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 |
|---|---|---|---|---|---|---|---|
& = \begin{cases}
\displaystyle
\frac{1}{\sigma_a^2 - \sigma_n^2}\left[(\sigma_a^2\mu_n - \sigma_n^2\mu_a) \pm \sigma_a\sigma_n\sqrt{(\mu_a - \mu_n)^2 + 2(\sigma_a^2 - \sigma_n^2)ln\frac{\sigma_a}{\sigma_n}}\right], & \sigma_a \ne \sigma_n\\
... | 1,801 | 4,908 | 682 | 1,433 | null | null | github_plus_top10pct_by_avg |
more, \[n2k-3\] & (\_2(x,E’,E)((x,\_+(E’,E,),E’))\
= & (x,E’,E)(x,\_+(E’,E,),E’)\
&+ \_2(x,E’,E)(\_)(x,\_+(E’,E,),E’) (E’,E,)\
&+ \_2(x,E’,E)(x,\_+(E’,E,),E’) and similarly for the last term in (\[n2k-2\]). Since $\mu_{22}(E,E)=\mu_{22,p}(E,E)=1$ we have \[n2k-4\] & ((\_[22,2]{})(x,,E’,E))\_[|E’=E]{}\
=& (x,E,E)(x,,E)\... | 1,802 | 306 | 1,375 | 1,841 | null | null | github_plus_top10pct_by_avg |
erally speaking, would you say that most people can be trusted, or that you can't be too careful in dealing with people?'. Institutional trust, the mediation variable, is obtained by combining respondents' reported trust in the parliament, legal system and parties (see [Table 2](#pone.0220160.t002){ref-type="table"} an... | 1,803 | 1,191 | 2,526 | 2,082 | null | null | github_plus_top10pct_by_avg |
\Gamma_{\pm})}\leq {\left\Vert \psi\right\Vert}_{\tilde{W}^2_{\mp}(G\times S\times I)}$$ and thus $\gamma_{\pm} :\tilde{W}^2_{\mp,0}(G\times S\times I)\to T^2(\Gamma_{\pm})$ is bounded.
Occasionally we work in the (energy independent) spaces $L^2(G\times S)$. The corresponding Hilbert spaces $W^2(G\times S)$, $T^2(\Ga... | 1,804 | 798 | 1,844 | 1,708 | null | null | github_plus_top10pct_by_avg |
. ]{}
[^17]: [\[Foot: 0=3Dphi\]If $y\notin\mathcal{R}(f)$ then $f^{-}(\{ y\}):=\emptyset$ which is true for any subset of $Y-\mathcal{R}(f)$. However from the set-theoretic definition of natural numbers that requires $0:=\emptyset$, $1=\{0\}$, $2=\{0,1\}$ to be defined recursively, it follows that $f^{-}(y)$ can be id... | 1,805 | 3,289 | 3,005 | 1,875 | 2,043 | 0.783245 | github_plus_top10pct_by_avg |
$$V_{2}(r) =
\displaystyle\frac{2\alpha^{2}}
{\biggl(r + r_{0} + \displaystyle\frac{1}{C\alpha}\biggr)^{2}}.
\label{eq.3.1.5}$$ We see, that this potential is finite in the whole region of its definition at any values of the parameters $C>0$ and $r_{0} \ge 0$. Thus, we have obtained the reflectionless potential... | 1,806 | 2,723 | 2,579 | 1,820 | 1,440 | 0.789549 | github_plus_top10pct_by_avg |
i_\mu })^t.
\label{eq:maid1}
\end{aligned}$$ Then $$\begin{aligned}
E_i F_{\beta _{n+1}}^{{b^{\chi}} (\beta _{n+1})-1}
F_{\beta _{n+2}}^{{b^{\chi}} (\beta _{n+2})-1}\cdots
F_{\beta _{n+\mu -1}}^{{b^{\chi}} (\beta _{n+\mu -1})-1} F_{\beta _\mu }^t \times
\qquad &
\\
F_{\beta _{\mu -1}}^{{b^... | 1,807 | 882 | 2,266 | 1,753 | null | null | github_plus_top10pct_by_avg |
consider the trace pairing ${\rm Tr}:\mathfrak{g}\times\mathfrak{g}\rightarrow\C^{\times}$. Define the Fourier transform $\calF^\mathfrak{g}:{\rm Fun}(\mathfrak{g})\rightarrow{\rm Fun}(\mathfrak{g})$ by the formula
$$\calF^\mathfrak{g}(f)(x)=\sum_{y\in\mathfrak{g}}\Psi\left({\rm Tr}\,(xy)\right)f(y)$$for all $f\in{\r... | 1,808 | 1,481 | 1,442 | 1,711 | null | null | github_plus_top10pct_by_avg |
Wisotzki, L., Barden, M., Beckwith, S. V. W., Bell, E. F., Borch, A., Caldwell, J. A. R., Häu[ß]{}ler, B., Jogee, S., McIntosh, D. H., Meisenheimer, K., Peng, C. Y., Rix, H.-W., Somerville, R. S., & Wolf, C. 2004, submitted to ApJ, astro-ph/0403645
Schade, D., Boyle, B. J., & Letawsky, M. 2000, MNRAS, 315, 498
Schle... | 1,809 | 1,429 | 3,309 | 2,147 | null | null | github_plus_top10pct_by_avg |
) = 0$ and $\lambda_2(\lL)=\lambda_3(\lL)=\cdots=\lambda_{\ld}(\lL)$. Therefore, using $\lambda_2(\lL) = \Tr(\lL)/(\ld-1) = n\ld$. Using the fact that $\E[\lM]$ and $\lL$ are symmetric matrices, we have, $$\begin{aligned}
\label{eq:lambda2_bottoml_expec}
\lambda_2(\E\big[\lM\big]) \geq \frac{e^{-4b}(1-\beta_1)^2(1 - ... | 1,810 | 1,307 | 1,560 | 1,733 | null | null | github_plus_top10pct_by_avg |
up to $N=8000$, see Fig. \[Results Figure\]. We see that the [*fairness condition*]{} $\mbox{sup}_m \bar{P}_{ch}(m) << 1$ is satisfied (for $N=600$ we got $\mbox{sup}_m \bar{P}_{ch}(m) = \bar{P}_{ch}(92)= 0.0811$, while for $N_A=N_R$ we have $\mbox{sup}_m \bar{P}_{ch}(m) = \bar{P}_{ch}(1455)= 0.0247$ for $N=8000$), whi... | 1,811 | 1,294 | 2,066 | 1,867 | 2,926 | 0.776036 | github_plus_top10pct_by_avg |
e ?
