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 |
|---|---|---|---|---|---|---|---|
_1(S)g_2(S)\}=g_2(S)\Dc_S g_1(S)+g_1(S) \Dc_S g_2(S).$$
Denote by $S=HLH^\top$ the eigenvalue decomposition of $S$, where $H=(h_{ij})$ is an orthogonal matrix of order $r$ and $L=\diag(\ell_1,\ldots,\ell_r)$ is a diagonal matrix of order $r$ with $\ell_1\geq \cdots\geq \ell_r$. The following lemma is provided by Stein... | 1,501 | 1,458 | 1,005 | 1,395 | 1,913 | 0.784437 | github_plus_top10pct_by_avg |
of an STM, the TAMR can be obtained by measuring the differential conductance (d$I$/d$V$) above an adatom or a dimer for two different magnetization directions. The TAMR is obtained from $$\begin{aligned}
\mathrm{TAMR} &= \frac{[(\mathrm{d}I/\mathrm{d}V)_\perp-(\mathrm{d}I/\mathrm{d}V)_\parallel]}{(\mathrm{d}I/\mathrm... | 1,502 | 473 | 1,633 | 1,765 | null | null | github_plus_top10pct_by_avg |
(1)}_{m,n}}{P^{(2)}_{m,n}} - 1,\end{gathered}$$ where $P^{(i)}_{m,n}$ are given in (\[eq:N5-P-G\]), maps system (\[eq:sys-5-red\]) to $$\begin{gathered}
\frac{\psi^{(0)}_{m,n+1}\psi^{(0)}_{m+1,n+1}+\psi^{(0)}_{m+1,n} \big(\psi^{(0)}_{m,n}+\psi^{(1)}_{m,n+1}\big)}{\psi^{(1)}_{m,n}+1} = \frac{\psi^{(0)}_{m,n+1}\psi^{(1)}... | 1,503 | 2,829 | 1,548 | 1,523 | null | null | github_plus_top10pct_by_avg |
an age was 34 years and participants had a mean body mass index (BMI) of 24.1 ± 2.78 kg/m^2^. A summary of the baseline characteristics of the participants is presented in [Table [1](#tbl1){ref-type="other"}](#tbl1){ref-type="other"}.
###### Demographic Data of Participants Included in the Study[a](#t1fn1){ref-type="t... | 1,504 | 197 | 2,761 | 1,732 | null | null | github_plus_top10pct_by_avg |
bottom panel of Android Studio). Based on the build variant you choose, you will have the BASE_WEB_URL correctly updating to the right value.
In your code you simply need to replace all your "127.0.0.1" and "10.0.2.2" with
BuildConfig.BASE_WEB_URL
Q:
Where Should Exception Messages be Stored
Since I can't use Micr... | 1,505 | 709 | 850 | 1,081 | null | null | github_plus_top10pct_by_avg |
ix instead of the Laplacian [@GW03]; the dynamics are then given by the unitary operator $U_t = e^{iHt}$ and the state of the walk at time $t$ is $\Psi_t = U_t \Psi_0$.
The following analysis makes use of the hypercube’s product graph structure; this structure will be useful again later when we consider the effects of... | 1,506 | 5,129 | 1,513 | 1,163 | 2,005 | 0.783597 | github_plus_top10pct_by_avg |
hen $$\begin{aligned}
d A_t^i A_t^j
=& d A_t^T \lrp{e_i e_j^T} A_t\\
=& A_t \lrp{e_i e_j^T} \lrp{\cm dV_t + M(x_s)^{-1} N(x_s) dW_s}^T + \lrp{\cm dV_t + M(x_s)^{-1} N(x_s) dW_s} \lrp{e_j e_i^T} a_t^T\\
&\qquad + \frac{1}{2} \tr\lrp{ \lrp{e_i e_j^T + e_j e_i^T}\lrp{c_m^2 M(x_s)^{-2} + M(x_s)^{-1} N(x_s)^... | 1,507 | 4,214 | 1,039 | 1,116 | null | null | github_plus_top10pct_by_avg |
the learning efficiency curves (Fig.\[fig3\]) up to cost $C_{min}$ and denote them as $S_n$ (area under the curve of number of characters v.s. cost like the ones in Fig. \[fig3\]a) and similarly $S_f$ (area under the curve of accumulated usage frequency v.s. cost like those in Fig. \[fig3\]b), respectively. The ratio b... | 1,508 | 484 | 1,305 | 1,248 | 2,498 | 0.779332 | github_plus_top10pct_by_avg |
res more computation labor. Although the LLL LR algorithm has known average complexity for some cases [@Daude1994; @Ling2007], its average complexity for our case is unknown, and the worst-case complexity could be unbounded [@Jalden2008].
- The quantized search (QS) method developed by Sakzad [*et al.*]{} in [@Sakza... | 1,509 | 574 | 907 | 1,302 | null | null | github_plus_top10pct_by_avg |
lphahat|_{Z_{(n)}}$ is trivial (the center $Z_{(n)}\simeq \F_q^{\times}$ consists of scalar matrices $a.I_n$).
We can show as for conjugacy classes [@hausel-letellier-villegas Lemma 2.1.2] that if the characteristic $p$ of $\F_q$ and $q$ are sufficiently large, generic tuples of irreducible characters of a given type ... | 1,510 | 1,064 | 1,649 | 1,392 | 4,080 | 0.768224 | github_plus_top10pct_by_avg |
n reboot instead of immediately, which would disconnect the provisioner.
