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$\theta_0,\lambda>0$ be fixed. We assume that for every $%
\kappa \geq 0$ and $q\in {\mathbb{N}}$ there exist $\pi (q,\kappa )$, $%
\theta _{1}\geq 0$ and $C_{q,\kappa}>0$ such that for every multi-indexes $%
\alpha $ and $\beta $ with $\left\vert \alpha \right\vert +\left\vert \beta
\right\vert \leq q$, $(x,y)\in {\mathbb{R}}^{d}\times {\mathbb{R}}^{d}$ and $%
t\in (0,1]$ one has $$(H_{3})\qquad \left\vert \partial _{x}^{\alpha }\partial _{y}^{\beta
}s_{t}(x,y)\right\vert \leq \frac{C_{q,\kappa}}{(\lambda
t)^{\theta_0(q+\theta_1)}}\times \frac{\psi _{\pi (q,\kappa )}(x)}{\psi
_{\kappa }(x-y)}. \label{h3}$$We also assume that $\pi (q,k)$ and $C_{q,\kappa}$ are both increasing in $q$ and $\kappa $.
This property will be used by means of the following lemma:
\[lemmaB\] Suppose that Assumption \[HH3\] holds.
$\mathbf{A.}$ For every $\kappa \geq 0$, $q\in {\mathbb{N}}$ and $p>1$ there exists $C>0$ such that for every $t\in(0,1]$ and $f$ one has $$\left\Vert S_{t}^{\ast }f\right\Vert _{q,\kappa ,p}\leq \frac{C}{(\lambda
t)^{\theta_0(q+\theta_1)}}\left\Vert f\right\Vert _{0,\nu ,1} \label{B1}$$where $\nu =\pi (q,\kappa +d)+\kappa +d$
$\mathbf{B.}$ For every $\kappa \geq 0$, $q_1,q_2\in{\mathbb{N}}$ there exists $C>0$ such that for every $t\in(0,1]$, for every multi-index $\alpha $ with $\left\vert \alpha \right\vert \leq q_{2}$ and $f$ one has $$\left\Vert \frac{1}{\psi _{\eta }}S_{t}(\psi _{\kappa }\partial ^{\alpha
}f)\right\Vert _{q_{1},\infty }\leq \frac{C}{(\lambda
t)^{\theta_0(q_1+q_2+\theta_1)}}\,\|f\|_\infty \label{B2}$$where $\eta =\pi (q_{1}+q_{2},\kappa +d+1)+\kappa$.
**Proof.** In the sequel, $C$ will denote a positive constant which may vary from a line to another and which may depend only on $\kappa$ and $q$ for the proof of **A.** and only on $\kappa, q_1$ and $q_2$ for the proof of **B.**
**A.** Using (\[h3\]) if $\left\vert \alpha \right\vert \leq q,$$$\left\vert \partial ^{\alpha }S_{t}^{\ast }f(x)\right\vert \leq \int
\left\vert \partial _{x}^{\alpha }s_{t}(y,x)\right\vert \times \left\vert
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the actual flux and noise distribution in real images. These simulations are described in detail by @sanc04a. We applied the peak subtraction nucleus removal as well as [[galfit]{}]{} to a set of $\sim$2000 quasar images created in this way. Comparing input and output parameter values yielded mean magnitude offsets as well as statistical errors for the individual host galaxy magnitudes (Fig. \[fig:corrs\_and\_errors\_z\]).
The simulations give reliability regions and error bars. The left panel in Figure \[fig:corrs\_and\_errors\_z\] shows which magnitudes are recovered for a given synthetic host galaxy. Since the input set covers a large range of different morphological configurations, scale lengths, nucleus-to-host ratios, etc., the recovered values will scatter. Close to the detection limit, the scatter and the corrections grow rapidly as a function of magnitude; additionally, the ability to differentiate between different morphological types will generally be lost. The combination of these effects is reflected in the spread of the output of the simulations. This measured spread is a direct estimate for the uncertainties of the total flux (right panel in Figure \[fig:corrs\_and\_errors\_z\]).
From these simulations we adopt approximate regions in brightness where host galaxy magnitudes can be reliably determined, with correction of 0.25 to a maximum of 0.6 mag. These regions go down to $\mathrm{{F606W}}=26.2$, $\mathrm{{F850LP}}=24.6$ for the peak subtraction method and the present data. Outside the corrections and errors increase. In three cases, marked in Table \[tab:results\_vz\] with ‘?’ in column $Z_\mathrm{hg}$, the observed magnitudes extend to outside these regions; here we continue using the derived corrections for these three objects, but the so derived host galaxies are more uncertain and their magnitudes should be taken with care. Notice that the [F850LP]{} band data are substantially shallower than the [F606W]{} band, mainly a consequence of the ACS detector sensitivity.
.\end{aligned}$$ This is reminiscent of other indications that large localized field excursions can be sustained around exponentially large sources [@nicolis].
Other similar dilatonic black hole solutions can be obtained, including different dilaton couplings $e^{-2a\phi}F^2$ and general dyonic charges [@Ivashchuk:1999jd; @Abishev:2015pqa] (see also [@Loges:2019jzs] for a recent analysis in the context of the WGC).[^4] In some cases simple analytic solutions are known. In the magnetic case, similar results are obtained. In the dyonic case, it is possible to have a finite dilaton excursion in the extremal limit. However, the curvature at the horizon is still controlled by the mass of the black hole and the amplitude of the dilaton excursion, in such a way that analogous bounds of the form $|\Delta\phi|<\log(M)$ still hold.
Discussion
==========
Both neutral and charged KK bubbles sample infinite distances in moduli space in finite spatial regions with size $R$ of order the bubble radius. Neutral bubbles are classically unstable, and we have argued that charged bubbles are destabilized in the presence of charged matter with $q/m\gtrsim 1$. Dilatonic black holes in a controlled EFT have a finite excursion $ |\Delta\phi|\lesssim |\log(M_{BH})|$.
We have not discussed KK monopoles [@sorkin; @GP], but they provide another interesting example. They are stable and sample an infinite distance in m
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tially. Although the error bars in the far regions, i.e., $d > 0.36L$, tend to increase, the average electric fields stay within the error bars shown in Fig.\[res\_ele\].
Based on these discussions, we believe that the simulation results are reliable at least in the intermediate regime, i.e., $1\lambda_{{\rm D}}\lesssim d\lesssim2.5\lambda_{{\rm D}}$. In this region, it is evident from Fig.\[res\_ele\] that the simulation results deviate from the theoretical prediction of the ODS attractive potential beyond $2\sigma$. The result also suggests that the electric fields acting on the grain are even larger than the standard Yukawa potential prediction. Although the large error bars make it difficult to draw conclusions from this result alone, the systematic deviation from the theoretical predictions suggests that the underlying assumptions made in the derivation of (\[yukawa-e\]) and (\[l-j-e\]) may be violated. In the next section, we discuss possible reasons for this discrepancy between the theory and simulations.
Discussion
==========
Our simulation results show that the force between two dust grains is repulsive and stronger than that predicted by the standard Yukawa potential Eq. (\[yukawa\]). At first, we discuss the validity of Eq. (\[yukawa\]). When the grain radius is negligible, the functional form of the Yukawa potential itself must be correct at large distances, where the shielding is nearly complete and the first-order expansion of the Boltzmann-type density distribution is appropriate. In fact, Poisson’s equation and the linearized Boltzmann distributions give $$\label{long}
q\phi\left(r\right)=\alpha\frac{q^{2}}{r}\exp{\left(-\frac{r}{\lambda_{{\rm D}}}\right)}.$$ However, the coefficient $\alpha$ (integration constant) in Eq. (\[long\]) is unknown and must be determined by the inner boundary condition. In standard textbooks, it is determined by assuming that the outer solution smoothly connects to the bare Coulomb potential at $r \rightarrow 0$, which gives $\alpha=1$.
On the other hand,
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onance. In the first extreme, we assume a featureless density of states in the vicinity of the Fermi energy over an interval $[-2\w_0,2\w_0]$, i. e. $\rho_{\mu,\mu'\sigma}(\w)\to {\rm const.}$ in , and $\beta\w_0\gg 1$ so that the Bose function can be ignored. Then $\tau^{(2)}_{\sigma}(\w) $ will be dominated by the Fermi functions, introducing two threshold contributions in the overall differential conductance $dI/dV$ at $\pm \w_0$. These are the typical $dI/dV$ steps that are often encountered in inelastic tunnel spectroscopy as shown in Fig. 1 of Ref. [@REED2008].
In the second extreme, we consider an electronic DOS of the sample system S which possesses a sharp spectral peak located at $\w=0$ with a width $\Gamma\ll \w_0$. Then $\tau^{(2)}_{\sigma}(\w)$ exhibits again a sharp threshold behavior at $\pm \w_0$, but instead of a plateau the spectral function decreases with increasing $|\w|$ on a scale given by the peak broadening $\Gamma$. In this case, two “replicas” of the peak at $\w=0$ can be found at $\pm \w_0$ in the overall differential conductance $dI/dV$. However, the Fermi function $f_{\rm S}(\w_0\pm\w)$ cuts away the halves of the replicas on the low-$|\w|$ side and modifies them to a threshold behavior: a minimal energy transfer for $eV=\pm\w_0$ is required to generate an inelastic contribution replicating the standard picture [@MolecularVibrationTunnel1968]. These truncated replicas of the $\w=0$ peak are generated by the inelastic tunneling process due to the change of the distance between system S and STM tip. In case of a large electron-phonon coupling in S, the approximation is invalid and the proper Green’s function $G_{\hat X_\nu d_{\mu \sigma}, \hat X_{\nu'} d^\dagger_{\mu' \sigma}}(t)$ must be calculated, along with $G_{X_\nu d_{\mu\sigma}, d^\dagger_{\mu'\sigma}}$ and $G_{d_{\mu\sigma}, X_\nu d^\dagger_{\mu'\sigma}}$.
### Discussion
The presented tunneling theory combines different limits [@MolecularVibrationTunnel1968; @Mahan81; @PerssonBaratoff1987; @LorentePersson2000; @Pa
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bar = \frac{h}{2\pi}.$$ Here $h$ is Planck’s constant, and the integral is thought of being over all paths with $\mathbf{x}(0)=0$ and $\mathbf{x}(t)=\mathbf{y}$.\
In the last fifty years there have been many approaches for giving a mathematically rigorous meaning to the Feynman integral by using e.g. analytic continuation,limits of finite dimensional approximations or Fresnel integrals. Instead of giving a complete list of publications concerning Feynman integrals we refer to [@AHKM08] and the references therein. Here we choose a white noise approach. white noise analysis is a mathematical framework which offers generalizations of concepts from finite-dimensional analysis, like differential operators and Fourier transform to an infinite-dimensional setting. We give a brief introduction to white noise analysis in Section 2, for more details see [@Hid80; @HKPS93; @Ob94; @BK95; @Kuo96]. Of special importance in white noise analysis are spaces of generalized functions and their characterizations. In this article we choose the space of Hida distributions, see Section 2.\
The idea of realizing Feynman integrals within the white noise framework goes back to [@HS83]. There the authors used exponentials of quadratic (generalized) functions in order to give meaning to the Feynman integral in configuration space representation $${\rm N}\int_{\mathbf{x}(0) =0, \mathbf{x}(t)=y} \exp\left(\frac{i}{\hbar} S(\mathbf{x}) \right) \, \prod_{0<\tau<t} \, d\mathbf{x}(\tau) ,\quad \hbar = \frac{h}{2\pi},$$ with the classical action $S(\mathbf{x})= \int_0^t \frac{1}{2} m \dot{\mathbf{x}}^2 -V(\mathbf{x})\, d\tau$. We use these concepts of quadratic actions in white noise analysis, which were further developed in [@GS98a] and [@BG10] to give a rigorous meaning to the Feynman integrand $$\begin{gathered}
\label{integrandpot}
I_V = {\rm Nexp}\left( \frac{i}{\hbar}\int_0^t \frac{m}{2} \dot{\mathbf{x}}(\tau)^2 d\tau +\frac{1}{2}\int_0^t \dot{\mathbf{x}}(\tau)^2 d\tau\right)\\
\times \exp\left(-\frac{i}{\hbar} \int_0^t V(\mathbf{x}(\ta
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L. upper limit on the Higgs mass in the SM is [@mh_smfits; @felcini] $$\label{mh_up}
{M_{\mathrm{H }}}< 220\,{\mbox{${\rm {GeV}}/c^2$} }\,,$$ if one makes due allowance for unknown higher loop uncertainties in the analysis. The corresponding central value is still rather imprecise: $${M_{\mathrm{H }}}= 71^{+75}_{-42}\pm5\,{\mbox{${\rm {GeV}}/c^2$} }\,.$$ The range given by Eq.\[mh\_up\] may be compared with the one derived from theoretical arguments [@hambye]. It is well known that in the SM with only one Higgs doublet a lower limit on the Higgs mass ${M_{\mathrm{H }}}$ can be derived from the requirement of vacuum stability. This limit is a function of the energy scale $\Lambda$ where the model breaks down and new physics appears. Similarly an upper bound on ${M_{\mathrm{H }}}$ is obtained from the requirement that up to the scale $\Lambda$ no Landau pole appears. If, for example, the SM has to remain valid up to the scale $\Lambda\simeq{\rm M_{GUT}}$, then it is required that $135<{M_{\mathrm{H }}}<180~{\mbox{${\rm {GeV}}/c^2$} }$.
In the MSSM two Higgs doublets are introduced, in order to give masses to the up-type quarks on the one hand and to the down-type quarks and charged leptons on the other. The Higgs particle spectrum therefore consists of five physical states: two CP-even neutral scalars (h,A), one CP-odd neutral pseudo-scalar (A) and a charged Higgs boson pair ($\rm{H}^{\pm}$). Of these, h and A could be detectable at LEP2 [@yellow]. In fact, at tree-level h is predicted to be lighter than the Z. However, radiative corrections to ${M_{\mathrm{h}}}$ [@ellis], which are proportional to the fourth power of the top mass, shift the upper limit of ${M_{\mathrm{h}}}$ to approximately 135 [ ]{}, depending on the MSSM parameters.
Higgs production and decay
==========================
At LEP2, the dominant mechanism for producing the standard model Higgs boson is the so-called Higgs-strahlung process ${\mathrm{e}^+\mathrm{e}^-}\to$ HZ [@khoze; @bjorken], with smaller contributions from the WW and ZZ f
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}{\text{\circle*{1.5}}}}}\oplus P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$, the map $\iota_s$ is just the difference map $a \oplus b \mapsto a - b$; on $P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes_AP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$, $\iota_s$ is our fixed homotopy $\iota:P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes_AP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\to P_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}+1}$. And similarly for the other surjection $s'$.
We leave it to the reader to check that if one computes $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(A_\#)\mid_{\Lambda_{\leq 2}}$ using this resolution $P^\#_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$, then one obtains exactly for the Connes’ differential $B$.
Discussion
==========
One of the most unpleasant features of the construction presented in Section \[cat\] is the strong assumptions we need to impose on the tensor category ${{\mathcal C}}$. In fact, the category one would really like to apply the construction to is the category ${\operatorname{End}}{{\mathcal B}}$ of endofunctors – whatever that means – of the category ${{\mathcal B}}$ of coherent sheaves on an algebraic variety $X$. But if $X$ is not affine, ${\operatorname{End}}{{\mathcal B}}$ certainly does not have enough projectives, so that Example \[cln.exa\] does not apply, and it is unlikely that ${\operatorname{End}}{{\mathcal B}}$ can be made homologically clean in the sense of Definition \[clean\]. We note that Definition \[clean\] has been arranged so as not impose anything more than strictly necessary for the proofs; but in practice, we do not know any examples which are not covered by Example \[cln.exa\].
As for the category ${\operatorname{End}}{{\mathcal B}}$, there is an even bigger problem with it: while there are ways to define endofunctors so that ${\operatorname{End}}{{\mathcal B}}$ is an abelian category with a right-exact tensor product, it cannot be equipped with a right-exact trace functor ${\operatornam
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figure \[fig:maxdEdt\_vortexCells\]. This analysis is performed using the level sets $\Gamma_{s}(F)\subset\Omega$ defined as $$\label{eq:levelSets}
\Gamma_{s}(F) := \{{\mathbf{x}}\in\Omega : F({\mathbf{x}}) = s \},$$ for a suitable function $F:\Omega\to\mathbb{R}$. In figures \[fig:maxdEdt\_vortexCells\](a–c) we choose $F({\mathbf{x}}) =
|{\bnabla\times}{\mathbf{u}}_1|({\mathbf{x}})$ with $s = 0.95||{\bnabla\times}{\mathbf{u}}_1||_{L_\infty}$. To complement this information, in figures \[fig:maxdEdt\_vortexCells\](d–f) we also plot the isosurfaces and cross-sectional distributions of the $x_1$ component of the field ${\mathbf{u}}_1$.
The fields shown in figure \[fig:maxdEdt\_vortexCells\] reveal interesting patterns involving well-defined “vortex cells”. More specifically, we see that in case (\[c1\]), given by equation and shown in figures \[fig:maxdEdt\_vortexCells\](a,d), the vortex cells are staggered with respect to the orientation of the cubic domain $\Omega$ in all three planes, whereas in case (\[c3\]), given by equation and shown in figures \[fig:maxdEdt\_vortexCells\](c,f), the vortex cells are aligned with the domain $\Omega$ in all three planes. On the other hand, in case (\[c2\]), given by equation and shown in figures \[fig:maxdEdt\_vortexCells\](b,e), the vortex cells are staggered in one plane and aligned in another with the arrangement in the third plane resulting from the arrangement in the first two. These geometric properties are also reflected in the $x_1$-component of the field ${\mathbf{u}}_1$ which is independent of $x_1$ in cases (\[c1\]) and (\[c2\]), but exhibits, respectively, a staggered and aligned arrangement of the cells in the $y-z$ plane in these two cases. In case (\[c3\]) the cells exhibit an aligned arrangement in all three planes. The geometric properties of the extreme vortex states obtained in the limit $\E_0 \rightarrow 0$ are summarized in Table \[tab:E0\]. We remark that an analogous structure of the optimal fields, featuring aligned and staggered arrangement
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_1$) introduced in Proposition \[prop5\].
$\ell_1$-spectral clustering algorithm {#section5}
======================================
Now, we only consider graphs with an exact cluster structure whose edges have been perturbed by a coefficient $p \in [0,1]$.
$\ell_1$-spectral clustering algorithm is developed in a Matlab software. Starting with the number of blocks $k$ of an adjacency matrix $A$ and the column index of one representative element of each block $I_{n-k+1}, \dots, I_n$, the pseudo-code for the algorithm is presented in Algorithm \[algo\].
Steps from $3$ to $14$ are dedicated to the recovery of the indicators of connected components. The minimization problem introduced in Section \[l1\] is solved using the $\ell_1$-eq function of the Matlab optimization package $\ell_1$-magic [@Candes05]. Vector $\tilde{v}_j$ contains the solution of the minimization problem (step 11).
To find the other connected components indicators, we add the constraint of being orthogonal to the previous computed vectors by deflating the matrix $A$ (step 13) and we do the same to estimate the other connected component indicators.
Let $F$ be the concatenation of the vectors $\tilde{v}_j$. As the algorithm is applied on a perturbed adjacency matrix, the elements in $F$ are not exactly equal to one or zero but are very close to one for the indices associated to edges belonging to a same cluster and to zero for the remaining ones. Therefore, we shrink the solution (steps 16 to 20):
For all $j=1,\dots,n$, for all $i=1,\dots k$,
$F_{ij}=\left\{
\begin{array}{ll}
1 \ \ \mbox{if} \ F_{ij} >\frac{1}{2}, \\
0 \ \ \mbox{if} \ F_{ij} \leq \frac{1}{2}.
\end{array}
\right.$
The indicators of the clusters are given by the $k$ column vectors of $F$.
number of clusters $k$ , adjacency matrix $A$, indices of representative elements $Index=[I_{n-k+1},\dots, I_n]$. Initialize $F=[]$. Eigen decomposition $[V,U]$ of $A$: $A=VU^tV$. Sort in ascending order the eigenvalues and the associated eigenvectors of $A$. Form the matrix $V_{2,n}$
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conjugate of $p$, that is, $q=p/(p-1)$ for $1<p<\infty$, $q=\infty$ for $p=1$ or $q=1$ for $p=\infty.$
Recently, Kara\[1\] has defined the Fibonacci difference sequence spaces $\ell_{p}(\widehat{F})$ and $\ell_{\infty}(\widehat{F})$ by$$\ell_{p}(\widehat{F})=\left \{ x=(x_{n})\in \omega:{\displaystyle \sum \limits_{n}}
\left \vert \frac{f_{n}}{f_{n+1}}x_{n}-\frac{f_{n+1}}{f_{n}}x_{n-1}\right \vert
^{p}<\infty \right \} ;\text{ }1\leq p<\infty$$ and$$\ell_{\infty}(\widehat{F})=\left \{ x=(x_{n})\in \omega:\sup_{n\in\mathbb{N}
}\left \vert \frac{f_{n}}{f_{n+1}}x_{n}-\frac{f_{n+1}}{f_{n}}x_{n-1}\right \vert
<\infty \right \} .\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$$
With the notation of (1.2), the sequence spaces $\ell_{p}(\widehat{F})$ and $\ell_{\infty}(\widehat{F})$ may be redefined by $$\ell_{p}(\widehat{F})=(\ell_{p})_{\widehat{F}}\text{ }(1\leq p<\infty)\text{
\ and \ }\ell_{\infty}(\widehat{F})=(\ell_{\infty})_{\widehat{F}}, \tag{2.1}$$ where the matrix $\widehat{F}=(\widehat{f}_{nk})$ is defined by$$\widehat{f}_{nk}=\left \{
\begin{array}
[c]{cc}-\frac{f_{n+1}}{f_{n}} & (k=n-1)\\
\frac{f_{n}}{f_{n+1}} & (k=n)\\
0 & (0\leq k<n-1)\text{ or }(k>n)
\end{array}
\right. ;\text{ }(n,k\in\mathbb{N}
). \tag{2.2}$$ Further, it is clear that the spaces $\ell_{p}(\widehat{F})$ and $\ell
_{\infty}(\widehat{F})$ are $BK$ spaces with the norms given by $$\left \Vert x\right \Vert _{\ell_{p}(\widehat{F})}=\left( \sum_{n}\left \vert
y_{n}(x)\right \vert ^{p}\right) ^{1/p}\text{ ; \ }(1\leq p<\infty)\text{
\ and }\left \Vert x\right \Vert _{\ell_{\infty}(\widehat{F})}=\sup_{n\in\mathbb{N}
}\left \vert y_{n}(x)\right \vert , \tag{2.3}$$ where the sequence $y=(y_{n})=(\widehat{F}_{n}(x))$ is the $\widehat{F}$-transform of a sequence $x=(x_{n})$, i.e., $$y_{n}=\widehat{F}_{n}(x)=\left \{
\begin{array}
[c]{cc}\frac{f_{0}}{f_{1}}x_{0}=x_{0}\text{ \ \ \ \ } & (n=0)\\
\frac{f_{n}}{f_{n+1}}x_{n}-\frac{f_{n+1}}{f_{n}}x_{n-1} & (n\geq1)
\end{array}
\right. \text{ };\text{ }(n\in\mathbb{N}
). \tag{2.4}
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3
\end{pmatrix},\quad
\begin{pmatrix}
t & 0 & 0\\
-t & t^3 & 0\\
0 & 0 & 1
\end{pmatrix},\quad
\begin{pmatrix}
t & 0 & 0\\
0 & 1 & 0\\
t & -1 & t^3
\end{pmatrix},\quad
\begin{pmatrix}
1 & 0 & 0\\
0 & \rho t & 0\\
0 & t & t^2
\end{pmatrix},\quad
\begin{pmatrix}
1 & 0 & 0\\
0 & \rho^2 t & 0\\
0 & t & t^2
\end{pmatrix}$$ (where $\rho$ is a primitive third root of unity). The latter two marker germs have the same center and lead to projectively equivalent limits, hence they contribute the same component of the PNC. The corresponding limits of ${{\mathscr C}}_1$ are given by $$xy^2z,\quad
x^2(8y^2-9xz),\quad
x(y+z)(y^2+yz+z^2),\quad
y(y^2z+x^3),\quad
z(yz^2+x^3),$$$$x(y^2z-x^3),\quad
y^2(y^2-(\rho+2)xz),\quad \text{and} \quad
y^2(y^2-(\rho^2+2)xz),$$ respectively: a triangle with one line doubled, a conic with a double tangent line, a fan with star centered at $(1:0:0)$, a cuspidal cubic with its cuspidal tangent (two limits), a cuspidal cubic with the line through the cusp and the inflection point, and finally a conic with a double transversal line (two limits). Schematically, the limits may be represented as follows:
According to Theorem \[main\], all limits of ${{\mathscr C}}_1$ (other than stars of lines) are projectively equivalent to one of these curves, or to limits of them (cf. §\[boundary\]).
\[extwo\] Consider the irreducible quartic ${{\mathscr C}}_2$ given by the equation $$(y^2-xz)^2=y^3z.$$ It has a ramphoid cusp at $(1:0:0)$, an ordinary cusp at $(0:0:1)$, and an ordinary inflection point at $(3^3 5{:}{-}2^6 3^2{:}{-}2^{12})$; there are no other singular or inflection points. The PNC for ${{\mathscr C}}_2$ has one component of type II, two components of type IV, corresponding to the inflection point and the ordinary cusp, and one component of type V, corresponding to the ramphoid cusp. (Note that there is no component of type IV corresponding to the ramphoid cusp.) Representative marker germs for the latter two components are $${\rm IV}:
\begin{pmatrix}
0 & t^3 &
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need to be the same. However, the results of the wave theory follow qualitatively those of Ref. [@qc-sb] while improving substantially the statistical convergence and increasing the length of the time interval spanned by a factor of $2-3$. Such results are particularly encorauging and suggest the possible application of the wave theory here proposed, for example, to the calculation of nonadiabatic rate constants of complex systems in the condensed phase [@ksreview].
Conclusions {#sec:conclusions}
===========
In this paper the approach to the quantum-classical mechanics of phase space dependent operators has been remodeled as a non-linear formalism for wave fields. It has been shown that two coupled non-linear equations for phase space dependent wave fields correspond to the single equation for the quantum-classical density matrix in the operator scheme of motion. The equations of motion for the wave fields have been re-expressed by means of a suitable bracket and it has been shown that the emerging formalism generalizes within a non-Hamiltonian framework the non-linear quantum mechanical formalism that has been proposed recently by Weinberg. Finally, the non-linear wave equations have been represented into the adiabatic basis and have been applied, after a suitable equilibrium approximation, to the numerical study of the adiabatic and nonadiabatic dynamics of the spin-boson model. Good results have been obtained. In particular, the time interval that can be spanned by the nonadiabatic calculation within the wave scheme of motion turns out to be a factor of two-three longer than that accessible within the operator scheme of motion. This encourages one to pursue the application of the wave scheme of motion to the calculation of correlation functions for systems in the condensed phase. Future works will be specifically devoted to such an issue.
[**Acknowledgment**]{}
I acknowledge Professor Kapral for suggesting the possibility of mapping the quantum-classical dynamics of operators into a wave scheme of motion
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{F_{p+1}, F_{p+2}, \ldots F_{v}\}
= \{T_{u+1}, T_{u+2}, \ldots, T_{u+(v-p)}\}$$ and $$\min F_{p+1} <\min F_{p+2} <\cdots <\min F_{v}.$$ Using these upper and lower sequences defined above, we can obtain a crank from expression in normal form called the [*standard expression*]{} of $w$.
Proof of Theorem 1.2
====================
In the previous section, we have defined the standard expression of a word in the alphabet ${\cal L}_n^1$ as a special expression of the crank form expressions in normal form. In this section, first we show that any two crank form expressions of a seat-plan $w$ are transformed to each other by finitely using the local moves shown in Section 2. Then we show that any word in the alphabet ${\cal L}_n^1$ is moved to a scalar multiple of one of the crank form expressions. Thus we can find that any word in the alphabet ${\cal L}_n^1$ is reduced to a scalar multiple of a the standard expression. Since the set of seat-plans makes a basis of $A_{n}(Q)$ and since every seat-plan has its standard expression, this proves that the partition algebra $A_{n}(Q)$ is characterized by the generators and relations in Theorem \[th:main\].
