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paired log likelihoods $$\begin{aligned}
\label{eq:likelihood_0}
\Lrb(\theta) &=&
\sum_{j=1}^n \sum_{a = 1}^{\ell_j}
\,\lambda_{j,a} \, \Big\{ \sum_{(i, \i) \in E_{j,a}}
\, \Big( \theta_{\i} - \log \Big(e^{\theta_i} + e^{\theta_{\i}}\Big) \,\Big)\, \Big\} \;,\end{aligned}$$ where $E_{j,a}$’s are defined as above via separators and different choices of the non-negative weights $\lambda_{j,a}$’s are possible and the performance depends on such choices. Each weight $\lambda_{j,a}$ determine how much we want to weigh the contribution of a corresponding rank-breaking graph $G_{j,a}$. We define the [*consistent rank-breaking estimate*]{} $\widehat\theta$ as the optimal solution of the convex program: $$\begin{aligned}
\label{eq:theta_ml}
\widehat{\theta} \;\; \in \;\; \arg\max_{\theta \in \Omega_b} \;\, \Lrb(\theta)\;. \end{aligned}$$ By changing how we weigh each rank-breaking graph (by choosing the $\lambda_{j,a}$’s), the convex program spans the entire set of consistent rank-breaking estimators, as characterized in [@APX14a]. However, only asymptotic consistency was known, which holds independent of the choice of the weights $\lambda_{j,a}$’s. Naturally, a uniform choice of $\lambda_{j,a}=\lambda$ was proposed in [@APX14a].
Note that this can be efficiently solved, since this is a simple convex optimization, in particular a logit regression, with only $O(\sum_{j=1}^n \,\ell_j\, \kappa_j)$ terms. For a special case of position-$p$ breaking, the $O(n \, (p-1)!)$ complexity of evaluating the objective function for the MLE is now significantly reduced to $O(n\,(\kappa-p))$ by rank-breaking. Given this potential exponential gain in efficiency, a natural question of interest is “what is the price we pay in the accuracy?”. We provide a sharp analysis of the performance of rank-breaking estimators in the finite sample regime, that quantifies the price of rank-breaking. Similarly, for a practitioner, a core problem of interest is how to choose the weights in the optimization in order to
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mes 1) \underline{M}^{\prime}(R)$ of $\underline{M}^{\prime}\otimes\kappa$};\\
\textit{the subfunctor $\tilde{M}^1:R\mapsto 1+\underline{\pi M^{\prime}}(R)$ of $\mathrm{Ker~}\tilde{\varphi}$}.
\end{array} \right.$$ Here, by $1+\underline{\pi M^{\prime}}(R)$, we mean the image of $\underline{\pi M^{\prime}}(R)$ inside $\underline{M}(R) (=\tilde{M}(R))$ under the morphism $1+$ at the level of $R$-points. That $1+\underline{\pi M^{\prime}}(R)$ is contained in $\mathrm{Ker~}\tilde{\varphi}(R)$ can easily be checked by observing the construction of $\tilde{\varphi}$. The multiplication on $\tilde{M}^1$ is as follows: for two elements $1+\pi x$ and $1+\pi y$ in $\tilde{M}^1(R)$, based on the above commutative diagram, the product of $1+\pi x$ and $1+\pi y$ is $$(1+\pi x)\cdot(1+\pi y)=1+\pi x\star \pi y=1+(\pi (x+y)+\pi^2(xy))=1+\pi (x+y).$$ Here, $\pi$ stands for $\pi\otimes 1 \in B\otimes_AR$. Then we have the following lemma.
\[la3\] (i) The functor $\tilde{M}^1$ is representable by a smooth, connected, unipotent group scheme over $\kappa$. Moreover, $\tilde{M}^1$ is a closed normal subgroup of $\mathrm{Ker~}\tilde{\varphi} $.
\(ii) The quotient group scheme $\mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$ represents the functor $$R\mapsto \mathrm{Ker~}\tilde{\varphi}(R)/\tilde{M}^1(R)$$ by Lemma \[la1\] and is smooth, connected, and unipotent.\
The proof is the same as that of Lemma A.3 of [@C2] and so we skip it.
This paragraph is a reproduction of 6.3.6 in [@GY]. Recall that there is a closed immersion $\tilde{G}\rightarrow \tilde{M}$. Notice that $\mathrm{Ker~}\varphi$ is the kernel of the composition $\tilde{G}\rightarrow \tilde{M} \rightarrow \tilde{M}/ \mathrm{Ker~}\tilde{\varphi}$. We define $\tilde{G}^1$ as the kernel of the composition $$\tilde{G}\rightarrow \tilde{M} \rightarrow \tilde{M}/ \tilde{M}^1.$$ Then $\tilde{G}^1$ is the kernel of the morphism $\mathrm{Ker~}\varphi\rightarrow \mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$ and, hence, is a closed normal subgroup of $\mathrm{Ker~}\varphi$. The induced
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cisely, the aforesaid regular domain is taken to be the half-space that contains $D$ with boundary hyperplane that is tangent to both the current maximal sphere and $D$. It is the use of a half-space that allows us to work with unbounded domains but which forces the assumption that $D$ is convex. With a little more care, we can remove the need for convexity without disturbing the main idea of the proof. However, this will come at the cost of insisting that $D$ is bounded. It does however, open the possibility that $D$ is not a connected domain. We give two results in this respect.
For the first one, we introduce the following definition, which has previously been used in the potential analysis of stable processes; see for example [@Chen-Song].
A domain $D$ in $\mathbb{R}^d$ is said to satisfy the [*uniform exterior-cone condition*]{}, henceforth written UECC, if there exist constants $\eta > 0$, $r > 0$ and a cone $${\rm Cone}(\eta) = \{x = (x_1,\dots,x_d) \in
\mathbb{R}^d\colon |x|<\eta x_1\}$$ such that, for every $z\in \partial D$, there is a cone $C_z$ with vertex $z$, isometric to ${\rm Cone}(\eta)$ satisfying $C_z \cap B(z,r) \subset D^{\mathrm{c}}$.
It is well known that, for example, bounded $C^{1,1}$ domains satisfy (UECC). We need a slightly more restrictive class of domains than those respecting UECC.
We say that $D$ satisfies the [*regularised uniform exterior-cone condition*]{}, written RUECC, if it is UECC and the following additional condition holds: for each $x\in D$, suppose that $\partial(x)$ is a closest point on the boundary of $D$ to $x$. Then the isometric cone that qualifies $D$ as UECC can be placed with its vertex at $\partial(x)$ and symmetrically oriented around the line that passes through $x$ and $\partial(x)$.
![A domain that satisfies the regularised uniform exterior-cone condition[]{data-label="fig:class_proof"}](cone_new){width="0.8\linewidth"}
Suppose that $D$ is open and bounded (but not necessarily connected) and satisfies RUECC. Then, for each $x\in D$, there
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yperbolic with respect to any vector ${\mathbf{e}}\in {\mathbb{R}}_{++}^n=(0,\infty)^n$: $$h(t{\mathbf{e}}-{\mathbf{x}}) = \prod_{j=1}^n (te_j-x_j).$$
2. Let $X=(x_{ij})_{i,j=1}^n$ be a matrix of $n(n+1)/2$ variables where we impose $x_{ij}=x_{ji}$. Then $\det(X)$ is hyperbolic with respect to $I=\diag(1, \ldots, 1)$. Indeed $t \mapsto \det(tI-X)$ is the characteristic polynomial of the symmetric matrix $X$, so it has only real zeros.
More generally we may consider complex hermitian $Z=(x_{jk}+iy_{jk})_{j,k=1}^n$ (where $i = \sqrt{-1}$) of $n^2$ real variables where we impose $x_{jk}=x_{kj}$ and $y_{jk}=-y_{kj}$, for all $1\leq j,k \leq n$. Then $\det(Z)$ is a real polynomial which is hyperbolic with respect to $I$.
3. Let $h({\mathbf{x}})=x_1^2-x_2^2-\cdots-x_n^2$. Then $h$ is hyperbolic with respect to $(1,0,\ldots,0)^T$.
Suppose $h$ is hyperbolic with respect to ${\mathbf{e}}\in {\mathbb{R}}^n$. We may write $$\label{dalambdas}
h(t{\mathbf{e}}-{\mathbf{x}}) = h({\mathbf{e}})\prod_{j=1}^d (t - \lambda_j({\mathbf{x}})),$$ where ${\lambda_{\rm max}}({\mathbf{x}})=\lambda_1({\mathbf{x}}) \geq \cdots \geq \lambda_d({\mathbf{x}})={\lambda_{\rm min}}({\mathbf{x}})$ are called the *eigenvalues* of ${\mathbf{x}}$ (with respect to ${\mathbf{e}}$), and $d$ is the degree of $h$. In particular $$\label{prolambda}
h({\mathbf{x}}) = h({\mathbf{e}})\lambda_1({\mathbf{x}}) \cdots \lambda_d({\mathbf{x}}).$$
By homogeneity $$\label{dilambdas}
\lambda_j(s{\mathbf{x}}+t{\mathbf{e}})=
\begin{cases}
s\lambda_j({\mathbf{x}})+t &\mbox{ if } s\geq 0 \mbox{ and } \\
s\lambda_{d-j}({\mathbf{x}})+t &\mbox{ if } s \leq 0
\end{cases},$$ for all $s,t \in {\mathbb{R}}$ and ${\mathbf{x}}\in {\mathbb{R}}^n$.
The (open) *hyperbolicity cone* is the set $$\Lambda_{\tiny{++}}= \Lambda_{\tiny{++}}({\mathbf{e}})= \{ {\mathbf{x}}\in {\mathbb{R}}^n : {\lambda_{\rm min}}({\mathbf{x}}) >0\}.$$ We denote its closure by $\Lambda_{\tiny{+}}= \Lambda_{\tiny{+}}({\mathbf{e}})=\{ {\mathbf{x}}\in {\mathbb{R}}^n : {\lambda_{\rm min}}({\mathbf{x}}) \
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with $f_\alpha\circ L_\alpha^{-1}$ on $a_{s\upharpoonright\alpha^+}\setminus n_\alpha$. Thus if $\beta\in L_\delta[a_s\setminus n_\delta]\cap L_{\alpha}[a_{s\upharpoonright\alpha^+}\setminus n_\alpha]$, then $h(\beta)=\dot{f}(\alpha)=\dot{f}(\delta)$. This completes the proof that the space is normal.
The strategy attempted in [@T3] was to expand a closed discrete subspace of a locally compact normal space to a discrete collection of compact $G_\delta$’s. There are limitations on such an approach, given by the following example.
MA$_{\omega_1}(S)[S]$ implies there is a locally compact space of character $\aleph_1$ which includes a normalized closed discrete set which does not have a normalized discrete expansion by compact $G_\delta$’s.
We modify the previous example. Let $\mathcal{A}_s$ denote the Boolean subalgebra of $\mathcal{P}(\omega)$ generated by $[\omega]^{<\omega}\cup\{a_s:s\in S\}$. In the forcing extension by $S$, let $x_g$ denote the member of the Stone space $\mathcal{S}(\mathcal{A}_s/\text{FIN})$ containing $\{a_s:s\in g\}$.
In the forcing extension, our space has the base set $(\omega_1\setminus C_0)\cup(C_0\times\mathcal{S}(\mathcal{A}_s))$. The points of $\omega_1\setminus C_0$ are isolated. For each $\delta\in C_0$ and $x\in\mathcal{S}(\mathcal{A}_s/\text{FIN})$, a neighborhood of $(\delta,x)$ must include $U_\delta(a)=L_\delta[a]\cup(\{\delta\}\times a^*)$ for some $a\in x$, where $a^* = \{p\in\mathcal{S}(\mathcal{A}_\mathcal{S}):a\in p\}$. Notice that $U_\delta(a)$ is disjoint from $\{\gamma\}\times\mathcal{S}(\mathcal{A}_s/\text{FIN})$, for all $\gamma\neq\delta$. It follows immediately that the sequence $D=\{(\gamma,x_g):\delta\in C_0\}$ is a closed discrete subset. It also follows from the proof of the normality of the previous example that $D$ is normalized.
Now we show that $D$ does not have a normalized discrete expansion by compact $G_\delta$’s, indeed by any $G_\delta$’s. Assume that $\{\dot{Z}_\delta:\delta\in C_0\}$ is a sequence of $S$-names so that $\dot{Z}_\delta$ i
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(\left\langle \mathbf{A}_{1,1}\mathbf{B}_{1,1}\right\rangle \!,\left\langle \mathbf{A}_{1,2}\mathbf{B}_{1,2}\right\rangle \!,\left\langle \mathbf{A}_{2,1}\mathbf{B}_{2,1}\right\rangle \!,\left\langle \mathbf{A}_{2,2}\mathbf{B}_{2,2}\right\rangle \right)\\
\le6-s_{1}\!\left(\left\langle \mathbf{A}_{1,1}\mathbf{A}_{1,2}\right\rangle \!,\left\langle \mathbf{A}_{2,1}\mathbf{A}_{2,2}\right\rangle \!,\left\langle \mathbf{B}_{1,1}\mathbf{B}_{2,1}\right\rangle \!,\left\langle \mathbf{B}_{1,2}\mathbf{B}_{2,2}\right\rangle \right),
\end{array}\label{eq:CbD-s0}$$ $$\begin{array}{l}
s_{1}\!\left(\left\langle \mathbf{A}_{1,1}\mathbf{B}_{1,1}\right\rangle \!,\left\langle \mathbf{A}_{1,2}\mathbf{B}_{1,2}\right\rangle \!,\left\langle \mathbf{A}_{2,1}\mathbf{B}_{2,1}\right\rangle \!,\left\langle \mathbf{A}_{2,2}\mathbf{B}_{2,2}\right\rangle \right)\\
\le6-s_{0}\!\left(\left\langle \mathbf{A}_{1,1}\mathbf{A}_{1,2}\right\rangle \!,\left\langle \mathbf{A}_{2,1}\mathbf{A}_{2,2}\right\rangle \!,\left\langle \mathbf{B}_{1,1}\mathbf{B}_{2,1}\right\rangle \!,\left\langle \mathbf{B}_{1,2}\mathbf{B}_{2,2}\right\rangle \right).
\end{array}\label{eq:CbD-s1}$$ where $$\begin{aligned}
s_{0}(x_{1},\dots,x_{n}) & =\max\{\pm x_{1}\pm\dots\pm x_{n}:\text{even \# of \ensuremath{-}'s}\},\\
s_{1}(x_{1},\dots,x_{n}) & =\max\{\pm x_{1}\pm\dots\pm x_{n}:\text{odd \# of \ensuremath{-}'s}\},\end{aligned}$$ and where we use the parameterization by the 12 expectation variables defined as $$\left\langle \mathbf{A}_{i,j}\mathbf{B}_{i,j}\right\rangle =\left(4p_{ij}-1\right)-\left(2a_{i}-1\right)-\left(2b_{j}-1\right),\label{eq:ABcorr}$$ $$\left\langle \mathbf{A}_{i,1}\mathbf{A}_{i,2}\right\rangle =1-4\alpha_{i}=1-2\Pr\left[\mathbf{A}_{i,1}\ne\mathbf{A}_{i,2}\right],\label{eq:Acorr}$$ $$\left\langle \mathbf{B}_{1,j}\mathbf{B}_{2,j}\right\rangle =1-4\beta_{j}=1-2\Pr\left[\mathbf{B}_{1,j}\ne\mathbf{B}_{2,j}\right],\label{eq:Bcorr}$$ $$\left\langle \mathbf{A}_{i}\right\rangle =2a_{i}-1,\label{eq:Amarginal}$$ $$\left\langle \mathbf{B}_{j}\right\rangle =2b_{j}-1,\la
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pha \in R_{H,+} \\ \alpha \not\perp \Delta_X}}} \frac {\langle \alpha, \mu \rangle} {\langle \alpha, \rho \rangle}
\right) k^{R_X} +
O(k^{R_X-1})
\end{aligned}$$ for the representations that occur in ${\mathbb C}[X]$. The coefficient $P(\mu) = \prod_{\alpha \in R_{H,+}, \alpha \not\perp \Delta_X} {\langle \alpha, \mu \rangle} / {\langle \alpha, \rho \rangle}$ is a polynomial function in $\mu$. Since $\Delta_X$ is compact, we can therefore find a constant $C > 0$ such that $$\dim V_{k \mu} \leq C \, k^{R_X} \qquad (\forall k, \mu \in \Delta_X \cap \frac 1 k \Lambda^*_{H,+}).$$ It follows that $$\begin{aligned}
&\sum_{\mathclap{\mu \in \Delta_X \cap \frac 1 k \Lambda^*_{H,+}}} m_{H,X,k}(k \mu) \, \dim V_{k \mu} \\
\leq~&C \, k^{R_X} \sum_{\mathclap{\mu \in \Delta_X \cap \frac 1 k \Lambda^*_{H,+}}} m_{H,X,k}(k \mu)
\sim
C \, k^{R_X + d_X} \int d\operatorname{DH}_X,
\end{aligned}$$ so that $\dim X \leq R_X + d_X$.
On the other hand, since $\operatorname{DH}_X$ is Lebesgue-absolutely continuous, the boundary of the moment polytope does not carry any measure. We can therefore find a compact set $K$ contained in the (relative) interior of the moment polytope which has positive measure with respect to $\operatorname{DH}_X$. Note that $P(\mu)$ is positive for all $\mu$ contained in the interior of the moment polytope (indeed, for all positive roots $\alpha$ with $\alpha \not\perp \Delta_X$ there exists $\nu \in \Delta_X$ such that $\langle \alpha, \nu \rangle > 0$; since we can always write $\mu$ as a proper convex combination of $\nu$ and some other point $\nu' \in \Delta_X$, it follows that $\langle \alpha, \mu \rangle > 0$). This implies that on the compact set $K$ we can bound $P(\mu)$ from below by a positive constant. Thus there exists a constant $D > 0$ (depending on $K$) such that $$\dim V_{k \mu} \geq D \, k^{R_X} \qquad (\forall \mu \in K \cap \frac 1 k \Lambda^*_{H,+}).$$ Consequently, $$\begin{aligned}
&\sum_{\mathclap{\mu \in \Delta_X \cap \frac 1 k \Lambda^*_{H,+}}} m_{H,
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e., for any quantity $X$, the perturbation is $\delta{\tilde X} = \delta X \exp\{{i(k_rr+k_z z)-i\omega t}\}$), we will suppose that the wavevectors obey three conditions: that $k = (k_z^2 + k_r^2)^{1/2} \ll \kappa \rho$; that $k_z \gg 1/h$; and that $k_r \gg 1/r$. The first limit means that the diffusion approximation applies, i.e. ${\bf p_r} = p_r{\bf I}$, so that $E = 3p_r$. The second is the WKB approximation, as applied to variations in both the radial and vertical directions. Note that we further restrict our attention to axisymmetric perturbations. The condition $k_z h \gg 1$ also means that we can ignore any gradients in the gravity or equilibrium radiation flux. In addition, assuming $k_r\gg 1/r$ allows us to neglect the terms in vector divergences arising from cylindrical geometry. For example, after Fourier-transforming, $\nabla\cdot\delta\vec v$ becomes $ik_r\delta v_r +\delta v_r/r +
ik_z\delta v_z \simeq ik_r\delta v_r + ik_z\delta v_z$. We then find: $$\begin{aligned}
-i\omega \delta\rho & + & \rho i\vec k \cdot \delta \vec v = 0 \\
-i\omega \rho \delta v_r & = & (\kappa \rho /c) \delta {\cal F}_r +
2\rho\Omega\delta v_\phi \\
-i\omega\rho \delta v_z & = & (\kappa \rho/c) \delta {\cal F}_z \\
-i\omega\rho \delta v_\phi & = & (\kappa\rho/c)\delta {\cal F}_\phi -
(1/2)\rho\Omega \delta v_r \\
-3i\omega \delta p_r - 3 \rho g \delta v_z & + & ik_z \delta {\cal F}_z +
ik_r\delta{\cal F}_r + 4p_r\left(i k_z \delta v_z + i k_r \delta v_r\right)
-i{2\omega\over c^2}\delta v_z F_z =
\delta Q \\
-i(\omega/c)\delta {\cal F}_r & + & ik_r c \delta p_r + i k_z (g/\kappa)
\delta v_r
- 4i(\omega/c)p_r \delta v_r = -\kappa\rho \delta {\cal F}_r \\
-i\left({\omega\over c}\right)\delta {\cal F}_z + ik_z c \delta p_r &
+ & {g\over\kappa}(2ik_z \delta v_z + ik_r \delta v_r) - 4i{\omega\over c}p_r \delta v_z =
-cg\delta\rho - \kappa\rho \delta{\cal F}_z
%-{g\over z\kappa}\delta v_z
\\
-i(\omega/c)\delta {\cal F}_\phi &+ & i(k_z/c){\cal F}_z\delta v_\phi -
4i(\omega/c)p_r\delta
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\Gamma'=\partial G\times S$ and $\Gamma'_{-}=\{(y,\omega)\in \partial G\times S\ |\ \omega\cdot\nu(y)<0\}$. Choose any $\eta\in C^1_0(I^\circ)$ and $\theta\in C^1(\ol G\times S)$ such that ${\rm supp}(\theta)\cap \Gamma'$ is a compact subset of $\Gamma_-'$. Then $w(x,\omega,E):=\theta(x,\omega)\eta(E)\in C^1(\ol{G}\times S\times I)$ and $w(\cdot,\cdot,0)=w(\cdot,\cdot,E_{\rm m})=0$, $w_{|\Gamma_+}=0$. Hence ${\left\langle}p_0,S_0(\cdot,0) w(\cdot,\cdot,0){\right\rangle}_{L^2(G\times S)}=0$ and ${\left\langle}q,\gamma_+(w){\right\rangle}_{T^2(\Gamma_+)}=0$, and so by (\[coexpr\]) for these $w$, \[coexpra\] &,S\_0[E]{}\_[L\^2(GSI)]{} -,\_x w\_[L\^2(GSI)]{}\
&+, CS\_0w\_[L\^2(GSI)]{}+, (\^\*-K\_C\^\*) w\_[L\^2(GSI)]{}\
&=[**f**]{},w\_[L\^2(GSI)]{}+[**g**]{}, \_-(w)\_[T\^2(\_-)]{}.
Since the solution $\phi$ obtained in part (i) belongs to ${\mathcal{H}}_P(G\times S\times I^\circ)$, we have by virtue of the Green’s formula and that \[coexpra-a\] & [**f**]{},w\_[L\^2(GSI)]{}+[**g**]{}, \_-(w)\_[T\^2(\_-)]{}\
=& [-]{}+\_x+CS\_0+(-K\_C),w\_[L\^2(GSI)]{} +\_-(),\_-(w) \_[T\^2(\_-)]{}\
=& [**f**]{},w\_[L\^2(GSI)]{} +\_-(),\_-(w) \_[T\^2(\_-)]{} and hence \[csda42d\] \_-(),\_-(w) \_[T\^2(\_-)]{} = [**g**]{},\_-(w) \_[T\^2(\_-)]{}, for any $w$ of the form as chosen above. This clearly implies that $\gamma_-(\phi)={\bf g}\in T^2(\Gamma_-)$.
Next, choose $\tilde{\eta}\in C^1(I)$ such that $\tilde{\eta}(0)=0$ and choose $\tilde{\theta}\in C_0^1(G)$. Let $\tilde{w}:=\tilde{\eta} \tilde{\theta}$. Then by similar calculation as above, we see that the Green’s formula (\[green-ex\]) and (\[coexpr\]) imply $${\left\langle}\phi(\cdot,\cdot,E_m),S_0(\cdot,E_{\rm m}) \tilde w(\cdot,\cdot,E_m){\right\rangle}_{L^2(G\times S)}=0$$ for these $\tilde{w}$. Consequently, $\phi(\cdot,\cdot,E_m)=0$ a.e. in $G\times S$ (since $S_0\geq\kappa>0$), as desired.
Finally, if $v\in C^1(\ol{G}\times S\times I)$, we have $$& {\left\langle}{\bf f},v{\right\rangle}_{L^2(G\times S\times I)}+{\left\langle}{\bf g},\gamma_-(v){\right\rangle}_{T^2(\Gamma_
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} {2^{14} d\beta^4\log\lrp{\frac{2^{14} d\beta^4}{\epsilon^4L^2}}}}$, assume additionally that $n=T/\delta$ is an integer.\
Let $\bx_t$ and $\bw_t$ have dynamics as defined in and respectively, and suppose that the initial conditions satisfy $\E{\lrn{\bx_0}_2^2}\leq R^2 + \beta^2/m$ and $\E{\lrn{\bw_0}_2^2}\leq R^2 + \beta^2/m$. Then there exists a coupling between $\bx_t$ and $\bw_t$ such that $$\begin{aligned}
\E{f(\bx_{i\delta} - \bw_{i\delta})} \leq e^{-\lambda i\delta} \E{f(\bx_{0} - \bw_{0})} + \frac{6}{\lambda} \lrp{L + \LN^2} \epsilon
\end{aligned}$$
From Lemma \[l:energy\_x\] and \[l:energy\_w\], our initial conditions imply that for all $t$, $\E{\|\bx_t\|_2^2} \leq 6\lrp{R^2 + \frac{\beta^2}{m}}$ and $\E{\|\bw_{k\delta}\|_2^2} \leq 8 \lrp{R^2 + \frac{\beta^2}{m}}$.
Consider an arbitrary $k$, and for $t\in[0,T)$, define $$\begin{aligned}
x_t := \bx_{kT+t} \quad \text{and} \quad
w_t := \bw_{kT+t} \numberthis \label{e:t:kasjnd}
\end{aligned}$$ Notice that as described above, $x_t$ and $w_t$ have dynamics described in and . Let $x_t,w_t$ have joint distribution as described in and , and let $(y_t,v_t)$ be the processes defined in and . Notice that the joint distribution between $x_t$ and $w_t$ equivalently describes a coupling between $\bx_t$ and $\bw_t$ over $t\in[kT, (k+1)T)$.
First, notice that the processes $\eqref{e:coupled_4_processes_x}$ and $\eqref{e:coupled_4_processes_y}$ have the same distribution as . We can thus apply Lemma \[l:gaussian\_contraction\]: $$\begin{aligned}
\E{f(x_{T} - y_{T})}
\leq& e^{-\lambda T} \E{f(x_0 - y_0)} + 6T (L+\LN^2) \epsilon
\end{aligned}$$
By Lemma \[l:non\_gaussian\_contraction\_stationary\], $$\begin{aligned}
\E{f(x_T - v_T)} - \E{f(x_T - y_T)}
\leq 4TL\epsilon
\end{aligned}$$ By Lemma \[l:non\_gaussian\_contraction\_anisotropic\], $$\begin{aligned}
\E{f(x_T - w_T)} - \E{f(x_T - v_T)}
\leq 4T(L+\LN^2)\epsi
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qn: lim filter}$$ and $${\textstyle \textrm{adh}(\mathcal{F})=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\forall F\in\mathcal{F})(F\bigcap N\neq\emptyset)\}}\label{Eqn: adh filter}$$
*are respectively the sets of* *limit points* *and* *adherent* *points* *of $\mathcal{F}$*[^28]*.$\qquad\square$*
A comparison of Eqs. (\[Eqn: lim net\]) and (\[Eqn: adh net2\]) with Eqs. (\[Eqn: lim filter\]) and (\[Eqn: adh filter\]) respectively demonstrate their formal similarity; this inter-relation between filters and nets will be made precise in Definitions A1.10 and A1.11 below. It should be clear from the preceding two equations that $$\textrm{lim}(\mathcal{F})\subseteq\textrm{adh}(\mathcal{F}),\label{Eqn: lim/adh(fil)}$$ with a similar result $$\textrm{lim}(\chi)\subseteq\textrm{adh}(\chi)\label{Eqn: lim/adh(net)}$$ holding for nets because of the duality between nets and filters as displayed by Defs. A1.9 and A1.10 below, with the equality in Eqs. (\[Eqn: lim/adh(fil)\]) and (\[Eqn: lim/adh(net)\]) being true (but not characterizing) for ultrafilters and ultranets respectively, see Example 4.2(3) for an account of this notion . It should be clear from the equations of Definition A1.8 that $$\textrm{adh}(\mathcal{F})=\{ x\in X\!:(\exists\textrm{ a finer filter }\mathcal{G}\supseteq\mathcal{F}\textrm{ on }X)\textrm{ }(\mathcal{G}\rightarrow x)\}\label{Eqn: filter adh}$$ consists of all the points of $X$ to which some finer filter $\mathcal{G}$ (in the sense that $\mathcal{F}\subseteq\mathcal{G}$ implies every element of $\mathcal{F}$ is also in $\mathcal{G}$) converges in $X$; thus $${\textstyle \textrm{adh}(\mathcal{F})=\bigcup\lim(\mathcal{G}\!:\mathcal{G}\supseteq\mathcal{F}),}$$ which corresponds to the net-result of Theorem A1.5 below, that a net *$\chi$* adheres at *$x$* iff there is some subnet of *$\chi$* that converges to *$x$* in *$X$*. Thus if $\zeta\preceq\chi$ is a subnet of $\chi$ and $\mathcal{F}\subseteq\mathcal{G}$ is a filter coarser than $\mathcal{G}$ then $$\begin{aligned}
\lim(\chi)\subseteq\lim(\zeta)
| 2,611
| 1,121
| 2,841
| 2,571
| 903
| 0.798076
|
github_plus_top10pct_by_avg
|
0.068
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
BLB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.002
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.068 0.042 0.048 0.074 0.060 0.080 0.076
2 K=50 0.044 0.080 0.038 0.060 0.060 0.060 0.052
K=100 0.032 0.066 0.032 0.054 0.066 0.066 0.046
K=150 0.030 0.054 0.032 0.050 0.064 0.058 0.058
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
BLB($n^{0.8}$) 0.000 0.002 0.002 0.000 0.000 0.000 0.004
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.044 0.078 0.040 0.064 0.072 0.082 0.080
3 K=50 0.066 0.060 0.042 0.054 0.062 0.052 0.060
K=100 0.054 0.060 0.042 0.062 0.054 0.046 0.046
K=150 0.060 0.056 0.048 0.064 0.054 0.050 0.048
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
B
| 2,612
| 6,013
| 725
| 1,530
| null | null |
github_plus_top10pct_by_avg
|
ithmetic, commonly found in Prolog implementations.
Given a program, containing integer arithmetics, and a class of queries, described using modes, we infer a subset of these queries for which we prove existential non-termination (i.e. the derivation tree for these queries contains an infinite path). The inference and proof are done in two phases. In the first phase, non-termination of the logic part of the program is proven by assuming that all comparisons between integer expressions succeed. We will show that only a minor adaption of our technique presented in [@DBLP:conf/iclp/VoetsS09] is needed to achieve this. In the second phase, given the moded query, integer arguments are identified and constraints over these arguments are formulated, such that solutions for these constraints correspond to non-terminating queries.
The paper is structured as follows. In the next section, we introduce some preliminaries concerning logic programs, integer arithmetics and we present the symbolic derivation trees used to abstract the computation. In Section 3, we introduce our non-termination condition for programs containing integer arithmetics. In Section 4, we describe our prototype analyzer and some results. Finally, we conclude in Section 5.
