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t basis $N_R$, keeping for simplicity the $N$ dependence implicit, and $q(N_R) = 2^{-N}\left( \begin{array}{c} N \\ N_R \end{array} \right)$ is the probability to have exactly $N_R$ states from the Reject basis. Note that $P_R(m;\alpha, N_R)$ is calculated under the assumption that the Accept observable is measured during the first $m$ steps, while after that the Reject observable is measured for the remaining $(N-m)$ steps.
For $m<(1-\alpha)N_R$ there is always a chance to reject the contract, thus $P_R(m;\alpha, N_R) = 1$. Otherwise, $$\label{reject}
P_R(m;\alpha, N_R) = \sum_{n=n^\prime}^{m^\prime} P(n \mbox{ in R};m, N_R)P_R(n \mbox{ in R};\alpha, N_R).$$ Here, the probability that exactly $n$ out of the first $m$ qubit states are from the Reject basis is given by[^3] $$P(n \mbox{ in R};m, N_R) = \left( \begin{array}{c} m \\ n \end{array} \right) \left( \begin{array}{c}N- m \\ N_R-n \end{array} \right) \left( \begin{array}{c} N \\ N_R \end{array} \right)^{-1},$$ while the probability of being able to reject the contract if $n$ qubits are from the Reject basis is given by[^4] $$P_R(n \mbox{ in R};\alpha, N_R) = 2^{-n} \sum_{i=0}^T \left( \begin{array}{c} n \\ i \end{array} \right),$$ where $T=(1-\alpha)N_R-1$ if $n\geq (1-\alpha)N_R$ and $T=n$ otherwise. Due to the constraint of having exactly $N_{A/R}$ qubits from the Accept/Reject basis, the range of the summation for $n$ in equation is given by $n^\prime=0$ for $m\leq N_A$ while $n^\prime=m-N_A$ otherwise, and $m^\prime=m$ for $m\leq N_R$ while $m^\prime=N_R$ otherwise.
Finally, the expected probability to cheat, with respect to a given probability distribution $p(\alpha)$ on the segment $I_{\alpha}$, is $$\bar{P}_{ch}(m) = \int_{I_{\alpha}}p(\alpha)P_{ch}(m;\alpha){\mathbf d\alpha}.$$
Using the simplest uniform probability $p(\alpha)=1/I_{\alpha}$ on the segment $I_{\alpha}=[0.9,0.99]$, we numerically evaluated the expected probability to cheat $\bar{P}_{ch}(m)$ for up to $N=600$, while for the “typical” case of $N_A=N_R$ we managed to evaluate it
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the inelastic tunnel current are classified by the power of the electron-phonon coupling $ \lambda^{\rm tip}_{\mu\nu}$ in the tunneling Hamiltonian. We note again that the additional inelastic contributions arise from phonon absorption and emission *during* the tunneling process, while all electron scattering processes within the system $S$ are included in $I_{\rm el}$.
In first order in $ \lambda^{\rm tip}_{\mu\nu}$, we obtain an inelastic tunnel current $$\begin{aligned}
I^{(1)}_{\rm inel}
&=&
\frac{2\pi e}{\hbar}
\sum_{\sigma}
\int_{-\infty}^{\infty} d\w \rho_{\sigma,\rm tip}(\w)
\tau^{(1)}_{\sigma}(\w)
\non
&& \times \left[
f_{\rm tip}(\w) - f_{S}(\w)
\right]
\label{eqn:inelasticCurrent}\end{aligned}$$ from Eq. , where following the general notation in this paper (see above) the transmission function $\tau^{(1)}_{\sigma}(\w)$ is defined as the spectral function of a composite Green’s function $G^{(1)}_{d\sigma}$ which in turn involves tunneling matrix elements $t_{\mu \sigma}$ and correlation functions $G_{X_\nu d_{\mu\sigma}, d^\dagger_{\mu'\sigma}}$ and $G_{d_{\mu\sigma}, X_\nu d^\dagger_{\mu'\sigma}}$ as $$\begin{aligned}
\label{eq:rho1explicit}
\tau^{(1)}_{\sigma}(\w)&=&
\frac{1}{\pi}\lim_{\delta\to 0^+} \Im G^{(1)}_{d\sigma} (\w-i\delta) \nonumber
\\
&=&\sum_{\mu\mu'} t_{\mu \sigma}t_{\mu' \sigma}
\\
\times &\bigg(&\sum_{\nu}^{N_\nu}\lambda^{\rm tip}_{\mu\nu}\lim_{\delta\to 0^+}
\frac{1}{\pi} \Im G_{\hat X_\nu d_{\mu\sigma}, d^\dagger_{\mu'\sigma}} (\w-i\delta) \nonumber
\\
&+& \sum_{\nu}^{N_\nu}\lambda^{\rm tip}_{\mu'\nu}\lim_{\delta\to 0^+}
\frac{1}{\pi} \Im G_{d_{\mu\sigma}, \hat X_\nu d^\dagger_{\mu'\sigma}} (\w-i\delta)\bigg).
\nonumber\end{aligned}$$ Since the expectation value of the anticommutator of a Green’s function $G_{A,B}(z)$ equals the frequency integral of the corresponding spectrum $\rho_{A,B}(\w)$, we can conclude that the spectra of $G_{d_{\mu \sigma}, \hat X_\nu d^\dagger_{\mu' \sigma}}$ and $G_{\hat X_\nu d_{\mu \sigma}, d^\dagger_{\mu' \sigma}}$ both individually
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-2+\widetilde{\alpha}_{i,i',\ell,\theta}$. We have, $$\begin{aligned}
\label{eq:posl_3}
&& \P_{\theta}\Big[\sigma^{-1}(i) = \ell, \sigma^{-1}(\i) > \ell \Big] \nonumber\\
&=& \sum_{\substack{j_1 \in S \\ j_1 \neq i,\i}} \Bigg(\frac{\exp(\theta_{j_1})}{W} \sum_{\substack{j_2 \in S \\ j_2 \neq i,\i,j_1}} \Bigg(\frac{\exp(\theta_{j_2})}{W-\exp(\theta_{j_1})}\cdots \nonumber \\
&& \Bigg( \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i,\i, \\ j_1,\cdots,j_{\ell-2}}} \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-2}}\exp(\theta_{k})}\frac{\exp(\theta_i)}{W-\sum_{k=j_1}^{j_{\ell-1}}\exp(\theta_{k})}\Bigg) \cdots\Bigg)\Bigg) \nonumber\\
&=&\frac{\exp(\theta_i)}{W} \,\sum_{\substack{j_1 \in S \\ j_1 \neq i,\i}} \Bigg( \frac{\exp(\theta_{j_1})}{W-\exp(\theta_{j_1})} \sum_{\substack{j_2 \in S \\ j_2 \neq i,\i,j_1}} \Bigg( \frac{\exp(\theta_{j_2})}{W-\exp(\theta_{j_1})-\exp(\theta_{j_2})}\cdots \nonumber\\
&& \sum_{\substack{j_{\ell-1} \in S \\ j_{\ell-1} \neq i,\i, \\ j_1,\cdots,j_{\ell-2}}} \Bigg( \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-1}}\exp(\theta_{k})} \Bigg)\cdots \Bigg)\Bigg) \nonumber\\ \end{aligned}$$ Consider the last summation term in the above equation and let $\Omega_\ell = S\setminus\{i,i',j_1,\ldots,j_{\ell-2}\}$. Observe that, $|\Omega_\ell| = \kappa-\ell$ and from equation , $\frac{\exp(\theta_{i})+\exp(\theta_{\i})}{\sum_{j \in \Omega_\ell} \exp(\theta_j)} \leq \frac{\widetilde{\alpha}_{i,i',\ell,\theta}}{\kappa-\ell}$. We have, $$\begin{aligned}
&&\sum_{j_{\ell-1} \in \Omega_\ell} \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-1}}\exp(\theta_{k})} \nonumber\\
&=& \sum_{j_{\ell-1} \in \Omega_\ell } \frac{\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-2}}\exp(\theta_{k}) - \exp(\theta_{j_{\ell-1}})} \nonumber\\
&\geq& \frac{\sum_{j_{\ell-1} \in \Omega_\ell}\exp(\theta_{j_{\ell-1}})}{W-\sum_{k=j_1}^{j_{\ell-2}}\exp(\theta_{k})-\big(\sum_{j_{\ell-1} \in \Omega_\ell}\exp(\theta_{j_{\ell-1}})\big)/|\Omega_\ell|} \label{eq:posl_jensen_ineq}\\
&=& \frac{\sum_{j_{\
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parentheses with some choice of operations. By induction we can define the *central letter* $c(w)$ of a dimonomial: if $w\in D$, then $c(w)=w$, otherwise $c(w_1{\mathbin\vdash}w_2)=c(w_2)$ and $c(w_1{\mathbin\dashv}w_2)=c(w_1)$. Let $c(w)=a_k$. If $D$ is 0-dialgebra, then $w=(a_1{\mathbin\vdash}\ldots{\mathbin\vdash}a_{k-1}{\mathbin\vdash}a_k{\mathbin\dashv}a_{k+1}{\mathbin\dashv}\ldots{\mathbin\dashv}a_n)$ for the same parenthesizing as in $(a_1\ldots a_n)$. We will denote this $w$ by $(a_1\ldots a_{k-1}\dot a_k a_{k+1}\ldots a_n)$. In an associative dialgebra parenthesizing does not matter, so it is reasonable to use the notation $w=a_1\ldots a_{k-1}\dot a_k a_{k+1}\ldots a_n$, where the dot indicates the central letter.
Let $X$ be a set of generators. Obviously, the basis of the free dialgebra ${\mathrm{Di}}{\mathrm{Alg}}\,\langle X\rangle$ generated by $X$ consists of dimonomials with a free placement of parentheses and a free choice of operations. It is clear that the basis of the free 0-dialgebra ${\mathrm{Di}}{\mathrm{Alg}}0\, \langle X\rangle$ is the set of dimonomials $(a_1\ldots a_{k-1}\dot a_k a_{k+1}\ldots a_n)$ where $k=1,\ldots,n$ and $a_1,\ldots,a_n\in X$. Finally, the basis of the free associative dialgebra ${\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ consists of dimonomials $a_1\ldots a_{k-1}\dot a_k a_{k+1}\ldots a_n$ (see [@Loday:01]).
If $X=\{x_1,\ldots,x_n\}$ then every dipolynomial $f\in
{\mathrm{Di}}{\mathrm{As}}\,\langle X\rangle$ can be presented as a sum $f=f_1+\ldots+f_n$, where each $f_i$ collects all those dimonomials with central letter $x_i$, $i=1,\dots, n$.
Jordan dialgebras
-----------------
Let us consider the class of Jordan dialgebras over a field $\Bbbk$ such that ${\mathop{\mathrm{char}}\nolimits}\Bbbk\not=2,3$. In this case, the variety of Jordan algebras ${\mathrm{Jord}}$ over the field $\Bbbk$ is defined by the following multilinear identities $$x_1x_2=x_2x_1,\
J(x_1,x_2,x_3,x_4)=0,$$ where $$\begin{gathered}
J(x_1,x_2,x_3,x_4)=x_1(x_2(x_3x_4))+(x_2(x_1x_3))x_4+x_3(
| 3,404
| 1,168
| 3,055
| 3,220
| 1,230
| 0.792364
|
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|
ceptible again. Therefore, individuals might change their state from susceptible to infected, and vice versa, repeatedly ($S\leftrightarrows I$).
- The *Susceptible-Infected-Removed* (SIR), which extends the SI model to take into account a removed state. Here, an individual can be infected just once because when the infected individual recovers, it becomes immune and will no longer pass the infection onto others ($S\rightarrow I \rightarrow R$).
As shown in Fig. \[fig:multfail\], the subset of cascading failures intersects the subset of epidemic failures. Cascading failures are common in most critical infrastructures such as telecommunications, electrical power, rail, and fuel distribution networks [@Strogatz2001]. In telecommunication networks, we consider cascading failures as an epidemic when it occurs due to a malfunctioning in one node of a network which eventually triggers a failure in its neighbors. Real cascading failures in telecommunication networks have been observed in the IP layer of the Internet and in the physical layer of BTNs [@wrap32818]. The propagation of a cascading failure happens gradually in phases: after the initial failure (e.g., a massive broadcast of a routing message with a bug), some of the neighboring nodes get overloaded and fail. This first step leads to further overloading of more nodes and their collapse, constituting the second step and so on. In this way, networks go through multiple stages of cascading failures before they finally stabilize and there are no more failures. It is worth noting that cascading failures in other critical infrastructures, such as power grids, do not necessarily propagate by the physical contact of nodes or links, but by the load balancing in the global network. In such cases, cascading failures are not similar to epidemics, and thus are out of the scope of this work.
A Failure Propagation Model for Telecommunication Networks\[epidemicsontelecom\]
================================================================================
Calle et al. pio
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|
olic polynomial and let ${\mathbf{v}}\in \Lambda_+$ be such that $D_{\mathbf{v}}h \not \equiv 0$. Then
1. $D_{\mathbf{v}}h$ is hyperbolic with hyperbolicity cone containing $\Lambda_{++}$.
2. The polynomial $h({\mathbf{x}})-yD_{\mathbf{v}}h({\mathbf{x}}) \in {\mathbb{R}}[x_1,\ldots, x_n,y]$ is hyperbolic with hyperbolicity cone containing $\Lambda_{++} \times \{y: y \leq 0\}$.
3. The rational function $${\mathbf{x}}\mapsto \frac {h({\mathbf{x}})}{D_{\mathbf{v}}h({\mathbf{x}})}$$ is concave on $\Lambda_{++}$.
(1). See [@BrOp Lemma 4].
(2). The polynomial $h({\mathbf{x}})y$ is hyperbolic with hyperbolicity cone containing $\Lambda_{++} \times \{y : y<0\}$. Hence so is $H({\mathbf{x}},y):= D_{({\mathbf{v}},-1)} h({\mathbf{x}})y= h({\mathbf{x}})- y D_{\mathbf{v}}h({\mathbf{x}})$ by (1). Since $H({\mathbf{e}}',0) = h({\mathbf{e}}') \neq 0$ for each ${\mathbf{e}}' \in \Lambda_{++}$, we see that also $\Lambda_{++}\times \{0\}$ is a subset of the hyperbolicity cone (by Theorem \[hypfund\] (2)) of $H$.
(3). If ${\mathbf{x}}\in \Lambda_{++}$, then (by Theorem \[hypfund\] (2)) $({\mathbf{x}},y)$ is in the closure of the hyperbolicity cone of $H({\mathbf{x}},y)$ if and only if $$y \leq \frac {h({\mathbf{x}})}{D_{\mathbf{v}}h({\mathbf{x}})}.$$ Since hyperbolicity cones are convex $$y_1 \leq \frac {h({\mathbf{x}}_1)}{D_{\mathbf{v}}h({\mathbf{x}}_1)} \mbox{ and } y_2 \leq \frac {h({\mathbf{x}}_2)}{D_{\mathbf{v}}h({\mathbf{x}}_2)} \mbox{ imply } y_1+y_2 \leq \frac {h({\mathbf{x}}_1+{\mathbf{x}}_2)}{D_{\mathbf{v}}h({\mathbf{x}}_1+{\mathbf{x}}_2)},$$ for all ${\mathbf{x}}_1,{\mathbf{x}}_2 \in \Lambda_{++}$, from which (3) follows.
\[rankalt\] Let $h$ be hyperbolic with hyperbolicity cone $\Lambda_{++}\subseteq {\mathbb{R}}^n$. The rank function does not depend on the choice of ${\mathbf{e}}\in \Lambda_{++}$, and $$\rk({\mathbf{v}})= \max\{ k : D_{\mathbf{v}}^kh \not \equiv 0\}, \quad \mbox{ for all } {\mathbf{v}}\in {\mathbb{R}}^n.$$
That the rank does not depend on the choice of ${\mathbf{e}}\in \Lambda_{++}$ is known,
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nvariant under translations in the independent variables $\MM{q}$. For each of these symmetries, Noether’s theorem yields a conservation law $$(L_j^\alpha z^j_{,\beta}-L\delta^\alpha_\beta)_{,\alpha}=0.$$ Such conservation laws can equally well be obtained by pulling back the quasi-conservation law (\[ofcl\]) to the base space of independent variables. Commonly, the independent variables are spatial position $\MM{x}$ and time $t$. Pulling back (\[ofcl\]) to these base coordinates yields the energy conservation law from the ${\mathrm{d}}{t}$ component, and the momentum conservation law from the remaining components. We shall see the form of these conservation laws for fluid dynamics in later sections.
The inverse map and Clebsch representation {#inverse map sec}
==========================================
Lagrangian fluid dynamics and the inverse map
---------------------------------------------
Lagrangian fluid dynamics provides evolution equations for particles moving with a fluid flow. This is typically done by writing down a flow map $\Phi$ from some reference configuration to the fluid domain $\Omega$ at each instance in time. As the fluid particles cannot cavitate, superimpose or jump, this map must be a diffeomorphism.
For an $n$-dimensional fluid flow, the flow map $\Phi:\,\mathbb{R}^n\times\mathbb{R}\mapsto\mathbb{R}^n$ given by $\MM{x}=\Phi(\MM{l},t)$ specifies the spatial position at time $t$ of the fluid particle that has *label* $\MM{l}=\Phi(\MM{x},0)$. The *inverse map* $\Phi^{-1}$ gives the label of the particle that occupies position $\MM{x}$ at time $t$ as the function $\MM{l}=\Phi^{-1}(\MM{x},t)$. The Eulerian velocity field $\MM{u}(\MM{x},t)$ gives the velocity of the fluid particle that occupies position $\MM{x}$ at time $t$ as follows: $$\MM{\dot{x}}(\MM{l},t)=\MM{u}(\MM{x}(\MM{l},t),t).$$ Each label component $l_k(\MM{x},t)$ satisfies the advection law $$\label{label eqn}
l_{k,t} + u_il_{k,i} = 0.$$ Here ${}_{,t}$ and ${}_{,i}$ denote differentiation with respect to $t$ and $x_i$ respect
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he estimation of the weight matrix in the RBFN is derived from the Lyapunov theory, which guarantees stability of the closed-loop system as presenting in the following theorem.
**Theorem 2.** Given the adaptation mechanism (\[eq34\]), the proposed control scheme (\[eq35\]) can guarantee the closed-loop DAR system (\[eq15\]) to be input-to-state stable [@RN28] with the attractor $$\label{eq36}
D=\left\{ {s}\in {{R}^{4}}|\left\| {s} \right\|> \frac{{{\varepsilon }_{N}}+\varsigma \frac{{{\left\| W \right\|}_{F}}^{2}}{4}}{{{c}_{3\min }}} \right\},$$ where ${c}_{3\min }$ is the minimum value of $c_3$, and ${\varepsilon }_{N}$ is a small positive number so that the approximation error $\varepsilon=\delta-\hat{\delta}$ satisfies $\left\| {\varepsilon} \right\|<{\varepsilon }_{N}$.
Let $$\tilde{W}=W-\hat{W}$$ define the error between the ideal weight $W$ and the estimated weight $\hat{W}$ Considering the Lyapunov function candidate $$\label{eq37}
{{V}_{2}}={{V}_{1}}+\frac{1}{2}{{{s}}^{T}}{s}+tr\left( {{{\tilde{W}}}^{T}}{{\Gamma }^{-1}}\tilde{W} \right)$$ and differentiating it with respect to time, one obtains $$\label{eq38}
\begin{array}{r@{}l@{\qquad}l}
{{{\dot{V}}}_{2}}=&{{{\dot{V}}}_{1}}+{{{{s}}}^{T}}\dot{{s}}+tr\left( {{{\tilde{W}}}^{T}}{{\Gamma }^{-1}}\dot{\tilde{W}} \right) \\
=&-{{{{z}}}_{1}}^{T}{{c}_{1}}{{{{z}}}_{1}}-{{{{s}}}^{T}}{{c}_{2}}sign\left( {{s}} \right)-{{{{s}}}^{T}}{{c}_{3}}{s}+{{{{s}}}^{T}}\left( {\delta }-{\hat{\delta }} \right) \\
&-tr\left( {{{\tilde{W}}}^{T}}{{\Gamma }^{-1}}\dot{\hat{W}} \right) \\
=&-{{{{z}}}_{1}}^{T}{{c}_{1}}{{{{z}}}_{1}}-{{{{s}}}^{T}}{{c}_{2}}sign\left( {{s}} \right)-{{{{s}}}^{T}}{{c}_{3}}{s}+{{{{s}}}^{T}}{{W}^{T}}{h}\\
&-{{{{s}}}^{T}}{{{\hat{W}}}^{T}}{h}-tr\left( {{{\tilde{W}}}^{T}}{{\Gamma }^{-1}}\dot{\hat{W}} \right) +{{{{s}}}^{T}}{\varepsilon} \\
=&-{{{{z}}}_{1}}^{T}{{c}_{1}}{{{{z}}}_{1}}-{{{{s}}}^{T}}{{c}_{2}}sign\left( {{s}} \right)-{{{{s}}}^{T}}{{c}_{3}}{s}+{{{{s}}}^{T}}{\varepsilon}\\
&+tr\left( {{{\tilde{W}}}^{T}}\left( {{{h}{{{s}}}^{T}-\Gamma }^{-1}}\dot{\h
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communicates with the three active neutrinos.
One of the authors (H.M.) thanks Enrique Fernandez-Martinez for interesting discussions about the relationship between high-scale and low-scale unitarity violation. He expresses a deep gratitude to Instituto Física Teórica, UAM/CSIC in Madrid, for its support via “Theoretical challenges of new high energy, astro and cosmo experimental data” project, Ref: 201650E082. He had been a member of Yachay Tech for 14 months, at that time the first research-oriented university in Ecuador [@Yachay-story], during which he was warmly supported by Ecuadorian people. He thanks kind supports by ICTP-SAIFR (FAPESP grant 2016/01343-7), UNICAMP (FAPESP grant 2014/19164-6), and PUC-Rio (CNPq) which enabled him to visit these institutions where part of this work was done. C.S.F. is supported by FAPESP under grants 2013/01792-8 and 2012/10995-7. H.N. is supported by CNPq. This work was supported in part by the Fermilab Neutrino Physics Center.
$\hat{S}$ matrix elements {#sec:hatS-elements}
=========================
The method of computing $\hat{S}$ matrix elements is outlined in section \[sec:hatS-matrix\]. In this appendix \[sec:hatS-elements\] we carry out this task. With the expression of $H_{1}$ in (\[H1-matrix\]) we compute perturbatively the matrix elements of $\Omega (x)$. Then, $\hat{S} (x) = e^{- i \hat{H}_{0} x} \Omega(x)$.
We denote computated results of the matrix elements of $\hat{S}$ as $\hat{S} [n]$ to indicate that it is the one that comes from $n$-th order contribution in $H_{1}$. Since the elements of $H_{1}$ are of order either $W$ or $W^2$, $\hat{S} [n]$ generally has order $W^n$ or higher. To show that a particular contribution is of order $W^m$ we use the superscript $``(m)''$. That is, $\hat{S}^{(m)} [n]$ denotes contribution to $\hat{S}$ that arizes from $n$-th order perturbative contribution in $H_{1}$ and is of order $W^m$.
Contribution to $\hat{S}$ matrix elements from zeroth and first order in $H_{1}$ {#sec:hatS-0th-1st}
--------------------------
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}}{\partial D^{(r)}}\frac{\partial D^{(r)}}{\partial v}\, ,$$ from which follows that, in the vicinity of the transition fixed point $(A^*, B^*, C^*, D^*)$ of the two-polymer system, the mean number of contacts $\langle M^{(r)}\rangle$, for large $r$, behaves as $\langle M^{(r)}\rangle\sim \lambda_{D}^r
\label{eq:lambdaD}$, where $$\label{svvrednost2}
\lambda_{D}={\left(\frac{\partial D^{(r+1)}}{\partial D^{(r)}}\right)}^*\>,$$ is relevant eigenvalue of RG equation (\[eq:RGA4\]), calculated at the transition fixed point. Knowing that $\langle {N_3^{(r)}}\rangle\sim \lambda_{\nu_3}^r$, one obtains $\ln \langle M^{(r)}\rangle/{\ln \langle N_3^{(r)}\rangle}\sim {\ln\lambda_{D}}/{\ln\lambda_{\nu_3}}$, [*i.e.*]{} the following scaling relation is satisfied $$\label{asawfi}
\langle M^{(r)}\rangle\sim \langle N_3^{(r)}\rangle^{\phi}\>,$$ where $$\phi=\frac{{\ln\lambda_{D}}}{\ln\lambda_{\nu_3}}\, , \label{eq:skaliranje}$$ is so-called contact critical exponent.
To establish the exact forms of RG equations, for each fractal, one needs to find the coefficients $a$, $b$, $c$, and $d$, that appear in (\[eq:RGA\])–(\[eq:RGA4\]). Using the computer facilities, by direct enumeration and classification of all possible SAW configurations on the first stage of fractal construction, it is feasible to find these coefficients for fractals labelled by $b=2,3$ and 4 (see \[app:ASAWsRG\]). Precise numerical analysis of the obtained RG equations (for $b=2,3$, and 4) reveals that two-polymer system can reside in several phases, depending on the values of the interaction parameters $u$ and $v$. In particular, for each value of $u$, there is a critical value $v=v_c(u)$, such that for $v<v_c(u)$ the two chains exist almost independently in the solution. This is indicated by the fact that $(A^*,B^*)$, and $C^*$ retain their fixed values that correspond to the solitary chain on 3D SG, and 2D SG, respectively (see table \[tab:avoiding\]), and confirmed by calculating the mean number of contacts $\langle
M^{(r)}\rangle$ between the chains,
| 3,410
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oposition 2. The strategy however is not doing this with one particular null geodesic, but with all the geodesics $\gamma_\Lambda(\lambda)$ generated by the points $\Lambda$ in $O'$. The resulting minimum for a given geodesic, denoted by $I_{\Lambda}$, is not necessarily a continuous function in $O'$ but, by Property 2, is bounded by below. As $O'$ is compact, there will exists an infimum value at a point $\Lambda_m\in O'$, with its corresponding smallest value $I_{\Lambda_m}$. By construction this value is smaller or equal than the corresponding to any other generic point $\Lambda$ in $O'$. Below, for notational simplicity, this value will be denoted by $I_m$ instead of $I_{\Lambda_m}$.
Once the value $I_m$ has ben found, choose a value $\lambda_i$ such that $\theta_0(\lambda_i)+c/2<I_m-\eta$ with $\eta> C>0$. The required value of $\lambda_i$ in $(0,\lambda_0)$ exists since, as discussed above, the expansion parameter $\theta_0(\lambda)$ for the reference geodesic $\gamma_0$ takes every real value when $\lambda$ varies in that closed interval. Then, from the minimality of $I_m$ it is clear that $\theta_0(\lambda_i)+c/2<I_m-\eta$ implies that $$\theta_0(\lambda_i)+\frac{c}{2}< \lim_{\lambda\to\infty} \text{inf}\int_{\lambda_i}^\lambda e^{-c \xi} [R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_0(\xi)
d\xi.$$ In these terms, one may choose an open $O$ inside $O'$ containing $\Lambda_0$, such that for every $\theta_\Lambda(\lambda)$ determined by a point $\Lambda$ in $O$ the inequality $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|<\epsilon$ holds. The validity of this statement may be seen from the continuity of $G(\lambda, \Lambda)$ and $G'(\lambda, \Lambda)$ in $O$, which implies the continuity of $\theta_\Lambda(\lambda)=G'(\Lambda,\lambda)/G(\Lambda, \lambda)$ with respect to $\Lambda$ if $G(\lambda, \Lambda)\neq 0$, that is, outside a conjugate point. This continuity property follows from the fact that $(A_\Lambda)_\mu^\nu$ satisfies the ordinary equation (\[smile\]), and thus $(A_\Lambda)_\mu^\nu$
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tures, the AOG makes it easier to transfer CNN patterns to other part-based tasks.
**Unsupervised/active learning:**[` `]{} Many methods have been developed to learn object models in an unsupervised or weakly supervised manner. Methods of [@Gpt_WeaklyCNN; @WeaklyMIL; @OurICCV15AoG; @ObjectDiscoveryCNN_2] learned with image-level annotations without labeling object bounding boxes. [@UnsuperCNN; @ChoDiscovery] did not require any annotations during the learning process. [@OnlineMetric] collected training data online from videos to incrementally learn models. [@Language2VideoAlign; @Language2ActionAlign] discovered objects and identified actions from language Instructions and videos. Inspired by active learning [@Active4; @i13; @Active2], the idea of learning from question-answering has been used to learn object models [@KB_Fei_Annotation; @KB_Fei_InteractionLabel; @TuQA]. Branson *et al.* [@ActivePart] used human-computer interactions to label object parts to learn part models. Instead of directly building new models from active QA, our method uses the QA to mine AOG part representations from CNN representations.
