2, i}'+2 k_{i+2}'$), for some $k_{i-2
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te its derivative: \^[a |a]{} &=& x STr(g\^[-1]{} t\^a g t\^[|a]{}) &=& x STr ( - g\^[-1]{} g g\^[-1]{} t\^a g t\^[|a]{} + g\^[-1]{} t\^a g g\^[-1]{} g t\^[|a]{} ) &=& x STr ( g\^[-1]{}\[t\^a,t\^d\] g t\^[|a]{} ) &=& - i[f\^a]{}\_[bc]{} \^[b |a]{}. We have left out the normal ordering symbols from the above classical calculation. The properties used in the calculation are that the supertrace is graded cyclic and the fact that the equation $g g^{-1}=1$ and its derivative hold true. We assume that the quantum theory is consistent with these two rules. In section \[primaries\] we will give a generic proof of equations and , valid up to a certain order in a semi-classical expansion (see equation ).
Notice that the relations and imply that $\partial(\kappa_{ab} \mathcal{A}^{a
\bar a} \mathcal{A}^{b \bar b})=0 =\bar \partial(\kappa_{ab} \mathcal{A}^{a \bar
a} \mathcal{A}^{b \bar b})$ (and identical equations with the barred indices contracted), and thus are compatible with the equations relating the adjoint primary to the identity and .
The left current - right current OPEs
-------------------------------------
We have collected the tools to calculate the left/right current operator product expansions. Thanks to equations and we only need the left current self OPEs as well as the OPE between the left current and the adjoint primary operator . As an example, we will explicitly compute the OPE $j^a_{L,z}(z) j^{\bar a}_{R,z}(w)$ at the order of the poles. We use the prescription of appendix \[compositeOPEs\]: $$\begin{aligned}
\label{jLjR1stStep} j^a_{L,z}&(z) j^{\bar a}_{R,z}(w) =
-j^a_{L,z}(z) \frac{c_-}{c_+} \kappa_{cb}:j^b_{L,z} \mathcal{A}^{c \bar a}:(w) \cr
%
& = -\frac{c_-}{c_+} \kappa_{cb} \lim_{:x \to w:}
\left[ j^a_{L,z}(z) j^b_{L,z}(x) \mathcal{A}^{c \bar a}(w) \right] \cr
%
& = -\frac{c_-}{c_+} \kappa_{cb} \lim_{:x \to w:} \left[
\left( \frac{c_1 \kappa^{ab}}{(z-x)^2}
+ \frac{c_2 {f^{ab}}_d j^d_{L,z}(x)}{z-x}
+ \frac{(c_2-g) {f^{ab}}_d j^d_{L,\bar z}(x)(\bar z - \bar x)}{(z-x)^2}
+ ... \right) \
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rt{\log n\ \log\log n}\right)})$.
We will work with polynomials over the ring $$\cR = \cR_{6,6}=\Z_6[\gamma]/(\gamma^6-1)$$ (see Section \[preliminaries\]). We will denote the vector $(\gamma^{z_1},\gamma^{z_2},\cdots,\gamma^{z_k})$ by $\gamma^\bz$ where $\bz=({z_1,\cdots,z_k}) \in \Z_6^k$. We will need to extend the notion of partial derivatives to polynomials in $\cR[x_1,\ldots,x_k]$. This will be a non standard definition, but it will satisfy all the properties we will need. Instead of defining each partial derivative separately, we define one operator that will include all of them.
Let $\cR$ be a commutative ring and let $F(\bx)=\sum c_{\bz}\bx^{\bz} \in \cR[x_1,\ldots,x_k]$. We define $F^{(1)} \in (\cR^k)[x_1,\ldots,x_k]$ to be $$\begin{aligned}
F^{(1)}(\bx)&:=\sum (c_{\bz}\cdot \bz) \bx^{\bz}
\end{aligned}$$
For example, when $F(x_1,x_2)=x_1^2x_2+4x_1x_2+3x_2^2$ (with integer coefficients), $$F^{(1)}(x_1,x_2)={\left[\begin{matrix}
2\\
1\\
\end{matrix}\right]}x_1^2x_2+{\left[\begin{matrix}
4\\
4\\
\end{matrix}\right]}x_1x_2+{\left[\begin{matrix}
0\\
6\\
\end{matrix}\right]}x_2^2$$ One can think of $F^{(1)}$ both as a polynomial with coefficients in $\cR^k$ as well as a $k$-tuple of polynomials in $\cR[x_1,\ldots,x_k]$. This will not matter much since the only operation we will perform on $F^{(1)}$ is to evaluate it at a point in $\cR^k$.
#### The Protocol:
Let $\ba=(a_1,a_2\cdots,a_n)\in {{\{0,1\}}}^n$ be an n-bit database shared by two servers $\cS_1$ and $\cS_2$. The user $\cU$ wants to find the bit $a_\tau$ without revealing any information about $\tau$ to either server. For the rest of this section, $\cR = \cR_{6,6} = \Z_6[\gamma]/(\gamma^6-1)$. The servers represent the database as a polynomial $F(\bx)\in \mathcal{R}[\bx]=\mathcal{R}[x_1,\cdots,x_k]$ given by $$F(\bx)=F(x_1,\cdots,x_k)=\sum_{i=1}^n a_i \bx^{\bu_i},$$ where $\cU = (\bu_1,\ldots,\bu_n)$ are given by the matching vector family $\cF = (\cU,\cV)$.
The user samples a uniformly random $\bz\in \Z_6^k$ and then sends $\bz+t_1\bv_\
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}}},{\widetilde{\mathbf{u}}}) - 2\nu{\Delta}^2{\widetilde{\mathbf{u}}}- \lambda{\Delta}{\widetilde{\mathbf{u}}}- \bnabla q & = 0 \qquad\mbox{in}\,\,\Omega , \label{eq:KKT_E_gradR}\\
\nabla\cdot{\widetilde{\mathbf{u}}}& = 0 \qquad\mbox{in}\,\,\Omega , \label{eq:KKT_E_divConstr}\\
\E({\widetilde{\mathbf{u}}}) - \E_0 & = 0, \label{eq:KKT_E_E0Constr}\end{aligned}$$
where $\lambda\in\mathbb{R}$ and $q:\Omega\to\mathbb{R}$ are the Lagrange multipliers associated with the constraints defining the manifold ${\mathcal{S}_{\E_0}}$, and ${\mathcal{B}}({\mathbf{u}},{\mathbf{v}})$, given by $${\mathcal{B}}({\mathbf{u}},{\mathbf{v}}) := {\Delta}\left( {\mathbf{u}}\cdot\bnabla{\mathbf{v}}\right) + (\bnabla{\mathbf{u}})^T{\Delta}{\mathbf{v}}-
{\mathbf{u}}\cdot\bnabla({\Delta}{\mathbf{v}}),$$ is the bilinear form from equation . Using the formal series expansions with $\alpha > 0$
\[eq:series3D\] $$\begin{aligned}
{\widetilde{\mathbf{u}}}& = {\mathbf{u}}_0 + \E_0^{\alpha}{\mathbf{u}}_1 + \E_0^{2\alpha}{\mathbf{u}}_2 + \ldots, \\
\lambda & = \lambda_0 + \E_0^{\alpha}\lambda_1 + \E_0^{2\alpha}\lambda_2 + \ldots, \\
q & = q_0 + \E_0^{\alpha}q_1 + \E_0^{2\alpha}q_2 + \ldots\end{aligned}$$
in and collecting terms proportional to different powers of $\E_0^{\alpha}$, it follows from that, at every order $m=1,2,\dots$ in $\E_0^{\alpha}$, we have $$\E_0^{m\alpha}: \qquad\sum_{j=0}^m {\mathcal{B}}({\mathbf{u}}_j,{\mathbf{u}}_{m-j}) - 2\nu{\Delta}^2{\mathbf{u}}_m -
\sum_{j=0}^m\lambda_j{\Delta}{\mathbf{u}}_{m-j} - \nabla q_m = 0 \quad\mbox{in}\,\,\Omega.$$ Similarly, equation leads to $$\label{eq:Incompressible_Uk}
\nabla\cdot{\mathbf{u}}_m = 0 \quad\mbox{in}\,\,\Omega$$ at every order $m$ in $\E_0^{\alpha}$. It then follows from equation that $$\begin{aligned}
\E({\mathbf{u}}) & = & \E({\mathbf{u}}_0) -\big\langle{\mathbf{u}}_0,{\Delta}{\mathbf{u}}_1\big\rangle_{L_2}\E_0^{\alpha} +
\left[ \E({\mathbf{u}}_1) - \big\langle{\mathbf{u}}_0,{\Delta}{\mathbf{u}}_2\big\rangle_{L_2}\right]\E_0^{2\alpha} + \ldots \\
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{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$+\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}\bigg|.$$ $$\leq (\lambda_e-\lambda_1)e^{2c\lambda_e} M \bigg|\bigg\{\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ -}| In the last expression $\lambda'$ is some value in the interval $[\lambda_1,\lambda_e]$, which comes from the mean value theorem for integrals, and $M$ denotes the maximum defined by $$M=\text{Max}\bigg|\bigg\{2+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$+\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}\bigg|_{O_1},$$ in an initial compact set $O_1\times [\lambda_1,\lambda_e]$. This maximum exists since the function $R_\lambda(\lambda)$ is never zero for $\lambda$ in $[\lambda_1,\lambda_e]$ and the integrals are convergent since the space time $(M, g_{\mu\nu})$ in consideration is such that $p_\Lambda(\lambda)=[R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\lambda)$ is not divergent at finite values of $\lambda$. The expression inside the brackets in the last step in (\[last\]) is then $$D_\Lambda=\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$-\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg],$$ where the denomination $D_\Lambda$ is chosen in order to emphasize that it represents a difference. Write this quantity as $$D_\Lambda=\bi
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space{{{\operatorname{dom}{\Phi}}}}} = \phi$.
\[N:SEQUENCE\_NOTATION\] A sequence in a set $S$ is some mapping $\sigma \colon {\mathbb{N}}\to S$ – that is, $\sigma \in S^{\mathbb{N}}$. The anonymous sequence convention allows reference to a sequence using the compound symbol $\lbrace s_n \rbrace$, understanding $s \in S$. Formally, the symbol $s_i$ denotes that term $(i, s_i) \in \lbrace s_n \rbrace$. The convention is clumsy expressing functional notation; for instance $s_i = \lbrace s_n \rbrace(i)$ means $i \stackrel{\lbrace s_n\!\rbrace}{\mapsto} s_i$.
\[D:SUCCESIVELY\_CONJOINT\] Let $\langle \Psi, \Phi \rangle$ be a basis with sequence of frames $\lbrace {\mathbf{f}}_n \rbrace \colon {\mathbb{N}}\to {\prod{\Psi}} \times {\prod{\Phi}}$. The sequence is *successively conjoint* if ${\mathbf{f}}_i$ conjoins ${\mathbf{f}}_{i+1}$ for each $i \geq 1$.
\[D:PROCESS\] With $\langle \Psi, \Phi \rangle$ a basis, a *process* is a successively conjoint sequence of frames ${\mathbb{N}}\to {\prod{\Psi}} \times {\prod{\Phi}}$.
\[D:ABSCISSA\_PROJECTION\] Let $\langle \Psi, \Phi \rangle$ be a basis with frame space ${\mathbf{F}} = {\prod{\Psi}} \times {\prod{\Phi}}$. Define the *abscissa* projection ${{\operatorname{absc}{}}}: {\mathbf{F}} \to {\prod{\Psi}}$ by $(\psi, \phi) \stackrel{{{\operatorname{absc}{}}}}{\mapsto} \psi$. Define the *ordinate* projection ${{\operatorname{ord}{}}}: {\mathbf{F}} \to {\prod{\Phi}}$ by $(\psi, \phi) \stackrel{{{\operatorname{ord}{}}}}{\mapsto} \phi$.
\[D:PERSISTENT\_VOLATILE\_COMPONENTS\] Let $\langle \Psi, \Phi \rangle$ be a basis with persistent-volatile partition $\Psi = \Phi\Xi$ (see appendix §\[S:STATE\_EVENT\_PRTN\]). Suppose ${\mathbf{f}}$ is a frame in ${\prod{\Psi}} \times {\prod{\Phi}}$. The *reactive* state of frame ${\mathbf{f}}$ is $\psi = \phi \xi = {{\operatorname{absc}{{\mathbf{f}}}}}$. The *event* or *volatile* excitation state of frame ${\mathbf{f}}$ is $\xi = {{({{\operatorname{absc}{{\mathbf{f}})}}}}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Xi}}}}}$. Simila
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er\end{aligned}$$ In the previous lines we only kept track of the operators that will lead to poles in the final result. We evaluated the operator $\phi$ at the point $w$ so that the action of the derivatives is easier to take care of: $$\begin{aligned}
= -\kappa_{ab}& t^a t^b \left(\frac{c_+}{c_++c_-} \right)^2
\lim_{:x \to w:}\sum_{n,\bar n=0}^{\infty}\frac{(z-x)^{n-1}}{n!}\frac{(\bar z-\bar x)^{\bar n}}{\bar n!}
\cr
&
\left( \sum_{m,\bar m=0}^{\infty} (-1) \frac{(x-w)^{m-n-1}}{m\ (m-n-1)!} \frac{(\bar x - \bar w)^{\bar m - \bar n}}{(\bar m - \bar n)!}\p^m \bar \p^{\bar m}\phi(w)
\right)- \kappa_{ab} t^a \frac{c_+}{c_++c_-} \frac{:j^b_{L,z} \phi:(w)}{w-z} + ... \nonumber\end{aligned}$$ The regular limit gives a non-zero result for the anti-holomorphic factor only if $\bar m - \bar n = 0$. For the holomorphic factor, one needs $m-n-1=0$. Notice that the terms with $n=m=0$ also contributes with a non-vanishing term. Eventually we obtain: $$\begin{aligned}
\kappa_{ab}& t^a t^b \left(\frac{c_+}{c_++c_-} \right)^2
\left( \sum_{n,\bar n=0}^{\infty}\frac{(z-w)^{n-1}}{(n+1)!}\frac{(\bar z-\bar x)^{\bar n}}{\bar n!} \p^{n+1} \bar \p^{\bar n}\phi(w) \right. \cr
& \qquad
\left. + \sum_{\bar n=0}^{\infty}\frac{1}{(z-w)^2} \frac{(\bar z-\bar x)^{\bar n}}{\bar n!} \bar \p^{\bar n}\phi(w)
\right)- \kappa_{ab} t^a \frac{c_+}{c_++c_-} \frac{:j^b_{L,z} \phi:(w)}{w-z} + ...\cr
%
& = \kappa_{ab} t^a t^b \left(\frac{c_+}{c_++c_-} \right)^2 \frac{\phi(z)}{(z-w)^2} - \kappa_{ab} t^a \frac{c_+}{c_++c_-} \frac{:j^b_{L,z} \phi:(w)}{w-z} + ...
\end{aligned}$$ This completes the evaluation of the first term in . The other terms are much easier to deal with. The only non-trivial part is the computation the OPE between a current $j^b_{L,z}(w)$ and the composite operators $:j^c_{L,z}\phi:(z)$ and $:j^c_{L,\bar
z}\phi:(z)$. Since the coefficients ${A^a}_c$ and ${B^a}_c$ already are of order $f^2$, we only need to know these OPEs at order $f^0$. We find: $$\begin{aligned}
j^b_{L,z}(w):j^c_{L,z}\phi:(z) = &
\left(c_1 \k
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|
(\oplus_i(\pi)e_i)\oplus M_2,$$ $$M_1''=\pi M_1\oplus M_3'=\pi M_1\oplus M_3, ~~~M_2''=M_4'=M_4.$$ Thus, $M_0''$ is *free of type II* as both $M_1''$ and $M_2''$ are *of type II*.
2. If $b\in A$ is a unit, then let $\sqrt{b}$ be an element of $A$ such that $\sqrt{b}^2\equiv b$ mod $2$. We choose a basis $(\pi a, \pi a+1/\sqrt{b}\cdot e)$ for the component $(\pi)a\oplus Be$ of $M_1'$ whose associated Gram matrix is $\begin{pmatrix} -2\delta& -2\delta+\pi/\sqrt{b} \\-2\delta-\pi/\sqrt{b}& -2\delta+2b/\sqrt{b}^2 \end{pmatrix}$. Here, the $(2,2)$-component $-2\delta+2b/\sqrt{b}^2$ is contained in the ideal $(4)$ since $\delta\equiv 1$ mod $2$. Thus, as in the above case (i), we have that $$Y(M_1')= (2)a\oplus B\left( \pi a+1/\sqrt{b}\cdot e \right) \oplus \pi M_1.$$ Here, $(2)a\oplus B\left( \pi a+1/\sqrt{b}\cdot e \right)$ is $\pi^2$-modular *of type II* and $\pi M_1$ is $\pi^3$-modular *of type II*. Since we rescale $Y(C(L^j))$ by $\xi^{-1}$, we have that $$M_0''=\left((2)a\oplus B\left( \pi a+1/\sqrt{b}\cdot e \right)\right) \oplus M_2'=\left((2)a\oplus B\left( \pi a+1/\sqrt{b}\cdot e \right)\right) \oplus (\oplus_i(\pi)e_i)\oplus M_2,$$ $$M_1''=\pi M_1\oplus M_3'=\pi M_1\oplus M_3, ~~~M_2''=M_4'=M_4.$$ Thus, $M_0''$ is *free of type II* as both $M_1''$ and $M_2''$ are *of type II*.\
Therefore, we conclude that the $\pi^0$-modular Jordan component of $Y(C(L^j))$, which is $M_0''$, is *free of type II*. The reason of our assumption that $L_{j+2}, L_{j+3}, L_{j+4}$ are *of type II*, while $L_j$ is *of type I*, is to make $M_0''$ *free of type II*. Let $G_j$ denote the special fiber of the smooth integral model associated to the hermitian lattice $(Y(C(L^j)), \xi^{-(m+1)}h)$. If $m$ is an element of the group of $R$-points of the naive integral model associated to the hermitian lattice $L$, for a flat $A$-algebra $R$, then $m$ stabilizes the hermitian lattice $(Y(C(L^j))\otimes_AR, \xi^{-(m+1)}h\otimes 1)$ as well. This fact induces a morphism from $\tilde{G}$ to $G_j$ (cf. the second paragraph of page 488 in
| 2,711
| 2,044
| 2,031
| 2,478
| 3,102
| 0.774864
|
github_plus_top10pct_by_avg
|
4 e^{-\phi_0} r_1 r_2 r_3 \, e^\alpha \wedge e^\xi \wedge \left( \nu_1 d\phi_1 + \nu_2 d\phi_2 + \nu_3 d\phi_3 \right) \ , \\
\widehat{F}_5 &= (1 +\star) \frac{4 e^{-\phi_0}}{\lambda} r_1 r_2 r_3 \, e^\alpha \wedge e^\xi \wedge d\phi_1 \wedge d\phi_2 \wedge d\phi_3 \ ,
\end{aligned}$$ in complete agreement with the results of [@Frolov:2005dj].
To close this section let us make a small observation. For the $\beta$-deformation $\nu_1 = \nu_2 = \nu_3 \equiv \gamma$ there a special simplification that happens when $\gamma = \frac{1}{n}$, $n\in \mathbb{Z}$. In this case the deformed gauge theory is equivalent to that of D3 branes on the discrete torsion orbifold $\mathbb{C}^3/\Gamma$ with $\Gamma = \mathbb{Z}_n \times \mathbb{Z}_n$. These cases are also special in the dualisation procedure above. Notice that the Lagrange multiplier $v$ corresponding to the central extension is inversely proportional to $\gamma$ and hence the orbifold points correspond to cases where $v$ is integer quantised. Moreover, recalling that non-abelian T-duality with respect to a centrally-extended $U(1)^2$ is equivalent to first adding a total derivative $B$-field, i.e. making a large gauge transformation, and then T-dualising with respect to $U(1)^2$, where the required total derivative is again given by the expression in footnote \[foot:bdiff\], we find that at the orbifold points ($\nu_1 = \nu_2 = \nu_3 \equiv \gamma = \frac1n$) the integral of this total derivative $$\frac1{4\pi^2} \int B_2 = \frac{n}{12\pi^2} \int (d\phi_2 \wedge d\phi_3 + d \phi_1 \wedge d \phi_2 + d\phi_3 \wedge d\phi_1) = n \ ,$$ is also integer quantised.
Application 3: Dipole Deformations {#ssec:app3}
----------------------------------
Dipole theories [@Bergman:2000cw; @Bergman:2001rw] are a class of non-local field theories obtained from regular (or even non-commutative) field theories by associating to each non-gauge field $\Phi_a$ a vector $L^\mu_a$ and replacing the product of fields with a non-commutative product $$(\Phi_1 \tilde\star \Phi_2 )(x) \equiv \P
| 2,712
| 1,252
| 1,529
| 2,649
| 3,991
| 0.768748
|
github_plus_top10pct_by_avg
|
x{\bf g}}, \bar{\mbox{\bf K}})$ form. This surprising result, as we shall see, arises from the behavior of the lapse function [@AAJWY98; @YorkFest].
The constraint equations on $\Sigma$ are, in vacuum, $$\begin{aligned}
\bar{\nabla}_j(\bar{K}^{i j}-\bar{K}\bar{g}^{i j})&=&0 \; , \label{Eq:MomCon}\\
R(\bar{g})-\bar{K}_{i j}\bar{K}^{i j}+\bar{K}^2&=&0\;, \label{Eq:HamCon}\end{aligned}$$ where $R(\bar{g})$ is the scalar curvature of $\bar{g}_{i j}$, $\bar{\nabla}_j$ is the Levi-Civita connection of $\bar{g}_{i j}$; and $\bar{K}$ is the trace of $\bar{K}_{i j}$, also called the “mean curvature” of the slice. (A review of this geometry is given in [@York79].) The overbar is used to denote quantities that satisfy the constraints.
The time derivative of the spatial metric $\bar{g}_{i j}$ is related to $\bar{K}_{i j}$, $\bar{N}$, and the shift vector $\bar{\beta}^{i}$ by $$\partial_t \bar{g}_{i j} \equiv \dot{\bar{g}}_{i j}
= -2\bar{N} \bar{K}_{i j} +
(\bar{\nabla}_i \bar{\beta}_{j}+\bar{\nabla}_j \bar{\beta}_{i}) \; ,
\label{Eq:gdot}$$ where $\bar{\beta}_{j}=\bar{g}_{j i} \bar{\beta}^{i}$. The fixed spatial coordinates $\vec{x}$ of a point on the “second” hypersurface, as evaluated on the “first” hypersurface, are displaced by $\bar{\beta}^i (\vec{x}) \delta t$ with respect to those on the first hypersurface, with an orthogonal link from the first to the second surface as a fiducial reference: $\bar{\beta}_{i}= \mbox{\boldmath $\frac{\partial}{\partial t} $} *
\mbox{\boldmath $\frac{\partial}{\partial x^i} $}$, where $*$ is the physical spacetime inner product of the indicated natural basis four-vectors. The essentially arbitrary direction of [$\frac{\partial}{\partial t}$]{} is why $\bar{N}(x)$ and $\bar{\beta}^{i}(x)$ appear in the TS formulation. In contrast, the tensor $\bar{K}_{i j}$ is always determined by the behavior of the unit normal on one slice and therefore does not possess the kinematical freedom, [*i.e.*]{} the gauge variance, of [$\frac{\partial}{\partial t}$]{}. Therefore, $\bar{N
| 2,713
| 4,692
| 2,023
| 2,223
| null | null |
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|
',\omega,E',E)d\omega' dE'\leq M_1,\quad
&& {\rm a.e.}\ G\times S\times I,\quad j=1,2,3,\nonumber\\[2mm]
&\sum_{k=1}^3\int_{S\times I} \sigma_{jk}(x,\omega,\omega',E,E')d\omega' dE'\leq M_2,\quad
&& {\rm a.e.}\ G\times S\times I,\quad j=1,2,3,\\[2mm]
&\sigma_{kj}\geq 0,\quad && {\rm a.e.}\ G\times S^2\times I^2,\quad k,j=1,2,3.\nonumber\end{aligned}$$
Define the *(restricted) scattering operator* $\Sigma_j$ and the *(restricted) collision operator* $K_j$ corresponding to the particle type $j$, for $j=1,2,3$ and $\psi_j\in
L^2(G\times S\times I)$, as follows \[scat\] (\_j\_j)(x,,E)=\_j(x,,E)\_j(x,,E), and for $\psi=(\psi_1,\psi_2,\psi_3)\in L^2(G\times S\times I)^3$, \[coll3\] (K\_j)(x,,E)=\_[k=1]{}\^3\_[SI]{}\_[kj]{}(x,’,,E’,E)\_k(x,’,E’)d’ dE’. Furthermore, we define for $\psi=(\psi_1,\psi_2,\psi_3)\in L^2(G\times S\times I)^3$, \[sda1\] =(\_1\_1,\_2\_2,\_3\_3) and \[sda2\] K=(K\_1,K\_2,K\_3). One immediately sees that $\Sigma:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ is a bounded linear operator. In addition, by applying Hölder’s inequality we have the following (cf. [@dautraylionsv6 pp. 227-228] and [@tervo14 Theorem 5.2] for $p=1$; see also , ).
\[skb\] The linear operator $K:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ is bounded and \[k-norm\] [K]{}& \_[j=1,2,3]{}[\_[k=1]{}\^3\_[SI]{}\_[kj]{}(,’,,E’,)d’ dE’]{}\_[L\^]{}\^[1/2]{} [\_[k=1]{}\^3\_[SI]{} \_[jk]{}(,,’,,E’)d’ dE’]{}\_[L\^]{}\^[1/2]{}\
& [M\_1]{}\^[1/2]{}[M\_2]{}\^[1/2]{}, where $L^\infty = L^\infty(G\times S\times I)$.
We assume that functions $S_j:\ol G\times I\to{\mathbb{R}},\ j=2,3$, the so-called *restricted stopping powers*, satisfy the following assumptions: $$\begin{aligned}
& S_j\in L^\infty(G\times I), \label{sda2a} \\
& {{\frac{\partial S_j }{\partial E}}}\in L^\infty(G\times I), \label{sda2a-2} \\
& \kappa_j:=\inf_{(x,E)\in \ol{G}\times I} S_j(x,E) > 0, \label{sda2b} \\
& \nabla_x S_j\in L^\infty(G\times I), \label{sda2c}\end{aligned}$$ Note that (\[sda2b\]) implies that in $\ol{
| 2,714
| 855
| 2,377
| 2,640
| null | null |
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|
_{\lambda\muhat}$ is non-zero for all $\lambda$ and $\muhat$. Define $$\label{v-1-defn}
v(\lambda,\mu):=v_q\left(\langle
h_{\mu}(\x),s_\lambda(\x\y)\rangle\right).$$
\[v-lambda\] We have $$-v(\lambda)=(2g-2+k) n(\lambda)+(g-1)n
-\sum_{i=1}^kv(\lambda,\mu^i).$$
Straightforward.
\[v-fmla-lemma\] For $\mu=(\mu_1,\mu_2,\ldots,\mu_r)\in \calP_n$ we have $$\label{v-fmla}
v(\lambda,\mu)= \min
\{n(\rho^1)+\cdots+n(\rho^r)\,\,|\,\, |\rho^p|=\mu_p, \,
\cup_p\rho^p \unlhd\lambda\}.$$
For $C_{\nu\mu}(\y)$ non-zero let $v_m(\nu,\mu):=v_q\left(C_{\nu\mu}(\y)\right)$. When $y_i=q^{i-1}$ we have $v_q(m_\rho(\y))= n(\rho)$ for any partition $\rho$. Hence by $$v_m(\nu,\mu)= \min \{n(\rho^1)+\cdots+n(\rho^r)\,\,|\,\, |\rho^p|=\mu_p, \,
\cup_p\rho^p=\nu\}.$$ Since $K_{\lambda\nu}\geq 0$ for any $\lambda,\nu$, $K_{\lambda\nu}>0$ if and only if $\nu\unlhd \lambda$ [@fulton Ex 2, p.26], and the coefficients of $C_{\nu\mu}(\y)$ are non-negative, our claim follows from .
For example, if $\lambda=(1^n)$ then necessarily $\rho^p=(1^{\mu_p})$ and hence $\rho^1\cup \cdots \cup \rho^r=\lambda$. We have then $$\label{v-example}
v\left((1^n\right),\mu)=\sum_{p=1}^r\binom{\mu_p} 2
= -\tfrac12 n +\tfrac12 \sum_{p=1}^r \mu_p^2.$$ Similarly, $$\label{v-example-1}
v(\lambda,(n))=n(\lambda)$$ by the next lemma.
\[n-ineq\] If $\beta\unlhd\alpha$ then $n(\alpha)\leq n(\beta)$ with equality if and only if $\alpha=\beta$.
We will use the raising operators $R_{ij}$ see [@macdonald I p.8]. Consider vectors $w$ with coefficients in $\Z$ and extend the function $n$ to them in the natural way $$n(w):=\sum_{i\geq 1}(i-1)w_i.$$ Applying a raising operator $R_{ij}$, where $i<j$, has the effect $$n(R_{ij}w)=n(w)+i-j.$$ Hence for any product $R$ of raising operators we have $n(Rw) < n(w)$ with equality if and only if $R$ is the identity operator. Now the claim follows from the fact that $\beta\unlhd\alpha$ implies there exist such and $R$ with $\alpha=R\beta$.
Recall [@macdonald (1.6)] that for any partition $\lambda$ we have $\langle\lambda,\lambda\ran
| 2,715
| 1,784
| 2,157
| 2,391
| null | null |
github_plus_top10pct_by_avg
|
_\tau x^\mu\partial_\tau x^\nu}\end{aligned}$$ where $m=TL$ for $L\gg1/\sqrt T$. Therefore, in our toy model we introduce a charged scalar with action $$\begin{aligned}
S[\Phi]=\int d^5x\sqrt{-G}\left(-G^{\mu\nu}(D_\mu\Phi)^*(D_\nu\Phi)-m^2 V |\Phi|^2\right),\end{aligned}$$ where $D_\mu=\partial_\mu+i q A_\mu$ for a particle of charge $q$. Here $q$ has mass dimension $-1/2$. We treat $\Phi$ as a probe, neglecting backreaction on the metric and gauge field.
The Klein-Gordon equation for $\Phi$ is $$\begin{aligned}
({\cal D}^2-m^2 V)\Phi
=(\Box+2iqA^t\partial_t-q^2 A^t A_t -m^2 V )\Phi=0\;,
\label{eq:KGfull}\end{aligned}$$ and the energy is $$\begin{aligned}
E&=\int d^4 x \sqrt{-G}\left[-G^{tt} |\partial_t\Phi|^2+G^{ii} |\partial_i\Phi|^2+\Phi^*\left(q^2G^{tt}A_t^2 +m^2 V\right)\Phi\right]\;.
