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 |
|---|---|---|---|---|---|---|---|
uthor:
- Da Rong Cheng
bibliography:
- 'compactness.bib'
title: 'A Compactness Result for Energy-minimizing Harmonic Maps with Rough Domain Metric'
---
---
abstract: 'Determining the properties of starbursts requires spectral diagnostics of their ultraviolet radiation fields, to test whether very massive stars are ... | 3,901 | 609 | 3,421 | 3,904 | null | null | github_plus_top10pct_by_avg |
-1}=R/P_i$ and let $P\in
\Supp({\mathcal F})$. Since there are no non-trivial inclusions between the prime ideals in $\Supp({\mathcal F})$ it follows that $M_P$ has a filtration $(0)=(M_0)_P\subset (M_1)_P\subset \cdots \subset (M_r)_P=M_P$ such that $$(M_i)_P/(M_{i-1})_P = \left\{ \begin{array}{lll} R_P/PR_P, & \text{... | 3,902 | 2,640 | 3,190 | 3,321 | null | null | github_plus_top10pct_by_avg |
a primary field $\phi$. This computation relies on the prescription of appendix \[compositeOPEs\], and on the current-current and current-primary OPEs and .Since we computed these OPEs up to order $f^2$, we will also obtain the stress-tensor OPE up to order $f^2$. $$\begin{aligned}
\label{phiT}
\phi(z) 2 c_1 T(w)
&= \l... | 3,903 | 2,577 | 3,055 | 3,554 | null | null | github_plus_top10pct_by_avg |
_0^z \frac{d z'}{(1 + z')\,\sqrt{\Omega_\Lambda + \Omega_m (1 + z')^3}}\right)~.$$ It is clear that, in order to for $n_{\rm short}(z)$ to be reduced significantly, e.g., by two orders of magnitude, at the epoch $z \simeq 0.9$ of the GRB 090510 [@grb090510], while keeping $n^\star ={\cal O}(1)$, one must consider the m... | 3,904 | 3,034 | 3,606 | 3,506 | null | null | github_plus_top10pct_by_avg |
ger $N$ a curve $\cD=\cD_1\cup...\cup\cD_r$ can be found in $\PP^2_w$ satisfying the following extra conditions:
1. \[prop:3b\] $\cD\cap \operatorname{Sing}\PP^2_w \subseteq\{O\}$ and $\operatorname{Sing}\cD=\{O\}\cup \cN$, where $\cN$ is a set of nodal points of $\cD$ (at smooth points of $\PP^2_w$),
2. \[prop:4\]... | 3,905 | 2,952 | 2,933 | 3,387 | null | null | github_plus_top10pct_by_avg |
root vectors. In Sect. \[sec:Verma\] we start to study Verma modules and special maps between them. Prop. \[pr:VTMiso\] gives a criterion for bijectivity of such maps, and Prop. \[pr:M=L\] identifies irreducible Verma modules. In Sect. \[sec:shapdet\] we study Shapovalov determinants following the approach in [@b-Jose... | 3,906 | 2,946 | 1,119 | 3,576 | 1,615 | 0.787597 | github_plus_top10pct_by_avg |
zero structure Fig.2(a).
We note that a back-bending dispersion was already observed in an early QMC study,[@ph97] where antiferromagnetic fluctuations were proposed as the origin of the pseudogap. Although our numerical data do not exclude the antiferromagnetic fluctuations from the possible mechanisms, the less-$\Ve... | 3,907 | 1,095 | 2,516 | 3,769 | 3,009 | 0.775468 | github_plus_top10pct_by_avg |
at $p={p_\text{c}}$.
It thus remains to show the bounds in [(\[eq:IR-xbd-so\])]{} at $p={p_\text{c}}$. These bounds are proved by adapting the model-independent bootstrapping argument in [@hhs03] (see the proof of [@hhs03 Proposition 2.2] for self-avoiding walk and percolation), together with the fact that $G(x)$ deca... | 3,908 | 2,645 | 2,512 | 3,589 | 1,958 | 0.784151 | github_plus_top10pct_by_avg |
{RM}_2(3,5))=32-17=15$ in accordance with Example 6.12 in [@HP].
Theorem \[thm:main\] is somewhat similar in spirit as Theorem 6.8 from [@HP] in the sense that in both theorems a certain representation in terms of dimensions of Reed–Muller codes is used to give an expression for $d_r({\mathrm{RM}_q}(d,m))$. Where we s... | 3,909 | 1,458 | 3,300 | 3,546 | null | null | github_plus_top10pct_by_avg |
ve Hamiltonian which has new Wilson coefficients and analyze the contributions coming from these coefficients within this model.\
The organization of this work as follows. In Section II, starting from the most general effective Hamiltonian, we compute the differential decay width of the $B \rar \pi\ell^+\ell^-$ decay,a... | 3,910 | 2,192 | 2,122 | 3,660 | null | null | github_plus_top10pct_by_avg |
tegers; thus every sequence is a net but the family[^5] indexed, for example, by $\mathbb{Z}$, the set of *all* integers, is a net and not a sequence) with a sequence and provide the necessary background and motivation for the concept of graphical convergence.
***Begin Tutorial2: Convergence of Functions***
This Tuto... | 3,911 | 4,256 | 4,141 | 3,423 | 2,798 | 0.776868 | github_plus_top10pct_by_avg |
Reference
AG 151 (15.7) 135 (10.0) Codominant 1.57 (1.20-2.05) \< 0.001
GG 11 (1.1) 3 (0.2) 6.18 (1.59-23.98)
AG + G... | 3,912 | 365 | 3,670 | 3,871 | null | null | github_plus_top10pct_by_avg |
i\ \bu_i,\bv_i \in \Z_m^k$. Then $\cF$ is called an $S$-matching vector family of size $n$ and dimension $k$ if $\forall\ i,j$, $$\begin{aligned}
{\langle \bu_i,\bv_j \rangle}\begin{cases}
= 0 & \mbox{if } i=j\\
\in S & \mbox{if } i\ne j
\end{cases}\end{aligned}$$ If $S$ is omitted, it implies that $S=\Z_m\setminus{\... | 3,913 | 4,279 | 3,909 | 3,495 | null | null | github_plus_top10pct_by_avg |
tarrows X$ is a [*finite*]{} equivalence relation if the maps $\sigma_1,\sigma_2$ are finite. In this case, $\sigma:R\to X\times_SX$ is also finite, hence a closed embedding (\[monom.defn\]).
