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 |
|---|---|---|---|---|---|---|---|
t. Suppose -4/9*n**2 + g + 4/3*n = 0. Calculate n.
-2, 5
Let t = 9370 - 327938/35. Let q(w) be the first derivative of 23 + 0*w**2 + t*w**5 + 0*w + 2/7*w**3 + 1/14*w**6 + 15/28*w**4. Suppose q(b) = 0. What is b?
-2, -1, 0
Let p = 725 - 731. Let b be (200/35 + p)/(4/(-6)). Find d such that b*d**2 + 75/7 - 30/7*d = 0.
5... | 2,901 | 1,888 | 2,150 | 2,353 | null | null | github_plus_top10pct_by_avg |
s the proof.
Proof of Proposition \[pro:dy\] {#sub:dy}
===============================
In this subsection we will prove two lemmas and then combine them to establish the proposition. The lemmas and their proofs are inspired by [@KP06 Lemma 2.1 and 2.2]. Recall that a subgraph of ${\mathcal{C}}_n$ is $c$-loaded if eve... | 2,902 | 1,229 | 2,248 | 2,681 | 701 | 0.802132 | github_plus_top10pct_by_avg |
ages implement a Cheddar program compiler and interpreter. Then scheduling simulation analysis is performed on AADL specifications with hierarchical schedulers.
- A two-level scheduler for RTSJ
The Real-Time Specification for Java (RTSJ) is a set of interfaces and behavioral specifications that allow for real-time ... | 2,903 | 1,425 | 1,617 | 2,048 | 739 | 0.80111 | github_plus_top10pct_by_avg |
it may so happen that operators like $\widetilde{H}_N $, acting on a suitably chosen function subspace can preserve the space partially. In such cases we introduce the term quasi-solvability. A linear differential operator $H_N$ of several variables $\{z_j \vert j=1,\dots,N\}$, is said to be quasi-solvable if it preser... | 2,904 | 1,970 | 2,963 | 2,916 | null | null | github_plus_top10pct_by_avg |
*m**2*s + m*s**2 + 5*s**2 wrt m.
-18*s**2 + 1770*s - 42
What is the second derivative of -4*c*m**5 + 34278*c*m**2 - 29*c*m - c - 2*m + 363 wrt m?
-80*c*m**3 + 68556*c
What is the third derivative of -822*b**6 - 2*b**5 + 8*b**4 + 72952*b**2?
-98640*b**3 - 120*b**2 + 192*b
What is the third derivative of 126472*w**6 + w*... | 2,905 | 622 | 2,729 | 2,776 | null | null | github_plus_top10pct_by_avg |
Assume that $i$ is even. A $\pi^i$-modular lattice $L$ is *of parity type I* if $n(L)=s(L)$, and *of parity type II* otherwise. The zero lattice is considered to be *of parity type II*. We caution that we do not assign a *parity type* to a $\pi^i$-modular lattice $L$ with $i$ odd.
\[r23\]
1. If $L$ is $\pi^i$-modu... | 2,906 | 1,816 | 1,253 | 3,121 | 4,114 | 0.768021 | github_plus_top10pct_by_avg |
nt: method and leads Age (SD) No. of males (%) No. of Sp. type 1 patients (%) No. of SCN5a positive patients (%) Endpoints Comparisons No. of patients with adverse events /without adverse events/%/% per year Follow‐up duration (months) Quality score Ref... | 2,907 | 2,286 | 2,797 | 2,971 | null | null | github_plus_top10pct_by_avg |
&Education Assistance8
For additional information about Enron Corp's Organizational Development and
Training offerings, contact:
Suzanne Gruber, Senior Director
713/345-8314
Email: suzanne.gruber@enron.com
Nothing with or pertaining to ASpen to the best of my knowledge.
Kay C. Young
Legal Specialist
Enron No... | 2,908 | 1,102 | 1,637 | 2,647 | null | null | github_plus_top10pct_by_avg |
er98]. \[pageref:quasi-poly\] That is, there exists a decomposition of ${\mathbb Z}^{r_G+r_H}$ into polyhedral chambers such that on each chamber $C$ the function $n(y)$ is given by a single quasi-polynomial, i.e., there exists a sublattice $L \subseteq {\mathbb Z}^{r_G+r_H}$ of finite index and polynomials $(p_z)$ wit... | 2,909 | 1,392 | 2,724 | 2,645 | 4,017 | 0.768629 | github_plus_top10pct_by_avg |
u$ final state, while in charm decays it generates to a very good approximation the same amount of $d \bar d$ and $s \bar s$ states.
We write the amplitudes very generally and up to a normalization factor as $${\cal A} = 1 + r a e^{i(\phi+\delta)}\,, \label{eq:generic-ampl}$$ such that $r$ is real and depends on CKM ... | 2,910 | 2,215 | 3,047 | 2,803 | 2,656 | 0.778067 | github_plus_top10pct_by_avg |
ig){\:\Dot{\cup}\:}{{\mathbb S}}_j\equiv F_{\vec\omega_k}^{{\scriptscriptstyle}(0)}({{\mathbb S}}_0){\:\Dot{\cup}\:}F_{\vec\omega_k}^{{\scriptscriptstyle}(j)}({{\mathbb S}}_j).\end{aligned}$$ Therefore, $\vec F_{\vec\omega_k}$ is a bijection from ${\mathfrak{S}}_{\vec\omega_k}$ to ${\mathfrak{S}}'_{\vec\omega_k}$. This... | 2,911 | 1,235 | 2,226 | 2,921 | null | null | github_plus_top10pct_by_avg |
aph topology effects the accuracy. When $\theta^*$ is chosen uniformly at random, the accuracy does not change with $d$ (left), and the accuracy is better for those graphs with larger spectral gap. However, for a certain worst-case $\theta^*$, the error increases with $d$ for the chain graph and the barbell-like graph,... | 2,912 | 1,793 | 1,103 | 3,022 | 1,249 | 0.792087 | github_plus_top10pct_by_avg |
$x$, $y\in D$. Let ${\mathrm{Var}}$ be a variety of ordinary algebras. In the paper [@Pozh:09] it is shown that $D\in{\mathrm{Di}}{\mathrm{Var}}$ if and only if $\bar D\in{\mathrm{Var}}$ and $D$ is a ${\mathrm{Var}}$-bimodule over $\bar D$ in the sense of Eilenberg, i. e., the split null extension $\widehat D=\bar D\o... | 2,913 | 2,332 | 2,503 | 2,659 | null | null | github_plus_top10pct_by_avg |
isc\]). At this spectral resolution, H$_\alpha$ is blended with the \[N[II]{}\] doublet, and a fit to the three lines must be obtained simultaneously in order to measure their line strengths. With the exception of close doublets (i.e. \[O[II]{}\], \[S[II]{}\]) the other lines in the spectrum are all comparitively unble... | 2,914 | 1,874 | 3,157 | 3,001 | 2,516 | 0.779217 | github_plus_top10pct_by_avg |
Phi$ and the Killing vector $I$ as in eq. of appendix \[app:sugra\].