\label{eq:energy}\end{aligned}$$ The kinetic, mass, and gradient terms in the energy are positive, although the mass term is suppressed by a factor of $V$ near the bubble wall. The $A_t^2$ term, which arises from the last term in the gauge-invariant charge density $J_t=i \Phi^* \partial_t\Phi-i \partial_t\Phi^* \Phi-2 qA_t|\Phi|^2$, is negative, indicating that small perturbations can lower the energy if this term dominates. Note, however, that the “potential energy operator" $-\nabla^2+q^2G^{tt}A_t^2 +m^2 V$ differs from the fluctuation operator appearing in the equation of motion by a term $ -2iqA^t\partial_t$. Thus negative energy perturbations do not immediately imply complex frequencies or the exponentially growing modes characteristic of classical instabilities.
We will look for $s$-wave solutions to Eq. (\[eq:KGfull\]). Setting $\Phi=\phi(\rho) e^{i\omega t}$ we obtain $$\begin{aligned}
h U \phi '' + \left(U
h'+\frac{h U'}{2}+\frac{3 h U}{\rho }\right)\phi '+ \left(\frac{\left(q A_t+\omega \right){}^2}{h}-m^2 U\right)\phi=0\;.
\label{KGeq}\end{aligned}$$ This equation admits bound states of finite Klein-Gordon norm (charge). The energy and charge of a bound mode is $$\begin{aligned}
E_\phi&=\int d^4
| 2,716
| 3,548
| 2,458
| 2,439
| null | null |
github_plus_top10pct_by_avg
|
kappa^2} + \frac{(p-2)(p-3)(p-3)!}{(p-1)!(\kappa-1)^2} + \cdots + \frac{2(p-3)!}{(p-1)!(\kappa-p+3)^2} \Bigg), \label{eq:crB3} \\
B_4 & \equiv & \frac{(p-3)!}{(p-1)!} \Bigg(\sum_{a,b \in [p-1], b \neq a} \bigg(\frac{1}{\kappa} + \frac{1}{\kappa -1} + \cdots + \frac{1}{\kappa - a+1} \bigg) \bigg(\frac{1}{\kappa} + \frac{1}{\kappa -1} + \cdots + \frac{1}{\kappa - b+1} \bigg) \Bigg) \,. \label{eq:crB4}\end{aligned}$$ Observe that, $$\begin{aligned}
\label{eq:cr6}
&&\frac{\partial^2\P(\theta)}{\partial\theta_i \partial \theta_{\i}}\bigg|_{\theta = {\boldsymbol{0}}} \nonumber\\
&=& \I_{\big\{\Omega^{-1}(i),\Omega^{-1}(\i) > p\big\}} A_1 \Big((-A_2)(-A_2) + B_1 \Big) \nonumber\\
&&+ \;\Big(\I_{\big\{\Omega^{-1}(i) > p, \Omega^{-1}(\i) = p\big\}} + \I_{\big\{\Omega^{-1}(i) = p, \Omega^{-1}(\i) > p\big\}} \Big) A_1 \Big((-A_2)(1-A_2) + B_1 \Big) \nonumber\\
&&+ \; \Big( \I_{\big\{\Omega^{-1}(i) = p, \Omega^{-1}(\i) < p\big\}} + \I_{\big\{\Omega^{-1}(i) < p, \Omega^{-1}(\i) = p\big\}}\Big) A_1 \Big((1-A_3) + (-A_2)(1-A_3) + B_2 \Big) \nonumber\\
&& + \; \Big(\I_{\big\{\Omega^{-1}(i) > p, \Omega^{-1}(\i) < p\big\}} + \I_{\big\{\Omega^{-1}(i) < p, \Omega^{-1}(\i) > p\big\}} \Big) A_1 \Big((-A_2)(1-A_3) + B_2 \Big) \nonumber\\
&& + \;\I_{\big\{\Omega^{-1}(i) < p, \Omega^{-1}(\i) < p\big\}} A_1 \Big((1-A_3) + (-A_3) + B_4 + B_3 \Big)\,.\end{aligned}$$ The claims in are easy to verify by combining Equations and with . Also, define constants $C_1$, $C_2$ and $C_3$ such that, $$\begin{aligned}
C_1 &\equiv &\Bigg( \frac{\kappa-1}{(\kappa)^2} + \frac{\kappa - 2}{(\kappa-1)^2} + \cdots + \frac{\kappa - p}{(\kappa-p+1)^2} \Bigg)\,,\label{eq:crC1}\\
C_2 & \equiv & \Bigg(\frac{(p-1)(p-2)!(\kappa-1)}{(p-1)!(\kappa)^2} + \frac{(p-2)(p-2)!(\kappa-2)}{(p-1)!(\kappa-1)^2} + \cdots + \frac{(p-2)!(\kappa-p+1)}{(p-1)!(\kappa-p+2)^2} \Bigg) \,, \label{eq:crC2}\\
C_3 &\equiv & \frac{(p-2)!}{(p-1)!} \Bigg(\sum_{a,b \in [p-1], b=a} \bigg(\frac{1}{\kappa} + \frac{1}{\kappa-1} + \cdots + \frac{1}{\kappa-a+1}\bigg) \bigg(\frac{1}{\kappa} + \f
| 2,717
| 2,542
| 2,894
| 2,542
| null | null |
github_plus_top10pct_by_avg
|
\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A W \right\}_{K K}
\left\{ W ^{\dagger} A (UX) \right\}_{K l}
\nonumber \\
&+&
\sum_{m}
\sum_{k \neq l } \sum_{K \neq L }
\frac{ 1 }{ ( h_{l} - h_{k} ) ( \Delta_{L} - \Delta_{K} ) ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} ) ( \Delta_{L} - h_{k} ) ( \Delta_{L} - h_{l} ) }
\nonumber \\
&\times&
\biggl[
( h_{l} - h_{k} )
\biggl\{ ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} )
e^{- i ( \Delta_{L} - h_{m} ) x} -
( \Delta_{L} - h_{k} ) ( \Delta_{L} - h_{l} )
e^{- i ( \Delta_{K} - h_{m} ) x}
\biggr\}
\nonumber \\
&+&
( \Delta_{L} - \Delta_{K} )
\biggl\{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{k} ) e^{- i ( h_{l} - h_{m} ) x} -
( \Delta_{K} - h_{l} ) ( \Delta_{L} - h_{l} ) e^{- i ( h_{k} - h_{m} ) x} \biggr\}
\biggl]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l}
(UX)^*_{\alpha m} (UX)_{\beta m}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A W \right\}_{K L}
\left\{ W ^{\dagger} A (UX) \right\}_{L l}
\biggr\}.
\label{P-beta-alpha-W4-H3-offdiag}\end{aligned}$$ $$\begin{aligned}
&& P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{4th-s}
\equiv
2 \mbox{Re} \left[ \left(
S^{(0)}_{\alpha \beta} \right)^{*}
S_{\alpha \beta}^{(4)} [4]_{ \text{ offdiag } } (\text{single})
\right]
\nonumber \\
&=&
2 \mbox{Re}
\biggl\{
\sum_{n}
\sum_{k \neq l }
\sum_{K}
\biggl[
\frac{ (ix) }{ ( \Delta_{K} - h_{k} )^2 ( h_{l} - h_{k} ) }
e^{ - i ( h_{k} - h_{n} ) x}
-
\frac{ (ix) }{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - h_{l} ) }
e^{ - i ( \Delta_{K} - h_{n} ) x}
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{l} )^2 ( h_{l} - h_{k} )^2 }
e^{ - i ( h_{l} - h_{n} ) x}
- \frac{ 1 }{ ( \Delta_{K} - h_{k} )^3 ( h_{l} - h_{k} )^2 }
\left( \Delta_{K} + 2 h_{l} - 3 h_{k} \right)
e^{ - i ( h_{k} - h_{n} ) x}
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{k} )^3 ( \Delta_{K} - h_{l} )^2 }
\left( h_{k} + 2 h_{l} - 3 \Delta_{K} \right)
e^{ - i ( \Delta_{K} - h_{n} ) x}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l}
| 2,718
| 3,471
| 2,796
| 2,729
| null | null |
github_plus_top10pct_by_avg
|
tary model can be written as $$\begin{aligned}
\delta_{\alpha \beta} =
\sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} + \sum_{J=4}^{N+3} W_{\alpha J} W^{*}_{\beta J}.
\label{unitarity0}\end{aligned}$$ Then, $\left| \sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} \right|^2 = \left| \sum_{J=4}^{N+3} W_{\alpha J} W^{*}_{\beta J} \right|^2$ in the appearance channel ($\alpha \neq \beta$), and $\left( \sum_{j=1}^{3} \vert U_{\alpha j} \vert^2 \right)^2 = \left( 1 - \sum_{J=4}^{N+3} \vert W_{\alpha J} \vert^2 \right)^2 = 1 - \mathcal{O} (W^2)$ in the disappearance channel ($\alpha = \beta$), which justifies the above statement.
We emphasize, therefore, that the probability leaking term $\mathcal{C}_{\alpha \beta}$ and the another constant term $\left| \sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} \right|^2$ in the oscillation probabilities are the same order, $\mathcal{O} (W^4)$, in the appearance channels. Hence, we do not see any good reasons why the former can be ignored, as was done in the existing literatures. It is also worth to note that $\mathcal{O} (W^2)$ difference in normalization in the disappearance channel would make detection of unitarity violation more feasible. It is one of the reasons for high sensitivity to unitarity violation that could be reached in disappearance measurement in the JUNO-like setting [@Fong:2016yyh].
Non-unitary evolution of neutrinos in matter {#sec:nonunitarity-matter}
--------------------------------------------
We have also examined in ref. [@Fong:2016yyh] the question of whether inclusion of the matter effect alters the above features of non-unitary evolution of neutrino states in vacuum. We have found that as far as we remain in the region of unitarity violating element $\vert W \vert \simeq 0.1$[^6] or larger, the matter effect does not alter the above features of the oscillation probability in (\[P-beta-alpha-ave-vac\]) under the same restriction on sterile masses. Notice that $\vert W \vert \simeq 0.1$ implies that the unitarity violating effect in the probability is of the o
| 2,719
| 1,158
| 3,169
| 2,578
| null | null |
github_plus_top10pct_by_avg
|
60 U/L
CK-MB 747↑ 0--25 U/L
Myoglobin 72↑ 0.72--4.49 nmol/L
Creatinine 48 15--77 μmol/L
CPK 3,876↑ 26--140 U/L
HBDH 580↑ 72--300 U/L
HCO~3~ ^−^ 28.7↑ 22.0--27.0 mmol/L
Hormones
ACTH 403.76↑ 0.00--90.16 pmol/L
LH 15.89↑ 0.33--6.10 IU/L
FSH 43.90↑ 1.37--6.97 IU/L
Cortisol~8:00am~ 0.06↓ 0.22--1.10 μmol/L
PRA \<0.09↓ 0.09--5.93 nmol/L/h
Progesterone 7.98↑ 1.27--3.82 nmol/L
Aldosterone 115.56↓ 168.88--723.78 pmol/L
17-OHP 0.10↓ 0.80--1.80 nmol/L
Androstenedione \<1.05↓ 1.05--11.57 nmol/L
Estradiol \<43.32↓ 136.76--315.00 pmol/L
Testosterone 0.59↓ 0.70--1.70 nmol/L
DHEA-S 0.25↓ 0.90--15.96 μmol/L
AST, aspartate aminotransferase; ALT, alanine aminotransferase; LDH, lactate dehydrogenase; CK, creatine phosphokinase; CK-MB, creatine kinase muscle/brain; CPK, creatine phosphokinase; HBDH, hydroxybutyrate dehydrogenase; ACTH, adrenocorticotropic hormone; PRA, plasma renin activity; LH, luteinizing hormone; FSH, follicle-stimulating hormone; 17-OHP, 17*α*-hydroxyprogesterone; DHEA-S, dehydroepiandrosterone sulfate.
{#f1}
Genetic Diagnosis {#s3_2}
-----------------
Genomic DNA was extracted from p
| 2,720
| 3,670
| 3,101
| 2,677
| null | null |
github_plus_top10pct_by_avg
|
appa^{bc} + \frac{c_-(c_2-g)-c_+c_2}{c_++c_-}{f^{bc}}_d t^d \right)\frac{\phi(z)}{(w-z)^2} \cr
& \quad - \frac{c_+}{c_++c_-} \frac{t^b :j^c_{L,z}\phi:(z)}{w-z} + \mathcal{O}(f^2) \\
%
j^b_{L,z}(w):j^c_{L,\bar z}\phi:(z) = &
\left(\tilde{c} \kappa^{bc} + \frac{c_-(c_2-g)+c_+(c_4-g)}{c_++c_-}{f^{bc}}_d t^d \right)\phi(z)2\pi \delta^{(2)}(w-z) \cr
& \quad - \frac{c_+}{c_++c_-} \frac{t^b :j^c_{L,\bar z}\phi:(z)}{w-z} + \mathcal{O}(f^2). \end{aligned}$$ All the terms that appear in these OPEs give zero once contracted either with ${A^a}_c \kappa_{ab}$ or with ${B^a}_c \kappa_{ab}$. In particular factors of the form ${f^a}_{bc}t^c t^b$ vanish since the dual Coxeter number is zero. Gathering everything we obtain: $$\begin{aligned}
\phi(z) 2 c_1 T(w)
&= \frac{c_+^2}{(c_++c_-)^2}t^a t^b \kappa_{ab} \frac{\phi(z)}{(z-w)^2} -\frac{2 c_+}{c_++c_-} \frac{\kappa_{ab}t^a :j^b_{L,z}\phi:(z)}{w-z}+ \mathcal{O}(z-w)^{0}+ \mathcal{O}(f^2).\end{aligned}$$ The previous result is true only up to terms of order $f^2$, since a term of order $f^4$ in the current-primary OPE may give a term of order $f^2$ once contracted with an additional current (see lemma ). We rewrite the result as: $$\begin{aligned}
T(w) \phi(z)
&= \frac{f^2}{2} \frac{t^a t^b \kappa_{ab} \phi(z)}{(z-w)^2} +\frac{1}{c_+} \frac{\kappa_{ab}t^a :j^b_{L,z}\phi:(z)}{w-z}+ \mathcal{O}(z-w)^{0}+ \mathcal{O}(f^4) \end{aligned}$$ This concludes the proof of equation in section \[primaries\].
The mode expansion on the cylinder {#commutators}
==================================
When the theory is defined on a cylinder we can expand the operators in modes by means of a Fourier transform along the compact coordinate. Then we can convert the current-current OPEs into graded commutation relations for the modes of the currents. This was done for the current algebra in [@Ashok:2009xx]. In this appendix we give the translation of the left current - right current OPEs (\[jLjR1\], \[jLjR2\]) in terms of commutation relations. We use the same techniques as in section 5 of [@Ashok:2009xx]
| 2,721
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|
ing (NYMEX + .055) . That's still being
negotiated.
1. Enron will warrant that we will have 15,000 of FT on Koch
2. Ormet pays the demand charge ($.05) on the transport on any volumes below
15,000/d
3. Volumes above 15,000/d up to 22/d will be supplied on IT. We will
utilize the same receipt/delivery path as the FT volumes in order to get the
discounted transport rate. If Koch cuts our IT, we will use best efforts to
get the gas there on another path or on Cypress. We will pass thru the
actual transport costs if they are above the discounted FT rate.
4. Ormet must supply us with accurate and timely consumption information
5. If Koch declares Force Majeure on our FT, we must use best efforts to
get gas and transport on Cypress before we declare FM
6. This is a full requirements contract (we supply all the gas that they
consume)
7. If Koch's tariff changes and they go away from monthly balancing, then we
will have a price re-opener.
8. If Ormet's loads change significantly (i.e. new cogen) we will have a
price re-opener.
I keep trying to email them, but it won't go thru. Maybe this time...
michael.mulligan@americas.bnpparibas.com
09/01/2000 02:24 PM
To: Tana.Jones@enron.com
cc:
Subject: Re: weather derivatives confirmation form
Hi Tana; I never did receive a copy of the Weather Derivative Confirmation.
Can
you send it again upon your return from the Labor Day weekend?
Thanks much. Kind regards.
---------------------- Forwarded by Michael
MULLIGAN/Corporate_Banking/US/PARIBAS on 09/01/2000 03:22 PM
---------------------------
Internet
From: Tana.Jones@enron.com on 08/24/2000 04:25 PM GMT
To: Michael MULLIGAN
cc:
bcc:
Subject: Re: weather derivatives confirmation form
Did you get the email of the documents I sent you, soon after we spoke?
michael.mulligan@americas.bnpp
aribas.com To:
Tana.Jones@enron.com
cc:
| 2,722
| 1,565
| 2,518
| 2,817
| null | null |
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|
sion, the random-walk representation (e.g., [@ffs92]) or the FK random-cluster representation (e.g., [@fk72]). In this paper, we use the random-current representation (Section \[ss:RCrepr\]), which applies to models in the Griffiths-Simon class (e.g., [@a82; @ag83]). This representation is similar in philosophy to the high-temperature expansion, but it turned out to be more efficient in investigating the critical phenomena [@a82; @abf87; @af86; @ag83]. The main advantage in this representation is the source-switching lemma (Lemma \[lmm:switching\] below in Section \[sss:2ndexp\]) by which we have an identity for ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda
-{{\langle \varphi_o\varphi_x \rangle}}_{{\cal A}}$ with “${{\cal A}}\subset\Lambda$” (the meaning will be explained in Section \[ss:RCrepr\]). We will repeatedly apply this identity to complete the lace expansion for ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda$ in Section \[sss:complexp\].
Random-current representation {#ss:RCrepr}
-----------------------------
In this subsection, we describe the random-current representation and introduce some notation that will be essential in the derivation of the lace expansion.
First we introduce some notions and notation. We call a pair of sites $b=\{u,v\}$ with $J_b\ne0$ a *bond*. So far we have used the notation $\Lambda\subset{{\mathbb Z}^d}$ for a site set. However, we will often abuse this notation to describe a *graph* that consists of sites of $\Lambda$ and are equipped with a certain bond set, which we denote by ${{\mathbb B}}_\Lambda$. Note that “$\{u,v\}\in{{\mathbb B}}_\Lambda$” always implies “$u,v\in\Lambda$”, but the latter does not necessarily imply the former. If we regard ${{\cal A}}$ and $\Lambda$ as graphs, then “${{\cal A}}\subset\Lambda$” means that ${{\cal A}}$ is a subset of $\Lambda$ as a site set, and that ${{\mathbb B}}_{{\cal A}}\subset{{\mathbb B}}_\Lambda$.
Now we consider the partition function $Z_{{\cal A}}$ on ${{\cal A}}\subset\Lambda$. By expanding the Bol
| 2,723
| 921
| 923
| 2,905
| 1,676
| 0.786891
|
github_plus_top10pct_by_avg
|
mmetry potential $D_{7,1}$. Similarly, the quadratic term in the fluxes $({\rm flux} \cdot {\rm flux})_4$ is mapped to the component $({\rm flux} \cdot {\rm flux})^a_3$ from the second line of the quadratic constraints in eqs. and , so that the full Chern-Simons term is mapped to $$\int D_{6\, a,a} \wedge ( {\rm flux} \cdot {\rm flux} )^a_3 \quad . \label{CSNSNSD71}$$ In this expression, the $a$ index of the potential after the comma in meant to be contracted with the upstairs index of the flux term, while the other ten indices are all different. Therefore, the three downstairs indices of the flux term are not along $a$, and in general by T-duality starting from the first Bianchi identity one can only reach components such that the upstairs indices are all different from the downstairs ones. This means that the constraints of eqs. and (as well as the ones in eq. ) are actually more than what ones gets by simply starting with the first constraints and applying T-dualities. As we will see, this point turns out to be crucial when we discuss the solutions of the quadratic constraints in the IIB and IIA orientifold models.[^7]
We can now study the solutions to the constraints in eqs. and for the IIB/O3 and IIA/O6 orientifolds. In the IIB/O3 case, only the second and fourth equations in are non-trivial, and can be schematically written as $$(Q \cdot H_3 -P_1^2 \cdot F_3)^{a}_{bcd}=0 \label{NSNSBianchiIIBO3}$$ and $$(Q \cdot Q-P^{1,4} \cdot F_3 )^{abc}_d=0 \quad . \label{NSNSBianchiIIBO3bis}$$ The relevant components of eq. with $a$ different from $b,c,d$ are $(Q \cdot H_3 -P_1^2 \cdot F_3)^{x^j}_{x^i y^i y^j}$ and $(Q\cdot H_3 -P_1^2 \cdot F_3)^{y^j}_{x^i y^i x^j}$, which would induce a charge for the KK-monopoles ([*i.e.*]{} $5_2^1$-branes) associated to the components $D_{4\, x^k y^k x^j, x^j}$ and $D_{4\, x^k y^k y^j , y^j}$ of the mixed-symmetry potential $D_{7,1}$. By substituting the symbols given in the first columns of Tables \[TableRRfluxes\], \[TableNSfluxes\] and \[allPfluxes\] (and considering for simplici
| 2,724
| 1,521
| 1,587
| 2,575
| 1,939
| 0.78427
|
github_plus_top10pct_by_avg
|
^lm_i$ is called a *$\rho$-generator* over $\mathbb{F}_q$ if $\rho$ is the smallest positive integer for which there are codewords $c_i(x)=(c_{i,1}(x),c_{i,2}(x), \ldots, c_{i,l}(x))$, $1\leq i \leq \rho$, in $C$ such that $C=Rc_1(x)+Rc_2(x)+\cdots +Rc_\rho(x)$.
Assume that the dimension of each $C_i$, $i=1,2,\ldots,s$, is $k_i$, and set ${\mathcal K}={\rm max} \\{\{ k_i \mid 1\leq i\leq s\}}$. Now by generalizing Theorem 3 of [@Esmaeili], we get
[**Theorem 3.5** ]{} *Let $C$ be a $\rho$-generator skew GQC code of block length $(m_1, m_2, \ldots, m_l)$ and length $\sum_{i=1}^lm_i$ over $\mathbb{F}_q$. Let $C=\bigoplus_{i=1}^sC_i$, where each $C_i$, $i=1,2,\ldots,s$, is with dimension $k_i$ and ${\mathcal K}={\rm max} {\{ k_i \mid 1\leq i\leq s\}}$. Then $\rho={\mathcal K}$. In fact, any skew GQC code $C$ with $C=\bigoplus_{i=1}^sC_i$, where each $C_i$, $i=1,2,\ldots,s$, is with dimension $k_i$ satisfying $\rho={\rm max }_{1\leq i\leq s} k_i$, is a $\rho$-generator skew GQC code.*
*Proof* Let $C$ be a $\rho$-generator skew GQC code generated by the elements $c^{(j)}(x)=(c_1^{(j)}(x), c_2^{(j)}, \ldots, c_l^{(j)}(x)) \in {\mathcal R}, ~j=1,2,\ldots,\rho$. Then for each $i=1,2,\ldots,s$, $C_i$ is spanned as a left $R$-module by $\widetilde{c}^{(j)}(x)=(\widetilde{c}_1^{(j)}(x), \widetilde{c}_2^{(j)}(x), \ldots, \widetilde{c}_l^{(j)}(x))$, where $\widetilde{c}_\nu^{(j)}(x)=c_\nu^{(j)}(x)~({\rm mod} g_i^*)$ if $g_i^*$ is a factor of $x^{m_i}-1$ and $\widetilde{c}_\nu^{(j)}(x)=0$ otherwise, $\nu=1,2,\ldots,l$. Hence $k_i\leq \rho$ for each $i$, and so ${\mathcal K}\leq \rho$. On the other hand, since ${\mathcal K}={\rm max}_{1\leq i\leq s} k_i$, there exist $q_i^{(j)}(x)\in R^l$, $1\leq j \leq {\mathcal K}$, such that $q_i^{(j)}(x)$ span $C_i$, $1\leq i \leq s$, as a left $R$-module. Then, by Theorem 3.3, for each $1\leq j \leq {\mathcal K}$, there exists $q^{(j)}(x)\in C$ such that $q_i^{(j)}(x)=q^{(j)}(x)~({\rm mod}~g_i^*)$ and $C$ is generated by $q_i^{(j)}(x)$, $1\leq j \leq{\mathcal K}$. Hence $\rho \leq {\math
| 2,725
| 948
| 1,160
| 2,791
| 2,817
| 0.776774
|
github_plus_top10pct_by_avg
|
_{r+s=n-1} \mathcal P(n) \otimes A^{\otimes r}
\otimes M \otimes A^{\otimes s}\right)_{S_{n}}$. Then $F_{\mathcal P,A} M$ is a module over $F_{\mathcal P}A$ over $\mathcal P$, which means that there are maps $\gamma^M:\bigoplus_{r+s=n-1}
\mathcal P(n)\otimes (F_{\mathcal P} A )^{\otimes r} \otimes
F_{\mathcal P,A} M \otimes (F_{\mathcal P} A )^{\otimes s} \to
F_{\mathcal P,A} M$ given by composition of elements of the operad and tensor product of elements of $A$. These maps satisfy the required module axioms, see [@GK (1.6.1)]. A module derivation $g\in \mathrm{Der}_d (F_{\mathcal P, A}M)$ over $d\in \mathrm{Der}(F_{\mathcal P}A)$, is define to be a map from $F_{\mathcal P, A}M$ to itself, making the following diagram commutative: $$\begin{CD}
\mathcal P(k) \otimes (F_{\mathcal P}A)^{\otimes (k-1)}\otimes
F_{\mathcal P, A}M & @>\gamma^M>> & F_{\mathcal P, A}M\\ @V\sum_{i<k} \mathrm{id}\otimes \mathrm{id}^{\otimes i}
\otimes d \otimes \mathrm{id}^{\otimes (k-i-1)}+ \mathrm{id}\otimes
\mathrm{id}^{\otimes (k-1)} \otimes g VV
& & @VVgV\\ \mathcal P(k) \otimes (F_{\mathcal P}A)^{\otimes
(k-1)}\otimes F_{\mathcal P, A}M & @>\gamma^M>> & F_{\mathcal P, A}M
\end{CD}$$
Finally, a module map $f\in \mathrm{Mod}(F_{\mathcal P, A}M, F_{\mathcal
P,A}N)$ is defined to be a map from $F_{\mathcal P,A}M$ to $F_{\mathcal P,A}N$ making the following diagram commutative: $$\begin{CD}
\mathcal P(k) \otimes (F_{\mathcal P}A)^{\otimes (k-1)}\otimes
F_{\mathcal P, A}M & @>\gamma^M>> & F_{\mathcal P, A}M\\
@V\mathrm{id}\otimes \mathrm{id}^{\otimes (k-1)} \otimes f VV & &
@VVfV\\ \mathcal P(k) \otimes (F_{\mathcal P}A)^{\otimes
(k-1)}\otimes F_{\mathcal P, A}N & @>\gamma^N>> & F_{\mathcal P, A}N
\end{CD}$$
If $\mathcal P$ is a cyclic operad, then we can use this extra datum to associate to every derivation $g$ over a free module $M$ a derivation $h$ over the free module on the dual space $M^\ast=Hom(M,k)$.
\[dual-module\] Let $\mathcal P$ be a cyclic operad, and let $A$ and $M$ be a vector space over $k$, which are finite dimensional i
| 2,726
| 1,535
| 2,453
| 2,497
| null | null |
github_plus_top10pct_by_avg
|
egory ${\mathsf{Repr}}(S)$. The coreflector is given by the functor $\Psi\Phi$.
Let $(X,\mu)$ be a non-strict $S$-set. Just as in the proof of Theorem \[th:equiv\], we have the map $\beta_{\mu}\colon \Psi\Phi(X,\mu)\to X$ given by . This map is surjective, and is injective if and only if $\mu$ is connected. We show that the functor $\Psi\Phi$ is a right adjoint to the functor ${\mathrm{i}}\colon {\mathsf{ConRepr}}(S) \to {\mathsf{Repr}}(S)$ where the maps $\beta_{\mu}$ are the components of the counit $\beta\colon i\circ \Psi\Phi \to {\mathrm{id}}_{{\mathsf{Repr}}(S)}$.
Let $(X,\mu)$ be any connected non-strict $S$-set, $(Y,\nu)$ be any non-strict $S$-set, and $$g\colon (X,\mu) \to (Y,\nu)$$ a morphism. To define the morphism $$f\colon (X,\mu) \to \Psi\Phi(Y,\nu),$$ let $x\in X$ and $e\in E(S)$ be such that $\mu(e)(x)$ is defined. Then it follows that $\nu(e)(f(x))$ is defined, as well. We set $$\label{eq:def_f}
f(x)=[e,g(x)]\in \Psi\Phi(Y,\nu).$$ For brevity, in this proof, we write $s\cdot x$ for $\mu(s)(x)$, $s\circ x$ for $\nu(s)(x)$ and $s*x$ for $\Psi\Phi(\nu)(s)(x)$. Note that if $h\cdot x$ is defined where $h\in E(S)$ then $h\circ f(x)$ is defined, and an induction shows that $(e,g(x))\sim (h,g(x))$ follows from $(e,x)\sim (h,x)$, where the latter equivalence holds because $\mu$ is connected. Therefore, the map $f$ is well-defined. Let us show that $f$ is a morphism of non-strict $S$-sets. Assume that $s\cdot x$ is defined. This is equivalent to that ${\mathbf{d}}(s)\cdot x$ is defined. It follows that $s*[{\mathbf{d}}(s),g(x)]$ is defined as well, and applying we have $$s*[{\mathbf{d}}(s),g(x)]=[{\mathbf{r}}(s),s\circ g(x)]=[{\mathbf{r}}(s),g(s\cdot x)].$$ On the other hand, $f(s\cdot x)=[{\mathbf{r}}(s),g(s\cdot x)]$ holds by . All that remains is to note that the equality $g=\beta_{\nu} f$ is a direct consequence of the definitions of $f$ and $\beta_{\nu}$.