\[setth.eq.rel.defn\] Let $X$ and $R$ be reduced $S$-schemes. We say that a morphism $\sigma:R\to X\times_SX$ is a [*set theore... | 3,914 | 3,955 | 3,261 | 3,499 | 1,724 | 0.786386 | github_plus_top10pct_by_avg |
\quad
\overline{\nabla}_i\overline{\nabla}_{\alpha}v=0.$$ where $i,j=1,2,\cdots,n$, and $\alpha,\beta=n+1,n+2,\cdots,m$, moreover, $$\label{warpedLaplacian}
\Delta_{\overline{M}}=\Delta_f+e^{-\frac{2f}{q}}\Delta_N.$$ Thus, if $u:M\to[0,\infty)$ is a positive solution to , then $u$ satisfies the following equation $$... | 3,915 | 2,944 | 2,089 | 3,430 | null | null | github_plus_top10pct_by_avg |
de K}{\partial E}}}(E)\phi$ is provided by Lemma \[le:K\_C1\]. By assumptions , , we thus see that $h_\phi$ is in $C^1(I,L^2(G\times S))$.
By Theorem \[evoth\] there exists a unique solution $\phi\in C(I,W^2_{-,0}(G\times S))\cap C^1(I,L^2(G\times S))$ of (\[ecsd7\]). Then $\psi(x,\omega,E):=
e^{C(E_m-E)}\phi(x,\omega... | 3,916 | 2,062 | 1,698 | 3,849 | null | null | github_plus_top10pct_by_avg |
he same conditions we have imposed in section \[sec:energy-denominator\] the first one in (\[expansion-parameters\]) is $\simeq 7.6 \times 10^{-4}$ for $\Delta m^2_{J i} = 0.1$ eV$^2$ and $\rho E = 10 \text{ (g/cm}^3) \text{GeV}$, while the second and the third, which are comparable at around the first oscillation maxi... | 3,917 | 2,333 | 3,834 | 3,741 | null | null | github_plus_top10pct_by_avg |
-P_{\sigma}}{2}
\left[ 1- \left(\dfrac{\varrho^{\mathrm{IS}}(\boldsymbol{r})}{\varrho_{0}}\right)^{\gamma} \right]
\delta(\boldsymbol{r}-\boldsymbol{r}^{\prime}). \label{eq:res_pp}$$ with $V_{0}=-390$ MeV $\cdot$fm$^{2}$ and $\varrho_{0}=0.16$ fm$^{-3}$, $\gamma=1$. Here, $\varrho^{\mathrm{IS}}(\boldsymbol{r})$ denote... | 3,918 | 1,100 | 2,504 | 3,913 | null | null | github_plus_top10pct_by_avg |
tum numbers $M = 1$ on the left and $M = 0$ on the right hand side. The top row shows data from the Pb, the bottom row data from the H$_2$ target. The wave intensities are dominated by a broad structure around 1.2[ $\text{GeV}\! / c^2$]{} which is the ${\ensuremath{a_1}}(1260)$.[]{data-label="fig:MDep"}](a1_M1_2004 "fi... | 3,919 | 2,093 | 2,287 | 3,400 | 149 | 0.822566 | github_plus_top10pct_by_avg |
\Delta_{J} - h_{i} )^2 e^{- i h_{k} x}
+ ( h_{i} - h_{k} )( h_{i} + h_{k} - 2 \Delta_{J} ) e^{- i \Delta_{J} x}
\biggr\}
\biggr]
\nonumber \\
&\times&
\left\{ W^{\dagger} A (UX) \right\}_{J k}
\left\{ (UX)^{\dagger} A W \right\}_{k J}
\left\{ W^{\dagger} A (UX) \right\}_{J i}
\nonumber \\
&+&
\sum_{K \neq J} ... | 3,920 | 2,353 | 2,449 | 3,687 | null | null | github_plus_top10pct_by_avg |
ism $(P_{F_i}+(f_i))/P_{F_i}\iso (f_i)/(f_i)P_{F_i}$ results from the fact that $(f_i)\sect P_{F_i}=(f_i)P_{F_i}$ since the set of variables dividing $f_i$ and the set of variables generating $P_{F_i}$ have no element in common. Thus we have shown
\[shelling numbers\] Let $\Delta$ be a shellable simplicial complex wit... | 3,921 | 2,363 | 3,176 | 3,525 | null | null | github_plus_top10pct_by_avg |
odel with many inflating branes along the fifth compact direction, thus extending on the recent models proposed by Randall-Sundrum and Oda \[1,2,3\]. Each brane is taken as a point in the fifth dimension (negligible thickness) $S^1$. They act as localized cosmological constants or alternatively as gravitational point s... | 3,922 | 3,465 | 3,789 | 3,456 | 3,306 | 0.773317 | github_plus_top10pct_by_avg |
a(y)\approx e^{-\sqrt{-\Lambda/6}\;y}\left( 1 - {1\over5}\left({128\pi
\rho_0 \over -\Lambda M^3}\right)^{2/5} e^{2\sqrt{-\Lambda/6}\;y} +\dots
\right).$$
Conclusions and discussion
==========================
We have shown that in brane-world scenarios with a warped extra dimension, it is in principle possible ... | 3,923 | 3,450 | 3,470 | 3,597 | 3,594 | 0.771318 | github_plus_top10pct_by_avg |
nt family of closed, countably compact subspaces each with unbounded range.
Note that the paracompact subspace is the topological sum of $\leq \aleph_1$ $\sigma$-compact subspaces.
An early version of [@DT1] used the axioms ${\mathbf{\mathop{\pmb{\sum}}}}^-$ (defined in Section 5), ${\mathbf{PPI}}$, and the $\aleph_1... | 3,924 | 2,012 | 3,744 | 3,467 | 733 | 0.801206 | github_plus_top10pct_by_avg |
8].
For $x\in X$, let $G_x\subset G$ denote the stabilizer. Let $x\in U_x\subset X$ be a $G_x$-invariant affine open subset. By shrinking $U_x$ we may assume that $G_{x'}\subset G_x$ for every $x'\in U_x$.