**AOG for knowledge transfer:** Transferring hidden patterns in the CNN to other tasks is important for neural networks. Typical research includes end-to-end fine-tuning and transferring CNN representations between different categories [@CNNAnalysis_2; @CNNSemantic] or datasets [@UnsuperTransferCNN]. In contrast, we believe that a good explanation and transparent representation of parts will create a new possibility of transferring part features. As in [@AllenAoG; @MiningAOG], the AOG is suitable to represent the semantic hierarchy, which enables semantic-level interactions between human and neural networks.
**Modeling “objects” vs. modeling “**parts**” in un-/weakly-supervised learning:**[` `]{} Generally speaking, in the scenario of un-/weakly-supervised learning, it is usually more difficult to model object parts than to represent entire objects. For example, object discovery [@ObjectD
| 3,412
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we obtain (see Figure \[fig:1stpiv\])
$$\raisebox{0.2pc}{\includegraphics[scale=0.17]{1stpiv1}}~~~=~~~
\raisebox{-0.5pc}{\includegraphics[scale=0.17]{1stpiv2}}~~~+~~
\sum_b~\raisebox{-1.7pc}{\includegraphics[scale=0.17]{1stpiv3}}$$
$$\begin{aligned}
{\label{eq:pre-1st-exp}}
{{\langle \varphi_o\varphi_x \rangle}}_\Lambda=\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)+\sum_{b\in
{{\mathbb B}}_\Lambda}\;\sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\text{ off }b\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n_b>0\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x
\text{ in }{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}\}$}}}.\end{aligned}$$
Next, we consider the sum over ${{\bf n}}$ in [(\[eq:pre-1st-exp\])]{}. Since $b$ is pivotal for $o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x$ from $o\,(\ne x$, due to the last indicator) and ${\partial}{{\bf n}}=o{\vartriangle}x$, in fact $n_b$ is an *odd* integer. We alternate the parity of $n_b$ by changing the source constraint into $o{\vartriangle}b{\vartriangle}x\equiv\{o\}{\,\triangle\,}\{{\underline{b}},{\overline{b}}\}{\,\triangle\,}\{x\}$ and multiplying by $$\begin{aligned}
\frac{\sum_{n\text{ odd}}(p J_b)^n/n!}{\sum_{n\text{ even}}(p
J_b)^n/n!}=\tanh(p J_b)\equiv\tau_b.\end{aligned}$$ Then, the sum over ${{\bf n}}$ in [(\[eq:pre-1st-exp\])]{} equals $$\begin{aligned}
{\label{eq:0th-summand1}}
\sum_{{\partial}{{\bf n}}=o{\vartriangle}b{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\text{ off }b\}$}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \
| 3,413
| 1,552
| 2,408
| 3,355
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punctatum* (CPC 18974). a, b. Sporulating colonies on potato-dextrose agar; c--h. macroconidiophores showing rachis; i, j. rachis showing pimple-like denticles; k. conidia. --- Scale bars = 10 μm.](per-28-113-g009){#F9}
{#F10}
######
Collection details and GenBank accession numbers of isolates for which novel sequences were generated in this study.
Species Strain number Substrate Country Collector GenBank Accession no.
------------------------------ ------------------------------------------------ ----------------------------------------------- --------------------------------------------------- ---------------------------------- ----------------------------------- ------------- ---------- ---------- ----------
*Dissoconium aciculare* 132079 18965 MA1 10B1a *Malus domestica* fruit surface Massachusetts, USA D. Cooley JQ622082 JQ622090 JQ622107
132080 18966 PEB1a *Malus domestica* fruit surface Iowa, USA M. Gleason JQ622083 JQ622091 JQ622108
132081 18967 CUB2c *Malus domestica* fruit surface Illinois, USA M. Gleason AY598874 JQ622097 JQ622114
132082
| 3,414
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tion with the mapping $J\to J'$ gives the required homomorphism ${\mathrm{Di}}{\mathrm{SJ}}\langle X\rangle\to J'$.
A *bar-unit* of a 0-dialgebra $D$ is an element $e\in D$ such that $x{\mathbin\dashv}e=e{\mathbin\vdash}x=x$ for every $x\in D$ and $e$ belongs to the associative center of $D$ that is $$(x,e,y)_\times=(e,x,y)_\dashv=(x,y,e)_\vdash=0$$ for all $x$, $y\in D$.
\[prop:EmbWithBarUnit\] For every associative dialgebra $D$ there exists an associative dialgebra $D_e$ with the bar-unit $e$ such that $D\hookrightarrow D_e$.
\[lemma:SpecUnitEmb\] Let $J$ be a special Jordan dialgebra. Then there exists a special Jordan dialgebra $J_e$ such that $J\hookrightarrow J_e$ and $\bar e$ is a unit in the algebra $\bar J_e$.
By the defintion of a special Jordan dialgebra it follows that $J=(J,{{}_{(\vdash)}},{{}_{(\dashv)}})$ is embedded into $D^{(+)}$ for some associative dialgebra $D=(D,\vdash,\dashv)$. By Proposition \[prop:EmbWithBarUnit\] we have an embedding $D^{(+)}\hookrightarrow D_e^{(+)}$ where $e$ is a bar-unit in $D_e$. Therefore, $J_e=D_e^{(+)}$ is the required dialgebra.
Further, $e{\mathbin\vdash}x=x{\mathbin\dashv}e=x$ holds for every $x\in D_e$, so in $J_e$ we have $e{\mathbin{{}_{(\vdash)}}}x=\frac{1}{2}(e{\mathbin\vdash}x+x{\mathbin\dashv}e)=x$, $x{\mathbin{{}_{(\dashv)}}}e=x$. Hence $\bar e\bar x=\bar x\bar e=\bar x$ in the quotient algebra $\bar J_e$, so $\bar e$ is a unit in $\bar J_e$.
\[lemma:SpecFact\] Let $J$ be a special Jordan dialgebra and such that $\bar{J}$ contains a unit. Then $\bar{J}$ is special.
Let $D$ be an associative dialgebra such that $J\hookrightarrow
D^{(+)}$. Denote $\langle D,D\rangle:={\mathop{\mathrm{Span}}\nolimits}\{a{\mathbin\vdash}b-a{\mathbin\dashv}b\mid
a,\,b\in D\}$, $[J,J]:={\mathop{\mathrm{Span}}\nolimits}\{a{\mathbin{{}_{(\vdash)}}}b-a{\mathbin{{}_{(\dashv)}}}b\mid a,\,b\in
J\}$. As before $\bar D=D/\langle D,D\rangle$ is an associative algebra. Since $J\subseteq D$ we have $[J,J]\subseteq\langle
D,D\rangle$. Then the homomorphism $\phi\colon \bar J\to\b
| 3,415
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loma of Gratitude from the Red Cross of Spain on p. 592. It is
left blank here as well.
In the Index and the text, mention of the "Duke of Parmella should have
been "Duke of Palmella", which appears correctly elsewhere. Both have been
corrected.
No systematic attempt was made to verify the accuracy of page references
as printed in the index or table of contents. However, one error has
been corrected. The final reference in the Contents to the section on
'Notes' was printed as p. 683. That section begins on p. 682, and has
been corrected.
Other issues are noted below and their resolutions described below.
p. 18 upon its humblest ministers
and assistants[.] Added.
p. 37 THE TREATY OF GENEVA.[.] Removed from
caption.
p. 50 shall render the [the] useful institution Removed.
p. 53 com[m]mit[t]ees of the different nations Removed/added.
p. 60 monarchial government _sic._
p. 64 rec[c]ommend Removed.
p. 68 less[o/e]n Corrected.
p. 79 p[o]eople Removed.
p. 80 theref[or/ro]m Transposed.
p. 88 Senator E. [P.] Lapham, _sic._ The
reference is to
Elbridge G.
Lapham.
p. 100 th[o]roughly Added.
p. 110 organ[i]zation Added.
p. 131 the mother said ["(/("]for it was a good,
strong house) Transposed.
p. 139 a grea[l/t] deal of unkind criticism Corrected.
p. 141 in the case[.] Lacking this Added.
p. 145 w[ie]rd _sic._
p. 176 'Oh, right enough, God be pr
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assuming success of all arithmetic calls. In a second phase, we characterize successful arithmetic calls as a constraint problem, the solution of which determines the non-terminating queries.
Keywords: non-termination analysis, numerical computation, constraint-based approach
author:
- |
Dean Voets[^1] $~~~~~$ Danny De Schreye\
Department of Computer Science, K.U.Leuven, Belgium\
Celestijnenlaan 200A, 3001 Heverlee\
{Dean.Voets, Danny.DeSchreye }@cs.kuleuven.be
bibliography:
- 'prolog.bib'
title: 'Non-termination Analysis of Logic Programs with integer arithmetics'
---
**Note:** This article has been published in *Theory and Practice of Logic Programming, volume 11, issue 4-5, pages 521-536, 2011*.
Introduction
============
The problem of proving termination has been studied extensively in Logic Programming. Since the early works on termination analysis in Logic Programming, see e.g. [@DBLP:journals/jlp/SchreyeD94], there has been a continued interest from the community for the topic. Lots of in-language and transformational tools have been developed, e.g. [@Giesl06aprove1.2] and [@DBLP:journals/corr/abs-0912-4360], and since 2004, there is an annual Termination Competition[^2] to compare the current analyzers on the basis of an extensive database of logic programs. In contrast with termination analysis, the dual problem, to detect non-terminating classes of queries, is a fairly new topic. The development of the first and most well-known non-termination analyzer, $NTI$ [@nti_06], was motivated by difficulties in obtaining precision results for termination analyzers. Since the halting problem is undecidable, one way of demonstrating the precision of a termination analyzer is with a non-termination analyzer. For $NTI$ it was already shown that for many examples one can partition queries in terminating and non-terminating. $NTI$ compares the consecutive calls in the program using binary unfoldings and proves non-termination by comparing the head and body of these binary clauses with a special
| 3,417
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_{e} \rho E \lsim 50 \, \text{ (g/cm}^3) \text{GeV}$. See e.g., figure 3 of ref. [@Minakata:2015gra]. Clearly, it excludes the interesting region of “IceCube resonance” due to sterile neutrino mass of eV scales [@Nunokawa:2003ep], for which an entirely different theoretical framework would be necessary.
Then, we notice that in a regime $W^2 \sim 10^{-2}$, the condition in (\[suppression-cond\]) is valid given the estimation (assuming $Y_{e} = 0.5$) $$\begin{aligned}
\biggl | \frac{ a }{ \Delta m^2_{J i} } \biggr | = 2.13 \times 10^{-3}
\left(\frac{ \Delta m^2_{J i} }{ 0.1~\mbox{eV}^2}\right)^{-1}
\left(\frac{\rho}{2.8 \,\text{g/cm}^3}\right) \left(\frac{E}{1~\mbox{GeV}}\right),
\label{rA-def-value}\end{aligned}$$ unless $\rho E \gsim 10 \, \text{ (g/cm}^3) \text{GeV}$. That is, the second-order matter dependent correction terms can be ignored in comparison with $\mathcal{O} (W^4)$ terms if $\Delta m^2_{J k} \gsim 0.1$ eV$^2$, which is already required in vacuum. If we want to treat the regime $W^2 \gsim 10^{-n}$, we need to limit the sterile masses to $\Delta m^2_{J k} \simeq m^2_{J} \gsim 10^{(n-3)}$ eV$^2$ to keep our $(3+N)$ space unitary model insensitive to details of the sterile sector [@Fong:2016yyh]. We note, however, that terms of order $W^4 \sim 10^{-4}$ may be the limit of exploration for near future neutrino oscillation experiments.
The condition (\[suppression-cond\]) is identical with the one obtained using the first order matter perturbation theory [@Fong:2016yyh], which may look strange to the readers. Let us understand the reason why taking care of all order matter effect does not alter the condition obtained by first-order treatment in matter perturbation theory. The matter-dependent term in the zeroth-order Hamiltonian $\tilde{H}_{0}$ only involves $U$ matrix, but no $W$ matrix. Since we treat $\tilde{H}_{0}$ in an unperturbed fashion it produces all-order effect of the matter potential which is however independent of $W$ matrix elements. On the other hand, perturbative effects that co
| 3,418
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| 3,609
| 3,425
| 3,817
| 0.769921
|
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|
ace the shift of the second peak. Curves are shifted upwards by 40 units for clarity.[]{data-label="fig:fkt2"}](fig5a "fig:"){width="4.25cm"} ![Colloid 1-droplet (a) and colloid 2-droplet (b) radial distribution functions, $g_{1\textrm{d}}(r)$ and $g_{2\textrm{d}}(r)$, respectively, as a function of the scaled distance $r/\sigma$ after $t=t_6$ ($4.0\times 10^{5}$ MC cycles). Results are shown for different energy ratios $k$. An asterisk is used as a guide to the eyes to trace the shift of the second peak. Curves are shifted upwards by 40 units for clarity.[]{data-label="fig:fkt2"}](fig5b "fig:"){width="4.25cm"}
Examples of the obtained cluster structures are shown in Fig. \[fig:cluster\](a) for $k=0.1$, Fig. \[fig:cluster\](b) for $k=0.5$ and Fig. \[fig:cluster\](c) for $k=1$. Clusters with colloid numbers between $n_c=4$ and $n_c=10$ are found. For the same number of constituent colloids $n_c$, clusters can have several distinct structures (isomers) [@Ingmar2011; @Bo2013]. It is convenient to use the bond-number $n_b$ as an indicator for the compactness of clusters. For a given value of $n_c$, the smaller the bond number $n_b$ is, the more open the structure is. As shown in Fig. \[fig:cluster\](a), open structures are obtained for $k=0.1$. In these isomers, the colloids of type 1 arrange themselves into symmetric structures, i.e, doublet, triplet, tetrahedron and triangular dipyramid. Increasing the energy ratio $k$, a larger number of isomers with different bond number $n_b$ are found. For example, for $n_c=4$ \[Fig. \[fig:cluster\](b)\] we find four different isomers with $n_b$ ranging from $3$ to $6$, corresponding to a transition from string-like clusters to more compact structures. Finally, for the special case $k=1$ the two colloidal species are identical and we find compact isomers with the largest $n_b$ \[Fig. \[fig:cluster\](c)\] such as $n_c=4,n_b=6$ (tetrahedron); $n_c=6,n_b=12$ (octahedron); $n_c=8,n_b=18$ (snub disphenoid); $n_c=10,n_b=22$ (gyreoelongate square dipyramid). These structures a
| 3,419
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| 3,310
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| 0.779365
|
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|
sigma_D}) + \int_0^{\sigma_{B_k}\wedge \sigma_D }{f}(X_s)\,{\rm d}s \\
= & \, M_k, \qquad k\geq 1,
\end{aligned}$$ where $\mathcal{G}_k = \mathcal{F}_{\sigma_{B_k} \wedge\sigma_D}$, $k\geq 1$. For consistency, we may define $M_0 = \mathbb{E}_x[M_k] = \hat{u}(x)$ thanks to .
Next, we appeal to the definition of $B^*$ and, in particular, that $\sigma_D\leq \sigma_{B^*}$, as well as the continuity of $\hat{u}$ to deduce that, for all $k\geq 0$, $$\begin{aligned}
& \left|\hat{u} (X_{\sigma_{B_k}\wedge \sigma_D}) + \int_0^{\sigma_{B_k}\wedge \sigma_D }{f}(X_s)\,{\rm d}s\right| \\
& \leq \left|\hat{u} (X_{\sigma_{B^*}})\right|\,\mathbf{1}_{\{\sigma_{B_k}\wedge \sigma_D = \sigma_{B^*}\}}
+\sup_{y\in B^*}\left|\hat{u} (y)\right|\,\mathbf{1}_{\{\sigma_{B_k}\wedge \sigma_D < \sigma_{B^*}\}}
+\sup_{y\in D} \left|{f}(y)\right|\,\sigma_{D}\\
& \leq \left|{g}(X_{\sigma_{B^*}})\right| + c_1 + c_2\sigma_{B^*},
\end{aligned}$$ where $c_1,c_2$ are constants. We know that for each fixed $x\in D$, $\mathbb{E}_x[\sigma_{B^*}]<\infty$ and, moreover, from Theorem \[BGR\], after scaling (see for example ), $\mathbb{E}_x[ |{g}(X_{\sigma_{B^*}})| ]<\infty$ as ${g}\in L^1_\alpha(D^\mathrm{c})$. Dominated convergence allows us to deduce that $(M_k, k\geq 0)$ is a uniformly integrable martingale such that, for each fixed $x\in D$, $$\begin{aligned}
\hat{u}(x) & = \lim_{k\to\infty}\mathbb{E}_x[M_k] \\
& = \mathbb{E}_x[\lim_{k\to\infty}M_k] \\
& = \mathbb{E}_x\left[\hat{u} (X_{\sigma_D}) + \int_0^{ \sigma_D }{f}(X_s)\,{\rm d}s\right] \\
& = \mathbb{E}_x\left[{g}(X_{\s
| 3,420
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| 2,715
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e set $\Omega_2$. Therefore, the term in Equation is the probability of the least two preference score items in the set $\Omega_2$, that is of size $(\kappa-{\left \lfloor{\ell\beta_1} \right \rfloor})$, being ranked in bottom $(\ell-{\left \lfloor{\ell\beta_1} \right \rfloor})$ positions.
The following lemma shows that the probability of the least two preference score items in a set being ranked at any two positions is lower bounded by their probability of being ranked at $1$st and $2$nd position.
\[lem:bl\_prob2\] Consider a set of items $S$ and a ranking $\sigma$ over it. Define $i_{\min_1} \equiv \arg \min_{i \in S} \theta_i$, $i_{\min_2} \equiv \arg \min_{i \in S\setminus i_{min_1}} \theta_i$. For all $ 1 \leq i_1, i_2 \leq |S|$, $i_1 \neq i_2$, following holds: $$\begin{aligned}
\P\Big[ \sigma^{-1}(i_{\min_1}) = i_1, \sigma^{-1}(i_{\min_2}) = i_2 \Big] \geq \P\Big[ \sigma^{-1}(i_{\min_1}) = 1, \sigma^{-1}(i_{\min_2}) = 2 \Big]. \end{aligned}$$
Using the fact that $\i = \arg \min_{j \in \Omega_2} \ltheta^*_j $, $i = \arg \min_{j \in \Omega_2 \setminus \i} \ltheta^*_j$, for all $1 \leq i_1, i_2 \leq \kappa-{\left \lfloor{\ell\beta_1} \right \rfloor}$, $i_1 \neq i_2$, we have that $$\begin{aligned}
\label{eq:bl_prob_5}
\P\Big[ \sigma^{-1}(\i) = i_1, \sigma^{-1}(i) = i_2 \Big] \geq \P\Big[ \sigma^{-1}(\i) = 1, \sigma^{-1}(i) = 2 \Big] \geq e^{-4b}\frac{1}{\kappa^2}\;, \end{aligned}$$ where the second inequality follows from the definition of the PL model and the fact that $\ltheta^* \in \lOmega_{2b}$. Together with Equation and the fact that there are a total of $(\ell-{\left \lfloor{\ell\beta} \right \rfloor})(\ell-{\left \lfloor{\ell\beta} \right \rfloor}-1) \geq (\ell-{\left \lfloor{\ell\beta} \right \rfloor})^2/2$ pair of positions that $i,\i$ can occupy in order to being ranked in bottom $(\ell-{\left \lfloor{\ell\beta} \right \rfloor})$, we have, $$\begin{aligned}
\P\Big[\sigma^{-1}(i),\sigma^{-1}(\i) > \kappa - \ell \Big] \geq \frac{e^{-4b}(1-{\left \lfloor{\ell\beta_1} \right \rfloor}/\ell)^2}{2}\fra
| 3,421
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ion, $h_\alpha L_i L_j$, should be present, which can lead to $H^-$ (on-shell or off-shell) decays into $l^-\nu$. Even in this case, lepton number is still conserved, just as in the Standard Model, since the vector quark and charged Higgs will naturally carry the lepton number ($L=\pm
2)$. Another way for $H$ to decay is to introduce a second Higgs doublet and let $H$ couple to two different Higgs doublets. In that case $H$ can decay into a neutral Higgs, plus a charged Higgs which in turn decays into ordinary quarks and leptons.
Spontaneously Broken CP symmetry {#spontaneously-broken-cp-symmetry .unnumbered}
================================
We shall comment on the corresponding model in which CP is broken spontaneously. This can be implemented by adding a CP-odd scalar, $a$, which develops a non-zero vacuum expectation value (VEV) and breaks CP. However, this scalar will in general couple to $\bar{Q_L}Q_R$ and give rise a complex tree level vector quark mass and, therefore, a tree level $\theta_{\rm QCD}$. To avoid this, one can add another CP-even scalar singlet, $s$, and impose discrete symmetries which change the signs of either or both $a$ and $s$ and nothing else. As a result, a term such as $ i a \bar Q \gamma_5 Q$ is forbidden and the only additional term relevant for CP violation is $i\left[ s \,a
\,({h_1}^{\dag} h_2 - {h_2}^{\dag} h_1) \right]$. This extra term will give rise to complex $(m^2)_{12}$ after both $s$ and $a$ develop VEVs and break CP. Note that before breaking CP spontaneously, there are two possible definitions of CP symmetry, depending on which of $s$ and $a$ are defined to be CP odd; this is why both must develop develop VEVs in order to break CP. The extra neutral Higgs will of course mix with the SM Higgs, but since $a$ does not couple to fermions directly, it will have scalar-pseudoscalar coupling to fermions only at the loop level. As a result, its contribution to any CP violating phenomenology will be small.
Conclusion {#conclusion .unnumbered}
==========
We have proposed a mod
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is with Lemma \[T:FT-isometry\](\[I:isometry\]) it follows that $\mu^{(n)}(f)$ is ${\left\vert G^{(n)} \right\vert}^{-1/2}$-Lipschitz as a function of $Y^{(n)}$, and so $${\mathbb{P}}\left[{\left\vert \mu^{(n)}(f) - {\mathbb{E}}\mu^{(n)}(f)\bigr) \right\vert}
\ge t \right] \le 2 e^{-c t \sqrt{{\left\vert G^{(n)} \right\vert}/ K}}.$$ Combined with the already known convergence in mean and the Borel–Cantelli lemma, this implies almost sure convergence of $\mu(f)$.
2. The proof is similar to the previous part, using instead Talagrand’s convex-distance concentration inequality for independent bounded random variables [@Talagrand-ihes Theorem 4.1.1] (see e.g. [@Meckes-jfa Corollary 4] for an explicit statement of a version that applies directly to complex random variables), cf. the proof of [@Meckes Theorem 2]).
3. The stated Lyapunov-type assumption yields upper bounds on all the $\delta_n$ quantities in the proofs above of order ${\left\vert G^{(n)} \right\vert}^{-\delta/2}$ for $0 < \delta \le 1$ (cf. [@BhRa Corollary 18.3]). Thus the assumption that $\sum_{n=1}^\infty {\left\vert G^{(n)} \right\vert}^{-\delta/2}$ allows the Borel–Cantelli lemma to be applied again.
4. The assumption that $p > 0$ implies that ${\left\vert G^{(n)} \right\vert}$ actually grows exponentially: since $p_2^{(n)}$ is always the reciprocal of an integer (by Lagrange’s theorem about the orders of subgroups of finite groups), $p_2^{(n)} \to p > 0$ implies that $p_2^{(n)}$ is eventually constant. By the classification of finite abelian groups, $$G \cong \left(\prod_{j = 1}^m {\mathbb{Z}}_{2^{k_j}} \right) \times H,$$ where $m \ge 0$, $k_j \ge 1$ for each $j$, and each nonidentity element of $H$ has odd order. (For simplicity of notation, we are again suppressing the dependence of all these on $n$.) In this notation, the number of $a \in G$ such that $a = a^{-1}$ is $2^m$, so that ${\left\vert G \right\vert} = 2^m / p_2$. The hypothesis that ${\left\vert G^{(n)} \right\vert}$ is strictly increasing thus implies that $
| 3,423
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re general form of the restriction on the inclusion that is needed for image continuity to behave well for subspaces of $Y$.
**Theorem A2.3.** *Let* $q\!:(X,\mathcal{U})\rightarrow(Y,\textrm{FT}\{\mathcal{U};q\})$ *be an image continuous* *function. For a subspace* $B$ of $(Y,\textrm{FT}\{\mathcal{U};q\})$,$$\textrm{FT}\{\textrm{IT}\{ j;\mathcal{U}\};q_{<}\}=\textrm{IT}\{ i;\textrm{FT}\{\mathcal{U};q\}\}$$
*where* $q_{<}\!:(q^{-}(B),\textrm{IT}\{ j;\mathcal{U}\})\rightarrow(B,\textrm{FT}\{\textrm{IT}\{ j;\mathcal{U}\};q_{<}\})$, *if either $q$ is an* *open map or $B$ is an open set of* $Y$.$\qquad\square$
In summary we have the useful result that an open preimage continuous function is image continuous and an open image continuous function is preimage continuous, where the second assertion follows on neglecting non-saturated open sets in $X$; this is permitted in as far as the generation of the final topology is concerned, as these sets produce the same images as their saturations. Hence *an image continuous function* $q\!:X\rightarrow Y$ *is preimage continuous iff every open set in $X$ is saturated with respect to $q$,* and *a preimage continuous function* $e\!:X\rightarrow Y$ *is image continuous iff the $e$-image of every open set of $X$ is open in $Y$.*
**A3. More on Topological Spaces**
This Appendix — which completes the review of those concepts of topological spaces begun in Tutorial4 that are needed for a proper understanding of this work — begins with the following summary of the different possibilities in the distribution of $\textrm{Der}(A)$ and $\textrm{Bdy}(A)$ between sets $A\subseteq X$ and its complement $X-A$, and follows it up with a few other important topological concepts that have been used, explicitly or otherwise, in this work.
**Definition A3.1.** ***Separation, Connected Space*.** *A* *separation* *(disconnection)* *of $X$ is a pair of mutually disjoint nonempty open (and therefore closed) subsets $H_{1}$ and $H_{2}$ such that $X=H_{1}\cup H_{2}$* *A space $X$ is said to be* *c
| 3,424
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| 0.774523
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|
_{{\mathbb{R}}^{d}}\left\vert s_{t}(x,y)\right\vert
_{q_{1}+q_{2}}\psi _{\kappa }(y)\times \left\vert f(y)\right\vert dy \\
&\leq \frac{C}{(\lambda t)^{\theta_0(q_1+q_2+\theta_1)}} \left\Vert
f\right\Vert _{\infty }\int_{{\mathbb{R}}^{d}}\frac{\psi _{\pi
(q_{1}+q_{2},\kappa +d+1)}(x)}{\psi _{\kappa +d+1}(x-y)}\times \psi _{\kappa
}(y)dy \\
&\leq \frac{C}{(\lambda t)^{\theta_0(q_1+q_2+\theta_1)}} \left\Vert
f\right\Vert _{\infty }\int_{{\mathbb{R}}^{d}}\frac{\psi _{\pi
(q_{1}+q_{2},\kappa +d+1)+\kappa }(x)}{\psi _{d+1}(x-y)}dy \\
&\leq \frac{C}{(\lambda t)^{\theta_0(q_1+q_2+\theta_1)}}\left\Vert
f\right\Vert _{\infty }\psi _{\pi (q_{1}+q_{2},\kappa +d+1)+\kappa }(x).\end{aligned}$$This implies (\[B2\]). $\square $
We are now able to give the regularity lemma. This is the core of our approach.
\[Reg\] Suppose that Assumption \[H1H\*1\], \[H2H\*2\] and \[HH3\] hold. We fix $t\in (0,1]$, $m\geq 1$ and $\delta_{i}>0$, $i=1,\ldots ,m$ such that $\sum_{i=1}^{m}\delta_{i}=t.$
$\mathbf{A.}$ There exists a function $\tilde{p}_{\delta_{1},...,\delta_{m}}%
\in C^{\infty }({\mathbb{R}}^{d}\times {\mathbb{R}}^{d})$ such that $$\prod_{i=1}^{m-1}(S_{\delta_{i}}U_{i})S_{\delta_{m}}f(x)=\int \tilde{p}%
_{\delta_{1},...,\delta_{m}}(x,y)f(y)dy. \label{h6}$$
$\mathbf{B.}$ We fix $q_{1},q_{2}\in {\mathbb{N}},\kappa \geq 0,p>1$ and we denote $q=q_{1}+q_{2}+(a+b)(m-1).$ One may find universal constants $C,\chi ,%
\bar{p}\geq 1$ (depending on $\kappa ,p$ and $q_{1}+q_{2})$ such that for every multi-index $\beta $ with $\left\vert \beta \right\vert \leq q_{2}$ and every $x\in {\mathbb{R}}^{d}$$$\left\Vert \partial _{x}^{\beta }\tilde{p}_{\delta _{1},...,\delta
_{m}}(x,\cdot )\right\Vert _{q_{1},\kappa ,p}\leq C\Big(\frac{2m}{\lambda t}%
\Big)^{\theta _{0}(q_{1}+q_{2}+d+2\theta _{1})}\Big(C_{q,\chi ,\bar{p}%
,\infty }(U,S)\Big(\frac{2m}{\lambda t}\Big)^{\theta _{0}(a+b)}\Big)%
^{m-1}\psi _{\chi }(x). \label{h7}$$
**Proof.** **A.** For $g=g(x,y)$, we denote $g^{x}(y):=g(x,y)$. By the very definition of $U_{i}^{\ast }$ one has$$S_{t}U
| 3,425
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_system}\\
\frac{d\E({\mathbf{u}})}{dt} & = \R({\mathbf{u}}). \label{eq:dEdt_system}\end{aligned}$$
A standard approach at this point is to try to upper-bound $d\E / dt$ and using standard techniques of functional analysis it is possible to obtain the following well-known estimate in terms of $\K$ and $\E$ [@d09] $$\label{eq:dEdt_estimate_KE}
\frac{d\E}{dt} \leq -\nu \frac{\E^2}{\K} + \frac{c}{\nu^3} \E^3$$ for $c$ an absolute constant. A related estimate expressed entirely in terms of the enstrophy $\E$ is given by $$\frac{d\E}{dt} \leq \frac{27}{8\,\pi^4\,\nu^3} \E^3.
