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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"}](outil_cas_i.jpg){width="3.5in"}
#### Case ii.
If $h$ is defined by (\[eq:4.17\].ii), then the corresponding nontrival solution of system (\[eq:4.1\]) takes the form
\[eq:4.27\] u(t,x,y)=&e(c\_3(t))+2,\
v(t,x,y)=&e(c\_3(t))+2,\
(t,x,y)=&-(\^[-1]{}\
&),
where the mean pressure $\sigma$ is given by (\[eq:4.3\]), where we substitute the values of functions $u,v,\theta$ given by (\[eq:4.21\]). The complex functions $c_i(t)$, $i=1,2,3$, and the real function $\sigma_0(t)$ which appear in this solution are arbitrary. For any time $t_0$, solution (\[eq:4.3\]), (\[eq:4.27\]) has a singularity at point $(x_0,y_0)$ which satisfies the equation $$x_0^2+y_0^2+2{\left( \mathcal{R}e(c_2(t_0))x_0+\mathcal{I}m(c_2(t_0))y_0 \right)}+|c_2(t_0)|^2=0.$$This singularity is stationary if the function $c_2(t)$ is constant. Otherwise, its position varies with time. If the functions $\sigma_0$ and $c_i$, $i=1,2,3$, are of the form (\[eq:4.25\]) defined in a region of the $xy$-plane on a time interval $[T_0,T)$ where the gradient catastrophe does not occur, then the solution is bounded and damped. In figure \[fig:2\], we have drawn the shape of an extrusion die corresponding to the solution (\[eq:4.27\]) for the following choice of parameters: $\mathcal{R}e{\left( c_1(t) \right)}=0$, $\mathcal{I}m{\left( c_1(t) \right)}=0$, $\mathcal{R}e{\left( c_2(t) \right)}=0$, $\mathcal{I}m{\left( c_2(t) \right)}=-0.5$, $\mathcal{R}e{\left( c_3(t) \right)}=-0.5$ and $\mathcal{I}m{\left( c_3(t) \right)}=0$. The feeding velocity has component $U_0=0.2$, $V_0=0.2$, and the extraction of material is performed at the velocity $U_1=0.2$, $V_1=-0.2$. This type of tool can be used to bend a rod by extrusion without having to fold it. Finally
| 1,901
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delta_{i+1}v_{i+1}\cdot m_{i+1, i}=\pi m_{i,i}^{\ast}$$ such that $m_{i,i}^{\ast} \in M_{1\times n_i}(B\otimes_AR)$. This equation is considered in $B\otimes_AR$ and $\pi$ stands for $\pi\otimes 1\in B\otimes_AR$. Here,
- $v_{i-1}=(0,\cdots, 0, 1)$ (resp. $v_{i-1}=(0,\cdots, 0, 1, 0)$) of size $1\times n_{i-1}$ if $L_{i-1}$ is *of type* $\textit{I}^o$ (resp. *of type* $\textit{I}^e$).
- $v_{i+1}=(0,\cdots, 0, 1)$ (resp. $v_{i+1}=(0,\cdots, 0, 1, 0)$) of size $1\times n_{i+1}$ if $L_{i+1}$ is *of type* $\textit{I}^o$ (resp. *of type* $\textit{I}^e$).
- Assume that $i$ is odd and that $L_i$ is *bound of type I*. Then $$\delta_{i-1}v_{i-1}\cdot {}^tm_{i, i-1}+\delta_{i+1}v_{i+1}\cdot {}^tm_{i, i+1} = \pi m_{i,i}^{\ast\ast}$$ such that $ m_{i,i}^{\ast\ast} \in M_{1\times n_i}(B\otimes_AR)$. This equation is considered in $B\otimes_AR$ and $\pi$ stands for $\pi\otimes 1\in B\otimes_AR$. Here, $v_{i-1}$ (resp. $v_{i+1}$)$=(0,\cdots, 0, 1)$ of size $1\times n_{i-1}$ (resp. $1\times n_{i+1}$).\
Let $$\tilde{M_i}= \mathrm{GL}_{B/\pi B}(B_i/Y_i) \textit{ for all $i$}.$$
Let $s_i=m_{i,i}$ if $L_i$ is *of type II* or if $L_i$ is *bound of type I* with $i$ odd in the above description of an element of $\tilde{M}(R)$. Then $s_i$ mod $\pi\otimes 1$ is an element of $\tilde{M}_i(R)$. Therefore, we have a surjective morphism of algebraic groups $$r : \tilde{M} \longrightarrow \prod\tilde{M}_i, ~~~~~~~ m \mapsto \prod \left(\textit{$s_i$ mod $\pi\otimes 1$}\right)$$ defined over $\kappa$. We now have the following easy lemma:
\[la2\] The kernel of $r$ is the unipotent radical $\tilde{M}^+$ of $\tilde{M}$, and $\prod\tilde{M}_i$ is the maximal reductive quotient of $\tilde{M}$.\
Since $\prod\tilde{M}_i$ is a reductive group, we only have to show that the kernel of $r$ is a connected smooth unipotent group. By the description of the morphism $r$ in terms of matrices explained above, the kernel of $r$ is isomorphic to an affine space as an algebraic variety over $\kappa$. Therefore, it is a connected smooth unipo
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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 & $-0.08$ & 0.45 (F5) & 0.64 & 2.01 & 2.65 & 8.0 2.5$\times 10^{-7}$ & 0.7 & 0.30 & 5500 & 16.71 & $+0.00$ & 0.53 (F8) & 0.56 & 1.76 & 2.32 & 7.5 1.0$\times 10^{-7}$ & 0.7 & 0.25 & 5000 & 16.02 & $+0.12$ & 0.66 (G4) & 0.44 & 1.38 & 1.82 & 6.9 5.0$\times 10^{-8}$ & 0.7 & 0.25 & 4600 & 15.45 & $+0.24$ & 0.78 (G9) & 0.32 & 1.01 & 1.33 & 6.7 2.5$\times 10^{-8}$ & 0.7 & 0.25 & 4200 & 14.72 & $+0.37$ & 0.91 (K2) & 0.19 & 0.60 & 0.79 & 6.1 1.0$\times 10^{-8}$ & 0.7 & 0.25 & 3700 & 13.58 & $+0.58$ & 1.13 (K5) & & & & 5.2
[lcccccccc]{} $L_{\rm WD,0}=2000 L_\odot$ & 6100 & 17.31 & $-0.09$ & 0.41 (F3) & 0.65 & 2.04 & 2.60 & 8.4 $L_{\rm WD,0}=4000 L_\odot$ & 6400 & 17.59 & $-0.13$ & 0.36 (F1) & 0.69 & 2.17 & 2.87 & 8.8 $\eta_{\rm ir,MS}=1.0$ & 5500 & 16.76 & $+0.00$ & 0.53 (F8) & 0.56 & 1.76 & 2.32 & 7.7 $\eta_{\rm ir,MS}=0.25$ & 5500 & 16.66 & $+0.00$ & 0.53 (F8) & 0.56 & 1.76 & 2.32 & 7.4 $\eta_{\rm ir,DK}=1.0$ & 6000 & 17.16 & $-0.06$ & 0.43 (F4) & 0.62 & 1.95 & 2.57 & 8.3 $\eta_{\rm ir,DK}=0.25$ & 5300 & 16.39 & $+0.08$ & 0.58 (F9) & 0.48 & 1.51 & 1.99 & 7.6 $\nu=3.0$ & 5600 & 16.90 & $-0.04$ & 0.51 (F7) & 0.60 & 1.88 & 2.48 & 7.6 $\nu=1.25$ & 5300 & 16.39 & $+0.09$ & 0.58 (F9) & 0.47 & 1.47 & 1.95 & 7.7
---
author:
- 'Aimeric Colléaux $^{1,}$ [^1] , Sergio Zerbini $^{2,}$'
title: |
Modified Gravity Models\
Admitting Second Order Equations of Motion
---
**Abstract** : The aim of this paper is to find higher order geometrical corrections to the Einstein-Hilbert action that can lead to only second order equations of motion. The metric formalism is used, and static spherically symmetric and Friedmann-Lemaître space-times are considered, in four dimensions. The FKWC-basis are introdu
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|
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 \[fig:piN-bd\]):
\[prp:diagram-bd\] For the ferromagnetic Ising model, we have $$\begin{aligned}
\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)\leq
\begin{cases}{\label{eq:piNbd}}
P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}(o,x)\equiv{{\langle \varphi_o\varphi_x \rangle}}_\Lambda^3
&(j=0),\\[5pt]
{\displaystyle}\sum_{\substack{b_1,\dots,b_j\\ v_1,\dots,v_j}}P_{\Lambda;v_1}^{\prime
{{\scriptscriptstyle}(0)}}(o,{\underline{b}}_1)\,\bigg(\prod_{i=1}^{j-1}\tau_{b_i}Q''_{\Lambda;v_i,v_{
i+1}}({\overline{b}}_i,{\underline{b}}_{i+1})\bigg)\,\tau_{b_j}Q'_{\Lambda;v_j}({\overline{b}}_j,x)&(j\ge1),
\end{cases}\end{aligned}$$ where, as well as in the rest of the paper, the empty product is regarded as 1 by convention.
$$\begin{aligned}
\pi^{{\scriptscriptstyle}(1)}_\Lambda(x)\lesssim\raisebox{-11pt}{\includegraphics[scale=
0.18]{pi1}}\qquad
\pi^{{\scriptscriptstyle}(2)}_\Lambda(x)\lesssim\raisebox{-20pt}{\includegraphics[scale=
0.18]{pi21}}+\raisebox{-20pt}{\includegraphics[scale=0.18]{pi22}}\end{aligned}$$
Bound on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ {#ss:pi0bd}
---------------------------------------------------
The key ingredient of the proof of Proposition \[prp:diagram-bd\] is Lemma \[lmm:GHS-BK\] below, which is an extension of the GHS idea used in the proof of Lemma \[lmm:switching\]. In this subsection, we demonstrate how this extension works to prove the bound on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ and the inequality $$\begin{aligned}
{\label{eq:pi0'-bd}}
\sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}
x\\}$}}}\,\cap\,\{o{\underset{\raiseb
| 1,904
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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 + 3*p + 344*t**3 + 601050*t**2 wrt t.
-202878*p + 2064
Find the third derivative of -2*r*
-12755, -12236, -11371, -10160, -8603?
173*j**2 - 12928
What is the b'th term of 132495, 265001, 397499, 529989, 662471, 794945, 927411?
-4*b**2 + 132518*b - 19
What is the y'th term of -253416, -253469, -253550, -253653, -253772, -253901, -254034, -254165?
y**3 - 20*y**2 - 253397
What is the t'th term of -21112274, -21112277, -21112280?
-3*t - 21112271
What is the s'th term of 19007, 19505, 19991, 20465, 20927, 21377, 21815?
-6*s**2 + 516*s + 18497
What is the c'th term of -50642650, -50642649, -50642648?
c - 50642651
What is the w'th term of -444044, -1775507, -3994610, -7101353?
-443820*w**2 - 3*w - 221
What is the u'th term of -5717689, -11435377, -17153065, -22870753?
-5717688*u - 1
What is the t'th term of -137966576, -275933149, -413899722, -551866295, -689832868?
-137966573*t - 3
What is the n'th term of -307908, -615824, -923738, -1231650, -1539560, -1847468, -2155374?
n**2 - 307919*n + 10
What is the c'th term of 82630, 661067, 2231096, 5288509, 10329098, 17848655, 28342972?
82632*c**3 + 4*c**2 + c - 7
What is the y'th term of -870, -1008, -1238, -1560?
-46*y**2 - 824
What is the q'th term of 44464, 178410, 401654, 714196, 1116036, 1607174, 2187610?
44649*q**2 - q - 184
What is the g'th term of 22367156, 44734325, 67101494?
22367169*g - 13
What is the u'th term of -111098348, -222196698, -333295048, -444393398?
-111098350*u + 2
What is the p'th term of -838, -3154, -6774, -11506, -17158?
32*p**3 - 844*p**2 - 8*p - 18
What is the i'th term of 355948, 356376, 357096, 358114, 359436, 361068, 363016?
i**3 + 140*i**2 + i + 355806
What is the r'th term of 3318, 6398, 11532, 18720, 27962?
1027*r**2 - r + 2
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|
+ \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} \leq 6\lrp{R^2 + \frac{\beta^2}{m}}
\end{aligned}$$
For our second claim that $\Ep{p^*}{\lrn{x}_2^2} \leq R^2 + \frac{\beta^2}{m}$, we can use the fact that if $x_0 \sim p^*$, then $\E{a(x_t)}$ does not change as $p^*$ is invariant, so that $$\begin{aligned}
0 = \ddt \E{a(x_t)} \leq -m\E{a(x_t)} + \beta^2
\end{aligned}$$ Thus $$\begin{aligned}
\E{a(x_t)} \leq \frac{\beta^2}{m}
\end{aligned}$$ Again, $$\begin{aligned}
\Ep{p^*}{\lrn{x}_2^2} = \E{\lrn{x_t}_2^2} \leq 2 \E{a(x_t)} + 4 R^2 \leq 4 \lrp{R^2 + \frac{\beta^2}{m}}
\end{aligned}$$
\[l:energy\_y\] Let the sequence $y_{k\delta}$ be as defined in . Assuming that $\delta \leq m/(16L^2)$ and $\E{\|y_0\|_2^2} \leq 2 \lrp{R^2 + \frac{\beta^2}{m}}$ Then for all $k$, $$\begin{aligned}
\E{\|y_{k\delta}\|_2^2} \leq 8 \lrp{R^2 + \frac{\beta^2}{m}}
\end{aligned}$$
Let $a(w) := \lrp{\|w\|_2 - R}_+^2$. We can verify that $$\begin{aligned}
\nabla a(w) =& \lrp{\|w\|_2 - R}_+\frac{w}{\|w\|_2}\\
\nabla^2 a(w) =& \ind{\|w\|_2 \geq R}\frac{ww^T}{\|w\|_2^2} + \lrp{\|w\|_2 - R}_+ \frac{1}{\|w\|_2} \lrp{I - \frac{ww^T}{\|w\|_2^2}}
\end{aligned}$$ Observe that
1. $\lrn{\nabla^2 a(w)}_2 \leq 2 \ind{\|w\|_2 \geq R} \leq 2$
2. $\lin{\nabla a(w), - \nabla U(w)} \leq -m a(w)$.
3. $a(w) \geq \frac{1}{2}\|w\|_2^2 - 2R^2$.
The proofs are identical to the proof at the start of Lemma \[l:energy\_w\], so we omit them here.
Using Taylor’s Theorem, and taking expectation of $y_{(k+1)\delta}$ conditioned on $y_{k\delta}$, $$\begin{aligned}
&\E{a(y_{(k+1)\delta})}\\
=& \E{a(y_{k\delta})} + \E{\lin{\nabla a(y_{k\delta}), y_{(k+1)\delta} - y_{k\delta}}}\\
&\
| 1,906
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|
})\right],
\label{eq:Phodge}$$ 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 the normalization $$\label{eq:FixE0_3D}
\mathbb{P}_{\mathcal{S}'_{\E_0}}({\mathbf{u}}) = \sqrt{\frac{\E_0}{\E\left({\mathbf{u}}\right)}}\,{\mathbf{u}}.$$
Thus, composing with , the projection onto the manifold ${\mathcal{S}_{\E_0}}$ defined in problem \[pb:maxdEdt\_E\] is constructed as $$\mathbb{P}_{\mathcal{S}_{\E_0}}({\mathbf{u}}) = \mathbb{P}_{\mathcal{S}'_{\E_0}}\Big( \mathbb{P}_{\mathcal{S}_0} ({\mathbf{u}})\Big).
\label{eq:P}$$ This approach, which was already successfully employed by [@ap11a; @ap13a], allows one to enforce the enstrophy constraint essentially with the machine precision.
For a given value of $\E_0$, the maximizer ${\widetilde{\mathbf{u}}_{\E_0}}$ can be found as ${\widetilde{\mathbf{u}}_{\E_0}}= \lim_{n\rightarrow \infty} {\mathbf{u}}_{\E_0}^{(n)}$ using the following iterative procedure representing a discretization of a gradient flow projected on $\mathcal{S}_{\E_0}$ $$\begin{aligned}
{\mathbf{u}}_{\E_0}^{(n+1)} & = \mathbb{P}_{\mathcal{S}_{\E_0}}\left(\;{\mathbf{u}}^{(n)}_{\E_0} + \tau_n \nabla\R\left({\mathbf{u}}^{(n)}_{\E_0}\right)\;\right), \\
{\mathbf{u}}_{\E_0}^{(1)} & = {\mathbf{u}}^0,
\end{aligned}
\label{eq:desc}$$ where ${\mathbf{u}}^{(n)}_{\E_0}$ is an approximation of the maximizer obtained at the $n$-th iteration, ${\mathbf{u}}^0$ is the initial guess and $\tau_n$ is the length of the step in the direction of the gradient. It is ensured that the maximizers ${\widetilde{\mathbf{u}}_{\E_0}}$ obtained for different values of $\E_0$ lie on the same [maximizing]{} branch by using the continuation approach, where the maximizer ${\widetilde{\mathbf{u}}_{\E_0}}$ is [employed]{} as the initial guess ${\mathbf{u}}^0$ to compute $\widetilde{\mathbf{u}}_{\E_0+\D
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|
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 top and bottom surfaces. We see in Fig. \[fig:mdoio\_Ds\] that modeled $\dot{M}_{\mathrm{outflow}}$ with the best-fit value of $f_{\mathrm{outflow}}=0.7$ reproduces the simulation result with remarkable agreement.
### Possible mass loss from neutral inflow inside the sink {#sec:mass_loss_inner}
As mentioned above, we neglect the possible mass loss from the innermost part of the flow masked by the sink. Since the size of the accretion disc is supposed to be much smaller than the sink radius $R_{\mathrm{in}}$ (see Fig. \[fig:acc\_whole\]), we neglect the centrifugal effect and assume the similar flow structure extends inward. As an upper limit for the mass-loss rate, we evaluate the integral with the same integrand as equation but for the different range of $0< r <
R_{\mathrm{in}}$, and obtain $$\begin{aligned}
\dot{M}_{\mathrm{loss}}(<R_{\mathrm{in}})
\lesssim 0.1 \left[\frac{R_{\mathrm{in}}}{(r_{\mathrm{B}}/7)}\right]^{1/2} \dot{M}_{\mathrm{B}}\,.
\label{eq:4}\end{aligned}$$ Here, we take $R_{\mathrm{in}} \approx r_{\mathrm{B}}/7$ of our simulation setup (see Table \[tab:model\]) as a reference value.
The mass supply rate to the accretion disc can be conservatively estimated by $\dot{M} - \dot{M}_{\mathrm{loss}}(<R_{\mathrm{in}})$, meaning that $\dot{M}$ measured in the simulation slightly overestimates the true value. This would be alleviated by taking a smaller value for $R_{\mathrm{in}}$. We have performed a test run with the smaller sink radius (see Appendix \[sec:res\_check\]), but found no remarkable differences of the accretion rate. Note that the effect of the angular momentum becomes important on the smaller scale. It is not allowed to take an arbitrary small sink radius without con
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|
\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$$ denotes the quasienergy splitting. Hence, the particle is coherently oscillating between the two defects; the transfer time $T_{\rm trans}$, after which the particle will be found at the other defect, is given by $$T_{\rm trans} = \frac{\pi \hbar}{\Delta \varepsilon^F}$$ and thus depends on the driving amplitude $F$.
When the amplitude changes sufficiently slowly in time, the system responds in an adiabatic manner [@BreuerHolthaus89]. Hence, under the influence of a slowly varying amplitude $F(t)$, the initial state (\[eq:ini\]) follows the instantaneous Floquet states and evolves into $$\begin{aligned}
\left|\psi(t)\right> & = & \frac1{\sqrt{2}}\left[ \left|u_1^{F(t)}(t)\right>\right.
\nonumber \\ & & \left.
\pm \left|u_2^{F(t)}(t)\right>
\exp\left(-\frac{{{\rm i}}}{\hbar} \int_0^t \! {{\rm d}}\tau \,
\Delta\varepsilon^{F(\tau)}\right) \right]
\nonumber \\ & & \times
\exp\left(-\frac{{{\rm i}}}{\hbar} \int_0^t \! {{\rm d}}\tau \,
\varepsilon_1^{F(\tau)}\right) \; . \end{aligned}$$ This implies that the transfer time from one defect to the other now is given by the relation $$\label{eq:int}
\frac{1}{\hbar} \int_0^{T_{\rm trans}} \! {{\rm d}}\tau \,
\Delta\varepsilon^{F(\tau)} = \pi \; ,$$ which constitutes an immediate analog of the $\pi$-pulse-condition known from two-level systems [@AllenEberly87]. This is the physics which will now be exploited for coherent control of population transfer between two communicating defects.
To this end, the driving amplitude $F(t)$ is shaped such that one has $eFd/(\hbar\omega) = j_{0,1}$ for $t\le 0$, so that the defect Floquet states are confined to their respective sites, and the communication between the two defects is effectively disrupted. Then the amplitude is adiabatically lowered such that the
| 1,909
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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 database 1989]; (3) [@Ferland:1992aa]; (4) [@Hummer:1998aa]; (5) [@Aldrovandi:1973aa]; (6) [@Draine:2011aa]; (7) [@Zygelman:1989aa]; (8) [@Kimura:1993aa]; (9) [@Palla:1983aa].
In Table \[tab:reaction\_rates\], we summarize the chemical reactions considered in this work, which are adopted following [@Glover:2007aa], [@Abel:1997aa] and [@Anninos:1997aa]. We adopt the Case B recombination rates for the recombination of ${\mathrm{H^+}}$, ${\mathrm{He^+}}$ and ${\mathrm{He^{2+}}}$. We neglect the ${\mathrm{He^+}}$ recombination through the quasi-stable triplet state $2^3S$ of ${\mathrm{He}}$, assuming that ${\mathrm{He}}$ in that state is easily photoionized by the BH irradiation [see, e.g., @Clegg:1989aa].
Cross sections {#sec:cross_sections}
--------------
No. Reaction Cross section ${\mathrm{[cm^2]}}$ Ref.
----- ---------------------------------------------------------------------- ----------------------------------- --------------------------------------------- ------
1 ${\mathrm{H}}+h\nu \rightarrow {\mathrm{H^+}} + {\mathrm{e}}$ $\sigma_{\nu,1}=$ $h\nu_{\mathrm{T,1}}= 13.60{\,\mathrm{eV}}$ 1
2 ${\mathrm{He}}+h\nu \rightarrow {\mathrm{He^+}} + {\mathrm{e}}$ $\sigma_{\nu,2}=$ $h\nu_{\mathrm{T,2}}= 24.58{\,\mathrm{eV}}$ 2
3 ${\mathrm{He^+}}+h\nu \rightarrow {\mathrm{He^{2+}}} + {\mathrm{e}}$ $\sigma_{\nu,3}=$ $h\nu_{\mathrm{T,3}}= 54.40{\,\mathrm{eV}}$ 1
\
REFERENCES. (1) [@Osterbrock:1989aa]; (2) [@Yan:1998aa].
In Table \[tab:cross\_sections\], we summarize the cross sections con
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|
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 species $X$ in units of ${\mathrm{cm^{-3}}}$. We calculate the photoionization heating rates as $\Gamma_i=\int (4\pi j_\nu/h\nu)
n(X_i)\sigma_{\nu,i} (h\nu - h\nu_{{\mathrm{T}},i}){\mathrm{d}}\nu$ (see Table \[tab:cross\_sections\]), with $X_1={\mathrm{H}}$, $X_2= {\mathrm{He}}$ and $X_3={\mathrm{He^{+}}}$.
Resolution check {#sec:res_check}
================
![Same as Fig. \[fig:mdot\] but for the runs in App. \[sec:res\_check\]. The physical parameters are the same as Dds run but the resolution is different in each run. See the text for details.[]{data-label="fig:mdot_rdep"}](figure/mdot_rdep.eps){width="8.5cm"}
To check the resolution dependence of our results, we here see how the evolution of $\dot{M}$ is affected by numerical settings, namely the number of grids and sink size $R_{\mathrm{in}}$. Taking the same physical parameters as Dds run, we perform additional simulations with different resolutions, as shown in Fig. \[fig:mdot\_rdep\]. Here, we take $N_r
\times N_\theta = 512\times 144$; $N_r \times N_\theta = 256\times 72$; $N_r \times N_\theta = 128\times 36$; $N_r \times N_\theta = 256\times
72$ with $R_{\mathrm{in}}$ halved and doubled from the fiducial value. Note that our main results are obtained with the high- and medium-resolution simulations with $N_r \times N_\theta = 512\times 144$ and $256\times
72$, respectively.
The dependence on the number of grids is checked by comparing the results with $N_r \times N_\theta = 512\times 144$, $256\times 72$, and $128\times 36$ (Fig. \[fig:mdot\_rdep\]). The strong variability of $\dot{M}$ for $t \lesssim 10^6{\,\mathrm{yr}}$ seen with the highest-resolution is smoothed out with the lower resolutions. However, the values of $\dot{M}$ a
| 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 postbleaching tooth sensitivity, making this esthetic clinical procedure safer and more comfortable to the patients.
The authors acknowledge the partial support by the Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq (Grant no. 301291/2010-1), Fundação Amparo à Pesquisa do Estado de São Paulo-FAPESP (Grant nos. 2011/15366-5 and 2011/12938-8) and FUNDUNESP (Grant no. 0024/021/13-PROPe-CDC).
Conflict of Interests
=====================
The authors have no conflict of interests.
######
Results of the viability of the MDPC-23 cells exposed to different hydrogen peroxide (H~2~O~2~) concentrations for determination of the IC-50.
H~2~O~2~concentration Cell viability (%)
----------------------- --------------------
0 100
0.035% 5
0.018% 41
0.009% 77
0.0045% 72
######
Control and experimental groups (*n* = 6) formed according to the treatment of the MDPC-23 cells with different alpha-tocopherol (*α*-T) concentrations followed by exposure or not to hydrogen peroxide (H~2~O~2~).
Groups Treatment
-------- ---------------------
G1 (*α*-T− H~2~O~2~−)
G2 (*α*-T− H~2~O~2~+)
G3 (1 mM+ H~2~O~2~−)
G4 (3 mM+ H~2~O~2~−)
G5 (5 mM+ H~2~O~2~−)
G6 (10 mM+ H~2~O~2~−)
G7 (1 mM+ H~2~O~2~+)
G8 (3 mM+ H~2~O~2~+)
G9 (5 mM+ H~2~O~2~+)
G10 (10 mM+ H~2~O~2~+)
######
Percentage of viability of MDPC-23 cells treated with different alpha-tocopherol concentrations followed by exposure or not to hydrogen peroxide.
H~2~O~2~ Alpha-tocopherol concentrations
| 1,912
| 2,976
| 1,806
| 1,929
| null | null |
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|
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 ${\delta }:V\to {\Bbbk }{\mathbb{Z}}^I{\otimes }V$ are defined by $$\begin{aligned}
K_i {\boldsymbol{\cdot}}E_j=q_{ij}E_j,\qquad {\delta }(E_i)=K_i{\otimes }E_i
\end{aligned}$$ for all $i,j\in I$. The algebra $U^+(\chi )$ is commonly known as the *Nichols algebra* of the module $V$.
\(ii) There are various descriptions of the ideal ${\mathcal{I}}^+(\chi )$, see *e.g.* [@inp-AndrSchn02]. In case of quantized enveloping algebras, see Sect. \[sec:Uqg\], Serre relations generate the ideal ${\mathcal{I}}^+(\chi )$. A more general case is studied by Angiono [@p-Angi08]. For quantized Lie superalgebras the defining relations are determined in [@a-Yam99; @a-Yam99e]. It is in general an open problem to give a nice set of generators of ${\mathcal{I}}^+(\chi )$, see [@inp-Andr02 Question5.9].
