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 |
|---|---|---|---|---|---|---|---|
the tool expels the matter at a velocity of $U_1=24.05$, $V_1=0$. The plasticity region at the opening is bounded by curve $C_1$ and at the exit by the curve $C_2$. This extrusion die can thin a plate or rod of ideal plastic material.
![Extrusion die corresponding to the solution (\[eq:4.19\]).[]{data-label="fig:1"}]... | 1,901 | 826 | 2,317 | 1,934 | 1,947 | 0.784198 | github_plus_top10pct_by_avg |
delta_{i+1}v_{i+1}\cdot m_{i+1, i}=\pi m_{i,i}^{\ast}$$ such that $m_{i,i}^{\ast} \in M_{1\times n_i}(B\otimes_AR)$. This equation is considered in $B\otimes_AR$ and $\pi$ stands for $\pi\otimes 1\in B\otimes_AR$. Here,
- $v_{i-1}=(0,\cdots, 0, 1)$ (resp. $v_{i-1}=(0,\cdots, 0, 1, 0)$) of size $1\times n_{i-1}$ ... | 1,902 | 817 | 1,386 | 1,894 | 3,996 | 0.768731 | github_plus_top10pct_by_avg |
Stars, (Cambridge: Cambridge University Press), chaps. 4 and 5
Webbink, R. F. 1985, Interacting binary stars, eds. J. E. Pringle & R. A. Wade (Cambridge: Cambridge Univ. Press), p.39
Webbink, R. F., Livio, M., Truran, J. W., & Orio, M. 1987, , 314, 653
[ccccccccccc]{} 5.0$\times 10^{-7}$ & 0.7 & 0.30 & 5900 & 17.18 ... | 1,903 | 366 | 3,197 | 2,139 | null | null | github_plus_top10pct_by_avg |
Lambda;u,v}(z,x){\nonumber}\\
&\quad+\sum_{v',z}\big(\delta_{y,v'}+\tilde G_\Lambda(y,v')\big)\,\tilde
G_\Lambda(v',z)\,P'_{\Lambda;u}(z,x)\,\psi_\Lambda(v',v).{\label{eq:Q''-def}}\end{aligned}$$
The following are the diagrammatic bounds on the expansion coefficients (see Figure \[fig:piN-bd\]):
\[prp:diagram-bd\] F... | 1,904 | 1,072 | 1,999 | 1,855 | 1,960 | 0.784141 | github_plus_top10pct_by_avg |
ve of -5037809*x - 4185552 wrt x?
-5037809
Find the third derivative of -4*n**4 + 4564388*n**3 + 2*n**2 - 56142*n + 29.
-96*n + 27386328
Find the third derivative of 101000734*j**3*z**3 + 488*j**3*z + j*l*z**2 + j*z**2 - l*z**4 - 128399*l wrt z.
606004404*j**3 - 24*l*z
Find the third derivative of -33813*p*t**3 - 2*p*t... | 1,905 | 1,448 | 1,810 | 1,619 | null | null | github_plus_top10pct_by_avg |
+ \frac{\beta^2}{m}$, then $\E{a(x_0)}\leq R^2 - \frac{\beta^2}{m}$, then $ \lrp{\E{a(x_0)} - R^2 - \frac{\beta^2}{m}} \leq 0$, and $\lrp{\E{a(x_t)} - R^2 + \frac{\beta^2}{m}} \leq e^{-mt} \cdot 0 \leq 0$ for all $t$. This implies that, for all $t$, $$\begin{aligned}
\E{\|x_t\|_2^2} \leq \E{2 a(x_t) + 4R^2} \le... | 1,906 | 2,909 | 1,506 | 1,634 | null | null | github_plus_top10pct_by_avg |
})\right],
\label{eq:Phodge}$$ where ${\Delta}^{-1}$ is the inverse Laplacian associated with the periodic boundary conditions; the operator $\mathbb{P}_{\mathcal{S}_0}$ is also known as the Leray-Helmholtz projector.
- $(\E_0)$-constraint: the projection onto the manifold $\mathcal{S}'_{\E_0}$ is calculated by... | 1,907 | 1,063 | 2,069 | 1,784 | null | null | github_plus_top10pct_by_avg |
n{cases}
4 r_{\mathrm{B}}^{3/2} \left(r^{1/2}-R_{\mathrm{in}}^{1/2}\right)
& r < r_{\mathrm{B}} \\
r^2 + 3r_{\mathrm{B}}^2 -4r_{\mathrm{B}}^{3/2}R_{\mathrm{in}}^{1/2}
&
r > r_{\mathrm{B}}
\end{cases}\,,
\label{eq:14}\end{aligned}$$ where a factor of two is multiplied in the first equality to take into account both to... | 1,908 | 2,625 | 2,202 | 1,844 | 2,736 | 0.777406 | github_plus_top10pct_by_avg |
\right>
\pm \left|u_2^{F}(t)\right> \exp(-{{\rm i}}\Delta\varepsilon^{F}t/\hbar) \Big)
\nonumber \\ & & \times
\exp(-{{\rm i}}\varepsilon_1^Ft/\hbar) \; ,\end{aligned}$$ where $$\Delta \varepsilon^F = \varepsilon_2^F - \varepsilon_1^F$$ denotes the quasienergy splitting. Hence, the particle is coherently oscilla... | 1,909 | 4,690 | 1,176 | 1,452 | 3,090 | 0.774931 | github_plus_top10pct_by_avg |
temperature in units of K and eV, respectively; $^{a}$Case B; $^{b}$radiative [Case B; singlet; our fit to @Hummer:1998aa] and dielectric [@Aldrovandi:1973aa] recombination; $^{c}$Case B [@Draine:2011aa with typo about the charge dependence corrected].\
REFERENCES. (1) [@Janev:1987aa]; (2) [@Abel:1997aa from Aladdin da... | 1,910 | 2,293 | 3,287 | 2,050 | null | null | github_plus_top10pct_by_avg |
NCES. (1) [@Ferland:1992aa]; (2) [@Hummer:1998aa]; (3) [@Black:1981aa] (4) [@Draine:2011aa]; (5) [@Cen:1992aa]; (6) [@Bray:2000aa]; (7) [@Anninos:1997aa]; (8) [@Shapiro:1987aa].