\label{eq:dEdt_estimate_E}$$ By simply integrating the differential inequality in with respect to time we obtain the finite-time bound $$\E(t) \leq \frac{\E(0)}{\sqrt{1 - \frac{27}{4\,\pi^4\,\nu^3}\,\E(0)^2\, t}}
\label{eq:Et_estimate_E0}$$ which clearly becomes infinite at time $t_0 = 4\,\pi^4\,\nu^3 /
[27\,\E(0)^2]$. Thus, based on estimate , it is not possible to establish the boundedness of the enstrophy $\E(t)$ globally in time and hence the regularity of solutions. Therefore, the question about the finite-time singularity formation can be recast in terms of whether or not estimate can be saturated. By this we mean the existence of initial data with enstrophy $\E_0 := \E(0)> 0$ such that the resulting time evolution realizes the largest growth of enstrophy $\E(t)$ allowed by the right-hand side (RHS) of estimate . A systematic search for such most singular initial data using variational optimization methods is the key theme of this study. Although different notions of sharpness of an estimate can be defined, e.g., sharpness with respect to constants or exponents in the case of estimates in the form of power laws, the precise notion of sharpness considered in this study is the following
\[def:NotionSharpness\] Given a parameter $p\in\mathbb{R}$ and maps $f,g:\mathbb{R}\to\mathbb{R}$, the estimate $$f(p) \leq g(p)$$ is declared sharp in the limit $p \to p_0\in\mathbb{R}$ if and only if $$\lim_{p \to p_0} \frac{f(p)}{g(p)} \sim \beta, \quad
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so $\b^2(x,y)=-k$, whereas the link $\{c(y,x),L_y\}$ is a Whitehead link so $\b^2(y,x)= -1$. We claim further that, as long as $k\neq 0$, for $\textbf{any}$ basis $\{X,Y\}$ of $H^1(M)$, $\b^2(X,Y)\neq 0$. It will then follow from Theorem \[linear\] that the first Betti number of $M$ will grow sub-linearly in **any** family of finite cyclic covers. A general basis, $\{V_X,V_Y\}$, of $H_2(M)$ can be represented as follows. Represent $V_X$ by $p$ parallel copies of $V_x$ together with $q$ parallel copies $V_y$, and represent $V_Y$ by $r$ parallel copies of $V_x$ together with $s$ parallel copies $V_y$, where $ps-qr=\pm 1$. Thus $c(X,Y)=-c(Y,X)$ is represented by $ps-qr$ parallel copies of $c(x,y)$. It follows that $\b^1(X,Y)=\b^1(x,y)=0$, reinforcing our above claim that $\b^1$ is independent of basis. Hence $V_{c(X,Y)}=\pm V_{c(x,y)}$ so $c(X,X,Y)$ is represented by $\pm pc(x,x,y)\mp qc(y,y,x)$. Since $\b^2(X,Y)$ is the self-linking number of this class, it can be evaluated to be $$p^2\b^2(x,y) + q^2\b^2(y,x) - 2pq{\ensuremath{\ell k}}(c(x,x,y),c(y,y,x))$$ but the latter mixed linking number is easily seen to be zero in this case. Hence $\b^2(X,Y)=-kp^2-q^2$ which is non-zero if $k$ is non-zero.
\[equivalence\] Suppose $c(1),\dots,c(n)$ have been defined as embedded oriented curves on $V_x$ arising as $c(1)=V_x\cap V_y$ and $$c(j) = V_x\cap V_{c(j-1)}\qquad2\le j\le n$$ where $V_{c(j)}$, $1\le j\le n-1$, is an embedded, oriented connected surface whose boundary is a positive multiple $k_j$ of $c^+(j)$ (in the sense above). Then $\b^j$ is defined for $1\le j\le n-1$ and the following are equivalent:
1. $\b^1,\dots,\b^n$ are defined using the given system of surfaces.
2. $\b^j$ is defined for $1\le j\le n$ and is [**zero**]{} for $1\le
j\le\[\f n2\]$
3. $c(n+1)$ exists
4. For all $s$, $t$ such that $1\le s\le t$ and $s+t\le n$ , ${\ensuremath{\ell k}}(c(s),c^+(t))=0$.
Assume $1\le j\le
n-1$. The hypotheses imply that a positive multiple of $c^+(j)$ is (homotopic to) the boundary of a surface so $c^+(j
| 3,427
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replacing ${\mathfrak{h}}\oplus{\mathfrak{h}}^*/ {{W}}$, $\operatorname{Hilb(n)}$ and ${\mathcal{P}}$ by $H_c{\text{-}{\textsf}{mod}}$, $B{\text{-}{\textsf}{qgr}}$ and $eH_c$, respectively. Then Corollary \[morrat-cor\] shows that $eH_c$ still induces a derived equivalence between the two categories. Indeed, it is even a equivalence of categories. The fact that derived equivalences in the commutative case can become full equivalences in the noncommutative case happens elsewhere and is in accord with the philosophy behind [@GK Conjecture 1.6] (see [@GK Remark 1.7]).
As will be justified in [@GS2], Corollary \[cohh-subsect\] therefore “sees” the equivalence $\xi$ and this provides some intriguing connections between sheaves on $\operatorname{Hilb(n)}$ and modules over $H_c$.
{#section-3}
If one considers Cherednik algebras in characteristic $p>0$, where $H_c$ is a finite module over its centre, then the relationship between $H_c$ and $\operatorname{Hilb(n)}$ becomes closer still. For example, [@BFG] shows that there is even a derived equivalence between $H_c$ and an Azumaya algebra over a Frobenius twist of $\operatorname{Hilb(n)}$. Similarly in characteristic zero, symplectic reflection algebras with parameter $t=0$ are finite modules over their centre, and [@GSm Theorem 1.2] shows that there are often derived equivalences between these algebras and varieties that deform Hilbert schemes.
Tensor product filtrations {#sect7}
==========================
{#sect701}
The tensor product decomposition of the $B_{ij}$ can be used to give a second filtration on that module by inducing a filtration on $B_{ij}$ from the $\operatorname{{\textsf}{ord}}$ filtration on the tensorands. It turns out that the main theorem is essentially equivalence to the assertion that the two filtrations are equal. In this short section we give the details behind this assertion. Analogues of this result also hold for the module $N(k)$ defined in and the module $M(k)=H_{c+k}eB_{k0} =H_{c+k}\delta e
B_{k-1,0}$ defined in and so we beg
| 3,428
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| 1,361
| 3,161
| 1,749
| 0.786063
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a more fair comparison, we conducted the second baseline based on two-stage fine-tuning, namely *Fast-RCNN (2 fts)*. This baseline first fine-tuned the VGG-16 using numerous object-box annotations in the target category, and then fine-tuned the VGG-16 using a few part annotations.
The third baseline was proposed in [@CNNSemanticPart], namely *CNN-PDD*. *CNN-PDD* selected a filter in a CNN (pre-trained using ImageNet ILSVRC 2012 dataset) to represent the part on well-cropped objects. Then, we slightly extended [@CNNSemanticPart] as the fourth baseline *CNN-PDD-ft*. *CNN-PDD-ft* first fine-tuned the VGG-16 using object bounding boxes, and then applied [@CNNSemanticPart] to learn object parts.
The strongly supervised DPM (*SS-DPM-Part*) [@SSDPM] and the approach of [@PLDPM] (*PL-DPM-Part*) were the fifth and sixth baselines. These methods learned DPMs for part localization. The graphical model proposed in [@SemanticPart] was selected as the seventh baseline, namely *Part-Graph*. The eighth baseline was the interactive learning for part localization [@ActivePart] (*Interactive-DPM*).
Without lots of training samples, “simple” methods are usually insensitive to the over-fitting problem. Thus, we designed the last four baselines as follows. We first fine-tuned the VGG-16 using object bounding boxes, and collected image patches from cropped objects based on the selective search [@SelectiveSearch]. We used the VGG-16 to extract *fc7* features from image patches. The two baselines (*i.e.* *fc7+linearSVM* and *fc7+RBF-SVM*) used a linear SVM and an RBF-SVM, respectively, to detect object parts. The other baselines *VAE+linearSVM* and *CoopNet+linearSVM* used features of the VAE network [@VAE] and the CoopNet [@CoopNet], respectively, instead of *fc7* features, for part detection.
The last baseline [@CNNAoG] learned AOGs without QA (*AOG w/o QA*). We randomly selected objects and annotated their parts for training.
Both object annotations and part annotations are used to learn models in all the thirteen basel
| 3,429
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v}$ is the monomial $\prod_{i\in I}X_i^{v_i}$ for some independent commuting variables $\{X_i\}_{i\in I}$. \[hua\].
Since $A_{\Gamma,\v}(q)\in\Z[q]$, we see by Theorem \[hua\] and Lemma \[Exp\], that $M_{\Gamma,\v}(q)$ also has integer coefficients.
Comet-shaped quivers {#comet}
--------------------
Fix strictly positive integers $g,k,s_1,\dots,s_k$ and consider the following (comet-shaped) quiver $\Gamma$ with $g$ loops on the central vertex and with set of vertices $I=\{0\}\cup\left\{[i,j]\,|\, i=1,\dots,k\,;\, j=1,\dots,s_i\right\}$.
0.1in
( 52.1000, 15.4500)( 4.0000,-17.0000)
(19.7000,-2.4500)[(0,0)[$[1,1]$]{}]{}(29.7000,-2.4000)[(0,0)[$[1,2]$]{}]{}(55.7000,-2.5000)[(0,0)[$[1,s_1]$]{}]{}(19.7000,-6.5500)[(0,0)[$[2,1]$]{}]{}(29.7000,-6.4500)[(0,0)[$[2,2]$]{}]{}(55.7000,-6.5500)[(0,0)[$[2,s_2]$]{}]{}(19.7000,-17.8500)[(0,0)[$[k,1]$]{}]{}(29.7000,-17.8500)[(0,0)[$[k,2]$]{}]{}(55.7000,-17.8500)[(0,0)[$[k,s_k]$]{}]{}(14.3000,-7.6000)[(0,0)[$0$]{}]{}
Let $\Omega^0$ denote the set of arrows $\gamma\in\Omega$ such that $h(\gamma)\neq t(\gamma)$.
Let $\K$ be any field. Let $\varphihat\in{\rm
Rep}_{\Gamma,\v}(\K)$ and assume that $v_0> 0$. If $\varphihat$ is indecomposable, then the linear maps $\varphi_\gamma$, with $\gamma\in\Omega^0$, are all injective. \[injective\]
If $\gamma$ is the arrow $[i,j]\rightarrow [i,j-1]$, with $j=1,\dots,s_i$ and with the convention that $[i,0]=0$, we use the notation $\varphi_{ij}:V_{[i,j]}\rightarrow V_{[i,j-1]}$ rather than $\varphi_\gamma:V_{t(\gamma)}\rightarrow V_{h(\gamma)}$. Assume that $\varphi_{ij}$ is not injective. We define a graded vector subspace $\Vhat'=\bigoplus_{i\in I}V_i'$ of $\Vhat=\bigoplus_{i\in I}V_i$ as follows.
If the vertex $i$ is not one of the vertices $[i,j],[i,j+1],\dots,[i,s_i]$, we put $V'_i:=\{0\}$. We put $V'_{[i,j]}:={\rm Ker}\, \varphi_{ij}$, $V'_{[i,j+1]}:=\varphi_{i(j+1)}^{-1}(V'_{[i,j]}),\dots,V'_{[i,s_i]}:=\varphi_{is_i}^{-1}(V'_{i(s_i-1)})$. Let $\v'$ be the dimension of the graded space $\Vhat'=\bigoplus_{i\in
I}V_i'$ which w
| 3,430
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[1](#gcbb12419-tbl-0001){ref-type="table-wrap"}). The genotypes used were *M. sinensis*,*M. sacchariflorus*,*M. sinensis* × *M. sacchariflorus* hybrids and an *M. sacchariflorus var robustus* which were representative of those in the mixed population and mapping family. At the end of each growing season (January--March), an area of each plot was harvested to give a yield for each genotype in t ha^−1^. The weight of each individual plant at each time point was then modelled as a % of the final harvest mass (Fig. [4](#gcbb12419-fig-0004){ref-type="fig"}a). Although autumn yields have been reported or calculated as a % of final yield in previous publications (Clifton‐Brown *et al*., [1998](#gcbb12419-bib-0008){ref-type="ref"}; Whittaker *et al*., [2016](#gcbb12419-bib-0047){ref-type="ref"}), July yields have not. The mean value of the four genotypes in July was 50% of the final harvest mass which was, on average, 30% of peak, autumn biomass (Fig. [4](#gcbb12419-fig-0004){ref-type="fig"}b). In October, yields were projected to be an average of 40% higher than harvest weight (Fig. [4](#gcbb12419-fig-0004){ref-type="fig"}a). This finding is in close agreement with Kiesel & Lewandowski ([2016](#gcbb12419-bib-0023){ref-type="ref"}) who observed that harvested biomass was 39% higher in October compared to February in *M. x giganteus* in Germany.
{#gcbb12419-fig-0004}
Projected NSC yields {#gcbb12419-sec-0022}
--------------------
Based on the modelled values and the final yield harvest the following spring (Figs [3](#gcbb12419-fig-0003){ref-type="fig"} and [4](#gcbb12419-fig-0004){ref-type="fig"}), the mass of plants at the two time points was calculated (Fig. [5](#gcbb12419-fig-0005){ref-type="fig"}) and the
| 3,431
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@derrida1986random; @keeling2005networks]. Therefore, $\psi$ is estimated as the rate of $N i (k-2)$ to the number of all edges $N k$ in the thermodynamic limit. That is, $\psi=i (k-2)/k$. Below, we focus on the case $k=3$.
Let $P(s,i,t)$ be the probability density of $s(t)=s$ and $i(t)=i$. Then, $P(s,i,t)$ obeys the master equation $$\begin{aligned}
\frac{\partial P(s,i,t)}{\partial t}
&=& N \left( i+\frac{1}{N} \right) P \left(s,i+\frac{1}{N},t \right)
-N i P \left(s,i,t \right) \nonumber \\
&+& N \lambda \left (s+\frac{1}{N}\right) \left(i-\frac{1}{N}\right)
P\left( s+\frac{1}{N},i-\frac{1}{N},t\right) \nonumber \\
&-& N\lambda s i P\left(s,i,t\right).
\label{MST}\end{aligned}$$ When $N$ is sufficiently large, the master equation for $P(s,i,t)$ can be expanded as $$\frac{\partial P}{\partial t}
+\partial_i J_i +\partial_s J_s+O\left( \frac{1}{N^2} \right)=0,
\label{sNMST}$$ with $$\begin{aligned}
J_i&=&
\left(\lambda s-1\right) i P
- \partial_i\left[ \frac{\left(\lambda s+1\right)i}{2N} P \right]
+ \partial_s \left(\frac{\lambda s i }{2N}P \right), \nonumber \\
J_s&=&
-\lambda s i P
- \partial_s \left( \frac{\lambda s i }{2N}P\right)
+\partial_i \left(\frac{\lambda s i}{2N} P \right).\end{aligned}$$ By assuming that $O(1/N^2)$ terms can be ignored, we obtain the Fokker-Planck equation [@gardiner2004handbook].
It can be confirmed by direct calculation that this Fokker-Planck equation (\[sNMST\]) describes the time evolution of the probability density for the following set of Langevin equations: $$\begin{aligned}
{\frac{d s}{d t}} &=& -\lambda s i -\sqrt{\frac{\lambda s i}{N}}\cdot \xi_1,
\label{s_lgv}
\\
{\frac{d i}{d t}} &=& \lambda s i - i +\sqrt{\frac{\lambda s i}{N}}\cdot \xi_1
+\sqrt{\frac{i}{N}}\cdot \xi_2,
\label{i_lgv} \end{aligned}$$ where $\xi_i$ is Gaussian white noise that satisfies $\left<\xi_i\left(t\right) \right>=0$ and $ \left<\xi_i\left(t\right) \xi_j\left(t'\right) \right>
=\delta_{i j}\delta\left(t-t'\right)$. The symbol “$\cdot$” in front of $\xi_1$ and $\xi_2$ i
| 3,432
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|
ne is denoted by ${\cal L}_i$ where we have separated a constant vacuum energy contribution, $V_{i}$. The following equations result from varying the action in Eqn.(2) with respect to the metric, \[1,2,3\]: $$\begin{aligned}
\frac{1}{f^2}\left[ (\frac{\dot{v}}{v})^2 \right] - \frac{1}{f^2}
\left[\frac{f''}{f} + (\frac{f'}{f})^2 \right] & = &
- \frac{\kappa^2}{3 f}[\sum_i({\cal L}_i + V_i)\delta(z - L_i)] +
\frac{\Lambda_5}{3}\nonumber \\
& & \nonumber\\
\frac{1}{f^2}\left[ (2\frac{\ddot{v}}{v})+ (\frac{\dot{v}}{v})^2\right]
- \frac{3}{f^2}\left[ \frac{f''}{f} + (\frac{f'}{f})^2\right] & = &
- \frac{\kappa^2}{f}\left[ \sum_i({\cal L}_i + V_i)\delta(z - L_i)\right]
+{\Lambda_{5}}\nonumber\\
& & \nonumber\\
\frac{1}{f^2}\left[ (\frac{2\ddot{v}}{v})+ (\frac{\dot{v}}{v})^2\right]
- \frac{2}{f^2}(\frac{f'}{f})^2 & = & \frac{\Lambda_5}{3} \end{aligned}$$ where prime and dot denote differentiation with respect to z and t respectively. It is straightforward from the Eqns. (4) above (as well as the component $G_{55}=0$), that $v(t)$ satisfies: $$\frac{\ddot{v}}{v} = \frac{\dot{v}^2}{v}$$ which gives for $v$: $$v(t) = v(0)exp[Ht]$$ Thus Eqns. (4) become: $$\begin{array}{lcc}
\frac{H^2}{f^2} - \frac{1}{f^2} \left[ \frac{f''}{f} +
(\frac{f'}{f})^2 \right] & = &
- \frac{\kappa^2}{3f} [\sum_i({\cal L}_i + V_i)]\delta(z - L_i) +
\frac{\Lambda_5}{3} \\
& & \\
\frac{H^2}{f^2} - \frac{2}{f^2}(\frac{f'}{f})^2
= \frac{\Lambda_5}{6}
\end{array}$$ The effective expansion rate is defined as $H_{eff} = H / f(z) $ . The effective expansion rate of each brane, located at $z = L_i$ in the extra dimension, is then given by $H_{ieff} = H / f(L_i)$. Thus each brane, depending on its position in the fifth dimension, sees a different expansion rate resulting from the fact that the canonical proper time has the following scaling on the position of the brane, $d \tau_{i} = f( L_{i}) dt$. Denote the absolute value of $\Lambda_5$ by $\Lambda$. Since we consider an $ADS_5$ geometry, $\Lambda$ is positive and satisfies the f
| 3,433
| 4,740
| 2,612
| 2,809
| null | null |
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$ the random variable parameterized by $\alpha$. The expected value $$E(Y)=\int p(\alpha) P_{bind}^{\cal{B}}(m,\alpha)[1-P_{bind}^{\cal{A}}(m,\alpha)]{\mathbf d\alpha} %\equiv \bar{P}_{ch}(m)$$ is nothing but the [*expected probability to cheat*]{} $\bar{P}_{ch}(m)$ (note that due to $P^{\cal{B}}_{bind}(m, \alpha) \approx P^{\cal{A}}_{bind}(m, \alpha)$, we have $\bar{P}_{ch}(m) \equiv \bar{P}^{\cal{B}}_{ch}(m) \approx \bar{P}^{\cal{A}}_{ch}(m)$ ). Using the above fairness criterion $\bar{P}_{ch}(m) < \varepsilon$, Chebyshev inequality, and putting $\delta^3=\varepsilon$, we obtain $\mbox{Prob}_\alpha[Y<\delta+\delta^3]\geq 1-\delta.$ Thus, the probability $\delta$ can be made arbitrarily small with arbitrarily high probability.
IV. Fairness of the protocol: Ideal case {#sec:fairness-ideal}
========================================
In the following, we show that our protocol is [*fair*]{}: both $|P^{\cal{B}}_{bind}(m, \alpha) - P^{\cal{A}}_{bind}(m, \alpha)|$ and $\bar{P}_{ch}(m)$ could be made arbitrarily low. In this section, we assume that only $\hat{A}$ or $\hat{R}$ are measured, and that no measurement errors or qubit state corruption occur. In the next section, we discuss general (one or multi-qubit) observables and real-life scenario of imperfect measurements and noisy channels.
In case Bob is cheating during the Exchange phase, he will be detected after a small number of steps, with probability growing exponentially in the number of qubits measured by Alice. Assume Bob’s cheating is detected after Alice measured $m$ qubits. Alice terminated the Exchange phase and participants proceed with the Binding phase, that can be delayed (Trent is offline). Meanwhile, participants are allowed to change their preferences and we would like to examine symmetry of their position. We are interested only in the situation when Alice wants to bind the protocol and Bob wants to reject, and vice versa.
In the former case Alice tries to do her best to bind the contract. This means she measures all unmeasured qubits in the
| 3,434
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|
e outcome of the sample splitting. The claimed results follows almost directly from , with few additional technicalities. The first difficulty is that the least squares estimator is not always well-defined under the bootstrap measure, which is the probability distribution of $n$ uniform draws with replacement from $\mathcal{D}_{2,n}$. In fact, any draw consisting of less than $d$ distinct elements of $\mathcal{D}_{2,n}$ will be such that the corresponding empirical covariance matrix will be rank deficient and therefore not invertible. On the other hand, because the distribution of $\mathcal{D}_{2,n}$ has a Lebesgue density by assumption, any set of $d$ or more points from $\mathcal{D}_{2,n}$ will be in general position and therefore will yield a unique set of least squares coefficients. To deal with such complication we will simply apply on the event that the bootstrap sample contains $d$ or more distinct elements of $\mathcal{D}_{2,n}$, whose complementary event, given the assumed scaling of $d$ and $n$, has probability is exponentially small in $n$, as shown next.
\[eq:lem.occupancy\] For $d \leq n/2$, the probability that sampling with replacement $n$ out of $n$ distinct objects will result in a set with less than $d$ distinct elements is no larger than $$\label{eq:occupancy}
\exp \left\{ - \frac{n (1/2 - e^{-1})^2}{2} \right\}.$$
[**Remark.**]{} The condition that $d \leq n/2$ can be replaced by the condition that $d \leq c n$, for any $ c \in (0, 1 - e^{-1})$.
Thus, we will assume that the event that the bootstrap sample contains $d$ or more distinct elements of $\mathcal{D}_{2,n}$. This will result in an extra term that is of smaller order than any of the other terms and therefore can be discarded by choosing a larger value of the leading constant.
At this point, the proof of the theorem is nearly identical to the proof of except for the way the term $A_3$ is handled. The assumptions that $n$ be large enough so that $v_n$ and $u_n$ are both positive implies, by and Weyl’s theorem, that, for each $P
| 3,435
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|
ameters increases exponentially with $k$ which may result in overfitting [@murphy] since we can always produce better fits to the data with more model parameters.]{}]{} To demonstrate this behavior, we produced a random navigational dataset by randomly (uniformly) picking a next click state out of a list of arbitrary states. One of these states determines that a path is finished and a new one begins. With this process we could generate a random path corpus that is close to one main dataset of this work (Wikigame topic dataset explained in the section called “”). Concretely, we as well chose 26 states and the same number of total clicks. Purely from our intuition, such a process should produce navigational patterns with an appropriate Markov chain order of zero or at maximum one. However, if we look at the log-likelihoods depicted in Figure \[fig:randomloglikelihood\] we can observe that the higher the order the higher the corresponding log likelihoods are.
This strongly suggests that – as previously explained – looking at the log-likelihoods is not enough for finding the appropriate Markov chain order. Hence, we first resort to a well-known statistical likelihood tool for comparing two models – the so-called *likelihood ratio test*.
This test is suited for comparing the fit of two composite hypothesis where one model – the so-called *null model* $k$ – is a special case of the *alternative model* $m$. The test is based on the log likelihood ratio, which expresses how much more likely the data is with the alternative model than with the null model. We follow the notation provided by Tong [@tong1975] and denote the ratio as ${_k}\eta{_m}$: $${_k}\eta{_m} = -2(\mathcal{L(P(D|}{\theta}_k)) -
\mathcal{L(P(D|\theta}_m)))
\label{eq:lr}$$
To address the overfitting problem we perform a significance test on this ratio. The significance test recognizes whether a better fit to data comes only from the increased number of parameters. The test calculates the p-value of the likelihood ratio distribution. Wh
| 3,436
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barrier are characterized by the formation of quasiregular mound structures differently from those obtained using the original model that exhibits irregular structures within the intervals of size and time we investigated. These plots also show a coarsening of the mounds represented by the first minimum displacement at the early growth times.
![Main plot: Correlation function for the DT model in two-dimensional substrates for distinct times shown in the legends and fixed $N_s=10$. Inset: Correlation function for DT model in two dimensions at a fixed time $t=10^5$ and different values of $N_s$ shown in legends. Curves correspond to averages over 100 independent samples.[]{data-label="h_corr_DT"}](gamma_dt2d_ts_Ns.pdf){width="0.8\linewidth"}
The effect of the parameter $N_s$ in WV model is shown in Fig. \[hh\_corr\_Ns\]. As indicated by the interface profiles shown in Figs. \[snapshot\_1d\] and \[snapshot\_2d\], the characteristic lateral length increases with $N_s$ in both dimensions. The correlation function for DT model follows a qualitative similar dependence with $N_s$, as can be seen in Fig. \[h\_corr\_DT\] where the effects of time and number of diffusion steps in the correlation function of the DT model are shown. However, the mounds are much less evident than those obtained in the WV model. However, the correlation functions still present the typical oscillatory behavior of mounded structures that is preserved after the averaging over 100 independent samples. Besides, the typical width of the mounds in the DT model are much smaller than those of WV. It is important to note that the correlation function of the original DT model also presents an irregular behavior as does the WV model.
![\[width\]Time evolution of the interface width $w$ for WV (main panels) and DT (insets) models grown on (a) one- and (b) two-dimensional substrates. Both simulations with the kinetic barrier (using $N_s$ values indicated in the legend) and the original version are shown. In (a), dashed and solid lines are power-laws w
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Assume moreover that there exist a constant $c>0$ and $s\in[0,1]$ such that for $u\in C_c^\infty({\mathbb{R}}^d)$ $$\label{additional AS Svhrödinger example}\int_{{\mathbb{R}}^d}m_1(\xi)|u(\xi)|^2 {\mathrm{d}}\xi\leq c\|u\|_{H^s({\mathbb{R}}^d)}.$$Consider the non-autonomous Cauchy problem $$\label{Schroedinger operator}
\left\{
\begin{aligned}
\dot {u}(t) - &{\mathop{}\!\mathbin\bigtriangleup}u(t)+m(t,\cdot)u(t) = 0,
\\ u(0)&=x\in V.
