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mportance. Let $G'\subset{\mathbb{R}}^3$ be an open bounded set such that $\ol{G'}$ is a $C^1$-manifold with boundary (i.e. $G'$ has the same regularity properties as $G$) and such that $\ol{G}\subset G'$. Let $G_{\rm e}:=G'\setminus \ol G$. Then $$\partial G_{\rm e}=\partial^1 G_{\rm e}\cup \partial^2G_{\rm e},$$ where $\partial^1 G_{\rm e}:=\partial G$ and $\partial^2 G_{\rm e}:=\partial G_{\rm e}\backslash \partial^1 G_{\rm e}$. Let $$\Gamma_{\rm e}:= (\partial G_{\rm e})\times S\times I
=
\Gamma_{\rm e,+}\cup \Gamma_{\rm e,-}\cup \Gamma_{\rm e,0},$$ where on the right hand side, the decomposition of $\Gamma_{\rm e}$ into the three disjoint subsets corresponds to $\Gamma=\Gamma_+\cup \Gamma_-\cup\Gamma_0$ when considering $G_{\rm e}\times S\times I$ instead of $G\times S\times I$. Finally, we can decompose $\Gamma_{\rm e,+}$ and $\Gamma_{\rm e,-}$ respectively as $$\Gamma_{\rm e,\pm} = \Gamma^1_{\rm e,\pm}\cup \Gamma^2_{\rm e,\pm},$$ where for $j=1,2$, $$\Gamma^j_{\rm e,\pm}:=\big((\partial^j G_{\rm e})\times S\times I\big)\cap \Gamma_{\rm e,\pm}.$$ Notice that $\Gamma^1_{\rm e,-}=\Gamma_+$, while $\Gamma^1_{\rm e,+}=\Gamma_-$.
We assume (for simplicity) that $f=0$. Consider the problem (\[csda1a\])-(\[csda3\]) that is, find $\psi=(\psi_1,\psi_2,\psi_3)$ that satisfies on $G\times S\times I$ the system of transport equations, $$\begin{gathered}
\omega\cdot\nabla_x\psi_1+\Sigma_1\psi_1-K_{1}\psi=0,\label{ref2}\\
-{{\frac{\partial (S_{j}\psi_j)}{\partial E}}}+\omega\cdot\nabla_x\psi_j+\Sigma_{j}\psi_j-K_{j}\psi=0,\quad j=2,3\label{ref3}\end{gathered}$$ under the (inflow) boundary condition on $\Gamma_-$, \[ref4\] [\_j]{}\_[|\_-]{}=g\_j,j=1,2,3, and the initial condition on $G\times S$, \[ref5\] \_j(,,E\_[m]{})=0,j=2,3.
Since $\Gamma^1_{\rm e,-}=\Gamma_+$, we find that the flux $\psi_{|\Gamma_+}$ is an inflow boundary source for the domain $G_{\rm e}$, on the part $\partial^1 G_{\rm e}=\partial G$ of its boundary. Suppose that $G_{\rm e}$ does not contain any extra internal or boundary sources. Then the transpo
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\sqrt{m} ({\widehat}{\theta}_{km} -\theta ) \notag \\
&=\frac{1}{\sqrt{K}}\sum_{k=1}^{K} W_{km}+ \frac{1}{\sqrt{K}}\sum_{k=1}^{K} R_{km},\end{aligned}$$ where $W_{km}=\frac{1}{\sqrt{m}}\sum_{i=1}^{m}\eta_{k,i}$. From Assumption \[assumption2\], we get the last term in (\[eqA1\]) is $o_{p}(1)$.
Now, we prove that $\frac{1}{\sqrt{K}}\sum_{k=1}^{K} W_{km} $ has the asymptotic normality distribution. Let $V_{km}=c^\top W_{km}$, then ${\mathbb{E}}(V_{km})=0, Var(V_{km})=c^\top \Sigma c=\sigma^2$. By the Cramér-Wold theorem, we only need to prove $$\frac{1}{\sqrt{K}}\sum_{k=1}^{K} V_{km} \stackrel{d}{\longrightarrow}N(0,\sigma^2)$$ for each fixed $c\in\mathbb{R}^p\setminus\{0\}$.
Since $V_{km}$ is a normalized sum of $K$ independent and identically distributed random variables, it follows from Linderberg’s CLT that $${\mathbb{E}}e^{\imath tV_{mk}(u)}=e^{-t^2\sigma^2/2}+o(t^2),$$ for any real $t\in {\mathbb{R}}$. Here $\imath=\sqrt{-1}$.
Hence, $$\begin{aligned}
&{\mathbb{E}}\exp\left\{ \imath t\frac{1}{\sqrt{K}}\sum_{k=1}^K V_{km}\right\}\\
&=\left({\mathbb{E}}e^{\imath tK^{-1/2} V_{km}}\right)^K\\
&=\Big(e^{-(tK^{-1/2})^2\sigma^2/2}+o(tK^{-1/2})^2\Big)^K\to e^{-t^2\sigma^2/2},\end{aligned}$$ as $K\to\infty$. The proof of Theorem \[theorem1\] is completed.
For proving Theorem \[theorem2\], we need the following two lemmas.
\[lem-1\] Let $Z_{K}=\max_{1\leq k\leq K} \|Y_{km}- \mu \|$. Under the conditions of Theorem \[theorem2\], we have $$Z_{K}=o_p(K^{1/2})$$ as $K, m\to \infty$.
Note that $$Y_{km} - \mu =\frac{1}{\sqrt{m}}\sum_{i=1}^m\eta_{ki}+R_{km}.$$ Since $\eta_{ki}$’s are independent and identically distributed random vectors with mean zero and finite fourth moment, $$\begin{aligned}
{\mathbb{P}}\Big(\max_{1\leq k\leq K} \Big\|\frac{1}{\sqrt{m}}\sum_{i=1}^m\eta_{ki}\Big\|>\epsilon \sqrt{K}\Big)
&\leq \sum_{k=1}^K {\mathbb{P}}\Big( \Big\|\frac{1}{\sqrt{m}}\sum_{i=1}^m\eta_{ki}\Big\|>\epsilon \sqrt{K} \Big) \\
&\leq K ( \epsilon \sqrt{K})^{-4} {\mathbb{E}}\Big\|\frac{1}{\sqrt{m}}\sum_{i=1}^m\eta
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rages by the delta method. First, we carry out a coordinate-wise Taylor expansion of $\widehat{\theta}$ around $\theta$. We then utilize a a high-dimensional Berry-Esseen theorem for polyhedral sets established in [@cherno2] (see below for details) to derive a Gaussian approximation to the linear part in the expansion, resulting in the error term $\Delta_{n,1}$. Finally, we bound the reminder term due to the non-linearity of the function $g$ with basic concentration arguments paired with the Gaussian anti-concentration bound due to [@nazarov1807maximal] (see in the Appendix), thus obtaining the second error term $\Delta_{n,2}$. Throughout, we keep track of the dependence on $v$ and $\underline{\sigma}$ in order to obtain rates with a leading constant dependent only on $A$ (assumed fixed) but not on any other term that may vary with $k$ or $b$.
### Asymptotically honest confidence sets: Normal approximation approach {#sec:berry.normal .unnumbered}
We now show how to use the high-dimensional central limit theorem to construct asymptotically honest confidence sets for $\theta$. We will first to obtain a consistent estimator of the covariance matrix $\Gamma=G(\psi) V(\psi) G(\psi)^\top$ of the linear approximation to $\hat{\theta} - \theta$. In conventional fixed-dimension asymptotics, we would appeal to Slutzky’s theorem and ignore the effect of replacing $\Gamma$ with a consistent estimate. But in computing Berry-Esseen bounds with increasing dimension we may not discard the effect of estimating $\Gamma$. As we will see below, this extra step will bring an additional error term that must be accounted for. We will estimate $\Gamma$ with the plug-in estimator $$\label{eq:hat.gamma.berry}
\hat\Gamma = G(\hat\psi) \hat V G(\hat\psi)^\top,$$ where $\hat{V} = \frac{1}{n} \sum_{i=1}^n W_i W_i^\top - \hat{\psi}
\hat{\psi}^\top$ is the empirical covariance matrix. Below, we bound the element-wise difference between $\Gamma$ and $\hat{\Gamma}$. Although this is in general a fairly weak notion of consistency in covari
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trates on formal specifying/developing/modeling separation kernels and *Verification* on formally verifying separation kernels. Some work aims at these two aspects together. The “Property” indicates the policies or properties specified or verified in each work. The “Formal Language” indicates what’s the formal language used when specifying or verifying the separation kernels. The “Approach” indicates the formal specification or verification approaches used. The “Size” shows the scale of the formal specification or verification proofs. The “Tools” shows the software tools used in each work.
[|p[3.5cm]{}|p[3.2cm]{}|p[1.5cm]{}|p[2.0cm]{}|p[2.8cm]{}|p[2.0cm]{}|p[1.8cm]{}|p[3.0cm]{}|]{} **Related Work** & **Target Kernel** & **Objective** & **Property** & **Formal Language** & **Approach** & **Size** & **Tools**\
Department of Defense [@Martin00; @Martin02] & MASK separation kernel &
---------------
Specification
Verification
---------------
& Data separation & SPECWARE &
-----------------
Refinement
Theorem proving
-----------------
& [$\divideontimes$]{}& SPECWARE environment\
GWV, GWVr2 and extensions [@Greve03; @Rushby04; @Alves04; @Tverdy11; @Richards10] & Applicable for generic separation kernels & Specification & Information flow security & ACL2, PVS & Theorem proving & [$\divideontimes$]{}& ACL2, PVS theorem prover\
Naval Research Lab [@Heitmeyer06; @Heitmeyer08] & ED separation kernel &
---------------
Specification
Verification
---------------
& Data separation & TAME &
-----------------
Refinement
Theorem proving
-----------------
& 368 LOC of TAME spec. & TAME, PVS theorem prover\
Craig’s book [@Craig06; @Craig07] & A separation kernel & Specification & & Z notation & Refinement & $\approx$100 pages & By hand\
Verified software project [@Velykis09; @Velykis10] & A separation kernel &
---------------
Specification
Verification
---------------
& PIFP & Z notation &
-----------------
Refinement
Theorem proving
----------------
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**a member of a filter belongs to the filter: $(F\in\mathcal{F})\wedge(F\subseteq G)\Rightarrow G\in\mathcal{F}$; in particular $X\in\mathcal{F}$.
**Example A1.4.** (1) The *indiscrete filter* is the smallest filter on $X$.
\(2) The neighbourhood system $\mathcal{N}_{x}$ is the important *neighbourhood filter at $x$ on $X$,* and any local base at $x$ is also a filter-base for $\mathcal{N}_{x}$. In general for any subset $A$ of $X$, $\{ N\subseteq X\!:A\subseteq\textrm{Int}(N)\}$ is a filter on $X$ at $A$.
\(3) All subsets of $X$ containing a point $x\in X$ is the *principal filter* $_{\textrm{F}}\mathcal{P}(x)$ *on $X$ at $x$.* More generally, if $\mathcal{F}$ consists of all supersets of a *nonempty* subset $A$ of $X$, then $\mathcal{F}$ is the *principal filter* $_{\textrm{F}}\mathcal{P}(A)=\{ N\subseteq X\!:A\subseteq\textrm{Int}(N)\}$ *at $A$. By adjoining the empty set to this filter give the $p$-inclusion and $A$-inclusion topologies on $X$ respectively.* The single element sets $\{\{ x\}\}$ and $\{ A\}$ are particularly simple examples of filter-bases that generate the principal filters at $x$ and $A$.
\(4) For an uncountable (resp. infinite) set $X$, all cocountable (resp. cofinite) subsets of $X$ constitute the *cocountable* (resp. *cofinite* or *Frechet*) filter on $X$. Again, adding to these filters the empty set gives the respective topologies.$\qquad\blacksquare$
Like the topological and local bases $_{\textrm{T}}\mathcal{B}$ and $\mathcal{B}_{x}$ respectively, a subclass of $\mathcal{F}$ may be used to define a filter-base $_{\textrm{F}}\mathcal{B}$ that in turn generate $\mathcal{F}$ on $X$, just as it is possible to define the concepts of limit and adherence sets for a filter to parallel those for nets that follow straightforwardly from Def. A1.7, taken with Def. A1.11.
**Definition A1.8.** *Let $(X,\mathcal{T})$ be a topological space and $\mathcal{F}$ a filter on $X$. Then*$$\textrm{lim}(\mathcal{F})=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\exists F\in\mathcal{F})(F\subseteq N)\}\label{E
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| 0.788211
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|
ed in Section \[section1\], random graph models, in general, are not always relevant to represent the structure of a graph that has been inferred from observations. To tackle this issue, we create a new random model with an underlying structure that is a randomized version of a deterministic graph with exact cluster structure.
### Ideal model
We consider that the graph $G_*(V,E)$ is the union of $k$ complete graphs that are disconnected from each other. We denote by $C_1, \dots, C_k$ the $k$ connected components of the graph, that match the $k$ clusters. We allow the number of vertices in each subgraph to be different. We denote by $c_1,\cdots,c_k $ $(\geq 2)$ their respective size ($\sum_{i=1}^k c_i=n$). To simplify, we assume that the nodes, labeled from $1$ to $n$, are ordered with respect to their block membership and in increasing order with respect to the size of the blocks.
From a matricial point of view, the associated adjacency matrix $A_*$ is a $k$-block diagonal matrix of size $n$ of the form:
$$A_*=
\left[
\begin{array}{c@{}c@{}c}
C_1 & & \mathbf{0} \\
& \ddots & \\
\mathbf{0} & & C_k \\
\end{array}\right]$$
where $C_1, \cdots, C_k$ are symmetric matrices of size $c_1 \times c_1, \cdots, c_k \times c_k$.
### Perturbed model
The reality is that we consider the graph $G_*$ but we observe a randomized version of this graph, denoted by $\tilde{G}$.
We introduce the Erdös–Rényi model of a graph [@Erdos59; @Stewart90], one of the oldest and best studied random graph model.
Given a set of $n$ vertices, we consider the variable $X_{ij}$ that indicates the presence/absence of an edge between vertices $i$ and $j$. Then, for all $ \left\{ X_{ij} \right\} i.i.d.$, we have $ X_{ij} \sim B(p)$. Some edges have been added between the clusters and others have been removed within the clusters independently with respect to the same probability $p$. The adjacency matrix $B$ of the Erdos-Renyi graph of size $n$, whose upper entries are realizations of independent Bernoulli variables, can be wri
| 3,506
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|
er $i,j$ will take values from $\{1,\dots,n\}$ and $k,l,m$ will take values from $\{1,\dots,r\}$. We simplify the above as $$\biggl(\sum_i c_i\biggr)\biggl(\sum_j\sum_{\substack{{k,l}\\k\neq l}}x_{jk}x_{jl}\biggr)
\leq c\biggl(\sum_{i,j}\sum_{\substack{{k,l}\\k\neq
l}}x_{ik}x_{jl}\biggr),$$ then we factor out and also utilize the symmetry in $k,l$ to arrive at the equivalent form $$\sum_{i,j}c_i\sum_{\substack{{k,l}\\k<l}}x_{jk}x_{jl}\leq
\sum_{i,j}c\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{jl}.$$ We distribute the terms in $i,j$ on both sides as follows: $$\sum_i c_i\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{il}+
\sum_{\substack{{i,j}\\i<j}}\biggl(c_i\sum_{\substack{{k,l}\\k<l}}x_{jk}x_{jl}
+c_j\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{il}\biggr)\leq \sum_i
c\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{il}+
\sum_{\substack{{i,j}\\i<j}}c\sum_{\substack{{k,l}\\k<l}}(x_{ik}x_{jl}+x_{jk}x_{il}).$$ It is clear that $$c_i\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{il}\leq c\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{il},\quad 1\leq i\leq n,$$ therefore it suffices to show that $$c_i\sum_{\substack{{k,l}\\k<l}}x_{jk}x_{jl}+c_j\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{il}
\leq c\sum_{\substack{{k,l}\\k<l}}(x_{ik}x_{jl}+x_{jk}x_{il}),\quad
1\leq i<j\leq n.$$ We will prove this in the stronger form $$c_i\sum_{\substack{{k,l}\\k<l}}x_{jk}x_{jl}+c_j\sum_{\substack{{k,l}\\k<l}}x_{ik}x_{il}
\leq
c_i\sum_{\substack{{k,l}\\k<l}}(x_{ik}x_{jl}+x_{jk}x_{il}),\quad
1\leq i<j\leq n.$$
We now fix $1\leq i<j\leq n$ and introduce $x_k:=x_{ik}$, $x'_k:=x_{jk}$. Then the previous inequality reads $$\biggl(\sum_m
x_m\biggr)\biggl(\sum_{\substack{{k,l}\\k<l}}x'_kx'_l\biggr)+
\biggl(\sum_m
x'_m\biggr)\biggl(\sum_{\substack{{k,l}\\k<l}}x_kx_l\biggr)\leq
\biggl(\sum_m
x_m\biggr)\sum_{\substack{{k,l}\\k<l}}(x_kx'_l+x'_kx_l),$$ that is, $$\sum_{\substack{{k,l,m}\\k<l}}(x_mx'_kx'_l+x_kx_lx'_m)\leq
\sum_{\substack{{k,l,m}\\k<l}}(x_kx_mx'_l+x_lx_mx'_k).$$ The right hand side equals $$\begin{aligned}
\sum_{\substack{{k,l,m}\\k<l}}(x_kx_mx'_l+x_lx_mx'_k)
&=\sum_{\substack{{k,l,m}\\l\neq
k}}
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$ TeV 88 1600 760
\(2) chirally-coupled scalar
$m_S=1$ TeV, $\Gamma_S=350$ GeV 100 570 430
\(3) O(2N) 90 470 350
\(4) chirally-coupled vector
a\. $m_V=1$ TeV, $\Gamma_V=0.4$ TeV 180 2400 280
b\. $m_V=1.2$ TeV, $\Gamma_V=0.5$ TeV 52 590 29
c\. $m_V=1.5$ TeV, $\Gamma_V=0.6$ TeV 88 120 40
LET 150 110 170
: \[table1\] The number of the signal events for the strong $W_L W_L$ scattering predicted by various models at $\gamma\gamma$ collider of $\sqrt{s}=1.5$ TeV. The acceptance cuts on the final $W_LW_L$ or $Z_LZ_L$ are: $m(WW,ZZ)>500$ GeV and $|y(W,Z)|<1.5$. The luminosity is assumed 100 fb$^{-1}$. No efficiencies are included here.
$|y(W)|<$ No cuts Tagging at least one $W$ Tagging both $W$’s
----------- --------- -------------------------- --------------------
- 14.7 - -
1.5 - 13.4 (91%) 6.16 (42%)
2.0 - 14.5 (98.5%) 12.1 (82%)
: \[table2\] Table showing the cross sections (fb) for the process $\gamma\gamma\to WWH$ with a SM Higgs boson of mass $m_H=1$ TeV at $\sqrt{s_{\gamma\gamma}}=1.5$ TeV, with and without imposing acceptance cuts on the final state $W$’s. The acceptance cuts are $p_T(W)>25$ GeV and $|y(W)|<1.5$ or 2. T
| 3,508
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| 3,330
| 4,157
| 0.76769
|
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|
\
Weyl Spinors {#Sec3.3}
-----------------------------------------
We have seen how, on the basis of the chiral SU(2)$_+\times$SU(2)$_-$ group, it is possible to readily identify the finite dimensional representation theory of the Lorentz group SO(1,3). Let us now discuss yet another construction of its two fundamental Weyl spinor representations, which is also of importance in the construction of supersymmetric field theories. The present discussion shall also make explicit why the universal covering group of the Lorentz group SO(1,3) is the group SL(2,$\mathbb C$) of complex 2$\times$2 matrices of unit determinant. We shall thus establish the relation, at the level of the corresponding Lie algebras, $$so(1,3)_{\mathbb C}=su(2)_+\oplus su(2)_-=sl(2,{\mathbb C})\ .$$
Let us introduce the notation $$\sigma_\mu=(\one,\sigma_i)\ \ \ ,\ \ \
\sigma^\mu=(\one,\sigma^i)=(\one,-\sigma_i)\ ,$$ where the space index $i$ carried by the usual Pauli matrices is raised and lowered according to our choice of signature for the Minkowski spacetime metric, namely $\eta_{\mu\nu}={\rm diag}\,(+---)$. Consider now an arbitrary spacetime 4-vector $x^\mu$, and construct the 2$\times$2 hermitian matrix $$X=x^\mu\sigma_\mu=\left(\begin{array}{c c}
x^0+x^3 & x^1-ix^2 \\
x^1+ix^2 & x^0-x^3
\end{array}\right)\ .$$ Note that conversely, any 2$\times$2 hermitian matrix $X=X^\dagger$ possesses such a decomposition, and may thus be associated to some spacetime 4-vector $x^\mu$ through the above relation. In particular, the determinant of any such matrix is equal to the Lorentz invariant inner product of the associated 4-vector with itself, $${\rm det}\,X=x^2=\eta_{\mu\nu}x^\mu x^\nu\ .$$
Consider now an arbitrary SL(2,$\mathbb Z$) group element $M$, thus of unit determinant, ${\rm det}\,M=1$, and its adjoint action on any hermitian matrix $X$ as $$X'=M\,X\,M^\dagger\ .$$ It should be clear that the transformed matrix itself is hermitian, ${X'}^\dagger=X'$, hence possesses a decomposition in terms of a 4-vector ${
| 3,509
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.0 573.5 48.0 340.5
Starch solution 12.4 9.3 11.4 10.6 9.9 16.2 14.4
PVA solution 7.7 16.2 10.1 13.4 15.1 19.6 17.7
Textile wastewater 40.6 52.5 68.6 73.5 60.9 56.7 93.1
polymers-12-00289-t002_Table 2
######
The zeta potential of the prepared microspheres.
Adsorbents Zeta Potential (mV)
------------ ---------------------
CCS 20.3
CCP 30.4
CCSP1 22.7
CCSP2 26.2
CCSP3 28.1
polymers-12-00289-t003_Table 3
######
Effect of bed height and flow rate on the column adsorption capacity of the mixture adsorbent (CCSP2 and AC) to treat the textile wastewater.
Bed Height (cm) Flow Rate (mL/min) The Adsorption Capacity (mg/g)
----------------- -------------------- --------------------------------
9.5 (10g) 2 93.1
7.6 (8g) 2 88.9
5.7 (6g) 2 80.5
9.5 (10g) 3 86.8
9.5 (10g) 4 77.0
polymers-12-00289-t004_Table 4
######
Characterization of the regenerated activated carbon (RAC) and AC.
Characterization BET Surface Area (m^2^/g) Pore Volume (cm^3^/g) Average Pore Diameter (nm)
------------------ --------------------------- ----------------------- ----------------------------
AC 386 0.39 4.12
RAC 581 0.53 3.13
American cutaneous leishmaniasis (ACL) is an infectious, noncontagious disease caused by different species of protozoa of the genus *Leishmania* Ross, 1903, that affects the skin, cartilage, and mucous membranes of the upper respiratory tract ([@B20]). Drugs used in the treatment of leishmaniasis have a number of drawbacks, such as high degrees of toxicity, the development of resistance o
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|
\psi({\widehat{S}},P)=
\left[ \begin{array}{c}
\mathrm{vech}(\Sigma_{{\widehat{S}}})\\
\alpha_{{\widehat{S}}}\\
\end{array} \right] \in \mathbb{R}^{b},$$ where $b = \frac{ k^2 + 3k}{2} $. Similarly, based on the sub-sample $\mathcal{D}_{2,n}$ we define the $n$ random vectors $$W_i =
\left[ \begin{array}{c}
\mathrm{vech}(X_i({\widehat{S}}) X_i({\widehat{S}})^\top)\\
Y_i \cdot X_i({\widehat{S}}) \\
\end{array}
\right] \in \mathbb{R}^b, \quad i \in \mathcal{I}_{2,n},$$ and their average $$\label{eq:hat.psi.beta}
\hat{\psi} = \hat{\psi}_{{\widehat{S}}} = \frac{1}{n} \sum_{i \in \mathcal{I}_{2,n}} W_i.$$ It is immediate to see that $\mathbb{E}_P[\hat{\psi}] = \psi$, uniformly over all $P \in \mathcal{P}_n^{\mathrm{OLS}}$.
We express both the projection parameter $\beta_{{\widehat{S}}}$ and the least square estimator $\hat{\beta}_{{\widehat{S}}}$ as non-linear functions of $\psi$ and $\hat{\psi}$, respectively, in the following way. Let $g \colon \mathbb{R}^b
\rightarrow \mathbb{R}^k$ be given by $$\label{eq:g.beta}
x = \left[ \begin{array}{c}
x_1\\
x_2\\
\end{array}
\right] \mapsto \left( \mathrm{math}(x_1) \right)^{-1} x_2,$$ where $x_1$ and $x_2$ correspond to the first $k(k+1)/2$ and the last $k$ coordinates of $x$, respectively, and $\mathrm{math}$ is the inverse mapping of $\mathrm{vech}$, i.e. $\mathrm{math}(x) = A$ if and only if $\mathrm{vech}(A) =
x$. Notice that $g$ is well-defined over the convex set $$\left\{ \left[ \begin{array}{c}
\mathrm{vech}(\Sigma)\\
x
\end{array}
\right] \colon \Sigma \in \mathcal{C}^+_{k}, x \in \mathbb{R}^k \right\}$$ where $\mathcal{C}^+_k$ is the cone of positive definite matrices of dimension $k$. It follows from our assumptions that, for each $P \in \mathcal{P}_n^{\mathrm{OLS}}$, $\psi$ is in the domain of $g$ and, as long as $n \geq d$, so is $\hat{\psi}$, almost surely. Thus, we may write $$\beta_{{\widehat{S}}} = g(\psi_{{\widehat{S}}}) \quad \text{and} \quad \hat{\beta}_{{\widehat{S}}} =
g
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|
ase 2* and a uniformizing element $\pi$ of $B$ and $\delta$ are fixed as explained above throughout this paper.
- Set $$\xi:=\pi\cdot\sigma(\pi).$$
- We consider a $B$-lattice $L$ with a hermitian form $$h : L \times L \rightarrow B,$$ where $h(a\cdot v, b \cdot w)=\sigma(a)b\cdot h(v,w)$ and $h(w,v)=\sigma(h(v,w))$. Here, $a, b \in B$ and $v, w \in L$. We denote by a pair $(L, h)$ a hermitian lattice. We assume that $V=L\otimes_AF$ is nondegenerate with respect to $h$.
- We denote by $(\epsilon)$ the $B$-lattice of rank 1 equipped with the hermitian form having Gram matrix $(\epsilon)$. We use the symbol $A(a, b, c)$ to denote the $B$-lattice $B\cdot e_1+B\cdot e_2$ with the hermitian form having Gram matrix $\begin{pmatrix} a & c \\ \sigma (c) & b \end{pmatrix}$. For each integer $i$, the lattice of rank 2 having Gram matrix $\begin{pmatrix} 0 & \pi^i \\ \sigma(\pi^i) & 0 \end{pmatrix}$ is called the hyperbolic plane and denoted by $H(i)$.
- A hermitian lattice $L$ is the orthogonal sum of sublattices $L_1$ and $L_2$, written $L=L_1\oplus L_2$, if $L_1\cap L_2=0$, $L_1$ is orthogonal to $L_2$ with respect to the hermitian form $h$, and $L_1$ and $L_2$ together span $L$.
- The ideal in $B$ generated by $h(x,x)$ as $x$ runs through $L$ will be called the norm of $L$ and written $n(L)$.
- By the scale $s(L)$ of $L$, we mean the ideal generated by the subset $h(L,L)$ of $B$.
- We define the dual lattice of $L$, denoted by $L^{\perp}$, as $$L^{\perp}=\{x \in L\otimes_A F : h(x, L) \subset B \}.$$
\[d1\] Let $L$ be a hermitian lattice. Then:
1. For any non-zero scalar $a$, define $aL=\{ ax|x\in L \}$. It is also a lattice in the space $L\otimes_AF$. Call a vector $x$ of $L$ maximal in $L$ if $x$ does not lie in $\pi L$.
2. The lattice $L$ will be called $\pi^i$-modular if the ideal generated by the subset $h(x, L)$ of $E$ is $\pi^iB$ for every maximal vector $x$ in $L$. Note that $L$ is $\pi^i$-modular if and only if $L^{\perp}=\pi^{-i}L$. We can also see that $H(i)$ is $\pi^i$-modular.
3.
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|
Y)$ is the smallest closed topological extension of $M=\textrm{Map}(X,Y)$ is the following, refer Thm. A1.4 and its proof. Let $(M,\mathcal{T}_{0})$ be a topological space and suppose that$${\textstyle \widehat{M}=M\bigcup\{\widehat{m}\}}$$
is obtained by adjoining an extra point to $M$; here $M=\textrm{Map}(X,Y)$ and $\widehat{m}\in\textrm{Cl}(M)$ is the multifunctional limit in $\widehat{M}=\textrm{Multi}_{\mid}(X,Y)$. Treat all open sets of $M$ generated by local bases of the type (\[Eqn: local\_base\]) with finite intersection property as a filter-base $_{\textrm{F}}\mathcal{B}$ on $X$ that induces a filter $\mathcal{F}$ on $M$ (by forming supersets of all elements of $_{\textrm{F}}\mathcal{B}$; see Appendix A1) and thereby the filter-base $${\textstyle \widehat{_{\textrm{F}}\mathcal{B}}=\{\widehat{B}=B\bigcup\{\widehat{m}\}\!:B\in\,_{\textrm{F}}\mathcal{B}\}}$$
on $\widehat{M}$; this filter-base at $m$ can also be obtained independently from Eq. (\[Eqn: multi\_bi\]). Obviously $\widehat{_{\textrm{F}}\mathcal{B}}$ is an extension of $_{\textrm{F}}\mathcal{B}$ on $\widehat{M}$ and $_{\textrm{F}}\mathcal{B}$ is the filter induced on $M$ by $\widehat{_{\textrm{F}}\mathcal{B}}$. We may also consider the filter-base to be a topological base on $M$ that defines a coarser topology $\mathcal{T}$ on $M$ (through all unions of members of $_{\textrm{F}}\mathcal{B}$) and hence the topology$${\textstyle \widehat{\mathcal{T}}=\{\widehat{G}=G\bigcup\{\widehat{m}\}\!:G\in\mathcal{T}\}}$$
on $\widehat{M}$ to be the topology associated with $\widehat{\mathcal{F}}$. A finer topology on $\widehat{M}$ may be obtained by adding to $\widehat{\mathcal{T}}$ all the discarded elements of $\mathcal{T}_{0}$ that do not satisfy FIP. It is clear that $\widehat{m}$ is on the boundary of $M$ because every neighbourhood of $\widehat{m}$ intersects $M$ by construction; thus $(M,\mathcal{T})$ is dense in $(\widehat{M,}\widehat{\mathcal{T}})$ which is the required topological extension of $(M,\mathcal{T}).$
In the present case, a filter-bas
| 3,513
| 2,999
| 3,690
| 3,236
| 2,781
| 0.777024
|
github_plus_top10pct_by_avg
|
d antibody incubation conditions are shown in [Table 2](#pone.0214536.t002){ref-type="table"}. The Western blots were also incubated with secondary antibodies only for detecting non-specific signals. For the chemiluminescence detection of immunocomplexes by ChemiDoc, the membranes were treated with Supersignal West Femto Maximum Sensitivity Substrate (Thermo Scientific). All preparations and immunoblottings were performed with samples from two independent culture processes. To ensure the detection results for NheC, a third cultivation of *B*. *toyonensis* BCT-7112^T^ was conducted and prepared.