In Table \[tab:cool\_rates\], we summarize the heating and cooling processes considered in this work. Here, $n(X)$ is the number density of s... | 1,911 | 1,244 | 2,876 | 2,144 | 486 | 0.80813 | github_plus_top10pct_by_avg |
ne against the toxic effects generated by hydrogen peroxide *in vitro*. These data can drive further *in vivo* studies with the purpose of establishing specific therapies capable of preventing or at least minimizing the pulpal damage caused by tooth bleaching techniques widely used in dentistry. This may avoid the post... | 1,912 | 2,976 | 1,806 | 1,929 | null | null | github_plus_top10pct_by_avg |
U(\chi ).$$
\[re:ideal\] (i) The vector space $V=\oplus _{i\in I}{\Bbbk }E_i$ is a module over the group algebra ${\Bbbk }{\mathbb{Z}}^I\simeq {\Bbbk }[K_i,K_i^{-1}\,|\,i\in I]\subset {{\mathcal{U}}^0}$, where the left action ${\boldsymbol{\cdot}}:{\Bbbk }{\mathbb{Z}}^I\otimes V\to V$ and the left coaction ${\delta }... | 1,913 | 2,131 | 1,930 | 1,648 | 3,916 | 0.769296 | github_plus_top10pct_by_avg |
es, whether in GR or beyond-GR theories.
We organize the paper as follows. In Sec. \[sec:kerr-nhek-limit\] we review the NHEK limit of the Kerr black hole, and elaborate on the structure of NHEK’s isometry Lie group ${\ensuremath{SL(2,\mathbb{R})\times U(1)}}$. In Sec. \[sec:high-lowest-weight\], we construct the high... | 1,914 | 3,753 | 2,167 | 1,734 | null | null | github_plus_top10pct_by_avg |
N_r=N_t=17, L_r=L_t=5, N_{\psi,{\mathrm{cl}}}=10, N_{\psi,{\mathrm{ry}}}=8,$ $\sigma_{\theta^r}=\sigma_{\theta^t}=3^\circ$, and $d/\lambda=1/2$.[]{data-label="fig:GainOut"}](GainOut){width=".9\columnwidth"}
We now present the outage throughput gain of employing the reconfigurable antennas. Figure \[fig:GainOut\] plots... | 1,915 | 255 | 1,917 | 2,020 | 1,209 | 0.792673 | github_plus_top10pct_by_avg |
be equipped with the inner product \[f17\] ,v\_[W\^2\_[\_1,\_2]{}(GSI)]{} =&,v\_[L\^2(GSI)]{}+ \_1\_x,\_1\_x v\_[L\^2(GSI)]{}\
&+\_2[E]{},\_2\_[L\^2(GSI)]{}, rendering $W^2_{\rho_1,\rho_2}(G\times S\times I)$ to a Hilbert space. Similar weighted spaces can be defined generalizing other spaces above. These spaces are ne... | 1,916 | 356 | 1,669 | 1,948 | null | null | github_plus_top10pct_by_avg |
)^2} + \frac{c_+}{c_++c_-} \frac{\bar \partial \mathcal{A}^{a \bar a}(w)}{\bar z-\bar w} \right)+ ... \cr
%
j^a_{L,z}(z) j^{\bar a}_{R,\bar z}(w) &=
-\frac{c^2_+}{c_++c_-} \left(\mathcal{A}^{a \bar a}(w) 2\pi \delta^{(2)}(z-w)
- \frac{c_-}{c_++c_-} \frac{\partial \mathcal{A}^{a \bar a}(w)}{\bar z - \bar w} + \frac{c_-... | 1,917 | 487 | 2,198 | 2,034 | null | null | github_plus_top10pct_by_avg |
ibizumab to aflibercept. Further studies are needed to confirm this finding, to identify the entire spectrum of influencing factors, and to investigate whether these factors are also applicable for switching eyes refractory to aflibercept to ranibizumab.
**Disclosure**
The authors report no conflicts of interest in t... | 1,918 | 726 | 2,022 | 2,246 | null | null | github_plus_top10pct_by_avg |
\Rq}{\aq\Rq^2 + 1}$. Let $\epsilon:= \frac{\lambda}{16 (L+\LN^2)} \exp\lrp{-\frac{7\aq\Rq^2}{3}} \hat{\epsilon}$. Let $T:= \min\lrbb{\frac{1}{16L}, \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}, \frac{\epsilon}{32\sqrt{L} \beta}, \frac{\epsilon^2}{128\beta^2}, \frac{\epsilon^4 \LN^2}{2^{14}\beta^2 \cm^2}}$ and let $\delta$... | 1,919 | 1,218 | 1,741 | 1,645 | null | null | github_plus_top10pct_by_avg |
morphism $\mathrm{Ker~}\varphi/\tilde{G}^1\rightarrow \mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$ is a monomorphism, and thus $\mathrm{Ker~}\varphi/\tilde{G}^1$ is a closed subgroup scheme of $ \mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$ by (Exp. $\mathrm{VI_B},$ Corollary 1.4.2 in [@SGA3]).
\[ta4\] $\tilde{G}^1$ is conn... | 1,920 | 1,322 | 1,675 | 1,740 | 2,085 | 0.782895 | github_plus_top10pct_by_avg |
ap sample contains more than $k$ distinct values. Thus, the bootstrap guarantees given below only holds on such event. Luckily, this is a matter of little consequence, since under our assumptions the probability that such event does not occur is exponentially small in $n$ (see below).
For a given $\alpha \in (0,1)$, l... | 1,921 | 3,682 | 1,279 | 1,432 | null | null | github_plus_top10pct_by_avg |
varepsilon$-visibility can also lead to a lower bound on the maximum load achieved by the balanced allocation process]{} on hypergraphs. ]{} [This theorem is proved in Appendix \[sec:lower-bound\].]{}
\[thm:lower-bound\] Let $s=s(n)=n^{\varepsilon}$, where $\varepsilon\in (0, 1)$ is an arbitrary small real number. The... | 1,922 | 373 | 1,004 | 1,953 | 1,344 | 0.790682 | github_plus_top10pct_by_avg |
all physics constraints.
Flow {#sec:flow}
====
We are now in the position to integrate the flow equation . To begin with, we can immediately integrate out the spatial gauge fields $\vec A_\bot$ for $Z_i=1$, that is the second line in . This part of the flow only carries an explicit dependence on the cut-off $k$, deta... | 1,923 | 845 | 1,151 | 1,829 | 1,179 | 0.793103 | github_plus_top10pct_by_avg |
icient to show the inclusions ${{\bf V}}_{fin}\subseteq\mathbf{wPN}_{ch}$ and $\mathbf{wPN}^{{\lambda}}_{ch}\subseteq {{\bf V}}^{{\lambda}}_{cb}$.