(Triangular decomposition) The map $$\mathrm{m}(\iota _-{\otimes }\iota _0{\otimes }\iota _+):
U^-(\chi ){\otimes }{{\mathcal{U}}^0}{\otimes }U^+(\chi )\to U(\chi )$$ is an isomorphism of ${\mathbb{Z}}^I$-graded vector spaces, where $\mathrm{m}$ denotes the multiplication map. \[pr:tridec\]
Following the convention in [@b-Joseph Sect.3.2.1], a skew-Hopf pairing ${\eta }:A\times B\to {\Bbbk }$, $(x,y)\mapsto {\eta }(x,y)$, of two Hopf algebras $A$, $B$ is a bilinear map satisfying the equations $$\begin{aligned}
\label{eq:sHp1}
{\eta }(1,y)=&\,{\varepsilon }(y),& {\eta }(x,1)=&\,{\varepsilon }(x),\\
\label{eq:sHp2}
{\eta }(xx',y)=&\,{\eta }(x',y_{(1)}){\eta }(x,y_{(2)}),&
{\eta }(x,yy')=&\,{\eta }(x_{(1)},y){\eta }(x_{(2)},y'),\\
\label{eq:sHp3}
&\qquad
\makebox[0pt][l]{${\eta }({S}(x),y)={\eta }(x,{S}^{-1}(y))$}\end{aligned}$$ for all $x,x'\in A$ and $y,y'\in B$.
\[pr:sHpdef\] (i) There exists a unique
| 1,913
| 2,131
| 1,930
| 1,648
| 3,916
| 0.769296
|
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|
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 highest-weight module for NHEK’s isometry group, and obtain the scalar/vector/symmetric tensor basis functions. In Sec. \[sec:orthogonality-basis\] we present a proof of orthogonality for the basis functions in global coordinates. In Sec. \[sec:separation-variables\] we show that with these bases, we can separate variables in the scalar Laplacian, Maxwell system, and linearized Einstein equation. Finally we conclude and discuss future work in Sec. \[sec:concl-future-work\].
Kerr and the NHEK limit {#sec:kerr-nhek-limit}
=======================
In this paper we choose geometric units $(G = c = 1)$ and signature $(-{}+{}+{}+)$ for our metric $g$ on the spacetime manifold $\mathcal{M}$. A rotating, asymptotically-flat black hole in vacuum general relativity is described by the Kerr metric [@Kerr:1963ud]. For simplicity we will set the mass to $M=1$. In BL coordinates $(t, r, \theta, \phi)$ the line element of the Kerr black hole is given by [@Boyer:1966qh] $$\begin{aligned}
\text{d}s^2 =
&- \frac{\Delta}{\Sigma} (\text{d}t - a\sin^2\theta\,\text{d}\phi)^2
+ \frac{\Sigma}{\Delta}\,\text{d}r^2
+ \Sigma\,\text{d}\theta^2 \\ \nonumber
&+ \frac{\sin^2\theta}{\Sigma}
\left[ (r^2 + a^2)\,\text{d}\phi - a\,\text{d}t\right]^2,\end{aligned}$$ where $\Delta = r^2 - 2r + a^2$ and $\Sigma = r^2 + a^2 \cos^2\theta$. The ranges of the BL coordinates are given by $t\in(-\infty, +\infty)$, $r\in(0,+\infty)$, $\theta\in[0,\pi]$, $\phi\in[0,2\pi)$. In this paper we focus on a particular scaling limit of Kerr. This limit is usually described by the scaling coordinates $(T,\Phi,R)$ introduced in [@Bardeen:1999px], which are related to the BL coordinates via $$\begin{aligned}
t &= \frac{2 T}{\lam
| 1,914
| 3,753
| 2,167
| 1,734
| null | null |
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|
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 the outage throughput gain, $G_{R^{\mathrm{out}}}$, versus the number of reconfiguration states, $\Psi$. Different outage levels are considered, i.e., $\epsilon=0.01, \epsilon=0.05$, and $\epsilon=0.1$. As the figure shows, $G_{R^{\mathrm{out}}}$ increases as $\Psi$ increases. Similar to the results in Figure \[fig:Gain\], we find that the dominant outage throughput gain of employing the reconfigurable antennas can be achieved by having a few number of reconfiguration states. To obtain the outage throughput gain of $G_{R^{\mathrm{out}}}=1.5$, we only need $\Psi=2$, $\Psi=3$, and $\Psi=4$ reconfiguration states for the systems requiring $\epsilon=0.01, \epsilon=0.05$, and $\epsilon=0.1$, respectively. In addition, we note that $G_{R^{\mathrm{out}}}$ increases as $\epsilon$ increases, which indicates that the outage throughput gain of employing the reconfigurable antennas is more significant when the required outage level becomes more stringent.
Finally, we examine the performance of the proposed algorithm for fast selection by evaluating the average throughput loss ratio, which is defined by $$\label{}
\Delta_R=\left(\bar{R}_{\mathrm{max}}-\bar{R}_{\mathrm{fast}}\right)/{\bar{R}_{\mathrm{opt}}},$$ where $\bar{R}_{\mathrm{max}}$ denotes the average throughput achieved by the exhaustive search and $\bar{R}_{\mathrm{fast}}$ denotes the average throughput achieved by the proposed fast selection algorithm. Figure \[fig:Algerror\] plots the throughput loss ratio, $\Delta_R$, versus the transmit power to noise ratio, $\rho$. Systems with different numbers of reconfiguration states are considered, i.e., $\Psi=2$, $\Psi=4$, and $\Psi=8$. As shown in the figure, the proposed fast selection algorithm ach
| 1,915
| 255
| 1,917
| 2,020
| 1,209
| 0.792673
|
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|
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 needed e.g. in the context of time-dependent transport equations (where $\rho_1=\rho_2=\sqrt{E}$).
On Collision Operators {#coll}
----------------------
The differential cross-sections may have singularities, or even hyper-singularities, which would lead to extra (partial differential and) pseudo-differential terms in the transport equation ([@hsiao Sec. 7.1, pp. 353-394]). Instead of explaining systematically the underlying theory, the following slightly informal description suffices for the purposes of this work.
First of all, in the case where $\sigma(x,\omega',\omega,E',E)$ has hyper-singularities (like Møller differential cross section given in the below example) the integral $\int_S\int_I$ occurring in the collision operator must be understood in the sense of [*Cauchy principal value*]{} ${\rm p.v.}\int_S\int_I$ or more generally in the sense of [*Hadamard finite part integral*]{} ${\mathrm{p.f.}}\int_S\int_I$ ([@hsiao Sec. 3.2], [@chan], [@martin-rizzo], [@schwartz pp. 104-105]). We remark that one encounters this kind of hyper-singularities frequently in physical models. In addition, we must assume that $E_0>0$ in the energy interval $I=[E_0,E_{\rm m}]$, because otherwise $K\psi$, for $\psi\in C_0^\infty(G\times S\times I^\circ)$, might turn out to be (strictly) a distribution, which would increase the complexity of what is presented here. In [@lorence p. 7], it is reported that the differential cross sections are not necessarily valid for very small energies which supports this assumption.
Consider the following partial hyper-singular integral operator; for clarity we denote by $S'$ and $I'$ the set $S$ and $I$ when its variable is $\omega'$ and $E'$, respectively, \[coll-1\] (K\_0)(x,,E
| 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_-}{c_++c_-} \frac{\bar \partial \mathcal{A}^{a \bar a}(w)}{z - w}\right) + ...\cr
%
j^a_{L,\bar z}(z) j^{\bar a}_{R, 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_+}{c_++c_-} \frac{\bar \partial \mathcal{A}^{a \bar a}(w)}{z - w}\right)+ ...\end{aligned}$$ The first two OPEs can be written in the alternative form: j\^a\_[L,z]{}(z) j\^[|a]{}\_[R,z]{}(w) = + ... j\^a\_[L,|z]{}(z) j\^[|a]{}\_[R,|z]{}(w) = + ...It is straightforward to show that the OPEs are compatible with current conservation and the Maurer-Cartan equation. These OPEs are also compatible with the fact that the stress-tensor can be written either in terms of the left-current or in terms of the right currents. As an example of these consistency checks, it is shown in appendix \[TRJL\] that when we express the energy-momentum tensor in terms of right currents, it satisfies the expected OPE with the left current: T(z) j\^a\_[L,z]{}(w) = \_[|c |b]{}:j\^[|b]{}\_[R,z]{}j\^[|c]{}\_[R,z]{}:(z) j\^a\_[L,z]{}(w) = + + ((z-w)\^0)
When the theory is defined on a cylinder we can Fourier expand the currents along the angular coordinate, at a given time. It was shown in [@Ashok:2009xx] that the modes of the time components of the left (or the right) currents generate an affine Lie algebra at level $k$. The full commutator algebra computed in appendix \[commutators\] shows that these two affine Lie algebras commute.
Summary {#summary .unnumbered}
-------
In this section we have determined the pole order parts of the left and right current operator product expansions. The algebra closes on the current components and the
| 1,917
| 487
| 2,198
| 2,034
| null | null |
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|
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 this work.
######
Univariate and multivariate analyses of the post-switch outcomes of injection number, change in BCVA, and change in CRT
Outcome Injection numbers post-switch BCVA change post-switch (+3M) CRT change post-switch (+3M)
------------------------------------------------------ --------------- ----------------------------------------- ---------------------------------------------------- ---------------------------------------------------- ----------------------------------------- ------ ------------ ----------------------------------------- ------------------------------------------------------ ------------------------------------------------------
Age Per 10 years 0.01±0.2 (*R*[@b2-opth-12-593] 0.00) 0.97 −1.1±0.8 (*R*[@b2-opth-12-593] −0.14) 0.18 −1.9±9.0 (*R*[@b2-opth-12-593] −0.02) 0.83
Gender Female, n=69 10.8±0.2 0.13 −0.4±0.7 0.50 −38.9±7.4 0.52
| 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 $\delta$ be a step size satisfying $$\begin{aligned}
\delta \leq \min\lrbb{\frac{T\epsilon^2L}{36 d\beta^2\log \lrp{ \frac{36 d\beta^2}{\epsilon^2L}}}, \frac{T\epsilon^4L^2} {2^{14} d\beta^4\log\lrp{\frac{2^{14} d\beta^4}{\epsilon^4L^2}}}}.
\end{aligned}$$
If we assume that $x_0 = w_0$, then there exists a coupling between $x_t$ and $w_t$ such that for any $k$, $$\begin{aligned}
\E{\lrn{x_{k\delta} - w_{k\delta}}_2} \leq \hat{\epsilon}.
\end{aligned}$$
Alternatively, if we assume that $n \geq \frac{3\aq\Rq^2}{\delta} \cdot \log \frac{R^2 + \beta^2/m}{\hat{\epsilon}}$, then $$\begin{aligned}
W_1\lrp{p^*, p^w_{n\delta}} \leq 2\hat{\epsilon},
\end{aligned}$$ where $p^w_t := \Law(w_t)$.
To achieve $W_1(p^*, p^w_{n\delta}) \leq \hat{\epsilon}$, the number of steps needed is of order $n= \tilde{O}\lrp{\frac{1}{\hat{\epsilon}^8} \cdot e^{29 \aq \Rq^2}}$. The $\hat{\epsilon}$ dependency is considerably worse than in Theorem \[t:main\_gaussian\]. This is because we need to take many steps of in order to approximate a single step of . For details, see the coupling construction in equations – of Appendix \[s:nongaussianproof\].
[Application to Stochastic Gradient Descent]{}\[s:sgd\] In this section, we will cast SGD in the form of . We consider an objective of the form $$\begin{aligned}
\numberthis
\label{e:example_U}
U(w) = \frac{1}{n} \sum_{i=1}^n U_i(w).\end{aligned}$$ We reserve the letter $\eta$ to denote a random minibatch from $\lrbb{1, \ldots, n}$, sampled with replacement, and define $\zeta(w,\eta)$ as follows: $$\label{e:def_zeta}
\zeta(w,\eta) := \nabla U(w) - \frac{1}{\lrabs{\eta}} \sum_{i\in \eta} \nabla U_i(w)$$ For a sample
| 1,919
| 1,218
| 1,741
| 1,645
| null | null |
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|
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 connected, smooth, and unipotent. Furthermore, the underlying algebraic variety of $\tilde{G}^1$ over $\kappa$ is an affine space of dimension $$\sum_{i<j}n_in_j+\sum_{i:\mathrm{even}}\frac{n_i^2+n_i}{2}+\sum_{i:\mathrm{odd}}\frac{n_i^2-n_i}{2}
+\#\{i:\textit{$i$ is odd and $L_i$ is free of type I}\}$$ $$-\#\{i:\textit{$i$ is even and $L_i$ is of type I}\}
+ \#\{i:\textit{$i$ is even, $L_i$ is of type I and $L_{i+2}$ is of type II}\}.$$
We prove this theorem by writing out a set of equations completely defining $\tilde{G}^1$ (after all there are so many different sets of equations defining $\tilde{G}^1$). We first introduce the following trick. Consider the polynomial ring $\kappa[x_1, \cdots, x_n]$ and its quotient ring $\kappa[x_1, \cdots, x_n]/(x_1+P(x_2, \cdots, x_n))$. Then the quotient ring $\kappa[x_1, \cdots, x_n]/(x_1+P(x_2, \cdots, x_n))$ is isomorphic to $\kappa[x_2, \cdots, x_n]$ and in this case we say that *$x_1$ can be eliminated by $x_2, \cdots, x_n$*.
Let $R$ be a $\kappa$-algebra. As explained in Remark \[r33\].(2), we consider the given hermitian form $h$ as an element of $\underline{H}(R)$ and write it as a formal matrix $h=\begin{pmatrix} \pi^{i}\cdot h_i\end{pmatrix}$ with $(\pi^{i}\cdot h_i)$ for the $(i,i)$-block and $0$ for the remaining blocks. We also write $h$ as $(f_{i, j}, a_i\cdots f_i)$. Recall that the notation $(f_{i, j}, a_i\cdots f_i)$ is defined and explained in Section \[h\] and explicit values of $(f_{i, j}, a_i\cdots f_i)$ for the $h$ are given in Remark \[r33\].(2).
We choose an element $m=(m_{i,j}, s_i\cdots w_i)\in (\mathrm{Ker~}\tilde{\varphi})(R)$ with a formal matrix interpretation $m= \begin{pmatrix} \pi^{max\{0,j-i\}}m_{i,j} \end{pmatrix}$, where the
| 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 $\alpha \in (0,1)$, let $\hat{t}^*_\alpha$ be the smallest positive number such that $$\mathbb{P}\left( \sqrt{n} \| \hat{\beta}^*_{{\widehat{S}}} - \hat{\beta}_{{\widehat{S}}}\|
\leq \hat{t}^*_\alpha \Big| \mathcal{D}_{2,n} \right) \geq 1 - \alpha.$$ Next, let $(\tilde{t}^*_j, j \in {\widehat{S}})$ be such that $$\mathbb{P}\left( \sqrt{n} | \hat{\beta}^*_{{\widehat{S}}}(j) - \hat{\beta}_{{\widehat{S}}}
(j) \leq \tilde{t}^*_j, \forall j \Big| \mathcal{D}_{2,n} \right) \geq 1 - \alpha.$$ By the union bound, each $\tilde{t}^*_j$ can be chosen to be the largest positive number such that $$\mathbb{P}\left( \sqrt{n} | \hat{\beta}^*_{{\widehat{S}}}(j) - \hat{\beta}_{{\widehat{S}}}
(j) > \tilde{t}^*_j, \Big| \mathcal{D}_{2,n} \right) \leq \frac{\alpha}{k}.$$ Consider the following two bootstrap confidence sets: $$\label{eq:ci.boot.beta}
\hat{C}^*_{{\widehat{S}}} = \left\{ \beta \in \mathbb{R}^{{\widehat{S}}} \colon \|
\beta - \hat{\beta}_{{\widehat{S}}}
\|_\infty \leq \frac{ \hat{t}^*_{\alpha}}{\sqrt{n}} \right\} \quad
\text{and} \quad
\tilde{C}^*_{{\widehat{S}}} = \left\{ \beta \in \mathbb{R}^{{\widehat{S}}} \colon |
\beta(j) - \hat{\beta}_{{\widehat{S}}}(j)
| \leq \frac{ \tilde{t}^*_{j}}{\sqrt{n}}, \forall j \in {\widehat{S}}\right\}$$
It is immediate to see that $\hat{C}^*_{{\widehat{S}}}$ and $\tilde{C}^*_{{\widehat{S}}}$ are just the bootstrap equivalent of the confidence sets of and , respectively.
\[theorem::beta.boot\] Let $$v_n = v - K_{1,n}, \quad \overline{v}_n = \overline{v} + K_{1,n}, \quad u_n = u - K_{2,n} \quad \text{and} \quad U_n = U + K_{2,n},$$ where $$K_{1,n} = C A^2 \sqrt{ b \overline{v}\frac{\log b + \log n}{n} } \quad \text{and}
| 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. There exists a dynamic $s$-uniform hypergraph, say $({\mathcal{H}}^{(1)},\ldots, {\mathcal{H}}^{(n)})$, which satisfies the balancedness [condition and (trivially) satisfies the $\varepsilon$-visibility condition. Let $2{\leqslant}d{\leqslant}s$ be any integer which is constant. Suppose that the balanced allocation process on $({\mathcal{H}}^{(1)},\ldots, {\mathcal{H}}^{(n)})$ has allocated $n$ balls, then the maximum load is at least $\min\{\Omega(1/\varepsilon),\, \Omega(\log n /\log\log n)\}$ with high probability. ]{}
### Balanced Allocation on Dynamic Graphs {#balanced-allocation-on-dynamic-graphs .unnumbered}
A dynamic graph is a special case of a dynamic hypergraph, where $s=s(n)=2$ for all $n$. Write $(G^{(1)},\ldots, G^{(n)})$ to denote a dynamic graph, where $G^{(t)}=([n],E_t)$ for $t=1,2,\ldots, n$. Theorem \[thm:d-choice\] does not cover the case of graphs ($s=2$), due to the size property. We will prove a result on balanced allocation for regular dynamic graphs.
Suppose that $(G^{(1)}, \ldots, G^{(n)})$ is a regular dynamic graph on vertex set $[n]$. The balanced allocation algorithm on $(G^{(1)},\ldots, G^{(n)})$ proceeds in rounds $(t=1,\ldots, n)$. In each round $t$, the $t$-th ball chooses an edge of $G^{(t)}$ uniformly at random, and the ball is then placed in one of the bins incident to the edge with a lesser load, with ties broken randomly.
Say that the dynamic graph is *regular* if $G^{(t)}$ is $\Delta_t$-regular for some positive integer $\Delta_t$ and all $t=1,2,\ldots, n$. For every pair of distinct bins $\{i,j\}\subset [n]$, we will assume that the visibility ${\ensuremath{\operatorname{\mathtt{vis}}(i,j)}}$ satisfies $${\ensuremath{\operatorname{\mathtt{vis}}(i, j)}}=|\{
| 1,922
| 373
| 1,004
| 1,953
| 1,344
| 0.790682
|
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|
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 $k$, details of the calculation can be found in Appendix \[app:intoutAI\]. It results in a non-trivial effective potential $V_{\bot,k}[A_0]$ that approaches the Weiss potential [@Weiss:1980rj] in the limit $k/T \to 0$, and falls off like $\exp(-\beta k_\bot (k)) \cos (g \beta A_0) $ in the UV limit $k/T\to
\infty$, see Fig. \[fig:VPreWeiss3D\]. In terms of the effective action, after the integration over $\vec A_\bot$, we are led to an effective action of $A_0$, $$\label{eq:truncated_eff_actioncopy}
\Gamma_{k}[A_0] = \beta \int d^3x \left(\frac{ Z_0}{2}
(\vec \partial A_0)^2 + \Delta V_{k}[A_0] + V_{\bot,k} [A_0] \right)\,.$$ follows from with $\Gamma_k[A_0]=\Gamma_k[A_0,\vec A_\bot =0]$, and $$\label{eq:Vk}
V_k[A_0]= \Delta
V_{k}[A_0] + V_{\bot,k} [A_0]\,.$$ The full effective potential is given by $V_{\mathrm{eff}}[A_0] =
\Delta V_{k=0}[A_0] + V_{\bot,k=0}[A_0]$. We are left with the task to determine $\Delta V_k$, which is the part of the effective potential induced by $A_0$-fluctuations. In Polyakov gauge, these fluctuations carry the confining properties of the Polyakov loop variable, whereas the spatial fluctuations generate a deconfining effective potential for $A_0$, see Appendix \[app:intoutAI\]. We emphasise that this structure is not present for spatial confinement which is necessarily also driven by the spatial fluctuations, and solely depends on these fluctuations in the high temperature limit. We hope to report on this matter in the near future.
Here we proceed with the integration of the flow for the potential $\Delta V_{k}$. To that end we reformulate the flow as a flow for $\Delta V_{k}$ with the external input $V_{\bot,k}$, see . The flow equation for $\Delta V_{k}$ reads $$\label{eq:de
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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 $G=(V, \Sigma, S, M)$ such that $L=L(G)$ and $ind(G)\leq k$. Without loss of generality we assume that $G$ is without repetitions. Let $R$ be the set of the rules of $M$. By Theorem 16 in [@tur], we can construct an $h$-Petri net controlled grammar $G'=(V, \Sigma, S, R, N_h)$, $N_h=(P\cup Q, T, F\cup E, \varphi, \zeta, \gamma, \mu_0, \tau)$, which is equivalent to the grammar $G$. By definition, for every sentential form $w\in (V\cup\Sigma)^*$ in the grammar $G$, $|w|_V\leq k$. It follows that $|w|_A\leq k$ for all $A\in V$. By bijection $\zeta:P\cup Q\to V\cup\{{\lambda}\}$ we have $\mu(p)=\mu(\zeta^{-1}(A))\leq k$ for all $p\in P$ and $\mu \in \mathcal{R}(N_h, \mu_0)$, i.e., the corresponding cf Petri net $(P, T, F, \phi, \beta, \gamma, \iota)$ is with $k$-place capacity. Therefore $G'$ is with weak place capacity.
On the other hand, the construction of an equivalent vector grammar for an $h$-Petri net controlled grammar, can be extended to the case of weak capacities just by assigning the capacities of the corresponding places to the nonterminal symbols of the grammar.
As regards strong capacities, there is no difference between weak and strong capacities for grammars controlled by $c$- and $s$-Petri nets because the number of tokens in every circle is limited by $1$. This yields:
\[lem:wPNx=sPNx\] For $z\in \{c,s\}$, ${{\bf MAT}}_{{\mathit{fin}}}=\mathbf{sPN}^{[{\lambda}]}_{cz}$.
The only families not characterized yet are $\mathbf{sPN}^{[{\lambda}]}_{ch}$. We conjecture that they are also equal to ${{\bf MAT}}_{{\mathit{fin}}}$.
Conclusions {#sec:conclusions}
===========
We have introduced grammars with capacity bounds and their Petri net controlled counterparts. In particular, we have
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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-invariant action
-------------------------------
By use of the string fields introduced in the previous subsection, the complete action for the WZW-like open superstring field theory is given by[@Kunitomo:2015usa] $$S\ =\ -\frac{1}{2}{\langle\!\langle}\Psi, YQ\Psi{\rangle\!\rangle}-\int_0^1 dt \langle A_t(t), QA_\eta(t)+(F(t)\Psi)^2\rangle\,,
\label{complete action}$$ and is invariant under the gauge transformations
\[full gauge\] $$\begin{aligned}
A_{\delta_g}\ =&\ D_\eta\Omega + Q\Lambda
+ \{F\Psi,F\Xi\{F\Psi,\Lambda\}\}
- \{F\Psi,F\Xi\lambda\}\,,
\label{gauge tf ns}\\
\delta_g\Psi\ =&\ -X\eta F\Xi[F\Psi, D_\eta\Lambda] + Q\lambda + X\eta F\lambda\,,
\label{gauge tf r}\end{aligned}$$
where we have introduced the one parameter extension $\Phi(t)$ of $\Phi$ $(t\in[0,1])$ satisfying the boundary condition $\Phi(1)=\Phi$ and $\Phi(0)=0$, and defined $$A_{\mathcal{O}}(t)\ =\ (\mathcal{O} e^{\Phi(t)})e^{-\Phi(t)}\,,$$ with $\mathcal{O}=\partial_t, \eta,$ or $\delta$, which are analogs of (components) of the right-invariant one form, satisfying the Maurer-Cartan-like equation $$\mathcal{O}_1A_{\mathcal{O}_2}(t)
-(-1)^{\mathcal{O}_1\mathcal{O}_2}\mathcal{O}_2A_{\mathcal{O}_1}(t)
-[\![A_{\mathcal{O}_1}(t),\,A_{\mathcal{O}_2}(t)]\!]
=\ 0\,,\label{MC}$$ where $[\![A_1,A_2]\!]$ is the graded commutator of the two string field $A_1$ and $A_2$: $[\![A_1,A_2]\!]=A_1A_2-(-1)^{A_1A_2}A_2A_1$. Using $A_\eta(t)$, the covariant derivative $D_\eta(t)$ is defined by the operator acting on the string field $A$ as $$D_\eta(t) A\ =\ \eta A - [\![A_\eta,\, A]\!]\,,
$$ which is nilpotent: $(D_\eta(t))^2=0$. Then the linear map $F(t)$ on a general string field $\Psi$ in the Ramond sector is defined by $$\begin{aligned}
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Q^\dagger$, one has for the general self-adjoint combination of supercharges $$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)\right)\right]\ .$$ Consequently, given the variations $\delta_\epsilon\psi(z,\theta)=iQ_\epsilon\psi(z,\theta)$, the bosonic and fermionic components of such wave functions are transformed according to the rules $$\delta_\epsilon\psi_B(z)=i\sqrt{\hbar\omega}\,z\epsilon\psi_F(z)
\ \ ,\ \
\delta_\epsilon\psi_F(z)=i\sqrt{\hbar\omega}\,
\epsilon^\dagger\partial_z\psi_B(x)\ .
\label{eq:SUSYvariation1}$$ These expressions thus provide the infinitesimal supersymmetry transformations of the wave functions of the system. We shall come back to these relations hereafter.
In order to identify which type of classical system corresponds to the present situation, let us now introduce the configuration and momentum space degrees of freedom through the usual relations,[@GovCOPRO2] $$\begin{array}{r c l}
a=\sqrt{\frac{m\omega}{2\hbar}}\left[x+\frac{i}{m\omega}p\right]\ \ &,&\ \
a^\dagger=\sqrt{\frac{m\omega}{2\hbar}}\left[x-\frac{i}{m\omega}p\right]\ ,\\
& & \\
b=\sqrt{\frac{m\omega}{2\hbar}}\left[\theta_1+i\theta_2\right]\ \ &,&\ \
b^\dagger=\sqrt{\frac{m\omega}{2\hbar}}\left[\theta_1-i\theta_2\right]\ .
\end{array}$$ Note well that the variables $x$, $p$, $\theta_1$ and $\theta_2$, which are assumed to be self-adjoint, $x^\dagger=x$, $p^\dagger=p$, $\theta^\dagger_1=\theta_1$, $\theta^\dagger_2=\theta_2$, are still operators at this stage. The decomposition of the fermionic operators $b$ and $b^\dagger$ in these terms is of course to maintain as manifest as possible the parallel between the bosonic and fermionic sectors of the system, which are exchanged under supersymmetry transformations. Given these operator redefinitions, it follows that the only nonvanishing (anti)commutators are (note that the operators $\
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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 almost simple group, which will be used as a tool.
