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 |
|---|---|---|---|---|---|---|---|
$-th ($k=1,2,\ldots,K$) conv-layer, where $K$ and $\{n_{k}\}$ are hyper-parameters. Let each latent pattern $u$ in the $k$-th conv-layer correspond to a square deformation range, which is located in the $D_{u}$-th slice of the conv-layer’s feature map. $\overline{\bf p}_{u}$ denotes the center of the range. As analyzed... | 2,001 | 815 | 574 | 2,046 | 1,204 | 0.792792 | github_plus_top10pct_by_avg |
\leq r-1$.
\[nonzerodivisorgammapower\] For any $t\in{\{0,1,\cdots,r-1\}}$, $\gamma^t$ is not a zero divisor of the ring $\cR_{m,r}$. This holds since the coefficients of $\gamma^t \cdot f(\gamma)$ are the same as those of $f(\gamma)$ (shifted cyclicly $t$ positions).
Matrices over Commutative Rings
----------------... | 2,002 | 4,263 | 3,131 | 1,860 | 3,160 | 0.774427 | github_plus_top10pct_by_avg |
local basis of the tangent space of $\partial G$ (and ${{\frac{\partial g}{\partial \tilde y_i}}}\in C(\Gamma_-)$ is to be understood in a local sense), and $\Sigma\in C(\ol G\times S\times I)$ such that ${{\frac{\partial \Sigma}{\partial x_j}}}\in C(\ol G\times S\times I),\ j=1,2,3$. Then the unique (classical) soluti... | 2,003 | 588 | 1,577 | 1,958 | null | null | github_plus_top10pct_by_avg |
care of the second statement. Denoting by $i\colon[1]\to\ulcorner$ the sieve classifying the horizontal morphism $(0,0)\to (1,0)$ and by $k'\colon[1]\to\square$ the functor classifying the vertical morphism $(1,0)\to (1,1)$, the cofiber morphism is given by $$\label{eq:cof}
{\mathsf{cof}}\colon{\sD}^{[1]}\stackrel{i_\... | 2,004 | 1,709 | 1,383 | 2,100 | 3,337 | 0.773016 | github_plus_top10pct_by_avg |
j - K\_[j,C]{},j=2,3, and define a (densily defined) closed linear operator $T_C:L^2(G\times S\times I)^3\to L^2(G\times S\times I)^3$ by setting D(T\_C):=&{L\^2(GSI)\^3 | T\_[j,C]{}L\^2(GSI), j=1,2,3}\
T\_C:=&(T\_[1,C]{},T\_[2,C]{},T\_[3,C]{}). Let $f\in L^2(G\times S\times I)^3$ and $g\in T^2(\Gamma_-)^3$. In the cas... | 2,005 | 489 | 2,127 | 2,056 | null | null | github_plus_top10pct_by_avg |
National Science Foundation Grants 200021-165977 and 200020-162884.
[^2]: Rényi Institute, Hungarian Academy of Sciences, P.O.Box 127 Budapest, 1364, Hungary; `tardos@renyi.hu`. Supported by the Cryptography “Lendület” project of the Hungarian Academy of Sciences and by the National Research, Development and Innovati... | 2,006 | 103 | 2,209 | 2,271 | null | null | github_plus_top10pct_by_avg |
s.
It has therefore recently been suggested to study the *asymptotic growth* of multiplicities (e.g., [@grochowrusek12 §2.2]). The natural object is the *Duistermaat–Heckman measure*, which is defined as the weak limit $$\label{definition duistermaat-heckman}
\operatorname{DH}_X := \lim_{k \rightarrow \infty} \frac... | 2,007 | 618 | 1,615 | 1,982 | null | null | github_plus_top10pct_by_avg |
}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
---
abstract: 'The family of left-to-right algorithms reduces input numbers by repeatedly subtracting the smaller number, or multiple of the smaller number, from the larger number. This paper describes how to extend any such algorithm... | 2,008 | 761 | 1,291 | 1,600 | null | null | github_plus_top10pct_by_avg |
tion, and the rest of $\kappa_j-p$ items that are ranked on the bottom. An example of a sample with position-4 ranking six items $\{a,b,c,d,e,f\}$ might be a partial ranking of $(\{a,b,d\}>\{e\}>\{c,f\})$. Since each sample has only one separator for $2<p$, Theorem \[thm:main2\] simplifies to the following Corollary.
... | 2,009 | 1,657 | 1,474 | 1,766 | null | null | github_plus_top10pct_by_avg |
one";
}
Is This what you are looking for? Anything needing to be changed to meet your needs?
Q:
Android: Gradle compilation error expects element value to be a constant expression - Feature Module
This is my first multi-module project.
This login activity exists in a Feature Module which gets many of its depende... | 2,010 | 599 | 1,377 | 2,219 | null | null | github_plus_top10pct_by_avg |
ar{F}}_{\beta _\kappa }\,|\,\mu <\kappa <\nu \rangle
\subset U^-(\chi )
\end{aligned}$$ for all $\mu ,\nu \in \{1,2,\dots ,n\}$ with $\mu <\nu $.
We prove the first relation for $\mu =1$ and all $\nu \in \{2,3,\dots ,n\}$. Then the first relation for $\mu >1$ follows from $$E_{\beta _\mu }E_{\beta _\nu }-\chi (\... | 2,011 | 3,340 | 1,814 | 1,810 | null | null | github_plus_top10pct_by_avg |
d simple closed curves that are dual to $a$.
Claim 1: $(g_a)_{\#}$ agrees with $\lambda$ on every vertex of $\{[a]\} \cup L_a \cup D_a$.
Proof of Claim 1: We already know that $(g_a)_{\#}$ agrees with $\lambda$ on every vertex of $\{[a]\} \cup L_a$. Let $d$ be a 1-sided simple closed curve that is dual to $a$. Let $T... | 2,012 | 1,144 | 1,588 | 2,058 | 3,150 | 0.774478 | github_plus_top10pct_by_avg |
silon$, denote $N:=c(1+\Delta w_2)>0$. Take $n_1\in \ZZ_{>0}$ such that $\frac{n_1}{N}\in [\lambda,\lambda+\varepsilon)$. Note that $\frac{n_1+j}{N}\in [\lambda,\lambda+\varepsilon)$ for all $j=0,...,w_2-1$, hence there is exactly one such $j$ for which $a:=n_1+j\equiv k \mod (w_2)$. This proves the claim.