As regards the first inclusion, let $L$ be a vector language of finite index (with or without erasing rules), and let $ind(L)=k$, $k\geq 1$. Then, there is a vector grammar... | 1,924 | 816 | 819 | 1,866 | 1,591 | 0.787831 | github_plus_top10pct_by_avg |
mma}_{11}$ is defined by using the zero-modes of the world-sheet fermion $\psi^\mu(z)$ as $$\hat{\Gamma}_{11}\ =\ 2^5\,\psi^{0}_0\psi^{1}_0\cdots\psi^{9}_0\,.
\label{gamma11}$$ We summarize the convention on how the zero modes $\psi^\mu_0$ act on the Ramond ground states in Appendix \[convention\].[^5]
Complete gauge-... | 1,925 | 1,466 | 2,217 | 1,792 | null | null | github_plus_top10pct_by_avg |
Q^\dagger$, one has for the general self-adjoint combination of supercharges $$Q_\epsilon=\epsilon Q+Q^\dagger\epsilon^\dagger=
\epsilon Q-\epsilon^\dagger Q^\dagger\ ,$$ the action $$Q_\epsilon\psi(z,\theta)=\sqrt{\hbar\omega}
\left[\left(z\epsilon\psi_F(z)\right)\ +\
\theta\left(\epsilon^\dagger\partial_z\psi_B(z)\ri... | 1,926 | 2,982 | 2,117 | 1,803 | null | null | github_plus_top10pct_by_avg |
conditions on the factors, as were considered in [@BCL; @FMOpi; @ZGS]. This means that, in contrast to some of the mentioned results whose proofs are elementary, the classification of finite simple groups (CFSG) has been used in our proof. In particular, we derive some results on the center of the prime graph of an alm... | 1,927 | 1,402 | 2,060 | 1,753 | null | null | github_plus_top10pct_by_avg |
bar{b}_{jj}+\bar{g}_{jk}\bar{b}_{ij}-\bar{f}_kb_{kj}+b_{kk}\bar{f}_j-\bar{b}_{ik}\bar{g}_{jj}-\bar{b}_{jk}\bar{g}_{ij}+\bar{h}_kg_{kj}=0\nonumber \\
& f_k\bar{b}_{kj}-g_{jk}b_{ij}-g_{ik}b_{jj}+\bar{g}_{kk}h_j-h_k\bar{g}_{kj}+b_{jk}g_{ij}+b_{ik}g_{jj}-\bar{b}_{kk}f_j=0\nonumber \\
& -2\bar{g}_{jk}b_{jj}+2\bar{f}_kh_j-g_... | 1,928 | 801 | 2,541 | 1,867 | null | null | github_plus_top10pct_by_avg |
1}^{N_{\psi,c}}\sigma^2_{\alpha,\psi,i}=\gamma_{\psi}$, where $\gamma_{\psi}$ is a normalization parameter to ensure that $\mathbb{E}\{\left\|{\mathbf{H}}_{\psi}\right\|^2_F\}=N_rN_t$. We also assume that $\theta^r_{\psi,i,l}$ are uniformly distributed with mean $\theta_{\psi,i}^r$ and a constant angular spread (standa... | 1,929 | 1,219 | 500 | 1,796 | 1,664 | 0.787053 | github_plus_top10pct_by_avg |
\chi}} (\beta _\nu )-1}
({\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu }-
\chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu })^{N_{\nu ,t}}$$ for some $N_{\nu ,t}\in {\mathbb{N}}_0$ and an element $f\in {\Bbbk }[K_i,L_i\,|\,i\in I]$ which is invertible on ${\mathbb{T}}$. In particular, $\det ^\chi _{\alpha }\not=0$. We fi... | 1,930 | 1,530 | 1,522 | 1,783 | null | null | github_plus_top10pct_by_avg |
}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\longleftrightarrow}}}v\}$}}}\big]&\leq\sum_z\big(\delta_{y,z}
+\tilde G_\Lambda(y,z)\big)\,\Theta''_{z,x,v;{{\cal A}}}{\nonumber}\\
&\quad+\sum_{v',z}\big(\delta_{y,v'}+\tilde G_\Lambda(y,v')\big)\,\til... | 1,931 | 685 | 1,819 | 2,013 | null | null | github_plus_top10pct_by_avg |
ible that the value ${a_{\text{today}}}/{a_{\text{f}}}$ can be lower than calculated before.
After this analysis we can return now to equation (\[modeeq\]). We need to use the definition of the quantum correction $D$ and effective mass $m^2_{\text{eff}}$ in the quantum regime. It is also useful to rescale the conforma... | 1,932 | 3,728 | 1,908 | 1,677 | null | null | github_plus_top10pct_by_avg |
DQ”4: $\forall \alpha \in S$,\
${{\rm NEXT}}((T,T'),\alpha) \in \sim_1$ and the set $\{ (\pi,\pi') \in {\cal R}\times{\cal R} \mid \alpha\cdot (\pi,\pi')\in S\}$ is full for ${{\rm NEXT}}((T,T'),\alpha)$.\
Every finite prefix of a strategy is a D-q-strategy. \[L-PD\_implies\_DQ\]
Let $S'$ be a D-strategy w.r.t. $(T,T... | 1,933 | 1,022 | 1,834 | 1,775 | null | null | github_plus_top10pct_by_avg |
d({\mathbb{R}}, dx)_{{\mathbb{C}}} \to L^2_d({\mathbb{R}}, dx)_{{\mathbb{C}}}$ be linear and continuous such that:
- $\mathbf{Id+K}$ is injective.
- There exists $p \in {\mathbb{N}}_0$ such that $(\mathbf{Id+K})(L^2_{d}({\mathbb{R}},\,dx)_{{\mathbb{C}}}) \subset H_{p,{\mathbb{C}}}$ is dense.
- There exist $q \... | 1,934 | 2,191 | 1,966 | 1,797 | null | null | github_plus_top10pct_by_avg |
phenomenology {#sec:pheno}
=============
Dark matter {#subsec:DM}
-----------
If $({\mathcal M}_0)_{\eta\eta} < ({\mathcal M}_0)_{ss}$, the dark matter candidate ${\mathcal H}_1^0$ is given by $\eta_r^0$ approximately because we assume small mixing. See, e.g., Ref. [@i-doublet] for studies about the inert doublet sc... | 1,935 | 714 | 2,815 | 1,919 | 3,673 | 0.770787 | github_plus_top10pct_by_avg |
beta_{\mathcal{C}_N,P_{N}}^{(kM)}
=\beta_{\mathcal{C},P}^{(k)}.$$ Again, for $M$ big enough the map $\sigma_{kM}$ is surjective and thus $\beta_{\mathcal{C},P}^{(k)}=0$.
