text large_stringlengths 384 2.05k | rank_avg float64 1 4.19k ⌀ | rank_max float64 1 8.21k ⌀ | rank_min float64 1 5.03k ⌀ | rank_median float64 1 4.21k ⌀ | rank_by_avgsim float64 1 4.19k ⌀ | avgsim_to_github float32 0.77 0.85 ⌀ | dataset large_stringclasses 1
value |
|---|---|---|---|---|---|---|---|
nction $f(x_T - y_T)$ contracts exponentially with rate $\lambda$, plus a discretization error term. The function $f$ is defined in Appendix \[s:defining-q\], and sandwiches $\lrn{x_T - y_T}_2$. In Corollary \[c:main\_gaussian:1\], we apply the results of Lemma \[l:gaussian\_contraction\] recursively over multiple step... | 2,701 | 1,961 | 2,167 | 2,408 | 1,538 | 0.788517 | github_plus_top10pct_by_avg |
in\sR^k$ there exists a constant vector $const$ (all elements are equal) such that $\vln\,\softmax(\vx)=\vx+const$. Furthermore, $\softmax(\vv+const)=\softmax(\vv)$ for any vector $\vv$ and any constant vector $const$. Therefore, $$\begin{aligned}
\muh_{DirLin}(\softmax(\vz); \frac{1}{t}\MI, \vzero)
&=\softmax(\frac{1}... | 2,702 | 1,275 | 1,178 | 2,816 | 3,191 | 0.774138 | github_plus_top10pct_by_avg |
- 'Saint Francis University, Loretto, PA 15940'
author:
- 'A. Fox'
- 'B. LaBuz'
- 'R. Laskowsky'
title: A coarse invariant
---
Introduction
============
A coarse function $f:X\to Y$ between metric spaces is a function that is bornologous and proper. $f$ is bornologous if for each $N>0$ there is an $M>0$ such that if ... | 2,703 | 1,222 | 841 | 2,354 | null | null | github_plus_top10pct_by_avg |
ion from $T_3(\bar{\kappa})$ to itself;
- $Y \mapsto \sigma({}^t Y) + Y$ defines a surjection $T_3(\bar{\kappa}) \rightarrow T_2(\bar{\kappa})$.
Here, all the above maps are interpreted as in Remark \[r35\] (if they are well-defined). Then $\rho_{\ast, m}$ is the composite of these three. Condition (3) is direct fr... | 2,704 | 1,400 | 954 | 2,764 | 2,707 | 0.777709 | github_plus_top10pct_by_avg |
te its derivative: \^[a |a]{} &=& x STr(g\^[-1]{} t\^a g t\^[|a]{}) &=& x STr ( - g\^[-1]{} g g\^[-1]{} t\^a g t\^[|a]{} + g\^[-1]{} t\^a g g\^[-1]{} g t\^[|a]{} ) &=& x STr ( g\^[-1]{}\[t\^a,t\^d\] g t\^[|a]{} ) &=& - i[f\^a]{}\_[bc]{} \^[b |a]{}. We have left out the normal ordering symbols from the above classical c... | 2,705 | 2,134 | 3,138 | 2,743 | null | null | github_plus_top10pct_by_avg |
rt{\log n\ \log\log n}\right)})$.
We will work with polynomials over the ring $$\cR = \cR_{6,6}=\Z_6[\gamma]/(\gamma^6-1)$$ (see Section \[preliminaries\]). We will denote the vector $(\gamma^{z_1},\gamma^{z_2},\cdots,\gamma^{z_k})$ by $\gamma^\bz$ where $\bz=({z_1,\cdots,z_k}) \in \Z_6^k$. We will need to extend the ... | 2,706 | 1,260 | 1,487 | 2,588 | 2,975 | 0.77567 | github_plus_top10pct_by_avg |
}}},{\widetilde{\mathbf{u}}}) - 2\nu{\Delta}^2{\widetilde{\mathbf{u}}}- \lambda{\Delta}{\widetilde{\mathbf{u}}}- \bnabla q & = 0 \qquad\mbox{in}\,\,\Omega , \label{eq:KKT_E_gradR}\\
\nabla\cdot{\widetilde{\mathbf{u}}}& = 0 \qquad\mbox{in}\,\,\Omega , \label{eq:KKT_E_divConstr}\\
\E({\widetilde{\mathbf{u}}}) - \E_0 & = ... | 2,707 | 3,573 | 2,810 | 2,472 | null | null | github_plus_top10pct_by_avg |
{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$+\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}\bigg|.$$ $$\leq (\lambda_e-\lambda_1)e^{2c\lambda_e} M... | 2,708 | 1,263 | 1,761 | 2,696 | null | null | github_plus_top10pct_by_avg |
space{{{\operatorname{dom}{\Phi}}}}} = \phi$.
\[N:SEQUENCE\_NOTATION\] A sequence in a set $S$ is some mapping $\sigma \colon {\mathbb{N}}\to S$ – that is, $\sigma \in S^{\mathbb{N}}$. The anonymous sequence convention allows reference to a sequence using the compound symbol $\lbrace s_n \rbrace$, understanding $s \in... | 2,709 | 655 | 2,402 | 2,559 | null | null | github_plus_top10pct_by_avg |
er\end{aligned}$$ In the previous lines we only kept track of the operators that will lead to poles in the final result. We evaluated the operator $\phi$ at the point $w$ so that the action of the derivatives is easier to take care of: $$\begin{aligned}
= -\kappa_{ab}& t^a t^b \left(\frac{c_+}{c_++c_-} \right)^2
\lim_... | 2,710 | 3,429 | 1,953 | 2,381 | 3,790 | 0.770071 | github_plus_top10pct_by_avg |
(\oplus_i(\pi)e_i)\oplus M_2,$$ $$M_1''=\pi M_1\oplus M_3'=\pi M_1\oplus M_3, ~~~M_2''=M_4'=M_4.$$ Thus, $M_0''$ is *free of type II* as both $M_1''$ and $M_2''$ are *of type II*.