Here is my c_cpp_properties.json :
{
"configurations": [
{
"name": "Win32",
"includePath": [
"${workspaceFolder}",
"C:\\Program Files (x86)\\mingw-w64\\i686-8.1.0-posix-dwarf-rt_v6-rev0\\mingw32\\lib\\gcc\\i686-w64-mingw32\\8.1.0\\include"
... | 1,812 | 2,082 | 25 | 1,949 | 123 | 0.824058 | github_plus_top10pct_by_avg |
f $\Gamma$ to be $$\Lambda_{\Kul}(\Gamma)=\Lambda(\Gamma)\cup L_2(\Gamma),$$
4. *Kulkarni’s discontinuity region* of $\Gamma$ to be $$\Omega_{\Kul}(\Gamma)=\mathbb{P}^n_{\mathbb{C}}\setminus\Lambda_{\Kul}(\Gamma).$$
Kulkarni’s limit set has the following properties. For a more detailed discussion of this in the two-... | 1,813 | 571 | 1,191 | 1,855 | null | null | github_plus_top10pct_by_avg |
tain a teleported state identical to the original input state. We call the channel $f_q$ reversible, if it is injective, that is, for different input state $|\Phi\rangle_A$ ($\||\Phi\rangle_A\|=1$) the corresponding output state $f_q(|\Phi\rangle_A)$ is different. We remark, that the reversibility of teleportation chan... | 1,814 | 4,930 | 381 | 1,216 | null | null | github_plus_top10pct_by_avg |
Y order due to the proximity to even layers. Here, however, we presume that SP scenario holds and $u_r$ vanishes at large $L$. Practically, simulations are performed at finite $u$ and we always control that the XY stiffness (helicity modulus) along the layers remains finite and independent of $L$.
Dual formulation
---... | 1,815 | 1,115 | 2,601 | 1,896 | 2,743 | 0.777352 | github_plus_top10pct_by_avg |
of the CW2 attack on the average of defense arrangements is investigated in Supplementary Material. While it may seem that our proposed defense scheme is easily fooled by a strong attack such as CW2, there are still ways of recovering from such attacks by using detection. In fact, there will always be perturbations th... | 1,816 | 1,648 | 2,667 | 1,831 | null | null | github_plus_top10pct_by_avg |
-1\] in the proof of Proposition \[pre-cohh\] and so we will just indicate how to modify the earlier proofs to work here.
{#B-freeC}
Since $M(k)$ is a $(H_{c+k},{U}_c)$-bimodule, the embeddings ${\mathbb{C}}[{\mathfrak{h}}]\hookrightarrow
H_{c+k}$ and ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}\hookrightarrow {U}_c$ make ... | 1,817 | 1,110 | 780 | 1,744 | 3,565 | 0.771522 | github_plus_top10pct_by_avg |
ng a vbo and vao from some raw data, in this case a pointer to an array of floats.
RawModel* Loader::loadToVao(float* positions, int sizeOfPositions) {
unsigned int vaoID = this->createVao();
this->storeDataInAttributeList(vaoID, positions, sizeOfPositions);
this->unbindVao();
return new RawModel(vaoID,... | 1,818 | 1,907 | 206 | 772 | 95 | 0.826729 | github_plus_top10pct_by_avg |
[ea3\]), (\[ea3’\]), (\[ea7\]), (\[ea8\]), (\[ea9\]), (\[ea10\]), (\[13\]), (\[ea14\]), (\[ea15\]), (\[ea16\]), (\[ea17\]), (\[ea19\])) stated in the proof of Theorem \[ta4\] in order to compute $h\circ (1+\pi y)$. Based on this, we enumerate equations which $m$ satisfies as follows:
1. Assume that $i<j$. By Equation... | 1,819 | 348 | 1,562 | 1,965 | 3,149 | 0.774483 | github_plus_top10pct_by_avg |
/{(\pi x)}} \exp{\left[ -ix +i{|\nu|\pi}/{2} +i{\pi}/{4} \right]}$. Classically the limit $x \gg 1 $ corresponds to advanced solution called the Bunch-Davies vacuum $e^{-ik\tau}/\sqrt{2k}$. Generally the limit obtained here differs from the classical one but can be restored taking $l=2$. Then to obtain the proper high ... | 1,820 | 3,530 | 1,688 | 1,620 | null | null | github_plus_top10pct_by_avg |
sumptions , and the operator $A_1(E)$ is bounded for any fixed $E\in I$, and collectively they obey a uniform bound, $$\begin{aligned}
\label{eq:A1E_unif_bound}
\sup_{E\in I} {\left\Vert A_1(E)\right\Vert}\leq \kappa^{-1}\Big({\left\Vert \tilde \Sigma\right\Vert}_{L^\infty(G\times S\times I)}+
{\left\Vert {{\frac{\par... | 1,821 | 1,017 | 1,149 | 1,644 | null | null | github_plus_top10pct_by_avg |
d by $$\|\Sigma^{-1}\|^2_{\mathrm{op}} \frac{\|
\hat{\Sigma} - \Sigma
\|_{\mathrm{op}} } { 1 - \|\hat{\Sigma} - \Sigma \|_{\mathrm{op}} \|
\Sigma^{-1}\|_{\mathrm{op}} }.$$ The matrix Bernstein inequality along with the assumption that $U \geq \eta > 0$ yield that, for a positive $C$ (which depends on ... | 1,822 | 1,732 | 1,712 | 1,538 | null | null | github_plus_top10pct_by_avg |
$ as a $\kappa$-variety so that the number of rational points is $2f$, where $f$ is the cardinality of $\kappa$.
Construction following our technique {#cfot}
------------------------------------
Let $q$ be the function defined over $L$ such that $$q : L\longrightarrow A, l\mapsto h(l,l).$$ If we write $l=x+\pi y$ suc... | 1,823 | 1,227 | 1,141 | 1,815 | null | null | github_plus_top10pct_by_avg |
by Eq. (\[Eqn: chaos1\]), serves to complete the significance of the tower by capping it with a “boundary” element that can be taken to bridge the classes of functional and non-functional relations on $X$.
We are now ready to define a *maximally ill-posed problem $f(x)=y$* for *$x,y\in X$* in terms of a *maximally no... | 1,824 | 2,784 | 2,835 | 1,658 | 2,449 | 0.779621 | github_plus_top10pct_by_avg |
TR pCmdLine, int nCmdShow)
{
// Register the window class.
const wchar_t CLASS_NAME[] = L"Sample Window Class";
WNDCLASS wc = { };
wc.lpfnWndProc = WindowProc;
wc.hInstance = hInstance;
wc.lpszClassName = CLASS_NAME;
RegisterClass(&wc);
// Create the window.
