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 |
|---|---|---|---|---|---|---|---|
dual configuration. The torus partition function can be recast into the one without vacuum charges$$\tilde{Z}(\alpha,\beta)=\int_{\Delta_0}^{\infty}\int_{\xi_0}^{\infty}e^{-\alpha\Delta-\beta\xi}\rho(\Delta,\xi)$$ where the $\tilde{Z}$ is the partition function defined by $$\tilde{Z}(\alpha,\beta)=e^{\alpha M_v+\beta ... | 3,301 | 1,437 | 2,889 | 3,057 | null | null | github_plus_top10pct_by_avg |
In our case, we will work in conventions in which our Fock vacuum $| 0 \rangle$ has the properties $$\lambda_{-,0}^{a} | 0 \rangle \: = \: 0 \: = \:
\psi_{+,0}^{\overline{\imath}} | 0 \rangle .$$
As before, reflecting the fact that the $\lambda$’s and $\psi$’s couple to nontrivial bundles, this Fock vacuum is itself ... | 3,302 | 2,431 | 2,766 | 2,969 | null | null | github_plus_top10pct_by_avg |
covering of $A_4$ [@Liu:2019khw], in which masses, mixing, and CP phases for quark and lepton are predicted [^1]. A possible correction from Kähler potential is also discussed in Ref. [@Chen:2019ewa]. Furthermore, a systematic approach to understand the origin of CP transformations is recently discussed in ref. [@Baur... | 3,303 | 1,651 | 2,799 | 3,090 | null | null | github_plus_top10pct_by_avg |
ght parameters can be extracted from eight observables that can be used to completely determine them. Additional observables can then be used in order to overconstrain the system. We divide the eight observables that we use to determine the system into four categories:
$(i)$ Branching ratio measurements (3 observables... | 3,304 | 3,076 | 3,160 | 2,927 | null | null | github_plus_top10pct_by_avg |
4 &&&&&\\
& & & & & \\
\hline
& & & & & \\
P\Omega_{2n}^{+}(q) & q_{2(n-1)} & q_{n-1} &\frac{(q^{n-1}+1)(q+1)}{(4, q^n-1)} &\frac{(q^{n-1}-1)(q-1)}{(4, q^n-1)} & n \mbox{ even } \\
& & &&&(n,q)\neq (4,2)\\
n \geq 4 & q_{2(n-1)} & q_{n} &\frac{(q^{n-1}+1)(q+1)}{(4, q^n-1)} & \frac{q^n-1}{(4, q^n-1)} & n \mbo... | 3,305 | 2,864 | 2,709 | 2,972 | null | null | github_plus_top10pct_by_avg |
]). It is important to remark that Eq. (\[hs\]) indeed describes quite well the system when $\omega_0 - \omega_L=\pm m\nu$ and $\Omega$ is moderately weak, which are conditions easily implemented in the laboratories [@meekhof1996; @roos]. Before the interaction with the laser, the trapped ion is found to be in thermal ... | 3,306 | 3,152 | 3,516 | 2,968 | 3,190 | 0.774145 | github_plus_top10pct_by_avg |
tarrow}}}x\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \tilde I_{\vec z_j,\vec z'_j}^{{\scriptscriptstyle}(j)}(y,
x)$}}},\end{aligned}$$ to which we can apply the bound discussed between [(\[eq:Theta’-2ndindbd2\])]{} and [(\[eq:Theta’-2ndindbd8\])]{}.
**(d-2)** If $v\notin{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$ ... | 3,307 | 1,567 | 2,282 | 3,185 | null | null | github_plus_top10pct_by_avg |
f the dissipative GPE coupled to the impurity, and the final section summarizes our conclusions.
Modeling approach {#sec:model}
=================
We model the interaction between the impurity and a two-dimensional BEC through a Gaussian repulsive potential which can be reduced to a delta-function limit similar to pre... | 3,308 | 873 | 2,505 | 3,084 | 2,852 | 0.776535 | github_plus_top10pct_by_avg |
Lambda )$, since ${\Omega }(u)v \in I^\chi (\Lambda )\subset
\oplus _{{\alpha }\not=0}M^\chi (\Lambda )_{\alpha }$. Thus by Eq. , $\Lambda {\mathrm{Sh}}$ induces a symmetric bilinear form on $L^\chi (\Lambda )$, also denoted by $\Lambda {\mathrm{Sh}}$. The radical of this form is a ${\mathbb{Z}}^I$-graded $U(\chi ){\ot... | 3,309 | 2,414 | 2,324 | 3,035 | null | null | github_plus_top10pct_by_avg |
when we apply the results after sample splitting (as is our goal), we need to define $u$ as $u = \min_{|S|\leq k} \lambda_{\rm min}(\Sigma_S)$. As $d$ increases, $u$ can get smaller and smaller even with fixed $k$. Hence, the usual fixed $k$ asymptotics may be misleading.
[**Remark:**]{} We only ever report inferences... | 3,310 | 1,643 | 2,021 | 3,152 | 2,499 | 0.779312 | github_plus_top10pct_by_avg |
ice product is a relation between $\prod\Theta \times \prod\Phi$ and $\prod\Theta\Phi$. For this relation to be a mapping, it must yet be established that any member of the domain is related to exactly one member of the co-domain.
Again with $(\theta, \phi) \in \prod\Theta \times \prod\Phi$, suppose $\alpha \in \prod\... | 3,311 | 1,590 | 3,529 | 3,093 | null | null | github_plus_top10pct_by_avg |
eta_k\|}_{p+1}^{p+1} + C(p,k){\|\theta_k\|}_p^p + C(k,p,M). \end{aligned}$$ Using the Gagliardo-Nirenberg-Sobolev inequality (Lemma \[lem:GNS\]) implies, $$\frac{d}{d\tau}{\|\theta_k\|}_p^p \leq -\frac{C(p)}{{\|\theta_k\|}^{\alpha_2}_1}{\|\theta_k\|}_{p+1}^{p+1} + C(p){\|\theta_k\|}_{p+1}^{p+1} + C(p,k){\|\theta_k\|}_p... | 3,312 | 2,216 | 820 | 3,329 | null | null | github_plus_top10pct_by_avg |
thermore, randomly and uniformly select 2000 sets of data with irradiance between *G* = 150 W/m^2^--1000 W/m² and temperature between 15--40 °C as the test set.
