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$-th ($k=1,2,\ldots,K$) conv-layer, where $K$ and $\{n_{k}\}$ are hyper-parameters. Let each latent pattern $u$ in the $k$-th conv-layer correspond to a square deformation range, which is located in the $D_{u}$-th slice of the conv-layer’s feature map. $\overline{\bf p}_{u}$ denotes the center of the range. As analyzed in the appendix, we only need to estimate the parameters of $D_{u},\overline{\bf p}_{u}$ for $u$.
**How to learn:**[` `]{} Just like the pattern pursuing in Fig. \[fig:rawMapToModel\], we mine the latent patterns by estimating their best locations $D_{u},\overline{\bf p}_{u}\in{\boldsymbol\theta}$ that maximize the following objective function, where ${\boldsymbol\theta}$ denotes the parameter set of the AOG. $$\begin{split}
{Loss}^{\textrm{AOG}}=\mathbb{E}_{I\in{\bf I}^{\textrm{ant}}}\big[-S_{top}+L(\Lambda_{top},\Lambda_{top}^{*})\big]\qquad\\
+\lambda^{\textrm{unant}}\mathbb{E}_{I\in{\bf I}^{\textrm{obj}}}\big[-S^{\textrm{unant}}_{\textrm{AOG}}+L^{\textrm{unant}}({\boldsymbol\Lambda}_{\textrm{AOG}})\big]
\end{split}
\label{eqn:LossAOG}$$ First, let us focus on the first half of the equation, which learns from part annotations. $S_{top}$ and $L(\Lambda_{top},\Lambda_{top}^{*})$ denote the final inference score of the AOG on image $I$ and the loss of part localization, respectively. Given annotations $(\Lambda_{top}^{*},v^{*})$ on $I$, we get $$\begin{split}
&S_{top}=\max_{v\in Child(top)}S_{v}\approx S_{v^{*}}\\
&L(\Lambda_{top},\Lambda_{top}^{*})=-\lambda_{v^{*}}\Vert{\bf p}_{top}-{\bf p}^{*}_{top}\Vert
\end{split}$$ where we approximate the ground-truth part template $v^{*}$ as the selected part template. We ignore the small probability of the AOG assigning an annotated image with an incorrect part template to simplify the computation. The part-localization loss $L(\Lambda_{top},\Lambda_{top}^{*})$ measures the localization error between the parsed part region ${\bf p}_{top}$ and the ground truth ${\bf p}^{*}_{top}={\bf p}(\Lambda_{top}^{*})$.
The second half of Equation (\[eqn:LossAOG\])
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\leq r-1$.
\[nonzerodivisorgammapower\] For any $t\in{\{0,1,\cdots,r-1\}}$, $\gamma^t$ is not a zero divisor of the ring $\cR_{m,r}$. This holds since the coefficients of $\gamma^t \cdot f(\gamma)$ are the same as those of $f(\gamma)$ (shifted cyclicly $t$ positions).
Matrices over Commutative Rings
-------------------------------
Let $\cR$ be a commutative ring (with unity). Let $M\in \cR^{n\times n}$ be an $n\times n$ matrix with entries from $\cR$. Most of the classical theory of determinants can be derived in this setting in exactly the same way as over fields. One particularly useful piece of this theory is the Adjugate (or Classical Adjoint) matrix. For an $n \times n$ matrix $M \in \cR^{n \times n}$ the Adjugate matrix is denoted by ${\mathrm{adj}}(M) \in \cR^{n \times n}$ and has the $(j,i)$ cofactor of $A$ as its $(i,j)$th entry (recall that the cofactor is the determinant of the matrix obtained from $M$ after removing the $i$th row and $j$th column multiplied by $(-1)^{i+j}$). A basic fact in matrix theory is the following identity.
\[adjointlemma\] Let $M \in \cR^{n \times n}$ with $\cR$ a commutative ring with identity. Then $M\cdot {\mathrm{adj}}(M)={\mathrm{adj}}(M)\cdot M=\det(M)\cdot I_n$ where $I_n$ is the $n\times n$ identity matrix.
The way we will use this fact is as follows:
\[remark-adjugate\] Suppose $M\in \cR^{n \times n}$ has non-zero determinant and let $\ba = (a_1,\ldots,a_n)^t \in \cR^n$ be some column vector where $a_1=0$ or $a_1=c$, where $c$ is not a zero-divisor. Then we can determine the value of $a_1$ (i.e., tell whether its $0$ or $c$) from the product $M \cdot \ba $. The way to do it is to multiply $M \cdot \ba$ from the left by ${\mathrm{adj}}(M)$ and to look at the first entry. This will give us $\det(M) \cdot a_1$ which is zero iff $a_1$ is (since $\det(M) \cdot c$ is always nonzero).
Matching Vector Families
------------------------
Let $S\subset\Z_m\setminus{\{0\}}$ and let $\cF=(\cU,\cV)$ where $\cU=({\bu_1,\cdots,\bu_n}),\cV=({\bv_1,\cdots,\bv_n})$ and $\forall
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local basis of the tangent space of $\partial G$ (and ${{\frac{\partial g}{\partial \tilde y_i}}}\in C(\Gamma_-)$ is to be understood in a local sense), and $\Sigma\in C(\ol G\times S\times I)$ such that ${{\frac{\partial \Sigma}{\partial x_j}}}\in C(\ol G\times S\times I),\ j=1,2,3$. Then the unique (classical) solution of the homogeneous *convection-scattering equation* (recall the definition of $D$ in ) \[trath1\] \_x+=0 D, satisfying the inhomogeneous inflow boundary condition \[trath2\] (y,,E)=g(y,,E)(y,,E)\_-, is given by ([@tervo14 Theorem 3.13]) \[trath3\] (x,,E)=e\^[-\_0\^[t(x,)]{}(x-s,,E)ds]{} g(x-t(x,),,E). We denote by $L_-g$ ($=\psi$) the solution of the problem (\[trath1\])-(\[trath2\]) that is, $L_-g$ is given by the right hand side of (\[trath3\]). Note that $L_-g$ is not generally even continuous (for non-convex $G$) even if $g$ happens to be smooth. We show, however, that the formula (\[trath3\]) gives a *weak solution* of the problem (\[trath1\])–(\[trath2\]) that is, $$\begin{aligned}
{3}
&
{\left\langle}\psi,-\omega\cdot \nabla_xv+\Sigma v{\right\rangle}_{L^2(G\times S\times I)}=0
\quad && {\rm for\ all}\ v\in C_0^\infty(G\times S\times I^\circ),\nonumber\\[2mm]
&
\psi(y,\omega,E)=g(y,\omega,E)\quad && {\rm for\ a.e.} \ (y,\omega,E)\in\Gamma_-,\end{aligned}$$ for any $g\in T_{\tau_-}^2(\Gamma_{-})$, and that $L_-g\in W^2(G\times S\times I)$.
We record the following Lemma for later use.
\[le:lift\] Assume that $\Sigma\in L^\infty(G\times S\times I)$ and that $\Sigma\geq 0$. Then for any $g\in T_{\tau_-}^2(\Gamma_{-})$ the function $L_-g$ defined by (\[trath3\]) (is measurable and) belongs to $L^2(G\times S\times I)$, and \[trpr9\] [L\_-g]{}\_[L\^2(GSI)]{}[g]{}\_[T\^2\_[\_-]{}(\_-)]{}. Moreover, equality holds here if $\Sigma=0$.
Write $L_{\Sigma,-}$ for the lift-operator defined in for a given $\Sigma\geq 0$, i.e. $\psi=L_{\Sigma,-} g$. Then $${\left\Vert L_{0,-} g\right\Vert}_{L^2(G\times S\times I)}^2
={}&\int_{G\times S\times I} g(x-t(x,\omega)\omega,\omega,E)^2 dx d\omega dE \\
={}&
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care of the second statement. Denoting by $i\colon[1]\to\ulcorner$ the sieve classifying the horizontal morphism $(0,0)\to (1,0)$ and by $k'\colon[1]\to\square$ the functor classifying the vertical morphism $(1,0)\to (1,1)$, the cofiber morphism is given by $$\label{eq:cof}
{\mathsf{cof}}\colon{\sD}^{[1]}\stackrel{i_\ast}{\to}{\sD}^\ulcorner\stackrel{(i_\ulcorner)_!}{\to}{\sD}^\square\stackrel{(k')^\ast}{\to}{\sD}^{[1]}.$$ Since the morphisms $i_\ast$ and $(k')^\ast$ are right adjoints, they preserve arbitrary right Kan extensions, hence homotopy finite limits. By assumption on , [@groth:can-can Prop. 3.9], and [@groth:can-can Lem. 4.9], also the morphism $(i_\ulcorner)_!$ preserves homotopy finite limits, and hence so does ${\mathsf{cof}}$ by [@groth:can-can Prop. 5.2]. An additional composition with the continuous evaluation morphism $1^\ast\colon{\sD}^{[1]}\to{\sD}$ establishes the corresponding result for $C$.
Given a pointed derivator , the derivator ${\sD}^\square={\sD}^{[1]\times[1]}$ admits cone and fiber morphisms in the first and the second coordinate, and these are respectively denoted by $$C_1, C_2\colon{\sD}^\square\to{\sD}^{[1]}\qquad\mbox{and}\qquad F_1,F_2\colon{\sD}^\square\to{\sD}^{[1]}.$$ Since these morphisms are pointed, for $X\in{\sD}^\square$ there is by [@groth:can-can Construction 9.7] a canonical comparison map $$\label{eq:cof-fib-comm}
C(F_2 X)\to F(C_1X).$$
\[cor:lim-comm\] Let be a derivator in which homotopy finite colimits and homotopy finite limits commute. Then is pointed and the canonical transformations are isomorphisms for every $X\in{\sD}^\square$.
This is immediate from .
As we show next, this property already implies that the derivator is stable. Together with we thus obtain the following more conceptual characterization of stability.
\[thm:stable-lim-II\] A derivator is stable if and only if homotopy finite colimits and homotopy finite limits commute.
By it suffices to show that a derivator is stable as soon as homotopy finite colimits and homotopy finite limits
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j - K\_[j,C]{},j=2,3, and define a (densily defined) closed linear operator $T_C:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ by setting D(T\_C):=&{L\^2(GSI)\^3 | T\_[j,C]{}L\^2(GSI), j=1,2,3}\
T\_C:=&(T\_[1,C]{},T\_[2,C]{},T\_[3,C]{}). Let $f\in L^2(G\times S\times I)^3$ and $g\in T^2(\Gamma_-)^3$. In the case where $\phi$ is regular enough, say $\phi\in
\tilde W^2(G\times S\times I)\times (\tilde W^2(G\times S\times I)\cap W_1^2(G\times S\times I))^2$ , the problem (\[desol10\]), (\[desol11\]), (\[desol12\]) can be expressed equivalently as \[comp5\] T\_C=[**f**]{},\_[|\_-]{}=[**g**]{},\_j(,,E\_m)=0,j=2,3, where ${\bf f}=e^{CE}f$, ${\bf g}=e^{CE}g$ as in Section \[cosyst\].
Assume that $g\in H^1(I,T^2(\Gamma'_-)^3)$ such that $g_j(E_{\rm m})=0$, $j=2,3$. Then ${\bf g}\in H^1(I,T^2(\Gamma'_-)^3)$ and ${\bf g}_j(E_{\rm m})=0,\ j=2,3$. Applying on ${\bf g}$ the lift operator $L$ given by $$\big((L{\bf g})(E)\big)(x,\omega):={\bf g}(E)(x-t(x,\omega)\omega,\omega)={\bf g}(x-t(x,\omega)\omega,\omega,E),$$ we have $L{\bf g}\in H^1(I,\tilde W^2(G\times S)^3)$, and it satisfies (cf. [@tervo14 Lemma 5.11]) $$\omega\cdot\nabla_x (L{\bf g})=0,\quad
(L{\bf g})_{|\Gamma_-}={\bf g}.$$ Furthermore, the condition $g_j(E_{\rm m})=0$ implies that \[comp7\] (L[**g**]{}\_j)(,,E\_[m]{})=0,j=2,3.
Denoting \[comp8\] P\_1(x,,E,D)\_1:=&\_x\_1+\_1\_1\
P\_[j,C]{}(x,,E,D)\_j:=&-[E]{}+\_x\_j+ CS\_j\_j+\_j\_j,j=2,3, and $${\bf P}_C(x,\omega,E,D)\phi:={}& \big(P_1(x,\omega,E,D)\phi_1,P_{2,C}(x,\omega,E,D)\phi_2,P_{3,C}(x,\omega,E,D)\phi_3\big), \\[2mm]
K_C\phi:={}&(K_{1,C}\phi,K_{2,C}\phi,K_{3,C}\phi),$$ we find that $T_C={\bf P}_C-K_C$. To simplify the notation, we shall write below $T=T_C$, $K=K_C$, and ${\bf P}={\bf P}_C$.
Let ${\bf P}_0$ be the densely defined linear operator acting in $L^2(G\times S\times I)^3$ such that D([**P**]{}\_0):=&{W\^2(GSI)(W\^2(GSI)W\_1\^2(GSI))\^2 |\
& \_[|\_-]{}=0, (,,E\_[m]{})=0},\
[**P**]{}\_0:=&[**P**]{}. Furthermore, let $\widetilde{\bf P}_0:L^2(G\times S\times I)^3\to L^2(G\times S\time
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National Science Foundation Grants 200021-165977 and 200020-162884.
[^2]: Rényi Institute, Hungarian Academy of Sciences, P.O.Box 127 Budapest, 1364, Hungary; `tardos@renyi.hu`. Supported by the Cryptography “Lendület” project of the Hungarian Academy of Sciences and by the National Research, Development and Innovation Office, NKFIH, projects K-116769 and SNN-117879.
[^3]: Rényi Institute, Hungarian Academy of Sciences, P.O.Box 127 Budapest, 1364, Hungary; `geza@renyi.hu`. Supported by National Research, Development and Innovation Office, NKFIH, K-111827.
---
abstract: 'In this paper, we show that the eccentricity of a planet on an inclined orbit with respect to a disc can be pumped up to high values by the gravitational potential of the disc, even when the orbit of the planet crosses the disc plane. This process is an extension of the Kozai effect. If the orbit of the planet is well inside the disc inner cavity, the process is formally identical to the classical Kozai effect. If the planet’s orbit crosses the disc but most of the disc mass is beyond the orbit, the eccentricity of the planet grows when the initial angle between the orbit and the disc is larger than some critical value which may be significantly smaller than the classical value of 39 degrees. Both the eccentricity and the inclination angle then vary periodically with time. When the period of the oscillations of the eccentricity is smaller than the disc lifetime, the planet may be left on an eccentric orbit as the disc dissipates.'
author:
- |
Caroline Terquem$^{1,2}$[^1] and Aikel Ajmia$^1$\
$^1$ Institut d’Astrophysique de Paris, UPMC Univ Paris 06, CNRS, UMR7095, 98 bis bd Arago, F-75014, Paris, France\
$^2$ Institut Universitaire de France
title: Eccentricity pumping of a planet on an inclined orbit by a disc
---
celestial mechanics — planetary systems — planetary systems: formation — planetary systems: protoplanetary discs — planets and satellites: general
Introduction {#sec:intro}
============
Among the
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s.
It has therefore recently been suggested to study the *asymptotic growth* of multiplicities (e.g., [@grochowrusek12 §2.2]). The natural object is the *Duistermaat–Heckman measure*, which is defined as the weak limit $$\label{definition duistermaat-heckman}
\operatorname{DH}_X := \lim_{k \rightarrow \infty} \frac 1 {k^{d_X}} \sum_{\mu \in \Lambda^*_{H,+}} m_{H,X,k}(\mu) \, \delta_{\mu/k},$$ where $d_X \in {\mathbb Z}_{\geq 0}$ is the appropriate exponent such that $\operatorname{DH}_X$ is a non-zero finite measure [@okounkov96]. The Duistermaat–Heckman measure has a continuous density function $f_X$ with respect to Lebesgue measure on the moment polytope; it is supported on the entire moment polytope (both statements follow from the main result of [@okounkov96]). For well-behaved varieties, Duistermaat–Heckman measures have a geometric interpretation [@heckman82; @guilleminsternberg82b; @sjamaar95; @meinrenken96; @meinrenkensjamaar99; @vergne98; @teleman00], which makes their computation potentially much more tractable [@guilleminlermansternberg88; @guilleminlermansternberg96; @christandldorankousidiswalter12] (this connection is however less clear in the singular cases relevant to geometric complexity theory). In this context, our main technical result is the following (see for the proof):
\[C\] The exponent $d_X$ is equal to $\dim X - R_X$, where $R_X$ is the number of positive roots of $H$ that are not orthogonal to all points of the moment polytope $\Delta_X$.
The significance of is that the order of growth of the “smoothed” multiplicities, as captured by the Duistermaat–Heckman measures, does only depend on the dimension of the orbit closures and on their moment polytopes.
Now suppose that we are in the situation that $X$ and $Y$ cannot be separated by using moment polytopes, i.e., $\Delta_X \subseteq \Delta_Y$. For the orbit closures $X$ and $Y$ that one tries to separate in geometric complexity theory, one can show that $\dim X < \dim Y$ [@burgisserlandsbergmaniveletal11; @burgisseri
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}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
---
abstract: 'The family of left-to-right algorithms reduces input numbers by repeatedly subtracting the smaller number, or multiple of the smaller number, from the larger number. This paper describes how to extend any such algorithm to compute the Jacobi symbol, using a single table lookup per reduction. For both quadratic time algorithms (Euclid, Lehmer) and subquadratic algorithms (Knuth, Schönhage, Möller), the additional cost is linear, roughly one table lookup per quotient in the quotient sequence. This method was used for the 2010 rewrite of the Jacobi symbol computation in .'
author:
- Niels Möller
bibliography:
- 'ref.bib'
date: 2019
title: Efficient computation of the Jacobi symbol
---
Introduction
============
The Legendre symbol and its generalizations, the Jacobi symbol and the Kronecker symbol, are important functions in number theory. For simplicity, in this paper we focus on computation of the Jacobi symbol, since the Kronecker symbol can be computed by the same function with a little preprocessing of the inputs.
Jacobi and GCD
--------------
Two quadratic algorithms for computing the Kronecker symbol (and hence also the Jacobi symbol) are described as Algorithm 1.4.10 and 1.4.12 in [@cohen]. These algorithms run in quadratic time, and consists of a series of reduction steps, related to Euclid’s algorithm and the binary algorithm, respectively. Both Kronecker algorithms share one property with the binary algorithm: The reduction steps examine the current pair of numbers in both ends. They examine the least significant end to cast out powers of two, and they examine the most significant end to determine a quotient (like in Euclid’s algorithm) or to determine which number is largest (like in the binary algorithm).
Fast, subquadratic, algorithms work by divide-and-conquer, where a substantial piece of the work is done by examining only one half of the input numbers. Fast left-to-right is related to f
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tion, and the rest of $\kappa_j-p$ items that are ranked on the bottom. An example of a sample with position-4 ranking six items $\{a,b,c,d,e,f\}$ might be a partial ranking of $(\{a,b,d\}>\{e\}>\{c,f\})$. Since each sample has only one separator for $2<p$, Theorem \[thm:main2\] simplifies to the following Corollary.
Under the hypotheses of Theorem \[thm:main2\], there exist positive constants $C$ and $c$ that only depend on $b$ such that if $n \geq C (\eta d \log d) /(\alpha^2\gamma^2\beta)$ then $$\begin{aligned}
\label{eq:main3}
\frac{1}{\sqrt{d}}\big\|\widehat{\theta} - \theta^* \big\|_2 \;\; \leq \;\; \frac{c}{\alpha\gamma} \sqrt{\frac{d\, \log d}{n }} \;.
\end{aligned}$$ \[coro:main2\]
Note that the error only depends on the position $p$ through $\gamma$ and $\eta$, and is not sensitive. To quantify the price of rank-breaking, we compare this result to a fundamental lower bound on the minimax rate in Theorem \[thm:cramer\_rao\_position\_p\]. We can compute a sharp lower bound on the minimax rate, using the Cramér-Rao bound, and a proof is provided in Section \[sec:proof\_cramer\_rao\_position\_p\].
\[thm:cramer\_rao\_position\_p\] Let $\mathcal{U}$ denote the set of all unbiased estimators of $\theta^*$ and suppose $b >0$, then $$\begin{aligned}
\inf_{\widehat{\theta} \in \mathcal{U}} \sup_{\theta^* \in \Omega_b} \E[{\|\widehat{\theta} - \theta^*\|}^2] &\geq&
\frac{1}{2p\log(\kappa_{\max})^2} \sum_{i = 2}^d \frac{1}{\lambda_i(L)}
\;\, \geq \;\, \frac{1}{2p\log(\kappa_{\max})^2} \frac{(d-1)^2}{n }\;,
\end{aligned}$$ where $\kappa_{\rm max} = \max_{j\in[n]} |S_j| $ and the second inequality follows from the Jensen’s inequality.
Note that the second inequality is tight up to a constant factor, when the graph is an expander with a large spectral gap. For expanders, $\alpha$ in the bound is also a strictly positive constant. This suggests that rank-breaking gains in computational efficiency by a super-exponential factor of $(p-1)!$, at the price of increased error by a fact
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one";
}
Is This what you are looking for? Anything needing to be changed to meet your needs?
Q:
Android: Gradle compilation error expects element value to be a constant expression - Feature Module
This is my first multi-module project.
This login activity exists in a Feature Module which gets many of its dependencies from a Base Feature Module.
I'm getting the error error: element value must be a constant expression on lines where I'm using @BindView with my TextInputEditText and Button.
Also, I noticed that the import for the R class is missing, yet none of the R.id. are in red. Why is that?
A:
Feature modules are similar to library modules. Once you move from an application module to a library/feature module, your R class fields aren't final constants anymore. That's why the issue happens. To fix this, you will need to switch to Butterknife's R2 class.
More info in the links below:
Official documentation: https://github.com/JakeWharton/butterknife#library-projects
Android: Why do we need to use R2 instead of R with butterknife?
https://github.com/JakeWharton/butterknife/issues/1113 <- This mentions you may need ButterKnife 9.0
Q:
Routing to a specific LAN gateway depending which application sends the request
I have a VPN set up with USAIP (usaip.eu) to get an IP in the US although being physically connected to the internet in Europe.
Now i want to route only specific requests via this VPN (you already guessed it, Hulu). Because Hulu's IPs are changing quite often, i would like to specifiy, that only traffic initiated by some applicatins (say firefox.exe) is routed thru the VPN. Can a software firewall do this (and if yes, which one do you recommend?) or what would be the best way to accomplish this?
A:
And you'd actually need to route DNS requests as that's Akamai's first round of routing.
A much easier way is to just use SSH as a socks proxy:
ssh -D 1080 <host>
Then tell your browser there's a SOCKS proxy on 127.0.0.1:1080 and things just work.
Even for apps that don't directly support SOCK
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ar{F}}_{\beta _\kappa }\,|\,\mu <\kappa <\nu \rangle
\subset U^-(\chi )
\end{aligned}$$ for all $\mu ,\nu \in \{1,2,\dots ,n\}$ with $\mu <\nu $.
We prove the first relation for $\mu =1$ and all $\nu \in \{2,3,\dots ,n\}$. Then the first relation for $\mu >1$ follows from $$E_{\beta _\mu }E_{\beta _\nu }-\chi (\beta _\mu ,\beta _\nu )
E_{\beta _\nu }E_{\beta _\mu }
={T}_{i_1}\cdots {T}_{i_{\mu -1}}(
E_{i_\mu }E'_\nu -\chi (\beta _\mu ,\beta _\nu )E'_\nu E_{i_\mu }),$$ where $E'_\nu ={T}_{i_\mu }\cdots {T}_{i_{\nu -1}}(E_{i_\nu })$, by using the case $\mu =1$, Eq. and the first relation in . The proof of the second relation of the theorem is similar. The third and fourth relations can be obtained from the first two by applying ${\Omega }$ and using the formulas $$\begin{aligned}
{\Omega }(E_{\beta _\kappa })\in {{\Bbbk }^\times }{\bar{F}}_{\beta _\kappa },\quad
{\Omega }({\bar{E}}_{\beta _\kappa })\in {{\Bbbk }^\times }F _{\beta _\kappa },
\qquad \kappa \in \{1,2,\dots ,n\},
\label{eq:aaaU(E)}
\end{aligned}$$ which follow from Thm. \[th:Liso\](iii).
Let $\nu \in \{2,3,\dots ,n\}$. For all $(m_1,m_2,\dots ,m_n)\in {\mathbb{N}}_0^I$ with $m_\kappa <{b^{\chi}} (\beta _\kappa )$ for all $\kappa \in \{1,2,\dots ,n\}$ let $a_{m_1,\dots ,m_n}\in {\Bbbk }$ such that $$\begin{aligned}
E_{i_1}E_{\beta _\nu }-\chi ({\alpha }_{i_1},\beta _\nu )E_{\beta _\nu }E_{i_1}
=\sum _{m_1,\dots ,m_n}a_{m_1,\dots ,m_n}E_{\beta _1}^{m_1}
\cdots E_{\beta _n}^{m_n}.
\label{eq:EE-EE}
\end{aligned}$$ The numbers $a_{m_1,\dots ,m_n}\in {\Bbbk }$ exist and are unique by Thm. \[th:PBW\]. Let $\chi _\nu =r_{i_\nu }\cdots r_{i_2}r_{i_1}(\chi )$. Apply to Eq. the isomorphism ${T}^-={T}^-_{i_\nu }\cdots {T}^-_{i_2}{T}^-_{i_1}\in {\mathrm{Hom}}(U(\chi ),
U(\chi _\nu ))$. For all $\kappa \in \{1,2,\dots ,\nu \}$, $$\begin{aligned}
{T}^-_{i_\nu }\cdots {T}^-_{i_2}{T}^-_{i_1}(E_{\beta _\kappa })
=&{T}^-_{i_\nu }\cdots {T}^-_{i_{\kappa +1}}
{T}^-_{i_\kappa }(E_{i_\kappa })\\
=&{T}^
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d simple closed curves that are dual to $a$.
Claim 1: $(g_a)_{\#}$ agrees with $\lambda$ on every vertex of $\{[a]\} \cup L_a \cup D_a$.
Proof of Claim 1: We already know that $(g_a)_{\#}$ agrees with $\lambda$ on every vertex of $\{[a]\} \cup L_a$. Let $d$ be a 1-sided simple closed curve that is dual to $a$. Let $T$ be a regular neighborhood of $a \cup d$. We see that $T$ is a real projective plane with two boundary components, say $x, y$. Since $(g_a)_{\#}^{-1} \circ \lambda$ is identity on ${[a]} \cup L_a$, $(g_a)_{\#}^{-1} \circ \lambda$ is identity on the complement of $T$. Since $(g_a)_{\#}^{-1} \circ \lambda$ is a superinjective simplicial map, there exists $d' \in \lambda([d])$ such that $d'$ is disjoint from the complement of $T$, so $d'$ is in $T$. Since $d$ is the only nontrivial simple closed curve in $T$ which is not isotopic to $a$ by Scharlemann’s Theorem in [@Sc], we see that $d'$ is isotopic to $d$. So, $((g_a)_{\#}^{-1} \circ \lambda) ([d]) = [d]$. Hence, $(g_a)_{\#}^{-1} \circ \lambda$ is identity on ${[a]} \cup L_a \cup D_a$.
=2.2in =2.2in
=2.2in =2.2in
=2.2in
Claim 2: Let $v$ be any 1-sided simple closed curve on $N$. Then, $(g_v)_{\#} = (g_a)_{\#}$ on $\mathcal{C}(N)$.
Proof of Claim 2: Since $g=1$, $\widetilde{X}(N)$ is connected by Theorem 3.10 in [@AK]. So, we can find a sequence $a \rightarrow a_1 \rightarrow a_2 \rightarrow \cdots \rightarrow a_n=v$ of 1-sided simple closed curves connecting $a$ to $v$ such that each consecutive pair is connected by an edge in $\widetilde{X}(N)$. By Claim 1, $(g_a)_{\#}$ agrees with $\lambda$ on $\{[a]\} \cup L_a \cup D_a$, and $(g_{a_1})_{\#}$ agrees with $\lambda$ on $\{[a_1]\} \cup L_{a_1} \cup D_{a_1}$. So, we see that $(g_a)_{\#} ^{-1} (g_{a_1})_{\#}$ fixes every vertex in $([a] \cup L_a \cup D_a) \cap ([a_1] \cup L_{a_1} \cup D_{a_1})$. We note that this is the same thing as fixing every vertex in $(St_a \cup D_a) \cap (St_{a_1} \cup D_{a_1})$ since the vertex sets of them are equal. By Theorem \[id\], we get $(g_a)_{\#} = (g_{a_1})_{\
| 2,012
| 1,144
| 1,588
| 2,058
| 3,150
| 0.774478
|
github_plus_top10pct_by_avg
|
silon$, denote $N:=c(1+\Delta w_2)>0$. Take $n_1\in \ZZ_{>0}$ such that $\frac{n_1}{N}\in [\lambda,\lambda+\varepsilon)$. Note that $\frac{n_1+j}{N}\in [\lambda,\lambda+\varepsilon)$ for all $j=0,...,w_2-1$, hence there is exactly one such $j$ for which $a:=n_1+j\equiv k \mod (w_2)$. This proves the claim.
Partial versus total resolutions
--------------------------------
This example illustrates the advantage of using $\Q$-resolutions. In addition, this can be used to obtain information about local invariants of the embedding of a curve in a cyclic quotient singularity.
Let $\mathcal{C}$ be the curve in $X:=\PP^2_{(9,5,2)}$ defined by $g = x^2 z + y^2 z^5 + y^4$. Consider the cyclic branched covering $\rho: \tilde{X} \to X$ of degree $d = 20$ ramifying on $\mathcal{C}$. This curve passes through the point $[1:0:0]$ which is of type $\frac{1}{9}(5,2)$ and it is quasi-smooth, since it is locally isomorphic to $z=0$. The curve is singular at $[0:0:1] \in \frac{1}{2}(1,1)$ and it defines a *quasi-node*, i.e. it is locally isomorphic to $xy=0$. The rest of the points are smooth and $[0:1:0] \not \in \mathcal{C}$. Therefore the curve itself is already a $\mathbb{Q}$-divisor with (non-simple!) $\Q$-normal crossings and Theorem \[thm:conucleo\_singular\] tells us that $H^1(\tilde{X},\CC)=0$ because the map $\pi^{(k)}$ is identically zero. Note that showing this fact is not obvious if one uses a standard good resolution.