To show that the dual background solves the modified supergravity equations we follow the derivation in [@Hoare:2016wsk]. After splitting the Lagrange multiplier as $v_a = u_a + y n_a$, it transpires that shifting $y$ is a symmetry of the dual backgr... | 2,915 | 1,641 | 1,086 | 2,826 | 3,086 | 0.774977 | github_plus_top10pct_by_avg |
\tilde{{\bf k}}| = - [k_b u^b(\tau)]$. Substituting the above expression in Eq.(\[gwhitmann\]) and upon performing the straightforward ${\bf k}$ integral, we can write a compact expression for $W_{\Theta_0}$ of the following form $$\begin{aligned}
W_{\Theta_0}(\tau,\tau^\prime) &=& \frac{-1}{16} \frac{\partial}{\partia... | 2,916 | 5,028 | 894 | 2,603 | null | null | github_plus_top10pct_by_avg |
here uses $\sum_{a = 1}^p \frac{1}{\kappa -a+1} \leq \log\big(\frac{\kappa}{\kappa-p}\big)$ and $C_3 \geq 0$. Equation follows from the fact that for any $x>0$, $\log(1+x) \leq x$. To prove , we have the first order partial derivative of $\P(\theta)$ given by $$\begin{aligned}
\label{eq:cr2}
\nabla_i \P(\theta) &=& \... | 2,917 | 2,359 | 2,514 | 2,755 | null | null | github_plus_top10pct_by_avg |
uld include another suitable criteria in general case which is still an active area of research.
The a priori estimates
----------------------
We are going to derive the a priori estimates for the geometric quantities. We fix a small constant $\delta>0$ and a constant $u_1\in(u_0,0)$ and denote $$\begin{aligned}
\mat... | 2,918 | 2,113 | 1,243 | 2,913 | null | null | github_plus_top10pct_by_avg |
m $v=cz+d$ with the integers $(c,d)$ co-prime.
Now given the positive lattice basis $v=cz+d$ and $v'=az+b$, form the integer matrix $\g=\begin{pmatrix} a&b\\ c&d\end{pmatrix}$ , which has $\det(\g)=+1$ since $\{v,v'\}$ form a positive basis of the lattice. Thus we get a matrix in the modular group $\G=SL_2(\Z)$. Then ... | 2,919 | 2,391 | 2,797 | 2,630 | 3,377 | 0.772767 | github_plus_top10pct_by_avg |
Since any unmixed sequentially Cohen-Macaulay module is Cohen-Macaulay, all assertions are proved.
As a consequence of the previous theorem we get
\[interesting\] Let $M$ be an $R$-module. If the non-zero factors of the dimension filtration of $M$ are clean, then $M$ is pretty clean.
Conversely assume that $R$ is a ... | 2,920 | 2,255 | 1,186 | 2,751 | null | null | github_plus_top10pct_by_avg |
chi (\Lambda )\cap (U^-(\chi ){\otimes }1)\big)\fiee$$ and that $L^\chi (\Lambda )$ is ${\mathbb{Z}}^I$-graded. For all ${\alpha }\in {\mathbb{Z}}^I$ let $$I^\chi (\Lambda )_{\alpha }=M^\chi (\Lambda )_{\alpha }\cap
I^\chi (\Lambda ), \quad
L^\chi (\Lambda )_{\alpha }=M^\chi (\Lambda )_{\alpha }/I^\chi (\Lambda )_{\alp... | 2,921 | 1,818 | 1,931 | 2,590 | null | null | github_plus_top10pct_by_avg |
\ell_i}
\in \mathcal{R}_n $ and $(z_j-z_k)/(z_j+z_k)(1-\widetilde{\Lambda})_{jk}\prod_i z_i^{\ell_i}
\in \mathcal{R}_n $. They can be rewritten as \[ratio1\] (1-\_[jk]{})\_i z\_i\^[\_i]{} =(\_[i(j,k)]{} z\_i\^[\_i]{})(z\_j+z\_k)(z\_jz\_k) \^[(\_j,\_k)]{}(\_j-\_k) \_[r=0]{}\^[\_j-\_k-1]{}z\_j\^[\_j -\_k-1-r]{}z\_k\^r an... | 2,922 | 3,277 | 3,808 | 2,762 | null | null | github_plus_top10pct_by_avg |
}_1,[\mathcal{O}_2,\cdots,
[\mathcal{O}_{i_0},\cdots,[\mathcal{O}_n,(p-X_0\tilde{p})]]]]\eta\Phi\
\nonumber\\
&\hspace{25mm}
+Q\eta(\xi_0(X_0)^p[\mathcal{O}_1,[\mathcal{O}_2,\cdots,
[\mathcal{O}_{i_0},\cdots,[\mathcal{O}_n,(p-X_0\tilde{p})]]]]\eta\Phi)\,.\end{aligned}$$ Using (\[p tilde p\]), it is easy to show that th... | 2,923 | 1,850 | 1,796 | 2,662 | null | null | github_plus_top10pct_by_avg |
a)$ is expected to explode at affine parameter value $\lambda_e$.
[^4]: This map is defined such that $H: O\times [0,\lambda_0]\to M$ where $O$ is an open in $S$.
---
abstract: 'The initial-value problem is posed by giving a conformal three-metric on each of two nearby spacelike hypersurfaces, their proper-time sepa... | 2,924 | 1,627 | 3,428 | 2,874 | null | null | github_plus_top10pct_by_avg |
rix} \right) }\in D({{{\mathcal{}}}A})$, we have $\psi(E)\in D(A)=\tilde W_{-,0}^2(G\times S)$, for any $E\geq 0$, and thus in particular, $\psi_{|\Gamma_-}=0$. We find by the first row of matrix equation (\[R7\]) that $\psi(0)=T(0)\phi+R(0)f=\phi$.