Two layers of MLP structure are used and the activation function is set as LeakyRelu. In the MLP model, the learning rate is set as 1.75 × 10^-3^, the learnin... | 3,313 | 2,298 | 1,303 | 3,006 | 270 | 0.815353 | github_plus_top10pct_by_avg |
e $SU(N)$ gauge symmetry.) The gauge algebra is non-Abelian even for the Abelian group $U(1)$.
To describe the noncommutative $U(N)$ gauge algebra properly, it is a necessity to use the generators (\[factor\]) including functional dependence on the base space. Nevertheless, the noncommutative $U(N)$ gauge symmetry sti... | 3,314 | 2,693 | 3,568 | 2,956 | null | null | github_plus_top10pct_by_avg |
d $E_{\mathrm{th}}=4.44$ and 4.51 MeV. []{data-label="28Ne_strength"}](fig8-2.eps "fig:")
The central panel in Fig. \[response\] shows the response function in $^{28}$Ne. In the low-energy region, we can see a two-bump structure at around 7 and 8 MeV. Because the deformation is small as in $^{26}$Ne, we cannot see a s... | 3,315 | 2,721 | 3,473 | 3,087 | null | null | github_plus_top10pct_by_avg |
the usual open-closed intervals in $\mathbb{R}$[^13]. The subbases $_{\textrm{T}}\mathcal{S}_{1}=\{(a,\infty),(-\infty,b)\}$, $_{\textrm{T}}\mathcal{S}_{2}=\{[a,\infty),(-\infty,b)\}$, $_{\textrm{T}}\mathcal{S}_{3}=\{(a,\infty),(-\infty,b]\}$ and $_{\textrm{T}}\mathcal{S}_{4}=\{[a,\infty),(-\infty,b]\}$ give the respe... | 3,316 | 4,200 | 3,993 | 3,131 | 2,642 | 0.778198 | github_plus_top10pct_by_avg |
following we derive the desired properties for applying Lemma \[thelemma\]. First we write also the potential term in in a quadratic way.
\[magneticL\] The operator matrix $$\begin{aligned}
{\mathbf{L}}=\mathbf{P}_{[0,t)}\left(\begin{matrix} 0 & ik\left(A-A^*\right)\\ik\left(A^* - A\right) & 0\end{matrix}\right)\math... | 3,317 | 3,146 | 2,647 | 3,101 | null | null | github_plus_top10pct_by_avg |
with the sole assumption (besides $g_p$ sufficiently small) of smallness of the unsteady and/or inhomogeneous part $\delta\psi_0$ of the wavefunction, which allows linearization. Eq. (\[eq:Fi\_th\]) assumes in addition weak inhomogeneities below scales $a$ and $\xi$, and finally Eqs. (\[eq:InertialSimpleDimless\]) and ... | 3,318 | 1,761 | 328 | 3,421 | null | null | github_plus_top10pct_by_avg |
relative to the KL loss (\[eqn:loss\]) and has a constant risk.
Recently, Matsuda and Komaki (2015) constructed an improved Bayesian predictive density on $\ph_U(Y\mid X)$ by using a prior density of the form $$\label{eqn:pr_em}
\pi_{EM}(\Th)=|\Th\Th^\top|^{-\al^{EM}/2},\quad \al^{EM}=q-r-1.$$ The prior (\[eqn:pr\_em... | 3,319 | 704 | 2,141 | 3,051 | 2,777 | 0.777049 | github_plus_top10pct_by_avg |
$p_0$ and $q_0$ and, by Proposition \[1stpart3rdrequire\], there is an open $O$ in $S$ containing $\Lambda_0$ such that, for all $\Lambda=(p, k^\mu)$ in $O$, there is a conjugate point $q$. By the proof of Proposition \[1stpart3rdrequire\] and the discussion above, one may find a value $\lambda_i$ such that for all the... | 3,320 | 3,567 | 3,909 | 3,010 | null | null | github_plus_top10pct_by_avg |
A \to A$ a map for all $i\in I$, and $C^a=(c^a_{jk})_{j,k \in I}$ a generalized Cartan matrix in ${\mathbb{Z}}^{I \times I}$ for all $a\in A$. The quadruple $${\mathcal{C}}= {\mathcal{C}}(I,A,({r}_i)_{i \in I}, (C^a)_{a \in A})$$ is called a *Cartan scheme* if
1. ${r}_i^2 = {\operatorname{id}}$ for all $i \in I$,
2... | 3,321 | 2,970 | 2,757 | 3,004 | null | null | github_plus_top10pct_by_avg |
in $A$ (similarly in $B$), its degree is at most $\frac n 2$, and at least $\frac n 2 - O(\sqrt{n})$. It has $\frac n 2 - O(\sqrt{n})$ neighbors in $B$, so the number of its neighbors in $A$, and the number of its non-neighbors in $B$ is $O(\sqrt{n})$. By deleting and adding $O(\sqrt{n})$ edges to each vertex, we get a... | 3,322 | 1,462 | 2,900 | 2,992 | 2,766 | 0.777161 | github_plus_top10pct_by_avg |
(\mathcal{L(H)},\subset )$ and/or the lattice $(\mathcal{L(S)},\subset )$ can be identified with Birkhoff and von Neumann’s lattice of *experimental propositions*, which was introduced in the 1936 paper that started the research on QL.$^{(31)}$ This identification is impossible, however, if $\mathcal{H}$ is not finite-... | 3,323 | 4,293 | 3,224 | 3,042 | null | null | github_plus_top10pct_by_avg |
ictive density is obtained by considering a proper hierarchical prior. In Section \[sec:superharmonic\], we utilize Stein’s (1973, 1981) ideas for deriving some minimax predictive densities with superharmonic priors. Section \[sec:MCstudies\] investigates numerical performance in risk of some Bayesian minimax predictiv... | 3,324 | 1,265 | 1,174 | 3,316 | null | null | github_plus_top10pct_by_avg |
stant $C_1>0$ such that *a priori* estimate \[diss-co-es-ad\] [\^\*]{}\_[L\^2(GSI)\^3]{} C\_1(\_[L\^2(GSI)\^3]{}+ \_[T\^2(\_+)H\^1(I,T\^2(\_+’))\^2]{}), holds.