We can use this fact to obtain a formula for the correction term map $$R_{X,P}: \operatorname{Weil}(X,P)/\operatorname{Cart}(X,P) \longrightarrow \QQ,$$ –see section \[sec:RR\]. We resolve the singularity of $X$ at the point $P=[0:1:0] \in \frac{1}{5}(1,3)$ so that $\sum_{Q \in \pi^{-1}(P)} R_{Y,Q}( L^{(k)} ) = 0$. Since the curve does not pass through this point, the second term in is zero and the formula $\beta_{\mathcal{C},P}^{(k)} = 0$ becomes $$R_{X,P}(-kH) = \alpha_{\mathcal{C},P}^{(k)}
:= \frac{1}{2} \sum_{i,j=1}^2 {\left \{ -k \b_i \right \}} \left( {\left \{ -k \b_i \right \}} + \nu_i
| 2,013
| 1,821
| 1,565
| 1,933
| 3,213
| 0.773994
|
github_plus_top10pct_by_avg
|
st be true, as the 2PI theory is built around the requirement that $\left\langle \varphi^A\varphi^B\right\rangle=G^{AB}$
The 1PIEA $\Gamma_{1PI}$ is recovered from the 2PIEA $\Gamma$ as
\_[1PI]{}=where the correlations $G_0$ are slaved to the mean field through
\_[,(AB)]{}=0 One further derivative shows that
G\^[CD]{}\_[0,E]{}=-\^[-1]{}\_[,(AB)E]{} Therefore the first and second derivatives of the 1PIEA are
\_[1PI,A]{}=\_[,A]{}
\_[1PI,AB]{}=\_[,AB]{}+\_[,A(CD)]{}G\^[CD]{}\_[0,B]{} The stochastic equation for the mean field fluctuations, as derived from the 1PIEA, is simply eq. (\[1pilan\]) with noise self-correlation (\[1pinoise\]). Now eq. (\[2pilan2\]) admits the solution
\^[CD]{}=G\^[CD]{}\_[0,E]{}\^E-12\^[-1]{}\_[AB]{} Using this into eq. (\[2pilan1\]) we get back eq. (\[1pilan\]) provided we identify
\^[1PI]{}\_A=\_A+12G\^[CD]{}\_[0,A]{}\_[CD]{} Indeed, both expressions have the same self-correlation.
Stochastic equations for the physical propagators
-------------------------------------------------
Finally, let us recover the noise self-correlation given in [@CalHu99]. The situation is similar to the one in the 1PI theory. In the 1PI theory, the $\phi^{-}$ field vanishes identically on-shell. However, in the stochastic approach we assign a nontrivial source $\xi^{1PI}_{-}$ to it. It is by eliminating this auxiliary field that we recover the usual approach, with a single stochastic source $\tilde{\xi}$ whose self-correlation is given by the noise kernel.
Similarly, in the quantum field theory problem the correlator $G^{--}=\left\langle \varphi^{-}\varphi^{-}\right\rangle$ vanishes identically, as a result of path ordering. However, in the stochastic approach, we consider it as an auxiliary field and couple a source to it. The authors of [@CalHu99] failed to recognize the violation of the constraint $G^{--}=0$, but compensated this oversight by forcing a sign change in their expression for the noise self-correlation. In the context of the proper theory, this sign change is due to the elimination of
| 2,014
| 543
| 2,527
| 2,011
| null | null |
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|
the data.
Guaranteeing positive MUTFs greater than some minimum [@Bro03; @Maj07] would require a change to the likelihood. The approach taken in @Rid06 is to specify independent left-truncated gamma prior distributions for the the expected MUTFs, $\mu_j$ for $j=1,\ldots,u$. However any such change would not lead to the tractable updates required for the concise sequential analysis described in Sections \[sec:Overview\] and \[sec:DetailObsProc\]. Within the constraints of the algorithm overviewed in Section \[sec:Overview\], the natural mechanism for preventing these undesirable scenarios is via post-processing: the conditional prior for ${\boldsymbol{\mu}}|\nu$ in is re-calibrated by truncating it to the region $M=[{\mu_{\min}},\infty)^u$ for some minimum MUTF ${\mu_{\min}}$. It follows that the re-calibrated marginal prior for ${\boldsymbol{\mu}}$ is: $$\begin{aligned}
\tilde{\pi}({\boldsymbol{\mu}}) & = \frac{1}{\pi(M)}\pi({\boldsymbol{\mu}})\mathbb{I}_M({\boldsymbol{\mu}}),\label{eq:TruncMU}\end{aligned}$$ where $\pi({\boldsymbol{\mu}})$ is the multivariate Student’s t-density centred at ${\mathbf{m}}_0$ with shape matrix $\frac{b_0}{a_0}C_0$ and $2a_0$ degrees of freedom, and, with a slight abuse of notation, $\pi(M)=\int_M\pi({\boldsymbol{\mu}})d{\boldsymbol{\mu}}$. The effect on the marginal likelihood from the prior re-calibration is examined by Proposition \[prop:AdjML\].
\[prop:AdjML\] Let $f(y_{1:T}|s_{1:T})$ denote the marginal likelihood defined in Section \[sec:Overview\]. The re-calibrated marginal likelihood, denoted by $\tilde{f}(y_{1:T}|s_{1:T})$, resulting from truncating the prior for ${\boldsymbol{\mu}}$ in to $M$ is: $$\begin{aligned}
\tilde{f}(y_{1:T}|s_{1:T}) = \frac{\pi(M|y_{1:T},s_{1:T})}{\pi(M)} f(y_{1:T}|s_{1:T}) = \frac{\pi(M|y_{1:T},s_{1:T})}{\pi(M|y_{1:\tau-1},s_{1:\tau-1})} f(y_{1:T}|s_{1:T}), \label{eq:recalib}\end{aligned}$$ where $\pi(M|y_{1:t},s_{1:t}) = \int_M \pi({\boldsymbol{\mu}}|y_{1:t},s_{1:t}) d{\boldsymbol{\mu}}$.
Expressing the re-calibrated marginal likeliho
| 2,015
| 2,734
| 2,595
| 1,759
| 1,010
| 0.795817
|
github_plus_top10pct_by_avg
|
have the following dynamical system:
$\dot{x_1}= -x_2 + (x_1(1-(x_1^2+x_2^2)^2))$ , $
\dot{x_2}= x_1 + (x_2(1-(x_1^2+x_2^2)^2))$,
$\dot{x_3}= \epsilon x_3$ .
I am required to work out the flow for this system. I have switched it to cylindrical coordinates obtaining $\dot{r}=r(1-r^4)$ , $\dot{\theta}=1$, $\dot{z}=\epsilon{z}$.
I assume in order to work out $r$ I must use partial fractions, but I'm not really sure how to proceed with this, as surely it gets a bit awkward. Have I made a mistake somewhere? Is this the right approach?
Thanks
A:
Have to write something otherwise it doesn't let me post the comment.
Q:
Control loading of kernel module automatically
When i do rmmod usb_storage and then dmesg i can see the output saying usb_storage deregistered. But when i attach a flash drive and then dmesg , it says
Initializing USB Mass Storage driver...
[16565.129239] scsi41 : usb-storage 1-1:1.0
[16565.130134] usbcore: registered new interface driver usb-storage
[16565.130172] USB Mass Storage support registered.
How do i manually insert/remove the module without it automatically getting loaded?
A:
Just blacklist the kernel driver:
echo "blacklist usb-storage" | sudo tee -a /etc/modprobe.d/blacklist.conf
After that, nobody can use a USB flash drive in your system.
You'll have to manually "insmod" your driver with sudo to use USB devices again or if you want to restore the initial behaviour remove the module from the blacklist file.
Q:
Visual C++ 2008 runtime error-- debug vs release exe problem?
I have a Windows executable (native, not .Net) project that I'm trying to pass along to a new team member. It's a graphics modeling tool that uses the Qt widget library and OpenGL.
The project runs fine on my box but when we buld and link it on this new member's machine and he tries deubugging it, here's what he sees (not all entries included, for brevity):
ModelingTool.exe': Loaded
'C:\ModelingTool\ModelingTool\ModelingTool\Debug\ModelingTool.exe', Symbols
loaded.
'ModelingTool.exe': Loaded
| 2,016
| 4,226
| 1,447
| 2,037
| 682
| 0.80236
|
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|
}$$ provided these expressions are well defined. We shall now show that the expressions are well defined provided $\Theta_0 \ne \pi/2$, and we give a more explicit formula for ${\dot {\cal F}}_{\Theta_0}$.
Suppose hence from now on that $\Theta_0 \ne \pi/2$. We work in global Minkowski coordinates in which points on Minkowski spacetime are represented by their position vectors, following the notation in [@schlicht]. The trajectory is written as $x^b(\tau)$. At each point on the trajectory, we introduce three spacelike unit vectors $e^b_{(\alpha)}(\tau)$, $\alpha = 1,2,3$, which are orthogonal to each other and to $u^b(\tau) = \frac{dx^b(\tau)}{d\tau}$, and are Fermi-Walker transported along the trajectory. We coordinatise the hyperplane orthogonal to $u^b(\tau)$ by ${\bm \xi} = (\xi^1, \xi^2, \xi^3)$ by $$\begin{aligned}
x^b(\tau, {\bf \xi}) = x^b(\tau) + \xi^\alpha e^b_{(\alpha)}(\tau)
\ . \end{aligned}$$
Using (\[angwhitmannfunction\]) and (\[smearedRARF\]), we obtain $$W_{\Theta_0}(\tau,\tau^\prime) = \frac{1}{(2 \pi)^3} \int \frac{d^3 {\bf k}}{2 \omega({\bf k})} \;
g_{\Theta_0} ( {\bf k}, \tau) \, g^*_{\Theta_0} ( {\bf k}, \tau^\prime)
\ ,
\label{gwhitmann}$$ where $\omega({\bf k}) = \sqrt{{\bf k}^2}$ and $$g_{\Theta_0} ( {\bf k}, \tau) = \int d^3\xi \; f_{\epsilon}({\bm \xi},\Theta_{0}) e^{i k_b x^b(\tau, \, {\bm \xi} )}
\ .
\label{gdef}$$ The index $\alpha$ refers to values $(1,2,3)$ and $e^b_{(\alpha)}(\tau)$ are the orthogonal Fermi unit-basis vectors in the spatial direction orthogonal to $u^b(\tau)$. Then defining $3$ - vector $\tilde{{\bf k}}$ having components $(\tilde{{\bf k}})_\alpha = k_b {\bf e}^b_{(\alpha)}(\tau)$ and working in spherical co-ordinates in the ${\bf \xi}$- space, we can recast Eq. (\[gdef\]) to get $$\begin{aligned}
g_{\Theta_0}( {\bf k}, \tau) &=& \frac{1}{\pi} e^{i k_b x^b(\tau)} \int_0^\infty d\xi \, \frac{\xi^2\epsilon}{(\xi^2+\epsilon^2)^2} e^{i \xi \cos\Theta_0 |\tilde{{\bf k}}|}
\notag
\\
& = & e^{i k_b x^b(\tau)} \left( I_R + I_M \right)
\ , \end{aligned}$$ whe
| 2,017
| 3,561
| 2,265
| 1,925
| null | null |
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|
htarrow}}}{\underline{b}}_{i+1}\}\cap\big\{n_{b_i}>0,~b_i\text{ is pivotal for }y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\big\}\Big\},\end{aligned}$$ where, by convention, ${\underline{b}}_{T+1}=x$. Then, by ${\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}\leq{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}$, we obtain $$\begin{aligned}
{\label{eq:Theta'-2ndindbd2}}
{(\ref{eq:contr-(c)})}&=\sum_{T\ge1}\sum_{\vec b_T}\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\,H_{{{\bf n}};\vec b_T}(y,x)
\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}{\nonumber}\\
&\leq\sum_{T\ge1}\sum_{\vec b_T}\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 n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,H_{{{\bf n}};\vec b_T}(y,x)\,\cap\,
\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}.\end{aligned}$$
On the event $H_{{{\bf n}};\vec b_T}(y,x)$, we denote the ${{\bf n}}$-double connections between the pivotal bonds $b_1,\dots,b_T$ by $$\begin{aligned}
{{\cal D}}_{{{\bf n}};i}=\begin{cases}
{{\cal C}}_{{\bf n}}^{b_1}(y)&(i=0),\\
| 2,018
| 971
| 1,928
| 1,961
| null | null |
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|
Recall that $\underline{G}$ is a closed subgroup scheme of $\underline{M}^{\ast}$ and $\tilde{G}$ is a closed subgroup scheme of $\tilde{M}$, where $\tilde{M}$ is the special fiber of $\underline{M}^{\ast}$. Thus we may consider an element of $\tilde{G}(\kappa_R)$ as an element of $\tilde{M}(\kappa_R)$. Based on Section \[m\], an element $m$ of $\tilde{G}(\kappa_R)$ may be written as, say, $(m_{i,j}, s_i \cdots w_i)$ and it has the following formal matrix description: $$m= \begin{pmatrix} \pi^{max\{0,j-i\}}m_{i,j} \end{pmatrix} \textit{ together with } z_i^{\ast}, m_{i,i}^{\ast}, m_{i,i}^{\ast\ast}.$$ Here, if $i$ is even and $L_i$ is *of type* $\textit{I}^o$ or *of type* $\textit{I}^e$, then $$m_{i,i}=\begin{pmatrix} s_i&\pi y_i\\ \pi v_i&1+\pi z_i \end{pmatrix} \textit{or}
\begin{pmatrix} s_i&r_i&\pi t_i\\ \pi y_i&1+\pi x_i&\pi z_i\\ v_i&u_i&1+\pi w_i \end{pmatrix},$$ respectively, where $s_i\in M_{(n_i-1)\times (n_i-1)}(B\otimes_A\kappa_R)$ (resp. $s_i\in M_{(n_i-2)\times (n_i-2)}(B\otimes_A\kappa_R)$), etc. We can write $m_{i, i}=(m_{i, i})_1+\pi\cdot (m_{i, i})_2$ when $L_i$ is *of type II* and for each block of $m_{i,i}$ when $L_i$ is *of type I*, $s_i=(s_i)_1+\pi\cdot (s_i)_2$ and so on. We can also write $m_{i, j}=(m_{i, j})_1+\pi\cdot (m_{i, j})_2$ when $i\neq j$. Here, $(m_{i, i})_1, (m_{i, i})_2\in M_{n_i\times n_i}(\kappa_R) \subset M_{n_i\times n_i}(B\otimes_A\kappa_R)$ when $L_i$ is *of type II* and so on, and $\pi$ stands for $\pi\otimes 1\in B\otimes_A\kappa_R$. Note that the description of the multiplication in $\tilde{M}(\kappa_R)$ given in Section \[m\] forces $(m_{i,i})_1$ (when $L_i$ is *of type II*) and $(s_i)_1$ to be invertible.
Then an element $m$ of $\tilde{G}(\kappa_R)$ maps to
$$\left\{
\begin{array}{l l}
\begin{pmatrix} (s_i)_1&0\\
\mathcal{X}_i&1 \end{pmatrix} & \quad \textit{if $L_i$ is \textit{of type} $\textit{I}^o$};\\
\begin{pmatrix} (s_i)_1&0\\
\mathcal{Y}_i&1 \end{pmatrix} & \quad \textit{if $L_i$ is \textit{of type} $\textit{I}^e$};\\
\begin{pmatrix} (m_{i,i})_1&0\
| 2,019
| 1,461
| 1,510
| 1,725
| 2,679
| 0.777923
|
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|
n{aligned}
\label{eq:hessian}
H(\theta) = -\sum_{j=1}^n \sum_{a=1}^{\ell_j} \sum_{i<\i \in S_j} \I_{\big\{(i,\i) \in G_{j,a}\big\}} \lambda_{j,a} \Bigg( (e_i - e_{\i})(e_i - e_{\i})^\top \frac{\exp(\theta_i + \theta_{\i})}{[\exp(\theta_i) + \exp(\theta_{\i})]^2}\Bigg).\end{aligned}$$ It follows from the definition that $-H(\theta)$ is positive semi-definite for any $\theta \in \reals^d$. The smallest eigenvalue of $-H(\theta)$ is equal to zero and the corresponding eigenvector is all-ones vector. The following lemma lower bounds its second smallest eigenvalue $\lambda_2(-H(\theta))$.
\[lem:hessian\_positionl\] Under the hypotheses of Theorem \[thm:main2\], if $$\begin{aligned}
\label{eq:posl_lam_cond}
\sum_{j = 1}^n \sum_{a = 1}^{\ell_j} \lambda_{j,a}(\kappa_j - p_{j,a}) \geq 2^{6}e^{18b} \frac{\eta\delta}{\alpha^2\beta\gamma^2\tau} d\log d
\end{aligned}$$ then with probability at least $ 1- d^{-3}$, the following holds for any $\theta\in\Omega_b$: $$\begin{aligned}
\label{eq:lambda2_bound_positionl}
\lambda_2(-H(\theta)) \;\geq\; \frac{e^{-4b}}{(1+e^{2b})^2}\frac{\alpha \gamma}{d-1} \sum_{j = 1}^n \sum_{a = 1}^{\ell_j} \lambda_{j,a}(\kappa_j - p_{j,a})\,.
\end{aligned}$$
Define $\Delta = \widehat{\theta} - \theta^*$. It follows from the definition that $\Delta$ is orthogonal to the all-ones vector. By the definition of $\hat{\theta}$ as the optimal solution of the optimization , we know that $\Lrb(\widehat{\theta}) \geq \Lrb(\theta^*)$ and thus $$\begin{aligned}
\Lrb(\widehat{\theta}) - \Lrb(\theta^*) - \langle\nabla\Lrb(\theta^*),\Delta\rangle \;\geq\; -\langle\nabla\Lrb(\theta^*),\Delta\rangle \;\geq\; -{\|\nabla\Lrb(\theta^*)\|}_2{\|\Delta\|}_2, \label{eq:thm_ml_1}\end{aligned}$$ where the last inequality follows from the Cauchy-Schwartz inequality. By the mean value theorem, there exists a $\theta = a\widehat{\theta} + (1-a)\theta^*$ for some $a \in [0,1]$ such that $\theta \in \Omega_b$ and $$\begin{aligned}
\label{eq:thm_ml_2}
\Lrb(\widehat{\theta}) - \Lrb(\theta^*) - \langle\nabla\Lr
| 2,020
| 1,234
| 1,797
| 1,923
| null | null |
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|
We are going to prove such $\delta_n, u_n$ are two sequences we need.
We hope to apply Theorem \[estimate\], so we compute, for each $n$, $$\begin{aligned}
C^2\delta_n|u_n|^{-1}|\varphi_n|\mathscr{W}_n^{\frac{1}{2}}=\frac{1}{4}C^2\Omega_n^{-2}\exp\left(-\frac{\varphi_n^2}{2^8c_1\Omega_n^4}\right)|\varphi_n|\mathscr{W}_n\end{aligned}$$ where $\mathscr{W}_n=\mathscr{W}(u_0,u_n)=\left\{1,\left|\log\frac{\Omega_n}{\Omega_0(u_0)}\right|\right\}$. We can see the right hand side tends to zero and therefore holds for $\delta=\delta_n$, $u=u_n$ for sufficiently large $n$ depending on $C$ and the initial bound of $L\phi$ on $C_{u_0}$. As a consequence, we have the following estimates for a sufficiently large $C\ge C_0$ and $(\ub,u)\in[0,\delta_n]\times\{u_n\}$:
- $\displaystyle |{\underline{h}}+1|\le C^{-1},\ \text{which implies}\ -{\underline{h}}\ge\frac{1}{2}$.
- $\displaystyle \Omega^{-2}\ge\frac{1}{4}\Omega_n^{-2}, 1\ge\frac{r}{2r_n}$.
- $|rL\phi-\varphi_n|\le c|rL\phi(\ub,u_0)-\varphi(u_0)|+cC^{-1}|\varphi_n|$ for some $c$ depending on the initial bound of $L\phi$ on $C_{u_0}$ which follows from and implies that $\displaystyle |rL\phi|>\frac{1}{2}|\varphi_n|$ for $n$ sufficiently large.
- $\displaystyle \Omega_n^2\delta_n\ge\Omega_n^2h_n\delta_n=\int_0^{\delta_n}\Omega_n^2h_n{\mathrm{d}}\ub\ge\frac{1}{4}\int_0^{\delta_n}\Omega^2h{\mathrm{d}}\ub=\frac{1}{4}(r-r_n)$, where we use $h_n=h(0,u_n)\ge h$ because of $Dh\le0$ from equation .
From , and the above all estaimtes, we have, for $n$ sufficiently large, $$\begin{aligned}
m-m_n=&\frac{1}{2}\int_0^{\delta_n}(-{\underline{h}})\Omega^{-2}(rL\phi)^2{\mathrm{d}}\ub\\
>&\frac{1}{2^6}\delta_n\Omega_n^{-2}\varphi_n^2\\
=&\frac{1}{2^6}\delta_n\Omega_n^{-2}\cdot2^8c_1\Omega_n^4\log\frac{r_n}{4\Omega_n^2\delta_n}\\
\ge&\frac{c_1r}{2r_n}\cdot4\Omega_n^2\delta_n\log\frac{r_n}{4\Omega_n^2\delta_n}\\
\ge&\frac{c_1r}{2r_n}(r-r_n)\log\frac{r_n}{r-r_n}\end{aligned}$$ which is the inequality in . The last inequality above is because the function $x\log\frac{r_n}{x}$ is mo
| 2,021
| 769
| 1,682
| 1,910
| null | null |
github_plus_top10pct_by_avg
|
a star unless $|\cI|=3$ and $\cI=\binom{K}{2}$ for some $K=\{x,y,z\}$. But then $\{\{x\},\{x,y\},\{x,z\}\}\subseteq \cH_x$, and so $|\cI|=|\cH_x|$, which is case (\[case:2\]) of the theorem with $M=\emptyset$.\
Thus we may assume that $\cH$ contains a set of size $3$ and, consequently, also contains a star of size $4$. Therefore $|\cI|\ge 4$. If $\cIo\not=\emptyset$ or $|\cIt|\ge 4$ then $\cI$ is a star and we are done; so we will assume that $\cIo=\emptyset$ and $|\cIt|\le 3$ (thus $\cIr\ne\emptyset$). Our proof splits into cases, based on $|\cIt|$.\
We first introduce some notation that we make use of below. Without loss of generality $\bigcup_{I\in\cI^2}I=[m]$ for some $m\le 4$. For $\emptyset\not=J\subset [m]$ define $\oJ=[m]\setminus J$, $\cA(J)=\{I\in\cIr\mid I\cap [m]=J\}$, and $C(J)=(\bigcup_{A\in\cA(J)}A)\setminus J$. In practice, we relax the notation somewhat to write $\cA(2,3)$ instead of $\cA(\{2,3\})$, and $C(\otwo)$ instead of $C(\overline{\{2\}})$, for example. Note that, when $m=3$, $|C(\oi)|=|\cA(\oi)|$ and $\cIr\setminus\bigcup_{i\in [3]}\cA(\oi)\subseteq\{[3]\}$.\
$|\cIt|=3$
----------
### $\cIt$ is a star
We may assume that $\cIt=\{\{1,2\},\{1,3\},\{1,4\}\}$. If $\cIr=\cIro$, then $\cI$ is a star. Otherwise, we must have $\cIr\setminus\cIro=\{\{2,3,4\}\}$. Therefore $(\cI\setminus\{\{2,3,4\}\})\cup\{\{1\}\}\cup\{\{1,j\}\mid j\in I\in\cIro\}$ is a star subfamily of $\cH$ that has size at least $|\cI|$ and, in fact, is larger unless $I\subseteq [4]$ for every $I\in\cIro$. Therefore we must have that $|\cIro|\le 3$.\
If $|\cIro|<3$ then, without loss of generality, $\cIro\subseteq\{\{1,2,3\},\{1,2,4\}\}$, and then $(\cI\setminus\{\{1,3\},\{1,4\}\})\cup\{\{2\},\{2,3\},\{2,4\}\}$ is a larger intersecting subfamily of $\cH$, a contradiction. So we are left with the case in which $|\cIro|=3$ and, consequently, $|\cI|=7$ and $\cH\supseteq\binom{[4]}{\le 3}$.\
If there is an $H\in\cH$ such that both $H\cap [4]\not=\emptyset$ and $H\setminus [4]\not=\emptyset$ then, by taking $h\in H\cap [4]$, we h
| 2,022
| 987
| 454
| 2,164
| null | null |
github_plus_top10pct_by_avg
|
the galactic core $1/\kappa(\rho)\sim
\lambda_{DE}[\Lambda_{DE}/8\pi\rho_H]^{(1+\alpha_\Lambda)/2}$, where $\rho_H$ is the core density. Even though $\lambda_{DE}=
14010^{+800}_{-810}$ Mpc, because $\rho_H\gg\Lambda_{DE}/2\pi$, $\alpha_\Lambda$ can be chosen so that $1/\kappa(r)$ is comparable to typical $r_H$. Doing so sets $\alpha_\Lambda\approx 3/2$.
Given a $v(r)$, $\mathbf{a}(r)$ can be found and $f(r)$ determined. We idealize the observed velocity curves as $v^{\hbox{\scriptsize ideal}}(r) =v_H r/r_H$ for $r \le
r_H$, while $v^{\hbox{\scriptsize ideal}}(r)=v_H$ for $r>r_H$, where $v_H$ is the observed asymptotic velocity. This $v^{\hbox{\scriptsize{ideal}}}(r)$ is more tractable than the pseudoisothermal velocity curve, $v^{\hbox{\scriptsize{p-iso}}}(r)$, used in [@Blok-1]. As it has the same limiting forms in both the $r\ll r_H$ and $r\gg r_H$ limits, $v^{\hbox{\scriptsize ideal}}(r)$ is also an idealization of $v^{\hbox{\scriptsize p-iso}}(r)$.
For cusp-like density profiles [@Silk], it is the density profile that is given. While it is possible to integrate the general density profile to find the corresponding curves $v_{\hbox{\scriptsize
cusp}}(r)$, both the maximum value of $v_{\hbox{\scriptsize cusp}}(r)$ and the size of the core are different depending on the profile. These core sizes would thus have to be scaled appropriately to compare one profile with another. Doing so is possible in principle, but would be analytically intractable in practice. We instead take $f(r) = \rho_H
\left(r_H/r\right)^\gamma $ if $r \le
r_H$, and $f(r) = \rho_H \left(r_H/r\right)^\beta/3 $ if $r>r_H$ for the density profiles. Here, $\gamma < 2$ and $\beta\ge 2$ agrees with the parameters for the generic cusp-like density profile [@Krav], with the core size set to $r_H$. The $\gamma=0, \beta=2$ case corresponds to the idealized psuodoisothermal profile.
Since $\rho\gg\Lambda_{DE}/2\pi$ in Regions I and II, Eq. $(\ref{rhoGEOM})$ minimizes $$\begin{aligned}
\mathcal{F}[\rho] =
\frac{\Lambda_{DE}c^2}{8\pi}\left(\chi^
| 2,023
| 4,009
| 2,415
| 2,023
| 2,231
| 0.781598
|
github_plus_top10pct_by_avg
|
127 to EV-B73 and EV-B strains.
Region Position ^1^ TO-127 versus EV-B73 ^2^ TO-127 versus EV-B ^3^
---------------- -------------- -------------------------- ------------------------ -------- ------------
Partial 5′ UTR 1--327 76--82 Non coding 81--83 Non coding
VP4 328--528 72--78 82--95 70--77 81--87
VP2 529--1311 77--81 91--97 66--72 76--86
VP3 1312--2022 76--82 93--98 66--71 74--81
VP1 2023--2889 76--82 89--96 58--65 61--70
2A 2890--3339 72--78 87--93 78--80 92--95
2B 3340--3636 69--79 82--95 79--82 83--98
2C 3637--4623 79--82 95--97 80--82 97--98
3A 4624--4890 73--78 94--95 77--78 93--95
3B 4891--4956 72--84 90--95 80--85 94--95
3C 4957--5505 79--82 95--97 81--82 96--97
3D 5506--6891 76--81 90--97 79--82 96--97
3' UTR 6895--6993 88--89 Non coding 88--91 Non coding
CDS 328--6891 79--80 94--96 74--76 87--90
Whole Genome 01--6993 79--81 Non coding 75--77 Non coding
^1^ Positions are numbered according to strain TO-127. ^2^ EV-B73 strains utilized for comparison and GenBank access number (088/SD/CNH/04 (KF874626) and CA55-1988 (AF241359)). ^3^ EV-B prototype strains utilized for comparison and GenBank access number (CA76-10392
| 2,024
| 5,238
| 1,676
| 1,379
| null | null |
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|
orresponds to a diagonal of $M$ in which the variance of the entries is $2$ instead of $1$. (See Theorem \[T:semicircle-law-general\] below and the discussion following it.)
On the other hand, the analogous results *are* simple in the case in which the characters $\chi \in \widehat{G}$ are all real-valued, so that the Fourier transform defines an isometry (up to scaling) between the *real* $\ell^2$ spaces on $G$ and $\widehat{G}$. By Lemma \[T:p2\], this is the case precisely when every $a \in G$ satisfies $a^2 = 1$, or in other words, when $G \cong ({\mathbb{Z}}_2)^n$ for some $n$. In this case a $G$-circulant matrix is automatically symmetric, so that there is no difference (except for scaling) between the “$G$-circulant real Ginibre ensemble” and the “$G$-circulant GOE”. The following results are proved in the same way as Proposition \[T:C-Ginibre-eigenvalues\] and Corollary \[T:C-Ginibre-limit\].
\[T:Z2-GOE-eigenvalues\] Let $G \cong ({\mathbb{Z}}_2)^n$ and let $\{Y_a \mid a \in G\}$ be independent, standard real Gaussian random variables. Then the eigenvalues $\bigl\{ \lambda_\chi \mid \chi \in \widehat{G} \bigr\}$ of $M$ given by are independent, standard real Gaussian random variables.
\[T:Z2-GOE-limit\] Suppose that for each $n$, $G^{(n)} \cong ({\mathbb{Z}}_2)^n$ and $\{Y_a^{(n)}
\mid a \in G^{(n)}\}$ are independent, standard real Gaussian random variables. Then ${\mathbb{E}}\mu^{(n)} = \gamma_{\mathbb{R}}$ for each $n$, and $\mu^{(n)} \to \gamma_{\mathbb{R}}$ weakly almost surely.
General matrix entries {#S:general}
======================
Our main results are stated under a Lindeberg-type condition on the random variables $Y_a^{(n)}$ used to generate the random matrices: $$\label{E:Lindeberg-condition}
\forall {\varepsilon}> 0 : \quad
\lim_{n \to \infty} \frac{1}{{\left\vert G^{(n)} \right\vert}}
\sum_{a\in G^{(n)}} {\mathbb{E}}\Bigl( \bigl\vert Y_a^{(n)}\bigr\vert^2
{\mathbbm{1}_{\vert Y_a^{(n)}\vert
\ge {\varepsilon}\sqrt{\vert G^{(n)} \vert }}} \Bigr)
= 0.$$ The usu
| 2,025
| 2,544
| 2,466
| 1,777
| null | null |
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|
A}_1\equiv \sum_i a_{1i}\hat{\Pi}_{1i},\qquad \hat{A}_2\equiv\sum_k a_{2k}\hat{\Pi}_{2k}.$$ The generating function for the moments ${\left<A_1^{n_1}A_2^{n_2}\right>}$ reads $$\begin{split}
& C{\left(\zeta_1,\zeta_2\right)}=\text{Tr}{\left(e^{i\zeta_1\hat{A}_1}e^{i\zeta_2\hat{A}_2}\hat{\rho}\right)}=\sum_{i,k}e^{i\zeta_1 a_{1i}}e^{i\zeta_2 a_{2k}}\text{Tr}{\left(\hat{\Pi}_{1i}\hat{\Pi}_{2k}\hat{\rho}\right)}\equiv\\
&\qquad\qquad \equiv \sum_{i,k}e^{i\zeta_1 a_{1i}}e^{i\zeta_2 a_{2k}}p_{ik}.