2. If $b\in A$ is a unit, then let $\sqrt{b}$ be an element of $A$ such that $\sqrt{b}^2\equiv b$ mod $2$. We choose a basis $(\pi a, \pi... | 2,711 | 2,044 | 2,031 | 2,478 | 3,102 | 0.774864 | github_plus_top10pct_by_avg |
4 e^{-\phi_0} r_1 r_2 r_3 \, e^\alpha \wedge e^\xi \wedge \left( \nu_1 d\phi_1 + \nu_2 d\phi_2 + \nu_3 d\phi_3 \right) \ , \\
\widehat{F}_5 &= (1 +\star) \frac{4 e^{-\phi_0}}{\lambda} r_1 r_2 r_3 \, e^\alpha \wedge e^\xi \wedge d\phi_1 \wedge d\phi_2 \wedge d\phi_3 \ ,
\end{aligned}$$ in complete agreement with the res... | 2,712 | 1,252 | 1,529 | 2,649 | 3,991 | 0.768748 | github_plus_top10pct_by_avg |
x{\bf g}}, \bar{\mbox{\bf K}})$ form. This surprising result, as we shall see, arises from the behavior of the lapse function [@AAJWY98; @YorkFest].
The constraint equations on $\Sigma$ are, in vacuum, $$\begin{aligned}
\bar{\nabla}_j(\bar{K}^{i j}-\bar{K}\bar{g}^{i j})&=&0 \; , \label{Eq:MomCon}\\
R(\bar{g})-\bar{K}_... | 2,713 | 4,692 | 2,023 | 2,223 | null | null | github_plus_top10pct_by_avg |
',\omega,E',E)d\omega' dE'\leq M_1,\quad
&& {\rm a.e.}\ G\times S\times I,\quad j=1,2,3,\nonumber\\[2mm]
&\sum_{k=1}^3\int_{S\times I} \sigma_{jk}(x,\omega,\omega',E,E')d\omega' dE'\leq M_2,\quad
&& {\rm a.e.}\ G\times S\times I,\quad j=1,2,3,\\[2mm]
&\sigma_{kj}\geq 0,\quad && {\rm a.e.}\ G\times S^2\times I^2,\quad k... | 2,714 | 855 | 2,377 | 2,640 | null | null | github_plus_top10pct_by_avg |
_{\lambda\muhat}$ is non-zero for all $\lambda$ and $\muhat$. Define $$\label{v-1-defn}
v(\lambda,\mu):=v_q\left(\langle
h_{\mu}(\x),s_\lambda(\x\y)\rangle\right).$$
\[v-lambda\] We have $$-v(\lambda)=(2g-2+k) n(\lambda)+(g-1)n
-\sum_{i=1}^kv(\lambda,\mu^i).$$
Straightforward.
\[v-fmla-lemma\] For $\mu=(\mu_1,\mu_2,... | 2,715 | 1,784 | 2,157 | 2,391 | null | null | github_plus_top10pct_by_avg |
_\tau x^\mu\partial_\tau x^\nu}\end{aligned}$$ where $m=TL$ for $L\gg1/\sqrt T$. Therefore, in our toy model we introduce a charged scalar with action $$\begin{aligned}
S[\Phi]=\int d^5x\sqrt{-G}\left(-G^{\mu\nu}(D_\mu\Phi)^*(D_\nu\Phi)-m^2 V |\Phi|^2\right),\end{aligned}$$ where $D_\mu=\partial_\mu+i q A_\mu$ for a pa... | 2,716 | 3,548 | 2,458 | 2,439 | null | null | github_plus_top10pct_by_avg |
kappa^2} + \frac{(p-2)(p-3)(p-3)!}{(p-1)!(\kappa-1)^2} + \cdots + \frac{2(p-3)!}{(p-1)!(\kappa-p+3)^2} \Bigg), \label{eq:crB3} \\
B_4 & \equiv & \frac{(p-3)!}{(p-1)!} \Bigg(\sum_{a,b \in [p-1], b \neq a} \bigg(\frac{1}{\kappa} + \frac{1}{\kappa -1} + \cdots + \frac{1}{\kappa - a+1} \bigg) \bigg(\frac{1}{\kappa} + \fra... | 2,717 | 2,542 | 2,894 | 2,542 | null | null | github_plus_top10pct_by_avg |
\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A W \right\}_{K K}
\left\{ W ^{\dagger} A (UX) \right\}_{K l}
\nonumber \\
&+&
\sum_{m}
\sum_{k \neq l } \sum_{K \neq L }
\frac{ 1 }{ ( h_{l} - h_{k} ) ( \Delta_{L} - \Delta_{K} ) ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} ) ( \Delta_{L} - h_{k} ) ( \Delta_{L} -... | 2,718 | 3,471 | 2,796 | 2,729 | null | null | github_plus_top10pct_by_avg |
tary model can be written as $$\begin{aligned}
\delta_{\alpha \beta} =
\sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} + \sum_{J=4}^{N+3} W_{\alpha J} W^{*}_{\beta J}.
\label{unitarity0}\end{aligned}$$ Then, $\left| \sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} \right|^2 = \left| \sum_{J=4}^{N+3} W_{\alpha J} W^{*}_{\beta ... | 2,719 | 1,158 | 3,169 | 2,578 | null | null | github_plus_top10pct_by_avg |
60 U/L
CK-MB 747↑ 0--25 U/L
Myoglobin 72↑ 0.72--4.49 nmol/L
Creatinine 48 15--77 μmol/L
CPK 3,876↑ 26--140 U/L
HBDH 580↑ 72--300 U/L
HCO~3~ ^−^ 28.7↑ 22.0--27.0 mmol/L
Hormones ... | 2,720 | 3,670 | 3,101 | 2,677 | null | null | github_plus_top10pct_by_avg |
appa^{bc} + \frac{c_-(c_2-g)-c_+c_2}{c_++c_-}{f^{bc}}_d t^d \right)\frac{\phi(z)}{(w-z)^2} \cr
& \quad - \frac{c_+}{c_++c_-} \frac{t^b :j^c_{L,z}\phi:(z)}{w-z} + \mathcal{O}(f^2) \\
%
j^b_{L,z}(w):j^c_{L,\bar z}\phi:(z) = &
\left(\tilde{c} \kappa^{bc} + \frac{c_-(c_2-g)+c_+(c_4-g)}{c_++c_-}{f^{bc}}_d t^d \right)\phi(z)... | 2,721 | 1,282 | 1,483 | 2,581 | null | null | github_plus_top10pct_by_avg |
ing (NYMEX + .055) . That's still being
negotiated.
