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 |
|---|---|---|---|---|---|---|---|
CP\CP$ annihilation with $S = 0$, and those of $\CP\CPC$, $\DM\CP$ with $S = 1$.
While gamma rays from the wino-like neutralino annihilation in the Galactic center [@Hisano:2004ds; @Hisano:2003ec] and anti-particles fluxes from that in Galactic Halo [@Hisano:2005ec] are evaluated including non-perturbative effects, th... | 3,001 | 3,291 | 3,404 | 3,013 | 2,889 | 0.776302 | github_plus_top10pct_by_avg |
MP3 to video MOV in OS X?
I need to combine a movie without sounds and a MP3 file. How can I do it in OS X? I have Adobe After Effects but it is not at all intuitive to use -- or better I need to find some button to render the sounds with the movie but now no time -- is there some easy fast-to-use software for OS X t... | 3,002 | 621 | 1,681 | 2,962 | 931 | 0.797457 | github_plus_top10pct_by_avg |
}_i)=x_i^q.$$
It is easy to see that these are independent of the choices made. Thus $F^q$ gives a functor from algebraic spaces over $S$ to algebraic spaces over $S$. One can define $X^{(q)}$ intrinsically as $$X^{(q)}=X\times_{S,F^q} S.$$
If $X$ is an algebraic space which is essentially of finite type over ${{\mat... | 3,003 | 2,334 | 2,867 | 2,635 | 3,156 | 0.774433 | github_plus_top10pct_by_avg |
s clear that $\psi$ is a $B_1$-shift in $N$. Now $\psi(N) | (C_1 \cup \wh{B} \cup W_2) = N|(C_1 \cup \wh{B} \cup W_2)$ and each $x \in B_1 \cup W_0 \cup W_1$ is a loop or is parallel to some element of $\wh{B}$, so $\psi(N)$ is a $\wh{B}$-clique. Now, since $M \del C_1 = N \del C_1$ and $\psi$ is a $B_1$-shift in $N$, ... | 3,004 | 2,541 | 1,397 | 2,807 | 3,591 | 0.771333 | github_plus_top10pct_by_avg |
as defined in Lemma \[l:hproperties\] (using parameter $\epsilon$). Then
1. 1. $\nabla g(z) = h'(\|z\|_2) \frac{z}{\|z\|_2}$
2. For $\|z\|_2 \geq 2\epsilon$, $\nabla g(z) = \frac{z}{\|z\|_2}$.
3. For any $\|z\|_2$, $\lrn{\nabla g(z)}_2\leq 1$
2. 1. $\nabla^2 g(z) = h''(\|z\|_2)\frac{zz^T}{\|z\|_2^2} ... | 3,005 | 4,603 | 1,829 | 2,327 | null | null | github_plus_top10pct_by_avg |
he intrinsic velocity dispersion of the galaxy spectra, we convolved this template with a Gaussian kernel as necessary to lower the resolution. He 2–10 and NGC 3077 were best fit with no convolution; NGC 3504 required a template with $150$ resolution, and NGC 4102 required a $175$ template. For NGC 4214 and NGC 4861,... | 3,006 | 2,642 | 3,091 | 3,039 | null | null | github_plus_top10pct_by_avg |
iments. Data were analyzed using one-way analysis of variance (ANOVA) followed by *post hoc* comparisons using the Dunnet\'s multiple comparison test or two-way ANOVA statistical analysis followed by a *post hoc* test. A probability of value of *p* \< 0.05 was considered as statistically significant.
Results
=======
... | 3,007 | 346 | 2,483 | 2,970 | null | null | github_plus_top10pct_by_avg |
nt colloidal spheres.
In previous work, Schwarz [*et al*.]{} [@Ingmar2011] studied cluster formation via droplet evaporation using Monte Carlo (MC) simulation with shrinking droplets. It was shown that a short-ranged attraction between colloidal particles can produce $M2$ nonminimal isomers and the fraction of isomers... | 3,008 | 1,002 | 2,591 | 2,760 | 3,814 | 0.769951 | github_plus_top10pct_by_avg |
-Klein Einstein-Maxwell-Dilaton theory containing two Maxwell fields, three neutral scalars and an axion in AdS, again using Eqs.. Note that the speed of sound and the other thermodynamic quantities as entropy, temperature and chemical potential agree too.
Finally, let us study whether the bound \[bound\] 2(-c\^2\_s) ... | 3,009 | 862 | 2,245 | 2,917 | null | null | github_plus_top10pct_by_avg |
y ${\mathfrak{A}}$ is an iterative operator (that is, if ${\mathit{s}} \in {\mathbb{S}}$, then $T_{{\mathfrak{A}}}({\mathit{s}}) \in {\mathbb{S}}$).
By definition \[D:ACTUATED\_AUTOMATON\], automaton ${\mathfrak{A}}$ consists of components $\langle \Psi, \Phi, {\mathscr{F}}\!, {\mathsf{A}}, \Lambda, \ell, \Delta \rang... | 3,010 | 1,527 | 2,073 | 2,822 | null | null | github_plus_top10pct_by_avg |
try algebra (\[commutator\]). In the representation (\[Teps\]), the gauge potential A\_(x) = \_[, p]{} \_\^[()]{}(p)T\_[()]{}(p) = A\_\^[()]{}(x) \_ should transform like A\_(x) = \[D\_, (x)\], \[A-transf\] where D\_ \_ + A\_(x) = (\_\^ + A\_\^[()]{})\_ is the covariant derivative and the gauge transformation parameter... | 3,011 | 682 | 3,251 | 2,891 | null | null | github_plus_top10pct_by_avg |
\circ{\hat\Theta_{T}}$ where $4\ls d<a$ and $T$ is a semistandard tableau having a $2$ and a $3$ in each row. $T$ contains a single $d$ and a single $d+1$. If these both lie in the same row of $T$, then ${\psi_{d,1}}\circ{\hat\Theta_{T}}=0$. Otherwise, ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ is a semistandard homomorphism ... | 3,012 | 1,582 | 1,877 | 2,711 | null | null | github_plus_top10pct_by_avg |
c}}}/Z_{{{\cal B}}{^{\rm c}}}$, we obtain $$\begin{aligned}
{\label{eq:Theta'-2ndindbd2.2}}
{(\ref{eq:Theta'-2ndindbd2})}=\sum_{{{\cal B}}\subset\Lambda}\,\sum_{T\ge1}
\sum_{\vec b_T}\sum_{\substack{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}\\ {\partial}{{\bf h}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm ... | 3,013 | 1,369 | 1,946 | 2,821 | null | null | github_plus_top10pct_by_avg |
\E{\lrn{y_{k\delta}}_2^2} \leq 2\E{a(y_{k\delta})} + 4 R^2 \leq 8\lrp{R^2 + \beta^2/m}
\end{aligned}$$ for all $k$.