10.1371/journal.pone.0214536.t002
###### Blocking and antibody incubation conditions.
{#pone.0214536.t002g}
Protein Blocking 2h rt[^a^](#t002fn001){ref-type="table-fn"} Primary antibody 4°C overnight Secondary antibody 1 h rt
--------- ----------------------------------------------------------- ---------------------------------- ---------------------------
Hbl L1 3% dry milk in PBST[^b^](#t002fn002){ref-type="table-fn"} 1:100 1:20,000
3% dry milk in PBST 3% dry milk in PBST
Hbl L2 Superblock[^c^](#t002fn003){ref-type="table-fn"} 1:100 1:20,000
Superblock 3% dry milk in PBST
NheB 5% dry milk in PBST 1:100 1:20,000
5% dry milk in PBST 5% dry milk in PBST
NheC 5% dry milk in PBST 1:200(pellet, conc. supernatant) 1:20,000
1:100 (pure supernatant) 5% dry milk in PBST
5% dry milk in PBST
^a^ Room temperature
^b^ Phosphate bu
| 3,514
| 106
| 3,410
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|
, \label{a2}$$where $p_\ast$ is the conjugate of $p$. So, does not matter the value of $\theta _{1},$ one may replace it by $\frac{d%
}{p_{\ast }}.$
**Proof** We take $n_{\ast }\in \N$ and we define $f_{n}=0$ for $n\leq
n_{\ast }$ and $f_{n}=\varphi p_t$ for $n>n_{\ast }.$ Notice that $d_{0}(\varphi p_{t},0)\leq m.$ Then (\[reg4\]) with $k=0$ gives ($C$ denoting a positive constant which may change from a line to another) $$\begin{aligned}
\left\Vert \varphi p_{t}\right\Vert _{q,p}
&\leq &C\Big(m\sum_{n=0}^{n_{\ast }}2^{n(q+\frac{d}{p_{\ast }})}+\left\Vert \varphi p_{t}\right\Vert _{2h+q,2h,p}\sum_{n=n_{\ast }+1}^{\infty }\frac{1%
}{2^{2nh}}\Big) \\
&\leq &C\Big(m2^{n_{\ast }(q+\frac{d}{p_{\ast }})}+\left\Vert \varphi p_{t}\right\Vert _{2h+q,2h,p}\frac{1}{2^{2n_{\ast }h}}\Big).\end{aligned}$$We denote $\rho _{h}=(q+\frac{d}{p_{\ast }})/2h.$ We optimize over $n_{\ast
}$ and we obtain $$\begin{aligned}
\left\Vert \varphi p_{t}\right\Vert _{q,p} &\leq &2C\times
m^{\frac{1}{1+\rho _{h}}}\times \left\Vert \varphi p_{t}\right\Vert _{2h+q,2h,p}^{\frac{\rho _{h}}{1+\rho _{h}}} \\
&\leq &2Cm^{\frac 1{1+\rho_h}}\times Ct^{-\theta _{0}(2h+q+\theta _{1})%
\frac{\rho _{h}}{1+\rho _{h}}}.\end{aligned}$$Since $\lim_{h\rightarrow \infty }\rho _{h}=0$ and $\lim_{h\rightarrow \infty }(2h+q+\theta _{1})\frac{\rho _{h}}{1+\rho
_{h}}=q+\frac{d}{p_{\ast }}$ the proof is completed, we choose $h$ large enough and we obtain (\[a2\]). $\square $
We will also use the following consequence of Lemma \[lemma-inter\].
\[REG\] Let $k,q,h\in {\mathbb{N}}$, with $h\geq 1$, and $p>1$ be given and set $$\rho _{h}:=\frac{k+q+d/p_{\ast }}{2h}. \label{reg5}$$We consider an increasing sequence $\theta (n)\geq 1,n\in {\mathbb{N}}$such that $\lim_{n}\theta (n)=\infty $ and $\theta (n+1)\leq \Theta \times
\theta (n)$ for some constant $\Theta \geq 1.$ Suppose that we may find a sequence of functions $f_{n}\in C^{2h+q}({\mathbb{R}}^{d}),n\in {\mathbb{N}}$ such that $$\left\Vert f_{n}\right\Vert _{2h+q,2h,p}\leq \theta (n) \label{reg9}$$and, with $\m
| 3,515
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| 3,212
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|
s filed on or about August 25, 2003, recorded under document
number 2003053083 of the Official Public Records of Brazoria County, Texas (See
Exhibit "A" attached hereto and incorporated herein for all purposes);
2. Notice of Lis Pendens filed on or about September 19, 2005, recorded under
document number 2005054235 of the Official Public Records of Brazoria County,
Texas by Plaintiff, Paula M. Miller (See Exhibit "B" attached hereto and incorporated
herein for all purposes)~
3. Notice of Lis Pendens filed on or about June I 0, 2013, recorded under document
number 2013027728 of the Official Public Records of Brazoria County, Texas by the
Defendant, Paula Miller (See Exhibit "C" attached hereto and incorporated herein for
all purposes); and
4. Affidavit of Adverse Possession executed by Paula M. Miller as a claim recorded
under document number 2011035377 of the Official Public Records of Brazoria
County, Texas purporting to supplement the Original Document filed with the wrong
title of Notice of Lis Pendens Lien documer:tt number 2005054235 (See Exhibit "D',
attached hereto· and incorporated herein for all purposes).
IT IS FURTHER ORDERED that the following three "Release of Lis Pendens" and the
Release of Affidavit of Adverse Possession all attached as Exhibit "E~' hereto shall be executed
by Paula M. Miller immediately after her receipt of notice of entry of this Order by the District
Clerk of Brazoria County, Texas.
IT IS FURTHER ORDERED that JAS Family Limited Partnership #4 LTD acting by and
through its managing member, James A. Prince shall tender into the registry of this Court the
amount equal to one .. half (1/2) of all sums due and payable to "JAS Family Limited Partnership
:.#!h!:fsgt ••
#4 LTD'' after deduction of closing costs, commi
| 3,516
| 4,475
| 2,759
| 2,489
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|
}\psi)(x,\omega,\cdot,E)\big)(E)$. We have by , \[csda5\] & [H]{}\_1((\_[22,1]{})(x,,,E))(E) = \_E\^[E\_m]{}[1]{}(\_[22,1]{})(x,,E’,E)dE’\
=& \_E\^[2E]{}[1]{}(\_[22,1]{})(x,,E’,E)dE’+ \_[2E]{}\^[E\_m]{}[1]{}(\_[22,1]{})(x,,E’,E)dE’\
& \_E\^[2E]{}[1]{}\_1(x,E’,E)(x,,E’)dE’+ \_[2E]{}\^[E\_m]{}[1]{}(\_[22,1]{})(x,,E’,E)dE’. Due to the Taylor’s formula, & \_E\^[2E]{}[1]{}\_1(x,E’,E)(x,,E’)dE’\
& \_E\^[2E]{}[1]{}(\_1(x,E,E)(x,,E)+ R\_[1,1]{}(E’)(E’-E))dE’ where \[csda5-b\] R\_[1,1]{}(E’)=\_0\^1 (\_1(x,,E)(x,,))(E+t(E’-E))dt. The [*second CSDA-type approximation*]{} (which is valid if $E\approx E'$) is that \[csda5-c\] R\_[1,1]{}(E’)(E’-E)=0, which gives by (\[hada1\]) that approximately \[csda5-a\] & [H]{}\_1((\_[22,1]{})(x,,,E))(E)\
& \_E\^[2E]{}[1]{}\_1(x,E,E)(x,,E)dE’\
& +\_[2E]{}\^[E\_m]{}[1]{}(\_[22,1]{})(x,,E’,E)dE’\
=& \_1(x,E,E)(x,,E)(E) +(K\_[22,1,2]{})(x,,E), where $$(K_{22,1,2}\psi)(x,\omega,E):=
\int_{2E}^{E_m}{1\over{E'-E}}(\ol {{{\mathcal{}}}K}_{22,1}\psi)(x,\omega,E',E)dE'.$$ Applying the approximation (\[d-approx\]), $K_{22,1,2}$ is approximately a partial Schur integral operator, say $\tilde K_{22,1,2}$.
Next, consider the term ${{{\mathcal{}}}H}_2((\ol{{{\mathcal{}}}K}_{22,2}\psi)(x,\omega,\cdot,E))(E)$. In virtue of the Taylor’s formula, $$\begin{gathered}
\label{csda6}
\hat\sigma_2(x,E',E)\psi(x,\omega,E')
=
\hat\sigma_2(x,E,E)\psi(x,\omega,E)
\\
+
{\partial\over{\partial E'}}\big(\hat\sigma_2(x,\cdot,E)\psi(x,\omega,\cdot)\big)(E)(E'-E)+R_{2,2}(E')(E'-E)^2, \end{gathered}$$ where $$R_{2,2}(E')={2\over{2!}}
\int_0^1(1-t){\partial^2\over{\partial E'^2}}\big(\hat\sigma_2(x,\cdot,E)\psi(x,\omega,\cdot)\big)(E+t(E'-E))dt$$ In the case where $E\approx E'$ we can omit the residual term that is, \[csda6-a\] R\_[2,2]{}(E’)(E’-E)\^2=0 which is the [*third CSDA-type approximation*]{}. Then we get by (\[hada1\]), (\[hada2\]) approximately \[csda8\] & [H]{}\_2((\_[22,2]{})(x,,,E))(E)\
& \_E\^[2E]{}[1]{}( \_2(x,E,E)(x,,E) + (\_2(x,,E)(x,,))(E))dE’\
& +\
| 3,517
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|
ainties in the anisotropic shadowing effect (Sec. \[sec:direction\_dep\]), we study the cases with different shadow opening angles $\theta_{\mathrm{shadow}}$ by reducing it from $45^\circ$ in “Dds run” (here we also call it “s100 run”) to $33.75^\circ$ (“s075 run”), $22.5^\circ$ (“s050 run”) and $11.25^\circ$ (“s025 run”). We call this series of runs as “s-series”. We take $M_{\mathrm{BH}}=10^3M_\odot$ and $n_\infty=10^5{\,\mathrm{cm^{-3}}}$ for the s-series.
Our main findings are as follows: in all the runs of the s-series, the overall flow structures are similar and the accretion rates $\dot{M}$ are much higher than in the cases without the shadow (i.e., Di and Ddn runs). The obtained accretion rates and opening angles of the equatorial neutral region at Bondi radius $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ are summarized in Table \[tab:s-model\]. The values of $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ agree well with the prediction by equation with only small offsets $\lesssim 3^\circ$. Note also that $\theta_{\mathrm{inflow}}(r_{\mathrm{B}}) \simeq \theta_{\mathrm{shadow}}$ despite the gradual transition between the shadowed and non-shadowed regions modeled as in equation .
The equatorial inflow rates $\dot{M}_{\mathrm{inflow}} (r)$ (see equation \[eq:16\]) are shown in Fig. \[fig:mdotio\_sdep\](a). The values of $\dot{M}_{\mathrm{inflow}}$ at $R_{\mathrm{in}}$ agree well with the rates estimated by the Bondi flow through the solid angle of $4\pi\sin\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ (equation \[eq:17\]; arrows in Fig. \[fig:mdotio\_sdep\] a), but with slight downward offset due to the photoevaporation mass loss. Fig. \[fig:mdotio\_sdep\](b) shows the outflow rates in polar directions $\dot{M}_{\mathrm{outflow}}(r)$ (again, see equation \[eq:16\]). As seen in Sec. \[sec:shadow\_rad\], the estimate by equation with $f_{\mathrm{outflow}}=0.7$ gives a good fit to the numerical results. Small differences of $\sim$ a few $\times$ 10 % among them are comparable to
| 3,518
| 1,385
| 2,872
| 3,375
| 2,317
| 0.780891
|
github_plus_top10pct_by_avg
|
n [(\[eq:Theta’-evdec\])]{} to $\Theta'_{y,x;{{\cal A}}}$ and $\Theta''_{y,x,v;{{\cal A}}}$, respectively.
**(a)** First we investigate the contribution to $\Theta'_{y,x;{{\cal A}}}$ from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}}}$: $$\begin{aligned}
{\label{eq:contr-(a)}}
\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}
({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}.\end{aligned}$$ For a set of events $E_1,\dots,E_N$, we define $E_1\circ\cdots\circ E_N$ to be the event that $E_1,\dots,E_N$ occur *bond*-disjointly. Then, we have $$\begin{aligned}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}\leq{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}\leq\sum_{u\in{{\cal A}}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}u
\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x},\end{aligned}$$ where the right-hand side does not depend on ${{\bf m}}$. T
| 3,519
| 2,344
| 2,392
| 3,198
| null | null |
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|
ns with left-sided pleural-based focus of confluent fluid attenuation that extends through the anterior chest wall and insinuates between the pectoralis major and minor muscles representing empyema necessitans (arrows) are shown.](CRIID2018-4906547.001){#fig1}
{#fig2}
######
Reported cases of empyema necessitans due to *S. aureus*.
Study Age of the patient Isolate/organism Risk factors Invasive procedures Treatment Outcomes and complications
------------------------------ -------------------- ---------------------------------------------------------- ------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------
Stallworth et al. \[[@B14]\] 8 months MRSA (blood and pleural fluid)
| 3,520
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| 3,688
| 3,524
| null | null |
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|
er
iB(-2I_{21}-q_0I_{11}+2I_{10}+q_0I){\vec{\sigma}_1}({\vec{q}}\times{\vec{p}})
\Big]\,.\end{aligned}$$ We have used the master integrals with $q_0=-\frac{M_\Lambda-M_N}{2}$, $q_0'=-M_\Lambda+M_N$ and ${\vec{q}}={\vec{p}}'-{\vec{p}}$.
![Second type of down-triangle involving the intermediate exchange of a $\Sigma$.[]{data-label="downtri2"}](downtriangle2g)
The second type of down-triangle diagram involves the intermediate exchange of the $\Sigma$ (Fig. \[downtri2\]). Its amplitude is $$\begin{aligned}
V_e=&
\frac{G_Fm_\pi^2D_s}{4\sqrt{3}f_\pi^3}
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon}
\nonumber\\\times&\nonumber\,
\frac{1}{(l+q)^2-m_\pi^2+i\epsilon}\,
\frac{(2l^\mu+q^\mu)l^\nu}{k_N^2-M_\Sigma^2+i\epsilon}
\\\times&\nonumber\,
{\overline{u}}_1({\overline{E}},{\vec{p}\,'})(A_{\Sigma}+B_{\Sigma}\gamma_5)({\cancel{k}_N}+M_\Sigma)
\gamma_{\nu}\gamma_5 u_1(E_p^\Lambda,{\vec{p}})
\\\times&\nonumber\,
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\mu u_2(E_p,-{\vec{p}}) \,.\end{aligned}$$ Using the heavy baryon expansion $$\begin{aligned}
V_e=&
-\frac{G_Fm_\pi^2D_s}{8\sqrt{3}M_N f_\pi^3}
\Big[
B_{\Sigma}\Big(-2I_{30}+(-5q_0-2\Delta M_\Sigma)I_{20}
\\+&\nonumber
2(3-\eta)I_{32}+2{\vec{q}}^2I_{33}+2{\vec{q}}^2I_{21}+{\vec{q}}^2I_{21}
\\+&\nonumber
q_0(-2q_0-\Delta M_\Sigma)I_{10}
+(3-\eta)q_0I_{22}+q_0{\vec{q}}^2I_{23}+q_0{\vec{q}}^2I_{11}\Big)
\\-&\nonumber
2A_{\Sigma} M_N(2I_{21}
+q_0I_{11})({\vec{\sigma}_1}\cdot{\vec{q}})
\Big]\,.\end{aligned}$$ The isospin is taken into account by replacing every $A_\Sigma$ and $B_\Sigma$ by $$\begin{aligned}
\frac{2}{3}\left(\sqrt3 A_{\Sigma\frac12}+
A_{\Sigma\frac32}\right)
\vec{\tau_1}\cdot\vec{\tau_2},
~~~
\frac{2}{3}\left(\sqrt3 B_{\Sigma\frac12}+
B_{\Sigma\frac32}\right)
\vec{\tau_1}\cdot\vec{\tau_2}\,,\end{aligned}$$ where, we have used the master integrals with $q_0=-\frac{M_\Lambda-M_N}{2}$, $q_0'=M_\Sigma-M_\Lambda$ and ${\vec{q}}={\vec{p}}'-{\vec{p}}$.
Box diagrams {#sec:boxs}
============
We have two kind of direct box diagrams and two cros
| 3,521
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|
iv\,
\frac{1}{i}
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m^2+i\epsilon} \,,$$ $$\begin{aligned}
A_{;\mu;\mu\nu}(q,q')\equiv\,&
\frac{1}{i}{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{(l+q)^2-m^2+i\epsilon}
\\&\times
\frac{1}{-l_0-q_0'+i\epsilon}(1;l_\mu;l_\mu l_\nu) \,,\end{aligned}$$ $$\begin{aligned}
C_{;\mu;\mu\nu;\mu\nu\rho}(q_0,q_0')\equiv\,&
\frac{1}{i}{\int\frac{d^4l}{(2\pi)^4}}\,
\frac{1}{(l+q)^2-m^2+i\epsilon}\,
\\&\times
\frac{1}{-l_0-q_0'+i\epsilon}
\frac{(1;l_\mu;l_\mu l_\nu;l_\mu l_\nu l_\rho)}{-l_0+i\epsilon} \,,\end{aligned}$$ $$\begin{aligned}
D_{;\mu;\mu\nu;\mu\nu\rho}(q_0,q_0')\equiv\,&
\frac{1}{i}{\int\frac{d^4l}{(2\pi)^4}}\,
\frac{1}{(l+q)^2-m^2+i\epsilon}
\\&\times
\frac{1}{-l_0-q_0'+i\epsilon}
\frac{(1;l_\mu;l_\mu l_\nu;l_\mu l_\nu l_\rho)}{l_0+i\epsilon} \,.\end{aligned}$$ The integrals can be divided depending on their subindexes being temporal or spatial. We show explicitly all the cases for the integrals $J$. The same definitions are used for all the other integrals. Therefore, to know any other integral one needs to replace in Eq. (\[eq:many\]) $J$ by $A$, $B$, $I$, etc. $$\begin{aligned}
J_\mu\equiv\,&
\delta_{\mu0}J_{10}+\delta_{\mu i}J_{11}{\vec{q}}_i \label{eq:many}\\\nonumber\\
J_{\mu\nu}\equiv\,&
\delta_{\mu0}\delta_{\nu0}J_{20}
+(\delta_{\mu0}\delta_{\nu i}
+\delta_{\mu i}\delta_{\nu 0})J_{21}{\vec{q}}_i
\nonumber\\&
+\delta_{\mu i}\delta_{\nu j}(J_{22}\delta_{ij}
+J_{23}{\vec{q}}_i{\vec{q}}_j)
\nonumber\\\nonumber\\
J_{\mu\nu\rho}\equiv\,&
\delta_{\mu0}\delta_{\nu0}\delta_{\rho0}J_{30}
+\delta\delta\delta_{\{\mu\nu\rho 00i\}}{\vec{q}}_iJ_{31}
\nonumber\\&
+\delta\delta\delta_{\{\mu\nu\rho 0ij\}}
(\delta_{ij}J_{32}+{\vec{q}}_i{\vec{q}}_jJ_{33})
\nonumber\\&
+\delta_{\mu i}\delta_{\nu j}\delta_{\rho k}
(\delta{\vec{q}}_{\{ijk\}}J_{34}+{\vec{q}}_i{\vec{q}}_j{\vec{q}}_kJ_{35})
\nonumber\\\nonumber\\
J_{\mu\nu\rho\sigma}\equiv\,&
\delta_{\mu0}\delta_{\nu0}\delta_{\rho0}\delta_{\sigma0}J_{40}
+\delta\delta\delta\delta_{\{\mu\nu\rho\sigma000i\}}{\vec{q}}_iJ_{41}
\nonumber\\&
+\delta\delta\delt
| 3,522
| 1,208
| 1,542
| 3,645
| null | null |
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|
f(y)\right\vert dy\leq \frac{C}{(\lambda t)^{\theta_0(q+\theta_1)}}\int
\frac{\psi _{\pi (q,\kappa +d)}(y)}{\psi _{\kappa +d}(x-y)}\times \left\vert
f(y)\right\vert dy.$$By (\[NOT3b\]) $\psi _{\kappa +d}(x)/\psi _{\kappa +d}(x-y)\leq C \psi
_{\kappa +d}(y)$ so that$$\begin{aligned}
\psi _{\kappa +d}(x)\left\vert \partial ^{\alpha }S_{t}^{\ast
}f(x)\right\vert &\leq \frac{C}{(\lambda t)^{\theta_0(q+\theta_1)}} \int
\frac{ \psi _{\kappa +d}(x)\psi _{\pi (q,\kappa +d)}(y)}{\psi _{\kappa
+d}(x-y)} \times \left\vert f(y)\right\vert dy \\
&\leq \frac{C}{(\lambda t)^{\theta_0(q+\theta_1)}} \int \psi _{\pi (q,\kappa
+d)+\kappa +d}(y)\times \left\vert f(y)\right\vert dy \\
&=\frac{C}{(\lambda t)^{\theta_0(q+\theta_1)}}\left\Vert f\right\Vert
_{0,\nu ,1}\end{aligned}$$We conclude that $$\left\Vert S_{t}^{\ast }f\right\Vert _{q,\kappa +d,\infty }\leq \frac{C}{%
(\lambda t)^{\theta_0(q+\theta_1)}}\left\Vert f\right\Vert _{0,\nu ,1}.$$ By (\[NOT5a\]) $\left\Vert S_{t}^{\ast }f\right\Vert _{q,\kappa ,p}\leq
C\left\Vert S_{t}^{\ast }f\right\Vert _{q,\kappa +d,\infty }$ so the proof of (\[B1\]) is completed.
**B.** Let $\gamma $ with $\left\vert \gamma \right\vert \leq q_{1}$. Using integration by parts$$\begin{aligned}
\partial ^{\gamma }S_{t}(\psi _{\kappa }\partial ^{\alpha }f)(x) &=&\int_{{%
\mathbb{R}}^{d}}\partial _{x}^{\gamma }s_{t}(x,y)\psi _{\kappa }(y)\partial
^{\alpha }f(y)dy \\
&=&(-1)^{\left\vert \alpha \right\vert }\int_{{\mathbb{R}}^{d}}\partial
_{y}^{\alpha }(\partial _{x}^{\gamma }s_{t}(x,y)\psi _{\kappa }(y))\times
f(y)dy.\end{aligned}$$Using (\[NOT3c\]), (\[h3\]) and (\[NOT3b\]), it follows that $$\begin{aligned}
\left\vert \partial ^{\gamma }S_{t}(\psi _{\kappa }\partial ^{\alpha
}f)(x)\right\vert &\leq \int_{{\mathbb{R}}^{d}}\left\vert \partial
_{y}^{\alpha }(\partial _{x}^{\gamma }s_{t}(x,y)\psi _{\kappa
}(y))\right\vert \times \left\vert f(y)\right\vert dy \\
&\leq \int_{{\mathbb{R}}^{d}}\left\vert s_{t}(x,y)\psi _{\kappa
}(y)\right\vert _{q_{1}+q_{2}}\times \left\vert f(y)\right\vert dy \\
&\leq C \int
| 3,523
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| 2,105
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|
ed transport system (\[intro10a\])-(\[intro12\]), which has not been studied in the literature, in $L^2(G\times S\times I)^3$-based spaces. We also show certain a priori estimates (needed e.g. in section \[irtpre\]) which in particular show (under specific assumptions) that the solution depends continuously on the data. Analogous results for the [*adjoint problem*]{} are formulated in section \[adjoint\]. Section \[comp\] considers certain computational aspects. The emphasis is on how to calculate numerical solution (for the forward problem), in principle, without inversion of (huge) matrices. Finally, in the last section \[irtpre\] we outline a related IRTP-problem but its thorough study remains open for future work.
Preliminaries {#pre}
=============
Notations, Assumptions and Introduction of Relevant Function Spaces {#fs}
-------------------------------------------------------------------
We assume that $G$ is an open bounded connected set in ${\mathbb{R}}^3$ such that $\ol{G}$ is a $C^1$-manifold with boundary (as a submanifold of ${\mathbb{R}}^3$; cf. [@lee]). In particular, it follows from this definition that $G$ lies on one side of its boundary.
The unit outward (with respect to $G$) pointing normal on $\partial G$ is denoted by $\nu$, and the surface measure (induced by the Lebesgue measure) on $\partial G$ is written as $\sigma$. We let $S=S_2$ be the unit sphere in ${\mathbb{R}}^3$ equipped with the usual rotationally invariant surface measure $\mu_{S}$. Here and in what follows we typically refer to the measures of interest to us, namely Lebesgue measure ${\mathcal{L}}^3$ in ${\mathbb{R}}^3$, and the above (surface) measures $\sigma$ and $\mu_{S}$ simply by $dx$, ${{d}\sigma}$ and ${{d}\omega}$ in the sense that for all sets $A$ measurable with respect to relevant one of them, $${\mathcal{L}}^3(A)=\int_A {d}x,
\quad
\sigma(A)=\int_A {{d}\sigma},
\quad
\mu_{S}(A)=\int_A {{d}\omega}.$$
Furthermore, let $I=[0,E_{\rm m}]$ where $0<E_{\rm m}<\infty$. We could replace $I$ by $I=[E_0,E_{\rm m}]$ or $I=[
| 3,524
| 1,980
| 918
| 3,407
| 3,929
| 0.769226
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|
d event $\xi \in \Xi$. Frame $\mho_{\mathbf{F}}({\mathit{s}})$ conjoins frame $\mho_{\mathbf{F}}({\mathfrak{A}}_\xi({\mathit{s}}))$.
By hypothesis $\xi \in {\prod{\Xi}}$ and ${\mathit{s}} \in {\mathbb{S}}$. By definition \[D:STEP\_SPACE\] of step space, there exist locus $\lambda \in \Lambda$, frame ${\mathbf{f}} = (\psi, \phi) \in {\prod{\Psi}} \times {\prod{\Phi}}$, and functionality ${\mathit{f}} \in {\mathscr{F}}$ such that ${\mathit{s}} = (\lambda, {\mathit{f}}, (\psi, \phi))$. Definition \[D:STEP\_SPACE\_PROJECTION\] establishes that $\mho_{\mathbf{F}}({\mathit{s}}) = (\psi, \phi)$.
By Theorem \[T:AUTOMATON\_OPERATOR\] the automaton induces an iterative operator, so there exists ${\mathfrak{A}}_\xi( {\mathit{s}}) = (\lambda', {\mathit{f}}',(\psi', \phi')) \in {\mathbb{S}}$. Again by definition \[D:STEP\_SPACE\_PROJECTION\], $\mho_{\mathbf{F}}({\mathfrak{A}}_\xi( {\mathit{s}})) = (\psi', \phi')$.
Definition \[D:ITERATIVE\_TRANSFORM\] evaluates ${\mathbf{f}}\,' = (\psi', \phi') = (\phi\xi, {\mathit{f}}'(\phi\xi))$ as the succeeding frame. Definition \[D:CONJOINT\] asserts that frame $(\psi, \phi)$ conjoins frame $(\psi', \phi')$ if ${{\psi'}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Phi}}}}} = \phi$. Here $\psi' = \phi\xi$, so ${{\phi\xi}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Phi}}}}} = \phi$ by virtue of persistent-volatile partition $\Psi = \Phi\Xi$. Thus we conclude $(\psi, \phi)$ conjoins $(\psi', \phi')$.
Since $\mho_{\mathbf{F}}({\mathit{s}}) = (\psi, \phi)$, $(\psi, \phi)$ conjoins $(\psi', \phi')$, and $(\psi', \phi') = \mho_{\mathbf{F}}({\mathfrak{A}}_\xi( {\mathit{s}}))$, then by transitivity $\mho_{\mathbf{F}}({\mathit{s}})$ conjoins $\mho_{\mathbf{F}}({\mathfrak{A}}_\xi( {\mathit{s}}))$.
\[T:AUTOMATON\_WALK\_PROCESS\] Let ${\mathfrak{A}}$ be an automaton and ${\mathbb{S}}$ be a step space with step ${\mathit{s}} \in {\mathbb{S}}$ and volatile excitation $\lbrace \xi_n \rbrace$. Sequential process projection $\overline{\mho}_{\mathbf{F}}({\mathfrak{A}}^{\mathbb{N}}_{\lbrace
| 3,525
| 2,080
| 2,546
| 3,184
| null | null |
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|
exists a random variable $\hat{\Gamma}$ such that $N\leq \hat{\Gamma}$ almost surely and $$\mathbb{P}(\hat{\Gamma} = k) = (1-\hat{q})^{k-1}\hat{q}, \qquad k\in \mathbb{N},$$ for some $\hat{q}=\hat{q}(\alpha, D)$.