The aim of this paper is then to prove the following result:
\[mainth\] Let the group $G = AB$ be the product of subgroups $A$ and $B$, and let $p$ be a prime. Then $p$ does not divide $i_G(x)$ for every $p$-regular element of prime power order $x\in A\cup B$ if and only if $G$ is $p$-decomposable, i.e. $G=O_p(G) \times O_{p'}(G)$.
Note that if $G$ is $p$-decomposable, then clearly the conditions on the indices hold. For the converse, the following lemma shows that only the existence of a unique Sylow $p$-subgroup should be proved.
\[pclos\] Let the group $G = AB$ be the product of the subgroups $A$ and $B$, and let $p$ be a prime. If $p$ does not divide $i_G(x)$ for every $p$-regular element of prime power order $x\in A\cup B$, then the following statements are equivalent:
- $G$ is $p$-closed, i.e. $G$ has a normal Sylow $p$-subgroup.
- $G$ is $p$-decomposable.
Clearly, it is enough to prove that (i) implies (ii). Let $P\in{{\operatorname}{Syl}_{p}\left(G\right)}$ and assume that $P \unlhd G$. Since $p$ does not divide $i_G(x)=|G : {{\operatorname}{C}_{G}(x)}|$ it follows that $P\leq {{\operatorname}{C}_{G}(x)}$ for every $p$-regular element of prime power order $x\in A\cup B$. Since $G$ is $p$-separable, by Lemma \[1.3.2\], we may consider $H$ a Hall $p'$-subgroup of $G$ such that $H= (H \cap A)(H \cap B)$. Hence, for every element $x\in (H \cap A) \cup (H \cap B)$ of prime power order, it holds that $P\leq {{\operatorname}{C}_{G}(x)}$. Therefore $[P, H]=1$ and (ii) follows.
As an inmediate consequence of the Main Theorem, we get:
\[all\] Let the group $G = AB$ be the product of subgroups $A$ and $B$, and let $p$ be a prime. Then $p$
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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_{kk}\bar{b}_{kj}+\bar{g}_{ik}b_{ij}+b_{kk}\bar{g}_{kj}-\bar{b}_{ik}g_{ij}+2\bar{b}_{jk}g_{jj}-2\bar{h}_kf_j=0\nonumber \\
& 2g_{ik}\bar{b}_{ij}-2\bar{g}_{kk}b_{kj}+f_k\bar{h}_j-g_{jk}\bar{b}_{jj}-h_k\bar{f}_j+b_{jk}\bar{g}_{jj}-2b_{ik}\bar{g}_{ij}+2\bar{b}_{kk}g_{kj}=0\nonumber \\
& 2\bar{g}_{jk}\bar{b}_{ij}-2\bar{f}_kb_{kj}+g_{kk}\bar{h}_j-\bar{g}_{ik}\bar{b}_{jj}-b_{kk}\bar{f}_j+\bar{b}_{ik}\bar{g}_{jj}-2\bar{b}_{jk}\bar{g}_{ij}+2\bar{h}_kg_{kj}=0\nonumber \\
& -2g_{ik}b_{jj}+2\bar{g}_{kk}h_j-f_k\bar{b}_{kj}+g_{jk}b_{ij}+h_k\bar{g}_{kj}-b_{jk}g_{ij}+2b_{ik}g_{jj}-2\bar{b}_{kk}f_j=0\nonumber \\
& g_{kk}b_{jj}-\bar{g}_{ik}h_j-b_{kk}g_{jj}+\bar{b}_{ik}f_j=0\nonumber\\
& \bar{g}_{jk}\bar{b}_{kj}-\bar{f}_kb_{ij}-\bar{b}_{jk}\bar{g}_{kj}+\bar{h}_kg_{ij}=0\nonumber \\
& -f_k\bar{b}_{ij}+g_{jk}b_{kj}+h_k\bar{g}_{ij}-b_{jk}g_{kj}=0\nonumber \\
& -g_{ij}\bar{h}_j+\bar{g}_{kk}\bar{b}_{jj}+b_{ik}\bar{f}_j-\bar{b}_{kk}\bar{g}_{jj}=0 \quad .\end{aligned}$$
Exactly as we have discussed in the previous subsection, what we have determined is not yet the full set of constraints. Indeed, the four-dimensional $\alpha=-3$ space-filling branes correspond to the 4-form potential $E_{4, MN \dot{\alpha}}$ belonging to the ‘tensor-spinor’ ${\overline{\bf 1728}}$ representation of $SO(6,6)$. Together with the potentials in eq. that are associated to the branes, there are additional potentials that must be included in order to generate the whole four-dimensional representation. Focusing on the IIB/O3 model, it turns out that in order to get all the possible constraints for $P_1^2 \cdot Q$ one has to introduce also the fields $E_{9,3}$ and $E_{10,2}$. These potentials occur in the IIB decomposition of $E_{11}$ and corres
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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 (standard deviation) $\sigma_{\theta^r}$. $\theta^t_{\psi,i,l}$ are uniformly distributed with mean $\theta_{\psi,i}^t$ and a constant angular spread (standard deviation) $\sigma_{\theta^t}$. We further assume that $\theta_{\psi,i}^r$ and $\theta_{\psi,i}^t$ are both uniformly distributed within the range of $[-\pi/2, \pi/2]$. Unless otherwise stated, the system parameters are $N_r=N_t=17, L_r=L_t=5, N_{\psi,{\mathrm{cl}}}=10, N_{\psi,{\mathrm{ry}}}=8,$ $\sigma_{\theta^r}=\sigma_{\theta^t}=3^\circ$, and $d/\lambda=1/2$. All average results are over 5,000 randomly generated channel realizations. Note that $N_{\psi,{\mathrm{cl}}}=10$ and $N_{\psi,{\mathrm{ry}}}=8$ are based on the existing observations at 60 GHz in the literature [@Gustafson_14_ommcacm; @Maltsev_10_cmf60gwsmodl], and practical mmWave channels at 28 GHz may have relatively small numbers of clusters and paths [@Raghavan8255763; @Raghavan8053813].
![PDF of $R_\psi$. The parameters are $N_r=N_t=17, L_r=L_t=5, \sigma_{\theta^r}=\sigma_{\theta^t}=3^\circ$, and $d/\lambda=1/2$.[]{data-label="fig:DisR"}](DisR){width=".9\columnwidth"}
\
![PDF of $R_\psi$. The parameters are $N_r=N_t=17, L_r=L_t=5, \sigma_{\theta^r}=\sigma_{\theta^t}=3^\circ$, and $d/\lambda=1/2$.[]{data-label="fig:DisR"}](DisR_c4p2){width=".9\columnwidth"}
We first demonstrate the accuracy of the Gaussian approximated probability density function (PDF) of $R_\psi$. Figure \[fig:DisR\] plots the simulated PDF and the Gaussian approximated PDF of $R_\psi$. Figure \[fig:DisR\](a) and Figure \[fig:DisR\](b) are for the case of $N_{\psi,{\mathrm{cl}}}=10$ and $N_{\psi,{\mathrm{ry}}}=8$ and the case of $N_{\psi,{\mathrm{cl}}}=4$ and $N_{\psi,{\mathrm{ry}}}=2$, respectively. Diff
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\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 finish the proof of the theorem by showing that $N_{\nu ,t}\ge {P}^\chi ({\alpha },\beta _\nu ;t)$ for all $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$. The essential ingredients will be Lemmata \[le:subfch\] and \[le:detXfactor\].
Let $\nu \in \{1,2,\dots ,n\}$ and $t\in \{1,2,\dots ,{b^{\chi}} (\beta _\nu )-1\}$. Let $w=1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_{\nu -1}}$. Then $${{\mathcal{U}}^0}={\Bbbk }[K_{w({\alpha }_j)},K_{w({\alpha }_j)}^{-1},
L_{w({\alpha }_j)},L_{w({\alpha }_j)}^{-1}\,|\,j\in I]$$ and $w({\alpha }_{i_\nu })=\beta _\nu $. Let $$B ={\Bbbk }[L_{\beta _\nu },L_{\beta _\nu }^{-1},
K_{w({\alpha }_j)},K_{w({\alpha }_j)}^{-1},
L_{w({\alpha }_j)},L_{w({\alpha }_j)}^{-1}\,|\,j\in I \setminus \{i_\nu \}]$$ and $x={\rho ^{\chi}} (\beta _\nu )K_{\beta _\nu }-
\chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu }$. Then $${{\mathcal{U}}^0}\simeq B[x,(x+
\chi (\beta _\nu ,\beta _\nu )^t L_{\beta _\nu })^{-1}].$$ Let $X'=(x'_{ij})_{i,j\in \{1,2,\dots ,k\}}\in ({{\mathcal{U}}^0})^{k\times k}$ with $$x'_{ij}={\mathrm{Sh}}(F'_i,F'_j) \quad \text{for all $i,j\in \{1,2,\dots ,k\}$.}$$ Let $l\in {\mathbb{Z}}$ such that $K_{\beta _\nu }^l X'\in B[x]^{k\times k}$, and let $X=K_{\beta _\nu }^l X'$. By Lemma \[le:subfch\] and Eq. there is a non-empty open subset of the variety of $B\simeq
{{\mathcal{U}}^0}/(x)$ such that $\mathrm{rk}\,X(0)_p\le k-{P}^\chi ({\alpha },\beta _\nu ;t)$ for all $p$ in this set. By Lemma \[le:detXfactor\], $\det X=x^{{P}^\chi ({\alpha },\beta _\nu ;t)}b'$ for some $b'\in
B[x]$. In particular, $x^{{P}^\chi ({\alpha },\beta _\nu ;t)}$ is a factor of $\det ^\chi _{\alpha }$, and the proof of the theorem is compl
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}\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)\,\tilde G_\Lambda(v',z)\,\Theta'_{z,x;
{{\cal A}}}\,\psi_\Lambda(v',v).{\label{eq:Theta[I]-bd}}\end{aligned}$$
\[lmm:Theta’Theta”bd\] For the ferromagnetic Ising model, we have $$\begin{aligned}
{\label{eq:Theta'Theta''bd}}
\Theta'_{y,x;{{\cal A}}}\leq\sum_{u\in{{\cal A}}}P'_{\Lambda;u}(y,x),&&
\Theta''_{y,x,v;{{\cal A}}}\leq\sum_{u\in{{\cal A}}}P''_{\Lambda;u,v}(y,x).\end{aligned}$$
We prove Lemma \[lmm:Thetabds\] in Section \[sss:chopping-off\], and Lemma \[lmm:Theta’Theta”bd\] in Section \[sss:dbconn\].
Recall [(\[eq:pij-def\])]{}. By [(\[eq:Theta\[1\]-bd\])]{}, [(\[eq:Theta’Theta”bd\])]{} and [(\[eq:Q’-def\])]{}, we obtain $$\begin{aligned}
{\label{eq:nest-diagbd}}
\Theta^{{\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}}\Big[\tau_{b_j}
\Theta^{{\scriptscriptstyle}(j)}_{{\overline{b}}_j,x;\tilde{{\cal C}}_{j-1}}\Big]&\leq\Theta^{
{\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}}\bigg[\sum_z\tau_{b_j}\big(
\delta_{{\overline{b}}_j,z}+\tilde G_\Lambda({\overline{b}}_j,z)\big)\sum_{v_j\in
\tilde{{\cal C}}_{j-1}}P'_{\Lambda;v_j}(z,x)\bigg]{\nonumber}\\
&\leq\sum_{v_j}\Theta^{{\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}}
\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}_{j-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\longleftrightarrow}}}v_j\}$}}}\big]\,\tau_{b_j}Q'_{\Lambda;v_j}
({\overline{b}}_j,x).\end{aligned}$$ For $j=1$, we use [(\[eq:pi0’-bd\])]{} and [(\[eq:nest-diagbd\])]{} to obtain $$\begin{aligned}
{\label{eq:pi0'-bd-appl}}
&\pi_\Lambda^{{\scriptscriptstyle}(1)}(x)\equiv\sum_{b_1}\Theta^{{\scriptscripts
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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 conformal time and introduce $-\eta=-\tau+\beta$. We will back later to the previous definition because an additional degree of freedom $\beta$ is necessary to fit properly the boundary solutions with numerical one. Equation (\[modeeq\]) takes the form $$\frac{d^2}{d\eta^2}f(k,\eta)+\left[ D_*\xi^n(-\eta)^{np}k^2-p(p-1) \right]f(k,\eta) = 0$$ and the general solution in terms of Bessel functions have the form $$f (k,\eta)= C_1\sqrt{-\eta}J_{|\nu|}(x)+ C_2\sqrt{-\eta}Y_{|\nu|}(x) \label{solmodes}$$ with $$\begin{aligned}
x &=& k \frac{2\sqrt{D_*\xi^n}}{|2+np|} (-\eta)^{(2+np)/2}, \\
\nu &=& -\frac{\sqrt{1+4p(p-1)}}{2+np}.\end{aligned}$$ With the use of the Wronskian condition (\[Wronskian\]) we can rewrite the solution (\[solmodes\]) to the form $$f (k,\eta) = \sqrt{\frac{\pi}{2|2+np|}}\sqrt{-\eta}\left[ D_1 H^{(1)}_{|\nu|}(x) + D_2 H^{(2)}_{|\nu|}(x) \right]
\label{solmodes2}$$ where we introduced Hankel functions defined as $$\begin{aligned}
H^{(1)}_{|\nu|}(x) &=& J_{|\nu|}(x) +i Y_{|\nu|}(x) \\
H^{(2)}_{|\nu|}(x) &=& J_{|\nu|}(x) -i Y_{|\nu|}(x).\end{aligned}$$ and the constants $D_1$ and $D_2$ enjoy the relation $|D_1|^2-|D_2|^2=1$. To fix values of the constants $D_1$ and $D_2$ we must consider the high energy limit, namely $x \gg 1 $. In this limit the Bessel functions behave as follow $$\begin{aligned}
J_{|\nu|}(x) \rightarrow \sqrt{\frac{2}{\pi x}} \sin \left(x- \frac{|\nu|\pi}{2} -\frac{\pi}{4} \right), \\
Y_{|\nu|}(x) \rightarrow \sqrt{\frac{2}{\pi x}} \cos \left(x- \frac{|\nu|\pi}{2} -\frac{\pi}{4} \right),\end{aligned}$$ what give us $H^{(1)}_{|\nu|}(x) \rightarrow \sqrt{{2}/{(\pi x)}} \exp{\left[ix - i{|\nu|\pi}/{2} -i{\pi}/{4} \right]}$ and $H^{(2)}_{|\nu|}(x) \rightarrow \sqrt{{2}
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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. $(T,T')$ and $$S= S'\cap ({\cal R}\times{\cal R})^{\leq n}$$ for some $n \in \N$, $S'$ D-strategy w.r.t. $(T,T')$.\
DQ1: Since $S'$ is non-empty and prefix-closed $(\varepsilon,\varepsilon) \in S'$, hence $(\varepsilon,\varepsilon) \in S'\cap S({\cal R}\times{\cal R})^{\leq n}$.\
DQ2: $S'$ and $({\cal R}\times{\cal R})^{\leq n}$ are both prefix-closed, hence their intersection is also prefix-closed.\
DQ3: $S'\subseteq {{\rm PLAYS}}(T,T')$ and $S \subseteq S'$, hence $S\subseteq {{\rm PLAYS}}(T,T')$\
DQ4: $\forall \alpha \in S$,\
${{\rm NEXT}}((T,T'),\alpha) \notin \sim_1$\
or \[${{\rm NEXT}}((T,T'),\alpha) \in \sim_1$ and the set $\{ (\pi,\pi') \in {\cal R}\times{\cal R} \mid \alpha\cdot (\pi,\pi')\in S'\}$ is full for ${{\rm NEXT}}((T,T'),\alpha)$\]. If $|\alpha| < n$, the above property holds in $S$.\
If $|\alpha| = n$, the property $\alpha \backslash S=\{(\varepsilon,\varepsilon)\}$ holds. In all cases DQ4 is fulfilled.\
We define the [*extension*]{} ordering over ${\cal P}(({\cal R}\times{\cal R})^*)$ as follows: for every $S_1,S_2 \in {\cal P}(({\cal R}\times{\cal R})^*)$, $S_1 \sqsubseteq S_2$ iff the two conditions below hold:\
E1- $S_1 \subseteq S_2$\
E2- $\forall \alpha \in S_2-S_1, \exists \beta \in S_1, \mbox{ which is maximal in } S_1
\mbox{ for the prefix ordering and such that },\\
\beta \preceq \alpha.$ \[def-extension\]
Let $T,T'\in \TERMS$. The extension ordering over the set of all D-q-strategies w.r.t. $(T,T')$, is inductive. \[L\_inclusion\_is\_inductive\]
We recall that a partial order $\leq$ over a set $E$ is [*inductive*]{} iff, every totally ordered subset of $E$ has some upper-bound.\
One can check that, if $P$ is a set of D-q-strategies w.r.t. $(T,T')$, which is totally o
| 1,933
| 1,022
| 1,834
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|
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 $q \in{\mathbb{N}}_0$ such that $\mathbf{(Id+K)^{-1}} :H_{p,{\mathbb{C}}} \to H_{-q,{\mathbb{C}}}$ is continuous with $p$ as in (ii).
Then we define the normalized exponential $$\label{Nexp}
{\rm{Nexp}}(- \frac{1}{2} \langle \cdot ,\mathbf{K} \cdot \rangle)$$ by $$T({\rm{Nexp}}(- \frac{1}{2} \langle \cdot ,\mathbf{K} \cdot \rangle))({\bf f}) := \exp(-\frac{1}{2} \langle {\bf f}, \mathbf{(Id+K)^{-1}} {\bf f} \rangle),\quad {\bf f} \in S_d({\mathbb{R}}).$$
The “normalization” of the exponential in the above definition can be regarded as a division of a divergent factor. In an informal way one can write $$\begin{gathered}
T({\rm{Nexp}}(- \frac{1}{2} \langle \cdot ,\mathbf{K} \cdot \rangle))({\mathbf f})=\frac{T(\exp(- \frac{1}{2} \langle \cdot ,\mathbf{K} \cdot \rangle))(\mathbf{f})}{T(\exp(- \frac{1}{2} \langle \cdot ,\mathbf{K} \cdot \rangle))(0)}\\
=\frac{T(\exp(- \frac{1}{2} \langle \cdot ,\mathbf{K} \cdot \rangle))(\mathbf{f})}{\sqrt{\det(\mathbf{Id+K})}} , \quad {\bf f} \in S_d({\mathbb{R}}), \end{gathered}$$ i.e. if the determinant in the Example \[Grotex\] above is not defined, we can still define the normalized exponential by the T-transform without the diverging prefactor. The assumptions in the above definition then guarantee the existence of the generalized Gauss kernel in .
\[pointprod\] For sufficiently “nice” operators $\mathbf{K}$ and $\mathbf{L}$ on $L^2_{d}({\mathbb{R}})_{{\mathbb{C}}}$ we can define the product $${\rm{Nexp}}\big( - \frac{1}{2} \langle \cdot,\mathbf{K} \cdot \rangle \big) \cdot \exp\big(-\frac{1}{2} \langle \cdot,\mathbf{L}\cdot \rangle \big)$$ of two square-integrable functions. Its $T$-transform is then given by $$\begin{gathered}
T\Big({\r
| 1,934
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|
phenomenology {#sec:pheno}
=============
Dark matter {#subsec:DM}
-----------
If $({\mathcal M}_0)_{\eta\eta} < ({\mathcal M}_0)_{ss}$, the dark matter candidate ${\mathcal H}_1^0$ is given by $\eta_r^0$ approximately because we assume small mixing. See, e.g., Ref. [@i-doublet] for studies about the inert doublet scalar. Let us assume $m_{{\mathcal H}_1^0}^{}\simeq 75\,{{\text{GeV}}}$ and $m_{{\mathcal A}^0}^{} \gtrsim 125\,{{\text{GeV}}}$. As shown in Ref. [@Lundstrom:2008ai], these values satisfy constraints from the LEP experiments [@:2005ema; @EspiritoSanto:2003by] and the WMAP experiment [@Komatsu:2008hk]. The mass splitting ($m_{{\mathcal A}^0}^{} - m_{{\mathcal H}_1^0}^{} \gtrsim 50\,{{\text{GeV}}}$) suppresses quasi-elastic scattering on nuclei (${\mathcal H}_1^0 N \to {\mathcal A}^0 N$ mediated by the $Z$ boson) enough to satisfy constraints from direct search experiments of the DM [@Aprile:2011hi]. By using eqs. and , we obtain $$\begin{aligned}
\frac{ \lambda_{s\Phi\eta}^2 v_s^2 }{ ({\mathcal M}_0)_{ss}^2 }
\simeq
\frac{ 2}{ v^2 }
\left( m_{{\mathcal A}^0}^2 - m_{{\mathcal H}_1^0}^2 \right)
\gtrsim 0.3 .
\label{eq:lambda}\end{aligned}$$ In order to be consistent with our assumption of small $\theta_0^\prime$ (e.g., $\simeq 0.1$), $({\mathcal M}_0)_{ss} \gtrsim 3\,{{\text{TeV}}}$ is required. The value in eq. results in $$\begin{aligned}
\frac{\mu}{\mu_\eta} \gtrsim 10^{-4} .
\label{eq:mu-50GeV}\end{aligned}$$ For the greater value of $m_{{\mathcal A}^0}^{}$, the larger $\mu/\mu_\eta$ is predicted. In particular, by taking $m_{{\mathcal A}^0}^{}$ to be the TeV scale, we obtain $\mu/\mu_\eta \sim 10^{-2}$, which yields $v_\Delta^{} \sim 1\,{{\text{GeV}}}$ for $\mu_\eta^{}$ and $m_\Delta^{}$ to be at the electroweak scale. Such a value for $v_\Delta^{}$ is suggested in the recent study of radiative corrections to the electroweak parameters [@Kanemura:2012rs].
On the contrary, if we take $m_{{\mathcal A}^0}^{} \simeq 83\,{{\text{GeV}}}$ which is allowed in a tiny region [@Lundstrom:2008ai]
| 1,935
| 714
| 2,815
| 1,919
| 3,673
| 0.770787
|
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|
beta_{\mathcal{C}_N,P_{N}}^{(kM)}
=\beta_{\mathcal{C},P}^{(k)}.$$ 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 [@Esnault-Viehweg82] about cyclic branched covering assuming the ambient surfaces involved have abelian quotient singularities and the divisors have simple $\Q$-normal crossings –this problem was also considered by Steenbrink [@Steenbrink77 Lemma 3.14] in the context of the semistable reduction associated with an embedded resolution. Let us start with a couple of preliminary results.
\[lemma:acyclic\] Let $\pi: Y \to X$ be a weighted blow-up at a cyclic quotient singular point and let $D$ be a Weil divisor on $X$. Then $R^i \pi_{*} \cO_Y( {\left \lfloor \pi^{*}D \right \rfloor} ) = 0$ for $i>0$.
For dimension reasons it is enough to prove the result for $i=1$. Consider $\rho: Z \to Y$ a resolution of singularities of $Y$ and denote by $\sigma$ the composition $\pi \circ \rho$ which is in turn a resolution of $X$. Let us denote $\cF := \cO_Z({\left \lfloor \sigma^{*}D \right \rfloor})$. By [@Blache95 §1.1], one has $R^1 \sigma_{*} \cF = 0$.
On the other hand, Grothendieck’s spectral sequence $E_2^{p,q} = (R^p \pi_* \circ R^q \rho_*) \cF
\Longrightarrow R^{p+q} \sigma_* \cF$ gives rise to the following $$R^1 \sigma_* \cF \simeq (R^0 \pi_* \circ R^1 \rho_*) \cF \oplus (R^1 \pi_* \circ R^0 \rho_*) \cF$$ and, in particular, $(R^1 \pi_* \circ R^0 \rho_*) \cF = 0$.
Finally, $$R^0 \rho_* \cF = \rho_* \cF = \rho_* \cO_Z({\left \lfloor \sigma^*D \right \rfloor})
= \rho_* \cO_Z ( {\left \lfloor \rho^* \pi^* D \right \rfloor} ) = \cO_Y ( {\left \lfloor \pi^*D \right \rfloor} ),$$ where the latest equality holds by the projection formula in Theorem \[thm:projection\].
\[prop:comparison\_cohomology\] Under the assumptions of Lemma[ \[lemma:acyclic\]]{}, one has the isomorphisms $$H^p (X, \
| 1,936
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| 1,979
| 3,298
| 0.773383
|
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|
r}}) \ .
\label{eq:psi1linComoving}\end{aligned}$$ 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 transform the condensate velocity (see below) to the new frame of reference, and account for the shift in kinetic energy. Indeed, such a combined transformation leaves the GPE unchanged at $\gamma = 0$ [@pismen1999] (but not for $\gamma>0$). The density perturbation $\delta\rho_1$ is already given correctly by $\delta\psi_1+\delta\psi_1^*$, where $\delta\psi_1({\boldsymbol{z}},t)$ is the solution of (\[eq:psi1linComoving\]), without the need of any additional phase factor. The velocity in the comoving frame would need to be corrected as $\delta\omega^{(1)}({\boldsymbol{z}},t)=\delta {\boldsymbol{v}}^{(1)}-{\boldsymbol{V}}_p$, with $\delta v^{(1)}$ given by expression (\[eq:velocity\_linear\]) in terms of he solution of (\[eq:psi1linComoving\]).
Eq. (\[eq:psi1linComoving\]) in the steady-state can be solved by using the Fourier transform $\delta\psi_1({\boldsymbol{z}})
=1/(2\pi)^2 \int d^2{\boldsymbol{k}} e^{i{\boldsymbol{k}}\cdot {\boldsymbol{z}}}
\delta\hat\psi_1({\boldsymbol{k}})$. It follows that the linear system of equations for $\delta\hat\psi_1({\boldsymbol{k}})$ and $\delta\hat\psi_1^*(-{\boldsymbol{k}})$ is given by $$\begin{aligned}
\left[-2i {\boldsymbol{k}}\cdot{\boldsymbol{V}}_p +(i+\gamma) (k^2+2)\right]&\delta\hat\psi_1& + 2(i+\gamma)\delta\hat\psi_1^* = \nonumber \\
&-& 2(i+\gamma) e^{-\frac{a^2k^2}{2}} ,\nonumber\\
\left[-2i {\boldsymbol{k}}\cdot{\boldsymbol{V}}_p +(-i+\gamma) (k^2+2)\right]&\delta\hat\psi_1^*& + 2(-i+\gamma)\delta\hat\psi_1 = \nonumber \\
&-& 2(-i+\gamma) e^{-\frac{a^2k^2}{2}} .\nonumber\\\end{aligned}$$ By solving these equations, we find $\delta\hat\psi_1({\boldsymbol{k}})$ and $\delta\hat\psi_1^*(-{\boldsymbol{k}})$, and the Fourier transform of the density pertu
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|
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 expected based on the results in [@varvia].
![Lowest, average, and the highest CI% as a function of training set size for GPR estimates.[]{data-label="fig:citrain"}](ci_nt.png){width="70mm"}
Discussion {#sec:disc}
----------
Conceptually, GPR is a non-parametric machine learning method that has similarities with kNN. Thus, many approaches proposed for improvement of kNN estimates within ABA could be also utilized to further improve GPR estimates. GPR seems to be insensitive to multicollinearity and quite large numbers of predictors can been used simultaneously [@alsgpr]. In this paper, fairly traditional ABA metrics were used, adding additional predictors, such as $\alpha$-shape [@alphashape] or composite metrics [@zhao2009], could potentially improve prediction performance. Additionally, dimension reduction, for example by using principal component analysis (PCA) [@junttila2015], would probably improve performance when using small training sets. Besides PCA, the deep belief network pretraining proposed in [@hinton2008] could be beneficial.
