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.
The solution of (\[desol20b\]) is (cf. [@dautraylionsv6 Ch. XXI, Sec. 3.2, pp. 233-235], or [@tervo14 proof of Theorem 6.3]; replace first $\eta$ by $r_{m,j}-\eta$) $$\begin{gathered}
\label{desol20ab}
v_{2,j}(x,\omega,\eta)
\\
=H(r_{m,j}-\eta-t(x,\omega))e^{\int_0^{t(x,\omega)}-\Sigma_j(x-s\omega,\omega)ds}
\tilde{g}_j(x-t(x,\omega)\omega,\omega,\eta+t(x,\omega)),\end{gathered}$$ where $H$ is the Heaviside function. By performing similar computations as in the proof of Lemma \[trathle1\], one sees that $v_{2,j}$ defined by is in fact a weak (distributional) solution of . Moreover, $v_{2,j}$ satisfies (in generalized sense) the inflow boundary condition, since $t(y,\omega)=0$ on $\tilde{\Gamma}_{-,j}$ (see Lemma \[le:esccont:1\]) and $r_{m,j}-\eta>0$ (therefore $H(r_{m,j}-\eta-t(y,\omega))=1$ on $\tilde{\Gamma}_{-,j}$), as well as the initial (energy) condition, since $t(x,\omega)>0$ on $G\times S$ (hence $H(r_{m,j}-\eta-t(x,\omega))=0$ for all $\eta$ close to $r_{m,j}$).
The solution of , on the other hand, is obtained as follows. Let $V_{1,j}(x,\omega,\eta):=v_{1,j}(x,\omega,r_{m,j}-\eta)$. Then the problem (\[desol20a\]) is equivalent to $$\begin{gathered}
{{\frac{\partial V_{1,j}}{\partial \eta}}}+\omega\cdot\nabla_x {V_{1,j}}+
\Sigma_j V_{1,j}
=F_j,\nonumber\\
{V_{1,j}}_{|\tilde\Gamma_{-,j}}=0,
\quad V_{1,j}(\cdot,\cdot,0)=0, \label{desol20aa}\end{gathered}$$ where $F_j(x,\omega,\eta):=\tilde f(x,\omega,r_{m,j}-\eta)$. Let $B_0:L^2(G\times S)\to L^2(G\times S)$ be a densily defined operator (as in section \[evcsd\]) such that $$&D(B_0)=\tilde{W}_{-,0}^2(G\times S),
\quad
B_0\psi=-\omega\cdot\nabla_x\psi.$$ Then $B_0$ generates a contraction $C^0$-semigroup $T(\eta)$, and in fact for $h\in L^2(G\times S)$ we have (cf. [@dautraylionsv6 Ch. XXI, Sec. 2.2, p. 222], or [@tervo14 proof of Theorem 5.15]) $$(T(\eta)h)(x,\omega)=H(t(x,\omega)-\eta)h(x-\eta\omega,\omega),$$ where $H$ is the Heaviside function. The problem (\[desol20aa\]) can be put into the abstract form \[desol20aaa\] -(B\_0-\_j) [V\_[1,j]{}]{} =
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latter differs by a total derivative.
Supergravity theories are constructed [@SUGRA] based on the YM-like theory for the gauge potential (\[usual\]). It will be interesting to consider the supersymmetrization of the gauge symmetry of diffeomorphism and to derive the supergravity theory as an alternative formulation of supergravity. We leave this project for future publications.
Scattering Amplitudes {#ScattAmp}
=====================
In recent years, there has been amazing progress in the techniques of calculating scattering amplitudes, as well the understanding of their structures. Among them, a very interesting and mysterious structure is the connection between YM theory and gravity through the so-called double-copy procedure, which utilizes the color-kinematics duality (also known as the BCJ duality) [@BCJ]. According to Ref.[@BCJ], certain gravity theories are double copies of YM theories: the scattering amplitudes of gravity theories can be obtained from those of YM theories with color factors replaced by kinematic factors. In many cases this connection can find its origin in the open-closed string duality (the KLT duality [@KLT]), although there are also other cases in which the string-theory origin is absent at this moment.
A complete off-shell field-theoretic explanation of this connection between gravity and YM theories, which applies only to on-shell amplitudes, may not be possible. But it is desirable to understand how much of the on-shell miracle can be understood in an off-shell theory. Earlier efforts in this direction include Refs. [@BjerrumBohr:2012mg; @Monteiro:2014cda]. We propose that the formulation of gravity as a YM theory we constructed above may shed some new light on this problem.
Heuristic Explanation {#heuristic}
---------------------
Let us first re-examine the double-copy procedure to illustrate our idea. As the simplest example, for the pure YM theory, the color-ordered 3-point amplitude at tree-level is f\_[abc]{} n\^[(3)]{}\_(p, q, -(p+q)) \[fn\] where $f_{abc}$ is the struct
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lowing result.
\[keyreduction\] Every germ as specified in §\[preamble\] is equivalent to one which, up to a parameter change, has matrix representation $$\begin{pmatrix}
1 & 0 & 0\\
q(t) & t^b & 0\\
r(t) & s(t)t^b & t^c
\end{pmatrix}$$ in suitable coordinates, with $1\le b\le c$ and $q,r,s$ polynomials such that $\deg(q)<b$, $\deg(r)<c$, $\deg(s)<c-b$, and $q(0)=r(0)=s(0)=0$.
A refined version of this statement is given in Lemma \[faber\].
We will deal with $3\times 3$ matrices with entries in ${{\mathbb{C}}}[[t]]$, that is, ${{\mathbb{C}}}[[t]]$-valued points of $\operatorname{Hom}(V,W)$, for $V$, $W$ 3-dimensional complex vector spaces with chosen bases. Every such matrix $\alpha(t)$ determines a germ in ${{\mathbb{P}}}^8$. A generator $F$ of the ideal of ${{\mathscr C}}$ will be viewed as an element of ${\text{\rm Sym}}^d W^*$, for $d=\deg{{\mathscr C}}$; the composition $F\circ\alpha(t)$, a ${{\mathbb{C}}}[[t]]$-valued point of ${\text{\rm Sym}}^d V^*$, generates the ideal of ${{\mathscr C}}\circ\alpha(t)$.
We will call matrices of the form $$\lambda(t)=\begin{pmatrix}
t^a & 0 & 0\\
0 & t^b & 0\\
0 & 0 & t^c
\end{pmatrix}$$ ‘1-PS’, as they correspond to 1-parameter subgroups of ${\text{\rm PGL}}(3)$.
We will say that two matrices $\alpha(t)$, $\beta(t)$ are equivalent if the corresponding germs are equivalent in the sense of Definition \[equivgermsnew\]. The following lemma will allow us to simplify matrix expressions of germs up to equivalence. Define the degree of the zero polynomial to be $-\infty$.
\[MPI\] Let $$h_1(t) =\begin{pmatrix}
u_1 & b_1 & c_1 \\
a_2 & u_2 & c_2 \\
a_3 & b_3 & u_3
\end{pmatrix}$$ be a matrix with entries in ${{\mathbb{C}}}[[t]]$, such that $h_1(0)=I$, and let $a\le b\le c$ be integers. Then $h_1(t)$ can be written as a product $h_1(t)=h(t)\cdot j(t)$, with $$h(t)=\begin{pmatrix}
1 & 0 & 0 \\
q & 1 & 0 \\
r & s & 1
\end{pmatrix}
\quad,\quad j(t)=\begin{pmatrix}
v_1 & e_1 & f_1 \\
d_2 & v_2 & f_2 \\
d_3 & e_3 & v_3
\end{pmatrix}$$ where $q$, $r$, $s$ are [*polynomial
| 3,603
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| 0.768608
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nalysis [@2018osc].
Lifetime limits are computed using a Bayesian method, assuming that the SK-I through SK-IV datasets have correlated systematic uncertainties [@2018pdg]. For the nucleon decay modes, the systematic uncertainties of the “High $P_{\text{tot}}$" and “Low $P_{\text{tot}}$" search boxes are treated as independent datasets with fully correlated systematic uncertainties. The conditional probability distribution for the decay rate is given by Eq. \[eq:probability\], where $\Gamma$ is the decay rate and for dataset $i$, $\lambda_i$ is the exposure (given in proton-years for nucleon decay and in oxygen-years for dinucleon decay), $\epsilon_i$ is the efficiency, $b_i$ is the number of background events, and $n_i$ is the number of candidate events. Since the systematic errors are correlated for all datasets, integrating the prior probability distribution up to $b$ in some dataset implies that we integrate the prior distribution in dataset $i$ up to $b_i(b)$.
We assume a Gaussian prior distribution $P(\lambda_i(\lambda) | \lambda_{i,0}, \sigma_{\lambda_{i,0}})$ for $\lambda_i$ with a mean value of $\lambda_{i,0}$ and $\sigma_{\lambda_{i,0}}$ given by the $1\%$ percent systematic uncertainty in exposure. We also assume Gaussian priors $P(\epsilon_i(\epsilon) | \epsilon_{i,0}, \sigma_{\epsilon_{i,0}})$ and $P(b_i(b) | b_{i,0}, \sigma_{b_{i,0}})$ for $\epsilon_i$ and $b_i$ with standard deviations set to the total percent systematic uncertainties in efficiency and background, respectively. To require a positive lifetime, $P(\Gamma)$ is 1 for $\Gamma \geq 0$ and otherwise 0. We calculate the upper bound of the decay rate $\Gamma_{\text{limit}}$ as in Eq. \[eq:conflevel\], with a 90% confidence level (CL):
$$\label{eq:conflevel}
\text{CL} = \frac{\int_{\Gamma=0}^{\Gamma_{\text{limit}}}\prod_{i=1}^{N} P(\Gamma|n_i)d\Gamma}{\int_{\Gamma=0}^{\infty}\prod_{i=1}^{N} P(\Gamma|n_i)d\Gamma}.$$
Therefore we obtain the lower bound on the partial lifetime limit of a decay mode: $\tau/\text{B} = 1/\Gamma_{\text{
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$.
Again, there will be a smallest $n_{2}\in{I \kern -4.5pt N}$ such that $d(y_{n},x)-d(y_{n_{1}},x)>2$ for all $n\geq n_{2}$. Set $w_{n}=y_{0}$ for $n_{1}\leq n<n_{2}$. Thus, for $n_{1}\leq n<n_{2}$, we have that $d(y_{n},x)-d(w_{n},x)>1$ and $d(w_{n},x)\geq 0$.
Once again, there will be a smallest $n_{3}\in{I \kern -4.5pt N}$ such that $d(y_{n},x)-d(y_{n_{2}},x)>3$ for all $n\geq n_{3}$. Set $w_{n}=y_{n_{1}}$ for $n_{2}\leq n<n_{3}$. Thus, for $n_{2}\leq n<n_{3}$, we have that $d(y_{n},x)-d(w_{n},x)>2$ and $d(w_{n},x)>1$. The last inequality follows from $d(y_{n_{1}},x)>d(y_{0},x)+1\geq 1$ for all $n\geq n_{1}$.
Continuing this way, we will have a smallest $n_{k}\in{I \kern -4.5pt N}$ such that $d(y_{n},x)-d(y_{n_{k-1}},x)>k$ for all $n\geq n_{k}$. Set $w_{n}=y_{n_{k-2}}$ for $n_{k-1}\leq n<n_{k}$. In this general case for $n_{k-1}\leq n<n_{k}$, we have that $d(y_{n},x)-d(w_{n},x)>k-1$ and $d(w_{n},x)> k-2$. The last inequality occurs because $d(y_{n_{k-2}},x)>d(y_{n_{k-3}},x)+k-2> k-2$ for all $n\geq n_{k-2}$.
Altogether then, $w_{n}$ is defined for all $n$. Moreover, $d(w_{n},x)$ increases monotonically, eventually becoming larger than $m$ for every $m\in{I \kern -4.5pt N}$. Therefore, ${\bf w}=[w_{n}]$ is in a galaxy $\Gamma_{a}$ different from the principal galaxy $\Gamma_{0}$. Furthermore, $d(y_{n},x)-d(w_{n},x)$ also increases monotonically in the same way. Consequently, the galaxy $\Gamma_{a}$ containing ${\bf w}=[w_{n}]$ is closer to $\Gamma_{0}$ than is the galaxy $\Gamma_{b }$ containing ${\bf y}=[y_{n}]$.
We can now repeat this argument with $\Gamma_{b}$ replaced by $\Gamma_{a}$ and with ${\bf w}=[w_{n}]$ playing the role that ${\bf y}=[y_{n}]$ played to find still another galaxy $\Gamma_{a}'$ different from $\Gamma_{0}$ and closer to $\Gamma_{0}$ than is $\Gamma_{a}$. Continual repetitions yield an infinite sequence of galaxies indexed by, say, the negative integers and totally ordered by their closeness to $\Gamma_{0}$.
The conclusion that there is an infinite sequence of galaxies progressive
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lux constraint*]{}.
In this section we use the positivity constraint to fix energy correlators completely in the case of extremal values of ${a \over c}$. We also show that an additional assumption of energy correlators finiteness is not consistent with the three-point function of the stress tensor being proportional to that of the free boson, free fermion or free vector field.
First, we use the zero flux constraint to reduce the number of possible functions in . Second, we use the positivity constraint to fix energy correlators completely. The positivity constraint is used in two steps as well. We start by imposing the positivity constraint at the points where the energy correlator is finite. We then impose the integrated positivity constraint in the isolated singular points.
It is absolutely crucial for the argument that the stress tensor has nonzero spin. Imagine a state created by a scalar operator. The two-point energy correlator takes the form The positivity constraint states that $f(\xi) \geq 0$ and clearly is not strong enough to fix the energy correlator.
Let us assume that the three-point function of the stress tensor is given by the purely bosonic structure $n_b \neq 0$, $n_f = 0$, $n_v = 0$. A priori nothing prevents us from thinking that there are interacting CFTs of this type. The one-point energy correlator takes the form
Henceforth it is more convenient to choose the reference frame $q = (q^0 , \vec 0)$ and $\hat n = (1,0,0,1)$ so we align the detector along the $z$-axis. We also switch to the state $T_{i j} \eps^{i j}$ where $\eps^{i i} = 0$ and Latin indices run from $1$ to $3$.
As a first step we find polarization tensors such that the one-point function is zero. This condition imposes $\eps_{3 3}=0$. Other than this the polarization tensor is arbitrary. We can use this polarization tensor to impose the zero flux constraint on the two-point energy correlator Let us explain how we impose the zero flux constraint. We choose $\hat n$ along the $z$-axis and $\eps_{33} = 0$. The cros
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athbf{v}}(t)= T_{k,d}(h(t{\mathbf{e}}-{\mathbf{v}}))$ where $T_{k,d}: {\mathbb{R}}[t] \rightarrow {\mathbb{R}}[t]$ is the linear operator defined by $$T_{k,d}\left(\sum_{j\geq 0} a_jt^j\right) = -\sum_{j=0}^d \left( \frac {j+1} {k+1} a_{j+1} +(d-1-j)a_j\right) (d-j)! \binom {k+1}{d-j}t^j.$$ Moreover if $f$ is a $[0,1/k]$–rooted polynomial of degree $d$, then $T_{k,d}(f)$ is real–rooted.
By $$\label{collect}
h(t{\mathbf{e}}-{\mathbf{v}})= \sum_{j=0}^d (-1)^j \frac 1 {j!} D_{\mathbf{v}}^jh({\mathbf{e}})t^{d-j}=:\sum_{j \geq 0} a_j t^j.$$ Note that $D_{\mathbf{v}}^j h(t{\mathbf{e}})= D_{\mathbf{v}}^j h({\mathbf{e}})t^{d-j}$ and $D_{\mathbf{v}}^j D_{\mathbf{e}}h(t{\mathbf{e}})= D_{\mathbf{e}}D_{\mathbf{v}}^j h(t{\mathbf{e}})= (d-j)D_{\mathbf{v}}^j h({\mathbf{e}})t^{d-j-1}$. Expanding and comparing coefficients with one sees that $g_{\mathbf{v}}(t)=T_{k,d}(h(t{\mathbf{e}}-{\mathbf{v}})$.
To prove the final statement of the lemma we may by Hurwitz’ theorem on the continuity of zeros assume that $f$ is a $(0,1/k)$–rooted polynomial of degree $d$. We may choose a hyperbolic degree $d$ polynomial $h$ and a vector ${\mathbf{v}}$ such that $f(t)= h(t{\mathbf{e}}-{\mathbf{v}})$, for example $h(x,y)= (-y)^df(-x/y)$, ${\mathbf{e}}=(1,0)$ and ${\mathbf{v}}=(0,1)$. Then ${\mathbf{v}}\in \Lambda_{++}$ and ${\mathbf{w}}={\mathbf{e}}-k{\mathbf{v}}\in \Lambda_{++}$ by e.g. . Hence $$T_{k,d}(f)(t) = \chi[{\mathbf{v}},{\mathbf{v}}, \ldots, {\mathbf{v}}, {\mathbf{w}}](t)$$ is real–rooted.
The *trace*, $\tr(f)$, of a non-constant polynomial is the sum of the the zeros of $f$ (counted with multiplicity). Let ${\mathcal{M}}_d$ be the affine space of all monic real polynomials of degree $d$.
\[affine\] Let $T : {\mathcal{M}}_d \rightarrow {\mathcal{M}}_m$ be an affine linear operator, and let $\epsilon>0$. Suppose $T$ sends $[a,b]$–rooted polynomials to real–rooted polynomials. Consider the problem of maximizing the largest zero of $T(f)$ over all $[a,b]$–rooted polynomials $f \in {\mathcal{M}}_d$ with $\tr(f) = \epsilo
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the rate of growth of enstrophy. We also provide a comprehensive characterization of the extreme vortex states which realize estimate together with the resulting flow evolutions.
The structure of the paper is as follows: in the next section we present analytic estimates [on]{} the instantaneous and finite-time growth of enstrophy in 3D flows. In §\[sec:3D\_InstOpt\] we formulate the variational optimization problems which will be solved to find the vortex states with the largest rate of enstrophy production and in §\[sec:3D\_InstOpt\_E0to0\] we provide an asymptotic representation for these optimal states in the limit of vanishing enstrophy. In §\[sec:3D\_InstOpt\_E\] we present numerically computed extreme vortex states corresponding to intermediate and large enstrophy values, while in §\[sec:timeEvolution\] we analyze the temporal evolution corresponding to different initial data in order to compare it with the predictions of estimates and . Our findings are discussed in §\[sec:discuss\], whereas conclusions and outlook are deferred to §\[sec:final\].
[l|c|c]{} &
[Estimate]{}\
&
[Realizability ]{}
\
& &\
& &\
& $\frac{d\P(t)}{dt} \le -\nu\frac{\P^2}{\E} + \frac{C_1}{\nu} \E\,\P$\
$\frac{d\P(t)}{dt} \le \frac{C_2}{\nu} \K^{1/2}\P^{3/2}$ &\
2D Navier-Stokes\
finite-time
& $\max_{t>0} \P(t) \le \P_0 + \frac{C_1}{2\nu^2}\E_0^2$\
$\max_{t>0} \P(t) \le \left(\P_0^{1/2} + \frac{C_2}{4\nu^2}\K_0^{1/2}\E_0\right)^2$ &\
& &\
3D Navier-Stokes\
finite-time
& $\E(t) \le \frac{\E(0)}{\sqrt{1 - 4 \frac{C \E(0)^2}{\nu^3} t}}$\
$\frac{1}{\E(0)} - \frac{1}{\E(t)} \leq \frac{27}{(2\pi\nu)^4}\left[\K(0) - \K(t) \right]$ &\
Bounds on the Growth of Enstrophy in 3D Navier-Stokes Flows {#sec:Bounds3DNS}
===========================================================
We consider the incompressible Navier-Stokes system defined on the 3D unit cube $\Omega = [0,1]^3$ with periodic boundary conditions
\[eq:NSE3D\] $$\begin{aligned}
{2}
\partial_t{\mathbf{u}}+ {\mathbf{u}}\cdot\bnabla{\mathbf{u}}+ \bnabla p - \nu{\Delta}{\
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eturned adjoint form ${\mathfrak{a}}_r^*$ inherits all properties of ${\mathfrak{a}}.$
In the following we establish that ${{U}}$ can be extended to a strongly continuous evolution family on $V'.$
\[Lemma EVF on V’\] Let ${\mathfrak{a}}$ be a non-autonomous sesquilinear form satisfying (\[eq:continuity-nonaut\])-(\[square property\]). Then ${{U}}$ can be extended to a strongly continuous evolution family on $V^\prime,$ which we still denote ${{U}}.$
Let $x\in H$ and $(t,s)\in\overline {\Delta}.$ Then using Proposition \[equalities: adjoint EVF and EVF\] and the fact that ${{U}}$ and ${{U}}_r^*$ define both strongly continuous evolution families on $V$ and $H$ we obtain that $$\begin{aligned}
\|{{U}}(t,s)x\|_{V'}&=\sup_{\underset{v\in V}{\|v\|_V=1}}{\, \vert \,}<{{U}}(t,s)x, v>{\, \vert \,}\\&=\sup_{\underset{v\in V}{\|v\|_V=1}}{\, \vert \,}({{U}}(t,s)x|v)_H{\, \vert \,}=\sup_{\underset{v\in V}{\|v\|_V=1}}{\, \vert \,}(x| {{U}}(t,s)^\prime v)_H{\, \vert \,}\\&=\sup_{\underset{v\in V}{\|v\|_V=1}}{\, \vert \,}(x|{{U}}_r^*(T-s,T-t)v)_H{\, \vert \,}\\&=\sup_{\underset{v\in V}{\|v\|_V=1}}{\, \vert \,}<x,{{U}}_r^*(T-s,T-t)v>{\, \vert \,}\\&\leq \|x\|_{V^\prime}\|{{U}}_r^*(T-s,T-t)\|_{{\mathcal{L}}(V)}
\\&\leq c\|x\|_{V^\prime}\end{aligned}$$ where $c>0$ is such that $\underset{t,s\in\Delta}{\sup}\|{{U}}_r^*(t,s)\|_{{\mathcal{L}}(V)}\leq c.$ Thus, the claim follows since $H$ is dense in $V'.$
Let $\Delta:=\{(t,s)\in\Delta{\, \vert \,}t\ge s\}.$ The following theorem is the main result of this paper
\[main result\] Let ${\mathfrak{a}}$ be a non-autonomous sesquilinear form satisfying (\[eq:continuity-nonaut\])-(\[square property\]). Let $\{U(t,s):\ (t,s)\in\Delta\}$ given by (\[evolution family\]). Then the function $(t,s)\mapsto {{U}}(t,s)$ is norm continuous on $\Delta$ into ${\mathcal{L}}(X)$ for $X=V, H$ and $V'.$
The norm continuity for ${{U}}$ in the case where $X=V$ follows from [@LH17 Theorem 2.7]. On the other hand, applying [@LH17 Theorem 2.7] to ${\mathfrak{a}}_r^*$ we obtain that ${{U}}_r^*$ is also no
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2_relu_v3.pdf){width="100.00000%"}
---
abstract: 'Let $X, Y$ be complete, simply connected Riemannian surfaces with pinched negative curvature $-b^2 \leq K \leq -1$. We show that if $f : \partial X \to \partial Y$ is a Moebius homeomorphism between the boundaries at infinity of $X, Y$, then $f$ extends to an isometry $F : X \to Y$. This can be viewed as a generalization of Otal’s marked length spectrum rigidity theorem for closed, negatively curved surfaces, in the sense that Otal’s theorem asserts that if $X, Y$ admit properly discontinuous, cocompact, free actions by groups of isometries and the boundary map $f$ is Moebius and equivariant with respect to these actions then it extends to an isometry. In our case there are no cocompactness or equivariance assumptions, indeed the isometry groups of $X, Y$ may be trivial.'
address: 'Indian Statistical Institute, Kolkata, India. Email: kingshook@isical.ac.in'
author:
- Kingshook Biswas
bibliography:
- 'moeb.bib'
title: 'Moebius rigidity for simply connected, negatively curved surfaces'
---
\[section\] \[theorem\][Proposition]{} \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Definition]{} \[theorem\][Conjecture]{} \[theorem\][Remark]{} \[theorem\][Claim]{} \[theorem\][Definition-Theorem]{}
Introduction
============
We continue in this article the study of Moebius maps between boundaries of CAT(-1) spaces undertaken in [@biswas3], [@biswas4], [@biswas5], [@biswas6], [@biswas7]. The principal question is whether a Moebius homeomorphism between the boundaries at infinity of two CAT(-1) spaces extends to an isometry between the spaces. We recall that the boundary $\partial X$ of a CAT(-1) space comes equipped with a positive function on the set of quadruples of distinct points in $\partial X$, called the cross-ratio, and a map $f : \partial X \to \partial Y$ between boundaries is said to be Moebius if it preserves cross-ratios.
Bourdon [@bourdon2] showed that if $X$ is a rank one symmetric space of noncompact type with the metric normalized so that t
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\quad \mbox{with}\quad s_{t}\in {\mathcal{S}}({%
\mathbb{R}}^{d}\times {\mathbb{R}}^{d}).$$
We define the formal adjoint operator $$S_{t}^{\ast }f(y)=\int_{{\mathbb{R}}^{d}}s_{t}(x,y)f(x)dx,\quad t>0.$$
\[H2H\*2\] If $f\in {\mathcal{S}({\mathbb{R}}^{d})}$ then $S_{t}f\in {\mathcal{S}({\mathbb{R}}^{d})}$. Moreover, there exist $b\in {\mathbb{N}}$ such that for every $q\in {\mathbb{%
N}},\kappa \geq 0$ and $p\in \lbrack 1,\infty )$ there exist constants $%
C_{q,\kappa ,p}(S)$ such that for every $t>0$, $$\begin{aligned}
(H_{2})& \qquad \left\Vert S_{t}f\right\Vert _{q,-\kappa ,\infty }\leq
C_{q,\kappa ,\infty }(S)\left\Vert f\right\Vert _{q+b,-\kappa ,\infty },
\label{h2} \\
(H_{2}^{\ast })& \qquad \left\Vert S_{t}^{\ast }f\right\Vert _{q,\kappa
,p}\leq C_{q,\kappa ,p}(S)\left\Vert f\right\Vert _{q+b,\kappa ,p}.
\label{h2'}\end{aligned}$$We assume that $C_{q,\kappa ,p}(S)$, $p\in \lbrack 1,\infty ]$, is non decreasing with respect to $q$ and $\kappa $.
We denote $$\begin{aligned}
C_{q,\kappa ,\infty }(U,S)& =C_{q,\kappa ,\infty }(U)C_{q,\kappa ,\infty
}(S),\quad C_{q,\kappa ,p}(U,S)=C_{q,\kappa ,p}(U)C_{q,\kappa ,p}(S),
\label{h3'} \\
C_{q,\kappa ,\infty ,p}(U,S)& =C_{q,\kappa ,\infty }(U,S)\vee C_{q,\kappa
,p}(U,S). \label{hh3'}\end{aligned}$$Under Assumption \[H1H\*1\] and \[H2H\*2\], one immediately obtains$$\begin{aligned}
\left\Vert (S_{t}U_{j})f\right\Vert _{q,-\kappa ,\infty }& \leq C_{q,\kappa
,\infty }(U,S)\left\Vert f\right\Vert _{q+a+b,-\kappa ,\infty }, \label{h}
\\
\left\Vert (S_{t}^{\ast }U_{j}^{\ast })f\right\Vert _{q,\kappa ,p}& \leq
C_{q,\kappa ,p}(U,S)\left\Vert f\right\Vert _{q+a+b,\kappa ,p}. \label{h'}\end{aligned}$$In fact these are the inequalities that we will employ in the following. We stress that the above constants $C_{q,\kappa ,\infty }(U,S)$ and $%
C_{q,\kappa ,p}(U,S)$ may depend on $a,b$ and are increasing w.r.t. $q$ and $%
\kappa $.
Finally we assume that the (possible) blow up of $s_{t}\rightarrow \infty $ as $t\rightarrow 0$ is controlled in the following way.
\[HH3\] Let
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igraph objects, rather than the list of names of the objects. Then your current code would work as-is, and you would have functions like lapply available to you.
Update: re your assigning code.
You could do this many ways, but the common element is that instead of storing your variables in A1, ..., A100 , you store them in a list A so that A[[1]] is your old A1.
e.g.
As <- lapply(names,
function (name) {
filepath <- ...
as.matrix(read.dta(filepath)) # this is Ai
})
# note As[[i]] is your old Ai
Bs <- lapply(As, graph.adjacency, mode="directed", weighted = NULL, diag = FALSE)
# now Bs[[i]] is your old Bi
And you could do e.g.
avg.lengths <- lapply(Bs, average.path.length)
degree <- lapply(Bs, degree)
and so on. Then for graph i you could use Bs[[i]] to get the graph, avg.lengths[[i]] to get the avg length, etc.
If average.path.length, diameter, transitivity, and degree all return SINGLE values, you could simply store them in a dataframe:
graph.properties <- data.frame(
graph=1:length(Bs),
average.path.length=sapply(Bs, average.path.length),
diameter=sapply(Bs, diameter),
...)
Then graph.properties$diameter[i] is the diameter of Bi. However you can't store the graphs themselves in a dataframe, as dataframe cells should contain single values only not complicated objects.
Also I think degree returns a numeric vector, so you're stuck with a list rather than dataframe (hence my use of lapply initially rather than sapply).
Q:
Catch exit code of external program in c++
I am running an external program with the command execvp, now I want to catch the exit code of the external program and if possible get the PID of it.
Is there anyway possible?(I know I can read $? in ubuntu and use ps faxu but these are dirty ways for that)
A:
The exec* functions does not return when the program has successfully run, so you can't get the return code via execvp. However,
| 3,612
| 6,701
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| 2,763
| 2,606
| 0.77849
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\colon{\sD}(B) \to {\sD}(B\times A)$. Both of these preserve colimits in both variables, on the left because colimits there are defined by postcomposition, and on the right because $F$ and $G$ preserve colimits.
This construction is universal in the sense that if is a monoidal left derivator, then to make into a -module is equivalent to giving a cocontinuous monoidal morphism ${\sV}\to{\mathsf{END}\ccsub}({\sD})$. Specifically, the latter assigns to each $X\in {\sV}(A)$ a morphism ${\sD}\to {\sD}^A$, which is the external tensor product with $X$. Monoidality of the morphism ${\sV}\to{\mathsf{END}\ccsub}({\sD})$ gives the associativity and unitality of the action, while its cocontinuity gives left cocontinuity of the action; right cocontinuity of the action comes from the fact that this morphism lands in ${\mathsf{END}\ccsub}({\sD}) = {\mathsf{HOM}\ccsub}({\sD},{\sD})$ rather than ${\mathsf{HOM}}({\sD},{\sD})$.