Transitive representations of $S$ as directed functors on $L(S)$ {#sub:3.2}
----------------------------------------------------------------
\
| 2,727
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|
} p_{\mu} [\widetilde{\Delta}_{c}(\mu)]$ for some $p_{\mu} \in {\mathbb{Z}}[v,v^{-1}]$. To calculate the $p_{\mu}$ note that, by , $Y\cong
{\mathbb{C}}[{\mathfrak{h}}]\otimes {\mathbb{C}}[{\mathfrak{h}}^*]^{\text{co}{{W}}}$. Applying $({\mathbb{C}}\otimes_{{\mathbb{C}}[{\mathfrak{h}}]}-)$ to the equation $ [Y] =\sum p_{\mu} [\widetilde{\Delta}_{c}(\mu)]$ therefore yields $[{\mathbb{C}}[{\mathfrak{h}}^*]^{\text{co}{{W}}}]
= \sum_{\mu} p_{\mu} [\mu].$ Thus implies that $p_{\mu} = f_{\mu}(v^{-1})$ (this is a polynomial in $v^{-1}$ rather than $v$ since ${\mathbb{C}}[{\mathfrak{h}}^*]$ is negatively ${\mathbf{E}}$-graded) and so, as an element of $G_0(\widetilde{{\mathcal{O}}}_{c})$, $$\label{grot22}
[Y] = \sum_{\mu} f_{\mu}(v^{-1}) [\widetilde{\Delta}_{c}(\mu)] .$$
Now consider $\underline{M(k)}$, which we can write as $H_{c+k}e\otimes_{U_{c+k}} B_{k0}\otimes_{U_c} eY$. By and Corollary \[morrat-cor\], $H_{c+k}e\otimes_{U_{c+k}} e\widetilde{\Delta}_{c+k}(\lambda)
\cong \widetilde{\Delta}_{c+k}(\lambda)$. Thus and Lemma \[standAAA\] combine to show that $$[\underline{M(k)} ] =
\sum_{\mu} f_{\mu}(v^{-1})v^{k(n(\mu)-n(\mu^t)}
[\widetilde{\Delta}_{c+k}(\mu)] .$$ As graded vector spaces, $\widetilde{\Delta}_{c+k}(\mu) \cong {\mathbb{C}}[{\mathfrak{h}}]\otimes \mu$ and so $p(\widetilde{\Delta}_{c+k}(\mu), v) = f_\mu(1) (1-v)^{-(n-1)}$ by . Therefore, $$\label{wrongsideformulaC}
p(\underline{M(k)}, v) =
\frac{\sum_{\mu} f_\mu(1)f_{\mu}(v^{-1})v^{k(n(\mu)-n(\mu^t)}}
{(1-v)^{(n-1)}}.$$
By parts (2) and (3) of Lemma \[Bbar-freeC\], a homogeneous basis for $\overline{M(k)}$ is given by lifting a homogeneous ${\mathbb{C}}$-basis for $\overline{M(k)}\otimes_{{\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}}
{\mathbb{C}}= {\mathbb{C}}\otimes_{{\mathbb{C}}[{\mathfrak{h}}]} {\underline{M(k)}}.$ Thus, combining with the formulæ $p({\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}, v) = \prod_{i=2}^n (1-v^{-i})^{-1}$ and $ p({\mathbb{C}}[{\mathfrak{h}}], v) = (1-v)^{n-1}$ gives $$\label{wrongsideformula2C}
p(\overline{M(k)}, v) =
\frac{\sum_{\mu}
| 2,728
| 1,872
| 2,424
| 2,589
| null | null |
github_plus_top10pct_by_avg
|
\
\mathcal{D}_{Ru} & -\frac{h u}{2 \left(u^2+1\right)} & -\frac{i m u}{\left(u^2+1\right)^2} \\
\mathcal{D}_{uu} & \frac{u^2 \left(u^2+3\right)}{\left(u^2+1\right)^3} & \frac{i (2 h+3) m}{2 \left(u^4-1\right)} \\
\mathcal{D}_{TR} & \frac{i m \left(u^2-1\right) \left(u^4+6 u^2-3\right)}{4 \left(u^2+1\right)^3} & \frac{8 \left(u^6-7 u^4+7 u^2-1\right)-m^2 \left(u^8+8 u^6+10 u^4-3\right)}{8 \left(u^2-1\right) \left(u^2+1\right)^3} \\
\mathcal{D}_{Tu} & -\frac{i m u \left(u^2-3\right)}{2 \left(u^2+1\right)^2} & -\frac{(h+2) u}{\left(u^2+1\right)^2} \\
\mathcal{D}_{\Phi R} & \frac{i m \left(u^2-1\right) \left(h \left(u^2+1\right)^2+2 \left(u^2-1\right)\right)}{2 \left(u^2+1\right)^3} & -\frac{m^2}{2 \left(u^2+1\right)} \\
\mathcal{D}_{\Phi u} & -\frac{i m u \left(u^2-1\right)}{\left(u^2+1\right)^2} & 0 \\
\end{array}
$
$
\begin{array}{c|c}
\mathcal{D}_{AB} & C_{\Phi \Phi }(u) \\
\noalign{\smallskip}
\hline \hline \noalign{\smallskip}
\mathcal{D}_{TT} & \frac{h^2 \left(u^2-1\right) \left(u^6+7 u^4+3 u^2-3\right)^2-2 \left(3 u^{12}+68 u^{10}-5 u^8-128 u^6+153 u^4-36 u^2+9\right)}{8 \left(u^2-1\right)^2 \left(u^2+1\right)^5} \\
\mathcal{D}_{T\Phi} & -\frac{-2 \left(u^8+8 u^6+10 u^4-3\right) h^2+\left(u^8+8 u^6+10 u^4-3\right) h+4 \left(9 u^6+13 u^4-9 u^2+3\right)}{4 \left(u^2+1\right)^5} \\
\mathcal{D}_{\Phi\Phi} & \frac{\left(u^2-1\right) \left(-3 u^4-6 u^2+2 h^2 \left(u^2+1\right)^2-2 h \left(u^2+1\right)^2+5\right)}{\left(u^2+1\right)^5} \\
\mathcal{D}_{RR} & \frac{2 \left(7 u^8-30 u^6+72 u^4-42 u^2+9\right)-h \left(u^2+1\right)^2 \left(u^6+5 u^4-9 u^2+3\right)}{8 \left(u^2-1\right)^2 \left(u^2+1\right)^3} \\
\mathcal{D}_{Ru} & -\frac{u \left(8 \left(u^4-4 u^2+3\right)+h \left(u^6+11 u^4-13 u^2+9\right)\right)}{8 \left(u^4-1\right)^2} \\
\mathcal{D}_{uu} & \frac{\left(u^8+8 u^6+10 u^4-3\right) h^2+\left(u^8+8 u^6+10 u^4-3\right) h+2 \left(7 u^6+3 u^4+9 u^2-3\right)}{8 \left(u^2-1\right)^2 \left(u^2+1\right)^3} \\
\mathcal{D}_{TR} & -\frac{i m \left(u^4+6 u^2-3\right)^2}{16 \left(u^2-1\right)
| 2,729
| 2,529
| 2,321
| 2,632
| null | null |
github_plus_top10pct_by_avg
|
metric spaces enjoying certain high regularity properties - having group structure with left-invariant metric. Using methods of harmonic analysis we show that, given a compact metrizable group $G$ equipped with an arbitrary compatible left-invariant metric $d$, the Lipschitz-free space over $G$, ${\mathcal{F}}(G,d)$, satisfies the metric approximation property. We show also that, given a finitely generated group $G$, with its word metric $d$, from a class of groups admitting a certain special type of combing, which includes all hyperbolic groups and Artin groups of large type, ${\mathcal{F}}(G,d)$ has a Schauder basis. Examples and applications are discussed. In particular, for any net $N$ in a real hyperbolic $n$-space $\mathbb{H}^n$, ${\mathcal{F}}(N)$ has a Schauder basis.'
address:
- |
Institute of Mathematics\
Czech Academy of Sciences\
Žitná 25\
115 67 Praha 1\
Czech Republic
- 'Instituto de Ciência e Tecnologia da Universidade federal de São Paulo, Av. Cesare Giulio Lattes, 1201, ZIP 12247-014 São José dos Campos/SP, Brasil'
author:
- Michal Doucha
- 'Pedro L. Kaufmann'
bibliography:
- 'references.bib'
title: 'Approximation properties in Lipschitz-free spaces over groups'
---
[^1]
Introduction
============
Lipschitz-free spaces form by now a fundamental class of Banach spaces, whose study has been revitalized since the appearance of the seminal paper of Godefroy and Kalton ([@godefroy2003lipschitz]). There are two main important properties that both characterize these spaces. Namely, they are free objects in the category of Banach spaces over the metric spaces. Second, they are canonical isometric preduals to the Banach spaces of pointed Lipschitz real-valued functions. Another appealing feature is that their study connects Banach space theory to several other areas of mathematics, including optimal transport and geometry, and, as we demonstrate here, also harmonic analysis. We recall some basic facts about Lipschitz-free spaces in Section \[section:preliminaries\].
Approxima
| 2,730
| 2,884
| 383
| 2,175
| null | null |
github_plus_top10pct_by_avg
|
(z_t)
=& q'(g(z_t)) \nabla g(z_t) \\
=& q'(g(z_t)) \frac{z_t}{\lrn{z_t}_2}\\
\nabla^2 f(z_t)
=& q''(g(z_t))\nabla g(z_t) \nabla g(z_t)^T + q'(g(z_t))\nabla^2 g(z_t)\\
=& q''(g(z_t)) \frac{z_t z_t^T}{\|z_t\|_2^2} + q'(g(z_t)) \frac{1}{\|z_t\|_2} \lrp{I - \frac{z_tz_t^T}{\|z_t\|_2^2}}
\end{aligned}$$
Once again, by Assumption \[ass:U\_properties\].3, $$\begin{aligned}
\circled{1} \leq q'(g(z_t))\lrn{\nabla_t}_2 \leq q'(g(z_t)) \cdot \LR \cdot \|z_t\|_2 \leq& L \cdot q'(g(z_t)) g(z_t) + 2 L \epsilon
\end{aligned}$$ Where the last inequality uses Lemma \[l:gproperties\].4. We can also verify that $$\begin{aligned}
\circled{2} \leq L \lrn{y_t - y_0}_2
\end{aligned}$$
Using the expression for $\nabla^2 f(z_t)$, $$\begin{aligned}
\circled{4}
= 2\cm^2 \tr\lrp{\nabla^2 f(z_t) \gamma_t \gamma_t^T} = 2\cm^2 \cdot q''(g(z_t))
\end{aligned}$$
Finally, $$\begin{aligned}
\circled{5}
=& \frac{1}{2}\tr\lrp{\nabla^2 f(z_t) \lrp{N_t+ N(y_t) - N(y_0)}^2}\\
=& \frac{1}{2}\tr\lrp{\lrp{q''(g(z_t)) \frac{z_t z_t^T}{\|z_t\|_2^2} + q'(g(z_t)) \frac{1}{\|z_t\|_2} \lrp{I - \frac{z_tz_t^T}{\|z_t\|_2^2}}} \lrp{N_t+ N(y_t) - N(y_0)}^2}\\
\leq& \frac{1}{2}\tr\lrp{\lrp{ q'(g(z_t)) \frac{1}{\|z_t\|_2} \lrp{I - \frac{z_tz_t^T}{\|z_t\|_2^2}}} \lrp{N_t+ N(y_t) - N(y_0)}^2}\\
\leq& \frac{q'(g(z_t))}{\|z_t\|_2}\cdot \lrp{\tr\lrp{N_t^2} + \tr\lrp{\lrp{N(y_t) - N(y_0)}^2}}\\
\leq& q'(g(z_t)) \cdot L_N^2 \|z_t\|_2 + \frac{L_N^2\|y_t-y_0\|_2^2}{2\epsilon}\\
\leq& q'(g(z_t)) \cdot L_N^2 g(z_t) + \frac{L_N^2\|y_t-y_0\|_2^2}{2\epsilon} + 2 L_N^2 \epsilon
\end{aligned}$$
The above uses multiples times the fact that $0 \leq q' \leq 1$ and $q'' \leq 0$ (proven in items 3 and 4 of Lemma \[l:qproperties\]). The second inequality is by Young’s inequality, the third inequality is by item 2 of Lemma \[l:N\_is\_regular\], the fourth inequality uses item 4 of Lemma \[l:gproperties\].
Summing thes
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ion of $n$, and $i,j,k$ are positive integers with $j\neq k$ and $\mu_j\gs\mu_k$. Suppose $T\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$, and let $\cals$ be the set of all $S\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$ such that:
- $S^j_i=T^j_i+T^k_i$;
- $S^j_l\ls T^j_l$ for every $l\neq i$;
- $S^l=T^l$ for all $l\neq j,k$.
Then $${\hat\Theta_{T}} = (-1)^{T^k_i}\sum_{S\in\cals}\prod_{l\gs1}\binom{S^k_l}{T^k_l}{\hat\Theta_{S}}.$$
Informally, a tableau in $\cals$ is a tableau obtained from $T$ by moving all the $i$s from row $k$ to row $j$, and moving some multiset of entries different from $i$ from row $j$ to row $k$.
One very simple case of Lemma \[lemma7\] which we shall apply frequently is the following: if $T\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$ and for some $i,j,k$ we have $T^j_i+T^k_i>\max\{\mu_j,\mu_k\}$, then ${\hat\Theta_{T}}=0$.
Lemma \[lemma7\] turns out to be very useful for expressing a tableau homomorphism in terms of semistandard homomorphisms. However, we shall occasionally need to use the following alternative.
\[newsemilem\] Suppose $\la$ and $\mu$ are partitions of $n$, and $T$ is a row-standard $\la$-tableau of type $\mu$. Suppose $i\gs1$, and $A,B,C$ are multisets of positive integers such that $|B|>\la_i$ and $A\sqcup B\sqcup C=T^i+T^{i+1}$. Let $\calb$ be the set of all pairs $(D,E)$ of multisets such that $|D|=\la_i-|A|$ and $B=D\sqcup E$. For each such pair $(D,E)$, define $T_{D,E}$ to be the row-standard tableau with $$T_{D,E}^j=
\begin{cases}
A\sqcup D&(j=i)\\
C\sqcup E&(j=i+1)\\
T^j&(\text{otherwise}).
\end{cases}$$ Then $$\sum_{(D,E)\in\calb}\prod_{i\gs1}\binom{A_i+D_i}{D_i}\binom{C_i+E_i}{E_i}{\hat\Theta_{T_{D,E}}}=0.$$
This lemma appears in the second author’s forthcoming paper [@garnir] where it is proved in the wider context of Iwahori–Hecke algebras; however, a considerably easier proof exists in the symmetric group case. In [@garnir], Lemma \[newsemilem\] is used to provide an explicit fast algorithm for writing a tableau homomorp
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|
reduction method and of the generalized method of characteristics. A variant of the conditional symmetry method for constructing this type of solution is proposed. A specific feature of that approach is an algebraic-geometric point of view, which allows the introduction of specific first-order side conditions consistent with the original system of PDEs, leading to a generalization of the Riemann invariant method for solving elliptic homogeneous systems of PDEs. A further generalization of the Riemann invariants method to the case of inhomogeneous systems based on the introduction of specific rotation matrices enabled us to weaken the integrability condition. It allows us to establish the connection between the structure of the set of integral elements and the possibility of the construction of specific classes of simple mode solutions. These theoretical considerations are illustrated by the examples of an ideal plastic flow in its elliptic region and a system describing a nonlinear interaction of waves and particles. Several new classes of solutions have been obtained in explicit form including the general integral for the latter system of equations.'
author:
- |
A.M. Grundland[^1],\
Centre de Recherche Mathématiques, Université du Montréal,\
C.P. 6128, Succc. Centre-ville, Montréal, (QC) H3C 3J7, Canada\
and Département de mathématiques et informatiques, Université du Québec,\
Trois-Rivières, (QC) G9A 5H7, Canada\
- |
V. Lamothe[^2],\
Département de Mathématiques et Statisque, Université de Montréal,\
C.P. 6128, Succc. Centre-ville, Montréal, (QC) H3C 3J7, Canada\
title: 'Multimode solutions of first-order quasilinear systems obtained from Riemann invariants. Part I. '
---
[Running Title: Multimode solutions of quasilinear systems. Part I.\
PACS numbers: 02.30.Jr; Secondary 02.70.-c\
Keywords: symmetry reduction method, generalized method of characteristics, Riemann invariants, multimode solutions]{}
Introduction {#intro}
============
Riemann waves represent a ve
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hi_i({\mathbf x}), \\
\phi_i({\mathbf x})& =0,
\end{split}
\end{cases}
\quad
\begin{split}
{\mathbf x}&\in\Omega, \\
{\mathbf x}&\in\partial\Omega.
\end{split}\end{aligned}$$ In two dimensions with $\Omega=[-L_1,L_1]\times[-L_2,L_2]$ we introduce the positive integers $i_1\le m_1$ and $i_2\le m_2$. The number of basis functions is then $m=m_1m_2$ and the solution to is given by
$$\begin{aligned}
\phi_i({\mathbf x}) &= \frac{1}{\sqrt{L_1L_2}}\sin\big(\varphi_{i_1}(x_1+L_1)\big)\sin\big(\varphi_{i_2}(x_2+L_2)\big) ,
\\
\lambda_i &= \varphi_{i_1}^2+\varphi_{i_2}^2, \quad \varphi_{i_1}=\frac{\pi i_1}{2L_1}, \quad \varphi_{i_2}=\frac{\pi i_2}{2L_2}, \end{aligned}$$
where $i=i_1+m_1(i_2-1)$. Let us now build the vector ${\boldsymbol\phi}_*\in{{\mathbb R}}^{m\times1}$, the matrix $\Phi\in{{\mathbb R}}^{m\times M}$ and the diagonal matrix $\Lambda\in{{\mathbb R}}^{m\times m}$ as
$$\begin{aligned}
({\boldsymbol\phi}_*)_i&=\phi_i({\mathbf x}_*), \\
\label{eq:Phi_entry}
\Phi_{ij} &= \int_{-R}^R \phi_i({\mathbf x}^0_j+s\hat{{\mathbf u}}_j)ds, \\
\Lambda_{ii} &= S(\sqrt{\lambda_i}). \end{aligned}$$
The entries $\Phi_{ij}$ can be computed in closed form with details given in \[app:compdet\]. Now we substitute $Q\approx\Phi^{\mathsf{T}}\Lambda\Phi$ and ${\mathbf q}_*\approx\Phi^{\mathsf{T}}\Lambda{\boldsymbol\phi}_*$ to obtain
\[eq:pred\_appr2\] $$\begin{aligned}
\mathbb{E}[f({\mathbf x}_*) \mid {\mathbf y}]
&\approx{\boldsymbol\phi}_{*}^{\mathsf{T}}\Lambda \Phi (\Phi^{\mathsf{T}}\Lambda \Phi + \sigma^2 I)^{-1}{\mathbf y},
\\
\mathbb{V}[f({\mathbf x}_*) \mid {\mathbf y}]
&\approx {\boldsymbol\phi}_{*}^{\mathsf{T}}\Lambda {\boldsymbol\phi}_{*} - {\boldsymbol\phi}_{*}^{\mathsf{T}}\Lambda \Phi (\Phi^{\mathsf{T}}\Lambda \Phi + \sigma^2I)^{-1} \Phi^{\mathsf{T}}\Lambda {\boldsymbol\phi}_{*}.\end{aligned}$$
When using the spectral densities corresponding to the classical regularization methods in and , the mean equation reduces to the classical solution (on the given basis). However, also fo
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ghtarrow \:
{\cal O}_{\Lambda}(2)^{n+1} \: \longrightarrow \:
T G {\mathbb P}^n \: \longrightarrow \: 0.$$ Using the isomorphisms above, we see this short exact sequence is the same as $$0 \: \longrightarrow \: \pi^* {\cal O} \: \longrightarrow \:
\pi^* {\cal O}(1)^{n+1} \: \longrightarrow \: T G {\mathbb P}^n \:
\longrightarrow
\: 0,$$ which is just $\pi^*$ of the Euler sequence for the tangent bundle $$0 \: \longrightarrow \: {\cal O} \: \longrightarrow \:
{\cal O}(1)^{n+1} \: \longrightarrow \: T {\mathbb P}^n \:
\longrightarrow \: 0.$$
For ${\mathbb Z}_k$ gerbes over ${\mathbb P}^n$ built as the weighted projective stack ${\mathbb P}^n_{[k,\cdots,k]}$, there is a closely analogous story. Here, coherent sheaves on $G {\mathbb P}^n$ decompose as $$\mbox{Coh}(G{\mathbb P}^n) \: = \: \cup_{\chi} \mbox{Coh}({\mathbb P}^n,
\chi(\alpha)),$$ where the union is over irreducible representations of ${\mathbb Z}_k$, and there are $k$ different pullbacks, first the canonical $$\pi^*: \: \mbox{Coh}({\mathbb P}^n) \: \stackrel{\sim}{\longrightarrow} \:
\mbox{Coh}({\mathbb P}^n, 1(\alpha)),$$ followed by $\pi_i^*(-) \equiv \pi^*(-) \otimes {\cal O}_{\Lambda}(i)$. Identifying $\pi_0^*$ with $\pi^*$, we have the general relation $$\pi_i^* {\cal O}(m) \: = \: {\cal O}_{\Lambda}(km+i).$$ An argument nearly identical to the one above shows that the tangent bundle $T G {\mathbb P}^n$ seen by a gauged linear sigma model is given by $\pi^* T {\mathbb P}^n$, exactly as must be true on general grounds.
Sheaf cohomology
----------------
On a global quotient stack $\mathfrak{X} = [X/G]$, for $G$ finite, given a vector bundle ${\cal E} \rightarrow \mathfrak{X}$, (equivalently, a $G$-equivariant bundle on $X$,) $$H^{\bullet}(\mathfrak{X}, {\cal E}) \: = \:
H^{\bullet}(X, {\cal E})^G.$$ In our discussion of massless spectra of heterotic strings on stacks, this is ultimately the reason why in orbifolds one gets $G$-invariants.
Now, nontrivial gerbes over projective spaces have a global quotient description as some $[X/G]$ for $G$ nonfi
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ut any reference to General Relativity. Second, by not tying the reduction to any particular fluid, we achieve a large degree of generality. Clearly, the method can be extended to the case in which the higher-dimensional fluid carries a particle number or some other property, but we will not pursue this in the present article.
Hydrodynamic Kaluza-Klein ansatz and\
reduction of the perfect fluid
=====================================
Let us consider a neutral relativistic fluid in flat space-time in $p+1$ dimensions. The hydrodynamical behaviour of this fluid is governed by the stress energy tensor conservation equations $\partial_A T^{AB}=0$. In general we split the stress energy tensor into a perfect fluid and a dissipative part, T\_[AB]{}=T\_[AB]{}\^[pf]{}+T\_[AB]{}\^[diss]{} \[eqn:generT\]. The perfect fluid part is given in terms of the energy density $\epsilon$, the pressure $P$ and the normalized velocity field $u^A$ by \[pf\] T\_[AB]{}\^[pf]{}=(+P) u\_A u\_B + P g\_[AB]{} while, to first derivative order, the dissipative part is \[eqn:diss\] T\_[AB]{}\^[diss]{}=-2\_[AB]{}-P\_[AB]{} where $\eta$ and $\zeta$ are the shear and bulk viscosities and $\theta$, $ \sigma_{AB}$ and $P_{AB}$ are defined as &=&\_A u\^A\[espa\],\
\_[AB]{}&=& P\_A\^C P\_B\^D\_[(C]{} u\_[D)]{}-P\_[AB]{}\[shear\],\
P\_[AB]{}&=&g\_[AB]{}+u\_A u\_B\[orto\]. A complete description of the fluid requires the specification of the equation of state, namely the relation between $P$ and $\epsilon$, and of the viscosities. For the most part we will keep them general, and will only specify them in sec. \[6\]. Furthermore, we naturally assume that this uncharged fluid is described in the Landau frame where $u^A T_{AB}^{diss}=0$.
We assume that the spacetime in which the fluid moves contains $N$ compact directions that form an $N$-torus $$\label{ansatz}
d\hat s^2= \sum_{j=1}^{N} dy_j^2 +\eta_{ab} d\sigma^a d\sigma^b \,,$$ where the metric $\eta_{ab}$ is the Minkowski metric in $p-N+1$ spacetime dimensions and the coordinates $y_j$ are identified
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techniques of the proof of Theorem \[tth\] and Example \[desolex1\]. If $S=S(x,E)$ depends also on $x$, we conjecture that it suffices only to assume that $S$ is regular enough.
Let $$P'(x,\omega,E,D)v=
S_0{{\frac{\partial v}{\partial E}}}-\omega\cdot\nabla_x v$$ be the formal transpose of $P(x,\omega,E,D)$. Making the assumption ${\bf TC}$ the (extended) Green formula \[green-ex\] & \_[GSI]{}(P(x,,E,D) )v dxddE -\_[GSI]{}(P’(x,,E,D) v) dxddE\
=& \_[GSI]{}() v dddE\
& + \_[GS]{}(S\_0(,0)(,,0)v(,,0)-S\_0(,E\_[m]{})(,,E\_[m]{})v(,,E\_[m]{}))dx dis valid for all $\phi,\ v\in {{{\mathcal{}}}H}_P(G\times S\times I^\circ)$ for which $({\rm supp}(v))\cap \partial (G\times S\times I)$ is a compact subset of $\Gamma_-\cup \Gamma_+\cup (G\times S\times\{E_m\})\cup(G\times S\times\{0\})$. Moreover, (\[green-ex\]) holds for $\phi,\ v\in {{{\mathcal{}}}H}_P(G\times S\times I^\circ)$ when $\gamma_{\pm}(\phi)\in T^2(\Gamma_{\pm})$ and $\gamma_{\rm m}(\phi),\ \gamma_0(\psi)\in L^2(G\times S)$. We omit the proof of both these claims.
We are now in position to formulate and prove the following theorem.
\[csdath3\] Suppose that the assumptions (\[ass1\]), (\[ass2\]), (\[ass3\]) (with $C=\frac{\max\{q,0\}}{\kappa}$ and $c>0$) and (\[csda9\]), (\[csda9aa\]), (\[csda9a\]) are valid. Let ${\bf f}\in L^2(G\times S\times I)$ and ${\bf g}\in T^2(\Gamma_-)$. Then the following assertions hold.
\(i) The variational equation (see , ) \[csda40a\] (,v)=F(v)vH\_2, has a solution $\tilde\phi=(\phi,q,p_0,p_{\rm m})\in H_1$.
Furthermore, $\phi \in {{{\mathcal{}}}H}_P(G\times S\times I^\circ)$ and it is a weak (distributional) solution of the equation (\[csda3A\]).
\(ii) Suppose that additionally the assumption ${\bf TC}$ holds. Then a solution $\phi$ of the equation obtained in part (i) is a solution of the problem , , .
In addition, we have $q_{|\Gamma_+}=\gamma_+(\phi)$ and $p_0=\phi(\cdot,\cdot,0)$, when $\tilde{\phi}=(\phi,q,p_0,p_m)$ is a solution in $H_1$ obtained in part (i).
\(iii) Under the assumptions imposed in part (ii),
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odel as well and have a principles means to tune the regularization parameters. Finally, the third contribution is to present methods for hyperparameter estimation that arise from the machine learning literature and apply the methodology to the tomographic reconstruction problem. In particular, the proposed methods are applied to simulated 2D chest phantom data available in <span style="font-variant:small-caps;">Matlab</span> and real carved cheese data measured with $\mu$CT system. The results show that the reconstruction images created using the proposed GP method outperforms the FBP reconstructions in terms of image quality measured as relative error and as peak signal to noise ratio.
Constructing the model
======================
The tomographic measurement data
--------------------------------
Consider a physical domain $\Omega \subset {{\mathbb R}}^2$ and an attenuation function $f:\Omega\rightarrow{{\mathbb R}}$. The x-rays travel through $\Omega$ along straight lines and we assume that the initial intensity (photons) of the x-ray is $I_0$ and the exiting x-ray intensity is $I_d$. If we denote a ray through the object as function $s \mapsto (x_1(s),x_2(s))$ Then the formula for the intensity loss of the x-ray within a small distance $ds$ is given as:
$$\label{calibration1}
\frac{dI(s)}{I(s)}= -f(x_1(s),x_2(s)) ds,$$
and by integrating both sides of , the following relationship is obtained $$\label{calibration2}
\int_{-R}^{R} f(x_1(s),x_2(s)) ds = \log\frac{I_0}{I_d},$$ where $R$ is the radius of the object or area being examined.
In x-ray tomographic imaging, the aim is to reconstruct $f$ using measurement data collected from the intensities $I_d$ of x-rays for all lines through the object taken from different angles of view. The problem can be expressed using the Radon transform, which can be expressed as $$\label{Measurement Model}
\mathcal{R} f(r,\theta) = \int f(x_1,x_2) d{\mathbf x}_L,
$$ where $d{\mathbf x}_L$ denotes the $1$-dimensional Lebesgue measure along the line defined by $L=\{(x_1,x_2)
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ce. For brevity we refer to a point in a choice space (that is, a choice mapping) simply as a *choice*.
\[T:PROTOSPACE\_INCLUDES\_CHOICESPACE\] The proto-space ${\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$ of ensemble $\Psi$ includes its choice space $\prod\Psi$ (that is, $\prod\Psi \subseteq {\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$).
Suppose $\chi \in \prod\Psi$. By definition \[D:CHOICE\], $\chi$ is a mapping ${{\operatorname{dom}{\Psi}}} \to \Psi_\heartsuit$. Then, by definition \[D:PROTOSPACE\], $\chi \in {\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$. Thus, any member of $\prod\Psi$ is also a member of ${\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$. From this we conclude $\prod\Psi \subseteq {\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$.
\[T:ENSEMBLE\_UNIQ\_SPACE\] An ensemble generates one unique choice space: let $\Theta$ and $\Phi$ be two ensembles generating choice spaces $\prod\Theta$ and $\prod\Phi$ respectively. If $\Theta = \Phi$, then $\prod\Theta = \prod\Phi$.
To show the contrapositive, suppose $\prod\Theta \not= \prod\Phi$. This premise can be true if either A: there is a choice $\alpha \in \prod\Theta$ such that $\alpha \notin \prod\Phi$, or if B: there is a choice $\beta \in \prod\Phi$ such that $\beta \notin \prod\Theta$.
In case A, $\alpha$ is a choice of $\Theta$ – that is, for each $k \in {{\operatorname{dom}{\Theta}}}$, $\alpha(k) \in \Theta(k)$. For hypothesis, assume $\Theta = \Phi$, so that ${{\operatorname{dom}{\Theta}}} = {{\operatorname{dom}{\Theta}}}$. Then for each $k \in {{\operatorname{dom}{\Theta}}}$, $\alpha(k) \in \Phi(k)$, since by equality hypothesis $\Theta(k) = \Phi(k)$ and $\alpha(k) \in \Theta(k)$. This means $\alpha$ is a choice of $\Phi$, that is, $\alpha \in \prod\Phi$. However, this conclusion contradicts the second part of the premises for case A, namely that $\alpha \notin \prod\Phi$. Thus the hypothesis $\Theta = \Phi$ is false, so $\Theta \not= \Phi$.