It might be worth attempting to generalize this method under less res... | 2,925 | 578 | 2,204 | 2,933 | null | null | github_plus_top10pct_by_avg |
igma_D}) + \int_0^{ \sigma_D }{f}(X_s)\,{\rm d}s\right],
\end{aligned}$$ where in the final equality we have used that $\hat{u} = {g}$ on $D^{\rm c}$. Uniqueness now follows. $\square$
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Mateusz Kwaśniki for pointing out ... | 2,926 | 1,291 | 1,750 | 2,733 | null | null | github_plus_top10pct_by_avg |
e believe one should) and, moreover, we are required to use empirical and U-process theory. We adhere to their notation as much as possible.
The first step is to notice that, if we define $\delta_n(t)$ by the equation $$\label{delta}
\delta_n(t)=\frac{\hat f^{1/2}(t;h_{1,n})-f^{1/2}(t)}{f^{1/2}(t)}=\frac{\hat f(t;h_{1... | 2,927 | 1,194 | 2,147 | 2,788 | 3,693 | 0.770624 | github_plus_top10pct_by_avg |
a-1)^2},$$ where the quantity $P_{\pm}$ corresponds to the value calculated for $r_{\pm}$.
From equation (\[s\_grav\_nonrot\_chrg\]) we find that the gravitational entropy is proportional to the area of the event horizon of the black hole, just as in the case of the Bekenstein-Hawking entropy. We can further check the... | 2,928 | 2,885 | 2,666 | 2,873 | null | null | github_plus_top10pct_by_avg |
ier result of the author. Combined with results of Breuillard–Green, Breuillard–Green–Tao, Gill–Helfgott–Pyber–Szabó, and the author, this leads to improved rank bounds in Freiman-type theorems in residually nilpotent groups and certain linear groups of bounded degree.'
address: 'Pembroke College, Cambridge, CB2 1RF, U... | 2,929 | 1,118 | 1,219 | 2,774 | null | null | github_plus_top10pct_by_avg |
gned}
{\dot {\cal F}}_{\Theta_0}(\omega) &=& \frac{\pi g^{2}}{32 {\left(1+b^2 \epsilon^2 \right)}^3} \; \; \frac { e^{\frac{2\omega }{g} \tan^{-1}\left( g|\cos\Theta_{0}|\epsilon \right)}}{\left( e^{\frac{2 \pi \omega}{g}} -1\right) } \nonumber \\
&& \times \bigg\{ \; \; \frac{16 \pi}{3} \left(3 b^2 \epsilon^2 + b... | 2,930 | 4,223 | 2,866 | 2,619 | null | null | github_plus_top10pct_by_avg |
$ for some $e \in E(\wh{M})$, so $N$ is an elementary projection of $M$ if and only if $M = \wh{M} \del e$ and $N = \wh{M} \con e$.
Suppose that $M$ has a $U_{s2^{4s},2s 2^{4s}}$-minor. If $N$ is an elementary projection of $M$, then note that $M$ has a $U_{s+1,s2^{4s}}$-minor $M \con C \del D$. Let $M_0 = \wh{M} \con... | 2,931 | 1,882 | 931 | 2,967 | 3,367 | 0.772847 | github_plus_top10pct_by_avg |
e the imaginary part (see Fig. \[fig.3\] (c)). Here, one can see also that at such choice of the real and imaginary parts of the partial components of the S-matrix the wave function leaves from its zero value at $r=0$.
![The dependence of the radial wave function (\[eq.3.2.1.2\]) from the wave vector $k$ and the param... | 2,932 | 1,499 | 563 | 3,304 | 1,435 | 0.789628 | github_plus_top10pct_by_avg |
be written as $$\rho_{t+1} = (1-p) U \rho_t U^{\dagger} + p \sum_{i} \mathbf{P_i}
U \rho_t U^{\dagger} \mathbf{P_i}$$ where $U$ is the unitary operator of the walk, $i$ runs over the dimensions where the decoherence occurs, and the $\mathbf{P_i}$ project in the usual “computational” basis [@KT03].
In the continuo... | 2,933 | 5,548 | 1,389 | 2,081 | 1,979 | 0.783927 | github_plus_top10pct_by_avg |
a _{2}^{2}$, the adsorption energy between colloids and droplets is much larger than the total repulsive energy. Therefore, we observe mostly closed structures \[Fig. \[fig:hist2\](c)\].
At a larger size asymmetry of $\sigma_{1}=2.0\sigma_{2}$, but at the same interfacial tension $\gamma=40, 100\, k_{\textrm{B}}T/\sig... | 2,934 | 628 | 2,694 | 3,015 | null | null | github_plus_top10pct_by_avg |
rch Foundation of Korea. The work of ECO on this project is supported by grant 510940 from the Simons Foundation. The computation of this work was supported by the Supercomputing Center/Korea Institute of Science and Technology Information with supercomputing resources including technical support (KSC-2018-C3-0015) and... | 2,935 | 2,244 | 3,507 | 2,860 | 2,503 | 0.779284 | github_plus_top10pct_by_avg |
_0 \epsilon _0/\epsilon _1$, and so the relation becomes explicitly, for all value of $(\alpha,\beta)$ : $$\begin{aligned}
\begin{split}
~& \sqrt{- \Bigg[ \Big(6\mathcal{L}_4 -12\mathcal{L}_6-\mathcal{L}_7 \Big)\alpha \big( 5\alpha +18\beta \big) +6 \big( \alpha +3 \beta \big) \Big( \alpha \curv{L}_5 + \beta \curv{L}... | 2,936 | 3,710 | 2,831 | 2,554 | 3,763 | 0.770256 | github_plus_top10pct_by_avg |
e response in the $A$ units, especially for training on faces generated with half of the original principal component standard deviation. The lower standard deviation had a similar effect in the $R$ units, although training on the original rotating faces actually led to a smaller caricature response than the initial we... | 2,937 | 604 | 2,375 | 2,441 | 674 | 0.802604 | github_plus_top10pct_by_avg |
i^T\mathbf{X}_i\mathbf{e}},$$ and notice that $\mathbf{b}_j^T\mathbf{e}=0$ (because $\mathbf{b}_j$ and $\mathbf{e}$ are eigenvectors corresponding to different eigenvalues of $\mathbf{B}$) to produce $$\mathbf{B}_i=\mathbf{B}-\sum_{j=1}^{i-1}\beta_j\mathbf{b}_j\mathbf{b}_j^T=\mathbf{B}_{i-1}-\beta_{i-1}\mathbf{b}_{i-1}... | 2,938 | 1,967 | 762 | 2,795 | 2,897 | 0.776238 | github_plus_top10pct_by_avg |
. The complexity is $O(L\sqrt{P\norm{\h}^2})$ for a given channel vector $\h\in\Rbb^L$ and signal power constraint $P$, and is of average value $O(P^{0.5}L^{1.5})$ for i.i.d. standard Gaussian channel entries.