Partial ver... | 2,013 | 1,821 | 1,565 | 1,933 | 3,213 | 0.773994 | github_plus_top10pct_by_avg |
st be true, as the 2PI theory is built around the requirement that $\left\langle \varphi^A\varphi^B\right\rangle=G^{AB}$
The 1PIEA $\Gamma_{1PI}$ is recovered from the 2PIEA $\Gamma$ as
\_[1PI]{}=where the correlations $G_0$ are slaved to the mean field through
\_[,(AB)]{}=0 One further derivative shows that
G\^[CD... | 2,014 | 543 | 2,527 | 2,011 | null | null | github_plus_top10pct_by_avg |
the data.
Guaranteeing positive MUTFs greater than some minimum [@Bro03; @Maj07] would require a change to the likelihood. The approach taken in @Rid06 is to specify independent left-truncated gamma prior distributions for the the expected MUTFs, $\mu_j$ for $j=1,\ldots,u$. However any such change would not lead to th... | 2,015 | 2,734 | 2,595 | 1,759 | 1,010 | 0.795817 | github_plus_top10pct_by_avg |
have the following dynamical system:
$\dot{x_1}= -x_2 + (x_1(1-(x_1^2+x_2^2)^2))$ , $
\dot{x_2}= x_1 + (x_2(1-(x_1^2+x_2^2)^2))$,
$\dot{x_3}= \epsilon x_3$ .
I am required to work out the flow for this system. I have switched it to cylindrical coordinates obtaining $\dot{r}=r(1-r^4)$ , $\dot{\theta}=1$, $\dot{z}=\e... | 2,016 | 4,226 | 1,447 | 2,037 | 682 | 0.80236 | github_plus_top10pct_by_avg |
}$$ provided these expressions are well defined. We shall now show that the expressions are well defined provided $\Theta_0 \ne \pi/2$, and we give a more explicit formula for ${\dot {\cal F}}_{\Theta_0}$.
Suppose hence from now on that $\Theta_0 \ne \pi/2$. We work in global Minkowski coordinates in which points on M... | 2,017 | 3,561 | 2,265 | 1,925 | null | null | github_plus_top10pct_by_avg |
htarrow}}}{\underline{b}}_{i+1}\}\cap\big\{n_{b_i}>0,~b_i\text{ is pivotal for }y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\big\}\Big\},\end{aligned}$$ where, by convention, ${\underline{b}}_{T+1}=x$. Then, by ${\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underse... | 2,018 | 971 | 1,928 | 1,961 | null | null | github_plus_top10pct_by_avg |
Recall that $\underline{G}$ is a closed subgroup scheme of $\underline{M}^{\ast}$ and $\tilde{G}$ is a closed subgroup scheme of $\tilde{M}$, where $\tilde{M}$ is the special fiber of $\underline{M}^{\ast}$. Thus we may consider an element of $\tilde{G}(\kappa_R)$ as an element of $\tilde{M}(\kappa_R)$. Based on Secti... | 2,019 | 1,461 | 1,510 | 1,725 | 2,679 | 0.777923 | github_plus_top10pct_by_avg |
n{aligned}
\label{eq:hessian}
H(\theta) = -\sum_{j=1}^n \sum_{a=1}^{\ell_j} \sum_{i<\i \in S_j} \I_{\big\{(i,\i) \in G_{j,a}\big\}} \lambda_{j,a} \Bigg( (e_i - e_{\i})(e_i - e_{\i})^\top \frac{\exp(\theta_i + \theta_{\i})}{[\exp(\theta_i) + \exp(\theta_{\i})]^2}\Bigg).\end{aligned}$$ It follows from the definition tha... | 2,020 | 1,234 | 1,797 | 1,923 | null | null | github_plus_top10pct_by_avg |
We are going to prove such $\delta_n, u_n$ are two sequences we need.
We hope to apply Theorem \[estimate\], so we compute, for each $n$, $$\begin{aligned}
C^2\delta_n|u_n|^{-1}|\varphi_n|\mathscr{W}_n^{\frac{1}{2}}=\frac{1}{4}C^2\Omega_n^{-2}\exp\left(-\frac{\varphi_n^2}{2^8c_1\Omega_n^4}\right)|\varphi_n|\mathscr{W... | 2,021 | 769 | 1,682 | 1,910 | null | null | github_plus_top10pct_by_avg |
a star unless $|\cI|=3$ and $\cI=\binom{K}{2}$ for some $K=\{x,y,z\}$. But then $\{\{x\},\{x,y\},\{x,z\}\}\subseteq \cH_x$, and so $|\cI|=|\cH_x|$, which is case (\[case:2\]) of the theorem with $M=\emptyset$.\
Thus we may assume that $\cH$ contains a set of size $3$ and, consequently, also contains a star of size $4$.... | 2,022 | 987 | 454 | 2,164 | null | null | github_plus_top10pct_by_avg |
the galactic core $1/\kappa(\rho)\sim
\lambda_{DE}[\Lambda_{DE}/8\pi\rho_H]^{(1+\alpha_\Lambda)/2}$, where $\rho_H$ is the core density. Even though $\lambda_{DE}=
14010^{+800}_{-810}$ Mpc, because $\rho_H\gg\Lambda_{DE}/2\pi$, $\alpha_\Lambda$ can be chosen so that $1/\kappa(r)$ is comparable to typical $r_H$. Doing s... | 2,023 | 4,009 | 2,415 | 2,023 | 2,231 | 0.781598 | github_plus_top10pct_by_avg |
127 to EV-B73 and EV-B strains.
Region Position ^1^ TO-127 versus EV-B73 ^2^ TO-127 versus EV-B ^3^
---------------- -------------- -------------------------- ------------------------ -------- ------------
Partial 5′ UTR 1--327 76--82 Non coding ... | 2,024 | 5,238 | 1,676 | 1,379 | null | null | github_plus_top10pct_by_avg |
orresponds to a diagonal of $M$ in which the variance of the entries is $2$ instead of $1$. (See Theorem \[T:semicircle-law-general\] below and the discussion following it.)
