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ties. Thus the net second order baryon number susceptibility is related to the second-order QNS as $\chi_B=\frac{1}{3}\chi$.
The strength of the magnetic field produced in non-central heavy-ion collision can be up to $(10-20)m_\pi^2$ at the time of collision [@Bzdak:2011yy]. However, it decreases very fast being inversely proportional to the square of time [@PhysRevLett.110.192301; @McLerran:2013hla]. But if one considers finite electric conductivity of the medium, then the magnetic field strength will not die out very fast [@Tuchin:2013bda; @Tuchin:2012mf; @Tuchin:2013ie]. We consider two different cases with strong and weak magnetic field in this article.
Strong magnetic field {#sfa}
=====================
In this section we consider strong field scale hierarchy $gT < T < \sqrt{eB}$. In presence of magnetic field, the energy of charged fermion becomes $E_n=\sqrt{k_3^2+m_f^2+2n q_fB}$ where $k_3$ is the momentum of fermion along the magnetic field direction, $m_f$ is the mass of the fermion and the Landau level, $n$, can vary from 0 to $\infty$. The transverse momentum of fermion becomes quantised. It can be shown that at very high magnetic field, the contribution from all the Landau levels except the lowest Landau level can be ignored [@Bandyopadhyay:2016fyd]. Consequently, the dynamics becomes $(1+1)$ dimensional when one considers only lowest Landau level (LLL). The general structures of quark and gluon self-energy in presence of magnetic field have been formulated in Ref. [@Karmakar:2019tdp] at finite temperature but for zero quark chemical potential. Here we extend it for the case of non-zero quark chemical potential. In the presence of strong magnetic field, the general structure of quark self-energy can be written as [@Karmakar:2019tdp] (p\_0,p\_3)&=& a u + b n + c\_5 u +d\_5 n , where the rest frame of heat bath velocity $u_\mu=(1,0,0,0)$ and the direction of magnetic field $n_\mu=(0,0,0,1)$. Now, the various form factors can be obtained as a&=&\[u\], \[a\_def\]\
b &=& -\[n\] , \[b\_def\]\
c&= &\
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${{\mathcal C}}$. Section 5 is a postface, or a “discussion” (as they do in medical journals) – we discuss some of the further things one might (and should) do with cyclic bimodules, and how to correct some deficiencies of the theory developed in Sections 2 and 4.
Acknowledgements. {#acknowledgements. .unnumbered}
-----------------
In the course of this work, I have benefited greatly from discussions with A. Beilinson, E. Getzler, V. Ginzburg, A. Kuznetsov, N. Markarian, D. Tamarkin, and B. Tsygan. I am grateful to Northwestern Univeristy, where part of this work was done, and where some of the results were presented in seminars, with great indulgence from the audience towards the unfinished state they were in. And, last but not least, it is a great pleasure and a great opportunity to dedicate the paper to Yuri Ivanovich Manin on his birthday. Besides all the usual things, I would like to stress that it is the book [@GM1], – and [@GM2], to a lesser extent – which shaped the way we look at homological algebra today, at least “we” of my generation and of Moscow school. Without Manin’s decisive influence, this paper certainly would not have appeared (as in fact at least a half of the papers I ever wrote).
Recollection on cyclic homology.
================================
We start by recalling, extremely briefly, A. Connes’ approach to cyclic homology, which was originally introduced in [@C] (for detailed overviews, see e.g. [@L Section 6] or [@FT Appendix]; a brief but complete exposition using the same language and notation as in this paper can be found in [@Ka Section 1]).
Connes’ approach relies on the technique of homology of small categories. Fix a base field $k$. Recall that for every small category $\Gamma$, the category ${\operatorname{Fun}}(\Gamma,k)$ of functors from $\Gamma$ to the category $k{\operatorname{\it\!-Vect}}$ of $k$-vector spaces is an abelian category with enough projectives and enough injectives, with derived category ${{\mathcal D}}(\Gamma,k)$. For any object $E \in
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not understand what is the proper cyclic bimodule context for higher-level infinitesemal extensions. Of course, if one is only interested in an $R$-deformation ${\widetilde}{A} = A_R$ over an Artin local base $R$, not in its cyclic bimodule generalizations, one can use Goodwillie’s Theorem: using the full cyclic object ${\widetilde}{A}_\#$ instead of its quotient $\overline{A_\#}$ in Proposition \[spl\] immediately gives a splitting $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A) \to HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_R/R)$ of the augmentation map $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_R/R) \to HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$, and this extends by $R$-linearity to an isomorphism $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_R/R) \cong
HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A) \otimes R$. However, this is not quite satisfactory from the conceptual point of view, and it does not work in positive characteristic (where Goodwillie’s Theorem is simply not true). If ${\operatorname{\sf char}}k \neq 2$, the latter can be cured by using ${\widetilde}{A}_\#/F^3{\widetilde}{A}_\#$, but the former remains. We plan to return to this elsewhere.
Categorical approach. {#cat}
=====================
Let us now try to define cyclic homology in a more general setting – we will attempt to replace $A{\operatorname{\!-\sf bimod}}$ with an arbitrary associative unital $k$-linear tensor category ${{\mathcal C}}$ with a unit object ${\operatorname{{\sf I}}}\in {{\mathcal C}}$. We do not assume that ${{\mathcal C}}$ is symmetric in any way. However, we will assume that the tensor product $- \otimes -$ is right-exact in each variable, and we will need to impose additional technical assumptions later on.
The first thing to do is to try to define Hochschild homology; so, let us look more closely at . The formula in the right-hand side looks symmetric, but this is an optical illusion – the two copies of $A$ are completely different objects: one is a left module over $A^{opp} \otime
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) \cap B(O,r) \right) = 1 ~.$$
- For any $k\in \{1,\ldots,5\}$ and for all integers $n$ such that $0\leq n\leq (R-r_k-1)/2\varepsilon$: $$\hat{N}\left(B\big((r_k+1+2n\varepsilon) e^{\i (2k\pi/5\pm2\varepsilon)},\varepsilon\big) \cap V_{\varepsilon}(r_{1},\ldots,r_{5}) \right)=1 ~.$$
- The previous points are the only ones of $\hat{N}$.
It is clear that the event $\hat{N}\in B_{\varepsilon}(r_{1},\ldots,r_{5})$ occurs with positive probability, for all $\varepsilon>0$. Roughly speaking, the points of $\hat{N}$ introduced in $(\clubsuit)$ form a chain from $re^{\i 2k\pi/5}$ to $(r_{k}-1)e^{\i 2k\pi/5}$, for any index $k$ such that $r_{k}>r$. See Figure \[fig:5arbres\].
![\[fig:5arbres\] [*RST of the PPP $N$ satisfying both events $A_{\varepsilon}(r_{1},\ldots,r_{5})$ and $B_{\varepsilon}(r_{1},\ldots,r_{5})$. Beware the fact that, in order not to overload the figure, the condition $(\diamondsuit)$ of $B_{\varepsilon}(r_{1},\ldots,r_{5})$ has not been represented. The two balls are centered at $O$ with radii $r=\min_{k\in \{1,\dots,5\}} r_{k}$ and $R=\max_{k\in \{1,\dots,5\}} r_{k}+1$. The $X_{k}$’s are represented by big gray squares while the other points of $N$ by small black circles.*]{}](5arbres.eps){width="11cm" height="11cm"}
On Figure \[fig:5arbres\], imagine that $R=r_{4}+1$ is much larger than $r=r_{5}$ (indeed, we have no control on the $r_{k}$’s). Henceforth, the semi-infinite path $\gamma_{5}$ could prefer to branch on the points of $\hat{N}$ introduced in $(\clubsuit)$ and with direction $8\pi/5$ rather than on $X_{5}$. To prevent this situation from occurring, we contain each path $\gamma_k$ in the cone $C_{2k\pi/5,3\varepsilon,r_{k}}$ thanks to the points of $\hat{N}$ introduced in $(\diamondsuit)$. These points form “landing runways” for the $\gamma_{k}$’s (they may also change slightly the $\gamma_{k}$’s).\
Let us denote by $A_{\varepsilon}$ and $B_{\varepsilon}$ the events $A_{\varepsilon}(r_{1},\ldots,r_{5})$ and $B_{\varepsilon}(r_{1},\ldots,r_{5})$. Then, $$\{ \widetilde{N
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noted that the gauge parameter in this form, $\lambda_{{\mathcal{S}}\tilde{p}}=X\eta\xi_0[{\mathcal{S}},\tilde{p}]\Phi
+ X\eta[{\mathcal{S}},M]\Phi$, is in the restricted small Hilbert space: $\eta\lambda_{{\mathcal{S}}\tilde{p}}=0$ and $XY\lambda_{{\mathcal{S}}\tilde{p}}=\lambda_{{\mathcal{S}}\tilde{p}}$.
In addition, a further extra transformation is produced by considering the commutator between $\delta_{\tilde{p}_1}$ and $\delta_{\tilde{p}_2}$, and this sequence of extra transformations does not terminate as long as the nested commutators, $[{\mathcal{O}},[{\mathcal{O}},{\mathcal{O}}]]$, $[{\mathcal{O}},[{\mathcal{O}},[{\mathcal{O}},{\mathcal{O}}]]]$, $\cdots$, with $\mathcal{O}=$ ${\mathcal{S}}$ or $\tilde{p}$, do not vanish. This complicates the structure of the algebra, but we can similarly show that all of these extra transformations act trivially on the physical S-matrix, as shown in Appendix \[app B\].
Summary and discussion
======================
In this paper, we have explicitly constructed a space-time supersymmetry transformation of the WZW-like open superstring field theory in flat ten-dimensional space-time. Under the GSO projections, we have extended a linear transformation expected from space-time supersymmetry in the first-quantized theory to a nonlinear transformation so as to be a symmetry of the complete action (\[complete action\]). We have also shown that the transformation satisfies the supersymmetry algebra up to gauge transformation, the equations of motion and a transformation $\delta_{\tilde{p}}$ acting trivially on the asymptotic physical states defined by the asymptotic string fields. This unphysical transformation produces a series of transformations $\delta_{[{\mathcal{S}},\tilde{p}]},\, \delta_{[\tilde{p}\tilde{p}]},\,\cdots$ by taking commutators with $\delta_{\mathcal{S}}$ or $\delta_{\tilde{p}}$ repeatedly. All of these symmetries also act trivially on the asymptotic physical states, and thus are unphysical, but it is interesting to clarify their complete structure, which
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” of . If *is* symmetric, then the duality $A\mapsto A\op$ extends to a self-duality of the bicategory $\cProf({\sV})$, from which the equivalence of \[item:sd1\] and \[item:sd1op\] follows formally; the proof given above shows that this equivalence remains true even in the non-symmetric case, due to this “centrality”.)
Now we can answer our first two questions from \[sec:galois\].
\[thm:stab-op\] For a derivator ${\sD}$ and a class of functors $\Phi$, the following are equivalent.
1. ${\sD}$ is left $\Phi$-stable, i.e. $\Phi$-colimits in commute with arbitrary limits.\[item:so1\]
2. For each $u\in\Phi$, the morphism $u_! :{\sD}^A \to{\sD}^B$ has a left adjoint.\[item:so2\]
3. ${\sD}$ is right $\Phi\op$-stable, i.e. $\Phi\op$-limits in commute with arbitrary colimits.\[item:so3\]
4. For each $u\in\Phi\op$, the morphism $(u\op)_\ast :{\sD}^{A\op} \to{\sD}^{B\op}$ has a right adjoint.\[item:so4\]
We have \[item:so2\] implies \[item:so1\], since all right Kan extensions exist in a derivator (as opposed to a left derivator), and are preserved by any right adjoint morphism. Dually, \[item:so4\] implies \[item:so3\]. We will prove that \[item:so3\] implies \[item:so2\]; by duality then also \[item:so1\] implies \[item:so4\] and we are done. If ${\sD}$ is right $\Phi\op$-stable, then we remarked above that ${\mathsf{END}\ccsub}({\sD})$ is right $\Phi\op$-stable, and ${\sD}$ is a ${\mathsf{END}\ccsub}({\sD})$-module. Therefore, by \[thm:stable-dual\], $u_!$ has a left adjoint (that is even a weighted colimit functor) for each $u\in\Phi$.
If $\Phi=\Phi\op$, then ${\mathsf{Stab}_L}(\Phi)={\mathsf{Stab}_R}(\Phi)$.
This explains the self-dual nature of pointedness, semiadditivity, and stability as due to the fact that $\Phi=\{\emptyset\}$, $\Phi=\mathsf{FINDISC}$, and $\Phi=\mathsf{FIN}$ are self-dual. Similarly, it explains the identity ${\mathsf{Stab}_L}(\{\emptyset,\ulcorner\}) = {\mathsf{Stab}_R}(\{\emptyset,\lrcorner\}) = \mathsf{STABLE}$, since $(\ulcorner)\op = \lrcorner$.
Stability via
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The rest of this section is devoted to the definition and basic properties of category $\mathcal{O}_c$. Since its structure depends upon the combinatorics of ${{W}}$-representations, we begin with the relevant notions from that theory.
We write a partition of $n$ as $\mu = (\mu_1\geq \mu_2 \geq \cdots \geq \mu_l > 0)$, with the understanding that $\mu_i =0$ for $i>l$. The *Ferrers diagram* of $\mu$ is the set of lattice points\[d-mu-defn\] $$d(\mu) = \{ (i,j)\in {\mathbb{N}}\times {\mathbb{N}}: j <
\mu_{i+1}\}.$$ Following the French style, the diagram is drawn with the $i$-axis vertical and the $j$-axis horizontal, so the parts of $\mu$ are the lengths of the rows, and $(0,0)$ is the lower left corner. The *arm* $a(x)$ and the *leg* $l(x)$ of a point $x\in d(\mu)$ denote the number of points strictly to the right of $x$ and above $x$, respectively. The [*hook length*]{} $h(x)$ is $1+a(x)+l(x)$. For example: $$\label{e:arm-leg-pix}
\mu =(5,5,4,3,1)\qquad
\begin{array}[c]{cccccc}
\bullet & \hbox to 0pt{\hss $\scriptstyle l(x)$\hss }\\
\cline{2-2}
\bullet & \multicolumn{1}{|c|}{\bullet }& \bullet \\
\bullet & \multicolumn{1}{|c|}{\bullet }& \bullet & \bullet \\
\cline{2-5}
\bullet & \multicolumn{1}{|c|}{\llap{${}_{x}$}\bullet } & \bullet &
\bullet& \multicolumn{1}{c|}{\bullet }& {\scriptstyle a(x)} \\
\cline{2-5}
\llap{${}_{(0,0)}$} \bullet & \bullet & \bullet &
\bullet & \bullet
\end{array}
\qquad a(x) = 3,\quad l(x) = 2, \quad h(x)=6.$$ The [*transpose partition $\mu^t$*]{} is obtained from $\mu$ by exchanging the rows and columns of $\mu$.
We will always use the [*dominance ordering*]{}\[dominance-defn\] of partitions as in [@MacD p.7]; thus if $\lambda$ and $\mu$ are partitions of $n$ then $\lambda\geq \mu$ if and only if $\sum_{i=1}^k \lambda_i \geq \sum_{i=1}^k\mu_i$ for all $k\geq 1$.
Let ${{\textsf}{Irrep}({{W}})}$\[irred-defn\] denote the set of simple ${{W}}$-modules, up to isomorphism. As usual, irreducible representations of ${{W}}$ will be parametrised by partitions of $n$.
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understood today. Other directions of research focus on BO in the presence of relaxation processes (spontaneous emission) [@PRA2], BO in 2D optical lattices [@PRL3], and BO in the presence of atom-atom interactions (‘BEC-regime’) [@Berg98; @Choi99; @Chio00; @Mors01]. The present Letter deals with the third problem, which is approached here by an ‘ab initio’ analysis of the dynamics of a system of many atoms. This distinguishes this work from previous studies of BO in the BEC regime [@Berg98; @Choi99; @Chio00], which were based on the a mean field approach using a nonlinear Schrödinger equation. A new effect, so far unaddressed by these earlier studies, is predicted: besides the usual Bloch dynamics, the atomic oscillations may exhibit another fundamental period, entirely defined by the strength of the atom-atom interactions.
Let us first recall some results on BO in the single-particle case. Using the tight-binding approximation [@Fuku73], the Hamiltonian of a single atom in an optical lattice has the form $$H = E_0\sum_l |l\rangle\langle l|
-\frac{J}{2}\left(\sum_l |l+1\rangle\langle l|+h.c.\right)$$ $$\label{1}
+dF\sum_l l |l\rangle\langle l| \;.$$ In Eq. (\[1\]), $|l\rangle$ denotes the $l$th Wannier state $\phi_l(x)$ corresponding to the energy level $E_0$ [@remark1], $J$ is the hopping matrix elements between neighbouring Wannier states, $d$ is the lattice period, and $F$ is the magnitude of the static force. The Hamiltonian (\[1\]) can be easily diagonalised, which yields the spectrum $E_l=E_0+dFl$ (the so-called Wannier-Stark ladder) and the eigenstates (Wannier-Stark states) $$\label{1a}
|\psi_l\rangle=\sum_m {\cal J}_{m-l}(J/dF)|m\rangle \;,\quad
\langle x|m\rangle=\phi_m(x) \;,$$ (here ${\cal J}_m(z)$ are the Bessel functions). As a direct consequence of the equidistant spectrum, the evolution of an arbitrary initial wave function is periodic in time, with the Bloch period $T_B=2\pi\hbar/dF$. In particular, we shall be interested in the time evolution of the Bloch states $|\psi_\kappa\rangl
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ion: $$P^f (k_\parallel) = A \, {\rm exp} [- {k_\parallel^2/
{k^s_\parallel}^2}] \int_{k_\parallel}^{\infty} \Biggl[
[2-0.7(\gamma-1)] + f_{\Omega} {k_\parallel^2 \over k^2} -
{{\gamma-1}\over 4} k_\parallel^2 b_{T_0}^2 \Biggr]^2 {\tilde P}^\rho (k)
e^{- {k^2 / k_F^2}}
{k dk \over 2 \pi}
\label{Pf}$$ where $f_{\Omega} = d\, {\rm ln} D/d\, {\rm ln} a$ with $D$ being the linear growth factor and $a$ the Hubble scale factor (see ). The three-dimensional isotropic real-space mass power spectrum is denoted by ${\tilde P}^\rho (k)$, and $k_F$ is the scale of smoothing due to baryon-pressure i.e. ${\tilde P}^\rho (k) \, {\rm exp} [- {k^2
/k_F^2}]$ gives the power spectrum of the baryons. As argued by Gnedin & Hui [-@gh98][^2], $k_F^{-1}$ should be given by $\sqrt 2 \bar H (1+\bar z)^{-1} f_J^{-1} x_J$, where $x_J$ is commonly known as the Jeans scale. The latter is equal to ${\gamma k_B T_0 / 4 \pi a^2 G
\bar \rho \mu}$, where $\mu$ is the mean mass per particle and $\bar \rho$ is the mean mass density. The numerical factor $f_J$ relating $k_F^{-1}$ and $x_J$ should be $O(1)$, its precise value depending somewhat on the reionization history (), but it should have an insignificant effect on our work here, because we are interested primarily in the large scale fluctuations.
The other smoothing scale $k^s_\parallel$ should be equal to ${\sqrt 2} /b_{T_0}$ due to thermal broadening, but we can allow it to be more general to include the effect of finite resolution as well: $$k^s_\parallel = {{\sqrt 2} \over {b_{\rm eff}}} \, \, , \, \, b^2_{\rm
eff} = b^2_{T_0} + {{\rm FWHM}^2 \over {4 \, {\rm ln} 2}}
\label{ks}$$ where ${\rm FWHM}$ is the resolution full-width-half-maximum.
The proportionality constant $A$ for eq. (\[Pf\]) should be equal to $\bar \tau^2$ within the context of linear theory. However, in the spirit of Croft et al. [-@croft98], we assumes the linear prediction gives the right shape but not necessarily the right amplitude for the power spectrum on large scales (see §\[conclude\] for discussions). Hence, $A$
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roposition are jointly distributed as $\sqrt{2}$ times the real parts of the eigenvalues of the random matrix defined in Proposition \[T:C-Ginibre-eigenvalues\], and are thus independent real standard normal random variables.
Observe that in the “$G$-circulant GUE” of Proposition \[T:GUE-eigenvalues\], every element $a \in G$ with $a = a^{-1}$ corresponds to a “diagonal” of $M$ in which the entries are constrained to be real.
The following corollary follows from Proposition \[T:GUE-eigenvalues\] in the same way that Corollary \[T:C-Ginibre-limit\] follows from Proposition \[T:C-Ginibre-eigenvalues\].
\[T:GUE-limit\] Suppose that for each $n$, $\bigl\{Y_a^{(n)} \mid a \in G^{(n)}
\bigr\}$ are real and complex Gaussian random variables as described in Proposition \[T:GUE-eigenvalues\]. Then ${\mathbb{E}}\mu^{(n)} =
\gamma_{\mathbb{R}}$ for each $n$, and $\mu^{(n)} \to \gamma_{\mathbb{R}}$ weakly in probability. Furthermore, if ${\left\vert G^{(n)} \right\vert} = \Omega(n^{{\varepsilon}})$ for some ${\varepsilon}> 0$, then $\mu^{(n)} \to \gamma_{\mathbb{R}}$ weakly almost surely.
The real Ginibre ensemble $X$ consists of a square matrix with independent, real standard Gaussian random variables. The Gaussian Orthogonal Ensemble (GOE) is distributed as $2^{-1/2}(X+X^t)$. Equivalently, the diagonal entries of the GOE are distributed as ${\mathcal{N}}(0,2)$ and the off-diagonal entries are distributed as ${\mathcal{N}}(0,1)$. In general the analogues of Propositions \[T:C-Ginibre-eigenvalues\] and \[T:GUE-eigenvalues\] for matrices with real entries are less elegant. In the nonsymmetric case the eigenvalues have a Gaussian joint distribution in a ${\left\vert G \right\vert}$-dimensional real subspace of ${\mathbb{C}}^{{\left\vert G \right\vert}}$, and in the symmetric case the ${\left\vert G \right\vert}$ eigenvalues are not independent in general. We will not state such results in general, but will note for future reference that in the “$G$-circulant GOE”, every element $a
\in G$ with $a = a^{-1}$ c
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able on $\mathbb{R}$ can be solved [@boyd2004fastest]: the case where $P_{mix}$ is symmetric and the case where $P_{mix}$ is reversible for a given fixed stationary distribution. Let us first consider the case where $P$ is symmetric. The minimisation problem takes the following form:
$$\label{eq:mixmix}
\left\{
\begin{array}{rcr}
\min\limits_{P \in S_n} \lambda(P) \\
P(i,j) \geq 0, P*\textbf{1}=\textbf{1} \\
A(i,j)=0 \Rightarrow P(i,j)=0\\
\end{array}
\right.$$
given the strict convexity of $\lambda$ and the compactness of the stochastic matrices, this problem admits an unique solution.
$P$ is symmetric thus $\textbf{1}$ is an eigenvector associated with the largest eigenvalue of $P$. Then the eigenvectors associated to $\lambda(P)$ are in the orthogonal of $ \textbf{1}$.The orthogonal projection on $\textbf{1}^{\perp}$ writes: $ I_d-\frac{1}{n}\textbf{1}\textbf{1}^t$
Moreover, if we take the matrix norm associated with the euclidiean norm i.e. for $M$ any matrix $|||M|||= \max \frac{ ||MX||_2}{||X||_2} \text{ } X \in \mathbb{R}^n \text{ } X\neq 0$ it is equal to the square root of the largest eigenvalue of $ MM^t$ and then if $M$ is symmetric it is equal to $\lambda(M)$.
Then the minimization problem can be rewritten:
$$\label{eq:mix2}
\left\{
\begin{array}{rcr}
\min\limits_{P \in S_n} ||| (I_d-\frac{1}{n}\textbf{1}\textbf{1}^t)P(I_d-\frac{1}{n}\textbf{1}\textbf{1}^t)|||=|||P-\frac{1}{n}\textbf{1}\textbf{1}^t|||\\
P(i,j) \geq 0, P*\textbf{1}=\textbf{1} \\
A(i,j)=0 \Rightarrow P(i,j)=0\\
\end{array}
\right.$$
We solve this constrained optimization problem (Karush-Kuhn-Tucker) with Matlab and we denote $P_{mix}$ the matrix which minimizes this system. We remark that the mixing time of $P_{KS}$ is smaller than the mixing time of $P_{mix}$. This is coherent because in order to calculate $P_{KS}$ we take the minimum on all the matrix space whereas to calculate $P_{mix}$ we restrict us to the symmetric matrix space. Nevertheless, we can go a step further and calculate, the stationary d
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immune to systematic market fluctuations.
There is an important caveat with respect to the foregoing statement. Recall that there is an inverse relationship between ${{V}}_{\mu}$, defined as the positive square root of ${{V}}_{\mu}^{2}$, and ${W}_{\mu}$, so that for highly leveraged portfolios which are characterized by the condition ${W}_{\mu} \ll 1$, the above argument would imply a principal variance far exceeding the original ones. Of course the condition ${W}_{\mu} \ll 1$ that implies such large variances also implies large expected returns, so that a more sensible comparative measure under such conditions is ${\check{V}}_{\mu}\stackrel{\rm
def}{=}{V}_{\mu}/{R}_{\mu}={v}_{\mu}/{\sum}_{i=1}^{N}{e}^{\mu}_
{i}{r}_{i}$, which may be called [*return-adjusted volatility*]{} of the principal portfolio. As expected, the relative weight ${W}_{\mu}$ is no longer present in this adjusted version of the volatility.
The return-adjusted volatility for the market-aligned portfolio, on the other hand, can be calculated from Eqs. (\[431\]), (\[439\]), and (\[440\]). It is given by $${\check{V}}_{N} \simeq \{ 1- [{({\bar{{\rho}^{2}}}_{mkt})}^{1 \over
2}/{\bar{\rho}}_{mkt}]{\sum}_{i=1}^{N}
{\gamma}_{i}{\hat{\beta}}_{i} \} {({\bar{{\rho}^{2}}}_{mkt})}^{1 \over
2}/{\bar{\rho}}_{mkt}, \label{442}$$ which is approximately equal to ${({\bar{{\rho}^{2}}}_{mkt})}^{1 \over
2}/{\bar{\rho}}_{mkt}$. This ratio is of course precisely what one would expect for the approximate value of the return-adjusted volatility of a portfolio which is aligned with the overall market price movements.
It is appropriate at this point to summarize the properties of the principal portfolios for the single-index model.
[**Proposition 1.**]{} [*The principal portfolios of the single-index model consist of a market-aligned portfolio, which is unleveraged and has a return-adjusted volatility $\simeq {({\bar{{\rho}^{2}}}_{mkt})}^{1 \over
2}/{\bar{\rho}}_{mkt}$ characteristic of market-driven price movements, and $N-1$ market-orthogonal portfolios which are
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9 Partial response 3.2 6.2 Progressed 6.5
DUB, duodenal malignant ulcer bleeding
######
Survival of pancreatic cancer patients treated with tomotherapy and concurrent capecitabine
Group Characteristics Median duration of survival (months)
------- -------------------------------------------------------------------------- --------------------------------------
I Locally advanced without metastasis (n = 10) 9.25 (2.00-18.4)
No previous chemotherapy (n = 8) 12.55 (6.50-18.4)
Previous chemotherapy (n = 2) 3.90 (2.00, 5.8)
II Locally relapsed without metastasis following complete resection (n = 1) 4.80 (4.80)
III Metastatic disease (n = 8) 4.25 (1.10-21.00)
De novo (n = 3) 4.40 (3.90-6.50)
Relapsed (n = 5) 4.10 (1.10-21.00)
Data in parentheses are ranges of survival times
Progression of disease outside the targeted tumor volume (defined as the out-field progression) occurred in 7 patient. The median time to out-field progression was 3.8 months (range 2.2-7.3) with or without systemic chemotherapy following CCRT.
Toxicity
--------
Acute toxicity is summarized in Table [6](#T6){ref-type="table"}. As shown, only minor toxicities developed. The most common acute toxicity was grade 1 or 2 fatigue that occurred 2 to 3 weeks after the start
| 3,213
| 2,809
| 2,135
| 2,977
| null | null |
github_plus_top10pct_by_avg
|
that $F_1=4$. Indeed, there are four ways to choose a root edge in a planar cubic map with two vertices: $$\begin{picture}(340,40) \put(15,20){\oval(30,30)} \put(15,5){\vector(0,1){30}}
\put(70,20){\oval(30,30)} \put(120,20){\oval(30,30)}
\put(85,20){\vector(1,0){20}} \put(175,20){\oval(30,30)}
\put(225,20){\oval(30,30)} \put(190,20){\line(1,0){20}}
\put(210,15){\vector(0,1){5}} \put(275,20){\oval(30,30)}
\put(290,20){\line(1,0){20}} \put(325,20){\oval(30,30)}
\put(310,25){\vector(0,-1){5}} \end{picture}$$ Also we have that $F_2=32$. Indeed, there are six cubic maps with 4 vertices (and 6 edges): $$\begin{picture}(300,50) \put(40,25){\oval(40,40)} \put(40,5){\line(0,1){40}}
\put(60,25){\line(1,0){20}} \put(95,25){\oval(30,30)} \put(5,20){\small 1)}
\put(155,25){\oval(30,30)} \put(215,25){\oval(30,30)}
\qbezier(155,40)(185,55)(215,40) \qbezier(155,10)(185,-5)(215,10)
\put(125,20){\small 2)}
\put(280,25){\oval(40,40)} \put(280,25){\line(0,1){20}} \put(245,20){\small 3)}
\put(280,25){\line(3,-2){17}} \put(280,25){\line(-3,-2){17}} \end{picture}$$ $$\begin{picture}(310,50) \put(20,35){\circle{10}} \put(20,5){\circle{10}}
\put(40,20){\line(-4,3){16}} \put(40,20){\line(-4,-3){16}}
\put(40,20){\line(1,0){20}} \put(65,20){\circle{10}} \put(0,17){\small 4)}
\put(110,20){\oval(20,20)} \put(120,20){\line(1,0){20}}
\put(150,20){\oval(20,20)} \put(160,20){\line(1,0){20}}
\put(190,20){\oval(20,20)} \put(85,17){\small 5)}
\put(240,20){\oval(20,20)} \put(250,20){\line(1,0){20}}
\put(290,20){\oval(40,40)} \put(290,20){\oval(20,20)}
\put(300,20){\line(1,0){10}} \put(215,17){\small 6)}
\end{picture}$$
[Figure 1]{}
Group of automorphisms of the first map is trivial, of the second has order 4, of the third has order 12, of the forth has order 3, of the fifth and the sixth has order 2. Thus, there are 12 ways to choose a root edge in the first map, 3 — in the second, 1 — in the third, 4 — in the forth, 6 — in the fifth and the sixth. All this gives us 32 edge-rooted maps.
