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ehat{\phi }^{2\kappa }}{\varepsilon (\phi )^{q(q+1)+1/p_{\ast }}}\times
(1\vee \left\Vert \phi \right\Vert _{1,q+2,\infty }^{2dq+1+2\kappa })\times
\left\Vert f\right\Vert _{q,\kappa ,p}. \label{J6} \\
&\end{aligned}$$This means that Assumption \[H1H\*1\] from Section \[sect:3.2\] hold uniformly in $z\in E$ and the constant given in (\[hh3’\]) is upper bounded by $$C_{q,\kappa ,\infty ,p}(U,P)\leq C\frac{1\vee \widehat{\phi }^{\kappa }}{%
\varepsilon (\phi )^{q(q+1)+1/p_{\ast }}}\times (1\vee \left\Vert \phi
\right\Vert _{1,q+2,\infty }^{2dq+1+2\kappa })\times (C_{q,\kappa ,\infty
}(P)\vee C_{q,\kappa ,p}(P)). \label{J6'}$$
We are now able to give our result (the proof being postponed for Section \[sect:proofJ\]):
\[J\] Suppose that $P_{t}$ satisfies assumptions \[H2H\*2-P\] and [H3]{}. Suppose moreover that $\phi $ satisfies (\[J4’\]), (\[J4”\]), (\[J3\]). Then $\overline{P}%
_{t}(x,dy)=\overline{p}_{t}(x,y)dy$ and we have the following estimates. Let $q\in {\mathbb{N}},\kappa \geq 0$ and $\delta >0$ be given. There exist some constants $C,\chi $ such that for every $\alpha ,\beta $ with $\left\vert
\alpha \right\vert +\left\vert \beta \right\vert \leq q$$$\left\vert \partial _{x}^{\alpha }\partial _{y}^{\beta }\overline{p}%
_{t}(x,y)\right\vert \leq \frac{C^{\rho }}{((\lambda t)^{\theta
_{0}(q+2d)(1+\delta )}}\times \frac{\psi _{\chi }(x)}{\psi _{\kappa }(x-y)}.
\label{J10}$$We stress that the constant $C$ depends on $C_{k,\kappa ,\infty ,p}(U,P)$ (see (\[J6’\])) and on $q,\kappa $ and $\delta $ **but not on** $%
t,\rho $ and $\lambda .$
This gives the following consequence concerning the semigroup $P_{t}$ itself:
\[Cor\]Suppose that (\[J6a\]),(\[J6b\]),(\[J6c\]) hold. Then, does not matter the value of $\theta _{1}$ in (\[J6c\]), the inequality ([J6c]{}) holds with $\theta _{1}^{\prime }=2d+\varepsilon $ for every $%
\varepsilon >0.$
**Proof.** Just take $\phi _{z}(x)=x.$ $\square $
Regularity results {#sect:reg}
==================
This section is devoted to some preliminary results allowing u
| 1,701
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_{C,P}$ (resp. $\nu_{D,P}$) be the $w$-multiplicity of $C$ (resp $D$) at $P$, (defined such that $\nu_{x,P}=a$ and $\nu_{y,P}=b$). Then the following equalities hold:
1. \[formula\_self-intersection1\] $\displaystyle \pi^{*}(C) = \widehat{C} + \frac{\nu_{C,P}}{e} E$,
2. \[formula\_self-intersection2\] $\displaystyle (E \cdot \widehat{C})_{\hat X} = \frac{e \nu_{C,P}}{a b d}$,
3. \[formula\_self-intersection3\] $\displaystyle (C \cdot D)_P=\widehat{C} \cdot \widehat{D}+ \frac{\nu_{C,P}\nu_{D,P}}{a b d}$,
4. \[formula\_self-intersection4\] $\displaystyle (E \cdot E)_{\hat X}=-\frac{e^2}{dab}$.
\[ex:QNC:resolution\] Consider $\pi_1$ the blow-up with $w=(1,1)$ of $X_1$ from Example \[ex:QNC\], using Proposition \[formula\_self-intersection\] one obtains $\pi_1^*D_1=\hat D_1 + \frac{1}{3} E$, $(\hat D_i\cdot E)_{\hat X_1}=1$, $(\hat D_1\cdot \hat D_2)_{\hat X_1}=0$, and $(E\cdot E)_{\tilde X_1}=-3$. Moreover, by $\tilde X_1$ is a smooth surface.
Rational divisors on normal surfaces
------------------------------------
Throughout this section we will assume $X$ is a reduced, irreducible, complex projective variety of dimension 2. Given any $D=\sum \alpha_iC_i \in \operatorname{Weil}_{\QQ}(X)$, where $C_i$ are irreducible curves and $\alpha_i\in \QQ$ we use the following notation: $$\array{l}
\left\lfloor D\right\rfloor=\sum \lfloor \alpha_i\rfloor C_i,\\
\left\lceil D\right\rceil=\sum \lceil \alpha_i\rceil C_i.\\
\endarray$$ If $\pi:Y\to X$ is a $\Q$-resolution of $X$, then the total transform of $D$ by $\pi$ can be written as $$\pi^*D=\hat D + \sum_{\v\in \Gamma} m_\v E_\v,$$ where $\Gamma$ is the dual graph of the $\Q$-resolution, $\hat D$ the strict transform of $D$, and $m_\v\in \QQ$ are uniquely determined by a solution to the linear system $$\label{eq:pullback}
(E_\v\cdot \pi^*D)_Y=0 \text{ for all } \v\in \Gamma.$$ Under these conditions, one can define $(D_1\cdot D_2)_X$ as the rational number $$\label{eq:intersection}
(D_1\cdot D_2)_X:=(\pi^*(D_1)\cdot \pi^*(D_2))_Y,$$ where the right-hand si
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s and reduced curves in $\PP^2$ introducing ideals of quasi-adjunction \[rem:qaideals\]. Here we will introduce the analogous objects for the cyclic covers of $\PP^2_w$ ramified along non-reduced curves. These objects are not ideals in general, but modules over the local rings $\cO_{\PP^2_w,P}$ which will be called quasi-adjunction modules.
Consider the local situation at $0\in X=\frac{1}{w_2}(w_0,w_1)\cong \CC^2/G$ a cyclic singularity, as defined in §\[subsec:CyclicQuotient\], fixing $\zeta\in \CC$ a $w_2$-th primitive root of unity. In §\[subsec:CyclicQuotient\] we defined the class $[g]:k\in\ZZ_{w_2}$ of a quasi-germ $g\in\cO_{X,\zeta^{k}}$ (which depends on the choice of $\zeta$). Also note that any $g\in \cO_{X,\zeta^{k}}$ defines a Weil divisor in $X$ denoted by $\operatorname{div}_X(g)$.
Let $C=\sum_{i=1}^r n_iC_i$ be a local Weil divisor at $0$ given by the set of zeros of quasi-germs $C_i=\{f_i=0\}$. Denote by $\pi: Y \to X$ a good $\Q$-resolution of $C\subset (X,0)$ and $\Gamma$ the dual graph associated with $\pi$. Then the total transform of $C$ and the relative canonical divisor can be written as $$\label{eq:notationEi-local}
\begin{aligned}
\pi^{*} C &= \hat{C} + \sum_{\v \in \Gamma} m_\v E_\v,
& \quad \pi^{*} C_i &= \hat{C}_i + \sum_{\v \in \Gamma} m_{\v i} E_\v,
& \quad K_\pi &= \sum_{\v \in \Gamma} (\nu_\v-1) E_\v,
\end{aligned}$$ where $m_\v=\sum_{i=1}^r n_im_{\v i}$.
\[def:M\] For any integer $k\in \ZZ$ and rational $\lambda\in \QQ$ we introduce the following *quasi-adjunction $\cO_X$-modules*: $$\label{eq:Mlocal}
\cM_\pi(C^\lambda,k)=
\left\{ g \in \cO_{X,\zeta^{\s_{\lambda,k}}}
\vphantom{\sum_{i=1}^r}\right.
\left|\
\operatorname{mult}_{E_\v} (\pi^{*}\operatorname{div}_X(g)) > \sum_{i=1}^r {\left \{ \lambda n_i \right \}} m_{\v i} - \nu_\v, \
\forall \v \in \Gamma
\right\}$$ where $\s_{\lambda,k}:=k - |w| - \sum_{i=1}^r {\left \lfloor \lambda n_i \right \rfloor} [f_i]$ and $|w|:=w_0+w_1+w_2$.
In section \[sec:examples\] (Proposition \[prop:indres\]) it will be shown that $\cM_\
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anti-parallel
vortex tubes $$-{\Delta}{\mathbf{u}}_0 = {\bnabla\times}{\bm{\omega}}_0, \quad {\bm{\omega}}_0 = \omega(x,y)\frac{\bm{\sigma'}}{|\bm{\sigma}'|}(s)$$ $$\omega(x(r,\theta),y(r,\theta)) = \frac{A}{(r/a)^{4} + 1}$$ $$\bm{\sigma}(s) = [2a, 2b/\cosh(s^2/c^2)-b, s]$$ Taken from
@k13.
$a= 0.05$, $b=a/2$,
$c=a$, $s$ is the arc-length parameter.
$A$ chosen so that $\E({\mathbf{u}}_0) = \E_0$.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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, and the Fourier-transformed emission function is defined as $\tilde S(q,K) = \int \d^4 x S(x,K) \exp(i q x).$
The measured $\lambda_*$ parameter of the correlation function is utilized to correct the core spectrum for long-lived resonance decays [@Csorgo:1994in]: $ N_1(k) = N_c(k)/{\sqrt{\lambda_{*}(k)}}. $ The emission function of the core is assumed to have a hydrodynamical form, $$S_c(x,k) d^4 x = \frac{g}{(2 \pi)^3}
\frac{ k^\nu d^4\Sigma_\nu(x)}{B(x,k) +s_q},$$ where $g$ is the degeneracy factor ($g = 1$ for pseudoscalar mesons, $g = 2$ for spin=1/2 barions). The particle flux over the freeze-out layers is given by a generalized Cooper–Frye factor: the freeze-out hypersurface depends parametrically on the freeze-out time $\tau$ and the probability to freeze-out at a certain value is proportional to $H(\tau)$, $
k^\nu d^4\Sigma_\nu(x) =
m_t \cosh(\eta - y)
H(\tau) d\tau \, \tau_0 d\eta \, dr_x \, dr_y.
$ Here $\eta = 0.5 \log[(t + r_z)/(t-r_z)]$, $\tau=\sqrt{t^2 - r_z^2}$, $ y = 0.5 \log[(E + k_z)/(E-k_z)]$ and $m_t=\sqrt{E^2 - k_z^2}$. The freeze-out time distribution $H(\tau)$ is approximated by a Gaussian, $
H(\tau) = \frac{1}{(2 \pi \Delta\tau^2)^{3/2}}
\exp\left[-\frac{(\tau - \tau_0)^2} {2 \Delta \tau^2} \right],
$ where $\tau_0$ is the mean freeze-out time, and the $\Delta\tau$ is the duration of particle emission, satisfying $\Delta\tau \ll \tau_0$. The (inverse) Boltzmann phase-space distribution, $B(x,k)$ is given by $$B(x,k)=
\exp\left( \frac{ k^\nu u_\nu(x)}{T(x)} -\frac{\mu(x)}{T(x)} \right),$$ and the term $s_q$ is $ 0$, $-1$, and $+1$ for Boltzmann, Bose-Einstein and Fermi-Dirac statistics, respectively. The flow four-velocity, $u^\nu(x)$, the chemical potential, $\mu(x)$, and the temperature, $T(x)$ distributions for axially symmetric collisions were determined from the principles of simplicity, analyticity and correspondence to hydrodynamical solutions in the limits when such solutions were known [@Csorgo:1995bi; @Csorgo:1995vf]. Rec
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eparation for a given set of parameters used in the simulations. The parameters ${\epsilon_{\mathrm{SW}}=9k_{\textrm{B}}T}$, ${\Delta=0.09\sigma _{2}}$ are the depth and the width of a short-ranged attractive square well, respectively, while the parameter $\epsilon_{\textrm{Y}}=24.6k_{\textrm{B}}T$ controls the strength of long-ranged repulsive Yukawa interaction with inverse Debye length $\kappa\sigma_{2}=10$.
A comparison between experimental quantities and simulation parameters can be found in Ref. [@Mani2010]. In principle, the strength of the attractive interaction is chosen so that physical bonds between colloids at the end of evaporation are irreversible. At the same time the repulsive barrier is chosen to be large enough to hinder spontaneous clustering. A wide range of simulation parameters satisfies the above conditions without qualitatively affecting the final results. The potential shape depicted in Fig. \[fig:pot\](a) is similar to that employed by Mani [*et al*.]{} [@Mani2010]. However, differently from their systematic investigation of the repulsive parameters ($\epsilon_{\textrm{Y}}$,$\kappa$) on the stability of colloidal shells, we restrict our consideration to a fixed value of both attractive and repulsive parameters between colloids but vary colloid-droplet energies in order to address competing interactions.
The droplet-droplet pair interaction is a hard-sphere potential, $$\Phi_{\textrm{dd}}(r)=\left \{
\begin{array}{ll}
\infty & r< \sigma_{\textrm{d}}+\sigma_{1}\\
0 & \text{otherwise,}
\end{array} \right .
\label{eq:phidd}$$ where the hard core droplet diameter $\sigma_{\textrm{d}}$ is added to the colloid diameter $\sigma _{1}$ such that no two droplets can share the same colloid (recall that $\sigma_{1}\geq\sigma_{2}$).
The colloid-droplet interaction is taken to model the Pickering effect [@Pieranski1980]. Since the droplets shrink, their diameter is recorded as a function of time, that is, $\sigma_{d}(t)$ may be larger or smaller than that of the colloids. Hence, if $\sig
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ents, we used the Docker virtualisation technology (Fig. [2](#Fig2){ref-type="fig"}). The docker-compose tool is used to define multi-containers, connections, and all necessary parameters.Fig. 2Deployment organization
TASKA services were developed following the characteristics and requirements of three complementary entities: a) Tasks, i.e., what to do; b) Workflows, how to do; and c) Users, who will do. These elements' functionalities allow a team to conduct any kind of study in TASKA.
Tasks {#Sec5}
-----
A task is the basic information unit in the system. Each task is organised in three main components: the input, e.g., the data files that are necessary for the task; the definition, i.e., what needs to be done in this task; and the output, the results of the assignment.
To address all the foreseen scenarios for our task/workflow management system, we created three distinct types of task:a *Simple task*, in which the description provides the instructions about what must be performed;a *Form task*, which allows the construction of a simple online questionnaire (text, multiple choices, etc.) that needs to be completed by each assignee. Each form is created using a drag-and-drop graphical user interface;a *Processing task*, which allows automatic execution of RESTful services provided by external systems. The task definition consists of describing the web-services end-point, and also the parameters that will be used.
Workflows {#Sec6}
---------
A workflow consists of combining a set of tasks in a hierarchical order (Fig. [3](#Fig3){ref-type="fig"}). The workflow begins with a single task, and then, in the following level, it can proceed with one or more parallel tasks. This process, distribution or aggregation, can be repeated up to the last layer, where the final task will collect the final results of the workflow. The web interface can create each workflow in a user-friendly manner. Each box, representing a task, can be configured according to its type. The dependencies between tasks are described as conn
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$ \dfrac{\pi}{2} $, the conformal infinity shifts towards infinity, giving rise to the $ \sim \dfrac{1}{r^2} $ behavior which diverges only at the ring singularity at $ r=0, \theta=\dfrac{\pi}{2} $. Moreover, we observe that except for $ \theta=\dfrac{\pi}{2} $, the gravitational entropy density remains finite at $ r=0 $ for all other values of $\theta$.
Rotating charged accelerating black hole
----------------------------------------
For the rotating charged black hole, the Weyl scalar $W$ is $$\begin{aligned}
\left.W\right. &=48\frac { \left( \alpha r\cos \left( \theta \right) -1 \right) ^{6}}
{ \left( {r}^{2}+{a}^{2} \left( \cos \left(\theta \right) \right) ^{2} \right) ^{6}} \times\nonumber\\
&( \left( {e}^{2}r\alpha+am \left( a\alpha+1 \right) \right) {
a}^{2}\cos^{3}\theta+ \left( 2a{e}
^{2}{r}^{2}\alpha+3{a}^{2}m \left( a\alpha-1 \right) r+{a}^{2}{e}^{2
} \right) \cos^{2}\theta \nonumber\\
&+ \left( -{e
}^{2}\alpha\,{r}^{3}-3am \left( a\alpha+1 \right) {r}^{2}+2 a{e}^{2
}r \right) \cos \left( \theta \right) -{r}^{2} \left( m \left( a\alpha
-1 \right) r+{e}^{2} \right) ) \nonumber\\
& ( \left( {e}^{2}r\alpha+a
m \left( a\alpha-1 \right) \right) {a}^{2} \left( \cos \left( \theta
\right) \right) ^{3}+ \left( -2a{e}^{2}{r}^{2}\alpha-3{a}^{2}m
\left( a\alpha+1 \right) r+{a}^{2}{e}^{2} \right) \left( \cos
\left( \theta \right) \right) ^{2} \nonumber\\
&+ \left( -{e}^{2}\alpha{r}^{3}-3
am \left( a\alpha-1 \right) {r}^{2}-2a{e}^{2}r \right) \cos
\left( \theta \right) +{r}^{2} \left( m \left( a\alpha+1 \right) r-{e
}^{2} \right) ),\end{aligned}$$ and the Kretschmann scalar $K$ is $$\begin{aligned}
\left.K\right. &= 48\frac {\left( \alpha r \cos \left( \theta \right)-1 \right)^{6}}{\left({r}^{2}+{a}^{2}\left(\cos\left(\theta \right)\right)^{2}\right)^{6}}({a}^{4}\left({a}^{4}{\alpha}^{2}{m}^{2}+2{a}^{2}{\alpha}^{2}{e}^{2}mr+7/6{\alpha}^{2}{e}^{4}{r}^{2}
-{a}^{2}{m}^{2} \right) \left( \cos \left( \theta \right) \right) ^{6} \nonumber\\
&+2 \left( {a}^{2}m+5/6{e}^{2}r \right) \left( {e}^{2}-
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e data.
In addition to the $L_\infty$ ball given in , we also construct a confidence set for $\beta_{{\widehat{S}}}$ to be a hyper-rectangle, with sides of different lengths in order to account for different variances in the covariates. This can be done using the set $$\label{eq:beta.hyper:CI}
\tilde C_{{\widehat{S}}} = \bigotimes_{j\in {\widehat{S}}} \tilde{C}(j),$$ where $$\tilde{C}(j) = \left[ \hat\beta_{{\widehat{S}}}(j) - z_{\alpha/(2k)}
\sqrt{\frac{ \hat\Gamma_{{\widehat{S}}}(j,j)}{n}}, \hat\beta_{{\widehat{S}}}(j) + z_{\alpha/(2k)}
\sqrt{\frac{ \hat\Gamma_{{\widehat{S}}}(j,j)}{n}}\right],$$ with $\hat\Gamma_{{\widehat{S}}}$ given by (\[eq::Ga\]) and $z_{\alpha/(2k)}$ the upper $1 -
\alpha/(2k)$ quantile of a standard Normal variate. Notice that we use a Bonferroni correction to guarantee a nominal coverage of $1-\alpha$.
\[thm::big-theorem\] Let $\hat{C}_{{\widehat{S}}}$ and $\tilde{C}_{{\widehat{S}}}$ the confidence sets defined in and , respectively. Let $$\label{eq:un}
u_n = u -K_{2,n},$$ where $$K_{2,n} = C A \sqrt{ k U \frac{\log k + \log n}{n} },$$ with $C = C(\eta)>0$ the universal constant in . Assume, in addition, that $n$ is large enough so that $ u_n $ is positive. Then, for a $C >0$ dependent on $A$ only, $$\label{eq:big-theorem.Linfty}
\inf_{w_n \in \mathcal{W}_n} \inf_{P\in {\cal P}_n^{\mathrm{OLS}}}\mathbb{P}(\beta \in \hat{C}_{{\widehat{S}}}) \geq 1-\alpha -
C \Big(\Delta_{n,1} + \Delta_{n,2}+\Delta_{n,3} \Big)$$ and $$\label{eq:big-theorem.hyper}
\inf_{w_n \in \mathcal{W}_n} \inf_{P\in {\cal P}_n^{\mathrm{OLS}}}\mathbb{P} (\beta \in \tilde{C}_{{\widehat{S}}}) \geq 1-\alpha -
C\Big(\Delta_{n,1} + \Delta_{n,2}+\tilde{\Delta}_{n,3} \Big),$$ where $$\Delta_{n,1} = \frac{1}{\sqrt{v}}\left( \frac{
\overline{v}^2 k^2 (\log kn)^7)}{n}\right)^{1/6} , \quad
\Delta_{n,2} = \frac{ U }{ \sqrt{v}} \sqrt{ \frac{k^4 \overline{v} \log^2n \log k}{n\,u_n^6}
},$$ $$\Delta_{n,3} = \left( \frac{ U^2 }{
v }\right)^{1/3} \left( \overline{v}^2 \frac{k^{5}}{u_n^{6} u^4}
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$i>0$, while $H^0(\operatorname{Hilb^n{\mathbb{C}}^2}, {\mathcal{P}}_1 \otimes {\mathcal{L}}_1^d) = {\mathbb{J}}^d$ by . Thus, in the notation of [@hai1 Section 3], $p({\mathbb{J}}^d, s, t)=\chi_{{\mathcal{P}}_1 \otimes {\mathcal{L}}_1^d}(s,t)$ and so, by [@hai1 Proposition 3.2], $$\label{bigr11}
p({\mathbb{J}}^d, s, t)=\sum_\mu p({\mathcal{P}}_1\otimes {\mathcal{L}}_1^d(I_\mu),s,t)\,\,
\Omega(\mu)^{-1}
=\sum_\mu p({\mathcal{P}}_1(I_{\mu}),s,t) p({\mathcal{L}}_1(I_\mu),s,t)^d\,\,
\Omega(\mu)^{-1}.$$ Here we have used the fact that, as $I_\mu$ defines a finite dimensional scheme, we can identify the sheaf ${\mathcal{P}}_1\otimes {\mathcal{L}}_1^d(I_\mu)$ with its global sections, and so $p({\mathcal{P}}_1\otimes {\mathcal{L}}_1^d(I_\mu),s,t)$ is naturally defined.
We now evaluate the right hand side of . It is proved in [@haidis (3.9)], using the notation of [@haidis (1.9)], that $p({\mathcal{L}}_1(I_{\mu}),s, t) = \prod_{x\in d(\mu)} s^{l'(x)}t^{a'(x)}=
s^{n(\mu)}t^{n(\mu^t)}$. On the other hand, by [@hai1 Proposition 3.4] (which is proved in [@hai3 Section 3.9] and uses the notation of [@hai1 (46)]), $p({\mathcal{P}}_1(I_{\mu}),s, t)
= P_{\mu}(s,t)$. Substituting these observations into shows that $$p({\mathbb{J}}^d, s, t)=\sum_\mu P_{\mu}(s,t)\, \Omega(\mu)^{-1}
s^{dn(\mu)}t^{dn(\mu^t)},$$ as required.
Blowing up $({\mathfrak{h}}\oplus {\mathfrak{h}}^*)/{{W}}$ {#hi-defn-sec}
----------------------------------------------------------
All the results described so far have natural analogues for the subvariety ${\mathfrak{h}}\oplus {\mathfrak{h}}^*$ of ${\mathbb{C}}^{2n}$. Geometrically, this follows from the observation that the natural additive action of ${\mathbb{C}}^2$ by translation on $\operatorname{Hilb^n{\mathbb{C}}^2}$ gives a decomposition $\operatorname{Hilb^n{\mathbb{C}}^2}= {\mathbb{C}}^2\times \left( \operatorname{Hilb^n{\mathbb{C}}^2}\right)/{\mathbb{C}}^2$ into a product of varieties [@Nak p.10]. Unravelling the actions shows that $ \operatorname{Hilb^n{\mathbb{C}}^2}/{\mathbb{C}}^2
| 1,710
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te vectors somehow. Suppose one is given a finite vector of $P^f (k_{\parallel})$, for $k^A_\parallel \le k_\parallel \le k^B_\parallel$ say. Eq. (\[projection3\]), in component form, can be rewritten as: $$P^f (k_\parallel) - \Delta = \sum_{k=k_\parallel}^{k^B_\parallel}
A(k_\parallel,k) \, \tilde P^\rho (k)
\label{projection4}$$ where $\Delta = \int_{k^B_\parallel}^\infty W^{f\rho}(k_{\parallel}/k,k)
{\tilde P^\rho} (k) {k dk /{2 \pi}}$, and $A(k_\parallel,k) = W^{f\rho} (k_{\parallel}/k,k) {k dk/2 \pi}$. $A(k_\parallel,k)$ can be regarded as a triangular matrix in the sense that $A(k_\parallel,k)$ can be set to zero for $k < k_\parallel$ by the virtue of the lower limit of summation in eq. (\[projection4\]).
By inverting eq. (\[projection4\]), we can in principle determine $\tilde P^\rho (k)$, with $\Delta$ left as a free parameter. We can do better, however, by the following observation: since $\tilde P^\rho (k)$ is generally a rapidly decreasing function of $k$ for sufficiently high $k$’s ($\sim k^{-3}$, or faster if $\rho$ is equated with the baryon density, see footnote in §\[largescales\]), assuming $W^{f\rho} (k_{\parallel}/k,k)$ does not increase significantly with $k$, one can see that $\Delta$ can be made small by choosing a sufficiently high truncation $k^B_\parallel$. Therefore, inverting eq. (\[projection4\]) by ignoring $\Delta$ altogether would still give accurate estimates of $\tilde P^\rho (k)$ for $k$’s sufficiently smaller than $k^B_\parallel$. We will illustrate this with an explicit example of ${\bf A}$ or $W^{f\rho}$ in the next section.
A Perturbative Example {#perturb}
----------------------
In this section, we will perform a linear calculation of $P^f$, and we will assume the actual shape of $P^f$ on large scales, even in the presence of nonlinearities on small scales, agrees with that of the linear prediction, while its amplitude might not. This is in the spirit of Croft et al. [-@croft98] who argued that, ignoring redshift-distortions, $P^f$ should be proportional to the line
| 1,711
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|
ed by $\Z_m[C_r]$ and referred to as the [*group ring*]{} of the cyclic group $C_r$ with coefficients in $\Z_m$. See e.g., [@KKS13; @HH11] for some recent applications of these rings in cryptography.
---
abstract: |
In this investigation we study extreme vortex states defined as incompressible velocity fields with prescribed enstrophy $\E_0$ which maximize the instantaneous rate of growth of enstrophy $d\E/dt$. We provide [an analytic]{} characterization of these extreme vortex states in the limit of vanishing enstrophy $\E_0$ and, in particular, show that the Taylor-Green vortex is in fact a local maximizer of $d\E / dt$ [in this limit]{}. For finite values of enstrophy, the extreme vortex states are computed numerically by solving a constrained variational optimization problem using a suitable gradient method. In combination with a continuation approach, this allows us to construct an entire family of maximizing vortex states parameterized by their enstrophy. We also confirm the findings of the seminal study by [@ld08] that these extreme vortex states saturate (up to a numerical prefactor) the fundamental bound $d\E / dt < C \, \E^3$, for some constant $C >
0$. The time evolution corresponding to these extreme vortex states leads to a larger growth of enstrophy than the growth achieved by any of the commonly used initial conditions with the same enstrophy $\E_0$. However, based on several different diagnostics, there is no evidence of any tendency towards singularity formation in finite time. Finally, we discuss possible physical reasons why the initially large growth of enstrophy is not sustained for longer times.
author:
- |
Diego Ayala$^{1,2}$ and Bartosz Protas$^{2,}$[^1]\
\
$^1$ Department of Mathematics, University of Michigan,\
Ann Arbor, MI 48109, USA\
\
$^2$ Department of Mathematics and Statistics, McMaster University\
Hamilton, Ontario, L8S 4K1, Canada
title: Extreme Vortex States and the Growth of Enstrophy in 3D Incompressible Flows
---
Keywords: Navier-Stok
| 1,712
| 265
| 1,194
| 1,794
| null | null |
github_plus_top10pct_by_avg
|
-----------------------
Subjects, *n* 177 73 14 29 27 3 31
Age, years 177 71.6 (5.9) 72.5 (5.5) 72.4 (7.2) 70.6 (8.9) 71.7 (8.4) 69.2 (5.8)
Education, years 177 9.6 (5.0) 9.9 (5.3) 9.8 (4.4) 8.2 (5.0) 9.3 (5.8) 6.8 (3.7)
Female sex 177 39 (53.4%\] 7 (50.0%\] 13 (44.8%\] 13 (48.1%) 0 (0.0%\] 19 (61.3%\]
APOE *∊* 4 carrier status 175 14 (19.2%) 8 (57.1%) 6 (20.7%) 18 (66.7%) 0 (0.0%) 20 (64.5%)
K-MMSE score, points 177 26.6 (2.6) 27.1 (2.9) 25.4 (3.5) 25.5 (3.8) 23.0 (2.6) 18.1 (5.4)
CDR 177 0.3 (0.2) 0.3 (0.3) 0.5 (0.0) 0.5 (0.0)
| 1,713
| 4,011
| 1,699
| 1,465
| null | null |
github_plus_top10pct_by_avg
|
ty the isotropic case), one gets the equations $$\begin{aligned}
& \bar{a}(\bar{b}+\bar{\beta})-\bar{h}a-\bar{f}e+b\bar{h}_0+gm-q(\bar{g}+\bar{\gamma})=0 \nonumber \\
& a(b+\beta)+e(\gamma+g)+\bar{b}h_0-\bar{a}h+e_0\bar{g}+fq=0 \quad ,\label{constraintsD71braneIIB}\end{aligned}$$ where the notation for the isotropic fluxes is as in eq. . Similarly, eq. , with the index $d$ different from $a,b,c$, leads to the two components $(Q \cdot Q-P^{1,4}\cdot F_3 )^{x^j y^j x^k}_{y^k}$ and $(Q\cdot Q-P^{1,4} \cdot F_3 )^{x^j y^j y^k}_{x^k}$, which would induce a charge for the $5_2^3$-branes associated to the components $D_{4\, x^i y^i x^j y^j x^k, x^j y^j x^k}$ and $D_{4\, x^i y^i x^j y^j y^k, x^j y^j y^k}$ of the mixed-symmetry potential $D_{9,3}$. In the isotropic case these constraints are $$\begin{aligned}
& -b(b+\beta)+h(\bar{b}+\bar{\beta})-f'q+e(g'+\gamma')-\bar{g}'e_0=0 \nonumber \\
& \bar{b}(\bar{b}+\bar{\beta})-\bar{h}(b+\beta)-q(\bar{\gamma}'+\bar{g}')-g'm+\bar{f}'e=0 \quad . \label{constraintsD93IIB}\end{aligned}$$ In the IIA/O6 case, the non-trivial constraints in eq. are the second, the third and the fourth, but actually only the second and the fourth are relevant for the components such that the upstairs indices are different from the downstairs ones. The second constraint is $$(Q\cdot H_3 + f \cdot f+P^1_1 \cdot F_2 -P^{1,1}\cdot F_4 +P_1^3 \cdot F_4+P^{1,3} \cdot F_6)^{a}_{bcd}=0 \quad ,\label{NSNSBianchiIIAO6}$$ which again would induce a charge for the $5_2^1$-branes associated to the same $D_{7,1}$ components as in IIB. The constraints in this case are the second equation in and the first in . The other non-trivial constraint is $$(R \cdot f +Q \cdot Q+ P_1^3 F_0-P^{1,3} \cdot F_2-P_1^5 \cdot F_2-P^{1,5}\cdot F_4)^{abc}_d=0 \quad , \label{NSNSBianchiIIAO6bis}$$ which would induce a charge for the same $5_2^3$-branes as in the IIB case, leading to the first equation in and the second in . We therefore have perfect match between the IIB and the IIA result.