1. Enron will warrant that we will have 15,000 of FT on Koch
2. Ormet pays the demand charge ($.05) on the transport on any volumes below
15,000/d
3. Volumes above 15,000/d up to 22/d will be supplied on IT. We will
utilize the same receipt/delivery path as th... | 2,722 | 1,565 | 2,518 | 2,817 | null | null | github_plus_top10pct_by_avg |
sion, the random-walk representation (e.g., [@ffs92]) or the FK random-cluster representation (e.g., [@fk72]). In this paper, we use the random-current representation (Section \[ss:RCrepr\]), which applies to models in the Griffiths-Simon class (e.g., [@a82; @ag83]). This representation is similar in philosophy to the ... | 2,723 | 921 | 923 | 2,905 | 1,676 | 0.786891 | github_plus_top10pct_by_avg |
mmetry potential $D_{7,1}$. Similarly, the quadratic term in the fluxes $({\rm flux} \cdot {\rm flux})_4$ is mapped to the component $({\rm flux} \cdot {\rm flux})^a_3$ from the second line of the quadratic constraints in eqs. and , so that the full Chern-Simons term is mapped to $$\int D_{6\, a,a} \wedge ( {\rm flux} ... | 2,724 | 1,521 | 1,587 | 2,575 | 1,939 | 0.78427 | github_plus_top10pct_by_avg |
^lm_i$ is called a *$\rho$-generator* over $\mathbb{F}_q$ if $\rho$ is the smallest positive integer for which there are codewords $c_i(x)=(c_{i,1}(x),c_{i,2}(x), \ldots, c_{i,l}(x))$, $1\leq i \leq \rho$, in $C$ such that $C=Rc_1(x)+Rc_2(x)+\cdots +Rc_\rho(x)$.
Assume that the dimension of each $C_i$, $i=1,2,\ldots,s... | 2,725 | 948 | 1,160 | 2,791 | 2,817 | 0.776774 | github_plus_top10pct_by_avg |
_{r+s=n-1} \mathcal P(n) \otimes A^{\otimes r}
\otimes M \otimes A^{\otimes s}\right)_{S_{n}}$. Then $F_{\mathcal P,A} M$ is a module over $F_{\mathcal P}A$ over $\mathcal P$, which means that there are maps $\gamma^M:\bigoplus_{r+s=n-1}
\mathcal P(n)\otimes (F_{\mathcal P} A )^{\otimes r} \otimes
F_{\mathcal P,A} M \o... | 2,726 | 1,535 | 2,453 | 2,497 | null | null | github_plus_top10pct_by_avg |
egory ${\mathsf{Repr}}(S)$. The coreflector is given by the functor $\Psi\Phi$.
Let $(X,\mu)$ be a non-strict $S$-set. Just as in the proof of Theorem \[th:equiv\], we have the map $\beta_{\mu}\colon \Psi\Phi(X,\mu)\to X$ given by . This map is surjective, and is injective if and only if $\mu$ is connected. We show t... | 2,727 | 1,859 | 1,391 | 2,676 | null | null | github_plus_top10pct_by_avg |
} p_{\mu} [\widetilde{\Delta}_{c}(\mu)]$ for some $p_{\mu} \in {\mathbb{Z}}[v,v^{-1}]$. To calculate the $p_{\mu}$ note that, by , $Y\cong
{\mathbb{C}}[{\mathfrak{h}}]\otimes {\mathbb{C}}[{\mathfrak{h}}^*]^{\text{co}{{W}}}$. Applying $({\mathbb{C}}\otimes_{{\mathbb{C}}[{\mathfrak{h}}]}-)$ to the equation $ [Y] =\sum p_... | 2,728 | 1,872 | 2,424 | 2,589 | null | null | github_plus_top10pct_by_avg |
\
\mathcal{D}_{Ru} & -\frac{h u}{2 \left(u^2+1\right)} & -\frac{i m u}{\left(u^2+1\right)^2} \\
\mathcal{D}_{uu} & \frac{u^2 \left(u^2+3\right)}{\left(u^2+1\right)^3} & \frac{i (2 h+3) m}{2 \left(u^4-1\right)} \\
\mathcal{D}_{TR} & \frac{i m \left(u^2-1\right) \left(u^4+6 u^2-3\right)}{4 \left(u^2+1\right)^3} & \... | 2,729 | 2,529 | 2,321 | 2,632 | null | null | github_plus_top10pct_by_avg |
metric spaces enjoying certain high regularity properties - having group structure with left-invariant metric. Using methods of harmonic analysis we show that, given a compact metrizable group $G$ equipped with an arbitrary compatible left-invariant metric $d$, the Lipschitz-free space over $G$, ${\mathcal{F}}(G,d)$, s... | 2,730 | 2,884 | 383 | 2,175 | null | null | github_plus_top10pct_by_avg |
(z_t)
=& q'(g(z_t)) \nabla g(z_t) \\
=& q'(g(z_t)) \frac{z_t}{\lrn{z_t}_2}\\
\nabla^2 f(z_t)
=& q''(g(z_t))\nabla g(z_t) \nabla g(z_t)^T + q'(g(z_t))\nabla^2 g(z_t)\\
=& q''(g(z_t)) \frac{z_t z_t^T}{\|z_t\|_2^2} + q'(g(z_t)) \frac{1}{\|z_t\|_2} \lrp{I - \frac{z_tz_t^T}{\|z_t\|_2^... | 2,731 | 2,417 | 2,490 | 2,358 | null | null | github_plus_top10pct_by_avg |
ion of $n$, and $i,j,k$ are positive integers with $j\neq k$ and $\mu_j\gs\mu_k$. Suppose $T\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$, and let $\cals$ be the set of all $S\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$ such that:
- $S^j_i=T^j_i+T^k_i$;
- $S^j_l\ls T^j_l$ for every $l\neq i$;
- ... | 2,732 | 2,390 | 1,234 | 2,487 | 3,170 | 0.774313 | github_plus_top10pct_by_avg |
reduction method and of the generalized method of characteristics. A variant of the conditional symmetry method for constructing this type of solution is proposed. A specific feature of that approach is an algebraic-geometric point of view, which allows the introduction of specific first-order side conditions consiste... | 2,733 | 583 | 2,117 | 2,363 | null | null | github_plus_top10pct_by_avg |
hi_i({\mathbf x}), \\
\phi_i({\mathbf x})& =0,
\end{split}
\end{cases}
\quad
\begin{split}
{\mathbf x}&\in\Omega, \\
{\mathbf x}&\in\partial\Omega.