Reviewing the proof of Theorem \[main\], we note that it suffices to prove that, in the context of Corollary \[indicators\], for each $x\in D$, there exists a Bernoulli random variable $\hat{J}_x$ with parameter $\hat{q}$ (independent of $x$) such that $\mathbb{P}_x(I_D\geq \hat{J}_x)=1$. To this end, we recall that, without loss of generality, we may choose our coordinate system such that $x = |x|{\rm\bf i}\in D$ is such that $\partial(x) = 0$. The assumption that $D$ is bounded implies that there exists a $\eta$ such that $|x|\leq \eta$. From the definition of RUECC, we know that there exists an $r>0$ and a cone, $C_{0}$, with vertex at $0$, a closest point on $\partial D$ to $x$, which is symmetrically oriented around the line passing through $x$ and $0$, such that $C_{0, r}\coloneqq C_{0} \cap B(0,r)\subset D^{\texttt{c}}$. We have $$\begin{aligned}
\mathbb{P}_{x}(X_{\sigma_{B_1}} \in C_{0, r})
& =\mathbb{P}_{\mathbf i}(X_{\sigma_{B(\mathbf{i},1)}}\in C_{0, r/|x|}) \\
& \geq \mathbb{P}_{\mathbf i}(X_{\sigma_{B(\mathbf{i},1)}}\in C_{0, r/\eta}) \\
& = \frac{\Gamma(d/2)}{\pi^{(d+2)/2}}\,\sin(\pi\alpha/2)\int_{C_{-{\rm\bf i}, (r/\eta)}}\left|1- |y|^2\right|^{-\alpha/2}|y|^{-d}\,{\rm d}y \\
& \eqqcolon\hat{q},
\end{aligned}$$ where $C_{z, u} \coloneqq [C_{0} \cap B(0,u)]-\{z\}$, for $z\in\mathbb{R}^d$ and $u>0$. Note that $\hat{q}$ is necessarily strictly positive. Taking account of scaling, we have $\mathbb{P}_x$-almost surely that $$I_D\geq \mathbf{1}_{\{|x|^{
| 3,526
| 3,840
| 3,034
| 3,046
| null | null |
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|
re trained to obtain the optimal training models. After the test set verification of the optimal training model, the results obtained are shown in [Figure 5](#sensors-20-02119-f005){ref-type="fig"}. [Figure 5](#sensors-20-02119-f005){ref-type="fig"}a,b indicate the linear fitting effect of I--V curves predicted by MLP model and CNN model. Compared with MLP, there are fewer cusps in the nonlinear fitting curves of CNN model, thus it shows that the fitting effect of CNN model is better than that of MLP model. The cusps will greatly reduce the prediction accuracy and increase the error and the degree of difficulty in finding the maximum power for PV modules.
The evaluation terms for the I--V curves are defined as Mean Absolute Error (MAE, seen in equation (14)) and Root Mean Square Error (RMSE) and *RMSE* is $$RMSE = \sqrt{\frac{1}{N}{\sum\limits_{i = 1}^{N}{(y_{i}^{\prime} - y_{i})}^{2}}}$$ where, *N* is set as 50 in this study. *MAE* shows the distance between the predicted value and the measured value, which is close to zero in ideal situations, while *RMSE* further demonstrates the degree of dispersion between the predicted value and the measured value. Obviously, smaller *RMSE* means higher aggregation of the errors. *RMSE* and *MAE* of the eight groups of I--V curves are shown in [Table 2](#sensors-20-02119-t002){ref-type="table"}.
It can be seen from [Table 2](#sensors-20-02119-t002){ref-type="table"} that, when the irradiation intensity is lower than 500 W/m^2^, *MAE* and *RMSE* of MLP and CNN are relatively low, while *MAE* and *RMSE* of MLP become high when he irradiation intensity is higher than 500 W/m^2^.Thus, accuracy of the CNN model is obviously higher than the MLP model. In addition, it indicates that the aggregation of the errors of CNN is higher than that of MLP.
4.2. Analysis of the Fitting Degree {#sec4dot2-sensors-20-02119}
-----------------------------------
The fitting degree of the I--V curves is analyzed in the following. As we all know, Euclidean distance is the distance between two po
| 3,527
| 1,925
| 1,690
| 3,284
| 2,144
| 0.782309
|
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|
ften used Henyey-Greenstein kernel.
The *dose* $D(x)=(D\psi)(x)$ is calculated from the solution of the problem $$\begin{gathered}
\omega\cdot\nabla_x\psi_1+\Sigma_1\psi_1-K_1\psi=f_1, \label{intro10a}\\
-{{\frac{\partial (S_{j,r}\psi_j)}{\partial E}}}+\omega\cdot\nabla_x\psi_j+\Sigma_{j,r}\psi_j-K_{j,r}\psi=f_j,\quad j=2,3,
\label{intro10} \\
{\psi_j}_{|\Gamma_-}=g_j,\quad j=1,2,3, \label{intro11} \\[2mm]
\psi_j(\cdot,\cdot,E_{\rm m})=0,\quad j=2,3, \label{intro12}\end{gathered}$$ by \[intro13\] D(x)=\_[j=1]{}\^3\_[SI]{}\_j(x,E)\_j(x,,E) [[d]{}]{}[[d]{}E]{}, where $\varsigma_j(x,E)$ are [*stopping powers*]{}, which in general can be different from the restricted stopping powers $S_{j,r}$. The dose calculation is a [*forward problem*]{}. The determination of the external particle flux $g=(g_1,g_2,g_3)$ and/or the distribution of internal source $f=(f_1,f_2,f_3)$ is called *inverse radiation treatment planning problem* (IRTP) which is an *inverse problem*. It always requires a dose calculation model. We refer to [@schepard], [@tervo14], [@webb] and references therein for some details concerning the IRTP-problem. In [@frank10] the IRTP-problem has been studied in the context of CSDA-equation for a single particle (when the stopping power is independent of $x$). See also [@boman] where related spatially $3$-dimensional numerical simulations (real case simulations applying finite element methods, FEM) have been explored.
This paper contains several novel contributions to the study of particle transport in tissues. In the beginning we discuss the preliminaries including details (many of which are reproductions of known results) of the so-called escape-time mapping and inflow trace theory. These tools are essential in the treated analysis. After that we consider the existence and uniqueness of solutions for a single (particle) CSDA equation $$\begin{gathered}
-{{\frac{\partial (S_{0}\psi)}{\partial E}}}+\omega\cdot\nabla_x\psi+\Sigma\psi-K\psi=f, \label{intro14} \\
\psi_{|\Gamma_-}=g, \label{intro15} \\[2mm]
\psi(\cd
| 3,528
| 1,193
| 1,799
| 3,482
| 2,149
| 0.782264
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|
3|\geq 28$. Since $28=(4-1)^3+1$, we can use Theorem \[kflemma\] to conclude that $\cI_3$ contains a $4$-flower. Let $k\geq 4$ be maximum such that $\cS$ is a $k$-flower in $\cI_3$, and let $C$ be the core of $\cS$. As $\cI_3$ is $3$-uniform and intersecting, every subfamily $\cG\sse \cI$ has $\tau(\cG)\leq 3$, which implies that $C\neq \mt$. Suppose first that $C=\{a\}$, and suppose $\cI$ is not a star centered at $a$. Let $A\in \cI$ be such that $a\notin A$. Consider the family $\cS_C$. As $\tau(\cS_C)\geq 4$, there exists some $S_1\in \cS_C$ such that $A\cap S_1=\mt$. Consequently, if $S^\pr=S_1\cup \{a\}$, then $S^\pr\in \cI$ and $A\cap S^\pr=\mt$, a contradiction. As a result, we may assume that $C=\{a,b\}$. This implies that $\cS_C$ is a family of singletons. Consequently, $\cS$ is a sunflower with at least $4$ petals.[^5] Additionally, for every $A\in \cI_3$, $A\cap \{a,b\}\neq\mt$.\
Let $\cA=\{A\in \cI_3:A\cap C=\{a\}\}$, and let $\cB=\{B\in \cI_3:B\cap C=\{b\}\}$. We have $|\cI_3|=|S|+|\cA|+|\cB|$. Let $\cAp=\{A-\{a\}:A\in \cA\}$, and $\cBp=\{B-\{b\}:B\in \cB\}$. If $\cAp=\mt$ or $\cBp=\mt$, we can conclude that $\cI_3$, and hence, $\cI$ is a star (centered at either $a$ or $b$), so suppose both are non-empty. Since $\cI$ is intersecting, $\cAp$ and $\cBp$ are cross-intersecting families, i.e. for any $A\in \cAp$ and $B\in \cBp$, $A\cap B\neq \mt.$ Let $V(\cAp)$ and $V(\cBp)$ be the vertex sets of $\cAp$ and $\cBp$ respectively, and let $n(\cX)=|V(\cX)|$ for $\cX\in \{\cAp,\cBp\}$. We first prove the following claims.
\[clm1\] If both $\cAp$ and $\cBp$ are intersecting, or $|\cAp|\geq 2$ and $|\cBp|\geq 2$, then, $|\cX|\leq 2+ n(\cX)$ for each $\cX\in \{\cAp,\cBp\}$.
If $\cAp$ is intersecting, it is either a triangle, or a star. In either case, the bound follows trivially. A similar argument works for $\cBp$, so suppose, without loss of generality that $\cAp$ has two disjoint edges, say $\{xy,x^\pr y^\pr\}$. $\cBp\sse \{xy^\pr,y^\pr y,yx^\pr,x^\pr x\}$, giving the required bound for $\cBp$. Now, if $\c
| 3,529
| 1,935
| 1,685
| 3,299
| null | null |
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|
{pspicture}};\varnothing), \\ G\subset \mathcal
F(\widehat{E})({
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}};\varnothing),\end{aligned}$$ where $G$ corresponds for a given coloring to the space $$G:= span \left<
\begin{pspicture}(0,0.2)(1,1)
\psline(0.5,0)(0.3,0.5)
\psline(0.5,0)(0.7,0.5)
\psline(0.3,0.5)(0.3,1)
\psline(0.3,0.7)(0.1,1)
\psline(0.3,0.7)(0.5,1)
\rput[b](0.7,0.8){$\alpha$}
\end{pspicture}
-
\begin{pspicture}(0,0.2)(2,1)
\psline(0.5,0)(0.3,0.5)
\psline(0.5,0)(0.7,0.5)
\psline(0.7,0.5)(0.7,1)
\psline(0.7,0.7)(0.9,1)
\psline(0.7,0.7)(0.5,1)
\rput[b](1.5,0.6){$\tau_3(\alpha)$}
\end{pspicture}
\text{ , for all } \alpha\in E^{{
\begin{pspicture}(0,0)(0.2,0.2)
\psline[linewidth=1pt](0.1,0)(0.1,0.2)
\end{pspicture}},{
\begin{pspicture}(0,0)(0.2,0.2)
\psline[linewidth=1pt](0.1,0)(0.1,0.2)
\end{pspicture}}}_{{
\begin{pspicture}(0,0)(0.2,0.2)
\psline[linewidth=1pt](0.1,0)(0.1,0.2)
\end{pspicture}}} \right>.$$
Koszulness of $\widehat{\mathcal O}$ {#quadrat-koszul}
====================================
This section is concerned with our main theorem, that Koszulness for $\mathcal O$ implies Koszulness for $\widehat{\mathcal O}$. To set up notation, we briefly recall the notion of quadratic dual, cobar dual and Koszulness of a (colored) operad.
Recall that if the vector space $V$ is an $S_n$-module and $sgn_n$ is the sign representation, then we defined $V^\vee$ to be $V^*\otimes sgn_n$, where $V^*=Hom(V,k)$ denotes the dual space.
Let $C$ be a set of colors. For every quadratic colored operad $\mathcal P$, we define the quadratic dual colored operad $\mathcal P^!:=\mathcal
F(E)^\vee/(R^\perp)$, where $(R^\perp)$ is the ideal in $\mathcal
F(E)^\vee$ generated by the orthogonal complement $R^\
| 3,530
| 2,557
| 1,421
| 3,328
| 1,449
| 0.789405
|
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|
gerbe with rank 8 bundle. Nearly the same analysis applies as in the Spin$(32)/{\mathbb Z}_2$ case. At the level of SCFT, before imposing the left GSO projections, the same duality argument we have just given suggests the gerbe theory should be dual to an $E_8$ bundle, as above. The left GSO for the corresponding bundle duplicates the gerbe ${\mathbb Z}_2$, and so should act trivially on the theory. The dual should be thus be interpreted as class I, and so the result should have the form of a disjoint union of two copies of an $E_8 \times E_8$ compactification. As the details are largely duplicative of the Spin$(32)/{\mathbb Z}_2$ case just discussed, and for which we will see examples below, we will not treat this case further.
We have discussed bundles with structure group $SU(n)$ embedded into ${\rm Spin}(32)/{\mathbb Z}_2$ and $E_8 \times E_8$ in the form of the standard worldsheet construction, but more general embeddings exist, and admit worldsheet descriptions [@anom]. One open question we leave for future work is to generalize the duality discussed here to more general embeddings.
Toroidal orbifold example {#sect:class2-ex1}
-------------------------
Consider a Spin$(32)/{\mathbb Z}_2$ heterotic string compactified on a ${\mathbb Z}_2$ gerbe over $[T^4/{\mathbb Z}_2]$, with a rank eight bundle, defined as follows. The ${\mathbb Z}_2$ gerbe is $[T^4/{\mathbb Z}_4]$, where the ${\mathbb Z}_4$ acts on the $T^4$ by $$x \: \mapsto \: \exp\left( \frac{2\pi i (2k)k}{4} \right) x
\: = \: (-)^k x,$$ so that there is a trivially-acting ${\mathbb Z}_2$ subgroup; only the sectors $k=1, 3$ have twisted bosons. (Mathematically, this is a nontrivial[^14] ${\mathbb Z}_2$ gerbe.) The bundle is the rank eight bundle ${\cal O}^{\oplus 8}$, on which the ${\mathbb Z}_4$ acts (effectively) by fourth roots of unity.
We will compute the spectrum, and discover not only that it is consistent, but in addition that it has the same form as the spectrum of a perturbative $E_8 \times E_8$ compactification on a space, as expected fr
| 3,531
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| 2,114
| 3,394
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The proof of Theorem \[morrat\] will be through a series of lemmas and we begin with the first equality in . Set $d=c+1$; thus $d\in {\mathbb{R}}_{\geq 1}$, with $d\notin \frac{1}{2}+{\mathbb{Z}}$.
Reduction to Category ${\mathcal{O}}$ {#subsec-4.2}
-------------------------------------
If $H_de_-H_d$ is a *proper* two-sided ideal of $H_d$ it must be contained in a primitive ideal, and hence, by [@ginz Generalized Duflo Theorem], annihilate an object from category $\mathcal{O}_d$. Thus it is enough to show that $e_-$ does not annihilate any simple module belonging to $\mathcal{O}_d$.
To do this we first show in Corollary \[poono\] that the composition factors of $\Delta_d(\mu)$ are of the form $L_d(\lambda)$ for $\lambda \leq \mu$. Under the ${\mathbb{Z}}$-strings ordering such a result is proved in [@guay] but as we work with the dominance ordering of partitions and representations, as defined in , this definitely requires work, see also . We then show that the lowest weight copy of the sign module for ${{W}}$ in $\Delta_d(\mu)$ does not occur in any standard module $\Delta_d(\lambda)$ for $\lambda < \mu$. Since $L_d(\mu)$ is the head (that is, the unique simple factor module) of $\Delta_d(\mu)$ it will follow that $e_-L_d(\mu)\neq 0$.
Lemma. {#basiccom}
------
[*Let $c\in {\mathbb{R}}_{\geq 0}$ with $c\notin \frac{1}{2}+\mathbb{Z}$. If $\mathrm{Hom}_{H_c}(\Delta_c(\lambda),\,\Delta_c(\mu))\not=0$ for $\lambda,\mu\in {{\textsf}{Irrep}({{W}})}$, then $\lambda \leq \mu$ in the dominance ordering.* ]{}
Let $S_q = S_q(n,n)$\[schur-defn\] be the $q$-Schur algebra defined in [@DJ Section 1], where $q = \exp(2\pi i c)$. It is conjectured in [@GGOR Remark 5.17] that $S_q{\text{-}{\textsf}{mod}}$ is equivalent to ${\mathcal{O}}_c$. We cannot prove this, but we will show that there is a relationship which implies the lemma.
For each $\mu\in {{\textsf}{Irrep}({{W}})}$ there is an $S_q$-module $W_q(\mu)$, \[q-weyl-defn\] called the [*$q$-Weyl module*]{}. By [@DJ Corollary 8.6], there is an isomorphism $$\label
| 3,532
| 2,512
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| 3,585
| 2,112
| 0.782654
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|
to view this structure is the following. One checks easily that for any cyclic $A$-bimodule $M_\#$, the restriction $j^*M_\# \in {\operatorname{Fun}}(\Delta^{opp},k)$ is canonically isomorphic to the simplicial $k$-vector space $M^\Delta_\#$ associated to the underlying $A$-bimodule $M$ as in . By adjunction, we have a natural map $$\tau_\#:j_!M^\Delta_\# \to M_\#.$$ Then $j_!M^\Delta_\#$ in this formula only depends on $M \in
A{\operatorname{\!-\sf bimod}}$, and all the structure maps which turn $M$ into the cyclic bimodule $M_\#$ are collected in the map $\tau_\#$.
We can now define cyclic homology with coefficients. The definition is rather tautological. We note that for any cyclic $A$-bimodule $M_\#$ – or in fact, for any $M_\# \in {\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$ – we can treat $M_\#$ as a cyclic vector space by forgetting the bimodule structure on its components $M_n$.
\[cycl.def\] The [*cyclic homology $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M_\#)$ with coefficients*]{} in a cyclic $A$-bimodule $M$ is equal to $H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda,M_\#)$.
Of course, , being valid for any cyclic $k$-vector space, also applies to $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M_\#)$, so that we automatically get the whole package – the Connes’ exact sequence, the periodicity endomorphism, and the periodic cyclic homology $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$. By Lemma \[hoch\], $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M_\#)$ coincides with $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$ as defined in .
Gauss-Manin connection. {#getz}
=======================
To illustate the usefulness of the notion of a cyclic bimodule, let us study the behavior of cyclic homology under deformations.
There are two types of deformation theory objects that one can study for an associative algebra $A$. The first is the notion of a [*square-zero extension*]{} of the algebra $A$ by a $A$-bimodule $M$. This is an associative algebra ${\
| 3,533
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| 0.785799
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|
gma}_K^2}{\sigma_K^2}\right)\mathcal{N}_K\notag\\
=&-2(\gamma-1)\sigma_K\int_M\left[|\nabla\nabla v|^2+({\rm Ric}+Kg)(\nabla v,\nabla v)+(\gamma-1)(\Delta v)^2\right]vu\,dV\notag\\
&+2(\gamma-1)\sigma_K\int_MK|\nabla v|^2vu\,dV+2\frac{\dot{\sigma}_K}{\sigma_K}\frac{d}{dt}\mathcal{N}_K
+\left(\frac{\ddot{\sigma}_K}{\sigma_K}
-\frac{2\dot{\sigma}_K^2}{\sigma_K^2}\right)\mathcal{N}_K\notag\\
\le&-2(\gamma-1)\sigma_K\int_M \left[|\nabla\nabla v|^2+({\rm Ric}+Kg)(\nabla v,\nabla v )+(\gamma-1)(\Delta v)^2\right]vu\,dV\notag\\
&+2\left(\frac{\dot{\sigma}_K}{\sigma_K}+{\kappa}\right)\frac{d}{dt}\mathcal{N}_K
+\left(\frac{\ddot{\sigma}_K}{\sigma_K}
-\frac{2\dot{\sigma}_K^2}{\sigma_K^2}-2{\kappa}\frac{\dot{\sigma}_K}{\sigma_K}\right)\mathcal{N}_K.\end{aligned}$$
Inspired by S. Li and X.-D. Li [@LiLi2](see also [@LiLi3] for a survey), we define the Perelman type $\mathcal{W}$-entropy by
$$\begin{aligned}
\label{KWentropy}
\mathcal{W}_K(t):=&\frac1{\dot{\alpha}_K(t)}\frac d{dt}(\alpha_K(t)\mathcal{N}_K(t))
=\mathcal{N}_K+\beta_K(t)\frac{d}{dt}\mathcal{N}_K\notag\\
=&-\sigma_K\int_M\Big[\beta_K(\gamma-1)\Delta v+\big(1+(\log\sigma_K)'\beta_K\big)\Big]vu\,dV,\notag\\
=&\sigma_K\beta_K\int_M\left[\gamma\frac{|\nabla v|^2}{v}-\left(\frac{1}{\beta_K}+\frac{\dot{\sigma}_K}{\sigma_K}\right)\right]vu\,dV,\end{aligned}$$
where $\beta_K(t)=\frac{\alpha_K}{\dot{\alpha}_K}$, then $$\frac{d}{dt}\mathcal{W}_K(t)=\beta_K\left(\frac{d^2}{dt^2}\mathcal{N}_K
+\frac{1+\dot{\beta}_K}{\beta_K}\frac{d}{dt}\mathcal{N}_K\right).$$ Combining and , we have
$$\begin{aligned}
\label{wkpment1}
\frac{d}{dt}\mathcal{W}_K(t)\le&
-2(\gamma-1)\sigma_K\beta_K\int_M \left[|\nabla\nabla v|^2+({\rm Ric}+Kg)(\nabla v,\nabla v )+(\gamma-1)(\Delta v)^2\right]vu\,dV\notag\\
&+2\beta_K\left(\frac{\dot{\sigma}_K}{\sigma_K}+\frac{1+\dot{\beta}_K}{2\beta_K}
+{\kappa}\right)\frac{d}{dt}\mathcal{N}_K
+\beta_K\left(\frac{\ddot{\sigma}_K}{\sigma_K}
-\frac{2\dot{\sigma}_K^2}{\sigma_K^2}
-2{\kappa}\frac{\dot{\sigma}_K}{\sigma_K}\right)\mathcal{N}_K.\end{aligned}$$
On th
| 3,534
| 2,564
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shielding effect.
![Comparison between simulation results and theoretical models including the OML correction for the electric field acting on the grain. Only results with $\Lambda\simeq16$ are shown.[]{data-label="com"}](comcom.eps){width="90mm"}
To determine the value of $\alpha$, Figure \[com\] compares the simulation results and a theoretical electric field around ${\it a}$ ${\it single}$ ${\it isolated}$ ${\it grain}$ including the OML correction. That is to say, the potential $\phi$ was determined by solving Poisson’s equation, $$\label{poi}
\nabla^{2}\phi=-4\pi e\left(n_{{\rm p}}-n_{{\rm e}}\right),$$ with the ion and electron densities given by Eqs. (\[oml\]) and (\[boltz-e\]), respectively. $n_{0}$ in Eqs. (\[oml\]) and (\[boltz-e\]) was approximated as $n_{0}=(n_{{\rm e}0}+n_{{\rm p}0})/2$ for simplicity. The plasma parameter was $\Lambda\simeq16$, which is almost the same as that in our simulations. The electric field $E$ was calculated by taking spatial derivatives of $\phi$. As we have already mentioned, the functional form of Eq. (\[long\]) should be valid far from the grain even if the OML correction is included. Therefore, Poisson’s equation was integrated from a large radial distance toward the inner region by taking $\alpha$ as a free parameter. We then tried to find the values of $\alpha$ for which this theoretical solution reasonably matched the simulation results. It is readily seen from Fig.\[com\] that the simulation results are well explained by this model with $\alpha\simeq 1.8-2.0$. Note again that the theoretical curve is for an isolated grain, whereas the simulation results are obtained with two dust grains. This means that the effect of ODSs is not observed, at least to a detectable level beyond the error bars of our simulations. This result is qualitatively consistent with the suggestion by Markes and Williams (2000). They have shown explicitly that the electrostatic force acting between two grains surrounded by a plasma is repulsive by solving Poisson’s equation. The
| 3,535
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ies $V_m(z,s)$ satisfies $$\label{straightforward}
(\Delta-s(1-s))V_m(z,s)=(2\pi m)^2V_m(z,s+2) \textrm{ when } {\operatorname{Re}}(s)>1,$$ since $f_s(z)=y^s e^{2\pi i m {\operatorname{Re}}z}$ satisfies this equation and because the Laplacian commutes with isometries, so does $V_m(z,s)$, being a sum of translates of $f_s$. Therefore $$\begin{aligned}
\nonumber (\Delta-s(1-s))&(V_m(z,s)-h(y)y^se(mx))\\ =(2\pi m)^2& (V_m(z,s+2)-h(y)y^{s+2}e(mx))\\ \nonumber &-h''(y)y^{s+2}e(mx)-2h'(y)y^{s+1}e(mx)\end{aligned}$$ is also square integrable, since the last two terms are compactly supported. We can therefore use the resolvent $({\Delta}-s(1-s))^{-1}$ to invert this and find $$V_m(z,s)- h(y)y^se(mx)=({\Delta}-s(1-s))^{-1}((2\pi m)^2V_m(z,s+2)-H(z,s))$$ where $$H(z,s)=(2\pi
m)^2h(y)y^{s+2}e(mx))+h''(y)y^{s+2}e(mx)+2h'(y)y^{s+1}e(mx)$$ This defines the meromorphic continuation of $V_m(z,s)$ to ${\operatorname{Re}}(s)>1/2$ by the meromorphicity of the resolvent (see e.g [@Faddeev:1967a]). The singular points are simple and contained in the set of $s\in \C$ such that $s(1-s)$ is an eigenvalue of ${\Delta}$. Since ${\Delta}$ is self-adjoint, these lie on the real line (when ${\operatorname{Re}}(s)>1/2$). The potential pole at $s=1$ has residue a constant times $$\int_{\GmodH}(2\pi m)^2V_m(z,3)-H(z,1)d\mu$$ The contribution from $h''(y)y^{s+2}e(mx)+2h'(y)y^{s+1}e(mx)$ is easily seen to be zero if $T$ is large enough using $\int_0^1
e(mx)dx=0$ when $m\neq 0$. To handle the rest we may unfold to get $$\begin{aligned}
(2\pi m)^2&\int_{\GmodH}(V_m(z,3)-h(y)y^3e(mx))d\mu(z)\\& = (2\pi
m)^2\int_0^\infty\int_0^1(y^3-h(y)y^3)e(mx)y^{-2}dxdy=0 \end{aligned}$$ so $V_m(z,s)$ is analytic at $s=1$. The claim about growth in vertical strips is proved as in [@PetridisRisager:2004a Lemma 3.1].
It is possible to extend the main idea of the proof of Proposition \[continuation\] to prove the meromorphic continuation of $V_m(z,s)$ to $s\in \C$. But since our main aim was to prove Theorem \[equidistribution\] we shall stop here.
[10]{}
Yves
| 3,536
| 2,327
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> 0$ and $x \geq 0$, we define $$\begin{aligned}
y_{2} (a, x) &:= a \g(a, x)^{2} - a \G(a)^{2} + 2 e^{-x} \G(2a, x); \\
y_{3} (a, x) &:= a x^{a - 1} \g(a, x) - \G(2a, x) - x^{2a - 1} e^{-x}; \\
y_{4} (a, x) &:= a (a - 1) \g(a, x) + x^{a} e^{-x} (2x + 1 - a). \end{aligned}$$ Then, we have $$\begin{aligned}
\frac{d}{dx} y_{1} (a, x) &= x^{a - 1} y_{2} (a, x); \\
\frac{d}{dx} y_{2} (a, x) &= 2 e^{-x} y_{3} (a, x); \\
\frac{d}{dx} y_{3} (a, x) &= x^{a - 2} y_{4} (a, x); \\
\frac{d}{dx} y_{4} (a, x) &= x^{a} e^{-x} (3a + 1 - 2x). \end{aligned}$$ From these relations, we find that the (positive or negative) signs of $\frac{d}{dx} y_{i}(a, x)$ and $y_{i + 1}(a, x)$ ($i = 1, 2, 3$) are equal to each other for $a > 0$ and $x > 0$. Let $p_{i} (a)$ ($i = 2, 3, 4$) be the value of $x$ satisfying $y_{i}(a, x) = 0$. It is easily verified that $\lim_{x \to 0+} \frac{d}{dx}y_{4}(a, x) = \lim_{x \to +\infty}\frac{d}{dx}y_{4}(a, x) = \lim_{x \to 0+} y_{4}(a, x) = 0$ and $\lim_{x \to +\infty}y_{4}(a, x) = a (a - 1) \G(a)$ for $a > 0$. Therefore, from the first derivative test, we obtain Tables $1$ and $2$. Moreover, using Lemmas $\ref{lem:4-2-2}$, $\ref{lem:4-2-3}$, and L’Hôpital’s rule, we obtain $$\begin{aligned}
&\lim_{x \to 0+} \frac{d}{dx} y_{3}(a, x)
=
\begin{cases}
\infty & (0 < a < 1), \\
0 & (a \geq 1),
\end{cases}
&
&\lim_{x \to +\infty} \frac{d}{dx} y_{3}(a, x)
=
\begin{cases}
0 & (0 < a < 2), \\
2 & (a = 2), \\
\infty & (a > 2),
\end{cases} \allowdisplaybreaks \\
&\lim_{x \to 0+} y_{3}(a, x)
= - \G(2a) \quad (a > 0),
&
&\lim_{x \to +\infty} y_{3}(a, x)
=
\begin{cases}
0 & (0 < a < 1), \\
1 & (a = 1), \\
\infty & (a > 1),
\end{cases} \allowdisplaybreaks \\
&\lim_{x \to 0+} \frac{d}{dx} y_{2}(a, x)
= - 2\G(2a) \quad (a > 0),
&
&\lim_{x \to +\infty} \frac{d}{dx} y_{2}(a, x)
= 0 \quad (a > 0), \allowdisplaybreaks \\
&\lim_{x \to 0+} y_{2}(a, x)
= 2 \G(2a) - a\G(a)^{2} \quad (a > 0),
&
&\lim_{x \to +\infty} y_{2}(a, x)
= 0 \quad (a > 0), \allowdisplaybreaks \\
&\lim_{x \to 0+} \frac{d}{dx} y_{1}(a,
| 3,537
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9.2\rm\,GHz$. Solid lines show the experimental results and the theoretical curves are dotted for experimental parameter (available only for the 50$\Omega$ load case) and dashed for fitting parameter. The corresponding parameter and the transmission coefficient for the measuring antenna $a$ are listed in Tab. \[tab:01\].](fig3b "fig:"){width=".95\columnwidth"}\
 Fidelity decay $|f(t)|^2$ for three types of perturbation: $\lambda_{\rm 50\Omega}$ (black); $\lambda_{\rm hw}$ (orange, light gray); $\lambda_{\rm oe}$ (green, dark gray) in three different frequency ranges: (i) $7.2-7.7\rm\,GHz$; (ii) $8.0-8.5\rm\,GHz$; (iii) $8.7-9.2\rm\,GHz$. Solid lines show the experimental results and the theoretical curves are dotted for experimental parameter (available only for the 50$\Omega$ load case) and dashed for fitting parameter. The corresponding parameter and the transmission coefficient for the measuring antenna $a$ are listed in Tab. \[tab:01\].](fig3c "fig:"){width=".95\columnwidth"}
For the perturbations with reflecting ends the systems are closed. The correct description for this situations is Eq. (\[eq:s14\]). With the total coupling constants $\lambda_{\rm oe}$ and $\lambda_{\rm hw}$ as a free fitting parameter we find again an agreement between experiment and theory (dashed lines), for both, real and imaginary part. As one would expect from theory for the case of reflecting ends we see a significant imaginary part of the fidelity amplitude $f_I(t)$ (green and orange lines), whereas in the case of an absorbing end (black line) the imaginary part is nearly zero.