Esnault–Viehweg’s theorem revisited {#sc:Esnault}
===================================
Here we present a generalization of Esnault–Viehweg’s results ... | 1,936 | 1,695 | 1,334 | 1,979 | 3,298 | 0.773383 | github_plus_top10pct_by_avg |
r}}) \ .
\label{eq:psi1linComoving}\end{aligned}$$ Note that such Galilean transformations of the GPE using a constant ${\boldsymbol{V}}_p$ are often accompanied by a multiplication of the transformed wavefunction by a phase factor $\exp(i{\boldsymbol{V}}_p \cdot {\boldsymbol{z}} + \frac i 2 V_p^2 t)$, in order to tran... | 1,937 | 1,787 | 2,729 | 2,057 | null | null | github_plus_top10pct_by_avg |
hat are close in the stand attribute space, but distant in the predictor space) in large training sets, which might partly explain the slight improvement of CI% when training set size decreases. With the Bayesian inference approach, on the contrary, a substantial drop in CI% in smaller training set sizes would be expec... | 1,938 | 1,572 | 2,621 | 1,616 | null | null | github_plus_top10pct_by_avg |
the previous paragraph. Choose any $s$ above $s_0$ and any $\delta\in C_2$ such that $s$ forces that $\delta\in\dot{E}$. Without loss of generality, the height of $s$ is $\geq\delta^+$, but note that $s\!\!\upharpoonright\!\!\delta$ forces a value on $\dot{n}_\alpha$, for all $\alpha<\delta$. This means that $s\!\!\up... | 1,939 | 1,723 | 2,753 | 2,012 | 2,843 | 0.776604 | github_plus_top10pct_by_avg |
m,j}(w) E^{\pm,i}(z_2) \nonumber\\
& &~~~~~+ E^{\pm,j}(w) E^{\pm,i}(z_1) E^{\pm,i}(z_2) +
(\mbox{replacement:}~z_1 \leftrightarrow z_2) = 0
~\mbox{for}~a_{ij} = -1, \label{8}\end{aligned}$$
where $a_{ij}$ are elements of the Cartan matrix of the type $A_{N-1}$ and
$$\begin{aligned}
\delta(x)= \sum_{n\in {\Bbb Z}} x^n... | 1,940 | 2,411 | 1,882 | 1,876 | null | null | github_plus_top10pct_by_avg |
peed 3)
(debug 0)
(safety 0)))
(format t "~a~%" (loop for i fixnum from 1 upto 100000000
sum i of-type fixnum)))
~~~
acdha
> The Python code can be made a lot faster by using xrange and also reduce.
This is a key ... | 1,941 | 3,279 | 2,225 | 1,481 | 1,711 | 0.786514 | github_plus_top10pct_by_avg |
line. Then $1$-integrally we have$$f_{k}^{(-1)}(x)=-\frac{1}{k}\cos kx=-\frac{1}{k}+\int_{0}^{x}\sin kx_{1}dx_{1},$$
which obviously converges to $0$ uniformly (and therefore in the mean) as $k\rightarrow\infty$. And herein lies the point: even though we cannot conclude about the exact nature of $\sin kx$ as $k$ incr... | 1,942 | 3,274 | 2,813 | 1,782 | 3,553 | 0.771576 | github_plus_top10pct_by_avg |
super-Eddington regime (equations \[eq:12\] and \[eq:3\]). The radius of the [H[ii]{} ]{}region thus varies as $r_{\mathrm{HII}}(\theta) \propto M_{\mathrm{BH}}^{2/3}$ (equation \[eq:13\]), while the Bondi radius follows $r_{\mathrm{B}} \propto
M_{\mathrm{BH}}$ (equation \[eq:2\]). Since $r_{\mathrm{HII}}(\theta)$ is a... | 1,943 | 3,065 | 2,938 | 2,030 | null | null | github_plus_top10pct_by_avg |
al remarks apply about the sufficiency of identical distribution or a Lyapunov-type condition: holds in the settings of Theorems \[T:circular-law-correlated\] and \[T:circular-law-uncorrelated\] if all the $Y_a^{(n)}$ are identically distributed, or have uniformly bounded $(2 + \delta)$ moments; it holds in the setting... | 1,944 | 2,383 | 2,035 | 1,748 | null | null | github_plus_top10pct_by_avg |
tor $f(t)\xi_0$: $$\{D_\eta(t),\, f(t)\xi_0\}\ =\ 1\,,\qquad
\langle f\xi_0 \Phi_1, \Phi_2\rangle\ =\
(-1)^{\Phi_1}\langle \Phi_1, f\xi_0 \Phi_2\rangle\,.
\label{BPZ homotopy NS}$$ We can define the projection operators $$\mathcal{P}_{NS}\ =\ D_\eta f\xi_0\,,\qquad
\mathcal{P}_{NS}^\perp\ =\ f\xi_0 D_\eta\,,\qquad
... | 1,945 | 849 | 1,526 | 1,730 | null | null | github_plus_top10pct_by_avg |
o confusion may arise, we drop the subindices $t,w$ from $g$. To estimate the bias of the ideal estimator, $\tilde f(t;h)-f(t)$, one develops $g(hw)$ about zero and integrates. For further reference, we record the first four derivatives of $g(u)$: by direct computation or e.g. from Novak (1999), we have, with $r(u)=f^{... | 1,946 | 1,505 | 1,664 | 1,781 | null | null | github_plus_top10pct_by_avg |
------+
| 5′ Cap analogue cost | 2500‐10 000 | USD/g | Scale and supplier purchase price | [53](#amp210060-bib-0053){ref-type="ref"} ... | 1,947 | 1,146 | 1,162 | 1,550 | null | null | github_plus_top10pct_by_avg |
lon
\end{aligned}$$
Summing the above three equations, $$\begin{aligned}
\E{f(x_T - w_T)}
\leq e^{-\lambda \delta} \E{f(x_0 - w_0)} + 14T (L+\LN^2)
\end{aligned}$$ Where we use the fact that $y_0 = w_0$ by construction in .
Recalling , this is equivalent to $$\begin{ali... | 1,948 | 2,311 | 1,473 | 1,984 | null | null | github_plus_top10pct_by_avg |
\] also fails for this value of $c$.