First we show that any two crank form expressions are transformed to each other. For $w\in\Sigma_n^1$, let ${\mathbb M} = (M_1, \ldots, M_u)$ and ${\mathbb F} = (F_1, \ldots, F_v)$ be sequences of the upper and the lower parts of $w$ respectively. Assume that the subsequence $\pi(\mathbb{M}) = (M_{i_1}, \ldots, M_{i_p})$ ($i_1<\cdots <i_p$) of $\mathbb{M}$ is the sequence of the upper propagating parts and $\pi(\mathbb{F}) = (F_{j_1}, \ldots, F_{j_p})$ ($j_1<\cdots <j_p$) is that of the lower propagating parts. Then there exists a permutation $\sigma$ of degree $p = |\pi(w)|$ which specifies how the propagating parts of $w$ are recovered from $\pi(\mathbb{M})$ and $\pi(\mathbb{F})$. Let $\mathbb{E} = (E_1, \ldots, E_s)$ be a sequence of the upper or lower parts. Suppose that $\tau\in{\mathfrak S}_s$ acts on $\mathbb{E}$ by $\tau\mathbb{E} = (E_{\tau^{-1}(1)}, \ldot
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Analysis
==================
The numerical values of the input parameters we used in our analysis are the following: $$\begin{aligned}
m_B=5.28\,GeV\,, m_{\pi}=0.14\,GeV\,, m_b=4.8\,GeV\,, m_c=1.4\,GeV\,,\nonumber\\
1/\alpha=129\,, G_F=1.6639\times10^{-5}\,GeV^2\,,
|V_{tb}V^*_{td}|=0.011\,.\nonumber\end{aligned}$$ The values of the Wilson coefficients within the SM are given in Table I, on the other hand we assume that all new Wilson coefficients are real in the numerical analysis. In order to complete the analysis we need the parametrization of the form factors. For the values of that, we have used the results of [@Ball]. According to these results the form factors can be parameterized as $$\begin{aligned}
F(\hat{s})=\frac{F(0)}{1-a_F\hat{s}+b_F\hat{s}^2}\,.\end{aligned}$$ where the parameters $F(0), a_F$ and $b_F$ for each form factor are given in Table 2.\
$$\begin{array}{|c c|| c c|}
\hline C_1 &-0.248 & C_6 &-0.031 \\
\hline
C_2 &1.107 &C_7 &-0.313 \\
\hline C_3 &0.011 &C_9 &4.344
\\ \hline C_4 &-0.026 &C_{10}
&-4.669 \\ \hline C_5 &0.007 & & \\
\hline
\end{array}$$
$$\begin{array}{|l l l l|}
\hline
& \phantom{-}F(0) &\phantom{-}a_F & \phantom{-}b_F \\ \hline
f_{+}&\phantom{-}0.305 &\phantom{-}1.29 &\phantom{-}0.206 \\
\hline f_0 &\phantom{-}0.305 &\phantom{-}0.266
&\phantom{-}-0.752
\\ \hline f_T &\phantom{-}0.296 &\phantom{-}1.28
&\phantom{-}0.193 \\ \hline
\end{array}$$
In our numerical analysis to investigate the dependence of the FB asymmetry,the normalized FB asymmetry and the branching ratio to the Wilson coefficients we consider two cases where one of the new coefficients has two values $\pm |C_{10}|$, the others are set to zero. From Eq.(13), after some calculations, one can see that all terms which contribute to the FB asymmetry have the form of product of any two new coefficients except one. According to this result, we plot the dependence of the FB asymmetry on $\hat{s}$ for the $B \rar \pi\ell^+\ell^-$ decay for $C_{TE}=\pm |C_{10}|$ in Fig.(1).\
At first glance, we see
| 3,815
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|
rossianGlobalMin10], not every case is covered. For instance, if ${\mathcal{K}}\in L^1({\mathbb R}^d)$ and $2-2/d < m < 2$, stationary solutions are only known to exist for sufficiently large mass, and the behavior of smaller solutions is unknown. Moreover, convergence to these stationary solutions is only known in certain cases [@KimYao11].
In what follows, we denote ${\|u\|}_p := {\|u\|}_{L^p({\mathbb R}^d)}$ where $L^p({\mathbb R}^d) := L^p$ is the standard Lebesgue space. We will often suppress the dependencies of functions on space and/or time to enhance readability. The standard characteristic function for some $S \subset {\mathbb R}^d$ is denoted $\mathbf{1}_{S}$ and we denote the ball $B_R(x_0) := {\left\{x \in {\mathbb R}^d : {\left\vertx - x_0\right\vert} < R\right\}}$. In formulas we use the notation $C(p,k,M,..)$ to denote a generic constant, which may be different from line to line or term to term in the same formula. In general, these constants will depend on more parameters than those listed, for instance those which are fixed by the problem, such as ${\mathcal{K}}$ and the dimension, but these dependencies are suppressed. We use the notation $f \lesssim_{p,k,...} g$ to denote $f \leq C(p,k,..)g$ where again, dependencies that are not relevant are suppressed.
Statement of Results {#sec:StatResults}
--------------------
We need the following definition from [@BRB10], which we restate here.
\[def:adm\] We say a kernel ${\mathcal{K}}\in C^3 \setminus {\left\{0\right\}}$ is *admissible* if ${\mathcal{K}}\in W^{1,1}_{loc}({\mathbb R}^d)$ and the following holds:
- ${\mathcal{K}}$ is radially symmetric, ${\mathcal{K}}(x) = k({\left\vertx\right\vert})$ and $k({\left\vertx\right\vert})$ is non-increasing.
- $k^{\prime\prime}(r)$ and $k^\prime(r)/r$ are monotone on $r \in (0,\delta)$ for some $\delta > 0$.
- ${\left\vertD^3{\mathcal{K}}(x)\right\vert} \lesssim {\left\vertx\right\vert}^{-d-1}$.
The definition ensures that the kernel is radially symmetric, attractive, reasonably well-behaved at
| 3,816
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ing non-unitary evolution in matter {#sec:analytical-numerical}
==============================================================================
In this section, we describe the numerical and analytical methods for calculating the neutrino oscillation probability by solving non-unitary evolution in matter.
Numerical method for calculating neutrino oscillation probability {#sec:numerical}
-------------------------------------------------------------------
We describe a numerical method for computing the oscillation probability in matter. This method can be used, assuming adiabaticity, in cases with varying matter density. We show that in zeroth order in $W$ the system simplifies to an evolution equation in the $3 \times 3$ active subspace.
We solve the Schrödinger equation in the vacuum mass eigenstate basis (“tilde basis”), $\tilde{\nu}_{z} = ({\bf U}^{\dagger})_{z \zeta} \nu_{\zeta}$ with Hamiltonian $\tilde{H}$ in (\[tilde-H\]): $$\begin{aligned}
i \frac{d}{dx}
\left[
\begin{array}{c}
\tilde{\nu}_{i} \\
\tilde{\nu}_{J} \\
\end{array}
\right] =
\left[
\begin{array}{cc}
{\bf \Delta_{a} } + U^{\dagger} A U & U^{\dagger} A W \\
W^{\dagger} A U & {\bf \Delta_{s} } + W^{\dagger} A W \\
\end{array}
\right] \left[
\begin{array}{c}
\tilde{\nu}_{i} \\
\tilde{\nu}_{J} \\
\end{array}
\right],
\label{Schroedinger-eq}\end{aligned}$$ where $i = 1,2,3$ and $J= 4,5,\cdot \cdot \cdot,3+N$ denote mostly active and mostly sterile neutrino mass eigenstate labels, respectively. The initial condition with only active component implies $$\begin{aligned}
\tilde{\nu}_{i} (0) &=& \sum_{\alpha} (U^{\dagger})_{i \alpha} \nu_{\alpha} (0), \nonumber \\
\tilde{\nu}_{J} (0) &=& \sum_{\alpha} (W^{\dagger})_{J \alpha} \nu_{\alpha} (0).
\label{initial-condition}\end{aligned}$$ Using the solution of equation (\[Schroedinger-eq\]), we need the wave function of active flavor component to calculate the probability at baseline $x=L$. $$\begin{aligned}
\nu_{\alpha} (L) &=&
\sum_{i} U_{\alpha i} \tilde{\nu}_{i} (L)
+ \sum_{J} W_{\alpha J}
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dual pairing of $S_d({\mathbb{R}})$ and $S'_d({\mathbb{R}})$ also by $\langle \cdot , \cdot \rangle$. Note that its restriction on $S_d({\mathbb{R}}) \times L_d^2({\mathbb{R}}, dx)$ is given by $(\cdot, \cdot )$. We also use the complexifications of these spaces denoted with the subindex ${\mathbb{C}}$ (as well as their inner products and norms). The dual pairing we extend in a bilinear way. Hence we have the relation $$\langle g,f \rangle = (\mathbf{g},\overline{\mathbf{f}}), \quad \mathbf{f},\mathbf{g} \in L_d^2({\mathbb{R}})_{{\mathbb{C}}},$$ where the overline denotes the complex conjugation.
White Noise Spaces
------------------
We consider on $S_d' ({\mathbb{R}})$ the $\sigma$-algebra ${\mathcal{C}}_{\sigma}(S_d' ({\mathbb{R}}))$ generated by the cylinder sets $\{ \omega \in S_d' ({\mathbb{R}}) | \langle \xi_1, \omega \rangle \in F_1, \dots ,\langle \xi_n, \omega \rangle \in F_n\} $, $\xi_i \in S_d({\mathbb{R}})$, $ F_i \in {\mathcal{B}}({\mathbb{R}}),\, 1\leq i \leq n,\, n\in {\mathbb{N}}$, where ${\mathcal{B}}({\mathbb{R}})$ denotes the Borel $\sigma$-algebra on ${\mathbb{R}}$.\
The canonical Gaussian measure $\mu$ on $C_{\sigma}(S_d'({\mathbb{R}}))$ is given via its characteristic function $$\begin{aligned}
\int_{S_d' ({\mathbb{R}})} \exp(i \langle {\bf f}, \boldsymbol{\omega} \rangle ) d\mu(\boldsymbol{\omega}) = \exp(- \tfrac{1}{2} \| {\bf f}\|^2 ), \;\;\; {\bf f} \in S_d({\mathbb{R}}),\end{aligned}$$ by the theorem of Bochner and Minlos, see e.g. [@Mi63], [@BK95 Chap. 2 Theo. 1. 11]. The space $(S_d'({\mathbb{R}}),{\mathcal{C}}_{\sigma}(S_d'({\mathbb{R}})), \mu)$ is the basic probability space in our setup. The central Gaussian spaces in our framework are the Hilbert spaces $(L^2):= L^2(S_d'({\mathbb{R}}),$ ${\mathcal{C}}_{\sigma}(S_d' ({\mathbb{R}})),\mu)$ of complex-valued square integrable functions w.r.t. the Gaussian measure $\mu$.\
Within this formalism a representation of a d-dimensional Brownian motion is given by $$\label{BrownianMotion}
{\bf B}_t ({\boldsymbol \omega}) :=(B_t(\omega_1
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no free 2D vortices in the limit $u=0$. There are, actually, two options: i) looking for a composite vortex characterized by phase windings $q_1$ and $q_2$ in odd and even layers, respectively, forming a string of length $N_z$ perpendicular to the layers; ii) considering independent vortices $q_1=\pm 1$ only in odd layers (characterized by smallest stiffness $K_{11}$) and characterized by the temperature $T_{(1,0)}= \pi/2$. As the analysis shows, the option when composite vortices form finite strings (say, $q_2=1$ in layers $z=1,3$ and $q_1=-2X$ in the layer $z=2$ by analogy with the $N_z=2$ case) costs larger energy than in the case ii). The option i) is characterized by the vortex energy $E_v= \pi K_{11} (N_z/2)[(q_1 +2Xq_2)^2 + (Y-4X^2)q_2^2]\propto N_z$. The minimum for a $2X$ integer is achieved at $q_1=-2Xq_2,\, q_2=\pm 1$. Thus, in order to compensate for the factor $N_z >>1$, the system must be very close to the instability $0<Y-4X^2 <2/N_z$. Simulations in this region turn out to be problematic. Thus, we will conduct simulations in the range $T_d<T<T_{(1,0)}$, provided the condition $T_d <T_{(1,0)}$ holds in the limit where $\delta=Y-4X^2$ remains finite for $N_z>>1$. Specifically, the condition $T_d <\pi/2$ reads \^[Nz/2 -1]{}\_[m=0]{} > 8. \[TdT\] It can surely be achieved for large enough $X$ in the limit $ N_z >>1$. Replacing the summation by integration in this limit and considering $\delta <<1$, Eq.(\[TdT\]) gives $ \delta < (X /4\sqrt{2})^2$. For the simulations we have chosen $\delta =0.3$ and $X=6$, which gives $T_d \approx 0.983$ with $T= 1.28$ chosen in the middle of the interval between $T_{(1,0)}=\pi/2\approx 1.57$ and $T_d$. The chosen value of $T_d$ corresponds to the limit $N_z \to \infty$, and for any finite $N_z$, the actual $T_d$ from Eq.(\[TdNz\]) is below this value.
At this point we note that for any finite $u$ the system is 3D and, strictly, speaking the notion of the BKT transition becomes inadequate for large enough $N_z$: even at $T>\pi/2$ the odd layers would still have X
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|
ively. We use Cartesian coordinates and the Euclidean inner product[^1], so we shall not generally distinguish between ‘up’ and ‘down’ indices; summation from 1 to $n$ is implied whenever an index is repeated.
Clebsch representation using the inverse map
--------------------------------------------
A canonical variational principle for fluid dynamics may be formulated by following the standard Clebsch procedure using the inverse map (Seliger & Whitham (1968), Holm & Kupershmidt, 1983). The Clebsch procedure begins with a functional $\ell[\MM{u}]$ of the Eulerian fluid velocity $\MM{u}$, which is known as the *reduced Lagrangian* in the context of Euler-Poincaré reduction (Holm *et al.*, 1998). One then enforces stationarity of the action $S=\int\ell[\MM{u}]{\mathrm{d}}t$ under the constraint that equation (\[label eqn\]) is satisfied by using a vector of $n$ Lagrange multipliers, which is denoted as $\MM{\pi}$. These Lagrange multipliers are the conjugate momenta to $\MM{l}$ in the course of the Legendre transformation to the Hamiltonian formulation. One may choose $\ell[\MM{u}]$ to be solely the kinetic energy, which depends only on $\MM{u}$. More generally, $\ell$ will also depend on thermodynamic Eulerian variables such as density, whose evolution may also be accommodated by introducing constraints. These constraints are often called the “Lin constraints” (Serrin, 1959). This idea was also used in reformulating London’s variational principle for superfluids (Lin, 1963).
\[clebsch\] The Clebsch variational principle using the inverse map is $$\delta \int_{t_0}^{t_1} \ell[\MM{u}] +
\int_{\Omega} \MM{\pi}\cdot(\MM{l}_t+\MM{u}\cdot\nabla\MM{l}){\mathrm{d}}V(\MM{x})
{\mathrm{d}}{t}=0,$$ where $\MM{\pi}(\MM{x},t)$ are Lagrange multipliers which enforce the constraint that particle labels $\MM{l}(\MM{x},t)$ are advected by the flow.
Taking the indicated variations leads to the following equations: $$\begin{aligned}
\label{momentum map}
\delta \MM{u}:&&
{\frac{\delta \ell}{\delta \MM{u}}} + (\nabla\
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s the highest cold dust temperature, with a maximum of $\sim$40K while the median temperature of the N158-N159-N160 complex is 26.9$^{\pm2.3}$K (28.2 if we restrict the analysis to ISM elements with a 3-$\sigma$ detection in the SPIRE bands). Based on observations made with the TopHat telescope combined with DIRBE data (100, 140, and 240 [$\mu$m]{}, 42), @Aguirre2003 derived temperatures of 25$^{\pm1.8}$K and 26.2$^{\pm2.3}$K for the LMC and 30Dor, using $\beta$= 1.33$^{\pm0.07}$ and 1.5$^{\pm0.08}$, respectively. From [[*Spitzer*]{}]{} data (thus no data above 160 [$\mu$m]{}) and the same emissivity values, @Bernard2008 obtained a colder temperature of 21.4 K for the LMC and 23 K for the 30 Dor region. Our values are thus close to the values derived for the 30Dor region by @Aguirre2003. We note however that different methods were used to derive the temperatures, making a direct comparison difficult. The N159S region is finally the coldest region of the whole complex, with a temperature $\sim$22K. This supports the argument that massive stars may still be at infancy and deeply embedded in the circumstellar material in this particular region.
In order to investigate more precisely the distribution of radiation field intensities across the region, we derive a map of the mass-weighted mean starlight heating intensity $<$U$>$ [@Draine_Li_2007] that develops into:
[ $$<U>=\left \{
\begin{array}{ll}
\vspace{7pt}
{\large \frac{\Delta U}{ln~(~1~+~\Delta U/U_{min}~)}} &if~\alpha = 1 \\
\vspace{7pt}
U_{min}~\frac{ln~(~1+~\Delta U/U_{min}~)}{\Delta U~/~(U_{min}~+~\Delta U)} &if~\alpha = 2 \\
\vspace{7pt}
\frac{1-\alpha}{2-\alpha}\frac{(U_{min}+\Delta U)^{2-\alpha}~-~U_{min}^{2-\alpha}}{(U_{min}+\Delta U)^{1-\alpha}~-~U_{min}^{1-\alpha}}& if~\alpha \ne 1~\&~\alpha \ne 2 \\
\end{array} \right .$$ ]{}
with $\Delta$U the range of starlight intensities, U$_{min}$ the minimum heating intensity and $\alpha$ the index describing the fraction of dust exposed to a given intensity, all three obtained from the resolved @Galliano
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| 3,864
| 0.769656
|
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|
ix given by an orthogonal system $({\boldsymbol\eta}_k)_{k=1,\dots J}$ of non–zero functions from $L^2_d({\mathbb{R}})$, $J\in {\mathbb{N}}$, under the bilinear form $\left( \cdot ,\mathbf{N}^{-1} \cdot \right)$, i.e. $(M_{\mathbf{N}^{-1}})_{i,j} = \left( {\boldsymbol\eta}_i ,\mathbf{N}^{-1} {\boldsymbol\eta}_j \right)$, $1\leq i,j \leq J$. Under the assumption that either $$\begin{aligned}
\Re(M_{\mathbf{N}^{-1}}) >0 \quad \text{ or }\quad \Re(M_{\mathbf{N}^{-1}})=0 \,\text{ and } \,\Im(M_{\mathbf{N}^{-1}}) \neq 0,\end{aligned}$$ where $M_{\mathbf{N}^{-1}}=\Re(M_{\mathbf{N}^{-1}}) + i \Im(M_{\mathbf{N}^{-1}})$ with real matrices $\Re(M_{\mathbf{N}^{-1}})$ and $\Im(M_{\mathbf{N}^{-1}})$,\
then $$\Phi_{\mathbf{K},\mathbf{L}}:={\rm Nexp}\big(-\frac{1}{2} \langle \cdot, \mathbf{K} \cdot \rangle \big) \cdot \exp\big(-\frac{1}{2} \langle \cdot, \mathbf{L} \cdot \rangle \big) \cdot \exp(i \langle \cdot, {\bf g} \rangle)
\cdot \prod_{i=1}^J \delta_0 (\langle \cdot, {\boldsymbol\eta}_k \rangle-y_k),$$ for ${\bf g} \in L^2_{d}({\mathbb{R}},{\mathbb{C}}),\, t>0,\, y_k \in {\mathbb{R}},\, k =1\dots,J$, exists as a Hida distribution.\
Moreover for ${\bf f} \in S_d({\mathbb{R}})$ $$\begin{gathered}
\label{magicformula}
T\Phi_{\mathbf{K},\mathbf{L}}({\bf f})=\frac{1}{\sqrt{(2\pi)^J \det((M_{\mathbf{N}^{-1}}))}}
\sqrt{\frac{1}{\det(\mathbf{Id}+\mathbf{L}(\mathbf{Id}+\mathbf{K})^{-1})}}\\
\times \exp\bigg(-\frac{1}{2} \big(({\bf f}+{\bf g}), \mathbf{N}^{-1} ({\bf f}+{\bf g})\big) \bigg)
\exp\bigg(-\frac{1}{2} (u,(M_{\mathbf{N}^{-1}})^{-1} u)\bigg),\end{gathered}$$ where $$u= \left( \big(iy_1 +({\boldsymbol\eta}_1,\mathbf{N}^{-1}({\bf f}+{\bf g})) \big), \dots, \big(iy_J +({\boldsymbol\eta}_J,\mathbf{N}^{-1}({\bf f}+{\bf g})) \big) \right).$$
The Feynman integrand for a Charged Particle in a Constant Magnetic Field
=========================================================================
In classical mechanics a charged particle moving through a magnetic field $\mathbf{H}=(0,0,H_3)$ has the Lagrangian $$L({\mathbf{x}}, \d
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|
given $H_3$ flux. More precisely, the components that are not connected by T-duality to a given $H_3$ flux are $f_{ab}^a$ and $Q_a^{ab}$ (with indices not summed). It is common procedure in the literature not to consider these fluxes, and we will also not consider them in this paper.
We then move to the representation of the $P$ fluxes, which is the ${ \bf 352}$ of ${ SO(6,6)}$. This is the vector-spinor ([*i.e.*]{} ‘gravitino’) representation ${\theta_{M \dot{\alpha}}}$. By decomposing the whole representation under ${GL(6,\mathbb{R})}$ one gets [@Bergshoeff:2015cba] $$\theta_{M \dot{\alpha}} \rightarrow \left\{ \begin{array}{ll} P_a \ \ P_a^{b_1 b_2}\ \ P_a^{b_1 ...b_4}\ \ P_a^{b_1 ...b_6}\ \ P^{a,b_1 b_2} \ \ P^{a, b_1 ...b_4} \ \ P^{a, b_1 ...b_6} \qquad ({\rm IIB})
\\
\\
P_a^b \ \ P_a^{b_1 b_2 b_3} \ \ P_a^{b_1 ...b_5} \ \ P^{a,b} \ \ P^{a,b_1 b_2 b_3} \ \ P^{a, b_1 ...b_5} \qquad \qquad \quad \ \ \! \ ({\rm IIA})
\end{array} \right. \quad , \label{decompositionofPfluxes}$$ where the convention for each of the two decompositions is fixed by the corresponding convention of the spinor, which is given in eq. . The flux $P_a^{b_1 b_2}$, which is the second flux in the IIB decomposition, is the S-dual of the $Q$ flux. In all the fluxes, the indices $b_1...b_p$ are completely antisymmetrised, and the representations with all upstairs indices $a , b_1 ...b_p$ are irreducible with vanishing completely antisymmetric part, while the representations with the $a$ index downstairs and some $b$ indices upstairs are reducible, with the condition that the singlet is always removed [@Bergshoeff:2015cba].
To conjecture how a single T-duality transformation should act on a given $P$ flux within the set of fluxes in eq. , we simply observe that in the embedding tensor $\theta_{M\dot{\alpha}}$ the vector index transforms as it should, namely a lower index in a given direction is raised if one performs a T-duality in that direction, while the spinor index decomposes in the set of
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|
BX}^\dagger)(\mathbf{BX}^\dagger)^T$$ and $\mathbf{c}_i\perp\mathbf{c}_j$ for $i\neq j$.
It is sufficient to prove that $\mathbf{m}_i\perp\mathbf{m}_j$ for $i\neq j$. From $\mathbf{A}=\mathbf{X}^T\mathbf{X}$ we have $$\mathbf{d}=\mathbf{Ae}=\mathbf{X}^T\mathbf{Xe},$$ $$2m=\mathbf{d}^T\mathbf{e}=\mathbf{e}^T\mathbf{X}^T\mathbf{Xe},$$ where $\mathbf{e}$ is a column vector with all ones. Therefore, $$\mathbf{B}=\mathbf{A}-\frac{\mathbf{d}\mathbf{d}^T}{2m}
=\mathbf{X}^T\mathbf{X}-\frac{(\mathbf{X}^T\mathbf{Xe})(\mathbf{X}^T\mathbf{Xe})^T}{\mathbf{e}^T\mathbf{X}^T\mathbf{Xe}}$$ $$=\mathbf{X}^T\mathbf{X}-\frac{\mathbf{X}^T\mathbf{Xe}\mathbf{e}^T\mathbf{X}^T\mathbf{X}}{\mathbf{e}^T\mathbf{X}^T\mathbf{Xe}}.$$ Since $\mathbf{X}^T\mathbf{X}\mathbf{X}^{\dagger}=\mathbf{X}^T$ is always true, we have $$\mathbf{BX}^\dagger=\mathbf{X}^T-\frac{\mathbf{X}^T\mathbf{Xe}\mathbf{e}^T\mathbf{X}^T}{\mathbf{e}^T\mathbf{X}^T\mathbf{Xe}}.$$ Consequently, $$(\mathbf{BX}^\dagger)(\mathbf{BX}^\dagger)^T$$ $$=\bigg(\mathbf{X}^T-\frac{\mathbf{X}^T\mathbf{Xe}\mathbf{e}^T\mathbf{X}^T}{\mathbf{e}^T\mathbf{X}^T\mathbf{Xe}}\bigg)\bigg(\mathbf{X}-\frac{\mathbf{X}\mathbf{ee}^T\mathbf{X}^T\mathbf{X}}{\mathbf{e}^T\mathbf{X}^T\mathbf{Xe}}\bigg)$$ $$=\mathbf{X}^T\mathbf{X}-\frac{2\mathbf{X}^T\mathbf{Xe}\mathbf{e}^T\mathbf{X}^T\mathbf{X}}{\mathbf{e}^T\mathbf{X}^T\mathbf{Xe}}$$ $$+\frac{(\mathbf{e}^T\mathbf{X}^T\mathbf{Xe})\mathbf{X}^T\mathbf{Xe}\mathbf{e}^T\mathbf{X}^T\mathbf{X}}{(\mathbf{e}^T\mathbf{X}^T\mathbf{Xe})^2}$$ $$=\mathbf{X}^T\mathbf{X}-\frac{\mathbf{X}^T\mathbf{Xe}\mathbf{e}^T\mathbf{X}^T\mathbf{X}}{\mathbf{e}^T\mathbf{X}^T\mathbf{Xe}}.$$ Therefore $\mathbf{B}=(\mathbf{BX}^\dagger)(\mathbf{BX}^\dagger)^T$. Since $\mathbf{B}\mathbf{b}_i=\lambda_i\mathbf{b}_i$, $\mathbf{B}\mathbf{b}_j=\lambda_j\mathbf{b}_j$, $\lambda_i\neq0$, $\lambda_j\neq0$, we have $$\mathbf{m}_i^T\mathbf{m}_j=(\mathbf{b}_i^T\mathbf{X}^\dagger)(\mathbf{b}_j^T\mathbf{X}^\dagger)^T$$ $$=\bigg(\frac{1}{\lambda_i}\mathbf{b}_i^T\mathbf{BX}^\dagger\bigg)\bigg(\frac{1}{\lambda_j}\m
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| null | null |
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|
nt paths on ${{\mathbb S}}_0$. In addition, ${\mathfrak{S}}'$ is trivially a subset of ${\mathfrak{S}}'_0$ in [(\[eq:Ssupset\])]{}. Therefore, we have $|{\mathfrak{S}}_0|\leq|{\mathfrak{S}}'_0|$. This completes the proof of [(\[eq:piNbd\])]{} for $j=0$.
Here, we summarize the basic steps that we have followed to bound $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ and which we generalize to prove [(\[eq:pi0’-bd\])]{} below and the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$ in Section \[sss:dbconn\].
(i) Count the (minimum) number, say, $k+1$, of *edge-disjoint* paths on ${{\mathbb G}}_{{\bf n}}$ that satisfy the source constraint (as well as other additional conditions, if there are) of the considered function $f(x)$. For example, $k=2$ for $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)\equiv\frac1{Z_\Lambda}\sum_{{\partial}{{\bf n}}=\{o,x\}}
w_\Lambda({{\bf n}})\,{\mathbbm{1}{\scriptstyle\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}}}$.
(ii) Multiply $f(x)$ by $(\frac{Z_\Lambda}{Z_\Lambda})^k=\prod_{i=1}^k
(\frac1{Z_\Lambda}\sum_{{\partial}{{\bf m}}_i={\varnothing}}w_\Lambda({{\bf m}}_i)) \,(\equiv1)$ and then overlap the $k$ dummies ${{\bf m}}_1,\dots,{{\bf m}}_k$ on the original current configuration ${{\bf n}}$. Choose $k$ paths $\omega_1,\dots,\omega_k$ among $k+1$ edge-disjoint paths on ${{\mathbb G}}_{{{\bf n}}+\sum_{i=1}^k{{\bf m}}_i}$.
(iii) Use Lemma \[lmm:GHS-BK\] to exchange the occupation status of edges on $\omega_i$ between ${{\mathbb G}}_{{\bf n}}$ and ${{\mathbb G}}_{{{\bf m}}_i}$ for every $i=1,\dots,k$. The current configurations after the mapping, denoted by $\tilde{{\bf n}},\tilde{{\bf m}}_1,\dots,\tilde{{\bf m}}_k$, satisfy ${\partial}\tilde{{\bf n}}={\partial}{{\bf n}}{\vartriangle}{\partial}\omega_1{\vartriangle}\cdots{\vartriangle}{\partial}\omega_k$ and ${\partial}\tilde{{\bf m}}_i={\partial}\omega_i$ for $i=1,\dots,k$.
If $y=o$ or $x$, then [(\[eq:pi0’-bd\])]{} is reduced to the inequality for
| 3,825
| 2,162
| 2,875
| 3,535
| 3,507
| 0.771862
|
github_plus_top10pct_by_avg
|
inition, $\underline{N(k)} = eM$ where $M = \widetilde{S}_{c+k-1}\circ \cdots \circ \widetilde{S}_{c}(X)$, in the notation of . By Proposition \[shiftonO\] $M$ also has a finite filtration by standard modules and so [@GGOR Proposition 2.21] shows that $M$ is a finitely generated free module over ${\mathbb{C}}[{\mathfrak{h}}]$ and hence over ${\mathbb{C}}[{\mathfrak{h}}]^{{{W}}}$. Thus, so is its summand $eM$.
\(4) We first show that $N(k)$ is a finitely generated right module over $R=({\mathbb{C}}[{\mathfrak{h}}]^{{W}})^{\mathrm{op}} \otimes_{\mathbb{C}}{\mathbb{C}}[{\mathfrak{h}}^*]$. By part (1), $B_{k0}\subseteq U_{c}$ and so $N(k)\subseteq eH_c$. Thus $\operatorname{{\textsf}{ogr}}N(k) \subseteq \operatorname{{\textsf}{ogr}}H_{c}
= {\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]\ast {{W}}$, which is certainly a noetherian ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}\otimes {\mathbb{C}}[{\mathfrak{h}}^*]$-module. Since the $\operatorname{{\textsf}{ord}}$ filtration on $N(k)$ is the one induced from $D({{\mathfrak{h}}^{\text{reg}}})\ast {{W}}$, the actions of ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$ and ${\mathbb{C}}[{\mathfrak{h}}^*]$ on $\operatorname{{\textsf}{ogr}}N(k)$ are the natural ones induced from the actions of those rings on $N(k)\subset D({{\mathfrak{h}}^{\text{reg}}})\ast{{W}}$. In other words, the given $R$-module structure of $\operatorname{{\textsf}{ogr}}N(k) $ is the one induced from the $R$-module structure of $N(k)$. Since the former module is finitely generated, so is the latter.