\end{split}$$ We are tempetd to define the joint probability as $$\begin{split}
& p{\left(a_1,a_2\right)}\equiv\frac{1}{4\pi^2}\int d\zeta_1 d\zeta_2 e^{-i{\left(\zeta_1a_1+\zeta_2a_2\right)}}C{\left(\zeta_1,\zeta_2\right)}=\\
&\qquad\qquad =\sum_{i,k}p_{ik}\delta{\left(a_1-a_1i\right)}\delta{\left(a_2-a_{2k}\right)}.\label{a1}
\end{split}$$ Due to $$\sum_{k}p_{ik}=\sum_k\text{Tr}{\left(\hat{\Pi}_{1i}\hat{\Pi}_{2k}\hat{\rho}\right)}=\text{Tr}{\left(\hat{\Pi}_{1i}{\left(\sum_k\hat{\Pi}_{2k}\right)}\hat{\rho}\right)}=\text{Tr}{\left(\hat{\Pi}_{1i}\hat{\rho}\right)}=p_{1i}$$ single probability densities can be obtained as marginals $$p_1{\left(a_1\right)}=\int da_2 p{\left(a_1,a_2\right)}.$$ To have the genuine probability distribution we must assume $p_{ik}\geq 0$. Then the last expression (\[a1\]) provides a finite positive measure on $\mathbb{R}^2$. Therefore, by Bochner theorem $C{\left(\zeta_1,\zeta_2\right)}$ is positive definite function [@reed]. In particular $$C{\left(\zeta_1,\zeta_2\right)}=\overline{C{\left(-\zeta_1,-\zeta_2\right)}}$$ or $$\text{Tr}{\left(e^{i\zeta_1\hat{A}_1}e^{i\zeta_2\hat{A}_2}\hat{\rho}\right)}=\text{Tr}{\left(e^{i\zeta_2\hat{A}_2}e^{i\zeta_1\hat{A}_1}\hat{\rho}\right)}.\label{a2}$$ Assuming that (\[a2\]) holds for all states $\hat{\rho}$ we find $$e^{i\zeta_1\hat{A}_1}e^{i\zeta_2\hat{A}_2}=e^{i\zeta_2\hat{A}_2}e^{i\zeta_1\hat{A}_1}$$ or ${\left[\hat{A}_1,\hat{A}_2\right]}=0$. We see that the joint probability can be defined only for commuting variables.
Taking into account Fine’s results one
| 2,026
| 1,446
| 1,562
| 1,890
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|
liver necroinflammation and fibrosis. All significant factors identified by the univariate analysis were entered into the multivariate models for identifying predictors associated with marked alterations of liver histology. A *P* value less than 0.05 was considered statistically significant.
RESULTS
=======
The baseline characteristics of 115 carriers with PNALT are shown in Table [1](#T1){ref-type="table"}. The mean age was 39.7 years (range, 21-67 years), and 62 (54.2%) subjects were male. Among the 115 patients, 86 (55.5%) were HBeAg-positive and 69 (44.5%) were HBeAg-negative.
######
Baseline demographic and clinical characteristics of all subjects with normal alanine aminotransferase level
**All (*n* = 155)** **HBeAg-positive (*n* = 86)** **HBeAg-negative (*n* = 69)**
-------------------- --------------------- ------------------------------- -------------------------------
Age (yr) 39.7 ± 9.9 42.9 ± 8.9 37.2 ± 9.9
Male, *n* (%) 84 (54.2) 52 (60.9) 34 (48.8)
HBV DNA (logIU/mL) 5.8 ± 2.1 4.3 ± 1.4 6.9 ± 1.9
WBC (10^9^/mL) 5.9 ± 3.1 5.9 ± 4.4 5.8 ± 1.5
PLT (10^9^/mL) 190.08 ± 62.9 182.29 ± 57.7 196.3 ± 66.5
ALT (U/L) 26.0 ± 8.7 25.0 ± 9.2 26.8 ± 8.1
AST (U/L) 24.6 ± 6.6 24.1 ± 6.1 25.1 ± 7.0
ALP (U/L) 68.1 ± 20.1 67.1 ± 17.2 68.9 ± 22.2
GGT (U/L) 21.8 ± 14.6 21.1 ± 11.9 22.3 ± 16.5
TB (g/L) 13.6 ± 6.7 13.6 ± 7.5 13.6 ± 6.1
PT (s) 11.4 ± 1.1 11.4 ± 1.0 11.4 ± 1.4
TT (s) 19.1 ± 1.7 18.8 ± 1.8 19.3 ± 1.5
A/G ratio 1.7 ± 0
| 2,027
| 331
| 2,106
| 2,369
| null | null |
github_plus_top10pct_by_avg
|
, with $H \cap A$ a Hall $\pi$-subgroup of $A$ and $H \cap B$ a Hall $\pi$-subgroup of $B$.
Next we record some arithmetical lemmas, that will be applied later.
\[SylowSym\] Let $G$ be the symmetric group of degree $k$ and let $s$ be a prime. If $s^{ N}$ is the largest power of $s$ dividing $|G|=k!$, then $N \leq \frac{k-1}{s-1}$.
\[cuentas\] Let $q,s,t$ be positive integers. Then:
1. $(q^s-1,q^t-1)=q^{(s,t)}-1$,
2. $(q^s+1,q^t+1)=\begin{cases} q^{(s,t)}+1 \quad \text{if both } s/(s,t) \text{ and } t/(s,t) \text{ are odd,}\\
(2,q+1) \quad \text{otherwise,}\end{cases}$
3. $(q^s-1,q^t+1)=\begin{cases} q^{(s,t)}+1 \quad \text{if } s/(s,t) \text{ is even and } t/(s,t) \text{ is odd,}\\
(2,q+1) \quad \text{otherwise.}\end{cases}$
We introduce now some additional terminology. Let $n$ be a positive integer and $p$ be a prime number. A prime $r$ is said to be *primitive with respect to the pair $(p, n)$* (or a *primitive prime divisor of $p^n-1$*) if $r$ divides $p^n-1$ but $r$ does not divide $p^k-1$ for every integer $k$ such that $1\leq k< n$.
\[Zsi\] Let $n$ be a positive integer and $p$ a prime. Then:
- If $n \geq 2$, then there exists a prime $r$ primitive with respect to the pair $(p, n)$ unless $n=2$ and $p$ is a Mersenne prime or $(p, n)=(2, 6)$.
- If the prime $r$ is primitive with respect to the pair $(p, n)$, then $r-1 \equiv 0\,(\mbox{mod }n)$. In particular, $r \geq n+1$.
The following lemmas are used when dealing with prime power order elements. We remark that the proof of the first one uses CFSG.
\[FKS\] Let $G$ be a group acting transitively on a set $\Omega$ with $|\Omega|>1$. Then there exists a prime power order element $x\in G$ which acts fixed-point-freely on $\Omega$.
\[feinkantor\] Let $H$ be a subgroup of a group $G$. If every prime power order element of $G$ lies in $\bigcup_{g\in G} H^g$, then $G=H$.
If $H$ is normal in $G$, then every prime power order element belongs to $H$, and since $G$ is generated by such elements, we get $G=H$. So we may assume that $H$ is no
| 2,028
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| 1,887
| null | null |
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|
=\|\phi(x)-\phi(y)\|_{\infty}$.
By the pigeonhole principle there exist two pairs, say $(x_1,y_1)$ and $(x_2,y_2)$, for which $j(x_1,y_1) = j(x_2,y_2)=j$. It is easy to verify that our assumptions on the four real numbers $\phi(x_1)_j$, $\phi(x_2)_j$, $\phi(y_1)_j$, $\phi(y_2)_j$, are contradictory. Thus $d(n,l_\infty) \geq \frac n 2 - 1$.
into $l_2$. {#l2}
-----------
\[momad2\] $\frac n 2 \leq d(n,l_2) \leq n$. Furthermore, for every $\delta > 0$, and every large enough $n$, almost no linear orders $\rho$ on ${[n] \choose 2}$ can be realized in dimension less than $\frac n {2+\delta}$.
The upper bound is apparently folklore. As we could not find a reference for it, we give a proof here.
Let $\rho$ be a linear order on ${[n] \choose 2}$. Let $\epsilon$ be a real symmetric matrix with the following properties:
- $\epsilon_{ii} = 0$ for all $i$.
- $\frac{1}{n} > \epsilon_{ij} > 0$, for all $i \neq j$.
- The numbers $\epsilon_{i,j}$ are consistent with the order $\rho$.
Since the sum of each row is strictly less than one, all eigenvalues of $\epsilon$ are in the open interval $(-1,1)$. It follows that the matrix $I - \epsilon$ is positive definite. Therefore, there exists a matrix $V$ such that $V V^t = I - \epsilon$. Denote the $i$’th row of $V$ by $v_i$. Clearly, the $v_i$’s are unit vectors, and $<v_i,v_j> = - \epsilon_{i,j}$ for $i \neq j$. Therefore, $||v_i - v_j||_2^2 = <v_i,v_i> + <v_j,v_j> - 2<v_i,v_j> = 2 + 2 \epsilon_{i,j}$. It follows that the map $\phi(i) = v_i$ is a realization of $\rho$, and the upper bound is proved. In fact, one can add another point without increasing the dimension, by mapping it to $0$, and perturbing the diagonal.\
For the lower bound, it is essentially known that if $X$ is the metric induced by $K_{n,n}$, then $d(X,l_2) \geq n$. We discuss this in more detail in the next section.
For the second part of the lemma we need a bound on the number of [*sign-patterns*]{} of a sequence of real polynomials. Let $p_1,...,p_m$ be real polynomials in $l$ variables of (to
| 2,029
| 3,697
| 2,733
| 1,837
| null | null |
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|
gation for different values of $k$. Only the top three categories with the highest transition probabilities are shown. With high consistency, the transition probabilities to the same topic increase while those to other categories decrease with ascending order $k$.[]{data-label="fig:sameornot_msnbc"}](sameornot_MSNBC){width="\columnwidth"}
Discussion {#subsec:discussion .unnumbered}
----------
Our findings and observations in this article show that simple likelihood investigations (see e.g., [@chierichetti]) may not be sufficient to select the appropriate order of Markov chains and to prove or falsify whether human navigation is memoryless or not. To ultimately answer this, we think it is inevitable to look deeper into the results obtained and to investigate them with a broader spectrum of model selection methods starting with the ones presented in this work.
By applying these methods to human navigational data, the results suggest that on the Wikigame page dataset a zero order model should be preferred. This is due to the rising complexity of higher order models and indicates that it is difficult to derive the appropriate order for finite datasets with a huge amount of distinct pages having only limited observations of human navigational behavior. In this article we presented and applied a variety of distinct model selection that all include (necessary) ways of penalizing the large number of parameters needed for higher order models. [[Yet, we do not necessarily know what would happen if we would apply the models to a much larger number of navigational paths over pages. Perhaps higher order models would then outperform lower ones. As it is unlikely to get hands on such an amount of data for large websites, a starting point to further test this could be to analyze a sub-domain with rich data; i.e., a large number of observations over just a very limited number of distinct pages. However, due to no current access to such data, we leave this open for future work.]{}]{}
On the other hand, the results on a topical
| 2,030
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| 1,500
| 1,177
| null | null |
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|
uad \textit{if $i$ is even and $L_i$ is \textit{of type $I^e$}};\\
\begin{pmatrix} \tilde{s}_i''&\pi \tilde{r}_i''& \tilde{t}_i''\\ \tilde{y}_i''&1+\pi \tilde{x}_i''& \tilde{u}_i''\\
\pi \tilde{v}_i''&\pi \tilde{z}_i''&1+\pi \tilde{w}_i'' \end{pmatrix} & \quad \textit{if $i$ is odd and $L_i$ is \textit{free of type $I$}}.
\end{array} \right.$$
Let $$\left\{
\begin{array}{l l}
(\tilde{z}_i^{\ast})''=1/\pi\left(\tilde{z}_i''+\delta_{i-2}\tilde{k}_{i-2, i}''+\delta_{i+2}\tilde{k}_{i+2, i}'' \right) & \quad \textit{if $i$ is even and $L_i$ is \textit{of type I}};\\
(\tilde{m}_{i,i}^{\ast})''=1/\pi\left(\delta_{i-1}v_{i-1}\cdot \tilde{m}_{i-1, i}''+\delta_{i+1}v_{i+1}\cdot \tilde{m}_{i+1, i}''\right)
& \quad \textit{if $i$ is odd and $L_i$ is \textit{bound of type I}};\\
(\tilde{m}_{i,i}^{\ast\ast})''=1/\pi\left(\delta_{i-1}v_{i-1}\cdot {}^t\tilde{m}_{i, i-1}''+\delta_{i+1}v_{i+1}\cdot {}^t\tilde{m}_{i, i+1}''\right)
& \quad \textit{if $i$ is odd and $L_i$ is \textit{bound of type I}}.
\end{array} \right.$$ Here, notations ($\tilde{z}_i''$, $\tilde{k}_{i-2, i}''$, and so on) in the right hand sides are as explained above in the description of an element of $\underline{M}(R)$ in terms of $\tilde{m}_{i,j}''$. These three equations should be interpreted as follows. We formally compute the right hand sides and then they are of the form $1/\pi(\pi X)$. Then the left hand sides are the same as $X$ in the right hand sides. Such $X$’s are formal polynomials about $(m_{i,j}, s_i\cdots w_i)$ and $(m_{i,j}', s_i'\cdots w_i')$. The precise description will be given below.
Let $(m_{i,j}'', s_i''\cdots w_i'')$ be formed by letting $\pi^2$ be zero in each entry of $(\tilde{m}_{i,j}'', \tilde{s}_i''\cdots \tilde{w}_i'')$. Then each matrix of $(m_{i,j}'', s_i''\cdots w_i'')$ has entries in $B\otimes_AR$ and so $(m_{i,j}'', s_i''\cdots w_i'')$ is an element of $\underline{M}(R)$ and is the product of $(m_{i,j}, s_i\cdots w_i)$ and $(m_{i,j}', s_i'\cdots w_i')$. More precisely,
1. If $i\neq
| 2,031
| 4,320
| 1,522
| 1,343
| 3,634
| 0.771124
|
github_plus_top10pct_by_avg
|
}$$ Hence combining (\[eqn:De1\]) and (\[eqn:De2\]) gives that $$\begin{aligned}
\label{eqn:De0}
\De(W;\pi_{GB}^J)&=\De_1(W;\pi_{GB}^J)-\De_2(W;\pi_{GB}^J)-(q-3r-3)\tr M_{GB} \non\\
&= -(q-3r+3-a-2b)\tr M_{GB} +\De_3+\De_4,\end{aligned}$$ where $$\begin{aligned}
\De_3 &= -2(b-2)\int_{\Rc_r}\tr[(I_r-\La)^{-1}\La]f_{GB}(\La;W)\dd\La \\
&\qquad +\frac{b-2}{m_{GB}}\int_{\Rc_r}\tr[M_{GB}(I_r-\La)^{-1}\La]f_{GB}(\La;W)\dd\La, \\
\De_4 &=-2(1-v_1)c\int_{\Rc_r}\tr[\{v_1I_r+(1-v_1)\La\}^{-1}\La^2]f_{GB}(\La;W)\dd\La \\
&\qquad +\frac{(1-v_1)c}{m_{GB}}\int_{\Rc_r}\tr[M_{GB}\{v_1I_r+(1-v_1)\La\}^{-1}\La]f_{GB}(\La;W)\dd\La.\end{aligned}$$ Here, it can easily be verified that $\De(W;\pi_{GB}^J)$ is finite for $q+a>0$ and $b>2$.
For notational simplicity, we use the notation $$\Er_\La[g(\La)]=\int_{\Rc_r}g(\La)f_{GB}(\La;W)\dd\La \Big/\int_{\Rc_r}f_{GB}(\La;W)\dd\La$$ for an integrable function $g(\La)$. Then from (\[eqn:De0\]), $$\begin{aligned}
\label{eqn:De00}
{\De(W;\pi_{GB}^J)\over m_{GB}}
=& (c-q+r+b-1)\tr \Er_\La(\La) \non\\
&+ (b-2)\Big[ \tr\big[ \Er_\La(\La)\Er_\La\{(I_r-\La)^{-1}\La\} \big] - 2\tr\big[ \Er_\La\{(I_r-\La)^{-1}\La\}\big]\Big]\non\\
&+(1-v_1)c\Big[ \tr\big[ \Er_\La(\La)\Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big]
\non\\
&\qquad\qquad\qquad -2\tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La^2\}\big]\Big]\end{aligned}$$ for $c=a+b+2r-2$. Note that $0_{r\times r} \preceq \La \preceq I_r$ and $I_r \preceq (I_r-\La)^{-1}$. Since $\Er_\La(\La) \preceq I_r$ and $\tr\big[ (I_r-\La)^{-1}\La \big] \geq \tr\La$, the second term in the r.h.s. of (\[eqn:De00\]) is evaluated as $$\begin{aligned}
\tr\big[ &\Er_\La(\La)\Er_\La\{(I_r-\La)^{-1}\La\}\big] - 2\tr\big[ \Er_\La\{(I_r-\La)^{-1}\La\}\big]
\\
&\leq - \tr\big[ \Er_\La\{(I_r-\La)^{-1}\La\}\big] \leq - \tr \Er_\La(\La).\end{aligned}$$ Since $b>2$, we have $$\begin{aligned}
\label{eqn:De01}
{\De(W;\pi_{GB}^J)\over m_{GB}}
\leq & (c-q+r+1)\tr \Er_\La(\La) \non\\
&+(1-v_1)c\Big[ \tr\big[ \Er_\La(\La)\Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big]
\non\\
&\qquad\qquad\qquad
| 2,032
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| 649
| 2,196
| null | null |
github_plus_top10pct_by_avg
|
x) = \langle U_x \zeta^U_j,\zeta^U_k\rangle$$ are called *coordinate functionals* (associated to $U$ and $\{\zeta^U_1,...,\zeta^U_{d_U}\}$). They satisfy the following property, as a consequence of [@hewitt2013abstract Theorem 27.20]: $$\begin{aligned}
\forall f\in C(G)\,\forall x\in G,\, f\ast \varphi^U_{jk}(x) = \sum_{r=1}^{d_U} \int f(y)\,\overline{\varphi^U_{rj}(y)}\,d\lambda(y)\, \varphi^U_{rk}(x).
\label{coordfunct}\end{aligned}$$
A *trigonometric polynomial* on $G$ is a linear combination of matrix coefficients. It is straightforward to see that all trigonometric polynomials are in fact linear combinations of coordinate functionals, independently of the choice of bases $\{\zeta^U_1,...,\zeta^U_{d_U}\}$, which are assumed to be fixed. Thus, for any given trigonometric polynomial $P$, there is a finite set $F$ of continuous irreducible unitary representations such that $P$ is in the linear span of $\{\varphi^U_{jk}| U \in F,\, j,k = 1,...,d_U\}$. It follows from (\[coordfunct\]) that the operator $f\in C(G) \mapsto f\ast P\in C(G)$ has its range contained in the linear span of $\{\varphi^U_{jk}| U \in F,\, j,k = 1,...,d_U\}$, thus in particular it is of finite rank.
The following proposition is our crucial tool from harmonic analysis.
\[crucialharmonic\] Suppose that $G$ is a compact Lie group. Then there exists a sequence $F_n$ of positive real functions on $G$ satisfying
1. each $F_n$ is a positive definite *central* (commutes under convolution with any function in $L_1(G)$) trigonometric polynomial,
2. $F_n(g^{-1}) = F_n(g)$, $g\in G$, for each $n$,
3. for each $n$, $\int F_n \,d\lambda =1$, and
4. $f\ast F_n(x) \rightarrow f(x)$ $\lambda$-almost everywhere for every $f\in L_p(G),\,1\leq p <\infty$.
By [@Knapp Corollary IV.4.22], every compact Lie group is isomorphic to a matrix group, so we may suppose that $G\subseteq \mathrm{GL}(n,\mathbb{C})$ for some $n\in {\mathbb{N}}$. By the following standard ‘unitarization trick’, we may assume that $G$ is a (necessarily closed) subgroup of the u
| 2,033
| 1,728
| 1,873
| 1,816
| null | null |
github_plus_top10pct_by_avg
|
athbb{R}}^{d})$, $j=1,\ldots,d$. We denote $\nabla \phi$ the $%
d\times d$ matrix field whose $(i,j)$ entry is $\partial_j\phi^i$ and $%
\sigma (\phi )=\nabla \phi (\nabla \phi )^{\ast }$.
We suppose that $\sigma (\phi )$ is invertible and we denote $\gamma (\phi
)=\sigma ^{-1}(\phi ).$ Then $$\int (\partial _{i}f)(\phi (x))g(x)dx=\int f(\phi (x))H_{i}(\phi ,g)(x)dx
\label{ip1}$$with$$H_{i}(\phi ,g)=-\sum_{k=1}^{d}\partial _{k}\left( g\sum_{j=1}^{d}\gamma
^{i,j}(\phi )\partial _{k}\phi ^{j}\right). \label{ip2}$$Moreover, for a multi-index $\alpha =(\alpha _{1},...,\alpha _{m})$ we define$$H_{\alpha }(\phi ,g)=H_{\alpha _{m}}(\phi ,H_{(\alpha _{1},...,\alpha
_{m-1})}(\phi ,g)) \label{ip3}$$and we obtain $$\int (\partial ^{\alpha }f)(\phi (x))g(x)dx=\int f(\phi (x))H_{\alpha }(\phi
,g)(x)dx \label{ip4}$$
**Proof**. The proof is standard: we use the chain rule and we obtain $%
\nabla (f(\phi ))=(\nabla \phi)^*(\nabla f)(\phi ) .$ By multiplying with $%
\nabla \phi $ first and with $\gamma (\phi )$ then, we get $(\nabla f)(\phi
)=\gamma(\phi )\nabla \phi \nabla (f(\phi ))$. Using standard integration by parts, (\[ip1\]) and (\[ip2\]) hold. And (\[ip3\]) follows by iteration. $\square $
Our aim now is to give estimates of $\left\vert H_{a}(\phi
,g)(x)\right\vert_q .$ We use the notation introduced in (\[NOT1\]) and for $q\in{\mathbb{N}}$, we denote$$C_{q}(\phi )(x)=\frac{1\vee \left\vert \phi (x)\right\vert _{1,q+2}^{2d-1}}{%
1\wedge (\det \sigma (\phi )(x))^{q+1}}. \label{ip5}$$
For every multi index $\alpha $ and every $q\in {\mathbb{N}}$ there exists a universal constant $C\geq 1$ such that$$\left\vert H_{\alpha }(\phi ,g)(x)\right\vert _{q}\leq C\left\vert
g(x)\right\vert _{q+\left\vert \alpha \right\vert }\times C_{q+\left\vert
\alpha \right\vert }^{\left\vert \alpha \right\vert }(\phi )(x). \label{ip6}$$
**Proof**. We begin with some simple computational rules:$$\begin{aligned}
\left\vert f(x)g(x)\right\vert _{q} &\leq &C\sum_{k_{1}+k_{2}=q}\left\vert
f(x)\right\vert _{k_{1}}\left\vert g(x)\right\vert
| 2,034
| 710
| 1,881
| 2,011
| null | null |
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|
\exp(\theta_{j_2})}{W-\exp(\theta_{j_1})}\cdots \Bigg( \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i, \\ j_1,\cdots,j_{\ell-2}}} \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-2}}\exp(\theta_{k})}\Bigg)\Bigg)\Bigg) \frac{e^{2b}}{\kappa-\ell+1} \label{eq:posl_upper2}\end{aligned}$$ Consider the last summation term in the equation , and let $\Omega_\ell = S\setminus\{i,j_1,\ldots,j_{\ell-2}\}$, such that $|\Omega_\ell| = \kappa-\ell+1$. Observe that from equation , $\frac{\exp(\theta_i)}{\sum_{j \in \Omega_\ell} \exp(\theta_j)} \geq \frac{\lalpha_{i,\ell,\theta}}{\kappa-\ell+1}$. We have, $$\begin{aligned}
\sum_{j_{\ell-1} \in \Omega_\ell} \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-2}}\exp(\theta_{k})} &=& \frac{\sum_{j_{\ell-1} \in \Omega_\ell} \exp(\theta_{j_{\ell-1}}) }{ \exp(\theta_i)+ \sum_{j_{\ell-1} \in \Omega_\ell} \exp(\theta_{j_{\ell-1}})} \nonumber\\
&\leq& \bigg( \frac{\lalpha_{i,\ell,\theta}}{\kappa-\ell+1} + 1\bigg)^{-1} \nonumber\\
&=& \frac{\kappa-\ell+1}{\lalpha_{i,\ell,\theta} + \kappa - \ell +1} \nonumber\\
&=& \sum_{j_{\ell-1} \in \Omega_\ell} \frac{\exp(\ltheta_{j_{\ell-1}})}{\lW-\sum_{k=j_1}^{j_{\ell-2}}\exp(\ltheta_{k})}, \label{eq:posl_upper3}\end{aligned}$$ where follows from the definition of $\{\ltheta\}_{j \in S}$.
Consider $\{\Omega_{\widetilde{\ell}}\}_{2 \leq \widetilde{\ell} \leq \ell - 1}$, $|\Omega_{\widetilde{\ell}}| = \kappa - \widetilde{\ell} +1$, corresponding to the subsequent summation terms in . Observe that $\frac{\exp(\theta_i)}{\sum_{j \in \Omega_{\widetilde{\ell}}} \exp(\theta_j)} \geq \lalpha_{i,\ell,\theta}/|\Omega_{\widetilde{\ell}}|$. Therefore, each summation term in equation can be lower bounded by the corresponding term where $\{\theta_j\}_{j \in S}$ is replaced by $\{\ltheta_j\}_{j \in S}$. Hence, we have $$\begin{aligned}
\label{eq:posl_upper4}
&&\P_{\theta}\Big[\sigma^{-1}(i) = \ell\Big] \nonumber\\
&\leq & \sum_{\substack{j_1 \in S \\ j_1 \neq i}} \Bigg(\frac{\exp(\ltheta_{j_1})}{\lW} \sum_{\substack{j_2 \in S \\ j_2 \neq i,
| 2,035
| 1,214
| 1,637
| 1,838
| null | null |
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|
and $U(Y_*) \simeq \log{\epsilon}$, we estimate the slope of the straight line connecting two points $\left(1,U(1)\right)$ and $\left(Y_*,U(Y_*)\right)$ in the $(Y,U)$ plane as $(U(Y_*)-U(1))/(Y_*-1)\simeq
\sqrt{{\epsilon}}(\log {\epsilon})$, which approaches zero in the limit ${\epsilon}\rightarrow 0$. Thus, the transition from $Y=1$ to $Y=Y_*$ may be assumed to be free Brownian motion with the diffusion constant $D=(\lambda+1)/8$. The transition rate from $Y=1$ to $Y_*$ is then estimated as $T=2D/Y_*^2 = {\epsilon}+ O({\epsilon}^2)$. We thus obtain $$\begin{aligned}
q(\lambda) &=& {\epsilon}+O({\epsilon}^2).
\label{qep}\end{aligned}$$ In Fig. \[fig-sy\_lgv-sp-Nxxx-fit\], we compare the theoretical result with those obtained in numerical simulations of and . We measured the probability that $\rho > 0.003$, which is denoted as $p(\rho > 0.003)$. Recall that $\lim_{N \to \infty} p(\rho >0.003) =q(\lambda)$ when $\rho_*(\lambda) > 0.003$. Since the experimental result suggests $p(\rho >0.003)={\epsilon}+O({\epsilon}^2)$ in the limit $N \to \infty$, we claim that the theoretical result (\[qep\]) is in good agreement with the experimental result.
Concluding remarks
==================
In this paper, we have achieved a novel understanding of the intrinsic unpredictability of epidemic outbreaks by analyzing the Langevin equation , which effectively describes this singular phenomenon. Further, trajectories in the outbreak phase are divided into two groups: trajectories in one group are absorbed into zero, and the others diverge in . The division corresponds to the non-trivial limiting density given in . On the basis of this description, we calculated the probability of an epidemic outbreak near the transition point. Before ending the paper, we make a few remarks. First, the probability $q(\lambda)$ was studied in the mathematical literature (see [@yan2008distribution] and [@britton2010stochastic] as reviews.) To the best of our knowledge, the method proposed in this paper has never been used in previous studies. It m
| 2,036
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| 2,106
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|
specific regions of the parameters space.
The HMF model is characterized by the following Hamiltonian $$\label{eq:ham}
H = \frac{1}{2} \sum_{j=1}^N p_j^2 + \frac{1}{2 N} \sum_{i,j=1}^N
\left[1 - \cos(\theta_j-\theta_i) \right]$$ where $\theta_j$ represents the orientation of the $j$-th rotor and $p_j$ is its conjugate momentum. To monitor the evolution of the system, it is customary to introduce the magnetization, a macroscopic order parameter defined as $M=|{\mathbf M}|=|\sum {\mathbf m_i}| /N$, where ${\mathbf m_i}=(\cos \theta_i,\sin \theta_i)$ stands for the microscopic magnetization vector. As previously reported [@antoni-95], after an initial transient, the system gets trapped into Quasi-Stationary States (QSSs), i.e. non-equilibrium dynamical regimes whose lifetime diverges when increasing the number of particles $N$. Importantly, when performing the mean-field limit ($N
\rightarrow \infty$) [*before*]{} the infinite time limit, the system cannot relax towards Boltzmann–Gibbs equilibrium and remains permanently confined in the intermediate QSSs. As mentioned above, this phenomenology is widely observed for systems with long-range interactions, including galaxy dynamics [@Padmanabhan], free electron lasers [@Barre], 2D electron plasmas [@kawahara].