\[l:energy\_w\] Let the sequence $w_{k\delta}$ be as defined in . Assuming that $\delta \leq m/(16L^2)$ and $\E{\|w_0\|_2^2} \leq 2 \lrp{R^2 + \frac{\beta^2}{m}}$ Then for all $k$, $$\begin{ali... | 3,014 | 3,347 | 2,259 | 2,693 | null | null | github_plus_top10pct_by_avg |
alars each in the $\wedge^2 {\bf
| \pm \pm \rangle$ \overline{8}} = {\bf \overline{28}}$ of $su(8)$
-----------------------------------------------------------------------------------------------------------------
Furthermore, copies of the states above occur at each f... | 3,015 | 3,625 | 2,757 | 2,699 | 3,268 | 0.773598 | github_plus_top10pct_by_avg |
]), with eigenvalue $\lambda_D$ being larger than one. Although there are large number of contacts between them, both chains also have large parts that are not interconnected. Even for $v>v_c(u)$ fixed value $(A^*,B^*)$ does not change, but then $C^*$ becomes equal to zero, meaning that the whole $P_2$ chain is covered... | 3,016 | 598 | 2,959 | 3,032 | 1,296 | 0.791438 | github_plus_top10pct_by_avg |
sponding subgroup in $\Z/2 \int \D_4$. We will describe the subgroups $\I_{2,j}(\Z/2 \oplus \D_4)
\subset \Z/2 \int \D_4$, $j=x,y,z$. We will describe the transformations of $\R^4$ in the standard base $(f_1,f_2,f_3,f_4)$ determined by generators of the groups.
Let us consider the subgroup $\I_{2,x}$. The generator $c... | 3,017 | 4,381 | 2,789 | 2,602 | null | null | github_plus_top10pct_by_avg |
[trath8\] (y,,E)=0 (y,,E)\_-, is given by (cf. [@dautraylionsv6], or [@tervo14 Section 3.3]) \[trath9\] (x,,E)=\_0\^[t(x,)]{}e\^[-\_0\^t(x-s,,E)ds]{} f(x-t,,E) dt.
We need the next lemma.
Assume that $G$ is bounded, $d:={\rm diag}(G)$, $\Sigma\in L^\infty(G\times S\times I)$ and that $\Sigma\geq 0$. Then for any $f\i... | 3,018 | 1,346 | 2,564 | 2,885 | null | null | github_plus_top10pct_by_avg |
cdots$ terms are of lowest order and do not contribute to the surface integrals. Hence, the boundary conditions are invariant under the full conformal group in two dimensions, generated by $T^{+}(x^{+})$ and $T^{-}(x^{-})$.
Surface integrals
-----------------
We shall compute the conserved (Virasoro) charges within t... | 3,019 | 1,129 | 2,320 | 2,877 | null | null | github_plus_top10pct_by_avg |
what contaminated in NGC 4861 and seriously contaminated in NGC 4214 (the lowest redshift galaxy in our sample). Accordingly, we will consider only in lieu of both and for these two galaxies.
Combining Spectra {#sec:combspec}
-----------------
For the H–band spectra, we observed targets for $0.5$ hr between calibr... | 3,020 | 2,916 | 3,677 | 3,192 | null | null | github_plus_top10pct_by_avg |
[\Dc_\La\{f_\pi(\La;W)\La\}M]=\tr[\Dc_\La\{f_\pi(\La;W)\La M\}]=\tr[\Dc_\La\{f_\pi(\La;W)M\La\}]\end{aligned}$$ because $M=M(W)$ is symmetric and does not depend on $\La$. It is observed that $\La^2$ and $\La M+M\La$ are symmetric for $\La\in\Rc_r$, so that $$\label{eqn:gdt1}
\int_{\Rc_r}\tr[\Dc_\La\{f_\pi(\La;W)\La^2\... | 3,021 | 923 | 1,349 | 3,076 | null | null | github_plus_top10pct_by_avg |
ness, $\varepsilon$-visibility and size properties. Fix $d=d(n)$ such that $2{\leqslant}d = o(\log n)$. There exists $\Theta(n){\leqslant}m{\leqslant}n$ such that after the balanced allocation process on $({\mathcal{H}}^{(1)},\ldots, {\mathcal{H}}^{(n)})$ has allocated $m$ balls, the maximum load is $\log_d\log n+{\mat... | 3,022 | 1,844 | 1,071 | 2,997 | 2,104 | 0.782712 | github_plus_top10pct_by_avg |
e with the orientifold for both the IIB and IIA theory. We start by considering the IIB superpotential with the $P_a^{bc}$ flux turned on, which is given in eq. . Performing three T-dualities along the $x$ directions and using the T-duality rules given in eq. we find a T-dual IIA expression which contains all the IIA $... | 3,023 | 2,462 | 2,954 | 2,853 | null | null | github_plus_top10pct_by_avg |
alino dark matter, including the non-perturbative effect. In the evaluation we have to include coannihilation processes in addition to the wino-like neutralino pair annihilation. We use the method developed in Ref. [@Griest:1990kh; @Gondolo:1990dk] for the calculation of the relic abundance including coannihilation eff... | 3,024 | 3,482 | 3,414 | 2,952 | 2,396 | 0.780216 | github_plus_top10pct_by_avg |
00000%"}
Non-rotating charged accelerating black hole
--------------------------------------------
The Kretschmann scalar for the non-rotating charged black hole given by the metric (\[cmetric1\]) is evaluated to be $$K=\dfrac{56\left(\alpha rcos\theta-1\right)^6 \left(cos^2\theta \alpha^2 e^4 r^2+\dfrac{10}{7} \left... | 3,025 | 4,988 | 2,297 | 2,493 | null | null | github_plus_top10pct_by_avg |
nergy and consequently the NL can be obtained.