Note that unlike ${\mathsf{HOM}}({\sD},{\sE})$, the left derivator ${\mathsf{HOM}\ccsub}({\sD},{\sE})$ is not a derivator even if is: since limits and colimits do not in general commute, the limit in ${\mathsf{HOM}}({\sD},{\sE})$ of cocontinuous morphisms need no longer be cocontinuous. However, we can say;
\[thm:ldh-ran\] If $u\colon A\to B$ is such that has right Kan extensions along $u$ that commute with arbitrary left Kan extensions, then so does ${\mathsf{HOM}\ccsub}({\sD},{\sE})$.
Right Kan extensions in ${\mathsf{HOM}}({\sD},{\sE})$ are defined by postcomposition; if $u_*$ is cocontinuous then ${\mathsf{HOM}\ccsub}({\sD},{\sE})$ is closed under such postcomposition. Since left Kan extensions are also defined by postcomposition, commutativity follows.
We now introduce the notion of weighted colimits. First note that the internal, external, and canceling versions of morphisms of two-variables can be combined. In particular, given a monoidal derivator and $A,B,C\in\cCat$, there is the **(homotopy) tensor product of functors** $$\otimes_{[B]}\colon {\sV}(A\times B\op)\times{\sV}(B\times C\op)\stackr
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| 2,615
| 3,308
| 3,335
| 0.773044
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nt may be considered to be *neutral.* Appendix A3 shows the various possibilities for the derived set and boundary of a subset $A$ of $X$.$\qquad\blacksquare$
Some useful properties of these concepts for a subset $A$ of a topological space $X$ are the following.
\(a) $\textrm{Bdy}_{X}(X)=\emptyset$,
\(b) $\textrm{Bdy}(A)=\textrm{Cl}(A)\bigcap\textrm{Cl}(X-A)$,
\(c) $\textrm{Int}(A)=X-\textrm{Cl}(X-A)=A-\textrm{Bdy}(A)=\textrm{Cl}(A)-\textrm{Bdy}(A)$,
\(d) $\textrm{Int}(A)\bigcap\textrm{Bdy}(A)=\emptyset$,
\(e) $X=\textrm{Int}(A)\bigcup\textrm{Bdy}(A)\bigcup\textrm{Int}(X-A)$,
\(f) $${\textstyle \textrm{Int}(A)=\bigcup\{ G\subseteq X\!:G\textrm{ is an open set of }X\textrm{ contained in }A\}}\label{Eqn: interior}$$
\(g) $${\textstyle \textrm{Cl}(A)=\bigcap\{ F\subseteq X\!:F\textrm{ is a closed set of }X\textrm{ containing }A\}}\label{Eqn: closure}$$
A straightforward consequence of property (b) is that the boundary of any subset $A$ of a topological space $X$ is closed in $X$; this significant result may also be demonstrated as follows. If $x\in X$ is not in the boundary of $A$ there is some neighbourhood $N$ of $x$ that does not intersect both $A$ and $X-A$. For each point $y\in N$, $N$ is a neighbourhood of that point that does not meet $A$ and $X-A$ simultaneously so that $N$ is contained wholly in $X-\textrm{Bdy}(A)$. We may now take $N$ to be open without any loss of generality implying thereby that $X-\textrm{Bdy}(A)$ is an open set of $X$ from which it follows that $\textrm{Bdy}(A)$ is closed in $X$.
Further material on topological spaces relevant to our work can be found in Appendix A3.
***End Tutorial4***
Working in a general topological space, we now recall the solution of an ill-posed problem $f(x)=y$ [@Sengupta1997] that leads to a multifunctional inverse $f^{-}$ through the generalized inverse $G$. Let $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ be a (nonlinear) function between two topological space $(X,\mathcal{U})$ and $(Y,\mathcal{V})$ that is neither one-one or onto. Since $f$ is
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| 3,124
| 2,790
| 0.776935
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|
\[-0.27, 0.16\] 0.61 -0.10 \[-0.35, 0.14\] 0.40 0.006 \[-1.30, 1.31\] 0.99
FES -- FSI -0.006 \[-0.10, 0.09\] 0.90 0.03 \[-0.03, 0.09\] 0.40 0.07 \[-0.003, 0.15\] 0.06 -0.004 \[-0.06, 0.06\] 0.91
Organization^2^ -0.16 \[-0.54, 0.21\] 0.40 -0.13 \[-0.40, 0.14\] 0.36 0.02 \[-0.30, 0.33\] 0.92 0.88 \[-1.03, 2.79\] 0.37
Control^2^ 0.006 \[-0.39, 0.40\] 0.98 0.19 \[-0.10, 0.48\] 0.20 0.20 \[-0.13, 0.53\] 0.24 -0.20 \[-1.52, 1.11\] 0.76
FES -- Norms^2^ -0.05 \[-0.42, 0.32\] 0.79 0.10 \[-0.18, 0.37\] 0.49 0.28 \[-0.03, 0.59\] 0.08
| 3,615
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|
tral limit theorem to show that the eigenvalues $\lambda_\chi$ are approximately distributed like uncorrelated Gaussian random variables. Even to prove asymptotic results, it is necessary here to apply some *quantitative* version of the central limit theorem, in order to achieve suitably uniform control over the $\lambda_\chi$. The approach taken here (and earlier in [@Meckes]) generalizes and extends the method used by Bose and Mitra in [@BoMi], which applied a multivariate version of the Berry–Esseen theorem and thus required the matrix entries to have uniformly bounded third moments. Here a quantitative, multivariate version of Lindeberg’s theorem is applied.
If $f : {\mathbb{R}}^d \to {\mathbb{R}}$ is bounded and Lipschitz with Lipschitz constant ${\left\vert f \right\vert}_L$, its bounded Lipschitz norm may be defined by $${\left\Vert f \right\Vert}_{BL} = \max\{ {\left\Vert f \right\Vert}_\infty, {\left\vert f \right\vert}_L\}.$$ The bounded Lipschitz distance between random vectors $X$ and $Y$ in ${\mathbb{R}}^d$ is defined by $$d_{BL}(X,Y) = \sup_{{\left\Vert f \right\Vert}_{BL} \le 1} {\left\vert {\mathbb{E}}f(X) - {\mathbb{E}}f(Y) \right\vert}.$$ It is well known (see e.g. [@Dudley section 11.3]) that the class of bounded Lipschitz functions is a convergence-determining class. The subclass of compactly supported such functions is furthermore separable with respect to the sup norm [@Dudley Corollary 11.2.5]. Thus to show that a sequence $\nu^{(n)}$ of probability measures on ${\mathbb{R}}^d$ converges weakly to $\nu$ in mean, in probability, or almost surely, it suffices to show that for each bounded Lipschitz function $f$, $\nu^{(n)}(f) \to \nu(f)$ in the same sense.
The following is a special case of [@BhRa Theorem 18.1] (cf. the proof of [@BhRa Corollary 18.2]).
\[T:Lindeberg\] Suppose that $X_1, \dotsc, X_k$ are independent mean $0$ random vectors in ${\mathbb{R}}^d$ such that $\frac{1}{k} \sum_{j=1}^k \operatorname{Cov}(X_j) =
I_d$. For ${\varepsilon}> 0$ let $$\theta({\varepsilon}) = \fr
| 3,616
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| 3,035
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|
m_\nu \lambda^{\rm tip}_{\mu\nu}\hat X_\nu (t'))
(1 +\sum_{\nu'} \lambda^{\rm tip}_{\mu\nu'}\hat X_{\nu'} (t) )
d_{\mu\sigma} (t') d^\dagger_{\mu'\sigma}(t)
%%
\Big \rangle_0
\Big \langle
c^\dagger_{0\sigma,\rm tip}(t') c_{0\sigma,\rm tip}(t)
\Big \rangle_0
\non
&&
-
\sum_{\mu\mu'\sigma}
t_{\mu \sigma}t_{\mu' \sigma}
\Big \langle
(1 + \sum_\nu \lambda^{\rm tip}_{\mu\nu}\hat X_\nu (t'))
(1 +\sum_{\nu'} \lambda^{\rm tip}_{\mu\nu'}\hat X_{\nu'} (t) )
d^\dagger_{\mu\sigma}(t') d_{\mu'\sigma}(t)
%%
\Big\rangle_0
\Big\langle
c_{0\sigma,\rm tip}(t') c^\dagger_{0\sigma,\rm tip}(t)
\Big\rangle_0,
\nonumber \\\end{aligned}$$
under the assumption that the system S and the STM tip are in a normal conducting state. This factorization does not require a Wick’s theorem, and, therefore, the Hamiltonian $\hat H_0$ remains fully general. Note that the electronic correlation function of the STM tip is spin-diagonal, and hence the double sum over $\sigma \sigma'$ in Eq. collapses to a single sum over $\sigma$ in Eq. . It is clear that the displacements terms $\hat X_\nu$ and the electronic orbital operators $d_{\mu \sigma}$ do not factorize, because we explicitly allow for a strong electron-phonon coupling and thus polaron formation in the system S. Finally, the terms of type $\langle d^\dagger_{\mu\sigma} d^\dagger_{\mu'\sigma }\rangle_0$ that we have neglected in Eq. must be included if either the STM tip or the sample system S are superconducting. In this case, our approach reproduces the well-known derivation of the Josephson current by Ambegaokar and Baratoff [@Ambegaokar1963].
Eq. can be divided into elastic and inelastic contributions. The former are obtained by setting all $ \lambda^{\rm tip}_{\mu\nu}=0$, while the inelastic terms are given by the difference between Eq. for non-vanishing $\lambda^{\rm tip}_{\mu\nu}$ and for $\lambda^{\rm tip}_{\mu\nu}=0$. Similarly, the total current decomposes into the sum $$\begin{aligned}
\label{eq:totalcurrent}
I_{\rm tot} &=& I_{\rm el} +I_{\rm inel}, \e
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| null | null |
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|
cO_X(D)) \simeq H^p (Y, \cO_Y( {\left \lfloor \pi^{*} D \right \rfloor} ))
\text{ and }H^p (X, \cO_X(D)) \simeq H^p (Y, \cO_Y( {\left \lceil \pi^{*} D + K_\pi \right \rceil} )).$$
Since $\cO_Y({\left \lfloor \pi^{*}D \right \rfloor})$ is acyclic for the functor $\pi_{*}$ by Lemma \[lemma:acyclic\], Leray’s spectral sequence $E_2^{p,q} = H^p(X,R^q \pi_{*} \cO_Y({\left \lfloor \pi^{*} D \right \rfloor}) ) \Longrightarrow
H^{p+q}(Y,\cO_Y({\left \lfloor \pi^*D \right \rfloor}))$ provides the isomorphism $H^p (X, R^0 \pi_{*} \cO_Y({\left \lfloor \pi^{*}D \right \rfloor})) \simeq
H^p(Y,\cO_Y({\left \lfloor \pi^{*}D \right \rfloor}))$. By the projection formula on normal surfaces Theorem \[thm:projection\], one obtains $\cO_X(D) = \pi_{*} \cO_Y({\left \lfloor \pi^{*} D \right \rfloor})$ and hence the first isomorphism holds.
The second isomorphism is a consequence of combining the first one with a generalization of Serre’s duality in this context [@Blache95 §3.1], as follows $$\begin{aligned}
& H^p(X,\cO_X(D)) = H^{2-p}(X,\cO_X(-D+K_X)) = H^{2-p}(Y,\cO_Y({\left \lfloor \pi^{*}(-D+K_X) \right \rfloor})) \\
&= H^p (Y, \cO_Y( - {\left \lfloor \pi^{*}(-D+K_X) \right \rfloor} + K_Y)) = H^p (Y, \cO_Y( - {\left \lfloor -\pi^{*} D + \pi^{*} K_X - K_Y \right \rfloor} )) \\
&= H^p (Y, \cO_Y( - {\left \lfloor -\pi^*D - K_\pi \right \rfloor} )) = H^p (Y, \cO_Y( {\left \lceil \pi^*D+K_\pi \right \rceil} )).\end{aligned}$$ Recall that $K_Y = \pi^{*} K_X + K_\pi$ and both canonical divisors are Weil divisors with integral coefficients.
Let $\rho: \tilde{X} \to X$ be a cyclic branched covering of $n$ sheets between two projective surfaces having at most abelian quotient singularities. In particular, $H^1(\tilde{X},\CC)$ is endowed with a pure Hodge structure compatible with the monodromy of the covering. Due to the construction of this Hodge structure $H^1(\tilde{X},\CC)$ is naturally isomorphic to $H^1(\tilde{X},\cO_{\tilde{X}}) \oplus H^0(\tilde{X},\Omega_{\tilde{X}}^1)$ where $H^0(\tilde{X},\Omega_{\tilde{X}}^1)$ is
| 3,618
| 3,143
| 3,582
| 3,174
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|
a_i} - p_i^{a_i -1}), \, & \mbox { if }&
4\!\not|
m\\
(n,2) (n, 2^{a_0 -2}) \prod_{i=1}^k (n, p_i^{a_i} - p_i^{a_i -1}),
\, & \mbox { if }& 4|m
\end{array} \right.$$ In particular $$|\varsigma (2, m) | = \left \{\begin{array}{rcl}
2^k, \, & \mbox {\rm if }& \, a_0\leq 1\\
2^{k+1}, \, & \mbox {\rm if }& \, a_0=2\\
2^{k+2}, \, & \mbox {\rm if }& \, a_0\geq 3
\end{array} \right.$$ and $$| \varsigma (p, m) | = \left \{\begin{array}{rcl}
\prod_{i=1}^k (p, p_i -1), \, & \mbox {\rm if }& \, p^2\!\not| m\\
p\prod_{i=1}^k (p, p_i -1), \, & \mbox {\rm if }& \, p^2|m
\end{array} \right.$$ for an odd prime $p$.
Let $m$ be a positive integer. Then $(C_2, C_m, \alpha, \beta)$ is a matched pair if and only if the action $\alpha$ is trivial and there exists a positive integer $t\in [m-1]$ such that $m|t^2 -1$ and $\beta = \beta_t : C_m \times C_2 \rightarrow C_m$ is given by $${\label{eq:2.4.6}}
\beta (b^i, a) = b^{it}, \quad \beta (b^i, 1) = b^i$$ for any $i= 0, \cdots, m-1$. In particular, there are $|\varsigma
(2, m)|$ matched pairs $(C_2, C_m, \alpha, \beta)$.
Indeed, as $\alpha$ is an action we get $b\triangleright a \neq 1
= b \triangleright 1$. Thus $b \triangleright a = a$ which implies that $\alpha$ is trivial. Thus $(C_2, C_m, \alpha, \beta)$ is a matched pair if and only if $\beta ' : C_2 \rightarrow {{\rm Aut}\,}(C_m)$, $\beta' (x) (y) := \beta (y, x)$ is a morphism of groups, so by letting $n=2$ in [(\[eq:2.4.399\])]{} we obtain that $(C_2, C_m,
\alpha, \beta)$ is a matched pair if and only if there exists $t\in [m-1]$ such that $m|t^2 -1$ and $\beta (b, a) = b^t$. The formula [(\[eq:2.4.6\])]{} follows as $\beta$ is an action.
In order to describe all matched pairs $(C_3, C_m, \alpha, \beta)$ we need the following observation.
[\[re:knitauto\]]{} Let $(H, G, \alpha, \beta)$ be a matched pair such that $\alpha$ is an action of $G$ on $H$ as group automorphisms. Then the compatibility condition [(\[eq:2\])]{} from the definition of a matched pair is equivalent to $(g\triangleleft {h_1})
\triangleright h_2 =
| 3,619
| 2,458
| 2,429
| 3,415
| null | null |
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|
every $\kappa \geq 0,q\in {\mathbb{N}}$ there exists $C\geq 1$ such that $$(A_{4})\quad \left\Vert P_{t}f\right\Vert _{q,-\kappa ,\infty }\leq
C\left\Vert f\right\Vert _{q,-\kappa ,\infty }. \label{R7}$$
For $\delta \geq 0$ we denote$$\Phi _{n}(\delta )=\varepsilon _{n}\Lambda _{n}\times \lambda _{n}^{-\theta
_{0}(a+b+\delta )}, \label{R7'}$$where $a$ and $b$ are the constants in Assumption \[A1A\*1\] and \[A2A\*2\] respectively. Notice that $$\Phi _{n}(\delta )\leq \gamma ^{1+\theta _{0}(a+b+\delta )}\Phi
_{n+1}(\delta ). \label{TRa}$$
And, for $\kappa \geq 0,\eta \geq 0$ we set $$\Psi _{\eta ,\kappa }(x,y):=\frac{\psi _{\kappa }(y)}{\psi _{\eta }(x)}%
,\quad (x,y)\in {\mathbb{R}}^{d}\times {\mathbb{R}}^{d}. \label{R7''}$$
Our first result concerns the regularity of the semigroup $P_{t}:$
\[Transfer\] Suppose that Assumption \[A1A\*1\], \[A2A\*2\] , \[A3\] and \[A5\] hold. Moreover we suppose there exists $\delta >0$ such that $$\limsup_{n}\Phi _{n}(\delta )<\infty , \label{TR6}$$$\Phi_n(\delta)$ being given in (\[R7’\]). Then the following statements hold.
**A**. $P_{t}f(x)=\int_{{\mathbb{R}}^{d}}p_{t}(x,y)dy$ with $p_{t}\in
C^{\infty }({\mathbb{R}}^{d}\times {\mathbb{R}}^{d}).$
**B**. Let $n\in {\mathbb{N}}$ and $\delta _{\ast }>0$ be such that $$\overline{\Phi }_{n}(\delta _{\ast }):=\sup_{n^{\prime }\geq n}\Phi
_{n^{\prime }}(\delta _{\ast })<\infty . \label{TR6a}$$We fix $q\in {\mathbb{N}}$, $p>1$, $\varepsilon _{\ast }>0,$ $\kappa \geq 0$ and we put $\mathfrak{m}=1+\frac{q+2d/p_{\ast }}{\delta _{\ast }}$ with $%
p_{\ast }$ the conjugate of $p$. There exist $C\geq 1$ and $\eta_0\geq 1$ (depending on $q,p,\varepsilon
_{\ast },\delta _{\ast },\kappa $ and $\gamma $) such that for every $\eta>\eta_0$ and $t>0$ $$\begin{aligned}
\left\Vert \Psi _{\eta ,\kappa }p_{t}\right\Vert _{q,p} &\leq &C\times
Q_{n}(q,\mathfrak{m})\times t^{-\theta _{0}((a+b)\mathfrak{m}+q+2d/p_{\ast
})(1+\varepsilon _{\ast })}\quad with \label{TR6'} \\
Q_{n}(q,\mathfrak{m}) &=&\Big(\frac{1}{\lambda _{n}^{\theta _{0}(a+b)
| 3,620
| 1,780
| 1,308
| 3,610
| null | null |
github_plus_top10pct_by_avg
|
etail in §\[details\].
For $a<b<c$ positive integers such that $\frac ca=C$ and $\frac ba=
\frac{C-\lambda_0}2+1$, let $$\alpha(t)=\begin{pmatrix}
1 & 0 & 0 \\
t^a & t^b & 0 \\
\underline{f(t^a)} & \underline{f'(t^a)t^b} & t^c
\end{pmatrix}$$ where $\underline{\cdots}$ denotes the truncation modulo $t^c$. The integer $a$ is chosen to be the minimal one for which all entries in this germ are polynomials. Then $\lim_{t\to 0}{{\mathscr C}}\circ \alpha(t)$ is given by $$x^{d-2S}\prod_{i=1}^S\left(zx-\frac {\lambda_0(\lambda_0-1)}2
\gamma_{\lambda_0}y^2 -\frac{\lambda_0+C}2
\gamma_{\frac{\lambda_0+C}2}yx-\gamma_C^{(i)}x^2\right)\quad,$$ where $S$ and $\gamma_C^{(i)}$ are defined in §\[details\].
These curves consist of [*at least two*]{} ‘quadritangent’ conics—that is, nonsingular conics meeting at exactly one point—and (possibly) a multiple kernel line. (Again, the picture drawn here does not capture the subtlety of the situation: these limits may occur already for irreducible singularities.) These curves are item (12) in the list in §\[appendix\], and are studied enumeratively in [@MR2002d:14083], §4.1. They are precisely the curves for which the maximal connected subgroup of the stabilizer is the additive group ${{\mathbb{G}}}_a$.
Details for types IV and V {#details}
--------------------------
[*Type IV:*]{} Let $p=\operatorname{im}\alpha(0)$ be a singular or inflection point of the support of ${{\mathscr C}}$; choose a line in the tangent cone to ${{\mathscr C}}$ at $p$, and choose coordinates $(x:y:z)$ as before, so that $p=(1:0:0)$ and the selected line in the tangent cone has equation $z=0$. The [*Newton polygon*]{} for ${{\mathscr C}}$ in the chosen coordinates is the boundary of the convex hull of the union of the positive quadrants with origin at the points $(j,k)$ for which the coefficient of $x^iy^jz^k$ in the generator $F$ for the ideal of ${{\mathscr C}}$ is nonzero (see [@MR88a:14001], p. 380). The part of the Newton polygon consisting of line segments wi
| 3,621
| 1,189
| 2,835
| 3,404
| 2,789
| 0.776938
|
github_plus_top10pct_by_avg
|
ned}$$ where $ w_t \in \Re^d $ is the state variable at time $t$, $\delta$ is the step size, $U:\Re^d\to\Re$ is a (possibly nonconvex) potential, $\xi:\Re^d \times \Omega \to \Re^d$ is the [*noise function*]{}, and $\eta_k$ are sampled i.i.d. according to some distribution over $\Omega$ (for example, in minibatch SGD, $\Omega$ is the set of subsets of indices in the training sample). Throughout this paper, we assume that $\Ep{\eta}{\xi(x,\eta)}=0$ for all $x$. We define a matrix-valued function $M(\cdot):\Re^d \to \Re^{d\times d}$ to be the square root of the covariance matrix of $\xi$; i.e., for all $x$, $M(x) := \sqrt{\Ep{\eta}{\xi(x,\eta) \xi(x,\eta)^T}}$, where for a positive semidefinite matrix $G$, $A=\sqrt{G}$ is the unique positive semidefinite matrix such that $A^2 = G$.
In studying the generalization behavior of SGD, earlier work [@jastrzkebski2017three; @he2019control] propose that be approximated by the stochastic process $y_{(k+1)\delta} = y_{k\delta} - \delta \nabla U(y_{k\delta}) + \sqrt{\delta} M(y_{k\delta}) \theta_k$ where $\theta_k\sim \N(0,I)$, or, equivalently: $$\begin{aligned}
&d y_t = -\nabla U(y_{k\delta}) dt + M(y_{k\delta}) dB_t
\numberthis \label{e:discrete-langevin}\\
&\quad \text{for } t\in[k\delta, (k+1)\delta],\end{aligned}$$ with $B_t$ denoting standard Brownian motion [@karatzas1998brownian]. Specifically, the non-Gaussian noise $\xi(\cdot,\eta)$ is approximated by a Gaussian variable $M(\cdot) \theta$ with the same covariance, via an assumption that the minibatch size is large and an appeal to the central limit theorem.
The process in can be seen as the Euler-Murayama discretization of the following SDE: $$\begin{aligned}
\numberthis \label{e:exact-sde}
d x_t = -\nabla U(x_t) dt + M(x_t) dB_t.\end{aligned}$$ We let $p^*$ denote the invariant distribution of .
We prove quantitative bounds on the discretization error between , and , as well as convergence rates of and to $p^*$. Our bounds are in Wasserstein-1 distance (denoted by $W_1(\cdot, \cdot)$ in the following). We pre
| 3,622
| 2,208
| 2,410
| 3,298
| 2,930
| 0.776006
|
github_plus_top10pct_by_avg
|
(cYBE and mcYBE) respectively. We adopt some notation $X\wedge Y = X\otimes Y - Y \otimes X$ and define e.g. $$r= T_1 \wedge T_2 + T_3 \wedge T_4 +\dots \ , \quad RX = {\operatorname{Tr}}_2 ( r (\mathbb{I}\otimes X)) \ .$$
We define an inner product by the matrix trace of generators, ${\operatorname{Tr}}(T_{A} T_{B})$, and lower and raise indices with this inner product and its inverse. In this way the $r$-matrix acts as $$\label{eq:rmatdef}
R(T_{A}) \equiv R_{A}{}^{B}T_{B} \ , \quad R_{A}{}^{B} = {\operatorname{Tr}}\left( {\operatorname{Tr}}_2 ( r (\mathbb{I}\otimes T_{A})) T^{B} \right) \ .$$
Suppose we have a $\mathbb{Z}_{2}$ grading $\mathfrak{f} = \mathfrak{g}\oplus \mathfrak{k}$ for a subgroup $\mathfrak{g}$. Let $T_{A}$ be generators for $\mathfrak{f}$, $T_{\alpha}$ those of $\mathfrak{g}$ and $T_{i}$ the remaining orthogonal generators of $\mathfrak{k}$. We introduce a projector to the coset defined by $P(T_{\alpha})= 0$ and $P(T_{i})= T_{i}$ or, in matrix form, $$P(T_{A}) \equiv P_{A}{}^{B}T_{B} \ , \quad P_{A}{}^{B} = {\operatorname{Tr}}\left( P( T_{A}) T^{B} \right) \ .$$ We also define the adjoint action for $g\in F$ by $${\operatorname{Ad}}_{g} T_{A} \equiv gT_{A} g^{-1} \equiv D_{A}{}^{B}(g) T_{B} \ , \quad D_{AB} = {\operatorname{Tr}}(g T_{A} g^{-1} T_{B} ) \ .$$
Let the two-dimensional worldsheet field $g$ be a coset representative for $F/G$ with which we define the currents $$J_\pm = J_{\pm}^{A}T_{A} = g^{-1} \partial_{\pm} g \ , \quad J_{\pm}^{A}= {\operatorname{Tr}}(g^{-1} \partial_{\pm} g T^{A}) \ ,$$ where we use light-cone coordinates $\partial_\pm = \partial_0 \pm \partial_1$.
The standard (bosonic) $\sigma$-model whose target is the coset space $F/G$ is $$\label{eq:cosetPCM}
{\cal L } = {\operatorname{Tr}}(J_+ P(J_-) ) \ .$$ To define the Yang-Baxter model first we let $$R_{g} = {\operatorname{Ad}}_{g^{-1}} R {\operatorname{Ad}}_{g} \ , \quad (R_{g})_{A}{}^{B} = D(g)_{A}{}^{C}R_{C}{}^{D}D(g^{-1})_{D}{}^{B} \ ,$$ and define the operator $${\cal O} = \mathbb{I} - \eta R_{g}P \ , \quad {
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)+\sum_{i=1}^k\chi(G[W_i])\le|P|+\sum_{i=1}^k|Z_i|^4$$ $$\le(k-1)\omega(G)+(\sum_{i=1}^k|Z_i|)^4\le(k-1)\omega(G)+(\omega(G))^4,$$ as required.
We can turn this estimate into a polynomial time algorithm as required, using the fact that the proof of Theorem 0 is constructive. In particular, we use that, given a family of segments in the plane, one can efficiently find a subfamily $K$ of pairwise disjoint segments and a proper coloring of the disjointness graph with at most $|K|^4$ colors. This readily follows from the proof of Theorem 0, based on the four easily computable (semi-algebraic) partial orders on the family of segments, introduced in [@LMPT94].
Our algorithm finds the sets $V_i$, as in the proof. However, finding $W_i$ and a maximum size clique $Z_i\subseteq W_i$ is a challenge. Instead, we use the constructive version of Theorem 0 to find $Z_i\subseteq W_i$ and a proper coloring of $G[W_i]$. The definition of $W_i$ remains unchanged. Next, the algorithm identifies the piercing points.
The algorithm outputs the clique $K=\bigcup Z_i$ and the coloring of $G$. The latter one is obtained by combining the previously constructed colorings of the subgraphs $G[W_i]$ (using disjoint sets of colors for different subgraphs), and coloring each remaining vertex by a previously unused color, associated with one of the piercing points the corresponding segment passes through. $\Box$
[**Proof of Theorem 1.**]{} Consider the set of all lines in the [ *projective*]{} space $\mathbb{P}^d$ that contain at least one segment belonging to $V(G)$. Let $\bar{G}'$ denote the disjointness graph of these lines. Obviously, we have $\omega(\bar{G}')\le\omega(G)$. Thus, Theorem 2(ii) implies that $$\chi(\bar{G}')\le (\omega(\bar{G}'))^2\le(\omega(G))^2.$$
Let $C$ be the set of lines corresponding to the vertices of a maximum complete subgraph in $\bar{G}'$. Fix an optimal proper coloring of $\bar{G}'$. Suppose that we used $k$ “planar” colors (each such color is given to a set of lines that lie in the same plane) and $\c
| 3,624
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|
S\times I)\to{\mathbb{R}}$ has an unique extension $\tilde B(\cdot,\cdot):H_1\times H_2\to{\mathbb{R}}$ which satisfies \[csda36\] |B(,v)|M\_[H\_1]{}[v]{}\_[H\_2]{}H\_1, vH\_2 and \[csda37\] B(v,v)c’[v]{}\_[H\_1]{}\^2vH\_2. We see that actually $$\begin{aligned}
\label{coex}
\tilde B(\tilde\phi,v)
={}&{\left\langle}\phi,S_0{{\frac{\partial v}{\partial E}}}{\right\rangle}_{L^2(G\times S\times I)}
-{\left\langle}\phi,\omega\cdot\nabla_x v{\right\rangle}_{L^2(G\times S\times I)} \nonumber\\
&+C{\left\langle}\phi,S_0v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}\phi,(\Sigma^*-K_C^*) v{\right\rangle}_{L^2(G\times S\times I)} \nonumber\\
&+{\left\langle}q_{|\Gamma_+},\gamma_+(v){\right\rangle}_{T^2(\Gamma_+)}+{\left\langle}p_0,S_0(\cdot,0) v(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)},\end{aligned}$$ when $\tilde\phi=(\phi,q,p_0,p_m)\in H_1$ and $v\in H_2$. In addition, since for $v\in C^1(\ol G\times S\times I)$ we have ${\left\Vert \gamma_-(v)\right\Vert}_{T^2(\Gamma_-)}\leq {\left\Vert \gamma(v)\right\Vert}_{T^2(\Gamma)}$, it follows that \[csda39\] |F(v)| & |[**f**]{},v\_[L\^2(GSI)]{}|+|[**g**]{},\_-(v)\_[T\^2(\_-)]{}|\
& \_[L\^2(GSI)]{}[v]{}\_[L\^2(GSI)]{}+\_[T\^2(\_-)]{}[(v)]{}\_[T\^2()]{}, and therefore, since $C^1(\ol{G}\times S\times I)$ is dense in $H_1$, the linear form $F:C^1(\ol{G}\times S\times I)\to{\mathbb{R}}$ has a unique bounded extension, which we still denote by $F$, $$\begin{aligned}
\label{Fex}
F:H_1\to{\mathbb{R}};\quad
F(\tilde{\phi})={\left\langle}{\bf f},\phi{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}{\bf g}, q{\right\rangle}_{T^2(\Gamma_-)},\end{aligned}$$ when $\tilde\phi=(\phi,q,p_0,p_m)\in H_1$. Recall also that the embedding $H_2\subset H_1$ is continuous.