The existence result analogous to Theorem \[coupthev\] (based on the theory of evolution operators) holds also for the adjoint problem, and it guarantees that ... | 3,325 | 1,236 | 2,252 | 3,265 | null | null | github_plus_top10pct_by_avg |
e globally for all $i=1,2,\cdots,I$ is assured leading to the conclusion that $f_{\alpha}(x_{i})\in V_{i}$ eventually for every $i=1,2,\cdots,I$. Hence $f_{\alpha}\in B((x_{i})_{i=1}^{I};(V_{i})_{i=1}^{I})$ eventually; this completes the demonstration that $f_{\alpha}\rightarrow f$ in $(\textrm{Map}(X,Y),\mathcal{T})$,... | 3,326 | 3,007 | 3,666 | 3,133 | 2,270 | 0.781272 | github_plus_top10pct_by_avg |
sheff and Scott Wilson for useful comments and remarks regarding this topic. The second author was partially supported by the Max-Planck Institute in Bonn.
$\widehat{\mathcal Comm}_\infty$ structure for Poincaré duality spaces {#Comm-section}
======================================================================
Befo... | 3,327 | 2,076 | 1,584 | 3,017 | 3,716 | 0.770527 | github_plus_top10pct_by_avg |
hen a set of parameters was tested that was very different from the last set, the experiment almost always produced no atoms, meaning we had to assign a default cost that did not provide meaningful gradient information to the learner. Once the next set of parameters is determined they are sent to the experiment to be t... | 3,328 | 684 | 3,601 | 3,157 | 3,928 | 0.769239 | github_plus_top10pct_by_avg |
t adjoint as soon as it is preserved by precomposition with $X$ (see for instance [@maclane Theorem X.7.2] or [@maysig:pht 16.4.12]). In our case when $X = (u\times\id)_!\lI_A$, precomposition with $X$ is just left Kan extension along $u$, which by our assumption of $u\op$-stability preserves the right Kan extension $(... | 3,329 | 2,530 | 3,052 | 2,993 | 2,032 | 0.783319 | github_plus_top10pct_by_avg |
therefore the group of equivariant automorphisms of $P_{\mathcal{E}}$ living over the identity on $M$. This is known as the gauge group, ${\mathcal{G}}(P_{\mathcal{E}})$.
From this viewpoint it becomes clear that there is a map of fibrations $ P_{\mathcal{E}}^{Ad} \to {\mathcal{G}}L_n({\mathcal{E}})$ over $M$ (after ... | 3,330 | 2,156 | 3,251 | 2,988 | null | null | github_plus_top10pct_by_avg |
m{OLS}}} \|\hat\beta_{{\widehat{S}}} - \beta_{{\widehat{S}}} \| \leq C
B_n,$$ with probability at last $1 - \frac{2}{n}$.
[**Remarks.**]{}
1. It is worth recalling that, in the result above as well as in all the result of the paper, the probability is with respect to joint distribution of the entire sample and of th... | 3,331 | 1,764 | 3,017 | 3,044 | null | null | github_plus_top10pct_by_avg |
from HPLC analysis of periorbital skin (*larger right panel*) and tropical mix seeds (*smaller left panel*). *Arrows* indicate lutein esters *1*, *2* and *3*
Effect of hormone treatment on plasma concentrations of steroids and cholesterol {#Sec14}
-----------------------------------------------------------------------... | 3,332 | 46 | 3,306 | 3,632 | null | null | github_plus_top10pct_by_avg |
i(\lambda([c]), \lambda([b])) \neq 0$ by superinjectivity. Since $i([c], [e]) = 0$ for all $e \in P \setminus \{a, b\}$, we have $i(\lambda([c]), \lambda([e])) = 0$ for all $e \in P \setminus \{a, b\}$. But this is not possible because $\lambda([c])$ has to intersect geometrically essentially with some isotopy class ot... | 3,333 | 1,014 | 1,845 | 3,123 | null | null | github_plus_top10pct_by_avg |
====================
As we now have a canonical variational principle for fluid dynamics *via* the inverse map, one may obtain its multisymplectic formulation by extending the phase space so that the Lagrangian is affine in the space and time derivatives. In this section we show how to do this for EPDiff($H^1$) as dis... | 3,334 | 1,869 | 704 | 3,620 | null | null | github_plus_top10pct_by_avg |
{ for any } 1\leq k\leq 5 , \mbox{ there exists a semi-infinite path } \gamma_{k} \\
\mbox{ included in the cone } C_{2k\pi/5,\varepsilon,r_{k}} \mbox{ and starting from } \\
\mbox{ a vertex } X_{k} \mbox{ satisfying } r_{k}<|X_{k}|\leq r_{k}+1
\end{array} \right\} ~,$$ we get that for all $\varepsilon>0$, there exist ... | 3,335 | 1,933 | 2,684 | 2,976 | 2,280 | 0.78117 | github_plus_top10pct_by_avg |
O(1)$ the above quantity $\gamma$ can be made arbitrarily close to one, for large enough problem size $d$. On the other hand, when $p_{j,\ell_j}$ is close to $\kappa_j$, the accuracy can degrade significantly as stronger alternatives might have small chance of showing up in the rank breaking. The value of $\gamma$ is q... | 3,336 | 1,682 | 793 | 3,529 | 1,872 | 0.784823 | github_plus_top10pct_by_avg |
x_kx_mx'_l=\sum_{\substack{{k,l,m}\\m\neq k}}x_kx_lx'_m
=\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,l,m}\\l\neq k\\m\neq k}}x_kx_lx'_m\\
&=\sum_{\substack{{k,m}\\m\neq
k}}x_k^2x'_m+\sum_{\substack{{k,m}\\m\neq k}}x_kx_mx'_m
+\sum_{\substack{{k,l,m}\\l\neq k\\m\neq k,l}}x_kx_lx'_m\\
&=\sum_{\substack{{... | 3,337 | 1,371 | 1,957 | 3,195 | null | null | github_plus_top10pct_by_avg |
hi,\Psi],{\mathcal{S}}\Phi]
\nonumber\\
&
+\frac{1}{2}\{\Psi,\Xi\{\eta\Phi,\Xi {\mathcal{S}}\Phi\}\}
+\frac{1}{2}\{\Psi,\Xi[\Phi,\Xi {\mathcal{S}}\eta\Phi]\}