On the other hand, the analogous results *are* simple in the case in which the characters $\chi \in \widehat{G}$ are all real-valued, so that the... | 2,025 | 2,544 | 2,466 | 1,777 | null | null | github_plus_top10pct_by_avg |
A}_1\equiv \sum_i a_{1i}\hat{\Pi}_{1i},\qquad \hat{A}_2\equiv\sum_k a_{2k}\hat{\Pi}_{2k}.$$ The generating function for the moments ${\left<A_1^{n_1}A_2^{n_2}\right>}$ reads $$\begin{split}
& C{\left(\zeta_1,\zeta_2\right)}=\text{Tr}{\left(e^{i\zeta_1\hat{A}_1}e^{i\zeta_2\hat{A}_2}\hat{\rho}\right)}=\sum_{i,k}e^{i\zeta... | 2,026 | 1,446 | 1,562 | 1,890 | null | null | github_plus_top10pct_by_avg |
liver necroinflammation and fibrosis. All significant factors identified by the univariate analysis were entered into the multivariate models for identifying predictors associated with marked alterations of liver histology. A *P* value less than 0.05 was considered statistically significant.
RESULTS
=======
The base... | 2,027 | 331 | 2,106 | 2,369 | null | null | github_plus_top10pct_by_avg |
, with $H \cap A$ a Hall $\pi$-subgroup of $A$ and $H \cap B$ a Hall $\pi$-subgroup of $B$.
Next we record some arithmetical lemmas, that will be applied later.
\[SylowSym\] Let $G$ be the symmetric group of degree $k$ and let $s$ be a prime. If $s^{ N}$ is the largest power of $s$ dividing $|G|=k!$, then $N \leq \fr... | 2,028 | 4,461 | 2,438 | 1,887 | null | null | github_plus_top10pct_by_avg |
=\|\phi(x)-\phi(y)\|_{\infty}$.
By the pigeonhole principle there exist two pairs, say $(x_1,y_1)$ and $(x_2,y_2)$, for which $j(x_1,y_1) = j(x_2,y_2)=j$. It is easy to verify that our assumptions on the four real numbers $\phi(x_1)_j$, $\phi(x_2)_j$, $\phi(y_1)_j$, $\phi(y_2)_j$, are contradictory. Thus $d(n,l_\infty... | 2,029 | 3,697 | 2,733 | 1,837 | null | null | github_plus_top10pct_by_avg |
gation for different values of $k$. Only the top three categories with the highest transition probabilities are shown. With high consistency, the transition probabilities to the same topic increase while those to other categories decrease with ascending order $k$.[]{data-label="fig:sameornot_msnbc"}](sameornot_MSNBC){w... | 2,030 | 415 | 1,500 | 1,177 | null | null | github_plus_top10pct_by_avg |
uad \textit{if $i$ is even and $L_i$ is \textit{of type $I^e$}};\\
\begin{pmatrix} \tilde{s}_i''&\pi \tilde{r}_i''& \tilde{t}_i''\\ \tilde{y}_i''&1+\pi \tilde{x}_i''& \tilde{u}_i''\\
\pi \tilde{v}_i''&\pi \tilde{z}_i''&1+\pi \tilde{w}_i'' \end{pmatrix} & \quad \textit{if $i$ is odd and $L_i$ is \textit{free of... | 2,031 | 4,320 | 1,522 | 1,343 | 3,634 | 0.771124 | github_plus_top10pct_by_avg |
}$$ Hence combining (\[eqn:De1\]) and (\[eqn:De2\]) gives that $$\begin{aligned}
\label{eqn:De0}
\De(W;\pi_{GB}^J)&=\De_1(W;\pi_{GB}^J)-\De_2(W;\pi_{GB}^J)-(q-3r-3)\tr M_{GB} \non\\
&= -(q-3r+3-a-2b)\tr M_{GB} +\De_3+\De_4,\end{aligned}$$ where $$\begin{aligned}
\De_3 &= -2(b-2)\int_{\Rc_r}\tr[(I_r-\La)^{-1}\La]f_{GB}(... | 2,032 | 880 | 649 | 2,196 | null | null | github_plus_top10pct_by_avg |
x) = \langle U_x \zeta^U_j,\zeta^U_k\rangle$$ are called *coordinate functionals* (associated to $U$ and $\{\zeta^U_1,...,\zeta^U_{d_U}\}$). They satisfy the following property, as a consequence of [@hewitt2013abstract Theorem 27.20]: $$\begin{aligned}
\forall f\in C(G)\,\forall x\in G,\, f\ast \varphi^U_{jk}(x) = \sum... | 2,033 | 1,728 | 1,873 | 1,816 | null | null | github_plus_top10pct_by_avg |
athbb{R}}^{d})$, $j=1,\ldots,d$. We denote $\nabla \phi$ the $%
d\times d$ matrix field whose $(i,j)$ entry is $\partial_j\phi^i$ and $%
\sigma (\phi )=\nabla \phi (\nabla \phi )^{\ast }$.
We suppose that $\sigma (\phi )$ is invertible and we denote $\gamma (\phi
)=\sigma ^{-1}(\phi ).$ Then $$\int (\partial _{i}f)(\p... | 2,034 | 710 | 1,881 | 2,011 | null | null | github_plus_top10pct_by_avg |
\exp(\theta_{j_2})}{W-\exp(\theta_{j_1})}\cdots \Bigg( \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i, \\ j_1,\cdots,j_{\ell-2}}} \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-2}}\exp(\theta_{k})}\Bigg)\Bigg)\Bigg) \frac{e^{2b}}{\kappa-\ell+1} \label{eq:posl_upper2}\end{aligned}$$ Consider the last su... | 2,035 | 1,214 | 1,637 | 1,838 | null | null | github_plus_top10pct_by_avg |
and $U(Y_*) \simeq \log{\epsilon}$, we estimate the slope of the straight line connecting two points $\left(1,U(1)\right)$ and $\left(Y_*,U(Y_*)\right)$ in the $(Y,U)$ plane as $(U(Y_*)-U(1))/(Y_*-1)\simeq
\sqrt{{\epsilon}}(\log {\epsilon})$, which approaches zero in the limit ${\epsilon}\rightarrow 0$. Thus, the tran... | 2,036 | 714 | 1,882 | 2,106 | null | null | github_plus_top10pct_by_avg |
specific regions of the parameters space.