Preliminaries
=============
Logic Programming
-----------------
We assume the reader is familiar with standard terminology of logic programs, in particular with SLD-resolution as described in [@Lloyd_foundations]. Variables are denoted by strings beginning with a capital letter. Predicates, functions and constant symbols are denoted by strings beginning with a lower case letter. We denote the set of terms constructible from a program $P$ by $Term_P$. Two atoms are called *variants* if they are equal up to variable renaming. An atom $A$ is *more general* than an atom $B$ and $B$ is an *instance* of $A$ if there exists a substitution $\theta$ such that $A\theta = B$.
We restrict our attention to definite logic programs. A logic program $P$ is a finite set of clauses of the form $H
| 2,613
| 791
| 1,726
| 2,003
| null | null |
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|
'}(G)}\cap A}(y)}={{\operatorname}{C}_{{{\operatorname}{O}_{p'}(A)}}(y)}$ and $A_0:=\langle y \rangle \times C$. Then $A=(N\cap A)A_{0}$ and $G=AN=A_{0}N$.
By the minimal choice of $G$, we deduce that $G/N\cong A/A\cap N$ is $p$-decomposable. Hence $[{{\operatorname}{O}_{p'}(G)}\cap A, \langle y\rangle ]\leq N\cap A$. Thus, by coprime action, ${{\operatorname}{O}_{p'}(G)}\cap A=[{{\operatorname}{O}_{p'}(G)}\cap A, \langle y\rangle]{{\operatorname}{C}_{{{\operatorname}{O}_{p'}(G)}\cap A}(y)}\leq (N\cap A){{\operatorname}{C}_{{{\operatorname}{O}_{p'}(G)}\cap A}(y)}$. Hence $A=(N\cap A){{\operatorname}{C}_{{{\operatorname}{O}_{p'}(G)}\cap A}(y)}\langle y\rangle=(N \cap A)A_0$, and the assertion follows.
Recall that $N=N_1\times N_2\times \cdots \times N_r$ with $N_i \cong N_1$ a non-abelian simple group, and set $\Omega:=\{N_1, N_2, \ldots, N_r\}$. By Lemma \[new\], $r > 1$. As $G=A_0N$, $A_0$ acts transitively on $\Omega$. We adapt here some arguments used in [@KMP3] and we claim some facts about this action:
- The orbits of $B$ on $\Omega$ are the same as those of $C$.
This is clear because ${{\operatorname}{O}_{p'}(G)}={{\operatorname}{O}_{p'}(A)}N=CN=BN$.
- Let $\Delta$ be an orbit of $C$ on $\Omega$ of minimal lenght. If $c\in C$, then $\Delta^{yc}=\Delta^{cy}=\Delta^y$, so $\Delta^y$ and $\Delta\cap \Delta^y$ are also orbits of $C$. Therefore, by the choice of $\Delta$, either $\Delta=\Delta^y$ (and hence $\Delta=\Delta^{y^{i}}$ for $i\in\{1,\ldots, p\}$), or $\Delta\cap \Delta^y=\emptyset$ (and hence $\Delta^{y^{i}}\cap \Delta^{y^{j}}=\emptyset$ for $i\neq j$, $i, j \in \{1,\ldots, p\}$). It follows that there is a partition of $\Omega$ of the form $$\Omega=\Delta_1 \cup \Delta_2 \cup \cdots \cup \Delta_k,$$ where $\Delta_i:=\Delta^{y^{i-1}}$ for $i\in\{1,\ldots, k\}$, and $k \in \{1, p\}$. Note that all $\Delta_i$ have the same length, say $m$, and $\{\Delta_1, \ldots, \Delta_k\}$ are all the $C$-orbits (and $B$-orbits) on $\Omega$. Note also that $m$ is a $p'$-number, since $C$ is a $p'$-sub
| 2,614
| 1,817
| 2,277
| 2,293
| null | null |
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|
o $L_n \subset {\mathbb{R}^2}$. Suppose that for a transformation ${\phi}_{n}$ on $L_n$ defined as $${\phi}_{n}(r \cos \frac{2k\pi}{n}, r \sin \frac{2k\pi}{n})=(r \cos \frac{2(k+1)\pi}{n}, r \sin \frac{2(k+1)\pi}{n})$$ there exists a $C^s$ diffeomorphism ${\Phi}_{n}$ satisfying ${\phi}_n \circ q=q \circ {\Phi}_n$ and that for one ${\phi}_{r}$ on $L_n$ defined as $${\phi}_{r}(r \cos \frac{2k\pi}{n}, r \sin \frac{2k\pi}{n})=(r \cos \frac{-2k\pi}{n}, r \sin \frac{-2k\pi}{n})$$ there exists a $C^s$ diffeomorphism ${\Phi}_{r}$ satisfying ${\phi}_r \circ q=q \circ {\Phi}_r$. Then $q$ is said to be [*$D_n$-symmetric*]{}.
\[thm:1\] There exist a class $\mathcal{C}$ of $(C^r,C^0)$ maps whose Reeb spaces are regarded as Reeb graphs and a class ${\mathcal{Q}}_{\mathcal{C}}$ of pseudo quotient maps of the class satisfying the following.
1. For maps of the class $\mathcal{C}$, inverse images of regular values are disjoint unions of standard spheres.
2. For maps of the class ${\mathcal{Q}}_{\mathcal{C}}$, around each vertex of degree $n>1$, the map is regarded as a $D_n$-symmetric map onto $L_n$ by applying a suitable identification.
3. For any finite connected graph which is not a single point, we can construct a map of the class ${\mathcal{Q}}_{\mathcal{C}}$ onto the graph.
4. For a map of the class ${\mathcal{Q}}_{\mathcal{C}}$, if for the target graph, for each vertex, the degree is at most $3$, then the map is realized as a map into $S^1$ or $\mathbb{R}$ of the class $\mathcal{C}$.
In the proof, we first perform local construction around each vertex (Step 1–3) and in Step 4, we complete the construction by constructing remaining parts. Last, we give the strict definitions of $\mathcal{C}$ and $\mathcal{Q}_{\mathcal{C}}$. We will see that this completes the proof except for the fourth condition. Last we discuss the fourth condition.
Step 1 Around a vertex of degree $2$.\
We consider a trivial $C^r$ bundle over $[-1,1]$ whose fiber is a standard sphere. We compose a surjective function over $[-1,1]$ defined by
| 2,615
| 988
| 1,839
| 2,355
| null | null |
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|
\\
\mathcal{D}_{Ru} & \frac{i m u \left(u^4+6 u^2-3\right)}{4 \left(u^2-1\right) \left(u^2+1\right)^2} & -\frac{i h m \left(u^4+6 u^2-3\right)}{8 \left(u^4-1\right)} \\
\mathcal{D}_{uu} & -\frac{i (h+1) m \left(u^4+6 u^2-3\right)}{4 \left(u^2-1\right)^2 \left(u^2+1\right)} & \frac{i m u \left(u^2+3\right)}{2 \left(u^4-1\right)} \\
\mathcal{D}_{TR} & -\frac{u^4-12 u^2+3}{\left(u^2+1\right)^3} & \frac{2 u \left(u^4-14 u^2+9\right)}{\left(u^2+1\right)^4} \\
\mathcal{D}_{Tu} & \frac{(h+2) u \left(u^2-3\right)}{\left(u^2-1\right) \left(u^2+1\right)^2} & -\frac{(h+2) \left(u^4+6 u^2-3\right)}{2 \left(u^2+1\right)^3} \\
\mathcal{D}_{\Phi R} & \frac{6 u^2-2}{\left(u^2+1\right)^3} & -\frac{2 u \left(h \left(u^2+1\right)^2-2 \left(u^4-6 u^2+5\right)\right)}{\left(u^2+1\right)^4} \\
\mathcal{D}_{\Phi u} & -\frac{2 (h+1) u}{\left(u^2-1\right) \left(u^2+1\right)^2} & -\frac{(h+1) \left(h \left(u^2+1\right)^2+4 \left(u^2-1\right)\right)}{2 \left(u^2+1\right)^3} \\
\end{array}
$
$
\begin{array}{c|cc}
\mathcal{D}_{AB} & C_{RR}(u) & C_{Ru}(u) \\
\noalign{\smallskip}
\hline \hline \noalign{\smallskip}
\mathcal{D}_{TT} & \frac{8 \left(u^{10}-2 u^8-6 u^6-8 u^4+21 u^2-6\right)-m^2 \left(u^6+7 u^4+3 u^2-3\right)^2}{8 \left(u^2-1\right) \left(u^2+1\right)^5} & -\frac{4 u \left((2 h+3) u^4+2 (h-6) u^2+9\right)}{\left(u^2+1\right)^4} \\
\mathcal{D}_{T\Phi} & \frac{-\left(u^8+8 u^6+10 u^4-3\right) m^2+2 h \left(u^2-1\right) \left(u^2+1\right)^2+8 \left(u^6+u^4-3 u^2+1\right)}{2 \left(u^2+1\right)^5} & -\frac{4 u \left(u^2-1\right) \left(h u^2+2 u^2+h-4\right)}{\left(u^2+1\right)^4} \\
\mathcal{D}_{\Phi \Phi } & \frac{2 \left(u^2-1\right) \left(-m^2 \left(u^2+1\right)^2+h \left(u^2+1\right)^2+2 \left(u^4+2 u^2-1\right)\right)}{\left(u^2+1\right)^5} & -\frac{4 (h+1) u \left(u^2-1\right)}{\left(u^2+1\right)^3} \\
\mathcal{D}_{RR} & \frac{u^2-1}{\left(u^2+1\right)^3} & \frac{4 u}{\left(u^2+1\right)^2} \\
\mathcal{D}_{Ru} & -\frac{u}{\left(u^2+1\right)^2} & \frac{8 \left(u^6+3 u^4-5 u^2+1\right)-m^2 \left(u^8+8 u^6+10 u^4
| 2,616
| 2,442
| 2,207
| 2,403
| null | null |
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|
ions are twofold: firstly, the choice of prior is necessary for certain tractable operations but does not reflect true prior belief; secondly, prior specification does not lead conveniently and efficiently to sequential inference yet a simple approximation achieves this goal. An overview of the methodology is first provided, with details about each part given subsequently. Adapting terminology from sequential inference, the re-ordered index $t$ shall henceforth be referred to as ‘time’.
Overview {#sec:Overview}
--------
The ultimate aim is to calculate and compare the posterior model probabilities for a range of models, each with a different number of MUs, $u$. Posterior model probabilities are straightforward to obtain once the marginal likelihood for each model is available. Hence, for a given model with $u$ MUs, the target for inference is its marginal likelihood, $f(y_{1:T}|s_{1:T})$; throughout this section, for notational simplicity, we suppress the dependence on $u$. This can be expressed as a product of sequential predictive factors with each defined by: $$\begin{aligned}
f\left(y_t |~ y_{1:t-1},~ s_{1:t}\right) & = \sum_{{\mathbf{x}}_{1:t-1}\in\mathcal{X}_{1:t-1}} f\left(y_t|~ {\mathbf{x}}_{1:t-1},~ y_{1:t-1},~ s_{1:t}\right) \mathbb{P}\left({\mathbf{x}}_{1:t-1} |~ y_{1:t-1},~ s_{1:t-1}\right)\label{eq:SeqPredFactor}\end{aligned}$$ where $\mathcal{X}_{1:t} = \{0,1\}^{ut}$ denotes the space for the sequence of vectors of historical firing events.
The inference scheme is based upon two key observations. Firstly, the observation and excitability parameters are conditionally independent given the set of firing events ${\mathbf{x}}_{1:T}$. Such an independence structure separates the observational and firing processes and simplifies the marginalisation of the parameter space for evaluating the marginal likelihood. Secondly, conditional on ${\mathbf{x}}_t$, the priors for the observation parameters in are nearly conjugate for the likelihood in . For a baseline measurement (which has ${\mathbf{x}}_t={\
| 2,617
| 938
| 2,623
| 2,475
| 1,133
| 0.79385
|
github_plus_top10pct_by_avg
|
B2‐A1** −0.41 −0.43 −0.37 −0.41 0.46 0.52 0.05 −0.17
**B2‐A2** −0.45 −0.41 −0.44 −0.42 −0.45 −0.37 −0.37 −0.34
**B3** 0.19 0.24 0.21 0.25 1.36 1.34 1.20 1.20
**C1** −0.03 −0.02 −0.03 −0.02 n/a n/a n/a n/a
**C2** −0.01 0.07 −0.01 0.07 n/a n/a n/a n/a
**D1** −0.10 −0.13 −0.10 −0.13 0.26 0.34 0.24 0.32
**D2** 0.13 0.12 0.13 0.12 0.46 0.49 0.45 0.46
See Table [1](#sim7930-tbl-0001){ref-type="table"} for full data generation details relating to each scenario. True value for θ is −9.66.
n/a = not applicable, since there is no τ ~β~ ^2^ to vary when a beta distribution is used for the intercept data generating mechanism. Options: ML, maximum likelihood estimation; REML, restricted maximum likelihood estimation.
######
Median percentage bias of the between‐trial variance of treatment effects ( $\left. {\hat{\tau}}^{2} \right)$, under different scenarios for the random treatment effect with normal and beta distributions for the intercept data generating mechanisms. Results shown separately for stratified and random intercept models, under each of the estimation options considered
Median Percentage Bias of ${
| 2,618
| 4,446
| 2,770
| 2,415
| 2,225
| 0.781668
|
github_plus_top10pct_by_avg
|
rt of particles in $G_{\rm e}$ is governed by the system of equations on $G_{\rm e}\times S\times I$, $$\begin{gathered}
\omega\cdot\nabla_x\Psi_1+\Sigma_{{\rm e},1}\Psi_1-K_{{\rm e},1}\Psi=0,\label{ref2a}\\
-{{\frac{\partial (S_{{\rm e},j}\Psi_j)}{\partial E}}}+\omega\cdot\nabla_x\Psi_j+\Sigma_{{\rm e},j}\Psi_j-K_{{\rm e},j}\Psi=0,\quad j=2,3,\label{ref3a}\end{gathered}$$ for $\Psi=(\Psi_1,\Psi_2,\Psi_3)$, along with the boundary conditions $$\begin{aligned}
{5}\label{ref4a}
& {\Psi_j}_{|\Gamma^1_{\rm e,-}}={\psi_j}_{|\Gamma_+}\quad && {\rm on}\ \Gamma^1_{\rm e,-}=\Gamma_+,\quad && \\
& {\Psi_j}_{|\Gamma^2_{\rm e,-}}=0\quad && {\rm on}\ \Gamma^2_{\rm e,-},\quad && j=1,2,3, \end{aligned}$$ and the initial condition \[ref5a\] \_j(x,,E\_[m]{})=0 G\_[e]{}SI,j=2,3. Above $\Sigma_{{\rm e},j}$, $\sigma_{{\rm e},kj}$ and $S_{{\rm e},j}$ are the (restricted) cross-sections and the (restricted) stopping powers for the medium inside $G_{\rm e}$, and $$(K_{{\rm e},j}\Psi)(x,\omega,E)
:=
\sum_{k=1}^3\int_{S\times I}\sigma_{{\rm e},kj}(x,\omega',\omega,E',E)\Psi_k(x,\omega',E')d\omega' dE',$$ for $\Psi\in L^2(G_{\rm e}\times S\times I)^3$.
In this setup, we define a reflection operator $R=R_{\rm b}$ by setting (recall that $\Gamma^1_{{\rm e},+}=\Gamma_-$) $$\begin{aligned}
& R_{\rm b}(\psi_{|\Gamma_+}):=\Psi_{|\Gamma^1_{{\rm e},+}}=\Psi_{|\Gamma_-}, \nonumber\\
& D(R_{\rm b}):=\{\psi_{|\Gamma_+}\ |\ \psi\ {\rm is\ a\ solution\ of\ (\ref{ref2})-(\ref{ref5})\ for \ some}\ g\in T^2(\Gamma_-)^3\}. \label{ref6}\end{aligned}$$ If the assumptions of Theorem \[cosystth2\] hold for $G'$ in place of $G$ (and for the respective cross-sections, stopping powers, $\Sigma'_j,\sigma'_{kj},S'_j$), the operator $R_{\rm b}$ is a linear operator $T^2(\Gamma_+)^3\to T^2(\Gamma_-)^3$ with domain of definition $D(R_{\rm b})\subset T^2(\Gamma_+)^3$. In general, $R_{\rm b}$ is not bounded.
The meaning of this definition is that $R_{\rm b}(\psi_{|\Gamma_+})$ models an extra source on $\Gamma_-$ (i.e. an inflow boundary source for $G$) due to backsc
| 2,619
| 1,665
| 1,153
| 2,696
| null | null |
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|
2 + {\mathbf{k}}_3, {\mathbf{k}}_2 - {\mathbf{k}}_3 \}$; the most general solution can be then constructed as $$\begin{aligned}
{\mathbf{u}}_1({\mathbf{x}}) & = & \mathbf{A}{\textrm{e}^{2\pi i[1,1,0]\cdot{\mathbf{x}}}} +
\mathbf{B}{\textrm{e}^{2\pi i[1,-1,0]\cdot{\mathbf{x}}}} +
\mathbf{C}{\textrm{e}^{2\pi i[1,0,1]\cdot{\mathbf{x}}}} + \nonumber \\
& & \mathbf{D}{\textrm{e}^{2\pi i[1,0,-1]\cdot{\mathbf{x}}}} +
\mathbf{E}{\textrm{e}^{2\pi i[0,1,1]\cdot{\mathbf{x}}}} +
\mathbf{F}{\textrm{e}^{2\pi i[0,1,-1]\cdot{\mathbf{x}}}} + \textrm{C.C.}
\label{eq:uvec_3D_k2}\end{aligned}$$ with the constants $\mathbf{A},\mathbf{B},\ldots,\mathbf{F}\in\mathbb{C}^3$ suitably chosen so that $\mathbf{A}\cdot[1,1,0] = 0$, $\mathbf{B}\cdot[1,-1,0]
= 0 ,\ldots,\mathbf{F}\cdot[0,1,-1] = 0$, which ensures that incompressibility condition is satisfied, and that $\E({\mathbf{u}}_1) = 1$; in this case, $|{\mathbf{k}}|^2 =
2$, $\forall\,{\mathbf{k}}\in{\mathcal{W}}_2$, and the optimal asymptotic value of $\R$ is $$\label{eq:R0_kvec_3D_k2}
\R({\widetilde{\mathbf{u}}}) \approx - 16\pi^2\nu\E_0,$$ \[c2\]
3. ${\mathcal{W}}_3 = {\mathcal{W}}\cup (-{\mathcal{W}})$ for ${\mathcal{W}}= \{
{\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3,-{\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3,{\mathbf{k}}_1-{\mathbf{k}}_2+{\mathbf{k}}_3,{\mathbf{k}}_1+{\mathbf{k}}_2-{\mathbf{k}}_3
\}$; the most general solution can then be constructed as $$\begin{aligned}
{\mathbf{u}}_1({\mathbf{x}}) & = & \mathbf{A}{\textrm{e}^{2\pi i[1,1,1]\cdot{\mathbf{x}}}} +
\mathbf{B}{\textrm{e}^{2\pi i[-1,1,1]\cdot{\mathbf{x}}}} + \nonumber \\
& & \mathbf{C}{\textrm{e}^{2\pi i[1,-1,1]\cdot{\mathbf{x}}}} +
\mathbf{D}{\textrm{e}^{2\pi i[1,1,-1]\cdot{\mathbf{x}}}} + \textrm{C.C.}
\label{eq:uvec_3D_k3}\end{aligned}$$ with the constants $\mathbf{A},\mathbf{B}
| 2,620
| 5,301
| 524
| 1,966
| null | null |
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|
proof of Theorem \[mult-thm\] for $\v_\muhat$ is indivisible is given in [@hausel-letellier-villegas] by expressing $\langle
\Lambda\otimes R_\muhat,1\rangle$ as the Poincaré polynomial of a comet-shaped quiver variety. This quiver variety exists only when $\v_\muhat$ is indivisible.
In [@Hiss] the authors discusses some results of Mattig who showed that the multiplicities $\langle\calX_1\otimes\calX_2,\calX_3\rangle$, with $\calX_1,\calX_2,\calX_3$ unipotent characters of $\GL_n$, are polynomials in $q$ with rational coefficients. Using calcultion with CHEVIE, he also observed when $n\leq 8$ that the coefficients of $\langle\calX_1\otimes\calX_2,\calX_3\rangle$ are non-negative integers. The following proposition confirms part of this prediction.
For a $k$-tuple of partitions $\muhat=(\mu^1,\ldots,\mu^k)$ of $n$ there exists a polynomial $U_\muhat\in \Z[T]$ such that $\langle
\Lambda\otimes \calU_{\mu^1}\otimes\cdots\otimes
\calU_{\mu^k},1\rangle=U_\muhat(q)$.
Since the complete symmetric functions $\{h_\lambda(\x)\}_\lambda$ forms a $\Z$-basis of the ring $\Lambda(\x)$, for any partitions $\mu,\lambda$, there exist integers $a_{\mu\lambda}$ such that $$s_\mu(\x)=\sum_\lambda a_{\mu\lambda}h_\lambda(\x).$$The matrix $(a_{\mu\lambda})_{\mu,\lambda}$ is the transpose inverse of the matrix $K=(K_{\mu\lambda})_{\mu,\lambda}$ of Kostka numbers.
If $\calU_\mu$ is the unipotent characters of $\GL_n$ corresponding to $\mu$, by Theorem \[Rtau\] we have $$\calU_\mu(C)=\left\langle
\tilde{H}_\omega(\x;q),s_{\mu}(\x)\right\rangle$$where $C$ is a conjugacy class of type $\omega$. Hence by Corollary \[R\], we deduce that $$\calU_\mu=\sum_\lambda
a_{\mu\lambda}R_{L_\lambda}^G(1).$$Hence $\left\langle\Lambda\otimes\calU_1\otimes\cdots\otimes\calU_k,1\right\rangle$ is a $\Z$-linear combination of multiplicities of the form $$\left\langle\Lambda\otimes
R_{L_{\lambda^1}}^G(1)\otimes\cdots\otimes
R_{L_{\lambda^k}}^G(1),1\right\rangle$$and so by Remark \[rem326\], it is a $\Z$-linear combination of polynomia
| 2,621
| 2,186
| 2,240
| 2,186
| null | null |
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|
c connective $N$ in $\mathcal{L}_{Q}^{P}$ (it has already been proved in Ref. 27 that this principle does not hold in the general language $\mathcal{L}^{P}$).
It is also interesting to note that the justification values of different elementary afs, say $\vdash E(x)$ and $\vdash F(x)$, must be different for some state $S$, since $\mathcal{S}_{E}\neq \mathcal{S}_{F}$ if $E\neq F$ (Sec. 2.2), hence $\mathcal{S}_{\vdash E(x)}\neq \mathcal{S}_{\vdash F(x)}$.
Finally, we remind that the general theory of $\mathcal{L}^{P}$ associates an assignment function $\sigma $ with a set $\Pi _{\sigma }$ of pragmatic evaluation functions (Sec. 3.1), hence this also occurs within $\mathcal{L}_{Q}^{P}$. One may then wonder whether $\Pi _{\sigma }$ is necessarily nonvoid and, if this is the case, whether it may contain more than one pragmatic evaluation function. In order to answer these questions, let us consider an interpretation $\xi $ of the variable $x$ that maps $x$ on a physical object in the state $S$. Then, $\xi $ determines a unique assignment function $\sigma (\xi )$ and a unique pragmatic evaluation function associated with it, that we have denoted by $\pi _{S}$, for it depends only on the state $S$. Since every assigment function in $\Sigma $ is induced by an interpretation $\xi $ because of A$_{4}$ in Sec. 3.2, this proves that $\Pi _{\sigma }$ is necessarily nonvoid for every $\sigma \in
\Sigma $. Moreover, note that an interpretation $\xi ^{\prime }$ of $x$ may exist within the SR interpretation of QM that maps $x$ on a physical object in the state $S^{\prime }$, with $S^{\prime }\neq S$, yet such that $\sigma
(\xi ^{\prime })=\sigma (\xi )$. The pragmatic evaluation functions $\pi
_{S} $ and $\pi _{S^{\prime }}$ are then different, but they are both associated with the assignment function $\sigma =\sigma (\xi )=\sigma (\xi
^{\prime })$, so that they both belong to $\Pi _{\sigma }$. Hence, $\Pi
_{\sigma }$ may contain many pragmatic evaluation functions.[^9]
Pragmatic validity and order in $\mathcal{L}_{Q}^{P}$
---
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$\ker\alpha(0)$. Fans and stars are studied in [@MR2002d:14084], and are the only kinds of curves with small orbit that consist of lines; they are items (1) through (5) in our classification of curves with small orbit, see §\[appendix\].
For types II—V we choose coordinates so that $p=(1:0:0)$ is a point of ${{\mathscr C}}$; for types II, IV, and V we further require that $z=0$ is a chosen component $\ell$ of the tangent cone to ${{\mathscr C}}$ at $p$.
[**Type II.**]{} Assume that $p$ is a nonsingular, non-inflectional point of the support ${{{{\mathscr C}}'}}$ of ${{\mathscr C}}$, contained in a nonlinear component, with tangent line $z=0$. Let $$\alpha(t)=\begin{pmatrix}
1 & 0 & 0 \\
0 & t & 0 \\
0 & 0 & t^2
\end{pmatrix}\quad.$$ Then the ideal of $\lim_{t\to 0}{{\mathscr C}}\circ \alpha(t)$ is generated by $$x^{d-2S}(y^2+\rho x z)^S\quad,$$ where $S$ is the multiplicity of the component in ${{\mathscr C}}$, and $\rho\ne 0$; that is, the limit consists of a (possibly multiple) nonsingular conic tangent to the kernel line, union (possibly) a multiple of the kernel line.
Such curves are items (6) and (7) in the classification reproduced in §\[appendix\]. The extra kernel line is present precisely when ${{\mathscr C}}$ is not itself a multiple nonsingular conic.
[**Type III.**]{} Assume that $p$ is a singular point of ${{{{\mathscr C}}'}}$ of multiplicity $m$ in ${{\mathscr C}}$, with tangent cone supported on at least three lines. Let $$\alpha(t)=\begin{pmatrix}
1 & 0 & 0 \\
0 & t & 0 \\
0 & 0 & t
\end{pmatrix}\quad.$$ Then $\lim_{t\to 0} {{\mathscr C}}\circ\alpha(t)$ is a fan consisting of a star centered at $(1:0:0)$ and projectively equivalent to the tangent cone to ${{\mathscr C}}$ at $p$, and of a residual $(d-m)$-fold line supported on the kernel line $x=0$.
[**Type IV.**]{} Assume that $p$ is a singular or inflection point of the support of ${{\mathscr C}}$. Germs of type IV are determined by the choice of the line $\ell$ in the tange
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\times {\mathbb Z}_2$ partition functions are very different, despite the fact that the theories differ by a trivially-acting gauged ${\mathbb Z}_2$.
In fact, in the example above, one can show that the partition function of the $D_4$ orbifold is the same as the partition function of a disjoint union of two ${\mathbb Z}_2 \times {\mathbb Z}_2$ orbifolds, one with and the other without discrete torsion. The one-loop partition function of a disjoint union is the sum of the partition functions of the components, and discrete torsion adds a sign to the $(\overline{a}, \overline{ab})$, $(\overline{b},\overline{ab})$ and $(\overline{a},\overline{b})$ sectors, so they cancel out of the partition function for the disjoint union. This is a simple example of the ‘decomposition conjecture’ we review in section \[sect:decomp-22review\].
Notions of twisting {#sect:twisting}
-------------------
Now that we have outlined gerbes and demonstrated their physical meaningfulness, let us turn to possible bundles over gerbes. A gerbe was defined by a trivial group action on the base space; however, that same group action can be nontrivial on the bundle. The resulting bundle is then interpreted as some sort of twisted bundle, in some sense, as we shall review here.
There are various notions of twisted bundles in the literature. One notion, discussed for example in [@cks], is of a twisted bundle in which the twisting refers to the fact that the transition functions do not quite close on triple overlaps: instead of $$g_{\alpha \beta } g_{\beta \gamma} g_{\gamma \alpha} \: = \: 1$$ the transition functions obey $$\label{cocyc1}
g_{\alpha \beta } g_{\beta \gamma} g_{\gamma \alpha} \: = \:
h_{\alpha \beta \gamma} I$$ for some cocycle $h_{\alpha \beta \gamma}$. At the level of the gauge field, such a twisting means that across coordinate patches, the gauge field receives an affine translation in addition to a gauge transformation. Such twisted bundles appear physically on D-branes. After all, under a gauge transformation of the $B$
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e additional non-trivial constraints that survive the orientifold projection but are not such that the upstairs and downstairs indices are all different. In the IIB case, from eq. one gets the components $(Q \cdot H_3-P_1^2 \cdot F_3)^{x^j}_{x^i y^i x^j}$ and $(Q \cdot H_3-P_1^2 \cdot F_3)^{y^j}_{x^i y^i y^j}$, which in eq. are associated to the components $D_{4\, x^k y^k y^j, x^j}$ and $D_{4\, x^k y^k x^j , y^j}$ of the mixed-symmetry potential $D_{7,1}$, and similarly from eq. . In the IIA case, neither eq. nor eq. lead to additional relations, while the non-trivial relations come from the third constraint in eq. , which after the orientifold projection becomes $$(-Q \cdot f-R\cdot H_3+P_1^3 \cdot F_2-P^{1,3} \cdot F_4 - P_1^5 \cdot F_4 + P^{1,5} \cdot F_6 )^{ab}_{cd}=0 \quad . \label{NSNSBianchiIIAO6ter}$$ What one finds is that the IIB and IIA constraints that one gets do not match, unless the additional constraints $$\begin{aligned}
& q(g+\gamma)+\bar{g}e-fm+gq+e(\bar{g}+\bar{\gamma})+e_0\bar{f}=0 \label{1c}\\
& -mf'-q(g'+\gamma')+\bar{g}'e-g'q+e(\bar{g}'+\bar{\gamma}')-\bar{f}'e_0=0 \label{2cond} \end{aligned}$$ are satisfied.
In order to understand and solve this mismatch, we remember that the fields listed in eq. , that are associated to the $\alpha=-2$ branes, in the four-dimensional theory belong to representations of $SO(6,6)$. In particular, the space-filling branes correspond to a 4-form potential $D_{4,MNPQ}$ in the ${\bf 495}$ representation [@stringsolitons]. This representation not only contains the fields in eq. , but also the potentials $D_8$, $D_{9,1}$, $D_{10}$ and $D_{10,2}$ [@stringsolitons]. The components of the mixed-symmetry potentials in with indices after the comma that are not parallel to any of the other indices are related by T-duality to these additional potentials.[^8] In particular, the 8-form field $D_8$ is the one that together to $C_8$ and $E_8$ forms the triplet of $SL(2,\mathbb{R})$ [@Meessen:1998qm]. In the IIB/O3 setup, the tadpole induced by the fluxes to this p
| 2,625
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| 0.780458
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particular, the bounds are valid unconditionally with respect to the joint distribution of the entire sample and of the splitting outcome.
Also, in the proof $C$ denotes a positive positive that may depend on $A$ only but not on any other variable, and whose value may change from line to line.