\end{aligned} \right.$$ Here $A(t)=-{\mathop{}\!\mathbin\bigtriangleup}+m(t,\cdot)$ is associated with the non-autonomous form ${\mathfrak{a}}:[0,T]\times V\times V{\longrightarrow}{\mathbb{C}}$ given by $$V:=\left\{u\in H^1({\mathbb{R}}^d): \int_{{\mathbb{R}}^d}m_0(\xi)|u(\xi)|^2 {\mathrm{d}}\xi <\infty \right\}$$and $${\mathfrak{a}}(t;u,v)=\int_{{\mathbb{R}}^d}\nabla u\cdot\nabla v {\rm d} \xi+\int_{{\mathbb{R}}^d}m(t,\xi)|u(\xi)|^2 {\mathrm{d}}\xi.$$ The form ${\mathfrak{a}}$ satisfies also (\[eq:continuity-nonaut\])-(\[square property\]) with $\gamma:=s$ and $\omega(t)=t^\alpha$ for $\alpha>\frac{s}{2}$ and $s\in[0,1].$
This example is taken from [@Ou15 Example 3.1]. Using our Theorem \[main result\] we have that the solution of Cauchy problem (\[Schroedinger operator\]) is governed by a norm continuous evolution family on $L^2({\mathbb{R}}^d), V$ and $V'.$
[999]{}
P. Acquistapace. Evolution operators and strong solutions of abstract linear parabolic equations. *Differential Integral Equations* 1 (1988), no. 4, 433-457. R. A. Adams, J. J. F. Fournier. *Sobolev spaces.* Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. W. Arendt. *Heat kernels.* $9^{th}$ Internet Seminar (ISEM) 2005/2006. Available at. https://www.uni-ulm.de/mawi/iaa/members/professors/arendt.html
W. Arendt, S. Monniaux. Maximal regularity for non-autonomous Robin boundary conditions. Math. Nachr. 1-16(2016) /DOI: 10.1002/mana.201400319
H. Brézis. *Functional Analysis, Sobolev Spaces and Partial Differential Equations*. Springer, Berlin 2011.
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**vol. 11 (1), (2009), 17-33. preprint: arXiv:0711.0540**
K. Gruher and P. Salvatore, *Generalized string topology operations* Proc. Lond. Math. Soc. (3) **96 (2008), 78Ð106.**
J.A. Lind, *Bundles of spectra and Algebraic $K$-theory*, preprint arXiv:1304567
J. P. May and J. Sigurdsson, [Parametrized homotopy theory]{}. Mathematical Surveys and Monographs, vol. 132, Amer. Math. Soc., 2006 Mathematical Surveys and Monographs, [**vol. 132**]{}, Amer. Math. Soc., 2006
D. McDuff, G.B. Segal, *Homology fibrations and the “group completion" theorem*, Invent. Math., **31**, (1976), 279 - 284.
[^1]: The first author was partially supported by a grant from the NSF.
---
abstract: 'The principal portfolios of the standard Capital Asset Pricing Model (CAPM) are analyzed and found to have remarkable hedging and leveraging properties. Principal portfolios implement a recasting of any *correlated* asset set of $N$ risky securities into an equivalent but *uncorrelated* set when short sales are allowed. While a determination of principal portfolios in general requires a detailed knowledge of the covariance matrix for the asset set, the rather simple structure of CAPM permits an accurate solution for any reasonably large asset set that reveals interesting universal properties. Thus for an asset set of size $N$, we find a *market-aligned* portfolio, corresponding to the *market* portfolio of CAPM, as well $N-1$ *market-orthogonal* portfolios which are market neutral and strongly leveraged. These results provide new insight into the return-volatility structure of CAPM, and demonstrate the effect of unbridled leveraging on volatility.'
author:
- 'M. Hossein Partovi'
title: 'Hedging and Leveraging: Principal Portfolios of the Capital Asset Pricing Model'
---
1. Introduction {#introduction .unnumbered}
===============
Modern investment theory dates back to the mean-variance analysis of Markowitz (1952, 1959), which is expected to hold if asset prices are normally distributed or the investor preferences are quadratic. Undoub
| 3,439
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unctions of $x$, be the maximum integer no greater than $x$ and the minimum integer no less than $x$, respectively. Let $\floor{\w}_\ell$ and $\ceil{\w}_\ell$ be the vectors generated from $\w$ by applying the corresponding operation on the $\ell$-th element only. $\0$ denotes an all-zero vector, and $\I$ denotes an identity matrix. ${\rm sign}(\w)$ returns the vector that contains the signs of the elements in $\w$. ${\rm abs}(\w)$ returns the vector whose elements are the absolute values of the elements in $\w$.
Problem Statement {#section:ProblemStatement}
=================
We consider additive white Gaussian noise (AWGN) networks [@Nazer2011] where sources, relays and destinations are connected by linear channels with AWGN. For the ease of explanation, we first develop our method for real-valued channels, and then demonstrate how to apply our method to complex-valued channels. An AWGN network with real-valued channels is defined as the following.
\[definition:RealChannelModel\] *(Real-Valued Channel Model)* In an AWGN network, each relay (indexed by $m=1,2,\cdots,M$) observes a noisy linear combination of the transmitted signals through the channel, $$\begin{aligned}
\label{equation:RealChannelModel}
\y_m = \sum_{\ell=1}^L \h_m(\ell)\x_\ell + \z_m,\end{aligned}$$ where $\x_\ell \in \Rbb^n$ with the power constraint $\frac{1}{n}\norm{\x_\ell}^2 \leq P$ is the transmitted codeword from source $\ell$ ($\ell = 1,2,\cdots,L$), $\h_m \in \Rbb^L$ is the channel vector to relay $m$, $\h_m(\ell)$ is the $\ell$-th entry of $\h_m$, $\z_m \in \Rbb^n$ is the noise vector with entries being i.i.d. Gaussian, i.e., $\z_m\!\sim\!\bigN\!\left(\0,\I\right)$, and $\y_m$ is the signal received at relay $m$.
In the sequel, we will focus on one relay and thus ignore the subscript “$m$” in $\h_m$, $\a_m$, etc.
In CF, rather than directly decode the received signal $\y$ as a codeword, a relay first applies to $\y$ an amplifying factor $\alpha$ such that $\alpha\h$ is close to an integer *coefficient vector* $\a$, and tries to
| 3,440
| 1,155
| 2,057
| 3,340
| 1,390
| 0.79019
|
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|
t: $$\begin{aligned}
m & \leq & \frac {tr A} {\sum_{i \neq j} |A_{ij}|} =
\frac {n(1+\alpha)}
{\delta n^2 (\beta - \alpha) + ((1-\delta) n^2 - n) \alpha} \\
& = &
\frac {n+ \frac {\delta n} {\lambda_2} - 1}
{\delta n (\frac {n+\delta n} {\lambda_2} - 1) + ((1-\delta) n - 1)
(\frac {\delta n} {\lambda_2} - 1)}
<
4 \frac {1 + \frac {\delta} {\lambda_2}} {\frac {\delta n} {\lambda_2}}
= 4 \frac {\lambda_2 + \delta} {\delta n}.\end{aligned}$$ In particular, $b-a = O(\frac {\lambda_2 + \delta} {\delta n})$.
In order to derive a non trivial result from Johnson-Lindenstrauss lemma, we need that $\frac 1 {m^2} \log n = o(n)$, and in particular that $\lambda_2=\Omega(\delta \sqrt{n \log n})$. The above shows that this can happen only if $\lambda_2 = \omega(\delta \sqrt{n \log n})$. On the other hand, Frankl and Maehara show that their method does give a non trivial bound when $\lambda_n = o(\sqrt{\frac n {\log n}})$. Consequently, we get that a $\delta n$-regular graph (think of $\delta$ as constant) can’t have both $\lambda_2=o(\sqrt{n \log n})$ and $\lambda_n = o(\sqrt{\frac n {\log n}})$. This is a bit more subtle than what one gets from the second moment argument, namely, that the graph can’t have both $\lambda_2=o(\sqrt{n})$ and $\lambda_n = o(\sqrt{n})$.
Lower Bound on Sphericity
-------------------------
\[our-bound\] Let $G$ be a $d$-regular graph with diameter $D$ and $\lambda_2$, the second largest eigenvalue of $G$’s adjacency matrix, is at least $d - {\frac 1 2}n$. Then $Sph(G) = \Omega(\frac {d - \lambda_2} {D^2(\lambda_2 + O(1))})$.
In the interesting range where $d \leq \frac n 2$, and $\lambda_2 \geq 1$ the bound is $Sph(G) = \Omega(\frac {d - \lambda_2} {D^2 \lambda_2})$. It will be useful to consider the following operation on matrices. Let $A$ be an $n \times n$ symmetric matrix, and denote by ${\vec{a}}$ the vector whose $i$-th coordinate is $A_{ii}$. Define $R(A)$ to be the $n \times n$ matrix with all rows equal to ${\vec{a}}$, and $C(A) = R(A)^t$. Define: $$\begin{aligned}
\breve{A} = 2A
| 3,441
| 2,971
| 3,113
| 3,012
| 3,820
| 0.769896
|
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|
= \!{(i\eta)}^{\!\!^m}\!\! \sqrt{\!\!\tfrac{ n!}{(m+n)!} } \,
\text{e}^{-\frac{\eta}{2}^{\!2}} L_{n}^{m}\!\left(\eta^{2}\right), \end{aligned}$$ where we used Eq. (\[Omegme\]). With the above two partition functions, we can calculate $\mathcal L$ in Eq. (\[nlfinal\]). Before the presentation of the simulations, we want to make the notation clear emphasizing that $\mathcal Z_{+}(\lambda_f)$ refers the JC-type Hamiltonians with $\omega_L = \omega_{0} - m\nu$, while $\mathcal Z_{-}(\lambda_f)$ refers to the AJC-type Hamiltonians with $\omega_L = \omega_{0} + m\nu$. Now we carry on to the numerical investigation of the NL. For that, it is important to have in mind the reality of the physical parameters to be used in the simulations. First, the initial thermal occupation numbers $\bar{n}$ will be considered relatively small in order to have quantum fluctuations playing some role. The experiments employ sophisticated and very efficient cooling techniques for that aim [@leibfried]. For the typical frequencies and coupling constants, we will be focusing on the experimental implementation of Eq. (\[hamrwa\]) using $\rm Ca^{+}$ ions [@roos]. In these experiments, the electronic level separation is about THz while the trap frequencies are set typically in some MHz, and one order of magnitude smaller or higher by adjusting the trap potentials. For the classical Rabi frequency $\Omega$, a few MHz is also a realistic choice. We would also like to emphasize that our analysis and results are suitable to be applied to other known experimental setups such as those involving $\rm Be^{+}$ [@meekhof1996] or $\rm Yb^{+}$ [@olmschenk]. The partition function in (\[partf1\]) is a sum of an infinity number of terms which cannot be reduced analytically to a closed expression. Thus, a truncation is necessary. The convergence criterion for performing the truncation is explained in the note [@footnote2]. Each plot required a different number of terms kept in the sum, but in all cases the same criterium is
| 3,442
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|
{\partial\ln \rho_{e}^{(0)\alpha\alpha'} }{\partial P}
&=&-\beta P\delta_{\alpha\alpha'}\;.\end{aligned}$$ Equations (\[eq:c\]) and (\[eq:cstar\]) become $$\begin{aligned}
\frac{d}{dt}{\psi}_{\alpha}(X,t)
&=&
-iE_{\alpha}\psi_{\alpha}(X,t)\nonumber\\
&-&\sum_{\beta}P\cdot d_{\alpha\beta}
\left(1-\frac{\beta}{2}E_{\alpha\beta}\right)
\psi_{\beta}(X,t)
\label{eq:c2}\\
%neweq
\frac{d}{dt}{\phi}_{\alpha^{\prime}}(X,t)
&=&
iE_{\alpha^{\prime}}\phi_{\alpha^{\prime}}(X,t)
\nonumber\\
&-&\sum_{\beta^{\prime}}P\cdot d_{\alpha^{\prime}\beta^{\prime}}
\left(1-\frac{\beta}{2}E_{\alpha^{\prime}\beta^{\prime}}\right)
\phi_{\beta^{\prime}}(X,t)
\;.
\nonumber\\
\label{eq:cstar2}\end{aligned}$$ In Eqs. (\[eq:cstar\]) and (\[eq:cstar2\]) the terms $\pm(\beta/2)P\cdot F_{\alpha} \psi_{\alpha}$ (and the analogous terms with $\xi_{\alpha'}$) cancel each other. In the adiabatic basis $d_{11}(R)=d_{22}(R)=0$. Hence, defining the matrix $$\begin{aligned}
\mbox{\boldmath$\Sigma$}
&=&
\left[\begin{array}{cc} -iE_1 & -P\cdot d_{12}\left(1-\frac{\beta}{2}E_{12}\right)\\
P\cdot d_{12}\left(1+\frac{\beta}{2}E_{12}\right) & -iE_2 \end{array}\right]\;,
\nonumber\\\end{aligned}$$ Equations (\[eq:c2\]) and (\[eq:cstar2\]) can be written as $$\begin{aligned}
\frac{d}{dt}\left[\begin{array}{c}{\psi}_1 \\ {\psi}_2\end{array}\right]
&=&
\mbox{\boldmath$\Sigma$}\cdot
\left[\begin{array}{c}\psi_1 \\ \psi_2\end{array}\right]
\;, \quad
\frac{d}{dt}\left[\begin{array}{c}{\phi}_1 \\ {\phi}_2\end{array}\right]
=
\mbox{\boldmath$\Sigma$}^*\cdot
\left[\begin{array}{c}\phi_1 \\ \phi_2\end{array}\right]
\;,\nonumber\\
\label{eq:matrixSigma}\end{aligned}$$ which can be integrated by means of the simple algorithm $\mbox{\boldmath$\Psi$}(X,d\tau)=\mbox{\boldmath$\Psi$}(X,0)+
d\tau\mbox{\boldmath$\Theta$}(X,0)\cdot
\mbox{\boldmath$\eta$}(X,0)$, where $\mbox{\boldmath$\Psi$}=
(\mbox{\boldmath$\psi$},\mbox{\boldmath$\phi$})$ and $\mbox{\boldmath$\Theta$}=(\mbox{\boldmath$\Sigma$},
\mbox{\boldmath$\Sigma$}^*)$. The phase space part of the initial values of $\mbo
| 3,443
| 1,017
| 958
| 3,718
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|
rder and a right I-order in $Q$, we say that $S$ is an *I-order* in $Q$ and $Q$ is a semigroup of *I-quotients* of $S$. It is clear that, if $S$ a left order in an inverse semigroup $Q$, then it is certainly a left I-order in $Q$; however, the converse is not true (see for example [@GG] Example 2.2).
A left I-order in an inverse semigroup $Q$ is *straight left I-order* if every element in $Q$ can be written as $a^{-1}b$ where $a,b \in S$ and $a\,\mathcal{R}\,b$ in $Q$; we also say that $Q$ is a *straight left I-quotients* of $S$. If $S$ is straight in $Q$, we have the advantage of controlling the product in $Q$.
In [@NG] the author has given the necessary and sufficient conditions for a semigroup $S$ to have a bisimple inverse $\omega$-semigroup left I-quotients, modulo left I-order in the bicyclic semigroup $\mathcal{B}$, which is the most straightforward example of the bisimple inverse $\omega$-semigroup. In fact, it is a semigroups with many remarkable properties. Left I-orders in the bicyclic semigroup are interesting in their own right. By describtions left I-order in $\mathcal{B}$, we obtain:
\[main\] Let $S$ be a subsemigroup of $\mathcal{B}$. If $S$ is a left I-order in $\mathcal{B}$, then it is straight.
In the preliminaries after introducing the necessary notation, we give some previous results giving the description of subsemigroups of $\mathcal{B}$.
We use the classification of subsemigroups of $\mathcal{B}$ in [@ruskuc] to investigate which of them are left I-orders in $\mathcal{B}$. Subsemigroups of $\mathcal{B}$ fall into three classes upper, lower and two-sided. In Sections 3, 4 and 5 we give the necessary and sufficient conditions for upper, lower and two-sided subsemigroups of $\mathcal{B}$ to be left I-orders in $\mathcal{B}$, respectively. In each case, such left I-orders are straight and this proves Theorem \[main\].
Preliminaries {#prelim}
=============
Throughout this article we shall follow the terminology and notation of [@clifford]. The symbol $\mathbb{N}$ will denote the set co
| 3,444
| 881
| 2,743
| 3,073
| 3,339
| 0.773007
|
github_plus_top10pct_by_avg
|
hat acts nontrivially on the base, acts nontrivially on a rank 8 bundle, that subgroup of the gauge group is locally duplicating the effect of one of the ten-dimensional left-moving GSO projections. If one starts with a Spin$(32)/{\mathbb Z}_2$ string, then the dual looks locally like an $E_8 \times E_8$ string.
In this section, we will describe[^12] a precise duality relating such Spin$(32)/{\mathbb Z}_2$ compactifications to ordinary $E_8 \times E_8$ compactifications, and discuss some examples.
First, let us consider the easiest case. If the ${\mathbb Z}_2$ gerbe is trivial, the result is automatic: the worldsheet left-moving GSO projection is duplicated exactly, not just locally. When the gerbe is nontrivial, one must think a little harder to find a precise duality.
We propose[^13] a duality to heterotic $E_8 \times E_8$ strings as follows. To set conventions, suppose our stack $\mathfrak{X} = [X/\tilde{G}]$, where $$1 \: \longrightarrow \: {\mathbb Z}_2 \: \longrightarrow \:
\tilde{G} \: \longrightarrow \: G \: \longrightarrow \: 1$$ and ${\mathbb Z}_2$ acts trivially on $X$. Suppose furthermore that ${\cal E}$ is a bundle on $\mathfrak{X}$, [*i.e.*]{} a $\tilde{G}$-equivariant bundle on $X$, whose embedding into $E_8$ is via the standard worldsheet fermionic construction, in which left-moving fermions are in the fundamental representation of the structure group. Suppose that the ${\mathbb Z}_2$ acts nontrivially on ${\cal E}$.
In general ${\cal E}$ will not admit a $G$-equivariant structure. Nevertheless, at least in the special case that the ${\mathbb Z}_2$ is central in $\tilde{G}$, the bundles ${\cal E}^* \otimes {\cal E}$ and $\wedge^{\rm even} {\cal E}$ will admit a $G$-equivariant structure, and so can be defined on $[X/G]$. Moreover, it was observed in [@dist-greene] that, for the ‘typical’ worldsheet embeddings of $SU(n)$ in $E_8$ (including the present one), massless spectra of heterotic compactifications on smooth spaces can be defined solely in terms of sheaf cohomology with coefficients
| 3,445
| 2,002
| 2,975
| 3,229
| null | null |
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|
]; we leave it to the reader to write down these conditions precisely. Thus $T$ has a trace-like property with respect to the product in ${{\mathcal C}}$, and this motivates our terminology.
Recall now that ${{\mathcal C}}$ is a $k$-linear abelian category. To define Hochschild homology, we have to assume that it is equipped with a right-exact trace functor ${\operatorname{\sf tr}}:{{\mathcal C}}\to k{\operatorname{\it\!-Vect}}$; then for any $M \in
{{\mathcal C}}$, we set $$\label{hh.def.gen}
HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M) = L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}(M).$$
The functor ${\operatorname{\sf tr}}:A{\operatorname{\!-\sf bimod}}\to k{\operatorname{\it\!-Vect}}$ canonically extends to a right-exact trace functor in the sense of Definition \[trace.defn\].
[[*Proof.*]{}]{} For any object $\langle [n],M_n \rangle \in A{\operatorname{\!-\sf bimod}}_\#$, $[n] \in \Lambda$, $M_n \in A^{\otimes n}{\operatorname{\!-\sf bimod}}$, let $${\operatorname{\sf tr}}(\langle [n],M_n \rangle) = M_n/\{ a_{v'}m - ma_v \mid v \in
V([n]), m \in M_n, a \in A \},$$ where $a_v = 1 \otimes 1 \otimes \dots \otimes a \otimes \dots
\otimes 1 \in A^{\otimes V([n])}$ has $a$ in the multiple corresponding to $v \in V([n])$, and $v' \in V([n])$ is the next marked point after $v$ counting clockwise. The compatibility with maps in the category $A{\operatorname{\!-\sf bimod}}_\#$ is obvious.
We note that here, in the case ${{\mathcal C}}=A{\operatorname{\!-\sf bimod}}$, the category $A{\operatorname{\!-\sf bimod}}_\#$ is actually larger than what we would have had purely from the monoidal structure on ${{\mathcal C}}$: $M_n$ is allowed to be an arbitrary $A^{\otimes n}$-bimodule, not a collection of $n$ $A$-bimodules. To do the same for general $k$-linear ${{\mathcal C}}$, we need to replace $A^{\otimes n}{\operatorname{\!-\sf bimod}}$ with some version of the tensor product ${{\mathcal C}}^{\otimes n}$. Here we have a difficulty: for various technical reasons, it is not clear how to define
| 3,446
| 2,747
| 2,092
| 3,239
| 1,707
| 0.786589
|
github_plus_top10pct_by_avg
|
6 24.66 40 penicillin-binding protein 3
\* 2648343 A:5 C:192 C:37 5 24.6 40 drug transporter phosphotransferase system, glucose-
\* 2674216 A:5 C:170 C:37 5 24.6 40 specific IIABC component
\* 1542366 T:6 G:129 G:37 6 24.5 40 hypothetical protein lantibiotic epidermin biosynthesis
\* 1950547 A:5 C:164 C:37 5 23.4 40 protein EpiC
\* 105211 A:6 G:132 G:37 8 20.3 40 hypothetical protein
\* 1857182 A:14 G:85 G:37 14 19.42 40 hypothetical protein\
| 3,447
| 6,129
| 1,183
| 2,222
| null | null |
github_plus_top10pct_by_avg
|
k=0}^{\infty} \frac {(-y)^k D_{\mathbf{v}}^k}{k!} \right) h({\mathbf{x}})= (1-yD_{{\mathbf{v}}})h({\mathbf{x}}),$$ from which the lemma follows.
Note that $({\mathbf{v}}_1,\ldots,{\mathbf{v}}_m) \mapsto h[{\mathbf{v}}_1,\ldots,{\mathbf{v}}_m]$ is affine linear in each coordinate, i.e., for all $p \in {\mathbb{R}}$ and $1\leq i \leq m$: $$\begin{aligned}
& h[{\mathbf{v}}_1,\ldots,(1-p){\mathbf{v}}_i+p{\mathbf{v}}_i',\ldots, {\mathbf{v}}_m] \\
= &(1-p)h[{\mathbf{v}}_1,\ldots,{\mathbf{v}}_i,\ldots, {\mathbf{v}}_m] +ph[{\mathbf{v}}_1,\ldots,{\mathbf{v}}_i',\ldots, {\mathbf{v}}_m].\end{aligned}$$ Hence if ${\mathsf{X}}_1, \ldots, {\mathsf{X}}_m$ are independent random variables in ${\mathbb{R}}^n$, then $$\label{mixedexp}
{\mathbb{E}}h[{\mathsf{X}}_1,\ldots,{\mathsf{X}}_m] = h[{\mathbb{E}}{\mathsf{X}}_1,\ldots,{\mathbb{E}}{\mathsf{X}}_m].$$
\[mixedchar\] Let $h({\mathbf{x}})$ be hyperbolic with respect to ${\mathbf{e}}\in {\mathbb{R}}^n$, let $V_1, \ldots, V_m$ be finite sets of vectors in $\Lambda_+$, and let ${\mathbf{w}}\in {\mathbb{R}}^{n+m}$. For ${\mathbf{V}}=({\mathbf{v}}_1,\ldots, {\mathbf{v}}_m) \in V_1\times \cdots \times V_m$, let $$f({\mathbf{V}};t) := h[{\mathbf{v}}_1,\ldots, {\mathbf{v}}_m](t{\mathbf{e}}+{\mathbf{w}}).$$ Then $\{f({\mathbf{V}};t)\}_{{\mathbf{V}}\in V_1\times \cdots \times V_m}$ is a compatible family.
In particular if in addition all vectors in $V_1 \cup \cdots \cup V_m$ have rank at most one, and $$g({\mathbf{V}};t) := h(t{\mathbf{e}}+{\mathbf{w}}- \alpha_1{\mathbf{v}}_1-\cdots-\alpha_m{\mathbf{v}}_m),$$ where ${\mathbf{w}}\in {\mathbb{R}}^n$ and $(\alpha_1,\ldots, \alpha_m)\in {\mathbb{R}}^m$, then $\{g({\mathbf{V}};t)\}_{{\mathbf{V}}\in V_1\times \cdots \times V_m}$ is a compatible family.
Let ${\mathsf{X}}_1 \in V_1, \ldots, {\mathsf{X}}_m \in V_m$ be independent random variables. Then the polynomial ${\mathbb{E}}h[{\mathsf{X}}_1, \ldots, {\mathsf{X}}_m]= h[{\mathbb{E}}{\mathsf{X}}_1,\ldots,{\mathbb{E}}{\mathsf{X}}_m]$ is hyperbolic with respect to $({\mathbf{e}}, 0,\ldots,0)$ b
| 3,448
| 2,435
| 2,488
| 3,094
| null | null |
github_plus_top10pct_by_avg
|
tioned on the rank breaking due to previous separators $\{G_{j,a'}\}_{a'<a}$ that are ranked higher (i.e. $a'<a$), which follows from the next lemma.
\[lem:consistency\] For a position-$p$ rank breaking graph $G_p$, defined over a set of items $S$, where $p \in [|S|-1]$, $$\begin{aligned}
\label{eq:grad_eq6}
\P\Big[\sigma^{-1}(i) < \sigma^{-1}(\i) \;\Big|\; \big(i,\i\big) \in G_p \Big] \;=\; \frac{\exp(\theta^*_{i})}{\exp(\theta^*_{i})+\exp(\theta^*_{i'})} \;,\end{aligned}$$ for all $i,i'\in S$ and also $$\begin{aligned}
\label{eq:grad_eq8}
\P\Big[\sigma^{-1}(i) < \sigma^{-1}(\i) \;\Big|\; \big(i,\i\big) \in G_p \text{ and } \{i'' \in S \,:\,\sigma^{-1}(i'')<p \} \Big] \;=\; \frac{\exp(\theta^*_{i})}{\exp(\theta^*_{i})+\exp(\theta^*_{i'})} \;.\end{aligned}$$
This is one of the key technical lemmas since it implies that the proposed rank-breaking is consistent, i.e. $\E_{\theta^*}[\nabla \Lrb(\theta^*)] = 0$. Throughout the proof of Theorem \[thm:main2\], this is the only place where the assumption on the proposed (consistent) rank-breaking is used. According to a companion theorem in [@APX14a Theorem 2], it also follows that any rank-breaking that is not union of position-$p$ rank-breakings results in inconsistency, i.e. $\E_{\theta^*}[\nabla \Lrb(\theta^*)] \neq 0$. We claim that for each rank-breaking graph $G_{j,a}$, ${\|\nabla\L_{G_{j,a}}(\theta^*)\|}_2^2 \leq (\lambda_{j,a})^2 (\kappa_j - p_{j,a})(\kappa_j- p_{j,a}+1)$. By Lemma \[lem:az\_gen\] which is a generalization of the vector version of the Azuma-Hoeffding inequality found in [@hayes2005large Theorem 1.8], we have $$\begin{aligned}
\P\big[\big\|\nabla\Lrb(\theta^*)\big\|_2 \geq \delta \big] \;\;\leq\;\; 2e^{3}\exp\Bigg(\frac{-\delta^2}{2\sum_{j=1}^n \sum_{a=1}^{\ell_j} \big(\lambda_{j,a}\big)^2 \big(\kappa_j - p_{j,a}\big)\big(\kappa_j- p_{j,a}+1\big)}\Bigg)\,,\end{aligned}$$ which implies the result.
\[lem:az\_gen\] Let $(X_1,X_2,\cdots, X_n)$ be real-valued martingale taking values in $\reals^d$ such that $X_0 = 0$ and for every $
| 3,449
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m by defining a new unknowns $v_j$, for $j=2,3$, by setting v\_j(x,,):=S\_j(R\_j\^[-1]{}())u\_j(x,,R\_j\^[-1]{}()), i.e. v\_j(x,,R\_j(E))=S\_j(E)u\_j(x,,E). Then we find that =R\_j’(E)=[1]{} =[1]{}, and so, after writing $$\begin{aligned}
{2}
\tilde{f}_j(x,\omega,\eta):=S_j(R_j^{-1}(\eta))f_j(x,\omega,R_j^{-1}(\eta)),
\quad & (x,\omega,\eta)\in G\times S\times [0,r_{m,j}], \\
\tilde{g}_j(y,\omega,\eta):=S_j(R_j^{-1}(\eta))g_j(y,\omega,R_j^{-1}(\eta)),
\quad & (y,\omega,\eta)\in \tilde\Gamma_{-,j},\end{aligned}$$ where $$\tilde\Gamma_{-,j}:=\{(y,\omega,\eta)\in \partial G\times S\times [0,r_{m,j}]\ |\ \omega\cdot\nu(y)<0\},$$ we see that the problem is equivalent to \[desol20\] &-+\_x v\_j+ \_j v\_j =\_j GS, subject to inflow boundary and initial value conditions, $$\begin{aligned}
{3}
{v_j}_{|\tilde\Gamma_{-,j}}&=\tilde{g}_j\quad && {\rm a.e.\ on}\ \tilde\Gamma_{-,j},\label{desol19a} \\
v_j(\cdot,\cdot,r_{m,j})&=0\quad && {\rm a.e.\ on}\ G\times S,\ j=2,3. \label{desol19aa}\end{aligned}$$ Notice that $\tilde{g}_j(r_{m,j})=0$ since $g_j(E_{\rm m})=0$. The original unknowns $u_j$, $j=2,3$, are given in terms of $v_j$ by \[desol21\] u\_j(x,,E)=[1]{}v\_j(x,, R\_j(E)).