A.A.C. acknowledges to “Coordenação de Aperfeiçoamento de Pessoal de Nível Superior” (CAPES). FN, FLS and MP are supported by the CNPq “Ciência sem Fronteiras” programme through the “Pesquisador Visitante Especial” initiative (Grant No. 401265/2012-9). MP acknowledges fin... | 3,026 | 641 | 2,846 | 2,992 | 3,003 | 0.7755 | github_plus_top10pct_by_avg |
nduced action of ${\mathbb{T}}^2$ and this is equivalent to a ${\mathbb{Z}}^2$-grading $M=\bigoplus M_{i,j}$; explicitly, an element $f\in M$ is homogeneous of weight $(i,j)$ if $(\alpha,\beta)f=\alpha^i\beta^jf$.
The ${\mathbb{T}}^2$–fixed points of $\operatorname{Hilb^n{\mathbb{C}}^2}$ are precisely the ideals $I_\m... | 3,027 | 2,034 | 2,113 | 2,746 | 3,424 | 0.77249 | github_plus_top10pct_by_avg |
C^1_0(\Gamma_-)$.
Let $\varphi\in C_0^\infty (G\times S\times I^\circ)$. Then by Fubini’s Theorem $$&-(\omega\cdot \nabla_x (L_- g))(\varphi)
=(L_- g)(\omega\cdot \nabla_x\varphi) \\
={}&\int_{G\times S\times I} e^{-\int_0^{t(x,\omega)}\Sigma(x-s\omega,\omega,E)ds}
g(x-t(x,\omega)\omega,\omega,E)(\omega\cdot \nabla_x... | 3,028 | 2,197 | 2,353 | 2,756 | null | null | github_plus_top10pct_by_avg |
Weyl spinors are not parity invariant representations of the Lorentz group. The fundamental parity invariant representation is obtained as the direct sum of a right- and a left-handed Weyl spinor, leading to the Dirac spinor, a 4-dimensional spinor representation of the Lorentz group, which is irreducible for the Loren... | 3,029 | 5,032 | 2,457 | 2,380 | null | null | github_plus_top10pct_by_avg |
ff metric ensures convergence of any class of compact subsets in $\mathbb{R}^{n}$. It appears eminently plausible that our multifunctional graphical convergence on $\textrm{Map}(\mathbb{R}^{n})$ implies Hausdorff convergence on $\mathbb{R}^{n}$: in fact pointwise biconvergence involves simultaneous convergence of image... | 3,030 | 3,123 | 3,001 | 2,889 | 2,106 | 0.782701 | github_plus_top10pct_by_avg |
_{G}\circ Btr : \mu_{R}(G)\to R$.
- We denote by $(-,-)_{\phi_{G}}$ the bilinear form on $\mu_{R}(G)$ defined by $(x,y)_{\phi_{G}}=tr_{\phi_{G}}(xy),$ for $x,y\in\mu_{R}(G)$.
- We denote by $b_{\phi_{G}}$ the bilinear form on $RB(G)$ defined by $b_{\phi_{G}}(X,Y):=\phi_{G}(XY)$.
The map $tr_{\phi}$ is a central ... | 3,031 | 1,730 | 2,383 | 2,842 | null | null | github_plus_top10pct_by_avg |
tion [2]{} to Stein’s equation [1]{}, we have
Define We have It is known that $g(s)$ is a bounded solution to and \[see, for example, (2.13) of [@ChGoSh10]\] This implies [25]{}.
It is known that $V(t,x):=E\varphi(x+\sqrt{t}Z)$ is the solution to the heat equation The next lemma relates the solution to the Stein equa... | 3,032 | 1,856 | 1,281 | 2,958 | null | null | github_plus_top10pct_by_avg |
=\int_{0}^{\infty} d\tau \langle\xi_{\rm eq}(t)\xi_{\rm eq}(t-\tau)
\rangle=\gamma_{\rm eq}KT$$
For the nonequilibrium version, Eq.(15) may be rearranged further to note that $$J_{nn}=\int_{0}^{\infty} d\tau \langle\xi_{\rm neq}(t)\xi_{\rm neq}(t-\tau)
\rangle=\gamma_{\rm neq}KT(1+r e^{-\frac{\gamma}{2} t})$$ where $r... | 3,033 | 2,402 | 2,936 | 2,773 | null | null | github_plus_top10pct_by_avg |
{aligned}
[E,F_i]={\partial ^K}_i(E)K_i-L_i{\partial ^L}_i(E)\quad
\text{for all $E\in {\mathcal{U}}^+(\chi )$.}
\end{aligned}$$ The maps ${\partial ^K}_i,{\partial ^L}_i\in {\mathrm{End}}_{\Bbbk }({\mathcal{U}}^+(\chi ))$ are skew-derivations. More precisely, $$\begin{gathered}
{\partial ^K}_... | 3,034 | 2,708 | 2,441 | 2,651 | null | null | github_plus_top10pct_by_avg |
ation $$(\nu-\mu)\phi(\mu,\nu)=\frac{c\nu}{2}\label{Eqn: case_eigen}$$ for the unknown function $\phi(\nu,\mu)$. Case then suggested, see @Case1967, the non-simple complete solution of this equation to be $$\phi(\mu,\nu)=\frac{c\nu}{2}\mathcal{P}\frac{1}{\nu-\mu}+\lambda(v)\delta(v-\mu),\label{Eqn: singular_eigen}$$
w... | 3,035 | 2,958 | 3,276 | 2,937 | null | null | github_plus_top10pct_by_avg |
e^{- i ( \Delta_{L} - h_{m} ) x}
- \left( \Delta_{K} - h_{m} \right) e^{- i ( \Delta_{L} - h_{k} ) x}
- ( h_{m} - h_{k} ) e^{- i ( \Delta_{L} - \Delta_{K} ) x}
\biggr]
\nonumber \\
&\times&
(UX)^*_{\alpha k} (UX)_{\beta m}
W_{\alpha L} W^*_{\beta L}
\left\{ (UX)^{\dagger} A W \right\}_{m K}
\left\{ W ^{\dagger} ... | 3,036 | 1,386 | 2,523 | 3,036 | null | null | github_plus_top10pct_by_avg |
iltration ${\mathcal F}$ of $M$ with $\Supp({\mathcal F})=\Ass(M)$. The filtration $\mathcal F$ will be the following refinement of the dimension filtration. Denote by $\bar{a}$ the residue class of an element $a\in L$ in $L/L\sect Q_1=D_2(M)/D_1(M)$. Then $(0)\subset
(\bar{z})\subset(\bar{z},\bar{v})\subset (\bar{z},\... | 3,037 | 1,858 | 1,513 | 2,769 | 2,658 | 0.778059 | github_plus_top10pct_by_avg |
taxa). The goal of this analysis is to detect clusters of OTUs at their family taxonomy level according to their abundance by patients ($53$ OTUs at this taxonomy). Our aim is to identify the associations between the different microbial families by reconstructing the ecological network and make a direct comparisons bet... | 3,038 | 159 | 4,078 | 2,856 | 3,018 | 0.775411 | github_plus_top10pct_by_avg |
he 1-BAP, we say that $X$ has the metric approximation property (MAP for short).