The argument for case B is the same as A, except reversing the roles of $\Theta$
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ently, the Buda-Lund hydro model lead to the discovery of a number of new, exact analytic solutions of hydrodynamics, both in the relativistic [@relsol-cyl; @relsol-ell] and in the non-relativistic domain [@nr-sol; @nr-ell; @nr-inf].
The expanding matter is assumed to follow a three-dimensional, relativistic flow, characterized by transverse and longitudinal Hubble constants, u\^(x) = ( , H\_t r\_x, H\_t r\_y, H\_z r\_z ), where $\gamma$ is given by the normalization condition $u^\nu(x) u_\nu(x) = 1$. In the original form, this four-velocity distribution $u^\nu(x)$ was written as a linear transverse flow, superposed on a scaling longitudinal Bjorken flow . The strength of the transverse flow was characterized by its value $\langle u_t\rangle$ at the “geometrical" radius $R_G$, see refs. [@Csorgo:1995bi; @Chapman:1994ax; @Ster:1998hu]: u\^(x) & = & ( , , , ),\
& = & r\_t / R\_G, with $ r_t = (r_x^2 + r_y^2)^{1/2}$. Such a flow profile, with a time-dependent radius parameter $R_G$, was recently shown to be an exact solution of the equations of relativistic hydrodynamics of a perfect fluid at a vanishing speed of sound, see refs. [@Biro:1999eh; @Biro:2000nj].
The Buda-Lund hydro model characterizes the inverse temperature $1/T(x)$, and fugacity, $\exp\left[\mu(x)/T(x)\right]$ distributions of an axially symmetric, finite hydrodynamically expanding system with the mean and the variance of these distributions, in particular & = & - -[ (- y\_0)\^2 2 \^2 ]{}, \[e:mu\]\
[1 T(x)]{} & = & [1 T\_0 ]{} ( 1 + [r\_t\^2 2 R\_s\^2]{} ) ( 1 + [(- \_0)\^2 2 \_s\^2 ]{} ). Here $R_G$ and $\Delta\eta$ characterize the spatial scales of variation of the fugacity distribution, $\exp\left[\mu(x)/T(x)\right]$, that control particle densities. Hence these scales are referred to as geometrical lengths. These are distinguished from the scales on which the inverse temperature distribution changes, the temperature drops to half if $r_x = r_y = R_s$ or if $\tau = \tau_0 + \sqrt{2} \Delta\tau_s$. These parameters can be considered as se
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E_p^\Lambda,{\vec{p}})
\\\times&\nonumber\,
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\nu\gamma_5({\cancel{r}_N}+M_N)\gamma_\mu\gamma_5
u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using heavy baryon expansion and the master integrals of Sec. \[sec:mi\], and redefining ${\vec{q}}\equiv{\vec{p}\,'}-{\vec{p}}$, $$\begin{aligned}
V_h=&
-\frac{G_Fm_\pi^2g_A^3}
{32M_N f_\pi^3}
(3+2{\vec{\tau}_1}\cdot{\vec{\tau}_2})\Big[
-2 iB J_{22}{\vec{\sigma}_2}\left({\vec{p}}\times {\vec{q}}\right)
\nonumber\\+&
2B J_{22} \left(-{\vec{p}}\cdot
{\vec{q}}+{\vec{q}}^2\right)
{\vec{\sigma}_1}\cdot {\vec{\sigma}_2}\\+&\nonumber
2 i B\left(J_{22}+{\vec{q}}^2 J_{23}+(5+\eta) J_{34}+{\vec{q}}^2
J_{35}\right)
{\vec{\sigma}_1}\cdot\left({\vec{p}}\times {\vec{q}}\right)
\\+&\nonumber
4i A J_{22} M_N\left({\vec{\sigma}_1}\times {\vec{\sigma}_2}\right){\vec{q}}\\+&\nonumber
4 AM_N \left({\vec{q}}^2 J_{23}+5 J_{34}+{\vec{q}}^2
J_{35}+J_{22}\right){\vec{\sigma}_1}\cdot{\vec{q}}\\+&\nonumber
2BJ_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot {\vec{p}})
-2BJ_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})
\\-&\nonumber
2B\left({\vec{q}}^2 q_0 J_{21}+\left(-{\vec{p}}\cdot {\vec{q}}+{\vec{q}}^2\right)
J_{22}+(-{\vec{p}}\cdot {\vec{q}}{\vec{q}}^2 +{\vec{q}}^4)
J_{23}
\right.\\-&\left.\nonumber
{\vec{q}}^2 J_{31}+(3-\eta) q_0 J_{32}+{\vec{q}}^2 q_0 J_{33}
\right.\\+&\left.\nonumber
(5-\eta)(-{\vec{p}}\cdot {\vec{q}}+2 {\vec{q}}^2 )J_{34}
+
(-{\vec{p}}\cdot {\vec{q}}{\vec{q}}^2 +2 {\vec{q}}^4 )J_{35}
\right.\\-&\left.\nonumber
(3-\eta) J_{42}
-{\vec{q}}^2 J_{43}+(15-8\eta)J_{46}
+
2(5-\eta) {\vec{q}}^2 J_{47}
\right.\\+&\left.\nonumber
{\vec{q}}^4 J_{48}\right)
\Big]\,.\end{aligned}$$ We have used the master integrals with $q_0=\frac{M_\Lambda-M_N}{2}$, $q_0'=-\frac{M_\Lambda-M_N}{2}$, and ${\vec{q}}={\vec{p}}'-{\vec{p}}$.
![Second crossed-box-type Feynman diagram \[xbox2\]](box4g)
The amplitude for the crossed-box diagram with a $\Sigma$ propagator is $$\begin{aligned}
V_i=&
-i\frac{G_Fm_\pi^2g_
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& {\mathcal N}_{\sigma, 1}(\gamma^b) & \cdots & {\mathcal N}_{\sigma, n-1}(\gamma^b)\\
1 & {\mathcal N}_{\sigma, 1}(\gamma^{b+1}) & \cdots & {\mathcal N}_{\sigma, n-1}(\gamma^{b+1})\\
\vdots & \vdots & \ddots & \vdots \\
1& {\mathcal N}_{\sigma, 1}(\gamma^{b+\delta-2}) & \cdots & {\mathcal N}_{\sigma, n-1}(\gamma^{b+\delta-2}) \\
\end{array}
\right)$$ is a parity-check matrix. Any $\delta-1$ columns of (\[matrix\]) form a $(\delta-1)\times (\delta-1)$ matrix and denote $D$ as its determinant. Since $D$ is a Vandermonde determinant, $D=0$ if and only if ${\mathcal N}_{\sigma, i}(\gamma)={\mathcal N}_{\sigma, j}(\gamma)$, for $i\neq j$. It is equivalent to $$\gamma^{\frac{q_0^i-q_0^j}{q_0-1}}=\gamma^{\frac{q_0^j(q_0^{i-j}-1)}{q_0-1}}=1.$$ In particular, $\gamma^{q_0^j(q_0^{i-j}-1)}=1$ implies that $(q_0^n-1)\mid q_0^j(q_0^{i-j}-1)$. Since ${\rm gcd}(q_0^n-1, q_0^j)=1$, it follows that $(q_0^n-1)\mid (q_0^{i-j}-1)$. Therefore, there exists a positive integer $l$ such that $i-j=nl$. It means that $\frac{q_0^{nl}-1}{q_0-1}=k(q_0^n-1)$ for some positive integer $k$. Thus $(q_0-1)\mid \frac{q_0^{nl}-1}{q_0^n-1}=\sum_{i=0}^{m-1}q_0^{ni}$. It implies that $\gamma^{q_0-1}\mid \gamma$, which is impossible. This shows that any $\delta-1$ columns are linearly independent, and hence the minimum Hamming distance of $C$ is is at least $ \delta$. $\Box$
[**Example 2.7** ]{} Consider $R=\mathbb{F}_{3^2}[x, \sigma]$, where $\sigma=\theta$ is a Frobenius automorphism of $\mathbb{F}_{3^2}$ over $\mathbb{F}_3$. The polynomial $g(x)=x-\alpha^2$ is a right factor of $x^4-1$, where $\alpha$ is a primitive element of $\mathbb{F}_{3^2}$. Since $\phi(x^4-1)=Y^{81}-Y$, it follows that $\phi(x^4-1)$ splits in $\mathbb{F}_{3^4}$. Let $\xi$ be a primitive element of $\mathbb{F}_{3^4}$. Then $\alpha=\xi^{20}$ and $\phi(g(x))=Y^3-\alpha^2 Y$ has a root $\xi^{20}$. Therefore, by Lemma 2.5, $(\xi^{20})^3/\xi^{20}=\xi^{40}$ is a right root of $g(x)$. Let $C$ be a skew cyclic code of len
| 2,742
| 2,838
| 2,781
| 2,612
| 4,126
| 0.767955
|
github_plus_top10pct_by_avg
|
e equivalent to the Taylor microscale $\lambda^2
= 15\int_\Omega |{\mathbf{u}}|^2 d{\mathbf{x}}/ \int_\Omega |{\bm{\omega}}|^2 d{\mathbf{x}}$ used in turbulence research [@davidson:turbulence]. Another length scale, better suited to the ring-like vortex structures shown in figures \[fig:RvsE0\_FixE\_large\](c)-(e), is the average radius $R_{\Pi}$ of one of the vortex rings calculated as $$\label{eq:VortexRadius_def}
R_{\Pi} := \frac{ \int_\Omega r({\mathbf{x}})\chi_{\Pi}({\mathbf{x}}) \,d{\mathbf{x}}}{ \int_\Omega \chi_{\Pi}({\mathbf{x}})d{\mathbf{x}}},
\ \text{where} \ \
r({\mathbf{x}}) = |{\mathbf{x}}- \overline{{\mathbf{x}}}|,\quad \overline{{\mathbf{x}}} = \frac{\int_{\Omega} {\mathbf{x}}\chi_{\Pi}({\mathbf{x}}) d{\mathbf{x}}}{\int_{\Omega} \chi_{\Pi}({\mathbf{x}})d{\mathbf{x}}},$$ and $\chi_{\Pi}$ is the characteristic function of the set $$\begin{aligned}
\Pi & = & \{ \Gamma_s( Q ) : s > 0.9|| Q ||_{L_\infty}\} \cap \\
& & \{ {\mathbf{x}}\in\Omega : {\mathbf{n}}\cdot({\mathbf{x}}-{\mathbf{x}}_0) > 0, \ {\mathbf{n}}= [1,1,1], \ {\mathbf{x}}_0 = [1/2,1/2,1/2] \}.\end{aligned}$$ In the above definition of the set $\Pi$, the intersection of the two regions is necessary to restrict the set $R_{\Pi}$ to only one of the two ring structures visible in figures \[fig:RvsE0\_FixE\_large\](c)–(e). The quantity $\overline{{\mathbf{x}}}$ can be therefore interpreted as the geometric centre of one of the vortex rings. The dependence of $\Lambda$ and $R_\Pi$ on $\E_0$ is shown in figures \[fig:ScalingLaws\_fixE\](c,d) in which the following power laws can be observed
$$\begin{aligned}
{4}
\Lambda &\sim \O(1)\quad\mbox{and}& \quad R_\Pi &\sim \O(1)&&& \quad &
\mbox{as } \,\E_0\to 0, \\
\Lambda &\sim C_1\E_0^{\alpha_1},& \quad
C_1 &= 10.96,& \alpha_1 &= -0.886 \pm 0.105,& \qquad &
\mbox{as } \, \E_0\to \infty, \label{eq:Lambda_powerLaw_largeE0} \\
R_\Pi &\sim C_2\E_0^{\alpha_2},& \quad
C_2 &= 2.692,& \quad \alpha_2 &= -1.01 \pm 0.16,& \quad &
\mbox{as } \, \E_0\to \infty.
\label{eq:Radius_powerLaw_largeE
| 2,743
| 2,633
| 2,671
| 2,596
| null | null |
github_plus_top10pct_by_avg
|
_q)$ for some $\w\in\mathcal{S}$. On the other hand, by Lemma \[injective\], ${\rm
Rep}_{\Gamma,\v}^*(\F_q)$ contains all the indecomposable representations in ${\rm Rep}_{\Gamma,\v}(\F_q)$. This implies the following identity $$\sum_{\v\in\mathcal{S}}M_{\Gamma,\v}^*(q)X^{\v}
=\prod_{\v\in\mathcal{S}-\{0\}}(1-X^{\v})^{-I_{\Gamma,\v}(q)},$$ where $X^{\v}$ denotes the monomial $\prod_{i\in I}X_i^{v_i}$ for some fixed independent commuting variables $\{X_i\}_{ i\in I}$. Exactly as Hua [@hua Proof of Lemma 4.5] does we show from this formal identity that $$\Log\left(\sum_{\v\in\mathcal{S}}M_{\Gamma,\v}^*(q)X^{\v}\right)
=\sum_{\v\in\mathcal{S}-\{0\}}A_{\Gamma,\v}(q)\,X^{\v}.$$
It follows from Proposition \[hua-inj\] that since $A_{\Gamma,\v}(T)\in\Z[T]$ the quantity $M_{\Gamma,\v}^*(q)$ is also the evaluation of a polynomial with integer coefficients at $T=q$.
Given a non-increasing sequence $u=(n_0\geq n_1\geq \cdots)$ of non-negative integers we let $\Delta u$ be the sequence of successive differences $n_0-n_1, n_1-n_2\ldots$. We extend the notation of §\[charGL\] and denote by $\calF_{\Delta u}$ the set of partial flags of $\F_q$-vector spaces $$\{0\}\subseteq E^r\subseteq\cdots\subseteq E^1\subseteq
E^0=(\F_q)^{n_0}$$ such that ${\rm dim}(E^i)=n_i$.
Assume that $\v\in\mathcal S$ and let $\muhat=(\mu^1,\dots,\mu^k)$, where $\mu^i$ is the partition obtained from $\Delta \v_i$ by reordering, where $\v_i:=(v_0\geq v_{[i,1]}\geq\cdots\geq
v_{[i,s_i]})$. Consider the set of orbits $${\mathfrak G}_\muhat(\F_q) :=\left.\left({\rm
Mat}_{n_0}(\F_q)^g\times\prod_{i=1}^k\calF_{\mu^i}(\F_q)\right)
\right/\GL_{v_0}(\F_q),$$ where $\GL_{v_0}(\F_q)$ acts by conjugation on the first $g$ coordinates and in the obvious way on each $\calF_{\mu^i}(\F_q)$.
Let $\varphihat\in {\rm Rep}_{\Gamma,\v}^*(\F_q)$ with underlying graded vector space $\Vhat=V_0\oplus\bigoplus_{i,j} V_{[i,j]}$. We choose a basis of $V_0$ and we identify $V_0$ with $(\F_q)^{v_0}$. In the chosen basis, the $g$ maps $\varphi_\gamma$, with $\gamma\i
| 2,744
| 2,188
| 2,203
| 2,438
| 3,917
| 0.769295
|
github_plus_top10pct_by_avg
|
ly for the other arguments. The total number of atoms can be computed from the partition function (\[Z2\]): $$\begin{aligned}
\langle N \rangle = -T\frac{\partial \ln Z_B^0}{\partial \mu}
-T\frac{\partial \ln Z_F^{eff}}{\partial \mu}.\end{aligned}$$ The first term is given by the usual expression for an ideal Bose gas $$\begin{aligned}
\langle N_B^0 \rangle=2L \int \frac{dK}{2\pi} \Big[ e^{\beta
\left(K^2/4m - (2\mu-\nu)\right)}-1\Big]^{-1}\end{aligned}$$ with $2\mu-\nu<0$. When $T\to 0$, the fraction of atoms that are bound into bare dimers is: $$\begin{aligned}
\frac{\langle N_B^0 \rangle}{N}&\simeq& \frac{2}{n}
e^{-\beta(\nu-2\mu)} \int \frac{dK}{2\pi} e^{-\beta K^2/4m}\nonumber
\\ &=&\frac{2}{n}\sqrt{\frac{m}{\pi \beta}}e^{-\beta(\nu-2\mu)}\to 0.\end{aligned}$$ Therefore $$\begin{aligned}
N=\langle N \rangle \simeq -T\frac{\partial \ln Z_F^{eff}}{\partial
\mu}\end{aligned}$$ which shows that $\mu$ is the chemical potential for the gas of atoms only.
In conclusion, before resonance and under the assumptions that the resonance is broad and that $\nu> |\epsilon_{\star}|$, the system is described by a single channel model of fermions with an action $$\begin{aligned}
S_F^{eff}&=&\int_0^{\beta}d\tau \int dx \bigg(
\sum_{\sigma={\uparrow,\downarrow}} \bar{\psi}_{\sigma}
\Big[\partial_{\tau}-\frac{\partial_x^2}{2m}-\mu_F\Big]\psi_{\sigma}
\nonumber \\ &+&g_1
\bar{\psi}_{\uparrow}\bar{\psi}_{\downarrow}\psi_{\downarrow}\psi_{\uparrow}
\bigg)\end{aligned}$$ where $\mu_F=\mu$ and $g_1=-g^2/\nu<0$. This is the action corresponding to the Gaudin-Yang model of 1D fermions interacting via an attractive delta potential [@GY]. The single dimensionless coupling constant is $\gamma\equiv mg_1/n$. In order to describe the BCS-BEC crossover, we will use the parameter $1/\gamma$ (see Appendix B), the BCS limit corresponding to $1/\gamma \to -\infty$ or $\nu \to +\infty$. Due to the condition $\nu>|\epsilon_{\star}|$, before resonance, the parameter $1/\gamma$ is restricted to: $$\begin{aligned}
-\infty < \frac{1}{\gamma} < -\frac
| 2,745
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| 2,631
| null | null |
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|
-2\tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La^2\}\big]\Big].\end{aligned}$$ It is here observed that $$\begin{aligned}
&(1-v_1)\{v_1I_r+(1-v_1)\La\}^{-1}\La \\
&\qquad = \{v_1I_r+(1-v_1)\La\}^{-1} \{ v_1I_r+(1-v_1)\La - v_1I_r\}\\
&\qquad= I_r - v_1 \{v_1I_r+(1-v_1)\La\}^{-1},
\\
&(1-v_1)\{v_1I_r+(1-v_1)\La\}^{-1}\La^2 \\
&\qquad= \La - v_1 \{v_1I_r+(1-v_1)\La\}^{-1}\La,\end{aligned}$$ which is used to get $$\begin{aligned}
&(1-v_1)c\Big[ \tr\big[ \Er_\La(\La)\Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big] \\
&\qquad\qquad -2\tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La^2\}\big]\Big] \\
&\qquad =-c\tr \Er_\La(\La)\\
&\qquad\qquad + cv_1 \Big[ 2 \tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big] \\
&\qquad\qquad\qquad-\tr\big[ \Er_\La(\La) \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\}\big]\Big].\end{aligned}$$ Substituting this quantity into (\[eqn:De01\]) gives $$\begin{aligned}
\label{eqn:De02}
{\De(W;\pi_{GB}^J)\over m_{GB}}
\leq & -(q-r-1)\tr \Er_\La(\La) \non\\
&+ cv_1 \Big[ 2 \tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big] \non\\
&\qquad\qquad -\tr\big[ \Er_\La(\La) \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\}\big]\Big].\end{aligned}$$
To evaluate the second term in the r.h.s. of (\[eqn:De02\]), note that $$I_r \preceq \{v_1I_r+(1-v_1)\La\}^{-1} \preceq v_1^{-1}I_r.
\label{eqn:inq}$$ In the case of $c\geq 0$, it is seen from (\[eqn:inq\]) that $$\begin{aligned}
cv_1 &\Big[ 2 \tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big]-\tr\big[ \Er_\La(\La) \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\}\big]\Big]
\\
&\leq cv_1 \Big\{ {2\over v_1}\tr \Er_\La(\La) - \tr \Er_\La(\La)\Big\}
= c (2-v_1) \tr \Er_\La(\La),\end{aligned}$$ which implies that $$\De(W;\pi_{GB}^J)/m_{GB}
\leq \{ - (q-r-1) + c(2-v_w/v_0)\}\tr \Er_\La(\La),$$ because $1>v_1=v/v_0\geq v_w/v_0>0$. It is noted that $c=a+b+2r-2>-q + 2r +2=- (q-r-1) + r+1$ because $a>2-q$ and $b>2$. Thus, one gets a sufficient condition given by $$\max \{0, - (q-r-1) + r+1\} \leq c \leq (q-r-1)/(2-v_w/v_0).
\label{eqn:sc1}$$ In the case of $c\leq 0$, it is seen from (\[eqn:inq\]) that $$\beg
| 2,746
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|
y $options = null)
{
/** @var \Zend\View\Renderer\PhpRenderer $renderer */
$renderer = $container->get('ViewRenderer')
return new MyCustomController($renderer);
}
}
In the Controller, require it be set in the __construct() function:
public function __construct(PhpRenderer $renderer)
{
// ... set it somewhere, e.g.:
$this->setRenderer($renderer);
}
Then use it in your function:
$view = new ViewModel();
$renderer = $this->getRenderer();
$view->setTemplate('tools/tools/alert');
$html = $renderer->render($view);
Why, you ask?
Because the Renderer is configured via the Zend Configuration. You can find that in the \Zend\Mvc\Service\ServiceManageFactory class. The alias configuration provided is the following:
'ViewPhpRenderer' => 'Zend\View\Renderer\PhpRenderer',
'ViewRenderer' => 'Zend\View\Renderer\PhpRenderer',
'Zend\View\Renderer\RendererInterface' => 'Zend\View\Renderer\PhpRenderer',
The alias'es are mapped to Factory:
'Zend\View\Renderer\PhpRenderer' => ViewPhpRendererFactory::class,
That Factory is:
class ViewPhpRendererFactory implements FactoryInterface
{
/**
* @param ContainerInterface $container
* @param string $name
* @param null|array $options
* @return PhpRenderer
*/
public function __invoke(ContainerInterface $container, $name, array $options = null)
{
$renderer = new PhpRenderer();
$renderer->setHelperPluginManager($container->get('ViewHelperManager'));
$renderer->setResolver($container->get('ViewResolver'));
return $renderer;
}
}
As such, it has some presets included when you use it with $this->getRenderer, namely it has the HelperPluginManager and the Resolver set. So it knows where to get additional resources (if needed) and it knows how to resolve (ie render) a View.
Q:
How to calculate the intersection of two date ranges in VBA?
I like to calculate how much one date range overlaps with another data range in VBA (MS-Access).
Sa
| 2,747
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| 0.8344
|
github_plus_top10pct_by_avg
|
theta_1$ and $\theta_2$ thus anticommute with one another, $\left\{\theta_1,\theta_2\right\}=0$) $$[x,p]=i\hbar\ \ ,\ \
\left\{\theta_1,\theta_1\right\}=\frac{\hbar}{m\omega}=
\left\{\theta_2,\theta_2\right\}\ .
\label{eq:quantumbrackets}$$ Furthermore, the Hamiltonian operators is then expressed as $$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2+im\omega^2\theta_1\theta_2\ ,
\label{eq:Hamiltonian}$$ leading to the operator equations of motion in the Heisenberg picture, $$\begin{array}{l c l}
i\hbar\dot{x}=\left[x,H\right]=i\hbar\frac{p}{m}\ \ \ &,&\ \ \
i\hbar\dot{p}=\left[p,H\right]=-i\hbar m\omega^2x\ ,\\
& & \\
i\hbar\dot{\theta}_1=\left[\theta_1,H\right]=i\hbar\omega\theta_2\ \ \ &,&\ \ \
i\hbar\dot{\theta}_2=\left[\theta_2,H\right]=-i\hbar\omega\theta_1\ .
\end{array}
\label{eq:quantumEM}$$ It is also possible to determine how the supercharges $Q$ and $Q^\dagger$act on the operators $x$, $p$, $\theta_1$ and $\theta_2$, an exercise left to the reader (of which the results are used hereafter).
Through the correspondence principle, the (anti)commutation relations (\[eq:quantumbrackets\]) are required to translate into the following classical Grassmann graded Poisson brackets for the associated degrees of freedom, $$\left\{x,p\right\}=1\ \ ,\ \
\left\{\theta_1,\theta_1\right\}=-\frac{i}{m\omega}=
\left\{\theta_2,\theta_2\right\}\ ,$$ with now all the variables $x$, $p$ , $\theta_1$ and $\theta_2$ real under complex conjugation, $x$ and $p$ being ordinary commuting Grassmann even degrees of freedom, but $\theta_1$ and $\theta_2$ being anticommuting Grassmann odd degrees of freedom associated to the fermionic sector of the system. At the classical level, the Hamiltonian is given by the same expression as in (\[eq:Hamiltonian\]). In particular, using these Grassmann graded Poisson brackets, at the classical level the same Hamiltonian equations of motion are recovered as those in (\[eq:quantumEM\]) for the quantum operators. These classical equations of motion follow through the variational principle from the fir
| 2,748
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|
see [@weinberg Section 8.7]; see also [@lorence Section VII]) \_[11]{}(x,’,,E’,E) =& \_[11]{}(x,E’,E)\_[11]{}(E,E’) (’-\_[11]{}(E’,E)), where $$\hat{\sigma}_{11}(x,E',E):={}&\sigma_0(x)\Big({{1}\over{E'}}\Big)^2\Big({{E'}\over{E}}+{{E}\over{E'}}-1+\mu_{11}(E',E)^2\Big)
\nonumber\\
\chi_{11}(E',E):={}&\chi_{{\mathbb{R}}_+}(E-E_0)\chi_{{\mathbb{R}}_+}\big(E-\frac{E'}{1+2E'}\big)\chi_{{\mathbb{R}}_+}(E'-E) \\
\mu_{11}(E',E):={}&1+{{1}\over{E'}}-{{1}\over{E}}.$$ Here $(\omega', E')$ and $(\omega, E)$ are, respectively, the (direction, energy) of the incident and the scattered (outgoing) photons. If the scattering angle is written as $\theta_{11}$, then $\omega\cdot\omega'=\cos(\theta_{11})=\mu_{11}(E',E)$, and this condition is enforced by the delta-distribution term in $\sigma_{11}$.
We point out that if one defines (for a presentation more or less in this way, see [@hensel Appendix A.1.]), $$\ol{\sigma}_{11}(x,\omega',\omega,E',E)
=&\sigma_0(x)\ol{\sigma}_{11}'(E',E)\delta(E-E'\hspace{0.5mm}\ol{P}(\omega,\omega',E')),$$ where $$& P(E',E):=\frac{1}{1+E'(1-\mu_{11}(E',E))}=\frac{E}{E'}, \\
& \ol{P}(\omega,\omega',E'):=\frac{1}{1+E'(1-\omega'\cdot\omega)}, \\
& \ol{\sigma}_{11}'(E',E)
:=P(E',E)^2\Big(P(E',E)+\frac{1}{P(E',E)}-1+\mu_{11}(E',E)^2\Big),$$ Then $\sigma_0(x)\ol{\sigma}_{11}'(E',E)=E^2\hat{\sigma}_{11}(x,E',E)$, using which one can further show that the collision operator, say $\ol{K}_{11}$, corresponding to $\ol{\sigma}_{11}$ is equal to the collision operator $K_{11}$ defined by $\sigma_{11}$, i.e. $K_{11}=\ol{K}_{11}$.
We find that the operator $\hat {{{\mathcal{}}}K}_{11}$ is given by &(\_[11]{})(x,,E’,E)\
=& \_[11]{}(x,E’,E)\_[11]{}(E’,E)\_[S’]{} (’-\_[11]{}(E’,E))(x,’,E’)d’\
=&\_[11]{}(x,E’,E)\_[11]{}(E’,E)\_[0]{}\^[2]{}(x,(s),E’)ds, where $\gamma=\gamma_{11}(E',E,\omega):[0,2\pi]\to S$ is a parametrization with (constant) speed $${\left\Vert \gamma'(s)\right\Vert}=\sqrt{1-\mu_{11}(E',E)^2},\quad s\in [0,2\pi],$$ of the curve $\Gamma(E',E,\omega)$ which is the int
| 2,749
| 876
| 2,792
| 2,604
| null | null |
github_plus_top10pct_by_avg
|
number of integral points in each of these polytopes in polynomial time [@barvinok94] (see also [@dyerkannan97; @barvinokpommersheim99]). This gives rise to the following polynomial-time algorithm for , thereby establishing :
\[main algorithm\] Let $f \colon H \rightarrow G$ be a homomorphism of compact connected Lie groups. Given as input two highest weights $\lambda \in \Lambda^*_G \cong {\mathbb Z}^{r_G}$ and $\mu \in \Lambda^*_H \cong {\mathbb Z}^{r_H}$, encoded as bitstrings containing their coordinates with respect to the fundamental weight bases fixed above, the following algorithm computes the multiplicity $m^\lambda_\mu$ in polynomial time in the bitlength of the input:
$m \gets 0$ $n \gets \# \left( \Delta_{\mathcal A,\mathcal B}(\lambda, \mu + \gamma) \cap {\mathbb Z}^{s+s'} \right)$ as computed by Barvinok’s algorithm (see discussion above) $m \gets m + c_\gamma n$ **return** $m$
Here, $\Delta_{\mathcal A,\mathcal B}(y)$ denotes the rational convex polytope defined in , and the finite index set $\Gamma_H \subseteq \Lambda^*_H$ as well as the coefficients $(c_\gamma)$ are defined in the statement of .
There are at least two software packages which have implemented Barvinok’s algorithm, namely <span style="font-variant:small-caps;">LattE</span> [@deloeradutrakoppeetal11] and <span style="font-variant:small-caps;">barvinok</span> [@verdoolaegeseghirbeylsetal07; @verdoolaegebruynooghe08]. In we have reported on the performance of our implementation of for computing Kronecker coefficients using the latter package.
The existence of a polynomial-time algorithm for in fact already follows abstractly from , since in order to compute $m^\lambda_\mu$ we merely have to evaluate a *fixed* piecewise quasi-polynomial function. This piecewise quasi-polynomial can be computed algorithmically by using a variant of Barvinok’s algorithm which is also implemented in the <span style="font-variant:small-caps;">barvinok</span> package; see [@verdoolaegeseghirbeylsetal07 Proposition 2] and also [@barvinokpommershei
| 2,750
| 1,035
| 2,614
| 2,532
| 807
| 0.799604
|
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|
easing lattice volume $V$. We have performed a fit as a function of $1/V$ and have seen that the large volume $\beta=3.92$ lattice gives results which are very close to the infinite volume limit. In Fig. \[fig:pert\_vs\_latt\] we plot $D(q^2)$ in the intermediate and ultraviolet regime and compare with the three-loop perturbative QCD form. We see that the above $q\sim 2$ GeV the agreement is excellent, but that below this momentum scale nonperturbative effects are becoming apparent.