\(2) This also shows that the hypothesis $c\notin (-2,0)_{\mathcal{C}}$ is serious. Indeed, for any $n\geq 2$, let $c=-m/n\in (-1,0)_{\mathcal C}$. Then one can prove that the factor module $V_c =\Delta_c(\operatorname{{\textsf}{sign}})/I_c$ considered in [@CE Theorem 3.2] does not... | 1,949 | 1,113 | 693 | 1,981 | 1,494 | 0.788856 | github_plus_top10pct_by_avg |
,\H_\muhat(-z,-w)=\H_\muhat(z,w). \label{HHduality}$$
We may recover $\Omega(z,w)$ from the $\H_\muhat(z,w)$’s by the formula:
(z,w)=(\_[\^k]{}m\_). \[exp\]
From Formula (\[eulermac\]) and Formula (\[H-specializ\]) we have:
With the specialization $y_i=q^{i-1}$,
$$\Omega\left(\sqrt{q},\frac{1}{\sqrt{q}}\right)=\su... | 1,950 | 673 | 1,284 | 1,853 | null | null | github_plus_top10pct_by_avg |
\<0.001 0.703 \<0.001 0.07 0.151
Mapping family Acid digestible carbohydrate Enzymatic carbohydrate release \% Digestibility ... | 1,951 | 5,357 | 1,118 | 977 | null | null | github_plus_top10pct_by_avg |
c to $(\mathcal{L(S)},\subset )$, as stated.
Let us come now to physical QL. We have seen in Sec. 2.2 that $(\mathcal{L(S)},\subset )$ is order-isomorphic to $(\mathcal{E},\prec )$. We can then conclude that $(\mathcal{E},\prec )$ is order-isomorphic to $(\phi
_{AD}^{Q}/\approx ,\prec )$, which provides the desired in... | 1,952 | 4,099 | 3,314 | 1,973 | null | null | github_plus_top10pct_by_avg |
$m$-dimensional manifold into the plane obtained in the explanation of FIGURE \[fig:2\] and $n$-copies of such a map. We deform these maps by scaling suitably and attach the maps as FIGURE \[fig:4\] on the arcs corresponding to ones including $(0,-1)$ in the original image and the inverse images. $D_k$ are the images ... | 1,953 | 369 | 2,587 | 1,898 | null | null | github_plus_top10pct_by_avg |
E-H_{\mathrm{eff}}^a)^{-1}]_{nm}\rangle=[(E/2)-i\sqrt{1-E^2/4}]\delta_{nm}$, where the imaginary part accounts for the famous Wigner’s semicircle law. This implies the following result (valid up to the terms of the order of $M/N$) for the average $S$-matrix [@ver85a; @sok89], $$\label{eq:s_aver}
\langle S_{ab} \rangle... | 1,954 | 1,782 | 2,399 | 1,963 | 2,030 | 0.783321 | github_plus_top10pct_by_avg |
ciety of the Pacific Conference Series, ed. [I. N. Evans, A. Accomazzi, D. J. Mink, & A. H. Rots]{}, 155–+
, B. [et al.]{} 2007, , 172, 615
, T. M. 1980, , 87, 152
, J. F. [et al.]{} 2006, , 131, 1
, J. R. & [Lehmann]{}, E. L. 1963, The Annals of Mathematical Statistics, 34, 598
, P. F., [Bundy]{}, K., [Hernquist]... | 1,955 | 383 | 3,131 | 2,095 | null | null | github_plus_top10pct_by_avg |
zation, L.J.F.; methodology, L.J.F., C.F., M.B.-F. and R.N.P.; formal analysis, J.E.; writing---original draft preparation, L.J.F., J.E. and R.N.P.; writing---review and editing, L.J.F., J.E. and R.N.P.
This research and the APC were partially funded by FEDER projects COMRDI16-1-0035-03 and RTI2018-097700-B-I00 from t... | 1,956 | 148 | 2,428 | 2,135 | null | null | github_plus_top10pct_by_avg |
a particles (ions and electrons) and two charged dust grains. For the time integration, the Coulomb force acting on each particle must be evaluated by taking the summation over all particles. Since the Coulomb interaction is a long-range interaction, convergence of the summation is very slow and the calculation of cont... | 1,957 | 3,084 | 2,753 | 1,981 | 3,051 | 0.775194 | github_plus_top10pct_by_avg |
A_\mu-v \bar{A}^\mu,\ \ \bar{A}^\mu\rightarrow\bar{A}^\mu.$$
Now we want to find the diffeomorphism of the geometry by considering an infinitesimal coordinate transformation, $$x\rightarrow x+\epsilon(x,y),\ \ \ y\rightarrow y+\xi(x,y).$$ The infinitesimal variations are, \[if1\] dx&=&\_x dx+\_y dy,\
dy&=&\_x dx+\_y ... | 1,958 | 3,160 | 2,533 | 1,712 | null | null | github_plus_top10pct_by_avg |
e $i\in I$, and a matrix $(q_{ij})_{i,j\in I}\in (\fienz)^{I\times I}$, such that $${\delta }(x_i)=g_i{\otimes }x_i,\quad g_i{\boldsymbol{\cdot}}x_j=q_{ij}x_j \quad
\text{for all $i,j\in I$.}$$ Assume that $\chi ({\alpha }_i,{\alpha }_j)=q_{ij}$ for all $i,j\in I$. For all ${\alpha }\in {\mathbb{Z}}^I$ define the “boun... | 1,959 | 2,294 | 1,441 | 1,786 | null | null | github_plus_top10pct_by_avg |
general.
We now examine the relation between the covariant derivative operators $\nabla_i$ of $g_{i j}$ and $\bar{\nabla}_i$ of $\bar{g}_{i j}$. The relation is determined by $$\bar{\Gamma}^i\mathstrut_{j k}(\bar{g}) = \Gamma^i\mathstrut_{j k}(g)
+ 2 \psi^{-1} \left( 2 \delta^i_{( j} \partial_{k )} \psi
- g^... | 1,960 | 4,135 | 1,943 | 1,603 | null | null | github_plus_top10pct_by_avg |
Lemma \[lemma7\]), we find that the homomorphism labelled by $$\gyoung(;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;2;2;{x_1}_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};{x_s},;d;d;{z_2};{... | 1,961 | 2,038 | 1,714 | 1,736 | 1,235 | 0.792287 | github_plus_top10pct_by_avg |
(\psi_1,\psi_2,\psi_3)=(\psi_1,\hat{\psi})$ is a solution of , , we have by equation , \[psi1\] \_1=T\_[1,0]{}\^[-1]{}(f\_1+K) and hence by the function $\hat\psi$ satisfies the equation, for $j=2,3$, \[pr5.5.7\] -[E]{}+\_x\_j+\_j\_j-K\_j-K\_[1,j]{}(T\_[1,0]{}\^[-1]{}(K)) =f\_j+f\_j, where we wrote $$\hat{f}_j:=\hat K_... | 1,962 | 336 | 1,858 | 1,962 | null | null | github_plus_top10pct_by_avg |
cal{Z}( \Gamma(G))=\{p \, | \, p \mbox{ is adjacent to } r, \forall r \in \pi(G)\}$.