The problem - can be solved explicitly, at least formally. The solution $v_j$ of is the sum $v_{1,j}+v_{2,j}$ of solutions $v_{1,j}$ and $v_{2,j}$ of the following problems $$\begin{gathered}
-{{\frac{\partial v_{1,j}}{\partial \eta}}}+\omega\cdot\nabla_x {v_{1,j}}+
\Sigma_j v_{1,j}
=\tilde f_j,\nonumber\\
{v_{1,j}}_{|\tilde\Gamma_{-,j}}=0,
\quad
v_{1,j}(\cdot,\cdot,r_{m,j})=0,\label{desol20a}\end{gathered}$$ and $$\begin{gathered}
-{{\frac{\partial v_{2,j}}{\partial \eta}}}+\omega\cdot\nabla_x {v_{2,j}}+
\Sigma_j v_{2,j}
=0,\nonumber\\
{v_{2,j}}_{|\tilde\Gamma_{-,j}}=\tilde g_j,
\quad
v_{2,j}(\cdot,\cdot,r_{m,j})=0,\label{desol20b}\end{gathered}$$ where the (partial differential) equations are to be satisfies on $G\times S\times [0,r_{m,j}]$, the (inflow) boundary conditions on $\tilde\Gamma_{-,j}$ and the initial (energy) conditions on $G\times S$
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2}\nu\nu_{0}}{4}X(-\nu)\nonumber \\
\int_{0}^{1}W(\mu)\phi(\mu,\nu^{\prime})\phi(\mu,-\nu)d\mu & = & \frac{c\nu^{\prime}}{2}(\nu_{0}+\nu)\phi(\nu^{\prime},-\nu)X(-\nu)\nonumber \end{aligned}$$
where the half-range weight function $W(\mu)$ is defined as
$$W(\mu)=\frac{c\mu}{2(1-c)(\nu_{0}+\mu)X(-\mu)}\label{Eqn: W(mu)}$$
in terms of the $X$-function $$X(-\mu)=\textrm{exp}-\left\{ \frac{c}{2}\int_{0}^{1}\frac{\nu}{N(\nu)}\left[1+\frac{c\nu^{2}}{1-\nu^{2}}\right]\ln(\nu+\mu)d\nu\right\} ,\qquad0\leq\mu\leq1,$$
that is conveniently obtained from a numerical solution of the nonlinear integral equation $$\Omega(-\mu)=1-\frac{c\mu}{2(1-c)}\int_{0}^{1}\frac{\nu_{0}^{2}(1-c)-\nu^{2}}{(\nu_{0}^{2}-\nu^{2})(\mu+\nu)\Omega(-\nu)}d\nu\label{Eqn: Omega(-mu)}$$
to yield $$X(-\mu)=\frac{\Omega(-\mu)}{\mu+\nu_{0}\sqrt{1-c}},$$
and the $X(\pm\nu_{0})$ satisfy $$X(\nu_{0})X(-\nu_{0})=\frac{\nu_{0}^{2}(1-c)-1}{2(1-c)v_{0}^{2}(\nu_{0}^{2}-1)}.$$ Two other useful relations involving the $W$-function are given by $\int_{0}^{1}W(\mu)\phi(\mu,\nu_{0})d\mu=c\nu_{0}/2$ and $\int_{0}^{1}W(\mu)\phi(\mu,\nu)d\mu=c\nu/2$.
The utility of these full and half range orthogonality relations lie in the fact that a suitable class of functions of the type that is involved here can always be expanded in terms of them, see @Case1967. An example of this for a full-range problem has been given above; we end this introduction to the generalized — traditionally known as singular in neutron transport theory — eigenfunction method with two examples of half-range orthogonality integrals to the half-space problems A and B of Sec. 5.
**Problem A: The Milne Problem.** In this case there is no incident flux of particles from outside the medium at $x=0$, but for large $x>0$ the neutron distribution inside the medium behaves like $e^{x/\nu_{0}}\phi(-\nu_{0},\mu)$. Hence the boundary condition (\[Eqn: BC\_HR\]) at $x=0$ reduces to $$-\phi(\mu,-\nu_{0})=a_{\textrm{A}}(\nu_{0})\phi(\mu,\nu_{0})+\int_{0}^{1}a_{\textrm{A}}(\nu)\phi(\mu,\nu)d\nu\qquad\mu\geq0.
| 3,451
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y regular function. The fractional Dirichlet problem and variants thereof appear in many applications, in particular in physical settings where anomalous dynamics occur and where the spread of mass grows faster than linearly in time. Examples include turbulent fluids, contaminant transport in fractured rocks, chaotic dynamics and disordered quantum ensembles; see [@FracDy; @AnTr; @flights]. The numerical analysis of is no less deserving than in the diffusive setting.
Just as with the classical Dirichlet setting, the solution to has a Feynman–Kac representation, expressed as an expectation at first exit from $D$ of the associated stable process. The theorem below is proved in this paper in a probabilistic way. Similar statements and proofs we found in the existing literature take a more analytical perspective. See for example the review in [@bucur] as well as the monographs [@BH], [@BV] and [@BBKRSV], the articles [@B99], [@R-O1] and [@R-O2] and references therein.
We say a real-valued function $\phi$ on a Borel set $S\subset \mathbb{R}^d$ belongs to $L^1_\alpha(S)$ if it is a measurable function that satisfies $$\int_{S}\frac{|\phi(x)|}{1+|x|^{\alpha + d}}\,{\rm d}x <\infty.
\label{i-test}$$
\[corr\] For dimension $d\geq 2$, suppose that $D$ is a bounded domain in $\mathbb{R}^d$ and that ${g}$ is a continuous function in $L^1_\alpha(D^\mathrm{c})$. Then there exists a unique continuous solution to in $L^1_\alpha(\mathbb{R}^d)$, which is given by
$$u(x) = \mathbb{E}_x[{g}(X_{\sigma_D})],\qquad x\in D,$$ where $X = (X_t, t\geq 0)$ is an isotropic stable Lévy process with index $\alpha$ and $\sigma_D = \inf\{t>0: X_t\not\in D\}$.
The case that $D$ is a ball can be found, for example, in Theorem 2.10 of [@bucur]. We exclude the case $d=1$ because convex domains are intervals for which exact solutions are known; see again [@bucur] or the forthcoming Theorem \[BGR\] lifted from [@BGR]. Theorem \[corr\] follows in fact as a corollary of a more general result stated later in Theorem \[hasacorr\], which is pr
| 3,452
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ce given by the postulated gravitational dual background constructed in [@Frolov:2005dj]. Upon setting all three deformation parameters equal this reduces to the $\beta$-deformation with enhanced ${\cal N}=1$ supersymmetry and hence we will proceed with the general case.
Rather remarkably the string $\sigma$-model in the $\gamma$-deformed target space can be obtained as Yang-Baxter $\sigma$-model [@Kyono:2016jqy; @Osten:2016dvf]. Let us consider the bosonic sector, restricting our attention to the five-sphere of $AdS_5\times S^5$; the $AdS$ factor plays no role in what follows. It is convenient to follow [@Frolov:2005dj] and parametrise the $S^5$ in coordinates adapted to the $U(1)^3$ isometry $$\label{eq:metgam}
ds_{S^5}^2 = d\alpha^2 + \S_\alpha^2 d\xi^2 + \C_\alpha^2 d\phi_1^2 + \S_\alpha^2 \C_\xi^2 d\phi_2^2 + \S_\alpha^2 \S_\xi^2 d\phi_3^2
= \sum_{i= 1\dots 3} dr_i^2 + r_i^2 d\phi_i^2 \ ,$$ where $r_1 = \C_\alpha$, $r_2= \S_\alpha \C_\xi$, $r_3= \S_\alpha \S_\xi$ with $\C_x$ and $\S_x$ denoting $\cos x$ and $\sin x$ respectively. The sphere can be realised as the coset $SU(4)/SO(5)$ for which a particular coset representative is given by $$\label{eq:paragam}
g = e^{\frac{1}{2} \sum_{m=1}^3 \phi^m h_m } e^{-\frac{\xi}{2} \gamma^{13}} e^{\frac{i}{2} \alpha \gamma^1} \ ,$$ where $ \gamma^{13}$ and $\gamma^1$ are certain $SU(4)$ generators (see appendix \[app:algconv\] for conventions) and $h_i$ are the three Cartan generators. Letting $P$ be the projector onto the coset and $J_\pm = g^{-1} \partial_\pm g$ pull backs of the left-invariant one-form, the $S^5$ $\sigma$-model Lagrangian is $${\cal L } = {\operatorname{Tr}}(J_+ P(J_-) ) \ ,$$ with the parametrisation giving the $\sigma$-model with target space metric .
Starting with the $r$-matrix $$r= \frac{\nu_1}{4} h_2 \wedge h_3 + \frac{\nu_3}{4} h_1 \wedge h_2 + \frac{\nu_2}{4} h_3 \wedge h_1 \ ,$$ it was shown in [@Matsumoto:2014nra; @vanTongeren:2015soa] that the NS sector of the Yang-Baxter $\sigma$-model matches the $\gamma$-deformed target space explicit
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effective mass plots of the two-point correlators and the extracted mass $E_1^-$ of the negative-parity state cannot be reliable. Leaving aside these failures, we try to extract $g_A^{0-}$. The result is added in the lower panel in Fig. \[AxialVectorC\] as a faint-colored symbol, which is consistent with those obtained at other $\kappa$’s. On the other hand, the axial charge $g_A^{1-}$ of $N^*(1650)$ is found to be about 0.55, which has almost no quark-mass dependence. The striking feature is that these axial charges, $g_A^{0-}\sim 0$ and $g_A^{1-}\sim 0.55$, are consistent with the naive nonrelativistic quark model calculations [@Nacher:1999vg; @Glozman:2008vg], $g_A^{0-}= -\frac19$ and $g_A^{1-}= \frac59$. Such values are obtained if we assume that the wave functions of $N^*(1535)$ and $N^*(1650)$ are $|l=1, S=\frac12\rangle$ and $|l=1, S=\frac32\rangle$ neglecting the possible state mixing. (Here, $l$ denotes the orbital angular momentum and $S$ the total spin.)
In the chiral doublet model [@DeTar:1988kn; @Glozman:2007jt], the small $g_A^{N^*N^*}$ is realized when the system is decoupled from the chiral condensate $\langle \bar \psi \psi \rangle$. The small $g_A^{0-}$ of $N^*(1535)$ then does not contradict with the possible and attempting scenario, the [*chiral restoration scenario in excited hadrons*]{} [@Glozman:2007jt]. If this scenario is the case, the origin of mass of $N^*(1535)$ (or excited nucleons) is essentially different from that of the positive-parity ground-state nucleon $N(940)$, which mainly arises from the spontaneous chiral symmetry breaking. However, the non-vanishing axial charge of $N^*(1650)$ unfortunately gives rise to doubts about the scenario.
In order to reveal the realistic chiral structure, studies with much lighter u,d quarks will be indispensable. A study of the axial charge of Roper, as well as the inclusion of strange sea quarks could also cast light on the low-energy chiral structure of baryons and the origin of mass.
[**Conclusions.**]{} In conclusion, we have perfo
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l heterotic string spectra. It is possible to apply the same methods to the A/2 model to formulate a mathematical theory of sheaf cohomology of orbifolds, and this has been done in [@manion-toappear].
Briefly, the A/2 model is a heterotic analogue of the A model topological field theory. If $X$ is a smooth space and ${\cal E} \rightarrow X$ a holomorphic vector bundle, then the A/2 model is well-defined if both[^24] $${\rm ch}_2({\cal E}) \: = \: {\rm ch}_2(TX)
\mbox{ and }
\det {\cal E}^* \: \cong \: K_X.$$ See [*e.g.*]{} [@katz-s; @ade; @s-b; @jock-ilarion; @dgks1; @dgks2; @mss; @tan1; @tan2] for more information on the A/2 and B/2 models. As this is no longer a physical theory, constraints on the dimension of $X$ and rank of ${\cal E}$ are dropped. When $X$ is smooth, the massless spectrum consists of sheaf cohomology groups of the form $$H^{\bullet} ( X, \wedge^{\bullet} {\cal E}^* )$$ When $X$ is a stack $\mathfrak{X}$, reference [@manion-toappear] applies methods similar to those in this appendix (modulo restricting to (R,R) sector states and omitting the GSO projections) to define a generalization, which broadly speaking adds in various sheaf cohomology groups associated to twisted sectors (nontrivial components of the inertia stack).
Line bundles on gerbes over projective spaces {#app:linebundles}
=============================================
For any stack $\mathfrak{X}$ presented as $\mathfrak{X} = [X/G]$ for some space $X$ and group $G$, a vector bundle (sheaf) on $\mathfrak{X}$ is the same as a $G$-equivariant vector bundle (sheaf) on $X$. Suppose that $G$ is an extension $$1 \: \longrightarrow \: K \: \longrightarrow \: G \: \longrightarrow \:
H \: \longrightarrow \: 1,$$ where $K$ acts trivially on $X$, and $G/K \cong H$ acts effectively. In this case, $\mathfrak{X} = [X/G]$ is a $K$-gerbe. A vector bundle on $\mathfrak{X}$ is a $G$-equivariant vector bundle on $X$, and as such, the $K$ action is defined by a representation of $K$ on the fibers of that vector bundle.
In this section, we will disc
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esearch interests include statistics, machine learning, and computer vision.
[Song-Chun Zhu]{} Song-Chun Zhu received a Ph.D. degree from Harvard University, and is a professor with the Department of Statistics and the Department of Computer Science at UCLA. His research interests include computer vision, statistical modeling and learning, cognition and AI, and visual arts. He received a number of honors, including the Marr Prize in 2003 with Z. Tu et. al. on image parsing,the Aggarwal prize from the Int’l Association of Pattern Recognition in 2008, twice Marr Prize honorary nominations in 1999 for texture modeling and 2007 for object modeling with Y.N. Wu et al., a Sloan Fellowship in 2001, the US NSF Career Award in 2001, and the US ONR Young Investigator Award in 2001. He is a Fellow of IEEE.
And-Or graph representations {#and-or-graph-representations-1 .unnumbered}
============================
Parameters for latent patterns {#parameters-for-latent-patterns .unnumbered}
------------------------------
We use the notation of ${\bf p}_{u}$ to denote the central position of an image region $\Lambda_{u}$. For simplification, all position variables ${\bf p}_{u}$ are measured based on the image coordinates by propagating the position of $\Lambda_{u}$ to the image plane.
Each latent pattern $u$ is defined by its location parameters $\{L_{u},D_{u},\overline{\bf p}_{u},\Delta{\bf p}_{u}\}\subset{\boldsymbol\theta}$, where ${\boldsymbol\theta}$ is the set of AOG parameters. It means that a latent pattern $u$ uses a square within the $D_{u}$-th channel of the $L_{u}$-th conv-layer’s feature map as its deformation range. The center position of the square is given as $\overline{\bf p}_{u}$. When latent pattern $u$ is extracted from the $k$-th conv-layer, $u$ has a fixed value of $L_{u}=k$.
$\Delta{\bf p}_{u}$ denotes the average displacement from $u$ and $u$’s parent part template $v$ among various images, and $\Delta{\bf p}_{u}$ is used to compute $S^{\textrm{inf}}(\Lambda_{u}|\Lambda_{v})$. Given parameter $\o
| 3,456
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| 3,187
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the difference in the method of bounding diagrams for the expansion coefficients. Take the $0^\text{th}$-expansion coefficient for example. For percolation, the BK inequality simply tells us that $$\begin{aligned}
{\label{eq:pi0perc-comp}}
\pi_p^{{\scriptscriptstyle}(0)}(x)\leq{{\mathbb P}}_p(o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\longleftrightarrow}}}x)^2.\end{aligned}$$ For the ferromagnetic Ising model, on the other hand, we first recall [(\[eq:pi0-def\])]{}, i.e., $$\begin{aligned}
\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)=\sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}$}}},\end{aligned}$$ where $w_\Lambda({{\bf n}})/Z_\Lambda\ge0$. Due to the indicator, every current configuration ${{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_\Lambda}$ that gives nonzero contribution has at least *two bond-disjoint* paths $\zeta_1,\zeta_2$ from $o$ to $x$ such that $n_b>0$ for all $b\in\zeta_1{\:\Dot{\cup}\:}\zeta_2$. Also, due to the source constraint, there should be at least one path $\zeta$ from $o$ to $x$ such that $n_b$ is odd for all $b\in\zeta$. Suppose, for example, that $\zeta=\zeta_1$ and that $n_b$ for $b\in\zeta_2$ are all positive-even. Since a positive-even integer can split into two odd integers, on the labeled graph ${{\mathbb G}}_{{\bf n}}$ with ${\partial}{{\mathbb G}}_{{\bf n}}=o{\vartriangle}x$ (recall the notation introduced above [(\[eq:Sbefore\])]{}) there are at least *three edge-disjoint* paths from $o$ to $x$. This observation naturally leads us to expect that $$\begin{aligned}
{\label{eq:pi0-comp}}
\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)\leq{{\langle \varphi_o\varphi_x \rangle}}_\Lambda^3\end{aligned}$$ holds for the ferromagnetic Ising model. This naive argument to justify [(\[eq:pi0-comp\])]{} will be made rigorous in Section \[s:bounds\] by taking account of partition functions.
The highe
| 3,457
| 1,474
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| 0.781754
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ere $\rm{diam}_D(H_n):=\sup_{g,h\in H_n} D(g,h)=\sup_{g\in H_n} D(g,1)$.
Suppose on the contrary that there exist $\varepsilon>0$ and a sequence $(h_n)_n\subseteq G$ such that $h_n\in H_n$ and $D(h_n,1)\geq \varepsilon$. Since $G$ is compact, the sequence without loss of generality converges to some $h\in G$. By the continuity of the metric, we have $D(h,1)\geq \varepsilon$, thus in particular $h\neq 1$. On the other hand, since the sequence $(H_n)_n$ is decreasing and each of the subgroups is closed, $h\in \bigcap_n H_n$. This contradicts that $\bigcap_n H_n=\{1\}$.
For each $n$, let $\mu_n$ be the normalized invariant Haar measure on the compact subgroup $H_n$, and let $P'_n$, resp. $P_n$ be the map, resp. projection from Proposition \[prop:projection\] applied to the groups $G$ and $H_n$ with the metric $d$ and the quotient metric $d_n$ on $G/H_n$.
[**Claim 2.**]{} For every $x\in {\mathcal{F}}(G)$, we have $x=\lim_{n\to\infty} P_n(x)$.
Suppose first that $x\in {\mathcal{F}}(G)$ is a finite linear combination of Dirac elements. That is, there are $m\in {\mathbb{N}}$, $g_1,\dots,g_m \in G$ and $\alpha_1,\dots,\alpha_m\in{\mathbb{R}}$ with $x=\sum_{i=1}^m \alpha_i \delta(g_i)$. For a fixed $n$, let us compute $\|x-P_n(x)\|$. We have $$\begin{aligned}
\|x-P_n(x)\|& = \|x-\int_{H_n} h\cdot x - (\sum_{i=1}^m \alpha_i)\delta(h)\,d\mu_n(h)\|\\
&\leq \int_{H_n}\|x- h\cdot x \|\,d\mu_n(h) + (\sum_{i=1}^m |\alpha_i|)\|\int_{H_n}\delta(h)\,d\mu_n(h)\|.\end{aligned}$$ Suppose that $\rm{diam}_D(H_n)= \varepsilon_n$. Then for every $h\in H_n$ we have $$\begin{aligned}
\|x-h\cdot x\|& =\|\sum_{i=1}^m \alpha_i \delta(g_i)-\sum_{i=1}^m \alpha_i \delta(h\cdot g_i)\|\leq \sum_{i=1}^m |\alpha_i| d(h\cdot g_i,g_i)\\
& \leq \sum_{i=1}^m |\alpha_i| D(h\cdot g_i,g_i)\leq \sum_{i=1}^m |\alpha_i| D(h,1)\leq \sum_{i=1}^m |\alpha_i| \varepsilon_n,\end{aligned}$$ and it follows that $\int_{H_n}\|x-h\cdot x\|d\mu_n(h)\leq \sum_{i=1}^m |\alpha_i| \varepsilon_n$. On the other hand, for each $f\in B_{\rm{Lip}_0(G)}$, $$|\langle \int_{H
| 3,458
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ere the orbital occupancy can be 0, 1 or 2, a core set where the orbitals are doubly occupied and a virtual set of empty orbitals then the total Hamilton, $\hat{H}$ and Fink’s Hamiltonian, $\hat{H}_{0}$ can be expressed in second quantization as,
$$\hat{H}=\sum_{ij}t_{ij}a_{i}^{\dagger}a_{j}+\sum_{ijkl}\Braket{ij|kl}a_{i}^{\dagger}a_{j}^{\dagger}a_{l}a_{k}\label{eq:H@ndQ}$$
$$\hat{H_{0}}=\sum_{{ij;\atop \Delta n=(0,0,0)}}t_{ij}a_{i}^{\dagger}a_{j}+\sum_{{ijkl;\atop \Delta n=(0,0,0)}}\Braket{ij|kl}a_{i}^{\dagger}a_{j}^{\dagger}a_{l}a_{k}\label{eq:HFink}$$
where $i,j,k,l$ refer to any orbitals and $\Delta n$ denotes the change in the total number of electrons between the three subsets of orbitals. The only operators belonging to $\hat{H_{0}}$ are the ones that do not transfer electrons between the three subsets.
The successive correction ($\Ket{\Psi_{m}}$) to the zeroth order wavefuntion can be computed by using the following equation, $$\left(\hat{H_{0}}-E_{0}\right)\Ket{\Psi_{m}}=-Q\left(\hat{V}\Ket{\Psi_{m-1}}-\sum_{k=1}^{m-1}E_{k}\Ket{\Psi_{m-k}}\right),\label{eq:bthorderwf}$$ where $Q$ is the projector onto the orthogonal space of the zeroth order wavefunction. Those sets of equation can be solved sequentially to compute the $m^{th}$ order of the wavefunction, $\Ket{\Psi_{m}}$. Once $\Ket{\Psi_{m}}$ is known the $2m$ and $2m+1$ energies can be computed thanks to Wigner’s rules: $$E_{2m}=\Braket{\Psi_{m-1}|V|\Psi_{m}}-\sum_{k=1}^{m}\sum_{j=1}^{m-1}E_{2m-k-j}\Braket{\Psi_{k}|\Psi_{j}},$$ $$E_{2m+1}=\Braket{\Psi_{m}|V|\Psi_{m}}-\sum_{k=1}^{m}\sum_{j=1}^{m}E_{2m+1-k-j}\Braket{\Psi_{k}|\Psi_{j}}.$$
Note that with the definition of the zeroth order Hamiltonian given in Eq.\[eq:HFink\], the first order energy $E_{1}=\Braket{\Psi_{0}|V|\Psi_{0}}$ vanishes.
In a recent paper[@sharma_multireference_2015], both the 0 order wavefunction and the successive corrections were expressed as matrix product states (MPS) and computed deterministically by functional minimization. Here we proposed an alternative approach w
| 3,459
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$HH_0(A,M)$); therefore the map $\rho_0$ extends to a map $\rho_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}:H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},M^\Delta_\#) \to HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$. To prove that $\rho_{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is an isomorphism for any $M$, it suffices to prove it when $M$ is free over $A^{opp} \otimes A$, or in fact, when $M = A^{opp} \otimes A$. Then on one hand, $HH_0(A,M)
= A$, and $HH_i(A,M) = 0$ for $i \geq 1$. And on the other hand, the standard complex associated to the simplicial $k$-vector space $(A^{opp} \otimes A)_\#^\Delta$ is just the usual bar resolution of the diagonal $A$-bimodule $A$.
It is more or less obvious that for an arbitrary $M \in A{\operatorname{\!-\sf bimod}}$, $M_\#^\Delta$ does not extend to a cyclic vector space – in order to be able to define $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$, we have to equip the bimodule $M$ with some additional structure. To do this, we want to use the tensor structure on $A{\operatorname{\!-\sf bimod}}$. The slogan is the following:
- To find a suitable category of coefficients for cyclic homology, we have to repeat the definition of the cyclic vector space $A_\# \in {\operatorname{Fun}}(\Lambda,k)$, but replace the associative algebra $A$ in this definition with the tensor category $A{\operatorname{\!-\sf bimod}}$.
Let us explain what this means.
First, consider an arbitrary associative unital monoidal category ${{\mathcal C}}$ with unit object $I$ (at this point, not necessarily abelian). For any integer $n$, we have the Cartesian product ${{\mathcal C}}^n =
{{\mathcal C}}\times {{\mathcal C}}\times \dots \times {{\mathcal C}}$. Moreover, the product on ${{\mathcal C}}$ induces a product functor $$m:{{\mathcal C}}^n \to {{\mathcal C}},$$ where if $n=0$, we let ${{\mathcal C}}^n={\operatorname{{\sf pt}}}$, the category with one object and one morphism, and let $m:{\operatorname{{\sf pt}}}\to {{\mathcal C}}$ be the embedding of the unit object. M
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rgin}{-69pt}
\begin{document}$$\begin{aligned} \varepsilon _{\pm }({\vec {k}}) =-2t_2\alpha _1({\vec {k}})\cos \phi \pm \sqrt{m({\vec {k}})^{2} + t_1^{2}|\Omega ({\vec {k}})|^{2}}\;. \end{aligned}$$\end{document}$$The size of the bands can be bounded by $\documentclass[12pt]{minimal}
\usepackage{amsmath}
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\begin{document}$$\max _{{\vec {k}}}\varepsilon _+({\vec {k}})- \min _{{\vec {k}}}\varepsilon _-({\vec {k}})$$\end{document}$, which we call the *bandwidth*. To make sure that the energy bands do not overlap, we assume that $\documentclass[12pt]{minimal}
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\begin{document}$$t_2/t_1<1/3$$\end{document}$. For $\documentclass[12pt]{minimal}
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\begin{document}$$L\rightarrow \infty $$\end{document}$, the two bands can touch only at the *Fermi points*$\documentclass[12pt]{minimal}
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\setl
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} x } } \\
\end{array}
\right].
\label{exp-H0}\end{aligned}$$ Then, the perturbed Hamiltonian is given by $$\begin{aligned}
\hat{H}_{1} &=& {\bf X}^{\dagger} \tilde{H}_{1} {\bf X}
=
\left[
\begin{array}{cc}
0 & (UX)^{\dagger} A W \\
W^{\dagger} A (UX) & W^{\dagger} A W \\
\end{array}
\right].
\label{hat-H1}\end{aligned}$$ The eigenvalues of $\tilde{H}_{0}$ is therefore $h_{1}$, $h_{2}$, $h_{3}$, and $\Delta_{J}$ ($J=4, \cdot \cdot \cdot, 3+N$). Therefore, the sterile neutrino masses are affected neither by the active states nor the matter potential in our zeroth-order unperturbed basis. It must be a good approximation because we have assumed that the sterile masses are much heavier than the active ones, and we are interested in the energy region $a \sim \Delta m^2_{31}$.
To do real calculations of the $S$ matrix elements we must solve the zeroth order Hamiltonian $\tilde{H}_{0}$. This task will be carried out in section \[sec:exact-solution-zeroth\], in which we derive explicit expressions of the eigenvalues $h_{i}$ and the unitary matrix $X$.
Now, we formulate perturbation theory with the hat basis Hamiltonian, $\hat{H}_{0}$ in (\[hat-H0\]) and $\hat{H}_{1}$ in (\[hat-H1\]) after a clarifying note in the next subsection.
### The relationship between quantities in various bases
So far we have introduced the tilde- and the hat-basis: $$\begin{aligned}
\tilde{H} = {\bf U}^{\dagger} H {\bf U},
\hspace{10mm}
\hat{H} = {\bf X}^{\dagger} \tilde{H} {\bf X},
\label{tilde-hat-relation}\end{aligned}$$ where ${\bf X}$ is given by eq. (\[bfX-def\]). Therefore, $$\begin{aligned}
\hat{H}
= \left( {\bf U} {\bf X} \right)^{\dagger} H \left( {\bf U} {\bf X} \right).
\label{flavor-hat-relation}\end{aligned}$$ Or $$\begin{aligned}
H = \left( {\bf U} {\bf X} \right) \hat{H} \left( {\bf U} {\bf X} \right)^{\dagger},
\hspace{10mm}
S = \left( {\bf U} {\bf X} \right) \hat{S} \left( {\bf U} {\bf X} \right)^{\dagger}.
\label{flavor-hat-relation2}\end{aligned}$$ Notice that both ${\bf U}$ and ${\bf X}$ are unitary, an
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11–143.
A. Sakai. Asymptotic behavior of the critical two-point function for spread-out Ising ferromagnets above four dimensions. *Unpublished manuscript* (2005).
A. Sakai. Diagrammatic estimates of the lace-expansion coefficients for finite-range Ising ferromagnets. *Unpublished notes* (2006).