[**Proof of .**]{} As usual, we condition on $\mathcal{D}_{1,n}$ and thus treat ${\widehat{S}}$ as a fixed subset of $\{1,\ldots,d\}$ of size $k$. Recalling the definitions of $\hat{\beta}_{{\widehat{S}}}$ and $\beta_{{\widehat{S}}}$ given in and , respectively, and dropping the dependence on ${\widehat{S}}$ in the notation for convenience, we have that $$\begin{aligned}
\| \hat{\beta}_{{\widehat{S}}} - \beta_{{\widehat{S}}}\| & = \left\| \left( \hat{\Sigma}^{-1}
- \Sigma^{-1} \right) \hat{\alpha} + \Sigma^{-1}\left( \hat{\alpha} -
\alpha \right) \right\|\\
& \leq \left\| \hat{\Sigma}^{-1} - \Sigma^{-1} \right\|_{\mathrm{op}}
\|\hat{\alpha} \| + \frac{1}{u} \| \hat{\alpha} - \alpha\|\\
& = T_1 + T_2.\end{aligned}$$ By the vector Bernstein inequality , $$\| \hat{\alpha} - \alpha \| \leq C A \sqrt{ \frac{k \log n}{n} },$$ with probability at least $1 - \frac{1}{n}$ and for some universal constant $C$ (independent of $A$). Since the smallest eigenvalue of $\Sigma$ is bounded from below by $u$, we have that $$T_1 \leq C \frac{1}{u} \sqrt{ \frac{k \log n}{n}}.$$ To bound $\left\| \hat{\Sigma}^{-1} - \Sigma^{-1} \right\|_{\mathrm{op}}$ in the term $T_2$ we write $\hat{\Sigma} = \Sigma + E$ and assume for the moment that $\|E\|_{\mathrm{op}} \|
\Sigma^{-1}\|_{\mathrm{op}} < 1 $ (which of course implies that $\| E \Sigma^{-1}
\|_{\mathrm{op}} < 1$). Since $E$ is symmetric, we have, by formula 5.8.2 in [@Horn:2012:MA:2422911], that $$\left\| \hat{\Sigma}^{-1} - \Sigma^{-1} \right\|_{\mathrm{op}} = \left\|
(\Sigma + E)^{-1} - \Sigma^{-1} \right\|_{\mathrm{op}} \leq \|
\Sigma^{-1}\|_{\mathrm{op}} \frac{\| E
\Sigma^{-1}\|_{\mathrm{op}} } { 1 - \| E
\Sigma^{-1}\|_{\mathrm{op}} },$$ which in turn is upper bounde
| 2,626
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|
ec{p}\,'}}{E'+M_N}
\end{array}\right)\,.\end{aligned}$$ The relativistic propagator of a baryon with mass $M_B$ and momentum $p$ reads $$\frac{i}{\cancel{p}-M_B+i\epsilon}
=\frac{i(\cancel{p}+M_B)}{p^2-M_B^2+i\epsilon} \,.$$ Making the heavy baryon expansion with these spinors and propagators introduces mass differences ($M_\Lambda-M_N$, $M_\Sigma-M_\Lambda$) in the baryonic propagators. A reasonable approach would be to consider these mass differences of order ${\cal O}\left({\vec{q}}^{\,2}/\Lambda^2\right)$ ($M_B={\overline{M}}+{\cal O}\left({\vec{q}}^{\,2}/\Lambda^2\right)$), and thus they would not enter in the loop diagrams. We have chosen to leave the physical masses in both the initial and final spinors and also in the intermediate propagators; i.e. we consider the mass differences as another scale in the heavy baryon expansion. The corresponding SU(3) symmetric limit is also given at the end of section \[ss:tped\], and can be easily obtained from our expressions by setting the mass differences, which we explicitly retain, to zero.
The procedure we follow to compute the different Feynman diagrams entering the transition amplitude is the following: first we write down the relativistic expressions for each diagram, and then afterwards, we perform the heavy baryon expansion.
In the next sections we will describe the LO and NLO contributions to the process $\Lambda N\to NN$, following the scheme presented here. The explicit expressions and details of the calculations are given in the Appendices.
Leading order Contributions {#ss:loc}
===========================
------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------
![One-pion and one-kaon exchange contributions to the transition.\[fig:loc\]](ope "fig:") ![One-pion and one-kaon exchange contributions to the transition.\[fig:loc
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It is useful to derive a closed analytic expression for the self-energy of the molecular orbital GF [@Kolodzeiski2017] which is used to increase the precision of the NRG GF [@BullaHewsonPruschke98] as well as analyze the results. We consider the system Hamiltonian $H_S$ $$\begin{aligned}
H_S&=& \sum_{\k\sigma} \e_{\k\sigma} c^\dagger_{\k\sigma} c_{\k\sigma}
+\w_0 b^\dagger_0 b_0
+
\sum_\sigma \e_{d\sigma} n^d_\sigma + U n^d_\uparrow n^d_\downarrow
\non
&& + \sum_{\k\sigma} V_{\k} ( c^\dagger_{\k\sigma} d_\sigma + d^\dagger_\sigma c_{\k\sigma} )
\\
&& \nonumber
+ \lambda_d \hat X_0 (\sum_\sigma n^d_\sigma - n_{d0})
+\lambda_c \hat X_0 (\sum_\sigma c^\dagger_{0\sigma} c_{0\sigma} -n_{c0})\end{aligned}$$ where we have defined $$\begin{aligned}
c_{0\sigma} &=& \frac{1}{V_0} \sum_{\k} V_{\k} c_{\k\sigma}
\\
V_0^2 &=& \sum_{\k} |V_{\k} |^2 \, .\end{aligned}$$ We start from the commutators $$\begin{aligned}
\, [d_\sigma, H_S] &=& \e_{d\sigma}d_\sigma + U n^d_{-\sigma} d_\sigma
+V_0 c_{0\sigma} +\lambda_d \hat X_0 d_\sigma \\
\, [c_{k\sigma}, H_S] &=& \e_{\k\sigma} c_{\k\sigma}
+\lambda_c \hat X_0 \frac{V_k}{V_0} c_{0\sigma} + V_k d_\sigma
\label{eq:commu-c0}\end{aligned}$$ and obtain the equation of motion (EOM) $$\begin{aligned}
(z-\e_d) G_{d_\sigma,d^\dagger_\sigma}(z) &=& 1
+ U F_\sigma(z) +\lambda_d M_\sigma(z) \\
&& \nonumber
+ \sum_k V_k G_{c_{\k\sigma} ,d^\dagger_\sigma}(z)\end{aligned}$$ after introducing the notation $$\begin{aligned}
F_\sigma(z) &=& G_{d_\sigma n_{-\sigma},d^\dagger_\sigma}(z) \\
M_\sigma(z) &=& G_{\hat X_0 d_\sigma,d^\dagger_\sigma}(z) .\end{aligned}$$ While the complex function $F_\sigma(z)$ contains the information about the local correlations between the electrons of different spins $\sigma$, the influence of the molecular vibration onto the equilibrium GF is account for by $M_\sigma(z)$ that also is relevant for the inelastic tunneling current – see Sec. \[sec:I-inelastic\]. In order to close the EOM, we use the commutator to derive $$\begin{aligned}
(z- \e_{\k\sigma}) G
| 2,628
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{\v \in \Gamma_P} (\nu_\v-1) E_\v.
\end{aligned}$$ Clearly one has the relations $\hat{\mathcal{C}} = \sum_{j=1}^r n_j \hat{\mathcal{C}}_j$ and $m_\v = \sum_{j=1}^r n_j m_{\v j}$. Finally consider the divisor $L^{(k)}$ corresponding to the covering $\rho_Y$, see Theorem \[thm:Esnault\].
\[thm:h2Lk\] The dual space $H^2(Y,\cO_{Y}(L^{(k)}))^{*}$ is isomorphic to the $\CC$-vector space $$\left\{ F \in \CC[x,y,z]_{w,s_k-|w|}
\vphantom{\sum_{j=1}^r}\right.
\left|\
\operatorname{mult}_{E_\v} \pi^{*} F > \sum_{j=1}^r {\left \{ \frac{kn_j}{d} \right \}} m_{\v j} - \nu_\v, \
\forall \v \in \Gamma_P, \, \forall P \in \Si
\right\}$$ where $\CC[x,y,z]_{w,l}$ is the vector space of $w$-homogeneous polynomials of degree $l$, $$s_k := \sum_{j=1}^r {\left \{ \frac{kn_j}{d} \right \}} d_j \in \mathbb{Z},\text{ and }|w|=w_0+w_1+w_2.$$ If $\mathcal{C}$ is reduced, then $s_k = k$ and $\displaystyle\sum_{j=1}^r {\left \{ \frac{kn_j}{d} \right \}} m_{\v j} = \frac{km_\v}{d}$.
By Serre’s duality $H^2(Y,\cO_{Y}(L^{(k)}))^{*} \cong H^0(Y,\cO_{Y}(K_Y-L^{(k)}))$. We plan to apply Proposition \[prop:H0YD\] to the divisor $D' := K_Y - L^{(k)}$. Recall that $K_Y = \pi^{*} K_X + K_{\pi}$.
According to the divisor $\pi^{*} \mathcal{C} - d \pi^{*} H$ is decomposed as $$- d \hat{H} + \sum_{j=1}^r n_j \hat{\mathcal{C}}_j + \sum_{P \in \Si}
\sum_{\v \in \Gamma_P} (m_\v - d \b_\v) E_\v$$ and hence by definition $L^{(k)}$ is the divisor $$L^{(k)} = -k \hat{H} + \sum_{j=1}^r {\left \lfloor \frac{kn_j}{d} \right \rfloor} \hat{\mathcal{C}}_j +
\sum_{P \in S} \sum_{\v \in \Gamma_P} {\left \lfloor \frac{k(m_\v - d\b_\v)}{d} \right \rfloor} E_\v,$$ that can be easily written as $$\label{eq:Lk}
L^{(k)} = \pi^{*} \bigg( -kH + \sum_{j=1}^r {\left \lfloor \frac{kn_j}{d} \right \rfloor} \mathcal{C}_j \bigg) +
\sum_{P \in S} \sum_{\v \in \Gamma_P} \bigg( {\left \lfloor \frac{k(m_\v-d\b_\v)}{d} \right \rfloor} + k \b_\v - e_{\v k} \bigg) E_\v,$$ where $e_{\v k} = \sum_{j=1}^r {\left \lfloor \frac{kn_j}{d} \right \rfloor} m_{\v j}$. Then $$\label{eq:div
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revious papers that the coherence between active and sterile, and sterile and sterile states are not maintained for sterile mass differences larger than $0.1$ eV$^2$. The effect of decoherence is taken into account by making average over the fast oscillations. We feel it desirable for the current treatment be replaced by the real quantum mechanical one using wave packets, in which the effect of decoherence would automatically come in. But, we do believe that our present framework is able to describe effectively the right physics derived from such improved treatment.
Smallness of expansion parameters and higher order corrections {#sec:higher-order}
----------------------------------------------------------------
Here, we discuss general structure of the perturbation series without recourse to averaging out the fast oscillations. The effective expansion parameters in our perturbative framework are the following four, $$\begin{aligned}
\frac{A W }{ \Delta_{J} - h_{i} },
\hspace{10mm}
\frac{A W }{ h_{j} - h_{i} },
\hspace{10mm}
A L W,
\hspace{6mm}
\text{and}
\hspace{6mm}
W.
\label{expansion-parameters}\end{aligned}$$ We already saw them, except for the last one, in the discussion in section \[sec:energy-denominator\], and it can be seen by inspecting the expressions of the oscillation probabilities up to the fourth orders given in section \[sec:probability-2nd\] and appendix \[sec:expression-probability-4th\]. Formally, the expansion parameter is the first one in (\[expansion-parameters\]) in view of (\[Omega-expand\]) with $\Omega [1]$, the kernel, in (\[Omega-1st-order\]). But, the spacial integration in (\[Omega-expand\]) produces different effective expansion parameters, the second and the third ones in (\[expansion-parameters\]). The extra factor of $W$’s without the kinematical factors is provided when transforming from the $\hat{S}$ to $S$ matrices, as seen in section \[sec:S-matrix\].
For simplicity of the discussion in this section, we limit ourselves to the case of $|W| \sim 0.1$. Under t
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hermore, assume that the stopping powers $S_j$, where $j=2,3$, satisfy $$\begin{aligned}
& S_j\in C^2(I,L^\infty(G)), \label{ass5} \\[2mm]
& \nabla_x S_j\in L^\infty(G\times I), \label{ass5n} \\[2mm]
& \kappa_j:=\inf_{(x,E)\in \ol{G}\times I} S_j(x,E) > 0. \label{ec4}\end{aligned}$$
Let $f\in C^1(I,L^2(G\times S)^3)$ and let $g\in C^2(I,T^2(\Gamma_-')^3)$ which satisfies the *compatibility condition* $$g_j(E_m)=0,\quad j=2,3.$$ Then the problem (\[csda1a\])-(\[csda3\]) has a unique solution $\psi\in \tilde W^2(G\times S\times I)\times \big(C(I,\tilde{W}^2(G\times S)^2)\cap C^1(I,L^2(G\times S)^2)\big)$. In particular, $\psi\in \tilde{W}^2(G\times S\times I)\times (\tilde{W}^2(G\times S\times I)\cap
W_1^2(G\times S\times I))^2$.
If in addition, for some $c>0$, the inequalities $$\begin{aligned}
\Sigma_j(x,\omega,E)-\sum_{k=1}^3\int_S\tilde\sigma_{jk}(x,\omega,\omega',E) d\omega'
\geq c, \label{ec8} \\
\Sigma_j(x,\omega,E)-\sum_{k=1}^3\int_S\tilde\sigma_{kj}(x,\omega',\omega,E) d\omega'
\geq c, \label{ec9}\end{aligned}$$ hold for $j=1,2,3$ and for a.e. $(x,\omega,E)\in G\times S\times I$, then the solution $\psi$ satisfies the estimate .
At first we notice that by the assumption (\[ass3-aa\]), for a.e. $(x,\omega,E)\in G\times S\times I$, $$\begin{aligned}
&\sum_{k=1}^3\int_S\tilde\sigma_{kj}(x,\omega',\omega,E)d\omega'
\leq
\sum_{k=1}^3\sup_{E\in I}{\left\Vert \tilde\sigma_{kj}(E)\right\Vert}_{L^\infty(G\times S,L^1(S'))}=:M_1<\infty, \label{m'} \\
&\sum_{k=1}^3\int_S\tilde\sigma_{jk}(x,\omega',\omega,E)d\omega
\leq
\sum_{k=1}^3
\sup_{E\in I}{\left\Vert \tilde\sigma_{jk}(E)\right\Vert}_{L^\infty(G\times S',L^1(S))}=:M_1'<\infty. \label{m''}\end{aligned}$$
We begin by treating the special case where $g=0$. Recall that the system of equations of interest on $G\times S\times I$ for $\psi=(\psi_1,\psi_2,\psi_3)$ is &\_x\_1+\_1\_1-K\_[1]{}=f\_1 \[proof1\]\
&-[E]{}+\_x\_j+\_[j]{}\_j-K\_[j]{}=f\_j,j=2,3.\[proof2\] Equation can be written as \[pr5.5.1\] \_x\_1+\_1\_1-K\_1\_1-K=f\_1, where $$\hat\psi:=(\psi_2,\psi_
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0$ sites keeping per block on average $400$ states for gapped phases and $600$ states for gapless ones), and bosonization techniques to unveil the low-energy behavior of model . For $j<1/4$, we employ standard bosonization transformations [@Giamarchi], with an additional oscillating factor $b_i\to (-1)^ie^{i\sqrt{\pi}\theta(x)}$, to obtain the low-energy effective theory, which is given by the sine-Gordon model $$\label{sine-Gordon}
{\mathcal H}\!=\! \frac{v_s}{2}\left[ \frac{(\partial_x \phi)^2}{K}+
K(\partial_x \theta)^2 \right]\!-\!{\mathcal M}
\cos[2\pi \bar n x-\sqrt{4\pi} \phi ],$$ where $\theta$ and $\partial_x\phi$ describe phase and density fluctuations of bosons respectively, $[\theta(x),\partial_y\phi]=i\delta(x-y)$, $v_s$ is the sound velocity and $K$ the Luttinger parameter. In the weak-coupling, $Um\ll 1$, hydrodynamic relations are expected to hold: $v_s(j)\sim \sqrt{\bar n U/m\pi^2}=v_s(0)\sqrt{1-4j}$ and $K(j)\sim \sqrt{\bar n\pi^2/ Um}=K(0) \sqrt{1-4j}$, clearly showing that $j$ enhances correlations. At $j=1/4$, $m$ diverges and the system enters a Mott-insulator (MI) even for vanishingly small $U$ (Fig. \[fig:2\](a)). The SF-MI transition takes place however in the strong-coupling regime in which $v_s$ and $K$ must be determined numerically. We obtain $K$ from the single-particle correlations $G_{ij} =\langle b_i b^{\dagger}_j\rangle$ which in the SF decay as $\sim (-1)^{i-j}{|i-j|^{-1/2K}}$. The value $K=2$ marks the boundary between SF ($U<U_c$, $K>2$) and MI ($U>U_c$, $K<2$, and ${\mathcal M}>0 $). The MI phase is characterized by a hidden parity order [@Berg2008], ${\cal O}^2_P=\lim_{|i-j|\rightarrow\infty}\langle
(-1)^{\sum_{i<l<j}\delta n_l}\rangle\sim\langle \cos \sqrt{\pi}
\phi \rangle^2$, which has been recently measured in site-resolved experiments [@Endres2011].
![Phase diagram for unconstrained bosons as a function of the frustration parameter $j$ and (a) the on-site interaction $U$ (with $U_3=0$) and (b) the three-body repulsion $U_3$ (and $U=0$). In the figures,
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X,k}(k \mu) \, \dim V_{k \mu}
\geq~\sum_{\mathclap{\mu \in K \cap \frac 1 k \Lambda^*_{H,+}}} m_{H,X,k}(k \mu) \, \dim V_{k \mu} \\
\geq~&D \, k^{R_X} \sum_{\mathclap{\mu \in K \cap \frac 1 k \Lambda^*_{H,+}}} m_{H,X,k}(k \mu)
\sim D \, k^{R_X + d_X} \int_K d\operatorname{DH}_X.
\end{aligned}$$ We conclude that also $\dim X \geq R_X + d_X$, hence we have equality.
Let us now elaborate on the argument presented at the end of the introduction, where we showed that Duistermaat–Heckman measures do not directly give rise to new complexity-theoretic obstructions. For this, we consider a pair of projective subvarieties $X$ and $Y$ with $\dim X < \dim Y$, as is the case for the orbit closures of relevance to GCT. Let us assume that $\Delta_X \subseteq \Delta_Y$, so that the moment polytopes alone do not already give rise to an obstruction. Clearly, this implies that $R_X \leq R_Y$.
\[polytope smaller lemma\] Let $\Delta_X \subseteq \Delta_Y$ and $R_X < R_Y$. Then, $\dim \Delta_X < \dim \Delta_Y$.
Note that we have $$\dim \Delta_X = \dim \operatorname{aff}\Delta_X \leq \dim \operatorname{aff}\Delta_Y = \dim \Delta_Y,$$ with equality if and only if the two affine hulls $\operatorname{aff}\Delta_X \subseteq \operatorname{aff}\Delta_Y$ are equal.
Now by assumption there exists a positive root $\alpha \in R_{H,+}$ that is orthogonal to all points in $\Delta_X$ (i.e., for all $p \in \Delta_X$, $\alpha \perp p$), but not to all points in $\Delta_Y$. It follows that $\alpha$ is also orthogonal to all points in the affine hull of $\Delta_X$, but not to all points in the affine hull of $\Delta_Y$. Therefore, we have $\operatorname{aff}\Delta_X \subsetneq \operatorname{aff}\Delta_Y$.
\[different lebesgue lemma\] Let $\dim \Delta_X < \dim \Delta_Y$. Then, $X \subseteq Y$ implies $d_X < d_Y$.
If $X \subseteq Y$ then it is immediate from and that $d_X \leq d_Y$. Let us suppose for a moment that in fact $d_X = d_Y$. Then it follows from that $$\int_{\Delta_X} d\operatorname{DH}_X(\mu) \, g(\mu) \leq
\int_{\Del
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corresponding to the $i$-th element of ${\bf f}_{I'}$.
[^3]: It is the “forehead” part for birds in the CUB200-2011 dataset.
[^4]: Two latent patterns may select the same neural unit
[^5]: We used part boxes annotated during the QA process to learn a fast-RCNN for part detection. Given the inference result $\Lambda_{v}$ of part template $v$ on image $I$, we define a new inference score for localization refinement $S_{v}^{\textrm{new}}(\Lambda_{v}^{\textrm{new}})=S_{v}+\lambda_1\Phi(\Lambda^{\textrm{new}}_{v})+\lambda_2\frac{\Vert{\bf p}_{v}-{\bf p}_{v}^{\textrm{new}}\Vert}{2\sigma^2}$, where $\sigma=70$ pixels, $\lambda_1=5$, and $\lambda_2=10$. $\Phi(\Lambda^{\textrm{new}}_{v})$ denotes the fast-RCNN’s detection score for the patch of $\Lambda^{\textrm{new}}_{v}$.
---
abstract: |
The Shapovalov determinant for a class of pointed Hopf algebras is calculated, including quantized enveloping algebras, Lusztig’s small quantum groups, and quantized Lie superalgebras. Our main tools are root systems, Weyl groupoids, and Lusztig type isomorphisms. We elaborate powerful novel techniques for the algebras at roots of unity, and pass to the general case using a density argument.
Key words: Hopf algebra, Nichols algebra, quantum group, representation
MSC: 16W30; 17B37, 81R50
address:
- 'István Heckenberger, Universität zu Köln, Mathematisches Institut, Weyertal 86–90, D-50931 Köln, Germany'
- 'Hiroyuki Yamane, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka 560-0043, Japan'
author:
- 'I. Heckenberger'
- 'H. Yamane'
title: ' Drinfel’d doubles and Shapovalov determinants'
---
Introduction
============
We study finite-dimensional representations of a large class of Hopf algebras $U(\chi )$, where $\chi $ is a bicharacter on ${\mathbb{Z}}^I$ for some finite index set $I$. These algebras emerged from a program of Andruskiewitsch and Schneider to classify pointed Hopf algebras [@a-AndrSchn98], [@p-Heck07b]. Promine
| 2,634
| 2,062
| 892
| 1,944
| 3,412
| 0.772575
|
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|
critical assumption in their model is that the ion and electron densities can be written as a function of the local electrostatic potential alone. Although this assumption sounds reasonable for instance in the collisionless limit where OML theory should apply, its validity must be tested carefully. On the other hand, our first principles approach free from such an assumption also demonstrates a repulsive nature for the electrostatic interaction. Furthermore, the fact that the electric field around the grain is consistent with the OML theory indicates the assumption made by Markes and Williams (2000) is indeed reasonable.
One might argue that the fact that $\alpha\neq1$ explains the discrepancy between the simulation results and ODS theory, but this is not the case. Assuming that linear superposition of the potential is also possible for $\alpha\neq1$, we can easily calculate the ODS attractive force for this case as well. The resulting attractive potential force may be written as $$\label{ml-j}
q{\phi_{{\rm ODS}}}\left(d\right)=\alpha\frac{q^{2}}{\lambda_{{\rm D}}}\left(\frac{\lambda_{{\rm D}}}{d}-\frac{\alpha}{2}\right)\exp\left(-\frac{d}{\lambda_{{\rm D}}} \right),$$ which is shown in Fig.\[mod\] for $\alpha=1, 1.2, 1.4$. It can be easily understood that the potential minimum moves inward and the depth increases as $\alpha$ increases. In fact, an easy analytical calculation confirms this tendency. Clearly, $\alpha\neq1$ does not help to explain the discrepancy.
![Modified ODS attractive potential given by Eq. (\[ml-j\]). Red, green, and blue lines represent $\alpha = 1, 1.2, 1.4$, respectively.[]{data-label="mod"}](modod.eps){width="90mm"}
Although it is not easy to analytically determine the value of $\alpha$ in general, we can estimate the upper and lower bounds as follows. We define $r_{{\rm c}}$ as a solution to the equation $$\label{exp-equ}
\frac{\alpha}{4\pi\Lambda}\frac{q}{e}\frac{\lambda_{{\rm D}}}{r_{{\rm c}}}\exp{\left(-\frac{r_{{\rm c}}}{\lambda_{{\rm D}}}\right)}=1,$$ where the left
| 2,635
| 2,747
| 3,732
| 2,746
| 3,366
| 0.77285
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|
abelian group $G$ in a way that naturally generalizes the relationship between circulant matrices and cyclic groups. It is shown that, under mild conditions, when the size of the group $G$ goes to infinity, the spectral measures of such random matrices approach a deterministic limit. Depending on some aspects of the structure of the groups, whether the matrices are constrained to be Hermitian, and a few details of the distributions of the matrix entries, the limit measure is either a (complex or real) Gaussian distribution or a mixture of two Gaussian distributions.'
address: 'Department of Mathematics, Case Western Reserve University, 10900 Euclid Ave., Cleveland, Ohio 44106, U.S.A.'
author:
- 'Mark W. Meckes'
bibliography:
- 'G-circ-abelian.bib'
title: 'The spectra of random abelian $G$-circulant matrices'
---
Introduction
============
Given a finite group $G$ and a function $f : G \to {\mathbb{C}}$, the matrix $M
= \bigl[f(ab^{-1})\bigr]_{a,b \in G}$ is called a $G$-circulant matrix by Diaconis [@Diaconis-book; @Diaconis-matrices]. This generalizes the classical notion of circulant matrices, which arise as the special case in which $G$ is a finite cyclic group. The action of such a matrix $M$ on the vector space $\{ g : G \to {\mathbb{C}}\}$ is as a convolution operator: for $g : G \to {\mathbb{C}}$ and $a \in G$, $$\label{E:convolution}
(Mg)(a) = \sum_{b \in G} f(ab^{-1}) g(b) =: (f*g)(a).$$
This paper considers the asymptotic behavior of the spectra of random $G$-circulant matrices, or equivalently random convolution operators on $G$, when $G$ is a large abelian group. (For the rest of this paper, $G$ will always stand for a *finite abelian* group.) Such random matrices will be generated by picking the values $f(a)$ independently, with or without imposing a constraint $f(a^{-1}) =
\overline{f(a)}$ which is equivalent to insisting that the matrix $M$ is Hermitian. This generalizes the study of random circulant matrices, whose theory has already been developed in [@BoMi; @BoSe; @BrSe; @Meckes; @BoHaSa] amo
| 2,636
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ever, it is impossible that a mapping $\xi ^{\prime }$ exists such that $\xi ^{\prime }(x)\in S^{\prime }$, with $S\neq S^{\prime }$ and $\sigma (\xi )=\sigma (\xi ^{\prime })$, since $\sigma (\xi )$ and $\sigma
(\xi ^{\prime })$ are defined on different domains ($\mathcal{E}_{S}\cup
\mathcal{E}_{S}^{\bot }$ and $\mathcal{E}_{S^{\prime }}\cup \mathcal{E}_{S^{\prime }}^{\bot }$, respectively). Hence, an assigment function $\sigma
$ is associated with a unique state $S$, and $\Pi _{\sigma }$ reduces to the singleton $\{\pi _{S}\}$.
---
abstract: |
In this paper, we prove Perelman type $\mathcal{W}$-entropy formulae and global differential Harnack estimates for positive solutions to porous medium equation on closed Riemannian manifolds with Ricci curvature bounded below. As applications, we derive Harnack inequalities and Laplacian estimates.
**Mathematics Subject Classification (2010)**. Primary 58J35, 35K92; Secondary 35B40,35K55
**Keywords**. Porous medium equation, Perelman type entropy formula, differential Harnack estimates, Bakry-Émery Ricci curvature.
address: 'School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, China'
author:
- 'Yu-Zhao Wang'
title: '$\mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds'
---
\[section\] \[theorem\][Corollary]{} \[theorem\][Lemma]{} \[theorem\][Definition]{} \[theorem\][Proposition]{} \[theorem\][Remark]{} \[theorem\][Example]{}
[^1]
Introduction and main results
=============================
Monotonicity formula and differential Harnack inequality are two important tools in geometric analysis. The more spectacular one is the entropy monotonicity formula discovered by G.Perelman [@P] and related differential Harnack inequality for the conjugate heat equation under the Ricci flow. More precisely, let $(M,g(t))$ be the closed $n$-dimensional Riemannian manifolds along the Ricci flow and $(g(t), f(t), \tau(t))$ be a solution to conjugate heat equation coup
| 2,637
| 2,262
| 747
| 2,326
| null | null |
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s for the problem considered remains an open question.
Single Continuous Slowing Down Equation {#single-eq}
=======================================
Preliminaries {#presingle-eq}
-------------
At first we consider a [*single CSDA transport equation*]{} given by \[se1\] -[E]{}+\_x+- K= f GSI, where the solution satisfies inflow boundary and initial value conditions $$\begin{aligned}
{3}
\psi_{|\Gamma_-}&=g\quad && \textrm{on}\ \Gamma_-, \label{se2} \\[2mm]
\psi(\cdot,\cdot,E_{\rm m})&=0\quad && \textrm{on}\ G\times S. \label{se3}\end{aligned}$$ We assume that \[ass1\] L\^(GSI),0 a.e. on $G\times S\times I$. Furthermore, the collision operator is given for $\psi\in L^2(G\times S\times I)$ by \[colla\] (K)(x,,E)=\_[SI]{}(x,’,,E’,E)(x,’,E’)d’ dE’, where $\sigma:G\times S^2\times I^2\to{\mathbb{R}}$ is a non-negative measurable function such that \[ass2\] &\_[SI]{}(x,’,,E’,E)d’ dE’M\_1,\
&\_[SI]{}(x,,’,E,E’)d’ dE’M\_2, for a.e. $(x,\omega,E)\in G\times S\times I$, and we assume that there exists $c\geq 0$ such that \[ass3\] &(x,,E)-\_[SI]{}(x,,’,E,E’)e\^[C(E’-E)]{} d’ dE’ c,\
&(x,,E)-\_[SI]{}(x,’,,E’,E)e\^[C(E-E’)]{} d’ dE’ c, for a.e. $(x,\omega,E)\in G\times S\times I$, and where the constant $c\geq 0$ is specified below (see ). The criterion (\[ass2\]) is called the [*Schur criterion*]{} (emerged from measure theory) and the corresponding collision operators are [*Schur operators*]{}. Note that in some cases we will assume $c$ to be strictly positive, $c>0$. This assumption has been relaxed in [@egger2014] (see also [@dautraylionsv6 Remark 15, pp. 241-242]). We have by Cauchy-Schwarz inequality and , \[k-norma\] [K]{}& [\_[SI]{}(,’,,E’,)d’ dE’]{}\_[L\^]{}\^[1/2]{} [\_[SI]{}(,,’,,E’)d’ dE’]{}\_[L\^]{}\^[1/2]{}\
& [M\_1]{}\^[1/2]{}[M\_2]{}\^[1/2]{}, where $L^\infty = L^\infty(G\times S\times I)$, ${\left\Vert K\right\Vert}$ is the norm of $K$ as an operator in $L^2(G\times S\times I)$, and $$\begin{aligned}
\label{eq:sigma_norms_Linfty_L1}
&{\left\Ver
| 2,638
| 312
| 2,447
| 2,732
| null | null |
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|
ned}
\label{equ:49-Gamma-eff}
\Gamma_\text{eff} &=&\Gamma_0 + \Im[\Sigma_\sigma(-i0^+)] \end{aligned}$$ must hold using the general property $G_{d_{0\sigma}, d^\dagger_{0\sigma} }(z)=[ z-\e_{d \sigma} -\Delta(z) -\Sigma_\sigma(z)]^{-1}$, where we have divided the total self-energy of the molecular orbital $\Sigma_{\rm tot}(z)=\Delta_\sigma(z)+\Sigma_\sigma(z)$ into the hybridization-induced part $\Delta_\sigma(z)$ for the non-interacting problem and all correlation-induced and electron-phonon induced corrections $\Sigma_\sigma(z)$. Since the imaginary part of the self-energy $\Sigma_\sigma(z)$ vanishes for $T,\w\to 0$ in a local Fermi liquid in the standard case of a non-interacting conduction band, the Green’s function is pinned to a fixed value $\rho_{d_{0\sigma},d^\dagger_{0\sigma}}(0)=\frac{1}{\pi} (\pi V^2\rho_0)^{-1}=(\pi \Gamma_0)^{-1}$ independent of the model parameters [@Langreth1966; @YoshimoriZawadowski1982; @Anders1991].