The situation is different if one considers th
| 1,714
| 834
| 1,019
| 1,764
| null | null |
github_plus_top10pct_by_avg
|
,n})}{2f(t)}+\frac{D(t;h_{1,n})+b(t;h_{1,n})}{2f(t)}
\frac{f^{1/2}(t)-\hat f^{1/2}(t;h_{1,n})}{\hat f^{1/2}(t;h_{1,n})+f^{1/2}(t)}\\
&:=&\frac{D(t;h_{1,n})}{2f(t)}+\frac{b(t;h_{1,n})}{2f(t)}+\delta_4(t),\end{aligned}$$ (where $\delta_4$ depends on $n$, but we do not display this dependence) and note that (again using $(a^{1/2}-b^{1/2}=(a-b)/(a^{1/2}+b^{1/2})$), $$\label{delta4}
\sup_{t\in D_r^\varepsilon}|\delta_4(t)|\le \frac{1}{3r^{3/2}}\sup_{t\in D_r^\varepsilon}\left[D(t;h_{1,n})+b(t;h_{1,n})\right]^2$$ which is small by (\[zero\]) (note that $\delta_n$ is $D+b$ divided by a quantity which is bounded away from zero on $D_r$) . Setting $$\label{eps1}
\varepsilon_1(t,h_{1,n},h_{2,n}):=\frac{1}{n h_{2,n}}\sum_{i=1}^{n}L\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right)
f^{-1/2}(X_i)D(X_i;h_{1,n})I(|t-X_{i}|<h_{2,n}B),$$ $$\label{eps2}
\varepsilon_2(t,h_{1,n},h_{2,n}):=\frac{1}{n h_{2,n}}\sum_{i=1}^{n}L\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right)
f^{-1/2}(X_i)b(X_i;h_{1,n})I(|t-X_{i}|<h_{2,n}B),$$ and $$\label{eps3}
\varepsilon_3(t,h_{1,n},h_{2,n}):=\delta_3(t)+\frac{1}{n h_{2,n}}\sum_{i=1}^{n}L\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right)
f^{1/2}(X_i)\delta_4(X_i;h_{1,n})\textbf{1}\{|t-X_{i}|<h_{2,n}B\},$$ we obtain (from (\[e1\])-(\[e4\])), $$\label{expans}
\hat f(t;h_{1,n}, h_{2,n})=\bar f(t;h_{2,n})+\frac{1}{2}\varepsilon_1(t)+\frac{1}{2}\varepsilon_2(t)+\varepsilon_3(t).$$ By the comments above, the $\varepsilon_2$ and $\varepsilon_3$ terms will be of smaller order than $\varepsilon_1$. $\varepsilon_1$ itself has a $U$-process structure, and the linear term in its Hoeffding decomposition will be the dominant term. This is the content of the lemmas that follow.
\[lemma1\] For $i=2,3$, $$\sup_{t\in D_r^\varepsilon}|\varepsilon_i(t,h_{1,n},h_{2,n})|=O_{\rm a.s.}(n^{-4/9}) \ \ uniformly\ in\ \ f\in{\cal P}_C$$ for all $C<\infty$.
We begin with $i=2$. Because the function $L$ is of bounded variation and $b(t;h_{1,n})$ satisfies inequality (\[classic2\]), it follows (see the Appendix) that the classes of functi
| 1,715
| 1,079
| 2,092
| 1,682
| 3,935
| 0.76918
|
github_plus_top10pct_by_avg
|
oup $S_{n+1}$ on $\mathcal O(n)$, which extends the given $S_n$-action, and satisfies, for $1\in\mathcal O(1)$, $\alpha\in \mathcal O(m)$, $\beta\in \mathcal O(n)$ the following relations: $$\begin{aligned}
\label{compos_cyclic1} \tau_2(1)&=&1,\\ \label{compos_cyclic2}
\tau_{m+n}(\alpha\circ_k \beta)&=&\tau_{m+1}(\alpha)\circ_{k+1}
\beta,\quad\quad\quad \text{ for } k<m \\ \label{compos_cyclic3}
\tau_{m+n}(\alpha\circ_m \beta)&=&\tau_{n+1}(\beta)\circ_1
\tau_{m+1}(\alpha),\end{aligned}$$ where $\tau_{j}\in S_{j}$ denotes the cyclic rotation of $j$ elements $\tau_{j} :=1\in{\mathbb{Z}}_{j}\subset S_{j}$.
\[def\_O\_hat\] Let $\mathcal O$ be a cyclic operad with $\mathcal O(1)=k$. For a sequence of $n$ input colors $\vec X=(x_1,\dots, x_n)$ and the output color $x$, where $x_1, \dots, x_n, x\in\{{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}},\varnothing\}$, let $$\widehat{\mathcal O}(\vec X;x):=
\begin{cases}
\mathcal O(n) & \text{if } x \text{ is ``full'', and } \vec X=({
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}},\ldots,{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}}),\\
\mathcal O(n) & \text{if } x \text{ is ``dashed'', and $\vec X$ has
exactly one ``dashed'' input}\\
\mathcal O(n-1) & \text{if } x=\varnothing \text{ and
$\vec X$ has exactly two ``dashed' inputs,} \\
\{0\} & \text{otherwise}.
\end{cases}$$ The definition of $\widehat{\mathcal O}(\vec X,\varnothing)$ is motivated by the idea that one considers trees with $n-1$ inputs and one output, and then uses the $S_{n+1}$ action to turn this output into a new input: $$\begin{pspicture}(0,0.8)(4,3.4)
\psline[linestyle=dashed](2,2)(1.4,3)
\psline(2,2)(1.8,3)
\psline(2,2)(2.2,3)
\psline(2,2)(2.6,3)
\psline[linestyle=dashed](2,2)(2,1)
\rput[b](2,3.2){$1\,\,\, 2
| 1,716
| 3,424
| 1,392
| 1,577
| 800
| 0.799832
|
github_plus_top10pct_by_avg
|
submanifold ${\mathbb{S}}_{2k}$ in the first jet space $J^1=J^1(X\times U)$.
####
Let us introduce a set of $p-2k$ linearly independent vectors $\xi_a:{\mathbb{R}}^q{\rightarrow}{\mathbb{C}}^p$ defined by
\[eq:3.13\] \_a(u)=[( \_a\^1(u),…,\_a\^p(u) )]{},a=1,…,p-2k,
satisfying the orthogonality conditions
\[eq:3.14\] \_i\^A\_a\^i=0,|\_i\^A\_a\^i=0,A=1,…,k,a=1,…,p-2k,
for a set of $2k$ linearly independent wave vectors ${\left\{ \lambda^1,\ldots,\lambda^k,
\bar{\lambda}^1,\ldots, \bar{\lambda}^k \right\}}$. It should be noted that the set ${\left\{ \lambda^1,\ldots,\lambda^k,\bar{\lambda^1},\ldots,\bar{\lambda}^k,\xi^1,\ldots,\xi^{p-2k} \right\}}$ forms a basis for the space of independent variables $X$. Note also, that the vectors $\xi_a$ are not uniquely defined since they obey the homogeneous conditions (\[eq:3.14\]). As a consequence of equation (\[eq:3.6\]) or (\[eq:3.7\]), the graph $\Gamma={\left\{ x,u(x) \right\}}$ is invariant under the family of first-order differential operators
\[eq:3.15i\] X\_a=\_a\^i(u),a=1,…,p-2 k,
defined on $X\times U$ space. Since the vector fields $X_a$ do not include vectors tangent to the direction of $u$, they form an Abelian distribution on $X\times U$ space, [*i.e.* ]{}
\[eq:3.16\] =0,a, b=1,…,p-2k.
The set ${\left\{ r^1,\ldots, r^k,\bar{r}^1,\ldots,\bar{r}^k,u^1,\ldots
u^q \right\}}$ constitutes a complete set of invariants of the Abelian algebra $\mathcal{L}$ generated by the vector fields (\[eq:3.15i\]). So geometrically, the characterization of the proposed solution (\[eq:3.5\]) of equations (\[eq:3.1\]) can be interpreted in the following way. If $u(x)$ is a $q$-component function defined on a neighborhood of the origin $x=0$ such that the graph of the solution $\Gamma={\left\{ (x,u(x)) \right\}}$ is invariant under a set of $p-2k$ vector fields $X_a$ with the orthogonality property (\[eq:3.14\]), then for some function $f$ the expression $u(x)$ is a solution of equation (\[eq:3.5\]). Hence the group-invariant solutions of the system (\[eq:3.1\])
| 1,717
| 2,147
| 2,804
| 1,787
| null | null |
github_plus_top10pct_by_avg
|
ence\_vt\] Let $v_t$ be as defined in , initialized at $v_0$. Then for any $T=n\delta$, $$\begin{aligned}
\E{\lrn{v_T - v_0}_2^2} \leq T^2 L^2 \E{\lrn{v_0}_2^2} + T\beta^2
\end{aligned}$$ If we additionally assume that $\E{\lrn{v_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$ and $T \leq \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}$, then $$\begin{aligned}
\E{\lrn{v_T - v_0}_2^2} \leq 2 T\beta^2
\end{aligned}$$
From , $$\begin{aligned}
v_T - v_0 = - T \nabla U(v_0) + \sqrt{\delta} \sum_{i=0}^{n-1} \xi(v_0, \eta_i)
\end{aligned}$$ Conditioned on the randomness up to time $i$, $\E{\xi(v_0,\eta_{i+1})}=0$. Thus $$\begin{aligned}
& \E{\lrn{v_T - v_0}_2^2}\\
=& T^2\E{\lrn{\nabla U(v_0)}_2^2} + \delta \sum_{i=0}^{n-1} \E{\lrn{\xi(v_0,\eta_i)}_2^2}\\
\leq& T^2 L^2 \E{\lrn{v_0}_2^2} + T \beta^2
\end{aligned}$$ where the inequality is by item 1 of Assumption \[ass:U\_properties\] and item 2 of Assumption \[ass:xi\_properties\].
\[l:divergence\_wt\] Let $w_t$ be as defined in , initialized at $w_0$. Then for any $T=n\delta$ such that $T \leq \frac{1}{2L}$, $$\begin{aligned}
\E{\lrn{w_T - w_0}_2^2} \leq 16 \lrp{T^2 L^2 \E{\lrn{w_0}_2^2} + T \beta^2}
\end{aligned}$$ If we additionally assume that $\E{\lrn{w_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$ and $T \leq \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}$, then $$\begin{aligned}
\E{\lrn{w_T - w_0}_2^2} \leq 32 T\beta^2
\end{aligned}$$
$$\begin{aligned}
& \E{\lrn{w_{(k+1)\delta} - w_0}_2^2}\\
=& \E{\lrn{w_{k\delta} - \delta \nabla U(w_{k\delta}) + \sqrt{\delta} \xi\lrp{w_{k\delta}, \eta_k} - w_0}_2^2}\\
=& \E{\lrn{w_{k\delta} - \delta \nabla U(w_{k\delta}) - w_0}_2^2} + \delta \E{\lrn{\xi\lrp{w_{k\delta}, \eta_k}}_2^2} \numberthis \label{e:t:sfddd}
\end{aligned}$$
We can bound $\delta \E{\lrn{\xi\lrp{w_{k\delta}, \eta_k}}_2^2} \leq \delta \beta^2$ by item 2 of Assumption \[ass:xi\_properties\]. $$\begin{aligned}
& \E{\lrn{w_{k\delta} - \delta \nabla U(w_{k\delta})
| 1,718
| 1,851
| 941
| 1,542
| null | null |
github_plus_top10pct_by_avg
|
|\Th\sim\Nc_{r\times q}(\Th, v_yI_r\otimes I_q)$ with $(v_x,v_y)=(0.1,\, 1),\ (1,\, 1)$ and $(1,\, 0.1)$. It has been assumed that a pair of the maximum and the minimum eigenvalues of $\Th\Th^\top$ is $(0,\, 0),\ (24,\, 0)$ or $(24,\, 24)$. Note that the best invariant predictive density $\ph_U(Y|X)$ has a constant risk and its risk is approximately given by $$R(\ph_U,\Th)=\frac{rq}{2}\log\frac{v_s}{v_y}\approx\begin{cases}
1.42 & \textup{for $(v_x,v_y)=(0.1,\ 1)$},\\
10.4 & \textup{for $(v_x,v_y)=(1,\ 1)$},\\
36.0 & \textup{for $(v_x,v_y)=(1,\ 0.1)$},
\end{cases}$$ when $r=2$ and $q=15$.
Denote by $\Bc(a,b)$ the matrix-variate beta distribution having the density (\[eqn:pr\_GB\]). Using (\[eqn:m(W)-1\]) with $\La=v_1\Om\{I_r-(1-v_1)\Om\}^{-1}$ and $v_1=v/v_0$, we can rewrite $\ph_{GB}(Y|X)$ as $$\ph_{GB}(Y|X)=\frac{\Er^{\Om}[g_{v_w}(\Om|W)]}{\Er^{\Om}[g_{v_x}(\Om|X)]}\ph_U(Y|X),$$ where $\Er^{\Om}$ indicates expectation with respect to $\Om\sim \Bc(a+q,b)$ and $$g_{v}(\Om|Z)=\Big|I_r-\Big(1-\frac{v}{v_0}\Big)\Om\Big|^{-q/2}\exp\Big[-\frac{1}{2v_0}\tr\Big[\Om\Big\{I_r-\Big(1-\frac{v}{v_0}\Big)\Om\Big\}^{-1}ZZ^\top\Big]\Big]$$ for an $r\times q$ matrix $Z$. Hence in our simulations, the expectation $\Er^{\Om}[g_{v}(\Om|Z)]$ was estimated by $j_0^{-1}\sum_{j=1}^{j_0} g_{v}(\Om_j|Z)$, where $j_0=100,000$ and the $\Om_j$ are independent replications from $\Bc(a+q,b)$.
$
\begin{array}{ccccccccc}
\hline
(v_x, v_y) & {\rm Eigenvalues}&{\rm Minimax}&\multicolumn{6}{c}{(a,b)}\\
\cline{4-9}
& {\rm of}\ \Th\Th^\top &{\rm risk}&(-11,3)&(-11,9)&(-11,15)&(-5,3)&(-5,9)&(1,3) \\
\hline
(0.1,1) &(\ 0,\ 0)&1.42& 0.47 & 0.96 & 1.20 & 0.38 & 0.80 & 0.33 \\
&(24,\ 0) & & 0.91 & 1.17 & 1.30 & 0.87 & 1.09 & 0.84 \\
&(24,24) & & 1.39 & 1.39 & 1.40 & 1.37 & 1.37 & 1.39 \\
[6pt]
(1, 1) &(\ 0,\ 0)&10.4& 5.3 & 6.9 & 7.7 & 2.8 & 4.6 & 1.6 \\
&(24,\ 0) & & 6.9 & 8.0 & 8.4 & 5.3 & 6.3 & 4.6 \\
&(24,24) & & 8.7 & 9.0 & 9.2 & 7.9 & 8.0 & 8.4 \\
[6pt]
| 1,719
| 649
| 1,385
| 1,641
| null | null |
github_plus_top10pct_by_avg
|
ing.
\[trathle2\] Assume that $\Sigma\in L^\infty(G\times S\times I)$ and that $\Sigma\geq 0$. Then $\psi$ defined by satisfies weakly in $G\times S\times I$, \[trath11\] \_x +=f, and the inflow boundary condition (\[trath8\]) is valid.
Due to (\[trath12\]) it suffices to show (\[trath11\]) only for $f\in C^1(\ol G\times S\times I)$. Using the notations from the proof of Lemma \[trathle1\], for $\varphi\in C_0^\infty(G\times S\times I^\circ)$ we get by the Fubin’s Theorem $$&-(\omega\cdot \nabla_x \psi)(\varphi)
=\psi(\omega\cdot\nabla_x\varphi) \\
={}&\int_{G\times S\times I} \psi(x,\omega,E) (\omega\cdot \nabla_x \varphi)(x,\omega,E){d}x {d}\omega {d}E \\
={}&\int_{S\times I}\int_{G_{\omega}}\sum_i \int_{J^i_{y,\omega}} \psi(y+\tau\omega,\omega,E) {\frac{{d}}{{d}\tau}}\varphi(y+\tau\omega,\omega,E){d}\tau{d}y{d}\omega{d}E \\
={}&\int_{S\times I}\int_{G_{\omega}}\sum_i \int_{J^i_{y,\omega}} \int_0^{\tau-a^i_{y,\omega}}e^{-\int_0^t\Sigma(y+(\tau-s)\omega,\omega,E)ds} f(y+(\tau-t)\omega,\omega,E)\\
&\cdot{\frac{{d}}{{d}\tau}}\varphi(y+\tau\omega,\omega,E) dt{d}\tau{d}y{d}\omega{d}E \\
={}&\int_{S\times I}\int_{G_{\omega}}\sum_i \int_{J^i_{y,\omega}} \int_{a^i_{y,\omega}}^\tau e^{-\int_0^{\tau-t}\Sigma(y+(\tau-s)\omega,\omega,E)ds} f(y+t\omega,\omega,E)\\
&\cdot{\frac{{d}}{{d}\tau}}\varphi(y+\tau\omega,\omega,E) dt{d}\tau{d}y{d}\omega{d}E \\
={}&\int_{S\times I}\int_{G_{\omega}}\sum_i \int_{J^i_{y,\omega}} f(y+t\omega,\omega,E) \int_t^{b^i_{y,\omega}} e^{-\int_{t}^\tau\Sigma(y+s\omega,\omega,E)ds} \\
&\cdot{\frac{{d}}{{d}\tau}}\varphi(y+\tau\omega,\omega,E) {d}\tau{d}t{d}y{d}\omega{d}E.$$ In the last step, we changed the order of integration $dtd\tau\to d\tau dt$, in which the domain of integration $$\{(\tau,t)\ |\ \tau\in J^i_{y,\omega}=]a^i_{y,\omega},b^i_{y,\omega}[,\ t\in ]a^i_{y,\omega},\tau[\}$$ changes into $$\{(t,\tau)\ |\ t\in J^i_{y,\omega}=]a^i_{y,\omega},b^i_{y,\omega}[,\ \tau\in ]t,b^i_{y,\omega}[\}$$ as usual.
Observing that $$&\int_t^{b^i_{y,\omega}} e^{-\int_{t}^\tau\Sigma(y+s\omega,\omega,E){d
| 1,720
| 757
| 1,083
| 1,585
| null | null |
github_plus_top10pct_by_avg
|
this.vgap;
}
}
}
/* these 3 methods need to be overridden properly */
@Override public Dimension minimumLayoutSize(Container parent) {
return new Dimension(0,0);
}
@Override public Dimension preferredLayoutSize(Container parent) {
return new Dimension(200,200);
}
@Override public Dimension maximumLayoutSize(Container target) {
return new Dimension(600,600);
}
@Override public void removeLayoutComponent(Component comp) {
this.components.remove(comp);
}
public static void main(String[] args) {
JFrame frame = new JFrame("VerticalFlowLayoutTest");
VerticalFlowLayout vfl = new VerticalFlowLayout();
JPanel panel = new JPanel(vfl);
vfl.setHGap(20);
vfl.setVGap(2);
int n = 19;
Random r = new Random(12345);
for (int i = 0; i < n; ++i)
{
JLabel label = new JLabel(labelName(i,r));
panel.add(label);
}
frame.setContentPane(panel);
frame.pack();
frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
frame.setVisible(true);
}
private static String labelName(int i, Random r) {
StringBuilder sb = new StringBuilder();
sb.append("label #");
sb.append(i);
sb.append(" ");
int n = r.nextInt(26);
for (int j = 0; j < n; ++j)
sb.append("_");
return sb.toString();
}
}
Q:
Google Analytics visitor flow shows drop-offs for homepage
I checked the visitors flow on my Google Analytics account. It shows drop-offs from the homepage and the group details don't really give much info. The group details for other pages do show that the reason for the drop-offs (at that particular node) are broken links.
Its only the homepage where I don't see any details. Could it be that other external sites are linking to the wrong page?
A:
The amount of drop off should, give-or-take, match the exit and bounce figures f
| 1,721
| 1,726
| 43
| 1,991
| 1,803
| 0.785591
|
github_plus_top10pct_by_avg
|
\tilde{\nu}_{J} (L).
\label{final-condition}\end{aligned}$$ Therefore, in the mass-basis formulation only $U$ and $W$ are involved, which is consistent with our experience in $W$ perturbation theory. An apparent contradiction to this property that one faces in the evolution equation in the flavor basis is resolved in appendix \[sec:flavor-basis-evolution\].
A drawback of this method is that we have to solve explicitly the evolution of the sterile states which are coupled to the active states. Then, we need to specify the sterile sector model, and have to know how to deal with averaging over the fast modes.
We notice, however, that in the zeroth-order in $W$ the system simplifies. Since the Hamiltonian $\tilde{H}$ is block-diagonal it suffices to solve the equation only in the $3 \times 3$ active neutrino subspace: $$\begin{aligned}
i \frac{d}{dx} \nu_{i} =
\sum_{j}
\left( {\bf \Delta_{a} } + U^{\dagger} A U \right)_{ij}
\nu_{j}.
\label{Schroedinger-eq-0th}\end{aligned}$$ The initial condition (\[initial-condition\]) and final reverse-back formula (\[final-condition\]) involve only $U$ matrix elements. Therefore, the oscillation probability in the zeroth-order in $W$ can be calculable in a manner independent of sterile sector models.[^15]
An exact solution of zeroth-order oscillation probability {#sec:exact-solution-zeroth}
-----------------------------------------------------------
Here, we describe a method for obtaining the analytical solution of the zeroth-order Hamiltonian. The exact solution, as well as the numerical one described in the previous section, provides the basis for computing the higher order corrections in $W$.
We calculate an exact form of the oscillation probability $P(\nu_\beta \rightarrow \nu_\alpha)$ in leading order in our perturbative framework, the one in (\[P-beta-alpha-final\]) except for $\mathcal{C}_{\alpha \beta}$, in the case of uniform matter density.
The zeroth-order $S$ matrix element $S_{\alpha \beta}^{(0)}$ in (\[S-alpha-beta-0th\]) can be written as $$\begin{alig
| 1,722
| 1,772
| 2,044
| 1,827
| null | null |
github_plus_top10pct_by_avg
|
\overline{v} b \frac{\log n}{n} } \right)\\
\label{eq:Ac}
& \leq \frac{1}{n},
\end{aligned}$$ where in the third identity we have used the definition of $\epsilon_n$ in and the final inequality inequality follows from the vector Bernstein inequality and by taking the constant $C$ in appropriately large. In fact, the bound on the probability of the event $\mathcal{E}_n^c$ holds uniformly over all $P \in \mathcal{P}_n$.
Next, for any $t > 0$ and uniformly in $P \in \mathcal{P}_n$, $$\begin{aligned}
\mathbb{P}( \sqrt{n}||\hat\theta - \theta||_\infty \leq t) &=
\mathbb{P}( \sqrt{n}||\hat\theta - \theta||_\infty \leq t,\ \mathcal{E}_n) +
\mathbb{P}( \sqrt{n}||\hat\theta - \theta||_\infty \leq t,\ \mathcal{E}_n^c)
\nonumber \\
& \leq
\mathbb{P}( \sqrt{n}||\hat\nu - \nu||_\infty \leq t+\epsilon_n) +
\mathbb{P}(\mathcal{E}_n^c) \nonumber \\
& =
\mathbb{P}( ||Z_n||_\infty \leq t+\epsilon_n) +
C \frac{1}{\sqrt{v}} \left( \frac{\overline{v}^2 b (\log 2bn)^7}{n}
\right)^{1/6} +
\mathbb{P}(\mathcal{E}_n^c)
\label{eq:sorryrogeryouaretigernow}\end{aligned}$$ where the inequality follows from and the fact that $\|
R \|_\infty \leq \epsilon_n $ on the event $\mathcal{E}_n$ and the second identity from the Berry-Esseen bound (\[eq::CLT\]). By the Gaussian anti-concentration inequality of , $$\mathbb{P}( ||Z_n||_\infty \leq t + \epsilon_n )\leq \mathbb{P}( ||Z_n||_\infty
\leq t) + \frac{\epsilon_n}{\underline{\sigma}} (\sqrt{2 \log b} +2).$$ Using the previous inequality on the first term of , we obtain that $$\begin{aligned}
\mathbb{P}( \sqrt{n}||\hat\theta - \theta||_\infty \leq t)& \leq
\mathbb{P}( ||Z_n||_\infty \leq t) + C \left[ \frac{\epsilon_n}{\underline{\sigma}} (\sqrt{2 \log b} +2) +
\frac{1}{\sqrt{v}} \left( \frac{\overline{v}^2 b (\log 2bn)^7}{n}
\right)^{1/6}
\right] +
\mathbb{P}(\mathcal{E}_n^c)\\
& \leq
\mathbb{P}( ||Z_n||_\infty \leq t) + C \left [\frac{\epsilon_n}{\underline{\sigma}} (\sqrt{2 \log b} +2) +
\frac{1}{\sqrt{v}} \left( \frac{\overline{v}^2 b (\log 2bn)^7}{n}
\ri
| 1,723
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| 2,021
| 1,518
| null | null |
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|
K}}$ and $T_{\mathrm{HII}} \simeq
7\times10^4{\,\mathrm{K}}$ are the temperature in the neutral and ionized regions, respectively. Although not very precise, this simple expression captures the qualitative features of the bipolar ionized outflows.
![The opening angle of the equatorial neutral inflow region $\theta_{\mathrm{inflow}}$ as a function of $r$ (solid red), along with the angle-dependence of the radius of the [[H[ii]{} ]{}]{} region $r=r_{\mathrm{HII}}(\theta)$ (dashed orange; equation \[eq:6\]). The dotted part of the red line marks a few innermost cells that are artificially ionized as we neglect the absorption within the sink (also see the text). []{data-label="fig:th_in_Ds"}](figure/th_in_Ds.eps){width="8.5cm"}
Fig. \[fig:th\_in\_Ds\] shows the opening angle of the equatorial neutral region $\theta_{\mathrm{inflow}}$ as a function of $r$. In practice, we define the neutral region as the region where the ionization degree of hydrogen is less than $50\%$. Although $\theta_{\mathrm{inflow}}$ decreases as $r$ decreases, the neutral inflow region reaches the inner boundary at $r = R_{\mathrm{in}}$ with a finite angle, unlike in the case with disc radiation without shadowing effect. The sharp drop of $\theta_{\mathrm{inflow}}$ around $R_{\mathrm{in}} = 2000$ AU is caused by photoionization of a few innermost cells due to our ignorance of the consumption of ionizing photons within the sink, although this does not affect our conclusion. The opening angle at the Bondi radius, $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})=40^\circ$, is similar to the assumed shadow opening angle $\theta_{\mathrm{shadow}} =45^\circ$ (equation \[eq:10\]).
In order to estimate $\theta_{\mathrm{inflow}}$, we calculate the radius of the [[H[ii]{} ]{}]{} region $r_{\mathrm{HII}}$ in each direction $\theta$ with the modeled density profiles. The supply rate of ionizing photons per unit solid angle is given by $\dot{N}_{\mathrm{ion}}\,\mathcal{F}(\theta)/4\pi$, where $\dot{N}_{\mathrm{ion}}$ is the total ionizing photon emissivity
| 1,724
| 1,079
| 3,021
| 2,026
| 2,912
| 0.776146
|
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|
nced and size properties, with $d{\leqslant}s$. Then there exists a constant $\alpha=\alpha(\beta)$, which depends only on $\beta$, and there exists $m=\Theta(n)$ with $m < n$, such that the balanced allocation process on $({\mathcal{H}}^{(1)},\ldots, {\mathcal{H}}^{(n)})$ is $(\alpha,m)$-uniform. [Specifically, we may take $\alpha = 44\beta$.]{} ]{}
We are ready to prove Lemma \[lem:col\].
\[lem:col1\] Fix $d=d(n)$ with $2{\leqslant}d = o(\log n)$. [Let $({\mathcal{H}}^{(1)},\ldots, {\mathcal{H}}^{(n)})$ be a dynamic hypergraph which satisfies the $\beta$-balanced, $\varepsilon$-visibility and $c_0$-size properties.]{} Suppose that $c {\geqslant}44\beta {\mathrm{e}}^2$ is a sufficiently large constant, and let $k=C\log n$ for some constant $C{\geqslant}1$. There exists $\Theta(n){\leqslant}m{\leqslant}n$ such that the probability that ${\mathcal{C}}_{m}$ contains a $c$-loaded $k$-vertex tree is at most $$\exp\Big\{4k\log({2\beta d})- c(d-1)(k-r-1) + \big(c_0 + 3 - r\varepsilon/2\big) \log(n)\Big\}$$ where $r$ is the number of red vertices in the blue-red coloring of the tree. Moreover, with high probability, [if ${\mathcal{C}}_{m}$ contains any such tree]{} then $r={\mathcal{O}}(1/\varepsilon)$.
[Fix $m=m(n)$ to equal the $m$ provided by [Lemma \[lem:uni\]]{}.]{} There are at most $4^k$ ordered trees with $k$ vertices. (Proposition \[pro:ordered\]). Fix such a tree, say $T$, and label the vertices $\{1,2,\ldots, k\}$ such that vertex $i$ is the $i$-th new vertex visited when performing depth-first search in $T$ starting from the root, and respecting the given ordering. In particular, the root of $T$ is vertex $1$. [Next, we will assign a $d$-choice to the root vertex of $T$, as a first step in describing trees which may be present in the witness graph $\mathcal{C}_m$.]{} Let $x$ count the number of [possible $d$-choices that can be assigned to the root of $T$]{}. Then $$x{\leqslant}\binom{{s}}{d}\cdot \left|\bigcup_{t=1}^{m}{\mathcal{E}}_t\right|\cdot m
{\leqslant}\binom{{s}}{d}\cdot n^{c_0+2},$$ where
| 1,725
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| 1,561
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| 0.78581
|
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|
haracteristics of two systems on the basis of Darboux transformations (and we obtain a construction of hierarchy of potentials as in [@Maydanyuk.2005.APNYA], see p. 443–445): $$\begin{array}{l}
H_{1} = A_{1}^{+} A_{1} =
-\displaystyle\frac{\hbar^{2}}{2m}
\displaystyle\frac{d^{2}}{dr^{2}}
+ \bar{V}_{1}(r), \\
H_{2} = A_{1} A_{1}^{+} =
-\displaystyle\frac{\hbar^{2}}{2m}
\displaystyle\frac{d^{2}}{dr^{2}}
+ \bar{V}_{2}(r).