In the $N \to \infty$ limit the discrete HMF dynamics reduces to the Vlasov equation $$\partial f / \partial t + p \, \partial f / \partial \theta \,\,
- (dV / d \theta ) \, \partial f / \partial p = 0 \, ,
%\frac{\partial f}{\partial t} + p\frac{\partial f}{\partial \theta} -
%\frac{d V}{d \theta} \frac{\partial f}{\partial p}=0\quad ,
\label{eq:VlasovHMF}$$ where $f(\theta,p,t)$ is the microscopic one-particle distribution function and $$\begin{aligned}
V(\theta)[f] &=& 1 - M_x[f] \cos(\theta) - M_y[f] \sin(\theta) ~, \\
M_x[f] &=& \int_{-\pi}^{\pi} \int_{-\infty}^{\infty} f(\theta,p,t) \, \cos{\theta} {\mathrm d}\theta
{\mathrm d}p\quad , \\
M_y[f] &=& \int_{-\pi}^{\pi} \int_{\infty}^{\infty} f(\theta,p,t) \, \sin{\theta}{\mathrm d}\theta
{\mathr
| 2,037
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| 2,689
| 1,721
| null | null |
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|
$ (or a renormalized one), see e.g., [@Minakata:2015gra] and the references therein. Possible interpretation of applicability of the perturbative framework to the region of solar level crossing has been discussed [@Xu:2015kma; @Ge:2016dlx]. Another example for the similar phenomena is the one at the small atmospheric mass splitting limit with additional expansion parameter $\sin \theta_{13}$. In this case it is observed that near the atmospheric resonance region not only the oscillation probability is finite but also its accuracy improves when the higher order terms to fourth order in $\sin \theta_{13}$ is added [@Asano:2011nj].
Then, one might ask if our small unitarity violation perturbation theory gives quantitatively accurate result at around the denominator with small solar mass splitting. However, we note that this problem is not relevant in our case because all these terms with apparent singularities vanish after averaging over the high-frequency oscillations and using the suppression by the sterile mass denominators. Yet, we must remark that if we investigate possible enhancement of the correction terms outside the condition (\[suppression-cond\]), as done in section \[sec:correction-terms\], the quantitative accuracy of the expression may become an issue.
Concluding remarks {#sec:conclusion}
==================
In this paper, we have presented a comprehensive treatment of the three active plus $N$ sterile neutrino model in the context of leptonic unitarity test. We have formulated an appropriate perturbative framework with expansion in small unitarity violating $W$ matrix elements, while keeping (non-$W$ suppressed) matter effect to all orders.
What we have done in this paper is mainly threefold:
- We have shown that the same condition on sterile state masses $0.1\, \text{eV}^2 \lsim m^2_{J} \lsim 1\, \text{MeV}^2$, as we imposed in vacuum, is sufficient to make the $(3+N)$ model sterile-sector model independent, apart from $N$ dependence in the lower bound on probability leaking constant $\mathca
| 2,038
| 456
| 2,004
| 2,054
| null | null |
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|
nu_{\tau}) \equiv P(\nu_{\mu} \rightarrow \nu_{\tau})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{\tau})_{ \text{non-unitary} }^{(0)}$ is presented. The parameters used are the same as in figure \[fig:Pmue\_energy\_dist\]. []{data-label="fig:Pmutau_energy_dist"}](Pmutau_energy_dist_non_unitary_small_size.jpeg "fig:"){width="85.00000%"} ![ In the upper panel (a), presented is the iso-contour of $P(\nu_{\mu} \rightarrow \nu_{\tau})_{ \text{non-unitary} }^{(0)}$ in $E-L$ space. In the lower panel (b), the iso-contour of the difference $\Delta P (\nu_{\mu} \rightarrow \nu_{\tau}) \equiv P(\nu_{\mu} \rightarrow \nu_{\tau})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{\tau})_{ \text{non-unitary} }^{(0)}$ is presented. The parameters used are the same as in figure \[fig:Pmue\_energy\_dist\]. []{data-label="fig:Pmutau_energy_dist"}](Pmutau_energy_dist_difference_small_size.jpeg "fig:"){width="85.00000%"}
![ In the upper panel (a), presented is the iso-contour of $P(\nu_{\mu} \rightarrow \nu_{\mu})_{ \text{non-unitary} }^{(0)}$ in $E-L$ space. In the lower panel (b), the iso-contour of the difference $\Delta P (\nu_{\mu} \rightarrow \nu_{\mu}) \equiv P(\nu_{\mu} \rightarrow \nu_{\mu})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{\mu})_{ \text{non-unitary} }^{(0)}$ is presented. The parameters used are the same as in figure \[fig:Pmue\_energy\_dist\]. []{data-label="fig:Pmumu_energy_dist"}](Pmumu_energy_dist_non_unitary_small_size.jpeg "fig:"){width="85.00000%"} ![ In the upper panel (a), presented is the iso-contour of $P(\nu_{\mu} \rightarrow \nu_{\mu})_{ \text{non-unitary} }^{(0)}$ in $E-L$ space. In the lower panel (b), the iso-contour of the difference $\Delta P (\nu_{\mu} \rightarrow \nu_{\mu}) \equiv P(\nu_{\mu} \rightarrow \nu_{\mu})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{\mu})_{ \text{non-unitary} }^{(0)}$ is presented. The parameters used are the same as in figure \[fig:Pmue\_energy\_dist\]. []{data-label="fig:Pmumu_energy_dist"}](Pmumu_energy_dist_difference_small_size.jpeg "fig:"
| 2,039
| 232
| 1,743
| 1,881
| 1,791
| 0.785673
|
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|
ick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;i;{b\!\!+\!\!3};{b\!\!+\!\!4},;2,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}\\
\intertext{and the tableaux}
T'[i,j]&={\text{\footnotesize$\gyoungx(1.2,;1;1;2;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{\hat\jmath};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};v;i;{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;j;{b\!\!+\!\!3};{b\!\!+\!\!4},;2,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}\tag*{for $2\ls j\ls v$.}\\
\intertext{Applying Lemma \ref{lemma7} to $T'[i]$ to move the $2$ from row $3$ to row $2$, we obtain three tableaux, but two of these are not dominated by $S$. The other one is}
T''[i]&={\text{\footnotesize$\gyoungx(1.2,;1;1;2;3_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;2;{b\!\!+\!\!3};{b\!\!+\!\!4},;i,;3,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,de
| 2,040
| 873
| 2,226
| 1,940
| 244
| 0.816382
|
github_plus_top10pct_by_avg
|
} (UX)_{\alpha k} (UX)^*_{\beta l}
\hat{S}_{kl}^{(2)}
+
\sum_{k L} (UX)_{\alpha k} W^*_{\beta L}
\hat{S}_{kL}^{(1)}
\nonumber \\
&+&
\sum_{K l} W_{\alpha K} (UX)^*_{\beta l}
\hat{S}_{K l}^{(1)}
+
\sum_{K L} W_{\alpha K} W^*_{\beta L}
\hat{S}_{KL}^{(0)},
\nonumber \\
S_{\alpha \beta}^{(4)} &=&
\sum_{k l} (UX)_{\alpha k} (UX)^*_{\beta l}
\hat{S}_{kl}^{(4)}
+
\sum_{k L} (UX)_{\alpha k} W^*_{\beta L}
\hat{S}_{kL}^{(3)}
\nonumber \\
&+&
\sum_{K l} W_{\alpha K} (UX)^*_{\beta l}
\hat{S}_{K l}^{(3)}
+
\sum_{K L} W_{\alpha K} W^*_{\beta L}
\hat{S}_{KL}^{(2)}.
\label{Sab-hatSab}\end{aligned}$$ Using (\[Sab-hatSab\]) the explicit expressions of $S$ matrix elements can be easily obtained with use of $\hat{S}$ matrix elements given in appendix \[sec:hatS-elements\]. For example, $S_{\alpha \beta}$ in zeroth and second orders in $W$ are given, respectively, by $$\begin{aligned}
S_{\alpha \beta}^{(0)} &=&
\sum_{k} (UX)_{\alpha k} (UX)^*_{\beta k}
e^{-i h_{k} x},
\label{S-alpha-beta-0th}\end{aligned}$$ and $$\begin{aligned}
&& S_{\alpha \beta}^{(2)} =
\sum_{k, K}
\frac{ 1 }{ \Delta_{K} - h_{k} }
\left[
(ix) e^{- i h_{k} x} + \frac{e^{- i \Delta_{K} x} - e^{- i h_{k} x} }{ ( \Delta_{K} - h_{k} ) }
\right]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta k}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\nonumber \\
&-&
\sum_{k \neq l} \sum_{K}
\frac{ 1 }{ ( h_{l} - h_{k} ) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{l}) }
\nonumber \\
&\times&
\biggl[
\left( \Delta_{K} - h_{k} \right) e^{- i h_{l} x}
- \left( \Delta_{K} - h_{l} \right) e^{- i h_{k} x}
- ( h_{l} - h_{k} ) e^{- i \Delta_{K} x}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K l}
\nonumber \\
&+&
\sum_{k, K}
\frac{e^{- i \Delta_{K} x} - e^{- i h_{k} x} }{ ( \Delta_{K} - h_{k} ) }
\biggl[
(UX)_{\alpha k} W^*_{\beta K}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
+
W_{\alpha K} (UX)^*_{\beta k}
\le
| 2,041
| 1,582
| 2,200
| 2,059
| null | null |
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|
[ecsd6a\]\
& S\_0C\^2(I,L\^(G)),\[ecsd6a-a\]\
& C\^1(I,L\^(GS,L\^1(S’)))C\^1(I,L\^(GS’,L\^1(S))).\[ecsd6a-b\]
Let $f\in C^1(I,L^2(G\times S))$ and let $g\in C^2(I,T^2(\Gamma_-))$ which satisfies the *compatibility condition* \[cc\] g(E\_m)=0. Then the problem (\[se1\]), (\[se2\]), (\[se3\]) has a unique solution $\psi\in C(I,\tilde W^2(G\times S))\cap C^1(I,L^2(G\times S))$.
If in addition the assumptions (\[ass2b\]) (with $c>0$) are also valid, the estimate \[evoest\] \_[[H\_1]{}]{} (\_[L\^2(GSI)]{}+\_[T\^2(\_-)]{}). holds. (The constants $\kappa$, $q$ and $c'$ were defined in , and , respectively.)
A. Assume at first that $g=0$. We make the change of variables and the change of unknown function as above by setting $\tilde\psi(x,\omega,E)=\psi(x,\omega,E_m-E)$ and $\phi=e^{-CE}\tilde\psi$. Choose $C=C_0+C_0'$ (see , ). Then, as observed above, the problem (\[se1\]), (\[se2\]), (\[se3\]) can be cast into an equivalent form (see ) \[ecsd7\] [E]{}-A\_C(E)=[**f**]{}(E),(0)=0, where the domain $D(A_C(E))=\tilde{W}^2_{-,0}(G\times S)=:D$ of definition of $A_C(E)$ is independent of $E$. We have, moreover, demonstrated (see Lemma \[le:ACE\_m\_diss\]) that the (densely defined) operator $A_C(E):L^2(G\times S)\to L^2(G\times S)$ is $m$-dissipative for any fixed $E\in I$.
The assumptions , , imply that for any fixed $\phi\in D$ the mapping $$h_{\phi}:I\to L^2(G\times S);\quad h_\phi(E):=A_C(E)\phi,$$ is differentiable and $$h_\phi'(E)=&-{\partial\over{{\partial E}}}\Big({1\over{\tilde S(E)}}\Big)\omega\cdot \nabla_x\phi
-{\partial\over{{\partial E}}}\Big({1\over{\tilde S(E)}}\tilde\Sigma(E)\Big)\phi
-{\partial\over{{\partial E}}}\Big({1\over{\tilde S(E)}}{{\frac{\partial \tilde S}{\partial E}}}\Big)\phi \\
&+{\partial\over{{\partial E}}}\Big({1\over{\tilde S(E)}}\Big)\tilde K(E)\phi
+{1\over{\tilde S(E)}}{{\frac{\partial \tilde K}{\partial E}}}(E)\phi,$$ where ${{\frac{\partial \tilde K}{\partial E}}}(E)\phi$ is defined in , and the derivative ${{{\partial}\over{\partial E}}}(\tilde{K}(E)\phi)={{\frac{\partial \til
| 2,042
| 440
| 1,272
| 2,071
| null | null |
github_plus_top10pct_by_avg
|
^{ - i ( h_{k} - h_{n} ) x}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l}
(UX)^*_{\alpha n} (UX)_{\beta n}
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K m}
\left\{ (UX)^{\dagger} A W \right\}_{m K}
\left\{ W ^{\dagger} A (UX) \right\}_{K l}
\biggr\}.
\label{P-beta-alpha-W4-H4-single}\end{aligned}$$ $$\begin{aligned}
&& P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{4th-d}
\equiv
2 \mbox{Re} \left[ \left(
S^{(0)}_{\alpha \beta} \right)^{*}
S_{\alpha \beta}^{(4)} [4]_{ \text{ offdiag } } (\text{double})
\right]
\nonumber \\
&=&
2 \mbox{Re}
\biggl\{
\sum_{n}
\sum_{k \neq l }
\sum_{K \neq L}
\biggl[
\frac{ (ix) }{ (\Delta_{K} - h_{k} ) (\Delta_{L} - h_{k} ) ( h_{l} - h_{k} ) }
e^{- i ( h_{k} - h_{n} ) x}
\nonumber \\
&-&
\frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{L} - h_{k} )^2 (\Delta_{L} - h_{l} ) }
e^{- i ( \Delta_{L} - h_{n} ) x}
+ \frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{K} - h_{k} )^2 (\Delta_{K} - h_{l} ) }
e^{- i ( \Delta_{K} - h_{n} ) x}
\nonumber \\
&+&
\frac{ 1 }{ (\Delta_{K} - h_{l} ) (\Delta_{L} - h_{l} ) ( h_{l} - h_{k} )^2 }
e^{- i ( h_{l} - h_{n} ) x}
\nonumber \\
&-&
\frac{ 1 }{ (\Delta_{K} - h_{k} )^2 (\Delta_{L} - h_{k} )^2 ( h_{l} - h_{k} )^2 }
\biggl\{
3 h_{k}^2 - 2 h_{k} h_{l} + \left( h_{l} - 2 h_{k} \right) (\Delta_{K} + \Delta_{L} ) + \Delta_{K} \Delta_{L}
\biggr\}
e^{- i ( h_{k} - h_{n} ) x}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l}
(UX)^*_{\alpha n} (UX)_{\beta n}
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\left\{ (UX)^{\dagger} A W \right\}_{k L}
\left\{ W ^{\dagger} A (UX) \right\}_{L l}
\nonumber \\
&+&
\sum_{n}
\sum_{k \neq l }
\sum_{K \neq L}
\biggl[
-
\frac{ (ix) }{ (\Delta_{K} - h_{l} ) (\Delta_{L} - h_{l} ) ( h_{l} - h_{k} ) }
e^{- i ( h_{l} - h_{n} ) x}
\nonumber \\
&+&
\frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{K} - h_{k} ) (\Delta_{K} - h_{l} )^2 }
e^
| 2,043
| 3,251
| 2,416
| 1,938
| null | null |
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|
bb{R})$ such that $\gamma_0\iota M\ddot{o}b(\hat{\Bbb{R}})\gamma^{-1}_0$ preserves $$\{[x,w,z]\in\Bbb{P}^2_{\Bbb{R}}:\vert y\vert^2+\vert w\vert^2<\vert x\vert^2\}.$$ Hence $\gamma_0\iota\PSL(2,\hat{\Bbb{R}})\gamma^{-1}_0=\PO^+(2,1)$. Part (\[l:con2\]) is now trivial.
\[t:liedim\] Let $\Gamma\subset\PSL(2,\Bbb{C})$ be a discrete non-elementary group. The group $\iota\Gamma$ is complex hyperbolic if and only if $ \Gamma$ is Fuchsian, [ i.e.]{} a subgroup of $\PSL(2,\Bbb{R})$.
Assume that $\iota\Gamma$ preserves a complex ball $B$. Then by Lemma \[c:liedim\] we deduce that $\Gamma$ preserves a circle $C$ in the Riemann sphere. Let $B^+$ and $B^-$ be the connected components of $\Bbb{P}_\Bbb{C}^1\setminus C$ and assume that there is a $\tau\in\Gamma$ such that $\tau(B)^+=B^-$. Let $x\in Ver\cap B$ and denote by $Aut^+(BV)$ the principal connected component of $Aut(BV)$ which contains the identity. Then by Lemma \[c:liedim\] we deduce $$\begin{array}{l}
Aut^+(BV)x= \psi\iota^{-1}Aut(BV)\psi^{-1}x=
\psi (B^{+}) \;\;\hbox {and}
\\
Aut^+(BV) \iota\tau(x)= \psi\iota^{-1}Aut(BV)\tau\psi^{-1}x=\psi(B^{-}).\\
\end{array}$$ Therefore $$Ver=Aut^+(BV)x\cup Aut^+(BV)\iota\tau(x)\cup C\subset\overline{\Bbb{H}}^2,$$ which is a contradiction. Clearly, this concludes the proof.
We arrive at the following theorem:
Let $\Gamma\subset\PSL(2,\Bbb{C})$. Then the following claims are equivalent:
1. The group $\Gamma$ is Fuchsian.
2. The group $\iota\Gamma$ is complex hyperbolic.
3. The group $\iota\Gamma$ is $\Bbb{R}$-Fuchsian
Subgroups of $\PSL(3,\Bbb{R})$ that Leave Invariant a Veronese Curve {#s:riv}
====================================================================
In this section we characterize those subgroups of $\PSL(3,\Bbb{R})$ which leave invariant a projective copy of $Ver$.
Let $\Gamma\subset \PSL(2,\Bbb{C})$ be a discrete subgroup. Then the following facts are equivalent
1. The group $\Gamma $ is conjugate to a subgroup of $Mob(\hat{\Bbb{R}})$.
2. The group $\iota\Gamma$ is conjugate to a subgroup of $
| 2,044
| 1,211
| 1,333
| 1,913
| null | null |
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|
\beta}\,,\qquad
\langle{}_{\dot{\alpha}}|\psi^\mu_0|{}_{\dot{\beta}}\rangle\ =\
-\frac{i}{\sqrt{2}}(C^T\gamma^\mu)_{\dot{\alpha}\dot{\beta}}\,.$$
Triviality of the extra unphysical symmetries at the linearized level {#app B}
=====================================================================
First, in order to show the triviality of (\[trans tilde ramond\]), it is useful to introduce the local inverse picture-changing operator $$Y(z_0)\ =\ -c(z_0)\delta'(\gamma(z_0))\,,$$ which also satisfies $$XY(z_0)X\ =\ X\,,\label{xyx local}$$ and in addition is commutative with $Q$: $[Q, Y(z_0)]=0$. The point $z_0$ can be chosen to be any point on the string, for example, the midpoint $z_0=i$. Due to (\[xyx local\]), we can define another projection operator $XY(z_0)$ that is commutative with $Q$, and acts identically with $XY$ in the restricted small Hilbert space: $$[Q, XY(z_0)]\ =\ 0\,,
$$ and if $XY\Psi=\Psi$ then $$XY(z_0)\Psi\ =\ XY(z_0)XY\Psi\ =\ XY\Psi\,.$$ Using this projection operator, the linearized transformation (\[trans tilde ramond\]) can be written as the a linearized gauge transformation, $$\begin{aligned}
\delta_{\tilde{p}}^{(0)}\Psi\
=&\
XY(z_0)\left(p(v) - X \tilde{p}(v)\right)\Psi\
=\ XY(z_0)\{Q, \tilde{M}(v)\}\,\Psi\
\nonumber\\
\cong&\ Q(XY(z_0)\tilde{M}(v)\Psi)\,,
\label{B4}\end{aligned}$$ up to the linearized equation of motion, $Q\Psi=0$, with $$\tilde{M}(v)\ =\ v^\mu
\oint\frac{dz}{2\pi i}(\xi(z)-\Xi)\psi_\mu(z)e^{-\phi(z)}\,.$$ We can see that the gauge parameter in (\[B4\]), $$\lambda_{\tilde{p}}\ =\ XY(z_0)\tilde{M}(v)\Psi\,,$$ is in the restricted small Hilbert space, $$\eta \lambda_{\tilde{p}}\ =\ 0\,,\qquad
XY \lambda_{\tilde{p}}\ =\ \lambda_{\tilde{p}}\,,$$ if we note that $\{\eta, \tilde{M}\}=0$.
As was mentioned in section \[extra symm\], the commutator $[\delta_{\tilde{p}_1},\delta_{\tilde{p}_2}]$ produces another unphysical transformation $\delta_{[\tilde{p},\tilde{p}]}$: $$[\delta_{\tilde{p}_1},\, \delta_{\tilde{p}_2}]\ \cong\
\delta_g + \delta_{[\tilde{p},\tilde{p}]_{12
| 2,045
| 907
| 1,818
| 1,976
| null | null |
github_plus_top10pct_by_avg
|
upersymmetric[@Gliozzi:1976qd].
For the Ramond sector, we use the string field $\Psi$ constrained on the restricted small Hilbert space satisfying the conditions[@Kunitomo:2015usa] $$\eta\Psi\ =\ 0\,,\qquad
XY\Psi\ =\ \Psi\,, \label{R constraints}$$ where $X$ and $Y$ are the picture-changing operator and its inverse acting on the states in the small Hilbert space with picture numbers $-3/2$ and $-1/2$, respectively. They are defined by $$X\ =\ -\delta(\beta_0)G_0 + \delta'(\beta_0)b_0\,,\qquad Y\ =\ -c_0\delta'(\gamma_0)\,,
\label{PCO}$$and satisfy $$XYX\ =\ X\,,\qquad YXY\ =\ Y\,, \qquad [Q,\,X]\ =\ 0\,.
\label{xyx}$$ The string field $\Psi$ is Grassmann odd, and has ghost number $1$ and picture number $-1/2$. The picture-changing operator $X$ is BRST exact in the large Hilbert space, and can be written using the Heaviside step function as $ X=\{Q,\Theta(\beta_0)\}$. Here, instead of $\Theta(\beta_0)$, we introduce $$\Xi\ =\ \xi_0 + (\Theta(\beta_0)\eta\xi_0 - \xi_0)P_{-3/2}
+ (\xi_0\eta\Theta(\beta_0) - \xi_0)P_{-1/2}\,,$$ and anew define $$X\ =\ \{Q,\ \Xi\}\,.
\label{X in Ramond}$$ This is identical to the one defined in (\[PCO\]) when it acts on the states in the small Hilbert space with picture number $-3/2$, but can act on the states in the large Hilbert space without the restriction on the picture number.[@Erler:2016ybs] The operator $\Xi$ is nilpotent ($\Xi^2=0$) and satisfies $\{\eta, \Xi\}=1$ [@Erler:2016ybs], from which, with $\{Q,\eta\}=0$, we can conclude $$\begin{aligned}
[\eta, X]\ =&\ [\eta,\{Q,\Xi\}]
\nonumber\\
=&\ -[Q,\{\Xi,\eta\}]-[\Xi,\{\eta,Q\}]\ =\ 0\,.\end{aligned}$$ We impose the BRST-invariant GSO projection as $$\Psi\ =\ \frac{1}{2}(1+\hat{\Gamma}_{11}(-1)^{G_R})\,\Psi\,,
\label{GSO Ramond}
$$ where $G_R$ is given by $$\begin{aligned}
G_R\ =&\ \sum_{n>0}(\psi^\mu_{-n}\psi_{n\mu}-\gamma_{-n}\beta_n+\beta_{-n}\gamma_n)
- \gamma_0\beta_0
\nonumber\\
\equiv&\ \sum_{n>0}\psi^\mu_{-n}\psi_{n\mu} + p_\phi + \frac{1}{2}\qquad (\textrm{mod}\ 2)\,.\end{aligned}$$ The gamma matrix $\hat{\Ga
| 2,046
| 1,120
| 1,768
| 1,972
| null | null |
github_plus_top10pct_by_avg
|
construct another ranking $\widetilde{\sigma}_1^\ell$ from $\widehat{\sigma}_1^\ell$ by replacing $i_{\min}$ with $i$-th item. Observe that $ \P[\widehat{\sigma}_1^\ell] \leq \P[\widetilde{\sigma}_1^\ell]$ and $\widetilde{\sigma}_1^\ell \in \Omega_2$. Moreover, such a construction gives a bijective mapping between $\Omega_1$ and $\Omega_2$. Hence, the first claim is proved. For the second claim, let $$\begin{aligned}
\widehat{\Omega}_1 = \Big\{ \widehat{\sigma}_1^\ell : {(\widehat{\sigma}_1^\ell)^{-1}(i_{\min}) = 1} \Big\} \;\; \text{and} \;\; \widehat{\Omega}_2 = \Big\{ \widehat{\sigma}_1^\ell : {(\widehat{\sigma}_1^\ell)^{-1}(i_{\min}) = \ell} \Big\}.\end{aligned}$$ We have $\mathbb{P}[\sigma^{-1}(i_{\min}) =1] - \mathbb{P}[\sigma^{-1}(i_{\min}) = \ell] =\sum_{\widehat{\sigma}_1^\ell \in \widehat{\Omega}_1}\P[\widehat{\sigma}_1^\ell]- \sum_{\widehat{\sigma}_1^\ell \in \widehat{\Omega}_2} \P[\widehat{\sigma}_1^\ell].$ Now, take any ranking $\widehat{\sigma}_1^\ell \in \widehat{\Omega}_1$ and construct another ranking $\widetilde{\sigma}_1^\ell$ from $\widehat{\sigma}_1^\ell$ by swapping items at $1$st position and $\ell$-th position. Observe that $ \P[\widehat{\sigma}_1^\ell] \leq \P[\widetilde{\sigma}_1^\ell]$ and $\widetilde{\sigma}_1^\ell \in \widehat{\Omega}_2$. Moreover, such a construction gives a bijective mapping between $\widehat{\Omega}_1$ and $\widehat{\Omega}_2$. Hence, the claim is proved.
Proof of Theorem \[thm:cramer\_rao\_topl\] {#sec:proof_cramer_rao_topl}
------------------------------------------
The first order partial derivative of $\L(\theta)$, Equation , is given by $$\begin{aligned}
&&\nabla_i\L(\theta) \nonumber\\
&=&\sum_{j:i\in S_j} \sum_{m =1}^{\ell_j} \I_{\{\sigma_j^{-1}(i) \geq m \}} \Big[ \I_{\{\sigma_j(m) = i\}} - \frac{\exp(\theta_i)}{\exp(\theta_{\sigma_j(m)})+\exp(\theta_{\sigma_j(m+1)})+ \cdots + \exp(\theta_{\sigma_j(\kappa_j)})} \Big], \; \forall i \in [d]\end{aligned}$$ and the Hessian matrix $H(\theta) \in \mathcal{S}^d$ with $H_{i\i}(\theta) = \frac{\partial^2\L(\th
| 2,047
| 845
| 758
| 2,030
| null | null |
github_plus_top10pct_by_avg
|
h{i.e.}} \quad \rho_t+\nabla\cdot(\rho\MM{u}) = 0. \,.
\label{advecD}$$ A more extensive list of different types of advected quantity is given in Holm *et al.* (1998).
We write the reduced Lagrangian $\ell$ as a functional of the Eulerian fluid variables $\MM{u}$ and $a$, and add further constraints to the action $S$ to account for their advection relations, $$\label{principle with advected qs}
S = \int \ell[\MM{u},a]\,{\mathrm{d}}t
+ \int {\mathrm{d}}t\int_{\Omega}\MM{\pi}\cdot(\MM{l}_t+
(\MM{u}\cdot\nabla)\MM{l}) + \phi(a_t+\mathcal{L}_{\MM{u}}a)
\,{\mathrm{d}}V(\MM{x}).$$ The Euler-Lagrange equations, which follow from the stationarity condition $\delta S=0$, are $$\begin{aligned}
\delta\MM{u}:&&
{\frac{\delta \ell}{\delta \MM{u}}} + (\nabla\MM{l})^T\cdot\MM{\pi}
+ \phi\diamond a = 0,
\label{EL advected 1}\\
\delta\MM{\pi}:&&
\MM{l}_t + (\MM{u}\cdot\nabla)\MM{l} = 0,
\nonumber\\
\delta\MM{l}:&&
-\MM{\pi}_t - \nabla\cdot(\MM{u}\MM{\pi}) = 0,
\label{EL advected 3}\\
\delta\phi:&&
a_t + \mathcal{L}_{\MM{u}}a = 0,
\nonumber \\
\delta{a}:&&
-\phi_t-\mathcal{L}_{\MM{u}}\phi + {\frac{\delta \ell}{\delta a}} = 0,
\label{EL advected 5}\end{aligned}$$ where the diamond operator ($\diamond$) is defined as the dual of the Lie derivative operation $\mathcal{L}_{\MM{u}}$ with respect to the $L^2$ pairing. Explicitly, under integration by parts, $$\int_\Omega (\phi\diamond a)\cdot\MM{u}\,{\mathrm{d}}{V}(\MM{x})
= -\int_\Omega (\phi\mathcal{L}_{\MM{u}}a)\,{\mathrm{d}}{V}(\MM{x}).$$
The map to the spatial momentum in equation (\[EL advected 1\]) $${\frac{\delta \ell}{\delta \MM{u}}} =: \MM{m}
= -\,\pi_A\nabla l^A -\, \phi\diamond a
\,,$$ is again a momentum map, this time for the semidirect-product action of the diffeomorphisms on $\Omega\times V^*$. Again the momentum map property allows the canonical variables to be eliminated in favour of the Eulerian quantities. As a result, eliminating the variables $\MM{l}$, $\MM{\pi}$ and $\phi$ leads to the Euler-Poincaré equation with advected quantities $a$.
The labels $\M
| 2,048
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|
consist of those functions $u=f(x)$ which satisfy the overdetermined system composed of the initial system (\[eq:3.1\]) together with the invariance conditions
\[eq:3.17i\] \_a\^iu\_i\^=0,i=1,…,p,a=1,…,p-2k,
ensuring that the characteristics of the vector fields $X_a$ are equal to zero.
####
It should be noted that, in general, the conditions (\[eq:3.17i\]) are weaker than the differential constraints (\[eq:2.4\]) required by the generalized method of characteristics, since the latter method is subjected to the algebraic conditions (\[eq:2.2\]). In fact, equations (\[eq:3.17i\]) imply that there exist complex-valued matrix functions $\Phi_A^\alpha(x,u)$ and $\overline{\Phi}_A^\alpha(x,u)$ defined on the first jet space $J=J(X\times U)$ such that all first derivatives of $u$ with respect to $x^i$ are decomposable in the following way
\[eq:3.18\] u\_i\^=\_A\^(x,u)\_i\^A+\_A\^(x,u)|\_i\^A,
where
\[eq:3.19\] \_A\^&=[( I\_q- )]{}\^[-1]{} [( + )]{},\
\_A\^&=[( I\_q- )]{}\^[-1]{} [( + )]{},
or
\[eq:3.20\] \_A\^&=[( + )]{}M\^1+[( + )]{}\^2,\
\_A\^&=[( + )]{}\^1+[( + )]{}M\^2.
The matrices $\Phi_A^\alpha\lambda_i^A$ and $\overline{\Phi}_A^\alpha\bar{\lambda}_i^A$ appearing in equation (\[eq:3.18\]) do not necessarily satisfy the wave relation (\[eq:2.2\]). As a result of this fact, the restrictions on the initial data at $t=0$ are eased, so we are able to consider more diverse types of modes in the superpositions than in the case of the generalized method of characteristics described in Section \[sec:2\].