With the above script in place, add something like this just below the trigger definitions that we added earlier:
config.vm.provision "shell", path: "./scripts/configure-static-ip.sh"
config.vm.provision :reload
4. Provision the VM
vagrant up a... | 1,511 | 3,112 | 255 | 1,570 | null | null | github_plus_top10pct_by_avg |
ein could be cannulated (red arrow) at a point close to the center of the left renal vein (white arrow) using CT guidance. Because the catheter was wedged, requiring a change in catheter to obtain a blood sample (B).](CCR3-5-482-g003){#ccr3875-fig-0003}
######
Adrenal venous sampling results after adrenocorticotropi... | 1,512 | 389 | 1,795 | 1,907 | null | null | github_plus_top10pct_by_avg |
{\rm sth}) &= \frac{1}{4\pi G (\delta y)^2} \Psi|_{j=1}, & \sigma_{i,k}({\rm nth}) &= \frac{1}{4\pi G (\delta y)^2} \Psi|_{j=N_y},\end{aligned}$$ where $\sigma_{i,j}({\rm bot})$ denotes the screening charge on the bottom boundary, etc. Note that the screening charges have units of mass density rather than surface densi... | 1,513 | 3,053 | 2,367 | 1,420 | null | null | github_plus_top10pct_by_avg |
ht\Vert}_{W^2(G\times S\times I)} =0$. In addition the trace mapping $\gamma_-:W^2(G\times S\times I)\to \ L^2_{\rm loc}(\Gamma_-,|\omega\cdot\nu|\ d\sigma d\omega dE)$ such that $\gamma_-(\psi)=\psi_{|\Gamma_-}$ is continuous. Similarly one has a continuous trace mapping $\gamma_+:W^2(G\times S\times I)\to \ L^2_{\rm ... | 1,514 | 451 | 1,492 | 1,570 | null | null | github_plus_top10pct_by_avg |
the complex conjugation of $H^1(\tilde{X},\cO_{\tilde{X}})$. The following result computes the Hodge structure of $H^q(\tilde{X},\cO_{\tilde{X}})$ providing the decomposition in terms of its invariant subspaces.
\[thm:Esnault\] Under the previous notation, assume the ramification set is given by a simple $\Q$-normal c... | 1,515 | 1,452 | 1,415 | 1,444 | null | null | github_plus_top10pct_by_avg |
R^{\mu\nu\alpha\beta} R_{\mu\nu\alpha\beta} =R T
\\ \mathcal{L}_5=R^{\mu\nu\alpha\beta}R_{\mu\alpha}R_{\nu\beta} \;\;\quad\;\; , \;\;\quad \mathcal{L}_6=R^{\mu\nu}R_{\nu\alpha}R^{\alpha}_{\;\,\mu}
\\ \mathcal{L}_7=R R^{\mu\nu} R_{\mu\nu} = R S \;\,\, , \,\;\quad \mathcal{L}_8=R^3
\end{array}
\right.
$\
\
\
$\le... | 1,516 | 792 | 1,112 | 1,403 | null | null | github_plus_top10pct_by_avg |
gradeName1 = "Craftmanship Lvl 1";
var UpgradePrice1 = 100;
var UpgradeContent1 = "+100% Gods Hands! <br> +20% Blessed Farmer!";
var Upgrade1 = 0;
var UpgradeName2 = "Craftmanship Lvl 2";
var UpgradePrice2 = 200;
var UpgradeContent2 = "+100% Gods Hands! <br> +20% Blessed Farmer!";
var Upgrade2 = 0;
And so on...
If i... | 1,517 | 524 | 312 | 932 | 78 | 0.82785 | github_plus_top10pct_by_avg |
(w_i,w_i) \in (4)$, where $w_i \in W_i\otimes_{A}R$, and $i=2m$.
5. $f(B_i, B_i^{\perp})\in B\otimes_{A}R$ and $f(a_i, b_i^{\prime})-h(a_i, b_i^{\prime})\in B\otimes_{A}R$, where $a_i\in A_i\otimes_{A}R$ and $b_i^{\prime}\in B_i^{\perp}\otimes_{A}R$, and $i$ is odd.
**
We interpret the above conditions in terms of ... | 1,518 | 1,067 | 1,270 | 1,483 | 3,647 | 0.770946 | github_plus_top10pct_by_avg |
e MI-phase as $v$ is varied ($F=0$, $L=N=7$).[]{data-label="fig1"}](fig1.eps){width="8cm"}
We proceed with the multi-particle case. A natural extension of the tight-binding model (\[1\]), which accounts for the repulsive interaction of the atoms, is given by the Bose-Hubbard model [@Fish89], $$H=-\frac{J}{2}\left(\sum... | 1,519 | 1,340 | 2,032 | 1,704 | 2,276 | 0.781207 | github_plus_top10pct_by_avg |
24 months 57.36 67.03 63.69 73.83 67.37 62.79 66.63 61.97 64.99 68.24
36 months 68.88 70.10 70.11 72.62 69.05 71.94 70.16 69.38 69.28 72.72
*p* 0.12 0.03 0.71 0.12 0.89 ... | 1,520 | 5,848 | 479 | 516 | null | null | github_plus_top10pct_by_avg |
al{L}_6 -7 \mathcal{L}_7\Big) }=\frac{3}{\sqrt{2}} \sqrt{- \curv{L}_8} + T \; \, \text{,}\end{aligned}$$ such that we can in fact consider a unique scalar (let us choose $\sqrt{- \curv{L}_8}$) made of order 6 scalars that leads to non vanishing second order differential equations. We can note that $T$ cannot be equal t... | 1,521 | 2,148 | 2,052 | 1,643 | null | null | github_plus_top10pct_by_avg |
The patients were classified into two groups on the basis of their average *CLU* mRNA levels. Chi-square test indicated that *CLU* level was closely related to tumor stage (*P* = 0.006) and lymph node metastasis (*P* = 0.002), but not to gender, age, tumor size, serum AFP, vascular invasion, cirrhosis or recurrence (*P... | 1,522 | 462 | 1,412 | 1,556 | null | null | github_plus_top10pct_by_avg |
ts,a_8,$ $b_1,\ldots,b_8$ and computing the coefficients $c(\alpha)$ defined in Sec. II. More precisely, for each example we write all probabilities entering the sums $S_1$, $S_2$, $S_3$ as the marginals of joint probability distribution. As a result we obtain the expressions of the form $$S=\sum_\alpha c(\alpha)p(\alp... | 1,523 | 5,248 | 1,216 | 975 | null | null | github_plus_top10pct_by_avg |
te any of the spaces $\ell_{\infty},$ $\ell_{1}$ or $\ell_{p}$. Then, we have $\left \Vert
a\right \Vert _{X}^{\ast}=\left \Vert a\right \Vert _{X^{\beta}}$ for all $a\in
X^{\beta}$, where $\left \Vert .\right \Vert _{X^{\beta}}$ is the natural norm on the dual space $X^{\beta}$.
\[[\[3, Theorem 1.23 (a)\]]{}\]Let $X$... | 1,524 | 790 | 1,165 | 1,526 | null | null | github_plus_top10pct_by_avg |
ance matrix estimation, it is all that is needed to apply the Gaussian comparison theorem \[thm:comparisons\], which will allow us to extend the Berry-Esseen bound established in to the case when $\Gamma$ is estimated.