\end{array}
\label{eq.2.1.8}$$
The interdependence between wave functions \[sec.2.3\]
------------------------------------------------------
We shall study two quantum systems, in each of which there is the scattering of the particle on the potential $V_{1}(r)$ or $V_{1}(r)$. Further, we shall not consider processes, concerned with loss of complete energy of systems (for example, dissipation, bremsstrahlung etc.). The energy spectra of these systems are *continuous*, and their lowest levels are *zero*. In accordence with (\[eq.2.1.8\]), we write: $$\begin{array}{l}
H_{1} \chi^{(1)}_{k,l} =
A_{1}^{+} A_{1} \chi^{(1)}_{k,l} =
E^{(1)}_{k,l} \chi^{(1)}_{k,l}, \\
H_{2} \chi^{(2)}_{k^{\prime},l^{\prime}} =
A_{1} A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}} =
E^{(2)}_{k^{\prime},l^{\prime}} \chi^{(2)}_{k^{\prime},l^{\prime}},
\end{array}
\label{eq.2.3.1}$$ where $E^{(1)}_{k, l}$ and $E^{(2)}_{k^{\prime}, l^{\prime}}$ are the energy levels of two systems with orbital quantum numbers $l$ and $l^{\prime}$, $\chi^{(1)}_{k,l}(x)$ and $\chi^{(2)}_{k^{\prime}, l^{\prime}}(x)$ are the radial components of wave functions, concerned with these levels, $k = \displaystyle\frac{1}{\hbar}\sqrt{2mE^{(1)}_{k,l}}$ and $k^{\prime} =
\displaystyle\frac{1}{\hbar}\sqrt{2mE^{(2)}_{k^{\prime},l^{\prime}}}$ are wave vectors corresponding to the levels $E^{(1)}_{k,l}$ and $E^{(2)}_{k^{\prime}, l^{\prime}}$. From (\[eq.2.3.1\]) we obtain: $$H_{2} (A_{1} \chi^{(1)}_{k,l}) =
A_{1} A_{1}^{+} (A_{1} \chi^{(1)}_{k,l}) =
A_{1} (A_{1}^{+} A_{1} \chi^{(1)}_{k,l}) =
A_{1} (E^{(1)}_{k,l} \chi^{(1)}_{k
| 1,726
| 2,995
| 2,584
| 1,778
| null | null |
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|
es the true character of $\delta$ to allow only a view of its integral manifestation on functions. This unfortunately is not general enough in the strongly nonlinear physical situations responsible for chaos, and is the main reason for constructing the multifunctional extension of function spaces that we use. ]{}
[^10]: [\[Foot: cont=3Dbound\]Recall that for a linear operator continuity and boundedness are equivalent concepts. ]{}
[^11]: [\[Foot: OrthoMatrix\]A real matrix $A$ is an orthogonal projector iff $A^{2}=A$ and $A=A^{\textrm{T}}$. ]{}
[^12]: [\[Foot: class\]In this sense, a]{} *class* [is a set of sets. ]{}
[^13]: [\[Foot: interval\]By definition, an interval $I$ in a totally ordered set $X$ is a subset of $X$ with the property $$(x_{1},x_{2}\in I)\wedge(x_{3}\in X\!:x_{1}\prec x_{3}\prec x_{2})\Longrightarrow x_{3}\in I$$ ]{}
[so that any element of $X$ lying between two elements of $I$ also belongs to $I$.]{}
[^14]: [\[Foot: entropy\]Although we do not pursue this point of view here, it is nonetheless tempting to speculate that the answer to the question]{} *“Why* [does the entropy of an isolated system increase?” may be found by exploiting this line of reasoning that seeks to explain the increase in terms of a visible component associated with the usual topology as against a different latent workplace topology that governs the dynamics of nature.]{}
[^15]: [\[Foot: subspace\]In a subspace $A$ of $X$, a subset $U_{A}$ of $A$ is open iff $U_{A}=A\bigcap U$ for some open set $U$ of $X$. The notion of subspace topology can be formalized with the help of the inclusion map $i\!:A\rightarrow(X,\mathcal{U})$ that puts every point of $A$ back to where it came from, thus $$\begin{array}{ccl}
\mathcal{U}_{A} & = & \{ U_{A}=A\bigcap U\!:U\in\mathcal{U}\}\\
& = & \{ i^{-}(U)\!:U\in\mathcal{U}\}.\end{array}$$ ]{}
[^16]: [\[Foot: assoc&embed\]A surjective function is an]{} *association* [iff it is image continuous and an injective function is an]{} *embedding* [iff it is preimage continuous
| 1,727
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| 2,035
| 1,676
| 2,066
| 0.783079
|
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|
d} \in \Sigma_{p,d,P}$. Hence $d|(m-n-d)$, that is, $m-n-d=td$ for some $t \in \mathbb{N}^0$ and so $m-n=(t+1)d$. Hence for any $a^ib^j \in S$ we have that $d|i-j$. By Lemma \[dimpact\], the first part of the following lemma is clear.
\[dequal1l\] If a two-sided subsemigroup $S=F_{D}\cup \widehat{F}\cup \widehat{\Lambda}_{I,p,d}\cup \Sigma_{p,d,P}$ of $\mathcal{B}$ is a left I-order in $\mathcal{B}$, then $d=1$ and $q=0$.
Suppose that $q\neq 0$, let $a^0b^k\in \mathcal{B}$ where $k\in \mathbb{N}$. Then $$a^0b^k=(a^ib^j)^{-1}(a^mb^n)=a^{j-i+t}b^{n-m+t}$$ where $t=\max\{m,i\}$, so that $0=j-i+t$. Hence we can deduce that $j=0$. If $i=0$, then $a^0b^k=a^mb^n$ so that $a^0b^k\in S$ a contradiction and so $i>0$. Hence $a^ib^0\in S$, but $a^ib^0 \in \widehat{\Lambda}_{0,p,1}\cup \widehat{F}$ as $\widehat{F}\subset T_{0,p}$ a contradiction again. Therefore $q=0$ as required.
\[L1\] In the case where $q=0$ it is easy to see that $a^pb^0\in S$. If $m\notin I$, then $a^ub^m\notin \widehat{F}$ for any $0\leq u<p$. For, if $a^ub^m \in \widehat{F}$, then $ a^pb^0a^ub^m =a^{p+u}b^m \in \widehat{\Lambda}_{I,p,d}$ a contradiction.
\[gcase\] The subsemigroup $S=F_{D}\cup \widehat{F}\cup \widehat{\Lambda}_{I,p,d}\cup \Sigma_{p,d,P}$ of $\mathcal{B}$ is a left I-order in $\mathcal{B}$ if and only if $d=1$ and $I=\{0,...,p-1\}$.
($\Longrightarrow$) Suppose that $S$ is a left I-order in $\mathcal{B}$. Then any element $q=a^mb^n \in \mathcal{B}$ can be written as $(a^ib^j)^{-1}(a^kb^l)$ for some $a^ib^j,a^kb^l \in S$. By Lemma \[dequal1l\], $d=1$ and $0\in I$. It is remains to show that $I=\{0,...,p-1\}$.
Let $0<m<p$. Then $$\begin{array}{rcl}
a^mb^m&=&(a^ib^j)^{-1}(a^kb^l)\\ &=& a^{j-i+t}b^{l-k+t}\end{array}$$ where $t=$max$\{i,k\}$, for some $a^ib^j,a^kb^l\in S$. Then $m=j$ or $m=l$; so that $a^ub^m \in S$ for some $u$. If $m \notin I$, so $u< p$, then $a^pb^0a^ub^m=a^{p+u}b^m \in S$, in contradiction to Remark \[L1\].
($\Longleftarrow$) Suppose that $d=1$ and $I=\{0,...,p-1\}$. Then for any $a^mb^n\in \mathcal{B}$ we hav
| 1,728
| 2,208
| 1,806
| 1,629
| 3,340
| 0.773006
|
github_plus_top10pct_by_avg
|
def}(\Omega_{\rm c})_{d[ab}(J_{\rm c}^2)_{c]def} \Big) \nonumber \\
& (P_1^5\cdot\Omega_{\rm c}\cdot J_{\rm c}^2)_{abc} =\tfrac{1}{3} \Big( \tfrac{1}{16} P^{d_1...d_5}_{[a}(\Omega_{\rm c})_{bc]d_1}(J_{\rm c}^2)_{d_2...d_5}+\tfrac{1}{8}P^{d_1...d_5}_{[a}(\Omega_{\rm c})_{| d_1 d_2 d_3}(J_{\rm c}^2)_{d_4 d_5|bc]}\nonumber \\
& \qquad \qquad - \tfrac{1}{4}P^{d_1...d_5}_{[a}(\Omega_{\rm c})_{|d_1 d_2|b}(J_{\rm c}^2)_{c]d_3 d_4 d_5} \Big)\nonumber \\
& (P^{1,5}\cdot\Omega_{\rm c}\cdot J_{\rm c}^3)_{abc} =\tfrac{1}{3} \Big(\tfrac{1}{8}P^{d,d e_1...e_4}(\Omega_{\rm c})_{d e_1 e_2}(J_{\rm c}^3)_{d e_3 e_4 abc} +\tfrac{1}{2}P^{d,d e_1...e_4}(\Omega_{\rm c})_{d e_1[a}(J_{\rm c}^3)_{bc]d e_2 e_3 e_4}\nonumber \\
&\qquad \qquad -\tfrac{1}{16}P^{d,d e_1...e_4}(\Omega_{\rm c})_{d[ab}(J_{\rm c}^3)_{c]d e_1 ... e_4}\Big) \ .\end{aligned}$$ Performing again three T-dualities along the $x$ directions on eq. , we come back to the IIB superpotential with all $P$ fluxes allowed by the projections. Namely, we find $$W_{\rm IIB/O3} = \int [( F_3 - i S H_3 ) + ( Q - i S P_1^2 ) \cdot \mathcal{J}_{\rm c} - P^{1,4} \cdot \mathcal{J}_{\rm c}^2 ]\wedge \Omega \quad ,
\label{Wballpfluxes}$$ where the contractions for $Q$ and $P_1^2$ are [@Shelton:2005cf; @Aldazabal:2006up] $$\begin{aligned}
& (Q \cdot \mathcal{J}_{\rm c})_{abc}=\tfrac{3}{2} Q^{de}_{[a}(\mathcal{J}_{\rm c})_{bc]de} \nonumber \\
& (P_1^2 \cdot \mathcal{J}_{\rm c})_{abc}=\tfrac{3}{2} P^{de}_{[a}(\mathcal{J}_{\rm c})_{bc]de}\quad ,\end{aligned}$$ while the last contraction is defined as $$(P^{1,4}\cdot{\mathcal{J}_{\rm c}^2})_{abc} = \tfrac{3}{4}P^{d,d e_1 e_2 e_3}(\mathcal{J}_{\rm c})_{d e_1[ab}(\mathcal{J}_{\rm c})_{c]d e_2 e_3} \quad ,$$ which is the only non-vanishing contraction we can introduce consistently with the orientifold.
The IIB/O3 superpotential we find coincides with the one derived in [@Aldazabal:2010ef] on the basis of generalised geometry considerations, and this is a positive test in favour of our T-duality rules for $P$ fluxes. As far as the
| 1,729
| 1,676
| 1,631
| 1,598
| 3,808
| 0.769987
|
github_plus_top10pct_by_avg
|
aybreaks \\
&= 2 \sum_{k = n + 1}^{2n} \frac{1}{k} \allowdisplaybreaks \\
&= 2 \sum_{k = 1}^{n} \frac{1}{k + n}. \end{aligned}$$ Therefore, we find $$\begin{aligned}
\lim_{n \rightarrow \infty} S_{n}
&= 2 \lim_{n \to \infty} \sum_{k = 1}^{n} \frac{1}{k + n} \allowdisplaybreaks \\
&= 2 \lim_{n \to \infty} \frac{1}{n} \sum_{k = 1}^{n} \frac{1}{1 + \frac{k}{n}}
\allowdisplaybreaks \\
&= \int_{0}^{1} \frac{1}{1 + x} dx \allowdisplaybreaks \\
&= 2\log{2}.\end{aligned}$$ The lemma is thus proved.
Now we can prove Lemma $\ref{lem:4-1-1}$.
Let $$\begin{aligned}
g(a) := \frac{4^{a} \G\left(a + \frac{1}{2} \right)}{\sqrt{\pi} \G(a + 1)}. \end{aligned}$$ To prove $g(a) > 1$ for $a > 0$, we use the following formula [@andrews_askey_roy_1999 p.13, Theorem 1.2.5]: $$\begin{aligned}
\frac{d}{dx} \log{\G(x)}
= \frac{\G^{'}(x)}{\G(x)}
= - \g_{0} + \sum_{n = 1}^{\infty} \left(\frac{1}{n} - \frac{1}{x + n - 1} \right), \end{aligned}$$ where $\g_{0}$ is Euler’s constant given by $$\begin{aligned}
\g_{0} := \lim_{n \to \infty} \left(\sum_{k = 1}^{n} \frac{1}{k} - \log{n} \right). \end{aligned}$$ Taking the logarithmic derivative of $g(a)$, from the above formula, we have $$\begin{aligned}
\frac{d}{da} \log{g(a)}
&= 2\log{2} + \frac{d}{da} \log{\G\left(a + \frac{1}{2} \right)}
- \frac{d}{da} \log{\G(a + 1)} \allowdisplaybreaks \\
&= 2\log{2} + \sum_{n = 1}^{\infty} \left(\frac{1}{n} - \frac{1}{a - \frac{1}{2} + n} \right)
- \sum_{n = 1}^{\infty} \left(\frac{1}{n} - \frac{1}{a + n} \right) \allowdisplaybreaks \\
&= 2\log{2} - \frac{1}{2} \sum_{n = 1}^{\infty} \frac{1}{(a + n) \left(a - \frac{1}{2} + n \right)}
\allowdisplaybreaks \\
&> 2\log{2} - \frac{1}{2} \sum_{n = 1}^{\infty} \frac{1}{n \left(n - \frac{1}{2} \right)}
\allowdisplaybreaks \\
&= 2\log{2} - \sum_{n = 1}^{\infty} \frac{1}{n (2n - 1)}\end{aligned}$$ for $a > 0$. Moreover, using Lemma $\ref{lem:4-1-2}$, we obtain $\frac{d}{da} \log{g(a)} > 0$ for $a > 0$. This leads to $\frac{d}{da} g(a) > 0$ for $a > 0$. The lemma follows from this and $g(0) = 1$.
Now, w
| 1,730
| 3,874
| 2,264
| 1,470
| null | null |
github_plus_top10pct_by_avg
|
h between this and alternatives such as the logistic curve will arise, chiefly, from tail events. Moreover, the Gaussian assumption allows a small, albeit potentially negligible, probability of a spurious firing event when no stimulus is applied. Given this contradiction with the experimental design, the following log-logistic form of the excitability is used: $$\begin{aligned}
F(s; \eta, \lambda) = \left[1 + \left(\frac{s}{\eta}\right)^{-4\eta/\lambda}\right]^{-1}. \label{eq:ECdef}\end{aligned}$$ Nonetheless, the inference method described in Section \[sec:DetailFireProc\] is applicable for any sigmoidal curve.
Prior distributions {#sec:PriorDist}
-------------------
The excitability parameters of individual MUs are assumed to be independent *a priori*. For some upper limits ${\eta_{\max}}$ and ${\lambda_{\max}}$, the excitability parameters are assigned vague independent beta prior distributions: $$\begin{aligned}
\frac{\eta}{{\eta_{\max}}} \sim \mathrm{Beta}(1.1,~ 1.1), \quad\quad \frac{\lambda}{{\lambda_{\max}}} \sim \mathrm{Beta}(1.1,~ 1.1). \label{eq:ECprior}\end{aligned}$$ The shape parameters are chosen so that the densities are uninformative yet tail off towards the boundaries. The location upper bound is conservatively set just greater than the supramaximal stimulus, ${\eta_{\max}}= 1.1s_\tau$. Evidence for specifying the upper bound ${\lambda_{\max}}$ is taken from @Hal04 where, for a Gaussian excitability curve, the coefficient of variation of a random variable whose cumulative distribution function is given by the excitability curve was estimated to be 1.65%. With the log-logistic curve this corresponds to $\lambda/\eta \approx 3.64\%$. Given that $\eta\le \eta_{max}=1.1s_{\tau}$, we deduce that $\lambda \le 0.04 s_\tau$. The limitations of the the study of @Hal04, commented on by @Maj07, indicate that a larger bound may be required than initially suggested, so sensitivity of MUNE to ${\lambda_{\max}}$ is investigated in Sections \[sec:SimStudy\_under\] and \[sec:CaseStudy\].
The following p
| 1,731
| 2,067
| 3,194
| 1,774
| 3,457
| 0.772227
|
github_plus_top10pct_by_avg
|
a^{-1}\int_{0}^{1}\int_{-\pi}^{\pi} {f}(\rho_n+r_n r(\cos\theta,\sin\theta)) \\
& \hspace{7cm}\times\left(1-I(r^2;1-\alpha/2,\alpha/2)\right)\frac{{\rm d}\theta}{2\pi}\times \alpha r^{\alpha-1}\,{\rm d}r.
\end{aligned}$$ We used the Monte Carlo approach for evaluating this integral. Consider independent random variables $\Theta\sim U(-\pi,\pi)$ and $R=X^{1/\alpha}$ such that $X\sim U(0,1)$. Then $R$ has the probability density function $f_R(r)=\alpha r^{\alpha-1}$ and we want to evaluate $$r_n^{\alpha}V_1(0,{f}(\rho_n+r_n\cdot))=a_{2,\alpha} r_n^\alpha \mathbb{E}\left[\left(1-I(R^2;1-\alpha/2,\alpha/2)\right){f}(\rho_n+r_n R(\cos\Theta,\sin\Theta))\right] \label{MCEXP}$$ with $a_{2,\alpha}=\alpha^{-1}2^{-\alpha+1}\Gamma(\alpha/2)^{-2}B(1-\alpha/2,\alpha/2)$. We simulate $n_{R,\Theta}$ samples of pairs $(R,\Theta)$ and compute the sample mean of the quantity in . The quantity is evaluated more efficiently by writing $${f}(\rho_n+r_n R(\cos\Theta,\sin\Theta))= {f}(\rho_n)+[{f}(\rho_n+r_n R(\cos\Theta,\sin\Theta))-{f}(\rho_n)]
\label{[]}$$ This gives two terms: one can be evaluated directly (by storing $\mathbb{E} [1-I(R^2; 1-\alpha/2, \alpha/2)]$) and the second can be evaluated using a Monte Carlo method, but with smaller variance (as the quantity in square brackets in is $\order{r_n}$). It is worth noting that a similar mixed approach using the trapezoidal rule over $\theta$ and randomising $r$ as earlier for evaluating the left-hand side of was also tested. However, results showed that the pure Monte Carlo approach is, in comparison to the mixed one, superior with regards to accuracy and computational cost. With this view, we decided to focus on the first one.
Accuracy of this algorithm and its feasibility of implementation was checked with model solutions to problems of the type and they are presented below in order.
Free-space Green’s function
---------------------------
The free-space Green’s function for the fractional Laplacian $(-\Delta)^{\alpha/2}$ is $$G(x, y)=c_{d,\alpha} \fr
| 1,732
| 1,603
| 2,063
| 1,686
| null | null |
github_plus_top10pct_by_avg
|
d currents realization of $U_q(\widehat{sl_N})$
------------------------------------------------------
$U_q(\widehat{sl_N})$ is an associative algebra generated by the Drinfeld generators $E^{\pm,i}_n~(n\in {\Bbb Z})$, $H^i_n~(n\in {\Bbb Z})~
(i=1,~2,~...,~N-1)$ and the center $\gamma$. Let
$$K_i = \mbox{exp}\left((q-q^{-1}) \frac{1}{2} H^i_0 \right),$$
then we can write the Drinfeld currents in the form of formal power series of the complex parameter $z$ with coefficients given by the above generators,
$$\begin{aligned}
& & H^i(z) = \sum_{n\in {\Bbb Z}}H^i_n z^{-n-1},~~
E^{\pm,i}(z) = \sum_{n \in {\Bbb Z}} E^{\pm,i}_n z^{-n-1},\nonumber\\
& & \psi^i_\pm(z) = \sum_{n \in {\Bbb Z}} \psi^i_{\pm,n} z^{-n}
\equiv K_i^{\pm 1} \mbox{exp} \left( \pm (q-q^{-1})
\sum_{\pm n > 0} H^i_n z^{-n} \right). \nonumber\end{aligned}$$
The generating relations for $U_q(\widehat{sl_N})$ in terms of these currents can be written as follows [@sln],
$$\begin{aligned}
& & [ \psi^i_\pm(z),~\psi^j_\pm(w) ] =0, \label{1} \\
& & (z-q^{a_{ij}} \gamma^{-1} w) (z-q^{-a_{ij}} \gamma w)
\psi^i_+(z) \psi^j_-(w) \nonumber\\
& &~~~~= (z-q^{a_{ij}} \gamma w) (z-q^{-a_{ij}} \gamma^{-1} w)
\psi^i_-(w) \psi^j_+(z), \label{2}\\
& & (z-q^{\pm a_{ij}} \gamma^{\mp \frac{1}{2}} w)
\psi^i_+(z) E^{\pm,j} (w) =
(q^{\pm a_{ij}} z- \gamma^{\mp \frac{1}{2}} w)
E^{\pm,j} (w) \psi^i_+(z), \label{3} \\
& & (z-q^{ \pm a_{ij}} \gamma^{\mp \frac{1}{2}} w)
E^{\pm,j} (z) \psi^i_-(w) =
(q^{ \pm a_{ij}} z- \gamma^{\mp \frac{1}{2}} w)
\psi^i_-(w) E^{\pm,j} (z), \label{4} \\
& & [ E^{+,i}(z),~E^{-,j}(w) ] = \frac{\delta^{ij}}{(q-q^{-1})zw}
\left( \delta(z^{-1}w \gamma) \psi^i_+( \gamma^{\frac{1}{2}}w)
- \delta(z^{-1}w \gamma^{-1}) \psi^i_-( \gamma^{- \frac{1}{2}}w)
\right), \label{5} \\
& & (z- q^{\pm a_{ij}} w ) E^{\pm,i}(z)E^{\pm,j}(w)
= (q^{\pm a_{ij}} z - w ) E^{\pm,j}(w)E^{\pm,i}(z), \label{6} \\
& & E^{\pm,i}(z)E^{\pm,j}(w) = E^{\pm,j}(w)E^{\pm,i}(z)
~~\mbox{for}~a_{ij}=0, \label{7} \\
& & E^{\pm,i}(z_1) E^{\pm,i}(z_2) E^{\pm,j}(w)
-(q+q^{-1}) E^{\pm,i}(z_1) E^{\p
| 1,733
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interested in the representation $V_{\operatorname{U}(abc),(k)} = \operatorname{Sym}^k({\mathbb C}^{abc})$, not in arbitrary irreducible representations of $\operatorname{U}(abc)$. By specializing the construction described in to this one-parameter family of representations, we obtain the following result:
\[optimized kronecker\] The multiplicity of a weight $\delta = (\delta^{A},\delta^B,\delta^C) \in {\mathbb Z}^a \oplus {\mathbb Z}^b \oplus {\mathbb Z}^c \cong \Lambda^*_H$ (we use the identifications fixed at the beginning of ) in the irreducible $G$-representation $\operatorname{Sym}^k({\mathbb C}^{a b c})$ is equal to the number of integral points in the rational convex polytope $$\begin{aligned}
&\Delta(k, \delta)
= \Big\{
(x_{l,m,n}) \in {\mathbb R}^{a b c}_{\geq 0} \;:\; \sum_{l,m,n} x_{l,m,n} = k, \\
&\quad \sum_{m,n} x_{l,m,n} = \delta^{A}_l,
\sum_{l,n} x_{l,m,n} = \delta^{B}_m,
\sum_{l,m} x_{l,m,n} = \delta^{C}_n
\Big\}.
\end{aligned}$$ It follows that the Kronecker coefficient for Young diagrams $\lambda, \mu, \nu$ with $k$ boxes and at most $a$, $b$ and $c$ rows, respectively, is given by the formula $$g_{\lambda,\mu,\nu} = \sum_{\gamma \in \Gamma_H} c_\gamma \, \# \left( \Delta(k, (\lambda,\mu,\nu)+\gamma) \cap {\mathbb Z}^{abc} \right),$$ where $\Gamma_H$ and $(c_\gamma)$ are defined as in the statement of .
It is well-known that the weight spaces for the action of $\operatorname{U}(d)$ on $\operatorname{Sym}^k({\mathbb C}^d)$ are all one-dimensional and that the set of weights corresponds to the integer vectors in the standard simplex rescaled by $k$ [@fulton97]. In our case, $d = a b c$, so that the weights are just the integral points of the polytope $$\Big\{ x = (x_{l,m,n})_{l \in [a], m \in [b], n \in [c]} \in {\mathbb R}^{a b c}_{\geq 0} \;:\; \sum_{l,m,n} x_{l,m,n} = k \Big\}.$$ Moreover, the dual map $F^* \colon \Lambda^*_{\operatorname{U}(abc)} \rightarrow \Lambda^*_{\operatorname{U}(a) \times \operatorname{U}(b) \times \operatorname{U}(c)}$ as define
| 1,734
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f $f$ is indispensable.
The fiber product $X\times_YX\subset X\times X$ defines an equivalence relation on $X$, and one might hope to reconstruct $Y$ as the quotient of $X$ by this equivalence relation. Our main interest is in the cases when $f$ is not flat. A typical example we have in mind is when $Y$ is not normal and $X$ is its normalization. In these cases, the fiber product $X\times_YX$ can be rather complicated. Even if $Y$ and $X$ are pure dimensional and CM, $X\times_YX$ can have irreducible components of different dimension and its connected components need not be pure dimensional. None of these difficulties appear if $f$ is flat [@MR0232781; @sga3] or if $Y$ is normal (\[quot.pure.dim.lem\]).
The aim of this note is to give many examples, review known results, pose questions and to prove a few theorems concerning finite equivalence relations.
Definition of equivalence relations
===================================
\[eq.rel.defn\] Let $X$ be an $S$-scheme and $\sigma:R\to X\times_SX$ a morphism (or $\sigma_1,\sigma_2:R\rightrightarrows X$ a pair of morphisms). We say that $R$ is an [*equivalence relation*]{} on $X$ if, for every scheme $T\to S$, we get a (set theoretic) equivalence relation $$\sigma(T):{\operatorname{Mor}}_S(T,R)\into
{\operatorname{Mor}}_S(T,X)\times {\operatorname{Mor}}_S(T,X).$$ Equivalently, the following conditions hold:
1. $\sigma$ is a monomorphism (\[monom.defn\])
2. (reflexive) $R$ contains the diagonal $\Delta_X$.
3. (symmetric) There is an involution $\tau_R$ on $R$ such that $\tau_{X\times X}\circ\sigma\circ\tau_R=\sigma$, where $\tau_{X\times X}$ denotes the involution which interchanges the two factors of $X\times X$.
4. (transitive) For $1\leq i<j\leq 3$ set $X_i:=X$ and let $R_{ij}:=R$ when it maps to $X_i\times_SX_j$. Then the coordinate projection of $R_{12}\times_{X_2}R_{23}$ to $X_1\times_SX_3$ factors through $R_{13}$: $$R_{12}\times_{X_2}R_{23}\to R_{13}\stackrel{\pi_{13}}{\longrightarrow}
X_1\times_SX_3.$$
We say that $\sigma_1,\sigma_2:R\rightrigh
| 1,735
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| 1,627
| 1,907
| 0.78451
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thcal{B}$. Then $$q=a^mb^n=(a^0b^m)^{-1}(a^0b^n),$$ so that $R_1$ is a straight left I-order in $\mathcal{B}$. In fact, it is a special case of Clifford’s result, mentioned in the Itroduction.
\[rems\] Any subsemigroup of $\mathcal{B}$ that contains $R_1$ is a straight left I-order in $\mathcal{B}$.
\[identity\] Let $S$ be a left I-order in $\mathcal{B}$. Then for any $\mathcal{L}$-class $L_k$ of $\mathcal{B}$, $S\cap L_k\neq \emptyset$.
Let $k\in \mathbb{N}^0$. Then
$$\begin{array}{rcl}a^kb^k&=&(a^ib^j)^{-1}(a^mb^n)\\ &=&a^jb^ia^mb^n\\ &=&
a^{j-i+t}b^{n-m+t}\end{array}$$ where $t=$max$\{i,m\}$, for some $a^ib^j,a^mb^n\in S$. Hence $k=j-i+t=n-m+t$, so that either $k=j$ or $k=n$. Thus $S\cap L_k\neq \emptyset$.
We conclude this section by the following lemma which plays a significant role in the next sections.
\[dimpact\] Let $S$ be a left I-order in $\mathcal{B}$ and let $d\in \mathbb{N}$. If for all $a^ib^j \in S$ we have $d|i-j$, then $d=1$.
Let $a^kb^l\in \mathcal{B}$. Then there exist $a^ib^j,a^mb^n \in S$ with
$$\begin{array}{rcl} a^kb^l&=&(a^ib^j)^{-1}(a^mb^n)\\ &=&a^jb^ia^mb^n\\ &=& a^{j-i+t}b^{n-m+t} \end{array}$$ where $t=$max$\{i,m\}$. Now $$\begin{array}{rcl} k-l&=&(j-i+t)-(n-m+t)\\ &=&(j-i)+(m-n)\equiv 0 (\mbox{mod} \ d).\end{array}$$ It follows that $d=1$.
Upper subsemigroups {#leftiupper}
===================
In this section we give necessary and sufficient conditions for an upper subsemigroup $S$ of $\mathcal{B}$ to be a left I-order in $\mathcal{B}$. The upper subsemigroups of $\mathcal{B}$ are those having all elements above the diagonal; that is, all elements satisfy: $a^{i}b^{j}, j \geq i$. Throughout this section $S$ is an upper subsemigroup of $\mathcal{B}$ having the form (3).($i$) in Proposition \[subbicyclic\]. We have already met one of them, which is the $\mathcal{R}$-class of the identity. By Lemma \[identity\], we deduce that any left I-order upper subsemigroup is a monoid.
The next example is of a subsemigroup bigger than $\mathcal{R}$-class of the identity. In fact, i
| 1,736
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}}\left[(1-p/\alpha) + e^{\frac{-\alpha d \log n}{2}}(1+p/\alpha)\right].$$ Suppose we choose $d$ such that $d > (p - \alpha)^{-1}$. Then $\gamma = o(n^{-1/2})$, which implies that $\|P^n - U^n\| = o(1)$. On the other hand, if $d < (p - \alpha)^{-1}$, then $\gamma = \omega(n^{-1/2})$ and there exists a constant $\epsilon$ such that $\|P^n - U^n\| \geq \epsilon > 0$. This shows that the walk mixes in time $\Theta(n \log n)$ when $p > 4k$. Notice that when $p = 4k$, $(p - \alpha)^{-1} = (4k)^{-1}$, so that the same technique easily extends to that case via equation (\[pequals4k\]). This completes the proof of Theorem 3.