In the affine case, quotients by finite groups are easy to get (\[inv.of.fin.gps\]); this is where the condition... | 3,925 | 4,596 | 3,918 | 3,477 | null | null | github_plus_top10pct_by_avg |
6 75.58 79.84 81.62 1.00 0.85 0.86 0.88
Sox2 1.82 4.23 4.59 4.65 98.37 90.34 90.57 92.93 1.00 0.96 0.98 0.98
Stat3 1.60 8.49 4.80 8.33 97.09 80.99 81.7... | 3,926 | 1,361 | 3,199 | 3,243 | null | null | github_plus_top10pct_by_avg |
small scale effects in the inversion procedure laid out in eq. (\[iteration\]) (see also eq. \[\[split\]\]). We “pretend” that we do not know the actual full distortion kernel (eq. \[\[Wfull\]\]), but instead assume only knowledge of the large scale distortion kernel (eq. \[\[Wlinear\]\]) when carrying out the inversio... | 3,927 | 1,693 | 2,700 | 3,731 | null | null | github_plus_top10pct_by_avg |
ymmetry
-----------------------------
As $\MM{\pi}$ and $\MM{l}$ are still continuous in the discretised equations, the multisymplectic integrator will have a discrete particle-relabelling symmetry analogous to the one given in Section \[inverse map EPDiff\], with the only difference being the discretisation of the co... | 3,928 | 2,499 | 303 | 3,897 | null | null | github_plus_top10pct_by_avg |
ere inscribed in $D$ that is centred at $\rho_{n-1}$. Calling this sphere $S_n$, we have that $S_n = \{y\in \mathbb{R}^d \colon |y-\rho_{n-1}| =r_n\}$. We now select $\rho_n$ to be a point that is uniformly positioned on $S_n$. Once again, we note that if $\rho_{n-1}\in D$ almost surely, then the uniform distribution o... | 3,929 | 3,685 | 3,325 | 3,516 | 3,185 | 0.774201 | github_plus_top10pct_by_avg |
\hat\Theta_{B}}=0$$ for $t=1,3$. Finally, we have ${\psi_{1,2}}\circ{\hat\Theta_{B}}=0$ if $v\equiv1\ppmod 4$, and ${\psi_{1,2}}\circ{\hat\Theta_{A}}=0$ if $a-v\equiv3\ppmod4$ or $v=b+3$ (where we apply Lemma \[lemma7\] in the latter case), and ${\psi_{1,2}}\circ{\hat\Theta_{A}}={\psi_{1,2}}\circ{\hat\Theta_{B}}$ if $a... | 3,930 | 2,423 | 1,593 | 3,850 | null | null | github_plus_top10pct_by_avg |
r\Omega^{-2}(L\phi)^2|{\mathrm{d}}\ub\lesssim\delta\Omega_0^{-2}|u|^{-1}\mathscr{F}^2\mathcal{A}^2,\end{aligned}$$ which is the desired estimate . Using , using and , we have $$\begin{aligned}
|{\underline{h}}+1|\lesssim\int_0^\delta\left|\frac{\Omega^2(1+h{\underline{h}})}{r}\right|{\mathrm{d}}\ub\lesssim\delta(1+\mat... | 3,931 | 2,220 | 2,711 | 3,570 | null | null | github_plus_top10pct_by_avg |
al V}_{i,j}, i=1,2,\ldots, \rho, j=0,1,\ldots, m-1\}\subseteq \mathbb{F}_q^l$ where each ${\mathcal V}_{i,j}$ is the vector corresponding to the coefficients $\widetilde{v}_{i,j} \in \mathbb{F}_{q^l}$ with respect to a $\mathbb{F}_q$-basis $\{1, \xi, \ldots, \xi^{l-1}\}$.* $\Box$
Define the Euclidean inner product of ... | 3,932 | 2,094 | 2,077 | 3,775 | 3,420 | 0.772533 | github_plus_top10pct_by_avg |
e broken vertical line)[]{data-label="fig2"}](f51_DOSall_Fermi_0.eps){width="\linewidth"}
![The (black curve) electronic density of states (DOS) and (orange drop lines) Inverse Participation Ratio (IPR) of the insulating model (a) and the metallized model (b). Energy axis for all datasets is shifted to have Fermi leve... | 3,933 | 1,169 | 2,873 | 3,932 | 1,295 | 0.791449 | github_plus_top10pct_by_avg |
’\^2 + 2’\^2 = 0.
We are interested in the behavior of this extended solution in the sector $u < 0$, $v > 0$, i.e. $x < 0$. In this sector, Eqs. (\[einx1\]), (\[phix\]) and (\[einx2\]) can be integrated to (-x)\^[3/2]{}’ & = & \_x\^0 (-x)\^[1/2]{}e\^[2]{}dx,\
(-x)\^[5/4]{}’ & = & 14 \_x\^0 (-x)\^[1/4]{}’ dx,\
-x’ & = ... | 3,934 | 2,165 | 3,332 | 3,774 | null | null | github_plus_top10pct_by_avg |
$$
Moreover, it is obvious by (2.2) that $\widehat{F}$ is a triangle. Thus, it has a unique inverse $\widehat{F}^{-1}$ which is also a triangle and the entries of $\widehat{F}^{-1}$ are given by$$\widehat{f}_{nk}^{-1}=\left \{
\begin{array}
[c]{cc}\frac{f_{n+1}^{2}}{f_{k}f_{k+1}} & (0\leq k\leq n)\\
0 & (k>n)
\end{arr... | 3,935 | 2,381 | 2,803 | 3,566 | null | null | github_plus_top10pct_by_avg |
\int d^3\xi \; f_{\epsilon} ({\bm \xi}) \, \phi(x(\tau, {\bm \xi}))
\ ,
\label{smearedoperator}$$ where ${\bm \xi} = (\xi^1, \xi^2, \xi^3)$ stands for the spatial coordinates associated with the local Fermi-Walker transported frame and $x(\tau, {\bm \xi})$ is a spacetime point written in terms of the Fermi-Walker coo... | 3,936 | 4,063 | 3,718 | 3,502 | 3,384 | 0.772737 | github_plus_top10pct_by_avg |
)=\delta_{o,u}\delta_{o,v}+(1-\delta_{
o,u}\delta_{o,v})\,P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(o,u)\leq\delta_{o,u}
\delta_{o,v}+\frac{O(\theta_0^2)}{{\vbu{|\!|\!|}}^{2q}{\vbv{|\!|\!|}}^q{\vbu-v{|\!|\!|}}^q}.\end{aligned}$$ In addition, instead of using [(\[eq:bb1-bd\])]{}, we use $$\begin{aligned}
{\label... | 3,937 | 1,993 | 3,088 | 3,840 | null | null | github_plus_top10pct_by_avg |
We consider a Markov semigroup $P_{t}$ on ${\mathcal{S}({\mathbb{R}}^{d})}$ with infinitesimal operator $L$ and a sequence $P_{t}^{n},n\in {\mathbb{N}}$ of Markov semigroups on ${\mathcal{S}({\mathbb{R}}^{d})}$ with infinitesimal operator $L_{n}.$ We suppose that ${\mathcal{S}({\mathbb{R%
}}^{d})}$ is included in the... | 3,938 | 2,548 | 2,660 | 3,620 | null | null | github_plus_top10pct_by_avg |
ideals $\{J_1,\ldots, J_r\}$ is uniquely determined. In fact, this set corresponds bijectively to the set of facets of the multicomplex associated with $I$.