\[lemma::upsilon\] Let $$\label{eq:aleph}
\aleph_n = \max \Big\{ \overline{H} B \overline{v} \sqrt{ ... | 1,525 | 2,239 | 872 | 1,511 | 3,798 | 0.770046 | github_plus_top10pct_by_avg |
ion $(1)$ of Lemma $\ref{lem:1.1}$, we have the following corollary:
\[cor:1.2\] We have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z)] - \operatorname{{E}}[\Pe(Z + C)]
&= \frac{(k_{1} + k_{2}) b}{2 \G(a)}
\g\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right), \\
\frac{\operatorname{{E}}[\Pe(Z + C)]}... | 1,526 | 1,917 | 1,753 | 1,469 | 3,130 | 0.774659 | github_plus_top10pct_by_avg |
2.4,1)(1.9,0.5)(1.5,0.9)
\end{pspicture}$$ Here $\tau_{k+l+2}^{l+1}$ denotes the $(l+1)$st iteration of $\tau_{k+l+2}$, and $\epsilon=(|m^*_1|+|a^*_1|+\dots+|a^*_k|)\cdot
(|m_2|+|a^*_{k+1}|+\dots+|a^*_{k+l}|)$.
A straightforward check shows that if $d^2=0$ and $g^2=0$, then it is also $h^2=0$.
The following propositi... | 1,527 | 1,761 | 1,489 | 1,459 | 2,356 | 0.7806 | github_plus_top10pct_by_avg |
cal{P}}= \rho_{\ast}{\mathcal{O}}_{X_n}$\[PP-defn\] is the [*Procesi bundle*]{} on $\operatorname{Hilb(n)}$ of rank $n!$ arising from the map $\rho : X_n \to \operatorname{Hilb(n)}$ while ${\mathcal{L}}$\[LL-A-defn\] is the canonical ample line bundle ${\mathcal{O}}_{\operatorname{Hilb(n)}}(1)$ associated to the presen... | 1,528 | 1,127 | 1,237 | 1,453 | 2,131 | 0.782535 | github_plus_top10pct_by_avg |
nt cone to ${{\mathscr C}}$ at $p$, and by the choice of a side of a corresponding Newton polygon, with slope strictly between $-1$ and $0$. This procedure is explained in more detail in §\[details\].
Let $b<c$ be relatively prime positive integers such that $-b/c$ is the slope of the chosen side. Let $$\alpha(t)=\beg... | 1,529 | 441 | 1,604 | 1,535 | 3,839 | 0.769758 | github_plus_top10pct_by_avg |
4.479 6.062 2.437 2.667
Standard Deviation 8.69 10.50 3.38 3.18
Min/Max 0/40 0/40 0/10 0/10
Proportion of zero sent ... | 1,530 | 5,350 | 2,222 | 1,291 | 4,048 | 0.768456 | github_plus_top10pct_by_avg |
at happens in a specific geographical location anchored to a specific time interval. Mathematically, given a multidimensional query : $$Q = \langle q, q_{time}, q_{geo}, g_{entity} \rangle,$$ and a subset of highly relevant documents $R \subseteq D$, the algorithm for this purpose $\textsc{EventDetect}$ should produce ... | 1,531 | 1,966 | 2,464 | 1,717 | 3,088 | 0.774957 | github_plus_top10pct_by_avg |
ave kept running our business as usual, and here's few examples to show you that we are, indeed, doing just that:
EnronOnline did 5,866 transactions with 302 counterparties on Friday. Transaction counts remain higher than average.
EES signed a three-year fixed price agreement with Home Depot to supply power to 115 ... | 1,532 | 1,270 | 715 | 2,088 | null | null | github_plus_top10pct_by_avg |
\hat{\theta}^*(j) - \hat{\theta}
(j) \leq \tilde{t}^*_j, \forall j \Big| (W_1,\ldots,W_n) \right) \geq 1 - \alpha.$$ By the union bound, each $\tilde{t}^*_j$ can be chosen to be the largest positive number such that $$\mathbb{P}\left( \sqrt{n} | \hat{\theta}^*(j) - \hat{\beta}
(j) > \tilde{t}^*_j, \Big| (W_1,... | 1,533 | 4,138 | 1,243 | 1,220 | null | null | github_plus_top10pct_by_avg |
$\bar G'$ into a clique of the same size in $G$, by picking an arbitrary segment from each of the pairwise disjoint lines.) $\Box$
Ramsey-type bounds in $\mathbb{R}^2$ vs. $\mathbb{R}^3$–Proof of Theorem 4
==========================================================================
As we have pointed out in the Introd... | 1,534 | 1,398 | 1,969 | 1,514 | 3,352 | 0.772971 | github_plus_top10pct_by_avg |
Women’s University\
Kita-Uoya Nishimachi\
Nara 630-8506, Japan
- |
Department of Mathematics\
Kyushu University\
33, Fukuoka 812-8581 Japan
- |
Department of Mathematical and Computing Sciences\
Tokyo Institute of Technology\
Ohokayama, Meguro\
Tokyo 152-8552 Japan
author:
- Yasus... | 1,535 | 338 | 1,677 | 1,544 | null | null | github_plus_top10pct_by_avg |
o sum up contributions that come from different order in $H_{1}$. For example, $\hat{S}_{i J}^{(3)} = \hat{S}_{i J}^{(3)} [3] + \hat{S}_{i J}^{(3)} [2]$.
- We present only the terms which are required to compute $S$ matrix elements to order $W^4$. In view of the relations between $\hat{S}$ and $S$ matrix elements gi... | 1,536 | 650 | 2,094 | 1,683 | null | null | github_plus_top10pct_by_avg |
the intersection of the domains. Since the presumption that the subtrahend and remainder are not disjoint arrives at a contradiction, then subtrahend $\Phi$ and remainder $\Psi \setminus \Phi$ are indeed disjoint.
The ensembles are complementary with respect to $\Psi$ by definition \[D:COMPLEMENTARY\_ENSEMBLES\] beca... | 1,537 | 268 | 1,773 | 1,546 | 1,755 | 0.785935 | github_plus_top10pct_by_avg |
are functionally independent in a neighborhood of $x=0$ and the matrix functions $\mathcal{A}^\mu$, ${\partial}f/{\partial}r$, ${\partial}f/{\partial}\bar{r}$, ${\partial}\bar{f}/{\partial}r$, ${\partial}\bar{f}/{\partial}\bar{r}$, $Q_a$ and $K_a$ depend on $r$ and $\bar{r}$ only. So, equations (\[eq:3.26\]) (or (\[eq:... | 1,538 | 3,782 | 2,934 | 1,578 | null | null | github_plus_top10pct_by_avg |
e same vector grammar with capacity function constantly $k$.