Let $y_1,\ldots ,y_{n-1}$ be the generators of ${\mathbb{C}}[{\mathfrak{h}}^*]$ and let $q_1, \ldots ,q_{n-1}$ be the fundamental invariants of ${\mathbb{C}}[{\mathfrak{h}}]^{{{W}}}$. By (2), the $\{y_j\}$ form an r-sequence in $N(k)$, while (3) implies that the $\{q_j\}$ form an r-sequence in the factor $\underline{N(k)}=N(k)/\sum N(k)y_j$ as a module over ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}=R/\sum y_jR$. Thus $\Sigma = \{y_\ell ,q_{m} : 1\leq \ell, m\leq n-1\}$ is a regular sequence for the right $R$-module $N(k)$. I
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|
gin{aligned}
\langle N_F^0 \rangle=L \int \frac{dk}{2\pi} \Big[ e^{\beta
\left(k^2/2m -\mu\right)}+1\Big]^{-1}.\end{aligned}$$ In the limit $T\to 0$ and using the fact that after resonance $\mu\simeq \epsilon_b/2$, the fraction of atoms that are unbound is exponentially small. Therefore, (\[N45\]) becomes $$\begin{aligned}
\frac{N}{2}=\frac{\langle N \rangle}{2} \simeq -T\frac{\partial \ln
Z_B^{eff}}{\partial 2\mu}\end{aligned}$$ which shows that $2\mu$ is the chemical potential for the gas of dimers only. We now shift the zero of energy of the many-body system by an amount $-N\epsilon_b/2$, and accordingly define $\mu_B\equiv 2\mu -\epsilon_b$ as the new chemical potential for dimers.
In conclusion, after resonance and under the assumption that the resonance is broad, the system is described by a single channel model of bosons (i.e. dimers) with an action $$\begin{aligned}
S_B^{eff}&=&\int_0^{\beta}d\tau \int dx \bigg(\bar{\psi}_{B}
\Big[\partial_{\tau}-\frac{\partial_x^2}{2m_B}-\mu_B\Big]\psi_{B}
\nonumber \\ &+&\frac{g_B}{2}
\bar{\psi}_{B}\bar{\psi}_{B}\psi_{B}\psi_{B} \bigg)\end{aligned}$$ where $m_B\equiv 2m$, $\mu_B=2\mu-\epsilon_B$ and $g_B=
3g^4\sqrt{m}/8|\epsilon_b|^{5/2}$ [@footnote4]. This is the action corresponding to the Lieb-Liniger model of $N_B\equiv N/2$ bosons of mass $m_B$ interacting via a repulsive delta potential [@LL]. The single dimensionless coupling constant is $\gamma\equiv mg_B/n$ and the BEC limit corresponds to $1/\gamma \to +\infty$ or $\nu \to -\infty$. Because $g_B \sim 1/mr_{\star}$ when $\nu \to 0^{-}$ (see Appendix B), the parameter $1/\gamma$ is restricted to: $$\begin{aligned}
n r_{\star} <\frac{1}{\gamma} < +\infty\end{aligned}$$ In the broad resonance limit, $nr_{\star} \to 0$ implying that apart from a vanishingly small region close to resonance the Boson-Fermion resonance model, after resonance, is equivalent to the single channel repulsive Lieb-Liniger model for dimers.
Discussion
==========
It was recently shown [@FRZ; @Tokatly] that interacting fermions in a quasi-
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^1]: Université de Lausanne, Faculté des lettres, Section de philosophie, 1015 Lausanne, Switzerland. E-mail: <Antonio.Vassallo@unil.ch>
[^2]: Ludwig-Maximilians-Universität München, Mathematisches Institut, Theresienstrasse 39, 80333 München, Germany. E-mail: <deckert@math.lmu.de>
[^3]: Université de Lausanne, Faculté des lettres, Section de philosophie, 1015 Lausanne, Switzerland. E-mail: <Michael-Andreas.Esfeld@unil.ch>
---
abstract: |
The evolution of number density, size and intrinsic colour is determined for a volume-limited sample of visually classified early-type galaxies selected from the HST/ACS images of the GOODS North and South fields (version 2). The sample comprises $457$ galaxies over $320$ arcmin$^2$ with stellar masses above $3\cdot
10^{10}$in the redshift range 0.4$<$z$<$1.2. Our data allow a simultaneous study of number density, intrinsic colour distribution and size. We find that the most massive systems (${\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}3\cdot
10^{11}M_\odot$) do not show any appreciable change in comoving number density or size in our data. Furthermore, when including the results from 2dFGRS, we find that the number density of massive early-type galaxies is consistent with no evolution between z=1.2 and 0, i.e. over an epoch spanning more than half of the current age of the Universe. Massive galaxies show very homogeneous [*intrinsic*]{} colour distributions, featuring red cores with small scatter. The distribution of half-light radii – when compared to z$\sim$0 and z$>$1 samples – is compatible with the predictions of semi-analytic models relating size evolution to the amount of dissipation during major mergers. However, in a more speculative fashion, the observations can also be interpreted as weak or even no evolution in comoving number density [*and size*]{} between 0.4$<$z$<$1.2, thus pushing major mergers of the most massive galaxies towards lower redshifts.
author:
- |
Ignacio Ferreras$^{1}$[^1], Thorsten Lisker$^2$, Anna Pasquali$^3$,
| 3,828
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{\operatorname{Fun}}(\Gamma,k)$, the homology $H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Gamma,E)$ of the category $\Gamma$ with coefficients in $E$ is by definition the derived functor of the direct limit functor $$\displaystyle\lim_{\overset{\to}{\Gamma}}:{\operatorname{Fun}}(\Gamma,k) \to k{\operatorname{\it\!-Vect}}.$$ Analogously, the cohomology $H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Lambda,E)$ is the derived functor of the inverse limit $\displaystyle\lim_{\overset{\gets}{\Gamma}}$. Equivalently, $$H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Gamma,E) = {\operatorname{Ext}}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(k,E),$$ where $k \in {\operatorname{Fun}}(\Gamma,k)$ is the constant functor (all objects in $\Gamma$ go to $k$, all maps go to identity). In particular, $H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Gamma,k)$ is an algebra. For any $E \in {\operatorname{Fun}}(\Gamma,k)$, the cohomology $H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Gamma,E)$ and the homology $H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Gamma,E)$ are modules over $H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Gamma,k)$.
We also note, although it is not needed for the definition of cyclic homology, that for any functor $\gamma:\Gamma' \to \Gamma$ between two small categories, we have the pullback functor $\gamma^*:{\operatorname{Fun}}(\Gamma,k) \to {\operatorname{Fun}}(\Gamma',k)$, and for any $E \in
{\operatorname{Fun}}(\Gamma,k)$, we have natural maps $$\label{dir.im}
H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Gamma',\gamma^*E) \to H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Gamma,E),\qquad
H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Gamma,E) \to H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Gamma',\gamma^*E).$$ Moreover, the pullback functor $\gamma^*$ has a left adjoint $\gamma_!:{\operatorname{Fun}}(\Gamma',k) \to {\operatorname{Fun}}(\Gamma,k)$ and a right-adjoint $f_*:{\operatorname{Fun}}(\Gamma',k) \to {\operatorname{Fun}}(\Gamma,k)$, known as the left and right Kan extensions. In general, $f_!$ i
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pear to want lag(). This is complicated, because your dates are not well-formed, but that is fixable:
select t1.entity, t1.submissionperiod,
lag(t1.submissionperiod) over (partition by t1.entity order by convert(date, '01-' + t1.submissionperiod)) as prev_submissionperiod,
t1.value,
lag(t1.value) over (partition by t1.entity order by convert(date, '01-' + t1.submissionperiod)) as prev_submissionperiod,
from table1 t1
Q:
Ejabberd Packet parsing using erlang
Ejabberd server receives packet like this:
{xmlel,<<"message">>,[{<<"from">>,<<"user1@localhost/resource">>},{<<"to">>,<<"user2@localhost">>},{<<"xml:lang">>,<<"en">>},{<<"id">>,<<"947yW-9">>}],[{xmlcdata,<<">">>},{xmlel,<<"body">>,[],[{xmlcdata,<<"Helllo wassup!">>}]}]}
I want to fetch data from this packet.
Needed data : Type, If the body has a certain parameter, say {<<"xml:lang">>,<<"en">>}
I am doing the following operations:
{_XMLEL, TYPE, DETAILS , _BODY} = Packet
This provides me the type : <<"message">> or <<"iq">> or <<"presence">>.
To check if DETAILS has {<<"xml:lang">>,<<"en">>} I do this:
Has_Attribute=lists:member({<<"xml:lang">>,<<"en">>},DETAILS)
Is there any better way to do this?
I also need the to and from attributes from the packet.
A:
Use a combination of pattern matching in the function head together with a fold over the details to extract everything you need.
The function below returns a list of key-value tuples, where the <<"type">> tuple is artificially created so the list is homogenous:
extract({xmlel, Type, Details, _}) ->
[{<<"type">>,Type} |
lists:foldl(fun(Key, Acc) ->
case lists:keyfind(Key, 1, Details) of
false -> Acc;
Pair -> [Pair|Acc]
end
end, [], [<<"from">>,<<"to">>,<<"xml:lang">>])];
extract(_) -> [].
The first clause matches the {xmlel, ...} tuple, extracting Type and Details. The return value consists of a list with head {<<"type">>,Type} followed by
| 3,830
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| 0.829519
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ial components of the S-matrixes and the phases for two systems are interdependent (also see [@Cooper.1995.PRPLC], p. 278–279): $$\begin{array}{cc}
S_{l}^{(1)}(k) = - S_{l}^{(2)}(k), &
\delta_{l}^{(1)}(k) = \delta_{l}^{(2)}(k) + \pi/2.
\end{array}
\label{eq.2.4.9}$$
Let’s consider a spherically symmetric quantum system with the radial potential, to which a zero amplitude of the reflection $R(k)$ of the wave function corresponds. The particle during its scattering in this field propagates into a center without the smallest reflection by the field. In particular, such is a nul radial potential. We shall name such quantum systems and their radial potentials as *reflectionless* or *absolutely transparent*. Then from (\[eq.2.4.5\]) one can see, that the potential-partner for the reflectionless potential is reflectionless also in that region, where it is finite. If such potential is finite on the whole region of its definition, then it is reflectionless completely (i. e. in standard definition of quantum mechanics). A series of the finite potentials of hierarchy, which contains the nul radial potential, should be reflectionless also. Using this simple idea and knowing a form of only one reflectionless potential, one can construct many new exactly solvable radial reflectionless potentials.
Spherically symmetric systems with absolute transparency \[sec.3\]
==================================================================
A radial reflectionless potentials with barriers \[sec.3.1\]
------------------------------------------------------------
In [@Maydanyuk.2005.APNYA] (see sec. 5.3.2, p. 459–462) an one-dimensional superpotential, defining a reflectionless potential which in semiaxis $0 < x < +\infty$ has one hole, one barrier and then with increasing of $x$ falls down monotonously to zero in asymptotic region, had found. As this superpotential is obtained on the basis of interdependence between two one-dimensional hamiltonians with continuous energy spectra, one can use it in the problem about scatte
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|
ional regularization in place is determining how to trade off classification accuracy and gradient orthogonality. Our defense framework requires little computational overhead to filter operations such as blurs and sharpens, and is not particularly computationally intensive when there are VAEs to train. Training a number of VAEs equal to the depth of a network in order to obtain an ensemble containing an exponentially large number of models can be computationally intensive, however, in critical mission scenarios, such as healthcare and autonomous driving, spending more time to train a robust system is certainly warranted and is a key to broad adoption.
Conclusion
==========
In this project, we presented a probabilistic framework that uses properties intrinsic to deep CNNs in order to defend against adversarial examples. Several experiments were performed to test the claims that such a setup would result in an exponential ensemble of models for just a linear computation cost. We demonstrated that our defense cleans the adversarial noise in the perturbed images and makes them more similar to natural images (Supplementary). Perhaps our most exciting result is that the cosine similarity of the gradients between defense arrangements is highly predictive of attack transferability which opens a lot of avenues for developing defense mechanisms of CNNs and DNNs in general. As proof of a concept regarding classification biases between models, we showed that the triplet network detector was quite effective at detecting adversarial examples, and was fooled by only a small fraction of the adversarial examples that fooled LeNet. To conclude, probabilistic defenses are able to substantially reduce adversarial attack success rates, while revealing interesting properties about existing models.
Supplementary Material {#supplementary-material .unnumbered}
======================
Reconstruction of Feature Embeddings {#appdx:reconstruct .unnumbered}
------------------------------------
Since using autoencoders to reconstruct activ
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|
d we show below that this multifunction is infact the Dirac delta “function” $\delta_{\nu}(\mu)$, usually written as $\delta(\mu-\nu)$. This suggests that in $\textrm{Multi}(V(\mu),\mathbb{R})$*, every $\nu\in V(\mu)$ is in the point spectrum of $\mu$*, so that *discontinuous functions that are pointwise limits of functions in function space can be replaced by graphically converged multifunctions in the space of multifunctions*. Completing the equivalence class of $0$ in Fig. \[Fig: GenInv\], gives the multifunctional solution of Eq. (\[Eqn: eigen\]).
From a comparison of the definition of ill-posedness (Sec. 2) and the spectrum (Table \[Table: spectrum\]), it is clear that $\mathscr L_{\lambda}(x)=y$ is ill-posed iff
\(1) $\mathscr L_{\lambda}$ not injective $\Leftrightarrow$ $\lambda\in P\sigma(\mathscr L_{\lambda})$, which corresponds to the first row of Table \[Table: spectrum\].
\(2) $\mathscr L_{\lambda}$ not surjective $\Leftrightarrow$ the values of $\lambda$ correspond to the second and third columns of Table \[Table: spectrum\].
\(3) $\mathscr L_{\lambda}$ is bijective but not open $\Leftrightarrow$ $\lambda\textrm{ is either in }C\sigma(\mathscr L_{\lambda})\textrm{ or }R\sigma(\mathscr L_{\lambda})$ corresponding to the second row of Table \[Table: spectrum\].
We verify in the three steps below that $X=L_{1}[-1,1]$ of integrable functions, $\nu\in V(\mu)=[-1,1]$ belongs to the continuous spectrum of $\mu$.
\(a) *$\mathcal{R}(\mu_{\nu})$ is dense, but not equal to $L_{1}$*. The set of functions $g(\mu)\in L_{1}$ such that $\mu_{\nu}^{-1}g\in L_{1}$ cannot be the whole of $L_{1}$. Thus, for example, the piecewise constant function $g=\textrm{const}\neq0$ on $\mid\mu-\nu\mid\leq\delta>0$ and $0$ otherwise is in $L_{1}$ but not in *$\mathcal{R}(\mu_{\nu})$* as $\mu_{\nu}^{-1}g\not\in L_{1}$. Nevertheless for any $g\in L_{1}$, we may choose the sequence of functions $$g_{n}(\mu)=\left\{ \begin{array}{ccl}
0, & & \textrm{if }\mid\mu-\nu\mid\leq1/n\\
g(\mu), & & \textrm{otherwise}\end{array}\righ
| 3,833
| 3,577
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astro-ph/0712.1202) Wang, W.-H., Cowie, L.L., Barger, A.J. 2004, [ApJ]{}, 613, 655 Wang, W.-H., Cowie, L.L., Barger, A.J. 2006, [ApJ]{}, 647, 74 Wang, W.-H., et al. 2007, [ApJ]{}, 670, L89 Webb, T. M. A., et al. 2006, [ApJ]{}, 636, L17 Wilson, G. W., et al. 2008a, [MNRAS]{}, in press (astro-ph/0801.2783) Wilson, G. W., et al. 2008b, [MNRAS]{}, submitted (astro-ph/0803.3462) Younger, J. D., Fazio, G. G., Huang, J.-S., Yun, M. S., Wilson, G. W. et al. 2007, [ApJ]{}, 671, 1531 Younger, J. D., et al. 2008, [MNRAS]{}, in press (astro-ph/0801.2764) Yun, M.S., Reddy, N., & Condon, J.J. 2001, [ApJ]{}, 554, 803 Yun, M.S., & Carilli, C.L. 2002, [ApJ]{}, 568, 88
\[lastpage\]
[^1]: E-mail: myun@astro.umass.edu
---
abstract: 'In this short article we develop recent proposals to relate Yang-Baxter sigma-models and non-abelian T-duality. We demonstrate explicitly that the holographic space-times associated to both (multi-parameter)-$\beta$-deformations and non-commutative deformations of ${\cal N}=4$ super Yang-Mills gauge theory including the RR fluxes can be obtained via the machinery of non-abelian T-duality in Type II supergravity.'
---
[**Marginal and non-commutative deformations\
via non-abelian T-duality**]{}
[Ben Hoare$^{a}$ and Daniel C. Thompson$^{b}$]{}
[*$^{a}$ Institut für Theoretische Physik, ETH Zürich,\
Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland.*]{}
[*$^{b}$ Theoretische Natuurkunde, Vrije Universiteit Brussel & The International Solvay Institutes,\
Pleinlaan 2, B-1050 Brussels, Belgium.*]{}
[*E-mail: *]{} [<bhoare@ethz.ch>, <Daniel.Thompson@vub.ac.be>]{}
Introduction {#sec:intro}
============
There is a rich interplay between the three ideas of T-duality, integrability and holography. Perhaps the most well studied example of this is the use of the TsT transformation to ascertain the gravitational dual space-times to certain marginal deformations of ${\cal N}=4$ super Yang-Mills gauge theory [@Lunin:2005jy]. Whilst this employs familiar T-dualities of $U(1)$ isometries in space-time, T-
| 3,834
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| 3,264
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|
cal Higgs states in addition to the light CP-even (SM like) Higgs boson. The coupling of the Higgs bosons to the bottom quark depends on the MSSM parameters, particularly, $\tanb$. In addition, the couplings also depend on the bottom quark threshold corrections and the effect of these corrections have been the subject of many works especially in the large $\tanb\ $ regime [@Carena:1999py; @Guasch:2001wv; @Belyaev:2002eq; @Guasch:2003cv; @Crivellin:2009ar; @Crivellin:2010er; @Crivellin:2012zz; @Baer:2011af; @Carena:2012rw; @Dittmaier:2014sva; @Carena:2013qia]. The low energy effective Lagrangian coupling the bottom quark with the up- and the down-type Higgs bosons in the MSSM including the supersymmetric threshold corrections can be written as \_[eff]{} = -\_b\^0 |[b]{}\_R\^0 b\_L\^0 + ,
where \_d\^0 &=& ( v\_d + H - h + i A -i G\^0 )\
\_u\^0 &=& ( v\_u + H + h + i A +i G\^0 ) .\[Eq:bare-eff-lag\]
Here $\Delta_2$ represents the coupling of the bottom quark to the “wrong" Higgs, which is generated by the radiative effects discussed in this paper. The corrections to the coupling of the bottom quark to the down-type Higgs are represented by $\Delta_1$. The $\Delta_2$ interactions are $\tanb$-enhanced while the $\Delta_1$ corrections are not. The expression in \[Eq:bare-eff-lag\] must be matched to the renormalized Lagrangian given by [@Guasch:2003cv] \_[eff]{} = -\_b |[b]{}\_R b\_L + , yielding the relations \_b = \_b\^0(1+\_1)\
\
= .\[Eq:deltas\]
Consider the gluino contribution in the approximate form of the threshold corrections given by \[Eq:common-app\]. The $\mu$-term, which is proportional to $\tanb$, is included in $\Delta_2$ while the $A_b$-term is included in $\Delta_1$. The $\Delta_1$ correction is typically found to be $\mathcal{O}$(1)% and is therefore often neglected [@Guasch:2003cv]. We point out here that neither the $B_1^{\widetilde{g}}$ terms from the gluino-sbottom contribution nor the contributions from the Higgses are proportional to $\tanb$, and therefore they enhance $\Delta_1$ by $\s
| 3,835
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|
{-\theta _{0}\xi _{1}(l)}\lambda _{n}^{-\theta _{0}\omega
_{1}(l)}\Phi _{n}(0))^{pm_{0}}\int_{{\mathbb{R}}^{d}}\frac{dx}{\psi _{p\eta
-l^{\prime }-p\chi }(x)}.\end{aligned}$$We conclude that $$\left\Vert \Psi _{\eta ,\kappa }\phi _{t}^{n,m_{0}}\right\Vert
_{2h+q,2h,p}\leq Ct^{-\theta _{0}\xi _{1}(q+2h)}\times \lambda _{n}^{-\theta
_{0}\omega _{1}(q+2h)}\Phi _{n}^{m_{0}}(0)=:\theta (n). \label{R6}$$By (\[TRa\]) $\theta (n)\uparrow +\infty $ and $\Theta \theta (n)\geq
\theta (n+1)$ with $$\Theta =\gamma ^{\theta _{0}((a+b)m_{0}+q+2h+d+2\theta _{1})+m_{0}}\geq 1.$$In the following we will choose $h$ sufficiently large, depending on $\delta
_{\ast },m_{0},q,d$ and $p.$ So $\Theta $ is a constant depending on $\delta
_{\ast },m_{0},q,d,a,b$,$\gamma $ and $p,$ as the constants considered in the statement of our theorem.
**Step 3: analysis of the remainder**. We study here $%
d_{m_{0}}(n):=d_{(a+b)m_{0}}(\mu ^{\eta ,\kappa },\mu _{n}^{\eta ,\kappa
,m_{0}})$ as required in (\[reg10\]): we prove that, if $\eta \geq \kappa
+d+1,$ then$$d_{m_{0}}(n)\leq C(\Lambda _{n}\varepsilon _{n})^{m_{0}}\leq \lambda
_{n}^{\theta _{0}(a+b+\delta _{\ast })m_{0}}\Phi _{n}^{m_{0}}(\delta _{\ast
}). \label{R9}$$
Using first $(A_{1})$ and $(A_{2})$ (see (\[TR3\]) and (\[TR2\])) and then $(A_{4})$ (see (\[R7\])) we obtain $$\left\Vert \prod_{i=0}^{m_{0}-1}(P_{t_{i}-t_{i+1}}^{n}\Delta
_{n})P_{t_{m_{0}}}f\right\Vert _{0,-\kappa ,\infty }\leq C\left\Vert
f\right\Vert _{(a+b)m_{0},-\kappa ,\infty }(\Lambda _{n}\varepsilon
_{n})^{m_{0}}$$which gives $$\left\Vert R_{n}^{m_{0}}f\right\Vert _{0,-\kappa ,\infty }\leq C\left\Vert
f\right\Vert _{(a+b)m_{0},-\kappa ,\infty }(\Lambda _{n}\varepsilon
_{n})^{m_{0}}.$$Using now the equivalence between (\[NOT6a\]) and (\[NOT6b\]) we obtain$$\left\Vert \frac{1}{\psi _{\kappa }}R_{n}^{m_{0}}(\psi _{\kappa
}f)\right\Vert _{\infty }\leq C\left\Vert f\right\Vert _{(a+b)m_{0},\infty
}(\Lambda _{n}\varepsilon _{n})^{m_{0}}. \label{R8}$$We take now $g\in C^{\infty }({\mathbb{R}}^{d}\times {\mathbb{R}}^{d}),$ we
| 3,836
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| 1,755
| 3,895
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|
or a fixed set of parameters, statistical data are collected by running 30 independent simulations. In each run, a maximum displacement step of colloids ${d_{c}=0.01\sigma _{2}}$ and droplets ${d_{d}=d_{c}\sqrt{\sigma _{2}/\sigma _{\textrm{d}}}}$ ensures that Monte Carlo simulations are approximately equivalent to Brownian dynamics simulations [@Sanz2010].
The total number of MC cycles per particle is $10^{6}$, with $5\times 10^{5}$ MC cycles used to shrink the droplets at a fixed rate. This shrinking rate is chosen such that the droplet diameter vanishes after $5\times 10^{5}$ MC steps. Another $5\times 10^{5}$ MC cycles are used to equilibrate the cluster configurations. As a test, for $k=0.1$ (open clusters), $k=0.5$ (intermediate clusters) and $k=1$ (closed clusters), we monitored the total energy and the obtained number of clusters $N_{n_{c}}$ (composing of $n_c$ colloids and $n_b$ bonds) for an additional $10^{6}$ cycles and found no changes in the results.
Our kinetic MC simulation uses sequential moves of individual particles and neglects the collective motion of particles in the cluster, i.e. collective translational and rotational cluster moves are absent. Such collective modes of motion only play a role in dense colloidal suspensions of strongly interacting overdamped particles [@Stephen2009; @Stephen2011].
We did not attempt to reproduce the correct experimental time scale of droplet evaporation, and the influence of colloid adsorption on the evaporation rate is neglected. The physical time corresponding to the MC time scale can be roughly estimated via the translational diffusion coefficient of clusters $D_{\textrm{cls}}$ defined by the Einstein relationship [@Huitema1999], $$\lim_{n\to\infty}\frac{\left\langle \triangle r_{\textrm{cls}}^{2}(n)\right\rangle }{n}=6D_{\textrm{cls}}\tau,$$ where $n$ is the number of MC cycles and $\tau$ is the physical time per MC cycle. The Stokes-Einstein equation for diffusion of spherical particles is ${D_{\textrm{cls}}=\frac{k_{B}T}{3\pi\eta\sigma_{\textrm{cl
| 3,837
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| 3,728
| 3,182
| 0.774219
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|
\[eq:SW:13\]). The remaining arbitrary quantities are arbitrary functions of the $x$’s and $u$’s, which have to be used to satisfy the conditions (\[eq:SW:18\]). However, in the particular case when the system (\[eq:SW:1\]) has two equations in two dependent variables, the two-dimensional matrix $L$ is defined by a single parameter and $\Omega$ is the only other parameter available to satisfy the condition (\[eq:SW:13\]). Thus, since the system is autonomous (it can be expressed solely in terms of the $u$’s and their derivatives), it is clear that, in that case, $\Omega$ and $L$ depend only on the dependent variables $u$’s.
- Supposing that $b$ is continous and assuming that conditions (i-v) are satisfied, the right side of equation (\[eq:SW:15\]) is continuous, which ensures the existence and uniqueness of the solution for $f(r)$ (see for example [@Ince]).
- A sufficient condition for the system (\[eq:SW:15\]) to be expressible in terms of $r$ only ([*i.e.* ]{}well-defined) is: $$\frac{{\partial}\lambda_i}{{\partial}u}L b=0,\qquad i=1,\ldots, p.$$This condition is trivially satisfied when the vector $\lambda$ is constant. More generally, it is sufficient to require that $\frac{{\partial}\lambda_i}{{\partial}u}Lbx^i$ be proportional to $r=\lambda_i x^i$, [*i.e.* ]{}that there exist $\beta(x,u)$ such that $$\frac{{\partial}\lambda_i}{{\partial}u}L b=\beta(x,u)\lambda_i,\qquad i=1,\ldots,p.$$
- If the matrix $A^i\lambda_i$ is invertible, then $$\frac{{\partial}\lambda_i}{{\partial}u}Lbx^i=\frac{{\partial}\lambda_i}{{\partial}u}(\mathrm{A}^j\lambda_j)^{-1}(\mathrm{A}^j\lambda_j) Lb x^i=\frac{1}{\Omega}\frac{{\partial}\lambda_i}{{\partial}u}(\mathcal{A}^j\lambda_j)^{-1}bx^i$$is satisfied due to condition (\[eq:SW:13\]). From the previous remark, we deduce $$\frac{1}{\Omega}\frac{{\partial}\lambda_i}{{\partial}u}(\mathcal{A}^ j\lambda_j)^{-1}b=\beta(u) \lambda_i,\quad i=1,\ldots,p,$$which is a sufficient condition where the rotation matrix $L$ is not involved when the matrix $A^j\lambda_j$ is invert
| 3,838
| 4,252
| 4,174
| 3,612
| 3,953
| 0.769023
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_2|\leq N$.
Let $V$ be the vase from the previous example. Then $V$ is not coarsely equivalent to $\mathbb R$.
According to the previous example $\sigma(V)=1$ and according to [@MMS Corollary 3.7] $\sigma(\mathbb R)=2$.
[99]{}
B. Miller, J. Moore, and L. Stibich. *An invariant of metric spaces under bornologous equivalences.* Mathematics Exchange. 7 (2010) 12–19.
J. Roe. *Lectures on coarse geometry, University lecture series, 31.* American Mathematical Society, Providence, RI, 2003.
---
author:
- 'Jan-Peter Calliess$^{1}$ [^1] [^2]'
title: '**Lipschitz Optimisation for Lipschitz Interpolation$^*$** '
---
[^1]: \*This paper is an extended version of a conference paper that will appear in the Proceedings of the American Control Conference (ACC 2017).