Quark Propagator
================
The Landau gauge quark propagator results summarized here have been presented and discussed in more detail elsewhere.[@quark_prop] All ${\cal O}(a)$ errors in the fermion action can be removed by adding appropriate terms to the Lagrangian[@Luscher:1996sc; @Dawson:1997gp]. It is then usual to perform appropriate field transformations to improve the quark operators as well.[@Heatlie:1991kg]
(14,7) (0,0)
(7,7)(-0.9,-0.4)[[ ]{}]{}
(7,0)
(7,7)(-0.9,-0.4)[[ ]{}]{}
In the continuum, the quark propagator has the following general form, $$S(p) = \frac{1}{i{\not\!p}A^c(p) + B^c(p)} \equiv
\frac{Z^c(p)}{i{\not\!p}+M^c(p)}.$$ We expect the lattice quark propagator to have a similar form, but with ${\not\!k}$ replacing ${\not\!p}$: $$S(p) = \frac{Z(p)}{i{\not\!k}+ M(p)}$$ where $k$ is a new ‘lattice momentum’, $k_\mu = \frac{1}{a}\sin(\hat p_\mu a)$. We do not have sufficient space here to describe the hybrid tree-level correction that was used for the quark propagator results presented here, but a detailed description has recently been given.[@quark_prop] We again use the cylinder cut to further reduce hypercubic discretization artefacts. As for the gluon propagator the results for $Z(p)$ are for the bare quantity only and contain an overall renormalization constant $Z_2(\mu,a)$. In Fig. \[fig:z\_np\_compare\] the vertical scales have been adjusted so that the two sets of results are renormalized and hence coincide at 2.1 GeV. In this figure we see the charactreistic dip in the infrared for $Z(p)$,
| 2,751
| 1,510
| 2,917
| 2,567
| 4,046
| 0.768475
|
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|
\in kB(G)$ and $[s(G)]$ is a system of representatives of conjugacy classes of subgroups of $G$. In this situation the set of the primitive orthogonal idempotents of $kB(G)$ is well known. These idempotents are in bijection with the conjugacy classes of subgroups of $G$. If $H$ is a subgroup of $G$, let us denote by $e^{G}_{H}$ the idempotent corresponding to the conjugacy class of $H$. For more details, see [@yoshida_idempotent],[@gluck_idempotent] or [@bouc_burnside] for a summary. Let us recall some important results about these idempotents:
Let $G$ be a finite group.
1. Let $H$ and $K$ be subgroups of $G$, then $|(e_{H}^{G})^{K}| = 1$ if $H$ is conjugate to $K$ and $0$ otherwise.
2. Let $X$ be a $G$-set and $H\leqslant G$, then $X.e_{H}^{G} = |X^{H}|e^{G}_{H}$.
3. Let $H\leqslant K$ be subgroups of $G$, then $Ind_{K}^{G}(e_{H}^{K})=\frac{|N_{G}(H)|}{|N_{K}(H)|}e_{H}^{G}.$
4. Let $H$ be a subgroup of $G$, then $$e_{H}^{G}=\frac{1}{|N_{G}(H)|} \sum_{K\leqslant H} |K|\mu(K,H) G/K.$$
\[lee1\]
1. Let $G$ be a finite group, then $\phi_{G}$ is a linear form.
2. The family $(\phi_{G})_{G}$ is stable by induction.
3. Let $G$ be a finite group, then $\phi_{G}(G/1)=1$.
The only non obvious assertion is the second. Since the map is linear it is enough to check this assertion on the basis elements of $kB(G)$. We use the basis consisting of the primitive orthogonal idempotents. Let $H\leqslant K\leqslant G$, then $$\begin{aligned}
\phi_{G}(Ind_{K}^{G}(e_{H}^{K}))&=\frac{|N_{G}(H)|}{|N_{K}(H)|}\phi_{G}(e^{G}_{H})\\
&=\frac{|N_{G}(H)|}{|N_{K}(H)|}\frac{1}{|N_{G}(H)|}\\
&=\frac{1}{|N_{K}(H)|}.\end{aligned}$$ In the other hand, $$\begin{aligned}
\phi_{K}(e_{H}^{K})=\frac{1}{|N_{K}(H)|}. \end{aligned}$$
\[prop1\] The determinant of this bilinear form $b_{\phi_{G}}$, in the basis consisting of the transitive $G$-sets is: $$det(b_{\phi})=\prod_{H\in [s(G)]} \frac{|N_{G}(H)|}{|H|^{2}}.$$ If $G$ is abelian, this determinant is equal to $1$.
We first compute the determinant of this bilinear form in the basis consi
| 2,752
| 2,290
| 2,338
| 2,437
| null | null |
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|
49685);
device.on('data', function(buf) {
var ch = buf.toString('hex').match(/.{1,2}/g).map(function(c) {
return parseInt(c, 16);
});
var position = ((ch[2] & 0x0f) << 6) + ((ch[1] & 0xfc) >> 2);
position = parseInt(position);sentData(position);
});
});
A:
The arduino code should look like this:
String data = '';
while(Serial.available() > 0) {
data = data + Serial.read();
}
Serial.println(data);
if(data == "2") {
//code
}
But, sorry, I can't see if there is problem in you node.js
Q:
BREACH - a new attack against HTTP. What can be done?
Following on from CRIME, now we have BREACH to be presented at Black Hat in Las Vegas Thursday (today). From the linked article, it suggests that this attack against compression will not be as simple to turn off as was done to deter CRIME. What can be done to mitigate this latest assault against HTTP?
EDIT: The presenters of BREACH have put up a website with further details. The listed mitigations are:
Disabling HTTP compression
Separating secrets from user input
Randomizing secrets per request
Masking secrets
Protecting vulnerable pages with CSRF
Length hiding
Rate-limiting requests
(note - also edited title and original question to clarify this attack is against HTTP which may be encrypted, not HTTPS specifically)
A:
Though the article is not full of details, we can infer a few things:
Attack uses compression with the same general principle as CRIME: the attacker can make a target system compress a sequence of characters which includes both a secret value (that the attacker tries to guess) and some characters that the attacker can choose. That's a chosen plaintext attack. The compressed length will depend on whether the attacker's string "looks like" the secret or not. The compressed length leaks through SSL encryption, because encryption hides contents, not length.
The article specifically speaks of "any secret that's [...] located in the body". So we are talking about HTTP-level compression, not SSL-level compression. HTTP compr
| 2,753
| 2,302
| 2,509
| 2,215
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|
66% 95% 0.06
Type of delivery: % Caesarean section 33% 5% 0.075
Gestational age 41.47 (±1.5) 41.15 (±2.2) 0.6
*Maternal representation of the baby (good/intermediate/poor)* [\*\*](#nt102){ref-type="table-fn"}
Third trimester 1/8/9 17/2/0 \<10^−5^
Birth 1/9/8 18/1/0 \<10^−5^
2 months postpartum 0/12/6 17/1/0 \<10^−5^
*Newborn Characteristics*
Infant Gender: Boy vs. Girl 66% vs. 33% 68% vs. 32% 1
Weight (g) 3483.95 (±376.3) 3348.33 (±551.5) 0.386
APGAR score 5′ 10 10 1
*Feeding Practices*
Bottle 44% 10% 0.053
Breast feeding one week and stop
| 2,754
| 5,754
| 1,590
| 1,453
| null | null |
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|
{}[\_j]{}\^2-(-1) \_ (- ) .
Integrability of the model Hamiltonian
======================================
The integrability of such types of Hamiltonian is established by constructing a complete set of commuting momentum operators that also commute with the model Hamiltonian. These operators were initially introduced in the study of Calogero-Sutherland model with periodic boundary conditions (both spin and classical cases) and are known as Dunkl operators. Similar operators have been used to study Calogero-Sutherland-type models derived from different root systems and are called Dunkl-type operators [@fin01; @buch94].
To construct commutative Dunkl-type operators we introduce variables, $z_j = \exp(2ix_j)$. Using this substitution, the anti-periodic Hamiltonian becomes, \[globe1\] H\_N\^[\^[ap]{}]{}=\_[j=1]{}\^N(z\_j\_j)\^2-(-1) \_ -(-1) \_ , z\_[jk]{}\^ = z\_j z\_k . Let us apply the following similarity transformation, \_N=\^[-1]{}H\_N\^[\^[ap]{}]{}where, =\_ , \_1()=,1-, \_2() = \[1\] . Thus, the anti-periodic Hamiltonian becomes \[sim2\] \_N=\_[j=1]{}\^N(z\_j\_j)\^2 + \_ (z\_j\_j-z\_k\_k) + \_ (z\_j\_j-z\_k\_k) The term $\mu_1(\lambda)$ is real for all $\lambda$, however, $\mu_2(\lambda)$ is real only for $1+4\lambda-4\lambda^2 \geq 0$, i.e., $\vert\lambda-\frac{1}{2}\vert \leq \frac{1}{\sqrt{2}}$. Under this restriction, the Hamiltonian $\widetilde{H}_N$ becomes hermitian. In the following, the integrability of the Hamiltonian is studied for different allowed values of $\lambda$.
Let us introduce the coordinate exchange operators $\{\Lambda_{jk}\vert j,k = 1,..N; j \neq k\}$ and the sign reversing operators $\{\Lambda_j\vert j,k = 1,..N\}$. The coordinate exchange operator acting on the coordinates of $j$-th and $k$-th particle may be defined by the operation $\Lambda_{jk}f(z_1,..,z_j,..,z_k,.., z_N)=
f(z_1,..,z_k,..,z_j,.., z_N)$. This operator is (i) self-adjoint, (ii) unitary, and satisfies (iii) $\Lambda_{ij}\Lambda_{jk}=\Lambda_{ik}\Lambda_{ij}
=\Lambda_{jk}\Lambda_{ik}$ , (iv) $\Lambda_{ij}\Lambda_
| 2,755
| 679
| 2,538
| 2,783
| 3,611
| 0.771235
|
github_plus_top10pct_by_avg
|
ptimization problem defined below.
Hereafter, $H^2(\Omega)$ will denote the Sobolev space of functions with square-integrable second derivatives endowed with the inner product [@af05] $$\forall\,\mathbf{z}_1, \mathbf{z}_2 \in H^2(\Omega) \qquad
\Big\langle \mathbf{z}_1, \mathbf{z}_2 \Big\rangle_{H^2(\Omega)}
= \int_{\Omega} \mathbf{z}_1 \cdot \mathbf{z}_2
+ \ell_1^2 \,\bnabla \mathbf{z}_1 \colon \bnabla \mathbf{z}_2
+ \ell_2^4 \,\Delta \mathbf{z}_1 \cdot \Delta \mathbf{z}_2 \, d{\mathbf{x}}, \label{eq:ipH2}$$ where $\ell_1,\ell_2\in \RR_+$ are parameters with the meaning of length scales (the reasons for introducing these parameters in the definition of the inner product will become clear below). The inner product in the space $L_2(\Omega)$ is obtained from by setting $\ell_1 = \ell_2 = 0$. The notation $H^2_0(\Omega)$ will refer to the Sobolev space $H^2(\Omega)$ of functions with zero mean. For every fixed value $\E_0$ of enstrophy we will look for a divergence-free vector field ${\widetilde{\mathbf{u}}_{\E_0}}$ maximizing the objective function $\R \; : \; H^2_0(\Omega) \rightarrow \RR$ defined in . We thus have the following
\[pb:maxdEdt\_E\] Given $\E_0\in\mathbb{R}_+$ and the objective functional $\R$ from equation , find $$\begin{aligned}
{\widetilde{\mathbf{u}}_{\E_0}}& = & \mathop{\arg\max}_{{\mathbf{u}}\in{\mathcal{S}_{\E_0}}} \, \R({\mathbf{u}}) \\
{\mathcal{S}_{\E_0}} & = & \left\{{\mathbf{u}}\in H_0^2(\Omega)\,\colon\,\nabla\cdot{\mathbf{u}}= 0, \; \E({\mathbf{u}}) = \E_0 \right\}\end{aligned}$$
which will be solved for enstrophy $\E_0$ spanning a broad range of values. This approach was originally proposed and investigated by [@ld08]. In the present study we extend and generalize these results by first showing how other fields considered in the context of the blow-up problem for both the Euler and Navier-Stokes system, namely the Taylor-Green vortex, also arise from variational problem \[pb:maxdEdt\_E\]. We then thoroughly analyze the time evolution corresponding to our extreme vortex states
| 2,756
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| 3,012
| 2,623
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|
ather than the CGF. The proper operation of the James’s method thus relies on the accurate calculation of the DGF, yet finding its analytic expressions in cylindrical coordinates is a daunting task. To our knowledge, the analytic DGF is available only in 2D Cartesian coordinates [@bune71]. In 3D Cartesian coordinates, @burk97 addressed the definition, existence, and uniqueness of the DGF and derived asymptotic expansion formulae, applicable at distances far from the source. In this section, we provide a working definition of the DGF and a numerical method to calculate $\Theta$ in Cartesian and cylindrical coordinates. We refer the reader to Appendix \[s:calc\_dgf\] for our method for the DGF.
### Cartesian Grid
The DGF, ${\cal G}_{i-i',j-j',k-k'}$, in Cartesian coordinates is the gravitational potential per unit mass due to a discrete point mass at $(i',j',k')$ and ought to satisfy $$\label{eq:def_car_green}
\left(\Delta_x^2 + \Delta_y^2 + \Delta_z^2\right){\cal G}_{i-i',j-j',k-k'} = 4\pi G \frac{\delta_{ii'}\delta_{jj'}\delta_{kk'}}{\cal V},$$ where the symbol $\delta_{ii'}$ is the Kronecker delta and ${\cal V}=\int_{z_{k'-1/2}}^{z_{k'+1/2}}\int_{y_{j'-1/2}}^{y_{j'+1/2}}\int_{x_{i'-1/2}}^{x_{i'+1/2}}dxdydz = \delta x\delta y\delta z$ is the volume of the $(i',j',k')$-th cell. Note that in writing the indices of ${\cal G}_{i-i',j-j',k-k'}$, we implicitly allow for the translational symmetry on a Cartesian grid. In Appendix \[s:calc\_dgf\_cart\], we follow @james77 to calculate the Cartesian DGF numerically.
The gravitational potential $\Theta_{i,j,k}$ generated by the screening charges $\sigma_{i,j,k}$ is given by $$\label{eq:car_bpot_by_dgf}
\Theta_{i,j,k} = \sum_{i'=0}^{N_x+1}\sum_{j'=0}^{N_y+1}\sum_{k'=0}^{N_z+1}{\cal G}_{i-i',j-j',k-k'}\sigma_{i',j',k'}{\cal V}.$$ Substituting Equation into Equation , one can easily check that $\Theta_{i,j,k}$ and $\sigma_{i,j,k}$ is a valid potential-density pair.
Because Equation involves a discrete convolution, it is efficient to use FFTs for computations. Making
| 2,757
| 3,647
| 3,022
| 2,722
| 1,795
| 0.785659
|
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|
{aligned}$$
$$\begin{aligned}
A_{\Sigma2}\to&
-\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32}
+\frac23(\sqrt3A_{\Sigma\frac12}+A_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\\
B_{\Sigma2}\to&
-\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32}
+\frac23(\sqrt3B_{\Sigma\frac12}+B_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\,.\end{aligned}$$
$$\begin{aligned}
A_{\Sigma3}\to&
-\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32}
-\frac23(\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\\
B_{\Sigma3}\to&
-\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32}
-\frac23(\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\,.\end{aligned}$$
Note that Eqs. (\[eq:va\]) and (\[eq:vc\]) only have physical meaning away from the SU(3) limit.
Brief comparison of LO and NLO contributions {#sec:bc}
============================================
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(UP) Medium-Long range part of the potentials for the one-pion-exchange, one-kaon-exchange, ball diagram and triangle diagrams. (DOWN) Medium-Long range part of the potentials for the one-pion-exchange, one-kaon-exchange, box and crossed box diagrams. \[fig:pot\]](pottriangles "fig:")![(UP) Medium-Long range part of the potentials for the one-pion-exchange, one-kaon-exchange, ball diagram and triangle diagrams. (DOWN) Medium-Long range part of the potentials for the one-pion-exchange, one-kaon-exchange, box and crossed box diagrams. \[fig:pot\]](potboxes "fig:")
-------------------------
| 2,758
| 1,772
| 1,055
| 2,848
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i(1+\pi m_{i,i}')+\pi^3(\ast).$$ Here, the nondiagonal entries of this equation are considered in $B\otimes_AR$ and each diagonal entry of $a_i'$ is of the form $\pi^3 x_i'$ with $x_i'\in R$. Now, the nondiagonal entries of $-\pi^2\cdot{}^tm_{i,i}'a_i m_{i,i}'+\pi^3(\ast)$ are all $0$ since they contain $\pi^2$ as a factor. In addition, the diagonal entries of $\pi^3(\ast)$ are $0$ since they contain $\pi^5$ as a factor, which can be verified by using Equation (\[ea72\]), and the diagonal entries of $-\pi^2\cdot{}^tm_{i,i}'a_i m_{i,i}'$ are also $0$ since they contain $\pi^4$ as a factor. Thus the above equation equals $$a_i'=a_i+\sigma(\pi)\cdot {}^tm_{i,i}'a_i+ \pi\cdot a_i m_{i,i}'.$$ By letting $a_i'=a_i$, we have the following equation $$\sigma(\pi)\cdot {}^tm_{i,i}'a_i+ \pi\cdot a_i m_{i,i}'=0.$$
Based on (3) of the description of an element of $\underline{H}(R)$ for a $\kappa$-algebra $R$, which is explained in Section \[h\], in order to investigate this equation, we need to consider the nondiagonal entries of $\sigma(\pi)\cdot {}^tm_{i,i}'a_i+ \pi\cdot a_i m_{i,i}'$ as elements of $B\otimes_AR$ and the diagonal entries of $\sigma(\pi)\cdot {}^tm_{i,i}'a_i+ \pi\cdot a_i m_{i,i}'$ as of the form $\pi^3 x_i$ with $x_i\in R$. Recall from Remark \[r33\].(2) that $$a_i=\begin{pmatrix} \begin{pmatrix} 0&1\\-1&0\end{pmatrix}& & \\ &\ddots & \\ & & \begin{pmatrix} 0&1\\-1&0\end{pmatrix}\end{pmatrix}.$$ Note that ${}^ta_i=-a_i$ and $\sigma(\pi)=-\pi$ so that $\sigma(\pi)\cdot {}^tm_{i,i}'a_i+ \pi\cdot a_i m_{i,i}'=\pi({}^t(a_i m_{i,i}')+a_i m_{i,i}')$. Then we can see that each diagonal entry as well as each nondiagonal (upper triangular) entry of $\sigma(\pi)\cdot {}^tm_{i,i}'a_i+ \pi\cdot a_i m_{i,i}'$ produces a linear equation. Thus there are exactly $(n_i^2+n_i)/2$ independent linear equations and $(n_i^2-n_i)/2$ entries of $m_{i,i}'$ determine all entries of $m_{i,i}'$.
For example, let $m_{i,i}'=\begin{pmatrix} x&y\\z&w\end{pmatrix}$ and $a_i=\begin{pmatrix} 0&1\\-1&0\end{pmatrix}$. Then $$\sigma
| 2,759
| 2,532
| 2,246
| 2,547
| null | null |
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|
m{Nexp}}( - \frac{1}{2} \langle \cdot,\mathbf{K} \cdot\rangle ) \cdot \exp( - \frac{1}{2} \langle \cdot,\mathbf{L} \cdot\rangle )\Big)({\bf f})\\
=\sqrt{\frac{1}{\det(\mathbf{Id+L(Id+K)^{-1}})}}
\exp(-\frac{1}{2} \langle {\bf f}, \mathbf{(Id+K+L)^{-1}} {\bf f} \rangle ),\quad {\bf f} \in S_{d}({\mathbb{R}}),
\end{gathered}$$ in the case the right hand side indeed is a U-funcional.
\[prodnexp\] Let $\mathbf{K}: L^2_{d}({\mathbb{R}}, dx)_{{\mathbb{C}}} \to L^2_{d}({\mathbb{R}}, dx)_{{\mathbb{C}}}$ be as in Definition \[D:Nexp\], i.e. ${\rm{Nexp}}(- \frac{1}{2} \langle \cdot ,\mathbf{K} \cdot \rangle)$ exists. Furthermore let $\mathbf{L}: L^2_d({\mathbb{R}}, dx)_{{\mathbb{C}}} \to L^2_d({\mathbb{R}}, dx)_{{\mathbb{C}}}$ be trace class. Then we define $${\rm{Nexp}}( - \frac{1}{2} \langle \cdot,\mathbf{K} \cdot\rangle ) \cdot \exp( - \frac{1}{2} \langle \cdot,\mathbf{L} \cdot\rangle )$$ via its $T$-transform, whenever $$\begin{gathered}
T\Big({\rm{Nexp}}( - \frac{1}{2} \langle \cdot,\mathbf{K} \cdot\rangle ) \cdot \exp( - \frac{1}{2} \langle \cdot,\mathbf{L} \cdot\rangle )\Big)({\bf f})\\
=\sqrt{\frac{1}{\det(\mathbf{Id+L(Id+K)^{-1}})}}
\exp(-\frac{1}{2} \langle {\bf f}, \mathbf{(Id+K+L)^{-1}} {\bf f} \rangle ),\quad {\bf f} \in S_{d}({\mathbb{R}}),
\end{gathered}$$ is a U-functional.
In the case $\mathbf{g} \in S_d({\mathbb{R}})$, $c\in{\mathbb{C}}$ the product between the Hida distribution $\Phi$ and the Hida test function $\exp(i \langle \mathbf{g},. \rangle + c)$ can be defined because $(S)$ is a continuous algebra under pointwise multiplication. The next definition is an extension of this product.
\[linexp\] The pointwise product of a Hida distribution $\Phi \in (S)'$ with an exponential of a linear term, i.e. $$\Phi \cdot \exp(i \langle {\bf g}, \cdot \rangle +c), \quad {\bf g} \in L^2_{d}({\mathbb{R}})_{{\mathbb{C}}}, \, c \in {\mathbb{C}},$$ is defined by $$T(\Phi \cdot \exp(i\langle {\bf g}, \cdot \
| 2,760
| 4,051
| 2,540
| 2,407
| null | null |
github_plus_top10pct_by_avg
|
\sum_{p_{A},p_{B},q_{A},r_{B}}\lambda\left(v\right)=\Pr\left[\mathbf{Q}_{C}=q_{C},\mathbf{R}_{C}=r_{C}\right].
\end{array}\label{eq:CbD exmaple}$$ The noncontextuality hypothesis for $\mathbf{P}_{A},\mathbf{Q}_{A},\mathbf{R}_{B}$ and $\mathbf{P}_{B},\mathbf{Q}_{C},\mathbf{R}_{C}$ is that among these jpds $\lambda$ we can find at least one for which $\Pr\left[\mathbf{P}_{A}\not=\mathbf{P}_{B}\right]=\Pr\left[\mathbf{Q}_{A}\not=\mathbf{Q}_{C}\right]=\Pr\left[\mathbf{R}_{B}\not=\mathbf{R}_{C}\right]=0,$ which is equivalent to $\Delta=\Pr\left[\mathbf{P}_{A}\not=\mathbf{P}_{B}\right]+\Pr\left[\mathbf{Q}_{A}\not=\mathbf{Q}_{C}\right]+\Pr\left[\mathbf{R}_{B}\not=\mathbf{R}_{C}\right]=0.$ Such a jpd need not exist, and then the smallest possible value $\Delta_{\min}$ of $\Delta$ for which a jpd of $\left(\mathbf{P}_{A},\mathbf{P}_{B},\mathbf{Q}_{A},\mathbf{Q}_{C},\mathbf{R}_{B},\mathbf{R}_{C}\right)$ exists can be taken as a measure of contextuality.[^3]
The CdB approach has its precursors in the literature: various aspects of the contextual indexation of random variables and probabilities of the kind shown are considered in Refs. [@larsson_kochen-specker_2002; @dzhafarov_qualified_2014; @Simon-Brukner-Zeilinger; @Winter2014; @svozil_how_2012; @dzhafarov_all-possible-couplings_2013; @Khr2005; @Khr2008; @Khr2009]. The principal difference, however, is in the use of minimization of $\Delta$ under the assumption that a jpd exists. This is a well-defined mathematical problem, solvable in principle for any set of distributions observed empirically. We will now compare and interrelate the two approaches, NP and CbD, by applying them to the Leggett-Garg and the EPR-Bell setups.
Leggett-Garg
=============
Let us consider Leggett and Garg’s $\pm1$-valued random variables, $\mathbf{Q}_{1}$, $\mathbf{Q}_{2}$, and $\mathbf{Q}_{3}$ [@leggett_quantum_1985]. Applying the NP approach, we seek signed probabilities $\mu$ for $\left(\mathbf{Q}_{1},\mathbf{Q}_{2},\mathbf{Q}_{3}\right)$ that are consistent with the observed correlati
| 2,761
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| 2,595
| 2,410
| null | null |
github_plus_top10pct_by_avg
|
D_d(M)/D_{d-1}(M)$ is clean as well.
\[conclusion\] Let $S=K[x_1,\ldots,x_n]$ be a the polynomial ring and $I\subset S$ a monomial ideal. Then the following conditions are equivalent:
1. $S/I$ is pretty clean;
2. $S/I$ is sequentially CM, and the non-zero factors in the dimension filtration of $S/I$ are clean;
3. the non-zero factors in the dimension filtration of $S/I$ are clean.
(a)(b): Since the associated prime ideals of $S/I$ are all generated by subsets of $\{x_1,\ldots,x_n\}$, all hypotheses Theorem \[sequentially\] and Corollary \[interesting\] are satisfied, so that the assertions follow.
(b)(c) is trivial.
(c)(a): The refinement of the dimension filtration by the clean filtrations of the non-zero factors gives us the desired pretty clean filtration of $S/I$.
*Let $S=K[x,z,u,v]$, and consider the ideals $L=(u,v,z)$, $Q_1=(x,z^2)$, $Q_2=(x,v^2,z^3)$ and $I=L\sect Q_1\sect Q_2$. We claim that the module $M=L/I$ is not pretty clean, but that $M$ has a prime filtration $\mathcal F$ with $\Supp({\mathcal F})=\Ass(M)$.*
Note that $(L\sect Q_1)\sect (L\sect Q_2)$ modulo $I$ is an irredundant primary decomposition of $(0)$ in $M$. Hence since $\emptyset \neq \Ass(L/L\sect
Q_i)\subset \Ass(S/Q_i)=\{P_i\}$ with $P_1=(x,z)$ and $P_2=(x,v,z)$ we see that $\Ass(M)=\{P_1,P_2\}$.
It follows from Corollary \[description\] that $D_1(M)=(L\sect Q_1)/I$ and that $D_2(M)=M$. We show that $D_2(M)/D_1(M)=L/L\sect Q_1$ is not clean. Indeed, suppose $L/L\sect Q_1$ is clean. Then, since $\Ass(L/L\sect Q_1)=\{P_1\}$, this module has a filtration with all factors isomorphic to $S/P_1$, and the number of these factors equals the length of the $S_{P_1}$-module $(L/L\sect Q_1)_{P_1}=S_{P_1}/Q_1S_{P_1}$. This length is obviously $2$. On the other hand, since $L/L\sect Q_1$ is generated by 3 elements, it cannot have a filtration with two factors, both of them being cyclic.
Knowing now that $D_2(M)/D_1(M)$ is not clean, we conclude from Corollary \[interesting\] that $M$ is not pretty clean.
Finally we construct a prime f
| 2,762
| 1,595
| 2,496
| 2,519
| 1,892
| 0.784654
|
github_plus_top10pct_by_avg
|
rangle + c))({\bf f}):= T\Phi({\bf f}+{\bf g})\exp(c),\quad {\bf f} \in S_d({\mathbb{R}}),$$ if $T\Phi$ has a continuous extension to $L^2_d({\mathbb{R}})_{{\mathbb{C}}}$ and the term on the right-hand side is a U-functional in ${\bf f} \in S_d({\mathbb{R}})$.
\[donsker\] Let $D \subset {\mathbb{R}}$ with $0 \in \overline{D}$. Under the assumption that $T\Phi$ has a continuous extension to $L^2_d({\mathbb{R}})_{{\mathbb{C}}}$, ${\boldsymbol\eta}\in L^2_d({\mathbb{R}})_{{\mathbb{C}}}$, $y \in {\mathbb{R}}$, $\lambda \in \gamma_{\alpha}:=\{\exp(-i\alpha)s|\, s \in {\mathbb{R}}\}$ and that the integrand $$\gamma_{\alpha} \ni \lambda \mapsto \exp(-i\lambda y)T\Phi({\bf f}+\lambda {\boldsymbol\eta}) \in {\mathbb{C}}$$ fulfills the conditions of Corollary \[intcor\] for all $\alpha \in D$. Then one can define the product $$\Phi \cdot \delta_0(\langle {\boldsymbol\eta}, \cdot \rangle-y),$$ by $$T(\Phi \cdot \delta_0(\langle {\boldsymbol\eta}, \cdot \rangle-y))({\bf f})
:= \lim_{\alpha \to 0} \int_{\gamma_{\alpha}} \exp(-i \lambda y) T\Phi({\bf f}+\lambda {\boldsymbol\eta}) \, d \lambda.$$ Of course under the assumption that the right-hand side converges in the sense of Corollary \[seqcor\], see e.g. [@GS98a].
This definition is motivated by the definition of Donsker’s delta, see Definition \[D:Donsker\].