HWND hwnd = CreateWi... | 1,825 | 6,640 | 52 | 1,355 | 369 | 0.811851 | github_plus_top10pct_by_avg |
over the previous best result (47% $\rightarrow$ 53%) on the [*Success Rate weighted by Path Length*]{} metric.'
author:
- |
Xiujun Li^$\spadesuit\diamondsuit$^Chunyuan Li^$\diamondsuit$^Qiaolin Xia^$\clubsuit$^Yonatan Bisk^$\spadesuit\diamondsuit\heartsuit$^\
**[Asli Celikyilmaz]{}^$\diamondsuit$^**[Jianfeng ... | 1,826 | 965 | 851 | 1,671 | null | null | github_plus_top10pct_by_avg |
distribution free framework. For notational convenience, in this section we let $\theta_{{\widehat{S}}}$ be any of the parameters of interest: $\beta_{{\widehat{S}}}$, $\gamma_{{\widehat{S}}}$, $\phi_{{\widehat{S}}}$ or $\rho_{{\widehat{S}}}$.
We will rely on sample splitting: assuming for notational convenience that ... | 1,827 | 3,870 | 1,930 | 1,503 | null | null | github_plus_top10pct_by_avg |
ernabei:2013cfa; @Aalseth:2012if; @Angloher:2011uu; @Agnese:2013rvf; @Fan:2013faa] – begin to set tight limits (with some conflicting signal hints) on the standard WIMP scenario with a contact interaction to quarks. This makes it necessary to look for a more complete set of DM models which are theoretically motivated w... | 1,828 | 616 | 2,657 | 1,859 | 1,656 | 0.787126 | github_plus_top10pct_by_avg |
± 4.3 326.0 ± 3.8 284.5 ± 4.6 242.1 ± 5.6 280.2 ± 4.7 79.4 ± 0.5 66.4 ± 4.3 103.7 ± 3.5 96.2 ± 3.0
Sin 7 413.2 ± 2.8 427.6 ± 12.3 240.0 ± 8.1 302.8 ± 1.0 330.7 ± 11.5 306.9 ± 12.2 247.6 ± 7.8 291.8 ± 4.9 80.0 ± 2.8 71.8 ± 2... | 1,829 | 5,177 | 879 | 998 | null | null | github_plus_top10pct_by_avg |
2}k_{j-2l+2, j-2l}'=0.$$ Then the sum of equations $$\sum_{0\leq l \leq m_j}\mathcal{Z}_{j-2l}'$$ is the same as $$\sum_{0\leq l \leq m_j}z_{j-2l}'=0$$ since $k_{j-2l+2, j-2l}'=k_{j-2l,j-2l+2}'$. Therefore, among Equations (\[ea19\]) for $j-2m_j \leq i \leq j$, only one of them is redundant. In conclusion, there are $$... | 1,830 | 666 | 1,278 | 1,767 | 4,006 | 0.768673 | github_plus_top10pct_by_avg |
lde s)}2\tilde K}.\end{aligned}$$ Moreover, and imply that $$\tilde A\subset N\prod\{A_1^{(1)},\ldots,A_{r_1}^{(1)},\ldots,A_1^{(r_0)},\ldots,A_{r_{r_0}}^{(r_0)},X_1^{(1)},\ldots,X_{\ell_1}^{(1)},\ldots,X_1^{(r_0)},\ldots,X_{\ell_{r_0}}^{(r_0)},X\}$$ with the product taken in some order. We also have $$\begin{aligned}
... | 1,831 | 760 | 1,584 | 1,726 | 3,179 | 0.774256 | github_plus_top10pct_by_avg |
e = d\^D x f\^(x) \_(x) f’\^(x) \[Killing-1\] up to an overall normalization constant factor. Here the measure $\sqrt{\hat{g}} = \det \hat{e}_{\mu}{}^a$ must be present to ensure that the integration is diffeomorphism-invariant.
The Killing form can be slightly simplified by a change of basis. Let us use the field-dep... | 1,832 | 1,054 | 2,371 | 1,804 | null | null | github_plus_top10pct_by_avg |
\in \Gamma_P\! \right\}\!.$$ Then, $$\dim H^1(Y,\mathcal{O}_Y(L^{(k)}))=\dim\operatorname{coker}\pi^{(k)}.$$ In particular, if $\cC$ is reduced, then $$\pi^{(k)}: H^0\left(\PP^2_w,\mathcal{O}_{\PP^2_w}\left( kH+K_{\PP^2_w}\right) \right)
\longrightarrow \bigoplus_{P \in S}
\frac{\mathcal{O}_{\PP^2_w,P}\left( kH+K_{\P... | 1,833 | 1,413 | 1,939 | 1,499 | null | null | github_plus_top10pct_by_avg |
the following compatibility condition: $${\label{eq:crossed}}
r \circ v^{-1}\bigl((v(g_{1}) \lhd r(g_{2}))v(g_{2})\bigl) =
r(g_{1})\bigl(v(g_{1}) \rhd r(g_{2})\bigl)$$ holds for all $g_{1}, g_{2} \in G$. On the set $G$ we define a new multiplication $*$ and two new actions $\beta' : G\times H
\rightarrow G$, $\alpha' :... | 1,834 | 3,310 | 1,737 | 1,713 | null | null | github_plus_top10pct_by_avg |
a$ by our choices; if $B> \frac{C-\lambda_0}2+1$ then the weight of the coefficient of $y^2$ exceeds $c$, so it does not survive the limiting process, and the limit is a line. If $B= \frac{C-\lambda_0}2+1$, the term in $y^2$ is dominant, and the limit is a conic. The explicit expressions given in the statement are obta... | 1,835 | 667 | 1,857 | 1,882 | 1,856 | 0.785042 | github_plus_top10pct_by_avg |
th the lowest $\sigma_\mathrm{{rms}}$ has $T_{\rmn{eff}}=12\,000$K, $M_*=0.585\,M_{\sun}$ and $M_\rmn{H}=10^{-5.0}\,M_*$, however, its $\sigma_\mathrm{{rms}}$ is relatively large ($6.8$s), which means that there are major differences between the observed and calculated periods. Table \[table:lp133params\] lists the ste... | 1,836 | 743 | 2,921 | 2,016 | null | null | github_plus_top10pct_by_avg |
either ${\mathbf{f}} \in {\mathit{f}}$ or ${\mathbf{f}} \notin {\mathit{f}}$.
\[D:BASIC\_COVERING\] Let ${\mathbf{f}}$ be a frame and ${\mathit{f}}$ be a functionality. The functionality *covers* the frame if ${\mathbf{f}} \in {\mathit{f}}$ (that is, ${\mathbf{f}} = (\psi, \phi) = (\psi, {\mathit{f}}(\psi))$).
\[D:CO... | 1,837 | 1,212 | 1,551 | 1,747 | null | null | github_plus_top10pct_by_avg |
ute**.