The following result on the center of the prime graph of alternating and symmetric groups will be used later:
\[angraph\] Let $n \geq 5, n\neq 6$ be a positive integer. Let $k$ be the largest positive integer such that $\{n, n-1, \... | 1,963 | 721 | 1,485 | 1,811 | null | null | github_plus_top10pct_by_avg |
r a noetherian ring $R$. Although easy, the next result provides the foundation for our approach to $U_c$: in order to study $U_c{\text{-}{\textsf}{mod}}$ it suffices to study $R_{\mathbb{Z}}{\text{-}{\textsf}{qgr}}$, for any Morita ${\mathbb{Z}}$-algebra $R_{\mathbb{Z}}$ with $R_0\cong U_c$.
Suppose that $R_{\mathbb{... | 1,964 | 1,261 | 1,422 | 1,742 | 3,995 | 0.768732 | github_plus_top10pct_by_avg |
ms. Note that (\[Eq:NewMomCon\]) and (\[Eq:NewHamCon\]) are not coupled if $K = \mbox{constant}$, [*i.e.*]{}, one solves (\[Eq:NewMomCon\]), then (\[Eq:NewHamCon\]). [*No tensor splittings*]{} [@York73; @York74] are needed in the new formulation of the constraints.
Thus, the free data are $\left\{g_{i j}, u_{i j}, N, ... | 1,965 | 2,718 | 3,056 | 1,871 | 2,180 | 0.781963 | github_plus_top10pct_by_avg |
� 7.2 ^\#\#\#^ 2.8 ± 0.8 8.0 ± 2.0 ^\#\#\#^
\#2-6(LT) 10 7.6 ± 6.8 ^\#\#\#^ 4.6 ± 1.3 ^\#\#\#^ 13.4 ± 6.7 ^\#\#\#^ 2.7 ± 0.5 11.7 ± 2.7 ^\#\#\#^
Selectedlines (T~2~) \#1-27(HT)L\#2 \- 4.9 6.1 2... | 1,966 | 3,572 | 2,464 | 2,026 | null | null | github_plus_top10pct_by_avg |
\[-3.40, 1.84\] 0.56 2.04 \[-0.02, 4.10\]... | 1,967 | 6,369 | 883 | 778 | null | null | github_plus_top10pct_by_avg |
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: MIL-STD-882E Severity Categories[]{data-label="Ta:SEVERITY_CATEGORY"}
Description Le... | 1,968 | 6,883 | 677 | 732 | null | null | github_plus_top10pct_by_avg |
}}\,,
\label{alg pp}$$ where the field-dependent parameters are given by $$\begin{aligned}
\Lambda_{\tilde{p}_1\tilde{p}_2}\ =&\
f\xi_0\Big((D_{\tilde{p}_1}f\xi_0 D_{\tilde{p}_2}
- D_{\tilde{p}_2}f\xi_0 D_{\tilde{p}_1})A_Q
+D_{\tilde{p}_1}f\xi_0[F\Psi, F\Xi D_{\tilde{p}_2}F\Psi]
\nonumber\\
&\
-D_{\tilde{p}_2}f\xi_0[... | 1,969 | 658 | 1,553 | 2,081 | null | null | github_plus_top10pct_by_avg |
delta \otimes \delta^{-1} p \delta em.$$ Since $\operatorname{{\mathbf{E}}\text{-deg}}\delta^{-1}p \delta=\operatorname{{\mathbf{E}}\text{-deg}}p \leq -t$, we have $\delta^{-1} p\delta em = 0$ by the hypothesis on $t$.
Therefore $p(he_- \delta \otimes em) = [p,h]e_- \delta \otimes em$ for any such $p$. Since the choic... | 1,970 | 2,246 | 2,151 | 1,751 | 3,846 | 0.769719 | github_plus_top10pct_by_avg |
bound $\sigma^2_k$ for the maximum variance of the functions in ${\cal F}_k$: $$\begin{aligned}
&&\frac{1}{h}\int_{\mathbb{R}}K^{2}\left(\frac{t-x}{h}f^{1/2}(x)\right) I(
|t-x|<hB)f^2(x)dx
\le\frac{1}{h}\int_{\mathbb{R}}K^{2}\left(\frac{t-x}{h}f^{1/2}(x)\right)
f^{2}(x)dx\nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~... | 1,971 | 664 | 1,349 | 1,843 | null | null | github_plus_top10pct_by_avg |
Radon\] For most purposes of this paper, we could have chosen, instead of the particular measure $\mu_S$, any positive Radon measure $\rho$ on (Borel sets of) the unit sphere $S$, as long as relevant additional assumptions are supposed, for example that $\Gamma_0$ has measure zero with respect to $\sigma\otimes \rho\ot... | 1,972 | 993 | 952 | 1,904 | null | null | github_plus_top10pct_by_avg |
tion between non-low protein and low protein CKD patients in different age groups.
Elderly Non-elderly
------------------------------ -------------- ---------------- ---------------- ------------------
Age 70.2 ± 6.8 70.2... | 1,973 | 6,679 | 646 | 406 | null | null | github_plus_top10pct_by_avg |
to a significant degree inspired by the contents of Ref. . Any further search through the SPIRES databasis ([http://www.slac.stanford.edu/spires/hep/]{}; UK mirror: [http://www-spires.dur.ac.uk/spires/hep/]{}) will quickly uncover many more useful reviews.
In Sec. \[Sec2\], we briefly recall the basic facts of relati... | 1,974 | 1,828 | 2,953 | 1,882 | null | null | github_plus_top10pct_by_avg |
romagnetic, the GKS inequalities [@Gri67a] are valid. Averages of products of spin variables are monotone non-decreasing functions of all variables $J(i,j)$ and $H$. Hence, for finite $L_x,\,
L_y$, and $L_z$, $m(i;K,\bbox{r}^{(\text{s})},h)$ is bounded by $m(i;K,r^<,h)$ from below and by $m(i;K,r^>,h)$ from above. We c... | 1,975 | 724 | 1,457 | 1,936 | 3,709 | 0.770551 | github_plus_top10pct_by_avg |
ot h\cdot m=h$ with $i\neq j$ are trivial and the nondiagonal blocks of $\sigma({}^tm)\cdot h\cdot m=h$ are also trivial. The $(j, j)$-block of $\sigma({}^tm)\cdot h\cdot m$ is $$\left\{
\begin{array}{l l}
\pi^j\cdot\begin{pmatrix}a_j&0\\0&(1+\sigma(2 z_j^{\ast}))\cdot (1+2\bar{\gamma}_j)\cdot (1+2 z_j^{\ast}) \end{pma... | 1,976 | 1,360 | 1,912 | 1,769 | null | null | github_plus_top10pct_by_avg |
is level of approximation neither compressibility nor dissipation effects appear explicitly in the inertial force, in analogy with classical compressible fluids [@parmar2012equation]. But these effects are indirectly present by determining the structure of the field ${\boldsymbol{v}}^{(0)}({\boldsymbol{r}},t)$.