The statements (a) and (b) are obviously equivalent, while the existence of of the irreducible ideals $J_i$ is just the multigraded version of Proposition \[prima... | 3,939 | 2,196 | 2,594 | 3,580 | null | null | github_plus_top10pct_by_avg |
`atacaattttgt` cation efflux family protein
... | 3,940 | 3,958 | 4,407 | 3,742 | null | null | github_plus_top10pct_by_avg |
irness condition finally reads as: for any given $\varepsilon$, $$\! \bar{P}^{\cal{B}}_{ch}(m) \! = \! \int p(\alpha) P_{bind}^{\cal{B}}(m,\alpha)[1-P_{bind}^{\cal{A}}(m,\alpha)]{\mathbf d\alpha} < \varepsilon.$$ The coefficient $\alpha$ is sampled randomly by Trent to achieve stronger security requirements. This assu... | 3,941 | 1,966 | 3,229 | 3,555 | null | null | github_plus_top10pct_by_avg |
\PSL(3,\Bbb{R})$.
Let $\Gamma$ be a subgroup of $Mob(\hat{\Bbb{R}})$ and $\gamma\in \Gamma$. Then $$\gamma=\left [\left [
\begin{array}{ll}
i & 0\\
0 & -i\\
\end{array}
\right ]\right ]
\left [\left [
\begin{array}{ll}
a & b\\
c & d\\
\end{array}
\right ]\right ]$$ where $a,b,c,d\in \Bbb{R}$ and $ad-bc=1$. A straightf... | 3,942 | 3,280 | 1,828 | 3,746 | null | null | github_plus_top10pct_by_avg |
acy classes of subgroups of $G$ such that $O^{p}(K)=_{G}J$.
Let $G$ be a finite group and $J$ be a $p$-perfect subgroup of $G$. If $p\mid \frac{|N_{G}(J)|}{|J|}$ and $p^{2}\nmid \frac{|N_{G}(J)|}{|J|}$, then there are exactly two conjugacy classes of subgroups $L$ of $G$ such that $O^{p}(L)=J$.
Let $S_{J}\leqslant N_... | 3,943 | 2,113 | 2,785 | 3,546 | null | null | github_plus_top10pct_by_avg |
ion range to represent this pattern.
**OR nodes:** Both the top semantic-part node and latent-pattern nodes in the third layer are OR nodes. The parsing process assigns each OR node $u$ with an image region $\Lambda_{u}$ and an inference score $S_{u}$. $S_{u}$ measures the fitness between the parsed region $\Lambda_{u... | 3,944 | 2,434 | 3,873 | 3,754 | 1,137 | 0.793782 | github_plus_top10pct_by_avg |
phonon coupling, we expect $\Im N_\sigma(-i0^+)\propto \lambda_c$, as confirmed by NRG calculations. Furthermore, the change in real part of correlation function $F_\sigma(z)$ must also depend quadratically on $\lambda_c$, because it scales with the polaron energy. Hence Eq. predicts an analytic form $1/(1+\alpha \la... | 3,945 | 1,975 | 2,806 | 3,690 | null | null | github_plus_top10pct_by_avg |
$ the vortex ring shrinks with respect to the domain $\Omega$ (cf. figure \[fig:ScalingLaws\_fixE\](d)) and the field ${\widetilde{\mathbf{u}}_{\E_0}}$ ultimately becomes axisymmetric (i.e., in this limit boundary effects vanish). At the same time, it is known that the 3D Navier-Stokes problem on an unbounded domain an... | 3,946 | 1,916 | 1,780 | 3,777 | 2,110 | 0.782673 | github_plus_top10pct_by_avg |
of the solar–metallicity models, and defer discussion of the low–mass, low–metallicity galaxies to § \[sec:lowZspecsynth\].
In figure \[fig:models\], as decreases, the line ratios decrease during the main sequence phase (because the ionizing spectrum softens), and the gap widens between the two phases of high line ra... | 3,947 | 3,113 | 4,084 | 3,810 | null | null | github_plus_top10pct_by_avg |
}{n_b!\;m'_b!\;m''_b!}=\bigg(\sum_{{\partial}{{\bf n}}=\{
o,x\}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\bigg)^3.\end{aligned}$$
It remains to show $|{\mathfrak{S}}_0|\leq|{\mathfrak{S}}'_0|$. To do so, we use the following lemma, in which we denote by $\Omega_{z\to z'}^{{{\bf N}}}$ the set of paths on ${{\mathbb G}}_{... | 3,948 | 2,365 | 2,753 | 3,763 | null | null | github_plus_top10pct_by_avg |
modular invariance. To see more clearly why this is the case we consider the decomposition of the $\bf{16}$ representation in the combinatorial notation of ref. [@xmap] $$\begin{aligned}
{\bf 16}
& \equiv &
\left[ \binom{5}{0} + \binom{5}{2} + \binom{5}{4} \right] \label{so1016}\\
& \equiv &
\left[ \binom{3}{0} + \bi... | 3,949 | 2,678 | 1,086 | 3,842 | null | null | github_plus_top10pct_by_avg |
s of $\Bbb{P}_\Bbb{C}^2\setminus T_{\psi(x)}Ver$, thus $\psi x\in \partial B$ and $T_{\psi(x)}Ver$ is tangent to $\partial(B)$ at $x$. This concludes the proof.\
Now let us prove part (\[l:3\]). Since $\Gamma$ preserves the ball $B$, there is a Hermitian matrix $A=(a_{ij})$ with signature $(2,1)$ such that $B=\{[x]\in\... | 3,950 | 1,822 | 2,196 | 3,648 | 2,185 | 0.781949 | github_plus_top10pct_by_avg |
C}}_{{\bf n}}^b(o){^{\rm c}}}=0$. As a result, $$\begin{aligned}
{\label{eq:0th-summand3}}
{(\ref{eq:0th-summand2})}~=\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}}\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$... | 3,951 | 1,836 | 2,877 | 3,853 | null | null | github_plus_top10pct_by_avg |
tails see Ref. [@bruno_doublegama]) $$\begin{aligned}
\sigma \left( h_1 h_2 \rightarrow h_1 \otimes V_1V_2 \otimes h_2 ;s \right)