The prediction step of GPR is not computationally much more costly than using kNN. The most computationally expensive part is the GPR model training, which requires computing the matrix inverse of a large matrix (see the equations and ). However, the matrix inverse can be precomputed for a given set of training data. After this, computing the prediction and the prediction interval only requires calculating matrix products. In the LOO case ($n_t=492$), computing the GPR prediction and intervals for a plot/cell took on average 345 ms in Matlab on a AMD Ryzen 1700X (3.4 GHz) processor, this is more expensive than kNN (1.5 ms), but still feasible for practice. For com
| 1,938
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|
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 $s\!\!\upharpoonright\!\!\delta^+$ forces that $\delta\in \dot{E}$, since it will also decide the value of $L_\delta[a_{g\upharpoonright\delta^+}]$. We also have that $s\!\!\upharpoonright\!\!\delta$ forces a value on $\dot{f}\!\!\upharpoonright\!\!\delta$ and so we can choose a value $e\in\{0,1\}$ so that $s\!\!\upharpoonright\!\!\delta$ forces that $L_\delta[a_{s\upharpoonright\delta^+}]$ intersected with $\{L_\alpha[a_{s\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]:\alpha<\delta\text{ and }\dot{f}(\alpha)=e\}$ is infinite. We now have a contradiction, since $s\!\!\upharpoonright\!\!\delta^+\cup\{(\delta^+,1-e)\}$ forces that the assigned neighborhood of $\delta$ must meet the assigned neighborhood of $\alpha$, for some $\alpha<\delta$ with $\dot{f}(\alpha)=e\neq\dot{f}(\delta)$.
Point-countable type
====================
There is another normal-implies-collectionwise-Hausdorff result holding in $L$ for which we don’t know whether it holds in our MM$(S)[S]$ model:
A space is of **point-countable type** if each point is a member of a compact subspace which has a countable outer neighbourhood base.
Spaces of point-countable type simultaneously generalize locally compact and first countable spaces, and V$=$L implies normal spaces of point-countable type are collectionwise Hausdorff [@W].
Does MM$(S)[S]$ imply normal spaces of point-countable type are $\aleph_1$-collectionwise Hausdorff?
The usual arguments would show that if so, in our front-loaded model of MM$(S)[S]$, normal spaces of point-countable type would be collectionwise Hausdorff.
**Acknowledgement.** We thank Peter Nyikos for catching errors in an earlier version of this manuscript.
[Alan Dow, Department of Mathematics and Statistics, U
| 1,939
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|
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|
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.\end{aligned}$$
In this paper we only consider $q$-affine algebra and Yangian double as associative algebras and do not care about the Hopf algebra aspect.
The Yangian double $DY_\hbar(sl_N)$
-----------------------------------
As an associative algebra, the Yangian double $DY_\hbar(sl_N)$ is generated by the Drinfeld generators $\{h_{il},~e^{\pm}_{il} |
i=1,~2,~...,~N-1;~l \in {\Bbb Z}_{ \geq 0} \}$ and the center $c$. In terms of the formal power series (Drinfeld currents)
$$\begin{aligned}
& & H^{+}_i(u) = 1 + \hbar \sum_{l \geq 0} h_{il} u^{-l-1},~
H^{-}_i(u) = 1 - \hbar \sum_{l < 0} h_{il} u^{-l-1}, \\
& & E^{\pm}(u) = \sum_{l \in {\Bbb Z}} e^{\pm}_{il} u^{-l-1}\end{aligned}$$
we can write the generating relations for $DY_\hbar(sl_N)$ as follows [@iohara],
$$\begin{aligned}
& & [ H_i^\pm(u),~H_j^\pm(v) ] =0, \label{y1}\\
& & (u_\mp-v_\pm + B_{ij} \hbar) (u_\pm-v_\mp - B_{ij} \hbar)
H_i^+(u) H_j^-(v) \nonumber\\
& &~~~~= (u_\mp-v_\pm - B_{ij} \hbar) (u_\pm-v_\mp + B_{ij} \hbar)
H_j^-(v) H_i^+(u), \label{y2} \\
& & (u_\pm-v \mp B_{ij} \hbar) H_i^+(u) E^{\pm}_j(v)
= (u_\pm-v \pm B_{ij} \hbar) E^{\pm}_j(v) H_i^+(u), \label{y3} \\
& & (u_\mp-v \mp B_{ij} \hbar) H_i^-(u) E^{\pm}_j(v)
= (u_\mp-v \pm B_{ij} \hbar) E^{\pm}_j(v) H_i^-(u), \label{y4} \\
& & (u-v \mp B_{ij} \hbar) E^{\pm}_i(u) E^{\pm}_j(v)
= (u-v \pm B_{ij} \hbar) E^{\pm}_j(v) E^{\pm}_i(u), \label{y5} \\
& & [ E^+_i(u),~E^-_j(v) ] =
\frac{1}{\hbar} \delta_{ij} \left(
\delta( u_- - v_+) H^+_i(v_+) - \delta( u_+ - v_-) H^-_i(v_-) \right),
\label{y6} \\
& & E^{\pm}_i(u_1) E^{\pm}_i(u_2) E^{\pm}_j(v)
-2 E^{\pm}_i(u_1) E^{\pm}_j(v) E^{\pm}_i(u_2) \nonumber\\
& &~~~~~+ E^{\pm}_j(v) E^{\pm}_i(u_1) E^{\pm}_i(u_2) +
(\mbox{replacement:}~u_1
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|
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 distinction since it reveals that most of the difference is due
to these programs doing different things.
Changing this to compare the same thing shows why this matters:
cadams@jupiter:~ $ sbcl --version
SBCL 1.4.0
cadams@jupiter:~ $ python2.7 --version
Python 2.7.14
cadams@jupiter:~ $ pypy --version
Python 2.7.13 (84a2f3e6a7f88f2fe698e473998755b3bd1a12e2, Oct 05 2017, 16:34:13)
[PyPy 5.9.0 with GCC 4.2.1 Compatible Apple LLVM 9.0.0 (clang-900.0.37)]
cadams@jupiter:~ $ time ./test.lisp
5000000050000000
real 0m0.209s
user 0m0.194s
sys 0m0.011s
cadams@jupiter:~ $ time python2.7 test.py
5000000050000000
real 0m0.758s
user 0m0.744s
sys 0m0.008s
cadams@jupiter:~ $ time pypy test.py
5000000050000000
real 0m0.123s
user 0m0.101s
sys 0m0.019s
So at the end of that we've discovered two things we already knew: an
interpreter is slower than a JIT given enough work to balance the startup
time, and that allocating a list with millions of items and then immediately
discarding it is more expensive than summing an iterator.
Since Python 3 made range() lazy by default, the core developers clearly agree
that this is better than allocating lists unless explicitly requested.
Journey to Python Part 2: Input, Output, and Documentation - noor420
http://www.tuxtips.org/?p=14
======
dazzawazza
python with 'end of scope' comments looks totally alien to me. I hope they are
there for educational reasons rather than for a coding 'style'.
Ask HN: What is the state of the art in text to speech? - th
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| 0.786514
|
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|
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 $k$ increases indefinitely (except that its oscillations become more and more pronounced), we may very definitely state that $\lim_{k\rightarrow\infty}(\cos kx)/k=0$ uniformly. Hence from$$f_{k}^{(-1)}(x)\longrightarrow0=0+\int_{0}^{x}\lim_{k\rightarrow\infty}\sin kx_{1}dx_{1}$$ it follows that $$\lim_{k\rightarrow\infty}\sin kx=0\label{Eqn: intsin}$$ $1$-integrally.
Continuing with the same sequence of functions, we now examine its test-functional convergence with respect to $\varphi\in\mathcal{C}_{0}^{1}(-\infty,\infty)$ that vanishes for all $x\notin(\alpha,\beta)$. Integrating by parts, $$\begin{aligned}
{\displaystyle {\displaystyle \int_{-\infty}^{\infty}f_{k}\varphi}}= & {\displaystyle \int_{\alpha}^{\beta}\varphi(x_{1})\sin kx_{1}dx_{1}}\\
= & -\frac{1}{k}\left[\varphi(x_{1})\cos kx_{1}\right]_{\alpha}^{\beta}-\frac{1}{k}\int_{\alpha}^{\beta}\varphi^{\prime}(x_{1})\cos kx_{1}dx_{1}\end{aligned}$$
The first integrated term is $0$ due to the conditions on $\varphi$ while the second also vanishes because $\varphi\in\mathcal{C}_{0}^{1}(-\infty,\infty)$. Hence $$\int_{-\infty}^{\infty}f_{k}\varphi\longrightarrow0=\int_{\alpha}^{\beta}\lim_{k\rightarrow\infty}\varphi(x_{1})\sin ksdx_{1}$$ for all $\varphi$, and leading to the conclusion that $$\lim_{k\rightarrow\infty}\sin kx=0\label{Eqn: testsin}$$ test-functionally.$\qquad\blacksquare$
This example illustrates the fact that if $\textrm{Supp}(\varphi)=[\alpha,\beta]\subseteq J$[^7], integrating by parts sufficiently large number of times so as to wipe out the pathological behaviour of $(f_{k})$ gives $$\begin{aligned}
\int_{J}f_{k}\varphi= & \int_{\alpha}^{\beta}f_{k}\varphi\\
= & \int_{\alpha}^{\beta}f_{k}^{(-1)}\varphi^{\prime}=\cdots=(-1)^{m}\int_{
| 1,942
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|
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|
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 an increasing function of $\theta$ (equation \[eq:6\]; see also Fig. \[fig:th\_in\_Ds\]), this means that $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$, obtained by solving $r_{\mathrm{B}}=r_{\mathrm{HII}}(\theta)$ with respect to $\theta$, increases with $M_{\mathrm{BH}}$. Equation indeed explains the variation of the opening angle $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ in the numerical results within the error of $4^\circ$. Similarly, the accretion rates $\dot{M}/\dot{M}_{\mathrm{B}}$ estimated by equation reproduce the results with errors $\lesssim10\%$.
### Dependence on ambient density {#sec:ndep}
run $n_\infty\,[{\mathrm{cm^{-3}}}]$ $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ $\dot{M}/\dot{M}_{\mathrm{B}}$
------------ ---------------------------------- -------------------------------------------- --------------------------------
n1e3 $10^3$ $36^\circ$ $59\%$
n1e4 $10^4$ $38^\circ$ $54\%$
n1e5 (Dds) $10^5$ $40^\circ$ $59\%$
n1e6 $10^6$ $44^\circ$ $71\%$
: Summary of the $n_\infty$ dependence.[]{data-label="tab:n-model"}
\
Motivated by a wide variety of the environment in the vicinity of BHs, we finally investigate the cases with different ambient densities, termed “n-series”, where $n_\infty$ is $10^3{\,\mathrm{cm^{-3}}}$ (“n1e3 run”), $10^4{\,\mathrm{cm^{-3}}}$ (“n1e4 run”), $10^5{\,\mathrm{cm^{-3}}}$ (“n1e5 run” ide
| 1,943
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| 2,030
| null | null |
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|
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 settings of Theorems \[T:semicircle-law-general\] and \[T:semicircle-law-special\] if all the random variables with a given variance assumption satisfy such assumptions.
We now state our main results, deferring the proofs until the end of the section.
\[T:circular-law-correlated\] Let $\alpha \in [0,1]$. Suppose that for each $n$, $\{Y_a^{(n)} \mid
a \in G^{(n)}\}$ are independent; that $${\mathbb{E}}Y_a^{(n)} = 0, \quad {\mathbb{E}}\bigl\vert Y_a^{(n)}\bigr\vert^2 = 1, \quad
\text{and} \quad {\mathbb{E}}\bigl(Y_a^{(n)}\bigr)^2 = \alpha$$ for every $a \in G^{(n)}$; and that holds. Suppose further that $\lim_{n\to \infty} p_2^{(n)} = p$ exists. Then $\mu^{(n)}$ converges, in mean and in probability, to $(1 - p) \gamma_{\mathbb{C}}+ p \gamma_\alpha$.
One of the main special cases of interest in Theorem \[T:circular-law-correlated\] is when $\alpha = 1$, that is, when the matrix entries are all real. In that case, the limiting spectral distribution of $M^{(n)}$ is complex Gaussian if the number of $a$ with $a^2 = 1$ is negligible for large $n$. On the other hand, if the fraction of such $a$ is asymptotically constant then, due to the presence of many real-valued characters $\chi$, the limiting spectral distribution will be a mixture of $\gamma_{\mathbb{C}}$ and $\gamma_{\mathbb{R}}$.
The other main special case of interest is when $\alpha = 0$, so that the matrix entries have uncorrelated real and imaginary parts. In that case, which generalizes the setting of Corollary \[T:C-Ginibre-limit\], one can remove the assumption that $p_2^{(n)}$ approaches a limit.
\[T:circular-law-uncorrelated\] Suppose that for each $n$, $\{Y_a^{(n)} \mid a \in G^{(n)}\}$ are independent; that $${\mathbb{E}}Y_a^{(n)} = 0,
| 1,944
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|
tor $f(t)\xi_0$: $$\{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
\label{proj ns}$$ onto the NS string field annihilated by $D_\eta$ and its orthogonal complement, respectively.
Space-time supersymmetry
========================
Now let us discuss how space-time supersymmetry is realized in the WZW-like formulation. Starting from a natural linearized transformation exchanging the NS string field $\Phi$ and the Ramond string field $\Psi$, we construct a nonlinear transformation that is a symmetry of the complete action (\[complete action\]). We show that the transformation satisfies the supersymmetry algebra, up to the equations of motion and gauge transformation, except for an unphysical symmetry.
Space-time supersymmetry transformation
---------------------------------------
At the linearized level, a natural space-time supersymmetry transformation of string fields in the small Hilbert space, $\eta\Phi$ and $\Psi$, is given by $$\delta^{(0)}_{{\mathcal{S}}(\epsilon)} \eta\Phi\
=\ {\mathcal{S}}(\epsilon)\Psi,\qquad
\delta^{(0)}_{{\mathcal{S}}(\epsilon)} \Psi\
=\ X{\mathcal{S}}(\epsilon)\eta\Phi\,,
\label{restricted linear}
$$ where $${\mathcal{S}}(\epsilon)\ =\
\epsilon_\alpha q^\alpha\ =\
\epsilon_\alpha
\oint\frac{dz}{2\pi i}S^\alpha(z) e^{-\phi(z)/2}\\,
\label{supercharge}$$ is the first-quantized space-time supersymmetry charge with the parameter $\epsilon_\alpha$. The spin operator $S^\alpha(z)$ in the matter sector can be constructed from $\psi^\mu(z)$ using the bosonization technique [@Friedan:1985ge]. This ${\mathcal{S}}(\epsilon)$ is a (Grassmann-even) derivation of the string product, and is commutative with $Q$, $\eta$ and $\xi_0$: $[Q,{\mathcal{S}}(\epsilon)]=[\eta,{\mathcal{S}}(\epsilon)]=[\xi_0, {\mathcal{S}}(\epsilon)]=0$. It satisfies the a
| 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 $r(u)=f^{3/2}(t+u)$ and $s(u)=wf^{1/2}(t+u)$, $$g(u)=r(u)K(s(u)),\ \ g'(u)=r'(u)K(s(u))+r(u)s'(u)K'(s(u)),$$ and, dropping the arguments for simplicity, $$\begin{aligned}
\label{deriv}
g''&=&r''K +(2r's'+rs'')K'+r(s')^2K'',\notag\\
g'''&=&r'''K+(3r''s'+3r's''+rs''')K'+3(r'(s')^2+rs's'')K''+r(s')^3K'''\notag\\
g^{(4)}&=&r^{(4)}K+(4r'''s'+6r''s''+4r's'''+rs^{(4)})K'+(6r''(s')^2+12r's's''+4rs's'''
\notag\\
&&~~~~~~~~~~~~~~+3r(s'')^2)K''+
(4r'(s')^3+6r(s')^2s'')K'''+r(s')^4K^{(4)}.\end{aligned}$$
\[biasid\] Under the hypotheses in Assumptions \[ass2\], if the constant $B$ in the definition of $\tilde f_n(t;h_n)$ satisfies $B\ge T/r^{1/2}$, then, for all $0<C<\infty$ and functions $z\ge0$ with $z(h)\searrow 0$ as $h\searrow 0$, we have $$\label{unibias}
\lim_{n\to\infty}\sup_{f\in{\cal D}_{C,z}}\sup_{t\in D_r}\left|\frac{\tilde f(t;h_n)-f(t)}{h_n^4}-H(t,f,K)\right|=0$$ and $$\label{H}
\sup_{f\in{\cal D}_{C,z}}\sup_{t\in D_r}|H(t,f,K)|<\infty,$$ where $$H(t,f,K)=\left[\frac{(f')^4(t)}{ f^5(t)}-\frac{3(f')^2(t)f''(t)}{ 2f^4(t)}+\frac{4f'(t)f'''(t)+3(f'')^2(t)}{ 12f^3(t)}-\frac{f^{(4)}(t)}{24 f^2(t)}\right]\int v^4K(v)dv.$$
Since $f$ and $K$ and their first four derivatives are continuous and $f>r/2$ on a neighborhood of $D_r$, it follows that, if $g_{tw}$ is as defined in (\[g\]), there exists $n_0<\infty$ such that, for all $t\in D_r$ and for all $w\in{\mathbb R}$, $g_{t,w}^{(4)}$ is continuous on $[-Bh_n,Bh_n]$ for all $n\ge n_0$: note that $g^{(4)}(u)$ is a linear combination of $K$ and its first four derivatives at $wf^{1/2}(t+u)$ whose coefficients are fractions that have products of powers of $w$ and powers of $f(t+u)$ and its derivatives in the numerator, and powers of $f(t+u)$ in the denominator (see(\[d
| 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"} |
+------------------------------------------------------------------+------------------+-------------+---------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| Downstream purification losses | 20‐50 | \% | Type of unit operations, process development | [48](#amp210060-bib-0048){ref-type="ref"}, [49](#amp210060-bib-0049){ref-type="ref"}, [60](#amp210060-bib-0060){ref-type="ref"} β |
+------------------------------------------------------------------+------------------+-------------+---------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| Raw material recycling[^a^](#amp210060-note-0002){ref-type="fn"} | 0‐8 | Fold | Stability of the materials, regulatory approval | α |
+------------------------------------------------------------------+------------------+-------------+---------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------
| 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{aligned}
\E{f(\bx_{(k+1)T} - \bw_{(k+1)T})}
\leq e^{-\lambda \delta} \E{f(\bx_{kT} - \bw_{kT})} + 14T (L+\LN^2)
\end{aligned}$$
Applying the above recursively gives, for any $i$ $$\begin{aligned}
\E{f(\bx_{iT} - \bw_{iT})} \leq e^{-\lambda iT} \E{f(\bx_{0} - \bw_{0})} + \frac{14}{\lambda} \lrp{L + \LN^2} \epsilon
\end{aligned}$$
[Proof of Theorem \[t:main\_nongaussian\]]{} \[ss:proof:t:main\_nongaussian\] For ease of reference, we re-state Theorem \[t:main\_nongaussian\] below as Theorem \[t:main\_nongaussian:restated\] below. We make a minor notational change: using the letters $\bx_t$ and $\by_t$ in Theorem \[t:main\_nongaussian:restated\], instead of the letters $x_t$ and $y_t$ in Theorem \[t:main\_nongaussian\]. This is to avoid some notation conflicts in the proof.
\[Equivalent to Theorem \[t:main\_nongaussian\]\] \[t:main\_nongaussian:restated\] Let $\bx_t$ and $w_t$ have dynamics as defined in and respectively, and suppose that the initial conditions satisfy $\E{\lrn{\bx_0}_2^2}\leq R^2 + \beta^2/m$ and $\E{\lrn{\bw_0}_2^2}\leq R^2 + \beta^2/m$. Let $\hat{\epsilon}$ be a target accuracy satisfying $\hat{\epsilon} \leq \lrp{\frac{16\lrp{L + \LN^2}}{\lambda}} \cdot \exp\lrp{7\aq\Rq/3} \cdot \frac{\Rq}{\aq\Rq^2 + 1}$. 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 $\delta$ be a step size satisfying $$\begin{aligned}
\delta \leq \min\lrbb{\frac{T\epsilon^2L}{36 d\beta^2\log \lrp{ \fr
| 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 contain a copy of the ${{W}}$-module $\operatorname{{\textsf}{triv}}$ (we thank Pavel Etingof for this fact). In particular $eV_c=0$ and so [*the functor $E_c$ is not an equivalence*]{}. If we further assume that $(m,n)=1$, then $V_c$ is the unique irreducible finite dimensional $H_c$-module by [@CE Corollary 3.3] and [@BEGfd Theorem 1.2(ii)]. Since $U_c=\mathrm{End}_{H_c}(eH_c)$, this implies that $U_c$ has no finite dimensional modules. However, by Corollary \[morrat-cor\](1) and [@BEGfd Theorem 1.2] $U_{c\pm 1}{\text{-}{\textsf}{mod}}$ does have such modules and so [*there is no equivalence between $U_c $ and $U_{c\pm 1} $*]{}.
Corollary {#gldim}
---------
[*Assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[morrat-hyp\]. Then $H_c$ and $U_c$ have finite homological global dimension and satisfy the Auslander-Gorenstein conditions and Cohen-Macaulay conditions of [@lev].*]{}
Since this result takes us a little far afield, the details of the proof are left to the interested reader. Standard techniques show that $\operatorname{{\textsf}{ogr}}H_c\cong {{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}\ast{{W}}$ and hence $H_c$ have the given properties (see, for example, [@Br Theorem 4.4]). By Corollary \[morrat-cor\], $U_c$ is Morita equivalent to $H_c$ and it follows that $U_c$ also has these properties.
The shift functor on $\mathcal{O}_c$ {#subsec-4.6}
------------------------------------
Many computations for ${U}_c$ reduce to computations in category $\mathcal{O}$ and so it is important to know that, under the hypotheses of Theorem \[morrat\], $S_c$ does provide an equivalence between the corresponding categories. This is the point of the next result.
\[shiftonO\] Assume
| 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 $y_i=q^{i-1}$,
$$\Omega\left(\sqrt{q},\frac{1}{\sqrt{q}}\right)=\sum_{\lambda\in\calP}q^{(1-q)|\lambda|}\left(q^{-n(\lambda)}H_\lambda(q)\right)^{2g+k-2}\prod_{i=1}^ks_\lambda(\x_i\y).$$
\[specializ\]
The rational function $\H_\muhat(z,w)$ is a polynomial with integer coefficients. It has degree
$$d_\muhat:=n^2(2g-2+k)-\sum_{i,j}(\mu^i_j)^2+2$$ in each variable and the coefficients of $\H_\muhat(-z,w)$ are non-negative. \[conjH\]
The function $\H_\muhat(z,w)$ is computed in many cases in [@hausel-letellier-villegas §1.5].
Characters and Fourier transforms {#finite-groups}
---------------------------------
### Characters of finite general linear groups {#charGL}
For a finite group $H$ let us denote by ${\rm Mod}_H$ the category of finite dimensional $\C[H]$ left modules. Let $K$ be an other finite group. By an *$H$-module-$K$* we mean a finite dimensional $\C$-vector space $M$ endowed with a left action of $H$ and with a right action of $K$ which commute together. Such a module $M$ defines a functor $R_K^H:{\rm Mod}_K\rightarrow{\rm Mod}_H$ by $V\mapsto M\otimes_{\C[K]} V$. Let $\C(H)$ denotes the $\C$-vector space of all functions $H\rightarrow\C$ which are constant on conjugacy classes. We continue to denote by $R_K^H$ the $\C$-linear map $\C(K)\rightarrow\C(H)$ induced by the functor $R_K^H$ (we first define it on irreducible characters and then extend it by linearity to the whole $\C(K)$). Then for any $f\in\C(K)$, we have
$$R_K^H(f)(g)=|K|^{-1}\sum_{k\in K}{\rm Trace}\,\left((g,k^{-1})\,|\, M\right)f(k).\label{bimod}$$
Let $G=\GL_n(\F_q)$ with $\F_q$ a finite field. Fix a partition $\lambda=(\lambda_1,\dots,\lambda_r)$ of $n$ and let $\calF_\lambda=\calF_\lambda(\F_q)$ be the variety of partial flags of $\F_q$-vector spaces $$\{0\}=E^r\subset E^{r-1
| 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
---------------- ------------------------------ -------------------------------- ------------------ -------------- -------------- -------------- -------------- -------------- ------------ ------------ -------------- -------------
Goliath 423.2 ± 6.8 404.9 ± 0.5 255.5 ± 9.0 271.0 ± 3.2 305.9 ± 6.1 257.6 ± 7.5 253.0 ± 12.6 300.5 ± 12.4 72.3 ± 1.3 63.6 ± 1.9 98.9 ± 1.5 111.1 ± 5.9
Hyb 5 465.4 ± 16.8 441.0 ± 3.5 258.6 ± 24.0 287.9 ± 9.6 318.5 ± 39.7 311.5 ± 9.7 264.0 ± 28.9 324.9 ± 2.9 68.4 ± 6.0 70.9 ± 2.6 102.0 ± 26.4 113.1 ± 4.9
Hyb 6 430.4 ± 2.6 419.0 ± 4.7 255.2 ± 10.7 288.4 ± 16.5 299.6 ± 15.5 273.9 ± 5.0 258.1 ± 23.7 308.3 ± 5.9 69.8 ± 3.2 65.4 ± 0.4 103.9 ± 11.3 107.2 ± 6.4
Hyb 7 416.8 ± 7.6 445.2 ± 4.0 286.8 ± 22.6 278.1 ± 3.0 353.8 ± 10.3 316.6 ± 9.1 289.3 ± 1.6 326.7 ± 6.0 85.0 ± 1.6 71.1 ± 1.8 100.8 ± 9.4 117.5 ± 2.8
Hyb 8 451.9 ± 13.1 405.9 ± 5.8 243.0 ± 10.3 277.0 ± 11.4 364.8 ± 6.3 320.9 ± 6.3 261.7 ± 5.9 305.5 ± 6.9 80.8 ± 1.5 79.5 ± 0.6 107.6 ± 3.8 110.5 ± 2.0
Hyb 9 417.3 ± 2.8 440.3 ± 6.5 2
| 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 interpretation of a physical QL into $\mathcal{L}_{QD}^{P}$.
Let us comment briefly on the pragmatic interpretation of physical QL provided above.
Firstly, we note that our interpretation maps $\mathcal{E}$ on the quotient set $\phi _{AD}^{Q}/\approx $, not onto $\phi _{AD}^{Q}$. Yet, the set of the (well formed) formulas of the lattice $(\mathcal{E},^{\bot },\Cap ,\Cup
) $ can be mapped bijectively onto $\phi _{AD}^{Q}$ by means of the mapping induced by the following formal correspondence.
\(i) $E\in \mathcal{E}$ $\longleftrightarrow \vdash E(x)\in \phi _{AD}^{Q}$.
\(ii) $^{\bot }\longleftrightarrow N$
\(iii) $\Cap \longleftrightarrow K$
\(iv) $\Cup \longleftrightarrow A_{Q}$.