\end{split}\end{aligned}$$ In two dimensions with $\Omega=[-L_1,L_1]\times[-L_2,L_2]$ we introduce the positive integers $i_1\le m_1$ and $i_2\le m_2$. T... | 2,734 | 1,740 | 2,146 | 2,488 | null | null | github_plus_top10pct_by_avg |
ghtarrow \:
{\cal O}_{\Lambda}(2)^{n+1} \: \longrightarrow \:
T G {\mathbb P}^n \: \longrightarrow \: 0.$$ Using the isomorphisms above, we see this short exact sequence is the same as $$0 \: \longrightarrow \: \pi^* {\cal O} \: \longrightarrow \:
\pi^* {\cal O}(1)^{n+1} \: \longrightarrow \: T G {\mathbb P}^n \:
\lon... | 2,735 | 2,129 | 2,412 | 2,509 | null | null | github_plus_top10pct_by_avg |
ut any reference to General Relativity. Second, by not tying the reduction to any particular fluid, we achieve a large degree of generality. Clearly, the method can be extended to the case in which the higher-dimensional fluid carries a particle number or some other property, but we will not pursue this in the present ... | 2,736 | 871 | 2,508 | 2,653 | null | null | github_plus_top10pct_by_avg |
techniques of the proof of Theorem \[tth\] and Example \[desolex1\]. If $S=S(x,E)$ depends also on $x$, we conjecture that it suffices only to assume that $S$ is regular enough.
Let $$P'(x,\omega,E,D)v=
S_0{{\frac{\partial v}{\partial E}}}-\omega\cdot\nabla_x v$$ be the formal transpose of $P(x,\omega,E,D)$. Making th... | 2,737 | 989 | 2,879 | 2,643 | null | null | github_plus_top10pct_by_avg |
odel as well and have a principles means to tune the regularization parameters. Finally, the third contribution is to present methods for hyperparameter estimation that arise from the machine learning literature and apply the methodology to the tomographic reconstruction problem. In particular, the proposed methods are... | 2,738 | 2,492 | 3,538 | 2,850 | 1,806 | 0.785573 | github_plus_top10pct_by_avg |
ce. For brevity we refer to a point in a choice space (that is, a choice mapping) simply as a *choice*.
\[T:PROTOSPACE\_INCLUDES\_CHOICESPACE\] The proto-space ${\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$ of ensemble $\Psi$ includes its choice space $\prod\Psi$ (that is, $\prod\Psi \subseteq {\Psi_\heartsuit}^{{... | 2,739 | 799 | 2,320 | 2,688 | null | null | github_plus_top10pct_by_avg |
ently, the Buda-Lund hydro model lead to the discovery of a number of new, exact analytic solutions of hydrodynamics, both in the relativistic [@relsol-cyl; @relsol-ell] and in the non-relativistic domain [@nr-sol; @nr-ell; @nr-inf].
The expanding matter is assumed to follow a three-dimensional, relativistic flow, cha... | 2,740 | 811 | 3,019 | 2,734 | null | null | github_plus_top10pct_by_avg |
E_p^\Lambda,{\vec{p}})
\\\times&\nonumber\,
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\nu\gamma_5({\cancel{r}_N}+M_N)\gamma_\mu\gamma_5
u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using heavy baryon expansion and the master integrals of Sec. \[sec:mi\], and redefining ${\vec{q}}\equiv{\vec{p}\,'}-{\vec{p}}$, $$... | 2,741 | 2,068 | 2,561 | 2,503 | null | null | github_plus_top10pct_by_avg |
& {\mathcal N}_{\sigma, 1}(\gamma^b) & \cdots & {\mathcal N}_{\sigma, n-1}(\gamma^b)\\
1 & {\mathcal N}_{\sigma, 1}(\gamma^{b+1}) & \cdots & {\mathcal N}_{\sigma, n-1}(\gamma^{b+1})\\
\vdots & \vdots & \ddots & \vdots \\
1& {\mathcal N}_{\sigma, 1}(\gamma^{b+\d... | 2,742 | 2,838 | 2,781 | 2,612 | 4,126 | 0.767955 | github_plus_top10pct_by_avg |
e equivalent to the Taylor microscale $\lambda^2
= 15\int_\Omega |{\mathbf{u}}|^2 d{\mathbf{x}}/ \int_\Omega |{\bm{\omega}}|^2 d{\mathbf{x}}$ used in turbulence research [@davidson:turbulence]. Another length scale, better suited to the ring-like vortex structures shown in figures \[fig:RvsE0\_FixE\_large\](c)-(e), is ... | 2,743 | 2,633 | 2,671 | 2,596 | null | null | github_plus_top10pct_by_avg |
_q)$ for some $\w\in\mathcal{S}$. On the other hand, by Lemma \[injective\], ${\rm
Rep}_{\Gamma,\v}^*(\F_q)$ contains all the indecomposable representations in ${\rm Rep}_{\Gamma,\v}(\F_q)$. This implies the following identity $$\sum_{\v\in\mathcal{S}}M_{\Gamma,\v}^*(q)X^{\v}
=\prod_{\v\in\mathcal{S}-\{0\}}(1-X^{\v... | 2,744 | 2,188 | 2,203 | 2,438 | 3,917 | 0.769295 | github_plus_top10pct_by_avg |
ly for the other arguments. The total number of atoms can be computed from the partition function (\[Z2\]): $$\begin{aligned}
\langle N \rangle = -T\frac{\partial \ln Z_B^0}{\partial \mu}
-T\frac{\partial \ln Z_F^{eff}}{\partial \mu}.\end{aligned}$$ The first term is given by the usual expression for an ideal Bose gas ... | 2,745 | 2,538 | 2,706 | 2,631 | null | null | github_plus_top10pct_by_avg |
-2\tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La^2\}\big]\Big].\end{aligned}$$ It is here observed that $$\begin{aligned}
&(1-v_1)\{v_1I_r+(1-v_1)\La\}^{-1}\La \\
&\qquad = \{v_1I_r+(1-v_1)\La\}^{-1} \{ v_1I_r+(1-v_1)\La - v_1I_r\}\\
&\qquad= I_r - v_1 \{v_1I_r+(1-v_1)\La\}^{-1},
\\
&(1-v_1)\{v_1I_r+(1-v_1)\La\}^{-1}\L... | 2,746 | 776 | 1,765 | 2,768 | null | null | github_plus_top10pct_by_avg |
y $options = null)
{
/** @var \Zend\View\Renderer\PhpRenderer $renderer */
$renderer = $container->get('ViewRenderer')
return new MyCustomController($renderer);
}
}
In the Controller, require it be set in the __construct() function:
public function __construct(PhpRenderer $renderer)
{
... | 2,747 | 3,413 | 167 | 1,746 | 34 | 0.8344 | github_plus_top10pct_by_avg |
theta_1$ and $\theta_2$ thus anticommute with one another, $\left\{\theta_1,\theta_2\right\}=0$) $$[x,p]=i\hbar\ \ ,\ \
\left\{\theta_1,\theta_1\right\}=\frac{\hbar}{m\omega}=
\left\{\theta_2,\theta_2\right\}\ .