[[@BG10]]{}\[thelemma\] Let $\mathbf{L}$ be a $d\times d$ block operator matrix on $L^2_{d}({\mathbb{R}})_{{\mathbb{C}}}$ acting componentwise such that all entries are bounded operators on $L^2({\mathbb{R}})_{{\mathbb{C}}}$. Let $\mathbf{K}$ be a d $\times d$ block operator matrix on $L^2_{d}({\mathbb{R}})_{{\mathbb{C}}}$, such that $\mathbf{Id+K}$ and $\mathbf{N}=\mathbf{Id}+\mathbf{K}+\mathbf{L}$ are bounded with bounded inverse. Furthermore assume that $\det(\mathbf{Id}+\mathbf{L}(\mathbf{Id}+\mathbf{K})^{-1})$ exists and is different from zero (this is e.g. the case if $\mathbf{L}$ is trace class and -1 in the resolvent set of $\mathbf{L}(\mathbf{Id}+\mathbf{K})^{-1}$). Let $M_{\mathbf{N}^{-1}}$ be the matr
| 2,763
| 1,537
| 2,073
| 2,466
| null | null |
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|
Together with the naive bound $G(x)\leq O(1){\vbx{|\!|\!|}}^{-q}$ (cf., [(\[eq:IR-xbd\])]{}) as well as Proposition \[prp:conv-star\](ii) (with $x=x'$ or $y=y'$), we also obtain $$\begin{aligned}
{\label{eq:GGpsi-bd}}
\sum_{v'}G(v'-y)\,G(z-v')\,\psi_\Lambda(v',v)&\leq G(v-y)\,G(z-v)+
\sum_{v'}\frac{O(\theta_0^2)}{{\vbv'-y{|\!|\!|}}^q{\vbz-v'{|\!|\!|}}^q{\vbv-v'{|\!|\!|}}^{2q}}
{\nonumber}\\
&\leq\frac{O(1)}{{\vbv-y{|\!|\!|}}^q{\vbz-v{|\!|\!|}}^q}.\end{aligned}$$ The $O(1)$ term in the right-hand side is replaced by $O(\theta_0)$ or $O(\theta_0^2)$ depending on the number of $G$’s on the left being replaced by $\tilde G_\Lambda$’s.
Since [(\[eq:pi0-1stbd\])]{}–[(\[eq:tildeG-bd\])]{} immediately imply the bound on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$, it suffices to prove the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)$ for $i\ge1$. To do so, we first estimate the building blocks of the diagrammatic bound [(\[eq:piNbd\])]{}: $\sum_{b:{\underline{b}}=y}\tau_b\,Q'_{\Lambda;u}({\overline{b}},x)$ and $\sum_{b:{\underline{b}}=y}\tau_b\,Q''_{\Lambda;u,v}({\overline{b}},x)$.
Recall [(\[eq:P’0-def\])]{}–[(\[eq:Q”-def\])]{}. First, by using $G(x)\leq O(1){\vbx{|\!|\!|}}^{-q}$ and [(\[eq:GGpsi-bd\])]{}, we obtain $$\begin{aligned}
P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y,x)&\leq\frac{O(1)}{{\vbx-y{|\!|\!|}}^{2q}
{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q},{\label{eq:P'0-bd}}\\
P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)&\leq\frac{O(1)}{{\vbx
-y{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^q}.{\label{eq:P''0-bd}}\end{aligned}$$ We will show at the end of this subsection that, for $j\ge1$, $$\begin{aligned}
P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(j)}}(y,x)&\leq\frac{O(j)\,O(\theta_0^2)^j}
{{\vbx-y{|\!|\!|}}^{2q}{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q},{\label{eq:P'j-bd}}\\
P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(j)}}(y,x)&\leq\frac{O(j^2)\,O
(\theta_0^2)^j}{{\vbx-y{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q{\v
| 2,764
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elocity field ${\mathbf{u}}$, and it is essential that the gradient be characterized by the required regularity, namely, $\nabla\R({\mathbf{u}}) \in H^2(\Omega)$. This is, in fact, guaranteed by the Riesz representation theorem [@l69] applicable because the Gâteaux differential $\R'({\mathbf{u}};\cdot) : H_0^2(\Omega) \rightarrow \RR$, defined as $\R'({\mathbf{u}};{\mathbf{u}}') := \lim_{\epsilon \rightarrow 0}
\epsilon^{-1}\left[\R({\mathbf{u}}+\epsilon {\mathbf{u}}') - \R({\mathbf{u}})\right]$ for some perturbation ${\mathbf{u}}' \in H_0^2(\Omega)$, is a bounded linear functional on $H_0^2(\Omega)$. The Gâteaux differential can be computed directly to give $$\R'({\mathbf{u}};{\mathbf{u}}') = \int_{\Omega}\left[{\mathbf{u}}'\cdot\bnabla{\mathbf{u}}\cdot{\Delta}{\mathbf{u}}+
{\mathbf{u}}\cdot\bnabla{\mathbf{u}}'\cdot{\Delta}{\mathbf{u}}+
{\mathbf{u}}\cdot\bnabla{\mathbf{u}}\cdot{\Delta}{\mathbf{u}}' \right]\,d{\mathbf{x}}-2\nu\int_{\Omega}{\Delta}^2{\mathbf{u}}\cdot{\mathbf{u}}'\,d{\mathbf{x}}\label{eq:dR}$$ from which, by the Riesz representation theorem, we obtain $$\R'({\mathbf{u}};{\mathbf{u}}')
= \Big\langle \nabla\R({\mathbf{u}}), {\mathbf{u}}' \Big\rangle_{H^2(\Omega)}
= \Big\langle \nabla^{L_2}\R({\mathbf{u}}), {\mathbf{u}}' \Big\rangle_{L_2(\Omega)}
\label{eq:riesz}$$ with the Riesz representers $\nabla\R({\mathbf{u}})$ and $\nabla^{L_2}\R({\mathbf{u}})$ being the gradients computed with respect to the $H^2$ and $L_2$ topology, respectively, and the inner products defined in . We remark that, while the $H^2$ gradient is used exclusively in the actual computations, cf. , the $L_2$ gradient is computed first as an intermediate step. Identifying the Gâteaux differential with the $L_2$ inner product and performing integration by parts yields $$\nabla^{L_2}\R({\mathbf{u}}) = {\Delta}\left( {\mathbf{u}}\cdot\bnabla{\mathbf{u}}\right) + (\bnabla{\mathbf{u}})^T{\Delta}{\mathbf{u}}-
{\mathbf{u}}\cdot\bnabla({\Delta}{\mathbf{u}}) - 2\nu{\Delta}^2{\mathbf{u}}.
\label{eq:gradRL2}$$ Similarly, identifying the
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H^1$ semi-norm and the $L^2$ norm of the errors are reported in Table \[tab:test1\] and Figure \[fig:test1\]. These errors are computed using a 5th order Gaussian quadrature on triangles. For quadrilateral elements, the errors can be conveniently computed by dividing the quadrilateral into two triangles and then applying the Gaussian quadrature. Our results show that the $H^1$ semi-norm has an approximate order of $O(h)$, while the $L^2$ norm has an approximate order of $O(h^2)$, as predicted by the theoretical analysis.
![Initial and refined mesh for test 1.[]{data-label="fig:mesh1"}](mesh1-1 "fig:"){width="6cm"}![Initial and refined mesh for test 1.[]{data-label="fig:mesh1"}](mesh1-2 "fig:"){width="6cm"}
------------------------- ---------------- ---------------- ---------------- ----------------- ----------------- ----------------
$h$ $\frac{1}{16}$ $\frac{1}{32}$ $\frac{1}{64}$ $\frac{1}{128}$ $\frac{1}{256}$ $O(h^r)$, $r=$
\[1mm\] $|u-u_h|_{1,h}$ 1.2006 0.5904 0.2917 0.1452 0.0725 1.0124
$\|u-u_h\|$ 0.0551 0.0159 0.0042 0.0011 0.0003 1.9270
------------------------- ---------------- ---------------- ---------------- ----------------- ----------------- ----------------
: Convergence rates for test 1.[]{data-label="tab:test1"}
![Convergence rates for test 1.[]{data-label="fig:test1"}](error1){width="8cm"}
In the second test, we consider a hybrid mesh containing mainly hexagons, but with a few quadrilaterals and pentagons. Indeed, it is derived by taking the dual mesh of a simple triangular mesh. In Figure \[fig:mesh2\], the initial triangular mesh and its dual mesh are shown. By refining the triangular mesh and computing its dual mesh, we get a sequence of hexagon hybrid meshes. Again, we solve the interior penalty discontinuous Galerkin formulation (\[eq:dg\]) on these hexagon hybrid meshes, with the local discrete spaces $V_K$ of $
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notation $(m_{i,j}, s_i\cdots w_i)$ is explained in Section \[m\]. Then $h\circ m$ is an element of $\underline{H}(R)$ and $(\mathrm{Ker~}\varphi)(R)$ is the set of $m$ such that $h\circ m=(f_{i, j}, a_i\cdots f_i)$. The action $h\circ m$ is explicitly described in Remark \[r35\]. Based on this, we need to write the matrix product $h\circ m=\sigma({}^tm)\cdot h\cdot m$ formally. To do that, we write each block of $\sigma({}^tm)\cdot h\cdot m$ as follows:
The diagonal $(i,i)$-block of the formal matrix product $\sigma({}^tm)\cdot h\cdot m$ is the following: $$\begin{gathered}
\label{ea1}
\pi^i\left(\sigma({}^tm_{i,i})h_im_{i,i}+\sigma(\pi)\cdot\sigma({}^tm_{i-1, i})h_{i-1}m_{i-1, i}+\pi\cdot\sigma({}^tm_{i+1, i})h_{i+1}m_{i+1, i}\right)+\\
\pi^i\left((\sigma\pi)^2\cdot\sigma({}^tm_{i-2, i})h_{i-2}m_{i-2, i}+
\pi^2\cdot\sigma({}^tm_{i+2, i})h_{i+2}m_{i+2, i}+\pi^3(\ast)\right),\end{gathered}$$ where $0\leq i < N$ and $(\ast)$ is a certain formal polynomial.\
The $(i,j)$-block of the formal matrix product $\sigma({}^tm)\cdot h\cdot m$, where $i<j$, is the following: $$\label{ea2}
\pi^j\left(\sum_{i\leq k \leq j} \sigma({}^tm_{k,i})h_km_{k,j}+\sigma(\pi)\cdot\sigma({}^tm_{i-1,i})h_{i-1}m_{i-1,j}+\pi\cdot\sigma({}^tm_{j+1,i})h_{j+1}m_{j+1,j}+\pi^2(\ast)\right),$$ where $0\leq i, j < N$ and $(\ast)$ is a certain formal polynomial. In the following computations, we always have in mind that $\sigma(\pi)=-\pi$. as mentioned at the beginning of Section \[Notations\].
Before studying $\tilde{G}^1$, we describe the conditions for an element $m\in \tilde{M}(R)$ as above to belong to the subgroup $\tilde{M}^1(R)$.
1. $m_{i,j}=\pi m_{i,j}^{\prime} \mathrm{~if~} i\neq j,$
2. If $i$ is even and $L_i$ is *of type* $\textit{I}^o$, then $$m_{i,i}= \begin{pmatrix} s_i&\pi y_i\\ \pi v_i&1+\pi z_i \end{pmatrix}=\begin{pmatrix} \mathrm{id}+\pi s_i^{\prime}&\pi^2 y_i^{\prime}\\ \pi^2 v_i^{\prime}&1+\pi^2 z_i^{\prime} \end{pmatrix}.$$
3. If $i$ is even and $L_i$ is *of type* $\textit{I}^e$, then $$m_{i,i}=\begin{pmatrix} s_i&r_i&
| 2,767
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our main result. We include here a new proof for the sake of completeness.
(of Proposition \[equalities: adjoint EVF and EVF\]) Let $\Lambda=(0=\lambda_0<\lambda_1<...<\lambda_{n+1}=T)$ be a subdivision of $[0,T].$ Let ${\mathfrak{a}}_k:V \times V \to \mathbb C\ \ \hbox{ for } k=0,1,...,n$ be given by $$\begin{aligned}
\ {\mathfrak{a}}_k(u,v):={\mathfrak{a}}_{k,\Lambda}(u,v):=\frac{1}{\lambda_{k+1}-\lambda_k}
\int_{\lambda_k}^{\lambda_{k+1}}&{\mathfrak{a}}(r;u,v){\rm d}r\ \hbox{ for } u,v\in V. \
\end{aligned}$$ All these forms satisfy (\[eq:continuity-nonaut\]) with the same constants $\alpha, M.$ The associated operators in $V'$ are denoted by ${\mathcal{A}}_k\in {\mathcal{L}}(V,V')$ and are given for all $u\in V$ and $k=0,1,...,n$ by $$\label{eq:op-moyen integrale}
{\mathcal{A}}_ku :={\mathcal{A}}_{k,\Lambda}:=\frac{1}{\lambda_{k+1}-\lambda_k}
\int_{\lambda_k}^{\lambda_{k+1}}{\mathcal{A}}(r)u{\rm d}r.\ \ $$ Consider the non-autonomous form ${\mathfrak{a}}_\Lambda:[0,T]\times V \times V \to {\mathbb{C}}$ defined by $$\label{form: approximation formula1}
{\mathfrak{a}}_{\Lambda}(t;\cdot,\cdot):=\begin{cases}
{\mathfrak{a}}_k(\cdot,\cdot)&\hbox{if }t\in [\lambda_k,\lambda_{k+1})\\
{\mathfrak{a}}_n(\cdot,\cdot)&\hbox{if }t=T\ .
\end{cases}$$ Its associated time dependent operator ${\mathcal{A}}_\Lambda(\cdot): [0,T]\subset {\mathcal{L}}(V,V')$ is given by $$\label{form: approximation formula1}
{\mathcal{A}}_{\Lambda}(t):=\begin{cases}
{\mathcal{A}}_k&\hbox{if }t\in [\lambda_k,\lambda_{k+1})\\
{\mathcal{A}}_n &\hbox{if }t=T\ .
\end{cases}$$ Next denote by $T_k$ the $C_0-$semigroup associated with ${\mathfrak{a}}_k$ in $H$ for all $k=0,1...n.$ Then applying Theorem \[wellposedness in V’2\]) to the form ${\mathfrak{a}}_\Lambda$ we obtain that in this case the associated evolution family ${{U}}_\Lambda(t,s)$ is given explicitly for $\lambda_{m-1}\leq s<\lambda_m<...<\lambda_{l-1}\leq t<\lambda_{l}$ by $$\label{promenade1}{{U}}_\Lambda (t,s):= T_{l-1}(t-\lambda_
| 2,768
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\w_0$ in the self-energy occur which also find their way into the tunneling spectra. Ref. [@JovchevAnders2013] discusses the deviations of the non-perturbatively calculated full electron-phonon self-energy from the lowest-order perturbative results.
In conclusion, we maintain the terminology of the elastic current for all current contributions where the electrons travel ballistically between the tip and sample system S. Internal many-body scattering processes within the system S are all included the spectral functions within $\tau^{(0)}_{\sigma}(\w)$ and no assumption of the strength of the internal interactions are required. Therefore, $I_{\rm el}$ describes the current for a static distance between the system S and the STM tip.
Modeling the system {#sec:Modeling the system}
===================
In the previous section, we have presented a tunneling theory which relies on three spectral functions: one contains the information on the elastic tunneling current, the other two are connected linearly and quadratically to vibrational displacements. While this tunneling theory is completely general, for its application we need to specify the Hamiltonian of the system $\hat H_{S}$ and thus also the spectral functions which enter the tunneling theory. In the present section, we specify and discuss a $\hat H_{S}$ which turns out to be of sufficient generality to describe the physical sample system which we investigate experimentally in section \[sec:experiment-NTCDA\].
Electronic degrees of freedom
-----------------------------
We employ a single-orbital single impurity Anderson model (SIAM) for the electronic degrees of freedom $$\begin{aligned}
\label{eq:H-e}
\hat H_{\rm e} &=&
\sum_{\k\sigma} \e_{\k\sigma} c^\dagger_{\k\sigma} c_{\k\sigma}
+
\sum_\sigma \e_{d\sigma} n^d_\sigma + U n^d_\uparrow n^d_\downarrow
\non
&& + \sum_{\k\sigma} V_{\k} ( c^\dagger_{\k\sigma} d_{0\sigma} + d^\dagger_{0\sigma} c_{\k\sigma} )
\label{eqn:SIAM}\end{aligned}$$ of the sample system S, thereby assuming that only one single molec
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ppendix {#appendix .unnumbered}
========
The Higgs potential of our model is given by $V=V_2+V_3+V_4$ where $$\begin{aligned}
V_2
&\equiv&
- m_{s_1}^2 |s_1^0|^2
+ \frac{1}{\,2\,} m_{s_2}^2 (s_2^0)^2
- m_\Phi^2\, \Phi^\dagger \Phi
+ m_\eta^2\, \eta^\dagger \eta
+ m_\Delta^2\, {{\text{tr}}}(\Delta^\dagger \Delta) ,\end{aligned}$$ $$\begin{aligned}
V_3
&\equiv&
( \mu_\eta^{}\, \eta^T\, i\sigma_2\, \Delta^\dagger\, \eta)
+ \text{h.c.} ,\end{aligned}$$ $$\begin{aligned}
V_4
&\equiv&
\lambda_{1\Phi}\, (\Phi^\dagger \Phi)^2
+ \lambda_{1\eta}\, (\eta^\dagger \eta)^2
+ \lambda_{1\Phi\Phi}\, (\Phi^\dagger \Phi) (\eta^\dagger \eta)
+ \lambda_{1\Phi\eta}\, (\Phi^\dagger \eta) (\eta^\dagger \Phi)
\nonumber\\
&&{}
+ \lambda_2\, [{{\text{tr}}}(\Delta^\dagger \Delta)]^2
+ \lambda_3\, {{\text{tr}}}[(\Delta^\dagger \Delta)^2]
\nonumber\\
&&{}
+ \lambda_{4\Phi}\,
(\Phi^\dagger \Phi)\, {{\text{tr}}}(\Delta^\dagger \Delta)
+ \lambda_{4\eta}\,
(\eta^\dagger \eta)\, {{\text{tr}}}(\Delta^\dagger \Delta)
\nonumber\\
&&{}
+ \lambda_{5\Phi}\,
(\Phi^\dagger \Delta \Delta^\dagger \Phi)
+ \lambda_{5\eta}\,
(\eta^\dagger \Delta \Delta^\dagger \eta)
\nonumber\\
&&{}
+ \lambda_{s1}\, |s_1^0|^4
+ \lambda_{s2}\, (s_2^0)^4
+ \lambda_{s3}\, |s_1^0|^2 (s_2^0)^2
\nonumber\\
&&{}
+ \lambda_{s\Phi 1}\, |s_1^0|^2\, (\Phi^\dagger \Phi)
+ \lambda_{s\Phi 2}\, (s_2^0)^2\, (\Phi^\dagger \Phi)
\nonumber\\
&&{}
+ \lambda_{s\eta 1}\, |s_1^0|^2\, (\eta^\dagger \eta)
+ \lambda_{s\eta 2}\, (s_2^0)^2\, (\eta^\dagger \eta)
+ \left\{
\lambda_{s\Phi\eta}\, s_1^0\, s_2^0\, (\eta^\dagger \Phi)
+ \text{h.c.}
\right\}
\nonumber\\
&&{}
+ \lambda_{s\Delta 1}\, |s_1^0|^2 {{\text{tr}}}(\Delta^\dagger \Delta)
+ \lambda_{s\Delta 2}\, (s_2^0)^2 {{\text{tr}}}(\Delta^\dagger \Delta) .\end{aligned}$$ All coupling constants are real because the phases of $\mu_\eta$ and $\lambda_{s\Phi\eta}$ can be absorbed by $\Delta$ and $s_1^0$, respectively.
Mass eigenstates of two $Z_2$-even CP-even neutral scalars which are composed of $s_{1r}
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hey are radially ordered in the reference plane. We will keep this point in mind without expressing the radial-ordering explicitly.
The vacuum is invariant under the global group discussed in the previous section[^6] $$\langle0|G=0,$$ where $G$ are the generators of the global subgroup. Consequently the correlation functions are invariant under the global transformations $$\langle0|GO(x_1,y_1)O(x_2,y_2)|0\rangle=0$$ where $$G\in\{L_{-1},\ L_0,\ L_1,\ M_{-d},\ \cdots,\ M_d\}.$$ Moving $G$ from the left to the right gives the constraints on the two-point functions. For example, the translation symmetries require that the correlation functions must depend only on $x=x_1-x_2$ and $y=y_1-y_2$.
Let us discuss case by case, setting $c=1$. The $d=0$ case is special, since the representation is special. As shown in [@Song:2017czq], there is $$\langle \mO_1(x,y)\mO_2(0,0)\rangle=d_\mO\delta_{h_1,h_2}\delta_{\xi_1,-\xi_2}\frac{1}{x^{2h_1}}e^{\xi y}.$$ For $d=1$, there are no descendant operators involved when doing the local transformations on the primary operators. The two-point function is different from the other cases[@Bagchi:2009ca] $$\langle \mO_1(x,y)\mO_2(0,0)\rangle=d_\mO\delta_{h_1,h_2}\delta_{\xi_1,\xi_2}\frac{1}{x^{2h_1}}e^{2\xi\frac{y}{x}}. \label{2ptb1}$$ For $d \geq 2$, the correlation functions become much more involved. The correlation functions of the descendant operators with the primary operators are not vanishing in such cases. Namely we have to consider the following correlation functions $$f(n,d)=\langle(M_{n}\mO_1)(x,y)\mO_2(0,0)\rangle.$$ Solving the constraints from the invariance of the two-point functions under the global transformations, one gets $$f(-d+1,d)=-\frac{1}{2}xf(-d,d),$$ $$f(n,d)=\frac{(d-1)!(d-n)!}{2(2d-1)!(-n)!}(-1)^{n+d} x^{n+d} f(-d,d),\ \ \ \mbox{for}\ n\in [-d+2,0].$$ In the end, one finds $$\langle \mO_1(x,y)\mO_2(0,0)\rangle=d_\mO\delta_{h_1,h_2}\delta_{\xi_1,(-1)^{d+1}\xi_2}\frac{1}{x^{2h_1}}e^{2C_{2d-1}^d(-1)^{d+1}\xi\frac{y}{x^d}},$$ where $C^n_m$ is the binomial coeffici
| 2,772
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ell-1} \in \Omega_\ell}\exp(\theta_{j_{\ell-1}})}{\exp(\theta_i)+\exp(\theta_{\i}) +\sum_{j_{\ell-1} \in \Omega_\ell}\exp(\theta_{j_{\ell-1}})-\big(\sum_{j_{\ell-1} \in \Omega_\ell}\exp(\theta_{j_{\ell-1}})\big)/|\Omega_\ell|} \nonumber\\
&=&\Bigg({\frac{\exp(\theta_{i})+\exp(\theta_{\i})}{\sum_{j_{\ell-1} \in \Omega_\ell} \exp(\theta_{j_{\ell-1}})} + 1 - \frac{1}{\kappa-\ell}}\Bigg)^{-1} \nonumber\\
&\geq& \Bigg(\frac{\widetilde{\alpha}_1}{\kappa-\ell} + 1 - \frac{1}{\kappa-\ell}\Bigg)^{-1} \label{eq:posl_1}\\
&=& \frac{\kappa-\ell}{\widetilde{\alpha}_1 + \kappa-\ell-1} \nonumber\\
&=& \sum_{j_{\ell-1} \in \Omega_\ell } \frac{\exp(\ltheta_{j_{\ell-1}})}{\widetilde{W}-\sum_{k=j_1}^{j_{\ell-2}}\exp(\ltheta_{k}) - \exp(\ltheta_{j_{\ell-1}})} \label{eq:posl_2}\;,\end{aligned}$$ where follows from the Jensen’s inequality and the fact that for any $c >0$, $0 < x < c$, $\frac{x}{c-x}$ is convex in $x$. Equation follows from the definition of $\widetilde{\alpha}_{i,i',\ell,\theta}$, , and the fact that $|\Omega_\ell| = \kappa-\ell$. Equation uses the definition of $\{\ltheta_j\}_{j \in S}$.
Consider $\{\Omega_{\widetilde{\ell}}\}_{2 \leq \widetilde{\ell} \leq \ell - 1}$, $|\Omega_{\widetilde{\ell}}| = \kappa - \widetilde{\ell}$, corresponding to the subsequent summation terms in . Observe that $\frac{\exp(\theta_i)+\exp(\theta_{\i})}{\sum_{j \in \Omega_{\widetilde{\ell}}} \exp(\theta_j)} \leq \widetilde{\alpha}_{i,i',\ell,\theta}/|\Omega_{\widetilde{\ell}}|$. Therefore, each summation term in equation can be lower bounded by the corresponding term where $\{\theta_j\}_{j \in S}$ is replaced by $\{\ltheta_j\}_{j \in S}$. Hence, we have $$\begin{aligned}
\label{eq:posl_4}
&&\P_{\theta}\Big[\sigma^{-1}(i) = \ell, \sigma^{-1}(\i) > \ell \Big] \nonumber\\
&\geq& \frac{\exp(\theta_i)}{W} \sum_{\substack{j_1 \in S \\ j_1 \neq i,\i}} \Bigg( \frac{\exp(\ltheta_{j_1})}{\widetilde{W}-\exp(\ltheta_{j_1})} \sum_{\substack{j_2 \in S \\ j_2 \neq i,\i,j_1}} \Bigg( \frac{\exp(\ltheta_{j_2})}{\widetilde{W}-\exp(\ltheta_{j_1})-\exp(\lt
| 2,773
| 1,711
| 2,039
| 2,515
| null | null |
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|
9.3 56.6 9.5 4.6
4--6 times/week 310 18.7 54.2 16.1 11.0
Daily 222 16.2 43.7 17.1 23.0
BMI (kg/m^2^) at 11y Mean (SD) 2858 18.3 (3.1) 18.7 (3.3) 18.7 (3.4) 18.7 (3.2) 0.04
BMI z-score at 11y Mean (SD) 2858 0.12 (1.20) 0.25 (1.29) 0.23 (1.34) 0.28 (1.25) 0.04
Weight status at 11y Normal 2266 35.0 49.9 9.2 5.9 0.06
Overweight 515 30.1 54.4 9.3 6.2
Obese 77 23.4 53.2 16.9 6.5
WC (cm) at 11y Mean (SD) 3184 66.4 (8.7) 66.8 (9.1) 67.3 (9.6) 66.9 (9.4) 0.13
BMI (kg/m^2^) at 12y Mean (SD) 2121 19.0 (3.0) 19.2 (3.4) 19.2 (3.6) 19.2 (3.2) 0.30
BMI z-score at 12y Mean (SD) 2121 0.13 (1.13) 0.18 (1.27) 0.15 (1.32) 0.24 (1.14) 0.36
BMI (kg/m^2^) at 13y Mean (SD) 2378 19.6 (3.0) 19.8 (3.5) 19.9 (3.5) 19.9 (3.3) 0.19
BMI z-score at 13y Mean (SD) 2378 0.07 (1.07) 0.09 (1.22) 0.14 (1.21) 0.15 (1.16) 0.29
BMI (kg/m^2^) at 14y Mean (SD) 1925 19.9 (2.9) 20.1 (3.4) 20.3 (3.4) 19.9 (2.9) 0.29
BMI z-score at 14y Mean (SD) 1925 −0.09 (1.02) −0.07 (1.15) 0.02 (1.21) −0.07 (1.07)
| 2,774
| 4,883
| 1,717
| 1,888
| null | null |
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|
Lemma \[lemma:SpecFact\] $\bar J_e$ is special, hence by Lemma \[lemma:SpecSplitNullExt\] $\widehat{J_e}$ is a special Jordan algebra and $J_e\in {\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$. Therefore, ${\mathrm{Di}}{\mathrm{SJ}}\langle X\rangle\in{\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$. Since the free algebra of the variety ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$ belongs to the variety ${\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$, the variety ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$ is embedded into ${\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$.
We prove the inclusion “$\supseteq$”. Let $J\in{\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$. By the definition of a variety of dialgebras in the sense of Eilenberg it means that $\widehat J\in{\mathcal{H}}{\mathrm{SJ}}$, hence by Lemma \[lemma:ifHomSJthenHomDiSJ\] we obtain $J\in{\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$.
s-identities in dialgebras
--------------------------
Let ${\mathrm{Var}}$ be a variety of algebras, $X=\{x_1, x_2, \ldots \}$ be a countable set. Consider a mapping $\phi_{\mathrm{Var}}\colon{\mathrm{Alg}}\,\langle X
\rangle \to {\mathrm{Var}}\,\langle X \rangle$ which maps $x_i\mapsto x_i$. Let $T_0({\mathrm{Var}})$ be a set of multilinear polynomials from $\ker\phi_{\mathrm{Var}}$, these are exactly all multilinear identities of ${\mathrm{Var}}$. We suppose that the variety is defined by multilinear identities that is ${\mathrm{Var}}=\{A\mid A\vDash T_0({\mathrm{Var}})\}$. There we use the denotation $A\vDash f$ which means that the identity $f(x_1,
\ldots, x_n)=0$ holds on the algebra $A$.
Further, let ${\mathrm{Di}}{\mathrm{Alg}}0\,\langle X\rangle$ be a free 0-dialgebra, $\phi_{{\mathrm{Di}}{\mathrm{Var}}}\colon{\mathrm{Di}}{\mathrm{Alg}}0\,\langle X \rangle \to
{\mathrm{Di}}{\mathrm{Var}}\,\langle X \rangle$, $T_0({\mathrm{Di}}{\mathrm{Var}})$ be a set of multilinear dipolynomials from $\ker \phi_{{\mathrm{Di}}{\mathrm{Var}}}$, i. e., all multilinear identities from ${\mathrm{Di}}{\mathrm{Var}}$.
In paper [@Pozh:09] the following theorem was proved
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|
^{\iota}(X)|\right.