B. Simon. Correlation inequalities and the decay of correlations in ferromagnets. (1980): 111–126.
G. Slade. The lace expansion and its applications. Springer Lecture Notes in Mathematics [**1879**]{} (2006).
A.D. Sokal. An alternate constructive approach to the $\varphi_3^4$ quantum field theory, and a possible destructive approach to $\varphi_4^4$. *Ann. Inst. Henri Poincaré Phys. Théorique* [**37**]{} (1982): 317–398.
[^1]: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. [a.sakai@bath.ac.uk]{}
[^2]: Updated: November 13, 2006
[^3]: In [(\[eq:IRbd-so\])]{} and [(\[eq:IRbd-sokal\])]{}, we also use the fact that, for $p<{p_\text{c}}$, our $G_p$ (i.e., the infinite-volume limit of the two-point function under the free-boundary condition) is equal to the infinite-volume limit of the two-point function under the periodic-boundary condition.
[^4]: The mean-field results in [@a82; @abf87; @af86; @ag83] are based on a couple of differential inequalities for $M_{p,h}$ and $\chi_p$ (under the periodic-boundary condition) using a certain random-walk representation. We can simplify the proof of the same differential inequalities (under the free-boundary condition as well) using Proposition \[prp:through\].
[^5]: Repeated applications of [(\[eq:G-delta-bd\])]{} to the translation-invariant models result in the random-walk bound: ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda\leq S_\tau(x)$ for $\Lambda\subset{{\mathbb Z}^d}$ and $\tau\leq1$.
---
author:
- 'Chee Sheng Fong$^{1}$'
- 'Hisakazu Minakata$^{2,3}$'
- 'Hiroshi Nunokawa$^{4}$'
title: 'Non-unitary evolution of neutrinos in matter and the leptonic unitarity test\'
---
IFT-UAM/CSIC-17-117
Introduction {#sec:introduction}
=======
| 3,463
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|
NONSYN T:6 G:140 G:37 `aaacctaacacg`
`attattgaacat` sulfatase family protein lipoate-protein ligase A family protein
1021087 `Query: 36 VKLAMEEYVLKN` `1`
NONSYN T:5 G:151 G:37 `+ LAMEEYVLKN`
`Sbjct: 14 LNLAMEEYVLKN` `25`
`gi 87162345 ref` `143` `AGIGRYLLNRVD`
`AGIGRYLLNR+D`
`AGIGRYLLNRLD`
`SRR022865_32431` `-37` `Ggagattcaatg` lantibiotic epidermin biosynthesis protein EpiC
`Cgtggattagta`
19505
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1}=\textrm{IT}\{ g;\mathcal{V}\}$.$\qquad\square$
As we need the second part of these theorems in our applications, their proofs are indicated below. The special significance of the first parts is that they ensure the converse of the usual result that the composition of two continuous functions is continuous, namely that one of the components of a composition is continuous whenever the composition is so.
**Proof of Theorem A2.1.** If $f$ be image continuous, $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:f^{-}(V_{1})\in\mathcal{U}_{1}\}$ and $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:q^{-}(U_{1})\in\mathcal{U}\}$ are the final topologies of $Y_{1}$ and $X_{1}$ based on the topologies of $X_{1}$ and $X$ respectively. Then $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:q^{-}f^{-}(V_{1})\in\mathcal{U}\}$ shows that $h$ is image continuous.
Conversely, when $h$ is image continuous, $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:h^{-}(V_{1})\}\in\mathcal{U}\}=\{ V_{1}\subseteq Y_{1}\!:q^{-}f^{-}(V_{1})\}\in\mathcal{U}\}$, with $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:q^{-}(U_{1})\in\mathcal{U}\}$, proves $f^{-}(V_{1})$ to be open in $X_{1}$ and thereby $f$ to be image continuous.
**Proof of Theorem A2.2.** If $f$ be preimage continuous, $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:V_{1}=e^{-}(V)\textrm{ if }V\in\mathcal{V}\}$ and $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:U_{1}=f^{-}(V_{1})\textrm{ if }V_{1}\in\mathcal{V}_{1}\}$ are the initial topologies of $Y_{1}$ and $X_{1}$ respectively. Hence from $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:U_{1}=f^{-}e^{-}(V)\textrm{ if }V\in\mathcal{V}\}$ it follows that $g$ is preimage continuous.
Conversely, when $g$ is preimage continuous, $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:U_{1}=g^{-}(V)\textrm{ if }V\in\mathcal{V}\textrm{ }\}=\{ U_{1}\subseteq X_{1}\!:U_{1}=f^{-}e^{-}(V)\textrm{ if }V\in\mathcal{V}\}$ and $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:V_{1}=e^{-}(V)\textrm{ if }V\in\mathcal{V}\}$ shows that $f$ is preimage continuous.$\qquad\blacksquare$
Since both Eqs. (\[Eq
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nchrotron radiation. The anti-DID impacts also the hit rate on the VXD due to beamstrahlung electrons, by reducing the number of backscattered electrons travelling backwards from further along the beam line.
[r]{}[0.5]{}
{width="0.5\columnwidth"}
The anti-DID reduces by roughly 30% the number of hits on the VXD, in particular the large hit time component, as can be seen in Figure \[Fig::occl1aD\]. This leads to a more homogeneous local distribution in $\phi$. The occupancy of the ILD vertex detector, which is a driving parameter of its requirements, has been evaluated with the latest version of the experimental apparatus, assuming a five-layer VXD geometry with 15 mm inner radius and a 3.5 T magnetic field. The evaluation was performed for two different sets of pixel characteristics, representative of the most mature sensor technologies under consideration. Both sets assume a continous read-out during the train. They differ by their read-out time, pixel pitch, cluster multiplicity and sensitive volume thickness.
Conclusion
==========
Occupancies of $\sim2\%$ and $\sim7\%$ were found in the innermost layer for the two sets. The average occupancy would be about 30% lower in presence of anti-DID, with a 50% decrease in one azimuthal sector. Accounting for the uncertainties on these predictions translates into upper limits on the occupancy in the innermost layer in the range 5-15%, depending on the sensor characteristics. These high rates plead for additional R&D on the sensors equipping this layer, in particular for shortening the read-out time significantly below 50 $\mu s$.
[99]{} Presentation:\
`http://ilcagenda.linearcollider.org/contributionDisplay.py?contribId=221&sessionId=21&confId=2628` R.De Masi [*et al.*]{}, ILC-note in preparation. D. Schulte, PhD Thesis, University of Hamburg, (1996). P.M. de Freitas, MOKKA, `http://mokka.in2p3.fr`. ILC software `http://ilcsoft.desy.de`. `http://iphc.in2p3.fr/-CMOS-ILC-.html`.
---
author:
- Mario Hamuy
title: The Standard Candle Method f
| 3,466
| 1,181
| 2,960
| 3,748
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an inverted parabolic barrier with the cross-section given by: $$\sigma = \frac{R_C^2}{2E}\hbar\omega \cdot ln \left \{ 1+exp\left [\frac{2\pi}{\hbar\omega}(E-V_C)\right] \right \}$$ where E is the incident energy, V$_C$ is the barrier height, R$_C$ is the radius of interaction and $\hbar$$\omega$ is the barrier curvature. The fit of the high resolution $^{18}$O data and the $^{16}$O data are indicated as the solid black and dashed black lines in Fig. \[fig:xsect\] respectively. The good agreement observed between the Wong fit of the high resolution $^{18}$O data and the $^{18}$O data measured in this experiment (blue points) underscores that there are no significant systematic errors associated with the present measurement. The solid red curve in Fig. \[fig:xsect\] depicts the fit of the $^{19}$O data. With the exception of the cross-section measured at E$_{cm}$ $\approx$ 12 MeV, the measured cross-sections are reasonably described by the Wong formalism. The extracted parameters for the $^{16}$O, $^{18}$O, and $^{19}$O reactions are summarized in Table 1. It is not surprising that the barrier height, V$_C$, remains essentially the same for all of the three reactions examined as the charge density distribution is unchanged.
V$_C$ (MeV) R$_C$ (fm) $\hbar$$\omega$ (MeV)
--------------------- ----------------- ----------------- -----------------------
$^{16}$O + $^{12}$C 7.93 $\pm$0.16 7.25 $\pm$ 0.25 2.95 $\pm$ 0.37
$^{18}$O + $^{12}$C 7.66 $\pm$ 0.10 7.39 $\pm$ 0.11 2.90 $\pm$ 0.18
$^{19}$O + $^{12}$C 7.73 $\pm$ 0.72 8.10 $\pm$ 0.47 6.38 $\pm$ 1.00
: \[tab:5/tc\]Wong fit parameters for the indicated fusion excitation functions. See text for details.
![\[fig:xs\_ratio\] Dependence of the ratio of $\sigma$($^{19}$O)/$\sigma$($^{18}$O) and $\sigma$($^{18}$O)/$\sigma$($^{16}$O) on E$_{c.m.}$. The shaded region depicts the uncertainty associated with the ratio for the $^{19}$O reaction. Inset: Dependence of the ratio of the barrier cur
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parental genotypes [@pone.0025810-Aleza1]. As triploid hybrids are sterile [@pone.0025810-Cameron1], only some of the diploid genotypes that are known to be cross-fertile with clementine mandarin were considered PPDs. Plot B was composed of a population of 477 hybrids belonging to a rootstock breeding program. These hybrids were randomly distributed within the plot, and all them were, in principle, potential pollinators of clementine.
10.1371/journal.pone.0025810.t001
###### Potential pollen donor (PPD) genotypes present at the study site, including their abbreviation codes, population sizes (number of adult trees) and relative amounts.
{#pone-0025810-t001-1}
Plot PPD Genotype code Population size Relative amount (%)
------ -------------------------------------------- ------ ----------------- ---------------------
T Pineapple sweet orange P 24 3.80
Carrizo citrange C 24 3.80
Mexican lime L 24 3.80
A Fortune mandarin F 34 5.38
Orlando tangelo ORL 10 1.58
Murcott mandarin MU 7 1.11
Nova tangor N 6 0.95
Ortanique tangor ORT 6 0.95
Willowleaf mandarin MC 6 0.95
Ellendale mandarin E 6 0.95
Kara mandarin K 4 0.63
Minneola tangelo MI 4 0.63
B King mandarin x *Poncirus trifoliata*
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hadowing effect.
For the inner anisotropy factor $f_{\mathrm{disc}}$, we simply assume $$\begin{aligned}
f_{\mathrm{disc}}(\theta)\propto\sin\theta\,,
\label{eq:11} \end{aligned}$$ which corresponds to radiation from an infinitely thin disc (recall that we define $\theta$ as the angle from the equatorial plane). Although numerical simulations suggest somewhat steeper $\theta$-dependence especially in the polar directions [e.g., @Ohsuga:2005aa; @Sc-adowski:2016aa], such deviations cause little effects on our results because the mass accretion predominantly occurs through the infalling region near the equatorial plane. For the disc radiation without the outer shadowing effect (i.e., $f_{\mathrm{shadow}}=1$), the normalized anisotropy factor is $\mathcal{F}(\theta) = 2\sin \theta$ (Fig. \[fig:dirdep\]).
We model the outer anisotropy factor $f_{\mathrm{shadow}}$ as $$\begin{aligned}
f_{\mathrm{shadow}}(\theta)
=
\begin{cases}
{\displaystyle}\exp\left[-\left(\frac{\theta-\tilde{\theta}_{\mathrm{shadow}}}{\delta\theta}\right)^2\right]
&0 < \theta < \tilde{\theta}_{\mathrm{shadow}}\\[0.4cm]
1 & \tilde{\theta}_{\mathrm{shadow}} < \theta < 90^\circ
\end{cases}
\label{eq:10}\end{aligned}$$ where $\tilde{\theta}_{\mathrm{shadow}}=\theta_{\mathrm{shadow}}+2\,\delta\theta$, $\theta_{\mathrm{shadow}}$ is the opening angle of the shadow, and $\delta\theta$ the thickness of the transition region. Here, we assume $f_{\mathrm{shadow}}$ is symmetric about the equatorial plane. We adopt the finite transition region setting $\delta\theta = 6^\circ$ to avoid artificial ionization structure that appears with $\delta\theta\rightarrow 0$. Our conclusions are independent of the arbitrary choice of a small value for $\delta\theta$. We show $\mathcal{F}(\theta)$ for the disc radiation with the outer shadowing effect with $\theta_{\mathrm{shadow}}=45^\circ$ and $22.5^\circ$ in Fig. \[fig:dirdep\]. With the expression given by equation , the outer anisotropy factor begins to decrease even for $\theta > \theta_{\mathrm{shadow}}$, an
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ult}_{E_{P,i}} \sigma^*\omega_P$.
An $n$-th cyclic covering ramified along $\cC$ induces an $n$-th cyclic cover ramified along the divisor $\sigma^*\cC$ in $Y$ . One can define the line bundles $\mathcal{L}^{(k)}=\cO_Y(L^{(k)})$ as in , where $H$ is the pull-back in $Y$ of a projective line. The announced description of $H^1(Y,\mathcal{L}^{(k)})$ admits the following form.
\[thm:conucleo\_liso\] For $0\leq k<n$, let $$\label{eq:defsigmak}
\sigma_k:
H^0(\PP^2,\mathcal{O}_{\PP^2}(k-3))\longrightarrow
\bigoplus_{P\in\operatorname{Sing}^*(\mathcal{C})}
\frac{\mathcal{O}_{\PP^2,P}}{\mathcal{J}_{\mathcal{C},P,\frac{k}{n}}}$$ be the natural map where $$\array{rcl}
\mathcal{J}_{\mathcal{C},P,\lambda}&:=&
\{h\in\mathcal{O}_{\PP^2,P}\mid\operatorname{mult}_{E_{P,i}}\sigma^*(h)\geq
\left\lfloor \lambda n_{P,i}\right\rfloor
-\nu_{P,i}+1\}\\
&=&
\{h\in\mathcal{O}_{\PP^2,P}\mid\operatorname{mult}_{E_{P,i}}\sigma^*(h)>
\lambda n_{P,i}-\nu_{P,i}\}.
\endarray$$ $$\dim H^1(Y,\mathcal{L}^{(k)})=\dim\operatorname{coker}\sigma_k.$$
The second goal of this work is to generalize this theorem to the weighted projective plane. In order to do so we present a different proof of Theorem \[thm:conucleo\_liso\] that allows for an extension to the singular case. This new proof is somewhat more conceptual than the original one from [@Artal94] and will be outlined here for exposition purposes and to help the reader have a basic idea of the strategy that will be used for the proof of the main Theorem \[thm:conucleo\_singular\].
The proof starts using the Riemann-Roch Theorem on the line bundles $\mathcal{L}^{(k)}=\cO_Y(L^{(k)})$: $$\label{eq:RRliso}
\array{c}
\chi(Y,\mathcal{L}^{(k)})=
\dim H^0(Y,\mathcal{L}^{(k)})
-\dim H^1(Y,\mathcal{L}^{(k)})+
\dim H^2(Y,\mathcal{L}^{(k)})\\
=\chi(Y,\mathcal{O}_Y)+
\dfrac{{L}^{(k)}\cdot({L}^{(k)}-K_Y)}{2}.
\endarray$$
Note that $\chi(Y,\mathcal{O}_Y)=1$ since $Y$ is a rational surface. It is not hard to check that $H^0(Y,\mathcal{L}^{(k)})$ is a subspace of $H^0(\PP^2,\mathcal{O}_{\PP^2}(-k))$ and he
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}\hbar\nabla_{\mathbf{r}}+\frac{e}{c}\vec{A}_{\mathbf{R}}(t)\right)\psi_{i,\mathbf{R}}(\vec{r},t) \nonumber \\
&& -\psi_{i,\mathbf{R}}(\vec{r},t)\left(-\mathrm{i}\hbar\nabla_{\mathbf{r}}-\frac{e}{c}\vec{A}_{\mathbf{R}}(t)\right)\psi_{i,\mathbf{R}}^{\ast}(\vec{r},t)\biggr\}.\end{aligned}$$ This microscopic current is averaged over a unit cell to define the macroscopic current ($\vec{J}_{\mathbf{R}}$) as: $$\label{eq:maccurrent}
\vec{J}_{\mathbf{R}}(t)=\frac{1}{\Omega}\int_{\Omega} \vec{j}_{\mathbf{R}}(\vec{r},t) \, \mathrm{d}\vec{r},$$ where $\Omega$ is the unit cell volume. It should be noted that there is also a contribution to the current from a nonlocal pseudopotential. The propagation of the laser pulse is described by the wave equation with the macroscopic current as its source term: $$\label{eq:we}
\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\vec{A}_{\mathbf{R}}(t)-\nabla^2_{\mathbf{R}}\vec{A}_{\mathbf{R}}(t)=-\frac{4\pi e}{c}\vec{J}_{\mathbf{R}}(t).$$ The vector potential obtained by solving Eq. (\[eq:we\]) is used to solve Eq. (\[eq:tdks\]) at the next time step. The interaction between a laser pulse and matter can be fully described by solving Eqs. (\[eq:tdks\]) and (\[eq:we\]) self-consistently via the macroscopic current and the vector potential [@Yabana:2012]. It should be noted that the electron motion is restricted to be within a unit cell; non-local processes such as electron transport among unit cells cannot be accounted for in the present method. Moreover, the ionic motion is neglected since the motion of ions is slow enough in comparison with electrons due to their large mass. However, these are beyond the scope of our interest since we consider laser intensities smaller than $10^{17} \, \mathrm{W/cm^{2}}$ and wavelength of the pulse in the near-visible region.
\[ssec:details\] Simulation details
-----------------------------------
For the sake of simplicity of the laser–matter interaction geometry, the case of normal incidence of the pulse was considered in the simulation. The $\left(\bar{2
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ective sample size is approximately $n_1\ell_1$, i.e. pairwise comparisons coming from small set size do not contribute without proper normalization. This gap in MSE corroborates bounds of Theorem \[thm:main\]. Normalization constant $C$ is $10^{3}/d^2$.
The Role of the Topology of the Data {#sec:role}
====================================
We study the role of topology of the data that provides a guideline for designing the collection of data when we do have some control, as in recommendation systems, designing surveys, and crowdsourcing. The core optimization problem of interest to the designer of such a system is to achieve the best accuracy while minimizing the number of questions.
The Role of the Graph Laplacian
-------------------------------
Using the same number of samples, comparison graphs with larger spectral gap achieve better accuracy, compared to those with smaller spectral gaps. To illustrate how graph topology effects the accuracy, we reproduce known spectral properties of canonical graphs, and numerically compare the performance of data-driven rank-breaking for several graph topologies. We follow the examples and experimental setup from [@SBB15] for a similar result with pairwise comparisons. Spectral properties of graphs have been a topic of wide interest for decades. We consider a scenario where we fix the size of offerings as $\kappa_j=\kappa=O(1)$ and each agent provides partial ranking with $\ell$ separators, positions of which are chosen uniformly at random. The resulting spectral gap $\alpha$ of different choices of the set $S_j$’s are provided below. The total number edges in the comparisons graph (counting hyper-edges as multiple edges) is defined as $|E|\equiv{\kappa \choose 2}\,n$.
- Complete graph: when $|E|$ is larger than ${d \choose 2}$, we can design the comparison graph to be a complete graph over $d$ nodes. The weight $A_{ii'}$ on each edge is $n\,\ell/(d(d-1))$, which is the effective number of samples divided by twice the number of edges. Resulting spectral gap is one,
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a group and $\sigma \in {{\rm Aut}\,}(H)$ an automorphism of $H$. We define the category $\mathcal{C}(H, \sigma)$ as follows: an object of $\mathcal{C}(H, \sigma)$ is a triple $(G, \alpha,
\beta)$ such that $(H, G, \alpha, \beta)$ is a matched pair of groups. A morphism $\psi : (G', \alpha', \beta') \rightarrow (G,
\alpha, \beta)$ in $\mathcal{C}(H, \sigma)$ is a morphism of groups $\psi: H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G'
\rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ such that the following diagram $${\label{eq:D1}}
\begin{CD}
H@>i_H>> H\, {}_{\alpha'}\bowtie_{\beta'}G' \\
@VV\sigma V @VV\psi V\\
H@>i_H>> H _{\alpha} \bowtie_{\beta} G
\end{CD}$$ is commutative. A (iso)morphism $\psi: H\, {}_{\alpha'}\!\!
\bowtie_{\beta'} \, G' \rightarrow H\, {}_{\alpha}\!\!
\bowtie_{\beta} \, G$ in the category $\mathcal{C}(H, \sigma)$ will be called a *$\sigma$-invariant (iso)morphism* between the two bicrossed products.
The following key proposition describes explicitly the morphisms of $\mathcal{C}(H, \sigma)$ and gives a necessary and sufficient condition for two bicrossed products $H\, {}_{\alpha'}\!\!
\bowtie_{\beta'} \, G'$ and $H\, {}_{\alpha}\!\! \bowtie_{\beta}
\, G$ to be isomorphic in the category $\mathcal{C}(H, \sigma)$. If $G'$ is a new group we shall denote by “$*$” the multiplication of $G'$ and $\alpha'(g',h) = g' \rhd' h$, $\beta' (g',h) = g'
\lhd' h$, for all $g'\in G'$ and $h\in H$.
[\[pr:1\]]{} Let $H$ be a group, $\sigma \in {{\rm Aut}\,}(H)$ and $(H, G, \alpha,
\beta)$, $(H, G', \alpha', \beta')$ two matched pairs. There exists a one to one correspondence between the set of all morphisms $\psi: H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G'
\rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ in the category $\mathcal{C}(H, \sigma)$ and the set of all pairs $(r,v)$, where $r: G' \rightarrow H$, $v: G' \rightarrow G$ are two maps such that: $$\begin{aligned}
\sigma(g' \rhd' h)r(g' \lhd' h) &=& r(g')\bigl(v(g') \rhd
\sigma(h)\bigl){\label{eq:p1}} \\
v(g' \lhd' h) &=& v(g') \lhd \sigma(
| 3,473
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imates are more relaxed but sufficient for giving another robust argument in proving the instability, in particular not by contradiction. In another related paper, we are able to prove instability theorems of the spherical symmetric naked singularities under certain isotropic gravitational perturbations without symmetries. The argument given in this paper plays a central role.'
address:
- |
Department of Mathematics, Sun Yat-sen University\
Guangzhou, China
- |
Department of Mathematics, Sun Yat-sen University\
Guangzhou, China
author:
- Jue Liu
- Junbin Li
title: A robust proof of the instability of naked singularities of a scalar field in spherical symmetry
---
[^1]
Introduction
============
In the paper [@Chr99], Christodoulou proved both the *weak cosmic censorship conjecture* and the *strong cosmic censorship conjecture* for spherically symmetric solutions of the Einstein equations coupled with a massless scalar field. The coupled system reads $$\begin{aligned}
\mathbf{Ric}_{\alpha\beta}-\frac{1}{2}\mathbf{R}g_{\alpha\beta}=\mathbf{T}_{\alpha\beta}=\nabla_\alpha\phi\nabla_\beta\phi-\frac{1}{2}g_{\alpha\beta}g^{\mu\nu}\nabla_\mu\phi\nabla_\nu\phi,\end{aligned}$$ which we call the Einstein-scalar field equations. The proof, which is by contradiction, contains sharp estimates which may not be easily obtained beyond spherical symmetry. In this paper, we will provide a robust proof which is not by contradiction and contains only relaxed estimates. The main advantage of this proof is that it has the potential to be extended beyond spherical symmetry.
Consider the characteristic initial value problem of the Einstein-scalar field equations in spherical symmetry. The initial data is given on a null cone $C_o$ issuing from a fixed point $o$ of the symmetry group $SO(3)$, and consists of a function $\alpha_0=\frac{\partial}{\partial r}(r\phi)\big|_{C_o}$ defining on $[0,+\infty)$, where $r$ is area radius of the orbit spherical sections of $C_o$, and $\phi$ is the scalar field function. Then what wa
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or $$\begin{aligned}
D_0(k,\omega)=\bigg[\omega-\frac{k^2}{4m}+2\mu-\nu+i0^+\bigg]^{-1}
\label{D0}\end{aligned}$$ and the “polarization”, i.e., self-energy of the closed channel propagator, $\Pi(k,\omega)$ is given by: $$\begin{aligned}
\Pi(k,\omega)=g^2 \int \frac{dk'}{2\pi}
\Big[\omega-k'^2/m-k^2/4m+2\mu+i0^+\Big]^{-1}.
\label{pi2}\end{aligned}$$ From Eq. (\[dysonvacuum\]), we can compute the dressed rest energy $\epsilon_b$ of a dimer, which is defined as being the $k=0$ pole of the molecular propagator when $\mu=0$: $$\begin{aligned}
D(0,\epsilon_b)^{-1}=D_0(0,\epsilon_b)^{-1}-\Pi(0,\epsilon_b)=0.
\label{formalBS}\end{aligned}$$ We find that Eq. (\[formalBS\]) admits a unique real negative solution $|\epsilon_b|=-\epsilon_b$ irrespective of the sign of the detuning $\nu$ $$\frac{|\epsilon_b|}{|\epsilon_{\star}|}-
\sqrt{\frac{|\epsilon_{\star}|}{|\epsilon_b|}}+
\frac{\nu}{|\epsilon_{\star}|}=0,
\label{bse}$$ where we have introduced the on-resonance ($\nu=0$) bound state energy $\epsilon_{\star}\equiv \epsilon_b(\nu=0)=-m^{1/3}g^{4/3}/2^{2/3}$. This has to be compared with the 3D BFRM where a bound state is present only when the detuning is negative [@footnote1].
According to standard scattering theory, the $T$-matrix is given by $T=g^2D$ (see, e.g., [@DS]). Therefore, in the BFRM, the Lippmann-Schwinger equation for atoms is equivalent to the closed channel Dyson equation for the molecular propagator in vacuum, equation (\[dysonvacuum\]). From the latter it is possible to show that the scattering between two atoms can be described as resulting from an effective contact potential $g_1\delta(x)$, which is a well defined 1D potential, with a bare scattering amplitude $$\begin{aligned}
g_1\equiv g^2 D_0(0,0)=-\frac{g^2}{\nu}.
\label{g1}\end{aligned}$$ When the detuning goes to zero, the bare scattering amplitude diverges: this corresponds to the resonance. Before resonance, we have $\nu>0$ and an attractive effective interaction $g_1<0$ between the atoms, while after resonance $\nu<0$ and the effective inter
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SC Customer Care 713-345-4727
The Portland Web Server will be going down for 20 min at 1:45 instead of
12:00 noon.
Regards,
Paul
Great - no changes
-----Original Message-----
From: Shah, Kal
Sent: Wednesday, February 06, 2002 4:14 PM
To: Kitchen, Louise
Subject: RE: Direct Mail Pack Letter
I've redrafted it. See attached. Thanks.
Kal
<< File: custletter_postlaunch_1.doc >>
-----Original Message-----
From: Kitchen, Louise
Sent: Wednesday, February 06, 2002 3:59 PM
To: Shah, Kal
Subject: RE: Direct Mail Pack Letter
Can you redraft in light of how I redrafted the other one - removing Enron's name from the first paragraph. And resend to me
Thanks
-----Original Message-----
From: Shah, Kal
Sent: Wednesday, February 06, 2002 3:31 PM
To: Kitchen, Louise
Subject: Direct Mail Pack Letter
Louise -- Could you review and approve the attached draft of the cover letter for the direct mail pack? The letter will be sent under your name. I will come by tomorrow to get your signature on a piece of paper. We will scan it and print on the letter electronically.
Thanks
Kal
<< File: custletter_postlaunch.doc >>
I've got an idea. Howzabout Ted cooks and you clean up? Seriously, though,
I'll try to host next time at my humble apartment.
Jacqueline Kelly <JKelly@FairIsaac.com>
11/09/2000 01:31 PM
To: "'Jeff.Dasovich@enron.com'" <Jeff.Dasovich@enron.com>
cc:
Subject: RE: spreadsheet
No problemo! Maybe I will cook for us next time...oh, that's right I don't
cook anymore...
I thought we did a good job of cranking things out. Have a good weekend!
-----Original Message-----
From: Jeff.Dasovich@enron.com [mailto:Jeff.Dasovich@enron.com]
Sent: Thursday, November 09, 2000 10:12 AM
To: Jacqueline Kelly
Subject: Re: spreadsheet
thanks again for hosting. most kind of you, and we seemed to make a lot of
progress. did you get enough money for dinner?