[^2]: $^{1}$Jan-Peter Calliess is with the Engineering Department, University of Cambridge, UK. [jpc73@cam.ac.uk]{}
---
abstract: |
We have performed [[*HST*]{}]{} imaging of a sample of 23 high-redshift ($1.8<z<2.75$) Active Galactic Nuclei, drawn from the [[combo-17]{}]{}survey. The sample contains moderately luminous quasars ($M_B \sim
-23$). The data are part of the [[gems]{}]{} imaging survey that provides high resolution optical images obtained with the Advanced Camera for Surveys in two bands ([F606W]{} and [F850LP]{}), sampling the rest-frame UV flux of the targets. To deblend the AGN images into nuclear and resolved (host galaxy) components we use a PSF subtraction technique that is strictly conservative with respect to the flux of the host galaxy. We resolve the host galaxies in both filter bands in 9 of the 23 AGN, whereas the remaining 14 objects are considered non-detections, with upper limits of less than 5 % of the nuclear flux. However, when we coadd the unresolved AGN images into a single high signal-to-noise composite image we find again an unambiguously resolved host galaxy. The recovered host galaxies have apparent magnitudes of $23.0<\mathrm{{F606W}}<26.0$ and $22.5<\mathrm{{F850LP}}<24.5$ with rest-frame UV colours in the
| 3,839
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|
mprises the inputs $(x_1,x_2)$, the outputs $(\delta_1,\delta_2,\delta_3,\delta_4)$ and a number of neurons in the hidden layers. If ${r}={{\left( {{{{x}}}_{1}}^{T},{{{{x}}}_{2}}^{T} \right)}^{T}}$ and $\delta=(\delta_1,\delta_2,\delta_3,\delta_4)^T$, then the output of the RBFN can be presented by $$\delta\left( r \right)={{W}^{T}}h\left( r \right),$$ where $W$ is the weight matrix, $h\left( r \right)={{({{h}_{1}}\left( r \right),{{h}_{2}}\left( r \right),...,{{h}_{l}}\left( r \right) })^{T}}$, where ${{h}_{i}}\left( r \right)$ is an activation function. The widely used activation function, which is also employed in this work, is Gaussian, $$\label{eq33}
{{h}_{i}}(r)=\frac{\exp \left( \frac{{{\left\| {{x}_{1}}-{{\rho }_{1i}} \right\|}^{2}}+{{\left\| {{x}_{2}}-{{\rho}_{2i}} \right\|}^{2}}}{{{b}_{i}}^{2}} \right)}{\sum\limits_{j=1}^{n}{\exp \left( -\frac{{{\left\| {{x}_{1}}-{{\rho}_{1j}} \right\|}^{2}}+{{\left\| {{x}_{2}}-{{\rho}_{2j}} \right\|}^{2}}}{{{b}_{j}}^{2}} \right)}}, \;\; i=1,2,...,n,$$ where $n$ is the number of neurons in the hidden layer, $\rho$ is the matrix of means and $b$ is the vector of variances.
If $\hat{W}$ denotes estimation of the weight matrix $W$, which is updated by the adaptation mechanism as follows, $$\label{eq34}
\dot{\hat{W}}=\Gamma \left( {h}{{{{s}}}^{T}}-\varsigma \left\| {{s}} \right\|\hat{W} \right)$$ where $\varsigma$ is positive and $\Gamma$ is the positive definite diagonal matrix of the adaptation constants, then the output of the RBFN $\delta(r)$ is approximated by $${\hat{\delta }}\left( r \right)={{\hat{W}}^{T}}{h}.$$ In this work, we employ the RBFN to adaptively estimate the uncertain dynamic $K$; therefore, the control input in (\[eq28\]) can be approximated by $$\label{eq35}
{u}=-M\left( {{c}_{2}}sign\left( {{s}} \right)+{{c}_{3}}{s} \right)-M\left( {{\lambda }}{{{\dot{{z}}}}_{1}}+{{M}^{-1}}{{\hat{W}}^{T}}{h}-{{{{\dot{\alpha }}}}_{2f}} \right) .$$
![Schematic diagram of RBF neural network.[]{data-label="fig4"}](noron.png){width="0.8\linewidth"}
It is noticed that t
| 3,840
| 2,337
| 3,483
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|
iii). We see that $b$ is a 1-sided curve and $c$ separates a genus two subsurface containing $a, b$. Complete $\{a, b, c\}$ to a top dimensional pair of pants decomposition $P$ on $N$. We assumed that $\lambda([a])=[a]$. Since $a, b, c$ are pairwise disjoint, there exist $b', c'$ some representatives of $\lambda(b)$ and $(\lambda(c)$ respectively, such that $a, b', c'$ are pairwise disjoint. Let $P'$ be the set of pairwise disjoint representatives of $\lambda([P])$ containing $a, b', c'$. Since $a$ is adjacent to $b$ and $c$ w.r.t. $P$, $a$ is adjacent to $b'$ and $c'$ w.r.t. $P'$ by Lemmma \[adjacent\]. Since $a$ is 1-sided there is a genus one subsurface with two boundary components, say $T$, such that $a$ is in $T$ and the boundary components of $T$ are $b', c'$. Since $b$ is adjacent to only $a, c$ w.r.t. $P$, and nonadjacency is preserved by $\lambda$ by Lemma \[nonadjacent\], we see that $b'$ must be only adjacent to and $a, c'$. We also know that $b'$ must be a 1-sided curve whose complement is nonorientable, as $b$ is such a curve (see Lemma \[1-sided-cn\]). This shows that $a, b', c'$ are as shown in the Figure \[bdcor\] (iv). Since $\lambda_a$ is induced by $(G_a)_{\#}$, they agree on $[b]$ and $[c]$. This shows that $G_a (\partial_a) = \partial_a$. We note that when $g \geq 3$, we can use the curves shown in Figure \[bdcor\] (iv) to get similar results.
Now we continue as follows: As in case (i), by composing $G_a$ with a homeomorphism isotopic to identity, we can assume that $G_a$ maps antipodal points on the boundary $\partial_a$ to antipodal points. So, $G_a$ induces a homeomorphism $g_a: N \rightarrow N$ such that $g_a (a)= a$ and $(g_a)_{\#}$ agrees with $\lambda$ on every vertex of $[a] \cup L_a$. So, $(g_a)_{\#}^{-1} \circ \lambda$ fixes every vertex in $[a] \cup L_a$.
=2.5in =2.5in
As in the proof of case i, we get $(g_{a})_{\#}$ agrees with $\lambda$ on $\{[a]\} \cup L_{a} \cup D_{a}$. Also as in that proof, we can see that if $v$ is any 1-sided simple closed curve $v$ such that $[v] \in P(
| 3,841
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| 2,469
| 3,520
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1\]. Assume the hypothesis in Theorem \[t1\], and form the polynomial $$P({\mathbf{x}})= h(x_1{\mathbf{u}}_1+\cdots+x_m{\mathbf{u}}_m)/h({\mathbf{e}}).$$ It follows that $P({\mathbf{x}})$ is hyperbolic with hyperbolicity cone containing the positive orthant. Since $\rk({\mathbf{u}}_i) \leq 1$ for all $1\leq i \leq m$ we may expand $P({\mathbf{x}})$ as $$P({\mathbf{x}})= \sum_{S \subseteq [m]} \mu(\{S\}) \prod_{j \in S}x_j,$$ where $\mu(\{S\}) \geq 0$ for all $S \subseteq [m]$. Since $\tr_h({\mathbf{u}}_i)= \tr_P({\mathbf{e}}_i)$ for all $1\leq i \leq m$, the conclusion in Theorem \[t1\] now follows from Theorem \[t11\].
The *support* of $\mu$ is $\{S: \mu(\{S\})>0\}$. Choe *et al.* [@COSW] proved that the support of a constant sum strong Rayleigh measure is the set of bases of matroid. Such matroids are called *weak half-plane property matroids*. The rank function, $r$, of such a matroid is given by $$r(S) = \rk\left(\sum_{i \in S}{\mathbf{e}}_i\right),$$ where $\rk$ is the rank function associated to the hyperbolic polynomial $P_\mu$ as defined in the introduction, see [@BrObs; @Gu]. Edmonds Base Packing Theorem [@Edm] characterizes, in terms of the rank function, when a matroid contains $k$ disjoint bases. Namely if and only if $$r(S) \geq d -\frac {n-|S|} k, \quad \mbox{ for all } S \subseteq [n],$$ where $r$ is the rank function of a rank $d$ matroid on $n$ elements. Using Theorem \[t11\] we may deduce a sufficient condition (of a totally different form) for a matroid with the weak half-plane property to have $k$ disjoint bases:
\[packing\] Let $k\geq 2$ be an integer. Suppose $\mu$ is a constant sum strong Rayleigh measure such that $${\mathbb{P}}[S : i \in S] \leq \left(\frac 1 {\sqrt{k-1}} - \frac 1 {\sqrt{k}}\right)^2$$ for all $1\leq i \leq n$. Then the support of $\mu$ contains $k$ disjoint bases.
Suppose $\tr({\mathbf{e}}_i) \leq \epsilon$ for all $i$. Let $S_1 \cup \cdots \cup S_k=[n]$ be a partition afforded by Theorem \[t11\], and let ${\mathbf{v}}_j= \sum_{i \in S_j}{\mathbf{e}}_i$ for each
| 3,842
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| 3,075
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|
ic attractor to be that on which the dynamics is chaotic in the sense of Defs. 4.1. and 4.2. Hence
**Definition 4.3.** ***Chaotic Attractor.*** *Let $A$ be a positively invariant subset of $X$. The attractor* $\textrm{Atr}(A)$ *is chaotic on $A$ if there is sensitive dependence on initial conditions for* all *$x\in A$. The sensitive dependence manifests itself as multifunctional graphical limits for all $x\in\mathcal{D}_{+}$ and as chaotic orbits when* $x\not\in\mathcal{D}_{+}$*.*$\qquad\square$
[$$f_{\textrm{f}}(x)=\left\{ \begin{array}{ccl}
2(1+x)/3, & & 0\leq x<1/2\\
2(1-x), & & 1/2\leq x\leq1\end{array}\right.$$ ]{}
The picture of chaotic attractors that emerge from the foregoing discussions and our characterization of chaos of Def. 4.1 is that it it is a subset of $X$ that is simultaneously “spiked” multifunctional on the $y$-axis and consists of a dense collection of singleton domains of attraction on the $x$-axis. This is illustrated in Figure \[Fig: attractor\] which shows some typical chaotic attractors. The first four diagrams (a)$-$(d) are for the logistic map with (b)$-$(d) showing the 4-, 2- and 1-piece attractors for $\lambda=3.575,\textrm{ }3.66,\textrm{ and }3.8$ respectively that are in qualitative agreement with the standard bifurcation diagram reproduced in (e). Figs. (b)$-$(d) have the advantage of clearly demonstrating how the attractors are formed by considering the graphically converged limit as the object of study unlike in Fig. (e) which shows the values of the 501-1001th iterates of $x_{0}=1/2$ as a function of $\lambda$. The difference in Figs. (a) and (b) for a change of $\lambda$ from [$\lambda>\lambda_{*}=3.5699456$]{} to 3.575 is significant as $\lambda=\lambda_{*}$ marks the boundary between the nonchaotic region for $\lambda<\lambda_{*}$ and the chaotic for $\lambda>\lambda_{*}$ (this is to be understood as being suitably modified by the appearance of the nonchaotic windows for some specific intervals in $\lambda>\lambda_{*}$). At $\lambda_{*}$ the generated fractal Canto
| 3,843
| 2,781
| 4,010
| 3,454
| 2,307
| 0.780984
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|
this section we give the proof of Theorem \[J\].
**Step 1.** Let$$\omega _{t}(dt_{1},...,dt_{m})=\frac{m!}{t^{m}}1_{\{0<t_{1}<...<t_{m}<t%
\}}dt_{1}....dt_{m}$$and (with $t_{m+1}=t$) $$I_{m}(f)(x)={\mathbb{E}}\Big(1_{\{N(t)=m\}}\int_{{\mathbb{R}}^m_+}\Big(%
\prod_{i=0}^{m-1}P_{t_{m-i+1}-t_{m-i}}U_{Z_{m-i}}\Big)P_{t_1}f(x) \omega
_{t}(dt_{1},...,dt_{m}))\Big),$$ Since, conditionally to $N(t)=m,$ the law of $(T_{1},...,T_{m})$ is given by $\omega _{t}(dt_{1},...,dt_{m}),$ it follows that$$\overline{P}_{t}f(x)=\sum_{m=0}^{\infty
}I_{m}(f)(x)=\sum_{m=0}^{m_{0}}I_{m}(f)(x)+R_{m_{0}}f(x)$$with$$R_{m_{0}}f(x)=\sum_{m=m_{0}+1}^{\infty }I_{m}(f)(x).$$
**Step 2.** We analyze first the regularity of $I_{m}(f).$ We apply Lemma \[Reg\]. Here $S_{t}=P_{t}$, so assumptions \[H2H\*2\] and \[HH3\] hold due to assumptions \[H2H\*2-P\] and \[H3\] respectively. Moreover, here $U_{i}=U_{m-Z_{i}}$, so Assumption \[H1H\*1\] is satisfied uniformly in $\omega $ as observed in Remark \[rem-J\]. Notice that $a=b=0$ in our case. Then Lemma \[Reg\] gives$$\Big(\prod_{i=0}^{m-1}P_{t_{m-i+1}-t_{m-i}}U_{Z_{m-i}}\Big)%
P_{t_{1}}f(x)=\int p_{t_{1},t_{2}-t_{1},...,t-t_{m}}(x,y)f(y)dy$$and, for $q_{1},q_{2}\in {\mathbb{N}},\kappa \geq 0,p>1$ and $\left\vert
\beta \right\vert \leq q_{2}$ we have $$\left\Vert \partial _{x}^{\beta }p_{t_{1},t_{2}-t_{1},...,t-t_{m}}(x,\cdot
)\right\Vert _{q_{1},\kappa ,p}\leq \theta _{q,t}(m)\times \psi _{\chi }(x).
\label{j1}$$with$$\theta _{q,t}(m)=\frac{C\times m^{q+d+2\theta _{1}}}{(\lambda t)^{\theta
_{0}(q+d+2\theta _{1})}}\times C_{q,\chi ,p,\infty }^{m-1}(P,U).$$Here $C$ and $\chi $ are constants which depend on $q_{1},q_{2}$ and $\kappa
.$ We notice that $$\theta _{q,t}(m+1)\leq C\times C_{q,\chi ,p,\infty }(P,U)\times \theta
_{q,t}(m).$$We summarize: for each fixed $q,\chi ,\kappa ,p$ and each $\delta >0$ there exists some constants $\Theta \geq 1$ and $Q\geq 1$ (depending on $q,\chi
,\kappa ,p$ and $\delta $ but not on $m$ and on $t)$ such that for every $%
m\in \N$ $$\begin{aligned}
\theta _{q,t}(m+1)
&\
| 3,844
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|
athbf{u}}) & := \int_{\Omega} {\mathbf{u}}\cdot\nabla{\mathbf{u}}\cdot{\Delta}{\mathbf{u}}\, d{\mathbf{x}}, \label{eq:Rcub}\end{aligned}$$
so that $\R({\mathbf{u}}) = \R_{\nu}({\mathbf{u}}) + \R_{\textrm{cub}}({\mathbf{u}})$. The values of $\R_{\textrm{cub}}({\widetilde{\mathbf{u}}_{\E_0}})$ are also plotted in figure \[fig:RvsE0\_FixE\_large\](a) and it is observed that this quantity exhibits a power-law behaviour of the form $$\label{eq:RcubvsE0_powerLaw}
\R_{\textrm{cub}}({\widetilde{\mathbf{u}}_{\E_0}}) = C''_1\E_0^{\,\alpha_2}, \qquad C''_1 = 1.38\times 10^{-2},\ \alpha_2 = 2.99 \pm 0.05.$$ While the value of $C''_1$ is slightly larger than the value of $C'_1$ in , it is still some six orders of magnitude smaller than the constant factor $C_1 = 27/(8\pi^4\nu^3)$ from estimate . These differences notwithstanding, we may conclude that estimate is sharp in the sense of definition \[def:NotionSharpness\]. The power laws from equations and are consistent with the results first presented by @l06 [@ld08], where the authors reported a power-law with exponent $\alpha_{LD} = 2.99$ and a constant of proportionality $C_{LD} = 8.97\times 10^{-4}$. The energy of the optimal fields $\K({\widetilde{\mathbf{u}}_{\E_0}})$ for large values of $\E_0$ is shown in figure \[fig:RvsE0\_FixE\_large\](b) in which we observe that the energy stops to increase at about $\E_0 \approx 20$. This transition justifies using $\E_0 = 20$ as the lower bound on the range of $\E_0$ where the power laws are determined via least-square fits.
Figures \[fig:RvsE0\_FixE\_large\](c)-(e) show the isosurfaces $\Gamma_{0}(Q - 0.5||Q||_{L_\infty})$ representing the optimal fields ${\widetilde{\mathbf{u}}_{\E_0}}$ for selected large values of $\E_0$. The formation of these localized vortex structures featuring two rings as $\E_0$ increases is evident in these figures. The formation process of localized vortex structures is also visible in figures \[fig:RvsE0\_FixE\_large\](f)–(h), where the component of vorticity normal to the plane $P_{xz} = \{{\math
| 3,845
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| 0.783215
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ond to noiseless (top), constant noise 20 (middle), and constant noise 40 (bottom), respectively.[]{data-label="fig:boidsefficiency"}](witkowski-fig7-efficiency.jpeg){width=".7\columnwidth"}
We find however that from a certain noise level, the cost to signal is fully compensated by the benefits of signaling, as it helps the foraging of agents. The average fitness becomes even higher as we increase the noise level, which suggest that the signaling behavior increases in efficiency for high levels of noise, allowing the agents to overcome imperfect information by forming swarms.
We also observe scale effects in the influence of the signal propagation on the average fitness of the population. For a smaller population, only middle values of signal propagation seem to bring about fitter behaviors, whereas this is not the case for larger sizes of population. On the contrary, larger populations are most efficient for lower levels of signal propagation. This may suggest a phase transition in the agents’ behavior for large populations, eventually in the way the swarming itself helps foraging.
Understanding criticality seems strongly related to a broad, fundamental theory for the physics of life as it could be, which still lacks a clear description of how it can arise and maintain itself in complex systems. The effects of criticality have recently been investigated futher by one of the authors, using a similar setup [@khajehabdollahi2018critical]. The results showed exploratory dynamics at criticality in the evolution of foraging swarms, and the tension between local and global scale adaptation.
Through this work, by increasing the number of simulated boids that maintain their own states, we may introduce more than the mere number. By allowing for many information exchanges between computing agents, the simulation can effectively take leaps of creativity. In Stanley and Lehman’s 2015 book [@stanley2015greatness], objective functions are presented as a distraction, as novelty and diversity might not be achieved by ha
| 3,846
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|
ds=\int\limits_{0}^t{\mathbf{1}}_{[0,\tau)}(s)f(s)\,ds\, \\
\text{and }\,\left(A^*f\right)(\tau)=\int\limits_\tau^t f(s)\,ds=\int\limits_{0}^t{\mathbf{1}}_{[\tau,t)}(s)f(s)\,ds.\end{gathered}$$
If ${\mathbf{1}}_{[0,\tau)}$ and ${\mathbf{1}}_{[\tau,t)}$ are Hilbert-Schmidt-kernels, the above integral operators $A$ and $A^*$ are compact operators on $L^2([0,t),dx)_{{\mathbb{C}}}$ and so are $M$ and $M^*$. Indeed
$$\begin{aligned}
&\int\limits_{0}^t\int\limits_{0}^t ({\mathbf{1}}_{[0,s)}(\tau))^2 \, d\tau ds=\int\limits_{0}^t\int\limits_{0}^t ({\mathbf{1}}_{[s,t)}(\tau))^2 \, d\tau ds=\frac12 t^2<\infty.\end{aligned}$$
$M$ as well as $M^*$can be written as the limit of a sequence of finite rank operators $(M_n)_{n\in {\mathbb{N}}}$ and $(M_n^*)_{n\in {\mathbb{N}}}$, respectively, in operator norm. Then: $$\begin{gathered}
\sup_{\|(f_1 , f_2)\|\leq 1} \left\| \Bigg(\left(\begin{array}{l l} 0 & M \\ M^* & 0 \end{array}\right)- \!\left(\begin{array}{l l} 0 & M_n \\ M_n^* & 0 \end{array}\right)\Bigg)\left(\begin{array}{l} f_1 \\ f_2 \end{array} \right)\right\|\\
\leq\!\sup\limits_{\|f_1\| \leq1}\|(M-M_n)f_1\|+\!\sup\limits_{\|f_2\| \leq1}\!\|(M^*-M^*_n)f_2\|,\end{gathered}$$ where the right hand side tends to zero as $n$ goes to $\infty$. Hence, $\left(\begin{array}{l l} 0 & M \\ M^* & 0 \end{array}\right)$ as the limit of finite rank operators is compact.\
It is left to show that $\ker\left({\mathbf{N}}_2\right)=\{0\}$. Let
$$\begin{aligned}
\left(\begin{matrix}
Id&M\\
M^*&Id
\end{matrix}\right)
\left(\begin{matrix}
f_1\vphantom{\left(A-A^*\right)}\\f_2\vphantom{\left(A-A^*\right)}
\end{matrix}
\right)=
\left(\begin{matrix}
0\vphantom{\left(A-A^*\right)}\\0\vphantom{\left(A-A^*\right)}
\end{matrix}
\right).\end{aligned}$$
This leads to the system $$\begin{aligned}
f_1(s)&=k\int\limits_{0}^sf_2(\tau)d\tau-k\int\limits_s^tf_2(\tau)d\tau,\quad s \in [0,t),\\
f_2(s)&=k\int\limits_s^tf_1(\tau)d\tau-k\int\limits_{0}^sf_1(\tau)d\tau,\quad s \in [0,t).\end{aligned}$$
An analogue calculation as in the proof of Propo
| 3,847
| 2,618
| 1,158
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| null | null |
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|
nal portfolios introduced in §4.1. Not surprisingly, the spectrum of ${\tilde{\sf \sigma}}_{ij}^{crv}$ is found to be highly degenerate. The eigenvector ${{\bf e}^{crv}}^{N}$ corresponding to the major eigenvalue is (exactly) equal to ${\hat{\beta}}_{i}$ \[cf. Eq. (\[438\])\], while the remaining $N-1$ minor eigenvectors are not uniquely determined[^6] and may be arbitrarily chosen to be any orthonormal set of $N-1$ vectors orthogonal to the major eigenvector ${\hat{\beta}}_{i}$. The expected return and volatility features of the $N-1$ market-orthogonal portfolios defined by this arbitrary choice, on the other hand, do depend on that choice, as the following analysis will show.
Since our main objective is the determination of the efficient frontier, we shall choose the remaining $N-1$ eigenvectors with respect to their volatility level, which, it may be recalled from §2, is given by ${{V}_{\mu}}^{2}={v}_{\mu}^{2}/{{W}_{\mu}}^{2}$. For the present case, minimizing ${V}_{\mu}$ amounts to maximizing ${W}_{\mu}$. Therefore, we will look for a unit vector $\bf e$ that is orthogonal to $\bm{{\hat{\beta}}}$ as stipulated above [*and*]{} maximizes ${\sum}_{i=1}^{N}{e}_{i}$. In terms of rescaled quantities, this problem appears as $${\max \;}_{\bf e} {\;} {\bf e} \cdot {\hat{{\bf u}}} {\:} {\;}
s.t. \; {{\bf e} \cdot {\bf e}}=1, {\:} {\:} {\bf e} \cdot
{\hat{\bm{\beta}}}=0, \label{445}$$ where ${\hat{{\bf u}}}_{i}\stackrel{\rm def}{=}{N}^{-{1 \over
2}}(1,1, \ldots,1)$ is an $N$-dimensional unit vector all of whose components are equal. The solution to Eq. (\[445\]) may be found by standard methods provided that ${\hat{{\bf u}}}$ and ${\hat{\bm{\beta}}}$ are not parallel, a condition whose violation is very improbable and will henceforth be assumed to hold. On the other hand, it is clear from geometric considerations that the solution must be that linear combination of ${\hat{{\bf u}}}$ and ${\hat{\bm{\beta}}}$ which is orthogonal to ${\hat{\bm{\beta}}}$. Designating the solution vector as ${{\bf e}^{crv}}^{1}$, we fi
| 3,848
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|
ng the angular momentum of the final state up to $l=0$ and $l=1$, respectively. ](fig3)
The angular distribution of the emitted neutrons can be also calculated using the decay amplitude, Eq. (\[amplitude1\]). The amplitude for emitting the two neutrons with spin components of $s_1$ and $s_2$ and momenta ${\mbox{\boldmath $k$}}_1$ and ${\mbox{\boldmath $k$}}_2$ reads [@EB92], $$\begin{aligned}
f_{s_1s_2}({\mbox{\boldmath $k$}}_1,{\mbox{\boldmath $k$}}_2)&=&
\sum_{j,l}e^{-il\pi}e^{i(\delta_1+\delta_2)}\,
M_{j,l,k_1,k_2} \nonumber \\
&&\times \langle [{\cal Y}_{jl}(\hat{{\mbox{\boldmath $k$}}}_1)
{\cal Y}_{jl}(\hat{{\mbox{\boldmath $k$}}}_2)]^{(00)}|\chi_{s_1}\chi_{s_2}\rangle,
\label{angularamplitude}\end{aligned}$$ where ${\cal Y}_{jlm}$ is the spin-spherical harmonics, $\chi_s$ is the spin wave function, and $\delta$ is the nuclear phase shift. The angular distribution is then obtained as $$\frac{dP}{d\theta_{12}}=4\pi\sum_{s_1,s_2}
\int dk_1dk_2\, |f_{s_1s_2}(k_1,\hat{{\mbox{\boldmath $k$}}}_1=0,k_2,
\hat{{\mbox{\boldmath $k$}}}_{2}=\theta_{12})|^2,
\label{angular}$$ where we have set $z$-axis to be parallel to ${\mbox{\boldmath $k$}}_1$ and evaluated the angular distribution as a function of the opening angle, $\theta_{12}$, of the two emitted neutrons.
The angular distribution obtained without including the final state $nn$ interaction is shown by the dotted line in Fig. 3. The main component in the initial wave function, $\Psi_i$, is the $d_{3/2}$ configuration, and the angular distribution is almost symmetric around $\theta_{12}=\pi/2$. In the presence of the final state $nn$ interaction, the angular distribution becomes highly asymmetric, in which the emission of two neutrons in the opposite direction (that is, $\theta_{12}=\pi$) is enhanced[@GMZ13], as is shown by the solid line. Notice that we have obtained the correlated distribution by evaluating Eq. (\[angular\]) only at $e_1+e_2=0.14$ and then normalize it, since it is hard to carry out the integrations in Eq. (\[angular\]) when the resonance width
| 3,849
| 2,883
| 3,939
| 3,564
| 2,860
| 0.776478
|
github_plus_top10pct_by_avg
|
that $c\in {\mathbb{C}}$ satisfies Hypothesis \[morrat-hyp\] and that $c\notin {\mathbb{Q}}_{\leq -1}$. Then the shift functor $\widetilde{S}_c$ restricts to an equivalence between $\mathcal{O}_{c}$ and $\mathcal{O}_{c+1}$ such that $\widetilde{S}_c(\Delta_{c}(\lambda)) \cong \Delta_{c+1}(\lambda)$ for all partitions $\lambda$ of $n$. Thus $S_c(e\Delta_{c}(\lambda))
=e\Delta_{c+1}(\lambda)$.
By Corollary \[morrat-cor\](2), an analogue of the proposition also holds when $c\in \mathbb{Q}_{\leq -1}$, provided that one shifts in a negative direction.
The final assertion of the proposition is immediate from the previous one combined with Corollary \[morrat-cor\](1).
We begin by showing that $\widetilde{S}_c$ restricts to an equivalence between $\mathcal{O}_{c}$ and $\mathcal{O}_{c+1}$. Fix $M\in \mathcal{O}_{c}$. Let $\mathcal{I}_t={\mathbb{C}}[{\mathfrak{h}}^*]^{{{W}}}_{\geq t}$ denote the ${{W}}$-invariant elements of ${\mathbb{C}}[{\mathfrak{h}}^*]$ of degree at least $t$ and set $I_t=\mathcal{I}_t{\mathbb{C}}[{\mathfrak{h}}^\ast ]$, Then ${\mathbb{C}}[{\mathfrak{h}}^\ast]/I_t$ is a finite dimensional algebra and so all homogeneous elements of ${\mathbb{C}}[{\mathfrak{h}}^*]$ of large degree belong to $I_t$. Thus it is enough to show that, if $\widetilde{m}=he_- \delta\otimes em \in
\widetilde{S}_c(M)= H_{c+1}e_-\delta \otimes_{{U}_c} eM$, for some $h\in H_{c+1}$ and $m\in M$, then $\widetilde{m}$ is annihilated by $\mathcal{I}_t$ for $t\gg 0$.
Recall the ${\mathbf{E}}$-grading on $H_c$ from . Since ${\mathbb{C}}[{\mathfrak{h}}^*]$ acts locally nilpotently on $M$, the PBW isomorphism shows that any homogeneous element of $H_c$ of sufficiently large negative ${\mathbf{E}}$-degree annihilates $m\in M$. Thus, assume that $qm=0$ for all $q\in H_c$ with $\operatorname{{\mathbf{E}}\text{-deg}}(q)\leq -t$ and let $p\in {\mathbb{C}}[{\mathfrak{h}}^*]^{{{W}}}_{\geq t}$. Then $$phe_-\delta \otimes em \ =\ [p,h]e_-\delta \otimes em +
h\delta \delta^{-1} pe_- \delta \otimes em \ =\ [p,h]e_-\delta \otimes em +
he_- \
| 3,850
| 2,890
| 2,186
| 3,507
| null | null |
github_plus_top10pct_by_avg
|
a certain subclass of finitely generated groups. We state the result and postpone the definition of the new notions to the corresponding section.
\[thm:intro2\] Let $G$ be a shortlex combable group with its word metric $d$. Then ${\mathcal{F}}(G,d)$ has the Schauder basis [*(see Theorem \[thm:shortlex\])*]{}.
