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 |
|---|---|---|---|---|---|---|---|
N}}$. The following construction generalizes the Poincaré-Birkhoff-Witt basis of quantized enveloping algebras given by Lusztig.
Let $i_1,i_2,\dots ,i_n\in I$ such that $\ell (1_{\chi }{\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_n})=n$. For all $\nu \in \{1,2,\dots,n\}$ let $$\begin{gathered}
\label{eq:betak}
... | 1,201 | 1,485 | 986 | 1,179 | null | null | github_plus_top10pct_by_avg |
x{\boldmath$\psi$}$ and $\mbox{\boldmath$\phi$}$ can be used as the weight for sampling the coordinates $X$ entering the classical integral in Eq. (\[eq:sigma-sb\]). Then, for each initial value $X$, Eqs. (\[eq:matrixSigma\]) must be integrated in time so that averages can be calculated. It is worth to note that in suc... | 1,202 | 313 | 2,074 | 1,243 | null | null | github_plus_top10pct_by_avg |
sidering the effect of the angular momentum.
If the accretion disc is spatially resolved, we expect further mass loss happens due to, e.g., the disc winds [e.g., @Blandford:1999aa; @Zahra-Zeraatgari:2016aa; @Begelman:2016aa] and/or jets from a close vicinity of the BH [e.g., @Ohsuga:2005aa; @Jiang:2014aa; @Yuan:2015aa... | 1,203 | 1,884 | 2,612 | 1,327 | null | null | github_plus_top10pct_by_avg |
beta} \left\{ \sum_{i = 1}^{n} L(r_{i}(\beta)) \right\}. \end{aligned}$$ The case of $L(r_{i}(\beta)) = r_{i}(\beta)^{2}$ is well-known (see, e.g., Refs. [@doi:10.1111/j.1751-5823.1998.tb00406.x], [@legendre1805nouvelles], and [@stigler1981]). In the case of an asymmetric loss function, we refer the reader to, e.g., Re... | 1,204 | 5,034 | 979 | 658 | 754 | 0.800756 | github_plus_top10pct_by_avg |
t,0){\right\rangle}_{L^2(G\times S)}
+
{\left\langle}\psi(\cdot,\cdot,E_m),v(\cdot,\cdot,E_m){\right\rangle}_{L^2(G\times S)}.\end{aligned}$$ The elements of $H_1$ are of the form $\tilde\psi=(\psi,q,p_0,p_m)\in
L^2(G\times S\times I)\times T^2(\Gamma)\times L^2(G\times S)^2$. More precisely, the elements of $H_1$ are... | 1,205 | 660 | 1,108 | 1,201 | null | null | github_plus_top10pct_by_avg |
s (g(N^{n-2k}) \cap (U_P))$ by $N^{n-2k}_{int}$, and the complement $N^{n-2k} \setminus
N^{n-2k}_{int}$ by $N^{n-2k}_{ext}$. The manifolds $N^{n-2k}_{ext}$, $N^{n-2k}_{int}$ are submanifolds in $N^{n-2k}$ of codimension 0 with the common boundary, this boundary is denoted by $N_Q^{n-2k-1}$. The self-intersection manifo... | 1,206 | 443 | 1,617 | 1,228 | 2,675 | 0.777931 | github_plus_top10pct_by_avg |
akli], with the result [@li] that they are effectively proportional to the coupling $g_{37}^2$ in (\[coupl-N\]). The amplitudes depend on kinematical invariants expressible in terms of the Mandelstam variables: $s=-(k_1 + k_2)^2$, $t=-(k_2 + k_3)^2$ and $u=-(k_1 + k_3)^2$, which satisfy $s + t + u =0$ for massless part... | 1,207 | 2,382 | 1,497 | 1,169 | null | null | github_plus_top10pct_by_avg |
a_\varphi$ is the largest relevant solution of the eigenvalue equation $$\label{csawlamdafi}
\mbox{det}\left| \left({\partial X^{(r+1)}_i\over \partial X^{(r)}_j}
\right)^{*}-
\lambda_\varphi\,\delta_{ij} \right|=0\>,$$ where the asterisk means that the derivatives should be taken at the tricritic... | 1,208 | 1,138 | 1,964 | 1,234 | null | null | github_plus_top10pct_by_avg |
+2b} z_j^{\ast} \end{pmatrix}
&0 \\ 0& 0 &id \end{pmatrix}.$$ Here, the $(2,2)$-block of the above formal matrix corresponds to $B(-2be_1+e_2)\oplus$ $B(-ae_1+e_3)\oplus$ $B(e_2+e_3)$ with the Gram matrix $A(2b(2b-1), a(a+1), a(2b-1))\oplus (a+2b)$ and $id$ in the $(1,1)$-block corresponds to the direct summand $(\... | 1,209 | 1,964 | 1,484 | 1,160 | 2,643 | 0.778197 | github_plus_top10pct_by_avg |
obi symbol properties
------------------------
The Jacobi symbol $(a | b)$ is defined for $b$ odd and positive, and arbitrary $a$. We work primarily with non-negative $a$, and make use of the following properties of the Jacobi symbol.
Assume that $a$ is positive and that $b$ is odd and positive. Then
(i) \[it:zero\]... | 1,210 | 5,821 | 587 | 590 | 2,748 | 0.777304 | github_plus_top10pct_by_avg |
m_j}\mathcal{F}_{_{j-2l}}(\tilde{m})$ is naturally identified with $\sum_{0\leq l \leq m_j}\mathcal{F}_{_{j-2l}}(m)$.