We need the following result, so called [*Lions-Lax-Milgram Theorem*]{} (generalized Lax-Milgram Theorem).
\[glm\] Let $X$ and $Y$ be Hilbert spaces, with $Y$ continuously embedded into $X$. Assume that $B(\cdot,\cdot):X\times Y\to{\mathbb{R}}$ is a bilinear form satisfying the following properties with $
| 3,625
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|
above would be equivalent to the Jacobi identify for $n^{(3)}$ interpreted as structure constants if the terms $m_i^{(4)}$ were absent. The presence of $m_i^{(4)}$ is the evidence that $n^{(3)}$ cannot be used as structure constants.
Often the relation between YM and GR indicated by the double-copy procedure is symbolically represented as (YM)${}^2 = $(GR). This expression is actually misleading, because neither the color factors or the propagators are squared in the gravity theory. Instead, the identification of color factors with kinematic factors (e.g. $f_{abc}$ with $n_{\lam\m\n}(p, q, -(p+q))$ for tree-level 3-point amplitudes) identifies YM directly with GR. It is more appropriate to use (YM)${}^{\prime}$ = GR as the symbolic representation of this connection. The prime on (YM) indicates the modification of YM theory by the replacement of color factors by kinematic factors.
Since the color factors are composed of structure constants of the gauge group, we are naturally led to consider the possibility of gauge symmetries with structure constants involving kinematic factors. This was precisely what we did in Sec.\[GaugeSymm\], which led to new formulations of gravity theories as generalized YM theories in Sec.\[YM\]
Apparently, the hope for a direct matching between structure constants and the kinematic factors is too naive. First, as the 3-point amplitude is defined on-shell, the structure constant can be different from $n_{\lam\m\n}(p, q, -(p+q))$ when it is off-shell. Secondly, even if the 3-point amplitudes agree with structure constant, it is not clear if higher-point amplitudes will automatically agree with the corresponding color factors, as there will be different on-shell conditions at work. (In fact, eq.(\[nn+m\]) says that the structure constants for 4-point amplitudes are not to be given by the structure constants for 3-point amplitudes due to the extra term $m^{(4)}_i$.) In general, as the BCJ duality only holds on-shell, the correspondence between structure constants and kinematic factors in
| 3,626
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|
ely poor since it considers only the largest gap between two distributions, which provides little information.
Discussion
----------
MGoF’s learning procedure of anomalous probability hypothesis is inefficient. To maintain a comprehensive knowledge of anomalies, MGoF has to reserve a single hypothesis entry for every type of them. But in reality, it is always the case that we face the heterogeneity of outliers. In the Koubei data set, there can be tens of anomalous distributions caused solely by centralized click farming. It takes a long time to discover every possible type of anomaly. Besides, if there happens to be more than $c_{th}$ anomalous distributions of the same type, later discovered collections will no longer declared to be anomalous any more.
However, in SDD-R and SDD-E, that is not a problem since it can map and gather all anomalies together and draw a universal boundary between them and all normal collections. These techniques are suitable to all typical divergence metrics and consume little computation power(except dynamic SDD-E). The only drawback is that they require comprehensive estimation of target distributions. Although other parameters need estimation as well, they are naturally addressable under big data circumstances.
Conclusion {#sec:conclusion}
==========
This paper proposes a series of collective anomaly detection techniques, which helps detect data manipulations in modern data pipelines and data centres. Different from existing algorithms designed for collective anomalies, our approach employs statistical distance as the similarity measurement. We explored several technical points involved in the design of the algorithm and performed a thorough experiment to test its efficiency. The comparison experiment also illustrated the advantages of our technique. It can be concluded that the our technique can efficiently discover anomalies within the data collections and the classifier is sensitive enough toward real world data manipulations.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]
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$\delta j_\mu (x)/\delta A^t_\nu
(x^{\prime\prime})|_{A^t=A^{ext}}=\Pi_{\mu\nu}(x,x^{\prime\prime}\vert
A_\mu^{ext})$. The external (constant and homogeneous) classical magnetic field is described by $A_\mu^{ext}(x)=1/2F_{\mu\nu}^{ext}x^\nu$, where the electromagnetic field tensor $F_{\mu \nu}^{ext}=\partial_\mu
A_\nu^{ext}-\partial_\nu A_\mu^{ext}=B (\delta_{\mu 1}\delta_{\nu
2}-\delta_{\mu 2}\delta_{\nu 1})$ and $F^*_{\mu
\nu}=\frac{i}{2}\epsilon_{\mu \nu \rho \kappa}F^{\rho \kappa}$ is its dual pseudotensor.
To understand what follows it is necessary to recall some basic results developed in refs. [@shabad1],[@shabad2]. The presence of the constant magnetic field creates, in addition to the photon momentum four-vector $C^{4}_\mu=k_\mu$, three other orthogonal four-vectors which we write as four-dimensional transverse $k_\mu C^{i\mu}=0$ for $i=1,2,3$. These are $C^{1}_\mu=
k^2 F^2_{\mu \lambda}k^\lambda-k_\mu (kF^2 k)$, $C^{2}_\mu=F^{*}_{\mu \lambda}k^\lambda$, $C^{3}_\mu=F_{\mu
\lambda}k^\lambda$ ($C^{1,2,3}_{\mu}k^{\mu}=0$). We have $C^{4}_\mu C^{4\nu}=k_\mu k^\nu=0$ on the light cone. One gets from these four-vectors three basic independent scalars $k^2$, $kF^2k$, $kF^{*2}k$, which in addition to the field invariant ${\cal F}=\frac{1}{4}F_{\mu \rho}F^{\rho \mu}=\frac{1}{2}B^2$, are a set of four basic scalars of our problem.
In momentum space it can be written the eigenvalue equation [@shabad1] $$\Pi_{\mu\nu}(k,k^{\prime\prime}\vert A_\mu^{ext})=\sum_i
\pi^{(i)}_{n,n^\prime} a^{(i) \nu }a^{(i)}_\mu/(a^{(i)\nu
}a^{(i)}_\nu ) \label{2}$$ In correspondence to each eigenvalue $\pi^{(i)}_{n,n^\prime}$ $i=1,2,3$ there is an eigenvector $a^{(i)\nu }$. The set $a^{(i)\nu }$ is obtained by simply normalizing the set of four vectors $C^{i}_\mu$. ($C^{4}_\mu=k_\mu$ leads to a vanishing eigenvalue due to the four-dimensional transversality property $\Pi_{\mu\nu}(k,k^{\prime\prime}\vert A_\mu^{ext})k_\mu=0$). The solution of the equation of motion (\[sdpmBF\]) can be written as a superposition of eigenwaves given by $$
| 3,628
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Next
.Save
End With
End Sub
coupled with a Custom Document Property named 'Counter', whose value you initially set to 0. Should you need to re-set the counter for some reason, simply edit the 'Counter' Custom Document Property accordingly.
Then, wherever you want the count to appear, add a DOCPROPERTY field that references the 'Counter' property. You can do this via Insert|Quick Parts|Field, or by pressing Ctrl-F9 to create a pair of field braces (ie '{}') and typing 'DOCPROPERTY Counter' between them, so that you end up with '{DOCPROPERTY Counter}'. You can use a numeric picture switch in the DOCPROPERTY field to format the number to have a particular lenght (e.g. 4 diigits). In that case the field code would appear as '{DOCPROPERTY Counter # 0000}'. When you're done, press F9 to update the field. You can then copy & paste the field to wherever else you might want also it to appear in your document.
If the documents you want this to apply to share a common template, but that template isn't used for other documents also, you could simply add the macro to that template; otherwise you might need to add it to each of the documents concerned and save them in the docm or doc format.
Q:
Microcontroller Interface with Modbus Serial
I'm a beginner and I'm trying to acquire data from a microcontroller with Modbus serial protocol in order to create an interface with LabVIEW. I'm using a Prolific Technology USB-RS232 converter and Windows 7 as OS.
I can't read data from the micro. I know that read only varibles are in registers from 100hex and 1FFhex and read/write variables are from 200hex and 2FFhex.
How does the memory of micro need to be used? How do I define holding registers and input registers?
Thanks in advance for any advice.
Andrea.
A:
Here are a few basic Modbus Commands you need to know:
0x01 - Read Coil Status
0x02 - Read Input Registers
0x03 - Read Holding Registers
Holding registers are READ ONLY registers. They typically hold values that the microcontroller measured or calculated. Input registe
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clusion $D\subset V(x)$. The formula for $p(\alpha,d)$ uses the coordinate system and scaling property of stable processes in Lemma \[scaled\] as well as the identity for the first exit from a sphere given by Theorem \[BGR\].
We are now ready to prove our main result.
Suppose we condition on the previous positions of the walk-on-spheres, $\rho_0,\dots, \rho_{k-1}$ as well as on the event $\{N>k-1\}$. Thanks to stationary and independent increments as well as isotropy in the law of a stable process, we can always choose a coordinate system, or equivalently reorient $D$ in such a way that $\rho_k = |\rho_k|{\rm\bf i}$. This has the implication that, with the aforesaid conditioning, the random variable $\mathbf{1}_{\{N = k\}}$ is independent of $\rho_0,\dots, \rho_{k-1}$ and equal in law to $I_D(\rho_{k-1})$, where we have abused our original notation to indicate the initial position of $X$ in the definition of $I_D$. Similarly, with the same abuse of notation, the event $I_V(\rho_{k-1})$ is independent of $\rho_0,\dots, \rho_{k-1}$ and equal in law to a Bernoulli random variable with probability of success $p = p(\alpha, d)$. In particular, the sequence $I_V(\rho_k)$, $k\geq 0$ is a sequence of Bernoulli trials. That is to say, if we define $$\Gamma = \min\{k\geq 1\colon I_V(\rho_k) = 1 \},$$ then it is geometrically distributed with parameter $p$. Thanks to Corollary \[indicators\], we also have that $\mathbb{P}_x(I_D\geq I_V)|_{x = \rho_k}=1$, $k< N$, that is to say, $\{I_V(\rho_k)=1\}$ almost surely implies $\{I_D(\rho_k)=1\}$, for $k<N$, and hence $$\min\{n\geq 1\colon I_D(\rho_k) = 1 \}\leq \min\{n\geq 1\colon I_V(\rho_k) = 1 \}$$ almost surely. In other words, we have $N\leq \Gamma$, almost surely, as required.
Non-convex domains {#non_convex}
==================
The key element in the proof of Theorem \[main\] is the comparison of the event that the next step of the walk-on-spheres exits the domain $D$ with the event that the next step of the walk-on-spheres exits a larger, more regular domain. More pre
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| 2,717
| 0.777613
|
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|
tilde {\bf P}_{C,0})$.
We assume that the stopping powers for $j=2,3$ satisfy $$\begin{aligned}
{}& S_j\in C^2(I, L^\infty(G)), \label{Sj-ass:1} \\[2mm]
{}& \kappa_j:=\inf_{(x,E)\in \ol{G}\times I}S_j(x,E)>0, \label{Sj-ass:2} \\[2mm]
{}& \nabla_x S_j\in L^\infty(G\times I). \label{Sj-ass:3}\end{aligned}$$
We give here only the following concluding result, which can be proven using the methods of section \[m-d\] and those of [@tervo14 Section 5.3]. Recall that by the conventions adopted above, ${\bf f}_j=e^{CE}f_j$ and ${\bf g}_j=e^{CE}g_j$, where $C$ is as defined in .
\[m-d-j-co1\] Suppose that the assumptions (\[scateh\]), (\[colleh\]), (\[csda3aa\]), (\[csda4aa\]) (with $c>0$) and , , are valid. Furthermore, suppose that ${f}\in L^2(G\times S\times I)^3$, and $g\in T^2(\Gamma_-)\times H^1(I,T^2(\Gamma_-'))^2$ is such that the compatibility condition \[comp-d-j\] g\_j(,,E\_[m]{})=0,j=2,3, holds. Then the problem - has a unique solution $\psi\in {{{\mathcal{}}}H}_{\bf P}(G\times S\times I^\circ)$. In addition, there exists a constant $C_1>0$ such that *a priori* estimate \[diss-co-es\] \_[L\^2(GSI)\^3]{} C\_1(\_[L\^2(GSI)\^3]{}+ \_[T\^2(\_-)H\^1(I,T\^2(\_-’))\^2]{}), holds.
Existence of Solutions Based on the Theory of Evolution Equations {#evo-op}
-----------------------------------------------------------------
For collision operator of special type, the existence result based on the theory of evolution operators is valid also for the coupled system. One of the important features of this approach is that it yields more regularity for the solution.
We assume that the collision operator $K=(K_1,K_2,K_3)$ is of the form (see Remark \[cosdare1\]) \[ec1moda\] (K\_j)(x,,E)=\_[k=1]{}\^3\_S\_[kj]{}(x,’,,E)\_k(x,’,E)d’,j=1,2,3.
For a fixed $E\in I$ we define bounded linear operators $L^2(G\times S)\to L^2(G\times S)$ by $$\begin{gathered}
(\Sigma_1(E)v)(x,\omega):=\Sigma_1(x,\omega,E)v(x,\omega), \\
(\ol K_1(E)v)(x,\omega):=\int_S\tilde\sigma_{11}(x,\omega',\omega,E)v(x,\omega') d\omega'.\end{gathered}$$
| 3,631
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|
lpha-\beta)\beta^{-1}\tau}I(\theta)^{1/2} \left(\int \theta {\left\verte^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\ast\theta\right\vert}^2 d\eta \right)^{1/2} \\
& \leq (1-e^{-\delta\tau})I(\theta) + Ce^{(2-2\alpha-2\beta)\beta^{-1}\tau + \delta\tau}. \end{aligned}$$ The rest of the proof follows similarly to the case $m > 1$ using Theorems \[thm:rel\_entropy\] and \[thm:CK\]. This concludes the proof of Theorem \[thm:IA\].
A generalization of Talagrand’s inequality [@CarrilloMcCannVillani03] shows that $\theta \rightarrow \theta_M$ also in the Euclidean Wasserstein distance.
Infinite Length-Scales
======================
We now turn to the proofs of Theorem \[thm:Decay\] and Theorem \[thm:IA2\] in the case ${\nabla}{\mathcal{K}}\not\in L^1$. In order to properly extend the work of the previous section, we must estimate the quantities ${\|e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\ast\theta\|}_p$ appearing in and the proof of Lemma \[lem:rescaled\_inftybdd\]. However, ${\nabla}{\mathcal{K}}\not\in L^1({\mathbb R}^d)$ and Young’s inequality is not sufficient; in fact we will not bound ${\|e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\ast\theta\|}_p$ uniformly in time but instead bound the rate at which it grows. We separately estimate the growth of the quantities ${\|\lambda^d{\nabla}{\mathcal{K}}(\lambda \cdot)\mathbf{1}_{B_1(0)}\|}_{1}$ and ${\|\lambda^d{\nabla}{\mathcal{K}}(\lambda \cdot)\mathbf{1}_{{\mathbb R}^d\setminus B_1(0)}\|}_{p}$ as $\lambda \rightarrow \infty$. Using ${\left\vert{\nabla}{\mathcal{K}}(x)\right\vert} \lesssim {\left\vertx\right\vert}^{-\gamma}$ for sufficiently large ${\left\vertx\right\vert}$, if $\gamma < d$, then for large $\lambda$, $$\begin{aligned}
\int \lambda^d{\left\vert{\nabla}{\mathcal{K}}(\lambda y)\right\vert}\mathbf{1}_{B_1(0)}({\left\verty\right\vert}) dy & = \int_{{\left\verty\right\vert} \leq \lambda} {\left\vert{\nabla}{\mathcal{K}}(y)\right\vert} dy \nonumber \\
& = \int_{S^{d-1}}\int_0^{\lambda} {\left\vert{\nabla}{\mathcal{K}}(\rho\omega)\right\vert}r\rho^
| 3,632
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mma < d$, $$\begin{aligned}
{\|e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\ast \theta\|}_\infty & \leq \left( {\| e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\mathbf{1}_{B_1(0)} \ast \theta\|}_\infty + {\| e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\mathbf{1}_{{\mathbb R}^d \setminus B_1(0)} \ast \theta\|}_\infty \right) \nonumber \\
& \lesssim \left(1 + e^{(d-\gamma)\tau}\right)\left({\|\theta\|}_\infty + M\right) \\ & \lesssim 1 + e^{(d-\gamma)\tau} \lesssim e^{(d-\gamma)\tau}. \label{ineq:velocity_bounded_IA2} \end{aligned}$$ Similarly, if $\gamma = d$ then, for all $\delta > 0$, $${\|e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\ast \theta\|}_\infty \lesssim 1 + \tau \lesssim_\delta e^{\delta \tau}.$$ The growth of in time is the source of the degraded convergence rate observed in . As noted above, this is a manifestation of slow decay in the kernel, which causes growth of $e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)$ in $L^1_{loc}$. Indeed, computing the decay of the relative entropy (with linear or nonlinear diffusion) as above with , $$\begin{aligned}
\frac{d}{d\tau}H(\theta(\tau)|\theta_M) & = \leq -I(\theta) + e^{(1-\alpha-\beta)\beta^{-1}\tau}I(\theta)^{1/2} \left(\int \theta {\left\verte^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\ast\theta\right\vert}^2 d\eta \right)^{1/2} \nonumber \\
& \leq (1-e^{-2\delta\tau})I(\theta) + Ce^{ (2(1-\alpha-\beta)\beta^{-1} + 2(d-\gamma) + 2\delta)\tau}. \end{aligned}$$ As before, Theorems \[thm:rel\_entropy\] and \[thm:CK\] imply, $${\|\theta(\tau) - \theta_M\|}_1 \lesssim e^{-\tau\min\left(1,1 + \gamma - \beta^{-1} - \delta\right)}.$$ Re-writing in terms of $x$ and $t$ and interpolating against completes the proof. The corresponding argument follows also for $\gamma = d$, absorbing the mild growth of ${\|e^{d\tau}{\mathcal{K}}(e^{\tau}\cdot)\ast\theta\|}_\infty$ into the $\delta$ already introduced.
Acknowledgments
===============
The author would like to thank Andrea Bertozzi, Thomas Laurent and Nancy Rodríguez for helpful discussions and guidance, a
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airs $(H, G, \alpha, \beta)$ and classify up to an isomorphism all bicrossed products $H\, {}_{\alpha}\!\! \bowtie_{\beta} \,
G$.*
The motivation for the above problem is triple: first of all, the problem presents an interest in itself in group theory. On the other hand the construction of the bicrossed product provides the easiest way of constructing finite quantum groups [@masu], hence the classification theorems from group level lead us to classification theorems for finite quantum groups. Finally, the bicrossed product construction at the level of groups served as a model for similar constructions in other fields of mathematics like: algebras [@cap], coalgebras [@CIMZ], groupoids [@AA], Hopf algebras [@Takeuchi], locally compact groups [@baaj] or locally compact quantum groups [@VV], Lie Algebras [@Mic] or Lie groups [@Kro]. Thus, the above problem can be easily formulated for each of the above different levels where the bicrossed product construction was made. For instance, at the level of algebras (the bicrossed product of two algebras is also called *twisted tensor product algebra*) the first steps were already made in the last years: the story started with [@CIMZ Examples 2.11] where all bicrossed product between two group algebras of dimension two are completely described and classified. Recently, the classification of all bicrossed product between the algebras $k^2$ and $k^m$ was finished in [@Pena] and the description of some bicrossed products between two polynomial algebras $k[X]$ and $k[Y]$ was started in [@gucci]. On the other hand, in [@Jara] only a sufficient condition for the isomorphism between two bicrossed products of algebras that fix one of the algebra is given under the name of *invariance under twisting* problem.
This paper is devoted to the classification part of the factorization problem at the group level. Namely we shall ask the following question: when are two bicrossed products $H\,
{}_{\alpha}\!\! \bowtie_{\beta} \, G$ and $H\, {}_{\alpha'}\!\!
\bowtie_{\beta'} \, G$ isomorphic?
| 3,634
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|
}=0,\quad
\psi(\cdot,\cdot,E_{\rm m})=0.\end{gathered}$$ For simplicity we denote ${1\over{S_0(E)}}f$ and ${1\over{S_0(E)}} K$ again by $f$ and $K$. Assume that $I=[0,\infty[{}=:{\mathbb{R}}_+$ and that $K$ is of the *Volterra type operator* (cf. [@engelnagel pp. 447-452]) $$(K\psi)(x,\omega,E)=\int_0^E\int_S
\sigma(x,\omega',\omega,E-E',E)\psi(x,\omega',E') d\omega' dE'.$$ In other words, $K\psi=\int_{S\times I} \tilde{\sigma}(\cdot,\omega',\cdot,E',\cdot)\psi(\cdot,\omega',E') d\omega' dE'$ for a differential cross section $\tilde{\sigma}$ of the form $\tilde{\sigma}(x,\omega',\omega,E',E)=\chi_I(E-E')\sigma(x,\omega',\omega,E-E',E)$, where $\chi_I$ is the characteristic function of $I$. Assume additionally that $\sigma$ has a decomposition $$\sigma(x,\omega',\omega,E-E',E)=\sigma_1(x,\omega',\omega,E)\sigma_2(E-E').$$ Let $K_1(E)$ (for a fixed $E$) be a linear operator $L^2(G\times S)\to L^2(G\times S)$ defined by $$(K_1(E)\phi)(x,\omega)=\int_S\sigma_1(x,\omega',\omega,E)\phi(x,\omega') d\omega'.$$ Then we find that $$(K\psi)(x,\omega,E)=\int_0^E
\sigma_2(E-E')K_1(E)\psi(E') dE'.$$
Define an [*extended space*]{} by ${{{\mathcal{}}}X}:=L^2(G\times S)\times L^1({\mathbb{R}}_+,L^2(G\times S))$ and a linear operator ${{{\mathcal{}}}A}(E):{{{\mathcal{}}}X}\to {{{\mathcal{}}}X}$ for a fixed $E$ by (here the argument of ${\mathbb{R}}_+$ is denoted by $s$) & D([A]{}(E))=D:=W\_[-,0]{}\^2(GS)H\^[1,1]{}(\_+,L\^2(GS)),\
& [A]{}(E):=
(
A(E)&\_0B(E)&[d]{}
)
, where $$H^{1,1}({\mathbb{R}}_+,L^2(G\times S)):= \{F\in L^1({\mathbb{R}}_+,L^2(G\times S))\ |\ F'\in L^1({\mathbb{R}}_+,L^2(G\times S))\},$$ is the domain of the derivative operator $\frac{d}{ds}:L^1({\mathbb{R}}_+,L^2(G\times S))\to L^1({\mathbb{R}}_+,L^2(G\times S))$, the linear operator $A(E):L^2(G\times S)\to L^2(G\times S)$ with domain $\tilde{W}^2_{-,0}(G\times S)$ (independent of $E$) is given by, $$A(E)\phi=-\Big({1\over { S_0(E)}}\omega\cdot\nabla_x\phi+{1\over { S_0(E)}}\Sigma(E)\phi
-{1\over{ S_0(E)}}{{\frac{\partial S_0}{\partial E}}}(E)\phi\Big),$$
| 3,635
| 1,996
| 2,458
| 3,359
| null | null |
github_plus_top10pct_by_avg
|
stributing the colloidal dumbbells in the simulation box and with random orientations. The initial distance between the colloid-1 and colloid-2 in a dumbbell is set smaller than $\lambda+\Delta$. In contrast, the initial distance between any two colloidal species that belongs to different dumbbells is larger than ${\sigma _{1}+\Delta}$. In this way, no two colloidal dumbbells bind together in the initial stage of the simulation. In addition, all colloids are located outside of the droplets. The initial droplet diameter is set to $8\sigma _{2}$ and shrunk at constant rate.
Results and Discussion
======================
Asymmetric wetting properties and symmetric sizes {#s:fluid1}
-------------------------------------------------
![Snapshots of the simulation for colloidal dumbbells with symmetric sizes and droplets at the energy ratio $k=0.1$. Results are shown at two different stages of the time evolution (a) after $2.5\times10^{5}$ MC cycles several colloidal dumbbells (bright yellow and dark red spheres) are trapped at the surface of the droplets (gray spheres) and (b) after $10^{6}$ MC cycles the stable clusters that are formed due to the droplets are composed of different colored colloids, that is, blue and green spheres represent colloidal species 1 and colloidal species 2, respectively. Open cluster structures with a compact core by colloid 1 and protruding arms by colloid 2 can be observed.[]{data-label="fig:snap"}](fig3){width="9cm"}
![Radial distribution functions, $g_{i\textrm{d}}(r)$ ($i=1,2$), for colloid 1-droplet (solid lines) and colloid 2-droplet (dashed lines) as a function of the scaled distance $r/\sigma$ at energy ratio $k=0.1$. Shown are results at different stages of the computer simulation. (See the notation in Table \[tab:peak-position\].)[]{data-label="fig:fkt1"}](fig4){width="9cm"}
----- ------- ---------------------- ------------------------- -------
$g_{1\textrm{d}}(r)$ $g_{2\textrm{d}}(r)$
| 3,636
| 2,190
| 1,645
| 3,808
| 2,977
| 0.77565
|
github_plus_top10pct_by_avg
|
s-box ones. Direct box diagrams usually present a pinch singularity. This is because the poles appearing in the baryonic propagators get infinitesimally close to one another. In our integrals the denominators appearing in the baryonic propagators also contain terms proportional to $M_\Lambda-M_N$ and $M_\Sigma-M_\Lambda$, and this avoids the singularity.
The integrals entering in the expression of the amplitudes are the $J$ and $K$ defined in Appendix \[sec:mi\]. The amplitude for the first type of box diagram (Fig. \[box1\]) is
![Box diagram contributing at NLO.[]{data-label="box1"}](box1g)
$$\begin{aligned}
V_f=&
i\frac{G_Fm_\pi^2g_A^3}{8f_\pi^3}
(3-2{\vec{\tau}_1}\cdot{\vec{\tau}_2})
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon}
\nonumber\\\times&\nonumber\,
\frac{1}{(l+q)^2-m_\pi^2+i\epsilon}\,
\frac{1}{k_N^2-M_N^2+i\epsilon}a
\\\times&\nonumber\,
\frac{(l^\rho+q^\rho)(l^\nu+q^\nu)l^\mu}{r_N^2-M_N^2+i\epsilon}
\\\times&\nonumber\,
{\overline{u}}_1({\overline{E}},{\vec{p}\,'})
\gamma_\rho\gamma_5
({\cancel{k}_N}+M_N)
(A+B\gamma_5)
u_1(E_p^\Lambda,{\vec{p}})
\\\times&\nonumber\,
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})
\gamma_\nu\gamma_5
({\cancel{r}_N}+M_N)
\gamma_\mu\gamma_5
u_2(E_p,-{\vec{p}}) \,.\end{aligned}$$
Using the heavy baryon expansion, $$\begin{aligned}
V_f&=-\frac{G_Fm_\pi^2g_A^3}{32M_N f_\pi^3}
(3-2{\vec{\tau}_1}\cdot{\vec{\tau}_2})\Bigg[
-4A M_N \left(4K_{22}
+ K_{11} {\vec{q}}^2
\nonumber\right.\\+&\left.\nonumber
2 K_{23} {\vec{q}}^2+K_{35} {\vec{q}}^2+(5-\eta)
K_{34}\right){\vec{\sigma}_1}\cdot {\vec{q}}\nonumber\\-&\nonumber
2B K_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot {\vec{q}})
+
2B K_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot {\vec{p}})
\nonumber\\-&\nonumber
4i A M_N K_{22} \left({\vec{\sigma}_1}\times{\vec{\sigma}_2}\right)\cdot{\vec{q}}-
2 B \left({\vec{p}}\cdot
{\vec{q}}-{\vec{q}}^2\right)
K_{22} {\vec{\sigma}_1}\cdot {\vec{\sigma}_2}\nonumber\\+&\nonumber
2 i B \left(K_{11} {\vec{q}}^2+2
K_{23} {\vec{q}}^2+K_{
| 3,637
| 1,228
| 2,534
| 3,511
| null | null |
github_plus_top10pct_by_avg
|
er surface whose Kodaira dimension is not equal to $- \infty$, then $$Y(M)=-4\sqrt{2}\pi\sqrt{(2\chi+3\sigma)(\tilde{M})},$$ where $\sigma$ denotes the signature and $\tilde{M}$ is the minimal model of $M$. Now based on this evidence, one can ask if the blowing-up does not change the Yamabe invariant of a closed orientable $4$-manifold with nonpostive Yamabe invariant, namely
\[ques1\] Let $M$ be a closed orientable $4$-manifold with $Y(M)\leq 0$. Is there an orientation of $M$ such that $Y(M\sharp\ l\
\overline{\Bbb CP^{2}})=Y(M)$ for any integer $l>0$? What about in higher dimensions?