&=& \int \hat{\sigma}\left(\gamma \gamma \rightarrow V_1V_2 ;
W_{\gamma \gamma} \right ) N\left(\omega_{1},{\mathbf b_{1}} \right )
N\left(\omega_{2},{\mathbf b_{2}} \right ) S^2_{ab... | 3,952 | 3,840 | 3,497 | 3,522 | null | null | github_plus_top10pct_by_avg |
The matrix element for the $b \rar d\ell^+\ell^-$ decay coming from the most general effective Hamiltonian reads as [@Aliev4] $$\begin{aligned}
{\cal
M}&=&\,\frac{G_F\alpha}{\sqrt{2}\pi}\,V_{tb}V^*_{td}\Bigg[C_{LL}\bar{d}_L\gamma_{\mu}b_L
\bar{\ell}_L\gamma^{\mu}\ell_L+C_{LR}\bar{d}_L\gamma_{\mu}b_L
\bar{\ell}_R\gamma^... | 3,953 | 2,500 | 2,607 | 3,666 | null | null | github_plus_top10pct_by_avg |
(1,0.1)&(\ 0,\ 0)&36.0& 15.2 & 24.0 & 28.2 & 9.6 & 17.7 & 6.8 \\
&(24,\ 0) & & 23.6 & 28.5 & 30.9 & 20.1 & 24.5 & 18.7 \\
&(24,24) & & 32.7 & 33.2 & 33.5 & 31.0 & 31.4 & 32.4 \\
\hline
\end{array}
$
The simulated results for risk of $\ph_{GB}(Y|X)$ are given in Table \[tab:1\]. When the pa... | 3,954 | 1,811 | 1,397 | 4,015 | 3,469 | 0.77213 | github_plus_top10pct_by_avg |
(X)$ is generated by $$s_{i}=e_{i}(x_{1}+x_{1}^{-1},\dots,x_{n}+x_{n}^{-1}),\, i=1,\dots,n-1$$ and $$\Delta_{n}^{\pm}=\frac{1}{2}\bigg(\, \prod_{i=1}^{n}(x_{i}+\dfrac{1}{x_{i}})
\pm
\prod_{i=1}^{n}(x_{i}-\frac{1}{x_{i}})\,\bigg).$$ Moreover, $\Delta_{n}^{-}\in \c[s_{1},\dots,s_{p-1},\Delta^{+}]_P$, where $P$ is some po... | 3,955 | 3,098 | 2,573 | 3,653 | null | null | github_plus_top10pct_by_avg |
potential in presence of magnetic field, calculate the second-order QNS of deconfined QCD matter in this two scale hierarchies.
The paper is organized as follows: in Sec. \[setup\] we present the setup to calculate second-order QNS. In Subsec. \[quark\_f\], one-loop HTL free-energy of quark in presence of strong magne... | 3,956 | 1,852 | 2,431 | 3,756 | null | null | github_plus_top10pct_by_avg |
($+,+,-$) ($+,-$) 1/6 2/3
($+,+,-$) ($-,+$) 1/6 -1/3
$\left( \, \textbf{4} \, , \textbf{2} , \, 0 \, \right)$ ... | 3,957 | 6,109 | 695 | 3,162 | null | null | github_plus_top10pct_by_avg |
\gamma_\alpha(f) \\
& \qquad + \frac{1}{{\left\vert G \right\vert}}\sum_{\chi
\neq \overline{\chi}} {\mathbb{E}}f\left(\frac{1}{\sqrt{{\left\vert G \right\vert}}} \sum_{a
\in G}
\chi(a) Y_a\right) - (1-p_2) \gamma_{\mathbb{C}}(f) \Biggr\vert \\
& \le p_2 \delta_n + (1-... | 3,958 | 5,103 | 1,976 | 3,313 | null | null | github_plus_top10pct_by_avg |
nitiated \[[@B9],[@B23]\], using a high protein high fibre weight loss diet (Table [1](#T1){ref-type="table"}), and fed according to manufacturer's instructions. The initial food allocation for weight loss was determined by first estimating maintenance energy requirement (MER = 440 kJ \[105 Kcal\] × body weight \[kg\]^... | 3,959 | 4,430 | 4,168 | 3,721 | null | null | github_plus_top10pct_by_avg |
ects and consider a single impurity in a two-dimensional BEC.
We rewrite the dGPE in dimensionless units by using the characteristic units of space and time in terms of the long-wavelength speed of sound $c=\sqrt{\mu/m}$ in the homogeneous condensate and the coherence length $\xi=\hbar/(m
c)=\hbar/\sqrt{m\mu}$. Space ... | 3,960 | 2,631 | 3,790 | 3,643 | 3,481 | 0.772049 | github_plus_top10pct_by_avg |
.5$\pm$1.8 11.9$\pm$0.8 4.9$\pm$0.3 0.8$\pm$0.1
21 05 41 40.44 -69 48 36.23 55 0.5$\pm$0.1 20.0$\pm$2.0 30.6$\pm$2.3 30.4$\pm$1.5 15.5$\pm$1.1 7.1$\pm$0.5 2.9$\pm$0.2 0.4$\pm$0.1
22 05 38 00.17 -69 42 32.98 55 0.7$... | 3,961 | 4,656 | 3,313 | 3,701 | null | null | github_plus_top10pct_by_avg |
A_{\alpha}$, it follows that $\bigcup A_{\alpha}$ is an upper bound of $\mathcal{A}$; this is also be the smallest of all such bounds because if $U$ is any other upper bound then every $A_{\alpha}$ must precede $U$ by Eq. (\[Eqn: upper bound\]) and therefore so must $\bigcup A_{\alpha}$ (because the union of a class of... | 3,962 | 4,981 | 4,080 | 3,616 | 2,350 | 0.780656 | github_plus_top10pct_by_avg |
tic moment in the region near the thresholds, and for fields $B \lesssim B_c$. The eigenvalues of the modes can be written approximately [@Hugo2] as $$\pi_{n,n^{\prime}}^{(i)}\approx-2\pi\phi_{n,n^{\prime}}^{(i)}/\vert\Lambda\vert
\label{eg5}$$ with $\vert\Lambda\vert=((k_\perp^{\prime 2 }-k_\perp^{\prime
\prime 2})(k_... | 3,963 | 1,274 | 2,886 | 3,779 | null | null | github_plus_top10pct_by_avg |
set{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}z'_h\}$}}}
\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg(\prod_{i\ne ... | 3,964 | 2,567 | 2,448 | 3,603 | null | null | github_plus_top10pct_by_avg |
erity; *p* \< 0.05). Men with CSB scored higher on the DRRI measure of relationship quality during deployment (*p* \< 0.05), as well as on the DRRI post-deployment stressors scale (*p* \< 0.05). Both findings are indicative of more negative experiences (i.e., poor relationship quality and more stressors).