However, this formula does not seem to have a
| 3,214
| 3,629
| 2,977
| 2,460
| null | null |
github_plus_top10pct_by_avg
|
ed}
\nonumber
K= W = 48{m}^{2} \left( \alpha r\cos \left( \theta \right) -1 \right) ^{6} \left( \left({a}^{4}\alpha+{a}^{3} \right) \cos^{3}\theta
+ 3{a}^{2}r \left( a \alpha-1 \right) \cos^{2}\theta - 3a {r}^{2} \left( a\alpha+1 \right) \cos \theta -{r}^{3} \left( a\alpha-1 \right) \right) \\
\times \dfrac{ \left( \left( {a}^{4}\alpha - {a}^{3} \right) \cos^{3}\theta - 3{a}^{2}r \left( a \alpha+1 \right) \cos^{2}\theta
- 3a {r}^{2}\left(a\alpha-1 \right) \cos \theta + {r}^{3} \left( a\alpha+1 \right) \right) } { \left( {r}^{2} + {a}^{2} \cos^{2}\theta \right) ^{6} }.\end{aligned}$$ Therefore $ P^{2}=\dfrac{W}{K}=1 $, i.e. $ P=+1 $. Hence the total gravitational entropy in this case is given by $$\label{s_grav_rot}
S_{grav}=k_{s}\int_{\sigma}\mathbf{\Psi}.\mathbf{d\sigma}=k_{s}\int_{\sigma}d\sigma=k_{s}\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi C}\sqrt{g_{\theta\theta}g_{\phi\phi}}d\theta d\phi.$$ The entropy evaluated at $r_{\pm}$ is obtained as $$\label{s_grav_pm}
S_{grav_{\pm}}=k_{s}\dfrac{4\pi C(r^{2}_{\pm}+a^{2})}{(1-\alpha^{2}r_{\pm}^{2})}=k_{s}\dfrac{4\pi (r^{2}_{\pm}+a^{2})}{(1-\alpha^{2}r_{\pm}^{2})(1+2\alpha m+\alpha^2a^2)}.$$ If we substitute $ a=0 $ in (\[s\_grav\_pm\]), then we get back the expression (\[s\_grav\_nonrot\]) for the entropy of the non-rotating accelerating black holes. We see that as the acceleration parameter vanishes, i.e., $ \alpha\rightarrow 0 $, the equation (\[s\_grav\_pm\]) reduces to the expression of gravitational entropy for Kerr black holes derived in [@entropy2]. However for this axisymmetric metric, it is not possible to evaluate the spatial metric using equation (\[sm\]) because the object is rotating, and so there is a nonzero contribution from the component of $ g_{t\phi} $, which changes the spatial positions of events in course of time. Therefore the entropy density is calculated by using the full four-dimensional metric determinant $ g $ in the expression involving the covariant derivative [@entropy2], and we get $$\label{enden1}
s=k_{s}|\mathbf{\nabla}.\mathbf{
| 3,215
| 4,609
| 1,787
| 2,960
| null | null |
github_plus_top10pct_by_avg
|
] \end{aligned}$$ and $$\begin{aligned}
\Omega
F
\left[
\begin{array}{cc}
- \Omega \otimes \Omega & 0 \\
0 & I_k
\end{array}
\right] & =
\Omega
\Big[0_{k \times k^2} \;\;\;\;\; I_k\Big]
\left[
\begin{array}{cc}
- \Omega \otimes \Omega & 0 \\
0 & I_k
\end{array}
\right] \\
& = \Omega \Big[ 0_{k \times k^2} \;\;\;\;\; I_k \Big] =
\Big[ 0_{k \times k^2} \;\;\;\;\; \Omega \Big] .\end{aligned}$$ Plugging the last two expressions into the initial formula for $D
g_j(\psi)$ we obtain that $$\begin{aligned}
\nonumber
D g_j(\psi) & = e^\top_j \Big( \left[ - \left( \alpha^\top \otimes I_k
\right) (\Omega \otimes \Omega) \;\;\;\;\; 0_{k \times k}\right] +
\left[ 0_{k \times k^2} \;\;\; \Omega \right] \Big)\\
& = \label{eq::gj}
e^\top_j \Big( \left[ - \left( \alpha^\top \otimes I_k
\right) (\Omega \otimes \Omega) \;\;\;\;\; \Omega\right] \Big). \end{aligned}$$ The gradient of $g_j$ at $\psi$ is just the transpose of $Dg_j(\psi)$. Thus, the Jacobian of the function $g$ is $$\label{eq::GG}
\beta(j)/d\psi = G = \left(
\begin{array}{c}
G_1^\top\\
\vdots\\
G_k^\top
\end{array}
\right).$$
Next, we compute $Hg_j (\psi)$, the $b \times b$ Hessian of $g_j$ at $\psi$. Using the chain rule, $$H g_j(\psi) = D (D g_j(\psi)) = (I_b \otimes e^\top_j) \frac{ d \; \mathrm{vec} \Big( \left[ - \left( \alpha^\top \otimes I_k
\right) (\Omega \otimes \Omega) \;\;\;\;\; \Omega\right] \Big)}{
d \psi},$$ where the first matrix is of dimension $b \times kb$ and the second matrix is of dimension $kb \times b$. Then, $$\label{eq:H}
\frac{ d \; \mathrm{vec} \Big( \left[ - \left( \alpha^\top \otimes I_k
\right) (\Omega \otimes \Omega) \;\;\;\;\; \Omega\right] \Big)}{
d \psi} = \left[
\begin{array}{c}
-\frac{d \left( \alpha^\top \otimes I_k
\right) (\Omega \otimes \Omega) }{d \psi}\\
\;\; \\
\frac{d \Omega}{d \psi}
\end{array}
\right].$$
The derivative at the bottom of the previous expression is $$\frac{d \Omega}{d \psi} = \frac{d \Omega}{d \Sigma} \frac{d
\Sigma}{d \psi} = -
| 3,216
| 3,293
| 3,172
| 2,828
| null | null |
github_plus_top10pct_by_avg
|
**No of interviewed (13)**
--------------------------------------------------------------- ----------------------------
**Gender**
Female 8 (5 had children)
Male 5 (2 had children)
**Age**
19--29 5
30--49 7
50+ 1
**Ethnic background**
1\. generation immigrant 4
2\. generation immigrant 3
Swedish 6
**Education**
Elementary school 3
Secondary school 6
Dropped out from school 4
**Duration on social assistance**
1--2 years 4
3--5 years 3
More than 5 years 6
**Study site**
1 Outer area 4
2 Disadvantaged area, high proportion on social assistance 3
3 Disadvantaged area, less than expected on social assistance 2
4 Affluent area 2
5 Diversity, mixed neighbourhood 1
6 Disadvantaged area, less than expected on social assistance 1
The length of the interviews varied from 40 minutes to two and a half hours. Bef
| 3,217
| 6,602
| 1,150
| 1,800
| null | null |
github_plus_top10pct_by_avg
|
="48.00000%"} {width="46.40000%"}
Deconfinement aspects of the transition
=======================================
The deconfinement phenomenon in pure gauge theory is governed by breaking of the $Z(N_c)$ symmetry. The order parameter is the renormalized Polyakov loop, obtained from the bare Polyakov loop as $$L_{ren}(T)=z(\beta)^{N_{\tau}} L_{bare}(\beta)=
z(\beta)^{N_{\tau}} \left\langle\frac{1}{N_c} {\rm Tr }
\prod_{x_0=0}^{N_{\tau}-1} U_0(x_0,\vec{x})\right\rangle,$$ where $z(\beta)=\exp(-c(\beta)/2)$. $c(\beta)$ is the additive normalization of the static potential chosen such that it coincides with the string potential at distance $r=1.5r_0$ with $r_0$ being the Sommer scale.
In QCD $Z(N_c)$ symmetry is explicitly broken by dynamical quarks, therefore there is no obvious reason for the Polyakov loop to be sensitive to the singular behavior close to the chiral limit. Indeed, the temperature dependence of the Polyakov loop in pure gauge theory and in QCD is quite different, as one can see from Fig. \[Lren\_absT\]. Also note, that in this purely gluonic observable there is very little sensitivity (through the sea quark loops) to the cut-off effects coming from the fermionic sector. While losing the status of the order parameter in QCD, the Polyakov loop is still a good probe of screening of static color charges in quark-gluon plasma [@okacz02; @digal03].
Other probes of deconfinement are fluctuations and correlations of various charges that can signal liberation of degrees of freedom with quantum numbers of quarks and gluons in the high-temperature phase. Here we consider quadratic fluctuations and correlations of conserved charges: $$\begin{aligned}
\frac{\chi_i(T)}{T^2}=
\left.\frac{1}{T^3 V}\frac{\partial^2 \ln Z(T,\mu_i)}{\partial (\mu_i/T)^2}
\right|_{\mu_i=0},\,\,\,\,\,
\frac{\chi_{11}^{ij}(T)}{T^2}=
\left.\frac{1}{T^3 V}\frac{\partial^2 \ln Z(T,\mu_i,\mu_j)}{\partial (\mu_i/T)
\partial (\mu_j/T)} \right|_{\mu_i=\mu_j=0}.\end{aligned}$$ Fluctuations are also sensitive to
| 3,218
| 1,603
| 3,334
| 3,030
| 2,183
| 0.781955
|
github_plus_top10pct_by_avg
|
cdots h({\mathbf{x}}^k) \in {\mathbb{R}}[{\mathbf{y}}],$$ which is hyperbolic with respect to ${\mathbf{e}}^1\oplus \cdots \oplus {\mathbf{e}}^k$, where ${\mathbf{e}}^i$ is a copy of ${\mathbf{e}}$ in the variables ${\mathbf{x}}^i$, for all $1 \leq i \leq k$. The hyperbolicity cone of $g$ is the direct sum $\Lambda_+:=\Lambda_+({\mathbf{e}}^1) \oplus \cdots \oplus \Lambda_+({\mathbf{e}}^k)$, where $\Lambda_+({\mathbf{e}}^i)$ is a copy of $\Lambda_+({\mathbf{e}})$ in the variables ${\mathbf{x}}^i$, for all $1 \leq i \leq k$.
Let ${\mathsf{X}}_1, \ldots, {\mathsf{X}}_m$ be independent random vectors in $\Lambda_+$ such that for all $1\leq i \leq k$ and $1\leq j \leq m$: $${\mathbb{P}}\left[ {\mathsf{X}}_j = k{\mathbf{u}}_j^i\right] = \frac 1 k,$$ where ${\mathbf{u}}_1^i, \ldots, {\mathbf{u}}_m^i$ are copies in $\Lambda_+({\mathbf{e}}^i)$ of ${\mathbf{u}}_1, \ldots, {\mathbf{u}}_m$. Then $$\begin{aligned}
{\mathbb{E}}{\mathsf{X}}_j &= {\mathbf{u}}_j^1 \oplus {\mathbf{u}}_j^2 \oplus \cdots \oplus {\mathbf{u}}_j^k, \\
\tr({\mathbb{E}}{\mathsf{X}}_j) &= k\tr({\mathbf{u}}_j) \leq k\epsilon, \mbox{ and } \\
{\mathbb{E}}\sum_{j=1}^m {\mathsf{X}}_j &= {\mathbf{e}}^1\oplus \cdots \oplus {\mathbf{e}}^k,\end{aligned}$$ for all $1\leq j \leq k$. By Theorem \[hypprob\] there is a partition $S_1\cup \cdots \cup S_k =[m]$ such that $${\lambda_{\rm max}}\left(\sum_{i \in S_1}k{\mathbf{u}}_i^1+\cdots + \sum_{i \in S_k}k{\mathbf{u}}_i^k \right)\leq \delta(k\epsilon,m).$$ However $${\lambda_{\rm max}}\! \left(\sum_{i \in S_1}k{\mathbf{u}}_i^1+\cdots + \sum_{i \in S_k}k{\mathbf{u}}_i^k \right) = k \! \max_{1\leq j \leq k} {\lambda_{\rm max}}\! \left(\sum_{i \in S_j}{\mathbf{u}}_i^j \right) =
k \! \max_{1\leq j \leq k} {\lambda_{\rm max}}\! \left(\sum_{i \in S_j}{\mathbf{u}}_i \right),$$ and the theorem follows.
On a conjecture on the optimal bound
====================================
We have seen that the core of the proof of Theorem \[t1\] is to bound the zeros of mixed characteristic polynomials. To achieve better bounds in T
| 3,219
| 1,646
| 2,086
| 3,049
| null | null |
github_plus_top10pct_by_avg
|
19.74, \< 0.01 45.11 (42.1, 50.19) 2.08 22.18, \< 0.01
Census median centered 0.32 (0.28, 0.36) 0.02 15.62, \< 0.01 0.82 (0.73, 0.82) 0.04 19.88, \< 0.01
A2D 0.13 (0.10, 0.16) 0.02 8.08, \< 0.01 0.14 (0.10, 0.18) 0.02 6.65, \< 0.01
PIT (post vs pre) −21.61 (−23.78, −19.44) 1.11 19.44, \< 0.01 −18.45 (−21.37, −15.52) 1.50 12.36, \< 0.01
LOSD
Intercept 260.7 (251.27, 270.13) 4.81 54.23, \< 0.01 223.47 (216.42, 230.52) 3.59 62.2, \< 0.01
Census median centered 0.66 (0.53, 0.78) 0.06 10.67, \< 0.01 0.90 (0.80, 1.01) 0.05 16.91, \< 0.01
A2D 0.25 (0.19, 0.31) 0.03 8.52, \< 0.01 0.21 (0.13, 0.28) 0.04 5.08, \< 0.01
PIT (post vs pre) −29.83 (−38.03, −21.68) 4.17 7.16, \< 0.01 −11.45 (−16.16, −4.77) 2.40 4.77, \< 0.01
*ED*, emergency department; *AED*, tertiary care academic emergency department; *CED*, community emergency department; *D2P*, arrival to being seen by physician; *LOSD*, total length of stay for discharged patients; *A2D*, admit request to departure for boarded patients awaiting hospital admission; *PIT*, physician in triage; *95% CI*, 95% confidence interval; *SE*, standard error.
Introduction {#Sec1}
============
Reproductive history, like parity, age at first birth and number of births, has consistently been shown to be associated with breast cancer risk \[[@CR1]\]. Women who have undergone a full time pregnancy before 20 years of age, for example, have a 50% reduced lifetime risk of developing breast cancer when compared to nulliparo
| 3,220
| 724
| 2,487
| 3,172
| null | null |
github_plus_top10pct_by_avg
|
all even or odd antisymmetric indices, and T-duality should remove or add an index according to whether it is already there or not. As a consequence, one derives the following T-duality rules $$\begin{aligned}
P_a^{b_1 ... b_p} \ & \overset{T_a}{\longleftrightarrow} \ P^{a, b_1 ... b_p a} \nonumber \\
P_a^{b_1 ... b_p} \ & \overset{T_{b_p}}{\longleftrightarrow} \ P_{a}^{b_1 ... b_{p-1}} \label{TdualityrulesPfluxes}\\
P^{a ,b_1 ... b_p} \ & \overset{T_{b_p}}{\longleftrightarrow} \ P^{a , b_1 ... b_{p-1}} \quad ,\nonumber \end{aligned}$$ which simply summarise the statements above, that is under the action of $T_a$ a downstairs $a$ index is raised and vice versa, while in the set of antisymmetric indices the rule is precisely as for the RR fluxes.
The components of the flux $P_a^{b_1 b_2}$ that one considers are such that $b_1$ and $b_2$ are different from $a$, precisely as for the $Q$ flux. Therefore, by applying the T-duality rules in eq. , one finds that by performing any chain of T-dualities one always ends up with components such that if the ${a}$ index is down, then it is different from any of the ${ b}$ indices, while if it is up it has to be parallel to the ${ b}$ indices. It is for this reason that in eq. we have not included the rule that maps the flux $P_a^{b_1 ... b_p a}$ to $P^{a, b_1 ...b_p}$ under $T_a$: both these components are not connected by T-duality transformations to the components of the flux $P_a^{b_1 b_2}$ we are considering.[^4] It should also be appreciated that all these rules actually apply to any dimension, although in this paper we are only interested in the four-dimensional case.
We can now apply these rules to determine all the $P$ fluxes that can be included in the four-dimensional $T^6/[\mathbb{Z}_2 \times \mathbb{Z}_2 ]$ orientifold model. In the case of the O3-orientifold of IIB, as we have already reviewed in the introduction all the fluxes $P_a^{bc}$ with each of the three indices along a direction of each of the three different tori can be turned on [@Aldazabal:2006up
| 3,221
| 1,564
| 3,217
| 2,843
| 3,654
| 0.770885
|
github_plus_top10pct_by_avg
|
d takes a value of $\sim 0.01$ at $\theta = \theta_{\mathrm{shadow}}$. Although we fix the shadowing profile $f_{\mathrm{shadow}}(\theta)$ during each simulation run for simplicity, it probably depends on accretion rates in reality.[^3] In view of large uncertainties in the shadowing effect, we perform a number of simulations varying $\theta_{\mathrm{shadow}}$ as a free parameter (see Sec. \[sec:parameters\]).
Cases considered {#sec:parameters}
----------------
[lccccccccccc]{} run& $M_{\mathrm{BH}}\,[M_\odot]$ & $n_\infty\,[{\mathrm{cm^{-3}}}]$ & $\theta_{\mathrm{shadow}}^{a}$& $N_r\times N_\theta$ & $R_{\mathrm{in}}\,[{\mathrm{AU}}]$ & $R_{\mathrm{out}}\,[{\mathrm{AU}}]$ & $t_{\mathrm{end}}\,[{\mathrm{yr}}]$\
Di & $10^3$ & $10^5$ & [**isotropic**]{}$^{b}$ & $512\times144$ & $3\times10^2$ & $6\times10^5$ & $5\times10^5$\
Ddn & $10^3$ & $10^5$ & [**disc**]{}$^{c}$ & $512\times144$ & $3\times10^2$ & $6\times10^5$ & $5\times10^5$\
Dds$^{d}$ & $10^3$ & $10^5$ & [$45^\circ$]{} & $512\times144$ & $2\times10^3$ & $3\times10^6$ & $2\times10^6$\
\
s075 & $10^3$ & $10^5$ & [$33.75^\circ$]{} & $256\times72$ & $2\times10^3$ & $3\times10^6$ & $2\times10^6$\
s050 & $10^3$ & $10^5$ & [$22.5^\circ$]{} & $256\times72$ & $2\times10^3$ & $3\times10^6$ & $2\times10^6$\
s025 & $10^3$ & $10^5$ & [$11.25^\circ$]{} & $256\times72$ & $2\times10^3$ & $3\times10^6$ & $2\times10^6$\
\
M1e2 & [$10^2$]{} & $10^5$ & $45^\circ$ & $256\times72$ & $2\times10^2$ & $1.5\times10^6$ & $2\times10^6$\
M1e4 & [$10^4$]{} & $10^5$ & $45^\circ$ & $256\times72$ & $2\times10^4$ & $2\times10^7$ & $2\times10^7$\
M1e5 & [$10^5$]{} & $10^5$ & $45^\circ$ & $256\times72$ & $2\times10^5$ & $1\times10^8$ & $5\times10^7$\
\
n1e3 & $10^3$ & [$10^3$]{} & $45^\circ$ & $256\times72$ & $2\times10^3$ & $1\times10^7$ & $5\times10^7$\
n1e4 & $10^3$ & [$10^4$]{} & $45^\circ$ & $256\times72$ & $2\times10^3$ & $6\times10^6$ & $2\times10^7$\
n1e6 & $10^3$ & [$10^6$]{} & $45^\circ$ & $256\times72$ & $2\times10^3$ & $2\times10^6$ & $2\times10^6$\
\
NOTES.$^{a}$Disc radiation
| 3,222
| 1,281
| 1,965
| 3,011
| null | null |
github_plus_top10pct_by_avg
|
]{} AGN selection for this redshift range. One object falls into our redshift range but has $R>24$, outside our selection limits.
[cccccccc]{} ID& Tile& RA (2000)& DEC (2000)& $z$& $R$ (Vega)& [F606W]{} & [F850LP]{}\
12325 &11& 033301.7& –275819& 1.843& 20.38& 20.13& 19.65\
19965 &23& 033145.2& –275436& 1.90 & 19.96& 20.59& 20.29\
30792 &82& 033243.3& –274914& 1.929& 22.36& 21.60& 22.06\
02006 &04& 033232.0& –280310& 1.966& 19.76& 19.59& 18.98\
04809 &08& 033136.3& –280150& 1.988& 22.31& 21.18& 20.91\
06817 &09& 033127.8& –280051& 1.988& 21.59& 21.61& 20.86\
18324 &19& 033300.9& –275522& 1.990& 22.58& 22.00& 21.33\
05498 &01& 033316.1& –280131& 2.075& 22.29& 22.91& 22.28\
11941 &10& 033326.3& –275830& 2.172& 20.51& 20.80& 20.40\
62127 &62& 033136.7& –273446& 2.175& 23.57& 24.91& 24.37\
51835 &55& 033140.1& –273917& 2.179& 22.92& 23.01& 22.41\
00784 &05& 033227.1& –280336& 2.282& 23.29& 23.28& 22.80\
36120 &39& 033149.4& –274634& 2.306& 22.37& 22.70& 22.23\
05696 &02& 033321.8& –280121& 2.386& 23.06& 22.73& 22.32\
05696 &03& $''$ & $''$ & $''$ & $''$ & 23.14& 22.68\
07671 &07& 033151.8& –280026& 2.436& 22.35& 22.35& 22.24\
07671 &15& $''$ & $''$ & $''$ & $''$ & 22.36& 22.22\
06735 &02& 033306.3& –280056& 2.444& 21.98& 22.14& 21.88\
01387 &08& 033144.0& –280320& 2.503& 23.24& 24.05& 23.17\
33644 &31& 033259.9& –274748& 2.538& 21.87& 21.28& 21.17\
11922 &11& 033309.1& –275827& 2.539& 22.25& 22.65& 21.93\
16621 &19& 033309.7& –275614& 2.540& 19.98& 20.41& 20.06\
15396 &21& 033216.2& –275644& 2.682& 22.64& 22.69& 22.41\
33630 &33& 033140.1& –274746& 2.719& 21.74& 22.21& 21.98\
42882 &45& 033201.6& –274328& 2.719& 23.18& 23.89& 23.48\
42882 &95& $''$ & $''$ & $''$ & $''$ & 24.04& 23.54\
Data analysis {#sec:analysis}
=============
Background and variances {#sec:bg}
------------------------
Even though space based, ACS shows a non-negligible background from stray light. In the reduction process already a global, outlier clipped median background was subtracted (Caldwell
| 3,223
| 502
| 2,110
| 3,500
| null | null |
github_plus_top10pct_by_avg
|
q is not an option.
The problem, in short, is when I query with WhereGreaterThan I get zero results even though my Asset.Data has a price greater than a value. This applies to WhereGreaterThanOrEquals, WhereLessThan, and WhereLessThanOrEquals as well.
public class AssetDataSearch : AbstractIndexCreationTask<Asset>
{
public AssetDataSearch()
{
Map = (docs) =>
from d in docs
select new
{
DataType = d.DataType,
_ = d.SearchableParameters.Select(s => CreateField(s.Key, s.Value))
};
}
}
public class Test
{
public void TestMethod()
{
var assets = new []
{
new Asset()
{
ID = Guid.NewGuid().ToString(),
Data = new ListingData()
{
Beds = 5,
Baths = 5,
ListingType = ListingTypeEnum.Condo,
Price = 100
}
},
new Asset()
{
ID = Guid.NewGuid().ToString(),
Data = new ListingData()
{
LotSize = 55,
SqFeet = 89,
YearBuilt = 1965,
Price = 200
}
},
};
RavenHelper.InitTestingStore();
using (var session = RavenDB.RavenUtility.OpenSession())
{
foreach(var a in assets)
session.Store(a);
session.SaveChanges();
var assetsInDb = session.Advanced.LuceneQuery<Asset>().WaitForNonStaleResults().ToArray();
var n = session.Advanced.LuceneQuery<Asset, AssetDataSearch>().WhereEquals("Price", 100).ToArray(); // returns expected results
var gt = session.Advanced.LuceneQuery<Asset, AssetDataSearch>().WhereGreaterThan("Price", 60).ToArray(); // returns nothing
var lt = session.Advanced.LuceneQuery<Asset, AssetDataSearch>().WhereLessThan("Price", 60).ToArray(
| 3,224
| 4,084
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| 2,836
| 28
| 0.836156
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github_plus_top10pct_by_avg
|
ition and multiplication) with entries in $(B\otimes_AR)/(\pi\otimes 1)(B\otimes_AR)$. Thus, the assignment $m\mapsto \mathcal{X}_{i,j}(m)$ is a polynomial in $m$. Furthermore, since $m$ actually belongs to $\mathrm{Ker~}\varphi(R)/\tilde{G}^1(R)$, we have the following equation by the argument made at the beginning of this paragraph: $$\mathcal{X}_{i,j}(m)=f_{i,j}\textit{ mod $(\pi\otimes 1)(B\otimes_AR)$}=0.$$ Thus we get an $n_i\times n_j$ matrix $\mathcal{X}_{i,j}$ of polynomials on $\mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$ defined by Equation (\[ea20\]), vanishing on the subscheme $\mathrm{Ker~}\varphi/\tilde{G}^1$. For example, if $j=i+1$, then $$\label{ea21}
\sigma({}^tm_{i,i}) \bar{h}_i m_{i,i+1}+\sigma({}^tm_{i+1,i}) \bar{h}_{i+1} m_{i+1,i+1}=0.$$
Before moving to the following steps, we fix notation. Let $m$ be an element in $(\mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1)(R)$ and $\tilde{m}\in \mathrm{Ker~}\tilde{\varphi}(R)$ be its lift. For any block $x_i$ of $m$, $\tilde{x}_i$ is denoted by the corresponding block of $\tilde{m}$ whose reduction is $x_i$. Since $x_i$ is a block of an element of $(\mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1)(R)$, it involves $(B\otimes_AR)/(\pi\otimes 1)(B\otimes_AR)$ as explained in Remark \[ra5\], whereas $\tilde{x}_i$ involves $B\otimes_AR$. In addition, for a block $a_i$ of $h$, $\bar{a}_i$ is denoted by the image of $a_i$ in $(B\otimes_AR)/(\pi\otimes 1)(B\otimes_AR)$.\
2. Assume that $i$ is odd and that $L_i$ is *bound of type I*. By Equation (\[ea3’\]) which involves an element of $\tilde{M}^1(R)$, each entry of $(f_{i,i}^{\ast})'$ has $\pi$ as a factor so that $(f_{i,i}^{\ast})'\equiv f_{i,i}^{\ast}=0$ mod $(\pi\otimes 1)(B\otimes_AR)$. Let $\tilde{m}\in \mathrm{Ker~}\tilde{\varphi}(R)$ be a lift of $m$. We write $$\pi\mathcal{X}_{i,i}^{\ast}(\tilde{m})=\delta_{i-1}(0,\cdots, 0, 1)\cdot \mathcal{X}_{i-1,i}(\tilde{m})+
\delta_{i+1}(0,\cdots, 0, 1)\cdot \mathcal{X}_{i+1, i}(\tilde{m})$$ formally, where $\mathcal{X}_{i,i}^{\ast}(\tilde{m}) \in M_{1\time
| 3,225
| 1,513
| 1,705
| 3,099
| 1,452
| 0.789369
|
github_plus_top10pct_by_avg
|
[]{data-label="ATSVsPower"}](Fig3.pdf)
The simulated of the reflection coefficient is shown in Fig. \[ATSVsPower\](b) with a simplified model by numerically solving Eq. (\[MasterEQ\]) in a steady state with the artificial atom parameters: $\omega_{eg}=2\pi\times3.379\,$GHz, $\omega_{fg}=2\pi\times12.173\,$GHz, $\gamma_{eg}=2\pi\times2\,$MHz, $\gamma_{fg}=2\pi\times4\,$MHz, $\gamma_{fe}=2\pi\times25\,$MHz, $\gamma_{ff}=2\pi\times1\,$MHz, and $\gamma_{ee}=2\pi\times20\,$MHz. The dephasing effect comes from $|f\rangle \leftrightarrow |g\rangle$ fluctuations, which induced by the photon shot noise inside the cavity, is also added to $\gamma_{ff}$ [@Schuster2005]. The other second-order effects in the $|f\rangle \leftrightarrow |g\rangle$ interaction with the resonator are neglected. We simplify our model for the numerical calculations because of limited computer resources insufficient to operate with the full Hamiltonian in the dissipative regime with many photons because of the too large Hilbert space. Note that although there is a weak signature of zero-one state photon splitting, it cannot be resolved in the experiment because of the measurement noise. In our experiment, it is not yet reach the reliable resolution of the two states, which can be expressed as $\Delta\omega_{10} > 2\lambda_{fg}$ – the half-spacing between two dips is larger than $\lambda_{fg}$ which is the decay rate of the atomic off-diagonal terms $\rho_{fg}$, where $\Delta\omega_{10}=g_0 (\sqrt{2}-1) = 2\pi\times$ 27 MHz and $\lambda_{fg}=2\pi\times$ 16 MHz(details are given in [@Suppl]).
