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um map is defined by the formula, $$\label{momentummapdef}
\mathbf{J} (\nu_q) \cdot \xi = \left\langle \nu_q, \xi_Q (q)
\right\rangle,$$ where $\nu_q \in T ^{\ast} _q Q $ and $\xi \in \mathfrak{g}$. In this formula $\xi _Q $ is the infinitesimal generator of the action of ${G}$ on $Q$ associated with the Lie algebra element $\xi$, and $\left\langle \nu_q, \xi_Q (q) \right\rangle$ is the natural pairing of an element of $T ^{\ast}_q Q $ with an element of $T _q Q $.
The Clebsch relation (\[momentum map\]) defines a momentum map for the right action $\operatorname{Diff} (\Omega)$ of the diffeomorphisms of the domain $\Omega$ on the back-to-labels map $\MM{l}$. [^2]
The spatial momentum in equation (\[momentum map\]) may be rewritten as a map $\MM{J}_\Omega:\,T^*\Omega\mapsto\mathfrak{X}^*(\Omega)$ from the cotangent bundle of $\Omega$ to the dual $\mathfrak{X}^*(\Omega)$ of the vector fields $\mathfrak{X}(\Omega)$ given by $$\begin{aligned}
\label{rightmommap}
\MM{J}_\Omega:\,\MM{m}\cdot {\mathrm{d}}\MM{x}
= - \Big( (\nabla\MM{l})^T\cdot\MM{\pi} \Big)\cdot {\mathrm{d}}\MM{x}
= - \,\MM{\pi} \cdot {\mathrm{d}}\MM{l}
=: - \,\pi_k {\mathrm{d}}l_k
\,.\end{aligned}$$ That is, $\MM{J}_\Omega$ maps the space of labels and their conjugate momenta $(\MM{l},\MM{\pi})\in T^*\Omega$ to the space of one-form densities $\MM{m}\in\mathfrak{X}^*(\Omega)$ on $\Omega$. The map (\[rightmommap\]) may be associated with the [*right action*]{} $\MM{l}\cdot\eta$ of smooth invertible maps (diffeomorphisms) $\eta$ of the back-to-labels maps $\MM{l}$ by composition of functions, as follows, $$\label{rightDiff}
\operatorname{Diff}(\Omega):\
\MM{l}\cdot\eta=\MM{l}\circ\eta
\,.$$ The infinitesimal generator of this right action is obtained from its definition, as $$\label{infgen-right}
X_{\Omega}(\MM{l})
:=
\frac{d}{ds}\Big|_{s=0}\Big(\MM{l}\circ\eta(s)\Big)
=
T\MM{l} \circ X
\,,$$ in which the vector field $X \in \mathfrak{X}(\Omega)$ is tangent to the curve of diffeomorphisms $\eta _s$ at the identity $s = 0$. Thus, pairing the map $\MM{
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\beta}\nabla^{\alpha} R^{\nu\beta}=0$. Therefore, for conformally invariant space-times, there are 1 relations coming from the corollary of the Lovelock theorem, and 7 coming from $W_{\mu\nu\alpha\beta}=0$, so there are 8 scalars less in the reduced basis, i.e. 8 scalars left.
General order 6 linear combination for FLRW
-------------------------------------------
The sum of all order six independent scalars, expressed in terms of $a(t)$ and its derivatives is : $$\begin{aligned}
\begin{split}
J=&\sum \Big( v_i \mathcal{L}_i +x_i \curv{L}_i \Big) =\frac{3}{a(t)^6} \Bigg( \sigma_1(v_i,x_i) \; \dot{a}(t)^6 + \sigma_2(v_i,x_i) \; a(t) \; \dot{a}(t)^4 \; \ddot{a}(t) + \sigma_3(v_i,x_i) \; a(t)^2 \; \dot{a}(t)^2 \; \ddot{a}(t)^2
\\& + \sigma_4(v_i,x_i) \; a(t)^3 \; \ddot{a}(t)^3 + \sigma_5(v_i,x_i) \; a(t)^2 \; \dot{a}(t)^3 \; a^{(3)}(t)+ \sigma_6(v_i,x_i) \; a(t)^3 \; \dot{a}(t) \; \ddot{a}(t) \; a^{(3)}(t)
\\& + \sigma_7(v_i,x_i) \; a(t)^4 \; a^{(3)}(t)^2+ \sigma_8(v_i,x_i) \; a(t)^3 \; \dot{a}(t)^2 \; a^{(4)}(t)+ \sigma_9(v_i,x_i) \; a(t)^4 \; \ddot{a}(t) \; a^{(4)}(t) \Bigg)
\end{split}\end{aligned}$$ with,
$\left\{
\begin{array}{l}
\sigma _1=4 (2 v_1-2 v_3+6 v_4+2 v_5+2 v_6+6 v_7+18 v_8+2
x_2+x_3+6 x_4\\
\quad\quad -6 x_5-5 x_6-8 x_7-12 x_8)\vspace{6pt}\\
\sigma _2=2 \big(-2 v_3+12 v_4+4 v_5+6 v_6+24 v_7+108 v_8+30 x_1+x_2\\
\quad\quad -3 x_3-18 x_4+20 x_5+18 x_6+32 x_7+24 x_8\big) \vspace{6pt}\
\\
\sigma _3= \big[ -6 v_2-4 v_3+24 v_4+14 v_5+6 v_6+48 v_7+216 v_8+48 x_1\\
\quad\quad +14 x_2+7 x_3+42 x_4-20 x_5-19 x_6-36 x_7-12 x_8 \big]\vspace{6pt}
\\
\sigma _4=8 v_1+2 v_2-8 v_3+24 v_4+6 v_5+10 v_6+24 v_7+72 v_8-12
x_1-x_3-6 x_4\vspace{6pt}
\\
\sigma _5=-2 \left(18 x_1+5 x_2+x_3+6 x_4-4 x_5-2 x_6-24 x_8\right)\vspace{6pt}
\\
\sigma _6=-\left(36 x_1+8 x_2+x_3+6 x_4-2 x_6-8 x_7+24 x_8\right)\vspace{6pt}
\\
\sigma _7=-\left(4 x_5+3 x_6+4 \left(x_7+3 x_8\right)\right) \vspace{6pt}
\\
\sigma _8=-2 \left(6 x_1+x_2\right)\vspace{6pt}
\\
\sigma _9=-\left(12 x_1+4 x_2+x_3+6 x_4\right)
\end{arra
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( (36)(10) \: + \: 162 \right) \: + \: 20 \: = \: 281, \\
n_H - n_V & = & -12 \: \neq \: 244,\end{aligned}$$ and so we see that this cannot satisfy anomaly cancellation, mechanically verifying our previous observation that this theory cannot be consistent.
More generally, any heterotic compactification on a gerbe, in which the bundle is twisted, will be of this same general type, unless the bundle has rank 8 and the trivially-acting group is ${\mathbb Z}_2$. Locally each theory will look like a compactification of a ten-dimensional theory with an altered GSO projection, and except for the case that the GSO projection switches between Spin$(32)/{\mathbb Z}_2$ and $E_8 \times E_8$, the resulting theory cannot be consistent.
For purposes of comparison, and to help illuminate the methods encoded in appendix \[app:spectra\], let us also outline the results in a closely related consistent compactification. If we did not orbifold, if we took a compactification of an $E_8 \times
E_8$ heterotic string on a smooth large-radius $K3$ with a rank 4 vector bundle, then from a similar computation we would find
$h^0(X, {\cal O})=1$ vector multiplets in the adjoint of ${\rm Spin}(10)$,
One vector multiplet in the adjoint of $E_8$,
$h^1(X, {\cal E}) = 16$ half-hypermultiplets in the ${\bf 16}$ of ${\rm Spin}(10)$,
$h^1(X, \Lambda^2 {\cal E}) = 36$ half-hypermultiplets in the ${\bf 10}$ of ${\rm Spin}(10)$,
$h^1(X, \Lambda^3 {\cal E} = {\cal E}^*) = 16$ half-hypermultiplets in the ${\bf 16}$ of ${\rm Spin}(10)$,
20 singlet hypermultiplets for $K3$ moduli,
$h^1({\rm End}\, {\cal E}) = 162$ singlet half-hypermultiplets for bundle moduli,
where the representations of ${\rm Spin}(10)$ are constructed in the same fashion. Altogether, we find that $$\begin{aligned}
n_V & = & 45 \: + \: 248 \: = \: 293, \\
n_H & = & (1/2)\left( (16)(16) \: + \: (36)(10) \: + \: (16)(16)
\: + \: 162 \right) \: + \: 20 \: = \: 537,
\\
n_H - n_V & = & 244,\end{aligned}$$ consistent with anomaly cancellation, in that standard compactification. Unfo
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| 3,091
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|
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|
}_{\tau^{i}_D}) = u(x),\qquad \text{almost surely.}
\label{FKMC}$$ For practical purposes, since it is impossible to take the limit, one truncates the series of estimates for large $n$ and the [central limit theorem]{}gives $\mathcal{O}(1/n)$ upper bounds on the variance of the $n$-term sum, which serves as a numerical error estimate. Although forming the fundamental basis of most Monte Carlo methods for diffusive Dirichlet-type problems, is an inefficient numerical approach. Least of all, this is because the Monte Carlo simulation of $u(x)$ is independent for each $x\in D$. Moreover, it is unclear how exactly to simulate the path of a Brownian motion on its first exit from $D$, that is to say, the quantity $W_{\tau_D}$. This is because of the fractal properties of Brownian motion, making its path difficult to simulate. This introduces additional numerical errors over and above that of Monte Carlo simulation.
A method proposed by [@Mu], for the case that $D$ is convex, sub-samples special points along the path of Brownian motion to the boundary of the domain $D$. The method does not require a complete simulation of its path and takes advantage of the distributional symmetry of Brownian motion. In order to describe the so-called ‘walk-on-spheres’, we need to first introduce some notation. We may thus set $\rho_0 = x$ for $x\in D$ and define $r_1$ to be the radius of the largest sphere inscribed in $D$ that is centred at $x$. This sphere we will call $S_1 = \{y\in\mathbb{R}^d\colon |y-\rho_0| = r_1\}$. To avoid special cases, we henceforth assume that the surface area of $S_1\cap \partial D$ is zero (this excludes, for example, the case that $x = 0$ and $D$ is a sphere centred at the origin).
Now set $\rho_1\in D$ to be a point uniformly distributed on $S_1$ and note that, given the assumption in the previous sentence, $\mathbb{P}_x(\rho_1\in \partial D) = 0$. Construct the remainder of the sequence $(\rho_n,\,n
\geq 1)$ inductively. Given $\rho_{n-1}$, we define the radius, $r_n$, of the largest sph
| 3,704
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ea}}^{-1}\cap \{x_{i_j}=0\}$ for $j = 1,\cdots, n$. We denote by $\Delta_{p,\theta}$ such a hyperbolic polyhedron. Then the conditions for $\theta$ as in the definition of $\Theta_n$ ensures us that the hyperbolic polyhedron $\Delta_{p,\theta}$ has exactly $n$ facets.
Let us denote by $(i_1 i_2)i_3 \ldots i_n$ or simply $(i_1 i_2)$ the face of $\Delta_{p,\theta}$ represented by $\Delta_{p,\theta} \cap \{x_{i_1}=0\}$ since it corresponds to the degenerate configurations where the points marked by $i_1$ and $i_2$ collide. Similarly we use $(i_1i_2)(i_3 i_4)i_5 \ldots i_n$ or $(i_1 i_2 i_3) i_4 \ldots i_n$, etc. to represent the codimension two faces of $\Delta_{p,\theta}$.
Now gluing $(n-1)!/2$ hyperbolic polyhedra $\Delta_{p, \theta}$ for all labels $p$ along the faces which represent the same degenerate configurations, we obtain $\overline{X_{n,\theta}}$ in which $X_{n, \theta}$ lies as an open dense subset.
The case $n=5$
===============
When $n=5$, $\Delta_{p, \theta}$ is a hyperbolic right pentagon where the edges are labeled by $(i_1i_2)$, $(i_3i_4)$, $(i_5i_1)$, $(i_2i_3)$, $(i_4i_5)$ cyclically. For the equal weight $\theta_0 =(2\pi/5, \cdots, 2\pi/5)$, $\Delta_{p, \theta_0}$ is a hyperbolic regular pentagon, i.e, with all edges having equal lengths.
For any $\theta \in \Theta_5$, the space $\overline{X_{5,\theta}}$ is a closed hyperbolic surface homeomorphic to ${\#}^5 {{\bold R}}{{\bold P}}^2$. The space of hyperbolic structures, which we called a Teichmüller space and denoted by ${\cal T}({\#}^5 {{\bold R}}{{\bold P}}^2)$, is parameterized by the lengths and twisting amounts for the 2-sided ones of a maximal family of mutually disjoint nonparallel simple closed curves on ${\#}^5 {{\bold R}}{{\bold P}}^2$. It is homeomorphic to ${{\bold R}}^9$. Thus we have a map $\Phi_5: \Theta_5 \to {\cal T}({\#}^5 {{\bold R}}{{\bold P}}^2)$ assigning to each $\theta$ the marked hyperbolic structure of $\overline{X_{5,\theta}}$.
On the other hand, our surface $\overline{X_{5,\theta}}$ has a geometric cell decomp
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integration conserves the total energy of the planet and $L_Z$ within 1 to 2%.
The planet is set on a circular orbit at the distance $r_p$ from the star. The initial inclination angle of the orbit with respect to the disc is $I_0$. In the simulations reported here we have taken $n=1/2$ in equation (\[sigma\]). The functional form of $\Sigma$ is shallower than what is usually used for discs, but that has no significant effect on the argument we develop here.
We first compare the numerical results with the analysis summarised in section \[sec:Kozai\] by setting up a case with $R_i \gg r_p$. In figure \[fig1\] we display the evolution of $e$ and $I$ for $M_p=10^{-3}$ M$_{\sun}$, $r_p=1$ au, $M_{\rm disc}=10^{-2}$ M$_{\sun}$, $R_o=100$ au, $R_i=50$ au and $I_0=42^{\circ}.3$. For this run, $L_Z$ is conserved within 2% but the energy of the planet is conserved only within 10%. We are here in the conditions of the analysis of section \[sec:Kozai\] with $\eta=0.5$. From equation (\[emax\]), we expect $e_{\rm max}=0.3$, which is a bit smaller than the value of 0.41 found in the simulation. Also the minimum value of $I$ should be $I_c=39^{\circ}.2$ and is observed to be $36^{\circ}.5$. Note that since the energy varies by about 10% in this run, we do not expect exact agreement between the numerical and the analytical results. We observe that the time it takes to reach $e_{\rm max}$ from the initial conditions is $2.8 \times 10^7$ years, which agrees well with $t_{\rm
evol}$ given by equation (\[tevol\]) provided we take $e_0 \simeq 2
\times 10^{-2}$. As mentioned above, $t_{\rm evol}$ becomes very long when $I_0$ is smaller than 45$^{\circ}$. As the disc lifetime is only a few Myr, $e$ would not have time to reach the maximum value in this case, if starting from a very small value.
Figure \[fig3\] shows the evolution of $e$ and $I$ for $M_p=10^{-3}$ M$_{\sun}$, $r_p=20$ au, $M_{\rm
disc}=10^{-2}$ M$_{\sun}$, $R_o=100$ au, $R_i=1$ au and $I_0=47^{\circ}.7$ (case A). We see that $e$ oscillates between $
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ith some $a_i\in\ZZ$ and some graded prime ideals $P_i$.
The module $M$ is called a [*graded (pretty) clean module*]{}, if it admits a (pretty) clean filtration which is a graded prime filtration.
Similarly we define multigraded filtrations and multigraded (pretty) clean modules.
We denote by $(N)_i$ the $i$th graded component of a graded $R$-module $N$, and by $$\Hilb(N)=\sum_i\dim_K(N)_it^i\in\ZZ[t,t^{-1}]$$ its Hilbert-series.
By the additivity of the Hilbert-series, one obtains for a module with a graded prime filtration as above the Hilbert-series $$\Hilb(M)=\sum_{i=1}^r\Hilb(R/P_i)t^{a_i}.$$
We now consider a more specific case
\[monomial\] Let $S=K[x_1,\ldots, x_n]$ be the polynomial ring, and $I\subset S$ a monomial ideal. Assume that $S/I$ is a graded pretty clean ring whose graded pretty clean filtration has the factors $M_j/M_{j-1}\iso S/P_j(-a_j)$ for $j=1,\ldots,
r$, $a_j\in \NN$ and $P_j\in\Ass(S/I)$. For all $k$ and $i$ set $$h_{ki}=|\{j\: a_j=k,\; \dim S/P_j=i\}|.$$ Then $$\Hilb(S/I)=\sum_iH_i(t) \quad \text{with}\quad H_i(t)=\frac{Q_i(t)}{(1-t)^i} \quad\text{where} \quad Q_i(t)=\sum_kh_{ki}t^k.$$
We have $$\begin{aligned}
\Hilb(S/I)&=&\sum_i \sum_{j\atop \dim S/P_j=i}\Hilb(S/P_j)t^{a_j}\\
&=& \sum_i(\sum_{j\atop \dim S/P_j=i}t^{a_j})/(1-t)^i.\end{aligned}$$ The last equality holds, since all associated prime ideals of $S/I$ are generated by subsets of the variables. Finally the desired formula follows, if we combine in the sum $\sum_{j,\; \dim S/P_j=i}t^{a_j}$ all powers of $t$ with the same exponent.
The attentive reader will notice the similarity of formula \[monomial\] with the formula of McMullen and Walkup for shellable simplicial complexes, see [@BH Corollary 5.1.14]. The precise relationship will become apparent in Section 8 where the numbers $a_j$ are interpreted as shelling numbers.
We now derive similar formulas for the modules $\Ext^i_S(M,\omega_S)$ when $M$ is a graded pretty clean module. Suppose $\dim S=n$. Using the graded version of the exact sequence (\[exact\]) in the
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X}^{(x)}(t)\not\in B(x,\abs{x}) }\notag \\
& = |x|^{\alpha}\,\inf\Bp{|x|^{-\alpha}t> 0\colon |x| \hat{X}^{(\mathbf{i})}(|x|^{-\alpha}t)\not\in B( x,\abs{x}) }\notag \\
& = |x|^{\alpha}\,\inf\Bp{u> 0\colon \hat{X}^{(\mathbf{i})}(u)\not\in B({\rm\bf i},1) }\notag \\
& \eqqcolon |x|^{\alpha}\,\hat{\sigma}_{B({\rm\bf i},1)}.
\label{tscale}
\end{aligned}$$ It follows that $$X^{(x)}_{\sigma_{B_1}} = |x| \hat{X}^{(\mathbf{i})}_{|x|^{-\alpha} |x|^{\alpha} \hat{\sigma}_{B({\rm\bf i},1) }}\,{\buildrel d \over =}\, |x| {X}^{(\mathbf{i})}_{\sigma_{B({\rm\bf i},1)}},
\label{scaleB1}$$ as required. The proof of the second claim follows the same steps and is omitted for the sake of brevity.
An important consequence of the previous result is the comparison between the first exit from the largest sphere in $D$ centred at $x$ and the first exit from the tangent hyperplane to the latter sphere. Recall that $B_n = \{z\in\mathbb{R}^d\colon |z - \rho_{n-1}|< r_n\}$ denotes the $n$th sphere.
\[indicators\]Suppose that $x\in D$ is such that $\partial V(x)$ is a tangent hyperplane to both $D$ and $B_1$. Define under $\mathbb{P}_x$ the indicator random variables $${I}_D = \mathbf{1}_{\{X_{\sigma_{B_1}} \not\in D \}}\quad\text{ and } \quad {I}_V=\mathbf{1}_{\{X_{\sigma_{B_1}} \not\in V(x) \}}.$$ Then $\mathbb{P}_x(I_D\geq I_V)= 1$ and, independently of $x\in D$, $\mathbb{P}_x(I_V = 1) =p(\alpha,d)$, where $$\begin{aligned}
p(\alpha,d) & \coloneqq \mathbb{P}_{\mathbf i}(X_{\sigma_{B({\rm\bf i},1)}}\not\in V({\rm\bf i})) \\
& =\frac{\Gamma(d/2)}{\pi^{(d+2)/2}}\,\sin(\pi\alpha/2)\,\int_{x_1<-1}\left|1- |x|^2\right|^{-\alpha/2}|x|^{-d}\,{\rm d}x,
\end{aligned}$$ which is a number in $(0,1)$.
The inequality follows from the in
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Model \[sec:model\]
================================
The glycerol concentration $p$ of the solution was controlled in our experiments, which led to a change in the viscosity $\mu$ shown in Appendix A. In this section, we consider a viscosity dependence of the camphor boat velocity. Now, the annular glass chamber used in our experiments is recognized as a one-dimensional channel with an infinite length.
The time evolution equation of the camphor boat in a one-dimensional system (The spatial coordinate is represented as $x$) is given as $$\begin{aligned}
m\frac{d^2X}{dt^2} = -h\frac{dX}{dt}+F,
\label{eq:motion}
\end{aligned}$$ where $m$, $X$, $h$ and $F$ are the mass, the center of mass, the friction coefficient of the camphor boat, and the driving force exerted on the moving camphor boat, respectively. We assume that $h$ is proportional to viscosity $\mu$ such as $h=K\mu$, where $K$ is a constant ($K > 0$). The assumption has been used in many previous papers [@Eric; @Koyano; @Nagayama; @Nakata; @Nishimori; @Suematsu; @Suematsu2; @Kohira; @Soh; @NishiWakai; @Heisler], and it was also reported that the viscous drag on the mobility of thin film in Newtonian fluid obeyed a linear relationship with the fluid viscosity [@stone]. Therefore, we considered that the assumption $h=K\mu$ is natural [@viscous; @drag]. The driving force $F$ is described as $$\begin{aligned}
F = w[\gamma(c(X+r+\ell))-\gamma(c(X-r))],
\label{eq:driving}
\end{aligned}$$ where $w$ is the width of the camphor disk. Here, we consider that the positions of the front and the back of the boat are shown as $x=X+r+\ell$ and $x=X-r$, where $r$ and $\ell$ are the radius of the disk and the size of the boat as defined in Fig. \[fig:model\]. The surface tension $\gamma$ depends on the concentration $c$ of the camphor molecules at the surface of the solution, and we assume the linear relation as $$\begin{aligned}
\gamma=\gamma_0-\Gamma c,
\label{eq:surface}
\end{aligned}$$ where $\gamma_0$ is the
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h $\Delta_{\theta}$ ($\Delta_{p}$), and zero otherwise. This choice corresponds to the so-called “water-bag" distribution which is fully specified by energy $h[f]=e$, momentum $P[f]=\sigma$ and the initial magnetization ${\mathbf
M_0}=(M_{x0}, M_{y0})$. The maximum entropy calculation is then performed analytically [@antoniazziPRL] and results in the following form of the QSS distribution $$\label{eq:barf} \bar{f}(\theta,p)= f_0\frac{e^{-\beta (p^2/2
- M_y[\bar{f}]\sin\theta
- M_x[\bar{f}]\cos\theta)-\lambda p-\mu}}
{1+e^{-\beta (p^2/2 - M_y[\bar{f}]\sin\theta
- M_x[\bar{f}]\cos\theta)-\lambda p-\mu}}$$ where $\beta/f_0$, $\lambda/f_0$ and $\mu/f_0$ are rescaled Lagrange multipliers, respectively associated to the energy, momentum and normalization. Inserting expression (\[eq:barf\]) into the above constraints and recalling the definition of $M_x[\bar{f}]$, $M_y[\bar{f}]$, one obtains an implicit system which can be solved numerically to determine the Lagrange multipliers and the expected magnetization in the QSS. Note that the distribution (\[eq:barf\]) differs from the usual Boltzmann-Gibbs expression because of the “fermionic” denominator. Numerically computed velocity distributions have been compared in [@antoniazziPRL] to the above theoretical predictions (where no free parameter is used), obtaining an overall good agreement. However, the central part of the distributions is modulated by the presence of two symmetric bumps, which are the signature of a collective dynamical phenomenon [@antoniazziPRL]. The presence of these bumps is not explained by our theory. Such discrepancies has been recently claimed to be an indirect proof of the fact that the Vlasov model holds only approximately true. We shall here demonstrate that this claim is not correct and that the deviations between theory and numerical observation are uniquely due to the approximations built in the Lynden-Bell approach.
A detailed analysis of the Lynden-Bell equilibrium (\[eq:barf\]) in the parameter plane $(M_{0},e)$ enabled us to unr
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lde{p}]\Psi\,,
\end{aligned}$$ and can further be rewritten in the form of a linearized gauge transformation: $$\begin{aligned}
\delta_{[\tilde{p},\tilde{p}]}\ \Phi\ =&\
\xi_0 Q [\tilde{p},\{Q, M\}]\Phi\
=\
\xi_0 Q [\tilde{p}, M]Q\Phi
\nonumber\\
\cong&\
-Q(\xi_0[\tilde{p}, M]Q\Phi)
+ \eta(\xi_0X_0[\tilde{p}, M]Q\Phi)\,,\end{aligned}$$ and $$\begin{aligned}
\delta_{[\tilde{p},\tilde{p}]}\ \Psi\
=&\
X\eta\Xi[\tilde{p},\{Q, \tilde{M}\}]\Psi\
\cong\
X\eta\Xi Q[\tilde{p}, \tilde{M}]\Psi
\nonumber\\
=&\
Q(X\eta\Xi[\tilde{p},\tilde{M}]\Psi)\,,\end{aligned}$$ up to the linearized equations of motion. The parameter $\lambda_{\tilde{p}\tilde{p}}=X\eta\Xi[\tilde{p},\tilde{M}]\Psi$ is in the restricted small Hilbert space: $\eta\lambda_{\tilde{p}\tilde{p}}=0$ and $XY\lambda_{\tilde{p}\tilde{p}}=\lambda_{\tilde{p}\tilde{p}}$.
Finally we show that all the extra symmetries obtained from the repeated commutators of $\delta_{\mathcal{S}}$’s and $\delta_{\tilde{p}}$’s act trivially on the physical states defined by the asymptotic string fields. For this purpose, it is enough to consider the transformations of $\eta\Phi$ and $\Psi$ at the linearized level for a similar reason to that discussed in Section \[sec algebra\]. Using the linearized form of (\[large small\]) for general variation, $$\delta\Phi\ =\ \xi_0\delta\eta\Phi + \eta(\xi_0\delta\Phi)\,,$$ we can show that if the transformation of $\eta\Phi$ has the form of a gauge transformation, $\delta\eta\Phi= - Q\eta\Lambda$, with some field-dependent parameter $\Lambda$, then the transformation of $\Phi$ also has the form of a gauge transformation: $$\begin{aligned}
\delta\Phi\ =&\ - \xi_0 Q\eta\Lambda + \eta\Omega
\nonumber\\
=&\ Q\Lambda + \eta(\Omega-\xi_0Q\Lambda)\,, \end{aligned}$$ with some field-dependent $\Omega$.
Starting from the linearized transformations $$\begin{aligned}
{3}
\delta_{\mathcal{S}}\eta\Phi\ =&\ {\mathcal{S}}\Psi\,,\qquad & \delta_{\mathcal{S}}\Psi\ =&\ X{\mathcal{S}}\eta\Phi\,,\\
\delta_{\tilde{p}}\eta\Phi\ =&\ (p-X_0\tilde{p})\eta\Ph
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f the event horizon of the black hole, as in the case of the Bekenstein-Hawking entropy [@SWH1; @Bekenstein]. Here $ C=\dfrac{1}{(1+2\alpha m)} $ is the deficiency factor in the limit of $ \phi $ as it runs from $ 0\rightarrow 2\pi C $ (as mentioned earlier). In FIG. \[plot3\], we have shown the variation of the total entropy on the horizon with the acceleration parameter $ \alpha $.
Similarly we can compute the entropy density as $$\label{s_nonrot}
s=k_{s}\frac{1}{\sqrt{h}}\frac{\partial}{\partial r}\left(\sqrt{h}\dfrac{P}{\sqrt{h_{rr}}} \right)=\dfrac{2k_{s}}{r}\sqrt{\left(1-\alpha^{2}r^{2}\right)\left(1-\frac{r_{h}}{r}\right)}.$$ In the above equation (\[s\_nonrot\]), inserting $ \alpha=0 $, we get the entropy density for the Schwarzschild black hole. In FIG. \[fig2\], the dependence of the gravitational entropy density corresponding to this metric on other relevant parameters have been indicated. From equation (\[s\_nonrot\]) we can see that the zeroes of the gravitational entropy density function are located at the acceleration horizon $ r=\dfrac{1}{\alpha} $, and at the event horizon $ r=2m $, which is clearly evident from FIG. \[fig2\]. Specifically, FIG. \[fig2\](a) shows that for $ \alpha=0 $, the acceleration horizon goes to infinity where the entropy density reduces to zero, and at the event horizon $r=2$, the entropy density becomes zero. Similarly for $ \alpha=0.5 $, the acceleration horizon and the event horizon coincide at $ r=2 $, where the entropy density becomes zero. FIG. \[fig2\](b) indicates that for $\alpha=0.25$, the acceleration horizon lies at $r=4$ and the event horizon is at $ r=2 $, the entropy density going to zero at both these places, and diverges at the singularity $ r=0 $ which is in agreement with equation (\[s\_nonrot\]).