We may take $\Phi$ to be a class of categories instead of functors, in which case we identify a category $A$ with the unique functor $A\to\bbone$. In the next section we will show that left $\Phi$-stability coincides with right $\Phi\op$-stability.
A derivator is pointed if and only if it is left $\emptyset$-s... | 1,838 | 1,759 | 1,360 | 1,790 | null | null | github_plus_top10pct_by_avg |
positionl\], we have $\lambda_2(-H(\theta)) \geq \frac{e^{2b}}{(1 + e^{2b})^2} \lambda_2(M)$, when $\lambda_{j,a} = 1/(\kappa_j-1)$ is substituted in the Hessian matrix $H(\theta)$, Equation . From Weyl’s inequality we have that $$\begin{aligned}
\label{eq:topl3}
\lambda_2(M) \;\; \geq \lambda_2(\E[M]) - {\|M - \E[M]\|... | 1,839 | 1,754 | 1,532 | 1,700 | null | null | github_plus_top10pct_by_avg |
sf}{ord}}^n D({\mathfrak{h}^{\text{reg}}})\ast {{W}}$ for all $n\geq 0$. We therefore obtain an induced grading, again called the ${\mathbf{E}}$-grading, on the associated graded ring $\operatorname{{\textsf}{ogr}}D({\mathfrak{h}^{\text{reg}}})\ast {{W}}\cong
{\mathbb{C}}[{\mathfrak{h}^{\text{reg}}}\oplus {\mathfrak{h}... | 1,840 | 984 | 1,144 | 1,857 | 3,472 | 0.772089 | github_plus_top10pct_by_avg |
n is commutativity $\Theta\Phi = \Phi\Theta$, since $\Theta \cup \Phi = \Phi \cup \Theta$.
\[T:DYADIC\_PRODUCT\_IS\_ENSEMBLE\] Let $\Theta$ and $\Phi$ be disjoint ensembles. Their dyadic product $\Upsilon = \Theta\Phi$ is an ensemble with domain $({{\operatorname{dom}{\Theta}}} \cup {{\operatorname{dom}{\Phi}}})$ and ... | 1,841 | 1,197 | 2,391 | 1,807 | null | null | github_plus_top10pct_by_avg |
re $I_R$ and $I_M$ are the real and imaginary parts of the integral. Here, $\tilde{{\bf k}}$ is oriented along the $z$ direction in the ${\bf \xi}$- space and the angle $\Theta_0$ is measured from the $z$ direction. The real part can be evaluated by contour integration, with the result $$\begin{aligned}
I_R &=& \frac{1... | 1,842 | 3,900 | 1,860 | 1,826 | null | null | github_plus_top10pct_by_avg |
l{K-defn}
\tilde{H}_{\lambda}(\x;q,t)=
\sum_{\nu}\tilde{K}_{\nu\lambda}(q,t)s_{\nu}(\x).$$ These are $(q,t)$ generalizations of the $\tilde{K}_{\nu\lambda}(q)$ Kostka-Foulkes polynomial in Macdonald [@macdonald III, (7.11)], which are obtained as $q^{n(\lambda)}K_{\nu\lambda}(q^{-1})=
\tilde{K}_{\nu\lambda}(q)=\tilde{K... | 1,843 | 991 | 1,634 | 1,550 | null | null | github_plus_top10pct_by_avg |
onary points.
In Fig. \[fig:Mrho0\] we illustrate the mass of the initial data as a function of bubble radius $\rho_0$ for $0\leq Q\leq Q_{max}$. Clearly even the perturbatively stable bubble at the local minimum is unstable against tunneling to larger radii. At $Q= Q_{max}$, the barrier disappears completely. On the ... | 1,844 | 1,500 | 2,245 | 1,880 | 2,451 | 0.779615 | github_plus_top10pct_by_avg |
mega_M|=1 \mathrm{~and~} \int_{H'}|\omega_H|=1.$$
We choose another nonzero, translation-invariant forms $\omega^{\prime}_M$ and $\omega^{\prime}_H$ on $\mathrm{End}_EV$ and $H$, respectively, with normalization $$\int_{\underline{M}(A)}|\omega^{\prime}_M|=1 \mathrm{~and~} \int_{\underline{H}(A)}|\omega^{\prime}_H|=... | 1,845 | 1,737 | 1,635 | 1,631 | null | null | github_plus_top10pct_by_avg |
}(x) = \sinh \left[\frac{\pi\omega /g}{4\pi^{4}g} \right]^{1/2} K_{i\omega /g} \left[ \frac{\sqrt{k_{\bot}^{2} + m^{2}}}{g e^{-g z}} \right] e^{ik_{\perp} \cdot x_{\perp} - i \omega t}
\label{modsol}$$ The smeared field operator defined in Eq.(\[smeared\]) can then be expressed as $$\phi(\tau) = \int d\omega \int d^2 ... | 1,846 | 3,911 | 1,906 | 1,633 | null | null | github_plus_top10pct_by_avg |
nguish these two SL(2,$\mathbb C$) representations, or equivalently the two Weyl spinors, the van der Waerden dotted and undotted index notation has been introduced. This notation proves particularly valuable for the construction of manifestly supersymmetric invariant Lagrangian densities.
The undotted indices $\alpha... | 1,847 | 1,254 | 2,022 | 1,670 | null | null | github_plus_top10pct_by_avg |
rtunately, our gerbe example is not so well-behaved.
Second cautionary example {#sect:class3-caution2}
-------------------------
For completeness, we give here a second cautionary example, here involving a heterotic Spin$(32)/{\mathbb Z}_2$ compactification on a nontrivial toroidal orbifold. This will involve a rank ... | 1,848 | 1,434 | 1,137 | 1,780 | null | null | github_plus_top10pct_by_avg |
xe] is in progress.
---
abstract: 'Linear perturbation theory is a powerful toolkit for studying black hole spacetimes. However, the perturbation equations are hard to solve unless we can use separation of variables. In the Kerr spacetime, metric perturbations do not separate, but curvature perturbations do. The cost... | 1,849 | 1,177 | 819 | 1,935 | null | null | github_plus_top10pct_by_avg |
l.com (match this)
I tried the following: (?!@)gmail\.com but this did not work. This is matching both the cases highlighted in the example above. Any suggestions?
A:
[^@\s]*(?<!@)\bgmail\.com\b
assuming you want to find strings in a longer text body, not validate entire strings.