We mention that the theorem applies in particular to hyperbolic groups and Artin groups of large type. One of the applications (see Corollary \[cor:hyperbolicnet\]) is that the Lipschitz-free space over any net in the real hyperbolic $n$-space ${\mathbb{H}}^n$ has the Schauder basis.\
The paper is organized as follows. In Section \[section:preliminaries\] we present a characterization of the $\lambda$-bounded approximation property tailored for Lipschitz-free spaces (Proposition \[tool\]). In Section \[section:cpt\] we prove Theorem \[thm:intro1\]; first we tackle Lie groups using harmonic analysis tools in Subsection, then in Subsection \[subsection:generalCpt\] we prove the general case by approximating compact groups by compact Lie ones. In this last subsection we also generalize of Theorem \[thm:intro1\] to compact homogeneous spaces equipped with quotient metrics (Theorem \[thm:homogeneousspace\]). Section \[section:fingengrps\] is dedicated to finitely generated groups; we prove Theorem \[thm:intro2\] and provide some examples and applications. We conclude with some remarks and questions in Section \[section:problems\], and presenting in Appendix \[appendSphere\] a generalization of Theorem \[thm:homogeneousspace\] for the specific case of the Euclidean sphere.
Preliminaries {#section:preliminaries}
=============
Lipschitz-free spaces
---------------------
Let $M$ be a metric space and $0$ be some distinguished point in $M$. Let $\mathrm{Lip}_0(M)$ denote the Banach space of real-valued Lipschitz functions defined on $M$ which vanish at $0$, equipped with the norm $\|\cdot\|_{\mathrm{Lip}}$ which assigns to each function its minimal Lipschitz constant. The Lipschitz-free space over $M$, denoted
| 3,851
| 2,841
| 1,454
| 3,586
| null | null |
github_plus_top10pct_by_avg
|
athbf{P}_{B},\mathbf{R}_{B}\right)$, and $\left(\mathbf{Q}_{C},\mathbf{R}_{C}\right)$, and in the CbD approach the joint distribution imposed on them allows, say, $\mathbf{P}_{A}$ and $\mathbf{P}_{B}$ to be unequal with some probability.
Here we compare the NP and CdB approaches applied to the simplest contextual case possible, given by three pairwise correlated random variables, and to the standard EPR-Bell experiment. We show that for such examples the two measures of contextuality are the same.
Negative Probabilities (NP)
============================
Using our above example, with $\mathbf{P},\mathbf{Q},\mathbf{R}$ observed in pairs, in the NP approach one ascribes to the vector $\left(\mathbf{P},\mathbf{Q},\mathbf{R}\right)$ a joint quasi-distribution by means of assigning to each possible combination $w=\left(p,q,r\right)$ a real number $\mu\left(w\right)$ (possibly negative), such that $$\begin{array}{c}
\sum_{r}\mu\left(w\right)=\Pr\left[\mathbf{P}=p,\mathbf{Q}=q\right],\\
\sum_{q}\mu\left(w\right)=\Pr\left[\mathbf{P}=p,\mathbf{R}=r\right],\\
\sum_{p}\mu\left(w\right)=\Pr\left[\mathbf{Q}=q,\mathbf{R}=r\right].
\end{array}\label{eq:NP exmaple}$$ Such $\mu$ exists if and only if the no-signaling condition (built into EPR paradigms with spacelike separation) is satisfied [@al-safi_simulating_2013; @oas_exploring_2014; @abramsky_operational_2014], i.e., the distribution of, say, $\mathbf{P}$ is the same in $\left(\mathbf{P},\mathbf{Q}\right)$ and in $\left(\mathbf{P},\mathbf{R}\right)$.[^2] The numbers $\mu\left(w\right)$ can then be interpreted as quasi-probabilities of events $\left\{ w\right\} $, with the quasi-probability of any other event (subset of $w$ values) being computed by additivity, inducing thereby a signed measure [@halmos_measure_1974] on the set of all events. The quasi-probability of the entire set of $w$ will then be necessarily equal to unity, because, e.g., $$1=\sum_{p,q}\Pr\left[\mathbf{P}=p,\mathbf{Q}=q\right]=\sum_{w}\mu\left(w\right).$$ The function $\mu$ is generally not unique. I
| 3,852
| 4,399
| 4,271
| 3,629
| 2,879
| 0.776374
|
github_plus_top10pct_by_avg
|
iven by M=-. \[magnetization\] The pressure becomes anisotropic [@Karmakar:2019tdp; @PerezMartinez:2007kw] due to the magnetization acquired by the system in presence of strong magnetic field which results in two different pressure along parallel and perpendicular to the magnetic field direction. The longitudinal pressure is given as P\_z=-F =-(F\_q\^r+F\_g\^r). and transverse pressure is given as P\_=-F-eB M. One gets two different second-order QNS, namely, along the longitudinal ($\chi_z$) and transverse ($\chi_{\perp}$) direction in the presence of the strong magnetic field. The longitudinal second-order QNS can be obtained as \_z= \_[=0]{}, whereas the transverse one can be obtained as \_= \_[=0]{}. The pressure of non-interacting quark-gluon gas in the presence of strong magnetic field is given as P\_[sf]{}= \_f N\_c N\_f q\_fB (1+12\^2)+(N\_c\^2-1). The second-order diagonal QNS for the ideal quark gluon plasma is given as \_[sf]{}=\_f N\_c N\_f .
![Variation of the longitudinal part of the second-order QNS scaled with that of free field value in presence of strong magnetic field with temperature (left panel) and magnetic field (right panel) strength for $N_f=3$.[]{data-label="QNS_sfa_long_T"}](chi2_sfa_long.pdf "fig:") ![Variation of the longitudinal part of the second-order QNS scaled with that of free field value in presence of strong magnetic field with temperature (left panel) and magnetic field (right panel) strength for $N_f=3$.[]{data-label="QNS_sfa_long_T"}](chi2_sfa_long_eB.pdf "fig:")
In the left panel of Fig. \[QNS\_sfa\_long\_T\] the variation of the longitudinal second-order QNS with temperature is displayed for two values of magnetic field strength. For a given magnetic field strength the longitudinal second-order QNS is found to increase with temperature and approaches the free field value at high temperature. On the other hand for a given temperature the longitudinal second-order QNS decreases with increase of the magnetic field strength as shown in the right panel of Fig. \[QNS\_sfa\_l
| 3,853
| 1,329
| 2,467
| 3,592
| null | null |
github_plus_top10pct_by_avg
|
± 0.0 19 ± 1.0 19.6 ± 2.0 19.6 ± 0.5 20.6 ± 2.0 21 ± 1.0 22.6 ± 3.0 22.3 ± 1.5
Rifampicin 25.6 ± 1.5 28.6 ± 0.5 27.6 ± 1.1 31.6 ± 2.5 28.3 ± 2.8 33.3 ± 1.1 30.3 ± 2.0 34.3 ± 2.0
25 50 75 100
Ampicillin 22.6 ± 1.1 28.3 ± 1.5 25 ± 1.0 31.3 ± 1.5 29.3 ± 0.5 33.6 ± 1.5 30.6 ± 1.1 35.3 ± 0.5
Zone of inhibition in mm and values represent mean ± SD of triplicates.
######
Down regulated genes of *S. pyogenes* under low shear modeled microgravity compared with the normal gravity.
Gene Fold change Function
------------ ------------- ------------------------------------------------------------
M6_Spy0007 1.52 Heat shock protein 15
M6_Spy0079 1.57 Phosphoribosylaminoimidazole carboxylase catalytic subunit
M6_Spy0101 1.52 50S ribosomal protein L29
M6_Spy0266 1.60 Hypothetical protein
M6_Spy0375 1.72 Hypothetical protein
M6_Spy0669 1.50 Hypothetical protein
M6_Spy0873 1.83 Hypothetical protein
M6_Spy0974 1.85 Signal peptidase I
M6_Spy1122 1.56 ComE operon protein 3
M6_Spy1556 1.62 Major tail protein
M6_Spy1696 1.58 Hypothetical protein
M6_Spy1699 1.56 Hypothetical cytosolic protein
M6_Spy1775 1.96 Histidine ammonia-lyase
M6_Spy1779 1.78 30S ribosomal protein S2
M6_Spy1780 1.65 Elongation factor Ts
M6_Spy1826 1.50 ArgR
M6_Spy1829 1.50 Hypothetical membrane spanning protein
M6_Spy1830 1.50 Hypothetical membrane spanning protein
M6_Spy1834 1.86 50S ribosomal protein L32
M6_Spy1835 1.67 50S ribosomal protein L
| 3,854
| 6,080
| 2,704
| 2,420
| null | null |
github_plus_top10pct_by_avg
|
sum_{i=l}^n x_i <
\sum_{i=l}^n
y_i\}.$$ Denote by $t_1$ the number of components in $x$ which are equal to $x_n$ while $t_2$ the number of components in $y$ which are equal to $y_n$. If $t_1=t_2=t$ , $x_nx_d\leq x_{n-t}^2$, and $y_ny_d\leq y_{n-t}^2$, then we can also deduce that $x\not\in
M(y)$.
Using the lemmas above, we can now prove that $T(y)\not =M(y)$ for some probability vector $y$ by deriving a sufficient condition under which $T(y)\not\subseteq M(y)$, as the following theorem states.
\[thm:mrdeterm\] Suppose $y$ is a nonincreasingly arranged $n$-dimensional probability vector. Denote by $t$ and $m$ the numbers of components which are equal to $y_1$ and which are equal to $y_n$, respectively. Let $d$ be the minimal index of the components which are less than $y_{t+1}$. That is, $$y_1=\cdots=y_{t}>y_{t+1}=\cdots=y_{d-1}>y_{d},$$ and $$y_{n-m}>y_{n-m+1}=\cdots=y_n.$$ If $d<n-m$ and $y_1y_{d}\geq y_{t+1}^2$, then $T(y)\not\subseteq
M(y)$.
[*Proof.*]{} Take a positive number $\epsilon$ such that $$\epsilon< \min\{\frac{d-t-1}{d-t}(y_{d-1}-y_{d}), \
\frac{m}{m+1}(y_{n-m}-y_{n-m+1})\}.$$ Define two $(n-t)$-dimensional nonnegative vectors $\bar{x}$ and $\bar{y}$ as follows $$\begin{array}{rcl}
\displaystyle\bar{x}&=&(y_{t+1}-\displaystyle\frac{\epsilon}{\triangle},
\ \cdots,\ y_{d-1}-\frac{\epsilon}{\triangle},\ y_{d}+\epsilon,\
y_{d+1},\ \cdots, \\ \\
&& y_{n-m-1},\ y_{n-m}-\epsilon,\ \displaystyle
y_{n-m+1}+\frac{\epsilon}{m}, \ \cdots,\ y_n+\frac{\epsilon}{m})
\end{array}$$ and $$\bar{y}= (y_{t+1},\ y_{t+2},\ \cdots,\ y_n).$$ Here $\triangle=d-t-1$. It is easy to check that $\bar{x}$ and $\bar{y}$ are both nonincreasingly arranged, and $\bar{x}\prec
\bar{y}$. Furthermore, $\bar{x}$ is in the interior of $T(\bar{y})$ by Lemma 1 in [@DK01] since $\bar{x}_1<\bar{y}_1$ and $\bar{x}_{n-t}>\bar{y}_{n-t}$. So there exists a sufficiently small but positive $\delta$ such that $\bar{x}'\in T(\bar{y})$ where $$\begin{array}{l}
\displaystyle\bar{x}'^{\da}=(y_{t+1}-\displaystyle\frac{\epsilon}{\triangle},
\ \cdots,\
| 3,855
| 3,039
| 3,024
| 3,514
| 3,972
| 0.768891
|
github_plus_top10pct_by_avg
|
begin{array}{ll}
S &\xRightarrow{r_1}ABCD\xRightarrow{(r_2r_3r_4r_6r_8r_9)^n}a^nABb^nc^nCD
\xRightarrow{r_2r_3}a^{n+1}EFb^{n+1}c^{n+1}AD \\
& \xRightarrow{r_5r_7}a^{n+1}FCb^{n+1}c^{n+1}ED\xRightarrow{r_{10}r_{11}r_{12}}a^nb^nc^n
\end{array}$$ (in the last phase, the sequences $r_{10}r_{12}r_{11}$ and $r_{12}r_{10}r_{11}$ could also be applied with the same result). Therefore, $L(G)=\{a^nb^nc^n{:}n\geq 1\}$.$\diamond$
\[exa:NBLnotCF2\] Let $G=(\{S,A,B,C\},\{a,b,c\},S,R,\mathbf{1})$ be the context-free capa-city-bounded grammar where $R$ consists of the rules $r_1: S\to aBbaAb$, $r_2: A\to aBb$, $r_3: B\to C$, $r_4: C\to A$, $r_5: A\to BC$, $r_6: A\to c$, and let $M$ be the regular set $M=\{a^*ccb^*a^*cb^*\}$. The derivations in $G$ generating words from $M$ are exactly those of the form $$\begin{array}{ll}
S &\xRightarrow{r_1}aBbaAb\xRightarrow{(r_3r_2r_4r_3r_2r_4)^n}a^nBb^na^nAb^n
\xRightarrow{r_6r_3r_4}a^nAb^na^ncb^n\\
&\xRightarrow{(r_2r_3r_4)^m}a^{n+m}Ab^{n+m}a^ncb^n
\xRightarrow{r_5r_4r_3r_6r_4r_6}a^{n+m}ccb^{n+m}a^ncb^n
\end{array}$$ (one can also apply $r_3r_6r_4$ in the third phase and $r_5r_4r_6r_3r_4r_6$ in the last phase with the same result). Hence, $
L(G)\cap M=\{a^nccb^na^mcb^m{:}n\geq m\geq 1\}\not\in \mathbf{CF},
$ implying that $L(G)$ is not context-free.$\diamond$
The above examples show that capacity-bounded grammars – in contrast to derivation bounded grammars – can generate non-context-free languages. The generative power of capacity-bounded grammars will be studied in detail in the following two sections.
The notions of finite index and bounded capacities can be extended to matrix, vector and semi-matrix grammars. The corresponding language families are denoted by ${{\bf MAT}}^{[{\lambda}]}_{{\mathit{fin}}}$, ${{\bf V}}^{[{\lambda}]}_{{\mathit{fin}}}$, ${{\bf sMAT}}^{[{\lambda}]}_{{\mathit{fin}}}$, ${{\bf MAT}}^{[{\lambda}]}_{cb}$, ${{\bf V}}^{[{\lambda}]}_{cb}$, ${{\bf sMAT}}^{[{\lambda}]}_{cb}$.
Also control by Petri nets can in a natural way be extended to Petri nets with place cap
| 3,856
| 2,484
| 3,252
| 3,487
| 2,302
| 0.781047
|
github_plus_top10pct_by_avg
|
{\rm [lat]}}$ for the positive- and negative-parity nucleons via three-point functions with the so-called sequential-source method [@Sasaki:2003jh]. In practice, we evaluate $g_{V,A}^{\pm{\rm [lat]}}(t)$ defined as $$g_{V,A}^{\pm{\rm [lat]}}(t)
=
\frac
{
{\rm Tr}\ \Gamma_{V,A}
\langle B(t_{\rm snk})
J_\mu^{V,A}(t)
\overline B(t_{\rm src}) \rangle
}
{
{\rm Tr}\ \Gamma_{V,A}
\langle B(t_{\rm snk})
\overline B(t_{\rm src}) \rangle
},$$ and extract $g_{V,A}^{\pm{\rm [lat]}}$ by the fit $g_{V,A}^{\pm{\rm [lat]}}=g_{V,A}^{\pm{\rm [lat]}}(t)$ in the plateau region. $B(t)$ denotes the (optimized) interpolating field for nucleons, and $\Gamma_{V,A}$ are $\gamma_\mu \frac{1+\gamma_4}{2}$ and $\gamma_\mu \gamma_5 \frac{1+\gamma_4}{2}$, respectively. $J_\mu^{V,A}(t)$ are the vector and the axial vector currents inserted at $t$. We show in Fig. \[3pointfunc\] $g_{A}^{-0{\rm [lat]}}(t)$ for $N^*(1535)$ as a function of the current insertion time $t$. They are rather stable around $t_{\rm src}<t<t_{\rm snk}$.
![\[3pointfunc\] The non-renormalized axial charge of $N^*(1535)$, $g_{A}^{-0{\rm [lat]}}(t)$, as a function of the current insertion time $t$. ](ga1535.eps)
We finally reach the renormalized charges $g_{A,V}^\pm=\widetilde Z_{A,V}g^{\pm{\rm [lat]}}_{A,V}$ with the prefactors $\widetilde Z_{A,V}
\equiv
2\kappa
u_0
Z_{V,A}
\left(
1+b_{V,A}\frac{m}{u_0}
\right)
$, which are estimated with the values listed in Ref. [@AliKhan:2001tx].
[1.0]{} [@cccccccccccccc]{} $\kappa$ & $L_1^+$ & $R_1^+$ & $L_1^-$ & $R_1^-$ & $L_2^+$ & $R_2^+$ & $L_2^-$ & $R_2^-$ & $M_\pi$ & $E_1^+$ & $E_1^-$ & $E_2^+$ & $E_2^-$\
0.1375 & $-$0.4341 & $-$0.4573 & 0.0355 & 0.0126 & $-$1353 & $-$314.1 & $-$1.432 & $-$1.302 & 0.8985(5) & 1.696(1) & 2.137(10) & 2.524(53) & 2.141(14)\
0.1390 & $-$0.4526 & $-$0.4552 & 0.1115 & $-$0.2036 & $-$845.9 & $-$228.1 & $-$2.729 & $-$1.084 & 0.7351(5) & 1.459(1) & 1.854(13) & 2.162(44) & 1.908(17)\
0.1400 & $-$0.1605 & $-$0.3552 & 0.0990 & $-$0.0151 & $-$408.9 & $-$143.6 & $-$1.510 & $-$1.038 & 0.6024(6) & 1.270(2) &
| 3,857
| 3,706
| 293
| 3,770
| null | null |
github_plus_top10pct_by_avg
|
/P}, & \text{if} & i=d\,\\
0, & \mbox{if} & i\neq d, \end{array} \right.$$ it follows that $\Ext_R^{n-i}(M,\omega_R)\iso \Ext_R^{n-i}(U,\omega_R)$ for all $i\neq d,d+1$. Thus for such $i$ we have $\Ext_R^{n-i}(M,\omega_R)$ is Cohen-Macaulay of dimension $i$ if not the zero module.
Moreover we have the exact sequence $$\begin{aligned}
0&\to &\Ext_R^{n-d-1}(M,\omega_R)\to \Ext_R^{n-d-1}(U,\omega_R)\to \Ext_R^{n-d}(R/P,\omega_R)\\
&\to& \Ext_R^{n-d}(M,\omega_R)\to
\Ext_R^{n-d}(U,\omega_R)\to 0.\end{aligned}$$
Suppose the map $\Ext_R^{n-d-1}(U,\omega_R)\to \Ext_R^{n-d}(R/P,\omega_R)\iso\omega_{R/P}$ is not the zero map. Then its image $C\subset \omega_{R/P}$ is not zero. Since $R/P$ is domain, $\omega_{R/P}$ may be identified with an ideal in $R/P$, see [@BH Proposition 3.3.18]. Hence also $C$ may be identified with an ideal in $R/P$. Again using that $R/P$ is a domain, we conclude that $CR_P\neq 0$. It follows that $\Ext_R^{n-d-1}(U,\omega_R)_P\neq 0$, and so the set $$\Ass_{R_P}(\Ext_R^{n-d-1}(U,\omega_R)_P)=\{QR_P\: Q\in \Ass_R(\Ext_R^{n-d-1}(U,\omega_R)),\; Q\subset P\}$$ is not empty. Thus there exists $Q\in \Ass_R(\Ext_R^{n-d-1}(U,\omega_R))$ with $Q\subset P$. By Corollary \[need\] we know that $\Ass_R(\Ext_R^{n-d-1}(U,\omega_R))\subset \Ass_R^{d+1} (U)$. Therefore, since $\dim R/P=d$, the inclusion $Q\subset P$ must be proper. But this contradicts the fact that ${\mathcal F}$ is a pretty clean filtration of $M$.
It follows now that $$\Ext_R^{n-d-1}(M,\omega_R)\iso \Ext_R^{n-d-1}(U,\omega_R),$$ and that the sequence $$\begin{aligned}
\label{exact}
0\To\omega_{R/P}\to \Ext_R^{n-d}(M,\omega_R)\To \Ext_R^{n-d}(U,\omega_R)\To 0\end{aligned}$$ is exact. Using the induction hypothesis we conclude that $\Ext_R^{n-d-1}(M,\omega_R)$ is either Cohen-Macaulay of dimension $d+1$ or the zero module, and that $\Ext_R^{n-d}(M,\omega_R)$ is Cohen-Macaulay of dimension $d$.
If $\dim R/P=\dim M$ for all $P\in \Supp({\mathcal F})$, then the pretty clean filtration ${\mathcal F}$ is necessarily clean, and $M$ is unmixed.
| 3,858
| 3,173
| 2,692
| 3,320
| 3,011
| 0.775458
|
github_plus_top10pct_by_avg
|
egin{array}{ccc}
z^4+\zeta ^2-r^2 \eta ^2 & -2 r z^2 \eta
& 2 r \zeta \eta \\
2 r z^2 \eta & z^4-\zeta ^2-r^2 \eta
^2 & - 2 z^2 \zeta \\
2 r \zeta \eta & 2 z^2 \zeta & z^4-\zeta ^2+r^2 \eta ^2 \\
\end{array}
\right) \ ,$$ with the corresponding spinor representation $$\Omega = \frac{1}{\sqrt{f} } \left( z^2 \mathbb{I} - r \eta \Gamma^{12} - \zeta \Gamma^{23} \right) \ .$$ This completes the IIB supergravity solution with the three-form and five-form flux $$\begin{aligned}
F_3 &=\frac{4 e^{-\phi_0} }{z^5 \zeta \eta }\left( \zeta dx_0\wedge dx_1 \wedge dz - r \eta dx_0 \wedge dr \wedge dz \right) \ , \\
F_5&= (1+\star) \frac{-4 e^{-\phi_0} r }{ z \zeta \eta f} dx_0\wedge dx_1 \wedge dr \wedge dz\wedge d\theta \ ,
\end{aligned}$$ in agreement with the expressions following from the Yang-Baxter $\sigma$-model [@Borsato:2016ose].
Non-unimodular r-matrix {#sapp:366a}
-----------------------
The final example we consider is an $r$-matrix that can be found by infinitely boosting the Drinfel’d-Jimbo solution to the modified classical Yang-Baxter equation for $\mathfrak{su}(2,2)$ [@Hoare:2016hwh] $$\label{eq:rmatnm}
r= \eta \left(\mathfrak{D} \wedge \mathfrak{P}_0 + \mathfrak{M}_{01}\wedge \mathfrak{P}_1 + \mathfrak{M}_{+2}\wedge \mathfrak{P}_2 + \mathfrak{M}_{+3}\wedge \mathfrak{P}_3 \right) \ .$$ This $r$-matrix of jordanian type and the corresponding deformations of the $AdS_5 \times S^5$ superstring were first studied in [@Kawaguchi:2014qwa; @Kawaguchi:2014fca]. Furthermore, the $r$-matrix is non-unimodular and the corresponding dualisation of $AdS_5$ with respect to the non-abelian subalgebra $$\mathfrak{h}= \{\mathfrak{D} , \mathfrak{P}_0 , \mathfrak{M}_{01}, \mathfrak{P}_1, \mathfrak{M}_{+2} , \mathfrak{P}_2 , \mathfrak{M}_{+3} , \mathfrak{P}_3\} \ ,$$ is afflicted with a mixed gravity/gauge anomaly (i.e. $n_a= f_{ab}{}^b \neq 0$) [@Elitzur:1994ri]. The algebra $\mathfrak{h}$ admits a single central extension with the commutator of each pair of generators in being extended by the same generator. Since all direct
| 3,859
| 3,005
| 1,126
| 3,744
| null | null |
github_plus_top10pct_by_avg
|
ment4}
\frac{d}{dt}\mathcal{W}_K(t)\le&
-\sigma_K\beta_K\int_M 2 (\gamma-1)\left[\left| \nabla_i\nabla_jv+\frac{\eta_K}{n(\gamma-1)}g_{ij}\right|^2+({\rm Ric}+Kg)(\nabla v,\nabla v )\right]vu\,dV\notag\\
&-\sigma_K\beta_K\int_M2\left[(\gamma-1)\Delta v+\eta_K\right]^2vu\,dV.\end{aligned}$$
Thus, when ${\rm Ric}\ge-Kg$, $K\ge0$, $\sigma_K>0,\eta_K>0$, Perelman type entropy is monotone decreasing along the porous medium equation .
In [@LiLi; @LiLi2], when $n\le m\in \mathbb{N}$, Li-Li gave another proof for entropy formula of Witten Laplacian and explained its geometric meaning by warped product, so we can also obtain the entropy formula for the weighted porous medium equation by the analogous method. In fact, once one can obtain the entropy formula on Riemannian manifold, the weighted version is a direct result by this warped product approach when $m>n$ is a positive integer.
Let $\overline{M}=M\times N$ be a warped product manifold equipped the metric $$g_{\overline{M}}=g_M+e^{-\frac{2f}q}g_N,\quad q={m-n},$$ where $m,n,q$ are the dimensions of $\overline{M}, M, N$ respectively, then the volume measure satisfies $$\label{volmeasure}
\,\,dV_{\overline{M}}=e^{-f}\,\,dV_M\otimes \,\,dV_N=\,d\mu\otimes \,\,dV_N.$$ where $(N^q,g_N)$ is a compact Riemannian manifold. Let $\pi:M\times N\to M$ be a nature projection map, $\overline{X}$ and $X$ are vector fields on $\overline{M}$ and $M$ respectively, then [@Besse] $${\rm Ric}_{\overline{M}}(\overline{X},\overline{X})=\pi^*\left({\rm Ric}_{M}(X,X)-qe^{\frac fq}\nabla\nabla e^{-\frac fq}(X,X)\right),$$ that is $$\label{mBER}
{\rm Ric}_f^m(X,X)=\pi_*\left({\rm Ric}_{\overline{M}}(\overline{X},\overline{X})\right).$$ Assume $V_N(N)=1$, $\overline{\nabla}$ denotes Levi-Civita connection on $(\overline{M},g_{\overline{M}})$, for any $v\in C^2(M)$, by S.Li and X.-D.Li [@LiLi], we know $$\label{warped}
\overline{\nabla}_i\overline{\nabla}_jv=\nabla_i\nabla_jv,\quad
\overline{\nabla}_{\alpha}\overline{\nabla}_{\beta}v=-\frac 1q g_{\alpha\beta}g^{kl}\partial_kv\partial_lf,
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g the following result.
\[thm:old\] Let $s\in{\mathbb{N}}$ and $K\ge1$. Let $G$ be an $s$-step nilpotent group $s$, and suppose that $A\subset G$ is a finite $K$-approximate group. Then there exist a subgroup $H$ of $G$ normalised by $A$ and a nilprogression $P_{\text{\textup{nil}}}(x;L)$ of rank at most $K^{O_s(1)}$ such that $$A\subset HP_{\text{\textup{nil}}}(x;L)\subset H\overline P(x;L)\subset A^{K^{O_s(1)}}.$$
In particular, $|H\overline P(x;L)|\le\exp(K^{O_s(1)})|A|$.
The aim of the present paper is to show that, like in the abelian case, if we ask for $HP$ to be dense in $A$, rather than the other way around, we can replace most of the polynomial bounds of \[thm:old\] with polylogarithmic bounds, as follows.
\[thm:new.gen\] Let $s\in{\mathbb{N}}$ and $K\ge1$. Let $G$ be an $s$-step nilpotent group, and suppose that $A\subset G$ is a finite $K$-approximate group. Then there exist a subgroup $H\subset A^{K^{e^{O(s)}}}$ normalised by $A$ and an ordered progression $P_{\text{\textup{ord}}}(x;L)\subset A^{e^{O(s^2)}\log^{O(s)}2K}$ of rank at most $e^{O(s^2)}\log^{O(s)}2K$ such that $$P_{\text{\textup{ord}}}(x;L)\subset P_{\text{\textup{nil}}}(x;L)\subset\overline P(x;L)\subset A^{e^{O(s^3)}\log^{O(s^2)}2K}$$ and $$|HP_{\text{\textup{ord}}}(x;L)|\ge\exp\left(-e^{O(s^2)}\log^{O(s)}2K\right)|AH|.$$
The proof of \[thm:old\] proceeds by an induction on the step $s$, in which \[thm:sanders\] features both in the base case $s=1$ and in the proof of the inductive step. The original proof used an earlier version of \[thm:sanders\], due to Green and Ruzsa, in which the bounds are polynomial rather than polylogarithmic. Let us emphasise, though, that losses elsewhere in the argument overwhelmed the bounds of \[thm:sanders\] to the extent that it made no difference to the shape of the final bounds to use the Green–Ruzsa result instead. In particular, proving \[thm:new.gen\] is not merely a case of substituting \[thm:sanders\] for the Green–Ruzsa result in the original proof: we also need to make the rest of the a
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mall category, and let $A\colon {\mathcal C}\to {\mathsf{Sets}}$ be a functor. We describe a construction to be found in [@MM] of a pair of adjoint functors $f^*\colon {\mathcal{B}}({\mathcal C})\to {\mathsf{Sets}}$ and $f_*\colon {\mathsf{Sets}}\to {\mathcal{B}}({\mathcal C})$. The functor $f_*$ is easier to define and thus we start from its description. We have $f_*=\underline{\mathrm{Hom}}_{\mathcal C}(A,-)$, where the latter is the presheaf defined for each set $R$ and $C\in {\mathcal C}$ by $$\underline{\mathrm{Hom}}_{\mathcal C}(A,R)(C)={\mathrm{Hom}}_{\mathsf{Sets}}(A(C),R).$$
For a presheaf $P\in {\mathcal{B}}({\mathcal C})$, we define $f^*(P)$ to be the colimit $$f^*(P)=\lim_{\longrightarrow}\left(\int_{\mathcal C}P\stackrel{\pi_1}{\to} {\mathcal C} \stackrel{A}{\to} {\mathsf{Sets}}\right),$$ where $\pi_1(C,p)=C$. This colimit is the set which we denote by $P\otimes_{\mathcal{C}}A$. It is the quotient of the set $\bigcup_{C\in {\mathcal{C}}}(P(C)\times A(C))$ by the equivalence relation $\sim$ generated by $$(pu,a')\sim (p,ua'), \, p\in P(C), u\colon C\to C', a'\in A(C'),$$ where we denote $pu=P(p)(u)$ and $ua'=A(u)(a')$. We denote the elements of $P\otimes_{\mathcal{C}}A$ by $p\otimes a$ and treat them as tensors where ${\mathcal{C}}$ ‘acts’ on $P$ on the right and on $A$ on the left.