\label{eq:quantumbrackets}$$ Furthermore, the Hamiltonian operators is then expressed as $$H=\frac{p^2}{2m}... | 2,748 | 3,221 | 2,971 | 2,615 | null | null | github_plus_top10pct_by_avg |
see [@weinberg Section 8.7]; see also [@lorence Section VII]) \_[11]{}(x,’,,E’,E) =& \_[11]{}(x,E’,E)\_[11]{}(E,E’) (’-\_[11]{}(E’,E)), where $$\hat{\sigma}_{11}(x,E',E):={}&\sigma_0(x)\Big({{1}\over{E'}}\Big)^2\Big({{E'}\over{E}}+{{E}\over{E'}}-1+\mu_{11}(E',E)^2\Big)
\nonumber\\
\chi_{11}(E',E):={}&\chi_{{\mathbb{R}}... | 2,749 | 876 | 2,792 | 2,604 | null | null | github_plus_top10pct_by_avg |
number of integral points in each of these polytopes in polynomial time [@barvinok94] (see also [@dyerkannan97; @barvinokpommersheim99]). This gives rise to the following polynomial-time algorithm for , thereby establishing :
\[main algorithm\] Let $f \colon H \rightarrow G$ be a homomorphism of compact connected Lie ... | 2,750 | 1,035 | 2,614 | 2,532 | 807 | 0.799604 | github_plus_top10pct_by_avg |
easing lattice volume $V$. We have performed a fit as a function of $1/V$ and have seen that the large volume $\beta=3.92$ lattice gives results which are very close to the infinite volume limit. In Fig. \[fig:pert\_vs\_latt\] we plot $D(q^2)$ in the intermediate and ultraviolet regime and compare with the three-loop p... | 2,751 | 1,510 | 2,917 | 2,567 | 4,046 | 0.768475 | github_plus_top10pct_by_avg |
\in kB(G)$ and $[s(G)]$ is a system of representatives of conjugacy classes of subgroups of $G$. In this situation the set of the primitive orthogonal idempotents of $kB(G)$ is well known. These idempotents are in bijection with the conjugacy classes of subgroups of $G$. If $H$ is a subgroup of $G$, let us denote by $e... | 2,752 | 2,290 | 2,338 | 2,437 | null | null | github_plus_top10pct_by_avg |
49685);
device.on('data', function(buf) {
var ch = buf.toString('hex').match(/.{1,2}/g).map(function(c) {
return parseInt(c, 16);
});
var position = ((ch[2] & 0x0f) << 6) + ((ch[1] & 0xfc) >> 2);
position = parseInt(position);sentData(position);
});
});
A:
The arduino code should look like ... | 2,753 | 2,302 | 2,509 | 2,215 | null | null | github_plus_top10pct_by_avg |
66% 95% 0.06
Type of delivery: % Caesarean section 33% 5% 0.075
Gestational age ... | 2,754 | 5,754 | 1,590 | 1,453 | null | null | github_plus_top10pct_by_avg |
{}[\_j]{}\^2-(-1) \_ (- ) .
Integrability of the model Hamiltonian
======================================
The integrability of such types of Hamiltonian is established by constructing a complete set of commuting momentum operators that also commute with the model Hamiltonian. These operators were initially introduced... | 2,755 | 679 | 2,538 | 2,783 | 3,611 | 0.771235 | github_plus_top10pct_by_avg |
ptimization problem defined below.
Hereafter, $H^2(\Omega)$ will denote the Sobolev space of functions with square-integrable second derivatives endowed with the inner product [@af05] $$\forall\,\mathbf{z}_1, \mathbf{z}_2 \in H^2(\Omega) \qquad
\Big\langle \mathbf{z}_1, \mathbf{z}_2 \Big\rangle_{H^2(\Omega)}
= \int_{... | 2,756 | 1,562 | 3,012 | 2,623 | null | null | github_plus_top10pct_by_avg |
ather than the CGF. The proper operation of the James’s method thus relies on the accurate calculation of the DGF, yet finding its analytic expressions in cylindrical coordinates is a daunting task. To our knowledge, the analytic DGF is available only in 2D Cartesian coordinates [@bune71]. In 3D Cartesian coordinates, ... | 2,757 | 3,647 | 3,022 | 2,722 | 1,795 | 0.785659 | github_plus_top10pct_by_avg |
{aligned}$$
$$\begin{aligned}
A_{\Sigma2}\to&
-\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32}
+\frac23(\sqrt3A_{\Sigma\frac12}+A_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\\
B_{\Sigma2}\to&
-\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32}
+\frac23(\sqrt3B_{\Sigma\frac12}+B_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_... | 2,758 | 1,772 | 1,055 | 2,848 | null | null | github_plus_top10pct_by_avg |
i(1+\pi m_{i,i}')+\pi^3(\ast).$$ Here, the nondiagonal entries of this equation are considered in $B\otimes_AR$ and each diagonal entry of $a_i'$ is of the form $\pi^3 x_i'$ with $x_i'\in R$. Now, the nondiagonal entries of $-\pi^2\cdot{}^tm_{i,i}'a_i m_{i,i}'+\pi^3(\ast)$ are all $0$ since they contain $\pi^2$ as a fa... | 2,759 | 2,532 | 2,246 | 2,547 | null | null | github_plus_top10pct_by_avg |
m{Nexp}}( - \frac{1}{2} \langle \cdot,\mathbf{K} \cdot\rangle ) \cdot \exp( - \frac{1}{2} \langle \cdot,\mathbf{L} \cdot\rangle )\Big)({\bf f})\\
=\sqrt{\frac{1}{\det(\mathbf{Id+L(Id+K)^{-1}})}}
\exp(-\frac{1}{2} \langle {\bf f}, \mathbf{(Id+K+L)^{-1}} {\bf f} \rangle ),\quad {\bf f} \in... | 2,760 | 4,051 | 2,540 | 2,407 | null | null | github_plus_top10pct_by_avg |
\sum_{p_{A},p_{B},q_{A},r_{B}}\lambda\left(v\right)=\Pr\left[\mathbf{Q}_{C}=q_{C},\mathbf{R}_{C}=r_{C}\right].