Further one can also ask whether the analogous statement holds true for the “quaternionic blow-up”, i.e. a connected sum with a quaternionic projective space $\Bbb HP^{n}$, or even a connected sum with a Cayley plane $CaP^{2}$. The purpose of this paper is to prove an affirmative answer to this:
\[th1\] Let $M$ be a closed $4k$-manifold with $Y(M)\leq 0$. Then $$Y(M\sharp\ l\ \Bbb HP^{k}\sharp \ m\ \overline{\Bbb HP^{k}})=Y(M),$$ where $l,m$ are nonnegative integers. When $k=4$, we also have $$Y(M\sharp\ l\ CaP^{2}\sharp \ m\ \overline{CaP^{2}})=Y(M).$$
Preliminaries
=============
A computationally useful formula for the Yamabe constant is $$|Y(M,[g])|=\inf_{\tilde{g}\in [g]}(\int_{M} {|s_{\tilde{g}}|}^{\frac{n}{2}}
d\mu_{\tilde{g}})^{\frac{2}{n}},$$ where the infimum is attained only by a Yamabe metric. (For a proof, see [@lb3; @sung3].) So when $Y(M,[g])\leq 0$, this implies that $$Y(M,[g]) =
-\inf_{\tilde{g}\in [g]}(\int_{M} |s^{-}_{\tilde{g}}|^{\frac{n}{2}}
d\mu_{\tilde{g}})^{\frac{2}{n}},$$ where $s_{g}^-$ is defined as $\min\{s_g,0\}$. Therefore when $Y(M)\leq 0$, $$\begin{aligned}
\label{form1}
Y(M)=-\inf_{g}(\int_{M} |s_{g}|^{\frac{n}{2}}
d\mu_{g})^{\frac{2}{n}}=-\inf_{g}(\int_{M} |s_{g}^-|^{\frac{n}{2}}
d\mu_{g})^{\frac{2}{n}}.\end{aligned}$$ Also essential is Kobayashi’s connected sum formula [@koba; @sung2]. $$Y(M_{1}\sharp M_{2}) \geq
\left\{
\begin{array}{ll} -( |Y(M_1)|^{\frac{n}{2}}+ |Y(M_2)|^{\frac{n}{
| 3,638
| 2,046
| 2,436
| 3,326
| null | null |
github_plus_top10pct_by_avg
|
Wigner tomography measurements. (b) and (c) The measured joint Wigner function $W_{12}$ of the Bell states ${\ensuremath{\left|\Phi_{+}\right\rangle}}$ and ${\ensuremath{\left|\Phi_{-}\right\rangle}}$ on the Re-Re and Im-Re planes, respectively. (d) and (e) Real parts of the density matrices of the states ${\ensuremath{\left|\Phi_{+}\right\rangle}}$ and ${\ensuremath{\left|\Phi_{-}\right\rangle}}$ measured with the decoding and state tomography sequence as shown in Fig. \[fig:fig3\](b), respectively. Solid black outlines are for the ideal density matrices. Measured imaginary parts for both states are smaller than 0.04 and not shown. The fidelities for ${\ensuremath{\left|\Phi_{\pm}\right\rangle}}$ are 0.957 and 0.930, respectively.[]{data-label="fig:fig4"}](Figure4_final.pdf)
We finally show that our conditional dynamics can be used to deterministically create high-fidelity single-photon Bell states ${\ensuremath{\left|\Phi_{\pm}\right\rangle}} = \left({\ensuremath{\left|01\right\rangle}}\pm{\ensuremath{\left|10\right\rangle}}\right)/\sqrt{2}$. The approach is an extension of the previously reported SNAP operation for universal control of one cavity [@Krastanov2015; @Heeres2015] to two cavities. When combined with the single-cavity SNAP gates, our method can be used to realize arbitrary universal multi-cavity control. The experimental sequence is shown in Fig. \[fig:fig4\](a), where a conditional $2\pi$ rotation on qubit $Q_3$ is sandwiched in between two pairs of phase-space displacements of the cavities. With help of the two ancillary qubits, joint Wigner tomography of the two cavities is performed and two slice cuts of the measured two-mode Wigner function are shown in Figs. \[fig:fig4\](b-c). The density matrices of ${\ensuremath{\left|\Phi_{+}\right\rangle}}$ and ${\ensuremath{\left|\Phi_{-}\right\rangle}}$, reconstructed by mapping the state of the two cavities to qubits $Q_1$ and $Q_2$ and then jointly measuring the state of these qubits \[as in the tomography measurement of Fig. \[fig:fig3\](b)\], a
| 3,639
| 1,117
| 2,526
| 3,515
| 2,210
| 0.781834
|
github_plus_top10pct_by_avg
|
(1,2)
+\frac{{\mbox{\boldmath $p$}}_1\cdot{\mbox{\boldmath $p$}}_2}{A_cm},
\label{3bh}$$ where $A_c$ is the mass number of the core nucleus, $m$ is the nucleon mass, and $\hat{h}_{nC}$ is the single-particle (s.p.) Hamiltonian for a valence neutron interacting with the core. The last term in Eq. (\[3bh\]) is the two-body part of the recoil kinetic energy of the core nucleus [@EBH97], while the one-body part is included in $\hat{h}_{nC}$. We use a contact interaction between the valence neutrons, $v$, given as[@BE91; @HS05; @EBH97], $$v({\mbox{\boldmath $r$}}_1,{\mbox{\boldmath $r$}}_2)=\delta({\mbox{\boldmath $r$}}_1-{\mbox{\boldmath $r$}}_2)
\left(v_0+\frac{v_\rho}{1+\exp[(r_1-R_\rho)/a_\rho]}\right).
\label{vnn}$$ Here, the strength $v_0$ is determined to be $-$857.2 MeV$\cdot$fm$^{3}$ from the scattering length for the $nn$ scattering together with the cutoff energy, which we take $E_{\rm cut}=30$ MeV. See Refs.[@HS05; @EBH97] for the details. The second term in Eq. (\[vnn\]) simulates the density dependence of the interaction. Taking $R_\rho=1.34\times A_c^{1/3}$ fm and $a_\rho$=0.72 fm, we adjust the value of $v_\rho$ to be 952.3 MeV$\cdot$fm$^{3}$ so as to reproduce the experimental two-neutron separation energy of $^{27}$F, $S_{\rm 2n}$=2.80(18) MeV[@JSM07].
We employ a Woods-Saxon form for the s.p. potential in $\hat{h}_{nC}$. For the $^{24}$O+$n+n$ system, we take $a=0.72$ fm and $R_0=1.25A_c^{1/3}$ fm with $A_c=24$, and determine the values of $V_0=-44.1$ MeV and $V_{ls}$=45.87 MeV$\cdot$fm$^2$ in order to reproduce the single-particle energies of $\epsilon_{2s_{1/2}}=-4.09(13)$ MeV and $\epsilon_{1d_{3/2}}=770^{+20}_{-10}$ keV [@H08]. This potential yields the width for the 1$d_{3/2}$ state of $\Gamma_{1d_{3/2}}=92.9$ keV, which is compared with the empirical value, $\Gamma_{1d_{3/2}}=172(30)$ keV [@H08]. For the $^{25}$F+$n+n$ system, one has to modify the Woods-Saxon potential in order to take into account the presence of the valence proton in the core nucleus. The important effect comes from the
| 3,640
| 1,480
| 2,958
| 3,422
| 2,233
| 0.781588
|
github_plus_top10pct_by_avg
|
$e^{O(n^2)}\log^{O(n)}2K$ and step at most $n$ such that $A$ is contained in the union of at most $\exp(K^{O_n(1)})$ left translates of $P$.
This compares with the bound of $K^{O_n(1)}$ on the rank of $P$ obtained by Gill, Helfgott, Pyber and Szabó using \[thm:old\].
If $K<2$ then $A$ is a finite subgroup and the corollary is trivial, so we may assume that $K\ge2$. It follows from [@gill-helf Theorem 3] that there exist subgroups $H\lhd N<GL_n({\mathbb{F}}_p)$ such that $H\subset A^{K^{O_n(1)}}$, such that $N/H$ is nilpotent of step at most $n$, and such that $A$ is contained in a union of at most $\exp(K^{O_n(1))}$ left cosets of $N$. \[lem:fibre.pigeonhole\] then implies that $A$ is contained in a union of at most $\exp(K^{O_n(1))}$ left translates of $A^2\cap N$, which is a $K^3$-approximate group by \[lem:slicing\]. The desired result therefore follows from applying \[cor:ruzsa\] to the image of $A^2\cap N$ in $N/H$.
One can obtain a similar result in characteristic zero by combining \[cor:chang.ag\] with a result of Breuillard, Green and Tao [@bgt.lin Theorem 2.5], as follows.
\[cor:bgt\] Let $n\in{\mathbb{N}}$ and $K\ge1$, and let $\Bbbk$ be a field of characterisic zero. Suppose that $A\subset GL_n(\Bbbk)$ is a finite $K$-approximate group. Then there is a coset nilprogression $P_1\subset A^{K^{O_n(1)}}$ of rank at most $e^{O(n^2)}\log^{O(n)}2K$ such that $A$ is contained in the union of at most $\exp(\log^{O_n(1)}2K)$ left translates of $P_1$, and a coset nilprogression $P_2\subset A^{K^{O_n(1)}}$ of rank at most $e^{O(n^2)}K^3\log^{O(n)}2K$ such that $A$ is contained in the union of at most $K^{O_n(1)}$ left translates of $P_2$.
If $K<2$ then $A$ is a finite subgroup and the corollary is trivial, so we may assume that $K\ge2$. It then follows from [@bgt.lin Theorem 2.5] that $A$ is contained in a union of at most $K^{O_n(1)}$ left cosets of a nilpotent subgroup $N$ of $GL_n(\Bbbk)$ of step at most $n-1$, and hence from \[lem:fibre.pigeonhole\] that $A$ is contained in a union of at most $K^{O_n(1)
| 3,641
| 2,539
| 1,880
| 3,419
| 2,254
| 0.781406
|
github_plus_top10pct_by_avg
|
derset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}$}}
{\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal A}}{^{\rm c}}\}$}}}{\nonumber}\\
&=\sum_{{{\cal A}}\subset\Lambda}\;\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}}\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\text{ off }b\\}$}}}\,\cap\,\{{{\cal C}}_{{\bf n}}^b(o)={{\cal A}}}
\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n_b\text{ even}\}$}}}\underbrace{\sum_{{\partial}{{\bf m}}={\overline{b}}{\vartriangle}x}\frac{
w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}$}}
{\overset{}{\longleftrightarrow}}}x\text{ (in }{{\cal A}}{^{\rm c}})\}$}}}}_{=\;{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}}}{\nonumber}\\
&=\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\text{ off }b\}$}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n_b\text{ even}\}$}}}\,{{\langle \varphi_{{\overline{b}}}
\varphi_x \rangle}}_{{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}}.\end{aligned}$$ Furthermore, “off $b$” and ${\mathbbm{1}{\scriptstyle\{n_b\text{ even}\}}}$ in the last line can be omitted, since $\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\}\setminus\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}$ off $b\}$ and $\{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}\}\cap\{n_b$ odd} are subsets of $\{{\overline{b}}\in{{\cal C}}_{{\bf n}}^b(o)\}$, on which ${{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal
| 3,642
| 2,078
| 1,914
| 3,445
| null | null |
github_plus_top10pct_by_avg
|
55 6.4$\pm$0.6 68.2$\pm$6.8 100.9$\pm$5.3 76.4$\pm$3.8 32.6$\pm$2.3 14.5$\pm$1.0 5.8$\pm$0.4 1.1$\pm$0.1
11 05 38 45.73 -69 27 53.10 55 3.7$\pm$0.4 45.2$\pm$4.5 78.0$\pm$4.2 65.0$\pm$3.3 30.1$\pm$2.1 14.2$\pm$1.0 5.9$\pm$0.4 0.8$\pm$0.1
12 05 38 13.31 -69 30 39.46 55 1.0$\pm$0.1 20.5$\pm$2.0 38.9$\pm$2.3 40.2$\pm$2.0 19.2$\pm$1.4 8.9$\pm$0.6 3.7$\pm$0.3 0.4$\pm$0.1
13 05 40 00.70 -69 31 04.85 55 2.0$\pm$0.2 44.9$\pm$4.5 66.8$\pm$3.6 59.6$\pm$3.0 27.3$\pm$1.9 12.3$\pm$0.9 4.9$\pm$0.4 0.6$\pm$0.1
14 05 40 09.44 -69 32 44.60 55 4.1$\pm$0.4 68.8$\pm$6.9 103.0$\pm$5.4 82.6$\pm$4.1 36.2$\pm$2.5 16.1$\pm$1.1 6.3$\pm$0.5 0.8$\pm$0.1
15 05 40 51.61 -69 32 20.98 55 1.5$\pm$0.1 30.1$\pm$3.0 38.6$\pm$2.3 33.2$\pm$1.7 15.7$\pm$1.1 7.0$\pm$0.5 2.8$\pm$0.2 0.3$\pm$0.1
16 05 38 17.88 -69 33 37.52 55 1.4$\pm$0.1 28.4$\pm$2.8 45.0$\pm$2.5 43.9$\pm$2.2 22.6$\pm$1.6 11.0$\pm$0.8 4.5$\pm$0.3 0.7$\pm$0.1
17 05 38 55.03 -69 34 36.05 55 2.6$\pm$0.3 53.8$\pm$5.4 86.8$\pm$4.6 78.3$\pm$3.9 37.6$\pm$2.6 17.2$\pm$1.2 6.8$\pm$0.5 1.0$\pm$0.1
18 05 40 54.56 -69 38 04.19 55 1.3$\pm$0.1 18.8$\pm$1.9 38.6$\pm$2.4 34.0$\pm$1.7 16.6$\pm$1.2 7.8$\pm$0.6 3.3$\pm$0.2 0.5$\pm$0.1
19 05 40 58.54 -69 42 04.22 55 0.7$\pm$0.1 26.5$\pm$2.6 34.7$\pm$2.1 33.1$\pm$1.7 17.2$\pm$1.2 8.2$\pm$0.6 3.5$\pm$0.3 0.3$\pm$0.1
20 05 41 03.54 -69 46 36.87 55 1.0$\pm$0.1 33.9$\pm$3.4 49.3$\pm$2.8 48.5$\pm$2.4 25
| 3,643
| 4,653
| 2,079
| 3,126
| null | null |
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|
imate equalities in assume that there is no accidental degeneracy among the sterile state masses. That is, we assume that the relation $|\Delta m^2_{JK}| \gg |\Delta m^2_{31}|$ always holds.
After averaging out the fast oscillations, $P(\nu_\beta \rightarrow \nu_\alpha)$ is given to second order in $W$ by $$\begin{aligned}
&& P(\nu_\beta \rightarrow \nu_\alpha)^{(0+2)}
\nonumber \\
&=&
\left| \sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} \right|^2 -
2 \sum_{j \neq k}
\mbox{Re}
\left[ (UX)_{\alpha j} (UX)_{\beta j}^* (UX)_{\alpha k}^* (UX)_{\beta k} \right]
\sin^2 \frac{ ( h_{k} - h_{j} ) x }{ 2 }
\nonumber\\
&-&
\sum_{j \neq k} \mbox{Im}
\left[ (UX)_{\alpha j} (UX)_{\beta j}^* (UX)_{\alpha k}^* (UX)_{\beta k} \right]
\sin ( h_{k} - h_{j} ) x
\nonumber \\
&+&
2 \mbox{Re}
\biggl\{
\sum_{m}
\sum_{k, K}
\frac{ 1 }{ \Delta_{K} - h_{k} }
\left[
(ix) e^{- i ( h_{k} - h_{m} ) x}
- \frac{ e^{- i ( h_{k} - h_{m} ) x} }{ ( \Delta_{K} - h_{k} ) }
\right]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta k}
(UX)^*_{\alpha m} (UX)_{\beta m}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\nonumber \\
&-&
\sum_{m}
\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} - h_{m} ) x}
- \left( \Delta_{K} - h_{l} \right) e^{- i ( h_{k} - h_{m} ) x}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l}
(UX)^*_{\alpha m} (UX)_{\beta m}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K l}
\nonumber \\
&-&
\sum_{m}
\sum_{k, K}
\frac{ e^{- i ( h_{k} - h_{m} ) x} }{ ( \Delta_{K} - h_{k} ) }
\biggl[
(UX)_{\alpha k} W^*_{\beta K}
(UX)^*_{\alpha m} (UX)_{\beta m}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\nonumber \\
&+&
W_{\alpha K} (UX)^*_{\beta k}
(UX)^*_{\alpha m} (UX)_{\beta m}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\biggr]
\biggr\},
\label{P-beta-alpha-2nd-averaged}\end{aligned}$$ where the z
| 3,644
| 2,617
| 3,326
| 3,486
| null | null |
github_plus_top10pct_by_avg
|
cr{C}}}(X|\mathcal{O}, H)$, which is evaluated through a set of matrix operations [@rasmussen_gaussian_2006] (see Appendix). As we are using a GP, we can also get the variance of the functions conditioned on $\mathcal{O}$ and $H$: $\sigma_{\hat{\mathscr{C}}}(X|\mathcal{O}, H)$ [@rasmussen_gaussian_2006]. Both of these estimates depend on the correlation lengths $H$, normally referred to as the *hyperparameters* of our estimate. We assume that $H$ is not known a priori and needs to be fitted online.
The correlation lengths $H$ control the sensitivity of the model to each of the parameters, and relates to how much a parameter needs to be changed before it has a significant effect on the cost (see Fig. 1). A standard approach to fit $H$ is maximum likelihood estimation [@rasmussen_gaussian_2006]. Here, the hyperparameters are globally optimized over the likelihood of the parameters $H$ given our observations $\mathcal{O}$, or $L(H | \mathcal{O})$ [@rasmussen_gaussian_2006] (see Appendix). However, when the data set is small there will often be multiple local optima for the hyperparameters whose likelihoods are comparable to the maximum. We term these hyperparameters the hypothesis set $\mathcal{H}= (H_1,\cdots,H_P)$ with corresponding likelihood set $\mathcal{L} = (L_1,\cdots,L_P)$.
To produce our final estimates for the mean function and variance we treat each hypothesis as a *particle* [@gramacy_particle_2011], and perform a weighted average over $\mathcal{H}$. The weighted mean function is now defined as $M_{\hat{\mathscr{C}}}(X|\mathcal{O}, \mathcal{H}) \equiv \sum_{i=1}^P w_i \mu_{\hat{\mathscr{C}}}(X|\mathcal{O}, H_i)$ and weighted variance of the functions is $\Sigma_{\hat{\mathscr{C}}}^2(X|\mathcal{O}, \mathcal{H}) \equiv \sum_{i=1}^P w_i(\sigma_{\hat{\mathscr{C}}}^2(X|\mathcal{O}, H_i) + \mu_{\hat{\mathscr{C}}}^2(X|\mathcal{O}, H_i)) - M_{\hat{\mathscr{C}}}^2(X|\mathcal{O}, \mathcal{H})$, where $w_i = L_i/\sum_{i=1}^P L_i$ are the relative weights for the hyperparameters. Now that we have our final estim
| 3,645
| 3,156
| 3,969
| 3,646
| 2,181
| 0.781956
|
github_plus_top10pct_by_avg
|
hrm{SJ}}\,\langle X\rangle$ is a subalgebra in ${{\mathrm{As}}\,\langle
X\rangle}^{(+)}$ generated by the set $X$. Similarly, ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle\hookrightarrow {{\mathrm{Di}}{\mathrm{As}}\,\langle
X\rangle}^{(+)}$.
An element from ${\mathrm{As}}\,\langle X\rangle$ is called a *Jordan polynomial* if it belongs to ${\mathrm{SJ}}\,\langle X\rangle$. By analogy, an element from ${\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ is called a *Jordan dipolynomial* if it belongs to ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle$.
\[lemma2\] For arbitrary $u\in{\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$, $v\in{\mathrm{Alg}}\,\langle
X\rangle$ we have $$u{\mathbin\dashv}\mathcal{J}(v)^\dashv=u{\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(v^\dashv)=u{\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(v^\vdash),$$ $$\mathcal{J}(v)^\vdash{\mathbin\vdash}u=\mathcal{J}_{{\mathrm{Di}}}(v^\vdash){\mathbin\vdash}u=\mathcal{J}_{{\mathrm{Di}}}(v^\dashv){\mathbin\vdash}u.$$
Use an induction on the length of the word $v$. A base is evident. Let $v=v_1 v_2$. Then $$\begin{gathered}
u{\mathbin\dashv}\mathcal{J}(v)^{\dashv} =u{\mathbin\dashv}\mathcal{J}(v_1
v_2)^{\dashv}=\frac{1}{2}u{\mathbin\dashv}(\mathcal{J}(v_1)^{\dashv}{\mathbin\dashv}\mathcal{J}(v_2)^{\dashv}
+\mathcal{J}(v_2)^{\dashv}{\mathbin\dashv}\mathcal{J}(v_1)^{\dashv}) \\
=\frac{1}{2}u{\mathbin\dashv}(\mathcal{J}_{{\mathrm{Di}}}(v_1^{\dashv})
{\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(v_2^{\dashv})
+\mathcal{J}_{{\mathrm{Di}}}(v_2^{\dashv}){\mathbin\vdash}\mathcal{J}_{{\mathrm{Di}}}(v_1^{\dashv}))=u{\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(v_1^{\dashv}{\mathbin\dashv}v_2^{\dashv})=u{\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(v^{\dashv}).\end{gathered}$$ All remaining equalities are proved in the same way.
\[lemma:CommutOperJPhi\] For all $\Phi\in {\mathrm{Alg}}_z\,\langle X\rangle$ we have $$\Psi^z_{{\mathrm{As}}}(\mathcal{J}(\Phi))=\mathcal{J}_{{\mathrm{Di}}}(\Psi^z_{{\mathrm{Alg}}}(\Phi)).$$
Since all mappings are linear, it is enough to prove the st
| 3,646
| 1,589
| 2,472
| 3,420
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|
is convenient to use the Landau gauge. In this gauge neither $\Sigma_n$ ($n=0,1,2...$) nor the contribution of Fig.1f contain the large logarithm $\ln(\lambda_C/r_0)$. Therefore, after the renormalization, taking zero momentum transfer as a reference point, the contribution of Fig.1f is of the form $\delta_f \sim Z\alpha^2(1+fZ\alpha/\pi+...)$, where we expect $f\sim 1$. Since $\Sigma $ does not contain logarithms, at $r \ll \lambda_C$ we have $\Sigma u =({{Z\alpha^2}/{\pi r}}){\cal D} u$, where ${\cal D} \sim1 $ is some matrix dependent on $Z\alpha$, and $u$ is the electron wave function. Substitution of $\Sigma u$ into the Dirac equation results in the $x$-independent terms of the order of $\sim \alpha/\pi$ in the right hand sides of Eqs. (\[fg2\]). As a result, the relative correction to the matrix element of the weak interaction due to the diagram Fig.1e contains logarithmically enhanced $Z^2\alpha^3$ terms. There is also a $Z\alpha^2$ contribution coming from the distances $r\sim \lambda_C$. All in all the total contribution of diagrams Fig.1e and Fig.1f is of the form $$\label{ef}
\delta_{ef} =Z\alpha^2\left[a_1+ a_2{{Z\alpha}\over{\pi}}
\ln(\lambda_C/r_0)\right].$$ Our preliminary estimate gives $a_1 \approx 0.15$, and we expect $a_2 \sim 1$. Therefore, the value of $\delta_{ef}$ for Cs is $\delta_{ef}\sim 0.1\%$. The calculation of the coefficient $a_2$ in Eq. (\[ef\]) is a very interesting and challenging problem. At this stage we can claim only that the contribution (\[ef\]) is much smaller than that of the Uehling potential (\[db\]) because it does not contain the big logarithm squared.
There is one more radiative correction that has never been considered before. This contribution is due to the virtual excitation of the nuclear giant dipole resonance $A^*$ shown in Fig.3.
=3.cm
Our estimate gives $$\label{giant}
\delta_{A^*}\approx -0.1Z^{2/3}\alpha^2.$$ So it is completely negligible.
Considering all the corrections, one has to remember about the contribution of the electron-electron weak interac
| 3,647
| 2,160
| 3,434
| 3,446
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|
e o so that 2/9*o**2 - 2/9 - v*o**3 + 2/3*o = 0.
-1, 1/3, 1
Let n(i) be the first derivative of 3*i**5/20 + i**4/16 - i**3/4 - i**2/8 - 1. Factor n(x).
x*(x - 1)*(x + 1)*(3*x + 1)/4
Suppose -4*h = f + 4, 58*h + f + 4 = 57*h. Factor -2/7 + 2/7*b**2 + h*b.
2*(b - 1)*(b + 1)/7
Let r(b) be the third derivative of b**7/1120 + b**6/240 + b**5/160 - 2*b**3/3 - 4*b**2. Let z(o) be the first derivative of r(o). Let z(y) = 0. Calculate y.
-1, 0
Let k(n) = -4*n**4 + 4*n**2 - 3*n + 3. Let u(y) = 11*y**4 - 11*y**2 + 8*y - 8. Suppose 0 = -0*h - 2*h - 6. Let p(g) = h*u(g) - 8*k(g). Factor p(s).
-s**2*(s - 1)*(s + 1)
Let a = 1/588 - -4/49. Let d(y) be the first derivative of 1/4*y - a*y**3 + 3 - 1/16*y**4 + 1/8*y**2. Let d(z) = 0. What is z?
-1, 1
Let w(t) = t - 1. Let c(i) = -2*i**2 + 3*i + 1. Let a(d) = -c(d) - w(d). Find v, given that a(v) = 0.
0, 2
Let h(v) be the second derivative of v**5/100 - v**4/20 + v**3/10 - v**2/10 - 4*v. Suppose h(x) = 0. What is x?
1
Let h = 153 - 150. Let x(g) be the second derivative of 7/24*g**4 + 0 + 9/4*g**2 + 5/4*g**h + 1/40*g**5 + 2*g. Factor x(t).
(t + 1)*(t + 3)**2/2
Let l be ((-5)/((-825)/22))/(2/72). Determine d, given that -18/5*d**3 - 3/5*d**4 - 36/5*d**2 - l*d + 0 = 0.
-2, 0
Let l(w) = 14*w**3 + w**2 - 3. Let d(v) = 9*v**3 + v**2 - 2. Suppose 5*i + 36 = 2*i. Let b = i + 7. Let r(o) = b*l(o) + 8*d(o). Factor r(k).
(k + 1)**2*(2*k - 1)
Factor 63*j**3 - 2*j + 4*j**4 - 67*j**3 + 2*j.
4*j**3*(j - 1)
Let m(k) be the second derivative of 0 - 5/24*k**4 - 1/40*k**5 - 2/3*k**3 - k**2 + 3*k. Factor m(g).
-(g + 1)*(g + 2)**2/2
Let x(c) = -c**3 + c**2. Let d(w) = -8*w**3 + 4*w**2 - 3*w - 1. Let t(q) = 3*d(q) - 21*x(q). Factor t(i).
-3*(i + 1)**3
Let i(f) = -7*f**3 + 2*f**2 + 4*f + 5. Let m(o) = 8*o**3 - 2*o**2 - 4*o - 6. Suppose -2*c = -4*j + 22, -c = -2 + 3. Let x(s) = j*m(s) + 6*i(s). Factor x(g).
-2*g*(g - 2)*(g + 1)
Let p(b) = -4*b**3 - 3*b**2 - 15*b - 11. Let w be 3*3*(-1)/(-3). Let f(y) = 11*y**3 + 10*y**2 + 44*y + 32. Let i(l) = w*f(l) + 8*p(l). Let i(h) = 0. What is h?
-2
Let q be (-62 - -
| 3,648
| 2,725
| 2,201
| 3,348
| null | null |
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|
48
28/10/2006 4037.0438 1800 149 2039
28/10/2006 4037.0836 1800 135 1849
28/10/2006 4037.1051 1800 127 1741
28/10/2006 4037.1758 1800 106 1448
28/10/2006 4037.1968 1800 112 1530
29/10/2006 4037.9357 1800 85 1162
29/10/2006 4037.9568 1800 88 1199
29/10/2006 4037.9901 1800 85 1156
29/10/2006 4038.0125 1800 83 1136
29/10/2006 4038.0335 1800 87 1188
29/10/2006 4038.0551 1800 89 1216
30/10/2006 4039.0799 1800 95 1295
28/11/2006 4067.9843 1500 84 1152
28/11/2006 4068.0029 1500 90 1234
28/11/2006 4068.1349 2400 95 1298
29/11/2006 4068.9424 1500 84 1141
29/11/2006 4068.9599 1500 83 1135
29/11/2006 4069.0729 1500 86 1167
29/11/2006 4069.0905 1500 81 1105
29/11/2006 4069.1080 1500 72 984
29/11/2006 4069.1311 2100 77 1049
30/11/2006 4069.9889 3000 126 1713
30/11/2006 4070.1327 3000 118 1606
01/12/2006 4070.9555 3600 58 794
01/12/2006 4071.0433 3600 48 669
07/12/2006 4077.0996 3600 85 1164
08/12/2006 4077.9564 3600 96 1314
08/12/2006 4078.1182 3600 45 621
09/12/2006 4078.9482 3600 105 1434
09/12/2006 4079.1085 3600 109 1488
10/12/2006 4079.9563 3600 146 1991
10/12/2006 4080.1041 3600 118 1615
11/12/2006 4080.9480 3600 121 1652
11/12/2006 4081.0727 3000 86 1170
11/12/2006 4081.1132 2400 55 751
------------ ----------- ------ ------- ------
: The full version of the observing log.[]{data-label="tab:log"}
The observations of SZ Psc were carried out on 2004 November 20–27, 2006 September 01–06, 2006 October 28–30, 2006 November 28–December 01 and 2006 December 07–11, using the Coudé echelle spectrograph [@zhao2001] mounted on the 2.16m telesco
| 3,649
| 3,227
| 2,832
| 3,657
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|
s ratio becomes $\xi = {1 - n^3 \over 2}$ and we keep it fixed. This allows us to think of $f(\xi)$ as constant numbers. We consider then all possible polarization tensors and $n^{1,2}$ components. One can check that the solution to is given by We are left with one functional degree of freedom given by $f_5$. Indeed, the special thing about ${\rm T}_5$ is that it is identically zero when $\eps_{3 3} = 0$. To further constrain the energy correlator we relax the condition that $\eps_{3 3} = 0$ and again by keeping $\xi$ fixed impose the positivity condition We first imagine that the energy correlator is finite for all values $0 < \xi \leq 1$. For example, we can choose the following polarization tensor $\eps_{11} = - \eps_{13} = 1$, $\eps_{12} = \eps_{21} = \alpha$ where $\alpha$ is real. It produces the following form that should be sign-definite for any $n_1$ and $n_2$ The only way to make this form sign-definite for arbitrary $\alpha$,$n^1$ and $n^2$ such that $(n^1)^2 + (n^2)^2 \leq 1 - (n^3)^2$ is to set $n^3 = \pm 1$. One can check that the same is true for the most general polarization tensor. Since we are not considering the case when detectors are on top of each other we are left with the only solution, $n^3 = - 1$, when detectors are triggering particles propagating in the opposite directions. In the covariant language it corresponds to $\xi = 1$.