As for Equation (\[ea20\]) of Step (1), we need to express $\sum_{0\leq l \leq m_j}\mathcal{F}_{_{j-2l}}(m)$ precisely. Each entry $\tilde{x}_i$ of $(\tilde{m_{i,j}}, \tilde{s_i} \cdots \tilde{w_i}... | 1,211 | 4,661 | 329 | 878 | null | null | github_plus_top10pct_by_avg |
\end{aligned}$$ and for the bundle ${\cal E}$, $$c_1^{\rm rep}({\cal E})|_{\alpha} \: = \:
\sum_a \frac{n_a}{k} J \alpha^{-n_a} \: - \: \frac{m}{k} J \alpha^{-m},$$ $$\begin{aligned}
{\rm ch}_2^{\rm rep}({\cal E})|_{\alpha} & = &
{\rm ch}_2^{\rm rep}( \oplus_a {\cal O}(n_a) )|_{\alpha} \: - \:
{\rm ch}_2^{\rm rep}( {\c... | 1,212 | 985 | 1,660 | 1,167 | null | null | github_plus_top10pct_by_avg |
1273.001){#fig1}
{#fig2}
![Demonstration of cataract and inferior posterior synechiae of the right eye. The cornea... | 1,213 | 629 | 1,197 | 1,613 | null | null | github_plus_top10pct_by_avg |
\le x_2$, $y_1 \le y \le y_2$, and $z_1 \le z \le z_2$ in a Cartesian grid. In a cylindrical grid, we consider a rectangular torus with density $\rho=1$, occupying the regions with $R_1 \le R \le R_2$, $\phi_1 \le \phi \le \phi_2$, and $z_1 \le z \le z_2$. Table \[tb:convergence\_test\] lists the parameters of the soli... | 1,214 | 1,824 | 2,638 | 1,477 | 3,245 | 0.77379 | github_plus_top10pct_by_avg |
ptions, and , the integrals $\int_{S\times I}\sigma_{jk}(x,\omega',\omega,E',E)d\omega' dE'$ (resp. $\int_{S\times I}\sigma_{jk}(x,\omega,\omega',E,E')d\omega' dE'$) over $S\times I$ are to be replaced with $\int_S\tilde\sigma_{kj}(x,\omega',\omega,E)d\omega'$ (resp. $\int_{S}\sigma_{jk}(x,\omega,\omega',E)d\omega'$) o... | 1,215 | 202 | 1,236 | 1,310 | 3,106 | 0.774856 | github_plus_top10pct_by_avg |
$q^{\frac{1}{2}(n^2-\sum_i\lambda_i^2)}R_{\mathfrak{l}_\lambda}^\mathfrak{g}(1)$ is a character of $(\mathfrak{g},+)$.
\[charab\]
Absolutely indecomposable representations
=========================================
Generalities on quiver representations {#genquiv}
--------------------------------------
Let $\Gamma$... | 1,216 | 998 | 1,127 | 1,163 | 3,949 | 0.769058 | github_plus_top10pct_by_avg |
e on-shell physical states defined by these asymptotic string fields, and thus the physical S-matrix. Thus the supersymmetry algebra is realized on the physical S-matrix, and we can identify the transformation (\[complete transformation\]) with space-time supersymmetry.
Extra unphysical symmetries {#extra symm}
------... | 1,217 | 361 | 1,376 | 1,253 | null | null | github_plus_top10pct_by_avg |
estimator
===================
In this section we obtain the asymptotic size of the uniform deviation of the ideal estimator (\[ideal0\]) from the density $f$, that is, we will consider the a.s. asymptotic size of $$\sup_{t\in D_r} |\bar f(t;h_n)-f(t)|:=\|\bar f(t;h_n)-f(t)\|_{D_r}$$ As usual this quantity is divided ... | 1,218 | 544 | 1,006 | 1,187 | 2,773 | 0.777065 | github_plus_top10pct_by_avg |
ance which is equal to a lattice constant (red bonds, weighted with $v$). On the other hand, in the case of CSAWs model (b), the polymers $P_3$ and $P_2$ are cross-linked at the two sites, so that each contact contributes the weight factor $w$, while the red bonds (marked by $t$) correspond to the interactions between ... | 1,219 | 1,305 | 2,022 | 1,391 | null | null | github_plus_top10pct_by_avg |
al dimension of such operators as a function of the two parameters $(k,f)$ of the supergroup sigma-model. At the WZW point these operators are descendants in the highest-weight representations of the left affine Lie algebra.
Operators of the form $:j_L \phi :$ {#operators-of-the-form-j_l-phi .unnumbered}
-------------... | 1,220 | 461 | 1,087 | 1,138 | null | null | github_plus_top10pct_by_avg |
size specified by plot title. For each baseline SGD run, we have a corresponding large-noise SGD run, denoted by $\diamond$ with the same color. As mentioned, these $\diamond$ runs are designed to match the noise covariance of SGD with larger step size or smaller batch size. In addition to $\times$ and $\diamond$, we a... | 1,221 | 699 | 468 | 1,209 | null | null | github_plus_top10pct_by_avg |
.54 92.71
36 months 47.71 81.45 68.13 79.76 66.71 84.34 62.36 77.73 66.18 73.17
*p* \<0.01 0.34 0.86 0.60 0.15
Adjusted for i... | 1,222 | 19 | 2,031 | 1,387 | null | null | github_plus_top10pct_by_avg |
that, by this definition, $r_i$ for $i\ge1$ equals $((d-2)\wedge(i+3))-{\epsilon}$ and increases until it reaches $d-2-{\epsilon}$. We prove below by induction that $\sum_x|x|^{r_i}|\Pi(x)|$ is finite for all $i\ge0$. This is sufficient for the finiteness of $\sum_x|x|^{{\scriptscriptstyle}\frac{d+2}3}|\Pi(x)|$, since ... | 1,223 | 347 | 369 | 1,333 | 1,146 | 0.79358 | github_plus_top10pct_by_avg |
_{i,i}')a_i(\pi m_{i,i}')$ since its nondiagonal entries contain $\pi^2$ as a factor and its diagonal entries contain $\pi^4$ as a factor. Thus the above equation equals $$a_i'=a_i+\sigma(\pi)\cdot {}^tm_{i,i}'a_i+ \pi\cdot a_i m_{i,i}'.$$ By letting $a_i'=a_i$, we have the following equation $$\sigma(\pi)\cdot {}^tm_{... | 1,224 | 1,613 | 1,320 | 1,166 | null | null | github_plus_top10pct_by_avg |
1,j} \right)
\ = \ \operatorname{{\textsf}{ogr}}B_{ij}.$$
\(3) When $k=0$, the assertion $eJ^k\delta^{k}
= \operatorname{{\textsf}{ogr}}N(k)$ is just the statement that $e{{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}= e({{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}\ast {{W}}) $. When $k>0$, part (i) a... | 1,225 | 1,283 | 1,403 | 1,188 | null | null | github_plus_top10pct_by_avg |
ss Type
0 __text 00000043 0000000100000f30 TEXT
[...]
You can also do the same trick to get the size of whole segments (as opposed to sections) by using the syntax segment$start$__TEXT / segment$end$__TEXT.
Q:
Why does new allocate 1040 extra bytes the first time?