It is convenient to continue our discussion in terms of the fidelity (not its amplitude) introduced in Eq. (\[eq:F\_ab\]). In Fig. \[fig:03\] we present the experimental and theoretical fidelity results for the situations 50$\Omega$ load (black lines), open-end (green lines), and hard-wall (orange lines) as perturbation, for three frequency ranges. In case of the closed channels the total phase shift $\varphi(\nu)$ increase
| 3,538
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nction $f_{\textrm{C}}$ such that $$g(x)=f_{\textrm{C}}(\mathscr{M}(x))\in\mathscr{M}(x)$$ is an immediate successor of $x$ chosen from the many possible in the set $\mathscr{M}(x)$. The basic idea in the proof of the first of the three-parts is to express the existence of a maximal element of a partially ordered set $X$ in terms of the existence of a fixed point in the set, which follows as a contradiction of the assumed hypothesis that every point in $X$ has an immediate successor. Our basic application of immediate successors in the following will be to classes $\mathcal{X}\subseteq(\mathcal{P}(X),\subseteq)$ of subsets of a set $X$ ordered by inclusion. In this case for any $A\in\mathcal{X}$, the function $g$ can be taken to be the superset $${\textstyle g(A)=A\bigcup f_{\textrm{C}}(\mathscr{G}(A)),\quad\textrm{where }\mathscr{G}(A)=\{ x\in X-A\!:A\bigcup\{ x\}\in\mathcal{X}\}}\label{Eqn: FilterTower}$$ of $A$. Repeated application of $g$ to $A$ then generates a principal filter, and hence an associated sequence, based at $A$.
**Theorem 4.1.** *Let $(X,\preceq)$ be a partially ordered set that satisfies*
(ST1) *There is a smallest element $x_{0}$ of $X$ which has no immediate predecessor in $X$.*
(ST2) *If $C\subseteq X$ is a totally ordered subset in $X$, then $c_{*}=\sup_{X}C$ is in $X$.*
*Then there exists a maximal element $x_{+}$ of $X$ which has no immediate successor in $X$.*$\qquad\square$
**Proof.** Let $T\subseteq(X,\preceq)$ be a subset of $X$. If the conclusion of the theorem is false then the alternative
(ST3) *Every element $x\in T$ has an immediate successor $g(x)$ in $T$*[^22]
leads, as shown below, to a contradiction that can be resolved only by the conclusion of the theorem. A subset $T$ of $(X,\preceq)$ satisfying conditions (ST1)$-$(ST3) is sometimes known as an $g$*-tower* or an $g$*-sequence:* an obvious example of a tower is $(X,\preceq)$ itself. If $${\textstyle _{\rightarrow}T=\bigcap\{ T\in\mathcal{T}\!:T\textrm{ is an }x_{0}-\textrm{tower}\}}$$ is the $(\mathcal{P}(X),\subset
| 3,539
| 4,378
| 4,349
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| 1,502
| 0.788802
|
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|
35} {\vec{q}}^2
\nonumber\right.\\+&\left.\nonumber
(4-\eta) K_{22}+(5-\eta) K_{34}\right)
{\vec{\sigma}_1}\cdot\left({\vec{p}}\times {\vec{q}}\right)
+2 i B K_{22}
{\vec{\sigma}_2}\cdot\left({\vec{p}}\times {\vec{q}}\right)
\nonumber\\-&\nonumber
2 B \Big(K_{11}{\vec{q}}^2
\left({\vec{p}}\cdot {\vec{q}}+2 q_0{}^2\right)+ K_{23}( 2{\vec{p}}\cdot
{\vec{q}}{\vec{q}}^2+2q_0^2{\vec{q}}^2+{\vec{q}}^4)
\nonumber\\+&\nonumber
K_{35}( {\vec{p}}\cdot {\vec{q}}{\vec{q}}^2+2{\vec{q}}^4)
+K_{22}((4-\eta) {\vec{p}}\cdot{\vec{q}}+{\vec{q}}^2+(6-2\eta)q_0^2)
\nonumber\\+&\nonumber
(5-\eta)K_{34}({\vec{p}}\cdot {\vec{q}}+2{\vec{q}}^2)
+K_{48} {\vec{q}}^4+K_{21}
{\vec{q}}^2 q_0+K_{33} {\vec{q}}^2 q_0
\nonumber\\-&\nonumber
K_{31} {\vec{q}}^2
-K_{43} {\vec{q}}^2
+2(5-\eta) K_{47} {\vec{q}}^2
+(3-\eta) K_{32}
q_0
\nonumber\\-&\nonumber
(3-\eta) K_{42}+(15-8\eta) K_{46}\Big)
\Bigg]\,,\end{aligned}$$ where we have used the master integrals with $q_0=-\frac{M_\Lambda-M_N}{2}$, $q_0'=M_N-M_\Lambda$, and ${\vec{q}}={\vec{p}}'-{\vec{p}}$.
![Second box-type Feynman diagram. \[box2\]](box3g)
The second box diagram (Fig. \[box2\]), which involves a $\Sigma$ propagator, contributes with $$\begin{aligned}
V_g=&
-i\frac{G_Fm_\pi^2g_A^2D_s}{4\sqrt{3} f_\pi^3}
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon}
\\\times&\nonumber\,
\frac{1}{(l+q)^2-m_\pi^2+i\epsilon}\,
\frac{1}{k_N^2-M_\Sigma^2+i\epsilon}
\\\times&\nonumber\,
\frac{l^\rho(l^\nu+q^\nu)l^\mu}{r_N^2-M_N^2+i\epsilon}
\\\times&\nonumber\,
{\overline{u}}_1({\overline{E}},{\vec{p}\,'})(A_\Sigma+B_{\Sigma}\gamma_5)({\cancel{k}_N}+M_N)\gamma_\rho\gamma_5 u_1(E_p^\Lambda,{\vec{p}})
\\\times&\nonumber\,
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\nu\gamma_5({\cancel{r}_N}+M_N)\gamma_\mu\gamma_5 u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using the heavy baryon expansion $$\begin{aligned}
V_g=&
\frac{G_Fm_\pi^2g_A^2D_s}{16\sqrt{3}M_N f_\pi^3}
\Big[
-2B_{\Sigma} K_{22} {\vec{q}}^2 {\vec{\sigma}_1}\cdot {\vec{\sigma}_2}\nonumber\\-&\nonum
| 3,540
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| 3,237
| null | null |
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|
im \mathcal{O} (1),
\label{assumption}\end{aligned}$$ where $L$ denotes baseline, $\Delta m^2_{ji} \equiv m_j^2 - m_i^2$, and $$\begin{aligned}
\frac{ a L }{2E}
&=& \sqrt{2} G_F N_e L
= 0.58
\left(\frac{\rho}{3 \text{g/cm}^3}\right)
\left(\frac{L}{1000 \mbox{km}}\right).
\label{aL-2E}\end{aligned}$$ They probably ensure that our oscillation probability formulas have applicability to the terrestrial LBL and atmospheric neutrino experiments with baseline up to $\sim 10^4$ km and energies from low to high, up to $E \sim 100$ GeV. More precise discussions on where our formulas are valid will be given in sections \[sec:energy-denominator\] and \[sec:higher-order\].
Vacuum mass eigenstate basis, or tilde basis {#sec:tilde-basis}
--------------------------------------------
To formulate perturbative treatment it is convenient to consider the vacuum mass eigenstate basis, the tilde basis, introduced in the previous section $$\begin{aligned}
\tilde{\nu}_{z} = ({\bf U}^{\dagger})_{z \zeta} \nu_{\zeta}.
\label{tilde-basis}\end{aligned}$$ The tilde basis Hamiltonian is related to the flavor basis one as $$\begin{aligned}
\tilde{H} = {\bf U}^{\dagger} H {\bf U}.
\label{tilde-hamiltonian}\end{aligned}$$ The explicit form of $\tilde{H}$ is given by $$\begin{aligned}
\tilde{H} &=& \tilde{H}_{ \text{vac} } + \tilde{H}_{ \text{matt} } =
\left[
\begin{array}{cc}
{\bf \Delta_{a} } & 0 \\
0 & {\bf \Delta_{s} } \\
\end{array}
\right]
+
{\bf U}^{\dagger}
\left[
\begin{array}{cc}
A & 0 \\
0 & 0 \\
\end{array}
\right]
{\bf U}.
\label{H-tilde-3+N}\end{aligned}$$ We parametrize the $(3+N) \times (3+N)$ dimensional flavor mixing matrix ${\bf U}$ as $$\begin{aligned}
{\bf U} = \left[
\begin{array}{cc}
U & W \\
Z & V \\
\end{array}
\right].
\label{U-parametrize}\end{aligned}$$ The matrix $U$ and $V$ are $3 \times 3$ and $N \times N$ matrices, respectively, and $W$ and $Z$ have sizes that just fill in the space. In our $(3+N)$ model, unitarity is obeyed in the whole $(3+N)$ state space: $$\begin{aligned}
{\bf U} {\bf U}^{\dagge
| 3,541
| 3,371
| 3,733
| 3,337
| 3,387
| 0.772732
|
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|
m in the potential comes from the exchange of photons, while the second one is from the exchange of $Z$ bosons. In the calculation of $\Gamma$, we consider only the final states of SM particles, and neglect their masses, since they are light enough compared to the wino-like neutralino we are discussing.
In both cases of the $\DM\CP$ annihilation processes with $S = 0$ and $1$, $V(r)$ is induced from the exchange of $W$ bosons, and both cases have the same form. On the other hand, the absorptive part $\Gamma$ is different each other. These are given by $$\begin{aligned}
V(r)
=
- \alpha_2 \frac{e^{-m_W r}}{r}~,
\quad
\Gamma_{(S = 0)}
=
\frac{1}{2} \frac{\pi\alpha_2^2}{m^2}~,
\quad
\Gamma_{(S = 1)}
=
\frac{25}{24} \frac{\pi\alpha_2^2}{m^2}~,\end{aligned}$$ where $m_W$ is the $W$ boson mass. The potential and absorptive terms in the $\CP\CPC$ annihilation with $S = 1$ are $$\begin{aligned}
V(r)
=
- \frac{\alpha}{r} - \alpha_2 c_W^2 \frac{e^{-m_Z r}}{r}~,
\qquad
\Gamma
=
\frac{25}{24} \frac{\pi \alpha_2^2}{m^2}~.\end{aligned}$$
The $\DM\DM$ two-bodies system is mixed with $\CP\CPC$ state with $S = 0$, in which mixing occurs through a $W$ boson exchange. Thus, the potential and absorptive terms are written by $2 \times 2$ matrices as $$\begin{aligned}
{\bf V}(r)
=
\left(
\begin{array}{cc}
2\delta m - \ds\frac{\alpha}{r} - \ds\alpha_2 c_W^2 \frac{e^{-m_Z r}}{r}
&
-\sqrt{2} \alpha_2 \ds\frac{e^{-m_W r}}{r}
\\
-\sqrt{2} \alpha_2 \ds\frac{e^{-m_W r}}{r}
&
0
\end{array}
\right),
~
{\bf \Gamma}
=
\frac{\pi \alpha_2^2}{2 m^2}
\left(
\begin{array}{cc}
3 & \sqrt{2}
\\
\sqrt{2} & 2
\end{array}
\right).\end{aligned}$$ Off-diagonal elements describe the transition between $\CP\CPC$ and $\DM\DM$ systems.
As seen in these potentials, all processes have attractive channels except for that of $\tilde \chi ^- \tilde \chi ^-$. The overlap between the wave functions of the incident particles are increased compared to the case without including the potentials, and
| 3,542
| 3,785
| 3,518
| 3,416
| 3,988
| 0.768756
|
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|
downarrow}_n$, we have $T^{\lambda}(y)=S^{\lambda}(y)$ (Theorem 10 of [@FD05a]), then $T^{\lambda}(y)= T^{\lambda-}(y)$ follows from the continuity of $S^{\lambda}(y)$ for any $0<\lambda<1$ (Theorem 3 of [@FD05a]).
Now we prove 3). The relation $T^{\lambda+}(y)\subseteq
T^{\lambda}(y)^{\circ}$ is easy from 1) and 2). To prove the reverse relation, we take any $x\in T^{\lambda}(y)^{\circ}$. Then ${{P({x}\otimes c\ra y\otimes
c)}}\geq \lambda$ for some $c$. There are two cases to consider.
Case 1. ${{P({x}\otimes c\ra y\otimes
c)}}>\lambda$. In this case, we know immediately that $x\in T^{\lambda'}(y)\subseteq T^{\lambda+}(y)$ for $\lambda'={{P({x}\otimes c\ra y\otimes
c)}}$.
Case 2. ${{P({x}\otimes c\ra y\otimes
c)}}=\lambda$. Since $x$ is an interior point of $T^{\lambda}(y)$, from Theorem 9 of [@FD05a] we have $x^{\downarrow}_n/y^{\downarrow}_n>\lambda$. Then ${{P({x}\otimes c\ra y\otimes
c)}}<\min\{1,x^{\downarrow}_n/y^{\downarrow}_n\}$. By Theorem 2 of [@FD04], there exists a catalyst $c'$ such that $$P(x\otimes c\otimes c'\ra y\otimes c\otimes c')>{{P({x}\otimes c\ra y\otimes
c)}}=\lambda.$$ So we also have $x\in T^{\lambda'}(y)\subseteq T^{\lambda+}(y)$ for $\lambda'=P(x\otimes c\otimes c'\ra y\otimes c\otimes c')$. $\Box$
Notice that we assume $y^\da_n>0$ in 2) of the above theorem. When $y^\da_n=0$, it is not clear till now whether or not the result still holds.
With similar techniques, we can prove a corresponding result of Theorem \[lem:tplus\] for probabilistic multiple-copy transformation.
\[lem:Mplus\] For any $y\in V^n$ and $0<\lambda<1$,
1\) $M^{\lambda+}(y)$ is open while $M^{\lambda-}(y)$ is closed,
2\) $M^{\lambda+}(y)\varsubsetneq M^{\lambda}(y)\subseteq
M^{\lambda-}(y)$, and when $y^{\downarrow}_n>0$, $M^{\lambda}(y)=
M^{\lambda-}(y)$ if and only if $y^{\downarrow}_2=y^{\downarrow}_n$,
3\) $M^{\lambda}(y)^{\circ}= M^{\lambda+}(y)$.
[*Proof.*]{} Similar to the proof of Theorem \[lem:tplus\]. $\Box$
Now we can show our main result of this section. Rather surprisingly, when the probabil
| 3,543
| 2,553
| 1,426
| 3,431
| 2,884
| 0.776323
|
github_plus_top10pct_by_avg
|
on of the intergrain distance $d$. Note that the distance is normalized by the Debye length defined with the kinetic energy measured at the equilibrium states rather than the initial temperature. The red and green lines are the theoretical curves expected from the standard Yukawa potential and the ODS attractive potential, respectively. Blue triangles and magenta diamonds show the results of our simulations with $\Lambda\simeq13$ and $\Lambda\simeq16$, respectively.[]{data-label="res_ele"}](reseses.eps){width="90mm"}
Figure \[res\_ele\] summarizes the results of our simulations. Blue triangles show the results for $2L^{3}n_{{\rm e}}=5000$, $2L^{3}n_{{\rm p}}=3000$, $\lambda_{{\rm D}}/L\simeq0.15L$, and $\Lambda\simeq13$. Individual triangles represent intergrain distances of $d=0.2L, 0.4L, 0.5L, 0.6L, 0.8L$. Simulations with a different set of parameters ($2L^{3}n_{{\rm e}}=10000$, $2L^{3}n_{{\rm p}}=8000$, $\lambda_{{\rm D}}/L\simeq0.12L$, and $\Lambda\simeq16$) were also run, and the results are shown by magenta diamonds; in this case, the intergrain distances were $d=0.1L, 0.25L, 0.4L, 0.5L, 0.6L, 0.75L, 0.9L$. In all runs, $q=1000e$ and $\lambda_{{\rm D}0}\simeq0.11L$. Note that $\lambda_{{\rm D}}$, which normalizes the intergrain distances in Fig.\[res\_ele\], was defined at the equilibrium states, and thus not necessarily the same in each run because the self-consistent increase in temperature depends on plasma densities, plasma parameters and intergrain distances $d$. The red and green lines in Fig.\[res\_ele\] show the theoretical curves expected from the standard Yukawa potential and the ODS attractive potential of Resendes et al. (1998), respectively, which are written as $$\label{yukawa-e}
qE_{{\rm Yukawa}}\left(d\right)=\frac{q^{2}}{\lambda_{{\rm D}}^{2}}\left[\left(\frac{\lambda_{{\rm D}}}{d}\right)^{2}+\frac{\lambda_{{\rm D}}}{d}\right]\exp\left(-\frac{d}{\lambda_{{\rm D}}}\right)$$ and $$\label{l-j-e}
qE_{{\rm ODS}}\left(d\right)=\frac{q^{2}}{\lambda_{{\rm D}}^{2}}\left[\left(\frac
| 3,544
| 1,178
| 3,287
| 3,615
| 1,788
| 0.785693
|
github_plus_top10pct_by_avg
|
one the same in the CbD approach as well: we could have defined $\text{\ensuremath{\Delta}}$ directly based on the joint of all eight variables without explicitly defining the connection correlations –, and then we would have obtained the result directly from the half-space representation, as we do in the negative probabilities approach.
Leggett–Garg: Contextuality-by-Default
--------------------------------------
The results for Leggett–Garg $\mathbf{Q}_{1},\mathbf{Q}_{2},\mathbf{Q}_{3}$ can be obtained in the same way as for the EPR-Bell systems. In the CbD approach, the observed correlations $\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{2,1}\right\rangle $, $\left\langle \mathbf{Q}_{1,3}\mathbf{Q}_{3,1}\right\rangle $, $\left\langle \mathbf{Q}_{2,3}\mathbf{Q}_{3,2}\right\rangle $, $\left\langle \mathbf{Q}_{1}\right\rangle =\left\langle \mathbf{Q}_{1,2}\right\rangle =\left\langle \mathbf{Q}_{1,3}\right\rangle $, $\left\langle \mathbf{Q}_{2}\right\rangle =\left\langle \mathbf{Q}_{2,1}\right\rangle =\left\langle \mathbf{Q}_{3,2}\right\rangle $, $\left\langle \mathbf{Q}_{3}\right\rangle =\left\langle \mathbf{Q}_{3,1}\right\rangle =\left\langle \mathbf{Q}_{3,2}\right\rangle $ are consistent with the connection correlations $\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{1,3}\right\rangle $, $\left\langle \mathbf{Q}_{2,1}\mathbf{Q}_{2,3}\right\rangle $, $\left\langle \mathbf{Q}_{3,1}\mathbf{Q}_{3,2}\right\rangle $ if and only if these connection correlations are realizable with the given marginals (i.e., each correlation $\left\langle \mathbf{AB}\right\rangle $ has to satisfy $-1+|\left\langle \mathbf{A}\right\rangle +\left\langle \mathbf{B}\right\rangle |\le\left\langle \mathbf{A}\mathbf{B}\right\rangle \le1-|\left\langle \mathbf{A}\right\rangle -\left\langle \mathbf{B}\right\rangle |$ as discussed in the EPR-Bell case above) and satisfy $$\begin{array}{l}
s_{0}\left(\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{2,1}\right\rangle ,\left\langle \mathbf{Q}_{1,3}\mathbf{Q}_{3,1}\right\rangle ,\left\langle \mathbf{Q}_{2,3}\ma
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=&K\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right)+L_1\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right)
\delta (X_i)+\delta_2(t,X_i),\end{aligned}$$ where $$\label{delta2}
\delta_2(t,X_i)=\frac{K^{\prime\prime}(\xi)}{2}
\frac{(t-X_i)^2}{h_{2,n}^2}f(X_i)\delta^2(X_i),$$ $\xi$ being a (random) number between $\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)$ and $\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)+\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\delta (X_i).$ Then, plugging this development and that of $\hat f^{1/2}$ in the definition (\[realest\]) of $\hat f$, we obtain $$\begin{aligned}
\hat f(t;h_{1,n}, h_{2,n})\!\!\!&=&\!\!\!\bar f(t;h_{2,n})\nonumber\\
&&\!\!\!\!+\frac{1}{n h_{2,n}}\sum_{i=1}^{n}L_1\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right)
f^{1/2}(X_i)\delta (X_i)I(|t-X_{i}|<h_{2,n}B)\nonumber\\
&&\!\!\!\!+\frac{1}{n h_{2,n}}\sum_{i=1}^{n}K\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right)
f^{1/2}(X_i)\delta (X_i)I(|t-X_{i}|<h_{2,n}B)\nonumber\\
&&\!\!\!\!+\frac{1}{n h_{2,n}}\sum_{i=1}^{n}f^{1/2}(X_i)\delta_2 (t,X_i))I(|t-X_{i}|<h_{2,n}B)\label{e1}\\
&&\!\!\!\!+\frac{1}{n h_{2,n}}\sum_{i=1}^{n}L_1\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right)
f^{1/2}(X_i)\delta^2 (X_i)I(|t-X_{i}|<h_{2,n}B)\label{e2}\\
&&\!\!\!\!+\frac{1}{n h_{2,n}}\sum_{i=1}^{n}
f^{1/2}(X_i)\delta (X_i)\delta_2 (t,X_i))I(|t-X_{i}|<h_{2,n}B)\label{e3}\\
\!\!\!&=&\!\!\!\bar{f}(t;h_{2,n})+\delta_3(t)\nonumber\\
&&\!\!\!\!+\frac{1}{n h_{2,n}}\sum_{i=1}^{n}L\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right)
f^{1/2}(X_i)\delta (X_i)I(|t-X_{i}|<h_{2,n}B),\label{e4}\end{aligned}$$ where $\delta_3(t)$ is the sum of the terms (\[e1\]), (\[e2\]) and (\[e3\]), which are of a smaller order than the term (\[e4\]) by (\[zero\]) and (\[delta2\]) for $t\in D_r$ (as we will readily check). Since by (\[classic1\]) and (\[classic2\]), $D(y;h_{1,n})$ dominates $b(y;h_{1,n})$ uniformly in $\mathbb R$, we should further decompose (\[e4\]) to display its $D$-part and its $b$-part. By the definitions of $\delta$, $D$ and $b$, we have $$\begin{aligned}
\delta(t)&=&\frac{D(t;h_{1,n})}{2f(t)}+\frac{b(t;h_{1
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y the definition of $C$ and assumption , we have $$C=\frac{\max\{q,0\}}{\kappa}\geq \frac{\max\{q,0\}}{S_0}\geq \frac{q}{S_0}$$ a.e. and hence $$C{\left\langle}\phi,S_0\phi{\right\rangle}_{L^2(G\times S\times I)}\geq q{\left\Vert \phi\right\Vert}_{L^2(G\times S\times I)}^2.$$
Taking Lemmas \[csdale0\] and \[csdale1a\] into account, one can thus estimate $$B(\phi,\phi)={}&
-{\left\langle}\phi,{{\frac{\partial (S_0\phi)}{\partial E}}}{\right\rangle}_{L^2(G\times S\times I)}
+{\left\langle}\phi(\cdot,\cdot,E_m),S_0(\cdot,E_m)\phi(\cdot,\cdot,E_m){\right\rangle}_{L^2(G\times S)}
\nonumber\\
&-\frac{1}{2}\big({\left\Vert \gamma_+(\phi)\right\Vert}_{T^2(\Gamma_+)}^2-{\left\Vert \gamma_-(\phi)\right\Vert}_{T^2(\Gamma_-)}^2\big)
+{\left\Vert \gamma_+(\phi)\right\Vert}_{T^2(\Gamma_+)}^2 \nonumber\\
&+C{\left\langle}\phi,S_0\phi{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}(\Sigma-K_C)\phi,\phi{\right\rangle}_{L^2(G\times S\times I)}
\nonumber\\
\geq {}&
-q{\left\Vert \phi\right\Vert}^2_{L^2(G\times S\times I)}
-\frac{1}{2}{\left\langle}\phi(\cdot,\cdot,E_m),S_0(\cdot,E_m)\phi(\cdot,\cdot,E_m){\right\rangle}_{L^2(G\times S)} \\
&+\frac{1}{2}{\left\langle}\phi(\cdot,\cdot,0),S_0(\cdot,0)\phi(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)}
+{\left\langle}\phi(\cdot,\cdot,E_m),S_0(\cdot,E_m)\phi(\cdot,\cdot,E_m){\right\rangle}_{L^2(G\times S)}\nonumber\\
&
+\frac{1}{2}{\left\Vert \gamma(\phi)\right\Vert}_{T^2(\Gamma)}^2
+q{\left\Vert \phi\right\Vert}_{L^2(G\times S\times I)}^2+c{\left\Vert \phi\right\Vert}^2_{L^2(G\times S\times I)} \\
\geq {}&
\frac{\kappa}{2}\big({\left\Vert \phi(\cdot,\cdot,0)\right\Vert}_{L^2(G\times S)}^2
+{\left\Vert \phi(\cdot,\cdot,E_m)\right\Vert}_{L^2(G\times S)}^2\big)
+\frac{1}{2}{\left\Vert \gamma(\phi)\right\Vert}_{T^2(\Gamma)}^2
+c{\left\Vert \phi\right\Vert}^2_{L^2(G\times S\times I)}.$$
Because $C^1(\ol G\times S\times I)\times C^1(\ol G\times S\times I)$ is dense in $H_1\times H_2$ and since holds, the bilinear form $B(\cdot,\cdot): C^1(\ol G\times S\times I)\times C^1(\ol G\times
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|
variant constant implies $$q_a\partial_\mu q^a=0,\ \ \ \bar{q}^a\partial_\mu \bar{q}_a=0,$$ which means that $$q_aq^a=\mbox{const.}, \ \ \ \bar{q}_a\bar{q}^a=\mbox{const.}$$ As at different points, the normalization should be the same, one can choose the constants to be unit.
The scaling structure is covariantly constant also implies the spin connection can be expressed as $$\omega_\mu=-\frac{1}{c+d}(c\bar{q}_a\partial_\mu q^a+dq^b\partial_\mu \bar{q}_b).$$ One should also impose that $$D_\mu q^a=0.$$ This means the weights of vectors do not change when being parallel transported. This condition implies $$\bar{q}_a\partial_\mu q^a=q^a\partial_\mu \bar{q}_a.$$ Then one reaches the conclusion that in the $(x,y)$ coordinates in the tangent space $$\omega_\mu=0,$$ and in turn $$R=0.$$ However, the affine connection and the torsion $$\Gamma^\rho_{\mu\nu}=e^\rho_a\partial_\mu e^a_\nu,\ \ \ T^a=de^a$$ are now not vanishing. This is the same as the warped geometry of warped CFTs.
Affine connection
-----------------
In this subsection, we discuss the various constraints to determine the affine connection without the help of the zweibein. The starting point is the Newton-Cartan geometry $(M,A_{\mu},G^{\mu\nu})$. $A_\mu$ is a temporal one-form which defines the local time direction, while $G^{\mu\nu}$ is the inverse metric on the spatial slice. One may define $$G^{\mu\nu}=\bar{A}^\mu\bar{A}^\nu,\ \ G_{\mu\nu}=A_\mu A_\nu,$$ and the antisymmetric tensor H\_=e\^a\_e\^b\_h\_[ab]{}=e\^a\_[\[]{} e\^b\_[\]]{} h\_[ab]{}=A\_[\[]{}|[A]{}\_[\]]{}. The velocity field $A^{\mu}$ is defined by A\_|[A]{}\^=0, |[A]{}\^A\_=0, where $\bar{A}_\mu$ is the dual one-form of $A^\mu$ $$\bar{A}_\nu=H_{\mu\nu}A^\mu.$$ The vectors and one-forms are related to the zweibein in the last subsection by $$\hat A=\hat e \cdot \hat q.$$ In components, we have $$\bar{A}^\mu=e_a^\mu\bar{q}^a,\ \ \ A_\mu=e^a_\mu q_a,$$ $$\bar{A}_\mu=e^a_\mu\bar{q}_a,\ \ \ A^\mu=e_a^\mu q^a.$$ The question is what conditions should be imposed to determine the geometry comple
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(E',E))\psi(x,\omega',E')d\omega'
=
\int_{0}^{2\pi}
\psi(x,\gamma_{\mu_{11}(E',E)}(s),E')ds,$$ as desired, since $\gamma_{\mu_{11}(E',E)}(s)=\gamma(s)$ ($=\gamma_{11}(E',E,\omega)(s)$).
It thus follows that (K\_[11]{})(x,,E) =&\_[I’]{} (\_[11]{} )(x,,E’,E)dE\
=& \_[I’]{}\_[11]{}(E’,E)\_[11]{}(x,E’,E) \_[0]{}\^[2]{}(x,(s),E’)ds.