### Quantum Zeno effect for large $p$
Recall from the previous section that the time required to mix when $p > 4k$ is $$t \geq \frac{n~ \log n}{p - \alpha}$$ which clearly increases with $p$. Further, for large $p$, $p/\alpha$ tends to $1$, and hence $\gamma$ tends to $1/2.$ Notice that $\gamma = 1/2$ corresponds to remaining at the initial state forever. We conclude that the mixing of the walk is retarded by the quantum Zeno effect, where measurement occurs so often that the system tends to remain in the initial state.
Alexander Russell gratefully acknowledges the support of the National Science Foundation, under the grants CAREER CCR-0093065, CCR-0220264, EIA-0218443, and ARO grant W911NF-04-R-0009. The authors are grateful to Viv Kendon for helpful suggestions.
[15]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, in ** (, ), pp. , .
, , , , , , in ** (, ), pp. , .
(), .
, in ** (), .
, ****, (), .
, in ** (, ), pp. , .
, , , **** (), .
, , , , in ** (, ), pp. , .
, in ** (, ), pp. , .
, ****, (), .
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, in ** (), .
, .
---
abstract: 'We study the process of single ionization of Li in collisions with H$^+$ and O$^{8+}$ projectile ions at 6 MeV and 1.5-MeV/amu impact energies, respectively. Using the frameworks of the independe
| 1,737
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property due to the input restrictions.
One can easily verify that the while loop in algorithm **** is iterated $$|u|-\sum_{i=1}^{h}(|c(w_{i},P_{n}(w_{1}w_{2}\ldots w_{i-1}))|-1)$$ times, where $w=w_{1}w_{2}\ldots w_{h}$, and $P_{n}$ is the function given in Theorem 2.1.
In practice, we can use only adaptive codes satisfying the equality ${\it Prefix}(c)={\it True}$, since designing a decoding algorithm for the other case requires additional information and more complicated techniques.
Data Compression using Adaptive Codes
=====================================
The construction of adaptive codes requires different approaches, depending on the structure of the input data strings. In this section, we focus on data compression using adaptive codes of order one.
Let $\Sigma$ be an alphabet and $w=w_{1}w_{2}\ldots w_{h}\in\Sigma^{\geq{2}}$, with $w_{i}\in\Sigma$, for all $i\in\{1,2,\ldots,h\}$. A subword $uu$ of $w$, with $u\in\Sigma$, is called a of w.
Let $\Sigma$ be an alphabet and $w=w_{1}w_{2}\ldots w_{h}\in\Sigma^{\geq{2}}$. It is useful to consider the following notations:
1. ${\it Pairs}(w)=\{i \mid 1\le i\le |w|-1, w_i=w_{i+1}\}$,
2. ${\it NRpairs}(w)=|{\it Pairs}(w)|$,
3. ${\it Prate}(w)=\frac{{\it NRpairs}(w)}{|w|}$.
The main goal of this section is to design an algorithm for constructing adaptive codes of order one, under the assumption that the input data strings have a large number of pairs. Let $\Sigma=\{\sigma_{1},\sigma_{2},\ldots,\sigma_{h}\}$ and $\Delta=\{0,1\}$ be alphabets, $c\in{{\it AC}(\Sigma,\Delta,1)}$ an adaptive code of order one, and $w\in\Sigma^{+}$. We denote by $A_{c}$ the matrix given by:
$
A_{c} =
\left(\begin{array}{ccccc}
c(\sigma_{1},\sigma_{1}) & c(\sigma_{1},\sigma_{2}) & \ldots & c(\sigma_{1},\sigma_{h}) & c(\sigma_{1},\lambda) \\
c(\sigma_{2},\sigma_{1}) & c(\sigma_{2},\sigma_{2}) & \ldots & c(\sigma_{2},\sigma_{h}) & c(\sigma_{2},\lambda) \\
& & \ldots & & \\
| 1,738
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|
depends on the thermal drag such that it vanishes at zero temperature. But it also depends non-trivially on the size of the impurity and it diverges in the limit of point-like particle. Faxén corrections involving derivatives of the unperturbed flow are not present here because of the decoupling between $\delta\psi_0$ and $\delta\psi_1$ arising in the perturbative approach leading to (\[eq:psi0lin\])-(\[eq:psi1lin\]).
Numerical results {#sec:numerics}
=================
{width="\textwidth"}
To test the analytical predictions of the inertial force and the self-induced drag deduced above from the total force expression Eq. (\[eq:fp2\]), we performed numerical simulations of the dGPE. Actually, our simulations are done in the co-moving frame of the impurity moving at constant velocity ${\boldsymbol{V}}_p$, so that the equation we solve is (see numerical details in the Appendix): $$\begin{aligned}
&\partial_t \psi - {\boldsymbol{V}}_p \cdot \nabla \psi =(i+\gamma)\left[\frac{1}{2}\nabla^2 \psi +\left(1-g_p\mathcal U_p - |\psi|^2\right)\psi\right], \nonumber \\
\label{eq:ComovingdGPE}\end{aligned}$$ where the impurity is described by the Gaussian potential of intensity $g_p=0.01$ and effective size $a=\xi=1$, and is situated in the middle of the domain with the coordinates $x/\xi= 128$ and $y/\xi= 64$. As an initial condition, we start with the condensate being at rest and in equilibrium with the impurity. This is done by imaginary time integration of Eq. (\[eq:ComovingdGPE\]) for $V_p=0$ and $\gamma=0$. Then, at $t=0$, we solve the full Eq. (\[eq:ComovingdGPE\]), and as a consequence, sound waves are emitted from the neighborhood of the impurity (the size of the impurity is below the critical size for vortex nucleation). Their speed is determined by the dispersion relation $\omega({\boldsymbol{k}})$ giving the frequency as a function of the wavenumber and can be obtained by looking for plane-wave solutions to Eq. (\[eq:psi0lin\]). If $\gamma=0$, $\omega({\boldsymbol{k}})$ is given by th
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s the energy of two monomers in contact, it turns out that the two chains cannot exist independently, even for extremely weak attraction ($|\epsilon_c|\ll k_BT$). Therefore, we define additional weight factor $t=e^{-\epsilon_t/k_BT}$, where $\epsilon_t>0$ is the energy associated with two sites, visited by different SAWs, and both neighbouring a crossing site (see figure \[fig:interakcije\](b)), so that unbinding transition can occur. To describe exactly all possible configurations of the two-chain polymer system, within this model we need to introduce nine restricted partition functions: $A^{(r)}$, $B^{(r)}$, $C^{(r)}$, $A_1^{(r)}$, $A_2^{(r)}$, $A_3^{(r)}$, $A_4^{(r)}$, $B_1^{(r)}$, and $B_2^{(r)}$.
![The six restricted generating functions used in the description of all possible inter-chain configurations for the CSAWs model of the two-polymer system, within the $r$-th stage of 3D SG fractal structure. The 3D chain is depicted by green line, while the 2D surface-adhered chain is depicted by yellow line.[]{data-label="figure4"}](figure4.eps)
Functions $A^{(r)}$, $B^{(r)}$ and $C^{(r)}$, which correspond to one-polymer configurations are the same as in the ASAWs model (see figure \[fig:RGparametri\], and RG relations (\[eq:RGA\]) and (\[eq:RGB\])), whereas the remaining six functions, which describe the inter-chain configurations, are depicted in figure \[figure4\], and they are defined as $$\begin{aligned}
A_i^{(r)}&=& \sum_{N_2,N_3,L,M,K}{\mathcal A}_i^{(r)}(N_2,N_3,L,M,K) x_2^{N_2}x_3^{N_3}u^L w^M t^K\, , \quad i=1,2,3,4\, ,\nonumber\\
B_i^{(r)}&=&\sum_{N_2,N_3,L,M,K}{\mathcal B}_i^{(r)}(N_2,N_3,L,M,K) x_2^{N_2}x_3^{N_3}u^L w^M t^K\, , \quad i=1,2\, ,\nonumber\end{aligned}$$ where ${\mathcal A}_i^{(r)}$ and ${\mathcal B}_i^{(r)}$ are the numbers of particular two-polymer configurations on the $r$-th fractal structure. For instance, ${\mathcal A}_4^{(r)}(N_2,N_3,L,M,K)$ is the number of configurations in which the $N_3$-step $P_3$ chain (with $L$ intra-chain contacts) and $N_2$-step $P_2$ chain (with diff
| 1,740
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|
ation of the PSF-DW and HI-MI transition we have additionally analyzed the energy-level-crossings with, respectively, periodic and twisted-boundary conditions. The KT transitions from SF to MI and HI have been determined by the extraction of the Luttinger parameter $K$ [@commentLuttinger].](bh3_pd.eps){width="8.0cm"}
\[fig:3\]
Small $U<0$ disfavors singly-occupied sites and thus enhances ${\cal O}_S^2$ and the bulk excitation gap of the HI phase (see Figs. \[fig:4\] and \[fig:5\]). However, since large $U<0$ removes singly occupied sites completely, just like strong nearest neighbour repulsion, it is expected that the HI phase eventually will transform for growing $|U|$ into a gapped density-wave (DW) phase via Ising phase transition [@DallaTorre2006], and string order will evolve into DW order (Fig. \[fig:4\] shows how ${\cal O}_s$ merges with ${\cal O}_{DW}\equiv \lim_{j\rightarrow\infty}(-1)^j \langle n_in_{i+j}\rangle$ for $U/t<-3$). The DW phase is characterized by an exponential decay of both $G_{ij}$ and $G_{ij}^{(2)}$ though a finite ${\cal O}_{DW}$. Our DMRG results confirm this scenario (see Fig. \[fig:4\]), showing that a DW phase is located between the above mentioned PSF regions (Fig. \[fig:3\]).
Interestingly the DW phase remains in between both PSF regions all the way into $U\rightarrow-\infty$. In that regime, we may project out singly-occupied sites, and introduce a pseudo-spin-$1/2$, identifying $|0 \rangle\to|\!\!\downarrow \rangle , |2\rangle\to|\!\!\uparrow \rangle $, and defining the spin operators $\tau_i^-\to (-1)^i b_i^2/\sqrt{2}$, $2\tau_i^z\to b_i^{\dagger}b_i-1$. The effective model to leading order in $1/|U|$ is a spin-$1/2$ chain: $$\label{effectivespinhalf}
H_{\frac{1}{2}}\!=\!J \!\sum_i \!\!\left[{{\boldsymbol}\tau}_i{{\boldsymbol}\tau}_{i+1}\!+\! j^2(\tau_i^z
\tau_{i+2}^z\!-\!\tau_i^x \tau_{i+2}^x\!-\!\tau_i^y \tau_{i+2}^y)
\right],$$ where $J={t^2}/{|U|}$. For $j=0$, this is a $SU(2)$ symmetric chain, whereas the $j^2J$ terms break the symmetry down to $U(1)$, movi
| 1,741
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[Energy Bounds]{} \[ss:energy\_bounds\]
\[l:energy\_x\] Consider $x_t$ as defined in . If $x_0$ satisfies $\E{\|x_0\|_2^2} \leq R^2 + \frac{\beta^2}{m}$, then Then for all $t$, $$\begin{aligned}
\E{\|x_t\|_2^2} \leq 6\lrp{R^2 + \frac{\beta^2}{m}}\\
\end{aligned}$$ We can also show that $$\begin{aligned}
\Ep{p^*}{\lrn{x}_2^2} \leq 4\lrp{R^2 + \frac{\beta^2}{m}}
\end{aligned}$$
We consider the potential function $a(x) = \lrp{\|x\|_2 - R}_+^2$ We verify that $$\begin{aligned}
\nabla a(x) =& (\|x\|_2 - R)_+ \frac{x}{\|x\|_2}\\
\nabla^2 a(x) =& \ind{\|x\|_2 \geq R} \frac{xx^T}{\|x\|_2^2} + \frac{(\|x\|_2 - R)_+}{\|x\|_2}\lrp{I - \frac{xx^T}{\|x\|_2^2}}
\end{aligned}$$ Observe that
1. $\lrn{\nabla^2 a(x)}_2 \leq 2 \ind{\|x\|_2 \geq R} \leq 2$
2. $\lin{\nabla a(x), - \nabla U(x)} \leq -m a(x)$. This can be verified by considering 2 cases. If $\|x\|_2 \leq R$, then $\nabla a(x) = 0$ and $a(x) = 0$. If $\|x\|_2 \geq R$, then by Assumption \[ass:U\_properties\], $$\begin{aligned}
\lin{\nabla a(x), -\nabla U(x)}
\leq - m \lrp{\|x\|_2 - R}_+ \|w\|_2
\leq - m \lrp{\|x\|_2 - R}_+^2 = -m \cdot a(x)
\end{aligned}$$
3. $a(x) \geq \frac{1}{2}\|x\|_2^2 - 2R^2$. One can first verify that $a(x) \geq (\lrn{x}_2 - R)^2 - R^2$. Next, by Young’s inequality, $(\lrn{x}_2 - R)^2 = \lrn{x}_2^2 + R^2 - 2\lrn{x}_2 R \geq \lrn{x}_2^2 + R^2 - \frac{1}{2} \lrn{x}_2^2 - 2R^2 = \frac{1}{2} \lrn{x}_2^2 - R^2$.
Therefore, $$\begin{aligned}
& \ddt \E{a(x_t)}
= \E{\lin{\nabla a(x_t), -\nabla U(x_t) dt}} + \frac{1}{2}\E{\tr\lrp{M(x_t)^2 \nabla^2 a(x)}}
\leq -m\E{a(x_t)} + \beta^2\\
\Rightarrow \qquad &
\ddt \lrp{\E{a(x_t)} - \frac{\beta^2}{m}} \leq - m \lrp{\E{a(x_t)} - \frac{\beta^2}{m}}\\
\Rightarrow \qquad &
\ddt \lrp{\E{a(x_t)} - R^2 - \frac{\beta^2}{m}} \leq - m \lrp{\E{a(x_t)} - R^2 - \frac{\beta^2}{m}}
\end{aligned}$$
Thus if $\E{\|x_0\|_2^2} \leq R^2
| 1,742
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its from the Configuration Element class.
UPDATE:
The above code sample I pasted in allows you to edit what already exists in the config file inside your custom section. In order to add a new item for example like the following:
FavsSection favconfig = (FavsSection)config.GetSection("FavouritesMenu");
ToolStripMenuItem menu = (ToolStripMenuItem)returnMenuComponents("favouritesToolStripMenuItem", form);
ToolStripItemCollection items = menu.DropDownItems;
for (int i = 0; i < items.Count; i++)
{
//favconfig.FavsItems[i].ID = i.ToString();
//favconfig.FavsItems[i].Path = items[i].Text;
favconfig.FavsItems[i] = new FavouriteElement()
{
ID = i.ToString(),
Path = items[i].Text
};
}
As you can see above, I am physically adding a new 'FavouriteElement' object into the collection returned by the property 'favconfig.FavItems'. In order to to do this, one property needs extending to support this.
public FavouriteElement this[int idx]
{
get
{
return (FavouriteElement)BaseGet(idx);
}
set
{
base.BaseAdd(value);
}
}
This indexer or paramterful property as 'Jeffrey Richter' calls them needs to have it's 'Set' accessor implemented as shown above in the code snippet. I have pasted it in here as it did not take long to figure out and most of the code is changed using a template I have used from Derik Whittaker's Article. Hopefully this will alow other coders to implement something similar.
Another solution would be to simply rather than 'getting' the collection all the time that 'lassoes' together all my 'FavouriteElements', you could implement the 'set' accessor for the related property. I have not tested this but I might be worth trying out.
Q:
ZF3 PhpRenderer not finding template path
I know there are dozen of questions about PHPRenderer not finding the path of a template, but I think the prob
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$\begin{aligned}
\label{eq:outthsin}
R_{{\widehat{\psi}}}^{{\mathrm{out}}}=\max~R,~~\mathrm{s.t.}~~{\mathbb{P}}(R_{{\widehat{\psi}}}<R)\le\epsilon\end{aligned}$$ and $$\begin{aligned}
\label{eq:outthrec}
R_{\psi}^{{\mathrm{out}}}=\max~R,~~\mathrm{s.t.}~~{\mathbb{P}}(R_{\psi}<R)\le\epsilon,
\end{aligned}$$ respectively. The outage throughput gain of employing the reconfigurable antennas at an outage level $\epsilon$ is given by $$\label{eq:Defth_gain_out}
G_{R^{\mathrm{out}}}={R_{{\widehat{\psi}}}^{{\mathrm{out}}}}/{R_{\psi}^{{\mathrm{out}}}}, $$ where $R_{{\widehat{\psi}}}^{{\mathrm{out}}}$ and $R_{\psi}^{{\mathrm{out}}}$ are given in and , respectively. The outage throughput gain of employing the reconfigurable antennas is given in the following proposition.
\[Prop:2\] The outage throughput gain of employing the reconfigurable antennas with $\Psi$ distinct reconfiguration states is approximated by $$\label{eq:th_gain_out}
G_{R^{\mathrm{out}}}\approx\frac{{\bar{R}_{\psi}}-\sqrt{2{\sigma^2_{R_\psi}}}\mathrm{erf}^{-1}\left(1-2\epsilon^{\frac{1}{\Psi}}\right)}{{\bar{R}_{\psi}}-\sqrt{2{\sigma^2_{R_\psi}}}\mathrm{erf}^{-1}\left(1-2\epsilon\right)}.$$
See Appendix \[App:proofoutGa\]
### $\Psi$ Is Large
We now investigate the limiting performance gain of reconfigurable antennas in terms of the outage throughput as $\Psi\rightarrow\infty$. Based on , we present the limiting behavior of the outage throughput gain when $\Psi$ becomes large in the following corollary.
\[Cor:outGlsngrO\] As $\Psi\rightarrow\infty$, $G_{R^{\mathrm{out}}}(\Psi)$ is asymptotically equivalent to $$\label{eq:goutlarpsiasy}
G_{R^{\mathrm{out}}}(\Psi)\sim\frac{\sqrt{2{\sigma^2_{R_\psi}}}}{{\bar{R}_{\psi}}-\sqrt{2{\sigma^2_{R_\psi}}}\mathrm{erf}^{-1}\left(1-2\epsilon\right)}\sqrt{\ln(\Psi)}.
$$
See Appendix \[App:proofCor:outGlsngrO\]
Similar to the finding for the average throughput gain, we find from that $G_{R^{\mathrm{out}}}(\Psi)=O\left(\sqrt{\ln(\Psi)}\right)$ as $\Psi\rightarrow\infty$. Thus, the growth of the outage throu
| 1,744
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|
------ -- -- -- --
Buda-Lund
v1.5
0 - 30 % 0 - 5(6) %
$T_0$ \[MeV\] 200 $\pm$ 9 214 $\pm$ 7
$T_{\mbox{\rm e}}$ \[MeV\] 127 $\pm$ 13 102 $\pm$ 11
$\mu_B$ \[MeV\] 61 $\pm$ 40 77 $\pm$ 38
$R_{G}$ \[fm\] 13.2 $\pm$ 1.3 28.0 $\pm$ 5.5
$R_{s}$ \[fm\] 11.6 $\pm$ 1.0 8.6 $\pm$ 0.4
$\langle u_t^\prime \rangle$ 1.5 $\pm$ 0.1 1.0 $\pm$ 0.1
$\tau_0$ \[fm/c\] 5.7 $\pm$ 0.2 6.0 $\pm$ 0.2
$\Delta\tau$ \[fm/c\] 1.9 $\pm$ 0.5 0.3 $\pm$ 1.2
$\Delta\eta$ 3.1 $\pm$ 0.1 2.4 $\pm$ 0.1
$\chi^2/\mbox{\rm NDF}$ 132 / 208 158.2 / 180
------------------------------ ------ ----------- ------- ----------- -- -- -- --
: The first column shows the source parameters from simultaneous fits of final BRAHMS and PHENIX data for 0 - 30 % most central $Au+Au$ collisions at $\sqrt{s_{\NN}} = 200$ GeV, as shown in Figs. 1 and 2, as obtained with the Buda-Lund hydro model, version 1.5. The errors on these parameters are still preliminary. The second column is the result of an identical analysis of BRAHMS, PHENIX, PHOBOS and STAR data for 0 - 5 % most central Au+Au collisions at $\sqrt{s_{\NN}}=130$ GeV, ref. [@ster-ismd03]. []{data-label="tab:results"}
![ \[fig:spectra\] [Solid line shows the simultaneous Buda-Lund v1.5 fit to final Au+Au data at $\sqrt{s_{\NN}} = 200$ GeV. The transverse mass distributions of identified particles are measured by PHENIX [@Adler:2003cb] the pseudorapidity distributions of
| 1,745
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after a new part annotation as $$\begin{split}
\tilde{\bf Q}(y=+1|I')=&\frac{1}{Z}\exp[\beta S_{top,I'}^{\textrm{new}}|_{\tilde{I}}]\\
S_{top,I'}^{\textrm{new}}|_{\tilde{I}}=&S_{top,I'}+\Delta S_{top,\tilde{I}}e^{-\alpha\cdot dist(I',\tilde{I})}\!\!\!\!\!\!\!\!\!\!
\end{split}
\label{eqn:predict}$$ where $S_{top,I'}$ indicates the current AOG’s inference score of $S_{top}$ on image $I'$. $S_{top,I'}^{\textrm{new}}|_{\tilde{I}}$ denotes the predicted inference score of $I'$ when people annotate $\tilde{I}$. We assume that if object $I'$ is similar to object $\tilde{I}$, the inference score of $I'$ will have an increase similar to that of $\tilde{I}$. [$\Delta S_{top,\tilde{I}}\!=\!\mathbb{E}_{I\in{\bf I}^{\textrm{ant}}}S_{top,I}-S_{top,\tilde{I}}$]{} denotes the score increase of $\tilde{I}$. $\alpha$ is a scalar weight. We formulate the appearance distance between $I'$ and $\tilde{I}$ as [$dist(I',\tilde{I})\!=\!1-\frac{\phi(I')^{T}\phi(\tilde{I})}{\vert\phi(I')\vert\cdot\vert\phi(\tilde{I})\vert}$]{}, where [$\phi(I')\!=\!{\bf M}\,{\bf f}_{I'}$]{}. ${\bf f}_{I'}$ denotes features of $I'$ at the top conv-layer after ReLU operation, and ${\bf M}$ is a diagonal matrix representing the prior reliability for each feature dimension[^2]. In addition, if $I'$ and $\tilde{I}$ are assigned with different part templates by the current AOG, we set an infinite distance between $I'$ and $\tilde{I}$ to achieve better performance. Based on Equation (\[eqn:predict\]), we can predict the changes of the KL divergence after the new annotation on $\tilde{I}$ as $$\Delta{\bf KL}(\tilde{I})=\lambda{\sum}_{I\in{\bf I}^{\textrm{obj}}}{\sum}_{y}{\bf P}(y|I)\log\frac{\tilde{\bf Q}(y|I)}{{\bf Q}(y|I)}
\label{eqn:delta}$$ Thus, in each step, our method selects and asks about the object that decreases the KL divergence the most. $$\hat{I}={\arg\!\max}_{I\in{\bf I}^{\textrm{unant}}}\Delta{\bf KL}(I)
\label{eqn:select}$$
**QA implementations:**[` `]{} In the beginning, for each object $I$, we initialize [${\bf P}(y\!=\!+1|I)\!=\!1$]{} and
| 1,746
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| 1,874
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| 0.782054
|
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|
\]. We take $\eta >\kappa +d$ and, using $(A_{4})$ (see ([R7]{})) we obtain $$\left\vert \mu ^{\eta ,\kappa }\right\vert =\int_{{\mathbb{R}}^{2}}\frac{%
\psi _{\eta }(x)}{\psi _{\kappa }(y)}P_{t}(x,dy)dx\leq C\int_{{\mathbb{R}}}%
\frac{dx}{\psi _{\kappa -\eta }(x)}<\infty .$$Then, $A(\delta )<C$ (see (\[reg12’\])). One also has $B(\varepsilon
)<\infty $ (see (\[reg12”\])) and finally (see (\[reg11\])) $$C_{h,n_{\ast }}(\varepsilon )\leq Ct^{-\theta _{0}\xi _{3}}\Phi
_{n}^{m_{0}(1+\varepsilon _{\ast })}(\delta _{\ast }).$$We have used here (\[R8’\]). For large $h$ we also have $$\theta (n)^{\rho _{h}}\leq C(\lambda _{n}t)^{-\theta _{0}((a+b)m_{0}+q+\frac{%
2d}{p_{\ast }})(1+\varepsilon _{\ast })}\Phi _{n}^{\varepsilon _{\ast }}(0).$$
Now (\[reg12\]) gives (\[TR6’\]). So **A** and **B** are proved.
**Step 5: proof of C.** We apply **B.** with $q$ replaced by $\bar{%
q}=q+1$, so $\Psi _{\eta ,\kappa }p_{t}\in W^{\bar{q},p}({\mathbb{R}}^{d}\times {%
\mathbb{R}}^{d})=W^{\bar{q},p}({\mathbb{R}}^{2d})$. Since $\bar{q}>2d/p$ (here the dimension is $2d$), we can use the Morrey’s inequality: for every $%
\alpha $, $\beta $ with $|\alpha |+|\beta |\leq \lfloor\bar{q}- 2d/p\rfloor
=q$, then $|\partial _{x}^{\alpha }\partial _{y}^{\beta }(\Psi _{\eta
,\kappa }p_{t})(x,y)|\leq C\Vert \Psi _{\eta ,\kappa }p_{t}\Vert _{\bar{q},p}
$. By (\[TR6’\]), one has (with $\bar{\mathfrak{m}}=1+\frac{\bar{q}+2d/p_{\ast }}{\delta _{\ast }}$) $$\left\vert \partial _{x}^{\alpha }\partial _{y}^{\beta }(\Psi _{\eta ,\kappa
}p_{t})(x,y)\right\vert \leq C\Big(1+\Big(\frac{1}{\lambda _{n}t}\Big)^{(a+b)%
\bar{\mathfrak{m}}+\bar{q}+2d/p_{\ast }}+\Phi _{t,n,r}^{\bar{\mathfrak{m}}}(\delta
_{\ast })\Big)^{(1+\varepsilon _{\ast })}$$i.e. (using (\[NOT3c\])), $$\left\vert \partial _{x}^{\alpha }\partial _{y}^{\beta
}p_{t}(x,y)\right\vert \leq C\Big(1+\Big(\frac{1}{\lambda _{n}t}\Big)^{(a+b)%
\bar{\mathfrak{m}}+\bar{q}+2d/p_{\ast }}+\Phi _{t,n,r}^{\bar{\mathfrak{m}}}(\delta
_{\ast })\Big)^{(1+\varepsilon _{\ast })}\times \
| 1,747
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|
se set in $QP(3,\Bbb{C})=(M(3,\Bbb{C})\setminus\{0 \})/\Bbb{C}^*$ called in [@CS] the space of pseudo-projective maps. Let $\widetilde{M}:\mathbb{C}^{3}\rightarrow\mathbb{C}^{3}$ be a non-zero linear transformation. Let $Ker(\widetilde M)$ be its kernel and $Ker([[\widetilde M]])$ denote its projectivization. Then $\widetilde{M}$ induces a well defined map $[[\widetilde M]]:\mathbb {P}^{2}_\mathbb {C}\setminus Ker([[\widetilde M]]) \rightarrow\mathbb {P}^{2}_\mathbb {C}$ by $$[[\widetilde M]]([v])=[\widetilde M(v)].$$ The following result provides a relation between convergence in $QP(3,\Bbb{C})$ viewed as points in a projective space and the convergence viewed as functions.
\[See [@CS]\] \[p:completes\] Let $(\gamma_m)_{m\in \mathbb {N}}\subset \PSL(3,\mathbb {C})$ be a sequence of distinct elements. Then:
1. There is a subsequence $(\tau_m)_{m\in \mathbb {N}}\subset(\gamma_m)_{m\in\mathbb{N}}$ and a $\tau_0\in M(3,\Bbb{C})\setminus\{0\}$ such that $\tau_m\xymatrix{\ar[r]_{m\rightarrow\infty}&}\tau_0$ as points in $QP(3,\Bbb{C})$.
2. If $(\tau_m)_{m\in \mathbb {N}}$ is the sequence given by the previous part of this lemma, then $\tau_m\xymatrix{\ar[r]_{m\rightarrow\infty}&}\tau_0$, as functions, uniformly on compact sets of $\mathbb{P}^n_\mathbb{C}\setminus Ker(\tau_0)$. Moreover, the equicontinuity set of $\{\tau_m\vert m\in \Bbb{N} \}$ is $\Bbb{P}^n\setminus Ker(\tau_0)$.