I was creating this simple te... | 1,226 | 6,794 | 86 | 388 | 11 | 0.841901 | github_plus_top10pct_by_avg |
80 (25.5) 1.26 (0.86, 1.85) A allele 88 (27.8) 16 (21.6) 1.40 (0.76, 2.56)
G allele 151 (70.0) 234 (74.5) 0.2416 1 G allele 228 (72.2) 58 (78.4) 0.2756 1
**rs6603797** **rs6603797** ... | 1,227 | 2,069 | 1,368 | 1,299 | null | null | github_plus_top10pct_by_avg |
potent group, and suppose that $A\subset G$ is a finite $K$-approximate group. Then there exist a subgroup $H\subset A^{K^{e^{O(s)}}}$ normalised by $A$ and a nilprogression $P_{\text{\textup{nil}}}(x;L)$ of rank at most $e^{O(s^2)}K\log^{O(s)}2K$ such that $$A\subset HP_{\text{\textup{nil}}}(x;L)\subset H\overline P(x... | 1,228 | 260 | 1,136 | 1,202 | 2,427 | 0.779834 | github_plus_top10pct_by_avg |
s is highly relevant for geometric complexity theory, since it was recently shown in [@kumar10b] and [@burgisserikenmeyer11] that the studied varieties (the orbit closures of the determinant and permanent on the one hand, and of the matrix multiplication tensor and the unit tensor on the other hand) are in fact never n... | 1,229 | 1,659 | 1,101 | 1,126 | null | null | github_plus_top10pct_by_avg |
\[fig2\] and Fig. \[fig3\], Fig. \[fig4\] depicts the mean momentum $p(t)$ of the atoms for two different values of the occupation number $\bar{n}=N/L$ (number of atoms per lattice cite) – $\bar{n}=1$ (upper panel) and $\bar{n}=2/7$ (lower panel). As to be expected, the dynamics of the system depends on the value of $... | 1,230 | 445 | 878 | 1,324 | 2,810 | 0.77681 | github_plus_top10pct_by_avg |
times of measurement to record random variations in detected x-ray intensity are acquired. However, in this work, the collected datasets are not supported by the aforementioned factors and they fall outside the scope of this paper. The results presented here are focusing on the implementation of a new algorithm to lim... | 1,231 | 4,280 | 1,577 | 1,073 | null | null | github_plus_top10pct_by_avg |
y order in a semi-classical expansion. We will show that the knowledge of the poles in these OPEs is enough to fix all the subleading terms. The idea driving the bootstrap is to ask for the compatibility of the elementary OPEs with both current conservation and the Maurer-Cartan equation.
Current-current OPEs {#curren... | 1,232 | 69 | 1,883 | 1,366 | 3,101 | 0.774883 | github_plus_top10pct_by_avg |
nd the periodicity of the KK circle at infinity $\chi\sim\chi+L$ will be determined by smoothness of the metric at $\rho=\rho_0$.
We now add electric 3-form flux to the bubbles, $C=\frac{Q_0}{2\pi^2}(\star \epsilon_3)$, where $\epsilon_3$ is the volume element of the spatial $S^3$. Concretely, the field strength is $$... | 1,233 | 751 | 1,971 | 1,419 | 3,583 | 0.77139 | github_plus_top10pct_by_avg |
u_i&1+\pi w_i \end{pmatrix}\in \mathrm{GL}_{n_i}(B\otimes_AR),$$ where $s_i$ is an $(n_i-2) \times (n_i-2)-$matrix, etc.\
3. Let $i=2m$ be even. Then $g$ stabilizes $Z_i$ and induces the identity on $W_i/(X_i\cap Z_i)$. For the proof, it is easy to show that $g$ stabilizes $Z_i$. To prove the latter, we choose an ele... | 1,234 | 2,510 | 1,421 | 1,176 | 1,617 | 0.787545 | github_plus_top10pct_by_avg |
u i}_\alpha(u) \right)}$ are $m\times q$ matrix functions of $u$ and we denote the partial derivatives by $u^\alpha_i={\partial}u^\alpha/{\partial}x^i$. The matrix $L_i^\alpha$ satisfying the conditions[@Burnat:1972]
\[eq:intelem\] u\_i\^[{ L\_i\^:\_\^[i]{}L\_i\^=0, =1,…,q }]{}
at some open given point $u_0\in U$ is ... | 1,235 | 2,446 | 2,801 | 1,320 | null | null | github_plus_top10pct_by_avg |
d B.-J. Schaefer for discussions. We thank O. Jahn for discussions and collaboration at an early state of this project. FM acknowledges financial support from the state of Baden-Württemberg and the Heidelberg Graduate School of Fundamental Physics.
Faddeev-Popov determinant {#app:FPdet}
=========================
From... | 1,236 | 4,893 | 208 | 834 | null | null | github_plus_top10pct_by_avg |
ssion. The application of an integer constructor labels the free variable as an integer variable. An integer condition, e.g. $\geq/2$, is applicable if both arguments are integer expressions. Since integer variables denote unknown integers, integer expressions are allowed to contain integer variables. Applications of i... | 1,237 | 6,138 | 1,098 | 740 | 253 | 0.815848 | github_plus_top10pct_by_avg |
}_{1}}+{{\theta}_{2}}), \\
& {{J}_{12}}=-{{L}_{1}}\cos {{\theta}_{1}}-{{L}_{2}}\cos ({{\theta}_{1}}+{{\theta}_{2}}), \\
& {{J}_{13}}={{J}_{14}}=0, \\
& {{J}_{21}}=-{{L}_{2}}\sin ({{\theta}_{1}}+{{\theta}_{2}}), \\
& {{J}_{22}}=-{{L}_{2}}\cos ({{\theta}_{1}}+{{\theta}_{2}}), \\
& {{J}_{23}}={{J}_{... | 1,238 | 2,630 | 2,096 | 1,402 | null | null | github_plus_top10pct_by_avg |
{\ast}=\underline{M}$. As a matrix, each element of $\underline{M}^{\ast}(R)$ for a flat $A$-algebra $R$ can be written as $\begin{pmatrix} 1+2 z \end{pmatrix}$.