Let us approximate $\delta$-distribution with a smooth function, say $\eta_\epsilon$ as above. Then $$(\hat {{{\mathcal{}}}K}_{11}\psi)(x,\omega,E',E)
\approx
\hat{\sigma}_{11}(x,E',E)\chi_{11}(E',E)\int_{S'}
\eta_\epsilon(\omega'\cdot\omega-\mu_{11}(E',E))\psi(x,\omega',E')d\omega'$$ and so &(K\_[11]{})(x,,E’,E) =\_I (\_[11]{} )(x,,E’,E)dE\
& \_[I’]{}\_[S’]{}\_[11]{}(x,E’,E)\_[11]{}(E’,E)\_(’-\_[11]{}(E’,E))(x,’,E’)d’ dE’\
=:& \_[I’]{}\_[S’]{}\_[11]{}(x,’,,E’,E)(x,’,E’)d’ dE’\
=:& (K\_[11,]{})(x,,E), where $K_{11,\epsilon}$ is a partial Schur integral operator satisfying the below assumptions (\[ass2\]).
In the case where $\psi$ has the first order derivatives (in the weak sense) with respect to $\omega$ that is, $\psi$ belongs to the mixed-norm Sobolev-Slobodevskij space $H^{2,(0,1,0)}(G\times S\times I^\circ)$. We conjecture that [K\_[11,]{}-K\_[11]{}]{}\_[L\^2(GSI)]{}C[\_-]{}\_[H\^[-1]{}()]{}\_[H\^[2,(0,1,0)]{}(GSI\^)]{} and so the approximation of $K_{11}$ with $K_{11,\epsilon}$ would be under control.
\[re:S\_Radon:example\] We make the following remark regarding the fact, as explained in Remark \[re:S\_Radon\], that the unit sphere $S$ could have been equipped with more general Radon measure $\rho$, instead of its typical (Lebesgue-measure induced) measure $\mu_S$. Letting $\rho=H^1$ be the 1-dimensional Hausdorff measure on $S$ (see [@falconer86 pp. 7–10]), and writing $$\underline{\sigma}_{11}(x,\omega',\omega,E',E)
=\chi_{11}(E',E)\frac{\sigma_{11}(x,E',E)}{\sqrt{1-\mu_{11}(E',E)^2}}\chi_{{\mathcal{M}}}(\omega',\omega,E',E),$$ where $\chi_{{\mathcal{M}}}$ is the characteristic function of the set $${\mathcal{M}}:=\{(\omega
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|
rs. The panels of Fig. \[fig:ccd\] show the variable and the comparison stars in the CCD fields. We applied low-order polynomial fits to the light curves to correct for the instrumental trends and for the atmospheric extinction. This method did not affect the pulsation frequency domains. Figure \[fig:lcshort\] shows two illustrative light curve segments of G 207-9 and LP 133-144. All the light curves obtained for both pulsators are presented in Appendix \[app:g207\] and in Appendix \[app:lp133\].
Frequency analyses of the light curves
======================================
We determined the frequency content of the datasets on daily, weekly or monthly, and yearly time bases. We analysed the daily observations with custom developed software tools, as the command-line light curve fitting program <span style="font-variant:small-caps;">LCfit</span> [@2012KOTN...15....1S]. <span style="font-variant:small-caps;">LCfit</span> has linear (amplitudes and phases) and nonlinear (amplitudes, phases and frequencies) least-squares fitting options, utilizing an implementation of the Levenberg-Marquardt least-squares fitting algorithm. The program can handle unequally spaced and gapped datasets. <span style="font-variant:small-caps;">LCfit</span> is scriptable easily, which made the analysis of the relatively large number of nightly datasets very effective.
We performed the standard Fourier analyses of the weekly or monthly data subsets and the whole light curves with the photometry modules of the Frequency Analysis and Mode Identification for Asteroseismology (<span style="font-variant:small-caps;">famias</span>) software package [@2008CoAst.155...17Z]. Following the traditional way, we accepted a frequency peak as significant if its amplitude reached the 4 signal-to-noise ratio (S/N). The noise level was calculated as the average amplitude in a $\pm1200\,\mu$Hz interval around the given frequency.
{width="16cm"}
{width="6cm"} {width="6cm"}
G 207-9
-------
#
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any finite field with $q$ elements $A_{\Gamma,\v}(q)=\#{\rm A}_{\Gamma,\v}(\F_q)$. We call $A_{\Gamma,\v}$ the *$A$-polynomial* of $(\Gamma,\v)$.
Let $\Phi(\Gamma)\subset \Z^I$ be the root system associated with the quiver $\Gamma$ following Kac [@kacconj] and let $\Phi(\Gamma)^+\subset \left(\N\right)^I$ be the subset of positive roots. Let ${\bf C}=(c_{ij})_{i,j}$ be the Cartan matrix of $\Gamma$, namely $$c_{ij}=\begin{cases} 2-2(\text{the number of edges joining $i$ to
itself})\hspace{.2cm}\text{if }i=j\\
- (\text{the number of edges joining $i$ to
$j$})\hspace{1.2cm}\text{ otherwise}.
\end{cases}$$
Then we have the following well-known theorem (see Kac [@kacconj]).
$A_{\Gamma,\v}(q)\neq 0$ if and only if $\v\in\Phi(\Gamma)^+$; $A_{\Gamma,\v}(q)=1$ if and only if $\v$ is a real root. The polynomial $A_{\Gamma,\v}$, if non-zero, is monic of degree $2-{^t}\v{\bf C}\v$. \[kactheo\]
We have the following conjecture due to Kac [@kacconj].
The polynomial $A_{\Gamma,\v}(T)$ has non-negative coefficients. \[kac-conj\]
We will say that a dimension vector $\v$ is *indivisible* if ${\rm gcd}\,\{v_i\}_{i\in I}=1$. Conjecture \[kac-conj\] was proved by Crawley-Boevey and van den Bergh [@crawley-boevey-etal] when the dimension vector $\v$ is indivisible. This was achieved by giving a cohomological interpretation of $A_{\Gamma,\v}(q)$. A more recent work by Mozgovoy [@mozgovoy2] proves Conjecture \[kac-conj\] for any dimension vector for quivers with at least one loop at each vertex. His proof is accomplished via work of Kontsevich-Soibelman [@kontsevich-soibelman] and Efimov [@efimov] on motivic Donaldson-Thomas invariants associated to quivers.
By Kac[@kacconj], there exists a polynomial $M_{\Gamma,\v}(q)\in\Q[T]$ such that $M_{\Gamma,\v}(q):=\#{\rm
M}_{\Gamma,\v}(\F_q)$ for any finite field $\F_q$. The following formula is a reformation of Hua’s formula [@hua].
We have $$\Log\left(\sum_{\v\in(\N)^I}M_{\Gamma,\v}(q)X^{\v}\right)
=\sum_{\v\in(\N)^I-\{0\}}A_{\Gamma,\v}(q)X^{\v},$$ where $X^{\
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|
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|
0.42 \[-2.74, 3.57\] 0.80 -0.67 \[-4.28, 2.95\] 0.72 0.50 \[-2.71, 3.70\] 0.76
Family situation (married vs. stepfamily) 2.50 \[-1.84, 6.84\] 0.26 0.11 \[-2.34, 2.56\] 0.93 -0.46 \[-3.26, 2.35\] 0.75 0.57 \[-1.92, 3.06\] 0.66
Age (of ill child at diagnosis) -0.22 \[-0.41, -0.02\] 0.03^\*^ 0.01 \[-0.10, 0.13\] 0.82 0.07 \[-3.26, 2.35\] 0.33 -0.14 \[-0.26, -0.02\] 0.03^\*^
Sex (female vs. male) 2.38 \[-0.07, 4.82\] 0.06 -0.24 \[-2.19, 1.70\] 0.81 1.04 \[-1.20, 3.28\] 0.36 0.46 \[-1.27, 2.19\] 0.60
| 3,552
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|
ned. So we propose to use EL as follows.
Since the blocks are disjoint, ${\widehat}{\theta}_{1m}, \dots, {\widehat}{\theta}_{Km}$ are independent. We can regard them as one sample and apply EL to make inference on $\theta$. For notational convenience, let $Y_{km}={\sqrt{m}}{\widehat}{\theta}_{k m}$ and $\mu=\sqrt{m}\theta$. Hence, the empirical likelihood ratio for $\mu$ is given by $$\label{eq20}
\mathcal{R}(\mu)=\max\left\{ \prod_{k=1}^{K}K\omega_{k} ~\Big|~~\sum_{k=1}^{K}\omega_{k}Y_{k m}= \mu, \omega_{k}\geq 0,\quad \sum_{k=1}^{K}\omega_{k}=1 \right\}.$$ By the Lagrange multipliers method, we can find the maximum point $$\omega_{k}= \frac{1}{K} \frac{1}{1 + \lambda^\top(Y_{km}- \mu)},$$ where $\lambda=\lambda(\mu)$ satisfies the equation given by $$\label{eq24}
0=\frac{1}{K}\sum_{k=1}^{K} \frac{Y_{km}- \mu}{1 + \lambda^\top(Y_{km}- \mu)}.$$ As in [@Owen1990], we can get the follow Wilks’ theorem.
\[theorem2\] Under Assumptions \[assumption1\]–\[assumption2\], we have $$-2 \log \mathcal{R}(\mu) \stackrel{d}{\longrightarrow} \chi^{2}_{p}$$ as $K, m\to \infty$.
The accuracy of each block estimator increases as $m$ increases. The power of EL increases as $K$ becomes greater. So there is a trade-off between $K$ and $m$. But we are studying massive data, $K$ and $m$ are large enough to guarantee the accuracy of each step’s inference. In simulations, we set $n=10^5$, $K=\{50, 100, 150\}$. The numerical results show that our proposed method is not sensitive to $K$.
Compared with the BLB and SDB, our method provides a specific asymptotic distribution to make inference on $\theta$. It is unnecessary to apply bootstrap to specify critical values. This reduces the computation burden a lot.
Now, we discuss the computational times of our proposed method, BLB and SDB. Let $t(m)$ be the computational time to estimate ${\widehat}{\theta}_{m}$ based on a sample of size $m$. $c(K)$ denotes the cost time of EL based on $K$ blocks. Table \[table1\] presents the comparison. In Table \[table1\], the column
| 3,553
| 1,197
| 2,846
| 3,334
| 4,088
| 0.768181
|
github_plus_top10pct_by_avg
|
by means of the trace. This is not at all trivial. Indeed, if for instance $M_\# \in {\operatorname{Shv}}({{\mathcal C}}_\#)$ is cocartesian, then, while $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}^\#M_\#$ lies in the subcategory ${{\mathcal D}}_{const}(\Lambda,k) \subset {{\mathcal D}}(\Lambda,k)$, the same is certainly not true for the object $M_\# \in {\operatorname{Fun}}(\Lambda,k)$ obtained by forgetting the bimodule structure on $M_n$.
Thus these two objects are different. However, they do become equal after taking cyclic (or Hochschild, or periodic cyclic) homology. Namely, for any $M_\# \in {\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$ we have a natural map $$\label{natu}
M_\# \to L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}^\#M_\#$$ in the derived category ${{\mathcal D}}(\Lambda,k)$, and we have the following result.
\[main\] For every $M_\# \in {\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$, the natural map induces isomorphisms $$\begin{aligned}
HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M_\#) &\cong HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}M_\#),\\
HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M_\#) &\cong HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}M_\#),\\
HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M_\#) &\cong HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}M_\#).\end{aligned}$$
[[*Proof.*]{}]{} By , it suffices to consider $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(-)$; as in the proof of Lemma \[b.ch\], it suffices to consider $M_\# =
I_{n!}E$ given in , with $E$ being the free bimodule $$E = (A^{opp} \otimes A)^{\otimes n} \in {\operatorname{Shv}}({{\mathcal C}}^n) = A^{\otimes
n}{\operatorname{\!-\sf bimod}}$$ for some fixed $n$. Explicitly, we have $$\label{ind.2}
I_{n!}E([n']) = \bigoplus_{f:[n] \t
| 3,554
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|
he peak subtraction method.
Detection sensitivity {#sec:detection}
---------------------
To determine the limits for detecting host galaxies we constructed a sample of 200 randomly selected unsaturated stars, 100 in each observed band, to mimic unresolved quasars. This way we could investigate how our nucleus-removal techniques responded to a undetectable host galaxy, and we could set limits on the size and shape of expected residuals, thus lower flux limits for detectable host galaxies. For these stars the PSFs were created in exactly the same way as for the AGNs. The object itself was always excluded in the PSF production thus each test star and its PSF were fully independent.
This set of simulated ‘naked quasars’ showed that in 88% (97%) of all cases, any residuals – which could be taken as spurious (g)host galaxy detections – had fluxes of less than 5% (10%) of the total object flux. From this we adopted the condition that a real detection should show a residual flux after peak subtraction of at least 5% of the total flux, corresponding to a maximum nucleus-to-host ratio of 20. Because of the systematic oversubtraction inherent in the procedure corrections for the flux have to be applied (see next section).
The final decision if a host galaxy is resolved is based on this criterion. In addition we visually inspected whether the detected flux indeed came from a host galaxy or whether other, unmasked structures were present, using the peak subtracted images and radial profile. If this could be ruled out we classified a host galaxy as detected.
{width="8cm"} {width="8cm"}
Systematic offsets and errors {#sec:errors}
-----------------------------
While the mere detection of an AGN host galaxy can be achieved with comparably little effort, the determination of flux error bars and systematic offsets is much more complicated. We performed extensive simulations of artificial quasar images, composed from empirical PSFs and host galaxy models plus artificial noise matching
| 3,555
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|
e force, describing correspondingly the collapse or the indefinite separation of the branes, just as happened in the Appelquist and Chodos calculation [@ac]. In this case, then, the stabilization of the interbrane distance cannot be due to quantum fluctuations of fields propagationg into the bulk.
AdS Spacetime
--------------
Now, let us consider the curved space case. Since the bulk dimension is odd, there is no conformal anomaly [@bida] and the energy momentum tensor is traceless in the curved case too.[^3] This tensor is related to the flat space one by (see e.g. [@bida]) $$<T^{\mu}_{\ \nu}>_g = a^{-D} <T^{\mu}_{\ \nu}>_{flat}.$$ Hence, the energy density is given by $$\rho = a^{-D} \rho_0.
\label{dilute}$$ The effective potential per unit physical volume on the positive tension brane is thus given by $$V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}(\lambda) =
2\ a_+^{1-D} \int a^D(z) \rho\, dz =
\mp \ell^{1-D}{A \lambda^{D-1} \over (1-\lambda)^{D-1}}.
\label{ve1}$$ Note that the background solution $a(z)=\ell/z$ has only been used in the very last step.
The previous expression for the effective potential takes into account the casimir energy of the bulk, but it is not complete because in general the effective potential receives additional contributions from both branes. We can always add to $V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}$ terms which correspond to finite renormalization of the tension on both branes. These are proportional to $\lambda^0$ and $\lambda^{D-1}$. The coefficients in front of these two powers of $\lambda$ cannot be determined from our calculation and can only be fixed by imposing suitable renormalization conditions which relate them to observables. Adding those terms and particularizing to the case of $D=5$, we have $$V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}(\lambda) = \mp \ell^{-4}\left[{A\lambda^4 \over (1-\lambda)^4} +
\alpha+\beta\lambda^4\right],
\label{confveff}$$ where $A\approx 2.46 \cdot 10^{-3}$. The values $\alpha$ and $\beta$ can be obtained from the obse
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23}$. This shows that $\dim H^1(\tilde{X}_{1},\CC) = 2$ while $\dim H^1(\tilde{X}_{\zeta},\CC) = 0$. Note that the expected dimension of the cokernel is 0 since both spaces have the same dimension. The existence of the special curve $\{F=0\}$ whose equation is in the kernel of this map, satisfying certain local properties at $P_1$, $P_2$, and $P_3$ determines the non-trivial irregularity for $\tilde{X}_{1}$. This phenomenon, classically referred to as *superabundance*, can be geometrically seen in Figure \[fig:superabundance\].
\(A) at (0,2); (B) at (2,0); (C) at (-2,0.2); at (A) [$P_3$]{}; (A) circle \[radius=.1cm\]; at (C) [$P_1$]{}; (C) circle \[radius=.1cm\]; at (B) [$P_2$]{}; (B) circle \[radius=.1cm\]; (A) to\[out=-90, in=150\] (B) to\[out=150,in=0\] ($(B)+(C)-.25*(A)$) to\[out=180,in=9\] (C) to\[out=0,in=-90\] (A); (C) to\[out=0, in=200\] ($(C)+(1.7,0.3)$) to\[out=30,in=-90\] (A) to\[out=90,in=-30\] (B) to\[out=150,in=0\] (C); ($1.3*(C)-.3*(A)$) – ($1.3*(A)-.3*(C)$);
[10]{}
E. Artal, *Sur les couples de [Z]{}ariski*, J. Algebraic Geom. **3** (1994), no. 2, 223–247.
E. Artal, J. Martín-Morales, and J. Ortigas-Galindo, *Intersection theory on abelian-quotient [$V$]{}-surfaces and [$\bf Q$]{}-resolutions*, J. Singul. **8** (2014), 11–30.
R. Blache, *Riemann-[R]{}och theorem for normal surfaces and applications*, Abh. Math. Sem. Univ. Hamburg **65** (1995), 307–340.
M. Blickle and R. Lazarsfeld, *An informal introduction to multiplier ideals*, Trends in commutative algebra, Math. Sci. Res. Inst. Publ., vol. 51, Cambridge Univ. Press, Cambridge, 2004, pp. 87–114.
J.I. Cogolludo-Agust[í]{}n and J. Martín-Morales, *The correction term for the [R]{}iemann–[R]{}och formula of cyclic quotient singularities and associated invariants*, Rev. Mat. Complut. **32** (2019), no. 2, 419–450.
I. Dolgachev, *Weighted projective varieties*, Group actions and vector fields ([V]{}ancouver, [B]{}.[C]{}., 1981), Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 34–71.
H. Esnau
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from the space covered by the training data. The joint distribution of $\mathbf{Y}$ is then $$\label{joint}
\begin{bmatrix} \mathbf{Y}_t \\ \mathbf{y}_* \end{bmatrix} \sim\mathcal{N}\left(0,
\begin{bmatrix} \mathbf{K}(\mathbf{X}_t,\mathbf{X}_t)+\mathbf{E} & \mathbf{K}(\mathbf{x}_*,\mathbf{X}_t)^T \\ \mathbf{K}(\mathbf{x}_*,\mathbf{X}_t) & \mathbf{K}(\mathbf{x}_*,\mathbf{x}_*)+\mathbf{E}_* \end{bmatrix}\right),$$ where $\mathbf{E}$ and $\mathbf{E}_*$ describe the covariance of the error $\mathbf{e}$, i.e. uncertainty of $\mathbf{y}$. In this work, we use $\mathbf{E}*=0.1\mathbf{D}$, where $\mathbf{D}$ is a diagonal matrix that contains the sample variances of the training data $\mathbf{Y}_t$ on the main diagonal. The error matrix $\mathbf{E}= 0.1\mathbf{D}\otimes \mathbf{I}$, where $\otimes$ is the Kronecker product and $\mathbf{I}\in\mathbb{R}^{n_t\times n_t}$ is an identity matrix. For brevity, following shorthand notations are introduced: $$\begin{aligned}
&\mathbf{K} = \mathbf{K}(\mathbf{X}_t,\mathbf{X}_t)\in\mathbb{R}^{n_yn_t\times n_yn_t} \\
&\mathbf{K}_* = \mathbf{K}(\mathbf{x}_*,\mathbf{X}_t)\in\mathbb{R}^{n_y\times n_yn_t}.\end{aligned}$$
In GPR, the kernel matrices $\mathbf{K}$ and $\mathbf{K}_* $ are constructed based on a covariance function. In this study we use stationary Matérn covariance function with $\nu=3/2$, fixed length scale $l=10$, and $\sigma=1$: $$\label{matern}
k(\mathbf{x},\mathbf{x^\prime}) = \left(1+\frac{\sqrt{3}d(\mathbf{x},\mathbf{x^\prime})}{10}\right)\exp\left(-\frac{\sqrt{3}d(\mathbf{x},\mathbf{x^\prime})}{10}\right),$$ where the distance metric $d(\mathbf{x},\mathbf{x^\prime})$ is the Euclidean distance. The covariance function $k(\mathbf{x},\mathbf{x^\prime})$ describes the covariance between the vectors $\mathbf{x}$ and $\mathbf{x^\prime}$ based on the distance between the vectors. The covariance function is the core component of GPR that specifies properties such as smoothness of the regressor.
The covariance function is used to construct univariate kernel matrices $$\be
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iv \: \rho(\tilde{h}^{-1} g \tilde{h} )$$ for all $g \in K$. If $K$ is abelian, this is well-defined. If $K$ is not abelian, then it can be shown (see [@summ]\[section 4\]) that there exists an operator intertwining the representations $h \cdot \rho$ defined by any two lifts, hence $h \cdot \rho$ is well-defined in $\hat{K}$.
Then, the decomposition conjecture for (2,2) theories states that a string on the gerbe $[X/G]$ is the same as a string on the disjoint union of spaces $[ (X \times \hat{K} )/H ]$, together with a flat $B$ field defined in [@summ]\[section 4\].
In the special case that the gerbe $[X/G]$ is banded, the description above simplifies. In this case, the $H$ action on $\hat{K}$ is trivial, and so the decomposition conjecture reduces to the statement that a string on the gerbe $[X/G]$ is the same as a string on a disjoint union of $| \hat{K} |$ copies of $[X/H]$, in which each copy comes with a flat $B$ field determined by acting on the characteristic class of the gerbe with the irreducible representation corresponding to that copy: $$\rho \in \hat{K}: \: H^2([X/H], Z(G)) \: \longrightarrow \:
H^2([X/H], U(1) ).$$
Extensive evidence was presented in [@summ] for this conjecture, ranging from computations of orbifold spectra and partition functions to GLSM results and quantum cohomology computations. Other results have appeared since. For reasons of brevity, we only list two below:
- This conjecture makes a prediction for Gromov-Witten invariants of stacks, namely that the Gromov-Witten invariants of gerbes are equivalent to Gromov-Witten invariants of disjoint unions of spaces. This was checked in the mathematics literature in [*e.g.*]{} [@ajt1; @ajt2; @ajt3; @t1; @gt1; @xt1].
- This conjecture plays an important role in understanding certain GLSM’s. Specifically, it was used in [@cdhps] to understand Landau-Ginzburg points of complete intersections of quadrics, resolving some old unanswered questions, and also providing examples of GLSM’s that realize geometry in a different way than a
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re equivalent, are the wave equation: $$\begin{aligned}
\label{DbLphi}{\underline{D}}(rL\phi)=&-\Omega^2h\Lb\phi,\\
\label{DLbphi}D(r\Lb\phi)=&-{\underline{h}}L\phi.\end{aligned}$$ Finally, we have the following equation about the mass function $m$: $$\begin{aligned}
\label{Dm}Dm=&-\frac{1}{2}{\underline{h}}\Omega^{-2}(rL\phi)^2,\\
\label{Dbm}{\underline{D}}m=&-\frac{1}{2}h(r\Lb\phi)^2.\end{aligned}$$
A priori bounds for the solution
================================
We begin the proof of Theorem \[main\]. Recall that we start from an arbitrary initial data $\alpha_0\in BV$ and the central line has a singular endpoint $e$, approaching which $\frac{2m}{r}\nrightarrow0$. The double null coordinate $(\ub,u)$ is chosen such that $\ub=0, u=-r$ on the boundary of the causal past of $e$, and $u=u_0=-r_0$ and $\ub$ increases towards the future on the initial null cone $C_o$ where $r_0$ is the area radius of $\Cb_e\bigcap C_o$.
Geometry on $\Cb_0$
-------------------
First of all we would like derive some identities on $\Cb_0$, the boundary of the causal past of $e$. We denote the restrictions on $\Cb_0$ of some geometric quantities, which are considered as functions of $u$: $$\psi=\psi(u)=r\Lb\phi\Big|_{\Cb_0},\ \varphi=\varphi(u)=rL\phi\Big|_{\Cb_0},\ \Omega_0=\Omega_0(u)=\Omega\Big|_{\Cb_0},\ h_0=h_0(u)=h\big|_{\Cb_0}.$$ From $u=-r$ on $\Cb_0$, we must have ${\underline{h}}\big|_{\Cb_0}\equiv-1$. Substituting this into , we find $$\begin{aligned}
\label{Omega_0}
\frac{\partial}{\partial u}\log\Omega_0=-\frac{1}{2}\frac{\psi^2}{|u|},\ \text{and hence}\ -\log\frac{\Omega_0^2(u)}{\Omega_0^2(u_0)}=\int_{u_0}^u\frac{\psi^2(u')}{|u'|}{\mathrm{d}}u'.\end{aligned}$$ From , we have $$\begin{aligned}
\label{Omega_0^2h_0}
\frac{\partial}{\partial u}(\Omega_0^2h_0)=-\frac{\Omega_0^2(1-h_0)}{|u|},\ \text{and hence}\ -\log\frac{\Omega_0^2(u)h_0(u)}{\Omega_0^2(u_0)h_0(u_0)}=\int_{u_0}^u\frac{1}{|u'|}\left(\frac{1}{h_0(u')}-1\right){\mathrm{d}}u'.\end{aligned}$$ Because $m\big|_{\Cb_0}\ge0$ and the apparent horizon does not interse
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on $\theta$ in the RST.
The idea of the proof of Proposition \[prop:<2\] is classical: see [@howardnewman2] for first passage percolation models defined from homogeneous PPP on $\mathbb{R}^{2}$ and [@FP] for a directed last passage percolation model on the lattice $\mathbb{Z}^{2}$. Thanks to Fubini’s theorem, we get that for Lebesgue almost every $\theta$ in $[0,2\pi)$, there is at most one semi-infinite path with asymptotic direction $\theta$ with probability $1$. Actually, this statement holds for all $\theta\in[0,2\pi)$ by the isotropic character of the PPP $N$.
Let us denote by $U(\theta)$ the event that there exist at least two different semi-infinite paths in the RST with asymptotic direction $\theta$. Now, assume the event $U(\theta)$ occurs and let $\gamma_{1}$ and $\gamma_{2}$ be two such semi-infinite paths. Let $X$ be a point of the PPP $N$ belonging to $\gamma_{1}$ but not to $\gamma_{2}$. Thus the semi-infinite sub-path of $\gamma_1$ rooted at $X$ belongs to $\mathcal{T}_X$. Then, one of the two semi-infinite paths $\underline{\gamma}_{X}$ and $\overline{\gamma}_{X}$ is trapped between $\gamma_{1}$ and $\gamma_{2}$, by planarity and since paths are non-intersecting (see Lemma \[lemm:croisement\] in appendix). So, it also admits $\theta$ as asymptotic direction.\
Let us denote by $\lambda$ the Lebesgue measure on $[0,2\pi)$. We are interested in the Lebesgue measure of the set $\{\theta ; U(\theta) \}$ of directions $\theta\in [0,2\pi)$ where the event $U(\theta)$ is satisfied. The previous remark implies: $$\begin{aligned}
{{\mathbb E}}\lambda \{\theta ; U(\theta) \} & = & \int_{\Omega} \int_{0}^{2\pi} {{\bf 1}}_{U(\theta)}(\omega) \; d\theta \; d{{\mathbb P}}(\omega) \\
& \leq & \int_{\Omega} \sum_{X\in N(\omega)} {{\bf 1}}_{{\scriptstyle \mathcal{T}_X\;\mbox{\small{unbounded}}}} \int_{0}^{2\pi} {{\bf 1}}_{{\scriptstyle \underline{\gamma}_{X} \;\mbox{\small{or}}\; \overline{\gamma}_{X} \;\mbox{\small{admits}}\; \theta \;\mbox{\small{as}}} \atop \scriptstyle \mbox{\small{asymptotic direction}}
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anks to the boundedness of $D$). In the inequality, we have used the fact that, on $\{\sigma_D <\sigma_{B^*}\}$, we have $X_{\sigma_D}\in B^*\backslash D$, moreover, that, as a continuous function on $\mathbb{R}^d$, ${g}$ is bounded in $B^*\backslash D$. In the second equality, we have used spatial homogeneity and the scaling property of stable processes. In the third equality, we have used Theorem \[BGR\]. The fourth equality follows by changing variables to $z = x + B^*y$ in the integral, appropriately estimating the denominator and the assumption that ${g}$ is continuous and in $L^1_\alpha(D^\mathrm{c})$.
The boundedness of ${f}$ on $D$ and the uniform finite mean of $\sigma_D$ ensures that the second expectation in the definition of $\upsilon$ is bounded on $\overline{D}$. We claim that $\upsilon$ is continuous in $\mathbb{R}^d$ and belongs to $L^1_\alpha(\mathbb{R}^d)$. Continuity of $\upsilon$ follows thanks to path regularity of $X$, the continuity of ${g}$, the openness of $D$ and the fact that $\omega\mapsto X_{\sigma_D}(\omega)$ and $\omega\mapsto\int_0^{\sigma_D(\omega)} {f}(X_s(\omega))\,{\rm d}s$ are continuous in the Skorohod topology (for which it is important that $\omega\mapsto\sigma_D(\omega)$ is finite). Continuity is also a consequence of the classical potential analytic point of view, seeing the identity for $\upsilon$ in in terms of Riesz potentials; see for example the classical texts of [@BH] or [@L]
To check that $\upsilon\in L^1_\alpha(\mathbb{R}^d)$, we need some estimates. For $x\in D^{\rm c}$, $\upsilon(x) = {g}(x)$ and hence, as ${g}\in L^1_\alpha(D^\mathrm{c})$, it suffices to check that $
\int_{D}|\upsilon(x)|/(1+|x|^{\alpha + d})\,{\rm d}x<\infty.$ However, this is trivial on account of the boundedness and continuity of $\upsilon$ on $\overline D$.
Now fix $x'\in D$ and let $B(x')$ be the largest ball centred at $x'$ that is contained in $D$. A simple application of the strong Markov property tells us that $$\begin{aligned}
\upsilon(x) & = \mathbb{E}_x\left[\mathbb{
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ceivable that the constraints from measurements using probes in the charged lepton sector play a dominant role. On the other hand, in the case of low-scale unitarity violation, neutrino oscillation experiments will play key role in constraining unitarity violation.