Figure \[ATSVsPower\](c) shows the dependence of reflection power with $\langle n\rangle = 0$, $\langle n\rangle = 1$ and $\langle n\rangle = 5$. The bottom panel shows the difference between the traces with $\langle n\rangle = 0$ and $\langle n\rangle = 1$. It demonstrates the reflection of propagating microwaves in the transmission line could be controlled by microwave fields in the resonator at the single-photon level. It may find potential applic
| 3,226
| 1,952
| 3,804
| 3,292
| 2,863
| 0.776466
|
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|
um_{z',x}P'_{\Lambda;o}(z',x)
=O(\theta_0)\sum_{z',x}\bigg(P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}(z',x)
+\sum_{j\ge1}P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(j)}}(z',x)\bigg).\end{aligned}$$ Similarly to [(\[eq:pi0-rthmombd\])]{} for $r=0$, the sum of $P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(z',x)$ is easily estimated as $1+O(\theta_0)$. We claim that the sum of $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(j)}}(z',x)$ for $j\geq1$ is $(2j-1)\,O(\theta_0)^j$, since $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(j)}}(z',x)$ is a sum of $2j-1$ terms, each of which contains $j$ chains of nonzero bubbles; each chain is $\psi_\Lambda(v,v')-\delta_{v,v'}$ for some $v,v'$ and satisfies $$\begin{aligned}
\sum_{v'}\big(\psi_\Lambda(v,v')-\delta_{v,v'}\big)\leq\sum_{l\ge1}
\Big(\tau^2\big(D*(D*G^{*2})\big)(o)\Big)^l=\sum_{l\ge1}
O(\theta_0)^l=O(\theta_0).\end{aligned}$$ For example, $$\begin{aligned}
\sum_{z',x}P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(4)}}(z',x)&=\raisebox{-1.2pc}
{\includegraphics[scale=0.15]{Pprime4}}~+6\text{ other possibilities},\end{aligned}$$ which can be estimated, by translation invariance, as $$\begin{aligned}
\raisebox{-1.5pc}{\includegraphics[scale=0.15]{Pprime4}}&\leq
~\raisebox{-1.8pc}{\includegraphics[scale=0.15]{Pprime4dec}}{\nonumber}\\[5pt]
&\leq\bigg(\sum_y\big(\psi_\Lambda(o,y)-\delta_{o,y}\big)\bigg)^4\big(
\bar W^{{\scriptscriptstyle}(0)}\big)^4=O(\theta_0)^4,\end{aligned}$$ where $\bar W^{{\scriptscriptstyle}(t)}$ is given by [(\[eq:GbarWbar\])]{}.
The sum of $\tau_{y,z}Q''_{\Lambda;o,v}(z,x)$ in [(\[eq:block-sumbd\])]{} is estimated similarly [@sNN]. We complete the proof of the bound on $\sum_x\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$.
To estimate $\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$, we recall that, in each bounding diagram, there are at least three distinct paths between $o$ and $x$: the uppermost path (i.e., $o\to
b_1\to v_2\to b_3\to\cdots\to x$ in [(\[eq:piNbd\])]{}; see also Figure \[fig:piN-bd\]), the lowermos
| 3,227
| 2,975
| 2,000
| 3,149
| 1,510
| 0.788733
|
github_plus_top10pct_by_avg
|
the last three states are also visualized in figure \[fig:SummaryIC\]. We comment that, with the exception of the Taylor-Green vortex which was shown in §\[sec:3D\_InstOpt\_E0to0\] to be a local maximizer of problem \[pb:maxdEdt\_E\] in the limit $\E_0 \rightarrow 0$, all these initial conditions were postulated based on rather ad-hoc physical arguments. We also add that, in order to ensure a fair comparison, the different initial conditions listed in Table \[tab:InitialConditions\] are rescaled to have the same enstrophy $\E_0$, which is different from the enstrophy values used in the original studies where these initial conditions were investigated [@opc12; @k13; @dggkpv13; @opmc14]. As regards our choices of the initial enstrophy $\E_0$, to illustrate different possible behaviours, we will consider initial data located in the two distinct regions of the phase space $\{\K,\E\}$ shown in figure \[fig:K0E0\], corresponding to values of $\K_0$ and $\E_0$ for which global regularity may or may not be a priori guaranteed according to estimates –.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$${\mathbf{u}}_0({\mathbf{x}}) = [u, v, w]$$
------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------
| 3,228
| 2,318
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| 3,458
| null | null |
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|
}}\,
\frac{1}{(\Delta_b-\Delta_\alpha)(\frac12(\Delta_b+\Delta_\alpha)+M_N)+{\vec{p}}^2-{\vec{l}}^2}
\\&=i\frac{G_Fm_\pi^2}{16\pi M_N}
(C_{S(s)}^{\alpha}+C_{T(s)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\,
(C_{S(w)}^{\alpha}+C_{T(w)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\,
\\&\times
\sqrt{
(\Delta_b-\Delta_\alpha)(\frac12(\Delta_b+\Delta_\alpha)+M_N)+{\vec{p}}^2}\end{aligned}$$
Ball diagrams {#sec:balls}
=============
In our calculation we have two different kind of ball diagrams depending on the position of the weak vertex, although only one of them actually contributes. Their contribution can be written in terms of the $B$ integrals defined in Appendix \[sec:mi\].
Here and in the following sections we first write the relativistic amplitude using $V=i \ M$ and then the corresponding heavy baryon expression.
![Kinematical variables of the first kind of ball-diagram.\[fball1\]](ball11)
For the first type of ball diagram, depicted in Fig. \[fball1\], we obtain the following contribution, $$\begin{aligned}
V_{\text{ball 1}}=& \frac{G_Fm_\pi^2 h_{2\pi}}{4f_\pi^4}
\delta_{ab}\ \epsilon^{abc}\tau^c
\nonumber\\
&\times {\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon}
\frac{1}{(l-q)^2-m_\pi^2+i\epsilon}\nonumber
\\&\times
{\overline{u}}_1({\overline{E}},{\vec{p}\,'})
u_1(E_p^\Lambda,{\vec{p}})\nonumber\\
&\times
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})
\gamma_\mu(q^\mu-2l^\mu)
u_2(E_p,-{\vec{p}})
\\=&0\,,\end{aligned}$$ which is shown to vanish due to the isospin factor, $\delta_{ab}\epsilon^{abc}\tau^c=0$.
![Kinematical variables of the second kind of ball-diagram.\[fball2g\]](ball2g)
The amplitude corresponding to the diagram in Fig. \[fball2g\] reads, $$\begin{aligned}
V_a&=&-i
\frac{G_Fm_\pi^2h_{\Lambda N}}{8f_\pi^4} ({\vec{\tau}_1}\cdot{\vec{\tau}_2}) \nonumber\\
&\times&{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon}
\,\frac{1}{(l+q)^2-m_\pi^2+i\epsilon} \nonumber\\
&\times&
\frac{(2l^\mu+q^\mu)(q^\nu+2l^\nu)}{k_N^2-M_N^2+i\epsilon}
\nonumber\\
&\times&
{\ov
| 3,229
| 1,198
| 1,492
| 3,399
| null | null |
github_plus_top10pct_by_avg
|
e
-\langle {\mathcal{S}}\Xi(e^{-\Phi}F\Psi e^\Phi), (e^{-\Phi}F\Psi e^\Phi)^2\rangle\,.
\label{I-1}\end{aligned}$$ Here the second term vanishes owing to (\[BPZ S\]) and (\[small to large\]): $$\begin{aligned}
- \langle {\mathcal{S}}\Xi(e^{-\Phi}F\Psi e^\Phi), (e^{-\Phi}F\Psi e^\Phi)^2\rangle\ =&\
{\langle\!\langle}(e^{-\Phi}F\Psi e^\Phi), \{(e^{-\Phi}F\Psi e^\Phi), {\mathcal{S}}(e^{-\Phi}F\Psi e^\Phi)\}{\rangle\!\rangle}\nonumber\\
=&\
\frac{2}{3}\Big(
{\langle\!\langle}{\mathcal{S}}(e^{-\Phi}F\Psi e^\Phi), (e^{-\Phi}F\Psi e^\Phi)^2{\rangle\!\rangle}\nonumber\\
&\hspace{10mm}
+ {\langle\!\langle}(e^{-\Phi}F\Psi e^\Phi), \{(e^{-\Phi}F\Psi e^\Phi), {\mathcal{S}}(e^{-\Phi}F\Psi e^\Phi)\}{\rangle\!\rangle}\Big)
\nonumber\\
=&\ 0\,.\end{aligned}$$ The first term in (\[I-1\]) can further be calculated as $$\begin{aligned}
\textrm{(I)} =&\
- \langle {\mathcal{S}}(e^{-\Phi}F\Psi e^\Phi), \widetilde{A}_Q\rangle\
=\
\langle F\Psi, e^\Phi({\mathcal{S}}\widetilde{A}_Q)e^{-\Phi}\rangle
\nonumber\\
=&\
\langle F\Psi, QA_{\mathcal{S}}\rangle\
=\ \langle A_{\mathcal{S}}, QF\Psi\rangle\,,
\label{I}\end{aligned}$$ where we have used the relation (\[dual relation\]) with $(\mathcal{O}_1,\mathcal{O}_2)=(Q,{\mathcal{S}})$, and the identity $$\begin{aligned}
\eta(e^{-\Phi}F\Psi e^\Phi)\ =&\ e^{-\Phi} (D_\eta F\Psi) e^\Phi\ =\ 0\,.
$$ Summing (\[II\]), (\[III\]), and (\[I\]), the variation of the action under the space-time supersymmetry transformation finally becomes $$\delta_{\mathcal{S}}S\ =\
\langle A_{\mathcal{S}}, \left(QF\Psi - F(Q\Psi+X\eta F\Psi)
+ F\Xi[QA_\eta+(F\Psi)^2, F\Psi]\right)\rangle\,,$$ which vanishes due to the identity (4.89) in Ref.: $\delta_{\mathcal{S}}S=0$. Hence the complete action (\[complete action\]) is invariant under the transformation (\[complete transformation\]).
Algebra of transformation {#sec algebra}
=========================
Starting from a natural linear transformation (\[linearized tf\]), we have constructed the nonlinear transformation (\[complete transformation\]) as a symmetry of the c
| 3,230
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|
ise-canceling effects, as demonstrated in Study 3. Larger flocks can also be a source of individual behavioral differentiation, when a higher order of organization emerges. The key is not the size nor the amount of new information, but rather the system promoting the invention of new coordination patterns within itself.
We have shown how collective intelligence has the ability to augment the creation of new and diverse solutions in a swarm, when given limited channels of communication, a concurrent evolution bottleneck and a large number of constrainted degrees of freedom. It come as an inspiration for scientists: a good way to build an open-ended system, able to indefinitely discover new inventions, seems not to reside in centralized computation, but rather in distributed systems, composed of large collectives of communicating agents.
Acknowledgements
================
The authors would like to thank their collaborators who contributed partially to this work: Nathanael Aubert-Kato, Aleksandr Drozd, Yasuhiro Hashimoto, Norihiro Maruyama, Yoh-ichi Mototake and Mizuki Oka.
[^1]: at ALIFE 2018, in Tokyo
[^2]: A swarm can be shown to act as a collective memory, either explicitly/statefully [@witkowski2016emergence] or dynamically/statelessly [@couzin2002collective].
---
abstract: 'Unsupervised bilingual word embedding (BWE) methods learn a linear transformation matrix that maps two monolingual embedding spaces that are separately trained with monolingual corpora. This method assumes that the two embedding spaces are structurally similar, which does not necessarily hold true in general. In this paper, we propose using a pseudo-parallel corpus generated by an unsupervised machine translation model to facilitate structural similarity of the two embedding spaces and improve the quality of BWEs in the mapping method. We show that our approach substantially outperforms baselines and other alternative approaches given the same amount of data, and, through detailed analysis, we argue that data augmentation with the pse
| 3,231
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|
{qsch}\operatorname{Hom}_{\mathcal{H}_q}(Sp_q(\mu), Sp_q(\lambda))
\cong
\operatorname{Hom}_{S_q}(W_q(\lambda),W_q(\mu)).$$
On the other hand, by and we have $$\label{tf} \operatorname{Hom}_{H_c}(\Delta_c(\lambda), \Delta_c(\mu)) \cong
\operatorname{Hom}_{\mathcal{H}_q}(Sp_q(\lambda)^{\ast}, Sp_q(\mu)^{\ast}) \cong
\operatorname{Hom}_{{\mathcal{H}_{q}}}(Sp_q(\mu), Sp_q(\lambda)).$$
Each $W_q(\nu)$ has a simple head $F_q(\nu)$,\[F-defn\] [@DJ Theorem 4.6] and $\{
F_q(\nu): \nu\in {{\textsf}{Irrep}({{W}})}\}$ is a complete, repetition-free list of the simple $S_q$-modules up to isomorphism, [@DJ Theorem 8.8]. Furthermore, $F_q(\lambda)$ is a composition factor of $W_q(\mu)$ only if $\lambda\leq
\mu$, [@DJ Corollary 8.9]. By and a non-zero homomorphism $\phi: \Delta_c(\lambda)\to \Delta_c(\mu)$ implies the existence of a non-zero homomorphism $\phi ':W_q(\lambda)\to W_q(\mu)$. Thus $F_q(\lambda)$ must be a composition factor of $W_q(\mu)$ and so $\lambda\leq
\mu$.
Corollary {#poono}
---------
[*Assume that $c\in {\mathbb{R}}_{\geq 0}$, with $c\notin \frac{1}{2}+\mathbb{Z}$. If $ [\Delta_c(\mu) : L_c(\lambda) ] \neq 0$ for $\lambda,\mu\in {{\textsf}{Irrep}({{W}})}$, then $ \lambda \leq \mu$ in the dominance ordering.* ]{}
\(1) For arbitrary $c$ and $\mu$, the unique occurrence of $L_c(\mu)$ as a composition factor of $\Delta_c(\mu)$ is as its head—see, for example, the discussion after Lemma 7 in [@guay Section 2].
\(2) Since $\operatorname{{\textsf}{sign}}$ is minimal in the dominance ordering, the lemma and the above remark imply that $\Delta_c(\operatorname{{\textsf}{sign}})$ is irreducible for all $c\in {\mathbb{R}}_{\geq 0}$. This can also be deduced from [@guay].
We argue by induction on $\mu$. More precisely, suppose that $[\Delta_c(\mu): L_c(\lambda)] \neq 0$ for some $\mu \neq \lambda$ and that the lemma holds for any $\nu < \mu$. (The induction starts since there are only finitely many $\sigma$ with $\sigma<\mu$.) Let $P_c(\lambda)$ \[P-defn\] denote the projective cover of $\Delta_c(\lambda)$
| 3,232
| 2,317
| 1,045
| 3,255
| null | null |
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|
s already been solved in Ref. [@konno]. For generic $N$, the desired expressions are rather complicated and our construction depend largely on the observation \[ob1\] and the result of [@sln]. One crucial difference of our construction from the one in [@sln] is that, in our case, the Yangian double $DY_\hbar(sl_N)$ should be realized through [*ordinary*]{} Heisenberg algebras (i.e. [*without*]{} deformation), whereas in Ref.[@sln], $U_q(\widehat{sl_N})$ was realized via a set of $q$-deformed Heisenberg algebras. Therefore our observation \[ob1\] has to be used in somewhat a nontrivial way (for example, the vertex operators and screening currents cannot be obtained using our correspondence principles).
Free bosons and Fock space
--------------------------
We introduce the following set of $N^2-1$ Heisenberg algebras with generators $a^i_n~(1 \leq i \leq N-1),~b^{ij}_n~\mbox{and}~c^{ij}_n~
(1 \leq i < j \leq N)$ with $ n \in {\Bbb Z} - \{ 0 \}$ and $p_{a^i},~q_{a^i}~(1 \leq i \leq N-1),~p_{b^{ij}},~q_{b^{ij}},~
p_{c^{ij}},~q_{c^{ij}}~(1 \leq i < j \leq N)$,
$$\begin{aligned}
\begin{array}{ll}
$$[ a^i_n,~a^j_m ] = (k+g) B_{ij} n \delta_{n+m,0},$$ &
$$[ p_{a^i},~q_{a^j} ] = (k+g) B_{ij},$$ \cr
$$[ b^{ij}_n,~b^{i'j'}_m ] = - n \delta^{i,i'} \delta^{i,j'}
\delta_{n+m,0},$$ &
$$[ p_{b^{ij}},~q_{b^{ij}} ] = -\delta^{i,i'} \delta^{i,j'},$$ \cr
$$[ c^{ij}_n,~c^{i'j'}_m ] = n \delta^{i,i'} \delta^{i,j'}
\delta_{n+m,0},$$ &
$$[ p_{c^{ij}},~q_{c^{ij}} ] = \delta^{i,i'} \delta^{i,j'},$$
\end{array}\end{aligned}$$
where $g=N$ is the dual Coexter number for the Cartan matrix of type $A_{N-1}$.
The Fock space corresponding to the above Heisenberg algebras can be specified as follows. Let $| 0 \rangle$ be the vacuum state defined by
$$\begin{aligned}
& & a^i_n| 0 \rangle = b^{ij}_n| 0 \rangle = c^{ij}_n| 0 \rangle
=0~ ( n >0),\\
& & p_{a^i}| 0 \rangle = p_{b^{ij}}| 0 \rangle = p_{c^{ij}}| 0 \rangle
=0.\end{aligned}$$
Define
$$\begin{aligned}
& & | l_a,l_b,l_c \rangle = \\
& &~~~~ \mbox{exp} \left(
\sum_{i,j=1}^{N-1} \sum_
| 3,233
| 2,189
| 2,579
| 2,948
| 3,989
| 0.768752
|
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|
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In Eqs. (\[eq:tots\]) and (\[eq:tots2\]) we provide the explicit momentum and spin structures arising from the different Feynman diagrams. Some features can be easily read off from the different terms. First, the ball (a) and first two triangle diagrams (b,c) only contribute to the parity conserving part of the transition potential. Most other diagrams have a non-trivial contribution, involving all allowed momenta and spin structures.
To provide a sample of the contribution of the different diagrams to the full amplitude, we consider one particular transition, $^3 S_1\rightarrow ^3S_1$. In particular, we compare the $\pi$ and $K$ exchanges with the ball, triangle and box diagrams for the $\Lambda n\rightarrow nn$ interaction. Since the transition is parity conserving, none of the parity violating structures of Table \[tab:contacts\] contribute. For structures of the type $({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})$ we have that $$\begin{aligned}
({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})=\frac{{\vec{q}}^{\,2}}{3}({\vec{\sigma}_1}\cdot{\vec{\sigma}_2})+\frac{{\vec{q}}^2}{3}\hat{S}_{12}(\hat{q}),\end{aligned}$$ where the tensor operator $\hat{S}_{12}(\hat{q})$ changes two units of angular momentum and does not contribute to this transition. The potential, therefore, depends only on the modulus of the momentum (or ${\vec{q}}^{\,2}$). To obtain the potential in position space we Fourier-transform the expressions for the one-meson-exchange contributions, Eqs. \[eq:pion\] and \[eq:kaon\], and the loop expressions
| 3,234
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| null | null |
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|
on between the ideal and the true estimators
====================================================
In this section we make the following assumptions on the kernel $K$, the densities $f$ and the band sequences:
\[ass3\] We assume that $K$ is supported by $[-T,T]$ for some $T<\infty$ and that it has a uniformly bounded second derivative. We also assume that the densities $f$ are bounded and have at least two bounded derivatives, $${pc}
f\in{\cal P}_C:=\{f\ {\rm is\ a\ density}:\|f^{(k)}\|_\infty\le C, 0\le k\le 2\}$$ for some $C<\infty$. We set $h_{1,n}=n^{-2/9}$ and $h_{2,n}=((\log n)/n)^{1/9}$, $n\in\mathbb N$.
Let $$\label{realest}
\hat f(t;h_{1,n}, h_{2,n})=\frac{1}{n h_{2,n}}\sum_{i=1}^{n}K\left(\frac{t-X_i}{h_{2,n}}\hat f^{1/2}(X_i;h_{1,n})\right)\hat f^{1/2}(X_i;h_{1,n})I(|t-X_{i}|<h_{2,n}B),$$ where $\hat f(x;h_{1,n})$ is the classical kernel density estimator $$\label{real}
\hat f(x;h_{1,n})=\frac{1}{n h_{1,n}}\sum_{i=1}^{n}K\left(\frac{x-X_i}{h_{1,n}}\right).$$
The object of this section consists in proving that $$\label{diff}
\hat f(t;h_{1,n}, h_{2,n})-\bar f(t;h_{2,n})$$ is asymptotically almost surely of the order of $\sqrt{(\log h_{2,n}^{-1})/(nh_{2,n})}$ uniformly in $t$ on the region $D_r$ defined in (\[region0\]), for any $r>0$, if we take $h_{2,n}=\left((\log n)/n\right)^{1/9}$ and $h_{1,n}=n^{-2/9}$. Note that $h_{2,n}$ is the optimal rate ‘up to a log’ given the order of the bias, whereas the preliminary estimator has a bandwidth sensibly smaller than the optimal $n^{-1/5}$ (it is less smooth than the optimal, ‘undersmoothed’) and therefore its bias will be negligible with respect to its variance term. The main result of this paper will follow from this analysis and the result from the ‘ideal’ estimator.
We follow the pattern in Hall and Marron (1988) and Hall, Hu and Marron (1995) for the linearization of (\[diff\]), with significant differences in order to account for the uniformity in $t$. For instance, they do not necessarily undersmooth the preliminary estimator (whereas w
| 3,235
| 1,288
| 2,626
| 2,959
| null | null |
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|
ks\right)-i\sin\left(2ks\right)=-:g_1(s)\label{f2},\quad s \in [0,t).\end{aligned}$$
[**Proof:**]{} We have to check that $$\begin{aligned}
-i\left(\begin{matrix}
Id&M\\
M^*&Id
\end{matrix}\right)
\left(\begin{matrix}
f_1\vphantom{\left(A-A^*\right)}\\f_2\vphantom{\left(A-A^*\right)}
\end{matrix}
\right)=
\left(\begin{matrix}
{\mathbf{1}}_{[0.t)}\vphantom{\left(A-A^*\right)}\\0\vphantom{\left(A-A^*\right)}
\end{matrix}
\right),\end{aligned}$$ see the proof of Lemma \[magneticinvertability\]. The corresponding system of equations reads
$$\begin{aligned}
-if_1+ik\left(Af_2 -A^*f_2\right)&=1\label{48}\\
ik\left(A^*f_1-Af_1\right)-if_2&=0\label{49}.\end{aligned}$$
Let $s \in [0,t)$, then
$$\left((A- A^*)\sin(2k\cdot)\right)(s) = \int_0^s \sin(2k\tau) \, d\tau -\int_s^t \sin(2k\tau) \, d\tau =-\frac{\cos(2ks)}{k} + \frac{1+ \cos(2kt)}{2k},$$ and $$\left((A- A^*)\cos(2ks)\right)(s) = \int_0^s \cos(2k\tau) \, d\tau - \int_s^t \cos(2k\tau) \, d\tau = \frac{\sin(2ks)}{k} - \frac{\sin(2kt)}{2k}.$$
Thus $$\begin{gathered}
\left((A- A^*) f_2\right)(s) \\
= i\frac{\sin\left(2kt\right)}{\cos\left(2kt\right)+1} \left((A- A^*)\cos\left(2k\cdot\right)\right)(s) -i\left((A- A^*)\sin\left(2k\cdot\right)\right)(s) \\
= i(\frac{\sin\left(2kt\right)}{\cos\left(2kt\right)+1} (\frac{\sin(2ks)}{k} - \frac{\sin(2kt)}{2k}) +\frac{\cos(2ks)}{k} - \frac{1+ \cos(2kt)}{2k}).\\\end{gathered}$$ So we get $$\begin{gathered}
-if_1(s)+ik\left(Af_2(s) -A^*f_2(s)\right)\\
= \cos(2ks) + \frac{\sin\left(2kt\right)}{\cos\left(2kt\right)+1}\sin\left(2ks\right) -\frac{\sin\left(2kt\right)}{\cos\left(2kt\right)+1}\sin(2ks) \\
+ \frac{\sin^2(2kt)}{2\cos\left(2kt\right)+1} - \cos(2ks) + \frac{1+ \cos(2kt)}{2} = 1.$$\end{gathered}$$ Furthermore $$\begin{gathered}
\left((A^*-A) f_1\right)(s)\\
=i\left((A^*- A)\cos\left(2k\cdot\right)\right)(s)+i\frac{\sin\left(2kt\right)}{\cos\left(2kt\right)+1}\left((A^*- A)\sin\left(2k\cdot\right)\right)(s)\\
= \frac{i}{k}\left( \frac{\sin(2kt)}{\cos(2kt)+1} \cos(2ks) -\sin(2k
| 3,236
| 2,017
| 2,071
| 3,018
| null | null |
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|
RRT\
Chronic renal failure: 38% 277/4869 (5.7%)
Verma et al. \[[@CIT0067]\] 2017 USA Patient with end-stage heart failure underwent LVAD placement during 2010--2013\ 169 Continuous Increase in SCr of 0.3 mg/dL in 48 hours or 1.5 times from baseline in the seven days, or the need for RRT. AKI\
Mean age: 57.8\ 70/169 (47.3%)\
F: 23.7%\
| 3,237
| 5,306
| 2,649
| 2,484
| null | null |
github_plus_top10pct_by_avg
|
lambda(\phi))
\right].
\label{effect}$$ In the following Section, we calculate the contributions to $V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}$ from conformally invariant bulk fields.
Massless scalar bulk fields
===========================
The effective potential induced by scalar fields with arbitrary coupling to the curvature or bulk mass and boundary mass can be addressed. It reduces to a similar calculation to minimal the coupling massless field case, which is sovled in [@gpt], and correponds to bulk gravitons. However, for the sake of simplicity, we shall only consider below the contribution to $V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}(\phi)$ from conformally coupled massless bulk fields. Technically, this is much simpler than finding the contribution from bulk gravitons and the problem of backreaction of the Casimir energy onto the background can be taken into consideration in this case. Here we are considering generalizations of the original RS proposal [@alex1; @alex2; @alex3] which allow several fields other than the graviton only (contributing as a minimally coupled scalar field).
A conformally coupled scalar $\chi$ obeys the equation of motion $$-\Box_g \chi + {D-2 \over 4 (D-1)}\ R\ \chi =0,
\label{confin}$$ $$\Box^{(0)} \hat\chi =0.
\label{fse}$$ Here $\Box^{(0)}$ is the [*flat space*]{} d’Alembertian. It is customary to impose $Z_2$ symmetry on the bulk fields, with some parity given. If we choose even parity for $\hat\chi$, this results in Neumann boundary conditions $$\partial_{z}\hat\chi = 0,$$ at $z_+$ and $z_-$. The eigenvalues of the d’Alembertian subject to these conditions are given by $$\label{flateigenvalues}
\lambda^2_{n,k}=\left({n \pi \over L}\right)^2+k^2,$$ where $n$ is a positive integer, $L=z_{-}-z_+$ is the coordinate distance between both branes and $k$ is the coordinate momentum parallel to the branes. [^1]
Similarly, we could consider the case of massless fermions in the RS background. The Dirac equation,[^2] $$\gamma^{n}e^a_{\ n}\nabla_a\,\psi=0.$$
| 3,238
| 1,611
| 2,107
| 3,165
| 2,911
| 0.776146
|
github_plus_top10pct_by_avg
|
i}) (\Delta_{K} - h_{k}) }
\nonumber \\
&\times&
\biggl[
( h_{k} - h_{i} )
\biggl\{(\Delta_{K} - h_{i}) (\Delta_{K} - h_{k}) e^{- i \Delta_{J} x}
- (\Delta_{J} - h_{i}) (\Delta_{J} - h_{k}) e^{- i \Delta_{K} x}
\biggr\}
\nonumber \\
&-&
(\Delta_{J} - \Delta_{K})
\biggl\{ (\Delta_{J} - h_{k}) (\Delta_{K} - h_{k}) e^{- i h_{i} x}
- (\Delta_{J} - h_{i}) (\Delta_{K} - h_{i}) e^{- i h_{k} x}
\biggr\}
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W^{\dagger} A (UX) \right\}_{K k}
\left\{ (UX)^{\dagger} A W \right\}_{k J},
\label{hatS-iJ-W3-H2+3}\end{aligned}$$ and similarly, for $\hat{S}_{J i}^{(3)}$ $$\begin{aligned}
&& \hat{S}_{J i}^{(3)} [2+3]
\nonumber \\
&=&
- \frac{ 1 }{ \Delta_{J} - h_{i} }
\left[
(ix) e^{- i \Delta_{J} x} +
\frac{ e^{- i \Delta_{J} x} - e^{ - i h_{i} x} }{ ( \Delta_{J} - h_{i} ) }
\right]
\left\{ W^{\dagger} A W \right\}_{J J}
\left\{ W^{\dagger} A (UX) \right\}_{J i}
\nonumber \\
&+&
\sum_{K \neq J}
\frac{ 1 }{ ( \Delta_{J} - \Delta_{K} ) ( \Delta_{J} - h_{i} ) ( \Delta_{K} - h_{i} ) }
\nonumber \\
&\times&
\biggl[
\left( \Delta_{K} - h_{i} \right) e^{- i \Delta_{J} x}
- \left( \Delta_{J} - h_{i} \right) e^{- i \Delta_{K} x}
- \left( \Delta_{K} - \Delta_{J} \right) e^{- i h_{i} x}
\biggr]
\nonumber \\
&\times&
\left\{ W^{\dagger} A W \right\}_{J K}
\left\{ W^{\dagger} A (UX) \right\}_{K i}
\nonumber \\
&-&
\frac{ 1 }{ (\Delta_{J} - h_{i})^2 }
\biggl[
(ix) \left( e^{- i h_{i} x} + e^{- i \Delta_{J} x} \right)
+ 2 \frac{e^{- i \Delta_{J} x} - e^{- i h_{i} x} }{ ( \Delta_{J} - h_{i} ) }
\biggr]
\nonumber \\
&\times&
\left\{ W^{\dagger} A (UX) \right\}_{J i}
\left\{ (UX)^{\dagger} A W \right\}_{i J}
\left\{ W^{\dagger} A (UX) \right\}_{J i}
\nonumber \\
&+&
\sum_{k \neq i}
\biggl[
- \frac{ (ix) e^{- i \Delta_{J} x} }{ ( \Delta_{J} - h_{i} )( \Delta_{J} - h_{k} ) }
+ \frac{ 1 }{ ( h_{i} - h_{k} ) (\Delta_{J} - h_{i} )^2 (\Delta_{J} - h_{k} )^2 }
\nonumber \\
&\times&
\biggl\{
(\Delta_{J} - h_{k} )^2 e^{- i h_{i} x}
- (
| 3,239
| 1,511
| 3,063
| 3,024
| null | null |
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|
\alpha}^{\beta}f_{k}^{(-m)}\varphi^{m}\end{aligned}$$
where $f_{k}^{(-m)}(x)=\pi_{k}(x)+\int_{c}^{x}dx_{1}\int_{c}^{x_{1}}dx_{2}\cdots\int_{c}^{x_{m-1}}dx_{m}f_{k}(x_{m})$ is an $m$-times arbitrary indefinite integral of $f_{k}$. If now it is true that $\int_{\alpha}^{\beta}f_{k}^{(-m)}\rightarrow\int_{\alpha}^{\beta}f^{(-m)}$, then it must also be true that $f_{k}^{(-m)}\varphi^{(m)}$ converges in the mean to $f^{(-m)}\varphi^{(m)}$ so that $$\int_{\alpha}^{\beta}f_{k}\varphi=(-1)^{m}\int_{\alpha}^{\beta}f_{k}^{(-m)}\varphi^{(m)}\longrightarrow(-1)^{m}\int_{\alpha}^{\beta}f^{(-m)}\varphi^{(m)}=\int_{\alpha}^{\beta}f\varphi.$$ In fact the converse also holds leading to the following Equivalences between $m$-convergence in the mean and convergence with respect to test-functions, [@Korevaar1968].