The described adjoint pair $(f^*, f_*)$ is not in general a geometric morphism between toposes. By definition, it is a geometric morphism if and only if the tensor product functor $f^*$ is left exact. If this condition holds, the functor $A$ is called [*flat*]{}. Flat functors can be characterized precisely as filtered functors [@MM Theorem VII.6.3].
Let ${\mathsf{Filt}}({\mathcal C})$ denote the category of filtered functors ${\mathcal C}\to {\mathsf{Sets}}$, where morphisms are natural transformations, and ${\mathsf{Geom}}({\mathsf{Sets}}, {\mathcal{B}}({\mathcal C}))$ the category of geometric morphisms from ${\mathsf{Sets}}$ to the classifying topos ${\mathcal{B}}({\mathcal C})$ of ${\mathcal C}$ (or, equivalent
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nitary group $U(n)$: let $\langle \cdot,\cdot\rangle'$ be an arbitrary inner product on $\mathbb{C}^n$. Define a new inner product $\langle \cdot,\cdot\rangle$ by setting $$\langle \xi,\eta\rangle:=\int_{g\in G} \langle g\xi,g\eta\rangle'd\mu(g),$$ where $\mu$ is an invariant probability Haar measure on $G$. It is standard and straightforward to check that $\langle \cdot,\cdot\rangle$ is still an inner product, which is moreover invariant by the action of $G$. That implies that $G$ is a subgroup of a finite-dimensional unitary group. Now by [@hewitt2013abstract Theorem 44.29], $G$ satisfies all conditions needed to apply directly [@hewitt2013abstract Theorem 44.25], which gives us positive real functions $F_n$ on $G$ satisfying the conditions (1)–(4).
Another ingredient we will need is the following version of Young’s convolution inequality, suitable for Lipschitz functions defined on a locally compact group:
Let $G$ be a locally compact group equipped with a compatible left-invariant metric $d$. Suppose that $f\in L^1(G)$ and $g\in \mathrm{Lip}(G)$. Then $f\ast g\in \mathrm{Lip}(G)$ and $$\|f\ast g \|_{\mathrm{Lip}} \leq \|f\|_{L^1}\|g\|_{\mathrm{Lip}}.$$
\[lipconv\]
Given arbitrary $x,y\in G$, $$\begin{aligned}
|f*g(x) - f*g(y)| & = \left| \int f(z)[g(z^{-1}x) - g(z^{-1}y)]\,d\mu(z) \right| \\
& \leq \int |f(z)| |g(z^{-1}x) - g(z^{-1}y)|\,\mu(z)\\
& \leq \|g\|_{\mathrm{Lip}}\int |f(z)| d(z^{-1}x,z^{-1}y)\, \mu(z)\\
& \leq \|g\|_{\mathrm{Lip}}\int |f(z)| d(x,y) \,\mu(z)\\
&\leq \|f\|_{L^1}\|g\|_{\mathrm{Lip}} d(x,y).\end{aligned}$$
We are now ready to prove the main result from this subsection. We emphasize that in the following we are equipping a Lie group with an *arbitrary* left-invariant metric inducing its topology, not Riemannian metric as it is common in Lie theory.
Suppose that $G$ is a compact Lie group equipped with a compatible left-invariant metric. Then ${\mathcal{F}}(G)$ has the MAP.
\[proptorus\]
Define $T_n: C(G) \rightarrow C(G)$ by $T_n(f) = f\ast F_n$, where $F_n$ are the functions
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chi_h\in V_h} \3bar v-\chi_h \3bar \le C_A \left(\sum_{K\in \mathcal{T}_h} h_K^{2s} \|v\|_{s+1,K}^2\right)^{1/2}.$$
The abstract theory of the interior penalty discontinuous Galerkin method can be entirely based on Assumptions [**[I1]{}**]{}-[**[I3]{}**]{}.
\[lem:wellposedness\] Assume [**[I1]{}**]{}-[**[I2]{}**]{} hold. The bilinear form $A(\cdot,\cdot)$ is bounded in $V(h)$, with respect to the norm $\3bar\cdot\3bar$. Indeed, $$A(u,v)\le \frac{1+\alpha}{\alpha} \3bar u\3bar\, \3bar v\3bar\qquad \textrm{for all } u,\, v\in V(h).$$ Furthermore, denote $C_1 = C_T(1+C_I)^2$. Then for any constant $0<C<1$ and $\alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}$, the bilinear form $A(\cdot,\cdot)$ is coercive on $V_h$. That is, $$A(v, v) \ge \frac{C}{1+C_1} \3bar v\3bar^2\qquad \textrm{for all } v\in V_h.$$
The boundedness of $A(\cdot,\cdot)$ follows immediately from the Schwarz inequality. Here we only prove the coercivity. First, notice that for all $v\in V_h$, by assumptions [**[I1]{}**]{}-[**[I2]{}**]{} and the fact that $h_e\le h_K$ for all $e\in \partial K\cap \mathcal{E}_h$, $$\begin{aligned}
\sum_{e\in\mathcal{E}_h}h_e\|\{\nabla v\}\|_e^2 & \le \sum_{K\in\mathcal{T}_h} \left(\sum_{e\in \partial K\cap \mathcal{E}_h} h_e \|\nabla v\|_e^2\right) \\
&\le \sum_{K\in\mathcal{T}_h} h_K \|\nabla v\|_{\partial K}^2 \\
& \le \sum_{K\in\mathcal{T}_h} h_K \bigg( C_T(1+C_I^2) h_K^{-1} \|\nabla v\|_K^2 \bigg) \\
& = C_1 \sum_{K\in\mathcal{T}_h} \|\nabla v\|_K^2.
\end{aligned}$$ Then, by the Schwarz inequality, the Young’s inequality and assumptions [**[I1]{}**]{}-[**[I2]{}**]{}, we have $$\begin{aligned}
\sum_{e\in\mathcal{E}_h}\langle\{\nabla v\},\, [v]\rangle_e &\le
\varepsilon \sum_{e\in\mathcal{E}_h}h_e\|\{\nabla v\}\|_e^2 + \frac{1}{4\varepsilon} \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \|[v]\|_e^2 \\
&\le \varepsilon C_1 \sum_{K\in\mathcal{T}_h} \|\nabla v\|_K^2 + \frac{1}{4\varepsilon\alpha} \left(\alpha \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \|[v]\|_e^2 \right),
\end{aligned}$$ where $\varepsilon$ is chosen to
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mented in an efficient way in $O(L)$ flops. Thus, the complexity of quantizing all the $K$ real-valued approximations is $O(KL)$. Selecting a coefficient vector from the quantized vector set in step \[item:outline:Select\] has a cost of $O(KL)$. Step \[item:outline:Recover\] takes $O(L)$ flops. In summary, the complexity of the method is $O(L(\log(L)+\log(K_u)+K))$.
However, the above analyzed complexity expression involves the experiment-based $K_u$, and its exact order with respect to the dimension $L$ is intractable. As an alternative, we use an upper bound to approximate the cost. According to , it is easy to see that $K$ and $K_u$ are at most of order $O(\sqrt{P\norm{\h}^2})$. Then, the complexity of our method is $O(L(\log(L)+\sqrt{P\norm{\h}^2}))$. We reserve the power $P$ in the expression since we may also care about how the complexity varies when the SNR gets large.
In the complexity expression above, since the square root function is strictly concave, it follows from Jensen’s inequality that $\mathbb{E}(\sqrt{\norm{\h}^2})\leq\sqrt{\mathbb{E}(\norm{\h}^2)}$. Specifically, for i.i.d. standard Gaussian channel entries, the expectation of $\norm{\h}^2$ is $L$, and thus the corresponding average complexity of the proposed method becomes $O(L\log(L)+P^{0.5}L^{1.5})$. It is easy to see that the complexity is of order 1.5 with respect to the dimension $L$.
Extension to the Complex-Valued Channel Model
---------------------------------------------
We now consider the complex-valued channel model of the AWGN networks, and demonstrate how to apply the proposed QP relaxation method for complex-valued channels. The complex-valued channel model is defined as below.
\[definition:ComplexChannelModel\] *(Complex-Valued Channel Model)* In an AWGN network, each relay (indexed by $m=1,2,\cdots,M$) observes a noisy linear combination of the transmitted signals through the channel, $$\begin{aligned}
\label{equation:ComplexChannelModel}
\y_m = \sum_{\ell=1}^L \h_m(\ell)\x_\ell + \z_m,\end{aligned}$$ where $\x_\ell
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$\rm 2$ to $\rm 18\,GHz$ with a resolution of $\rm 0.1\,MHz$. We measured the reflection $S$-matrix element $S_{aa}$ first for the unperturbed system, which corresponds to the situation, where no additional antenna is inserted at position $c$. Then we perturbed the system by inserting another antenna at position $c$ which was terminated consecutively in three different ways:
1. connection to the VNA (total absorption),\
2. standard open (open end reflection),\
3. standard short (hard wall reflection),\
and again measured the corresponding reflection at antenna $a$ for each case. The connection of antenna $c$ to the VNA corresponds to a termination of antenna $c$ with a $50\,\Omega$ load. The terminators for the cases (b) and (c) have been taken from the standard calibration kit (Agilent 85052C Precision Calibration Kit) being part of our microwave equipment. For case (a) the reflection amplitude $S_{cc}$ was also measured. From this measurement the coupling strength of antenna $c$ can be obtained, see Eq. (\[eq:Tc\]) below. For all four cases we measured 18 different realizations by rotating an ellipse (see Fig. \[fig:01\]) to perform ensemble averages.
An alternative to the coupling of an antenna with variable end is an open wave guide whose coupling to the billiard can be varied by a variable slit. It showed up that, contrary to intuition, for this setup the main effect of the variation of the slit does not correspond to a change of the coupling to the outside, but to a distortion of the wave functions in the billiard, thus corresponding more to the case of a local scattering fidelity [@hoeh08a]. This system is discussed in Appendix \[app:Exp\].
Theory {#sec:theory}
======
Generalized VWZ approach to fidelity {#subsec:VWZ}
------------------------------------
The general case of $M$ scattering channels connected to $N$ levels of the closed cavity can be described in terms of the following effective non-Hermitian Hamiltonian $$\label{eq:Heff}
H_{\mathrm{eff}} = H - i\sum_{k=1}^{M}\lambda_k V_kV_k^
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After using the Poisson identity for each integer and performing the integrations over $\phi_i$ and $A$, the resulting expression becomes Z=\_[{J\_[1,ij]{}},{J\_[2,ij]{}}, {J\_[z,i]{}}]{} [e]{}\^[-H\_J]{}, \[Z2\] H\_J= \_[ij ; a,b]{} (K\^[-1]{})\_[ab]{} J\_[a,ij]{} J\_[b,ij]{} + \_i J\_i\^2, \[H\_J\] where $a,b=1,2$ labels layers and $(K^{-1})_{ab}$ is the matrix inverse to $K_{ab}$ introduced in Eqs.(\[2N0\]),(\[Ktg\]); The summation runs over the integer bond currents $ J_{1,ij}, \, J_{2,ij}$ (oriented from site $i$ to site $j$ so that $ J_{a,ij}=- J_{a,ji},\, a=1,2$) within each corresponding layer 1,2 as well as over the integer currents $J_{i}$ oriented along the bond connecting the site $i$ in the layer 1 to the site $i$ in the layer 2. All the configurations are restricted by the Kirchhoff’s current conservation rule – the total of all J-currents incoming to any site must be equal to the total of all outcoming currents from the same site.
It is useful to note that Eqs.(\[Z2\],\[H\_J\]) can also be obtained directly from Eq.(\[2N0\]) by reinstating the compact nature of the variables: $ \nabla_{ij} \phi_1 \to \nabla_{ij} \phi_1 +2\pi m_{1, ij}$ and $ \nabla_{ij} \phi_2 \to \nabla_{ij} \phi_2 +2\pi m_{2, ij}$ so that the matrix $K_{ab} $ is viewed as being independent from the parameters in the action (\[2N\]).
The system (\[Z2\],\[H\_J\]) features statistics of closed loops. If $u=0$, there are two sorts of loops – one in each layer. Thus, each configuration is characterized by definite values of the windings $W_{a, \alpha}$ in the $a$th layer along the $\alpha=\hat{x},\hat{y}$ directions of the planes. It is straightforward to show that statistics of these windings determine the renormalized values $\tilde{K}_{ab}$ of the matrix $K_{ab}$ along the line of the approach [@Ceperley]. More specifically \_[ab]{} = \_[=,]{} W\_[a,]{} W\_[b,]{}. \[KR\] This expressions are valid for periodic boundary conditions (PBC). It is important to note that $\tilde{K}_{ab}$ represents an exact linear response (at
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his also means that the Dirac Lagrangian is already in Hamiltonian form.[@GovBook; @FJ]
[^22]: Again, this conclusion is in perfect analogy with what happens for a real and a complex scalar field.[@GovCOPRO2]
[^23]: Note that up to a total time derivative term this function is indeed real under complex conjugation, because of the Grassmann odd character of the fermionic degree of freedom $\theta(t)$. Some total derivative terms in time have been ignored to reach this expression, and to bring it into such a form that no time derivatives of order strictly larger than unity appear in the action.
[^24]: Compared to the previous parametrisation, a factor $(-\sqrt{\hbar\omega}/2)$ has been absorbed into the normalisation of the supersymmetry constant parameters $\epsilon$ and $\epsilon^\dagger$ or supercharges $Q$ and $Q^\dagger$. Note also that these expressions are consistent with the properties under complex conjugation of the different degrees of freedom as well as their Grassmann parity.
[^25]: In such an analysis, one should beware of the surface terms induced by the supersymmetry transformation applied to the action, which also contribute to the definition of the Noether charges.[@GovBook]
[^26]: Some properties have to be met in the whole construction, such as preserving under supersymmetry transformations the real character under complex conjugation of the superfield considered hereafter. This leaves open a series of possible choices, essentially related to possible phase factors in the combinations defining the superspace differential operators introduced hereafter.
---
abstract: 'Using the most general effective Hamiltonian comprising scalar,vector and tensor type interactions, we have written the branching ratio, the forward-backward (FB) asymmetry and the normalized FB asymmetry as functions of the new Wilson coefficients. It is found that the branching ratio depends on all new coefficients,but the dependence of asymmetries on coefficients could be analyzed only for one Wilson coefficient.'
---
**T
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ed D\]) and (\[1st quantized alg\]), and denoted $\tilde{p}(v_{12})=\tilde{p}_{12}$. Comparing with (\[gauge tf r\]), we find that the second line has the form of the gauge transformation with the parameter $$\begin{aligned}
D_\eta\Lambda_{{\mathcal{S}}_1{\mathcal{S}}_2}\
=&\
- D_\eta\Big(D_{{\mathcal{S}}_1}F\Xi A_{{\mathcal{S}}_2}-D_{{\mathcal{S}}_2}F\Xi A_{{\mathcal{S}}_1}
+ [ F\Xi A_{{\mathcal{S}}_1},\, D_\eta F\Xi A_{{\mathcal{S}}_2}]\Big)
\nonumber\\
=&\
- A_{\tilde{p}_{12}}
+ ({\mathcal{S}}_1F\Xi{\mathcal{S}}_2-{\mathcal{S}}_1F\Xi{\mathcal{S}}_1)A_\eta
-[F\Xi{\mathcal{S}}_1 A_\eta,\, F\Xi{\mathcal{S}}_2A_\eta]\,.
\label{Lambda ss}\end{aligned}$$ The second form can be obtained using (\[gen MC\]), and will be used below.
In order to calculate the algebra on $A_\eta$, we first calculate the transformation of $F\Psi$ using (\[variation F\]): $$\begin{aligned}
\delta_{\mathcal{S}}F\Psi\ =&\
F\Xi\{\delta_{\mathcal{S}}A_\eta, F\Psi\} + F\delta_{\mathcal{S}}\Psi
\nonumber\\
=&\
FX\eta F\Xi{\mathcal{S}}A_\eta + F\Xi{\mathcal{S}}(F\Psi)^2
+ F\Xi[(F\Psi)^2, F\Xi{\mathcal{S}}A_\eta]
\nonumber\\
=&\ QF\Xi {\mathcal{S}}A_\eta
+ F\Xi {\mathcal{S}}\left(QA_\eta + (F\Psi)^2\right)
+F\Xi[QA_\eta + (F\Psi)^2, F\Xi {\mathcal{S}}A_\eta]
\nonumber\\
\cong&\ QF\Xi {\mathcal{S}}A_\eta\,,
\label{susy on F psi} \end{aligned}$$ where the third equality follows from (\[Q and FXi\]), and the symbol $\cong$ denotes an equation which holds up to the equations of motion. Then the commutator of two transformations on $A_\eta$ $$[\delta_{{\mathcal{S}}_1}, \delta_{{\mathcal{S}}_2}]\,A_\eta\
=\ \delta_{{\mathcal{S}}_1}\big({\mathcal{S}}_2 F\Psi+[F\Psi, F\Xi {\mathcal{S}}_2 A_\eta]\big)
- (1\leftrightarrow2)\,,$$ can be calculated similarly to that on $\Psi$. Since the first term can be calculated as $$\begin{aligned}
\delta_{{\mathcal{S}}_1}\big({\mathcal{S}}_2 F\Psi+[F\Psi, F\Xi {\mathcal{S}}_2 A_\eta]\big)
=&\
{\mathcal{S}}_2(\delta_{{\mathcal{S}}_1}F\Psi)+[(\delta_{{\mathcal{S}}_1}F\Psi), F\Xi{\mathcal{S}}_2A_\eta]
\nonumber\\
&\
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t is interesting that for $b=2$ and 4, the smallest value of $\phi$ is obtained for $u=u_\theta$, which is not the case for $b=3$ fractal.
Finally, one should note that in the case $b=3$, for the globular state of a solitary 3D chain ($u>u_\theta$), the coordinates of the corresponding fixed point are $A_G=0$ and $B_G=\infty$. Furthermore, a numerical analysis of function $D^{(r)}$, in the range $v\ge v_c(u)$ reveals that $D^*=\infty$. Nevertheless, the relation $\langle M^{(r)}\rangle\sim\lambda_D^r$ and formula (\[eq:skaliranje\]) are applicable, but with different meaning of $\lambda_D$. For the globule state of $b=3$ fractal, it was demonstrated [@Knezevic] that equations (\[eq:Ab3\]) and (\[eq:Bb3\]) in the vicinity of the corresponding fixed point $(0,\infty)$ have the following approximate form $$\label{knez1}
A^{(r+1)}= 320 (A^{(r)})^3 (B^{(r)})^6\, , \quad B^{(r+1)}= 4308 (A^{(r)})^2 (B^{(r)})^8\>,$$ from which it follows $\lambda_{\nu_3}=\frac{\sqrt{73}+11}2=9.772$ and $\nu_3^G=\ln 3/\ln 9.772=0.4819$. Besides, for $v\ge v_c(u)$, the inequality $D^{(r)}\ll B^{(r)}$ is valid, so that RG equation (\[eq:A4b3\]) obtains the approximate form $$\label{jedn2}
D^{(r+1)}=320 A^{(r)} (B^{(r)})^6 C^{(r)} (D^{(r)})^2\, .$$ Then, from equations (\[eq:srednjeMASAWs\]) and (\[dda\]), follows $
\langle M^{(r+1)}\rangle=2\frac v{D^{(r)}}\frac{\partial D^{(r)}}{\partial v}=2\langle M^{(r)}\rangle
$, implying that $
\langle M^{(r)}\rangle\sim \lambda_D^r$ (for large $r$), with $\lambda_D=2$. Finally, from (\[eq:skaliranje\]), one obtains $\phi={\ln2}/{\ln9.772}=0.3041$.
The model of crossing walks {#CSAWs}
===========================
In order to describe the physical situation when closer contact between the two polymers is possible, in this section we analyze the CSAWs model in which chains $P_2$ and $P_3$ can cross each other [@ZivicJSTAT]. If we assume that chains interact only at the crossing sites, and, similarly as in the ASAWs case, introduce the weight factor $w=e^{-\epsilon_c/k_BT}$, where $\epsilon_c\leq0$ i
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[@C2]). Moreover, since $M_0^{\prime\prime}$ is *free of type II* and nonzero, we have a morphism from $G_j$ to the even orthogonal group associated to $M_0^{\prime\prime}$ as explained in Section \[red\]. Thus, the Dickson invariant of this orthogonal group induces the morphism $$\psi_j : \tilde{G} \longrightarrow \mathbb{Z}/2\mathbb{Z}.$$ **
\(2) We next assume that $M_0$ is *of type* $\textit{I}^o$. We choose a Jordan splitting for the hermitian lattice $(C(L^j), \xi^{-m}h)$ as follows: $$C(L^j)=\bigoplus_{i \geq 0} M_i^{\prime},$$ where $$M_0^{\prime}= Be, ~~~M_1^{\prime}=M_1, ~~~ M_2^{\prime}=(\oplus_i(\pi)e_i)\oplus M_2, ~~~ \mathrm{and}~ M_k^{\prime}=M_k \mathrm{~if~} k\geq 3.$$ Here, $M_i^{\prime}$ is $\pi^i$-modular and $(\pi)$ is the ideal of $B$ generated by a uniformizer $\pi$. Notice that the rank of the $\pi^0$-modular lattice $M_0^{\prime}$ is 1 and that all of the lattices $M_2^{\prime}, M_3^{\prime}, M_4^{\prime}$ are *of type II*. If $G_j$ denotes the special fiber of the smooth integral model associated to the hermitian lattice $(C(L^j), \xi^{-m}h)$, then we have a morphism from $\tilde{G}$ to $G_j$ as in the above argument (1).
We now consider the new hermitian lattice $M_0^{\prime}\oplus C(L^j)$. Then for a flat $A$-algebra $R$, there is a natural embedding from the group of $R$-points of the naive integral model associated to the hermitian lattice $(C(L^j), \xi^{-m}h)$ to that of the hermitian lattice $M_0^{\prime}\oplus C(L^j)$ such that $m$ maps to $\begin{pmatrix} 1&0 \\ 0&m \end{pmatrix}$, where $m$ is an element of the former group. This fact induces a morphism from the smooth integral model associated to the hermitian lattice $(C(L^j), \xi^{-m}h)$ to the smooth integral model associated to the hermitian lattice $M_0^{\prime}\oplus C(L^j)$ (cf. from the last paragraph of page 488 to the first paragraph of page 489 in [@C2]). In Remark \[r410\], we describe this morphism explicitly in terms of matrices.
Thus we have a morphism from the special fiber $G_j$ of the smooth integral model
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|
2}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg(\prod_{\substack{i,i'\ge
2\\ i\ne i'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_{i'})=
{\varnothing}\}$}}}\bigg){\nonumber}\\
&\qquad\times\sum_{{\partial}{{\bf m}}'={\partial}{{\bf k}}'={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{
{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf m}}')}{Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,
\frac{w_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf k}}')}{Z_{\tilde
{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}'+{{\bf k}}'$}}
{\overset{}{\longleftrightarrow}}}z'_1\}$}}}
\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}'+{{\bf k}}'$}}
{\overset{}{\longleftrightarrow}}}v\}$}}},\end{aligned}$$ where the second line is bounded by [(\[eq:Theta”-2ndindbd1:j=1bd\])]{} for $j=1$, and then the first line is bounded by $\prod_{i=2}^j\sum_{l\ge1}(\tilde G_\Lambda^2)^{*(2l-1)}(z_i,z'_i)$, due to [(\[eq:lace-edges\])]{}–[(\[eq:lace-edgesbd\])]{}.
Summarizing the above bounds, we have (cf., [(\[eq:Theta’-2ndindbd5\])]{}) $$\begin{aligned}
{\label{eq:Theta''-2ndindbd2.2}}
{(\ref{eq:Theta''-2ndindbd1})}\leq\sum_{j\ge1}\sum_{\substack{z_1,\dots,
z_j\\ z'_1,\dots,z'_j}}\bigg(&\sum_{h=1}^j\sum_{v'}g_{\Lambda;v'}
(z_h,z'_h)\,\psi_\Lambda(v',v)\prod_{i\ne h}\sum_{l\ge1}\big(
\tilde G_\Lambda^2\big)^{*(2l-1)}(z_i,z'_i)\bigg){\nonumber}\\
&\times\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{
y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrigh
| 3,872
| 2,544
| 3,055
| 3,549
| null | null |
github_plus_top10pct_by_avg
|
s A$, and the other is a right module ($A$ just happens to have both structures at the same time). It is better to separate them and introduce the functor $${\operatorname{\sf tr}}:A{\operatorname{\!-\sf bimod}}\to k{\operatorname{\it\!-Vect}}$$ by ${\operatorname{\sf tr}}(M) = M \otimes_{A^{opp} \otimes A} A$ – or, equivalently, by $$\label{tr.A}
{\operatorname{\sf tr}}(M) = M/\{ am-ma \mid a \in A, m \in M \}.$$ Then ${\operatorname{\sf tr}}$ is a right-exact functor, and we have $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M) =
L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}(M)$.
We want to emphasize that the functor ${\operatorname{\sf tr}}$ can not be recovered from the tensor structure on $A{\operatorname{\!-\sf bimod}}$ – this really is an extra piece of data. For a general tensor category ${{\mathcal C}}$, it does not exist a priori; we have to impose it as an additional structure.
Let us axiomatize the situation. First, forget for the moment about the $k$-linear and abelian structure on ${{\mathcal C}}$ – let us treat it simply as a monoidal category. Assume given some other category ${{\mathcal B}}$ and a functor $T:{{\mathcal C}}\to {{\mathcal B}}$.
\[trace.defn\] The functor $T:{{\mathcal C}}\to {{\mathcal B}}$ is a [*trace functor*]{} if it is extended to a functor ${{\mathcal C}}_\# \to {{\mathcal B}}$ which sends any cocartesian map $f:M \to
M'$ in ${{\mathcal C}}_\#$ to an invertible map.
Another way to say the same thing is the following: the categories ${\operatorname{Fun}}({{\mathcal C}}^n,{{\mathcal B}})$ of functors from ${{\mathcal C}}^n$ to ${{\mathcal B}}$ form a fibered category over $\Lambda$, and a trace functor is a cartesian section of this fibration. Explicitly, a trace functor is defined by $T:{{\mathcal C}}\to {{\mathcal B}}$ and a collection of isomorphisms $$T(M \otimes M') \to T(M' \otimes M)$$ for any $M,M' \in {{\mathcal C}}$ which are functorial in $M$ and $M'$ and satisfy some compatibility conditions analogous to those in Lemma \[cycl.str\
| 3,873
| 2,722
| 1,900
| 3,551
| 4,103
| 0.768088
|
github_plus_top10pct_by_avg
|
imes\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m'_b,n'_b\text{ even}\}$}}}\sum_{\substack{
{\partial}{{\bf m}}''={\varnothing}\\ {\partial}{{\bf n}}''={\overline{b}}{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}
({{\bf m}}'')}{Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf n}}'')}
{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}''+{{\bf n}}''$}}
{\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal B}}{^{\rm c}}\}$}}}{\nonumber}\\
&=\sum_{{{\cal B}}\subset\Lambda}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}{\underline{b}}}}
\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}
\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};{{\cal A}})\text{ off }b\\}$}}}\,\cap\,\{{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v)=
{{\cal B}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b\text{ even}\}$}}}\,{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal B}}{^{\rm c}}}{\nonumber}\\
&=\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}{\underline{b}}}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}
{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};
{{\cal A}})\text{ off }b\}$}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b\text{ even}\}$}}}\,{{\langle \varphi_{{\overline{b}}}
\varphi_x \rangle}}_{{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v){^{\rm c}}},\end{aligned}$$ where we have been able to perform the sum over ${{\bf m}}''$ and ${{\bf n}}''$ independently, due to the fact that ${\mathbbm{1}{\scriptstyle\{{\overline{b}}\,{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}''+{{\bf n}}''$}}
{\overset{}{\longleftrightarrow}}}\,x
\text{ in }{{\cal B}}{^{\rm c
| 3,874
| 2,531
| 2,141
| 3,553
| null | null |
github_plus_top10pct_by_avg
|
fined as the number of examples of class $j$ that were classified as class $i$ by the binary classifier. In other words, a class is assigned to $\mathcal{C}_{i1}$ if it is less frequently confused with $c_1$ than with $c_2$, and to $\mathcal{C}_{i2}$ otherwise. Finally, the binary classifier is re-trained on the new meta-classes $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$. This way, each binary split is more easily separable for the base learner than a completely random split, but also exhibits a degree of randomness, which leads to diverse and high-performing ensembles.
Due to the fact that the size of the sample space of nested dichotomies under random-pair selection is dependent on the dataset and base learner (different initial random pairs may lead to the same split), it is not possible to provide an exact expression for the growth function $T_{RP}(n)$; using logistic regression as the base learner [@leathart2016building], it has been empirically estimated to be $$\begin{aligned}
T_{RP}(n) = p(n)T_{RP}(\frac{n}{3})T_{RP}(\frac{2n}{3}) \label{eqn:random_pair_estimation}\end{aligned}$$ where $$\begin{aligned}
p(n) = 0.3812n^2 - 1.4979n + 2.9027\end{aligned}$$ and $T_{RP}(2) = T_{RP}(1) = 1$.