The presence of the unusual Holstein coupling $\lambda_c$ however, leads to modifications of this picture. In appendix \[sec:EOM-GF\], we derive the exact analytic expression of the correlation-induced self-energy $\Sigma_\sigma(z)$ of the molecular orbital using the exact equation for motion (EOM) for the Green’s functions [@BullaHewsonPruschke98]. The result is $$\begin{aligned}
\label{eq:self-energy}
\Sigma_\sigma(z) &=&
\frac{U F_\sigma(z) +\lambda_d M_\sigma(z) +
\frac{ \lambda_c}{V_0}\Delta(z) N_\sigma(z) }{G_{d_{0\sigma},d^\dagger_{0\sigma}}(z)} ,\end{aligned}$$ with the definitions
\[eq-def-corr-eom\] $$\begin{aligned}
\label{equationa}
F_\sigma(z) &= &G_{d_{0\sigma} n^d_{-\sigma},d^\dagger_{0\sigma}}(z),\\
M_\sigma(z) &= & G_{\hat X_0 d_{0\sigma},d^\dagger_{0\sigma}}(z),
\label{equationb}
\\
N_\sigma(z) &= & G_{\hat X_{0} c_{0\sigma} ,d^\dagger_{0\sigma}}(z).
\label{equationc}\end{aligned}$$
We explicitly use Eq. to obtain the Green’s function of the molecular orbital from the NRG solution which provides $F_\sigma(z)$, $M_\sigma(z)$, $N_\sigma(z)$ and $G_{d_{
| 2,639
| 1,175
| 1,800
| 2,496
| null | null |
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|
$S_j$ and $S_k$ are the other geodesics besides $S_i$.
The notion of hyperbolicity makes sense even for metric spaces which are not literally geodesic, but when a reasonable notion of geodesic segment can be defined. This is the case e.g. for finitely generated groups with word metrics, where a geodesic segment between elements $x,y\in G$ is a sequence $g_0=x,\ldots,g_n=y$, where $n=d(x,y)$, and $d(g_i,g_{i+1})=1$, for $i<n$.
Let $G$ be a finitely generated hyperbolic group (with hyperbolicity constant $K$), generated by elements $a_1,\ldots,a_n$. We claim that $G$ with the ordered generating set $a_1\leq a^{-1}_1\leq\ldots\leq a_n\leq a_n^{-1}$ is shortlex combable. Indeed, pick $g,h\in G$ with $d(g,h)=1$, and $i<\max\{|g|,|h|\}$. We show that $d(g_i,h_i)\leq 2K+2$. We have two cases.
1. Either $g_i=g$ or $h_i=h$ (or both). Then it is clear that $d(g_i,h_i)\leq 2$.
2. We have $g_i\neq g$ and $h_i\neq h$. There are geodesic segments $g_0=1_G,\ldots,g_i,\ldots,g$, $h_0=1_G,\ldots,h_i,\ldots,h$, and $g,h$ (of length $1$). We consider thr triple of points $1_G,x,y$ and the geodesic segments between them as above. By definition of hyperbolicity with constant $K$, there exists point $z\in\{1_G=h_0,\ldots, h,g\}$ such that $d(g_i,z)\leq K$. Assume first that $z=h_j$, for some $j\leq |h|$.
- $j\geq i$: Since $d(g_i,h_j)\leq K$, by triangle inequality we get that $j\leq i+K$, therefore $d(g_i,h_i)\leq 2K$.
- $j<i$: Again by triangle inequality we get that $j\geq i-K$, so $d(g_i,h_i)\leq 2K$.
Finally, if $z=g$, then $d(g_i,h)\leq K+1$, so again by triangle inequality we get $|h|\leq i+K+1$, so $d(g_i,h_i)\leq 2(K+1)$.\
[*Large-type Artin groups.*]{} Holt and Rees in [@HoRe] prove that large-type Artin groups are shortlex automatic which immediately from the definition implies being shortlex combable. Artin groups in general belong to one of the most studied classes of groups in geometric group theory. We refer the reader to [@HoRe] for the notion of shortlex automaticity and for the definition o
| 2,640
| 4,279
| 2,699
| 2,214
| 4,104
| 0.768083
|
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|
ory as a generalized YM theory.
The translation symmetry implies that the commutators are of the form = \_[\_3]{} f(\_1, p\_1, \_2, p\_2, \_3) T(\_3, p\_1 + p\_2), \[ansatz\] and the Lorentz symmetry implies that the structure constants are Lorentz-invariant functions of the vectors $\teps_1, \teps_2, \teps_3, p_1, p_2$. (The summation over $\tilde{\eps}_3$ is a summation over an orthonormal basis of vectors.) The most general ansatz consistent with all assumptions is thus f(\_1, p\_1, \_2, p\_2, \_3) = (\_1\_3) - (\_2\_3) + (\_1\_2) , where $\a, \b, \g$ are constant parameters.
It follows from the ansatz that $$\begin{aligned}
&[[T(1), T(2)], T(3)] + [[T(2), T(3)], T(1)] + [[T(3), T(1)], T(2)]
= \sum_{\teps_4} \Big\{
\nn \\
&(\teps_1\cdot\teps_4)\left[
- \a\b (\teps_2\cdot\teps_3)(p_1\cdot(p_2-p_3))
+ \g [(\teps_2\cdot p_2)(\teps_3\cdot (\a p_1 + \g p_2)
- (\teps_3\cdot p_3)(\teps_2\cdot (\a p_1 + \g p_3)]
\right]
\nn \\
&+ \b (\teps_1\cdot\teps_2)\left[
(\teps_3\cdot(p_1 - p_2))
(\teps_4\cdot[(\a-\b)(p_1 + p_2) - \b(p_1 - p_2)])
- \g (\teps_3\cdot p_3)(\teps_4\cdot(p_1 - p_2))
\right]
\nn \\
&+ \mbox{cyclic permutations of $(1, 2, 3)$}
\Big\},\end{aligned}$$ where we have used $T(1)$ to represent $T(\teps_1, p_1)$, $T(2)$ to represent $T(\teps_2, p_2)$, etc. In order to satisfy the Jacobi identity, we have to set $\b = \g = 0$. The most general solution is thus equivalent to = \_[\_3]{} i T(3), \[commutator\] by scaling $\a$ to $-i$. More explicitly, it is = \_ iT\_[()]{}(p\_1+p\_2). \[commutator-2\]
Incidentally, it is consistent to allow $\a$ to depend on the momenta. For example, Jacobi identity is satisfied for (p\_1, p\_2) = c e\^[p\_1p\_2]{} for arbitrary constant parameters $c$ and $\lam$. It is equivalent to the scaling of the generators by $T(\eps, p) \rightarrow T'(\eps, p) \equiv c\, e^{\lam p^2/2} T(\eps, p)$.
Representations
---------------
A representation of the algebra constructed above is given by T(, p) (p’) = i (p’) (p’+p), on a linear space with the basis $\{\tilde{\phi}(p)\}$. Th
| 2,641
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|
.$$ Let us prove the reverse inclusion.\
Let $\alpha \in S' \cap ({\cal R}\times{\cal R})^{\leq n}$. Let $\beta$ be the longuest word in ${{\rm PREF}}(\alpha) \cap S$.\
If $\beta = \alpha$, then $\alpha \in S$, as required.\
Otherwise $\alpha \in S'-S$. By condition E2 of definition \[def-extension\], there exists some $\beta \in S$, which is maximal in $S$ for the prefix ordering and such that $$\beta \prec \alpha.$$ Maximality of $\beta$ implies, by condition (2) of the lemma, that, $$|\beta| = n \mbox{ or } {{\rm NEXT}}((T,T').\beta) \notin \sim_1.$$ Since $\beta \prec \alpha$ we are sure that $|\beta| < n$ so that $${{\rm NEXT}}((T,T').\beta) \notin \sim_1.$$ This last statement contradicts the fact that $\beta \backslash S'$ is a D-strategy, w.r.t ${{\rm NEXT}}((T,T').\beta)$ which is non-reduced to $\{(\varepsilon,\varepsilon)\}$ (since it posesses $\beta^{-1}\alpha$).\
We can conclude that $\alpha \in S$. Finally: $$S = S' \cap ({\cal R}\times{\cal R})^{\leq n}.$$
Let $T,T'\in \TERMS$ and let $S \subseteq ({\cal R}\times{\cal R})^*$ be finite. One can check whether $S$ is a finite prefix of a D-strategy w.r.t. $(T,T')$ \[L-decidability\_PDstrategies\]
This follows immediately from the characterisation given by Lemma \[L-characterisation\_PDstrategies\].
Formal systems {#subsec_formal_systems}
--------------
For every $T_0,T'_0 \in \TERMS$, $S_0$ finite prefix of strategy w.r.t $(T_0,T_0)$ and finite ${\cal B} \subseteq \TERMS \times \TERMS,$ is defined a formal system $${\cal J}(T_0,T'_0,S_0,{\cal B})$$ The set of judgments of all the systems are the same. But the axiom and one rule (namely R7), is depending on the parameters $(T_0,T'_0,S_0,{\cal B})$.
Judgments
---------
A [*judgment*]{} has one of the three forms:\
[**FORM 1**]{}:\
$$m {\:|\!\!\!=\!\!\!\!=\:}(T,T',S)$$ where $m \in \N$, and $T,T'\in \TERMS$ are regular terms and $S$ is a finite prefix of a strategy. w.r.t. $(T,T')$ (D-strategies are defined p.20, lines 27-30; finite prefixes are mentionned, though in a fuzzy way. at p. 23, line 11;
| 2,642
| 1,606
| 389
| 2,922
| 1,990
| 0.78378
|
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|
\hat{\mathcal{H}}_{+}^{[2]}$$ with $$\label{hblocks+}
\begin{aligned}
& \hat{\mathcal{H}}^{[1]}_{+} =
\bigoplus_{n = 0}^{m - 1} \langle n , g |\hat{\mathcal{H}}^{(m)}_{+} | n , g \rangle
= \hbar
\bigoplus_{n = 0}^{m - 1} \left( \nu n - \tfrac{\omega_0}{2} \right), \\
& \hat{\mathcal{H}}^{[2]}_{+} \!=\!
\bigoplus_{n = 0}^{\infty} \!\!\begin{pmatrix}
\!\! \langle n , e |\hat{\mathcal{H}}^{(m)}_{+} | n , e \rangle \!\! & \!\!
\!\! \langle n\! + \! m, g |\hat{\mathcal{H}}^{(m)}_{+} | n , e \rangle \!\!\\
\!\! \langle n , e |\hat{\mathcal{H}}^{(m)}_{+} | n\! + \! m , g \rangle \!\! &
\!\! \langle n\! + \! m, g |\hat{\mathcal{H}}^{(m)}_{+} | n\! + \! m , g \rangle \!\!
\end{pmatrix} \\
& \,\,\,\,\, = \hbar
\bigoplus_{n = 0}^{\infty}
\begin{pmatrix}
\nu n + \frac{\omega_{0}}{2} &
\text{e}^{-i\omega_{L}t}\Omega f_n^m \\
\text{e}^{i\omega_{L}t} \Omega f_n^{m\ast} &
\nu (n+m)-\frac{\omega_{0}}{2}
\end{pmatrix},
\end{aligned}$$ with $f_n^m$ defined in Eq. (\[auxf1\]).
The above block structure enables us to diagonalize the Hamiltonian by the diagonalization of each block. The first $m$ blocks in Eq. (\[hblocks+\]) are matrices of only one element having eigenvalues and eigenvectors, respectively, given by $$\label{eigval1+}
\zeta_{+}^{(n,m)} = \hbar\nu n-\frac{\hbar\omega_{0}}{2}, \,\,
\left|\zeta_{+}^{(n,m)}\right\rangle = \left|n,g\right\rangle$$ for each $n = 0,...,m-1$ for a given $m$. The following blocks in the diagonal block structure of (\[hblocks+0\]) are $2 \times 2$ matrices, which can be diagonalized to give for all $n, m$ the eigenvalues $$\l
| 2,643
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|
tSQLvar = 0;
string strSeperator = string.Empty;
foreach (KeyValuePair<int, string> benefit in benefits)
{
sb.AppendFormat(" {0} BenefitID=@benerfit{1}", strSeperator, intSQLvar);
intSQLvar++;
strSeperator = "OR";
}
SqlConnection con = new SqlConnection(m_strDBConnectionString);
SqlDataAdapter sqlDataAdapter = new SqlDataAdapter();
SqlCommand cmd = new SqlCommand();
cmd.Connection = con;
cmd.CommandText = sb.ToString();
intSQLvar = 0;
foreach (KeyValuePair<int, string> benefit in benefits)
{
cmd.Parameters.Add(string.Format("@benerfit{0}", intSQLvar), SqlDbType.Int, 32).Value = benefit.Key.ToString();
intSQLvar++;
}
DataSet ds = new DataSet();
try
{
con.Open();
sqlDataAdapter.SelectCommand = cmd;
sqlDataAdapter.Fill(ds);
}
finally
{
if (con != null)
{
con.Close();
con = null;
}
}
List<ProvisionDetails> lstProvisions = new List<ProvisionDetails>();
if (ds != null && ds.Tables[0].Rows.Count > 0)
{
lstProvisions = (from r in ds.Tables[0].AsEnumerable()
select new ProvisionDetails()
{
ID = r.Field<int>("ProvisionID"),
Name = r.Field<string>("ProvisionName"),
BenefitID = r.Field<int>("ProvisionID"),
OptionValue = r.Field<int>("ProvisionID")
}).ToList();
if (benefits.Count == 1)
{
return (from p in lstProvisions
select new FieldConfiguration()
{
Name = p.Name,
ProvisionFieldID = p.ID.ToString(),
Fi
| 2,644
| 3,652
| 42
| 2,130
| 45
| 0.832029
|
github_plus_top10pct_by_avg
|
0\sigma},d^\dagger_{0\sigma}}(z)$ [@BullaHewsonPruschke98].
As shown by Hewson and Meyer [@HewsonMeyer02], the self-energy $\Sigma_\sigma(z)= [UF_\sigma(z) +\lambda_d M_\sigma(z)] /G_{d_{0\sigma},d^\dagger_{0\sigma}}(z)$ maintains Fermi liquid properties and its imaginary part vanishes for $T,\w\to 0$ for a coupling of the orbital to a free electron gas. This can be understood from the topology of a Feynman diagram expansion of these correlation functions independent of the analytic shape of $G_{d_{0\sigma},d^\dagger_{0\sigma}}(z)$. In the presence of a finite $\lambda_c$, this statement does not hold any longer: the imaginary part of the self-energy acquires a negative offset which we will quantify in the following.
Applying the EOM to $F_\sigma(z)$ reveals that this composite Green’s function also contains additional self-energy corrections in the presence of a finite $\lambda_c$. Therefore, the self-energy contribution $\Sigma^U(z)= UF_\sigma(z) /G_{d_\sigma,d^\dagger_\sigma}(z)$ cannot be identified by the same skeleton expansion [@LuttingerWard1960] as for the $\lambda_c=0$ case.
Another modification stems from the third term in the nominator of Eq. , $$\begin{aligned}
\Delta \Sigma(z) &=& \frac{ \lambda_c}{V_0 |G_{d_\sigma,d^\dagger_\sigma}(z)|^2}\Delta(z) N_\sigma(z) G_{d_\sigma,d^\dagger_\sigma}^*(z) .\end{aligned}$$ Assuming particle-hole symmetry and $T\to 0$, and using that the real part of $G_{d_\sigma,d^\dagger_\sigma}(z)$ as well as the real part of $N_\sigma(-i0^+)$ vanish yields $$\begin{aligned}
\label{eq:27}
\Delta \Sigma(-i0^+)
&=& i \frac{ \lambda_c \Gamma_0}{V_0 \pi \rho_{d_{0\sigma},d^\dagger_{0\sigma}}(0)}
\Im N_\sigma(-i0^+) .\end{aligned}$$ $N_\sigma(z)$ is an off-diagonal Green’s function and its spectral integral is zero. Therefore, spectrum has equal positive and negative spectral weight in different frequency regions. The NRG calculation shows that $\Im N_\sigma(-i0^+)<0$ for $\lambda_c>0$ and $\Im N_\sigma(-i0^+)\propto \lambda_c$ in leading order.
![Symmetric singl
| 2,645
| 1,024
| 2,868
| 2,441
| 3,859
| 0.769669
|
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|
e convenient than from the computational point of view and will be used in the present study. We note, however, that since the RHS of this inequality cannot be expressed entirely in terms of properties of the initial data, this is [*not*]{} in fact an a priori estimate. Estimate also allows us to obtain a condition on the size of the initial data, given in terms of its energy $\K(0)$ and enstrophy $\E(0)$, which guarantees that smooth solutions will exist globally in time, namely, $$\label{eq:Cond_for_globalReg}
\mathop{\max}_{t \geq 0} \E(t) \leq \frac{\E(0)}{1 - \frac{27}{(2\pi\nu)^4}\K(0)\E(0)}$$ from which it follows that $$\label{eq:K0E0}
\K(0)\E(0) < \frac{(2\pi\nu)^4}{27}.$$ Thus, flows with energy and enstrophy satisfying inequality are guaranteed to be smooth for all time, in agreement with [the]{} regularity results [available under the assumption of]{} small initial data [[@lady69]]{}.
Instantaneously Optimal Growth of Enstrophy {#sec:3D_InstOpt}
===========================================
Sharpness of instantaneous estimate , in the sense of definition \[def:NotionSharpness\], can be probed by constructing a family of “extreme vortex states” ${\widetilde{\mathbf{u}}_{\E_0}}$ which, for each $\E_0 > 0$, have prescribed enstrophy $\E({\widetilde{\mathbf{u}}_{\E_0}}) =
\E_0$ and produce the largest possible rate of growth of enstrophy $\R({\widetilde{\mathbf{u}}_{\E_0}})$. Given the form of , the fields ${\widetilde{\mathbf{u}}_{\E_0}}$ can be expected to exhibit (at least piecewise) smooth dependence on $\E_0$ and we will refer to the mapping $\E_0
\longmapsto {\widetilde{\mathbf{u}}_{\E_0}}$ as a “[maximizing]{} branch”. Thus, information about the sharpness of estimate can be deduced by analyzing the relation $\E_0$ versus $\R({\widetilde{\mathbf{u}}_{\E_0}})$ obtained for a possibly broad range of enstrophy values. A [maximizing]{} branch is constructed by finding, for different values of $\E_0$, the extreme vortex states ${\widetilde{\mathbf{u}}_{\E_0}}$ as solutions of a variational o
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|
ines (including those without fine-tuning). *Fast-RCNN (1 ft)* and *CNN-PDD* used the cropped objects as the input of the CNN; *SS-DPM-Part*, *PL-DPM-Part*, *Part-Graph*, and *Interactive-DPM* used object boxes and part boxes to learn models. *CNN-PDD-ft*, *Fast-RCNN (2 fts)*, and methods based on *fc7* features used object bounding boxes for fine-tuning.
Evaluation metric
-----------------
As discussed in [@SemanticPart; @CNNAoG], a fair evaluation of part localization requires removing factors of object detection. Thus, we used ground-truth object bounding boxes to crop objects as testing images. Given an object image, some competing methods (*e.g.* *Fast-RCNN (1 ft)*, *Part-Graph*, and *SS-DPM-Part*) estimate several bounding boxes for the part with different confidences. We followed [@CNNSemanticPart; @SemanticPart; @ObjectDiscoveryCNN_1; @CNNAoG] to take the most confident bounding box per image as the part-localization result. Given part-localization results of a category, we applied the *normalized distance* [@CNNSemanticPart] and the *percentage of correctly localized parts* (PCP) [@fineGrained1; @fineGrained2; @fineGrained3] to evaluate the localization accuracy. We measured the distance between the predicted part center and the ground-truth part center, and then normalized the distance using the diagonal length of the object as the normalized distance. For the PCP, we used the typical metric of “$IoU\geq0.5$” [@FastRCNN] to identify correct part localizations.
See Table \[tab:imgnet\] for the introduction of the 2nd and 3rd columns. The 4th column shows the number of questions for training. The 4th column indicates whether the baseline used all object annotations (*more than part annotations*) in the category to pre-fine-tune a CNN before learning the part.
The 3rd and 4th columns show the number of part annotations and the average number of questions for training.
Experimental results
--------------------
We learned AOGs for the head, the neck, and the nose/muzzle/beak parts of the six an
| 2,647
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|
which gives a contradiction.
- Since $k=p>1$, then $A\cap M_{\Delta}=1={{\operatorname}{C}_{{{\operatorname}{O}_{p'}(A)}}(M_{\Delta})}$.
Let $x\in A\cap M_{\Delta}$ of prime power order, which is a $p$-regular element. Then by hypotheses there exists $n\in N$ with $x\in{{\operatorname}{C}_{G}(\langle y\rangle^n)}$. Hence $x^{y^n}=x\in M_{\Delta}\cap M_{\Delta}^{y^n}=M_{\Delta}\cap M_{\Delta}^{y}=1$. We deduce that $A\cap M_{\Delta}=1$.
Let $x\in {{\operatorname}{C}_{{{\operatorname}{O}_{p'}(A)}}(M_{\Delta})}$ of prime power order. Then there exists $n\in N$ such that $[\langle y \rangle^n, x]=1$. Therefore $[x, M_{\Delta}]=1=[x^{(y^j)^n}, M_{\Delta}^{(y^j)^n}]=[x, M_{\Delta}^{y^j}]$, for every $j\in\{1, \ldots, p-1\}$. So $x\in{{\operatorname}{C}_{G}(N)}=1$. Hence ${{\operatorname}{C}_{{{\operatorname}{O}_{p'}(A)}}(M_{\Delta})}=1$.
- Without loss of generality, let $\nabla =\{N_1, \ldots, N_p\}$. Set $M_{\nabla}:=N_1\times \cdots\times N_p$. Then $M_{\nabla}$ is a minimal normal subgroup of $N\langle y \rangle$, and if $1\neq R\leq N$ with $R\unlhd N\langle y\rangle$, then there exist $\{d_1,\ldots, d_t\}\subseteq\{a_1,\ldots, a_m\} \subseteq {{\operatorname}{O}_{p'}(A)}$ such that $R=M_{\nabla}^{d_1}\times \cdots \times M_{\nabla}^{d_t}$.
Moreover, if we set $F_1:=N_2\times \cdots \times N_p$, $F_i:=F_1^{a_i}$ for each $2\leq i\leq m$, and $F_{\nabla}:=F_1\times \cdots \times F_m$, then $F_{\nabla}\cap {{\operatorname}{O}_{p'}(A)}=1=F_{\nabla}\cap B$.
The first assertion is clear. If $x\in F_{\nabla}\cap {{\operatorname}{O}_{p'}(A)}$ is of prime power order, then there exists $n\in N$ such that $\langle y \rangle^n$ centralises $x$, so for every $1\leq j \leq p$ we get $x=x^{(y^j)^n}\in F_{\nabla}\cap F_{\nabla}^{y^j}\leq E_{\nabla}:=\cap_{g\in\langle y \rangle} F_{\nabla}^g$. It follows that $F_{\nabla}\cap {{\operatorname}{O}_{p'}(A)}\leq E_{\nabla}$. Note that $E_{\nabla}\leq N$ and it is normal in $N\langle y \rangle$, hence we deduce from the above that $E_{\nabla}=1$, and s
| 2,648
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|
\tau \; \langle \nabla\cdot F_1(t) \;
\exp(-\tau \nabla\cdot F_0) \; \nabla\cdot F_1(t-\tau) \rangle
\nonumber \\
\exp(\tau\nabla\cdot F_0) \} \; p(u,t) \; \; .\end{aligned}$$
The operator $\exp(-\tau\nabla\cdot F_0)$ in the above equation provides the solution of the equation $$\frac{\partial f(u,t)}{\partial t} = - \nabla \cdot F_0 \; f(u,t) \; \; ,$$
($f$ signifies the unperturbed part of $\rho$) which can be found explicitly in terms of characteristics curves. The equation $$\dot{u} = F_0(u)$$
for fixed $t$ determines a mapping from $u(\tau=0)$ to $u(\tau)$, i.e., $u\rightarrow u^\tau$ with inverse $(u^\tau)^{-\tau}=u$. The solution of Eq.(24) is $$f(u,t) = f(u^{-t},0) \left | \frac{d (u^{-t})}{d(u)} \right | =
\exp(-t \nabla\cdot F_0) f(u,0) \; \; ,$$
$\left | \frac{d (u^{-t})}{d(u)} \right |$ being a Jacobian determinant. The effect of $\exp(-t\nabla\cdot F_0)$ on $f(u)$ is as follows ; $$\exp(-t\nabla\cdot F_0) \; f(u,0) = f(u^{-t},0)
\left | \frac{d (u^{-t})}{d(u)} \right | \; \; .$$ The above simplification when put in Eq.(23) yields $$\begin{aligned}
\frac{\partial}{\partial t}p(u,t)=\nabla\cdot \left\{-F_{0}+\epsilon^{2}
\int_{0}^{\infty}\left|\frac{d(u^{-\tau})}{d(u)}\right|
\langle F_{1}(u,t)\nabla_{-\tau}\cdot F_{1}(u^{-\tau},t-\tau)
\rangle\right.\nonumber\\
\left.\left|\frac{d(u)}{d(u^{-\tau})}\right| d\tau\right\} p(u,t)\hspace{0.2cm}.\end{aligned}$$ $\nabla_{-\tau}$ denotes differentiation with respect to $u_{-\tau}$. We put $\epsilon = 1$ for the rest of the treatment. We now identify, $$\left.\begin{array}{l}
u_{1}=x\\
u_{2}=v\\
F_{01}=v\hspace{0.2cm},\hspace{0.2cm}F_{11}=0\\
F_{02}=-\Gamma(x)v-\tilde{V}'(x)\hspace{0.2cm},\hspace{0.2cm}
F_{12}=\xi_{\rm eq}(t)+g'(x)\xi_{\rm neq}(t)
\end{array} \right \} \; \; .$$ In this notation Eq.(28) now reduces to $$\begin{aligned}
\frac{\partial p}{\partial t}=-\frac{\partial}{\partial x}(vp)+\frac{\partial}
{\partial v}\left\{\Gamma v+\tilde{V}'(x)\right\}p\hspace{7.5cm}\nonumber\\
\nonumber\\
+\frac{\partial}{\partial v}\int_{0}^{\infty}d\tau\lan
| 2,649
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|
ap.
I was wondering, why can it be caused? I posted another post a while ago because I wanted to include this problem in a JSFiddle but I couldn't work out how to include the fonts files in the JSFiddle so the icons don't appear there.
Does anyone has a slight idea about what can I do to fix it?
.containerOfSites {
display: block;
height: auto;
float: none !important;
margin: 0 auto;
}
.wrapPortalItem {
padding: 0;
}
.insideWrapItem {
text-align: center;
border: 3px solid black;
padding: 15px;
position: relative;
}
.portalItem {
background-color: rgba(255, 255, 255, 0.75);
padding-top: 3%;
padding-bottom: 10%;
font-family: 'Open Sans', sans-serif;
position: relative;
}
.portalItem p {
margin-bottom: 40%;
font-size: 30px;
padding: 5px;
}
.portalItem p>span {
font-weight: 900;
}
.portalItem span.viewMorePs {
text-decoration: none;
font-size: 18px !important;
z-index: 999;
}
.portalItem h1 {
color: #B5D803;
font-weight: 900;
text-transform: uppercase;
text-shadow: 2px 2px 0 #fff;
}
.insideWrapItem span[class^="iconI-"] {
position: absolute;
color: white;
bottom: 12%;
left: 26%; /* <- */
font-size: 14em !important;
}
<div id="portalPage" class="col-md-24">
<div class="containerOfSites col-md-18 col-sm-24">
<div class="wrapPortalItem col-md-8">
<div class="insideWrapItem">
<span class="iconI-iconDA_automotive img-responsive"></span>
<div class="portalItem portA ">
<h1>AUTOMOTIVE</h1>
<p>web sites<br /> for the<br /> <span> automotive<br /> </span> market</p>
<a href="http://motors06.denison-automotive.co.uk/denison/"><span class="viewMorePsGreen">GO</span></a>
</div>
</div>
</div>
<div class="wrapPortalItem col-md-8">
<div class="insideWrapItem">
<span class="iconI-iconDA_web"></span>
<div class="portalItem">
<h1>DESIGN</h1>
<p>web sites<br /> for the
| 2,650
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aystyle\frac{\epsilon}{m}, \ \cdots,\ \lambda
y_{m}-\frac{\epsilon}{m},\ \lambda y_{m+1}+\epsilon+\delta,\\ \\
&&\lambda y_{m+2}, \ \cdots,\lambda y_{d-1},\ \lambda
y_{d}-\epsilon-\delta,\ \displaystyle \lambda
y_{d+1}+\frac{\epsilon}{\triangle}, \\ \\
&& \cdots,\ \lambda
y_{n-t}+\displaystyle\frac{\epsilon}{\triangle}).
\end{array}$$ Now define $x= (x', \lambda y_{n-t+1},\cdots, \lambda y_{n})$, where $x'=(x'_1,\bar{x}'^\da_2,\dots,\bar{x}'^\da_{n-t})$ and $x'_1=y_1+(1-\lambda)\sum_{i=2}^{n} y_i-\epsilon/m$. By Lemma 4 and 5 we can similarly prove that $x \in T^{\lambda}(y)$ but $x\not\in
M^{\lambda}(y)$. $\Box$
To draw a clearer picture of the relation between catalyst-assisted transformation and multiple-copy transformation in purely probabilistic setting, we investigate the limit properties of $T^{\lambda}(y)$ and $M^{\lambda}(y)$ about $\lambda$. Since $T^{\lambda'}(y)\subseteq T^{\lambda}(y)$ for any $\lambda'>\lambda$, we can define $$T^{\lambda-}(y)=\bigcap_{\lambda'<\lambda}
T^{\lambda'}(y),\ \ \ \ \ \
T^{\lambda+}(y)=\bigcup_{\lambda'>\lambda}
T^{\lambda'}(y)$$ which denote respectively the left limit and right limit of the set-valued function $T^{\lambda}(y)$ at the point $\lambda$. Similar notions can be defined for $M^{\lambda+}(y)$ and $M^{\lambda-}(y)$. It is direct from the definition that $$\begin{aligned}
T^{\lambda-}(y)&=&\{\ x\ |\ \sup_c P(x\otimes c\ra y\otimes
c)\geq \lambda \},\\
M^{\lambda-}(y)&=&\{\ x\ |\ \sup_k P(x^{\otimes{k}}\ra
y^{{\otimes{k}}})^{1/k}\geq \lambda \},\end{aligned}$$ and we have shown in [@DF05a] that $T^{\lambda-}(y)=M^{\lambda-}(y)$ for any $\lambda\in [0,1]$.