Essence of the present and the previous papers {#sec:essence}
==============================================
In this section, we present essence of the present and the previous [@Fong:2016yyh] papers in which we try to construct an adequate formulation to describe unitarity violation at low energies, $E \ll m_{W}$.
Unitary 3 active + $N$ sterile neutrino system {#sec:3+N-system}
----------------------------------------------
We have shown in the $(3+N)$ space unitary model that the active neutrino oscillation probability can be written in a sterile sector model-independent way under the constraint on sterile state masses to $0.1\, \text{eV}^2 \lsim m^2_{J} \lsim 1\, \text{MeV}^2$ both in vacuum and to first order in matter effect [@Fong:2016yyh]. The lower limit of the sterile mass range comes from the requirement that fast oscillations due to active-sterile and sterile-sterile mass differences are averaged out due to decoherence. Whereas the upper limit is for sterile neutrinos to take part in reactor neutrino experiments, which may be relaxed to $m^2_{J} \lsim (1 - 10) \,\text{GeV}^2$ for accelerator neutrinos. Then, the obvious question is whether the conclusion remains the same when all order effect of matter is taken into account. We will answer the question in the positive in this paper.
Non-unitary evolution of neutrinos in vacuum {#sec:nonunitarity-vacuum}
--------------------------------------------
Despite large state space of $(3+N)$ dimensions the resultant expression of the oscillation probability has a simple form under the above stated restriction on the sterile neutrino mass spectrum. In vacuum it has the form $$\begin{aligned}
P(\nu_\beta \rightarrow \nu_\alpha) &=&
\mathcal{C}_{\alpha \beta} +
\left| \sum_{j=1}^{3} U_{\alpha
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comp-dd\] [**g**]{}(,,E\_[m]{})=0 holds. Then the problem $$\begin{gathered}
-{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi+CS_0\phi+\Sigma\phi -K_C\phi={\bf f},\ \nonumber\\
{\phi}_{|\Gamma_-}={\bf g},\quad \phi(\cdot,\cdot,E_{\rm m})=0, \label{co3aa-dd}\end{gathered}$$ has a unique solution \[solin\] \_[P\_[C,0]{}]{}(GSI\^)+ L(H\^1(I,T\^2(\_-’)))\_[P\_C]{}(GSI\^),
Substitute $\phi$ in the problem (\[co3aa-dd\]) by $u:=\phi-L{\bf g}$ to obtain $$&
-{{\frac{\partial (S_0u)}{\partial E}}}+ \omega\cdot\nabla_x u+CS_0u+\Sigma u-K_C u,
\nonumber\\
={}&
{\bf f}+{{\frac{\partial (S_0L{\bf g})}{\partial E}}}
-CS_0(L{\bf g})-\Sigma(L{\bf g}) +K_C(L {\bf g})=:\tilde{\bf f}(x,\omega,E),\nonumber$$ where we have used the fact that $\omega\cdot\nabla_x(L{\bf g})=0$ (see Lemma \[trathle1\] with $\Sigma=0$). On the other hand, $$u_{|\Gamma_-}={\phi}_{|\Gamma_-}-(L{\bf g})_{|\Gamma_-}={\bf g}-{\bf g}=0,$$ and, using the compatibility condition (\[comp-dd\]), $$u(\cdot,\cdot,E_{\rm m})=\phi(\cdot,\cdot,E_{\rm m})-{\bf g}(x-t(x,\omega)\omega,\omega,E_{\rm m})=0.$$
We find that the assumptions guarantee that $\tilde {\bf f}\in L^2(G\times S\times I)$ (for more details, see the proof of Corollary \[cdd\] below), and hence by Theorem \[coth2-d\] the problem (for $u$) $$\begin{gathered}
-{{\frac{\partial (S_0u)}{\partial E}}}+ \omega\cdot\nabla_x u+CS_0u+\Sigma u -K_C u=\tilde{\bf f}, \\
u_{|\Gamma_-}=0,\quad u(\cdot,\cdot,E_{\rm m})=0,\end{gathered}$$ has a unique solution $u\in {{{\mathcal{}}}H}_{P_{C,0}}(G\times S\times I^\circ)$. It follows that then $$\phi:=u+L{\bf g}\in {{{\mathcal{}}}H}_{P_{C,0}}(G\times S\times I^\circ) + L\big(H^1(I,T^2(\Gamma_-'))\big),$$ is the wanted unique solution of (\[co3aa-dd\]).
Finally, the last inclusion in is justified by the inclusion (see , , ) $${{{\mathcal{}}}H}_{P_{C,0}}(G\times S\times I^\circ)\subset {{{\mathcal{}}}H}_{P_{C}}(G\times S\times I^\circ),$$ and by the fact that (see the proof of Corollary \[cdd\] below) $$P_C(x,\omega,E,D)L\tilde{{\bf g}}
=
-{{\frac{\p
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ss of stock $H^-\approx 0.51-0.52$. This means that intraday fluctuations of traded value are nearly uncorrelated.
2. For long time fluctuations the data are correlated, but the strength of correlations depends strongly on the liquidity of the stock. As one moves to groups of larger $\ev{f}$, the strength of correlations ($H^+$) grows, up to $H^+\approx 0.8$.
3. If one shuffles the time series, correlations are destroyed, and $H_{\rm shuff} = 0.5$.
The same phenomenon can be characterized by directly plotting the dependence of $H^\pm$ on $\ev{f}$, as done in Fig. \[fig:hurst\]. Such a dependence is well described by a logarithmic law: $$\begin{aligned}
H_i^{\pm} = H_0^\pm + \gamma^\pm \log \ev{f_i},
\label{eq:hurst_scaling}\end{aligned}$$ where $\gamma^-=0.00\pm0.01$, and $\gamma^+=0.053\pm0.01$. For the shuffled time series $\gamma_{\rm shuff}=0$.
These results indicate, at least in the case of traded value, the absence of universal behavior. Liquidity (or, analogously, company size) is a relevant quantity, which acts as a *continuous* parameter of empirical observables, in particular the strength of correlations and the distribution of $f$. Related results can be found in Refs. [@eisler.liquidity; @bonanno.dynsec; @ivanov.itt; @eisler.sizematters].
![The dependence of the scaling exponent $\alpha$ on the window size $\Delta t$ for the years $1994-1995$. The lighter shaded intervals have well-defined Hurst exponents and values of $\gamma$, the crossover is indicated with a darker background. Without shuffling ($\blacksquare$) there are two linear regimes: For shorter windows $\alpha = 0.74 \pm 0.02$, the slope is $\gamma^-=\gamma(\Delta t<60$ min$)=0.00\pm 0.01$ (solid line), while for longer windows $\alpha$ grows logaritmically, with a slope $\gamma^+=\gamma(\Delta t>390$ min$)=0.052\pm 0.01$ (dashed line). For shuffled data (O) the exponent is independent of window size, $\alpha (\Delta t)=0.74\pm0.02$.[]{data-label="fig:alpha"}](EislerFig4){width="255pt"}
Fluctuation scaling {#sec:alpha}
============
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ted entanglement transformation is strictly more powerful than multiple-copy one in either deterministic or probabilistic setting. For purely probabilistic setting, however, we can prove that these two kinds of transformations are geometrically equivalent in the sense that the two sets $T^{\lambda}(y)$ and $M^{\lambda}(y)$, denoting the sets of bipartite pure states which can be converted into a given state with Schmidt coefficient vector $y$ with maximal probabilities not less than $\lambda$ by catalyst-assisted transformation and by multiple-copy transformation, respectively, have the same closure. The limit properties of $T^{\lambda}(y)$ and $M^{\lambda}(y)$ as set-valued functions about $\lambda$ are also discussed.
The results about the relation between catalyst-assisted transformation and multiple-copy transformation shown in this paper and our previous works can be described by the following diagrams: $$\begin{array}{ccccccccc}
M^{\lambda+}(y)&=&M^{\lambda}(y)^\circ&\varsubsetneq&M^{\lambda}(y)&\varsubsetneq&\overline{M^{\lambda}(y)}&{\ensuremath {\underset{\mbox{\sout{\tiny{\,?\,}}}}{\subset}}}&
M^{\lambda-}(y)\\
\rotatebox{90}{=}&&\rotatebox{90}{=}&&\rotatebox{90}{$\varsupsetneq$}& &\rotatebox{90}{=}& & \rotatebox{90}{=}\\
T^{\lambda+}(y)&=&T^{\lambda}(y)^\circ&\varsubsetneq&T^{\lambda}(y)&\varsubsetneq&\overline{T^{\lambda}(y)}&{\ensuremath {\underset{\mbox{\sout{\tiny{\,?\,}}}}{\subset}}}&
T^{\lambda-}(y)
\end{array}$$ for purely probabilistic case ($\lambda<1$) and $$\begin{array}{ccccccccc}
M(y)^\circ&\varsubsetneq&M(y)&\varsubsetneq&\overline{M(y)}&
{\ensuremath {\underset{\mbox{\sout{\tiny{\,?\,}}}}{\subset}}}&
M^{-}(y)\\
\rotatebox{90}{{\ensuremath {\underset{\mbox{\sout{\tiny{\,?\,}}}}{\supset}}}}&&\rotatebox{90}{$\varsupsetneq$}& &\rotatebox{90}{{\ensuremath {\underset{\mbox{\sout{\tiny{\,?\,}}}}{\supset}}}}& & \rotatebox{90}{=}\\
T(y)^\circ&\varsubsetneq&T(y)&\varsubsetneq&\overline{T(y)}&{\ensuremath {\underset{\mbox{\sout{\tiny{\,?\,}}}}{\subset}}}&
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s paper I use these four objects to improve the calibration of the Hubble diagram, and solve for the value of the Hubble constant.
The Luminosity-Velocity Relation {#lv_sec}
================================
The SCM is based on the luminosity-velocity relation, which permits one to standardize the relative luminosities of SNe IIP. Figure \[L\_v.fig\] shows the latest version, based on 24 genuine SNe IIP. This plot reveals the well-known fact that SNe IIP encompass a wide range ($\sim$5 mag) in luminosities. This correlation reflects the fact that while the explosion energy increases, so do the kinetic energy and internal energies. Also plotted in this figure with open circles are the explosion models computed by [@litvinova83] and [@litvinova85] for stars with $M_{ZAMS}$ $\geq$ 8 $M_\odot$, which reveals a reasonable agreement with observations.
![Envelope velocity versus absolute plateau $V$ magnitude for 24 SNe IIP, both measured in the middle of the plateau (day 50) (filled circles). The expansion velocities were obtained from the minimum of the Fe II $\lambda$5169 lines. The absolute magnitudes were derived from redshift-based distances and observed magnitudes corrected for dust extinction. Open circles correspond to explosion models computed by [@litvinova83] and [@litvinova85] for stars with $M_{ZAMS}$ $\geq$ 8 $M_\odot$. []{data-label="L_v.fig"}](L_v.ps){height="75mm"}
The Hubble Diagram
==================
In a uniform and isotropic Universe we expect locally a linear relation between distance and redshift. A perfect standard candle should describe a straight line in the magnitude-log($z$) Hubble diagram, so the observed scatter is a measure of how standard the candle is. Next I assess the performance of the SCM based on the Hubble diagram constructed with the magnitudes and redshifts given in Table \[SN.tab\] for 24 SNe.
-------- --------------- -------------- --------------- ----------- ----------- ---------------
SN $v_{CMB}$ $A_{GAL}(V)$ $A_{host}(V)$ $V_{50}$ $I_{50}$
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Phys. A [**49**]{} (2016) no.44, 445403 \[[[arXiv:1607.00795](http://arxiv.org/abs/1607.00795)]{}\].
[^1]: A more precise field theoretic explanation of what this limit means has been proposed in [@Lozano:2016kum].
[^2]: In some cases it can be that this doesn’t fully fix the gauge and additional fixing should be imposed on the Lagrange multipliers $ V= v_a \tilde{H}^a$, details of this are discussed in [@Lozano:2011kb].
[^3]: An explicit demonstration of the RR transformation law in the context of supersymmetry in $SU(2)$ non-Abelian T-duality can be found in[@Kelekci:2014ima].
[^4]: Care needs be taken in the interpretation of this deformation. Away from the supersymmetric point the $\gamma_i$ deformation is not conformal due a running coupling of a double-trace operator [@Fokken:2013aea] and indeed the gravitational dual has a tachyon [@Spradlin:2005sv].
[^5]: As with the previous example the $B$-field obtained by the central extension dualisation procedure differs by a closed piece $\Delta B = \frac{1}{\nu_1^2+ \nu_2^2 + \nu_3^2} \left( \nu_1 d\phi_2\wedge d\phi_3 + \nu_3 d\phi_1\wedge d\phi_2+ \nu_2 d\phi_3\wedge d\phi_1 \right)$.\[foot:bdiff\]
---
abstract: 'Core to the vision-and-language navigation (VLN) challenge is building robust instruction representations and action decoding schemes, which can generalize well to previously unseen instructions and environments. In this paper, we report two simple but highly effective methods to address these challenges and lead to a new state-of-the-art performance. First, we adapt large-scale pretrained language models to learn text representations that generalize better to previously unseen instructions. Second, we propose a stochastic sampling scheme to reduce the considerable gap between the expert actions in training and sampled actions in test, so that the agent can learn to correct its own mistakes during long sequential action decoding. Combining the two techniques, we achieve a new state of the art on the Room-to-Room benchmark with 6% absolute gain
| 3,568
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|
\textrm{sup}}$ with $x_{1}\in C_{1}$ and $x_{2}\in C_{2}$, then from the $\subseteq$-comparability of $C_{1}$ and $C_{2}$ we may choose $x_{1},x_{2}\in C_{1}\supseteq C_{2}$, say. Thus $x_{1}$ and $x_{2}$ are $\preceq$-comparable as $C_{1}$ is a chain in $(X,\preceq)$; $C_{*}\in\mathcal{X}$ is therefore a chain in $(X,\preceq)$ which establishes that the supremum of a chain of $(\mathcal{X},\subseteq)$ is a chain in $(X,\preceq)$.
The tower Theorem 4.1 can now applied to $(\mathcal{X},\subseteq)$ with $C_{0}$ as its smallest element to construct a $g$-sequentially towered fully ordered subset of $\mathcal{X}$ consisting of chains in $X$ $$\mathcal{C}_{\textrm{T}}=\{ C_{i}\in\mathcal{P}(X)\!:C_{i}\subseteq C_{j}\textrm{ for }i\leq j\in\mathbb{N}\}=\,_{\rightarrow}\mathcal{T}\subseteq\mathcal{P}(X)$$ of $(\mathcal{X},\subseteq)$ — consisting of the common elements of all $g$-sequential towers $\mathcal{T}\in\mathfrak{T}$ of $(\mathcal{X},\subseteq)$ — that infact is a principal filter base of chained subsets of $(X,\preceq)$ at $C_{0}$. The supremum (chain in $X$) $C_{\leftarrow}$ of $\mathcal{C}_{\textrm{T}}$ in $\mathcal{C}_{\textrm{T}}$ must now satisfy, by Thm. 4.1, the fixed point $g$-chain of $X$ $$\sup_{\mathcal{C}_{\textrm{T}}}(\mathcal{C}_{\textrm{T}})=C_{\leftarrow}=g(C_{\leftarrow})\in\mathcal{C}_{\textrm{T}}\subseteq\mathcal{P}(X),$$ where the chain $g(C)=C\bigcup f_{\textrm{C}}(\mathscr{G}(C)-C)$ with $\mathscr{G}(C)=\{ x\in X-C\!:C\bigcup\{ x\}\in\mathcal{X}\}$, is an immediate successor of $C$ obtained by choosing one point $x=f_{\textrm{C}}(\mathscr{G}(C)-C)$ from the many possible in $\mathscr{G}(C)-C$ such that the resulting $g(C)=C\bigcup\{ x\}$ is a strict successor of the chain $C$ with no others lying between it and $C$. Note that $C_{\leftarrow}\in(\mathcal{X},\subseteq)$ is only one of the many maximal fully ordered subsets possible in $(X,\preceq)$.$\qquad\blacksquare$
With the assurance of the existence of a maximal chain $C_{\leftarrow}$ among all fully ordered subsets of a partiall
| 3,569
| 2,175
| 3,355
| 3,235
| 2,877
| 0.776382
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ghput from adding reconfiguration states becomes small when the number of reconfiguration states is already large. Comparing the limiting behaviors of the average throughput gain and the outage throughput gain, we further find that the growth rate of the average throughput and the growth rate of the outage throughput have the same order when the number of reconfiguration states is large.
Fast Selection Algorithm {#sec:fastalgrb}
========================
In the previous section, we analyzed the throughput gain of employing the reconfigurable antennas when the optimal reconfiguration state and beams are selected, while the problem of how to select the optimal reconfiguration state and beams has not been considered. In fact, as will be discussed later, selecting the optimal reconfiguration state and beams among all possible selections is extremely complicated and challenging for practical applications. To overcome the challenge, in this section, we propose a fast selection algorithm with low complexity and near-optimal throughput performance in the sparse mmWave MIMO environment. The objective of selecting the optimal reconfiguration state and beams is to obtain the corresponding optimal ${\widetilde{\mathbf{H}}_{\psi,V}}$ that maximizes the system throughput given in . The design problem of selecting ${\widetilde{\mathbf{H}}_{\psi,V}}$ is formulated as[^7] $$\label{eq:problemselect}
\max_{\psi\in\left\{1,\cdots,\Psi\right\}}\max_{{\widetilde{\mathbf{H}}_{\psi,V}}\in\left\{\tilde{\mathcal{H}}_\psi\right\}}\left|{\mathbf{I}}_{L_r}+\frac{\rho}{L_t}{\widetilde{\mathbf{H}}_{\psi,V}}{\widetilde{\mathbf{H}}_{\psi,V}}^H\right|.$$
A straightforward method to obtain the optimal ${\widetilde{\mathbf{H}}_{\psi,V}}$ is the exhaustive search among all possible selections of ${\widetilde{\mathbf{H}}_{\psi,V}}$. That is, we first search for the optimal beam selection for each reconfiguration state to obtain ${\widehat{\widetilde{\mathbf{H}}}_{\psi,V}}$, i.e., the optimal low-dimensional virtual channel of ${\mathbf{H}_{\psi,
| 3,570
| 1,837
| 1,267
| 3,527
| 3,751
| 0.770325
|
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|
Yes Yes
The proband underwent a series of genetic tests including oligo array, fragile X, *MECP2*, *CDKL5*, *SLC9A6*, and *COH1*, all of which were negative. The sibling also underwent genetic testing (*ZEB2* and *MECP2*). Both sisters had metabolic studies. After these tests failed to provide a molecular diagnosis, the family underwent research exome sequencing in 2014.
TECHNICAL ANALYSIS AND METHODS {#s2}
==============================
The proband, her affected sibling, and both parents underwent exome sequencing as follows. Exome capture was performed using Agilent SureSelect v5 reagents according to manufacturer protocols. Exome libraries underwent paired-end sequencing (2 × 100 bp) on an Illumina HiSeq 2500 instrument. We generated ∼6.2 Gbp of uniquely mapped reads per sample ([Table 2](#MCS002410KOBTB2){ref-type="table"}). Reads were mapped to the GRCh37 reference sequence and secondary data analysis was performed using Churchill ([@MCS002410KOBC4]), which implements the GATK "best practices" workflow for alignment, variant discovery and genotyping. Sequencing metrics are provided in [Supplemental Table 1](http://www.molecularcasestudies.org/lookup/suppl/doi:10.1101/mcs.a002410/-/DC1). Variants were called in all four samples simultaneously, yielding 612,356 variants of which 574,390 (506,121 single nucleotide variants and 68,269 indels) passed minimum quality filters (QUAL \> 100). Family relatedness was confirmed using the KING algorithm (v2.0; see [Supplemental Table 2](http://www.molecularcasestudies.org/lookup/suppl/doi:10.1101/mcs.a002410/-/DC1); [@MCS002410KOBC7]).
######
Exome sequencing metrics for the proband, sibling, and parents described in this study
Sample Reads Mapped Duplicate Map rate (%) Dup. rate Avg. depth
--------- ------------ ------------ ----------- -------------- ----------- ------------
Proband 56,019,520 55,815,413 1,577,081 99.64 2.83% 55.0
Mother 66,0
| 3,571
| 520
| 3,193
| 3,559
| null | null |
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|
js; @Li:2012cfa; @Atwood:2012ac; @Grossman:2012ry; @Brod:2011re]. Another example for category (3) is the $\Delta I=1/2$ rule in the kaon sector which is further discussed in sections \[sec:DeltaI12inKDB\] and \[sec:UandIrules\].
The current data, Eq. (\[eq:mainresult\]), is consistent with category (2). In the SM picture, the measurement of $\Delta a_{CP}^{\mathrm{dir}}$ proves the non-perturbative nature of the $\Delta U=0$ matrix elements with a mild enhancement from $\mathcal{O}(1)$ rescattering effects. This is the $\Delta U=0$ rule for charm.
Note that the predictions for $\Delta a_{CP}^{\mathrm{dir}}$ of category (i) and (ii) differ by $\mathcal{O}(10)$, although category (ii) contains only an $\mathcal{O}(1)$ nonperturbative enhancement with respect to the no QCD limit $\tilde{p}_0=1$. We emphasize that a measure for a QCD enhancement is not necessarily its impact on an observable, but the amplitude level comparison with the absence of QCD effects.
We also mention that we do not need SU(3)$_F$ breaking effects to explain the data. Yet, the observation of $\vert \tilde{s}_1\vert > \vert \tilde{t}_1\vert$ in Eqs. (\[eq:result-ret1tilde\])–(\[eq:result-res1tilde\]) provide additional supporting evidence that rescattering is significant. Though no proof of the $\Delta U=0$ rule on its own, this matches its upshot and is indicative of the importance of rescattering effects also in the broken penguin which is contained in $\tilde{s}_1$.
With future data on the phases $\delta_{KK}$ and $\delta_{\pi\pi}$ we will be able to determine the strong phase $\delta$ of Eq. (\[eq:defC\]). In that way it will be possible to completely determine the characteristics of the emerging $\Delta U=0$ rule.
$\Delta I=1/2$ Rules in $K$, $D$ and $B$ Decays \[sec:DeltaI12inKDB\]
=====================================================================
It is instructive to compare the $\Delta U=0$ rule in charm with the $\Delta I=1/2$ rule in kaon physics, and furthermore also to the corresponding ratios of isospin matrix eleme
| 3,572
| 1,166
| 3,073
| 3,289
| 1,346
| 0.790664
|
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|
is to generalize this result to normal surfaces, see Theorem \[thm:Esnault\].
Cyclic coverings of P2 branched along a curve
---------------------------------------------
In the particular case when $X=\PP^2$ the complex projective plane and $\mathcal{C}$ is a reduced projective plane curve of degree $n$, the description of the $n$-th cyclic cover of $\PP^2$ ramified along $\mathcal{C}$ can be given in very specific terms based on the dimension of the space of curves of a certain degree with a prescribed behavior at the singular points of $\cC$. Such a relation appears in several works and has its origin in Zariski (cf. [@Zariski-irregularity; @Libgober-alexander; @es:82; @Loeser-Vaquie-Alexander; @Sabbah-Alexander; @Artal94; @Libgober-characteristic]).
Before we can give a more precise statement we need to introduce some notation. Let $\sigma:Y\to\PP^2$ be an embedded resolution of $\cC$ (in fact, we can take the minimal one and we may resolve only the set $\operatorname{Sing}^*(\mathcal{C})$ of singular points of $\mathcal{C}$ other than ordinary double points whose branches belong to different irreducible components of $\mathcal{C}$). As a consequence $\sigma^*(\mathcal{C})$ is a *simple normal crossing divisor*. Let $\mathcal{C}_1,\dots,\mathcal{C}_r$ be the irreducible components of $\mathcal{C}$. Then, one can write $$\label{eq:ttC}
\sigma^*(\mathcal{C})=
\sum_{i=1}^r \hat{\mathcal{C}}_i+
\sum_{P\in\operatorname{Sing}(\mathcal{C})}
\sum_{i=1}^{r_P} n_{P,i} E_{P,i}$$ where $\hat{\mathcal{C}}_i$ is the strict transform of $\mathcal{C}_i$ under $\sigma$ and $E_{P,i}$, $i=1,...,r_P$ accounts for the exceptional components in $\sigma^{-1}(P)$. Given $h\in \cO_{X,P}$ the germ of a holomorphic function at $P$, we use the following standard notation $\operatorname{mult}_{E_{P,i}} \sigma^*h$ to describe the order of vanishing of the total transform of $h$ along $E_{P,i}$. Analogously, given $\omega_P$ the germ of a holomorphic $2$-form not vanishing at $P$, we use the notation $\nu_{P,i}:=1+\operatorname{m
| 3,573
| 2,502
| 2,238
| 3,353
| 2,059
| 0.783146
|
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|
y the same, as in the case of the surface-interacting polymer chain in a poor solvent in Euclidean spaces [@r1; @r2; @r3]. On the other hand, the obtained phase diagrams for CSAWs model, resemble the phase diagrams of the same surface-interacting chain problem, in fractal containers [@EZM]. This similarity is not surprising, since in both models studied, one of the two interacting polymers is adhered to one of four fractal surfaces, and its monomers appear as a part of interacting surface (in the surface-interacting polymer problem). Here we may conclude that, our findings should be useful in making the corresponding 3D models of the system of several interacting polymer chains in porous media. Besides, our results may serve inspiring in advancing theories of mutually interacting polymers, as well as for polymers interacting with boundary surfaces of homogeneous 3D lattices, in which case so far (to the best of our knowledge) an exact approach has not been yet made.
Renormalization group equations for the ASAWs model \[app:ASAWsRG\]
===================================================================
In this Appendix we give explicit RG equations for the model in which two chains avoid each other, for the cases $b=2$, and $b=3$ of 3D SG fractals. Equations for “bulk" parameters $A$ and $B$, as well as for the “surface" parameter $C$, were found in earlier works, and we give them here only for the sake of completeness.
First, we give the RG equations for $b=2$ 3D SG fractal $$\begin{aligned}
A'&=&A^2 + 2\,A^3 + 2\,A^4 + 4\,A^3\,B + 6\,A^2\,B^2\, , \label{eq:Ab2}\\
B'&=&A^4 + 4\,A^3\,B + 22\,B^4\, ,\label{eq:Bb2}\\
C'&=&C^2 + C^3\, ,\label{eq:Cb2}\\
D'&=& 2\,{ D}^3\,B + 6\,{ D}^2\,B^2 + 2\,A^2\, D\,C +
A^2\,C^2 + A\, D\,C^2\, . \label{eq:Db2}\end{aligned}$$ We note that first three equations were established for the first time in [@dhar78].
Next, we present RG equations for the $b=3$ case $$\begin{aligned}
\fl A'&=&A^3+6 A^4+16 A^5+34 A^6+76 A^7+112 A^8+112 A^9+ 64
A^{10}+ 8 A^4 B+ 36 A^5 B\nonumber\\ \fl
| 3,574
| 2,670
| 3,680
| 3,248
| null | null |
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|
gg(\frac{1}{R_\Lambda(\lambda')}-\frac{1}{R_0(\lambda')}\bigg)\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$+\frac{1}{R_0(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$-\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]$$ By taking into account (\[andros\]), it is already seen that the last quantity is bounded since the integral is finite. In any case, by simplifying some terms, the last expression becomes $$D_\Lambda=\frac{1}{R_0(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)+\theta_0(\lambda_i)+\int_{\lambda_1}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi-\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}$$ $$+\bigg(\frac{1}{R_\Lambda(\lambda')}-\frac{1}{R_0(\lambda')}\bigg)\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg],$$ From the continuity of $\theta_\Lambda(\lambda)$ it follows that $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|\leq \epsilon'/4$ by choosing a suitable open $O\subset O_1$ containing $\Lambda_0$. In addition, one has that $$\bigg|\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi-\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi)d\xi \bigg|\leq (\lambda_e-\lambda_i)\delta p^M_\Lambda,$$ where $\delta p^M_\Lambda$ denote the maximum value of $\delta p_\Lambda=e^{-c\lambda_i}(p_\Lambda(\lambda)-p_0(\lambda))$. Again, by a Cantor-Heine argument it can be seen that this maximum can be made arbitrarily small by making $O_1$ small enough. In particular, it can be made smaller than $\epsilon'/4(\lambda_e-\lambda_1)$. The minimum value of $R_0(\lambda')$ is $R_0(\lambda_1)$, as (\[prima2\]) shows that this quantity is monotone increasingly with $\lambda$. Thus by selecting $\epsilon'=\epsilon R_
| 3,575
| 853
| 2,599
| 3,537
| null | null |
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|
to the degree of $h(x)$.* $\Box$
Let $h_1(x)$ and $h_2(x)$ be the parity-check polynomials of $1$-generator skew GQC codes $C_1$ and $C_2$, respectively. If $C_1=C_2$, then $h_1(x)=h_2(x)$, which implies that ${\rm deg }(h_1(x))={\rm deg}(h_2(x))$. It means that $R/(h_1(x))_r=R/(h_2(x))_r$. Conversely, suppose $h_1(x)$ and $h_2(x)$ are similar. Then we have $R/(h_1(x))_r\cong R/(h_2(x))_r$, which implies that $C_1= C_2$. Then from the discussion above, we have $C_1=C_2$ if and only if $h_1(x)\sim h_2(x)$, i.e., any $1$-generator skew GQC code has a unique parity-check polynomial up to similarity.