-[\Xi[\Phi,\Psi],\Xi {\mathcal{S}}\eta\Phi]
\nonumber\\
&
-\frac{1}{2}{\mathcal{S}}\Xi[\Phi,\Xi\{\eta\Phi,\Psi\}]
-\frac{1}{2}{\mathcal{S}}\Xi[\eta\Phi,\Xi[\Phi,... | 3,338 | 700 | 1,719 | 3,520 | null | null | github_plus_top10pct_by_avg |
6.0 (6.0--11.9) 142.8 (113.0--142.8)
**Reference interval partitioned based on season**
Blood urea nitrogen mmol/L ... | 3,339 | 6,267 | 1,240 | 1,978 | null | null | github_plus_top10pct_by_avg |
l objective of chaotic dynamics is to generate a topology in $X$ with respect to which elements of the set can be grouped together in as large equivalence classes as possible in the sense that if a net converges simultaneously to points $x\neq y\in X$ then $x\sim y$: $x$ is of course equivalent to itself while $x,y,z$ ... | 3,340 | 3,387 | 3,750 | 3,154 | null | null | github_plus_top10pct_by_avg |
6--400), 3, 0 0-150 0.065
*Albumin (g/L)* 31 (27--36), 1, 0 29 (26--34), 0, 0 23-35 \<0.001
*Cholesterol (mmol/L)* 5.6 (2.5-9.1), 6, 1 5.0 (1.9-7.7), 4, 2 3.5-7.0 0.040
*Creatinine (μmol/L)* ... | 3,341 | 2,572 | 3,453 | 3,233 | null | null | github_plus_top10pct_by_avg |
[|a |b]{}I \[AdjAdj=I\] \_[|b |a]{} \^[a |a]{} \^[b |b]{} = \^[a b]{}I where $I$ is the identity at least as acting upon the current algebra. One can argue more generically that these bilinears are proportional to the unit operator by using the definition of the primary adjoint in terms of the supertrace, and using com... | 3,342 | 763 | 1,764 | 3,374 | null | null | github_plus_top10pct_by_avg |
the same purpose). We can use $C_1={\mathrm}{Fix}(C_0)$ to define a partition $\dot{f}$ of $\omega_1$ so that for each $\xi\in\omega_1$ and each $s\in S_{\xi^+(C_1)}$, $s^\smallfrown j$ forces that $\dot{f}(\xi)=j$. Now we choose two (names of) functions $\dot{h}_1$ and $\dot{h}_2$ witnessing normality as follows:
- ... | 3,343 | 2,555 | 2,738 | 2,912 | null | null | github_plus_top10pct_by_avg |
in\mathbb{N}}$ that is not the constant sequence $(x_{0})$ at a fixed point? As $i\in\mathbb{N}$ increases, points are added to $\{ x_{0},f(x_{0}),\cdots,f^{I}(x_{0})\}$ not, as would be the case in a normal sequence, as a piled-up Cauchy tail, but as points generally lying between those already present; recall a typic... | 3,344 | 4,186 | 3,845 | 3,096 | 2,314 | 0.780919 | github_plus_top10pct_by_avg |
s rank-breaking estimators. This provides guidelines for choosing the weights in the estimator to achieve optimal performance, and also explicitly shows how the accuracy depends on the topology of the data.
This paper provides the first analytical result in the sample complexity of rank-breaking estimators, and quanti... | 3,345 | 2,621 | 2,221 | 2,861 | null | null | github_plus_top10pct_by_avg |
_x[{g}(X_{\tau_D})^2]<\infty$. However, if we consider the computation in (\[worksforsquared\]) of the Appendix, which shows that $\mathbb{E}_x[{g}(X_{\tau_D})]<\infty$ when ${g}$ is continuous and in $L^1_{\alpha}(D^\mathrm{c})$, then it is easy to see that the same statement holds replacing ${g}$ by ${g}^2$. Under fi... | 3,346 | 2,493 | 2,820 | 2,932 | null | null | github_plus_top10pct_by_avg |
ad T_{3 }= \left( \begin{array}{ccc}
0 & 0 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{array} \right) \ ,$$ and the group element $$g= \exp \left[ x_{1} T_{1} + x_{2}T_{2} + (x_{3}-\frac{1}{2} x_{1}x_{2} ) T_{3}\right] = \left( \begin{array}{ccc}
1& x_{1} &x_{3} \\
0 &1 & x_{2}\\
0 & 0 & 1 \\
\end{array} \right) \ .$$ The le... | 3,347 | 2,028 | 2,846 | 2,895 | 4,057 | 0.768412 | github_plus_top10pct_by_avg |
eGa\]
To the best of our knowledge, the expression for $G_{\bar{R}}$ in cannot be simplified for general values of $\Psi$. Thus, the calculation of the average throughput gain for a general number of reconfiguration states involves an infinite integral of a complicated function. We then further consider two special c... | 3,348 | 4,413 | 2,781 | 2,697 | null | null | github_plus_top10pct_by_avg |
frac{2}{\pi}\sin(\pi\alpha/2)\left(r^2-r_n^2\right)^{-\alpha/2} r_n^{\alpha}\,\frac{{\rm d}r}{r} \times \frac{{\rm d}\theta}{2\pi}\,,\qquad r>r_n.\label{DISPOL}
\end{aligned}$$ From , we see that the angle $\theta$ is sampled uniformly on $[0,2\pi]$ whereas we can sample the radius $r$ via the inverse-transform sa... | 3,349 | 4,820 | 2,240 | 2,613 | null | null | github_plus_top10pct_by_avg |
0$ as $u\geq 0$, it follows from the part A above that $w\geq 0$, where $w$ is the solution of the problem . This allows us to conclude that $\phi=w+u\geq 0$, and so $\psi\geq 0$ as desired.
The same argument is valid for the coupled system considered below.
\[re:general\_positivity\_of\_u\] We sketch here a more gen... | 3,350 | 2,134 | 2,570 | 2,981 | null | null | github_plus_top10pct_by_avg |
frac{ 1 }{ \Delta_{J} - \Delta_{I} }
\left\{ W^{\dagger} A W \right\}_{I I}
\left\{ W^{\dagger} A W \right\}_{I J}
\nonumber \\
&-&
(ix) e^{- i \Delta_{J} x} \frac{ 1 }{ \Delta_{J} - \Delta_{I} }
\left\{ W^{\dagger} A W \right\}_{I J}
\left\{ W^{\dagger} A W \right\}_{J J}
\nonumber \\
&+&
\frac{e^{- i \Delta_{... | 3,351 | 4,465 | 2,916 | 2,974 | null | null | github_plus_top10pct_by_avg |
ft[dz^2 +d{\bf x}^2\right]
=dy^2+a^2(z)d{\bf x}^2.