We define another functor from the category of commutative flat $A$-algebras to the category of sets as follows. For any commutative flat $A$-algebra $R$, le... | 1,239 | 619 | 1,035 | 1,249 | null | null | github_plus_top10pct_by_avg |
imal categories in the Pascal VOC Part dataset. For the ILSVRC 2013 DET Animal-Part dataset and the CUB200-2011 dataset, we learned an AOG for the head part[^3] of each category. It is because all categories in the two datasets contain the head part. We did not train human annotators. Shape differences between two part... | 1,240 | 654 | 272 | 1,017 | 1,027 | 0.795439 | github_plus_top10pct_by_avg |
phic to $\PP^2_w$ and the isomorphism is induced by the branched covering $$\label{covering}
\PP^2\ni [X_0:X_1:X_2] \overset{\phi}{\longmapsto} [X_0^{w_0}:X_1^{w_1}:X_2^{w_2}]_w \in\PP^2_w.$$
Note that this branched covering is unramified over $$\PP^2_{w} \setminus \{ [X_0,X_1,X_2]_{w} \mid X_0\cdot X_1\cdot X_2 =... | 1,241 | 384 | 785 | 1,295 | null | null | github_plus_top10pct_by_avg |
\[r31\]. Thus $\tilde{T}$ is an element of $\underline{M}_j^{\ast}(R)$, where $\underline{M}_j^{\ast}$ is the group scheme in Section \[m\] associated to $M_0^{\prime}\oplus C(L^j)$ so that $\underline{G}_j'$ is defined as the closed subgroup scheme of $\underline{M}_j^{\ast}$ stabilizing the hermitian form on $M_0^{\... | 1,242 | 1,165 | 1,270 | 1,219 | 2,122 | 0.782583 | github_plus_top10pct_by_avg |
\hat{b}^{il}_{-}
( u - \frac{1}{2} (k+l) \hbar )
- \sum_{l=i+2}^{N} \hat{b}^{i+1,l}_{-}
( u- \frac{1}{2} (k+l-1) \hbar) \right): \nonumber\\
& & ~~~~- :\mbox{exp} \left( (b+c)^{i,i+1}
( u + \frac{1}{2} (k+i) \hbar ) \right) \nonumber\\
& & ~~~~~~~~\times \mbox{exp} \left( \hat{a}^i_+ (u)
+ \sum_{l=i+1}^N \hat{b}^{il}_... | 1,243 | 1,383 | 983 | 1,208 | null | null | github_plus_top10pct_by_avg |
=3$ system, i.e., equation (\[eq:Bog\]). Other systems with similar behaviour have been presented in [@BW].
Acknowledgements {#acknowledgements .unnumbered}
----------------
PX acknowledges support from the EPSRC grant [*Structure of partial difference equations with continuous symmetries and conservation laws*]{}, E... | 1,244 | 50 | 2,211 | 1,437 | 2,455 | 0.779594 | github_plus_top10pct_by_avg |
P51884 Lumican 26.62 35.50 38.4 6.61
P01876 Immunoglobulin heavy constant alpha 1 25.67 29.18 37.6 6.51
Q08380 Galectin-3-binding protein 25.15 17.78 65.3 5.27
P67936 Tropomyo... | 1,245 | 4,474 | 659 | 767 | null | null | github_plus_top10pct_by_avg |
a trial‐specific adjustment term for the baseline outcome value (here, centered at the mean for each trial ( ${\bar{Y}}_{\mathit{Bi}}$) to aid interpretation of the trial‐specific intercepts). For example, when there are *K* = 10 trials, there would be 10 *β* ~*i*~ terms and 10 *λ* ~*i*~ terms. Of main interest is an ... | 1,246 | 766 | 2,329 | 1,231 | 344 | 0.812502 | github_plus_top10pct_by_avg |
+ \int_0^t -\nabla U(w_0) ds + \int_0^t \cm \lrp{I - 2\gamma_s \gamma_s^t} dV_s + \int_0^T N(x_s) dW_s
\end{aligned}$$
Where $\gamma_t := \frac{x_t - y_t}{\|x_t-y_t\|_2} \cdot \ind{\|x_t-y_t\|_2 \in [2\epsilon, \Rq)}$. The coupling $(x_t,y_t)$ defined in and is identical to the coupling in (with ... | 1,247 | 1,977 | 1,354 | 1,199 | 3,892 | 0.769479 | github_plus_top10pct_by_avg |
cO_{X}(D))$ is isomorphic to $\CC[x,y,z]_{w,d}$, the $w$-homogeneous polynomials of degree $d := \deg_w({\left \lfloor D \right \rfloor})$.
It is a well-known result for integral Weil divisors. The general rational case follows from the fact that by definition $\cO_X(D) = \cO_X({\left \lfloor D \right \rfloor})$. The ... | 1,248 | 980 | 1,221 | 1,158 | 2,709 | 0.777697 | github_plus_top10pct_by_avg |
36 (32.7) 40 (27.0) 1.31 (0.77, 2.25) A allele 43 (28.7) 6 (23.1) 1.34 (0.50, 3.56)
G allele 74 (67.3) 108 (73.0) 0.3206 1 G allele 107 (71.3) 20 (76.9) 0.5572 1
**rs6603797** **r... | 1,249 | 4,800 | 308 | 669 | null | null | github_plus_top10pct_by_avg |
r to the literature for general results.
We begin by fixing a field $\bbf$; all our modules will be modules for the group algebra $\bbf{\mathfrak{S}_}n$. We assume familiarity with James’s book [@j2]; in particular, we refer the reader there for the definitions of partitions, the dominance order, the permutation modul... | 1,250 | 982 | 1,031 | 1,207 | 1,657 | 0.787113 | github_plus_top10pct_by_avg |
on_{\star}=-2/mr_{\star}^2$. After resonance when $\delta' \to +\infty$, the CI bound state energy behaves as $\epsilon_b'\simeq -1/ma^2$, which translates into $\epsilon_b'/\epsilon_{\star}'\simeq (\delta'+A)^2/2\simeq
\delta^{'2}/2$ in terms of the parameter $\delta'$ introduced above. Similarly, in the BFRM, the bou... | 1,251 | 417 | 1,277 | 1,331 | null | null | github_plus_top10pct_by_avg |
any solution $\phi\in {\mathcal{H}}_P(G\times S\times I^\circ)$ of the problem , , that further satisfies \[asscl\] \_[|\_+]{}T\^2(\_+)(,,0)L\^2(GS), is unique and obeys the estimate \[csda40aa\] \_[[H\_1]{}]{} (\_[L\^2(GSI)]{}+\_[T\^2(\_-)]{}), where $c'$ is given in .