The HMF model is characterized by the following Hamiltonian $$\label{eq:ham}
H = \frac{1}{2} \sum_{j=1}^N p_j^2 + \frac{1}{2 N} \sum_{i,j=1}^N
\left[1 - \cos(\theta_j-\theta_i) \right]$$ where $\theta_j$ represents the orientation of the $j$-th rotor and $p_j$ is its conjuga... | 2,037 | 4,158 | 2,689 | 1,721 | null | null | github_plus_top10pct_by_avg |
$ (or a renormalized one), see e.g., [@Minakata:2015gra] and the references therein. Possible interpretation of applicability of the perturbative framework to the region of solar level crossing has been discussed [@Xu:2015kma; @Ge:2016dlx]. Another example for the similar phenomena is the one at the small atmospheric m... | 2,038 | 456 | 2,004 | 2,054 | null | null | github_plus_top10pct_by_avg |
nu_{\tau}) \equiv P(\nu_{\mu} \rightarrow \nu_{\tau})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{\tau})_{ \text{non-unitary} }^{(0)}$ is presented. The parameters used are the same as in figure \[fig:Pmue\_energy\_dist\]. []{data-label="fig:Pmutau_energy_dist"}](Pmutau_energy_dist_non_unitary_small_size.jpeg "fi... | 2,039 | 232 | 1,743 | 1,881 | 1,791 | 0.785673 | github_plus_top10pct_by_avg |
ick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;i;{b\!\!+\!\!3};{b\!\!+\!\!4},;2,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end... | 2,040 | 873 | 2,226 | 1,940 | 244 | 0.816382 | github_plus_top10pct_by_avg |
} (UX)_{\alpha k} (UX)^*_{\beta l}
\hat{S}_{kl}^{(2)}
+
\sum_{k L} (UX)_{\alpha k} W^*_{\beta L}
\hat{S}_{kL}^{(1)}
\nonumber \\
&+&
\sum_{K l} W_{\alpha K} (UX)^*_{\beta l}
\hat{S}_{K l}^{(1)}
+
\sum_{K L} W_{\alpha K} W^*_{\beta L}
\hat{S}_{KL}^{(0)},
\nonumber \\
S_{\alpha \beta}^{(4)} &=&
\sum_{k l} (UX)_{\al... | 2,041 | 1,582 | 2,200 | 2,059 | null | null | github_plus_top10pct_by_avg |
[ecsd6a\]\
& S\_0C\^2(I,L\^(G)),\[ecsd6a-a\]\
& C\^1(I,L\^(GS,L\^1(S’)))C\^1(I,L\^(GS’,L\^1(S))).\[ecsd6a-b\]
Let $f\in C^1(I,L^2(G\times S))$ and let $g\in C^2(I,T^2(\Gamma_-))$ which satisfies the *compatibility condition* \[cc\] g(E\_m)=0. Then the problem (\[se1\]), (\[se2\]), (\[se3\]) has a unique solution $\psi... | 2,042 | 440 | 1,272 | 2,071 | null | null | github_plus_top10pct_by_avg |
^{ - i ( h_{k} - h_{n} ) x}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l}
(UX)^*_{\alpha n} (UX)_{\beta n}
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K m}
\left\{ (UX)^{\dagger} A W \right\}_{m K}
\left\{ W ^{\dagger} A (UX) \right\}_{... | 2,043 | 3,251 | 2,416 | 1,938 | null | null | github_plus_top10pct_by_avg |
bb{R})$ such that $\gamma_0\iota M\ddot{o}b(\hat{\Bbb{R}})\gamma^{-1}_0$ preserves $$\{[x,w,z]\in\Bbb{P}^2_{\Bbb{R}}:\vert y\vert^2+\vert w\vert^2<\vert x\vert^2\}.$$ Hence $\gamma_0\iota\PSL(2,\hat{\Bbb{R}})\gamma^{-1}_0=\PO^+(2,1)$. Part (\[l:con2\]) is now trivial.
\[t:liedim\] Let $\Gamma\subset\PSL(2,\Bbb{C})$ be... | 2,044 | 1,211 | 1,333 | 1,913 | null | null | github_plus_top10pct_by_avg |
\beta}\,,\qquad
\langle{}_{\dot{\alpha}}|\psi^\mu_0|{}_{\dot{\beta}}\rangle\ =\
-\frac{i}{\sqrt{2}}(C^T\gamma^\mu)_{\dot{\alpha}\dot{\beta}}\,.$$
Triviality of the extra unphysical symmetries at the linearized level {#app B}
=====================================================================
First, in order to sho... | 2,045 | 907 | 1,818 | 1,976 | null | null | github_plus_top10pct_by_avg |
upersymmetric[@Gliozzi:1976qd].
For the Ramond sector, we use the string field $\Psi$ constrained on the restricted small Hilbert space satisfying the conditions[@Kunitomo:2015usa] $$\eta\Psi\ =\ 0\,,\qquad
XY\Psi\ =\ \Psi\,, \label{R constraints}$$ where $X$ and $Y$ are the picture-changing operator and its inverse ... | 2,046 | 1,120 | 1,768 | 1,972 | null | null | github_plus_top10pct_by_avg |
construct another ranking $\widetilde{\sigma}_1^\ell$ from $\widehat{\sigma}_1^\ell$ by replacing $i_{\min}$ with $i$-th item. Observe that $ \P[\widehat{\sigma}_1^\ell] \leq \P[\widetilde{\sigma}_1^\ell]$ and $\widetilde{\sigma}_1^\ell \in \Omega_2$. Moreover, such a construction gives a bijective mapping between $\O... | 2,047 | 845 | 758 | 2,030 | null | null | github_plus_top10pct_by_avg |
h{i.e.}} \quad \rho_t+\nabla\cdot(\rho\MM{u}) = 0. \,.
\label{advecD}$$ A more extensive list of different types of advected quantity is given in Holm *et al.* (1998).
We write the reduced Lagrangian $\ell$ as a functional of the Eulerian fluid variables $\MM{u}$ and $a$, and add further constraints to the action $S$ ... | 2,048 | 3,230 | 2,334 | 1,951 | null | null | github_plus_top10pct_by_avg |
consist of those functions $u=f(x)$ which satisfy the overdetermined system composed of the initial system (\[eq:3.1\]) together with the invariance conditions
\[eq:3.17i\] \_a\^iu\_i\^=0,i=1,…,p,a=1,…,p-2k,
ensuring that the characteristics of the vector fields $X_a$ are equal to zero.
####
It should be noted th... | 2,049 | 481 | 2,719 | 2,143 | null | null | github_plus_top10pct_by_avg |
el{rho-beta}$$ where $\nu_0 = \left[2(1+3\alpha_\Lambda)/(1+\alpha_\Lambda)^2 -
1/4\right]^{1/2}$, $C_{\cos}$ and $C_{\sin}$ are determined by boundary conditions, and $A_\beta=1$ for $\beta = 2,3$. The first part, $\rho_{\hbox{\scriptsize{asymp}}}(r)$, of $\rho_{II}(r)$ corresponds to a background density. *It is un... | 2,050 | 4,360 | 1,539 | 1,568 | null | null | github_plus_top10pct_by_avg |
$ whose ratio to ${\gamma}_{i}^{2}$ vanishes in the $N \rightarrow \infty$ limit will be neglected.