Self-i... | 1,977 | 570 | 1,674 | 2,076 | null | null | github_plus_top10pct_by_avg |
su(2)$,
1 hypermultiplet in the $({\bf 1},{\bf 56},{\bf 2})$ of $e_7 \times e_7 \times su(2)$,
4 singlet hypermultiplets.
It is straightforward to compute that there are 272 vector multiplets and 228 hypermultiplets. Since the difference is not 244, this six-dimensional theory is anomalous.
Third cautionary example... | 1,978 | 1,221 | 1,617 | 1,819 | null | null | github_plus_top10pct_by_avg |
72.3 ± 7.4 293.5 ± 2.2 299.8 ± 13.9 284.8 ± 18.3 257.8 ± 10.7 315.4 ± 13.6 71.9 ± 2.7 65.5 ± 4.0 94.6 ± 3.2 107.4 ± 4.2
Hyb 10 452.6 ± 7.2 438.0 ± 11.0 258.7 ± 9.3 297.0 ± 5.7 317.4 ± 2.3 265.8 ± 3.4 269.3 ± 9.1 310.2 ± 3.7 ... | 1,979 | 4,236 | 1,583 | 1,554 | null | null | github_plus_top10pct_by_avg |
\displaystyle\frac{d f^{\pm}(r)}{dr} =
\mp \displaystyle\frac{i}
{k \biggl(\bar{r} +
\displaystyle\frac{1}{C\alpha}\biggr)^{2}}, &
W_{2}(r) =
\displaystyle\frac{\alpha}{\bar{r}}
\displaystyle\frac{1 - 6C\alpha\bar{r}^{3}}
{1 + 3C\alpha\bar{r}^{3}}.
\end... | 1,980 | 1,704 | 2,907 | 1,975 | null | null | github_plus_top10pct_by_avg |
), \dots, B_t(\omega_d)):= ( \langle {\mathbf{1}}_{[0,t)},\omega_1 \rangle, \dots \langle {\mathbf{1}}_{[0,t)},\omega_d \rangle),$$ with ${\boldsymbol \omega}=(\omega_1,\dots, \omega_d) \in S'_d({\mathbb{R}}),\quad t \geq 0,$ in the sense of an $(L^2)$-limit. Here ${\mathbf{1}}_A$ denotes the indicator function of a... | 1,981 | 1,617 | 1,477 | 1,771 | null | null | github_plus_top10pct_by_avg |
\boldsymbol\eta}, {\boldsymbol\eta}\rangle}} \exp\left( -\frac{1}{2\langle {\boldsymbol\eta},{\boldsymbol\eta} \rangle}(i\langle {\boldsymbol\eta},{\bf f} \rangle - x)^2 -\frac{1}{2}\langle {\bf f},{\bf f}\rangle \right), \, \, \mathbf{f} \in S_d({\mathbb{R}}).$$
Generalized Gauss Kernels
-------------------------
He... | 1,982 | 945 | 1,953 | 1,906 | null | null | github_plus_top10pct_by_avg |
}](fig24-AsymSpectra_Ld){width="50.00000%"}
A comment is in order with regards to particle-hole asymmetry. While for particle-hole symmetry, the resonance in the spectral function remains pinned to zero-frequency, a particle-hole asymmetry allows for a continuous change of the scattering phase [@Langreth1966; @Yoshimo... | 1,983 | 230 | 1,737 | 2,078 | 4,022 | 0.768597 | github_plus_top10pct_by_avg |
\end{matrix} \right]}^{T}}, \nonumber
\end{array}$$ where $b_i$ is the viscous friction at the $i^{th}$ joint. $M(\theta)$ is a $4\times4$ matrix of the mass moment of inertia, whose components are specified by $$\begin{array}{r@{}l@{\qquad}l}
& {{m}_{11}}={{A}_{1}}+{{A}_{2}}+2{{A}_{3}}\cos {{\th... | 1,984 | 5,008 | 291 | 1,664 | null | null | github_plus_top10pct_by_avg |
\to e\gamma)\lesssim 4.2\times 10^{-13}$ [@TheMEG:2016wtm; @Renga:2018fpd]. Meanwhile, our theoretical formula is given by $$\begin{aligned}
{\rm BR}(\mu\to e\gamma)&= \frac{3\alpha_{\rm em}}{16\pi{\rm G_F^2}}
\left| G(M_D,m_{\eta_1})\left( \sum_{a=1}^3 y_{N_{R_{1a}}} y^\dag_{N_{R_{a2}}} \right)
+
G(M_D,m_{\eta_2}) \l... | 1,985 | 2,001 | 2,308 | 1,885 | 3,397 | 0.772695 | github_plus_top10pct_by_avg |
q {\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]$, certainly ${\mathcal{J}}[\delta^{-2}] = e{\mathbb{C}}[{\mathfrak{h}^{\text{reg}}}\oplus {\mathfrak{h}}^{\ast}]=
e( {\mathbb{C}}[{\mathfrak{h}^{\text{reg}}}\oplus {\mathfrak{h}}^{\ast}]\ast {{W}})$. Since $\Theta$ is given by inclusion, $ \Theta[\delta^{-2}]$ is th... | 1,986 | 1,131 | 915 | 2,098 | null | null | github_plus_top10pct_by_avg |
\quad {\mathbb{E}}\bigl\vert Y_a^{(n)}\bigr\vert^2 = 1,
\quad \text{and} \quad {\mathbb{E}}\bigl(Y_a^{(n)}\bigr)^2 = 0$$ for every $a \in G^{(n)}$; and that holds. Then $\mu^{(n)}$ converges, in mean and in probability, to $\gamma_{\mathbb{C}}$.
The special case of Theorem \[T:circular-law-uncorrelated\] for classi... | 1,987 | 4,191 | 1,886 | 1,347 | null | null | github_plus_top10pct_by_avg |
quality of (\[Eq:Bargdot\]), that the conformal invariance of $\beta^k$ ($=\bar{\beta}^k$) and $K$ ($= \bar{K}$) are fully consistent, having led to the precisely geometrically correct form of $\dot{\bar{g}}_{i j}$ by virtue also of $\bar{N} = \psi^6 N$.