- For the complex-valued channels, we demonstrate how to apply our method in an efficient way to find the c... | 2,939 | 1,328 | 1,309 | 2,758 | 3,411 | 0.772579 | github_plus_top10pct_by_avg |
rrow\F, x\mapsto x^q$ is the Frobenius endomorphism.
We have $$\log\left(\sum_\muhat G_\muhat(q)m_\muhat\right)=\sum_{d=1}^\infty \phi_d(q)\cdot\log\left(\Omega\left(\x_1^d,\dots,\x_k^d;0,q^{d/2}\right)\right)$$where $\phi_n(q)=\frac{1}{n}\sum_{d|n}\mu(d)(q^{n/d}-1)$ is the number of $\langle f\rangle$-orbits of $\F^{... | 2,940 | 2,343 | 2,108 | 2,600 | null | null | github_plus_top10pct_by_avg |
ta(1,2),\dots,\theta(m-1,m),\theta(m,1))$ of asymptotic directions of the interfaces, given that there are $m$ unbounded trees and assuming that the latter are labeled by following the trigonometric sense.
For this purpose, it is equivalent to study the distribution of the sectors $(\phi(i+1):=\theta(i+1,i+2)-\theta(i... | 2,941 | 3,696 | 2,919 | 2,712 | null | null | github_plus_top10pct_by_avg |
The fact that these filtrations are induced from that of $D({\mathfrak{h}^{\text{reg}}})\ast{{W}}$ ensures that the associated graded object $$\operatorname{{\textsf}{ogr}}B = \bigoplus_{i\geq j\geq 0}\operatorname{{\textsf}{ogr}}B_{ij}$$ is also a ${\mathbb{Z}}$-algebra. Similarly, recall from the ${\mathbb{N}}$-grade... | 2,942 | 2,732 | 2,083 | 2,703 | null | null | github_plus_top10pct_by_avg |
hi(\bar{G'})-k$ “pointed” colors (each given to the vertices corresponding to a set of lines passing through a common point).
Consider now $G$, the disjointness graph of the segments. Let $G_0$ denote the subgraph of $G$ induced by the set of segments whose supporting lines received one of the $k$ planar colors in the... | 2,943 | 4,032 | 3,992 | 2,664 | 3,895 | 0.769433 | github_plus_top10pct_by_avg |
.$$ The state in argument is obtained as a partial inner product of its index state and the maximally entangled state $|\Psi^+\rangle$. The mapping creating the index state from the original state, $$\label{eq:Lpsip}
L_{|\Psi^+\rangle}: {\cal H}_A\rightarrow {\cal H}_B,\quad
L_{|\Psi^+\rangle}|\Psi\rangle_A = |\Ps... | 2,944 | 4,094 | 2,333 | 2,730 | 1,896 | 0.784626 | github_plus_top10pct_by_avg |
omega,E',E) d\omega' dE'
\geq c, \label{csda4aa}\end{aligned}$$ hold for a.e. $(x,\omega,E)\in G\times S\times I$.
Note that if $\sigma_{kj}$ were (cf. Remark \[cosdare1\]) of the form $\sigma_{kj}(x,\omega',\omega,E',E)=\tilde\sigma_{kj}(x,\omega',\omega,E)\delta(E-E')$ then $$\int_{S\times I}\sigma_{kj,C}(x,\omega',... | 2,945 | 548 | 2,246 | 2,800 | null | null | github_plus_top10pct_by_avg |
$3^9, 27$ $9, 29$
$2,2,2,4,6$ $1^8, 5^2$ $9, 13$
$2,2,2,2,6$ $1^9, 9$ $3, 15$
$2,2,2,4,4$ $1^9, 15$ $5,19$
: This table lists combinatorial data for anomaly-free (0,2) GLSM’s describing rank 8 bundles over ${\mathbb Z}_2$ gerbes on Calabi-Yau’s.[]{data-... | 2,946 | 1,570 | 2,680 | 2,621 | null | null | github_plus_top10pct_by_avg |
perature distribution $\theta_{0}(x)$, the inverse problem calculates $\theta_{0}(x)$ from the integral equation $$\theta_{T}(x)=\frac{2}{L}\int_{0}^{a}k(x,x^{\prime})\theta_{0}(x^{\prime})dx^{\prime},\qquad0\leq x\leq L,$$
when this final temperature $\theta_{T}$ is known, and $$k(x,x^{\prime})=\sum_{n=1}^{\infty}\si... | 2,947 | 4,046 | 3,979 | 2,833 | null | null | github_plus_top10pct_by_avg |
ly the role of a proper choice of the index set $\mathbb{D}$ in the description of convergence.
**Example A1.3.** (1) Let $\gamma\in\mathbb{D}$. The eventually constant net $\chi(\delta)=x$ for $\delta\succeq\gamma$ converges to $x$.