\label{rsmetric}$$ Here $a(z)=\ell/z$, where $\ell$ is the AdS radius. The branes are placed at arbitrary locations which we shall denote by $z_+$ and $z_-$, where the positive and negative signs refer to the positive and negative tension branes respectively ($z_+ ... | 3,352 | 3,650 | 3,248 | 2,985 | 3,133 | 0.774625 | github_plus_top10pct_by_avg |
at this time the equations do not impose $\partial_{+}f_{--}=0$ and furthermore, the transformation rule of $f_{--}$ also differs from the one found off the chiral point.
Conserved **charges for** $\mu l=1$
-----------------------------------
Evaluating the variation of the surface charges using the expressions of [@... | 3,353 | 1,112 | 2,715 | 3,072 | null | null | github_plus_top10pct_by_avg |
\hat{\Gamma}(j,j)} > \sqrt{\underline{\sigma}^2 - C \aleph_n} >0$ and, by Weyl’s inequality, the minimal eigenvalue of $\hat{V}$ is no smaller than $v - C
\daleth_n > 0$. In particular, the error terms $\Delta^*_{n,1}$ and $\Delta^*_{n,2}$ are well-defined (i.e. positive). Thus we have that $$\label{eq:A3}
A_3 \leq C \... | 3,354 | 5,019 | 1,443 | 2,707 | null | null | github_plus_top10pct_by_avg |
hin the fiducial volume (two meters inward from the detector walls),]{}
2. [There must be two Cherenkov rings,]{}
3. [Both rings must be showering for the $pp \rightarrow e^{+}e^{+}$, $nn \rightarrow e^{+}e^{-}$, $nn \rightarrow \gamma\gamma$ and $p \rightarrow e^{+}\gamma$ modes; one ring must be showering and one... | 3,355 | 800 | 3,168 | 3,271 | 3,137 | 0.774598 | github_plus_top10pct_by_avg |
ed}
\begin{split}
~& 8 \big(\gamma +2 \delta \big) \Big( -1+A(r)\Big) A(r) B(r)^3+4 r \big(5 \gamma -2 \delta
\big) B(r)^3 A'(r)-18 r^2 \gamma A(r) B(r) B'(r)^2
\\& +8 r \big(2 \gamma
+\delta \big) B(r)^2 \Big(-r A'(r) B'(r)+A(r) \left(B'(r)+r
B''(r)\right)\Big)
\\& +r^3 \big(\gamma -\delta \big) B'(r)... | 3,356 | 5,414 | 675 | 2,840 | null | null | github_plus_top10pct_by_avg |
ended detectors
---
Introduction
============
The Unruh effect [@Fulling:1972md; @Davies:1974th; @Unruh:1976db] states that an observer of negligible spatial size on a worldline of uniform linear acceleration in Minkowski spacetime reacts to the Minkowski vacuum of a relativistic quantum field by thermal excitations ... | 3,357 | 2,648 | 1,416 | 3,115 | null | null | github_plus_top10pct_by_avg |
)
position = [605, 6];
// OR
/*
var position = ((left >= 0 && left <= 80) ? [20, 1] :
((left >= 81 && left <= 198) ? [137, 1] :
((left >= 199 && left <= 315) ? [254, 3] :
((left >= 316 && left <= 430) ? [371, 4] :
((left >= 431 && left <= 548) ? [488, 5] :
((left >= 549) ?... | 3,358 | 4,147 | 58 | 2,500 | 10 | 0.842453 | github_plus_top10pct_by_avg |
label="fig:fsf"}](5.eps)
![$e_is_{i}=e_if_ie_{i+1} = s_ie_{i+1}$ ($E4''$)[]{data-label="fig:efe"}](6.eps)
As we noted in the previous paper [@Ko4], these basic relations are invariant under the transpositions of indices $i\leftrightarrow n-i+1$ as well as the $\mathbb{Z}[Q]$-linear involution $*$ defined by $(xy)^{*}... | 3,359 | 319 | 3,219 | 2,548 | 386 | 0.811329 | github_plus_top10pct_by_avg |
the last inequality follows from the size property and the inequality $m{\leqslant}n$. [ Therefore, there are $x$ possibilities for the root and hence there are at most $4^k\cdot \binom{{s}}{d}\cdot n^{c_0+2}$ ordered trees with the specified root. Fix an arbitrary $d$-choice $D_t$ as the root for $T$.]{}
Next we fix... | 3,360 | 2,055 | 2,001 | 3,137 | 2,124 | 0.782568 | github_plus_top10pct_by_avg |
r\\
\label{eq:jad}\end{aligned}$$ where $d_{\alpha\beta}=\langle\alpha;R\vert(\partial/\partial R)
\vert\beta;R\rangle$ is the nonadiabatic coupling vector and $$S_{\alpha\beta}=E_{\alpha\beta}d_{\alpha\beta}
\left(\frac{P}{M}\cdot d_{\alpha\beta}\right)^{-1}\;.$$ Using Eqs. (\[eq:ilad\]) and (\[eq:jad\]), the equatio... | 3,361 | 1,418 | 849 | 3,635 | null | null | github_plus_top10pct_by_avg |
of the corresponding families, and the descriptions given in the statement are immediately verified in these cases.
The second part of Lemma \[PNCtolimits\] may be viewed as the analogue in our context of an observation of Pinkham (‘sweeping out the cone with hyperplane sections’, [@MR0376672], p. 46).
\[eluding\] De... | 3,362 | 1,553 | 2,286 | 3,063 | 3,807 | 0.769994 | github_plus_top10pct_by_avg |
reen’s formula (\[green-ex\]), which in combination with the fact that $$P'(x,\omega,E,D)\phi=-P(x,\omega,E,D)\phi-{\frac{\partial S_0}{\partial E}}\phi,$$ leads us to $${\left\langle}P(x,\omega,E,D)\phi,\phi{\right\rangle}_{L^2(G\times S\times I)}
={}&
-{\left\langle}P(x,\omega,E,D)\phi,\phi{\right\rangle}_{L^2(G\time... | 3,363 | 946 | 1,735 | 3,322 | null | null | github_plus_top10pct_by_avg |
nario is if the adversary creates malicious examples when noise removing operations are turned on in all possible locations. It is possible that such adversarial examples would also fool the classifier when the defense is only applied in a subset of the layers. Fortunately, we note that for FGS, IGS, and CW2, transfera... | 3,364 | 1,456 | 2,939 | 3,109 | null | null | github_plus_top10pct_by_avg |
ation method to find the coefficient vectors, and calculate the corresponding average computation rate.