![Plot showing the variation of the total gravitational entropy for the accelerating non-rotating BH with respect to the acceleration parameter $ \alpha $, where we have taken $m=1 \: \textrm{and} \: k_{s}=1$.[]{data-label="plot3"}](Salphanew1.png){width="34.
| 3,712
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| 3,628
| 3,455
| 0.772247
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|
,\mathcal{V})$ ***denoted by* ***$f_{\alpha}\overset{\mathbf{G}}\longrightarrow\mathscr{M}$, is the subset of $\mathcal{D}_{-}\times\mathcal{R}_{-}$that is the union of the graphs of the function $F$ and the multifunction $G^{-}$ $$\mathbf{G}_{\mathscr{M}}=\mathbf{G}_{F}\bigcup\mathbf{G}_{G^{-}}$$*
*where $$\mathbf{G}_{G^{-}}=\{(x,y)\in X\times Y\!:(y,x)\in\mathbf{G}_{G}\subseteq Y\times X\}.\qquad\square$$*
***Begin Tutorial6: Graphical Convergence***
The following two examples are basic to the understanding of the graphical convergence of functions to multifunctions and were the examples that motivated our search of an acceptable technique that did not require vertical portions of limit relations to disappear simply because they were non-functions: the disturbing question that needed an answer was how not to mathematically sacrifice these extremely significant physical components of the limiting correspondences. Furthermore, it appears to be quite plausible to expect a physical interaction between two spaces $X$ and $Y$ to be a consequence of both the direct interaction represented by $f\!:X\rightarrow Y$ and also the inverse interaction $f^{-}\!:Y\rightarrow X$, and our formulation of pointwise biconvergence is a formalization of this idea. Thus the basic examples (1) and (2) below produce multifunctions instead of discontinuous functions that would be obtained by the usual pointwise limit.
**Example 3.1.** (1)
$$f_{n}(x)=\left\{ \begin{array}{lc}
0 & -1\leq x\leq0\\
nx & 0\leq x\leq1/n\\
1 & 1/n\leq x\leq1\end{array}\right.:\quad[-1,1]\rightarrow[0,1]$$ $$g_{n}(y)=y/n:\quad[0,1]\rightarrow[0,1/n]$$
Then$$F(x)=\left\{ \begin{array}{cc}
0 & -1\leq x\leq0\\
1 & 0<x\leq1\end{array}\right.\qquad\mathrm{on}\qquad\mathcal{D}_{-}=\mathcal{D}_{+}=[-1,0]\bigcup(0,1]$$
$$G(y)=0\quad\mathrm{on}\quad\mathcal{R}_{-}=[0,1]=\mathcal{R}_{+}.$$
The graphical limit is $([-1,0],0)\bigcup(0,[0,1])\bigcup((0,1],1)$.
\(2) $f_{n}(x)=nx$ for $x\in[0,1/n]$ gives $g_{n}(y)=y/n:[0,1]\rightarrow[0,1/n].$ Then $$F(x)=0\quad\mathr
| 3,713
| 3,407
| 3,826
| 3,315
| 1,613
| 0.787636
|
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|
breaking, $$\begin{aligned}
\nonumber
T<T_c: &\qquad \langle L(\vec x)\rangle = 0\,,\quad F_q=\infty\,, \\
T>T_c: &\qquad \langle L(\vec x)\rangle \neq 0\,, \quad F_q<\infty\,.
\label{eq:orderdisorder}\end{aligned}$$ The expectation value of the Polyakov loop can be deduced from the equations of motion of its effective potential $V_L[\langle
L\rangle]$. We shall argue, that the computation of the latter greatly simplifies within an appropriate choice of gauge. Indeed, gauge fixing is nothing but the choice of a specific parameterisation of the path integral, and a conveniently chosen parameterisation can simplify the task of computing physical observables.
In the present case our choice of gauge is guided by the demand of a particularly simple representation of the Polyakov loop variable . A gauge ensuring time-independent $A_0$ leads to both, a trivial integration in and renders the path ordering irrelevant. Having done this one can still rotate the Polyakov loop operator ${\cal P}(\vec x)$, , into the Cartan subgroup. The above properties are achieved for time-independent gauge field configurations in the Cartan subalgebra, i.e. $A_0(t_0,\vec x)=A_0^c(\vec x)$. For $SU(2)$ this reads $$\begin{aligned}
\label{eq:A0}
A_0(t_0,\vec x)=A_0(\vec x)\, \0{\sigma_3}{2}\end{aligned}$$ and entails a particularly simple relation between $A_0$ and $L$, $$\label{eq:polsu(2)}
L(\vec x) = \cos\, \012 g \beta A_0(\vec x) \,,$$ Note that this simple relation is not valid on the level of expectation values of $L$ and $A_0$, in $SU(2)$ we have $\langle
L\rangle \neq \cos\, \012 g \beta \langle A_0\rangle$. However, in the present work we consider an approach that gives direct access to the effective potential $V_{\rm eff}[ \langle A_0\rangle]$ for the gauge field, as distinguished to those for the Polyakov loop, $U_{\rm
eff}[\langle L\rangle]$ [^1].
Here, we argue that $ L[\langle A_0\rangle ]$ also serves as an order parameter: to that end we show that the order parameter $ \langle L[
A_0 ]\rangle $ is bounded from abov
| 3,714
| 3,327
| 3,411
| 3,241
| 2,997
| 0.775538
|
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trial 1/0 indicator) and *treat* ~*ij*~ (treatment group 1/0 indicator) value for each of 100 participants in each of 10 trials.
Next, based on the previous meta‐analysis,[22](#sim7930-bib-0022){ref-type="ref"} we set the true parameter values for this simulation to be as follows: θ = −9.66 (summary treatment effect; negative value favors treatment group), = 7.79 (between trial variation in the treatment effect), *β* = 159.73 (mean blood pressure response in control group), $\tau_{\beta}^{2}$ = 233.99 (between trial variation in the intercept), and *σ* ^*2*^ = 333.74 (residual variance).
We then used these parameter values to generate further terms, beginning with using *σ* ^2^ to generate an error term *e* ~*i j*~, for the *j*th participant from the *i*th trial $$e_{\mathit{ij}} \sim N\left( {0,\sigma^{2}} \right).$$ Then, we generated the trial level values for the random parts of the intercept and treatment effect terms, *u* ~1*i*~ and *u* ~2*i*~, respectively, $$\begin{matrix}
{u_{1i} \sim N\left( {0,\tau_{\beta}^{2}} \right)} \\
{u_{2i} \sim N\left( {0,\tau^{2}} \right).\mspace{900mu}} \\
\end{matrix}$$ Finally, with all the parameters defined (*β*, *u* ~1*i*~, θ, *u* ~2*i*~, *treat* ~*i j*~, and *e* ~*i j*~), we generated the end‐of‐trial continuous outcome value *Y* ~*Fi j*~, under the random intercept model (2) (with no baseline adjustment term and assuming a common residual variance) $$Y_{\mathit{Fij}} = \left( {\beta + u_{1i}} \right) + \left( {\theta + u_{2i}} \right)\textit{treat}_{\mathit{ij}} + e_{\mathit{ij}}.$$ This gave one complete IPD meta‐analysis dataset of 1000 total participants, containing 100 participants in each of 10 trials, consisting of the following data for each individual: a trial indicator (*trial* ~*i*~), a treatment group indicator (*treat* ~*i j*~), and an end‐of‐trial continuous outcome value (*Y* ~*Fi j*~).
*Step 2: Model fit and replication*
Using the generated data, we fitted a stratified intercept model (1) and a random intercept mo
| 3,715
| 1,981
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| 1,480
| 0.789047
|
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|
atement for the case when $\Phi$ is a word. If $\Phi=z$ then the claim is evident. If $\Phi=uv$ then $z$ can belong to either $u$ or $v$. Let $z$ belongs to $u$. Then using Lemma \[lemma2\] we obtain $$\begin{gathered}
\Psi^z_{{\mathrm{As}}}(\mathcal{J}(\Phi))=\Psi^z_{{\mathrm{As}}}(\frac{1}{2}[\mathcal{J}(u)\mathcal{J}(v)
+\mathcal{J}(v)\mathcal{J}(u)]) \\
=\frac{1}{2}[\Psi^z_{{\mathrm{As}}}(\mathcal{J}(u)){\mathbin\dashv}\mathcal{J}(v)^{\dashv}
+\mathcal{J}(v)^{\vdash}{\mathbin\vdash}\Psi^z_{{\mathrm{As}}}(\mathcal{J}(u))] \\
=\frac{1}{2}[\mathcal{J}_{{\mathrm{Di}}}(\Psi^z_{{\mathrm{Alg}}}(u)){\mathbin\dashv}\mathcal{J}_{{\mathrm{Di}}}(v^{\dashv})
+\mathcal{J}_{{\mathrm{Di}}}(v^{\dashv}){\mathbin\vdash}\mathcal{J}_{{\mathrm{Di}}}(\Psi^z_{{\mathrm{Alg}}}(u))] \\
=\mathcal{J}_{{\mathrm{Di}}}(\Psi^z_{{\mathrm{Alg}}}(u){\mathbin\dashv}v^{\dashv})
=\mathcal{J}_{{\mathrm{Di}}}(\Psi^z_{{\mathrm{Alg}}}(\Phi)).\end{gathered}$$ The case when $z$ belongs to $v$ is proved similarly.
We recall about the quotient $\bar D=D/D_0$ that has been defined in Section \[subsec:DefDialg\]. This quotient compares every 0-dialgebra with an ordinary algebra. The quotient $\bar D$ of a dialgebra $D$ generated by a set $X=\{x_i \mid i\in I\}$ is an algebra generated by the set $\bar X =\{\bar x_i \mid i\in I\}$. In order to simplify notation, we will further denote $\bar x\in \bar X$ by $x$. Following this convention we obtain, for example, $\overline{{\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle}={\mathrm{As}}\,\langle X\rangle$.
\[prop:SJDiSJ\] Let $f\in {\mathrm{Di}}{\mathrm{As}}_z\,\langle X\rangle$. Then $$f\in {\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle \Leftrightarrow \bar f\in {\mathrm{SJ}}\,\langle X\rangle.$$
“$\Rightarrow$”. Let $f\in {\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle$ that is $f=\mathcal{J}_{{\mathrm{Di}}}(\Phi)$ for some $\Phi\in{\mathrm{Di}}{\mathrm{Alg}}\,\langle
X\rangle$. Then $\bar
f=\overline{\mathcal{J}_{{\mathrm{Di}}}(\Phi)}=\mathcal{J}(\bar\Phi)$, so $\bar f\in {\mathrm{SJ}}\,\langle X\rangle$. T
| 3,716
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|
tal derivative $B$-field constructed from a 2-cocycle on the isometry group with respect to which we dualise and then dualising.
In this note we develop this line of reasoning. We begin by outlining the essential features of Yang-Baxter $\sigma$-models and the technology of non-abelian T-duality in Type II supergravity. After demonstrating that a centrally-extended T-duality can be reinterpreted as as non-abelian T-duality of a coset based on the Heisenberg algebra, we show how the machinery of non-abelian T-duality developed for Type II backgrounds can be readily applied to the construction of [@Hoare:2016wsk; @Borsato:2016pas]. We confirm that the centrally-extended non-abelian T-duals produce the full Type II supergravity backgrounds corresponding to $\beta$-deformations (when the duality takes place in the $S^5$ factor of $AdS_5\times S^5$), non-commutative deformations (when performed in the Poincaré patch of $AdS_5$) and dipole deformations (when performed in both the $S^{5}$ and $AdS^{5}$ simultaneously). In appendices \[app:sugra\] and \[app:algconv\] we outline our conventions for supergravity and certain relevant algebras respectively. As a third appendix \[app:furtherexamples\] we include some additional worked examples including one for which the non-abelian T-duality is anomalous and the target space solves the generalised supergravity equations.
The supergravity backgrounds in this note have appeared in the literature in the past but the derivation and technique presented here is both novel, simple and, we hope may have utility in the construction of more general supergravity backgrounds.
Yang Baxter sigma-models {#sec:yangbaxter}
========================
Given a semi-simple Lie algebra $\mathfrak{f}$ (and corresponding group $F$) we define an antisymmetric operator $R$ obeying $$\label{eq:cybe}
[R X , R Y] - R\left([R X, Y]+ [X,RY] \right) = c [ X, Y] \ , \quad X,Y \in \mathfrak{f} \ ,$$ where the cases $c=\pm 1$ and $c=0$ are known as the classical and modified classical Yang Baxter equations
| 3,717
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|
inite paths, say $\gamma$ and $\gamma'$, respectively below and above the horizontal axis, and satisfying the following property: there is no semi-infinite path in the RST, different from $\gamma$ and $\gamma'$, and trapped between them (in the trigonometric sense). Parts $(i)$ and $(ii)$ of Theorem \[HN1\] force $\gamma$ and $\gamma'$ to have the same asymptotic direction, namely $0$. Such a situation never happens by Proposition \[prop:<2\]. In other words, $${{\mathbb P}}\left( \limsup_{r\to\infty} \widetilde{\chi}_r \geq 1 \right) \, = \, 1 ~.$$ $\Box$
From vertices of $\gamma_{0}$ (different from $O$), some paths (finite or not) emanate, forming together an unbounded subtree of the RST $\mathcal{T}$ for which $\gamma_{0}$ can be understood as the spine. The next results describe the skeleton of this subtree.\
Let us denote by $V_{\infty}^{+}$ and $V_{\infty}^{-}$ the set of points $X\in N\cap \gamma_{0}\setminus\{O\}$ from which (at least) another semi-infinite path emanates, respectively above and below $\gamma_{0}$. Of course, $V_{\infty}^{+}$ and $V_{\infty}^{-}$ may have a nonempty intersection.
\[corol:skeleton\]
1. Almost surely, $V_{\infty}^{+}$ and $V_{\infty}^{-}$ are of infinite cardinality.
2. For $r>0$, let us denote by $D_r$ the set of directions $\alpha\in[0,2\pi)$ with a semi-infinite path starting from a point $X$ in $V_{\infty}^{+}\cup V_{\infty}^{-}$ with modulus $|X|>r$. Then, there a.s. exist two nonincreasing sequences $(\alpha_{r})_{r>0}$ and $(\beta_{r})_{r>0}$ of positive r.v.’s such that $$D_r = [-\alpha_{r},\beta_{r}] \; \mbox{ (modulo $2\pi$) and } \; \lim_{r\to+\infty} \alpha_{r} = \lim_{r\to+\infty} \beta_{r} = 0 ~.$$
3. Let $v_{\infty}^{r}$ be the cardinality of $( V_{\infty}^{+}\cup V_{\infty}^{-} )\cap B(O,r)$. Then, $$\lim_{r\to\infty} {{\mathbb E}}\frac{v_{\infty}^{r}}{r} = 0 ~.$$
The first two assertions of Corollary \[corol:skeleton\] say that an infinite number of unbounded subtrees emanate from the semi-infinite path $\gamma_{0}$. Each of them covers a who
| 3,718
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| 3,318
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| 0.772794
|
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|
{true_inequality}$$ A superficial look at the instance (\[prop17\]) of Proposition 17 can induce the idea that, for [*every*]{} D-strategy ${\cal S}$ (in particular for $S_6$), the inequality $$EqLv(E(L_1),E(L_1)) \leq EqLv(E(\bot),E(\bot),{\cal S})
\label{false_inequation}$$ holds. In fact, what shows Proposition 17, is that inequality (\[false\_inequation\]) does hold but, only for strategies ${\cal S}$ which are [*optimal*]{} for the defender, hence realizing exactly the equivalence level of $(E(\bot),E(\bot))$.
[Jan10]{}
P. Jancar. Short decidability proofs for dpda language equivalence and 1st order grammar bisimilarity. , pages 1–35, 2010.
[^1]: mailing adress:LaBRI and UFR Math-info, Université Bordeaux1\
351 Cours de la libération -33405- Talence Cedex.\
email:ges@labri.u-bordeaux.fr; fax: 05-40-00-66-69;\
URL:http://dept-info.labri.u-bordeaux.fr/$\sim$ges/
---
abstract: 'X-ray polarimetry in astronomy has not been exploited well, despite its importance. The recent innovation of instruments is changing this situation. We focus on a complementary MOS (CMOS) pixel detector with small pixel size and employ it as an x-ray photoelectron tracking polarimeter. The CMOS detector we employ is developed by GPixel Inc., and has a pixel size of 2.5$\mathrm{\mu}$m $\times$ 2.5 $\mathrm{\mu}$m. Although it is designed for visible light, we succeed in detecting x-ray photons with an energy resolution of 176eV (FWHM) at 5.9keV at room temperature and the atmospheric condition. We measure the x-ray detection efficiency and polarimetry sensitivity by irradiating polarized monochromatic x-rays at BL20B2 in SPring-8, the synchrotron radiation facility in Japan. We obtain modulation factors of 7.63% $\pm$ 0.07% and 15.5% $\pm$ 0.4% at 12.4keV and 24.8keV, respectively. It demonstrates that this sensor can be used as an x-ray imaging spectrometer and polarimeter with the highest spatial resolution ever tested.'
author:
- Kazunori Asakura
- Kiyoshi Hayashida
- Takashi Hanasaka
- Tomoki Kawabata
- Tomok
| 3,719
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$X$, and $G/K \cong H$ acts effectively. In this case, $\mathfrak{X} = [X/G]$ is a $K$-gerbe. A vector bundle on $\mathfrak{X}$ is a $G$-equivariant vector bundle on $X$, and as such, the $K$ action is defined by a representation of $K$ on the fibers of that vector bundle. This is the more general picture of the second notion of twisting. Any bundle on the gerbe that is not a pullback from the base, has a nontrivial action of $K$.
These two notions of twisting are not unrelated. Mathematically, it is a standard result that the category of sheaves on a gerbe decomposes into different sectors containing twisted sheaves on the underlying space, twisted by flat $B$ fields. Moreover, this decomposition is complete: there are no nonzero Ext groups between sheaves in different sectors on the same gerbe. This fact was one of the inspirations for the ‘decomposition conjecture’ presented in [@summ], which said that conformal field theories describing strings on gerbes should factorize in the same way, that the CFT’s are the same as CFT’s on disjoint unions of spaces. The resulting factorization of D-branes reflects the mathematical result above on factorization of sheaves on gerbes.
For completeness, let us discuss this decomposition for the special case of ${\cal O}(1/k) \rightarrow {\mathbb P}^N_{[k,\cdots,k]}$. To be twisted in the first sense we discussed, one can show that the rank of the twisted bundle must be divisible by the order of the twisting cocycle’s cohomology class. Here, since ${\cal O}(1/k)$ has rank one, the order of the cocycle must be one. Indeed, the twistings of ${\cal O}(1/k)$ appearing involve cocycles with trivial cohomology, so there is no rank restriction.
Class I: Gauge bundle a pullback from the base {#sect:het-decomp}
==============================================
We have classified heterotic string compactifications on gerbes into three fundamental classes or ‘building blocks,’ from which more general compactifications can be built. In this and the next two sections, we wil
| 3,720
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|
k} - h_{m} ) (\Delta_{L} - h_{k} )^2 (\Delta_{L} - h_{m} )^2 }
\nonumber \\
&\times&
\biggl\{
(\Delta_{L} - h_{m} )^2
e^{- i ( h_{k} - h_{n} ) x}
- (\Delta_{L} - h_{k} )^2
e^{- i ( h_{m} - h_{n} ) x}
+ ( h_{k} - h_{m} )( h_{k} + h_{m} - 2 \Delta_{L} ) e^{- i ( \Delta_{L} - h_{n} ) x}
\biggr\}
\biggr]
\nonumber \\
&\times&
W_{\alpha L} (UX)^*_{\beta k}
(UX)^*_{\alpha n} (UX)_{\beta n}
\left\{ W^{\dagger} A (UX) \right\}_{L m}
\left\{ (UX)^{\dagger} A W \right\}_{m L}
\left\{ W^{\dagger} A (UX) \right\}_{L k}
\nonumber \\
&+&
\sum_{n}
\sum_{L k}
\sum_{K \neq L}
\biggl[
- \frac{ (ix) e^{- i ( h_{k} - h_{n} ) x} }{ (\Delta_{L} - h_{k}) (\Delta_{K} - h_{k}) }
+ \frac{ 1 }{ (\Delta_{L} - \Delta_{K}) (\Delta_{L} - h_{k})^2 ( \Delta_{K} - h_{k} )^2 }
\nonumber \\
&\times&
\biggl\{
( \Delta_{K} - h_{k} )^2
e^{- i ( \Delta_{L} - h_{n} ) x}
- (\Delta_{L} - h_{k})^2
e^{- i ( \Delta_{K} - h_{n} ) x}
+
(\Delta_{L} - \Delta_{K}) (\Delta_{L} + \Delta_{K} - 2 h_{k} ) e^{- i ( h_{k} - h_{n} ) x}
\biggr\}
\biggr]
\nonumber \\
&\times&
W_{\alpha L} (UX)^*_{\beta k}
(UX)^*_{\alpha n} (UX)_{\beta n}
\left\{ W^{\dagger} A (UX) \right\}_{L k}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W^{\dagger} A (UX) \right\}_{K k}
\nonumber \\
&+&
\sum_{n}
\sum_{L k}
\sum_{K \neq L}
\sum_{m \neq k}
\frac{ 1 }{ (\Delta_{L} - \Delta_{K}) ( h_{m} - h_{k} ) (\Delta_{L} - h_{k}) (\Delta_{L} - h_{m}) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{m}) }
\nonumber \\
&\times&
\biggl[
( h_{m} - h_{k} )
\biggl\{ (\Delta_{K} - h_{k}) (\Delta_{K} - h_{m})
e^{- i ( \Delta_{L} - h_{n} ) x}
- (\Delta_{L} - h_{k}) (\Delta_{L} - h_{m})
e^{- i ( \Delta_{K} - h_{n} ) x}
\biggr\}
\nonumber \\
&-&
(\Delta_{L} - \Delta_{K})
\biggl\{ (\Delta_{L} - h_{m}) (\Delta_{K} - h_{m})
e^{- i ( h_{k} - h_{n} ) x}
- (\Delta_{L} - h_{k}) (\Delta_{K} - h_{k})
e^{- i ( h_{m} - h_{n} ) x}
\biggr\}
\biggr]
\nonumber \\
&\times&
W_{\alpha L} (UX)^*_{\beta k}
(UX)^*_{\alpha n} (UX)_{\beta n}
\left\{ W^{\dagger} A (UX) \right\}_{L m}
\l
| 3,721
| 1,085
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| 3,626
| null | null |
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|
e classification of tissue degeneration {#Sec11}
--------------------------------------------------
Variable reduction and classification based on PCA show that the samples can be grouped into two linearly separable classes based on variations in their NIR spectral data using the 1^st^ and 2^nd^ principal components scores (PC~1~ and PC~2~). The scores can be observed to group the samples according to level of degeneration along PC~1~, while samples within each group cluster along both PC~1~ and PC~2~.
"Class 1" consists of samples with low Mankin score (\<=2) and relatively high GAG content (\>23 μg/mg), and is representative of cartilage with mild degeneration (sham, weeks 1 and 2); while "class 2" consists of samples with relatively high Mankin score (=\>3) and low GAG content (\<23 μg/mg), indicative of advanced tissue degeneration (weeks 4 and 6) (Fig. [3](#Fig3){ref-type="fig"}). Although PCA was performed on the spectra with and without pre-processing (multiplicative scatter correction (MSC) and derivative (1^st^ and 2^nd^)), optimal classification was obtained without pre-processing. SVM shows that a decision boundary that optimally demarcates both classes in the PCA score plot can be obtained, since the classes are linearly separable (Fig. [3a](#Fig3){ref-type="fig"}). The SVM model classified all samples with advanced degeneration correctly, but misclassified two samples with mild degeneration (Fig. [3b](#Fig3){ref-type="fig"}). No significant difference (p = 0.0588) in tissue degeneration (via the Mankin score) was observed between the samples in class 1; however, statistically significant difference (p = 0.009) was observed between the samples in class 2.Figure 3(**a**) PCA score plot of the NIR spectral data of the samples showing classification into two distinct groups. "Class 1" consists of sham and samples from weeks 1 and 2 post-injury, class 2 consists of samples from weeks 4 and 6 post-injury. (**b**) SVM decision boundary showing the optimal demarcation of both classes.\[w1
| 3,722
| 2,185
| 2,730
| 2,766
| null | null |
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|
r in Krakow, to the effect that the need for ethical approval is waived for the experiments.
During all experiments we manipulated one main variable, namely the visibility level (amount of smoke was increasingly greater in consecutive experiments). Additionally, during experiment 3 we asked participants to obtain *the best individual evacuation time*, and as a consequence, competitive evacuation was observed. During the last experiment we changed bus stopping place, in order to impede the localization task for participants, and simultaneously two doors of the bus were open during evacuation.
Regarding the methodological point of view, we applied a direct observational method, namely structured observation, which is located between a naturalistic observation and lab experiments. The whole staff were neutral during experiments: the bus driver was instructed not to suggest any solutions to participants, firefighters securing the safety of the participants were hidden and did not participate directly in the experiment.
4 Results {#sec007}
=========
4.1 The decision to start the evacuation {#sec008}
----------------------------------------
We analyze the decision making process of individuals on starting the evacuation in experiment 1, since this is the only experiment when participants unfamiliar with the tunnel encountered a new, emergency situation (see [Table 1](#pone.0201732.t001){ref-type="table"}). Before entering the tunnel participants knew only that they will take part in an evacuation experiment in the tunnel, they were not informed about experiment details and did not expect artificial smoke. From the participants' point of view, the sequence of events was as follows:
- bus enters the tunnel and stops in the middle,
- then smoke begins to surround the bus,
- after a moment a siren starts to sound,
- voice alarm messages in Polish and English are initialized: (*Attention please, attention please. Fire alarm. Leave your car and go to the nearest emergency exit*).
After the bus had stopped,
| 3,723
| 1,343
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| 3,144
| null | null |
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|
ion parameters do not require linear estimators. For example we can define $$\gamma_{{\widehat{S}}}(j) = \mathbb{E}_{X,Y}\Biggl[ |Y-\hat\mu_{{\widehat{S}}(j)}(X_{{{\widehat{S}}}(j)})| -
|Y-\hat\mu_{{\widehat{S}}}(X_{{\widehat{S}}})|\ \Biggr], \quad j \in {\widehat{S}},$$ where $\hat\mu_{{\widehat{S}}}$ is any regression estimator restricted to the coordinates in ${\widehat{S}}$ and $\hat\mu_{{\widehat{S}}(j)}$ is the estimator obtained after performing a new model selection process and then refitting without covariate $j
\in {\widehat{S}}$. Similarly, we could have $$\rho_{{\widehat{S}}} = \mathbb{E}_{X,Y}\Bigl[| Y - \hat{\mu}_{{\widehat{S}}}(X_{{\widehat{S}}})|\,
\Bigr],$$ for an arbitrary estimator $\hat{\mu}_{{\widehat{S}}}$. For simplicity, we will focus on linear estimators, although our results about the LOCO and prediction parameters hold even in this more general setting.
2. It is worth reiterating that the projection and LOCO parameters are only defined over the coordinates in ${\widehat{S}}$, the set of variables that are chosen in the model selection phase. If a variable is not selected then the corresponding parameter is set to be identically zero and is not the target of any inference.
There is another version of the projection parameter defined as follows. For the moment, suppose that $d < n$ and that there is no model selection. Let $\beta_n = (\mathbb{X}^\top \mathbb{X})^{-1}\mathbb{X}^\top \mu_n$ where $\mathbb{X}$ is the $n\times d$ design matrix, whose columns are the $n$ vector of covariates $X_1,\ldots, X_n$, and $\mu_n = (\mu_n(1),\ldots, \mu_n(n))^\top$, with $\mu_n(i) = \mathbb{E}[Y_i | X_1,\ldots, X_n]$. This is just the conditional mean of the least squares estimator given $X_1,\ldots, X_n$. We call this the [*conditional projection parameter*]{}. The meaning of this parameter when the linear model is false is not clear. It is a data dependent parameter, even in the absence of model selection. [@buja2015models] have devoted a whole paper to this issue. Quoting from the
| 3,724
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| 0.786328
|
github_plus_top10pct_by_avg
|
neral one-dimensional tiles, Gruslys, Leader and Tan [@gltan16] conjectured that there is a bound on the dimension in terms of the size of the tile:
For any positive integer $t$, there exists a number $d$ such that any tile $T \subset \mathbb{Z}$ with $|T| \leq t$ tiles $\mathbb{Z}^d$.