To show ${{\bf V}}^{{\lambda}}_{{\mathit{cb}}}\subseteq {{\bf V}}^{{\lambda}}_{{\mathit{fin}}}$, consider a capacity-bounded vector grammar $$G=(\{A_0,A_1,\ldots,A_m\},\Sigma,A_0,M,\mathbf{1}).$$ (The proof that it suffices to consider the capacity function ... | 1,539 | 1,854 | 1,893 | 1,491 | 1,822 | 0.785424 | github_plus_top10pct_by_avg |
erior probabilities in the box corresponding to the true class. In Figure \[fig:nb:pos:scores:class:adas:car:uncal\] it is possible to see that the first class (*acceptable*) is missclassified as belonging to the third class (*unacceptable*) with high probability values, while Dirichlet Calibration is able to alleviate... | 1,540 | 1,654 | 379 | 1,124 | null | null | github_plus_top10pct_by_avg |
Indeed, the dominant diagrams for the CKM-induced quark EDMs have a rainbow topology. For instance, for the d-quark EDM:
![CKM-induced d-quark EDM[]{data-label="fig:CKMquarkEDM"}](figures/FigCKMQuarkEDM)
This EDM is tuned by the imaginary part of the 1-1 entry of a non-invariant commutator $Im(\textbf{X}^{dd}_{q})$, ... | 1,541 | 231 | 1,678 | 1,592 | 3,251 | 0.77375 | github_plus_top10pct_by_avg |
angle \;
e^{i\omega \tau} \nonumber \\
& = & \frac{1}{2} KT \; + \; e^{-\gamma t/2} \;
\left [ {\cal U}(\omega,0) - \frac{1}{2} KT \right ] \; \; ,\end{aligned}$$ $\left [ {\cal U}(\omega,0) - \frac{1}{2} KT \right ] $ is a measure of departure of energy density from thermal average at $t=0$. The exponential term impl... | 1,542 | 764 | 1,497 | 1,560 | null | null | github_plus_top10pct_by_avg |
riond; @del_moriond; @l3_moriond; @op_moriond].
$n_{\rm obs}$ $n_{\rm back}$ $n_{\rm sig}$
--------------------------------------- --------------- ---------------- ---------------
ALEPH 53 44.8 13.8
... | 1,543 | 2,301 | 2,713 | 1,571 | 3,139 | 0.774571 | github_plus_top10pct_by_avg |
omega''_i\in\Omega^{{{\bf n}}}_{v'\to a'}$, followed by the summation over $i=3,\dots,2j+1$ (cf., Figure \[fig:eventI\]). Finally, we apply Lemma \[lmm:GHS-BK\] to obtain the desired bound on the last line of [(\[eq:Theta”-2ndindbd5\])]{}.
Summarizing the above (d-1) and (d-2), we obtain $$\begin{aligned}
{(\ref{eq:Th... | 1,544 | 446 | 1,235 | 1,532 | 2,504 | 0.779277 | github_plus_top10pct_by_avg |
the self-dual case for $N=3$, $N=5$ and $N=7$. In each case, we give the lowest order symmetry $X^1$. However, this symmetry does [*not*]{} reduce to the case of (\[self-dual-equn\]), but the second symmetry, $X^2$, of the hierarchy generated by the master symmetries (\[eq:phi-sys-msym\]), is a symmetry of the reduced ... | 1,545 | 2,138 | 1,814 | 1,640 | null | null | github_plus_top10pct_by_avg |
eads the torus. Fig. 3 is analogous to Fig. 1 as described above with the notable exception that the $\nu$ splitting is non-trivial (hence fewer distinct states are shown), and level crossings occur at larger values. There is no plot analogous to Fig. 2 for larger values of $\tau_1$ with $\tau_0 = 0$ because $\phi$ dep... | 1,546 | 1,594 | 1,736 | 1,558 | 3,452 | 0.772267 | github_plus_top10pct_by_avg |
space to minimize numerical error on the marginal likelihood estimate. Although adaptive sparse grids [@Bun03] have the potential to be more beneficial in terms of resource management and precision, care would be needed in automating the grid refinements, and it is likely that a unique grid would be associated with eac... | 1,547 | 164 | 2,492 | 1,639 | 1,665 | 0.787044 | github_plus_top10pct_by_avg |
\rho_q(d-j(q-1)-\ell,m-j-1) + \sum_{i=1}^b\rho_q(i,m-a-1).$$
This follows using Lemma \[lem:help\] repeatedly. First applying the lemma to each sum within the double summation on the right-hand side, we see that $$\begin{gathered}
\sum_{j=0}^{a-1} \sum_{\ell=0}^{q-2} \rho_q(d-j(q-1)-\ell,m-j-1) = \\
\sum_{j=0}^{a-1} \... | 1,548 | 3,423 | 1,426 | 1,484 | null | null | github_plus_top10pct_by_avg |
he geometric quotient $X/R$ exists.
Proof. Consider first the case when $\dim X=1$. Let $\pi:X^n\to X$ be the normalization. We construct $X^n/R^{nd}$ as in (\[induct.plan\]). Note that since $Z$ is zero dimensional, it is finite over $S$. Let $V\subset S$ be its image. Next we make a different choice for $Z_1$. Inste... | 1,549 | 732 | 957 | 1,509 | 3,008 | 0.775481 | github_plus_top10pct_by_avg |
>0$ let $\delta_1=\varepsilon/4$ and $\delta_2=\varepsilon/(2\|K\|_V\|f\|^{1/2})$. Then, if the collections of functions $k_1^{(1)},\dots, k_{N_1}^{(1)}$ and $k_1^{(2)},\dots, k_{N_2}^{(2)}$ are $L_2(Q)$ $\delta_1$-dense respectively in the classes ${\cal K}_1$, ${\cal K}_2$, and $g_1,\dots,g_{N_3}$ are $L_1(Q)$ $\delt... | 1,550 | 675 | 1,049 | 1,522 | 4,154 | 0.767698 | github_plus_top10pct_by_avg |
from naive considerations; it is confirmed by the study of the blow-ups mentioned above. Slightly more refined phenomena (for example, involving multiplicities of the components) are not represented in this figure; in general, they can be easily established by applying the results of this paper or by analyzing the blo... | 1,551 | 407 | 930 | 1,341 | 3,725 | 0.770492 | github_plus_top10pct_by_avg |
ecisely, the crucial new step is:
\[thm41\] Suppose there is a model in which there is a Souslin tree $S$ and in which ****, ****, and $2^{\aleph_1}=\aleph_2$ hold. Then $S$ forces that locally compact normal spaces are $\aleph_1$-collectionwise Hausdorff.