Thus, the formal language of QL, for which the lattice $(\mathcal{L(S)},\subset )$ can be considered as an *algebraic semantics*,$^{(3)}$ can be substituted by the language $\mathcal{L}_{QD}^{P}$, for which $(\mathcal{L(S)},\subset )$ can be considered as an *algebraic pragmatics* (by the way, we also note that the above correspondence makes $I_{Q}$ correspond to a *Sasaki hook*, the role of which is well known in QL). This reinterpretation is relevant from a philosophical viewpoint, since it avoids all problems following from the standard concept of quantum truth (Sec. 2.4) considering physical QL as formalizing properties of a quantum concept of justification rather than a quantum concept of truth. This makes physical QL consistent also with the classical concept of truth adopted with the SR interpretation of QM (Sec. 2.5). Furthermore, as we have already observed in the Introduction, it places physical QL within a general *integrated perspective*, according to which non-Tarskian theories of truth can be integrated wi
| 1,952
| 4,099
| 3,314
| 1,973
| null | null |
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|
$m$-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 of the maps: maps are suitably scaled so that $(0,-1)$ goes to $(0,0)$ and that the angle formed by the pairs of the arcs including $(0,0)$ are equal, for example. We can obtain a local map around the vertex satisfying the first two conditions. Note that the measure zero set of the local function is obtained by identifying the equations of $n$-copies of $S^{m-1}$.
![Attacing copies of a map in FIGURE \[fig:2\].[]{data-label="fig:4"}](zahyothree.eps){width="30mm"}
\
Step 3 Around a vertex of degree $1$.\
We consider a natural height function on a unit disc of dimension $m>1$ whose image is $[0,1]$ and $0$ or $1$ is the only one vertex. The function is also a Morse function with just $1$ singular point in the interior.\
\
Step 4 Completing the construction.\
For the interior of each edge, we construct a trivial $C^r$ bundle whose fiber is a standard sphere. Last we glue all the constructed local maps together to obtain a global map.\
\
Step 5 Define $\mathcal{C}$ and $\mathcal{Q}_{\mathcal{C}}$.\
Through Steps 1–4, we construct a map for arbitrary finite graphs which are not single points. Last, we explain about the classes $\mathcal{C}$ and $\mathcal{Q}_{\mathcal{C}}$. $\mathcal{Q}_{\mathcal{C}}$ are naturally defined as a class locally of a form of the obtained forms. We define $\mathcal{C}$ as a map such that around each singular value corresponding to a vertex of degree not $1$, the map is represented as the composition of a presented local map onto a regular neighborhood of a graph, regarded as a map into the plane, a homeomorphism on the plane satisfying the following (we define such a map as an [*almost smooth generalized rotation with reflection*]{}) and a projection onto a straight
| 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], $$\label{eq:s_aver}
\langle S_{ab} \rangle = \frac{1-\lambda_a^*}{1+\lambda_a}\delta_{ab}$$ Here, we set $E=0$ as usual. As a result, the transmission coefficient takes the following form: $$\label{eq:Tc}
T_a \equiv 1- |\langle S_{aa}\rangle|^2 = \frac{4\,\mathrm{Re}(\lambda_a)}{|1+\lambda_a|^2}\,,\quad a=1,\ldots,M,$$ which is in agreement with the result of Ref. [@ver85a] obtained by a supersymmetry calculation. It is worth noting that in the case of real $\lambda$ one gets $T=\frac{4\lambda}{(1+\lambda)^2} \in [0,1]$. In the case of purely imaginary $\lambda$ corresponding to perfect reflection, the channel is closed, $T=0$.
Coupling fidelity
-----------------
We now proceed with the discussion of the scattering fidelity. Its amplitude is defined in terms of the $S$-matrix elements Fourier transformed into the time domain as follows [@sch05b]: $$\label{eq:f_ab}
f_{ab}(t) = \frac{\langle \hat{S}_{ab}(t)\hat{S}_{ab}^{\prime *}(t)\rangle}{ \sqrt{ \langle\hat{S}_{ab}(t)\hat{S}_{ab}^{*}(t)\rangle\langle \hat{S}_{ab}^{\prime}(t)\hat{S}_{ab}^{\prime *}(t)\rangle }}\,.$$ The prime indicates a change of the effective Hamiltonian of the original system after a small perturbation for the backward time evolution. Definition (\[eq:f\_ab\]) guarantees that $f_{ab}(0)=1$. Furthermore, an overall decay of the correlation functions due to absorption drops out, provided the decay is the same for the parametric cross-correlation functions in the nominator and the autocorrelation functions in the denominator [@sch05b]. The scattering fidelity itself is $$\label{eq:F_ab}
F_{ab}(t) = |f_{ab}(t)|^2\,.$$ Note that in contrast to the original definition of the scattering fidelity, we allow for a change in the channel vectors as well.
As it is explained
| 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]{}, L., [Wuyts]{}, S., & [Cox]{}, T. J. 2010, , 401, 1099
, P. F., [Bundy]{}, K., [Murray]{}, N., [Quataert]{}, E., [Lauer]{}, T. R., & [Ma]{}, C.-P. 2009, , 398, 898
, P. F., [Hernquist]{}, L., [Cox]{}, T. J., [Di Matteo]{}, T., [Robertson]{}, B., & [Springel]{}, V. 2006, , 163, 1
, P. F., [Hernquist]{}, L., [Cox]{}, T. J., [Keres]{}, D., & [Wuyts]{}, S. 2009, , 691, 1424
, P. F., [Lidz]{}, A., [Hernquist]{}, L., [Coil]{}, A. L., [Myers]{}, A. D., [Cox]{}, T. J., & [Spergel]{}, D. N. 2007, , 662, 110
, P. F., [Murray]{}, N., [Quataert]{}, E., & [Thompson]{}, T. A. 2010, , 401, L19
, S., [Dickinson]{}, M., [Alexander]{}, D. M., & [Salim]{}, S. 2011, , 736, 104
, G., [White]{}, S. D. M., [Heckman]{}, T. M., [M[é]{}nard]{}, B., [Brinchmann]{}, J., [Charlot]{}, S., [Tremonti]{}, C., & [Brinkmann]{}, J. 2004, , 353, 713
, S. & [Silk]{}, J. 2006, , 648, L21
, K. [et al.]{} 2010, , 718, 86
, M. [et al.]{} 2006, , 649, L71
, I. [et al.]{} 2005, , 624, L81
, L. [et al.]{} 2007, , 660, L51
—. 2010, , 718, 1158
, F. S., [Xia]{}, X. Y., [Mao]{}, S., [Wu]{}, H., & [Deng]{}, Z. G. 2008, , 385, 23
, M. [et al.]{} 2005, , 361, 897
, J. M. [et al.]{} 2008, , 672, 177
, D. T. [et al.]{} 2010, , 402, 282
, H. B. & [Whitney]{}, D. R. 1947, The Annals of Mathematical Statistics, 18, 50
, D. H., [Guo]{}, Y., [Hertzberg]{}, J., [Katz]{}, N., [Mo]{}, H. J., [van den Bosch]{}, F. C., & [Yang]{}, X. 2008, , 388, 1537
, A. D. [et al.]{} 2009, , 392, 125
, S. M., [Ellis]{}, R. S., [Treu]{}, T., [Smail]{}, I., [Dressler]{}, A., [Coil]{}, A. L., & [Smith]{}, G. P. 2005, , 634, 977
, A. D., [Richards]{}, G. T., [Brunner]{}, R. J., [Schneider]{},
| 1,955
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| 2,095
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|
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 the Spanish Ministry of Science, Innovation, and Universities.
The authors declare no conflicts of interest.
{#viruses-11-00706-f001}
viruses-11-00706-t001_Table 1
######
Information about the candidate genes and polymorphisms examined in the present study.
Marker Gene Acronym Gene Function Polymorphism Gene Location Chromosomal Location ^1^ Reference
-------------- -------------- ------------------------------------------------------------------------------- -------------- ----------------------------- -------------------------- -----------------------------
rs80800372 *GBP1* Interferon-induced guanylate binding protein with known antiviral functions A\>G 3'UTR SSC4 \[[@B7-viruses-11-00706]\]
rs340943904 *GBP5* Inflammasome assembly, innate immunity G\>T Intron 5 SSC4 \[[@B9-viruses-11-00706]\]
c.3534C\>T *CD163* Macrophage-specific scavenger receptor, mediates PRRSV entry into macrophages C\>T 3'UTR SSC5 \[[@B11-viruses-11-00706]\]
rs1107556229 G\>A Exon 10 SSC5 \[[@B11-viruses-11-00706]\]
| 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 contributions from many particles at long distances significantly increases the number of operations required.
We thus adopt the Ewald method, which allows us to accelerate the summation by dividing it into two parts: one in real space and the other in wavenumber space. For instance, the electrostatic potential may be calculated as follows: $$\label{ewald}
U=U_{{\rm real}}+U_{{\rm wave}}-U_{{\rm self}},$$ $$\label{real}
U_{{\rm real}}=\frac{1}{2}\sum_{i,j}\sum_{n}{\frac{q_{i}q_{j}}{r_{ijn}} {\rm erfc}\left(\frac{r_{ijn}}{\sigma} \right)},$$ $$\label{wave}
U_{{\rm wave}}=\frac{1}{2}\sum_{i,j}\sum_{{\bm k}\neq 0}{q_{i}q_{j}\frac{\exp\left[{-\pi^{2}\sigma^{2}k^{2}+2\pi i{\bm k}\cdot\left({\bm r}_{i}-{\bm r}_{j}\right)}\right]}{\pi V k^{2}}},$$ $$\label{self}
U_{{\rm self}}=\frac{1}{\sqrt{\pi}\sigma}\sum_{i}{q_{i}^{2}}.$$ Here, $n$ represents the labels of boxes, $r_{ijn}$ is the distance between particles $i$ and $j$ in box $n$, $q_{i}$ is the charge of particle $i$, ${\bm k}$ is the wavenumber vector, and $V$ is the volume of the box. The parameter $\sigma$ gives a cut-off radius beyond which the direct summation in real space, Eq. (\[real\]), is replaced by that in wavenumber space, Eq. (\[wave\]). Note that in Eq. (\[real\]), the term $n=0$ has to be excluded for $i=j$. This method approximates long-wavelength modes associated with the long-range nature of the Coulomb interaction in wavenumber space with the aid of the Fourier transform, whereas short-wavelength components arising from close encounters between particles are accurately calculated. The electric field is given by the spatial derivatives of Eqs. (\[real\]) and (\[wave\]) and is calculated in the same way.[@Deserno98-1; @Pol
| 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 dy. This should be the same as the one arisen from the Galilean boost and anisotropic transformations locally, \[if2\] dx&=&dx,\
dy&=&ddy+v dx. Comparing with , we get the constraints on the transformations, $$d\partial_x \epsilon(x,y)=\partial_y \xi(x,y),$$ $$\partial_y\epsilon(x,y)=0.$$ The allowed infinitesimal transformations are $$x\rightarrow x+\epsilon(x),\ \ \ y\rightarrow (1+d\epsilon'(x))y, \label{epsilon1}$$ $$x\rightarrow x,\ \ \ y\rightarrow y+\xi(x), \label{epsilon2}$$ It turns out the allowed finite symmetry transformations are $$x\rightarrow f(x) ,\hs{3ex}y\rightarrow f'(x)^dy,$$ and $$x\rightarrow x,\hs{3ex}y\rightarrow y+g(x).$$
From the infinitesimal transformations ,, we read the generators $$l_n=-x^{n+1}\partial_x-d(n+1)x^n y\partial_y,$$ $$m_n=x^{n+d}\partial_y,$$ which satisfy the algebra $$\begin{aligned}
&&\left[l_n,l_m\right]= (n-m)l_{n+m},\\
&&\left[l_n,m_m\right]= (dn-m)m_{n+m} ,\\
&&\left[m_n,m_m\right] =0.\end{aligned}$$ This algebra is analogous to the Witt algebra, and it is called the spin-$d$ Galilean algebra. The central extended one $\tilde{g}=g\oplus C$ has been discussed in section 2.
Now we require that the action of the theory is invariant under the symmetries above $$\delta S[\delta A_\mu,\delta \bar{A}_\mu]=0$$ where $$\delta A_\mu=\lambda A_\mu,\ \ \ \delta \bar{A}_\mu=d\lambda \bar{A}_\mu+v A_\mu.$$ The corresponding currents can be read from $$\delta S[\delta A_\mu,\delta \bar{A}_\mu]=\int H (J^\mu \delta A_\mu+\bar{J}^\mu\delta \bar{A}_\mu),$$ with $$\bar{J}^\mu A_\mu=0,\ \ \ J^\mu A_\mu+d\bar{J}^\mu \bar{A}_\mu=0.$$ In the canonical coordinate $(x,y)$, $$(\star\bar{J})_x=\bar{h}_x,\ \ (\star\bar{J})_y=\bar{h}_y,\ \ (\star J)_x=h_x,\ \ (\star J)_y=h
| 1,958
| 3,160
| 2,533
| 1,712
| null | null |
github_plus_top10pct_by_avg
|
e $i\in I$, and a matrix $(q_{ij})_{i,j\in I}\in (\fienz)^{I\times I}$, such that $${\delta }(x_i)=g_i{\otimes }x_i,\quad g_i{\boldsymbol{\cdot}}x_j=q_{ij}x_j \quad
\text{for all $i,j\in I$.}$$ Assume that $\chi ({\alpha }_i,{\alpha }_j)=q_{ij}$ for all $i,j\in I$. For all ${\alpha }\in {\mathbb{Z}}^I$ define the “bound function” $$\begin{aligned}
{b^{\chi}} ({\alpha })=&
\begin{cases}
\min \{ m\in {\mathbb{N}}\,|\,
\qnum{m}{\chi ({\alpha },{\alpha })}=0\} &
\text{if $\qnum{m}{\chi ({\alpha },{\alpha })}=0$}\\
& \text{for some $m\in {\mathbb{N}}$,}\\
\infty & \text{otherwise}.
\end{cases}
\label{eq:height}\end{aligned}$$ If $p\in I$ such that $\chi $ is $p$-finite, then $$\begin{aligned}
\bfun{r_p(\chi )}({\sigma }_p^\chi ({\alpha }))={b^{\chi}} ({\alpha })\quad
\text{for all ${\alpha }\in {\mathbb{Z}}^I$}
\label{eq:hghtrpchi}\end{aligned}$$ by Eq. .
The tensor algebra $T(V)$ admits a universal braided Hopf algebra quotient ${\mathfrak{B}(V)}$, called the *Nichols algebra of* $V$. As an algebra, ${\mathfrak{B}(V)}$ has a unique ${\mathbb{Z}}^I$-grading $$\begin{aligned}
{\mathfrak{B}(V)}=\oplus _{{\alpha }\in {\mathbb{Z}}^I}{\mathfrak{B}(V)} _{\alpha }\label{eq:NAVgrading}\end{aligned}$$ such that $\deg x_i={\alpha }_i$ for all $i\in I$. This is also a coalgebra grading. There exists a totally ordered index set $(L,\le )$ and a family $(y_l)_{l\in L}$ of ${\mathbb{Z}}^I$-homogeneous elements $y_l\in {\mathfrak{B}(V)}$ such that the set $$\begin{aligned}
\{ y_{l_1}^{m_1}y_{l_2}^{m_2}\cdots y_{l_k}^{m_k}\,|\,
&k\ge 0,\,l_1,\dots ,l_k\in L,\,l_1>l_2>\cdots >l_k,\\
&m_i\in {\mathbb{N}},\,m_i<{b^{\chi}} (\deg y_{l_i})
\quad \text{for all $i\in I$}\}
\end{aligned}
\label{eq:PBWbasis}$$ forms a vector space basis of ${\mathfrak{B}(V)}$. The set $$\begin{aligned}
\label{eq:roots}
R^\chi _+=\{\deg y_l\,|\,l\in L\}\subset {\mathbb{Z}}^I\end{aligned}$$ depends on the matrix $(q_{ij})_{i,j\in I}$, but not on the choice of the basis $\{x_i\,|\,i\in I\}$, the set $L$, and t
| 1,959
| 2,294
| 1,441
| 1,786
| null | null |
github_plus_top10pct_by_avg
|
general.
We now examine the relation between the covariant derivative operators $\nabla_i$ of $g_{i j}$ and $\bar{\nabla}_i$ of $\bar{g}_{i j}$. The relation is determined by $$\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^{i l} g_{j k} \partial_l \psi \right) \; ,$$ from which follows the scalar curvature relation first used in an initial-value problem by Lichnerowicz [@Lich], $$R(\bar{g}) = \psi^{-4} R(g) - 8 \psi^{-5} \Delta_g \psi \; ,$$ where $\Delta_g \psi \equiv g^{k l} \nabla_k \nabla_l \psi$ is the “rough” scalar Laplacian associated with $g_{i j}$.
Next, we solve (\[Eq:gdot\]) for its traceless part $$\dot{\bar{g}}_{i j} - \frac{1}{3} \bar{g}_{i j} \bar{g}^{k l}
\dot{\bar{g}}_{k l} \equiv \bar{u}_{i j}
= -2 \bar{N} \bar{A}_{i j} + (\bar{L} \bar{\beta})_{i j}
\label{Eq:traceless}$$ with $\bar{A}_{i j} \equiv \bar{K}_{i j} - \frac{1}{3} \bar{K} \bar{g}_{i j}$ and $$(\bar{L} \bar{\beta})_{i j} \equiv \bar{\nabla}_i \bar{\beta}_j
+ \bar{\nabla}_j \bar{\beta}_i - (2/3) \bar{g}_{i j}
\bar{\nabla}^k \bar{\beta}_k \; .
\label{Eq:LB}$$
Expression (\[Eq:LB\]) vanishes, for non-vanishing $\bar{\beta}^i$, if and only if $\bar{g}_{i j}$ admits a conformal Killing vector $\bar{\beta}^i = k^i$. Clearly, $k^i$ would also be a conformal Killing vector of $g_{i j}$, or of any metric conformally equivalent to $\bar{g}_{i j}$, with no scaling of $k^i$. This teaches us that in general $\bar{\beta}^i = \beta^i$, while $\bar{\beta}_i = \bar{g}_{i j} \bar{\beta}^j = \psi^4 g_{i j} \beta^j=\psi^4 \beta_i$. That $\bar{\beta}^i=\beta^i$ also follows because $\beta^i$, generator of a spatial diffeomorphism, is not a dynamical variable. The latter “rule” was inferred as a matter of principle.
It is clear in (\[Eq:traceless\]) that the left hand side $\bar{u}_{i j}$ satisfies $\bar{u}_{i j} = \psi^4 u_{i j}$ because the terms in $\dot{\psi}$ cancel out. Furthermore, a straightforward calculation shows tha
| 1,960
| 4,135
| 1,943
| 1,603
| null | null |
github_plus_top10pct_by_avg
|
Lemma \[lemma7\]), we find that the homomorphism labelled by $$\gyoung(;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;2;2;{x_1}_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};{x_s},;d;d;{z_2};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{z_t},;{z_1};k)$$ occurs in ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ but not in ${\psi_{d,1}}\circ{\hat\Theta_{T'}}$ for any other $T'$.
So in $\theta$ the coefficient of ${\hat\Theta_{T}}$ is zero for any $d$-bad tableau. In particular, this means that for any ${\hat\Theta_{T}}$ occurring in $\theta$, ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ is either zero or a single semistandard homomorphism. Now we claim that the coefficient of ${\hat\Theta_{T}}$ is zero whenever $T$ has two or three $2$s in its first row. Supposing this is false, take a $T$ with at least two $2$s in its first row such that ${\hat\Theta_{T}}$ appears with non-zero coefficient in $\theta$, and suppose that $T$ is minimal (with respect to the dominance order) subject to this property. Suppose the $(3,1)$-entry of $T$ is greater than $3$; then this entry equals $d+1$ for some $d\gs3$, and the entry equal to $d$ cannot be the $(2,1)$-entry (because $T$ is not $d$-bad). So the $d$ and the $d+1$ in $T$ lie in different rows and different columns, and ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ is the semistandard homomorphism obtained by replacing the $d+1$ with a $d$. The only other semistandard tableau $T'$ such that ${\psi_{d,1}}\circ{\hat\Theta_{T'}}={\psi_{d,1}}\circ{\hat\Theta_{T}}$ is the tableau obtained by interchanging the $d$ and the $d+1$ in $T$, so ${\hat\Theta_{T'}}$ must also occur with non-zero coefficient. But $T\dom T'$, contradicting the choice of $T$.
So the $(3,1)$-entry in $T$ must be $3$ (and hence the $(2,1)$-entry is $2$). Now consider the $(3,2)$-entry; call this $d+1$. Then $d\gs4$, and the $d$ in $T$ cannot occur in the $(
| 1,961
| 2,038
| 1,714
| 1,736
| 1,235
| 0.792287
|
github_plus_top10pct_by_avg
|
(\psi_1,\psi_2,\psi_3)=(\psi_1,\hat{\psi})$ 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 $$\hat{f}_j:=\hat K_{1,j}(T_{1,0}^{-1}f_1),\quad j=2,3.$$
Consider the term $\hat K_{1,j}(T^{-1}_{1,0}(\ol K\hat\psi))$. We find that for a fixed $E\in I$ $$\hat K_{1,j}(T^{-1}_{1,0}(\ol K\hat\psi))=\hat K_{1,j}(E)T_{1,0}(E)^{-1}(\ol K(E)\hat\psi(E))$$ where for every $E\in I$ the linear operator $T_{1,0}(E):L^2(G\times S)\to L^2(G\times S)$ with domain $D(T_{1,0}(E))$ is defined to be $$&
D(T_{1,0}(E)):=\tilde{W}^2_{-,0}(G\times S)\nonumber\\
&
T_{1,0}(E)v:=\omega\cdot\nabla_x v+\Sigma_1(E)v-\ol K_1(E)v.$$ By (the proof of) Lemma , the operator $T_{1,0}(E)$ is invertible and $$\begin{aligned}
\label{eq:T10Einv_bound}
{\left\Vert T_{1,0}(E)^{-1}\tilde{q}\right\Vert}_{L^2(G\times S)}\leq {1\over c}{\left\Vert \tilde{q}\right\Vert}_{L^2(G\times S)},\quad \forall\tilde{q}\in L^2(G\times S).\end{aligned}$$
Define linear operators $Q_j(E):L^2(G\times S)^2\to L^2(G\times S)$, $j=2,3$, $E\in I$, by setting $$Q_j(E)u:=\hat{K}_{1,j}(E)T_{1,0}(E)^{-1}(\ol{K}(E)u),$$ and let $Q(E):=(Q_1(E),Q_2(E))$. Then we have by and , (see e.g. ) that for all $E\in I$, [Q\_j(E)]{}[\_[1,j]{}(E)]{}[T\_[1,0]{}(E)\^[-1]{}]{}[K(E)]{} , where the operator norms used are taken, in an obvious way, with respect to the space $L^2(G\times S)$ and its product $L^2(G\times S)^2$.
Using assumption and Lemma \[coupevle2\], we find that for any fixed $u\in L^2(G\times S)^2$ the mapping $h_u:I\to L^2(G\times S)^2$ given by $h_u(E):=Q(E)u$ belongs to $C^1(I,L^2(G\times S)^2)$. Similarly, we find that $\hat{f}:=(\hat{f}_1,\hat{f}_2)\in C^1(I,L^2(G\times S)^2)$, since we have $$\hat{f}_j(E)=\hat{K}_{1,j}(E)T_{1,0}(E)^{-1}(f_1(E)),\quad E\in I,\ j=2,3,$$ where $\hat{f}_j(E)(x,\omega)=\hat{f}_j(x,\omega,E)$.
Let $C:=\max\{C_2,C_3\}$ where for $j=2,3$, $$\begin{
| 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 $\{n, n-1, \ldots, n-k+1\}$ are consecutive composite numbers. If $k=1$, then both $\Gamma(A_n)$ and $\Gamma(\Sigma_n)$ are non-connected. For $k \geq 2$, let $t$ be the largest prime number such that $t \leq k$. Then:
- If $k=2$, then $\Gamma(A_n)$ is non-connected and ${{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}=\{2\}$.
- If $k=3$, then ${{\operatorname}{\mathcal{Z}}(\Gamma(A_n))}=\{3\}$ and ${{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}=\{2,3\}$.
- If $k\geq 4$, then ${{\operatorname}{\mathcal{Z}}(\Gamma(A_n))}={{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}=\{s\in\pi(\Sigma_n)\, | \, 2\leq s \leq t\}$.
It is well known that two odd primes $s, u$ are adjacent in $\Gamma(A_n)$, and so in $\Gamma(\Sigma_n)$, if and only if $s+u \leq n$. On the other hand, if $s$ is an odd prime, $s$ is adjacent to $2$ in $\Gamma(A_n)$ if and only if $s+4 \leq n$, and $2,s$ are adjacent in $\Gamma(\Sigma_n)$ only when $s+2 \leq n$. It is then clear that if $k=1$, then both $\Gamma(A_n)$ and $\Gamma(\Sigma_n)$ are non-connected.
Consider the prime $r:=n-k$, which is the largest prime divisor of $n!$ by the choice of $k$. Clearly, $r>\frac{n}{2}>k=n-r \geq t$. Thus $r+t \leq n$, and we deduce that $t\in {\operatorname}{Z}(\Gamma(\Sigma_n))$.
If $k=2$, then $r=n-2$, and so ${{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}=\{2\}$. Since $r+4>n$, then $\Gamma(A_n)$ is non-connected.
If $k=3$, then $r=n-3$. It follows that ${{\operatorname}{\mathcal{Z}}(\Gamma(\Sigma_n))}=\{2, 3\}$ and ${{\operatorname}{\mathcal{Z}}(\Gamma(A_n))}=\{3\}$.
Finally, let us suppose that $k\geq 4$, so $n\geq 11$. Take a prime $s\in\pi(\Sigma_n)$. If $s\leq t$, then $r+s\leq r+t\leq r+k=n$, and so $s$ lies in ${{\operatorname}{\mathc
| 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 $R_{\mathbb{Z}}$ is the Morita ${\mathbb{Z}}$-algebra associated to the data $\{R_n,P_n : n\in {\mathbb{N}}\}$, where $R_0$ is noetherian.
1. Each finitely generated graded left $R_{\mathbb{Z}}$-module is noetherian.
2. The association $\phi: M \mapsto \bigoplus_{n\in {\mathbb{N}}}R_{n0}\otimes_{R_0}M$ induces an equivalence of categories between $R_0{\text{-}{\textsf}{mod}}$ and $R_{\mathbb{Z}}{\text{-}{\textsf}{qgr}}$.
\(1) Any finitely generated graded left $R_{\mathbb{Z}}$-module $M$ is a graded image of $\bigoplus_{a_i} \big(\bigoplus_{j\geq a_i} R_{ja_i}\big)
\otimes_{R_{a_i}} R_{a_i},$ for some $a_i\in {\mathbb{N}}$ and so we may assume that $M= \bigoplus_{j\geq a} R_{ja},$ for some $a\geq 0$. Let $L\subseteq M$ be a graded submodule and write $R^*_{ij}$ for the dual of the progenerator $R_{ij}$. Then $$X(j)=R^*_{ja}\otimes_{R_j}L_j \subseteq
R^*_{ja}\otimes_{R_j}M_j =
R^*_{ja}\otimes R_{ja}
\xrightarrow{\sim} R_a,\qquad \text{for}\ j\geq a.$$ As $R_a$ is Morita equivalent to $R_0$, it is noetherian and so $\sum_{j\geq
a}X(j)=\sum_{i=a}^b X(i)$, for some $b\geq a$. Now, $$L_k = R_{ka}X(k) \subseteq \sum_{i=a}^b R_{ka}X(i) =
\sum_{i=a}^b R_{ki}R_{ia}X(i) =\sum_{i=a}^b R_{ki}L_i
\qquad\text{for}\ k\geq a.$$ Thus $L$ is generated by $L_j$ for $b\geq j\geq a$. Finally, as each $L_i$ is a submodule of the noetherian left $R_i$-module $R_{ia}$, it is finitely generated and hence so is $L$.