----- Forwarded by Mark Taylor/HOU/ECT on 09/06/2000 11:23 AM -----
Andrea Bertone@ENRON
08/15/2000 10:43 AM
To: Mark Ta
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\sigma = 1$) and Tikhonov (using $\sigma = 0.2$) covariance functions, respectively. (d) & (e) reconstructions using CV with Laplacian and Tikhonov covariance functions, respectively []{data-label="fig:Parameter Choice Methods"}](LcurveChestLaplacian_sigman1 "fig:"){width="6.3cm"}]{} (306,138)[![(a) A ground truth of 2D chest phantom. (b) & (c) reconstructions using L-curve parameter choice method with Laplacian (using $\sigma = 1$) and Tikhonov (using $\sigma = 0.2$) covariance functions, respectively. (d) & (e) reconstructions using CV with Laplacian and Tikhonov covariance functions, respectively []{data-label="fig:Parameter Choice Methods"}](LcurveChestTikhonov_sigman02 "fig:"){width="6.3cm"}]{} (8,3)[![(a) A ground truth of 2D chest phantom. (b) & (c) reconstructions using L-curve parameter choice method with Laplacian (using $\sigma = 1$) and Tikhonov (using $\sigma = 0.2$) covariance functions, respectively. (d) & (e) reconstructions using CV with Laplacian and Tikhonov covariance functions, respectively []{data-label="fig:Parameter Choice Methods"}](CVChestLaplacian "fig:"){width="6.3cm"}]{} (155,3)[![(a) A ground truth of 2D chest phantom. (b) & (c) reconstructions using L-curve parameter choice method with Laplacian (using $\sigma = 1$) and Tikhonov (using $\sigma = 0.2$) covariance functions, respectively. (d) & (e) reconstructions using CV with Laplacian and Tikhonov covariance functions, respectively []{data-label="fig:Parameter Choice Methods"}](CVChestTikhonov "fig:"){width="6.3cm"}]{} (87,137)[(a)]{} (240,137)[(b)]{} (393,137)[(c)]{} (87,0)[(d)]{} (240,0)[(e)]{}
Real data: Carved cheese {#RealData}
------------------------
We now consider a real-world example using the tomographic x-ray data of a carved cheese slice measured with a custom-built CT device available at the University of Helsinki, Finland. The dataset is available online [@fips_dataset]. For a detailed documentation of the acquisition setup—including the specifications of the x-ray systems—see [@bubba2017tomographic]. We use th
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---------------------------------
RR flux & IIB & IIA\
------------------------------------------------------------------------
$-m$ & $F_{x^1 x^2 x^3}$ & ${F}$\
------------------------------------------------------------------------
$-q_i$ & $F_{y^i x^j x^k}$ & $F_{x^i y^i}$\
------------------------------------------------------------------------
$e_i$ & $ F_{x^i y^j y^k}$ & $F_{x^j y^j x^k y^k}$\
------------------------------------------------------------------------
$-e_0$ & $F_{y^1 y^2 y^3 }$ & $F_{x^1 y^1 x^2 y^2 x^3 y^3}$\
In other words, the IIA and IIB theory give rise to the same model provided that the fluxes of the two theories are identified as in Table \[TableRRfluxes\]. The fluxes are precisely those that are allowed by the orientifold projection in each of the two theories.
The situation is different when NS-NS geometric fluxes are turned on. In this case, the $H_3$ flux in gives a term which is $S$ times a cubic polynomial in the $U$ moduli for the IIB theory, while in the IIA theory the $H_3$ flux in gives a term linear in $S$ and one linear in $U$, while the $f$ flux gives a term linear is $ST$ and one linear in $UT$. Thus obviously the two models cannot be identified by mirror symmetry. This is not surprising, because indeed these fluxes are related to the non-geometric $Q$ and $R$ fluxes by the chain of T-dualities [@Shelton:2005cf; @Aldazabal:2006up] $$H_{abc} \overset{T_c}{\longleftrightarrow} -f_{ab}^c \overset{T_b}{\longleftrightarrow} -Q_a^{bc} \overset{T_a}{\longleftrightarrow} R^{abc} \quad . \label{TdualityruleNSfluxes}$$ In particular, in our model this implies that in IIB both the $H$ and $Q$ fluxes can be turned on, and they are related by T-duality to the IIA fluxes as in Table \[TableNSfluxes\]. The superpotential for IIB then becomes [@Shelton:2005cf; @Aldazabal:2006up] [^2] $${ W_{\rm IIB/O3}= \int ( F_3 - i S H_3 + Q \cdot {\mathcal J}_{\rm c}) \wedge \Omega } \quad , \label{NSnongeomIIBsuperpot}$$ while in IIA one has $$W_{\rm IIA/O6} = \int [ e^{J_{\rm c}}
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ta_{24} = 0.01$, and $\sin^2\theta_{34} = 0.1$ for $\Delta m^2_{41} = 0.1$ eV$^2$, and set all the CP phases to zero. Then, we cut out the $3\times3$ active neutrino mixing matrix, which is non-unitary.[^18] For the Standard Model mixing parameters in $U_{\text{\tiny PDG}}$, we take $\sin^2\theta_{12} = 0.3$, $\sin^2\theta_{23} = 0.5$, $\sin^2 (2\theta_{13}) = 0.09$, and the mass squared differences $\Delta m_{21}^2 = 7.4 \times 10^{-5}$ eV$^2$ and $\Delta m_{31}^2 = 2.4 \times 10^{-3}$ eV$^2$, and set the CP phase $\delta_{\text{CP}}$ to zero.
![ In the upper panel (a), presented is the iso-contour of $P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{non-unitary} }^{(0)}$ in space spanned by neutrino energy $E$ and baseline $L$. In the lower panel (b), the iso-contour of the difference $\Delta P (\nu_{\mu} \rightarrow \nu_{e}) \equiv P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{non-unitary} }^{(0)}$ is presented. For the values of unitarity-violating as well as the standard mixing parameters taken, see the text. []{data-label="fig:Pmue_energy_dist"}](Pmue_energy_dist_non_unitary_small_size.jpeg "fig:"){width="85.00000%"} ![ In the upper panel (a), presented is the iso-contour of $P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{non-unitary} }^{(0)}$ in space spanned by neutrino energy $E$ and baseline $L$. In the lower panel (b), the iso-contour of the difference $\Delta P (\nu_{\mu} \rightarrow \nu_{e}) \equiv P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{non-unitary} }^{(0)}$ is presented. For the values of unitarity-violating as well as the standard mixing parameters taken, see the text. []{data-label="fig:Pmue_energy_dist"}](Pmue_energy_dist_difference_small_size.jpeg "fig:"){width="85.00000%"}
![ In the upper panel (a), presented is the iso-contour of $P(\nu_{\mu} \rightarrow \nu_{\tau})_{ \text{non-unitary} }^{(0)}$ in $E-L$ space. In the lower panel (b), the iso-contour of the difference $\Delta P (\nu_{\mu} \rightarrow \
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| 3,688
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rating $d\tau_{xz}/dz$ across the interface gives $$S = -\lim\limits_{\epsilon \to 0} \int \limits_{1-\epsilon}^{1+\epsilon}dz\langle v_{1x}v_{1z} \rangle = -\lim\limits_{\epsilon \to 0} \int \limits_{1-\epsilon}^{1+\epsilon}dz\frac{d}{dz}\langle \frac{d\Phi}{dz}\frac{\partial \Phi}{\partial x}\rangle$$ where $\langle \rangle$ denotes averaging in $x$, while $\mathbf{v}_1$ is the perturbed velocity. Taking $\Phi = \mathcal{F}^{-1}[\hat{\phi}]$ with $\hat{\phi} = \beta_1\phi_1 + \beta_2\phi_2$ gives $$\label{shear}
\begin{split}
S = \int \limits_{-\infty}^{\infty}\frac{dk}{2\pi}4k^2e^{2|k|} \bigg[ &\mathrm{Im}(\omega_1^*)|\beta_1|^2 + \mathrm{Im}(\omega_2^*)|\beta_2|^2\\
+ &\mathrm{Im}\left[(\omega_2^*+k)\beta_1\beta_2^*\right] + \mathrm{Im}\left[(\omega_1^*+k)\beta_2\beta_1^*\right]\bigg].
\end{split}$$ When the stable modes are ignored, only the first term contributes to $S$. The coefficient $4k^2e^{2|k|}$ is positive, and Eq. shows that $\mathrm{Im}(\omega^*_1) \leq 0$ and $\mathrm{Im}(\omega^*_2) = -\mathrm{Im}(\omega^*_1)$, indicating that the transport due to unstable modes alone is negative, and the second term acts against the first to reduce $|S|$. Clearly the amplitude of $\beta_2(k)$ relative to $\beta_1(k)$ has a significant impact on the momentum transport in this system. The relative phase between $\beta_2(k)$ and $\beta_1(k)$ determines the contribution of the last two terms. If $|\beta_2(k)| = |\beta_1(k)|$, then the first two terms cancel and the transport is entirely determined by the last two terms. Analysis of other systems shows there are situations where eigenmode cross correlations significantly affect transport[@Baver; @Terry2009].
To determine the actual properties of $S$, it is necessary to solve Eqs. and for $\beta_j(k)$ and integrate Eq. , either analytically or numerically. This is beyond the scope of the present paper, but will be considered in the future. In lieu of such solutions, we construct an estimate of the ratio $|\beta_2(k)|/|\beta_1(k)|$ using the threshold parameter
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uad 1\leq j\leq n.$$
We claim that $c_{ji}=0$ for $i<j$ and that $|c_{ji}|>0$ whenever $i>j$ (and $c_{ji}\not=0$). Since $|u_i|\leq |u_{i+1}|$, we have $|a_i|\geq |a_{i+1}|$ for each $i$. Also $|c_{ji}|=|u_i|-|u_j|$ for all $i,j$ and so $c_{ji}=0$ if $|u_i|<|u_j|$. Thus both parts of the claim are clear when $|u_i|\not= |u_{j}|$; equivalently, when $|a_i|\not= |a_{j}|$. So, suppose that $|a_i|=|a_j|$, for some $i\not =j$ and that $c_{ji}\not=0$. Then $c_{ji}\in k^*$ and so expresses $a_i$ as a left linear combination of the other $a_\ell$. This contradicts the initial minimality assumption on $n$ and proves the claim. Note that $c_{jj}=1$ for all $j$, since otherwise would express $a_j$ as a left linear combination of the other $a_\ell$.
The last paragraph implies that $C=(c_{ji})$ is an upper triangular matrix, with units on the diagonal and so it is invertible. In particular, $\{p_1,\dots,p_n\}\cup \{u_{n+\ell} : \ell >0\}$ is a basis for $F$. Thus $G=\sum_{i=1}^n p_iA$ is a graded-free direct summand of $F$ contained in $P$. Thus $G$ is also a graded-free direct summand of $P$ which, by , contains $x$.
{#section-5}
The proof of the theorem follows from the sublemmas by an easy induction. By Sublemma \[graded-proj-sublemma1\] we may assume that $P$ is countably generated, say by homogeneous elements $z_i $ for $i\in \mathbb N$. By induction, suppose that there is a graded decomposition $P=Q_1\oplus\cdots \oplus Q_n\oplus R_n$, where each $Q_i$ is graded-free and $z_i\in Q_1\oplus\cdots \oplus Q_i$, for $1\leq
i\leq n$. By Sublemma \[graded-proj-sublemma\] this does hold when $n=1$. Write $z_{n+1} = q+r$ as a homogeneous sum, where $
q\in \sum Q_j$ and $r\in R_n$. Since $R_n$ also satisfies the hypotheses of Sublemma \[graded-proj-sublemma\], $R_n$ has a graded-free summand $Q_{n+1}$ containing $r$, completing the inductive step. Finally, $$\widetilde{P}\ = \ \lim_{n\to \infty} \big( Q_1\oplus\cdots \oplus
Q_n\big)
\ \cong \ \bigoplus_{i= 1}^\infty Q_i$$ is a graded-free submodule of $P$ that contains ea
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3.2.1.8}$$ One can see, that two components $\chi_{l=0}^{(\pm)}(k,r)$ in (\[eq.3.2.1.2\]) represent convergent and divergent waves, that can be useful for analysis of propagation of the particle in the field $V_{2}(r)$. Thus, we have found an *exact analytical division of the total radial wave function into its convergent and divergent components* (as for regular and singular Coulomb functions for the known Coulomb potential) in the description of scattering of the particle in the inverse power potential (\[eq.3.1.5\]).
If for the convergent and divergent waves to define radial flows as: $$j^{\pm} (k,r) =
\displaystyle\frac{i\hbar}{2m}
\biggl( \chi_{l=0}^{(\pm)}(k,r)
\displaystyle\frac{d \chi_{l=0}^{(\pm), *}(k,r)}{dr}-
\chi_{l=0}^{(\pm), *}(k,r)
\displaystyle\frac{d \chi_{l=0}^{(\pm)}(k,r)}{dr} \biggr),
\label{eq.3.2.1.9}$$ then for both waves we obtain coincided absolute values of their flows: $$j^{\pm} (k,r) = \pm\displaystyle\frac{\hbar k}{m} |\bar{N}_{2}|^{2}.
\label{eq.3.2.1.10}$$ We see, that the flows do not vary in dependence on $r$, and this gives a fulfillment of a conservation law for the flows from each wave and the total flow. Therefore, the convergent wave $\chi_{l=0}^{(-)}(k,r)$ propagates into the center without the smallest reflection by the field, because it is defined and is continuous on the whole region of the definition of the potential (\[eq.3.1.5\]) and it forms the constant radial flow $j^{-}(r)$. Now we can tell with confidence, that the *inverse power radial potential (\[eq.3.1.5\]), for which we have found the radial wave function (\[eq.3.2.1.2\])–(\[eq.3.2.1.4\]) for scattering, is reflectionless at $l=0$*.
Further, one can find the radial wave functions at $l \ne 0$ on the basis of the same analysis, if for the radial wave function (\[eq.3.2.1.1\]) for the potential with zero value to use spherical Hankel functions instead of factors $\exp{(\pm ikr)}$.
### Wave functions for the reflectionless potential with the barrier \[sec.3.2.2\]
One can use Exp.�
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|
psi}\gamma^\mu\partial^\nu\psi\,-\,
\partial^\nu\overline{\psi}\gamma^\mu\psi\right]\ ,$$ while fermion normal ordering is implicit of course. It then follows that the 1-particle states obtained by acting with the creation operators $b^\dagger(\vec{k},s)$ and $d^\dagger(\vec{k},s)$ on the Fock vacuum $|0\rangle$ are energy-momentum eigenstates of momentum $\vec{k}$ and mass $m$, possessing spin or helicity 1/2.
In the same way as for the scalar field,[@GovCOPRO2] it is possible to compute the Feynman propagator of the Dirac field, namely the causal probability amplitude for seeing a particle created at a given point in spacetime and annihilated at some other such point. This time-ordered amplitude is thus defined by the 2-point correlation function $$\begin{array}{r c l}
\langle 0|T\psi_\alpha(x)\overline{\psi}_\beta(y)|0\rangle&=&
\theta(x^0-y^0)\langle 0|\psi_\alpha(x)\overline{\psi}_\beta(y)|0\rangle\,-\,\\
& & \\
&&-\theta(y^0-x^0)\langle 0|\overline{\psi}_\beta(y)\psi_\alpha(x)|0\rangle\ ,
\end{array}$$ where the anticommuting nature of the spinor is accounted for through the negative sign in the second contribution in the r.h.s. ($\theta(x)$ denotes the usual step function, $\theta(x>0)=1$ and $\theta(x<0)=0$). In the case of the free Dirac field, a direct substitution of the mode expansion (\[eq:solution\]) leads to the integral representation, $$\langle 0|T\psi_\alpha(x)\overline{\psi}_\beta(y)|0\rangle=
\int\frac{d^4k}{(2\pi)^4}\,e^{-ik\cdot(x-y)}\,
\left(\frac{i}{\gamma^\mu k_\mu-m+i\epsilon}\right)_{\alpha\beta}\ ,$$ where, as usual,[@GovCOPRO2] $\epsilon>0$ corresponds to an infinitesimal imaginary part in the denominator of the momentum-space propagator introduced to specify the contour integration in the complex $k^0$ energy plane in order to pick up the correct pole contributions associated to the positive- and negative-frequency components of the Dirac spinor mode expansion. This Dirac propagator is the basis for perturbation theory involving Dirac spinors, in the same way that the Feynman propa
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o [(\[eq:IR-xbdNN\])]{} and the diagrammatic bound [(\[eq:piNbd\])]{}. It thus remains to show [(\[eq:pi-kbd\])]{} for $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ with $x\ne o$ and $j\ge1$.
The idea of the proof is somewhat similar to that of Proposition \[prp:exp-bootstrap\](iii) explained above. First, we take $|a_m|\equiv\max_n|a_n|$ from the lowermost path and $|a'_l|\equiv\max_n|a'_n|$ from the uppermost path of a bounding diagram. Note that, by [(\[eq:|x|-max\])]{}, $|a_m|$ and $|a'_l|$ are both bigger than $\frac1{j+1}|x|$. That is, $|a_m|^{-q}$ and $|a'_l|^{-q}$ are both bounded from above by $(j+1)^q|x|^{-q}$. If the path corresponding to $a_m$ in the $m^\text{th}$ block consists of $N$ segments, we take the “longest” segment whose end-to-end distance is therefore bigger than $\frac1{N(j+1)}|x|$. That is, the corresponding two-point function is bounded by $\lambda_0
N^q(j+1)^q|x|^{-q}$. Here, $N$ depends on the parity of $m$, as well as on $i\ge0$ for $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(i)}}$ (or $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(i)}}$ if $m=0$ or $j$) and the location of $u,v$ in each diagram, and is at most $N\leq O(i+1)$. However, the number of nonzero chains of bubbles contained in each diagram of $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(i)}}$ and $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(i)}}$ is $O(i)$, and hence their contribution would be $O(\theta_0)^{O(i)}$. This compensates the growing factor of $N^q$, and therefore we will not have to take the effect of $N$ seriously. The same is true for $a'_l$, and we refrain from repeating the same argument.
Next, we take the “longest” segment, denoted $a''$, among those which together with $a'_l$ (or a part of it) form a “loop”; a similar observation was used to obtain [(\[eq:IRSchwarz-bdii\])]{}. The loop consists of segments contained in the $l^\text{th}$ block and possibly in the $(l-1)^\text{st}$ block, and hence the number of choices for $a''$ is at most $O(i_{l-1}+i_l+1)$, where $i_l$ is the index
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{P}_2]= - \mathfrak{P}_3$, but since $[\mathfrak{M}_{23}, \mathfrak{P}_1]= [\mathfrak{P}_{2}, \mathfrak{P}_3]=0$ it is unimodular. In [@Borsato:2016ose] it was shown that the corresponding deformation is nevertheless equivalent to two non-commuting TsT transformations, with a non-linear coordinate redefinition in between. On the other hand it was discussed from the perspective of non-abelian T-duality in [@Hoare:2016wsk] where the relevant subalgebra was $\mathfrak{h}= \{ \mathfrak{M}_{23}, \mathfrak{P}_1 , \mathfrak{P}_2, \mathfrak{P}_3 \}$. The gauge freedom can be used to fix the coset representative in eq. to $\hat{g} = e^{-x_{0 }\mathfrak{P}_{0}} z^{\mathfrak{D}}$, but there remains one residual gauge symmetry which is used to fix a Lagrange multiplier to zero. The Lagrange multipliers are parametrised by $$v_1= -\frac{x_1}{\eta} + \frac{r^2}{2 \zeta} \ , \quad v_2= \frac{\theta}{\eta}\ , \quad v_3=\frac{r}{\zeta}\ ,\quad v_4= 0\ , \quad v_5= \frac{1}{\eta}\ , \quad v_6= \frac{1}{\zeta} \ ,$$ where $v_5$ and $v_6$ correspond to the two central generators and $r$ and $\theta$ are polar coordinates on the $x_2 ,x_3$ plane. Applying the non-abelian T-duality technology one finds the dual geometry is $$\begin{aligned}
\widehat{ds}^2 &=\frac{1}{z^2} \left( dz^2 - dx_0^2\right) + \widehat{e}_\pm \cdot \widehat{e}_\pm + ds^2_{S^5} \ ,
\\
\widehat{e}^{\,1}_+ &= \frac{dx_1 \left(\zeta ^2+z^4\right)+\eta r \left(-\zeta dr +r z^2 d\theta \right)}{z f } \ ,
\\ \widehat{e}^{\,2}_+& = \frac{z \left(\zeta dr+\eta r dx_1-r z^2 d \theta
\right)}{f} \ , \\
\widehat{e}^{\,3}_+&= \frac{-dr \left(\eta ^2 r^2+z^4\right)+\zeta \eta r
dx_1-\zeta r z^2 d\theta }{z f}\ ,
\end{aligned}$$ where $f= \zeta ^2+\eta ^2 r^2+z^4$, while the remaining NS fields are $$\widehat{B} = \frac{-\zeta \eta r dr \wedge d\theta + \left(\zeta ^2+z^4\right) dx_1\wedge d \theta }{\eta f } \ , \quad
e^{-2(\widehat{\Phi} - \phi_0)} = \frac{f}{\zeta ^2 \eta ^2 z^4} \ .$$ The Lorentz rotation $\Lambda e_- = e_+$ is given by $$\Lambda = \frac{1}{f} \left(
\b
| 3,485
| 2,232
| 2,628
| 3,065
| null | null |
github_plus_top10pct_by_avg
|
imal stimulus and the average MUTF provides a count estimate. However, there is no guarantee that a particular single-stepped increase in response corresponds to a new, previously latent, MU, since it may instead be due to a phenomenon called alternation [@Bro76]. This occurs when two or more MUs have similar activation thresholds such that different combinations of MUs may fire in reaction to two identical stimuli. Consequently, the incremental technique tends to underestimate the average MUTF and hence overestimate the number of MUs. A number of improvements both experimentally [@Kad76; @Sta94 e.g.] and empirically [@Dau95; @Maj07 e.g.] have been proposed to try to deal with the alternation problem but, despite these improvements, each method oversimplifies the data generating mechanism and there is no gold-standard averaging approach; @Bro07 and @Goo14 provide thorough discussions on these approaches to MUNE.
![Stimulus-response curve from a rat tibial muscle using 10sec (left) and 50sec duration stimuli. Histogram inserts represent the frequency in the absolute difference of twitch forces when ordered by stimulus.[]{data-label="fig:RatData"}](figures/SRcurve_rat.pdf){width="80.00000%"}
Motor units are more diverse than simple replicates of the ‘average’ MU, with many factors influencing their function. A desire for a more complete model for the data generating mechanism motivated the comprehensive approach to MUNE in @Rid06, which proposed three assumptions:
- MUs fire independently of each other and of previous stimuli in an all-or-nothing response. Each MU fires precisely when the stimulus intensity exceeds a random threshold whose distribution is unique to that MU, with a sigmoidal cumulative distribution function, called an excitability curve.
- The firing of a MU is characterised by a MUTF which is independent of the size of the stimulus that caused it to fire, and has a Gaussian distribution with an expectation specific to that MU and a variance common to all MUs.
- The measured WMTF is t
| 3,486
| 3,808
| 3,949
| 3,294
| null | null |
github_plus_top10pct_by_avg
|
|x\|$. For notational convenience, we drop the dependence on $\psi$, since all our bounds hold uniformly over all $\psi \in \mathcal{S}_n$. The first bound in on the norm of the gradient of $g_j$ is straightforward: $$\begin{aligned}
\nonumber
||G_j|| & \leq ||e_j|| \times \sigma_1\left( \left[ - \left( \alpha^\top \otimes I_k
\right) (\Omega \otimes \Omega) \;\;\;\;\; \Omega\right] \right)\\
\nonumber
& \leq \Big(
||\alpha||\times \sigma_1(\Omega)^2 + \sigma_1(\Omega) \Big)\\
\label{eq:Gj.constants}
& \leq \frac{A^2 \sqrt{ k}}{u^2} + \frac{1}{u}\\
\nonumber
& \leq
C \frac{\sqrt{k}}{u^2},\end{aligned}$$ since $\sigma_1(\Omega) \leq
\frac{1}{u}$, $\| \alpha \| \leq \sqrt{A^2 \mathrm{tr}(\Sigma)} \leq
A^2 \sqrt{k}$, and we assume that $k \geq u^2$.
Turning to the second bound in , we will bound the largest singular values of the individual terms in . First, for the lower block matrix in , we have that $$\sigma_1([ \Omega\otimes\Omega \;\;\;\;\; 0_{k^2 \times k}]) =
\sigma_1( \Omega\otimes\Omega) = \sigma_1^2(\Omega) = 1/u^2.$$ Next, we consider the two matrix in the upper block part of . For the first matrix we have that $$\begin{aligned}
\label{eq:mammamia}
\sigma_1 \Big( (\Omega\otimes\Omega\otimes I_k) \Big[ 0_{k^3 \times k^2 } \;\;\;\; I_k \otimes {\rm
vec}(I_k) \Big] \Big)&=
\sigma_1 \Big( \Big[ 0_{k^3 \times k^2} \;\;\;\; \Omega \otimes {\rm vec}(\Omega)
\Big]
\Big)\\
\nonumber
& = \sigma_1\left( \Omega \otimes \mathrm{vec}(\Omega)\right) \\
\nonumber
&=
\sigma_1 ( \Omega) \sigma_1({\rm vec}(\Omega))\\
\nonumber
& \leq \frac{\sqrt{k}}{u^2},\end{aligned}$$ since $$\sigma_1({\rm vec}(\Omega)) = ||\Omega||_F =
\sqrt{\sum_{i=1}^k \sigma_i^2(\Omega)}\leq \sqrt{k}\sigma_1(\Omega) = \frac{\sqrt{k}}{u}.$$ The identity in is established using the following facts, valid for conformal matrices $A$, $B$, $C$, $D$ and $X$:
- $(A \otimes B)(C \otimes D) = AC \otimes BD$ , with $A =
\Omega$, $B = \Omega \otimes I_k$, $C = I_k$ and $D =
\mathrm{vec}(\Omega)$, and
- $AXB = C$ is equi
| 3,487
| 2,372
| 2,673
| 3,115
| null | null |
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|
*4 - 7*b**2 + 3 wrt b?
-17928*b
Find the first derivative of -26*l*x**4 - 5072*l - 4*x**2 - x wrt x.
-104*l*x**3 - 8*x - 1
What is the third derivative of l*n**2*x**3 + 3*l*n**2 + l*n - 34*l*x**3 + 4*l*x**2 + 2 wrt x?
6*l*n**2 - 204*l
Find the third derivative of a**2*q**3*s**2 + 160*a**2*q**2*s**2 + 3*a**2*s - 540*a*q**3*s**2 wrt q.
6*a**2*s**2 - 3240*a*s**2
What is the third derivative of 572*o**6 + 17*o**2 + 24*o wrt o?
68640*o**3
Find the second derivative of 55*a*x**4 - a*x**3 - 2*a*x - 10*x**5 + 447*x wrt x.
660*a*x**2 - 6*a*x - 200*x**3
Find the first derivative of 1036*h**2*w - 434*h**2 wrt w.
1036*h**2
Find the second derivative of -26*g**3 + g**2*r**2 + 2*g*r**2 + 2*g + 11*r**2 wrt g.
-156*g + 2*r**2
Find the first derivative of 1144*o - 1116.
1144
Find the second derivative of -8*j**4 + j**3 - 711*j wrt j.
-96*j**2 + 6*j
What is the second derivative of 2*b**3*g**3 + 18*b**3*g - b**2*g**2 - 24*b*g**2 - 3*g**3 wrt b?
12*b*g**3 + 108*b*g - 2*g**2
What is the third derivative of -19*s**6 - s**5 + 20*s**3 - 75*s**2 - 8*s?
-2280*s**3 - 60*s**2 + 120
Find the third derivative of 11*i**3*o**3*u + 8*i**3*o**2*u**3 + 3*i**2*o**3*u**3 - 2*i**2*u**3 - i*o**3*u + 2*i*u**3 wrt i.
66*o**3*u + 48*o**2*u**3
What is the third derivative of -336*h*q**2*t**3 - 17*h*t**3 + 3*h*t + 114*q**2*t**2 - 5*q*t**2 wrt t?
-2016*h*q**2 - 102*h
What is the second derivative of o**3 - 762*o**2*p - 2*o**2 + 3*o*p + p - 271 wrt o?
6*o - 1524*p - 4
Find the second derivative of 36*b*w**3 + 2*b*w - 2*w**3 + 112*w + 2 wrt w.
216*b*w - 12*w
What is the second derivative of 6*h**3*r**2 - h**2*l*r**2 + 34*h**2*l - 5*h*l*r**2 - 11*h*l + 2*l*r wrt h?
36*h*r**2 - 2*l*r**2 + 68*l
Find the second derivative of -16807*d**3 - 25426*d wrt d.
-100842*d
Find the second derivative of 4*b**5*n - 989*b**3*n + 2*b**2*n + b*n + 41*b + 6*n wrt b.
80*b**3*n - 5934*b*n + 4*n
Differentiate -3916*a**2*f*m**2 + 134*a**2*m**2 - 2*a**2*m - 8*a*m**2 with respect to f.
-3916*a**2*m**2
What is the second derivative of 387*q**2 + 2891*q wrt q?