[**Theorem 4.3**]{} *Let $C$ be a $1$-generator skew GQC code of block length $(m_1, m_2,\ldots, m_l)$ and length $\sum_{i=1}^lm_i$ generated by $c(x)=(c_1(x), c_2(x), \ldots,c_l(x))\in {\mathcal R}$. Suppose $h_i(x)$ is given as in Theorem 4.2 and $h(x)={\rm lclm}\{h_1(x),h_2(x),\ldots,h_l(x)\}$. Let $\delta_i$ denote the number of consecutive powers of a primitive $m_i$-th root of unity that among the right zeros of $(x^{m_i}-1)/h_i(x)$. Then\
(i) $d_{\rm H}(C)\geq \sum_{{i}\not \in K}(\delta_i+1)$, where $K\subseteq \{1,2,\ldots,l\}$ is a set of maximum size such that ${\rm lclm}_{i\in K}h_i(x)\neq h(x)$.\
(ii) If $h_1(x)=h_2(x)=\cdots =h_l(x)$, then $d_{\rm H}(C)\geq\sum_{i=1}^l(\delta_i+1)$.*
*Proof* Let $a(x)\in C$ be a nonzero codeword. Then there exists a polynomial $f(x)\in R$ such that $a(x)=f(x)c(x)$. Since for each $i=1,2,\ldots,l$, the $i$-th component is zero if and only if $(x^{m_i}-1)\mid f(x)c_i(x)$, i.e., if and only if $h_i(x)\mid f(x)$. Therefore $a(x)=0$ if and only if $h(x)\mid f(x)$. So $a(x)\neq 0$ if and only if $h(x)\nmid f(x)$. This implies that $c(x)\neq 0$ has the most number of zero blocks whenever $h(x)\neq {\rm lclm}_{i\in K}h_i(x)$, where ${\rm lclm}_{i\in K}h_i(x)\mid f(x)$, and $K$ is a maximal subset of $\{1,2,\ldots,l\}$ having this property. Thus, $d_{\rm H}(C)\geq \sum_{i\notin K}d_i$, where $d_i=d_{\rm H}(\varphi_i(C))\geq \delta_i+1$. Clearly, $K=\emptyset$ if and only if $h
| 3,576
| 1,495
| 2,331
| 3,323
| 2,148
| 0.782266
|
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|
rn hidden inside a neural unit. Lots of visualization methods have been used in the literature.
Gradient-based visualization [@CNNVisualization_1; @CNNVisualization_2; @CNNVisualization_3] estimates the input image that maximizes the activation score of a neural unit. Dosovitskiy *et al.* [@FeaVisual] proposed up-convolutional nets to invert feature maps of conv-layers to images. Unlike gradient-based methods, up-convolutional nets cannot mathematically ensure the visualization result reflects actual neural representations. In recent years, [@olah2017feature] provided a reliable tool to visualize filters in different conv-layers of a CNN.
Zhou *et al.* [@CNNSemanticDeep] proposed a method to accurately compute the image-resolution receptive field of neural activations in a feature map. Theoretically, the actual receptive field of a neural activation is smaller than that computed using the filter size. The accurate estimation of the receptive field is crucial to understand a filter’s representations.
Unlike network visualization, our mining part representations from conv-layers is another choice to interpret CNN representations.
**Active network diagnosis:**[` `]{} Going beyond “passive” visualization, some methods “actively” diagnose a pre-trained CNN to obtain insight understanding of CNN representations.
[@CNNAnalysis_1] explored semantic meanings of convolutional filters. [@CNNAnalysis_2] evaluated the transferability of filters in intermediate conv-layers. [@CNNAnalysis_3; @CNNVisualization_5] computed feature distributions of different categories in the CNN feature space. Methods of [@visualCNN_grad; @visualCNN_grad_2] propagated gradients of feature maps *w.r.t.* the CNN loss back to the image, in order to estimate the image regions that directly contribute the network output. [@trust] proposed a LIME model to extract image regions that are used by a CNN to predict a label (or an attribute).
Network-attack methods [@pixelAttack; @CNNInfluence; @CNNAnalysis_1] diagnosed network represen
| 3,577
| 650
| 3,817
| 3,301
| 3,379
| 0.772763
|
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|
null and satisfying the normalization (\[norma\]).
The null geodesic corresponding to the element $\Lambda=(p, k^\mu)$ will be denoted as $\gamma_\Lambda(\lambda)$ in the following. All the quantities depending on this curve such as $G_\gamma(\lambda)$ will be subsequently denoted as $G_\Lambda(\lambda)$ and so on. The reason for this notational change is the desire to study continuity properties of these quantities as functions on $S$.
Proof of the *third requirement* for GR with Null Conditions
------------------------------------------------------------
The proof for the *third requirement* when the underlying theory is GR and the Null Energy and Null Generic Conditions are fulfilled is given in [@gaowald]. As this is one of the theorems to be generalized here, it is convenient to sketch the original argument. Consider a null geodesic $\gamma_0(\lambda)$ with $p_0=\gamma_0(0)$ and $q_0=\gamma_0(\lambda_0)$ conjugate points along it, with $\lambda_0>0$. Then $G_0(0)=G_0(\lambda_0)=0$ and $G_0(\lambda)>0$ for all $\lambda$ in the interval $0<\lambda<\lambda_0$. The Null Energy Condition $T_{\mu\nu}k^\mu k^\nu\geq 0$ implies, in the context of General Relativity, that $R_{\mu\nu}k^\mu k^\nu\geq 0$. This, together with (\[smile2\]) shows that $G_0''(\lambda)<0$ in the interval $0<\lambda<\lambda_0$. The mean value theorem applied to $G_0$ shows that $G_0'(\lambda_1)=-C^2$ for some value $\lambda_1$ in the interval and furthermore $G_0'(\lambda_1)<-C^2$ for $\lambda_1<\lambda<\lambda_0$, with $C^2$ a positive constant. By choosing $\lambda_0-\delta<\lambda<\lambda_0$ one has that $$\frac{G_0(\lambda_1)}{|G'_0(\lambda_1)|}<\delta,$$ since $|G_0'(\lambda)|$ is larger than $C^2$ if $\delta$ is small enough.
Consider now a small open $O\subset S$ around the point $\Lambda_0=(p_0, k^\mu)$ generating $\gamma_0(\lambda)$. As $G_\Lambda(\lambda)$ and its derivatives are continuous when moving in this open, then $G'_\Lambda(\lambda)<0$ and $$\frac{G_\Lambda(\lambda_1)}{|G'_\Lambda(\lambda_1)|}<\delta,$$ for all the $\L
| 3,578
| 2,481
| 2,959
| 3,319
| null | null |
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|
a^2}\over Z} \sum_{l,m,s=\pm} \left| \sum_{n \leq l}
v_{m,s}^n
\sqrt{{l! \over n!}}\sum_{n'=0}^n {n \choose n'}
{\alpha^{n-n'}(-\alpha)^{l-n'} \over(l-n'!)} \right| ^2 \nonumber \\
&&(e^{-\beta \varepsilon_{m,s}}+
e^{-\beta(\varepsilon_0+E_0+l\hbar\omega_0-\varepsilon_{BP})})
\delta(\omega+\varepsilon_{m,s}-\varepsilon_0 -E_0
\nonumber \\
&& -l\hbar\omega_0+\varepsilon_{BP})\quad.\end{aligned}$$ Here $Z_F=\frac{1}{Z} (1+e^{-\beta \varepsilon_0})(1+e^{-\beta
(\varepsilon_0+E_0-\varepsilon_{BP})})
n_B(\hbar \omega_0)$ represents the spectral weight of the non-bonding contributions which accounts for the coherent part of the photoemission spectrum, unaffected by any coupling to the Bosons and hence to the phonons ($n_B(\omega)$ denotes the Bose distribution function). The second and third contribution to the spectral function $I(\omega)$ account for the incoherent part of the spectrum. We illustrate in Fig.2 the photoemission spectral intensity $I_{PES}(\omega)$ for different temperatures (in units of $D$). For high temperatures ($T \simeq 0.06$) we observe a very much broadened spectral function which in shape comes close to that of a typical Fermi liquid. Upon lowering the temperature this spectral function starts exhibiting a pseudogap and at the same time a broad incoherent contribution (coming from the second and third term of the expression for $I(\omega)$ in Eq.(5)) emerges. The incoherent part of the spectrum extends over a region in energy which is of the order of the half band width ($\simeq 0.5 eV$) and is practically temperature independent at low temperatures, which seems to be confirmed experimentally[@Norman-97]. The closing up of the pseudogap (measured as the difference in energy between the chemical potential at $\omega=0$ and the midpoint of the leading edge of the photoemission spectrum) as we increase the temperature is illustrated in the inset of Fig.2. The pseudogap has a zero temperature limit of $0.085D\simeq 40\,meV$ and closes up at a characteristic temperature $T^* \simeq 0.06D\sim
| 3,579
| 3,477
| 3,633
| 3,298
| null | null |
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|
\alpha$" prescription to local fluctuations, one might expect that ${\cal Q}_p \simeq \alpha$.
The Dispersion Relation
=======================
General considerations
----------------------
In terms of these dimensionless quantities, the dispersion relation is: $$\begin{aligned}
\lefteqn{3\omega_{*}^5
+ \left[i k_{*}^2 /(G\tau) - 3 k_{*z}G/\tau -i {\cal Q}_p \right]\omega_{*}^4}\\
& - & \left(4k_{*}^2 P + 3\Omega_{*}^2 + ik_{*z}^3/\tau^2
- ik_{*z} G {\cal Q}_p /\tau\right)\omega_{*}^3 \\
& + & \left[4k_{*z}^3PG/\tau - ik_{*}^2 \Omega_{*}^2/(G\tau)
+ i{\cal Q}_p (-ik_{*z} + \Omega_{*}^2)
- i k_{*}^2 {\cal Q}_\rho \right]\omega_{*}^2 \\
& + & \left[4k_{*z}^2P\Omega_{*}^2 - 3k_{*r}^2 + k_{*z}^2 G(ik_{*z}{\cal Q}_\rho
- {\cal Q}_p)/\tau \right]\omega_* \\
& + & k_{*z}k_{*r}^2\Omega_{*}^2/(G\tau) + k_{*z}\Omega_{*}^2 (ik_{*z} {\cal Q}_\rho
- {\cal Q}_p) = 0 .\\\end{aligned}$$
In the limit of $\tau \rightarrow \infty$ and ${\cal Q}_p = {\cal Q}_\rho
= 0$, this dispersion relation simplifies to $$\omega_{*}^5 - \left[(4/3)k_{*}^2 P + \Omega_{*}^2\right]\omega_{*}^3 +
\left[ (4/3) k_{*z}^2 P\Omega_{*}^2 - k_{*r}^2\right] \omega_* = 0.$$ One root is clearly $\omega_* = 0$. The other four are given by: $$\omega_{*}^2 = \cases{ (4/3)k_{*}^2 P \cr
\left[2 k_{*z}^2 \Omega_{*}^2 - (3/2)k_{*r}^2/P \right]/ k_{*}^2 \cr}$$ The first pair of roots are the familiar radiation-supported sound waves. The second pair describe buoyancy behavior (cf. Balbus 1999). If $k_{*z}^2 \Omega_{*}^2 \geq (3/4) k_{*r}^2/P$, there are two neutrally stable gravity (epicyclic) waves; on the other hand, if $k_{*z}^2 \Omega_{*}^2 < (3/4) k_{*r}^2/P$, one mode is damped, but the other grows exponentially with essentially no oscillation. It is this last mode that corresponds to convection. Although rotation tends to have a stabilizing effect, when $\Omega_*>0$ one can always find a mode with $k_r/k_z$ large enough to satisfy this criterion. As a consequence, when radiation pressure dominates gas pressure in a very optically thick disk,
| 3,580
| 3,594
| 3,588
| 3,325
| null | null |
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|
f\right\Vert _{q,p}. \label{A38}$$
**Proof**. We will prove (\[A38\]) first. Let $\alpha $ with $%
\left\vert \alpha \right\vert \leq q.$ By (\[A36\]) $$\begin{aligned}
\left\vert \partial ^{\alpha }(\psi _{k}P_{t}^{\ast }(f/\psi
_{k})(x))\right\vert &\leq &C\psi _{k}(x)\sum_{\left\vert \gamma \right\vert
\leq q}\left\vert \partial ^{\gamma }(P_{t}^{\ast }(f/\psi
_{k})(x))\right\vert \\
&\leq &CD_{(q)}^{\ast }(\rho )\psi _{k}(x)\sum_{\left\vert \beta \right\vert
\leq q}(P_{t}^{\ast }(\left\vert \partial ^{\beta }(f/\psi _{k})\right\vert
^{\rho })(x))^{1/\rho } \\
&=&CD_{(q)}^{\ast }(\rho )\sum_{\left\vert \beta \right\vert \leq q}(\psi
_{\rho k}(x)P_{t}^{\ast }(\left\vert \partial ^{\beta }(f/\psi
_{k})\right\vert ^{\rho })(x))^{1/\rho } \\
&=&CD_{(q)}^{\ast }(\rho )\sum_{\left\vert \beta \right\vert \leq q}(\psi
_{\rho k}(x)P_{t}^{\ast }(g/\psi _{\rho k})(x))^{1/\rho }\end{aligned}$$with$$g(x)=\psi _{\rho k}(x)\left\vert \partial ^{\beta }(f/\psi
_{k})(x)\right\vert ^{\rho }=\left\vert \psi _{k}(x)\partial ^{\beta
}(f/\psi _{k})(x)\right\vert ^{\rho }.$$Using (\[A34\]) $$\left\Vert (\psi _{\rho k}P_{t}^{\ast }(g/\psi _{\rho k}))^{1/\rho
}\right\Vert _{p}=\left\Vert \psi _{\rho k}P_{t}^{\ast }(g/\psi _{\rho
k})\right\Vert _{p/\rho }^{1/\rho }\leq K_{k\rho p}^{1/p}Q^{(p-\rho )/\rho
p}\left\Vert g\right\Vert _{p/\rho }^{1/\rho }.$$And we have$$\left\Vert g\right\Vert _{p/\rho }^{1/\rho }=(\int \left\vert \psi
_{k}(x)\partial ^{\beta }(f/\psi _{k})(x)\right\vert ^{p}dx)^{1/p}\leq
C\sum_{\left\vert \gamma \right\vert \leq q}(\int \left\vert \partial
^{\gamma }f(x)\right\vert ^{p}dx)^{1/p}=C\left\Vert f\right\Vert _{q,p}.$$We conclude that $$\left\Vert \psi _{k}P_{t}^{\ast }(f/\psi _{k})\right\Vert _{q,p}\leq
CK_{k\rho p}^{1/p}Q^{(p-\rho )/\rho p}D_{(q)}^{\ast }(\rho )\left\Vert
f\right\Vert _{q,p}.$$$\square $
Integration by parts {#app:ibp}
--------------------
We consider a function $\phi \in C^{\infty }({\mathbb{R}}^{d},{\
\mathbb{R}}^{d})$ such that $\partial_j\phi \in C_{b}^{\infty }({\mathbb{R}}%
^{d},{\m
| 3,581
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| 1,707
| 3,562
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|
bility density $p(x,v)$ and the current $j_{s}$ as $$p(x,v,\infty)=
N\left[\left(\frac{\pi KT}{2 \alpha}\right)^{\frac{1}{2}}
+\int_{0}^{v-|a|x}dz \exp\left(-\frac{\alpha z^{2}}{2KT}\right)\right]
\exp\left[ -\frac{\frac{v^{2}}{2}+\tilde{V}(x)}{KT}\right]$$ with $$\begin{aligned}
F_{s} =
N\left[\left(\frac{\pi KT}{2 \alpha}\right)^{\frac{1}{2}}
+\int_{0}^{v-|a|x}dz \exp\left(-\frac{\alpha z^{2}}{2KT}\right)\right] \; \; ,\end{aligned}$$ (here the subscript $s$ in $F_s$ refers to steady state $F$) and $$j_{s}=\int_{-\infty}^{+\infty}dv \hspace{0.1cm}vp(x,v)
=N(KT)^{\frac{3}{2}}\left(\frac{2\pi}{\alpha+1}\right)^{\frac{1}{2}}
\exp\left(-\frac{E_b}{KT}\right)\hspace{0.2cm},$$ where we have used the linearized version of $\tilde{V}(x)$ near the top of the barrier at $x=0$, $$\begin{aligned}
\tilde{V}(x)=\bar{E}_{b}-\frac{1}{2}\bar{\omega}_{b}^{2}x^{2}\hspace{0.2cm},\end{aligned}$$ with ${\bar{E}}_b = E_b$ and ${\bar{\omega}}_b$ is as given in Eq.(40) and $N$ is the normalization constant.
Employing the asymptotic distribution (just before the system is subjected to the shock at $t=0$) of $P_{w}(x,v)$ for $x\rightarrow -\infty$ and at $t=0_-$ from $p(x,v,t)$, where $P_{w}(x,v)=p(x\rightarrow -\infty,v;t=0)$ \[see Sec. V for calculation of $p(x,v,t)$\], one obtains the total number of particles in the well, $$n_{a}=N \int_{-\infty}^{+\infty}dv \int_{-\infty}^{+\infty}dx P_{w}(x,v)
=N \frac{2\pi KT}{\omega_{0}}\left(\frac{2\pi KT}{\alpha}\right)^{\frac{1}{2}}
\hspace{0.2cm}.$$ Here $\omega_{0}$ is the frequency at the bottom of the left well. We have set the potential energy at the bottom of the left well equal to zero, for convenience.
The final result for the rate of escape in the steady state is given by $$k=\frac{j_{s}}{n_{a}}=\frac{\omega_{0}\lambda}{2\pi\bar{\omega}_{b}}
e^{-E_{b}/KT}\hspace{0.2cm},$$ where $$\lambda=\left[\left\{\left(\frac{\Gamma}{2}\right)^{2}+\bar{\omega}_{b}
^{2}\right\}^{\frac{1}{2}}-\frac{\Gamma}{2}\right]\hspace{0.2cm}.$$
It is evident that $\lambda$ is reminiscent of the ‘reactive f
| 3,582
| 3,466
| 3,388
| 3,248
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|
-contrbd\])]{}, we easily obtain $$\begin{aligned}
{\label{eq:EE'E''predec3}}
\Theta_{y,x;{{\cal A}}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle R_1(b)$}}}\big]&=\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{
{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E''_{{{\bf m}}+{{\bf n}}}({\overline{b}},x,v;{{\cal A}})\text{ off }b\}$}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b
\text{ even}\}$}}}\,{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_{{{\cal A}}{^{\rm c}}\cap\,
{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}}}{\nonumber}\\
&\leq{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b\sum_{\substack{
{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{
{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E''_{{{\bf m}}+{{\bf n}}}({\overline{b}},x,v;{{\cal A}})$}}}={{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b
\,\Theta''_{{\overline{b}},x,v;{{\cal A}}}.\end{aligned}$$ Similarly, we have $$\begin{aligned}
{\label{eq:EE'E''decpre3}}
\Theta_{y,x;{{\cal A}}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle R_2(b)$}}}\big]&=\sum_{{{\cal B}}\subset
\Lambda}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}
\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda
({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }
b\\}$}}}\,\cap\,\{{{\cal C}}^b_{{{\bf m}}+{{\bf n}}}(x)={{\cal B}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b
\text{ even}\}$}}}{\nonumber}\\
&\qquad\qquad\times\sum_{\substack{{\partial}{{\bf h}}={\varnothing}\\ {\partial}{{\bf k}}=y{\vartriangle}{\underline{b}}
| 3,583
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| 2,187
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|
to multi-partitions: $\muhat=
(\mu^1,\ldots,\mu^k) \in \left(\calP_n\right)^k$ is rectangular if each $\mu^i$ is (the $\mu^i$’s are not required to be of the same length). Note that $\muhat$ is rectangular if and only if the associated dimension vector $\v$ satisfies $(\e_{[i,j]},\v)= 0$ for all $[i,j]$ by .
\[sigma-ineq\] For $\muhat \in \left(\calP_n\right)^k$ we have $$\sigma(\muhat) \geq 0$$ with equality if and only if $\muhat$ is rectangular.
For any $\mu\in \calP_n$ we have $n\mu_1=\mu_1\sum_j\mu_j \geq
\sum_j\mu_j^2$ and equality holds if and only if $\mu_1=\mu_j$.
Since $$\label{Delta-sigma}
2\Delta(\muhat)=n\,\delta(\muhat)+\sigma(\muhat)$$ we find that $$\label{dim-ineq}
d_\muhat \geq n\,\delta(\muhat)+2$$ and in particular $d_\muhat \geq 2$ if $\delta(\muhat)\geq 0$.
If $\Gamma$ is affine it is known that the positive imaginary roots are of the form $t\v^*$ for an integer $t\geq 1$ and some $\v^*$. We will call $\v^*$ the [*basic positive imaginary root*]{} of $\Gamma$. The affine star-shaped quivers are given in the table below; their basic positive imaginary root is the dimension vector associated to the indicated multi-partition $\muhat^*$. These $\muhat^*$, and hence also any scaled version $t\muhat^*$ for $t\geq
1$, are rectangular. Moreover, $\Delta(\muhat^*)=0$ and in fact, $\muhat^*$ generates the one-dimensional radical of the quadratic form $\Delta$ so that $\Delta(\muhat^*,\nuhat)=0$ for all $\nuhat$.
\[affine-descrip\] Suppose that $\muhat = (\mu^1,\ldots,\mu^k) \in
\left(\calP_n\right)^k$ has $\delta(\muhat)\geq 0$. Then $d_\muhat=2$ if and only if $\Gamma$ is of affine type, i.e., $\Gamma$ is either the Jordan quiver $J$ (one loop on one vertex), $\tilde D_4,\tilde
E_6,\tilde E_7$ or $\tilde E_8$, and $\muhat=t\muhat^*$ (all parts scaled by $t$) for some $t\geq 1$, where $\muhat^*$, given in the table below, corresponds to the basic imaginary root of $\Gamma$.
By and Lemma \[sigma-ineq\] $d_\muhat=2$ when $\delta(\muhat)\geq 0$ if and only if $\delta(\muhat)=0$ and $\muhat$ is
| 3,584
| 1,901
| 3,219
| 3,217
| 3,468
| 0.772153
|
github_plus_top10pct_by_avg
|
ctly the one of a bicrossed product on the set $ H \times (G,
*)$ associated to the actions $\alpha'$ and $\beta'$. In other words, we have to prove that $$\bigl( h( g \rhd' h'), \, (g \lhd' h')g'\bigl) = \psi^{-1}
\Bigl(\psi(h,g) \cdot \psi(h',g')\Bigl)$$ for all $h$, $h' \in H$, $g$, $g' \in G$ or equivalently, as $\psi$ is bijective $${\label{eq:def4}}
\psi\bigl((h,g)\cdot (h',g')\bigl) = \psi(h,g) \cdot \psi(h',g')$$ for all $h$, $h' \in H$, $g$, $g' \in G$. This reduces to proving the following two conditions: $${\label{eq:c11}}
\sigma\bigl(h(g \rhd' h')\bigl) r\circ v^{-1}\bigl[\bigl(v(g \lhd'
h') \lhd r(g')\bigl)v(g')\bigl] = \sigma(h) r(g)\bigl(v(g) \rhd
(\sigma(h')r(g'))\bigl)$$ and $${\label{eq:c2}}
\bigl(v(g \lhd' h') \lhd r(g')\bigl)v(g') = \bigl(v(g) \lhd
(\sigma(h')r(g'))\bigl)v(g')$$ for any $h$, $h' \in H$, $g$, $g' \in G$. We have: $$\begin{aligned}
\sigma(h) r(g)\bigl(v(g) \rhd
(\sigma(h')r(g'))\bigl)&\stackrel{{(\ref{eq:2})}} {=}& \sigma(h)
r(g)\bigl(v(g) \rhd \sigma(h')\bigl)\bigl((v (g) \lhd
\sigma(h')) \rhd r(g')\bigl)\\
&\stackrel{{(\ref{eq:def2})}} {=}& \sigma(h) r(g)\bigl(v(g) \rhd
\sigma(h')\bigl)\bigl(v (g \lhd'h') \rhd r(g')\bigl)\\
&{=}& \sigma(h) r(g)\bigl(v(g) \rhd \sigma(h')\bigl)r(g \lhd'
h')^{-1}r(g \lhd' h')\\
&&\bigl(v (g \lhd'h') \rhd r(g')\bigl)\\
&\stackrel{{(\ref{eq:crossed})}} {=}& \sigma(h)
r(g)\bigl(v(g) \rhd \sigma(h')\bigl) r(g \lhd' h')^{-1}r\bigl((g\lhd' h') * g'\bigl)\\
&\stackrel{{(\ref{eq:def2})},{(\ref{eq:def1})}} {=}& \sigma(h)
r(g)\bigl(v(g) \rhd \sigma(h')\bigl) r\circ v^{-1}\bigl(v(g) \lhd
\sigma(h)\bigl)^{-1}\\
&& r\circ v^{-1}\bigl[\bigl(v (g
\lhd'h')\lhd r(g')\bigl)v(g')\bigl]\\
&\stackrel{{(\ref{eq:def3})}} {=}& \sigma(h)\sigma(g \rhd' h') r\circ
v^{-1}\bigl[\bigl(v(g \lhd' h')\lhd r(g')\bigl)v(g')\bigl]\\
&{=}& \sigma\bigl(h(g \rhd' h')\bigl) r\circ v^{-1}\bigl[\bigl(v(g
\lhd' h') \lhd r(g')\bigl)v(g')\bigl]\end{aligned}$$ hence [(\[eq:c11\])]{} holds. Moreover: $$\bigl(v(g \lhd' h') \lhd r(g')\bigl) \stackrel{{(\ref{eq:def2})}} {=}
\bigl[\bigl(v(g) \lhd
| 3,585
| 2,041
| 3,109
| 3,306
| null | null |
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|
Soft Markers (N = 19) Control (N = 19) *p*
---------------------------------------------------------------------------------------------------- ----------------------- ---------------------- ----------
*Socio-demographic Characteristics*
Mother\'s age (years): mean (±SD) 32.3 (±4.2) 32.2 (±3.9) 0.912
Couple Status: unmarried / married 61% / 33% 57% / 42% 0.737
Education Level: Completed A-level vs. Some University vs. Completed University \<5% vs. 11% vs. 83% \<5% vs. 31% vs. 63% 0.328
*Pregnancy Characteristics*
Minor Obstetrical History[\*](#nt101){ref-type="table-fn"} 50% yes / 50% no 36% yes/64% no 0.635
Minor Medico-chirurgical History[\*](#nt101){ref-type="table-fn"} 27% yes / 73% no 31% yes / 69% no 1
Para 0.7 (±0.8) 0.9 (±0.7) 0.624
Gesture 1.9 (±0.9) 2.1 (±1.1) 0.733
Life Events Number 7.57 (±3.5) 8.43 (±4.8) 0.595
*Delivery Characteristics*
Type of delivery: % Vaginal
| 3,586
| 4,752
| 3,104
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| null | null |
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|
------------------------
Let $D$ be an associative dialgebra. If we define on the set $D$ new operations $$\label{eq:QuasiJordanProduct} a{\mathbin{{}_{(\vdash)}}}b=\frac{1}{2}(a{\mathbin\vdash}b+b{\mathbin\dashv}a),\ a{\mathbin{{}_{(\dashv)}}}b=\frac{1}{2}(a{\mathbin\dashv}b+b{\mathbin\vdash}a)$$ then we obtain a new dialgebra which is denoted by $D^{(+)}$. It is easy to check that this dialgebra is Jordan [@Br:08].
A dialgebra $J$ is called *special*, if $J\hookrightarrow
D^{(+)}$ for some associative dialgebra $D$. Jordan dialgebras that are not special we call *exceptional*. Further, we will denote the operations in a special Jordan dialgebra through ${\mathbin{{}_{(\vdash)}}}$ and ${\mathbin{{}_{(\dashv)}}}$. These operations are expressed through associative operations by the formula (\[eq:QuasiJordanProduct\]).
The definition of special Jordan dialgebras has been introduced by the analogy with ordinary algebras, where a Jordan algebra $J$ is called special, if $J\hookrightarrow A^{(+)}$ for some associative algebra $A$ and the product in $A^{(+)}$ is given by the formula $$\label{eq:JordanProduct}
a\circ b=\frac{1}{2}(ab+ba).$$
Let now $D$ be an associative dialgebra. The mapping $*\colon D\to
D$ is called an *involution* (involution of the second type [@Pozh:09]) of the dialgebra $D$, if $*$ is linear and $$\label{eq:DefOfInvolution}
(a^*)^*=a,\quad (a{\mathbin\vdash}b)^*=b^*{\mathbin\dashv}a^*,\quad (a{\mathbin\dashv}b)^*=b^*{\mathbin\vdash}a^*$$ for all $a$, $b\in D$.
The set $H(D,*)=\{x\in D\mid x=x^*\}$ of symmetric elements with respect to $*$ is closed relative to operations (\[eq:QuasiJordanProduct\]). This set is a subalgebra of the algebra $D^{(+)}$. So, $H(D,*)$ is a special Jordan dialgebra.
We now construct an example of an exceptional Jordan dialgebra.
\[prop:ExampleExceptionalDialgebra\] Let $(J,\circ)$ be an exceptional Jordan algebra and suppose the condition $x\circ J=0$, $x\in J$, implies $x=0$. Then $J$ as a dialgebra with equal operations $x{\mathbin{{}_{(\vdash)}}}y:=x\circ y$
| 3,587
| 1,994
| 3,539
| 3,173
| 1,322
| 0.791025
|
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|
mpling using the starspot model and revealed that the variations of the shape of light curves are mainly accounted for by spot evolution and migration on the K star of SZ Psc. @eaton2007 suggested that the cooler component have many small starspots rather than a few large ones, because its line profiles lack large distortions.
In order to investigate the starspot activities on active close binaries, we have carried out a series of high-resolution spectroscopic observations on targets with various stellar parameters and evolutionary stages [@gu2003; @xiang2014; @xiang2015]. In this work, we have derived the surface images of the K subgiant component of SZ Psc for 2004 November, 2006 September, October, November and December, through Doppler imaging technique. To our knowledge, there is no Doppler image for SZ Psc before, which could offer us a more detailed distribution of starspots than light-curve modelling. We shall describe the observations and data reduction in Section 2. The Doppler images will be given and discussed in Section 3 and 4, respectively. In Section 5, we shall summarize the present work.
Observations and data reduction
===============================
------------ ----------- ------ ------- ------
UT Date HJD Exp. S/N S/N
2450000+ (s) Input LSD
20/11/2004 3330.1055 2400 141 1911
20/11/2004 3330.1365 2400 134 1818
21/11/2004 3331.1073 2400 87 1180
27/11/2004 3337.1254 2400 103 1407
01/09/2006 3980.1557 1800 55 750
01/09/2006 3980.1769 1800 62 847
04/09/2006 3983.1880 2100 83 1131
04/09/2006 3983.2125 2100 92 1262
05/09/2006 3984.1295 1800 101 1377
05/09/2006 3984.1505 1800 105 1430
06/09/2006 3985.1228 1800 121 1654
06/09/2006 3985.1437 1800 130 1776
28/10/2006 4036.9567 1800 131 1794
28/10/2006 4036.9777 1800 132 1811
28/10/2006 4037.0227 1800 142 19
| 3,588
| 4,557
| 3,313
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| null | null |
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|
text of the main neutrinosphere models proposed.