\[defBAP\]
Formulations (1) and (2) are classic; their equivalence with (3) is shown for instance in [@kim2008characterizations]. Recall that a Banach space $X$ has the $\lambda$-BAP if and only if, for each $\delta>0$, $X$ has the $(\la... | 3,039 | 2,179 | 1,897 | 2,740 | 3,914 | 0.769306 | github_plus_top10pct_by_avg |
pha^2 & 1\\
\end{array}
\right).$$ $C$ is an optimal $[12, 8, 4]$ skew GQC code over $\mathbb{F}_{3^2}$ actually.
[**Example 4.6**]{} Let $R=\mathbb{F}_{3^2}[x,\sigma]$, where $\sigma$ is the Frobenius automorphism of $\mathbb{F}_{3^2}$ over $\mathbb{F}_3$. The polynomial $g(x)=x-\alpha^2$ is a right divisor of $x^... | 3,040 | 1,936 | 2,845 | 2,914 | 2,537 | 0.779079 | github_plus_top10pct_by_avg |
ll n \; s \; t \; p . \mathbf{reachable} \; s \wedge \mathbf{reachable} \; t \; \wedge \\
s \stackrel{PSched}{\sim} t \wedge s \stackrel{\mathbf{sources} \; n \; s \; p}{\approx} t \longrightarrow s \stackrel{p}{\sim}_n t
\end{aligned}$$
It states that for two arbitrary and reachable states $s$ and $t$, if the two sta... | 3,041 | 2,279 | 3,624 | 2,834 | 299 | 0.814424 | github_plus_top10pct_by_avg |
on $b_r = 4r +n -2$ if $g = 2r+1$, or $b_r = 4r +n -4$ if $g=2r$, we will call it a [*top dimensional maximal simplex*]{}. Since a superinjective map $\lambda$ is injective, $\lambda$ sends top dimensional maximal simplices to top dimensional maximal simplices. In the following lemmas we will see that adjacency and non... | 3,042 | 1,527 | 1,970 | 2,883 | null | null | github_plus_top10pct_by_avg |
-1)(a^2(a^2\alpha m+a m)\cos^3(\theta)+ \nonumber\\
&(3a^3\alpha m r-3a^2mr)\cos^{2}(\theta)+(-3 a^2\alpha m r^2-3 a m r^2)\cos(\theta)-r^3 a m\alpha+r^3 m)\cos(\theta)\Big)\bigg)\Bigg|\Bigg)\end{aligned}$$
In FIG. \[fig8\](a), we indicate the variation of the gravitational entropy density with radial distance and the... | 3,043 | 2,447 | 2,505 | 2,975 | null | null | github_plus_top10pct_by_avg |
ivalent to a category which is finite, skeletal, and has no non-trivial endomorphisms, i.e., to a category whose nerve is a finite simplicial set.) Since Kan extensions in derivators are pointwise, these characterizations admit various improvements in terms of the commutativity of Kan extensions. This gives :
The foll... | 3,044 | 4,249 | 3,457 | 2,819 | null | null | github_plus_top10pct_by_avg |
\leq \sigma_{B^*}$ almost surely, irrespective of the initial position of $X$, where $\sigma_{B^*} = \inf\{t>0\colon X_t\not\in B^*\}$. In particular, thanks to stationary and independent increments, this upper bound for $\sigma_D$ does not depend on $X_0$ in law and $\sup_{x\in D}\mathbb{E}_x[\sigma_D]\leq \mathbb{E}_... | 3,045 | 2,614 | 2,254 | 2,654 | null | null | github_plus_top10pct_by_avg |
mials with coefficients from $A^d$. Similarly, ${\mathbb{J}}^d = J^d[\mathbf{z},\mathbf{z}^*]$.
[(2)]{} Each $J^d$ is a free module over ${\mathbb{C}}[{\mathfrak{h}}]$ and ${{\mathbb{C}}[{\mathfrak{h}}]}^{{{W}}}$.
\(1) By definition, ${\mathbb{A}}^1=\left({{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}[\mathbf{... | 3,046 | 2,457 | 1,833 | 2,717 | null | null | github_plus_top10pct_by_avg |
hi_{m,n} \frac{1}{P^{(1)}_{m,n}} \left( \frac{1}{P^{(1)}_{m+1,n}}-\frac{1}{P^{(1)}_{m-1,n}}\right),\end{gathered}$$ where $$\begin{gathered}
\label{eq:N3-P-G}
P^{(0)}_{m,n} = \phi_{m+1,n} \phi_{m,n} \phi_{m-1,n} , \qquad P^{(1)}_{m,n} = 2 + P^{(0)}_{m,n},\end{gathered}$$ first given in [@14-6]. Despite the $t_2$ notat... | 3,047 | 1,954 | 3,192 | 3,079 | null | null | github_plus_top10pct_by_avg |
ld solution, we now study linear perturbations of this solution. Our treatment will follow the analysis of perturbations of critical solutions in the case of scalar field collapse in 3+1 dimensions [@fro; @hay].