Kulkarni’s limit set
--------------------
When we look at the action of a group acting on a general topological space, in general there is no natural notion of limit set. A nice starting point is Kulkarni’s limit set.
\[d:lim\] Let $\Gamma\subset\PSL(n+1,\mathbb{C})$ be a subgroup. We define
1. the set $\Lambda(\Gamma)$ to be the closure of the set of cluster points of $\Gamma z$ as $z$ runs over $\mathbb{P}^n_{\mathbb{C}}$,
2. the set $L_2(\Gamma)$ to be the closure of cluster points of $\Gamma K$ as $K$ runs over all the compact sets in $\mathbb{P}^n_{\mathbb{C}}\setminus \Lambda(\Gamma)$,
3. and *Kulkarni’s limit set* o
| 1,748
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|
}(v_1,v'_2)=\raisebox{-9pt}{\includegraphics[scale=.1]
{P1}}\qquad
P_\Lambda^{{\scriptscriptstyle}(3)}(v_1,v'_3)=\raisebox{-15pt}{\includegraphics[scale=.1]
{P2}}\\[5pt]
P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1)=\raisebox{-7pt}{\includegraphics
[scale=.1]{P0p}}\hspace{5pc}
P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1)=\raisebox{-18pt}
{\includegraphics[scale=.1]{P0pp}}+~
\raisebox{-18pt}{\includegraphics[scale=.1]{P0pp2}}\\[5pt]
P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y,x)=\raisebox{-12pt}{\includegraphics
[scale=.1]{P0prime}}\hspace{5pc}
P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)=\raisebox{-18pt}
{\includegraphics[scale=.1]{P0primeprime}}\end{gathered}$$
Next, we define $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(j)}}(v_1,v'_j)$ by replacing one of the $2j-1$ two-point functions on the right-hand side of [(\[eq:P1-def\])]{}–[(\[eq:Pj-def\])]{} by the product of *two* two-point functions, such as replacing ${{\langle \varphi_z\varphi_{z'} \rangle}}_\Lambda$ by ${{\langle \varphi_z\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_{z'} \rangle}}_\Lambda$, and then summing over all $2j-1$ choices of this replacement. For example, we define (see the second line in Figure \[fig:P-def\]) $$\begin{aligned}
{\label{eq:P'1-def}}
P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1)=2\big(\psi_\Lambda(v_1,v'_1)-
\delta_{v_1,v'_1}\big){{\langle \varphi_{v_1}\varphi_u \rangle}}_\Lambda{{\langle \varphi_u
\varphi_{v'_1} \rangle}}_\Lambda,\end{aligned}$$ and $$\begin{aligned}
P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(2)}}(v_1,v'_2)=\sum_{v_2,v'_1}\bigg(\prod_{i
=1}^2\big(\psi_\Lambda(v_i,v'_i)-\delta_{v_i,v'_i}\big)\bigg)\Big({{\langle
\varphi_{v_1}\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_{v_2} \rangle}}_\Lambda
{{\langle \varphi_{v_2}\varphi_{v'_1} \rangle}}_\Lambda{{\langle \varphi_{v'_1}\varphi_{
v'_2} \rangle}}_\Lambda&{\nonumber}\\
+{{\langle \varphi_{v_1}\varphi_{v_2} \rangle}}_\Lambda{{\langle \varphi_{v_2}\varphi_u
\
| 1,749
| 2,735
| 1,825
| 1,768
| 1,648
| 0.787241
|
github_plus_top10pct_by_avg
|
ltaV_FRGeq}
\beta\, \partial_{t} {\Delta V}_{k} =
\frac{1}{2} \int \0{d^3 p}{(2 \pi)^3} \frac{ \partial_{t}
R_{0,k}}{Z_0\vec p{\,}^2 +
\partial^2_{A_0}( \Delta V_{k} + V_{\bot,k}) +
R_{0,k}} \,.$$ With the specific regulator $R_k$ in we can perform the momentum integration analytically. We also introduce the scalar field $\varphi = g \beta A_0$, and arrive at $$\begin{aligned}
\label{eq:preflowV}
\beta \partial_k \Delta V_k
= \frac{2}{3 (2 \pi)^2} \frac{(1+\eta_0/5) k^2 }{1+\frac{ g_{k}^2
\beta^2}{ k^2 }
\partial^2_{\varphi} ( V_{\bot,k} + \Delta V_k)}\,, \end{aligned}$$ where the coupling $g_k^2$ has to run with the effective cut-off scale $k_{\rm phys}$, and is estimated by an appropriate choice of the running coupling $\alpha_s$, $$\label{eq:runningg}
g_k^2=\frac{g^2}{Z_0}\,,\qquad {\rm with}\qquad g_k^2=4 \pi
\alpha_s(k_{\rm phys}^2)\,,$$ see also Appendix \[app:intoutAI\]. This also entails that the anomalous dimension $\eta_0$ is linked to the running coupling by $$\label{eq:eta0}
\eta_0=-\partial_t \log \alpha_s(k_{\rm phys}^2)\,.$$ At its root is an equation for the dimensionless effective potential $\hat V = \beta^4 V_k$ in terms of $\hat V_\bot=
\beta^4 V_{\bot,k}$ and $\hat \Delta V=\beta^4 \Delta V_k$. The infrared RG-scale $k$ naturally turns into the modified RG-scale $\hat
k = k \beta$, that is all scales are measured in temperature units. Then the flow equation is of the form $$\begin{aligned}
\label{eq:inter}
\partial_{\hat k} \Delta \hat{ V} = \frac{2 }{3 (2 \pi)^2} \frac{
(1+\eta_0/5) \hat k^2 }{1+\frac{ g_{k}^2 }{ \hat k^2 }
\partial^2_\varphi ( \hat{V}_{\bot} + \Delta \hat{ V})}\,.\end{aligned}$$ The potential $\hat V$ and hence $\hat\Delta V$ has a field-independent contribution which is related to the pressure. For the present purpose it is irrelevant and we can conveniently normalise the flow such that it vanishes at fields where $\partial_\varphi^2(\hat{V}_{\bot} + \Delta \hat{ V})=0$. This is achieved by subtracting $2(1+\eta_0/5)\, \hat k^2 / (3 (2\pi)^2)
| 1,750
| 3,825
| 2,031
| 1,482
| 3,210
| 0.77401
|
github_plus_top10pct_by_avg
|
) \le t' k_2$. Let $M'= M \con (F \cup E(S))$. By the maximality of $h$, every restriction of $M'$ with rank at most $t'$ is $\GF(q)$-representable. In particular, every rank-$(s-1)$ restriction of $M'$ is $\GF(q)$-representable and every nonloop $f$ of $M'$ is spanned by a $\PG(s-2,q)$-restriction $P_f$ of $M'$ contained in $E(R)$, so every such $f$ is in fact parallel to some element of $P_f$, as otherwise $M'|(E(P_f) \cup \{f\})$ is not $\GF(q)$-representable. Now $r_M(F \cup E(S)) \le r_M(F) + r_{M \con F}(E(S)) \le k_1 + s + t'k_2$, so any basis $C$ for $M|(F \cup E(S))$ will satisfy the lemma.
We now state and prove a stronger version of Theorem \[main2\].
\[maingeom\] Let $s\ge 2$ and $n \ge 2$ be integers and let $q$ be a prime power. Let $k = f_{\ref{localrep}}(s,n,s-1)$. If $M$ is a rank-$r$ matroid with a $\PG(r-1,q)$-restriction, no $U_{s,2s}$-minor, and no $\PG(n-1,q')$-minor for any $q' > q$, then there is a matroid $\wh{M}$ such that $\si(\wh{M}) \cong \PG(r-1,q)$ and $\dist(M,\wh{M}) \le 4k$. Moreover, there is a set $C \subseteq E(M)$ for which $|C| \le k$ and $\wh{M}$ is a $C$-shift of $M$.
Let $R$ be a $\PG(r-1,q)$-restriction of $M$. By Lemma \[localrep\], there exists a set $C \subseteq E(M)$ so that $|C| \le k$ and every nonloop of $M \con C$ is parallel to an element of $E(R)$. Let $\psi\colon E(M) - E(R) \to E(R) \cup \{\phi\}$ be defined so that $\psi(\cl_M(C) - E(R)) = \{\phi\}$ and each $x \in E(M) - (E(R) \cup \cl_M(C))$ is parallel to $\psi(x)$ in $M \con C$. Clearly $\psi$ is a $C$-shift and $\si(\psi(M)) \cong R$; the theorem follows.
Bicircular Cliques {#bicircsection}
==================
We now prove Theorem \[bicirc\], first showing that a spanning framed bicircular clique restriction together with a spanning projective geometry restriction gives a large uniform minor. We implicitly use the well-known fact that $B^+(H)$ is a minor of $B^+(G)$ whenever $H$ is a minor of $G$.
\[bicircpg\] There is a function $f_{\ref{bicircpg}}\colon \bZ \to \bZ$ so that, for every integer
| 1,751
| 1,528
| 783
| 1,707
| null | null |
github_plus_top10pct_by_avg
|
11 (37%) 28 (36%) 73 (42%)
Withdrawal by sponsor/ lack of funding 0 5 (23%) 0 5 (17%) 3 (4%) 2 (1%)
Benefit 0 0 0 0 6 (8%) 3 (2%)
Harm 1 (50%) 2 (95%) 0 3 (10%) 7 (9%) 17 (10%)
Futility 0 3 (14%) 0 3 (10%) 16 (21%) 21 (12%)
Other reasons[^3^](#t002fn004){ref-type="table-fn"} 0 6 (27%) 2 (33%) 8 (27%) 17 (22%) 35 (20%)
Unclear/missing 0 2 0 2 1 24
^1^ approved by 5 research ethics committees in Switzerland and Canada \[[@pone.0165605.ref001]\].
The numbers in brackets are proportions (column %) based on complete cases (excluding unclear/missing)
^2^ Please see [Table 3](#pone.0165605.t003){ref-type="table"} for proportions
^3^ Other reasons: administrative; retirement; change of institute of applicant; lack of staff resources; lack of flexibility of the system; logistical problems, i.e. interdisciplinary study organisation and logistics; number of participants too small; studies conducted by other researcher were larger and more meaningful; pilot phase failed, test material was insufficient; study data were deleted during maintenance works by the technician; change of study design; change of topic; termination of the liver transplant programme at the University Medical Center Freiburg by the government; shifting of the research focus of the study investigator
Abbreviations: NPSs, non-randomised prospective studies; RCTs, randomised controlled trials.
10.1371/
| 1,752
| 3,210
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| 1,814
| null | null |
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|
\widetilde{\Phi}_{i+1}^{mn} = 4\pi G \rho^{mn}_i$$ in uniform cylindrical coordinates, and $$\label{eq:tridiagonal_log}
\frac{1}{(R_i\ln f)^2} \widetilde{\Phi}_{i-1}^{mn} + \left[ \lambda_\phi^m + \lambda^n_z - \frac{2}{(R_i\ln f)^2} \right] \widetilde{\Phi}_i^{mn} + \frac{1}{(R_i\ln f)^2} \widetilde{\Phi}_{i+1}^{mn} = 4\pi G \rho^{mn}_i$$ in logarithmic cylindrical coordinates. Note that Equations and are tridiagonal matrix equations subject to the zero boundary conditions of $\widetilde{\Phi}_{0}^{mn} = \widetilde{\Phi}_{N_R+1}^{mn} = 0$, which can easily be solved via the Thomas algorithm involving back substitutions (e.g., @nr).
Therefore, the Poisson equation in cylindrical coordinates can be solved by the following three steps:
1. Perform a forward transform $\rho_{i,j,k}\to\rho^{mn}_i$ using Equation : ${\cal O}(N_RN_\phi N_z\log_2[N_\phi N_z])$.
2. Solve Equation or for $\widetilde{\Phi}^{mn}_i$ : ${\cal O}(N_RN_\phi N_z)$.
3. Perform a backward transform $\widetilde{\Phi}^{mn}_i\to\widetilde{\Phi}_{i,j,k}$ using Equation : ${\cal O}(N_RN_\phi N_z\log_2[N_\phi N_z])$.
In actual computation, the discrete transforms in Equations – can be carried out efficiently with an FFT algorithm. For transforms involving ${\cal P}^m_j$, we use the real-to-complex transform in [FFTW]{}, which halves the size of the output by utilizing the Hermitian symmetry. For transforms involving ${\cal Z}^n_k$, we use the sine transform as in Cartesian coordinates. For parallel computations, we employ the pencil decomposition technique along with the Steve Plimpton’s parallel transpose routines, similarly to the Cartesian solver.
CALCULATION OF THE BOUNDARY POTENTIAL {#s:bc}
=====================================
Overview of the James Algorithm {#s:James_overview}
-------------------------------
We adopt the James algorithm to calculate the boundary potential $\Phi^{\rm B}$ which is second-order accurate. As explained in Introduction, James’s method first solves for the preliminary potential $\Psi=\Phi+\Theta$ wi
| 1,753
| 919
| 1,713
| 1,939
| 3,545
| 0.771637
|
github_plus_top10pct_by_avg
|
ate as claimed.
For the original problem we get
\[m-d-co1\] Suppose that the assumptions (\[ass1\]), (\[ass2\]), (\[ass3\]), (\[evo16\]), (\[evo8-a\]) and (\[evo9-a\]) are valid with $C=\frac{\max\{q,0\}}{\kappa}$ and $c>0$. Furthermore, suppose that ${f}\in L^2(G\times S\times I)$, and $g\in H^1(I,T^2(\Gamma_-'))$ is such that \[comp-d-d\] g(,,E\_[m]{})=0. Then the problem $$\begin{gathered}
-{{\frac{\partial (S_0\psi)}{\partial E}}}+\omega\cdot\nabla_x\psi+\Sigma\psi -K\psi=f,\ \nonumber\\
{\psi}_{|\Gamma_-}=g,\quad \psi(\cdot,\cdot,E_{\rm m})=0, \label{co3aa-ddd}\end{gathered}$$ has a unique solution $\psi\in {{{\mathcal{}}}H}_P(G\times S\times I^\circ)$. In addition, there exists a constant $C_1'>0$ such that *a priori* estimate \[csda40aaa-dd\] \_[L\^2(GSI)]{}C\_1’( \_[L\^2(GSI)]{}+\_[H\^1(I,T\^2(\_-’))]{}), holds.
Recalling (see , ) that $\psi$ is a solution of the problem with data $(f,g)$ if and only if $\phi=e^{CE}\psi$ is a solution of the problem with data $({\bf f}, {\bf g})=(e^{CE}f, e^{CE}g)$, we deduce the claims from Theorem \[coth3-dd\] and Corollary \[cdd\].
On the Non-negativity of Solutions {#possol}
----------------------------------
We will outline an argument to show that the solution $\psi$ obtained by Corollary \[m-d-co1\], when the assumptions of it are valid, is non-negative if the data $f$ and $g$ are non-negative. Using a change of unknown $\phi=e^{CE}\psi$, with $C$ given in , we return the problem to the one considered in Theorem \[coth3-dd\], that is $$\begin{aligned}
{3}
-{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi+CS_0\phi+\Sigma\phi-K_C\phi={}&{\bf f} &&\quad \textrm{on}\ G\times S\times I, \label{p-s1-b} \\
\phi_{|\Gamma_-}={}&{\bf g} &&\quad \textrm{on}\ \Gamma_-,\label{p-s2-b} \\[2mm]
\phi(\cdot,\cdot,E_{\rm m})={}&0 &&\quad \textrm{on}\ G\times S. \label{p-s3-b}\end{aligned}$$ Clearly, the positivity condition on data, $f\geq 0$ and $g\geq 0$, is equivalent to having ${\bf f}\geq 0$ and ${\bf g}\geq 0$. Recall that $$P_C(x,\omega,E,D)\phi:=-{{\
| 1,754
| 378
| 852
| 1,846
| null | null |
github_plus_top10pct_by_avg
|
v\in {\mathcal{U}}(\chi )$.}
\label{eq:Shfprop}\end{aligned}$$ Moreover, by definition of ${\mathrm{Sh}}$ and ${\theta ^{\chi}} $, $$\begin{aligned}
{\mathrm{Sh}}(u,v)=0 \quad \text{if $u\in \sum _{i\in I}{\mathcal{U}}(\chi )E_i$ or
$v\in \sum _{i\in I}{\mathcal{U}}(\chi )E_i$.}
\label{eq:Shfprop2}\end{aligned}$$ Recall the definitions of $U(\chi )$, ${\mathcal{I}}^+(\chi )$ and ${\mathcal{I}}^-(\chi
)$ from Sect. \[sec:DD\]. Since ${\mathcal{U}}(\chi ){\mathcal{I}}^+(\chi ){\mathcal{U}}(\chi )+
{\mathcal{U}}(\chi ){\mathcal{I}}^-(\chi ){\mathcal{U}}(\chi )\subset \ker {\theta ^{\chi}} $, ${\theta ^{\chi}} $ and ${\mathrm{Sh}}$ induce maps $${\theta ^{\chi}} : U(\chi )\to {{\mathcal{U}}^0},\qquad
{\mathrm{Sh}}: U(\chi )\times U(\chi )\to {{\mathcal{U}}^0}.$$ The map ${\theta ^{\chi}} $ is ${\mathbb{Z}}^I$-homogeneous, that is, ${\theta ^{\chi}} (u)=0$ for all $u\in U(\chi )_{\alpha }$, where ${\alpha }\in {\mathbb{Z}}^I\setminus \{0\}$. The map ${\Omega }$ reverses degrees, that is, ${\Omega }(u)\in U(\chi)_{-{\alpha }}$ for all $u\in U(\chi )_{\alpha }$, where ${\alpha }\in {\mathbb{Z}}^I$. Therefore, for all ${\alpha },\beta \in {\mathbb{Z}}^I$, where $\al
\not=\beta $, we get $$\begin{aligned}
{\mathrm{Sh}}(u,v)=0 \quad \text{for all $u\in U(\chi )_{\alpha }$,
$v\in U(\chi )_\beta $ \quad $({\alpha }\not=\beta )$.}
\label{eq:Shfprop3}\end{aligned}$$
\[de:Shapdet\] The family of determinants $$\det \nolimits ^\chi _{\alpha }=
\det {\mathrm{Sh}}(F'_i,F'_j)_{i,j\in \{1,\dots ,k\}}\in {{\mathcal{U}}^0}/{{\Bbbk }^\times },$$ where ${\alpha }\in {\mathbb{N}}_0^I$, $k=\dim U^-(\chi )_{-{\alpha }}$, and $\{F'_1,F'_2,\dots ,F'_k\}$ is a basis of $U^-(\chi )_{-{\alpha }}$, is called the *Shapovalov determinant* of $U(\chi )$.
\[re:Shdet\] Let ${\alpha }\in {\mathbb{N}}_0^I$ and $k=\dim U^-(\chi )_{-{\alpha }}$. By the above considerations, ${\mathrm{Sh}}:U^-(\chi )_{-{\alpha }}\times U^-(\chi )_{-{\alpha }}\to {{\mathcal{U}}^0}$ is a symmetric bilinear form for all ${\alpha }\in {\mathbb{N}}_0^I$. Le
| 1,755
| 1,443
| 941
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, ${\mathbb C}^{\times}$ invariants only exist in the case that $k$ divides $m$, and in that case, are counted by degree $m/k$ polynomials in $n+1$ variables.
Now, let us compare to the original claim. It is a standard result that for $\ell > 0$, $$H^i({\mathbb P}^n, {\cal O}_{{\mathbb P}^n}(\ell)) \: = \:
\left\{ \begin{array}{cl}
0 & i \neq 0, \\
{\rm Sym}^{\ell} \mathbb{C}^{n+1} & i=0.
\end{array} \right.$$ In other words, the only nonzero cohomology is in degree zero, and in that degree, it is counted by homogeneous polynomials of degree $\ell$ in $n+1$ variables. The desired result follows.
Chern classes on the inertia stack {#app:chern-reps}
==================================
As we are manipulating bundles on stacks, it is worth spending a little time reviewing corresponding Chern classes. It is possible to define Chern classes on a stack itself; for example, Chern classes of a vector bundle ${\cal E}$ on a quotient stack $[X/G]$ are simply $G$-equivariant Chern classes of ${\cal E}$ on $X$. However, these Chern classes do not always behave well under mathematical manipulations, and in any event a different notion of Chern classes and Chern characters, denoted $c^{\rm rep}$ and ${\rm ch}^{\rm rep}$, exists and is relevant for index theory. These alternative notions of Chern classes do not live in the cohomology of the original stack, but rather of the inertia stack, which encodes twisted sectors of string orbifolds. (See appendix \[app:spectra\] for more information on the inertia stack.)
In this section, we will illustrate how to compute such Chern classes and characters (denoted $c^{\rm rep}$ and ${\rm ch}^{\rm rep}$) and describe their appearance in index theory in some examples. It is tempting to wonder whether one could derive extra anomaly constraints on orbifolds from these stack Chern classes over nontrivial components of the inertia stack, but we argue that does not seem to happen in heterotic compactifications in section \[sect:possible-anomcanc\] (though see section \[app:spectra:fockconst
| 1,756
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| 2,809
| 0.776817
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ence between the wave functions for the same energy levels of two systems. The coefficients $N_{1}$ and $N_{2}$ can be calculated from a normalization conditions for the wave functions (for the continuous energy spectra), and boundary condition are defined by scattering or decay process.
The interdependence between amplitudes of transittion and reflection \[sec.2.4\]
--------------------------------------------------------------------------------
For scattering the radial superpotential $W_{1}(r)$, the potentials $V_{1}(r)$ and $V_{2}(r)$ are finite in the whole spatial region of their definition and in asymptotic they tend to zero: $$\begin{array}{ll}
W_{1} (r \to +\infty) = 0, &
V_{1} (r \to +\infty) = V_{2} (r \to +\infty) = 0.
\end{array}
\label{eq.2.4.1}$$ Let’s find an interdependence between resonant and potential components of S-matrixes of these systems (for example, also see [@Andrianov.hep-th/9404061]).
One can describe the particle motion in the direction to zero inside the fields $V_{1}(r)$ and $V_{2}(r)$ with use of plane waves $e^{-ikr}$ (we assume, that the plane waves of both systems have the same wave vectors $k$). In spatial asymptotic regions we obtain transmitted waves $T_{1}(k)e^{ikx}$ and $T_{2}(k)e^{ikx}$, which are formed in result of total propagation (with possible tunneling) through the potentials and describe the resonant scattering of the particle on the potentials, and reflected waves $R_{1}(k)e^{ikx}$ and $R_{2}(k)e^{ikx}$, which are formed in result of reflection from the potentials and describe the potential scattering of the particle on the potentials. For each process of scattering one can write components of wave functions, which are formed in result of the transmission through the potential and the reflection from it: $$\begin{array}{ll}
\chi_{inc+ref}^{(1)}(k, r \to +\infty) =
\bar{N}_{1} (e^{-ikr} + R_{1} e^{ikr}), &
\chi_{tr}^{(1)}(k, r \to +\infty) \to
\bar{N}_{1} T_{1} e^{ikr}, \\
\chi_{inc+ref}^{(2)}(k, r \to +\infty) =
\bar{N}_{2} (e^{-ikr}
| 1,757
| 2,715
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| 0.776433
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aligned}
\rho_{\sigma,\rm tip}(\w) = \lim_{\delta\to 0^+} \frac{1}{\pi} \Im G_{c_{0\sigma,\rm tip}, c^\dagger_{0\sigma,\rm tip}}
(\w-i\delta).\end{aligned}$$ As usual, $G_{A,B}(z)$ refers to the equilibrium Green’s function [@Rickayzen1980] defined in the complex frequency plane $z$ except on the real axis. Throughout the paper we moreover use the notation $\rho_{A,B}(\w)\equiv \Im G_{A,B}(\w-i0^+)/\pi$ to connect a Green’s function of the two operators $A,B$ with its spectral function $\rho_{A,B}(\w)$. Note, however, that we have written Eq. in terms of a transmission function $\tau^{(0)}_{\sigma}(\w)$ which includes the tunnel matrix elements $t_{\mu \sigma}$ as well as the spectral properties $\rho_{\mu \sigma, \mu' \sigma}(\w)$ of S, the advantage being that then all contributions to the current in Eq. , i.e. Eqs. , and have the same overall structure. In fact, the transmission function $\tau^{(0)}_{\sigma}(\w) $ can also be interpreted as the fermionic Green’s function of the operator $A_\sigma= \sum_{\mu}^{M-1} t_{\mu \sigma}d_{\mu\sigma}$.
The usual assumption in STM experiments is that the density of states $\rho_{\sigma,\rm tip}(\w)$ is featureless in the energy (voltage) interval of interest. Then, it can be replaced by a constant $\rho_{\sigma,\rm STM}$ that only enters the prefactor in Eq. . This confirms that a detailed knowledge of the expansion coefficients in Eq. is not required, since these coefficients can be absorbed into this prefactor.
We note that the result in Eq. also allows for interferences among more than one elastic transport channels. For example, it is straightforward to show that Eq. reproduces the result of Eq. (6) of Ref. [@SchillerHershfield2000a], if we set $M=2$ and replace $d_{1\sigma}$ by the local surface conduction electron operator $\psi(\vec{R}_s)$. The Fano resonance [@FanoResonance1961] is generated by the quantum interference between the two or more elastic tunneling channels.
### Inelastic tunnel current {#sec:I-inelastic}
The contributions to
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|
in appendix \[sec:spectral\], assuming familiarity with standard methods of the spectral theory of automorphic forms.
[**Acknowledgements:**]{} We thank Peter Sarnak for his comments on an earlier version and for alerting us to Good’s work.
A geometric argument {#sec:Geom}
====================
We start with a basis $\{v,v'\}$ for the lattice $L$ which is oriented positively, that is ${\operatorname{Im}}(v'/v)>0$. For a given $v$, $v'$ is unique up to addition of an integer multiple of $v$. Consider the parallelogram $P(v,v')$ spanned by $v$ and $v'$. Since $\{v,v'\}$ form a basis of the lattice $L$, $P(v,v')$ is a fundamental domain for the lattice and the area of $P(v,v')$ depends only on $L$, not on $v$ and $v'$: ${\operatorname{area}}(P(v,v'))={\operatorname{area}}(L)$.
Let $\mu(L)>0$ be the minimal length of a nonzero vector in $L$: $$\mu(L)=\min\{ |v|:0\neq v\in L\}\;.$$
Any minimal vector $v'$ satisfies $$\label{upper bd on v'}
|v'|^2\leq (\frac{|v|}2 )^2 + (\frac {{\operatorname{area}}(L)} {|v|})^2 \;.$$ Moreover, if $|v|>2{\operatorname{area}}(L)/\mu(L)$ then the minimal vector $v'$ is unique up to sign.
To see , note that the height of the parallelogram $P$ spanned by $v$ and $v'$ is ${\operatorname{area}}(P)/|v| =
{\operatorname{area}}(L)/|v|$. If $h$ is the height vector, then the vector $v'$ thus lies on the affine line $h+\R v$ so is of the form $h+tv$. After adding an integer multiple of $v$ we may assume that $|t|\leq 1/2$, a choice that minimizes $|v'|$, and then $$|v'|^2 = t^2|v|^2+ |h|^2\leq \frac 14 |v|^2 +
(\frac{{\operatorname{area}}(L)}{|v|})^2 \;.$$
We now show that for $|v|\gg_L 1$, the minimal choice of $v'$ is unique if we assume ${\operatorname{Im}}(v'/v)>0$, and up to sign otherwise: Indeed, writing the minimal $v'$ as above in the form $v'=h+tv$ with $|t|\leq 1/2$, the choice of $t$ is unique unless we can take $t=1/2$, in which case we have the two choices $v'=h\pm
v/2$. To see that $t=\pm 1/2$ cannot occur for $|v|$ sufficiently large, we argue that if $v'=h+v/2$ then we
| 1,759
| 3,495
| 2,121
| 1,593
| 1,908
| 0.784508
|
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|
x)}
(1 + \Delta_i)^{3/2}(1 + \Delta_j)^{3/2}
\exp[-x(\Delta_i + \Delta_j)]~,
\nonumber \\
\langle\sigma_{ij}v\rangle
&=&
\left(\frac{m}{4\pi T}\right)^{3/2}
\int 4\pi v^2dv~\left(\sigma_{ij}v\right)
\exp\left(-\frac{mv^2}{4T}\right)~,
\label{effective CS}\end{aligned}$$ where $i,j = \DM$, $\CP$ and $\CPC$, $\Delta_{\CP} = \Delta_{\CPC} =
\delta m / m$, $\Delta_{\DM} = 0$, and $\sigma_{ij}$ is the annihilation cross section between $i$ and $j$. In Fig. \[CS\], the enhancement ratio of the averaged cross section, $\langle\sigma_{\rm
eff}v\rangle$, to that in the perturbative calculation is shown as a function of $m$ with fixed $m/T = 20, ~200, ~2000$ (left figure) and as a function of $T$ with fixed $m = 2.8$ TeV (right figure). For comparison, the cross section in a perturbative calculation is also shown as a dotted line in the right figure. Note that the perturbative cross section is constant in time. The little drop at $x
\sim 10^5$ is due to the decoupling of $\tilde \chi ^\pm$. In the calculation of the cross section, we used the running gauge coupling constant at the $m$ and $m_Z$ in the absorptive terms and the potentials, respectively.
(-120,138)[$\langle \sigma_{\rm eff} \rangle /
\langle \sigma_{\rm eff} \rangle_{\rm Tree}$]{} (-177,134)[Thermal averaged cross section, $\langle \sigma_{\rm eff} \rangle$]{}
In these figures, large enhancement of the cross section is found due to the non-perturbative effect when $m$ is larger than $\sim 1$ TeV. A significant enhancement is shown at $m\sim$ 2.4 GeV. This originates in the bound state composed of $\DM\DM$ and $\CP\CPC$ pairs [@Hisano:2004ds; @Hisano:2003ec]. The enhancement by the non-perturbative effect is more significant for lower temperature. Since $\DM$ and $\CP$ are more non-relativistic for lower temperature, the long-range force acting between these particles strongly modifies their wave functions and alters the cross section significantly.
(-115,136)[$Y / Y_{\rm Tree}$]{} (-165,136)[Thermal relic abundance, $\Omega_{\rm DM} h^2$]{}
| 1,760
| 2,959
| 2,699
| 1,751
| 2,064
| 0.783104
|
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|
(M_\infty)$. Since $[\tl c_{1}]\in{\operatorname{image}}\p_*$, it is $(t-1)$-torsion so $\b$ is $(t-1)^{N+1}$-torsion. Moreover $\b$ lies in the submodule $A\subset
H_1(M_\infty,\BQ)$ consisting of elements annihilated by some power of $t-1$, so, by choice of $N$, $(t-1)^N\b=0=[\tl c_{1}]=0$ as desired. This completes the verification of Step 4.
[**Step 5**]{}: C$\Rightarrow$A Let $\{x,y\}$ be as in the hypotheses of C and let $M_\infty$ correspond to the class $x$. Let $N$ be the positive integer as above. If $\b^{(N+1}$ can be defined, we know in particular that there exists some system of surfaces $\{V_x,V_y,...,V_{c(N)}\}$ that defines $\{c(j)\}$, $1\leq j\leq (N+1)$. Choose a preferred lift $\wt V_x$, of $V_x$ to $M_\infty$ and a preferred fundamental domain $\wt Y$ as above lying on the positive side of $\wt V_x$. Consider any $m, 1\leq m \leq N$. Since $c(m)$ and $c(m+1)$ lie on $V_x$, they lift to oriented 1-manifolds $\tl c(m)$ and $\tl c(m+1)$ in $\wt V_x$. Similarly $c^+(m)$ lifts to $\tl c^+(m)$, which is a push-off of $\tl c(m)$ lying in $\wt Y$. Recall that $c(m+1)=V_{c(m)}\cap V_x$ where $\p
V_{c(m)}=k_mc^+_{(m)}$ for some positive integer $k_m$. Letting $\wt V_{c(m)}$ be $V_{c(m)}$ cut open along $c(m+1)$ we observe that $\wt V_{c(m)}$ can be lifted to $\wt Y$ and viewed as a 2-chain showing that $k_m[\tl c^+(m)]=(t-1)[\tl c(m+1)]$ in $H_1(M_\infty;\BQ)$, as in Figure \[stepfive\].
(177,84) (10,10)[![[]{data-label="stepfive"}](stepfive.eps "fig:")]{} (15,64)[$\wt c(m+1)$]{} (108,72)[$k_m\wt c^+(m)$]{} (65,34)[$\wt V_{c(m)}$]{} (0,12)[$\wt V_x$]{} (173,12)[$t\wt V_x$]{} (179,59)[$t\wt c(m+1)$]{} (83,-5)[$\wt Y$]{}
Thus $[\tl c^+(1)]=(t-1)(1/k_1)[\tl c(2)]=(t-1)^2(1/k_1)(1/k_2)[\tl c(3)]$, et cetera, showing that $[\tl c(1)]$ is divisible by $(t-1)^N$. By Steps 1 through 4, this implies A of Theorem \[linear\], completing Step 5. [**Step 6**]{}: A$\Rightarrow$B We assume that there is a primitive class $x\in H^1(M;\BZ)$ corresponding to $M_\infty$ and $\{M_n\}$ where $\b_1(M_n)$ grows linearly.
| 1,761
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| 1,685
| 1,861
| 0.784953
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|
h compositions, and it is well-defined since for any $v \in V([n])$, its preimage $f^{-1} \subset V([m])$ carries a natural linear order induced by the orientation of the circle $S^1$.
\[alg.def\] For any associative unital algebra $A$ over $k$, its Hochschild, cyclic and periodic cyclic homology $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$, $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$, $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$ is defined as the corresponding homology of the cyclic $k$-vector space $A_\#$: $$HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A) {\overset{\text{\sf\tiny def}}{=}}HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_\#),\quad
HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A) {\overset{\text{\sf\tiny def}}{=}}HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_\#),\quad
HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(P) {\overset{\text{\sf\tiny def}}{=}}HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_\#).$$
Cyclic bimodules. {#naive}
=================
Among all the homology functors introduced in Definition \[alg.def\], Hochschild homology is the most accesible, and this is because it has another definition: for any associative unital algebra $A$ over $k$, we have $$\label{hh.def}
HH_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}} = {\operatorname{Tor}}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_{A^{opp} \otimes A}(A,A),$$ where ${\operatorname{Tor}}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is taken over the algebra $A^{opp} \otimes A$ (here $A^{opp}$ denotes $A$ with the multiplication taken in the opposite direction).