It will be convenient to consider the following intermediate ... | 1,552 | 537 | 1,949 | 1,760 | 2,547 | 0.778982 | github_plus_top10pct_by_avg |
{\Lambda_H}}.$$ The following is well-known and easily follows from the definitions:
\[weight restriction\] Let $V$ be a representation of $G$ and $v \in V$ a weight vector of weight $\beta \in \Lambda^*_G$. If we restrict the action to $H$ via then $v$ is a weight vector of weight $F^*(\beta) \in \Lambda^*_H$.
Let u... | 1,553 | 2,285 | 2,085 | 1,413 | null | null | github_plus_top10pct_by_avg |
\cdot (z_j^{\ast})_2$ (resp. $x_j=(x_j)_1+\pi \cdot (x_j)_2$) as explained in the paragraph before Equation (\[e42\]). The Dickson invariant of $T_1$ is the same as that of $\begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0& (x_j)_1 &1& (z_j^{\ast})_1\\0&0& 0 & 1 \end{pmatrix}$. Here, we consider $\begin{pmatrix} 1&0&0&0 \\ 0&1... | 1,554 | 657 | 1,369 | 1,431 | null | null | github_plus_top10pct_by_avg |
} | \leq t_j, \forall j) - \mathbb{P}( |\hat Z_{j,n}| \leq
t, \forall j |\hat\Gamma)\right|; \mathcal{F}_n \right] +
\mathbb{P}(\mathcal{F}_n^c) \leq C \Delta_{3,n} + \frac{1}{n}.$$ In the above expression the constant $C$ is the same as in and $\mathcal{F}_n$ is the event that $\{ \max_{j,k}
|\widehat{\Gamma} - \Gamm... | 1,555 | 3,969 | 1,627 | 1,368 | null | null | github_plus_top10pct_by_avg |
ght)^4}{8 \left(u^2-1\right) \left(u^2+1\right)^3} & \frac{-3 u^4+30 u^2+h \left(u^2+1\right)^2-7}{2 \left(u^2+1\right)^3} \\
\mathcal{D}_{Ru} & -\frac{(h+1) u}{\left(u^2+1\right)^2} & \frac{2 u \left(u^2+h \left(u^2-1\right)-2\right)}{\left(u^2-1\right) \left(u^2+1\right)^2} \\
\mathcal{D}_{uu} & \frac{m^2 \left(u... | 1,556 | 3,198 | 1,667 | 1,458 | null | null | github_plus_top10pct_by_avg |
perp$ of $R$ as a subspace of $\left(\bigoplus_{w,x,y,z\in C} \mathcal F(E)(w,x,y;z) \right) ^\vee$. Notice that $\mathcal F(E)(\vec X;x)^\vee = \mathcal F(E^\vee) (\vec X;x)$, so that $\mathcal P^!$ is generated by $E^\vee$ with relations $R^\perp$, see [@GK (2.1.9)].
Now if $\mathcal P=\mathcal F(E)/(R)$ is a quadra... | 1,557 | 1,471 | 1,602 | 1,250 | 3,788 | 0.770087 | github_plus_top10pct_by_avg |
\}
\,.\end{aligned}$$ This entire equation is proportional to the basis function $F^{(m\,h\,k)}$, which can thus be divided out, leaving an ODE for one function, $C_{mhk}(u)$.
Specializing to the homogeneous (source-free) case, we find the ODE $$\begin{aligned}
\frac{d}{du}
\left[
(1-u^{2}) \frac{d}{du}
C_{mh... | 1,558 | 4,697 | 1,062 | 1,050 | null | null | github_plus_top10pct_by_avg |
mathbf{u}}}) \approx - 8\pi^2\nu|{\mathbf{k}}|^2\E_0.$$
Since the fields ${\mathbf{u}}_1$ are real-valued, their Fourier modes must satisfy $\widehat{{\mathbf{u}}}_1(-{\mathbf{k}}) =
\overline{\widehat{{\mathbf{u}}}_1({\mathbf{k}})}$, where $\overline{z}$ denotes the complex conjugate (C.C.) of $z\in\mathbb{C}$. Depen... | 1,559 | 4,037 | 1,201 | 1,162 | null | null | github_plus_top10pct_by_avg |
delta-AliceBob}$$
We have the same simple coincidence the two measures as in the case of the Leggett-Garg systems, $$\Delta_{\min}=\Gamma_{\min}.$$
Final remarks
==============
We have discussed two ways to measure contextuality. The direct approach, named Contextuality-by-Default (CbD), assigns to each random vari... | 1,560 | 687 | 2,785 | 1,537 | null | null | github_plus_top10pct_by_avg |
l}^{\ast}+\sum_{l=0}^{k_j}
\bar{\gamma}_{j-l}u_{j-l}^{\ast}=0$ for any $j\in \mathcal{B}_1$, and the equation $\sum_{l=0}^{k_j}z_{j-l}^{\ast}+\sum_{l=0}^{k_j}
\bar{\gamma}_{j-l}u_{j-l}^{\ast}=1$ for any $j\in \mathcal{B}_2$ yield that for each $j\in \mathcal{B}$, only one of equations of type Equations (\[24’\]... | 1,561 | 1,812 | 1,410 | 1,487 | 1,546 | 0.788441 | github_plus_top10pct_by_avg |
$I^o$*) or the $(3,3)$-block (when $L_i$ is *of type $I^e$*) of the $(i, i)$-block of $h\circ \tilde{m}=\sigma({}^t\tilde{m})\cdot h\cdot \tilde{m}$. Note that $c_i'$ and $z_i'$ are indeed contained in $R$ and $R$ is a $\kappa$-algebra. Thus $c_i'$ and $z_i'$ mod $(\pi\otimes 1)(B\otimes_AR)$ are naturally identified ... | 1,562 | 972 | 1,232 | 1,610 | 2,541 | 0.779027 | github_plus_top10pct_by_avg |
M}'$ is a quotient of group schemes with respect to the addition, the operation $\star$ is well-defined on $(\underline{M}'\otimes\kappa)/ \underline{\pi M}'$. For the proof, see the first two paragraphs from below in page 511 and the first two paragraphs in page 512 in [@C2].