[^4]: This is the approximation in which any contribution to ${W}_{N}$ whose ratio to ${\hat{\beta}}_{i}$ vanishes in the $N \rightarrow \infty$ limit will be neglected.
[^5]: These market-orthogonal portfolios essenti... | 2,051 | 1,583 | 2,054 | 2,136 | null | null | github_plus_top10pct_by_avg |
athfrak{h}}^*/{{W}}. \end{CD}$$ and the map $\rho$ is flat of degree $n!$.
\(1) Recall from that $\operatorname{Hilb^n{\mathbb{C}}^2}= \operatorname{Proj}({\mathbb{A}})$. By Lemma \[hi-basic-lem\], ${\mathbb{A}}= A[\mathbf{z},\mathbf{z}^*]$. The maps $A\hookrightarrow {\mathbb{A}}$ and ${\mathbb{C}}[\mathbf{z},\mathbf... | 2,052 | 1,068 | 1,441 | 1,861 | null | null | github_plus_top10pct_by_avg |
---------------------------------- ------------------------------ --------------
**Gender** \
\ Male 3922 (49.00)
\ Female 4002 (... | 2,053 | 5,660 | 326 | 1,153 | null | null | github_plus_top10pct_by_avg |
{\beta}}\\
-{(C^T)_{\dot{\alpha}}}^\beta & 0
\end{pmatrix}\,.$$ The matrices $\mathcal{C}\Gamma^\mu$ are symmetric, or equivalently $$(C\bar{\gamma}^\mu)^{\alpha\beta}\ =\ (C\bar{\gamma}^\mu)^{\beta\alpha}\,,\qquad
(C^T\gamma^\mu)_{\dot{\alpha}\dot{\beta}}\ =\ (C^T\gamma^\mu)_{\dot{\beta}\dot{\alpha}}\,.$$
The wor... | 2,054 | 3,608 | 2,274 | 1,851 | null | null | github_plus_top10pct_by_avg |
\Psi}|=\dfrac{k_{s}}{\sqrt{-g}}\left(\dfrac{\partial}{\partial r}\sqrt{-g}P\right)=2k_{s}\dfrac{(2\cos^{3}\theta a^{2}\alpha+\cos\theta \alpha r^{2}+r)}{(1-\alpha r \cos\theta)(r^{2}+a^{2}cos^{2}\theta)},$$ where $ g=-\sin^{2}\theta \dfrac{(a^{2}\cos^{2}\theta+r^{2})^{2}}{(\alpha r \cos\theta -1)^{8}} $.
From equation... | 2,055 | 2,407 | 2,820 | 2,073 | 3,880 | 0.769547 | github_plus_top10pct_by_avg |
Each task can be attributed to multiple users, and each user can be assigned to several tasks. In the workflow, the SM may ensure that a task is completed only when all assignees finalise the assignment. After this configuration step, i.e., all the tasks have been assigned, the SM must decide if the study should start... | 2,056 | 677 | 2,065 | 2,772 | null | null | github_plus_top10pct_by_avg |
lde{p}]}F\Psi]\big)\,,
\label{tf sp ramond}\end{aligned}$$
where $[{\mathcal{S}},\tilde{p}]$ denotes the first-quantized charge defined by the commutator $[q^\alpha,\tilde{p}^\mu]$ with the parameter $\zeta_{\mu\alpha}$, $$[{\mathcal{S}},\tilde{p}]\ =\ \zeta_{\mu\alpha} [q^\alpha,\tilde{p}^\mu]\,,$$ and in particular ... | 2,057 | 1,052 | 1,626 | 2,200 | null | null | github_plus_top10pct_by_avg |
-Higgs phenomenology, even in the heavy squark regime.
[**Acknowledgements:**]{} We thank Michael Spira and Jaume Guasch for clarifying how the threshold corrections contribute to the Higgs couplings to bottom quarks and pointing out earlier works. We also thank Borut Bajc, Stéphane Lavignac, and Timon Mede for useful... | 2,058 | 251 | 2,331 | 2,102 | null | null | github_plus_top10pct_by_avg |
_0)\in S$. Therefore, the infinitesimal symmetry condition is a necessary and sufficient condition for the existence of the symmetry group $G$ of the overdetermined system (\[eq:3.21\]). Since the vector fields $X_a$ form an Abelian distribution on $X\times U$, it follows that conditions (\[eq:3.30\]) and (\[eq:3.34\])... | 2,059 | 248 | 1,904 | 2,163 | 3,460 | 0.772197 | github_plus_top10pct_by_avg |
al mass and momentum distributions are smeared out in a “tail" due to the correlated motion of a nearby nucleon. For both nucleons and pairs of nucleons, we assume that 10$\%$ of such decays are affected by the correlated motion of an additional nucleon [@1999corr]. Lepton rescattering within the nucleus is negligible.... | 2,060 | 153 | 2,996 | 2,275 | null | null | github_plus_top10pct_by_avg |
) \lrp{N_t+ N(y_t) - N(y_0)}^2}}_{\circled{5}} dt
\numberthis \label{e:t:asldka}
\end{aligned}$$
$\circled{3}$ goes to $0$ when we take expectation, so we will focus on $\circled{1}, \circled{2}, \circled{4}, \circled{5}$. We will consider 3 cases
**Case 1: $\|z_t\|_2 \leq 2\epsilon$**\
From item 1(c)... | 2,061 | 1,311 | 1,922 | 2,027 | null | null | github_plus_top10pct_by_avg |
where $$\begin{aligned}
\mathbf{A}
&=& \frac{t}{n}[(\identity \otimes in\sigma_x) - (in\sigma_x \otimes \identity) - p({\identity} \otimes {\identity})\\
&&+ p(\Pi_1 \otimes \Pi_1) + p(\Pi_0 \otimes \Pi_0)] \\
&=& \frac{t}{n}\left( \begin{matrix} 0 & ik & -ik & 0 \\
ik & -p & 0 & -i... | 2,062 | 5,050 | 875 | 1,436 | 2,549 | 0.778972 | github_plus_top10pct_by_avg |
t indicator to the existence of leptonic unitarity violation. Existence of second and higher order corrections in $W$, if detected, uniquely identifies the case for low-scale unitarity violation.