\end{array}\label{eq:CbD exmaple}$$ The noncontextuality hypothesis for $\mathbf{P}_{A},\mathbf{Q}_{A},\mathbf{R}_{B}$ and $\mathbf{P}_{B},\mathbf{Q}_{C},\mathbf{R}_{C}$ is that among these jpds $\lambda$ we ... | 2,761 | 513 | 2,595 | 2,410 | null | null | github_plus_top10pct_by_avg |
D_d(M)/D_{d-1}(M)$ is clean as well.
\[conclusion\] Let $S=K[x_1,\ldots,x_n]$ be a the polynomial ring and $I\subset S$ a monomial ideal. Then the following conditions are equivalent:
1. $S/I$ is pretty clean;
2. $S/I$ is sequentially CM, and the non-zero factors in the dimension filtration of $S/I$ are clean;
3.... | 2,762 | 1,595 | 2,496 | 2,519 | 1,892 | 0.784654 | github_plus_top10pct_by_avg |
rangle + c))({\bf f}):= T\Phi({\bf f}+{\bf g})\exp(c),\quad {\bf f} \in S_d({\mathbb{R}}),$$ if $T\Phi$ has a continuous extension to $L^2_d({\mathbb{R}})_{{\mathbb{C}}}$ and the term on the right-hand side is a U-functional in ${\bf f} \in S_d({\mathbb{R}})$.
\[donsker\] Let $D \subset {\mathbb{R}}$ with $0 \in \ove... | 2,763 | 1,537 | 2,073 | 2,466 | null | null | github_plus_top10pct_by_avg |
Together with the naive bound $G(x)\leq O(1){\vbx{|\!|\!|}}^{-q}$ (cf., [(\[eq:IR-xbd\])]{}) as well as Proposition \[prp:conv-star\](ii) (with $x=x'$ or $y=y'$), we also obtain $$\begin{aligned}
{\label{eq:GGpsi-bd}}
\sum_{v'}G(v'-y)\,G(z-v')\,\psi_\Lambda(v',v)&\leq G(v-y)\,G(z-v)+
\sum_{v'}\frac{O(\theta_0^2)}{{\v... | 2,764 | 2,007 | 2,249 | 2,718 | null | null | github_plus_top10pct_by_avg |
elocity field ${\mathbf{u}}$, and it is essential that the gradient be characterized by the required regularity, namely, $\nabla\R({\mathbf{u}}) \in H^2(\Omega)$. This is, in fact, guaranteed by the Riesz representation theorem [@l69] applicable because the Gâteaux differential $\R'({\mathbf{u}};\cdot) : H_0^2(\Omega) ... | 2,765 | 807 | 2,080 | 2,777 | null | null | github_plus_top10pct_by_avg |
H^1$ semi-norm and the $L^2$ norm of the errors are reported in Table \[tab:test1\] and Figure \[fig:test1\]. These errors are computed using a 5th order Gaussian quadrature on triangles. For quadrilateral elements, the errors can be conveniently computed by dividing the quadrilateral into two triangles and then applyi... | 2,766 | 236 | 2,624 | 2,569 | null | null | github_plus_top10pct_by_avg |
notation $(m_{i,j}, s_i\cdots w_i)$ is explained in Section \[m\]. Then $h\circ m$ is an element of $\underline{H}(R)$ and $(\mathrm{Ker~}\varphi)(R)$ is the set of $m$ such that $h\circ m=(f_{i, j}, a_i\cdots f_i)$. The action $h\circ m$ is explicitly described in Remark \[r35\]. Based on this, we need to write the ma... | 2,767 | 1,249 | 1,509 | 2,597 | 3,525 | 0.771768 | github_plus_top10pct_by_avg |
our main result. We include here a new proof for the sake of completeness.
(of Proposition \[equalities: adjoint EVF and EVF\]) Let $\Lambda=(0=\lambda_0<\lambda_1<...<\lambda_{n+1}=T)$ be a subdivision of $[0,T].$ Let ${\mathfrak{a}}_k:V \times V \to \mathbb C\ \ \hbox{ for } k=0,1,...,n$ be given by $$\begin{aligned... | 2,768 | 1,999 | 2,116 | 2,573 | null | null | github_plus_top10pct_by_avg |
ackground: url(http://88t.eu/Pictures/sh/1/intr_17.jpg) no-repeat center center;
}
.pic-18 {
animation-delay: 102s;
-o-animation-delay: 102s;
-moz--animation-delay: 102s;
-webkit-animation-delay: 102s;
background: url(http://88t.eu/Pictures/sh/1/intr_18.jpg) no-repeat center center;
}
.pic-1... | 2,769 | 6,533 | 614 | 1,265 | 25 | 0.837027 | github_plus_top10pct_by_avg |
\w_0$ in the self-energy occur which also find their way into the tunneling spectra. Ref. [@JovchevAnders2013] discusses the deviations of the non-perturbatively calculated full electron-phonon self-energy from the lowest-order perturbative results.
In conclusion, we maintain the terminology of the elastic current for... | 2,770 | 880 | 2,907 | 2,723 | null | null | github_plus_top10pct_by_avg |
ppendix {#appendix .unnumbered}
========
The Higgs potential of our model is given by $V=V_2+V_3+V_4$ where $$\begin{aligned}
V_2
&\equiv&
- m_{s_1}^2 |s_1^0|^2
+ \frac{1}{\,2\,} m_{s_2}^2 (s_2^0)^2
- m_\Phi^2\, \Phi^\dagger \Phi
+ m_\eta^2\, \eta^\dagger \eta
+ m_\Delta^2\, {{\text{tr}}}(\Delta^\dagger \Delta) ... | 2,771 | 1,642 | 926 | 2,799 | null | null | github_plus_top10pct_by_avg |
hey are radially ordered in the reference plane. We will keep this point in mind without expressing the radial-ordering explicitly.
The vacuum is invariant under the global group discussed in the previous section[^6] $$\langle0|G=0,$$ where $G$ are the generators of the global subgroup. Consequently the correlation fu... | 2,772 | 1,253 | 2,229 | 2,608 | 3,243 | 0.773797 | github_plus_top10pct_by_avg |
ell-1} \in \Omega_\ell}\exp(\theta_{j_{\ell-1}})}{\exp(\theta_i)+\exp(\theta_{\i}) +\sum_{j_{\ell-1} \in \Omega_\ell}\exp(\theta_{j_{\ell-1}})-\big(\sum_{j_{\ell-1} \in \Omega_\ell}\exp(\theta_{j_{\ell-1}})\big)/|\Omega_\ell|} \nonumber\\
&=&\Bigg({\frac{\exp(\theta_{i})+\exp(\theta_{\i})}{\sum_{j_{\ell-1} \in \Omega_\... | 2,773 | 1,711 | 2,039 | 2,515 | null | null | github_plus_top10pct_by_avg |
9.3 56.6 9.5 4.6
4--6 times/week 310 18.7 54.2 16.1 11.0
Daily 222 16.2 43.7 ... | 2,774 | 4,883 | 1,717 | 1,888 | null | null | github_plus_top10pct_by_avg |
Lemma \[lemma:SpecFact\] $\bar J_e$ is special, hence by Lemma \[lemma:SpecSplitNullExt\] $\widehat{J_e}$ is a special Jordan algebra and $J_e\in {\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$. Therefore, ${\mathrm{Di}}{\mathrm{SJ}}\langle X\rangle\in{\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$. Since the free algebra of the var... | 2,775 | 1,319 | 1,990 | 2,673 | null | null | github_plus_top10pct_by_avg |
^{\iota}(X)|\right.