At this moment it is clear that if we assume the finiteness we have to conclude that the two-point energy correlator is identically zero. Indeed, we cannot have a smooth function that is zero everywhere but a single point. As we explain below it is not consistent with the momentum conservation and the fact that the one-point energy correlator is nonzero.
In any CFT a flow of energy is accompanied by a flow of momentum as if it was carried by massless particles Consider again the reference frame where $\vec q =0$. If one detector triggers an energy flow and, therefore, a momentum flow, by momentum conservation there should be a nonzero flow of energy in other
| 3,650
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|
“Reality of Superstring Field Theory Action,” JHEP [**1611**]{}, 014 (2016) doi:10.1007/JHEP11(2016)014 \[arXiv:1606.03455 \[hep-th\]\]. A. Sen, “Unitarity of Superstring Field Theory,” arXiv:1607.08244 \[hep-th\]. A. Sen, “Wilsonian Effective Action of Superstring Theory,” arXiv:1609.00459 \[hep-th\]. A. Sen, “Equivalence of Two Contour Prescriptions in Superstring Perturbation Theory,” arXiv:1610.00443 \[hep-th\].
N. Ishibashi, “Light-cone gauge superstring field theory in linear dilaton background,” arXiv:1605.04666 \[hep-th\]. N. Ishibashi and K. Murakami, “Multiloop Amplitudes of Light-cone Gauge NSR String Field Theory in Noncritical Dimensions,” arXiv:1611.06340 \[hep-th\].
[^1]: E-mail: [kunitomo@yukawa.kyoto-u.ac.jp]{}
[^2]: Space-time supersymmetry in the homotopy-algebra-based formulation has recently been studied by Erler.[@Erler:2016rxg]
[^3]: We further assume asymptotic completeness in this paper.
[^4]: This BRST-invariant GSO projection and that for the Ramond sector to be introduced shortly were first given in Ref. . The operators $G_{NS}$ and $G_R$ are none other than world-sheet fermion number operators in the total Hilbert space including the ghost sectors.
[^5]: In the context of string field theory, the GSO projections are also needed to make the Grassmann properties of string fields $\Phi$ and $\Psi$ consistent with those of the coefficient space-time fields.
[^6]: In this subsection, the symbol $\cong$ denotes an equation that holds up to the linearized equations of motion, $Q\eta\Phi=Q\Psi=0$.
[^7]: For such analyses of superstring field theory, see, for example, Refs. -.
---
abstract: 'I show that a particle structure in conformal field theory is incompatible with interactions. As a substitute one has particle-like exitations whose interpolating fields have in addition to their canonical dimension an anomalous contribution. The spectra of anomalous dimension is given in terms of the Lorentz invariant quadratic invariant (compact mass operator)
| 3,651
| 1,580
| 3,018
| 3,344
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|
=\diag(\la_1,\ldots,\la_r)$. Denote by $\Ga=(\ga_{ij})$ an $r\times r$ orthogonal matrix such that $\Ga^\top\Si \Ga=\La$. Assume that $g(\Si)$ is orthogonally invariant, namely, $g(\Si)=g(P\Si P^\top)$ for any orthogonal matrix $P$. Then, we can assume that $g(\Si)=g(\La)$ without loss of generality.
\[prp:condition2\] Assume that $g(\Si)=g(\La)$ and $g(\La)$ is a twice differentiable function of $\La$. Then $\ph_\pi$ with $\pi(\Th)=g(\La)$ is minimax relative to the KL loss $(\ref{eqn:loss})$ if $$\begin{aligned}
&\tr[\nabla_\Th\nabla_\Th^\top\pi(\Th)]\\
&=2\sum_{i=1}^r\bigg\{(q-r+1)\phi_i(\La)+\sum_{j\ne i}^r\frac{\la_i\phi_i(\La)-\la_j\phi_j(\La)}{\la_i-\la_j}+2\la_i\frac{\partial\phi_i(\La)}{\partial\la_i}\bigg\}\leq 0,\end{aligned}$$ where $\phi_i(\La)=\partial g(\La)/\partial\la_i$.
[**Proof.**]{} Since from (i) of Lemma \[lem:diff2\] $$\{\Dc_\Si\}_{ij}\la_k=\ga_{ik}\ga_{jk},$$ it is observed that by the chain rule $$\{\Dc_\Si\}_{ij} g(\La)=\sum_{k=1}^r \frac{\partial g(\La)}{\partial\la_k}\{\Dc_\Si\}_{ij}\la_k
=\{\Ga\Phi(\La)\Ga^\top\}_{ij},$$ where $\Phi(\La)=\diag(\phi_1(\La),\ldots,\phi_r(\La))$. Using Lemma \[lem:condition1\] and (ii) of Lemma \[lem:diff2\] gives that $$\begin{aligned}
&\tr[\nabla_\Th\nabla_\Th^\top\pi(\Th)] \\
&=2[(q-r-1)\tr\{\Ga\Phi(\La)\Ga^\top\}+2\tr\{\Dc_\Si \Ga\La\Phi(\La)\Ga^\top\}] \\
&=2\sum_{i=1}^r\bigg[(q-r-1)\phi_i(\La)+\sum_{j\ne i}^r\frac{\la_i\phi_i(\La)-\la_j\phi_j(\La)}{\la_i-\la_j}+2\frac{\partial}{\partial\la_i}\{\la_i\phi_i(\La)\}\bigg]\\
&=2\sum_{i=1}^r\bigg\{(q-r+1)\phi_i(\La)+\sum_{j\ne i}^r\frac{\la_i\phi_i(\La)-\la_j\phi_j(\La)}{\la_i-\la_j}+2\la_i\frac{\partial\phi_i(\La)}{\partial\la_i}\bigg\}.\end{aligned}$$ Hence the proof is complete.
Using Proposition \[prp:condition2\], we give some examples of Bayesian predictive densities with respect to superharmonic priors. Consider a class of shrinkage prior densities, $$\pi_{SH}(\Th) = \{\tr(\Th\Th^\top)\}^{-\be/2}\prod_{i=1}^r \la_i^{-\al_i/2}
=\bigg\{\sum_{i=1}^r \la_i\bigg\}^{-\be/2}\prod_{i=1}^r \la_i^{
| 3,652
| 2,622
| 2,061
| 3,359
| null | null |
github_plus_top10pct_by_avg
|
\hat{\epsilon}}$, then $$\begin{aligned}
W_1\lrp{p^*, p^y_{n\delta}} \leq 2\hat{\epsilon}
\end{aligned}$$ where $p^y_t := \Law(y_t)$.
Note that $m,L,R$ are from Assumption \[ass:U\_properties\], $L_N$ is from , $\cm, \beta,L_\xi$ are from Assumption \[ass:xi\_properties\]).
Finding a suitable $y_0$ can be done very quickly using gradient descent wrt $U(\cdot)$. The convergence rate to the ball of radius $R$ is very fast, due to Assumption \[ass:U\_properties\].3.
After some algebraic simplifications, we see that for a sufficiently small $\hat{\epsilon}$, achieving $W_1(p^y_{n\delta}, p^*) \leq \hat{\epsilon}$ requires number of steps $$\begin{aligned}
n = \tilde{O}\left(\frac{\beta^2}{\hat{\epsilon}^2} \cdot \exp\left(\frac{14}{3} \cdot \lrp{\frac{\LR}{\cm^2} + \frac{16\beta^2 L_\xi^2}{\cm^4}} \right. \right.\\
\left.\left. \cdot\max\lrbb{R^2, \frac{2^{12} \beta^6 L_\xi^2}{m^2 \cm^4}}\right)\right).\end{aligned}$$
The convergence rate contains a term $e^{R^2}$; this term is also present in all of the work cited in the previous section under Remark 1. Given our assumptions, including the “convexity outside a ball of radius $R$” assumption, this dependence is unavoidable as it describes the time to transit between two modes of the invariant distribution. It can be verified to be tight by considering a simple double-well potential.
As illustrated in Section \[ss:example\_ass\], the $m$ from Assumption \[ass:xi\_properties\].3 should be thought of as a regularization term which can be set arbitrarily large. In the following discussion, we will assume that $ \max\lrbb{R^2, \frac{\beta^6 L_\xi^2}{m^2 \cm^4}}$ is dominated by the $R^2$ term.
To gain intuition about this term, let’s consider what it looks like under a sequence of increasingly weaker assumptions:
**a. Strongly convex, constant noise**: $U(x)$ $m$-strongly convex, $L$-smooth, $\xi(x,\eta)\sim \N(0,I)$ for all $x$. (In reality we need to consider a truncated Gaussian so as not to violate Assumption \[ass:xi\_properties\].2, but
| 3,653
| 3,247
| 3,109
| 3,145
| null | null |
github_plus_top10pct_by_avg
|
ft( \sup_{x\in Q}\left \Vert
(I-P_{m})(x)\right \Vert _{\ell_{p}}\right) ,$$ where $I$ is the identity operator on $\ell_{p}.$
Let $X$ and $Y$ be Banach spaces. Then, a linear operator $L:X\rightarrow Y$ is said to be $\mathit{compact}$ if the domain of $L$ is all of $X$ and $L(Q)$ is a totally bounded subset of $Y$ for every $Q\in M_{X}$. Equivalently, we say that $L$ is compact if its domain is all of $X$ and for every bounded sequence $\left( x_{n}\right) $ in $X,$ the sequence $\left( L\left(
x_{n}\right) \right) $ has a convergent subsequence in $Y.$
The idea of compact operators between Banach spaces is closely related to the Hausdorff measure of noncompactness, and it can be given as follows:
Let $X$ and $Y$ be Banach spaces and $L\in B(X,Y)$. Then, the Hausdorff measure of noncompactness of $L$, is denoted by $\left \Vert L\right \Vert
_{\chi}$, can be given by$$\left \Vert L\right \Vert _{\chi}=\chi(L(S_{X})) \tag{1.4}$$ and we have $$L\text{ is compact if and only if }\left \Vert L\right \Vert _{\chi}=0.
\tag{1.5}$$
The Hausdorff measure of noncompactness has various applications in the theory of sequence spaces, one of them is to obtain necessary and sufficient conditions for matrix operators between $BK$ spaces to be compact. Recently, several authors have studied compact operators on the sequence spaces and given very important results related to the Hausdorff measure of noncompactness of a linear operator. For example \[25-38\].
In this paper, we derive some identities for the Hausdorff measure of noncompactness on the Fibonacci difference sequence spaces $\ell_{p}(\widehat{F})$ and $\ell_{\infty}(\widehat{F})$ defined by Kara \[1\]. We also apply the Hausdorff measure of noncompactness to obtain the necessary and sufficient conditions for such operators to be compact.
** The Fibonacci Difference Sequence Spaces** $\ell
_{p}(\widehat{F})$ **and** $\ell_{\infty}(\widehat{F})$
=======================================================
Throughout, let $1\leq p\leq \infty$ and $q$ denote the
| 3,654
| 1,914
| 2,279
| 3,577
| 3,837
| 0.769762
|
github_plus_top10pct_by_avg
|
}_\#)
= H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},\gamma_{n!}\gamma^*_nE_n^{\Delta}) =
H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},E_n^{\Delta} \otimes \gamma_{n!}k),$$ where we have used the projection formula in the right-hand side. The homology of the category $\Delta^{opp}$ can be computed by the standard complex; then by the Künneth formula, the right-hand side is isomorphic to $$H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},E_n^{\Delta}) \otimes
H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},\gamma_{n!}k)
\cong H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},E_n^{\Delta}) \otimes
H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda_{[n]},k).$$ By Lemma \[hoch\], $$H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},E_n^{\Delta}) \cong HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,E_n) \cong
A^{\otimes n}.$$ Since the category $\Lambda_{[n]}$ has an initial object $[n] \in
\Lambda_{[n]}$, we have $k = i_{n!k}$, so that the second multiple $H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda_{[n]},k)$ is just $k$ in degree $0$.
The essential point of Proposition \[main\] is the following: the cyclic object $A_\#$ associated to an algebra $A$ inconveniently contains two things at the same time – the cyclic structure, which seems to be essential to the problem, and the bar resolution, which is needed only to compute the Hochschild homology $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$. Replacing $A_\#$ with the cyclic complex $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#A_\# \in {{\mathcal D}}(\Lambda,k)$ disentagles these two.
We note that while one still has to prove that this does not change the final answer, the construction itself looks pretty straightforward – if one wants to remove the non-essential bar resolution from the definition of the cyclic homology, Definition \[cycl.cat\] seems to be the obvious thing to try. However, it was actually arrived at by a sort of a reverse engeneering proc
| 3,655
| 3,203
| 2,923
| 3,203
| 3,386
| 0.772734
|
github_plus_top10pct_by_avg
|
59.35 58.56 56.34 65.68 58.15 60.30 58.00 52.20 57.61 53.42
18 months 63.46 55.91 56.43 58.95 58.02 55.56 59.95 41.67 54.66 57.49
24 months 57.13 56.58 53.89 61.26 56.96 55.55 56.83 47.92 55.44 53.63
36 months 75.15 67.13 66.12 73.26 70.91 62.56 77.47 65.80 74.89 71.65
*p* 0.50 0.02 0.45 0.01 0.67
Height
At birth 50.08 50.49 49.69 51.40 50.34 50.43 50.13 46.00 50.81 44.22
6 months 56.95 57.78 57.27 59.43 59.66 51.82 55.24 56.08 56.55 57.97
12 months 58.53 55.73 55.20 58.32 56.86 55.11 53.96 41.24 53.61 45.98
18 months 60.36 56.24 54.57 60.99 59.13 50.64 59.32 43.03 57.97 34.25
24 months 57.75 60.07 60.27 56.21 60.92 56.51 61.01 52.47 59.07 59.55
36 months 87.95 74.44 73.84 84.19 79.46 70.41 89.38 76.11 86.31 86.39
*p* 0.25 0.16 0.02 0.01 0.13
Head circumference
At birth 55.87 55.94 56.56 56.05 56.29 54.93 54.72 51.33 54.49 52.26
6 months 52.67 69.99 65.18 72.49 66.43 70.02 72.11 55.57 65.99 71.58
12 months 62.90 68.08 63.92 75.89 65.40 73.33 67.95 49.60 67.63 57.05
18 months 69.46 66.14 65.31 69.97 67.21 64.87 60.90 65.61 64.22 69.67
| 3,656
| 6,560
| 907
| 2,040
| null | null |
github_plus_top10pct_by_avg
|
ows from Proposition \[qisom\] that $F : X \to Y$ is an isometry. $\diamond$
Proof of main theorem
=====================
Let $X, Y$ be complete, simply connected Riemannian surfaces of pinched negative curvature $-b^2 \leq K \leq -1$, and let $f : \partial X \to \partial Y$ be a Moebius homeomorphism with circumcenter extension $F : X \to Y$. All the tools are now in hand for the proof of the main theorem:
[**Proof of Theorem \[mainthm\]:**]{} As mentioned in the previous section, for any $x \in X$ there exists a probability measure $\mu_x$ on $\partial X$ with support contained in $K_x$ such that $\mu_x$ is balanced at $x \in X$, and $f_* \mu_x$ is balanced at $F(x) \in Y$. As shown in [@biswas6], this is equivalent to the fact that the convex hull in $T_x X$ of the compact $\{ \overrightarrow{x\xi} : \xi \in K_x \}$ contains the origin of $T_x X$ and the convex hull in $T_{F(x)} Y$ of the compact $\{ \overrightarrow{F(x)f(\xi)} : \xi \in K_x \}$ contains the origin of $T_{F(x)} Y$. By the classical Caratheodory theorem on convex hulls, since $X$ is of dimension two this implies that there exists $1 \leq k \leq 3$ and distinct points $\xi_1, \dots, \xi_k \in K_x$ and $\alpha_1, \dots, \alpha_k > 0$ (all depending on $x$) such that $$\alpha_1 \overrightarrow{x\xi_1} + \dots + \alpha_k \overrightarrow{x\xi_k} = 0$$ and $\alpha_1 + \dots + \alpha_k = 1$. Since the vectors $\overrightarrow{x\xi_i}$ are non-zero we must have $2 \leq k \leq 3$. Now if any two of the vectors $\overrightarrow{x\xi_i}, \overrightarrow{x\xi_j}$ for some $i \neq j$ are linearly dependent, then since they are distinct unit norm vectors we must have $\overrightarrow{x\xi_i} = -\overrightarrow{x\xi_j}$, hence $\xi_j = i_x(\xi_i)$. Thus $\xi_i, i_x(\xi_i) \in K_x$, and it follows from Lemma \[antisom\] that $F$ is an isometry and we are done. In particular if $k = 2$ then we are done. Thus we may as well assume that for any $x \in X$, there exist distinct points $\xi_1, \xi_2, \xi_3 \in K_x$ and $\alpha_1, \alpha_2, \alpha_3 > 0$ (all depen
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{B}$, where $$\label{eq3}
\mathbf{B}=\mathbf{X}^T\mathbf{X}-\frac{\mathbf{d}\mathbf{d}^T}{2m}.$$ Under the assumptions in Lemma \[thm3\], let $$\mathbf{m}_i^T=\mathbf{b}_i^T\mathbf{X}^\dagger=\sum_{j=1}^k\frac{\gamma_{ij}}{\sigma_j}\mathbf{u}_j^T.$$ The $i$-th modularity component is defined to be $$\mathbf{c}_i=\frac{\mathbf{m}_i}{\|\mathbf{m}_i\|_2}.$$
By the two lemmas, it can be seen that as long as the assumptions in Lemma \[thm3\] are met, the modularity components are well-defined, and the definition of $\mathbf{c}_i$ is based on the linear combination of $\mathbf{b}_i^T\mathbf{X}^\dagger$ in terms of the $\mathbf{u}_i$. In the next section some important properties of the modularity components are established.
Properties of the Modularity Components
=======================================
In this section some properties of modularity components will be discussed. It will be seen that the properties of modularity components are similar to the ones of principal components. First we will prove that the modularity components, as long as they are well-defined, are perpendicular to each other. Then we will prove that if we project the uncentered data onto the span of the modularity components, then the projection will be a scalar multiple of the modularity vectors. Finally, we will prove that the ‘importance’ of each modularity component is given by its corresponding eigenvalue of $\mathbf{B}$. The first modularity component has the largest modularity, and the $i$-th modularity component has the largest modularity with the constraint that it is perpendicular to the preceding $i-1$ modularity components.
\[thm5\] With the assumptions in Lemma \[thm3\], suppose $\mathbf{X}_{p\times n}$ is the unnormalized data matrix, $\mathbf{A}=\mathbf{X}^T\mathbf{X}$, $\mathbf{B}=\mathbf{A}-\mathbf{d}\mathbf{d}^T/(2m)$. Suppose $\mathbf{b}_i$, $\mathbf{b}_j$ are the eigenvectors of $\mathbf{B}$ corresponding to eigenvalues $\lambda_i$ and $\lambda_j$, $1\le i,j\le k-1$, respectively. Then we have $$\mathbf{B}=(\mathbf{
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thbf{Q}_{3,2}\right\rangle \right)\\
+s_{1}\left(\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{1,3}\right\rangle ,\left\langle \mathbf{Q}_{2,1}\mathbf{Q}_{2,3}\right\rangle ,\left\langle \mathbf{Q}_{3,1}\mathbf{Q}_{3,2}\right\rangle \right)\le4,
\end{array}$$ $$\begin{array}{l}
s_{1}\left(\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{2,1}\right\rangle ,\left\langle \mathbf{Q}_{1,3}\mathbf{Q}_{3,1}\right\rangle ,\left\langle \mathbf{Q}_{2,3}\mathbf{Q}_{3,2}\right\rangle \right)\\
+s_{0}\left(\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{1,3}\right\rangle ,\left\langle \mathbf{Q}_{2,1}\mathbf{Q}_{2,3}\right\rangle ,\left\langle \mathbf{Q}_{3,1}\mathbf{Q}_{3,2}\right\rangle \right)\le4
\end{array}$$ These two inequalities expand to 32 linear inequalities and there are 21 trivial constraints.
Denoting $$\begin{aligned}
\Delta & =\Pr\left[\mathbf{Q}_{1,2}\!\ne\!\mathbf{Q}_{1,3}\right]+\Pr\left[\mathbf{Q}_{2,1}\!\ne\!\mathbf{Q}_{2,3}\right]+\Pr\left[\mathbf{Q}_{3,1}\!\ne\!\mathbf{Q}_{3,2}\right]\\
& =\frac{3}{2}-\frac{1}{2}\left(\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{1,3}\right\rangle +\left\langle \mathbf{Q}_{2,1}\mathbf{Q}_{2,3}\right\rangle +\left\langle \mathbf{Q}_{3,1}\mathbf{Q}_{3,2}\right\rangle \right)\end{aligned}$$ and eliminating the connection correlations from the system using the Fourier–Motzkin algorithm, we obtain the system $$\begin{aligned}
-\frac{1}{2}+\frac{1}{2}S_{LG}\le\Delta & \le3-\left[-\frac{1}{2}+\frac{1}{2}S_{LG}^{0}\right],\\
0\le\Delta & \le3-\left|\left\langle \mathbf{Q}_{1}\right\rangle \right|-\left|\left\langle \mathbf{Q}_{2}\right\rangle \right|-\left|\left\langle \mathbf{Q}_{3}\right\rangle \right|,\end{aligned}$$ where we denote $$\begin{aligned}
S_{LG} & =s_{1}\left(\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{2,1}\right\rangle ,\left\langle \mathbf{Q}_{1,3}\mathbf{Q}_{3,1}\right\rangle ,\left\langle \mathbf{Q}_{2,3}\mathbf{Q}_{3,2}\right\rangle \right),\\
S_{LG}^{0} & =s_{0}\left(\left\langle \mathbf{Q}_{1,2}\mathbf{Q}_{2,1}\right\rangle ,\left\langle \mathbf{Q}_{1,3}\mathbf{Q}_{3,1}\right\ran
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rresponds to the one of (at least) two semi-infinite paths, as in Part (iii) of Theorem \[HN1\].
It is worth pointing out here that our proofs strongly rely on the planarity of the RST and the non-crossing property of its branches (see Lemma \[lemm:croisement\] in appendix). They cannot be carried to an arbitrary dimension.\
The paper is organized as follows. In Section \[section:sublinear\], the sublinear character of the expected number of intersection points between semi-infinite paths and the sphere with radius $r$ is established. Section \[section:directiondeterministe\] contains results on semi-infinite paths with deterministic directions. The colored RST and competition intefaces are defined in Section \[section:coloredRST\]. Finally, open questions and numerical studies are gathered in Section \[section:conjectures\].
Sublinearity of the number of semi-infinite paths {#section:sublinear}
=================================================
Let $r$ be a positive real number. Let us denote by $\chi_{r}$ the number of intersection points of the sphere $S(O,r)=\{r e^{\i \theta},\theta\in [0,2\pi)\}$ with the semi-infinite paths of the RST. The main result of this section states that the expectation of $\chi_{r}$ is sublinear.
\[theo:sublin\] The following limit holds: $$\lim_{r\to\infty} {{\mathbb E}}\Big(\frac{\chi_{r}}{r}\Big) = 0 ~.$$
The idea of the proof is as follows. For $r>0$, we introduce the points $A_r=r e^{\i / r}$ and $B_r=r e^{-\i /r}$ of the sphere $S(O,r)$. In the sequel, we will denote by $[A_r,B_r]$ the line segment with extremities $A_r$ and $B_r$ and by $a(A_r, B_r)=\{re^{\i \theta}, \theta\in [-1/r,1/r]\}$ the arc of $S(O,r)$ with extremities $A_r$ and $B_r$ and containing the point $(r,0)$. This arc is by construction of length 2. We denote by $\widetilde{\chi}_{r}$ the number of intersection points between semi-infinite paths of the RST and $a(A_r, B_r)$. By the rotational invariance of the PPP $N$, ${{\mathbb E}}(\chi_r)=\pi r \ {{\mathbb E}}(\widetilde{\chi}_r)$. Hence, using an add
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0 --physdev-is-bridged
ACCEPT all -- anywhere 192.168.122.0/24 state RELATED,ESTABLISHED
ACCEPT all -- 192.168.122.0/24 anywhere
ACCEPT all -- anywhere anywhere
REJECT all -- anywhere anywhere reject-with icmp-port-unreachable
REJECT all -- anywhere anywhere reject-with icmp-port-unreachable
Chain OUTPUT (policy ACCEPT)
target prot opt source destination
Finally, in /etc/xen/xend.conf we see the following options enabled:
(network-script network-bridge)
(vif-script vif-bridge)
Which according to the documentation is all you should need. These are Ubuntu boxes, btw.
Being new to Xen, the behaviour I would expect would be that eth0 in domU would be assigned a 192.168.0.* address from the office's dhcp server - this is how other virtualisation products I've used in the past behave (i.e. Virtualbox + VMWare).
Could someone please shed some light on this?
Cheers!
A:
Found the solution, it turned out we had some interference from Qemu. Observe the following file:
# cat /etc/libvirt/qemu/networks/autostart/default.xml
<network>
<name>default</name>
<bridge name="virbr0" />
<forward/>
<ip address="192.168.122.1" netmask="255.255.255.0">
<dhcp>
<range start="192.168.122.2" end="192.168.122.254" />
</dhcp>
</ip>
</network>
This was messing with our ability to create a bridged network, and instead was forcing NAT instead. The fix was simple - remove the file and reboot! Following this our interfaces look like this:
eth0 Link encap:Ethernet HWaddr 54:04:a6:19:25:77
inet addr:192.168.0.107 Bcast:192.168.0.255 Mask:255.255.255.0
inet6 addr: fe80::5604:a6ff:fe19:2577/64 Scope:Link
UP BROADCAST RUNNING MULTICAST MTU:1500 Metric:1
RX packets:11544 errors:0 dropped:0 overruns:0 frame:0
TX packets:316 errors:0 dropped:0 overruns:0 carrier:0
collisions:0 txqueuelen:0
RX b
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ite (\[X1\]) and the corresponding $n$-direction symmetry in the potential variables (\[eq:dLP-gen-ph-1\]), we obtain
\[eq:phi-sys-sym\] $$\begin{gathered}
\partial_t \phi^{(i)}_{m,n} = \alpha^{-1} q^{(i-\ell_1)}_{m,n} \phi_{m-1,n}^{(i+k_1)} -\frac{\phi^{(i)}_{m,n}}{N\alpha^{N}} , \label{eq:phi-sys-sym-1} \\
\partial_s \phi^{(i)}_{m,n} = \beta^{-1} q^{(i-\ell_2)}_{m,n} \phi_{m,n-1}^{(i+k_2)} -\frac{\phi^{(i)}_{m,n}}{N\beta^{N}}. \label{eq:phi-sys-sym-2}\end{gathered}$$
The symmetry (\[eq:phi-sys-sym-1\]) is a combination of the “generalised symmetry” (\[X1\]) and a simple scaling symmetry, with coefficient chosen so that the vector field is [*tangent*]{} to the level surfaces $\prod\limits_{i=0}^{N-1} \phi^{(i)}_{m,n}=\mbox{const}$, so this symmetry survives the reduction to $N-1$ components, which we always make in our examples. The symmetry (\[eq:phi-sys-sym-2\]) is similarly related to (\[n-sym\]) and also survives the reduction to $N-1$ components.
The [*master symmetries*]{} are similarly adjusted, to give
\[eq:phi-sys-msym\] $$\begin{gathered}
\partial_{\tau} \phi^{(i)}_{m,n} = m \alpha^{-1} q^{(i-\ell_1)}_{m,n} \phi_{m-1,n}^{(i+k_1)}
-\frac{m \phi^{(i)}_{m,n}}{N\alpha^{N}} , \label{dtauphim} \\
\partial_{\sigma} \phi^{(i)}_{m,n} = n \beta^{-1} q^{(i-\ell_2)}_{m,n} \phi_{m,n-1}^{(i+k_2)}
-\frac{n \phi^{(i)}_{m,n}}{N\beta^{N}}, \label{dsigmaphim}\end{gathered}$$ where $\partial_\tau \alpha = 1/\big(N \alpha^{N-1}\big)$ and $\partial_\sigma \beta = 1/\big(N \beta^{N-1}\big)$.
The self-dual case {#sect:selfdual}
==================
In [@f17-2] we give a number of equivalence relations for our general discrete system. For the case with $(k_2,\ell_2)=(k_1,\ell_1)=(k,\ell)$ the mapping
\[sd\] $$\begin{gathered}
\label{sd-kl}
(k,\ell)\mapsto \big(\tilde k,\tilde \ell\big) = (N-\ell,N-k)\end{gathered}$$ is an involution on the parameters, so we refer to such systems as [*dual*]{}. The [*self-dual*]{} case is when $(\tilde k,\tilde \ell)=(k,\ell)$, giving $k+\ell=N$. In particular, we consider the case
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. This curve has only one singular point at $P = [0:0:1]$ and does not pass through the singular points of $\PP^2_{w}$. Resolving the singularity of the curve with a $(3,2)$-blow up gives rise to one exceptional divisor $E$ with self-intersection number $-1/6$ and $\pi^* \mathcal{C} = \hat{\mathcal{C}} + 6 E$, $K_{\pi} = 4E$. Since $H^0(\PP^2_w, \mathcal{O}_{\PP^2_{w}}(k-6)) = 0$ for $k = 0,\dots,5$, the cokernel of $\pi^{(k)}$ is $$\operatorname{coker}\pi^{(k)} = \frac{\mathcal{O}_{\CC^2,0}}{\{ g \mid \operatorname{mult}_{E} \pi^{*} h > k - 5 \}},$$ whose dimension is $1$ for $k=5$ and $0$ for $k=0,\ldots,4$. According to Theorem \[thm:conucleo\_singular\], $\dim H^1(\tilde{X},\CC) = 2$ and the characteristic polynomial of the monodromy of the covering acting on $H^1(\tilde{X},\CC)$ is $(t-e^{\frac{2\pi i}{6}})(t-e^{\frac{2\pi i 5}{6}}) = t^2 - t + 1$.