\(2) Let $\mathcal{N}_{x}$ be a neighbourhood system at a point $x$ in $X$ and suppo... | 2,948 | 4,402 | 3,369 | 2,792 | 2,161 | 0.782136 | github_plus_top10pct_by_avg |
a)$, where $f: M^{n-k}
\looparrowright \R^n$ is an immersion with the prescribed isomorphism $\Xi: \nu(g) \cong k \kappa$, called a skew-framing, $\nu(f)$ is the normal bundle of $f$, $\kappa$ is the given line bundle over $M^{m-k}$ with the characteristic class $w_1(\kappa)
\in H^1(M^{m-k};\Z/2)$. The cobordism relati... | 2,949 | 2,560 | 2,412 | 2,538 | null | null | github_plus_top10pct_by_avg |
},h_{2,n})-\bar f(t;h_{2,n})-T(t;h_{1;n},h_{2,n})\right|=o_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right)\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C.$$ and in particular, $$\sup_{t\in D_r}\left|\hat f(t;h_{1,n},h_{2,n})-\bar f(t;h_{2,n})\right|=O_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right)\ \ {... | 2,950 | 1,231 | 1,982 | 2,940 | null | null | github_plus_top10pct_by_avg |
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\nonumber \\
&-&
\sum_{m}
\sum_{k \neq l} \sum_{K}
\frac{ 1 }{ ( h_{l} - h_{k} ) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{l}) }
\nonumber \\
&\times&
\biggl[
\left( \Delta_{K} - h_{k} \right) e^{- i ( h_{l} - h_{m} ) x}
- \left( \Delta_{K} - h_{l} \right) e^{- i ( h_{k} - ... | 2,951 | 4,153 | 2,954 | 2,815 | null | null | github_plus_top10pct_by_avg |
8.39 ± 0.91 8.37 ± 0.85 0.846
Post transfusion hematocrit - mean (SD) 26.54 ± 2.96 26.34 ± 2.93 1.000
Post transfusion RBC count - mean (SD) 3.24 ± 0.41 3.09 ± 0.54 0.571
Hemoglobin recheck time after t... | 2,952 | 6,136 | 733 | 1,197 | null | null | github_plus_top10pct_by_avg |
full virtual channel matrix. The low-dimensional virtual channel matrix is defined by $$\label{eq:sHv1}
{\widetilde{\mathbf{H}}_{\psi,V}}=\left[{\mathbf{H}_{\psi,V}}(i,j)\right]_{i\in{\mathcal{M}_{\psi,r}},j\in\mathcal{M}_{\psi,t}},$$ where $\mathcal{M}_{\psi,r}=\left\{i:(i,j)\in\mathcal{M}_{\psi}\right\}$, $\mathcal... | 2,953 | 1,975 | 2,668 | 2,779 | 2,482 | 0.779405 | github_plus_top10pct_by_avg |
[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture... | 2,954 | 1,457 | 1,412 | 2,943 | 186 | 0.819898 | github_plus_top10pct_by_avg |
limits_{n\in N}}
A_{n}\right \Vert _{X}^{\ast}\right) .$$
Now, we prove the following result:
Let $1<p<\infty$ and $q=p/(p-1)$. If $A\in(\ell_{p}(\widehat{F}),\ell_{1})$, then$$\lim_{m}\left \Vert A\right \Vert _{(\ell_{p}(\widehat{F}),\ell_{1})}^{(m)}\leq \left \Vert L_{A}\right \Vert _{\chi}\leq4.\lim_{m}\left \Ve... | 2,955 | 1,860 | 2,315 | 2,595 | null | null | github_plus_top10pct_by_avg |
lassical Rabi frequency, $\hat a$ the annihilation operator for the CM motion, $\hat \sigma_z = | e \rangle \! \langle e | - | g \rangle \! \langle g |$, $\hat\sigma_+ = \hat\sigma^\dag_- = | e \rangle \! \langle g | $, and $\eta$ the Lamb-Dicke parameter defined as $$\begin{aligned}
\label{etatrue}
\eta=\frac{\omega_... | 2,956 | 3,176 | 3,293 | 2,878 | null | null | github_plus_top10pct_by_avg |
long the direction $\alpha$.
Our focus here on the renormalized value $u_r$ of the Josephson coupling $u$ in the SP regime. If the periodic boundary conditions are also imposed perpendicular to the layers (along $z$-direction), the inter-layer response $u_r$ is given by windings $W_z$ along $z$-direction: u\_r=W\^2\_z... | 2,957 | 1,617 | 3,609 | 3,104 | 4,146 | 0.767779 | github_plus_top10pct_by_avg |
}^q{\vbu_1-v_1{|\!|\!|}}^q}&\bigg(
\prod_{i=1}^{j-1}\frac{O(\theta_0)}{{\vbv_{i+1}-u_i{|\!|\!|}}^q{\vbu_{i+1}-
v_{i+1}{|\!|\!|}}^q{\vbu_{i+1}-v_i{|\!|\!|}}^{2q}}\bigg){\nonumber}\\
&\times\frac{O(\theta_0)}{{\vbx-u_j{|\!|\!|}}^q{\vbx-v_j{|\!|\!|}}^{2q}}\qquad(j\ge2).\end{aligned}$$ First, we consider the sum over $u_... | 2,958 | 3,061 | 1,708 | 2,665 | null | null | github_plus_top10pct_by_avg |
@SugiyamaDIE06; @SugiyamaADE07]. For instance, an iteration lemma due to Kowalczyk [@Kowalczyk05] may be extended easily to the case ${\mathbb R}^d$, $d \geq 2$ and to include the ${\nabla}\cdot (\eta \theta)$ term in [@CalvezCarrillo06].
Fix $p > d$. Then by Lemma \[lem:finite\_p\_bounded\], for sufficiently small $... | 2,959 | 2,055 | 433 | 3,039 | null | null | github_plus_top10pct_by_avg |
ties to bind the contract in the next section. Here, we only note that we do not discuss cases when after step $m$ both clients want the same: if the protocol is fair for the cases when clients’ wishes are opposite (a conservative assumption), it will be when they wish the same.
Averaging over all possible Bob’s strat... | 2,960 | 3,580 | 3,227 | 2,748 | 3,080 | 0.774985 | github_plus_top10pct_by_avg |
}$ and $\{\dot xxyz\}$.
Homomorphic images of special Jordan dialgebras
-----------------------------------------------
In this section we construct the example of an exceptional two-generated Jordan dialgebra which is a homomorphic image of a special Jordan dialgebra.