\nonumber\\
&-&\left.
|\psi^{\iota}(X)\rangle\langle\psi^{\iota}(X)|
\left\{\ln(\hat{\rho}),\hat{H}\right\}_{\mbox{\tiny\boldmath$\cal B$}}
\right)\;.\nonumber\\
\label{eq:wavematrix}\end{aligned}$$ Equation (\[eq:wavematrix\]) can be written as a system of two coupled equations for the wave fields [@fckrk]: $$\begin{aligned}
i\hbar\frac{d}{dt}\vert\psi^{\iota}_{(X,t)}\rangle & =&
\left(\hat{H}-\frac{\hbar}{2i}
\left\{\hat{H},\ln(\hat{\rho}_{(X,t)})\right\}_{\mbox{\tiny\boldmath$\cal B$}}
\right)\vert\psi^{\iota}_{(X,t)}\rangle\nonumber\\
%%%%%%%%
-i\hbar\langle\psi^{\iota}_{(X,t)}\vert\overleftarrow{\frac{d}{dt}}
&=&
\langle\psi^{\iota}_{(X,t)}\vert\left(\hat{H}
-\frac{\hbar}{2i}
\left\{
\ln(\hat{\rho}_{(X,t)}),\hat{H}\right\}_{\mbox{\tiny\boldmath$\cal B$}}
\right)
\;.\nonumber\\
\label{eq:fckrk}\end{aligned}$$ Equations (\[eq:fckrk\]), which are obeyed by the wave fields, are non-linear since their solution depends self-consistently from the density matrix defined in Eq. (\[eq:rho-ansatz\]). These equations are also non-Hermitian since the operators $\left\{\hat{H},\ln(\hat{\rho})\right\}_{\mbox{\tiny\boldmath$\cal B$}}$ and $\left\{\ln(\hat{\rho}),\hat{H}\right\}_{\mbox{\tiny\boldmath$\cal B$}}$ are not Hermitian. However, this does not cause problems for the conservation of probability. The wave fields $\vert\psi^{\iota}\rangle$ and $\langle\psi^{\iota}\vert$ evolve according to the different propagators $$\begin{aligned}
\overrightarrow{\cal U}_{{\mbox{\tiny\boldmath$\cal B$}},[\hat{\rho}]}(t)
&=&
\exp\left[-\frac{it}{\hbar}\left(\hat{H}
-\frac{\hbar}{2i}
\left\{\hat{H},\ln(\hat{\rho})\right\}_{\mbox{\tiny\boldmath$\cal B$}}
\right)\right]\;,\nonumber\\
&&\\
\overleftarrow{\cal U}_{{\mbox{\tiny\boldmath$\cal B$}},[\hat{\rho}]}(t)
&=&
\exp\left[-\frac{it}{\hbar}
\left(\hat{H}
-\frac{\hbar}{2i}
\left\{\ln(\hat{\rho}),\hat{H}\right\}_{\mbox{\tiny\boldmath$\cal B$}}
\right)\right]\;,\nonumber\\\end{aligned}$$ so that time-propagating wave fields are defined by $$\begin{aligned}
\vert\psi
| 2,776
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| 3,025
| null | null |
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|
t statistic is equal to 176.49 and the adequation with the Dirichlet distribution is rejected. However, we can see that as conjectured, the simulated sample looks like a simulated sample from a Dirichlet distribution.
Appendix: non-crossing property for the paths of the RST
========================================================
\[lemm:croisement\] Any two paths $\gamma$ and $\gamma'$ of the RST (finite or not) cannot cross: $$\forall X \in \gamma , \; \forall X' \in \gamma' , \; (X,\A(X)) \cap (X',\A(X')) = \emptyset$$ (where $(a,b)$ denotes the segment $[a,b]$ in $\mathbb{R}^{2}$ without its endpoints).
Let us assume there exists a point $I$ belonging to both $(X,\A(X))$ and $(Y,\A(Y))$. It is easy to check that this assumption and the construction rule of the RST force $X, Y, \A(X)$ and $\A(Y)$ to be four different points. The same is true for their Euclidean norms with probability one. Moreover, without loss of generality, we can also assume that $|Y|<|X|$. Then, two cases can be distinguished.
*First case:* If $|\A(X)|<|Y|$ then $Y$ is closer to $\A(Y)$ than $\A(X)$: $|\A(Y) - Y| < |\A(X) - Y|$. In the same way, the inequality $|\A(Y)|<|Y|<|X|$ implies $
|\A(X) - X| < |\A(Y) - X|$. Now, the triangular inequality leads to a contradiction: $$\begin{aligned}
|\A(Y) - Y| + |\A(X) - X| & < & |\A(X) - Y| + |\A(Y) - X| \\
& < & |\A(X) - I| + |I - Y| + |\A(Y) - I| + |I - X| \\
& < & |\A(Y) - Y| + |\A(X) - X| ~.\end{aligned}$$
*Second case:* We now assume that $|Y|<|\A(X)|$ and refer to Fig \[fig:croisement2\]. The points $X$ and $\A(X)$ do not belong to the open ball $B(O,|Y|)$ which contains $\A(Y)$ by definition. Hence the existence of the point $I$ forces the segment $(X,\A(X))$ to intersect $S(O,|Y|)$ at two distinct points, say $T_1$ and $T_2$, dividing the closed ball $\overline{B}(O,|Y|)$ in two non overlapping sets, say $U$ and $V$. By hypothesis, each of these two sets contains (exactly) one of the two points $Y$ and $\A(Y)$. Since $|T_1-X|$ and $|T_2-X|$ are smaller than $|X-\A(X)|$ by construction,
| 2,777
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|
lta S=2}$ to the mass matrix. From the input parameters $ B_K=0.75, F_K=160 \,{\hbox{MeV}}, m_K= 498\,{\hbox{MeV}} ,
\Delta m_K = 3.51 \times 10^{-15} \hbox{GeV} $ and the relation $$M_{12}^R = \frac{1}{2m_K}
\langle \bar{K^0} | {\cal H}^{\Delta S=2} |K^0\rangle^{*}
= \frac{G_F^2 m_W^2}{16\pi^2} \frac{1}{2m_K} C_{\Delta S=2}^{R*}(\mu)
\langle \bar{K^0} | O^R_{\Delta S=2}(\mu) |K^0\rangle^{*} \ ,$$ $$\langle \bar{K^0} | O^R_{\Delta S=2}(\mu) |K^0\rangle =
\hbox{${8\over3}$}\alpha_s(\mu)^{2/9} B_K F_K^2 m_K^2 \ ,$$ follows the numerical prediction $ M_{12}^R / \Delta m_K = 1.2 \times 10^4 \ C_{\Delta S=2}^R(M_Q) $. Demanding that the imaginary part of $M_{12}^R$ gives enough contribution to $\epsilon$ and the corresponding real part gives just a fraction $\cal F$ of the mass difference $\Delta m_K$ ([*i.e.*]{} $ 2 \hbox{Re}( M_{12}^R ) = {\cal F} \Delta m_K$), we obtain constraints on the Wilson coefficients: $\hbox{Im}\ C_{\Delta S=2}^R(M_Q)= 2.7 \times 10^{-7} $ and $\hbox{Re}\ C_{\Delta S=2}^R(M_Q)= 4.2 \times 10^{-5}{\cal F} $. Again, with $m_2 \gg m_1, m_1=M_Q$, we then find $$\hbox{Im} \left({\cal A}_{sd} / (0.049)^2 \right)^2 R_Q^2 =1 \ ,\quad
\hbox{Re} \left({\cal A}_{sd} / (0.049)^2 \right)^2 R_Q^2 =156 {\cal F}\ ;
\label{eq:dmineq}$$ where $R_Q = 300 \hbox{ GeV}/M_Q$. The reasonable constraint $|{\cal F}|
< 1$ can be easily satisfied.
Constraints from $(\epsilon'/\epsilon)$ and $B^0$–$\bar {B^0}$ mixing {#constraints-from-epsilonepsilon-and-b0bar-b0-mixing .unnumbered}
=====================================================================
The parameter $\epsilon'$ describes direct CP violation in the kaon system. It is given in terms of the $2\pi$ decay amplitudes $A_{0,2}= {\cal A}(K \to (\pi\pi)_{0,2})$, where the subscript indicates the isospin of the outgoing state. With $\omega= |A_2/A_0|=0.045$, $\xi=\hbox{Im}A_0/\hbox{Re}A_0$, $\Phi\approx \pi/4$, and $\Omega=(1/\omega )\cdot (\hbox{Im}A_2/\hbox{Im}A_0)$, $$\epsilon' = -\frac{\omega}{\sqrt 2} \xi (1-\Omega) \exp(i\Phi)
| 2,778
| 2,505
| 3,014
| 2,622
| 3,324
| 0.773161
|
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|
ac{1}{|x-y|^{\alpha-d}}$$ for a constant $c_{d,\alpha}$ for $d>1$ and $\alpha\in(0,2$); see [@bucur]. If the point $y$ is chosen outside a domain $D$, then we can construct $G$ as an exact solution to the homogeneous version of the fractional Dirichlet problem in ; that is, $u(x)=G(x,y)$ for $x\in D$ and ${g}(x)=G(x,y)$ for $x\not\in D$. Figure \[fig:test4\] shows the results of applying the walk-on-spheres algorithm to evaluate $u(0.6, 0.6)$ with $10^6$ samples, where $D$ is a unit ball in $\mathbb{R}^2$ centred at the origin and $y=(2,0)$. We observe the samples ${g}(\rho_{N})$ have larger variance when $\alpha$ is small and a larger error results from the same number of samples.
![Example simulation for with exterior data ${g}(x)=G(x,y)$ with $y=(2,0)$ on the domain given by the unit ball centred at the origin, based on $10^6$ samples. The left-hand plot shows the relative error and the right-hand plot shows the sample variance. The sample variance is larger for small $\alpha$ as the process stops further away from the boundary and can see the singularity at $(2,0)$ in the exterior data. Accordingly, the relative error is higher as we are using a fixed number of samples.[]{data-label="fig:test4"}](test4_fig5 "fig:") ![Example simulation for with exterior data ${g}(x)=G(x,y)$ with $y=(2,0)$ on the domain given by the unit ball centred at the origin, based on $10^6$ samples. The left-hand plot shows the relative error and the right-hand plot shows the sample variance. The sample variance is larger for small $\alpha$ as the process stops further away from the boundary and can see the singularity at $(2,0)$ in the exterior data. Accordingly, the relative error is higher as we are using a fixed number of samples.[]{data-label="fig:test4"}](test4_fig4 "fig:")\
Gaussian data
-------------
For the Poisson problem , we take $D$ to be the unit ball in $\mathbb{R}^2$, exterior data $${g}(x)=\exp(-| x-y|^2), \qquad x\in D^{\rm c},$$ for a given $y\in \mathbb{R}^2$, and zero source term ${f}=0$. We can represent the sol
| 2,779
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| 1,627
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| 1,023
| 0.795497
|
github_plus_top10pct_by_avg
|
mmatic functions consisting of two-point functions. Let $$\begin{aligned}
{\label{eq:tildeG-def}}
\tilde G_\Lambda(y,x)
=\sum_{b:{\overline{b}}=x}{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b,\end{aligned}$$ which satisfies[^5] $$\begin{aligned}
{\label{eq:G-delta-bd}}
{{\langle \varphi_y\varphi_x \rangle}}_\Lambda\leq\delta_{y,x}+\sum_{b:{\overline{b}}=x}\,
\sum_{\substack{{\partial}{{\bf n}}=y{\vartriangle}x\\ n_b\text{ odd}}}\frac{w_\Lambda
({{\bf n}})}{Z_\Lambda}=\delta_{y,x}+\sum_{b:{\overline{b}}=x}\tau_b\sum_{\substack{
{\partial}{{\bf n}}=y{\vartriangle}{\underline{b}}\\ n_b\text{ even}}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}
\leq\delta_{y,x}+\tilde G_\Lambda(y,x).\end{aligned}$$ Let $$\begin{aligned}
{\label{eq:psi-def}}
\psi_\Lambda(y,x)=\sum_{j=0}^\infty\big(\tilde G_\Lambda^2\big)^{*j}(y,x)
&\equiv\delta_{y,x}+\sum_{j=1}^\infty\sum_{\substack{u_0,\dots,u_j\\ u_0=
y,\;u_j=x}}\prod_{l=1}^j\tilde G_\Lambda(u_{l-1},u_l)^2,\end{aligned}$$ and define (see the first line in Figure \[fig:P-def\]) $$\begin{aligned}
P_\Lambda^{{\scriptscriptstyle}(1)}(v_1,v'_1)&=2\big(\psi_\Lambda(v_1,v'_1)-
\delta_{v_1,v'_1}\big)\,{{\langle \varphi_{v_1}\varphi_{v'_1} \rangle}}_\Lambda,
{\label{eq:P1-def}}\\[5pt]
P_\Lambda^{{\scriptscriptstyle}(j)}(v_1,v'_j)&=\sum_{\substack{v_2,\dots,v_j\\
v'_1,\dots,v'_{j-1}}}\bigg(\prod_{i=1}^j\big(\psi_\Lambda(v_i,
v'_i)-\delta_{v_i,v'_i}\big)\bigg){{\langle \varphi_{v_1}\varphi_{
v_2} \rangle}}_\Lambda{{\langle \varphi_{v_2}\varphi_{v'_1} \rangle}}_\Lambda{\nonumber}\\
&\qquad\qquad\times\bigg(\prod_{i=2}^{j-1}{{\langle \varphi_{v'_{i
-1}}\varphi_{v_{i+1}} \rangle}}_\Lambda{{\langle \varphi_{v_{i+1}}\varphi_{
v'_i} \rangle}}_\Lambda\bigg){{\langle \varphi_{v'_{j-1}}\varphi_{v'_j} \rangle}}
_\Lambda\qquad(j\ge2),{\label{eq:Pj-def}}\end{aligned}$$ where the empty product for $j=2$ is regarded as 1.
$$\begin{gathered}
P_\Lambda^{{\scriptscriptstyle}(1)}(v_1,v'_1)=\raisebox{-7pt}{\includegraphics[scale=.1]
{P0}}\qquad
P_\Lambda^{{\scriptscriptstyle}(2)
| 2,780
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|
ls of the form $G_\muhat(q)$ which have integer coefficients (see Remark \[G\]).
Example: Hilbert Scheme of $n$ points on $\C^\times\times\C^\times$ {#Hilbert}
===================================================================
Throughout this section we will have $g=k=1$ and $\muhat$ will be either the partition $(n)$ or $(n-1,1)$.
In this section we illustrate our conjectures and formulas in these cases.
Hilbert schemes: Review
-----------------------
For a nonsingular complex surface $S$ we denote by $S^{[n]}$ the Hilbert scheme of $n$ points in $S$. Recall that $S^{[n]}$ is nonsingular and has dimension $2n$.
We denote by $Y^{[n]}$ the Hilbert scheme of $n$ points in $\C^2$.
Recall (see for instance [@NakHilbert §5.2]) that $h_c^i(Y^{[n]})=0$ unless $i$ is even and that the compactly supported Poincaré polynomial $P_c(Y^{[n]};q):=\sum_ih_c^{2i}(Y^{[n]})q^i$ is given by the following explicit formula
$$\sum_{n\geq 0}P_c(Y^{[n]};q)T^n=\prod_{m\geq 1}\frac{1}{1-q^{m+1}T^m}.\label{Yn}$$
which is equivalent to
$$\Log\left(\sum_{n\geq 0}q^{-n}\cdot P_c(Y^{[n]};q)T^n\right)=\sum_{n\geq 1}q T^n.\label{Ynbis}$$
For $n\geq 2$, consider the partition $\mu=(n-1,1)$ of $n$ and let $C$ be a semisimple adjoint orbit of $\gl_n(\C)$ with characteristic polynomial of the form $(-1)^n(x-\alpha)^{n-1}(x-\beta)$ with $\beta=-(n-1)\alpha$ and $\alpha\neq 0$. Consider the variety
$$\calV_{(n-1,1)}=\{(a,b,X)\in (\gl_n)^2\times C\,|\,
[a,b]+X=0\}.$$
The group $\GL_n$ acts on $\calV_{(n-1,1)}$ diagonally by conjugating the coordinates. This action induces a free action of $\PGL_n$ on $\calV_{(n-1,1)}$ and we put $$\calQ_{(n-1,1)}:=\V_{(n-1,1)}/\!/\PGL_n={\rm
Spec}\left(\C[\V_{(n-1,1)}]^{\PGL_n}\right).$$ The variety $\calQ_{(n-1,1)}$ is known to be nonsingular of dimension $2n$ (see for instance [@hausel-letellier-villegas §2.2] and the references therein).
We have the following well-known theorem.
The two varieties $\calQ_{(n-1,1)}$ and $Y^{[n]}$ have isomorphic cohomology supporting pure mixed Hodge structures. \
| 2,781
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|
obtain the equations $$(z_j)_1+(z_j)_1^2=0, ~~~ (x_j)_1=0, ~~~(z_j)_1+(x_j)_2=0.$$ By combining all these, we see that $F_j$ is isomorphic to $ \mathbb{A}^{1} \times \mathbb{Z}/2\mathbb{Z}$ as a $\kappa$-variety.
**
We finally prove Lemma \[l46\].
We start with the following short exact sequence $$1\rightarrow \tilde{G}^1 \rightarrow \mathrm{Ker~}\varphi\rightarrow\mathrm{Ker~}\varphi/\tilde{G}^1\rightarrow 1.$$ It is obvious that $\mathrm{Ker~}\varphi$ is smooth by Theorems \[ta4\] and \[ta6\]. $\mathrm{Ker~}\varphi$ is also unipotent since it is a subgroup of a unipotent group $\tilde{M}^+$. Since $\tilde{G}^1$ is connected by Theorem \[ta4\], the component group of $\mathrm{Ker~}\varphi$ is the same as that of $\mathrm{Ker~}\varphi/\tilde{G}^1$ by Lemma A.10 of [@C2]. Moveover, the dimension of $\mathrm{Ker~}\varphi$ is the sum of the dimension of $\tilde{G}^1$ and the dimension of $\mathrm{Ker~}\varphi/\tilde{G}^1$. This completes the proof.
Examples {#App:AppendixB}
========
In this appendix, we provide an example with a unimodular lattice $(L, h)$ of rank 1. The structure of this appendix is parallel to that of Appendix B of [@C2] and thus many sentences of loc. cit. are repeated without comment. Let $L$ be $B\textit{e}$ of rank 1 hermitian lattice with hermitian form $h(le, l'e)=\sigma(l)l'$. With this lattice, we construct the smooth integral model and its special fiber and compute the local density.
Naive construction (without using our technique) {#nc}
------------------------------------------------
We first construct the smooth integral model and its special fiber, without using any technique introduced in this paper. If we write an element of $L$ as $x+\pi y$ where $x, y\in A$, then it is easy to see that a naive integral model $\underline{G}'$ is $\mathrm{Spec~}A[x,y]/(x^2+(\pi +\sigma(\pi))xy+\pi\sigma(\pi)y^2-1)$. As mentioned in Section \[Notations\], we may assume that $\pi +\sigma(\pi)=0$ and $\pi\sigma(\pi)=-2\delta$ for a unit $\delta\in A$ such that $\delta\equiv 1 \mathrm{~mod~}2$.
| 2,782
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Adult 26 (96) 148 (94) 47 (84) 221 (92) 283 (92) 648 (91)
Paediatric 1 (4) 10 (6) 9 (16) 20 (8) 23 (8) 63 (9)
**Participants**
Patients 18 (67) 142 (90) 45 (80) 205 (85) 282 (92) 612 (86)
Healthy volunteers 8 (30) 11 (7) 3 (5) 22 (9) 24 (8) 99 (14)
Both 1 (4) 5 (3) 8 (14) 14 (6) 0 0
**Published in peer-reviewed journal** 15 (56) 74 (47) 29 (52) 118 (49) 180 (59) 387 (54)
^1^ approved by 5 research ethics committees in Switzerland and Canada \[[@pone.0165605.ref001]\].
^2^ includes 37 RCTs and 13 non-RCTs with industry funding but no industry involvement, i. e. in study planning, management or analysis of data.
All percentages (in brackets) refer to the total number of the respective column
Abbreviations: NPSs, non-randomised prospective studies; RCTs, randomised controlled trials; IQR, interquartile range.
Study discontinuation {#sec011}
---------------------
Overall, NPSs were less frequently discontinued than RCTs (14% versus 27%, missing excluded, p\<0.001) (Tables [2](#pone.0165605.t002){ref-type="table"} and [3](#pone.0165605.t003){ref-type="table"}). Sensitivity analyses using different assumption for missing data only slightly changed these proportions and the differences between N
| 2,783
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| 0.773818
|
github_plus_top10pct_by_avg
|
cture grammar. We construct the grammar $G'=(V',\Sigma,(S,1),R')$ with capacity function $\mathbf{1}$ and $$\begin{aligned}
V'&=& \{(A,i) {:}A \in V, 1\leq i\leq \kappa(A)\},\\
R'&=& \{\alpha' \to \beta' {:}\alpha' \in h(\alpha), \beta' \in h(\beta), \mbox{ for some } \alpha \to \beta \in R\},
\end{aligned}$$ where $h:(V\cup \Sigma)^* \to (V' \cup \Sigma)^*$ is the finite substitution defined by $h(a)=\{a\}$, for $a \in \Sigma$, and , for $A \in V$.
It can be shown by induction on the number of derivation steps that $S \!{\Rightarrow}^*_{G,\kappa}\! \alpha$ holds iff , for some $\alpha' \in h(\alpha)$.
\[lem:GScbSubsetMATfin\] $\mathbf{GS}_{{\mathit{cb}}}\subseteq \mathbf{MAT}_{{\mathit{fin}}}$.
Consider some language $L\in \mathbf{GS}_{{\mathit{cb}}}$ and let $G=(V,\Sigma,S,R,\mathbf{1})$ be a capacity-bounded phrase structure grammar (due to Ginsburg and Spanier) such that $L=L(G)$. A word $\alpha\in (V\cup \Sigma)^*$ can be uniquely decomposed as $$\alpha=x_1 \beta_1 x_2 \beta_2 \cdots x_n \beta_n x_{n+1}, x_1,x_{n+1} \in \Sigma^*, x_2,\ldots,x_n \in \Sigma^+, \beta_1,\ldots, \beta_n\in V^+.$$ The subwords $\beta_i$ are referred to as the *maximal nonterminal blocks* of $\alpha$. Note that the length of a maximal block in any sentential form of a derivation in $G$ is bounded by $|V|$. We will first construct a capacity-bounded grammar $G'$ with $L(G')=L$ such that all words of $L$ can be derived in $G'$ by rewriting a maximal nonterminal block in every step. Let $G'=(V,\Sigma,S,R',\mathbf{1})$ where $$\begin{aligned}
R'&=& \{\alpha_1 \alpha \alpha_2 \to \alpha_1 \beta \alpha_2 {:}\alpha \to \beta \in R, \alpha_1,\alpha_2 \in V^*,
|\alpha_1 \alpha \alpha_2|_A \leq 1, \mbox{ for all } A\in V\}.
\end{aligned}$$ The inclusion $L(G) \subseteq L(G')$ is obvious since $R\subseteq R'$. On the other hand, any derivation step in $G'$ can be written as $\gamma_1 \underline{\alpha_1 \alpha \alpha_2} \gamma_2 {\Rightarrow}_{G'}
\gamma_1 \underline{\alpha_1 \beta \alpha_2} \gamma_2$, where $\alpha
| 2,784
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| 3,598
| 0.7713
|
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|
{erf}^{-1}\left(1-\frac{2}{e\Psi}\right)\right),\end{aligned}$$ where $\beta\simeq0.5772$ denotes the Euler’s constant.
See Appendix \[App:proofaveGlsn\]
Based on , we further obtain the limiting behavior of the average throughput gain when $\Psi$ becomes large in the following corollary.
\[Cor:aveGlsngrO\] As $\Psi\rightarrow\infty$, $G_{\bar{R}}(\Psi)$ is asymptotically equivalent to $$\label{eq:galarpsiasy}
G_{\bar{R}}(\Psi)\sim\frac{\sqrt{2{\sigma^2_{R_\psi}}}}{{\bar{R}_{\psi}}}\sqrt{\ln(\Psi)}.
$$
See Appendix \[App:proofCor:aveGlsngrO\]
From we find that $G_{\bar{R}}(\Psi)=O\left(\sqrt{\ln(\Psi)}\right)$ as $\Psi\rightarrow\infty$. Thus, the growth of the average throughput from adding reconfiguration states becomes small when the number of reconfiguration states is already large, although having more distinct reconfiguration states always benefits the average throughput.
Outage Throughput Gain
----------------------
In the above analysis, we focused on the performance gain in terms of the average throughput. However, it is insufficient to use the average throughput as the sole measure of the rate performance of the systems with multiple antennas. For scenarios where the channel remains (quasi) static during the transmission, it is appropriate to evaluate the system performance by the outage throughput, since every possible target transmission rate is associated with an unavoidable probability of outage. In the following, we analyze the performance gain of employing the reconfigurable antennas in terms of the outage throughput.
For a given target rate $R$, an outage event happens when the maximum achievable throughput is less than the target rate, and the outage probabilities for the systems without and with the reconfigurable antennas are given by ${\mathbb{P}}(R_{{\widehat{\psi}}}<R)$ and ${\mathbb{P}}(R_{\psi}<R)$, respectively. At a required outage level $0<\epsilon<1$, the outage throughputs for the systems with and without the reconfigurable antennas are given by [@Tse_05_Fundamentals] $
| 2,785
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| 1,064
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| 1,905
| 0.784541
|
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|
\-
Hospital stay
\<48 h 47 7 (14.9) 40 (85.1) 63.67 (26.3-179.2) 107.96 (31.37-371.60)
48-96 h 19 12 (63.2) 7 (36.8) 6.50 (2.05-20.33) 11.66 (3.05-44.52)
96+ h 255 234 (91.8) 21 (8.2) 1 (Ref) 1 (Ref)
Systolic BP
≥ 140 mmHg 126 115 (91.3) 11 (8.7) 1 (Ref) 1 (Ref)
\<140 mmHg 193 137 (71.0) 56 (29.0) 4.27 (2.05-9.09) 5.39 (1.92-15.17)
NA 2 1 (50) 1 (50) \- \-
No deaths in this category, results inconsistent, NA: Data not available, Adjusted odds ratio obtained by Binary Logistic regression analysis
AMI patients reporting after more than 24 h of onset were 4.27 times more likely to die during hospital stay compared to those reporting within 6 h of onset of AMI. Those having 'at admission' systolic blood pressure of less than 140 mmHg were 5.39 times more likely to die than those with at admission blood pressure measurement of more than 140 mmHg.
Kakade *et al.*([@CIT1]) found that on logistic regression analysis; age, gender, place of residence, time gap in treatment, and hospital treatment were the significant variables. Jiang *et al.*([@CIT2]) using a multivariate logistic regression model, identified age, history of hypertension, and dia
| 2,786
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| null | null |
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|
ns (e.g. [@GM1 Exercize II.1.6], [@L Section 6], or [@FT A.2], retold in [@Ka Section 1.4]). All the descriptions are equivalent. Objects in ${\operatorname{Fun}}(\Lambda,k)$ are usually called [*cyclic vector spaces*]{}.
The cyclic category $\Lambda$ is related to the more familiar [*simplicial category*]{} $\Delta^{opp}$, the opposite to the category $\Delta$ of finite non-empty linearly ordered sets. To understand the relation, consider the discrete cofibration $\Lambda_{[1]}/\Lambda$ associated to the functor $V:\Lambda \to
{\operatorname{Sets}}$ – equivalently, $\Lambda_{[1]}$ is the category of objects $[n]$ in $\Lambda$ eqipped with a map $[1] \to [n]$. Then it is easy to check that $\Lambda_{[1]}$ is equivalent to the $\Delta^{opp}$. From now on, we will abuse the notation and identify $\Lambda_{[1]}$ and $\Delta^{opp}$. We then have a natural projection $\Delta^{opp} = \Lambda_{[1]} \to \Lambda$, $\langle
[n],v \rangle \mapsto [n]$, which we denote by $j:\Delta^{opp} \to
\Lambda$.
For any cyclic $k$-vector space $E \in {\operatorname{Fun}}(\Lambda,k)$, we have its restriction $j^*E \in {\operatorname{Fun}}(\Delta^{opp},E)$, a simplicial vector space. One defines the cyclic homology $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E)$ and the Hochschild homology $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$ of $E$ by $$HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E) {\overset{\text{\sf\tiny def}}{=}}H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda,E), \qquad HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E) {\overset{\text{\sf\tiny def}}{=}}H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},j^*E).$$ By , we have a natural map $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E) \to
HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E)$ (moreover, since $j:\Delta^{opp} \to \Lambda$ is a discrete cofibration, the Kan extension $j_!$ is exact, so that we have $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E) \cong HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(j_!j^*E)$, and the natural map is induced
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|
tion of a particle with mass $m$ in the spherically symmetric potential field (also see [@Andrianov.hep-th/9404061; @Bagrov.quant-ph/9804032]). The spherical symmetry of the potential allows to reduce this problem to the one-dimensional problem about the motion of this particle in the radial field $V(r)$, defined on the positive semiaxis of $r$, where wave function of such system looks like: $$\psi(r, \theta, \varphi) =
\displaystyle\frac{\chi_{nl}(r)}{r}
Y_{lm} (\theta, \varphi),
\label{eq.2.1.1}$$ and the radial Schrödinger equation has a form: $$H \chi_{nl}(r) =
-\displaystyle\frac{\hbar^{2}}{2m}
\displaystyle\frac{d^{2} \chi_{nl}(r)}{dx^{2}} +
\biggl(V_{n}(r) +
\displaystyle\frac{l(l+1) \hbar^{2}}{2mr^{2}} \biggr)
\chi_{nl}(r) =
E_{n} \chi_{nl}(r)
\label{eq.2.1.2}$$ and differs from the one-dimensional Schrödinger equation by a presence of a centrifugal term. One can reduce this equation to one-dimensional one by replacement: $$\bar{V}_{n}(r) =
V_{n}(r) + \displaystyle\frac{l(l+1) \hbar^{2}}{2mr^{2}}.
\label{eq.2.1.3}$$
As in the one-dimensional case, one can introduce operators $A_{1}$ and $A_{1}^{+}$: $$\begin{array}{ll}
A_{1} =
\displaystyle\frac{\hbar}{\sqrt{2m}}
\displaystyle\frac{d}{dr}
+ W_{1}(r), &
A_{1}^{+} =
-\displaystyle\frac{\hbar}{\sqrt{2m}}
\displaystyle\frac{d}{dr}
+ W_{1}(r),
\end{array}
\label{eq.2.1.4}$$ where $W_{1}(r)$ is a function, defined in the positive semiaxis $0 \le r < +\infty$ and continuous in it with an exception of some possible points of discontinuity. Then one can determine an interdependence between two hamiltonians of the propagation of the particle with mass $m$ in the fields $\bar{V}_{1}(r)$ and $\bar{V}_{2}(r)$: $$\begin{array}{l}
H_{1} = A_{1}^{+} A_{1} + C_{1} =
-\displaystyle\frac{\hbar^{2}}{2m}
\displaystyle\frac{d^{2}}{dr^{2}}
+ \bar{V}_{1}(r), \\
H_{2} = A_{1} A_{1}^{+} + C_{1} =
-\displaystyle\frac{\hbar^{2}}{2m}
\displaystyle\frac{d^{2}}{dr^{2}}
+ \bar{V}_{2}(r),
\end{array}
\label{eq.2.1.5}$$ wher
| 2,788
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| 2,259
| 4,153
| 0.767717
|
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|
utting all the previous calculations , , , , together, one obtains $A = - \sum_{P \in S} \beta_{\cC,P}^{(k)}$ where $$\label{eq:beta}
\beta_{\cC,P}^{(k)}\! =\! \alpha_{\cC,P}^{(k)}
+ \dim\! \frac{\mathcal{O}_{\PP^2_w,P}\left( kH\! +\! K_{\PP^2_w}\! -\! \mathcal{C}^{(k)} \right)}{\mathcal{M}_{\mathcal{C},P}^{(k)}}
+ \!\!\!\!\sum_{Q \in \pi^{-1}(P)}\!\!\!\!\!\! R_{Y,Q}( L^{(k)} )
- R_{{\PP^2_w},P}\! \left(\! -kH \!+ \!\mathcal{C}^{(k)}\! \right)\!,$$ and, finally, $$\label{eq:H1-coker-beta}
\dim H^1(Y,\cO_Y(L^{(k)})) = \dim\operatorname{coker}\pi^{(k)} - \sum_{P \in S} \beta_{\cC,P}^{(k)}.$$
The second part of the proof consists in showing that $\beta_{\cC,P}^{(k)}=0$ for any $P\in S$. Without loss of generality one can assume $P=[0:0:1]$. The proof is analogous for the other singular points of $\PP^2_w$. For the remaining points in $S$ the same proof works changing $w_2$ by 1.
\[step1\] Note that $\beta_{\cC,P}^{(k)}$ only depends on the topological type of $(\mathcal{C},P) \subset (\PP^2_w,P) = \frac{1}{w_2}(w_0,w_1)$, $\frac{k}{d}\in\QQ$, and $k$ (resp. $d_1,\ldots,d_r,|w|$) modulo $w_2$.