Note that the $6$th-cyclic cover of $\PP^2$ ramified along any curve of degree $6$ having just a singular point with the topological type of $x^2+y^3$ does not have any irregularity. Therefore the singular points of the ambient space might affect the irregularity of the covering, even though they have codimension $2$.
A Zariski pair of curves on the weighted projective plane {#sec:zariski-pair}
---------------------------------------------------------
Let $\zeta$ be a $3$th-primitive root of unity and for $\lambda = 1, \zeta$, consider the curve $\mathcal{C}_{\lambda} \subset S=\PP^2_{(1,1,3)}$ defined by the polynomial $$G_{\lambda}(x,y,z) = (\lambda y z + x z - x^{a} y^{b})^3 + (y z - x z + x^{a} y^{b})^3
+ (- \lambda y z + \lambda x z + x^{a} y^{b})^3$$ with $a+b=4$ so that it is quasi-homogeneous. As above let $\rho_\lambda: \tilde{X}_\lambda \to S$ be the cyclic branched covering of degree $d=12$ ramifying on $\mathcal{C}_\lambda$. Using $H^1(\tilde{X}_\lambda,\CC)$, which is an invariant of the pair $(S,\mathcal{C}_\lambda)$, we will see that $(S,\mathcal{C}_1)$ and $(S,\mathcal{C}_{\zeta})$ provide a Zariski pair for $(a,b)=(1,3)$. However, one cannot
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tely. In the following, we review the fact that metric compatibility and the torsion free condition cannot determine the affine connection uniquely. The covariant derivative acts on the tensor as $$D_\mu V^\nu_\rho=\partial_\mu V^\nu_\rho+\Gamma^\nu_{\sigma\mu}V^\sigma_\rho-\Gamma^\sigma_{\rho\mu}V^\nu_\sigma.$$ The torsion is $$T^\mu_{\nu\rho}=\Gamma^\mu_{\nu\rho}-\Gamma^{\mu}_{\rho\nu},$$ and the curvature is defined as usual. Then the constancy condition of $A_\mu$ implies $$D_\mu A_\nu=\partial_\mu A_\nu-\Gamma^\rho_{\mu\nu}A_\rho=0$$ which gives constraints on the temporal affine connection $$\Gamma^\rho_{\mu\nu}A_\rho=\partial_\mu A_\nu.$$ Along with the torsion free condition $$T^\mu_{\nu\rho}=\Gamma^\mu_{\nu\rho}-\Gamma^{\mu}_{\rho\nu}=0,$$ one gets the point that the temporal one-form is closed $$\partial_\mu A_\nu-\partial_\nu A_\mu=0.$$ Considering the constancy of $\bar{A}^\mu$, one finds the affine connection $$\Gamma^\mu_{\nu\rho}=\bar{A}^\mu\bar{A}^\sigma(\bar{A_{(\rho}}\partial_{\nu)}\bar{A}_\sigma-\bar{A}_{(\nu|}\partial_\sigma\bar{A}_{|\rho)})+\bar{A}^\mu\partial_{(\nu}\bar{A}_{\rho)}+\bar{A}^\mu\bar{A}^\sigma A_{(\nu}F_{\rho)\sigma},$$ where $F_{\mu\nu}$ is an arbitrary anti-symmetric tensor. Moreover we impose the condition that the scaling structure is covariant constant, which implies that the parallel transport keeps the scaling weight of the vectors invariant. This fact implies that $$D_\mu A^\nu=0,$$ and then $$F_{\mu\nu}=0.$$ The requirement that the scaling structure is covariantly constant implies also $$\bar{A}^\mu\bar{A}_\mu=\mbox{const}.$$ This in turn determines $A^\mu$ and the affine connection $$\Gamma^\rho_{\mu\nu}=0.$$ Actually the conditions are too strong to allow interesting geometry.
To get the non-vanishing affine connection, one may relax the torsion free condition. The only constraints we impose are the metricity and the condition that the scaling structure is covariantly constant. Then the affine connection reads $$\Gamma^\rho_{\mu\nu}=A^{\rho}\partial_{\mu}A_\nu+\bar{
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$\alpha $ and $\beta $ such that $$\begin{aligned}
\label{eq:lambda2_L2}
\alpha \equiv \frac{\lambda_2(L )(d-1)}{\Tr(L)} = \frac{\lambda_2(L)(d-1)}{ \sum_{j = 1}^n \tau_{j}\ell_j} \;\; \text{and} \;\; \beta \equiv \frac{\Tr(L)}{d D_{\max}} = \frac{ \sum_{j = 1}^n \tau_{j}\ell_j}{d D_{\max}} \;.
\end{aligned}$$
For the proposed choice of $\lambda_{j,a} = 1/(\kappa_j-p_{j,a})$, we have $\tau_j = 1$ and the definitions of $\H$, $L$, $\alpha$, and $\beta$ reduce to those defined in Definition \[def:comparison\_graph1\]. We are left to prove an upper bound, $\delta\leq 32 (\log(\ell_{\max}+2))^2$, which implies the sufficient condition in and finishes the proof of Theorem \[thm:main2\]. We have, $$\begin{aligned}
\label{eq:main4}
\delta_{j,1} = \max_{a \in [\ell_j]} \Big\{\lambda_{j,a}(\kappa_j - p_{j,a})\Big\} + \sum_{a = 1}^{\ell_j} \lambda_{j,a}
&=& 1 + \sum_{a = 1}^{\ell_j} \frac{1}{\kappa_j - p_{j,a}} \nonumber\\
&\leq& 1 + \sum_{a=1}^{\ell_j} \frac{1}{a}\nonumber\\
&\leq& 2\log(\ell_j+2) \,, \end{aligned}$$ where in the first inequality follows from taking the worst case for the positions, i.e. $p_{j,a}= \kappa_j-\ell_j+a-1$ Using the fact that for any integer $x$, $\sum_{a=0}^{\ell-1} 1/(x+a) \leq \log((x +\ell -1)/(x -1))$, we also have $$\begin{aligned}
\label{eq:main5}
\frac{\delta_{j,2}\kappa_j}{\eta_j\ell_j} &\leq& \sum_{a = 1}^{\ell_j} \frac{1}{\kappa_j - p_{j,a}} \frac{\max{\{\ell_j,\kappa_j - p_{j,\ell_j}\}}}{\ell_j}\nonumber\\
&\leq& \min\Big\{\,\log(\ell_j+2) \,,\, \log\Big(\frac{\kappa_j-p_{j,\ell_j}+\ell_j -1}{\kappa_j-p_{j,\ell_j} -1 }\Big)\,\Big\} \frac{\max{\{\ell_j,\kappa_j - p_{j,\ell_j}\}}}{\ell_j}
\nonumber\\
&\leq& \frac{\log(\ell_j+2)\ell_j}{\max{\{\ell_j,\kappa_j - p_{j,\ell_j} -1}\}} \frac{\max{\{\ell_j,\kappa_j - p_{j,\ell_j}\}}}{\ell_j}\nonumber\\
&\leq& 2\log(\ell_j+2)\,,\end{aligned}$$ where the first inequality follows from the definition of $\eta_j$, Equation . From , , and the fact that $\delta_{j,2}\leq\log(\ell_j+2)$, we have $$\begin{aligned}
\label{eq:main6}
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y caused by metallicity variations from SN to SN). An uncertainty of $\pm$0.3 mag can be assigned to this technique based on the reddening difference yielded by both colors.
The ejecta velocities come from the minimum of the Fe II $\lambda$5169 lines interpolated to day 50, which is good to $\pm$300 km s$^{-1}$ [@hamuy01]. In the four cases where I had to extrapolate velocities I adopted an uncertainty of $\pm$2000 km s$^{-1}$.
![(bottom) Raw Hubble diagram from SNe II plateau $V$ magnitudes. (top) Hubble diagram from $V$ magnitudes corrected for envelope expansion velocities. []{data-label="hd3.fig"}](hd3.ps){height="75mm" width="75mm"}
The bottom panel of Fig. \[hd3.fig\] shows the Hubble diagram in the $V$ band, after correcting the apparent magnitudes for the reddening values, while the top panel shows the same magnitudes after correction for expansion velocities. A least-squares fit to the data in the top panel yields the following solution,
$$V_{50} - A_{V} + 6.564(\pm0.88)~log (v_{50}/5000) = 5~log(cz) - 1.478(\pm0.11).
\label{veqn_1}$$
The scatter drops from 0.91 mag to 0.38 mag, thus demonstrating that the correction for ejecta velocities standardizes the luminosities of SNe IIP significantly. It is interesting to note that part of the spread comes from the nearby SNe which are potentially more affected by peculiar motions of their host galaxies. When the sample is restricted to the eight objects with $cz$$>$3,000 km s$^{-1}$, the scatter drops to only 0.33 mag. The corresponding fit for the restricted sample is,
$$V_{50} - A_{V} + 6.249(\pm1.35)~log (v_{50}/5000) = 5~log(cz) - 1.464(\pm0.15).
\label{veqn_2}$$
![(bottom) Raw Hubble diagram from SNe II plateau $I$ magnitudes. (top) Hubble diagram from $I$ magnitudes corrected for envelope expansion velocities. []{data-label="hd4.fig"}](hd4.ps){height="75mm" width="75mm"}
Figure \[hd4.fig\] shows the same analysis but in the $I$ band. In this case the scatter in the raw Hubble diagram is 0.83 mag, which drops to 0.32 mag after correction for e
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for $j = i,f$. More details about the statistical meaning of work can be found in [@WGF] and references therein. One of the most important results of nonequilibrium statistical mechanics is the Jarzinsky equality [@jarzinsky], from which one can directly obtain a fundamental inequality involving average work $\left\langle W \right\rangle$ and Helmholtz free energy $F$ $$\begin{aligned}
\label{ji}
\left\langle W \right\rangle \ge \Delta F,\end{aligned}$$ where $\Delta F \equiv F(\lambda_f, \beta) - F(\lambda_i,\beta) $ is the difference between the free energies of the system. Explicitly, $$\label{hfe}
F(\lambda_j, \beta) = - \frac{1}{\beta} {\rm ln}\, \mathcal Z(\lambda_j),
\,\,\,
{\mathcal Z} (\lambda_j) \equiv
{\rm Tr} \, {\rm e}^{-\beta \hat {\mathcal H}(\lambda_j) },$$ with $j=i,f$. The equality in Eq. (\[ji\]) is only achieved by an isothermal quasistatic process, which is [*reversible*]{} [@WGF].
The indicator of irreversibility used in this work can then be defined considering what has just been exposed. Based on Eq. (\[ji\]), one defines [@crooks; @daffner] $$\label{nl1}
\mathcal L \equiv \beta ( \langle W \rangle - \Delta F),$$ as an indicator of irreversibility in the sense that the work protocol is reversible only when $\mathcal L=0$. What is reversible or irreversible for this indicator is the work protocol realized in an initially equilibrated system. The idea is that a backwards run of the work protocol after the system starts attempting thermal reequilibration will not, in general, bring the system and environment to their initial state. The quantity between the parentheses is known as irreversible work [@crooks], and $\mathcal L$ is usually called “nonequilibrium lag” (NL) as it gives an idea of how the system state, after the work protocol, lags behind an equilibrium thermal state fixed by the final Hamiltonian and inverse temperature $\beta$. Remarkably, it has been shown that the NL is ex
| 3,667
| 4,337
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matroids of smaller rank. Let $e \in B$. We may assume that $M \con e$ has no $U_{a+1,b}$-minor $U$ in which $E(U) \cap (B-\{e\})$ is a basis, so $\tau_{a}(M \con e) \le \binom{b}{a}^{r-a-1}$. Let $\cF$ be a cover of $M \con e$ with $\binom{b}{a}^{r-a-1}$ rank-$a$ sets. Since $\binom{b}{a}^{r-a} \le \tau_{a}(M) \le \sum_{F \in \cF}\tau_{a}(M|(F \cup e))$, there is some $F \in \cF$ such that $\tau_{a}(M|(F \cup \{e\})) \ge \binom{b}{a}$. Note that $r_M(F \cup \{e\}) = a+1$. By contracting a maximal subset of $B$ that is skew to $F \cup \{e\}$ in $M$, we obtain a rank-$(a+1)$ minor $N$ of $M$ so that $E(N) \cap B$ is a basis of $N$, and $M \con (F \cup \{e\})$ is a restriction of $N$. Now $\tau_{a}(N) \ge \tau_a(M|(F \cup \{e\})) \ge \binom{b}{a}$, and the lemma follows by the inductive hypothesis.
The next two results, which find a projective geometry minor or large uniform minor whenever the covering number is large, are special cases of the main theorems of \[\[covering1\]\] and \[\[covering2\]\] respectively.
\[pgdensity\] There is a function $f_{\ref{pgdensity}}\colon \bZ^2 \to \bZ$ so that, for all integers $s,n \ge 2$, if $M$ is a matroid with $r(M) > 1$ and $\tau_{s-1}(M) \ge r(M)^{f_{\ref{pgdensity}}(s,n)}$, then $M$ has a $U_{s,2s}$-minor or a rank-$n$ projective geometry minor.
\[exppgdensity\] There is a function $f_{\ref{exppgdensity}}\colon \bZ^3 \to \bZ$ so that, for all integers $s,n \ge 2$, and every prime power $q$, if $M$ is a matroid with $\tau_{s-1}(M) \ge q^{r(M) + f_{\ref{exppgdensity}}(s,q,n)}$, then $M$ has a $U_{s,2s}$-minor or a $\PG(n-1,q')$-minor for some $q' > q$.
Disjoint Bases {#selfdualsection}
==============
For $t \ge 0$, let $tU_{1,2}$ denote the direct sum of $t$ copies of $U_{1,2}$. Both $tU_{1,2}$ and $U_{s,2s}$ are the union of two disjoint bases. In this section we show that any large matroid with two disjoint bases has one of these two as a minor.
\[goodbasis\] Let $M$ be a rank-$r$ matroid with ground set $\{x_1, \dotsc, x_n\}$. There exists $t \in \{0,\dotsc,n\}$ so
| 3,668
| 2,696
| 2,317
| 3,449
| 3,687
| 0.770681
|
github_plus_top10pct_by_avg
|
ays identify a probability vector with the bipartite pure state represented by it.
Deterministic case
==================
In this section, we study the relation between catalyst-assisted transformation and multiple-copy transformation in deterministic case. First, we introduce some notations.
Denote by $V^n$ the set of all $n$-dimensional nonnegative vectors and let $x,y,\cdots$ range over $V^n$. Let $$S(y)=\{x\in V^n\ |\ x\prec y\}$$ be the set of states that can be transformed into $y$ by LOCC directly, $$T(y)=\{x\in V^n\ |\ \exists \mbox{ probability vector } c,\ x\otimes c\prec y\otimes c\}$$ be the set of states that can be transformed into $y$ by LOCC with the aid of some catalyst, and $$M(y)=\{x\in V^n\ |\ \exists \mbox{ integer }k\ \geq 1,\ x^{\otimes{k}}\prec y^{{\otimes{k}}}\}$$ the set of states which, when some appropriate number of copies are provided, can be transformed into the same number of $y$ by LOCC.
Suppose $x\in T(y)$ and $x'\in T(y')$. Then $\bar{x}\in T(\bar{y})$ where $\bar{x}= x\oplus x'$ and $\bar{y}=
y\oplus y'$.
[*Proof.*]{} By definition, $x\in T(y)$ and $x'\in T(y')$ imply that there exist $c$ and $c'$ such that $x\otimes c\prec y\otimes c$ and $x'\otimes c'\prec y'\otimes c'$. It can be easily checked that the vector $c\otimes c'$ serves as a catalyst for the transformation from $\bar{x}$ to $\bar{y}$, that is, $\bar{x}\otimes c \otimes
c'\prec \bar{y}\otimes c\otimes c'$. Thus $\bar{x}\in
T(\bar{y})$.$\Box$
The following lemma, important in its own right, is a powerful tool which gives us a sufficient condition on $x$ and $y$ such that they are incomparable in $any$ multiple-copy transformations. In other words, $any$ number of $x$ cannot be collectively transformed into the same number of $y$ using LOCC.
Suppose $x$ and $y$ are two nonincreasingly arranged $n$-dimensional probability vectors, $x_1=y_1$ but $x\nprec y$. Let $$d=\min\{l : 1\leq l \leq n,\ \sum_{i=1}^l x_i > \sum_{i=1}^l
y_i\}.$$ Denote by $t_1$ the number of components in $x$ which are equal to $x_1$, while $
| 3,669
| 3,299
| 3,510
| 3,373
| 1,779
| 0.785787
|
github_plus_top10pct_by_avg
|
e $x_i=y_i$. Let $\pi$ be the stationary distribution of the following random walk on $\Gamma(n, R)$: at each step, the walker stays at the current vertex with probability $p$, and otherwise chooses a neighbour randomly and moves to that neighbour. The transition probability from vertex $u$ to a neighbouring vertex $w$ is $(1-p)/(2R)$, where $2R$ is the degree of vertex $u$ in $\Gamma(n,R)$. Now place $n$ agents on vertices of $\Gamma(n,R)$ independently, each according to the distribution $\pi$. At each time step, each agent independently performs a step of the random walk described above (For random walks on a torus we refer the interested reader to [@LPW06]). For every pair of distinct agents $a$ and $b$, let $d_t(a,b)$ denote the Manhattan distance (in $\Gamma$) of the locations of $a$ and $b$ at time $t$. For a given $r{\geqslant}1$, we define the *communication graph process* $\{G^{(t)}_r \mid t=0,1,\ldots\}$ over the set of agents, say $A$, so that for every $t{\geqslant}0$, agents $a$ and $b$ are connected if and only if $d_t(a,b){\leqslant}r$. The model has been thoroughly studied when $R=2$ in the context of information spreading [@CMPS11]. We present the following result regarding the pair visibility of the communication graph process, proved in Appendix \[sec:pop\].
\[pro:gmn\] Fix $r=r(n) = n^{o(1)}$. Also let $\{G_r^{(t)}=(A, E_t) ~|~ 1{\leqslant}t{\leqslant}n\}$ be the communication graph process defined on an $R$-dimensional torus $\Gamma(n, R)$. Then there exists constant $\varepsilon> 0$ such that for every pair of agents, say $\{a, b\}\subset A$, $${\ensuremath{\operatorname{\mathtt{vis}}(a, b)}}=|\{t \in \{1,2,\ldots, n\} \mid \{a,b\}\in E_t\}|={\mathcal{O}}(n^{1-\varepsilon}).$$
Related Works
-------------
As we discussed, in the standard balls-into-bins, each ball picks a set of $d$ choices from $n$ bins, independently and uniformly at random. One of the first algorithms considering a different distribution over the bins is called *always-go-left* proposed by Vöcking [@V03]. In
| 3,670
| 1,290
| 2,754
| 3,440
| 1,696
| 0.78669
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|
e define a map $P':G\rightarrow {\mathcal{F}}(G)$ by setting for any $g\in G$ $$\begin{aligned}
P'(g):=\int_H \delta(g\cdot h)-\delta(h)d\mu(h).
\label{P'}\end{aligned}$$In case only (i) holds, we define for any $g\in G$ $$P'(g):=\int_H \delta(h\cdot g)-\delta(h)d\mu(h).$$ We will treat only the case where (ii) or (iii) holds and $P'$ is defined as in , the remainder case is completely analogous.
We claim that $P'$ is a $1$-Lipschitz map preserving the distinguished point. The latter is clear, we show that it is $1$-Lipschitz. Pick $g,f\in G$, we have $$\|P'(g)-P'(f)\|=\|\int_H \delta(g\cdot h)-\delta(f\cdot h)d\mu(h)\|\leq \int_H \|\delta(g\cdot h)-\delta(f\cdot h)\|d\mu(h)=d(g,f),$$ where in the last equality we used that $\mu$ is probability and $d$ is invariant. It follows that $P'$ extends to a norm one linear operator $P:{\mathcal{F}}(G)\rightarrow {\mathcal{F}}(G)$. We claim it is the desired projection.
First we show that it is a projection. For every $h\in H$ let $P'_h:G\rightarrow {\mathcal{F}}(G)$ be the map defined for every $g\in G$ by $P'_h(g):=\delta(g\cdot h)-\delta(h)$. The following are straightforward to verify:
- $P'_h$ is an isometry with $P'_h(1)=0$, thus it extends to a norm one linear map $P_h:{\mathcal{F}}(G)\rightarrow {\mathcal{F}}(G)$.
- For every $g\in G$ we have $P'(g)=\int_H P'_h(g)d\mu(h)$ and so also for every $x\in{\mathcal{F}}(G)$ we have $P(x)=\int_H P_h(x)d\mu(h)$.
It follows that in order to show that $P^2=P$ it suffices to check that for every $g\in G$ and $h\in H$ we have $P_h\circ P(\delta(g))=P(\delta(g))$. Indeed, the previous equality implies $$P^2(\delta(g))=\int_H P_h\circ P(\delta(g))d\mu(h)=\int_H P(\delta(g))d\mu(h)=P(\delta(g)).$$ Since the set $\{x\in{\mathcal{F}}(G)\colon P^2(x)=P(x)\}$ is a closed linear subspace, we get that $P^2(x)=P(x)$ for all $x\in{\mathcal{F}}(G)$ since ${\mathcal{F}}(G)$ is the closed linear span of $\{\delta(g)\colon g\in G\}$. Let us thus fix $g\in G$ and $h\in H$. We have $$\begin{aligned}
P_h\circ P(\delta(g))&=P_h\circ\int_
| 3,671
| 2,305
| 2,591
| 3,302
| null | null |
github_plus_top10pct_by_avg
|
428.7 ± 6.5 417.5 ± 5.2 275.4 ± 13.5 298.3 ± 4.1 290.8 ± 21.3 279.2 ± 6.4 271.5 ± 18.6 298.1 ± 7.9 67.5 ± 3.9 67.3 ± 1.5 98.6 ± 7.2 99.8 ± 4.1
Hyb 18 483.9 ± 4.8 406.2 ± 8.3 273.9 ± 4.0 275.7 ± 4.2 382.8 ± 5.0 296.7 ± 3.2 277.5 ± 4.9 301.4 ± 6.8 79.1 ± 1.6 73.2 ± 2.1 101.6 ± 1.6 109.5 ± 3.6
Hyb 19 460.1 ± 3.4 423.8 ± 15.0 254.2 ± 6.3 285.1 ± 15.0 322.3 ± 8.2 267.3 ± 14.4 264.3 ± 9.3 306.2 ± 13.2 70.1 ± 1.7 63.0 ± 4.1 104.0 ± 1.4 107.4 ± 2.2
Hyb 20 456.4 ± 20.4 407.4 ± 8.6 274.1 ± 20.5 266.0 ± 5.3 320.1 ± 11.2 282.0 ± 3.7 283.9 ± 12.4 297.0 ± 11.4 70.3 ± 2.5 69.4 ± 1.3 103.7 ± 12.0 111.6 ± 5.8
Hyb 21 463.5 ± 10.8 437.7 ± 7.0 235.4 ± 4.8 296.9 ± 3.5 365.0 ± 3.8 329.2 ± 8.6 241.9 ± 7.7 294.1 ± 4.2 78.7 ± 1.3 75.2 ± 1.7 103.3 ± 1.2 100.1 ± 1.7
Hyb 22 473.7 ± 8.9 435.7 ± 11.4 253.8 ± 7.1 280.4 ± 11.8 396.9 ± 3.7 297.5 ± 14.4 256.6 ± 8.9 312.8 ± 10.7 83.7 ± 1.3 68.3 ± 5.6 101.4 ± 0.7 111.7 ± 2.3
Hyb 23 471.2 ± 6.5 407.1 ± 23.9 254.7 ± 2.9 272.5 ± 2.5 320.9 ± 1.9 260.0 ± 15.8 264.6 ± 9.4 289.6 ± 12.7 68.1 ± 1.4 64.0 ± 6.0 103.9 ± 2.5 106.4 ± 3.5
Average 451.1 ± 4.5 425.4 ± 3.0 261.2 ± 3.4 282.7 ± 2.3 334.2 ± 6.4 293.0 ± 5.8 267.3 ± 2.5 306.5 ± 2.4
| 3,672
| 5,229
| 2,221
| 3,037
| null | null |
github_plus_top10pct_by_avg
|
r) = \alpha.$$
Output: $\hat{C}^*_{{\widehat{S}}} = \{ \beta\in\mathbb{R}^k:\ ||\beta-\hat\beta_{{\widehat{S}}}||_\infty\leq \hat{t}_\alpha/\sqrt{n}\}$.
For $\gamma_{{\widehat{S}}}$:
Get $\hat\beta_{{\widehat{S}}}$ from ${\cal D}_{1,n}$. This can be any estimator. For $j\in \hat{S}$ let $\hat\gamma_{\hat{S}}(j) = \frac{1}{n}\sum_{i=1}^n r_i$ where $r_i = (\delta_i(j) + \epsilon \xi_i(j))$, $\delta_i(j) = |Y_i - \hat\beta_{{\widehat{S}},j}^\top X_i| - |Y_i - \hat\beta_{{\widehat{S}}}^\top X_i|$ and $\xi_i(j)\sim {\rm Unif}(-1,1)$. Let $\hat\gamma_{{\widehat{S}}} = (\hat\gamma_{{\widehat{S}}}(j):\ j\in {\widehat{S}})$.
Draw $(X_1^*,Y_1^*),\ldots, (X_n^*,Y_n^*) \sim P_n$.
Let $\hat\gamma_{{\widehat{S}}}(j) = \frac{1}{n}\sum_{i=1}^n r_i^*$. Let $\hat\gamma_{{\widehat{S}}}^* = (\hat\gamma_{{\widehat{S}}}^*(j):\ j\in {\widehat{S}})$.
Repeat $B$ times to get $\hat\gamma_{{\widehat{S}},1}^*, \ldots, \hat\gamma_{{\widehat{S}},B}^*$.
Define $\hat{u}_\alpha$ by $$\frac{1}{B}\sum_{b=1}^B I\Bigl(\sqrt{n}||\hat\gamma_{{\widehat{S}},b}^* - \hat\gamma_{{\widehat{S}}}||_\infty > \hat{u}_\alpha\Bigr) = \alpha.$$
Output: $\hat{D}^*_{{\widehat{S}}} = \{ \gamma_{{\widehat{S}}} \in\mathbb{R}^k:\ ||\gamma_{{\widehat{S}}}-\hat\gamma_{{\widehat{S}}}||_\infty\leq \hat{u}_\alpha/\sqrt{n}\}$.
------------------------------------------------------------------------
------------------------------------------------------------------------
[Normal-Split]{}
[Input]{}: Data ${\cal D} = \{(X_1,Y_1),\ldots, (X_{2n},Y_{2n})\}$. Confidence parameter $\alpha$. Threshold and variance parameters $\tau$ and $\epsilon$ (only for $\gamma_{{\widehat{S}}}$).\
[Output]{}: Confidence set $\hat{C}_{{\widehat{S}}}$ for $\beta_{{\widehat{S}}}$ and $\hat{D}_{{\widehat{S}}}$ for $\gamma_{{\widehat{S}}}$.
Randomly split the data into two halves ${\cal D}_{1,n}$ and ${\cal D}_{2,n}$.
Use ${\cal D}_{1,n}$ to select a subset of variables ${\widehat{S}}$. This can be forward stepwise, the lasso, or any other method. Let $k= |{\widehat{S}}|$.
For $\beta_
| 3,673
| 2,227
| 2,088
| 3,395
| null | null |
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|
e $Z_2$.
The Yukawa interactions are the same as those in the HTM. The Higgs potential is given as $$\begin{aligned}
V
&=&
\frac{1}{\,2\,} m_{s_2^0}^2 (s_2^0)^2
+ \left\{
\mu_\eta^{}\, \eta^T\, i\sigma_2\, \Delta^\dagger\, \eta
+ \text{h.c.}
\right\}
+ \left\{
\lambda_{s\Phi\eta}\, s_1^0\, s_2^0\, (\eta^\dagger\, \Phi)
+ \text{h.c.}
\right\}
+ \cdots .\end{aligned}$$ Here we show only relevant parts for radiative generation of the $\mu$-term. See Appendix for the other terms. Vacuum expectation values $v$ and $v_s$ \[$= \sqrt{2}\,\langle s_1^0 \rangle$\] are given by $$\begin{aligned}
\begin{pmatrix}
v^2\\
v_s^2
\end{pmatrix}
=
\frac{2}{ 4 \lambda_{1\Phi} \lambda_{s1} - \lambda_{s\Phi 1}^2 }
\begin{pmatrix}
2 \lambda_{s1} & -\lambda_{s\Phi 1}\\
-\lambda_{s\Phi 1} & 2 \lambda_{1\Phi}
\end{pmatrix}
\begin{pmatrix}
m_\Phi^2\\
m_{s_1}^2
\end{pmatrix} .\end{aligned}$$ The $Z_2$-odd scalars in this model are two CP-even neutral ones (${\mathcal H}_1^0$ and ${\mathcal H}_2^0$), a CP-odd neutral one (${\mathcal A}^0 = \eta_i^0$), and a charged pair (${\mathcal H}^\pm = \eta^\pm$). The CP-even scalars are defined as $$\begin{aligned}
\begin{pmatrix}
{\mathcal H}_1^0\\
{\mathcal H}_2^0
\end{pmatrix}
=
\begin{pmatrix}
\cos\theta_0^\prime & -\sin\theta_0^\prime\\
\sin\theta_0^\prime & \cos\theta_0^\prime
\end{pmatrix}
\begin{pmatrix}
\eta_r^0\\
s_2^0
\end{pmatrix} , \quad
\tan{2\theta_0^\prime}
=
\frac{ \sqrt{2}\, \lambda_{s\Phi\eta}\, v\, v_s }
{
({\mathcal M}_0)_{ss}^2 - ({\mathcal M}_0)_{\eta\eta}^2
} ,\end{aligned}$$ where $({\mathcal M}_0)_{\eta\eta}^2 \equiv
m_\eta^2
+ ( \lambda_{1\Phi\Phi} + \lambda_{1\Phi\eta} )\, v^2/2
+ \lambda_{s\eta 1}\, v_s^2/2$ and $({\mathcal M}_0)_{ss}^2 \equiv
m_{s_2^0}^2
+ \lambda_{s3}\, v_s^2
+ \lambda_{s\Phi 2}\, v^2$. Squared masses of these scalars are given by $$\begin{aligned}
m_{{\mathcal H}_1^0}^2
&=&
\frac{1}{\,2\,}
\left\{
({\mathcal M}_0)_{\eta\eta}^2
+ ({\mathcal M}_0)_{ss}^2
- \sqrt{
\bigl\{
| 3,674
| 1,663
| 2,515
| 3,494
| null | null |
github_plus_top10pct_by_avg
|
the horizontal axis on $W_r$. On the right: The segment $[X,\mathcal{A}(X)]$ crosses $I_{r}$ (in bold) on $J(X)$. On this picture, $X$ is outside $I_{r}\oplus B(0,c)$.*]{}](Chi_r-c.eps "fig:"){width="6.5cm" height="5cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The number of semi-infinite paths that cross $a(A_r,B_r)$ is upper bounded by the number $\check{\chi}_r+\widehat{\chi}_r$ of edges of the RST which intersect $[A_r,W_r]\cup [W_r,B_r]$, where $\check{\chi}_r$ (resp. $\widehat{\chi}_r$) denotes the number of edges crossing $[A_r,W_r]$ (resp. $[W_r,B_r]$) and whose ancestors belong to the same half plane delimited by the line supporting $[A_r,W_r]$ (resp. $[W_r,B_r]$) as $O$. Since $\check{\chi}_r$ and $\widehat{\chi}_r$ are identically distributed: $${{\mathbb E}}(\widetilde{\chi}_{r}^{2})\leq {{\mathbb E}}((\check{\chi}_{r}+\widehat{\chi}_r)^{2})\leq 4 {{\mathbb E}}(\check{\chi}_r^2),\label{etape5}$$ and it is sufficient to show that $\limsup_{r\to\infty}{{\mathbb E}}(\check{\chi}_r^2)$ is finite. By rotational invariance, the distribution of $\check{\chi}_r$ is also the distribution of the number of edges with ancestors of smaller abscissa and that cross the vertical segment $I_r=[(r,r\tan(1/r)),(r,0)]$. With an abuse of notation,
| 3,675
| 1,995
| 968
| 3,830
| null | null |
github_plus_top10pct_by_avg
|
ke inference on parameters of population without assuming the form of the underlying distribution, such as mean, quantiles and regression parameters. We will take advantage of DAC and EL. Compared with BLB and SDB, we not only take full data information, but also save the cost computation. Our method is very simple and efficient. It has two steps. In the first step, we split the sample into random subsets and the estimate of each subset is obtained. In the second step, the estimates are regarded as one sample from a population so that one can apply EL to this simplified sample.