- Presence (low-$E$) or absence (high-$E$) of the probability leaking term $\mathcal{C}_{\alpha \beta}$ in (\[Cab\]) rema... | 2,063 | 1,944 | 3,119 | 2,234 | 4,179 | 0.76753 | github_plus_top10pct_by_avg |
{c}{b} \right\rvert^{\frac{1}{a}} \right) \\
&\quad + \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} \G(3a)}{2 \G(a)}
+ \operatorname{{sgn}}(c) \frac{(k_{1}^{2} - k_{2}^{2}) b^{2}}{2 \G(a)}
\g\left(3a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right). \end{aligned}$$ Also, from $(\ref{E[Pe(Z+c)]-2})$, we have $$\beg... | 2,064 | 2,497 | 2,213 | 1,911 | null | null | github_plus_top10pct_by_avg |
pes .unnumbered}
--------------------
The same logic applies to the computation of the current-primary OPEs. Current conservation links the $j^a_{L,z} \phi$ and $j^a_{L,\bar
z} \phi$ OPEs : \[phiCC\] (z) = 0. When the above equation is valid, the Maurer-Cartan constraint can be rewritten as: \[phimodMC\] (z) |j\^a\_... | 2,065 | 1,187 | 704 | 2,066 | null | null | github_plus_top10pct_by_avg |
egory, with tensor product $- \otimes_A -$ and the unit object $A$. Hochschild homology is a homological functor from $A{\operatorname{\!-\sf bimod}}$ to $k{\operatorname{\it\!-Vect}}$.
To obtain a small category interpretation of $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$, one notes that for any $n,n' \geq ... | 2,066 | 1,465 | 1,566 | 1,809 | 3,624 | 0.771189 | github_plus_top10pct_by_avg |
rline{v} \left( 2A \sqrt{b} \right)^k,\end{aligned}$$ where the first inequality follows from the bound $| v^\top_l (W_i - \psi) |\leq \|
W_i - \psi\|$ (as each $v_l$ is of unit norm), the second from the fact that the coordinates of $W_i$ are bounded in absolute value by $A$ and the third by the fact that $\overline{... | 2,067 | 3,626 | 1,738 | 1,728 | null | null | github_plus_top10pct_by_avg |
= -0*m
the third derivative of -21688*l**2*z**3 - 232*l**2 - 13*z**3 - 4*z**2 + 320*z - 1 wrt z.
-130128*l**2 - 78
What is the third derivative of -19292*n**5 + 131*n**3 - 27728*n**2 - 804*n?
-1157520*n**2 + 786
Differentiate 21431042*c**3*t**3 + 319*c**3 - 16092*c**2 wrt t.
64293126*c**3*t**2
What is the second deri... | 2,068 | 643 | 1,794 | 1,791 | null | null | github_plus_top10pct_by_avg |
point have definite values (and $\lambda_\varphi$ can be directly calculated from linearizied RG equations for $A_i$ and $B_i$), in the $b=3$ case some fixed point coordinates diverge, and calculation of $\lambda_\varphi$ requires an additional effort. To be more specific, a numerical analysis of RG equations (\[eq:RGA... | 2,069 | 526 | 1,790 | 2,126 | 2,287 | 0.781149 | github_plus_top10pct_by_avg |
bound on the magnitude of the reminder term in the Taylor series expansion of $g(\hat{\psi})$ around $g(\psi)$, as detailed in the proof of below. Of course, we may relax the requirement that holds almost everywhere to the requirement that it holds on an event of high probability. This is indeed the strategy we use whe... | 2,070 | 1,436 | 1,708 | 1,802 | null | null | github_plus_top10pct_by_avg |
-------- ----------------------------- -- ----------------------------- -------------- ----------------------------- -------------- ----------------------------- -------------- ---------------------- --
0 $\cO_X$ $\supsetneq$ $\langle xy,x^5,y^5\rangle$ ... | 2,071 | 2,011 | 1,636 | 1,927 | null | null | github_plus_top10pct_by_avg |
iction that requires $y_{1}=y_{2}$ whenever $(x,y_{1})\in f$ and $(x,y_{2})\in f$. In this subset notation, $(x,y)\in f\Leftrightarrow y=f(x)$. ]{}
[^4]: [\[Foot: EquivRel\]An useful alternate way of expressing these properties for a relation $\mathscr{M}$ on $X$ are]{}
[$\quad$(ER2) $\mathscr{M}$ is symmetric if... | 2,072 | 3,446 | 2,770 | 1,920 | 3,294 | 0.773394 | github_plus_top10pct_by_avg |
$\begin{aligned}
{\ensuremath{\operatorname{\mathbf{Pr}}\left[Y_k{\leqslant}{\ensuremath{\operatorname{\mathbf{E}}\left[Y_k\right]}}/2\right]}}{\leqslant}\exp({-{\ensuremath{\operatorname{\mathbf{E}}\left[Y_k\right]}}}/{8})=\exp(-\omega(\log n)).
\end{aligned}$$
[It follows that there exists a $d$-subset]{} $... | 2,073 | 627 | 1,245 | 2,141 | 3,737 | 0.770428 | github_plus_top10pct_by_avg |
t morphism ${\sD}^A\to{\sD}^B,A,B\in\cCat,$ preserves right homotopy finite right Kan extensions.\[item:sl4b\]
3. The derivator is pointed and $C\colon{\sD}^{[1]}\to{\sD}$ preserves right homotopy finite right Kan extensions.\[item:sl5\]
4. The derivator is pointed and $C\colon{\sD}^{[1]}\to{\sD}$ preserves homotop... | 2,074 | 1,483 | 1,759 | 1,942 | 3,378 | 0.772763 | github_plus_top10pct_by_avg |
en by $$F_{\mu }^{ext}=-(\partial _{\nu }A_{\mu }-\partial _{\mu }A_{\nu })v^{\nu }
\label{12}$$ where the quantity $A_{\mu }$ is the vector potential given in units of $m_{0}c/e$. For a linearly polarized laser pulse, $$A_{\mu }\equiv (\Phi /c,{\bf A}),\;\;\;\;{\bf A}=\hat{x}A_{x}(\phi
),\;\;\;\;\Phi =0 \label{16}$$ ... | 2,075 | 3,940 | 1,866 | 1,732 | null | null | github_plus_top10pct_by_avg |
m$ is eventually strictly increasing, and hence ${\left\vert G^{(n)} \right\vert}$ is eventually *exponentially* increasing. Therefore the previous part of the theorem applies.