######
Bas... | 3,965 | 6,389 | 2,423 | 1,817 | null | null | github_plus_top10pct_by_avg |
$L$ and let $P_{t}^{n},n\in {%
\mathbb{N}}$, be a sequence of Markov semigroups on ${\mathcal{S}({\mathbb{R}%
}^{d})}$ with infinitesimal operators $L_{n}$, $n\in {\mathbb{N}}$. Classical results (Trotter Kato theorem, see e.g. [@EK]) assert that, if $L_{n}\rightarrow L$ then $P_{t}^{n}\rightarrow P_{t}.$ The problem t... | 3,966 | 2,855 | 1,183 | 3,846 | null | null | github_plus_top10pct_by_avg |
the $\alpha=-3$ branes, whose corresponding mixed-symmetry potentials are [@branesandwrappingrules] $$\begin{aligned}
& E_8 \ \ E_{8,2} \ \ E_{8,4} \ \ E_{9,2,1} \ \ E_{8,6} \ \ E_{9,4,1} \ \ E_{10,2,2} \ \ E_{10,4,2} \ \ E_{10,6,2} \ \ \ \ ({\rm IIB}) \nonumber \\
& E_{8,1} \ \ E_{8,3} \ \ E_{9,1,1} \ \ E_{8,5} \ ... | 3,967 | 2,597 | 3,878 | 3,667 | 3,659 | 0.770865 | github_plus_top10pct_by_avg |
at{W}}\right) \right).
\end{array}$$ If we substitute (\[eq35\]) into the derivative of the Lyapunov function (\[eq38\]), it yields $$\begin{aligned}
\label{eq39}
{{\dot{V}}_{2}}=&-{{{z}}_{1}}^{T}{{c}_{1}}{{{z}}_{1}}-{{{s}}^{T}}{{c}_{2}}sign\left( {s} \right)-{{{s}}^{T}}{{c}_{3}}{s}+{{{s}}^{T}}{\varepsilon}\\ \nonumb... | 3,968 | 924 | 1,979 | 4,049 | null | null | github_plus_top10pct_by_avg |
s frames are given by $$\widehat e_+ = \left(\begin{array}{c} \frac{hu}{a^4} ( a^2 u^2 dx_2 + dx_3) \\ \frac{hu}{a^4}( a^2 u^2 dx_3 - dx_2) \end{array}\right) \ , \quad
\widehat e_- = \left(\begin{array}{c} \frac{hu}{a^4} (- a^2 u^2 dx_2 + dx_3) \\ \frac{hu}{a^4}(- a^2 u^2 dx_3 - dx_2) \end{array}\right) \ , \quad
h = ... | 3,969 | 2,927 | 3,410 | 3,444 | null | null | github_plus_top10pct_by_avg |
of $G$, and the class of all $p$-closed groups is a saturated formation, we deduce that ${{\operatorname}{\Phi}(G)}=1$ and that $G$ has a unique minimal normal subgroup, say $N$. Since $G/N$ has a normal Sylow $p$-subgroup, then ${{\operatorname}{O}_{p}(G)}=1$, and so $N$ is not a $p$-group. Also this implies that $PN ... | 3,970 | 2,459 | 2,565 | 3,584 | null | null | github_plus_top10pct_by_avg |
cr
& \left. \qquad
+ :j^a_{L,z}(z) j^b_{L,z}(w): + ...
\right) j^c_{L,\bar z}(x) \cr
%
& = \frac{c_1 \kappa^{ab}j^c_{L,\bar z}(x)}{(z-w)^2} + \frac{c_3(c_2-g) {f^{abc}} (\bar z - \bar w)}{(z-w)^2(\bar w - \bar x)^2} + :j^a_{L,z}(z) j^b_{L,z}(w):j^c_{L,\bar z}(x) + ...\end{aligned}$$ The OPE involving the composite ope... | 3,971 | 1,496 | 3,464 | 3,700 | null | null | github_plus_top10pct_by_avg |
2k$ and $r(M^*) \ge r(M)$, then $M$ has a minor isomorphic to one of $M(K_n)$, $B(K_n)$ or $U_{n,2n}$.
The following conjecture, which is essentially posed in \[\[highlyconnected\]\], states that any highly vertically connected matroid omitting a given uniform minor is close to having one of three specific structures ... | 3,972 | 2,498 | 2,361 | 3,592 | null | null | github_plus_top10pct_by_avg |
h these frames depend on $\nu_i$ the overall metric remains the round $S^5$ independent of $\nu_i$. The advantage of this basis is that the T-dualisation acts only on the $e_1$ and $e_2$ directions. We non-abelian T-dualise with respect to the central extension of $\tilde{h}_1$ and $\tilde{h}_2$ making the gauge fixing... | 3,973 | 3,373 | 3,705 | 3,539 | null | null | github_plus_top10pct_by_avg |
[@SGA]. A [*section*]{} of this projection is a functor $\sigma:\Lambda \to {{\mathcal C}}_\#$ such that $\tau
\circ \sigma = {\operatorname{\sf id}}$ (since $\Lambda$ is small, there is no harm in requiring that two functors from $\Lambda$ to itself are equal, not just isomorphic). These sections obviously form a cat... | 3,974 | 2,106 | 3,491 | 3,622 | 2,268 | 0.781275 | github_plus_top10pct_by_avg |
ucted experiments in simulation environment. To simulate the DAR protocol, the two robot arms were first to track the desired trajectories to reach the object. The reference trajectories in the first 2 seconds are mathematically specified by $$\label{eq45}
\begin{array}{r@{}l@{\qquad}l}
{{x}_{a1}}(t)&={{x}_{f1}}+({... | 3,975 | 3,686 | 4,342 | 3,636 | null | null | github_plus_top10pct_by_avg |
{\delta _{1},...,\delta
_{m}}(x,z)=Q_{2}^{\ast }p_{1}^{\beta ,x}(z)=\prod_{i=1}^{m-j}(S_{\delta
_{m-i+1}}^{\ast }U_{m-i}^{\ast })S_{\frac{1}{2}\delta _{j}}^{\ast
}p_{1}^{\beta ,x}(z).$$We will use (\[h’\]) $m-j$ times first and (\[B1\]) then. We denote $$q_{1}^{\prime }=q_{1}+(m-j)(a+b)$$and we write $$\begin{array}{rl... | 3,976 | 2,073 | 835 | 4,033 | null | null | github_plus_top10pct_by_avg |
x_2(x_1x_4))\\
-(x_1x_2)(x_3x_4)-(x_1x_3)(x_2x_4)-(x_3x_2)(x_1x_4)\end{gathered}$$ is the Jordan identity in a multilinear form [@Zhevl:78].