The following theorem tells us that generally, the function $T^{\lambda}(y)$ is neither left continuous nor right continuous at any point $\lambda\in (0,1)$, although it is ‘almost’ right continuous in the sense that the right limit at $\lambda$ shares the same interior points with $T^{\lambda}(y)$.
\[lem:tplus\] For any $y\in V^n$ and $0<\lambda<1$,
1\) $T^{\lambda+}(y)$ is open while $T^{\lambda-}(y)$
| 2,651
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|
ion 4, we compute the entropy bounds for this algorithm. A natural generalization of adaptive codes is presented in Section 5. Finally, the last section contains a few concluding remarks. Before ending this introductory section, let us present some useful notation used throughout the paper [@rs1; @as1], and then review some basic concepts. We denote by $|S|$ the *cardinality* of a set $S$; if $x$ is a string of finite length, then $|x|$ denotes the length of $x$. The *empty string* is denoted by $\lambda$. For an alphabet $\Sigma$, we denote by $\Sigma^{*}$ the set $\bigcup_{n=0}^{\infty}\Sigma^{n}$, and by $\Sigma^{+}$ the set $\bigcup_{n=1}^{\infty}\Sigma^{n}$, where $\Sigma^{0}$ is defined by $\{\lambda\}$. Let us denote by $\Sigma^{\leq n}$ the set $\bigcup_{i=0}^{n}\Sigma^{i}$ and by $\Sigma^{\geq n}$ the set $\bigcup_{i=n}^{\infty}\Sigma^{i}$. Let $X$ be a finite and nonempty subset of $\Sigma^{+}$, and $w\in\Sigma^{+}$. A *decomposition of w* over $X$ is any sequence of words $u_{1}, u_{2}, \ldots, u_{h}$ with $u_{i}\in X$, $1\leq i\leq h$, such that $w=u_{1}u_{2}\ldots u_{h}$. A *code* over $\Sigma$ is any nonempty set $C\subseteq\Sigma^{+}$ such that each word $w\in\Sigma^{+}$ has at most one decomposition over $C$. A *prefix code* over $\Sigma$ is any code $C$ over $\Sigma$ such that no word in $C$ is proper prefix of another word in $C$.
Adaptive Codes
==============
In this section we introduce a new class of non-standard variable-length codes, called adaptive codes. These codes are based on adaptive mechanisms, that is, the variable-length codeword associated to the symbol being encoded depends on the previous symbols in the input data string.
Let $\Sigma$ and $\Delta$ be two alphabets. A function ${c:\Sigma\times\Sigma^{\leq{n}}\rightarrow\Delta^{+}}$, with $n\geq{1}$, is called an if its unique homomorphic extension ${\overline{c}:\Sigma^{*}\rightarrow\Delta^{*}}$ given by:
- $\overline{c}(\lambda)=\lambda$,
- $\overline{c}({\sigma_{1}\sigma_{2}\ldots\sigma_{m}})=$ $c(\sigma_{1},\lambda)$
| 2,652
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|
(UX)^*_{\alpha n} (UX)_{\beta n}
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K l}
\nonumber \\
&+&
\sum_{n}
\sum_{k \neq l }
\sum_{K}
\biggl[
-
\frac{ (ix) }{ ( \Delta_{K} - h_{l} )^2 ( h_{l} - h_{k} ) }
e^{ - i ( h_{l} - h_{n} ) x}
-
\frac{ (ix) }{ ( \Delta_{K} - h_{l} )^2 ( \Delta_{K} - h_{k} ) }
e^{ - i ( \Delta_{K} - h_{n} ) x}
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - h_{l} )^3 }
\left( h_{l} + 2 h_{k} - 3 \Delta_{K} \right)
e^{ - i ( \Delta_{K} - h_{n} ) x}
\nonumber \\
&-&
\frac{ 1 }{ ( \Delta_{K} - h_{l} )^3 ( h_{l} - h_{k} )^2 }
\left( \Delta_{K} + 2 h_{k} - 3 h_{l} \right)
e^{ - i ( h_{l} - h_{n} ) x}
+ \frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 ( h_{l} - h_{k} )^2 }
e^{ - i ( h_{k} - h_{n} ) x}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l}
(UX)^*_{\alpha n} (UX)_{\beta n}
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K l}
\left\{ (UX)^{\dagger} A W \right\}_{l K}
\left\{ W ^{\dagger} A (UX) \right\}_{K l}
\nonumber \\
&+&
\sum_{n}
\sum_{k \neq l }
\sum_{K} \sum_{m \neq k, l}
\biggl[
\frac{ (ix) }{ ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} ) ( \Delta_{K} - h_{m} ) }
e^{ - i ( \Delta_{K} - h_{n} ) x}
\nonumber \\
&-&
\frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - h_{l} )^2 ( \Delta_{K} - h_{m} )^2 }
\nonumber \\
&\times&
\left\{
3 \Delta_{K}^2 - 2 \Delta_{K} \left( h_{k} + h_{l} + h_{m} \right) + \left( h_{k} h_{l}+ h_{l} h_{m} + h_{m} h_{k} \right)
\right\}
e^{ - i ( \Delta_{K} - h_{n} ) x}
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{m} )^2 ( h_{m} - h_{k} ) ( h_{m} - h_{l} ) }
e^{ - i ( h_{m} - h_{n} ) x}
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{l} )^2 ( h_{l} - h_{m} ) ( h_{l} - h_{k} ) }
e^{ - i ( h_{l} - h_{n} ) x}
-
\frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 ( h_{k} - h_{m} ) ( h_{l} - h_{k} ) }
e
| 2,653
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| 2,932
| 2,671
| null | null |
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|
lem:3.1}$, we have the following: $$\begin{aligned}
\operatorname{{E}}[\Pe(Z)]
&= \frac{(k_{1} + k_{2}) b}{2 \G(a)}
\G\left( 2a \right), \label{E[L(Z)]} \\
\operatorname{{V}}[\Pe(Z)]
&= \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} \G(3a)}{2 \G(a)}
- \frac{(k_{1} + k_{2})^{2} b^{2} \G(2a)^{2}}{4 \G(a)^{2}}. \label{V[L(Z)]}\end{aligned}$$
Let $\operatorname{{erf}}(x)$ be the error function defined by $$\begin{aligned}
\operatorname{{erf}}(x) := \frac{2}{\sqrt{\pi}} \int_{0}^{x} \exp{\left( - t^{2} \right)} dt \end{aligned}$$ for any $x \in \mathbb{R}$. We give two examples of $\operatorname{{E}}[\Pe(Z + c)]$ and $\operatorname{{V}}[\Pe(Z + c)]$.
\[rei:3.2\] In the case of ${\rm Laplace}(0, b)$, since $a = 1$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]
&= \Pe(c) + \frac{(k_{1} + k_{2}) b}{2} \exp{\left(- \left\lvert \frac{c}{b} \right\rvert \right)}, \\
\operatorname{{V}}[\Pe(Z + c)]
&= \left\{ k_{1}^{2} + k_{2}^{2} + \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2} ) \right\} b^{2} \\
&\quad - \operatorname{{sgn}}(c) (k_{1} + k_{2}) \left\{ \Pe(c) + b (k_{1} - k_{2}) \right\} b \exp{\left(- \left\lvert \frac{c}{b} \right\rvert \right)} \\
&\quad - \frac{(k_{1} + k_{2})^{2} b^{2}}{4} \exp{\left(- 2 \left\lvert \frac{c}{b} \right\rvert \right)}. \end{aligned}$$ In the case of $\mathcal{N}(0, \frac{1}{2} b^{2})$, since $a = \frac{1}{2}$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]
&= \frac{(k_{1} - k_{2}) c}{2}
+ \frac{(k_{1} + k_{2}) c}{2} \operatorname{{erf}}{\left(\frac{c}{b} \right)}
+ \frac{(k_{1} + k_{2}) b}{2 \sqrt{\pi}} \exp{\left(- \frac{c^{2}}{b^{2}} \right)}, \\
\operatorname{{V}}[\Pe(Z + c)]
&= \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} }{4}
+ \frac{(k_{1} + k_{2})^{2} c^{2} }{4}
+ \frac{(k_{1}^{2} - k_{2}^{2}) b^{2} }{4} \operatorname{{erf}}{\left( \frac{c}{b} \right)}
- \frac{(k_{1} + k_{2})^{2} c^{2} }{4} \operatorname{{erf}}^{2}{\left( \frac{c}{b} \right)} \\
&\quad - \frac{(k_{1} + k_{2})^{2} b c}{2 \sqrt{\pi}} \operatorname{{erf}}{\left(\frac{c}{b} \right)} \exp{\left(
| 2,654
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| 2,179
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|
e other hand, $$\begin{aligned}
\label{wkpment2}
(\gamma-1)\left|\nabla_i\nabla_jv+\frac{\eta_K(t)}{n(\gamma-1)}g_{ij}\right|^2
=&(\gamma-1)|\nabla\nabla v|^2+\frac{2\eta_K}{n}\Delta v+\frac{\eta_K^2}{n(\gamma-1)}.\end{aligned}$$ Putting into , we get
$$\begin{aligned}
\label{wkpment3}
&\frac{d}{dt}\mathcal{W}_K(t)\notag\\
\le&
-\sigma_K\beta_K\int_M 2 (\gamma-1)\left[\left|\nabla_i\nabla_jv+\frac{\eta_K}{n(\gamma-1)}g_{ij}\right|^2+({\rm Ric}+Kg)(\nabla v,\nabla v )\right]vu\,dV\notag\\
&-\sigma_K\beta_K\int_M\left[2(\gamma-1)^2(\Delta v)^2+2(\gamma-1)\left((\log\sigma_K)'+\frac{1+\dot{\beta}_K}{2\beta_K}+{\kappa}
-\frac{2\eta_K}{n(\gamma-1)}\right)\Delta v\right]vu\,dV\notag\\
&-\sigma_K\beta_K\int_M\left[(\log\sigma_K)''+ ((\log\sigma_K)')^2+\frac{1+\dot{\beta}_K}{\beta_K}(\log\sigma_K)'-\frac{2\eta_K^2}{n(\gamma-1)}\right]vu\,dV.\end{aligned}$$
Set $\lambda=(\log\sigma_K)'$ and choose a proper function $\eta_{K}(t)$ such that $$\label{etabetak}
\left\{
\begin{array}{l}
2\eta_K=\lambda+\frac{1+\dot{\beta}_K}{2\beta_K}+{\kappa}
-\frac{2}{n(\gamma-1)}\eta_K \\
2\eta_K^2=\lambda'+ \lambda^2+\frac{1+\dot{\beta}_K}{\beta_K}\lambda-\frac{2}{n(\gamma-1)}\eta_K^2,
\end{array}
\right.$$ which is equivalent to $$\begin{aligned}
\label{etak}
0=&\eta^2_K-2\lambda\eta_K+\frac{a}{a+1}
\left(\lambda^2-\lambda'+2{\kappa}\lambda\right)\notag\\
=&(\eta_K-\lambda)^2-\frac{1}{a+1}\left(\lambda^2
+a\left(\lambda'-2{\kappa}\lambda\right)\right),\end{aligned}$$ where $a=\frac{n(\gamma-1)}{n(\gamma-1)+2}$. Solving the equation , we get a special solution $$\label{lambdaetak}
\lambda=\eta_K=\frac{2a{\kappa}}{1-e^{-2{\kappa}t}}.$$ Putting back to system , we have $$\frac{1+\dot{\beta}_K}{\beta_K}=2\kappa\coth (kt),\quad \beta_K=\frac{\sinh(2{\kappa}t)}{2{\kappa}}$$ and $$\alpha_K={\kappa}\tanh({\kappa}t),\quad
\sigma_K=\left(e^{{\kappa}t}\frac{\sinh({\kappa}t)}{\kappa}\right)^{a}=\left(\frac{e^{2{\kappa}t}-1}{2\kappa}\right)^{a}.$$
Therefore, from , we obtain the entropy monotonicity formula,
$$\begin{aligned}
\label{wkp
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|
grightarrow \underline{H} .$$
We start with any $m\in \underline{M}^{\ast}(R)$ and $f\in \underline{H}(R)$. In order to show that $\underline{M}^{\ast}(R)$ acts on the right of $\underline{H}(R)$ by $f\circ m = \sigma({}^tm)\cdot f\cdot m$, it suffices to show that $f\circ m$ satisfies conditions (a) to (e) given in the beginning of Section \[h\]. The proof that $f\circ m$ satisfies conditions (a) to (c) is similar to the proof of Theorem 3.4 in [@C2] and so we skip it.\
For condition (d), it suffices to show that $$\xi^{-m} f(mw_i, mw_i)-\xi^{-m}h(w_i,w_i) \in (4),$$ where $w_i\in W_i$. Since $m$ induces the identity on $W_i/(X_i\cap Z_i)$, we can write $mw_i=w_i+x_i$ where $x_i\in X_i\cap Z_i$. Thus it suffices to show that $$\xi^{-m} \left( f(w_i, x_i)+f(x_i, w_i)+f(x_i, x_i) \right) \in (4).$$ Since $\xi^{-m} f(w_i, x_i)\equiv \xi^{-m} h(w_i, x_i)$ mod $\pi$ by condition (c) and $\xi^{-m} h(w_i, x_i) \in (\pi)$ by the definition of $X_i$, we can see that $$\xi^{-m} f(w_i, x_i) \in (\pi) \textit{ and so }
\xi^{-m} \left( f(w_i, x_i)+f(x_i, w_i) \right) \in (4).$$ Furthermore, since $x_i\in Z_i$ and clearly $x_i\in W_i$, we can see that $$\xi^{-m} f(x_i, x_i) - \xi^{-m} h(x_i, x_i) \in (4) \textit{ and }
\xi^{-m} h(x_i, x_i) \in (4).$$ This completes the proof of condition (d).\
For condition (e), it suffices to show that $$f(ma_i, mb_i^{\prime})-h(a_i, b_i^{\prime})\in B,
\textit{ where $a_i\in A_i$ and $b_i^{\prime}\in B_i^{\perp}$.}$$ We write $ma_i=a_i+b_i$ and $mb_i^{\prime}=b_i^{\prime}+a_i^{\prime}$, where $b_i\in B_i$ and $a_i^{\prime} \in A_i^{\perp}$. Hence it suffices to show $$f(a_i+b_i, a_i^{\prime})+f(b_i, b_i^{\prime})\in B.$$ Since $a_i+b_i, b_i\in A_i$ and $a_i', b_i'\in B_i^{\perp}$, we can see that $$\left(f(a_i+b_i, a_i^{\prime})+f(b_i, b_i^{\prime})\right)-\left(h(a_i+b_i, a_i^{\prime})+h(b_i, b_i^{\prime})\right)\in B
\textit{ and } h(a_i+b_i, a_i^{\prime})+h(b_i, b_i^{\prime})\in B.$$ This completes the proof of condition (e).
The proof of the representability of this action
| 2,656
| 2,787
| 2,411
| 2,386
| 3,371
| 0.772823
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bel{eq:Bmarginal}$$ for $i,j\in\text{\{1,2\}}.$
Writing the inequalities and in terms of these expectations rather than in terms of probabilities is the most economic way of presenting the 128 non-trivial inequalities of the system, as the marginal probabilities $a_{1},a_{2},b_{1},b_{2}$ (or expectations $\left\langle \mathbf{A}_{1}\right\rangle ,\left\langle \mathbf{A}_{2}\right\rangle ,\left\langle \mathbf{B}_{1}\right\rangle ,\left\langle \mathbf{B}_{2}\right\rangle $) vanish in this form. However, it should be noted that in addition to these 128 inequalities, the form of the observed distributions and connections itself imposes further 28 trivial constraints on the 12 expectation variables of the system: the probabilities within each $2\times2$ matrix in and should be nonnegative and sum to one. 16 of these trivial constraints pertain to the observed distributions and 12 to the connections. In terms of the expectations, these trivial constraints correspond to $$-1+|\left\langle \mathbf{A}\right\rangle +\left\langle \mathbf{B}\right\rangle |\le\left\langle \mathbf{A}\mathbf{B}\right\rangle \le1-|\left\langle \mathbf{A}\right\rangle -\left\langle \mathbf{B}\right\rangle |,\label{eq:trivial}$$ for given marginals for each pair $(\mathbf{A},\mathbf{B})$ of random variables in –. This expands to four inequalities for each of the observed distributions and to three inequalities for each of the connections (the two upper bounds in coincide when $\left\langle \mathbf{A}\right\rangle =\left\langle \mathbf{B}\right\rangle $). Although these trivial constraints can usually be assumed implicitly, it is important to keep them explicitly in the system for the next step.
Adding the equation $$\begin{aligned}
\Delta= & \Pr\left[\mathbf{A}_{1,1}\ne\mathbf{A}_{1,2}\right]+\Pr\left[\mathbf{A}_{2,1}\ne\mathbf{A}_{2,2}\right]\\
& +\Pr\left[\mathbf{B}_{1,1}\ne\mathbf{B}_{2,1}\right]+\Pr\left[\mathbf{B}_{1,2}\ne\mathbf{B}_{2,2}\right]\\
= & 2-\frac{1}{2}\left(\left\langle \mathbf{A}_{1,1}\mathbf{A}_{1,2}\right\rangle +\left\la
| 2,657
| 3,154
| 3,449
| 2,646
| null | null |
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Typical program for hospital bed linen 50 ppm Chlorine, 54 ppm peracid, 100 ppm peroxid *Clostridium difficile* spores Hellickson & Owens, 2007 \[[@B18-ijerph-09-03330]\]
3. Reports on the Presence of Microorganisms on Hospital Textiles
=================================================================
[Table 2](#ijerph-09-03330-t002){ref-type="table"} summarizes reports of articles on the presence of microorganisms on hospital textiles.
ijerph-09-03330-t002_Table 2
######
Reports on the presence of microorganisms on hospital textiles.
Surviving microorganism Hospital textile Time Reference
--------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------ ---------------------------------------- -------------------------------------------------------
Moulds Sheets, pyjamas After use by patients Bureau-Chalot *et al*. 2004 \[[@B3-ijerph-09-03330]\]
Coagulase-negative staphylococci, *Bacillus* spp., *Corynebacterium* spp., saprophytic Gram negative bacilli Sheets, pyjamas, uniforms After laundering in hospital laundry Fijan *et al*. 2005 \[[@B6-ijerph-09-03330]\]
*Staphylococcus aureus, Clostridium difficile* and vancomycin resistant enterococci Nurses' uniforms After 24 h shift
| 2,658
| 3,813
| 2,485
| 2,429
| null | null |
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|
g.
\[prop:normalize\] A crank form expression of a seat-plan is moved to its standard expression.
Now we prove that any word in the alphabet ${\cal L}_n^1$ is moved to a crank form expression. By the above proposition, we will find that any word can be moved to its standard expression.
If ${\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is the standard expression of a seat-plan $w$, then $s_i{\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is moved to a crank form expression of $s_iw$.
If $i$ and $i+1$ are both included one of the (upper) parts of $w$, say $M_k$, then we have $$\sum_{j=1}^{k-1}|M_j|
<x^{-1}_{\overline{\mathbb{M}}}(i) < x^{-1}_{\overline{\mathbb{M}}}(i+1)
= x^{-1}_{\overline{\mathbb{M}}}(i)+1
\leq\sum_{j=1}^{k}|M_j|$$ and $$(x^{-1}_{\overline{\mathbb{M}}}(i), x^{-1}_{\overline{\mathbb{M}}}(i+1))
C_{\mathbb{M}}[k]
= (x^{-1}_{\overline{\mathbb{M}}}(i), x^{-1}_{\overline{\mathbb{M}}}(i)+1)
C_{\mathbb{M}}[k]
= {\cal C}_{\mathbb{M}}[k].$$ Since $$s_ix_{\overline{\mathbb{M}}} = (i,i+1)x_{\overline{\mathbb{M}}}
= x_{\overline{\mathbb{M}}}
(x^{-1}_{\overline{\mathbb{M}}}(i), x^{-1}_{\overline{\mathbb{M}}}(i+1)),$$ we find that $s_i{\cal C}(\mathbb{M}, \sigma, \mathbb{F})
= {\cal C}(\mathbb{M}, \sigma, \mathbb{F})$ is a crank form expression.
If $i$ is included in $M_j$ and $i+1$ is included in $M_k$ ($j\neq k$), then we have $s_ix_{\overline{\mathbb{M}}} = x_{\overline{\mathbb{M}'}}$. Here $\mathbb{M}'$ is the sequence of the upper parts obtained from $\mathbb{M}=(M_1,\ldots, M_u)$ by replacing $M_j$ with $M_j' = (M_j\setminus\{i\})\cup\{i+1\}$ and $M_k$ with $M_k' = (M_k\setminus\{i+1\})\cup\{i\}$.
Hence we find that $s_i{\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is moved to ${\cal C}(\mathbb{M}',\sigma,\mathbb{F})$, a crank form expression. In particular this expression again becomes the standard expression, unless $k = j+1$, $t(M_j) = t(M_{j+1})$, and $i = \min M_j$, $i+1 = \min M_{j+1}$.
If ${\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is the standard expression of a seat-plan $w$, then $f{\cal C}(\mathbb{M},\sigma,\ma
| 2,659
| 1,159
| 1,966
| 2,361
| 3,831
| 0.769798
|
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|
tively. A label [$y\in\{+1,-1\}$]{} indicates whether $I$ contains the target part. The AOG estimates the probability of object $I$ containing the target part as [${\bf Q}(y\!=\!+1|I)\!=\!\frac{1}{Z}\exp[\beta S_{top}]$]{}, where $Z$ and $\beta$ are parameters for scaling (see Section \[sec:implement\] for details); [${\bf Q}(y=-1|I)\!=\!1-{\bf Q}(y=+1|I)$]{}. Let [${\bf I}^{\textrm{ant}}$]{} denote the set of objects without being asked during previous QA. For each asked object [$I\in{\bf I}^{\textrm{ant}}$]{}, we set its prior distribution [${\bf P}(y=+1|I)=1$]{} if $I$ contains the target part; [${\bf P}(y=+1|I)=0$]{} otherwise. For each un-asked object [$I\in{\bf I}^{\textrm{unant}}$]{}, we set its prior distribution based on statistics of previous answers, [${\bf P}(y=+1|I)=\mathbb{E}_{I'\in{\bf I}^{\textrm{ant}}}{\bf P}(y=+1|I')$]{}. Therefore, we formulate the loss function as the KL divergence between the prior distribution [${\bf P}$]{} and the estimated distribution [${\bf Q}$]{}. $$\begin{split}
\!\!{Loss}^{\textrm{QA}}\!\!\!=\!{\bf KL}({\bf P}\Vert{\bf Q})\!=\!&\sum_{I\in{\bf I}^{\textrm{obj}}}\sum_{y}{\bf P}(y,I)\log\frac{{\bf P}(y,I)}{{\bf Q}(y,I)}\!\!\!\!\!\!\!\\
=&\lambda\sum_{I\in{\bf I}^{\textrm{obj}}}\sum_{y}{\bf P}(y|I)\log\frac{{\bf P}(y|I)}{{\bf Q}(y|I)}
\end{split}$$ where [${\bf P}(y,I)\!=\!{\bf P}(y|I)P(I)$; ${\bf Q}(y,I)\!=\!{\bf Q}(y|I)P(I)$; $\lambda=P(I)\!=\!1/\vert{\bf I}^{\textrm{obj}}\vert$]{} is a constant prior probability for $I$.
We keep modifying both the prior distribution ${\bf P}$ and the estimated distribution ${\bf Q}$ during the QA process. Let the algorithm select an unannotated object [$\tilde{I}\in{\bf I}^{\textrm{unant}}={\bf I}^{\textrm{obj}}\setminus{\bf I}^{\textrm{ant}}$]{} and ask people to label its part. The annotation would encode part representations of $\tilde{I}$ into the AOG and significantly change the estimated distribution for objects that are similar to $\tilde{I}$. For each object $I'\in{\bf I}^{\textrm{obj}}$, we predict its estimated distribution
| 2,660
| 2,467
| 2,553
| 2,553
| 2,981
| 0.775625
|
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|
d}}_\#)$ are taken pointwise: for every $n$, we have an exact sequence $$0 \to ({\operatorname{{\sf Ker}}}\phi)([n]) \to M_\#([n]) \overset{\phi}{\to}
M'_\#([n]) \to ({\operatorname{{\sf Coker}}}\phi)([n]) \to 0.$$ The transtition maps $\iota_f$ for ${\operatorname{{\sf Ker}}}\phi$ are obtained by restriction from those for $M_\#$; for ${\operatorname{{\sf Coker}}}\phi$, one uses the fact that the functors $f_!$ are right-exact.
A [*cyclic bimodule $M$*]{} over a unital associative algebra $A$ is a cocartesian section $M_\# \in {\operatorname{\sf Sec}}_{cart}(A{\operatorname{\!-\sf bimod}}_\#)$. A [*complex of cyclic bimodules $M_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$*]{} over $A$ is an object in the derived category ${{\mathcal D}}({\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#))$ whose homology objects are cocartsian.
Complexes of cyclic bimodules obviously form a full triangulated subcategory in ${{\mathcal D}}({\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#))$; consistent notation for this category would be ${{\mathcal D}}_{cart}({\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#))$, but for simplicity we will denote it ${{\mathcal D}}\Lambda(A{\operatorname{\!-\sf bimod}})$. We have to define complexes separately for the following reasons:
1. The category ${\operatorname{\sf Sec}}_{cart}(A{\operatorname{\!-\sf bimod}}_\#) \subset
{\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$ need not be abelian – since the transition functors $f_!$ are only right-exact, the condition of being cocartesian need not be preserved when passing to kernels.
2. Even if ${\operatorname{\sf Sec}}_{cart}(A{\operatorname{\!-\sf bimod}}_\#)$ is abelian, its derived category might be much smaller than ${{\mathcal D}}\Lambda(A{\operatorname{\!-\sf bimod}})$.
\[const.exa\] An extreme example of is the case $A = k$: in this case ${\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$ is just the category of cyclic vector spaces, ${\operatorname{Fun}}(\Lambda,k)$, and $E
| 2,661
| 2,110
| 2,225
| 2,417
| 2,340
| 0.780741
|
github_plus_top10pct_by_avg
|
e each potential is expressed through one function $W_{1}(r)$: $$\begin{array}{ll}
\bar{V}_{1}(r) =
W_{1}^{2}(r) - \displaystyle\frac{\hbar}{\sqrt{2m}}
\displaystyle\frac{d W_{1}(r)}{dr} + C_{1}, &
\bar{V}_{2}(r) =
W_{1}^{2}(x) + \displaystyle\frac{\hbar}{\sqrt{2m}}
\displaystyle\frac{d W_{1}(r)}{dr} + C_{1}.
\end{array}
\label{eq.2.1.6}$$ One can find: $$\bar{V}_{2} (r) - \bar{V}_{1}(r) =
V_{2} (r) - V_{1}(r) =
2\displaystyle\frac{\hbar}{\sqrt{2m}}
\displaystyle\frac{d W_{1}(r)}{dr}.
\label{eq.2.1.7}$$ The determination of the potentials $V_{1}(r)$ and $V_{2}(r)$ of two quantum systems on the basis of one function $W_{1}(r)$ establishes the interdependence between spectral characteristics (spectra of energy, wave functions, S-matrixes) of these systems. We shall consider this interdependence, as the interdependence given by Darboux transformations in the radial problem, and we shall name $W_{1}(r)$ as *superpotential*, potentials $V_{1}(r)$ and $V_{2}(r)$ as *supersymmetric potentials-partners*. Note, that there is a constant $C_{1}$ in the definition (\[eq.2.1.2\]) of the hamiltonians of two quantum systems. If to choose $C_{1}=E^{(1)}_{0}, E^{(2)}_{0} \ne E^{(1)}_{0}$ ($E^{(1)}_{0}$ and $E^{(2)}_{0}$ are the lowest levels of energy spectra of the first and second hamiltonians $H_{1}$ and $H_{2}$), then we obtain the most widely used construction two hamiltonians $H_{1}$ and $H_{2}$ in the one-dimensional case on the basis of the operators $A_{1}$ and $A_{1}^{+}$ (for example, see p. 287–289 in [@Cooper.1995.PRPLC]). However, this case corresponds to bound states in the discrete regions of the energy spectra of two studied quantum systems. For study of scattering, decay or synthesis processes in the radial consideration usually we deal with unbound states with the continuous region of the energy spectra (with the lowest energy levels $C_{1} = E^{(1)}_{0} = E^{(2)}_{0} = 0$) of quantum systems. Therefore, one need to use $C_{1}=0$ for obtaining the interdependence between the spectral c
| 2,662
| 2,952
| 3,046
| 2,507
| 2,428
| 0.779831
|
github_plus_top10pct_by_avg
|
or instance, the situation where there is no background cosmological constant ($\Lambda=0$). In this case we easily obtain $$a^3(z)={6\pi G A \over (z_- -z_+)^5}(C-z)^2
={{3\over 4}\pi^3 \zeta'_R(-4) G_5 }{(z_0-z)^2\over(z_- -z_+)^5},$$ where the brane tensions are given by $$2\pi G \sigma_{\pm}=\pm (C-z_{\pm})^{-1}$$ and $C$ is a constant. This is a self–consistent solution where the warp in the extra dimension is entirely due to the Casimir energy.
Of course, the conformal interbrane distance $(z_--z_+)$ is different from the physical $d$, although they are related. For instance, imposing $a(z_+)=1$, which we can rewrite as $$6\pi A G_5 = \left({z_--z_+\over z_0-z_+}\right)^2 (z_--z_+)^3$$ and we get the relation $$d=(z_--z_+)\left[{3\over5}\sqrt{(z_--z_+)^3 \over 6\pi G_5 A}\left(
1-\biggl(1-\sqrt{6\pi G_5 A\over(z_--z_+)^3} \;\biggr)^{5/3}\right)\right].$$ Here we can see that the case of negligible Casimir energy, ${6\pi G_5 A/(z_--z_+)^3} \ll 1$, indeed corresponds to the flat case, in which the conformal and the physical distances coincide.
We can also integrate Eq. (\[friedmann\]) in the general case [@tesina], and get $$a(y)=\left({16\pi A M^3 \over{-\Lambda
(z_--z_+)^5}}\right)^{1/5}\sinh^{2/5}\left({5\over2}\sqrt{-\Lambda/6}\;(y_0-y)\right),$$ with brane tensions given by $$\sigma_{\pm}=\pm
{3\over{4\pi}}{\sqrt{-\Lambda/6}\over
G_5}\coth\left({5\over2}\sqrt{-\Lambda/6}\;(y_0-y_\pm)\right).$$ Here we are assuming $\Lambda<0$, and $y_0$ is an integration constant. Moreover we can explicitely check how this reduces to RS solution in the limit of small Casimir energy compared to the cosmological constant, [*i.e.*]{}, when $${16\pi G_5 \over 3} \rho_0 \ll {\Lambda \over 6}.$$ Again fixing $a(z_+)=1$ we find $$y_0={2\over5}\sqrt{-6\over\Lambda}\; {\rm arcsinh}\left(\left({32\pi
\rho_0\over-\Lambda M^3}\right)^{-1/2}\right) >> 1$$ since $(32\pi \rho_0/(-\Lambda M^3)) << 1 $, so that we can write the warp factor as a power series in the parameter $(32\pi \rho_0/(-\Lambda M^3))^{1/5} \ll 1$: $$
| 2,663
| 4,273
| 2,674
| 2,307
| 3,803
| 0.770014
|
github_plus_top10pct_by_avg
|
s in this context, in accordance with section \[m-d\].