The result is trivial for the last summand of . The key point for the first two summands is to show that the term ${\left \lfloor \frac{k m_\v}{d}-k\b_\v \right \rfloor}+k\b_\v$ only depends on $\frac{k}{d}$, $m_\v$, $\b_\v$ (which depend on the local topological type both of the surface and the curve), and $k$ modulo $w_2$ as opposed to $k$. The latter is a consequence of the fact that $\b_\v\in \frac{1}{w_2}\ZZ$. Finally, for the third summand in one needs to use .
For any given $P\in S$ we can apply Lemma \[lemma:global-realization\] to $(\cC,P)$ and obtain a global generic curve $\cD\subset \PP^2_{w}$ such that $(\cD,P)=(\cC,P)$.
One applies to the new curve $\cD$. Note that the only singularity contributing to the right-hand side is $P\in \cD$ and $(\cD,P)$ has the same topological type as $(\cC,P)$. For a curve $\cD$ of big enough degree, the cokernel in is zero. In addition, by [@Nori-zariski Theorem II,
| 2,789
| 2,426
| 1,783
| 2,556
| null | null |
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|
rho _{h}(1+\delta )}(c(r_h)\rho )\leq
\frac{Q^{\rho_h+\varepsilon}}{(\lambda t)^{(2h+q+d/p_{\ast })\rho_h(1+\delta )}}
\leq
C\times \frac{e^{2c_\kappa \rho}}{(\lambda t)^{(q+2d/p_{\ast })(1+\delta )}}.$$Since $\rho \geq 1$ we conclude that $$\left\Vert p^{\eta ,\kappa }\right\Vert _{q,p}\leq
C\times \frac{e^{2c_\kappa \rho}}{(\lambda t)^{(q+2d/p_{\ast })(1+\delta )}},$$$C$ denoting a constant which is independent of $\rho$. We take now $p=2d+\varepsilon $ and, using now Morrey’s inequality $$\left\Vert p^{\eta ,\kappa }\right\Vert _{q,\infty }\leq \left\Vert p^{\eta
,\kappa }\right\Vert _{q+1,p}\leq
C\times \frac{e^{2c_\kappa \rho}}{(\lambda t)^{(q+2d)(1+\delta )}}.$$This proves (\[J10\]). $\square $
Appendix
========
Weights {#app:weights}
-------
We denote$$\psi _{k}(x)=(1+\left\vert x\right\vert ^{2})^{k}. \label{n1}$$
\[Psy1\]For every multi-index $\alpha $ there exists a constant $%
C_{\alpha }$ such that $$\Big|\partial ^{\alpha }\Big(\frac{1}{\psi _{k}}\Big)\Big\vert \leq \frac{%
C_{\alpha }}{\psi _{k}}. \label{n2}$$Moreover, for every $q$ there is a constant $C_{q}\geq 1$ such that for every $f\in C_{b}^{\infty }({\mathbb{R}}^{d})$$$\frac{1}{C_{q}}\sum_{0\leq |\alpha | \leq q}\Big\vert \partial ^{\alpha }%
\Big(\frac{f}{\psi _{k}}\Big)\Big\vert \leq \sum_{0\leq \left\vert \alpha
\right\vert \leq q}\frac{1}{\psi _{k}}\left\vert \partial ^{\alpha
}f\right\vert \leq C_{q}\sum_{0\leq \left\vert \alpha \right\vert \leq q}%
\Big\vert \partial ^{\alpha }\Big(\frac{f}{\psi _{k}}\Big)\Big\vert .
\label{n3}$$
**Proof**. One checks by recurrence that $$\partial ^{\alpha }\Big(\frac{1}{\psi _{k}}\Big)=\sum_{q=1}^{\left\vert
\alpha \right\vert }\frac{P_{\alpha ,q}}{\psi _{k+q}}$$where $P_{\alpha ,q}$ is a polynomial of order $q.$ And since$$\frac{(1+\left\vert x\right\vert )^{q}}{(1+\left\vert x\right\vert
^{2})^{q+k}}\leq \frac{C}{(1+\left\vert x\right\vert ^{2})^{k}}$$the proof (\[n2\]) is completed. In order to prove (\[n3\]) we write$$\partial ^{\alpha }\Big(\frac{f}{\psi _{k}}\Big)=\frac{1}{\psi _
| 2,790
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inary results about regularity. Namely, in Section [sect:3.1]{} we recall and develop some results concerning regularity of probability measures, based on interpolation type arguments, coming from [@[BC]]. These are the main instruments used in the paper. In Section \[sect:3.2\] we prove a regularity result which is a key point in our approach. In fact, it allows to handle the multiple integrals coming from the application of a Lindeberg method for the decomposition of $P_t-P^n_t$. The results stated in Section \[sect:NotRes\] are then proved in the subsections in which Section \[sect:proofs\] is split. Finally, in Appendix \[app:weights\], \[app:semi\] and \[app:ibp\] we prove some technical results used in the paper.
Notation and main results {#sect:NotRes}
=========================
Notation {#sect:notation}
--------
For a multi-index $\alpha =(\alpha _{1},...,\alpha _{m})\in \{1,...,d\}^{m}$ we denote $\left\vert \alpha \right\vert =m$ (the length of the multi-index) and $\,\partial ^{\alpha }$ is the derivative corresponding to $\alpha ,$ that is $\partial ^{\alpha _{m}}\cdots\partial ^{\alpha _{1}}$, with $%
\partial^{\alpha_i}=\partial_{x_{\alpha_i}}$. For $f\in C^{\infty }({\mathbb{%
R}}^{d}\times {\mathbb{R}}^{d})$, $(x,y)\in {\mathbb{R}}^{d}\times {\mathbb{R%
}}^{d}$ and two multi-indexes $\alpha$ and $\beta$, we denote by $\partial
_{x}^{\alpha }$ the derivative with respect to $x$ and by $\partial
_{y}^{\alpha }$ the derivative with respect to $y$.
Moreover, for $f\in C^{\infty }({\mathbb{R}}^{d})$ and $q\in {\mathbb{N}}$ we denote$$\left\vert f\right\vert _{q}(x)=\sum_{0\leq \left\vert \alpha \right\vert
\leq q}\left\vert \partial ^{\alpha }f(x)\right\vert . \label{NOT1}$$If $f$ is not a scalar function, that is, $f=(f^{i})_{i=1,\ldots ,d}$ or $%
f=(f^{i,j})_{i,j=1,\ldots ,d}$, we denote $\left\vert f\right\vert
_{q}=\sum_{i=1}^{d}\left\vert f^{i}\right\vert _{q}$ respectively $%
\left\vert f\right\vert _{q}=\sum_{i,j=1}^{d}\left\vert f^{i,j}\right\vert
_{q}.$
We will work with the weights $$\ps
| 2,791
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|
ation maps is an unconventional task, we visualize the reconstructions in order to inspect their quality. For the activations obtained from the first convolutional layer seen in Figure \[fig:conv1\_layer\_reconstructions\], it is obvious that the VAEs are effective at reconstructing the activation maps. The only potential issue is that the background for some of the reconstructions is slightly more gray than in the original activation maps. For the most part, this is also the case for the second convolutional layer activation maps. However, in the first, fourth, and sixth rows of Figure \[fig:conv2\_layer\_reconstructions\], there is an obvious addition of arbitrary pixels that were not present in the original activation maps.
![First convolutional layer feature map visualization for LeNet on MNIST. Original feature maps are on the left, VAE reconstructed features maps are on the right. As is seen, reconstructions are of very high quality.[]{data-label="fig:conv1_layer_reconstructions"}](figures/conv1_layer_reconstructions.pdf){width="5cm" height="15cm"}
![Second convolutional layer feature map visualization for LeNet on MNIST. Original feature maps are on the left, VAE reconstructed features maps are on the right. As is seen, reconstructions are of very high quality.[]{data-label="fig:conv2_layer_reconstructions"}](figures/conv2_layer_reconstructions.pdf){width="5cm" height="15cm"}
Cleaning Adversarial Examples {#cleaning-adversarial-examples .unnumbered}
-----------------------------
The intuitive notion that VAEs or filters remove adversarial noise can be tested empirically by comparing the distance between adversarial examples and their unperturbed counterparts. In figure \[fig:lenet\_fgs\_distances\], the evolution of distances between normal an adversarial examples can be seen. When the classifier is undefended, the distance increases significantly with the depth of the network, and this confirms the hypothesis that affine transformations amplify noise. However, it is clear that applying our defense has
| 2,792
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dots,a_n)=T\bar{f_i}(a_1,\ldots,a_n).$$
From the last equality we obtain $\bar{f_i}(a_1,\ldots,a_n)=0$, so $A\vDash\bar{f_i}$ and $\bar{f_i}\in T_0({\mathrm{Var}})$. By the previous proposition $f_i\in T_0({\mathrm{Di}}{\mathrm{Var}})$.
We recall that $f$ is called a multilinear s-identity (in the case of ordinary algebras) if $$f\in T_0({\mathcal{H}}{\mathrm{SJ}})\setminus T_0({\mathrm{Jord}}):={\mathrm{SId}}.$$
A similar notion can be introduced for dialgebras [@Br:09] $$f\in T_0({\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}})\setminus T_0({\mathrm{Di}}{\mathrm{Jord}}):={\mathrm{Di}}{\mathrm{SId}}.$$
\[thm:CorrespSId\]
1. Let $g = g(x_1,\ldots,x_n)\in{\mathrm{SId}}$. Then $\Psi^{x_i}_{\mathrm{Alg}}\,g\in{\mathrm{Di}}{\mathrm{SId}}$ for all $i=1,\ldots,n$.
2. Let $f=f(x_1,\ldots,x_n)\in{\mathrm{Di}}{\mathrm{SId}}$, $f=f_1+\ldots+f_n$ by a central letter. Then there exists $j\in\{1,\,\ldots,\,n\}$ such that $\bar{f_j}\in{\mathrm{SId}}$.
We prove the statement 1. Let $g\in{\mathrm{SId}}$, hence by the definition ${\mathrm{SId}}$ we have $g\in T_0({\mathcal{H}}{\mathrm{SJ}})$ and $g\not\in T_0({\mathrm{Jord}})$. Proposition \[prop:PsiDiVarVar\] implies $\Psi^{x_i}_{\mathrm{Alg}}\,g\in T_0({\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}})$, $\Psi^{x_i}_{\mathrm{Alg}}\,g\not\in
T_0({\mathrm{Di}}{\mathrm{Jord}})$. It follows from the equality of varieties ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}={\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$ that $\Psi^{x_i}_{\mathrm{Alg}}\,g\in {\mathrm{Di}}{\mathrm{SId}}$.
For proving the statement 2 consider $f\in{\mathrm{Di}}{\mathrm{SId}}$. By the definition of ${\mathrm{Di}}{\mathrm{SId}}$ and Theorem \[thm:EqOfVarDialg\] we have $f\in T_0({\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}})=T_0({\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}})$ and $f\not\in
T_0({\mathrm{Di}}{\mathrm{Jord}})$. It follows from $f\in T_0({\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}})$ by Proposition \[prop:f1fnDiVarDiVar\] that $f_i\in T_0({\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}})$ for all $i$. It follows from $f\not\in T_0({\mathrm{Di
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roups nor virtually affine groups.
This paper is organized as follows: in Section \[s:recall\] we review some general facts and introduce the notation used throughout the text. In Section \[s:gever\] we describe some properties of the Veronese curve which are useful for our purposes. In Section \[s:chvh\] we characterize the complex hyperbolic subgroups that leave invariant a Veronese curve. In Section \[s:riv\] we depict those real hyperbolic subgroups leaving invariant a Veronese curve. Finally, in Section \[s:rep\] we show that every discrete compact surface group in $\PO(2,1)^+$ admits a deformation in $\PSL(3,\Bbb{C})$ which is not conjugate to a complex hyperbolic subgroup and has non-empty Kulkarni region of discontinuity.
Preliminaries {#s:recall}
=============
Projective geometry
-------------------
The complex projective space $\mathbb{P}^2_{\mathbb {C}}$ is defined as $$\mathbb{P}^{2}_{\mathbb {C}}=(\mathbb {C}^{3}\setminus \{0\})/\Bbb{C}^*,$$ where $\Bbb{C}^*$ acts by the usual scalar multiplication. This is a compact connected complex $2$-dimensional manifold. If $[\mbox{}]:\mathbb{C}^{3}\setminus\{0\}\rightarrow\mathbb{P}^{2}_{\mathbb{C}}$ is the quotient map, then a non-empty set $H\subset\mathbb{P}^2_{\mathbb{C}}$ is said to be a line if there is a $\mathbb{C}$-linear subspace $\widetilde{H}$ in $\mathbb{C}^{3}$ of dimension $2$ such that $[\widetilde{H}\setminus \{0\}]=H$. If $p,q$ are distinct points then $\overleftrightarrow{p,q}$ is the unique complex line passing through them. In this article, $e_1,e_2,e_{3}$ will denote the standard basis for $\Bbb{C}^{3}$.
Projective transformations
----------------------------
The group of projective automorphisms of $\mathbb{P}^{2}_{\mathbb{C}}$ is defined as $$\PSL(3, \mathbb {C}) \,:=\, \GL({3}, \Bbb{C})/\Bbb{C}^*,$$ where $\Bbb{C}^*$ acts by the usual scalar multiplication. Then $\PSL(3, \mathbb{C})$ is a Lie group acting by biholomorphisms on $\Bbb{P}^2_{\Bbb{C}}$; its elements are called projective transformations. We denote by $[[\mbox{ }]]
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entries of $\Sigma$. Assume that $k \geq u^2$. Then, $$\label{eq::B-and-lambda}
B= \sup_{P \in \mathcal{P}_n^{\mathrm{OLS}} } \max_j \|G_j(\psi(P)) \| \leq C \frac{ \sqrt{k} }{u^2},\ \ \
\overline{H}=\max_j \sup_{P \in \mathcal{P}_n^{\mathrm{OLS}}} \| H_j(\psi(P))\|_{\mathrm{op}} \leq C
\frac{k}{u^3},$$ and $$\label{eq:sigmamin}
\underline{\sigma} = \inf_{P \in \mathcal{P}^{\mathrm{OLS}}_n} \min_j \sqrt{ G_j V G_j^\top}
\geq \frac{ \sqrt{v } }{ U },$$ where $C>0$ depends on $A$ only.
[**Remark.**]{} The assumption that $k \geq u^2$ is not actually needed but this is the most common case and it simplifies the expressions a bit.
[**Proof of .**]{} The maximal length is of the sides of $\tilde{C}_n$ is $$2 \max_{j \in {\widehat{S}}} z_{\alpha/(2k)}
\sqrt{\frac{ \hat\Gamma_{{\widehat{S}}}(j,j)}{n}} \leq 2 \max_{j \in {\widehat{S}}} z_{\alpha/(2k)}
\sqrt{\frac{ \Gamma_{{\widehat{S}}}(j,j) + \left| \hat\Gamma(j,j)-\Gamma(j,j)\right|}{n}}.$$ By and Equation , the event that $$\max_{ j,l \in {\widehat{S}}} \left| \hat\Gamma(j,l)-\Gamma(j,l)\right| \leq C \frac{k^{3/2}}{u_n^3 u^2} \overline{v} \sqrt{ \frac{k^2 \log n}{n}}$$ holds with probability at least $1 - \frac{2}{n}$ and for each $P \in \mathcal{P}_n^{\mathrm{OLS}}$, where $C > 0$ depends on $A$ only. Next, letting $G = G(\psi_{{\widehat{S}}})$ and $V = V_{{\widehat{S}}}$, we have that, for each $j \in {\widehat{S}}$ and $P \in \mathcal{P}_n^{\mathrm{OLS}}$, $$\Gamma_{{\widehat{S}}}(j,j) = G_j V G_j^\top \leq \|G_j\|^2 \lambda_{\max}(V) \leq B^2 \overline{v} \leq C \frac{k }{u^4} \overline{v}$$ where $G_j$ denotes the $j^{\mathrm{th}}$ row of $G$ and, as usual, $C>0$ depends on $A$ only. The second inequality in the last display follows from property 3. in and by the definition of $B$ in , while the third inequality uses the first bound in Equation . The result follows from combining the previous bounds and the fact that $z_{\alpha/(2k)} = O \left( \sqrt{ \log k} \right)$. $\Box$
[**Proof of .**]{} We condition on $\mathcal{D}_{1,n}$ and th
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|
be estimated as $$\begin{aligned}
{\label{eq:2nddec-bd:n=jbd}}
&\sup_y\sum_{z,z',x}|x|^2\tau_{y,z}\big(\delta_{z,z'}+\tilde
G_\Lambda(z,z')\big)P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}(z',x){\nonumber}\\
&=\sup_y\sum_{z,z',x}\tau_{y,z}\big(\delta_{z,z'}+\tilde G_\Lambda(
z,z')\big)\,{{\langle \varphi_{z'}\varphi_o \rangle}}_\Lambda\,{{\langle \varphi_{z'}
\varphi_x \rangle}}_\Lambda^2\,|x|^2{{\langle \varphi_o\varphi_x \rangle}}_\Lambda{\nonumber}\\
&\leq\sup_y\Big((\tau D*G)(y)+(\tau D*G)^{*2}(y)\Big)\,G^{*2}(o)\,
\bar G^{{\scriptscriptstyle}(2)}=d\sigma^2O(\theta_0)^2,\end{aligned}$$ where $\bar G^{{\scriptscriptstyle}(s)}$ is given by [(\[eq:GbarWbar\])]{}. The other contributions from $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(i)}}(z',x)$ for $i\ge1$ and from the even-$j$ case can be estimated similarly; if $j$ is even, then, by using $|x-y|^2\leq2|z'-y|^2+2|x-z'|^2$ and estimating the contributions from $|z'-y|^2$ and $|x-z'|^2$ separately, we obtain that the supremum in [(\[eq:2nddec-bd:n=j\])]{} is $d\sigma^2O(\theta_0)$. Consequently, [(\[eq:2nddec-bd:n=j\])]{} is $d\sigma^2O(\theta_0)^{2\lfloor{{\scriptscriptstyle}\frac{j+1}2}\rfloor}$.
\(ii) To bound the contributions to $\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ from $|a_n|^2$ for $n<j$, we define (cf., Figure \[fig:tildeQ”\]) $$\begin{aligned}
{\label{eq:tildeQ''-def}}
\tilde Q''_{\Lambda;u,v}(y,x)=\sum_b\bigg(P''_{\Lambda;u,v}(y,
{\underline{b}})+\sum_{y'}\tilde G_\Lambda(y,y')\,P'_{\Lambda;u}(y',{\underline{b}})
\,\psi_\Lambda(y,v)\bigg)\,\tau_b\big(\delta_{{\overline{b}},x}+\tilde
G_\Lambda({\overline{b}},x)\big).\end{aligned}$$ By translation invariance and a similar argument to show [(\[eq:block-sumbd\])]{}, we can easily prove $$\begin{aligned}
{\label{eq:tildeQ''-bd}}
\sup_z\sum_{y,v}\tilde Q''_{\Lambda;o,v}(y,v+z)=\sum_{y,v}\tilde
Q''_{\Lambda;v,o}(y,z)=O(\theta_0).\end{aligned}$$ Therefore, the contribution from $|a_0|^2$ to $\sum_x|x|^2
\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ is bounded by $$\begin{aligned}
| 2,796
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| 0.771517
|
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|
density of $\Th$ given $\Om$ and $W$. To make it easy to derive sufficient conditions that $\ph_H$ is minimax, we show the following lemma.
\[lem:alter\_m(W)\] The marginal density $m(W)$ can alternatively be represented as $$m(W)=\int_{\Rc_r}f_\pi(\La;W)\dd\La,$$ where $$f_\pi(\La;W)=(2\pi v)^{-qr/2}|\La|^{q/2}\pi_2^J(\La)\exp\Big[-\frac{1}{2v}\tr(\La WW^\top)\Big]$$ with $$\pi_2^J(\La)
=v_1^{r(r+1)/2}|v_1I_r+(1-v_1)\La|^{-r-1}\pi_2[\La\{v_1I_r+(1-v_1)\La\}^{-1}].$$
[**Proof.**]{} Let $$\La(I_r-\La)^{-1}=v_1\Om(I_r-\Om)^{-1},\quad v_1=\frac{v}{v_0},$$ where $0_{r\times r}\prec\La\prec I_r$. Since $v^{-1}(I_r-\La)^{-1}=v^{-1}I_r+v_0^{-1}\Om(I_r-\Om)^{-1}$, we observe that $$\begin{aligned}
&\frac{1}{v}\Vert W-\Th\Vert^2+\frac{1}{v_0}\tr\{\Om(I_r-\Om)^{-1}\Th\Th^\top\}\\
&=\frac{1}{v}\tr\Big[(I_r-\La)^{-1}\{\Th-(I_r-\La)W\}\{\Th-(I_r-\La)W\}^\top\Big]
+\frac{1}{v}\tr(\La WW^\top),\end{aligned}$$ so $\pi(\Th|\Om,W)$ is proportional to $$\pi(\Th|\Om,W)
\propto \exp\Big[-\frac{1}{2v}\tr\Big[(I_r-\La)^{-1}\{\Th-(I_r-\La)W\}\{\Th-(I_r-\La)W\}^\top\Big]\Big],$$ namely, $\Th|\Om,W\sim\Nc_{r\times q}((I_r-\La)W,v(I_r-\La)\otimes I_q)$. Integrating out (\[eqn:m(W)\]) with respect to $\Th$ gives that $$\label{eqn:m(W)-1}
m(W)=(2\pi v)^{-qr/2}\int_{\Rc_r} |\La|^{q/2}\pi_2(\Om)\exp\Big[-\frac{1}{2v}\tr(\La WW^\top)\Big]\dd\Om.$$ Note that $\Om=\La\{v_1I_r+(1-v_1)\La\}^{-1}$ and the Jacobian of the transformation from $\Om$ to $\La$ is given by $$J[\Om\to \La]=v_1^{r(r+1)/2}|v_1I_r+(1-v_1)\La)|^{-r-1}.$$ Hence making the transformation from $\Om$ to $\La$ for (\[eqn:m(W)-1\]) completes the proof.
Let $\Dc_\La$ be an $r\times r$ symmetric matrix of differentiation operators with respect to $\La=(\la_{ij})$, where the $(i,j)$-th element of $\Dc_\La$ is $$\{\Dc_\La\}_{ij}=\frac{1+\de_{ij}}{2}\frac{\partial}{\partial\la_{ij}}.$$ Proposition \[prp:cond\_mini\] and Lemma \[lem:alter\_m(W)\] are utilized to get sufficient conditions for minimaxity of $\ph_H$.
\[thm:faith\] Let $f_\pi(\La;W)$ and $\pi_2^J(\La)$ be defined as
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tit{of type I}};\\
0 & \quad \textit{if $L_j$ is \textit{of type II}}.
\end{array} \right.$$ Recall from the beginning of Section 2 that we can choose a uniformizer $\pi$ in $B$ such that $\sigma(\pi)=-\pi$ and $\pi^2=2\delta$ with $\delta (\in A) \equiv 1$ mod 2.
We reproduce the beginning of Section 3 of [@C2] to explain our goal. Let $\underline{G}^{\prime}$ be the naive integral model of the unitary group $\mathrm{U}(V, h)$, where $V=L\otimes_AF$, such that for any commutative $A$-algebra $R$, $$\underline{G}^{\prime}(R)=\mathrm{Aut}_{B\otimes_AR}(L\otimes_AR, h\otimes_AR).$$ The scheme $\underline{G}^{\prime}$ is then an (possibly non-smooth) affine group scheme over $A$ with the smooth generic fiber $\mathrm{U}(V, h)$. Then by Proposition 3.7 in [@GY], there exists a unique smooth integral model, denoted by $\underline{G}$, with the generic fiber $\mathrm{U}(V, h)$, characterized by $$\underline{G}(R)=\underline{G}^{\prime}(R)$$ for any étale $A$-algebra $R$. Note that every étale $A$-algebra is a finite product of finite unramified extensions of $A$. This section, Section 4 and Appendix A are devoted to gaining an explicit knowledge of the smooth integral model $\underline{G}$ in *Case 2*, which will be used in Section 5 to compute the local density of $(L, h)$ (again, in *Case 2*). For a detailed exposition of the relation between the local density of $(L, h)$ and $\underline{G}$, see Section 3 of [@GY].
In this section, we give an explicit construction of the smooth integral model $\underline{G}$ when $E/F$ satisfies *Case 2*. The construction of $\underline{G}$ is based on that of Section 5 in [@GY] and Section 3 in [@C2]. Since the functor $R \mapsto \underline{G}(R)$ restricted to étale $A$-algebras $R$ determines $\underline{G}$, we first list out some properties that are satisfied by each element of $\underline{G}(R)=\underline{G}^{\prime}(R)$.
We choose an element $g\in \underline{G}(R)$ for an étale $A$-algebra $R$. Then $g$ is an element of $\mathrm{Aut}_{B\otimes_AR}(L\otimes_
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en there is a tuple ${\mathbf{s}}=(s_1, \ldots, s_n) \in S_1 \times \cdots \times S_m$, with ${\mathbb{P}}[{\mathsf{X}}_i=s_i]>0$ for each $1\leq i \leq m$, such that the largest zero of $f(s_1,\ldots, s_m;t)$ is smaller or equal to the largest zero of ${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_m;t)$.
The proof is by induction over $m$. The case when $m=1$ is Lemma \[largestz\], so suppose $m>1$. If $S_m=\{c_1,\ldots, c_k\}$, then $${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_m;t)= \sum_{i=1}^k q_i {\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_{m-1}, c_i;t),$$ for some $q_i \geq 0$. However $$\sum_{i=1}^k p_i {\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_{m-1}, c_i;t)$$ is real–rooted for all choices of $p_i \geq 0$ such that $\sum_ip_i=1$. By Lemma \[largestz\] and Theorem \[CS\] there is an index $j$ with $q_j>0$ such that ${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_{m-1}, c_j;t) \not \equiv 0$ and such that the largest zero of ${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_{m-1}, c_j;t)$ is no larger than the largest zero of ${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_m;t)$. The theorem now follows by induction.
Mixed hyperbolic polynomials
============================
Recall that the *directional derivative* of $h({\mathbf{x}}) \in {\mathbb{R}}[x_1,\ldots, x_n]$ with respect to ${\mathbf{v}}=(v_1,\ldots, v_n)^T \in {\mathbb{R}}^n$ is defined as $$D_{\mathbf{v}}h({\mathbf{x}}) := \sum_{k=0}^n v_k \frac{ \partial h }{\partial x_k}({\mathbf{x}}),$$ and note that $$\label{dvalt}
(D_{\mathbf{v}}h)({\mathbf{x}}+t{\mathbf{v}}) = \frac d {dt} h({\mathbf{x}}+ t {\mathbf{v}}) .$$ If $h$ is hyperbolic with respect to ${\mathbf{e}}$, then $$\tr({\mathbf{v}})= \frac {D_{\mathbf{v}}h({\mathbf{e}})}{h({\mathbf{e}})},$$ by . Hence ${\mathbf{v}}\rightarrow \tr({\mathbf{v}})$ is linear.
The following theorem is essentially known, see e.g. [@BGLS; @Ga; @Ren]. However we need slightly more general results, so we provide proofs below, when necessary.
\[direct\] Let $h$ be a hyperb
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=====================================================
Here we note that a standard periodization technique, which substitutes $Q=\S$ to Eq. (\[eq:periodize\]), cannot reproduce the Mott gap for large $U$. Because Im$G=0$ in the Mott gap, Im$\S$ must be 0 or $\infty$ in the whole Brillouin zone. However, this situation does never occur in Eq. (\[eq:periodize\]) with $Q=\S$ unless $^\forall i,j \in \text{C}, {\rm Im}\S_{ij}^{\text{C}}=0$ or $^\exists (ij), \S_{ij}^{\text{C}}=\infty$ for all $\w$ inside the gap, and these conditions are never satisfied as far as both $t$ and $U$ are nonzero and finite.
As a matter of fact, we see the electronic structure shown in Fig. \[fig:speriodize\](a) when we use the $\S$-periodization procedure for the same Mott insulator shown in Fig. 1(a) in Ref. \[and reproduced in Fig. \[fig:speriodize\](b) for comparison\]. We see that zeros of $G$ exist only at the Fermi level without a dispersion, and that poles of $G$ extend to the Fermi level around $(0,0)$ from the positive $\w$ side and around $(\pi,\pi)$ from the negative $\w$ side, making the density of states half metallic. This failure of the $\S$ periodization in the Mott insulator is ascribed to the fact that $\S$ is not localized within the $2\times 2$ cluster.[@sk06] The nonlocality of $\S$ is a direct consequence of the presence of dispersive zeros of $G$, i.e., momentum-dependent divergence of $\S$.
As we have discussed so far, zeros of $G$ still persist in doped Mott insulators up to a critical doping level beyond which the Fermi liquid emerges. Therefore $\S$ should be highly nonlocal also in the non-Fermi-liquid region and the $\S$ periodization again breaks down there. On the contrary, the cumulant $M$ is well localized within the $2\times 2$ cluster in this region, as we mentioned in Sec. \[sec:method\]. In APPENDIX B we give another numerical evidence for the locality of the cumulant in doped Mott insulators.
APPENDIX B: Locality of cumulant in doped Mott insulators {#appendix-b-locality-of-cumulant-in-do
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