We first show that as stated in Remark \[remark:KPractical\], for high dimension and large power, the number of real-valued approximations $K$ can be set to a rather small value without degrading the rate apparently... | 3,365 | 1,447 | 1,915 | 3,043 | null | null | github_plus_top10pct_by_avg |
th slope strictly between $-1$ and $0$ does not depend on the choice of coordinates fixing the flag $z=0$, $p=(1:0:0)$.
The limit curves are then obtained by choosing a side of the polygon with slope strictly between $-1$ and $0$, and setting to $0$ the coefficients of the monomials in $F$ [*not*]{} on that side. Thes... | 3,366 | 1,717 | 2,920 | 3,098 | 1,453 | 0.789361 | github_plus_top10pct_by_avg |
1}^{-1})h_{1}$$
Thus $g \rhd h_{1}^{-1} = h_{1}^{-1}$, i.e. $h_{1}^{-1} \in {\rm
Fix}(H)$. In a similar way we can show that ${\rm Fix}(G)$ is a subgroup of $G$. Using the compatibility condition [(\[eq:2\])]{} we obtain that the map given by: $$\varphi_{\rhd}: {\rm Fix}(G) \rightarrow {\rm Aut}(H), \quad
\varphi_{\rh... | 3,367 | 2,324 | 2,496 | 3,097 | null | null | github_plus_top10pct_by_avg |
group.
Moreover, $\langle y \rangle$ acts transitively on $\{\Delta_1, \Delta_2, \ldots, \Delta_k\}$.
- The length of an orbit $\nabla$ of $\langle y \rangle$ on $\Omega$ is $k=p$, and there are $m$ orbits $\nabla_1:=\nabla$, $\nabla_2\dots, \nabla_m$. Hence there is a partition $$\Omega=\nabla_1\cup \cdots \cu... | 3,368 | 2,075 | 2,252 | 2,986 | null | null | github_plus_top10pct_by_avg |
scriptstyle}(1)}}(v_1,v'_1){\nonumber}\\
=\sum_{u',u'',v'}\bigg(&2\psi_\Lambda(v_1,u')\,\tilde G_\Lambda(u',u'')
\Big({{\langle \varphi_{u'}\varphi_u \rangle}}_\Lambda\tilde G_\Lambda(u,u'')+\tilde
G_\Lambda(u',u'')\,\delta_{u,u''}\Big)\,\psi_\Lambda(u'',v'_1){\nonumber}\\
&\times{{\langle \varphi_{v_1}\varphi_{v'} \... | 3,369 | 3,733 | 3,041 | 2,994 | null | null | github_plus_top10pct_by_avg |
t normal in $G$. Note that $G$ acts on $\Omega:=\{H^g\: : \: g\in G\}$ transitively. If $H<G$, then certainly $|\Omega|>1$ and, by Lemma \[FKS\], there exists a prime power order element $x\in G$ acting fixed-point-freely on $\Omega$. But the hypotheses imply that $x\in H^z$ for some $z\in G$, so $H^{zx}=H^z$ and this ... | 3,370 | 1,631 | 1,397 | 3,257 | null | null | github_plus_top10pct_by_avg |
in{aligned}
cv_1 &\Big[ 2 \tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big]-\tr\big[ \Er_\La(\La) \Er_\La(v_1I_r+(1-v_1)\La)^{-1}\big]\Big]
\\
&\leq cv_1 \Big\{ 2\tr \Er_\La(\La) - {1\over v_1}\tr \Er_\La(\La)\Big\}
= c (2v_1-1) \tr \Er_\La(\La),\end{aligned}$$ which implies that $$\De(W;\pi_{GB}^J)/m_{GB}
\leq \{ -... | 3,371 | 1,228 | 1,595 | 3,367 | 2,196 | 0.781873 | github_plus_top10pct_by_avg |
0.003 $-0.081$
(k) $\nu[321]3/2$ $\nu[211]1/2$ 7.14 0.001 0.106
(l) $\pi[220]1/2$ $\pi[101]1/2$ 7.96 0.037 0.0095
(m) $\pi[21... | 3,372 | 2,033 | 3,828 | 3,127 | null | null | github_plus_top10pct_by_avg |
0$ for all $m\in \{0,1,\dots ,{b}-1-t\}$. Hence $\ker {\hat{T}}^{\chi }_{p,\Lambda }=U^-(\chi )F_p^{{b}-t}{\otimes }{\mathbb{K}}_{\Lambda '}$ and ${\operatorname{Im}}{\hat{T}}^{\chi }_{p,\Lambda } = U^-(\chi )F_p^t {\otimes }{\mathbb{K}}_\Lambda $.
For all $w\in {\mathrm{Aut}}({\mathbb{Z}}^I)$ and ${\alpha }\in {\math... | 3,373 | 1,571 | 2,625 | 3,080 | null | null | github_plus_top10pct_by_avg |
t explicitly discuss in this article) and Bayesians. The categorization used here is motivated by a short blog post (see <http://labstats.net/articles/overview.html>). ]{}]{}
[^4]: [[Note that this method also utilizes mechanisms usually known from the frequentist school; i.e., hypothesis testing.]{}]{}
[^5]: [[We co... | 3,374 | 213 | 3,195 | 3,170 | null | null | github_plus_top10pct_by_avg |
\mathbb{C}}[{\mathbb{C}}^{2n}]{\mathbb{A}}^1$, $J^1= {\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]A^1$, ,
$L_c(\mu)$, simple factor of $\Delta_c(\mu)$,
${\mathcal{L}}_1 ={\mathcal{O}}_{\operatorname{Hilb^n{\mathbb{C}}^2}}(1)$, ${\mathcal{L}}={\mathcal{O}}_{\operatorname{Hilb(n)}}(1)$, ,
$\mathcal{O}_c$, catego... | 3,375 | 2,419 | 1,113 | 3,333 | 3,593 | 0.771324 | github_plus_top10pct_by_avg |
given in (\[eqn:JS\]), when $\al_1=\cdots=\al_r=0$ and $\be=\be^{JS}$, and to $\Thh_{EM}$, given in (\[eqn:EM\]), when $\al_1=\cdots=\al_r=\al^{EM}$ and $\be=0$. In estimation of the normal mean matrix relative to the quadratic loss $L_Q$, $\Thh_{JS}$ and $\Thh_{EM}$ are minimax.