The rest of this article is organized as follows. In Section \[sec2\], we explain our method in details, and establish its theoretical property. In Section \[sec3\], we assess the finite sample performance of proposed method via Monte Carlo simulations. A real data set is analyzed in Section \[sec4\]. All technical proofs of main results are postponed to Appendix.
Methodology {#sec2}
===========
Let $\mathcal{X}_{n}=\{X_{1}, \dots, X_{n}\}$ be a sample consisting of independent and identically distributed observations form some unknown $q$ dimensional distribution $F$. The parameter of interest is $\theta=\theta(F)\in {\mathbb{R}}^{p}$. Its estimator is ${\widehat}{\theta}_{n}={\widehat}{\theta}(\mathcal{X}_{n})$, which could be maximum likelihood estimator, M-estimator, sample correlation coefficient, U-statistics and many others. In this paper, we mainly focus on the inference of $\theta$. Here is our method.
We first divide the full data set into $K$ blocks randomly, say $\mathcal{X}_{1n_{1}},\dots, \mathcal{X}_{Kn_{K}}$, and then compute $\{ {\widehat}{\theta}_{1n_{1}}={\widehat}{\theta}(\mathcal{X}_{1n_{1}}), \dots {\widehat}{\theta}_{Kn_{K}}={\widehat}{\theta}(\mathcal{X}_{Kn_{K}})\}$. For simplicity, we assume $n_{j}=m$ for all $1\leq j\leq K$. The DAC estimator is defined by $$\widetilde{\theta}_{n}=\frac{1}{K}\sum_{j=1}^{K} {\widehat}{\theta}_{jm}.$$
Now, we discuss the asymptotic properties of $\widetilde{\theta}_n$. We ass
| 3,676
| 1,616
| 3,089
| 3,364
| 3,100
| 0.774883
|
github_plus_top10pct_by_avg
|
{(4)} [3]
\nonumber \\
&=&
- \sum_{K}
\biggl[
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 }
\{ (ix) e^{- i \Delta_{K} x} + (ix) e^{- i h_{i} x} \}
+
\frac{ 2 }{ ( \Delta_{K} - h_{i} )^3 }
\left( e^{- i \Delta_{K} x} - e^{- i h_{i} x} \right)
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A W \right\}_{K K}
\left\{ W ^{\dagger} A (UX) \right\}_{K i}
\nonumber \\
&+&
\sum_{K \neq L}
\biggl[
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - \Delta_{L} )}
e^{- i \Delta_{K} x}
-
\frac{ 1 }{ ( \Delta_{L} - h_{i} )^2 ( \Delta_{K} - \Delta_{L} )}
e^{- i \Delta_{L} x}
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{L} - h_{i} )^2 }
\left( \Delta_{K} + \Delta_{L} - 2 h_{i} \right)
e^{- i h_{i} x}
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A W \right\}_{K L}
\left\{ W ^{\dagger} A (UX) \right\}_{L i}.
\label{hatS-3rd-order-ii-T-transf}\end{aligned}$$ $\hat{S} _{i i}^{(4)} [4]$ is given by $$\begin{aligned}
&& \hat{S} _{i i}^{(4)} [4]
\nonumber \\
&=&
\sum_{K}
\biggl[
- \frac{x^2}{2} e^{- i h_{i} x}
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 }
-
(ix) e^{- i h_{i} x}
\frac{ 2 }{ ( \Delta_{K} - h_{i} )^3 }
-
(ix) e^{- i \Delta_{K} x}
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^3 }
\nonumber \\
&-&
\frac{ 3 }{ ( \Delta_{K} - h_{i} )^4 }
\left( e^{- i \Delta_{K} x} - e^{- i h_{i} x} \right)
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A (UX) \right\}_{K i}
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A (UX) \right\}_{K i}
\nonumber \\
&+&
\sum_{K} \sum_{k \neq i}
\biggl[
(ix) e^{- i h_{i} x}
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( h_{k} - h_{i} ) }
-
(ix) e^{- i \Delta_{K} x}
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - h_{k} ) }
\nonumber \\
&+&
\frac{ ( h_{i} + 2 h_{k} - 3 \Delta_{K} ) }{ ( \Delta_{K} - h_{i} )^3 ( \Delta_{K} - h_{k} )^2 }
e^{- i \Delta_{K} x}
+
\frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 ( h_{k} -
| 3,677
| 1,569
| 3,242
| 3,563
| null | null |
github_plus_top10pct_by_avg
|
me $\delta>0$. Let $\rho$ be the maximal zero in Problem \[central\]. Then the function $$(-\delta, 1+\delta) \ni s \mapsto \chi[{\mathbf{v}}_1(s),{\mathbf{v}}_2(s), {\mathbf{v}}_3, \ldots, {\mathbf{v}}_m](\rho)$$ is a degree at most two polynomial which has local minima at $s=0$ and $s=1$. Hence this function is identically zero, and thus $$\chi[{\mathbf{v}}_1(1/2),{\mathbf{v}}_2(1/2), {\mathbf{v}}_3, \ldots, {\mathbf{v}}_m](\rho) =0$$ as desired.
\[infav\] Suppose ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m$ is a solution to Problem \[central\] such that ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_k \in \Lambda_{++}$ all have the same trace, and let $${\mathbf{v}}= \frac 1 k ({\mathbf{v}}_1 + \cdots + {\mathbf{v}}_k).$$ By applying Lemma \[average\] infinitely many times (and invoking Hurwitz’ theorem on the continuity of zeros) we see that also ${\mathbf{v}}, \ldots, {\mathbf{v}}, {\mathbf{v}}_{k+1}, \ldots, {\mathbf{v}}_m$ is a solution to Problem \[central\].
Clearly Conjecture \[maxmax\] implies Conjecture \[maxmax2\]. To prove the other implication assume Conjecture \[maxmax2\]. Then by Proposition \[righttrace\] and Remark \[infav\] we may assume that we have a solution of the form ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m$, where ${\mathbf{v}}_1=\cdots={\mathbf{v}}_k={\mathbf{v}}$, ${\mathbf{v}}_{k+1}={\mathbf{e}}-k{\mathbf{v}}$, ${\mathbf{v}}_{k+2}=\cdots={\mathbf{v}}_m=0$, and where ${\mathbf{v}}, {\mathbf{e}}-k{\mathbf{v}}\in \Lambda_{++}$ and $\tr({\mathbf{v}})= \epsilon$ and $0<d-k\epsilon = \tr({\mathbf{e}}-k{\mathbf{v}})<\epsilon$. Hence we want to maximize the largest zero of $$\label{gv}
g_{\mathbf{v}}(t):=(1-D_{\mathbf{v}})^{k} (1-D_{\mathbf{e}}+kD_{\mathbf{v}}) h(t{\mathbf{e}})$$ where
- ${\mathbf{v}}, {\mathbf{e}}-k{\mathbf{v}}\in \Lambda_{++}$
- $\tr({\mathbf{v}})=\epsilon$, where $0<d-k\epsilon <\epsilon$.
Let $I \subseteq {\mathbb{R}}$ be an interval. We say that a univariate polynomial is $I$–*rooted* if all its zeros lie in $I$.
\[tgv\] Let $g_{\mathbf{v}}(t)$ be given by . Then $g_{\m
| 3,678
| 2,155
| 1,824
| 3,538
| null | null |
github_plus_top10pct_by_avg
|
bda(u_1,u_2)^2}.{\label{eq:nsum-2ndbd}}\end{gathered}$$ We repeat this computation until all indicators for ${{\bf k}}$ are used up. We also apply the same argument to the sum over ${{\bf h}}$ in [(\[eq:Psi-def\])]{}. Summarizing these bounds with [(\[eq:psi-delta\])]{} and [(\[eq:psi-delta-G2\])]{}, and replacing $u_0$ in [(\[eq:ind-bd\])]{}–[(\[eq:nsum-1stbd\])]{} by $v'$, we obtain [(\[eq:Psi-bd\])]{}. This completes the proof of [(\[eq:Theta\[I\]-bd\])]{}.
### Proof of Lemma \[lmm:Theta’Theta”bd\] {#sss:dbconn}
We note that the common factor ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}}}$ in $\Theta'_{y,x;{{\cal A}}}$ and $\Theta''_{y,x,v;{{\cal A}}}$ can be decomposed as $$\begin{aligned}
{\label{eq:Theta'-evdec}}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}$}}}={\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}$}}}+{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\\}$}}}
\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}.\end{aligned}$$ We estimate the contributions from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}}}$ to $\Theta'_{y,x;{{\cal A}}}$ and $\Theta''_{y,x,v;{{\cal A}}}$ in the following paragraphs (a) and (b), respectively. Then, in the paragraphs (c) and (d) below, we will estimate the contributions from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\\}}}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}$ i
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oor \pi^* D \right \rfloor} \geq 0$ is equivalent to $(\tilde h) + \pi^{*} D \geq 0$ because $(\tilde h)$ is an integral divisor. Writing $\tilde h = \pi^{*} h$ where $h \in K(X)$, one can split the previous condition $(\pi^{*} h) + \pi^{*} D \geq 0$ into two different ones, $$\label{eq:global_sections}
(h) + D \geq 0 \quad \text{ and} \quad \operatorname{ord}_E \left( (\pi^{*} h) + \pi^{*} D \right) \geq 0,$$ for all $E$ exceptional divisors of $\pi$.
Note that if $A = (a_{ij})_{i,j}$ is a negative-definite real matrix such that $a_{ij} \geq 0$, $i\neq j$, then all entries of $-A^{-1}$ are non-negative. This implies that the pull-back of an effective divisor is also an effective divisor. Hence one can easily observe that the second condition in follows from the first one and the proof is complete.
Finally, we will define the concept of linear equivalence for $\QQ$-divisors. Given $D_1, D_2\in \operatorname{Weil}_\QQ(X)$ we say $D_1$ is [*linearly equivalent*]{} to $D_2$ and denote it by $D_1\sim D_2$ if $D_1-D_2$ is a principal divisor defined by a meromorphic function on $X$. Note that: $$D_1\sim D_2 \Leftrightarrow
\begin{cases}
D_1-D_2 \text{ is an integer divisor and}\\
\cO_X(D_1)\cong \cO_X(D_2).
\end{cases}$$
\[ex:QNC:LEquiv\] Following Example \[ex:QNC:pullback\] consider $D_{11}=\frac{1}{3}D_1$ and $D_{12}=\frac{2}{3}D_1$. Note that $D_1\sim D_2$ since $\frac{x}{y}$ is a meromorphic function on $X_1$. However $D_{11}\not\sim D_{12}$ despite $\cO_{X_1}(D_{11})=\cO_{X_1}(D_{12})=\cO_{X_1}.$ The reason here is that $D_{11}-D_{12}=\frac{1}{3}D_1-\frac{1}{3}D_2$ is not a Weil divisor.
Also, note that $\hat D_1 + \lambda E\sim \hat D_2+\lambda E$ for any $\lambda\in \QQ$ since the difference is a Weil divisor and $\hat D_1- \hat D_2$ is defined by composing $\frac{x}{y}$ with the blow-up $\pi_1$.
Finally, in $X_2$ the two quasi-smooth germs $D_1$ and $D_2$ are not linearly equivalent since $D_1\sim D_2$ implies that $[D_1]=[D_2]$ however $[D_1]=1$ and $[D_2]=2$.
Riemann-Roch formula for normal surfa
| 3,680
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dition (\[eq:3\]) says that, for any $\epsilon > 0$, there exists an $n_0$ such that for all $n > n_0$, $P^B_c - \frac{1}{2} <
\epsilon$ and $P^A_c < \epsilon$, to which we may refer as $\epsilon$-[*concealing*]{} and $\epsilon$-[*binding*]{}. These cheating probabilities are to be computed purely on the basis of logical and physical laws, and thus would survive any change in technology, including an increase in computational power. In general, one can write down explicitly the optimal $P^B_c$, $$P^B_c = \frac{1}{4}\left(2 + {\| \rho^B_0 - \rho^B_1 \|_1}\right),
\label{eq:4}$$ where ${\| \cdot \|_1}$ is the trace norm, ${\| \tau \|_1} \equiv \tr
(\tau^\dag \tau)^{1/2}$ for a trace-class operator $\tau$.
The entanglement cheating mechanism is explicitly spelled out in the impossibility proof. Under perfect concealing $P^B_c=\frac{1}{2}$, it follows from (\[eq:4\]) that the state $\rho^B_b$ at Bob’s possession obeys $\rho^B_0=\rho^B_1$. Hence by the Schmidt decomposition Alice can turn $\ket{\phi_{0i}}$ into $\ket{\phi_{1i}}$ by a unitary transformation on $\cH^A$ in her possession, thus succeeds in cheating perfectly. Under approximate concealing, an explicit transformation on $\cH^A$ can be similarly identified [@yuen2]-[@yuec] which leads to $$4(1-P^B_c)^2 \le P^A_c \le 2 \sqrt{P^B_c
(1-P^B_c)}.
\label{eq:5}$$ The lower bound in (\[eq:5\]) yields the following impossibility result, $$\lim_n P^B_c = \frac{1}{2} \,\, \Rightarrow
\,\, \lim_n P^A_c = 1
\label{eq:6}$$ Note that the impossibility proof makes a stronger statement than the mere impossibility of unconditional security, i.e., (\[eq:6\]) is stronger than (\[eq:3\]) not being possible.
The assumption in the impossibility formulation that $\ket{\Phi_{\sb }}$ are openly known has been challenged. In a multi-pass protocol where Alice and Bob exchange states, each $\ket{\phi_{\sb i}}$ becomes of the form $\ket{\phi_{\sb i k}}$ [@sr] $$\ket{\phi_{\sb ik}} = U^A_{\sb i_n} \ldots U^A_{\sb i_2} U^B_{k_1}
U^A_{\sb i_1} \ket{\phi_0}.
\label{eq:7}$$ where $U^A
| 3,681
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| 3,347
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| 0.780927
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uniform linear acceleration. We find that the transition rate is non-thermal, and angle dependent. Thermality is however restored in the low and high frequency regimes, and also in the regime of high acceleration compared with the inverse of the detector’s spatial extent.
In section \[rindlerframesection\] we analyse a profile defined in the Rindler frame of a Rindler trajectory, and confined to the Rindler wedge, following De Bievre and Merkli [@DeBievre:2006pys]. We find that the transition rate is isotropic and thermal at the usual Unruh temperature.
In section \[discsection\] we discuss and resolve the discrepancy of these two outcomes. The key property responsible for the non-thermality and anisotropy for the Lorentz-function profile is that this profile leaks outside the Rindler wedge, past the Rindler horizon. The leaking is an unphysical side effect of a detector model with a noncompact spatial profile, and it is unlikely to have a counterpart in spatially extended detectors with a more fundamental microphysical description. We leave the development of such spatially extended detector models subject to future work.
Spatially isotropic Lorentz-function profile\[schlichtsection\]
===============================================================
In this section we briefly review Schlicht’s generalisation [@schlicht] of a two-level Unruh-DeWitt detector [@Unruh:1976db; @DeWitt:1979] to a nonzero spatial size.
We consider a massless scalar field $\phi$ in four-dimensional Minkowski spacetime, and a two-level quantum system, a detector, localised around a timelike worldline $x(\tau)$, parametrised in terms of the proper time $\tau$. The interaction Hamiltonian reads $H_{int} = c \, m(\tau) \,\chi(\tau) \phi(\tau)$, where $c$ is a coupling constant, $m(\tau)$ is the detector’s monopole moment operator, $\chi(\tau)$ is the switching function that specifies how the interaction is turned on and off, and $\phi(\tau)$ is the spatially smeared field operator. The formula for $\phi(\tau)$ is $$\phi(\tau) =
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A^2D_s}{16\sqrt{3}M_N^2f_\pi^3}
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{(l+q)^2-m_\pi^2+i\epsilon}
\nonumber\\\times&\,
\frac{1}{l^2-m_\pi^2+i\epsilon}\,
\frac{1}{r_N^2-M_N^2+i\epsilon}
\\\times&\nonumber\,
\frac{(l^\rho+{\vec{q}}^\rho)(l^\nu+q^\nu)(l^\mu)}{k_N^2-M_\Sigma^2+i\epsilon}
\\\times&\nonumber\,
{\overline{u}}_1({\overline{E}},{\vec{p}\,'})(A_\Sigma+B_\Sigma\gamma_5)({\cancel{k}_N}+M_\Sigma)\gamma_\rho\gamma_5
u_1(E_p^\Lambda,{\vec{p}})
\\\times&\nonumber\,
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\nu\gamma_5({\cancel{r}_N}+M_N)\gamma_\mu\gamma_5
u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using heavy baryon expansion and the master integrals of Sec. \[sec:mi\], and redefining ${\vec{q}}\equiv{\vec{p}\,'}-{\vec{p}}$, $$\begin{aligned}
V_i=&
\frac{G_Fm_\pi^2g_A^2D_s}{16\sqrt{3}M_N f_\pi^3}
\Big[
2B_\Sigma J_{22}{\vec{q}}^2 {\vec{\sigma}_1}\cdot {\vec{\sigma}_2}\\-&\nonumber
2iA_\Sigma J_{22} M_N
\left({\vec{\sigma}_1}\times {\vec{\sigma}_2}\text{)$\cdot$}{\vec{q}}\right.
\\-&\nonumber
A_\Sigma M_N \left({\vec{q}}^2 J_{11}+2 {\vec{q}}^2 J_{23}+5
J_{34}+{\vec{q}}^2 J_{35}+4J_{22}\right){\vec{\sigma}_1}\cdot{\vec{q}}\\-&\nonumber
2B_\Sigma J_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot {\vec{q}})
\\+&\nonumber
2B_\Sigma \left(
({\vec{q}}^2-(3-\eta)q_0(q_0+\Delta M_\Sigma) J_{22}
\right.\\+&\left.\nonumber
({\vec{q}}^4-{\vec{q}}^2q_0^2-{\vec{q}}^2q_0\Delta M_\Sigma)J_{23}
-{\vec{q}}^2 J_{31}
\right.\\-&\left.\nonumber
(3-\eta)(2q_0+\Delta M_\Sigma)J_{32}
-(2 {\vec{q}}^2 q_0 +{\vec{q}}^2\Delta M_\Sigma)J_{33}
\right.\\+&\left.\nonumber
2(5-\eta)
{\vec{q}}^2 J_{34}
+2 {\vec{q}}^4 J_{35}-(3-\eta) J_{42}-{\vec{q}}^2 J_{43}
\right.\\+&\left.\nonumber
(15-8\eta)J_{46}+
2(5-\eta) {\vec{q}}^2 J_{47}+{\vec{q}}^4 J_{48}
\right.\\-&\left.\nonumber
{\vec{q}}^2 q_0 J_{11}
\left(q_0+\text{$\Delta $M}_{\Sigma }\right)-{\vec{q}}^2 J_{21}
\left(2 q_0+\text{$\Delta $M}_{\Sigma }\right)\right)
\Big]\,.\end{aligned}$$ To take into account the isospin we must replace every $A_\Sigma$ and $B_
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f $a_n$ is just a high probability bound on the maximal element-wise difference between $V$ and $\hat{V}$, valid for each $ P \in \mathcal{P}_n^{\mathrm{OLS}}$.
Next, recall that $\beta_S = g(\psi_S)$. Now define $$C_n = \Biggl\{ g(\psi) :\ \psi \in H_n \Biggr\}.$$ We call $C_n$ the [*image bootstrap confidence set*]{} as it is just the nonlinear function $g$ applied to the confidence set $H_n$. Then, by , $$\inf_{P\in {\cal P}_n'}\mathbb{P}(\beta \in C_n) \geq 1-\alpha - \frac{C}{a_n}\left( \frac{\log k}{n}\right)^{1/6}.$$ In particular, the implied confidence set for $\beta(j)$ is $$C_j = \Biggl[\inf_{\psi \in H_n}g(\psi),\ \sup_{\psi \in H_n}g(\psi)\Biggr].$$
Remarkably, in the coverage accuracy of the image-bootstrap the dimension $k$ enters only logarithmically. This is in stark contrast with the coverage accuracy guarantees for the projection parameters from , which depend polynomially in $k$ and on the other eigenvalue parameters.
The image bootstrap is usually avoided because it generally leads to conservative confidence sets. Below we derive bounds on the accuracy of the image bootstrap.
\[thm:beta.accuracy\] Let $u_n$ be as in and assume that $k \geq u_n^2$. Then, for each $P \in \mathcal{P}_n^{\mathrm{OLS}}$, with probability at least $\frac{1}{n}$, the diameter of the image bootstrap confidence set $H_n$ is bounded by $$C
\frac{k^{3/2}}{u_n^2}\sqrt{ \frac{\log k + \log n}{n}}.$$ where $C>0$ depends on $A$ only.
[**Remark.**]{} The assumption that $k \geq u_n^2$ is not necessary and can be relaxed, resulting in a slightly more general bound.
Assuming non-vanishing $u$, the diameter tends uniformly to $0$ if $k (\log k)^{1/3} = o(n^{1/3})$. Interestingly, this is the same condition required in [@portnoy1987central] although the setting is quite different.
Currently, we do not have a computationally efficient method to find the supremum and infimum. A crude approximation is given by taking a random sample $\psi_1,\ldots, \psi_N$ from $H_n$ and taking $$a(j) \approx \min_{j} g(\hat\psi_j),\ \
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chi \coloneqq {g}(\rho_{N}) + \sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))
, \qquad x\in D,$$ which is again justified by an obvious [strong law of large numbers]{}and the [central limit theorem]{}in the spirit of Corollary \[rate1\].
\[rate2\] When $D$ is bounded and convex, ${g}$ is continuous and in $ L^1_\alpha(D^\mathrm{c})$ and ${f}$ is a function in $ C^{\alpha +\varepsilon}(\overline{D})$ for some $\varepsilon>0$, then $$\label{WoSMC3}
\lim_{n \to\infty} \frac{1}{n}\sum_{i = 1}^n \chi^{i} = \mathbb{E}_x\left[ {g}(\rho_{N}) + \sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))\right] = u(x),$$ almost surely where $\chi^{i}$, $i\geq 1$ are [*iid*]{}copies of $\chi$ and $u(x)$ is the solution to (\[aDirichlet\_g\]). Moreover, when $$\int_{D^\mathrm{c}}\frac{{g}(x)^2}{1+|x|^{\alpha + d}}\,{\rm d}x<\infty.
\label{f22}$$ then $\operatorname{Var}(\chi)<\infty$ and, in the sense of weak convergence, $$\lim_{n\to\infty}n^{1/2}\left(\frac 1n \sum_{i=1}^n \chi^{i}- u(x)\right)= \operatorname{Normal}(0, \operatorname{Var}(\chi)).$$
Theorem \[hasacorr\] and Lemma \[integral\] ensure that the [strong law of large numbers]{}may be invoked. For the [central limit theorem]{}, we need $\mathbb{E}_x[\chi^2]<\infty$. Taking account of the fact that $\chi$ is the sum of two terms, the Cauchy–Schwarz inequality ensures that $\mathbb{E}_x[\chi^2] $ is finite if $\mathbb{E}_x\left[{g}(\rho_{N})^2\right] $ and $ \mathbb{E}_x\left[\left( \sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))\right)^2\right]$ are finite. Recall that $\mathbb{E}_x\left[{g}(\rho_{N})^2\right] = \mathbb{E}_x[{g}(X_{\sigma_D})^2]$ and, from Corollary \[rate1\], that is sufficient to ensure that this expectation is bounded.
Now note that, on account of the fact that ${f}$ is bounded, there exists a constant $\kappa\in(0,1)$, such that, for each $n\leq N$, appealing to (\[timescale\]), we have $r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))\leq \kappa \sigma_n$
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h x\in \dot{ \mathcal{X}}$. It is clear that $\mathcal{X}$ is a stationary subset of $[\kappa]^\omega$ because $s_0$ forces that $\mathcal{X}$ meets every cub. Now apply **Axiom R** to choose $Y\in \mathcal{C}$ so that $\mathcal{X}\cap [Y]^\omega$ is a stationary subset of $Y$.
Now we obtain a contradiction (and thus a proof) by showing that there is an extension $s\in S$ of $s_0$ that forces that $\dot{ \mathcal{X}}\cap [Y]^\omega$ is stationary. Let $\{ y_\alpha :\alpha\in \omega_1\}$ be an enumeration of $Y$. Let $\mathcal{E}$ be the set of $\delta\in \omega_1$ such that $x_\delta = \{y_\alpha :\alpha\in \delta\}\in \mathcal{X}$. Notice that $\{ \{ y_\alpha : \alpha \in \delta \} : \delta\in \omega_1\}$ is a cub in $[Y]^\omega$. Thus it follows that $\mathcal{E}$ is stationary. In fact, if $\mathcal{E}'$ is any stationary subset of $\mathcal{E}$, then $\mathcal{E}'$ is also a stationary subset of $[Y]^\omega$.
For each $\delta\in \mathcal{E}$ choose $s_\delta\in S$ above $s_0$ so that $s_\delta\Vdash x_\delta\in \dot{ \mathcal{X}}$ (as per the definition of $ \mathcal{X}$). Now we have a name $\dot{ \mathcal{E}} = \{ (x_\delta, s_\delta) :
\delta\in \omega_1\}$. We prove that there is some $s\in S$ above $s_0$ that forces that $\dot{ \mathcal{E}}$ is stationary. Thus such an $s$ forces that $\dot{ \mathcal{X}}\cap [Y]^\omega$ is stationary as required.
Let $s_0$ be on level $\alpha_0$ of $S$. There is a $\gamma>\alpha_0$ so that each member of $S_\gamma$ decides if $\dot{ \mathcal{E}}$ is stationary. Also, for each $\bar s\in S_\gamma$ that forces $\dot{ \mathcal{E}}$ is not stationary, there is a cub $\mathcal{C}_{\bar
s}$ of $\omega_1$ that $\bar s$ forces is disjoint from $\dot{ \mathcal{E}}$. Choose any $\delta $ in the intersection of those countably many cubs that is also in $\mathcal{E}$. Clearly if $\bar s\in S_\gamma$ is compatible with $s_\delta$, then $\mathcal{C}_{\bar s}$ did not exist since $\bar s \cap s_\delta$ would force that $\delta \in
\mathcal{C}_{\bar s}\cap \dot{ \mathcal{E}}$. T
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tate at 8.9 MeV in $^{28}$Ne has a similar structure to that in $^{26}$Ne and $K^{\pi}=0^{-}$ state in $^{28}$Ne, corresponding mainly to the neutron two-quasiparticle excitation of $\nu(2s^{-1}_{1/2}2p_{3/2})$, with 64.0% contribution. In addition to this neutron p-h like excitation, the following excitations have an appreciable contribution; $\nu(1d^{-1}_{5/2}1f_{7/2})$ (13.9%), $\nu(2s^{-1}_{1/2}1f_{7/2})$ (2.8%), $\pi(1p^{-1}_{1/2}1d_{5/2})$ (7.9%) and $\pi(1p_{3/2}\otimes 1d_{5/2})$ (1.4%).
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Same as Fig. \[trans\_density\] but for the $K^{\pi}=0^{-}$ state at 8.1 MeV and for the $K^{\pi}=1^{-}$ state at 6.7 MeV in $^{30}$Ne. []{data-label="30Ne_trans_density"}](fig9-1.eps "fig:") ![Same as Fig. \[trans\_density\] but for the $K^{\pi}=0^{-}$ state at 8.1 MeV and for the $K^{\pi}=1^{-}$ state at 6.7 MeV in $^{30}$Ne. []{data-label="30Ne_trans_density"}](fig9-2.eps "fig:")
![Same as Fig. \[trans\_density\] but for the $K^{\pi}=0^{-}$ state at 8.1 MeV and for the $K^{\pi}=1^{-}$ state at 6.7 MeV in $^{30}$Ne. []{data-label="30Ne_trans_density"}](fig9-3.eps "fig:") ![Same as Fig. \[trans\_density\] but for the $K^{\pi}=0^{-}$ state at 8.1 MeV and for the $K^{\pi}=1^{-}$ state at 6.7 MeV in $^{30}$Ne. []{data-label="30Ne_trans_density"}](fig9-4.eps "fig:")
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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get the following: In the complement of $a'$ there can be at most a $3(r-1) + (n+ 2) - 4 = 3r + n -5$ dimensional simplex. Hence, there can be at most a $3r + n - 4$ dimensional simplex containing $a'$ on $N$. Since dim $\lambda(\Delta) = 4r + n -4 > 3r + n - 4$, we get a contradiction (since $r \geq 1$).