This has a version with coefficients: if $M$ is a left module over $A^{opp} \otimes A$, – in other words, an $A$-bimodule, – one defines Hochschild homology of $A$ with coefficients in $M$ by $$\label{hoch.coeff}
HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M) = {\operatorname{Tor}}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_{A^{opp} \otimes A}(M,A).$$ The category $A{\operatorname{\!-\sf bimod}}$ of $A$-bimodules is a unital (non-symmetric) tensor cat
| 1,762
| 1,230
| 1,691
| 1,664
| 2,723
| 0.777553
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|
ions in vacuum can be found in Christodoulou’s work [@Chr] on the formation of black holes.
[^3]: The notation $A\lesssim B$ means $A\le cB$ for some universal constant $c$.
[^4]: Because is used very frequently in a similar manner, we will not point it out again when we use in the rest of the paper. Because $\mathscr{W}\ge1$, will also be used in the form $C^2\delta|u_1|^{-1}\mathscr{F}\mathcal{A}\le1$.
[^5]: Similar to , the estimates are used frequently and we will not point this out in the argument.
---
abstract: 'We study the best arm identification ([<span style="font-variant:small-caps;">Best-$1$-Arm</span>]{}) problem, which is defined as follows. We are given $n$ stochastic bandit arms. The $i$th arm has a reward distribution ${\mathcal{D}}_i$ with an unknown mean $\mu_{i}$. Upon each play of the $i$th arm, we can get a reward, sampled i.i.d. from ${\mathcal{D}}_i$. We would like to identify the arm with the largest mean with probability at least $1-\delta$, using as few samples as possible. We provide a nontrivial algorithm for [<span style="font-variant:small-caps;">Best-$1$-Arm</span>]{}, which improves upon several prior upper bounds on the same problem. We also study an important special case where there are only two arms, which we call the [<span style="font-variant:small-caps;">Sign</span>-$\xi$]{} problem. We provide a new lower bound of [<span style="font-variant:small-caps;">Sign</span>-$\xi$]{}, simplifying and significantly extending a classical result by Farrell in 1964, with a completely new proof. Using the new lower bound for [<span style="font-variant:small-caps;">Sign</span>-$\xi$]{}, we obtain the first lower bound for [<span style="font-variant:small-caps;">Best-$1$-Arm</span>]{} that goes beyond the classic Mannor-Tsitsiklis lower bound, by an interesting reduction from [<span style="font-variant:small-caps;">Sign</span>-$\xi$]{} to [<span style="font-variant:small-caps;">Best-$1$-Arm</span>]{}. We propose an interesting conjecture concerning the optimal sample complexity o
| 1,763
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| 1,118
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| 0.800809
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|
ed to describe the relevant quasienergy splitting when the periodic force is turned on. Sec. \[sec:twodefs\] explains in detail the strategy for achieving controlled population transfer between the two defects. Conclusions are drawn in the final Sec. \[sec:conclusion\].
Localization at a single defect {#sec:single}
===============================
A single particle on a one-dimensional, infinite, regular lattice with matrix elements connecting neighboring sites only, as is appropriate in the tight-binding limit, is described by the Hamiltonian $$\label{eq:h0}
\hat{H}_0
=
-\frac{W}4 \sum_{\ell = -\infty}^{\infty}
\left\{{{\left|\ell\right.\rangle}}{{\langle \left.\ell+1\right|}} + {{\left|\ell+1\right.\rangle}}{{\langle \left.\ell\right|}}\right\}\,,$$ where ${{\left|\ell\right.\rangle}}$ is the Wannier state localized at the $\ell$-th site, adopting the normalization ${{\langle \left.\ell'\right|}}\ell\rangle=\delta_{\ell',\ell}$. Denoting the lattice constant by $d$, its eigenstates $$\label{eq:chi}
\left|\chi_k\right>
=
\sum_{\ell = -\infty}^{\infty} {{\rm e}}^{{{\rm i}}k\ell d}{{\left|\ell\right.\rangle}}$$ are extended Bloch waves, labeled by the wave number $k$. The hopping matrix elements $-W/4$ in Eq. (\[eq:h0\]) have been chosen such that the energy dispersion reads $$\label{eq:band}
E(k) = -\frac{W}2\cos(kd)\,,$$ corresponding to a band of width $|W|$. For positive $W$, its minimum lies at $k = 0$. In order to introduce a defect into the ideal system (\[eq:h0\]), the on-site energy at the site $\gamma$ is now altered by an amount $\nu$, giving rise to the perturbation $$\label{eq:def}
\hat{V}_{\rm r} = {{\left|\gamma\right.\rangle}}\nu{{\langle \left.\gamma\right|}}\;,\quad \gamma>0 \;.$$ The single-defect Hamiltonian $$\label{eq:h1}
\hat{H} = \hat{H}_0 + \hat{V}_{\rm r}$$ then admits a localized state with energy $$\label{eq:E_0}
E_0=
p\,\frac W2 \sqrt{\frac{4\nu^2}{W^2}+1}\;,$$ where $$\label{eq:defp}
p=\left\{\begin{array}{r@{\quad:\quad}l}
1&\frac
| 1,764
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| 0.774026
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|
he infinite equivalent set [\[]{}0[\]]{}. While Fig. \[Fig: tent4\] was a comparison of the first four iterates of the tent and absolute sine maps, Fig. [\[Fig: tent17\]]{} following shows the “converged” graphical limits for after 17 iterations.
***4.1. The chaotic attractor***
One of the most fascinating characteristics of chaos in dynamical systems is the appearance of attractors the dynamics on which are chaotic. **For a subset $A$ of a topological space $(X,\mathcal{U})$ such that $\mathcal{R}(f(A))$ is contained in $A$ — in this section, unless otherwise stated to the contrary, $f(A)$ *will* *denote the* *graph and not the range (image)* *of* $f$ — which ensures that the iteration process can be carried out in $A$, let $$\begin{array}{ccl}
{\displaystyle f_{\mathbb{R}_{i}}(A)} & = & {\displaystyle \bigcup_{j\geq i\in\mathbb{N}}f^{j}(A)}\\
& = & {\displaystyle \bigcup_{j\geq i\in\mathbb{N}}\left(\bigcup_{x\in A}f^{j}(x)\right)}\end{array}\label{Eqn: absorbing set}$$ generate the filter-base $_{\textrm{F}}\mathcal{B}$ with $A_{i}:=f_{\mathbb{R}_{i}}(A)\in\,_{\textrm{F}}\mathcal{B}$ being decreasingly nested, $A_{i+1}\subseteq A_{i}$ for all $i\in\mathbb{N}$, in accordance with Def. A1.1. The existence of a maximal chain with a corresponding maximal element as asssured by the Hausdorff Maximal Principle and Zorn’s Lemma respectively implies a nonempty core of $_{\textrm{F}}\mathcal{B}$. As in Sec. 3 following Def. 3.3, we now identify the filterbase with the neighbourhood base at $f^{\infty}$ which allows us to define $$\begin{array}{ccl}
{\displaystyle \textrm{Atr}(A_{1})} & \overset{\textrm{def}}= & \textrm{adh}(\,_{\textrm{F}}\mathcal{B})\\
& = & {\displaystyle \bigcap_{A_{i}\in\,_{\textrm{F}}\mathcal{B}}\textrm{Cl}(A_{i})}\end{array}\label{Eqn: attractor_adherence}$$ as the attractor of the set $A_{1}$, where the last equality follows from Eqs.(\[Eqn: Def: omega(A)\]) and (\[Eqn: Def: Closure\]) and the closure is with respect to the topology induced by the neighbourhood filter base $_{\textr
| 1,765
| 3,219
| 2,492
| 1,834
| 1,433
| 0.789655
|
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|
f monoid schemes. Namely, we claim the following commutative diagram of schemes: $$\xymatrixcolsep{5pc}\xymatrix{
\underline{M}^{\prime}\times \underline{M}^{\prime} \ar[d]^{\star} \ar[r]^{(1+)\times (1+)} &\underline{M}\times
\underline{M}\ar[d]^{multiplication}\\
\underline{M}^{\prime} \ar[r]^{1+} &\underline{M}}$$ Since all schemes are irreducible and smooth, it suffices to check the commutativity of the diagram at the level of flat $A$-points as explained in the third paragraph from below in Remark \[r32\] of [@C2], and this is obvious.
Since $\underline{M}^{\ast}$ is an open subscheme of $\underline{M}$, $(1+)^{-1}(\underline{M}^{\ast})$ is an open subscheme of $\underline{M}^{\prime}$. The composite of the following three morphisms $$\xymatrixcolsep{5pc}\xymatrix{
(1+)^{-1}(\underline{M}^{\ast}) \ar[r]^{(1+)} &\underline{M}^{\ast} \ar[r]^{inverse} &\underline{M}^{\ast}\ar[r]^{(1+)^{-1}}&
(1+)^{-1}(\underline{M}^{\ast})}$$ defines the inverse morphism on the scheme of monoids $(1+)^{-1}(\underline{M}^{\ast})$ with respect to the operation $\star$. Thus we can see that $(1+)^{-1}(\underline{M}^{\ast})$ is a group scheme with respect to $\star$ and the morphism $1+$ is an isomorphism of group schemes between $(1+)^{-1}(\underline{M}^{\ast})$ and $\underline{M}^{\ast}$.
Let $R$ be a $\kappa$-algebra. Since the morphism $1+$ is an isomorphism of monoid schemes between $\underline{M}^{\prime}$ and $\underline{M}$, we can write each element of $\underline{M}(R)$ as $1+x$ with $x \in \underline{M}^{\prime}(R)$. Here, $1+x$ means the image of $x$ under the morphism $1+$ at the level of $R$-points. Note that $\underline{M}^{\prime}(R)$ is a $B\otimes_AR$-algebra for any $A$-algebra $R$ with respect to the original multiplication on it, not the operation $\star$. In particular, $\underline{M}^{\prime}(R)$ is a $(B/2B)\otimes_AR$-algebra for any $\kappa$-algebra $R$. Therefore, we consider the following two functors: $$\left\{
\begin{array}{l}
\textit{the subfunctor $\underline{\pi M^{\prime}} : R \mapsto (\pi\oti
| 1,766
| 1,773
| 1,490
| 1,584
| null | null |
github_plus_top10pct_by_avg
|
ome (32% for males and 47% for females).
{#pone.0153583.t003g}
------------------------- ------------------------- -------------------
**Males (N = 69)**
**4-MA4-MM** \<0.6 m/s (dismobility) ≥0.6 m/s (normal)
\<0.6 m/s (dismobility) 6 (8.7%) 0
≥0.6 m/s (normal) 13 (18.8%) 50 (72.5%)
**Females (N = 103)**
**4-MA4-MM** \<0.6 m/s (dismobility) ≥0.6 m/s (normal)
\<0.6 m/s (dismobility) 14 (13.6%) 8 (7.8%)
≥0.6 m/s (normal) 16 (15.5%) 65 (63.1%)
------------------------- ------------------------- -------------------
The correlations of 4-MM and 4-MA with other measures of functional performance are also shown in [Table 2](#pone.0153583.t002){ref-type="table"}. A significant positive correlation with 6MWT was found for both 4-MM and 4-MA in men and women (4-MM men r = 0.59, p\<0.001; women r = 0.49, p\<0.001; 4-MA men r = 0.50, p = 0.0004; women r = 0.22, p = 0.048).
4-MA showed a significant positive correlation with handgrip strength (r = 0.40, p = 0.005 in men; r = 0.29, p = 0.01 in women), as also 4-MM (r = 0.51, p\<0.001 in men; r = 0.38, p = 0.0001 in women).
Discussion {#sec016}
==========
In a cohort of community-dwelling older individuals, we found a significant correlation between the assessment of gait speed using a manual (i.e., stopwatch) and technological (i.e., accelerometer) technique. However, our results suggest that the concordance of two tests is less strong than anticipated and might be suboptimal in the classification of single subjects. This is the first study investigating the correlation between these two assessment modalities of gait speed.
There is a wide range of methods available for assessing physical function in both research and clinical practice \[[@pone.0153583.ref024]\]. The final choice on the best measurement should take into account the inter-rater and
| 1,767
| 57
| 2,144
| 2,086
| null | null |
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|
ac{\mu(s)\Psi(s)}{\psi(s)}ds}
\end{aligned}$$
For $r\in[0,\Rq]$, $\tau'(r) = r$, so that $\psi'(r) = \psi(r)(-\aq r)$. Thus $$\begin{aligned}
q'(r)
=& \psi(r) \nu(r)\\
q''(r)
=& \psi'(r) \nu(r) + \psi(r) \nu'(r)\\
=& \psi(r) \nu(r) (-\aq r) + \psi(r) \nu'(r)\\
=& -\aq r \nu'(r) + \psi(r) \nu'(r)\\
q''(r) + \aq r q'(r)
=& \psi(r) \nu'(r)\\
=& - \frac{1}{2}\frac{\mu(r)\Psi(r)}{\int_0^{4\Rq}\frac{\mu(s)\Psi(s)}{\psi(s)} ds}\\
=& - \frac{1}{2}\frac{\Psi(r)}{\int_0^{4\Rq}\frac{\mu(s)\Psi(s)}{\psi(s)} ds}
\end{aligned}$$ Where the last equality is by definition of $\mu(r)$ in Lemma \[l:mu\] and the fact that $r\leq \Rq$.
We can upper bound $$\begin{aligned}
\int_0^{4\Rq} \frac{\mu(s) \Psi(s)}{\psi(s)} ds
\leq \int_0^{4\Rq} \frac{\Psi(s)}{\psi(s)} ds
\leq \frac{\int_0^{4\Rq} s ds}{\psi(4\Rq)}
= \frac{16\Rq^2}{\psi(4\Rq)}
\leq& 16 \Rq^2 \cdot \exp\lrp{\frac{7\aq\Rq^2}{3}}
\end{aligned}$$ Where the first inequality is by Lemma \[l:mu\], the second inequality is by the fact that $\psi(s)$ is monotonically decreasing, the third inequality is by Lemma \[l:tau\].
Thus $$\begin{aligned}
q''(r) + \aq r q'(r)
\leq& -\frac{1}{2} \lrp{\frac{\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{16\Rq^2}} \Psi(r)\\
\leq& -\frac{\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{32\Rq^2} q(r)
\end{aligned}$$ Where the last inequality is by $\Psi(r) \geq q(r)$.
**Proof of \[f:q(r)\_bounds\]** Notice first that $\nu(r) \geq \frac{1}{2}$ for all $r$. Thus $$\begin{aligned}
q(r)
:=& \int_0^r \psi(s) \nu(s) ds\\
\geq& \frac{1}{2} \int_0^r \psi(s) ds\\
\geq& \frac{\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{2}\cdot r
\end{aligned}$$ Where the last inequality is by Lemma \[l:tau\].
**Proof of \[f:q’(r)\_bounds\]** By definition of $f$, $q'(r) = \psi(r) \nu(r) $, and $
| 1,768
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and accuracy of the variant algorithm in Section \[section5\] through experiments on simulated and real medical dataset (Section \[section6\]).
New random graph model {#section2}
======================
Notations
---------
This work considers the framework of an unweighted undirected graph $G(V,E)$ with no self-loops consisting of vertices $V=\left\{ 1,\dots,n \right\}$ and $p$ edges connecting each pair of vertices. An edge $e \in E$ that connects a node $i$ and a node $j$ is denoted by $e = (i,j)$. In this paper, we consider that the existence of a link between two nodes in an interaction network is already inferred from the estimation of a statistical dependance measure. The graph $G$ is represented hereusing an adjacency matrix $A = (A_{ij})_{(i,j) \in V^2}$ defined by
$A_{i,j} = \left\{
\begin{array}{ll}
1 \ \ \mbox{if there is an edge between } i \mbox{ and } j,\\
0 \ \ \mbox{otherwise.}
\end{array}
\right.$
Since the graph is undirected and with no self-loops, $A \in \mathbb{M}_n(\mathbb{R})$ is a symmetric matrix with coefficients zero on the diagonal.
For each node $i$, the degree $d_i$ is defined as the number of edges incident to $i$ and is equal to : $d_i =\sum\limits_{j=1}^n A_{ij}$. We denote by $D$ the diagonal degree matrix containing $(d_1,\dots ,d_n)$ on the diagonal and zero elsewhere.
A subset $C \in V$ of a graph is said to be connected if any two vertices in $C$ are connected by a path in $C$. Non empty sets $ C_1, \dots, C_k $ form a partition of the graph $G(V,E)$ if $C_i \cap C_j =\emptyset$ and $C_1\cup \cdots \cup C_k=V$. In addition, $C_i$ are called connected components if there are no connections between vertices in $C_i$ and $\overline{C_i}$ for all i in $ \left\{1, \dots, k \right\}$.
We define the indicators of connected components $\textbf{1}_{C_i}$ whose entries are defined by:
$(\textbf{1}_{C_i})_j = \left\{
\begin{array}{ll}
1 \ \ \mbox{if vertex } j \mbox{ belong to } C_i,\\
0 \ \ \mbox{otherwise.}
\end{array}
\right.$
Graph models
------------
As mention
| 1,769
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ith uniform weights by a constant factor. We fix $\kappa = 128$ and $n=128000$. As illustrated by a figure on the right, the position of the separators are chosen such that there is one separator at position one, and the rest of $\ell-1$ separators are at the bottom. Precisely, $(p_{j,1},p_{j,2},p_{j,3},\ldots,p_{j,\ell})=(1,128-\ell+1,128-\ell+2,\ldots,127)$. We consider this scenario to emphasize the gain of optimal weights. Observe that the MSE does not decrease at a rate of $1/\ell$ in this case. The parameter $\gamma$ which appears in the bound of Theorem \[thm:main2\] is very small when the breaking positions $p_{j,a}$ are of the order $\kappa_j$ as is the case here, when $\ell$ is small. Normalization constant $C$ is $n/d^2$.
![The gain of choosing optimal $\lambda_{j,a}$’s is significant when $\kappa_j$’s are highly heterogeneous. []{data-label="fig:lambda_impact2"}](Plot4-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-180,50) (-104,-7)
The gain of optimal weights is significant when the size of $S_j$’s are highly heterogeneous. Figure \[fig:lambda\_impact2\] compares performance of the proposed algorithm, for the optimal choice and uniform choice of weights $\lambda_{j,a}$ when the comparison sets $S_j$’s are of different sizes. We consider the case when $n_1$ agents provide their top-$\ell_1$ choices over the sets of size $\kappa_1$, and $n_2$ agents provide their top-$1$ choice over the sets of size $\kappa_2$. We take $n_1 = 1024$, $\ell_1 = 8$, and $n_2 = 10n_1\ell_1$. Figure \[fig:lambda\_impact2\] shows MSE for the two choice of weights, when we fix $\kappa_1 = 128$, and vary $\kappa_2$ from $2$ to $128$. As predicted from our bounds, when optimal choice of $\lambda_{j,a}$ is used MSE is not sensitive to sample set sizes $\kappa_2$. The error decays at the rate proportional to the inverse of the effective sample size, which is $n_1\ell_1 + n_2\ell_2 = 11n_1\ell_1$. However, with $\lambda_{j,a} = 1$ when $\kappa_2 = 2$, the MSE is roughly $10$ times worse. Which reflects that the eff
| 1,770
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|
-------------------- -------------- -------------- -------------- ---------
Sex Girls 1863 37.5 48.2 8.4 5.9 \<0.001
Boys 1765 28.4 53.4 10.4 7.8
Maternal age at birth \< = 24y 346 27.2 51.7 11.0 10.1 0.04
25--29y 1129 32.6 51.4 9.6 6.5
30--34y 1438 35.2 49.5 9.5 5.8
\> = 35y 713 32.4 51.9 8.0 7.7
Mother's birthplace Non-HK 1419 38.7 47.1 7.8 6.4 \<0.001
HK 2200 29.5 53.2 10.4 7.0
Highest parental education Grade 9 or below 1029 32.4 50.9 9.2 7.5 0.64
Grade 10--11 1558 32.9 50.3 9.7 7.1
Grade 12 or above 1041 34.2 51.2 9.0 5.6
Highest parental occupation Professional 842 34.0 51.1 9.6 5.3 0.10
Managerial 469 34.3 48.4 10.7 6.6
Non-manual skilled 918 29.6 54.1 9.4 6.9
| 1,771
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|
(6)$ $0.43(1)$
CTTP $0.64(1)$ $0.78(3)$ $1.55(5)$ $0.43(1)$
Manna $0.64(1)$ $0.78(2)$ $1.57(4)$ $0.42(1)$
DP $0.583(4)$ $0.80(1)$ $1.766(2)$ $0.451(1)$
$\tau_s$ $D$ $z$ $\eta$ $\delta$
CLG $1.29(1)$ $2.75(1)$ $1.53(2)$ $0.29(1)$ $0.49(1)$
CTTP $1.28(1)$ $2.76(1)$ $1.54(1)$ $0.30(3)$ $0.49(1)$
Manna $1.28(1)$ $2.76(1)$ $1.55(1)$ $0.30(3)$ $0.48(2)$
DP $1.268(1)$ $2.968(1)$ $1.766(2)$ $0.230(1)$ $0.451(1)$
------- ------------ ------------------- ------------ ------------ ------------
: Critical exponents for spreading and steady state experiments. Figures in parenthesis indicate the statistical uncertainty in the last digit. Steady state Manna exponents from Ref. [@bigfes]. []{data-label="table"}
In order to provide further evidence for the existence of a general universality class, we have simulated several other models exhibiting an APT in the presence of a conserved field. The first is a conserved threshold transfer process (CTTP). In the CTTP, the sites of a lattice can be vacant, singly occupied, or doubly occupied by particles, corresponding to a dynamic variable $n_i=0,1$ or $2$, respectively. Values $n_i>2$ are strictly forbidden. Dynamics affects only doubly occupied sites: every site with $n_i=2$ tries to transfer both its particles to randomly selected nearest neighbors with $n_j<2$. Singly occupied sites are, on the other hand, inert. The total number of particles $N=\sum_i n_i$ is thus constant in time. Results from simulations are obtained along the lines shown for the CLG and are reported in Table \[table\]; in this case, the largest sizes used are $L=512$ for the stationary exponents and $L=1024$ for the spreading exponents. As an example of our simulations, in Fig. \[fig:steady3\] we plot $\rho_{a,al
| 1,772
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|
polynomial $f$ so that $f(x)=\sum_{i=1}^q \alpha_{i}x^{i-1}$. Since we defined $t_i=i-1$, we have $f(x)=\sum_{i=1}^q \alpha_{i}x^{t_i}$. Define $\beta_i=-\alpha_i$ for all $i\in [q]$. Our final construction of $h$ is thus $$h(\ell)=f(\gamma^\ell)-\ell f(\gamma^\ell)$$
$h(\ell)=0\ \forall \ell\in S$
Since $0\notin S$, $\ell\ne 0$. We will look at $h(\ell)$ modulo each of the primes. $$\begin{aligned}
h(\ell) \mod p_i = f_i(\gamma^\ell)-(\ell\mod p_i) f_i(\gamma^\ell)=
\begin{cases}
f_i(\gamma^\ell)= 0 & \mbox{if}\ \ell=0 \mod p_i\\
f_i(\gamma^\ell)-f_i(\gamma^\ell) =0 & \mbox{if}\ \ell=1 \mod p_i
\end{cases}\end{aligned}$$ Therefore, using Chinese Remaindering, $h(\ell)=0\ \forall \ell\in S$.
$(h(0) \mod p_j)\ne 0$ for all $j\in [r]$
Suppose in contradiction that $(h(0) \mod p_j)= 0$, then $$h(0) \mod p_j=f_j(1)=\prod_{\ell\in S,\ \ell=0\mod p_j}(1-\gamma^\ell)=0.$$ The above equation holds in the ring $\left(\Z_{p_j}[\gamma]/(\gamma^m-1)\right)$.Therefore, if we consider what happens in the ring $\Z_{p_i}[\gamma] \cong \F_{p_i}[x]$ (we replace the formal variable $\gamma$ with $x$ to highlight the fact that $x$ does not satisfy any relation) we get that $$\label{eq-identity}
\prod_{\ell\in S,\ \ell=0\mod p_j}(1-x^\ell)=(x^m-1)\theta(x)$$ for some polynomial $\theta(x)\in \F_{p_j}[x]$. The above equation is an identity in the ring $\F_{p_j}[x]$. So we can check its validity by substituting values for $x$ from the algebraic closure of $\F_{p_j}$. Let $m' = m/p_j$ and let $\zeta$ be an element in the algebraic closure of $\F_{p_j}$ of order $m'$ (so $\zeta^\ell=1$ iff $m'$ divides $\ell$). Since $m'$ and $p_j$ are co-prime, such an element exists by Lemma \[lem-order\]. If we substitute $\zeta$ into Eq. \[eq-identity\], the RHS is zero (since $m'$ divides $m$). However, each term in the LHS product is nonzero, since if $\ell =0 \mod p_j$ and $m'$ divides $\ell$ then $\ell = 0 \mod m$ but we know that $0\notin S$. Since we are working over the algebraic closure of $\F_{p_j}$ which is a field, the product of
| 1,773
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| 1,709
| null | null |
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|
\end{aligned}$$ as $n\to\infty$. This, together with (\[rem\]) and (\[rem2\]) prove the proposition.
.1truein This proposition is similar to Theorem 3.1 of Hall, Hu and Marron (1995) and to Theorem 1 of Novak (1999), who do not consider uniformity in $t$ or $f$, and our proof is somewhat adapted from the latter reference (which deals with a slightly different estimator). See also Hall (1990), Terrell and Scott (1992) and McKay (1993).
Combining Propositions \[varid\] and \[biasid\] we obtain the following result for the ‘ideal’ estimator.
\[unifidealthm\] Under the Assumptions \[ass2\] and with $h_n= ((\log n)/n)^{1/9}$, we have, for every $0<C<\infty$ and function $Z$ such that $z(h)\searrow 0$ as $h\searrow 0$, for all $r>0$ and constant $B\ge T/r^{1/2}$ in the definition of $\bar f_n(t;h)$ in (\[ideal0\]), $$\label{main4}
\sup_{t\in D_r}\left|\bar f(t;h_n)-f(t)\right|=O_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right)\ \ {\rm uniformly \ in}\ f\in{\cal D}_{C,z}.$$
[The limit (\[comp\]) is straightforward, but lengthy to compute. By way of illustration we indicate how to prove a ‘small piece’ of it. Let us consider, for example, the term in $f^{(4)}$ from the first summand $r^{(4)}K$ in the expression for $g^{(4)}$ in (\[deriv\]). It is $(3/2)f^{1/2}(t+u)f^{(4)}(t+u)K(wf^{1/2}(t+u))$. Then, $$|f^{1/2}(t+u)f^{(4)}(t+u)K(s(t+u))-f^{3/2}(t)f^{(4)}(t)K(s(t))|
\le \|f\|_\infty^{1/2}\|K\|_\infty|f^{(4)}(t+u)-f^{(4)}(t)|$$ $$+\|f^{(4)}\|_\infty\|K\|_\infty|f^{1/2}(t+u)-f^{1/2}(t)|+\|f^{(4)}\|_\infty \|f\|_\infty^{1/2}|K(s(t+u))-K(s(t))|.$$ And we have, for the first summand, $$|f^{(4)}(t+\tau h_nw)-f^{(4)}(t)|\le z(Bh_n)\to 0$$ uniformly in $t$ and $f$ (recall $|\tau|\le 1$, $|w|\le B$). For the second summand, for $n$ large enough, $$|f^{1/2}(t+u\tau h_nw)-f^{1/2}(t)|\le\frac{|f(t+u\tau h_nw)-f(t)|}{r^{1/2}}\le\frac{CBh_n}{r^{1/2}}\to0,$$ and the limit zero for the third follows directly by uniform continuity of $K$ and the common Lipschitz constant $C$ for all $f\in{\cal D}_{C,z}$.]{}
Comparis
| 1,774
| 1,172
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| 1,723
| 3,747
| 0.770382
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|
only the case $n=1$. The significance of this Corollary is that it has been previously shown by Ivanov and Katz (\[IK, Theorem 9.2 and Cor.9.3\]) that the conclusion of Corollary \[systole\] is sufficient to guarantee a certain optimal systolic inequality for $M$. The interested reader is referred to those works.
Suppose $c$ and $d$ are disjointly embedded oriented circles in $M$ that are zero in $H_1(M;\BQ)$. Then the [**linking number of $c$ with $d$**]{}, ${\ensuremath{\ell k}}(c,d)\in\BQ$ is defined as follows. Choose an embedded oriented surface $V_d$ whose boundary is “$m$ times $d$” (i.e. a circle in a regular neighborhood $N$ of $d$ that is homotopic in $N$ to $md$) for some positive integer $m$, and set: $${\ensuremath{\ell k}}(c,d) = \f1m(V_d\cd c).$$ Given this, the invariants $\b^n(x,y)$ are defined as follows. Let $\{V_x,V_y\}$ be embedded, oriented connected surfaces that are Poincaré Dual to $\{x,y\}$ and meet transversely in an oriented circle that we call $c(x,y)$ (by the proof of \[C1, Theorem 4.1\] we may assume that $c(x,y)$ is connected). Let $c^+(x,y)$ denote a parallel of $c(x,y)$ in the direction given by $V_y$. Note that $\{V_x,V_y\}$ induce two maps $\psi_x$, $\psi_y$ from $M$ to $S^1$ wherein the surfaces arise as inverse images of a regular value. The product of these maps yields a map $\psi:M\to S^1\x S^1$ that induces an isomorphism on $H_1$/torsion. Since $c(x,y)$ and $c^+(x,y)$ are mapped to points under $\psi$, they represent the zero class in $H_1(M;\BQ)$. Therefore we may define $\b^1(x,y)={\ensuremath{\ell k}}(c(x,y)$, $c^+(x,y))$. In fact, $-\b^1(x,y)\cd|{\operatorname{Tor}}H_1(M;\BZ)|$ is precisely Lescop’s invariant of $M$ \[Les; p.90-94\]. An example is shown in Figure \[satolevine\] of a manifold with $\b^1(x,y)= -k$.
(105,92) (10,10)[![Example of $\b^1(x,y)= -k$[]{data-label="satolevine"}](satolevine.eps "fig:")]{} (11,63)[$0$]{} (92,20)[$0$]{} (76,61)[$k$]{} (39,0)[$M$]{}
The idea of the higher invariants is to iterate this process as long as possible (compar
| 1,775
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| 0.776844
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g Equation , but we only keep four 3D arrays ${\cal G}^m_{i,i'}({\rm top \to top}) = {\cal G}^m_{i,i',0}$, ${\cal G}^m_{i,i'}({\rm top \to bot}) = {\cal G}^m_{i,i',-N_z-1}$, ${\cal G}^m_{k,i'}({\rm top \to inn}) = {\cal G}^m_{0,i',k-N_z-1}$, and ${\cal G}^m_{k,i'}({\rm top \to out}) = {\cal G}^m_{N_R+1,i',k-N_z-1}$, corresponding to four DGFs due to the point sources at the top boundary. We repeat the above calculations by varying $i' \in [1, N_R]$ to fill all the components of the four DGFs.