Explanation:
[^@\s]* # match any... | 1,850 | 1,701 | 541 | 1,274 | 2,670 | 0.777959 | github_plus_top10pct_by_avg |
the compatibility conditions of the system (\[eq:ms:12\]).
Example. Nonlinear interaction of waves and particles. {#sec:7}
======================================================
Let us consider the inhomogeneous system describing the propagation of shock waves intensity in nonlinear interaction of waves and particles... | 1,851 | 795 | 323 | 2,161 | null | null | github_plus_top10pct_by_avg |
M$, where $\alpha \geq 2$ will be determined shortly. This takes care of the all constraints except for $A \in PSD$. Note that since $M$ is regular, its eigenvectors are also eigenvectors of $A$. Moreover, if $M u = \lambda u$ for a non constant $u$, then $A u = \alpha d - n - \alpha \lambda$ (and $A \vec{1} = 0$). S... | 1,852 | 2,142 | 2,354 | 1,824 | 2,796 | 0.776869 | github_plus_top10pct_by_avg |
T}}_{m-1,r}(\lambda_{m}-s){\mathcal{T}}_{m,r}(\lambda_{m+1}-\lambda_{m})...{\mathcal{T}}_{l-2,r}(\lambda_{l-1}-\lambda_{l-2}){\mathcal{T}}_{l-1,r}(t-\lambda_{l-1})
\\&={\mathcal{T}}_{m-1}\Big((T-s)-(T-\lambda_{m})\Big){\mathcal{T}}_{m}\Big((T-\lambda_m)-(T-\lambda_{m+1})\Big)...{\mathcal{T}}_{l-1}\Big((T-\lambda_{l-1})... | 1,853 | 661 | 516 | 2,084 | null | null | github_plus_top10pct_by_avg |
skew-Hopf pairing ${\eta }$ of ${\mathcal{V}}^+(\chi )$ and ${\mathcal{V}}^-(\chi )$ such that for all $i,j\in I$ one has $$\begin{aligned}
{\eta }(E_i,F_j)=-\delta _{i,j},\quad
{\eta }(E_i,L_j)=0,\quad
{\eta }(K_i,F_j)=0,\quad
{\eta }(K_i,L_j)=q_{ij}.
\end{aligned}$$
\(ii) The skew-Hopf pairing ${\... | 1,854 | 1,042 | 2,089 | 1,724 | null | null | github_plus_top10pct_by_avg |
X/R)^{cat}$ for affine schemes.
\[basic.quot.exmp\] Let $f:X\to Y$ be a finite and surjective morphism. Set $R:={\operatorname{red}}(X\times_YX)\subset X\times X$ and let $\sigma_i:R\to X$ denote the coordinate projections. Then the geometric quotient $X/R$ exists and $X/R\to Y$ is a finite and universal homeomorphism... | 1,855 | 1,407 | 2,155 | 1,812 | null | null | github_plus_top10pct_by_avg |
ock notation (the “advanced-retarded” ordering of supermatrix elements), $\mathcal{F}_M$ reads $$\mathcal{F}_M = \prod_{k=1}^M \exp\left\{-\frac{1}{2} \mathrm{trg} \ln \left[ 1 + i
\left(\begin{array}{cc} \lambda_k & 0 \\ 0 & \lambda_k'^{*} \end{array}\right)
\sigma_G L \right]\right\},$$ where $\lambda_k=\lambda_k... | 1,856 | 2,638 | 2,494 | 1,788 | 2,983 | 0.775613 | github_plus_top10pct_by_avg |
the (context-free or extended) Petri nets with place capacity.
Quite obviously, a context-free Petri net with place capacity regulates the defining grammar by permitting only those derivations where the number of each nonterminal in each sentential form is bounded by its capacity. A similar mechanism was discussed in... | 1,857 | 4,452 | 2,020 | 1,901 | 1,651 | 0.787225 | github_plus_top10pct_by_avg |
ince each component of the inertia stack $I_{\mathfrak{X}}$ is isomorphic to the original stack $\mathfrak{X}$, the normal bundle $N_q$ vanishes, and each component of ${\rm ch}(d(\lambda_q))$ is $1$. Furthermore, as $\mathfrak{X}$ is essentially a $k$-fold quotient of ${\mathbb P}^n$, $$\int_{\mathfrak{X}} \: = \: \fr... | 1,858 | 1,377 | 2,070 | 1,618 | null | null | github_plus_top10pct_by_avg |
\Psi$ and $\Phi$, we write $\Phi \subseteq \Psi$ to express that $\Phi$ is contained in $\Psi$, following ordinary set theory that term $(i,P) \in \Phi$ implies $(i,P) \in \Psi$.
\[D:ENSEMBLE\_DIFFERENCE\] Let $\Psi$ and $\Phi$ be ensembles such that $\Phi \subseteq \Psi$. In classification of difference between $\Psi... | 1,859 | 3,960 | 3,078 | 1,915 | null | null | github_plus_top10pct_by_avg |
ndence.
Let $g = (g_1,\ldots,g_s)^\top \colon \mathbb{R}^b \rightarrow \mathbb{R}^s$ be a twice-continuously differentiable vector-valued function defined over an open, convex subset $\mathcal{S}_n$ of $[-A,A]^b$ such that, for all $P \in
\mathcal{P}_n$, $\psi = \psi(P) = \mathbb{E}[W_1] \in \mathcal{S}_n$. Let $\wi... | 1,860 | 2,700 | 1,570 | 1,722 | 4,113 | 0.768023 | github_plus_top10pct_by_avg |
, $\cS(Y)=\emptyset$, $|\cS|=|\cS^*|+2$, and $|\cR|\ge 5+|Y|=5$. Using \[prop:rrstar\], equation (\[eq:general\]) becomes $$|\cR|\le 2+|\cS^*|-\frac{1}{2}|\cR^*|\le 2+3-0=5.$$ Thus $|\cR|=5$, $|\cS^*|=3$, $\cR^*=\emptyset$, $|\cS|=5$, and all inequalities hold with equality in inequality (\[eq:individual\]).\
We may as... | 1,861 | 645 | 1,035 | 1,921 | null | null | github_plus_top10pct_by_avg |
arameter $\eta$, defined through $D_\eta=4-\eta$, and on the renormalization scale $\mu$, for which we have taken $\mu=m_\pi$. In the following we use, $$\begin{aligned}
R=&-\frac{2}{\eta}-1+\gamma-\log(4\pi) \,,
\\
q_0''=&q_0'-q_0 \,.\end{aligned}$$ The integrals $A(m)$, $A(q_0,q_0')$ and $B(q_0,|{\vec{q}}|)$ appear, ... | 1,862 | 551 | 1,145 | 2,021 | null | null | github_plus_top10pct_by_avg |
ow. The parameters in the potential of $\chi$ are thus guaranteed to be independent of the parameters of our brane.