This conjecture remains unresolved. The authors of [@gltan16] showed that if $d$ always exists then $d \to \infty$ as $t \to \infty$, by exhibiting a tile of size $3d-1$ that does not tile $\mathbb{Z}^d$. This gives a simple lower bound on $d$; better bounds would be of great interest.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank Vytautas Gruslys for suggesting this problem and for many helpful discussions, and Imre Leader for his encouragement and useful comments.
[99]{}
A. Adler and F. C. Holroyd, ‘Some results on one-dimensional tilings’, *Geom. Dedicata* 10 (1981) 49–58.
R. Berger, ‘The undecidability of the domino problem’, *Mem. Amer. Math. Soc.* 66 (1966) 1–72.
J. H. Conway and J. C. Lagarias, ‘Tiling with polyominoes and combinatorial group theory’, *J. Combin. Theory Ser. A* 53 (1990) 183–208.
K. Dahkle, ‘Tiling rectangles with polyominoes’, http://eklhad.net/polyomino/index.html (retrieved 7 May 2018)
E. Friedman, ‘Problem of the Month (February 1999)’,\
https://www2.stetson.edu/\~efriedma/mathmagic/0299.html (retrieved 7 May 2018)
S. W. Golomb, ‘Tiling with sets of polyominoes’, *J. Combin. Theory* 9 (1970) 60–71.
V. Gruslys, ‘Decomposing the vertex set of a hypercube into isomorphic subgraphs’, arXiv:1611.02021.
V. Gruslys, I. Leader, T. S. Tan, ‘Tiling with arbitrary tiles’, *Proc. London Math. Soc.* (3) 112 (2016) 1019–1039.
V. Gruslys, I. Leader, I. Tomon, ‘Partitioning the Boolean lattice into copies of a poset’, arXiv:1609.02520.
V. Gruslys, S. Letzter, ‘Almost partitioning the hypercube into copies of a graph’, arXiv:1612.04603.
A. U. O. Kisisel, ‘Polyomino convolutions and tiling problems’, *J. Combin.
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|
J}_\Omega$ with the vector field $X \in
\mathfrak{X}(\Omega)$ yields $$\begin{aligned}
\left\langle \MM{J}_\Omega (\MM{l}, \MM{\pi} ), X \right\rangle & =
-\,\langle\, \MM{\pi} \cdot {\mathrm{d}}\MM{l} \,, X \,\rangle
\\& =
-\,\int_S
\pi_kl_{k,j}X_j (\MM{x})
\,{\mathrm{d}}V(\MM{x})
\\& =
-\,\left\langle (\MM{l}, \MM{\pi}),
T\MM{l}\cdot X \right\rangle
\\& =
-\,\left\langle (\MM{l}, \MM{\pi}),
X_{\Omega}(\MM{l}) \right\rangle
\,,\end{aligned}$$ where $\left\langle \,\cdot\,, \,\cdot\,\right\rangle:\,T
^{\ast}_{\MM{l}}\Omega\times T_{\MM{l}}\Omega\mapsto\mathbb{R}$ is the $L^2$ pairing of an element of $T ^{\ast}_{\MM{l}}\Omega $ (a one-form density) with an element of $T_{\MM{l}}\Omega $ (a vector field).\
Consequently, the Clebsch map (\[momentum map\]) satisfies the defining relation (\[momentummapdef\]) to be a momentum map, $$\label{momentummap-JOmega}
\MM{J}(\MM{l}, \MM{\pi})
=
-\,\MM{\pi}
\cdot {\mathrm{d}}\MM{l}
\,,$$ with the $L^2$ pairing of the one-form density $-\,\MM{\pi}
\cdot {\mathrm{d}}\MM{l}$ with the vector field $X$.
Being the cotangent lift of the action of $\operatorname{Diff}
(\Omega)$, the momentum map $\mathbf{J}_\Omega$ in (\[rightmommap\]) is equivariant and Poisson. That is, substituting the canonical Poisson bracket into relation yields the Lie-Poisson bracket on the space of $\MM{m}$’s. See, for example, Holm & Kupershmidt (1983) and Marsden & Weinstein (1983) for more explanation, discussion and applications. The momentum map property of the Clebsch representation guarantees that the canonically conjugate variables $(\MM{l},\MM{\pi})$ may be eliminated in favour of the spatial momentum $\MM{m}$. Before its momentum map property was understood, the use of the Clebsch representation to eliminate the canonical variables in favour of Eulerian fluid variables was a tantalising mystery (Seliger & Whitham, 1968).
Note that the right action of $\operatorname{Diff}(\Omega)$ on the inverse map is not a symmetry. In fact, as we shall see, the right action of $\operatorname{Diff}(\Omega)$ o
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|
egrees of freedom, the naive choice for the supercharges would be $Q=-i\partial_\eta$ and $Q^\dagger=i\partial_{\eta^\dagger}$. However, a quick check then finds that all anticommutators $\left\{Q,Q,\right\}$, $\left\{Q^\dagger,Q^\dagger\right\}$ and $\left\{Q,Q^\dagger\right\}$ vanish, thus not reproducing the supersymmetry algebra in (\[eq:SUSYalgebra\]). Hence, in order that the anticommutator of $Q$ and $Q^\dagger$ also reproduces the Hamiltonian, it is necessary that while a translation is performed in $\eta$ and $\eta^\dagger$, a translation in $t$ be also included in an amount proportional to the Grassmann odd coordinates in superspace. It turns out that an appropriate choice is given by[^26] $$Q=-i\partial_\eta+\frac{2}{\omega}\eta^\dagger\,\partial_t\ \ \ ,\ \ \
Q^\dagger=i\partial_{\eta^\dagger}-\frac{2}{\omega}\eta\,\partial_t\ .$$ A direct calculation finds that these operators obey the supersymmetry algebra $$\left\{Q,Q\right\}=0=\left\{Q^\dagger,Q^\dagger\right\}\ \ ,\ \
\left\{Q,Q^\dagger\right\}=\left(-\frac{2}{\sqrt{\hbar\omega}}\right)^2\,
\left(i\hbar\partial_t\right)\ ,$$ in perfect correspondence with the abstract algebra in (\[eq:SUSYalgebra\]) (one should recall that a rescaling by a factor $(-\sqrt{\hbar\omega}/2)$ of the supersymmetry parameters $\epsilon$ and $\epsilon^\dagger$ or the supercharges $Q$ and $Q^\dagger$ has been applied in the intervening discussion).
In order to readily construct manifestly supersymmetric invariant Lagrange functions, it proves necessary to also use another pair of superspace differential operators, that anticommute with the supercharges, and define so-called superspace covariant derivatives. These supercovariant derivatives thus enable one to take derivatives of superfields in a manner consistent with supersymmetry transformations. Again, a convenient choice turns out to be $$D=\partial_\eta-\frac{2i}{\omega}\eta^\dagger\,\partial_t\ \ \ ,\ \ \
D^\dagger=-\partial_{\eta^\dagger}+\frac{2i}{\omega}\eta\,\partial_t\ ,$$ leading to the algebra $$\left\{D,
| 3,727
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|
tilde{\varphi}/\tilde{M}^1$ by using points of the scheme $(\underline{M}'\otimes\kappa)/ \underline{\pi M}'$, based on Lemma \[la3\]. To do that, we take the argument in pages 511-512 of [@C2].
Recall from two paragraphs before Lemma \[la3\] that $(1+)^{-1}(\underline{M}^{\ast})$, which is an open subscheme of $\underline{M}'$, is a group scheme with the operation $\star$. Let $\tilde{M}'$ be the special fiber of $(1+)^{-1}(\underline{M}^{\ast})$. Since $\tilde{M}^{1}$ is a closed normal subgroup of $\tilde{M} (=\underline{M}^{\ast}\otimes\kappa)$ (cf. Lemma \[la3\].(i)), $\underline{\pi M'}$, which is the inverse image of $\tilde{M}^{1}$ under the isomorphism $1+$, is a closed normal subgroup of $\tilde{M}'$. Therefore, the morphism $1+$ induces the following isomorphism of group schemes, which is also denoted by $1+$, $$1+ : \tilde{M}'/\underline{\pi M'}\longrightarrow \tilde{M}/\tilde{M}^{1}.$$ Note that $\tilde{M}'/\underline{\pi M'}(R)=\tilde{M}'(R)/\underline{\pi M'}(R)$ by Lemma \[la1\]. Thus each element of $(\mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1)(R)$ is uniquely written as $1+\bar{x}$, where $\bar{x}\in \tilde{M}'(R)/ \underline{\pi M}'(R)$. Here, by $1+\bar{x}$, we mean the image of $\bar{x}$ under the morphism $1+$ at the level of $R$-points.
We still need a better description of an element of $(\mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1)(R)$ by using a point of the scheme $(\underline{M}'\otimes\kappa)/ \underline{\pi M}'$. Note that $(\underline{M}'\otimes\kappa)/ \underline{\pi M}'$ is a quotient of group schemes with respect to the addition, whereas $\tilde{M}'/ \underline{\pi M}'$ is a quotient of group schemes with respect to the operation $\star$.
The open immersion $\iota : \tilde{M}' \rightarrow \underline{M}'\otimes\kappa, x\mapsto x$ induces a monomorphism of monoid schemes preserving the operation $\star$: $$\bar{\iota} : \tilde{M}'/ \underline{\pi M}' \rightarrow (\underline{M}'\otimes\kappa)/ \underline{\pi M}'.$$ Note that although $(\underline{M}'\otimes\kappa)/ \underline{\pi
| 3,728
| 2,970
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|
H \delta(g\cdot f)-\delta(f)d\mu(f)\\
&=\int_H \Big(\delta(g\cdot f\cdot h)-\delta(h)\Big)-\Big(\delta(f\cdot h)-\delta(h)\Big)d\mu(f)\\
&=\int_H \delta(g\cdot f\cdot h)-\delta(f\cdot h)d\mu(f\cdot h)=P(\delta(g)),\end{aligned}$$ which finishes the claim.
Let $X$ denote the range of $P$ and let us show that it is linearly isometric with ${\mathcal{F}}(G/H,D)$. We define a map $T':G/H\rightarrow {\mathcal{F}}(G)$ by setting for any left coset $gH$ $$T'(gH):=P'(g).$$ We check that it is correctly defined and $1$-Lipschitz. For the former, we need to check that for any $g\in G$ and $h\in H$ we have $P'(g)=P'(gh)$, i.e. $\int_H \delta(g\cdot f)d\mu(f)=\int_H \delta(gh\cdot f)d\mu(f)$. But the equality follows from the invariance of $\mu$. To check that $T'$ is $1$-Lipschitz, pick two cosets $g_1H$ and $g_2H$ and suppose that $f\in H$ is such that $D(g_1H,g_2H)=d(g_1,g_2f)$. Then we have $$\begin{aligned}
\|T'(g_1H)-T'(g_2H)\|&=\|\int_H \delta(g_1\cdot h)-\delta(g_2 f\cdot h)d\mu(h)\| \\
&=\int_H D(g_1H,g_2H)d\mu(f)=D(g_1H,g_2H),\end{aligned}$$ showing that $T'$ is actually isometric. It follows that $T'$ extends to a norm one linear surjection $T:{\mathcal{F}}(G/H)\rightarrow X$. In order to show that $T$ is isometric, it suffices to prove that for any finite linear combination $x=\sum_i \alpha_i \delta(g_i H)$ we have $\|x\|_{{\mathcal{F}}(G/H)}=\|T(x)\|_{{\mathcal{F}}(G)}$. One inequality already follows from the fact that $\|T\|=1$, so we only need to prove $\|x\|_{{\mathcal{F}}(G/H)}\leq\|T(x)\|_{{\mathcal{F}}(G)}$. Let $f\in\rm{Lip}_0(G/H)$ be a $1$-Lipschitz function satisfying $\|x\|_{{\mathcal{F}}(G/H)}=|\sum_i \alpha_i f(g_iH)|$. Let $\tilde f$ denote its lift to $G$. That is, for any $g\in G$ and $h\in H$, $\tilde f(gh)=f(gH)$. It is clear that $\tilde f$ is $1$-Lipschitz. In the following, we shall not notationally distinguish between $\rm{Lip}_0(G/H)$-functions and their unique extension to linear functionals.
Since we have $\|T(x)\|\geq \tilde f(T(x))$, it suffices to check that $\tilde f(T(x))=f(x)$.
| 3,729
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| null | null |
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|
igma}_n^2\partial^2_{xx} v_n&=&0, \ (t,x)\in (0,1]\times\mathbb{R},\\
v_n(0,x)&=& \varphi (x).\end{aligned}$$ As $\varphi$ vanishes at infinity, $$\mathbf{M}(R):=\mathop{\max_{|x|\ge R;}}_{1\ge t\ge 0}\big\{|u(t,x)|, |v_n(t,x)|: \ n\in\mathbb{N} \big\}$$ approaches zero as $R$ approaches $+\infty$. Also, we have $$\mathbf{m}(\epsilon):=\max_{(t,x)\in [0,\epsilon]\times\mathbb{R}}\big\{|u(t,x)-\varphi(x)|, |v_n(t,x)-\varphi(x)|: \ n\in\mathbb{N} \big\}$$ goes to zero as $\epsilon$ goes to $0$. Set $w_n=u-v_n$ and $\varepsilon_n=\widetilde{\sigma}_n^2-\widetilde{\sigma}_\varphi^2$. Then, $w_n$, which is nonnegative, satisfies $$\begin{aligned}
\partial_t w_n-\frac{1}{2}\widetilde{\sigma}_n^2\partial^2_{xx} w_n&=&\frac{1}{2}\varepsilon_n\partial_{xx}^2u, \ (t,x)\in (0,1]\times\mathbb{R},\\
w_n(0,x)&=&0.\end{aligned}$$ According to the Aleksandrov-Bakel’man-Pucci-Krylov maximum principle (see, for instance, Theorem 7.1 of [@Lie]), $$\sup\limits_{(t,x)\in (\epsilon,1]\times\mathbf{B}(R)}w_n\le 2\mathbf{M}(R)+2\mathbf{m}(\epsilon)+c_0 (\frac{R}{\underline{\sigma}})^{1/2}\|\varepsilon_n\partial_{xx}^2u\|_{L^2([\epsilon,1]\times \mathbf{B}(R))},$$ where $c_0$ is a universal constant. Note that, following the interior regularity of $G$-heat equation, $$\|\varepsilon_n\partial_{xx}^2u\|_{L^2([\epsilon,1]\times \mathbf{B}(R))}\le 2\overline{\sigma}\|\partial_{xx}^2u\|_{\infty;[\epsilon,1]\times\mathbb{R}}\|\sigma_n-\sigma_\varphi\|_{L^2([0,1]\times \mathbf{B}(R))}\rightarrow0$$ as $n$ approaches $+\infty$. Thus, $$\mathbf{O}(R,\epsilon):=\limsup_{n\rightarrow\infty}\Big(\sup\limits_{(t,x)\in (\epsilon,1]\times\mathbf{B}(R)}w_n\Big)\le 2(\mathbf{M}(R)+\mathbf{m}(\epsilon))$$ and $$\mathbf{O}(R,\epsilon)\le\lim_{R\rightarrow\infty, \epsilon\rightarrow0}\mathbf{O}(R,\epsilon)\le\lim_{R\rightarrow\infty, \epsilon\rightarrow0} 2(\mathbf{M}(R)+\mathbf{m}(\epsilon))\le0.$$ In particular, we have $$\mathcal{N}_G[\varphi]=u(0,1)=\lim_{n\rightarrow\infty}v_n(0,1)=\lim_{n\rightarrow\in
| 3,730
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|
} &=&-\frac{v_{x}}{u}AG(\phi )sin\phi +\varepsilon
\left[ u\left( \frac{d^{2}v_{z}}{d\phi ^{2}}-v_{z}Q\right) +\frac{v_{x}}{\gamma }\frac{dv_{x}}{d\phi }\frac{dv_{z}}{d\phi }-\left( 1-\frac{v_{z}}{\gamma }\right) \left( \frac{dv_{z}}{d\phi }\right) ^{2}\right] \label{52}\end{aligned}$$
From the above equations we derive the zeroth order for the derivatives of the components of the acceleration
$$\begin{aligned}
\left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{0} &=&-A\frac{d}{d\phi }[G(\phi )sin\phi ] \label{54} \\
\left( \frac{d^{2}v_{z}}{d\phi ^{2}}\right) _{0} &=&-A\frac{d}{d\phi }[\frac{v_{x}}{u}G(\phi )sin\phi ] \label{55}\end{aligned}$$
Substituting these relations back into $\left( \ref{50}\right) $ and $\left(
\ref{52}\right) $, we obtain the first order approximation for the components of the acceleration $\left( \frac{dv_{x}}{d\phi }\right)
_{1},\left( \frac{dv_{z}}{d\phi }\right) _{1}.$ On the other hand, the zeroth order of the second derivative of the components of the acceleration are given by
$$\begin{aligned}
\left( \frac{d^{3}v_{x}}{d\phi ^{3}}\right) _{0} &=&-A\frac{d^{2}}{d\phi ^{2}}[G(\phi )sin\phi ] \label{56} \\
\left( \frac{d^{3}v_{z}}{d\phi ^{3}}\right) _{0} &=&-A\frac{d^{2}}{d\phi ^{2}}[\frac{v_{x}}{u}AG(\phi )sin\phi ] \label{58}\end{aligned}$$
Taking now the derivatives of $\left( \ref{50}\right) $ and $\left( \ref{52}\right) $ we find the rather lengthy relations
$$\begin{aligned}
\left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{1} &=&\left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{0}+\varepsilon \{u\left[ \left( \frac{d^{3}v_{x}}{d\phi
^{3}}\right) _{0}-v_{x}Q^{\prime }-\left( \frac{dv_{x}}{d\phi }\right)
_{1}Q\right] +u^{\prime }\left[ \left( \frac{d^{2}v_{x}}{d\phi ^{2}}\right)
_{0}-v_{x}Q\right] + \nonumber \\
&&+2\frac{v_{x}}{\gamma }\left( \frac{dv_{x}}{d\phi }\right) _{1}\left(
\frac{d^{2}v_{x}}{d\phi ^{2}}\right) _{0}+\frac{1}{\gamma ^{2}}\left[ \gamma
\left( \frac{dv_{x}}{d\phi }\right) _{1}-v_{x}\gamma ^{\prime }\right]
\left( \frac{dv_{x}}{d\phi }\right) _{1}^{2}-\lef
| 3,731
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| null | null |
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|
od}}$;
- the associated graded ${\mathbb{Z}}$-algebra $\operatorname{gr}B$ has $\operatorname{gr}B{\text{-}{\textsf}{qgr}}\simeq \operatorname{{\textsf}{Coh} }\operatorname{Hilb(n)}$, the category of coherent sheaves on the Hilbert scheme of points in the plane.
This can be regarded as saying that $U_c$ simultaneously gives a noncommutative deformation of ${\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}}$ and of its resolution of singularities $\operatorname{Hilb(n)}\to {\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}}$. As the companion paper [@GS2] shows, this result is a powerful tool for studying the representation theory of $H_c$ and its relationship to $\operatorname{Hilb(n)}$.
address:
- 'Department of Mathematics, Glasgow University, Glasgow G12 8QW, Scotland'
- 'Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA.'
author:
- 'I. Gordon'
- 'J. T. Stafford'
title: Rational Cherednik algebras and Hilbert schemes
---
[^1]
Introduction
============
{#sec101}
This is the first of two closely related papers on rational Cherednik algebras.
In their short history, Cherednik algebras have been influential in a surprising range of subjects: for example they have been used to answer questions in integrable systems, combinatorics, and symplectic quotient singularities (see [@BEGqi; @gordc; @BFG; @GK]). In this paper we strengthen the connections between Cherednik algebras and geometry by showing that they can be regarded as noncommutative deformations of Hilbert schemes of points in the plane. In the sequel [@GS2] this will be used to show the close relationship between modules over the Cherednik algebra and sheaves on the Hilbert scheme as well as to answer various open problems about these modules.
{#intro-1.2}
Fix $c\in {\mathbb{C}}$. We assume throughout the paper that $c\notin \frac{1}{2} + {\mathbb{Z}}$ and, for simplicity, we will also assume that $c\not\in \mathbb{R}_{\leq 0}$ in this introduction, see and for the more general case.
Let $H_c= H_{1,c}$ be the rational Cher
| 3,732
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| null | null |
github_plus_top10pct_by_avg
|
or which we can have singularities (irregularities) that integrate to a finite number . In other words, we think about energy correlators as if they are regular functions up to a set of isolated points $\{ \xi^*_i \}$ where they are defined only in the functional sense.
Even though the terms like $\delta(1-\xi)$ do appear in free theories and in perturbation theory, available strong coupling data as well as resummed perturbation theory analysis in QCD show that delta functions disappear when all orders are included. So, it is reasonable to assume that in interacting CFTs energy correlators are finite when detectors are at non-coincident points. This is an assumption of [*finiteness*]{}. We will not use it unless stated otherwise.
There is good evidence that energy correlators in unitary CFTs are non-negative for any $\Psi$. This condition looks physically reasonable and holds in all known examples, though we are not aware of its general proof.
In light of the previous section we need to specify what do we exactly mean by . Indeed, for the configuration of detectors such that the energy correlator is finite or under assumption of finiteness we can use ; however, in the case when the energy correlator is not regular we need to impose the positivity of the integrated energy correlator with $g(\delta \Omega) \geq 0$ .
In it was shown that the positivity of the one-point function implies the following inequalities for $n_{b,f,v}$
A weaker version of this constraint was quoted in the introduction , where the value ${a \over c} = {1 \over 3}$ corresponds to $n_b \neq 0$,$ n_f = n_v =0$; the case ${a \over c} = {31 \over 18}$, on the other hand, is given by $n_v \neq 0$,$ n_b = n_f =0$. One can ask if it is possible to derive more constraints from the positivity of higher point energy correlators?
Let us consider, for example, the two-point energy correlator. We can fix the position of one detector and vary the position of the second detector and the polarization tensor such that the cross ratio $\xi$ is fixed
| 3,733
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| null | null |
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|
gned}$$
Similarly, we observe that from Lemmas \[lem:diff1\] and \[lem:diff3\] $$\begin{aligned}
\tr[\Dc_\La\{f_\pi(\La;W)\La\}M]
&=\tr[\La M\Dc_\La f_\pi(\La;W)]+f_\pi(\La;W)\tr[M\Dc_\La\La]\\
&=\frac{1}{2}\Big[(q+r+1)\tr M+\frac{2}{\pi_2^J(\La)}\tr[\La M\Dc_\La \pi_2^J(\La)] \\
&\qquad -\frac{1}{v}\tr(MWW^\top\La)\Big]f_\pi(\La;W),\end{aligned}$$ which leads to $$\begin{aligned}
\label{eqn:E2}
E_2(W)
&=(q+r+1)m(W)\tr M +m(W)\De_2(W;\pi_2^J) \non\\
&\qquad -2\int_{\Rc_r}\tr[\Dc_\La\{f_\pi(\La;W)\La\}M]\dd\La .\end{aligned}$$ Combining (\[eqn:d2\_mw1\]), (\[eqn:E1\]) and (\[eqn:E2\]) gives that $$\begin{aligned}
\label{eqn:d2_mw2}
\De
&= \frac{\De(W;\pi_2^J)}{v} -\frac{4}{v}\int_{\Rc_r}\tr[\Dc_\La\{f_\pi(\La;W)\La^2\}]\dd\La \non\\
&\qquad +\frac{2}{vm(W)}\int_{\Rc_r}\tr[\Dc_\La\{f_\pi(\La;W)\La\}M]\dd\La.\end{aligned}$$ If we can show that two integrals in (\[eqn:d2\_mw2\]) are, respectively, equal to zero, then the proof is complete.
Let $G=(g_{ij})$ be an $r\times r$ symmetric matrix such that all the elements of $G$ are differentiable functions of $\La\in\Rc_r$. Denote $$\vec(G)=(g_{11},g_{12},\ldots,g_{1r},g_{22},g_{23},\ldots,g_{r-1,r-1},g_{r-1,r},g_{rr})^\top,$$ which is a $\{2^{-1}r(r+1)\}$-dimensional column vector. Denote an outward unit normal vector at a point $\La$ on $\partial\Rc_r$ by $$\nu=\nu(\La)=(\nu_{11},\nu_{12},\ldots,\nu_{1r},\nu_{22},\nu_{23},\ldots,\nu_{r-1,r-1},\nu_{r-1,r},\nu_{rr})^\top.$$ If $\tr(\Dc_\La G)$ is integrable on $\Rc_r$ then it is seen that $$\int_{\Rc_r}\tr(\Dc_\La G)\dd\La
=\int_{\Rc_r}\sum_{i=1}^r\sum_{j=1}^r\frac{1+\de_{ij}}{2}\frac{\partial g_{ji}}{\partial\la_{ij}}\dd\La
=\int_{\Rc_r}\sum_{i=1}^r\sum_{j=i}^r\frac{\partial g_{ij}}{\partial\la_{ij}}\dd\La$$ by symmetry of $\La$ and $G$. From the Gauss divergence theorem, we obtain $$\int_{\Rc_r}\tr(\Dc_\La G)\dd\La=\int_{\partial\Rc_r}\sum_{i=1}^r\sum_{j=i}^r\nu_{ij} g_{ij}\dd\si=\int_{\partial\Rc_r}\nu^\top\vec(G)\dd\si,$$ where $\si$ stands for Lebesgue measure on $\partial\Rc_r$.
Note that $$\begin{aligned}
\tr
| 3,734
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|
n $M_0''$ is *free of type II* and so we have a morphism from $G_{j-1}$ to the even orthogonal group associated to $M_0^{\prime\prime}$. Here, $G_{j-1}$ is the special fiber of the smooth integral model associated to $Y(C(L^{j-1}))$. Thus, the Dickson invariant of this orthogonal group induces the morphism $$\psi_j : \tilde{G} \longrightarrow \mathbb{Z}/2\mathbb{Z}.$$
2. If $M_0$ is *of type II*, then we follow the argument (1) with $j-1$. Namely, if we consider the lattice $Y(C(L^{j-1}))=\bigoplus_{i \geq 0} M_i^{\prime\prime}$, then it is easy to show that $M_0''$ is *free of type II* by using the similar argument used in Step (1). As in the above case, we have a morphism from $G_{j-1}$ to the even orthogonal group associated to $M_0^{\prime\prime}$. Here, $G_{j-1}$ is the special fiber of the smooth integral model associated to $Y(C(L^{j-1}))$. Thus, the Dickson invariant of this orthogonal group induces the morphism $$\psi_j : \tilde{G} \longrightarrow \mathbb{Z}/2\mathbb{Z}.$$
3. If $M_0$ is *of type $I^o$*, then we follow the argument (2) with $j-1$. We briefly summarize it below. As in the above case, since $M_0$ is *of type $I$* and $n(M_1)=(2)$, we choose another basis for $M_0\oplus M_1$ whose associate Jordan splitting is $M_0'\oplus M_1'$ with $n(M_1')=(4)$. Consider two lattices $C(L^{j-1})=\bigoplus_{i \geq 0} M_i^{\prime}$ and $M_0^{\prime}\oplus C(L^{j-1})$. Here the rank of the $\pi^0$-modular lattice $M_0^{\prime}$ is 1. Then we can assign the even orthogonal group to the $\pi^0$-modular Jordan component of $Y\left(C(M_0^{\prime}\oplus C(L^{j-1}))\right)$. Thus, the Dickson invariant of this orthogonal group induces the morphism $$\psi_j : \tilde{G} \longrightarrow \mathbb{Z}/2\mathbb{Z}.$$
\(4) Combining all cases, we have the morphism $$\psi=\prod_j \psi_j : \tilde{G} \longrightarrow (\mathbb{Z}/2\mathbb{Z})^{\beta},$$ where $\beta$ is the number of integers $j$ such that $L_j$ is *of type I* and $L_{j+2}, L_{j+3}, L_{j+4}$ (resp. $L_{j-1},$ $L_{j+1}, L_{j+2}, L_{j+3}$) are *of type II* (
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from the bracket equation for the quantum-classical density matrix (\[eq:rhoW\]), by dealing in a suitable manner with the Poisson bracket terms, the most simple way to find the representation of the wave equations (\[eq:fckrk\]) in the adiabatic basis is to first represent Eq. (\[eq:rhoW\]) in such a basis and then deal with the terms arising from the Poisson brackets. The adiabatic representation of Eq. (\[eq:rhoW\]) is [@kcmqc] $$\begin{aligned}
\partial_t \rho_{\alpha\alpha^{\prime}}(X,t)
&=&-\sum_{\beta\beta^{\prime}}
i{\cal L}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}
\rho_{\beta\beta^{\prime}}(X,t) \;,\end{aligned}$$ where $$\begin{aligned}
i{\cal L}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}
&=&
i{\cal L}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}^{(0)}
\delta_{\alpha\beta}\delta_{\alpha^{\prime}\beta^{\prime}}
-J_{\alpha\alpha^{\prime},\beta\beta^{\prime}}\nonumber\\
&=&\left(i\omega_{\alpha\alpha^{\prime}}
+iL_{\alpha\alpha^{\prime}}\right)
\delta_{\alpha\beta}\delta_{\alpha^{\prime}\beta^{\prime}}
-J_{\alpha\alpha^{\prime},\beta\beta^{\prime}}\;.