For the potential generated by point sources at the inner radial boundary, we place a point mass at $(i',j',k')=(0,1,k')$ for any $k' \in [1, N_z]$. We then solve Equation and apply Fourier transform to obtain ${\cal G}_{i,0,k-k'}^m$. We keep only four 3D arrays ${\cal G}^m_{i,k'}({\rm inn\to top}) = {\cal G}^m_{i,0,N_z+1-k'}$, ${\cal G}^m_{i,k'}({\rm inn\to bot}) = {\cal G}^m_{i,0,-k'}$, ${\cal G}^m_{k-k'}({\rm inn\to inn}) = {\cal G}^m_{0,0,k-k'}$, and ${\cal G}^m_{k-k'}({\rm inn\to out}) = {\cal G}^m_{N_R+1,0,k-k'}$[^6], corresponding to four DGFs due to the point sources at the inner radial boundary. We repeat the above calculations for all $k'\in[1,N_z]$ to completely fill the elements of the arrays. We follow a similar procedure to obtain the remaining eight DGFs due to points sources at the bottom and outer radial boundaries.
The boundary condition in solving Equation can be obtained by requiring that the DGF at a large distance from the source is approximately equal to $$\label{eq:cyl_asymptotic_green}
{\cal G}_{i,i',j-j',k-k'} \approx - \sum_{p=0}^{P-1} \frac{G}{\sqrt{R_i^2 + R_{i'}^2 - 2R_iR_{i'}\cos(\phi_j - \phi_{j'} - pL_\phi) + (z_k - z_{k'})^2 }}\quad\text{(far from the source)},$$ The summation over $p$ in Equation is to add all the contributions from the periodic images of the point mass when the mass distribution holds $P$-fold symmetry in the $\phi$-direction.
As discussed in Section \[s:calc\_dgf\_cart\], Equation remains valid as long as the distance between the cells $(i,j,k)$ and $(i',j',k')$ is s
| 1,776
| 1,390
| 2,306
| 1,916
| 3,470
| 0.77211
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|
ming from the Lagrangians in Eq. (\[eq:weakl\]). The weak vertex is represented by a solid black circle. \[vf2\]](nspw "fig:") ![Weak vertices for the $\Lambda N\pi$, $\Sigma N\pi$ and $NNK$ stemming from the Lagrangians in Eq. (\[eq:weakl\]). The weak vertex is represented by a solid black circle. \[vf2\]](nnkw "fig:")
The weak interaction between the $\Sigma$, $\Lambda$ and $N$ baryons and the pseudoscalar $\pi$ and $K$ mesons is described by the phenomenological Lagrangians:
$$\begin{aligned}
\label{eq:weakl}
{\mathcal{L}}_{\Lambda N\pi}^w=&-iG_Fm_\pi^2\overline{\Psi}_N(A+B\gamma^5)
{\vec{\tau}}\cdot\vec{\pi}\Psi_\Lambda
\\\nonumber
{\mathcal{L}}_{\Sigma N\pi}^w=&
-iG_Fm_\pi^2\overline{\Psi}_N(\vec{A}_{\Sigma_i}+\vec{B}_{\Sigma_i}\gamma^5)
\cdot\vec{\pi}\Psi_{\Sigma_i}\,,
\\\nonumber
{\cal L}^{w}_{NN K} =&
-iG_Fm_\pi^2 \, \left[ \, \overline{\psi}_{N} \left( ^0_1 \right)
\,\,( C_{K}^{PV} + C_{K}^{PC} \gamma_5) \,\,(\phi^{K})^\dagger
\psi_{N} \right.
\\ \nonumber
& \left. + \, \overline{\psi}_{N} \psi_{N}
\,\,( D_{K}^{PV}
+ D_{K}^{PC}
\gamma_5 ) \,\,(\phi^{K})^\dagger \,\,
\left( ^0_1 \right) \right] \ ,\end{aligned}$$
where $G_Fm_\pi^2=2.21\times10^{-7}$ is the weak Fermi coupling constant, $\gamma$ are the Dirac matrices and $\tau$ the Pauli matrices. The index $i$ appearing in the $\Sigma$ field refers to the different isospurion states for the $\Sigma$ hyperon: $$\Psi_{\Sigma\frac12}=
\left(\begin{array}{c}-\sqrt{\frac23}\Sigma_+\\\frac{1}{\sqrt3}\Sigma_0\end{array}\right)\,,
~~
\Psi_{\Sigma\frac32}=
\left(\begin{array}{c}0\\-\frac{1}{\sqrt3}\Sigma_+\\\sqrt{\frac23}\Sigma_0\\\Sigma_-\,
\end{array}\right)\,.$$ The PV and PC structures, $\vec{A}_{\Sigma_i}$ and $\vec{B}_{\Sigma_i}$ contain the corresponding weak coupling constants together with the isospin operators $\tau^a$ for $\frac12\to\frac12$ transitions and $T^a$ for $\frac12\to\frac32$ transitions. The weak couplings $A=1.05$, $B=-7.15$, $A_{\Sigma\frac12}=-0.59$, $A_{\Sigma\frac32}=2.00$, $B_{\Sigma\frac12}=-15.68$, and $B_{\Sigma\frac32}=-0.26
| 1,777
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|
\
b(j) \approx \max_{j} g(\hat\psi_j).$$
[**Proof of .**]{} We will establish the claims by bound the quantity $\left\|
\hat\beta_{S} - \beta_{S}
\right\| $ uniformly over all $\beta_S \in H_n$.\
Our proof relies on a first order Taylor series expansion of of $g$ and on the uniform bound on the norm of the gradient of each $g_j$ given in. Recall that, by conditioning on $\mathcal{D}_{1,n}$, we can regard $S$ and $\beta_{S}$ as a fixed. Then, letting $G(x)$ be the $|S| \times b$-dimensional Jacobian of $g$ at $x$ and using the mean value theorem, we have that $$\begin{aligned}
\left\|
\hat\beta_{S} - \beta_{S}
\right\| & =
\left\|
\left(
\int_0^1 G\bigl( (1-t)\psi_{S} + ut \hat \psi_{S} \bigr) dt
\right)
(\hat\psi_{S} - \psi_{S})
\right\|\\
& \leq
\left\| \int_0^1 G\bigl( (1-t)\psi_{S} + t \hat
\psi_{S}\bigr) dt \right\|_{\mathrm{op}} \left\| \hat{\psi}_{S} - \psi_{S} \right\|.
$$ To further bound the previous expression we use the fact, established in the proof of , that $\| \hat{\psi}_{S} - \psi_{S} \| \leq C k \sqrt{ \frac{ \log n + \log k}{n}
}$ with probability at least $1/n$, where $C$ depends on $A$, for each $P \in \mathcal{P}_n^{\mathrm{OLS}}$. Next, $$\left\| \int_0^1 G\bigl( (1-t)\psi_{S} + t \hat
\psi_{S} \bigr) dt \right\|_{\mathrm{op}} \leq
\sup_{t \in (0,1)} \left\| G\left( 1-t)\psi_{S} + t \hat
\psi_{S} \right) \right\|_{\mathrm{op}} \leq \sup_{t \in (0,1)} \max_{j \in S} \left\| G\left( 1-t)\psi_{S} + t \hat
\psi_{S} \right) \right\|$$ where $G_i(\psi)$ is the $j^{\mathrm{th}}$ row of $G(\psi)$, which is the gradient of $g_j$ at $\psi$. Above, the first inequality relies on the convexity of the operator norm and the second inequality uses that the fact that the operator norm of a matrix is bounded by the maximal Euclidean norm of the rows. For each $P \in \mathcal{P}_n^{\mathrm{OLS}}$ and each $t \in (0,1)$ and $j \in S$, the bound in yields that, for a $C>0$ depending on $A$ only, $$\left\| G\left( 1-t)\psi_{S} + t \hat
\psi_{S} \right) \right\| \leq
| 1,778
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| 1,660
| null | null |
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|
ral static bubble. We also see from the Hamiltonian constraint that for small $Q/L^2$, curvatures near the bubble are of order $1/Q$. Therefore there is a minimum $Q$, controlled by the cutoff, for which we can study this geometry classically.
Adding Charged Matter
---------------------
We would like to study the stability of the flux-stabilized bubble against the introduction of probe charges. In the 6D description, charged objects are wound strings. To simplify the analysis, we consider only the lowest states of a string with winding number one and zero KK excitations. Formally, we can dimensionally reduce over the circle to obtain a 5D geometry with ordinary electromagnetic flux, and we introduce a massive scalar particle with charge $q$ to represent the wound string. Near the bubble wall the scalar mass decreases with the radion.
In this toy model we can study the single-particle ground states of positive and negative charge as a function of $q$. For $|q|/m\gtrsim 1$, the electrostatic potential energy of a negatively charged particle near the bubble wall is sufficient to compensate for its rest mass energy at infinity. The naïve vacuum in the zero-charge sector is then unstable against spontaneous pair creation, and in the subsequent section we argue that the bubble discharge rate is unsuppressed.
We begin by parametrizing the 6D spacetime (\[eq:dsfull\]) as $$\begin{aligned}
ds^2=G_{\mu\nu}dx^\mu dx^\nu+Vd\chi^2\;
\label{eq:fivemetric}\end{aligned}$$ where $V=U(\rho)$ by comparison with (\[eq:spatial\]). The dimensional reduction of the three-form flux gives rise to an ordinary 5D electric field, and we choose a gauge where the field arises from a scalar potential vanishing at infinity, $$\begin{aligned}
A_t=\frac{\sqrt{L}Q_0}{4\pi^2\rho^2}\;.
\label{eq:At}\end{aligned}$$
Dimensionally reducing the worldsheet action for a wound string of tension $T$ with no $\chi$ excitations, the corresponding worldline action for the free particle is $$\begin{aligned}
m \int d\tau \sqrt{V}\sqrt{-G_{\mu\nu}\partial
| 1,779
| 785
| 1,950
| 1,903
| null | null |
github_plus_top10pct_by_avg
|
- e^{- i h_{k} x} \right)
\left( e^{+ i \Delta_{L} x} - e^{+ i h_{l} x} \right)
\nonumber \\
&\times&
\biggl[
(UX)_{\alpha k} W^*_{\beta K}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
+
W_{\alpha K} (UX)^*_{\beta k}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\biggr]
\nonumber \\
&\times&
\biggl[
(UX)_{\alpha l}^* W_{\beta L}
\left\{ W ^{\dagger} A (UX) \right\}_{L l}
+
W_{\alpha L}^* (UX)_{\beta l}
\left\{ (UX)^{\dagger} A W \right\}_{l L}
\biggr]
\nonumber \\
&+&
\sum_{K}
\vert W_{\alpha K} \vert^2 \vert W_{\beta K} \vert^2
+ \sum_{K \neq L}
e^{- i ( \Delta_{K} - \Delta_{L} ) x}
W_{\alpha K} W^*_{\beta K} W_{\alpha L}^* W_{\beta L}.
\label{S(2)-squared-1}\end{aligned}$$ Except for the first term in (\[S(2)-squared-1\]) we did not try to unify the two exponential factors because the expressions become cumbersome. Apart from the last line in (\[S(2)-squared-1\]) all the terms are suppressed by the two energy denominators with $\Delta m^2_{J k}$ which doubly suppress the active-sterile state transition. The first term in the last line is the probability leaking term mentioned in section \[sec:nonunitarity-vacuum\].
The second term of $\left| S^{(2)}_{\alpha \beta} \right|^2$ (interference terms) is given by $$\begin{aligned}
&& \left| S^{(2)}_{\alpha \beta} \right|^2_{\text{2nd}} =
- 2 \mbox{Re}
\biggl\{
\sum_{k, K}
\sum_{l \neq m} \sum_{L}
\frac{ 1 }{ ( \Delta_{K} - h_{k} ) (\Delta_{L} - h_{l}) (\Delta_{L} - h_{m}) ( h_{m} - h_{l} ) }
\nonumber \\
&\times&
\biggl[
\left( \Delta_{L} - h_{l} \right) e^{- i h_{m} x}
- \left( \Delta_{L} - h_{m} \right) e^{- i h_{l} x}
- ( h_{m} - h_{l} ) e^{- i \Delta_{L} x}
\biggr]
\left[
- (ix) e^{+ i h_{k} x} + \frac{e^{+ i \Delta_{K} x} - e^{+ i h_{k} x} }{ ( \Delta_{K} - h_{k} ) }
\right]
\nonumber \\
&\times&
(UX)_{\alpha l} (UX)^*_{\beta m}
(UX)_{\alpha k}^* (UX)_{\beta k}
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{l L}
\left\{ W ^{\dagger} A (UX) \right\}_{L m}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A
| 1,780
| 1,877
| 2,206
| 1,806
| null | null |
github_plus_top10pct_by_avg
|
, etc..);
Foo foo(bar);
But if bar is a big object (for example, contains an array), then it would be inefficient to pass bar by value. The only way is to have bar passed by reference or pointer, but I also want the user to be able to use bar with parameterized constructor, and hence my confusion began.
Regarding to the auto variable scope, it would be safe if a user do something like:
int main() {
Bar1 bar1(1,2,3);
Foo foo(&bar1);
return 0;
}
As long as Foo does not delete bar1.
A:
You can't detect the difference between stack and heap allocation (the standard doesn't even mention a stack), but that's not the issue here.
Foo should not be deleting things that it does not own. If it wants a copy for itself then it should just make a copy:
template <class T = Bar>
class Foo
{
T t_;
public:
Foo() {}
Foo(const T& t) : t_(t) {}
};
Or if you need it to be a pointer:
template <class T = Bar>
class Foo
{
T* t_;
public:
Foo() : t_(new T()) {}
Foo(const T& t) : t_(new T(t)) {}
};
You can't just go around deleting things that people give you, and that's regardless of whether you know if it's stack or heap allocated. What if your Foo goes and deletes it but the calling code still wants to use it? Or what if that same object is passed into two of your Foo objects and they both delete it?
Your option is either to make a copy, or not delete it.
An alternative to doing the copy would be to mark whether you are using your own or someone else's:
template <class T = Bar>
class Foo
{
T* t_;
bool owned;
public:
Foo() : t_(new T()), owned(true) {}
Foo(T* t) : t_(t), owned(false) {}
~Foo() { if (owned) delete t_; }
};
In case you are wondering, t is passed by const-ref in my previous solutions, not by value, so there is no excessive expense, although there could be an expense in the copying.
A:
Use a smart pointer and you won't have to worry about memory management. If you really have to, I guess you should have the user of your class handle the memory they pass in, your class should be only respon
| 1,781
| 2,099
| 708
| 1,351
| 858
| 0.798823
|
github_plus_top10pct_by_avg
|
um_{ij\in \text{C}}\sum_n
\left|\left[\hat{g}_0(i\w_n)\right]_{ij}
-\left[\hat{g}_{0,N_{\text{B}}}(i\w_n)\right]_{ij}\right|^2
e^{-\w_n/t},
\label{eq:distance}\end{aligned}$$ where $\hat{g}_{0,N_{\text{B}}}$ is the non-interacting Green’s function for the effective Hamiltonian and we have introduced the exponential weight factor $e^{-\w_n/t}$ with $\w_n\equiv (2n+1)\pi/\b'$ to reproduce more precisely the important low-frequency part. We have examined several other types of distance functions and confirmed that qualitative feature of the results obtained in this paper does not depend on the choice.
An advantage of the Lanczos method is that Green’s function is obtained as a function of real frequency $\w$ by the continued-fraction expansion, $$\begin{aligned}
G(\w)&=\frac{\langle 0| c c^\dagger | 0\rangle}
{\w+i\eta-a_0^>-\displaystyle{ \frac{{b_1^>}^2}{\w+i\eta-a_1^>
- \displaystyle{\frac{{b_2^>}^2}{\w+i\eta-a_2^>-\cdots}}}}}\nonumber\\
&+\frac{\langle 0| c^\dagger c | 0\rangle}
{\w+i\eta-a_0^<-\displaystyle{\frac{{b_1^<}^2}{\w+i\eta-a_1^<
-\displaystyle{\frac{{b_2^<}^2}{\w+i\eta-a_2^<-\cdots}}}}},
\label{eq:cf}\end{aligned}$$ where $|0\rangle$ is the ground-state vector and the coefficients $a_i^{>,<}$ and $b_i^{>,<}$ are the elements of the tridiagonal matrix appearing in the Lanczos algorithm.[@gb87] We take account of up to 2000th order in the expansion. A small positive $\eta$ is introduced to satisfy the causality. In principle, $\eta$ is taken to be infinitesimal, but in practice, it is useful to consider $\eta$ as a parameter, which serves as a resolution in energy or mimics an infinite-size effect not incorporated into the ED calculation.
In the following study, we pursue a further benefit of the parameter $\eta$: We use $\eta$ as a mimic of the source of incoherence in the electronic structure, such as thermal or impurity scattering. This is based on the observation that $\eta$ does not substantially change the location of poles and zeros of $G$ but changes only t
| 1,782
| 3,916
| 2,062
| 1,424
| 3,104
| 0.77486
|
github_plus_top10pct_by_avg
|
18}\cap[G,G]\right)\\
&\subset\left(A^{18}\cap\pi^{-1}(H)\right)\prod_{i=1}^r\left(A^{24}\cap\pi^{-1}(\langle x_i\rangle)\right).\end{aligned}$$ Since $a$ was an arbitrary element of $\pi^{-1}(HP)\cap A^6$, the proposition then follows from .
It is at this point that we diverge from the original proof of \[thm:old\].
\[prop:ind.tor-free.post.chang\] Let $m>0$ and $s\ge\tilde s\ge2$ be integers, and let $K,\tilde K\ge2$. Let $G$ be an $s$-step nilpotent group generated by a finite $K$-approximate group $A$, and let $\tilde A\subset A^m$ be a $\tilde K$-approximate group that generates an $\tilde s$-step nilpotent subgroup $\tilde G$ of $G$. Then there exist a normal subgroup $N\lhd G$ with $N\subset A^{K^{e^{O(s)}m}}$; an integer $r\le\log^{O(1)}2\tilde K$; finite $\tilde K^{O(1)}$-approximate groups $A_1,\ldots,A_r\subset\tilde A^{O(1)}$ such that, writing $\rho:G\to G/N$ for the quotient homomorphism, each group $\langle\rho(A_i)\rangle$ has step less than $\tilde s$; and a set $X\subset\tilde A$ of size at most $\exp(\log^{O(1)}2\tilde K)$ such that $\tilde A\subset XA_1\cdots A_r$.
This is immediate from \[prop:key.general,lem:covering\].
Using \[prop:ind.tor-free.post.chang\] to induct on the step, we arrive at the following result.
\[prop:tor-free.post.induc\] Let $m>0$ and $s\ge\tilde s\ge1$ be integers, and let $K,\tilde K\ge2$. Let $G$ be an $s$-step nilpotent group generated by a finite $K$-approximate group $A$, and let $\tilde A\subset A^m$ be a $\tilde K$-approximate group that generates an $\tilde s$-step nilpotent subgroup $\tilde G$ of $G$. Then there exist integers $r,\ell\le e^{O(\tilde s^2)}\log^{O(\tilde s)}2\tilde K$; a normal subgroup $N\lhd G$ satisfying $$\label{eq:tor-free.post.induc.N}
N\subset A^{e^{O(\tilde s^2)}K^{e^{O(s)}m}\log^{O(\tilde s)}2\tilde K};$$ finite $\tilde K^{e^{O(\tilde s)}}$-approximate groups $A_1,\ldots,A_r\subset\tilde A^{e^{O(\tilde s)}}$ such that, writing $\pi:G\to G/N$ for the quotient homomorphism, each group $\langle\pi(A_i)\rangle$ is abelian; and sets
| 1,783
| 1,103
| 1,180
| 1,850
| 4,170
| 0.767605
|
github_plus_top10pct_by_avg
|
})$ would be a good target for PINGU extensions of IceCube and KM3NeT-ORCA [@TheIceCube-Gen2:2016cap; @Adrian-Martinez:2016zzs]. $P(\nu_{\mu} \rightarrow \nu_{\tau})$ and $P(\nu_{\mu} \rightarrow \nu_{\mu})$ would be explored by them, with possibility of seeing anticorrelation between $\mu$ and $\tau$ yields. Although it is very interesting to investigate these experimental prospects, a detailed examination of these questions is beyond the scope of this paper.
The probability leaking and $W$ correction terms {#sec:correction-terms}
--------------------------------------------------
As we stated in section \[sec:low-vs-high\], uncovering the effect of $W$ correction or the probability leaking term $\mathcal{C}_{\alpha \beta}$ in eq. (\[Cab\]) would offer another way of distinguishing low-scale unitarity violation from high-scale one. Let us give a brief sketch of how and where we might see visible effect.
### A further note on the parameter choice {#sec:parameter-choice2}
To discuss $W$ correction and the probability leaking term we have to determine the $W$ matrix. Given the non-unitary $U$ matrix there is a way to construct the $W$ matrix. In general, it is given by $$W = S \sqrt{w} R,
\label{W-construction}$$ where $S$ is a $3\times 3$ matrix which diagonalizes ${\bf 1}_{3\times 3} - UU^\dagger$, $w$ is diagonal matrix which consists of eigenvalues of ${\bf 1}_{3\times 3} - UU^\dagger$, and $R$ is an arbitrary $3\times N$ complex matrix obeying $RR^\dagger = \bf{1}_{3 \times 3}$. The construction makes sense for $N \geq 3$. Therefore, for a given $N$ there is a large arbitrariness on the choice of the $W$ matrix, and hence on the sizes of the $W$ corrections and $\mathcal{C}_{\alpha \beta}$.
Lacking a guiding principle of how to choose the $R$ matrix in (\[W-construction\]), we examine the cases with largest and smallest possible values of $\mathcal{C}_{\alpha \beta}$ for given values of unitarity violation $1 - \sum_{j=1}^3 |U_{\alpha j}|^2$ ($\alpha=e,\mu,\tau$). It is shown that in the $(3+N)$ mo
| 1,784
| 1,940
| 3,123
| 1,950
| 1,928
| 0.784326
|
github_plus_top10pct_by_avg
|
_\mathrm{vir}$), whereas it suddenly drops to below $10^5$ K around $0.2 R_{\rm vir}$ when metal-line cooling is included. Thus, the catastrophic cooling flow of the diffuse, hot component in the inner haloes is due to metals. Indeed, while the median hot-mode radial peculiar velocity within $0.2 R_{\rm vir}$ is positive without metal-line cooling, it becomes negative (i.e. infalling) when metal-line cooling is included.
Evolution: Milky Way-sized haloes at $z=0$ {#sec:z0}
==========================================
Fig. \[fig:haloradz0\] is identical to Fig. \[fig:haloradz2\] except that it shows profiles for $z=0$ rather than $z=2$. For comparison, the dotted curves in Fig. \[fig:haloradz0\] show the corresponding $z=2$ results. As we are again focusing on $10^{11.5}$ M$_\odot<M_\mathrm{halo}<10^{12.5}$ M$_\odot$, the results are directly relevant for the Milky Way galaxy.
Comparing Fig. \[fig:haloradz2\] to Fig. \[fig:haloradz0\] (or comparing solid and dotted curves in Fig. \[fig:haloradz0\]), we see that the picture for $z=0$ looks much the same as it did for $z=2$. There are, however, a few notable differences. The overdensity profiles hardly evolve, although the difference between the cold and hot modes is slightly smaller at lower redshift. However, a constant overdensity implies a strongly evolving proper density ($\rho \propto (1+z)^3$) and thus also a strongly evolving cooling rate. The large decrease in the proper density caused by the expansion of the Universe also results in a large drop in the pressure and a large increase in the entropy.
The lower cooling rate shifts the peak of the cold-mode temperature profile from about 2$R_{\rm vir}$ to about $R_{\rm vir}$. While there is only a small drop in the temperature of the hot-mode gas, consistent with the mild evolution of the virial temperature of a halo of fixed mass ($T_{\rm vir} \propto (1+z)$; eq. \[\[eqn:virialtemperature\]\]), the evolution in the median
| 1,785
| 958
| 1,976
| 1,877
| null | null |
github_plus_top10pct_by_avg
|
a$ is generated by the one-loop diagram to which $Z_2$-odd particles contribute. The lightest $Z_2$-odd scalar boson can be a candidate for the dark matter. We briefly discuss a characteristic signal of our model at the LHC.'
author:
- Shinya Kanemura
- Hiroaki Sugiyama
title: |
Dark matter and a suppression mechanism for neutrino masses\
in the Higgs triplet model
---
introduction {#sec:intro}
============
Existence of dark matter (DM) has been established, and its thermal relic abundance has been determined by the WMAP experiment [@WMAP; @Komatsu:2008hk]. If the essence of DM is an elementary particle, the weakly interacting massive particle (WIMP) would be a promising candidate. It is desired to have a viable candidate for the dark matter in models beyond the standard model (SM). The WIMP dark matter candidate can be accommodated economically by introducing only an inert scalar field [@Silveira:1985rk; @Deshpande:1977rw; @i-nplet], where we use “inert” for the $Z_2$-odd property. The imposed $Z_2$ parity ensures the stability of the DM candidate. Phenomenology in such models have been studied in, e.g., Refs. [@c-i-singlet; @r-i-singlet; @i-doublet; @Lundstrom:2008ai; @Araki:2011hm; @THDM-iSDM; @HTM-iSDM].
On the other hand, it has been confirmed by neutrino oscillation measurements that neutrinos have nonzero but tiny masses as compared to the electroweak scale [@solar; @atm; @acc; @reactor-S; @reactor-L]. The different flavor structure of neutrinos from that of quarks and leptons may indicate that neutrino masses are of Majorana type. In order to explain tiny neutrino masses, many models have been proposed. The seesaw mechanism is the simplest way to explain tiny neutrino masses, in which right-handed neutrinos are introduced with large Majorana masses [@seesaw; @Mohapatra:1979ia]. Another simple model for generating neutrino masses is the Higgs Triplet Model (HTM) [@Mohapatra:1979ia; @HTM]. However, these scenarios do not contain dark matter candidate in themselves.
In a class of m
| 1,786
| 200
| 2,703
| 1,868
| null | null |
github_plus_top10pct_by_avg
|
nerally become sparser the higher the order. Hence, we come up with many more ties and the chance is higher that we assign higher ranks for observed transitions in the testing data. The most extreme case happens when we do not have any information available for observations in the testing set (which frequently happens for higher orders); then we assign the maximum rank (i.e., the number of states) to all states. We finally average the ranks over all folds for a given order and suggest the model with the lowest average rank.[^6]
This method requires priors (i.e., fake counts; see the section named “”) – otherwise prediction of unseen states is not possible. It also resorts to the maximum likelihood estimate for calculating the parameters of the models as described in the section entitled “”. Also, as shown in the previous section called “” cross validation has asymptotic equivalence to AIC.
One disadvantage of cross validation methods usually is that the results are dependent on how one splits the data. However, by using our stratified k-fold cross validation approach, we counteract this problem as it matters less of how the data is divided. Yet, by doing so we need to rerun the complete evaluation k times, which leads to high computational expenses compared to the other model selection techniques described earlier and we have to manually decide of which k to use. One main advantage of this method is that eventually each observation is used for both training and testing.
-- ----------------------------------------------------- -- --
**[Wikigame]{} & **[Wikispeedia]{} & **[MSNBC]{}\
\#Page Ids & 360,417 & n/a & n/a\
\#Topics & 25 & 15 & 17\
\#Paths & 1,799,015 & 43,772 & 624,383\
\#Visited nodes & 10,758,242 & 259,019 & 4,333,359\
******
-- ----------------------------------------------------- -- --
: **[Dataset statistics]
| 1,787
| 6,575
| 976
| 592
| null | null |
github_plus_top10pct_by_avg
|
- C(A) - R(A) + J\end{aligned}$$ First note that the rank of $\breve{A}$ and that of $A$ can differ by at most 3. Now, consider the case where $A$ is the Gram matrix of some vectors $v_1,...,v_n \in R^d$. Then all diagonal entries of $\breve{A}$ equal one, and the $(i,j)$ entry is 2$<v_i,v_j> - <v_i,v_i> - <v_j,v_j> + 1 = 1 - ||v_i-v_j||^2$.
We will need the following lemma (see [@HoJo], p.175):
\[d-lemma\] Let $X$ be a real symmetric matrix, then $rank(X) \geq \frac {(tr X)^2} {\sum_{i,j}X_{i,j}^2}$
Applying this to $\breve{A}$, we conclude that: $$\begin{aligned}
\label{d-bound}
rank(\breve{A}) \geq \frac {n^2} {n +
\sum_{i \neq j} (1 - ||v_i - v_j||^2)^2 }\end{aligned}$$
Let $v_1,...,v_n \in \R^d$ be an embedding of $G$. By the discussion above it is enough to show that $$\begin{aligned}
\label{denom}
\sum_{i \neq j} (1-||v_i - v_j||^2)^2 = O(D^2 n^2 \frac {\lambda_2}
{d - \lambda_2}).\end{aligned}$$ By the triangle inequality $||v_i - v_j|| \leq D$ for any two vertices. So the LHS of (\[denom\]) is bigger by at most a factor of $D^2$ than: $$\begin{aligned}
&& \sum_{(i,j) \notin E} (||v_i - v_j||^2 - 1) +
\sum_{(i,j) \in E} (1 - ||v_i - v_j||^2) =\end{aligned}$$ $$\begin{aligned}
\label{hs-norm}
&& \sum_{(i,j) \notin E} ||v_i - v_j||^2 -
\sum_{(i,j) \in E} ||v_i - v_j||^2 - {n \choose 2} + nd\end{aligned}$$ We can bound this sum from above, by solving the following SDP: $$\begin{aligned}
& \max & \sum_{(i,j) \notin E} (V_{ii} + V_{jj} - 2V_{ij}) +
\sum_{(i,j) \in E} (- V_{ii} - V_{jj} + 2V_{ij}) - {n \choose 2} + nd\\
& s.t. & V \in PSD \\
& \forall (i,j) \in E & V_{ii} + V_{jj} - 2V_{ij} \leq 1 \\
& \forall (i,j) \notin E & V_{ii} + V_{jj} - 2V_{ij} \geq 1\end{aligned}$$ The dual problem is: $$\begin{aligned}
& \min & {\frac 1 2}tr A \\
& s.t. & A \in PSD \\
& \forall (i,j) \in E & A_{ij} \leq -1 \\
& \forall (i,j) \notin E, i\neq j & A_{ij} \geq 1 \\
& \forall i \in [n] & \sum_{j=1,...,n} A_{ij} = 0\end{aligned}$$ Let $M$ by the adjacency matrix of the graph, and set $A = (\alpha d - n)I + J - \alpha
| 1,788
| 3,935
| 2,387
| 1,636
| 3,677
| 0.770774
|
github_plus_top10pct_by_avg
|
{Q-1}
\end{pmatrix}.\end{aligned}$$
Definition of $\rho_{{\mbox{\boldmath $\alpha$}}}(f_i)$
-------------------------------------------------------
Next, we define a linear map for $f_i$.