\(2) Certainly $\phi(M)\in R_{\mathbb{Z}}{{\textsf}{\text{-}Grmod}}$ and, as $\phi(M)$ is finitely generated by the generators of ${}_{R_0}M$, one has $\phi(M)\in R_{\mathbb{Z}}{{\textsf}{\text{-}grmod}}$. Thus $\Phi (M)=\pi\phi(M)\in R_{{\mathbb{Z}}}{\text{-}{\textsf}{qgr}}$. Since $\Phi$ sends $R_0$-module homomorph
| 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 $\left\{g_{i j}, u_{i j}, N, K\right\}$ and the solution is $\left\{ \psi, \bar{\beta}^i \right\}$. Mathematical analysis of the corresponding elliptic system (\[Eq:NewMomCon\], \[Eq:NewHamCon\]) has been carried out elsewhere, for example, [@CBYHeld; @OMY; @Isenberg; @CBBrill; @CBIJWY], and will not be repeated here. The corresponding situation in the $(\Sigma,\bar{\mbox{\bf g}}, \bar{\mbox{\bf K}})$ analysis is that the free data are $\left\{g_{i j}, A_{i j}, K\right\}$ and the solution is $\left\{ \varphi, W^i \right\}$, where $W^i$ is obtained from a tensor splitting of $A_{i j}$ [@York73; @York74]. Note that $\varphi \neq \psi$ and $W^i \neq \bar{\beta}^i$. Only part of $A_{i j}$, found in the splitting, is free. The conformal covariance of the new method, [*i.e.*]{}, starting with different representatives of a given conformal equivalence class is [*unique*]{} and clear. On the other hand, that of the $(\Sigma,\bar{\mbox{\bf g}}, \bar{\mbox{\bf K}})$ analysis can follow two inequivalent routes because there are two slightly different conformal analyses possible for construction of $\left( \Sigma, \bar{\mbox{\bf g}}, \bar{\mbox{\bf K}} \right)$. This non-uniqueness arises because conformal scaling and tensor splittings are not commutative in a straightforward way. The method of tensor splitting in [@York79] gives the Hamiltonian constraint in the form of (\[Eq:NewHamCon\]).
These data are not in perfect analogy to those conjectured by BSW, because $K$ and $N$ can be thought of as belonging to the thin sandwich as a whole. The role of $K$ has been described. The role of $N = g^{1/2} \alpha$ is to give the thickness of the sandwich, $\bar{N} \delta t$, in proper time measured orthogonally from $t=t^\prime$ to $t=t
| 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 29.6 2.4 9.4
\#2-17(HT)\#2-1 \- 3.7 6.5 20.0 3.1 10.4
\#2-1(LT)\#2-4 \- 5.3 4.3 29.4 3.1 9.1
\#2-6(LT)\#2-2 \- 3.1 4.9 21.1 2.7 13.2
T~3~ WT 6 11.1 ± 4.1 1.5 ± 0.5 3.3 ± 2.1 1.9 ± 0.5 4.2 ± 1.6
\#1-27(HT)L\#2 10 2.5 ± 0.6 ^\#\#\#^ 4.3 ± 0.4 ^\#\#\#^ 7.7 ± 1.9 ^\#\#^ 1.2 ± 0.1 ^\#\#\#^ 5.1 ± 0.7
\#2-17(HT)\#2-1 12 1.8 ± 0.5 ^\#\#\#^ 2.9 ± 0.5 ^\#\#\#^ 8.1 ± 2.3 ^\#\#\#^ 1.1 ± 0.2 ^\#\#\#^ 4.1 ± 0.8
Mean value of the alkaloid content (% dry weight) with standard deviation (mean ± SD) for each line and the alkaloid content of selected lines are summarized. nd \*: Not detected; ^\#^*p* \< 0.05; ^\#\#^*p* \< 0.005; and ^\#\#\#^*p* \< 0.001 *vs.* WT.
The thebaine content in T~1~ plants varied widely, from 0.3% to 26.5%. From these plants, two high thebaine lines, \#1-27(HT) (thebaine content: 23.1%) and \#2-17(HT) (24.4%), and two low thebaine lines, \#2-1(LT) (0.3%) and \#2-6(LT) (1.0%), were selected and subjected to analysis of the T~2~ progeny. Interestingly, most of the progeny plants from both the HT and LT lines showed the high thebaine phenotype. From the T~2~ lines, two lines, \#1-27(HT)L\#2 (thebaine content: 29.6%) and \
| 1,966
| 3,572
| 2,464
| 2,026
| null | null |
github_plus_top10pct_by_avg
|
\[-3.40, 1.84\] 0.56 2.04 \[-0.02, 4.10\] 0.05 -0.04 \[-2.42, 2.34\] 0.97 -1.98 \[-4.01, 0.04\] 0.06
Family member (Sibling vs. Patient) -2.72 \[-5.29, -0.15\] 0.04^\*^ 0.60 \[-1.40, 2.60\] 0.56 1.56 \[-0.70, 3.82\] 0.18 -5.37 \[-7.48, -3.26\] \<0.001^\*\*^
Diagnosis (AML vs. ALL) 0.31 \[-2.93, 3.56\] 0.85 0.05 \[-1.87, 1.98\] 0.96 -0.38 \[-2.59, 1.83\] 0.74 1.37 \[-0.58, 3.32\] 0.17
Diagnosis (CML vs. ALL) 1.37 \[-4.57, 7.31\] 0.65 2.81 \[-0.31, 5.93\] 0.09 -0.43 \[-3.94, 3.08\] 0.81 0.14 \[-3.15, 3.43\] 0.93
Diagnosis (Non-Hodgkin vs. ALL) 1.39 \[-1.04, 3.82\]
| 1,967
| 6,369
| 883
| 778
| null | null |
github_plus_top10pct_by_avg
|
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: MIL-STD-882E Severity Categories[]{data-label="Ta:SEVERITY_CATEGORY"}
Description Level Specific Individual Item Fleet or Inventory
------------- ------- ------------------------------------------------------------------------------------------ ----------------------------------------------------
Frequent A Likely to occur often in the life of an item. Continuously experienced.
Probable B Likely to occur often in the life of an item. Will occur frequently.
Occasional C Likely to occur sometime in the life of an item. Will occur several times.
Remote D Unlikely, but possible to occur in the life of an item. Unlikely, but can reasonably be expected to occur.
Improbable E So unlikely, it can be assumed occurrence may not be experienced in the life of an item. Unlikely to occur, but possible.
Eliminated F
: MIL-STD-882E Probability Levels[]{data-label="Ta:PROBABILITY_LEVELS"}
Table 2 above is a qualitative description of levels. Table 3 below, appearing in MIL-STD-882E Appendix A, outlines certain pitfalls in accomplishing the same task quantitatively. Numerical expression of the intensity or rate of occurrence is generally preferable to mere qualitative phrasing. For quantitative description, the intensity is the ratio of mishaps (numerator) to some measure of exposure (denominator).
Description Level Individual Item
| 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[F\Psi, F\Xi D_{\tilde{p}_1}F\Psi]
+\{F\Psi, F\Xi(D_{\tilde{p}_1}F\Xi D_{\tilde{p}_2}
- D_{\tilde{p}_2}F\Xi D_{\tilde{p}_1})F\Psi\}
\nonumber\\
&\
-[F\Xi D_{\tilde{p}_2}F\Psi, F\Xi D_{\tilde{p}_2}F\Psi]\Big)\,,\\
\lambda_{\tilde{p}_1\tilde{p}_2}\ =&\
-X\eta F\Xi(D_{\tilde{p}_1}F\Xi D_{\tilde{p}_2}
- D_{\tilde{p}_2}F\Xi D_{\tilde{p}_1})F\Psi\,,\end{aligned}$$ and $\Omega_{\tilde{p}_1\tilde{p}_2}$ in (\[Omega IJ\]). The unphysical transformation $\delta_{[\tilde{p},\tilde{p}]}$ is defined by
\[tf pp\] $$\begin{aligned}
A_{\delta_{[\tilde{p},\tilde{p}]}}\ =&\
- f\xi_0\Bigg(Qf\xi_0\big(
QA_{[\tilde{p},\tilde{p}]}+[F\Psi, \,F\Xi D_{[\tilde{p},\tilde{p}]}]F\Psi]\big)
\nonumber\\
&\
+ [F\Psi,\,F\Xi\Big(
QF\Xi D_{[\tilde{p},\tilde{p}]}F\Psi
+[F\Psi,f\xi_0\big(
QA_{[\tilde{p},\tilde{p}]}+[F\Psi, F\Xi D_{[\tilde{p},\tilde{p}]}F\Psi]\big)]
\Big)]\Bigg)\,,
\label{tf pp ns}\\
\delta_{[\tilde{p},\tilde{p}]}\Psi\ =&\
-X\eta F\Xi\Big(
QF\Xi D_{[\tilde{p},\tilde{p}]}F\Psi
+[F\Psi,\,f\xi_0\big(
QA_{[\tilde{p},\tilde{p}]}+[F\Psi, \,F\Xi D_{[\tilde{p},\tilde{p}]}F\Psi]\big)]
\Big)\,.
\label{tf pp ramond}\end{aligned}$$
The first-quantized charge $[\tilde{p},\tilde{p}]$ is defined by $$[\tilde{p},\tilde{p}]\ =\ w_{\mu\nu}[\tilde{p}^\mu,\tilde{p}^\nu]\,,$$ with the parameter $w_{\mu\nu}\,(=-w_{\nu\mu})$, and $[\tilde{p},\tilde{p}]_{12}
=[\tilde{p},\tilde{p}](w_{12}=(v_1v_2-v_2v_1)/2)$ in (\[alg pp\]). At the linearized level, the transformation (\[tf pp\]) becomes $$\begin{aligned}
\delta_{[\tilde{p},\tilde{p}]}\ \Phi\ =&\
-\xi_0Q\xi_0Q[\tilde{p},\tilde{p}]\Phi\
=\ -\xi_0QX_0[\tilde{p},\tilde{p}]\Phi\,,\\
\delta_{[\tilde{p},\tilde{p}]}\ \Psi\ =&\
-X\eta\Xi Q\Xi[\tilde{p},\tilde{p}]\Psi\
=\ -X\eta\Xi X[\tilde{p},\ti
| 1,969
| 658
| 1,553
| 2,081
| null | null |
github_plus_top10pct_by_avg
|
delta \otimes \delta^{-1} p \delta em.$$ 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 choice of $t$ was independent of $h$, this implies that $p^r (he_- \delta \otimes em) =
\operatorname{ad}(p)^r(h)(e_- \delta \otimes em)$, for any $r>0$. Now, $p$ commutes with both ${\mathbb{C}}[{{W}}]$ and ${\mathbb{C}}[{\mathfrak{h}}^*]$, and so the defining relations of $H_{c+1}$ from ensure that the adjoint action of $p\in {\mathbb{C}}[{\mathfrak{h}}^*]^{{{W}}}$ on $H_{c+1}$ is locally nilpotent (see also [@BEGqi Lemma 3.3(v)]). Therefore a sufficiently large power of $p$ annihilates $he_-\delta \otimes em$. Thus $\widetilde{S}_c(M)\in \mathcal{O}_{c+1}$ and $\widetilde{S}_c$ does restrict to the desired equivalence.
It remains to compute $\widetilde{S}_c(\Delta_{c}(\lambda))$ and we begin with the analogous problem on ${H_{c+1}^{\text{reg}}}$. In the notation of , $${\widetilde{S}_c(\Delta_{c}(\lambda))^{\text{reg}}} =
{H_{c+1}^{\text{reg}}}e_-\delta \otimes_{\delta^{-1}{U}^-_{c+1}\delta }
e\Delta_c(\lambda).$$ By , ${H_{c+1}^{\text{reg}}} \cong A=D({\mathfrak{h}^{\text{reg}}})\ast W$ and so ${\widetilde{S}_c(\Delta_{c}(\lambda))^{\text{reg}}}
\cong Ae_-\delta \otimes_{B} {e\Delta_{c} (\lambda)^{\text{reg}}},$ where $B=\delta^{-1} e_-Ae_-\delta $. On the other hand, induces an isomorphism $$\theta: Ae_-\delta
\otimes_{B} e{\Delta_c(\lambda)^{\text{reg}}}
\longrightarrow Ae \otimes_{eAe} e{\Delta_c(\lambda)^{\text{reg}}};
\qquad ae_-\delta \otimes em\mapsto a\delta e\otimes em.$$ Combined with the identity $H_ceH_c=H_c$ from Corollary \[morrat-cor\](1), this implies that $$\label{nonzeromap1}
{\widetilde{S}_c(\Delta_{c}(\lambda))^{\text{reg}}} \cong
A e \otimes_{eA e} {e\Delta_{c}(\lambda)^{\text{reg}}}
\cong {\left(H_{c}e \otimes_{{U}_c} e\Delta_{c}(\lambda)\right)^{\text{reg}}}
\cong
| 1,970
| 2,246
| 2,151
| 1,751
| 3,846
| 0.769719
|
github_plus_top10pct_by_avg
|
bound $\sigma^2_k$ for the maximum variance of the functions in ${\cal F}_k$: $$\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\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\int_{\mathbb{R}}K^{2}\left(uf^{1/2}(t-hu)\right)f^{2}(t-hu)du \nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\le\int_{\mathbb{R}}(||K||_{\infty}^{2}||f||_{\infty}^{2})\wedge (||K||_{\infty}^{2}
(T/|u|)^4)du \notag\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=2||K||_{\infty}^{2}\left[||f||_{\infty}^{2}\int_{0}^{T/||f||_{\infty}^{1/2}}du
+\int_{T/||f||_{\infty}^{1/2}}^{\infty}
\left(\frac{T}{u}\right)^4du\right]\nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{8}{3}T||K||_{\infty}^{2}||f||_{\infty}^{3/2}.\end{aligned}$$ So, we can take $\sigma_k^2:=\frac{8}{3}T||K||_{\infty}^{2}||f||_{\infty}^{3/2}2h_{2^{k}}$ (using the fourth condition in (\[band\])). $U_k=W$ is eventually much larger than $\sigma_k$ and $$\sqrt{2^k}\sigma_k\sqrt{\log \frac{U_k}{\sigma_k}}<<2^k\sigma_k^2$$ by the second condition in (\[band\]) (here and elsewhere, the sign $<<$ should be read as ‘of smaller order than’ when the indexing variable, in this case $k$, tends to infinity). If $\lambda$ in (\[eq1\]) is taken to be large enough so that $$\label{tcond}
C_1\sqrt{2^k}\sigma_k\sqrt{\log \frac{RU_k}{\sigma_k}}<\lambda\sqrt{2^{k-1} h_{2^k}\log h_{2^k}^{-1}}<<2^k\sigma_k^2,$$ where $C_1$ is one of the constants in Talagrand’s inequality (\[tal\]), then, this inequality applied to the inequalities (\[eq1\]), gives $$\label{tineq}
\Pr\left\{\max_{2^{k-1}<n\le 2^{k}}\sqrt{\frac{n h_{n}}{\log h_{n}^{-1}}}
||\bar{f}_n-E\bar{f}_n||_{\infty}>\lambda\right\}\le C_2\exp\left(-\frac{C_3\lambda^2 2^{k-1} h_{2^k}\log h_{2^k}^{-1}}{2^k\frac{16}{3}T||K||_{\infty}^{2}||f||_{\infty}^{3/2}h_{2^k}}\right).$$ Set $\lambda=L\sqrt{T}\|K\|_\infty C^{3/4}$. Then we can choose $L$ large enough such that inequality (\[tcond\]) is satisfied for all $k
| 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 $\sigma\otimes \rho\otimes {\mathcal{L}}^1$, i.e. $\int_{\Gamma} \chi_{\Gamma_0}(y,\omega,E)d\sigma(y)d\rho(\omega)dE=0$. Note that it follows from this assumption that $N_0$ has measure zero as well.
Especially, the main existence and uniqueness results for the solutions of the CSDA Boltzmann transport problem, as given in Sections \[single-eq\] and \[cosyst\] (Theorems \[csdath3\], \[coth2-d\], \[evoth1\], \[cosystth2\], \[m-d-j-co1\], \[coupthev\], along with their corollaries), as well as the non-negativity result of Subsection \[possol\], remain essentially true even with this more general choice of measure on $S$.
Below the maps $t:G\times S\to [0,\infty]$ and $\tau_{\pm}:\partial G\times S\to [0,\infty]$, will often be interpret as maps $t:G\times S\times I\to [0,\infty]$ and $\tau_{\pm}:\Gamma\to [0,\infty]$, by dropping off the energy variable, e.g. $t(x,\omega,E)=t(x,\omega)$.
\[le:esccont\] Let $(y_0,\omega_0)\in \partial G\times S$ be such that there exists a bounded open cone $C:=\{x\in{\mathbb{R}}^3\ |\ \Big|{x\over{{\left\Vert x\right\Vert}}}-{a\over{{\left\Vert a\right\Vert}}}\Big|<r, 0<{\left\Vert x\right\Vert}<r\}\subset{\mathbb{R}}^3,\ a\not=0$ containing $-\omega_0$ such that for some $\lambda_0>0$ one has $y_0+\lambda_0 C\subset {\mathbb{R}}^3\backslash G$. Then we have the following:
\(i) The map $t(\cdot,\cdot)$ is continuous at any point $(x_0,\omega_0)\in G\times S$ such that $y_0=x_0-t(x_0,\omega_0)\omega_0$.
\(ii) Letting $y_+ = y_0 + \tau_-(y_0,\omega_0)\omega_0$, one has the following limits $$& \lim_{(x,\omega)\to (y_0,\omega_0)} t(x,\omega)=0, \\
& \lim_{(x,\omega)\to (y_+,\omega_0)} t(x,\omega)=\tau_-(y_0,\omega_0),$$ where $(x,\omega)\in G\times S$ when taking the limits.
By using c
| 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 ± 7.3 50.7 ± 8.9^\*^ 46.8 ± 9.5^\*^
Male (n, %) 57 ( 72 % ) 15 ( 63 % ) 23 ( 54 % ) 7 ( 54 % )
DM( n, %) 21 ( 27 % ) 5 ( 21 % ) 9 ( 21 % ) 2 ( 15 % )
Systolic BP (mmHg) 132 ± 15 131 ± 18 127 ± 13 125 ± 16
Diastolic BP (mmHg) 66 ± 9 67 ± 9 71 ± 7^†^ 67.7 ± 8.0
Protein intake / IBW(g/kg) 1.01 ± 0.17 0.71 ± 0.06^‡^ 0.95 ± 0.13 0.71 ± 0.05^‡^
Energy intake / IBW(kcal/kg) 29.0 ± 4.2 23.3 ± 2.5^‡^ 27.9 ± 2.9 22.9 ± 2.4^‡^
BMI (kg/m^2^) 24.1 ± 3.1 25.4 ± 2.9 23.9 ± 3.9 23.4 ± 4.4
waist circumference (cm) 85.9 ± 9.0 84.7 ± 11.6 83.3 ± 12.1 83.6 ± 15.7
eGFR (mL/min/1.73m^2^) 25.7 ± 11.9 23.9 ± 11.8 24.5 ±10.2 19.5 ± 11.1
Serum albumin (g/dL) 4.2 ± 0.3 4.2 ± 0.3 4.3 ± 0.3 4.5 ± 0.3^†^
Hemoglobin level ( g/dL) 11.6 ± 1.9 11.4 ± 1.7 11.6 ± 2.1 10.4 ± 1.7
HbA1c ( % ) (Diabetes) 6.7 ± 0.6 6.7 ± 0.9 7.1 ± 1.5 6.0 ± 0.4
Total cholesterol (mg/dL) 168\. ± 25.6 170.4 ± 30.9 175.3 ± 33.2 147.9 ± 26.5^§†^
HDL (mg/dL) 57.3 ± 16.5 53.6 ± 14.8 54.0 ± 15.3 52.0 ± 11.6
LDL (mg/dL) 88.7 ± 21.5 92.6 ± 32.7 93.8 ± 29.4 78.1 ± 24.9
TG (mg/dL) 128.8 ± 82.3 96.6 ± 29.7 154.7 ± 118.4 116.0 ± 42.9
**Body composition**
Fat %
| 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 relativistic quantum field theory for bosonic degrees of freedom, discussed at greater length in Ref. , in order to explain why such systems are the natural framework for describing relativistic quantum point-particles. The same considerations are then developed in Sec. \[Sec3\] in the case of fermionic degrees of freedom associated to particles of half-integer spin, based on a discussion of the theory of finite dimensional representations of the Lorentz group, leading in particular to the free Dirac equation for the description of spin 1/2 particles without interactions. Section \[Sec4\] then considers, as a simple introductory illustration of some facts essential and generic to supersymmetric field theories, and much in the same spirit as that of the discussion in Sec. \[Sec2\], the $\mathcal N=1$ supersymmetric harmonic oscillator which already displays quite a number of interesting properties. Section \[Sec5\] then concludes with a series of final remarks related to the actual construction of supersymmetric field theories based on the general concepts of the Lie symmetry algebraic structures inherent to such relativistic invariant quantum field theories and their manifest realisations through specific choices of field content, indeed the underlying theme to both these lectures and the previous ones.[@GovCOPRO2]
Basics of Quantum Field Theory: A Compendium for Scalar Fields {#Sec2}
==============================================================
Within a relativistic classical framework,[^2] material reality consists, on the one hand, of dynamical fields, and on the other hand, of point-particles. Fields act on particles through forces that they develop, such as the Lorentz force of the electrom
| 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 choose $i\equiv i_s$ to be a surface site, take the thermodynamic limit (first) and then let $H\to 0^+$. The bounds converge towards the respective values of $m_1(K,r,0^+)$, the spontaneous magnetization of the surface layers per site, for $r=r^<$ and $r^>$. Thus we obtain $$\label{Grifineq}
m_1(K,r^<,0^+)\le m(i_s;K,\bbox{r}^{(\text{s})},0^+)\le
m_1(K,r^>,0^+)\;.$$
The following limiting forms of $m_1$ are well established [@PS98; @Die86a; @LB90a; @rigres; @BD94]: $$\label{limform}
m_1=\cases{C_1|\tau|^{\beta_1^{\text{ord}}}[1+o(\tau)]&
as $\tau\to 0^-$ at fixed $r<r_c$,\cr
C_1'|\tau|^{\beta_1^{\text{sp}}}[1+o(\tau)]
&as $\tau\to 0^-$ at fixed $r=r_c$,\cr
m_{1c}+O(\tau)
&as $\tau\to 0^\pm$ at fixed $r>r_c$,}$$ where $r_c\simeq 1.50 $ [@LB90a] is the critical value associated with the special transition. The quantities $m_{1c}>0$, $C_1$, and $C'_1$ are nonuniversal, whence the first two depend on $r$.
Consider first the case $r^><r_c$. Let $C^<$ and $C^>$ be the values of $C_1$ for $r=r^<$ and $r=r^>$, respectively. (These satisfy $0<C^<\le C^><\infty$ provided $0<J<\infty$ and $0<J_1^<\le J_1^><\infty$.) It follows that there exists a number $\epsilon >0$ independent of the disorder configuration $\bbox{r}^{\text{(s)}}$ such that $$\label{ineq}
C^>\le m[i_s;K(\tau),\bbox{r}^{\text{(s)}},0^+]\,
\,|\tau|^{-\beta_1^{\text{ord}}}\le C^>$$ whenever $-\epsilon <\tau <0$. We denote the average of a quantity $Q$ over all choices of the random variables $\bbox{r}^{\text{(s)}}$ as $\overline{Q}$. Upon averaging $m(i_s;.)$ to obtain the disorder-averaged surface magnetization $\overline{m_1}$, we see that the inequality (\[ineq\]) holds for $\overline{m_1}\,|\tau|^{-\beta_1^{\text{ord}}}$ as well. An eleme
| 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 $$\left\{
\begin{array}{l l}
\pi^j\cdot\begin{pmatrix}a_j&0\\0&(1+\sigma(2 z_j^{\ast}))\cdot (1+2\bar{\gamma}_j)\cdot (1+2 z_j^{\ast}) \end{pmatrix}
& \quad \textit{$L_j$ : of type $I^o$ with j even};\\
\pi^j\cdot\begin{pmatrix}a_j&0&0\\0&(1+\sigma(\pi x_j))(1+\pi x_j)&(1+\sigma(\pi x_j))(1+2 z_j^{\ast})\\
0&(1+\sigma(2 z_j^{\ast}))(1+\pi x_j)&(1+2 z_j^{\ast})\sigma(2 z_j^{\ast})+2 z_j^{\ast}+2\bar{\gamma}_j \end{pmatrix} & \quad
\textit{$L_j$ : of type $I^e$ with j even};\\
\pi^j\cdot\begin{pmatrix}a_j&0&0\\0&\pi^3\bar{\gamma}_j+\pi^3\left((z_j)_1+(z_j)_1^2\right)&1+\sigma(\pi x_j)+\pi\cdot\sigma(\pi z_j) \\
0&-1-\pi x_j+\pi^2 z_j&\pi \end{pmatrix} & \quad
\textit{$L_j$ : free of type $I$ with j odd}.
\end{array}\right.$$ We write $z_j^{\ast}=(z_j^{\ast})_1+\pi (z_j^{\ast})_2$, $x_j=(x_j)_1+\pi (x_j)_2$, and $z_j=(z_j)_1+\pi (z_j)_2$, where $(z_j^{\ast})_1, (z_j^{\ast})_2,$ $(x_j)_1,$ $(x_j)_2,$ $(z_j)_1,$ $(z_j)_2 \in R \subset R\otimes_AB$ and $\pi$ stands for $1\otimes \pi\in R\otimes_AB$. When $L_j$ is *of type $I^o$*, by considering the $(2, 2)$-block of the matrix above, we obtain the equation $$(z_j^{\ast})_1+(z_j^{\ast})_1^2=0.$$ Therefore, in this case, $F_j$ is isomorphic to $ \mathbb{A}^{1} \times \mathbb{Z}/2\mathbb{Z}$ as a $\kappa$-variety.
When $L_j$ is *of type $I^e$*, by considering the $(2, 2)$-block of the matrix above, we obtain the equation $$(x_j)_1^2=0.$$ We also consider the $(2, 3)$-block of the matrix above, and we obtain two equations $$(x_j)_1=0, ~~~ (x_j)_2+(z_j^{\ast})_1=0.$$ By considering the $(3, 3)$-block of the matrix above, we obtain the equation $$(z_j^{\ast})_1+(z_j^{\ast})_1^2=0.$$ By combining all these, we see that $F_j$ is isomorphic to $ \mathbb{A}^{1} \times \mathbb{Z}/2\mathbb{Z}$ as a $\kappa$-variety.
By using a similar argument used above, when $L_j$ is *free of type $I$* with $i$ odd, we
| 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-induced drag force {#sec:drag}
-----------------------
In classical fluids, consideration of the self-induced force on a particle moving at arbitrary time-dependent velocity, coming from the perturbation in produces in the flow, leads to different terms, namely [@maxey1983equation; @parmar2012equation] the viscous (Stokes) drag, the unsteady-inviscid term that in the incompressible case becomes the added-mass force, and the unsteady-viscous term that in the incompressible case becomes the Basset history force. They are expressed in terms of the undisturbed velocity flow ${\boldsymbol{v}}^{(0)}$ and the particle velocity ${\boldsymbol{V}}_p(t)$. Here, for the BEC case, we are able to obtain the self-induced force only for a particle moving at constant speed on the condensate. For the classical fluid case, in this situation the only non-vanishing force is the Stokes drag, so that this is the force we have to compare our quantum result with. We note that the condensate itself in the absence of the particle perturbation can be in any state of (weak) motion since in our perturbative approach summarized in Eqs (\[eq:psi0lin\])-(\[eq:psi1lin\]), the inhomogeneity $\delta\psi_0$ and the $g_p$-perturbation $\delta\psi_1$ are uncoupled.