\label{eq:qc-l}\end{aligned}$$ Here, $\omega_{\alpha\alpha^{\prime}}=
\left(E_{\alpha}(R)-E_{\alpha^{\prime}}(R)\right)/\hbar
\equiv E_{\alpha\alpha^{\prime}}/\hbar$ and $$iL_{\alpha\alpha^{\prime}}
=\frac{P}{M}\cdot\frac{\partial}{\partial R}
+\frac{1}{2}\left(F_{\alpha}+F_{\alpha^{\prime}}\right)
\frac{\partial}{\partial P}\;,
\label{eq:ilad}$$ where $$F_{\alpha}=
-\langle\alpha;R\vert\frac{\partial\hat{h}(R)}{\partial R}
\vert\alpha;R\rangle$$ is the Hellmann-Feynman force for state $\alpha$. The operator $J$ that describes nonadiabatic effects is $$\begin{aligned}
J_{\alpha\alpha^{\prime},\beta\beta^{\prime}}
&=&-\frac{P}{M}\cdot d_{\alpha\beta}
\left(1+\frac{1}{2}S_{\alpha\beta}\cdot
\frac{\partial}{\partial P}\right)\delta_{\alpha^{\prime}\beta^{\prime}}
\nonumber\\
&&-\frac{P}{M}\cdot d_{\alpha^{\prime}\beta^{\prime}}^*
\left(1+\frac{1}{2}S_{\alpha^{\prime}\beta^{\prime}}^*\cdot
\frac{\partial}{\partial P}\right)\delta_{\alpha\beta}
\;,\nonumbe
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nfalling gas remains $\sim10^2$, and while the cold-mode temperature remains $\sim 10^4~$K, the temperature of infalling gas increases with halo mass. Both of these differences can be explained by noting that, around the virial radius, hot-mode gas accounts for a greater fraction of the infall in higher mass haloes (see the bottom-middle panel of Fig. \[fig:halomassz2\]).
As was the case for the cold-mode gas, the radial peculiar velocity of infalling gas scales like the escape velocity (bottom-middle panel). Interestingly, although the mass flux-weighted median outflowing velocity is almost independent of halo mass, the high-velocity tail is much more prominent for low-mass haloes. Because the potential wells in these haloes are shallow and because the gas pressure is lower, the outflows are not slowed down as much before they reach the virial radius. The flux-weighted outflow velocities are larger than the inflow velocities for $M_\mathrm{halo}<10^{11.5}$ M$_\odot$, whereas the opposite is the case for higher-mass haloes.
Finally, the last panel of Fig. \[fig:halomassflux\] shows that the fraction of the gas that is outflowing around $R_{\rm vir}$ is relatively stable at about 30–40 per cent. Although the accretion rate is negative for $10^{10}$ M$_\odot<M_\mathrm{halo}<10^{11}$ M$_\odot$, which indicates net outflow, less than half of the gas is outflowing.
Effect of metal-line cooling and outflows driven by supernovae and AGN {#sec:SNAGN}
======================================================================
Fig. \[fig:halodiff\] shows images of the same $10^{12}$ M$_\odot$ halo as Fig. \[fig:halo\] for five different high-resolution (*L025N512*) simulations at $z=2$. Each row shows a different property, in the same order as the panels in the previous figures. Different columns show different simulations, with the strength of galactic winds increasing from left to r
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conformal Field Theories from Principal 3-Bundles over Twistor Space,” arXiv:1305.4870 \[hep-th\]. S. Palmer and C. Sämann, “Six-Dimensional (1,0) Superconformal Models and Higher Gauge Theory,” J. Math. Phys. [**54**]{}, 113509 (2013) \[arXiv:1308.2622 \[hep-th\]\]. S. Palmer, “Higher Gauge Theory and M-Theory,” arXiv:1407.0298 \[hep-th\]. B. Jurco, C. Saemann and M. Wolf, “Semistrict Higher Gauge Theory,” arXiv:1403.7185 \[hep-th\].
[^1]: e-mail address: pmho@phys.ntu.edu.tw
[^2]: Here $p\th p'$ stands for $p_{\m}\th^{\m\n}p'_{\n}$.
[^3]: More explicitly [@Zhu:1980sz], n\_s\^[(4)]{} &=& \_1\^\_2\^\_3\^\_4\^ n\^[(3)]{}\_(-p\_4, -p\_3, p\_3+p\_4)n\^[(3)]{}\_\^(p\_1,p\_2,-p\_1-p\_2) + m\_s\^[(4)]{}, where $p_i^{\m}, \eps_i^{\m}$ ($i = 1, 2, 3, 4$) are the momenta and polarization vectors of the external legs, $s = (p_1+p_2)^2$ and m\_s\^[(4)]{} s\[(\_1\_4)(\_2\_3)-(\_1\_3)(\_2\_4)\]. (The choice of $m_s^{(4)}$ is not unique.) The other two kinematic factors $n^{(4)}_t, n^{(4)}_u$ can be obtained by permutations of external legs.
[^4]: As we have learned from pure GR, the trace part of the fluctuation of the metric is not a physical propagating mode. Hence we should identify the trace part of $\hat{A}_{ab}$ (and $A_{ab}$) as an independent scalar field.
[^5]: The mathematical structure for the symmetry of a 2-form gauge potential is called a non-Abelian gerbe. But there are different versions of its definition.
---
abstract: 'We experimentally study a vacuum-induced Autler-Townes doublet in a superconducting three-level artificial atom strongly coupled to a coplanar waveguide resonator and simultaneously to a transmission line. The Autler-Townes splitting is observed in the reflection spectrum from the three-level atom in a transition between the ground state and the second excited state when the transition between the two excited states is resonant with a resonator. By applying a driving field to the resonator, we observe a change in the regime of the Autler-Townes splitting from qu
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}{{\mathbb S}}={\varnothing},~{{\mathbb S}}_{{\cal A}}={\varnothing}\}.\end{aligned}$$ We note that $|{\mathfrak{S}}|$ is zero when there are no paths on ${{\mathbb G}}_{{\bf N}}$ between $v$ and $x$ consisting of edges whose endvertices are both in ${{\cal A}}{^{\rm c}}$, while $|{\mathfrak{S}}'|$ may not be zero. The identity [(\[eq:switching\])]{} reads that $|{\mathfrak{S}}|$ equals $|{\mathfrak{S}}'|$ if we compensate for this discrepancy.
Suppose that there is a path (i.e., a ) $\omega$ from $v$ to $x$ consisting of edges in ${{\mathbb G}}_{{\bf N}}$ whose endvertices are both in ${{\cal A}}{^{\rm c}}$. Then, the map $$\begin{aligned}
{\label{eq:bijection}}
{{\mathbb S}}\in{\mathfrak{S}}~\mapsto~{{\mathbb S}}{\,\triangle\,}\omega\in{\mathfrak{S}}',\end{aligned}$$ is a bijection [@a82; @ghs70], and therefore $|{\mathfrak{S}}|=|{\mathfrak{S}}'|$. Here and in the rest of the paper, the symmetric difference between graphs is only in terms of *edges*. For example, ${{\mathbb S}}{\,\triangle\,}\omega$ is the result of adding or deleting edges (not vertices) contained in $\omega$. This completes the proof of [(\[eq:switching\])]{}.
We now start with the second stage of the expansion by using Proposition \[prp:through\] and applying inclusion-exclusion as in the first stage of the expansion in Section \[sss:1stexp\]. First, we decompose the indicator in [(\[eq:lmm-through\])]{} into two parts depending on whether or not there is a pivotal bond $b$ for $v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x$ from $v$ such that $v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}{\underline{b}}$. Let $$\begin{aligned}
{\label{eq:E-def}}
E_{{\bf N}}(v,x;{{\cal A}})=\{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\cap\{\nexists
\text{ pivotal bond }b\text{ for }v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{}{\l
| 3,739
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| 2,910
| 0.776148
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$ for all $t\in [0,\ol{\tau}]$.
We write this solution as $\gamma(x,\omega,E,t)$, and write $\ol{\tau}(x,\omega,E)$ for the end-point of its maximal interval of existence. Because $S_0\geq \kappa>0$ on $G\times I$ (by ), we have $$\begin{aligned}
\label{eq:inequality_gamma}
\gamma(x,\omega,E,t')\geq \gamma(x,\omega,E,t)+\kappa(t'-t),\end{aligned}$$ whenever $t\leq t'$, and therefore one can see that (i) or (ii) (or both) below holds: $$\begin{aligned}
\label{eq:alpha_endpoint}
\textrm{(i)}\ \gamma\big(x,\omega,E,\ol{\tau}(x,\omega,E)\big)=E_m,
\quad\textrm{or}\quad \textrm{(ii)}\ \ol{\tau}(x,\omega,E)=t(x,\omega).\end{aligned}$$
Denote for $(x,\omega,E,t)\in G\times S\times I\times [0,\ol{\tau}(x,\omega,E)]$, $$\Gamma(x,\omega,E,t):=(x-t\omega, \omega, \gamma(x,\omega,E,t)),$$ and notice that $\Gamma(x,\omega,E,0)=(x,\omega,E)$. It follows from that $$\begin{aligned}
\label{eq:T_boundary}
\beta(x,\omega,E):=\Gamma(x,\omega,E,\ol{\tau}(x,\omega,E))\in \Gamma_-\cup (G\times S\times \{E_m\}),\end{aligned}$$ for all $(x,\omega,E)\in G\times S\times I$.
Below we shall understand that $S_0(\Gamma(x,\omega,E,t))$ means $S_0(x-t\omega, \gamma(x,\omega,E,t))$, and similarly for ${\frac{\partial S_0}{\partial E}}$, since $S_0(x,E)$ is assumed not to depend on $\omega$.
Finally, if $\varphi:G\times S\times I\to{\mathbb{R}}$ is smooth enough (say $C^1$), then $${{{\partial}\over{\partial t}}} \Big(\varphi\big(\Gamma(x,\omega,E,t)\big)\Big)
=
S_0\big(\Gamma(x,\omega,E,t)\big){\frac{\partial \varphi}{\partial E}}\big(\Gamma(x,\omega,E,t)\big)-(\omega\cdot\nabla_x\varphi)\big(\Gamma(x,\omega,E,t)\big),$$ from which one can deduce that the solution $u$ of the problem is given by $$\begin{aligned}
\label{eq:general_explicit_u}
u(x,\omega,E)=h(x,\omega,E)e^{-\int_0^{\ol{\tau}(x,\omega,E)}W(\Gamma(x,\omega,E,t))dt},\end{aligned}$$ where $$W(x,\omega,E):=-{\frac{\partial S_0}{\partial E}}(x,E)+CS_0(x,E)+\Sigma(x,\omega,E),$$ and (recall ) $$h(x,\omega,E)
:=\begin{cases}
0, & {\rm if}\ \beta(x,\omega,E)\in G\times S\times \{E_m\}
| 3,740
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| 2,508
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ete.
Shapovalov determinants for bicharacters with finite root systems {#sec:shapdetgen}
=================================================================
In Sect. \[sec:shapdet\] we mainly considered bicharacters $\chi \in {\mathcal{X}}_5$. Here we extend our results to all $\chi \in {\mathcal{X}}_3$ with $\chi (\beta ,\beta )\not=1$ for all $\beta \in R^\chi _+$.
In what follows let ${\overline{{\mathcal{X}}}}$ denote the set of ${{\bar{{\Bbbk }}}^\times }$-valued bicharacters on ${\mathbb{Z}}^I$. Identify ${\overline{{\mathcal{X}}}}$ with $({{\bar{{\Bbbk }}}^\times })^{I\times I}$ via $\chi \mapsto (\chi ({\alpha }_i,{\alpha }_j) )_{i,j\in I}$ for all $\chi \in {\overline{{\mathcal{X}}}}$. For all $i\in \{1,2,3,4,5\}$ define ${\overline{{\mathcal{X}}}}_i\subset {\overline{{\mathcal{X}}}}$ in analogy to Eqs. –. Note that ${\mathcal{X}}_i={\mathcal{X}}\cap {\overline{{\mathcal{X}}}}_i$ for all $i\in \{1,2,3,4,5\}$.
For all $\beta ,\beta '\in {\mathbb{Z}}^I$ let $f_{\beta ,\beta '}$ be the rational function on the affine variety ${\overline{{\mathcal{X}}}}=({{\bar{{\Bbbk }}}^\times })^{I\times I}$ such that $$f_{\beta ,\beta '}(\chi )=\chi (\beta ,\beta ') \quad
\text{for all $\chi \in {\overline{{\mathcal{X}}}}$.}$$ Clearly, the functions $f_{\beta ,\beta '}$ with $\beta ,\beta '\in \{\al
_i,-{\alpha }_i\,|\,i\in I\}$ generate the algebra ${\bar{{\Bbbk }}}[{\overline{{\mathcal{X}}}}]$. Recall that a subset of ${\overline{{\mathcal{X}}}}$ is locally closed, if it is the intersection of an open and a closed subset of ${\overline{{\mathcal{X}}}}$.
\[pr:Vchi\] Let $\chi \in {\overline{{\mathcal{X}}}}_3$. Assume that $\chi (\beta ,\beta )\not=1$ for all $\beta \in R^\chi _+$. Let $\underline{n}=(n_\beta )_{\beta \in R^\chi _{+\infty }}$ with $n_\beta \in {\mathbb{N}}$ for all $\beta \in R^\chi _{+\infty }$. Then there exists an ideal $J\subsetneq {\bar{{\Bbbk }}}[{\overline{{\mathcal{X}}}}]$ generated by products of polynomials of the form $$q-\prod _{i,j\in I}f_{{\alpha }_i,{\alpha }_j}^{m_{ij}},\quad
\te
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e interval. An example would be : *Usain Bolt won the gold medal at the 2008 summer Olympics in Bejing*. With the availability of annotators that can provide us with accurate semantic annotations in form of named entities, geographic locations, and temporal expressions; we can leverage the growing number of knowledge resources such as Wikipedia [@wiki] and ontologies such as Freebase [@freebase] to understand natural language text and mine important events. Formally the central hypothesis can be stated as follows:
As a toy example consider the following text snippet [^1] with demonstrative semantic annotations in Figure \[fig:text\] :
In the text snippet (Figure \[fig:text\]), we obtain the named entity whose mention has been identified and disambiguated to point to an external knowledge source. Also identified is a geographical location - , which is disambiguated and resolved to its geographical coordinates. Likewise the temporal expression has also been resolved to time range. Having these semantic annotations we can now devise algorithms that can deduce that the event is that of Usain Bolt winning Olympic competition in Beijing, China.
The goal of the proposed research is to leverage the semantic annotations for mining important events and use them to navigate text corpora. The research will find application in many domains of research such as *digital humanities*, in which social scientists are interested in computational history in large digital-born text collections. Anthropologists are interested in cultural and linguistic shifts that occur in such collections. Collectively we can allow *computational culturomics* [@culturomics] on corpora to study cultural trends. Events can also be used to link information in multiple and diverse text collections. In short, important events provide a way to create a gist from semantically annotated corpora, which otherwise is not possible through manual human effort.
**Outline**. The article consists of:
- a literature survey (Section \[sec:background\]);
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ee effective interactions have been tested and compared in RHB plus proton-neutron relativistic QRPA calculations of $\beta$-decay half-lives for the isotopic chains: Fe, Ni, Zn, Cd, Sn and Te. The nuclear ground-states have been calculated in the RHB model with the DD-ME1, D$^{3}$C, and D$^3$C effective interactions in the particle-hole channel, and the pairing part of the Gogny force, $$V^{pp}(1,2)~=~\sum_{i=1,2}e^{-((\mathbf{r}_{1}-\mathbf{r}_{2})/{\mu _{i}})^{2}}\,(W_{i}~+~B_{i}P^{\sigma }-H_{i}P^{\tau }-M_{i}P^{\sigma }P^{\tau })
\label{Gogny}$$ in the particle-particle channel, with the set D1S [@BGG.91] for the parameters $\mu _{i}$, $W_{i}$, $B_{i} $, $H_{i}$ and $M_{i}$ $(i=1,2)$. This force has been very carefully adjusted to pairing properties of finite nuclei all over the periodic table. In particular, the basic advantage of the Gogny force is the finite range, which automatically guarantees a proper cut-off in momentum space. In the following calculations we have also used the Gogny interaction in the $T=1$ $pp$-channel of the PN-RQRPA.
The RHB ground-state solution determines the single-nucleon canonical basis, i.e. the configuration space in which the matrix equations of the relativistic QRPA are expressed (see Refs. [@Paa.03; @Paa.04] for a detailed presentation of the formalism). The particle-hole residual interaction of the PN-RQRPA is derived from the following Lagrangian density: $$\mathcal{L}_{\pi + \rho}^{int} =
- g_\rho \bar{\psi}\gamma^{\mu}\vec{\rho}_\mu \vec{\tau} \psi
- \frac{f_\pi}{m_\pi}\bar{\psi}\gamma_5\gamma^{\mu}\partial_{\mu}
\vec{\pi}\vec{\tau} \psi \; .
\label{lagrres}$$ The coupling between the $\rho$-meson and the nucleon is already contained in the RHB effective Lagrangian, and the same interaction is consistently used in the isovector channel of the QRPA. The direct one-pion contribution to the ground-state RHB solution vanishes because of parity-conservation, but it must be included in the calculation of the Gamow-Teller strength. For the pseudovec
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hbar \omega_{ge}\sigma _{ee} \\ \nonumber
&&+\hbar g_{0}\left( a^{\dag}\sigma _{ef}+a\sigma _{fe}\right) + 2\hbar \Omega(\sigma_{fg}+\sigma_{gf}) \cos\omega_p t.\end{aligned}$$ Here, the atomic operator $\sigma_{jk}$ is defined as $\sigma_{jk}=|j\rangle\langle k|$ with $\{|j\rangle,|k\rangle\}=\{|g\rangle,|e\rangle, |f\rangle\}$ and $a^\dag$ and $a$ are the photon creation and annihilation operators in the single-mode resonator, respectively. The first four terms of Eq. (1) represent the Jaynes-Cummings Hamiltonian for the two-level system with the following specific features: it presents the interaction of two excited states ($|e\rangle$ and $|f\rangle$) with the resonator rather than the ground and excited states. Nevertheless, the Jaynes-Cummings physics can be applied here [@JC1963; @Haroche2006; @Rempe1987]. Namely, in the vicinity of the $|e\rangle \leftrightarrow |f\rangle$ resonance with the resonator, $|f\rangle$ level splits due to the interaction with the resonator vacuum mode, which can be described in terms of zero-photon dressed states as $|v_{0}\rangle=\cos\theta|f0\rangle-\sin\theta|e1\rangle$ and $|u_{0}\rangle=\sin\theta|f0\rangle+\cos\theta|e1\rangle$, where $\tan{2\theta} = 2g_{0}/(\omega_{r}-\omega_{f}+\omega_{e})$ and the state $|jn\rangle=|j\rangle\otimes|n\rangle$ with the two quantum numbers $j$ and $n$, denoting the three-level atomic states $|j\rangle$ and the Fock states of resonator $|n\rangle=\lbrace|0\rangle,|1\rangle,|2\rangle,...\rbrace$, respectively. The dressed states are schematically shown in the left panel of Fig. \[picture\](d), while the right panel shows the general case of a system with $n$ photons in the resonator. The last term represents the probing field which couples the ground and the second excited states ($|g\rangle \leftrightarrow |f\rangle$, where the vacuum mode $|0\rangle$ is omitted for simplification).
![(a) Spectrum of the artificial atom vs biased flux plotted as a reflection coefficient $|r|$. The transition $|e\rangle \leftrightarrow |f\rangle$ is v
| 3,744
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=\bigoplus \operatorname{{\textsf}{ogr}}^n H_c$, where $\operatorname{{\textsf}{ogr}}^n H_c = \operatorname{{\textsf}{ord}}^n H_c/\operatorname{{\textsf}{ord}}^{n-1}H_c$. Then is equivalent to the assertion that $\operatorname{{\textsf}{ogr}}H_c$ is isomorphic to the skew group ring ${\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]\ast {{W}}$ defined by $\sigma f = \sigma(f) \sigma$, for $\sigma\in {{W}}$ and $f\in {\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]$.
The Dunkl-Cherednik representation {#dunch}
----------------------------------
Let $\delta \in {\mathbb{C}}[{\mathfrak{h}}]$\[delta-defn\] denote the discriminant polynomial $\delta = \prod_{s\in \mathcal{S}} \alpha_s$. Thus $\delta$ transforms under ${{W}}$ by the sign representation and ${\mathfrak{h}^{\text{reg}}}={\mathfrak{h}}\setminus \{\delta=0\}$ is the subset of ${\mathfrak{h}}$ on which the action of ${{W}}$ is free. By [@EG Proposition 4.5] there is an injective algebra morphism $\theta_c : H_c \to D ({\mathfrak{h}^{\text{reg}}}) \ast {{W}},
$\[theta-defn\] where $D(Z)$ denotes the [*ring of differential operators*]{} on an affine variety $Z$. Under $\theta_c$ the elements of ${\mathbb{C}}[{\mathfrak{h}}]$ are identified with the multiplication operators while, by [@EG p.280] and in the notation of , $y_i\in{\mathfrak{h}}$ is sent to the [*Dunkl operator* ]{} $$\label{dunkop}\theta_c (y_i) =
\partial_{i} - \sum_{s\in S} c\alpha_s(y_i)\alpha_s^{-1}(1-s),
\qquad \text{where}\ \partial_{i}= \partial/\partial x_i.$$
Since $\delta$ acts ad-nilpotently on $D({\mathfrak{h}^{\text{reg}}})\ast{{W}}$, the set $\{\delta^n\}$ forms an Ore set in that ring. As observed in [@BEGqi p.288]), $\theta_c$ becomes an isomorphism on inverting $\delta$; that is, $$\label{locdunk} {H_c^{\text{reg}}} = H_c[\delta^{-1}]
\cong D({\mathfrak{h}^{\text{reg}}})\ast {{W}}.$$
For any variety $Z$, there is a natural filtration on $D(Z)$ by order of operators and this induces a filtration on $D({\mathfrak{h}^{\text{reg}}})\ast {{W}}$ and its subalgebras by defini
| 3,745
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=0}^{\infty}f^{-i}(\textrm{Per}(f))$ (where $\textrm{Per}(f)$ denotes the set of periodic points of $f$) are totally disconnected, it is expected that at any point on this complement the behaviour of the limit will be similar to that on $\mathcal{D}_{+}$: these points are special as they tie up the iterates on $\textrm{Per}(f)$ to yield the multifunctions. Therefore in any neighbourhood $U$ of a $\mathcal{D}_{+}$-point, there is an $x_{0}$ at which the *forward orbit $\{ f^{i}(x_{0})\}_{i\geq0}$ is chaotic* in the sense that
\(a) the sequence neither diverges nor does it converge in the image space of $f$ to a periodic orbit of any period, and
\(b) the Liapunov exponent given by
$$\begin{aligned}
\lambda(x_{0}) & = & \lim_{n\rightarrow\infty}\ln\left|\frac{df^{n}(x_{0})}{dx}\right|^{1/n}\\
& = & {\displaystyle \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}\ln\left|\frac{df(x_{i})}{dx}\right|,\, x_{i}=f^{i}(x_{0}),}\end{aligned}$$
which is a measure of the average slope of an orbit at $x_{0}$ or equivalently of the average loss of information of the position of a point after one iteration, is positive. Thus *an orbit with positive Liapunov exponent is chaotic if it is not* *asymptotic* (that is neither convergent nor adherent, having no convergent suborbit in the sense of Appendix A1) *to an unstable periodic orbit* *or to any other limit set on which the dynamics is simple.* A basic example of a chaotic orbit is that of an irrational in $[0,1]$ under the shift map and that of the chaotic set its closure, the full unit interval.
Let $f\in\textrm{Map}((X,\mathcal{U}))$ and suppose that $A=\{ f^{j}(x_{0})\}_{j\in\mathbb{N}}$ is a sequential set corresponding to the orbit $\textrm{Orb}(x_{0})=(f^{j}(x_{0}))_{j\in\mathbb{N}}$, and let $f_{\mathbb{R}_{i}}(x_{0})=\bigcup_{j\geq i}f^{j}(x_{0})$ be the $i$-residual of the sequence $(f^{j}(x_{0}))_{j\in\mathbb{N}}$, with $_{\textrm{F}}\mathcal{B}_{x_{0}}=\{ f_{\mathbb{R}_{i}}(x_{0})\!:\textrm{Res}(\mathbb{N})\rightarrow X\textrm{ for all }i\in\mathbb{N}\}$ bein
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rder of $\vert W \vert^4 \sim 10^{-4}$, except for the $\mathcal{O} (W^2)$ difference in normalization constant in the disappearance probability. It is practically the limit of order of magnitude that can be explored by the next generation neutrino oscillation experiments.
We must note, however, that this conclusion is based on our treatment to first order in matter perturbation theory. Hence, the statement is not a very convincing one. Given the fact that setup of some of the next generation long-baseline (LBL) experiments require consideration of matter effect of comparable size with the vacuum effect, it is clear that a better treatment is necessary to understand the influence of the matter effect in the $(3+N)$ model. In particular, the conditions that allows us to make the active space oscillation probabilities sterile-sector model independent have to be worked out. This task will be carried out in section \[sec:formulation\].
As an outcome of the honest computation using the small unitarity-violation perturbation theory in the $(3+N)$ space unitary model to order $W^4$, we postulate the following theorem:
[**Uniqueness theorem**]{}
- All the matter dependent perturbative corrections in $W$ in the oscillation probability vanish or can be ignored after averaging over the fast oscillations and using the suppression due to the large energy denominators, leaving only the probability leakage term $\mathcal{C}_{\alpha \beta}$, the first term in (\[P-beta-alpha-ave-vac\]).
It must be remarked here that unitarity violation effects which are hidden in non-unitary active space mixing matrix $U$ produces zeroth- to higher order effects of $W$. The above theorem is only about the terms generated by explicit perturbative corrections in $W$.[^7]
As a result of the explicit computation, we show that the oscillation probability in matter between active flavor neutrinos in the $(3+N)$ space unitary model to fourth order in $W$ in our small unitarity-violation perturbation theory can be written as $$\begin{aligned}
P
| 3,747
| 2,389
| 3,649
| 3,417
| null | null |
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|
es}. \end{aligned}$$ If $J$ is a $p$-perfect subgroup of $G$ such that $p\mid |N_{G}(J)/J|$, then there are two conjugacy classes of subgroups $L$ of $G$ such that $O^{p}(L)=J$. We denote by $S_{J}$ a subgroup of $G$ such that $J\subset S_{J}$ and $O^{p}(S_{J})=J$. The block matrix indexed by $J$ is of size $2$. The first diagonal entry is: $$\begin{aligned}
b_{\phi_{G}}(G/Jf_{J}^{G},G/Jf_{J}^{G})=\frac{|N_{G}(J)|}{|J|}. \end{aligned}$$ the anti-diagonal entries are: $$\begin{aligned}
b_{\phi_{G}}(G/J\times G/S_{J} f_{J}^{G})&=\sum_{g\in [S_{J}\backslash G/J]} \phi(G/S_{J}\cap J^{g} f_{J}^{G})\\
&=\sum_{g\in [S_{J}\backslash N_{G}(J)/J]} 1\\
&=\frac{|N_{G}(J)|}{|S_{J}|}.\end{aligned}$$ Finally, the second diagonal element is: $$\begin{aligned}
a:=b_{\phi_{G}}(G/S_{J}f_{J}^{G},G/S_{J}f_{J}^{G})= \sum_{g\in [S_{J}\backslash G/S_J]} \phi_{G}(G/S_{J}\cap S_{J}^{g} f_{J}^{G}).\end{aligned}$$ Now, if $g\notin N_{G}(J)$ we have $G/S_{J}\cap S_{J}^{g} f_{J}^{G}=0$ and if $g\in N_{G}(S_{J})$, we have $$\phi_{G}(G/S_{J}f_{J}^{G})=0.$$ For the computation of $a$, we work in $\mathbb{Q}$. Then we have: $$\begin{aligned}
a&=\sum_{g\in [S_{J}\backslash G/S_J]} \phi_{G}(G/S_{J}\cap S_{J}^{g} f_{J}^{G})\\
&= \sum_{g\in N_{G}(J)\backslash N_{G}(S_{J})} \frac{|S_{J}\cap S_{J}^{g}|}{|S_{J}|^{2}} \phi_{G}(G/J f_{J}^{G})\\
&=\sum_{g\in N_{G}(J)\backslash N_{G}(S_{J})} \frac{|J|}{|S_{J}|^2}\\
&= \frac{|J|}{|S_{J}|^2}\big(|N_{G}(J)|-|N_{G}(S_{J})|).\end{aligned}$$ The determinant of each of these blocks is: $$\begin{aligned}
&\frac{|N_{G}(J)|}{|J|}\times \Big(\frac{|J|}{|S_{J}|^2}\big(|N_{G}(J)|-|N_{G}(S_{J})|)\Big)- \frac{|N_{G}(J)|^2}{|S_{J}^2|}\\
& = -\frac{|N_{G}(J)|\times |N_{G}(S_{J})|}{S_{J}^{2}}\in R^{\times}.\end{aligned}$$ This determinant is invertible in $R$, so the bilinear form $b_{\phi_{G}}$ is non degenerate.