If $\Thh_{MS}$ with certain specified ... | 3,376 | 2,106 | 1,061 | 3,286 | 3,359 | 0.772882 | github_plus_top10pct_by_avg |
uations should be compared with the corresponding exact ones of Appendix A4. We shall see that the net of functions (\[Eqn: phieps\]) converges graphically to the multifunction Eq. (\[Eqn: singular\_eigen\]) as $\varepsilon\rightarrow0$.
In the discretized spectral approximation., the singular eigenfunction $\phi(\mu,... | 3,377 | 2,951 | 3,715 | 3,285 | 3,851 | 0.769696 | github_plus_top10pct_by_avg |
of $E_{2}$ converge with the initiator approximation.
In Fig.\[fig1:compinit-alpha\] we plotted the second order energy for the Carbon dimer with the cc-pVQZ basis computed with the QMC-LCC framework. The black line correspond to the simulations presented in fig.\[fig1:swalknum\] i.e. 50k walkers and a initiator thre... | 3,378 | 2,781 | 1,298 | 3,002 | null | null | github_plus_top10pct_by_avg |
is in the ballpark of the redshift range where the expansion of the Universe apparently made a transition from deceleration to acceleration [@decel].
The result (\[total3\]) was derived in an oversimplified case, where the possible effects of other branes and orientifolds were not taken into account. However, as we n... | 3,379 | 2,411 | 4,068 | 3,250 | null | null | github_plus_top10pct_by_avg |
C^c$ OPE will look like: $$\begin{aligned}
\dots \kappa^{ac} :j_{e\bar z} j_{z}^e :
+
\dots (:j^a_{\bar z} j^c_{ z} :
+
(-1)^{ac} :j^c_{\bar z} j^a_{ z} :).\end{aligned}$$ The first term is proportional to a component of the energy-momentum tensor (and to the kinetic term in the Lagrangian). The other term indicates t... | 3,380 | 1,279 | 1,293 | 3,245 | 3,521 | 0.771804 | github_plus_top10pct_by_avg |
ction of $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(A_\#)$ to $\Lambda_{\leq 2}
\subset \Lambda$. In other words, we do not need to compute the full $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(A_\#)$ and to construct a full resolution $P^\#_{{\:\raisebox{1pt}{\te... | 3,381 | 2,983 | 1,886 | 3,193 | 3,391 | 0.77271 | github_plus_top10pct_by_avg |
he remaining variables becomes H\_0= \_[ij]{} K\_[ab]{} \_[ij]{} \_a \_[ij]{} \_b - \_i u(\_2-\_1), \[2N0\] where the matrix $K_{ab}, \, a,b =1,2$ is related to the original parameters as K\_[11]{}&=& , K\_[22]{}= ,\
K\_[12]{} &=& - . \[Ktg\] . The stability of $H_0$ is guaranteed by K\_[11]{}K\_[22]{} - K\_[12]{}\^2= ... | 3,382 | 916 | 3,200 | 3,191 | 3,835 | 0.769785 | github_plus_top10pct_by_avg |
care whether the number of points in a neighborhood is finite or infinite.
Previous construction
=====================
We recall the construction from [@MMS]. Fix a basepoint $x_0\in X$. Given $N>0$, an $N$-sequence in $X$ based at $x_0$ is an infinite list $x_0,x_1,\ldots$ of points in $X$ with $d(x_i,x_{i+1})\leq N... | 3,383 | 3,436 | 3,749 | 3,118 | 1,535 | 0.788533 | github_plus_top10pct_by_avg |
a_1\eta_2+h.c.,\end{aligned}$$ where $$\begin{aligned}
y_{N_R} &=
\left[\begin{array}{ccc}
\alpha_\nu &0 & 0 \\
0 & \beta_\nu &0 \\
0& 0 & \gamma_\nu \\
\end{array}\right]
\left[\begin{array}{ccc}
y_{1} & y_{3} &y_{2} \\
y_{3} & y_{2} &y_{1} \\
y_{2} & y_{1} & y_{3} \\
\end{array}\right],\\
y_{N_L} &=
\left[... | 3,384 | 4,215 | 2,569 | 3,003 | null | null | github_plus_top10pct_by_avg |
g\}.\end{aligned}$$ Therefore, we obtain $$\begin{aligned}
{\label{eq:Theta[1]-rewr}}
&\Theta_{y,x;{{\cal A}}}-\Theta'_{y,x;{{\cal A}}}{\nonumber}\\
&=\sum_b\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{
w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\... | 3,385 | 1,440 | 1,774 | 3,171 | null | null | github_plus_top10pct_by_avg |
{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\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W^{\... | 3,386 | 1,831 | 3,115 | 3,223 | null | null | github_plus_top10pct_by_avg |
\oint_{C(t)}\frac{\MM{m}}{\rho}\cdot{\mathrm{d}}\MM{x}=
\oint_{C(t)}\frac{1}{\rho}{\frac{\partial \ell}{\partial a}}\diamond a\cdot {\mathrm{d}}\MM{x}.$$ For the incompressible Euler equations, $a$ is the relative density $\rho$, so $${\frac{\partial \ell}{\partial a}}\diamond a = \rho\nabla{\frac{\partial \ell}{\parti... | 3,387 | 455 | 2,744 | 3,166 | null | null | github_plus_top10pct_by_avg |
4]. The character $\calU_\tau$ of $L_\lambda$ decomposes as,
$$\calU_\tau=|W_\lambda|^{-1}\sum_{w\in W_\lambda}\chi^\tau_w\cdot R_{T_w}^{L_\lambda}(1)$$where $\chi^\tau_w$ denotes the value of $\chi^\tau$ at $w$. Applying the Harish-Chandra induction $R_{L_\lambda}^G$ to both side and using the transitivity of inducti... | 3,388 | 2,568 | 2,609 | 3,016 | null | null | github_plus_top10pct_by_avg |
M\geq 0,\ c>0$, \[csda38\] |B(u,v)|M[u]{}\_[X]{}[v]{}\_[Y]{}uX, vY ([boundedness]{}) and \[csda37a\] B(v,v)c[v]{}\_[X]{}\^2vY ([coercivity]{}). Suppose that $F:X\to{\mathbb{R}}$ is a bounded linear form. Then there exists $u\in X$ (possibly non-unique) such that \[csda38-a\] B(u,v)=F(v)vY.