It is convenient to transform the problem to the frame of reference moving with the particle $({\boldsymbol{r}},t)\rightarrow ({\boldsymbol{z}},t)$ with ${\boldsymbol{z}}={\boldsymbol{r}} - {\boldsymbol{r}}_p(t)$, so that Eq. (\[eq:psi1lin\]) becomes $$\begin{aligned}
\partial_t \delta\psi_1 - {\boldsymbol{V}}_p \cdot \nabla \delta\psi_1 &=&
(i+\gamma)\left(\frac{1}{2}\nabla^2-1\right)\delta\psi_1 \nonumber \\
&-&(i+\gamma)\delta\psi_1^* - (i+\gamma) \mathcal U({\boldsymbol{
| 1,977
| 570
| 1,674
| 2,076
| null | null |
github_plus_top10pct_by_avg
|
su(2)$,
1 hypermultiplet in the $({\bf 1},{\bf 56},{\bf 2})$ of $e_7 \times e_7 \times su(2)$,
4 singlet hypermultiplets.
It is straightforward to compute that there are 272 vector multiplets and 228 hypermultiplets. Since the difference is not 244, this six-dimensional theory is anomalous.
Third cautionary example {#sect:class3-caution4}
------------------------
Now consider an $E_8 \times E_8$ string on a ${\mathbb Z}_3$ gerbe over a different $[T^4/{\mathbb Z}_2]$, constructed as $[T^4/{\mathbb Z}_6]$. Let the generator $g$ of ${\mathbb Z}_6$ act on the $T^4$ with coordinates $(X^3, X^4)$ as $$g: \: \left( X^3, X^4 \right) \: \mapsto \:
\left( \exp(+4 \pi i/3), \exp(-4 \pi i/3) \right).$$ Define a rank 2 bundle over this stack by taking ${\cal O}^{\oplus 2}$ over $T^4$, and let $g$ act with eigenvalues $$\left( \exp(-2 \pi i/3), \exp(-4 \pi i/3) \right).$$ It is straightforward to check that this satisfies anomaly cancellation in the sense of [@freedvafa], and also the constraints in appendix \[app:spectra:fockconstraints\].
In an $E_8 \times E_8$ compactification, we can describe this as the following action on fields: $$\begin{aligned}
g \cdot X^{1-2} & = & + X^{1-2}, \\
g \cdot X^3 & = & \exp(+4 \pi i/3) X^3, \\
g \cdot X^4 & = & \exp(-4 \pi i/3) X^4, \\
g \cdot \psi^{1-2} & = & + \psi^{1-2}, \\
g \cdot \psi^{3} & = & \exp(+ 4 \pi i/3) \psi^3, \\
g \cdot \psi^4 & = & \exp(-4 \pi i/3) \psi^4, \\
g \cdot \lambda^{1-6} & = & + \lambda^{1-6}, \\
g \cdot \lambda^7 & = & \exp(-2 \pi i/3) \lambda^7, \\
g \cdot \lambda^8 & = & \exp(-4 \pi i/3) \lambda^8.\end{aligned}$$
Let us now outline the massless spectrum.
In the untwisted sector, there are massless states in the (NS,NS) sector. It is straightforward to compute that $E_{\rm left} = -1$, $E_{\rm right} = -1/2$, and one has ${\mathbb Z}_6$-invariant states of the form
--------------------------------------------------------------------------------------------------------------------------------
State
| 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 70.1 ± 1.5 60.7 ± 2.5 106.6 ± 1.8 104.5 ± 0.8
Hyb 11 457.2 ± 15.1 436.9 ± 5.8 265.4 ± 11.7 280.2 ± 6.0 322.0 ± 3.8 273.8 ± 10.2 263.7 ± 11.3 303.4 ± 23.0 70.5 ± 2.5 62.6 ± 2.3 99.8 ± 2.3 108.6 ± 7.9
Hyb 12 466.4 ± 10.4 431.7 ± 6.6 267.1 ± 5.7 284.6 ± 2.1 350.4 ± 7.2 310.7 ± 5.9 277.9 ± 8.5 321.2 ± 15.1 75.1 ± 3.4 72.0 ± 1.3 104.1 ± 1.3 112.8 ± 6.4
Hyb 13 463.7 ± 28.3 425.4 ± 22.3 279.2 ± 4.9 292.4 ± 5.1 329.2 ± 26.6 276.7 ± 5.0 274.7 ± 8.9 306.3 ± 12.9 71.1 ± 2.6 65.2 ± 2.5 98.4 ± 3.4 104.8 ± 2.8
Hyb 14 428.8 ± 5.8 422.1 ± 9.8 226.1 ± 15.4 288.5 ± 5.9 334.1 ± 7.4 285.7 ± 3.3 263.9 ± 16.5 298.6 ± 9.4 78.2 ± 1.6 67.7 ± 2.4 116.9 ± 5.2 103.6 ± 4.8
Hyb 15 450.0 ± 6.5 425.8 ± 4.4 274.5 ± 4.4 264.5 ± 4.1 349.5 ± 3.3 314.2 ± 6.8 279.0 ± 4.7 290.7 ± 11.3 77.9 ± 1.1 73.8 ± 1.1 101.6 ± 1.1 110.0 ± 3.6
Hyb 16 461.5 ± 10.7 436.7 ± 12.4 259.5 ± 8.1 275.8 ± 6.6 339.4 ± 2.5 355.8 ± 6.4 273.5 ± 6.3 318.7 ± 15.8 70.9 ± 2.3 81.5 ± 2.1 64.3 ± 1.3 116.1 ± 4.6
Hyb 17
| 1,979
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\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{array}
\label{eq.3.2.2.4}$$ Substituting these expressions into (\[eq.3.2.2.2\]), one can find the total radial wave function for the reflectionless potential with the barrier. The value of the partial component of the S-matrix $S_{l=0}^{(2)}$ can be found from a boundary condition of this wave function at point $r=0$, as it was made in the previous paragraph for the inverse power reflectionless potential (\[eq.3.1.5\]).
Conclusions \[sec.conclusions\]
===============================
In finishing we note main conclusion and new results.
- The new exactly solvable radial reflectionless potential with barrier, which in the spatial semiaxis of radial coordinate $r$ has one hole and one barrier, after which it falls down monotonously to zero with increasing of $r$, is proposed. It has shown, that at its shape such potential looks qualitatively like radial scattering potentials in two-partial description of collision between particles and nuclei or radial decay potentials in the two-partial description of decay of compound spherical nuclear systems.
- The found reflectionless potential with the barrier depends on parameters $\gamma_{n}$ and $C$. One can deform the shape of this potential: by discrete values of $\gamma_{n}$ (from the sequence (\[eq.3.1.7\])) and by continuous values of $C$. The parameter $\gamma_{n}$ at its variation does not displace visibly a maximum of the barrier and a minimum of the hole along the semiaxis $r$, but it changes their absolute values. The parameter $C$ allows to displace continuously both the barrier maximum and the hole minimum.
- A new approach for construction of a hierarchy of the radial reflectionless potentials with barriers is proposed.
- An exact
| 1,980
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|
), \dots, B_t(\omega_d)):= ( \langle {\mathbf{1}}_{[0,t)},\omega_1 \rangle, \dots \langle {\mathbf{1}}_{[0,t)},\omega_d \rangle),$$ with ${\boldsymbol \omega}=(\omega_1,\dots, \omega_d) \in S'_d({\mathbb{R}}),\quad t \geq 0,$ in the sense of an $(L^2)$-limit. Here ${\mathbf{1}}_A$ denotes the indicator function of a set $A$.
The Hida triple
---------------
Let us now consider the Hilbert space $(L^2)$ and the corresponding Gel’fand triple $$(S) \subset (L^2) \subset (S)'.$$ Here $(S)$ denotes the space of Hida test functions and $(S)'$ the space of Hida distributions. In the following we denote the dual pairing between elements of $(S)$ and $(S)'$ by $\langle \! \langle \cdot , \cdot \rangle \!\rangle$. Instead of reproducing the construction of $(S)'$ here we give its characterization in terms of the $T$-transform.\
We define the $T$-transform of $\Phi \in (S)'$ by $$T\Phi({\bf f}) := \langle\!\langle \exp(i \langle {\bf f}, \cdot \rangle),\Phi \rangle\!\rangle, \quad {\bf f}:= ({ f_1}, \dots ,{ f_d }) \in S_{d}({\mathbb{R}}).$$
- Since $\exp(i \langle {\bf f},\cdot \rangle) \in (S)$ for all $f \in S_d({\mathbb{R}})$, the $T$-transform of a Hida distribution is well-defined.
- For ${\bf f} = 0$ the above expression yields $\langle\!\langle \Phi, 1 \rangle\!\rangle$, therefore $T\Phi(0)$ is called the generalized expectation of $\Phi \in (S)'$.
In order to characterize the space $(S)'$ by the $T$-transform we need the following definition.
A mapping $F:S_{d}({\mathbb{R}}) \to {\mathbb{C}}$ is called a [*U*-functional]{} if it satisfies the following conditions:
- For all ${\bf{f, g}} \in S_{d}({\mathbb{R}})$ the mapping ${\mathbb{R}}\ni \lambda \mapsto F(\lambda {\bf f} +{\bf g} ) \in {\mathbb{C}}$ has an analytic continuation to $\lambda \in {\mathbb{C}}$ ([**[ray analyticity]{}**]{}).
- There exist constants $0<C,D<\infty$ and a $p \in {\mathbb{N}}_0$ such that $$|F(z{\bf f})|\leq C\exp(D|z|^2 \|{\bf f} \|_p^2),$$ for all $z \in {\mathbb{C}}$ and ${\bf f} \in S_{d}({\mathbb{R}})$ ([**[gr
| 1,981
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|
\boldsymbol\eta}, {\boldsymbol\eta}\rangle}} \exp\left( -\frac{1}{2\langle {\boldsymbol\eta},{\boldsymbol\eta} \rangle}(i\langle {\boldsymbol\eta},{\bf f} \rangle - x)^2 -\frac{1}{2}\langle {\bf f},{\bf f}\rangle \right), \, \, \mathbf{f} \in S_d({\mathbb{R}}).$$
Generalized Gauss Kernels
-------------------------
Here we review a special class of Hida distributions which are defined by their T-transform, see e.g. [@HS83],[@HKPS93],[@GS98a]. Proofs and more details for can be found in [@BG10]. Let $\mathcal{B}$ be the set of all continuous bilinear mappings $B:S_{d}({\mathbb{R}}) \times S_{d}({\mathbb{R}}) \to {\mathbb{C}}$. Then the functions $$S_d({\mathbb{R}})\ni \mathbf{f} \mapsto \exp\left(-\frac{1}{2} B({\bf f},{\bf f})\right) \in {\mathbb{C}}$$ for all $B\in \mathcal{B}$ are U-functionals. Therefore, by using the characterization of Hida distributions in Theorem \[charthm\], the inverse T-transform of these functions $$\Phi_B:=T^{-1} \exp\left(-\frac{1}{2} B\right)$$ are elements of $(S)'$.
\[GGK\] The set of [**[generalized Gauss kernels]{}**]{} is defined by $$GGK:= \{ \Phi_B,\; B\in \mathcal{B} \}.$$
[[@GS98a]]{} \[Grotex\] We consider a symmetric trace class operator $\mathbf{K}$ on $L^2_{d}({\mathbb{R}})$ such that $-\frac{1}{2}<\mathbf{K}\leq 0$, then $$\begin{aligned}
\int_{S'_{d}({\mathbb{R}})} \exp\left(- \langle \omega,\mathbf{K} \omega\rangle \right) \, d\mu(\boldsymbol{\omega})
= \left( \det(\mathbf{Id +2K})\right)^{-\frac{1}{2}} < \infty.\end{aligned}$$ For the definition of $\langle \cdot,\mathbf{K} \cdot \rangle$ see the remark below. Here $\mathbf{Id}$ denotes the identity operator on the Hilbert space $L^2_{d}({\mathbb{R}})$, and $\det(\mathbf{A})$ of a symmetric trace class operator $\mathbf{A}$ on $L^2_{d}({\mathbb{R}})$ denotes the infinite product of its eigenvalues, if it exists. In the present situation we have $\det(\mathbf{Id +2K})\neq 0$. Therefore we obtain that the exponential $g= \exp(-\frac{1}{2} \langle \cdot,\mathbf{K} \cdot \rangle)$ is square-integrable and its T-tra
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|
}](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; @YoshimoriZawadowski1982] of the low energy quasiparticles. In order to understand the spectra in this regime for $\lambda_d> \lambda_{d}^{c}$, we can perform a particle-hole transformation of one spin species to convert an attractive U back to a repulsive $U$ in the transformed model. Starting from the impurity Hamiltonian in the absence of an external magnetic field $ \e_{d\sigma} = \e_{d}$ and replacing $n_\uparrow = (1 - d_\uparrow d^\dagger_\uparrow) = 1 -\bar n_\uparrow$, where $\bar n_\uparrow$ is the number operator of the holes, we derive $$\begin{aligned}
\sum_\sigma \e_{d\sigma} n^d_\sigma + U n^d_\uparrow n^d_\downarrow
=
\non
\sum_\sigma \left(\frac{U}{2}-\sigma \Delta \e \right) \bar n^d_\sigma - U \bar n^d_\uparrow \bar n^d_\downarrow
+ \e_{d} \, ,\end{aligned}$$ where $\Delta \e = \e_d + U/2$ serves as a measure of the particle-hole asymmetry [@KrishWilWilson80b], and $\bar n^d_\downarrow= n^d_\downarrow$. A negative $U$ model describes the same physics as the positive $U$ model after the particle-hole transformation but with $\Delta \e$ acting as effective magnetic field. The increasing $(-U_{\rm eff})$ leads to a decreasing charge Kondo temperature $T^c_K$ since the hybridization $\Gamma_0\to\Gamma_{\rm eff}$ is also reduced [@Mahan81; @LangFirsov1962; @EidelsteinSchiller2013]. Consequently the dimensionless magnetic field $\Delta\e/T_K(\lambda_d)$ increasing with further increasing of $\lambda_d$.
Therefore, we can understand the evolution of the spectra in Fig. \[NRG\_fig-ph-asymmetry\] for two values of $\Delta \e$ in terms of this analysis. The zero-bias Kondo resonance is shifted to a finite value $\Delta\e$, representing the effective magnetic field in the transformed model. F
| 1,983
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| 2,078
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|
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|
\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 {{\theta}_{2}}, \\ \nonumber
& {{m}_{12}}={{m}_{21}}={{A}_{2}}+{{A}_{3}}\cos {{\theta}_{2}}, \\ \nonumber
& {{m}_{22}}={{A}_{2}}, \\ \nonumber
& {{m}_{13}}={{m}_{14}}={{m}_{23}}={{m}_{24}}=0, \\ \nonumber
& {{m}_{33}}={{A}_{4}}+{{A}_{5}}+2{{A}_{6}}\cos {{\theta}_{4}}, \\ \nonumber
& {{m}_{34}}={{m}_{43}}={{A}_{5}}+{{A}_{6}}\cos {{\theta}_{4}},\\ \nonumber
& {{m}_{44}}={{A}_{5}}, \\ \nonumber
& {{m}_{31}}={{m}_{32}}={{m}_{41}}={{m}_{42}}=0 \nonumber
\end{array}$$ with $$\begin{array}{r@{}l@{\qquad}l}
& {{A}_{1}}={{m}_{1}}k_{1}^{2}+{{m}_{2}}l_{1}^{2}+{{I}_{1}}, \\
& {{A}_{2}}={{m}_{2}}k_{2}^{2}+{{I}_{2}}, \\
& {{A}_{3}}={{m}_{2}}{{l}_{1}}{{k}_{2}}, \\
& {{A}_{4}}={{m}_{3}}k_{3}^{2}+{{m}_{4}}l_{3}^{2}+{{I}_{3}}, \\
& {{A}_{5}}={{m}_{4}}{{k}_{4}}^{2}+{{I}_{4}}, \\
& {{A}_{6}}={{m}_{4}}{{l}_{3}}{{k}_{4}}.
\end{array}$$ $C(\theta,\dot{\theta})$ is a $4\times1$ Coriolis-centripetal vector, whose elements are computed by $$\begin{array}{r@{}l@{\qquad}l}
& {{c}_{11}}=-{{A}_{3}}\sin {{\theta}_{2}}(\dot{\theta}_{2}^{2}+{{{\dot{\theta}}}_{1}}{{{\dot{\theta}}}_{2}})+{{b}_{1}}{{{\dot{\theta}}}_{1}}, \\
& {{c}_{21}}={{A}_{3}}\dot{\theta}_{1}^{2}\sin {{\theta}_{2}}+{{b}_{2}}{{{\dot{\theta}}}_{2}}, \\
& {{c}_{31}}=-{{A}_{6}}\sin {{\theta}_{4}}(\dot{\theta}_{4}^{2}+{{{\dot{\theta}}}_{3}}{{{\dot{\theta}}}_{4}})+{{b}_{3}}{{{\dot{\theta}}}_{3}}, \\
& {{c}_{41}}={{A}_{6}}\dot{\theta}_{3}^{2}\sin {{\theta}_{4}}+{{b}_{2}}{{{\dot{\theta}}}_{4}}.
\end{array}$$ Furthermore, $J$ is a $4\times4$ Jacobian matrix with the elements obtained by $$\begin{array}{r@{}l@{\qquad}l}
& {{J}_{11}}=-{{L}_{1}}\sin {{\theta}_{1}}-{{L}_{2}}\sin ({{\theta
| 1,984
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|
\to e\gamma)\lesssim 4.2\times 10^{-13}$ [@TheMEG:2016wtm; @Renga:2018fpd]. Meanwhile, our theoretical formula is given by $$\begin{aligned}
{\rm BR}(\mu\to e\gamma)&= \frac{3\alpha_{\rm em}}{16\pi{\rm G_F^2}}
\left| G(M_D,m_{\eta_1})\left( \sum_{a=1}^3 y_{N_{R_{1a}}} y^\dag_{N_{R_{a2}}} \right)
+
G(M_D,m_{\eta_2}) \left( \sum_{a=1}^3 y_{N_{L_{a1}}} y^\dag_{N_{L_{2a}}}\right)
\right|^2,\\
G(m_a,m_b)&\approx
\frac{2 m_a^6+3 m_a^4 m_b^2-6 m_a^2 m_b^4+m_b^6 +12 m_a^4 m_b^2 \ln\left[\frac{m_b}{m_a}\right]} {12 (m_a^2-m_b^2)^4},\end{aligned}$$ where ${\rm G_F}\approx 1.17\times10^{-5}$ GeV$^{-2}$ is the Fermi constant and $\alpha_{\rm em}\approx1/129$ is the fine structure constant. When the masses of $m_{\eta_{1,2}}, M_D$ are of the order of 100 GeV, the constraints for Yukawa couplings are found as $y_{N_R}\approx y_{N_L}={\cal O}(0.01)$ [^2].
Numerical analysis {#sec:num_analysis}
==================
Here, we search for the allowed regions that satisfy the constraints on the neutrino oscillation data given in Eqs. (\[eq:nuexp\_NH\]) and (\[eq:nuexp\_IH\]) and those on the LFVs. First, we set the ranges of input parameters as follows: $$\begin{aligned}
&\tau=[-0.5+0.1i, 0.5+3i],\quad [\alpha_\nu,\beta_\nu,\gamma_\nu,a_\nu^{(')},b_\nu^{(')},c_\nu^{(')}]=[-1,1],\\
&[M_D,m_{\eta_1},m_{\eta_2}]=[1,10]\ {\rm TeV},\\
&{\rm NH}: \Delta m^2_{\rm atm}=[2.431, 2.622]\times 10^{-3}\ {\rm eV}^2,\
{\rm IH}: \Delta m^2_{\rm atm}=[2.413, 2.606]\times 10^{-3}\ {\rm eV}^2.\end{aligned}$$ Here, the real part of $\tau$ has a periodicity of $1$ and the 3$\sigma$ interval of $\Delta m^2_{\rm atm}$ is used for the range of the scan. Fig. \[fig:1\] shows the allowed region of $\tau$. The left panel is for the NH case and the right one is for the IH case. In both cases, the allowed regions have the similar shape, [*i.e.*]{} having four peaks at around ${\rm Re}[\tau]=\pm0.1,\pm0.5$. A smaller ${\rm Im}[\tau]$ is preferred in the NH case and a larger ${\rm
Im}[\tau]$ is preferred in the IH case. Here, we emphasize that the region
| 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 therefore an isomorphism.
\(2) By Lemma \[filter-injA\], $\theta $ and hence $ \theta[\delta^{-2}] $ are graded maps under the ${\mathbf{E}}$-gradation and filtered under the order filtration. Since $\operatorname{gr}( \theta[\delta^{-2}] )= \Theta[\delta^{-2}] $ is an isomorphism, necessarily $\theta [\delta^{-2}] $ is a filtered isomorphism.
Notation {#eqpoi-sect}
--------
As in , set ${\mathcal{J}}=eJ^k\delta^k$, ${\mathcal{N}}=N(k)$ and write $\theta({\mathcal{J}})^m=\operatorname{{\textsf}{ord}}^m \theta({\mathcal{J}})=\theta({\mathcal{J}})\cap {\mathcal{N}}^m$ for all $m\geq 0$. We rewrite the ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-basis of $\theta({\mathcal{J}})$ constructed in the proof of Lemma \[filter-injA\] as $\{a_{g\ell m}\}$, where each $a_{g\ell m}$ is $g$-homogeneous under the ${\mathbf{E}}$-gradation and has order exactly $\ell$. Since these were induced from the bases $\{ a_{c\ell }\}$ of $\operatorname{{\textsf}{ogr}}^\ell {\mathcal{J}}$, the set $\{a_{g\ell m} : \ell\leq t\}$ does give a basis of $\theta({\mathcal{J}})^t$.
By Lemma \[Bbar-freeA\](2), ${\mathcal{N}}$ is a free left ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-module and it is certainly graded. Thus, by Theorem \[graded-proj-thm\], it is graded-free. We may therefore pick a $
{\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-basis $\{ b_{g u}\}$ of ${\mathcal{N}}$ where, again, each $b_{g u} $ is ${\mathbf{E}}$-homogeneous of degree $g$ but of unspecified order. This basis is far from unique and one cannot expect that $\{b_{gu} : b_{gu} \in {\mathcal{N}}^m\}$ forms a basis of ${\mathcal{N}}^m$; indeed at this stage we do not even know that ${\mathcal{N}}^m$ is a free $\mathbb
C[{\mathfrak{h}}]^W$-module.
{#subsec-step42}
We are
| 1,986
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|
\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 classical circulant matrices (that is, when the $G^{(n)}$ are cyclic groups) was proved by the author in [@Meckes].
\[T:semicircle-law-general\] Let $\alpha \in [0,1]$, $\beta > 0$. Suppose that for each $n$, $\bigl\{Y_a^{(n)} \mid a \in G^{(n)} \bigr\}$ are mean $0$ and independent except for the constraint $Y_{a^{-1}}^{(n)} =
\overline{Y_a^{(n)}}$; that $${\mathbb{E}}Y_a^{(n)} Y_b^{(n)} = \begin{cases}
1 & \text{ if } a = b^{-1} \neq a^{-1}, \\
\alpha & \text{ if } a = b \neq a^{-1}, \\
\beta & \text{ if } a = b = a^{-1}, \\
0 & \text{ otherwise,} \end{cases}$$ for $a, b \in G^{(n)}$; and that holds. Assume further that $\lim_{n\to \infty} p_2^{(n)} = p$ exists. Then $\mu^{(n)}$ converges, in mean and in probability, to $$(1-p) {\mathcal{N}}\bigl(0, 1 + p (\beta - \alpha - 1)\bigr)
+ p {\mathcal{N}}\bigl(0, 1 + \alpha + p (\beta - \alpha - 1) \bigr).$$ if $p<1$ and to $
{\mathcal{N}}\bigl(0, \beta \bigr)
$ if $p=1$.
Observe that by Lagrange’s theorem on orders of subgroups, $1/p_2^{(n)}$ is an integer, which implies that if $p<1$ then in fact $p\le 1/2$, and therefore the stated variances of the normal distributions named above are indeed positive.
The most obvious (though not necessarily, as we shall see, the most natural) special case of interest in Theorem \[T:semicircle-law-general\] is when the $Y_a^{(n)}$ are real and i.i.d. (except for the symmetry constraint), so that $\alpha = \beta
= 1$. In that case the limiting spectral distribution is the mixture distribution $$\label{E:iid-limit}
(1-p) {\mathcal{N}}(0,1-p) + p {\mathcal{N}}(0,2-p).$$
Two other special cases are suggested by considering the analogy with the GOE and GUE. The $G$-circulant analogue of the GOE,
| 1,987
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|
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 system has interesting differences from earlier ones because the data and solutions are related more simply to the spacetime metric, though not in the manner that would be implied by ordinary conformal transformations of the spacetime metric. In this “conformal” TS form one can see explicitly the role of every part of the metric. The new formulation shows that the one-hypersurface and two-hypersurfaces initial-value problems are both viable once the full implications in general relativity of the “dynamical conformal structures” are understood. The two viewpoints can be thought of as roughly analogous to a Hamiltonian and to a Lagrangian view of the constraints; the former because using $\bar{K}_{i j}$ directly [@CBYHeld; @OMY; @York79] is equivalent to using the initial canonical momentum $\bar{\pi}^{i j} = \bar{g}^{1/2}
\left( \bar{K} \bar{g}^{i j} - \bar{K}^{i j} \right)$, and the latter because $\dot{\bar{g}}_{i j}$ is the initial velocity. This striking correspondence hangs on the subtle role of the lapse function through the Choquet-Bruhat relation $\bar{N} = \bar{g}^{1/2} \alpha$ and on the corresponding conformal invariance of $K$ postulated by the author [@York73] in going beyond Lichnerowicz’s choice $K=0$. The “conformal thin sandwich” aspect of the results reflects Wheeler’s approach.
I am grateful to A. Anderson, J. David Brown, N. O’Murchadha, and especially Y. Choquet-Bruhat for encouragement and to Sarah and Mark Rupright for technical assistance. I owe special thanks to Dean Jerry Whitten of the College of Physical and Mathematical Sciences of North Carolina State University for making my Leave of Absence possible, and to J. A. Isenberg for his advice on the presen
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|
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 the generalised Chern-Simons term $$\frac{1}{2} \int E_{ 9,abc,a} \times ( {\rm flux} \cdot {\rm flux} )^{abc,a}_1 \quad ,$$ where the $( {\rm flux} \cdot {\rm flux} )$ term is given by $$\begin{aligned}
( {\rm flux} \cdot {\rm flux} )^{abc,a}_d & =-2P^{a,a[b|e|}f^{c]}_{de}+f^a_{cd}P^{c,abc}+f^a_{bd}P^{b,abc}+P^{a,a}Q^{bc}_{d}+Q^{ae}_{d}P_e^{abc}\nonumber \\ &-2P^{a[b|e}_{d}Q^{a|c]}_e+\tfrac{1}{2}P^{abcef}_{d} f^a_{ef}+P^a_{d}R^{abc}+\tfrac{1}{2}P^{a,abcef}H_{def} \quad . \label{quadratic931}\end{aligned}$$ As we have discussed in the previous subsection, since the $5_3^{2,1}$-branes correspond to the components of the potentials such that the indices $abc$ have to be inside the first $9$ indices, this implies that for these components the index $d$ in eq. differs from $a,b,c$. In particular, the components $
E_{4\,x^iy^ix^jy^jx^k,x^ix^jx^k,x^k}$ couple to the terms $( {\rm flux} \cdot {\rm flux} )^{x^i x^j x^k, x^k}_{y^k}$, which lead precisely to the tadpole conditions equivalent to eq. .