Let $G$ be a finite group. Then the Mackey algebra $\mu_{R}(G)$ is a symmetric algebra if and only if $p^2\nmid |G|$.
If $p^2\nmid |G|$, the fact that $\mu_{R}(G)$ is a symmetric algebra follows from Theorem \[m
| 3,748
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|
o make it a global and standardised therapy to treat COVID-19 infection.
RECOMMENDATIONS
===============
Through this review, we recommend a few guidelines which could help to speed up the process of viral deactivation and production of the vaccine. Also, these recommendations will help the COVID-19 patients to recover soon and help the recovered patients be safe against this virus in the future. The guidelines are as follows.
- Based on the global data available, it is very unfortunate that the spread of the virus is still ongoing and the impact of the infection is still on the rise, despite the various preventive and precautionary interventions carried out by us.
- It is mandatory that the infected or possibly infected SARS-CoV-2 patients should immediately contact the nearby health-care professional to check on their health status to safeguard both their families and their societies.
- It is necessary to follow the rules and regulations provided by the health-care professional and government officials, which include quarantine, national lockdown, and social distancing as a measure to control the spread of the infection.
- It is highly encouraged and recommended that only vigorous obedience to the guidelines amended by the government and maintaining the preventive and precautionary measures will yield the desired result of containing this viral infection.
- We should co-operate with the health-care officials if they need to test our samples by providing them with the desired specimens to confirm the presence or absence of the infection in a particular individual.
- Based on the efficiency of the hydroxychloroquine and chloroquine drugs, it should be approved soon as the one-stop remedy to treat the SARS-CoV-2 infection.
- As per the government order, more state hospitals should take up the charge to carry out the clinical trials on convalescent plasma therapy as a remedial approach for this viral infection.
- The recovered patients should come forward to provide their blood samples to
| 3,749
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| 3,924
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|
}-[E]{}=0,U(0)=f. The solution of can be written in the form (cf. Example \[desolex1\] below) $$\begin{gathered}
U(x,\omega,E,t)
=
(U(t))(x,\omega,E) \\
=
H(R_x(E_m)-R_x(E)-t)
{{S_0(x,R_x^{-1}(R_x(E)+t))}\over{S_0(x,E)}} f(x,\omega,R_x^{-1}(R_x(E)+t)),\end{gathered}$$ where $H$ is the Heaviside function and $$R_x(E):=\int_0^E{1\over{S_0(x,\tau)}}d\tau,
\quad E\in [0,E_m].$$
In other words, $$\begin{gathered}
(T_{B_0}(t)f)(x,\omega,E) \\
=
H(R_x(E_m)-R_x(E)-t)
{{S_0(x,R_x^{-1}(R_x(E)+t))}\over{S_0(x,E)}} f(x,\omega,R_x^{-1}(R_x(E)+t)),\end{gathered}$$ and therefore $T_{B_0}(t)$ is evidently of positive type.
Due to and there is $M>0$ such that $0<\kappa\leq S_0\leq M$ a.e. on $\ol{G}\times I$. For fixed $E\in I$, letting $s_E(t):=R_x^{-1}(R_x(E)+t)-E$, we have $s_E(0)=0$, and $s_E'(t)=S_0(x,s_E(t)+E)$, and hence the estimates $$\kappa t\leq s_E(t)\leq Mt,\quad \forall t\geq 0$$ hold, and thus in particular, $$E\leq \kappa t+E\leq R_x^{-1}(R_x(E)+t).$$ Assumption , written in the form $\frac{1}{S_0}{\frac{\partial S_0}{\partial E}}\leq 2C$ yields after integration from $E$ to $E'$, where $E\leq E'$, $$\frac{S_0(x,E')}{S_0(x,E)}\leq e^{2C(E'-E)},$$ and hence by the above, $$\frac{S_0(x,R_x^{-1}(R_x(E)+t))}{S_0(x,E)}\leq e^{2Cs_E(t)}\leq e^{2CMt},$$ On the other hand, $$J_{x,t}(E):={{{\partial}\over{\partial E}}} R_x^{-1}(R_x(E)+t)=\frac{R_x'(E)}{R_x'\big(R_x^{-1}(R_x(E)+t)\big)}=\frac{S_0(x,R_x^{-1}(R_x(E)+t))}{S_0(x,E)},$$ we therefore we obtain the following estimate for $T_{B_0}(t)$, $t\geq 0$, $${\left\Vert T_{B_0}(t)f\right\Vert}_{L^2(G\times S\times I)}^2
\leq{}&
e^{2CMt}\int_{G\times S\times I} |\ol f(x,\omega,R_x^{-1}(R_x(E)+t))|^2 J_{x,t}(E) dxd\omega dE \\
\leq {}&e^{2CMt}\int_{G\times S\times I} |f(x,\omega,E')|^2 dxd\omega dE'
=e^{2CMt}{\left\Vert f\right\Vert}_{L^2(G\times S\times I)}^2$$ where $\ol f$ is the extension by zero of $f$ onto $G\times S\times [0,\infty[$.
The above computations show that for any $n\in{\mathbb{N}}$, we have $${\left\Vert \Big[T_{B_0}(t/n)T_{A_0}(t/n)T_{-(\Sigma+CS_0 I)}(t
| 3,750
| 2,880
| 1,874
| 3,530
| null | null |
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|
analogous argument it is easy to prove that the element $\aleph= \sum_{j=1}^{j_0} \alpha_j$ is $\Z/2 \int \D_4$-framed cobordant to the manifold obtained by gluing the union $-OP[\tilde L^{n-4k}_x] \cup 2^j(-OP)^{j-1}[\tilde L^{n-4k}_y]$ by an $\I_3$-manifold along the boundary. Moreover, this cobordism is relative with respect to all copies of $\tilde L^{n-4k}_z$ (with various orientations). If $j_0$ is great enough, the manifold (with $\I_3$-framed boundary) $2^j(-OP)^{j_0-1}[\tilde
L^{n-4k}_y]$ is cobordant relative to the boundary to an $\I_3$-framed manifold.
Therefore the manifold $L^{n-4k}_y$ is $\Z/2 \int \D_4$-framed cobordant relative to the boundary to the union of an $\I_3$-framed manifold with the same boundary and a closed manifold that is the double cover with respect to $\omega$ over a $\Z/2 \int
\D_4$-framed manifold. This cobordism is realized as a cobordism of the self-intersection of a $\D_4$-framed immersion with support inside $U^{reg}_{\Delta}$. This cobordism joins the immersion $g_3$ with a $\D_4$–framed immersion $g_4$. After an additional deformation of $g_4$ inside a larger neighborhood of $\Delta^{reg}$ the relative $\I_b$-submanifold of the self-intersection manifold of $g_4$ is deformed outside of $U^{reg}_{\Delta}$. The $\D_4$-framed immersion obtained as the result of this cobordism admits an $\I_b$-control. The Theorem 1 is proved.
An $\I_4$-structure (a cyclic structure) of a $\D_4$-framed immersion
======================================================================
Let us describe the subgroup $\I_4 \subset \Z/2 \int \D_4$. This subgroup is isomorphic to the group $\Z/2 \oplus \Z/4$. Let us recall that the group $ \Z/2 \int \D_4$ is the transformation group of $\R^4$ that permutes the $4$-tuple of the coordinate lines and two planes $(f_1, f_2)$, $(f_3, f_4)$ spanned by the vectors of the standard base $(f_1, f_2, f_3, f_4)$ (the planes can remin fixed or be permuted by a transformation).
Let us denote the generators of $\Z/2 \oplus \Z/4$ by $l$, $r$ correspondingly.
| 3,751
| 3,150
| 1,713
| 3,466
| 3,937
| 0.769147
|
github_plus_top10pct_by_avg
|
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$ \lambda_{i} \bigl((A+H)\widetilde{B}^{-1} \bigr) \ge \textstyle\begin{cases} \frac{\lambda_{\ell}(AB^{-1} )+\lambda_{\hbar} (HB^{-1} )}{1+\mu}, &\lambda_{i}(A+H)\ge0;\\ \frac{\lambda_{j} (AB^{-1} )+\lambda_{k} (HB^{-1} )}{1-\mu}, &\lambda_{i}(A+H)< 0. \end{cases} $$\end{document}$$ The proof of Theorem [2.1](#FPar8){ref-type="sec"} is complete. □
Corollary 2.1 {#FPar10}
-------------
*Let* $\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$A,B,H,E \in\mathbb{C}^{n\times n}$\end{document}$ *be Hermitian matrices*, *B* *be a positive definite Hermitian matrix*, *and* $\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\widetilde{B}=B+E$\end{document}$. *If* $\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\mu =\frac{\|E\|_{2}}{\lambda_{n}(B)}<1$\end{docume
| 3,752
| 3,885
| 236
| 3,709
| 1,024
| 0.795493
|
github_plus_top10pct_by_avg
|
rdered by $\sqsubseteq$, then the set $$S := \bigcup_{s \in P} s$$ is still a D-q-strategy and fulfills: $$\forall s \in P, s \sqsubseteq S.$$ Hence the extension ordering over the set of D-q-strategies w.r.t. $(T,T')$ is inductive.
Let $S \subseteq ({\cal R}\times{\cal R})^*$ be finite and let $n:= \max\{ |\alpha| \mid
\alpha \in S\}$.\
$S$ is a finite prefix of a D-strategy w.r.t. $(T,T')$ iff\
(1) $S$ is a D-q-strategy w.r.t. $(T,T')$\
(2) $\forall \beta \in S, [ \beta \backslash S = \{(\varepsilon,\varepsilon)\}
\Rightarrow (|\beta| = n \mbox{ or } {{\rm NEXT}}((T,T'),\beta) \notin \sim_1] $). \[L-characterisation\_PDstrategies\]
[**Direct implication**]{}:\
Let $S'$ be a D-strategy w.r.t. $(T,T')$ and $$S= S'\cap ({\cal R}\times{\cal R})^{\leq n}$$ for some $n \in \N$ and some $S'$ which is a D-strategy w.r.t. $(T,T')$.\
1- By Lemma \[L-PD\_implies\_DQ\] $S$ is a D-q-strategy w.r.t. $(T,T')$.\
2- Suppose that $\beta \in S, \beta \backslash S = \{(\varepsilon,\varepsilon)\}$ and $|\beta| < n$. Then $\beta \backslash S' = \{(\varepsilon,\varepsilon)\}$ too. Since $S'$ is a D-strategy w.r.t. $(T,T')$, this implies that ${{\rm NEXT}}((T,T').\beta) \notin \sim_1$.\
[**Converse**]{}:\
Suppose that $S$ fulfills conditions (1)(2). By Lemma \[L\_inclusion\_is\_inductive\], Zorn’s lemma applies on the set of D-q-strategies w.r.t. $(T,T')$: there exists a maximal D-q-strategy $S'$ (for the extension ordering) such that $S \sqsubseteq S'$. Since $S'$ is maximal, if $\alpha \in S'$ and $\alpha \backslash S=\{(\varepsilon,\varepsilon)\}$, ${{\rm NEXT}}((T,T'),\alpha) \notin \sim_1$. Thus, instead of the weak property DQ4, $S'$ fulfills the property: $$\forall \alpha \in S',
{{\rm NEXT}}((T,T'),\alpha) \notin \sim_1\;\;
\mbox{ or }$$ $$[{{\rm NEXT}}((T,T'),\alpha) \in \sim_1 \mbox{ and }\{ (\pi,\pi') \in {\cal R}\times{\cal R} \mid \alpha\cdot (\pi,\pi')\in S\} \mbox{ is full for } {{\rm NEXT}}((T,T'),\alpha)].$$ Hence $S'$ is a strategy w.r.t. $(T,T')$.\
Clearly $$S \subseteq S' \cap ({\cal R}\times{\cal R})^{\leq n}
| 3,753
| 1,262
| 2,690
| 3,442
| 3,225
| 0.773905
|
github_plus_top10pct_by_avg
|
}
For either model (stratified or random intercept), under any given scenario and data generating mechanism, using ML always produced more downwardly biased estimates than REML (Table [3](#sim7930-tbl-0003){ref-type="table"}), as expected.([14](#sim7930-bib-0014){ref-type="ref"}, [15](#sim7930-bib-0015){ref-type="ref"}, [16](#sim7930-bib-0016){ref-type="ref"}, [17](#sim7930-bib-0017){ref-type="ref"}, [18](#sim7930-bib-0018){ref-type="ref"}) For example, for the base case scenario with the random intercept model, under the normal intercept data generating mechanism, the median percentage bias using REML estimation was −15.9% compared to −41.5% using ML estimation. The bias was worse when using a stratified intercept model (due to the extra number of parameters to estimate), as ML estimation often produced a downward median bias of 100%.
When using REML estimation, there were generally only small differences between random and stratified intercept models in terms of bias of the between‐trial variance of treatment effects; however, while better than ML, downward bias was not removed entirely with REML. Furthermore, the overall size of the bias was typically greater in the beta distribution intercept case than in the normal distribution intercept case, regardless of which model was used.
### 3.2.4. Empirical SE and MSE of summary treatment effect estimate {#sim7930-sec-0014}
There were negligible differences in empirical SE or MSE of $\hat{\theta}$ between the two models (stratified or random intercept), under any given scenario and data generating mechanism (Web Tables C.VIII to C.X).
### 3.2.5. Coverage of summary treatment effect estimate {#sim7930-sec-0015}
There were marked differences observed in the coverage of $\hat{\theta}$ across the different estimation approaches (ML or REML) and CI derivations (standard, KR, or Satterthwaite), as now explained.
*(i) Under a normal distribution intercept generating mechanism*
We consider first the normal distribution intercept generating mechanism (Figure
| 3,754
| 653
| 4,444
| 3,293
| 1,888
| 0.784669
|
github_plus_top10pct_by_avg
|
as a product of pairwise coprime t.s.m elements in $R$. Since ${\rm gcd}(\widehat{f}_i^*, f_i^*)=1$, there exist polynomials $b_i, c_i \in R$ such that $b_i\widehat{f}_i^*+c_if_i^*=1$. Let $e_i=b_i\widehat{f}_i^* \in R$. Then\
(i) $e_1, e_2, \ldots, e_t$ are mutually orthogonal in $R$;\
(ii) $e_1+e_2+\cdots +e_t=1$ in $R$;\
(iii) $R_i=(e_i)$ is a two-sided ideal of $R$ and $e_i$ is the identity in $(e_i)$;\
(iv) $R=R_1\bigoplus R_2 \bigoplus \cdots \bigoplus R_t$;\
(v) For each $i=1,2,\ldots,t$, the map $$\psi:~R/(f_i^*)\rightarrow R_i$$ $$g+(f_i^*)\mapsto (g+(x^n-1))e_i$$ is a well-defined isomorphism of rings;\
(vi) $R\cong R/(f_1^*)\bigoplus R/(f_2^*)\bigoplus \cdots \bigoplus R/(f_t^*)$.*
*Proof* (i) Suppose $e_i=0$ for some $i=1,2,\ldots,t$, i.e., $b_i\widehat{f}_i^*\in (x^n-1)$ in $R$. Then $b_i\widehat{f}_i^*\in (f_i^*)$. Thus $1=b_i\widehat{f}_i^*+c_if_i^*\in (f_i^*)$, which is a contradiction. Hence, for each $i=1,2,\ldots,t$, $e_i\neq 0$. Thus we have $b_i\widehat{f}_i^*b_j\widehat{f}_j^* \in (x^n-1)$ for $i\neq j$. This implies that $e_ie_j=0$ in $R$. (ii) We have $b_1\widehat{f}_1^*+\cdots +b_t\widehat{f}_t^*-1 \in (f_i^*)$, for all $i=1,2,\ldots,t$. Therefore $b_1\widehat{f}_1^*+\cdots +b_t\widehat{f}_t^*-1 \in (x^n-1)$. Thus $e_1+\cdots +e_t=1$ in $R$. (iii) Let $Re_i=(e_i)_l$. Then $(e_i)_l\subseteq (\widehat{f}_i^*)$. On the other hand, $\widehat{f}_i^*=\widehat{f}_i^*(b_i\widehat{f}_i^*+c_if_i^*)=\widehat{f}_i^*b_i\widehat{f}_i^*$ in $R$, which implies $(\widehat{f}_i^*)\subseteq (e_i)_l$. Therefore $(e_i)_l=(\widehat{f}_i^*)$. Similarly, one can prove that $e_iR=(e_i)_r=(\widehat{f}_i^*)$, which implies that $(e_i)$ is a two-sided ideal of $R$. Clearly, $e_i$ is the identity in $(e_i)$. (iv) For any $a\in R$, $a$ can be represented as $a=ae_1+ae_2+\cdots +ae_t$. Since $ae_i\in (e_i)$, $R=(e_1)+(e_2)+\cdots +(e_t)$. Assume that $a_1+a_2+\cdots+a_t=0$, where $a_i\in (e_i)$. Multiplying on the left (or on the right) by $e_i$, we obtain that $a_1
| 3,755
| 4,319
| 3,972
| 3,412
| 2,964
| 0.775793
|
github_plus_top10pct_by_avg
|
25 1
**Flexibility of multiplexing** Low---requires manual dispensing bioreagents High---Printing nL/µL size dots or multi-line Medium---test line configuration and positioning of antibodies has an influence.\ Medium---test line configuration and positioning of antibodies has an influence.\
**Non-Expert Ease of Use** Easy Challenging Easy Easy
**False positives in blank RB**\ Y\ Y\ N\ N\
(*n* = number of tested samples within assay working range) (*n* = 3) (*n* = 10) (*n* = 10) (*n* = 10)
**False negatives in spiked RB**\ N\ N\ N\ N\
(*n* = number of tested samples within assay working range) (*n* = 3) (*n* = 21) (*n* = 24) (*n* = 18)
**Equipment used**
| 3,756
| 7,465
| 1,330
| 1,527
| 2,008
| 0.78357
|
github_plus_top10pct_by_avg
|
tions in consideration of predictions errors. Actually, in some cases, it is best not to act as predicted because of prediction errors. For example, the paper [@Yamaguchi2018] formulates a method for minimizing the expected value of the procurement cost of electricity in two popular spot markets: [*day-ahead*]{} and [*intra-day*]{}, under the assumption that the expected value of the unit prices and the distributions of the prediction errors for the electricity demand traded in two markets are known. The paper showed that if the procurement is increased or decreased from the prediction, in some cases, the expected value of the procurement cost is reduced. (2) In recent years, prediction methods have been black boxed by the big data and machine learning (see, e.g., Ref. [@10.1145/3236009]). The day will soon come, when we must minimize the objective function by using predictions obtained by such black boxed methods. In our method, even if we do not know the prediction $\hat{y}$, we can determine the parameter $C$ if we know the prediction error distribution $f$ and asymmetric loss function $L$.
To obtain $y^{*}$, we derive $\operatorname{{E}}[\Pe(Z + c)]$ for any $c \in \mathbb{R}$. Let $\G(a, x)$ and $\g(a, x)$ be the upper and the lower incomplete gamma functions, respectively (see, e.g., Ref. [@doi:10.1142/0653]). The expected value and the variance of $\Pe(Z + c)$ are as follows:
\[lem:1.1\] For any $c \in \mathbb{R}$, we have $$\begin{aligned}
(1)\quad
\operatorname{{E}}[\Pe(Z + c)]
&= \frac{(k_{1} - k_{2}) c}{2}
+ \frac{(k_{1} + k_{2}) \lvert c \rvert}{2 \G(a)}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
+ \frac{(k_{1} + k_{2}) b}{2 \G(a)}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right), \\
(2)\quad
\operatorname{{V}}[\Pe(Z + c)]
&= \frac{(k_{1} + k_{2})^{2} c^{2}}{4}
+ \frac{(k_{1}^{2} - k_{2}^{2}) b c}{2 \G(a)}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \nonumber \\
&\quad - \frac{(k_{1} + k_{2})^{2} b \lvert c \rve
| 3,757
| 3,287
| 3,301
| 3,341
| null | null |
github_plus_top10pct_by_avg
|
Recruiting University of Hong Kong, Queen Mary Hospital
2\. Ribavirin Hong Kong, Hong Kong
3\. Interferon Beta-1B
Efficacy of Chloroquine and Lopinavir/ Ritonavir in mild/general novel coronavirus (CoVID-19) infections: a prospective, open-label, multileft randomized controlled clinical study 1\. Chloroquine \- The Fifth Affiliated Hospital Sun Yat-Sen University
2\. Lopinavir/ Ritonavir
| 3,758
| 5,617
| 2,989
| 2,632
| null | null |
github_plus_top10pct_by_avg
|
rovided is a step‐by‐step guide to our simulation study. For simplicity, and to considerably speed up the many simulations, we removed the baseline adjustment term in models (1) and (2), such that it does not exist in any of the data generating mechanisms or models fitted in our simulations. In other words, we generate data without baseline imbalances and thus analyze the data according to a final score IPD meta‐analysis model, which is appropriate in this situation~.~ [21](#sim7930-bib-0021){ref-type="ref"} For similar reasons of simplicity and computational complexity, we assumed a common residual variance across trials (both in data generation and models fitted). Extension to different residual variances is considered in our discussion (Section [5](#sim7930-sec-0019){ref-type="sec"}). To inform the true parameter values for the simulation, we used a previous IPD meta‐analysis of treatment for lower blood pressure outcomes.[22](#sim7930-bib-0022){ref-type="ref"}
All analyses were conducted using Stata v.14.2 (Stata Corporation, TX, USA).[23](#sim7930-bib-0023){ref-type="ref"}
### 3.1.1. Scenario 1 (base case) {#sim7930-sec-0008}
The simulation process is now explained, in the context of an initial base case scenario with IPD from 10 trials and a relatively simple data generating mechanism. Extensions to other more complex scenarios are described afterwards.
*Step 1: Data generating mechanism for one IPD meta‐analysis of 10 trials*
Consider that an IPD meta‐analysis of *i* = 1 to *K* related trials is of interest, with the goal to summarize a treatment effect on a continuous outcome. To generate such data for the base case of this simulation study, we started by setting the number of trials, *K*, to 10. We set a fixed number of participants, *n* = 100 in each trial, and assumed a fixed randomization of 1:1 in each trial; that is, on average, 50% of participants within any given trial are allocated to a treatment group, and the remaining 50% to a control group. This gave us a *trial* ~*i*~ (
| 3,759
| 587
| 3,638
| 3,556
| null | null |
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|
hcal{V},\mathcal{E})}}_0 \rangle}=\bigotimes_{\{i,j\} \in \mathcal{E}} CZ_{ij}{| {g_{(\mathcal{V})}}_0 \rangle}.$$
![\[Graph\] (Color online) Example of a mathematical graph associated to a physical graph state. We have displayed a possible partition of this graph, splitting the system in three parts $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$. The vertices and edges in grey corresponds to the *boundary qubits* and the *boundary-crossing edges* respectively.](graph2){width="0.7\linewidth"}
An example of such graph is shown in Fig. \[Graph\], where the system is divided into three regions, $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$. We call all edges that go from one region to the other the *boundary-crossing edges* and label the subset of all such edges by $\mathcal{X}$. All qubits connected by the boundary-crossing edges are in turn called the *boundary qubits* and the subset composed of all of these is called $\mathcal{Y}$.
*Open-system dynamics.–* Our ultimate goal is to quantify the entanglement in any partition of arbitrary graph states undergoing a generic physical process during a time interval $t$. The action of such process on an initial density operator $\rho$ can be described by a completely-positive trace-preserving map $\Lambda$ as $\rho_t=\Lambda(\rho)$, where $\rho_t$ is the evolved density matrix after time $t$. All such maps can be expressed in a Kraus representation, $\Lambda(\rho)=\sum_\mu p_\mu K_\mu \rho
K_\mu^\dag$, where $\sqrt{p_\mu}K_\mu$ are called the Kraus operators (each of which appearing with probability $p_\mu$), which satisfy the normalization conditions Tr$[K_\mu^\dag K_\mu]=1$ and $\sum_\mu p_\mu =1$ [@nielsen]. The Kraus representation guarantees that the map is (completely) positive and preserves trace normalization. When the map can be factorized as the composition of individual maps acting independently on each qubit, the noise is said to be individual (or independent); if not, it is said to be collective.
A very important class of processes is described by the
| 3,760
| 1,141
| 2,250
| 3,735
| 3,274
| 0.773549
|
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|
ft\{ W ^{\dagger} A (UX) \right\}_{K k}
\biggr]
\nonumber \\
&+&
\sum_{K}
e^{- i \Delta_{K} x}
W_{\alpha K} W^*_{\beta K}.
\label{S-alpha-beta-2nd}\end{aligned}$$
The oscillation probability to second order in $W$ {#sec:probability-2nd}
--------------------------------------------------
In this section, we discuss the oscillation probability to second order in $W$. It is to illuminate the principle of calculation, how averaging over the fast oscillation works, and to show which constraints are obtained on the sterile state masses by the requirement of suppression by the energy denominator to make these sterile-sector model dependent terms negligible.
Of course, we will calculate in this paper all the oscillation probabilities $P(\nu_\beta \rightarrow \nu_\alpha)$ in matter to fourth order in $W$ to keep the necessary term, the probability leaking term $\mathcal{C}_{\alpha \beta}$, as mentioned earlier. The key features of the fourth-order terms will be described in the next section \[sec:probability-4th\].
The following formulas include the cases of both disappearance ($\alpha=\beta$) and appearance ($\alpha \neq \beta$) channels. The oscillation probability $P(\nu_\beta \rightarrow \nu_\alpha)$ is given to second order in $W$ as $$\begin{aligned}
&&P(\nu_\beta \rightarrow \nu_\alpha)^{(0+2)} =
\left| S^{(0)}_{\alpha \beta} \right|^2
+ 2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S^{(2)}_{\alpha \beta} \right]
\nonumber \\
&=&
\sum_{k} (UX)_{\alpha k} (UX)^*_{\beta k}
(UX)^*_{\alpha k} (UX)_{\beta k}
+ \sum_{k \neq l} (UX)_{\alpha k} (UX)^*_{\beta k}
(UX)^*_{\alpha l} (UX)_{\beta l}
e^{-i ( h_{k} - h_{l} ) x}
\nonumber \\
&+&
2 \mbox{Re}
\biggl\{
\sum_{m}
\sum_{k, K}
\frac{ 1 }{ \Delta_{K} - h_{k} }
\left[
(ix) e^{- i ( h_{k} - h_{m} ) x} + \frac{e^{- i ( \Delta_{K} - h_{m} ) x} - e^{- i ( h_{k} - h_{m} ) x} }{ ( \Delta_{K} - h_{k} ) }
\right]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta k}
(UX)^*_{\alpha m} (UX)_{\beta m}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
| 3,761
| 3,051
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|
behavior at large distances ($r\sim a_B$) [@Kh]. In the leading approximation the PNC interaction
=8.cm
related to the weak charge is due to Z-bozon exchange, see Fig.1a. Calculation of the corresponding weak interaction matrix element gives [@Kh] $$\label{pnc}
<p_{1/2}|H_{W}|s_{1/2}>_0=M_0\propto (F_sG_p-G_sF_p)|_{r=r_0}.$$ At $r_0 \to 0$ this matrix element is divergent, $M_0\propto r_0^{2\gamma-2}$. As a result, the relativistic enhancement factor is $R \approx$3 for Cs and $R \approx 9$ for Tl, Pb, and Bi [@Kh]. In the present paper we show that this divergence results in the double logarithmic enhancement of the radiative corrections.