See e.g. [@treves p. 403] or... | 3,389 | 1,135 | 2,757 | 3,189 | null | null | github_plus_top10pct_by_avg |
entrally-extended two-dimensional abelian algebra, demonstrating that this is equivalent to a TsT transformation of the full supergravity background. There are additional classes of deformations that can be constructed as non-abelian T-duals. These come from considering particular non-semisimple subalgebras of $\mathfr... | 3,390 | 1,928 | 2,018 | 3,088 | null | null | github_plus_top10pct_by_avg |
6mr\right) {a}^{4}\alpha \left(\cos\left(\theta \right)\right)^{5} \nonumber\\
& +\left(15{a}^{4}{m}^{2}{r}^{2}-20{a}^{4}{\alpha}^{2}{e}^{2}m{r}^{3}-15{a}^{6}{\alpha}^{2}{m}^{2}{r}^{2}-{\frac {17{a}^{2}{\alpha}^{2}{e}^{4}{r}^{4}}{3}}-10{a}^{4}{e}^{2}mr+7/6{a}^{4}{e}^{4}\right) \left( \cos \left( \theta \right) \right... | 3,391 | 5,314 | 1,564 | 2,768 | null | null | github_plus_top10pct_by_avg |
RRT\
... | 3,392 | 1,551 | 2,969 | 3,366 | null | null | github_plus_top10pct_by_avg |
------------------------------------------
We now apply the highest-/lowest-weight formalism to NHEK. In the Schwarzschild spacetime, the orbit space of the isometry $SO(3)$ is $S^2$, therefore we expect a $2+2$ decomposition of the whole manifold. Similarly, in the NHEK spacetime, the isometry group ${\ensuremath{SL(... | 3,393 | 1,794 | 3,191 | 2,944 | null | null | github_plus_top10pct_by_avg |
\beta (\gamma_3+ i \gamma_4) \big) \big] \ ,$$ where $$r = 1+ |\alpha|^2 + |\beta|^2 \ , \quad s^2 = \frac{1}{2\sqrt{r} }(1+ \sqrt{r}) \ , \quad t^2 = \frac{1}{2\sqrt{r} (1+ \sqrt{r})} \ .$$ These coordinates give a metric on $S^5$ that makes manifest the structure of $S^5$ as a $U(1)$ fibration over $\mathbb{C}\mathbf... | 3,394 | 1,859 | 1,248 | 3,303 | 4,073 | 0.768255 | github_plus_top10pct_by_avg |
name{{sgn}}(c) \frac{(k_{1}^{2} - k_{2}^{2}) b^{2}}{2 \G(a)}
\g\left(3a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right). \nonumber\end{aligned}$$
Naoya Yamaguchi\
Office for Establishment of an Information-related School\
Nagasaki University\
1-14 Bunkyo, Nagasaki City 852-8521\
Japan\
yamaguchi@nagasaki-... | 3,395 | 3,428 | 135 | 3,244 | null | null | github_plus_top10pct_by_avg |
{} (1426, 14)[$w$]{} (376,-1186)[$0$]{} (3226,-1186)[$1$]{} (676,314)[Im]{} (2251,-1186)[$Q^2$]{} (4576,-811)[$\theta_{i_5}$]{} (5401,-1186)[$\theta_{i_1}$]{} (5701,-886)[$\theta_{i_2}$]{} (4651,-136)[$\theta_{i_4}$]{} (3526,-1186)[Re]{} (5626,-211)[$\theta_{i_3}$]{} (751,-586)[$e_0$]{} (1276,-736)[$e_P$]{} (1876,-736)... | 3,396 | 2,164 | 3,117 | 3,260 | 3,062 | 0.775094 | github_plus_top10pct_by_avg |
fields and the scalar fields of the model, their representations under the $A_4\times Z_3$ symmetry and their modular weights are given in Tab. \[tab:fields\]. We also show the representations of the Yukawa couplings in Tab. \[tab:couplings\]. Under these symmetries, we write the renormalizable Lagrangian for the lepto... | 3,397 | 2,565 | 3,375 | 3,201 | null | null | github_plus_top10pct_by_avg |
ft(\chi(a)^2 Y_a^2 + {\left\vert \chi(a) \right\vert}^2 {\left\vert Y_a \right\vert}^2 \right)\right] \\
&= \frac{1}{2}\left( {\left\vert G \right\vert} + \alpha \sum_{a \in G} \chi^2(a) \right)
= \frac{{\left\vert G \right\vert}}{2}\bigl( 1 + \alpha {\mathbbm{1}_{\chi = \overline{\chi}}}\bigr).
\end{aligned}... | 3,398 | 4,700 | 2,018 | 2,665 | null | null | github_plus_top10pct_by_avg |
}V\_[ab]{}(z-z’) J\^[(a)]{}\_[z,ij]{}J\^[(b)]{}\_[z’,ij’]{}\
&+& \_[i,z]{} J\^2\_[z,i]{}, \[Hdual\] where the matrix $V_{ab}(z-z')$ is defined in terms of the matrix $K_{ab}$. It reflects the asymmetry between odd and even layers. Explicitly, $V_{11} (z)=YV_{22}(z)$, for $z=z-z'$ being even, describes the interaction b... | 3,399 | 1,985 | 3,329 | 3,196 | 3,027 | 0.775365 | github_plus_top10pct_by_avg |
dots \le m_{d-1},$
3. $m_{i+(q-1)} > m_i$ for all $1 \le i \le d-q.$
Clearly this implies that $N=\sum_{i=1}^{d} \rho_q(i,m_i),$ but it is not clear a priori that $m_1,\dots,m_d$ satisfy conditions 2 and 3 as well. Conditions 2 and 3 would follow once we show that $m_d \ge m_{d-1}$ and either $m_d > m_{d-q+1}$ or $m... | 3,400 | 1,758 | 3,064 | 3,100 | null | null | github_plus_top10pct_by_avg |
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