\nonumber\\
&-&\left.
|\psi^{\iota}(X)\rangle\langle\psi^{\iota}(X)|
\left\{\ln(\hat{\rho}),\hat{H}\right\}_{\mbox{\tiny\boldmath$\cal B$}}
\right)\;.\nonumber\\
\label{eq:wavematrix}\end{aligned}$$ Equation (\[eq:wavematrix\]) can be written as a system of two coupled equations for the wave fields ... | 2,776 | 739 | 1,686 | 3,025 | null | null | github_plus_top10pct_by_avg |
t statistic is equal to 176.49 and the adequation with the Dirichlet distribution is rejected. However, we can see that as conjectured, the simulated sample looks like a simulated sample from a Dirichlet distribution.
Appendix: non-crossing property for the paths of the RST
============================================... | 2,777 | 2,603 | 2,825 | 2,552 | null | null | github_plus_top10pct_by_avg |
lta S=2}$ to the mass matrix. From the input parameters $ B_K=0.75, F_K=160 \,{\hbox{MeV}}, m_K= 498\,{\hbox{MeV}} ,
\Delta m_K = 3.51 \times 10^{-15} \hbox{GeV} $ and the relation $$M_{12}^R = \frac{1}{2m_K}
\langle \bar{K^0} | {\cal H}^{\Delta S=2} |K^0\rangle^{*}
= \frac{G_F^2 m_W^2}{16\pi^2} \frac{1}{2m_K}... | 2,778 | 2,505 | 3,014 | 2,622 | 3,324 | 0.773161 | github_plus_top10pct_by_avg |
ac{1}{|x-y|^{\alpha-d}}$$ for a constant $c_{d,\alpha}$ for $d>1$ and $\alpha\in(0,2$); see [@bucur]. If the point $y$ is chosen outside a domain $D$, then we can construct $G$ as an exact solution to the homogeneous version of the fractional Dirichlet problem in ; that is, $u(x)=G(x,y)$ for $x\in D$ and ${g}(x)=G(x,y)... | 2,779 | 874 | 1,627 | 2,289 | 1,023 | 0.795497 | github_plus_top10pct_by_avg |
mmatic functions consisting of two-point functions. Let $$\begin{aligned}
{\label{eq:tildeG-def}}
\tilde G_\Lambda(y,x)
=\sum_{b:{\overline{b}}=x}{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b,\end{aligned}$$ which satisfies[^5] $$\begin{aligned}
{\label{eq:G-delta-bd}}
{{\langle \varphi_y\varphi... | 2,780 | 3,272 | 2,849 | 2,667 | null | null | github_plus_top10pct_by_avg |
ls of the form $G_\muhat(q)$ which have integer coefficients (see Remark \[G\]).
Example: Hilbert Scheme of $n$ points on $\C^\times\times\C^\times$ {#Hilbert}
===================================================================
Throughout this section we will have $g=k=1$ and $\muhat$ will be either the partition $(n... | 2,781 | 2,351 | 2,181 | 2,605 | 3,695 | 0.770614 | github_plus_top10pct_by_avg |
obtain the equations $$(z_j)_1+(z_j)_1^2=0, ~~~ (x_j)_1=0, ~~~(z_j)_1+(x_j)_2=0.$$ By combining all these, we see that $F_j$ is isomorphic to $ \mathbb{A}^{1} \times \mathbb{Z}/2\mathbb{Z}$ as a $\kappa$-variety.
**
We finally prove Lemma \[l46\].
We start with the following short exact sequence $$1\rightarrow \til... | 2,782 | 1,620 | 520 | 2,916 | 1,330 | 0.790897 | github_plus_top10pct_by_avg |
Adult 26 (96) 148 (94) 47 (84) 221 (92) 283 (92) 648 (91)
Paediatric 1 (4) 10 (6) ... | 2,783 | 1,365 | 3,131 | 2,884 | 3,240 | 0.773818 | github_plus_top10pct_by_avg |
cture grammar. We construct the grammar $G'=(V',\Sigma,(S,1),R')$ with capacity function $\mathbf{1}$ and $$\begin{aligned}
V'&=& \{(A,i) {:}A \in V, 1\leq i\leq \kappa(A)\},\\
R'&=& \{\alpha' \to \beta' {:}\alpha' \in h(\alpha), \beta' \in h(\beta), \mbox{ for some } \alpha \to \beta \in R\},
\end{aligned}$$... | 2,784 | 3,808 | 2,255 | 2,611 | 3,598 | 0.7713 | github_plus_top10pct_by_avg |
{erf}^{-1}\left(1-\frac{2}{e\Psi}\right)\right),\end{aligned}$$ where $\beta\simeq0.5772$ denotes the Euler’s constant.
See Appendix \[App:proofaveGlsn\]
Based on , we further obtain the limiting behavior of the average throughput gain when $\Psi$ becomes large in the following corollary.
\[Cor:aveGlsngrO\] As $\Psi... | 2,785 | 1,326 | 1,064 | 2,966 | 1,905 | 0.784541 | github_plus_top10pct_by_avg |
\-
Hospital stay
\<48 h 47 7 (14.9) 40 (85.1) 63.67 (26.3-179.2) 107.96 (31.37... | 2,786 | 4,646 | 2,061 | 1,937 | null | null | github_plus_top10pct_by_avg |
ns (e.g. [@GM1 Exercize II.1.6], [@L Section 6], or [@FT A.2], retold in [@Ka Section 1.4]). All the descriptions are equivalent. Objects in ${\operatorname{Fun}}(\Lambda,k)$ are usually called [*cyclic vector spaces*]{}.