Denote by $\widehat I$ the ideal of ${\mathrm{D... | 2,961 | 1,939 | 1,351 | 2,964 | null | null | github_plus_top10pct_by_avg |
nciple inquires both into mechanics of transduction, and how transducible values come into being. The automaton of the software mechanism answers the latter question. If software hazard is to be evaluated starting at points of transduction and proceeding backwards through internal logic, then the automaton must support... | 2,962 | 964 | 2,461 | 2,570 | null | null | github_plus_top10pct_by_avg |
the following formula: $$\Theta_{\Z/2 \int \D_4}^k(h,\Psi,\zeta) = <w_2(\bar \eta)^{\frac{n-4k}{2}};[L^{n-4k}]>.$$ $$$$
This new invariant is a homomorphism $\Theta_{\Z/2 \int \D_4}^k:
Imm^{\Z/2 \int \D_4}(n,n-4k) \to \Z/2$ included into the following commutative diagram:
$$\begin{array}{ccc}
Imm^{\D_4}(n-2k,2k) & \... | 2,963 | 2,947 | 2,707 | 2,463 | null | null | github_plus_top10pct_by_avg |
nteractions. A common approach for the QCD dynamics in $\gamma \gamma$ and $\gamma h$ interactions is important to minimize the theoretical uncertainty and to perform a realistic comparison between the predictions of the two different mechanisms for the double vector production. In order to describe the vector meson pr... | 2,964 | 1,490 | 3,240 | 2,797 | 4,044 | 0.768481 | github_plus_top10pct_by_avg |
11.0 (4.0--17.0) 5.0 (5.0--6.0) 16.0 (16.0--17.0)
Glucose mmol/L 191 5.8 (2.5--12.9) 3.7 (3.1--4.1) 9.1 (8.3--10.2)
Aspartate aminotransferase ... | 2,965 | 3,743 | 2,034 | 2,659 | null | null | github_plus_top10pct_by_avg |
47 NONSYN A:5 C:164 C:37 `Tgagatgataat`
... | 2,966 | 1,888 | 3,558 | 2,920 | null | null | github_plus_top10pct_by_avg |
ak and the step-like feature indicate the same energy scale – we can thus conclude that the step is also a manifestation of the Kondo effect that leads to the peak recorded at the CH edge. Table \[tab:fwhm\_and\_q\] also reveals that $q$ is significantly smaller in the center of the molecule, indicating that there the ... | 2,967 | 3,688 | 3,619 | 3,089 | null | null | github_plus_top10pct_by_avg |
- 1$. Since $k$ divides this size, there must be an element in $\F_{p^w}$ of order $k$.
Review of $O(n^{1/3})$ cost 2-server PIR {#oldconstruction}
========================================
There are several known constructions of 2-server PIR with $O(n^{1/3})$ communication cost. We will recall here in detail a parti... | 2,968 | 2,658 | 3,286 | 2,918 | 1,371 | 0.790368 | github_plus_top10pct_by_avg |
he parameters $c_m$ are also selected randomly from a Gaussian distribution with zero mean and a variance we will specify. Under standard manipulations (Born, Markov and secular approximations), we find a master equation diagonal in the basis of eigenstates $\dot{P}_i = (\mathbb{W})_{ij} P_j$, where $P_i$ is the occupa... | 2,969 | 2,183 | 3,228 | 2,908 | 2,204 | 0.781852 | github_plus_top10pct_by_avg |
ed volatility for these portfolios, on the other hand, need not (and in typical cases will not) diverge at all.
As stated earlier, the efficient frontier in the presence of a riskless asset has a simple allocation rule which requires that each principal portfolio be included in inverse proportion to its variance. For ... | 2,970 | 1,318 | 3,223 | 2,799 | null | null | github_plus_top10pct_by_avg |
m more accurate starburst diagnostics than a forbidden/recombination pair like \[\]/. To reduce reddening effects, we select lines close in wavelength to H lines. Unfortunately, the helium lines are weak: [ $6678$]{} saturates at $0.014$ of the strength of , and [ $4471$]{} saturates at $0.05$ of . As such, in the sp... | 2,971 | 3,337 | 2,874 | 2,695 | null | null | github_plus_top10pct_by_avg |
\delta_a \mid a \in G\}$ is an orthonormal basis of $\ell^2(G)$. Then $\widehat{\delta_a}(\chi) = \chi(a)$ for each $\chi \in \widehat{G}$. By Lemma \[T:FT-isometry\](\[I:isometry\]), the number of $a \in G$ such that $a^2 = 1$ is equal to $$\begin{aligned}
\sum_{a \in G} {\left\langle \delta_a, \delta_{a^{-1}} \r... | 2,972 | 5,199 | 1,504 | 2,195 | null | null | github_plus_top10pct_by_avg |
74.0 ± 1.2 69.0 ± 1.3 100.8 ± 2.2 108.7 ± 1.1
Genotype 0.105 0.476 \<0.001 0.812 \<0.001 0.963
Date \<0.001 ... | 2,973 | 1,191 | 2,998 | 3,070 | null | null | github_plus_top10pct_by_avg |
he quantum-classical dynamics of operators is transformed into a theory for phase space dependent wave fields evolving in time. Such a theory for wave fields is also expressed by means of suitable non-Hamiltonian brackets: in this way a link is found with the generalization of Weinberg’s non-linear formalism given in A... | 2,974 | 1,918 | 1,497 | 2,623 | null | null | github_plus_top10pct_by_avg |
d we find that the discharge rate is fast if the WGC is satisfied by a light particle of mass $m\ll1/M$. For sufficiently large black holes $M\gtrsim {\rm max}(e^{|\Delta\phi|},1/m)$ the rate is slow.
We conclude with a loose conjecture: quantum gravity in asymptotically flat space requires a general bound on large lo... | 2,975 | 1,587 | 1,762 | 2,949 | null | null | github_plus_top10pct_by_avg |
to be 0.089 events/Mton$\cdot$yr; we take double this value (0.18 $\pm$ 0.18 events/Mton$\cdot$yr) as a conservative estimate of the background rate for this decay mode. Similarly, we extrapolate for all of the dinucleon decay modes, finding background rates of 0.008 ($NN \rightarrow ee$), 0.033 ($NN \rightarrow e \mu$... | 2,976 | 360 | 3,633 | 2,989 | null | null | github_plus_top10pct_by_avg |
rom \[prop:partition\], \[prop:ysmall\], and \[prop:ylarge\]) $|\cR\setminus\cR^*|\le 2$ and (from \[prop:bi\]) $|\cR^*|\le 2$. But this means that $|\cR|\le 4< 5+|Y|$, a contradiction. Therefore we know that if $|Y_x|\ge 2$ then $Y_{\hat{x}}=\emptyset$.\
If, for some $x\in X^*$, we have $|Y_x|=|Y_{\hat{x}}|=1$ (we may... | 2,977 | 1,470 | 1,659 | 2,924 | null | null | github_plus_top10pct_by_avg |
sum_{i,j}(\mu^i_j)^2+2$.