774
Find the first deriva
| 3,488
| 1,644
| 1,583
| 3,071
| null | null |
github_plus_top10pct_by_avg
|
n for dotted spinors, the convention is that the contraction is taken from bottom-left to top-right, namely $$\begin{array}{r c l}
\overline{\psi}\,\overline{\chi}&=&
\overline{\psi}_{\dot{\alpha}}\,\overline{\chi}^{\dot{\alpha}}=
\epsilon_{\dot{\alpha}\dot{\beta}}\,\overline{\psi}^{\dot{\beta}}\,
\overline{\chi}^{\dot{\alpha}}
=-\epsilon_{\dot{\alpha}\dot{\beta}}\,\overline{\psi}^{\dot{\alpha}}\,
\overline{\chi}^{\dot{\beta}}\\
& & \\
&=&-\overline{\psi}^{\dot{\alpha}}\,\overline{\chi}_{\dot{\alpha}}=
\overline{\chi}_{\dot{\alpha}}\,\overline{\psi}^{\dot{\alpha}}=
\overline{\chi}\,\overline{\psi}\ .
\end{array}$$ Further identities that may be established in a likewise manner are, $$\left(\psi\,\chi\right)^\dagger=
\overline{\chi}\,\overline{\psi}=
\overline{\psi}\,\overline{\chi}\ \ ,\ \
\left(\overline{\psi}\,\overline{\chi}\right)^\dagger=
\chi\,\psi=\psi\,\chi\ .$$
For the construction of Lorentz covariant spinor bilinears, one has to also involve the matrices $\sigma_\mu$ and $\overline{\sigma}_\mu$. Thus for instance, we have the quantities transforming as 4-vectors under Lorentz transformations, $$\psi\,\sigma^\mu\,\overline{\chi}=
\psi^\alpha\,{\sigma^\mu}_{\alpha\dot{\beta}}\,\overline{\chi}^{\dot{\beta}}
\ \ ,\ \
\overline{\psi}\,\overline{\sigma}^\mu\,\chi=
\overline{\psi}_{\dot{\alpha}}\,\overline{\sigma}^{\mu\dot{\alpha}\beta}\,
\chi_\beta\ .$$ Such quantities also obey a series of identities, for instance, $$\chi\,\sigma^\mu\,\overline{\psi}=-
\overline{\psi}\,\overline{\sigma}^\mu\,\chi\ \ ,\ \
\chi\,\sigma^\mu\,\overline{\sigma}^\nu\,\psi=
\psi\,\sigma^\nu\,\overline{\sigma}^\mu\,\chi\ ,$$ $$\left(\chi\,\sigma^\mu\,\overline{\psi}\right)^\dagger=
\psi\,\sigma^\mu\,\overline{\chi}\ \ ,\ \
\left(\chi\,\sigma^\mu\,\overline{\sigma}^\nu\,\psi\right)^\dagger=
\overline{\psi}\,\overline{\sigma}^\nu\,\sigma^\mu\,\overline{\chi}\ .$$ Identities of this type enter the explicit construction of supersymmetric invariant field theories.
The Dirac Spinor {#Sec3.4}
----------------
As mentioned earlier,
| 3,489
| 1,746
| 1,645
| 3,297
| null | null |
github_plus_top10pct_by_avg
|
e denote $\theta_0(\eta) := \theta(\eta,0) = u(x,0)$.
\[lem:finite\_p\_bounded\] For all $\overline{q} \leq p < \infty$, there exists $C_{\overline{q}} = C_{\overline{q}}(p,M)$ and $C_M = C_M(p,{\|\theta_0\|}_{\overline{q}})$ such that if ${\|\theta_0\|}_{\overline{q}} < C_{\overline{q}}$ and $M < C_M$, then ${\|\theta(\tau)\|}_{p} \in L_\tau^\infty({\mathbb R}^+)$.
Define $$\mathcal{I} = \int \theta^{m-1}{\left\vert{\nabla}\theta^{p/2}\right\vert}^2 dx.$$ We estimate the time evolution of ${\|\theta\|}_p$ using integration by parts, Hölder’s inequality and Lemma \[lem:CZ\_rescale\] in the appendix, $$\begin{aligned}
\frac{d}{d\tau} {\|\theta\|}_{p}^{p} & = -\frac{4mp}{(p+1)^2}\mathcal{I} + (p-1)e^{(1-\alpha-\beta)\beta^{-1}\tau}\int\theta^p{\nabla}\cdot (e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau} \cdot) \ast \theta) d\eta + d(p-1){\|\theta\|}_p^p \nonumber \\
& \leq -C(p)\mathcal{I} + C(p)e^{(1-\alpha-\beta)\beta^{-1}\tau}{\|\theta\|}_{p+1}^p{\|{\nabla}(e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau} \cdot)\ast \theta)\|}_{p+1} + C(p){\|\theta\|}_p^p \nonumber \\
& \leq -C(p)\mathcal{I} + C(p)e^{(1-\alpha)\beta^{-1}\tau}{\|\theta\|}_{p+1}^{p+1} + C(p){\|\theta\|}_p^p. \label{ineq:Lpevo}\end{aligned}$$ We bound the second term using the using the homogeneous Gagliardo-Nirenberg-Sobolev inequality (Lemma \[lem:GNS\] in appendix), $${\|\theta\|}_{p+1}^{p+1} \lesssim {\|\theta\|}_{\overline{q}}^{\alpha_2(p+1)}\mathcal{I}^{\alpha_1(p+1)/2}, \label{ineq:GNS1}$$ where $\alpha_2 = 1 - \alpha_1(p+m-1)/2$ and $$\alpha_1 = \frac{2d(\overline{q}-p-1)}{(p+1)\left(\overline{q}(d-2) - d(p+m-1)\right)}.$$ By the definition of $\overline{q}$ we have that, $$\frac{\alpha_1(p+1)}{2} = \frac{d(\overline{q}-p-1)}{\overline{q}(d-2) - d(p+m-1)} = 1.$$ We also estimate the second term in using Lemma \[lem:GNS\], $${\|\theta\|}_{p}^{p} \lesssim M^{\beta_2 p}\mathcal{I}^{\beta_1p/2}, \label{ineq:gns_below}$$ where $\beta_2 = 1 - \beta_1p/2$ and, $$\frac{\beta_1 p}{2} = \frac{d(p-1)}{2-d+d(p+m-1)} < 1,$$ by $1 - 2/d < m$. Then applying weighte
| 3,490
| 2,442
| 1,061
| 3,336
| null | null |
github_plus_top10pct_by_avg
|
20.7 51.7 Hysterectomy at 14 months Multiple myomas ranging from 0.6 to 3.4 cm; focal adenomyosis
1 Intramural 1.8
**4** 1 Intramural 2.0 62.5 28.1 10.3 60.3 Supracervical hysterectomy at 23 months Only morcellated tissue available. Findings: adenomyosis, leiomyomata, proliferative endometrium
1 Submucosal 2.0 1
Undefined 1.9
**5** 5 Intramurals 2.7; 2.7; 2.6; 8.5; 6.7 75.0 56.3 17.2 28.4 Hysteroscopic myomectomy by resection at 16 months Focal degenerative changes and features suggestive of polyp
1 Undefined 3.2
**6** 1 Intramural 2.0 53.1 62.5 37.1 43.1 Hysterectomy at 23.5 months Adenomyosis; multiple myomas ranging in size from 0.4 cm to 1.2 cm
1 Subserosal 2.0
| 3,491
| 3,889
| 2,925
| 3,267
| null | null |
github_plus_top10pct_by_avg
|
ing that such an $M$ is ‘close’ to a $\GF(q)$-representable matroid.'
address: 'Department of Combinatorics and Optimization, University of Waterloo, Canada'
author:
- Jim Geelen
- Peter Nelson
title: The Structure of Matroids with a Spanning Clique or Projective Geometry
---
[^1]
Introduction
============
In \[\[highlyconnected\]\], Geelen, Gerards, and Whittle describe the structure of highly-connected matroids in minor-closed classes of matroids represented over a fixed finite field. In the same paper they conjecture extensions of their results to minor-closed classes of matroids omitting a fixed uniform minor. The main results in this paper are motivated by those conjectures, which we shall restate at the end of this introduction. Here we are primarily concerned with the structure of matroids having either the cycle-matroid of a complete graph or a projective geometry as spanning restriction.
An *elementary projection* of a matroid $M$ is a matroid obtained from an extension of $M$ by contracting the new element, and an *elementary lift* of $M$ is one obtained from a coextension by deleting the new element. Given two matroids $M$ and $N$ on the same ground set, we say that $N$ is a [*distance-$k$ perturbation*]{} of $M$ if $N$ can be obtained from $M$ by a sequence of $k$ elementary lifts and elementary projections. Perturbations play a natural role in considering minor-closed classes of matroids omiting a uniform matroid. In particular, if $\cM$ is a minor-closed class of matroids that omits a uniform matroid, then the set of matroids that are distance-$k$ perturbations of matroids in $\cM$ is also minor-closed and omits a uniform matroid; see Theorem \[perturbthm\]. Note that the uniform matroid $U_{r,n}$ is contained as a minor of $U_{s,2s}$ where $s=\max(r,n-r)$, so it suffices to consider classes omitting ‘balanced’ uniform matroids $U_{s,2s}$.
We start with the easier of our two main results which concerns matroids with a spanning projective geometry restriction.
\[main2\] For all intege
| 3,492
| 2,088
| 1,524
| 3,145
| null | null |
github_plus_top10pct_by_avg
|
}{ ( \Delta_{J} - \Delta_{I} ) } -
\frac{e^{- i h_{k} x} - e^{- i \Delta_{I} x} }{ ( h_{k} - \Delta_{I} ) }
\right]
\left\{ W^{\dagger} A (UX) \right\}_{I k}
\frac{ 1 }{ \Delta_{J} - h_{k} }
\left\{ (UX)^{\dagger} A W \right\}_{k J}
\nonumber \\
&+&
\left[ (ix) e^{- i \Delta_{I} x} +
\frac{e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} }{ ( \Delta_{J} - \Delta_{I} ) }
\right]
\left\{ W^{\dagger} A W \right\}_{I I}
\frac{ 1 }{ \Delta_{J} - \Delta_{I} }
\left\{ W^{\dagger} A W \right\}_{I J}
\nonumber \\
&-&
\left[ (ix) e^{- i \Delta_{J} x} +
\frac{e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} }{ ( \Delta_{J} - \Delta_{I} ) }
\right]
\left\{ W^{\dagger} A W \right\}_{I J}
\frac{ 1 }{ \Delta_{J} - \Delta_{I} }
\left\{ W^{\dagger} A W \right\}_{J J}
\nonumber \\
&+&
\sum_{K \neq I, J}
\left[ \frac{e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} }{ ( \Delta_{J} - \Delta_{I} ) } -
\frac{e^{- i \Delta_{K} x} - e^{- i \Delta_{I} x} }{ ( \Delta_{K} - \Delta_{I} ) }
\right]
\left\{ W^{\dagger} A W \right\}_{I K}
\frac{ 1 }{ \Delta_{J} - \Delta_{K} }
\left\{ W^{\dagger} A W \right\}_{K J},
\nonumber \\
\hat{S}_{I I}^{(2+4)} [2] &=&
- \sum_{k}
\left[ (ix) e^{- i \Delta_{I} x} +
\frac{e^{- i h_{k} x} - e^{- i \Delta_{I} x} }{ h_{k} - \Delta_{I} }
\right]
\left\{ W^{\dagger} A (UX) \right\}_{I k}
\frac{ 1 }{ \Delta_{I} - h_{k} }
\left\{ (UX)^{\dagger} A W \right\}_{k I}
\nonumber \\
&+& \frac{ (- ix)^2 }{ 2 } e^{- i \Delta_{I} x}
\left( \left\{ W^{\dagger} A W \right\}_{I I} \right)^2
\nonumber \\
&-&
\sum_{K \neq I}
\left[ (ix) e^{- i \Delta_{I} x} +
\frac{e^{- i \Delta_{K} x} - e^{- i \Delta_{I} x} }{ \Delta_{K} - \Delta_{I} }
\right]
\left\{ W^{\dagger} A W \right\}_{I K}
\frac{ 1 }{ \Delta_{I} - \Delta_{K} }
\left\{ W^{\dagger} A W \right\}_{K I}.
\label{hatS-2nd-order}\end{aligned}$$ We now discuss generalized T invariance of second order $\hat{S}$ matrix elements. Let us first examine $\hat{S}_{i j} \vert_{i \neq j} [2]$. We first note that the matrix element transforms under generalized
| 3,493
| 1,533
| 2,951
| 3,135
| null | null |
github_plus_top10pct_by_avg
|
s and complexity in all likely and unlikely places, and possibly because of it, it is necessary that we have a clear mathematically-physical understanding of these notions that are supposedly reshaping our view of nature. This paper is an attempt to contribute to this goal. To make this account essentially self-contained we include here, as far as this is practicable, the basics of the background material needed to understand the paper in the form of *Tutorials* and an extended *Appendix.*
The paradigm of chaos of the kneading of the dough is considered to provide an intuitive basis of the mathematics of chaos [@Peitgen1992], and one of our fundamental objectives here is to recount the mathematical framework of this process in terms of the theory of ill-posed problems arising from non-injectivity [@Sengupta1997], *maximal ill-posedness,* and *graphical convergence* of functions [@Sengupta2000]. A natural mathematical formulation of the kneading of the dough in the form of *stretch-cut-and-paste* and *stretch-cut-and-fold* operations is in the ill-posed problem arising from the increasing non-injectivity of the function $f$ modeling the kneading operation.
***Begin Tutorial1: Functions and Multifunctions***
A *relation,* or *correspondence,* between two sets $X$ and $Y$, written $\mathscr{M}\!:X\qquad Y$, is basically a rule that associates subsets of $X$ to subsets of $Y$; this is often expressed as $(A,B)\in\mathscr{M}$ where $A\subset X$ and $B\subset Y$ and $(A,B)$ is an ordered pair of sets. The domain $$\mathcal{D}(\mathscr{M})\overset{\textrm{def}}=\{ A\subset X\!:(\!\exists Z\in\mathscr{M})(\pi_{X}(Z)=A)\}$$ and range $$\mathcal{R}(\mathscr{M})\overset{\textrm{def}}=\{ B\subset Y\!:(\!\exists Z\in\mathscr{M})(\pi_{Y}(Z)=B)\}$$
of $\mathscr{M}$ are respectively the sets of $X$ which under $\mathscr{M}$ corresponds to sets in $Y$; here $\pi_{X}$ and $\pi_{Y}$ are the projections of $Z$ on $X$ and $Y$ respectively. Equivalently, $\mathcal{D}(\mathcal{M})=\{ x\in X\!:\mathscr{M}(x)\neq\emptyset\}$ and $\ma
| 3,494
| 4,719
| 3,956
| 3,242
| 3,934
| 0.769188
|
github_plus_top10pct_by_avg
|
otential overfitting), they are always a better fit for the data [@murphy].]{}]{} Thus, higher order models are naturally favored by their improvements in likelihoods. A more comprehensive view on this issue shows that there exists a broad range of established model comparison techniques that also take into the account the complexity of a model in question [@akaike; @bartlett; @gates1976; @katz; @schwarz; @Strelioff; @tong1975]. Moreover, the principle objects of interest in the majority of the past studies are transitions between Web pages. Only a few studies [@cadez; @kumar2010; @west] investigate navigation as transitions between Web page features, such as the content or context of those Web pages.
Methods {#sec:methodology .unnumbered}
=======
[[ In the following, we briefly introduce Markov chains before discussing an expanded set of methods for order selection, including *likelihood*, *Bayesian*, *information-theoretic* and *cross validation* model selection techniques. ]{}]{}
Markov Chains {#subsec:markovchains .unnumbered}
-------------
Formally, a discrete (time and space) finite Markov chain is a stochastic process which amounts to a sequence of random variables $X_1, X_2, ..., X_n$. For a Markov chain of the first order, i.e., for a chain that satisfies the memoryless Markov property the following holds: $$\begin{aligned}
\nonumber P(X_{n+1} = x_{n+1} | X_1 = x_1, X_2 = x_2, ..., X_n = x_n) & = \\
P(X_{n+1} = x_{n+1} | X_n = x_n)\end{aligned}$$
[[ This classic first order Markov chain model is usually also called a *memoryless model* as we only use the current information for deriving the future and do not look into the past.]{}]{} For all our models we assume *time-homogeneity* – the probabilities do not change as a function of time. To simplify the notation we denote data as a sequence $D=(x_1, x_2, ..., x_n)$ with states from a finite set $S$. With this simplified notation we write the Markov property as: $$p(x_{n+1}|x_1, x_2, ..., x_n)=p(x_{n+1}|x_n)$$
[[ As we are also interested in highe
| 3,495
| 4,982
| 2,275
| 3,066
| 923
| 0.797575
|
github_plus_top10pct_by_avg
|
f the whole unseen testing set during the competition, while the accuracy score of predictions over the private subset was kept inaccessible for even testing until the end of the competition. Upon the end of the competition, the default ML frameworks proposed by the teams were tested over the entire testing set. Adopting such an evaluation strategy primarily aimed to first avoid the overfitting stemming from the teams’ behavior of exhaustive submissions to Kaggle to monitor the accuracy results of their ML models, and second evaluate the generalizability of the designed models on an extra testing dataset they did not have access to for model training and testing.
Performance Metrics
-------------------
We evaluated the performance of ML pipelines for ASD/NC classification by using the results derived from the public and private testing subsets in terms of accuracy, sensitivity, and specificity metrics. The public test set was partially accessible to the competitors during the competition so that they could check their model accuracy, whereas the private one was completely kept hidden to assess the generalizability of the models after the competition. By using these two test sets, public and private accuracy, sensitivity, and specificity values were computed for the method of each team, and then those two results were averaged for all three metrics. All participating teams were ranked in three different ways, each of which takes the basis of one type of average measurement, and the sum of three ranks determined the final rank of each team in the competition. Table \[tab:results\_table\] displays the classification results for the top 20 competitors and Table \[tab:methods\_details\] details the components of the 20 designed ML pipelines.
Best performing methodologies
-----------------------------
Evaluation scores of ML pipelines designed for ASD/NC classification are illustrated in Figure \[fig:avgmeas\]. These performance measurements comprise accuracy, sensitivity, and specificity metrics. On the other
| 3,496
| 1,836
| 2,226
| 2,882
| null | null |
github_plus_top10pct_by_avg
|
on of the operators $b_\nu$. We therefore need to analyze our theory in this respect. Specifically, we show in this section that the total current STS spectra calculated in our theory do not depend on the choice of the basis for the operators $b_\nu$. Interestingly, however, this choice of basis does determine the partitioning between elastic and the inelastic contributions to the total current. Inelastic and elastic currents are therefore not physical observables, but an interpretation based on a model-dependent partitioning of the total current.
Let us assume that we have made a particular choice $\hat X$ of the oscillator basis and find a non-zero $\langle \hat X_0 \rangle=x_0$ for the mode $\w_0$ (for which $\hat H_{\rm S}$ foresees an electron-phonon coupling). For simplicity, we assume that the other $N_\nu-1$ vibrational modes $\langle \hat X_\nu \rangle=0$ holds. Then we can define a new bosonic operator $$\begin{aligned}
\label{eq:XX}
\bar b_0 = b_0 - \frac{1}{2}x_0 \end{aligned}$$ such that $\langle \hat{\bar{X_0}}=\bar b_0+\bar b_0^\dagger\rangle=0$. Substituting this expression into $\hat H_{\rm ph} + \hat H_{\rm e-ph}$ (Eqs. ,) and ignoring all vibrational modes except $\omega_0$ yields $$\begin{aligned}
\label{eq:35}
\hat H_{\rm ph} &+& \hat H_{\rm e-ph} = \sum_{\nu=1}^{N_\nu-1} \w_\nu b^\dagger_\nu b_\nu \\
&+& \w_0 \bar b^\dagger_0 \bar b_0 +\lambda_d x_0 N_d + \lambda_c x_ 0\hat N_{c} + E_0
\nonumber\\
&+& \hat{\bar{X_0}}
\left( \lambda_d (\hat N_d - n_{d0}) +\lambda_c(\hat N_c -n_{c0}) +\frac{\w_0 x_0}{2}\right)\,
\nonumber\end{aligned}$$ where we define $\hat N_d \equiv \sum_\sigma \hat n^d_\sigma$, $\hat N_c\equiv \sum_\sigma c^\dagger_{0\sigma} c_{0\sigma}$ and absorb all constants into $E_0$. To keep the Hamiltonian $\hat H_{S}=\hat H_{\rm e}+\hat H_{\rm ph} + \hat H_{\rm e-ph}$ invariant under the basis set change of the bosonic operator, we substitute $\e_d\to \e_d + \lambda_d x_0$ and define a single-particle energy $\e_c=\lambda_c x_0$ for the orbital $c_{0\sigma}$. In case o
| 3,497
| 4,180
| 2,850
| 3,181
| null | null |
github_plus_top10pct_by_avg
|
equency range, see Fig. \[fig:02\]. The solid lines show the experimental results, derived from the Fourier transform of the measured $S_{aa}(\nu)$ and $S^{\prime}_{aa}(\nu)$ via relation (\[eq:f\_ab\]) for the situations (a) 50$\Omega$ load (black), (b) open-end reflection (green, dark gray) and (c) hard-wall reflection (orange, light gray). For the case (a) we are able to calculate the corresponding theoretical curve (black dotted line) without any free parameter, since the coupling constant $\lambda_W$ can be determined directly from the additional reflection measurement at antenna $c$ (see Sec. \[sec:exper\]) using relation (\[eq:Tc\]). According to Eq. (\[eq:s13\]) we expect a purely imaginary coupling with $\lambda_{\rm 50\Omega}$. For our antenna $\lambda_W$ varies from 0.1 to 0.4 in a range from 6 to $10\,\rm GHz$. Using the experimentally determined $\lambda_W$, one gets already very good agreement between experiment and theory for the $\rm 50\,\Omega$ load without any fit. A fit of $\lambda_W$ to the experimental curves only marginally improves the correspondence.
 Fidelity decay $|f(t)|^2$ for three types of perturbation: $\lambda_{\rm 50\Omega}$ (black); $\lambda_{\rm hw}$ (orange, light gray); $\lambda_{\rm oe}$ (green, dark gray) in three different frequency ranges: (i) $7.2-7.7\rm\,GHz$; (ii) $8.0-8.5\rm\,GHz$; (iii) $8.7-9.2\rm\,GHz$. Solid lines show the experimental results and the theoretical curves are dotted for experimental parameter (available only for the 50$\Omega$ load case) and dashed for fitting parameter. The corresponding parameter and the transmission coefficient for the measuring antenna $a$ are listed in Tab. \[tab:01\].](fig3a "fig:"){width=".95\columnwidth"}\
 Fidelity decay $|f(t)|^2$ for three types of perturbation: $\lambda_{\rm 50\Omega}$ (black); $\lambda_{\rm hw}$ (orange, light gray); $\lambda_{\rm oe}$ (green, dark gray) in three different frequency ranges: (i) $7.2-7.7\rm\,GHz$; (ii) $8.0-8.5\rm\,GHz$; (iii) $8.7-
| 3,498
| 480
| 2,303
| 3,392
| 805
| 0.799679
|
github_plus_top10pct_by_avg
|
ubsequence $(x_{i_{k}})_{k\in\mathbb{N}}$ converging to $x$, then a more direct line of reasoning proceeds as follows. Since the subsequence converges to $x$, its tail $(x_{i_{k}})_{k\geq j}$ must be in every neighbourhood $N$ of $x$. But as the number of such terms is infinite whereas $\{ i_{k}\!:k<j\}$ is only finite, it is necessary that for any given $n\in\mathbb{N}$, cofinitely many elements of the sequence $(x_{i_{k}})_{i_{k}\geq n}$ be in $N$. Hence $x\in\textrm{adh}((x_{i})_{i\in\mathbb{N}})$. ]{}
[^30]: \[Foot: seq xxx\][This is uncountable because interchanging any two eventual terms of the sequence does not alter the sequence. ]{}
[^31]: [Note that $\{ x\}$ is a $1$-point set but $(x)$ is an uncountable sequence.]{}
[^32]: \[Foot: e&q\][We adopt the convention of denoting arbitrary preimage and image continuous functions by $e$ and $q$ respectively even though they are not be injective or surjective; recall that the embedding $e\!:X\supseteq A\rightarrow Y$ and the association $q\!:X\rightarrow f(X)$ are $1:1$ and onto respectively. ]{}
[^33]: \[Foot: fil-nbd\][This is of course a triviality if we identify each $\chi(\mathbb{R}_{\beta})$ (or $F$ in the proof of the converse that follows) with a neighbourhood $N$ of $X$ that generates a topology on $X$.]{}
[^34]: **Nested-set theorem.** *If $(E_{n})$ is a decreasing sequence of nonempty, closed, subsets of a complete metric space $(X,d)$ such that* [$\lim_{n\rightarrow\infty}\textrm{dia}(E_{n})=0$]{}*, then there is a unique point* [$$x\in\bigcap_{n=0}^{\infty}E_{n}.$$ The uniqueness arises because the limiting condition on the diameters of $E_{n}$ imply, from property (H1), that $(X,d)$ is a Hausdorff space. ]{}
---
abstract: 'Rank aggregation systems collect ordinal preferences from individuals to produce a global ranking that represents the social preference. Rank-breaking is a common practice to reduce the computational complexity of learning the global ranking. The individual preferences are broken into pairwise comparisons and applied to ef
| 3,499
| 1,004
| 2,392
| 3,264
| 2,399
| 0.780174
|
github_plus_top10pct_by_avg
|
n - 9. Let k(v) = -2*v + 2. Let b be k(7). Let p(m) = 4*m - 11. Let w be p(4). Let d(r) = 10*r**3 + 12*r - 22. Calculate b*j(x) + w*d(x).
2*x**3 - 2
Let d(m) = 21*m. Let i(u) = -2*u**3 + 3*u**2 - 5*u + 6. Let r be i(2). Let j(q) = 7*q. Calculate r*j(v) + 3*d(v).
7*v
Let o be 32/(-40) + 68/10. Let n(b) = -6*b - 1 + 0*b**3 - 4 - 2*b**3 + 7*b**2. Let d(a) = 2*a**3 - 6*a**2 + 5*a + 4. What is o*d(m) + 5*n(m)?
2*m**3 - m**2 - 1
Let f(a) = -11*a + 3. Let k(s) = 12*s - 4. Calculate 5*f(x) + 4*k(x).
-7*x - 1
Let a(j) = j + 5. Let v(z) = -5*z - 30. Let u(p) = p + 6. Let c(o) = -11*u(o) - 2*v(o). Determine 7*a(n) + 6*c(n).
n - 1
Let c(j) be the second derivative of j**3/6 - j**2/2 - j. Let l(f) be the second derivative of -f**3/2 + f**2/2 - f. Calculate -2*c(v) - l(v).
v + 1
Let s(a) = -3*a**3 + 2*a**2 - 4. Let x(q) = -3*q**3 + 3*q**2 - 4. Let d(i) = 2*s(i) - 3*x(i). Let g(r) = r**3 - 2*r**2 + 2. Determine 2*d(v) - 5*g(v).
v**3 - 2
Let p(z) = 5*z**3 - 3*z**2 - 3*z - 4. Let k = 15 + -5. Let l(c) = -c**2 + 11*c - 7. Let d be l(k). Let v(o) = 4*o**3 - 2*o**2 - 2*o - 3. Give d*p(h) - 4*v(h).
-h**3 - h**2 - h
Let y(w) = 5*w + 2*w - 6*w. Let a(p) = -2*p + 1. Let j(v) = -a(v) + 2*y(v). Let h(r) = r. Calculate -3*h(f) + j(f).
f - 1
Suppose z + 18 = 4*z. Let u(r) = -17*r + 9. Let c(d) = -7 + 11 - 6 + 4*d. Calculate z*u(p) + 26*c(p).
2*p + 2
Let z(q) be the third derivative of -q**6/120 - q**5/60 - q**3/6 + 6*q**2. Let l(c) = 8*c**3 + 5*c**2 + 5. Calculate l(i) + 5*z(i).
3*i**3
Let h(d) = -6*d**3 + 29*d**2 - 7*d + 7. Let m(i) = i**3 - i**2 + i - 1. Calculate -2*h(x) - 14*m(x).
-2*x**3 - 44*x**2
Let l(t) = 4*t**2 - t + 9. Let i(b) = -6*b**2 + 2*b - 14. Give 5*i(a) + 8*l(a).
2*a**2 + 2*a + 2
Let a = -2 - -9. Let l(x) = -5*x**2 - 5*x + 4. Let z(w) = 11*w**2 + 11*w - 9. Determine a*l(b) + 3*z(b).
-2*b**2 - 2*b + 1
Suppose 2*y = -2*y - 100. Let f be y - (-1 - -3) - -1. Let h(t) = -t. Let s(u) = -4*u. Give f*h(z) + 6*s(z).
2*z
Let r(u) = 7*u**2 + 4*u + 3. Let f(o) = -6*o**2 - 3*o - 2. Let v(z) = 3*z**2 + 13*z + 6. Let q be v(-4). Give q*r
| 3,500
| 2,624
| 1,850
| 2,914
| null | null |
github_plus_top10pct_by_avg
|
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