**Type 1 Equivalence.** If $f$ and $(f_{k})$ are functions on $J$ that are integrable on every interior subinterval, then the following are equivalent statements.
\(a) For every interior subinterval $I$ of $J$ there is an integer $m_{I}\geq0$, and hence a smallest integer $m\geq0$, such that certain indefinite integrals $f_{k}^{(-m)}$ of the functions $f_{k}$ converge in the mean on $I$ to an indefinite integral $f^{(-m)}$; thus $\int_{I}|f_{k}^{(-m)}-f^{(-m)}|\rightarrow0.$
\(b) $\int_{J}(f_{k}-f)\varphi\rightarrow0$ for every $\varphi\in\mathcal{C}_{0}^{\infty}(J)$.
A significant generalization of this Equivalence is obtained by dropping the restriction that the limit object $f$ be a function. The need for this generalization arises because metric function spaces are known not to be complete: Consider the sequence of functions (Fig. \[Fig: FuncSpace\](a)) $$\begin{aligned}
f_{k}(x)= & \left\{ \begin{array}{lcl}
0 & \textrm{} & \textrm{if }a\leq x\leq0\\
kx & \textrm{} & \textrm{if }0\leq x\leq1/k\\
1 & \textrm{} & \textrm{if }1/k\leq x\leq b\end{array}\right.\label{Eqn: Lp[a,b]}\end{aligned}$$
which is not Cauchy in the uniform metric $\rho(f_{j},f_{k})=\sup_{a\leq x\leq b}|f_{j}(x)-f_{k}(x)|$ but is Cauchy in the
| 3,240
| 3,206
| 3,090
| 3,029
| null | null |
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|
ox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}y}\leq P_{\Lambda;y}^{\prime{{\scriptscriptstyle}(0)}}(o,x),\end{aligned}$$ which will be used in Section \[ss:pijbd\] to obtain the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$.
Since the inequality is trivial if $x=o$, we restrict our attention to the case of $x\ne o$.
First we note that, for each current configuration ${{\bf n}}$ with ${\partial}{{\bf n}}=\{o,x\}$ and ${\mathbbm{1}{\scriptstyle\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}}}=1$, there are at least *three edge-disjoint* paths on ${{\mathbb G}}_{{\bf n}}$ between $o$ and $x$. See, for example, the first term on the right-hand side in Figure \[fig:1stpiv\]. Suppose that the thick line in that picture, referred to as $\zeta_1$ and split into $\zeta_{11}{\:\Dot{\cup}\:}\zeta_{12}{\:\Dot{\cup}\:}\zeta_{13}$ from $o$ to $x$, consists of bonds $b$ with $n_b=1$, and that the thin lines, referred to as $\zeta_2$ and $\zeta_3$ that terminate at $o$ and $x$ respectively, consist of bonds $b'$ with $n_{b'}=2$. Let $\zeta'_i$, for $i=2,3$, be the duplication of $\zeta_i$. Then, the three paths $\zeta_2{\:\Dot{\cup}\:}\zeta_{13}$, $\zeta'_2{\:\Dot{\cup}\:}\zeta_{12}{\:\Dot{\cup}\:}\zeta_3$ and $\zeta_{11}{\:\Dot{\cup}\:}\zeta'_3$ are edge-disjoint.
Then, by multiplying $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ by *two* dummies $(Z_\Lambda/Z_\Lambda)^2\,(\equiv1)$, we obtain $$\begin{aligned}
{\label{eq:pi0*Z2}}
\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)&=\sum_{\substack{{\partial}{{\bf n}}=\{o,x\}\\ {\partial}{{\bf m}}'={\partial}{{\bf m}}''
={\varnothing}}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,\frac{w_\Lambda({{\bf m}}')}{Z_\Lambda}
\,\frac{w_\Lambda({{\bf m}}'')}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}$}}}{\nonumber}\\
&=\sum_{{\partial}{{\bf N}}=\{o,x\}}\frac{w_\Lambda({{\bf N}})}{Z_
| 3,241
| 1,601
| 1,036
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| null | null |
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|
{E}[||\hat \beta - \beta||_\infty]$. Let $P_0$ be multivariate Normal with mean $(0,\ldots, 0)$ and identity covariance. For $j=1,\ldots, D$ let $P_j$ be multivariate Normal with mean $\mu_j=(0,\ldots,0,a,0, 0)$ and identity covariance where $a = \sqrt{ \log D/(16n)}$. Then $$\begin{aligned}
\inf_{\hat\beta}\sup_{w\in {\cal W}_n}\sup_{P\in {\cal P}_n}\mathbb{E}[|\hat \beta(J) - \beta(J)|] &\geq
\inf_{\hat\beta}\sup_{P\in M}\mathbb{E}[|\hat \beta(J) - \beta(J)|]\\
&=
\inf_{\hat\beta}\sup_{P\in M}\mathbb{E}[||\hat \beta - \beta||_\infty]\end{aligned}$$ where $J= w_0(Y)$ and $M = \{P_0,P_1,\ldots,P_D\}$. It is easy to see that $${\rm KL}(P_0,P_j) \leq \frac{\log D}{16 n}$$ where KL denotes the Kullback-Leibler distance. Also, $||\mu_j - \mu_k||_\infty \geq a/2$ for each pair. By Theorem 2.5 of [@tsybakov2009introduction], $$\inf_{\hat\beta}\sup_{P\in M}\mathbb{E}[||\hat \mu - \mu ||_\infty] \geq \frac{a}{2}$$ which completes the proof. $\Box$
[**Proof of Lemma \[lemma::contiguity\].**]{} We use a contiguity argument like that in [@leeb2008can]. Let $Z_1,\ldots, Z_D \sim N(0,1)$. Note that $\hat\beta(j) \stackrel{d}{=} \beta(j)+ Z_j/\sqrt{n}$. Then $$\begin{aligned}
\psi_n(\beta) &= \mathbb{P}(\sqrt{n}(\hat\beta(S) - \beta(S))\leq t) =
\sum_j \mathbb{P}(\sqrt{n}(\hat\beta(j) - \beta(j))\leq t,\ \hat\beta(j) > \max_{s\neq j}\hat\beta_s)\\
&=
\sum_j \mathbb{P}(\max_{s\neq j}Z_s + \sqrt{n}(\beta(s)-\beta(j)) < Z_j < t) = \sum_j \Phi(A_j)\end{aligned}$$ where $\Phi$ is the $d$-dimensional standard Gaussian measure and $$A_j = \Bigl\{ \max_{s\neq j}Z_s + \sqrt{n}(\beta(s)-\beta(j) < Z_j < t \Bigr\}.$$ Consider the case where $\beta = (0,\ldots, 0)$. Then $$\psi_n(0)= D\, \Phi(\max_{s\neq 1}Z_s < Z_1 < t) \equiv b(0).$$ Next consider $\beta_n = (a/\sqrt{n},0,0,\ldots, 0)$ where $a>0$ is any fixed constant. Then $$\begin{aligned}
\psi(\beta_n) &= \Phi( (\max_{s\neq 1}Z_s )-a < Z_1 < t)\\
&\ \ \ \ \ +
\sum_{j=2}^D \Phi(\max\{Z_1+a,Z_2,\ldots, Z_{j-1},Z_{j+1},\ldots, Z_D\} < Z_j < t)\\
&\equiv b(a).\end{aligned}$$ Suppos
| 3,242
| 2,059
| 1,981
| 2,897
| 3,862
| 0.769659
|
github_plus_top10pct_by_avg
|
ac{n}{2}+32}/i
\times \RP^{95}, \dots$, $X_j = S^{\frac{n}{2} - 32(j-2)-1}/i
\times \RP^{32(j+2)-1}, \dots$, $X_{\frac{n+2}{64}} = S^{63}/i
\times \RP^{\frac{n}{2}+64}$. The embedding of the corresponding manifold in $Z$ is defined by the Cartesian product of the two standard embeddings.
The union of the submanifolds $\{X_i\}$ is a stratified submanifold (with singularities) $X \subset Z$ of the dimension $\frac{n}{2}+127$, the codimension of maximal singular strata in $X$ is equal to $64$. The covering $p_X: \hat X \to X$, induced from the covering $p_Z: \hat Z \to Z$ by the inclusion $X \subset
Z$, is well-defined. The covering space $\hat X$ is a stratified manifold (with singularities) and decomposes into the union of the submanifolds $\hat X_0 = \RP^{\frac{n}{2}+64} \times \RP^{63},
\dots, \hat X_j = \RP^{\frac{n}{2} - 32(j-2)} \times \RP^{32(j+2)
-1}, \dots, \hat X_{\frac{n+2}{64}}=\RP^{63} \times
\RP^{\frac{n}{2}+64}$. Each manifold $\hat X_i$ of the family is the $2$-sheeted covering space over the manifold $X_i$ over the first coordinate. Let us define $d_1(j)= \frac{n}{2} - 32(j-2)$, $d_2(j)=32(j+2)-1$. Then the formula for $X_i$ is the following: $X_j=\RP^{d_1(j)} \times
\RP^{d_2(j)}$.
The cohomology classes $\rho_{X,1} \in H^1(X;\Z/4)$, $\kappa_{X,2}\in H^1(X;\Z/2)$ are well-defined. These classes are induced from the generators of the groups $H^1(Z;\Z/4)$, $H^1(Z;\Z/2)$. Analogously, the cohomology classes $\kappa_{\hat
X,i}\in H^1(\hat X;\Z/4)$, $i=1,2$ are well-defined. The cohomology class $\kappa_{\hat X,1}$ is induced from the class $\rho_{X,1} \in H^1(X;\Z/4)$ my means of the transfer homomorphim, and $\kappa_{\hat X,2} = (p_X)^{\ast}(\kappa_{X,2})$.
Let us define for an arbitrary $j= 0, \dots, (\frac{n+2}{64})$ the space $J_j$ and the mapping $\varphi_j: X_j \to J_j$. We denote by $Y_1(k)$ the space $S^{31}/i \ast \dots
\ast S^{31}/i$ of the join of $k$ copies, $k=1, \dots
,(\frac{n+2}{64}+1)$, of the standard lens space $S^{31}/i$. Let us denote by $Y_2(k)$, $k=2, \dots, (\frac{n+2}{64}+2)
| 3,243
| 2,896
| 2,705
| 2,881
| 3,584
| 0.771384
|
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|
ng in $y_\chi$. These ingredients are difficult to consistently implement in other model constructions without violating constraints on light force carriers.
The effect of strong differences between proton and neutron coupling to DM have been explored by [@Feng:2011vu]. To concentrate on the kinematics we shall therefore assume the operator $\bar q q \phi \phi^*/\Lambda$ is flavor-blind in the quark mass basis. Above the electroweak symmetry breaking scale this operator is realized as $\bar Q_L H q_R \phi\phi^*/M^2$. It can be generated by integrating out heavy vector-like quarks which couple to the SM and $\phi$ [@dmDM], giving $1/\Lambda = y_Q^2 y_h v/M_Q^2$. This UV completion allows for large direct detection cross sections without being in conflict with collider bounds, but may be still probed at the 14 TeV LHC.
-------------------------------------------------------------------------------------------------------
$\scriptstyle m_\chi = 10 \gev, m_\phi = 0.2 \kev, v = 400 \kmpers $
$\scriptstyle m_N \ = \ \textcolor{blue}{28}, \ \textcolor{red}{73}, \ \textcolor{purple}{131} \gev$
-------------------------------------------------------------------------------------------------------
\
![ Nuclear recoil spectra of dmDM (without nuclear/nucleus form factors and coherent scattering enhancement) for $y_\chi = 1, \Lambda = 1 \tev$ in a Silicon, Germanium and Xenon target. The dashed lines are spectra of standard WIMP scattering (via operator $\bar q q \bar \chi \chi/\tilde \Lambda^2$, with $\tilde \Lambda = 7 \tev$) shown for comparison. dmDM spectra computed with `MadGraph5` [@Alwall:2011uj] and `FeynRules1.4` [@Ask:2012sm]. []{data-label="f.partonleveldistribution"}](raw2to3spectrum_mN_28_73_131_mX_10_mphi_02_v_400__2 "fig:"){width="7cm"}
Nuclear Recoil Spectrum
=======================
We start by examining the novel $2\rightarrow3$ regime of dmDM. The DM-nucleus collision is inelastic, not by introducing a new mass scale like a splitting, but by virtue of the process topology. The nuclear recoil
| 3,244
| 1,643
| 1,697
| 3,112
| null | null |
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|
ntity corresponds to the momentum distribution $P(k)=\rho(k,k)$ of the atoms, directly measured in the experiment. It is seen in Fig. \[fig1\] that, around $v=15$, there is a qualitative change in the momentum distribution, in close analogy with that observed in the experiment [@Grei02]. It should be noted, however, that this qualitative change of the momentum distribution alone does not yet prove the occurrence of a phase transition. A more reliable indication of a SF-MI phase transition are the fluctuations of the number of atoms in a single well, which drops from $\langle\Delta n^2\rangle\approx0.72$ at $v=5$ to $\langle\Delta n^2\rangle\approx0$ at $v=35$ [@remark3].
![Bloch oscillations of the atoms, induced by the static force $F=1/2\pi$ ($v=10$). One Bloch period is shown.[]{data-label="fig2"}](fig2.eps){width="8cm"}
![Dephasing of Bloch oscillations due to the atom-atom interaction. The period $T_W=2\pi F/W$ is clearly seen. ($F=1/2\pi$, $v=10$, $W=0.1\int dx \phi_l^4(x)=0.0324$.)[]{data-label="fig3"}](fig3.eps){width="8cm"}
Let us now discuss the effect of the static force. Figure \[fig2\] shows the dynamics of the momentum distribution of the atoms (which were initially in SF-phase) in presence of a force $F=1/2\pi$ [@remark2]. This numerical simulation illustrates atomic BO as observed in laboratory experiments [@Daha96; @Mors01]. It is seen that after one Bloch period the initial momentum distribution practically coincides with the final distribution. A small difference, which can be noticed by closer inspection of the figure, is obviously due to the atom-atom interaction [@remark4]. This difference becomes evident once the system evolved over several Bloch periods. In Fig. \[fig3\], the momentum distribution $P(k)$ at integer multiples of the Bloch period is shown. A periodic change of the distribution from SF to MI-like and back is clearly seen. (The use of the term ‘MI-like’ stresses the fact that the variance $\langle\Delta n^2\rangle$ does not change as time evolves.) In addition to Fig.
| 3,245
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| 3,054
| null | null |
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|
M(1)={\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]\delta e$. On the other hand, for the given decomposition Lemmas \[thetainjA\] and \[abstract-products\] imply that $\operatorname{{\textsf}{tgr}}H_c$ is a homomorphic image of $ T=\operatorname{{\textsf}{ogr}}H_{c+1}e\otimes_{U_{c+1}}\operatorname{{\textsf}{ogr}}Q_c^{c+1}\cong
{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]e\otimes_{A^0} A^1\delta e.$ Clearly the image of $T$ in $\operatorname{{\textsf}{ogr}}M(1)$ is just ${\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]e A^1\delta e=J^1\delta e$. By the argument of the second paragraph of , this is also the image of $\operatorname{{\textsf}{tgr}}M(1)$ in $\operatorname{{\textsf}{ogr}}M(1)$.
Graded projective modules {#app-a}
=========================
{#section-4}
The aim of this appendix is to prove the following graded analogue of a well-know result of Kaplansky [@Kap Theorem 2], for which we do not know a reference.
\[graded-proj-thm\] Let $A=\bigoplus_{i\geq 0} A_i$ be a connected ${\mathbb{N}}$-graded $k$-algebra (thus $A_0=k$). Let $P$ be a right $A$-module that is both graded and projective. Then $P$ is a *graded-free* $A$-module in the sense that $P$ has a free basis of homogeneous elements.
Throughout this proof all graded maps are graded maps of degree zero. We will write the degree of a homogeneous element $x\in P$ as $|x|$.
An observation of Eilenberg [@Eil Section 1] shows that $P$ is graded projective in the sense that there is a graded isomorphism $F\cong
P\oplus Q$, for some $A$-module $Q$ and graded-free $A$-module $F$. We need a minor variant on this result, so we give the proof. Take a graded surjection $\phi: F=\bigoplus f_iA\twoheadrightarrow P$ and an ungraded splitting $\theta: P\to F$. If $p_i=\phi(f_i)$, then write $\theta(p_i)=g_i+h_i$, where $g_i$ is the homogeneous component of $\theta(p_i)$ with $|g_i|=|p_i|$. Then check that the map $p_i\mapsto g_i$ also splits $\phi$. This proof also shows that, if $P$ is countably generated, then we can take $F$ to be a count
| 3,246
| 2,626
| 1,713
| 3,140
| 3,638
| 0.771027
|
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|
p\left(-i\frac{W}{2F}\sum_{l=1}^L\langle {\bf n}|
\hat{n}_l(\hat{n_l}-1)|{\bf n}\rangle \right) \;.$$ Finally, by noting that the quantity $\langle {\bf n}|\hat{n}_l(\hat{n_l}-1)|{\bf n}\rangle$ is always an even integer, one comes to the conclusion that, besides the Bloch period, there is additional period, $$\label{7}
T_W=2\pi F/W \;,$$ which characterises the dynamics of the system.
Further analytical results can be obtained if we approximate the ground state of the system for $F=0$ by the product of $N$ Bloch waves with quasimomentum $\kappa=0$, i.e., $$\label{8}
|\Psi\rangle=\frac{1}{\sqrt{N!}}\left(
\frac{1}{\sqrt{L}}\sum_{l=1}^L \hat{a}^\dag_l\right)^N
|0\ldots0\rangle \;.$$ Indeed, let us consider, for example, the dynamics of the mean momentum. Using the interaction representation (now with respect to the Stark energy term) the mean momentum is given by $$\label{9}
p(t)=J\; {\rm Im}\left(\langle \Psi U^\dag_W(t)|
\sum_{l=1}^L \hat{a}^\dag_{l+1}\hat{a}_l
|U_W(t) \Psi \rangle e^{-i2\pi Ft}\right) \;,$$ where $U_W(t)$ is the continuous-time version ($T_B\rightarrow t$) of the diagonal unitary matrix (\[6\]). Substituting Eq. (\[8\]) and Eq. (\[6\]) into Eq. (\[9\]), we obtain the following exact expression, $$\label{10}
\frac{p(t)}{NJ}=\frac{L}{N} {\rm Im}\left(
\sum_{n,n'} n {\cal P}(n,n')e^{i(n'-n+1)Wt}e^{-i2\pi Ft}\right) \;,$$ where ${\cal P}(n,n')$ is the joint probability to find $n$ and $n'$ atoms in two neighbouring wells. In the thermodynamic limit $N,L\rightarrow\infty$, $N/L=\bar{n}$, the function ${\cal P}(n,n')$ factorises into a product of the Poisson distributions ${\cal P}(n)=\bar{n}^n\exp(-\bar{n})/n!$, and the double sum in Eq. (\[10\]) converges to the positive periodic function, $f(t)=\exp(-2\bar{n}[1-\cos(Wt)])$, indicated in Fig. \[fig4\] by the dashed line. Good agreement between the envelope of $p(t)$ and the dashed line proves that in the numerical simulation presented above the convergence was indeed achieved.
In summary, Bloch oscillations of interacting cold atoms have b
| 3,247
| 2,270
| 2,812
| 3,023
| 4,109
| 0.768044
|
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|
acities. Since an extended cf Petri net $N_z$, $z\in\{h, c, s\}$, has two kinds of places, i.e., places labeled by nonterminal symbols and *control* places, it is interesting to consider two types of place capacities in the Petri net: first, we demand that only the places labeled by nonterminal symbols are with capacities (*weak capacity*), and second, all places of the net are with capacities (*strong capacity*).
A $z$-Petri net $N_z=(P\cup Q, T, F\cup E, \varphi, \zeta, \gamma, \mu_0, \tau)$ is with *weak capacity* if the corresponding cf Petri net $(P, T, F, \phi, \iota)$ is with place capacity, and *strong capacity* if the Petri net $(P\cup Q, T, F\cup E, \varphi, \mu_0)$ is with place capacity. A grammar controlled by a $z$-Petri net with *weak* (*strong*) *capacity* is a $z$-Petri net controlled grammar $G = (V, \Sigma, S, R, N_z)$ where $N_z$ is with weak (strong) place capacity. We denote the families of languages generated by grammars (with erasing rules) controlled by $z$-Petri nets with weak and strong place capacities by $\mathbf{wPN}_{cz}$, $\mathbf{sPN}_{cz}$ ($\mathbf{wPN}^{\lambda}_{cz}$, $\mathbf{sPN}^{\lambda}_{cz}$), respectively, where $z\in\{h, c, s\}$.
The power of arbitrary grammars with capacities {#sec:power-gs}
===============================================
It will be shown in this section that arbitrary grammars (due to Ginsburg and Spanier) with capacity generate exactly the family of matrix languages of finite index. This is in contrast to derivation bounded grammars which generate only context-free languages of finite index.
First we show that we can restrict to grammars with capacities bounded by $1$. Let $\mathbf{CF}_{{\mathit{cb}}}^{1}$ and $\mathbf{GS}_{{\mathit{cb}}}^{1}$ be the language families generated by context-free and arbitrary grammars with capacity function $\mathbf{1}$.
$\mathbf{CF}_{{\mathit{cb}}}=\mathbf{CF}_{{\mathit{cb}}}^{1}$ and $\mathbf{GS}_{{\mathit{cb}}}=\mathbf{GS}_{{\mathit{cb}}}^{1}$.
Let $G=(V,\Sigma,S,R, \kappa)$ be a capacity-bounded phrase stru
| 3,248
| 1,473
| 3,400
| 3,219
| 792
| 0.79992
|
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preliminaries on Moebius maps and the circumcenter extension, and then in section 3 we prove the main theorem.
Preliminaries
=============
For details and proofs of the assertions made in this section we refer to [@biswas3], [@biswas5], [@biswas6], [@biswas7].
Moebius metrics and visual metrics
----------------------------------
Let $(Z, \rho_0)$ be a compact metric space of diameter one. For a metric $\rho$ on $Z$, the cross-ratio with respect to the metric $\rho$ is the function of quadruples of distinct points in $Z$ defined by $$[\xi, \xi', \eta, \eta']_{\rho} := \frac{\rho(\xi, \eta)\rho(\xi', \eta')}{\rho(\xi, \eta')\rho(\xi', \eta)}$$ A metric $\rho$ on $Z$ is said to be antipodal if it has diameter one and for any $\xi \in Z$ there exists $\eta \in Z$ such that $\rho(\xi, \eta) = 1$. We assume that the metric $\rho_0$ is antipodal. We say that two metrics $\rho_1, \rho_2$ on $Z$ are Moebius equivalent if for all quadruples of distinct points in $Z$, the cross-ratios with respect to the two metrics are equal. We let $\mathcal{M}(Z, \rho_0)$ denote the set of all antipodal metrics on $Z$ which are Moebius equivalent to $\rho_0$. For any $\rho_1, \rho_2 \in \mathcal{M}(Z, \rho_0)$, there exists a positive continuous function on $Z$ called the derivative of $\rho_2$ with respect to $\rho_1$, denoted by $\frac{d\rho_2}{d\rho_1}$, such that $$\rho_2(\xi, \eta)^2 = \frac{d\rho_2}{d\rho_1}(\xi)\frac{d\rho_2}{d\rho_1}(\eta) \rho_1(\xi,\eta)^2$$ for all $\xi, \eta \in Z$, and such that $$\frac{d\rho_2}{d\rho_1}(\xi) = \lim_{\eta \to \xi} \frac{\rho_2(\xi, \eta)}{\rho_1(\xi, \eta)}$$ for all non-isolated points $\xi$ of $Z$. Moreover, $$\left( \max_{\xi \in Z} \frac{d\rho_2}{d\rho_1}(\xi) \right) \cdot \left( \min_{\xi \in Z} \frac{d\rho_2}{d\rho_1}(\xi) \right) = 1$$ The set $\mathcal{M}(Z, \rho_0)$ admits a natural metric defined by $$d_{\mathcal{M}}(\rho_1, \rho_2) = \sup_{\xi \in Z} \log \frac{d\rho_2}{d\rho_1}(\xi)$$ The metric space $(\mathcal{M}(Z, \rho_0), d_{\mathcal{M}})$ is proper.
Let $X$ be a prope
| 3,249
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| 1,520
| 3,176
| null | null |
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|
hen $H_y$ satisfies the hypotheses (H2). If ${\ensuremath{\left| H_y \right|}}<{\ensuremath{\left| G \right|}}$, by minimality we obtain that $H_y$ has a normal Sylow $p$-subgroup, and so $[y, {{\operatorname}{O}_{p'}(G)}]=1$. If this holds for every $y\in (P\cap A)\cup (P\cap B)$, then $[P, {{\operatorname}{O}_{p'}(G)}]=1$, a contradiction. Hence we may suppose that, for instance, there exists $y\in P\cap A$ with $H_y={{\operatorname}{O}_{p'}(G)}\langle y \rangle=G$. Further, since we are assuming that ${\ensuremath{\left| A \right|}}+{\ensuremath{\left| B \right|}}$ is minimal, then we deduce that $A=({{\operatorname}{O}_{p'}(G)}\cap A)\langle y \rangle$ and $B={{\operatorname}{O}_{p'}(G)}\cap B$.
Now, by coprime action and minimality, ${{\operatorname}{O}_{p'}(G)}=[{{\operatorname}{O}_{p'}(G)}, y]{{\operatorname}{C}_{{{\operatorname}{O}_{p'}(G)}}(y)}\leq N{{\operatorname}{C}_{G}(y)}.$ Thus $G={{\operatorname}{O}_{p'}(G)}\langle y\rangle = N{{\operatorname}{C}_{G}(y)}$.
\[ncapa\] $N \cap A \neq 1$.
Assume that $N\cap A=1$. By Lemma \[1\], we know that $G/N=\langle y \rangle N/N \times {{\operatorname}{O}_{p'}(G)}/N$. Hence $[\langle y \rangle, {{\operatorname}{O}_{p'}(G)}\cap A]\leq N\cap A=1$, so $\langle y\rangle$ is a Sylow $p$-subgroup of $G$ which is normal in $A$. Now, since $B$ is a $p'$-group, we have that for any $b \in B$ of prime power order, there exists $g \in G=AB$ such that $\langle y\rangle^g \leq C_G(b)$. This implies that $\langle y\rangle^{b_1} \leq C_G(b)$ for some $b_1 \in B$, since $\langle y\rangle^a=\langle y\rangle$ for any $a \in A$. It follows that each element of prime power order of $B$ lies in $\underset{x\in B}{\cup}{{\operatorname}{C}_{B}(\langle y\rangle)}^x$ and so, by Lemma \[feinkantor\], we deduce $[B, \langle y\rangle]=1$, a contradiction which proves our claim.
\[5\] $N$ is a non-abelian group, so $N=N_1\times N_2\times \cdots \times N_r$, with $N_i \cong N_1$ a non-abelian simple group.
Assume that $N$ is abelian. Therefore ${{\operatorname}{C}_{N}(y)}\leq N$ i
| 3,250
| 1,919
| 1,577
| 3,062
| null | null |
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|
urbations, Teukolsky adopted the Newman-Penrose (NP) formalism [@Newman:1961qr] and obtained a separable wave equation for Weyl curvature tensor components $\Psi_0$ and $\Psi_4$. The spin-weighted version of this equation, known as the Teukolsky equation, not only works for gravitational perturbations, i.e. tensor fields, but can also be applied to scalar, vector, and spinor fields. To obtain the other Weyl scalars and recover the perturbed metric, one has to go through a complicated metric reconstruction procedure. The methods were independently developed by Chrzanowski [@Chrzanowski:1975wv] and by Cohen and Kegeles [@Kegeles:1979an], in which they obtain the perturbed metric via an analogue of Hertz potentials. However, these methods only apply to certain gauge choices and vacuum or highly-restricted source terms [@Whiting:2005hr].