####
Let us now proceed to solve the overdetermined system composed of equations (\[eq:3.1\]) and differential constraints (\[eq:3.17i\])
\[eq:3.21\] \^[i]{}\_(u)u\_i\^=0,\_a\^i(u)u\_i\^=0.
Substituting (\[eq:3.6\]) or (\[eq:3.7\]) into (\[eq:3.1\]) yields the trace condition
\[eq:3.22\] [( \^\^[-1]{} [( [( + )]{}+c.c. )]{} )]{}=0,
or
\[eq:3.23\] [( \^ )]{}=0,
on the wave vectors $\lambda$ and $\bar{\lambda}$ and on the functions $f$ and $\bar{f}$, where $\mathcal{A}^1,\ldots,\mathcal{A}^q$ are $
| 2,049
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|
el{rho-beta}$$ where $\nu_0 = \left[2(1+3\alpha_\Lambda)/(1+\alpha_\Lambda)^2 -
1/4\right]^{1/2}$, $C_{\cos}$ and $C_{\sin}$ are determined by boundary conditions, and $A_\beta=1$ for $\beta = 2,3$. The first part, $\rho_{\hbox{\scriptsize{asymp}}}(r)$, of $\rho_{II}(r)$ corresponds to a background density. *It is universal, and has the same form irrespective of the detailed structure of the galaxy.* The second part, $\rho_{II}^1(r)$, gives the structural details.
The free energy, ${}^{II}\mathcal{F}$, for Region II separates into the sum of three parts. The first part depends only on $\rho_{\hbox{\scriptsize{asymp}}}$; it is positive, and is independent of $\beta$. The second part is $$\frac{{}^{II}\mathcal{F}_{\hbox{\scriptsize{asymp}}-\beta}}
{(\chi^{1/2}\lambda_{DE})^3}
= c^2 \int_{D_{II}} d^3\mathbf{u}
f(u)\left(\frac{\Lambda_{DE}}{8\pi\rho_{\hbox{\scriptsize{asymp}}}}
\right)^{\alpha_\Lambda}
+\frac{8\pi \alpha_\Lambda c^2}{\Lambda_{DE}\Sigma^{2(1+\alpha_\Lambda)}}
\int_{\partial D_{II}}
u^4\rho_{II}^1(u)\mathbf{\nabla}\rho_{\hbox{\scriptsize{asymp}}}\cdot d\mathbf{S},$$ where $D_{II}$ is Region II. It is negative because the minimum $\rho$ must be positive. Indeed, we find that ${}^{II}\mathcal{F}_{\hbox{\scriptsize{asymp}}-\beta} \sim -
(r_H/r_{II})^{\beta}$ for $\beta < 5/2$; ${}^{II}\mathcal{F}_{\hbox{\scriptsize{asymp}}-\beta} \sim -
(r_H/r_{II})^{5/2}$ for $5/2 \le
\beta < 5-2/(1+\alpha_\Lambda)$; and ${}^{II}\mathcal{F}_{\hbox{\scriptsize{asymp}}-\beta} \sim \pm
(r_H/\chi^{1/2}\lambda_{DE})^{5-2/(1+\alpha_\Lambda)}$ for $5-2/(1+\alpha_\Lambda)<\beta$. Clearly, free energy is lowest for $\beta=2$. The third part depends on $(\rho_{II}^1(r))^2$, and is negligibly small. The total free energy in this region is thus smaller for $\beta = 2 $ than for $\beta >2$. Combined with the calculation for ${}^{I}\mathcal{F}$, we conclude that the pseudoisothermal density profile has the lowest free energy, and is the preferred state of the system. We thus take $\gamma=0$ and $\beta=2$ in the fol
| 2,050
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| null | null |
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|
$ whose ratio to ${\gamma}_{i}^{2}$ vanishes in the $N \rightarrow \infty$ limit will be neglected.
[^4]: This is the approximation in which any contribution to ${W}_{N}$ whose ratio to ${\hat{\beta}}_{i}$ vanishes in the $N \rightarrow \infty$ limit will be neglected.
[^5]: These market-orthogonal portfolios essentially eliminate what is referred to as “market risk” in the single-index model jargon
[^6]: This is of course the exceptional case of spectrum degeneracy mentioned in §2.
---
abstract: 'In a D-brane model of space-time foam, there are contributions to the dark energy that depend on the D-brane velocities and on the density of D-particle defects. The latter may also reduce the speeds of photons [*linearly*]{} with their energies, establishing a phenomenological connection with astrophysical probes of the universality of the velocity of light. Specifically, the cosmological dark energy density measured at the present epoch may be linked to the apparent retardation of energetic photons propagating from nearby AGNs. However, this nascent field of ‘D-foam phenomenology’ may be complicated by a dependence of the D-particle density on the cosmological epoch. A reduced density of D-particles at redshifts $z \sim 1$ - a ‘D-void’ - would increase the dark energy while suppressing the vacuum refractive index, and thereby might reconcile the AGN measurements with the relatively small retardation seen for the energetic photons propagating from GRB 090510, as measured by the Fermi satellite.'
author:
- John Ellis
- 'Nick E. Mavromatos'
- 'Dimitri V. Nanopoulos'
title: 'D-Foam Phenomenology: Dark Energy, the Velocity of Light and a Possible D-Void'
---
Introduction to D-Phenomenology
===============================
The most promising framework for a quantum theory of gravity is string theory, particularly in its non-perturbative formulation known as M-theory. This contains solitonic configurations such as D-branes [@polchinski], including D-particle defects in space-time. One of the most challeng
| 2,051
| 1,583
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| 2,136
| null | null |
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|
athfrak{h}}^*/{{W}}. \end{CD}$$ and the map $\rho$ is flat of degree $n!$.
\(1) Recall from that $\operatorname{Hilb^n{\mathbb{C}}^2}= \operatorname{Proj}({\mathbb{A}})$. By Lemma \[hi-basic-lem\], ${\mathbb{A}}= A[\mathbf{z},\mathbf{z}^*]$. The maps $A\hookrightarrow {\mathbb{A}}$ and ${\mathbb{C}}[\mathbf{z},\mathbf{z}^*] \hookrightarrow {\mathbb{A}}$ give maps $\operatorname{Hilb^n{\mathbb{C}}^2}\to \operatorname{Proj}(A)$ and $\operatorname{Hilb^n{\mathbb{C}}^2}\to\mathrm{Spec}({\mathbb{C}}[\mathbf{z},\mathbf{z}^*])
\cong {\mathbb{C}}^2$ and hence, by universality, a map $\operatorname{Hilb^n{\mathbb{C}}^2}\to \operatorname{Proj}(A)\times {\mathbb{C}}^2$. It is easy to check that this is an isomorphism locally and hence globally. The identification of ${\mathfrak{h}}\oplus{\mathfrak{h}}^*$ with the subvariety $\mathbf{z}=0=\mathbf{z}^*$ of ${\mathbb{C}}^{2n}$ easily yields $\operatorname{Hilb(n)}=\operatorname{Proj}(A)$ and so $\operatorname{Hilb^n{\mathbb{C}}^2}= \operatorname{Hilb(n)}\times {\mathbb{C}}^2$. Since $\operatorname{Hilb^n{\mathbb{C}}^2}$ is a resolution of singularities of ${\mathbb{C}}^2/{{W}}$, the result follows.
\(2) By [@har Exercise II.8.3(b)] $\omega_{\operatorname{Hilb^n{\mathbb{C}}^2}} \cong
\omega_{\operatorname{Hilb(n)}} \boxtimes \omega_{{\mathbb{C}}^2}$, the external tensor product on $\operatorname{Hilb^n{\mathbb{C}}^2}= \operatorname{Hilb(n)}\times {\mathbb{C}}^2$. Now (2) follows since $\omega_{\operatorname{Hilb^n{\mathbb{C}}^2}}
\cong {\mathcal{O}}_{\operatorname{Hilb^n{\mathbb{C}}^2}}$ by [@hai3 Proposition 3.6.3].
\(3) As in part (1), $\mathbb{S}= \bigoplus {\mathbb{J}}^d
=S[\mathbf{z},\mathbf{z}^*]$ and $\operatorname{Proj}(\mathbb{S}) \cong
\operatorname{Proj}(S)\times {\mathbb{C}}^2$. The assertions of the corollary now follow from the corresponding results for ${\mathbb{X}}=\operatorname{Proj}(\mathbb{S})$ that were stated in .
We also have analogues of ${\mathcal{P}}_1$ and ${\mathcal{L}}_1$ for $\operatorname{Hilb(n)}$. These are defined in the same way: ${\math
| 2,052
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|
---------------------------------- ------------------------------ --------------
**Gender** \
\ Male 3922 (49.00)
\ Female 4002 (50.00)
\ Other 80 (1.00)
**Age in years** \
\ 18-34 2561 (32.00)
\ 35-44 1521 (19.00)
\ 45-65 3922 (49.00)
**Region type** \
\ City 3202 (40.00)
\ Town/urban area 2721 (34.00)
\ Countryside 1761 (22.00)
\ Do not know 320 (4.00)
**Education** \
\ Compulsory education 2001 (25.00)
\ Academic education 1121 (14.01)
\ Other^b^ 4642 (58.00)
\ Do not know 240 (3.00)
**Occupational group or status** \
\ At school or student 480 (6.00)
\ Worker 2162 (27.01)
\ Self-employed or sole trader 480 (6.00)
\ Junior white collar 880 (10.99)
\ Managerial position/senior 1361 (17.00)
\ Pensioner
| 2,053
| 5,660
| 326
| 1,153
| null | null |
github_plus_top10pct_by_avg
|
{\beta}}\\
-{(C^T)_{\dot{\alpha}}}^\beta & 0
\end{pmatrix}\,.$$ The matrices $\mathcal{C}\Gamma^\mu$ are symmetric, or equivalently $$(C\bar{\gamma}^\mu)^{\alpha\beta}\ =\ (C\bar{\gamma}^\mu)^{\beta\alpha}\,,\qquad
(C^T\gamma^\mu)_{\dot{\alpha}\dot{\beta}}\ =\ (C^T\gamma^\mu)_{\dot{\beta}\dot{\alpha}}\,.$$
The world-sheet fermion $\psi^\mu(z)$ in the Ramond sector has zero-modes that satisfy the $SO(1,9)$ Clifford algebra $$\{\psi^\mu_0, \psi^\nu_0\}\ =\ 0\,.$$ The degenerate ground states therefore become the space-time spinor, on which $\psi^\mu_0$ act as space-time gamma matrices. We summarize here the related convention. We denote the ground state spinor as $\left(\begin{matrix}|{}^\alpha\rangle\\ |{}_{\dot{\alpha}}\rangle\end{matrix} \right)$, on which $\psi^\mu_0$ acts as $$\psi^\mu_0|{}^\alpha\rangle\ =\
|{}_{\dot{\alpha}}\rangle\frac{1}{\sqrt{2}}(\bar{\gamma}^\mu)^{\dot{\alpha}\alpha}\,,\qquad
\psi^\mu_0|{}_{\dot{\alpha}}\rangle\ =\
|{}^\alpha\rangle\frac{1}{\sqrt{2}}(\gamma^\mu)_{\alpha\dot{\alpha}}\,.$$ Then $\hat{\Gamma}_{11}$ defined by (\[gamma11\]) acts on the ground states as $$\hat{\Gamma}_{11}|{}^\alpha\rangle\ =\ |{}^\alpha\rangle\,,\qquad
\hat{\Gamma}_{11}|{}_{\dot{\alpha}}\rangle\ =\ -|{}_{\dot{\alpha}}\rangle\,,$$ by which the definition of the GSO projection (\[GSO Ramond\]) is supplemented. Similarly, the BPZ conjugate of the ground state spinor $(\langle{}^\alpha|,\langle{}_{\dot{\alpha}}|)$ satisfies $$\langle{}^\alpha|\psi^\mu_0\ =\
\frac{i}{\sqrt{2}}(\bar{\gamma}^\mu)^{\dot{\alpha}\alpha}\langle{}_{\dot{\alpha}}|\,,\qquad
\langle{}_{\dot{\alpha}}|\psi^\mu_0\ =\
- \frac{i}{\sqrt{2}}(\gamma^\mu)_{\alpha\dot{\alpha}}\langle{}^\alpha|\,,$$ with the normalization $$\langle{}^\alpha|{}_{\dot{\alpha}}\rangle\ =\ {C^\alpha}_{\dot{\alpha}}\,,\qquad
\langle{}_{\dot{\alpha}}|{}^\alpha\rangle\ =\
-i{C^\alpha}_{\dot{\alpha}}\,.$$ The nontrivial matrix elements of $\psi^\mu_0$ are then given by $$\langle{}^\alpha|\psi^\mu_0|{}^\beta\rangle\
=\ \frac{1}{\sqrt{2}}(C\bar{\gamma}^\mu)^{\alpha
| 2,054
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| 1,851
| null | null |
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|
\Psi}|=\dfrac{k_{s}}{\sqrt{-g}}\left(\dfrac{\partial}{\partial r}\sqrt{-g}P\right)=2k_{s}\dfrac{(2\cos^{3}\theta a^{2}\alpha+\cos\theta \alpha r^{2}+r)}{(1-\alpha r \cos\theta)(r^{2}+a^{2}cos^{2}\theta)},$$ where $ g=-\sin^{2}\theta \dfrac{(a^{2}\cos^{2}\theta+r^{2})^{2}}{(\alpha r \cos\theta -1)^{8}} $.
From equation (\[enden1\]) we see that the entropy density diverges at the ring singularity and at $ r=\dfrac{1}{\alpha\cos\theta}, $ which is the conformal infinity in this spherical type coordinate system, as is evident from the metric (\[htn\]). This can also be further verified from the expressions of the Kretschmann scalar and the Weyl scalar in this case, since they vanish at the conformal infinity but diverge at the ring singularity. To compute the zeroes of the entropy density function we only need to find the roots of the numerator in (\[enden1\]), which is a quadratic function in $ r $.
Substituting $ \alpha=0 $ in the above expression of entropy density, we get the entropy density for the Kerr black hole: $$s_{kerr}=\dfrac{2k_{s}r}{(r^{2}+a^{2}\cos^{2}\theta)}.$$
![Plot showing the variation of the gravitational entropy density for the accelerating non-rotating charged BH with respect to the radial coordinate $ r $, where $ m=1, \; k_{s}=1, \; \alpha=0.25, \; e=0.5, \; \textrm{and} \; \theta=\dfrac{\pi}{2} $.[]{data-label="plot4a"}](chcsralphav.png){width="55.00000%"}
![Plot showing the variation of the gravitational entropy density for the accelerating rotating BH with respect to the radial coordinate $ r $ and the angular coordinate $ \theta $, where $ \alpha=0.45, \: m=1, \: {\color{blue}{a=0.5,}} \: \textrm{and} \: k_{s}=1 $. This figure clearly indicates that at the ring singularity $\left( r=0, \: \theta=\dfrac{\pi}{2} \right) $ the gravitational entropy density diverges.[]{data-label="plot4b"}](htnsrthetanew1.png){width="40.00000%"}
{ref-type="fig"} presents a running study, where it is clear which tasks were already completed and which remain to be done. This study manager view can also be shared with the study team.Fig. 5A study representation
### Assignees {#Sec9}
To build the assignee perspective, we were inspired by the typical email interfaces, i.e., a workspace with a list of requests (tasks), classified according to their priority: pending, solved and rejected. This workspace is the starting point of user activities (Fig. [6](#Fig6){ref-type="fig"}).Fig. 6Example of activities in the assignees' workspace: **a**) list of current tasks. **b**) task details
TASKA sends an email to all the assignees, whenever they are invited to, or included in, a study. Besides the email interactions, all the requests are listed in the user workspace, where they can accept, ask for further details, or complete each task.
This question and answer interaction, between the assignee and the SM, can be kept private or public for the whole team, so that the other participants in the task can see the discussion, avoiding dupl
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|
lde{p}]}F\Psi]\big)\,,
\label{tf sp ramond}\end{aligned}$$
where $[{\mathcal{S}},\tilde{p}]$ denotes the first-quantized charge defined by the commutator $[q^\alpha,\tilde{p}^\mu]$ with the parameter $\zeta_{\mu\alpha}$, $$[{\mathcal{S}},\tilde{p}]\ =\ \zeta_{\mu\alpha} [q^\alpha,\tilde{p}^\mu]\,,$$ and in particular $\zeta_{\mu\alpha}=\epsilon_\alpha v_\mu$ on the right-hand side of (\[alg sp\]). This new symmetry is also unphysical in a similar sense to $\delta_{\tilde{p}}$. At the linearized level, the transformation (\[tf sp\]) becomes[^6]
$$\begin{aligned}
\delta_{[{\mathcal{S}},\tilde{p}]}\Phi\ =&\
\xi_0 Q \xi_0 [{\mathcal{S}},\tilde{p}]\Psi\
=\ \xi_0 X_0[{\mathcal{S}},\tilde{p}]\Psi\,,
\label{tf sp ns at linear}\\
\delta_{[{\mathcal{S}},\tilde{p}]}\Psi\ =&\
X\eta\Xi Q[{\mathcal{S}},\tilde{p}]\Phi\
\cong\ XQ[{\mathcal{S}},\tilde{p}]\Phi\,,
\label{tf sp ramond at linear} \end{aligned}$$
where we have used the fact that ${\mathcal{S}}$, $\tilde{p}$, and thus $[{\mathcal{S}}, \tilde{p}]$ are commutative with $Q$ and $\eta$. If we note that $[{\mathcal{S}},p]=0$ and $$[{\mathcal{S}},X_0]\ =\ [{\mathcal{S}},\{Q,\xi_0\}]\ =\
\{Q,[{\mathcal{S}},\xi_0]\}+\{\xi_0,[{\mathcal{S}},Q]\}\ =\ 0\,,$$ the transformation of $\Phi$, (\[tf sp ns\]), can further be rewritten in the form of a linearized gauge transformation: $$\begin{aligned}
\delta_{[{\mathcal{S}},\tilde{p}]}\Phi\
=&\ - \xi_0[{\mathcal{S}}, (p-X_0\tilde{p})]\,\Psi\
=\ -\xi_0[{\mathcal{S}}, \{Q, M\}]\,\Psi
\nonumber\\
\cong&\ -\xi_0 Q[{\mathcal{S}}, M]\,\Psi
\nonumber\\
=&\ Q(\xi_0[{\mathcal{S}}, M]\Psi)-\eta(\xi_0X_0[{\mathcal{S}},M]\Psi)\,.\end{aligned}$$ Similarly the transformation of $\Psi$ (\[tf sp ramond\]) can also be written as $$\begin{aligned}
\delta_{[{\mathcal{S}},\tilde{p}]}\Psi\
\cong&\
X\eta\xi_0Q[{\mathcal{S}},\tilde{p}]\Phi
\nonumber\\
=&\
Q(X\eta\xi_0[{\mathcal{S}},\tilde{p}]\Phi)
+ X\eta X_0[{\mathcal{S}},\tilde{p}]\Phi
\nonumber\\
\cong&\
Q(X\eta\xi_0[{\mathcal{S}},\tilde{p}]\Phi
+ X\eta[{\mathcal{S}},M]\Phi)\,.
$$ It should be
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-Higgs phenomenology, even in the heavy squark regime.
[**Acknowledgements:**]{} We thank Michael Spira and Jaume Guasch for clarifying how the threshold corrections contribute to the Higgs couplings to bottom quarks and pointing out earlier works. We also thank Borut Bajc, Stéphane Lavignac, and Timon Mede for useful comments. AA is supported by the Ohio State University Presidential Fellowship. SR is supported by DOE/ DE-SC0011726.
Gluino-sbottom
==============
Gluinos couple with the down-type squarks and quarks proportional to the SU(3) gauge coupling $g_3$ and hence contribute large corrections to the bottom quark mass. The corrections are dominant when the squarks belong to the third family since the inter-generational mixings between the squarks are typically (and by assumption in this study) small. We will now calculate the individual diagrams shown in \[gluinocorrections\] considering the contributions from the two bottom squarks.
\
The three diagrams correct the inverse propagator S(p) = ,
where $-i \Sigma $ is the sum of the three diagrams in \[gluinocorrections\]: -i (p) = -i B\_[LR]{} -i p | A\_L -i p . A\_R .
The Lagrangian after including the corrections from the diagrams can be written as = b\^\* i b (1 - A\_L) + |[b]{}\^\* i |[b]{} (1-A\_R) + |[b]{} b (m\_[b0]{} + B\_[LR]{}) .
By rescaling $b$ and $\bar{b}$ by $\frac{1}{\sqrt{1-A_L}}$ and $\frac{1}{\sqrt{1-A_R}}$, respectively, the corrected bottom quark mass can be written as m\_b &=&\
&& m\_[b0]{} + B\_[LR]{} + (A\_L + A\_R)\
[$\Rightarrow$]{}m\_b = m\_b - m\_[b0]{} &=& B\_[LR]{} + (A\_L + A\_R) \[delmb\] .
We evaluate the loop integrals in each of the diagrams in \[gluinocorrections\]: -i B\_[LR]{}\^x &=& ( i g\_3 \_R\^x T\_l\^[aj]{} ) ( -i g\_3 (\_L\^x)\^ T\_j\^[ak]{} )\
&=& - g\_3\^2 \_R\^x (\_L\^x )\^\
-i p | A\_[L]{}\^x &=& (-i g\_3 \_L\^x T\_l\^[aj]{} ) ( -i g\_3 (\_L\^x)\^ T\_j\^[ak]{} )\
&=& -i g\_3\^2 \_L\^x (\_L\^x )\^\
-i p A\_[R]{}\^x &=& (i g\_3 \_R\^x T\_l\^[aj]{} ) ( i g\_3 (\_R\^x)\^ T\_j\^[ak]{} )\
&=& -i g\_3\^2 \
| 2,058
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|
_0)\in S$. Therefore, the infinitesimal symmetry condition is a necessary and sufficient condition for the existence of the symmetry group $G$ of the overdetermined system (\[eq:3.21\]). Since the vector fields $X_a$ form an Abelian distribution on $X\times U$, it follows that conditions (\[eq:3.30\]) and (\[eq:3.34\]) are satisfied. The solutions of the overdetermined system (\[eq:3.21\]) are invariant under the algebra $\mathcal{L}$ generated by the $p-2k$ vector fields $X_1,\ldots,X_{p-2k}$. The invariants of the group $G$ of such vector fields are provided by the functions ${\left\{ r^1,\ldots, r^k,\bar{r}^1,\ldots, \bar{r}^k,u^1,\ldots,u^q \right\}}$. So the general multimode solution of (\[eq:3.1\]) takes the required form (\[eq:3.5\]). $\Box$
The ideal plastic flow. {#sec:4}
=======================
In this section we would like to illustrate the proposed approach for constructing multimode solutions with the example of the ideal nonstationary irrotational planar ideal plastic flow subjected to an external force due to a work function $V$ (potential if $V_t=0$). Under the above assumptions the examined model is governed by a quasilinear elliptic homogenous system of five equations in (2+1) dimensions of the form [@Chak:2006; @Hill:1998; @Kat:1]
\[eq:4.1\] &(a) &&\_x- [( \_x 2+\_y 2)]{}+[( V\_x-u\_t - u u\_x - v u\_y )]{}=0,\
&(b) &&\_y- [( \_x2- \_y 2)]{}+[( V\_y-v\_t - u v\_x - v v\_y )]{}=0,\
&(c) &&(u\_y+v\_x)2+ (u\_x-v\_y)2=0,\
&(d) && u\_x+v\_y=0,\
&(e) && u\_y-v\_x=0.
Equations (\[eq:4.1\].a) and (\[eq:4.1\].b) involve the independent variables $\sigma, \theta,
u, v$ and the potential $V$ is a given function of $(t,x,y)\in{\mathbb{R}}^3$. The stress tensor is defined by the mean pressure $\sigma$ and the angle $\theta$ relative to the $x$-axis minus $\pi/4$. Equation (\[eq:4.1\].c) represents the Saint-Venant-Von Mises plasticity equation. Equations (\[eq:4.1\].d) and (\[eq:4.1\].e) for the velocities $u$ and $v$ (along the $x$-axis and $y$-axis respectively) correspond to the incompressibility
| 2,059
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| 0.772197
|
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|
al mass and momentum distributions are smeared out in a “tail" due to the correlated motion of a nearby nucleon. For both nucleons and pairs of nucleons, we assume that 10$\%$ of such decays are affected by the correlated motion of an additional nucleon [@1999corr]. Lepton rescattering within the nucleus is negligible.
The atmospheric $\nu$ MC sample corresponds to an exposure of 500 years for each of the four SK periods, 2000 years in total. Events in this sample are weighted assuming two-flavor mixing as is done in recent dinucleon and nucleon analyses [@2017mine; @2016miura; @2015jeff]. Details of the cross-sections and flux modeling used in this sample are discussed in recent SK nucleon decay analyses [@2016miura; @2017mine]. Event rates obtained from this sample are normalized to the relevant SK detector livetime.
Details of the neutron simulation and neutron tagging algorithm used for both the signal and atmospheric $\nu$ MC samples can be found in Ref. [@2016miura]. Neutron tagging can only be done for the SK-IV period; it reduces the expected number of background events by about 50% for our searches, and impacts signal efficiency by only a few percent.
![(color online) Total mass ($M_{tot}$) and total momentum ($P_{tot}$) projections for $p \rightarrow \mu^+ \gamma$ after cut (A4). The red histogram shows atmospheric $\nu$ MC corresponding to 2000 years of SK exposure normalized to SK-I through SK-IV data. The selection criteria are indicated by the vertical blue lines.[]{data-label="fig:projection"}](Fig3.png)
Although the selection criteria for all ten modes are similar, the two single-nucleon decay modes have more background due to their lower total mass. We adapt our strategy, as is done in Ref. [@2016miura], to perform a two-box analysis which allows us to study free and bound protons separately.
The following selection criteria are applied to signal MC, atmospheric $\nu$ MC, and data:
1. [Events must be fully contained in the inner detector with the event vertex wit
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|
) \lrp{N_t+ N(y_t) - N(y_0)}^2}}_{\circled{5}} dt
\numberthis \label{e:t:asldka}
\end{aligned}$$
$\circled{3}$ goes to $0$ when we take expectation, so we will focus on $\circled{1}, \circled{2}, \circled{4}, \circled{5}$. We will consider 3 cases
**Case 1: $\|z_t\|_2 \leq 2\epsilon$**\
From item 1(c) of Lemma \[l:fproperties\], $\lrn{\nabla f(z)}_2 \leq 1$. Using Assumption \[ass:U\_properties\].1, $\lrn{\nabla_t} \leq L\lrn{z_t}_2$, so that $$\begin{aligned}
\circled{1} \leq \lrn{\nabla_t}_2 \leq L \lrn{z_t}_2 \leq 2 L \epsilon
\end{aligned}$$
Also by Cauchy Schwarz, $$\begin{aligned}
\circled{2}
=& \lin{\nabla f(z_t), \Delta_t}
\leq \lrn{\Delta_t}_2 \leq L \lrn{y_t - y_0}_2
\end{aligned}$$
Since $\gamma_t = 0$ in this case by definition in , $\circled{4}=0$.
Using Lemma \[l:fproperties\].2.c. $ \lrn{\nabla^2 f(z_t)}_2 \leq \frac{2}{\epsilon}$, so that $$\begin{aligned}
\circled{5}
\leq& \frac{1}{\epsilon} \lrp{\tr\lrp{N_t^2 + N(y_t) - N(y_0)}^2}\\
\leq& \frac{2}{\epsilon} \lrp{\tr\lrp{N_t^2} + \tr\lrp{\lrp{N(y_t) - N(y_0)}^2}}\\
\leq& \frac{2L_N^2}{\epsilon} \lrp{\lrn{z_t}_2^2 + \lrn{y_t - y_0}_2^2}\\
\leq& 4L_N^2 \epsilon + \frac{2L_N^2}{\epsilon} \|y_t-y_0\|_2^2
\end{aligned}$$
Where the second inequality is by Young’s inequality, the third inequality is by item 2 of Lemma \[l:N\_is\_regular\], the fourth inequality is by our assumption that $\lrn{z_t}_2 \leq 2\epsilon$.
Summing these, $$\begin{aligned}
\circled{1} + \circled{2} + \circled{4} + \circled{5}
\leq 4 \lrp{L + L_N^2}\epsilon + L \lrn{y_t - y_0}_2 + \frac{2L_N^2}{\epsilon}\|y_t-y_0\|_2^2
\end{aligned}$$
**Case 2: $\|z_t\|_2\in (2\epsilon, \Rq)$**\
In this case, $\gamma_t = \frac{z_t}{\|z_t\|_2}$. Let $q$ be as defined in and $g$ be as defined in Lemma \[l:gproperties\]. By items 1(b) and 2(b) of Lemma \[l:fproperties\] and items 1(b) and 2(b) of Lemma \[l:gproperties\], $$\begin{aligned}
\nabla f
| 2,061
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|
where $$\begin{aligned}
\mathbf{A}
&=& \frac{t}{n}[(\identity \otimes in\sigma_x) - (in\sigma_x \otimes \identity) - p({\identity} \otimes {\identity})\\
&&+ p(\Pi_1 \otimes \Pi_1) + p(\Pi_0 \otimes \Pi_0)] \\
&=& \frac{t}{n}\left( \begin{matrix} 0 & ik & -ik & 0 \\
ik & -p & 0 & -ik \\
-ik & 0 & -p & ik \\
0 & -ik & ik & 0 \\
\end{matrix} \right).\end{aligned}$$ Notice that for $p = 0$, $\left[e^{\mathbf{A}}\right]^{\otimes n}
= \left[e^{-it\sigma_x} \otimes e^{it\sigma_x}\right]^{\otimes n}$, which is exactly the superoperator formulation of the dynamics of the non-decohering walk.
Small-system behavior and analysis of the walk
==============================================
So far we have shown that the walk with decoherence is still equivalent to $n$ non-interacting single-qubit systems. We now analyze the behavior of a single-qubit system under the superoperator $e^\mathbf{A}$. The structure of this single particle walk will allow us to then immediately draw conclusions about the entire system.
The eigenvalues of $\mathbf{A}$ are $0$, $- \frac{pt}{n}$, $\frac{-p t
- \alpha t}{2n}$ and $\frac{-p t + \alpha t}{2n}$. Here $\alpha =
\sqrt{p^2-16k^2}$ is a complex constant that will later turn out to be important in determining the behavior of the system as a function of the rate of decoherence $p$ and the energy $k$. The matrix exponential of $\mathbf{A}$ in this spectral basis can then be computed by inspection. To see how our superoperator acts on a density matrix $\rho_0$, we may change $\rho_0$ to the spectral basis, apply the diagonal superoperator to yield $\rho_t$, and finally change $\rho_t$ back to the computational basis. At that point we can apply the usual projectors $\Pi_0$ and $\Pi_1$ to determine the probabilities of measuring $0$ or $1$ in terms of time.