The proof is based on “variations” and it is qu... | 1,252 | 528 | 1,763 | 1,319 | null | null | github_plus_top10pct_by_avg |
k$ vertices, there are at most $\binom{k}{2}$ edges whose addition may produce a cycle. [This includes edges already present in the component, as an edge with multiplicity 2 (double edge) forms a 2-cycle.]{} Thus, the $p$ edges can be chosen in $\binom{k}{2}^p<k^{2p}$ ways. Let $\{e_1,e_2,\ldots, e_p\}$ denote a set of... | 1,253 | 1,423 | 1,281 | 1,213 | 2,961 | 0.775804 | github_plus_top10pct_by_avg |
s. Here's the code.
int main (void)
{
const char *alp = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
char *ptr = &alp[3];
printf("%s\n", ptr);
return 0;
}
Edit- Sorry for not mentioning the errors. The thing is I get tons of different errors depending on where I put different asterisks and ampersands. There is... | 1,254 | 6,702 | 225 | 196 | 42 | 0.832662 | github_plus_top10pct_by_avg |
.
\[thm:DefDiVar\] Let $D\in{\mathrm{Di}}{\mathrm{Alg}}0$. Then the following conditions are equivalent:
1. $D\in{\mathrm{Di}}{\mathrm{Var}}$
2. $\widehat D=\bar D\oplus D\in{\mathrm{Var}}$ the definition in the sense of Eilenberg
3. $D\vDash \Psi^{x_i}_{\mathrm{Alg}}\,f$ for every $f\in T_0({\mathrm{Var}})$, $\... | 1,255 | 1,290 | 1,085 | 1,190 | 2,469 | 0.779498 | github_plus_top10pct_by_avg |
e cannot be sure that system (\[eq:SW:15\]) for $f(r)$ is well-defined in the sense that it represents a system for $f(r)$ express in terms of $r$ only. To ensure this, we begin by introducing vector fields orthogonal to the wave vector $\lambda$, that is, vector fields of the form
\[eq:SW:16\] X\_a=\^i\_a(u)\_[x\^i]{... | 1,256 | 2,200 | 2,909 | 1,352 | null | null | github_plus_top10pct_by_avg |
t $D^{(k)} = (d_{ij}^{(k)}-\delta_{i,1}\cdot \delta_{kj})_{i,j=1,\dots,n}$. Expressing $\lambda$ as linear combination of the dual basis then leads to the system of linear equations
$D^{(k)} x = 0$ $(k=1,...,n)$. As pointed out by Mariano it has a unique solution (up to scalars). Again, this is no closed formula, but ... | 1,257 | 929 | 309 | 632 | 909 | 0.797879 | github_plus_top10pct_by_avg |
$, one has $\bar{a}\bar{b} = [a+F^{n-1}R][b+F^{m-1}R]
\subseteq [ab+F^{n+m-1}R]$. Since $\bar{a}\bar{b}\not=0$, $ab\in F^{n+m}R\smallsetminus F^{n+m-1}R$, whence $\bar{a}\bar{b}=\sigma(ab)$ is the image of $ab$ in $\operatorname{gr}_F(AB)$.
\(2) Define a map $\rho: \operatorname{gr}_FA\times \operatorname{gr}_FB \to \... | 1,258 | 2,177 | 1,131 | 1,173 | null | null | github_plus_top10pct_by_avg |
restate and prove Theorem \[bicirc\].
\[bicirctech\] There is a function $f_{\ref{bicirctech}}\colon \bZ \to \bZ$ so that, for every integer $s \ge 2$, if $M$ is a matroid without a $U_{s,2s}$-minor and with a $B^+(K_{r(M)})$-restriction framed by $B$, then there is a set $\wh{B} \subseteq B$ and a $\wh{B}$-clique $\w... | 1,259 | 494 | 537 | 1,192 | 3,154 | 0.77444 | github_plus_top10pct_by_avg |
ring sites
*wR*(*F*^2^) = 0.089 H-atom parameters constrained
*S* = 1.14 *w* = 1/\[σ^2^(*F*~o~^2^) + (0.0341*P*)^2^ + 13.8363*P*\] where *P* = (*F*~o~^2^ + 2*F*~c~^2^)/3
19520 reflections (Δ/σ)~max~ = 0.009
950 parameters Δρ~... | 1,260 | 134 | 1,436 | 1,536 | null | null | github_plus_top10pct_by_avg |
required to be large enough, but still independent of the mesh size $h$, in order to guarantee the well-posedness of the discontinuous Galerkin formulation. Details will be given later.
It is clear that the exact solution $u$ to Equation (\[eq:ellipticeq\]) satisfies $$\label{eq:dg-exactsol}
A(u,v) = (f,v)\qquad\text... | 1,261 | 729 | 786 | 1,303 | null | null | github_plus_top10pct_by_avg |
t \int_{S\times I}\sigma(\cdot,\omega',\cdot,E',\cdot)d\omega' dE'\right\Vert}_{L^\infty(G\times S\times I)}={\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,\omega,E)\in G\times S\times I}\int_{S\times I}\sigma(x,\omega',\omega,E',E)d\omega' dE', \nonumber\\
& {\left\Vert \int_{S\times I}\sigma(\cdot,\cdot,\omega',\cdot,E')... | 1,262 | 135 | 2,005 | 1,393 | null | null | github_plus_top10pct_by_avg |
\bar{b}_{ki}-f_j\bar{h}_i=0 \label{nobraneE84} \\
& \bar{g}_{ki}\bar{b}_{jj}-\bar{f}_ib_{ij}-\bar{g}_{jj}\bar{b}_{ki}+g_{ij}\bar{h}_i=0 \nonumber \\
& -g_{ji}b_{kj}+\bar{g}_{ii}h_j+g_{kj}b_{ji}-f_j\bar{b}_{ii}=0 \nonumber \\
& g_{ji}\bar{b}_{jj}-\bar{g}_{ii}b_{ij}-\bar{g}_{jj}b_{ji}+g_{ij}\bar{b}_{ii}=0 \quad .\nonumbe... | 1,263 | 927 | 1,537 | 1,293 | null | null | github_plus_top10pct_by_avg |
finition \[D:DISJOINT\_ENSEMBLES\], disjointness entails that ${{\operatorname{dom}{\Psi}}}\thickspace\cap\thickspace{{\operatorname{dom}{\Phi}}} = \varnothing$. Thus, if $i \in {{\operatorname{dom}{\Upsilon}}}$, exactly one of two cases hold: either A: $i \in {{\operatorname{dom}{\Psi}}}$ and $i \notin {{\operatorname... | 1,264 | 534 | 1,816 | 1,259 | null | null | github_plus_top10pct_by_avg |
Big(\PP^2_w, \mathcal{O}_{\PP^2_w}\left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)} \right)\Big) =
\chi \left( \PP^2_w,\mathcal{O}_{\PP^2_w} \left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)} \right) \right) \\
& =1 + \frac{1}{2} \left( kH+K_{\PP^2_w} - \mathcal{C}^{(k)} \right)
\cdot \left( kH - \mathcal{C}^{(k)} \right)
+ R_{\PP^2_w}... | 1,265 | 500 | 1,440 | 1,245 | null | null | github_plus_top10pct_by_avg |
\begin{aligned}
\hat{u}_{ {\bf k} }(\tau) &=& \hat{a}_{{\bf k} } f(k,\tau)+
\hat{a}_{-{\bf k} }^{\dagger} f^{*}(k,\tau), \label{sol11} \\
\hat{\pi}_{{\bf k} }(\tau) &=& \hat{a}_{{\bf k} } g(k,\tau)+
\hat{a}_{-{\bf k} }^{\dagger} g^{*}(k,\tau). \label{sol22}\end{aligned}$$ where $f(k,\tau)'=g(k,\tau)$. When we ... | 1,266 | 3,959 | 1,449 | 895 | null | null | github_plus_top10pct_by_avg |
ion parameters.