The relevant time parameter in critical collapse being the retarded time $U = -(1/2)\ln(-u)$ (the “scaling ... | 3,048 | 1,734 | 2,875 | 3,056 | null | null | github_plus_top10pct_by_avg |
ting \_j:=e\^[CE]{}\_j,j=1,2,3, where the constant $C$ will be fixed below. This transforms the problem (\[csda1a\])-(\[csda3\]) into an equivalent form, with transport equation on $G\times S\times I$, $$\begin{gathered}
\omega\cdot\nabla_x\phi_1+\Sigma_1\phi_1- K_{1,C}\phi = {\bf f}_1,\label{cosyst1}\\
-{{\frac{\parti... | 3,049 | 1,037 | 3,171 | 3,027 | null | null | github_plus_top10pct_by_avg |
sor is defined by a germ in $\cO_X$. Also, any two effective Weil divisors $D_1=(f_1)$, $D_2=(f_2)$, with $f_1,f_2\in \cO_{X,\zeta_d^k}$, are linearly equivalent, since $\frac{f_1}{f_2}$ is a meromorphic function. Since in the local context all Cartier divisors are principal, the divisor class group of $X$ is $\text{We... | 3,050 | 1,758 | 2,781 | 2,727 | 3,625 | 0.771187 | github_plus_top10pct_by_avg |
st-order Hamiltonian action $$S[x,p,\theta_1,\theta_2]=\int dt\left\{
\frac{1}{2}\left[\dot{x}p-\dot{p}x\right]
-\frac{1}{2}im\omega\left[\dot{\theta}_1\theta_1+\dot{\theta}_2\theta_2\right]
-H\right\}\ .$$
Using the Hamiltonian equation of motion for $x$ in order to reduce its conjugate momentum $p$, namely $p=m\dot{... | 3,051 | 3,106 | 2,935 | 2,846 | null | null | github_plus_top10pct_by_avg |
geq 0\}$. Since $h(t{\mathbf{e}}-{\mathbf{e}})=h({\mathbf{e}})(t-1)^d$ we see that ${\mathbf{e}}\in \Lambda_{\tiny{++}}$. The hyperbolicity cones for the examples above are:
1. $\Lambda_{\tiny{++}}({\mathbf{e}})= {\mathbb{R}}_{++}^n$.
2. $\Lambda_{\tiny{++}}(I)$ is the cone of positive definite matrices.
3. $\Lam... | 3,052 | 1,509 | 2,681 | 2,827 | null | null | github_plus_top10pct_by_avg |
1. The images $\kappa(\bar x_1), \kappa(\bar x_2)$ are $\varepsilon_2$-close in $\RP^s$ and the pairwise distances between the images $\kappa(\bar x_1)$ (or $\kappa(\bar x_2)$), $\kappa(\bar x_3)$ and $\kappa(\bar x_4)$) are greater than the caliber $\varepsilon_2$ of the approximation.
Type 2. The two pairs $(\kappa(... | 3,053 | 1,950 | 1,737 | 3,021 | 4,058 | 0.76841 | github_plus_top10pct_by_avg |
aligned}
\frac{d\omega}{dq}\approx -\frac{Q}{2\rho_0^2}\approx -1.\end{aligned}$$
Denoting the lowest mode in the negative charge sector by $\omega_0$, for some small value of $q$, the ground state energy $\omega_0\rightarrow 0$. At this point the negative charge state has binding energy that completely cancels its as... | 3,054 | 4,042 | 3,763 | 3,047 | null | null | github_plus_top10pct_by_avg |
Long date : 1428498595000
Converted date : Sun, 26 Apr 47237 13:16:40 (After parsing)
[Notice the year]
When the online converter is used (example) : http://www.onlineconversion.com/unix_time.htm , the same output is reproduced.
My purpose is to get the dates sorted in ascending order, but unfortunately, as the year i... | 3,055 | 6,224 | 88 | 2,526 | 282 | 0.814795 | github_plus_top10pct_by_avg |
h_{m} ) }
+ \frac{ 1 }{ ( h_{k} - h_{m} ) (\Delta_{L} - h_{k} )^2 (\Delta_{L} - h_{m} )^2 }
\nonumber \\
&\times&
\biggl\{
(\Delta_{L} - h_{m} )^2
e^{- i ( h_{k} - h_{n} ) x}
- (\Delta_{L} - h_{k} )^2
e^{- i ( h_{m} - h_{n} ) x}
+ ( h_{k} - h_{m} )( h_{k} + h_{m} - 2 \Delta_{L} ) e^{- i ( \Delta_{L} - h_{n} ) x... | 3,056 | 615 | 2,508 | 3,033 | null | null | github_plus_top10pct_by_avg |
um_{k}
\sum_{m}
\biggl[
\frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 }
\left\{ (ix) + \frac{ 2 }{ ( \Delta_{K} - h_{k} ) } \right\}
\left( e^{- i ( \Delta_{K} - h_{m} ) x} + e^{- i ( h_{k} - h_{m} ) x} \right)
\biggr]
\nonumber \\
&\times&
(UX)^*_{\alpha m} (UX)_{\beta m}
(UX)_{\alpha k} (UX)^*_{\beta k}
\left\{ (UX)^{... | 3,057 | 3,417 | 3,034 | 2,787 | null | null | github_plus_top10pct_by_avg |
(a_1){\varphi}(a_2){\varphi}(a_3)^{-1}={\varphi}(\alpha_1(x)){\varphi}(\alpha_2(x)){\varphi}(\alpha_3(x))$, and so $a_1a_2\in Y{\varphi}(A)$ as claimed.
Proof of the main result {#sec:details}
========================
Before we prove \[thm:new.gen\], let us remark that at various points we make the seemingly unnecess... | 3,058 | 1,917 | 1,311 | 3,116 | 1,558 | 0.78826 | github_plus_top10pct_by_avg |
kenmeyer11]. Then, $X \subseteq Y$ would imply that $d_X < d_Y$ (). But this means that we *cannot* deduce from and a criterion of the form $$X \subseteq Y
\;\Rightarrow\;
f_X(\mu) \leq f_Y(\mu)
\qquad
(\forall \mu),$$ since in order to take the weak limit we need to divide by different powers of $k$. Therefore... | 3,059 | 2,002 | 3,016 | 2,680 | null | null | github_plus_top10pct_by_avg |
very assignment function* $\sigma $* defined on* $\psi _{R}^{Q}$* is induced by an interpretation* $\xi $* of the variable x that appears in the rfs into a universe* $\mathcal{U}$* of physical objects, hence* $\sigma =\sigma (\xi )$* and the values of* $\sigma $ *on* $\psi _{R}^{Q}$ *are consistent with (not necessaril... | 3,060 | 3,439 | 3,094 | 2,757 | 3,809 | 0.769984 | github_plus_top10pct_by_avg |
e number of the Reject qubits among the first $m$ qubits, and let $k_a$ be the number of measurements of $\hat{K}$ on $m_a$ qubits from the Accept basis, and analogously for $k_r$. Bob’s measurements of $\hat{K}$ are equivalent to[^5] (for simplicity, we omit writing the $\alpha$ and $N_R$ dependences):
- $q_a\cdot ... | 3,061 | 2,731 | 3,001 | 2,995 | 2,033 | 0.783307 | github_plus_top10pct_by_avg |
another sign of cluster decomposition issues. These multiple dimension zero operators are (discrete Fourier transforms of) identity operators counting the number of components in the corresponding disjoint union of spaces [@summ].