The desire for separable equations, the complication of metric reconstruction along with gauge- and source-restrictions, motivate us to try to develop a new formalism for studying metric perturbations in the Kerr spacetime, in a covariant, gauge-invariant way.
The metric perturbation equation may not be separable in Kerr, but Schwarzschild perturbations have long been known as separable due to the time translation invariance and spherical symmetry [@Regge:1957td; @Vishveshwara:1970cc; @Zerilli:1970se; @Zerilli:1971wd]. The gauge-independent language of Schwarzschild perturbations was started by Sarbach and Tiglio [@Sarbach:2001qq], and brought to fruition by Martel and Poisson [@Martel:2005ir]. In the Schwarzschild background, metric perturbations are expanded in scalar, vector, and symmetric tensor spherical harmonics. These basis functions naturally lead to separation of variables in the LEE.
Schematically, the separation of variables in some differential equations of motion, such as the scalar wave equation, Maxwell’s equations, and the linearized Einstein equations, can all be understood via $$\begin{aligned}
\mathcal{D}_{x}\!
\Big[
\Big(
\parbox{1.1cm}{\cent
| 3,251
| 2,004
| 3,428
| 3,061
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|
nd{aligned}$$ comprising an elastic and an inelastic current. This naturally defines the terminology used throughout the rest of the paper.
In summary, we have related the total current to products involving a greater Green’s function $G^>\sim
\langle C(\lambda^{\rm tip}) d^\dagger_{\mu\sigma}(t') d_{\mu'\sigma}(t) \rangle_0$ of the system S and a lesser Green’s function $G^< \sim \langle c_{0\sigma,\rm tip}(t') c^\dagger_{0\sigma,\rm tip}(t)\rangle_0$ of the STM tip and vice versa [@Keldysh65]. Most importantly, the Keldysh Green’s functions of a fully interacting system entangling, in general, vibrational and electronic operators, are employed for the system S. Therefore, the expressions derived and analyzed in the following sections go well beyond the standard literature.
### Elastic tunnel current {#sec:elastic tunnel current}
Since we are interested in the asymptotic steady-state current, we perform the limit $t_0\to -\infty$ and calculate the current at the time $t=0$. Evaluating the greater and lesser Green functions with the equilibrium density operator in Eq. and calculating the steady-state current for $\lambda^{\rm tip}_{\mu\nu}=0$ in Eq. , we obtain the well-known expression for the elastic tunnel current $$\begin{aligned}
\label{eqn:el-current}
I_{\rm el}(t=0) &=&
\frac{2\pi e}{\hbar}
\sum_{\sigma}
\int_{-\infty}^{\infty} d\w \rho_{\sigma,\rm tip}(\w) \tau^{(0)}_{\sigma}(\w)
\nonumber \\
&&\times
\left[
f_{\rm tip}(\w) - f_{S}(\w)
\right],
\label{eqn:elasticCurrent}\end{aligned}$$ where $f_{\rm tip}(\w)=f(\w-eV)$ and $f_{S}(\w)=f(\w)$, with $f(\w)=[\exp(\beta \w)+1]^{-1}$ being the Fermi function. $$\begin{aligned}
\tau^{(0)}_{\sigma}(\w) &=& \sum_{\mu\mu'}^M t_{\mu \sigma}t_{\mu' \sigma}
\lim_{\delta\to 0^+}
\frac{1}{\pi} \Im G_{d_{\mu\sigma}, d^\dagger_{\mu'\sigma}} (\w-i\delta),
\label{eqn:tauzero}\end{aligned}$$ is the spin-dependent transmission function from the STM tip to the system S, and $\rho_{\sigma,\rm STM}(\w)$ denotes the spectral density of the STM tip, $$\begin{
| 3,252
| 3,046
| 3,205
| 2,925
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a monoidal left derivator, then any shift ${\sV}^A$ is also a -module.
We also have the following universal construction:
For any left derivators ${\sD},{\sE}$, define ${\mathsf{HOM}}({\sD},{\sE})$ by $${\mathsf{HOM}}({\sD},{\sE})(A) = \cDER({\sD},{\sE}^A)$$ where a functor $u:A\to B$ induces the restriction functor $${\mathsf{HOM}}({\sD},{\sE})(B) = \cDER({\sD},{\sE}^B) \to \cDER({\sD},{\sE}^A) = {\mathsf{HOM}}({\sD},{\sE})(A)$$ by postcomposition with $u^* \colon {\sE}^B \to {\sE}^A$. This makes ${\mathsf{HOM}}({\sD},{\sE})$ into a left derivator, and indeed a derivator if is one; its Kan extension functors are also simply given by postcomposition. In this way becomes a cartesian closed 2-category in an appropriate weak sense. In particular, ${\mathsf{HOM}}({\sD},{\sD})$ is a pseudo-monoid under composition, and there is a canonical action ${\mathsf{HOM}}({\sD},{\sD}) \times {\sD}\to {\sD}$. However, this monoidal structure and action do not preserve colimits in the right variable, hence do not make into a ${\mathsf{HOM}}({\sD},{\sD})$-module.
Thus, we define a new left derivator ${\mathsf{HOM}\ccsub}({\sD},{\sE})$, for which ${\mathsf{HOM}\ccsub}({\sD},{\sE})(A)$ is the category of *cocontinuous* morphisms ${\sD}\to{\sE}^A$. Since restriction and left Kan extension are cocontinuous morphisms, this is again a left derivator. The endomorphism object ${\mathsf{HOM}\ccsub}({\sD},{\sD})$, which we denote ${\mathsf{END}\ccsub}({\sD})$, *is* a monoidal left derivator under composition, and its action ${\mathsf{END}\ccsub}({\sD})\times{\sD}\to{\sD}$ does make into an ${\mathsf{END}\ccsub}({\sD})$-module.
Explicitly, the external monoidal product of $F\colon{\sD}\to{\sD}^A$ and $G\colon{\sD}\to{\sD}^B$ is the morphism $GF\colon{\sD}\to{\sD}^{A\times B}$ whose component ${\sD}(C) \to {\sD}(C\times A\times B)$ is the composite ${\sD}(C) \xto{F^C} {\sD}(C\times A) \xto{G^{C\times A}} {\sD}(C\times A\times B)$. Similarly, the external action of $F\colon{\sD}\to{\sD}^A$ on $X\in {\sD}(B)$ is the image of $X$ under $F^B
| 3,253
| 2,765
| 2,136
| 2,954
| null | null |
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|
-7.66); *p-*value \<0.01), and only 26% (4 of 15) of LTR were alive at the conclusion of the study.
***Conclusion.*** CDI is associated with increased mortality in LTR. LTR have many risk factors that predispose them to development of CDI. Strategies to decrease the risk of CDI are needed to improve survival in this patient population.
***Disclosures.*** **All authors:** No reported disclosures.
[^1]: **Session:** 47. Transplant Infectious Diseases
[^2]: Thursday, October 9, 2014: 12:30 PM
###### Strengths and limitations of this study
- The pilot study will test the use of administrative data in two ways:
- The use of health register data as a sampling frame.
- Linkage of the survey to a range of administrative datasets.
```{=html}
<!-- -->
```
- The use of health register data as a sampling frame is novel, and the pilot will establish the feasibility of this approach for future waves of the study, in particular, the optimisation of fieldwork time and costs and implications of survey non-response.
- The linkage of the survey to administrative datasets will:
- Test the feasibility and acceptability of linking a social survey of older people to administrative data by relevant ethics committees, administrative data custodians (eg, the Public Benefit and Privacy Panel) and study participants.
- Produce a longitudinal/survival dataset when linked to the cross-sectional pilot survey data.
```{=html}
<!-- -->
```
- The harmonised pilot survey data offers powerful cross-country comparisons across the social, economic and health domains that will be relevant for national and international policy debate from the outset.
- The benefits of data linkage are subject to the successful matching of the survey respondents to their administrative health records; therefore, this may limit the use of such data as a sample frame.
Introduction {#s1}
============
Around the world people are living longer.[@R1] This success brings both opportunities and risks to individuals, familie
| 3,254
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| 1,989
| 3,042
| null | null |
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|
. In atomic and molecular physics (Clebsch–Gordan series), as well as in high-energy physics, this problem has been studied extensively [@weyl50; @wigner59; @wigner73], perhaps most famously in Ne’eman and Gell-Mann’s eight-fold way of elementary particles [@neeman; @gellmann2; @gellmann]. In pure mathematics, the combinatorial resolution of the problem of decomposing tensor products of irreducible representations of the unitary group by Knutson and Tao has been a recent highlight with a long history of research [@fulton00; @knutsontao99]. More recently, the theories of quantum information [@keylwerner01; @christandlmitchison06; @klyachko06], computation and complexity [@baconchuangharrow07], as well as the geometric complexity theory approach to the ${\mathbf P}$ vs. ${\mathbf{NP}}$ problem [@mulmuleysohoni01; @mulmuleysohoni08; @burgisserlandsbergmaniveletal11] have brought the representation theory of Lie groups to the attention of the computer science community.
In this paper, we study the problem of computing multiplicities of Lie group representations:
\[main problem\] Let $f \colon H \rightarrow G$ be a homomorphism between compact connected Lie groups $H$ and $G$. The *subgroup restriction problem for $f$* is to determine the multiplicity $m^\lambda_\mu$ of the irreducible $H$-representation $V_{H,\mu}$ in the irreducible $G$-representation $V_{G,\lambda}$ when given as input the highest weights $\mu$ and $\lambda$ (specified as bitstrings containing their coordinates with respect to fixed bases of fundamental weights, see ).
The name *subgroup restriction problem* comes from the archetypical case where the map $f$ is induced by the inclusion of a subgroup $H \subseteq G$. is also known as the *branching problem*. The main result of this paper is a polynomial-time algorithm for :
\[A\] For any homomorphism $f \colon H \rightarrow G$ between compact connected Lie groups $H$ and $G$, there is a polynomial-time algorithm for the subgroup restriction problem for $f$.
Indeed, we describe a concrete
| 3,255
| 3,828
| 3,783
| 3,164
| null | null |
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|
i}\phi_{\varepsilon}(\mu,\nu_{i})=\psi_{+}(\mu)+\psi_{-}(\mu),\qquad\mu\in[-1,1],\textrm{ }\nu_{i}\geq0\label{Eqn: HRFR_Discrete}$$
of the discretized boundary condition Eq. (\[Eqn: BC\]), where $\psi_{+}(\mu)$ is by definition the incident flux $\psi(\mu)$ for $\mu\in[0,1]$ and $0$ if $\mu\in[-1,0]$, while $$\psi_{-}(\mu)=\left\{ \begin{array}{ccl}
{\displaystyle \sum_{i=0}^{N}b_{i}^{-}\phi_{\varepsilon}(\mu,\nu_{i})} & & \textrm{if }\mu\in[-1,0],\textrm{ }\nu_{i}\geq0\textrm{ }\\
0 & & \textrm{if }\mu\in[0,1]\end{array}\right.$$ is the the emergent angular distribution out of the medium. Equation (\[Eqn: HRFR\_Discrete\]) corresponds to the full-range $\mu\in[-1,1],\textrm{ }\nu_{i}\geq0$ form $$b(\nu_{0})\phi(\mu,\nu_{0})+\int_{0}^{1}b(\nu)\phi(\mu,\nu)d\nu=\psi_{+}(\mu)+\left(b^{-}(\nu_{0})\phi(\mu,\nu_{0})+\int_{0}^{1}b^{-}(\nu)\phi(\mu,\nu)d\nu\right)\label{Eqn: HRFR}$$ of boundary condition (\[Eqn: BC\_HR\]) with the first and second terms on the right having the same interpretation as for Eq. (\[Eqn: HRFR\_Discrete\]). This full-range simulation merely states that the solution (\[Eqn: CaseSolution\_HR\]) of Eq. (\[Eqn: NeutronTransport\]) holds for all $\mu\in[-1,1]$, $x\geq0$, although it was obtained, unlike in the regular full-range case, from the given radiation $\psi(\mu)$ incident on the boundary at $x=0$ over only half the interval $\mu\in[0,1]$. To obtain the simulated full-range coefficients $\{ b_{i}\}$ and $\{ b_{i}^{-}\}$ of the half-range problem, we observe that there are effectively only half the number of coefficients as compared to a normal full-range problem because $\nu$ is now only over half the full interval. This allows us to generate two sets of equations from (\[Eqn: HRFR\]) by integrating with respect to $\mu\in[-1,1]$ with $\nu$ in the half intervals $[-1,0]$ and $[0,1]$ to obtain the two sets of coefficients $b^{-}$ and $b$ respectively. Accordingly we get from Eq. (\[Eqn: HRFR\_Discrete\]) with $\textrm{ }j=0,1,\cdots,N$ the sets of equations
$$\begin{array}{c}
{\displaysty
| 3,256
| 3,110
| 3,822
| 3,238
| 4,118
| 0.768005
|
github_plus_top10pct_by_avg
|
\
-1/4\\
3/4\end{array}\right)\begin{array}{rr}
-2 & 1\\
0 & -1\\
1 & 0\\
0 & 1\end{array}\right)=\left(\begin{array}{rrrr}
1 & 0 & 0 & 0\\
-3 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
3/2 & -1/4 & 0 & 0\\
1/2 & 3/4 & 0 & 0\end{array}\right).$$
The second matrix on the left is invertible as its rank is $4$. This gives $${\displaystyle G_{\textrm{MP}}=\left(\begin{array}{rrrr}
9/275 & -1/275 & 18/275 & -2/55\\
-27/275 & 3/275 & -54/275 & 6/55\\
-10/143 & 6/143 & -20/143 & 16/143\\
238/3575 & -57/3575 & 476/3575 & -59/715\\
-129/3575 & 106/3575 & -258/3575 & 47/715\end{array}\right)}\label{Eqn: MPEx5}$$
as the Moore-Penrose inverse of $A$ that readily verifies all the four conditions of Eqs. (\[Eqn: MPInverse\]). The basic point here is that, as in the case of a bijective map, $G_{\textrm{MP}}A$ and $AG_{\textrm{MP}}$ are identities on the row and column spaces of $A$ that define its rank. For later use — when we return to this example for a simpler inverse $G$ — given below are the orthonormal bases of the four fundamental subspaces with respect to which $G_{\textrm{MP}}$ is a representation of the generalized inverse of $A$; these calculations were done by MATLAB. The basis for
\(a) the column space of $A$ consists of the first $2$ columns of the eigenvectors of $AA^{\textrm{T}}$: $$\begin{array}{c}
(-1633/2585,-363/892,\textrm{ }3317/6387,\textrm{ }363/892)^{\textrm{T}}\\
(-929/1435,\textrm{ }709/1319,\textrm{ }346/6299,-709/1319)^{\textrm{T}}\end{array}$$
\(b) the null space of $A^{\textrm{T}}$ consists of the last $2$ columns of the eigenvectors of $AA^{\textrm{T}}$:$$\begin{array}{c}
(-3185/8306,\textrm{ }293/2493,-3185/4153,\textrm{ }1777/3547)^{\textrm{T}}\\
(323/1732,\textrm{ }533/731,\textrm{ }323/866,\textrm{ }1037/1911)^{\textrm{T}}\end{array}$$
\(c) the row space of $A$ consists of the first $2$ columns of the eigenvectors of $A^{\textrm{T}}A$: $$\begin{array}{c}
(421/13823,\textrm{ }44/14895,-569/918,-659/2526,\textrm{ }1036/1401)\\
(661/690,\textrm{ }412/1775,\textrm{ }59/2960,-1523/10221,-303/3974)\end
| 3,257
| 4,959
| 3,914
| 3,082
| 650
| 0.803158
|
github_plus_top10pct_by_avg
|
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
vaccines-07-00065-t002_Table 2
######
Environmental Sciences and Research Limited Notification Classification System and criteria \[[@B46-vaccines-07-00065]\].
Classification Definition
---------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Suspect Idiopathic presentation of any paroxysmal cough with whoop, vomit or apnoea
Probable Presentation clinically compatible with pertussis with a high *B. pertussis* IgA test or a significant (fourfold increase in titres) in antibody levels between paired sera at the same laboratory
Confirmed Clinically compatible presentation with either laboratory confirmed pertussis infection (*B. pertussis* only) or epidemiologically linked to a confirmed case
vaccines-07-00065-t003_Table 3
######
ICD10 AM diagnostic codes for pertussis.
ICD10-AM Code Description
--------------- --------------------------------------------
A37.0 Whooping cough due to *B. pertussis*
A37.8 Whooping cough due to *Bordetella* species
A37.9 Whooping cough, unspecified
vaccines-07-00065-t004_Table 4
######
Power calculation.
Exposed (Deprived, NZDep13 Deciles 7--10) Not-Exposed (Not Deprived, NZDep13 Deciles 1--6)
----------------- ------------------------------------------- --------------------------------
| 3,258
| 5,847
| 1,264
| 2,633
| null | null |
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|
uestion is: what is $C$? As at this time no method is available in order to calculate $C$ with a well-defined theoretical uncertainty, we do not employ here a dynamical calculation in order to provide a SM prediction for $C$ and $\Delta a_{CP}^{\mathrm{dir}}$. We rather show the different principal possibilities and how to interpret them in view of the current data. In order to do so we measure the order of magnitude of the QCD correction term $C$ relative to the no QCD limit $\tilde{p}_0=1$. Relative to that limit, we differentiate between three cases
1. $C = \mathcal{O}(\alpha_s/\pi)$: Perturbative corrections to $\tilde p_0$.
2. $C = \mathcal{O}(1)$: Non-perturbative corrections that produce strong phases from rescattering but do not significantly change the magnitude of $\tilde p_0$.
3. $C \gg \mathcal{O}(1)$: Large non-perturbative effects with significant magnitude changes and strong phases from rescattering to $\tilde p_0$.
Note that category (2) and (3) are in principle not different, as they both include non-perturbative effects, which differ only in their size.
Some perturbative results concluded that $C=\mathcal{O}(\alpha_s/\pi)$, leading to $\Delta a_{CP}^{\mathrm{dir}}\sim 10^{-4}$ [@Grossman:2006jg; @Bigi:2011re]. Note that the value $\Delta a_{CP}^{\mathrm{dir}} = 1\times 10^{-4}$, assuming $O(1)$ strong phase, would correspond numerically to $C \sim 0.04$. We conclude that if there is a good argument that $C$ is of category (1), the measurement of $\Delta a_{CP}^{\mathrm{dir}}$ would be a sign of beyond the SM (BSM) physics, because it would indicate a relative $\mathcal{O}(10)$ enhancement.
If the value of $\Delta a_{CP}^{\mathrm{dir}}$ would have turned out as large as suggested by the central value of some (statistically unsignificant) earlier measurements [@Aaij:2011in; @Collaboration:2012qw], we would clearly need category (3) in order to explain that, i.e. penguin diagrams that are enhanced in magnitude, see e.g. Refs. [@Brod:2012ud; @Hiller:2012xm; @Cheng:2012wr; @Feldmann:2012
| 3,259
| 1,863
| 3,416
| 2,916
| 1,554
| 0.78833
|
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|
r, thereby counteracting the effect of the unstable modes. One may similarly expect that stable modes can affect other transport channels such as matter entrainment and heat transport. This line of inquiry will be left for future investigations.
The authors would like to thank F. Waleffe for valuable discussions and insights. Partial support for this work was provided by the Wisconsin Alumni Research Foundation and the U S Department of Energy, Office of Science, Fusion Energy Sciences, under award No. DE-FG02-89ER53291.
Coupling Coefficients
=====================
In Eqs. and , the nonlinear coupling coefficients $C_j,D_j$, which are obtained by expressing the nonlinearities of Eq. in terms of the eigenmode amplitudes $\beta_j$, are as follows: $$\begin{split}
C_1 &= \alpha \left[ ( b_2b_1' + b_1'' ) e^{2|k''|} + ( b_2b_1'' + b_1' ) e^{2|k'|} \right]\\
C_2 &= \alpha \left[ ( b_2b_1' + b_2'' ) e^{2|k''|} + ( b_2b_2'' + b_1' ) e^{2|k'|} \right]\\
C_3 &= \alpha \left[ ( b_2b_2' + b_1'' ) e^{2|k''|} + ( b_2b_1'' + b_2' ) e^{2|k'|} \right]\\
C_4 &= \alpha \left[ ( b_2b_2' + b_2'' ) e^{2|k''|} + ( b_2b_2'' + b_2' ) e^{2|k'|} \right]\\
D_1 &= -\alpha \left[ ( b_1b_1' + b_1'' ) e^{2|k''|} + ( b_1b_1'' + b_1' ) e^{2|k'|} \right]\\
D_2 &= -\alpha \left[ ( b_1b_1' + b_2'' ) e^{2|k''|} + ( b_1b_2'' + b_1' ) e^{2|k'|} \right]\\
D_3 &= -\alpha \left[ ( b_1b_2' + b_1'' ) e^{2|k''|} + ( b_1b_1'' + b_2' ) e^{2|k'|} \right]\\
D_4 &= -\alpha \left[ ( b_1b_2' + b_2'' ) e^{2|k''|} + ( b_1b_2'' + b_2' ) e^{2|k'|} \right],
\end{split}$$ where $$\alpha = \frac{ik|k'||k''|e^{-|k|-|k'|-|k''|}}{2|k|(b_1-b_2)},$$ with $b_j' \equiv b_j(k')$ and $b_j'' \equiv b_j(k'')$. For convenience, the definition of $b_j(k)$ is repeated here: $$b_j = e^{2|k|}\frac{2|k|(\omega_j+k)-k}{k}.$$ Notice that $\alpha(k,k') = \alpha(k,k-k')$ and $C_3(k,k') = C_2(k,k-k')$. Thus, changing the integration variable to $k'' = k-k'$ in the $C_3$ integral yields $$\begin{split}
\int_{-\infty}^{\infty}\frac{dk'}{2\pi}C_3(k,k')\beta_1(k'')\beta_2(k') &= \int_{-\i
| 3,260
| 3,077
| 3,769
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github_plus_top10pct_by_avg
|
_y.$$ The conditions are simply $$\bar{h}_y=0,\ \ h_y=-d\bar{h}_x,$$ which are exactly the relations and .
The other condition is the conservation of the currents[@Hofman:2014loa] $$D_\mu J^\mu_a=0.$$ With $$J^\mu=q^aJ^\mu_a,\ \ \ \bar{J}^\mu=\bar{q}^aJ^\mu_a,$$ we have $$\nabla_\mu J^\mu=0,\ \ \ \nabla_\mu\bar{J}^\mu=0.$$ This implies $$\partial_y \bar{h}_x=0,\ \ \ \partial_y h_x+\partial_x h_y=0.$$ This allows us to define infinitely many conserved charges as in equations and .
In summary, we have shown that the field theory defined on the Newton-Cartan geometry with anisotropic scaling and boost symmetry indeed possess the conservation currents and charges we need. In the following discussion, we denote $\bar{h}_x=M(x)$ and $h_x=T(x,y)$.
Quantization
============
In this section, we consider how to define the theories on the geometry discussed above. We will use the language in terms of operators in the discussion, and we focus on the case with $\ell=d/c$ being integer[^5]. For simplicity, we set $c=1$ such that $d$ is just an integer.
Cylinder Interpretation
-----------------------
The starting point is the so-called canonical cylinder characterized by a spatial circle $\phi$ and a temporal direction $t$ $$(\phi,t)\sim(\phi+2\pi,t).$$ One can get other kinds of spatial circles by tilting $t\rightarrow t+g(x)$. The compactified coordinate is considered in order to eliminate any potential infrared divergence. Now, we define the ‘lightcone coordinates’, $$x=t+\phi,\ \ y=t-\phi.$$ We impose the symmetry on the $x,\ y$ directions as discussed before $$x\rightarrow f(x),\ \ y\rightarrow f'(x)^{d}y$$ and $$x\to x, \ \ y\rightarrow y+g(x),$$ with $f(x)$ and $g(x)$ being arbitrary smooth functions of $x$. Consider the following complex transformation which maps the canonical cylinder to the reference plane $$z=e^{ix}=e^{t_E-i\phi},\ \ \ \tilde{y}=(iz)^dy,$$ where $t_E=-it$ is the Wick-rotated time. We have not considered the tilting of $y$ direction yet. The real time cylinder is capped off at $t=0$ by a ref
| 3,261
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|
]{}, . , , , , , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , , , , , , , & (). . , [ ** ]{}, . , , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , , & (). . , [ ** ]{}, . , , & (). . , [ ** ]{}, . , & (). . , [**]{}, . , & (). . , [ ** ]{}, . (). . , [ ** ]{}, . , , , & (). . , [ ** ]{}, . , , & (). . , [ ** ]{}, . , , , , & (). . , [ ** ]{}, . , , , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , & (). . , [**]{}, . , , , , , , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , & (). . , [**]{}, . , & (). . , [ ** ]{}, . , , & (). . , [ ** ]{}, . (). , [**]{}, . , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , , & (). . , [ ** ]{}, . (). . , [**]{}, . , , , , , , , & (). . , [ ** ]{}, . , , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , , , , & (). . , [ ** ]{}, . , & (). . , [**]{}, . , & (). . , [**]{}, . , & (). . , [**]{}, . , & (). . , [ ** ]{}, . , , , & (). . , [**]{}, . , & (). . , [**]{}, . , , , & (). . , [ ** ]{}, . , & (). . , [**]{}, . , , & (). . , [**]{}, . , & (). . , [ ** ]{}, . , , & (). , [**]{}, . (). . In , & (Eds.), [**]{} (pp. ). : volume . , , , & (). . , [**]{}, . , , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . (). . , [**]{}, . , & (). . , [**]{}, . , & (). . , [ ** ]{}, . (). . , [ ** ]{}, . , , & (). . , [**]{}, . (). . , [ ** ]{}, . , & (). . , [**]{}, . , & (). . , [ ** ]{}, . , , & (). . , [ ** ]{}, . , , & (). . , [ ** ]{}, . , , & (). . , [ ** ]{}, . , , , , , , , & (). . (pp. ). volume . , , , , , , , & (). . , [**]{}, . (). . , [ ** ]{}, . (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , , , , , , , , , & (). . , [ ** ]{}, . , , , , , , , , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , & (). . , [**]{}, . , , , & (). . , [ ** ]{}, . , , , , & (). . , [**]{}, . , , & (). . , [ ** ]{}, . , , & (). . , [ ** ]{}, . , & (). . , [**]{}, . , , & (). . , [**]{}, . , , & (). . , [ ** ]{}, . (). . , [ ** ]{}, . , , & (). . , [**]{}, . , , & (). . , [ ** ]{}, . ()
| 3,262
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|
ned}
\nonumber
\hspace{-.5cm}Z[J]&=&\int dA_0\,dA_{\bot,i} \,\exp\Bigl\{-S_{\rm eff}[A]\\
& &\hspace{.3cm}
+\int d^3 x\,J_0 A_0+\int_T d^4 x \,
J_{\bot,i} A_{\bot,i} \Bigr\}\,.
\label{eq:Zpol}\end{aligned}$$ In we have normalised the temporal component $J_0$ of the current with a factor $\beta$. The classical action $S_{\rm eff}$ is inherently non-local as is contains one-loop terms, the Faddeev-Popov determinant as well as the integration over the longitudinal gauge fields.
Instead of computing $Z[J]$ in we shall compute the effective action $\Gamma$ within a functional renormalisation group approach [@Wetterich:1993yh; @Litim:1998nf; @Schaefer:2006sr; @Berges:2000ew; @Bagnuls:2000ae; @Pawlowski:2005xe]. To that end we introduce an infrared cut-off for the transversal spatial gauge fields and in the temporal gauge fields by modifying the action, $S\to S_{\rm eff}+\Delta S_k[A_0]+\Delta S_{\bot,k}[\vec
A_{\bot}]$, with infrared scale $k$, and cut-off terms $$\begin{aligned}
\nonumber
\Delta S_k[A_0]&=&\012 \beta \int d^3 x \,
A_0\, R_{0,k}\, A_0\\
\Delta S_{k,\bot}[\vec A_{\bot}]&=&
\int_T d^4 x\, A^a_{\bot, i}\,
R_{\bot,k}\, A^a_{\bot, i}\,.
\label{eq:Cutoff} \end{aligned}$$ The regulators $R_k$ in are chosen to be momentum-dependent and required to provide masses at low momenta and to vanish at large momenta. For $k\to 0$ they vanish identically.
They can be written as one single regulator $R_{A,{\mu\nu}}$, which is a block-diagonal matrix in field space with entries $R_{A,{00}}=R_{0,k}$ and $R_{A,{ij}}=R_{\bot,k}
\Pi_{\bot,ij}$, where the transversal projector is defined by $$\begin{aligned}
\label{eq:transverse}
\Pi_{\bot,ij} = \delta_{ij} -\frac{p_i p_j}{\vec p^2}\,.\end{aligned}$$ The above structure is induced by the fact the $A_{\bot,i}$ are transversal, and hence $R_{\bot,k}$ only couples to the transversal degrees of freedom.
The flow of the cut-off dependent effective action $\Gamma_k$ is then given by Wetterich’s equation [@Wetterich:1993yh; @Berges:2000ew; @Bagnuls:2000ae] for Yang-
| 3,263
| 2,445
| 3,105
| 2,814
| 2,391
| 0.780256
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|
s problem has been recently discussed in the literature Ref. [@Hart], which provided numerical solutions of the Lorentz-Dirac equation. The integration was performed backwards in time so that the unphysical, exponentially growing homogeneous solutions of LD would damp out, resulting in a numerical stable solution.
We are going to show that Lorentz-Dirac equation of motion can be integrated forward in time with conditions specified at $t=0$. The idea is to construct the series solution of LD equation. The initial acceleration is then provided by replacing in the series the velocity by their initial value at the instant when the external force is applied. It is easy to show that the solution obtained with this procedure, when extrapolated to the distant future, satisfies Rohrlich condition. However, we still have to cope with the existence of the unphysical runaway solutions which although formally eliminated troubles the process of numerical integration. Then, by combining the recursive use of the series solution with implicit methods of numerical integration we show that the process of integration forward in time can be performed.