The scalar potential in our brane excluding $V(\chi)$ is now given by $$\begin{aligned}
V &=& m_1^2 \Phi^\dagger \Phi + m_2^2 \sigma^\dagger \sigma + m_3^2
\eta^\dagger \eta + {1 \over 2} \lambda_1 (\Ph... | 1,863 | 4,605 | 515 | 1,491 | null | null | github_plus_top10pct_by_avg |
other one composed of cross terms. The first one is given by $$\begin{aligned}
&& \left| S^{(2)}_{\alpha \beta} \right|^2_{\text{1st}} =
\sum_{k, K} \sum_{l, L}
\frac{ 1 }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{l} ) }
\nonumber \\
&\times&
\biggl[
x^2 e^{- i ( h_{k} - h_{l} ) x}
- (ix) \frac{e^{- i ( \Delta_{K... | 1,864 | 1,779 | 1,547 | 1,998 | null | null | github_plus_top10pct_by_avg |
) 0.7 Peritoneum Pancreas
10 F 80 Body, tail No No T3N1M0(III) 2.3 Pancreas
11 F 64 Tail ... | 1,865 | 5,459 | 1,214 | 1,096 | null | null | github_plus_top10pct_by_avg |
*residuals* $\textrm{Res}(\mathbb{D})$ *in $\mathbb{D}$ given by* $$\textrm{Res}(\mathbb{D})=\{\mathbb{R}_{\alpha}\in\mathcal{P}(\mathbb{D})\!:\mathbb{R}_{\alpha}=\{\beta\in\mathbb{D}\textrm{ for all }\beta\succeq\alpha\in\mathbb{D}\}\}.\label{Eqn: residual}$$
*The net* *adheres at* *$x\in X$*[^27] *if it is frequentl... | 1,866 | 728 | 2,006 | 1,773 | 2,875 | 0.776387 | github_plus_top10pct_by_avg |
s definition can be taken to suggest that their numerical plurality is a primitive fact. But this is just an artifact of the set-theoretical language. The elements of a set $\mathcal{M}$ *qua* set-theoretical objects have to be numerically distinct for $\mathcal{M}$ to be a well-defined set of $N$ objects. However, the... | 1,867 | 3,569 | 1,936 | 1,623 | null | null | github_plus_top10pct_by_avg |
t claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on $\omega_1$, as well as of a strong form of Chang’s Conjecture. Together ... | 1,868 | 638 | 871 | 1,350 | null | null | github_plus_top10pct_by_avg |
ivalent to $\mathcal{C}$. If $K_{\mathbb{P}_w^2}$ denotes the canonical divisor of $\mathbb{P}_w^2$ and $$\mathcal{C}^{(k)} = \sum_{j=1}^r {\left \lfloor \frac{kn_j}{d} \right \rfloor} \mathcal{C}_j, \qquad 0 \leq k < d,$$ then these dimensions are given as the cokernel of the evaluation linear maps $$\pi^{(k)}: H^0\le... | 1,869 | 1,517 | 1,645 | 1,562 | null | null | github_plus_top10pct_by_avg |
rators. Precisely, we assume the set of offerings $S_j$, the number of separators $\ell_j$, and their respective positions $\cP_j=\{p_{j,1},\ldots,p_{j,\ell_j}\}$ are predetermined. Each user draws the ranking of items from the PL model, and provides the partial ranking according to the separators of the form of $\{a>\... | 1,870 | 1,151 | 1,419 | 1,848 | 2,255 | 0.781386 | github_plus_top10pct_by_avg |
surface tension of the base solution without camphor and $\Gamma$ is a positive constant.
![\[fig:model\] Illustration of side view of a camphor boat.](model.eps){width="7cm"}
The time evolution on the camphor concentration $c$ is shown as $$\begin{aligned}
\frac{\partial c}{\partial t} = D \frac{\partial^2 ... | 1,871 | 3,066 | 3,147 | 2,108 | 3,881 | 0.769545 | github_plus_top10pct_by_avg |
\vert_{i \neq j}~(\text{single sum}) +
\hat{S}_{ij}^{(4)} [4] \vert_{i \neq j}~(\text{double sum})
\label{hatS-4th-order-ij-T-transf}\end{aligned}$$ where $$\begin{aligned}
&& \hat{S}_{ij}^{(4)} [4] \vert_{i \neq j}~(\text{single sum})
\nonumber \\
&=&
\sum_{K}
\biggl[
(ix) e^{ - i h_{i} x}
\frac{ 1 }{ ( \Delta_{K}... | 1,872 | 1,474 | 2,200 | 1,897 | null | null | github_plus_top10pct_by_avg |
&= \tau_D \times \frac{1}{16\pi m_D^3} \sqrt{
(m_D^2 - (m_{P_1} - m_{P_2})^2) (m_D^2 - ( m_{P_1} + m_{P_2})^2 )
}\,.\end{aligned}$$ The direct CP asymmetries are [@Golden:1989qx; @Pirtskhalava:2011va; @Nierste:2017cua] $$\begin{aligned}
a_{CP}^{\mathrm{dir}} &= \mathrm{Im}\left(\frac{\lambda_b}{\Sigma}\right) \mathr... | 1,873 | 2,303 | 2,505 | 1,737 | 2,067 | 0.783078 | github_plus_top10pct_by_avg |
mathit{f}}') \in {\mathbb{S}}$. By definition \[D:STEP\_SPACE\_PROJECTION\], $\mho_{\mathbf{F}}({\mathfrak{A}}_\xi({\mathit{s}})) = {\mathbf{f}}\,' = (\psi', \phi')$ and $\mho_{\mathscr{F}}({\mathfrak{A}}_\xi({\mathit{s}})) = {\mathit{f}}'$
Definition \[D:ITERATIVE\_TRANSFORM\] evaluates ${\mathbf{f}}\,' = (\psi', \ph... | 1,874 | 890 | 1,287 | 1,642 | null | null | github_plus_top10pct_by_avg |
e definition of $\Delta^2$ depends on the coordinates and the desired level of approximation. This section defines our computational domains and the finite-difference representations of $\Delta^2$, which are second-order accurate, in uniform Cartesian, uniform cylindrical, and logarithmic cylindrical coordinate systems... | 1,875 | 4,105 | 2,870 | 1,906 | 2,712 | 0.777638 | github_plus_top10pct_by_avg |
tting $p=0$ and expanding these terms, B\_[LR\_i]{}\^x &=& - B\_0(0,[M\_[\_[i]{}\^]{}]{},m\_[\_x]{})\
&=& B\_0(0,[M\_[\_[i]{}\^]{}]{},m\_[\_x]{}) .