Hence using the general definition of a variety of dialgebras [@Kol:08] we obtain that the class of Jordan dialgebras is defined by two 0-identities (\[eq:0-DialgebraDef\]) and t... | 3,977 | 1,355 | 1,842 | 4,041 | 1,517 | 0.788661 | github_plus_top10pct_by_avg |
symmetric KK theories. As usual we assume that the classical bubble solutions provide useful approximations to solutions in theories with moduli stabilized by additional fluxes or other objects. Ref. [@Dine:2004uw], for example, found that neutral bubble solutions persist after adding simple stabilizing potentials.
[^... | 3,978 | 3,322 | 1,483 | 3,105 | 4,128 | 0.767936 | github_plus_top10pct_by_avg |
the bar chart.
Pizza_bar <- ggplot(Pizza_Data_Research_Rockstar, aes(Number_of_times_eaten_pizza))
Times_eaten_pizza_7_days_bar <- Pizza_bar + geom_bar()
Times_eaten_pizza_7_days_bar
Don't know how to automatically pick scale for object of type tbl_df/tbl/data.frame. Defaulting to continuous.
The challenge becomes la... | 3,979 | 8,165 | 249 | 2,039 | 65 | 0.829179 | github_plus_top10pct_by_avg |
, for some $i\in\{1,2\}$, we have that $|\cA(i)|\le |\cA(1,2)|$ then $(\cI\cup\{A\setminus \{i\}\mid A\in\cA(1,2)\}\cup\{\{1,2\}\setminus\{i\}\})\setminus\cA(i)$ is a star-subfamily of $\cH$ of size larger than $\cI$, which is a contradiction. Thus we know that $|\cA(1,2)|<\min(|\cA(1)|,|\cA(2)|)$.\
If $|\cA(1)|=|\cA(2... | 3,980 | 2,842 | 1,182 | 3,844 | null | null | github_plus_top10pct_by_avg |
the correlator. By Lorentz invariance the expansion takes the schematic form ${1 \over (x-y. \bar n)^4 (x -y. n)^4}$ which scales like $\lambda^{-4}$ under rescaling of $n$. The additional factor of $\lambda$ comes from the measure $\int_{- \infty}^{\infty} d (x .n)$.
[c)]{} Momentum rescaling This follows from the fa... | 3,981 | 2,867 | 3,709 | 3,450 | null | null | github_plus_top10pct_by_avg |
n$. Then this (maximal) zero is achieved for some $T(f)$, where $f$ has at most one distinct zero in $(a,b)$.
Moreover, if the maximal zero above is achieved for some $T(f)$, where $f \in {\mathcal{M}}_d$ is $(a,b)$–rooted, then the maximal zero is also achieved for $T ((t-\epsilon/d)^d)$.
Let ${\mathcal{A}}={\mathca... | 3,982 | 4,003 | 4,042 | 3,669 | null | null | github_plus_top10pct_by_avg |
tandard pairs of $I$ as defined by Sturmfels, Trung and Vogel in [@STV]. However the main justification of the definition is given by Proposition \[multiprimary\] where we show that a pretty clean filtration of $S/I$ determines uniquely the facets of $\Gamma$. This result finally leads us to the definition of shellable... | 3,983 | 2,455 | 1,880 | 3,677 | 2,529 | 0.779161 | github_plus_top10pct_by_avg |
M_1 \oplus (\bigoplus_{i\geq 2} M_i)$$ and $C(L^{j})$ is spanned by $$(\langle \pi e_i\rangle, e) ~(resp.~ (\langle \pi e_i\rangle, \pi a, e)) \textit{~and~} M_1 \oplus (\bigoplus_{i\geq 2} M_i).$$\
We now construct a morphism $\psi_j : \tilde{G} \rightarrow \mathbb{Z}/2\mathbb{Z}$ as follows. (There are 3 cases.)\
... | 3,984 | 2,998 | 2,786 | 3,489 | 3,039 | 0.775288 | github_plus_top10pct_by_avg |
)\bar{\tau}^{(1)}_\sigma
+\bar{\tau}^{(2)}_\sigma
\nonumber\end{aligned}$$ The purely fermionic Green’s function $G_{d_{\mu \sigma}, d^\dagger_{\mu' \sigma}}(z)$ picks up the factor $(1+\lambda^{\rm tip} x_0)^2$ which therefore appears as a prefactor in the elastic density of states and in the corresponding tunnel... | 3,985 | 4,061 | 3,389 | 3,588 | null | null | github_plus_top10pct_by_avg |
els, where spacetime gauge symmetries typically appear as worldsheet global symmetries visible in the UV, but is not contradicted by any physics we know. In any event, spectrum computations at Landau-Ginzburg points in these theories have not proven insightful.
Class III: Twisted bundles {#sect:type3:twisted}
========... | 3,986 | 2,212 | 2,340 | 3,457 | null | null | github_plus_top10pct_by_avg |
$\overline{\Sigma}_L^{\mu\nu}\ :\ \
{\left(\overline{\Sigma}_L^{\mu\nu}\right)^{\dot{\alpha}}}_{\dot{\beta}}=
\frac{1}{4}i\left[
\overline{\sigma}_L^{\mu\dot{\alpha}\gamma}\,
{\sigma^\nu}_{\gamma\dot{\beta}}\,-\,
\overline{\sigma}^{\nu\dot{\alpha}\gamma}\,
{\sigma^\mu}_{\gamma\dot{\beta}}\right]\ .$$
Given these diff... | 3,987 | 2,908 | 3,504 | 3,637 | null | null | github_plus_top10pct_by_avg |
==================
In this section, we consider a class of hierarchical priors inspired by Faith (1978) and derive a sufficient condition for minimaxity of the resulting Bayesian predictive density. Also, a proper Bayes and minimax predictive density is provided.