Similarly to what we did in , we derive two approximate confidence sets: one is an $L_\infty$ ball and the other is a hyper-rectangle whose $j^{\mathrm{th}}$ side length is proportional to the standard deviation of the $j^{\mathrm{th}}$ coordinate of $\hat{\gamma}_{{\widehat{S}}}$. Both sets are center... | 1,267 | 2,821 | 1,549 | 1,214 | null | null | github_plus_top10pct_by_avg |
ion $\varphi^a=G_{ret}^{ab}q_{c}F^c_b$ we get
\^a\^bG\_s\^[ab]{}=G\_[ret]{}\^[ac]{}F\^d\_cG\_[ret]{}\^[be]{}F\^f\_eQ\_[df]{}=G\_[ret]{}\^[ac]{}G\_[ret]{}\^[be]{}N\_[ce]{} \[ne21\]
From effective actions to Langevin equations
---------------------------------------------
Let us now investigate a scalar field theory ... | 1,268 | 367 | 2,263 | 1,284 | null | null | github_plus_top10pct_by_avg |
LB($n^{0.8}$) 0.000 0.002 0.002 0.000 0.002 0.000 0.000
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 ... | 1,269 | 5,338 | 258 | 752 | null | null | github_plus_top10pct_by_avg |
make the following approximation:
- When assimilating a non-baseline observation, ${\bar{\mathcal{A}}}$ is kept fixed at its previous value, and for updating ${\mathcal{A}}$ it is assumed that ${\bar{\nu}}^{-1}=0$.
Approximation B2 implies that for ${\mathbf{x}}_t\neq{\mathbf{0}}$, $$\begin{aligned}
Y_t|~ {\bar{... | 1,270 | 2,913 | 1,708 | 1,206 | null | null | github_plus_top10pct_by_avg |
, where $\sigma_n$ is the time it takes for the walk-on-spheres to exit the $n$th sphere. Thus $\sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_n\cdot))\leq\kappa \sum_{n=0}^{N-1} \sigma_n = \kappa\,\sigma_D$. We thus have that $$\mathbb{E}_x\left[\left( \sum_{n=0}^{N-1} r_n^{\alpha}\, V_{1}(0, {f}(\rho_n + r_... | 1,271 | 963 | 1,183 | 1,200 | 3,682 | 0.770724 | github_plus_top10pct_by_avg |
$$
The Poincaré series of $N(k)$ is now easy to compute. First, in the [*canonical grading*]{}, shows that $$p(\Delta_d(\mu), v, W) = v^{D(d,\mu)}\frac{\sum_{\lambda}
f_{\lambda}(v) [\lambda\otimes \mu]}{\prod_{i=2}^n(1-v^i)}
\qquad\mathrm{and\ so}\qquad p(e\Delta_d(\mu), v)=
v^{D(d,\mu)} \frac{f_{\mu}(v)}{\prod_{i=... | 1,272 | 1,206 | 1,336 | 1,198 | 4,064 | 0.768366 | github_plus_top10pct_by_avg |
ivalence relations $R$ on $X={{\mathbb A}}^2$ (in any characteristic) such that the geometric quotient $X/R$ exists yet $R$ is strictly smaller than the fiber product $X\times_{X/R}X$. Closely related examples are in [@venken; @philippe].
In characteristic zero, this leaves open the following:
\[sch.th.quot.quest\] L... | 1,273 | 2,373 | 1,929 | 1,230 | 1,328 | 0.790923 | github_plus_top10pct_by_avg |
I have no idea how to read what address is currently stored in whatever OpenGL has currently bound as the vertex attrib array, so I cannot test whether or not it is pointing to the vertex data. I'm pretty sure that it's pointing to address 0 for some reason though.
Edit:
It turns out it was not the hard-coded 0 that w... | 1,274 | 4,233 | 412 | 1,195 | 676 | 0.802583 | github_plus_top10pct_by_avg |
rmation up to an additive term. $$\label{transform}
G_2 \left( \frac{a\tau +b}{c\tau +d}\right)
= (c\tau +d)^2 G_2 (\tau) - \frac{c}{4\pi i} (c\tau +d).$$ The ring $\Q [G_2,\ G_4,\ G_6]$ is called the ring of *quasi-modular* forms (see [@kaneko-zagier]).
We have $$1+\sum_{n\geq
1}\H_{(n-1,1)}\left(e^{u/2},e^{-u... | 1,275 | 3,715 | 1,769 | 1,082 | 2,887 | 0.776306 | github_plus_top10pct_by_avg |
entity \otimes \identity + p(\mathbf{P})]\enspace.\end{aligned}$$ The solution is $$\label{superop-soln}
S_t = \exp\left([i(\identity \otimes H - H \otimes \identity) - p \identity \otimes \identity + p(\mathbf{P})]t\right)\enspace.$$
We now define the decoherence operator $\mathbf P$. This operator will corresp... | 1,276 | 2,948 | 1,530 | 1,174 | 3,655 | 0.770882 | github_plus_top10pct_by_avg |
ary morphisms since all morphisms preserve left Kan extensions along left adjoint functors (see [@groth:can-can Prop. 5.7] and [@groth:can-can Rmk. 6.11]). As for the second statement, there is an adjoint triple $0\dashv\pi_{[1]}\dashv 1$ and hence an induced adjoint triple $1^\ast\dashv \pi_{[1]}^\ast\dashv 0^\ast$. T... | 1,277 | 608 | 1,101 | 1,247 | 1,988 | 0.78379 | github_plus_top10pct_by_avg |
f({\mathbf x}) &\sim {\GP \left(m({\mathbf x}),\,\, k({\mathbf x},{\mathbf x}') \right)}, \\
y_i &= \mathcal{H}_{{\mathbf x},i} f({\mathbf x})+\varepsilon_i.