The cyclic category $\Lambda$ is related to the more familiar [*simplicial category*]{} $\Delta^{... | 2,787 | 1,535 | 2,353 | 2,617 | 3,369 | 0.77283 | github_plus_top10pct_by_avg |
tion of a particle with mass $m$ in the spherically symmetric potential field (also see [@Andrianov.hep-th/9404061; @Bagrov.quant-ph/9804032]). The spherical symmetry of the potential allows to reduce this problem to the one-dimensional problem about the motion of this particle in the radial field $V(r)$, defined on th... | 2,788 | 5,205 | 653 | 2,259 | 4,153 | 0.767717 | github_plus_top10pct_by_avg |
utting all the previous calculations , , , , together, one obtains $A = - \sum_{P \in S} \beta_{\cC,P}^{(k)}$ where $$\label{eq:beta}
\beta_{\cC,P}^{(k)}\! =\! \alpha_{\cC,P}^{(k)}
+ \dim\! \frac{\mathcal{O}_{\PP^2_w,P}\left( kH\! +\! K_{\PP^2_w}\! -\! \mathcal{C}^{(k)} \right)}{\mathcal{M}_{\mathcal{C},P}^{(k)}}
+ \!\... | 2,789 | 2,426 | 1,783 | 2,556 | null | null | github_plus_top10pct_by_avg |
rho _{h}(1+\delta )}(c(r_h)\rho )\leq
\frac{Q^{\rho_h+\varepsilon}}{(\lambda t)^{(2h+q+d/p_{\ast })\rho_h(1+\delta )}}
\leq
C\times \frac{e^{2c_\kappa \rho}}{(\lambda t)^{(q+2d/p_{\ast })(1+\delta )}}.$$Since $\rho \geq 1$ we conclude that $$\left\Vert p^{\eta ,\kappa }\right\Vert _{q,p}\leq
C\times \frac{e^{2c_\kapp... | 2,790 | 1,675 | 711 | 2,938 | null | null | github_plus_top10pct_by_avg |
inary results about regularity. Namely, in Section [sect:3.1]{} we recall and develop some results concerning regularity of probability measures, based on interpolation type arguments, coming from [@[BC]]. These are the main instruments used in the paper. In Section \[sect:3.2\] we prove a regularity result which is a ... | 2,791 | 1,617 | 688 | 2,767 | null | null | github_plus_top10pct_by_avg |
ation maps is an unconventional task, we visualize the reconstructions in order to inspect their quality. For the activations obtained from the first convolutional layer seen in Figure \[fig:conv1\_layer\_reconstructions\], it is obvious that the VAEs are effective at reconstructing the activation maps. The only potent... | 2,792 | 674 | 2,506 | 1,667 | null | null | github_plus_top10pct_by_avg |
dots,a_n)=T\bar{f_i}(a_1,\ldots,a_n).$$
From the last equality we obtain $\bar{f_i}(a_1,\ldots,a_n)=0$, so $A\vDash\bar{f_i}$ and $\bar{f_i}\in T_0({\mathrm{Var}})$. By the previous proposition $f_i\in T_0({\mathrm{Di}}{\mathrm{Var}})$.
We recall that $f$ is called a multilinear s-identity (in the case of ordinary al... | 2,793 | 1,546 | 2,255 | 2,499 | null | null | github_plus_top10pct_by_avg |
roups nor virtually affine groups.
This paper is organized as follows: in Section \[s:recall\] we review some general facts and introduce the notation used throughout the text. In Section \[s:gever\] we describe some properties of the Veronese curve which are useful for our purposes. In Section \[s:chvh\] we character... | 2,794 | 1,224 | 1,324 | 2,868 | null | null | github_plus_top10pct_by_avg |
entries of $\Sigma$. Assume that $k \geq u^2$. Then, $$\label{eq::B-and-lambda}
B= \sup_{P \in \mathcal{P}_n^{\mathrm{OLS}} } \max_j \|G_j(\psi(P)) \| \leq C \frac{ \sqrt{k} }{u^2},\ \ \
\overline{H}=\max_j \sup_{P \in \mathcal{P}_n^{\mathrm{OLS}}} \| H_j(\psi(P))\|_{\mathrm{op}} \leq C
\frac{k}{u^3},$$ and $$\label... | 2,795 | 2,396 | 1,840 | 2,506 | null | null | github_plus_top10pct_by_avg |
be estimated as $$\begin{aligned}
{\label{eq:2nddec-bd:n=jbd}}
&\sup_y\sum_{z,z',x}|x|^2\tau_{y,z}\big(\delta_{z,z'}+\tilde
G_\Lambda(z,z')\big)P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}(z',x){\nonumber}\\
&=\sup_y\sum_{z,z',x}\tau_{y,z}\big(\delta_{z,z'}+\tilde G_\Lambda(
z,z')\big)\,{{\langle \varphi_{z'}\var... | 2,796 | 2,125 | 2,180 | 2,811 | 3,566 | 0.771517 | github_plus_top10pct_by_avg |
density of $\Th$ given $\Om$ and $W$. To make it easy to derive sufficient conditions that $\ph_H$ is minimax, we show the following lemma.
\[lem:alter\_m(W)\] The marginal density $m(W)$ can alternatively be represented as $$m(W)=\int_{\Rc_r}f_\pi(\La;W)\dd\La,$$ where $$f_\pi(\La;W)=(2\pi v)^{-qr/2}|\La|^{q/2}\pi_2... | 2,797 | 1,299 | 1,686 | 2,672 | null | null | github_plus_top10pct_by_avg |
tit{of type I}};\\
0 & \quad \textit{if $L_j$ is \textit{of type II}}.
\end{array} \right.$$ Recall from the beginning of Section 2 that we can choose a uniformizer $\pi$ in $B$ such that $\sigma(\pi)=-\pi$ and $\pi^2=2\delta$ with $\delta (\in A) \equiv 1$ mod 2.
We reproduce the beginning of Section 3 of... | 2,798 | 2,798 | 2,037 | 2,584 | null | null | github_plus_top10pct_by_avg |
en there is a tuple ${\mathbf{s}}=(s_1, \ldots, s_n) \in S_1 \times \cdots \times S_m$, with ${\mathbb{P}}[{\mathsf{X}}_i=s_i]>0$ for each $1\leq i \leq m$, such that the largest zero of $f(s_1,\ldots, s_m;t)$ is smaller or equal to the largest zero of ${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_m;t)$.
The proo... | 2,799 | 1,604 | 2,620 | 2,574 | 4,159 | 0.767674 | github_plus_top10pct_by_avg |
=====================================================
Here we note that a standard periodization technique, which substitutes $Q=\S$ to Eq. (\[eq:periodize\]), cannot reproduce the Mott gap for large $U$. Because Im$G=0$ in the Mott gap, Im$\S$ must be 0 or $\infty$ in the whole Brillouin zone. However, this situation... | 2,800 | 1,122 | 2,012 | 2,904 | null | null | github_plus_top10pct_by_avg |
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