A class of hierarchical prior distributions
----------... | 3,988 | 2,118 | 1,719 | 3,866 | null | null | github_plus_top10pct_by_avg |
1\in\tilde{{\cal D}}$ and ${{\bf k}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_{\tilde{{\cal D}}{^{\rm c}}}}$, this chain of bubbles starts and ends with ${{\bf m}}$-connected clusters (possibly with a single ${{\bf m}}$-connected cluster), not with ${{\bf k}}$-connected clusters. Therefore, by following the argument around [(\... | 3,989 | 1,810 | 2,844 | 3,851 | null | null | github_plus_top10pct_by_avg |
\delta$. To prove [(\[eq:P”j-bd\])]{} for $j=1$, we first recall the definition [(\[eq:P”1-def\])]{} of $P_{\Lambda;u,v}^{\prime\prime{\scriptscriptstyle}(1)}$ (and Figure \[fig:P-def\]). Note that, by [(\[eq:GGpsi-bd\])]{}, $\sum_{v'}G(v'-y)\,G(z-v')\,\psi_\Lambda(v',v)$ obeys the same bound on $\sum_{v'}G(v'-y)\,G(z-... | 3,990 | 1,618 | 3,106 | 3,847 | null | null | github_plus_top10pct_by_avg |
-------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------
![Box diagrams which contribute to the process at NLO. The solid circle stands for the weak... | 3,991 | 3,087 | 761 | 3,999 | null | null | github_plus_top10pct_by_avg |
times \R^{d}$$$\left\vert \partial _{x}^{\alpha }\partial _{y}^{\beta
}p_{t}(x,y)\right\vert \leq \frac{C}{t^{\theta _{0}(1+\frac{a+b}{\delta }%
)(\left\vert \alpha \right\vert +\left\vert \beta \right\vert
+2d+\varepsilon )}}\times \frac{(1+\left\vert x\right\vert
^{2})^{\pi(\kappa) }}{(1+\left\vert x-y\right\vert
^{2... | 3,992 | 2,412 | 2,319 | 3,756 | null | null | github_plus_top10pct_by_avg |
can define a non-strict $S$-set $X$, $(s,x)\mapsto s\cdot x$, where defined, to be [*connected*]{} if for any $x\in X$ and any $e,f\in E$ such that $e\cdot x$ and $f\cdot x$ are defined, there is a sequence of idempotents $e=e_1,e_2,\dots, e_k=f$, called a [*connecting sequence over*]{} $x$, such that $e_i\cdot x$ is ... | 3,993 | 2,481 | 2,957 | 3,617 | 1,809 | 0.785504 | github_plus_top10pct_by_avg |
}}
\frac{w_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}({{\bf h}})}{Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}}
{\overset... | 3,994 | 2,583 | 2,417 | 3,758 | null | null | github_plus_top10pct_by_avg |
psi_{|\Gamma}$, which belongs to $T^2(\Gamma)=T^2(\Gamma_{e})$. Let $\tilde\Psi\in \tilde W^2(G_e\times S\times I)$ given in Remark \[wla\], Part C. Define ${{{\mathcal{}}}E}\psi$ by \[cg5\] [E]{}:=
& [on]{} GSI\
\_()=\_[e,]{}() & [on]{} \_\
& [on]{} G\_eSI
Then ${{{\mathcal{}}}E}\psi$ is in $W^2({\mathbb{R}}^3\tim... | 3,995 | 1,191 | 3,093 | 3,654 | null | null | github_plus_top10pct_by_avg |
follow the usual conventions of nonstandard analysis. ${I \kern -4.5pt N}= \{0,1,2,\ldots\}$ is the set of natural numbers, and $^{*}\! {I \kern -4.5pt N}$ is the set of hypernaturals. The standard hypernaturals are (i.e., can be identified with) the natural numbers. Also, $\langle a_{n}\rangle$ or $\langle a_{n}\!: n... | 3,996 | 5,228 | 3,784 | 3,702 | 1,922 | 0.784347 | github_plus_top10pct_by_avg |
{L}})$ given by thinking of an automorphism of $\oplus_n P$ as an automorphism of $\oplus_{n+1} P$ by taking the direct sum with the identity map on the last factor.
Now for any principal bundle $G \to P \to M$, consider the Atiyah-Bott equivalence [@atiyahbott] mentioned in the proof of Theorem \[main\]: $$\beta : B{... | 3,997 | 2,767 | 3,151 | 3,761 | null | null | github_plus_top10pct_by_avg |
ntro\](2). This is derived from an analogous result about the associated graded module of $B_{k0}\otimes_{U_c}eH_c$ that also implies Corollary \[cohh-intro\]. Section \[sect7\] then gives a reinterpretation of Theorem \[mainthm-intro\] in terms of a tensor product filtration of $B_{ij}$. In Appendix \[app-a\] we prove... | 3,998 | 2,833 | 855 | 3,765 | 2,935 | 0.775984 | github_plus_top10pct_by_avg |
bel{A.bar}
\begin{CD}
0 @>>> j_!M^\Delta_\# @>>> \overline{A_\#} @>>> A_\# @>>> 0
\end{CD}$$ of cyclic $k$-vector spaces.
Now assume in addition that $M$ is equipped with a structure of a cyclic $A$-bimodule $M_\#$, so that $M^\Delta_\# \cong
j^*M_\#$, and we have the structure map $\tau_\#:j_!M^\Delta_\# \to
M_\#$. T... | 3,999 | 3,672 | 2,023 | 3,720 | 1,816 | 0.785458 | github_plus_top10pct_by_avg |
space, for which the matter sector is described by string coordinates $X^\mu(z)$ and their partners $\psi^\mu(z)$ $(\mu=0,1,\cdots,9)$. The reparametrization ghost sector and superconformal ghost sector are described by a fermion pair $(b(z),c(z))$ and a boson pair $(\beta(z),\gamma(z))$, respectively. The superconform... | 4,000 | 1,976 | 2,816 | 3,823 | null | null | github_plus_top10pct_by_avg |
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