These ideas have also been recently been applied to four-dimensional supergravity theori... | 3,062 | 1,106 | 2,472 | 2,823 | null | null | github_plus_top10pct_by_avg |
========================
In this Appendix we estimate the impact of a non-zero orbital energy of the radial orbit of the cloud towards the IMBH.
For the $E_{orb}<0$ case, the most extreme configuration is that the cloud had zero velocity at $d_{BH,0}$ equal to the current position of the tail of the cloud. This simpl... | 3,063 | 4,414 | 2,471 | 3,019 | 2,888 | 0.776302 | github_plus_top10pct_by_avg |
rom the source to the target node determined by MLE. The node size corresponds to the sum of the incoming transition probabilities from all other nodes to that source node. In the left figure the top four categories with the highest incoming transition probabilities are illustrated for an order of $k=1$. For those node... | 3,064 | 613 | 2,983 | 2,100 | 877 | 0.798418 | github_plus_top10pct_by_avg |
O}_{ {\mathbb P}^n }(m/k))$, again matching the result of the computation.
Another example[^26] will be handy to understand.
Take $\mathfrak{X} = [T^4/{\mathbb Z}_2]$, where the ${\mathbb Z}_2$ acts by sign flips (and so has 16 fixed points). Let us compute $$\chi\left( {\cal O}_{\mathfrak{X}}[0] \right), \: \: \:
\c... | 3,065 | 3,195 | 2,767 | 2,695 | 2,661 | 0.778045 | github_plus_top10pct_by_avg |
left for the reader to verify.
\[tech\] $$\xi_i[h-\partial_jh]=\xi_i[h]-\frac {\partial_j \xi_i[h]\cdot \xi_j[\partial_ih]}{\xi_j[\partial_ih]-1}.$$
\[engine\] If ${\mathbf{x}}\in \Lambda_{++}$, $1\leq i,j \leq n$, $\delta > 1$ and $$\xi_j[h]({\mathbf{x}}) \geq \frac \delta {\delta-1},$$ then $$\xi_i[h-\partial_jh]... | 3,066 | 2,456 | 2,279 | 2,841 | null | null | github_plus_top10pct_by_avg |
ides $\omega_2\,\cap M$ into $\aleph_1$ disjoint intervals. Choose any one of these intervals $J$. There is a family $\mathcal{F}_J=\{F_\gamma:\gamma<\omega_1\}$ in $M$ consisting of $A_\alpha$’s indexed in the interval $J$, with diagonal union including $E$, mod ${\mathrm}{NS}_{\omega_1}$. Then there is a cub $D_J$ in... | 3,067 | 925 | 2,825 | 2,825 | 2,330 | 0.78081 | github_plus_top10pct_by_avg |
ntelligencia UrlRewriter with RegEx is not working
I have the following rewrite rule:
<rewrite url="^/Membership/(.+)/(.+)/(.+)/(.+)" to="/Membership/Index.aspx?parentf=$3&f=$4"/>
which I am expecting should work with this URL:
/Membership/Benefits/Member-Groups/Sub-Groups/Motorcycle-Live.aspx
However, in my pag... | 3,068 | 754 | 1,094 | 2,416 | 14 | 0.840589 | github_plus_top10pct_by_avg |
al{Z}}(\Gamma(\Sigma_n))}$. Assume now $s>t$, so $s>k$. It is known that there exist two primes $\frac{n}{2}<r_1<r_2\leq n$, and we may take $r_2=r$. If $s\neq r$, then $s+r=s+n-k>n$. If $s=r$, then $s+r_1>\frac{n}{2}+\frac{n}{2}=n$. Hence, in both cases $s\notin {{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}$. This ... | 3,069 | 2,046 | 1,399 | 2,990 | null | null | github_plus_top10pct_by_avg |
s of $C$. Let us write the set of eigenvalues of $C$ as a disjoint union
$$\{\gamma_1,\gamma_1^q,\dots\}\coprod\{\gamma_2,\gamma_2^q,\dots\}\coprod\cdots\coprod\{\gamma_r,\gamma_r^q,\dots\}$$of $\langle f\rangle$-orbits, and let $m_i$ be the multiplicity of $\gamma_i$. The unipotent part of an element of $C$ defines a... | 3,070 | 2,822 | 2,448 | 2,690 | 2,510 | 0.779238 | github_plus_top10pct_by_avg |
g\{\al_i^2-(q-r-1)\al_i-2(r-i)\al_i+2\sum_{j>i}^r\al_j\bigg\}\frac{1}{\la_i}\\
&=\pi_{ST}(\Th)\sum_{i=1}^r\bigg\{\al_i^2-(q+r-2i-1)\al_i+2\sum_{j>i}^r\al_j\bigg\}\frac{1}{\la_i}.\end{aligned}$$ Here, assume additionally that $\al_i\leq\al_i^{ST}/2$ with $\al_i^{ST}=q+r-2i-1$ for $i=1,\ldots,r$. For each $i$ we observe ... | 3,071 | 1,718 | 1,800 | 2,863 | null | null | github_plus_top10pct_by_avg |
$$\begin{aligned}
\label{eq:detsummands}
\det \nolimits ^\chi _{\alpha }\in
\sum _{\beta ,\gamma \in {\mathbb{N}}_0^I,\beta +\gamma =k{\alpha }}
{\Bbbk }K_\beta L_\gamma .
\end{aligned}$$ The polynomials $${\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu }
-\chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu }={T}_... | 3,072 | 2,166 | 2,573 | 2,672 | null | null | github_plus_top10pct_by_avg |
translation invariance, we have $$\begin{aligned}
{\label{eq:dec-bd}}
\sum_x\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq\bigg(\sum_{v,x}P_{\Lambda;v}
^{\prime{{\scriptscriptstyle}(0)}}(o,x)\bigg)\bigg(\sup_y\sum_{z,v,x}\tau_{y,z}
Q''_{\Lambda;o,v}(z,x)\bigg)^{i-1}\bigg(\sup_y\sum_{z,x}\tau_{y,
z}Q'_{\Lambda;o}(z,x)... | 3,073 | 1,455 | 2,372 | 2,939 | 3,183 | 0.77421 | github_plus_top10pct_by_avg |
bol{r}}_p(t)}.