We have
$$\Log\left(\sum_\muhat q^{-\frac{1}{2}(d_\muhat-2)}V_\muhat(q)m_\muhat\right)=\frac{q}{q-1}\sum_\muhat A_\muhat(q)m_\muhat.$$ \[theohua\]
By Lemma \[moz\] and Formula (\[ExpA\]) we are reduced to prove the following.
We have $$\log\,\left(\sum_\muhat q^{-\frac{1}{2}(d_\muhat-2)}V_... | 2,978 | 1,887 | 2,148 | 2,614 | null | null | github_plus_top10pct_by_avg |
); // returns nothing
}
}
}
What am I doing wrong?
A:
I found the answer in the Raven Google group. It turns out I have to query with the same data type as the data. So, in this case since "Price" is a decimal, I have to pass in 60M to the where clause:
var gt = session.Advanced.LuceneQuery<Asset, A... | 2,979 | 2,867 | 125 | 2,021 | 2,452 | 0.779602 | github_plus_top10pct_by_avg |
a_{o,x}|\leq\theta\delta_{o,x}+\frac{\lambda(1
-\delta_{o,x})}{|x|^{d+2+\rho}},\end{aligned}$$ for any $p\leq{p_\text{c}}$ and any $x\in{{\mathbb Z}^d}$, where $(f*g)(x)=\sum_{y\in{{\mathbb Z}^d}}f(y)\,g(x-y)$. We note that the identity in [(\[eq:Ising-lace-Zdlim\])]{} is similar to the recursion equation for the rand... | 2,980 | 1,924 | 2,500 | 2,725 | 2,604 | 0.778511 | github_plus_top10pct_by_avg |
as ΔCt mean ± SEM are shown in [Table 3](#pone.0214536.t003){ref-type="table"} (raw data are shown in [S2](#pone.0214536.s002){ref-type="supplementary-material"} and [S3](#pone.0214536.s003){ref-type="supplementary-material"} Tables). The results showed that the expression levels of the Hbl and Nhe toxin genes were co... | 2,981 | 439 | 1,643 | 2,934 | null | null | github_plus_top10pct_by_avg |
ligned}
&H_0\, = T\,\partial_T - R\,\partial_R, \\ \nonumber
&H_+ = \partial_T, \\ \nonumber
&H_- = (T^2 + \frac{1}{R^2})\,\partial_T - 2\,TR\,\partial_R
- \frac{2}{R}\,\partial_\Phi, \\ \nonumber
&Q_0\,\, = \partial_\Phi.\end{aligned}$$ $H_0$ is the infinitesimal generator of *dilation*, whic... | 2,982 | 5,271 | 1,222 | 2,323 | null | null | github_plus_top10pct_by_avg |
.41)
Number of friends in the Netherlands (log) −0.06 (0.04) 0.08 (0.08)
Number of family members in the Netherlands (log) 0.03 (0.06) ... | 2,983 | 542 | 2,457 | 2,901 | 454 | 0.809399 | github_plus_top10pct_by_avg |
a broad resonance. Specifically, the realization using a CI resonance in a tight waveguide requires a sufficiently dilute gas with $na_{\perp}\ll 1$. Taking typical values of order 50 nm for the transverse oscillator length which have been realized very recently in bosonic 1D gases [@Esslinger; @BlochKinoshita], this r... | 2,984 | 889 | 2,793 | 2,824 | null | null | github_plus_top10pct_by_avg |
(s+\lambda^{\varphi,n}_\epsilon(s)\frac{X_{1}}{\sqrt{n}})]\ge\mathbb{E}[\varphi(s+\frac{\xi_{1}}{\sqrt{n}})]-\epsilon.$$ For any $\sigma\in\Sigma^{\mathbb{N}}_G$ with $\sigma_{i+1}(s)=\lambda^{\varphi,n}_\epsilon(s)$, we have $$E[\varphi(W^{\sigma}_{i+1,n})]=E[E[\varphi(s+\sigma_{i+1}(s)\frac{X_{i+1}}{\sqrt{n}})]\big|_... | 2,985 | 1,903 | 1,655 | 2,889 | null | null | github_plus_top10pct_by_avg |
Csiszar-Kullback inequality [@Csiszar67; @Kullback67] relates the relative entropy to the $L^1$ norm.
\[thm:CK\] Let $f\in L_+^1({\mathbb R}^d)$ with ${\|f\|}_1 = M$ and let $\theta_M$ be the ground state Barenblatt solution with mass $M$. Then, $${\|f - \theta_M\|}_1 \lesssim H(f|\theta_M)^{\min\left(\frac{1}{2},\fr... | 2,986 | 1,966 | 833 | 3,053 | 2,813 | 0.776801 | github_plus_top10pct_by_avg |
D\right\}=0=\left\{D^\dagger,D^\dagger\right\}\ \ ,\ \
\left\{D,D^\dagger\right\}=\left(-\frac{2}{\sqrt{\hbar\omega}}\right)^2\,
\left(i\hbar\partial_t\right)\ ,$$ as well as the required properties $$\left\{Q,D\right\}=0\ ,\
\left\{Q,D^\dagger\right\}=0\ ,\
\left\{Q^\dagger,D\right\}=0\ ,\
\left\{Q^\dagger,D^\dagger\... | 2,987 | 1,388 | 3,134 | 2,808 | null | null | github_plus_top10pct_by_avg |
60*b**3/3 - 110*b**2 - 858*b. Suppose v(l) = 0. Calculate l.
-22, 1, 2
Suppose -73*t = -4*m - 68*t - 17, -2*m = t - 9. Let y(k) be the first derivative of -1/8*k**4 + 0*k - 1/6*k**3 + 0*k**m + 15. Solve y(d) = 0.