Let $\Psi_0 = \vert 0 \rangle$ and $\rho_0 = \vert \Psi_0 \rangle
\langle \Psi_0 \vert$. In the diagonal basis, $$\rho_0 = \left[\begin{matrix}
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github_plus_top10pct_by_avg
|
t indicator to the existence of leptonic unitarity violation. Existence of second and higher order corrections in $W$, if detected, uniquely identifies the case for low-scale unitarity violation.
- Presence (low-$E$) or absence (high-$E$) of the probability leaking term $\mathcal{C}_{\alpha \beta}$ in (\[Cab\]) remains to be the best possible way to distinguish between high- and low-scale unitarity violation.
Oscillation probabilities with and without unitarity violation {#sec:probabilities}
----------------------------------------------------------------
Under the hope to uncover in which region of parameter space the effect of non-unitarity is most prominent, we first compare the oscillation probabilities with and without unitarity violation. We examine the three channels $\nu_{\mu} \rightarrow \nu_{e}$, $\nu_{\mu} \rightarrow \nu_{\tau}$, and $\nu_{\mu} \rightarrow \nu_{\mu}$. We do not intend to do quantitative analyses, nor attempt to cover the whole parameter space. Yet, we try to give the readers a feeling on how large and where are the effects of unitarity violation. Therefore, we present the results just for a specific choice of the parameters. Also, the use of uniform matter density $\rho = 3.2~{\rm g\,cm}^{-3} $ over the entire baseline is not realistic.[^17]
Notice that the results given in section \[sec:probabilities\] apply to both high-scale unitarity violation as well as low-scale one in its leading order in $W$. The probability leaking term is not included in the analysis in this section \[sec:probabilities\], but the effect is discussed in section \[sec:correction-terms\].
### A brief note on the parameter choice {#sec:parameter-choice}
Let us start by explaining briefly how the unitarity violation parameters are chosen. We go to the $(3+1)$ model in which the constraints on the parameters are best understood [@Kopp:2013vaa; @deGouvea:2015euy; @Fernandez-Martinez:2016lgt; @TheIceCube:2016oqi]. In consistent with the current ones we have chosen: $\sin^2\theta_{14} = 0.02$, $\sin^2\the
| 2,063
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| 0.76753
|
github_plus_top10pct_by_avg
|
{c}{b} \right\rvert^{\frac{1}{a}} \right) \\
&\quad + \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} \G(3a)}{2 \G(a)}
+ \operatorname{{sgn}}(c) \frac{(k_{1}^{2} - k_{2}^{2}) b^{2}}{2 \G(a)}
\g\left(3a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right). \end{aligned}$$ Also, from $(\ref{E[Pe(Z+c)]-2})$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]^{2}
&= \frac{(k_{1} - k_{2})^{2} c^{2}}{4}
+ \frac{(k_{1}^{2} - k_{2}^{2}) c \lvert c \rvert}{2 \G(a)}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
+ \frac{(k_{1}^{2} - k_{2}^{2}) b c}{2 \G(a)}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \\
&\quad + \frac{(k_{1} + k_{2})^{2} b \lvert c \rvert}{2 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \\
&\quad + \frac{(k_{1} + k_{2})^{2} c^{2}}{4 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2}
+ \frac{(k_{1} + k_{2})^{2} b^{2}}{4 \G(a)^{2}}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2}. \end{aligned}$$ Therefore, for any $c \in \mathbb{R}$, we have $$\begin{aligned}
\operatorname{{V}}[\Pe(Z + c)]
&= \operatorname{{E}}[\Pe(Z + c)^{2}] - \operatorname{{E}}[\Pe(Z + c)]^{2} \\
&= \frac{(k_{1} + k_{2})^{2} c^{2}}{4}
+ \frac{(k_{1}^{2} - k_{2}^{2}) b c}{2 \G(a)}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \nonumber \\
&\quad - \frac{(k_{1} + k_{2})^{2} b \lvert c \rvert}{2 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \nonumber \\
&\quad - \frac{(k_{1} + k_{2})^{2} c^{2}}{4 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2}
- \frac{(k_{1} + k_{2})^{2} b^{2}}{4 \G(a)^{2}}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2} \nonumber \\
&\quad + \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} \G(3a)}{2 \G(a)}
+ \operator
| 2,064
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|
pes .unnumbered}
--------------------
The same logic applies to the computation of the current-primary OPEs. Current conservation links the $j^a_{L,z} \phi$ and $j^a_{L,\bar
z} \phi$ OPEs : \[phiCC\] (z) = 0. When the above equation is valid, the Maurer-Cartan constraint can be rewritten as: \[phimodMC\] (z) |j\^a\_[L,z]{}(w) = f\^2 (z) i [f\^a]{}\_[cd]{}:j\^d\_[L,z]{} j\^c\_[L,|z]{}:(w). Again the discussion of appendix \[XXOPEs\] gives an ansatz for the current-primary OPE at a given order in $f^2$: the terms appearing in the current-primary OPE at order $f^{2n}$ are composites of at most $n$ currents with the primary field $\phi$. When we plug this ansatz in equation we obtain the value of the coefficients. This method is illustrated in appendix \[AppjPhi\] where we compute the current-primary OPE up to order $f^2$.
Further remarks {#further-remarks .unnumbered}
---------------
This perturbative approach squares well with the observation that the most singular terms in the current-current and current-primary OPEs come with the lower power of $f^2$. This is explained in appendix \[XXOPEs\]. Thus performing a computation up to a certain order in $f^2$ allows to truncate the current-current and current-primary OPEs at a certain order in the distance between the insertion points of the operators.
The consistency of this perturbative approach demands that the addition of higher-order terms to the elementary OPEs does not spoil their compatibility both with current conservation and with the Maurer-Cartan equation at lower order in $f^2$. That this is the case is proven in appendix \[consistentPertOPEs\].
One may hope to obtain a closed formula for the full current-current and current-primary OPEs thanks to this algebraic bootstrap.
Composite operators and their conformal dimension {#confdimcomp}
=================================================
In this section we consider operators that are composites of one or more currents with a primary operator. We are mostly interested in the computation of the conform
| 2,065
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|
egory, with tensor product $- \otimes_A -$ and the unit object $A$. Hochschild homology is a homological functor from $A{\operatorname{\!-\sf bimod}}$ to $k{\operatorname{\it\!-Vect}}$.
To obtain a small category interpretation of $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$, one notes that for any $n,n' \geq 0$, the $A$-bimodule structure on $M$ induces a multiplication map $$A^{\otimes n} \otimes M \otimes A^{\otimes n'} \to M.$$ Therefore, if to any $\langle [n],v \rangle \in \Delta^{opp}$ we associate the $k$-vector space $$\label{M.Delta}
M^\Delta_\#([n]) = M \otimes A^{\otimes (V([n]) \setminus \{v\})},$$ with $M$ filling the place corresponding to $v \in V([n])$, then make perfect sense for those maps $f:[n'] \to [n]$ which preserve the distinguished points. Thus to any $M \in
A{\operatorname{\!-\sf bimod}}$, we can associate a simplicial $k$-vector space $M^\Delta_\# \in {\operatorname{Fun}}(\Delta^{opp},k)$. In the particular case $M=A$, we have $A_\#^\Delta = j^*A_\#$.
\[hoch\] For any $M \in A{\operatorname{\!-\sf bimod}}$, we have a canonical isomorphism $$\label{hh.iso}
HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M) \cong H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},M^\Delta_\#).$$
[[*Proof.*]{}]{} It is well-known that for any simplicial $k$-vector space $E$, the homology $H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},E)$ can be computed by the standard complex of $E$ (that is, the complex with terms $E([n])$ and the differential $d = \sum_i(-1)^id_i$, where $d_i$ are the face maps). In particular, $H_0(\Delta^{opp},M^\Delta_\#)$ is the cokernel of the map $d:A \otimes M \to M$ given by $d(a \otimes m) =
am-ma$. The natural projection $M \to M \otimes_{A^{opp} \otimes A}
A$ obviously factors through this cokernel, so that we have a natural map $$\rho_0:H_0(\Delta^{opp},M^\Delta_\#) \to HH_0(A,M).$$ Both sides of are homological functors in $M$, and $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$ is a universal homological functor (=the derived functor of
| 2,066
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|
rline{v} \left( 2A \sqrt{b} \right)^k,\end{aligned}$$ where the first inequality follows from the bound $| v^\top_l (W_i - \psi) |\leq \|
W_i - \psi\|$ (as each $v_l$ is of unit norm), the second from the fact that the coordinates of $W_i$ are bounded in absolute value by $A$ and the third by the fact that $\overline{v}$ is the largest eigenvalue of $V$.
Thus we see that by setting $B_n = \overline{v} \left( 2A \sqrt{b} \right)$, condition (M2’) is satisfied (here we have used the fact that $\overline{v} \geq 1$). Finally, condition (E1’) is easily satisfied, possibly by increasing the constant in the term $B_n$.
Thus, gives $$\sup_{A \in \mathcal{A}} \left | \mathbb{P}(\sqrt{n} (\hat{\psi}
- \psi)
\in A)
- \mathbb{P}( \tilde{Z}_n \in A) \right|
\leq C \frac{1}{\sqrt{\lambda_{\min}(V)}} \left(
\frac{ \overline{v}^2 b (\log 2bn)^7}{n} \right)^{1/6},$$ and the result follows from , the fact that the choice of $G = G(\psi)$ is arbitrary and the fact that $\lambda_{\rm min}(V(P)) \geq v$ for all $P\in {\cal P}_n$, by assumption. $\Box$
\[lem:operator\] Let $X_1,\ldots,X_n$ be independent, mean-zero vectors in $\mathbb{R}^p$, where $p
\leq n$, such that $\max_{i=1\dots,n} \|X_i \|_\infty \leq K$ almost surely for some $K>0$ with common covariance matrix $\Sigma$ with $\lambda_{\max}(\Sigma)
\leq U$. Then, there exists a universal constant $C>0$ such that $$\label{eq:vector.bernstein.simple}
\mathbb{P}\left( \frac{1}{n} \left\| \sum_{i=1}^n X_i \right\| \leq C K \sqrt{ p
\frac{\log n }{n}} \right) \geq 1 - \frac{1}{n}.$$ Letting $\hat{\Sigma} = \frac{1}{n} \sum_{i=1}^n X_i X_i^\top$, if $U
\geq \eta > 0$, then there exists a $C>0$, dependent on $\eta$ only, such that $$\label{eq:matrix.bernstein.simple.2}
\mathbb{P}\left( \| \widehat{\Sigma} - \Sigma \|_{\mathrm{op}} \leq C K
\sqrt{p U \frac{ \log p + \log n}{n} } \right) \geq 1 - \frac{1}{n}.$$
[**Proof of .**]{} Since $\| X_i \| \leq
| 2,067
| 3,626
| 1,738
| 1,728
| null | null |
github_plus_top10pct_by_avg
|
= -0*m
the third derivative of -21688*l**2*z**3 - 232*l**2 - 13*z**3 - 4*z**2 + 320*z - 1 wrt z.
-130128*l**2 - 78
What is the third derivative of -19292*n**5 + 131*n**3 - 27728*n**2 - 804*n?
-1157520*n**2 + 786
Differentiate 21431042*c**3*t**3 + 319*c**3 - 16092*c**2 wrt t.
64293126*c**3*t**2
What is the second derivative of -4730*c*v**4 + 3*c*v**3 - 93*c*v**2 + 939*c*v - 23*c + 8 wrt v?
-56760*c*v**2 + 18*c*v - 186*c
Find the third derivative of -1509*c*t**4 - c*t**3 + 2*c*t**2 - 4*c*t - 219365*c + 9646*t**4 + 1 wrt t.
-36216*c*t - 6*c + 231504*t
Find the third derivative of 230293*k**5 - 5*k**4 + k**3 + k**2 - 186746*k - 1 wrt k.
13817580*k**2 - 120*k + 6
What is the derivative of 108*c**2*m*r**2 - 1404459*c**2*m*r - 553693*c**2*r - c*r**2 - 30*r**2 wrt m?
108*c**2*r**2 - 1404459*c**2*r
Find the third derivative of -12649173*p**6 + 3463644*p**2 + 2*p - 2.
-1517900760*p**3
What is the derivative of -60550*i**3 - 414*i**2 - 4*i + 1466427089 wrt i?
-181650*i**2 - 828*i - 4
What is the second derivative of -600241024*h**5 - 565292232*h?
-12004820480*h**3
What is the second derivative of 35*m**3*u**2 - 151*m**3*u + m**2*u**2 - 31*m**2*u + 2*m**2 - m*u**2 + 4856*u**2 + 124*u - 1 wrt u?
70*m**3 + 2*m**2 - 2*m + 9712
What is the second derivative of -7628629*s**3*v - s**2*v + 240283*s*v + v + 13 wrt s?
-45771774*s*v - 2*v
What is the third derivative of 510*z**5 - 1467*z**3 - 4*z**2 - 927095?
30600*z**2 - 8802
What is the second derivative of -22513*f**2*s**2 - 4*f**2*s - 2*f**2 - 2*f*s + 15*f - 890*s**3 + s**2 + 671*s - 2 wrt s?
-45026*f**2 - 5340*s + 2
What is the second derivative of 1489728*f**2*i*s**2 - 37537*f**2*i*s + 3*f*s + 18*i*s**2 wrt s?
2979456*f**2*i + 36*i
Differentiate 14946165*m**4 - 15776104 wrt m.
59784660*m**3
What is the second derivative of -244*k**3*y**3 + 1530*k**2*y**2 + 2*k*y + 182*k - y**3 - 31 wrt k?
-1464*k*y**3 + 3060*y**2
What is the second derivative of -3177*g**3*j**2 + 138*g**2*j**3 + 3*g - 17*j**3 - 7879 wrt g?
-19062*g*j**2 + 276*j**3
Differentiate 182860*r**3 + 447*r**2 - 6740936
| 2,068
| 643
| 1,794
| 1,791
| null | null |
github_plus_top10pct_by_avg
|
point have definite values (and $\lambda_\varphi$ can be directly calculated from linearizied RG equations for $A_i$ and $B_i$), in the $b=3$ case some fixed point coordinates diverge, and calculation of $\lambda_\varphi$ requires an additional effort. To be more specific, a numerical analysis of RG equations (\[eq:RGAi\]) and (\[eq:RGBi\]) reveals that $A_i^{(r)}\approx A^{(r)}C^*\to 0$, $B_i^{(r)}\approx B^{(r)}C^*\to \infty$. In this situation the appropriate eigenvalue $\lambda_\varphi$ can also be calculated, using the following transformation. If we write the relation (\[nov1\]) in the form $$\label{nov2}
\fl{1\over X_i^{(r+1)}}\frac{\partial X_i^{(r+1)}}{\partial w}=\sum_{j=1}^6 \left({X_j^{(r)}\over X_i^{(r+1)}}\frac{\partial X_i^{(r+1)}}{\partial X_j^{(r)}}\right){1\over X_j^{(r)}}\frac{\partial X_j^{(r)}}{\partial w}\>,\quad i=1,\ldots,6\>,$$ it can be shown that, when we keep only dominant terms in the RG equations, and in the derivatives $\frac{\partial
X_i^{(r+1)}}{\partial X_j^{(r)}}$, then, the matrix elements ${\left(\frac{X_j^{(r)}}{X_i^{(r+1)}}\frac{\partial
X_i^{(r+1)}}{\partial X_j^{(r)}}\right)}^*$ of the new eigenvalue problem are either equal to zero or to some finite constants (depending on $C^*$), from which we find $\lambda_\varphi=3.9919$, and therefrom $\varphi={\ln\lambda_\varphi}/{\ln\lambda_{\nu_3}}=0.6073$.
- Strong inter-chain attraction $w>w_c(u,t)$ destroys the globule and completely attaches the 3D chain to the 2D chain. This entangled phase is again characterized by the fixed point (\[fp5\]).
In table \[tab:CSAWs\] we have presented the numerical results for the crossover fixed points and the corresponding values of the contact exponent $\varphi$, obtained for the unbinding transitions from entangled two-polymer phase to segregated phases of 2D and 3D chains on the $b=2$ and $b=3$ 3D SG fractals. These values are correct for all studied cases of $t$ in the interval $(0,1)$. Varying the parameter $t$ in this interval changes only the particular values of $w_c(u
| 2,069
| 526
| 1,790
| 2,126
| 2,287
| 0.781149
|
github_plus_top10pct_by_avg
|
bound on the magnitude of the reminder term in the Taylor series expansion of $g(\hat{\psi})$ around $g(\psi)$, as detailed in the proof of below. Of course, we may relax the requirement that holds almost everywhere to the requirement that it holds on an event of high probability. This is indeed the strategy we use when in applying the present results to the projection parameters in .
The covariance matrix of the linear approximation of $g(\psi) -
g(\hat{\psi})$, which, for any $P \in
\mathcal{P}_n$, is given by $$\label{eq:Gamma}
\Gamma = \Gamma(\psi(P),P)=G(\psi(P)) V(P) G(\psi(P))^\top,$$ plays a crucial role in our analysis. In particular, our results will depend on the smallest variance of the linear approximation to $g(\psi) - g(\hat{\psi})$: $$\label{eq:sigma}
\underline{\sigma}^2 = \inf_{ P \in \mathcal{P}_n}\min_{j =1,\ldots,s}
G^\top_j(\psi(P)) V(P) G_j(\psi(P)).$$
With these definitions in place we are now ready to prove the following high-dimensional Berry-Esseen bound.
\[theorem::deltamethod\] Assume that $W_1,\ldots, W_n$ is an i.i.d. sample from some $P \in {\cal P}_n$ and let $Z_n \sim N(0,\Gamma)$. Then, there exists a $C>0$, dependent on $A$ only, such that $$\sup_{P\in {\cal P}_n}
\sup_{t > 0}
\Bigl|\mathbb{P}( \sqrt{n}||\hat\theta - \theta||_\infty \leq t) -\mathbb{P}(
||Z_n||_\infty \leq t)\Bigr| \leq C \Big(
\Delta_{n,1} + \Delta_{n,2} \Big),$$ where $$\begin{aligned}
\label{eq::Delta}
\Delta_{n,1} &=
\frac{1}{\sqrt{v}} \left( \frac{ \overline{v}^2 b (\log 2bn)^7}{n} \right)^{1/6}
\\
\Delta_{n,2} &=
\frac{1}{\underline{\sigma}}\sqrt{\frac{ b \overline{v} \overline{H}^2 (\log n)^2 \log b}{n}}.\end{aligned}$$
[**Remarks**]{} The assumption that the support of $P$ is compact is made for simplicity, and can be modified by assuming that the coordinates of the vectors $W_i$ have sub-exponential behavior. Notice also that the coordinates of the $W_i$’s need not be independent.
The proof of resembles the classic proof of the asymptotic normality of non-linear functions of ave
| 2,070
| 1,436
| 1,708
| 1,802
| null | null |
github_plus_top10pct_by_avg
|
-------- ----------------------------- -- ----------------------------- -------------- ----------------------------- -------------- ----------------------------- -------------- ---------------------- --
0 $\cO_X$ $\supsetneq$ $\langle xy,x^5,y^5\rangle$
1 $\cO_{X,\zeta^1}$
2 $\cO_{X,\zeta^2}$ $\supsetneq$ $\langle y^4\rangle$
3 $\cO_{X,\zeta^3}$ $\supsetneq$ $\langle x^4\rangle$
4 $\cO_{X,\zeta^4}$ $\supsetneq$ $\langle y^3\rangle$
: Stratification of $\cM(C^\lambda,k)$ for $\lambda\in [0,1)$[]{data-label="tab:MX"}
\[ex:non-reduced\] As a second example we will consider a non-reduced curve in a smooth surface. Take the divisor $\cC=\cC_1+2\cC_2$ in $(\CC^2,0)$, where $\cC_1$ is an ordinary cusp $\{y^2+x^3=0\}$, and $\cC_2$ is a smooth germ tangent to $\cC_1$, say $\{y=0\}$. Following the notation in , $\Gamma=\{\v \}$ since one weighted blow-up with $w=(2,3)$ resolves the curve. Note that $m_{\v 1}=6$, $m_{\v 2}=3$, $n_{1}=1$, $n_2=2$, and $\nu_{\v}=5$, then according to Definition \[def:M\] one obtains the following condition $$\operatorname{mult}_{E_\v}\pi^*x^iy^j=2i+3j>6\{\lambda\}+3\{2\lam
| 2,071
| 2,011
| 1,636
| 1,927
| null | null |
github_plus_top10pct_by_avg
|
iction that requires $y_{1}=y_{2}$ whenever $(x,y_{1})\in f$ and $(x,y_{2})\in f$. In this subset notation, $(x,y)\in f\Leftrightarrow y=f(x)$. ]{}
[^4]: [\[Foot: EquivRel\]An useful alternate way of expressing these properties for a relation $\mathscr{M}$ on $X$ are]{}
[$\quad$(ER2) $\mathscr{M}$ is symmetric iff $\mathscr{M}=\mathscr M^{-}$ ]{}
[$\quad$(ER3) $\mathscr{M}$ is transitive iff $\mathscr{M}\circ\mathscr{M}\subseteq\mathscr{M}$, ]{}
[with $\mathscr{M}$ an equivalence relation only if $\mathscr{M}\circ\mathscr{M}=\mathscr{M}$, where for $\mathscr{M}\subseteq X\times Y$ and $\mathscr{N}\subseteq Y\times Z$, the composition $\mathscr{N}\circ\mathscr{M}:=\{(x,z)\in X\times Z\!:(\exists y\in Y)\textrm{ }((x,y)\in\mathscr{M})\wedge((y,z)\in\mathscr{N})\}$]{}
[^5]: [\[Foot: family\]A function $\chi\!:\mathbb{D}\rightarrow X$ will be called a]{} *family* [in $X$ indexed by $\mathbb{D}$ when reference to the domain $\mathbb{D}$ is of interest, and a]{} *net* [when it is required to focus attention on its values in $X$.]{}
[^6]: [\[Foot: extension\]Observe that it is]{} *not* [being claimed that $f$ belongs to the same class as $(f_{k})$. This is the single most important cornerstone on which this paper is based: the need to “complete” spaces that are topologically “incomplete”. The classical high-school example of the related problem of having to enlarge, or extend, spaces that are not big enough is the solution space of algebraic equations with real coefficients like $x^{2}+1=0$. ]{}
[^7]: [\[Foot: support\]By definition, the support (or supporting interval) of $\varphi(x)\in\mathcal{C}_{0}^{\infty}[\alpha,\beta]$ is $[\alpha,\beta]$ if $\varphi$ and all its derivatives vanish for $x\leq\alpha$ and $x\geq\beta$. ]{}
[^8]: [\[Foot: integral\]Both Riemann and Lebesgue integrals can be formulated in terms of the so-called]{} *step functions* [$s(x)$, which are piecewise constant functions with values $(\sigma_{i})_{i=1}^{I}$on a finite number of bounded subintervals $(J_{i})_{i=1}^{I
| 2,072
| 3,446
| 2,770
| 1,920
| 3,294
| 0.773394
|
github_plus_top10pct_by_avg
|
$\begin{aligned}
{\ensuremath{\operatorname{\mathbf{Pr}}\left[Y_k{\leqslant}{\ensuremath{\operatorname{\mathbf{E}}\left[Y_k\right]}}/2\right]}}{\leqslant}\exp({-{\ensuremath{\operatorname{\mathbf{E}}\left[Y_k\right]}}}/{8})=\exp(-\omega(\log n)).
\end{aligned}$$
[It follows that there exists a $d$-subset]{} $D$ which is chosen by at least $k$ balls and hence there is at least one bin in $D$ whose load is at least $k/d$.
Proof of Proposition \[pro:gmn\] {#sec:pop}
================================
In this section we prove Proposition \[pro:gmn\]. First we restate a useful theorem from [@CLLM12].
*[@CLLM12 Theorem 3]* \[chernof\] Let $M$ be an ergodic Markov chain with finite state space $\Omega$ and stationary distribution $\pi$. Let $T = T(\varepsilon)$ be its $\varepsilon$-mixing time for $\varepsilon<1/8$. Let $(Z_1,\ldots, Z_t)$ denote a $t$-step random walk on $M$ starting from an initial distribution $\rho$ on $\Omega$ (that is, $Z_1$ is distributed according to $\rho$). For some positive constant $\mu$ and every $i \in [t]$, let $f_i:\Omega\to [0, 1]$ be a weight function at step $i$ such that the expected weight ${\ensuremath{\operatorname{\mathbf{E}}_{\pi}\left[f_i(v)\right]}} = \sum_{v\in\Omega} \pi(v) f_i(v)$ satisfies ${\ensuremath{\operatorname{\mathbf{E}}_{\pi}\left[f_i(v)\right]}} = \mu$ for all $i$. Define the total weight of the walk $(Z_1, . . . , Z_t)$ by $X=\sum_{i=1}^t f_i(Z_i)$. Write $||\rho||_{\pi}=\sqrt{\sum_{x\in \Omega} \rho_x^2/\pi_x}$. Then there exists some positive constant $c$ $($independent of $\mu$ and $\varepsilon)$ such that for all $\alpha{\geqslant}0$,
1. ${\ensuremath{\operatorname{\mathbf{Pr}}\left[X{\geqslant}(1+\alpha)\mu t\right]}}{\leqslant}c||\rho||_\pi\, {\mathrm{e}}^{-\alpha^2\mu t/72 T}$ for $0{\leqslant}\alpha{\leqslant}1$.
2. ${\ensuremath{\operatorname{\mathbf{Pr}}\left[X{\geqslant}(1+\alpha)\mu t\right]}}{\leqslant}c||\rho||_\pi \,{\mathrm{e}}^{-\alpha\mu t/72 T}$ for $\alpha>1$.
3. ${\ensuremath{\operatorname{\mathbf{Pr}}\left[X{\leqslant}(
| 2,073
| 627
| 1,245
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| 3,737
| 0.770428
|
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|
t morphism ${\sD}^A\to{\sD}^B,A,B\in\cCat,$ preserves right homotopy finite right Kan extensions.\[item:sl4b\]
3. The derivator is pointed and $C\colon{\sD}^{[1]}\to{\sD}$ preserves right homotopy finite right Kan extensions.\[item:sl5\]
4. The derivator is pointed and $C\colon{\sD}^{[1]}\to{\sD}$ preserves homotopy finite limits.\[item:sl6\]
5. The derivator is pointed and $C\colon{\sD}^{[1]}\to{\sD}$ preserves $F$.\[item:sl7\]
If is stable, then also the shifted derivators ${\sD}^A,A\in\cCat,$ are stable [@groth:ptstab Prop. 4.3]. Consequently, every left exact morphism ${\sD}^A\to{\sD}^B$ is also right exact [@groth:can-can Prop. 9.8] and it hence preserves left homotopy finite left Kan extensions [@groth:can-can Thm. 9.14]. This and a dual argument shows that statement \[item:sl1\] implies statements \[item:sl4a\] and \[item:sl4b\]. Since right Kan extension morphisms are right adjoint morphisms and hence left exact, the implications \[item:sl4a\] implies \[item:sl3a\] and \[item:sl3a\] implies \[item:sl2\] are immediate. Moreover, \[item:sl2\] implies \[item:sl1\] by , and, by duality, it remains to incorporate the three final statements. Statement \[item:sl1\] implies statement \[item:sl5\] since $C$ is left exact in this case and it hence preserves right homotopy finite right homotopy Kan extensions [@groth:can-can Thm. 9.14]. The implications \[item:sl5\] implies \[item:sl6\] and \[item:sl6\] implies \[item:sl7\] being trivial, it remains to show that \[item:sl7\] implies \[item:sl1\] which is already taken care of by the proof of .
There are, of course, various additional minor variants of the characterizations in obtained, for example, by replacing $C$ by ${\mathsf{cof}}\colon{\sD}^{[1]}\to{\sD}^{[1]}$.
\[rmk:interpretation\] A typical slogan is that spectra are obtained from pointed topological spaces if one forces the suspension to become an equivalence. This slogan is made precise by and the fact that the derivator of spectra is the stabilization of the derivator of point
| 2,074
| 1,483
| 1,759
| 1,942
| 3,378
| 0.772763
|
github_plus_top10pct_by_avg
|
en by $$F_{\mu }^{ext}=-(\partial _{\nu }A_{\mu }-\partial _{\mu }A_{\nu })v^{\nu }
\label{12}$$ where the quantity $A_{\mu }$ is the vector potential given in units of $m_{0}c/e$. For a linearly polarized laser pulse, $$A_{\mu }\equiv (\Phi /c,{\bf A}),\;\;\;\;{\bf A}=\hat{x}A_{x}(\phi
),\;\;\;\;\Phi =0 \label{16}$$ which is a function of the invariant phase of the traveling wave, $$\phi =k^{\mu }x_{\mu }(\tau )=x_{0}-z\;, \label{20}$$ where $k^{\mu }=(1,0,0,1)$ is the dimensionless laser wave number.
Following Ref. [@Hart] we use $\phi $ as the independent variable to recast the Dirac Lorentz equation as $$\begin{aligned}
\frac{d^{2}v_{x}}{d\phi ^{2}} &=&v_{x}\left( \left( \frac{d{\bf v}}{d\phi }\right) ^{2}-\left( \frac{d\gamma }{d\phi }\right) ^{2}\right) +\frac{1}{u}\left( \frac{dv_{x}}{d\phi }\left( \frac{1}{\varepsilon }-\frac{du}{d\phi }\right) +\frac{A}{\varepsilon }G(\phi )sin\phi \right) \label{28} \\
\frac{d^{2}v_{z}}{d\phi ^{2}} &=&v_{z}\left( \left( \frac{d{\bf v}}{d\phi }\right) ^{2}-\left( \frac{d\gamma }{d\phi }\right) ^{2}\right) +\frac{1}{u}\left( \frac{dv_{z}}{d\phi }\left( \frac{1}{\varepsilon }-\frac{du}{d\phi }\right) +\frac{v_{x}}{u}\frac{A}{\varepsilon }G(\phi )sin\phi \right)
\label{30}\end{aligned}$$ where the laser pulse electric field $$\frac{dA_{x}(\phi )}{d\phi }=AG(\phi )sin\phi :, \label{32}$$ has been introduced, with $A$ being the maximum amplitude of the pulse, $G(\phi )=e^{-(\phi /\Delta \phi )^{2}}$ is a unit Gaussian envelope of width $\Delta \phi $. Note that $$\left( \frac{d{\bf v}}{d\phi }\right) ^{2}=\left( \frac{dv_{x}}{d\phi }\right) ^{2}+\left( \frac{dv_{z}}{d\phi }\right) ^{2}\;, \label{33a}$$
$$\frac{d\gamma }{d\phi }=\frac{v_{x}}{\gamma }\frac{dv_{x}}{d\phi }+\frac{v_{x}}{\gamma }\frac{dv_{z}}{d\phi } \label{33b}$$
and
$$\frac{du}{d\phi }=\frac{d\gamma }{d\phi }-\frac{dv_{z}}{d\phi }. \label{33c}$$
Substituting these relations into Eqs. (\[28\] ) and (\[30\]) we find $$\begin{aligned}
\frac{d^{2}v_{x}}{d\phi ^{2}} &=&v_{x}Q+\frac{1}{u}\left( \frac{1
| 2,075
| 3,940
| 1,866
| 1,732
| null | null |
github_plus_top10pct_by_avg
|
m$ is eventually strictly increasing, and hence ${\left\vert G^{(n)} \right\vert}$ is eventually *exponentially* increasing. Therefore the previous part of the theorem applies.