Multiple Subset Evaluation\[sec:multiple\_subset\_selection\]
=============================================================
In existing class subset selection methods, at each internal node $i$, a single class split $(\mathcal{C}_{i1}, \mathcal{C}_{i2})$ of $\mathcal{C}_i$ is considered, produced by some splitting function $S(\mathcal{C}_i) : \mathbb{N}^n \rightarrow \mathbb{N}^a \times \mathbb{N}^b$ where $a+b=n$. Our approach for improving the predictive power of nested dichotomies is a simple extension. We propose to, at each internal node $i$, consider $\lambda$ subsets $\{(\mathcal{C}_{i1}, \mathcal{C}_{i2})_1 \dots (\mathcal{C}_{i1}, \mathcal{C}_{i2})_\lambda\}$ and choose the split for which the corresponding model has the lowest training root mean squared error (RMSE). The RMSE is defined as the squar
| 3,875
| 4,720
| 3,862
| 3,608
| 564
| 0.805549
|
github_plus_top10pct_by_avg
|
CP asymmetry is different depending on the value of $r$. For $r a\ll 1$ increasing $a$ results in enhancement of the CP asymmetry, while for $r a \gg 1$ it is suppressed. These two cases correspond to the charm and kaon cases, respectively. It follows that the $\Delta I=1/2$ rule in kaons reduces CP violating effects, while the $\Delta U=0$ rule in charm enhances them.
Conclusions \[sec:conclusions\]
===============================
From the recent determination of $\Delta a_{CP}^{\mathrm{dir}}$ we derive the ratio of $\Delta U=0$ over $\Delta U=1$ amplitudes as $$\begin{aligned}
\vert \tilde{p}_0\vert \sin(\delta_{\mathrm{strong}}) &= 0.65 \pm 0.12\,.\label{eq:mainresult-conclusion}\end{aligned}$$ In principle two options are possible in order to explain this result: In the perturbative picture beyond the SM (BSM) physics is necessary to explain Eq. (\[eq:mainresult-conclusion\]). On the other hand, in the SM picture, we find that all that is required in order to explain the result is a mild non-perturbative enhancement due to rescattering effects. Therefore, it is hard to argue that BSM physics is required.
Our interpretation of the result is that the measurement of $\Delta a_{CP}^{\mathrm{dir}}$ provides a proof for the $\Delta U=0$ rule in charm. The enhancement of the $\Delta U=0$ amplitude is not as significant as the one present in the $\Delta I=1/2$ rule for kaons. In the future, with more information on the strong phase of $\tilde{p}_0$ from time-dependent measurements or measurements of correlated $D^0\overline{D}^0$ decays, we will be able to completely determine the extent of the $\Delta U=0$ rule.
Interpreting the result within the SM implies that we expect a moderate non-perturbative effect and nominal $SU(3)_F$ breaking. The former fact implies that we expect U-spin invariant strong phases to be $\mathcal{O}(1)$. The latter implies that we anticipate the yet to be determined $SU(3)_F$ breaking effects not to be large. Thus, there are two qualitative predictions we can make $$\begin{aligned}
\de
| 3,876
| 1,081
| 2,700
| 3,738
| null | null |
github_plus_top10pct_by_avg
|
x}}-{\mathbf{x}}_0) = 0
\}$ for $\mathbf{n} = [1,0,0]$ and ${\mathbf{x}}_0 = [1/2,1/2,1/2]$ is shown in figures \[fig:RvsE0\_FixE\_small\](f)-(h), where the transition from cellular structures to a localized vortex structure as enstrophy increases is evident.
The results corresponding to large values of $\E_0$ are shown in figure \[fig:RvsE0\_FixE\_large\] with the maximum rate of growth of enstrophy $\R_{\E_0}$ plotted as a function of $\E_0$ in figure \[fig:RvsE0\_FixE\_large\](a). We observe that, as $\E_0$ increases, this relation approaches a power law of the form $\R_{\E_0} = C_1' \,\E_0^{\alpha_1}$. In order to determine the prefactor $C_1'$ and the exponent $\alpha_1$ we perform a local least-squares fit of the power law to the actual relation $\R_{\E_0}$ versus $\E_0$ for increasing values of $\E_0$ starting with $\E_{0} = 20$ (this particular choice the starting value is justified below). Then, the exponent $\alpha_1$ is computed as the average of the exponents obtained from the local fits with their standard deviation providing the error bars, so that we obtain $$\label{eq:RvsE0_powerLaw}
\R_{\E_0} = C'_1\E_0^{\,\alpha_1}, \qquad C'_1 = 3.72 \times 10^{-3} , \ \alpha_1 = 2.97 \pm 0.02$$ (the same approach is also used to determine the exponents in other power-law relations detected in this study). We note that the exponent $\alpha_1$ obtained in is in fact very close to 3 which is the exponent in estimate . For the value of the viscosity coefficient used in the computations ($\nu=0.01$), the constant factor $C_1 = 27/(8\pi^4\nu^3)$ in estimate has the value $C_1 \approx 3.465 \times 10^4$ which is approximately seven orders of magnitude larger than $C'_1$ given in . To shed more light at the source of this discrepancy, the objective functional $\R$ from equation can be separated into a negative-definite viscous part $\R_{\nu}$ and a cubic part $\R_{\textrm{cub}}$ defined as
$$\begin{aligned}
\R_{\nu}({\mathbf{u}}) & := -\nu\int_\Omega |{\Delta}{\mathbf{u}}|^2\,d{\mathbf{x}}, \\
\R_{\textrm{cub}}({\m
| 3,877
| 2,361
| 3,707
| 3,794
| 3,249
| 0.773763
|
github_plus_top10pct_by_avg
|
\text{-}Grmod}}$ consisting of ${\mathbb{N}}$-graded $S$-modules $M=\bigoplus_{i\in{\mathbb{N}}}M_i$. It is immediate from the definitions that the identity map $\iota: M=\bigoplus_{i\in{\mathbb{N}}}M_i\mapsto M=\bigoplus_{i\in{\mathbb{N}}}M_i$ gives equivalences of categories $S{{\textsf}{\text{-}Grmod}}_{\geq 0}\simeq \widehat{S}{{\textsf}{\text{-}Grmod}}$ and $S{{\textsf}{\text{-}grmod}}_{\geq 0}\simeq\widehat{S}{{\textsf}{\text{-}grmod}}$. For any module $M\in S{{\textsf}{\text{-}Grmod}}$, one has $\pi(M)=\pi(M_{\geq 0})$ in $S{\text{-}{\textsf}{Qgr}}$ and so $\iota$ induces category equivalences $$\label{zalgex11}
S{\text{-}{\textsf}{Qgr}}\simeq \widehat{S}{\text{-}{\textsf}{Qgr}}\qquad \text{and} \qquad
S{\text{-}{\textsf}{qgr}}\simeq \widehat{S}{\text{-}{\textsf}{qgr}}.$$
{#zalgex2}
For the second class of examples, suppose that we are given noetherian algebras $R_n$ for $n\in {\mathbb{N}}$ with $(R_i, R_j)$-bimodules $R_{ij}$, for $i> j\geq 0$. Assume, moreover, that there are morphisms $\theta_{ij}^{jk} : R_{ij}\otimes_{R_j}R_{jk}\to R_{ik}$ satisfying the the obvious associativity conditions. Then we can define a ${\mathbb{Z}}$-algebra $R_{\mathbb{Z}}$ by $R_{\mathbb{Z}}=\bigoplus_{i\geq j\geq 0}R_{ij}$, where $R_{ii}=R_i$ for all $i$.
A particular example of this construction is the one that interests us. Suppose that $\{R_n : n\in {\mathbb{N}}\}$ are Morita equivalent algebras, with the equivalence induced from the progenerative $(R_{n+1},R_{n})$-bimodules $P_n$. Define $R_{ij} =
P_{i-1}\otimes_{R_{i-1}}\otimes\cdots \otimes_{R_{j+2}} P_{j+1}\otimes_{R_{j+1}}P_j$ and $R_{jj}=R_j$, for $i>j\geq 0$. Tensor products provide the isomorphisms $\theta_\bullet^\bullet$ and associativity is automatic. The corresponding ${\mathbb{Z}}$-algebra $R_{\mathbb{Z}}=\bigoplus_{i\geq j\geq 0} R_{ij}$ will be called the [*Morita ${\mathbb{Z}}$-algebra associated to the data $\{R_n,P_n : n\in {\mathbb{N}}\}$*]{}.
{#Zalgequiv}
Write $R{\text{-}{\textsf}{mod}}$ for the category of finitely generated left modules ove
| 3,878
| 3,100
| 1,911
| 3,624
| 3,295
| 0.773392
|
github_plus_top10pct_by_avg
|
_i^{n-2}-+\cdots =0.$$ Thus $A[r_1,\dots, r_m]$ is integral over $A\bigl[\sigma_{ij}\bigr]$, hence also over the larger ring $A[r_1,\dots, r_m]^G$.
By assumption $R=U^{-1}A[r_1,\dots, r_m]$ where $U$ is a subgroup of units in $A[r_1,\dots, r_m]$. We may assume that $U$ is $G$-invariant. If $r/u\in R$ where $r\in A[r_1,\dots, r_m]$ and $u$ a unit in $A[r_1,\dots, r_m]$, then $$\frac{r}{u}=\frac{r\prod_{g\neq 1} g(u)}{u\prod_{g\neq 1} g(u)},$$ where the product is over the non-identity elements of $G$. Thus $r/u=r'/u'$ where $r'\in A[r_1,\dots, r_m]$ and $u'$ is a $G$-invariant unit in $A[r_1,\dots, r_m]$. Therefore, $$R=\bigl(U^G\bigr)^{-1}A[r_1,\dots, r_m]
{\quad\mbox{is finite over}\quad}
\bigl(U^G\bigr)^{-1}A\bigl[\sigma_{ij}\bigr].$$ Since $R^G$ is an $\bigl(U^G\bigr)^{-1}A\bigl[\sigma_{ij}\bigr]$-submodule of $R$, it is also finite over $\bigl(U^G\bigr)^{-1}A\bigl[\sigma_{ij}\bigr]$, hence the localization of a finitely generated algebra.
Assume next that $|G|$ is invertible in $A$. We claim that $JR\cap R^G=J$ for any ideal $J\subset R^G$. Indeed, if $a_i\in R^G$, $r_i\in R$ and $\sum r_ia_i\in R^G$ then $$|G|\cdot \sum_i r_ia_i=\sum_{g\in G}\sum_ig(r_i)g(a_i)=
\sum_ia_i\sum_{g\in G}g(r_i)\in \sum_ia_iR^G.$$ If $|G|$ is invertible, this gives that $R^G\cap\sum a_iR =\sum a_iR^G$. Thus the map $J\mapsto JR$ from the ideals of $R^G$ to the ideals of $R$ is an injection which preserves inclusions. Therefore $R^G$ is Noetherian if $R$ is.
If $R$ is an integral domain, then $R$ is finite over $R^G$ by (\[r/rg.finite.general\]). The general case, which we do not use, is left to the reader.
The arguments in case (3) are quite involved, see [@fogarty].
\[r/rg.finite.general\] Let $R$ be an integral domain and $G$ a finite group of automorphisms of $R$. Then $R$ is contained in a finite $R^G$-module. Thus, if $R^G$ is Noetherian, then $R$ is finite over $R^G$.
Proof. Let $K\supset R$ and $K^G\supset R^G$ denote the quotient fields. $K/K^G$ is a Galois extension with group $G$. Pick $r_1,\dots, r_n\in R$ that fo
| 3,879
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github_plus_top10pct_by_avg
|
for U.S. Stock, U.S. Bond and world FX market participants. ? 1999-2001 Earnings.com, Inc., All rights reserved about us | contact us | webmaster | site map privacy policy | terms of service
Anna:
I know that you have been negotiating with Cheryl Nelson about the new ECI
account. I am receiving urgent calls from ECI traders and hope you give me
some answers since Cheryl is in a meeting.
Can we execute the ECI agreement today?
Have we agreed on the assignment?
Does the assignment also provide for the transfer of the Master Securities
Loan Agreement?
Regarding the new ENE guaranty, Cheryl has told me that this was already
requested. I'll check on the expedited delivery date.
Please call me when you receive this email. Thanks very much for your
attention to this matter.
Sara Shackleton
Enron North America Corp.
1400 Smith Street, EB 3801a
Houston, Texas 77002
713-853-5620 (phone)
713-646-3490 (fax)
sara.shackleton@enron.com
I will forward this to Dutch.....
Dutch try it now on your NT machine...... call if probs.... x57862
-----Original Message-----
From: Pechersky, Julie
Sent: Tuesday, April 17, 2001 1:23 PM
To: Sieckman, John
Subject: FW:
John,
Becky re-enabled Dutch Quigley so that he is on both the old servers and the Digital. What is the next step?
-----Original Message-----
From: "Becky McGraw" <becky@cqg.com>@ENRON [mailto:IMCEANOTES-+22Becky+20McGraw+22+20+3Cbecky+40cqg+2Ecom+3E+40ENRON@ENRON.com]
Sent: Tuesday, April 17, 2001 1:20 PM
To: Pechersky, Julie
Subject: RE:
Julie,
I re-enabled his old system that went off last night. Both systems will be
functional with the old one going off on April 30th. If you need longer,
just let me know. I have been putting the latest upgrades to allow both
systems to run for at least two weeks before cancellation of the old one.
If you would like that to be a 30 day parallel, then I can do that to be on
the safe side....actually I will just start doing that so that there is
plenty of time for the transition.
Becky
-----Original
| 3,880
| 2,586
| 2,438
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| null | null |
github_plus_top10pct_by_avg
|
left| P \right|}}$ divides ${\ensuremath{\left| \Sigma_{n-r} \right|}}$, and so ${\ensuremath{\left| \Sigma_n:\Sigma_{n-r} \right|}}= n(n-1)\cdots (n-r+1)$ should be a $p'$-number. But this is a contradiction, since $p<r$.
If $p=k=2$, then, by Lemma \[angraph\], it should be $G=\Sigma_n$, so the above reasonings work as well. Finally, if $p=2$ and $k\geq 3$, then $r\geq 5$ and we can argue as above to get a contradiction since ${\ensuremath{\left| \Sigma_n:\Sigma_{n-r} \right|}}= n(n-1)\cdots (n-r+1)$ is divisible by $4$.
If $n=6$, by Lemma \[a6\], the only case to be considered is $G={\operatorname}{Aut}(N)$, and since ${{\operatorname}{\mathcal{Z}}(\Gamma(G))}=\{2\}$ it should be $p=2$. But a Sylow $2$-subgroup of $G$ is self-centralising, so we get a contradiction.
\[notsp\] $N$ is not an sporadic group.
Assume that $N$ is an sporadic group. Since $p \in {{\operatorname}{\mathcal{Z}}(\Gamma(G))}$, we may assume, by Lemma \[sporgraph\], that either $N= J_2$ or $N=McL$, $G={{\operatorname}{\textup{Aut}}({N})}$, and $p=2$. Now Lemma \[cent\] implies that $2$ is adjacent in $N$ to any prime $r\neq 2$, but this is a contradiction since $N$ has a self-centralising Sylow $s$-subgroup (take $s=7$ for $N=J_2$ and $s=11$ for $N=McL$; see [@Atl]).
$N$ is not a simple group of Lie type.
If $N$ is a simple group of Lie type of characteristic $t$, first notice that the prime $p$ such that $\pi(G)=\pi(C_G(P))$ should be different from $t$, because it is well known that a Sylow $t$-subgroup is self-centralising in $G$. Moreover, since ${\ensuremath{\left| N \right|}}{\ensuremath{\left| A\cap B \right|}}={\ensuremath{\left| \frac{G}{N} \right|}}{\ensuremath{\left| N\cap A \right|}}{\ensuremath{\left| N\cap B \right|}}$ and $|N|_t > |{{\operatorname}{\textup{Out}}({N})}|_t$, we get that $t \in \pi((N \cap A) \cup (N \cap B)) \subseteq \pi(C_{N}(P))$. This means that $t$ should be adjacent to $p$ in $\Gamma(N)$.
Now, we derive from Lemmas \[class\] and \[excep\] that, apart from some exceptional cases that we consider be
| 3,881
| 2,425
| 2,048
| 3,652
| 2,826
| 0.776747
|
github_plus_top10pct_by_avg
|
Eq. (\[eq:concentration\]) provides $$\begin{aligned}
-v\frac{dc}{d\xi} = D \frac{d^2 c}{d\xi^2}-ac+f(\xi).
\label{eq:concentration1}
\end{aligned}$$ Equation (\[eq:concentration1\]) leads to the following solutions $$\begin{aligned}
c(\xi) = \begin{cases}
\beta_1 \exp \big(\lambda_-(\xi-r)\big), & ({\xi > r}),\\
\dfrac{f_0}{a}+\alpha_2 \exp\big(\lambda_+\xi)+\beta_2
\exp\big(\lambda_-\xi\big), & ({-r <\xi < r}),\\
\alpha_3 \exp \big(\lambda_+(\xi+r)\big), & ({\xi < -r}),
\end{cases}
\label{eq:provide}
\end{aligned}$$ where $$\begin{aligned}
\lambda_\pm= -\frac{v}{2D}\pm\frac{\sqrt{v^2+4Da}}{2D},
\label{eq:lambda_pm}
\end{aligned}$$ $$\begin{aligned}
\beta_1 = \frac{f_0\lambda_+}{a(\lambda_+-\lambda_-)}(1-\exp(2\lambda_-r)),
\label{eq:beta1}
\end{aligned}$$ $$\begin{aligned}
\alpha_2= \frac{f_0\lambda_-\exp(-\lambda_+r)}{a(\lambda_+-\lambda_-)},
\label{eq:alpha2}
\end{aligned}$$ $$\begin{aligned}
\beta_2 = \frac{f_0\lambda_+\exp(-\lambda_-r)}{a(\lambda_+-\lambda_-)},
\label{eq:beta2}
\end{aligned}$$ $$\begin{aligned}
\alpha_3 = -\frac{f_0\lambda_-}{a(\lambda_+-\lambda_-)}(1-\exp(-2\lambda_+r)).
\label{eq:alpha1}
\end{aligned}$$
Equations (\[eq:provide\])-(\[eq:alpha1\]) provide $$\begin{aligned}
F =& -\Gamma w \left[\beta_1 \exp \left(\lambda_-\ell \right)-\alpha_3
\right]
\nonumber \\
=&-\frac{\Gamma w f_0}{a \left(\lambda_+-\lambda_-\right)}
\left[\lambda_+ \left(1-\exp \left(2\lambda_-r \right) \right) \exp
\left(\lambda_-\ell \right) \right. \nonumber\\
& \left. + \lambda_- \left(1-\exp \left(-2\lambda_+r\right) \right) \right].
\end{aligned}$$
As $v$ is sufficiently large in our experiments, we assume $r\ll1/\lambda_+$ and $\ell\gg1/\left|\lambda_-\right|$. Then, $\lambda_+\sim a/v$ and $\lambda_-\sim -v/D$, which lead to $$\begin{aligned}
| 3,882
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| 3,390
| null | null |
github_plus_top10pct_by_avg
|
tes and whose signals therefore are $\propto g_d^2$ or $\propto \frac{1}{f_a^2}$. Measuring an amplitude and not a rate makes it much easier to push the sensitivity up to high $f_a$ (low axion couplings).
Further, the actual size of the EDM is set by the product $g_d a$, where $a$ is the local dark matter, axion or ALP, field. As discussed in Section \[Sec:Overview\] this is approximately $a \approx a_0 \cos \left( m_a t \right)$. The amplitude of this field, $a_0$, is known if we require that this field makes up (all of) the local dark matter density $$\label{eqn: dm abundance}
\rho_\text{DM} = \frac{1}{2} m_a^2 a_0^2 \approx 0.3 \frac{{\text{GeV}}}{{\text{cm}}^3}$$ since the field $a$ is essentially a free scalar field with this mass term as the leading term in its potential. This then determines the nucleon EDM generated by ALP dark matter from Eqs. and to be $$\label{eqn: ALP EDM}
d_n = g_d \frac{\sqrt{2 \, \rho_\text{DM}}}{m_a} \cos \left( m_a t \right) \approx \left( 1.4 \times 10^{-25} \, e \cdot {\text{cm}}\right) \left( \frac{\text{eV} }{m_a} \right) \left( g_d \, {\text{GeV}}^2 \right) \cos \left( m_a t \right)$$
For the QCD axion, since the axion mass from Eq. scales as $m_a \propto \frac{1}{f_a}$, taking the axion to be all of the dark matter fixes the effective $\theta$ angle of the axion to be independent of $f_a$: $$\label{qcd axion theta}
\frac{a_0}{f_a} \approx 3.6 \times 10^{-19}$$ So for the QCD axion dark matter, the nucleon EDM it induces from Eq. , is actually independent of $f_a$ $$\label{eqn: qcd axion edm}
d_n^\text{QCD} \approx \left( 9 \times 10^{-35} \, e \cdot {\text{cm}}\right) \cos \left( m_a t \right).$$ Thus we have found a physical effect that does not decouple as $f_a$ increases. This useful fact allows experiments searching for this EDM to probe high $f_a$ axions.
Note that the EDM induced by the axion from Eq. is small. Of course the EDM induced by a general ALP, Eq. , is arbitrary. For the QCD axion though, the EDM is about eight orders of magnitude smaller than th
| 3,883
| 3,502
| 4,241
| 3,619
| 2,230
| 0.781612
|
github_plus_top10pct_by_avg
|
. To continue the computation we combine current conservation with the Maurer-Cartan equation to write the anti-holomorphic derivative of the $z$-component of the current in terms of a bilinear : |j\_[L,z]{}\^a = -i f\^2 [f\^a]{}\_[bc]{} :j\^c\_[L,z]{} j\^b\_[L,|z]{}:. Since all the poles in the OPE between $j^c_{L,z}$ and $ j^b_{L,\bar z}$ vanish when contracted with the structure constant ${f^a}_{bc}$, we can also write : |j\_[L,z]{}\^a = -i f\^2 [f\^a]{}\_[bc]{} :j\^b\_[L,|z]{} j\^c\_[L,z]{}:. Thus using successively the last two equations we obtain: |T(z) = f\_[abc]{} ( : :j\^c\_[L,z]{} j\^b\_[L,|z]{}: j\_[L,z]{}\^a:(z) + :j\_[L,z]{}\^a :j\^b\_[L,|z]{} j\^c\_[L,z]{} ::(z) ). Now let us consider the composite operator $::j^c_{L,z} j^b_{L,\bar z}: j_{L,z}^a:(z)$. It is defined as the regular term in the OPE between $
:j^c_{L,z} j^b_{L,\bar z}:$ and $j_{L,z}^a$. We will show that we have : \[f::::=f::\] f\_[abc]{}::j\^c\_[L,z]{} j\^b\_[L,|z]{}: j\_[L,z]{}\^a:(z) = f\_[abc]{}:j\^c\_[L,z]{} j\^b\_[L,|z]{} j\_[L,z]{}\^a:(z) where the operator $:j^c_{L,z} j^b_{L,\bar z} j_{L,z}^a:$ is defined as the regular term in the OPE of the three currents $j^c_{L,z}$, $j^b_{L,\bar z}$ and $j_{L,z}^a$. The difference between the operators on the left-hand side and the right-hand side of equation comes from the non-regular terms in the OPE between $j^c_{L,z}$ and $j^b_{L,\bar z}$. The crucial point is that all these terms vanish when contracted with the structure constant $f_{abc}$: f\_[abc]{} \[j\^c\_[L,z]{}(z) j\^b\_[L,|z]{}(w) - :j\^c\_[L,z]{}(z) j\^b\_[L,|z]{}(w):\] = 0. This can be checked via the current algebra OPEs order by order in $f^2$. In equation the current algebra is given up to terms of order $f^4$, and thus one can prove the previous statement up to terms of order $f^4$. Indeed, all tensors that appear in the current algebra vanish upon double contraction with a structure constant: f\_[abc]{} = 0 The non-degenerate metric $\kappa^{cb}$ and the tensors ${A^{cb}}_{de},\ {B^{cb}}_{de},\ {C^{cb}}_{de}$ are graded-sy
| 3,884
| 464
| 3,440
| 3,552
| null | null |
github_plus_top10pct_by_avg
|
4,\ \#\#\#\#^ 3: 8.3, 4: 75.0, 5: 16.7 8.3 47.0 ± 13.4 ^\#\#\#\#^
\*^1^: Days after transplanting; \*^2^: n = 5; \*^3^: n = 9; \*^4^: n = 11; ^\#^*p* \< 0.05; ^\#\#^*p* \< 0.01; ^\#\#\#^*p* \< 0.005; and ^\#\#\#\#^*p* \< 0.001 *vs.* WT.
2.2. Alkaloid Composition in the PsM1-2 Mutants
-----------------------------------------------
The soil-cultivated PsM1-2 T~0~ primary mutant accumulated 16.3% (% dry weight) of thebaine as a major opium alkaloid in the latex, which was not detected in the WT ([Figure 2](#pharmaceuticals-05-00133-f002){ref-type="fig"}; [Table 2](#pharmaceuticals-05-00133-t002){ref-type="table"}). The morphine content in the mutant was 1.3%, which was *ca.* one tenth of that in the WT, and the codeine content was 4.2% in the mutant, *vs*. 1.3% in the WT.
{#pharmaceuticals-05-00133-f001}
{#pharmaceuticals-05-00133-f002}
The alkaloid compositions in the dried opium of selected progenies are summarized in [Table 2](#pharmaceuticals-05-00133-t002){ref-type="table"}, and the morphine and thebaine contents of the T~1~, T~2~ and T~3~ plants are plotted on a scatter diagram ([Figure 3](#pharmaceuticals-05-00133-f003){ref-type="fig"}). The HPLC chromatograms of the representative lines of the T~1~ plants, WT plant, and authentic standards are shown in the [Supplementary Figure 1](#pharmaceuticals-05-00133-s001){ref-type="supplementary-material"}.
pharmaceuticals-05-00133-t002_Table 2
######
Opium alkaloid contents in PsM1-2 T~0~ mutant and selfed progenies.
Progenies Lines Number of plants Morphine
| 3,885
| 1,767
| 2,260
| 3,582
| 2,956
| 0.775841
|
github_plus_top10pct_by_avg
|
distribution of energy spent per resetting event. Red disks come from experiments and the theoretical prediction of [Eq. (\[Eq:Energy\])]{} is plotted as a solid blue line. d) Normalized energy spent per resetting event at constant power vs. the normalized radial return velocity as given by [Eq. (\[minmax\])]{}. The minimal energy is attained at a maximal velocity for which the trap is just barely strong enough to overcome the fluid drag force and prevent the particle from escaping the trap.[]{data-label="Fig:work"}](work_const_v.pdf)
In our experiments, work is done by the laser to capture the particle in an optical trap and drag it back to the origin. The total energy spent per resetting event is then simply given by $E=\mathcal{P}\tau(R)$, where $\mathcal{P}$ is the laser power fixed at 1W and $\tau(R)$ is the time required for the laser to trap the particle at a distance $R$ and bring it back to the origin. As the particle’s distance at the resetting epoch fluctuates randomly from one resetting event to another ([Fig. \[Fig:work\]]{}a), the energy spent per resetting event is also random ([Fig. \[Fig:work\]]{}b). To compute its distribution, we note that $E$ is proportional to the return time whose probability density function is in turn given by [@SM] (t)=\_[-]{}\^ d\_0\^ dR R \_0\^dt’ G\_0(R,t’)f(t’) , where $f(t)=re^{-rt}$ is the resetting time density and $G_0(R,t)=\frac{1}{4\pi Dt}e^{-R^2/4Dt}$ is the diffusion propagator in polar coordinates. For the case of constant radial return velocity, $v$, we have $\tau(R)=R/v$. A simple derivation then yields the probability density of the energy spent per resetting event [@SM] (E)= K\_0(E/E\_0) ,\[Eq:Energy\] with $E_0=\alpha_0^{-1}v^{-1}\mathcal{P}$; and note that this is a special case of the K-distribution [@Redding; @Long]. The mean energy spent per resetting event can be computed directly from [Eq. (\[Eq:Energy\])]{} and is given by $\langle E \rangle=\pi E_0/2$. Equation (\[Eq:Energy\]) demonstrates good agreement with experimental da
| 3,886
| 2,607
| 4,379
| 3,703
| 4,038
| 0.7685
|
github_plus_top10pct_by_avg
|
in Lemma \[lem:alter\_m(W)\]. Let $$M=M(W)=\int_{\Rc_r}\La f_\pi(\La;W)\dd\La.$$ Assume that $$f_\pi(\La;W)=0 \quad \textup{for all $\La\in\partial\Rc_r$}.$$ Then $\ph_H$ is minimax relative to the KL loss $(\ref{eqn:loss})$ if $\De(W;\pi_2^J)\leq 0$, where $$\De(W;\pi_2^J)=\De_1(W;\pi_2^J)-\De_2(W;\pi_2^J)-(q-3r-3)\tr M$$ with $$\begin{aligned}
\De_1(W;\pi_2^J)&=4\int_{\Rc_r}\frac{1}{\pi_2^J(\La)}\tr\{\La^2\Dc_\La \pi_2^J(\La)\}f_\pi(\La;W)\dd\La,\\
\De_2(W;\pi_2^J)&=\frac{2}{m(W)}\int_{\Rc_r}\frac{1}{\pi_2^J(\La)}\tr\{\La M\Dc_\La \pi_2^J(\La)\}f_\pi(\La;W)\dd\La,\end{aligned}$$ provided all the integrals are finite.