\label{eq:fp3}\end{aligned}$$
Perturbation analysis {#sec:perturbation}
=====================
For a weakly-interacting impurity, the condensate wavefunction $\psi$ can be decomposed into an unperturbed wavefunction $\psi_0({\boldsymbol{r}})$ describing the motion and density of the fluid in the absence... | 3,074 | 1,734 | 2,835 | 2,851 | null | null | github_plus_top10pct_by_avg |
ef_cyclic_compos}}}{=} (\tau_{n+1}(\beta)
\circ_1 \alpha)\circ_j \gamma \stackrel{\mathit{op.comp}}{=}
(\tau_{n+1}(\beta) \circ_{j-m+1} \gamma)\circ_1 \alpha
\stackrel{\eqref{compos_cyclic2}}{=}\\ = \tau_{n+p}(\beta
\circ_{j-m}\gamma)\circ_1\alpha \stackrel{\mathit{
\eqref{def_cyclic_compos}}}{=} \alpha\circ_{m+1}
(\b... | 3,075 | 2,238 | 2,419 | 2,627 | null | null | github_plus_top10pct_by_avg |
n\Omega-\Omega^0$, give an element in ${\rm
Mat}_{v_0}(\F_q)^g$. For each $i=1,\dots,k$, we obtain a partial flag by taking the images in $(\F_q)^{v_0}$ of the $V_{[i,j]}$’s via the compositions of the $\varphi_\gamma$’s where $\gamma$ runs over the arrows of the $i$-th leg of $\Gamma$. We thus have defined a map $${... | 3,076 | 2,876 | 2,396 | 2,684 | null | null | github_plus_top10pct_by_avg |
h(Y\mid X)}\bigg] \non\\
&=\int_{\Re^{r\times q}} p(Y\mid \Th)\log {p(Y\mid\Th)\over\ph(Y\mid X)}\dd Y.\end{aligned}$$ The performance of a predictive density $\ph$ is evaluated by the risk function with respect to the KL loss (\[eqn:loss\]), $$\begin{aligned}
R_{KL}(\Th,\ph)&=\Er^{X|\Th}[L_{KL}(\Th,\ph)]\\
&=\int_{\Re... | 3,077 | 1,823 | 2,453 | 2,865 | null | null | github_plus_top10pct_by_avg |
b(\theta^*),\Delta\rangle \; =\; \frac{1}{2}\Delta^\top H(\theta)\Delta \leq -\frac{1}{2}\lambda_2(-H(\theta)){\|\Delta\|}_2^2,\end{aligned}$$ where the last inequality holds because the Hessian matrix $-H(\theta)$ is positive semi-definite with $H(\theta){\boldsymbol{1}} = {\boldsymbol{0}}$ and $\Delta^\top{\boldsymbo... | 3,078 | 1,721 | 2,132 | 2,838 | null | null | github_plus_top10pct_by_avg |
i}}} \cup {{\operatorname{dom}{\Phi}}}$, from which transitivity of equality provides ${{\operatorname{dom}{\upsilon}}} = {{\operatorname{dom}{\Psi}}} \cup {{\operatorname{dom}{\Phi}}}$.
From lemma \[L:ENSEMBLE\_PROD\_SUBSETS\] we conclude $\Psi \subseteq \Upsilon$ and $\Phi \subseteq \Upsilon$. Since these relations ... | 3,079 | 670 | 1,571 | 3,020 | null | null | github_plus_top10pct_by_avg |
}-
{\gamma}_{i}^{2})]. \label{436}$$ This equation can be rearranged as an $N$th-order polynomial equation in the variable ${\tilde{v}}_{\mu}^{2}$, the $\mu$th eigenvalue of ${{\sf \sigma}}$ divided by $N{b}^{2}{\bar{{\rho}^{2}}}_{mkt}$, and is guaranteed to have $N$ real, positive roots (with multiple roots counted a... | 3,080 | 3,485 | 3,762 | 2,979 | null | null | github_plus_top10pct_by_avg |
)=(\psi_1,\hat{\psi})$ for homogeneous (inflow) boundary, and initial condition data, that is $\psi|_{\Gamma_-}=0$, $\psi_j(\cdot,\cdot,E_m)=0$ for $j=2,3$, which is what we were looking for.
By applying the lifts, the existence of a unique solution of the problem (\[csda1a\])-(\[csda3\]) satisfying the inhomogeneous ... | 3,081 | 671 | 1,673 | 3,074 | 3,361 | 0.772861 | github_plus_top10pct_by_avg |
pivot.
In fact, inverting a pivot is, in principle, a very general approach. We could even use inversion in the nonparametric framework as follows. For any $P\in {\cal P}$ and any $j$ define $t(j,P)$ by $$\mathbb{P}( \sqrt{n}|\hat\beta_S(j) - \beta_S(j)| > t(j,P)) = \alpha.$$ Note that, in principle, $t(j,P)$ is know... | 3,082 | 4,286 | 3,641 | 2,936 | null | null | github_plus_top10pct_by_avg |
\in \mathcal{P}_{n}^{\mathrm{OLS}}$ and with probability at least $
1- \frac{2}{n}$ with respect to the distribution of $\mathcal{D}_{2,n}$, the bootstrap distribution belongs to the class $\mathcal{P}_n^*$ of probability distributions for the pair $(X,Y)$ that satisfy the properties of the probability distributions in... | 3,083 | 1,986 | 1,423 | 2,973 | 3,346 | 0.772987 | github_plus_top10pct_by_avg |
an profile. Firstly, we estimated the radial velocity of the third body by cross-correlating the residuals with the template telluric profile for each observing run. In our case, we assumed that the intrinsic velocity of the third body was fixed in one observing run and only changed the velocity depending on the helioc... | 3,084 | 1,480 | 2,979 | 3,177 | null | null | github_plus_top10pct_by_avg |
(centre line), the third quartile (upper edge) and the maximum (upper whisker)).](polymers-08-00205-g002){#polymers-08-00205-f002}
.$$
Let $(-,-)_{B}$ be the bilinear map $\mu_{R}(G)\times \mu_{R}(G) \to RB(G)$ defined by
$(x,y)_{B}:=Btr(xy)$ for $x,y\in \mu_{R}(G)$.
\[bl\] In the basis of Proposition \[basis\] the matrix $M$ of the bilinear form $(-,-)_{B}$ i... | 3,086 | 3,433 | 2,990 | 2,806 | null | null | github_plus_top10pct_by_avg |
Denmark NSCLC Nivolumab Not reported ([@B12])
49 M Taiwan Stage 4 squamous cell carcinoma of hard palate Nivolumab Died ([@B13])
59 ... | 3,087 | 547 | 2,204 | 3,206 | null | null | github_plus_top10pct_by_avg |
4 spacetime scalars,
\left( \overline{\lambda}^{9-16}_{-1/2} \right)^2 \right)
\otimes \left( \psi^{3-4}_{-1/2}, \overline{\psi}^{3-4}_{-1/2} \right)$
valued in $\wedge^2 {\bf 8} = {\bf 28}$... | 3,088 | 4,077 | 2,665 | 2,609 | null | null | github_plus_top10pct_by_avg |
anish for $\lambda=\lambda_i$ but $\lambda'>\lambda_1>\lambda_i$. Thus, $R_\Lambda(\lambda')$ is never vanishing. On the other hand, one may show by use of analysis methods that the remaining terms are as small as possible by restricting $O$ to be is small enough. Thus $\Delta \lambda_e$ will be also very small, and th... | 3,089 | 994 | 2,613 | 2,900 | null | null | github_plus_top10pct_by_avg |
ent results, which lends further credibility to the SCM.