Given ${\bf f}=({\bf f}_1, {\bf f}_2, {\bf f}_3)\in L^2(G\times S\times I)^3$, and ${\bf g}\in T^2(\Gamma_-)\times H^1(I,T^2(\Gamma_-'))^2$, find $\phi=(\phi_1,\phi_2,\phi_3)\in L^2(G\times S\times I)^3$ which satisfies the system of equations of $G\times S\times I$, $$\begin{aligned}
\omega\cdot\nabla_x\phi_1+\Sigma_1\phi_1- K_{1,C}\phi = {}& {\bf f}_1,\label{codiss1}\\
-{{\frac{\partial (S_j\phi_j)}{\partial E}}}+\omega\cdot\nabla_x\phi_j+CS_j\phi_j+\Sigma_{j}\phi_j
-{K}_{j,C}\phi = {}& {\bf f}_j,\quad j=2,3,\label{codiss2}\end{aligned}$$ the boundary condition on $\Gamma_-$, \[codiss3\] \_[|\_-]{}=[**g**]{}, and the initial condition on $G\times S$, \[codiss4\] \_j(,,E\_m)=0,j=2,3.
Let $$P_{1}(x,\omega,E,D)\phi_1:={}&\omega\cdot\nabla_x\phi_1, \\[2mm]
P_{C,j}(x,\omega,E,D)\phi_j:={}&-{{\frac{\partial (S_j\phi_j)}{\partial E}}}+\omega\cdot\nabla_x\phi_j
+CS_j\phi_j,\quad j=2,3, \\[2mm]
{\bf P}_{C}(x,\omega,E,D)\phi:={}&\big(P_{1}(x,\omega,E,D)\phi_1,
P_{C,2}(x,\omega,E,D)\phi_2,P_{C,2}(x,\omega,E,D)\phi_3\big).$$ When $C=0$ we write ${\bf P}(x,\omega,E,D):={\bf P}_0(x,\omega,E,D)$, and define $$\begin{gathered}
{{{\mathcal{}}}H}_{\bf P}(G\times S\times I^\circ)
:=\{\psi\in L^2(G\times S\times I)^3\ | \\
{\bf P}(x,\omega,E,D)\psi\in
L^2(G\times S\times I)^3\ \textrm{in the weak sense}\}. \label{eq:H_bfP}\end{gathered}$$
The operator $\tilde {\bf P}_{C,0}$ is defined in the same way as $\tilde {P}_{C,0}$ in section \[m-d\], namely it is the smallest closed extension (closure) of ${\bf P}_{C,0}$, where $$D({\bf P}_{C,0}):={}&\big\{
\phi\in \tilde{W}^2(G\times S\times I)\times \big(\tilde{W}^2(G\times S\times I)\cap H^1(I,L^2(G\times S)\big)^2
\ \big| \\[2mm]
&\hspace{2.5mm} \phi_{|\Gamma_-}=0,\ \phi_j(\cdot,\cdot,E_m)=0,\ j=2,3\big\} \\[2mm]
{\bf P}_{C,0}\phi:={}&{\bf P}_{C}(x,\omega,E,D)\phi.$$
Using these notations, the problem - with ${\bf g}=0$, in the strong sense, is equivalent to $$(\tilde {\bf P}_{C,0}+\Sigma-K_C)\phi={\bf f},$$ where $\phi\in D(\
| 2,664
| 1,600
| 2,458
| 2,555
| null | null |
github_plus_top10pct_by_avg
|
}-D \delta^{ij} \partial_i u \partial_j u -m^2_{\text{eff}}u^2 ]$$ where $$m^2_{\text{eff}} = - \frac{\sqrt{D}}{a} \left( \frac{a}{\sqrt{D}} \right)^{''} .$$ Till now the considerations of the gravitational waves has been purely classical. The next step is the quantisation of the classical gravitational waves what brings us the concept of gravitons. To quantise the field $u$ we need to firstly calculate conjugated momenta $$\pi(\tau,{\bf x})=\frac{\delta S_{\text{t}}}{\delta u'} = u '.$$ The procedure of quantisation is the simple change of fields $u$ and $\pi$ for the operators just adding hats and to introduce the relations of commutation. We decompose operators considered for the Fourier modes $$\begin{aligned}
\hat{u}(\tau,{\bf x} ) &=& \frac{1}{2(2\pi)^{3/2}} \int d^3{\bf k } \left[ \hat{u}_{{\bf k}}(\tau) e^{i{\bf k}\cdot {\bf x}} +
\hat{u}_{{\bf k}}^{\dagger}(\tau) e^{-i{\bf k}\cdot {\bf x}} \right], \label{decomp1} \\
\hat{\pi}(\tau,{\bf x} ) &=& \frac{1}{2(2\pi)^{3/2}} \int d^3{\bf k }
\left[ \hat{\pi}_{{\bf k}}(\tau) e^{i{\bf k}\cdot {\bf x}} +
\hat{\pi}_{{\bf k}}^{\dagger}(\tau) e^{-i{\bf k}\cdot {\bf x}} \right], \label{decomp2} \end{aligned}$$ where the Fourier components fulfil the relations of commutation $$\begin{aligned}
\left[ \hat{u}_{{\bf k}}(\tau) ,\hat{\pi}_{{\bf p}}^{\dagger}(\tau) \right] &=& i \delta^{(3)}({\bf k} - {\bf p}),\label{com1}\\
\left[ \hat{u}_{{\bf k}}(\tau)^{\dagger} ,\hat{\pi}_{{\bf p}}(\tau) \right] &=& i \delta^{(3)}({\bf k} - {\bf p}),\label{com2} \\
\left[ \hat{u}_{{\bf k}}(\tau) ,\hat{\pi}_{{\bf p}}(\tau) \right] &=& i \delta^{(3)}({\bf k} + {\bf p}), \label{com3} \\
\left[ \hat{u}_{{\bf k}}(\tau)^{\dagger} ,\hat{\pi}_{{\bf p}}^{\dagger}(\tau) \right] &=& i \delta^{(3)}({\bf k} + {\bf p}).
\label{com4}\end{aligned}$$
To express the Fourier modes in terms of the annihilation and creation operators we need to solve the quantum Hamilton equations $$\begin{aligned}
\hat{u}^{'} &=& i [ \hat{H}_{\text{t}}, \hat{u} ], \label{Ham1} \\
\h
| 2,665
| 4,080
| 2,540
| 2,298
| null | null |
github_plus_top10pct_by_avg
|
of $(\textrm{Map}(X,Y),\mathcal{T})$ to be $$\begin{gathered}
B((x_{i}),(V_{i});(y_{i}),(U_{i}))=\{ g\in\mathrm{Map}(X,Y)\!:\\
(g(x_{i})\in V_{i})\wedge(g^{-}(y_{i})\bigcap U_{i}\neq\emptyset)\textrm{ },i=1,2,\cdots,I\},\label{Eqn: func_bi}\end{gathered}$$
where $(x_{i})_{i=1}^{I},(V_{i})_{i=1}^{I}$ are as in that example, $(y_{i})_{i=1}^{I}\in Y$, and the corresponding open sets $(U_{i})_{i=1}^{I}$ in $X$ are chosen arbitrarily[^20]. A local base at $f$, for $(x_{i},y_{i})\in\mathbf{G}_{f}$, is the set of functions of (\[Eqn: func\_bi\]) with $y_{i}=f(x_{i})$ and the collection of all local bases $$B_{\alpha}=B((x_{i})_{i=1}^{I_{\alpha}},(V_{i})_{i=1}^{I_{\alpha}};(y_{i})_{i=1}^{I_{\alpha}},(U_{i})_{i=1}^{I_{\alpha}}),\label{Eqn: local_base}$$ for every choice of $\alpha\in\mathbb{D}$, is a base $_{\textrm{T}}\mathcal{B}$ of $(\textrm{Map}(X,Y),\mathcal{T})$. Here the directed set $\mathbb{D}$ is used as an indexing tool because, as pointed out in Example A1.1, the topology of pointwise convergence is not first countable.
In a manner similar to Eq. (\[Eqn: func\_bi\]), the open sets of $(\mathrm{Multi}(X,Y),\widehat{\mathcal{T}})$, where $\textrm{Multi}(X,Y)$ are multifunctions with only countably many values in $Y$ for every point of $X$ (so that we exclude continuous regions from our discussion except for the “vertical lines” of $\textrm{Multi}_{\mid}(X,Y)$), can be defined by $$\begin{gathered}
\widehat{B}((x_{i}),(V_{i});(y_{i}),(U_{i}))=\{\mathscr{G}\in\mathrm{Multi}(X,Y)\!:(\mathscr{G}(x_{i})\bigcap V_{i}\neq\emptyset)\wedge(\mathscr G^{-}(y_{i})\bigcap U_{i}\neq\emptyset)\}\label{Eqn: multi_bi}\end{gathered}$$
where $$\mathscr G^{-}(y)=\{ x\in X\!:y\in\mathscr{G}(x)\}.$$
and $(x_{i})_{i=1}^{I}\in\mathcal{D}(\mathscr{G}),(V_{i})_{i=1}^{I};(y_{i})_{i=1}^{I}\in\mathcal{R}(\mathscr{G}),(U_{i})_{i=1}^{I}$ are chosen as in the above. The topology $\widehat{\mathcal{T}}$ of $\textrm{Multi}(X,Y)$ is generated by the collection of all local bases $\widehat{B_{\alpha}}$ for every choice of $\alpha\in\mathbb
| 2,666
| 1,602
| 2,474
| 2,494
| 2,296
| 0.781099
|
github_plus_top10pct_by_avg
|
ement at time $t$, and $U$ the uniform distribution: $$P(0) = \frac{1}{2}+ \gamma, \qquad P(1) = \frac{1}{2}- \gamma, \qquad U(0) = U(1) = \frac{1}{2},$$ where $$\gamma = \frac{1}{4}\left[e^{\frac{(-p-\alpha)t}{2n}}(1-p/\alpha) + e^{\frac{(\alpha-p)t}{2n}}(1+p/\alpha)\right].$$ For $x = (x_1, \dots, x_n) \in \mathbb{Z}_2^n$, $$P^n(x) = \prod_{i=1}^n P(x_i) \qquad\text{and}\qquad U^n(x) = 2^{-n}$$ are the analogous product distributions in the $n$-dimensional case. To analyze the limiting mixing behavior of the walk, we will consider the total variation distance $\|P^n - U^n\| =
\sum_x |P^n(x) - U^n(x)|$ between these distributions. In order to give bounds for total variation, we will use *Hellinger distance* [@ASYMP], defined as follows: $$H(A, B)^2 = \sum_{x} \left(\sqrt{A(x)} - \sqrt{B(x)}\right)^2 = 1 - \sum_{x}\sqrt{A(x)B(x)}.$$ We will make use of the following two properties of Hellinger distance: $$1 - H(A^n, B^n)^2 = (1-H(A,B)^2)^n\enspace,$$ and $$\|A - B\| \leq 2H(A, B) \leq 2\|A - B\|^{1/2}.
\label{helltv}$$ The first property makes it easy to work with product distributions. The second gives a nice relationship between Hellinger distance and total variation distance. In our case, $$\begin{aligned}
H(P^n,U^n)^2
& = 1 - (1-H(P,U)^2)^n \\
& = 1 - \left(\frac{1}{2}\sqrt{1+2\gamma} + \frac{1}{2}\sqrt{1-2\gamma}\right)^n\\
& = 1 - \left(1 - \frac{\gamma^2}{2} + O(\gamma^3)\right)^n.\end{aligned}$$ And hence, by (\[helltv\]), $$\|P_n - U_n\|^2 \leq 4 - 4\left(1 - \frac{\gamma^2}{2} + O(\gamma^3)\right)^n.$$ Consider the walk with decoherence rate $p > 4k$. We have $\alpha = \sqrt {p^2 - 16k^2} < p$, where $\alpha$ and $p$ are positive and real. It follows that for a fixed $p > 4k$, $\gamma \to 0$ and $\|P^n - U^n\|
\to 0$ as $t \to \infty$. Hence the walk does indeed mix eventually, and the measurement distribution in fact converges to the uniform distribution. Let $t = d \cdot n\log n$ where $d > 0$ is a constant, and rewrite $\gamma$ as follows: $$\gamma = \frac{1}{4}e^{-(p-\alpha)\frac{d \log n}{2
| 2,667
| 4,292
| 2,691
| 2,345
| null | null |
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|
mathcal R}, k=1,2,\ldots,s$. Then $\phi$ is an $R$-module isomorphism from ${\mathcal R}$ onto ${\mathcal M}_1\times\cdots\times{\mathcal M}_s$. For any left $R$-module $M_j$, it is obvious that $M_1\times\cdots\times M_s$ is a left $R$-submodule of ${\mathcal M}_1\times\cdots\times{\mathcal M}_s$. Therefore there is a unique left $R$-submodule $C$ of ${\mathcal R}$ such that $\phi(C)=M_1\times\cdots\times M_s$. $\Box$
Since ${\mathcal M_k}=(R/(g_k^*))^{n_k}=\bigoplus_{i=1}^lR/(g_k^{*d_{ik}})$ is up to an $R$-module isomorphism, Theorem 3.2 can lead to a canonical decomposition of skew GQC codes as follows.
[**Theorem 3.3** ]{} *Let $C$ be a skew GQC code of block length $(m_1, m_2, \ldots, m_l)$ and length $\sum_{i=1}^lm_i$ over $\mathbb{F}_q$. Then $$C=\bigoplus_{i=1}^sC_i$$ where $C_i$, $1\leq i\leq s$, is a linear code of length $l$ over $R/(g_i^{*d_{ik}})$ and each $j$-th, $1\leq j\leq l$, component in $C_i$ is zero if $d_{ji}=0$ and an element of the ring $R/(g_i^*)$ otherwise.* $\Box$
Let $m_1=m_2=\cdots m_l=m$. Then a skew GQC code $C$ is a *skew quasi-cyclic* (QC) code of length $ml$ over $\mathbb{F}_q$. From Theorems 3.2 and 3.3, we have the following result.
[**Corollary 3.4** ]{} *Let $R=\mathbb{F}_q[x, \sigma]$, ${\rm gcd}(m,q)=1$ and $x^m-1=g_1^*g_2^*\cdots g_s^*$, where $g_1^*, g_2^*, \ldots, g_s^*$ are pairwise coprime monic t.s.m elements in $R$. Then we have\
(i) There is an $R$-module isomorphism $\phi$ from ${\mathcal R}=(R/(x^m-1))^l, ~l\geq 1$, onto $(R/(g_1^*))^l\times (R/(g_2^*))^l\times \cdots \times (R/(g_s^*))^l$.\
(ii) $C$ is a skew QC code of length $ml$ over $\mathbb{F}_q$ if and only if there is a left $R$-submodule $M_i$ of $(R/(g_i^*))^l, ~i=1,2,\ldots,s$, such that $\phi(C)=M_1\times M_2\times\cdots \times M_s$.\
(iii) A skew QC code $C$ of length $ml$ can be decomposed as $C=\bigoplus_{i=1}^sC_i$, where each $C_i$ is a linear code of length $l$ over $R/(g_i^*)$, $i=1,2,\ldots,s$.* $\Box$
A skew GQC code $C$ of block length $(m_1, m_2, \ldots, m_l)$ and length $\sum_{i=1}
| 2,668
| 1,412
| 2,429
| 2,411
| 1,784
| 0.785711
|
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|
detected in Energy Import/Export schedule.
Variances detected in SC Trades schedule.
LOG MESSAGES:
PARSING FILE -->> O:\Portland\WestDesk\California Scheduling\ISO Final
Schedules\2001041523.txt
---- Energy Import/Export Schedule ----
$$$ Variance found in table tblINTCHG_IMPEXP.
Details: (Hour: 23 / Preferred: 12.00 / Final: 11.98)
TRANS_TYPE: FINAL
SC_ID: ECTRT
MKT_TYPE: 2
TRANS_DATE: 4/15/01
TIE_POINT: PVERDE_5_DEVERS
INTERCHG_ID: EPMI_CISO_LUCKY
ENGY_TYPE: WHEEL
---- SC Trades Schedule ----
*** Final schedule not found for preferred schedule.
Details:
TRANS_TYPE: FINAL
SC_ID: EPMI
MKT_TYPE: 2
TRANS_DATE: 4/15/01
TRADING_SC: SETC
PNT_OF_INTRC: NP15
SCHED_TYPE: ENGY
PURCH_SALE: 1
DEAL_NO: 1
+++ Hour 23 - bad data from ISO.
TRANS_TYPE: FINAL
SC_ID: EPMI
MKT_TYPE: 2
TRANS_DATE: 4/15/01
TRADING_SC: PGAE
PNT_OF_INTRC: NP15
SCHED_TYPE: ENGY
PURCH_SALE: 2
DEAL_NO: 1
Attached are the referenced swaps. They are complete with the prior comments
of K&E incorporated, except in each case for the maximum limit on share
settlement.
I will be on vacation next week. Angela Davis at ex. 58347 will assist you
as needed.
Cordially,
Mary Cook
Enron North America Corp.
1400 Smith, 38th Floor, Legal
Houston, Texas 77002-7361
(713) 345-7732 (phone)
(713) 646-3490 (fax)
mary.cook@enron.com
Ruth, remember that moorning you "allegedly" called me from Shemin's office last week? You said you were meeting with Shemin and wouldn't be in for awhile. I told you to ask Shemin "What the hell is going on with Dominion? What's the latest?". Well that's what this is. Never got me an answer on that now did you? I'm betting you and Shemin were at the Waffle House smokin' cig's, drinking coffee, and eating hash browns that were "covered" with cheese and filled with "chunks" of ham
Shemin, how about an update on Dominion? What's the word?
-----Original Message-----
From: "Proctor, Shemin V." <sproctor@akllp.com>@ENRON
Sent: Friday, April 05, 2002
| 2,669
| 1,380
| 1,944
| 2,996
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|
us turn to an example which does not have such a realization, but which is relevant to (0,2) GLSMs. Take $\mathfrak{X} = {\mathbb P}^4_{[1,1,1,2,2]}$, with bundle $$0 \: \longrightarrow \: {\cal E} \: \longrightarrow \:
\oplus_a {\cal O}(n_a) \: \stackrel{F_a}{\longrightarrow} \: {\cal O}(m)
\: \longrightarrow \: 0$$ where $\det {\cal E}^* \cong K_{\mathfrak{X}}$: $$\sum n_a \: - \: m \: = \: 7,$$ and second Chern classes match: $$\sum n_a^2 \: - \: m^2 \: = \: 11.$$ This is not Calabi-Yau, so it would not be directly useful for a string compactification, but can help illuminate some general aspects. This stack has a ${\mathbb P}^1$ of ${\mathbb Z}_2$ orbifolds, so the inertia stack has the form $$I_{\mathfrak{X}} \: = \: \mathfrak{X} \amalg
{\mathbb P}^1_{[2,2]}.$$ On the nontrivial component ${\mathbb P}^1_{[2,2]}$, call it $\alpha$, we can work out the decomposition of the gauge bundle. Suppose, for example, that $m$ is odd. For any given $a$, if $n_a$ is even, then $F_a$ is odd, so $F_a = 0$; if $n_a$ is even on the other hand, there is no constraint on $F_a$. In this case, we can decompose $$q^* {\cal E} |_{\alpha} \: = \: {\cal E}_+ \oplus {\cal E}_-,$$ where ${\cal E}_+$ is invariant, ${\cal E}_-$ anti-invariant under ${\mathbb Z}_2$, and specifically $$\begin{aligned}
{\cal E}_+ & = & \oplus {\cal O}(n_a \, {\rm even}),
\\
{\cal E}_- & = & {\rm ker}\left( \oplus {\cal O}(n_a \, {\rm odd}) \:
\longrightarrow \: {\cal O}(m) \right).\end{aligned}$$ A closely related decomposition exists for $m$ even.
Now that we have illuminated the definitions, let us return to our description of the general procedure for spectrum computation. At this point, the computation of spectra becomes more or less identical to that in an ordinary global orbifold by a finite group, if we think of $\alpha$ as denoting a twisted sector. We will walk through the details, as there are a few important subtleties for general cases not usually discussed in the literature, especially regarding Fock vacua, but the rest of the computation i
| 2,670
| 1,744
| 2,634
| 2,362
| 1,457
| 0.789324
|
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|
artial (S_0L\tilde{{\bf g}})}{\partial E}}}+CS_0(L\tilde{{\bf g}})\in L^2(G\times S\times I),$$ where the equality $\omega\cdot\nabla_x(L \tilde{{\bf g}})=0$ has been used again.
We additionally obtain the following a priori estimate.
\[cdd\] Under the assumptions of Theorem \[coth3-dd\] the solution $\phi$ of the problem (\[co3aa-dd\]) satisfies, with a constant $C_1\geq 0$, the estimate \[ess1-d\] \_[L\^2(GSI)]{}C\_1(\_[L\^2(GSI)]{} +\_[H\^1(I,T\^2(\_-’))]{}).
By estimate (\[bestim-d\]), we have \[ess1-da\] \_[L\^2(GSI)]{} =[u+L[**g**]{}]{}\_[L\^2(GSI)]{} \_[L\^2(GSI)]{} +[L[**g**]{}]{}\_[L\^2(GSI)]{}. By Lemma \[le:H1\_lift\], $${\left\Vert L{\bf g}\right\Vert}_{L^2(G\times S\times I)}={\left\Vert L{\bf g}\right\Vert}_{L^2(I,L^2(G\times S))}
\leq {\left\Vert L{\bf g}\right\Vert}_{H^1(I, L^2(G\times S))}\leq \sqrt{d}{\left\Vert {\bf g}\right\Vert}_{H^1(I,T^2(\Gamma'_-))},$$ where $d=\mathrm{diam}(G)<\infty$, and similarly, $${\left\Vert {\frac{\partial (L{\bf g})}{\partial E}}\right\Vert}_{L^2(G\times S\times I)}
={\left\Vert {\frac{\partial (L{\bf g})}{\partial E}}\right\Vert}_{L^2(I,L^2(G\times S))}
\leq {\left\Vert L{\bf g}\right\Vert}_{H^1(I,L^2(G\times S))}
\leq \sqrt{d}{\left\Vert {\bf g}\right\Vert}_{H^1(I,T^2(\Gamma'_-))}.$$ Finally, due to these estimates, Lemma \[csdale1a\], and the fact that $\omega\cdot\nabla_x(L{\bf g})=0$, one has $${\left\Vert \tilde{{\bf f}}\right\Vert}_{L^2}
\leq {}&
{\left\Vert {\bf f}\right\Vert}_{L^2}+{\left\Vert S_0{\frac{\partial (L{\bf g})}{\partial E}}+\Big({\frac{\partial S_0}{\partial E}}
-CS_0\Big)L{\bf g}-(\Sigma-K_C)L{\bf g}\right\Vert}_{L^2} \\
\leq {}&
{\left\Vert {\bf f}\right\Vert}_{L^2}
+\sqrt{d}\Big({\left\Vert {\frac{\partial S_0}{\partial E}}\right\Vert}_{L^\infty}+(C+1){\left\Vert S_0\right\Vert}_{L^\infty}+{\left\Vert \Sigma-K_C\right\Vert}\Big){\left\Vert {\bf g}\right\Vert}_{H^1(I,T^2(\Gamma'_-))},$$ where we wrote, unambiguously, $L^2=L^2(G\times S\times I)$ and $L^\infty=L^\infty(G\times I)$ in order to compress the formulas. This proves the estim
| 2,671
| 650
| 2,534
| 2,567
| null | null |
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|
e diagrams for the averaged ranking results of the metric Log-loss.](figures/results/crit_diff_loss_v2 "fig:"){width="\linewidth"}
The same process is applied to each of the $11$ classifiers for every metric. Table \[table:loss\] shows the final average results of all classifiers. Notice that the row corresponding to naive Bayes has the rounded average rankings from Figure \[fig:cd:nbayes:loss\].
Results {#sec:res}
=======
In this Section we present all the final results, including ranking tables for every metric, critical difference diagrams, the best hyperparameters selected for Dirichlet calibration with L2 regularisation, Frequency binning and Width binning; a comparison of how calibrated the $11$ classifiers are, and additional results on deep neural networks.
Final ranking tables for all metrics {#sec:res:rank}
------------------------------------
We present here all the final ranking tables for all metrics (Tables \[table:acc\], \[table:loss\], \[table:brier\], \[table:mce\], \[table:conf-ece\], \[table:cw-ece\], \[table:p-conf-ece\], and \[table:p-cw-ece\]). For each ranking, a lower value is indicative of a better metric value (eg. a higher accuracy corresponds to a lower ranking, while a lower log-loss corresponds to a lower ranking as well). Additional details on how to interpret the tables can be found in Section \[sec:exp:example\].
Final critical difference diagrams for every metric
---------------------------------------------------
In order to perform a final comparison between calibration methods, we considered every combination of dataset and classifier as a group $n = \#datasets \times \#classifiers$, and ranked the results of the $k$ calibration methods. With this setting, we have performed the Friedman statistical test followed by the one-tailed Bonferroni-Dunn test to obtain critical differences (CDs) for every metric (See Figure \[fig:multi:cd:all\]). The results showed Dirichlet L2 as the best calibration method for the measures accuracy, log-loss and p-cw-ece with statistical signi
| 2,672
| 2,066
| 483
| 2,359
| null | null |
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|
mall. Taking the matter potential of CC reaction and the earth diameter, $AL = 6.2 \left(\frac{\rho}{5 \text{g/cm}^3}\right) \left(\frac{L}{6,400 \mbox{km}}\right)$. Therefore, $ALW^2$ can be order unity for $|W| \simeq 0.4$.
[^24]: This statement applies also to the original expression (\[P-beta-alpha-0th+2nd\]).
[**About neutral mesons and particle oscillations in the light of field-theoretical prescriptions of Weinberg**]{}
L.M. Slad[^1]\
[*Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia*]{}
The postulated universality of the Weinberg’s prescriptions on the diagonalization of the mass term of the Lagrangian without increasing the total number of entities leads to the following conclusions: the set of neutral $K$-mesons consists of two elements, $K_{S}^{0}$ and $K_{L}^{0}$; the states $K^{0}$ and $\bar{K}^{0}$ do not exist as physical objects (in the form of particles or “particle mixtures”); the absence of the states $K^{0}$ and $\bar{K}^{0}$ destroys grounds for introducing the notion of their oscillations. The conclusions concerning the neutral $K$-mesons are also applicable to the neutral $D$-, $B$- and $B_{s}$-mesons.A theoretical and experimental vulnerability of the neutrino oscillation concept is noted.
The initial judgments about the family of four neutral $K$-mesons still remain almost unchanged and, furthermore, extend on the families of neutral $D$- and $B$-mesons. The concept of mutual transition of $K^{0}$- and $\bar{K}^{0}$-mesons in vacuum, originated long ago and retained up to present day, has served initially [@1] and continues to serve now [@2] as the only theoretical argument in favor of the hypothesis of neutrino oscillations by analogy with the letter.
In the present paper, we propose to put the status of neutral $K$-mesons in full accordance with field-theoretical prescriptions of Weinberg [@3] that have led to the prodigious gauge theory of electroweak interactions by making use of the diagonalization of the mass term in the
| 2,673
| 3,954
| 3,765
| 2,679
| null | null |
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|
bar \chi \chi \bar N N$ contact operator, but with an additional $1/E_r$ suppression in the cross section. This gives a similar phenomenology as a light mediator being exchanged at tree-level with derivative coupling.
Note that the relative importance of these two scattering processes is highly model dependent. For example, if $n_\phi = 2$ the dominant scalar-DM coupling could be $\bar q q \phi_1 \phi_2^*/\Lambda_{12}$. In that case, the $2\to2$ operator above is $\propto y_\chi^{\phi_1} y_\chi^{\phi_2}$ and can be suppressed without reducing the $2\to3$ rate by taking $y_\chi^{\phi_2} \gg y_\chi^{\phi_1}$. The scattering behavior of both the $2\rightarrow3$ and $2\rightarrow2$ regimes necessitates a re-interpretation of all DM direct detection bounds. We will do this below.
Indirect Constraints {#s.indirectconstraints}
====================
Direct detection experiments probe the ratio $y_\chi/\Lambda$ and $y_\chi^2/\Lambda$ for $2\to3$ and $2\to2$ scattering respectively. However, indirect constraints on dmDM from cosmology, stellar astrophysics and collider experiments are sensitive to the Yukawa coupling and $\Lambda$ separately. In [@dmDM] we conduct an extensive study of these bounds, including the first systematic exploration of constraints on the $\bar q q \phi \phi^*/\Lambda$ operator with light scalars $\phi$. Since these constraints (in particular, Eqns. \[e.NScoolingbound\] and \[e.thermalrelic\] below) provide important context for our results on direct detection, we summarize the two most important results here. For details we refer the reader to [@dmDM].
The scalar mediator(s) of dmDM are most stringently constrained from stellar astrophysics and cosmology:
- Avoiding overclosure requires $m_\phi \lesssim \ev$ [@Kolb:1990vq], so we take the heaviest stable $\phi$ to be essentially massless, making it a very subdominant dark matter component. This also satisfies structure formation, computed for light sterile neutrinos in [@Wyman:2013lza]. Measurements by the Planck satellite [@Ade:2013zuv] re
| 2,674
| 603
| 2,054
| 2,520
| null | null |
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|
ons $$\label{qu}
{\cal Q}_n:=\left\{Q(x)=L\left(\frac{t-x}{h_{2,n}}f^{1/2}(x)\right)
f^{-1/2}(x)b(x;h_{1,n})I(|t-x|<h_{2,n}B):t\in D_r\right\}$$ are of VC type with the same characteristics $A$ and $v$, for envelopes of the order of $M(K,r)\|f''\|_\infty h_{1,n}^2$, where $M$ depends on $r$ and $K$ only (in particular, through $L$). If we set $$Q_i(t)=L\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right)
f^{-1/2}(X_i)b(X_i;h_{1,n})I(|t-X_i|<h_{2,n}B)$$ it then follows (by the bound (\[classic2\]) on $b$, boundedness of $L$ and boundedness away from zero of $f$ on $D_r$), that $$\sup_{t\in D_r}E|Q_i(t)|\lessim \|f''\|_\infty h_{1,n}^2h_{2,n}=\|f''\|_\infty n^{-5/9}(\log n)^{1/9},$$ $$\sup_{t\in D_r}EQ_i^2(t)\lessim \|f''\|^2_\infty h_{1,n}^4h_{2,n}\le \|f''\|^2_\infty n^{-1}(\log n)^{1/9},\ \ \sup_{t\in D_r}|Q_i(t)|\lessim \|f''\|_\infty h_{1,n}^2=\|f''\|_\infty n^{-4/9},$$ where in these bounds we ignore multiplicative constants that do not depend on $f$. We have $$\begin{aligned}
\sup_{t\in D_r}\left|\epsilon_2(t;h_{1,n}, h_{2,n})\right|
&\le &\sup_{t\in D_r}\left|\frac{1}{nh_{2,n}}\sum_{i=1}^{n} [Q_i(t)-EQ_i(t)]\right|+ \sup_{t\in D_r}\frac{1}{h_{2,n}}|EQ_1(t)|\\
&\lessim & \sup_{t\in D_r}\left|\frac{1}{nh_{2,n}}\sum_{i=1}^{n} [Q_i(t)-EQ_i(t)] \right|+\|f''\|_\infty n^{-4/9},\end{aligned}$$ and Talagrand’s inequality (\[tal\]) gives that for $0<\delta\le 4/9$, $$\sum_n\sup_{f\in{\cal P}_C}{\Pr}_f\left\{\sup_{t\in D_r}\left|\sum_{i=1}^{n} [Q_i(t)-EQ_i(t)] \right|\ge n^{\delta}\right\}\le
C_2\sum_n\exp\left(-\frac{C_3n^{2\delta}}{C^2(\log n)^{1/9}}\right)<\infty.$$ Since $n^{\delta}<nh_{2,n}n^{-4/9}$, we conclude $$\sup_{t\in D_r}\left|\epsilon_2(t;h_{1,n}, h_{2,n})\right|=O_{\rm a.s.}(n^{-4/9})\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C$$ proving the lemma for $\varepsilon_2$. Note that $h_{1,n}^2\simeq n^{-4/9}$ plays a critical role in this estimation.