-1, 0
Let x(w) = 20*w - 240. Let z be ((-24)/(-10))/(14/(-280)*-4). Let r be x(z). Factor 0*m + 2/9*m**3 + ... | 2,988 | 1,317 | 2,609 | 2,745 | null | null | github_plus_top10pct_by_avg |
\_linear\]). Since the force formulae require to obtain the condensate density in a neighborhood of the particle position, it is convenient to move to a coordinate frame with center always at the (possibly moving) particle location ${\boldsymbol{r}}={\boldsymbol{r}}_p(t)$. Thus we change variables from $({\boldsymbol{r... | 2,989 | 2,111 | 2,706 | 2,870 | null | null | github_plus_top10pct_by_avg |
$and given $h\in {\mathbb{N}}$ we denote $$\rho _{h}=\frac{(a+b)m_{0}+q+2d/p_{\ast }}{2h}. \label{H5'}$$Notice that this is equal to the constant $\rho _{h}$ defined in (\[reg5\]) corresponding to $k=(a+b)m_{0}$ and $q$ and to $2d$ (instead of $d).$
**Step 1: a Lindeberg-type method to decompose $P_t-P^n_t$.** We fix... | 2,990 | 1,025 | 1,196 | 3,230 | null | null | github_plus_top10pct_by_avg |
rem \[evoth1\] are valid, and $\tilde{\sigma}\geq 0$. Let $f\in H^2(I,L^2(G\times S))$ and $g\in H^3(I,T^2(\Gamma_-))$ which satisfies the compatibility condition g(E\_m)=0. Then the problem (\[se1\]), (\[se2\]), (\[se3\]) has a unique solution $\psi\in C(I,\tilde W^2(G\times S))\cap C^1(I,L^2(G\times S))$.
If in addi... | 2,991 | 1,205 | 1,778 | 2,941 | null | null | github_plus_top10pct_by_avg |
,\tau_3$.
We can apply a similar argument in which we consider ${\psi_{3,1}}\circ{\hat\Theta_{T}}$ for $T\in{{\calt_{\hspace{-2pt}0}}(\mu,\la)}$. Again ${\psi_{3,1}}\circ{\hat\Theta_{T}}$ is either zero or a semistandard homomorphism; and if it is non-zero, then the only other $T'$ having ${\psi_{3,1}}\circ{\hat\Theta... | 2,992 | 1,888 | 1,040 | 3,000 | 3,158 | 0.774429 | github_plus_top10pct_by_avg |
m{2s}{s-1} \ge 4^{-s}n + s$ (this last inequality follows from $n \ge 4s$ and $\binom{2s}{s-1} \le \tfrac{1}{2}4^s$). Let $Y$ be a basis for $B''$ in $M \con A'$ and let $B_1 = B''-Y$, so $|B_1| \ge 4^{-s}n \ge 4^{s(t-1)}$. Let $M' = (M \con Y)|(A \cup B_0 \cup B_1).$ Note that, since they are bases for $M$, both $A$ a... | 2,993 | 2,081 | 2,389 | 2,691 | null | null | github_plus_top10pct_by_avg |
002ny] the value of this parameter is quantised according to $l_k=1-(2k)^{-1} \geq 1/2 , \ k \in \mathbb{N} $. For the further investigations we choose the representative value $l=3/4$. In the semi-classical region $ a_* \gg a \gg a_i$ expression (\[correction\]) simplify to the form $$D=D_*a^n$$ where $$D_* = \left( ... | 2,994 | 3,365 | 3,138 | 2,863 | null | null | github_plus_top10pct_by_avg |
at(q)=q^{-\frac{d_\muhat}{2}}PH_c(\M_\muhat;q),$$where $PH_c(\M_\muhat;q):=\sum_ih_c^{i,i;2i}(\M_\muhat)q^i$ is the *pure part* of $H_c(\M_\muhat;q,t)$. \[purconj\]
Conjecture \[purconj\] implies Kac’s conjecture [@kacconj] for comet shaped quivers, namely, $A_\muhat(q)$ is a polynomial in $q$ with non-negative coeffi... | 2,995 | 1,560 | 2,438 | 2,582 | 3,012 | 0.775448 | github_plus_top10pct_by_avg |
than the other parameters. In particular, we find it necessary to impose uniform bounds on the largest and smallest eigenvalues of the covariance matrices of all $k$ marginals of the $d$ covariates, as well as bounds on the higher moments of $X$ and on the mixed moments of $X$ and $Y$. We will further assume, in most ... | 2,996 | 1,066 | 2,406 | 2,668 | null | null | github_plus_top10pct_by_avg |
a generic Weil divisor on $(X,P)$ of class $k$, that is, a generic germ in $\cO_{X,\zeta^k}$. In [@jiJM-correction] it is shown that $$A_{X,P}(D)=\delta^{\operatorname{top}}(D)-\kappa_P(D).$$ When applied this formula for generic germs one obtains a combinatorial way to calculate $A_{X,P}(D)$, that is, $$A_{X,P}(D)=\d... | 2,997 | 2,360 | 1,989 | 2,898 | 2,614 | 0.778426 | github_plus_top10pct_by_avg |
can compute the tadpole conditions for the $6_3^1$ and $4_3^{3,2}$-branes in the IIA/O6 theory.
As an interesting application of our results, we now consider the IIB/O3 theory for the particular case in which $P^{1,4}=0$, and look at all the constraints related to $P_1^2 \cdot Q$ in the presence of exotic branes. From... | 2,998 | 1,286 | 1,653 | 2,987 | null | null | github_plus_top10pct_by_avg |
[@gopi.volume]. That work finds that the cumulative distribution function of traded volume for time windows of $\Delta t = 15$ minutes decays as a power-law with a tail exponent $\lambda = 1.7 \pm 0.1$ for a wide range of stocks. This is the so called *inverse half cube law*, and it can be written as $${\mathbb P}_{\De... | 2,999 | 672 | 3,146 | 2,857 | null | null | github_plus_top10pct_by_avg |
then the eigenvalues of $M$ are precisely the values $\bigl\{\widehat{f}(\chi) \mid \chi
\in \widehat{G}\bigr\}$ of the Fourier transform of $f$, and the characters of $G$ are eigenvectors of $M$. (For generalizations of these facts for nonabelian $G$, see [@Diaconis-book; @Diaconis-matrices].) Observe that every $G$-c... | 3,000 | 3,140 | 2,694 | 2,639 | null | null | github_plus_top10pct_by_avg |
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