To summarize, the morphism $1+ : \tilde{... | 1,563 | 975 | 1,835 | 1,580 | 3,036 | 0.775306 | github_plus_top10pct_by_avg |
t accuracy, and there is a linear trend between the test accuracy and the logarithm. We also highlight the fact that conditioned on the noise covariance, the test accuracy is not significantly correlated with either the step size or the minibatch size. In other words, similar to the observations in prior work [@jastrzk... | 1,564 | 900 | 681 | 1,425 | 1,299 | 0.791399 | github_plus_top10pct_by_avg |
ts Motion to Dismiss this untimely
appeal.
Respectfully submitted,
/s/ Michael M. Phillips
______________________________
Michael M. Phi... | 1,565 | 602 | 1,104 | 1,588 | null | null | github_plus_top10pct_by_avg |
- \frac{c_+}{c_++c_-} \frac{t^c \phi(w)}{x-w} + :j^c_{L,z} \phi:(w) %\right. \right. \cr
%& \quad \quad \left. + {A^c}_d \log|x-w|^2 :j^d_{L,z}\phi:(w) + {B^c}_d \frac{\bar x - \bar w}{x-w} :j^d_{L,z}\phi:(w)
+%AA \mathcal{O}(f^2)
... \right] j^b_{L,\bar z}(z) \right. \cr
& \quad \left. + j^c_{L,z}(x) \left[ -\frac{... | 1,566 | 804 | 1,336 | 1,548 | null | null | github_plus_top10pct_by_avg |
ill see shortly that $\calA_{\lambda\muhat}$ is nonzero for all $\lambda,
\muhat$; let $v(\lambda) :=v_q\left(\calA_{\lambda\muhat}(q)\right)$. The first main step toward the proof of the connectedness is the following theorem.
\[minim\] Let $\muhat=(\mu^1,\mu^2,\ldots,\mu^k)\in {\P_n}^k$ with $\delta(\muhat) \geq 0$.... | 1,567 | 2,949 | 1,625 | 1,490 | null | null | github_plus_top10pct_by_avg |
a I delta J\] $$\begin{aligned}
[\delta_I, \delta_J] A_\eta\ =&\ D_\eta A_{[\delta_I, \delta_J]}\,,
\label{delta I delta J 1}\\
A_{[\delta_I, \delta_J]}\ =&\
f\xi_0[\delta_I, \delta_J] A_\eta + D_\eta\Omega_{IJ}\,,
\label{delta I delta J 2}\end{aligned}$$
with $$\begin{aligned}
\Omega_{IJ}\ =&\
-f\xi_0[f\xi_0\del... | 1,568 | 967 | 1,174 | 1,678 | null | null | github_plus_top10pct_by_avg |
+\delta_5\{\dot xxyx^2\}+\delta_5\{x\dot xyx^2\}-\delta_5\{\dot
yy^2x^2\}-\delta_5\{y\dot yyx^2\} \\
-\delta_5\{\dot xxy^3\}-\delta_5\{x\dot xy^3\}+\delta_5\{\dot
yy^4\}+\delta_5\{y\dot yy^3\}.
\end{gathered}$$
This expression must coincide with $f = \{x^3\dot x y \} -
\{y^2x\dot x y\} $. In particular, a sum of all... | 1,569 | 3,214 | 1,820 | 1,463 | null | null | github_plus_top10pct_by_avg |
th, the results of the FNAL-E731 measurement of $(7.4 \pm 5.9) \times
10^{-4}$[@fnalE731], but further from agreement with the CERN-NA31 result of $(23\pm 7)\times 10^{-4}$[@cernNA31]. If we relax the constraint on the contribution to $\Delta m_K$ to allow ${\cal F}=-0.3$ (reflecting the uncertainty due to the large lo... | 1,570 | 2,475 | 2,717 | 1,654 | 2,114 | 0.782635 | github_plus_top10pct_by_avg |
ned}
\operatorname{{E}}[\Pe(Z + c)^{2}]
&= (k_{1}^{2} + k_{2}^{2}) b^{3} \beta
\int_{0}^{+\infty} z^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2}) b^{3} \beta
\int_{0}^{\lvert c / b \rvert} z^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + 2 (k_{1... | 1,571 | 2,552 | 1,240 | 1,573 | null | null | github_plus_top10pct_by_avg |
(Z_\Lambda/Z_\Lambda)^{2j+2}$, following Step (ii) of the strategy described in Section \[ss:pi0bd\]. Overlapping the $2j+3$ current configurations and using Lemma \[lmm:GHS-BK\] with ${{\cal V}}=\{y,x\}$ and $k=2j+2$, we obtain $$\begin{aligned}
{\label{eq:Theta'-2ndindbd6}}
\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}... | 1,572 | 2,944 | 1,054 | 1,539 | null | null | github_plus_top10pct_by_avg |
r bound ${\|\nabla\Lrb(\ltheta^*)\|}_2$. It might be possible to tighten the upper bound, given that $\ld \leq d$. However, for $\ell \ll \kappa$, for the smallest preference score item, $i_{\min} \equiv \arg \min_{i \in [d]} \ltheta^*_i$, the upper bound $\P[\sigma^{-1}(i_{\min}) > \kappa-\ell] \leq 1$ is tight upto c... | 1,573 | 727 | 572 | 1,663 | null | null | github_plus_top10pct_by_avg |
ith exponents $3/8$ and $1/2$, respectively, in both main panels and insets. In (b), the slopes of the dashed and solid lines are $1/4$ and $1/2$, respectively.](w_wvdt_1d.pdf "fig:"){width="0.8\linewidth"}\
![\[width\]Time evolution of the interface width $w$ for WV (main panels) and DT (insets) models grown on (a) on... | 1,574 | 171 | 2,651 | 1,590 | 838 | 0.799112 | github_plus_top10pct_by_avg |