Next, from (\[eps3\]) we see that $\varepsilon_3$ consists of four sums, the three that define $\delta_3$ and one involving $\delta_4$ (multiplied by bounded ter
| 2,675
| 1,876
| 2,617
| 2,360
| null | null |
github_plus_top10pct_by_avg
|
=\frac{\left\langle \sum_{j\in{\cal H}_{1}}N_{j}\left[\sum_{i\in{\cal H}_{0}}N_{i}H_{ij}\right]\right\rangle }{\left\langle \sum_{i\in{\cal H}_{0}}N_{i}^{2}\right\rangle },\label{eq:E2}$$
through the spawning process, this strategy has already been used for the calculation of reduced density matrices[@overy_unbiased_2014]. When a successful spawning from a determinant $\Ket{D_{i}}$ in replica 0 to a determinant $\Ket{D_{j}}$ in replica 1 occurs, the product of the matrix element and of the number of walkers on $\Ket{D_{i}}$, $N_{i}H_{ij}$ is communicated along the spawned walker to the processor holding the child determinant $\Ket{D_{j}}$. Once all the spawning attempt have been done, the processor that keep track of the $\Ket{Dj}$ walkers population will also contains all the $N_{i}H_{ij}$ contribution from all the determinants that spawned to $\Ket{Dj}$ this iteration. This strategy does not cause any noticeable increase of the computational cost. As the contribution of a $\Ket{D_{i}},\Ket{D_{j}}$ pair of determinants to the $E_{2}$ energy is only taken into account when a successful spawning step is actually happening this contribution should be rescaled by the normalized probability of spawning at least one child (of any weight) onto $\Ket{D_{j}}$ from $\Ket{D_{i}}$ during the current iteration. More details on how to compute this probability can be found in [@overy_unbiased_2014]. To avoid double counting of $C_{i}C_{j}$ contribution, it is necessary to carefully check for the rare but still possible case of multiple spawning from the same determinant $\Ket{D_{i}}$ to the same determinant $\Ket{D_{j}}$. Thus the $N_{i}H_{ij}$ contribution is only communicated to the processor holding $\Ket{D_{j}}$ in the first occurrence of such an event.
** Finally, it is necessary to rescale the $\Ket{\Psi_{1}}$ function by the $\alpha$ factor to compute the $E_{2}$ energy.
With this efficient way of computing the second order energy, we can go back to the example of Fig. \[fig1:swalknum\] and examine how the estimation
| 2,676
| 439
| 2,107
| 2,724
| 687
| 0.802339
|
github_plus_top10pct_by_avg
|
]_T
\,
\sigma_{d d}({\mbox{\boldmath $r$}}_1, {\mbox{\boldmath $r$}}_2,Y)
\, ,
\label{sigmatot}\end{aligned}$$ where $\Psi^{\gamma}$ and $\Psi^{V_i}$ are the light-cone wave functions of the photon and vector meson, respectively, and $T$ the transverse polarization. The variable ${\mbox{\boldmath $r$}}_1$ defines the relative transverse separation of the pair (dipole) and $z_1$ $(1-z_1)$ is the longitudinal momentum fraction of the quark (antiquark). Similar definitions are valid for ${\mbox{\boldmath $r$}}_2$ and $z_2$. Moreover, $\sigma_{d d}$ is the dipole - dipole cross section, which can be estimated taking into account the nonlinear effects in the QCD dynamics. In what follows, we assume the Gauss-LC model for the vector meson wave functions and estimate $\sigma_{d d}$ using the approach proposed in Refs. [@nosfofo; @bruno_doublegama], which is based on the CGC physics. We refer the reader to the Ref. [@bruno_doublegama] for more details about the double vector meson production in $\gamma \gamma$ interactions.
-- -- -- --
-- -- -- --
----------------- ----------------- ----------------------------- ---------------------------- -------------------------- ---------------------------- -----------------------------
Final state Mechanism $PbPb$ $PbPb$ $pPb$ $pp$ $pp$
$\sqrt{s}=2.76\,\mbox{TeV}$ $\sqrt{s}=5.5\,\mbox{TeV}$ $\sqrt{s}=5\,\mbox{TeV}$ $\sqrt{s}=7\,\mbox{TeV}$ $\sqrt{s}=14\,\mbox{TeV}$
$J/\Psi J/\Psi$ DSM 402.301 nb 1054.951 nb 28.473 pb 3.223 $\times$10$^{-4}$ pb 7.256$\times$10$^{-4}$ pb
$\gamma \gamma$ 235.565 nb 658.589 nb 310.194 pb 0.2412 pb 0.4793 pb
$\rho \rho$ DSM 21.150 mb
| 2,677
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| 1,920
| 2,523
| null | null |
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|
alization of the latter. In this subsection we shall review two early attempts in this direction and point out their shortcomings. The proper stochastic formulation shall be presented in next Section.
As we have said in Section II, the peculiar structure of the 1PI EA allows to associate to a field theory problem an equivalent Langevin equation, whereby, for example, the Hadamard propagator may be obtained as a stochastic average. One possible strategy is to try to cast the 2PI EA in a similar framework. This line of thought is pursued in [@CalHu95a]. Success is found only under special, and restrictive, assumptions on the structure of the propagators, and therefore it is unsuitable as a general foundation for the formalism.
In [@CalHu99] the same authors follow a different strategy, which may be regarded as a nonequilibrium generalization of the fluctuation - dissipation theorem. We shall consider here only the case of a field theory with no background mean fields. The general case will be discussed in next Section.
We assume there are stochastic kernels $G_s^{AB}=G^{AB}+\gamma^{AB}$ such that the stochastic averages of these kernels reproduce the quantum averages of products of the composite operator $\Phi^A\Phi^B$. These kernels obey the Langevin equation
\_[,(AB)]{}=2\_[AB]{} \[ne72\] After linearization, this becomes
\_[,(AB)(CD)]{}\^[CD]{}=2\_[AB]{} \[ne73\]
We have two ways of computing the self correlation for the stochastic kernels. By assumption
G\_s\^[AB]{}G\_s\^[CD]{}=\_H\^A\_H\^B\_H\^C\_H\^D=G\^[AB]{}G\^[CD]{}-4iW\^[,(AB)(CD)]{} \[ne74\] while from the explicit solution of the linearized equations we get
G\_s\^[AB]{}G\_s\^[CD]{}&=&G\^[AB]{}G\^[CD]{}+\^[AB]{}\^[CD]{}&=&G\^[AB]{}G\^[CD]{}+14 \^[-1]{}\^[-1]{}\_[EF]{}\_[GH]{}\[ne75\] Asking both computations to agree we get
\_[IJ]{}\_[KL]{}=-16i\_[,(IJ)(AB)]{}\_[,(KL)(CD)]{}W\^[,(AB)(CD)]{}=4i\_[,(IJ)(KL)]{} \[ne76\] As we shall see in next Section, this is the correct result. However, the authors of [@CalHu99] provide some contrived argument to
| 2,678
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ectral gap is larger, as evidenced in previous learning to rank results in simpler settings [@NOS14; @SBB15; @HOX14]. This is made precise in , and in the main result of Theorem \[thm:main2\], we appropriately rescale the spectral gap and use $\alpha\in[0,1]$ defined as $$\begin{aligned}
\label{eq:lambda2_L1}
\alpha &\equiv& \frac{\lambda_2(L)(d-1)}{\Tr(L)} \;\;=\;\; \frac{\lambda_2(L)(d-1)}{\sum_{j = 1}^n \ell_j } \;. \end{aligned}$$ The accuracy also depends on the topology via the maximum weighted degree defined as $D_{\max} \equiv \max_{i \in [d]} D_{ii} = \max_{i \in [d]} \{ \sum_{j: i \in S_j} \ell_j/\kappa_j\}$. Note that the average weighted degree is $\sum_i D_{ii}/ d = \Tr(L)/d$, and we rescale it by $D_{\rm max}$ such that $$\begin{aligned}
\label{eq:lambda2_L1beta}
\beta &\equiv& \frac{\Tr(L)}{d D_{\max}} \;\;=\;\; \frac{\sum_{j = 1}^n \ell_j }{d D_{\max}} \;. \end{aligned}$$ We will show that the performance of rank breaking estimator depends on the topology of the graph through these two parameters. The larger the spectral gap $\alpha$ the smaller error we get with the same effective sample size. The degree imbalance $\beta\in[0,1]$ determines how many samples are required for the analysis to hold. We need smaller number of samples if the weighted degrees are balanced, which happens if $\beta$ is large (close to one).
The following quantity also determines the convexity of the objective function. $$\begin{aligned}
\label{eq:gamma_def}
\gamma \;\equiv\; \min_{j \in [n]} \Bigg\{ \Bigg(1 - \frac{p_{j,\ell_j}}{\kappa_j} \Bigg)^{{\left \lceil{2e^{2b}} \right \rceil}-2}\Bigg\} \;.\;
\end{aligned}$$ Note that $\gamma$ is between zero and one, and a larger value is desired as the objective function becomes more concave and a better accuracy follows. When we are collecting data where the size of the offerings $\kappa_j$’s are increasing with $d$ but the position of the separators are close to the top, such that $\kappa_j = \omega(d)$ and $p_{j,\ell_j} = O(1)$, then for $b=
| 2,679
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and let $P_{\mathcal{L}}\to M$ denote the principal $GL_1R$-bundle defined by the classifying map of ${\mathcal{L}}$, $\gamma_{\mathcal{L}}: M \to BGL_1(R)$. Let $\oplus_n P_{\mathcal{L}}$ denote the principal bundle classified by $$M {\xrightarrow}{\Delta} \prod_n M {\xrightarrow}{\gamma_{\mathcal{L}}^n } \prod_n BGL_1(R) {\xrightarrow}{\mu} BGL_n(R).$$ Here $\mu$ is the usual block addition. This is the principal $GL_n(R)$-bundle associated to the $n$-fold Whitney sum $\oplus_n {\mathcal{L}}$.
Combining equivalence (\[autgl\]) with Lemma \[GLN\] defines an equivalence of group-like $A_\infty$-spaces, $$\begin{CD}
\Psi_n : {\mathcal{G}}(\oplus_n P_{\mathcal{L}}) @>\phi > \simeq > hAut^R(\oplus_n {\mathcal{L}}) @>\lambda >\simeq > GL_n(End^R({\mathcal{L}})).
\end{CD}$$
Notice that these equivalences respect the standard inclusions of wreath products, that makes the following diagrams commute:
$$\label{wreath}
\begin{CD}
\Sigma_k \int {\mathcal{G}}(\oplus_n P_{\mathcal{L}}) @>{\hookrightarrow}>> {\mathcal{G}}((\oplus_{nk} P_{\mathcal{L}}) \\
@V 1 \times \Psi_n^k VV @VV\Psi_{nk} V \\
\Sigma \int (GL_n(End^R({\mathcal{L}}))) @>>{\hookrightarrow}> GL_{nk}(End^R({\mathcal{L}})).
\end{CD}$$
Using the Barratt-Eccles $E_\infty$-operad, we can conclude the following:
The disjoint unions $\coprod_{n \geq 0} B{\mathcal{G}}(\oplus_n P_{\mathcal{L}})$ and $\coprod_{n\geq 0} BGL_n(End^R({\mathcal{L}}))$ have $E_\infty$-algebra structures, and the equivalence $$\sqcup B\Psi_n : \coprod_{n \geq 0} B{\mathcal{G}}(\oplus_n P_{\mathcal{L}}) \to \coprod_{n\geq 0} BGL_n(End^R({\mathcal{L}}))$$ is an equivalence of $E_\infty$-spaces.
We are now ready to deduce $K$-theoretic information from the above analysis.
If $X$ is an $E_\infty$ space, let $X^+$ denote the corresponding infinite loop space given by its group completion. In other words, $$X^+ = \Omega B(X).$$
Given a ring spectrum $S$, let $S_0$ denote its connective cover. We denote by $K_{conn}(S)$ the algebraic $K$-theory spe
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\{4,5\}\}$. Denote $\cR^*=\{R\in\cR\mid R\subseteq X^*\}$. Since $|\cR^*|\le 4<|Y|+5$, we know that $\cR\setminus\cR^*\ne\emptyset$.\
For each $x\in X^*$ we set $\hat{x}$ to be the integer and $C_x$ to be the $2$-set such that $\{\{x,\hat{x}\},C_x\}=\{\{2,3\},\{4,5\}\}$ (so, in particular, $C_x=C_{\hat{x}}$). Also define $Y_x=\{y\in Y\mid \{1,x,y\}\in\cS\}$. For $i\in\{2,3\}$ and $j\in\{4,5\}$ let $\cR(i,j)=\{R\in\cR\mid R\cap X^*=\{i,j\}\}$ and let $R(i,j)=\{y\mid \{i,j,y\}\in\cR(i,j)\}$. Note that $R(i,j)\sse Y$. The following properties are easy to see.\
1. \[prop:partition\] The collection $\{\cR(i,j)\mid i\in\{2,3\},j\in\{4,5\}\}$ partitions $\cR\setminus\cR^*$; in particular, at least one of these sets is nonempty.
2. \[prop:sstar\] If $\{1,\hat{i},\hat{j}\}\in\cS^*$ then $\cR(i,j)=\emptyset$. (Since no element of $\cR(i,j)$ intersects $\{1,\hat{i},\hat{j}\}$.)
3. \[prop:rrstar\] $|\cS^*|\le 3$. (This follows from \[prop:partition\] and \[prop:sstar\].)
4. \[prop:qij\] If $\min(|R(i,j)|,|R(\hat{i},\hat{j})|)\ge 1$ then $R(i,j)=R(\hat{i},\hat{j})$ with $|R(i,j)|=1$. Therefore if $\min(|\cR(i,j)|,$ $|\cR(\hat{i},\hat{j})|)\ge 1$ then $|\cR(i,j)|=|\cR(\hat{i},\hat{j})|)=1$. (Since elements of $\cR(i,j)$ and $\cR(\hat{i},\hat{j}) $ can intersect in at most one element.)
5. \[prop:bi\] If $X^*\setminus\{x\}\in\cR^*$ then $Y_x=\emptyset$. (Since $X^*\setminus\{x\}$ does not intersect sets of the form $\{1,x,y\}$ for $y\in Y$.)
6. \[prop:ysmall\] If $y\in Y_x$ then, for $j\in C_x$, we have $\cR(\hat{x},j)\subseteq\{\{\hat{x},j,y\}\}$. (Since $\{1,x,y\}\in\cS$.)
7. \[prop:ylarge\] If $|Y_x|\ge 2$ then $\bigcup_{j\in C_x} \cR(\hat{x},j)=\emptyset$. (This follows from \[prop:ysmall\].)
8. \[prop:ymin\] Since $\cR^*\ne\cR$, we have $\min(|Y_x|,|Y_{\hat{x}}|)\le 1$ for every $x\in X^*$. (This follows from \[prop:partition\] and \[prop:ylarge\].)
9. \[prop:last\] If $|\cS^*|=3$ then $\cS_x^*\ne\emptyset$ for all $x\in X^*$.
If, for some $x\in X^*$, we have $|Y_x|\ge 2$, and $|Y_{\hat{x}}|= 1$, then (f
| 2,681
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dictive densities with superharmonic priors.
Let $\ph_\pi=\ph_\pi(Y|X)$ be a Bayesian predictive density with respect to a prior $\pi(\Th)$, where $\pi(\Th)$ is twice differentiable and the marginal density $m_\pi(X;v_x)$ is finite. All the results in this section are based on the following key lemma.
\[lem:superharmonic\] Denote by $\nabla_\Th=(\partial/\partial\th_{ij})$ the $r\times q$ differentiation operator matrix with respect to $\Th$. Then $\ph_\pi$ is minimax relative to the KL loss $(\ref{eqn:loss})$ if $\pi(\Th)$ is superharmonic, namely, $$\tr[\nabla_\Th\nabla_\Th^\top \pi(\Th)]=\sum_{i=1}^r\sum_{j=1}^q\frac{\partial^2 \pi(\Th)}{\partial \th_{ij}^2}\leq 0.$$
[**Proof.**]{} This lemma can be proved along the same arguments as in Stein (1981). See also George et al. (2006) and Brown et al. (2008). $\Box$
Define a class of prior densities as $$\pi(\Th)=g(\Si),\quad \Si=\Th\Th^\top,$$ where $g$ is twice differentiable with respect to $\Si$. Let $\Dc_\Si$ be an $r\times r$ matrix of differential operator with respect to $\Si=(\si_{ij})$ such that the $(i,j)$ element of $\Dc_\Si$ is $$\{\Dc_\Si\}_{ij}=\frac{1+\de_{ij}}{2}\frac{\partial}{\partial \si_{ij}},$$ where $\de_{ij}$ stands for the Kronecker delta. Let $$G=(g_{ij})=G(\Si)=\Dc_\Si g(\Si),$$ namely, $G$ is an $r\times r$ symmetric matrix such that $g_{ij}=\{\Dc_\Si\}_{ij} g(\Si)$.
\[lem:condition1\] $\ph_\pi$ with respect to $\pi(\Th)=g(\Si)$ is minimax relative to the KL loss $(\ref{eqn:loss})$ if $$\tr[\nabla_\Th\nabla_\Th^\top\pi(\Th)]=2[(q-r-1)\tr(G)+2\tr(\Dc_\Si \Si G)]\leq 0,$$ where $G=\Dc_\Si g(\Si)$.
[**Proof.**]{} Using (i) and (ii) in Lemma \[lem:diff4\] gives that $$\begin{aligned}
\tr[\nabla_\Th\nabla_\Th^\top \pi(\Th)]
&=2\tr(\nabla_\Th \Th^\top \Dc_\Si g(\Si))=2\tr(\nabla_\Th \Th^\top G)\\
&=2\big[(q-r-1)\tr(G)+2\tr(\Dc_\Si \Si G)\big].\end{aligned}$$ From Lemma \[lem:superharmonic\], the proof is complete.
Let $\la_1,\ldots,\la_r$ be ordered eigenvalues of $\Si=\Th\Th^\top$, where $\la_1\geq\cdots\geq\la_r$, and let $\La
| 2,682
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gebra $\C[\gl_n(\F_q)]$.
If $\mu=(\mu_1,\mu_2,\dots,\mu_r)$ is a partition of $n$, an irreducible character of $\GL_n(\F_q)$ is said to be of type $\mu$ if it is of the form $R_{L_\mu}^{GL_n}(\alpha)$ where $L_\mu=\GL_{\mu_1}\times\GL_{\mu_2}\times\cdots\times\GL_{\mu_r}$ and where $\alpha$ is a *regular* linear character of $L_\mu(\F_q)$, see §\[applichar\] for definitions. Characters of this form are called *semisimple split*.
In [@hausel-letellier-villegas] we prove that for a generic tuple $(\calX_1,\dots,\calX_k)$ of semisimple split irreducible characters of $\GL_n(\F_q)$ of type $\muhat$, we have $$\langle
\Lambda\otimes\calX_1\otimes\cdots\otimes\calX_k,1\rangle
=\H_\muhat(0,\sqrt{q})\label{multi0}.$$ Note that in particular this implies that the left hand side only depends on the combinatorial type $\muhat$ not on the specific choice of characters.
Together with Formula (\[purity0\]) we deduce the following formula.
We have $$\langle
\Lambda\otimes\calX_1\otimes\cdots\otimes\calX_k,1\rangle=A_\muhat(q).$$
Using Kac’s results on quiver representations (see §\[genquiv\]) the above theorem has the following consequence.
Let $\Phi(\Gamma_\muhat)$ denote the root system associated with $\Gamma_\muhat$ and let $(\calX_1,\dots,\calX_k)$ be a generic $k$-tuple of irreducible characters of $\GL_n(\F_q)$ of type $\muhat$.
We have $\langle
\Lambda\otimes\calX_1\otimes\cdots\otimes\calX_k,1\rangle\neq 0$ if and only if $\v_\muhat\in\Phi(\Gamma_\muhat)$. Moreover $\langle
\Lambda\otimes\calX_1\otimes\cdots\otimes\calX_k,1\rangle=1$ if and only if $\v_\muhat$ is a real root. \[coromulti0\]
In [@letellier] the second author proves that Corollary \[coromulti0\] extends to any type of generic tuples of irreducible characters of $\GL_n(\F_q)$ (not necessarily semisimple split).
Recall that there is a natural parametrization, $\mu\mapsto
\calU_\mu$, of the unipotent characters of $\GL_n(\F_q)$ by partitions of $n$, fixed by requiring that $\calU_{(1^n)}$ is trivial. Using again quiver representations
| 2,683
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Gâteaux differential with the $H^2$ inner product , integrating by parts and using , we obtain the required $H^2$ gradient $\nabla\R$ as a solution of the following elliptic boundary-value problem $$\begin{aligned}
&\left[ {\operatorname{Id}}\, - \,\ell_1^2 \,\Delta + \,\ell_2^4 \,\Delta^2 \right] \nabla\R
= \nabla^{L_2} \R \qquad \text{in} \ \Omega, \\
& \text{Periodic Boundary Conditions}.
\end{aligned}
\label{eq:gradRH2}$$ The gradient fields $\nabla^{L_2}\R({\mathbf{u}})$ and $\nabla\R({\mathbf{u}})$ can be interpreted as infinite-dimensional sensitivities of the objective function $\R({\mathbf{u}})$, cf. , with respect to perturbations of the field ${\mathbf{u}}$. While these two gradients may point towards the same local maximizer, they represent distinct “directions”, since they are defined with respect to different topologies ($L_2$ vs. $H^2$). As shown by @pbh04, extraction of gradients in spaces of smoother functions such as $H^2(\Omega)$ can be interpreted as low-pass filtering of the $L_2$ gradients with parameters $\ell_1$ and $\ell_2$ acting as cut-off length-scales and the choice of their numerical values will be discussed in §\[sec:3D\_InstOpt\_E\].
The step size $\tau_n$ in algorithm is computed as $$\label{eq:tau_n}
\tau_n = \mathop{{\operatorname{argmax}}}_{\tau>0} \left\{ \R\left[\mathbb{P}_{\mathcal{S}_{\E_0}}
\left( \;{\mathbf{u}}^{(n)} + \tau\,\nabla\R({\mathbf{u}}^{(n)}) \;\right)\right] \right\}$$ which is done using a suitable derivative-free line-search algorithm [@r06]. Equation can be interpreted as a modification of a standard line search method where the optimization is performed following an arc (a geodesic) lying on the constraint manifold $\mathcal{S}_{\E_0}$, rather than a straight line. This approach was already successfully employed to solve similar problems in @ap11a [@ap13a].
It ought to be emphasized here that the approach presented above in which the projections – and gradients – are obtained based on the infinite-dimensional (continuous) formulation to be d
| 2,684
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{k}}\partial
^{\alpha }f+\sum_{\substack{ (\beta ,\gamma )=\alpha \\ \left\vert \beta
\right\vert \geq 1}}c(\beta ,\gamma )\partial ^{\beta }\Big(\frac{1}{\psi
_{k}}\Big)\partial ^{\gamma }f.$$This, together with (\[n2\]) implies $$\Big\vert \partial ^{\alpha }\Big(\frac{f}{\psi _{k}}\Big)\Big\vert \leq
C\sum_{0\leq \left\vert \gamma \right\vert \leq \left\vert \alpha
\right\vert }\frac{1}{\psi _{k}}\left\vert \partial ^{\gamma }f\right\vert$$so the first inequality in (\[n3\]) is proved. In order to prove the second inequality we proceed by recurrence on $q$. The inequality is true for $q=0.$ Suppose that it is true for $q-1.$ Then we write$$\frac{1}{\psi _{k}}\partial ^{\alpha }f=\partial ^{\alpha }\Big(\frac{f}{%
\psi _{k}}\Big)-\sum_{\substack{ (\beta ,\gamma )=\alpha \\ \left\vert
\beta \right\vert \geq 1}}c(\beta ,\gamma )\partial ^{\beta }\Big(\frac{1}{%
\psi _{k}}\Big)\partial ^{\gamma }f$$and we use again (\[n2\]) in order to obtain$$\frac{1}{\psi _{k}}\left\vert \partial ^{\alpha }f\right\vert \leq \Big\vert %
\partial ^{\alpha }\Big(\frac{f}{\psi _{k}}\Big)\Big\vert +C\sum_{\left\vert
\gamma \right\vert <\left\vert \alpha \right\vert }\frac{1}{\psi _{k}}%
\left\vert \partial ^{\gamma }f\right\vert \leq C\sum_{0\leq \left\vert
\beta \right\vert \leq q}\Big\vert \partial ^{\beta }\Big(\frac{f}{\psi _{k}}%
\Big)\Big\vert$$the second inequality being a consequence of the recurrence hypothesis. $%
\square $
The assertion is false if we define $\psi _{k}(x)=(1+\left\vert x\right\vert
)^{k}$ because $\partial _{i}\partial _{j}\left\vert x\right\vert =\frac{%
\delta _{i,j}}{\left\vert x\right\vert }-\frac{x_{i}x_{j}}{\left\vert
x\right\vert ^{2}}$ blows up in zero.
We look now to $\psi _{k}$ itself.
\[Psy2\]For every multi-index $\alpha $ there exists a constant $%
C_{\alpha }$ such that $$\left\vert \partial ^{\alpha }\psi _{k}\right\vert \leq C_{\alpha }\psi _{k}.
\label{n4}$$Moreover, for every $q$ there is a constant $C_{q}\geq 1$ such that for every $f\in C_{b}^{\infty }({\mathbb{R}}^{d})$$$\frac{1}{
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\mathcal{O}_{S_3}$, and $\mathcal{O}_{S_3,\zeta^2} = \langle x_3^2, x_3y_3, y_3^2 \rangle\mathcal{O}_{S_3}$.
For $P_2 = [0:1:0]$ one uses the change of coordinates on $G_\lambda(x,1,z)$, $(x, z) = (x_2, z_2-x_2)$ to get a polynomial of the form $\beta_\lambda x_2^4 + z_2^3 + \sum_{ij} b_{\lambda ij} x_2^i z_2^j$ with $3i+4j>12$ and $\beta_\lambda \neq 0$. The blow-up with respect to $(3,4)$ resolves the singular point. The exceptional divisor is denoted by $E_2$ and $m_2 = 12$, $\nu_2 = 7$. The second summand in is $$\label{eq:M_P2}
\frac{\mathcal{O}_{\CC^2}}{\mathcal{M}_{\mathcal{C}_{\lambda},P_2}^{(k)}}
= \frac{\mathcal{O}_{\CC^2}}{ \left\{ g \ \big| \ \operatorname{mult}_{E_3} \pi^{*} g > k - 7 \right\}} \cong
\begin{cases}
0 & \text{ if }\ \ k = 0,\ldots,6, \\
\CC & \text{ if } \ \ k = 7,8,9, \\
\langle 1,x_2 \rangle_\CC & \text{ if } \ \ k = 10, \\
\langle 1,x_2,z_2 \rangle_\CC & \text{ if } \ \ k = 11.
\end{cases}$$
Finally, for $P_1 = [1:0:0]$ one blows up $G_\lambda(1,y_1, \lambda^2(z_1-y_1^3)) =
\alpha_\lambda y_1^{10} + z_1^3 + \sum_{ij} a_{\lambda ij} y_1^i z_1^j$ with $3i+10j>30$ and $\alpha_\lambda \neq 0$ with weight vector $(3,10)$ to resolve the singular point. The exceptional divisor $E_1$ gives rise to $m_1=30$ and $\nu_1=13$. The first vector space in is $$\label{eq:M_P1}
\frac{\mathcal{O}_{\CC^2}}{ \left\{ g \ \big| \ \operatorname{mult}_{E_3} \pi^{*} g > \dfrac{5k}{2} - 13 \right\}}
\cong
\begin{cases}
0 & \text{ if } \ \ k = 0,\ldots,5, \\
\CC & \text{ if } \ \ k = 6, \\
\langle 1,y_1 \rangle_\CC & \text{ if } \ \ k = 7, \\
\langle 1,y_1,y_1^2 \rangle_\CC & \text{ if } \ \ k = 8, \\
\langle 1,y_1,y_1^2,y_1^3 \rangle_\CC & \text{ if } \ \ k = 9, \\
\langle 1,y_1,y_1^2,y_1^3,y_1^4,z_1 \rangle_\CC & \text{ if } \ \ k = 10, \\
\langle 1,y_1,y_1^2,y_1^3,y_1^4,z_1, y_1 z_1 \rangle_\CC & \text{ if } \ \ k = 11.
\end{cases}$$ Using the previous calculations one can check that $H^1(Y,\cO_Y(L^{(k)})) = 0$ for any $k \neq 10$ and $\lambda = 1,\zeta$ by studying the cokernel of $\pi^{(k)}$.
To
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d:1985] for hyperbolic systems of equations (\[eq:2.1\]) allows the construction of certain classes of $k$-wave solutions admitting $k$ arbitrary functions of one variable. The replacement of the matrix of derivatives $u_i^\alpha$ in the system of equations (\[eq:2.1\]) by the simple real element $L_i^\alpha$ allows us to construct more general classes of solutions by replacing the real elements with complex ones. A specific form of solution $u(x)$ of an elliptic system (\[eq:2.1\]) is postulated for which the tangent mapping $du(x)$ is a sum of a complex element and its complex conjugate
\[eq:2.4\] du\^(x)=(x)\^(u)\_i(u)dx\^i+|(x)|\^(u)|\_i(u)dx\^i,=1,…,q,
where $\gamma=(\gamma^1,\ldots, \gamma^q)\in{\mathbb{C}}^q$ and $\lambda=(\lambda_1,\ldots,\lambda_p)\in {\mathbb{C}}^p$ satisfy
\[eq:star\] \_i\_\^[i]{}(u)\^=0,|\_i\_\^[i]{}(u)|\^=0.