[**Proof.**]{} From Proposition \[prp:cond\_mini\], $\ph_H$ is minimax when $$\De=2\tr[\nabla_W\nabla_W^\top m(W)]-\frac{\Vert\nabla_W m(W)\Vert^2}{m(W)}\leq 0.$$
It is seen from Lemma \[lem:alter\_m(W)\] that $$\nabla_W f_\pi(\La;W)=-\frac{1}{v}\La W f_\pi(\La;W)$$ and $$\nabla_W\nabla_W^\top f_\pi(\La;W)=\Big(\frac{1}{v^2}\La WW^\top\La-\frac{q}{v}\La\Big)f_\pi(\La;W).$$ Hence we obtain $$\label{eqn:d2_mw1}
\De=\frac{1}{v}[2E_1(W)-\{m(W)\}^{-1}E_2(W)],$$ where $$\begin{aligned}
E_1(W)&=\int_{\Rc_r}\Big[\frac{1}{v}\tr(WW^\top\La^2)-q\tr\La\Big]f_\pi(\La;W)\dd\La,\\
E_2(W)&=\frac{1}{v}\tr\bigg[WW^\top\bigg\{\int_{\Rc_r}\La f_\pi(\La;W)\dd\La\bigg\}^2\bigg] \\
&=\frac{1}{v}\int_{\Rc_r}\tr(MWW^\top\La)f_\pi(\La;W)\dd\La.\end{aligned}$$
Using Lemmas \[lem:diff1\] and \[lem:diff3\] yields that $$\Dc_\La f_\pi(\La;W)=\frac{1}{2}\Big[q\La^{-1}+\frac{2}{\pi_2^J(\La)}\Dc_\La \pi_2^J(\La)-\frac{1}{v}WW^\top\Big]f_\pi(\La;W),$$ so that $$\begin{aligned}
\tr[\Dc_\La\{f_\pi(\La;W)\La^2\}]&=\tr[\La^2\Dc_\La f_\pi(\La;W)]+f_\pi(\La;W)\tr[\Dc_\La\La^2]\\
&=\frac{1}{2}\Big[\frac{2}{\pi_2^J(\La)}\tr[\La^2\Dc_\La \pi_2^J(\La)]-\Big\{\frac{1}{v}\tr(WW^\top\La^2)-q\tr\La\Big\} \\
&\qquad +2(r+1)\tr\La \Big]f_\pi(\La;W).\end{aligned}$$ Thus $E_1(W)$ can be expressed as $$\begin{aligned}
\label{eqn:E1}
E_1(W)
&=2(r+1)\tr M+\frac{1}{2}\De_1(W;\pi_2^J) -2\int_{\Rc_r}\tr[\Dc_\La\{f_\pi(\La;W)\La^2\}]\dd\La.\end{ali
| 3,887
| 2,858
| 1,567
| 3,817
| null | null |
github_plus_top10pct_by_avg
|
’]{} P\^b\_a\_b()\[eqn:kappagen\],\
-u\^a u\^b T\_[ab]{}\^[diss]{}+\_[j=1]{}\^[N]{}u\^a T\_[ay\_j]{}\^[diss]{} =-\[eqn:zetagen\]. Using these we can extract the viscosities $\hat\eta$, $\hat\zeta$ and the matrix of conductivities $\hat\kappa_{jj'}$. The derivative $\partial \hat P/ \partial \hat\epsilon$ is evaluated at constant charges, while $\partial \hat P/\partial \hat q_j$ are evaluated keeping fixed the energy density and the other charges $q_{k\neq j}$.\
We obtain =V \_[i=1]{}\^[N]{} \_i\[eqn:etaflui\], \_[jj]{}&=& VT 1-\_[i=1]{}\^[N]{} \_i\[eqn:kappafluidjj\],\
\_[jk]{}&=&\_[kj]{}=-VT\_[l=1]{}\^[k-1]{} \_l\[eqn:kappafluidjk\], kj, &=&2V\_[i=1]{}\^[N]{} \_i \[eqn:zetafluid\]\
&+&\
&-&\
&+&V.These are the main results of this article.
The transport coefficients can be rewritten in terms of the independent thermodynamic variables of the reduced theory, the temperature and the chemical potentials, using Eq. and Eq. functions of the rapidities.
Observe that the viscosity to entropy density ratio remains constant under the reduction, \[KSS\] =.
Furthermore, since the entropy current for our charged fluid in a canonical form is J\_s\^a=su\^a- T\^[ab]{}\_[diss]{}-\_[j=1]{}\^N \_j T\^[ay\_j]{}\_[diss]{} using the relations in Eq. and substituting the values of the chemical potentials Eq.(\[mu\]), it is easy to see that J\_s\^a=su\^a. Comparing this result with the entropy density of our neutral initial fluid J\_s\^A=s u\^A multiplied by the volume factor $V$, we recover the result obtain from the Euler relation in Eq.(\[redentropy\]).
Finally, the speed of sound is given by \[sos\] c\_s\^2==, where the derivative is considered at fixed $\hat s/\hat q_j$ for every $ \hat q_j$.\
Charged black brane/fluid duals {#6}
===============================
The previous analysis can be applied to the case of the fluid dual to a black p-brane. Let us consider a black p-brane in $D = p + n + 3$ dimensions with $p+1$ worldvolume coordinates of the p-brane and $n+2$ coordinates in directions transverse to that. Since we
| 3,888
| 1,032
| 3,717
| 3,684
| null | null |
github_plus_top10pct_by_avg
|
rences in the appraisal of the cancer diagnosis, the perception of family functioning, cancer-related emotions and perceived quality of life across members within one family.
Materials and Methods {#s1}
=====================
Participants
------------
The sample consisted of 115 families where one child has been diagnosed with leukemia or non-Hodgkin lymphoma. All families were Caucasian and living in the Flemish part of Belgium. Across the families, time since diagnosis varied from 0 to 33 months (*M* = 6,90, *SD* = 8,05). The ill child's mean age was 6,60 (*SD* = 4,84; Range = 0--19). In 24 families (21%), the diagnosed child was the only child. The remaining families had either two (52 families; 45%), three (28 families; 24%), four (9 families; 8%) or five (2 families; 2%) children.
Due to the questionnaires' age limits (e.g., the Family Environment Scale (FES) is only applicable for children aged 11 and above) and the willingness of the different family members to participate, data from 60 ill children, 172 parents and 78 siblings were included in the present study. More details on the sample are listed in [Table 1](#T1){ref-type="table"}. Ethical approval from the University Hospitals of Ghent, Brussels, Antwerp, and Louvain had been secured for the study. Written informed consent forms were obtained from all the participating parents in this study, as well as all the participating children above the age of 12. Parental consent was obtained for all participating children under the age of 16.
######
Background characteristics of the study sample.
Demographic variable
------------------- ----------- ---------------------------------------------- ------------------------------------ ---------------------
Families *N* 115
Age ill child, mean (*SD*)
| 3,889
| 1,157
| 3,188
| 3,725
| null | null |
github_plus_top10pct_by_avg
|
$v_{50}$
(km s$^{-1}$) (km s$^{-1}$)
$\pm$187 $\pm$0.06 $\pm$0.3
1968L 321 0.219 0.00 12.03(08) ... 4020(300)
1969L 784 0.205 0.00 13.35(06) ... 4841(300)
1970G 580 0.028 0.00 12.10(15) ... 5041(300)
1973R 808 0.107 1.40 14.56(05) ... 5092(300)
1986I 1333 0.129 0.20 14.55(20) 14.05(09) 3623(300)
1986L 1466 0.099 0.30 14.57(05) ... 4150(300)
1988A 1332 0.136 0.00 15.00(05) ... 4613(300)
1989L 1332 0.123 0.15 15.47(05) 14.54(05) 3529(300)
1990E 1426 0.082 1.45 15.90(20) 14.56(20) 5324(300)
1990K 1818 0.047 0.20 14.50(20) 13.90(05) 6142(2000)
1991al 4484 0.168 0.00 16.62(05) 16.16(05) 7330(2000)
1991G 1152 0.065 0.00 15.53(07) 15.05(09) 3347(500)
1992H 2305 0.054 0.00 14.99(04) ... 5463(300)
1992af 5438 0.171 0.00 17.06(20) 16.56(20) 5322(2000)
1992am 14009 0.164 0.28 18.44(05) 17.99(05) 7868(300)
1992ba 1192 0.193 0.00 15.43(05) 14.76(05) 3523(300)
1993A 8933 0.572 0.05 19.64(05) 18.89(05) 4290(300)
1993S 9649 0.054 0.70 18.96(05) 18.25(05) 4569(300)
1999br 848 0.078 0.65 17.58(05) 16.71(05) 1545(300)
1999ca 3105 0.361 0.
| 3,890
| 5,658
| 1,460
| 3,269
| null | null |
github_plus_top10pct_by_avg
|
providers operating SDNTNs. As observed, the CP can be either centralized (providers A and B) or distributed (provider C) [@Gringeri2013SDNTN; @distributed2013sdtns]. As shown in the bottom left, the SDN-controller might host different applications that run on a Network Operating System (NOS). In addition, there is a component called FlowVisor that acts as a transparent proxy between the DP nodes and the SDN-controller. FlowVisor is a network virtualization layer and its main objective is to make sure that each user of the SDN-controller controls his own virtual network [@sherwood2009flowvisor].
![Possible scenario of SDN transport networks.[]{data-label="fig:sdntn"}](sdntn.pdf)
The emergence of SDNTNs implies an increased reliance on software. On the one hand, this aspect brings many benefits such as network programmability and control logic centralization. On the other hand however, it is also the source of major security concerns. It has been argued that SDN introduces new threat vectors with respect to traditional networks [@vectorsvulnerable2013SDN]. For instance, software faults hold a pivotal role in the reliability of SDNTNs, and significant efforts are made to define tools able to detect SW bugs or errors [@Canini2012NICE]. Additionally, other type of failures could occur due to policy (also called rules) conflicts, given that several users might work on the same physical network, each one on his own virtualized network. In such cases, FlowVisor is in charge to ensure compatibility among all policies.
\
There are several cases in which epidemic-like failures could occur in SDNTNs. We classify these scenarios according to the initial event that triggers the propagation, which can be either a fault or an attack. First, in order to illustrate some potential scenarios in SDNTNs, we assume that a DDoS attack can be launched, for instance, by following the method provided in [@attacking2013SDN]. Consequently, we propose two propagation scenarios:
- *Vertical propagation*, which can be bottom-top or top
| 3,891
| 408
| 4,061
| 3,349
| null | null |
github_plus_top10pct_by_avg
|
efan-Boltzmann law for the neutrino flux at the resonance surface, a neutrino emitted in a direction ${\bf \hat{p }}$ has a momentum $p=E_{o}(1+4h_{T}^{-1}\delta r)$, where $E_{o}=E(r_{o})$. Therefore it carries an angular momentum $${\bf l=}r_{o}E_{o}({\bf \hat{r}}\times {\bf \hat{p})}\left[
1+4h_{T}^{-1}\delta r\right] \,.$$ By integrating at each point of the resonance surface over all directions and also over all the points, we compute the angular momentum gained by the star. Because of the symmetry of the system the resulting angular acceleration points along the rotational axis. The time derivative of the total angular momentum can be expressed as $$\dot{L}=\frac{C\Lambda }{3\pi A}\frac{h_{N_{e}}}{h_{T}}{\dot{{\cal E}}}
\Omega r_{o}\frac{\int_{0}^{\pi }d\theta \sin \theta\int_{0}^ {\frac{\pi }{2}
}d\theta^{\prime }\int_{0}^{2\pi }d\varphi ^{\prime } \sin \theta ^{\prime
}r_{o}\left( {\bf \hat{\Omega}}\times {\bf \hat{r}\cdot \hat{p}} \right)
^{2} }{\int_{0}^{\pi }d\theta \sin \theta \int_{0}^{\frac{\pi }{2} }d\theta
^{\prime }\sin \theta ^{\prime }}\,, \label{L}$$ where ${\dot{{\cal E}}}$ is the energy carried by the neutrinos per time unit, and a factor $\frac{1}{6}$ has been included to take into account that although six neutrino and antineutrino species are radiated, only one comes from the distorted neutrinosphere. In the latter expression $\theta $ is the angle between the radius vector ${\bf \hat{r}}$ and the angular velocity ${\bf \Omega }$, while $\theta ^{\prime }$ and $\varphi ^{\prime }$ are the spherical coordinates for ${\bf \hat{p}}$ taking ${\bf \hat{r}}$ as the $z$ axis. From Eq. (\[L\]) $$\Omega (t)=\Omega _{o}\exp \left( \frac{4r_{o}^{2}}{27}\int_{t_{0}}^{t}\frac{
C\Lambda }{AI}\frac{h_{N_{e}}}{h_{T}}{\dot{{\cal E}}}dt\right) \,,$$ where $I$ and $\Omega _{o}$ are the momentum of inertia and the initial angular velocity of the protostar. It should be noted that the rotational kick does not require a velocity ${\bf v}$ associated to a preferred frame.
As an example, let us consider the de
| 3,892
| 4,315
| 3,540
| 3,618
| null | null |
github_plus_top10pct_by_avg
|
er she want to bind or to reject the contract. In the former case she measures all unmeasured qubits in the Accept basis, in the latter in the Reject base. Both parties then report Trent for each respective qubit whether they measured it in the Accept or Reject basis, and submit respective measurement outcomes. Trent verifies whether their measurement outcomes correspond to their claims. If there is a mismatch in measurements of, say Bob, he is declared as cheater and Trent considers only Alice’s measurement outcomes. Let $N_A^{\cal{A}}$ ($N_R^{\cal{A}}$) denote the number of Alice’s qubits prepared in the Accept (Reject) basis, and analogously $N^{\cal{B}}_A$ and $N_R^{\cal{B}}$ for Bob. The contract is declared as valid if Alice presents at least $\alpha N_A^{\cal{A}}$ accept results and Bob presents less than $\alpha N_R^{\cal{B}}$ reject results, or when Bob presents at least $\alpha N_A^{\cal{B}}$ accept results and Alice presents less than $\alpha N_R^{\cal{A}}$ reject results. In case a client, say Bob, supplied incorrect measurement outcomes (see above), Trent declares the contract to be valid if Alice presents at least $\alpha N_A^{\cal{A}}$ accept results. In all other cases the contract is declared as invalid.*]{}
III. Fairness conditions {#sec:fairness-conditions}
========================
As noted in the previous section, our protocol is (probabilistically) viable: if both clients are honest, the probability to bind the contract is exponentially close to one. Therefore, it is also optimistic: honest clients do not need to contact Trent in order to obtain the verdict that, with exponentially high probability, they already know. In case a client, say Bob, is not honest, i.e. is not measuring the Accept observable in every step of the protocol, we say that he is cheating. Any cheating strategy will inevitably have a non-zero probability of producing a wrong result on qubits from the Accept basis, thus allowing Alice to detect Bob’s cheating and move on to the Binding phase.
In case the Exchange
| 3,893
| 1,375
| 3,519
| 3,642
| null | null |
github_plus_top10pct_by_avg
|
Cleft( of District Court~a Co., Texas
BY DEPUlY
JAS FAMILY LIMITED PARTNERSHIP § IN THE DISTRICT COURT .
#4LTD §
§
vs. § BRAZORIA COUNTY, TEXAS
§
§
PAULAM. MILLER § 149TH JUDICIAL DISTRICT
OR.DER
On the 12th day of November, 2013, the Court held a hearing on Plaintiff's Motion for
Emergency Relief recited in Plaintiff's Second Amended Petition and a Motion to Abate filed by
the Defendant, Paula M. Miller. JAS Family Limited Partnership #4 LTD appeared through its
duly authorized representative, James A. Prince and its counsel, Michael M. Phillips, and
announced ready.. Paula M. Miller, Defendant, appeared in person and prose. After review of
the pleadings, presentation of evidence, and argument of counsel, it is the opinion of this Court
that the Motion to Abate is not well taken and should be DENIED.
IT IS ORDERED that Paula M. Miller's Motion to Abate this proceeding is hereby
DENIED.
The Court further finds JAS Family Limited Partnership #4 LTD's Motion for Emergency
Relief regarding three filed Lis Pendens and an Amendment of the September 19, 2005, Lis
Pendens by an Affidavit of Adverse Possession filed on August 29, 2011, is well taken and
should be GRANTED.
IT IS THEREFORE ORDERED that the notices of Ljs Pendens that were filed by Paula
M. Miller should be released and be of no further force and effect, described as follows:
14132
1. Notice of Lis Penden
| 3,894
| 830
| 3,198
| 3,941
| null | null |
github_plus_top10pct_by_avg
|
---------------------- ---------------------------- --------- ------ --------------------
1 31 Female None Negative Fever Disturbance of consciousness Bacterial meningitis Sacroiliitis Blood culture Cerebrospina fluid l Positive None Penicillin for 42 days\ 8 Cure 1997 [@b8-idr-11-1043]
Oral penicillin for 14\
days
2 47 Female None Unknown Left-sided lower back pain Sacroiliitis Pneumonia Pus culture of percutaneous drainage Unknown Percutaneous Benzylpenicillin for 5 days\ 12 Cure 1998 [@b9-idr-11-1043]
Oral amoxycillin for 79 days
3 62 Male None Unknown Right buttock pain, fever Sacroiliitis Osteomyelitis Pyomyositis Blood culture Culture of the muscle biopsy Positive Surgical Ceftriaxone 7 days\ 6 Cure 2001 [@b10-idr-11-1043]
| 3,895
| 5,653
| 2,274
| 2,744
| null | null |
github_plus_top10pct_by_avg
|
atisfy reflection positivity, their mean-field behavior cannot necessarily be established, even in high dimensions. If we believe in universality, we expect that finite-range models exhibit the same mean-field behavior as soon as $d>4$. Therefore, it has been desirable to have approaches that do not assume reflection positivity.
The lace expansion has been used successfully to investigate mean-field behavior for self-avoiding walk, percolation, lattice trees/animals and the contact process, above the upper-critical dimension: 4, 6 (4 for oriented percolation), 8 and 4, respectively (see, e.g., [@s04]). One of the advantages in the application of the lace expansion is that we do not have to require reflection positivity to prove a Gaussian infrared bound and mean-field behavior. Another advantage is the possibility to show an asymptotic result for the decay of correlation. Our goal in this paper is to prove the lace-expansion results for the Ising model.
Main results
------------
From now on, we fix $h=0$ and abbreviate, e.g., ${{\langle \varphi_o\varphi_x \rangle}}_{p,h=0;\Lambda}$ to ${{\langle \varphi_o\varphi_x \rangle}}_{p;\Lambda}$. In this paper, we prove the following lace expansion for the two-point function, in which we use the notation $$\begin{aligned}
\tau_{x,y}=\tanh(pJ_{x,y}).\end{aligned}$$
\[prp:Ising-lace\] For any $p\ge0$ and any $\Lambda\subset{{\mathbb Z}^d}$, there exist $\pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)$ and $R_{p;\Lambda}^{{\scriptscriptstyle}(j+1)}(x)$ for $x\in\Lambda$ and $j\ge0$ such that $$\begin{aligned}
{\label{eq:Ising-lace}}
{{\langle \varphi_o\varphi_x \rangle}}_{p;\Lambda}=\Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)
+\sum_{u,v}\Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(u)\,\tau_{u,v}{{\langle
\varphi_v\varphi_x \rangle}}_{p;\Lambda}+(-1)^{j+1}R_{p;\Lambda}^{{\scriptscriptstyle}(j
+1)}(x),\end{aligned}$$ where $$\begin{aligned}
{\label{eq:Pij-def}}
\Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)&=\sum_{i=0}^j(-1)^i\,\pi_{p;
\Lambda}^{{\scriptscriptstyle}(i)}(x).\
| 3,896
| 2,652
| 3,811
| 3,388
| 2,615
| 0.778417
|
github_plus_top10pct_by_avg
|
tal magnitudes of $m_{\mathrm{{F606W}},\mathrm{tot}}=21.5$ and $m_{\mathrm{{F850LP}},\mathrm{tot}}=21.6$, respectively. This image is shown in Figure \[fig:unresplot\]. With the higher sensitivity we now indeed find a host galaxy component in the [F606W]{}-band image after PSF subtraction of 4.4% of the total flux. The radial surface brightness profile also shows a small excess over an unresolved point source. In both cases this flux is highly significant as we confirmed using a bootstrap simulation for the composition of the coadded frame from the 15 frames. In the bootstrap simulation we constructed 100 new sets of 15 frames each, drawn with repetition from the original 15 frames, coadded the images in each set and did the flux analysis as above. The uncertainty in the total flux estimated from these 100 realisations is $\sigma=1.05$% of the total flux, or 25% in host galaxy flux. All realisations yielded substantial positive fluxes. The error is resulting from a combination of PSF uncertainty and the noise inside the scaling aperture of 4 pixel diameter. We show the uncertainties in the radial surface brightness determined from bootstrapping as error bars for the derived host galaxy in Figure \[fig:unresplot\]. The so extracted magnitudes for host galaxy and nucleus in the [F606W]{}-band are listed in the last row of Table \[tab:results\_vz\] (the ‘stack’ object). The [F850LP]{}-band stack, however, with its lower sensitivity showed a much weaker signal than the [F606W]{}-band, too faint to reliably be classified as resolved.
[cccccccccccc]{} ID&Tile& $V_\mathrm{tot}$& $V_\mathrm{hg}$& $V_\mathrm{hg,cor}$& $V_\mathrm{nuc,cor}$& N/H$_{V,\mathrm{cor}}$& $Z_\mathrm{tot}$& $Z_\mathrm{hg}$& $Z_\mathrm{hg,cor}$& $Z_\mathrm{nuc,cor}$& N/H$_{Z,\mathrm{cor}}$\
19965&23& 20.59& 23.9& 23.3$\pm0.05$& 20.7& 11.3& 20.29& 23.8 & 23.5 $\pm0.2$& 20.4&17.5\
30792&82& 21.60& 24.6& 24.4$\pm0.2$ & 21.7& 12.0& 22.06& 24.2 & 23.9 $\pm0.2$& 22.3& 4.2\
18324&19& 22.00& 23.6& 23.0$\pm0.05$& 22.5& 1.6& 21.33& 22.7 & 22.4 $\pm0.2
| 3,897
| 2,258
| 4,197
| 3,787
| 1,085
| 0.794585
|
github_plus_top10pct_by_avg
|
integrate to $\langle \hat X_\nu\rangle \delta_{\mu,\mu'}$. Hence, for a vanishing displacement $\langle\hat X_\nu \rangle=0$, either the Green’s function $G^{(1)}_{d\sigma}$ is identically zero, or its spectrum (i.e., the transmission function $\tau^{(1)}_{\sigma}(\w)$) has equal positive and negative spectral contributions. The first statement is true in the limit of vanishing electron-phonon coupling. For non-vanishing electron-phonon coupling, however, the correlation function $G^{(1)}_{d\sigma}$ does not vanish, since quantum fluctuations and hence non-zero correlators $G_{d_{\mu \sigma}, \hat X_\nu d^\dagger_{\mu' \sigma}}$ and $G_{\hat X_\nu d_{\mu \sigma}, d^\dagger_{\mu' \sigma}}(z)$ are allowed even if $\langle\hat X_\nu \rangle=0$.
In second-order in $\lambda^{\rm tip}_{\mu \nu}$, the inelastic tunnel current $$\begin{aligned}
I^{(2)}_{\rm inel}
&=&
\frac{2\pi e}{\hbar}
\sum_{\sigma}
\int_{-\infty}^{\infty} d\w \rho_{\sigma,\rm tip}(\w)
\tau^{(2)}_{\sigma}(\w)
\non
&& \times \left[
f_{\rm tip}(\w) - f_{S}(\w)
\right],
\label{eq:second-order-inelastic-contribution}\end{aligned}$$ involves the transmission function $$\begin{aligned}
\label{eq:rho2explicit}
\tau^{(2)}_{\sigma}(\w) &= & \frac{1}{\pi} \lim_{\delta\to 0^+} \Im G^{(2)}_{d\sigma} (\w-i\delta) \nonumber
\\
&=&\sum_{\mu\mu'}^M t_{\mu \sigma}t_{\mu' \sigma}
\\
&&\times \sum_{\nu\nu'}^{N_\nu}\lambda^{\rm tip}_{\mu\nu}\lambda^{\rm tip}_{\mu'\nu'}\lim_{\delta\to 0^+}
\frac{1}{\pi} \Im G_{\hat X_\nu d_{\mu\sigma}, \hat X_{\nu'}d^\dagger_{\mu'\sigma}} (\w-i\delta). \nonumber\end{aligned}$$ Up to second-order, the total inelastic contribution to the tunneling current is thus given by $I_{\rm inel} = I^{(1)}_{\rm inel} +I^{(2)}_{\rm inel}$. Again, the spectral sum rule of $G_{\hat X_\nu d_{\mu \sigma}, \hat X_{\nu'} d^\dagger_{\mu' \sigma}}(z)$ is related to the expectation value of the anticommutator, i. e. $\langle \hat X_\nu \hat X_{\nu'} \rangle \delta_{\mu,\mu'}$.
### The limit of vanishing electron-phonon coupling in the system
| 3,898
| 2,526
| 2,655
| 3,558
| null | null |
github_plus_top10pct_by_avg
|
}(\omega) \; d\theta \; d{{\mathbb P}}(\omega)\end{aligned}$$ ($N$ is a.s. countable). For a given point $X\in N$ and a given $\omega\in \Omega$, the indicator function $${{\bf 1}}_{{\scriptstyle \underline{\gamma}_{X} \;\mbox{\small{or}}\; \overline{\gamma}_{X} \;\mbox{\small{admits}}\; \theta \;\mbox{\small{as}}} \atop \scriptstyle \mbox{\small{asymptotic direction}}}(\omega)$$ is equal to $1$ for at most two different angles in $[0,2\pi)$. Its integral is then equal to zero. Using the Fubini’s theorem, $$\int_{0}^{2\pi} {{\mathbb P}}(U(\theta)) \; d\theta = {{\mathbb E}}\lambda \{\theta ; U(\theta) \} = 0 ~.$$ So, the probability ${{\mathbb P}}(U(\theta))$ is zero for Lebesgue a.e. $\theta$ in $[0,2\pi)$. Actually, this is true for every $\theta$ in $[0,2\pi)$ thanks to the isotropic character of the PPP $N$. Combining this with Part $(ii)$ of Theorem \[HN1\], the announced result follows. $\Box$
Further description of the semi-infinite path with direction 0 {#section:description0}
--------------------------------------------------------------
In the rest of this section, we discuss some consequences of Proposition \[prop:<2\]. For any given $\theta\in [0,2\pi)$, let us denote by $\gamma_{\theta}$ the semi-infinite path of the RST, started at the origin and with asymptotic direction $\theta$. It is a.s. well defined by Proposition \[prop:<2\]. Since the distribution of the RST is invariant by rotation, we will henceforth assume that $\theta=0$.\
Let us recall that $\widetilde{\chi}_r$ denotes the number of intersection points of $a(A_r, B_r)$ with the semi-infinite paths of the RST and that $\widetilde{\chi}_r\rightarrow 0$ in probability, by . We have
$\limsup_{r\rightarrow \infty}\widetilde{\chi}_r\geq 1$ a.s.
Assume that there exists with positive probability a (random) radius $r_{0}$ such that $\widetilde{\chi}_r=0$ whenever $r>r_{0}$. Let us work on the set where this event is realized. In this case, no semi-infinite path crosses the abscissa axis after $r_0$. Then, we can exhibit two semi-inf
| 3,899
| 2,406
| 1,802
| 3,669
| 3,290
| 0.773419
|
github_plus_top10pct_by_avg
|
{\left \lfloor{\ell\beta_1} \right \rfloor} \}$. Since set $S$ is chosen randomly, $i,\i$ and $j \in \Omega$ are random. Throughout this section, we condition on the random indices $i,\i$ and the set $\Omega$ such that event $E_{\beta_1}$ holds. To get a lower bound on $\P[\sigma^{-1}(i),\sigma^{-1}(\i) > \kappa - \ell ]$, define independent exponential random variables $X_j \sim \exp(e^{\ltheta^*_j})$ for $j \in S$. Observe that given event $E_{\beta_1}$ holds, there exists a set $\Omega_1 \subseteq \Omega$ such that $$\begin{aligned}
\label{bl_prob_8}
\Omega_1 = \Big\{j\in S :\ltheta^*_i \leq \ltheta^*_j \Big\}\;,\end{aligned}$$ and $|\Omega_1| = \kappa-{\left \lfloor{\ell\beta_1} \right \rfloor} -2$. In fact there can be many such sets, and for the purpose of the proof we can choose one such set arbitrarily. Note that ${\left \lfloor{\ell\beta_1} \right \rfloor} +2 \leq \ell$ by assumption on $\beta_1$, so $|\Omega_1| \geq \kappa-\ell$. From the Random Utility Model (RUM) interpretation of the PL model, we know that the PL model is equivalent to ordering the items as per [*random cost*]{} of each item drawn from exponential random variable with mean $e^{\tilde\theta^*_i}$. That is, we rank items according to $X_j$’s such that the lower cost items are ranked higher. From this interpretation, we have that $$\begin{aligned}
\label{eq:bl_prob_4}
\P\Big[\sigma^{-1}(i),\sigma^{-1}(\i) > \kappa - \ell \Big] &=& \P\Big[ \sum_{j \in \Omega} \I_{\big\{\min\{X_i, X_{\i}\}\; > \;X_j\big\}} \geq \kappa-\ell \Big] \nonumber\\
&>& \P\Big[ \sum_{\j \in \Omega_1} \I_{\big\{\min\{X_i, X_{\i}\} \;>\; X_{\j}\big\}} \geq \kappa-\ell \Big] \end{aligned}$$ The above inequality follows from the fact that $\Omega_1 \subseteq \Omega$ and $|\Omega_1| \geq \kappa -\ell$. It excludes some of the rankings over the items of the set $S$ that constitute the event $\{\sigma^{-1}(i),\sigma^{-1}(\i) > \kappa - \ell \}$. Define $\Omega_2 = \{\Omega_1,i,\i\}$. Observe that items $i,\i$ have the least preference scores among all the items in th
| 3,900
| 1,971
| 1,938
| 3,755
| null | null |
github_plus_top10pct_by_avg
|
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