contributes fractional fermion number $$\frac{\theta}{2\pi} \: - \: \left[ \frac{\theta}{2 \pi} \right]
\: - \: \frac{1}{2}$$ and a complex right-moving fermion $\psi$ with the same boundary conditions contributes fractional fermion number $$- \left(
\frac{\theta}{2\pi} \: - \: \left[ \frac{\theta}{2 \pi} \right]
\: -... | 1,575 | 1,496 | 1,894 | 1,443 | null | null | github_plus_top10pct_by_avg |
nsists of the applied stimulus $s_t$ and resulting WMTF $y_t$. The data set is re-ordered such that the observation $y_1, \ldots, y_{\tau-1}$ define baseline measurements with $s_t=0$ for $t=1, \ldots, \tau-1$, followed by an overall WMTF $y_\tau$ corresponding to the supramaximal stimulus $s_\tau=\max_t(s_t)$ where al... | 1,576 | 1,264 | 2,387 | 1,612 | 1,484 | 0.78896 | github_plus_top10pct_by_avg |
roblem, which is equivalent to the $\mathbf{VP}$ vs. $\mathbf{VNP}$ problem [@valiant79], as well as the complexity of matrix multiplication [@strassen69] can be formulated in this framework [@mulmuleysohoni01; @mulmuleysohoni08; @burgisserikenmeyer11; @burgisserlandsbergmaniveletal11]. More concretely, let us denote b... | 1,577 | 1,805 | 2,700 | 1,521 | 2,326 | 0.780836 | github_plus_top10pct_by_avg |
-_{i_\nu }\cdots {T}^-_{i_{\kappa +1}}
(K_{i_\kappa }^{-1}F_{i_\kappa })
\in U^-(\chi _\nu ){{\mathcal{U}}^0}&
\end{aligned}$$ by Eq. . Hence $$\begin{aligned}
&\sum _{m_1,\dots ,m_n}a_{m_1,\dots ,m_n}
{T}^-(E_{\beta _1}^{m_1}\cdots E_{\beta _\nu }^{m_\nu })
{T}^-(E_{\beta _{\nu +1}}^{m_{\nu +1}}\... | 1,578 | 3,238 | 1,920 | 1,357 | null | null | github_plus_top10pct_by_avg |
for non-deterministic first-order grammars, and show that its conclusion is semantically false. We then locate and analyze the flawed argument in the soundness (meta)-proof of [@Jan10].\
The grammar
===========
We consider the alphabet of actions ${\cal A}$, an intermediate alphabet of labels ${\cal T}$ and a map ${{... | 1,579 | 755 | 942 | 1,628 | 3,589 | 0.771345 | github_plus_top10pct_by_avg |
rgument more efficient.
The one bound that is still polynomial in \[thm:new.gen\] is the bound $H\subset A^{K^{e^{O(s)}}}$; it appears that a new idea would be required to improve this any further (see \[rem:poly.bound\], below, for further details). Note, though, that in the case where the ambient group has no torsio... | 1,580 | 740 | 1,304 | 1,531 | null | null | github_plus_top10pct_by_avg |
)$, with $ \hat{\Sigma}_{{\widehat{S}}}$ given in \[eq:Sigma.loco\]. Notice that $\hat{\Sigma}_{{\widehat{S}}}$ is almost surely positive definite, a consequence of adding extra noise in the definition of $\gamma_{{\widehat{S}}}$ and $\hat{\gamma}_{{\widehat{S}}}$. Then, using Theorem 2.1 in [@cherno2], there exists a ... | 1,581 | 784 | 2,192 | 1,476 | null | null | github_plus_top10pct_by_avg |
of Example. $$\begin{picture}(160,200) \put(20,70){\line(2,-3){40}}
\put(20,70){\line(1,0){80}} \put(20,70){\line(2,3){80}}
\put(60,10){\line(2,3){80}} \put(60,130){\line(2,-3){40}}
\put(60,130){\line(1,0){80}} \put(100,190){\line(2,-3){40}}
\linethickness{0.6mm} \put(60,50){\line(0,1){40}}
\qbezier(60,90)(80,100)(100,... | 1,582 | 4,455 | 710 | 1,204 | 2,369 | 0.780449 | github_plus_top10pct_by_avg |
or the base clause, the case $i = 1$ is true by hypothesis, since the initial step ${\mathfrak{A}}^{\mathbb{N}}(1) = {\mathit{s}}$ is presumed consistent. For the recursive clause, definition \[D:ITERATIVE\_OPERATOR\_WALK\] provides that ${\mathit{s}}_{i+1} = {\mathfrak{A}}(\xi_{i}, {\mathit{s}}_i)$ for each $i \ge 1$.... | 1,583 | 293 | 1,376 | 1,706 | null | null | github_plus_top10pct_by_avg |
adjoint field. Under the assumptions on the quantum theory stated above, the coefficients of the algebra are exact[^3]. We now move from the determination of the left-right symmetry algebra of the model to the study of the vertex operators.
The primaries {#primaries}
=============
In this section we define the concep... | 1,584 | 779 | 489 | 1,551 | null | null | github_plus_top10pct_by_avg |
as one can see from Eq. (\[439\]), the variance of this principal portfolio in the leading order is given by $(N{b}^{2}/{{W}_{N}}^{2}){\bar{{\rho}^{2}}}_{mkt}$, which is of the same order of magnitude as ${\bar{{\rho}^{2}}}_{mkt}$ (recall that $b$ is of the same approximate magnitude as a typical $\beta$ and that ${W}... | 1,585 | 1,497 | 1,785 | 1,583 | null | null | github_plus_top10pct_by_avg |
ts.