In section 3 we have seen that the additional $5_3^{2,1}$-branes that can be included in the IIA/O6 theory are mapped in the IIB/O3 theory to the $3_3^4$, $6_3^{1,1}$ and $5_3^{2,2}$-branes associated to the potentials $E_{8,4}$, $E_{9,2,1}$ and $E_{10,4,2}$ respectively. The tadpole conditions for all these branes can be easily determined using our rules. We write schematically the flux contributions to the tadpole conditions for all these branes as $$\begin{aligned}
&P_1^2 \cdot H_3 \longleftrightarrow E_{8} \nonumber \\
&P_1^2 \cdot Q \longleftrightarrow E_{8,4},E_{9,2,1}\label{tadpolesforallPbranesIIBO3}\\
&P^{1,4} \cdot Q \longleftrightarrow E_{10,4,2} \nonumber \quad .\end{aligned}$$ Similarly, one
| 1,989
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| 0.778499
|
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|
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 of densities $${\cal D}_{C,z}:=\Bigg\{f: f\ {\rm is\ a\ density,}\ \|f^{(k)}\|_\infty\le C, \ 0\le k\le 4,~~~~~~~~~~~~~~~~~~~~~~$$ $$\label{De}
~~~~~~~~~~~~~~~~~~~~~~~{\rm and}\ \sup_{{t\in\mathbb R}\atop {|u|\le \delta}}\left|f^{(4)}(t+u)-f^{(4)}(t)\right|\le z(\delta), \ 0<\delta\le 1\Bigg\}$$
Here is the stronger version of Theorem \[main0\] that we prove in this article.
\[mainu\] Under the hypotheses of Theorem \[main0\] we have $$\label{main1'}
\sup_{t\in D_r(f)}\left|\hat f(t;h_{1,n}, h_{2,n})-f(t)\right|=O_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right)\ \ {\rm uniformly\ in}\ f\in{\cal D}_{C,z}$$ and $$\label{main2'}
\sup_{t\in \hat D_r^n}\left|\hat f(t;h_{1,n}, h_{2,n})-f(t)\right|=O_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right)\ \ {\rm uniformly\ in}\ f\in{\cal D}_{C,z}$$ for all $0<C<\infty$ and function $z\ge 0$ such that $z(h)\searrow 0$ as $h\searrow 0$.
It is natural that, as shown by Hall, Hu and Marron (1995), the estimator (\[realest0\]) be locally (that is, at each point $t$) asymptotically better than the classical kernel estimator that it modifies because, after all, it is obtained from the classical one by [*local or spatial adaptation*]{} of the bandwidth. This theorem shows that, up to a logarithmic factor, the improvement is not only local but holds uniformly over all $t$ for which $f(t)$ is slightly above zero, and uniformly as well over large classes of densities with four continuous derivatives. This may seem surprising and is certainly desirable. See the comments by Donoho, Johnstone, Kerkyacharian and Picard (1995) about the scarcity of theoretical results on ‘spatially adaptive’ estimators.
We do not know of any other non-negative est
| 1,990
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|
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 assumption that $\sum_{j = 1}^n \ell_j \geq 2^{12} d\log d$ and $\beta \leq 1$.
### Proof of Lemma \[lem:prob\_toplbound\]
Define $i_{\min} \equiv \arg \min_{i \in S} \theta_i$. We claim the following. For all $i \in S$ and any $1 \leq \ell \leq |S|-1$, $$\begin{aligned}
\label{eq:prob_topl_eq}
\mathbb{P}[\sigma^{-1}(i) > \ell] \;\leq\; \mathbb{P}[\sigma^{-1}(i_{\min}) > \ell] \;\; \text{and} \;\; \mathbb{P}[\sigma^{-1}(i_{\min}) = \ell] \; \geq \; \mathbb{P}[\sigma^{-1}(i_{\min}) = 1]\,.\end{aligned}$$ Therefore $\mathbb{P}[\sigma^{-1}(i) \leq \ell] \;\geq\; \mathbb{P}[\sigma^{-1}(i_{\min}) \leq \ell]$. Using $\mathbb{P}[\sigma^{-1}(i_{\min}) = 1] > e^{-2b}/\kappa$, we get the desired bound $\mathbb{P}[\sigma^{-1}(i) \leq \ell] > e^{-2b} \ell/\kappa$.
To prove the claim , let $\widehat{\sigma}_1^\ell$ denote a ranking of top-$\ell$ items of the set $S$ and $\P[\widehat{\sigma}_1^\ell]$ be the probability of observing $\widehat{\sigma}_1^\ell$. Let ${i \in (\widehat{\sigma}_1^\ell)^{-1}}$ denote that $i = (\widehat{\sigma}_1^\ell)^{-1}(j)$ for some $1 \leq j \leq \ell$. Let $$\begin{aligned}
\Omega_1 = \Big\{ \widehat{\sigma}_1^\ell : {i \notin (\widehat{\sigma}_1^\ell)^{-1}}, {i_{\min} \in (\widehat{\sigma}_1^\ell)^{-1}} \Big\} \;\; \text{and} \;\; \Omega_2 = \Big\{ \widehat{\sigma}_1^\ell : {i \in (\widehat{\sigma}_1^\ell)^{-1}}, {i_{\min} \notin (\widehat{\sigma}_1^\ell)^{-1}} \Big\}.\end{aligned}$$ We have $\mathbb{P}[\sigma^{-1}(i) > \ell] - \mathbb{P}[\sigma^{-1}(i_{\min}) > \ell] = \sum_{\widehat{\sigma}_1^\ell \in \Omega_1}\P[\widehat{\sigma}_1^\ell] - \sum_{\widehat{\sigma}_1^\ell \in \Omega_2} \P[\widehat{\sigma}_1^\ell].$ Now, take any ranking $\widehat{\sigma}_1^\ell \in \Omega_1$ and
| 1,991
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| 2,092
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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 of the form
\[eq:dLP-gen\] $$\begin{gathered}
\Psi_{m+1,n} = L_{m,n} \Psi_{m,n} \equiv \big( U_{m,n} + \lambda \Omega^{\ell_1}\big) \Psi_{m,n}, \label{eq:dLP-gen-L} \\
\Psi_{m,n+1} = M_{m,n} \Psi_{m,n} \equiv \big( V_{m,n} + \lambda \Omega^{\ell_2}\big) \Psi_{m,n}, \label{eq:dLP-gen-M}\end{gathered}$$ where $$\begin{gathered}
\label{eq:A-B-entries}
U_{m,n} = \operatorname{diag}\big(u^{(0)}_{m,n},\dots,u^{(N-1)}_{m,n}\big) \Omega^{k_1},\qquad
V_{m,n} = \operatorname{diag}\big(v^{(0)}_{m,n},\dots,v^{(N-1)}_{m,n}\big) \Omega^{k_2},\end{gathered}$$ and $$\begin{gathered}
(\Omega)_{i,j} = \delta_{j-i,1} + \delta_{i-j,N-1}.\end{gathered}$$
The matrix $\Omega$ defines a grading and the four matrices of (\[eq:dLP-gen\]) are said to be of respective levels $k_i$, $\ell_i$, with $\ell_i\neq k_i$ (for each $i$). The Lax pair is characterised by the quadruple $(k_1,\ell_1;k_2,\ell_2)$, which we refer to as [*the level structure*]{} of the system, and for consistency, we require $$\begin{gathered}
k_1 + \ell_2 \equiv k_2 + \ell_1 \quad (\bmod N).\end{gathered}$$ Since matrices $U$, $V$ and $\Omega$ are independent of $\lambda$, the compatibility condition of (\[eq:dLP-gen\]), $$\begin{gathered}
L_{m,n+1} M_{m,n} = M_{m+1,n} L_{m,n},\end{gathered}$$ splits into the system
\[eq:dLP-gen-scc\] $$\begin{gathered}
U_{m,n+1} V_{m,n} = V_{m+1,n} U_{m,n} , \label{eq:dLP-gen-scc-1}\\
U_{m,n+1} \Omega^{\ell_2} - \Omega^{\ell_2} U_{m,n} = V_{m+1,n} \Omega^{\ell_1} - \Omega^{\ell_1} V_{m,n}, \label{eq:dLP-gen-scc-2}\end{gathered}$$
which can be written explicitly as
\[eq:dLP-ex-cc\] $$\begin{gathered}
u^{(i)}_{m,n+1} v_{m,n}^{(i+k_1)} = v^{(i)}_{m+1,n} u^{(i+k_2)}_{m,n} , \label{eq:dLP-ex-cc-1}\\
u^{(i)}_{m,n+1
| 1,992
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| 1,968
| 2,273
| 0.781219
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ed (see Sec. \[sec:mass\_loss\_inner\]).
### Dependence on BH mass {#sec:Mdep}
run $M_{\mathrm{BH}}\,[M_\odot]$ $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$ $\dot{M}/\dot{M}_{\mathrm{B}}$
------------ ------------------------------ -------------------------------------------- --------------------------------
M1e2 $10^2$ $38^\circ$ $55\%$
M1e3 (Dds) $10^3$ $40^\circ$ $59\%$
M1e4 $10^4$ $43^\circ$ $61\%$
M1e5 $10^5$ $48^\circ$ $67\%$
: Summary of the $M_{\mathrm{BH}}$ dependence.[]{data-label="tab:M-model"}
\
Next, we study the dependence on the BH mass $M_{\mathrm{BH}}$ by performing a set of simulations termed “M-series”, where $M_{\mathrm{BH}}=10^2\,M_\odot$ (“M1e2 run”), $10^3\,M_\odot$ (“M1e3 run” identical to “Dds run”), $10^4\,M_\odot$ (“M1e4 run” ) and $10^5\,M_\odot$ (“M1e5 run”). The other parameters are set to $n_\infty=10^5{\,\mathrm{cm^{-3}}}$ and $\theta_{\mathrm{shadow}}=45^\circ$.
We find that the accretion proceeds roughly at the Bondi rate in a quasi-steady fashion for all the runs. Flow properties at the end of calculation are summarized in Table \[tab:M-model\]. We see that, for all the runs, the neutral region spans the opening angle $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})\simeq \theta_{\mathrm{shadow}} (=45^\circ)$ at the Bondi radius, and that the Bondi-like accretion proceeds through this solid angle with the rate $\dot{M} \simeq 0.5-0.7\,
\dot{M}_{\mathrm{B}}$.
Note, however, that the opening angle and thus the accretion rate increase gradually with the BH mass. This dependence can be understood as follows. Recall that the luminosity $L$ is approximately proportional to the Eddington value or the mass $M_{\mathrm{BH}}$ in the
| 1,993
| 2,947
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| 2,090
| 4,184
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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 to translate research into effective public health policies and practices.
According to recent data, COVID-19 is not only linked to person to person transmission, as indirect transmission may also occur. Recent scientific evidence seems to suggest that environmental factors and changes may act as extrinsic determinants in the epidemiology of COVID-19 and of other human and animal coronaviruses.
Based on the available scientific literature, we applied the EnvID framework ([@B8]) to facilitate the identification of the environment-disease relationships and connections that may impact on disease burden ([Figure 2](#F2){ref-type="fig"}). The EnvID framework encompasses 3 interlocking components including: environment, transmission and disease, and it defines three transmission groups: group I, including directly transmitted diseases; group II, including vector-borne diseases; and group III, including environmentally mediated diseases with non-human host. At first, based on the available data, we attempted to attribute COVID-19 to one of these three groups.
{#F2}
In agreement with the prevalent scientific evidence, COVID-19 might be classified as group I, as it can be transmitted person to person being mainly affected by social processes, such as over-crowding. Social distancing has in fact shown to prevent transmission from symptomatic and non-symptomatic cases, hence flattening the epidemic and delaying the peak.
However, SARS-CoV-2 is also quite resistant in the e
| 1,994
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|
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 particular, the limits $\lim_{t\to 0}{{\mathscr C}}\circ \alpha(t)$, $\lim_{t\to 0}{{\mathscr C}}\circ \beta(t)$ are equal.
If $\alpha$ and $\beta$ are equivalent germs, note that $\alpha(0)=
\beta(0)$; by Lemma \[stequivnew\] it follows that, for every curve ${{\mathscr C}}$, $\alpha$ and $\beta$ lift to germs in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ centered at the same point.
### Summary of the argument
The general plan for the rest of this section is as follows: we will show that every contributing $\alpha(t)$ centered at a rank-1 matrix is equivalent (in suitable coordinates, and possibly up to a parameter change) to one of the form $$\alpha(t)=\begin{pmatrix}
1 & 0 & 0\\
0 & t^b & 0\\
0 & 0 & t^c
\end{pmatrix}\quad\text{or}\quad
\begin{pmatrix}
1 & 0 & 0 \\
t^a & t^b & 0 \\
\underline{f(t^a)} & \underline{f'(t^a) t^b} & t^c\end{pmatrix}
\quad,$$ where $b\le c$ resp. $a<b\le c$ are positive integers, $z=f(y)$ is a formal branch for ${{\mathscr C}}$ at $(1:0:0)$, and $\underline{\cdots}$ denotes the truncation modulo $t^c$ (cf. §\[germlist\] and §\[details\]).
The main theorem will follow from further analyses of these forms, identifying which do [*not*]{} contribute components to the PNC, and leading to the restrictions explained in §\[germlist\] and §\[details\]. Specifically, the germs on the left lead to components of type II, III, and IV (§\[1PS\]); those on the right lead to components of type V. The latter germs require a subtle study, performed in §\[typeVcomps\], leading to the definition of ‘characteristics’ and to the description given in §\[details\] (cf. Proposition \[typeV\]).
Linear algebra
--------------
###
This subsection is devoted to the proof of the fol
| 1,995
| 610
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| 2,034
| 0.783305
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|
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}\\
\leq& \E{a(y_{k\delta})} + \E{\lin{\nabla a(y_{k\delta}), - \delta \nabla U(y_{k\delta})}} + 2\delta^2 \lrn{\nabla U(y_{k\delta})}_2^2 + 2\delta \E{\tr\lrp{M(y_{k\delta})^2}}\\
\leq& \E{a(y_{k\delta})} -m\delta \E{a(y_{k\delta})} + 2\delta^2 \E{\lrn{\nabla U(y_{k\delta})}_2^2} + 2\delta \E{\tr\lrp{M(y_{k\delta})^2}}\\
\leq& \E{a(y_{k\delta})} -m\delta \E{a(y_{k\delta})} + 2\delta^2L^2 \E{\lrn{y_{k\delta}}_2^2} + 2\delta \beta^2\\
\leq& \E{a(y_{k\delta})} -m\delta \E{a(y_{k\delta})} + 4\delta^2L^2 \E{a(y_{k\delta})} + 8\delta^2 L^2 R^2 + 2\delta \beta^2\\
\leq& (1-m\delta/2) \E{a(y_{k\delta})} + {m\delta} R^2 + 2\delta \beta^2
\end{aligned}$$ Where the first inequality uses the upper bound on $\lrn{\nabla^2 a(y)}_2$ above, the second inequality uses the fact that $y_{(k+1)\delta} \sim \N\lrp{y_{k\delta} - \delta \nabla U(y_{k\delta}), \delta M(y_{k\delta})^2}$, the third inequality uses claim 2. at the start of this proof, the fourth inequality uses item 2 of Assumption \[ass:xi\_properties\]. The fifth inequality uses claim 3. above, the sixth inequality uses our assumption that $\delta \leq \frac{m}{16L^2}$.
Taking expectation wrt $y_{k\delta}$, $$\begin{aligned}
& \E{a(y_{(k+1)\delta})} \leq \E{a(y_{k})}-m\delta \lrp{\E{a(y_{k\delta})} - 2R^2 + 2\beta^2/m}\\
\Rightarrow \qquad &
\E{a(y_{(k+1)\delta})} - (2R^2/2 + 2\beta^2/m) \leq (1-m\delta) \lrp{\E{a(y_{k\delta})} - (2R^2 + 2\beta^2/m}
\end{aligned}$$
Thus, if $\E{\|y_0\|_2^2} \leq 2R^2 + 2\beta^2/m$, then $\E{a(y_0)} - \lrp{2R^2 + 2\beta^2/m} \leq 0 $, then $\E{a(y_{k\delta})} - \lrp{2R^2 + 2\beta^2/m}\leq 0$ for all $k$, which implies that $$\begin{aligned}
| 1,996
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|
redes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch and I. Bloch, Nature [**429**]{}, 277 (2004); Toshiya Kinoshita, Trevor Wenger, and David S. Weiss, Science [**305**]{}, 1125 (2004).
G. Thalhammer, M. Theis, K. Winkler, R. Grimm, J. H. Denschlag, available at arXiv.org/ cond-mat/0409552.
---
abstract: |
[ We study the regularity of a Markov semigroup $(P_t)_{t>0}$, that is, when $P_t(x,dy)=p_t(x,y)dy$ for a suitable smooth function $p_t(x,y)$. This is done by transferring the regularity from an approximating Markov semigroup sequence $%
(P^n_t)_{t>0}$, $n\in{\mathbb{\N}}$, whose associated densities $p^n_t(x,y)$ are smooth and can blow up as $n\to\infty$. We use an interpolation type result and we show that if there exists a good equilibrium between the blow up and the speed of convergence, then $P_{t}(x,dy)=p_{t}(x,y)dy$ and $p_{t}$ has some regularity properties. ]{}
author:
- |
[Vlad Bally]{}[^1]\
[Lucia Caramellino]{}[^2]
title: Transfer of regularity for Markov semigroups
---
*Keywords:* Markov semigroups; regularity of probability laws; interpolation spaces.
*2010 MSC:* 60J25, 46B70.
Introduction
============
In this paper we study Markov semigroups, that is, strongly continuous and positive semigroups $P_{t}$, ${t\geq 0}$, such $P_t1=1$. We set the domain equal to the Schwartz space ${\mathcal{S}({\mathbb{R}}^d)}$ of the $C^\infty(%
{\mathbb{R}}^d)$ functions all of whose derivatives are rapidly decreasing.
The link with Markov processes gives the representation $$P_{t}f(x)=\int_{{\mathbb{R}}^d}f(y)P_t(x,dy),\quad t\geq 0,\ f\in {\mathcal{S%
}({\mathbb{R}}^d)}.$$ We study here the regularity of a Markov semigroup, which is the property $%
P_t(x,dy)=p_t(x,y)dy$, $t>0$, for a suitable smooth function $p_t(x,y)$, by transferring the regularity from an approximating Markov semigroup sequence $%
P^n_t$, $n\in{\mathbb{N}}$.
To be more precise, let $P_{t}$ be a Markov semigroup on ${\mathcal{S}({%
\mathbb{R}}^{d})}$ with infinitesimal operator
| 1,997
| 1,334
| 982
| 1,930
| 1,293
| 0.791463
|
github_plus_top10pct_by_avg
|
with $$\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 $$\begin{gathered}
\label{sd-phi}
\big(\phi^{(i)}_{m,n},\alpha,\beta\big) \mapsto \big(\widetilde\phi^{(i)}_{m,n},\widetilde\alpha,\widetilde\beta\big),\qquad\mbox{where}\quad \widetilde\alpha \alpha =1,\quad \widetilde\beta \beta =1,\quad \widetilde{\phi}^{(i)}_{m,n} \phi^{(2k-1-i)}_{m,n} = 1.\end{gathered}$$
The self-dual case admits the reduction $\widetilde{\phi}^{(i)}_{m,n}=\phi^{(i)}_{m,n}$, when $\alpha=-\beta$ ($=1$, without loss of generality), which we write as $$\begin{gathered}
\phi^{(i+k)}_{m,n} \phi^{(k-1-i)}_{m,n} = 1,\qquad i=0,\dots ,k-1.\end{gathered}$$ The condition $\prod\limits_{i=0}^{N-1} \phi^{(i)}_{m,n} = 1$ then implies $\phi^{(N-1)}_{m,n}=1$. Therefore the matrices $U_{m,n}$ and $V_{m,n}$ are built from $k$ components: $$\begin{gathered}
U_{m,n} = \operatorname{diag}\Bigg(\phi^{(0)}_{m+1,n}\phi^{(k-1)}_{m,n},\dots,\phi^{(k-1)}_{m+1,n}\phi^{(0)}_{m,n},
\frac{1}{\phi^{(k-1)}_{m+1,n}},\frac{1}{\phi^{(0)}_{m,n}\phi^{(k-2)}_{m+1,n}},\dots \\
\hphantom{U_{m,n} = \operatorname{diag}\bigg(}{}\dots ,\frac{1}{\phi^{(k-2)}_{m,n}\phi^{(0)}_{m+1,n}}, \frac{1}{\phi^{(k-1)}_{m,n}}\Bigg)\Omega^k ,\end{gathered}$$ with $V_{m,n}$ given by the same formula, but with $(m+1,n)$ replaced by $(m,n+1)$. In this case the system (\[eq:dLP-gen-sys-1-a\]) reduces to $$\begin{gathered}
\label{self-dual-equn}
\phi^{(i)}_{m+1,n+1} \phi^{(i)}_{m,n} = \frac{1}{\phi^{(k-i-2)}_{m+1,n}\phi^{(k-i-2)}_{m,n+1}} \left(\frac{\phi^{(k-i-2)}_{m+1,n}+\phi^{(k-i-2)}_{m,n+1}}{\phi^{(k-i-1)}_{m+1,n}+\phi^{(k-i-1)}_{m,n+1}}\right),
\quad\mbox{for}\quad i=0,1,\dots , k-1.\!\!\!\!\end{gathered}$$
This reduction has $\frac{N-1}{2}$ components and is represented by an $N\times N$ Lax pair, but is [*not*]{} $3D$ consistent.
### Symmetries {#symmetries .unnumbered}
Below we give the explicit forms of
| 1,998
| 639
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| 1,993
| 3,483
| 0.772035
|
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|
$\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 $$\xi _1(\beta )=\chi ({\alpha }_p,\beta )^{{b}-1}\chi (\beta ,\al
_p)^{{b}-1},
\qquad
\xi _2(\beta )=\frac{\rhomap{r_p(\chi )}({\sigma }_p^\chi (\beta ))}{
\rhomap\chi (\beta )}.$$ Then $\xi _1,\xi _2\in \ZIdual$. Thus it suffices to prove that $\xi _1({\alpha }_j)=\xi _2({\alpha }_j)$ for all $j\in I$. Let $q_{jk}=\chi ({\alpha }_j,{\alpha }_k)$ for all $j,k\in I$. Then $q_{pp}^{b}=1$ since $\qnum{{b}}{q_{pp}}=0$. Moreover, $$\begin{aligned}
\rhomap{r_p(\chi )}({\alpha }_j)=&r_p(\chi )({\alpha }_j,{\alpha }_j)\\
=&\chi ({\alpha }_j-c_{pj}^\chi {\alpha }_p,{\alpha }_j-c_{pj}^\chi {\alpha }_p)
=q_{jj}(q_{pj}q_{jp})^{-c_{pj}^\chi }q_{pp}^{c_{pj}^\chi c_{pj}^\chi }
\end{aligned}$$ for all $j\in I$.
By assumption, $\chi $ is $p$-finite, and hence for all $j\in I\setminus \{p\}$ we have $q_{pp}^{c_{pj}^\chi }=q_{pj}q_{jp}$ or $q_{pp}^{1-c_{pj}^\chi }=1$. Let $j\in I$. If $q_{pp}^{c_{pj}^\chi }=q_{pj}q_{jp}$, then $$\begin{gathered}
\rhomap{r_p(\chi )}({\alpha }_j)=q_{jj},\\
\xi _2({\alpha }_j)=q_{jj}^{-1}\rhomap{r_p(\chi )}({\sigma }_p^\chi ({\alpha }_j))=
q_{jj}^{-1}\rhomap{r_p(\chi )}({\alpha }_j-c_{pj}^\chi {\alpha }_p)
=q_{pp}^{-c_{pj}^\chi},\\
\xi _1({\alpha }_j)=q_{pj}^{{b}-1} q_{jp}^{{b}-1}=q_{pp}^{({b}-1)c_{pj}^\chi }
=q_{pp}^{-c_{pj}^\chi },
\end{gathered}$$ and hence $\xi _1({\alpha }_j)=\xi _2({\alpha }_j)$. Otherwise ${b}=1-c_{pj}^\chi $, $q_{pp}^{c_{pj}^\chi }=q_{pp}$, and then $$\begin{gathered}
\rhomap{r_p(\chi )}({\alpha }_p)=q_{pp},\quad
\rhomap{r_p(\chi )}({\alpha }_j)=q_{jj}(q_{pj}q_{jp})^{-c_{pj}^\chi }q_{pp},\\
\xi _2({\alpha }_j)=q_{jj}^{-1}\rhomap{r_p(\chi )}({\sigma }_p^\chi ({\alpha }_j))=
q_{jj}^{-1}\rhomap{r_p(\chi )}({\alpha }_j-c_{pj}^\chi {\a
| 1,999
| 2,815
| 1,641
| 1,869
| null | null |
github_plus_top10pct_by_avg
|
al associated with the surface screening charges. The DGF is pre-computed once at the beginning of any simulation. In this Appendix we outline the numerical method used to evaluate the DGF.
Cartesian Grid {#s:calc_dgf_cart}
--------------
We calculate the DGF ${\cal G}_{i-i',j-j',k-k'}$ in Cartesian coordinates by solving Equation for a point source with unit mass located in a cell at $(i',j',k')=(1,1,1)$. Once we obtain ${\cal G}_{i-1,j-1,k-1}$ for $1\le i \le N_x$, $1\le j \le N_y$, and $1\le k \le N_z$, the values for other indices can be computed from the symmetry requirement $$\label{eq:car_symm}
{\cal G}_{i-i',j-j',k-k'} = {\cal G}_{|i-i'|,|j-j'|,|k-k'|},$$ for $| i-i'| \le N_x-1$, $|j-j' | \le N_y-1$, and $ | k-k' | \le N_z-1$.
As we use the method presented in Section \[s:interior\_solver\] to obtain the numerical solution of Equation , we must supply an appropriate boundary condition a priori.
It is reasonable to assume that far from the source, the DGF asymptotes to the CGF: $$\label{eq:car_asymptotic_green}
{\cal G}_{i-1,j-1,k-1} \approx -\frac{G}{\sqrt{(x_i - x_{1})^2 + (y_j - y_{1})^2 + (z_k - z_{1})^2 }}\quad\text{(far from the source).}$$ Since high-order derivatives of the CGF are non-negligible close to the source, the near-field DGF that results from a second-order finite-difference approximation to the Poisson equation would deviate greatly from the CGF. How far should the boundary be away from the source to safely apply Equation as a proper boundary condition? The answer of @james77 to this question was 16 cells. In fact, one can compare the first two terms in the asymptotic expansion of the DGF in @burk97 to verify that the relative deviation between the DGF and CGF is $0.1\%$ at a distance of $16$ cells, regardless of the total number of cells or the grid spacing.
To ensure that all boundaries are sufficiently away from the point source at $(i',j',k') = (1,1,1)$, for the calculation of the DGF only, we extend the computational domain by adding 16 additional cells to one side in each d
| 2,000
| 2,371
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| 2,002
| 0.783625
|
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|
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