---
author:
- 'Peter Beelen[^1]'
title: 'A note on the generalized Hamming weights of Reed–Muller codes'
---
Preliminaries {#sec:in}
=======... | 2,076 | 586 | 1,364 | 2,051 | 3,765 | 0.770208 | github_plus_top10pct_by_avg |
given coordinates using a convolution with a narrow gaussian, not by shifting the PSF by means of rebinning to a new position. The latter is better for the application to AGN decomposition and is now incorporated in later versions of [[galfit]{}]{}.
---
abstract: 'The high velocity gradient observed in the compact c... | 2,077 | 124 | 3,082 | 2,388 | null | null | github_plus_top10pct_by_avg |
valued in $({\bf 12},{\bf 4})$ of $so(12) \times so(4)$
$\overline{\partial} X^{3-4}_{-1} 16 scalars (toroidal moduli)
\otimes \left( \psi^{3-4}_{-1/2}, \over... | 2,078 | 4,487 | 1,450 | 1,415 | null | null | github_plus_top10pct_by_avg |
ed}
v_{T}^{(m\,h\,0)} &= c_1\,, &
v_{\Phi}^{(m\,h\,0)} &= c_2\,, &
v_{R}^{(m\,h\,0)} &= c_3\,, &\end{aligned}$$ and $$\begin{aligned}
v_{T}^{(m\,h\,1)} &= -2 [c_3+c_1 (h R T+i m)]\,, \\ {\nonumber}v_{\Phi}^{(m\,h\,1)} &= -2 c_2 (h R T+i m)\,, \\ {\nonumber}v_{R}^{(m\,h\,1)} &= -2 [c_3 (h R T+i m)+c_1-c_2]\,.\end{align... | 2,079 | 4,739 | 684 | 1,662 | null | null | github_plus_top10pct_by_avg |
-algebra generated by the $q$-th powers of all elements. The proof is by Noetherian induction.
First consider the case when $B$ is Artinian. The residue field of $B$ is finite and purely inseparable over the residue field of $B_1$, hence $B^q$ is contained in a field of representatives of $B_1$ for large enough $q$.
... | 2,080 | 1,475 | 2,241 | 1,879 | null | null | github_plus_top10pct_by_avg |
ave $Y^{q_0}-\alpha^{q_0}/\alpha Y$ is a factor of $\phi (f(x))$. Therefore $\alpha$ is the root of $\phi(f(x))$. Conversely, suppose $\alpha$ is the root of $\phi(f(x))$. Let $f(x)=k(x)(x-\alpha^{q_0}/\alpha)+r$, where $r\in \mathbb{F}_q$. Then $\phi(f(x))=\phi(k(x))\circ \phi(x-\alpha^{q_0}/\alpha)+\phi(r)$. From the... | 2,081 | 2,358 | 2,174 | 1,928 | null | null | github_plus_top10pct_by_avg |
is expression allows us to interpret $\tilde{\phi}(p)$ as the Fourier modes of a scalar field $\phi(x)$ and $T(\teps, p)$ as the generator of a coordinate transformation with x\^ = \^[()]{} e\^[- i px]{}. That is, T(, p) = e\^[ip\_ x\^]{} \^[()]{}\_. Indices are contracted according to Einstein’s summation convention r... | 2,082 | 567 | 2,357 | 2,155 | null | null | github_plus_top10pct_by_avg |
dr]|{\ddots}&\ar@{}[dr]|{\ddots}&&&&&&\\
\ar@{}[dr]|{\ddots}&\Omega Ff\ar[r]\ar[d]\pullbackcorner&\Omega x\ar[r]\ar[d]\pullbackcorner&0\ar[d]&&&&&\\
&0\ar[r]&\Omega y\ar[r]\ar[d]\pushoutcorner\pullbackcorner&Ff\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[d]&&&&\\
&&0\ar[r]&x\ar[r]^-f\ar[d]\pushoutcorner\pullbackcorne... | 2,083 | 1,542 | 2,200 | 1,895 | null | null | github_plus_top10pct_by_avg |
ear train movement accomplished in one stage of operation becomes input to the next. A formalism called a process captures this notion of sequential inheritance. We assemble processes from a simple unit called the frame, which is two-part structure consisting of starting and ending conditions. A process is a sequence o... | 2,084 | 4,464 | 2,817 | 1,506 | 1,551 | 0.788346 | github_plus_top10pct_by_avg |
m99 (5.3.1)].
Kronecker Coefficients {#section:kronecker}
======================
As explained in the introduction, the Kronecker coefficients play an important role in geometric complexity theory and quantum information theory. In this section, we will describe precisely how they can be computed using our methods.
L... | 2,085 | 3,119 | 2,994 | 1,952 | 1,628 | 0.787411 | github_plus_top10pct_by_avg |
12} x_j \mathcal{M}_j \, .\end{aligned}$$ We derive its associated equation of motion and see what a simultaneous cancellation of all the higher order terms implies for the coefficients $(v_i , x_j)$ : we find that the unique linear combination of order 8 scalars for FLRW space-time that leads to second order equatio... | 2,086 | 3,553 | 2,156 | 1,863 | null | null | github_plus_top10pct_by_avg |
all when $|\lambda-\lambda'|<\delta$ and $O_c$ is small enough, independently on the value of $\lambda'(\Lambda)$. This implies in particular that, by making $O'$ and consequently $O_c$ and $O$ small enough one may chose $\delta \theta_m$ such that $$|\delta \theta_m|< \frac{R_m^2\epsilon}{(\lambda_1-\lambda_i)|2\theta... | 2,087 | 1,243 | 1,261 | 2,120 | null | null | github_plus_top10pct_by_avg |
(\[angtransitionrate\]) and (\[Wfinal\]) is to be valid for detector trajectories not involving casual horizons.
On the other hand, If the support of the detector’s profile is contained in the Rindler wedge, then the corresponding transition rate is KMS in the usual Unruh temperature of the Rindler trajectory as is ev... | 2,088 | 3,623 | 2,756 | 1,858 | null | null | github_plus_top10pct_by_avg |
N=1+[X]$. At higher temperatures $T>T_n$ the composite field $\Psi$ becomes disordered.