The first correction is shown in Fig.1b. It corresponds to a modification of the electron wave function because of the vacuum polarization. In the leading $Z\alpha$ approximation the vacuum polarization results in the Uehling potential [@Ueh]. At $r \ll \lambda_C$, this potential is of the form $V(r)\approx 2Z\alpha^2[\ln(r/\lambda_C)+C+5/6]/(3\pi r)$, where $C\approx 0.577$ is the Euler constant. Account of higher in $Z\alpha$ corrections in the vacuum polarization leads to a modification of the constant: $C \to C + 0.092Z^2\alpha^2+...$, see Ref. [@Mil]. However, this correction is small and can be neglected even for $Z\alpha \sim 1$. The potential $V(r)$ modifies the Coulomb interaction in Eqs. (\[fg\]) $-Z\alpha/r \to -Z\alpha/r + V(r)$. It is convenient to search for solution of the modified Eqs. (\[fg\]) in the following form ${\cal F}=F(1+F^{(1)})$, ${\cal G}=G(1+F^{(1)})$, where $F$ and $G$ are given by (\[fg1\]). The functions $F_{s,p}^{(1)}$ and $G_{s,p}^{(1)}$ satisfy the following equations $$\begin{aligned}
\label{fg2}
&& \frac{1}{\gamma+\kappa}\,\frac{d F^{(1)}}{d x}- F^{(1)}+
G^{(1)}=-\frac{2\alpha}{3\pi}\, x \nonumber \\
&&\frac{1}{\gamma-\kappa}\frac{d G^{(1)}}{d x}-G^{(1)}+ F^{(1)}
=-\frac{2\alpha}{3\pi}\, x \quad ,\end{aligned}$$ where $x=\ln(\lambda/r)$, and $\lambda=\lambda_C\exp(-C-5/6)$. The solution of these equations reads $$\begin{aligned}
\labe
| 3,762
| 2,387
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| null | null |
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|
^2(\alpha r cos\theta-1)^6}{r^6}.$$ The Weyl scalar is defined by $$W=C_{abcd}C^{abcd},$$ where the $C_{abcd} $ is the Weyl curvature tensor. For the $C$-metric (non-rotating black hole) the Weyl scalar is evaluated as $$\label{non-rot_Wscalar}
W_{c}=\dfrac{48m^2(\alpha r cos\theta-1)^6}{r^6}.$$ This result is expected since the Ricci tensor for this metric turns out to be zero. As the Riemann tensor can be decomposed into the Ricci and the Weyl parts according to equation (\[decom\]), the vanishing Ricci component renders the Riemann and Weyl tensors identical as evident from equations (\[non-rot\_Kscalar\]) and (\[non-rot\_Wscalar\]). The scalar function $P$ is defined by the relation (\[P\_sq\]) as $$P^2=\dfrac{C_{abcd}C^{abcd}}{R_{abcd}R^{abcd}}.$$ For this $C$-metric, we get $ P^2=1 $. Therefore we assume that $ P=+1 $ for our entropy calculations, since the entropy must be non-negative.
Now the *spatial section* corresponding to this metric is $$h_{ij}=diag\left[\dfrac{1}{( 1 - \alpha r cos\theta )^2 (1-2m/r) (1-\alpha^2 r^2)},
\dfrac{r^2}{(( 1 - \alpha r cos\theta )^2 ( 1 - 2\alpha m cos\theta ))},
\dfrac{r^2 sin^2 \theta ( 1 - 2 \alpha m cos\theta )}{(1 - \alpha r cos\theta )^2}\right],$$ with the determinant given by $$h=\dfrac{sin^2(\theta)r^5}{(\alpha^2 r^2-1)(-r+2m)(\alpha r cos(\theta)-1)^6}.$$ Therefore, the infinitesimal surface element has the form $$d\sigma=\dfrac{\sqrt{h}}{\sqrt{h_{rr}}}d\theta d\phi=\dfrac{r^2 sin\theta}{(\alpha r cos\theta-1)^2} d\theta d\phi.$$
We are now in a position to calculate the magnitude of the gravitational entropy on the event horizon $H_{0}$ at the location $ r_{h}=2m $ for this metric, which is $$\label{s_grav_nonrot}
S_{grav}=k_{s}r_{h}^2\int_{\theta=0}^{\pi}\dfrac{sin\theta}{(\alpha r_{h} cos\theta-1)^2}d\theta \int_{\phi=0}^{2\pi C} d\phi=k_{s}\dfrac{4 \pi C r_{h}^2}{(1-r_{h}^2\alpha^2)}=k_{s}\dfrac{4 \pi r_{h}^2}{(1-r_{h}^2\alpha^2)(1+2\alpha m)}.$$ From equation (\[s\_grav\_nonrot\]) it is evident that the gravitational entropy is proportional to the area o
| 3,763
| 4,472
| 3,736
| 3,476
| null | null |
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|
of the form $I_{n!}E$, $[n] \in \Lambda$, $E \in {\operatorname{Shv}}({{\mathcal C}}^n)$ admissible, and to prove the first claim, it suffices to consider $M_\#=I_{n!}E$ of this form. In degree $0$, is the definition of the functor ${\operatorname{\sf tr}}_\#$, and the higher degree terms in the right-hand side vanish by Definition \[clean\] . Therefore it suffices to prove that $M_\#=I_{n!}E$ is acyclic for the functor ${\operatorname{\sf tr}}_\#$. This is obvious: applying ${\operatorname{\sf tr}}_\#$ to any short exact sequence $$\begin{CD}
0 @>>> M'_\# @>>> M''_\# @>>> M_\# @>>> 0
\end{CD}$$ in ${\operatorname{Shv}}({{\mathcal C}}_\#)$, we see that, since $M_\#([n'])$ is acyclic for any $[n'] \in \Lambda$, the sequence $$\begin{CD}
0 @>>> {\operatorname{\sf tr}}M'_\#([n']) @>>> {\operatorname{\sf tr}}M''_\# ([n']) @>>> {\operatorname{\sf tr}}M_\# ([n']) @>>> 0
\end{CD}$$ is exact; this means that $$\begin{CD}
0 @>>> {\operatorname{\sf tr}}M'_\# @>>> {\operatorname{\sf tr}}M''_\# @>>> {\operatorname{\sf tr}}M_\# @>>> 0
\end{CD}$$ is an exact sequence in ${\operatorname{Fun}}(\Lambda,k)$, and this means that $M_\#$ is indeed acyclic for ${\operatorname{\sf tr}}_\#$.
With the first claim proved, the second amounts to showing that the natural map $$L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}\circ L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(f_!)_!(E) \to L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}(E)$$ is a quasiismorphism for any $f:[n] \to [n']$ and any $E \in
{\operatorname{Shv}}({{\mathcal C}}^n)$. It suffices to prove it for admissible $M$; then the higher derived functors vanish, and the isomorphism ${\operatorname{\sf tr}}\circ
(f_!)_! \cong {\operatorname{\sf tr}}$ is Definition \[trace.defn\].
In the assumptions of Lemma \[b.ch\], for any complex $M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_\#
\in {{\mathcal D}}\Lambda({{\mathcal C}})$ with the first component $M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}=
M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}
| 3,764
| 3,201
| 1,253
| 3,578
| 2,848
| 0.776548
|
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|
anyway. Of course, this concern applies to the scalar ALP model we have considered, other fields or models could change this. Having ALP dark matter with the coupling we are considering changes the status of the nucleon EDM from a fundamental constant of nature to a parameter dependent on the local field value. Thus we see that the nucleon EDM may be expected to change in time (and space), likely oscillating at high frequencies $\sim$ kHz to GHz. It is thus important to consider the limits that existing experiments put on the parameter space in Figure \[Fig:EDM\]. Further, this parameter space has not been considered before. Therefore it is also important to design experiments which are optimized to search for this signal. Beyond the cold molecule [@Graham:2011qk] and NMR techniques [@NMR; @paper] that we have considered there could be many possibilities for other experiments, for example using proton storage rings [@Semertzidis:2011qv; @Orlov:2006su; @Semertzidis:2003iq; @yannis].
Axial Nuclear Moment {#Sec: axial nuclear}
====================
The third operator in gives rise to the coupling
$$\label{eqn:gaNN}
\mathcal{L} \supset g_\text{aNN} \left( \partial_\mu a \right) \bar{N} \gamma^\mu \gamma_5 N$$
between the ALP and the axial nuclear current. For the QCD axion, this coupling usually exists to both protons and neutrons and is approximately $g_\text{aNN} \sim \frac{1}{f_a}$. Current bounds on this operator arise from two sources. First, this operator allows an accelerated nucleon to lose energy through ALP emission. These emissive processes are constrained by observations of the cooling rates of supernova, imposing an upper bound on $g_{aNN} \lessapprox 10^{-9} \text{ GeV}^{-1}$ [@Raffelt:2006cw]. Second, this operator leads to a force between nucleons through the exchange of ALPs. This force is spin dependent with a range $\sim m_a^{-1}$ [@Moody:1984ba]. Such spin-spin interactions have been searched for using a variety of spin polarized targets, but the limits on $g_{aNN}$ from them [@Vasilakis:2008yn
| 3,765
| 1,763
| 3,433
| 3,596
| null | null |
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|
Since $\langle W \rangle = 0$, the second moment is also the variance of the work distribution. The third moment is given by $$\label{3mom}
\left\langle W^{3}\right\rangle = \frac{\hbar^{3}\Omega^{2}}{4}
\left[\nu\eta^{2} +
\omega_{0}
\tanh \tfrac{\beta\hbar\omega_{0}}{2}
\right],$$ in which appears the dependence on the temperature. From the second and third moments, we can determine the skewness of the work distribution $\left\langle W^{3}\right\rangle/\left\langle W^{2} \right\rangle^{3/2} $. This turns out to be inversely proportional to the magnitude of the work parameter. Consequently, the stronger the laser, the more symmetric the distribution is around the mean value $\left\langle W\right\rangle = 0$. Since $\left\langle W^{3} \right\rangle > 0$, as seen from Eq. (\[3mom\]), the work distribution is biased towards negative values of work. All these facts about the first moments of the work distribution, obtained with the full Hamiltonian Eq. (\[hf\]), tell us that negative work (internal energy descrease) is more likely than the equivalent positive work (internal energy increase) at the very first instant of interaction with the laser field. Note also that the asymmetry around the mean value decreases with the temperature while it increases with $\eta$. Finally, according to Eq. (\[etatrue\]), $\left\langle W^{3}\right\rangle$ and the skewness are actually independent of the trap frequency $\nu$.
Now we turn our attention to the sideband Hamiltonians in Eq. (\[hs\]) and to the irreversibility of the work protocol consisting of the sudden quench of system Hamiltonian due to laser interaction. As said before, these effective Hamiltonians are obtained from the full Hamiltonian Eq. (\[hf\]) by setting resonance $\omega_{0} = \omega_{L} \pm m \nu$ and performing a rotating wave approximation. We will see that a thermodynamic analysis is able to reveal t
| 3,766
| 4,302
| 3,368
| 3,382
| null | null |
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|
r dimensions, by taking the coordination number sufficiently large.
Organization
------------
In the rest of this paper, we focus our attention on the model-dependent ingredients: the lace expansion for the Ising model (Proposition \[prp:Ising-lace\]) and the bounds on (the alternating sum of) the expansion coefficients for the ferromagnetic models (Proposition \[prp:Pij-Rj-bd\]). In Section \[s:laceexp\], we prove Proposition \[prp:Ising-lace\]. In Section \[s:reduction\], we reduce Proposition \[prp:Pij-Rj-bd\] to a few other propositions, which are then results of the aforementioned diagrammatic bounds on the expansion coefficients. We prove these diagrammatic bounds in Section \[s:bounds\]. As soon as the composition of the diagrams in terms of two-point functions is understood, it is not so hard to establish key elements of the above reduced propositions. We will prove these elements in Section \[ss:proof-so\] for the spread-out model and in Section \[ss:proof-nn\] for the nearest-neighbor model.
Lace expansion for the Ising model {#s:laceexp}
==================================
The lace expansion was initiated by Brydges and Spencer [@bs85] to investigate weakly self-avoiding walk for $d>4$. Later, it was developed for various stochastic-geometrical models, such as strictly self-avoiding walk for $d>4$ (e.g., [@hs92]), lattice trees/animals for $d>8$ (e.g., [@hs90]), unoriented percolation for $d>6$ (e.g., [@hs90']), oriented percolation for $d>4$ (e.g., [@ny93]) and the contact process for $d>4$ (e.g., [@s01]). See [@s04] for an extensive list of references. This is the first lace-expansion paper for the Ising model.
In this section, we prove the lace expansion [(\[eq:Ising-lace\])]{} for the Ising model. From now on, we fix $p\ge0$ and abbreviate, e.g., $\pi_{p;\Lambda}^{{\scriptscriptstyle}(i)}(x)$ to $\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)$.
There may be several ways to derive the lace expansion for ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda$, using, e.g., the high-temperature expan
| 3,767
| 2,653
| 1,321
| 3,491
| 503
| 0.807716
|
github_plus_top10pct_by_avg
|
8.57cm"}
Plotted in Fig. \[fig1\] are snapshots at different times of three trajectories at different disorder strengths $d$, all prepared in the same local initial state. At small $d$ it is clear that the exciton moves almost uniformly in space and time, with the lattice having been occupied uniformly after short times. Conversely, as $d$ is increased we find the exploration of the lattice becomes far from uniform in time, with large dwell times in certain regions and quick jumps between other regions. This effect becomes increasingly pronounced as $d$ is increased and it will be studied in greater detail later in the paper.
First we look at the statistics for jumps between states, captured by the probability $\pi_t(K)$ that there are $K$ jumps between states in a time $t$. Shown in Fig. \[fig2\] are histograms reflecting this distribution for long time intervals $t=300/J$ at different values of $d$. While at small $d=1$, the distribution is close to Poissonian, as $d$ is increased, it becomes progressively broad, indicating that *rare trajectories*, where $K$ is much smaller or larger than the mean, becoming increasingly likely.
We note that at long times the probability distribution takes the large-deviation form [@Touchette2009] $\pi_t(K) \simeq e^{-t\varphi(K/t)}$ where $\varphi(k)$ is a large-deviation function of the average jump rate, or *activity*, $k=K/t$. The associated moment generating function is also of large-deviation form $Z_t(s) = \sum_K \pi_t(K) e^{-sK} \simeq e^{t\theta(s)}$, with $s$ a conjugate field to the number of jumps $K$. The function $\theta(s)$ is analogous to (minus) a free energy for trajectories, with discontinuities in the derivatives of $\theta(s)$ corresponding to dynamical (or trajectory) phase transitions [@Garrahan2007; @Lecomte2007; @Garrahan2010]. The activity $k_s = -\partial_s\theta(s)$ will be used as an order parameter, with $k_{s=0}=k$ the average jump rate of the physical problem (*i.e.* with no $s$ field applied). We can find $\theta(s)$ as largest eigenvalue
| 3,768
| 2,210
| 4,112
| 3,581
| null | null |
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|
W14] also considered balanced allocation on graphs where the number of balls $m$ can be much larger than $n$ (i.e., $m\gg n$) and the graph is not necessarily regular and dense. Then, they established upper bound ${\mathcal{O}}(\log n/\sigma)$ for the gap between the maximum and the minimum loaded bin after allocating $m$ balls, where $\sigma$ is the edge expansion of the graph. Bogdan et al. [@BSS013] studied a model where each ball picks a random vertex and performs a local search from the vertex to find a vertex with local minimum load, where it is finally placed. They showed that when the graph is a constant degree expander, the local search guarantees a maximum load of $\Theta(\log\log n)$. Pourmiri [@PJ19] substitutes the local search by non-backtracking random walks of length $\ell=o(\log n)$ to sample the choices and then the ball is allocated to a least-loaded bin. Provided the underlying graph has sufficiently large girth and $\ell$, he showed the maximum load is a constant. [In the context of hashing (e.g., [@Aamand18; @Dahlgaard16]), authors apply the witness graph techniques to analyze the maximum load in the balls-into-[bins process]{} where the bins are picked based on tabulation.]{}
Balanced Allocation on Dynamic Hypergraphs {#sec:hyp}
============================================
In this section we establish an upper bound for the maximum load attained by the balanced allocation on hypergraphs (i.e., Theorem \[thm:d-choice\]). In order to analyze the process let us first define a *conflict graph*. We write $D_t$ for the set of $d$ bins chosen by the $t$-th ball, and sometimes refer to $D_t$ as the $d$-*choice* of the $t$-th ball. We will slightly abuse the notation and write $D_u\cap D_t$, $D_u\cup D_t$ to denote the set of common bins, and the union of bins, chosen by balls $u$ and $t$, respectively.
For $m=1,\ldots, n$, the conflict graph ${\mathcal{C}}_m$ is a simple graph with vertex set $\{D_1,D_2,\ldots,D_m\}$. Vertices $D_u$ and $D_t$ are connected by an edge in ${\mathcal{C}}_m$ if
| 3,769
| 1,128
| 2,734
| 3,314
| 1,355
| 0.790548
|
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|
$, $Y_2(k) =
\RP^{31} \ast \dots \ast \RP^{31}$ the joins of the $k$ copies of the standard projective space $\RP^{31}$. Let us define $J_j
=Y_1(\frac{n+2}{64}-j+2)) \times Y_2(j+2)$ $Q =
Y_1(\frac{n+2}{64}+2) \times Y_2(\frac{n+2}{64}+2)$. For a given $j$ the natural inclusions $J_j \subset Q$ are well-defined. Let us denote the union of the considered inclusions by $J$.
The mapping $\varphi_j: X_j \to J_j$ is well-defined as the Cartesian product of the two following mappings. On the first coordinate the mapping is defined as the composition of the standard 2-sheeted covering $\RP^{d_1(j)} \to S^{\frac{n}{2}-64(j-1)}/i$ and the natural projection $S^{d_1(j)}/i \to Y_1(d_1(j))$. On the second coordinate the mapping is defined by the natural projection $\RP^{d_2(j)} \to Y_2(j+1)$.
The family of mappings $\varphi_j$ determines the mapping $\varphi: \hat X \to J$, because the restrictions of any two mappings to the common subspace in the origin coincide.
For $n+2 \ge 2^{13}$ the space $J$ embeddable into the Euclidean $n$-space by an embedding $i_J: J \subset \R^n$. Each space $Y_1(k)$, $Y_2(k)$ in the family is embeddable into the Euclidean $(2^6k -1-k)$–space. Therefore for an arbitrary $j$ the space $J_j$ is embaddable into the Euclidean space of dimension $n+126-\frac{n+2}{64}$. In particular, if $n+2 \ge
2^{13}$ the space $J_j$ is embeddable into $\R^n$. The image of an arbitrary intersection of the two embeddings in the family belongs to the standard coordinate subspace. Therefore the required embedding $i_J$ is defined by the gluing of embeddings in the family.
Let us describe the mapping $\hat h: \hat X \to \R^n$. By $\varepsilon$ we denote the radius of a (stratified) regular neighborhood of the subpolyhedron $i_J(J) \subset \R^n$. Let us consider a small positive $\varepsilon_1$, $\varepsilon_1 <<
\varepsilon$, (this constant will be defined below in the proof of Lemma 4) and let us consider a generic $PL$ $\varepsilon_1$–deformation of the mapping $i_J \circ \varphi:
\hat X \to J \subset \R^n$. T
| 3,770
| 1,791
| 3,303
| 3,342
| 3,964
| 0.768934
|
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|
to examine whether [Fe [i]{} ]{}abundances show any trend with line strength for different values of $\xi$. Since $\xi$ for the samples of OAO, [Fran[ç]{}ois ]{}, and Clegg et al. were estimated using the empirical relation of Edvardsson et al. (1993), the uncertainty of $\Delta\xi \simeq 0.22$ [${\rm km \: s^{-1}}$]{} was estimated for the above uncertainties of [$T_{\rm eff}$ ]{}and [log $g$ ]{}. The rms scatter for this relation was suggested to be about 0.3 [${\rm km \: s^{-1}}$]{} by Edvardsson et al. (1993), so that the total uncertainty became $\pm 0.37$ [${\rm km \: s^{-1}}$]{}. We then adopted $\Delta\xi = \pm 0.5$ [${\rm km \: s^{-1}}$]{} as the total uncertainty allowing for other possible errors.
The abundance errors of S and Fe for the HIRES sample and HD 111721 were calculated for these uncertainties, and are given in Table 6 as the error bars on the \[S/Fe\] values. The abundance errors for the remaining dwarfs sample were also evaluated and listed in Table 8. We regard the combined error of $\pm 0.16$ dex as a typical error for \[S/Fe [i]{}\] values in the sample of dwarfs.
Results and Discussion
======================
The results of abundance analyses of S and Fe are summarized in Tables 6 and 7 for six stars of the HIRES sample and the giant star HD 111721 and for 61 dwarfs, including the Sun, respectively. Since nine stars of the OAO sample overlap with those of the samples of [Fran[ç]{}ois ]{}(1987, 1988; four stars) and Clegg et al. (1981; five stars), the abundance results of the OAO sample were preferentially adopted for these stars. We will describe these results below and discuss the results for S and Fe abundances of all our samples.
NLTE Corrections
----------------
While the NLTE corrections of the S abundances, $\Delta$(S), are found to be in the range of $-0.09$ – 0.00 dex for all of our sample, most of them concentrate on the range of $-0.01$ – $-0.03$ dex. Consequently, neglect of NLTE effect does not produce significant errors leading to the wrong conclusions
| 3,771
| 1,501
| 3,235
| 3,644
| null | null |
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|
3.18 2.35 0--10 22,602 (122)
Trust in the Legal System: 0 = No trust at all; 10 = Complete trust 4.47 2.65 0--10 22,461 (122)
Trust in the Parliament: 0 = No trust at all; 10 = Complete trust 3.95 2.54 0--10 22,505 (122)
Age 49.82 17.96 18--96 23,008 (122)
Male 0.46 0.50 0--1 23,037 (122)
Unemployment (not being employed in the last 7 days) 0.08 0.28 0--1 23,042 (122)
Member of a discriminated group 0.06 0.24 0--1 22,914 (122)
Homicides per 100,000 residents: Number of homicides reported to police/ Resident Population \* 100,000 1.21 0.60 0.28--3.10 21,253 (110)
Property crimes per 100,000 residents: Number of robberies and vehicle thefts reported to police/ Resident Population \* 100,000 249.55 221.99 23.27--1,088.17 23,042 (122)
European Quality of Government---Corruption Pillar -0.38 1.19 -2.26--2.22 15,299 (82)
European Quality of Government---Impartiality Pillar -0.45 1.10 -2.58--2.04 15,299 (82)
Long-term Unemployment: Number of registered unemployed for 12 months or more/Number of active people 4
| 3,772
| 7,202
| 1,402
| 1,132
| null | null |
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|
een the layers has immediate implication for the windings along $z$-direction – the minimal length $M$ of the element $J_i$ must be $M=2$ in Eq.(\[GenM\]). Thus, the stiffness $u_r$ in the limit $u<<1$ becomes u\_r = 4[e]{}\^[-1/u\_V]{}= u\^2, \[UR3\] where the asymptotic expression $ u_V=\frac{1}{2 \ln (2/u)}$ [@Villain; @Kleinert] has been used. The results of the simulations is shown in Fig.\[figN2\].
-8mm
-5mm
The first striking feature to notice is that $u_r$ does not depend on the layers size $L$ over 2 orders of magnitude of $L$ and over 7 orders of $u_V$ (which is actually $\sim \ln(1/u)$ of the bare coupling). Second, the numerically found value $u_r$ follows the analytical result (\[UR3\]) with high accuracy – even for values $u_V \sim 1$. Both features are in the striking conflict with the RG prediction stating that $u_r$ should decay as $\propto L^{2(1-T/T_d)}\approx L^{-1.28} \to 0$ in the SP regime ($T>T_d$).
It should be also noted that the stiffness along the layers (\[stif22\]) remains finite and much larger than $u_r$, that is, $\rho=32.3 \pm 0.1$ for all simulated sizes from $L=8$ to $L=960$. This justifies the validity of Eq.(\[UR3\]) even in the case $u_V \sim 1$.
Extending the two-layer model to arbitrary number $N_z/2$ of pairs of layers {#Sec:Nz}
============================================================================
As it became evident from the previous analysis, no SP can occur in the double layer. Referring to the sketch of the possibilities, Fig. \[PD\], the option (b) is realized. Here we will address a possibility of SP in a $N_z$-layers setup. In other words, we will be looking for a behavior where the renormalized stiffness (the inter-layer Josephson coupling) $u_r$ decays as a function of $N_z$ in the limit $L \to \infty$, while the stiffness along planes remains finite.\[This would be a “weaker” version of the SP\].
RG solution {#sec:RGNz}
-----------
We consider the PBC setup: the odd $z=1,3,5,7,...$ and the even $z=2,4,6,..$ layers are characterized by th
| 3,773
| 1,690
| 1,080
| 3,795
| 1,699
| 0.786657
|
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|
1 290 0
S4 86.43 1316 1 729 1
MEAN 89.09 534 \<1 278 \<1
**NINAPRO data** **COMPLETE** **REDUCED**
**Accuracy** **SVs** **FSM errors** **SVs** **FSM errors**
S1 90.46 142 0 116 0
S2 93.49 116 4 107 4
S3 92.89 132 0 67 0
S4 92.15 226 0 169 0
S5 90.15 158 0 114 0
MEAN 91.83 155 \<1 112 \<1
---------------------- -------------- ---------------- --------- ---------------- -----
1. Introduction {#s0005}
===============
Papillomaviruses are circular double-stranded DNA viruses approximately 8 kb in length that are present in humans and many animal species [@bib1], [@bib2]. These viruses are host and site specific [@bib1], [@bib2]. They infect keratinocytes at either mucosal or cutaneous sites and most often cause benign proliferations, such as papillomas or plaques [@bib1], [@bib2]. Typically, these lesions regress, although rarely they can persist and progress to cancer [@bib2], [@bib3], [@bib4]. Human mucosal papillomaviruses are known to cause essentially all cases of cervical cancer [@bib4]. They are divided into the low-risk and high-risk types, with the high-risk types associated with a higher risk of cancer development [@bib2]. With human high-risk mucosal papillomaviruses, such as human papillomavirus (HPV) 16 and 18, viral integration into the host genome is a critical event in carcinogenesis, although the underlying mechanism is not entirely clear [@bib3], [@bib5]. Unlike human mucosal high-risk papillomavirus
| 3,774
| 2,639
| 3,463
| 3,685
| null | null |
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|
0.001
MCP-1 (pg/mL) 51.3 41.04 0.1
TNF-α (pg/mL) 50.06 44.3 0.3
i-PTH (pg/mL) 47.3 48.9 0.8
SBP (mmHg) 135.1 ± 24.6 130.4 ± 23.5 0.4
MBP (mmHg) 99.4 ± 14.8 97.2 ± 13.6 0.5
c-fPWV (m/s) 12.04 ± 1.9 \* 11.02 ± 1.7 0.01
Augmentation index (Aix, %) 25.04 ± 2.2 23.8 ± 2.3 0.03
Pulse pressure (PP, mmHg) 64.5 ± 22.7 \* 54.4 ± 19.1 0.03
ABI ±0.5 ±0.4 0.4
EF (%) 25.8 ± 2.7 \* 58.4 ± 3.07 0.001
E/A ratio 31.5 \* 54.8 0.001
Thickness of interventricular septum (mm) 13.7 ± 2.0 \*
| 3,775
| 5,541
| 1,049
| 3,309
| null | null |
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|
s $2$-complete with joint-set $B'$. Now, for each $i \in \{1,2,3\}$ we have $M' \con (B'-B_i')|(B_i' \cup X') \cong m'U_{1,2}$ and $B_i'$ is a basis of $M \con (B-B_i')$, so for each $x \in X'$, there exists a unique $b_i(x) \in B_i'$ such that $\{x,b_i(x)\}$ is a parallel pair of $M \con (B'-B_i')$. Let $B(x) = \{b_1(x),b_2(x),b_3(x)\}$. It follows for each $x \in X'$ that $B(x) \cup \{x\}$ is a circuit of $M'$. Moreover, since $b_i(x) \ne b_i(y)$ for $x \ne y$, the sets in $\cB = \{B(x): x \in X'\}$ are pairwise disjoint. Let $h = \binom{n_0}{3}$. Since $r(M') = 3m' \ge n_0 + 3h$, there exist $x_1, \dotsc, x_h \in X'$ and a set $B_0 \subseteq B'$ with $|B_0| = n_0$ such that $B(x_1), \dotsc, B(x_h)$ are disjoint from $B_0$. By Lemma \[buildcomplete\] with $a = 2$, the matroid $M'$ has a $3$-complete minor of rank $n_0$. The lemma now follows from the definition of $n_0$ and Lemma \[3completewincor\].