We write Lorentz-Dirac (LD) equation of motion as $$F_{\mu }^{ext}=a_{\mu }-\epsilon \left( \frac{d^{2}v_{\mu }}{d\tau ^{2}}+v_{\mu }a_{\lambda }a^{\lambda }\right) \label{2}$$ where $v_{\mu }$,$a_{\mu }$ and $F_{\mu }^{ext}$ are, respectively, the four-vector components of the velocity,acceleration and of the external force given explicitly by
$$v_{\mu }=\gamma \left( 1,{\bf \beta }\right) , \label{4}$$
$$v^{\mu }=\gamma \left( 1,-{\bf \beta }\right) , \label{4a}$$
$$a_\mu =\frac{dv_\mu }{d\tau }$$
and
$$F_{\mu }^{ext}=\gamma \left( {\bf \beta }\cdot {\bf F}_{ext},{\bf F}_{ext}\right) . \label{6}$$
In these equations, $\tau $ is the dimensionless proper time $d\tau =\omega
_{0}dt/\gamma $, $\gamma $ is the relativistic factor $\gamma =\frac{1}{\sqrt{1-\beta ^{2}}}$ with ${\bf \beta }=\frac{1}{c}\frac{d{\bf r}}{dt}$ and $\omega _{0}$ is the frequency of the laser pulse wit
| 3,264
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|
0.30 (0.14)[\*](#table-fn11-0192513X17710773){ref-type="table-fn"} 0.25
Years in the Netherlands 0.11 (0.05)[\*](#table-fn11-0192513X17710773){ref-type="table-fn"} 0.28
Housing^[e](#table-fn9-0192513X17710773){ref-type="table-fn"}^ 0.28 (0.45) 0.07
Number of friends in the Netherlands 0.05 (0.11) 0.04
Number of family members in the Netherlands −0.15 (0.20) −0.08
Dutch proficiency −0.07 (0.21) −0.04
Child \< 8 years of age^[f](#table-fn9-0192513X17710773){ref-type="table-fn"}^ −0.01 (0.46) −0.00
Amount of contact with child −0.41 (0.16)[\*](#table-fn11-0192513X17710773){ref-type="table-fn"} −0.28
*R* ^2^ 0.44
*Note. SE* = standard error. Superscripts indicate reference categories that include (a) unhappy; (b) no work-to-family conflict; (c) male; (d) married/in a relationship; (e) room, student housing, institution, other; and (f) no children \< 8 years of age.
*Source*. TCRAf-Eu Angolan parent survey, The Netherlands 2010-2011.
*p* \< .05. \*\**p* \< .01. \*\*\**p* \< .001 (one-tailed test).
Discussion and Conclusions {#section12-0192513X17710773}
==========================
This article aimed to complement research on transnational families by investigating absenteeism and job stability of transnational parents. While the growing b
| 3,265
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| 3,070
| 3,363
| 2,003
| 0.783612
|
github_plus_top10pct_by_avg
|
\psi
_{\mu }\partial ^{\rho }g_{z})\big\Vert_{q_{2},\infty } \\
& \leq C\psi _{\eta ^{\prime }}(x)\sup_{z\in {\mathbb{R}}^{d}}\big\Vert %
Q_{1}(\psi _{\mu }\partial ^{\rho }g_{z})\big\Vert_{q_{2},-\eta ,\infty }.\end{aligned}$$Using (\[h\]) $j-1$ times and (\[B2\]) (with $\kappa =\mu )$ we get $$\begin{aligned}
\left\Vert Q_{1}(\psi _{\mu }\partial ^{\rho }g_{z})\right\Vert
_{q_{2},-\eta ,\infty } &\leq &C_{q_{2}^{\prime },\eta ,\infty
}^{j-1}(U,S)\Vert S_{\frac{1}{2}\delta _{j}}(\psi _{\mu }\partial ^{\rho
}g_{z})\Vert _{q_{2}^{\prime },-\eta ,\infty } \\
&\leq &C_{q_{2}^{\prime },\eta ,\infty }^{j-1}(U,S)\left\Vert
g_{z}\right\Vert _{\infty }\,C\Big(\frac{2m}{\lambda t}\Big)^{\theta
_{0}(q_{2}^{\prime }+d+\theta _{1})}.\end{aligned}$$Since $\left\Vert g_{z}\right\Vert _{\infty }=1$ we obtain$$\Vert p_{1}^{\beta ,x}\Vert _{0,\nu ,1}\leq \psi _{\eta }(x)C_{q_{2}^{\prime
},\eta ,\infty }^{j-1}(U,S)\,C\Big(\frac{2m}{\lambda t}\Big)^{\theta
_{0}(q_{2}^{\prime }+d+\theta _{1})}.$$By inserting in (\[h9\]) we obtain (\[h7\]), so the proof is completed. $%
\square $
Proofs of the main results {#sect:proofs}
==========================
In the present section, we use the results in Section \[sect:reg\] in order to prove Theorem \[Transfer\] (Section \[sect:proofTransfer\]) and Theorem \[J\] (Section \[sect:proofJ\]).
Proof of Theorem \[Transfer\] {#sect:proofTransfer}
-----------------------------
**Step 0: constants and parameters set-up.** In this step we will choose some parameters which will be used in the following steps. To begin we stress that we work with measures on ${\mathbb{R}}^{d}\times {\mathbb{R}}%
^{d}$ so the dimension of the space is $2d$ (and not $d).$ We recall that in our statement the quantities $q,d,p,\delta _{\ast },\varepsilon _{\ast
},\kappa $ and $n$ are given and fixed. In the following we will denote by $%
C $ a constant depending on all these parameters and which may change from a line to another. We define $$m_{0}=1+\Big\lfloor\frac{q+2d/p_{\ast }}{\delta _{\ast }}\Big\rfloor>0
\label{H4}$
| 3,266
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| null | null |
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|
valent to $\left( B^\top \otimes A\right)
\mathrm{vec}(X) = \mathrm{vec}(C)$, with $B=C = \Omega$ and $X = A =
I_k$.
We now bound $\sigma_1 \Big([I_{k^2}\otimes \alpha^\top \otimes I_k] \; J \; [ \Omega\otimes\Omega
\;\;\;\; 0_{k^2\times k}] \Big)$, the second matrix in the upper block in . We have that $$\begin{aligned}
\sigma_1(J) & \leq
2\sigma_1( (I_k \otimes \Omega \otimes I_{k^2})(I_k \otimes K_{k,k}\otimes I_k)(I_{k^2}\otimes {\rm vec}(I_k))\\
&=
2\sigma_1(\Omega)||I_k||_F\\
&= 2\sqrt{k}\sigma_1(\Omega), \end{aligned}$$ since $\sigma_1(K_{k,k}) = 1$. Hence, using the fact that $\sigma_1([I_{k^2}\otimes \alpha^\top \otimes I_k]) = ||\alpha||$, $$\sigma_1 \Big([I_{k^2}\otimes \alpha^\top \otimes I_k] \; J \; [ \Omega\otimes\Omega
\;\;\;\; 0_{k^2\times k}] \Big) \leq
2\sqrt{k} ||\alpha|| \sigma_1^3(\Omega) \leq 2 \sqrt{A U} \frac{k}{u^3},$$ since $\| \alpha \| \leq \sqrt{A U k}$. Thus, we have obtained the following bound for the largest singular value of the matrix $H$ in : $$\sigma_1(H)\leq
C \Big( \frac{1}{u^2} + \frac{\sqrt{k}}{u^2}+ \frac{k}{u^3} \Big),$$ where $C$ is a positive number depending on $A$ only. Putting all the pieces together, $$\begin{aligned}
\sigma_1(H_j) &=
\sigma_1\left( \frac{1}{2}((I_b \otimes e_j)H + H^\top (I_b\otimes e_j))\right)\\
& \leq
\sigma_1((I_b \otimes e_j)H)\\
& \leq
\sigma_1(I_b)\sigma_1(e_j)\sigma_1(H)\\
& \leq C \Big( \frac{1}{u^2} + \frac{\sqrt{k}}{u^2}+ \frac{k}{u^3} \Big).\end{aligned}$$ Whenever $u \leq \sqrt{k}$, the dominant term in the above expression is $\frac{k}{u^3}$. This gives the bound on $\overline{H}$ in (\[eq::B-and-lambda\]). The bound on $\underline{\sigma}$ given in follows from . Indeed, for every $P \in \mathcal{P}^{\mathrm{OLS}}$ $$\min_j \sqrt{ G_j V G_j^\top}
\geq \sqrt{v} \min_j \| G_j \|.$$ Then, using , $$\min_j \| G_j \| \geq \min_j \| \Omega_j \| \geq \lambda_{\min}(\Omega) = \frac{1 }{
U },$$ where $\Omega_j$ denotes the $j^{\mathrm{th}}$ row of $\Omega$.
The final value of the constant $C$ depends only
| 3,267
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|
$} for cavity $S_2$. The corresponding process fidelity $F_\mathrm{CNOT\_ED}$ ($F_\mathrm{ED}$) is 0.829 (0.857).[]{data-label="fig:fig3"}](Figure3_final.pdf)
The experiments presented in this work are based on two circuit quantum electrodynamics (QED) devices [@Wallraff; @Clarke2008Superconducting; @You2011Atomic; @Schoelkopf2013; @Gu2017Microwave]. Device A, on which single-cavity geometric phase gates are performed, consists of two transmon qubits simultaneously dispersively coupled to two three-dimensional cavities [@Paik; @Kirchmair; @Vlastakis; @Liu2017; @Wang2017]. The parameters and architecture setup are described in Ref. [@Xu2018]. Device B, on which two-cavity geometric phase gates are performed, consists of three transmon qubits dispersively coupled to two cylindrical cavities [@Reagor2016] and three stripline readout cavities [@axline2016an]. The device parameters are described in Ref. [@Supplement]. In Device A, the coupling between the qubit ($Q_1$) used to produce the geometric phase and the cavity used to encode this phase is described by the Hamiltonian $$H=-\hbar \chi _{\mathrm{qs}}a^{\dagger }a{\ensuremath{\left|e\right\rangle}}{\ensuremath{\left\langlee\right|}},$$ where $\chi_{\mathrm{qs}}$ denotes the qubit frequency shift induced by per photon, $a^{\dagger }$ and $a$ are the creation and annihilation operators for the particular cavity field respectively, and ${\ensuremath{\left|e\right\rangle}}$ $({\ensuremath{\left|g\right\rangle}})$ is the excited (ground) state of the qubit. In Device B, the qubit, commonly coupled to two cavities used to store the photonic qubits, undergoes a frequency shift dependent on the photon numbers of both cavities.
The geometric manipulation technique is well exemplified with the even cat state $\left({\ensuremath{\left|\alpha\right\rangle}} + {\ensuremath{\left|-\alpha\right\rangle}} \right)/\sqrt{2}$, where ${\ensuremath{\left|\alpha\right\rangle}}$ and ${\ensuremath{\left|-\alpha\right\rangle}}$ are coherent states with $\left\langle \alpha | -\a
| 3,268
| 3,086
| 3,570
| 3,282
| 1,427
| 0.789732
|
github_plus_top10pct_by_avg
|
{t\})$ and maximal tori $T_1$ and $T_2$ of $N$, of orders divisible by $r$ and $s$, respectively, with $( |T_1|, |T_2|)=1$, as stated in Table 3.
In the cases denoted by $(\star)$, $r$ and $s$ denote the largest prime divisor of $|T_1|$ and $|T_2|$, respectively.
The Tits group $N=F_4(2)'$ contains a Sylow 13-subgroup of order 13 which is self-centralising.
$$\begin{array}{c|c|c|c|c|c}
\hline
& & & & & \\
N& r &s& |T_1|& |T_2| & Remarks \\
& & & & & \\
\hline
& & & & & \\
G_2(q)& q_3 &q_{6} & q^2+q+1&q^2-q+1& q\neq 4\\
q > 2 & r=7 & & & & q=4\\
& & & & & \\
\hline
& & & & & \\
F_{4}(q) & q_{8} & q_{12} &q^{4}+1 & q^4-q^2+1& \\
& & & & & \\
\hline
& & & & & \\
E_6(q)& q_{9} & q_{12} & \frac{q^{6}+q^3+1}{(3, q-1)}& \frac{(q^4-q^2+1)(q^2+q+1)}{(3, q-1)}& \\
& & & & & \\
\hline
& & & & & \\
E_7(q)& q_{9} & q_{14} & \frac{(q^6+q^3+1)(q-1)}{(2, q-1)} & \frac{q^{7}+1}{(2, q-1)}& \\
& & & & & \\
\hline
& & & & & \\
E_8(q)& q_{20} & q_{24} & q^8-q^{6}+q^4-q^2+1& q^8-q^4+1& \\
& & & & & \\
\hline
& & & & & \\
{}^3D_4(q)& q_{3} & q_{12} &(q^{3}-1)(q+1)& q^4-q^2+1& \\
& & & & & \\
\hline
& & & & & \\
{}^2B_2(q) & r & s & q+\sqrt{2q}+1& q-\sqrt{2q}+1& (\star) \\
q=2^{2m+1} >2 &&&&&\\
& & & & & \\
\hline
& & & & & \\
{}^2G_2(q) & r & s & q+\sqrt{3q}+1& q-\sqrt{3q}+1& (\star) \\
q=3^{2m+1} >3 &&&&&\\
& & & & & \\
\hline
& & & & & \\
{}^2F_4(q) & r & s & \small{q^2+q \sqrt {2q}+q+\sqrt{2q}+1}& (q-\sqrt{2q}+1)(q-1)& (\star) \\
q=2^{2m+1} >2 &&&&&\\
& & & & & \\
\hline
& & & & & \\
{}^{2}E_6(q)& q_{18} & q_{12} & \frac{q^{6}-q^3+1}{(3, q+1)}& \frac{(q^4-q^2+1)(q^2-q+1)}{(3, q+1)}& \\
& & & & & \\
\hline
\end{array}$$
The existence of the subgroups $T_1$ and $T_2$ appearing in Table 3 can be derived from the information about the maximal tori in these groups (see [@VV Lemma 1.3] and [@VV2 Lemma 2.6]).
The fact that they are coprime can be deduced from Lemma \[cuentas\] having in mind that $|T_i|$ divides $q^n-1$ when we state that $q_n \in \pi(T_i)$, $i=1, 2$, (for the case ${}^3D_
| 3,269
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github_plus_top10pct_by_avg
|
formReady(); //ERROR: The method formReady is not defined for the class _addImageState
},
)
);
}
}
How do i call the function formReady from the widget addImage ?
EDIT:
I tried this code, and removed all compile errors however still not getting the desired result:
in addImage.dart
TextField(
onChanged: (text){
addImage().formReady;
}
)
in Itembought.dart
addImage(formReady: (){
ItemBought().formReady;
}
in task.dart
ItemBought(formReady: (){
this.formReady();
}),
A:
class task extends StatelessWidget {
@override
....
}
}
class MyHomePage extends StatefulWidget {
MyHomePage({Key key, this.title}) : super(key: key);
final String title;
@override
_MyHomePageState createState() => _MyHomePageState();
}
class _MyHomePageState extends State<MyHomePage> {
@override
Widget build(BuildContext context) {
return Scaffold(
appBar: AppBar(
title: Text(widget.title),
),
body: Center(
child: Column(
mainAxisAlignment: MainAxisAlignment.center,
children: <Widget>[
new ItemBought(callback: (data){
String receivedData = data;
},)
],
),
),
);
}
}
class ItemBought extends StatefulWidget {
final void Function(String data) callback;
const ItemBought({Key key, this.callback}) : super(key: key);
@override
_ItemBoughtState createState() => _ItemBoughtState();
}
class _ItemBoughtState extends State<ItemBought> {
@override
Widget build(BuildContext context) {
return Container(
child: new FlatButton(onPressed: (){
widget.callback("Item Id Or name");
}, child: new Text("Bought Item"),),
);
}
}
Q:
Safe way to create singleton with init method in Objective-C
I would like to take the GCD approach of using s
| 3,270
| 3,202
| 157
| 1,944
| 143
| 0.822858
|
github_plus_top10pct_by_avg
|
times S$ \[et\] \_[(x,)(y,),(x,)GS]{}t(x,)=y,+|y,|. For $(y,\omega)\in \Gamma'_+$ the projection ${\left\langle}y,\omega{\right\rangle}$ is non-negative (and then the limit (\[et\]) is $\tau_+(y,\omega)$) and for $(y,\omega)\in \Gamma'_-$ it is non-positive and then the limit (\[et\]) is $0$). Hence $\ol t:\ol G\times S\to{\mathbb{R}}$ is continuous.
Due to the Proposition \[prop-ex\] we can (almost everywhere) uniquely set $t(y,\omega)=0$ for $(y,\omega,E)\in\Gamma_{-}$. For $(y,\omega,E)\in \Gamma_+$ we set $t(y,\omega):=\tau_+(y,\omega)$. For further (unexplained) notations we refer to [@tervo14].
In the sequel we denote for $k\in{\mathbb{N}}_0$ $$C^k(\ol G\times S\times I):=\{\psi\in C^k(G\times S\times I^\circ)\ |\ \psi\ {\rm has\ continuous\ partial\ derivatives\ on}\ \ol G\times S\times I\}$$ and $$D^k(\ol G\times S\times I):=\{\psi\in C^k(G\times S\times I^\circ)\ |\ \psi=f_{|G\times S\times I^\circ},\ f\in C_0^k({\mathbb{R}}^n\times S\times{\mathbb{R}})\}.$$ It is well known that these spaces are equal, i.e. $C^k(\ol G\times S\times I)=D^k(\ol G\times S\times I)$, for a given $k$, if the boundary $\partial G$ of $G$ is of class $C^k$ (see [@friedman Part 1, Lemma 5.2]). Thus, in particular, since our standing assumption in this paper is that $\ol{G}$ is (at least) of class $C^1$, we have $$C^1(\ol G\times S\times I)=D^1(\ol G\times S\times I).$$
Define the (Sobolev) space $W^2(G\times S\times I)$ by \[fseq1\] W\^2(GSI) ={L\^2(GSI) | \_x L\^2(GSI) } and its subspace $W^2_1(G\times S\times I)$ by \[fseq2\] W\^2\_1(GSI) ={W\^2(GSI) | L\^2(GSI)}. Here $\omega\cdot\nabla_x\psi$ and ${{\frac{\partial \psi}{\partial E}}}$ are understood in the distributional sense. In what follows, $\omega\cdot\nabla_x\psi$ will stand for (the distribution) $\Omega\cdot\nabla_x \psi$, where $\Omega:G\times S\times I\to{\mathbb{R}}^3$; $\Omega(x,\omega,E)=\omega$. The spaces $W^2(G\times S\times I)$, $W^2_1(G\times S\times I)$ are equipped with the inner products, respectively \[fs4\] ,v\_[W\^2(GSI)]{}=,v\_[L\^2(GSI)]{}+
| 3,271
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|
athbb{C}}} W_{\alpha}(M).$
This observation has two useful consequences. First, if $\theta:M_1\rightarrow M_2$ is an $H_c$-module homomorphism with $M_i\in {\mathcal{O}}_c$, then $\theta
(W_{\alpha}(M_1)) \subseteq W_{\alpha}(M_2)$ for each $\alpha\in{\mathbb{C}}$. Secondly, if $p\in H_c$ has $\operatorname{{\mathbf{E}}\text{-deg}}p = t$, then , implies that $p\cdot W_{\alpha}(M) \subseteq
W_{\alpha+t}(M)$. Note that the standard module $\Delta_c(\mu)$ is therefore a lowest weight module since it is generated as a ${\mathbb{C}}[{\mathfrak{h}}]$-module by the space $1\otimes \mu$.
{#fkdeg}
To describe the graded structure of the standard modules we need a little notation. Recall that the space of coinvariants ${\mathbb{C}}[{\mathfrak{h}}]^{\text{co} {{W}}} = {{\mathbb{C}}[{\mathfrak{h}}]}\big/{{\mathbb{C}}[{\mathfrak{h}}]_+^{{{W}}}{\mathbb{C}}[{\mathfrak{h}}]}$ is a finite dimensional graded algebra isomorphic as a ${{W}}$-module to the regular representation. As in [@op], the polynomials $$\label{fakedegrees} f_{\mu}(v) = \sum_{i\geq 0}
[{\mathbb{C}}[{\mathfrak{h}}]^{\text{co} {{W}}}_i : \mu] v^i$$ are called the *fake degrees*\[fake-defn\] of $\mu\in{{\textsf}{Irrep}({{W}})}$. We define $n(\mu)$ to be the lowest power of $v$ appearing in $f_{\mu}(v)$; thus, $f_{\mu}(v) = a
v^{n(\mu) }+ \text{higher order terms.}$ In the notation of [@hai3], $n(\mu)$ is equal to the [*partition statistic*]{} $ \sum_{i}\mu_i (i-1)$ (see the proof of [@babyv Theorem 6.4]). Finally, implies that $$\label{fakedegrees2}
f_{\mu^t}(1)=\dim \mu^t = \dim \mu = f_\mu(1)
\qquad\rm{for}\ \mu\in {{\textsf}{Irrep}({{W}})}.$$
{#subsec-3.10}
Given a graded ${{W}}$-module $M=\sum_{\alpha\in{\mathbb{C}}}
W_\alpha(M)$ we define its [*graded Poincaré series*]{} to be $$p(M,v,{{W}}) = \sum_{\alpha\in {\mathbb{C}}} v^\alpha \sum_{\lambda\in {{\textsf}{Irrep}({{W}})}}
[W_\alpha(M) : \lambda][\lambda].$$\[W-poincare\] This is easily determined for standard modules.
[(1)]{} Under the canonical grading, the subspace $1\otimes \mu$ of $\Delta_
| 3,272
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| 0.772153
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|
\quad K_{2,n} = C A \sqrt{ k U \frac{\log k + \log n}{n} },$$ with $C = C(\eta)>0$ the constant in . Assume that $n$ is large enough so that $v_n = v - K_{1,n}$ and $u_n = u -K_{2,n}$ are both positive. Then, for a constant $C = C(A)>0$, $$ \inf_{w_n \in \mathcal{W}_n} \inf_{P\in {\cal
P}^{\mathrm{OLS}}_n}\mathbb{P}(\beta_{{\widehat{S}}} \in C^*_{{\widehat{S}}}) \geq 1-\alpha -
C\left(\Delta^*_{n,1} + \Delta^*_{n,2} + \Delta_{n,3} \right),$$ where $C^*_{{\widehat{S}}}$ is either one of the bootstrap confidence sets in , $$\Delta^*_{n,1} = \frac{1}{\sqrt{v_n}}\left( \frac{
k^2 \overline{v}_n^2 (\log kn)^7)}{n}\right)^{1/6} , \quad \Delta^*_{n,2}
= \frac{ U_n }{ \sqrt{v_n}} \sqrt{ \frac{k^4 \overline{v}_n \log^2n \log
k}{n\,u_n^6}}$$ and $\Delta_{n,3}$ is as in .
[**Remark.**]{} The term $\Delta_{n,3}$ remains unchanged from the Normal approximating case since it arises from the Gaussian comparison part, which does not depend on the bootstrap distribution.
[**Remark.**]{} It is important that we use the pairs bootstrap — where each pair $Z_i=(X_i,Y_i)$, $i=\mathcal{I}_{2,n}$, is treated as one observation — rather than a residual based bootstrap. In fact, the validity of the residual bootstrap requires the underlying regression function to be linear, which we do not assume. See [@buja2015models] for more discussion on this point. In both cases, the Berry-Esseen theorem for simple convex sets (polyhedra with a limited number of faces) with increasing dimension due to [@cherno1; @cherno2] justifies the method. In the case of $\beta_{{\widehat{S}}}$ we also need a Taylor approximation followed by an application of the Gaussian anti-concentration result from the same reference.
The coverage rates from are of course no better than the ones obtained in , and are consistent with the results of [@el2015can] who found that, even when the linear model is correct, the bootstrap does poorly when $k$ increases.
The coverage accuracy can also be improved by changing the bootstrap procedure; see Sect
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| 0.770167
|
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|
}, \overrightarrow{F(x)f(\xi_3)} > \\
c_1 = < \overrightarrow{x\xi_2}, \overrightarrow{x\xi_3} > \ , & \ c_2 = < \overrightarrow{F(x)f(\xi_2)}, \overrightarrow{F(x)f(\xi_3)} > \\\end{aligned}$$ Taking inner products of the left-hand side of equation (\[no1\]) above with the vectors $\overrightarrow{x\xi_i}, i = 1,2,3$, and taking inner products of the left-hand side of equation (\[no2\]) above with the vectors $\overrightarrow{F(x)f(\xi_i)}, i = 1,2,3$, we find that the vectors $\overrightarrow{u_i} := (a_i, b_i, c_i), i = 1,2$ both satisfy the same linear system of equations $$T \overrightarrow{u} = \overrightarrow{w}$$ where $T$ is the $3 \times 3$ matrix $$T = \begin{pmatrix}
\alpha_2 & \alpha_3 & 0 \\
\alpha_1 & 0 & \alpha_3 \\
0 & \alpha_1 & \alpha_2
\end{pmatrix}$$ and $\overrightarrow{w}$ is the column vector $$\overrightarrow{w} = \begin{pmatrix}
-\alpha_1 \\
-\alpha_2 \\
-\alpha_3
\end{pmatrix}$$ A computation gives $\det(T) = -2\alpha_1 \alpha_2 \alpha_3 < 0$, so $T$ is nonsingular, and it follows that $\overrightarrow{u_1} = \overrightarrow{u_2}$. We thus have $$< \overrightarrow{x\xi_i}, \overrightarrow{x\xi_j} > = < \overrightarrow{F(x)f(\xi_i)}, \overrightarrow{F(x)f(\xi_j)} >$$ for all $1 \leq i,j \leq 3$. On the other hand, by Proposition \[dFstar\], we have $$< \overrightarrow{x\xi_i}, \overrightarrow{x\xi_j} > = < dF_x(\overrightarrow{x\xi_i}), \overrightarrow{F(x)f(\xi_j)} >$$ for all $1 \leq i,j \leq 3$, thus $$\label{no3}
< dF_x(\overrightarrow{x\xi_i}), \overrightarrow{F(x)f(\xi_j)} > = < \overrightarrow{F(x)f(\xi_i)}, \overrightarrow{F(x)f(\xi_j)} >$$ for all $1 \leq i,j \leq 3$. Since the dimension of $X$ is two, the span of the vectors $\overrightarrow{x\xi_1}, \overrightarrow{x\xi_2}$ equals $T_x X$, thus since $dF^*_x$ is an isomorphism it follows from Proposition \[dFstar\] that the span of the vectors $\overrightarrow{F(x)f(\xi_1)}, \overrightarrow{F(x)f(\xi_2)}$ equals $T_{F(x)} Y
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| 0.774533
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|
_N=\c[t_1, \ldots, t_N]*\mathbb Z^N$, where the group $\mathbb Z^N$ is generated by the elements $\sigma_i$, $i=1, \ldots, N$ as above. For each $i=1, \ldots, N$ and any $c\in k$ consider the involutions $\epsilon_{R_N,c,i}^{\pm}$ on $R_N$ defined by $\epsilon_{R_N,c,i}^{\pm}(\sigma_i)=\pm\,\sigma_i^{-1}$, $\epsilon_{R_N,c,i}^{\pm}(\sigma_j)=\sigma_j$ if $i\neq j$, $\epsilon_{R_N,c,i}^{\pm}(t_i)=c-t_i$ and $\epsilon_{R_N,c,i}^{\pm}(t_j)=t_j$ if $j\neq i$.
\[lemma-on-transformation-inversion-mult-inversion-add\] Let $\Tilde{A_{N}}$ be the localisation of the Weyl algebra $A_N$ by the multiplicative set generaled by $\{x_1, \ldots, x_N\}$. The homomorphism $\phi^{\pm}_{c}:\Tilde{A_{N}}\rightarrow R_N$, given by $$\phi^{\pm}_{c}(x_i)=\sigma_i,\ \phi^{\pm}_{c}(\partial_i)=\big(t_i+1-\dfrac{c}{2}\big)\sigma_i^{-1}+ (1\mp\sigma_i^{-2}),$$ $i=1, \ldots, N$ is an isomorphism of algebras with involutions.
It is sufficient to consider the case $N=1$. Set $R=R_1$. Since $\phi^{\pm}_{c}(\partial x - x \partial)=t-\sigma t\sigma^{-1}=1$, $\phi^{\pm}_{c}$ are homomorphisms, $(\phi^{\pm}_{c})^{-1}(\sigma)=x$ and $(\phi^{\pm}_{c})^{-1}(t)=\big(\partial x+\dfrac{1}{2}\big)+(\dfrac{1}{x}\mp x).$ We also have $$\epsilon_{R,c}^{\pm}\phi^{\pm}_{c}(\partial)=\mp\big(t-1-\dfrac{c}{2}\big)\sigma+ (1\mp\sigma^{2})$$ and $$\phi^{\pm}_{c}\epsilon_{\tilde{A}_{1},c}^{\pm}(\partial)= \mp\sigma^{2}
\bigg(\big(t+1-\dfrac{c}{2}\big)\sigma^{-1}+ (1\mp\sigma^{-2}) \bigg)=\mp\big(t-1-\dfrac{c}{2}\big)\sigma+ (1\mp\sigma^{2}).$$
For integers $n\geq 1$, $m\geq 0$ denote $A_{n,m}$ the $n$-th Weyl algebra over the field of rational functions $\c(z_1, \ldots, z_m)$. Then $A_{n,m}$ admits the skew field of fractions $F_{n, m}\simeq F_n\otimes \c(z_1, \ldots, z_m)$. We have
\[lem-skew-main\] Let $\mathcal K=(L*\mathcal M)^G$ be a linear Galois algebra where $G=G_N$ is a classical Weyl group and
- $L=\c(t_{1},\dots, t_{N})$;
- $G$ acts naturally on $\mathcal K$;
- $\mathcal M\simeq \mathbb Z^{n}$ acts by shifts on $t_{1},\dots, t_{n}$, $n\leq
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| 2,779
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|
t take into account the change of angle for the (new) primary electron during transport. Hence the angular derivative ($\nabla_\omega$) is missing from it. On the other hand, CSDA-Focker-Plank approximation contains also second order partial derivatives (with respect to angle) which do not show up in (\[n2k-6\]).