--------- ----------- --------------- -------------------------- --------------------------
SN Distance Reference $H_0(V)$ $H_0(I)$
Modulus (km s$^{-1}$ Mpc$^{-1}$) (km s$^{-1}$ Mp... | 3,090 | 5,012 | 2,198 | 2,923 | null | null | github_plus_top10pct_by_avg |
ine $B_n$-invariant subvariety of $X$ and the action of $B_n$ on $U$ is free. In particular, the projection $\pi:U\mapsto U/B_n$ is etale.
\[proof-of-the-proposition-action-in-odd-case\] For $(i)$, consider the lexicographical order on Laurent monomials. Let $\pi=(k_{1},\dots,k_{n})$ be a sequence of integers with the... | 3,091 | 2,535 | 2,323 | 2,674 | 3,660 | 0.770864 | github_plus_top10pct_by_avg |
aly-free (0,2) GLSM describing a bundle ${\cal E}'$, say, $$0 \: \longrightarrow \: {\cal E}' \: \longrightarrow \:
\oplus_a {\cal O}(n_a) \: \stackrel{F}{\longrightarrow} \:
\oplus_i {\cal O}(m_i) \: \longrightarrow \: 0,$$ over a hypersurface in a weighted projective space ${\mathbb P}^d_{w_0, \cdots, w_d}[w_0 + \cdo... | 3,092 | 2,296 | 723 | 3,194 | 2,305 | 0.781022 | github_plus_top10pct_by_avg |
t|=a{\leqslant}d$. Then the total number of $d$-element subsets of $H_t$ which share only one bin with $D_{t_j}$ is [$a\binom{{s}-a}{d-1}{\leqslant}d\binom{{s}-1}{d-1}$]{}. Thus, we get $$\begin{aligned}
\label{pr:blu}
\sum_{t=1}^m q_i(t, \text{blue}){\leqslant}\sum_{t=1}^m {\frac{\beta d{s}}{n}\cdot
d \fra... | 3,093 | 4,673 | 1,641 | 2,534 | null | null | github_plus_top10pct_by_avg |
ground returns (i.e. returns with $z>2$ m), and metrics computed from the LiDAR intensity. From the aerial images, the mean values of each channel were used along with two spectral vegetation indices [@packalen2009].
Methods {#sec:methods}
=======
Let us denote a vector consisting of the stand attributes by $\mathbf... | 3,094 | 2,354 | 3,573 | 2,946 | null | null | github_plus_top10pct_by_avg |
nite, and the simple description of sheaf cohomology above in terms of $G$-invariants is only valid for $G$ finite, so for general cases a different approach is required. For example, let $\mathfrak{X} = {\mathbb P}^n_{[k,\cdots,k]}$, and ${\cal O}_{\mathfrak{X}}(m)$ as above, then $$H^i(\mathfrak{X}, {\cal O}_{\mathfr... | 3,095 | 2,704 | 3,266 | 2,716 | null | null | github_plus_top10pct_by_avg |
the origin give the states, $$\mO(0,0)|0\rangle\rightarrow |h_\mO,\xi_\mO \rangle.$$ This gives a bijection between the states in the Hilbert space at infinitely past and the operators insertion at the origin on the reference plane. Using the commutation relations, the states above fill the representation of the algeb... | 3,096 | 1,896 | 2,587 | 2,830 | null | null | github_plus_top10pct_by_avg |
*P* = .001), and IS (*P* = .01) muscles ([Table 2](#table2-2325967120913036){ref-type="table"}). Interestingly, even in the group with isolated SSc tendon repair, 7 patients (14.0%) had GC grade ≥ 2 changes in the IS muscle. No difference in muscle quality was observed between shoulders with partial or complete SSc ten... | 3,097 | 770 | 2,398 | 3,354 | null | null | github_plus_top10pct_by_avg |
u}}_{\E_0}})$ as $\E_0 \rightarrow 0$ is correctly captured by the numerically computed optimal states. In particular, we note that $\R_{\E_0}$ is negative for $0 \le
\E_0 \lessapprox 7$ and exhibits the same trend as predicted in for $\E_0 \rightarrow 0$. For larger values of $\E_0$ the rate of growth of enstrophy b... | 3,098 | 1,259 | 3,073 | 2,907 | 3,537 | 0.771696 | github_plus_top10pct_by_avg |
}}\geq \mathbf{1}_{\{X^{(x)}_{\sigma_{B_1}} \not\in B(x,\delta) \}} = \mathbf{1}_{\{|x|^{-1}X^{(x)}_{\sigma_{B_1}} \not\in B({\rm\bf i},\delta/|x|) \}} \geq \mathbf{1}_{\{|x|^{-1}X^{(x)}_{\sigma_{B_1}} \not\in B({\rm\bf i},\delta/\varepsilon) \}}. $$ Recall, however, from (\[scaleB1\]) that $|x|^{-1}X^{(x)}_{\sig... | 3,099 | 1,707 | 2,420 | 2,858 | null | null | github_plus_top10pct_by_avg |
rs more sensibly as follows. The $so(12)\times so(4) \cong so(12) \times su(2) \times su(2)$ is enhanced to an $e_7 \times su(2)$, using the fact that the adjoint representation of $e_7$ decomposes under $so(12) \times su(2)$ as [@slansky]\[table 52\] $${\bf 133} \: = \: ({\bf 66},{\bf 1}) \oplus
({\bf 32},{\bf 2}) \op... | 3,100 | 2,315 | 2,241 | 2,721 | 1,973 | 0.784062 | github_plus_top10pct_by_avg |
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