**Open Access** This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Appendix {#Sec21}
========
######
Original values of ... | 1,586 | 927 | 2,311 | 1,900 | null | null | github_plus_top10pct_by_avg |
*sacchariflorus* 4 Japan Davey *et al*., ([2016](#gcbb12419-bib-0503){ref-type="ref"}), Purdy *et al*. ([2013](#gcbb12419-bib-0037){ref-type="ref"}, [2014](#gcbb12419-bib-0038){ref-type="ref"}, [2015](#gcbb12419-bib-0039){ref-... | 1,587 | 2,888 | 1,646 | 1,440 | 2,347 | 0.780703 | github_plus_top10pct_by_avg |
G^{-1}\equiv G_0^{-1}+\Delta\end{aligned}$$ where $\Delta$ is given by: $$\begin{aligned}
&&\Delta(k,\tilde{\omega};k',\tilde{\omega}) \equiv \nonumber\\
&-&g\left(\begin{array}{cc}
0&\psi_B(k+k',\tilde{\omega}+\tilde{\omega}')\\
\bar{\psi}_B(k+k',\tilde{\omega}+\tilde{\omega}')&0\\
\end{array}\right).\end{aligned}$$ ... | 1,588 | 928 | 1,852 | 1,701 | null | null | github_plus_top10pct_by_avg |
m_\nu \ge t,\, 0\le m_\mu <{b^{\chi}} (\beta _\mu )
\,\text{for all $\mu $}}}
F_{\beta _1}^{m_1}F_{\beta _2}^{m_2}\cdots F_{\beta _n}^{m_n}.$$
\[le:P2\] For all ${\alpha }\in {\mathbb{N}}_0^I$, $$\begin{aligned}
{\alpha }\dim U^-(\chi )_{-{\alpha }}=
\sum _{\nu =1}^n \sum _{t=1}^{{b^{\chi}} (\beta _\nu )-1... | 1,589 | 1,030 | 1,231 | 1,497 | null | null | github_plus_top10pct_by_avg |
begin{aligned}
T_n^{\alpha^{-1}} & \cong & T_{-n}^{\alpha}, \: \: \:
T_0^{\alpha^{-1}} \: \cong \: T_0^{\alpha}, \\
{\cal E}_n^{\alpha^{-1}} & \cong & {\cal E}_{-n}^{\alpha}, \: \: \:
{\cal E}^{\alpha^{-1}}_0 \: \cong \: {\cal E}^{\alpha}_0,\end{aligned}$$ (in conventions where $-n$ denotes the component associated to ... | 1,590 | 2,516 | 1,734 | 1,486 | null | null | github_plus_top10pct_by_avg |
above are given by \[shift-defn\] $$S_c: {U}_c {\text{-}{\textsf}{mod}}\to {U}_{c+1}{\text{-}{\textsf}{mod}}: \qquad
N\mapsto Q_c^{c+1} \otimes_{{U}_c} N$$ and $$\widetilde{S}_c: H_{c} {\text{-}{\textsf}{mod}}\to H_{c+1}{\text{-}{\textsf}{mod}}: \qquad
M\mapsto H_{c+1}e_-\delta \otimes_{{U}_{c}} eM.$$
{#subsec-4.1}... | 1,591 | 1,060 | 1,628 | 1,400 | 3,570 | 0.771486 | github_plus_top10pct_by_avg |
array}{cll}
-1-\lambda x, & & x<-1\\
(1-\lambda)x, & & -1\leq x\leq1\\
1-\lambda x, & & 1<x,\end{array}\right.\\
\\f_{\lambda\textrm{b}}(x) & = & \left\{ \begin{array}{cll}
-\lambda x, & & x<1\\
(1-\lambda x)-1, & & 1\leq x\leq2\\
1-\lambda x, & & 2<x\end{array}\right.\\
\\f_{\lambda\textrm{c}}(x) & = & \left\{ \... | 1,592 | 2,100 | 2,497 | 1,589 | null | null | github_plus_top10pct_by_avg |
24.3 59.0 1006.5
6 653.4 22.8 22.1 88.3 1007.9
7 726.9 ... | 1,593 | 6,170 | 736 | 634 | null | null | github_plus_top10pct_by_avg |
nsform is given by $$Tg({\bf f}) = \left( \det(\mathbf{Id+K}) \right)^{-\frac{1}{2}} \exp\left(-\frac{1}{2} ({\bf f}, \mathbf{(Id+K)^{-1}} {\bf f})\right), \quad {\bf f} \in S_{d}({\mathbb{R}}).$$ Therefore $\left( \det(\mathbf{Id+K}) \right)^{\frac{1}{2}}g$ is a generalized Gauss kernel.
- \[traceL2\] Since a trace... | 1,594 | 1,820 | 1,533 | 1,451 | null | null | github_plus_top10pct_by_avg |
, as in [@GGOR Section 3.5], and write $K$ for the kernel of the associated homomorphism $\phi: P_c(\lambda) \to \Delta_c(\mu)$. By [@guay Proposition 13] there is a $\Delta$-filtration of $P_c(\lambda)$ $$P_c(\nu) = M_0 \supset M_1
\supset \cdots \supset M_t = 0$$ with each factor $M_j/M_{j+1}$ of the form $\Delta_c(\... | 1,595 | 650 | 404 | 1,749 | 2,141 | 0.782345 | github_plus_top10pct_by_avg |
ier, even more in order to define a consistent heterotic string compactification.
For completeness, let us now consider some possible examples.
One example is described in the paper [@dopr]. (See also [@Donagi:2000zf; @Donagi:2000zs; @Donagi:2000fw; @Ovrut:2002jk; @Ovrut:2003zj; @Braun:2004xv].) In that paper, the au... | 1,596 | 911 | 1,500 | 1,492 | 2,886 | 0.776309 | github_plus_top10pct_by_avg |
1 \leq i \leq n$, ${\|X_i-X_{i-1}\|}_2 \leq c_i$, for some non-negative constant $c_i$. Then for every $\delta > 0$, $$\begin{aligned}
\P[{\|X_n\|}_2 \geq \delta] & \leq & 2e^{3}e^{-\frac{\delta^2}{2\sum_{i=1}^n c_i^2}}\,.\end{aligned}$$
It follows from the upper bound on ${\|\nabla\L_{G_{j,a}}(\theta^*)\|}_2^2 \leq c... | 1,597 | 2,517 | 2,228 | 1,496 | null | null | github_plus_top10pct_by_avg |
bilayer distance. However, for smaller distances, we then encounter phases involving strong interlayer pairing [@pikovski2010; @zinner_10; @baranov2011] and the system would instead be better described in terms of interlayer bosonic dimers, as we discuss later.
![(Color online) Critical wave vector $q_c/k_F$ for the $... | 1,598 | 689 | 2,039 | 1,738 | null | null | github_plus_top10pct_by_avg |
define the direction dependence by writing the expression for $\phi(\tau)$ in Eq.(\[smearedoperator\]) as $$\frac{d \phi(\tau)}{d \Omega} = \int d\xi \; f_{\epsilon} ({\bm \xi}) \, \phi(x(\tau, {\bm \xi})) = \phi_\Omega(\tau)
\ ,
\label{smearedoperatorang}$$ such that the ${\bm \xi}$ points in the direction of $\Thet... | 1,599 | 3,819 | 2,611 | 1,630 | null | null | github_plus_top10pct_by_avg |
=\ - \langle A_\delta, QA_\eta+(F\Psi)^2\rangle
- {\langle\!\langle}\delta\Psi, Y(Q\Psi+X\eta F\Psi){\rangle\!\rangle}\,,
\label{general variation}$$ from which we find the equations of motion, $$QA_\eta + (F\Psi)^2\ =\ 0\,,\qquad
Q\Psi + X\eta F\Psi\ =\ 0\,.
\label{equations of motion}$$
Before closing this section,... | 1,600 | 2,756 | 1,315 | 1,534 | null | null | github_plus_top10pct_by_avg |
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