\end{aligned}$$
As discussed, for example, in [@Sarkka:2011; @SolinSarkka2015] the GP regression equations can be extended to this kind of mode... | 1,278 | 2,003 | 1,335 | 1,267 | 3,495 | 0.771947 | github_plus_top10pct_by_avg |
------------------------------------------------------
The zeroth and first order $\hat{S}$ matrix elements can be calculated as follows: $$\begin{aligned}
\hat{S}_{ii}^{(0+1)} &=&
\left( e^{-i \hat{H}_{0} x} \right)_{i k} (\Omega_{k i}) +
\left( e^{-i \hat{H}_{0} x} \right)_{i K} (\Omega_{K i}) =
e^{-i h_{i} x} (\O... | 1,279 | 2,269 | 1,900 | 1,380 | null | null | github_plus_top10pct_by_avg |
2,2)$-position (because $T$ is not $d$-bad). So we can repeat the above argument and show that that there is a tableau $T'\domby T$ such that ${\hat\Theta_{T'}}$ occurs in $\theta$; contradiction.
We now know that every semistandard homomorphism occurring in $\theta$ has at least two $2$s in the second row. This means... | 1,280 | 1,455 | 1,177 | 1,189 | 681 | 0.80236 | github_plus_top10pct_by_avg |
by stacking the $n-k+j-1$ eigenvectors associated to the smallest eigenvalues. Computation of constraints of the $l_1$ minimization problem Compute $T=V_{2,n}^t$ Compute $W=T^{-Index[j]}$ and $w=T^{Index[j]}$(where $T^{Index[j]}$ is the column $Index[j]$ of $T$ and $V^{-I_j}$ the matrix $T$ wihtout the column $Index[j]... | 1,281 | 577 | 723 | 1,372 | 1,857 | 0.785017 | github_plus_top10pct_by_avg |
x_k\tilde x_l\biggr)+
\biggl(\sum_m \tilde
x_m\biggr)\biggl(\sum_{\substack{{k,l}\\k<l}}x_k
x_l\biggr)\leq\sum_k x_k^2\biggl(\sum_{\substack{m\\m\neq k}}\tilde
x_m\biggr)+\sum_{\substack{{k,l}\\k<l}}x_kx_l\biggl(\sum_{\substack{m\\m\neq
k}}\tilde x_m+\sum_{\substack{m\\m\neq l}}\tilde x_m\biggr).$$ The right hand side ... | 1,282 | 1,287 | 1,491 | 1,332 | null | null | github_plus_top10pct_by_avg |
and $33$% for $A_1$ units ($p < 0.0005$). For $R_1$ units, the median %LS decrease was $2$% ($p=0.18$). Indeed, the average $R_1$ response did not exhibit much length suppression (see Supplement), though, there were particular examples with a strong effect.
#### Sequence Learning Effects in Visual Cortex
Predictions... | 1,283 | 879 | 1,923 | 886 | 3,527 | 0.771752 | github_plus_top10pct_by_avg |
_{c_{\k\sigma} ,d^\dagger_\sigma}(z)
&=& V_k G_{d_\sigma,d^\dagger_\sigma}(z) \\
&& \nonumber
+\lambda_c \frac{V_k}{V_0} N_\sigma(z)
.\end{aligned}$$ The off-diagonal composite correlation function $$\begin{aligned}
N_\sigma(z) &=& G_{\hat X_0 c_{0\sigma} ,d^\dagger_\sigma}(z)\end{aligned}$$ accounts for the correla... | 1,284 | 1,419 | 1,553 | 1,229 | 2,677 | 0.777926 | github_plus_top10pct_by_avg |
of the diagonal with two copies of $Z$, one of which maps as $$(p_1,p_2):Z\to Y_1\times Y_2\subset \bigl( Y_1\amalg Y_2\bigr)
\times
\bigl( Y_1\amalg Y_2\bigr),$$ the other its symmetric pair. The categorical quotient $\bigl(\bigl( Y_1\amalg Y_2\bigr)/R\bigr)^{cat}$ is also the universal push-out of $Y_1\stackrel{p_1}... | 1,285 | 1,374 | 1,310 | 1,215 | 2,491 | 0.779352 | github_plus_top10pct_by_avg |
ae} j_{\bar z}^d (w)
\nonumber \\
& = - c_- {f^{ac}}_g (c_2-g) j_{\bar z}^g (w)+ c_+ {f^{ac}}_g c_4 j_{\bar z}^g(w)
-i (-1)^a (-1)^a {f^{ac}}_{g} \tilde{c} j_{\bar z}^g (w)\end{aligned}$$ which also vanishes thanks to the relation : $$\begin{aligned}
-c_- (c_2-g) + c_+ c_4 - i \tilde{c} &=& 0.\end{aligned}$$
3\. Ther... | 1,286 | 1,965 | 1,499 | 1,244 | null | null | github_plus_top10pct_by_avg |
isms to graded $R_{\mathbb{Z}}$-module homomorphisms, $\Phi$ is a functor.
Conversely, suppose that $\widetilde{N}\in R_{\mathbb{Z}}{\text{-}{\textsf}{qgr}}$ and pick a preimage $N\in
R_{\mathbb{Z}}{{\textsf}{\text{-}grmod}}$. Then $N$ is generated by $\bigoplus_{i=0}^a N_i$, for some $a$, and so $N_j=R_{ja}N_a$, for ... | 1,287 | 1,036 | 844 | 1,268 | 3,569 | 0.771502 | github_plus_top10pct_by_avg |
eq{\vbx'-z{|\!|\!|}}\vee{\vbz-y'{|\!|\!|}}$. Suppose that ${\vbx-z{|\!|\!|}}\leq{\vbz-y{|\!|\!|}}$ and ${\vbx'-z{|\!|\!|}}\leq{\vbz-y'{|\!|\!|}}$. Then, by [(\[eq:conv\])]{} with $a=b=q$, the contribution from this case is bounded by $$\begin{aligned}
\frac{2^{2q}}{{\vbx-y{|\!|\!|}}^q{\vbx'-y'{|\!|\!|}}^q}\sum_z\frac1{... | 1,288 | 918 | 994 | 1,389 | 3,970 | 0.768903 | github_plus_top10pct_by_avg |
mm} \put(20,40){\line (1,0){20}}
\put(80,40){\line(1,0){20}} \qbezier(40,40)(40,10)(70,10)
\qbezier(70,10)(100,10)(100,40)
\put(160,40){\line (1,0){20}} \put(220,40){\line(1,0){20}}
\qbezier(180,40)(180,70)(210,70) \qbezier(210,70)(240,70)(240,40)
\end{picture}$$ with order two group of automorphisms each. Thus, they ... | 1,289 | 596 | 1,924 | 1,401 | null | null | github_plus_top10pct_by_avg |
in by giving a general context for all three results.