Neglecting terms proportional to $g_2$ and summing over the stops and charginos yields \_[i=1]{}\^2 \_[x=1]{}\^2 B\_[LR\_i]{}\^x &&\
&+& .\[Eq:chargino-B0\]
For $|\mu| > ... | 1,876 | 191 | 2,023 | 1,692 | 1,691 | 0.786728 | github_plus_top10pct_by_avg |
E_0,\infty[$ where $E_0\geq 0$ but we omit this generalization here. We shall denote by $I^\circ$ the interior $]0,E_m[$ of $I$. The interval $I$ in equipped with the 1-dimensional Lebesgue measure ${\mathcal{L}}^1$, which we typically write as $dE$ in the sense that ${\mathcal{L}}^1(A)=\int_A dE$.
For $(x,\omega)\in ... | 1,877 | 517 | 1,732 | 1,755 | null | null | github_plus_top10pct_by_avg |
nts were found to be between $\sim2$ (close to the resolution limit) and $14\,\mu$Hz, but we mark these findings uncertain because of the effect of the 1d$^{-1}$ aliasing. The 2nd–6th panels of Fig. \[fig:g207FTa\] shows the FTs of the weekly datasets. We found only slight amplitude variations from one week to another.... | 1,878 | 2,752 | 2,484 | 2,133 | null | null | github_plus_top10pct_by_avg |
is conformally invariant [@bida], and the conformally rescaled components of the fermion obey the flat space equation (\[fse\]) with Neumann boundary conditions. Thus, the spectrum (\[flateigenvalues\]) is also valid for massless fermions.
Flat Spacetime
---------------
Let us now consider the Casimir energy densit... | 1,879 | 983 | 2,177 | 1,795 | null | null | github_plus_top10pct_by_avg |
pinorial $\bf{16}$ and $\bf{\overline{16}}$ representations of $SO(10)$, under the $SU(4) \times SU(2)_L \times U(1)_L$ gauge group is given as follows: $$\begin{aligned}
\textbf{16} &= &\left({\textbf{4}},{\textbf{2}},
0\right) + \left(\overline{{\textbf{4}}},{\textbf{1}},
-1\right) + \left(\overline{{\textbf{4}}},{\t... | 1,880 | 2,480 | 2,955 | 1,980 | null | null | github_plus_top10pct_by_avg |
t(\sum_{\lambda}f_{\lambda}(v^{-1})
f_{\lambda}(1)\right) f_{\mu^t}(1) f_{\mu^t}(v^{-1}) v^N
v^{-d(n(\mu^t) - n(\mu))}.$$ The standard formula $\sum \dim
{\mathbb{C}}[{\mathfrak{h}}]^{\text{co}{{W}}}_iv^{-i}=[n]_{v^{-1}}!$ shows that the fake degrees satisfy the identity $$\sum_{\lambda} f_{\lambda}(v^{-1}) f_{\lam... | 1,881 | 1,099 | 2,144 | 1,801 | 3,355 | 0.772956 | github_plus_top10pct_by_avg |
\pi^{*} ((h') + {\left \lfloor D \right \rfloor}) + \pi^{*} {\left \{ D \right \}} + \sum_{i} m_i E_i \geq 0.$$ Let $F$ be the $w$-homogeneous polynomial defined by the effective $(h')+{\left \lfloor D \right \rfloor}$. Using the isomorphism $H^0(X,\cO_X(D)) \cong \CC[x,y,z]_{w,d}$ described in Lemma \[lemma:h0-weighte... | 1,882 | 1,656 | 544 | 1,931 | null | null | github_plus_top10pct_by_avg |
We use the ordering on ${{\textsf}{Irrep}({{W}})}$ arising from the dominance ordering; thus, as in [@MacD Example 1, p.116], the [trivial representation]{} $\operatorname{{\textsf}{triv}}$\[triv-defn\] is labelled by $(n)$ while the [sign representation]{} $\operatorname{{\textsf}{sign}}$\[sign-defn\] is parametrised... | 1,883 | 1,039 | 534 | 1,909 | 1,532 | 0.788564 | github_plus_top10pct_by_avg |
Of course this means that $s_{\beta_l}$ forces that $\dot{U}(\eta,0)$ meets $\dot{Z}(\xi, C_{\zeta(\gamma_\alpha)})$ for cofinally many $\xi<\delta$ such that $s_{\beta_l}\upharpoonright\gamma_\alpha\Vdash
\dot{f}_{\gamma_\alpha}(\xi)=n_l$. But $\bar{s}$ has already decided the value of $\dot{f}_{\gamma_\alpha}\upharpo... | 1,884 | 1,018 | 1,554 | 1,735 | 1,849 | 0.785091 | github_plus_top10pct_by_avg |
rintf. That's where the Undefined Behaviour1 happens. No such thing in the C++ example.
1Do note that the C example is not guaranteed to SEGFAULT. Undefined behaviour is undefined.
Q:
Selectively record commands and output from terminal to MySQL table
In some situations I would like to store complete commands and... | 1,885 | 3,316 | 982 | 1,722 | 2,479 | 0.779454 | github_plus_top10pct_by_avg |
\frac{2k}{\alpha}\right)}.$$
Let $\hat{E}_{{\widehat{S}}} = \bigotimes_{j\in S} E(j)$.
------------------------------------------------------------------------
[^1]: For simplicity, we assume that the data are split into two parts of equal size. The problem of determining the optimal size of the split is not conside... | 1,886 | 866 | 873 | 2,032 | 2,133 | 0.782497 | github_plus_top10pct_by_avg |
ac{d}{dt} \hat{\chi}(X,t)&=&
\frac{i}{\hbar}
\left[\hat{H},\hat{\chi}(X,t)\right]_{\mbox{\tiny\boldmath$\cal B$}}
-\frac{1}{2}\left\{\hat{H},\hat{\chi}(X,t)\right\}_{\mbox{\tiny\boldmath$\cal B$}}
\nonumber\\
&+&\frac{1}{2}\left\{\hat{\chi}(X,t),\hat{H}\right\}_{\mbox{\tiny\boldmath$\cal B$}}
%\nonumber \\
=\left(\hat{... | 1,887 | 1,104 | 2,100 | 1,858 | null | null | github_plus_top10pct_by_avg |
y^2\dot x \} \\
= \{ y^2x\dot x y\}.