This interpretation of the semi-linear elliptic constraint syste... | 1,988 | 1,904 | 3,276 | 1,849 | null | null | github_plus_top10pct_by_avg |
dex $j$ is summed. In the IIA theory, these conditions are mapped to the conditions for the $5_3^{2,1}$-branes associated to the components $
E_{4\,x^iy^ix^jy^jx^k,x^ix^jx^k,x^k}$ of the potential $E_{9,3,1}$ (see Table \[Ebranestable\]). This can be shown by evaluating for these components the constraints coming from ... | 1,989 | 953 | 2,006 | 1,989 | 2,605 | 0.778499 | github_plus_top10pct_by_avg |
y}\sup_{f\in {\cal D}}{\Pr}_f\left\{\sup_{n\ge k}\frac{1}{a_n}|Z_n(X_1,\dots,X_n,f)|>C\right\}=0,$$ and $o_{\rm a.s.}(a_n)$ uniformly in $f\in\cal D$ if the limit (\[defunif\]) holds for every $C>0$.
For $0<C<\infty$ and non-negative function $z$ such that $z(\delta)\searrow 0$ as $\delta\searrow 0$, define the class ... | 1,990 | 1,295 | 1,813 | 1,931 | null | null | github_plus_top10pct_by_avg |
ith probability at least $1 - d^{-3}$, we have, $$\begin{aligned}
\label{eq:topl_error}
{\|M - \E[M]\|} \leq 22e^{b}\sqrt{\frac{\log d}{\beta d} \sum_{j=1}^n \ell_j} +\frac{64 \log d}{3} \leq 32e^{b}\sqrt{\frac{\log d}{\beta d}\sum_{j=1}^n \ell_j} \;,\end{aligned}$$ where the second inequality follows from the assumpt... | 1,991 | 2,792 | 2,092 | 1,912 | null | null | github_plus_top10pct_by_avg |
nd discusses the general formulation of these transformations for any dimension $N$.
$\boldsymbol{{\mathbb{Z}}_N}$-graded Lax pairs {#sec:ZN-LP}
==============================================
We now consider the specific discrete Lax pairs, which we introduced in [@f14-3; @f17-2]. Consider a pair of matrix equations ... | 1,992 | 2,970 | 2,339 | 1,968 | 2,273 | 0.781219 | github_plus_top10pct_by_avg |
ed (see Sec. \[sec:mass\_loss\_inner\]).
### Dependence on BH mass {#sec:Mdep}
run $M_{\mathrm{BH}}\,[M_\odot]$ $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ $\dot{M}/\dot{M}_{\mathrm{B}}$
------------ ------------------------------ -------------------------------------------- -------------------------... | 1,993 | 2,947 | 3,211 | 2,090 | 4,184 | 0.767491 | github_plus_top10pct_by_avg |
lay the peak and reduce the final size of the epidemic ([@B80]). However, the course of the COVID-19 epidemic is defined by a series of further key factors, some of which are still poorly understood ([@B1]). The amount of scientific data produced during the COVID-19 pandemic is amazingly huge, and this can be crucial t... | 1,994 | 1,923 | 3,545 | 1,845 | null | null | github_plus_top10pct_by_avg |
e will frequently encounter this situation.
\[stequivnew\] Let ${{\mathscr C}}$ be any plane curve, with defining homogeneous ideal $(F(x,y,z))$. If $\alpha(t)$, $\beta(t)$ are equivalent germs, then the initial terms in $F\circ\alpha(t)$, $F\circ\beta(t)$ coincide up to a nonzero multiplicative constant; in particula... | 1,995 | 610 | 2,547 | 2,083 | 2,034 | 0.783305 | github_plus_top10pct_by_avg |
quad + \E{\int_0^1 \int_0^t \lin{\nabla^2 a(y_{k\delta} + s(y_{(k+1)\delta} - y_{k\delta}), (y_{(k+1)\delta} - y_{k\delta})(y_{(k+1)\delta} - y_{k\delta})^T} dt ds}\\
\leq& \E{a(y_{k\delta})} + \E{\lin{\nabla a(y_{k\delta}), y_{(k+1)\delta} - y_{k\delta}}} + \E{\lrn{(y_{(k+1)\delta} - y_{k\delta})}_2^2 ds}\\
... | 1,996 | 2,737 | 1,108 | 1,822 | null | null | github_plus_top10pct_by_avg |
redes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch and I. Bloch, Nature [**429**]{}, 277 (2004); Toshiya Kinoshita, Trevor Wenger, and David S. Weiss, Science [**305**]{}, 1125 (2004).
G. Thalhammer, M. Theis, K. Winkler, R. Grimm, J. H. Denschlag, available at arXiv.org/ cond-... | 1,997 | 1,334 | 982 | 1,930 | 1,293 | 0.791463 | github_plus_top10pct_by_avg |
with $$\begin{gathered}
\label{s-dual}
k+\ell=N,\qquad \ell-k =1 \quad\Rightarrow\quad N=2k+1,\end{gathered}$$ so we require that $N$ is [*odd*]{}. In this case, we have that Equations (\[eq:dLP-gen-sys-1\]) are invariant under the change $$\begin{gathered}
\label{sd-phi}
\big(\phi^{(i)}_{m,n},\alpha,\beta\big) \mapsto... | 1,998 | 639 | 1,599 | 1,993 | 3,483 | 0.772035 | github_plus_top10pct_by_avg |
$\chi $ is $p$-finite and $$\chi ({\alpha }_p,\beta )^{{b}-1}\chi (\beta ,{\alpha }_p)^{{b}-1}=
\frac{\rhomap{r_p(\chi )}({\sigma }_p^\chi (\beta ))}{\rhomap\chi (\beta )}$$ for all $\beta \in {\mathbb{Z}}^I$. \[le:rho\]
Define $\xi _1,\xi _2:{\mathbb{Z}}^I\to {{\Bbbk }^\times }$ by $$\xi _1(\beta )=\chi ({\alpha }_... | 1,999 | 2,815 | 1,641 | 1,869 | null | null | github_plus_top10pct_by_avg |
al associated with the surface screening charges. The DGF is pre-computed once at the beginning of any simulation. In this Appendix we outline the numerical method used to evaluate the DGF.
Cartesian Grid {#s:calc_dgf_cart}
--------------
We calculate the DGF ${\cal G}_{i-i',j-j',k-k'}$ in Cartesian coordinates by so... | 2,000 | 2,371 | 3,466 | 2,185 | 2,002 | 0.783625 | github_plus_top10pct_by_avg |
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