We now prove a result that will imply Theorem \[main1\].
\[maintech\] Let $s,n \ge 2$ be integers and let $h = f_{\ref{threenonsingular}}(s,n)$. If $M$ is a matroid with a spanning $B$-clique restriction having no $U_{s,2s}$-minor and no rank-$n$ projective geometry minor, then there is a set $\wh{B} \subseteq B$ and a $\wh{B}$-clique $\wh{M}$ such that $\dist(M,\wh{M}) \le 7 h$. Moreover, there are disjoint sets $C_1,C_2 \subseteq E(M)$ satisfying $|C_1| \le 3h$ and $|C_2| \le h$ such that $\wh{M}$ is a $C_2$-shift of $M \dcon C_1$.
Let $E = E(M)$ and let $X \subseteq E$ be maximal so that there exist disjoint sets $B_1,B_2,B_3 \subseteq B$ for which $|B_i| = |X|$ and $X$ is independent in $M \con (B-B_i)$ for each $i \in \{1,2,3\}$. Since $M$ is $2$-complete with joint-set $B$, Lemma \[threenonsingular\] gives $|X| \le h$. Let $\bar{B} = B_1 \cup B_2 \cup B_3$ and $\wh{B} = B - \bar{B}$; note that $\wh{B}$ is a basis of $M \con (X \cup (\bar{B} - B_i))$ for all $i,j \in \{1,2,3\}$. For each $f \in (E(M) - B \cup X)$, let $H_i(f) \subseteq \wh{B}$ be such that $H_i(f) \cup \{f\}$ is the fundamental circuit
| 3,776
| 3,185
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Other modifications we have added are summarized as follows.
### Chemical and thermal processes {#sec:chemistry}
To solve the chemical and thermal processes, we use the same methods developed in [@Hosokawa:2016aa] with several modifications. Unlike in [@Hosokawa:2016aa], we omit ${\mathrm{H}}_2$ chemistry assuming that ${\mathrm{H}}_2$ is completely photo-dissociated by the central FUV irradiation.[^2] We have added the ${\mathrm{He}}$ chemistry, since hard UV photons from BH accretion discs create a large helium photoionized region embedded in an [H[ii]{} ]{}region.
In summary, we solve the chemical network with six species: ${\mathrm{H}}$, ${\mathrm{H^+}}$, ${\mathrm{e}}$, ${\mathrm{He}}$, ${\mathrm{He^+}}$, and ${\mathrm{He^{2+}}}$, which consists of the following chemical processes: photoionization of ${\mathrm{H}}$, ${\mathrm{He}}$ and ${\mathrm{He^+}}$; collisional ionization of ${\mathrm{H}}$, ${\mathrm{He}}$ and ${\mathrm{He^+}}$; recombination of ${\mathrm{H^+}}$, ${\mathrm{He^+}}$ and ${\mathrm{He^{2+}}}$. Accordingly we consider the following thermal processes: photoionization heating of ${\mathrm{H}}$, ${\mathrm{He}}$, and ${\mathrm{He^+}}$; recombination cooling of ${\mathrm{H^+}}$, ${\mathrm{He^+}}$, and ${\mathrm{He^{2+}}}$; excitation cooling of ${\mathrm{H}}$, ${\mathrm{He}}$, and ${\mathrm{He^+}}$; collisional ionization cooling of ${\mathrm{H}}$, ${\mathrm{He}}$, and ${\mathrm{He^+}}$; free-free cooling of ${\mathrm{H}}$, ${\mathrm{He}}$, and ${\mathrm{He^{+}}}$; Compton cooling by cosmic microwave background (CMB) photons. Complete lists of our adopted chemical and thermal processes are available in Appendix \[sec:chem\_detail\].
We turn off the cooling when the temperature falls below $10^4{\,\mathrm{K}}$, as in the previous 2D simulations [e.g., @Park:2011aa]. We neglect secondary ionization and heating caused by X-ray photoionization [@Shull:1979ab; @Shull:1985aa; @Ricotti:2002aa]. We have confirmed with test calculations that these processes hardly affect the gas dynamics though the i
| 3,777
| 2,745
| 4,293
| 3,767
| 3,674
| 0.770782
|
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|
een the RCTs approved in Freiburg and RCTs approved in Canada (Hamilton) and Switzerland (Basel, Lausanne, Zurich, and Lucerne) at the same time period. Details about this cohort of 1017 RCTs were reported earlier \[[@pone.0165605.ref001], [@pone.0165605.ref012]\].
Results {#sec008}
=======
Included studies {#sec009}
----------------
We identified 917 studies that were approved by the REC in Freiburg ([Fig 2](#pone.0165605.g002){ref-type="fig"}). After excluding studies that were never started, still on-going, in vitro or other non-human studies, or of cross-sectional or retrospective design, our final data set for analysis comprised 547 prospective longitudinal studies. Of those, 306 were RCTs and 241 were NPSs (27 controlled trials, 158 single arm trials, and 56 cohort studies).
{#pone.0165605.g002}
Study characteristics {#sec010}
---------------------
Most NPSs (92%) and RCTs (92%) were conducted in adults. NPSs were on average smaller than RCTs, more frequently single centre and pilot studies, and less frequently industry-sponsored ([Table 1](#pone.0165605.t001){ref-type="table"}). Most NPSs and RCTs were in oncology (20%; 22%) and neurology (11%; 9%). NPSs were less often conducted in cardiovascular medicine (3%; 9%) and more often in dental medicine (6%; 2%). Fourty-nine precent of NPSs were published compared to 59% of RCTs.
10.1371/journal.pone.0165605.t001
###### Study characteristics.
{#pone.0165605.t001g}
Study characteristics REC Freiburg Other RECs[^1^](#t001fn001){ref-type="table-fn"}
------------------------------------------------ -------------- -------------------------------------------------- ------------- ------------- -------------- --------------
**Total n** 27 (100) 158 (100) 56 (100) 241 (100) 306 (100) 711 (100)
Sample size provided
| 3,778
| 2,276
| 1,227
| 3,488
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|
oduction functional by $$I(\theta) = \int \theta {\left\vert{\nabla}\log \theta + \eta\right\vert}^2 d\eta. \label{def:I_linear}$$ In the nonlinear case $m > 1$, the corresponding quantities are, $$H(\theta) = \frac{1}{m-1}\int \theta^m d\eta + \frac{1}{2}\int {\left\vert\eta\right\vert}^2 \theta d\eta, \label{def:H}$$ and the entropy production functional, $$I(\theta) = \int u{\left\vert\frac{m}{m-1}{\nabla}u^{m-1} + \eta\right\vert}^2 d\eta. \label{def:I}$$ In the nonlinear case, these entropies were originally introduced for studying in [@Newman84; @Ralston84]. Both and are displacement convex [@McCann97] and in fact, is a gradient flow for or in the Euclidean Wasserstein distance [@Otto01; @AmbrosioGigliSavare; @CarrilloMcCannVillani03; @CarrilloMcCannVillani06], and if $f(\tau,\eta)$ solves , then $$\frac{d}{d\tau}H(f(\tau)) = -I(f(\tau)).$$ For a given mass $M$, has a unique non-negative minimizer which is the ground state $\theta_M$. That is, if we define the relative entropy $$H(\theta|\theta_M) = H(\theta) - H(\theta_M), \label{def:relativeH}$$ then $H(\theta|\theta_M) \geq 0$ with equality if and only if $\theta = \theta_M$ [@CarrilloToscani00; @DelPinoDolbeault02]. In order to estimate a convergence rate, it is therefore sensible to measure how quickly $H(\theta|\theta_M) \rightarrow 0$. Following the methods of [@CarrilloToscani00; @CarrilloEntDiss01; @CarrilloMcCannVillani03; @CarrilloMcCannVillani06], this is made possible by the following two theorems. The first relates the entropy production functional to the relative entropy . This represents a generalization of the Gross logarithmic inequality [@Gross75] (see also [@DelPinoDolbeault02]).
\[thm:rel\_entropy\] Let $f\in L_+^1({\mathbb R}^d)$ with ${\|f\|}_1 = M$ and let $\theta_M$ be the ground state Barenblatt solution with mass $M$. Then, $$H(f|\theta_M) \leq \frac{1}{2} I(f). \label{ineq:relative_entropy}$$
For the Fokker-Plank equation , Theorem \[thm:rel\_entropy\] implies $H(\theta(\tau)|\theta_M) \lesssim e^{-2\tau}$. The (generalized)
| 3,779
| 2,230
| 1,758
| 3,630
| 2,019
| 0.783472
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^k-B^k_j)$ for all $j \in \{k+1,\dotsc,3\}$, and so that $M^k \con (B^k-B^k_j)|(B^k_j \cup X^k) \cong m_kU_{1,2}$ for all $j \in \{1, \dotsc, k\}$. This clearly holds for $k = 0$, and setting $k = 3$ will give the claim. The argument essentially consists of three aplications of Theorem \[selfdual\].
Suppose that these $M^k,B^k,B^k_i$ and $X^k$ exist for some $k \in \{0,1,2\}$. Consider the matroid $P = M^k \con (B^k - B^k_{k+1}) | (B^k_{k+1} \cup X^k)$. The sets $B^k_{k+1}$ and $X^k$ are disjoint bases of $P$, and $r(P) \ge m_k = 4^{(s4^s)^{m_{k+1}}}$ so by Theorem \[selfdual\] there are disjoint sets $C,D \subseteq E(P)$ such that $P \con C \del D \cong m_{k+1}U_{1,2}$ and $P \con C \del D$ has both $B^k_{k+1}-C \cup D$ and $X^k - C \cup D$ as bases. Choose $C$ independent and $D$ co-independent in $P$, so $|C| = |D| = m_k - m_{k+1}$. Let $X^{k+1} = X^k - (C \cup D)$ and $B^{k+1}_{k+1} = B^k_{k+1} - (C \cup D)$.
For each $j \in \{1,2,3\}$, let $Q^j = (M^k \con C) \con (B^k - B^k_j) | ((B^k_j-D) \cup X^{k+1})$. For $j \le k$, the matroid $Q^j$ is a minor of $M^k \con (B^k - B^k_j) | (B^k_j \cup X^{k}) \cong m_kU_{1,2}$ obtained by removing $(m_k - m_{k+1})$ elements of $X^k$. If $B^{k+1}_j$ is defined to be the set of the $m_{k+1}$ elements of $B^k_j$ that are not parallel to any of these removed elements, then we have $Q^j \con (B^k_j - B^{k+1}_j) \cong m_{k+1}U_{1,2}$. For $j = k+1$ we have $Q^j \cong m_{k+1}U_{1,2}$ by definition. For $j > k+1$, we have $C \subseteq (B^k-B^k_j) \cup (X^k-X^{k+1})$, and $X^k$ is independent in $M^k \con (B^k - B^k_j)$, so $X_{k+1}$ is independent in $Q^j$. Thus, there is an $m_{k+1}$-element set $B^{k+1}_j \subseteq B^k_j$ for which both $B^{k+1}_j$ and $X^{k+1}$ are independent in $Q^j \con (B^k_j - B^{k+1}_j)$. Now setting $B^{k+1} = \cup_{i=1}^3 B^{k+1}_i$ and $M^{k+1} = (M^k \con C) \con (B^k - B^{k+1})$, together with the $B^{k+1}_j$ and $X^{k+1}$ as defined, gives the claim.
Note, since $M'$ is a contraction-minor of $M$ and $B' \subseteq B$, that the matroid $M'$ i
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and (c) $\rho_{d_{0\sigma},c^\dagger_{0\sigma}}(\w)$. The spectral functions have been calculated with NRG. We set $U/\Gamma_0=10$, $\omega_0/\Gamma_0=0.1$ and different colors indicate various unconventional Holstein couplings $\lambda_c$. []{data-label="NRG_fig2"}](fig17-Elastic-Lc){width="50.00000%"}
The elastic part of the transmission function $\tau^{(0)}_\sigma(\omega)$ comprises three different contributions $$\begin{aligned}
\label{eq:transmission_elastic-two_channel_model}
\tau_\sigma^{(0)}(\omega)& =&
t_{d}^2\, \rho_{d_{0\sigma},d^\dagger_{0\sigma}}(\w)
+t_{c}^2 \rho_{c_{0\sigma}, c_{0\sigma}^\dagger}(\w)
\nonumber\\
&&
+t_{d} t_{c}\big[ \rho_{c_{0\sigma},d^\dagger_{0\sigma}}(\w)+ \rho_{d_{0\sigma},c_{0\sigma}^\dagger}(\w)\big],\end{aligned}$$ stemming from the tunneling into the molecular orbital $d_{0}$ and into the effective local surface orbital $c_0$, introduced in Eq. and implying $d_{1\sigma} =c_{0\sigma}$ in $H_T$, Eq. . The three relevant spectral functions are plotted versus frequency for two different values of $\lambda_c$ and a fixed $U$ in Fig. \[NRG\_fig2\]. For $ \rho_{d_{0\sigma},d^\dagger_{0\sigma}}(\w)$, displayed in panel (a), the narrowing of the Kondo resonance with increasing $\lambda_c$ is illustrated. The increase of the peak height is connected to the reduction of $\Gamma_{\rm eff}$, as discussed extensively in the previous section. The corresponding anti-resonance in $\rho_{c_{0\sigma}, c_{0\sigma}^\dagger}(\w)$ is clearly visible in panel (b) of Fig. \[NRG\_fig2\]. This anti-resonance can be associated with the contribution to the Kondo screening by the electrons in local substrate orbital. The mixed contribution $ \rho_{c_{0\sigma},d^\dagger_{0\sigma}}(\w)= \rho_{d_{0\sigma},c_{0\sigma}^\dagger}(\w)$ in panel (c) is an antisymmetric function and thus its integrated spectral weight vanishes. This contribution captures the interference between the two possible tunneling paths and generates Fano lineshapes in Eq. \[eq:transmission\_elastic-two\_channel\_model\]. We
| 3,781
| 1,179
| 3,218
| 3,797
| 3,921
| 0.769288
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\quad x{\mathbin\vdash}y{\mathbin\vdash}z + y{\mathbin\vdash}x{\mathbin\vdash}z,\ x>y; \quad x{\mathbin\vdash}x{\mathbin\vdash}y; \quad x{\mathbin\dashv}y{\mathbin\dashv}y$$ is a Gröbner-Shirshov basis. Reduced words are of the form $$\dot x_1x_2\dots x_k,\ k\ge 1,\ x_2<x_3<\dots < x_k,$$ and the set of all reduced words by [@BokutChenLiu:08] is a linear basis of the algebra $\Lambda_X$.
Define an involution $*$ on ${\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ in the following way: $$(x_{i_1}\ldots \dot{x}_{i_k}\ldots
x_{i_n})^*=x_{i_n}\ldots \dot{x}_{i_k}\ldots x_{i_1},$$ and extend to dipolynomials by linearity. This mapping reverses words and signs of dialgebraic operations. It is easy to check that the mapping $*$ satisfies properties of an involution (\[eq:DefOfInvolution\]). Through ${\mathrm{Di}}{\mathrm{H}}\,\langle X\rangle$ we denote the Jordan dialgebra $H({\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle,*)$ of symmetric elements from ${\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ with respect to $*$ with the product (\[eq:QuasiJordanProduct\]).
Further $\{u\}$ denotes $ u+u^*$ where $u$ is a basic word from ${\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$. Note that $\{u\}=\{u^*\}$.
An analogous involution on ${\mathrm{As}}\,\langle X\rangle$ we denote by $*$ too. It acts like as $$(x_{i_1}\ldots x_{i_k}\ldots x_{i_n})^*=x_{i_n}\ldots x_{i_k}\ldots x_{i_1},$$ on monomials and extends to polynomials by linearity.
The next theorem is an analogue of the classical Cohn’s Theorem [@Cohn:54 Theorems 4.1 and 4.2] that describes Jordan polynomials from $\le 3$ variables as symmetric elements of the free associative algebra.
\[thm:CohnForDialgebra\] For any set $X$ we have ${\mathrm{Di}}{\mathrm{SJ}}\,\langle
X\rangle\subseteq{\mathrm{Di}}{\mathrm{H}}\,\langle X\rangle$. If $|X|\le 2$ then there is an equality, if $|X|>2$ then there is a strict inclusion. Also, for any $X$ we have that ${\mathrm{Di}}{\mathrm{H}}\,\langle X\rangle$ is generated by $X$ and dotted tetrads $\{\dot xyzt\}$, $\{\dot xxyz\}$, wher
| 3,782
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of the complement of the ramification locus (a projective curve) and their influence on the topology of the original surface as a branched covering of the projective plane. He realized that not only the type of singularities of the branched locus was relevant, but their position as well ([@Zariski-irregularity]). In particular, he proved that the cyclic branched cover of an irreducible curve of degree $6d$ with only nodes and cusps is irregular, it has non-trivial first cohomology group, if the *effective dimension* of the space of curves of degree $5d-3$ passing through the cusps is larger than its *expected* (or *virtual*) dimension. This difference was called *superabundance*. More precise descriptions of the irregularity of cyclic branched coverings of curves in $\PP^2$ have been given by Libgober [@Libgober-alexander], Esnault [@es:82], Loeser-Vaquie [@Loeser-Vaquie-Alexander], and Sabbah [@Sabbah-Alexander], in general smooth surfaces by Esnault-Viehweg [@Esnault-Viehweg82] or even for general abelian branched coverings by Libgober [@Libgober-characteristic]. It is worth pointing out that very concrete formulas were given in [@Artal94; @Libgober-characteristic] for the particular case of curves on $\PP^2$. These formulas combine a local ingredient coming from a resolution of the singularities of the branched locus and a global one measuring the superabundance of a certain linear system of curves on $\PP^2$.
In this paper we address the problem of describing the irregularity of cyclic branched coverings of singular surfaces and use this description to find formulas for the particular case of the weighted projective plane. The main result of this paper is presented in Theorem \[thm:conucleo\_singular\], where we describe the dimension of the equivariant spaces of the first cohomology of a $d$-cyclic cover $\rho:\tilde X\to \PP^2_w$ ramified along a (not necessarily reduced) curve $\mathcal{C} = \sum_j n_j \mathcal{C}_j$. The cover $\rho$ naturally defines a divisor $H$ such that $dH$ is linearly equ
| 3,783
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| 4,081
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ons, they have turned to theorem proving approach and used the PVS theorem prover to analyze the DEOS scheduler [@Ha04]. They model the operations of the scheduler in PVS and the execution timeline of DEOS using a discrete time state-transition system. Properties of time partitioning (TP) are formulated as predicates on the set of states and proved to hold for all reachable states. The corresponding PVS proofs consist of the base step and the inductive step as follows.
$$init\_invariant: init(s) \longrightarrow TP(s)$$ $$transition\_invariant: TP(ps) \wedge transition(ps, s) \longrightarrow TP(s)$$
The $TP$ predicate is defined as follows. $$\begin{aligned}
good&Commitment(s,period) \equiv \\
& commitment(s,period) \leq remainTime(s,period)\end{aligned}$$ $$\begin{aligned}
TP(s,period) \equiv & goodCommitment(s,period) \ \vee \\
& \forall t. \ period \ (threadWithId(s,t)) \leq period \\
& \longrightarrow satisfied(s,t)\end{aligned}$$ $$TP(s) \equiv \forall period. \ TP(s,period)$$
The entire collection of theories has a total 1648 lines of PVS code and 212 lemmas. In addition to the inductive proofs of the time partitioning invariants, they use a feature-based technique to model state-transition systems and formulate inductive invariants. This technique facilitates an incremental approach to theorem proving that scales well to models of increasing complexity.
- A two-level scheduler for VxWorks kernel
In [@Asberg11], a hierarchical scheduler executing in the WindRiver VxWorks kernel has been modeled using task automata and model checked using the Times tool. The two-level hierarchical scheduler uses periodic/polling servers (PS) and fixed priority preemptive scheduling (FPPS) of periodic tasks for integrating real-time applications. In their framework, the *Global scheduler* responds for distributing the CPU capacity to the servers (the schedulable entity of a subsystem). Servers are allocated a defined time (budget) of every predefined period. Each server comprises a *Local scheduler* which schedules the
| 3,784
| 5,051
| 3,799
| 3,463
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|
^2-2b^2=1\}$ for a flat $A$-algebra $R$. Thus we cannot guarantee that $a-1$ is contained in the ideal $(2)$, which should be necessary in order that $(a, b)$ is an element of $\underline{G}(R)$.
The special fiber of the smooth integral model {#sf}
==============================================
In this section, we will determine the structure of the special fiber $\tilde{G}$ of $\underline{G}$ by determining the maximal reductive quotient and the component group when $E/F$ satisfies *Case 2*, by adapting the approach of Section 4 of [@C1] and Section 4 of [@C2]. From this section to the end, the identity matrix is denoted by id.
The reductive quotient of the special fiber {#red}
-------------------------------------------
Assume that $i=2m$ is even. Recall that $Z_i$ is the sublattice of $B_i$ such that $Z_i/\pi B_i$ is the radical of the quadratic form $\frac{1}{2^{m+1}}q$ mod 2 on $B_i/\pi B_i$, where $\frac{1}{2^{m+1}}q(x)=\frac{1}{2^{m+1}}h(x,x)$.
\[t43\] Assume that $i=2m$ is even. Let $\bar{q}_i$ denote the nonsingular quadratic form $\frac{1}{2^{m+1}}q$ mod 2 on $B_i/Z_i$. Then there exists a unique morphism of algebraic groups $$\varphi_i:\tilde{G}\longrightarrow \mathrm{O}(B_i/Z_i, \bar{q}_i)_{\mathrm{red}}$$ defined over $\kappa$, where $\mathrm{O}(B_i/Z_i, \bar{q}_i)_{\mathrm{red}}$ is the reduced subgroup scheme of $\mathrm{O}(B_i/Z_i, \bar{q}_i)$, such that for all étale local $A$-algebras $R$ with residue field $\kappa_R$ and every $\tilde{m} \in \underline{G}(R)$ with reduction $m\in \tilde{G}(\kappa_R)$, $\varphi_i(m)\in \mathrm{GL}(B_i\otimes_AR/Z_i\otimes_AR)$ is induced by the action of $\tilde{m}$ on $L\otimes_AR$ (which preserves $B_i\otimes_AR$ and $Z_i\otimes_AR$ by the construction of $\underline{M}$).
The proof of this theorem is similar to that of Theorem 4.3 of [@C2]. Thus we only provide the image of an element $m$ of $\tilde{G}(\kappa_R)$ in $\mathrm{O}(A_i/Z_i, \bar{q}_i)_{\mathrm{red}}(\kappa_R)$, where $R$ is an étale local $A$-algebra with $\kappa_R$ as its residue field.
| 3,785
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| 903
| 3,571
| null | null |
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|
prob(b>\{c,d\})\, \,\prob(c>d) \nonumber \\
&=& \frac{e^{\theta_a}}{(e^{\theta_a}+e^{\theta_b}+e^{\theta_c}+e^{\theta_d} ) } \,
\frac{e^{\theta_b}}{(e^{\theta_b}+e^{\theta_c}+e^{\theta_d} )} \,
\frac{e^{\theta_c}}{(e^{\theta_c}+e^{\theta_d})} \, \;.\end{aligned}$$ We use the notation $(a>b)$ to denote the event that $a$ is preferred over $b$. In general, for user $j$ presented with offerings $S_j$, the probability that the revealed preference is a total ordering $\sigma_j$ is $\prob(\sigma_j) = \prod_{i\in \{1,\ldots,\kappa_j-1\}} (e^{\theta_{\sigma^{-1}(i)}})/(\sum_{i'=i}^{\kappa_j}e^{\theta_{\sigma^{-1}(i')}}) $. We consider the true utility $\theta^* \in \Omega_b$, where we define $\Omega_b$ as $$\begin{aligned}
\Omega_b &\equiv& \Big\{ \, \theta \in \reals^d \,\big|\, \sum_{i\in[d]} \theta_i=0 \,,\, |\theta_i| \leq b \text{ for all $i\in[d]$ } \,\Big\} \;.\end{aligned}$$ Note that by definition, the PL model is invariant under shifting the utility $\theta_i$’s. Hence, the centering ensures uniqueness of the parameters for each PL model. The bound $b$ on the dynamic range is not a restriction, but is written explicitly to capture the dependence of the accuracy in our main results.
We have $n$ users each providing a partial ordering of a set of offerings $S_j$ according to the PL model. Let ${\cG}_j$ denote both the DAG representing the partial ordering from user $j$’s preferences. With a slight abuse of notations, we also let $\cG_j$ denote the set of rankings that are consistent with this DAG. For general partial orderings, the probability of observing $\cG_j$ is the sum of all total orderings that is consistent with the observation, i.e. $\prob(\cG_j)=\sum_{\sigma \in \cG_j} \prob(\sigma)$. The goal is to efficiently learn the true utility $\theta^*\in\Omega_b$, from the $n$ sampled partial orderings. One popular approach is to compute the maximum likelihood estimate (MLE) by solving the following optimization: $$\begin{aligned}
\underset{\theta \in \Omega_b }{\text{
| 3,786
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| 3,099
| 3,398
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directions as well so that the two-point energy correlator cannot be identically zero at non-coincident points. Thus, we have a contradiction and we are compelled to conclude that in CFTs with finite energy correlators the three-point function of the stress tensor cannot be purely bosonic.
Let us now relax the finiteness condition and allow for integrable singularities of the type described above. Our analysis of positivity above is valid only for the points where the energy correlator is finite. We concluded that at those points the energy correlator is necessarily zero. Thus, the energy correlator is possibly nonzero only in the set of isolated points where it is singular. Combined with our assumption that energy correlators are distributions it implies that at each of those points the functions of the cross ratio $f_j(\xi)$ are given by $\sum_{n=0}^{N} c^n_j \delta^{(n)}(\xi - \xi_i^*)$ with some finite $N$ (see e.g. ). In appendix C we show that only the $\delta(1- \xi)$ term is consistent with the positivity of energy correlators. Thus, in the $\vec q = 0$ reference frame the two-point energy correlator takes the form
Due to momentum conservation both detectors trigger the same momentum at each event and, thus, the same energy . We conclude that the following should be true where ${q^0 \over 2}$ is the energy measured by each detector in every event and we integrate over a small region $\delta \Omega$ around the point $\vec n_1 \vec n_2 = -1$. Together with the previous constraints it fixes the two-point energy correlator to be the one of the free boson theory In the covariant language $\xi = 1$ corresponds to $n_2^{\mu} \propto q^{\mu} - {q^2 \over 2 q.n_1} n_1^{\mu}$.
We can proceed and fix higher point energy correlators using the total flux bound. Notice that for any $\vec n_1 \neq \vec n_2 \neq \vec n_3$ we always have a pair of detectors for which $\vec n_i . \vec n_j \neq -1$. Without a loss of generality we can think of them as being $\vec n_2$ and $\vec n_3$. But then we have for any $\delta \O
| 3,787
| 2,889
| 3,307
| 3,215
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\- -0.020 (0.022)
Age Median \- -0.042 (0.032) \- -0.051 (0.027)
Area in Km2 \- -0.002 (0.002) \- 0.001 (0.002)
Long-term Unemployment \- -0.008 (0.025) \- 0.045 (0.025)
**Impartiality Pillar** \- **0.304**[\*\*\*](#t004fn004){ref-type="table-fn"} **(0.063)** \- \-
**Corruption Pillar** \- \- \- **0.477**[\*\*\*](#t004fn004){ref-type="table-fn"} **(0.070)**
*SOCIAL TRUST ON*
\% Migrants from outside EU -6.292 (3.840) 0.539 (4.627) -1.859 (3.381) 3.623 (4.253)
\% Migrants from inside EU -1.238 (5.325) 0.138 (5.454) 1.622 (6.001)
| 3,788
| 5,772
| 865
| 3,060
| 3,992
| 0.768748
|
github_plus_top10pct_by_avg
|
o feel lonely. This was the case for all family members. Finally, there was also a significant effect of the *age of the ill child at diagnosis* \[χ^2^(1) = 4.58, *p* = 0.03\]: the older the ill child was at diagnosis, the less all family members reported to feel lonely. None of the other variables were significantly related to loneliness (all χ^2^ \< 3.7, all *p* \> 0.05).