As an application of the above analysis we derive a CSDA-type approximation for the [Møller]{} scattering. We apply the hyper-singular integral form of the collision operator $K_{22}$ that is (recall \[k22\])) $$\begin{gathered}
\label{csda1}
(K_{22}\psi)(x,\omega,E)=
{{{\mathcal{}}}H}_2((\ol{{{\mathcal{}}}K}_{22,2}\psi)(x,\omega,\cdot,E))(E) \\
+
{{{\mathcal{}}}H}_1((\ol{{{\mathcal{}}}K}_{22,1}\psi)(x,\omega,\cdot,E))(E)
+
\int_{I'}(\hat{{{\mathcal{}}}K}_{22,0}\psi)(x,\omega,E',E)
dE'.\end{gathered}$$ Recall also that the operators $\ol {{{\mathcal{}}}K}_j$, $j=0,1,2$ are $$(\ol {{{\mathcal{}}}K}_{22,j}\psi)(x,\omega,E',E)=
\hat\sigma_j(x,E',E)
\int_{S'}
\delta(\omega'\cdot\omega-\mu_{22}(E,E'))\psi(x,\omega',E')d\omega'$$ and $$(\hat {{{\mathcal{}}}K}_{22,j}\psi)(x,\omega,E',E)
=\chi_{22}(E',E)(\ol {{{\mathcal{}}}K}_{22,j}\psi)(x,\omega,E',E).$$
In the case where $E\approx E'$ (as in the case of forward peaked primary electrons) we have \[csda2-a\] \_[22,p]{}(E’,E)1 and then for $E\leq E'\leq 2E$ \[csda3-aa\] (\_[22,j]{})(x,,E’,E) & \_j(x,E’,E) \_[S’]{} (’- 1)(x,’,E’)d’\
=& \_j(x,E’,E) (x,,E’). Assuming (\[csda2-a\]), we have an approximation $$\begin{gathered}
\label{csda4}
(K_{22,0}\psi)(x,\omega,E)\approx (\tilde K_{22,0}\psi)(x,\omega,E)
\\
:=\int_{E}^{2E}\hat\sigma_0(x,E',E)
\psi(x,\omega,E')dE'+\int_{2E}^{E_m}(\ol {{{\mathcal{}}}K}_{22,0}\psi)(x,\omega,E',E).\end{gathered}$$ Eq. is the [*first CSDA-type approximation*]{}. When we apply the approximation \[d-approx\] (’-\_[22]{}(E,E’))\_(’-\_[22]{}(E,E’)) we immediately see that $\tilde K_{22,0}$ is the usual partial Schur integral operator.
Consider now the term ${{{\mathcal{}}}H}_1\big((\ol{{{\mathcal{}}}K}_{22,1
| 3,276
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|
ic Oxide Gas Not yet recruiting \-
Efficacy and Safety of IFN-a2b in the Treatment of Novel Coronavirus Patients Recombinant human interferon α1β Not yet recruiting \-
Evaluating and Comparing the Safety and Efficiency of ASC09/Ritonavir and Lopinavir/Ritonavir for Novel Coronavirus Infection 1\. ASC09/ritonavir group Not yet recruiting \-
2\. Lopinavir/ritonavir group
Safety and Immunogenicity Study of 2019-nCoV Vaccine (mRNA-1273) to Prevent SARS-CoV-2 Infection mRNA-1273
| 3,277
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|
}}$, $\chi(i) \in \Psi(i)$. Since $R \subseteq {{\operatorname{dom}{\Psi}}}$, then for each $r \in R$, $\xi(r) \in \Psi(r)$. Consider ${{\Psi}\negmedspace\mid\negmedspace{R}}$, for which ${{\operatorname{dom}{({{\Psi}\negmedspace\mid\negmedspace{R}})}}} = R$. By definition of restriction, for $r \in R$, $({{\Psi}\negmedspace\mid\negmedspace{R}})(r) = \Psi(r)$. Since $\xi(r) \in \Psi(r)$ and $\Psi(r) = ({{\Psi}\negmedspace\mid\negmedspace{R}})(r)$, then for any $r \in R$, $\xi(r) \in ({{\Psi}\negmedspace\mid\negmedspace{R}})(r)$ – that is, $\xi$ is a choice of ${{\Psi}\negmedspace\mid\negmedspace{R}}$. From the preceding, $\xi \in (\thinspace\prod\Psi) \mid R$ implies $\xi \in \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$, or $(\thinspace\prod\Psi) \mid R \subseteq \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$.
Next suppose $\xi \in \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$. Then, by definitions \[D:CHOICE\] and \[D:CHOICE\_SPACE\] covering Cartesian products, for each $r \in R$, $\xi(r) \in ({{\Psi}\negmedspace\mid\negmedspace{R}})(r)$. The ensemble $\Psi$ coincides with its restriction ${{\Psi}\negmedspace\mid\negmedspace{R}}$ on $R$. A restatement of this is $({{\Psi}\negmedspace\mid\negmedspace{R}})(r) = \Psi(r)$ for $r \in R$. Substituting $\Psi(r)$ for $({{\Psi}\negmedspace\mid\negmedspace{R}})(r)$ yields $\xi(r) \in \Psi(r)$ for each $r \in R$. From this it follows that $\xi \in (\thinspace\prod\Psi) \mid R$, with the further implication that $\prod ({{\Psi}\negmedspace\mid\negmedspace{R}}) \subseteq (\thinspace\prod\Psi) \mid R$.
We conclude equality $(\thinspace\prod\Psi) \mid R = \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$ after establishing that each of these two sets is a subset of the other.
\[L:SUBSPACE\_SUBSET\] Let $\Psi$ and $\Phi$ be ensembles. If ${\prod{\Phi}}$ is a subspace of ${\prod{\Psi}}$, then $\Phi \subseteq \Psi$.
Let ${\prod{\Phi}}$ be a subspace of ${\prod{\Psi}}$. By definition \[D:SUBSPACE\], there exists $R \subseteq {{\operatorname{dom}{\Psi}}}$ such that
| 3,278
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| 2,254
| 3,077
| null | null |
github_plus_top10pct_by_avg
|
w[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}\tag*{for $v<i\ls b+2$;}\\
U[i]&={\text{\footnotesize$\gyoungx(1.2,;1;2;3_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!3};{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;1;i;{b\!\!+\!\!4},;2,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;{\hat\imath},|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}\tag*{for $2\ls i\ls b+2$;}\\
V[i]&={\text{\footnotesize$\gyoungx(1.2,;1;2;3_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!4};{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;1;i;{b\!\!+\!\!3},;2,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;{\hat\imath},|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}\tag*{for $2\ls i\ls b+2$.}\end{aligned}$$]{} As usual, the ${\hat\imath}$ in the first column indicates that $i$ does not appear in that column.
First consider the tableau $T[i]$, and apply Lemma \[lemma7\] to move the $1$ from row $2$ to row $1$. Of the tableaux appearing in the resulting expression, the only ones dominated by $S$ are [$$\begin{aligned}
T'[i]&={\text{\footnotesize$\gyoungx(1.2,;1;1;2;3_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[th
| 3,279
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| 2,738
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| 0.817103
|
github_plus_top10pct_by_avg
|
times k}$, and $D_0\in B[x]^{k\times k}$ such that $\det D_1,\det D_2\not=0$ and Eq. holds. Let $V$ be a non-empty Zariski open subset of the affine variety of $B\simeq B[x]/(x)$ such that $(\det D_1)_p,(\det D_2)_p,
b_p\not=0$ and $\mathrm{rk}\,X(0)_p\le r$ for all $p\in V$. This exists by the assumption on $r$ and since the variety of $B$ is irreducible. Then $s\le r$ by Eq. in the points $(p,0)$ of the variety of $B[x]$, where $p\in V$. Therefore $$\det D_1\, \det X \det D_2=x^{k-r}b'$$ for some $b'\in B[x]$. Since $\det D_1,\det D_2\in B$ and $B$ is an integral domain, we conclude that $\det X\in x^{k-r}B[x]$.
For all $k\in {\mathbb{N}}$ let ${\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$ denote the ring of Laurent polynomials in $k$ variables. For all $M=(m_{ij})_{i,j\in \{1,2,\dots ,k\}}\in \mathrm{GL}(k,{\mathbb{Z}})$ let $$X^{(M)} _i=\prod _{j=1}^kx_j^{m_{ij}},\quad 1\le i\le k.$$ Then the ring endomorphism of ${\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$ given by $x_i\mapsto X^{(M)}_i$ for all $i\in \{1,2,\dots ,k\}$ is an isomorphism with inverse map given by $x_i\mapsto X^{(M^{-1})}_i$ for all $i\in \{1,2,\dots ,k\}$.
Let $k\in {\mathbb{N}}$. Let $J\subsetneq {\bar{{\Bbbk }}}[x_i,x_i^{-1}\,|\,1\le i\le k]$ be an ideal generated by elements of the form $q-\prod _{i=1}^k x_i^{m_i}$, where $m_1,\dots ,m_k\in {\mathbb{Z}}$ and $q\in {{\bar{{\Bbbk }}}^\times }$ is a root of $1$. Then $J$ is a finite intersection of ideals of the form $$\begin{aligned}
( X_1^{(M)}-q_1, X_2^{(M)}-q_2,\dots ,X_l^{(M)}-q_l),
\label{eq:primeid}
\end{aligned}$$ where $l\in \{0,1,\dots ,k\}$, $q_1,\dots ,q_l\in {{\bar{{\Bbbk }}}^\times }$ are roots of $1$, and $M\in \mathrm{GL}(k,{\mathbb{Z}})$. \[le:torusideal\]
Proceed by induction on $k$. If $J$ is empty, then the claim is true. Assume now that $q-\prod _{i=1}^kx_i^{m_i}$ is one of the generators of $J$, where $q$ is a root of $1$ and $(m_1,\dots ,m_k)\in {\mathbb{Z}}^k\setminus \{0\}$. Let $m_0=\mathrm{gcd}(m_1,\dots ,m_k)$. Let $M'\in \mathrm{GL}(k,{\ma
| 3,280
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|
v \text{ if either }|w|<|v|, \text{ or }|w|=|v|\text{ and }w\preceq' v.$$
For an element $g\in G$, by $W_g$ we denote the set $\{w\in W\colon w_G=g\}$ and by $w_g$ the minimal element of the set $W_g$, which is easily verified to exist, in $\preceq$. If there is no danger of confusion, for an element $g\in G$ we shall denote by $|g|$ the number $|w_g|$ which is equal to $d_S(g,1_G)$.
Finally, we use the linear order $\preceq$ on $W$ to define a linear order $\leq$ on $G$. For $g,h\in G$ we set $$g\leq h\text{ if }w_g\preceq w_h.$$
We call $G$ *shortlex combable*, with respect to a fixed symmetric generating set $S$ and a linear order on $S$, if there exists a constant $K\geq 1$ such that for every $g,h\in G$ with $d_S(g,h)=1$ and for every $i\leq \min\{d_S(g,1_G),d_S(h,1_G)\}$ we have $$d_S((w_g(\leq i))_G, (w_h(\leq i))_G)\leq K.$$
The constant $K$ will be called the *combability constant* of $G$.
First we show how such groups are useful for our purposes. Then we provide examples and show some applications. The following is the main result of this section.
\[thm:shortlex\] Let $G$ be a finitely generated shortlex combable group (with respect to $S$ equipped with some linear order). Then ${\mathcal{F}}(G,d_S)$ has a Schauder basis.
Since the linear order $\leq$ on $G$ is clearly isomorphic to the standard order on ${\mathbb{N}}$, we can use it to enumerate $G$ as $(g_n)_{n\in{\mathbb{N}}}$. For each $n\in{\mathbb{N}}$, set $G_n:=\{g_i\colon i\leq n\}$. For each $n$ we now define maps $P_n: G\rightarrow G_n$ as follows. First, set $m=\max\{|h|\colon h\in G_n\}$, then for $g\in G$, set $$P_n(g):=\begin{cases} g & \text{if }g\in G_n\\
(w_g(\leq m))_G & \text{if }g\notin G_n, (w_g(\leq m))_G\in G_n\\
(w_g(\leq m-1))_G & \text{otherwise}.
\end{cases}$$ We leave to the reader the straightforward verification that $P_n$ is well defined.\
[**Claim.**]{} The maps $(P_n)_{n\in{\mathbb{N}}}$ are uniformly bounded Lipschitz commutting retractions.\
First we check that for every $n,m\in{\mathbb{N}}$, $P_n\circ P_m=P_m
| 3,281
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underlying Newton-Cartan geometry has been studied in [@Hofman:2014loa]. For the case at hand, we need to introduce a Newton-Cartan geometry with a different scaling structure however.
Flat Geometry
-------------
We start with the geometry similar to the flat Euclidean geometry. Such geometry admits the following symmetries $$H:x\rightarrow x'=x+\delta x,$$ $$\bar{H}:y\rightarrow y'=y+\delta y,$$ $$B:y\rightarrow y'=y+v x.$$ Note that for different scalings $c,d$ , the flat geometries are the same.
The invariant vector and one-form are respectively $$\bar{q}^a=\left(
\begin{aligned}
& 0\\
& 1
\end{aligned}
\right )\ ,\ \ \ \ q_a=(0\ \ \ 1),\hs{3ex}a=1,2.$$ Similarly, there is a metric $$g_{ab}=q_aq_b=\left(
\begin{aligned}
1\ \ & 0\\
0\ \ & 0
\end{aligned}
\right )$$ which is flat and invariant under boost transformation $$g=BgB^{-1}.$$ The metric is degenerate, and it is orthogonal to the invariant one-form. It has one positive eigenvalue and one vanishing eigenvalue. Besides, there is an antisymmetric tensor $h_{ab}$ to lower the index $$q_a=h_{ab}\bar{q}^b.$$ It is invariant under the boost transformation as well. It is invertible with $h^{ab}h_{bc}=\d^a_c$, and its inverse helps us to raise the index |[q]{}\^a=h\^[ab]{}q\_b. With $h^{ab}$, we can obtain the upper index metric |[g]{}\^[ab]{}=|[q]{}\^a|[q]{}\^b=h\^[ac]{}h\^[bd]{}q\_c q\_d=h\^[ac]{}h\^[bd]{}g\_[cd]{}.
Curved Geometry
---------------
In the previous subsection, the vector space and the dual 1-form space are introduced to define the geometry. The antisymmetric tensor $h_{ab}$ maps the vectors to one-forms, and the metric $g_{ab}$ defines the inner product of the vectors. This is in contrast with the usual Riemannian geometry, in which the metric serves also as a tool to map the vectors to the one-forms.
The curved geometry is defined by ‘gluing flat geometry’, in the sense that the tangent space is flat with the map determined by the zweibein. One needs to define the connection properly. The zweibein is required to map the space-time ve
| 3,282
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otation implicitly contains the assumption that the identity of Alice’s measurements as random variables does not depend on Bob’s settings, and vice versa. It is well known [@fine_hidden_1982] that under the no-signaling conditions the existence of the jpd is equivalent to the CHSH inequalities being satisfied. Applying the NP approach, the minimal L1 norm of the probability distribution is given by [@oas_exploring_2014] $$\Gamma_{\min}=\max\left\{ 0,\frac{1}{2}S_{CHSH}-1\right\} ,\label{eq:M-inequality-AliceBob}$$ where $$\begin{array}{r}
S_{CHSH}=\raisebox{0pt}[0pt][0pt]{\ensuremath{{\displaystyle \max_{\#^{-}=1,3}}}}\{\pm\left\langle \mathbf{A}_{1,1}\mathbf{B}_{1,1}\right\rangle \pm\left\langle \mathbf{A}_{1,2}\mathbf{B}_{1,2}\right\rangle \phantom{\mbox{\}}.}\\
\pm\left\langle \mathbf{A}_{2,1}\mathbf{B}_{2,1}\right\rangle \pm\left\langle \mathbf{A}_{2,2}\mathbf{B}_{2,2}\right\rangle \mbox{\}}.
\end{array}$$ Here $\Gamma_{\min}=0$ corresponds to the CHSH inequalities, and $\Gamma_{\min}>0$ to contextuality.
Turning now to the CbD approach, we have four pairs of random variables, $$\left(\mathbf{A}_{1,1},\mathbf{B}_{1,1}\right),\left(\mathbf{A}_{1,2},\mathbf{B}_{1,2}\right),\left(\mathbf{A}_{2,1},\mathbf{B}_{2,1}\right),\left(\mathbf{A}_{2,2},\mathbf{B}_{2,2}\right).\label{eq:AB pairs}$$ Here, $\mathbf{A}_{i,j}$ denotes Alice’s measurement under her setting $i=1,2$ when Bob’s setting is $j=1,2$, and analogously for $\mathbf{B}_{i,j}$. We seek a jpd with the smallest value $\Delta_{\min}$ of $$\begin{array}{r}
\Pr\left[\mathbf{A}_{1,1}\neq\mathbf{A}_{1,2}\right]+\Pr\left[\mathbf{A}_{2,1}\neq\mathbf{A}_{2,2}\right]+\Pr\left[\mathbf{B}_{1,1}\neq\mathbf{B}_{2,1}\right]+\Pr\left[\mathbf{B}_{1,2}\neq\mathbf{B}_{2,2}\right].\end{array}$$ No contextuality means $\Delta_{\min}=0$. A computer assisted Fourier-Motzkin elimination algorithm yields (this is a special case of the result in Ref. [@dzhafarov_generalizing_2014; @DKL2014; @KDL2014]) $$\Delta_{\min}=\max\left\{ 0,\frac{1}{2}S_{CHSH}-1\right\} .\label{eq:
| 3,283
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| 3,017
| 0.775413
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|
*)=\max_{i=1}^k\alpha(G[W_i])\le\alpha(G)\le m.$$
By our assumption, $G^*$ has at most $f(m)$ vertices, so that $\sum_{i=1}^k|W_i|\le f(m).$ As we have seen in the proof of Theorem 3.1, the total number of piercing points is at most $(k-1)\sum_{i=1}^k|Z_i|\le(k-1)\omega(G)<km$, and each segment in $V_0$ contains at least one of them. Each piercing point is contained in at most $m$ segments, because these segments induce an independent set in $G$. Thus, we have $|V_0|<km^2$ and $$|V(G)|=|V_0|+\bigcup_{i=1}^k|W_i|<km^2+|V(G^*)|\le km^2+f(m).$$
Now we turn to the general case, where there is no bound on the number of planes containing the segments. As in the proof of Theorem 1, we consider the disjointness graph $\bar{G}'$ of the supporting lines of the segments in the projective space $\mathbb{P}^3$. Clearly, we have $\omega(\bar{G}')\le\omega(G)\le m$, so by Theorem 1 we have $\chi(\bar{G}')\le m^2$. Following the proof of Theorem 1, take an optimal coloring of $\bar{G}'$, and let $G_0$ denote the subgraph of $G$ induced by the segments whose supporting lines received one of the planar colors. Letting $k$ denote the number of planar colors, for every $i, 1\le
i\le \omega(\bar{G}')-k,$ let $G_i$ denote the subgraph of $G$ induced by the set of segments whose supporting lines received the $i$th pointed color. As in the proof of Theorem 1, every $G_i, i\ge 1$ is perfect and, hence, its number of vertices satisfies $$|V(G_i)|\le\chi(G_i)\alpha(G_i)\le\omega(G_i)\alpha(G_i)\le\omega(G)\alpha(G)\le
m^2.$$ The segments belonging to $V(G_0)$ lie in at most $k$ planes. In view of the previous paragraph, $|V(G_0)|\le km^2+f(m)$ vertices. Combining the above bounds, we obtain $$|V(G)|=|V(G_0)|+\sum_{i=1}^{\chi(\bar{G}')-k}|V(G_i)|
\le km^2+f(m)+(\chi(\bar{G}')-k)m^2$$ $$\le km^2+f(m)+(m^2-k)m^2\le f(m)+m^4,$$ which completes the proof. $\Box$
Constructions–Proof of Theorem 5
================================
The aim of this section is to describe various arrangements of geometric objects in 2, 3, and 4 dimensions wit
| 3,284
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| 2,816
| 2,982
| 3,936
| 0.769166
|
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|
by cell $x$ is going to change the state of cell $y$, the change in the state of $y$ depends only on the states of $x$ and $y$. In other words, the new state of $y$ is a function of the previous states of $x$ and $y$.
- ED Data Separation Properties
To provide evidence for a CC evaluation of the ED (Embedded Devices) separation kernel to enforce data separation, five subproperties, namely, No-Exfiltration, No-Infiltration, Temporal Separation, Separation of Control, and Kernel Integrity are proposed to verify the kernel [@Heitmeyer06; @Heitmeyer08]. The Top-Level Specification (TLS) is used to provide a precise and understandable description of the allowed security-relevant external behavior and to make the assumptions on which the TLS is explicitly based. TLS is also to provide a formal context and precise vocabulary to define data separation properties. In TLS, the state machine representing the kernel behavior is defined in terms of an input alphabet, a set of states, an initial state and a transform relation describing the allowed state transitions. The input alphabet contains internal events (cause the kernel to invoke some process) and external events (performed by an external host). The state consists of the id of a partition processing data, the values of the partition’s memory areas and a flag to indicate sanitization of each memory area.
The No-Exfiltration Property states that data processing in any partition cannot influence data stored outside the partition, which is formulated as follows.
$$\begin{aligned}
& s,s' \in S \wedge s' = T(s,e) \; \wedge \\
& e \in P_j \cup E^{In}_j \cup E^{Out}_j \; \wedge \\
& a \in \mathcal{M} \wedge a_s \neq a_{s'} \\
& \Rightarrow a \in A_j
\end{aligned}$$
where $s$ and $s'$ are states and $s'$ is the next state of $s$ transited by an event $e$ in the partition $j$. $P_j$ is the internal event set of the partition $j$. $E^{In}_j$ is the set of external events writing into or clearing the input buffers of the partition $j$. $E^{Out}_j$ is the set of external
| 3,285
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| 2,683
| 880
| 0.798387
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|
----------
The average throughput gain of employing the reconfigurable antennas is given by $$\label{eq:th_gain}
G_{\bar{R}}={\bar{R}_{{\widehat{\psi}}}}/{\bar{R}_{\psi}}, $$ where $\bar{R}_{{\widehat{\psi}}}=\mathbb{E}\{R_{{\widehat{\psi}}}\}$, $\bar{R}_{\psi}=\mathbb{E}\{R_{\psi}\}$, and the expectation is over different channel realizations. As mentioned before, we assume that the channel matrices for different reconfiguration states have the same average channel power, and hence, $\bar{R}_{1}=\cdots=\bar{R}_{\Psi}$.
With the key property of VCR, each entry of ${\widetilde{\mathbf{H}}_{\psi,V}}$, i.e., ${\widetilde{\mathbf{H}}_{\psi,V}}(i,j)$, is associated with a set of physical paths [@Sayeed_02_Deconstuctingmfc], and it is approximated equal to the sum of the complex gains of the corresponding paths [@Sayeed_07_maxMcsparseRAA]. When the number of distinct paths associated with ${\widetilde{\mathbf{H}}_{\psi,V}}(i,j)$ is sufficiently large, we note from the central limit theorem that ${\widetilde{\mathbf{H}}_{\psi,V}}(i,j)$ tends toward a complex Gaussian random variable. As observed in [@Gustafson_14_ommcacm] for the practical mmWave propagation environment at 60 GHz, the average number of distinct clusters is 10, and the average number of rays in each cluster is 9. The 802.11ad model has a fixed value of 18 clusters for the 60 GHz WLAN systems [@Maltsev_10_cmf60gwsmodl]. With the aforementioned numbers of clusters and rays, the entries of ${\widetilde{\mathbf{H}}_{\psi,V}}$ can be approximated by zero-mean complex Gaussian variables. Different from the rich scattering environment for low-frequency communication, the associated groups of paths to different entries of ${\widetilde{\mathbf{H}}_{\psi,V}}$ may be correlated in the mmWave environment. As a result, the entries of ${\widetilde{\mathbf{H}}_{\psi,V}}$ can be correlated, and the entries of ${\widetilde{\mathbf{H}}_{\psi,V}}$ are then approximated by correlated zero-mean complex Gaussian variables. In the literature, it has been shown that the
| 3,286
| 1,041
| 2,911
| 3,031
| 1,569
| 0.788176
|
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|
chieved in our model as soon as $v$ is different from zero.
We shall in the following solve the generalized $BFM$ in the atomic limit (i.e., putting the second term in Eq.(2) equal to zero) for a grand canonical ensemble. In this case the eigenstates of the Hamiltonian are $$\begin{aligned}
|0,l \rangle& =& |~0\rangle \otimes |0) \otimes |\Phi(X)\rangle_l
\nonumber \\
|1,l \rangle& =& |\uparrow\rangle \otimes |0) \otimes
|\Phi(X)\rangle_l
\nonumber \\
|2,l \rangle& =& |\downarrow\rangle \otimes |0) \otimes
|\Phi(X)\rangle_l
\nonumber \\
|3,l \rangle& =& u_{l,+}|\uparrow \downarrow \rangle \otimes |0)
\otimes |\Phi(X)\rangle_{u_{l,+}}
\nonumber \\
&& \quad\qquad\qquad +v_{l,+}|0\rangle \otimes |1) \otimes
|\Phi(X)\rangle_{v_{l,+}}
\nonumber \\
|4,l \rangle& =& u_{l,-}|\uparrow \downarrow \rangle \otimes |0)
\otimes |\Phi(X)\rangle_{u_{l,-}}
\nonumber \\
&& \quad\qquad\qquad +v_{l,-}|0\rangle \otimes |1) \otimes
|\Phi(X)\rangle_{v_{l,-}}
\nonumber \\
|5,l \rangle& =& |\uparrow \rangle \otimes |1) \otimes
|\Phi(X-X_0)\rangle_l
\nonumber \\
|6,l \rangle& =& | \downarrow \rangle \otimes |1) \otimes
|\Phi(X-X_0)\rangle_l
\nonumber \\
|7,l \rangle& =& |\uparrow \downarrow \rangle \otimes |1) \otimes
|\Phi(X-X_0)\rangle_l
\quad ,\end{aligned}$$ where $|\sigma\rangle$ denotes a site occupied by an electron with spin $\sigma$ and $|\!\uparrow\downarrow\rangle$ a site occupied by a pair of electrons with spin up and down. $|0)$ and $|1)$ denote a site unoccupied and, respectively, occupied by a Boson. $|\Phi(X)\rangle_l$ denotes the $l$-th excited oscillator state and $|\Phi(X-X_0)\rangle_l= (a^+-\alpha)^l/\sqrt{l!}
\,exp(\alpha(a-a^+))|\Phi(x)\rangle_0$ the $l$-th excited shifted oscillator state. These two sets of oscillator states are sufficient to describe all the states listed in Eq.(3) except for the states $|3,l \rangle$ and $|4,l \rangle$ for which the corresponding oscillator states are given by $|\Phi(X)\rangle_{u_{l,\pm}}$ and $|\Phi(X)\rangle_{v_{l,\pm}}$. The latter are determined by num
| 3,287
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| 2,772
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| null | null |
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|
{1}$ and since $A\in(\ell_{1}(\widehat{F}),\ell_{p})$, we obtain from Lemma 2.3 that $\bar{A}\in(\ell_{1},\ell_{p})$ and $Ax=\bar{A}y.$ Thus, we have for every $m\in\mathbb{N}
$ that$$\begin{aligned}
\left \Vert (I-P_{m})(Ax)\right \Vert _{\ell_{p}} & =\left \Vert (I-P_{m})(\bar{A}y)\right \Vert _{\ell_{p}}\\
& =\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert \bar{A}_{n}(y)\right \vert ^{p}\right) ^{1/p}\\
& =\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert
{\displaystyle \sum \limits_{k}}
\bar{a}_{nk}y_{k}\right \vert ^{p}\right) ^{1/p}\\
& \leq{\displaystyle \sum \limits_{k}}
\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert \bar{a}_{nk}y_{k}\right \vert ^{p}\right) ^{1/p}\\
& \leq \left \Vert y\right \Vert _{\ell_{1}}\left( \sup_{k}\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}\right) \\
& =\left \Vert x\right \Vert _{\ell_{1}(\widehat{F})}\left( \sup_{k}\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}\right) .\end{aligned}$$ This yields that $$\sup_{x\in S}\left \Vert (I-P_{m})(Ax)\right \Vert _{\ell_{p}}\leq \sup
_{k}\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}\text{; \ }(m\in\mathbb{N}
).$$ Therefore, we deduce from (3.15) that $$\left \Vert L_{A}\right \Vert _{\chi}\leq \lim_{m}\left( \sup_{k}\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}\right) . \tag{3.16}$$
To prove the converse inequality, let $c^{(k)}\in \ell_{1}(\widehat{F})$ be such that $\widehat{F}c^{(k)}=e^{(k)}$ $(k\in\mathbb{N}
)$, that is, $e^{(k)}$ is the $\widehat{F}$-transform of $c^{(k)}$ for each $k\in\mathbb{N}
$. Then, we have by Lemma 2.3 that $Ac^{(k)}=\bar{A}e^{(k)}$ for every $k\in\mathbb{N}
$.