---
author:
- 'Peter Beelen[^1]'
title: 'A note on the generalized Hamming weights of Reed–Muller codes'
---
Preliminaries {#sec:in}
=============
Let ${\mathbb{F}_q}$ be the finite field with $q$ elements and denote by $\AA^m:=\AA^m({\mathbb{F}_q})$ the $m$-dimensional affine space defined over ${\mathbb{F}_q}$. This space consists of $q^m$ points $(a_1,\dots,a_m)$ with $a_1,\dots,a_m \in {\mathbb{F}_q}$. Let $T(m):={\mathbb{F}_q}[x_1, \dots , x_m]$ denote the ring of polynomials in $m$ variables and coefficients in ${\mathbb{F}_q}.$ Further let $T_{\le d}(m)$ be the set of polynomials in $T(m)$ of total degree at most $d$. A monomial $X_1^{\alpha_1}\cdots X_m^{\alpha_m}$ is called reduced if $(\alpha_1,\dots,\alpha_m) \in \{0,1,\dots,q-1\}^m$. Similarly a polynomial $f \in T(m)$ is called reduced if it is an ${\mathbb{F}_q}$-linear combination of reduced monomials. We denote the set of reduced polynomials by $T^{\mathrm{red}}(m)$ and define $T^{\mathrm{red}}_{\le d}(m):=^T_{\le d}(m) \cap T^{{\mathrm{red}}}(m)$.
One reason for considering reduced polynomials comes from coding theory. Indeed Reed–Muller codes are obtained by evaluating certain polynomials in the points of $\AA^m$, but the evaluation map $${\mathrm{Ev}}: T(m) \to {\mathbb{F}_q}^{q^m}, \ \makebox{defined by} \ {\mathrm{Ev}}(f)=(f(P))_{P \in \AA}$$ is not injective. However, its restriction to $T^{\mathrm{red}}(m)$ is. In fact the kernel of ${\mathrm{Ev}}$ consists precisely of the ideal $I \subset T(m)$ generated by the polynomials $x_i^q-x_i$ ($1 \le i \le m$). Working with reduced polynomials is simply a convenient way to take this into account, since for two reduced polynomials $f_1,f_2 \in T(m)$ the equality $f_1+I=f_2+I$ holds if and only if $f_1=f_2$.
The Reed–Muller code ${\mathrm{RM}_q}(d,m)$ is the set of vectors from ${\mathbb{F}_q}^{q^m}$ obtained by
| 2,076
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| 1,364
| 2,051
| 3,765
| 0.770208
|
github_plus_top10pct_by_avg
|
given coordinates using a convolution with a narrow gaussian, not by shifting the PSF by means of rebinning to a new position. The latter is better for the application to AGN decomposition and is now incorporated in later versions of [[galfit]{}]{}.
---
abstract: 'The high velocity gradient observed in the compact cloud CO-0.40-0.22, at a projected distance of 60 pc from the centre of the Milky Way, has led its discoverers to identify the closeby mm continuum emitter, CO-0.40-0.22\*, with an intermediate mass black hole (IMBH) candidate. We describe the interaction between CO-0.40-0.22 and the IMBH, by means of a simple analytical model and of hydrodynamical simulations. Through such calculation, we obtain a lower limit to the mass of CO-0.40-0.22\* of few $10^4 \times \; M_{\odot}$. This result tends to exclude the formation of such massive black hole in the proximity of the Galactic Centre. On the other hand, CO-0.40-0.22\* might have been brought to such distances in cosmological timescales, if it was born in a dark matter halo or globular cluster around the Milky Way.'
author:
- |
A. Ballone$^{1}$, M. Mapelli$^{1,2,3}$, M. Pasquato$^{1}$\
$^{1}$INAF, Osservatorio Astronomico di Padova, vicolo dell’Osservatorio 5, I-35122 Padova, Italy\
$^{2}$Institute für Astro- und Teilchen Physik, Universität Innsbruck, Technikerstrasse 25/8, A-6020 Innsbruck, Austria\
$^{3}$INFN, Milano Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
bibliography:
- 'mylit.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'Weighing the IMBH candidate CO-0.40-0.22\* in the Galactic Centre'
---
\[firstpage\]
black hole physics – Galaxy: centre – ISM: clouds
Introduction {#intro}
============
Intermediate-mass black holes (IMBHs), with masses $M_{BH}=10^2\div 10^5 \; M_{\odot}$, represent an “hollow” in the mass distribution of detected black holes. Yet, they might even be the missing link between stellar mass and supermassive black holes.
The LIGO and VIRGO interferometers have i
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|
valued in $({\bf 12},{\bf 4})$ of $so(12) \times so(4)$
$\overline{\partial} X^{3-4}_{-1} 16 scalars (toroidal moduli)
\otimes \left( \psi^{3-4}_{-1/2}, \overline{\psi}^{3-4}_{-1/2}
\right)$
$\left( \left( \lambda^{7-14}_{-1/2} \right)^2, 4 sets of scalars,
\left( \overline{\lambda}^{7-14}_{-1/2} \right)^2 \right)
\otimes \left( \psi^{3-4}_{-1/2}, \overline{\psi}^{3-4}_{-1/2}
\right)$
valued in $\wedge^2 {\bf 8} = {\bf 28}$, $\wedge^2 {\bf \overline{8}} = {\bf \overline{28}}$ of $su(8)$
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
There are no massless states in the untwisted (R,NS) sector, and in fact also no massless states in the $k=1$ or $k=3$ sectors.
All of the remaining massless states are in the $k=2$ sector. In the (NS,NS) sector, fields have the following boundary conditions: $$\begin{aligned}
X^{1-2}(\sigma + 2 \pi) & = & + X^{1-2}(\sigma), \\
X^{3-4}(\sigma + 2 \pi) & = & + X^{3-4}(\sigma), \\
\psi^{1-2}(\sigma + 2 \pi) & = & - \psi^{1-2}(\sigma), \\
\psi^{3-4}(\sigma + 2 \pi) & = & - \psi^{3-4}(\sigma), \\
\lambda^{1-6}(\sigma + 2 \pi) & = & - \lambda^{1-6}(\sigma), \\
\lambda^{7-14}(\sigma + 2 \pi) & = & - \exp\left( 2 \pi i
\frac{2}{4} \right) \lambda^{7-14}(\sigma) \: = \:
+ \lambda^{7-14}(\sigma), \\
\lambda^{15-16}(\sigma + 2 \pi) & = & - \lambda^{15-16}(\sigma).\end{aligned}$$ It is straightforward to compute that $E_{\rm left} = 0$, $E_{\rm right} =
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ed}
v_{T}^{(m\,h\,0)} &= c_1\,, &
v_{\Phi}^{(m\,h\,0)} &= c_2\,, &
v_{R}^{(m\,h\,0)} &= c_3\,, &\end{aligned}$$ and $$\begin{aligned}
v_{T}^{(m\,h\,1)} &= -2 [c_3+c_1 (h R T+i m)]\,, \\ {\nonumber}v_{\Phi}^{(m\,h\,1)} &= -2 c_2 (h R T+i m)\,, \\ {\nonumber}v_{R}^{(m\,h\,1)} &= -2 [c_3 (h R T+i m)+c_1-c_2]\,.\end{aligned}$$ Notice that there are three unknown coefficients $c_1$, $c_2$, and $c_3$. They endow us the freedom of choosing a 3-dimensional basis for covectors. In particular, we introduce a specific set of covector bases labeled by $B$ where $B\in\{T,\Phi,R\}$. They are defined by $$\begin{aligned}
\mathbf{V}_T^{(m\,h\,k)}&=\mathbf{V}^{(m\,h\,k)}|_{c_2=c_3=0}\,, \\ {\nonumber}\mathbf{V}_{\Phi}^{(m\,h\,k)}&=\mathbf{V}^{(m\,h\,k)}|_{c_1=c_3=0}\,, \\ {\nonumber}\mathbf{V}_R^{(m\,h\,k)}&=\mathbf{V}^{(m\,h\,k)}|_{c_1=c_2=0}\,.
\label{eq:specific-choice-of-vector}\end{aligned}$$
### Symmetric tensor bases {#app:tensor-basis-Poincare}
The symmetric tensor bases in Poincaré coordinates can be decomposed using the dual basis one-forms $\{\text{d}T,\text{d}\Phi,\text{d}R\}$ via $$\mathbf{W}^{(m\,h\,k)} = W^{(m\,h\,k)}_{ij}\,\text{d}x^i\otimes \text{d}x^j,\quad x\in\{T,\Phi,R\}\,.$$ The tensor components are given by $$\begin{aligned}
W_{ij}^{(m\,h\,k)} \propto
\left[
\begin{array}{ccc}
R^{+2}w^{(m\,h\,k)}_{TT}
& R^{+1} w^{(m\,h\,k)}_{T\Phi}
& R^{+0} w^{(m\,h\,k)}_{TR} \\
* & R^{+0} w^{(m\,h\,k)}_{\Phi\Phi}
& R^{-1} w^{(m\,h\,k)}_{R\Phi} \\
* &
* & R^{-2} w^{(m\,h\,k)}_{RR}
\end{array}
\right] \times {\nonumber}\\
\times R^{h-k} e^{im\Phi}\,,
\end{aligned}$$ where $$\begin{aligned}
w_{TT}^{(m\,h\,0)} &=c_1\,, &
w_{\Phi\Phi}^{(m\,h\,0)} &= c_2\,,&
w_{RR}^{(m\,h\,0)} &= c_3\,,\\ {\nonumber}w_{T\Phi}^{(m\,h\,0)} &= c_4\,, &
w_{\Phi R}^{(m\,h\,0)} &= c_5\,,&
w_{RT}^{(m\,h\,0)} &= c_6\,.\end{aligned}$$ Notice that there are six unknown $c$-coefficients. They endow us the freedom of choosing the six tensor bases. In particular, we introduce a specific set of highest-weight tensor bas
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-algebra generated by the $q$-th powers of all elements. The proof is by Noetherian induction.
First consider the case when $B$ is Artinian. The residue field of $B$ is finite and purely inseparable over the residue field of $B_1$, hence $B^q$ is contained in a field of representatives of $B_1$ for large enough $q$.
In the general, we can use the Artinian case over the generic points to obtain that $B_1\subset B_1B^q$ is an isomorphism at all generic points for $q\gg 1$. Let $I\subset B_1$ denote the conductor of this extension. That is, $IB_1B^q=I$. By induction we know that there is a $q'$ such that $(B_1B^q/I)^{q'}\subset B_1/I$. Thus we get that $$B^{(qq')}\to B^{qq'}\subset (B_1B^q)^{q'}\subset B_1+IB_1B^q=B_1.$$
Next consider $A\twoheadrightarrow B_1$. Here we have to make a good choice. The kernel is a nilpotent ideal $I\subset A$, say $I^m=0$. Choose $q'$ such that $q'\geq
m$. For $b_1\in B_1$ let $b_1'\in A$ be any preimage. Then $(b'_1)^{q'}$ depends only on $b_1$. The map $$b_1\mapsto (b'_1)^{q'}{\quad\mbox{defines a factorization}\quad}
B_1^{(q')}\to A\to B_1.$$ Combining the map $B^{(q)}\to B_1$ with $B_1^{(q')}\to A$ we obtain $B^{(qq')}\to A$.
The question is local on $S$, hence we may assume that $S$ is affine. $X$ and $R$ are defined over a finitely generated subring of ${{\mathcal O}}_S$, hence we may assume that $S$ is of finite type over ${{\mathbb F}}_p$.
The proof is by induction on $\dim X$. We follow the inductive plan in (\[induct.plan\]) and use its notation.
If $\dim X=0$ then $X$ is finite over $S$ and the assertion follows from (\[quot.X/S.finite.lem\]).
In going from dimension $d-1$ to $d$, the assumption (\[induct.plan\].2.1) holds by induction. Thus (\[induct.plan\].2.3) shows that $X^n/R^n$ exists.
Let $X^*\subset (X^n/R^n)\times_S X$ be the image of $X^n$ under the diagonal morphism. As we noted in (\[induct.plan\]), $X^*\to X$ is a finite and universal homeomorphism. Thus, by (\[factor.through.frob.prop\]), there is a factorization $$X^*\to X\to {X^*}^{(q)}\to \bigl(X^
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ave $Y^{q_0}-\alpha^{q_0}/\alpha Y$ is a factor of $\phi (f(x))$. Therefore $\alpha$ is the root of $\phi(f(x))$. Conversely, suppose $\alpha$ is the root of $\phi(f(x))$. Let $f(x)=k(x)(x-\alpha^{q_0}/\alpha)+r$, where $r\in \mathbb{F}_q$. Then $\phi(f(x))=\phi(k(x))\circ \phi(x-\alpha^{q_0}/\alpha)+\phi(r)$. From the discussion above, $\alpha$ is the root of $\phi(x-\alpha^{q_0}/\alpha)$. Therefore $\alpha$ is also the root of $\phi(r)$, i.e., $r\alpha=0$. Since $\alpha$ is a non-zero element in $\mathbb{F}_{q^s}$, we have $r=0$. This implies that $\alpha^{q_0}/\alpha$ is a right root of $f(x)$. $\Box$
Since $\phi(x^n-1)=Y^{q_0^n}-Y$ splits into linear factors in $\mathbb{F}_{q^s}$, it follows from Lemma 2.3 that $x^n-1$ also splits into linear factors in $\mathbb{F}_{q^s}[x, \sigma]$. It is well known that the non-zero roots of $Y^{q_0^n}-Y$ are precisely the elements of $\{1, \gamma, \ldots, \gamma^{q_0^n-2}\}$, where $\gamma$ is a primitive element of $\mathbb{F}_{q_0^n}$. Therefore, by Lemma 2.5, $x-(\gamma^i)^{q_0}/\gamma^i$ is the right factor of the skew polynomial $x^n-1$. It means that there are several different factorizations of the skew polynomial $x^n-1$.
In the following, we give the BCH-type bound for the skew cyclic code over $\mathbb{F}_q$.
[**Theorem 2.6** ]{} *Let $C$ be a skew cyclic code of length $n$ generated by a monic right factor $g(x)$ of the skew polynomial $x^n-1$ in $R$. If $x-\gamma^j$ is a right divisor of $g(x)$ for all $j=b, b+1, \ldots, b+\delta-2$, where $b\geq 0$ and $\delta \geq 1$, then the minimum Hamming distance of $C$ is at least $ \delta$.*
*Proof* Let $c(x)=\sum_{i=0}^{n-1}c_ix^i$ be a codeword of $C$. Then $c(x)$ is a left multiple of $g(x)$, and hence $x-\gamma^j$ is a right divisor of $c(x)$, for all $0\leq j \leq b+\delta-2$. From Lemma 2.4, $x-\gamma^j$ is a right divisor of $c(x)$ if and only if $\sum_{i=0}^{n-1}c_i{\mathcal N}_{\sigma, i}(\gamma^j)=0$, $j=b, b+1, \ldots, b+\delta-2$. Therefore the matrix $$\label{matrix}
\left(
\begin{array}{cccc}
1
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is expression allows us to interpret $\tilde{\phi}(p)$ as the Fourier modes of a scalar field $\phi(x)$ and $T(\teps, p)$ as the generator of a coordinate transformation with x\^ = \^[()]{} e\^[- i px]{}. That is, T(, p) = e\^[ip\_ x\^]{} \^[()]{}\_. Indices are contracted according to Einstein’s summation convention regardless of whether they are in parentheses or not. In this representation, it is clear that this algebra is that of space-time diffeomorphism. The gauge symmetry of GR arises as the most general gauge symmetry with Lie algebra of the form (\[ansatz\]), assuming that the structure constants are linear in momentum and that they respect Poincare symmetry.
A generic element of the Lie algebra is a superposition d\^D p \^[()]{}(p) T\_[()]{}(p) in $D$ dimensions, which can be written as T\_ \^[()]{}(x) \_, \[Teps\] where $\eps^{(\m)}(x)$ is the inverse Fourier transform of $\teps^{(m)}(p)$.
In view of this representation (\[Teps\]), it is tempting to interpret the algebra constructed above as merely the result of taking $T_a$’s to be derivatives $\del_a$’s in (\[factor\]) for a traditional gauge symmetry. But if we were really dealing with a traditional gauge symmetry, we would have obtained an Abelian gauge symmetry because $[T_a, T_b] = [\del_a, \del_b] = 0$. The need to generalize the notion of gauge symmetry here is due to the fact that $T_a = \del_a$ does not commute with space-time functions. Incidentally, the traditional interpretation of the torsion in teleparallel gravity is indeed the field strength of an Abelian gauge theory [@Cho:1975dh]. (See eq.(\[torsion-field-strength\]) below.)
Matter fields in the gauge theory are classified as representations of the gauge symmetry. Since the gauge symmetry under consideration is the diffeomorphism, we know all about other representations of different spins.
Gauge Field of Diffeomorphism {#YM}
=============================
Gauge Potential vs Vielbein {#vielbein}
---------------------------
We can define a gauge potential for the gauge symme
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dr]|{\ddots}&\ar@{}[dr]|{\ddots}&&&&&&\\
\ar@{}[dr]|{\ddots}&\Omega Ff\ar[r]\ar[d]\pullbackcorner&\Omega x\ar[r]\ar[d]\pullbackcorner&0\ar[d]&&&&&\\
&0\ar[r]&\Omega y\ar[r]\ar[d]\pushoutcorner\pullbackcorner&Ff\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[d]&&&&\\
&&0\ar[r]&x\ar[r]^-f\ar[d]\pushoutcorner\pullbackcorner&y\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[d]&&&\\
&&&0\ar[r]&Cf\ar[r]\ar[d]\pushoutcorner\pullbackcorner&\Sigma x\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[d]\ar@{}[rd]|{\ddots}&&\\
&&&&0\ar[r]\ar@{}[dr]|{\ddots}&\Sigma y\ar[r]\ar@{}[rd]|{\ddots}\pushoutcorner&\Sigma Cf\ar@{}[dr]|{\ddots}\pushoutcorner&\\
&&&&&&&&
}
}$$
(It turns out that $BP$ defines an equivalence of derivators (see [@gst:Dynkin-A Thm. 4.5]).)
Now, one half of the morphisms in the doubly-infinite chain simply amount to traveling in the Barratt–Puppe sequence in . If we imagine to sit on the morphism $f$ in $BP(f)$, then for every $n\geq 1$ an application of the $(2n\text{-}1)$-th left adjoint of $\pi^\ast$ to $f$ amounts to traveling $n$ steps in the positive direction. For low values this yields $y,Cf,\Sigma x,\Sigma y,$ and so on. There is a similar interpretation of the iterated right adjoints to $\pi^\ast$.
In order to obtain a similar visualization of the remaining adjoints, let us consider the Barratt–Puppe sequence $BP(\pi_{[1]}^\ast x), x\in{\sD}$, of a constant morphism which then looks like .
$$\vcenter{
\xymatrix@-1pc{
\ar@{}[dr]|{\ddots}&\ar@{}[dr]|{\ddots}&\ar@{}[dr]|{\ddots}&&&&&&\\
\ar@{}[dr]|{\ddots}&0\ar[r]\ar[d]\pullbackcorner&\Omega x\ar[r]\ar[d]\pullbackcorner&0\ar[d]&&&&&\\
&0\ar[r]&\Omega x\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[d]&&&&\\
&&0\ar[r]&x\ar[r]^-\id\ar[d]\pushoutcorner\pullbackcorner&x\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[d]&&&\\
&&&0\ar[r]&0\ar[r]\ar[d]\pushoutcorner\pullbackcorner&\Sigma x\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[d]\ar@{}[rd]|{\ddots}&&\\
&&&&0\ar[r]\ar@{}[dr]|{\ddots}&\Sigma x\ar[r]\ar@{}[rd]|{\
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ear train movement accomplished in one stage of operation becomes input to the next. A formalism called a process captures this notion of sequential inheritance. We assemble processes from a simple unit called the frame, which is two-part structure consisting of starting and ending conditions. A process is a sequence of frames such that the starting condition of each frame subsumes the ending condition of its predecessor frame. Interpreted in systems language, a frame’s starting condition is a stimulus and its ending condition is a response. Current response re-appears as part of future stimulus.
Definitions of *ensemble* and related basic concepts appear in Groundwork, Appendix \[S:ENSEMBLE\] ff.
\[D:BASIS\] The pair of ensembles $\langle \Psi, \Phi \rangle$ is a *basis* if $\Phi \subseteq \Psi$.
It is necessary to represent states (variables) which are used but not set – so-called [volatile]{} variables. For example, such variables can hold the transient values of sensors. The remainder $\Psi \setminus \Phi$ is the generating ensemble of volatile variables (see terminology following definition \[D:CHOICE\_SPACE\]).
\[D:FRAME\_SPACE\] The *frame space* ${\mathbf{F}}$ of basis $\langle \Psi, \Phi \rangle$ is the set ${\prod{\Psi}} \times {\prod{\Phi}}$. A member ${\mathbf{f}} \in {\mathbf{F}}$ is a *frame*.
Let ${\mathbf{f}} = (\psi, \phi) \in {\prod{\Psi}} \times {\prod{\Phi}}$ be a frame. The choice $\psi \in {\prod{\Psi}}$ is the frame’s starting condition (abscissa) and $\phi \in {\prod{\Phi}}$ is the frame’s ending condition (ordinate).
Two frames may be related such that the ending condition of one frame is embedded within the next frame’s starting condition. This stipulation is conveniently expressed as a mapping restriction:
\[D:CONJOINT\] Let $\langle \Psi, \Phi \rangle$ be a basis with frames ${\mathbf{f}} = (\psi, \phi)$, ${\mathbf{f}}\,' = (\psi', \phi') \in {\prod{\Psi}} \times {\prod{\Phi}}$. Frame ${\mathbf{f}}$ *conjoins* frame ${\mathbf{f}}\,'$ if ${{\psi'}\negmedspace\mid\negmed
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m99 (5.3.1)].
Kronecker Coefficients {#section:kronecker}
======================
As explained in the introduction, the Kronecker coefficients play an important role in geometric complexity theory and quantum information theory. In this section, we will describe precisely how they can be computed using our methods.
Let us recall the language of Young diagrams which is commonly used in this context [@fulton97]. A *Young diagram* with $r$ rows and $k$ boxes is given by an ordered list of integers $\lambda_1 \geq \ldots \geq \lambda_r > 0$ with $\sum_i \lambda_i = k$. It can be visualized as an arrangement of $k$ boxes in $r$ rows with $\lambda_j$ boxes in the $j$-th row. We set $\lambda_j = 0$ for all $j > r$. We will now consider the *unitary group* $\operatorname{U}(d)$, which consists of the unitary $d \times d$-matrices. Let us fix a system of positive roots and denote the corresponding basis of fundamental weights by $(\omega_j)$. To each Young diagram $\lambda$ with at most $d$ rows we associate the irreducible representation of $\operatorname{U}(d)$ with highest weight equal to $\sum_{j=1}^d \left( \lambda_j - \lambda_{j+1} \right) \omega_j$. Every polynomial irreducible representation of $\operatorname{U}(d)$ arises in this way. By a slight abuse of notation, we identify Young diagrams with the corresponding highest weights. More generally, we can associate to every integer vector $\beta \in {\mathbb Z}^d$ the weight $\sum_{j=1}^d \left( \beta_j - \beta_{j+1} \right) \omega_j$, where we set $\beta_{d+1} = 0$. This defines a bijection between ${\mathbb Z}^d$ and the weight lattice $\Lambda^*_{\operatorname{U}(d)}$ of $\operatorname{U}(d)$. In particular, the positive roots fixed above correspond to the integer vectors of the form $(\ldots,0,1,0,\ldots,0,-1,0,\ldots)$.
The *Kronecker coefficient* $g_{\lambda,\mu,\nu}$ associated with triples of Young diagrams $\lambda$, $\mu$ and $\nu$ with $k$ boxes each and at most $a$, $b$ and $c$ rows, respectively, can then be defined in terms of the following subgrou
| 2,085
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12} x_j \mathcal{M}_j \, .\end{aligned}$$ We derive its associated equation of motion and see what a simultaneous cancellation of all the higher order terms implies for the coefficients $(v_i , x_j)$ : we find that the unique linear combination of order 8 scalars for FLRW space-time that leads to second order equation is : $$\begin{aligned}
J_{4}=\mathcal{K}_{1} -48 \, \mathcal{K}_{11} -9 \, \mathcal{K}_{12} =1728 \, \Big( \, H^8 +2 \dot{H}H^6 \, \Big).\end{aligned}$$ Therefore, one may consider the action : $$\begin{aligned}
\begin{split}
S_{3} = \int d^4x \sqrt{-g} \; \Bigg( \frac{1}{16 \pi}\bigg[ R +\nu \Big( R^4 -48 \, R\,R^{\mu\nu\alpha\beta}R_{\mu\;\,\alpha}^{\;\,\sigma\;\,\rho}R_{\nu\sigma\beta\rho} -9 \, \big( R^{\mu\nu\alpha\beta} R_{\mu\nu\alpha\beta} \big)^2 \Big) \;\bigg] + \curv{L}_m \Bigg) ,
\end{split}\end{aligned}$$ and see that it brings an $H^8$ correction to the Friedmann equation. We note that this correction involves only contraction of curvature tensors like for the order 6 case.
### Non-polynomial gravity. $H^4$ correction.
#### Correction to the Einstein-Hilbert action
\
To find non-polynomial second order models from order 8 scalars, we follow exactly what we did for the order 6 and find the same kind of result : there are only two classes of perfect squares in this case. Those for which the square-roots give topological scalars, that does not give any contribution to the equation of motion, and a class of equivalent scalars with respect to the equation of motion, up to the topological scalars of the first class. Therefore in this case also, we can consider a unique perfect square that contributes to the dynamics.
To begin, the more general square made of order 8 scalars has the form : $$\begin{aligned}
\big( \alpha H(t)^4+\beta H(t)^2 \dot{H}(t)+\gamma \dot{H}(t)^2+\delta H(t)
\ddot{H}(t)+\sigma H^{(3)}(t) \big)^2.\end{aligned}$$ And its square-root gives the following equation of motion: $$\begin{aligned}
\begin{split}
~& 3 \big( \alpha -\beta +\gamma +2 \delta
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all when $|\lambda-\lambda'|<\delta$ and $O_c$ is small enough, independently on the value of $\lambda'(\Lambda)$. This implies in particular that, by making $O'$ and consequently $O_c$ and $O$ small enough one may chose $\delta \theta_m$ such that $$|\delta \theta_m|< \frac{R_m^2\epsilon}{(\lambda_1-\lambda_i)|2\theta_m+2c|}.$$ in $O$, which implies (\[socia\]) that |-| , for all $\Lambda \in O$. This prove that the left side of (\[artdeco\]) can be made arbitrarily small.
On the other hand, the absolute value of the sum of those terms of the right side of (\[artdeco\]) which are independent on $\Delta \lambda_e$ can be converted by use of a mean value theorem into $$I=\bigg|\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_\Lambda(\beta)}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]\bigg\}^2e^{c\beta}d\beta$$ $$-\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_0(\beta)}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_0(\xi) \bigg]\bigg\}^2e^{c\beta}d\beta\bigg|$$ $$=\bigg|(\lambda_e-\lambda_1)\bigg\{1+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]\bigg\}^2e^{c\lambda'}$$ $$-(\lambda_e-\lambda_1)\bigg\{1+\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}^2e^{c\lambda'}\bigg|$$ $$=(\lambda_e-\lambda_1)e^{2c\lambda'}\bigg|\bigg\{\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$-\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}$$ $$\times \bigg\{2+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac
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(\[angtransitionrate\]) and (\[Wfinal\]) is to be valid for detector trajectories not involving casual horizons.
On the other hand, If the support of the detector’s profile is contained in the Rindler wedge, then the corresponding transition rate is KMS in the usual Unruh temperature of the Rindler trajectory as is evident from the results of the second model of the detector defined in the Rindler wedge. This is regardless whether the peak of the spatial profile chosen coincides with the reference trajectory of the detector. The underlying reason is that the monopole interaction defines the energy gap of the detector with respect to the proper time of the reference trajectory, even when the proper time at the peak of the profile may be quite different from the proper time at the reference trajectory, which is the Rindler trajectory in the present case. One could question whether this way to define the interaction is a reasonable model of the microphysics of an extended body. When the profile has a peak, perhaps a more reasonable model would be to choose the reference trajectory to coincide with the peak of the profile. Thus, based on the second model defined in the Rindler wedge, we conclude that the Unruh effect is directionally isotropic with the usual Unruh temperature for spatially extended direction dependent detectors.
Nevertheless, there could be some interest in getting a more quantitative control of what happens for the Schlicht type detector model when the profile leaks outside the Rindler wedge, beyond Schlicht’s Lorentz-function profile. Schlicht’s profile gives KMS spectrum for the case discussed in the section \[schlichtsection\] but at a different temperature; greater than the usual Unruh temperature. One could question whether other leaking profiles still give KMS (at some temperature) or does any deviation from Schlicht’s profile necessarily break KMS.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Jorma Louko for helpful discussions and useful comments on the
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N=1+[X]$. At higher temperatures $T>T_n$ the composite field $\Psi$ becomes disordered.
-8mm
-5mm
Dual representation {#sec:dual}
-------------------
The partition function $Z$ can be evaluated by the high-temperature expansion method (see e.g. in [@Parisi]) in terms of $t_1,t_2,u$ and the explicit integration over the variables. This approach allows obtaining $Z$ in terms of the integer bond variables – powers of the corresponding Taylor series. Since the resulting configurational space consists of closed loops of the bond currents, further simulations can be effectively performed by the Worm Algorithm [@WA]. As will be also shown, the language of loops also allows obtaining analytic expressions for the renormalized Josephson coupling $u_r$ which are exact in the asymptotic limit.
We will be utilizing the Villain approximation [@Villain] for the cosines to obtain the so called J-current version [@Jcurr] of Eqs.(\[ZZ2\]),(\[2N\]): Z=\_[{m\_[a,ij]{}, m\_i}]{} DDA [e]{}\^[-H\_V]{}, \[Vill\] H\_V&=& \_[ij]{} \[ (\_[ij]{} \_1 - A\_[ij]{} +2m\_[1,ij]{})\^2\
&+& (\_[ij]{} \_2 -g\_2 A\_[ij]{} +2m\_[1,ij]{} )\^2 + A\^2\_[ij]{}\]\
&+& \_i (\_2(i)-\_1(i) +2m\_i)\^2, \[2Nv\] where $m_{a,ij}= - m_{a,ji}=0,\pm 1, \pm 2,...$ ($a=1,2$) are integer numbers defined along bonds between two nearest sites $i$ and $j$ along the planes and $m_i=0,\pm 1, \pm2, ...$ is an integer assigned to a site $i$ and oriented from the layer 1 to the layer 2.