### Uncertainty
The interaction effects between *family functioning (FRI and FSI)* and *family member* \[*FRI:*χ^2^(3) = 0.92, *p* = 0.82; *FSI:*χ^2^(3) = 2.55, *p* = 0.47\], between *cancer appraisal* and *family member* \[χ^2^(3) = 2.82, *p* = 0.42\] and between *family functioning (FRI and FSI)* and *cancer appraisal* \[*FRI:*χ^2^(1) = 1.08, *p* = 0.30; *FSI:*χ^2^(1) = 1.60, *p* = 0.21\] were not significant and were subsequently left out of the final model. In the final model, 18% of the variance in *uncertainty* was attributable to differences between family members (regardless of which family one belonged to) and 0% was attributable to differences between families.
There was a significant effect of *cancer appraisal* upon uncertainty in all family members \[χ^2^(1) = 118.66, *p* \< 0.001\]: the more one perceived the illness as uncontrollable and the less as manageable, the more s/he reported to feel insecure. There was also a significant effect of *time since diagnosis* \[χ^2^(1) = 8.20, *p* = 0.004\], indicating that participants reported less uncertainty if more time had passed since diagnosis. Finally, there was also a significant effect of *family member* \[χ^2^(3) = 9.99, *p* = 0.02\]. This will be explained below (see section "Similarities and Differences Across Members Within One Family"). None of the other variables were significantly related to uncertainty (all χ^2^ \< 1.0, all *p* \> 0.30).
### Helplessness
The interaction effects between *family functioning (FRI and FSI)* and *family member* \[*FRI:*χ^2^(3) = 3.42, *p* = 0.33; *FSI:*χ^2^(3) = 3.47, *p* = 0.32\], between *cancer appraisal* and *family member*
| 3,789
| 1,273
| 3,928
| 3,637
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|
field, of the form $$B \: \mapsto \: B \: + \: d \Lambda,$$ the Chan-Paton gauge field must necessarily transform as $$A \: \mapsto \: A \: - \: \Lambda$$ in order to preserve gauge-invariance on the open string worldsheet, and such affine translations correspond, in terms of transition functions, to the modified overlap condition equation (\[cocyc1\]). However, although such twistings are possible for D-branes, no such twisting is ordinarily possible in heterotic strings, because the heterotic gauge field never picks up affine translations across coordinate patches – the heterotic gauge field and the heterotic $B$ field are related in a very different fashion than in D-branes.
A second notion of twisting appears when discussing gerbes. Consider the weighted projective stack ${\mathbb P}^N_{[k,\cdots,k]}$, a ${\mathbb Z}_k$ gerbe on ${\mathbb P}^N$, described physically by an analogue of the supersymmetric ${\mathbb P}^N$ model in which chiral superfields have charge $k$ instead of $1$, as discussed earlier. Now, the total space of a line bundle ${\cal O}(-n) \rightarrow
{\mathbb P}^N$ can be described as a quotient of $N+1$ fields $\phi_i$ and one field $p$ of charges $1$, $-n$, respectively. Consider instead a quotient of the fields above in which the $\phi_i$ have charge $k$ (and so describe ${\mathbb P}^N_{[k,\cdots,k]}$), and the field $p$ has charge $-1$. This quotient is the total space of a line bundle on the gerbe sometimes denoted ${\cal O}(-1/k)$. (We will discuss line bundles on gerbes in more detail in appendix \[app:linebundles\].)
We can understand this second notion of twisting in much greater generality, as follows. First, for any stack $\mathfrak{X}$ presented as $\mathfrak{X} = [X/G]$ for some space $X$ and group $G$, a vector bundle (sheaf) on $\mathfrak{X}$ is the same as a $G$-equivariant vector bundle (sheaf) on $X$. Now, suppose that $G$ is an extension $$1 \: \longrightarrow \: K \: \longrightarrow \: G \: \longrightarrow \:
H \: \longrightarrow \: 1,$$ where $K$ acts trivially on
| 3,790
| 2,463
| 3,503
| 3,438
| null | null |
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ve where and
Theorem \[t2\] is a direct consequence of the following multivariate version, which will be proved in Section 5.1.
\[t4\] Let $X_1, X_2, \dots$ be an i.i.d. sequence of $d$-dimensional random vectors under a sublinear expectation $\E$ such that for a family of linear expectations $\{E_\theta: \theta\in \Theta\}$. Let be all possible means of $X_1$. Let $\mathcal{P}$ be the convex hull of the closure of $\mathcal{M}_1$. We have, for $\varphi: \mathbb{R}^d\to \mathbb{R}$ differentiable such that the gradient $D\varphi: \mathbb{R}^d\to \mathbb{R}^d$ is a Lipschitz function, where $\mu_i:=\operatorname*{argsup}_{\mu\in \mathcal{P}}\big\{\mu\cdot D\varphi\big[\frac{\sum_{j=1}^{i-1}(X_j-\mu_j)}{n}\big]\big\}$ (if the $\operatorname*{argsup}$ is not unique, choose any value), $\lambda_*$ is the supremum norm of the operator norm of the Hessian $D^2\varphi$, and $diam(\mathcal{P})$ denotes the diameter of $\mathcal{P}$.
Central limit theorem with rate of convergence
==============================================
As explained in the Introduction, in the special case where $\varphi$ is a convex or concave test function, the limit in Peng’s CLT in [002]{} is a usual normal distribution. We first provide a rate of convergence for this special case. Moreover, unlike in [002]{}, we do not need to impose the [*identically distributed*]{} assumption.
\[t6\] Suppose $X_1,\dots, X_n$ are independent under a sublinear expectation $\E$ with Let For convex test functions $\varphi(\cdot)\in lip(\mathbb{R})$, we have where $Z$ is a standard Gaussian random variable. For concave functions $\varphi$, if we let then
The proof of Theorem \[t6\] follows from a similar and simpler proof of Theorem \[t1\] below and is deferred to Section 5.2. Theorem \[t6\] has the following corollary if the $X_i$’s are assumed to be i.i.d.
\[t0\] Under the conditions of Theorem \[t6\], suppose further that $X_1,\dots, X_n$ are i.i.d., and denote Then, for a convex test function $\varphi\in lip(\mathbb{R})$, we have where $Z$ is a
| 3,791
| 2,117
| 1,142
| 3,674
| null | null |
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|
& & \lim(\mathcal{F})\subseteq\lim(\mathcal{G})\\
\textrm{adh}(\zeta)\subseteq\textrm{adh}(\chi) & & \textrm{adh}(\mathcal{G})\subseteq\textrm{adh}(\mathcal{F});\end{aligned}$$
a filter $\mathcal{G}$ finer than a given filter $\mathcal{F}$ corresponds to a subnet $\zeta$ of a given net $\chi$. The implication of this correspondence should be clear from the association between nets and filters contained in Definitions A1.10 and A1.11.
A filter-base in $X$ is a *nonempty* family $(B_{\alpha})_{\alpha\in\mathbb{D}}=\!\,_{\textrm{F}}\mathcal{B}$ of subsets of $X$ characterized by
(FB1) There are no empty sets in the collection $_{\textrm{F}}\mathcal{B}$: $(\forall\alpha\in\mathbb{D})(B_{\alpha}\neq\emptyset)$
(FB2) The intersection of any two members of **$_{\textrm{F}}\mathcal{B}$ **contains another member of $_{\textrm{F}}\mathcal{B}$: $B_{\alpha},B_{\beta}\in\,_{\textrm{F}}\mathcal{B}\Rightarrow(\exists B\in\,_{\textrm{F}}\mathcal{B}\!:B\subseteq B_{\alpha}\bigcap B_{\beta})$;
hence any class of subsets of $X$ that does not contain the empty set and is closed under finite intersections is a base for a unique filter on $X$; compare the properties (NB1) and (NB2) of a local basis given at the beginning of this Appendix. Similar to Def. A1.1 for the local base, it is possible to define
**Definition A1.9.** *A filter-base* $_{\textrm{F}}\mathcal{B}$ *in a set $X$ is a subcollection of the filter* $\mathcal{F}$ *on $X$ having the property that each $F\in\mathcal{F}$ contains some member of* $_{\textrm{F}}\mathcal{B}$*.* *Thus* $$_{\textrm{F}}\mathcal{B}\overset{\textrm{def}}=\{ B\in\mathcal{F}\!:B\subseteq F\textrm{ for each }F\in\mathcal{F}\}\label{Eqn: FB}$$ *determines the filter* $$\mathcal{F}=\{ F\subseteq X\!:B\subseteq F\textrm{ for some }B\textrm{ }\in\!\,_{\textrm{F}}\mathcal{B}\}\label{Eqn: filter_base}$$
*reciprocally as all supersets of the basic elements.$\qquad\square$*
This is the smallest filter on $X$ that contains $_{\textrm{F}}\mathcal{B}$ and is said to be *the filter generated by its filt
| 3,792
| 3,989
| 3,478
| 3,381
| 1,526
| 0.788619
|
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|
c sigma model on the Hitchin moduli space ${\cal M}_H(G,C)$, with a twisted bundle over that moduli space, twisted by an element of $H^2(Z(G))$. Since the Hitchin moduli space is defined by modding out the adjoint action of $G$, the center is trivial, and so one could naturally replace the Hitchin moduli space with a moduli stack which is a $Z(G)$ gerbe, just as in [@summ]\[section 12.3\]. A heterotic sigma model on such a stack would appear to be a sigma model on the moduli space but with a restriction on allowed maps, exactly as described in [@anton1]\[section 3.1\].
As these two-dimensional (0,2) theories are constructed by dimensional reduction of a consistent four-dimensional theory, it is difficult to believe that they are not consistent.
Other naively-consistent examples can be constructed in (0,2) GLSM’s. For example, consider the two examples:
- The rank 9 bundle $$0 \: \longrightarrow \: {\cal E} \: \longrightarrow \:
\bigoplus_1^9 {\cal O}(1) \oplus {\cal O}(10) \: \longrightarrow \:
{\cal O}(19) \: \longrightarrow \: 0$$ over ${\mathbb P}^4_{[3,3,3,3,6]}[18]$, a ${\mathbb Z}_3$ gerbe over ${\mathbb P}^4_{[1,1,1,1,2]}[6]$,
- The rank 9 bundle $$0 \: \longrightarrow \: {\cal E} \: \longrightarrow \:
\bigoplus_1^9 {\cal O}(1) \oplus {\cal O}(13) \: \longrightarrow \:
{\cal O}(22) \: \longrightarrow \: 0$$ over ${\mathbb P}^3_{[3,3,6,9]}[21]$, a ${\mathbb Z}_3$ gerbe over ${\mathbb P}^3_{[1,1,2,3]}[7]$.
It is straightforward to check, just at the level of combinatorics, that they satisfy the usual conditions for a GLSM to be anomaly-free. However, the usual danger with GLSM’s is that we do not have perfect control over the RG flow – although we have described them in terms of data associated to twisted bundles, along the RG flow they might pick up ‘phases’ (as suggested earlier), for example.
In the next subsections, we shall show explicitly that examples of this form do not yield consistent supersymmetric heterotic string compactifications, unfortunately.
Cautionary e
| 3,793
| 2,979
| 3,490
| 3,226
| 2,821
| 0.776768
|
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d_{\mathcal{P}}(\overline{X}_n)]\le \frac{\big(7\pi^2R+\sqrt{\overline{\sigma}^2+16R^2}\big)}{n^{\frac{2}{5}}}.$$
Statistical inference for uncertain distributions
-------------------------------------------------
The upper bound in Theorem \[t7\] provides us with a quantitative version of the fact that for large $n$, the sample mean is sufficiently concentrated inside the interval $[\underline{\mu}, \overline{\mu}]$. This is related to the estimation of $\underline{\mu}$ and $\overline{\mu}$ described below.
Given an i.i.d. sequence of random variables $X_1, \dots, X_N$ under linear expectations, the usual estimator for their mean is Here, we consider a statistical estimation under sublinear expectations.
Let $X_1,\dots, X_N$ be an i.i.d. sequence of random variables under a sublinear expectation $\E$ such that for a family of linear expectations $\{E_\theta: \theta\in \Theta\}$. Suppose that $N=nk$ and the data are expressed as follows: [@JiPe16] proposed to estimate the lower mean $\underline{\mu}$ and the upper mean $\overline{\mu}$ of $X_1$ by and respectively. Applying Theorem \[t7\] and the union bound, we have the following result.
\[p1\] Suppose $\E[X_1^2]<\infty$. We have where $C$ is a constant depending only on $\underline{\mu}, \overline{\mu}$ and $\overline{\sigma}^2$ in [120]{}.
Define We have, by the union bound and Theorem \[t7\], The second inequality follows from the same argument.
Proposition \[p1\] ensures that as $n\to \infty$ and $k=o(n)$, the estimators by [@JiPe16] are sufficiently concentrated inside $[\underline{\mu}, \overline{\mu}]$.
A new law of large numbers
--------------------------
We first formulate a new law of large numbers for the one-dimensional case.
\[t2\] Let $\{X_i\}_{i=1}^\infty$ be an i.i.d. sequence of random variables under a sublinear expectation $\E$ such that for a family of linear expectations $\{E_\theta: \theta\in \Theta\}$. Suppose that $\E[ X_1^2]<\infty$. Denote Then, for $\varphi$ differentiable such that $\varphi'\in lip(\mathbb{R})$, we ha
| 3,794
| 2,681
| 1,024
| 3,706
| 3,791
| 0.770068
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they are the positivity sets of functions from the linear space of functions of the two variables $u$ and $x$ spanned by $f^{1/2}(x)$, $xf^{1/2}(x)$ and $K_1^{-1}(u/f^{1/2}(x))$. Hence, by a result of Dudley (e.g. Proposition 5.1.12 in de la Peña and Giné (1999)) the subgraphs of ${\cal K}_1$ are VC of index 4. If the set $\{x:f(x)= 0\}$ is not empty, the same argument above shows that the class of subsets of $S=\{x:f(x)>0\}\times\mathbb R$, $\left\{(x,u)\in S:K_{1}\left(\frac{t-x}{h}f^{1/2}(x)\right)f^{1/2}(x)
\ge u \right\}$ is VC of index 4, and therefore so is the class of subgraphs of ${\cal K}|_1$, which is obtained from this one by taking the union of each of these sets with the set $\{x:f(x)=0\}\times\{u\le 0\}$ (which is disjoint with all of them). Therefore, in either case, by the Dudley-Pollard entropy theorem for VC-subgraph classes (e.g., loc. cit. Theorem 5.1.5), we have $$N({\cal K}_1, L_2(P), \varepsilon)\le \left(\frac{A\|K_1\|_\infty\|f\|_\infty^{1/2}}{\varepsilon}\right)^8,\ \ 0<\varepsilon\le \|K_1\|_\infty\|f\|_\infty^{1/2}$$ where $A$ is a universal constant, hence, $$\label{vc2}
N({\cal K}_1, L_2(P), \varepsilon)\le \left(\frac{A\|K\|_+\|f\|_\infty^{1/2}}{\varepsilon}\right)^8,\ \ 0<\varepsilon\le \|K\|_+\|f\|_\infty^{1/2}$$ where $\|K\|_+$ is the positive variation seminorm of $K$. The analogous bound holds for ${\cal K}_2$, defined with $K_2$ replacing $K$ in $\cal F$. Since, as is well known, the set ${\cal J}$ of all indicator functions of intervals in $\mathbb R$ is $VC$ of order 3, we also have $$\label{vc3}
N({\cal J}, L_2(P),\varepsilon)\le\left(\frac{\bar A }{\varepsilon}\right)^6, \ \ 0<\varepsilon\le 1.$$ for another universal constant $\bar A$. Now, any $H\in\cal F$ can be written as $H=k_1g-k_2g$ for $k_i\in{\cal K}_i$ and $g\in{\cal J}$, so that, for any probability measure $Q$ we have $$\begin{aligned}
Q(H-\bar H)^2&=&Q((k_1-k_2)g-(\bar k_1-\bar k_2)\bar g)^2\\
&\le&4Q(k_1-\bar k_1)^2+4Q(k_2-\bar k_2)^2+2\|K\|_V^2\|f\|_\infty Q(g-\bar g)^2.\end{aligned}$$ Given $\varepsilon
| 3,795
| 2,816
| 1,943
| 3,547
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−0.55 0.41 −1.36, 0.26
Daily 90 −0.58 0.84 −2.23, 1.08 90 −0.004 0.006 −0.016, 0.008 91 −0.37 0.54 −1.43, 0.69
BFP, body fat percentage; SE: standard error; CI: confidence interval; OR: odds ratio; SSB, sugar-sweetened beverage; WC, waist circumference; WHR, waist-to-hip ratio. All values were derived from multivariable linear regression models. Model 1 adjusted for age and sex; Model 2 additionally adjusted for maternal age at birth, maternal birthplace, parental highest education level, parental highest occupation, household income per head, and interaction of maternal birthplace with parental highest education level; Model 3 additionally adjusted for main caregiver, general health, and physical activity; Model 4 additionally adjusted for fruit, vegetable, and meat consumption and BMI z-score at 11 years.
nutrients-12-01015-t004_Table 4
######
Associations of sugar-sweetened beverage (SSB) consumption at 13 years with body mass index (BMI) z-score and overweight from 14--18 years and with waist circumference (WC), waist-to-hip ratio (WHR) and body fat percentage (BFP) at 16--19 years in Hong Kong's "Children of 1997" birth cohort.
Outcomes SSB Consumption *n* Beta/OR \* SE ^†^ 95% CI
-------------------------- ----------------- ------ ------------ -------- ---------------
BMI z-score \<weekly 924 Ref
1--3 times/week 1230 −0.04 0.05 −0.13, 0.06
4--6 times/week 283 0.04 0.07 −0.11, 0.18
Daily 180 0.00 0.09 −0.18, 0.18
Overweight \<weekly 924 Ref
(including obesity) ^\#^ 1--3 times/week 1230 −0.08 0.12 −0.32, 0.16
4--6 times/week 283 −0.15 0.20 −0.53, 0.24
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Thurston’s polyhedralization is constant, that is $n$. Under this assumption, the topology of a deformation will be almost constant.
In [@KojimaNishiYamashita], we discussed local behavior of the deformations appeared in our setting near the equal weight. When $n = 5$, the deformations are topologically a connected sum of five copies of the real projective space, ${\#}^5 {{\bold R}}{{\bold P}}^2$. The assignment of the hyperbolic structure of a deformation to each weight was shown to be a local embedding at the equal weight. When $n = 6$ and with the equal weight, the result of hyperbolization is a $3$-dimensional hyperbolic manifold of finite volume with ten cusps, which we denoted by $\overline{X_6}$. Any deformation induced by a variation of the weights can be regarded as some Dehn filled resultant of $\overline{X_6}$. The assignment of the deformation to each weight was also shown to be a local embedding at the equal weight.
In this paper, we prove the global injectivity of the above assignment. Namely, we show that $\Theta_n$ is mapped by the above assignment injectively to the deformation space in Theorem 1 when $n = 5$, and in Theorem 2 when $n = 6$. The local injectivity in [@KojimaNishiYamashita] is proven by computing the derivative of the map at the equal weight. The proof of the global injectivity we present here is based on rather geometric observation for variation of polygons developed in [@KojimaYamashita; @AharaYamada], and independent of the argument in [@KojimaNishiYamashita].
We review some of materials in [@KojimaNishiYamashita] to set up the notations in the next section, and prove the theorems in the sections after.
Preliminaries
=============
We here very briefly recall the hyperbolization in [@KojimaNishiYamashita].
The configuration space of $n$ marked points in the real projective line ${{\bold R}}{{\bold P}}^1$ is, by definition, the quotient of $({{\bold R}}{{\bold P}}^1)^n$ minus the big diagonal set by the diagonal action of the projective linear group ${\operatorname{PGL}}
| 3,797
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|
gned}$$
Combining with , one has $$\label{BERic}
{\rm \overline{Ric}}(\overline{\nabla} v,\overline{\nabla} v)={\rm Ric}^m_f(\nabla v,\nabla v),$$ and $$\label{WLaplacian}
\bar{\Delta}v=\Delta_{f}v.$$ Put , and into , we can get the desired result .
Differential Harnack estimates and applications
===============================================
In this section, we prove a Qian type differential Harnack estimates [@QianB] for the porous medium equation on the compact Riemannian manifolds with lower bound of Ricci curvature, which generalize the Lu-Ni-Vazquez-Villani’s estimate [@LNVV] and Huang-Huang-Li’s estimate [@HHL], Harnack inequalities and Laplacian estimate are derived as applications.
Let $\sigma(t)$ be a functions of $t$ and satisfy the assumption (A1)(A2) and $\alpha(t), \varphi(t)$ are defined in . Suppose $u$ be a smooth solution to and $v=\frac{\gamma }{\gamma-1}u^{\gamma-1}$ the pressure funciton. Define $$F_{\alpha}:=\alpha(t)\frac{v_t}{v}-\frac{|\nabla v|^2}{v}+\varphi(t).$$ Set ${\kappa}=K\sup\limits_{M\times(0,T]}(u^{\gamma-1})$, for any $\gamma>1$, we have
$$\begin{aligned}
\label{GWDBochner1}
\square F_{\alpha}\ge&2\gamma \left\langle \nabla v ,\nabla F_{\alpha}\right\rangle+\frac1a\left((\gamma-1)\Delta v+\frac{a\sigma'}{2\sigma}+{a\gamma\kappa}\right)^2-\frac{\sigma'}{\sigma}F_{\alpha}
+(\alpha-1)\Big(\frac{v_t}v\Big)^2,\end{aligned}$$
$$\begin{aligned}
\label{GWDBochner2}
\square(\sigma F_{\alpha})\ge&2\gamma\sigma \left\langle \nabla v ,\nabla F_{\alpha}\right\rangle+\frac{\sigma}a\left((\gamma-1)\Delta v+\frac{a\sigma'}{2\sigma}+{a\gamma\kappa}\right)^2
+(\alpha-1)\sigma\Big(\frac{v_t}v\Big)^2.\end{aligned}$$
By using of , we have $$\begin{aligned}
\label{WDBochner4}
\square F_{\alpha}=&2\gamma\left\langle \nabla v ,\nabla F_{\alpha}\right\rangle+2(\gamma-1)\Big(|\nabla\nabla v|^2+{\rm Ric}(\nabla v,\nabla v)\Big)\notag\\
&+\left(((\gamma-1)\Delta v)^2+(\alpha-1)\Big(\frac{v_t}v\Big)^2\right)
+\alpha'\Big(\frac{v_t}{v}\Big)+\varphi'.\end{aligned}$$ Applying Cauchy-Schw
| 3,798
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|
ions are dualised the coset representative is fully fixed to $\hat{g}=1$ leaving three further gauge fixings to be made on the dynamical Lagrange multipliers. We parametrise these as $$v_1= \frac{x_0}{\eta} \ , \quad v_2= \frac{-1+z}{\eta} \ , \quad v_3 = \frac{x_1}{\eta} \ , \quad v_5 + i v_7 = \frac{r e^{i\theta}}{\eta} \ , \quad v_4=v_6=v_8=0 \ , \quad v_9 = \frac{1}{\eta} \ ,$$ where $v_9$ corresponds to the central direction. The dual metric is given by $$\begin{aligned}
&\widehat{ds}^2 =e_\pm^i \eta_{ij} e_\pm^i + \widehat{ds}^2_{S^5} \ , \quad \eta_{ij} = \textrm{diag} (1,-1,1, 1,1) \ , \\
&\widehat{e}^{\,1}_+= \frac{1}{p}(-\eta dx_0 + z dz) \ , \quad \widehat{e}^{\,2}_+= \frac{1}{p}(-z dx_0 + \eta dz) \ , \quad \widehat{e}^{\,3}_+= -\frac{z}{q}(z^2 dx_1 + r \eta dr) \ ,\\
& \widehat{e}^{\,4}_+ + i \widehat{e}^{\,5}_+ = \frac{e^{i\theta}}{q}\left( r z \eta dx_1 -z^3 dr - \frac{i q r}{z} d\theta \right) \ ,
\end{aligned}$$ where $p= z^2-\eta^2$ and $q= z^4+ r^2 \eta^2$. The remaining NS fields are $$\widehat{B} = \frac{z}{p \eta}dz\wedge dx_0 + \frac{r\eta}{q} dr \wedge dx_1 \ , \quad e^{-2(\widehat{\Phi}-\phi_0)} = \frac{p q z^2}{\eta^8} \ .$$
The $SO(1,4)$ Lorentz rotation has a block diagonal decomposition $\Lambda = \Lambda_1 \oplus \Lambda_2$ with $$\Lambda_1 = \frac{1}{p}\left(\begin{array}{cc} z^2+\eta^2 & 2 z \eta \\ 2 z \eta & z^2 +\eta^2 \end{array} \right) \ , \ \ \
\Lambda_2= \frac{1}{q}\left(\begin{array}{ccc} z^4 -r^2 \eta^2 & 2 r z^2 \eta {\cal C}_\theta & 2 r z^2 \eta {\cal S}_\theta \\
-2 r z^2 \eta {\cal C}_\theta & z^4 - r^2 \eta^2 {\cal C}_{2\theta} & -r^2 \eta^2 {\cal S}_{2\theta} \\
-2 r z^2 \eta {\cal S}_\theta &- r^2 \eta^2 {\cal S}_{2\theta} & z^4 + r^2 \eta^2 {\cal C}_{2\theta}
\end{array}\right) \ .$$ The corresponding spinor rotation $\Omega = \Omega_1\cdot\Omega_2$ is given by (recalling the signature is such that $(\Gamma^{2})^{2}= - \mathbb{I} $ whilst the remaining $(\Gamma^{i})^{2}= \mathbb{I}$) $$\begin{aligned}
\Omega_1 = \frac{1}{\sqrt{p}} \left( z \mathbb{I} + \eta \Gam
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theorem can also be found in [@wilkinson1965algebraic].
\[thm1\] Let $\mathbf{C}=\mathbf{D}+\rho\mathbf{vv}^T$, where $\mathbf{D}$ is diagonal, $\|\mathbf{v}\|_2=1$. Let $d_1\le d_2\le \cdots\le d_n$ be the eigenvalues of $\mathbf{D}$, and let $\tilde{d}_1\le \tilde{d}_2\le \cdots\le \tilde{d}_n$ be the eigenvalues of $\mathbf{C}$. Then $\tilde{d}_1\le d_1\le \tilde{d}_2\le d_2\le\cdots \le\tilde{d}_n\le d_n$ if $\rho<0$. If the $d_i$ are distinct and all the elements of $\mathbf{v}$ are nonzero, then the eigenvalues of $\mathbf{C}$ strictly separate those of $\mathbf{D}$.
\[thm2\] With the notations in Theorem \[thm1\], the eigenvector of $\mathbf{C}$ corresponding to the eigenvalue $\tilde{d}_i$ is given by $(\mathbf{D}-\tilde{d}_i\mathbf{I})^{-1}\mathbf{v}$.
Theorem \[thm1\] tells us the eigenvalues of a DPR1 matrix are interlaced with the eigenvalues of the original diagonal matrix. Next we will write the eigenvector corresponding to the positive eigenvalues of a modularity matrix as a linear combination of the eigenvectors of the corresponding adjacency matrix.\
With the notations in Section 1, since $\mathbf{A}=\mathbf{X}^T\mathbf{X}$, then if the SVD of $\mathbf{X}$ is $\mathbf{X}=\mathbf{U\Sigma V}^T$, then $$\mathbf{A}=\mathbf{V}\mathbf{\Sigma^T\Sigma}\mathbf{V}^T=\mathbf{V}\mathbf{\Sigma_\mathbf{A}}\mathbf{V}^T,$$ where $\Sigma_\mathbf{A}$ is an $n\times n$ diagonal matrix. Suppose the rows and columns of $\mathbf{A}$ are ordered such that $\mathbf{\Sigma_\mathbf{A}}=diag(\alpha_1, \alpha_2, \cdots, \alpha_n)$, where $\alpha_1>\alpha_2>\cdots>\alpha_k>\alpha_{k+1}=\cdots=\alpha_n=0$. Let $\mathbf{V}=\begin{pmatrix}\mathbf{v}_1&\mathbf{v}_2&\cdots&\mathbf{v}_n\end{pmatrix}$. Similarly, since $\mathbf{B}$ is symmetric, it is orthogonally similar to a diagonal matrix. Suppose the eigenvalues of $\mathbf{B}$ are $\beta_1,\beta_2,\cdots,\beta_n$ with largest $k-1$ eigenvalues $\beta_1>\beta_2>\cdots>\beta_{k-1}$.
Since $\mathbf{B}=\mathbf{A}-\mathbf{d}\mathbf{d}^T/(2m)$, we have $$\mathbf{B}=\mathbf{A}-
| 3,800
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