For a tableaux $P = ({\mbox{\boldmath $\alpha$}}^{(0)}, {\mbox{\boldmath $\alpha$}}^{(1/2)}, \ldots, {\mbox{\boldmath $\alpha$}}^{(n)})$ of ${\mathbb T}({\mbox{\boldmath $\alpha$}})$, we define $\rho_{{\mbox{\boldmath $\alpha$}}}(f_i)(v_P)
= \sum_{Q \in {\mathbb T}({\mbox{\boldmath $\alpha$}})}(F_i)_{QP}v_Q$. Let $Q = ({\mbox{\boldmath $\alpha$}}^{\prime(0)}, {\mbox{\boldmath $\alpha$}}^{\prime(1/2)},
\ldots, {\mbox{\boldmath $\alpha$}}^{\prime(n)})$.
If there is an $i_0 \in \{1/2, 1, \ldots, n-1/2 \} \setminus \{i\}$ such that ${\mbox{\boldmath $\alpha$}}^{(i_0)}\neq {\mbox{\boldmath $\alpha$}}^{\prime(i_0)}$, then we put $$(F_i)_{QP} = 0.$$ In the following, we consider the case that ${\mbox{\boldmath $\alpha$}}^{(i_0)} = {\mbox{\boldmath $\alpha$}}^{\prime(i_0)}$ for $i_0\in\{0, 1/2, 1, \ldots, n-1/2\}\setminus\{i\}$.
If ${\mbox{\boldmath $\alpha$}}^{(i-1/2)}$ and ${\mbox{\boldmath $\alpha$}}^{(i+1/2)}$ are not labeled by the same Young diagram, then we put $$(F_i)_{QP} = 0.$$
We consider the case ${\mbox{\boldmath $\alpha$}}^{(i-1/2)}$ and ${\mbox{\boldmath $\alpha$}}^{(i+1/2)}$ have the same label $\widehat{\mu}$. In this case, the possible vertices as ${\mbox{\boldmath $\alpha$}}^{(i)}$ have labels $\{\widehat{\mu}^{+}_{(r)}\}$, which are obtained by adding one box to $\widetilde{\mu}$. Suppose that ${\mbox{\boldmath $\alpha$}}^{(i)}$, the $i$-th coordinate of $P$, has its label $\widetilde{\mu}^{+}_{(r_0)}$. Let $Q$ be a tableau obtained from $P$ by replacing ${\mbox{\boldmath $\alpha$}}^{(i)}$ with one of $\{\widehat{\mu}^{+}_{(r)}\}$.
Then we define $(F_i)_{QP}$ to be $$(F_i)_{Q_rP} = \frac{h(\widehat{\mu})}{h(\widehat{\mu}^{+}_{(r_0)})}.$$
Let $v(\mu^+_{(r)}, \mu)$ be the standard vector which corresponds to a tableau whose $(i-1/2)$-th, $i$-th and $(i+1/2)$-th coordinate $({\mbox{\boldmath $\alpha$}}^{(i-1/2)}
| 1,789
| 1,911
| 1,903
| 1,605
| 2,278
| 0.781193
|
github_plus_top10pct_by_avg
|
ametricstable}
\dot{\beta}_2(k) = i\omega_2(k)\beta_2(k) + \int \limits_{-\infty}^{\infty}\frac{dk'}{2\pi}D_1(k,k')\beta_1(k')\beta_1(k'').$$ Note that for these times $\beta_2 \ll \beta_1$ therefore the $D_1$ terms are the largest of the $D_j$ terms. Since the $C_j$ nonlinearities have not reached the amplitudes of the growing linear terms, $\beta_1$ can be approximated as $\beta_i \exp[i \omega_1 t]$. These approximations are referred to as the parametric instability approximations[@Terry2006]. Then Eq. is solved by $$\label{parametricsolution}
\beta_2(k,t) = \int \limits_{-\infty}^{\infty}\frac{dk'}{2\pi} \frac{D_1(k,k')\beta_i^2}{i\left(-\omega_2(k)+\omega_1(k')+\omega_1(k-k')\right)} \left[ e^{i(\omega_1(k')+\omega_1(k-k'))t} - e^{i\omega_2(k)t} \right] + \beta_i e^{i \omega_2(k) t}.$$ In assessing $P_t$ the above integral is only taken over unstable wavenumbers, as they drive $\beta_2$ more strongly than marginally stable modes.
![Nonlinear terms in Eq. at saturation for $k=0.4$ and $\beta_i=0.01,$ with $k'$ and $k-k'$ ranging from $-0.6$ to $0.6$. The $C_1$ term is responsible for the Kolmogorov-like saturation of the instability by energy transfer to unstable modes at smaller wavelengths. The $C_2$ term represents the previously-neglected coupling between unstable modes at $k$ and $k'$ with stable modes at $k''$. The threshold parameter is evaluated by dividing the peak value of the $C_2$ term by the peak value of the $C_1$ term. Here we find $P_t \approx 6$, indicating that stable modes are important in KH saturation.[]{data-label="PTscan"}](Fig4.eps){width="16cm"}
To evaluate $P_t$, the ratio of the largest $\beta_1\beta_2$ term and the largest $\beta_1\beta_1$ term in Eq. is taken at the time of saturation $t_s$: $$\label{Ptnew}
P_t = \left[\frac{\max|2C_2\beta_1(k')\beta_2(k'')|}{\max|C_1\beta_1(k')\beta_1(k'')|}\right]_{t=t_s},$$ where $t_s$ is defined as the time at which one of the nonlinearities in Eq. reaches the same amplitude as the linear term. Figure \[PTscan\] shows the size of the
| 1,790
| 1,778
| 3,191
| 1,932
| 2,101
| 0.782718
|
github_plus_top10pct_by_avg
|
hat \in
\P^k$ as above define $$a_\muhat:=a_{\mu^1}(\x_1)\cdots a_{\mu^k}(\x_k).$$ Let $\langle\cdot ,\cdot \rangle$ be the Hall pairing on $\Lambda(\x),$ extend its definition to $\Lambda(\x_1,\ldots,\x_k)$ by setting \[extendedhall\] a\_1(\_1)a\_k(\_k), b\_1(\_1)b\_k(\_k)= a\_1, b\_1 a\_k, b\_k , for any $a_1,\ldots,a_k;b_1,\ldots,b_k\in \Lambda(\x)$ and to formal series by linearity.
### Cauchy identity {#Cauchy}
Given a partition $\lambda\in\calP_n$ we define the genus $g$ [*hook function*]{} $\calH_{\lambda}(z,w)$ by $$\calH_{\lambda}(z,w):=
\prod_{s\in \lambda}\frac{(z^{2a(s)+1}-w^{2l(s)+1})^{2g}}
{(z^{2a(s)+2}-w^{2l(s)})(z^{2a(s)}-w^{2l(s)+2})},$$ where the product is over all cells $s$ of $\lambda$ with $a(s)$ and $l(s)$ its arm and leg length, respectively. For details on the hook function we refer the reader to [@hausel-villegas].
Recall the specialization (cf. [@hausel-letellier-villegas §2.3.5]) $$\calH_\lambda(0,\sqrt{q})=\frac{q^{g\langle\lambda,\lambda\rangle}}{a_\lambda(q)}\label{alambda}$$where $a_\lambda(q)$ is the cardinality of the centralizer of a unipotent element of $\GL_n(\F_q)$ with Jordan form of type $\lambda$.
It is also not difficult to verify that the Euler specialization of $\calH_\lambda$ is
$$\calH_\lambda(\sqrt{q},1/\sqrt{q})=\left(q^{-\frac{1}{2}\langle\lambda,\lambda\rangle}H_\lambda(q)\right)^{2g-2}.
\label{H-specializ}$$
We have $$\calH_\lambda(z,w)=\calH_{\lambda'}(w,z)\,\,\,{\rm and }\,\,\,\calH_\lambda(-z,-w)=\calH_\lambda(z,w).\label{hook-duality}$$
Let $$\Omega(z,w)=\Omega(\x_1,\dots,\x_k;z,w):=\sum_{\lambda\in \calP} \calH_{\lambda}(z,w)
\prod_{i=1}^k\tilde{H_\lambda}(\x_i;z^2,w^2).$$By (\[Hduality\]) and (\[hook-duality\]) we have $$\Omega(z,w)=\Omega(w,z)\,\,\,{\rm and }\,\,\,\Omega(-z,-w)=\Omega(z,w).\label{Oduality}$$
For $\muhat=(\mu^1,\cdots,\mu^k)\in\P^k$, we let
$$\H_\muhat(z,w):=(z^2-1)(1-w^2)\left\langle\Log\,\Omega(z,w),h_\muhat\right\rangle.
\label{H}$$
By (\[Oduality\]) we have the symmetries
$$\H_\muhat(z,w)=\H_\muhat(w,z) \,\,\,{\rm and}\,\,\
| 1,791
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| 1,895
| 1,739
| 3,908
| 0.769355
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to test the hypothesis: $\beta_{j}=\beta_{j0}$ for some $1\leq j\leq p$, or $\beta=\beta_{0}$.
We let $\beta$ be a $7\times 1$ vector with all coordinates equal to 0.2. $Z_i$ comes from seven distributions which were used in [@Shi2018].
- $N(0, \Sigma)$, $\Sigma=(\rho_{ij})$ with $\rho_{ij}=0.5^{I(i\neq j)}$, where $I(\cdot)$ is the indicator function.
- $N(1.5, \Sigma)$.
- $0.5 N(1, \Sigma) + 0.5 N(-1, \Sigma)$.
- The multivariate $t$ distribution $t_{3}(0, \Sigma)/10$, with degrees of freedom 3.
- The multivariate exponential distribution whose components are independent and each has an exponential distribution with a rate parameter of 2.
- $0.5 N(-2.14, \Sigma) + 0.5 N(-2.9, \Sigma)$.
Here, Cases 2 and 5 produce imbalanced data. Case 6 produces rare events data.
Tables \[table4\]-\[table7\] show the empirical sizes and powers. When we consider powers of test, the null hypothesis is that the parameter $\beta_j$ is zero. Tables \[table8\] and \[table9\] summarize the lengths of confidence intervals. We draw the following conclusions.
1. Regardless of imbalanced data or the rare events data, the empirical sizes of our proposed method are close to the nominal level, which implies our method performs well. Moreover, the empirical power is very close to 1. The differences among three values of K is not significantly.
2. As $\gamma$ increases, the performance of BLB and SDB becomes better. However, they are worse than our method. TB slightly inflated rejection probabilities under the null hypothesis. From Tables \[table8\] and \[table9\], the length of confidence intervals decreases as $\gamma$ increases. When $\gamma=0.6$, it is ten times as long as TB, which results in the lower empirical size.
3. In terms of empirical powers, mVC and mMSE outperform BLB and SDB. Compared with mVC and mMSE, our method is better, especially in the case imbalanced data and rare events data in terms of empirical sizes and powers.
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\bet
| 1,792
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can be identified with a (weighted) *colimit* functor, then of course it commutes with all other colimits. It also explains the left-right duality in the first theorem as due to the fact that the class $\mathsf{FIN}$ of finite categories is closed under taking opposites. Thus we can say:
**Answer \#2:** The homotopy theory of spectra is obtained from that of spaces by universally forcing homotopy finite limits to be weighted *colimits*, and dually.
There is one fly in the ointment: the “enrichment” in \[item:ie\] is rather weak: it has only tensors and not cotensors or hom-objects (so it is more properly called simply a “-module” rather than a “-enriched derivator”), and moreover is not itself a derivator, only a “left derivator” (having left homotopy Kan extensions but not right ones). This can be remedied by working with locally presentable $\infty$-categories rather than derivators, which we plan to do in [@gs:enriched]. However, this depends on rather more technical machinery, so it is interesting how much can be done purely in the realm of derivators.
In [@gs:enriched] we will also show more, namely that given $\Phi$ there is a *universal* choice of ${\sV}$ in \[item:ie\], with pointed spaces and spectra being particular examples. The construction again depends on the good behavior of local presentability, so it seems unlikely to hold in general for derivators. However, as noted above, in particular cases such a universal derivator does exist, such as pointed spaces and spectra for the cases $\Phi=\{\emptyset\}$ and $\Phi=\mathsf{FIN}$ respectively. For $\Phi=\mathsf{FINDISC}$ we expect that the universal consists of $E_\infty$-spaces, though we have not proven this.
This paper belongs to a project aiming for an abstract study of stability, and can be thought of as a sequel to [@groth:ptstab; @gps:mayer; @groth:can-can; @ps:linearity] and as a prequel to [@gs:enriched]. This abstract study of stability was developed in a different direction in the series of papers on abstract represen
| 1,793
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es the Randall-Sundrum brane world model [@20]. These two corrections are regular cosmological solutions and allow to avoid the Big-Bang singularity [@19; @16]. One should also note that the above equation may be obtained within other approches (see [@21; @22]).
Static Spherically Symmetric Space-times
========================================
In this section, we study static spherically symmetric space-times, defined by the general metric: $$\begin{aligned}
ds^2 = -B(r) dt^2 + A(r) dr^2 + r^2 d\Omega^2.\end{aligned}$$ First, following exactly the same procedure as for the FLRW case, it is possible to show that there is no second order linear combination made of order 6 scalars. The same conclusion is valid for the classes $\mathcal{R}_{2,2}^0$ and $\mathcal{R}_{8,4}^0$.
Now for order 4 scalars (and more generally for all orders, considering only monomials of the curvature tensor), the result of Deser and Ryzhov [@18] shows that the most general second order action is : $$\begin{aligned}
S_5 = \int d^4x \sqrt{-g} \; \Big( R +\, \sqrt{3} \, \sigma \, \sqrt{ W^{\mu\nu\alpha\beta}W_{\mu\nu\alpha\beta}} \; \Big).\end{aligned}$$ Moreover, there is no other perfect squares than $W^{\mu\nu\alpha\beta}W_{\mu\nu\alpha\beta}$ and $R^2$. Concerning order 4, all the scalars that are perfect squares lead to second order equations of motion inside the square.
Now, starting from order 6 scalars, it is possible to show that there are again only two perfect squares. Thus, we can consider the action : $$\begin{aligned}
S_6 = \int d^4x \sqrt{-g} \; \Bigg( \delta \, \sqrt{ \nabla_\sigma R \nabla^\sigma R } + \, \sqrt{3} \, \gamma \, \sqrt{ C^{\mu\nu\alpha}C_{\mu\nu\alpha}} \; \Bigg),\end{aligned}$$ here $C_{\mu\nu \alpha}$ is the Cotton tensor, expressed in terms of the Weyl tensor as $C_{\mu\nu \alpha} = -2 \nabla^{\sigma} W_{\mu \nu\alpha\sigma} $. Its square may be written in our basis as $C^{\mu\nu\alpha}C_{\mu\nu\alpha}=2 \big( \curv{L}_5-\curv{L}_6 \big) - \frac{1}{6} \curv{L}_8$.
In our search for second order
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mponents indicators are solution of some specific problem.
\[prop2\] The minimization problem ($\mathcal{P}_0$)
$\underset{v\in \mathcal{V}_{1,k}^0 \backslash\{0\}}{\arg\min} {\|v\|}_0 $
has a unique solution (up to a constant) given by $\textbf{1}_{C_{n-k+1}}$.
In other words, $\textbf{1}_{C_{n-k+1}}$ is the sparsest non-zero eigenvector in the space spanned by the eigenvectors associated to the $k$ largest eigenvectors.
We recall that $\|v\|_0=| \left\{ j : v_j \ne 0 \right\}|$. Let $v \in \mathcal{V}_{1,k}^0 \backslash \left\{ 0 \right\}$. Therefore, as $\textbf{1}_{C_{n-k+1}} \in \mathcal{V}_{1,k}^0$, $v$ can be decomposed as $v=\sum\limits_{j=n-k+1}^n \alpha_j \textbf{1}_{C_j}$ where $\alpha=(\alpha_{n-k+1},\dots ,\alpha_n)\in \mathbb{R}^k \ $ and $ \exists j, \ \alpha_j \ne 0$.
The connected components of sizes $c_{n-k+1},\dots, c_n$ are sorted in increasing order of size. Therefore, by Proposition \[prop1\], $\|v\|_0=\textbf{1}_{\alpha_{n-k+1} \ne 0}c_{n-k+1}+\dots +\textbf{1}_{\alpha_n \ne 0} c_n$. The solution of ($\mathcal{P}_0$) is given by the vector in $\mathcal{V}_{1,k}\backslash\{0\}$ with the smallest $\ell_0$-norm such that $\alpha=(\alpha_{n-k+1} , 0, \dots, 0)$ where $\alpha_{n-k+1} \ne 0$.
We can generalize Proposition \[prop2\] to find, iteratively and with sparsity constraint, the other following indicators of connected components.
For $i=n-k+2,\dots,n$, let $\mathcal{V}_{1,k}^i =\left\{v\in \mathcal{V}_{1,k} : v \perp \textbf{1}_{C_l}, l=n-k+1, \dots ,i-1 \right\}$.
\[prop3\] The minimization problem ($\mathcal{P}_i$)
$\underset{v\in \mathcal{V}_{1,k}^i \backslash\{0\}}{\arg\min} {\|v\|}_0 $
has a unique solution (up to a constant) given by $\textbf{1}_{C_i}$.
Solving ($\mathcal{P}_0$) (Proposition \[prop2\]) is a NP-hard problem. In order to tackle this issue, we replace the ${\ell_0}$-norm by its convex relaxation ${\ell_1}$-norm. We can show that the solution of the $\ell_0$ optimization problem is still the same by replacing the $\ell_0$-norm by the $\ell_1$-norm, if
| 1,795
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|
= \kappa$, are chosen uniformly at random from the set of $d$ items for all $j \in [n]$. The rank-breaking log likelihood function $\Lrb(\ltheta)$ for the set of items $[\ld]$ is given by $$\begin{aligned}
\label{eq:likelihood_bl_0}
\Lrb(\ltheta) &=&
\sum_{j=1}^n \sum_{a = 1}^{\ell_j}
\,\lambda_{j,a} \, \Big\{ \sum_{(i, \i) \in E_{j,a}}
\, \I_{\big\{i, \i \in [\ld] \big\}} \Big( \theta_{\i} - \log \Big(e^{\theta_i} + e^{\theta_{\i}}\Big) \,\Big)\, \Big\} \;.\end{aligned}$$ We analyze the rank-breaking estimator $$\begin{aligned}
\label{eq:theta_ml_bl}
\widehat{\ltheta} \;\; \equiv \;\; \max_{\ltheta \in \lOmega_{2b}} \Lrb(\ltheta)\;.\end{aligned}$$ We further simplify notations by fixing $\lambda_{j,a} = 1$, since from Equation , we know that the error increases by at most a factor of $4$ due to this sub-optimal choice of the weights, under the special scenario studied in this theorem.
\[thm:bottoml\_upperbound\] Under the bottom-$\ell$ separators scenario and the PL model, $S_j$’s are chosen uniformly at random of size $\kappa$ and $n$ partial orderings are sampled over $d$ items parametrized by $\theta^* \in \Omega_b$. For $\ld=\ell d / (2 \kappa)$ and any $\ell\geq 4$, if the effective sample size is large enough such that $$\begin{aligned}
\label{eq:bottoml_1}
n\ell \;\; \geq \;\; \bigg(\frac{2^{14}e^{8b}}{\chi^2 }\frac{\kappa^3}{\ell^3}\bigg) d\log d\;,
\end{aligned}$$ where $$\begin{aligned}
\chi & \equiv & \frac14 \Bigg(1 - \exp\bigg(-\frac{ 2}{9(\kappa-2)} \,\bigg)\,\Bigg),
\end{aligned}$$ then the [*rank-breaking*]{} estimator in achieves $$\begin{aligned}
\label{eq:bottoml_3}
\frac{1}{\sqrt{\ld}}\big\|\widehat{\ltheta} - \ltheta^*\big\|_2 \; \leq \; \frac{128(1+ e^{4b})^2}{\chi}\frac{\kappa^{3/2}}{{\ell}^{3/2}}\sqrt{\frac{d\log d}{n\ell} }\;,
\end{aligned}$$ with probability at least $1 - 3e^3 d^{-3}$.
Consider a scenario where $\kappa=O(1)$ and $\ell=\Theta(\kappa)$. Then, $\chi$ is a strictly positive constant, and also $\kappa/\ell$ is s f
| 1,796
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|
u))}\,{\rm d}u\notag\\
& \eqqcolon\Lambda({\varepsilon}/{\zeta_0})\,\sqrt{\zeta_0},
\label{ineq}
\end{aligned}$$ where $\mathbf{1}_S$ denotes the indicator function on the set $S$. Using the primitive ${\displaystyle\int} \sqrt{1- \cos(u)}\,{\rm d}u$ $ =-2\, \sqrt{1-\cos(u)}\,\cot(u/2)$, we have $$\Lambda(u) =\sqrt{u}\,\frac{\arccos(1-u)}{\pi}
+\frac{2}{\pi}\sqrt{u} \,\cot\pp{\frac{\arccos(1-u)}{2}}.$$ One easily verifies that there is a constant $\lambda\in(0,1)$ such that $\sup_{u\in [0,1]}\Lambda(u)<\lambda$.
Next we move to the second part of the proof. At each step of the walk-on-spheres, we can construct the quantities $\zeta_{n+1}$, the orthogonal distance of $\rho_{n+1}$ to the tangential hyperplane that passes through the closest point on $\partial D$ to $\rho_n$; and $\theta_n$, the angle that is subtended at $\rho_n$ between the aforesaid point and $\rho_{n+1}$. Note that $\varepsilon$ is an absorbing state for the sequence $(\zeta_n, n\geq 0)$ in the sense that, if $\zeta_n = \varepsilon$, then $\zeta_{n+k} = \varepsilon$ for all $k\geq 0$. We may thus write $N(\varepsilon) \leq N'(\varepsilon):=\min\{n\geq 0: \zeta_n = \varepsilon\}$.
By the strong Markov property and the spatial homogeneity of Brownian motion given the analysis leading to (\[ineq\]), we have, on $\{n<N(\varepsilon)\}$, $$\mathbb{E}\left[\left.\sqrt{\zeta_{(n+1)\wedge N(\varepsilon)}}\,\right|\zeta_0, \dots, \zeta_n\right] = \mathbb{E}\left[\left.\sqrt{\zeta_{(n+1)\wedge N(\varepsilon)}}\,\right| \zeta_n\right] \leq \Lambda({\varepsilon}/{\zeta_n})\sqrt{\zeta_n}< \lambda \sqrt{\zeta_n}.$$ As a consequence the process $\left(\lambda^{-(n\wedge N(\varepsilon))}\sqrt{\zeta_{n\wedge N(\varepsilon)}}, n\geq 0\right)$ is a supermartingale. The optional-sampling theorem and Jensen’s inequality give us $$\varepsilon \lambda^{-\mathbb{E}_x[N'(\varepsilon)]}\geq \mathbb{E}_{x}[\lambda^{-N'(\varepsilon)}\varepsilon]\leq \sq
| 1,797
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| 1,348
| 1,619
| null | null |
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|
1/2 \\ 0 \\ \frac{1}{4}(-1 + \frac{p}{\alpha}) \\ \frac{1}{4}(-1 - \frac{p}{\alpha}) \end{matrix} \right]$$ and thus at time $t$ we have $$\rho_t = e^{\mathbf{A}}\rho_0 = \left[\begin{matrix} 1/2 \\ 0 \\
\frac{1}{4}e^{\frac{-p t - \alpha t}{2n}}(-1 + \frac{p}{\alpha})
\\ \frac{1}{4}e^{\frac{-p t + \alpha t}{2n}}(-1 - \frac{p}{\alpha}) \end{matrix} \right].$$ If we then change back to the computational basis and project by $\Pi_0$ and $\Pi_1$, we may compute the probabilities of measuring $0$ and $1$ at a particular time $t$: $$P[0] = \frac{1}{4}\left[2 + e^{\frac{-p t - \alpha t}{2n}}(1 - p/ \alpha)
+ e^{\frac{-p t + \alpha t}{2n}}(1 + p/ \alpha)\right]$$ $$P[1] = \frac{1}{4}\left[2 - e^{\frac{-p t - \alpha t}{2n}}(1 - p/ \alpha)
- e^{\frac{-p t + \alpha t}{2n}}(1 + p/ \alpha)\right]$$ which can be simplified somewhat to $$P[0] = \frac{1}{2} + \frac{1}{2} e^{-\frac{pt}{2n}} \left[\cos\left(\frac{\beta t}{2n}\right) +
\frac{p}{\beta}~\sin\left(\frac{\beta t}{2n}\right) \right]$$ $$P[1] = \frac{1}{2} - \frac{1}{2} e^{-\frac{pt}{2n}} \left[\cos\left(\frac{\beta t}{2n}\right) +
\frac{p}{\beta}~\sin\left(\frac{\beta t}{2n}\right) \right].$$ Here we have let $\beta = -i \alpha = \sqrt{16k^2-p^2}$ for simplicity. A quick check shows that when $p = 0,$ $P[0] =
\cos^2\left(\frac{kt}{n}\right)$ and $P[1] =
\sin^2\left(\frac{kt}{n}\right)$, which are exactly the dynamics of the non-decohering walk. The probabilities for this non-decohering case are shown in Figure \[fig1\].
![The $p=0$ case - no decoherence: a plot of $P[0]$ and $P[1]$ versus time, for $k =1$, $n=5$, $p = 0$[]{data-label="fig1"}](pequals0){width="3in"}
The three regimes mentioned before are immediately apparent. For $p < 4k$, $\beta$ is real. When $p = 4k$, we have $\beta = 0$, which appears to be a serious problem at first glance. Finally, for $p > 4k$, $\beta$ is imaginary. We now address each of these three situations in detail.
The ca
| 1,798
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| null | null |
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|
n: R+}$$
It is to be noted that the conditions $\mathcal{D}_{+}=\mathcal{D}_{-}$ and $\mathcal{R}_{+}=\mathcal{R}_{-}$ are necessary and sufficient for the Kuratowski convergence to exist. Since $\mathcal{D}_{+}$ and $\mathcal{R}_{+}$ differ from $\mathcal{D}_{-}$ and $\mathcal{R}_{-}$ only in having cofinal subsets of $D$ replaced by residual ones, and since residual sets are also cofinal, it follows that $\mathcal{D}_{-}\subseteq\mathcal{D}_{+}$ and $\mathcal{R}_{-}\subseteq\mathcal{R}_{+}$. The sets $\mathcal{D}_{-}$ and $\mathcal{R}_{-}$ serve for the convergence of a net of functions just as $\mathcal{D}_{+}$ and $\mathcal{R}_{+}$ are for the convergence of subnets of the nets (*adherence*). The later sets are needed when subsequences are to be considered as sequences in their own right as, for example, in dynamical systems theory in the case of $\omega$-limit sets.
As an illustration of these definitions, consider the sequence of injective functions on the interval $[0,1]$ $f_{n}(x)=2^{n}x$, for $x\in\left[0,1/2^{n}\right],\textrm{ }n=0,1,2\cdots$. Then $\mathbb{D}_{0.2}$ is the set $\{0,1,2\}$ and only $\mathbb{D}_{0}$ is eventual in $\mathbb{D}$. Hence $\mathcal{D}_{-}$ is the single point set {0}. On the other hand $\mathbb{D}_{y}$ is eventual in $\mathbb{D}$ for all $y$ and $\mathcal{R}_{-}$ is $[0,1]$.
**Definition** **[<span style="font-variant:small-caps;">3.1</span>]{}***[<span style="font-variant:small-caps;">.</span>]{}* ***Graphical Convergence of a net of functions.*** *A net of functions $(f_{\alpha})_{\alpha\in D}\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ is said to* *converge graphically* *if either $\mathcal{D}_{-}\neq\emptyset$ or $\mathcal{R}_{-}\neq\emptyset$; in this case let $F\!:\mathcal{D}_{-}\rightarrow Y$ and $G:\mathcal{R}_{-}\rightarrow X$ be the entire collection of limit functions. Because of the assumed Hausdorffness of $X$ and $Y$, these limits are well defined.*
*The graph of the* *graphical limit* $\mathscr{M}$ *of the net* $(f_{\alpha})\!:(X,\mathcal{U})\rightarrow(Y
| 1,799
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|
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|
I*. If $L_i$ is *free of type I*, then $\pi^i f_{i,i}$ is of the form $$\xi^{(i-1)/2}\cdot \pi\begin{pmatrix} a_i&\pi b_i& e_i\\ -\sigma(\pi \cdot {}^tb_i) &\pi^3f_i&1+\pi d_i \\
-\sigma({}^te_i) &-\sigma(1+\pi d_i) &\pi+\pi^3c_i \end{pmatrix}.$$ Here, the diagonal entries of $a_i$ are are divisible by $\pi^3$, where $a_i$ is an $(n_i-2) \times (n_i-2)$-matrix with entries in $B\otimes_AR$, etc. If $L_i$ is *bound of type I*, then each entry of $$\delta_{i-1}(0,\cdots, 0, 1)\cdot f_{i-1,i}+\delta_{i+1}(0,\cdots, 0, 1)\cdot f_{i+1,i}$$ lies in the ideal $(\pi)$.
7. Since $\begin{pmatrix}\pi^{max\{i,j\}}f_{i,j}\end{pmatrix}$ is a hermitian matrix, its diagonal entries are fixed by the nontrivial Galois action over $E/F$ and hence belong to $R$.\
The functor $\underline{H}$ is represented by a flat $A$-scheme which is isomorphic to an affine space of dimension $n^2$. The proof of this is similar to that in Section 3C of [@C2] and so we skip it.
Now suppose that $R$ is any (not necessarily flat) $A$-algebra. We again use $\sigma$ to mean the automorphism of $B\otimes_AR$ given by $b\otimes r \mapsto \sigma(b)\otimes r$. Note that $\sigma(\pi)=-\pi$. By choosing a $B$-basis of $L$ as explained in Theorem \[210\] and Remark \[r211\].(a), we describe each element of $\underline{H}(R)$ formally as a matrix $\begin{pmatrix}\pi^{max\{i,j\}}f_{i,j}\end{pmatrix}$ with matrices $f_{i,i}^{\ast}$ satisfying the following:
1. When $i\neq j$, $f_{i,j}$ is an $(n_i \times n_j)$-matrix with entries in $B\otimes_AR$ and $(-1)^{max\{i,j\}}\sigma({}^tf_{i,j})=f_{j,i}$.
2. Assume that $i=j$ is even. Then $$\pi^if_{i,i}=\left\{
\begin{array}{l l}
\xi^{i/2}\begin{pmatrix} a_i&\pi b_i\\ \sigma(\pi\cdot {}^t b_i) &1+2\gamma_i +4c_i \end{pmatrix} & \quad \textit{if $L_i$ is \textit{of type $I^o$}};\\
\xi^{i/2}\begin{pmatrix} a_i&b_i&\pi e_i\\ \sigma({}^tb_i) &1+2f_i&1+\pi d_i \\ \sigma(\pi \cdot {}^te_i) &\sigma(1+\pi d_i) &2\gamma_i+4c_i \end{pmatrix} & \quad \textit{if $L_i$ is \textit{of type $I^e
| 1,800
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| null | null |
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|
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