The Villain approximation proves to be very accurate for establishing the transition points as well as in general if the effective constants $\tilde{t}_1,\tilde{t}_2, u_V$ are properly expressed in terms of the corresponding bare values $t_1,t_2,u$ (see in Ref.[@Kleinert]). The “renormalization” can be essentially ignored for $t_1,t_2 \geq 1$, so that in what follows we will be using $\tilde{t}_1=t_1,\, \tilde{t}_2=t_2$. Similarly, for the Josephson coupling $u \sim 1$ one should take $u_V=u$ and, if $u<<1$, the corresponding relation is $u_V=1/(2 \ln (2/u))$ [@Villain; @Kleinert].
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(0,2) SCFT’s. Although it seems there are new consistent (0,2) SCFT’s, we will argue that they do not seem to define new consistent supersymmetric heterotic string compactifications.
In hindsight, we can understand that result as follows[^5]. In an ineffective orbifold (one in which part of the orbifold group acts trivially on the space), the twisted sectors contain massless states whose wavefunctions have support over the entire space. This would seem to imply that there are ‘extra’ ten-dimensional massless states, but this would be a contradiction, since the ten-dimensional supergravity theory is known and fixed. Furthermore, so long as we work at low energies and close to large-radius limits, a ten-dimensional supergravity analysis should be applicable.
In type II strings, this conundrum was implicity solved by the decomposition conjecture [@summ]: strings on gerbes are the same as strings on disjoint unions of spaces. The ‘extra’ states are there, but simply fill out copies of the supergravity theory.
In heterotic strings, we will see a mix of several solutions: in some cases, an analogue of the decomposition conjecture exists; in other cases, the theory is dual to a compactification on a manifold; in yet other cases, the compactification does not seem to be consistent.
Generalities {#sect:generalreview}
============
Strings on stacks
-----------------
Stacks are a form of ‘generalized spaces,’ admitting smooth structures, metrics, bundles, and other structures needed to define sigma models. In particular, stacks are defined by the incoming maps from other spaces, making them a natural setting for defining sigma models.
Stacks have been discussed as target ‘spaces’ for nonlinear sigma models in a number of references, including[^6] [@kps; @nr; @msx; @glsm; @summ; @cdhps; @karp1; @karp2; @ps5; @me-tex; @me-qts] for two-dimensional (2,2) supersymmetric and [@git-sugrav; @sugrav-g] for four-dimensional ${\cal N}=1$ supersymmetric sigma models. References for physicists on the mathemat
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gathered}
C_j:={1\over 2}\kappa_j^{-2}{\left\Vert \nabla_x S_j\right\Vert}_{L^\infty(G\times I)}
\\
+\kappa_j^{-1}\Big({\left\Vert \Sigma_j\right\Vert}_{L^\infty(G\times S\times I)}
+{\left\Vert {{\frac{\partial S_j}{\partial E}}}\right\Vert}_{L^\infty(G\times I)}+\sqrt{M_1M_1'}
+{{M_1M_1'}\over c}\Big).\end{gathered}$$ Replacing $\hat\psi$ with $\hat\phi(x,\omega,E):=e^{-CE}\hat\psi(x,\omega,E_{\rm m}-E)$ (as in section \[evcsd\]) we find that the system (\[pr5.5.7\]) is equivalent to \[proof7\] [E]{}-[**A**]{}\_C(E)=F(E),(0)=0, where $$\hat{\phi}&=(\phi_2,\phi_3), \\[2mm]
{\bf A}_C(E)\hat\phi&=({A}_{C,2}(E)\hat\phi,{A}_{C,3}(E)\hat\phi), \\[2mm]
F&=(F_2,F_3),$$ and for $j=2,3$, $$F_j(E):={1\over{\tilde S_j(E)}}e^{-CE}\big(\tilde f_j(x,\omega,E)+\tilde{\hat f}_j(x,\omega,E)\big),$$ and $$\begin{gathered}
\label{proof9}
A_{C,j}(E)\hat\phi
:=-\Big({1\over {\tilde S_j(E)}}\omega\cdot\nabla_x\phi_j+C \phi_j+{1\over {\tilde S_j(E)}}\tilde\Sigma_j(E)\phi_j
+{1\over{\tilde S_j(E)}}{{\frac{\partial \tilde S_j}{\partial E}}}(E)\phi_j
\\
-{1\over{\tilde S_j(E)}} \tilde{\hat{K}}_j(E)\hat\phi
-{1\over{\tilde S_j(E)}} \tilde Q_{j}(E)\hat\phi\Big).\end{gathered}$$ Here $\tilde S_j(x,E):=S_j(x,E_m-E)$ and similarly for other expressions equipped with “tilde”.
Considering ${\bf A}_C(E)$ as an (unbounded) operator $L^2(G\times S)^2\to L^2(G\times S)^2$ with domain $$D({\bf A}_C(E))=\tilde W^2_{-,0}(G\times S)\times \tilde W^2_{-,0}(G\times S),$$ we get by applying Theorem \[evoth\] along with computations analogous to the ones done in section \[evcsd\], that the system (\[proof7\]) has a unique solution $\hat{\phi}\in C(I,\tilde W^2_{-,0}(G\times S)^2)\cap C^1(I,L^2(G\times S)^2)$, which satisfies the homogeneous boundary and initial conditions, $\hat{\phi}_{|\Gamma_-}=0$, $\hat{\phi}(\cdot,\cdot,0)=0$.
Then $\hat{\psi}(x,\omega,E)=e^{C(E_{\rm m}-E)}\hat\phi(x,\omega,E_{\rm m}-E)$, and $\psi_1$ is obtained from (\[psi1\]) that is, $\psi_1=T_{1,0}^{-1}(f_1+\ol K\hat\psi)$, giving us the solution $\psi=(\psi_1,\psi_2,\psi_3
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_{n\geq 1}\frac{\psi_n(f)}{n},$$ and $\Exp:\Lambda_k[[T]]^+\rightarrow 1+\Lambda_k[[T]]^+$ by
$$\Exp(f)=\exp(\Psi(f)).$$
The inverse $\Psi^{-1}:\Lambda_k[[T]]^+\rightarrow\Lambda_k[[T]]^+$ of $\Psi$ is given by
$$\Psi^{-1}(f)=\sum_{n\geq 1}\mu(n)\frac{\psi_n(f)}{n}$$where $\mu$ is the ordinary Möbius function.
The inverse $\Log:1+\Lambda_k[[T]]\rightarrow\Lambda_k[[T]]$ of $\Exp$ is given by
$$\Log(f)=\Psi^{-1}(\log(f)).$$
Let $f=1+\sum_{n\geq 1}f_nT^n\in 1+\Lambda_k[[T]]^+$. If we write
$$\log\,(f)=\sum_{n\geq 1}\frac{1}{n}U_nT^n,\hspace{1cm}\Log\,(f)=\sum_{n\geq 1}V_nT^n,$$ then
$$V_r=\frac{1}{r}\sum_{d | r}\mu(d)\psi_d(U_{r/d}).$$
We have the following propositions (details may be found for instance in Mozgovoy [@mozgovoy]).
For $g\in \Lambda_k$ and $n\geq 1$ we put
$$g_n:=\frac{1}{n}\sum_{d | n}\mu(d)\psi_{\frac{n}{d}}(g).$$ This is the Möbius inversion formula of $\psi_n(g)=\sum_{d| n}d\cdot g_d$.
Let $g\in \Lambda_k$ and $f_1,f_2\in 1+\Lambda_k[[T]]^+$ such that
$$\log\,(f_1)=\sum_{d=1}^\infty g_d\cdot\log\,(\psi_d(f_2)).$$ Then $$\Log\,(f_1)=g\cdot \Log\,(f_2).$$
\[moz\]
Assume that $f\in 1+\Lambda_k[[T]]^+$ belongs to $\Lambda(\x_1,\ldots,\x_k)\otimes_\Z\Z[q,t]$, then $\Exp(f)\in
\Lambda(\x_1,\ldots,\x_k)\otimes_\Z\Z[q,t]$.
\[Exp\]
### Types
We choose once and for all a total ordering $\geq$ on $\calP$ (e.g. the lexicographic ordering) and we continue to denote by $\geq$ the total ordering defined on the set of pairs $\N^*\times\calP^*$ as follows: If $\lambda\neq\mu$ and $\lambda\geq\mu$, then $(d,\lambda)\geq(d',\mu)$, and $(d,\lambda)\geq(d',\lambda)$ if $d\geq
d'$. We denote by $\mathbf{T}$ the set of non-increasing sequences $\omega=(d_1,\omega^1)\geq (d_2,\omega^2)\geq \cdots \geq
(d_r,\omega^r)$, which we will call a [*type*]{}. To alleviate the notation we will then omitt the symbol $\geq$ and write simply $\omega=(d_1,\omega^1)(d_2,\omega^2)\cdots (d_r,\omega^r)$. The [ *size*]{} of a type $\omega$ is $|\omega|:=\sum_id_i|\lambda^i|$. We denote by $\mathbf{T}_n$ the set of t
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omic levels. It can be referred to as a $m$-phonon Jaynes-Cummings (JC) model. On the other hand, Hamiltonian $\hat{\mathcal{H}}^{(m)}_{-} $ is obtained with $\omega_0 - \omega_L= - m \nu $ and it can be referred to as a $m$-phonon anti-Jaynes-Cummings (AJC) model. The case $m=0$ can be studied using either $\hat{\mathcal{H}}^{(m)}_{+} $ or $\hat{\mathcal{H}}^{(m)}_{-} $, $$\label{hamct}
\!\!\hat{\mathcal{H}}^{(0)}=\hat{\mathcal{H}}^{(0)}_{\pm} =
\hat{\mathcal{H}}_{0} +
\frac{\hbar}{2} \left(\! \text{e}^{-i\omega_{L} t }\hat\Omega_0^+ \hat{\sigma}_{+} +
\text{e}^{ i\omega_{L} t } \hat\Omega_0^-\hat{\sigma}_{-}\! \right),$$ and it describes Rabi oscillations between electronic levels, i.e., the carrier transitions [@leibfried; @orszag].
For what comes next, it is useful to present now the matrix elements of $\hat{\Omega}_{m}^\pm$ in the Fock basis of the CM harmonic motion $$\begin{aligned}
\label{Omegme}
\left\langle n\right|\hat{\Omega}_{m}^+\left|n'\right\rangle &=&
\left\langle n'\right|\hat{\Omega}_{m}^{-}\left|n\right\rangle^\ast \\
&=& \frac{\Omega(i\eta)}{2}^{\!\!^m}
\sqrt{ \frac{ n!}{(m+n)!} } \, \text{e}^{-\eta^{2}/2}
L_{n}^{m}\!\left(\eta^{2}\right) \delta_{n'\,n+m}, \nonumber \end{aligned}$$ with the associated Laguerre polynomials [@gradshteyn] $$L_{n}^{m}\left( x \right) =
\sum_{k=0}^{n}\left(-1\right)^{k}
\frac{(n + m)!}{(m + k)!(n-k)!}
\frac{x^{k}}{k!} .$$ As it can be seen from Eq. (\[Omegme\]), the quantum Rabi frequencies, $\left\langle n\right|\hat{\Omega}_{m}^+\left|n'\right\rangle$, have a strong dependence on the Lamb-Dicke parameter $\eta$. For small values of $\eta$, they present a quasilinear dependence on $n$, typical of $m$-photon Jaynes-Cummings models in the context of cavity quantum electrodynamics (cQED) [@multi]. However, for the ionic system, it is possible to induce considerable nonlinearities in
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number \\
& 6R^{[ab|e|}Q^{cd]}_e+{F}_a P^{a,abcd}+{F}_b P^{b,abcd}+{F}_c P^{c,abcd}+{F}_d P^{d,abcd}+\tfrac{1}{2}P^{a,bcdaef}{F}_{aef}\nonumber \\
& \qquad \qquad +\tfrac{1}{2}P^{b,bcdaef}{F}_{bef}+\tfrac{1}{2}P^{c,bcdaef}{F}_{cef}+\tfrac{1}{2}P^{d,bcdaef}{F}_{def} =0 \nonumber \end{aligned}$$ in the IIB case, and $$\begin{aligned}
& 6f^{e}_{[ab}H_{cd]e}+4P^e_{[a}F_{bcd]e}=0 \nonumber \\
& 3Q^{ae}_{[b}H_{cd]e}+3f^{e}_{[bc}f^{a}_{d]e}-3P^{a}_{[b}F_{cd]}-P^{a,a}F_{bcda}+\tfrac{1}{2}P^{a,aef}F_{bcdaef}+\tfrac{3}{2} P^{aef}_{[b}F_{cd]ef}=0 \nonumber \\
&-Q^{ab}_e f^e_{cd}-4Q^{[a|e|}_{[c}f^{b]}_{d]e}-R^{abe}H_{cde}
+2P^{abe}_{[c}{F}_{d]e}-P^{a,abe}{F}_{cdae}-P^{b,abe}{F}_{cdbe} \nonumber \\
& \qquad \qquad -\tfrac{1}{3} P^{abefg}_{[c} F_{d]efg} + \tfrac{1}{6} P^{a,abefg} F_{cdaefg} +\tfrac{1}{6} P^{b,abefg} F_{cdbefg} =0 \label{NSNSBianchiIIAwithP} \\
& 3R^{[ab|e|}f^{c]}_{de}+3Q^{[ab}_eQ^{c]e}_d+ {F} P^{abc}_d-P^{a,abc}{F}_{ad}-P^{b,abc}{F}_{bd}-P^{c,abc}{F}_{cd} \nonumber \\
& \qquad \qquad -\tfrac{1}{2} P^{abcef}_d {F}_{ef}- \tfrac{1}{2} P^{a,bcaef}{F}_{daef}- \tfrac{1}{2} P^{b,bcaef}{F}_{dbef}- \tfrac{1}{2} P^{c,bcaef}{F}_{dcef}=0\nonumber \\
& 6R^{[ab|e|}Q^{cd]}_e-{F}_{de}P^{d,abcde}-{F}_{ce}P^{c,abcde}-{F}_{be}P^{b,abcde}-{F}_{ae}P^{a,abcde}=0 \nonumber \end{aligned}$$ in the IIA case.
The NS-NS quadratic constraints can be relaxed by the inclusion of sources [@Villadoro:2007tb; @andriot]. This obviously also applies to the constraints modified by the inclusion of $P$ fluxes in eqs. and . In particular, relaxing the first constraints (that we schematically write as $( {\rm flux} \cdot {\rm flux} )_4 =0$) in both equations induces a charge for the NS5-brane coming from the generalised Chern-Simons term $$\int D_6 \wedge ( {\rm flux} \cdot {\rm flux} )_4 \quad .$$ As we have discussed in section 3, if we now consider a particular component for $D_6$, and we perform a T-duality $T_a$ in a direction whose index $a$ is not contained in $D_6$, this is mapped to $D_{6 \, a,a}$ which is a component of the mixed-sy
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{{\cal C}}_{{\bf n}}^{b_{i+1}}(y)\setminus{{\cal C}}_{{\bf n}}^{b_i}(y)&(i=1,\dots,T-1),\\
{{\cal C}}_{{\bf n}}(y)\setminus{{\cal C}}_{{\bf n}}^{b_T}(y)&(i=T).
\end{cases}\end{aligned}$$ As in Figure \[fig:lace-edges\], we can think of ${{\cal C}}_{{\bf n}}(y)$ as the interval $[0,T]$, where each integer $i\in[0,T]$ corresponds to ${{\cal D}}_{{{\bf n}};i}$ and the unit interval $(i-1,i)\subset[0,T]$ corresponds to the pivotal bond $b_i$. Since $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x$, we see that, for every $b_i$, there must be an $({{\bf m}}+{{\bf n}})$-bypath (i.e., an $({{\bf m}}+{{\bf n}})$-connection that does not go through $b_i$) from some $z\in{{\cal D}}_{{{\bf n}};s}$ with $s<i$ to some $z'\in{{\cal D}}_{{{\bf n}};t}$ with $t\ge i$. We abbreviate $\{s,t\}$ to $st$ if there is no confusion. Let ${{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(1)}=\{\{0T\}\}$, ${{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(2)}=\{\{0t_1,s_2T\}:0<s_2\leq t_1<T\}$ and generally for $j\leq T$ (see Figure \[fig:lace-edges\]),
![\[fig:lace-edges\]An element in ${{\cal L}}_{[0,8]}^{{\scriptscriptstyle}(4)}$, which consists of $s_1t_1=\{0,3\}$, $s_2t_2=\{2,4\}$, $s_3t_3=\{4,6\}$ and $s_4t_4=\{5,8\}$.](lace-edges)
$$\begin{aligned}
{{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(j)}=\big\{\{s_it_i\}_{i=1}^j:0=s_1<s_2\leq t_1<s_3
\leq\cdots\leq t_{j-2}<s_j\leq t_{j-1}<t_j=T\big\}.\end{aligned}$$
For every $j\in\{1,\dots,T\}$, we have $\bigcup_{st\in\Gamma}[s,t]=
[0,T]$ for any $\Gamma\in{{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(j)}$, which implies double connection. Conditioning on ${{\cal C}}_{{\bf n}}(y)\equiv\bigcup_{i=0}^{
\raisebox{-3pt}{$\scriptstyle T$}}{{\cal D}}_{{{\bf n}};i}={{\cal B}}$ (and denoting ${{\bf k}}={{\bf n}}|_{{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$, ${{\bf h}}={{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$ and ${{\cal D}}_{{{\bf n}};i}\equiv{{\cal D}}_{{{\bf h}};i}={{\cal B}}_i$) and multiplying by $Z_{{{\cal B}}{^{\rm
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_\muhat G_\muhat(q)m_\muhat&=\sum_{\alpha\in \calP^{{\bf F}^{\times}}}\calH_{\omega(\alpha)}(0,\sqrt{q})\prod_{i=1}^k\tilde{H}_{\omega(\alpha)}(\x_i,q)\\
&=\prod_{c\in{\bf F}^{\times}}\Omega\left(\x_1^{d(c)},\dots,\x_k^{d(c)};0,q^{d(c)/2}\right)\\
&=\prod_{d=1}^\infty\Omega\left(\x_1^d,\dots,\x_k^d;0,q^{d/2}\right)^{\phi_d(q)}\end{aligned}$$
The second formula displayed in the proof of Proposition \[sumM\] shows that
$$G_\muhat(q)=\left\langle\Lambda\otimes
R_\muhat(1),1\right\rangle$$where $R_\muhat(1):=R_{L_{\mu^1}}^G(1)\otimes\cdots\otimes
R_{L_{\mu^k}}^G(1)$. \[rem326\]
We have $$A_\muhat(q)=\H_\muhat(0,\sqrt{q}).$$ \[purity\]
From Formula (\[exp\]) we have
$$\sum_\muhat\H_\muhat(0,\sqrt{q})\,m_\muhat=(q-1)\, \Log\,\left(\Omega(0,\sqrt{q})\right).$$ We thus need to see that \_A\_(q)m\_=(q-1) ((0,)). \[ExpA\]
From Theorem \[MA\] we are reduced to prove that
$$\Log\left(\sum_\muhat G_\muhat(q)m_\muhat\right)=(q-1)\Log\left(\Omega(0,\sqrt{q})\right).$$
But this follows from Lemma \[moz\] and Proposition \[sumM\].
Another formula for the $A$-polynomial
--------------------------------------
When the dimension vector $\v_\muhat$ is indivisible, it is known by Crawley-Boevey and van den Bergh [@crawley-boevey-etal] that the polynomial $A_\muhat(q)$ equals (up to some power of $q$) to the polynomial which counts the number of points of some quiver variety over $\F_q$.
Here we prove some relation between $A_\muhat(q)$ and some variety which is closely related to quiver varieties. This relation holds for any $\muhat$ (in particular $\v_\muhat$ can be divisible).
We continue to use the notation $G$, $P_\lambda$, $L_\lambda$, $U_\lambda$, $\calF_\lambda$ of §\[finite-groups\] and the notation $\mathfrak{g}$, $\mathfrak{p}_\lambda$, $\mathfrak{l}_\lambda$, $\mathfrak{u}_\lambda$ of §\[Fourier\].
For a partition $\lambda$ of $n$, define $$\X_\lambda:=\left\{(X,gP_\lambda)\in \mathfrak{g}\times(G/P_\lambda)\,\left|\, g^{-1}Xg\in\mathfrak{u}_\lambda\right\}\right.$$It is well-known that the image of the pro
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]$. Since $z', t'$ are not in $P'$ and they are disjoint from $a'$ and $x'$, they both have to lie in the subsurface, $M$, which is a projective plane with two boundary components containing $y'$, and having $x'$ as a boundary component as shown in the figure. This gives a contradiction as there are no two nontrivial simple closed curves with distinct isotopy classes such that each of them intersects $y'$ essentially in $M$, (see [@Sc]).
=3.7in =3.7in
Case (iii): Suppose that $n > 4$. The argument is similar. We complete $a$ to a pair of pants decomposition $P= \{a, x_1, \cdots, x_{n-2}\}$ as shown in Figure \[Fig2\] (i). $P$ corresponds to a top dimensional maximal simplex. Let $P'$ be a set of pairwise disjoint elements of $\lambda([P])$ containing $a'$. Let $x_1' \in \lambda([x_1]) \cap P'$. Since $a$ is adjacent to only $x_1$ w.r.t. $P$, by Lemma \[adjacent\] and Lemma \[nonadjacent\], $a'$ should be adjacent to only $x_1'$ w.r.t $P'$. Since $g=1$, $n > 4$, $a'$ is separating and $a'$ is adjacent to only $x_1'$ w.r.t $P'$, we see that there is a subsurface $T \subseteq N$ such that it is homeomorphic to sphere with four boundary components where three of the boundary components of $T$ are the boundary components of $N$, $x_1'$ is a boundary component on $T$, and $a'$ divides $T$ into two pair of pants as shown in Figure \[Fig2\] (ii). Let $x_i' \in \lambda([x_i]) \cap P'$ for each $i= 2, \cdots, n-2$. Since $x_1$ is adjacent to only $x_2$, $x_1'$ is adjacent to only $x_2'$. Similar to the previous case using that adjacency and nonadjacency are preserved, $n > 4$ and $g=1$, we have elements of $P'$ and $N$ as shown in Figure \[Fig2\] (ii). But this will give us a contradiction as in the previous case (consider the curves $z, t$ as shown in Figure \[Fig2\] (i)).
We showed that $a'$ cannot be a 2-sided simple closed curve on $N$. Hence, $a'$ is a 1-sided simple closed curve on $N$.
=2.5in
\[1-sided-cn\] Let $g \geq 2$. Suppose that $(g, n) = (3, 0)$ or $g+n \geq 4$. Let $\lambda : \mathcal{C}(N) \rightarrow
| 2,097
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$\epsilon_n $ is positive and to replace $\epsilon$ with $\epsilon_n$. The price for this extra step is a factor of $\frac{1}{n}$, which upper bounds $\mathbb{P}(\mathcal{E}_n^c)$. Putting all the pieced together we arrive at the bound $$A_3 \leq C \mathrm{E}^*_{1,n} + \frac{1}{n}.$$ Finally notice that $\mathrm{E}_{1,n} \leq \mathrm{E}^*_{1,n}$ since $\epsilon_n \leq \epsilon$.
The very same arguments apply to the other bootstrap confidence set $\tilde{C}^*_\alpha$, producing the very same bound. We omit the proof for brevity but refer the reader to the proof of for details.
All the bounds obtained so far are conditionally on the outcome of the sample splitting and on $\mathcal{D}_{1,n}$ but are not functions of those random variables. Thus, the same bounds hold also unconditionally, for each $P \in \mathcal{P}_n^{\mathrm{LOCO}}$.
$\Box$
Let $F_{n,j}$ denote the empirical cumulative distribution function of $\{
\delta_i(j), i \in \mathcal{I}_{2,n}\}$ and $F_j$ the true cumulative distribution function of $\delta_i(j)$. Thus, setting $\beta_l = l/n$ and $\beta_u = u/n$, we see that $\delta_{(l)}(j) = F_{n,j}^{-1}(\beta_l)$ and $\delta_{(u)}(j) =
F_{n,j}^{-1}(\beta_u)$ and, furthermore, that $F_{n,j}(F_{n,j}^{-1}(\beta_l)) = \beta_l$ and $F_{n,j}F(_{n,j}^{-1}(\beta_u)) = \beta_u$. In particular notice that $\beta_l$ is smaller than $ \frac{1}{2} -
\sqrt{\frac{1}{2n}\log\left(\frac{2k}{\alpha}\right)}$ by at most $1/n$ and, similarly, $\beta_u$ is larger than $ \frac{1}{2} +
\sqrt{\frac{1}{2n}\log\left(\frac{2k}{\alpha}\right)}$ by at most $1/n$. By assumption, the median $\mu_j = F_j^{-1}(1/2)$ of $\delta_i(j)$ is unique and the derivative of $F_j$ is larger than $M$ at all points within a distance of $\eta$ from $\mu_j$. Thus, by the mean value theorem, we must have that, for all $x \in \mathbb{R}$ such that $| x -\mu_j | < \eta$, $$M |x - \mu_j| \leq | F_j(x) -F_j(\mu_j)|.$$ As a result, if $$\label{eq:M.inverse}
| F_j(x) - F_j(\mu_j) | \leq M \eta,$$ it is the case that $|x- \mu_j| \leq \eta$, an
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ible.
We note that the approach presented above generalizes the results obtained in [@GrundlandZelazny:1983] where the simple states have been constructed with wave vectors $\lambda$ of constant direction for hyperbolic inhomogeneous systems (\[eq:SW:1\]).
Simple mode solutions for inhomogeneous quasilinear system. {#sec:6}
===========================================================
We now generalize the concept of simple wave solutions for inhomogeneous, quasilinear, systems of form (\[eq:SW:1\]). As in the case of the simple wave, the system can be either hyperbolic or elliptic. We look for a real solution, in terms of a Riemann invariant $r$ and its complex conjugate $\bar{r}$, of the form
\[eq:ms:1\] u=f(r,|[r]{}),r(u,x)=\_i(u)x\^i,|[r]{}(u,x)=|\_i(u)x\^i,i=1,…, p,
where $\lambda(u)={\left( \lambda_1(u),\ldots,\lambda_p(u) \right)}$ is a complex wave vector and $\bar{\lambda}(u)$ its complex conjugate. The Jacobian matrix takes the form
\[eq:ms:2\] u=\^[-1]{}[( + | )]{}\^[qp]{},
where we assume that the matrix
\[eq:ms:3\] =[( I\_q-- )]{}\^[qq]{}.
is invertible. Replacing the Jacobian matrix (\[eq:ms:2\]) into the system (\[eq:SW:1\]), we obtain
\[eq:ms:4\] \^[i]{}\^[-1]{}[( + | )]{}=b,\^i=[( A\^[i]{}\_)]{}\^[qq]{}.
We introduce a rotation matrix $L=L(x,u)\in SO(q,{\mathbb{C}})$ and its complex conjuguate $\bar{L}=\bar{L}(x,u)\in SO(q,{\mathbb{C}})$ such that the relations
\[eq:ms:5\] \^[-1]{}=L b + ,\^[-1]{}=| |[L]{} b+ |,
hold, where $\Omega(x,u)$ and its conjugate $\bar{\Omega}$ are scalar complex functions, while $\tau(x,u)$ and its complex conjugate vector $\bar{\tau}(x,u)$ satisfy
\[eq:ms:7\] \^i\_i +\^i|\_i |=0.
The vectors $\tau$ and $\bar{\tau}$ can be seen as characteristic vectors of the homogeneous part of equation (\[eq:ms:4\]). We eliminate the vectors $\Phi^{-1}({\partial}f/{\partial}r)$ and $\Phi^{-1}({\partial}f/{\partial}\bar{r})$ from equation (\[eq:ms:4\]) using equations (\[eq:ms:5\]). Considering the equation (\[eq:ms:7\]), we obtain, as a condition on functions $\Omega$,
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ngle \mathbf{A}_{2,1}\mathbf{A}_{2,2}\right\rangle \right.\\
& +\left.\left\langle \mathbf{B}_{1,1}\mathbf{B}_{2,1}\right\rangle +\left\langle \mathbf{B}_{1,2}\mathbf{B}_{2,2}\right\rangle \right)\end{aligned}$$ to the system and then eliminating the connection correlations $\left\langle \mathbf{A}_{1,1}\mathbf{A}_{1,2}\right\rangle $, $\left\langle \mathbf{A}_{2,1}\mathbf{A}_{2,2}\right\rangle $, $\left\langle \mathbf{B}_{1,1}\mathbf{B}_{2,1}\right\rangle $, $\left\langle \mathbf{B}_{1,2}\mathbf{B}_{2,2}\right\rangle $ from the system using the Fourier–Motzkin elimination algorithm, we obtain the system $$\begin{aligned}
& -1+\frac{1}{2}S_{CHSH}\le\Delta\le4-\left[-1+\frac{1}{2}S_{CHSH}\right],\label{eq:delta1}\\
& 0\le\Delta\le4-\left(\left|\left\langle \mathbf{A}_{1}\right\rangle \right|+\left|\left\langle \mathbf{A}_{2}\right\rangle \right|+\left|\left\langle \mathbf{B}_{1}\right\rangle \right|+\left|\left\langle \mathbf{B}_{2}\right\rangle \right|\right),\label{eq:delta2}\end{aligned}$$ where we denote
$$\begin{array}{l}
S_{CHSH}=s_{1}\big(\left\langle \mathbf{A}_{1,1}\mathbf{B}_{1,1}\right\rangle ,\left\langle \mathbf{A}_{1,2}\mathbf{B}_{1,2}\right\rangle ,\\
\phantom{S_{CHSH}=s_{1}\big(}\left\langle \mathbf{A}_{2,1}\mathbf{B}_{2,1}\right\rangle ,\left\langle \mathbf{A}_{2,2}\mathbf{B}_{2,2}\right\rangle \big)
\end{array}$$
as in the main text. This means that $\Delta$ is compatible with the given observed probabilities if and only if the above inequalities are satisfied. Since the set of possible values of $\Delta$ constrained by (\[eq:delta1\]) and (\[eq:delta2\]) is known to be nonempty, it follows that the minimum value of $\Delta$ is always given by $$\Delta_{\min}=\max\left\{ 0,\frac{1}{2}S_{CHSH}-1\right\} .$$
EPR-Bell: Negative probabilities
--------------------------------
The analogous result for the negative probabilities approach is that the observable distributions are obtained as the marginals of some negative probability joint of $\mathbf{A}_{1}=\mathbf{A}_{1,1}=\mathbf{A}_{1,2},$ $\
| 2,100
| 1,803
| 2,639
| 2,089
| null | null |
github_plus_top10pct_by_avg
|
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