-8mm
-5mm
Dual representation {#sec:dual}
-------------------
The partition function $Z$ can be evaluated by the high-temperature expansion method (see e.g. in [@Parisi]) in terms of $t_1,t_2,u$ and the explicit integration over... | 2,089 | 258 | 1,183 | 2,077 | 2,355 | 0.780609 | github_plus_top10pct_by_avg |
(0,2) SCFT’s. Although it seems there are new consistent (0,2) SCFT’s, we will argue that they do not seem to define new consistent supersymmetric heterotic string compactifications.
In hindsight, we can understand that result as follows[^5]. In an ineffective orbifold (one in which part of the orbifold group acts tr... | 2,090 | 400 | 2,840 | 1,902 | null | null | github_plus_top10pct_by_avg |
gathered}
C_j:={1\over 2}\kappa_j^{-2}{\left\Vert \nabla_x S_j\right\Vert}_{L^\infty(G\times I)}
\\
+\kappa_j^{-1}\Big({\left\Vert \Sigma_j\right\Vert}_{L^\infty(G\times S\times I)}
+{\left\Vert {{\frac{\partial S_j}{\partial E}}}\right\Vert}_{L^\infty(G\times I)}+\sqrt{M_1M_1'}
+{{M_1M_1'}\over c}\Big).\end{gathered}$... | 2,091 | 761 | 869 | 2,104 | null | null | github_plus_top10pct_by_avg |
_{n\geq 1}\frac{\psi_n(f)}{n},$$ and $\Exp:\Lambda_k[[T]]^+\rightarrow 1+\Lambda_k[[T]]^+$ by
$$\Exp(f)=\exp(\Psi(f)).$$
The inverse $\Psi^{-1}:\Lambda_k[[T]]^+\rightarrow\Lambda_k[[T]]^+$ of $\Psi$ is given by
$$\Psi^{-1}(f)=\sum_{n\geq 1}\mu(n)\frac{\psi_n(f)}{n}$$where $\mu$ is the ordinary Möbius function.
The ... | 2,092 | 1,802 | 2,192 | 2,052 | 2,472 | 0.779475 | github_plus_top10pct_by_avg |
omic levels. It can be referred to as a $m$-phonon Jaynes-Cummings (JC) model. On the other hand, Hamiltonian $\hat{\mathcal{H}}^{(m)}_{-} $ is obtained with $\omega_0 - \omega_L= - m \nu $ and it can be referred to as a $m$-phonon anti-Jaynes-Cummings (AJC) model. The case $m=0$ can be studied using either $\hat{\math... | 2,093 | 3,381 | 2,068 | 1,918 | null | null | github_plus_top10pct_by_avg |
number \\
& 6R^{[ab|e|}Q^{cd]}_e+{F}_a P^{a,abcd}+{F}_b P^{b,abcd}+{F}_c P^{c,abcd}+{F}_d P^{d,abcd}+\tfrac{1}{2}P^{a,bcdaef}{F}_{aef}\nonumber \\
& \qquad \qquad +\tfrac{1}{2}P^{b,bcdaef}{F}_{bef}+\tfrac{1}{2}P^{c,bcdaef}{F}_{cef}+\tfrac{1}{2}P^{d,bcdaef}{F}_{def} =0 \nonumber \end{aligned}$$ in the IIB case, and $$\b... | 2,094 | 1,630 | 2,788 | 2,110 | null | null | github_plus_top10pct_by_avg |
{{\cal C}}_{{\bf n}}^{b_{i+1}}(y)\setminus{{\cal C}}_{{\bf n}}^{b_i}(y)&(i=1,\dots,T-1),\\
{{\cal C}}_{{\bf n}}(y)\setminus{{\cal C}}_{{\bf n}}^{b_T}(y)&(i=T).
\end{cases}\end{aligned}$$ As in Figure \[fig:lace-edges\], we can think of ${{\cal C}}_{{\bf n}}(y)$ as the interval $[0,T]$, where each integer $i\in[0,T]$ ... | 2,095 | 686 | 1,520 | 2,125 | null | null | github_plus_top10pct_by_avg |
_\muhat G_\muhat(q)m_\muhat&=\sum_{\alpha\in \calP^{{\bf F}^{\times}}}\calH_{\omega(\alpha)}(0,\sqrt{q})\prod_{i=1}^k\tilde{H}_{\omega(\alpha)}(\x_i,q)\\
&=\prod_{c\in{\bf F}^{\times}}\Omega\left(\x_1^{d(c)},\dots,\x_k^{d(c)};0,q^{d(c)/2}\right)\\
&=\prod_{d=1}^\infty\Omega\left(\x_1^d,\dots,\x_k^d;0,q^{d/2}\right)^{\p... | 2,096 | 1,021 | 1,767 | 1,854 | null | null | github_plus_top10pct_by_avg |
]$. Since $z', t'$ are not in $P'$ and they are disjoint from $a'$ and $x'$, they both have to lie in the subsurface, $M$, which is a projective plane with two boundary components containing $y'$, and having $x'$ as a boundary component as shown in the figure. This gives a contradiction as there are no two nontrivial s... | 2,097 | 1,256 | 1,944 | 1,999 | null | null | github_plus_top10pct_by_avg |
$\epsilon_n $ is positive and to replace $\epsilon$ with $\epsilon_n$. The price for this extra step is a factor of $\frac{1}{n}$, which upper bounds $\mathbb{P}(\mathcal{E}_n^c)$. Putting all the pieced together we arrive at the bound $$A_3 \leq C \mathrm{E}^*_{1,n} + \frac{1}{n}.$$ Finally notice that $\mathrm{E}_{1,... | 2,098 | 4,023 | 1,786 | 1,579 | null | null | github_plus_top10pct_by_avg |
ible.
We note that the approach presented above generalizes the results obtained in [@GrundlandZelazny:1983] where the simple states have been constructed with wave vectors $\lambda$ of constant direction for hyperbolic inhomogeneous systems (\[eq:SW:1\]).
Simple mode solutions for inhomogeneous quasilinear system. {... | 2,099 | 1,800 | 2,850 | 2,253 | null | null | github_plus_top10pct_by_avg |
ngle \mathbf{A}_{2,1}\mathbf{A}_{2,2}\right\rangle \right.\\
& +\left.\left\langle \mathbf{B}_{1,1}\mathbf{B}_{2,1}\right\rangle +\left\langle \mathbf{B}_{1,2}\mathbf{B}_{2,2}\right\rangle \right)\end{aligned}$$ to the system and then eliminating the connection correlations $\left\langle \mathbf{A}_{1,1}\mathbf{A}_{1,... | 2,100 | 1,803 | 2,639 | 2,089 | null | null | github_plus_top10pct_by_avg |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.