{#tens-defn-sect}
For fixed $i\geq j\geq 0$ we are interested in the following tensor product decompositions $$\label{tensor-1}
B_{ij}\cong Q_{c+i-1}^{c+i}\otimes Q_{c+i-2}^{c+i-1}\otimes\cdots \otimes
Q_{c+j}^{c+j+1},$$ $$\label{tensor-101}
N(i)\cong Q_{c+i-1}^... | 1,290 | 833 | 1,479 | 1,201 | 1,766 | 0.785863 | github_plus_top10pct_by_avg |
e{R}}$.
From ${{\Psi}\negmedspace\mid\negmedspace{R}} \subseteq \Phi$ and $\Phi \subseteq {{\Psi}\negmedspace\mid\negmedspace{R}}$ we infer $\Phi = {{\Psi}\negmedspace\mid\negmedspace{R}}$.
\[L:SUBSET\_SUBSPACE\] Let $\Psi$ and $\Phi$ be ensembles. If $\Phi \subseteq \Psi$, then ${\prod{\Phi}}$ is a subspace of ${\pr... | 1,291 | 504 | 961 | 1,342 | null | null | github_plus_top10pct_by_avg |
tcome, two died without requiring dialysis, and their uNGAL levels decreased within 48 h (details in Table [3](#Tab3){ref-type="table"} and Supplemental Table [7](#MOESM2){ref-type="media"}).Table 3Association between tracheotomy and discharge modality with regard to patients suffering from AKI with and without require... | 1,292 | 3 | 1,379 | 1,622 | 3,511 | 0.771854 | github_plus_top10pct_by_avg |
at\beta(\hat{S})=\overline{Y}(\hat{S})$ for $2n$ (non-splitting) and $n$ (splitting). In this example we see that indeed, the splitting estimator suffers a larger risk. In this example, $D=1,000$, $n=50$, and $\beta = (a,0,\ldots, 0)$. The horizontal axis is $a$ which is the gap between the largest and second largest m... | 1,293 | 91 | 1,801 | 1,427 | null | null | github_plus_top10pct_by_avg |
E}\left[\left.{g}(X_{\sigma_D}) + \int_0^{\sigma_{B(x')}} {f}(X_s)\,{\rm d}s
+ \int_{\sigma_{B(x')}}^{\sigma_D} {f}(X_s)\,{\rm d}s\right|\mathcal{F}_{\sigma_{B(x')}}\right]\right] \notag\\
& = \mathbb{E}_x\left[\upsilon (X_{\sigma_{B(x')}}) + \int_0^{\sigma_{B(x')}} {f}(X_s)\... | 1,294 | 2,546 | 1,164 | 1,205 | 3,195 | 0.774113 | github_plus_top10pct_by_avg |
instantaneous capacity of a MIMO system whose channel matrix has correlated zero-mean complex Gaussian entries can be approximated by a Gaussian variable [@Moustakas_03_Mctccitpocian; @Martin_03_aedacfcucf]. Based on the discussion above, the distribution of $R_{\psi}$ is approximated by a Gaussian distribution, and th... | 1,295 | 769 | 1,853 | 1,344 | 2,464 | 0.779547 | github_plus_top10pct_by_avg |
] Let $G^{\ddag}$ be the subfunctor of $ \mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$ consisting of those $m$ satisfying Equations (\[ea20\]), (\[ea22\]), (\[24\]), (\[24’\]), (\[ea25\]), (\[ea27\]), and (\[ea32\]). Note that such $m$ also satisfies Equation (\[32’\]). Then $G^{\ddag}$ is represented by a smooth closed su... | 1,296 | 581 | 1,890 | 1,340 | 2,756 | 0.777245 | github_plus_top10pct_by_avg |
{{\widehat{S}}}$:
Get $\hat\beta_{{\widehat{S}}}$ from ${\cal D}_{2,n}$ by least squares.
Output $\hat{C}_{{\widehat{S}}} = \bigotimes_{j\in {\widehat{S}}} C(j)$ where $C(j) = \hat\beta_{{\widehat{S}}}(j) \pm z_{\alpha/(2k)} \sqrt{\hat\Gamma_n(j,j)}$ where $\hat\Gamma$ is given by (\[eq::Ga\]).
For $\gamma_{{\wideha... | 1,297 | 694 | 1,044 | 1,239 | null | null | github_plus_top10pct_by_avg |
\^[i]{}\_[( \^[-1]{} )]{}\^\_\_i=b\^.
Note that the expression $\Phi^{-1}\frac{{\partial}f}{{\partial}r}\in {\mathbb{R}}^q$ is a contravariant vector as well as $b\in{\mathbb{R}}^q$. Hence there exists a nonzero scalar function $\Omega=\Omega(x,u)$, a rotation matrix $L=L(x,u)\in SO(q)$ and a vector $\tau=\tau(x,u)\i... | 1,298 | 1,934 | 2,176 | 1,361 | null | null | github_plus_top10pct_by_avg |
,0}$ have degree 0, 2 or 4 in $x$. Since $f$ contains summands of only 2-nd and 4-th degree in $x$, we have $\phi_{1,1}+\phi_{1,3}+\phi_{2,1}=0$.
Therefore, $\phi=\phi_{1,0}+\phi_{1,2}+\phi_{2,0}$.
Since $x$ is the central letter of the dipolynomial $f$, central letters of dimonomials from $\phi$ can be variables $x$... | 1,299 | 2,761 | 1,717 | 1,285 | null | null | github_plus_top10pct_by_avg |
s a normal subgroup of $G=N{{\operatorname}{C}_{G}(y)}$. Since $N$ is a minimal normal subgroup of $G$, we deduce that either $N={{\operatorname}{C}_{N}(y)}$ or ${{\operatorname}{C}_{N}(y)}=1$. The first case yields to the contradiction $G={{\operatorname}{C}_{G}(y)}$. So we may assume ${{\operatorname}{C}_{N}(y)}=1$. ... | 1,300 | 594 | 1,045 | 1,183 | null | null | github_plus_top10pct_by_avg |
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