Now, let $U=\{c^{(k)}:$ $k\in\mathbb{N}
\}$. Then $U\subset S$ and hence $AU\subset AS$ which implies that $\chi(AU)\leq \chi(AS)
| 3,288
| 2,122
| 2,064
| 3,007
| null | null |
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|
on to the Planck energy scale relevant to quantum gravity, $10^{19}$ GeV) and yet not exactly vanishing cosmological constant of our universe.[@Lambda] If only from that perspective, dynamical spontaneous symmetry breaking of supersymmetry is thus an extremely fascinating issue in the quest for a fundamental unification.[@Witten2]
The degeneracy between the bosonic states $|n,0\rangle$ and the fermionic ones $|n,1\rangle$ suggests that there exists a symmetry — a supersymmetry — relating these two sectors of the system. We need to construct the operators generating such transformations, by creating a fermion and annihilating a boson, or vice-versa, thus mapping between bosonic and fermionic states degenerate in energy. Clearly these operators are given by $$Q=\sqrt{\hbar\omega}\ a^\dagger b\ \ \ ,\ \ \
Q^\dagger=\sqrt{\hbar\omega}\ ab^\dagger\ ,$$ acting as $$\begin{array}{r c l}
Q|n,0\rangle=0\ \ &,&\ \
Q|n,1\rangle=\sqrt{\hbar\omega}\ \sqrt{n+1}|n+1,0\rangle\ ,\\
& & \\
Q^\dagger|n,0\rangle=\sqrt{\hbar\omega}\ \sqrt{n}|n-1,1\rangle\ \ &,&\ \
Q^\dagger|n,1\rangle=0\ .
\end{array}$$ Note that the vacuum $|n=0,0\rangle$ is the single state which is annihilated by both $Q$ and $Q^\dagger$, as it must since it is not degenerate in energy with any other state. The operators $Q$ and $Q^\dagger$ are thus the generators of a supersymmetry present in this system. Their algebra is given by $$\left\{Q,Q\right\}=0=\left\{Q^\dagger,Q^\dagger\right\}\ \ ,\ \
\left\{Q,Q^\dagger\right\}=H\ \ ,\ \
\left[Q,H\right]=0=\left[Q^\dagger,H\right]\ .
\label{eq:SUSYalgebra}$$ The fact that they define a symmetry is confirmed by their vanishing commutation relations with the Hamiltonian $H$.
Once again, we uncover here a general feature of supersymmetry algebras, namely the fact that acting twice with a supersymmetry generator, in fact one gets an identically vanishing result, $Q^2=0={Q^\dagger}^2$, a property directly reminiscent of cohomology classes of differential forms in differential geometry.[@Witten3] In addition, th
| 3,289
| 4,978
| 3,132
| 2,902
| null | null |
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|
j)$ are respectively defined as the infimum and the supremum of the set $\{r>0 , \theta_r(i,j) \mbox{ exists} \}$.
![\[fig:angle\_ij\] [*On the unit sphere, the black squares are points of $\mathbf{T}_{r}(i)$ while black circles are points of $\mathbf{T}_{r}(j)$. The arc $a(\theta,\theta')$ is divided in two equal parts by the line $\Delta$ whose angle (represented in grey) is $\theta_{r}(i,j)$.*]{}](angle_ij.eps){width="6cm" height="6cm"}
From $\beta(i,j)$ to $\partial(i,j)$, the trees ${{\mathbf T}}(i)$ and ${{\mathbf T}}(j)$ evolve in the plane side by side, separated by the competition interface $\varphi(i,j)$. The real numbers $\beta(i,j)$ and $\partial(i,j)$ can respectively be interpreted as the birth and death times of the competition interface $\varphi(i,j)$. When $\partial(i,j)=+\infty$, both sets ${{\mathbf T}}(i)$ and ${{\mathbf T}}(j)$ are unbounded. When $\partial(i,j)<+\infty$, one of the two sets ${{\mathbf T}}(i)$ and ${{\mathbf T}}(j)$ is included in the closed ball $\overline{B}(O,\partial(i,j))$, say ${{\mathbf T}}(j)$. In this case, $\partial(i,j)$ coincides with another death time $\partial(j,k)$ and two situations may occur according to the color $k$. Either $k=i$ which means $i$ is the only existing color outside the ball $\overline{B}(O,\partial(i,j))$ and there is no competition interface beyond that ball. Or $k$ is a third color (different from $i$ and $j$). Then, the competition interface $\varphi(i,k)$ extends $\varphi(i,j)$ and $\varphi(j,k)$ (until its de! ath time $\partial(k,j)$). Its birth time satisfies: $$\beta(i,k) = \partial(i,j) = \partial(j,k) > 0 ~.$$ Let us remark that the application $r\mapsto\theta_{r}(i,j)$ may be discontinuous. Finally, notice that $\theta_r(i,j)\not= \theta_r(j,i)$ and that one may exist and the other not. So, we distinguish the interfaces $\varphi(i,j)$ and $\varphi(j,i)$.\
Our first result states there can be up to five unbounded competition interfaces with positive probability.
\[arbresinfinis\] For any $m\in\{1,2,3,4,5\}$, there exist (exactly
| 3,290
| 865
| 1,870
| 3,179
| 1,844
| 0.785137
|
github_plus_top10pct_by_avg
|
other confusing issue that had to be straightened out to make sense of strings on stacks. Briefly, the answer is that the theory decomposes into a union of theories on ordinary spaces, see [*e.g.*]{} [@summ; @cdhps; @sugrav-g] for discussions in two and four-dimensional theories. We will return to this in section \[sect:het-gsomods\].
[^11]: Only if $K$ lies in the center of $G$ would the tangent bundle have a trivial extension over $\hat{K}$.
[^12]: We have not been able to locate this particular duality in the literature, but would not be surprised if it has been discussed somewhere previously, presumably in a different context. The closest of which we are aware is old work on T-duality in toroidally compactified heterotic strings, relating Spin$(32)/{\mathbb Z}_2$ strings and $E_8 \times E_8$ strings after the gauge group has been Higgsed to a common subgroup, see for example [@ginsparghet].
[^13]: We would link to thank J. Distler for suggesting this construction to us.
[^14]: This gerbe is the obstruction to lifting the principal ${\mathbb Z}_2$ bundle $T^4 \rightarrow [T^4/{\mathbb Z}_2]$ to a principal ${\mathbb Z}_4$ bundle on $[T^4/{\mathbb Z}_2]$. But a principal ${\mathbb Z}_k$ bundle on any space $X$ is the same thing as a homomorphism $\pi_1(X) \rightarrow {\mathbb Z}_k$. Therefore, we can study nontriviality of the gerbe as a question about lifts of group homomorphisms. The bundle $T^4 \rightarrow
[T^4/{\mathbb Z}_2]$ corresponds to a homomorphism $$\phi: \: \pi_1\left( [T^4/{\mathbb Z}_2] \right) \: \longrightarrow \:
{\mathbb Z}_2.$$ (In particular, since $T^4 \rightarrow [T^4/{\mathbb Z}_2]$ is a principal ${\mathbb Z}_2$ bundle, we have a long exact sequence with relevant part $$\pi_1(T^4) \: \longrightarrow \: \pi_1([T^4/{\mathbb Z}_2) \:
\stackrel{\phi}{\longrightarrow}
\: \pi_0\left( {\mathbb Z}_2 \right) \: \left( \cong \: {\mathbb Z}_2 \right)
\: \longrightarrow \: \pi_0(T^4),$$ and as $T^4$ is connected, we see that $\phi$ is surjective.) We want to understand wh
| 3,291
| 3,542
| 2,648
| 2,872
| 2,649
| 0.778164
|
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|
(\[h6\]) we obtain$$I_{n}^{m}f(x)=\int_{0}^{t}dt_{1}...\int_{0}^{t_{m-1}}dt_{m}\int
p_{t-t_{1},t_{1}-t_{2},...,t_{m}}^{n,m}(x,y)f(y)dy.$$We denote$$\phi
_{t}^{n,m_{0}}(x,y)=p_{t}^{n}(x,y)+\sum_{m=1}^{m_{0}-1}\int_{0}^{t}dt_{1}...%
\int_{0}^{t_{m-1}}dt_{m}p_{t-t_{1},t_{2}-t_{1},...,t_{m}}^{n,m}(x,y)$$so that (\[R2\]) reads$$\int f(y)P_{t}(x,dy)=\int f(y)\phi _{t}^{n,m_{0}}(x,y)dy+R_{n}^{m_{0}}f(x).$$We recall that $\Psi _{\eta ,\kappa }$ is defined in (\[R7”\]) and we define the measures on ${\mathbb{R}}^{d}\times {\mathbb{R}}^{d}$ defined by $$\mu ^{\eta ,\kappa }(dx,dy)=\Psi _{\eta ,\kappa }(x,y)P_{t}(x,dy)dx\quad %
\mbox{and}\quad \mu _{n}^{\eta ,\kappa ,m_{0}}(dx,dy)=\Psi _{\eta ,\kappa
}(x,y)\phi _{t}^{n,m_{0}}(x,y)dxdy.$$So, the proof consists in applying Lemma \[REG\] to $\mu =\mu ^{\eta
,\kappa }$ and $\mu _{n}=\mu _{n}^{\eta ,\kappa ,m_{0}}$.
**Step 2: analysis of the principal term.** We study here the estimates for $f_{n}(x,y)=\Psi _{\eta ,\kappa }\phi _{t}^{n,m_{0}}(x,y)$ which are required in (\[reg9\]).
We first use (\[h7\]) in order to get estimates for $%
p_{t-t_{1},t_{2}-t_{1},...,t_{m}}^{n,m}(x,y)$. We fix $q_{1},q_{2}\in {\
\mathbb{N}},\kappa \geq 0,p>1$ and we recall that in Lemma \[Reg\] we introduced $\overline{q}=q_{1}+q_{2}+(a+b)(m_{0}-1).$ Moreover in Lemma [Reg]{} one produces $\chi $ such that (\[h7\]) holds true: for every multi-index $\beta $ with $\left\vert \beta \right\vert \leq q_{2}$$$\begin{array}{l}
\left\Vert \psi _{\kappa }\partial _{x}^{\beta
}p_{t-t_{1},t_{1}-t_{2},...,t_{m}}^{n,m}(x,\cdot )\right\Vert
_{q_{1},p}\smallskip \\
\displaystyle\quad \leq C\Big(\frac{1}{\lambda _{n}t}\Big)^{\theta
_{0}(q_{1}+q_{2}+d+2\theta _{1})}\times \left( \varepsilon _{n}\Lambda _{n}%
\Big(\frac{1}{\lambda _{n}t}\Big)^{\theta _{0}(a+b)}\right) ^{m}\psi _{\chi
}(x).%
\end{array}%$$We recall the constant defined in (\[R7’\]): $$\Phi _{n}(\delta )=\varepsilon _{n}\Lambda _{n}\times \frac{1}{\lambda
_{n}^{\theta _{0}(a+b+\delta )}}.$$Denote$$\xi _{1}(q)=q+d+2\theta _{1}+m_{0}(a+b),\qquad
| 3,292
| 1,920
| 1,808
| 3,205
| null | null |
github_plus_top10pct_by_avg
|
on evolves through a set of stochastic processes that mimic the projector of Eq.\[eq:projector\]:
1. A cloning/death step, in which the walker population on each determinant is increased/reduced with a probability $\left(H_{ii}-S\right)\Delta\tau$. $S$ is a shift parameter that is used to control the total walker population.
2. A spawning step. For each walker on a determinant $\Ket{D_{i}}$ a singly or doubly connected determinant $\Ket{D_{j}}$ is generated with a probability $p_{gen}^{ij}$. A signed child is actually generated on the determinant $\Ket{D_{j}}$ with a spawning probability $$p_{spawn}^{ij}=\frac{\left|H_{ij}\right|\Delta\tau}{p_{gen}^{ij}}.\label{eq:pspawn}$$ The sign of the newly spawned walker is the same as the sign of the parent if $H_{ij}>0$, it is of opposite sign otherwise.
3. Each pairs of negative and positive newly spawned walkers lying on the same determinant are removed during an Annihilation step. This avoid the growth of an infinite noise due to the so-called sign problem.
We propose here a modification of the FCIQMC algorithm in order to stochastically sample simultaneously the zeroth order wavefunction and the successive order of the perturbation of Eq.\[eq:bthorderwf\]. Even if in principle any order of perturbation can be reached by this technique, in this article we will only consider the calculation of the fist order correction to the wave function, that is given through Eq.\[eq:bthorderwf\] by $$\Ket{\Psi_{1}}=\left(\hat{H}_{0}-E_{0}\right)^{-1}Q\hat{V}\Ket{\Psi_{0}}\label{eq:ps1equ}$$
In that case the problem is simpler since the Hilbert space on which the zeroth and first order wavefunctions are expanded is limited to the CASSD space. Moreover this space can be expressed as a direct sum of two subspaces ${\cal H}={\cal H}_{0}\oplus{\cal H}_{1}$, where ${\cal H}_{0}$ correspond to the CAS space and ${\cal H}_{1}$ is its orthogonal compliment which contains all the determinants that are single or double excitations from the ones belonging to ${\cal H}_{0}$. Applying $\h
| 3,293
| 3,136
| 3,965
| 2,942
| 1,936
| 0.784299
|
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|
\leftrightarrow u_2) = 0
~\mbox{for}~|i-j|=1, \label{y7}\\
& & E^{\pm}_i(u) E^{\pm}_j(v)
= E^{\pm}_j(v) E^{\pm}_i(u) ~~\mbox{for}~|i-j|>1, \label{y8}\end{aligned}$$
where
$$u_\pm = u \pm \frac{1}{4} \hbar c$$
and
$$B_{ij} = \frac{1}{2} a_{ij}.$$
$q$-affine-Yangian double correspondence
----------------------------------------
Our central goal is to establish a free boson representation of the Yangian double $DY_\hbar(sl_N)$. For this we would like to use the known results [@sln] for the $q$-affine algebra $U_q(\widehat{sl_N})$ by establishing a correspondence principle between these two algebras. Such a correspondence principle has been expected for some time and was “quite mysterious” as stated in Ref.[@iohara].
For the present authors, however, such a correspondence is rather obvious by making use of the Drinfeld current realizations for both $U_q(\widehat{sl_N})$ and $DY_\hbar(sl_N)$. For other realizations no such an obvious observation could be obtained. We give the following
\[ob1\] ($q$-affine-Yangian double correspondence). The following gives a simple correspondence between $U_q(\widehat{sl_N})$ and $DY_\hbar(sl_N)$ as associative algebras
$$\begin{aligned}
& & q \rightarrow \mbox{e}^{\frac{\hbar}{2}},~~~
\gamma \rightarrow \mbox{e}^{\frac{\hbar c}{2}},\\
& & z \rightarrow \mbox{e}^{u} ,\\
& & \psi^i_{\pm}(z) \rightarrow H^{\pm}_i(u),\\
& & z E^{\pm,i}(z) \rightarrow E^{\pm}_i( u)\end{aligned}$$
in the limit $\hbar \rightarrow 0,~u \rightarrow 0$ up to the linear approximation in $\hbar$ and $u$.
We remark that the above observation only gives a rule for obtaining equations (\[y1\]-\[y8\]) from (\[1\]-\[8\]) and does not imply any more fundamental Hopf algebraic or algebraic relations.
Free boson representation of $DY_\hbar(sl_N)$ with arbitrary level
==================================================================
In this section we shall consider our central problem–the establishment of a free boson representation of $DY_\hbar(sl_N)$ with arbitrary level. For $N=2$ this problem ha
| 3,294
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| 3,163
| 1,622
| 0.787516
|
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|
######
**Four single nucleotide polymorphisms in the*GNB1*gene in 265 HCV-1 and 195 HCV-2 infected patients receiving PEG-IFNα-RBV therapy with or without a RVR in a Chinese population in Taiwan**
**SNP** **Position in*GNB1*** **Chromosome position**^**a**^ **Alleles** **HCV-1** **HCV-2**
----------------- ----------------------- -------------------------------- ------------- ----------- ----------- -------- -------- -------- --------
rs10907185 (S1) Intron 7 1733219 A/G 0.2132 0.3009 0.2548 0.465 0.2785 0.2162
rs6603797 (S2) Intron 2 1765583 C/T 0.5854 0.1065 0.1401 0.4007 0.0981 0.0946
rs4648727 (S3) Intron 1 1776269 A/C 0.1309 0.3241 0.3429 0.8709 0.3513 0.2361
rs12126768 (S4) Intron 1 1778090 G/T 0.7439 0.2083 0.2756 1 0.2373 0.2162
Abbreviations: SNP, single nucleotide polymorphism; HWE, Hardy--Weinberg equilibrium; MAF, minor allele frequency.
^a^ Chromosome positions refer to the sequence in the NCBI database (build 37.3).
Association between tagging SNPs of *GNB1* and therapeutic response, RVR
------------------------------------------------------------------------
The genotype frequencies of each SNP showing responsiveness to PEG-IFNα-RBV therapy are shown in Table [3](#T3){ref-type="table"}. In the genotype association tests, none of the genotypes was associated with RVR in HCV-1 infected patients (Table [3](#T3){ref-type="table"}). However, the combination of genotypes G/G and G/T of rs12126768 was significantly inversely correlated with RVR responsiveness (*P* = 0.0330, OR = 0.58, 95% CI = 0.35, 0.96). In HCV-2 infected patients, the polymorphism at position rs4648727 in the *GNB1* gene was statistically associated with RVR (*P* = 0.0194). For the A/A +
| 3,295
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| 2,964
| 3,278
| null | null |
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|
converges to $x\in X$ with all such $x$ defining the closure $\textrm{Cl}(A)$ of $A$. Furthermore taking the directed set to be $${\textstyle _{\mathbb{D}}N_{t}=\{(N,t)\!:(N\in\mathcal{N}_{x})(t\in N\bigcap A-\{ x\})\}}\label{Eqn: Der_Directed}$$ which, unlike Eq. (\[Eqn: Closure\_Directed\]), excludes the point $x$ that may or may not be in the subset $A$ of $X$, induces the net $\chi(N,t)=t\in A-\{ x\}$ converging to $x\in X$, with the set of all such $x$ yielding the derived set $\textrm{Der}(A)$ of $A$. In contrast, Eq. (\[Eqn: Closure\_Directed\]) also includes the isolated points $t=x$ of $A$ so as to generate its closure. Observe how neighbourhoods of a point, which define convergence of nets and filters in a topological space $X$, double up here as index sets to yield a self-consistent tool for the description of convergence.
As compared with sequences where, the index set is restricted to positive integers, the considerable freedom in the choice of directed sets as is abundantly borne out by the two preceding examples, is not without its associated drawbacks. Thus as a trade-off, the wide range of choice of the directed sets may imply that induction methods, so common in the analysis of sequences, need no longer apply to arbitrary nets.
\(4) The non-convergent nets (actually these are sequences)
\(a) $(1,-1,1,-1,\cdots)$ adheres at $1$ and $-1$ and
\(b) $\begin{array}{ccl}
x_{n} & = & {\displaystyle \left\{ \begin{array}{lcl}
n & & \textrm{if }n\textrm{ is odd}\\
1-1/(1+n) & & \textrm{if }n\textrm{ is even}\end{array},\right.}\end{array}$ adheres at $1$ for its even terms, but is unbounded in the odd terms.$\qquad\blacksquare$
A converging sequence or net is also adhering but, as examples (4) show, the converse is false. Nevertheless it is true, as again is evident from examples (4), that in a first countable space where sequences suffice, a sequence $(x_{n})$ adheres at $x$ iff some subsequence $(x_{n_{m}})_{m\in\mathbb{N}}$ of $(x_{n})$ converges to $x$. If the space is not first countable this
| 3,296
| 6,175
| 2,712
| 2,598
| 1,662
| 0.78707
|
github_plus_top10pct_by_avg
|
tic c-number kernel $G_s^{AB}=G^{AB}+\gamma^{AB}$. We assume the identity
\_s\^A\_s\^B=\^A\^B+G\^[AB]{}+\^A\^B+\^A\^B+\^[AB]{} The required expectation values are
\_H\^C\_H\^E=\^C\^E+\^C\^E
\_H\^C\_H\^D\_H\^E&=&\_H\^C(\_H\^D\_H\^E)&=&\^C\^D\^E+\^CG\^[DE]{}+\^DG\^[CE]{}+\^EG\^[CD]{}+\^C\^[DE]{}Observe that this implies that $\left\langle \varphi^C\gamma^{DE}\right\rangle$ is totally symmetric.
\_H\^C\_H\^D\_H\^E\_H\^F&=&(\_H\^C\_H\^D)(\_H\^E\_H\^F)&=&(\^C\^D+G\^[CD]{})(\^E\^F+G\^[EF]{})&+&\^C&+&\^D&+&\^E\^F\^[CD]{}+\^F\^E\^[CD]{}+\^[CD]{}\^[EF]{}We can now relate the derivatives of the mean fields with respect to the sources to stochastic averages
\^[C,E]{}=i\^C\^E
G\^[CD,E]{}=i\^C\^[DE]{}
\^[C,(EF)]{}=i2{\^C\^[EF]{}+\^EG\^[CF]{}+\^FG\^[CE]{}}
G\^[CD,(EF)]{}=i2{\^E\^F\^[CD]{}+\^F\^E\^[CD]{}+\^[CD]{}\^[EF]{}} Whereby we get the identities
\^C\^E+\_[,A(CD)]{}\^[CD]{}\^E=i\_A\^E
&&{\^C\^[EF]{}+\^EG\^[CF]{}+\^FG\^[CE]{}}&+&\_[,A(CD)]{}{\^E\^F\^[CD]{}+\^F\^E\^[CD]{}+\^[CD]{}\^[EF]{}}=2i\_[(AB)]{}\^[(EF)]{}\^B which reduces to
\^C\^[EF]{}+\_[,A(CD)]{}\^[CD]{}\^[EF]{}=0
\_[,(AB)C]{}\^C\^E+\_[,(AB)(CD)]{}\^[CD]{}\^E=0
\_[,(AB)C]{}\^C\^[EF]{}+\_[,(AB)(CD)]{}\^[CD]{}\^[EF]{}=i2{\_A\^E\_B\^F+\_A\^F\_B\^E} Assuming the mean field equation $\Gamma_{,\left(AC\right)}=0$ these equation suggest a stochastic dynamics for the $\varphi$, $\gamma$ fields
\_[,AC]{} \^C+\_[,A(CD)]{}\^[CD]{}=-\_A \[2pilan1\] Observe that a possible term $\kappa_{AB}\phi^B$ is absent,
\_[,(AB)C]{} \^C+\_[,(AB)(CD)]{} \^[CD]{}=2\_[AB]{} \[2pilan2\] provided
\_A\^E=-i\_A\^E
\_A\^[CD]{}=0
\_[AB]{}\^E=0
\_[AB]{}\^[EF]{}=-i{\_A\^E\_B\^F+\_A\^F\_B\^E} Multiplying the Langevin equations by the sources and using these expectation values, we get
\_A\_B=i\_[,AB]{}
\_A\_[CD]{}=2i\_[,A(CD)]{}
\_[CD]{}\_[AB]{}=4i\_[,(AB)(CD)]{}
Recovery of the 1PI stochastic theory from the 2PI one
------------------------------------------------------
Let us check that the 2PI and 1PI theories agree as far as the mean field fluctuations are concerned. This mu
| 3,297
| 3,032
| 4,746
| 2,947
| null | null |
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|
n: IT\]) and (\[Eqn: FT’\]) are in terms of inverse images (the first of which constitutes a direct, and the second an inverse, problem) the image $f(U)=\textrm{comp}(V)$ for $V\in\mathcal{V}$ is of interest as it indicates the relationship of the openness of $f$ with its continuity. This, and other related concepts are examined below, where the range space $f(X)$ is always taken to be a subspace of $Y$. Openness of a function *$f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$* is the “inverse” of continuity, when images of open sets of $X$ are required to be open in $Y$; such a function is said to be *open.* Following are two of the important properties of open functions.
\(1) *If $f\!:(X,\mathcal{U})\rightarrow(Y,f(\mathcal{U}))$ is an open function, then so is* $f_{<}\!:(X,\mathcal{U})\rightarrow(f(X),\textrm{IT}\{ i;f(\mathcal{U})\})$*. The converse is true if $f(X)$ is an open set of $Y$; thus openness of* $f_{<}\!:(X,\mathcal{U})\rightarrow(f(X),f_{<}(\mathcal{U}))$ *implies tha*t *of $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ whenever $f(X)$ is open in $Y$ such that $f_{<}(U)\in\mathcal{V}$ for $U\in\mathcal{U}$.* The truth of this last assertion follows easily from the fact that if $f_{<}(U)$ is an open set of $f(X)\subset Y$, then necessarily $f_{<}(U)=V\bigcap f(X)$ for some $V\in\mathcal{V}$, and the intersection of two open sets of $Y$ is again an open set of $Y$.
\(2) *If $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ and $g\!:(Y,\mathcal{V})\rightarrow(Z,\mathcal{W})$ are open functions then $g\circ f\!:(X,\mathcal{U})\rightarrow(Z,\mathcal{W})$* *is also open.* It follows that the condition in (1) on $f(X)$ can be replaced by the requirement that the inclusion $i\!:(f(X),\textrm{IT}\{ i;\mathcal{V}\})\rightarrow(Y,\mathcal{V})$ be an open map. This interchange of $f(X)$ with its inclusion $i\!:f(X)\rightarrow Y$ into $Y$ is a basic result that finds application in many situations.
Collected below are some useful properties of the initial and final topologies that we need in this work.
| 3,298
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| 3,068
| 2,543
| 0.779
|
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|
, W. C. [Stwalley]{}, and P. L. [Gould]{}, , 9689 (1996).
K. [Aikawa]{} [*et al.*]{}, Physical Review Letters [**105**]{}, 203001 (2010), 1008.5034.
A. Wakim, P. Zabawa, M. Haruza, and N. P. Bigelow, Opt. Expr. [**20**]{}, 16083 (2012).
M. [Zeppenfeld]{}, M. [Motsch]{}, P. W. H. [Pinkse]{}, and G. [Rempe]{}, , 041401 (2009), 0904.4144.
F. [Robicheaux]{}, Journal of Physics B Atomic Molecular Physics [**42**]{}, 195301 (2009).
H. [Perrin]{} [*et al.*]{}, Comptes Rendus Physique [**12**]{}, 417 (2011), 1102.1327.
---
address: |
Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3,\
53 avenue des Martyrs,\
38026 Grenoble Cedex, France
author:
- SELIM TOUATI
title: ELECTRIC DIPOLE MOMENTS AND NEUTRINO MASS MODELS
---
EDMs generated by the CKM phase
===============================
In the standard model (SM), the only source of weak CP-violation is the complex phase of the CKM matrix. In order to measure the strength of CP-violation, one can construct a flavor invariant (basis-independent) which is sensitive to this phase, called the Jarlskog invariant [@Jarl]. A non-vanishing Jarlskog invariant is a necessary condition for having CP-violation. In the SM, all CP-violating effects are proportional to this invariant. However, this invariant is adequate for estimating CP-violation from closed fermion loops. For example, let us consider the CKM-induced lepton EDMs. Because the leptons cannot feel directly the complex phase of the CKM matrix, we need to go through a closed quark loop. The dominant diagram is:
![CKM-induced lepton EDM[]{data-label="fig:CKMleptonEDM"}](figures/FigCKMLeptonEDM)
This EDM is tuned by the Jarlskog invariant $\det[Y_{u}^{\dagger}Y_{u},Y_{d}^{\dagger}Y_{d}]$ which is proportional to the imaginary part of a quartet $Im(V_{us}V_{cb}V_{ub}^{\ast}V_{cs}^{\ast})$. As for the quarks, they can feel directly the complex phase of the CKM matrix and then there are non-invariants structures which arise from rainbow-like processes.
| 3,299
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| 3,532
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G \rightarrow H$ are not morphisms of groups.
We will fix two groups $H$, $G$ and $\beta: G \times H \rightarrow
G$ a right action of the group $H$ on the set $G$. We define $${\rm Ker}(\beta):= \{h \in H ~|~ g \lhd h = g, \forall g \in G\}$$ We denote by $MP_{\beta}(H,G):= \{\alpha ~|~ (H, G, \alpha, \beta)
{\rm~is~}{\rm~a~}{\rm~matched~}{\rm~pair~}\}$. Let $B_{2}^{\beta}(H,G)$ be the category having $MP_{\beta}(H,G)$ as the set of objects and the morphisms defined as follows: $\psi:
\alpha' \rightarrow \alpha$ is a morphism in $B_{2}^{\beta}(H,G)$ if and only if $\psi: H\, {}_{\alpha'}\!\! \bowtie_{\beta} \, G
\rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ is a morphism of groups such that $${\label{eq:diag2}}
\psi \circ i_H = i_H ~~~{\rm and}~~~ \pi_G \circ \psi = \pi_G$$
[\[pr:2\]]{} Let $(H, G, \alpha', \beta)$, $(H, G, \alpha, \beta)$ be two matched pairs. There exists a one to one correspondence between the set of all morphisms $\psi: \alpha' \rightarrow \alpha$ in the category $B_{2}^{\beta}(H,G)$ and the set of all maps $r:G
\rightarrow {\rm Ker} (\beta)$ such that : $$\begin{aligned}
(g \rhd' h)r(g \lhd h) &=& r(g)(g \rhd
h){\label{eq:p1'}} \\
r(g_{1} g_{2}) &=& r(g_{1})\bigl(g_{1} \rhd r(g_{2})\bigl)
{\label{eq:p3'}}\end{aligned}$$ for all $g, g_{1}, g_{2} \in G$, $h \in H$. Through the above bijection the morphism $\psi$ is given by $${\label{eq:p5'}}
\psi(h,g) = \bigl(hr(g), g\bigl)$$ for all $h \in H$, $g \in G$ and $\psi$ is an isomorphism of groups i.e. $B_{2}^{\beta}(H,G)$ is a grupoid [^2].
For any morphism of groups $\psi: H\, {}_{\alpha'}\!\!
\bowtie_{\beta} \, G \rightarrow H\, {}_{\alpha}\!\!
\bowtie_{\beta} \, G$ such that [(\[eq:diag2\])]{} hold there exists a unique map $r: G\to H$ such that $\psi (h, g) = (h r(g), g)$, for all $h\in H$ and $g\in G$. Now we are in a position to use [Proposition \[pr:1\]]{} for $G' = G$, $\beta' = \beta$ and $v = Id_G$. We obtain [(\[eq:p1’\])]{} and [(\[eq:p3’\])]{} by considering $v = Id_{G}$ in [(\[eq:p1\])]{}, respectively [(\[eq:p3\])]
| 3,300
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| 3,158
| null | null |
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