[**(ii)**]{} Suppose the genus of $N$ is odd, $g =2r + 1 $ for some $r \in \mathbb{Z}$. In this case complement of $a'$ is nonorientable. It has genus $2r-1$, and it has $n+2$ boundary components. By using Lemma \[dim\] we get the following: We can put $a$ into a maximal simplex $\Delta$ of dim $4r+n-2$. Then $\lambda(\Delta)$ has dimension $4r+n-2$. In the complement of $a'$ there can be at most a $4(r-1) + (n+ 2) - 2 = 4r + n - 4$ dimensional simplex by Lemma \[dim\]. Hence, there can be at most a $4r + n - 3$ dimensional simplex containing $a'$ on $N$. Since dim $ \lambda(\Delta) = 4r + n - 2 > 4r + n - 3$, we get a contradiction.
=1.8in =1.55in
[**Case 3:**]{} Suppose $a'$ is a separating simple closed curve on $N$. This case does not occur when $(g, n)=(3, 0)$ as there are no such nontrivial simple closed curves. So, we assume that $g \geq 2$ and $g+n\geq 4$.
[**(i)**]{} Suppose $g \geq 4$. If $(g,n) = (4, 0)$, we complete $a$ to a pants decomposition $P$ as shown in Figure \[Fig4a\]. Let $P'$ be a set of pairwise disjoint elements in $\lambda([P])$ containing $a'$. Let $c' \in \lambda([c]) \cap P'$. If $a'$ is a separating simple closed curve on $N$, we get a contradiction by Lemma \[adjacent\], as $c'$ will be in one of the connected components in the complement of $a'$, and hence $c'$ cannot be adjacent to four curves w.r.t. $P'$ even though $c$ is adjacent to four curves w.r.t. $P$.
Assume $(g, n) \neq (4, 0)$. We can choose a 1-sided curve $b$ on $N$ intersecting $a$ exactly once such that a regular neighborhood $R$ of $a \cup b$ will be a projective plane with two boundary components $x, y$ such that $N \setminus R$ is connected and there is a curve $z$ as shown in Figure \[Fig3\] such that $a, x
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ation. We define as $\mathcal{T}_{X}$ the subtree of $\mathcal{T}$ consisting of $X\not=O$ and all its descendants, i.e. all the vertices of $\mathcal{T}$ that have $X$ in their ancestry. This tree is naturally rooted at $X$.\
If $\mathcal{T}_X$ is unbounded, then we can construct two particular semi-infinite paths that we call the right-most and left-most semi-infinite paths, $\underline{\gamma}_X$ and $\overline{\gamma}_X$, of $\mathcal{T}_X$. The construction follows.\
Put $X_{0}=X$. Let $$K_0={{\rm Card}}\{Y\in N,\ \mathcal{A}(Y)=X_0\mbox{ and }\mathcal{T}_Y\mbox{ is unbounded }\}$$ be the number children of $X_0$ with infinite descendance. Since the number of children of a given vertex is a.s. finite (see [@baccellibordenave Section 3.3.2.]) and since $X_{0}$ has infinitely many descendants, $K_0\geq 1$ a.s. It is possible to rank these offspring $X_0^1,\dots,X_0^{K_0}$ by increasing order of the oriented angles $\widehat{\mathcal{A}(X_0)X_0 X_0^k}$ for $k\in \{1,\dots,K_0\}$. Define $X_{1}$ as the child of $X_{0}$ corresponding to the largest value of these angles. Iterating this construction, a semi-infinite path $\overline{\gamma}_{X}=(X_{n})_{n\in\mathbb{N}}$ rooted at $X$ is built.\
In the same way, a semi-infinite path $\underline{\gamma}_{X}$ rooted at $X$ is constructed such that, among the semi-infinite paths of $\mathcal{T}_{X}$, $\underline{\gamma}_{X}$ is the lowest one (in the trigonometric sense). Consequently, any given semi-infinite path in $\mathcal{T}_{X}$ is trapped between $\underline{\gamma}_{X}$ and $\overline{\gamma}_{X}$ (in the trigonometric sense).
Uniqueness {#section:unique}
----------
Part $(iii)$ of Theorem \[HN1\] ensures the existence of random directions with at least two semi-infinite paths. However, there is no more than one semi-infinite path with a deterministic direction (Proposition \[prop:<2\]). This result completes Part $(ii)$ of Theorem \[HN1\].
\[prop:<2\] For all $\theta\in[0,2\pi)$, there a.s. exists exactly one semi-infinite path with asymptotic directi
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of $H(\theta)$, e.g. $S$) such that for all $\alpha$, $N_\alpha\cap\kappa\in S$ if and only if there is a countable $M\prec H(\theta)$ such that $N_\alpha\subseteq M$, $M\cap\omega_1=N_\alpha\cap\omega_1$, and $M\cap\kappa\in S$.
Let $\mathcal{S}$ and $\mathcal{C}$ be as in **Axiom R**. Choose $\theta$ sufficiently large so that $\mathcal{S},\mathcal{C}\in H(\theta)$ and so that $\theta^{\aleph_1}=\theta$. Let $\{\mathcal{S},\mathcal{C}\}\in N_0$ and let $\{N_\alpha:\alpha\in\omega_1\}$ be as in **SRP**. By induction on $\alpha\in\omega_1$, choose $Y_\alpha\in C\cap N_{\alpha+1}$ so that $\bigcup (\mathcal{C}\cap N_\alpha)\subseteq Y_\alpha$. Then $\{Y_\alpha:\alpha\in\omega_1\}$ is an increasing chain in $\mathcal{C}$. Therefore $Y=\bigcup_{\alpha\in\omega_1}(N_\alpha\cap\kappa)$ is in $\mathcal{C}$.
$\mathcal{S}^+=\{M\prec H(\theta):M\cap \kappa\in\mathcal{S}\}$ is a stationary subset of $[H(\theta)]^\omega$. This is proved in the same way as 1) of Claim 1.12 on page 196 of [@S]. Since $\{N_\alpha:\alpha\in\omega_1\}$ is an element of $H(\theta)$, there is an $M\in\mathcal{S}^+$ such that $\{N_\alpha:\alpha\in\omega_1\}\in M$. Let ${M\cap\omega_1=\delta}$. Obviously $M\cap\kappa\in\mathcal{S}$, and, by continuity, $N_\delta\subseteq M$ and $M\cap\omega_1= N_\delta\cap\omega_1$. It then follows from **SRP** that $N_\delta\in\mathcal{S}$.
This actually proves that $\{\alpha\in\omega_1:N_\alpha\cap\kappa\in\mathcal{S}\}$ is a stationary subset of $\omega_1$, because we could have put any cub of $\omega_1$ as an element of $M$. Now assume that $\mathfrak{Z}\subseteq[Y]^\omega$ is a cub of $[Y]^\omega$. Choose a strictly increasing $g:\omega_1\to\omega_1$ such that for each $\alpha$, there is a $Z_\alpha\in \mathfrak{Z}$ such that $N_\alpha\cap \kappa\subseteq Z_\alpha\subseteq N_{g(\alpha)}$. If limit $\delta$ satisfies that $g(\alpha)<\delta$ for all $\alpha<\delta$, then we have that $N_\delta\cap\kappa\in\mathfrak{Z}$. This finishes the proof that $\mathcal{S}\cap[Y]^\omega$ is stationary.
Suppose we have a
| 3,690
| 1,578
| 1,890
| 3,440
| null | null |
github_plus_top10pct_by_avg
|
onal file [2](#MOESM2){ref-type="media"}).
During July and August from 2006 to 2013, 17,017 emergency hospitalizations due to cardiovascular diseases registered by SIDIAP fulfilled the inclusion criteria. The mean age at study entry was 70.1 years (sd = 13.9), 7,615 (45%) were women, 11,508 (68%) had one single cardiovascular event and 5,244 (31%) died before the conclusion of the study (31 August 2013). Median follow-up was 434 days (Inter Quartile Range: 310 to 496 days), what equals to 7 summers---considering only July and August. On average, the daily maximum temperature was 28.1 °C ± 5.3 °C during the study period and raised to 33.8 °C ± 4.9 °C during heatwaves. The median of the heatwave threshold, i.e. the 95^th^ percentile of the maximum daily temperature in each weather station, was 34 °C (Inter Quartile Range: 32 °C to 35.6 °C). Annual descriptives of temperature, heatwaves occurrence and duration, and frequency of emergency hospitalizations due to cardiovascular diseases are shown in Table [1](#Tab1){ref-type="table"}.
The effect of heatwaves on overall cardiovascular hospitalizations was not significantly different from the null (Fig. [2](#Fig2){ref-type="fig"}). No significant differences were observed when stratifying by sex, age or cardiovascular type categories (Fig. [2](#Fig2){ref-type="fig"}). Finally, sensitivity analyses did not modify the study findings (Additional file [3](#MOESM3){ref-type="media"}).Fig. 2Incident rate ratios (*IRR*) for cardiovascular emergency hospitalization during heatwaves. All models were adjusted by age, fortnight and air pollution. Results are shown stratified by gender, age groups and cardiovascular (*CV*) event type. ^*a*^Age groups used for adjustment in models restricted to population \< 65 years old: 18--35, 36--55; 56--64. ^*b*^Age groups used for adjustment in models restricted to population ≥65 years old; 65--70, 71--75, 76--80, \>80
Discussion {#Sec11}
==========
This study offers a comprehensive examination of
| 3,691
| 590
| 2,395
| 3,403
| null | null |
github_plus_top10pct_by_avg
|
the metric has the form $$\begin{aligned}
-2\Omega^2({\mathrm{d}}\ub\otimes{\mathrm{d}}u+{\mathrm{d}}u\otimes{\mathrm{d}}\ub)+r^2{\mathrm{d}}\sigma_{\mathbb{S}^2}\end{aligned}$$ where the area radius function $r=r(\ub,u)$ is defined by $$\begin{aligned}
\text{Area}(S_{\ub,u})=4\pi r^2,\end{aligned}$$ and ${\mathrm{d}}\sigma_{\mathbb{S}^2}$ is the standard metric of the unit sphere.
Equations
---------
From the form of the metric, the unknowns of the Einstein-scalar field equations are $r$, $\Omega$ and the scalar field function $\phi$. What we really concern are their derivatives. We define the null expansions relative to the normalized pair of null vectors $\Omega^{-2}L$, $\Lb$ and the mass function $m$ by $$\begin{aligned}
h=\Omega^{-2}D r,\ {\underline{h}}={\underline{D}}r,\ m=\frac{r}{2}(1+h{\underline{h}}),\end{aligned}$$ where $D$ and ${\underline{D}}$ are the restrictions on the orbit spheres of the Lie derivatives along $L$ and $\Lb$. When applying on functions, $D$ and ${\underline{D}}$ are simply the ordinary derivatives. We then define the $D$ derivative of the lapse $\Omega$$$\begin{aligned}
\omega=D\log\Omega,\end{aligned}$$ while its ${\underline{D}}$ derivative is not needed in this paper. Finally, we also need the derivatives of the scalar field function $\phi$: $$\begin{aligned}
L\phi=\frac{\partial}{\partial\ub}\phi,\ \Lb\phi=\frac{\partial}{\partial u}\phi.\end{aligned}$$
We then list below all the equations which are satisfied by the above quantities above and are needed in this paper. First of all, we have the following five null structure equations:[^2] $$\begin{aligned}
\label{Dh}Dh=&-r\Omega^{-2}(L\phi)^2,\\
\label{Dbh}{\underline{D}}(\Omega^2h)=&-\frac{\Omega^2(1+h{\underline{h}})}{r},\\
\label{Dhb}D{\underline{h}}=&-\frac{\Omega^2(1+h{\underline{h}})}{r},\\
\label{Dbhb}{\underline{D}}(\Omega^{-2}{\underline{h}})=&-r\Omega^{-2}(\Lb\phi)^2,\\
\label{Dbomega}{\underline{D}}\omega=&\frac{\Omega^2(1+h{\underline{h}})}{r^2}-L\phi\Lb\phi.\end{aligned}$$ The following two equations, which a
| 3,692
| 1,534
| 3,126
| 3,472
| null | null |
github_plus_top10pct_by_avg
|
h_{i} )^2 ( \Delta_{K} - h_{j} )^2 ( \Delta_{K} - h_{k} ) }
\left( h_{i} + h_{j} - 2 \Delta_{K} \right)
e^{- i \Delta_{K} x}
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{j} )^2 ( \Delta_{K} - h_{k} ) ( h_{j} - h_{i} ) }
e^{ - i h_{j} x}
-
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - h_{k} ) ( h_{j} - h_{i} ) }
e^{ - i h_{i} x}
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K j},
\label{hatS-ij-W4-H4-single}\end{aligned}$$ and $$\begin{aligned}
&& \hat{S}_{ij}^{(4)} [4] \vert_{i \neq j}~(\text{double sum})
\nonumber \\
&=&
\sum_{K \neq L}
\biggl[
(ix) e^{- i h_{i} x}
\frac{ 1 }{ (\Delta_{K} - h_{i} ) (\Delta_{L} - h_{i} ) ( h_{j} - h_{i} ) }
\nonumber \\
&-&
\frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{L} - h_{i} )^2 (\Delta_{L} - h_{j} ) }
e^{- i \Delta_{L} x}
+ \frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{K} - h_{i} )^2 (\Delta_{K} - h_{j} ) }
e^{- i \Delta_{K} x}
\nonumber \\
&+&
\frac{ 1 }{ (\Delta_{K} - h_{j} ) (\Delta_{L} - h_{j} ) ( h_{j} - h_{i} )^2 } e^{- i h_{j} x}
\nonumber \\
&-&
\frac{ 1 }{ (\Delta_{K} - h_{i} )^2 (\Delta_{L} - h_{i} )^2 ( h_{j} - h_{i} )^2 }
\biggl\{
3 h_{i}^2 - 2 h_{i} h_{j} + \left( h_{j} - 2 h_{i} \right) (\Delta_{K} + \Delta_{L} ) + \Delta_{K} \Delta_{L}
\biggr\}
e^{- i h_{i} x}
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A (UX) \right\}_{K i}
\left\{ (UX)^{\dagger} A W \right\}_{i L}
\left\{ W ^{\dagger} A (UX) \right\}_{L j}
\nonumber \\
&+&
\sum_{K \neq L}
\biggl[
- (ix) e^{- i h_{j} x}
\frac{ 1 }{ (\Delta_{K} - h_{j} ) (\Delta_{L} - h_{j} ) ( h_{j} - h_{i} ) }
\nonumber \\
&+&
\frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{K} - h_{i} ) (\Delta_{K} - h_{j} )^2 }
e^{- i \Delta_{K} x}
- \frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{L} - h_{i} ) (\Delta_{L} - h_{j} )^2 }
e^{- i \Delta_{L} x}
\nonumber \\
&+&
\frac{ 1 }{ (\D
| 3,693
| 3,364
| 3,040
| 3,494
| null | null |
github_plus_top10pct_by_avg
|
), with $N_z=1$. This leads to Eq.(\[GenM\]) with $M=1$: u\_r= . \[urXY\] Later we will present the numerical evidence that the stiffness perpendicular to the layers of a simple XY layered model in the asymptotic limit can well be described by the above equation. Below we will show that for the case of the two asymmetric layers, the effective value of $M$ is $M=2$ in Eq.(\[GenM\]).
Numerical results for $N_z=2$ {#sec:num}
-----------------------------
Here we present the results of Monte Carlo simulations of the bilayer in the regime of SP in the limit $u_V<<1$. The action (\[H\_J\]) can be represented in the notations $T,X,Y,\delta$ as H\_J= \_[ij]{}+ \_i , \[H12\] where $J_{1,ij}$ and $J_{2,ij}$ refer to the inplane bond currents in the layers 1 and 2, respectively; the actual values of the parameters used in the simulations have been discussed at the end of Sec.\[sec:RG\]: $X=5, \delta=1/2, T=(T_d + T_{(X,-1)})/2 \approx 0.565$.
The structure of the loops is determined by the energy of creating a J-current element along a given direction. A typical energy to create an additional J-current element in the plane 2 can be estimated as $ \delta E_2 \approx T/(2\delta) \approx 0.5$. Thus, large loops with a typical values $|\vec{J}_2|=1$ can exist in the plane 2. In contrast, the energy to create an isolated element in the plane 1 (with no $J_2$ currents along the same bond in the layer 2) costs much larger energy: $\delta E_1 \approx T(1+ X^2/\delta)/2 \approx 15 $ . Thus, the probability to create such an element is exponentially suppressed as $\sim \exp(-15)$, and no entropy contribution (due to 4 optional directions along the plane) can compensate for such a low value. This implies that no large isolated loops can exist in the layer 1. The only option to create a large loop in the layer 1 is if each element $J_{1,ij}$ is coupled to currents $J_{2,ij}=XJ_{1,ij}$ along the same bonds in the layer 2. A typical energy of this combined element is $\delta E_{12} \approx T/2 \approx 0.25$. This strong asymmetry betw
| 3,694
| 1,599
| 3,720
| 3,494
| 3,913
| 0.769312
|
github_plus_top10pct_by_avg
|
and $x{\mathbin{{}_{(\dashv)}}}y:=x\circ
y$ is an exceptional Jordan dialgebra.
Assume the opposite. Let $J\hookrightarrow D^{(+)}$ where $(D,\vdash,\dashv)$ is an associative dialgebra and the product in $D^{(+)}$ is given by the formula (\[eq:QuasiJordanProduct\]). Consider $I={\mathop{\mathrm{Span}}\nolimits}\{a{\mathbin\vdash}b-a{\mathbin\dashv}b\mid a,\,b\in D\}$ that is an ideal of $D$. Then $\bar D=D/I$ is an ordinary associative algebra and $\phi\colon D^{(+)}\to\bar D^{(+)}$ is the natural homomorphism of a Jordan dialgebra on its quotient algebra. The composition of the embedding $\hookrightarrow$ and $\phi$ is a homomorphism too, we denote this homomorphism through $\psi$. It is clear that $K:=\ker\psi$ is an ideal of $J$. Since $\psi$ is a restriction $\phi$ on $J$ so $K=\ker\psi\subseteq\ker\phi=I$. We have $I{\mathbin\vdash}J=J{\mathbin\dashv}I=0$, this is a consequence of the 0-identity. Hence $I\circ J=I{\mathbin{{}_{(\vdash)}}}J=\frac{1}{2}(I{\mathbin\vdash}J+J{\mathbin\dashv}I)=0$, from conditions of the proposition we obtain $I=0$ therefore and $K=0$. So $\psi$ is an embedding and $J\hookrightarrow\bar
D^{(+)}$, i. e., $J$ is exceptional.
Let $\mathbf{C}$ be the Cayley-Dickson algebra over the field $\Bbbk$, ${\mathop{\mathrm{char}}\nolimits}\Bbbk\not=2$. Consider an algebra $H(\mathbf{C}_3)$ of those $3\times 3$ matrices over $\mathbf{C}$ that are symmetric relative the involution in $\mathbf{C}$. This is so called Albert algebra. It is well-known that $J=H(\mathbf{C}_3)$ is a simple exceptional Jordan algebra, so $J$ satisfies the conditions of Proposition \[prop:ExampleExceptionalDialgebra\]. Therefore,
The Albert algebra is exceptional as a Jordan dialgebra.
Symmetric and Jordan polynomials
--------------------------------
Let ${\mathrm{Alg}}\,\langle X\rangle$ be a free non-associative algebra generated by $X$, ${\mathrm{As}}\,\langle X\rangle$ be a free associative algebra, ${\mathrm{Di}}{\mathrm{Alg}}\,\langle X\rangle$ be a free non-associative dialgebra, ${\mathrm{Di}}{\mathrm{As}
| 3,695
| 2,220
| 1,058
| 3,587
| 2,629
| 0.778295
|
github_plus_top10pct_by_avg
|
for $(\mu, \sigma^2)\in \mathcal{M}_2$ in [18]{}. It is straightforward to show, by checking the values of the supremum in [14]{} at the boundary points below and by the fact that $\sup_{(\mu, \sigma^2)\in \mathcal{M}_2} [f_{i-1, b}(\mu, \sigma^2) ]$ is continuous for $b$ in a compact set in $(0,\infty)$, that in [14]{}, if $\partial^2_{xx} V_{i-1}{\geqslant}0$, then and if $\partial^2_{xx} V_{i-1}< 0$, then Therefore, $\sigma_i^2$ is well-defined and is bounded below by $\sigma_0^2$ in [51]{}.
In Theorem \[t1\], if we assume that $\overline{\mu}=\underline{\mu}=:\mu$, then it is easy to check that If we assume further that $\varphi$ is a convex (concave resp.) function and hence $V(t,\cdot)=E\varphi(\cdot+\sqrt{t}Z)$ is convex (concave resp.), then $\sigma_i$ is further reduced to $\overline{\sigma}_{\mu}$ ($\underline{\sigma}_{\mu} $ resp.). In this special case, Theorem \[t1\] reduces to Corollary \[t0\] except for the constant.
Representation of $G$-normal distribution
=========================================
Under the sublinear expectation, the $G$-normal distribution $\mathcal{N}_G$ plays the same role as the classical normal distribution does in a probability space (cf. [002]{}). However, since $\mathcal{N}_G$ is linked with a fully nonlinear PDE, which is called $G$-heat equation, generally we cannot give an explicit expression for $\mathcal{N}_G[\varphi]$ like the linear case. So it would be important to give a representation or approximation for $\mathcal{N}_G[\varphi] $ using random variables or processes in a probability space.
Theorem \[t1\] shows that under a certain normalization, the partial sum of i.i.d random variables in a sublinear expectation space converges to the standard normal distribution. Motivated by this, in this section, we give an approximation of the $G$-normal distribution by using a suitably normalized partial sum of i.i.d. random variables in a probability space. Moreover, the continuous-time counterpart provides a representation of the $G$-normal distribution using (non-
| 3,696
| 1,731
| 2,351
| 3,376
| null | null |
github_plus_top10pct_by_avg
|
ct of dimension and the eigenvalues of $\Sigma$, while leveraging results from [@cherno1; @cherno2] on high dimensional central limit theorems for simple convex sets.
We derive a general result on the accuracy of the Normal approximation over hyper-rectangles for nonlinear parameters. We make use of three findings from [@cherno2; @chernozhukov2015comparison] and [@nazarov1807maximal]: the Gaussian anti-concentration theorem, the high-dimensional central limit theorem for sparely convex sets, and the Gaussian comparison theorem,reported in the appendix as Theorems \[thm:anti.concentration\], \[thm:high.dim.clt\] and \[thm:comparisons\], respectively. In fact, in the appendix we re-state these results in a slightly different form than they appear in the original papers. We do this because we need to keep track of certain constants that affect our results.
Let $W_1,\ldots, W_n$ be an independent sample from a distribution $P$ on $\mathbb{R}^b$ belonging to the class ${\cal P}_n$ of probability distribution supported on a subset of $[-A,A]^b$, for some fixed $A>0$ and such that $$v = \inf_{ P \in \mathcal{P}_n} \lambda_{\min}(V(P))) \quad
\text{and} \quad \overline{v} = \sup_{ P \in \mathcal{P}_n}
\lambda_{\max}(V(P))) \geq 1,$$ where $V(P) = \mathbb{E}_P[ (W_i-\psi)(W_i-\psi)^\top]$. We allow the class $\mathcal{P}_n$ to change with $n$, so that $b$, $v$ and $\overline{v}$ – but not $A$ – are to be regarded as functions of $n$, although we do not express such dependence in our notation for ease of readability. Notice that, in the worse case, $\overline{v}$ can be of order $b$.
[**Remark.**]{} The assumption that $\overline{v} \geq 1$ is made out of convenience and is used in the proof of in the Appendix. Our results remain valid if we assume that $\overline{v}$ is bounded away from $0$ uniformly in $n$, i.e. that $\overline{v} \geq \eta$ for some $\eta > 0$ and all $n$. Then, the term $\eta$ would then appear as another quantity affecting the bounds. We have not kept track of this additional depe
| 3,697
| 2,300
| 1,725
| 3,600
| 2,640
| 0.778206
|
github_plus_top10pct_by_avg
|
disorder (past yr.) 87 (40.7) 24 (55.8) 1.83 0.067
No
Yes 140 (66.2) 27 (62.8) Ref.
72 (33.8) 16 (37.2) 1.15 0.668
*Psychiatric co-morbidity, substance use*
Alcohol dependence (past yr.)
No 168 (79.0) 30 (69.8) Ref.
Yes 45 (21.0) 13 (30.2) 1.62 0.188
Drug dependence (past yr.)^a^
No 202 (94.9) 41 (95.3) Ref.
Yes 11 (5.1) 2 (4.7) 0.90 0.894
*Other psychiatric variables*
Probable TBI
No 141 (67.0) 27 (62.8) Ref.
Yes 70 (33.0) 16 (37.2) 1.19 0.596
*TV/Internet use*
Daily TV (hours)
0--1 62 (29.0) 8 (18.6) Ref.
1.5--2 74 (35.1) 18 (41.9) 1.89 0.167
\>2
| 3,698
| 7,001
| 960
| 1,106
| null | null |
github_plus_top10pct_by_avg
|
ion (\[explota\]) is obtained.
Proof of the *third requirement* as a consequence of Properties 1 and 2
=======================================================================
The purpose of this section is to show that Property 1 and Property 2 given above imply the *third requirement*. This leads to the conclusion that the Gao-Wald theorem holds for space times satisfying these properties.
Before making formal statements, it is worth to give the intuition behind the proof of *third requirement* given below. One considers a reference null geodesic $\gamma_0(\lambda)$ which, by Property 1, has two conjugate points let’s say $p_0$ and $q_0$. Remark \[theta\_i\] shows that (\[tang2\]) is satisfied for some point in the middle $s_0$. That is, for this point $s_0=\gamma_0(\lambda_i)$ one has $$\theta_0(\lambda_i)+\frac{c_\Lambda}{2}< \lim_{\lambda\to\infty} \text{inf}\int_{\lambda_i}^\lambda e^{-c_\Lambda \xi} p_0(\xi)
d\xi.$$ The strategy is to show that, for some geodesics emanating from points $p$ close to $p_0$, the condition (\[tang2\]) is also satisfied for the points $s=\gamma_\Lambda(\lambda_i)$, which are close to $s_0$. That is $$\theta_\Lambda(\lambda_i)+\frac{c_\Lambda}{2}< \lim_{\lambda\to\infty} \text{inf}\int_{\lambda_i}^\lambda e^{-c_\Lambda \xi} p_\Lambda(\xi)
d\xi,$$ for all these geodesics. The Proposition \[prop1\] will allow to conclude that $\theta_\Lambda(\lambda)$ also is going to tend to $-\infty$ at a finite value of the affine parameter $\lambda$. Therefore, these points $p$ will have a conjugate point $q$ joined by the null geodesic $\gamma_\Lambda(\lambda)$. This leads to the first part of the *third requirement*.
The second part is more subtle. The point is that, by assumption, the reference geodesic $\gamma_0(\lambda)$ has two conjugate points $p_0$ and $q_0$. At these points the determinant $G_0(\lambda)$ defined in (\[smile2\]) vanishes. Thus $G_0(0)=0$ and $G_0(\lambda_e)=0$. By continuity in $S$, one may work by analogy with section 2.3 and try to show that $|G_\Lambda(\lambd
| 3,699
| 1,660
| 2,876
| 3,544
| null | null |
github_plus_top10pct_by_avg
|
e anticommutator of a supersymmetry generator with its adjoint gives the Hamiltonian of the system. In a certain sense thus, making a system supersymmetric amounts to taking a square-root of its Hamiltonian. Put differently, the square-root of the Klein–Gordon equation is the Dirac equation, when this correspondence is extended to field theories. From these simply remarks it already transpires that supersymmetry algebras provide powerful new tools with which to explore mathematics questions within a context which may draw on a lot of insight and intuition from quantum physics.[@MATH; @Witten3] Results have indeed been very rewarding already, and many more are still to be established along such lines.
To complete the algebraic relations in (\[eq:SUSYalgebra\]), it is also useful to display the supersymmetry action on the creation and annihilation operators, $$\begin{array}{r c l}
[Q,a]=-\sqrt{\hbar\omega}\,b\ \ ,\ \
[Q,a^\dagger]=0\ \ &,&\ \
[Q^\dagger,a]=0\ \ ,\ \
[Q^\dagger,a^\dagger]=\sqrt{\hbar\omega}\,b^\dagger\ ,\\
& \\
\left\{Q,b\right\}=0\ \ ,\ \
\left\{Q,b^\dagger\right\}=\sqrt{\hbar\omega}\,a^\dagger\ \ &,&\ \
\left\{Q^\dagger,b\right\}=\sqrt{\hbar\omega}\,a\ \ ,\ \
\left\{Q^\dagger,b^\dagger\right\}=0\ .
\end{array}
\label{eq:QFock}$$
The properties $Q^2=0={Q^\dagger}^2$ also suggest that it should be possible to obtain wave function representations of the fermionic and supersymmetry algebras using complex valued Grassmann odd variables $\theta$, such that $\theta_1\theta_2=-\theta_2\theta_1$ and thus $\theta^2_1=0=\theta^2_2$, in the same way that the bosonic Fock algebra possesses wave function representations in terms of commuting coordinates, a configuration space coordinate $x$ and its conjugate momentum $p$, obeying the Heisenberg algebra.[@GovCOPRO2] In the latter case, these two variables may be combined into a single complex commuting variable $z$, leading for instance to the usual holomorphic representation in the bosonic sector, $$a=\frac{\partial}{\partial z}\ \ \ ,\ \ \ a^\dagger=z
| 3,700
| 1,956
| 2,951
| 3,291
| null | null |
github_plus_top10pct_by_avg
|
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