\end{gathered}$$
The last two equalities imply $\alpha _2 =1$ and other coefficients are equal to zero. Therefore, $$\begin{gathered}
\phi (x,y,z) = \{y\dot xxz\}-2\delta_3\{yxx\dot z\}
-2\delta_5\{xxy\dot z\}-2\delta_4\{y\dot zxx\} \\
+2(\delta_3+\delta_5)\{\dot zyyy\}+2\delta... | 1,888 | 515 | 1,554 | 1,852 | null | null | github_plus_top10pct_by_avg |
ion \[section::improving\].
[**Remark.**]{} Our results concern the bootstrap distribution and assume the ability to determine the quantities $\hat{t}^*_\alpha$ and $(\tilde{t}^*_j, j \in {\widehat{S}})$ in Equation . Of course, they can be approximated to an arbitrary level of precision by drawing a large enough numb... | 1,889 | 1,745 | 2,901 | 1,905 | 3,442 | 0.772357 | github_plus_top10pct_by_avg |
\_x,\_x v\_[L\^2(GSI)]{} and \[fs5\] ,v\_[W\^2\_1(GSI)]{}=,v\_[L\^2(GSI)]{}+ \_x,\_x v\_[L\^2(GSI)]{} +,\_[L\^2(GSI)]{}.
These spaces are Hilbert spaces. Let C\^1(GSI):={\_[| GSI]{} | C\_0\^1(\^3S)}. We have (cf. [@friedrichs], [@bardos]. The proof can also be shown by the similar considerations as in [@friedman pp. ... | 1,890 | 634 | 1,853 | 1,764 | null | null | github_plus_top10pct_by_avg |
n ${{\operatorname{edge}{{\mathcal{C}}}}}$, whereas software correctness examines only the structure within cone ${\mathcal{C}}$. The operational profile asserts the importance of relative excitational intensity to safety analysis. An accident that occurs more frequently is worse than an accident that happens less freq... | 1,891 | 5,521 | 2,484 | 1,081 | 2,957 | 0.775834 | github_plus_top10pct_by_avg |
ATGACGAAGAGGATTAAGTATCTCGTGTAGGCTGGAGCTGCTTC
Rev cassette generation GAATCGTTAAAAAAGCGCGGCCAGAGGCGTTCTGACCGCATGCTTTGCTACATATGAATATCCTCCTTAG
Upstream CATCTGCGAC... | 1,892 | 3,792 | 1,775 | 1,854 | null | null | github_plus_top10pct_by_avg |
to be normalized. We employed the ‘range’ method where each component of the data vector is normalized to lie in the intravel \[0,1\].
3. [**SOM Training:**]{} Select an input vector $x$ from the data set randomly. A best matching unit (BMU) for this input vector, is found in the map by the following metric $$\left\|... | 1,893 | 4,281 | 2,454 | 1,594 | null | null | github_plus_top10pct_by_avg |
finterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr
\rightarrowfill\cr\noalign{\kern.3ex}
\rightarrowfill\cr}}}\limits^{\!(q,hse)}_{\!(q,hte)}}} (q,y)$$ in the category $\int_{L(S)}\Phi(X,\mu)$. This means that $hsea=htea$. Then $hse{\mathbf r}(a)=hte{\mathbf r}(a)$ and also $$hse{\mathbf r}(a)\cdot y =... | 1,894 | 1,410 | 2,215 | 1,824 | 2,297 | 0.781088 | github_plus_top10pct_by_avg |
ge symmetry of general coordinate transformations.
The teleparallel gravity action (\[action-TP\]) is equivalent to the action (\[general-action-0\]) for the choice of parameters $\lam = 1/2, \alpha = 1, \beta = 1$. It is S\_[TP]{} = d\^D x . \[action-TP-2\] The first term is the YM action (\[YM-action\]). The rest of... | 1,895 | 676 | 2,188 | 1,914 | null | null | github_plus_top10pct_by_avg |
Let us describe the transformations of $\R^4$ given by each generator. Consider a new base $(e_1, e_2,
e_3, e_4)$, given by $e_1=f_1+f_2$, $e_2=f_1-f_2$, $e_3=f_3+f_4$, $e_4=f_3-f_4$. The generator $r$ of order 4 is represented by the rotation in the plane $(e_2,e_4)$ through the angle $\frac{\pi}{2}$ and the reflecti... | 1,896 | 1,808 | 2,300 | 1,865 | null | null | github_plus_top10pct_by_avg |
, as a direct application of the one-dimensional Berry-Esseen theorem, that:
Let $\hat{C}_{{\widehat{S}}}$ be the splitting-based confidence set. Then, $$\label{eq:lem12a}
\inf_{P\in {\cal P}_{n}}\mathbb{P}(\beta_{\hat{S}}\in \hat{C}_{{\widehat{S}}}) = 1-\alpha - \frac{c}{\sqrt{n}}$$ for some $c$. Also, $$\label{e... | 1,897 | 1,959 | 1,403 | 1,655 | 3,606 | 0.771251 | github_plus_top10pct_by_avg |
(X,\mathcal{U})\rightarrow(h(X),\textrm{IT}\{ i;\textrm{FT}\{\mathcal{U};h\}))$ for a subspace $h(X)$ of $Y_{1}$, the converse need not be true unless — entirely like open functions again — either $h(X)$ is an open set of $Y_{1}$ or $i\!:(h(X),\textrm{IT}\{ i;\textrm{FT}\{\mathcal{U};h\}))\rightarrow(X,\textrm{FT}\{\ma... | 1,898 | 3,139 | 2,605 | 1,798 | null | null | github_plus_top10pct_by_avg |
(UX) \right\}_{K k}
\nonumber \\
&+&
2 \mbox{Re}
\biggl\{
\sum_{k, K} \sum_{l, L}
\left[
- (ix) e^{+ i h_{k} x} + \frac{e^{+ i \Delta_{K} x} - e^{+ i h_{k} x} }{ ( \Delta_{K} - h_{k} ) }
\right]
\frac{e^{- i \Delta_{L} x} - e^{- i h_{l} x} }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{l} ) }
\nonumber \\
&\times... | 1,899 | 322 | 1,661 | 1,908 | null | null | github_plus_top10pct_by_avg |
^{\prime\prime}$: $$\bar{N} \delta t = (\bar{g}^{1/2} \alpha) \delta t
= (\psi^6 g^{1/2}) \alpha \delta t = \psi^6 (N \delta t) \; .$$
The final relationships between the two physical Riemannian metrics $\bar{g}_{i j}$ and $\bar{g}^\prime_{i j} = \bar{g}_{i j} +
\dot{\bar{g}}_{i j} \delta t$ and the given data ... | 1,900 | 5,165 | 281 | 1,410 | null | null | github_plus_top10pct_by_avg |
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