For a hard neutrinosphere model in thermal equilibrium as considered in Refs. [@kusegre; @kusegref; @qian], the momentum asymmetry in the ${\bf v}$ direction is generated by the emission at points with different temperatures on the resonance surface: $\Delta p/p\approx \frac{2}{9}h_{T}^{-1}\delta $, where $h_{T}^{-1}=\frac{d}{dr}\ln T $. In the case of a quasi-degenerate gas of relativistic electrons with a constant chemical potential $\mu _{e}\approx
\left( 3\pi ^{2}N_{e}\right) ^{1/3}$and $\frac{dN_{e}}{dT}=\frac{2}{3}T\mu
_{e}$. Then $$\frac{\Delta p}{p}\approx Q\frac{Bv}{A}\,, \label{asym}$$ with $Q=\frac{\eta ^{2}\Lambda }{9\pi ^{2}}$, where $\eta =\mu _{e}/T$ is the degeneracy parameter for the electrons and $\Lambda
=h_{A}/(h_{A}-h_{N_{e}})$. Another possibility is to assume that the electron fraction $\ Y_{e}$ remains constant and $\rho \sim
T^{3}$[@qian]. In this case $h_{N_{e}}\sim h_{T}/3$, and $Q=\frac{2}{27}\Lambda $.
A different kick model in the literature uses a soft neutrinosphere[@kusegref; @raffelt]. In such a case there is an important reduction in the anisotropy given by the ratio ${\rho _{o}}/{\rho _{c}}$ of the density at the resonance and the density at the core. The momentum asymmetry can also be written as in Eq.(\[asym\]), with $Q=\rho _{o}h_{N_{e}}\Lambda
/18m_{c}$, where $m_{c}= \int_{r_{c}}^{r_{s}}\rho \,dr$ is the integral of the mass density between the central core and the surface of the star. In all the cases considered above the adimensional parameter $Q$ depends only on the specific model and the remaining factors contain the PPN parameters.
For a quantitative estimation of the effects of VEP on the neutrinosphere, we use the density profile $\rho (r)=\rho _{c}$ for $r<r_{c}$ and $\rho
(r)=\rho _{c}\left( r_{c}/r\right) ^{n}$ for $r>r_{c}$[@horvat]. We take $\rho _{c}=8\times 10^{14}\;g/cm^{3}$, $r_{c}=10\;km$, and $5\leq n\leq 7$ that give a good description of the supernova SN1987A[@parametros]. The resonance surface has to
| 3,589
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|
e to generate three-dimensional GS states.
[**Acknowledgments**]{}
The author would like to thank B. Etemadi for useful discussions and F. Sales-Mayor for a careful reading of the text.
[**Appendix**]{}
This appendix gives closed form expressions for the matrix elements needed to construct the matrix $H_{\tau \bar{K} K
\bar{\nu} \nu}^{pq}= \big < \bar{K}^{\pm} \bar{\nu} |H_\tau|
K^{\pm} \nu \big >$. First let $$P_1 = 1 + {3 \over 2}\alpha^2$$ $$P_2 = 3\alpha + + {3 \over 4}\alpha^3$$ $$P_3 = {3 \over 2} \alpha^2$$ $$P_4 = {\alpha^3 \over 4}$$ $$f(\alpha) = {{\sqrt{1-\alpha^2}-1} \over \alpha}$$ and define $$\delta_{J,K} \equiv \Delta_{J-K}$$ Each operator in Eq. (9) will connect either only like parity states or opposite parity states; no single operator will do both. The matrix elements that connect like positive parities are
$$\bigg < \bar{K}^+ \bar{\nu} \bigg | {\partial^2 \over \partial^2
\theta} \bigg | K^+ \nu \bigg > =\pi
\sum_{m=0}^{\bar{K}}\sum_{n=0}^{K} c_{\bar{K}m} c_{Kn} (-n^2)
\Delta_{\bar{\nu}-\nu}\big[ (\Delta_{m+n}-\Delta_{m-n}) +$$ $${\alpha \over 2}
(\Delta_{m+n+1}+\Delta_{m-n+1}+\Delta_{m+n-1}+\Delta_{m-n-1})\big]
\eqno(A1)$$
$$\bigg < \bar{K}^+ \bar{\nu} \bigg | -{\alpha \over F} \ {\rm sin}
\theta {\partial \over
\partial \theta} \bigg | K^+ \nu \bigg > = {\alpha \pi \over 2}
\sum_{m=0}^{\bar{K}}\sum_{n=0}^{K} c_{\bar{K}m} c_{Kn}\ n \
\Delta_{\bar{\nu}-\nu}
(\Delta_{m+n-1}+\Delta_{m-n+1}-\Delta_{m-n-1}) \eqno(A2)$$
$$\bigg < \bar{K}^+ \bar{\nu} \bigg | {\alpha^2 \over
F^2}{\partial^2 \over \partial^2 \phi} \bigg | K^+ \nu \bigg > =
{\alpha^2 \pi \over \sqrt{1-\alpha^2}}
\sum_{m=0}^{\bar{K}}\sum_{n=0}^{K} c_{\bar{K}m} c_{Kn}\ (-\nu^2) \
\Delta_{\bar{\nu}-\nu} [f^{n+m}(\alpha)+f^{|n-m|}(\alpha)]
\eqno(A3)$$
$$\bigg < \bar{K}^+ \bar{\nu} \bigg | i \tau_0 \alpha^2{\partial
\over
\partial \phi} \bigg | K^+ \nu \bigg > =\pi \tau_0 \alpha^2
\sum_{m=0}^{\bar{K}}\sum_{n=0}^{K} c_{\bar{K}m} c_{Kn} (-\nu)
\Delta_{\bar{\nu}-\nu}\big[ (\Delta_{m+n}+\Delta_{m-n}) +$$ $${\alpha \over 2}
(
| 3,590
| 4,119
| 2,622
| 3,091
| null | null |
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|
up to logarithmic corrections as $q^2\to\infty$ because of asymptotic freedom. The continuum tree-level propagator is $1/q^2$. We also expect asymptotic freedom on the lattice despite finite lattice spacing artefacts. We [*define*]{} the lattice $q_\mu$ such that the lattice $D^{\rm tree}(q)\equiv 1/q^2$, and use this momentum throughout. This is referred to as tree-level correction and we have seen that it significantly reduces discretization arrors at large momenta. For the two actions considered here, this means that we work with the momentum variables defined as $$q_{\mu}^W \equiv \frac{2}{a} \sin\frac{{\hat{q}}_{\mu} a}{2},
\hskip1cm
q_\mu^I \equiv \frac{2}{a}\sqrt{ \sin^2
\Bigl( \frac{{\hat{q}}_\mu a}{2} \Bigr)
+ \frac{1}{3}\sin^4 \Bigl( \frac{{\hat{q}}_\mu a}{2} \Bigr)
} \, ,
\label{eq:latt_momenta}$$ for the Wilson and improved actions respectively. All figures (quark and gluon propagators) have a cylinder cut imposed upon them, i.e. all momenta must lie close to the lattice diagonal. In Table \[table:latlist\] we show the various lattices that we have studied for the gluon propagator. In Fig. \[fig:Comp1i\_6\] we plot $q^2 D(q^2)$ for a fine unimproved Wilson action and for our finest improved action. Despite having very different lattice spacings the agreement is excellent for the entire intermediate and high-momentum regime. The small discrepancy in the deep infrared due to finite volume effects is not apparent in this way of plotting that data. We plot $D(q^2)$ for five different lattice in Fig. \[fig:AllProps\] and see pleasing agreement for the results. Note that we are plotting bare quantities only and there is thus an overall wavefunction renormalization for the gluon propagator (i.e., $Z_3(\mu,a)$ for the renomalization point $\mu$). The vertical scale is thus unimportant and only the variation with momentum is relevant. This way of presenting the data shows that there is a small residual finite volume dependence, where the infrared gluon propagator is [*decreasing*]{} with incr
| 3,591
| 2,008
| 3,488
| 3,210
| null | null |
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|
with similar $T_\mathrm{eff}$ (about 1 dex), combined with the large error bars on theoretical predictions.
Since in most of the cases the models with $\alpha_\mathrm{PMS}=\alpha_\mathrm{MS}$ disagree with the data, and given the high sensitivity of $^7$Li surface abundance predictions to the convection efficiency, it is worth exploring the possibility that the mixing length parameter value varies from the pre-MS to the MS phases. Indeed, a possible dependence of $\alpha$ on the evolutionary phase (and/or gravity, $T_\mathrm{eff}$, mass) is suggested from both observations and hydrodynamical simulations, as discussed in the introduction. Thus, we computed models with different values of $\alpha_\mathrm{PMS}$, namely, $\alpha_\mathrm{PMS} = 1.0$, 1.2, 1.4, 1.68, and 1.9, once $\alpha_\mathrm{MS}$ and the ages have been fixed by the comparison in the CMD. Figure \[fig:litio\] shows the comparison between our ‘best fit’ models and $^7$Li data for each cluster (dotted lines and filled red squares). The theoretical error bars computed for each cluster are also shown.
{width="\columnwidth"} {width="\columnwidth"}\
{width="\columnwidth"} {width="\columnwidth"}\
{width="\columnwidth"} {width="\columnwidth"}
We emphasize that a satisfactory agreement with all the clusters in the sample (with the exception of the Pleiades) can be achieved by assuming the same pre-MS convection efficiency, namely $\alpha_\mathrm{PMS}=1.0$. Such low-convection efficiency models are able to reproduce, within the error bars, the mean depletion profile even for low-mass stars, especially in the case of Ic 2602 and $\alpha$ Per.
As shown in Fig. \[fig:litio\], the poorest match between theory and data is achieved for the Pleiades. The hottest stars are nearly compatible within the error bars with the observations, which sh
| 3,592
| 1,109
| 3,549
| 3,663
| 1,011
| 0.795809
|
github_plus_top10pct_by_avg
|
o [n']}\bigotimes_{v' \in V([n'])}
A^{opp} \otimes A^{\otimes f^{-1}(v')}$$ for any $[n'] \in \Lambda$. Then $L^p{\operatorname{\sf tr}}_\#I_{n!}E = 0$ for $p \geq 1$, and one checks easily that $${\operatorname{\sf tr}}_\# I_{n!}E = i_{n!}{\operatorname{\sf tr}}E = i_{n!}A^{\otimes n} \in {\operatorname{Fun}}(\Lambda,k),$$ where $i_n:{\operatorname{{\sf pt}}}\to \Lambda$ is the embedding of the object $[n] \in
\Lambda$ (${\operatorname{{\sf pt}}}$ is the category with one object and one morphism). Therefore $$HC_0(L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#I_{n!}E) = H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda,i_{n!}A^{\otimes n}) =
A^{\otimes n},$$ and $HC_p(L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#i_{n!}E) = 0$ for $p \geq 1$. We have to compare it with $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(i_{n!}E)$.
To do this, consider the category $\Lambda_{[n]}$ of objects $[n']
\in \Lambda$ equipped with a map $[n] \to [n']$, and let $j_n:\Lambda_{[n]} \to \Lambda$ be the forgetful functor. Then $j_n$ is obviously a discrete cofibration. Comparing and , we see that $$I_{n!}E = j_{n!}E_\#^{[n]}$$ for some $E_\#^{[n]} \in {\operatorname{Fun}}(\Lambda_{[n]})$. Moreover, fix once and for all a map $[1] \to [n]$. Then we see that the discrete cofibration $j_n:\Lambda_{[n]} \to \Lambda$ factors through the discrete cofibration $j:\Lambda_{[1]} = \Delta^{opp} \to \Lambda$ by means of a discrete cobifbration $\gamma_n:\Lambda_{[n]} \to
\Lambda_{[1]}$, and we observe that $$E^{[n]}_\#([n']) = (A^{opp})^{\otimes n'} \otimes A^{\otimes n}$$ only depends on $\gamma_n([n']) \in \Delta^{opp}$. More precisely, we have $E^{[n]}_\# = \gamma^*_nE_n^{\Delta}$, where $E_n^{\Delta}
\in {\operatorname{Fun}}(\Delta^{opp},k)$ is as in , and $E_n$ is the free $A$-bimodule $$E_n = A^{opp} \otimes A^{\otimes (n-1)} \otimes A.$$ The conclusion: we have $$HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(I_{n!}E) = H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda_{[n]},E^{[n]
| 3,593
| 3,355
| 2,677
| 3,190
| 2,771
| 0.777091
|
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|
of this blow-up. We have the following commutative diagram: $$\xymatrix@M=10pt{
E \ar@{->>}[d] \ar@<-.5ex>@{^(->}[r] & {{{{\widetilde{{{\mathbb{P}}}}}}}}^8 \ar@{->>}[d]^\pi
\ar@<-.5ex>@{^(->}[r] & {{\mathbb{P}}}^8\times {{\mathbb{P}}}^N \ar@{->>}[d] \\
{{\mathscr S}}\ar@<-.5ex>@{^(->}[r] & {{\mathbb{P}}}^8 \ar@{-->}[r]^c & {{\mathbb{P}}}^N
}$$ Therefore, as a subset of ${{\mathbb{P}}}^8\times{{\mathbb{P}}}^N$, the support of the PNC is $$\begin{gathered}
|E|=\{(\alpha,{{\mathscr X}})\in {{\mathbb{P}}}^8\times{{\mathbb{P}}}^N : \text{${{\mathscr X}}$ is a limit of
${{\mathscr C}}\circ \alpha(t)$}\\
\text{for some germ $\alpha(t)$ centered at
$\alpha\in {{\mathscr S}}$ and not contained in ${{\mathscr S}}$}\}\quad.\end{gathered}$$
\[PNCtolimits\] The set of limits of ${{\mathscr C}}$ consists of the image of the PNC in ${{\mathbb{P}}}^N$, and of limits of families ${{\mathscr C}}\circ \alpha(t)$ with $\alpha=\alpha(0)$ a singular matrix whose image is not contained in ${{\mathscr C}}$.
In the latter case: if $\alpha$ has rank 1, the limit consists of a multiple line supported on $\ker\alpha$; if $\alpha$ has rank 2, the limit consists of a star of lines through $\ker\alpha$, reproducing projectively the tuple of points cut out by ${{\mathscr C}}$ on the image of $\alpha$.
The PNC dominates the set of limits of families ${{\mathscr C}}\circ \alpha(t)$ for which $\alpha(t)$ is centered at a point of indeterminacy of $c$. This gives the first statement.
To verify the second assertion, assume that $\alpha(t)$ is centered at a singular matrix $\alpha$ at which $c$ [*is*]{} defined; $\alpha$ is then a rank-1 or rank-2 matrix such that $F(\alpha(x,y,z))\not\equiv 0$. After a coordinate change we may assume without loss of generality that $$\alpha=\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
\quad\text{or}\quad
\alpha=\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{pmatrix}$$ and $F(x,0,0)$, resp. $F(x,y,0)$ are not identically zero. These are then the forms defining the limits
| 3,594
| 2,675
| 3,085
| 3,132
| null | null |
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|
the fidelity decay.
Derivation of EQ. (\[eq:Teff\]) and qualitative discussion {#app:theo}
==========================================================
The calculation of Eq. (\[eq:ss\_ft\]) proceeds along the same lines as in [@ver85a], hence we indicate below only the main steps and essential differences. First, we make use of the representation of resolvents and thus $S$-matrix elements \[Eq. (\[eq:s\_cc1\])\] in terms of Gaussian integrals over auxiliary “supervectors” consisting of both commuting and anticommuting (Grassmann) variables. This allows us to perform statistical averaging over GOE exactly. Then in the RMT limit $N\to\infty$, the remaining integration over the auxiliary field can be done in the saddle-point approximation. It turns out that there exists a nontrivial saddle-point manifold [@efe83] over which one has to integrate exactly. As a result, the two-point correlation function of the $S$-matrix elements in the energy domain acquires the form of a certain expectation value in field theory (nonlinear supersymmetric $\sigma$-model), $\langle(\cdots)\rangle=\int\mathrm{d}[\sigma_G]e^{\mathcal{L}(\varepsilon)}\mathcal{F}_{M}(\cdots)$, cf. Eq. (7.13) of Ref. [@ver85a]. In the notations of this paper, the effective Lagrangian reads $\mathcal{L}(\varepsilon)=\frac{1}{4}N\varepsilon\,\mathrm{trg}(\sigma_GL)$, with $\varepsilon$ being the energy difference. Definitions of the supertrace, $\mathrm{trg}$, as well as of the supermatrices $\sigma_G$ and $L$ can also be found there (see [@efe96] for a general reference). The pre-exponential terms omitted above depend on the coupling constants in the channels $a$ and $b$ ($a,b\neq c$), being thus the same as considered in [@ver85a]. They finally correspond to the expressions appearing explicitly in Eqs. (\[eq:J\]) and (\[eq:P\]). At last, the so-called channel factor $\mathcal{F}_M$ accounts for the coupling to all the channels. It is the term that requires modification due to both generally complex and varied coupling constants. In the \[1,2\] bl
| 3,595
| 1,630
| 1,543
| 3,275
| 3,029
| 0.775351
|
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|
the auxiliary field $G^{--}$, keeping only the physical degrees of freedom.
For simplicity, and as in [@CalHu99], let us assume a symmetric scalar field theory, meaning that the background fields vanish and also $\Gamma_{,A\left(BC\right)}=0$. Then the 2PI EA can be written as
=12S\_[AB]{}\^[AB]{}-i2+\_Q
The first derivatives of the 2PI CTP EA yield the mean field equations of motion
S\_[AB]{}-i\^[-1]{}\_[AB]{}+2\_[Q,(AB)]{}=0 we shall call ${G}^{AB}$ the on-shell propagators. We adopt the $\left(\pm ,a\right)$ indexes, as before. In this representation, we have the identifications eq. (\[onshellprop\]) and (\[onshelleq\]). We now replace the generic kernels $\mathbf{G}^{AB}$ by the stochastic kernels ${G_s}^{AB}={G}^{AB}+\gamma^{AB}$ and expand the effective action to second order. For simplicity, we only keep linear terms in $\Gamma_Q$. We show in [@CalHu99] that this is enough to study the fluctuation terms in the kinetic field theory limit. The relevant quadratic terms in the effective action are
\^[(2)]{}=i4\^2 where, as in eq. (\[onshelleq\])
\^[-1]{}=(
[cc]{} 0&-i\_[b,a]{}\
-i\_[a,b]{}&\_[ab]{}
) therefore
\^[-1]{}=(
[cc]{} -i\_[b,a]{}\^[(-b)(+c)]{}&-i\_[b,a]{}\^[(-b)(-c)]{}\
-i\_[a,b]{}\^[(+b)(+c)]{}+\_[ab]{}\^[(-b)(+c)]{}&-i\_[a,b]{}\^[(+b)(-c)]{}+\_[ab]{}\^[(-b)(-c)]{}
) and
\^[(2)]{}&=&4{\_[b,a]{}\^[(-b)(+c)]{}\_[d,c]{}\^[(-d)(+a)]{}.&+&.2\_[b,a]{}\^[(-b)(-c)]{}.&+&.} The stochastic equations for the propagators are
2\_[b,a]{}=2\_[(-d)(+a)]{}
2\_[c,d]{}=2\_[(+d)(-a)]{}
&&2{\_[b,a]{}.&+&.i\_[ab]{}}=2\_[(-b)(-c)]{}
2\_[b,a]{}\^[(-b)(-c)]{}\_[c,d]{}=2\_[(+d)(+a)]{} This last equation implies
\^[(-b)(-c)]{}=-iG\_[adv]{}\^[cd]{}G\_[adv]{}\^[ba]{}\_[(+d)(+a)]{} Observe that $\gamma^{\left(-b\right)\left(-c\right)}$ is not zero in the equivalent stochastic problem. The equations for the physical stochastic propagators are obtained eliminating $\gamma^{\left(-b\right)\left(-c\right)}$ throughout. For example, for the fluctuations in the Hadamard propagator we obtain
\_[c,d]{}\^[(+d)(+a)]{}\_
| 3,596
| 1,485
| 3,032
| 3,447
| null | null |
github_plus_top10pct_by_avg
|
Furthermore, when implemented with many particles, the learner can be used to make estimates of the cost landscape to determine what parameters most contributed to BEC production and aid better experimental design. In future work we will apply MLOO to atomic species with more exotic scattering properties [@altin_collapse_2011] in order to find novel cooling ramps that find optimal solutions in the competition between poorly characterized complex dynamical processes. Our approach is generic and available [@hush_m-loop_2015] for use on other scientific experiments, ultra-cold atom based or otherwise.
MRH acknowledges funding from an Australian Research Council (ARC) Discovery Project (project number DP140101779). JJH acknowledges support of an ARC Future Fellowship (FT120100291). AL would like to thank the South Australian Government through the Premier’s Science and Research Fund for supporting this work.
Appendix
========
*Gaussian process evaluation:* In practice, evaluating a Gaussian process (GP) reduces to a set of matrix operations whose derivation is given by Rasmussen *et al.* [@rasmussen_gaussian_2006] in section 2.7. Consider $N$ previous experiments have been performed with parameter sets $\mathcal{X} = (X_1,\cdots X_N)$ (each $X_j = (x_{1,j}, \cdots x_{M,j})$), measured costs $\mathcal{C} = (C_1,\cdots, C_N)$ and uncertainties $\mathcal{U} = (U_1,\cdots, U_N)$. We refer to the set of this data as our observations $\mathcal{O} = (\mathcal{X},\mathcal{C},\mathcal{U})$. We fit a GP to these observations with constant function offset $\beta$ and covariance defined by a squared exponential correlation function $K(X_p,X_q,H) = e^{-\sum_{j=1}^M (x_{j,p}- x_{j,q})^2/h_j^2}$ where $H = (h_1, \cdots, h_M)$ are the hyperparameters of the model.
The mean function and variance of the functions are: $$\begin{aligned}
\mu_{\hat{\mathscr{C}}}(X|\mathcal{O},H) = & \beta + r(X)^{T}\gamma \\
\sigma_{\hat{\mathscr{C}}}^2 (X|\mathcal{O},H) = & \sigma_{\mathcal{C}}^2 ( 1 - r(X)^T R^{-1} r(X) {\nonumber}\\
& + (j^T R^
| 3,597
| 2,780
| 3,947
| 3,433
| 2,374
| 0.780429
|
github_plus_top10pct_by_avg
|
$ if and only if $\mu=\nu^1+\cdots+\nu^s$ up to permutation of the parts of each $\nu^p$ for $p=1,\ldots,s$.
It follows immediately from the definition of the monomial symmetric function.
Let $\v$ be the dimension vector associated to $\muhat$.
\[connectedness1\] If $\v$ is in the fundamental set of imaginary roots of $\Gamma$ then the character variety $\M_\muhat$ is non-empty and connected.
Assume $\v$ is in the fundamental set of roots of $\Gamma$. By Lemma \[fund-set\] this is equivalent to $\delta(\muhat)\geq 0$.
Note that $m_\nu(\x^d)=m_{d\nu}(\x)$ for any partition $\nu$ and positive integer $d$. Suppose $\omhat=(d,\omhat^1)\cdots(d,\omhat^s)$ is a multi-type for which $\gamma_{\muhat\omhat}$ is non-zero. Let $\nuhat^p=d\omhat^p$ for $p=1,\ldots, s$ (scale every part by $d$). These multi-partitions are then exactly in the hypothesis of Proposition \[Delta-ineq\] by Lemma \[perm-sum-lemma\]. Hence $$\label{conn-ineq}
d\sum_{p=1}^s
\Delta(\omhat^p)\leq d^2\sum_{p=1}^s
\Delta(\omhat^p)=\sum_{p=1}^s\Delta(\nuhat^p) \leq\Delta(\muhat).$$ Suppose $\Gamma$ is not affine. Then by Proposition \[Delta-ineq\] we have equality of the endpoints in if and only if $s=1$, $\nuhat^1=\muhat$ and $d=1$, in other words, if and only if $\omhat=(1,\muhat)$. Hence, since $C_{(1,\,\muhat)}^0=1$, the coefficient of the lowest power of $q$ in $\H_\muhat\left(\sqrt{q},1/\sqrt{q}\right)$ equals the coefficient of the lowest power of $q$ in $P_\muhat(q)$ which is $1$ by Lemma \[lowest\] and Theorem \[minim\], Case II. This proves our claim in this case.
Suppose now $\Gamma$ is affine. Then by Proposition \[Delta-ineq\] we have equality of the endpoints in if and only if $\muhat=t\muhat^*$ and $\omhat=(1,t_1\muhat^*),\ldots,(1,t_s\muhat^*)$ for a partition $(t_1,t_2,\ldots, t_s)$ of $t$ and $d=1$. Combining this with Lemma \[lowest\] and Theorem \[minim\], Case I we see that the lowest order terms in $q$ in $\Log\left(\Omega\left(\sqrt{q},1/\sqrt{q}\right)\right)$ are $$L:=\sum C^0_\omhat p(t_1)\cdots
p(t_s) \,m_{t
| 3,598
| 3,224
| 2,750
| 3,168
| 4,015
| 0.768644
|
github_plus_top10pct_by_avg
|
nd to Inwon Kim for helpful discussions as well suggesting the problem. This work was in part supported by NSF grant DMS-0907931.
Appendix
========
\[lem:GNS\] Let $d \geq 2$ and $f:{\mathbb R}^d \rightarrow {\mathbb R}$ satisfy $f \in L^p\cap L^q$ and ${\nabla}f^k \in L^r$. Moreover let $1 \leq p \leq rk \leq dk$, $k < q < rkd/(d-r)$ and $$\frac{1}{r} - \frac{k}{q} - \frac{s}{d} < 0. \label{cond:GNS}$$ Then there exists a constant $C_{GNS}$ which depends on $s,p,q,r,d$ such that $${\|f\|}_{L^q} \leq C_{GNS}{\|f\|}^{\alpha_2}_{L^p} {\|f^k\|}^{\alpha_1}_{\dot{W}^{s,r}}, \label{eq:GNS}$$ where $0 < \alpha_i$ satisfy $$1 = \alpha_1 k + \alpha_2,$$ and $$\frac{1}{q} - \frac{1}{p} = \alpha_1(\frac{-s}{d} + \frac{1}{r} - \frac{k}{p}).$$
The following lemma verifies that the distributions defined by the second derivatives of admissible kernels behave as expected under mass-invariant scalings.
\[lem:CZ\_rescale\] Let ${\mathcal{K}}$ be admissible. Then $\forall \, p, \; 1 < p < \infty$, $u \in L^p$ and $t > 0$, we have $${\|{\nabla}\left(t^d{\nabla}{\mathcal{K}}(t \cdot)\ast u \right)\|}_p \lesssim_p t {\|u\|}_p. \label{ineq:CZ_rescale}$$
We take the second derivative in the sense of distributions. Let $\phi \in C_c^\infty$, then by the dominated convergence theorem, $$\begin{aligned}
\int t^d\partial_{x_i}{\mathcal{K}}(tx) \partial_{x_j} \phi(x) dx & = \lim_{\epsilon \rightarrow 0} \int_{{\left\vertx\right\vert} \geq \epsilon}t^d \partial_{x_i}{\mathcal{K}}(tx) \partial_{x_j}\phi(x) dx \\
& = -t\lim_{\epsilon \rightarrow 0}\int_{{\left\vertx\right\vert} = \epsilon} t^{d-1}\partial_{x_y}{\mathcal{K}}(tx)\frac{x_j}{{\left\vertx\right\vert}}\phi(x) dS - t\textup{PV} \int t^{d}\partial_{x_i,x_j}{\mathcal{K}}(tx) \phi(x) dx. \end{aligned}$$ By ${\nabla}{\mathcal{K}}\in {L^{d/(d-1),\infty}}$, we have ${\nabla}{\mathcal{K}}= \mathcal{O}({\left\vertx\right\vert}^{1-d})$ as $x \rightarrow 0$. Therefore for $\epsilon$ sufficiently small, there exists $C > 0$ such that, $$\begin{aligned}
{\left\vertt\int_{{\left\vertx\right\
| 3,599
| 2,717
| 2,194
| 3,317
| null | null |
github_plus_top10pct_by_avg
|
of a conformal generator $R_{\mu }$ with pure discrete spectrum. The perturbative reading of $R_{0\text{ }}$as a Hamiltonian in its own right i.e. associated with an action in a functional integral setting naturally leads to the AdS formulation. The formal service role of AdS in order to access CQFT by a standard perturbative formalism (without being forced to understand first massive theories and then taking their scale-invariant limit) vastly increases the realm of conventionally accessible 4-dim. CQFT beyond those for which one had to use Lagrangians with supersymmetry in order to have a vanishing Beta-function.'
author:
- |
Bert Schroer\
presently CBPF, Rua Dr. Xavier Sigaud, 22290-180 Rio de Janeiro, Brazil\
email: schroer@cbpf.br\
Prof. emeritus of the Institut für Theoretische Physik\
FU-Berlin, Arnimallee 14, 14195 Berlin, Germany
date: 'May 9, 2000'
title: Particle versus Field Structure in Conformal Quantum Field Theories
---
A few introductory remarks
==========================
Ideas about the use of conformal quantum field theory entered particle physics for the first time at the height of the Kramers-Kronig dispersion relations [@Kastrup]. They were met with reactions ranging from doubts to outright rejection and the subject lay dormant for another 10 years when it reemerged on the statistical mechanics side in connection with second order phase transitions.
In the next section we will show that these early doubts of the old-time particle physicists were partially justified, because the particle structure in CQFT is indeed incompatible with interactions. However far from supplying a coffin nail for its utility in high energy physics, this no-go theorem also contains the message that one must use finer concepts in order preserve the usefulness of conformal quantum field theory as a theoretical laboratory for particle physics. There are massive particle-like objects (“infraparticles” [@Bu]) which have a continuous mass distribution with an accumulation of spectral weight at
| 3,600
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| 3,199
| 2,991
| null | null |
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|
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