Here the quantity $\xi(x)\neq 0$ is treated as a complex function of the real variables $x$. In what follows we assume that the vectors $\gamma$ and $\bar{\gamma}$ are linearly independent. The proposed form of solution (\[eq:2.4\]) is more general than the one proposed in [@Jeffrey:1976; @Rozdestvenski:1983] for which the derivatives $u^\alpha_i$ are represented by a real simple element leading to a simple Riemann wave solution. To distinguish this situation from the one proposed in (\[eq:2.4\]), we call the real-valued solution associated with a complex element and its complex conjugate a simple mode solution in accordance with [@Perad:1985]. This means that all first-order derivatives of $u^\alpha$ with respect to $x^i$ are decomposable in the following way
\[eq:star2\] =(x)\^(u)\_i(u)+|(x)|\^(u)|\_i(u),
or alternatively $$\frac{{\partial}u^\alpha}{{\partial}x^i}=i{\left( \xi(x) \gamma^\alpha(u)\lambda_i(u)-\bar{\xi}(x)\bar{\gamma}(u)^\alpha\bar{\lambda}_i(u) \right)},$$where a set of functions $(\lambda,\gamma)$ and their conjugates $(\bar{\lambda},\bar{\gamma})$ on $U$ satisfy the wave relation (\[eq:star\]). Similarly, as in the case of $k$-waves, for hyperbolic sys
| 2,687
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mathcal{A}^{c \bar a}(w) \right. \cr
& \qquad + \left. j^b_{L,z}(x)
\left( \frac{c_+}{c_++c_-} \frac{i{f^{ac}}_d
\mathcal{A}^{d \bar a}(w)}{z-w} + ... \right) \right] \cr
%
& = -\frac{c_-}{c_+} \left[ \frac{c_1 \mathcal{A}^{a \bar a}(w)}{(z-w)^2} + \left(-c_2 + \frac{i c_+}{c_++c_-} \right) \frac{{f^a}_{bc}:j^b_{L,z}\mathcal{A}^{c \bar a}:(w)}{z-w} \right. \cr
& \qquad \left. - (c_2-g) \frac{{f^a}_{bc}:j^b_{L,\bar z}\mathcal{A}^{c \bar a}:(w)(\bar z - \bar w)}{(z-w)^2} + ... \right]\end{aligned}$$ In principle the second- and first-order poles that we obtain in the last line may receive corrections from the lower-order terms that we neglected in the penultimate line. We will now argue that it is not the case. Let us consider the first term in the last line (the second-order pole). This term may receive corrections of the form ${T^a}_b \mathcal{A}^{b \bar a}$, where the tensor ${T^a}_b$ contains at least one structure constant. Such a tensor vanishes by using properties of the Lie super algebras under consideration [@Bershadsky:1999hk]. Let us now consider the second term (the holomorphic simple pole). It could receive corrections of the form ${T^a}_{bc}:j^b_{L,z}\mathcal{A}^{c \bar a}$, where ${T^a}_{bc}$ contains at least two structure constants. Again, according to [@Bershadsky:1999hk], this tensor vanishes because traceless four-tensors invariantly contracted with structure constants over two indices vanish. The third term receives no higher order corrections for the same reason. Thus the terms written in the last line of are not corrected. Using equations and we finally obtain: \[jLjR1\] j\^a\_[L,z]{}(z) j\^[|a]{}\_[R,z]{}(w) = ( + + ) + ... where the ellipses refer to terms of order zero or more in the distance between the two operators. Similarly we can compute: $$\begin{aligned}
\label{jLjR2}
j^a_{L,\bar z}(z) j^{\bar a}_{R,\bar z}(w) &=
\frac{c_+ c_-}{c_++c_-}\left( \frac{\mathcal{A}^{a \bar a}(w)}{(\bar z-\bar w)^2}
+ \frac{c_+}{c_++c_-} \frac{\partial \mathcal{A}^{a \bar a}(w)(z-w)}{(\bar z-\bar w
| 2,688
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y Eq. , ${\mathrm{ch}\,M}^\chi (\Lambda )$ does not depend on $\Lambda $. Hence by Prop. \[pr:VTMiso\] we may assume that ${\hat{T}}_p$ is an isomorphism. Then Eq. follows from Eq. .
\[le:chsub\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ and $t\in \{1,2,\dots ,{b}-1\}$. Assume that $\Lambda (K_pL_p^{-1})=\chi ({\alpha }_p,{\alpha }_p)^{t-1}$. Then $V=U^-(\chi )F_p^t {\otimes }{\mathbb{K}}_\Lambda $ is a $U(\chi ){\otimes }{\mathbb{K}}$-submodule of $M^\chi (\Lambda )$ with $$\fch{V}=\fch{M^\chi }(\Lambda )
\frac{e^{-t{\alpha }_p}-e^{-{b}{\alpha }_p}}{1-e^{-{b}{\alpha }_p}}.$$
The formal character of the subspace $\oplus _{m=t}^{{b}-1}F_p^m$ of $M^\chi (\Lambda )$ is $$e^{-t{\alpha }_p}+e^{-(t+1){\alpha }_p}+\cdots +e^{-({b}-1){\alpha }_p}=
\frac{e^{-t{\alpha }_p}-e^{-{b}{\alpha }_p}}{1-e^{-{\alpha }_p}}.$$ Thus the lemma is a consequence of Lemmata \[le:hwvector\], \[le:MLiso\] and Eqs. , .
*In the rest of this section assume that $\chi \in {\mathcal{X}}_4$.* Let $n=|R^\chi _+|$ and $i_1,\dots ,i_n\in I$ with $\ell (1_\chi {\sigma }_{i_1}\cdots \s_{i_n})=n$. Recall the definitions of $\beta _\nu $ and $F_{\beta _\nu }$, where $1\le \nu \le n$, from Eq. . Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. We characterize irreducible Verma modules (see also Lemma \[le:subfch\]).
\[le:Fhweight\] Let $m\in \{0,1,\dots ,n\}$, $\chi '=r_{i_m}\cdots r_{i_2}r_{i_1}(\chi )$ and $w={\sigma }_{i_m}\cdots {\sigma }_{i_2}{\sigma }_{i_1}^\chi $. Then $$\begin{aligned}
K_{\alpha }&L_{\alpha }^{-1}
F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots
F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda \\
&=\frac{{\rho ^{\chi}} ({\alpha })}{{\rho ^{\chi '}}(w({\alpha }))}
\Lambda (K_{{\alpha }}L_{{\alpha }}^{-1})
F_{\beta _m}^{{b^{\chi}} (\beta _m)-1}\cdots
F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda .
\end{aligned}$$ for all ${\alpha }\in {\mathbb{Z
| 2,689
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g the two parts, we now have ${\mathit{u}}_{i - n} = {\mathit{v}}_{i - n}$ for each $i$. In other words, the two localized predecessor walks are actually the same walk: ${\mathit{u}} = {\mathit{v}}$. Thus $\test$ is a bijection.
#### Volatile variables preservation {#S:VOLATILE_VARIABLES}
The danger in reversed thinking about tests is inadvertently conceptualizing volatile variables as free. This is untrue, as the volatile variables at any stage of a predecessor chain are fixed, and the [next]{} stage considers the set of what previous conditions may have led to the current stage. Thus, predecessor walks are chains of a poset of steps, which include the settings of volatile variables. One must be mindful to reproduce all volatile stimuli of the localized predecessor walk in its analogous test.
### Outcome {#S:OUTCOME}
The outcome of a test, pass or fail, will be regarded as a Bernoulli random event, $P_\rho = \rho^n {(1 - \rho)}^{1 - n}$, for $n = 1$ (pass) or $n = 0$ (fail). These probabilities are statistically independent of the bias involved with drawing the sample from the operational profile. This bias affects the origin of discovered failures, but not how many failures are found. In other words, the total statistical power of the sampling plan is not affected by sampling bias.
Sums of independent Bernoulli random variables are binomial. That is, the probability of finding $n$ failures collectively among $N$ sample items is binomial, $\binom{N}{n} \rho^n {(1 - \rho)}^{N - n}$.
### Physics {#S:PHYSICS}
In the real world, tests pass or fail depending on whether the information transduced at step ${\mathit{s}}_\text{crux}$ meets all safety constraints. Such engineering requirements are varied, ultimately involving position, timing, voltage, insulation, dimensional tolerance, toxicity, temperature, mechanical shielding, luminosity, and hydrostatic pressure – just to name a few areas. Review of a test offers a last chance to discover a missed constraint (requirement). Another possibility is that the cha
| 2,690
| 6,086
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chouk polynomial of order $s$ over $\Z_2^k$. For simplicity we assume that $k$ is odd.
1. For any $x \in \Z_2^k$ with $|x|=i$, $\sum_{z \in \Z_2^k\\|z|=s}(-1)^{<x,z>} = K_s^{(k)}(i)$.
2. $\sum_{s=0}^l K_s^{(k)}(i) = K_l^{(k-1)}(i-1)$.
3. For any $s$ and $k$, $\max_{i=0,\dots,n} |K_s^{(k)}(i)| = K_s^{(k)}(0) = {k \choose s}$.
Observe that $G$ is a Cayely graph for the group $\Z_2^k$ with generator set $\{g \in \Z_2^k : |g| \leq \frac k 2\}$. Since $\Z_2^k$ is abelian, the eigenvectors of the graphs are independent of the generators, and are simply the characters of the group written as the vector of their values. Namely, corresponding each $y \in \Z_2^k$ we have an eigenvector $v^y$, such that $v^y_x = (-1)^{<x,y>}$. For every $y$, $v^y_0 = 1$, so to figure out the eigenvalue corresponding to $v^y$, we simply need to sum the value of $v^y$ on the neighbors of $0$. Note that for $y=0$ we get the all $1$s vector, which corresponds to the largest eigenvalue. So we are interested in $y$’s such that $|y| > 0$. By the first two facts above we have: $$\lambda_y = \sum_{g \in \Z_2^k, |g| \leq \frac k 2}(-1)^{<y,g>} = \sum_{s=0}^{\frac {k-1} 2} K_s^{(k)}(|y|) = K_{\frac {k-1} 2}^{(k-1)}(|y|-1).$$ By the third fact, this is at most ${{k-1} \choose {\frac {k-1} 2}}
\approx \frac {2^{k-1}} {\sqrt{k-1}} =
o(2^{k-1})$.
Graphs with bounded $\lambda_2$ {#lambda2-cons}
===============================
Theorem \[our-bound\] suggests families of graphs that have linear sphericity. Namely, for $0 < \delta \leq {\frac 1 2}$, and $\lambda_2 > 0$, the theorem says that $\delta n$-regular graphs with second eigenvalue at most $\lambda_2$ have linear sphericity. In this section we characterize such graphs. We prove that for $\delta = {\frac 1 2}$ such graphs are nearly complete bipartite, and that for other values, only finitely many graphs exist.\
It is worth noting that graphs with bounded second eigenvalue have been previously studied. The apex of these works is probably that of Cameron, Goethals, Seidel and Shult, who charac
| 2,691
| 2,026
| 2,989
| 2,595
| 3,171
| 0.774311
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rm a $K^G$-basis of $K$. Then any $r\in R$ can be written as $$r=\sum_i a_i r_i{\quad\mbox{where $a_i\in K^G$.}\quad}$$ Applying any $g\in G$ to it, we get a system of equations $$\sum_i g(r_i)a_i= g(r){\quad\mbox{for $g\in G$.}\quad}$$ We can view these as linear equations with unknowns $a_i$. The system determinant is $D:=\det_{i,g}\bigl(g(r_i)\bigr)$, which is nonzero since its square is the discriminant of $K/K^G$. $D$ is $G$-invariant up to sign, thus $D^2$ is $G$-invariant hence in $R^G$. By Kramer’s rule, $a_i\in D^{-2}R^G$, hence $R\subset D^{-2}\sum_i r_i R^G$.
In the opposite case, when the equivalence relation is nontrivial only on a proper subscheme, we have the following general result.
\[quot.by.pushout\] Let $X$ be a reduced scheme, $Z\subset X$ a closed, reduced subscheme and $R\rightrightarrows X$ a finite, set theoretic equivalence relation. Assume that $R$ is the identity on $R\setminus Z$ and that the geometric quotient $Z/R|_Z$ exists. Then $X/R$ exists and is given by the universal push-out diagram $$\begin{array}{ccc}
Z & \into & X\\
\downarrow && \downarrow \\
Z/R|_Z & \into & X/R.
\end{array}$$
Proof. Let $Y$ denote the universal push-out (\[glue.thm.asp\]). Then $X\to Y$ is finite and so $X/R$ exists and we have a natural map $X/R\to Y$ by (\[quot.X/S.finite.lem\]). On the other hand, there is a natural map $Z/R|_Z\to X/R$ by (\[quot.of.sub\]), hence the universal property of the push-out gives the inverse $Y\to X/R$.
Inductive plan for constructing quotients {#induct.plan.sect}
=========================================
\[up-down.eq.defn\] Let $R\rightrightarrows X$ be a finite, set theoretic equivalence relation and $g:Y\to X$ a finite morphism. Then $$g^*R:= R\times_{(X\times X)}(Y\times Y)\rightrightarrows Y$$ defines a finite, set theoretic equivalence relation on $Y$. It is called the [*pull-back*]{} of $R\rightrightarrows X$. (Strictly speaking, it should be denoted by $(g\times g)^*R$.)
Note that the $g^*R$-equivalence classes on the geometric points of $Y$ map injectivel
| 2,692
| 2,216
| 2,504
| 2,432
| 2,557
| 0.778884
|
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xpression as that valid for the standard asymptotic behavior. The Virasoro charges are then easily integrated to yield $$\label{once}Q_{\pm}[T^{\pm}]=\frac{2}{l}\left( 1\pm\frac{1}{\mu l}\right)
\int T^{\pm}f_{\pm\pm}d\phi\$$ (up to additive constants). The details will be given in [@HMTfuture]. What happens is that the diverging pieces associated with the slower fall-off $h_{--}$ or $h_{++}$ disappear in $\delta Q_{\pm}[T^{\pm}]$ so that $Q_{\pm}$ is given by (\[once\]), and hence the charges acquire no correction involving the terms associated with the relaxed behavior. One can then view $h_{--}$ (or $h_{++}$), which cannot be gauged away, as defining a kind of hair.“ This situation is analogous to the one found for a scalar field with mass $m$ in the range $m_{ \rm BF}^{2}<m^{2}<m_{\rm BF}^{2}+1/l^{2}$ (where $m_{\rm BF}$ is the Breitenlohner-Freedman bound [@BF]). There are then two possible admissible behaviors (two branches”) for the scalar field, and the analysis proceeds as here when only the branch with slower behavior is switched on [@HMTZ3][^1].
Under an asymptotic conformal transformation (\[Asympt KV\]), $f_{++}$ and $f_{--}$ are straightforwardly found to transform as $$\begin{aligned}
\delta_{\eta}f_{++} & =2f_{++}\partial_{+}T^{+}+T^{-}\partial_{-}%
f_{++}+T^{+}\partial_{+}f_{++}
-l^{2}\left( \partial_{+}T^{+}+\partial_{+}^{3}T^{+}\right)
\!/2\ ,\label{deltaf++}\\
\delta_{\eta}f_{--} & =2f_{--}\partial_{-}T^{-}+T^{-}\partial_{-}%
f_{--}+T^{+}\partial_{+}f_{--}
-l^{2}\left( \partial_{-}T^{-}+\partial_{-}^{3}T^{-}\right) \!/2\ .
\label{deltaf--}%\end{aligned}$$ On shell, one verifies that $$\partial_{+}f_{--}=0=\partial_{-}f_{++}%$$ and so (\[deltaf++\]) and (\[deltaf–\]) reduce to $$\begin{aligned}
\delta_{\eta}f_{++} & =2f_{++}\partial_{+}T^{+}+T^{+}\partial_{+}%
f_{++}-\frac{l^{2}}{2}\left( \partial_{+}T^{+}+\partial_{+}^{3}T^{+}\right)
,\label{delta2h++}\\
\delta_{\eta}f_{--} & =2f_{--}\partial_{-}T^{-}+T^{-}\partial_{-}%
f_{--}-\frac{l^{2}}{2}\left( \partial_{-}T^{-}+\p
| 2,693
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| null | null |
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|
\in{\mathfrak S}_p$ such that $\{M_{i_{\sigma(k)}}\sqcup F_{j_{k}}\ ;\ k = 1, \ldots, p \}
= \pi(w)$. Then the product $${\cal C}(\mathbb{M},\sigma,\mathbb{F})
=
x_{\mathbb{\overline{M}}}
{C}[\mathbb{M}]C^{\mathbb{M}}_{\mathbb{F}}[\sigma]C^*[\mathbb{F}]
x^*_{\overline{F}}$$ becomes a presentation of $w$. We call this presentation a [*crank form expression of $w$ defined by ${\mathbb M}$ and ${\mathbb F}$*]{}. If a crank form expression is made from sequences $(M_1, \ldots, M_u)$ and $(F_1, \ldots, F_v)$ such that
1. $M_1, \ldots, M_p$ and $F_1, \ldots, F_p$ are propagating,
2. $M_{p+1}, \ldots, M_{u}$ and $F_{p+1}, \ldots, F_{v}$ are defective.
then we call it [*in normal form*]{}.
Finally, we define the [*standard expression*]{} of $w$, as a special expression of crank form expressions in normal form by properly choosing the sequences $(M_1, \ldots, M_u)$ and $(F_1, \ldots, F_v)$. For this purpose first we sort the parts $T_1, \ldots, T_s$ of $w$ so that they satisfy:
1. $\pi(w) = \{T_1, T_2, \ldots, T_p\},$
2. $\{T_i\ |\ i = p+1, p+2, \ldots, u\}$ is the set of all upper defective parts,
3. $\{T_i\ |\ i = u+1, u+2, \ldots, u+(v-p)\}$ is the set of all lower defective parts.
For an ordered set $E$, let $\min E$ be the minimum element in $E$. Let $T_1, T_{2}, \ldots, T_{p}$ be the parts of $\pi(w)$. Define $(M_1, M_{2}, \ldots, M_{p})$ so that they satisfy $$\{M_1, M_{2}, \ldots, M_{p}\}
= \{T_1^M, T_{2}^M, \ldots, T_{p}^M\}$$ and $$\min M_1 <\min M_{2} <\cdots <\min M_{p}.$$ Similarly $(F_1, F_{2}, \ldots, F_{p})$ are defined using the lower parts of $\pi(w)$. In such a method, the sequences of the upper parts $(M_1,\ldots, M_p)$ and the lower parts $(F_1, \ldots, F_p)$ are uniquely defined from a seat-plan $w$.
Now we define $(M_{p+1}, \ldots, M_{u})$ so that they satisfy $$\{M_{p+1}, M_{p+2}, \ldots, M_{u}\}
= \{T_{p+1}, T_{p+2}, \ldots, T_{u}\}$$ and $$\min M_{p+1} <\min M_{p+2} <\cdots <\min M_{u}.$$ Similarly we define $(F_{p+1}, \ldots, F_{v})$ so that they satisfy $$\
| 2,694
| 4,172
| 3,005
| 2,457
| 1,824
| 0.785383
|
github_plus_top10pct_by_avg
|
e normalized distribution was universal, then the ratio of the standard deviation and the mean would have to obey $\sigma_i/\ev{f_i}=h$, where $h$ is a constant independent of the stock. Equivalently, a relationship $$\sigma_i \propto \ev{f_i}^\alpha
\label{eq:alpha_first}$$ would have to hold with an exponent $\alpha = 1$, at least on average. Even though one finds a monotonic dependence between the two quantities \[as shown in Fig. \[fig:distrib\](right)\], the exponent is significantly less than $1$. This means, that the ratio $\sigma/\ev{f}$ decreases with growing $\ev{f}$, i.e., the normalized distribution of $f$ is narrower for larger stocks, so their trading exhibits smaller relative fluctuations. We will return to this observation in Section \[sec:alpha\].
{height="205pt"}{height="205pt"}
Non-universality of correlations in traded value time series {#sec:correl}
============================================================
One of the classical tools of both financial analysis and physics is the measurement of the correlation properties of time series [@bouchaud.book; @stanley.book; @tumminello]. In particular, scaling methods [@dfa] have a long tradition in the study of physical systems, where the Hurst exponent $H_i$ is often calculated. For the traded value time series $f_i^{\Delta t}(t)$ of stock $i$ this is defined as $$\label{eq:hurst}
\sigma_i^2(\Delta t) = \ev{\left [f_i^{\Delta t}(t)-\ev{f_i^{\Delta t}(t)} \right ]^2}\propto\Delta t^{2H_i},$$ Note that it follows from the results of Section \[sec:value\] that the variance on the left hand side exists regardless of stock and for any window size $\Delta t$.
The measurements were carried out for all $2474$ stocks that were continuously available on the market during $1994-1995$ [^3]. Then we sorted the stocks into $6$ groups according to the order of magnitude of their average traded value: $0\leq \ev{f}\leq 10^4$, $10^4\leq \ev{f}\leq 10^5$, …, $10^8\leq\ev{f}$, all values in USD/min. Finally we average
| 2,695
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| 983
| 2,777
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| 0.780445
|
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|
tandard model of particle interactions to include an extra $U(1)_A$ gauge symmetry and an extra $U(1)_{PQ}$ global symmetry. All standard-model particles are trivial under these two new symmetries. We then introduce a new heavy quark singlet $\psi$ and two scalar singlets $\sigma$ and $\eta$ with $U(1)_A$ and $U(1)_{PQ}$ charges as shown in Table 1. All fields except $\chi$ are confined to our brane.
-------------------- ---------------------------------------- ---------- -------------
Fields $SU(3)_C \times SU(2)_L \times U(1)_Y$ $U(1)_A$ $U(1)_{PQ}$
$(u_i, d_i)_L$ (3,2,1/6) 0 0
$u_{iR}$ (3,1,2/3) 0 0
$d_{iR}$ (3,1,$-$1/3) 0 0
$(\nu_i, e_i)_L$ (1,2,$-$1/2) 0 0
$e_{iR} $ (1,1,$-$1) 0 0
$\psi_L$ (3,1,–1/3) 1 $k$
$\psi_R$ (3,1,–1/3) –1 $-k$
$(\phi^+, \phi^0)$ (1,2,1/2) 0 0
$\sigma$ (1,1,0) 2 $2k$
$\eta $ (1,1,0) 2 $2k-2$
$\chi$ (1,1,0) 0 2
--------
| 2,696
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| 1,884
| 2,945
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|
s forced to be a $G_\delta$ containing $(\delta,x_g)$. There is a cub $C_1$ such that for each $\alpha\in C_0$ and each $s\in S_{\alpha^+}$ (again, $\alpha^+$ is the minimal element of $C_1$ above $\alpha$), $s$ forces that $\dot{Z}_\alpha$ contains $\{\alpha\}\times a_s^*$. Since $S$ is ccc, the cub $C_1$ can be chosen to be a member of the PFA$(S)$ model.
We use $C_1$ to define a partition of $C_0$: for each $\alpha\in C_0$, we define $\dot{f}(\alpha)$ to equal the value $g(\alpha^+)$ (i.e. the element of $S_{\alpha^+}$ that $g$ picks). Thus if $\delta$ is a limit of $C_1$ and $s\in S_\delta$, then $s$ forces a value for $\dot{f}\!\!\upharpoonright\!\!\delta$. Then a potential normalizing expansion would consist of a sequence $\{\dot{n}_\alpha:\alpha\in C_0\}$ of $S$-names of integers for which $L_\alpha[a_{g\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]\cup(\{\alpha\}\times a_{g\upharpoonright\alpha^+}^*)$ is an open neighborhood of $\dot{Z}_\alpha$. There is a cub $C_2\subseteq C_1$ so that for each $\delta\in C_2$ and each $s\in S_\delta$, $s$ forces a value on $\dot{n}_\alpha$ for all $\alpha<\delta$. We may choose any $s_0\in g$ so that $s_0$ forces that $L_\alpha[a_{g\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]\cap L_\delta[a_{g\upharpoonright\delta^+}\setminus\dot{n}_\delta]$ is empty whenever $\dot{f}(\alpha)\neq\dot{f}(\delta)$. Working in $V[g]$, we prove there is a stationary $E$ satisfying that $L_\delta[a_{g\upharpoonright\delta^+}]\cap\bigcup\{L_\alpha[a_{g\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]:\alpha\in\delta\}$ is infinite, for all $\delta\in E$. If not, then there would be an assignment $\langle m_\delta:\delta\in C\rangle$ (for some cub $C$) so that $L_\delta[a_{g\upharpoonright\delta^+}\setminus m_\delta]$ would be disjoint from $\bigcup\{L_\alpha[a_{g\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]:\alpha\in\delta\}$, for all $\delta\in C$. Pressing down, we would arrive at a contradiction.
Let $\dot{E}$ denote the $S$-name of the stationary set whose existence was shown in
| 2,697
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| 1,982
| 2,592
| 3,986
| 0.76879
|
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|
more information on the inertia stack.) In general, points in the inertia stack are pairs $(x,\alpha)$, where $x$ is a point of $\mathfrak{X}$, and $\alpha$ is an automorphism of $x$, which for an orbifold $[Y/G]$ by $G$ a finite group, would define the twisted sectors. In the $[{\mathbb C}^3/{\mathbb Z}_3]$ example, if $g$ generates ${\mathbb Z}_3$, then the two copies of $[{\rm point}/{\mathbb Z}_3]$ correspond to $\alpha = g, g^2$. The inertia stack $I_{\mathfrak{X}}$ always contains a copy of $\mathfrak{X}$ as one component, corresponding to $\alpha = {\rm Id}$.
Let us describe how to compute the spectrum on each component $\alpha$ of $I_{\mathfrak{X}}$. (We will use $\alpha$ to denote both a component of $I_{\mathfrak{X}}$ and the automorphism defining that component.)
First, let $q: I_{\mathfrak{X}} \rightarrow \mathfrak{X}$ denote the natural projection onto a single component, and for $\alpha \neq {\rm Id}$, decompose the pullback bundles into eigenbundles[^19] of $\langle \alpha
\rangle$: $$\begin{aligned}
q^* T\mathfrak{X}|_{\alpha} & = & \oplus_n T_n^{\alpha}, \\
q^* {\cal E} |_{\alpha} & = & \oplus_n {\cal E}_n^{\alpha}.\end{aligned}$$ Define $t_{\alpha}$ to be the order of the corresponding automorphism, and take $T_n^{\alpha}$ and ${\cal E}_n^{\alpha}$ to be associated with character $$\exp(2 \pi i n / t_{\alpha}).$$ By this we mean that the (R-sector) worldsheet fermions corresponding to $T_n^{\alpha}$ and ${\cal E}_n^{\alpha}$ have boundary conditions of the form $$\psi(\sigma + 2 \pi) \: = \: \exp(2 \pi i n / t_{\alpha})
\psi(\sigma).$$ We will denote fermions couplings to $T_n^{\alpha}$ (respectively, ${\cal E}_n^{\alpha}$) by $\psi_{+,n}$ (respectively, $\lambda_{-,n}$).
Let us pause to briefly discuss some concrete examples, to illuminate these abstract definitions. For global orbifolds by finite groups, it should hopefully be clear that the description above is an abstraction of the standard prescription for distinguishing various worldsheet fermions with different boundary conditions. Let
| 2,698
| 2,487
| 2,530
| 2,323
| 2,603
| 0.778523
|
github_plus_top10pct_by_avg
|
= (\lambda^{(1)}, \lambda^{(2)}, \lambda^{(3)},
\lambda^{(4)}, \lambda^{(5)})$$ to mean the tableau $p$ goes through $\lambda^{(1)}$, $\lambda^{(2)}$, $\lambda^{(3)}$, $\lambda^{(4)}$, $\lambda^{(5)}$ at the 1-st, the $(2-\frac{1}{2})$-th, the 2-nd, the $(3-\frac{1}{2})$-th and the 3-rd coordinates respectively.
Suppose that $$\begin{array}{lr}
\begin{array}{rcl}
q_1 &=& (\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{\emptyset}, \widehat{\emptyset}, \widetilde{\emptyset}),\\
q_2 &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset},
\widetilde{\emptyset}, \widehat{\emptyset}, \widetilde{\emptyset}),\\
q_3 &=& (\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset}, \widetilde{\emptyset}),
\end{array}
&
\begin{array}{rcl}
q_4 &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset}, \widetilde{\emptyset}),\\
q_5 &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset}, \widetilde{\emptyset}).
\end{array}
\end{array}$$ Then for the standard vectors $(v_j)_{j=1}^5$ which correspond to $(q_j)_{j=1}^5$ we define $\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)(v_1\ v_2\ v_3\ v_4\ v_5)$ by $$\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)(v_1\ v_2\ v_3\ v_4\ v_5)
= (v_1\ v_2\ v_3\ v_4\ v_5)
\begin{pmatrix}
1 & 0 & 0& 0 &0\\
0 & 0 & 0& 1 &1\\
0 & 0 & 1& 0 &0\\
0 & \frac{1}{Q-1} & 0& \frac{Q-2}{Q-1} &\frac{-1}{Q-1}\\
0 & \frac{Q-2}{Q-1} & 0& -\frac{Q-2}{Q-1} &\frac{1}{Q-1}
\end{pmatrix}.$$ Assume that $$\begin{array}{lr}
\begin{array}{rcl}
q_6 &=& (\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{\emptyset}, \widehat{\emptyset}, \widetilde{{\mbox{\tiny\yng(1)}}}),\\
q_7 &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset},
\widetilde{\emptyset}, \widehat{\emptyset}, \widetilde{{\mbox{\tiny\yng(1)}}}),\\
q_8 &=& (\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset}, \wid
| 2,699
| 1,232
| 2,519
| 2,648
| null | null |
github_plus_top10pct_by_avg
|
hcal{L}}, \oplus_n {\mathcal{L}})$, where $Mor^R$ refers to the $R$-module morphisms in the category of parameterized spectra over $X$. Thus when $S = End^R({\mathcal{L}})$, we have a natural equivalence $$End^S(\vee_n S) \simeq \prod_n(Mor^R({\mathcal{L}}, \oplus_n {\mathcal{L}}) = End^R (\oplus_n {\mathcal{L}}).$$ Furthermore this equivalence clearly preserves the ring structure. It therefore induces an equivalence of their group-like monoids of units, $$GL_1(End^S(\vee_n S)) \simeq GL_1(End^R(\oplus_n {\mathcal{L}}).$$ The left side is by definition $GL_n (End^R({\mathcal{L}}))$, and the right side is by definition ${hAut^R}(\oplus_n {\mathcal{L}})$.
We now notice that Theorem \[main\] and Lemma \[GLN\] together imply the following.
\[bgln\] There is a homotopy equivalence $$BGL_n({End^R_X {\mathcal{L}}}) \simeq Map_{\oplus_n{\mathcal{L}}}(X, BGL_nR).$$
When $G \to P \to M$ is a principal bundle over a manifold and ${\mathcal{L}}= \Sigma^\infty_M (P_+)$, we obtain the following result about the general linear groups of the string topology spectrum,
\[string\] There is a homotopy equivalence $$\beta : BGL_n ({\mathcal{S}}(P)) {\xrightarrow}{\simeq} Map_{\oplus_n {\mathcal{L}}}(M, BGL_n(\Sigma^\infty(G_+))$$ where ${\mathcal{L}}$ is the line bundle $\Sigma^\infty (G_+) \to \Sigma^\infty_M(P_+) \to M$. In particular there is an equivalence $$BGL_n(LM^{-TM}) \simeq Map_{\iota_n}(M, BGL_n(\Sigma^\infty(\Omega M_+)).$$
$K$-theoretic implications
==========================
The goal of this section is to use Corollary \[string\] to prove Theorem \[ktheory\] as stated in the introduction. We will then use it to derive descriptions of the $K$-theory of the connective covers of the Spanier-Whitehead dual $D(M) = Map (\Sigma^\infty (M_+), {\mathbb{S}})$ and of the string topology ring spectrum ${LM^{-TM}}$.
To do this we need to understand the equivalence given in Corollary \[bgln\] more carefully, so that we can deduce $K$-theoretic consequences. As in that corollary, let ${\mathcal{L}}\to M$ be an $R$-line bundle
| 2,700
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| 2,634
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| 0.768028
|
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|
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